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Use of Aspergillus oryzae during sorghum malting to enhance yield and quality of gluten-free lager beers
Monica Rubio-Flores1,
Arnulfo Ricardo García-Arellano ORCID: orcid.org/0000-0001-8395-742X1,
Esther Perez-Carrillo ORCID: orcid.org/0000-0003-2636-62811 &
Sergio O. Serna-Saldivar ORCID: orcid.org/0000-0002-9713-29281
Sorghum has been used for brewing European beers but its malt generally lower beer yields and alcohol contents. The aim of this research was to produce lager beers using worts from sorghum malted with and without Aspergillus oryzae inoculation. Worts adjusted to 15° Plato from the sorghum malt inoculated with 1% A. oryzae yielded 21.5% and 5% more volume compared to sorghum malt and barley malt worts, respectively. The main fermentable carbohydrate in all worts was maltose. Glucose was present in higher amounts in both sorghum worts compared to barley malt worts. Sorghum–A. oryzae beer had similar specific gravity and alcohol compared to the barley malt beer. Sorghum–A. oryzae beer contained lower amounts of hydrogen sulfide, methanethiol, butanedione, and pentanedione compared to barley malt beer. Sorghum–A. oryzae lager beer had similar yield and alcohol content compared to the barley malt beer but differed in color, key volatiles and aromatic compounds.
Typically, barley has been the most relevant raw material for beer production. Malt production is considered one of the oldest and complex examples of applied enzymology (Gupta et al. 2010). However, the growing interest for gluten-free products and need for efficient cultivars, especially in terms of water usage and drought resistance, have led to consider crops like sorghum. Sorghum is the fifth most important cereal worldwide with an annual production exceeding 57 million tons in 2017 (Food Agriculture Organization 2019). This cereal crop, widely adapted to arid and subtropical ecosystems around the globe, can play a dual role as refined brewing adjuncts or as a source of diastatic malt. For centuries, indigenous sorghum beers, also known as opaque or kaffir, have occupied a prominent place in the diets of many African people, and the industrialization of this kind of beer started more than 70 years ago (Bogdan and Kordialik-Bogacka 2017; Odibo et al. 2002). The main problems when brewing with sorghum are the lower diastatic power of its malt, especially deficient in β-amylase activity, and the comparatively higher gelatinization temperature of sorghum starch compared to barley starch (Serna-Saldívar 2010). In a previous research, Espinosa-Ramírez et al. (2013a, 2013b) successfully produced lager beers from different types of sorghum malts and gluten-free adjuncts, supplemented with β-amylase or amyloglucosidase.
On the other hand, Aspergillus oryzae, commonly known as Koji, is a fungus important for the production of traditional fermented foods and beverages in Japan and China due to its ability to secrete large amounts of amylolytic, lipolytic and proteolytic enzymes. These features have facilitated the use of A. oryzae in modern biotechnology (Machida et al. 2005; Barbesgaard et al. 1992). In food fermentation, A. oryzae secretes significant amounts of amylases and proteases to breakdown complex starches to sugar and proteins to peptides, which are further fermented by yeast and lactic acid bacteria (Kobayashi et al. 2007). Both A. oryzae and its enzymes are accepted as constituents of foods (FAO and WHO 1988). A previous study by Heredia-Olea et al. (2017) has analyzed the use of A. oryzae with sorghum malt. The research evaluated the use of A. oryzae (Koji) as a supplement for sorghum malt to increase the enzyme content of the malt without the direct addition of exogenous enzymes. The results of this study show that the sorghum malt originally inoculated with 1% of A. oryzae improved sorghum malt quality in terms of α-amylase and amyloglucosidase activities. Consequently, worts contained higher amounts of Free Amino Nitrogen (FAN) and fermentable carbohydrates. All this, without the generation of waste or additional byproducts that are generated with the addition of purified enzymes, to our knowledge there is no information about the production of lager beers using sorghum malt inoculated with A. oryzae. Thus, the main objective of this investigation was to produce and estimate yields of lager beers from optimally malted sorghum with and without A. oryzae inoculation and to assess properties of resulting beers in terms of extraction, alcohol content and physicochemical attributes.
The white sorghum was donated by INIFAP Rio Bravo whereas the A. oryzae 22788 was acquired from the American Type Culture Collection (ATCC). Barley malt for the control mash, hops (Humulus lupus) and fresh lager brewing yeast of the Saccharomyces spp. strain for all experimental treatments were donated by Cuauhtémoc-Moctezuma–Heineken Brewery. Native cornstarch and yellow maize grits with a particle size of 40–60 United States (US) mesh were obtained from Industrias Mexstarch SAPI de CV and Agroindustrias Integradas del Norte SA de CV.
Sorghum characterization
The thousand-sorghum kernel weight was determined by weighing 100 randomly selected whole kernels (Serna-Saldivar 2012). Test weight was measured with a Winchester Bushel Meter (Seedburo Equipment Company) according to official U.S. grain standard procedures (AACC 2000). Moisture (method 44-15), protein (method 46-13.01), ash (method 08-01) (AACC 2000) and starch (method 996.11) (AOAC 1980) were assayed based on the respective methods.
Oryzae culture
Aspergillus oryzae was cultured in potato dextrose agar plates at 30 °C for 5 days. Then, spores were collected and counted in a Neubauer chamber and inoculated into potato dextrose broth and allowed to grow for 5 additional days (Kammoun et al. 2008). Media was centrifuged at 1000g for 10 min in sterile tubes to concentrate mycelium.
Malting process
For steeping, sorghum grains were mixed with two parts of water (28 °C) containing 0.01% formaldehyde for 30 h in a cabinet set at 28 °C. Steeping was conducted under aeration to enhance aerobiosis and respiration. Sorghum samples previously steeped were inoculated with 1% w/w A. oryzae mycelium and then placed in plastic trays in a germination cabinet set at 28 °C with 90% relative humidity. Moisture losses due to evaporation were controlled by the spraying of water periodically. Samples were collected every 24 h throughout the 96 h germination time. Starch (method 996.11) and free amino nitrogen or FAN (method 945.30) were measured daily according to the Association of Official Analytical Collaboration (AOAC) 1980. Malting losses were determined by comparing the weight of the dry malt to the dry weight of the original amount of sorghum kernels. The germinated grains were dehydrated for 24 h in a forced convection oven set at 50 °C. The malts were coarsely milled using the break rolls of the Chopin mill. Malt losses were obtained using Eq. 1
$$ {\text{Malting}}\;{\text{loss}} \left( \% \right) = \frac{{{\text{Sorghum}}\;{\text{dry}}\;{\text{matter}} - {\text{Malt}}\;{\text{dry}}\;{\text{matter}} }}{{{\text{Sorghum}}\;{\text{dry}}\;{\text{matter}}}}. $$
Mashing procedure
The grist formulation consisted of 60% malt (barley, sorghum or sorghum–A. oryzae), 20% native cornstarch and 20% maize grits. The brewing adjuncts were mixed with 10% of the total amount of malt, and then mixed with deionized water in a 1:2.3 ratio. The double mashing was performed according to the procedure of Heredia-Olea et al. (2017). The ° Plato and volume of resulting wort were measured to calculate the amount of water needed to adjust the wort to 15° P. The final adjusted volume was recorded as wort yield. Cascade hop pellets were added to the sweet wort adjusted to 15° Plato in a rate of 0.35 g/L. In a vessel, 90% of the hops were added to the sweet wort before heating to boiling for 50 min. The 10% remaining hops were added 10 min before discontinuing heat. The hopped wort was centrifuged at 1800g for 10 min to remove the spent hops and trub solids by decantation. The resulting hopped worts were readjusted with sterile water to 15° P.
Prior to the wort inoculation, the percent of yeast solids was determined using a Spin-Down Method (Bendiak 1997). The sample was homogenized by placing the yeast in a glass with agitation for 30 min to remove gas bubbles. The glass containing the sample was maintained in contact with iced water during the preparation. A yeast cell counter Nucleocounter® YC-100TM (Chemometec) was used to measure the total cell count and viability of cells in suspension. The system uses a cassette pre-coated with propidium iodide dye, which stains the cells' nuclei and is then measured by an automatic fluorescence microscope. This method was carried out before introducing the yeast to the fermentation tanks and after the yeast removal. For a lager analysis, the parameter of the Nucleocounter® YC-100TM was set to diploid. Worts from three different malt treatments were obtained from previously malted sorghum, barley and sorghum–A. oryzae. The wort placed in sterilized liter jugs was poured into a 6-L flask inside a microbiological hood furnished with a gas-fired Bunsen burner to prevent cross-contamination. A previously sterilized oxygenator was introduced to the wort, transferring oxygen through a sterilized tube from a valve. The oxygenation time was set to 5 min. Inside the sterilized area, the oxygenated wort was then transferred to a sterilized 3-L fermentation laboratory-scale bioreactor (Applikon®). Once the wort temperature reached 12 °C the yeast was pitched to the tank using a sterile pipette. Yeast was added at a rate of 10 mL/L wort equivalent to 6 × 106 yeast cells/mL. The pH of wort and beer was determined using a potentiometer sensor installed in the fermentation tank. Measurements were taken every minute during the programmed 120 h fermentation. The lager fermentation lasted 120 h. The first 24 h was set at 12 °C and afterward at 15 °C for the remaining time. Samples were collected at 0, 12, 24, 48 and 120 h for analyses. After 120 h of fermentation, the maturation procedure took place. The temperature of the bioreactors was set at 2 °C for 24 h. After the maturation step, the yeast sediment was removed from the fermentation tanks by opening the lower valve located in the cone structure. The yeast was recollected in a flask for further viability analysis. Once all of the yeast was removed, the beer was drained from the fermentation tanks into flasks for further yeast filtration. The remaining yeast was precipitated by centrifugation at 1800g during 10 min and the beer by decantation.
The total volume of the filtered beer was measured using a graduated cylinder. The filtered beer was stored in sterile glass jugs at 2 °C with minimal headspace to prevent oxidation and deterioration until further analysis.
Analytical assays
Sorghum grain germination capacity was determined according to Serna-Saldivar (2012). In brief, a representative sample of 100 caryopses was taken randomly and soaked in excess water for 30 h with aeration. Then, the soaked kernels were placed on Petri dishes with soaked filter paper for controlled germination for 3 days in a cabinet set at 28 °C. Germinated kernels were considered those that developed rootlets and/or acospires (plumulae). Values were expressed on percentage. Starch was measured according to the colorimetric Method 996.11 of the AOAC (1980) whereas FAN of wort and beer by the spectrophotometry methods 8.10 and 9.10 (EBC 2008) based on the ninhydrin reaction. Specific gravity was determined at 20 °C with a digital density meter of the oscillation type using standard method 8.2.2 for worts and 9.43.2 for beers (EBC 2008). The extracts of worts were determined with a density meter following method 8.3 (EBC 2008). In beers, the calculations of original, real and apparent extracts were based on the original and specific gravity determinations following the method 9.4 (EBC 2008). Color of hopped wort was measured by spectrophotometry according to the European Brewery Convention (EBC) and pH with a potentiometer previously calibrated with 4.0 and 7.0 pH buffers (EBC 2008). The yield was determined by measuring of wort volume (15° Brix) obtained per kilogram of dry materials (Cortes-Ceballos et al. 2015). Spent grains and solids lost after centrifugation were determined gravimetrically.
The contents of fermentable carbohydrates fructose, glucose, disaccharides and trisaccharides were determined using a High-Performance Liquid Chromatograph (HPLC) equipped with a Refractive Index detector following the method 8.7 for worts and 9.27 for beers (EBC 2008). The ethanol in beer was determined using the Near Infrared Spectroscopy (NIR) rapid method EBC 9.2.6 (2008). The determination of lower boiling point volatile compounds in beer (alcohols, esters, acetaldehyde, and dimethyl sulfide) was measured by automatic headspace gas chromatography (method 9.39 of the EBC 2008) equipped with a chemically bonded fused silica capillary column and flame ionization detector. Volatile compounds were determined comparing with authentic international standards. Colors of worts and beers were determined according to methods 8.5 and 9.6, respectively (EBC 2008). The measurement of volatile sulfur compounds in beer was determined by Gas Chromatography equipped with a Sulfur Chemiluminescence Detector (ASBC 1994 Method Beer-44). The determination of vicinal diketones (VKD), 2,3-butanedione (diacetyl), 2,3-pentanedione and their precursors in beers was measured by gas chromatography coupled with an electron capture detector following the headspace method 9.24.2 (EBC 2008).
The Minitab 16 statistical software (Minitab, State College, PA, U.S.A.) was employed to identify statistically significant differences among treatments and compare means with Tukey tests (P ≤ 0.05).
The white sorghum had a thousand-kernel weight of 24.42 ± 0.49 g, apparent bulk density of 79.09 ± 2.75 kg/HL and a subjective endosperm texture of 2.12 ± 0.37 (1 = totally corneous or vitreous, 2.5 = intermediate endosperm texture and 5 = totally floury or chalky). In terms of chemical composition, the sorghum contained 10.07 ± 0.03% moisture, 9.73 ± 0.05% protein, 69.27 ± 0.52% starch, 5.09 ± 0.16% crude fiber and 2.03 ± 0.13% ash expressed on a dry matter basis. The physical and chemical properties fall within expected values for this cereal grain (Serna-Saldivar and Espinosa-Ramirez 2018; Serna-Saldivar 2010).
The addition of 1% (w/w) A. oryzae mycelium to sorghum kernels before malting did not affect the sorghum germination capacity, 98.33 ± 1.03% vs 98.75% ± 1.98% for the control sorghum malt. After 4 days of germination, the regular and sorghum–A. oryzae malts lost 9.9% and 14.8% of their dry matter, respectively. The granulation of particle size distribution is critically important because it affects lautering or filtration rate and the availability of malt reserve components to enzymes during mashing. Table 1 compares the particle size distributions of the three malts. There was a significant difference between the barley malt and both sorghum malts especially in terms of particles retained by the biggest sieve (US mesh No 20). This large sieve retained most of the glumes or husks of the barley malt. This is because the sorghum kernel is considered a naked caryopsis where the glumes usually detach during harvesting. A distinguished difference can also be noted between the barley malt and both the sorghum malts in the smallest mash (− 100). This difference can be attributed to the difference in endosperm texture, which was harder for sorghum. The barley malt normally has a floury and friable endosperm, which is more prone to milling and therefore generate fine particles composed mostly of starch granules that pass the 100 mesh sieve. The barley malt contained less FAN (1.088 mg/g malt) compared to the sorghum malts. Figure 1 shows the assayable starch and FAN contents of malts during germination. At the same day of malting, there were not differences between starch contents. Total starch content gradually diminished throughout germination in both treatments. The highest starch change was observed after the first 24 h of germination with a reduction of 4 units. The starch content of malts germinated for 96 h decreased 5 and 6 units in regular sorghum and sorghum–A. oryzae malts, respectively. The FAN values between malts were about the same between the original grains and malts germinated for only 24 h. However, the FAN contents greatly increased when malts were germinated for more than 48 h. FAN values gradually increased with germination time so the regular sorghum and sorghum–A. oryzae malts germinated for 96 h contained at least 6 and 8 times more FAN compared to the unmalted sorghum kernels. The higher FAN content of the sorghum–A oryzae malt is attributed to the presence of additional exogenous proteolytic enzymes synthesized by the mold.
Table 1 Particle size distributions of malts ground using a Chopin roller mill
Changes in starch (dry basis) and Free Amino Nitrogen (mg/g) contents during germination of sorghum with or without A. oryzae
Wort properties
Table 2 data show that the worts obtained after the double mashing, lautering and the hop boil processes were within the specific gravity and dissolved solid contents expected for lager beers (Gialleli et al. 2017). The volume of 15° Plato wort was used as an indicator to quantify differences in wort yields from the various malts. The barley and sorghum worts yielded lower wort volumes compared to the wort obtained from the grist containing the sorghum–A. oryzae malt (Table 2). The regular sorghum malt yielded 17.2% less volume compared to the barley malt. Interestingly, the sorghum–A. oryzae treatment yielded 6.5% more wort volume and 27.6% compared with worts produced by the barley malt and sorghum malt, respectively. This relevant finding indicates that the proposed strategy to enhance enzymatic activity of the sorghum malt was effective and promising for the production of gluten-free beers. The amount of dry brewers spent grains obtained after lautering and centrifugation indicated that the regular sorghum yielded the highest amounts (Table 2). The amounts of spent grains generated by the barley malt and sorghum–A. oryzae malt were approximately 25 and 35% less compared to the regular sorghum. In terms of specific gravity, pH, original extract and fructose and maltotriose contents, there were not differences among the three worts. In all worts, maltose was the most abundant fermentable sugar followed by glucose. However, the malt treatment affected maltose and glucose concentrations in worts. The regular sorghum and barley worts contained the highest and lowest glucose values (Table 2). In case of maltose concentrations, the wort from barley contained approximately 35% and 29% more maltose compared to the regular sorghum and sorghum–A. oryzae worts, respectively. The wort produced from sorghum–A.oryzae malt contained 21.7 and 6.8% more FAN concentrations compared to the barley and regular sorghum worts. The color of worts (Table 2) was significantly affected by the sort of malt. Compared to the barley malt wort, both sorghum worts showed almost twofold color values.
Table 2 Quality parameters, yields and carbohydrates profiles of worts (15° Plato) produced using malts form barley, and sorghum without or with A. oryzae
Fermentation analyses
There were not differences between treatments in terms of yeast in suspension (10.55 ± 0.33%) and dead cells (1.89 ± 0.21%). Figure 2 depicts changes of fermentable sugars during 120 h fermentation of the different worts. The charts display that glucose and fructose were totally consumed after 50 h. On the other hand, maltose and maltotriose were gradually utilized throughout the whole fermentation process. Maltose was totally fermented in all worts after 120 h of fermentation whereas small amounts of maltotriose remained in the barley and sorghum–A. oryzae worts. For fructose, an increment in its concentration was visualized after 15 h of fermentation.
Fermentable sugar depletion in worts from barley, sorghum or sorghum with A. oryzae fermented during 120 h. a Maltose (%); b glucose (%); c maltotriose (%); d fructose (%)
There were not differences between pH, FAN, superior alcohols, acetaldehyde, ethyl acetate, isoamyl acetate, dimethylsulfide and S-methyl acetate among treatments (Table 3). As expected there were close relationships among beer specific gravity, attenuation and ethanol concentrations (Table 3). The regular sorghum beer specific gravity was about 0.49% higher compared to the barley malt and A. oryzeae–sorghum beers. The observed differences are attributed to the almost 11% higher ethanol concentrations observed in the barley and A. oryzae–sorghum beers. As a result, the apparent and true attenuation values of the regular sorghum treatment were about 11.5% and 7% lower compared to the barley beer and sorghum–A. oryzae beers, respectively (Table 3). Interestingly, the beer produced with sorghum contained 37% less ethyl hexanoate compared to the counterpart produced with barley malt. In terms of sulfur compounds, hydrogen sulfide content was highest for the barley beer and lowest for the sorghum–A. oryzae beer. On the other hand, methanethiol content was comparatively higher for barley malt beer than the sorghum–A. oryzae beer. Large differences between the concentration of VKD compounds 2,3 butanedione and 2,3 pentanedione were observed when both sorghum beers were compared with the barley beer. The amounts of these compounds were approximately 10% of the concentrations assayed in the barley malt beer (Table 2). Likewise, the EBC color of the barley beer was significantly higher compared to the two experimental sorghum beers.
Table 3 Composition and characteristics of beers from barley, and sorghum without or with A. oryzae
Malt properties
The main chemical changes that grains undergo during malting are the reduction in assayable starch and hemicellulose contents. During this physiological process, the cell bound components weaken due to the synthesis of degrading enzymes like cellulases and arabinoxylanases. The hydrolyzed cell walls allow the entry of other relevant enzymes that will degrade protein bodies and matrix and starch granules (Boulton and Quain 2001). There are many investigations which clearly conclude that malted sorghum has significantly lower diastatic activity compared to barley malt especially in terms of maltose-producing β-amylase. These deficiencies have been counteracted by the addition of exogenous enzymes like α and β-amylases, glucoamylase or amyloglucosidase and even proteolytic enzymes that improve the exposure of the starch granules to amylolytic enzymes (Serna-Saldivar and Rubio-Flores 2016). The idea of inoculating A. oryzae during the malting process of sorghum is based in the fact that this mold synthesizes relevant enzymes like amylases and proteases that enhance the conversion of starch and proteins into fermentable carbohydrates and simpler and soluble nitrogenous compounds, respectively. Heredia-Olea et al. (2017) observed that α-amylase activity of sorghum malt was positively affected by addition of A. oryzae but it did not affect β-amylase activity. The conversion of starch into linear and branched dextrins explains the significant reduction of starch content during germination. In addition, the proteases generated are distributed to the entire caryopsis and hydrolyze conjugated proteins associated with amylases in order to activate starch-degrading enzymes. The proteases also degrade germ and endosperm proteins, solubilizing approximately 30% of the total protein. The sorghum malt contained higher amounts of readily assimilable amino acids (FAN) compared to the barley malt and other cereals such as wheat (Hill and Stewart 2019) likely enhancing yeast nutrition during beer fermentation and production of relevant fusel alcohols.
The regular sorghum malt generated less 15° P wort yield compared to the barley malt (Table 2). Similar difference was reported by Espinosa-Ramírez et al. (2013a) who observed that barley malt yielded was up to 24% more 12 °P wort and similar FAN and pH values compared with their counterparts produced with white or red sorghum malts. According to Espinosa-Ramírez et al. (2014) and Heredia-Olea et al. (2017), the use of regular sorghum malt usually yields worts with lesser amounts of fermentable carbohydrates and FAN contents and their fermented beers with lower alcohol contents. These differences, especially in terms of generation of maltose and glucose, may be attributed to the higher β-amylase and amyloglucosidase activities of the barley and sorghum–A. oryzae malts. It is well known that malt and adjunct starches are hydrolyzed to dextrins and sugars during mashing (Serna-Saldivar and Espinosa-Ramirez 2018). In terms of the preferred carbon source, Carlsen and Nielsen (2001) studied the influence of maltose and maltodextrins differing in chain length, glucose, fructose, galactose, sucrose, glycerol, mannitol and acetate on α-amylase production by A. oryzae. Productivity was found to be higher during growth on maltose and maltodextrins, which are present in relatively high amounts in typical worts. Maltose was the most abundant carbohydrate in the control and experimental worts (Table 2). This is due to the concerted enzymatic action of α- and β-amylases. The sorghum worts contained more glucose and less maltose than the barley wort. This is related to the higher β-amylase activity reported in the barley malt. Even though the worts were adjusted to 15° Plato, the total amount of fermentable sugars did not total 15% due to the presence of dextrins. Dextrins commonly account for 90% of the residual carbohydrate of beer because regular yeast is not capable of fermenting these carbohydrates. Approximately, 40–50% of the dextrins are oligosaccharides containing 4–9 glucose units, and the remaining 50–60% are higher dextrins with 10 or more glucose units (Boulton and Quain 2001). Espinosa-Ramírez et al. (2013a) enhanced the amounts of fermentable sugars in the sorghum malt worts with the addition of β-amylase. When amyloglucosidase was added, the total sugar content increased 20% and consequently the glucose content was five times higher compared with worts without exogenous enzymes. It is well known that A. oryzae is a mold that synthesizes large amounts of amylolytic, proteolytic and lipolytic enzymes. In fact, commercial mold cultures are used to produce and isolate enzymes widely used by the food industries. The main disadvantage of sorghum malt is its relatively low production of β-amylase, key enzyme in brewing operations because it complements the activity of α-amylase. Furthermore, the color of the beer is influenced by Maillard reactions that occur between sugars and amino-compounds (including the amino acids) during the hop-boil step, which gave rise to colored and flavored substances. The proportions of the flavored fermentation products made by yeast are dependent on the nitrogenous substances that are present. Nitrogenous components are present in the form of amino acids, small peptides, and proteins. Recommended FAN concentrations of wort range from 150 to 200 mg/L (Boulton and Quain 2001). Table 1 clearly depicts that the barley wort contained slightly lower FAN values compared to the sorghum with and without A. oryzae counterparts. In fact, the sorghum wort with A. oryzae contained 27% more FAN compared to the control barley counterpart. The significant difference is attributed to the synthesis and action of proteases (mainly carboxypeptidases) produced by the Aspergillus. Malt carboxypeptidases have optimum activity at temperatures between 40 and 60 °C and are inactivated at 70 °C. Likely the double mashing process gave these enzymes the opportunity to hydrolyze proteins into FAN components (Boulton and Quain 2001). The proteases generated by the mold facilitated the entrance of amylases associated to the sorghum malt and A. oryzae. This mold is known to express high amounts of amyloglucosidases that convert linear and branched dextrins into glucose (Heredia-Olea et al., 2017). The hydrolyzed peptides are converted to higher alcohols during fermentation (Barredo-Moguel et al. 2001). Briggs et al. (2004) indicate that 100 to 140 mg/L is regarded as the minimum level of FAN needed for trouble-free fermentations. The soluble proteins and polypeptides that remain in the fermented wort contribute to the `body' and `mouth-feel' of the beer, its foaming properties, and its susceptibility to haze formation.
Fermentation and beer parameters
Glucose and fructose consumption profiles during fermentation followed the expected trend. According to Cason et al. (1987), glucose and fructose are taken up by the same membrane transport system which explains the similar consumption profiles. The maltose consumption profiles were the same for all treatments. Cason et al. (1987) observed that utilization rates of maltose were identical independently of the adjunct concentration. The rapid consumption of glucose and fructose exerted catabolite repression on the maltose membrane transport system or any of the subsequent metabolic steps of maltose catabolism to glucose. Therefore, maltose started to be utilized when glucose or fructose levels fall to a cut-off point below which catabolite repression did not occur (Cason et al. 1987). The metabolism of maltose and maltotriose is highly interconnected. Both sugars are α-glucosides transported by the activated α-glucoside-Hc symporter encoded by gen AGT1. This permease, which is maltose inducible, has the same affinity for maltose and maltotriose (Zastrow et al. 2000). These authors observed a single exponential growth phase of maltotriose fermented by Saccharomyces cereviseae grown in medium containing glucose, maltose or maltotriose as carbon and energy sources indicating that the metabolism of this particular fermentable sugar was oxidative. In terms of sugar consumption, fermented beers for the three malt treatments did not show any significant differences.
Final specific gravity value in beer is closely related to the final ethanol content (Esslinger 2009) and this relationship was observed in all beers. In case of attenuation, the values obtained herein were lower compared to the ones reported by Esslinger (2009) for regular barley and sorghum beers where a final attenuation of 82.1% and 79.7% was observed, respectively. In terms of beer specific gravity, pH, ethanol, and FAN contents, the use of sorghum–A. oryzae improved values compared to the use of only sorghum malt and similar to the barley malt beer. Espinosa-Ramírez et al. (2014) obtained similar ethanol contents when barley malt beers were compared with sorghum malt beers produced with exogenous amyloglucosidase. The superior alcohols' yeast metabolism was not affected by the malt treatment despite dissimilarities in FAN concentrations. Esters are the products of the enzymatic catalysis of organic acids, ethanol and higher alcohols. Their formation is closely related to yeast propagation and lipid metabolism (Pires et al. 2014). Beer contains more than 50 different esters, from which three are of higher relevance because they greatly affect beer flavor: ethyl acetate, iso-amyl acetate, iso-butyl acetate. Since the sorghum–A. oryzae treatment had sixfold and fourfold less hydrogen sulfide than barley beer and sorghum beer, respectively, the difference can be attributed to its higher rate of catabolism mediated by the mold. In terms of hydrogen sulfide production, all beer treatments were statistically different, where only the sorghum–A. oryzae beer was under the threshold level of 5 ppb. It arises through yeast autolysis at the end of fermentation or during maturation. The best understood carbonyl flavor compounds associated with yeast fermentation are the VKD, which include diacetyl and 2,3 pentanedione. This led to high levels of VKD, during fermentation owing to effects on the regulation of valine synthesis by the yeast. According to Esslinger (2009) the beer taste thresholds of diacetyl and 2,3 pentanedione range from 0.08 to 0.2 ppm and from 0.5 to 0.6 ppm. Beer from barley contained significantly higher levels of both butanedione and 2,3 pentanedione whereas beer manufactured with the sorghum–A. oryzae malt contained about 96% lower content of VDK. This important difference in volatile compounds needs to be further researched especially in terms of beer stability and sensory analysis. According to the EBC scale, the sorghum beers are classified as pale whereas the slightly darker yellowish barley beer as Pilsner. The final beer color is influenced by the type of malt, color of brewing adjuncts, concentration and type of hops and pH (Esslinger 2009). The almost twice as high color score observed in the barley beer is attributed to the utilization of a malt rich in glumes that contain phenolic compounds that lixiviated into the wort and the production of Maillard type of compounds during boiling (Granato et al. 2011). It is worth mentioning that both sorghum beers were produced from naked caryopses of white sorghum that do not contain tannins and were low in phenolic compounds (Serna-Saldivar and Espinosa-Ramirez, 2018). Differences of ethyl hexanoate, hydrogen sulfide, methanethiol and VKD compounds could have also affected beer color. According to Dack et al. (2017), Maillard reaction products inhibit the synthesis of esters due to possible suppression of enzymes and/or gene expression linked to ester synthesis. The impact of FAN on formation of flavor and aroma compounds during fermentation has been previously studied. Initial wort FAN content and the amino acid and ammonium ion equilibrium in the medium impact the formation of esters, aldehydes, VKD, superior alcohols and acids, as well as sulfur compounds. Even small differences in wort composition can exert significant effect on the flavor of the resulting beer (Hill and Stewart 2019). According to Taylor et al. (2013), and Kobayashi et al. (2008), there is some indication that the differences in the free amino acid profile of sorghum malt worts compared with barley malt worts could influence beer flavor by affecting yeast metabolism. Sorghum malt worts were found to contain low levels of branched chain valine. The amino acid profiles of barley and sorghum differed because the first commonly contained 2.2 g of methionine and 1.8 g of cysteine per 100 g of protein whereas the second 1 g of methionine and 1.6 g of cysteine per 100 g of protein (Serna-Saldivar 2010).
Simultaneous solid-state fermentation with 1% A. oryzae of sorghum undergoing germination produced a malt that increased wort and beer yield (12% more ethanol) compared to the regular sorghum malt. Beers produced with the A. oryzae–sorghum malt had similar yield and alcohol content compared to the barley malt beer but differed in color, key volatile and aromatic compounds. Further investigations into changes in protein solubility, amino acids' profile changes during solid-state fermentation of sorghum malt with A. oryzae and it relation with changes in superior alcohols and other flavor beer components is in need.
The data that support the findings of this study are available from the corresponding author Sergio O. Serna-Saldivar (SOSS), upon reasonable request.
A. oryzae :
Aspergillus oryzae
AACC:
American Association of Cereal Chemists
ABSC:
American Society of Brewing Chemists
AOAC:
Association of Official Analytical Collaboration
ATCC:
American Type Culture Collection
EBC:
European Brewing Convention
Free amino nitrogen
Food Agriculture Organization
HPLC:
High-performance liquid chromatograph
VKD:
Vicinal diketones
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Cortes-Ceballos E, Nava-Valdez Y, Pérez-Carrillo E, Serna-Saldívar SO (2015) Effect of the use of thermoplastic extruded corn or sorghum starches on the brewing performance of lager beers. J Am Soc Brew Chem 73(4):318–322. https://doi.org/10.1094/ASBCJ-2015-1002-01
Dack RE, Black GW, Koutsides G, Usher SJ (2017) The effect of Maillard reaction products and yeast strain on the synthesis of key higher alcohols and esters in beer fermentation. Food Chem 232:595–601. https://doi.org/10.1016/j.foodchem.2017.04.043
EBC (2008) Analtytica European brewing convention. Fachverlag Hans Carl Gmbh, Nuremberg
Espinosa-Ramírez J, Pérez-Carrillo E, Serna-Saldívar SO (2013a) Production of lager beers from different types of sorghum malts and adjuncts supplemented with β-amylase or amyloglucosidase. J Am Soc Brew Chem 71:208–213. https://doi.org/10.1094/ASBCJ-2013-0914-01
Espinosa-Ramírez J, Pérez-Carrillo E, Serna-Saldívar SO (2013b) Production of brewing worts from different types of sorghum malts and adjuncts supplemented with β-amylase or amyloglucosidase. J Am Soc Brew Chem 71:49–56. https://doi.org/10.1094/ASBCJ-2013-0125-01
Espinosa-Ramírez J, Pérez-Carrillo E, Serna-Saldívar SO (2014) Maltose and glucose during fermentation of barley and sorghum lager beers as affected by β-amylase or amyloglucosidase addition. J Cereal Sci 60:602–609. https://doi.org/10.1016/j.jcs.2014.07.008
Esslinger HM (2009) Handbook of brewing: processes, technology, markets. Wiley VCH, Weinheim
FAO, WHO (1988) Joint FAO/WHO expert committee on food additives. WHO Tech Rep Ser 759:16–17
Food Agriculture Organization (2019) Statistical database. Rome. http://faostat.fao.org. Accessed 15 Feb 2019
Gialleli A-I, Ganatsios V, Terpou A, Kanalleki M, Bekatorou A, Koutinas AA, Dimitrellou D (2017) Technological development of brewing in domestic refrigerator using freeze-dried raw materials. Food Technol Biotechnol 55:325–332. https://doi.org/10.17113/ftb.55.03.17.4907
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Gupta M, Abu-Ghannam N, Gallaghar E (2010) Barley for brewing: characteristic changes during malting, brewing and applications of its by-products. Comp Rev Food Sci Food Saf 9:318–329. https://doi.org/10.1111/j.1541-4337.2010.00112.x
Heredia-Olea E, Cortés-Ceballos E, Serna-Saldívar SO (2017) Malting sorghum with Aspergillus oryzae enhances gluten-free wort yield and extract. J Am Soc Brew Chem 75:116–121. https://doi.org/10.1094/ASBCJ-2017-2481-01
Hill AE, Stewart GG (2019) Free amino nitrogen in brewing. Fermentation 5(1):22. https://doi.org/10.3390/fermentation5010022
Kammoun R, Naili B, Bejar S (2008) Application of statistical design to the optimization of parameters and culture medium for a-amylases productiono of Aspergillus oryzae CBS 819.72 grown on gruel (wheat grinding by-product). Bioresource Technology. 99(13):5602–5609
Kobayashi T, Abe K, Asai K, Gomi K, Juvvdi PR, Kato M, Kitamoto K, Takeuchi M, Machida M (2007) Genomics of Aspergillus oryzae. Biosci Biotechnol Biochem 7:649–670. https://doi.org/10.1271/bbb.60550
Kobayashi M, Shimizu H, Shioya S (2008) Beer volatile compounds and their application to low-malt beer fermentation. J Biosci Bioeng 106:317–323. https://doi.org/10.1263/jbb.106.317
Machida M, Asai K, Sano M, Tanaka T, Kumagai T, Terai G, Kusumot K-I, Arima T, Akita O, Kashiwagi Y, Abe K, Gomi K, Horiuchi H (2005) Genome sequencing and analysis of Aspergillus oryzae. Nat Lett 438:1157–1161. https://doi.org/10.1038/nature04300
Odibo FJC, Nwankwo LN, Agu RC (2002) Production of malt extract and beer from Nigerian sorghum varieties. Process Biochem 37:851–855. https://doi.org/10.1016/S0032-9592(01)00286-2
Pires EJ, Teixeira JA, Brányik T, Vicente AA (2014) Yeast: the soul of beer's aroma—a review of flavour-active esters and higher alcohols produced by the brewing yeast. Appl Microbiol Biotechnol 98:1937–1949. https://doi.org/10.1007/s00253-013-5470-0
Serna-Saldivar SO (2010) Cereal grains properties, processing and nutritional attributes. CRC Press, Boca Raton
Serna-Saldivar SO (2012) Cereal grains: laboratory reference and procedures manual. CRC Press, Taylor & Francis Group, Boca Raton
Serna-Saldivar SO, Espinosa-Ramirez J (2018) Grain structure and grain chemical composition. Chapter 5 In: Sorghum & millets: chemistry, technology and nutritional attributes. In: Taylor JRN, Duodu KG (editors). 2nd ed. Elsevier (Woodhead Publishing), AACC-International, pp 85–129
Serna-Saldivar SO, Rubio-Flores M (2016) Role of intrinsic and supplemented enzymes in brewing and beer properties. Chapter 15 In: Microbial enzyme technology and food applications. Ray RC, Rosell CM, (eds). CRC Press Taylor & Francis Group, Boca Raton, pp 273–298
Taylor JR, Dlamini BC, Kruger J (2013) 125th Anniversary Review: The science of the tropical cereals sorghum, maize and rice in relation to lager beer brewing. J Inst Brew 119:1–14. https://doi.org/10.1002/jib.68
Zastrow C, Mattos M, Hollatz C, Stambuk BU (2000) Maltotriose metabolism by Saccharomyces cerevisiae. Biotechnol Lett 22:455–459. https://doi.org/10.1023/A:1005691031880
The authors would like to thank Cervecería Cuahutemoc Moctezuma.
This research was supported by the research group of Nutriomics and Emerging Technologies from Tecnologico de Monterrey.
Tecnológico de Monterrey, Escuela de Ingeniería y Ciencias, Centro de Biotecnología FEMSA, Ave. Eugenio Garza Sada 2501 Sur, CP 64849, Monterrey, N.L., Mexico
Monica Rubio-Flores, Arnulfo Ricardo García-Arellano, Esther Perez-Carrillo & Sergio O. Serna-Saldivar
Monica Rubio-Flores
Arnulfo Ricardo García-Arellano
Esther Perez-Carrillo
Sergio O. Serna-Saldivar
SOSS conceptualize the idea, designed the work. MRF and ARGA performed experiments. All authors discussed the results. MRF, SOSS, and EPC cowrote the paper. All authors read and approved the final manuscript.
Correspondence to Sergio O. Serna-Saldivar.
This article does not contain any studies with human participants or animals.
I, Sergio O. Serna-Saldivar (SOSS), the corresponding author, hereby declare that it is my study and I developed the manuscript titled "Use of Aspergillus oryzae during sorghum malting to enhance yield and quality of gluten-free lager beers".
Rubio-Flores, M., García-Arellano, A.R., Perez-Carrillo, E. et al. Use of Aspergillus oryzae during sorghum malting to enhance yield and quality of gluten-free lager beers. Bioresour. Bioprocess. 7, 40 (2020). https://doi.org/10.1186/s40643-020-00330-w
DOI: https://doi.org/10.1186/s40643-020-00330-w | CommonCrawl |
Find the word definition
Enter the word
What is "mutation"
Crossword clues for mutation
Sport of a sort
An organism that has characteristics resulting from chromosomal alteration
The event consisting of a change in genetic structure
COLLOCATIONS FROM CORPUS
■ ADJECTIVE
▪ Now, suppose that deleterious mutations reduce survival below this optimal value.
▪ Senescence of clones is probably caused by the accumulation of deleterious mutations.
▪ Furthermore, the total rate of production of deleterious mutations and their pattern of age-specificity are unknown.
▪ Suppose now that deleterious mutations are age-specific.
▪ Andean vultures become avid for the life-giving molecule with quite a different set of mutations.
▪ Over 40 different mutations of the phenylalanine hydroxylase gene have been identified.
▪ Ironically, a similar belief prevails today, in a slightly different mutation.
▪ This is a worthwhile approach, though one should bear in mind that different mutagens give qualitatively different kinds of mutation.
▪ Fears that radiotherapy would cause genetic mutations leading to handicaps in offspring appear to be groundless, according to studies among 3,000 survivors.
▪ But in the 1980s, scientists found that a genetic mutation was responsible.
▪ Variations occur within a population, explicable as genetic mutations or the results of mixing of genetic material.
▪ On the other hand, most cases of the disease seem to develop without genetic mutations, Gibbs said.
▪ With enough genetic mutations at hand, the behaviour could perhaps have evolved independently in each species.
▪ Some people may have a predisposition to the genetic mutations that lead to disease.
▪ Evolution requires genetic change, mutation.
▪ Trichloroethene, a probable human carcinogen, can cause liver damage and genetic mutations in both human and animal populations.
▪ Long-inbred populations might be useful for assessing the effects of new mutations.
▪ It is people like Dawkins who are the new genetic mutations which will spread only if they have superior survival value.
▪ In outbred populations, selection acts on new mutations mainly through their heterozygous effects.
▪ It presses new mutations into service as they arise and is just as ready to make do with what is already around.
▪ When necessary, bacteria cheat to ensure a supply of new mutations.
▪ To begin with, this was the only such pattern in existence, but then a new mutation arose.
▪ Whether his fainting goats were a new mutation or part of an older breed remains unclear.
▪ In splurge-weed, the new mutation can arise in any cell, in any branch of the plant.
▪ When they seemed to resemble each other rather too closely, he introduced random mutations in the offspring.
▪ The genes that cause the elaborate ornament or long tail to appear are subject to random mutation.
▪ Our random mutation is essential because it is unnecessary.
▪ The more elaborate the ornament, the more likely that a random mutation will make the ornament less elaborate, not more.
▪ We want them to emerge solely as a result of cumulative selection of random mutations.
▪ It follows that some process other than random mutation and selection must be involved.
■ NOUN
▪ K-ras gene mutations, by contrast with p53, clearly occur at an earlier stage in the neoplastic sequence.
▪ Other workers have found ras gene mutations in between 39% and 47% of colorectal cancers.
▪ Enough premalignant cells are present in the bulk of stool to permit the analysis of tumour suppressor gene mutations by this technique.
▪ Loss of tumour suppressor function requires inactivation of both alleles, usually by chromosomal deletion or point mutation, or both.
▪ The use of immunohistochemical staining as a marker of point mutation has been critically reviewed by Wyndord-Thomas.
▪ Nineth five percent of large bowel cancers showing loss of heterozygosity for 17p alleles also contain a point mutation.
▪ Thus, both of the point mutations characterized here would be expected to eliminate kinase activity of the proposed atk protein product.
▪ Apart from point mutation, another way in which oncogenes can be activated is by over expression.
▪ If mutagens like cosmic rays are present then all normal mutation rates are boosted.
▪ It strives for a mutation rate of zero.
▪ For the last 20 years researchers have been able to calculate genome sizes and mutation rates.
▪ By far the shakiest part of the calculation is the average mutation rate.
▪ Moreover, mutation rates seem to change with the physiological state of the organism.
▪ Yes, this is the inverse of what is known as the mutation rate, and it can be measured.
▪ That is, the rate of substitution equals the neutral mutation rate.
▪ They are free to evolve at the mutation rate.
■ VERB
▪ Tests have shown them to restrict growth and cause mutations in micro-organisms.
▪ Each digestion included a positive colorectal carcinoma or adenoma control known to contain a mutation at codon 12.
▪ If this shows a lethal mutation it is classified as a presumed mammalian mutagen.
EXAMPLES FROM CORPUS
▪ A minor mutation should be deemed an eccentricity and nothing more.
▪ A second bug experienced a mutation that allowed it to make use of the acetate excreted from the first.
▪ Living organisms have a similar tradeoff in deciding how much mutation and innovation is needed to keep up with a changing environment.
▪ Previously, disease-causing mutations have been linked to rare or incurable disorders, providing often debatable benefits to small numbers of people.
▪ The other mode depends on occasional small mutations, like the changes in the parameters of protozoa.
The Collaborative International Dictionary
mutation \mu*ta"tion\ (m[-u]*t[=a]"sh[u^]n), n. [L. mutatio, fr. mutare to change: cf. F. mutation. See Mutable.] Change; alteration, either in form or qualities. The vicissitude or mutations in the superior globe are no fit matter for this present argument. --Bacon. 2. (Biol.) Gradual definitely tending variation, such as may be observed in a group of organisms in the fossils of successive geological levels. 3. (Biol.)
As now employed (first by de Vries), a cellular process resulting in a sudden inheritable variation (the offspring differing from its parents in some well-marked character or characters) as distinguished from a gradual variation in which the new characters become fully developed only in the course of many generations. The occurrence of mutations, the selection of strains carrying mutations permitting enhanced survival under prevailing conditions, and the mechanism of hereditary of the characters so appearing, are well-established facts; whether and to what extent the mutation process has played the most important part in the evolution of the existing species and other groups of organisms is an unresolved question.
The result of the above process; a suddenly produced variation.
Note: Mutations can occur by a change in the fundamental coding sequence of the hereditary material, which in most organisms is DNA, but in some viruses is RNA. It can also occur by rearrangement of an organism's chromosomes. Specific mutations due to a change in DNA sequence have been recognized as causing certain specific hereditary diseases. Certain processes which produce variation in the genotype of an organism, such as sexual mixing of chromosomes in offspring, or artificially induced recombination or introduction of novel genetic material into an organism, are not referred to as mutation.
4. (Biol.) a variant strain of an organism in which the hereditary variant property is caused by a mutation[3].
Related phrases:
ocher mutation
Douglas Harper's Etymology Dictionary
late 14c., "action of changing," from Old French mutacion (13c.), and directly from Latin mutationem (nominative mutatio) "a changing, alteration, a turn for the worse," noun of action from past participle stem of mutare "to change" (see mutable). Genetic sense is from 1894.
n. 1 Any alteration or change. 2 (context genetics English) Any heritable change of the base-pair sequence of genetic material.
point mutation
mixed mutation
nasal mutation
nasal mutations
point mutations
soft mutation
soft mutations
aspirate mutation
frameshift mutation
frameshift mutations
hard mutation
missense mutation
missense mutations
nonsense mutation
nonsense mutations
n. (biology) an organism that has characteristics resulting from chromosomal alteration [syn: mutant, variation, sport]
(genetics) any event that changes genetic structure; any alteration in the inherited nucleic acid sequence of the genotype of an organism [syn: genetic mutation, chromosomal mutation]
a change or alteration in form or qualities
gene mutation
chromosomal mutation
genetic mutation
In biology, a mutation is the permanent alteration of the nucleotide sequence of the genome of an organism, virus, or extrachromosomal DNA or other genetic elements. Mutations result from damage to DNA which then may undergo error-prone repair (especially microhomology-mediated end joining), or cause an error during other forms of repair, or else may cause an error during replication ( translesion synthesis). Mutations may also result from insertion or deletion of segments of DNA due to mobile genetic elements. Mutations may or may not produce discernible changes in the observable characteristics ( phenotype) of an organism. Mutations play a part in both normal and abnormal biological processes including: evolution, cancer, and the development of the immune system, including junctional diversity.
Mutation can result in many different types of change in sequences. Mutations in genes can either have no effect, alter the product of a gene, or prevent the gene from functioning properly or completely. Mutations can also occur in nongenic regions. One study on genetic variations between different species of Drosophila suggests that, if a mutation changes a protein produced by a gene, the result is likely to be harmful, with an estimated 70 percent of amino acid polymorphisms that have damaging effects, and the remainder being either neutral or marginally beneficial. Due to the damaging effects that mutations can have on genes, organisms have mechanisms such as DNA repair to prevent or correct mutations by reverting the mutated sequence back to its original state.
Mutation (disambiguation)
A mutation is a change in the sequence of an organism's genetic material.
Mutation may also refer to:
Mutation (novel)
Mutation is a book written by Robin Cook about the ethics of genetic engineering. It brings up the benefits, risks, and consequences.
Mutation (algebra)
In the theory of algebras over a field, mutation is a construction of a new binary operation related to the multiplication of the algebra. In specific cases the resulting algebra may be referred to as a homotope or an isotope of the original.
Mutation (genetic algorithm)
Mutation is a genetic operator used to maintain genetic diversity from one generation of a population of genetic algorithm chromosomes to the next. It is analogous to biological mutation. Mutation alters one or more gene values in a chromosome from its initial state. In mutation, the solution may change entirely from the previous solution. Hence GA can come to better solution by using mutation. Mutation occurs during evolution according to a user-definable mutation probability. This probability should be set low. If it is set too high, the search will turn into a primitive random search.
The classic example of a mutation operator involves a probability that an arbitrary bit in a genetic sequence will be changed from its original state. A common method of implementing the mutation operator involves generating a random variable for each bit in a sequence. This random variable tells whether or not a particular bit will be modified. This mutation procedure, based on the biological point mutation, is called single point mutation. Other types are inversion and floating point mutation. When the gene encoding is restrictive as in permutation problems, mutations are swaps, inversions, and scrambles.
The purpose of mutation in GAs is preserving and introducing diversity. Mutation should allow the algorithm to avoid local minima by preventing the population of chromosomes from becoming too similar to each other, thus slowing or even stopping evolution. This reasoning also explains the fact that most GA systems avoid only taking the fittest of the population in generating the next but rather a random (or semi-random) selection with a weighting toward those that are fitter.
For different genome types, different mutation types are suitable:
Bit string mutation
The mutation of bit strings ensue through bit flips at random positions. Example:
|1 || 0 || 1 || 0 || 0 || 1 || 0 |- | || || || || ↓ || || |- |1 || 0 || 1 || 0 || 1 || 1 || 0 |}
The probability of a mutation of a bit is $\frac{1}{l}$, where l is the length of the binary vector. Thus, a mutation rate of 1 per mutation and individual selected for mutation is reached.
Flip Bit
This mutation operator takes the chosen genome and inverts the bits (i.e. if the genome bit is 1, it is changed to 0 and vice versa).
This mutation operator replaces the genome with either lower or upper bound randomly. This can be used for integer and float genes.
Non-Uniform
The probability that amount of mutation will go to 0 with the next generation is increased by using non-uniform mutation operator. It keeps the population from stagnating in the early stages of the evolution. It tunes solution in later stages of evolution. This mutation operator can only be used for integer and float genes.
This operator replaces the value of the chosen gene with a uniform random value selected between the user-specified upper and lower bounds for that gene. This mutation operator can only be used for integer and float genes.
This operator adds a unit Gaussian distributed random value to the chosen gene. If it falls outside of the user-specified lower or upper bounds for that gene, the new gene value is clipped. This mutation operator can only be used for integer and float genes.
Mutation (knot theory)
In the mathematical field of knot theory, a mutation is an operation on a knot that can produce different knots. Suppose K is a knot given in the form of a knot diagram. Consider a disc D in the projection plane of the diagram whose boundary circle intersects K exactly four times. We may suppose that (after planar isotopy) the disc is geometrically round and the four points of intersection on its boundary with K are equally spaced. The part of the knot inside the disc is a tangle. There are two reflections that switch pairs of endpoints of the tangle. There is also a rotation that results from composition of the reflections. A mutation replaces the original tangle by a tangle given by any of these operations. The result will always be a knot and is called a mutant of K.
Mutants can be difficult to distinguish as they have a number of the same invariants. They have the same hyperbolic volume (by a result of Ruberman), and have the same HOMFLY polynomials.
Mutation (Jordan algebra)
In mathematics, a mutation, also called a homotope, of a unital Jordan algebra is a new Jordan algebra defined by a given element of the Jordan algebra. The mutation has a unit if and only if the given element is invertible, in which case the mutation is called a proper mutation or an isotope. Mutations were first introduced by Max Koecher in his Jordan algebraic approach to Hermitian symmetric spaces and bounded symmetric domains of tube type. Their functorial properties allow an explicit construction of the corresponding Hermitian symmetric space of compact type as a compactification of a finite-dimensional complex semisimple Jordan algebra. The automorphism group of the compactification becomes a complex subgroup, the complexification of its maximal compact subgroup. Both groups act transitively on the compactification. The theory has been extended to cover all Hermitian symmetric spaces using the theory of Jordan pairs or Jordan triple systems. Koecher obtained the results in the more general case directly from the Jordan algebra case using the fact that only Jordan pairs associated with period two automorphisms of Jordan algebras are required.
Back mutation
Usage examples of "mutation".
He went to the bathroom to wash his hands, but this time he did not ask the mirror, metaphysically, What can this be, he had recovered his scientific outlook, the fact that agnosia and amaurosis are identified and defined with great precision in books and in practice, did not preclude the appearance of variations, mutations, if the word is appropriate, and that day seemed to have arrived.
So perhaps he had an edited cerebral chemistry, or an adaptive aural processing mutation in his derivative Kido lineage.
On Gris we send biogenetic teams to Earth every five years to check our own mutation rate.
From the undoubted fact that gene mutations like the Tay-Sachs mutation or chromosomal abnormalities like the extra chromosome causing Down syndrome are the sources of pathological variation, human geneticists have assumed that heart disease, diabetes, breast cancer, and bipolar syndrome must also be genetic variants.
He has to admit, however, that atoms do not aggregate of their own accord, and rather than believe in a superior law and, finally, in the destiny he wishes to deny, he accepts the concept of a purely fortuitous mutation, the clinamen, in which the atoms meet and group themselves together.
This was a type of dwarfism that resulted from a spontaneous mutation.
Ultraviolet gives us hereditary mutation and the euchromatin contains the genes that transmit heredity.
Most new genes that arise, either by mutation or reassortment or immigration, are quickly penalized by natural selection: the evolutionarily stable set is restored.
What counts are mutations in the gametes, the eggs and sperm cells, which are the agents of sexual reproduction.
And the first glob, you know, was itself a mutation, and against stiff statistical odds.
It was an enormous snow-white mutation derived from an arctic gyrfalcon the bird that indeed was reserved for kings.
A very slight variation in haplotype number, the kind of subtle, meaningless mutation that happened in the DNA of a germ cell.
Through green hyaline panels he could see the lift of the EAMH, the Experimental Applied Mutation Hospital, moving, leaving him here isolated.
Tools have always functioned as human prostheses, integrated into our bodies through our laboring practices as a kind of anthropological mutation both in individual terms and in terms of collective social life.
Accidentally useful mutations provide the working material for biological evolution-as, for example, a mutation for melanin in certain moths, which changes their color from white to black.
words rhyming with mutation, words from word "mutation", words starting with "m", words starting with "mu", words starting with "mut", words starting with "muta", words starting with "mutat", words ending with "n", words ending with "on", words ending with "ion", words ending with "tion", words containing "u", words containing "ut", words containing "uta", words containing "utat", | CommonCrawl |
John Mackintosh Howie
John Mackintosh Howie CBE FRSE (23 May 1936 – 26 December 2011) was a Scottish mathematician and prominent semigroup theorist.[1]
John Mackintosh Howie
Born23 May 1936
Died26 December 2011(2011-12-26) (aged 75)
NationalityScottish
Alma materUniversity of Oxford
AwardsKeith Prize
Scientific career
FieldsMathematics
InstitutionsUniversity of St Andrews
ThesisSome Problems on the Theory of Semigroups (1962)
Doctoral advisorGraham Higman
Biography
Howie was educated at Robert Gordon's College, Aberdeen, the University of Aberdeen and Balliol College, Oxford, where he wrote a Ph.D. thesis under the direction of Graham Higman.
In 1966 the University of Stirling was established with Walter D. Munn (fr) at head of the department of mathematics. Munn recruited Howie to teach there.[2]: 290
According to Christopher Hollings,
...a 'British school' of semigroup theory cannot be said to have taken off properly until the mid-1960s when John M. Howie completed an Oxford DPhil in semigroup theory (partly under Preston's influence) and Munn began to supervise research students in semigroups (most notably, Norman R. Reilly).[2]
He won the Keith Prize of the Royal Society of Edinburgh, 1979–81. He was Regius Professor of Mathematics at the University of St Andrews from 1970–1997. No successor to this chair was named until 2015 when Igor Rivin was appointed.
Howie was charged with reviewing universal, comprehensive secondary education in Scotland, which was viewed as failing its students. Impressed with education in Denmark, his committee proposed a tracking scheme to improve academic outcomes, and communicated recommendations in Upper Secondary Education in Scotland (1992).[3][4]
Public appointments
• Mathematics Panel, Scottish Examination Board 1967–1973; Convener from 1970
• Chairman, Scottish Central Committee for Mathematics 1975–1981
• President, Edinburgh Mathematical Society 1973–74
• London Mathematical Society:
• Council 1982–1988, 1989–1992
• Vice-president 1986–1988, 1990–1992
• Chairman of Education Committee 1985–1989
• Chairman of Public Affairs Committee 1990–1992
• Member of Dunning Committee 1975–1977
• Chairman of Governors, Dundee College of Education 1983–1987
• Governor, Northern College of Education 1987–2001
• Chairman, Scottish Mathematical Council 1987–1993
• Chairman, Committee to review fifth and sixth years (Howie Committee) 1990–1992
• Council, Royal Society of Edinburgh 1992–1995
• Chairman, Steering Committee, International Centre for Mathematical Sciences 1991–1997
Books
• 1976: An Introduction to Semigroup Theory, Academic Press MR0466355
• 1991: Automata and Languages, Clarendon Press ISBN 0-19-853424-8 MR1254435
• 1992: Upper Secondary Education in Scotland (Howie Report)[5]
• 1995: Fundamentals of Semigroup Theory, Clarendon Press ISBN 0-19-851194-9 MR1455373
• 2001: Real Analysis, Springer books ISBN 1-85233-314-6 MR1826051
• 2003: Complex Analysis, Springer books ISBN 1-85233-733-8 MR1975725
• 2006: Fields and Galois Theory, Springer books ISBN 978-1-85233-986-9 MR2180311
References
1. Published on Monday 23 January 2012 00:00. "Obituary: Professor John Howie, academic who helped reform Scottish education - Obituaries". Scotsman.com. Retrieved 23 January 2012.
2. Hollings, Christopher (2014), Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups, American Mathematical Society, p. 185, ISBN 978-1-4704-1493-1, Zbl 1317.20001.
3. "Howie Report" (PDF). University of Edinburgh.
4. Andrew McPherson (1992) "The Howie Committee on Post-compulsory Schooling" (PDF). Scottish Government Yearbook.
5. O'Connor, John J.; Robertson, Edmund F., "Howie Committee", MacTutor History of Mathematics Archive, University of St Andrews
External links
• O'Connor, John J.; Robertson, Edmund F., "John Mackintosh Howie", MacTutor History of Mathematics Archive, University of St Andrews
• John Mackintosh Howie at the Mathematics Genealogy Project
• "Short CV at St. Andrews". Archived from the original on 27 September 2007.
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\author{ \IEEEauthorblockN{Nguyen Thi Thanh Van, Nguyen Cong Luong, Feng Shaohan, Huy T. Nguyen, Kun Zhu, Thien Van Luong, and Dusit Niyato
}
}
\maketitle \begin{abstract} In this paper, we address dynamic network selection problems of mobile users in an Intelligent Reflecting Surface (IRS)-enabled wireless network. In particular, the users dynamically select different Service Providers (SPs) and network services over time. The network services are composed of IRS resources and transmit power resources. To formulate the SP and network service selection, we adopt an evolutionary game in which the users are able to adapt their network selections depending on the utilities that they achieve. For this, the replicator dynamics is used to model the service selection adaptation of the users. To allow the users to take their past service experiences into account their decisions, we further adopt an enhanced version of the evolutionary game, namely fractional evolutionary game, to study the SP and network service selection. The fractional evolutionary game incorporates the memory effect that captures the users' memory on their decisions. We theoretically prove that both the game approaches have a unique equilibrium. Finally, we provide numerical results to demonstrate the effectiveness of our proposed game approaches. In particular, we have reveal some important finding, for instance, with the memory effect, the users can achieve the utility higher than that without the memory effect.
\end{abstract} \begin{IEEEkeywords} Intelligent reflecting surface, next-generation wireless network, evolutionary game, fractional game, dynamic network service selection. \end{IEEEkeywords}
\section{Introduction}
Intelligent Reflecting Surface (IRS) is an emerging technology for the development of the next-generation wireless networks~\cite{wu2019towards}, \cite{di2019smart}. IRS consists of passive elements that can reflect incident signals by intelligently adjusting their phase-shifts corresponding to wireless channels. The signals reflected by the IRS are added constructively with non-reflected signals, i.e., the Line-of-Sight (LoS) signals, at the user receivers to boost the received signal power and enhance the data rate at the users. As a result, IRS has recently been proposed to be integrated with next-generation wireless technologies such as terahertz (THz) communications. The reason is that THz communication is able to provide data transmission rate up to terabit per second (Tbps), but this technology is limited in distance due to the fact that the THz waves are vulnerability to blockage and have severe path attenuation. For this, network service providers (SPs) deploy multiple IRSs in the THz networks to extend the network coverage and enhance the Quality of Service (QoS) of the mobile users. This results in a high density of the IRSs and base stations (BSs) in the THz networks, and the mobile users will more frequently handover among IRSs, BSs, and even SPs to achieve their desired QoS with low cost. In this case, the dynamic network selection of the mobile users in the THz networks becomes critical.
Although there are some works, i.e.,~\cite{chen2019sum},~\cite{ma2020intelligent}, \cite{ma2020joint}, \cite{hao2020robust}, and~\cite{Ma2020archive}, that have recently investigated the IRS-enabled THz networks, they do not focus on the network selection of the mobile users. In particular, the work in \cite{ma2020joint} is proposed to determine phase shifts of IRS to maximize the data rate. Extending the work in \cite{ma2020joint}, the work in~\cite{chen2019sum} aims to jointly optimize the IRS phase shifts, beamforming at the BS, and spectrum allocation to maximize the data rate. Similar to~\cite{chen2019sum}, the work in \cite{hao2020robust} aims to maximize the overall network throughput by jointly optimizing the phase-shits of the IRS and beamforming at the BS.
Also, there are some works that have recently investigated the network service selection. However, these works are not considered under the THz networks, and they do not account fir the dynamics of the network service selection. In particular, the authors in~\cite{gao2020stackelberg} and \cite {gao2020resource} consider an IRS-enabled network in which the BS is owned by an SP and the IRS belongs to a different SP. The Stackelberg game is then adopted to maximize the individual utilities of the SPs. Accordingly, the SP of IRS as the leader offers reflection modules as network resources and decides their prices. Note that the authors consider the allocation of reflection modules, i.e. instead of all the reflection elements, to the users since triggering all the reflection elements frequently results in an increased latency of adjusting phase-shift as well as the implementation complexity. Given the price, the SP of BS as the follower selects the best trigger reflection modules and determines their phase shifts and the transmit beamforming at the BS. Although the proposed game approach is demonstrated by the simulation results to be effective, the dynamics of the network service selections are not modeled in the work. Therefore, a more effective approach needs to be adopted to study the dynamic network service selection.
Evolutionary game~\cite{hofbauer2003evolutionary} as an effective tool can be adopted to study the dynamic selection and adaptation decision of a population of agents or players. This game has significant advantages~\cite{han2012game},\cite{quijano2017role} compared with the traditional games. In particular, the traditional games allow players, e.g. the mobile users in this work, to choose the desired solution immediately, while in the evolutionary game, the players are able to gradually adjust their strategies until they achieve a refined equilibrium solution. Especially, the evolutionary game is able to track and capture the strategy dynamics of the players as well as the strategy trends and behaviors of the players over time. Therefore, the evolutionary game has been widely used to study the dynamics of selection behaviors of users. The authors in \cite{liu2018evolutionary} investigate the mining pool selection of miners in a blockchain system. Accordingly, the miners compete to solve a crypto-puzzle to win the reward of mining new blocks. Due to the difficulty of the crypto-puzzle, the miners are willing to select mining pools for their secure stable profits. To study the dynamic selection of mining pool of the miners, the evolutionary game with the replicator dynamics is adopted for modeling the strategy evolution of the miners. It is demonstrated by both theory analysis and simulation results that in the case of two mining pools, the evolutionary game approach exists a unique Nash equilibrium at which no miner has an incentive to switch its pool selection since this will undermine some other miner's utility.
Next-generation wireless networks are expected to deploy different wireless access technologies, and the wireless network technology selection of the mobile users is crucial that impacts their QoS. For this, the evolutionary game is adopted to effectively study the dynamic network selection of the mobile users as proposed in~\cite{niyato2008dynamics}. The evolutionary game is also adopted to model the network service selection of secondary transmitters in a backscatter-based cognitive network~\cite{gao2019dynamic}.~In particular, the system model includes multiple access points serving multiple secondary transmitters. Each access point provides three network services, namely harvest-then-transmit (HTT), backscatter, and HTT-backscatter, to the mobile users. The secondary transmitters receive different utilities when choosing network services from different access points. To model the access point and service adaptation of the secondary transmitters, a series of ordinary differential equations is used to formulate the replicator dynamic process. Both the theory and numerical results show that the evolutionary game approach exists a unique equilibrium at which the secondary transmitters achieve the same utility even if they select different access points and network services. This demonstrates that with the evolutionary game, the users can adapt their selections gradually to reach the equilibrium. Especially, the complexity of algorithm to implement the evolutionary game is low. In particular, as analyzed in~~\cite{gao2019dynamic}, the complexity of strategy adaptation at each secondary transmitter is $O(1)$, that is suitable for the dynamic strategy selections of the users.
Given the aforementioned advantages, in this paper, we adopt the evolutionary game theory to study the dynamic service selection strategies of mobile users in the IRS-enabled terahertz network. The considered network consists of multiple SPs that deploy BSs along with multiple IRSs to provide network services to multiple mobile users. In particular, the SPs offer combinations of IRS and transmit power resources as network services that the users can select for their data transmissions. The network is thus considered to be a user-centric network. To satisfy different QoS requirements of the mobile users, similar to \cite{gao2020reflection},~\cite{gao2020stackelberg}, we assume that the SP divides its IRSs into reflection modules. Furthermore, the SPs have different transmit power levels that the mobile users can select. To model the SP and service adaptation of the users in the network, we leverage the replicator dynamic process that is expressed as a series of ordinary differential equations. Note that with the classical evolutionary game, the users only consider the instantaneous utility for their decision-making, i.e., the SP and service adaptation. This is not natural and practical due to the fact that the user is typically aware of its past network service experience when making the network selection. In other words, the awareness of the users' memory needs to be accounted. To address the limitation of the classical evolutionary game, we further adopt the fractional evolutionary game as a memory-aware economic process. The fractional evolutionary game enables the users to incorporate the instantaneous and past experiences of the users for their decisions. Both theoretical analysis and simulation results show the effectiveness of the proposed game approaches.
The main contributions of the paper include the followings: \begin{itemize} \item We consider the IRS-enabled terahertz network in which multiple SPs deploy IRSs to serve the mobile users. The SP offers IRS and transmit power resources as network resources to the mobile users. Different combinations of the network resources constitute different network services provided by the SPs. The network is a user-centric network in which the users can select and adapt the network services provided by the SPs over time to achieve their desired utility. The IRS-enabled terahertz network introduces new transmission scheme that makes the utility function of the users more complicated, and a new solution is required to model the network service adaptation of the users. To model the network service adaptation, we adopt the evolutionary game. \item We consider the scenario in which the users use the delayed information for their decisions. Such a delay can cause instability in the decision making process. In this scenario, the delayed replicator dynamics is adopted to model the SP and network service adaptation. We analyze the equilibrium region of the delayed replicator dynamics and show in the simulation results that the evolutionary game approach still reach an equilibrium with a small delay. \item To capture the users' memory on their decision-making, we incorporate the users' memory effect to reformulate the SP and network service selection problem into a fractional evolutionary game. We then compare the network selection strategies of the users between the classical game and fractional game. \item We theoretically prove that the fractional evolutionary game processes a unique equilibrium. Then, the simulation results with the direction field of the replicator dynamics are provided to verify the stability of the equilibrium. \item We provide performance evaluation to demonstrate the consistency with the analytical results and to validate both the proposed game approaches. The performance comparison between the two game approaches are also discussed and analyzed. \end{itemize}
The rest of the paper is organized as follows. In Section~\ref{system_model}, we present the IRS-enabled terahertz system and utility functions of the mobile users. In Section~\ref{classical_evol_game}, we formulate the dynamic SP and network service selection problem as the classical evolutionary game and analytically derive the stability region of the delayed replicator dynamics. In Section~\ref{frac_game}, we reformulate the dynamic SP and network service selection problem as a fractional evolutionary game, followed by the proofs of the existence, uniqueness, the stability of the equilibrium of the game. The simulation results and discussions are presented in Section~\ref{perform_eval}, and the conclusions are given in Section~\ref{conclu}.
\section{System Model} \label{system_model}
This section presents the system model, channel models, and utility functions of the users in the network. Typical notations used in this paper are summarized in Table~\ref{table:major_notation}.
\subsection{Network Model}
\begin{table}[h!] \caption{List of frequency symbols used in this paper.} \label{table:major_notation} \footnotesize \centering
\begin{tabular}{l|l} \hline\hline \textbf{Notation} & \textbf{Description} \\ [1ex] \hline
$M,N$ & Number of SPs, number of users \\
\hline
$I_m, J_{m,j}$ & Number of IRS of SP $m$, power level $j$ offered by SP $m$ \\
\hline
$K_m$ & Number of reflection elements of each IRS of SP $m$ \\ \hline
$Q_m$ & Number of modules of each IRS of SP $m$ \\
\hline
$\theta_{m,l,k,e}$ & Phase shift of element $e$ of subset $k$ in IRS $l$ of SP $m$ \\
\hline
$N_{m,l,k,j}$ & Number of users selecting power level $J_{m,j}$ and subset $k$ in IRS $l$ of SP $m$ \\ \hline
$p_{m,l,k,j,i}$ & Probability that user $i$ selects power level $J_{m,j}$ and subset $k$ in IRS $l$ of SP $m$ \\
\hline
$\mathbf{h}_{m,i}$ & Channel from BS $m$ to user $i$ \\
\hline
$\mathbf{h}_{m,l,k,i}^{\rm{IU}}$ & Channel from subset $k$ in IRS $l$ of SP $m$ to user $i$ \\
\hline
$\mathbf{G}_{m,l,k}$ & Channel from BS $m$ to subset $k$ in IRS $l$ of SP $m$ \\
\hline
$\mathbf{w}_{m,j,i}$ & Beamforming vector associated with user $i$ that selects power level $J_{m,j}$ of SP $m$ \\
\hline
$f, \mu,\beta$ & Carrier frequency, learning rate, order of the Caputo fractional derivative \\
\hline
$\gamma_m^I, \gamma_m^P$ & Price per IRS element, price per power unit \\ \hline \end{tabular} \label{table:parameters} \end{table}
\begin{figure}
\caption{A system model with multiple SPs with multiple IRSs serving multiple users.}
\label{IRS_model_selection}
\end{figure}
The system model is an IRS-enabled terahertz MIMO system as shown in~Fig.~\ref{IRS_model_selection}. The system model consists of a set $\mathcal{M}$ of $M$ SPs and a set $\mathcal{N}$ of $N$ single-antenna users. Without loss of generality, each SP $m \in \mathcal{M}$ deploys a BS, i.e., BS $m$, that is equipped with $L_m$ antennas. Denote $B_m$ as the bandwidth allocated to SP $m$, i.e., BS $m$. To provide flexible services to the users, BS $m$ has a set $\mathcal{P}_m$ of $P_m$ power levels, denoted by $\{J_{m,1},\ldots,J_{m,P_m}\}$, that the users can select for their transmissions. Note that the assumption of the discrete power levels is reasonable since in real networks, transmit power control algorithms choose steps of power increment/decrement~\cite{wu2001distributed}. Also, the number of power levels can be increased to match with the real system implementation without causing much more complexity. We also assume that $J_{m,1} <J_{m,2}< \dots < J_{m,P_m}$, where $ J_{m,P_m}$ is the maximum power of BS $m$.~To improve the QoS for the users, SP $m$ deploys a set $\mathcal{I}_m$ of $I_m$ IRSs. Let $l \in \mathbb{Z} $ denote the index of IRS, and $1\leq l \leq \max_{m} \{I_m\}$. We assume that IRSs belonging to the same SP have the same size, and IRS $l$ of SP $m$ has $K_{m,l}= K_{m}$ reflection elements. IRS $l$ of SP $m$ is divided into $Q_{m,l}=Q_m$ modules that are controlled by parallel switches. Each module in IRS $l$ consists of $E_{m,l}=E_m$ elements. Note that during a time slot, one BS-IRS pair of the corresponding SP can serve multiple users, but the user is associated with one BS-IRS pair. Moreover, the user can select one or multiple modules, i.e., a subset of modules, of the selected IRS. A network service is defined as a combination of a power level and a subset of modules. In general, the data throughput achieved by the user, say user $i$, depends on 1) the power level that the user selects, 2) the bandwidth allocated to the BS that the user selects, 3) the location of the selected IRS, 4) the number of modules of the selected IRS, and 5) the interference caused by other users selecting the same BS with user $i$. Note that the data throughput does not depend on indexes of the modules of the selected IRS. As such, each IRS $l$ of SP $m$ has a set $\mathcal{Q}_{m,l}$ including $Q_{m,l}$ of potential subsets of modules that the user can select, and subset $k, 1 \leq k \leq Q_{m,l},$ of the IRS has $k$ modules. Denote $\boldsymbol{\Theta}_{m,l,k}$ as the phase-shift matrix corresponding to the subset that the user selects, i.e., subset $k$ of IRS $l$ of SP $m$. Then, $\boldsymbol{\Theta}_{m,l,k}$ is a diagonal matrix in which its main diagonal consists of phase-shifts of $kE_{m,l}$ reflection elements of IRS $l$ of SP $m$. In particular, we have $\boldsymbol{\Theta}_{m,l,k}=\text{diag}(\theta_{m,l,k,1},\ldots,\theta_{m,l,k, kE_{m,l}})$, where $\theta_{m,l,k,e}$ is the phase-shift of reflection element $e$ of subset $k$ in IRS $l$ of SP $m$, $\theta_{m,l,k,e}=e^{j\varphi_{m,l,k,e}}, \varphi_{m,l,k,e} \in [0, 2\pi), e=\{1,\ldots,kE_{m,l}\}$. With the assistance of subset $k$ of IRS $l$ of SP $m$, the signal received at each user $i$ is the sum of 1) the received signal via the direct link, 2) the received signal via the IRS-assisted link, and 3) the intra-interference caused by other users that select the same BS with user $i$.~Let $\mathcal{N}_m$ denote the set of $N_m$ users selecting SP $m$, i.e., and also BS $m$. Then, the received signal at user $i$ when selecting subset $k$ of IRS $l$ and power level $J_{m,j}$ offered by SP $m$ is determined as follows: \begin{align} \notag y_{i}=&\big{(}\mathbf{h}^{\text{H}}_{m,i} + (\mathbf{h}^{\rm{IU}}_{m,l,k,i})^{\text{H}}\boldsymbol{\Theta}^{\text{H}}_{m,l,k}\mathbf{G}_{m,l,k}\big{)}\mathbf{w}_{m,j,i}s_{i} + \\ &+ \sum_{n\in \mathcal{N}_m, n \neq i} \big{(}\mathbf{h}^{\text{H}}_{m,i} + (\mathbf{h}^{\rm{IU}}_{m,l,k,i})^{\text{H}}\boldsymbol{\Theta}^{\text{H}}_{m,l,k}\mathbf{G}_{m,l,k}\big{)}\mathbf{w}_{m,j,n}s_{n} + \omega_i, \label{received_signal_user} \end{align} where $s_{i}$ is the data symbol intended to user $i$, $\mathbf{w}_{m,j,i} \in \mathbb{C}^{L_m\times1}$ is the beamforming vector associated with $s_{i}$ containing power level $J_{m,j}$ that the user selects, $\mathbf{h}_{m,i} \in \mathbb{C}^{L_m\times1}$ is the channel from BS $m$ to user $i$, $\mathbf{h}^{\rm{IU}}_{m,l,k,i} \in \mathbb{C}^{K_{m,l}\times1}$ is the channel from subset $k$ of IRS $l$ of SP $m$ to user $i$, $\mathbf{G}_{m,l,k} \in \mathbb{C}^{K_{m,l} \times L_m}$ is the vector of channels from BS $m$ to subset $k$ of IRS $l$, and $\omega_i$ is the Gaussian noise at user $i$, $\omega_i \sim \mathcal{CN}(0,\sigma_0^2)$, where $\sigma_0^2$ is the variance. Similar to \cite{wu2019intelligent} and~\cite{zhou2020intelligent}, we assume that each BS $m$ has a perfect knowledge of channel state information (CSI) of the channels. Note that we aim to model network selection strategies of the users which is not influenced by this assumption. In fact, CSI estimation algorithms such as \cite{liaskos2019joint} can be used to obtain the full CSI at all the BSs with low training overhead. Since IRSs are typically deployed in static environments, we can also assume that the quasi-static flat-fading model or even static flat-fading model is applied for all channels~\cite{wu2019intelligent}. The signal-to-interference-plus-noise ratio (SINR) over bandwidth $B_m$ of user $i$ is defined as follows: \begin{align}
\eta_{m,l,k,j,i} = \frac{\left |(\mathbf{h}^{\text{H}}_{m,i} + (\mathbf{h}^{\rm{IU}}_{m,l,k,i})^{\text{H}}\boldsymbol{\Theta}^{\text{H}}_{m,l,k}\mathbf{G}_{m,l,k})\mathbf{w}_{m,j,i}\right|^2} {\left |\sum_{n\in \mathcal{N}_m, n \neq i} \big{(}\mathbf{h}^{\text{H}}_{m,i} + (\mathbf{h}^{\rm{IU}}_{m,l,k,i})^{\text{H}}\boldsymbol{\Theta}^{\text{H}}_{m,l,k}\mathbf{G}_{m,l,k}\big{)}\mathbf{w}_{m,j,n}\right|^2+ B_m\sigma_0^2}. \label{SINR_user_i} \end{align}
To remove the intra-interference among the users, the BSs can use time-division multiple access for their users. In this case, $\eta_{m,l,k,i}$ can be expressed by \begin{equation}
\eta_{m,l,k,j,i}=\frac{\left|(\mathbf{h}^{\text{H}}_{m,i} +(\mathbf{h}^{\rm{IU}}_{m,l,k,i})^{\text{H}}\boldsymbol{\Theta}^{\text{H}}_{m,l,k}\mathbf{G}_{m,l,k})\mathbf{w}_{m,j,i}\right |^2} {B_m\sigma_0^2}. \label{SINR_user_ii} \end{equation}
\subsection{Channel Model}
In this section, we describe the channel models for the IRS-enabled THz network. The channel models for each user include the channel between the BS that the user selects and the user, and the cascaded channel of the IRS-aided link. Here, the cascaded channel of the IRS-aided link includes (1) the channel between the BS and the subset of IRS modules that the user selects and (2) the channel between the subset of IRS modules and the user. To model the channels in the THz network, we adopt the Saleh-Valenzuela channel model~\cite{lin2015indoor}. Without loss of generality, we model the channels between user $i$ when it selects BS $m$, subset $k$ of IRS $l$ of SP $m$. In particular, we determine models of channels $\mathbf{h}_{m,i}$, $\mathbf{h}^{\rm{IU}}_{m,l,k,i}$, and $\mathbf{G}_{m,l,k}$ that are given in (\ref{SINR_user_i}). For an ease of presentation, we remove the indices $m, l, k, i$ from the channels, and thus $\mathbf{h}_{m,i}$, $\mathbf{h}^{\rm{IU}}_{m,l,k,i}$, and $\mathbf{G}_{m,l,k}$ can be expressed by $\mathbf{h}$, $\mathbf{h}^{\rm{IU}}$, and $\mathbf{G}$, respectively. Also, we assume that BS $m$ has $L$ antennas, subset $k$ of IRS $l$ that the user selects has $K$ reflection elements.
\subsubsection{BS-user channel}
The channel between the BS and the user is expressed by
\begin{equation} \mathbf{h} = \kappa^{(0)}\mathbf{a}(\Phi^{(0)})+ \sum_{l=1}^{\mathcal{L}} \kappa^{(l)}\mathbf{a}(\Phi^{(l)}), \label{Saleh_channel} \end{equation} where $\kappa^{(0)}\mathbf{a}(\Phi^{(0)})$ is the LoS element in which $\kappa^{(0)}$ is the gain and $\mathbf{a}(\Phi^{(0)})$ is the spatial direction, and $\kappa^{(l)}\mathbf{a}(\Phi^{(l)}), 1 \leq l \leq \mathcal{L}$, is one of $\mathcal{L}$ non-LoS (NLoS) elements. $\mathbf{a}(\Phi^{(l)})$ is the $L \times 1$ array steering vector corresponding to th-$l$ element that is defined as follows:
\begin{equation} \mathbf{a}(\Phi^{(l)})= \frac{1}{\sqrt{L}} \big{[} e^{-j2\pi\Phi^{(l)}(-\frac{N-1}{2})},\ldots, e^{-j2\pi\Phi^{(l)}(\frac{N-1}{2})} \big{]},
\end{equation} where $\Phi^{(l)}$ is the spatial direction of the signal corresponding to component $l$, that is defined as $\Phi^{(l)}= \frac{d}{\lambda}\sin\xi^{(l)} $, where $\xi^{(l)} \in [ -\pi/2, \pi/2]$ is the angle-of-departure (AoD) of path $l$ corresponding to the BS and user, $\lambda$ is the signal wavelength, and $d$ is the distance between adjacent antennas of the BS or the distance between adjacent IRS elements of the IRS that is typically defined as $d=\lambda/2$.
\subsubsection{BS-IRS-user channel}
The channel between the BS and the subset of modules of IRS that the user selects can be modeled as
\begin{equation} \mathbf{G}= \sqrt{ \frac{LK}{\mathcal{L}_{\text{BS-I}}} }\sum_{l=1}^{\mathcal{L}_{\text{BS-I}}}\kappa_{\text{BS-I}}^{(l)}\mathbf{a}_{\text{I}}(\Phi^{(l)}_{\text{AOA}})\mathbf{a}_{\text{BS}}^{\text{H}}(\Phi^{(l)}_{\text{AOD}}),
\end{equation}
where $\mathcal{L}_{\text{BS-I}}$ denotes the scattering paths between the BS and the subset of IRS that the user selects, $\kappa^{(l)}_{\text{BS-I}}$ is the complex gain of path $l$, and $\Phi^{(l)}_{\text{AOD}}$ and $\Phi^{(l)}_{\text{AOA}}$ are the spatial directions of path $l$ corresponding to the BS and the subset of IRS, respectively. We consider the BS's antennas and the IRS's reflection elements as uniform linear arrays (ULAs), and thus we can determine $\mathbf{a}_{\text{BS}}(\Phi^{(l)}_{\text{AoD}}) \in \mathcal{C}^{L}$ and $\mathbf{a}_\text{I}(\Phi^{(l)}_{\text{AOA}}) \in \mathcal{C}^{K}$ as follows: $\mathbf{a}_{\text{BS}}(\Phi^{(l)}_{\text{AOD}}) = \frac{1}{\sqrt{L}}\big{[} e^{-j2\pi\Phi^{(l)}_{\text{BS}} (-\frac{L-1}{2}) },\ldots, e^{-j2\pi\Phi^{(l)}_{\text{BS}}(\frac{L-1}{2})} \big{]}$, and $\mathbf{a}_{\text{I}}(\Phi^{(l)}_{\text{AOA}}) = \frac{1}{\sqrt{K}}\big{[} e^{-j2\pi\Phi^{(l)}_{\text{I}} (-\frac{K-1}{2}) },\ldots, e^{-j2\pi\Phi^{(l)}_{\text{I}}(\frac{K-1}{2})} \big{]}$. Here, $\Phi^{(l)}_{\text{I}}= \frac{d}{\lambda}\sin\xi^{(l)}_{\text{I}}$ and $\Phi^{(l)}_{\text{BS}}= \frac{d}{\lambda}\sin\xi^{(l)}_{\text{BS}} $, where $\xi^{(l)}_{\text{BS}} \in [ -\pi/2, \pi/2]$ and $\xi^{(l)}_{\text{I}} \in [ -\pi/2, \pi/2]$ are the AoD and the angle-of-arrival (AoA) of path $l$ corresponding to the BS and the subset of IRS, respectively.
Similarly, we can determine the channel between the subset of IRS and the user as follows:
\begin{equation} \mathbf{h}^\text{IU} = \sum_{l=0}^{\mathcal{L}_\text{I-U}} \kappa^{(l)}_{\text{I-U}}\mathbf{a}_{\text{I-U}}(\Phi^{(l)}_{\text{I-U}}), \label{Saleh_channel} \end{equation} where $\mathcal{L}_{\text{I-U}}$ denotes the scattering paths between the subset of IRS and the user, $\kappa^{(l)}_{\text{I-U}}$ is the complex gain of path $l$, and $\mathbf{a}_{\text{I}}(\Phi^{(l)}_{\text{I-U}}) = \frac{1}{\sqrt{K}}\big{[} e^{-j2\pi\Phi^{(l)}_{\text{I-U}} (-\frac{K-1}{2}) },\ldots, e^{-j2\pi\Phi^{(l)}_{\text{I-U}}(\frac{K-1}{2})} \big{]}$, where $\Phi^{(l)}_{\text{I-U}}= \frac{d}{\lambda}\sin\xi^{(l)}_{\text{I-U}} $ with $\xi^{(l)}_{\text{I-U}} \in [ -\pi/2, \pi/2]$ being the AoD of path $l$ corresponding to the subset of IRS.
\subsubsection{Path loss}
In THz communication systems, the non-LoS elements is proved to be much weaker than the LoS element, i.e., lower than $20$ dB~\cite{han2014multi}. Therefore, in the IRS-enabled THz network, we consider the LoS elements of the involved channels. Without loss of generality, we calculate the path loss of the LoS element between the BS and the user. This can be applied to calculating the path loss of the LoS elements between the BS and the subset of IRS as well as that between the subset of IRS and the user. The path loss of the LoS element, denoted by $\chi_\text{LoS}$, is a function of spreading loss and molecular absorption loss, denoted by $\chi_\text{abs}$. Then, the path loss is determined as~\cite{han2014multi} \begin{equation} \chi_\text{LoS}(f)= \chi_\text{spr}(f)\chi_\text{abs}(f)e^{-j2\pi f\tau_{\text{LoS}}}, \end{equation} where $f$ is the carrier frequency, and $\tau_{\text{LoS}}=r/c$ is the LoS propagation time, $r$ is the distance between the BS and the user, and $c$ is the speed of light. $\chi_\text{spr}$ is the spreading loss that is determined by \begin{equation} \chi_\text{spr}(f)= \frac{c}{4\pi f r}. \end{equation}
While, $\chi_\text{abs}(f)$ is the molecular absorption loss that is determined as follows: \begin{equation} \chi_\text{abs}(f)= e^{-0.5\zeta(f)r}, \end{equation} where $\zeta(f)$ is the medium absorption coefficient that depends on carrier frequency and the composition of the transmission medium at a molecular level. For example, given $f=2$ THz, the molecular absorption coefficient $\zeta=2.3\times10^{-5}$ m$^{-1}$ for oxygen (O$_{2}$)~\cite{jornet2011channel}.
\subsection{Utility Functions}
This section presents the utility functions of the users when they select different SP and network services. There are totally $N$ users, $M$ BSs, $P_m$ power levels, and $\sum_{m=1}^MI_m$ IRSs in the network. Users selecting the same SP, the same subset of the IRS, and the same power level are grouped into a group. Thus, there are totally $G$ groups in the network, where $G=\sum_{m=1}^M\sum_{l=1}^{I_m}P_mI_mQ_{m,l}$.~Without loss of generality, we can assume that group $g,1 \leq g \leq G$, consists of users that select SP $m$, subset $k$ of IRS $l$, i.e., the corresponding phase-shift matrix $\boldsymbol{\Theta}_{m,l,k}$, and power level $J_{m,j}$. For this, we can denote $g$ as the combination of indexes $(m,l,k,j)$ for the expression simplification. Let $\mathcal{N}_{m,l,k,j}$, i.e., $\mathcal{N}_{g}$, be a set of $N_{m,l,k,j}$, i.e., $N_{g}$, of users in group $g$. We have $\sum_{m=1}^M\sum_{l=1}^{I_m}\sum_{k=1}^{Q_{m,l}}\sum_{j=1}^{P_m}N_{m, l,k,j}=N$, and user $i \in \mathcal{N}_{g}$ selects SP $m$, IRS $l$, $\boldsymbol{\Theta}_{m,l,k}$, and $J_{m,j}$ at a probability of $p_{m,l,k,j,i}=N_{m,l,k,j}/N$. As the expected number of the users selecting SP $m$, IRS $l$, $\boldsymbol{\Theta}_{m,l,k}$, and $J_{m,j}$ is $p_{m,l,k,j,i}N$ and BS $m$ adopts the time-division multiple access, each user in the group will access the channel with a probability of $\frac{1}{p_{m,l,k,j,i}N}$ for every time slot. Therefore, the expected data rate that the user in the group can achieve is
\begin{equation}
\overline{R}_{m,l,k,j,i}=\frac{B_m}{p_{m,l,k,j,i}N}
\log_2\big{(}1+\eta_{m,l,k,j,i}\big{)},
\label{expected_data_rate}
\end{equation}
where $\eta_{m,l,k,j,i}$ is given in~(\ref{SINR_user_ii}). Let $v_{m,l,k,j,i}$ denote the value of unit data to user $i$ in group $g$ when selecting SP $m$, subset $k$ of IRS $l$, and power level $J_{m,j}$. Denote $\gamma_m^I$ as the price per element in IRSs of SP $m$ and $\gamma_m^P$ as the price per unit power. The prices, i.e., $\gamma_m^I$ and $\gamma_m^P$, are set by SP $m$ that are constant. Since users in each group share the same resources, they should share the resource cost. Then, the utility of the user is given by \begin{equation}
u_{m,l,k,j,i} = v_{m,l,k,j,i} \overline{R}_{m,l,k,j,i} - \frac{\gamma_m^I \norm{\boldsymbol{\Theta}_{m,l,k}}_0 - \gamma_m^P J_{m,j}} {p_{m,l,k,j,i}N},
\label{utility_user} \end{equation} where the $l_0$-norm is used to count the number of non-zero elements of a diagonal matrix that here refers to the number of active reflection elements of IRS $l$ of SP $m$ that the user selects.
\section{Evolutionary Game Formulation} \label{classical_evol_game}
In this section, we leverage the evolutionary game to model the dynamic SP and network service selection of the users. We prove that the game can achieve the evolutionary equilibrium at which no user has an incentive to change their network service strategy.
\subsection{Game Formulation} \label{classical_evol_game_form}
Each user in the network is able to adapt their network selection over time, and it can achieve different utility at different time points. Thus, by taking the SP and network service selection strategies, the expected or average utility of user $i$ at time $t$ is \begin{equation} \overline{u}_{i}=\sum_{m=1}^M\sum_{l=1}^{I_m}\sum_{k=1}^{Q_{m,l}}\sum_{j=1}^{P_m}p_{m, l,k,j,i} u_{m,l, k,j,i}.
\label{utility_user_average} \end{equation}
To model the SP and service adaptation of the users, we leverage the replicator dynamic process that is expressed as a series of ordinary differential equations as follows~\cite{gao2019dynamic}, \cite{feng2020dynamic}: \begin{align} \notag \dot{p}_{m,l, k,j,i}(t) =&\mu p_{m, l,k,j,i}(t)\left [u_{m, l,k,j,i}(t)-\overline{u}_{i}(t)\right ], \\ & m \in \mathcal{M}, l \in \mathcal{I}_m, k \in \mathcal{Q}_{m,l}, j \in \mathcal{P}_m, i \in \mathcal{N}_g, \forall t, \label{replicator_evolu} \end{align} where $\dot{p}_{m, l,k,j,i}(t)$ represents the first derivative of $p_{m,l, k,j,i}$ with respect to $t$, and $p_{m,l, k,j,i}(t_0)=p^0_{m, l,k,j,i}$ is the initial strategy of the user in group $g$ at $t_0$. The factor $\mu$ is the learning rate of the users that evaluates the strategy adaptation rate. The replicator dynamics process given in (\ref{replicator_evolu}) represents the population strategy evolution of the users in the network. That is, the population of users evolves over time, and the game converges to the evolutionary equilibrium. This means that the users select an SP and its service with higher utility over time, and the evolutionary equilibrium can be defined as the set of stable fixed points of the replicator dynamics.
To show the existence of the evolutionary equilibrium of the game as defined in (\ref{replicator_evolu}), we use the following theorem. First, we let $f_{g,i}(t,p_{g,i})=\mu p_{g,i}(t)\left[u_{g,i}(t)-\overline{u}_i(t)\right]$, where $g$ denotes as the combination of indexes $(m,l,k,j)$, and then we rewrite equation (\ref{replicator_evolu}) as follows: \begin{equation} \dot{p}_{g,i}(t) =f_{g,i}(t,p_{g,i}), \text{ } p_{g,i}(t_0)=p^0_{g,i}, \text{ } g \in \{1,\ldots,G\}, i \in \mathcal{N}_g. \label{replicator_evolu_2} \end{equation}
\begin{theorem}~\cite{picard_theorem} Suppose that functions $f_{g,i}(t,p_{g,i})$ $\frac{\partial f_{g,i}}{\partial p_{g,i}}(t,p_{g,i})$ are continuous in some open rectangle $\left\{ (t,p_{g,i}): 0 \leq t \leq \tau, 0 < p_{g,i} \leq 1\right\}$ that contains point $(t_0,p^0_{g,i})$. Then, the problem in (\ref{replicator_evolu_2}) has a unique solution in the interval of $I=\left[t_0-\varsigma, t_0+\varsigma\right]$, where $\varsigma> 0$. Moreover, the Picard iteration defined by \begin{equation} p^{<z+1>}_{g,i}(t) =p^0_{g,i} + \int_{t_0}^t f_{g,i}(t,p^{<z>}_{g,i}(t) ) \;\mathrm{d}t \label{Picard_iteration} \end{equation} produces a sequence of functions ${ p^{<z>}_{g,i}(t)}$ that converges to the solution uniformly on $I$. \label{theorem_picard} \end{theorem}
\begin{proof} Theorem \ref{theorem_picard} means that problem in (\ref{replicator_evolu_2}) converges to a unique solution, i.e., the equilibrium of the game defined in~(\ref{replicator_evolu_2}), given that function $f_{g,i}(t,p_{g,i})$ and its derivative are continuous with respect to time $t$. Therefore, we first show that $f_{g,i}(t,p_{g,i})$ and $\frac{\partial f_{g,i}}{\partial p_{g,i}}(t,p_{g,i})$ are continuous functions in the rectange $\left\{ (t,p_{g,i}): 0 \leq t \leq \tau, 0\leq p_{g,i} \leq 1\right \}$. Indeed, it is clear that function $p_{g,i}(t)=\frac{N_{g,i}(t)}{N}$ is continuous at every $t_0 \in [ 0, \tau]$. Moreover, due to the static flat-fading channel model, variables $\mathbf{h}_{m,i}(t)$, $\mathbf{h}_{g,i}^{\rm{IU}}(t)$, and $\mathbf{G}_{m,l,k}(t)$ are constant and thereby continuous at every $t_0 \in [ 0, \tau]$. Correspondingly, $\eta_{g,i}(t)$ is continuous at every $t_0 \in [ 0, \tau]$, and functions $\overline{R}_{g,i}(t), u_{g,i}(t)$, and $\overline{u}_i(t)$ are also continuous at every $t_0 \in [ 0, \tau]$ if $p_{g,i}(t_0)\neq 0$. Since $f_{g,i}(t,p_{g,i})=\mu p_{g,i}(t)[u_{g,i}(t)-\overline{u}_i(t)]$ and $\frac{\partial f_{g,i}}{\partial p_{g,i}}(t,p_{g,i})=\mu\left[u_{g,i}(t)-\overline{u}_i(t)\right]$, then $f_{g,i}(t,p_{g,i})$ and $\frac{\partial f_{g,i}}{\partial p_{g,i}}(t,p_{g,i})$ are continuous functions in the open rectangle $\left\{ (t,p_{g,i}): 0 \leq t \leq \tau, 0 < p_{g,i} \leq 1\right\}$.
Given that the continuity of $f_{g,i}(t,p_{g,i})$ and its derivative, there are many ways to prove Theorem~\ref{theorem_picard}. One of them is leveraging the Banach Fixed Point Theorem (BFPT)~\cite{ciesielski2007stefan} to approximate a solution, i.e., a fixed point, to (\ref{replicator_evolu_2}) by constructing a sequence of functions that converges to a unique solution. The proof of Theorem~\ref{theorem_picard} using the BFPT is well explained and presented in~\cite{picard_theorem}. The unique solution refers to the game equilibrium at which 1) all the users achieve the same utility and 2) no user has an incentive to change its network service selection. \end{proof}
The overall process of the network service selection is summarized as follows. Initially, each user randomly selects an SP and a service of the SP. Given the user selection, the SP determines the optimal phase-shift and beamforming for its associated users according to Algorithm~\ref{phase_shift}. The user computes its utility according to (\ref{utility_user}) and transmits the utility information to the SP. The user compares its utility and the expected utility determined by (\ref{utility_user_average}) and can change its network service selection to achieve a higher utility value. After all the users achieve the same utility by choosing any strategies, then no user has an incentive to change its network service selection and the game converges to the evolutionary equilibrium.
\iffalse \begin{algorithm} \footnotesize
\caption{\footnotesize Algorithm of SP and service selection}\label{evolutionary_game}
\begin{algorithmic}[1]
\State Output: Selection strategy of each user;
\State Initialize: Users randomly select an SP and its network service;
\For {$t=1$ to T}
\State Each SP determines the optimal phase-shift $\mathbf{G}_{m,k}$ and beamforming $\mathbf{w}_{m,j}$ for its user;
\State Each user calculates $u_{m,k,j}$ and send the utility information to SP;
\State Each SP computes $\bar{u}_{m,k,j}$ and sends the expected utility and a set of selection strategies to the users with $u_{m,k,j} > \bar{u}_{m,k,j}$;
\For {all the users with $u_{m,k,j}<\bar{u}_{m,k,j}$}
\State Select SP $m'$ and its service if $u_{m',k',j'}> u_{m,k,j}$;
\EndFor
\If {There is any user changing its selection strategy}
\State Go to Step 4;
\EndIf
\EndFor
\end{algorithmic}
\end{algorithm}
\fi
\begin{algorithm} \footnotesize
\caption{\footnotesize Optimizing phase-shift and beamforming for user $i$ when selecting SP $m$, subset $k$ of IRS $l$, and power level $J_{m,j}$~\cite{yu2019miso} (in this algorithm, function $\rm{unt}(\mathbf{a})$ is defined as $\rm{unt}(\mathbf{a})=[a_1/|a_1|,\ldots, a_n/|a_n|]$)}\label{phase_shift}
\begin{algorithmic}[1]
\State Output: $\mathbf{w}_{m,j}$ and $\mathbf{\Theta}_{m,l,k}$ for each user;
\State Initialize: $t=0$, $\epsilon_1=0$, $\mathbf{\Theta}_{m,l,k}^0$;
\State Calculate $\mathbf{R}_{11}=\text{diag}(({\mathbf{h}^{\rm{IU}}_{m,l,k,i})^{\rm{H}})\mathbf{G}_{m,l,k}\mathbf{G}_{m,l,k}^{\rm{H}}\text{diag}(\mathbf{h}^{\rm{IU}}_{m,l,k,i}}) $;
\State Construct $\mathbf{R}$ = \[
\begin{bmatrix} \mathbf{R}_{11}& \text{diag}(({\mathbf{h}^{\rm{IU}}_{m,l,k,i}})^{\rm{H}})\mathbf{G}_{m,l,k}\mathbf{h}_{m,l,k,i} \\
\mathbf{h}_{m,i}^{\rm{H}}\mathbf{G}_{m,l,k}^{\rm{H}}\text{diag}(\mathbf{h}^{\rm{IU}}_{m,l,k,i}) & 0
\end{bmatrix} \]
\Repeat
\State $\mathbf{v}^{(t)}=[\rm{diag}(\mathbf{\Theta}_{m,l,k}), t]^{\top}$;
\State Calculate $\mathbf{v}^{(t+1)}$ according by $\mathbf{v}^{(t+1)}$=unt$(\mathbf{Rv}^{(t)})$;
\State $t \gets t+1$;
\Until{$\parallel{\mathbf{Rv}^{(t+1)}\parallel_{1}-\parallel\mathbf{Rv}^{(t)}\parallel_{1}} \leq \epsilon$};
\State Take first $kE_{m,l}$ elements of ${(\mathbf{v}^{t+1})}^{*}$ as the main diagonal of $\mathbf{\Theta}_{m,l,k}$;
\State Compute $\mathbf{w}_{m,j,i}=\sqrt{J_{m,j}}\frac{\mathbf{G}_{m,l,k}^{\rm{H}}\rm{diag}(\mathbf{h}^{\rm{IU}}_{m,l,k,i})\mathbf{\Theta}^{\rm{H}}_{m,l,k}+\mathbf{h}_{m,l,k,i}}{\parallel\mathbf{G}_{m,l,k}^{\rm{H}}\rm{diag}(\mathbf{h}^{\rm{IU}}_{m,l,k,i})\mathbf{\Theta}^{\rm{H}}_{m,l,k}+\mathbf{h}_{m,i}\parallel}$.
\end{algorithmic}
\end{algorithm}
The computational complexity of the algorithm is mainly caused from 1) the phase-shift and beamforming optimization implemented at the SPs (BSs) side and 2) the utility computation implemented at the user side. When the users select a network service of SP $m$, the SP optimizes the phase-shift matrix and beamforming for each user using Algorithm~\ref{phase_shift} that requires $5K_m^2L_m + K_m^2(5L_m+2) + K_m(L_m+2)+ 2L_m+1$ multiplications and additions. When the size increases to infinite, the complexity of the algorithm for each user is $\mathcal{O}(n^3)$. Each user calculates its own utility based on the prices and network services that it selects. Thus, the computational complexity of the user does not increase with the total numbers of users and the SPs. Thus, the complexity of the algorithm implemented at each user is $\mathcal{O}(1)$. This implies that the game approach is computationally efficient and highly scalable.
\subsection{Delay in Replicator Dynamics}
In the game model discussed in the previous section, to make the decision on SP and service selections, the users need information about the average utility, i.e., $\overline{u}_{i}$, and the proportion of users choosing different strategies, i.e., $p_{m,l,k,j,i}$, from the BSs. However, the up-to-date information may not be available at the users due to the communication latency. Thus, at time instance $t$, the users may need to use the information at time $t-\delta$, i.e., delay for $\delta$ time units, to make the SP and service selections. Thus, the delayed replicator dynamic process is expressed as \begin{align} \label{replicator_evolu_delay} \notag \dot{p}_{m, l,k,j,i}(t) =&\mu p_{m, l,k,j,i}(t-\delta)\left[u_{m, l,k,j,i}(t-\delta)-\overline{u}_i(t-\delta)\right], \\ & m \in \mathcal{M}, l \in \mathcal{L}_m, k \in \mathcal{Q}_{m,l}, j \in \mathcal{P}_m, i \in \mathcal{N}_g ,\forall t. \end{align}
Note that as delay $\delta$ is large, the decisions of the users based on the outdated information tend to be inaccurate. In this case, the SP and service selections may not converge. How to determine $\delta^*$ such that the selections converge is challenging. As an example, consider a simple scenario with $M=2$, and SP $m$ offers one service including subset $\boldsymbol{\Theta}_{m}$ and power level $J_m$: \begin{theorem} The evolutionary game can converge to a stable equilibrium if the value of $\delta$ is satisfied the following condition: \begin{equation}
\delta^* < \frac{\pi}{2\mu \sum_{m\in \mathcal{M}} \frac{v_mB_m\log_2(1 + \eta_m) -\gamma_m^I||\boldsymbol{\Theta}_{m}||_0-\gamma_m^PJ_m }{N} }. \label{stability} \end{equation} \label{theorem_stability} \end{theorem} \begin{proof} The delayed replicator dynamics in (\ref{replicator_evolu_delay}) can be rewritten as \begin{equation} \mathbf{\dot{p}}(t) =\mathbf{A} \mathbf{p}(t-\delta)+\mathbf{c}, \end{equation} where $\mathbf{\dot{p}}(t) ={[{\dot{p}_{1}}(t),\ldots, {\dot{p}_{M}}(t)]}^{\top}$, $\mathbf{p}(t) ={[{p}_{1}(t-\delta), \ldots, {p}_{M}(t-\delta)]}^{\top}$, $\mathbf{c} ={[\frac{\mu a_{1}}{N}, \ldots, \frac{\mu a_{M}}{N}]}^{\top}$ with
$a_{m} = \frac{v_mB_m\log_2(1 + \eta_m) -\gamma_m^I||\boldsymbol{\Theta}_{m}||_0-\gamma_m^PJ_m }{N}$, and $\mathbf{A}=-\kappa\mathbf{I}$. Here, $I$ is the identity matrix of size $M$, and $\kappa$ is defined as \begin{align} \notag
\kappa=\mu \sum_{m\in \mathcal{M}} \frac{v_mB_m\log_2(1 + \eta_m) -\gamma_m^I||\boldsymbol{\Theta}_{m}||_0-\gamma_m^PJ_m }{N}. \end{align}
Otherwise, the evolutionary game with the delayed replicator dynamics can converge to a stable equilibrium if the real parts of all the roots are negative ~\cite{gopalsamy2013stability}. This is equivalently the condition $\kappa\delta<\pi$, and thus we have \begin{align} \notag
\delta^* < \frac{\pi}{2\kappa} = \frac{\pi}{2\mu \sum_{m\in \mathcal{M}} \frac{v_mB_m\log_2(1 + \eta_m) -\gamma_m^I||\boldsymbol{\Theta}_{m}||_0-\gamma_m^PJ_m }{N} }. \end{align} \end{proof}
Theorem~\ref{theorem_stability} means that the evolutionary game is guaranteed to converge to the equilibrium as the users use information at $t< \delta^*$ for their decisions.
\section{Fractional Evolutionary Game Formulation} \label{frac_game}
In this section, we discuss the use of the fractional evolutionary game to model the SP and network service selection of the users with memory effect in the IRS-enabled terahertz system. In particular, we first present the concept of memory-aware economic process. Then, we present how to cast the evolutionary game that describes the SP and network service selection into a fractional evolutionary game by using the memory-aware economic processes. Finally, we analyze the equilibrium of the game.
\subsection{Memory-aware Economic Process}
With the classical evolutionary game as presented in Section~\ref{classical_evol_game}, each user, say user $i$, decides on the SP and network service selection according to its instantaneously achievable utility functions, i.e., functions $u_{m, l,k,j,i}(t)$ and $\overline{u}_{i}(t)$ at time instant $t$. In reality, the users take into account their memory, i.e., of service experience, on their strategy decisions. Specifically, the selection decision of the users at time $t$ is based not only on the information about the state of the process at time $t$, but also on the information about the process states at previous time instants $\tau \in [0,t]$.~This is considered to be a memory-aware economic process~\cite{tarasova2018concept},~\cite{tarasova2016generalization}. To describe the memory-aware economic process, we consider a typical economic model with two variables, namely \textit{exogenous variable} and \textit{endogenous variable}. The exogenous variable and endogenous variable are the input and output of the economic model, respectively. This means that the endogenous variable depends on the exogenous variable, and they are similar to the independent and dependent variables, respectively. Denote $X(t)$ as the exogenous variable and $Y(t)$ as the endogenous variable variable, in which the exogenous variable changes depends on the changes of the endogenous variable. Then, the economic process is typically expressed by $Y(t)=F^t_0(X(\tau)) + Y_0$, where $\tau \in [0,t]$, $Y_0$ is the initial state of the output of the process, and $F^t_0$ is an operator. To enable the memory awareness of the economic process, the operator is defined as $F^t_0(X(\tau)):= \int_0^tM_{\beta}(t-\tau)X(\tau)\rm{d}t$, where $M_{\beta}(t-\tau)$ is the weight function that represents how the input $X(\tau)$ at time $\tau$ impacts on the output $Y(t)$ at time $t$. In general, function $M_{\beta}(t-\tau)$ changes with respect to $\tau$ so as to capture the dynamic characteristic of the memory. Furthermore, by taking the time derivative of $Y(t)$, we have $\frac{\rm{d}}{\rm{d}t}Y(t)= M_{\beta}(t)X(0)+ \int_0^tM_{\beta}(t-\tau) [\frac{\rm{d}}{\rm{d}t}Y(t) ] X(\tau)\rm{d}\tau$ that depends on both $X(t)$ and $X(\tau)$ with $\tau \in [0,t)$. The formulation of $M_{\beta}(t-\tau)$ is $M_{\beta}(t-\tau)=\frac{1}{\Gamma(\beta)(t-\tau)^{1-\beta}}$, where $\Gamma(\cdot)$ is the gamma function that is defined by $\Gamma(\beta)= \int_0^{+\infty}x^{\beta-1}e^{-x}\rm{d}x$.
Since $Y(t)$ depends on both $X(t)$ and $X(\tau)$, the economic process is namely \textit{memory-aware economic process} that can be expressed in the fractional equation by taking the derivation of $Y (t)$ at the order of $\beta$ through the left-sided Caputo fractional derivative as follows: \begin{equation}
{}^C_0D^{\beta}_{t} Y(t)=X(t), \end{equation} where $Y (0) = Y_0$ is the initial state, and ${}^C_0D^{\beta}_{t} Y(t)$ is the left-sided Caputo fractional derivative~\cite{tarasova2017logistic} of $Y (t)$ at the order of $\beta$ that is given by: \begin{equation} {}^C_0D^{\beta}_{t} Y(t)=\frac{1}{\Gamma(\ceil{\beta}-\beta)}\int^t_0\frac{Y^{(\ceil{\beta})}(\tau)}{(t-\tau)^{\beta+1-\ceil{\beta}}}\rm{d}\tau, \label{fractional_derivative} \end{equation} where $\ceil{\cdot}$ is the ceiling function.
The memory-aware economic process given in (\ref{fractional_derivative}) has two key properties. First, the past experiences of the user at different time instances have different impacts on its decision-making so as to capture dynamically the memory of the user. Second, the user is affected by the experience within the memory rather than that at the current time, and consequently the memory-aware users can make network selection decisions differently from the memory-unaware users. Given the properties, we incorporate the memory awareness of the economic process when modeling the SP and network service selection of the users. The memory-aware economic process can be modeled as the fractional evolutionary game that is presented in the next section.
\subsection{Fractional Game Formulation}
For convenience, we rewrite the replicator dynamic process of the users in the classical evolutionary game as expressed in (\ref{replicator_evolu}) as follows: \begin{align} \notag \dot{p}_{m,l, k,j,i}(t) =&\mu p_{m, l,k,j,i}(t)\left[u_{m, l,k,j,i}(t)-\overline{u}_{i}(t)\right], \\ & m \in \mathcal{M}, l \in \mathcal{I}_m, k \in \mathcal{Q}_{m,l}, j \in \mathcal{P}_m, i \in \mathcal{N}_g, \forall t. \label{replicator_evolu_re} \end{align}
Then, given the utility functions and the average utility of the users defined in~(\ref{utility_user}) and~(\ref{utility_user_average}), respectively, and by incorporating the memory characteristic of the users, we can formulate the fractional evolutionary game as follows: \begin{align} \label{replicator_evolu_fractional} \notag {}^C_0D^{\beta}_{t} p_{m, l,k,j,i}(t) =&\mu p_{m, l,k,j,i}(t)[u_{m, l,k,j,i}(t)-\overline{u}_i(t)], \\ & m \in \mathcal{M}, l \in \mathcal{L}_m, k \in \mathcal{Q}_{m,l}, j \in \mathcal{P}_m, i \in \mathcal{N}_g, \forall t, \end{align} where $\beta \in (0,2), \beta \neq 1$ is the order of the Caputo fractional derivative, and it is called memory effect coefficient.~The physical meaning of the left-sided Caputo is further explained and discussed in Section~\ref{perform_eval}. The equilibrium analysis of the fractional evolutionary game is presented in the next section.
\subsection{Equilibrium Analysis}
In this section, we theoretically discuss the existence and the uniqueness of the equilibrium, and the unique and stable equilibrium is admitted as the solution of the fractional evolutionary game defined in~(\ref{replicator_evolu_fractional}). The specific steps are as follows. First, we transfer the fractional game defined in~(\ref{replicator_evolu_fractional}) into an equivalent problem, i.e,~(\ref{Picard_iteration_fractional_game_equilibrium}), and the equivalence between which is verified in Theorem~\ref{th:equilivent_problem_FEG}. Then, to prove the existence and uniqueness of equilibrium of the game defined in~(\ref{replicator_evolu_fractional}), we provide the proof of the uniqueness of the solution to the equivalent problem defined in ~(\ref{Picard_iteration_fractional_game_equilibrium}).
For the ease of presentation, we let $\mathbf{P}(t)=[p_{m,l,k,j,i}\left(t\right)]_{ m \in \mathcal{M}, l \in \mathcal{L}_m, k \in \mathcal{Q}_{m,l}, j \in \mathcal{P}_m, i \in \mathcal{N}_g}$ and $\mathbf{E}(\mathbf{P}(t))= \big{[}\mu p_{m, l,k,j,i}(t)[u_{m, l,k,j,i}(t)-\overline{u}_i(t)]\big{]}_{ m \in \mathcal{M}, l \in \mathcal{L}_m, k \in \mathcal{Q}_{m,l}, j \in \mathcal{P}_m, i \in \mathcal{N}_g}$, and reorganize the fractional evolutionary game defined in~(\ref{replicator_evolu_fractional}) as follows:
\begin{equation} \label{replicator_evolu_delay_power_simplied} {}^C_0D^{\beta}_{t} \mathbf{P}(t)=\mathbf{E}(\mathbf{P}(t)), \end{equation} with the initial strategy $\mathbf{P}(0)=\mathbf{P}^0=[p^0_{m,l,k,j,i}]_{ m \in \mathcal{M}, l \in \mathcal{L}_m, k \in \mathcal{Q}_{m,l}, j \in \mathcal{P}_m, i \in \mathcal{N}_g}$ and the time horizon $\mathcal{T}=[0,T]$.
\begin{theorem}\label{th:equilivent_problem_FEG}
If all the elements of vector $\mathbf{E}$ in~(\ref{replicator_evolu_delay_power_simplied}), i.e., $e_n$ (element $n$ of vector $\mathbf{E}$) for all $n \in \left\{\left. \left(m,l,k,j,i\right) \right| m \in \mathcal{M}, l \in \mathcal{L}_m, k \in \mathcal{Q}_{m,l}, j \in \mathcal{P}_m, i \in \mathcal{N}_g \right\}$, can satisfy the following two conditions: \begin{itemize} \item $e_n \in {\cal{C}}^2$ with ${\cal{C}}^2$ being the set of the twice differentiable functions;
\item $\frac{\partial} {\partial p_{m,l,k,j,i} }e_n$ exists and is bounded for all $m \in \mathcal{M}, l \in \mathcal{L}_m, k \in \mathcal{Q}_{m,l}, j \in \mathcal{P}_m,i \in \mathcal{N}_g$. \end{itemize} Then,~(\ref{replicator_evolu_delay_power_simplied}) can be equivalently transformed into the following problem \begin{equation} \mathbf{P}(t)= \mathbf{P}^0 + {}_0\rm{I}^\beta_t\mathbf{E}(\mathbf{P}(t)), \forall t\in \mathcal{T}. \label{Picard_iteration_fractional_game_equilibrium} \end{equation}
The second condition means that for all $n \in \left\{\left. \left(m,l,k,j,i\right) \right| m \in \mathcal{M}, l \in \mathcal{L}_m, k \in \mathcal{Q}_{m,l}, j \in \mathcal{P}_m, i \in \mathcal{N}_g \right\}$, there exists $L \in \mathbb{R}^+$ such that $|e_n({\hat{\mathbf{P}}}(t)) - e_n({\tilde{\mathbf{P}}}(t))| < L||({\hat{\mathbf{P}}}(t)) - {\tilde{\mathbf{P}}}(t)||_{\mathcal{L}_1}$, which implies the satisfaction of the Lipschitz condition.
\end{theorem}
\begin{proof} According to~(\ref{Picard_iteration_fractional_game_equilibrium}), the $\ceil{\beta}$-th derivative of $\mathbf{P}(t)$ with respect to $t$ is as follows: \begin{equation} \frac{\rm{d}^{\ceil{\beta}}}{\rm{d}t^{\ceil{\beta}}} \mathbf{P}(t)= \frac{\rm{d}^{\ceil{\beta}}}{\rm{d}t^{\ceil{\beta}}} \big{[} {}_0\rm{I}^\beta_t\mathbf{E}(\mathbf{P}(t)) \big{]} = {}^{RL}_0D^{\beta - \ceil{\beta}}_{t} {\mathbf{E}}\left(\mathbf{P}(t)\right), \label{Picard_iteration_fractional_game_equilibrium_proof_1} \end{equation} with ${}^{RL}_0D^{\beta}_{t} {\mathbf{E}}\left(\mathbf{P}(t)\right)$ being defined as the left-sided Riemann-Liouville fractional derivative with respect to $t$, and the following derivation is satisfied \begin{equation}\label{eq:RL_upper_bound} \begin{aligned} {}^{RL}_0D^{\ceil{\beta} - \beta}_{t} {\mathbf{E}}\left(\mathbf{P}(t)\right) &= \frac{\rm{d}}{\rm{d}t} \Big{[} \frac{1}{\Gamma(1-\ceil{\beta}+\beta)} \int^t_0 \frac{\mathbf{E}(\mathbf{P}(\tau))} { (t-\tau)^{\ceil{\beta}-\beta} } \rm{d}\tau \Big{]}\\ &= \frac{1}{\Gamma(1-\ceil{\beta}+\beta)} \frac{\rm{d}}{\rm{d}t} \int^t_0\theta^{\beta -\ceil{\beta}} \mathbf{E}(\mathbf{P}(t-\theta)) \rm{d}\theta\\ &= \frac{1}{\Gamma(1-\ceil{\beta}+\beta)}\Big{[}t^{\beta-\ceil{\beta}} \mathbf{E}(\mathbf{P}^0) + \int^t_0\theta^{\beta-\ceil{\beta}} \frac{\rm{d}}{\rm{d}t} \mathbf{E}(\mathbf{P}(t-\theta)) \rm{d}\theta \Big{]}\\ &= \frac{1}{\Gamma(1-\ceil{\beta}+\beta)}\Big{[}t^{\beta-\ceil{\beta}} \mathbf{E}(\mathbf{P}^0) + \int^t_0(t-\tau)^{\beta-\ceil{\beta}} \frac{\rm{d}}{\rm{d}\tau} \mathbf{E}(\mathbf{P}(\tau)) \rm{d} \tau \Big{]} \\ &= \frac{ t^{\beta-\ceil{\beta}} }{ \Gamma(1-\ceil{\beta}+\beta) }\mathbf{E}(\mathbf{P}^0) + {}_0\rm{I}^{\beta}_t \Big{[}\frac{\rm{d}^{\ceil{\beta}} }{\rm{d}t^{\ceil{\beta}} }\mathbf{E}(\mathbf{P}(t)) \Big{]}. \end{aligned} \end{equation}
Let $\sigma \in (0,t)$, the $\mathcal{L}^1$ norm of $\frac{ t^{\beta-\ceil{\beta}} }{ \Gamma(1-\ceil{\beta}+\beta) }\mathbf{E}(\mathbf{P}^0)$ exists an upper bound that is derived as follows: \begin{equation}\label{eq:RL_residual_term} \Vert \frac{ t^{\beta-\ceil{\beta}} }{ \Gamma(1-\ceil{\beta}+\beta) }\mathbf{E}(\mathbf{P}^0) \Vert_{\mathcal{L}^1} \le \Vert \frac{ \sigma^{\beta-\ceil{\beta}} }{ \Gamma(1-\ceil{\beta}+\beta) }\mathbf{E}(\mathbf{P}^0) \Vert_{\mathcal{L}^1}. \end{equation}
Using the condition in Theorem~\ref{th:equilivent_problem_FEG} that $\frac{\delta} {\delta p_{m,l,k,j,i} }e_n$ exists and is bounded for all $m \in \mathcal{M}, l \in \mathcal{L}_m, k \in \mathcal{Q}_{m,l}, j \in \mathcal{P}_m,i \in \mathcal{N}_g$, and~(\ref{Picard_iteration_fractional_game_equilibrium_proof_1}) and~(\ref{eq:RL_upper_bound}) as well as~(\ref{eq:RL_residual_term}), we have \begin{equation} \label{eq:integer_derivative_P}
\begin{aligned} \Big{\Vert} \frac{\rm{d}^{\ceil{\beta}}}{\rm{d}t^{\ceil{\beta}}} \mathbf{P}(t) \Big{\Vert}_{\mathcal{T}} &< \Vert \frac{ \sigma^{\beta-\ceil{\beta}} }{ \Gamma(1-\ceil{\beta}+\beta) }\mathbf{E}(\mathbf{P}^0) \Vert_{\mathcal{L}^1} + \Big{\Vert} {}_0\rm{I}_t^\beta \frac{\rm{d}^{\ceil{\beta}}} {\rm{d}t^{\ceil{\beta}}} \mathbf{E}(\mathbf{P}(t)) \Big{\Vert}_{\mathcal{T}}\\ &< \Vert \frac{ \sigma^{\beta-\ceil{\beta}} }{ \Gamma(1-\ceil{\beta}+\beta) }\mathbf{E}(\mathbf{P}^0) \Vert_{\mathcal{L}^1} + \Big{\Vert} {}_0\rm{I}_t^\beta \frac{\rm{d}^{\ceil{\beta}}} {\rm{d}t^{\ceil{\beta}}} \mathbf{P}(t) \Big{\Vert}_{\mathcal{T}} A L,
\end{aligned} \end{equation}
where $\Vert z \Vert_{\mathcal{T}} = \int_{\mathcal{T}} \exp\left(-\mu t\right)\Vert z\Vert_{\mathcal{L}^1}\rm{d}t $ and $A$ is the cardinality of $\left\{\left. \left(m,l,k,j,i\right) \right| m \in \mathcal{M}, l \in \mathcal{L}_m, k \in \mathcal{Q}_{m,l},\right.$ $\left. j \in \mathcal{P}_m, i \in \mathcal{N}_g \right\}$. For the last term in~(\ref{eq:integer_derivative_P}), we have \begin{equation}\label{eq:fractional_integral_integer_derivative_P} \begin{aligned}
\Big{\Vert} {}_0\rm{I}_t^\beta \frac{\rm{d}^{\ceil{\beta}}} {\rm{d}t^{\ceil{\beta}}} \mathbf{P}(t) \Big{\Vert}_{\mathcal{T}}&= \int^T_0 \exp\left(-\mu t\right) \Big{\Vert} {}_0\rm{I}_t^\beta \frac{\rm{d}^{\ceil{\beta}}} {\rm{d}t^{\ceil{\beta}}} \mathbf{P}(t) \Big{\Vert} \rm{d}t \leq \int_0^T \exp\left(-\mu t\right) \int^t_0 \frac{1}{\Gamma(\beta)} \frac{ \Vert \frac{ \rm{d}^{\ceil{\beta}} }{\rm{d}s^{\ceil{\beta}}} \mathbf{P}(s) \Vert } {(t-s)^{1-\beta}}\rm{d}s\rm{d}t\\
&= \int_0^T \frac{\exp\left(-\mu s\right)} {\Gamma(\beta)} \Vert \frac{ \rm{d}^{\ceil{\beta}} }{\rm{d}s^{\ceil{\beta}}} \mathbf{P}(s) \Vert \int^T_s \exp\left(-\mu(t-s)\right)(t-s)^{\beta-1}\rm{d}t\rm{d}s\\
&= \int_0^T \frac{\exp\left(-\mu s\right)} {\Gamma(\beta)} \Vert \frac{ \rm{d}^{\ceil{\beta}} }{\rm{d}s^{\ceil{\beta}}} \mathbf{P}(s) \Vert \int^{\mu(T-s)}_0 \exp\left(-\Psi\right) (\frac{\Psi}{\mu})^{\beta-1} \rm{d}(\frac{\Psi}{\mu}) \rm{d}s\\ & < \frac{1} {\mu^\beta \Gamma(\beta)}\int_0^T \exp\left(-\mu s\right) \Vert \frac{ \rm{d}^{\ceil{\beta}} }{\rm{d}s^{\ceil{\beta}}} \mathbf{P}(s) \Vert \rm{d}s \int^{+\infty}_0 \exp\left(-\Psi\right) \Psi^{\beta-1} \rm{d}\Psi \\ & = \frac{1} {\mu^\beta}\Vert \frac{ \rm{d}^{\ceil{\beta}} }{\rm{d}s^{\ceil{\beta}}} \mathbf{P}(t) \Vert_{\cal{T}}. \end{aligned} \end{equation} Then, we substitute~(\ref{eq:fractional_integral_integer_derivative_P}) into~(\ref{eq:integer_derivative_P}) as follows \begin{equation}
\begin{aligned} &\Big{\Vert} \frac{\rm{d}^{\ceil{\beta}}}{\rm{d}t^{\ceil{\beta}}} \mathbf{P}(t) \Big{\Vert}_{\mathcal{T}} < \Vert \frac{ \sigma^{\beta-\ceil{\beta}} }{ \Gamma(1-\ceil{\beta}+\beta) }\mathbf{E}(\mathbf{P}^0) \Vert_{\mathcal{L}^1} + \frac{AL} {\mu^\beta}\Vert \frac{ \rm{d}^{\ceil{\beta}} }{\rm{d}s^{\ceil{\beta}}} \mathbf{P}(t) \Vert_{\cal{T}}\\ \Leftrightarrow & \Big{\Vert} \frac{\rm{d}^{\ceil{\beta}}}{\rm{d}t^{\ceil{\beta}}} \mathbf{P}(t) \Big{\Vert}_{\mathcal{T}} < \frac{1}{1 - \frac{AL} {\mu^\beta}}\Vert \frac{ \sigma^{\beta-\ceil{\beta}} }{ \Gamma(1-\ceil{\beta}+\beta) }\mathbf{E}(\mathbf{P}^0) \Vert_{\mathcal{L}^1},
\end{aligned} \end{equation} which implies that if $\mu$ is sufficiently large such that $\frac{AL} {\mu^\beta} < 1$, $\Big{\Vert} \frac{\rm{d}^{\ceil{\beta}}}{\rm{d}t^{\ceil{\beta}}} \mathbf{P}(t) \Big{\Vert}_{\mathcal{T}}$ has an upper bound. In this case, the fractional derivative of $\mathbf{P}(t)$ with the order of $\beta\in \left(0,1\right)\cup\left(1,2\right)$ exists and can be obtained in the following: \begin{equation} {}^C_0D^{\beta}_{t} \mathbf{P}(t) = {}_0\rm{I}^{{\ceil{\beta}} - \beta}_t \frac{\rm{d}^{\ceil{\beta}}}{\rm{d}t^{\ceil{\beta}}} \mathbf{P}(t) = {}_0\rm{I}^{{\ceil{\beta}} - \beta}_t \left\{\frac{ t^{\beta-\ceil{\beta}} }{ \Gamma(1-\ceil{\beta}+\beta) }\mathbf{E}(\mathbf{P}^0) + {}_0\rm{I}^{\beta}_t \Big{[}\frac{\rm{d}^{\ceil{\beta}} }{\rm{d}t^{\ceil{\beta}} }\mathbf{E}(\mathbf{P}(t)) \Big{]} \right\} = \mathbf{E}(\mathbf{P}(t)) , \end{equation} \end{proof} and hence the equivalence between~(\ref{replicator_evolu_delay_power_simplied}) and~(\ref{Picard_iteration_fractional_game_equilibrium}) has been verified, which completes the proof.
\begin{theorem} \label{th:existence_unique_equlivent_problem_FEG} Given the conditions in Theorem~\ref{th:equilivent_problem_FEG}, the uniqueness of the solution to the problem defined in~(\ref{Picard_iteration_fractional_game_equilibrium}) can be guaranteed. \label{admit_solution} \end{theorem}
\begin{proof} First, by defining an operator $\Lambda: \mathcal{D}_{\mathbf{P}} \mapsto \mathcal{D}_{\mathbf{P}}$, where $\mathcal{D}_{\mathbf{P}}$ is the feasible domain of ${\mathbf{P}}$, there exists an inequality expression as follows: \begin{equation}
\Vert \Lambda {\hat{\mathbf{P}}}(t) - \Lambda {\tilde{\mathbf{P}}}(t)\Vert_{\mathcal{T}} < \frac{AL}{\mu^\beta}\Vert {\hat{\mathbf{P}}}(t) - {\tilde{\mathbf{P}}}(t)\Vert_{\mathcal{T}},
\label{inequality_operator} \end{equation} and the specific derivation of which has been shown as follows \begin{equation}\label{eq:bounded_operator} \begin{aligned} & \Vert\Lambda {\hat{\mathbf{P}}}(t) - \Lambda {\tilde{\mathbf{P}}}(t)\Vert_{\mathcal{T}} = \int^T_0 \exp\left(-\mu t\right)\Vert {}_0\mathbf{I}^\beta_t \mathbf{E}({\hat{\mathbf{P}}}(t))-{}_0\mathbf{I}^\beta_t \mathbf{E}({\tilde{\mathbf{P}}}(t)) \Vert\rm{dt} \\ &< AL \Big{[} \int^T_0 \exp\left(-\mu t\right)\Vert {}_0\mathbf{I}^\beta_t {\hat{\mathbf{P}}}(t)-{}_0\mathbf{I}^\beta_t {\tilde{\mathbf{P}}}(t) \Vert\rm{dt} \Big{]}\\
&\leq AL \Big{[} \int^T_0 \exp\left(-\mu t\right) \int^t_0 \frac{ \Vert {\hat{\mathbf{P}}}(s)- {\tilde{\mathbf{P}}}(s)\Vert} {\Gamma(\beta) (t-s)^{1-\beta} } {\rm{d}}s{\rm{d}}t \Big{]} \\
&= \frac{AL}{\Gamma(\beta)} \Big{[} \int^T_0 \int^T_s \exp\left(-\mu t\right) \frac{ \Vert {\hat{\mathbf{P}}}(s)- {\tilde{\mathbf{P}}}(s)\Vert} { (t-s)^{1-\beta} } \rm{dtds} \Big{]}\\
&= \frac{AL}{\Gamma(\beta)} \Big{[} \int^T_0 \exp\left(-\mu s\right) \Vert {\hat{\mathbf{P}}}(s)- {\tilde{\mathbf{P}}}(s) \Vert \int^{T}_s\frac{ \exp\left(-\mu (t- s)\right) } { (t-s)^{1-\beta} } {\rm{d}}s{\rm{d}}t \Big{]}\\
&= \frac{AL}{\mu^\beta \Gamma(\beta)} \Big{[} \int^T_0 \exp\left(-\mu s\right) \Vert {\hat{\mathbf{P}}}(s)- {\tilde{\mathbf{P}}}(s) \Vert \int^{\mu(T-s)}_0 \exp\left(-\psi\right)\psi^{\beta-1} {\rm{d}}\psi {\rm{d}}s \Big{]}\\
&< \frac{AL}{\mu^{\beta} \Gamma(\beta)}\Vert {\hat{\mathbf{P}}}(t)- {\tilde{\mathbf{P}}}(t) \Vert_\mathcal{T} \int^{+\infty}_0\exp\left(-\sigma\right)\sigma^{\beta -1}\rm{d}\sigma = \frac{AL}{\mu^\beta} \Vert {\hat{\mathbf{P}}}(t)- {\tilde{\mathbf{P}}}(t) \Vert_\mathcal{T}. \end{aligned} \end{equation}
Based on~(\ref{eq:bounded_operator}), we can conclude that $\Vert\Lambda {\hat{\mathbf{P}}}(t) - \Lambda {\tilde{\mathbf{P}}}(t)\Vert_{\mathcal{T}} < \Vert {\hat{\mathbf{P}}}(t)- {\tilde{\mathbf{P}}}(t) \Vert_\mathcal{T} $ if $\mu^\beta \ge AL$. In this case, the operator $\Lambda$ satisfies the fixed point theorem, which indicates the uniqueness of the solution to~(\ref{Picard_iteration_fractional_game_equilibrium}). By following this, there exists a unique solution to the fractional evolutionary game defined in~(\ref{replicator_evolu_delay_power_simplied}), which completes this proof.
\end{proof}
\section{Performance Evaluation} \label{perform_eval}
= 0:7 and
= 1:3 with that obtained from the classical evolutionary game, i.e.,
= 1 In this section, we present and discuss simulation results obtained by the proposed evolutionary game approaches. To evaluate the game approaches, we consider three cases, i.e., $\beta=1$ corresponding to the classical evolutionary game, and $\beta=1.1$ and $0.8$ corresponding to the fractional evolutionary games. For the comparison purpose, we consider a network that consists of two SPs, namely SP 1 and SP 2, and $100$ users. Each SP deploys a BS that is equipped with $4$ antennas. SP 1 deploys 2 IRSs, namely IRS $1_1$ and $1_2$, and SP 2 deploys 1 IRS, namely IRS $2$. SP 1 divides each IRS into two modules and offers $1$ power level, i.e., $J_{1,1}=30$ dBm. SP 2 does not divide its IRS and offers $2$ power levels, i.e., $J_{2,1}=25$ dBm and $J_{2,2}=35$ dBm. As such, SP 1 offers 4 services, and SP 2 offers 2 services that the users can select. Correspondingly, $100$ users are divided into $6$ groups. Note that the SPs can offer more services and our proposed game approaches are scalable since the complexity of the algorithm implemented at each user is $\mathcal{O}(1)$ as analyzed in Section~\ref{classical_evol_game_form}. The locations of the BSs, IRSs, and users are shown in Fig.~\ref{Location}. The simulation parameters are provided in Table~\ref{table:parameters}. In particular for the involved channels, the non-LoS elements is proved to be much weaker than the LoS element, i.e., lower than $20$ dB~\cite{han2014multi}, and thus similar to~\cite{gao2016fast},~we mainly consider the channel with only LoS element, i.e., $\mathcal{L}=\mathcal{L}_{\text{BS-I}}=\mathcal{L}_{\text{I-U}}=1$. \begin{figure}
\caption{Coordinates of BSs, IRSs, and users.}
\label{Location}
\end{figure}
\begin{table}[t] \caption{\small Simulation parameters.} \label{table:parameters_CRN} \footnotesize \centering
\begin{tabular}{lc|lc|lc|lc} \hline\hline {\em Parameters} & {\em Value} & {\em Parameters} & {\em Value} & {\em Parameters} & {\em Value}& {\em BSs, IRSs and user} & {\em Coordinate $(m)$ }\\ [1ex] \hline
$M$ & $2$ & $B_1=B_2$ & $2$ MHz & $v_1$, $v_2$ & $10^{-6}$ & BS $1$ & [100 20] \\ \hline
$L_1=L_2$ & $4$ & $f$& $0.3$ THz & $\gamma^I_1$ & $2.5\times10^{-4}$& BS 2 & [0 20] \\ \hline $P_1$ & $1$ & $c$& $3 \times 10^8$ m/s & $\gamma^I_2$ & $10^{-3}$ &IRS $1_1$ & [80 20] \\ \hline $P_2$ & $ 2 $ & $N$ & $100$ & $\gamma^P_1$ & $0.05$ &IRS $1_2$ & [40 20] \\ \hline $J_{1,1}$ & $30$ dBm & $Q_{1,1}=Q_{1,2}$ & $2 $ & $\gamma^P_2$ & $0.03$& IRS $2_1 $ & [30 20] \\ \hline $J_{2,1}$ & $25$ dBm & $Q_{2,1}$ & $1$& $\gamma^P_2$ & $0.03$ &User group & [50 0] \\ \hline $J_{2,2}$ & $35$ dBm & $E_{1,1}$, $E_{1,2}$, $E_{2,1}$ & $8$ & $(\mu, \zeta)$ & $(e^{-2},0)$ & & \\ \hline \end{tabular} \label{table:parameters} \end{table}
\begin{figure*}
\caption{Proportion of users selecting different SPs and services.}
\label{proportion1}
\end{figure*}
First, we discuss strategies that the users choose different SPs and services over evolutionary time. Figures~\ref{proportion1}(a), (b), and (c) illustrate the results for the evolutionary games with $\beta=1, 1.1$ and $0.8$, respectively. As seen, the strategies of the users choosing different SPs and services eventually converge to an equilibrium point over time. Moreover, during the initial phase, the users' strategies in the evolutionary games with $\beta=1$ and $0.8$ fluctuate in a range smaller than those in the evolutionary game with $\beta=1.1$. The results indicate that the adaptations of the users' strategies in the game with $\beta = 1.1$ are faster than those in the games with $\beta = 1$ and $0.8$. These results are further verified in Fig.~\ref{proportion1}(d) in which the strategy adaptation frequency of the users in the game with $\beta = 1.1$ is higher than those in the games with $\beta = 1$ and $0.8$. Note that as the users' strategies have a larger fluctuation, the convergence speed can be slower. Thus, as we can observe from Fig.~\ref{proportion1}(d), the game with $\beta = 1.1$ converges to the equilibrium more slowly than those in the games with $\beta=1$ and $0.8$.
\begin{figure*}
\caption{The impact of $\beta$ on the evolutionary games.}
\label{impact_beta}
\end{figure*}
Now, we discuss how the memory effect coefficient, i.e., $\beta$, impacts on the evolutionary games. As shown in Fig.~\ref{impact_beta}(a), with $\beta < 1$ and $\beta > 1$, as $\beta$ increases, the convergence time of the user's strategy is shorter. This implies that as $\beta$ increases, the replicator dynamics converges faster. Moreover, as shown in Fig.~\ref{impact_beta}(b), as $\beta$ increases, the rate that the corresponding games converge to the vicinity of the equilibrium is faster. This means that the adaptation rate of the user's strategy increases with the increase of $\beta$. This results is also consistent with descriptions in Figs.~\ref{proportion1}(a), (b), (c), and (d).
Next, it is important to show the utility that the users can achieve when different games are used. For this, we vary the values of $\beta$, i.e., the memory effect, and we evaluate the total utility the the users achieve. As shown in Fig.~\ref{impact_beta}(b), the utility for the users with $\beta=0.8 < 1$ is worse than that for the users with $\beta=1$. Meanwhile, with $ \beta = 1.1 > 1$, the users achieve higher utility values when the users have no memory effect, i.e., $\beta=1$. Since the users achieve higher utility values with $\beta = 1.1$, we can say that the memory effect with $\beta > 1$ is a positive effect. This further implies that to achieve a higher utility value, the users should incorporate both the past and instantaneously achievable experiences for their network selection. In addition, as seen from Fig.~\ref{impact_beta}(b), the total utility of users with IRSs in the games, i.e., $\beta = 0.8$, $1$, and $1.1$ is much higher than that of users without IRSs. This result demonstrates that deploying IRSs increases the throughput of the users.
\begin{figure*}
\caption{Direction field of the replicator dynamics when $t = 200$.}
\label{direction_field}
\end{figure*}
To show that the users' strategies in the proposed game approaches can be stabilized at the equilibrium, we present the direction field of the replicator dynamics. As illustrated in Fig.~\ref{direction_field}, the strategies of the users eventually reach the equilibrium strategy after a certain time, i.e., $t = 200$, that is represented by the black circles. We can take the results shown in Fig~\ref{direction_field}(a) as an example. In the figure, we show the replicator dynamics of selection strategy of SP 1, service 1 and SP 2, service 1 $(p_{1,1}$ and $p_{2,1}$). Assuming that the strategies the users select services provided by SP 2 (i.e., $p_{2,1}$ and $p_{2,2}$) achieve the equilibrium, and $p_{1,1}= 1 - p_{1,2} - p_{1,3} - p_{1,4} - p_{2,1} - p_{2,2}$. As seen, the users are able to adapt their strategies by following the directions of the arrows. Furthermore, any initial strategy eventually reach the equilibrium that verifies the stability of our proposed game approaches. That is similar to Figs.~\ref{direction_field}(b), (c), (d), (e), and (f).
\begin{figure*}
\caption{Utility of user groups.}
\label{utility1}
\end{figure*}
Now, we discuss how the utilities of the users obtained at the equilibrium. Figures~\ref{utility1}(a), (b), and (c) show the utilities that the users achieve by selecting different SPs and network services over evolutionary time. As seen, the utilities of the users vary until the equilibrium is reached. At the equilibrium, the users have the same utility even if they select different SPs and services. The reason is that the evolutionary equilibrium is reached only when the utilities of the users choosing any SP and any service are equal to their expected utility.
\begin{figure*}
\caption{Proportion of users selecting different service as the size of IRS provided by SP 2 varies.}
\label{proportioin_change_size_IRS}
\end{figure*}
Next, we investigate the impact of sizes of IRSs on the proportions of users selecting different SPs and services. Figures~\ref{proportioin_change_size_IRS} (a), (b), and (c) show the results obtained from the evolutionary game with $\beta=1, 1.1$ and $0.8$, respectively. In particular, we vary the size $K_{2}$ of IRS 2 of SP 2. As shown in Fig.~\ref{proportioin_change_size_IRS}(a), as $K_{2}$ increases, the proportions of users selecting services provided by SP 2 increase since the throughput and utility obtained by the users selecting services provided by SP 2 increase. However, as the size of IRS 2 is large, the increasing rate tends to be slower. This is because of that the users pay a very high resource cost if they select the services provided by SP 2, and thus they tend to select the services provided by SP 1. Figures~\ref{proportioin_change_size_IRS}(b) and (c) have the same pattern as Fig. 8(a), and the results can be explained in the same way.
\begin{figure*}
\caption{Impact of the number of users on the total utility and the time to reach the equilibrium.}
\label{learning_rate}
\end{figure*}
Note that the time to reach the equilibrium can be different depending on the learning rate $\mu$ and the number of users. Figures~\ref{learning_rate}(a), (b), and (c) show the results obtained from the fractional evolutionary games with $\beta=1, 1.1$ and $0.8$, respectively. As seen, the evolutionary equilibrium is reached faster as $\mu$ is higher since the frequency of the strategy adaptation of the users is higher. Moreover, the games need more time to converge to the equilibrium as the number of users $N$ increases. Note that as $N$ increases, the total utility of the users decreases. This can be explained based on~(\ref{expected_data_rate}), more users share the fixed amount of bandwidth that results in reducing the throughput of the users. Moreover, from Fig.~\ref{learning_rate}(a), the total utility of users with IRSs is much higher than that of users without IRSs. The reason is that deploying IRSs increases the throughput of the users.
\begin{figure*}
\caption{Proportions of users selecting different SPs vs. distance.}
\label{proportion_change_distance}
\end{figure*}
Next, we discuss how the mobility of the users impacts the selection strategies of the users. In particular, we evaluate the proportions of users selecting different SPs as the distance between the users and IRS 1 provided by SP 1 varies. The results for the evolutionary games with $\beta=1, 1.1$ and $0.8$ are respectively shown in Figs.~\ref{proportion_change_distance}(a), (b), and (c). As observed from Figs.~\ref{proportion_change_distance}(a), (b), and (c), the proportion of users selecting SP 2 increases as the distance between the users and IRS 1 of SP 1 increases. This is because of that the throughput obtained by the users selecting services of SP 1 decreases. Thus, the users are willing to select services of SP 2. In addition, we consider the case as SP 1 deploys one IRS. As observed from Figs.~\ref{proportion_change_distance}(a), (b), and (c), as SP 1 deploys 1 IRS, the proportions that the users select SP 1 is lower than those that SP 1 deploys 2 IRSs. Especially, decreases more slowly than those that SP 1 deploys 2 IRSs. Interestingly, as SP 1 deploys 1 IRS, the proportions that the users select SP 1 decrease faster than those that SP 1 deploys 2 IRSs. This implies that by deploying more IRSs, the SP can further improve the QoS of the users and can prevent the users to select the network service of other SPs.
\begin{figure*}
\caption{Proportion of the users selecting SP1 and Service 1 with different delays.}
\label{proportion_delay}
\end{figure*}
Next, we discuss how the proportions of users selecting different SPs and services depend on the information delay $\delta$. For the evaluation purpose, we consider the proportion of the users selecting Service 1 provided by SP 1 as shown in Figs.~\ref{proportion_delay}(a) (b), and (c). First, we discuss the results obtained by the classical evolutionary game shown in Figs.~\ref{proportion_delay}(a). This figure shows the proportion of the users choosing Service 1 of SP 1 when the users use information for their decisions at $\delta= 0, 40$ and $80$. As seen, when $\delta > 0$, there is a fluctuating dynamics of strategy adaptation. In particular, as $\delta=40$, the game can sill converge to the equilibrium that is the same as the case when $\delta = 0$. However, as $\delta = 80$, the game cannot reach the equilibrium. These results mean that when the users use information with a small delay for their decisions, the game is still guaranteed for the convergence. In addition, we find in the figure that as $\delta=40$, the proportions of the users have more fluctuations and the game needs more time to reach the equilibrium, compared with the game where $\delta=0$. This implies that the convergence speed is slower as the users use information with larger delay. The results obtained by the fractional games with $\beta=0.8$ and $1.1$ are the same as that obtained by the classical evolutionary game. However, it seems to be that the service selection cannot reach the evolutionary equilibrium even when $\delta$ is small. For example, for fractional evolutionary game with $\beta=1.1$, as $\delta \leq 2$, the service selection cannot reach the evolutionary equilibrium. This is also a shortcoming of the fractional game in which the very outdated information may not be used for the user decisions.
\begin{figure*}
\caption{Proportion of the users selecting different SPs as the size of IRS and the price of IRS element provided by SP2 varies.}
\label{proportion_price}
\end{figure*}
We finally discuss how the sizes of IRSs and the price of IRS 2 elements impact on the SP and service selection of the users. Figure~\ref{proportion_price}(a) shows that, for a given price, the proportion of users selecting SP 2 increases as the number of elements of IRS 2 increases. This is because of that the throughput obtained by the users selecting services of SP 2 increases. Thus, the users are willing to select services of SP 2. This figure also shows the proportion of users selecting SP 1 increases as the price of IRS 2 elements increases. This is because of that the utility of users choose SP 2 will fall down as the price of IRS 2 elements increases. Therefore, the users are willing choose services of SP 1. The same explanations can be applied to the results obtained as the values of $\beta$ are $0.8$ and $1.1$ (Figs.~\ref{proportion_price}(b) and (c)).
\section{Conclusions} \label{conclu}
We have proposed dynamic game frameworks for modeling the dynamic network selection of mobile users in the IRS-enabled terahertz network. First, we have adopted the classical evolutionary game in which the SP and service adaptation of the users is modeled as replicator dynamics. We have further considered the scenario in which the users use delayed information for their decision-making. In this scenario, we have analyzed the stability region of the delayed replicator dynamics. Furthermore, we have adopted the fractional evolutionary game that incorporates the memory effect to model the SP and service adaptation of the users. The proof of the existence and uniqueness of the game equilibrium has been provided. We have finally provided simulation results obtained by the proposed evolutionary game approaches. In addition, we have further discussed the selection behaviors of the users and compared the performance obtained by the proposed evolutionary games. For future work, we will study the SP and network service selections in a heterogeneous network that includes different types of relaying, i.e., IRS and active relay devices, and different communication technologies, i.e., terahertz and millimeter wave communications.
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\end{document} | arXiv |
\begin{document}
\author{}
\title{Unbounded Bell violations for quantum genuine multipartite non-locality}
\author{Abderram\'an Amr $^1$} \author{Carlos Palazuelos $^{1,3}$} \author{Julio I. de Vicente$^2$} \address[$^1$]{Departamento de An\'alisis Matem\'atico y Matem\'atica Aplicada, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain} \address[$^2$]{Departamento de Matem\'aticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Legan\'es, Madrid, Spain} \address[$^3$]{Instituto de Ciencias Matem\'aticas (ICMAT), C/ Nicol\'as Cabrera, Campus de Cantoblanco, 28049 Madrid, Spain} \email[$^1$]{[email protected]} \email[$^2$]{[email protected]} \email[$^3$]{[email protected]}
\maketitle
\begin{abstract} The violations of Bell inequalities by measurements on quantum states give rise to the phenomenon of quantum non-locality and express the advantage of using quantum resources over classical ones for certain information-theoretic tasks. The relative degree of quantum violations has been well studied in the bipartite scenario and in the multipartite scenario with respect to fully local behaviours. However, the multipartite setting entails a more complex classification in which different notions on non-locality can be established. In particular, genuine multipartite non-local distributions apprehend truly multipartite effects, given that these behaviours cannot be reproduced by bilocal models that allow correlations among strict subsets of the parties beyond a local common cause. We show here that, while in the so-called correlation scenario the relative violation of bilocal Bell inequalities by quantum resources is bounded, i.e. it does not grow arbitrarily with the number of inputs, it turns out to be unbounded in the general case. We identify Bell functionals that take the form of non-local games for which the ratio of the quantum and bilocal values grows unboundedly as a function of the number of inputs and outputs. \end{abstract}
\section{Introduction}
Bell's theorem establishes that correlations among the results of spatially separated measurements on composite quantum systems are incompatible with a local variable model \cite {Bell}. Besides its crucial implications in the foundations of quantum mechanics, this phenomenon, known as \emph{quantum non-locality}, is behind many important applications in quantum information theory, such as quantum cryptography \cite{Ekert91,ABGMPS,Vazirani14}, communication complexity \cite{CleveBuhrman97}, randomness expansion \cite{Pironio10} and randomness amplification \cite{Colbeck12,Gallego13}. The simplest scenario that enables quantum non-locality considers two isolated parties and has been thoroughly studied over the last three decades (see e.g.\ the review \cite{BCPSW}). However, multipartite scenarios have a greater complexity and offer a much richer source of correlations. The potential applications of these phenomena in the context of quantum networks or many-body physics has triggered notable interest in the study of quantum non-locality in the multipartite setting in the last years.
The most natural extension of the notion of locality to the multipartite domain is that of full locality, in which the only source of correlations among the parties is a local common cause. However, the verification of non-fully-local correlations does not necessarily imply non-locality shared among all parties as this resource distributed among just two parties can be enough to falsify these models. Thus, as it also happens in the study of quantum entanglement, a notion of genuine multipartite non-locality (GMNL) can be established in order to capture the idea that the non-local correlations must be truly shared among all parties. In particular, in his celebrated paper \cite{sve}, Svetlichny proved that there exist tripartite quantum correlations that cannot be reproduced by a larger class of models in which, in addition to the local variable, arbitrary (even signalling) correlations are allowed within a strict subset of the parties. This idea, which was extended to an arbitrary number of parties in \cite{Collins02,svegen}, leads to the definition of what we will call general bilocal models. The impossibility of such a model to reproduce certain correlations thus enables the verification of GMNL. Interestingly, this introduces a rich theoretical structure because in these hybrid models one can impose different conditions to the kind of correlations that are allowed within the subsets of parties \cite{Jones05,Bancal09}. In particular, as observed in \cite{Gallego12,Bancal13}, in certain scenarios allowing for signalling correlations in the definition of bilocality leads to ill-defined resource theories and grandfather-style paradoxes. A natural way to get rid of these problems is to consider bilocal models in which correlations among subsets of parties are required to be non-signalling (see e.g.\ \cite{Almeida10,Curchod19}). We will refer to these models as non-signalling bilocal. They introduce an alternative weaker notion of GMNL as non-signalling bilocal models are general bilocal but not necessarily the other way around.
One of the main aims in the study of non-locality from an information-theoretical perspective is to identify and quantify possible advantages in the use of quantum non-local correlations over local ones. Most of these potential protocols boil down mathematically to linear functionals acting on the space of joint conditional probability distributions for the parties and, hence, this leads to the investigation of quantum violations of Bell inequalities. Thus, a natural and often used quantity to understand the difference between local and quantum non-local behaviours in a resource-theoretic way is the relative ratio of violation optimized over all possible Bell inequalities, a question that is moreover related to classical problems in functional analysis. Motivated by the seminal work of Tsirelson \cite{Tsirelson}, a series of papers have been devoted to studying the asymptotic behaviour of this ratio and have shown that this study is very suitable to tackle different problems involving quantum non-locality \cite{BuhrmanRSW12, JungeP11low, JPPVW}. In particular, this finds immediate application to dimension witnesses, communication complexity and entangled games \cite{JPPVW2, PV-Survey}. In this context, the classical result of Tsirelson \cite{Tsirelson}, which states that the above quantity is upper bounded by a universal constant (Grothendieck's constant) when one considers two-output correlations independently of the number of inputs and the Hilbert space dimension, can be understood as a limitation of the advantages of quantum mechanics. This motivated the question, posed by Tsirelson himself, of whether a similar result was true for tripartite correlations and which was answered in \cite{PerezWPVJ08tripartite} in the negative comparing non-local and fully local behaviours. In this sense, tripartite quantum correlations lead to unbounded Bell violations over fully local ones, something that suggests, at least on the theoretical level, the idea of ``unlimited advantage''. Beyond the two-output correlation scenario, it is also known that Bell violations are unbounded for general bipartite probability distributions (see e.g.\ \cite{BuhrmanRSW12,JungeP11low}).
According to the more general notions of non-locality described above that arise in the multipartite scenario, one can wonder whether the main result in \cite{PerezWPVJ08tripartite} is still true in this context. Thus, the goal of this paper is to study the relative violation of Bell inequalities for quantum multipartite behaviours with respect to bilocal ones. Our first result is that in the two-output correlation scenario, contrary to the full-locality case and on the analogy of Tsirelson's result for the bipartite case, there is a universal constant which prevents from having unbounded violations irrespectively of the notion of bilocality used (general bilocal models boil down to non-signalling bilocal models in this case). Next, we consider the same question for general probability distributions. Our main result is that in this case quantum GMNL systems lead to unbounded Bell violations with respect to general bilocal models and, hence, with respect to all other bilocal models since this is the strongest notion of bilocality. In more detail, we provide an instance of a Bell functional, which happens to be a three-prover one-round game, for which the ratio of the quantum value over bilocal models grows with the number of inputs and ouputs. Interestingly, for this it is already enough to consider tripartite systems and, hence, the result extends trivially to an arbitrary number of parties. Thus, although most of our claims are easily generalized to general multipartite systems, we stick throughout the text to the tripartite case, which has the additional benefit of alleviating considerably the notation. The techniques we use are elementary and should be accessible to all readers familiar with quantum non-locality. It turns out that the constraints of the bipartite case can be naturally extended to study relative Bell violations in the genuine multipartite setting. In particular, the aforementioned games that we consider are built as tensor products of bipartite games in the flavour of parallel repetition.
\section{General definitions}
We will consider here the standard Bell scenario in which $k$ parties produce outputs $\{a_i\}_{i=1}^k$ upon receiving inputs $\{x_i\}_{i=1}^k$ according to the joint conditional probability distribution, also called behaviour, \begin{equation}\label{behaviour}
(P(a_1,\cdots,a_k|x_1,\cdots , x_k))_{x_1,\ldots, x_k}^{a_1, \ldots, a_k}. \end{equation}
For simplicity, we will assume that both the input and output alphabets have the same cardinality for all parties. We will denote by $N$ the number of possible inputs and by $K$ the number of possible outputs. A distribution (\ref{behaviour}) is said to be fully local if \begin{align}\label{Def NL Tri}
P(a_1,\cdots,a_k|x_1,\cdots , x_k)=\sum_\lambda p_\lambda P_1(a_1|x_1,\lambda)\cdots P_k(a_k|x_k,\lambda), \end{align}
where $(p_\lambda)_\lambda$ denotes a probability distribution and $(P_i(a_i|x_i,\lambda))_{x_i,a_i}$ is a conditional probability distribution in the $i^{th}$ party for every value of the local variable $\lambda$. We will denote by $\mathcal{L}^k(N,K)$ the set of $k$-partite fully local probability distributions for $N$ inputs and $K$ outputs per party. On the other hand, bilocal behaviours admit a more general model of the form \begin{align}\label{Def Sve}
&P(a_1,\cdots,a_k|x_1,\cdots , x_k)\\\nonumber&=\sum_{M,\lambda}p_{M,\lambda}P_M((a_i)_{i\in M}|(x_i)_{i\in M},\lambda)P_{\bar{M}}((a_i)_{i\in \bar{M}}|(x_i)_{i\in \bar{M}},\lambda), \end{align}
where $M$ runs over all strict non-empty subsets of $\{1,\cdots, k\}$, $\bar{M}=\{1,\cdots, k\}\setminus M$, $(p_{M,\lambda})_\lambda$ is a probability distribution $\forall M$ and for each $M$, $P_M((a_i)_{i\in M}|(x_i)_{i\in M},\lambda)$ and $P_{\bar{M}}((a_i)_{i\in \bar{M}}|(x_i)_{i\in \bar{M}},\lambda)$ are conditional probability distributions on the parties in $M$ and in $\bar{M}$ respectively for all values of the local variable $\lambda$. If no restriction is added to these conditional probability distributions for the subsets of parties, we have Svetlichny's notion of bilocality. We will denote the set of such behaviours by $\mathcal{BL}^k_{\mathcal{G}}(N,K)$ and we will refer to them as general bilocal. If, on the other hand, the conditional probability distributions for the subsets are required to be non-signalling, we will refer to these behaviours as non-signalling bilocal and we will denote the corresponding set by $\mathcal{BL}^k_{\mathcal{NS}}(N,K)$. We recall that non-signalling conditional probability distributions are such that each party's marginal conditional probability distribution is independent of the other parties' inputs. That is, a bipartite conditional probability distribution $P(a,b|x,y))_{x,y}^{a,b}$ is said to be non-signalling if it verifies \begin{align}\label{nonsignalling}
\sum_{a=1}^KP(a,b|x,y)=\sum_{a=1}^KP(a,b|x'y)\text{ for all }a,b,y,x\neq x',\\
\nonumber \sum_{b=1}^KP(a,b|x,y)=\sum_{b=1}^KP(a,b|xy')\text{ for all }a,b,x,y\neq y'. \end{align}
These two conditions can be generalized in the obvious way to any number of parties and we will denote by $\mathcal{NS}^k(N, K)$ the set of $k$-partite non-signalling probability distributions with $N$ inputs and $K$ outputs per party (see \cite[Definition 1]{LW} for the explicit statement). Finally, a behaviour (\ref{behaviour}) is quantum if \begin{align}\label{Def quantum}
P(a_1,\cdots,a_k|x_1,\cdots , x_k)=\langle \psi|E_{a_1,x_1}^{1}\otimes \cdots \otimes E_{a_k,x_k}^{k}|\psi\rangle, \end{align}
where $(E_{a_i,x_i}^{i})_{x_i, a_i}$ is a family of measurements for the i$^{th}$-party (that is, for each party $i$ $E_{a_i,x_i}^{i}\geq0$ $\forall a_i,x_i$ and $\sum_{a_i}E_{a_i,x_i}^{i}=\mbox{$1 \hspace{-1.0mm} {\bf l}$}$ $\forall x_i$) and $|\psi\rangle$ is a $k$-partite pure quantum state. We will denote the set of behaviours of this form by $\mathcal{Q}^k(N,K)$.
There are several known relations among these sets. For instance, $\mathcal{L}^k(N,K)\subsetneq \mathcal{Q}^k(N,K)\subsetneq \mathcal{NS}^k(N,K)$. The first strict inclusion is the content of Bell's theorem while the second is due to Tsirelson \cite{Tsirelson} and Popescu and Rohrlich \cite{PR}. It readily follows from the definitions that $\mathcal{L}^k(N,K)\subset \mathcal{BL}^k_{\mathcal{NS}}(N,K)\subset \mathcal{BL}^k_{\mathcal{G}}(N,K)$. Svetlichny's result states that $\mathcal{Q}^k(N,K)\nsubseteq \mathcal{BL}^k_{\mathcal{G}}(N,K)$. Notice that it also holds that $\mathcal{BL}^k_{\mathcal{NS}}(N,K)\nsubseteq \mathcal{Q}^k(N,K)$.
Given any linear functional $M$ acting on the set of $k$-partite joint conditional probability distributions characterized by real numbers $\{M_{x_1 \ldots x_k}^{a_1 \ldots a_k}\}$ we will write
$$\langle M,P\rangle=\sum_{a_1, \ldots, a_k}\sum_{x_1, \ldots, x_k}M_{x_1 \ldots x_k}^{a_1 \ldots a_k}P(a_1,\cdots,a_k|x_1,\cdots , x_k).$$
Many information-theoretic problems boil down to optimizing a linear functional over particular sets of behaviours and, as explained in the introduction, a good way of understanding the relative power as resources of two such sets is to consider the relative ratio of violation optimized over all linear functionals. More precisely, if $\mathcal A_1$ and $\mathcal A_2$ are certain sets of behaviours like those defined above and $M$ is a linear functional on them, we define $\omega_{\mathcal{A}_i}(M)=\sup_{P\in\mathcal{A}_i}|\langle M,P\rangle|$, $i=1,2$ and also \begin{align}\label{Larg viol} LV(\mathcal A_1, \mathcal A_2)=\sup_M \frac{\omega_{\mathcal{A}_1}(M)}{\omega_{\mathcal{A}_2}(M)}. \end{align} This quantity will be our major object of study here in order to compare $\mathcal{Q}^k(N,K)$ with the sets of bilocal behaviours. As discussed in the introduction, Eq.\ (\ref{Larg viol}) gives a quantitative notion of the relative power as a resource of the behaviours in $\mathcal{A}_1$ compared to those in $\mathcal{A}_2$. This quantification is particularly clear when the Bell inequality $M$ is a two-prover one-round game, where Eq. (\ref{Larg viol}) is exactly the quotient of the winning probability of the game by using strategies defined by $\mathcal {A}_1$ over the the winning probability of the game by using strategies defined by $\mathcal A_2$.
We note that, in order for Eq.\ (\ref{Larg viol}) to make sense, we must require that the set $\mathcal A_1$ is contained in the affine hull of the set $\mathcal{A}_2$ (otherwise, we could find examples where $0=\omega_{\mathcal{A}_2}(M)<\omega_{\mathcal{A}_1}(M)$, so the quotient is infinity) and we must also define $0/0=0$ if we want to allow for general Bell functionals $M$ in the equation. We refer to \cite[Section 5]{JungeP11low} for a complete study of the geometric interpretation of Eq.\ (\ref{Larg viol}) in the bipartite case. On the other hand, this restriction is not needed anymore if we restrict to the case of correlations (see the following section) or to Bell functionals $M$ with positive coefficients (in particular, two-prover one-round games), since in both cases $\omega_{\mathcal A}(M)>0$ for every $M$, for all the sets $\mathcal A$ we will consider.
\section{Correlation scenario}\label{Sec: Bi-local correlations}
A particularly simple setting that has been thoroughly studied in the literature is the so-called correlation scenario. This arises when all outputs are binary $a_i\in \{-1,1\}$ $\forall i$ (i.e.\ $K=2$) and, instead of considering the full joint conditional probability distribution (\ref{behaviour}), only correlations -- that is, expectations over the product of the outputs -- are considered. Then, we define the correlation associated to a behaviour (\ref{behaviour}), $$\gamma=(\gamma_{x_1\ldots x_k})_{x_1,\ldots, x_k}\in\mathbb{R}^{N^k},$$ as \begin{align*}
\gamma_{x_1\ldots x_k}&=\mathbb{E}[a_1 \ldots a_k|x_1,\ldots, x_k]=\sum_{a_1, \ldots, a_k}a_1\cdots a_k P(a_1, \ldots, a_k|x_1,\ldots, x_k)\\&=P(a_1\cdots a_k{=}{1}|x_1,\ldots, x_k)-P(a_1\cdots a_k{=}{-}1|x_1,\ldots, x_k). \end{align*}
We will say that a certain correlation is local (resp. quantum, non-signalling bilocal, general bilocal and non-signalling) if there exists a local (resp. quantum, non-signalling bilocal, general bilocal and non-signalling) joint conditional probability distribution (\ref{behaviour}) such that $\gamma$ is the correlation associated to it. In this way, we will denote, in correspondence with the definitions of the previous section, $\mathcal{L}_{cor}^k(N)$ (resp. $\mathcal{Q}_{cor}^k(N)$, $\mathcal{BL}^k_{cor, \mathcal{NS}}(N)$, $\mathcal{BL}^k_{cor, \mathcal{G}}(N)$ and $\mathcal{NS}_{cor}^k(N)$) the set of local (resp. quantum, non-signalling bilocal, general bilocal and non-signalling) correlations with $N$ inputs per party.
It is well known that a given correlation $\gamma$ is in $\mathcal{L}_{cor}^k(N)$ if $$\gamma \in \text{conv}\{(a_{x_1}\cdots a_{x_k})_{x_1,\ldots, x_k}: a_{x_i}=\pm 1, x_i=1,\cdots, N, i=1,\cdots , k\},$$where $\text{conv}$ denotes the convex hull. On the other hand, $\gamma$ is in $\mathcal{Q}_{cor}^k(N)$ if \begin{align*}
\gamma_{x_1\ldots x_k}=\langle \psi|A^1_{x_1}\otimes \ldots \otimes A^k_{x_k}|\psi\rangle \hspace{0.2 cm}\text{ for every $x_1,\ldots, x_k$,}
\end{align*}where $A^i_{x_i}$ is a norm-one selfadjoint operator for every $i$ and for every $x_i=1,\ldots, N$ and $|\psi\rangle$ is a $k$-partite pure quantum state.
A characterization of the set $\mathcal{NS}_{cor}^k(N)$ can be done by the following lemma (see \cite[Proposition 1,2]{communicationcomplexityns} for the proof in the case $k=2$) \begin{lemma}\label{nscorrelation}
Given a correlation $\gamma$, it is in $\mathcal{NS}_{cor}^k(N)$ if and only if $|\gamma_{x_1\ldots x_k}|\leq 1$ for every $x_1,\ldots, x_k$. \end{lemma} \begin{proof}
Proving that $|\gamma_{x_1\ldots x_k}|$ is less than or equal to 1 if it is in $\mathcal{NS}_{cor}^k(N)$ follows from the definition. For the other implication, given a correlation $\gamma_{x_1,\ldots,x_k}$ satisfying $|\gamma_{x_1\ldots x_k}|\leq 1$ for every $x_1,\ldots, x_k$, consider the probability distribution defined as:
$$P(a_1,\ldots,a_k|x_1,\ldots,x_k)=\left\{ \begin{array}{ccc} \frac{1+\gamma_{x_1\ldots x_k}}{2^k}& \text{if} & a_1a_2\ldots a_k=1,\\ \frac{1-\gamma_{x_1\ldots x_k}}{2^k}& \text{if} & a_1a_2\ldots a_k=-1. \end{array} \right.$$
This probability distribution can be easily seen to be in $\mathcal{NS}^k(N,2)$ as a consequence that for all $i$ we have \begin{align*}
\sum_{a_i}&P(a_1,\ldots,a_k|x_1,\ldots,x_k)\\
&=P(a_1,\ldots,\underbrace{1}_i,\ldots,a_k|x_1,\ldots,x_k)+P(a_1,\ldots,\underbrace{-1}_i,\ldots,a_k|x_1,\ldots,x_k)\\ &=\frac{1\pm\gamma_{x_1,\ldots,x_k}}{2^k}+\frac{1\mp\gamma_{x_1,\ldots,x_k}}{2^k}=\frac{1}{2^{k-1}}. \end{align*} \end{proof} In particular, notice that Lemma \ref{nscorrelation} implies that correlations associated to nonsignalling probability distributions are the same as correlations associated to general probability distributions. So we have $$\mathcal{BL}^k_{cor, \mathcal{NS}}(N)=\mathcal{BL}^k_{cor, \mathcal{G}}(N).$$ Let us then just denote $\mathcal{BL}^k_{cor}(N)$ in this case.
In the following proposition we characterize correlations associated to bilocal probability distributions. Since we will only consider the case of tripartite distributions, we will restrict to this case. However, this result generalizes to an arbitrary number of parties trivially.
\begin{proposition}\label{Prop correlation bilocal} A correlation $(\gamma_{xyz})_{x,y,z}$ is in $\mathcal{BL}^3_{cor}(N)$ if and only if $$\gamma\in \text{conv}\big\{(\alpha_{xy}c_z)_{x,y,z}, \, (\beta_{yz}a_x)_{x,y,z}, \, (\gamma_{xz}b_y)_{x,y,z}\big\},$$ where $(\alpha_{xy})_{x,y}$, $(\beta_{yz})_{y,z}$, $(\gamma_{xz})_{x,z}$ are elements in $\mathcal{NS}_{cor}^2(N)$ and $a_x,b_y,c_z=\pm1$ for every $x,y,z$. \end{proposition}
\begin{proof} An extremal probability distribution $P$ of the set $\mathcal{BL}^3_{\mathcal{G}}(N,2)$ will have one of the following forms: \begin{align}\label{3 cases}
(Q(a,b|x,y)R(c|z))_{x,y,z}^{a,b,c},\, (Q(b,c|y,z)R(a|x)))_{x,y,z}^{a,b,c},\, (Q(a,c|x,z)R(b|y)))_{x,y,z}^{a,b,c},
\end{align}where in all cases $Q$ and $R$ are general probability distributions. Let us assume that this extremal element has the form $P=(Q(a,b|x,y)R(c|z))_{x,y,z}^{a,b,c}$ and denote $\beta$ and $\alpha$ the corresponding correlations from the probability distributions $(Q(a,b|x,y))_{x,y}^{a,b}$ and $(R(c|z))_{z}^c$, respectively. Then, given $x,y,z$, we have
\begin{align*}
\gamma_{xyz}&=\mathbb{E}[a\cdot b\cdot c|x,y,z]=\sum_{a,b,c}abc P(a,b,c|x,y,z)=\sum_{a,b,c}abc Q(a,b|x,y)R(c|z)\\
&=\Big(\sum_{a,b}ab Q(a,b|x,y)\Big)\Big(\sum_cc R(c|z)\Big)=\mathbb{E}[a\cdot b|x,y]\mathbb{E}[c|z]=\beta_{xy}\alpha_{z}. \end{align*}
By definition, $(\beta_{xy})_{xy}$ is in $\mathcal{NS}^2_{cor}$ whenever $Q$ is in $\mathcal{NS}^2$ and, clearly, $|\mathbb{E}[c|z]|\leq 1$. Since the other two cases in (\ref{3 cases}) are completely analogous, the result follows by convexity. \end{proof}
We are now in the position to address the main aim of this section: characterize the relative Bell violations for the quantum and bilocal sets in the correlation scenario using the figure of merit defined in Eq.\ (\ref{Larg viol}). Before presenting these results, let us first briefly recall several known results in this direction. The result of Tsirelson (\cite{Tsirelson}) mentioned in the introduction states that \begin{align}\label{Tsirelosn} LV\big(\mathcal{Q}_{cor}^2(N), \mathcal{L}_{cor}^2(N)\big)\leq K_G, \end{align}for every number of inputs $N$, where $K_G$ denotes the real Grothendieck's constant. On the other hand, it is not difficult to show (see for instance \cite[Ex. 29]{LCPW}) that \begin{align}\label{Eq. upper bound NS-CL} LV\big(\mathcal{NS}_{cor}^2(N), \mathcal{L}_{cor}^2(N)\big)\leq \sqrt{2N}, \end{align}and that this upper bound is essentially optimal. That is, there exits $M_0\in\mathbb{R}^{N^2}$ such that $\omega_{\mathcal{NS}_{cor}^2(N)}(M_0)\geq \sqrt{N/2}\omega_{\mathcal{L}_{cor}^2(N)}(M_0)$. As mentioned in the introduction, the relative violation between the quantum and the fully local set turns out to be unbounded as shown in \cite{PerezWPVJ08tripartite}. Later, the estimates proved therein were improved in \cite{briet11}, by showing that \begin{align*} LV\big(\mathcal{Q}_{cor}^3(N), \mathcal{L}_{cor}^3(N)\big)\geq CN^{\frac{1}{4}}, \end{align*} where $C$ is a universal constant. Moreover, it is known that this estimate is not far from being optimal, since the following inequality holds for every $N$ (see \cite{briet11}): \begin{align*} LV\big(\mathcal{Q}_{cor}^3(N), \mathcal{L}_{cor}^3(N)\big)\leq C\sqrt{N}, \end{align*}where $C$ is a universal constant. Note that, throughout this work, $C$ will always denote a general constant, not necessarily the same one each time that it appears.
It turns out that quantum behaviours cannot lead to an unbounded violation with respect to bilocal behaviours in the correlation scenario, i.e. it cannot hold that $$\lim_{N\rightarrow \infty}LV\big(\mathcal{Q}_{cor}^3(N), \mathcal{BL}^3_{cor}(N)\big)=\infty.$$ In the following proposition, we show that Tsirelson's result already prevents from having such a behaviour (see item (\ref{lv2}) in the following result). We also analyze the ratio between the set of bilocal correlations and the sets of fully local and quantum correlations.
\begin{proposition} Given $N$, the following inequalities hold: \begin{enumerate} \item \label{lv2} $LV\big(\mathcal{Q}_{cor}^3(N), \mathcal{BL}^3_{cor}(N)\big)\leq K_G$. \item \label{lv3} $LV\big(\mathcal{BL}^3_{cor}(N), \mathcal{L}_{cor}^3(N)\big)\leq \sqrt{2N}$. This implies $LV\big(\mathcal{BL}^3_{cor}(N), \mathcal{Q}_{cor}^3(N)\big)\leq \sqrt{2N}$. Moreover, the order $\sqrt{N}$ is optimal in these inequalities, since, in particular, $LV\big(\mathcal{BL}^3_{cor}(N), \mathcal{Q}_{cor}^3(N)\big)\geq \sqrt{N}/(4K_G)$. \end{enumerate} \end{proposition}
\begin{proof}
To prove (\ref{lv2}) consider a general Bell functional $M=(M_{xyz})_{x,y,z=1}^N$. Then, for a given tripartite quantum correlation of the form $$\gamma=\Big(\langle\psi|A_x\otimes B_y\otimes C_z|\psi\rangle\Big)_{x,y,z=1}^N,$$ we have \begin{align*}
\langle M, \gamma\rangle&=\Big|\sum_{x,y,z}M_{xyz}\langle\psi|A_x\otimes B_y\otimes C_z|\psi\rangle\Big|= \Big|\sum_{x,y,z}M_{xyz}\langle\psi|A_x\otimes D_{yz}|\psi\rangle \Big|\\
&\leq K_G \sup_{a_{x}=\pm1,b_{yz}=\pm1}\Big|\sum_{x,y,z}M_{xyz}a_x b_{yz}\Big|\leq K_G \sup_{\delta\in\mathcal{BL}^3_{cor}(N)}|\langle M,\delta\rangle|, \end{align*}where in the second equality we have denoted $D_{yz}=B_y\otimes C_z$, which is a self adjoint norm one operator, in the first inequality we have used Eq. (\ref{Tsirelosn}) and in the last inequality we have used Proposition \ref{Prop correlation bilocal}.
To prove (\ref{lv3}) consider a generic extremal strategy for the set $\mathcal{BL}^3_{cor}(N)$, which we will assume, without loss of generality, that is of the form $$\gamma=(\gamma_{xy}^{\mathcal{G}}\gamma_z)_{x,y,z=1}^N,$$ where $\gamma^{\mathcal{G}}=(\gamma_{xy}^{\mathcal{G}})_{xy}\in \mathcal{NS}_{cor}^2(N)$. Then, \begin{align*}
\langle M, \gamma \rangle&=\Big|\sum_{x,y,z}M_{xyz}\gamma_{xy}^{\mathcal{G}}\gamma_z\Big|=\sqrt{2N}\Big|\sum_{x,y,z}M_{xyz}\Big(\frac{\gamma_{xy}^{\mathcal{G}}}{\sqrt{2N}}\Big)\gamma_z\Big|\\
&=\sqrt{2N}\Big|\sum_{x,y,z}M_{xyz}\widetilde{\gamma}_{xy}^{\mathcal{C}}\gamma_z\Big|\leq \sqrt{2N} \sup_{ \mathcal{L}_{cor}^3(N)}|\langle M,\gamma\rangle|. \end{align*} Here, we have that $(\widetilde{\gamma}_{xy}^{\mathcal{C}})_{x,y}:=\Big(\frac{\gamma^{\mathcal{G}}_{xy}}{\sqrt{2N}}\Big)_{x,y}\in\mathcal{L}_{cor}^2$ and so $\Big(\widetilde{\gamma}_{xy}^{\mathcal{C}}\gamma_z\Big)_{x,y,z}\in \mathcal{L}_{cor}^3$. This follows from Eq. (\ref{Eq. upper bound NS-CL}). This proves the first inequality in (\ref{lv3}).
The second inequality in (\ref{lv3}) is straighfordward from the first one and the fact that $ \mathcal{L}_{cor}^3(N) \subset \mathcal{Q}_{cor}^3(N)$.
Finally, let us show the optimality of the order $\sqrt{N}$. To this end consider $n$ such that $N/2< 2^n\leq N$ and let $H_{2^n}=(h_{xy})_{x,1=1}^{2^n}$ be a Hadamard matrix, i.e. it has the property $H_{2^n}H_{2^n}^T=2^n\mathbbm{1}$\footnote{The restriction on the dimension to be a power of 2 guarantees that a Hadamard matrix exists (see e. g. \cite{hadamard}).}. Then, define the Bell functional $M=(M_{xyz})$ as $M_{xyz}=h_{xy}$ for $1\leq x,y\leq 2^n$, $z=1$ and $M_{xyz}=0$ otherwise. We will study the values $\omega_{\mathcal{BL}^3_{cor}(N)}(M)$ and $\omega_{\mathcal{Q}_{cor}^3(N)}(M)$.
The element $\gamma=(\gamma_{xyz})_{xyz}$ defined by $\gamma_{xyz}=M_{xyz}$ for all $x,y,z$ is clearly in $\mathcal{BL}^3_{cor}(N)$ (since $|\gamma_{xyz}|\leq 1$ for every $x$, $y$, $z$). Then,
$$\omega_{\mathcal{BL}^3_{cor}(N)}(M)\geq \Big|\sum_{x,y,z}M_{xyz}\gamma_{xyz}\Big|=\Big|\sum_{x,y=1}^{2^n}h_{xy}^2\Big|=2^{2n}>\frac{N^2}{4}.$$
On the other hand, for every quantum correlation $\gamma=\Big(\langle\psi|A_x\otimes B_y\otimes C_z|\psi\rangle\Big)_{x,y,z=1}^N,$ we have
$$\Big|\sum_{x,y,z}M_{xyz}\gamma_{xyz}\Big|=\Big|\sum_{x,y}h_{xy}\langle\psi|A_x\otimes B_y\otimes C_1|\psi\rangle \Big|=\Big|\sum_{x,y}h_{xy}\langle u_x| v_y\rangle\Big|,$$where we have defined $|u_x\rangle=A_x\otimes\mathbbm{1}\otimes C_{1}|\psi\rangle$ and $|u_y\rangle=\mathbbm{1}\otimes B_y\otimes\mathbbm{1}|\psi\rangle$. We can now apply Eq. (\ref{Tsirelosn}) to upper bound the previous expression by
$$K_G\sup_{a_x,b_y=\pm1}\Big|\sum_{x,y=1}^{2^n}h_{xy}a_xb_y\Big|\leq K_G(2^{n})^{3/2}\leq K_G N^{3/2},$$ where the last inequality is proved in \cite[Ex. 29]{LCPW}.
Since the previous estimate holds for all quantum correlations, the upper bound $\omega_{\mathcal{Q}_{cor}^3(N)}(M)\leq K_G N^{3/2}$ follows. Hence, we deduce $$LV\big(\mathcal{BL}^3_{cor}(N), \mathcal{Q}_{cor}^3(N)\big)\geq \frac{\sqrt{N}}{4K_G}.$$ \end{proof}
\section{General probability distributions}
The previous section motivates the question of whether we can obtain unbounded violations of tripartite quantum probability distributions over bilocal probability distributions. As we have seen, this is impossible in the setting of correlations and we want to investigate here if $$\lim_{\substack{N\rightarrow \infty\\K\rightarrow \infty}}LV\big(\mathcal{Q}^3(N,K), \mathcal{BL}^3_{\mathcal{A}}(N,K)\big)=\infty,\text{ for $\mathcal{A}=\mathcal{NS}$ or $\mathcal{G}$,}$$ holds. Note that, according to the comments after Eq. (\ref{Larg viol}), the previous question is well posed since it is well known that the set of $k$-partite quantum probability distributions is contained in the affine hull of the set of $k$-partite fully local probability distributions. We will show that this is in fact true in the strongest case, $\mathcal{A}=\mathcal{G}$, using $k$-prover one-round games, which are particular Bell functionals. Indeed, one such a game is a Bell functional whose coefficients are of the form \begin{align}\label{2P1R-games} G_{x_1\ldots x_k}^{a_1\ldots a_k}=\pi(x_1,\ldots , x_k)V^{a_1\ldots a_k}_{x_1\ldots x_k}, \end{align} where $(\pi(x_1\ldots , x_k))_{x_1,\ldots , x_k}$ is a probability distribution and $V$ is a predicate function taking values one or zero. Note that, in particular, $G$ has non-negative coefficients. Games describe a setting where each of $k$ players is asked a certain question $x_i$ according to the probability distribution $\pi$ and must answer a certain output $a_i$, being the condition of winning the game that the questions and answers verify $V^{a_1\ldots a_k}_{x_1\ldots x_k}=1$. In this context, the quantity $\omega_{\mathcal{A}}(G)=\sup_{P\in\mathcal{A}}\langle G,P\rangle$ represents the winning probability of the game if the players are restricted to the use of strategies defined by the set $\mathcal{A}$.
Nevertheless, for the purpose of this work, we will consider a slightly more general definition for games and we will treat them as functionals $G$ with non negative coefficients such that \begin{align}\label{normalization condition} \sum_{x_1,\ldots , x_k}\max_{a_1,\ldots, a_k}G_{x_1\ldots x_k}^{a_1\ldots a_k}\leq 1 \hspace{0.4 cm} \text{(normalization condition).} \end{align}
Let us first recall that, given a bipartite game $G=(G_{xy}^{ab})_{x,y,a,b}$ with $N$ inputs and $K$ outputs per party, we denote by $G^{\otimes_2}$ the bipartite game with $N^2$ inputs and $K^2$ outputs per party, whose coefficients are: $$G_{x_1x_2y_1 y_2}^{a_1 a_2b_1 b_2}=G_{x_1y_1}^{a_1 b_1}G_{x_2y_2}^{a_2 b_2}.$$ That is, Alice's and Bob's inputs are $(x_1, x_2)$ and $(y_1, y_2)$, respectively, and Alice's and Bob's outputs are $(a_1, a_2)$ and $(b_1, b_2)$, respectively. This means that the parties are playing two instances of the game simultaneously. Studying the classical value $G^{\otimes_2}$ for a given game $G$ is the core of the results about parallel repetition theorems, which are of great relevance in computer science.
To make the following result more intuitive, let us explain that our aim is to define a tripartite game $\widetilde{G}$ from a bipartite one $G$. Although it will not be presented here, the first (and somehow easiest) construction we considered was based on two instances of the bipartite game, one for Alice and Bob and the other for Bob and Charlie. In this situation, Alice receives input $x$ and she outputs $a$, Bob receives $(y_1,y_2)$ and outputs $(b_1,b_2)$ (the first is for the game he is playing with Alice and the second, with Charlie) and, finally, Charlie receives input $z$ and outputs $c$. Then the coefficients have the form: $$\widetilde{G}_{xy_1y_2z}^{ab_1b_2c}=G_{xy_1}^{ab_1} G_{y_2z}^{b_2c}.$$
In order to find an example such that it does not only give unbounded violations between $\mathcal{Q}^3$ and $\mathcal{BL}_{\mathcal{G}}^3$, but it is also optimal in some parameters, we will present here another construction using three instead of two instances of the game. More precisely, one instance of the game will be asked to Alice and Bob, another to Bob and Charlie and another to Charlie and Alice. Hence, the coefficients of the new game will have the following form:
$$\hat{G}_{x_1x_2y_1y_2z_1z_2}^{a_1a_2b_1b_2c_1c_2}=G_{x_1y_1}^{a_1b_1}G_{x_2z_1}^{a_2c_1}G_{y_2z_2}^{b_2c_2}.$$
\begin{theorem}\label{thm2} Let $G$ be a bipartite game with $N$ inputs and $K$ outputs per party. Then, the construction of the paragraph above leads to a tripartite game $\hat{G}$ with $N^2$ inputs and $K^2$ outputs per player, such that \begin{align}\label{Estimate in them2} \frac{\omega_{\mathcal{Q}^3}(\hat{G})}{\omega_{\mathcal{BL}_{\mathcal G}^3}(\hat{G})}\geq \frac{\omega_{\mathcal{Q}^2}(G)^3}{\omega_{\mathcal{L}^2}(G^{\otimes_2})}. \end{align}
Moreover, if $\omega_{\mathcal{Q}^2}(G)$ is attained with local dimension $d$, then Eq. (\ref{Estimate in them2}) is attained with local dimension $d^2$. \end{theorem}
\begin{proof}
To show that $\omega_{\mathcal{Q}^3}(\hat{G})\geq \omega_{\mathcal{Q}^2}(G)^3$, first note that there must exist a quantum strategy which uses a quantum state $|\phi\rangle$ in some Hilbert space $\mathcal{H}_1\otimes\mathcal{H}_2$ and POVMs $\{\Pi_x^a\}_{a=1}^n$ and $\{\Lambda_y^b\}_{b=1}^n$ acting on the Hilbert spaces $\mathcal{H}_1$ and $\mathcal{H}_2$ in such a way that\footnote{Although the value $\omega_{\mathcal{Q}^2}(G)$ could be not attained, we can find a quantum strategy up to arbitrarily high precision. We avoid writing inequalities up to $\epsilon$.}:
$$\Big|\sum_{x,y,a,b}G_{xy}^{ab}\langle\phi| \Pi_x^a\otimes \Lambda_y^b|\phi\rangle\Big|=\omega_{\mathcal{Q}^2}(G).$$
Then we can consider the tripartite quantum state $|\psi\rangle=|\phi\rangle|\phi\rangle|\phi\rangle\in\big(\mathcal{H}_1\otimes\mathcal{H}_2\big)\otimes\big(\mathcal{H}_1\otimes\mathcal{H}_2\big)\otimes\big(\mathcal{H}_1\otimes\mathcal{H}_2\big)$ and we can define the operators $E_{x_1,x_2}^{a_1,a_2}=\Pi_{x_1}^{a_1}\otimes \Pi_{x_2}^{a_2}$, $F_{y_1,y_2}^{b_1,b_2}=\Lambda_{y_1}^{b_1}\otimes\Pi_{y_2}^{b_2}$ and $G_{z_1,z_2}^{c_1,c_2}=\Lambda_{z_1}^{c_1}\otimes\Lambda_{z_2}^{c_2}$. It is clear that $\{E_{x_1,x_2}^{a_1,a_2}\}_{a_1,a_2}$, $\{F_{y_1,y_2}^{b_1,b_2}\}_{b_1,b_2}$ and $\{G_{z_1,z_2}^{c_1,c_2}\}_{c_1,c_2}$ are POVMs for all $x_1$, $x_2$, $y_1$, $y_2$, $z_1$, $z_2$. Moreover, \begin{align*}
\omega_{\mathcal{Q}^3}(\hat{G})&\geq \sum\hat{G}_{x_1x_2y_1y_2z_1z_2}^{a_1a_2b_1b_2c_1c_2}\langle\psi|E_{x_1,x_2}^{a_1,a_2}\otimes F_{y_1,y_2}^{b_1,b_2}\otimes G_{z_1,z_2}^{c_1,c_2}|\psi\rangle \\
&=\Big(\sum_{x_1,y_1,a_1,b_1}G_{x_1y_1}^{a_1b_1}\langle\phi| \Pi_{x_1}^{a_1}\otimes \Lambda_{y_1}^{b_1}|\phi\rangle\Big)\times\Big(\sum_{x_2,z_1,a_2,c_1}G_{x_2z_1}^{a_2c_1}\langle\phi| \Pi_{x_2}^{a_2}\otimes \Lambda_{z_1}^{c_1}|\phi\rangle\Big)\\
&\times\Big(\sum_{y_2,z_2,b_2,c_2}G_{y_2z_2}^{b_2c_2}\langle\phi| \Pi_{y_2}^{b_2}\otimes \Lambda_{z_2}^{c_2}|\phi\rangle\Big)=\omega_{\mathcal{Q}^2}(G)^3, \end{align*}where the first sum runs over all indices.
Note also that, if we assume $\dim \mathcal{H}_1=\dim \mathcal{H}_2=d$, then, by construction, the local dimension of the quantum state $|\psi\rangle$ is $d^2$.
In order to prove the corresponding upper bound for the classical value, let us consider a bilocal probability distribution $P$ of the form $$\big(P_1(a_1,a_2,b_1,b_2|x_1,x_2,y_1,y_2)P_2(c_1,c_2|z_1,z_2)\big)_{x_1,x_2,y_1,y_2, z_1,z_2}^{a_1,a_2,b_1,b_2, c_1,c_2}$$ and the other two cases will follow by symmetry.
First of all, notice that, given a certain positive pointwise element $(f(a_2,b_2,x_2,y_2))_{a_2,b_2,x_2,y_2}$ such that $\sum_{a_2,b_2}f(a_2,b_2,x_2,y_2)\leq1$ for all $x_2$ and $y_2$, we can find a probability distribution $\widetilde{P}$ for which all its components are greater than or equal to those of $f$ by defining:
\[ \widetilde{P}(a_2,b_2|x_2,y_2)=\begin{cases}
f(a_2,b_2,x_2,y_2) & \text{if } 1\leq a_2,b_2\leq K,\, (a_2,b_2)\neq (K,K),\\
1-\sum_{(a'_2,b'_2)\neq (K,K)}f(a_2',b_2',x_2,y_2) & \text{if } a_2=b_2=K.
\end{cases} \]
Then, using the upper bound for the classical value of $G^{\otimes_2}$, we can write \begin{align}\label{aux I}
&\sum_{x_2,z_1,a_2,c_1,y_2,z_2,b_2,c_2}G_{x_2z_1}^{a_2c_1}G_{y_2z_2}^{b_2c_2} f(a_2,b_2,x_2,y_2)P(c_1,c_2|z_1,z_2)\\&\nonumber\leq \sum_{x_2,z_1,a_2,c_1,y_2,z_2,b_2,c_2}G_{x_2z_1}^{a_2c_1}G_{y_2z_2}^{b_2c_2} \widetilde{P}(a_2,b_2|x_2,y_2)P(c_1,c_2|z_1,z_2)\leq \omega_{\mathcal{L}^2}(G^{\otimes_2}). \end{align}
Hence, we have \begin{align}\label{classical three games}
\langle{\hat{G}}, P\rangle&=\sum G_{x_1y_1}^{a_1b_1}G_{x_2z_1}^{a_2c_1}G_{y_2z_2}^{b_2c_2}P(a_1,a_2,b_1,b_2|x_1,x_2,y_1,y_2)P(c_1,c_2|z_1,z_2)\\\nonumber
&=\sum_{x_2,z_1,a_2,c_1,y_2,z_2,b_2,c_2} G_{x_2z_1}^{a_2c_1}G_{y_2z_2}^{b_2c_2}\Big(\sum_{x_1,y_1,a_1,b_1} G_{x_1y_1}^{a_1b_1}P(a_1,a_2,b_1,b_2|x_1,x_2,y_1,y_2)\Big)P(c_1,c_2|z_1,z_2)\\ \nonumber
&=\sum_{x_2,z_1,a_2,c_1,y_2,z_2,b_2,c_2} G_{x_2z_1}^{a_2c_1}G_{y_2z_2}^{b_2c_2} f(a_2,b_2,x_2,y_2)P(c_1,c_2|z_1,z_2)\leq\omega_{\mathcal{L}^2}(G^{\otimes_2}),
\end{align}where the first sum runs over all indices, we have defined $$f(a_2,b_2,x_2,y_2)=\sum_{x_1,y_1,a_1,b_1} G_{x_1y_1}^{a_1b_1}P(a_1,a_2,b_1,b_2|x_1,x_2,y_1,y_2)$$and the last inequality in Eq. (\ref{classical three games}) follows from Eq. (\ref{aux I}) and the fact that $\sum_{a_2,b_2}f(a_2,b_2,x_2,y_2)\leq1$ for all $x_2$ and $y_2$. To show this last claim, fix $x_2$ and $y_2$, and write \begin{align*}
&\sum_{a_2,b_2}\sum_{x_1,y_1,a_1,b_1}G_{x_1y_1}^{a_1b_1}P(a_1,a_2,b_1,b_2|x_1,x_2,y_1,y_2)\\
&=\sum_{x_1,y_1,a_1,b_1}G_{x_1y_1}^{a_1b_1}\sum_{a_2, b_2}P(a_1,a_2,b_1,b_2|x_1,x_2,y_1,y_2)\\&
\leq \sum_{x_1, y_1}\max_{a_1,b_1}G_{x_1y_1}^{a_1b_1}\sum_{a_1,a_2, b_1, b_2,}P(a_1,a_2,b_1,b_2|x_1,x_2,y_1,y_2)\\&=\sum_{x_1, y_1}\max_{a_1,b_1}G_{x_1y_1}^{a_1b_1}\leq 1, \end{align*} because of Eq. (\ref{normalization condition}). \end{proof}
There are two interesting applications of the previous theorem. The first one comes from the application to pseudotelepathy games. That is, those bipartite games which can be won perfectly with quantum strategies but not with classical ones (as it is, for instance, the magic square game \cite{CHTW04}). As a consequence, our construction leads to the existence of pseudotelepathy against bilocality.
\begin{corollary} Let $G$ be a pseudotelepathy game. Applying the construction of Theorem \ref{thm2} we obtain a tripartite game $\hat{G}$ such that $\omega_{\mathcal{Q}^3}(\hat{G})=1$ and $\omega_{\mathcal{BL}^3_{\mathcal G}}(\hat{G})<1$. \end{corollary}
The second, and more important, application is to obtain an unbounded violation between tripartite quantum and bilocal conditional probability distributions. For that purpose we will use the \emph{Khot-Vishnoi game}, $G_{KV}$ or KV game \cite{KhVi}, which we briefly explain here. For any $n=2^l$ with $l\in\N$ and every $\eta\in [0,\frac{1}{2}]$ we consider the group $\{0,1\}^n$ and the Hadamard subgroup $H$. Then, we consider the quotient group $G=\{0,1\}^n/H$ which is formed by $\frac{2^n}{n}$ cosets $[x]$ each with $n$ elements. The questions of the games $(x,y)$ are associated to the cosets whereas the answers $a$ and $b$ are indexed in $[n]$. The referee chooses a uniformly random coset $[x]$ and one element $z\in \{0,1\}^n$ according to the probability distribution $pr(z(i)=1)=\eta$, $pr((z(i)=0)=1-\eta$ independently of $i$. Then, the referee asks question $[x]$ to Alice and question $[x\oplus z]$ to Bob. Alice and Bob must answer one element of their corresponding cosets and they win the game if and only if $a\oplus b=z$. Although the KV game is not a two-prover one-round game in the sense of Eq. (\ref{2P1R-games}), it is very easy to see that it verifies the normalization condition given in Eq. (\ref{normalization condition}).
Hence, the Khot-Vishnoi game has $N=2^n/n$ inputs and $K=n$ outputs per player and it can be proved (\cite[Theorem 7]{BuhrmanRSW12}) that \begin{align}\label{estimate KV} \omega_{\mathcal{L}^2}(G_{KV})\leq C/n \hspace{0.2 cm}\text{ and } \hspace{0.2 cm} \omega_{\mathcal{Q}^2}(G_{KV})\geq D/\ln^2 n, \end{align} for some universal constants $C$ and $D$.
The next lemma is necessary in order to apply the Khot-Vishnoi to Theorem \ref{thm2} and it essentially shows that the classical value of the game is multiplicative.
\begin{lemma}\label{tensor product KV} Let $G_{KV}$ be the Khot-Vishnoi game. Then, $$\omega_{\mathcal L^2}(G_{KV}^{\otimes_2})\leq C\frac{1}{n^2},$$where $C$ is a universal constant. \end{lemma}
\begin{proof} The proof of this result follows exactly the same steps as in the proof of \cite[Theorem 4.1]{BuhrmanRSW12}. As it is explained there, a deterministic strategy (which corresponds to an extremal classical probability distribution) can be identified with a couple of boolean functions $A, B:\{0,1\}^n\rightarrow \{0,1\}$ such that each of them verifies that, restricted to each coset $[x]$ (see explanation of the game right before this lemma), takes value one for one of the elements and zero for the rest. Then, the winning probability of the game can be written as $$n\mathbb E_u\mathbb E_z [A(u)B(u\oplus z)],$$where $u$ is sampled uniformly at random in $\{0,1\}^n$ and $z\in \{0,1\}^n$ is sampled pointwise independently according to the probability distribution $pr(z(i)=1)=\eta$, $pr((z(i)=0)=1-\eta$. We fix here $\eta=1/2-1/ \log n$. Then, Cauchy-Schwarz inequality followed by a use of the hypercontractive inequality lead to the classical upper bound stated in Eq. (\ref{estimate KV}).
In the case of $G_{KV}^{\otimes_2}$, a deterministic strategy can be identified with a couple of boolean functions $A, B:\{0,1\}^{2n}=\{0,1\}^n\times \{0,1\}^n\rightarrow \{0,1\}$ such that each of them verifies that, restricted to each pair $[x]\times [y]$, takes value one for one of the elements and zero for the rest. Then, the key point is that sampling $u=(u_1,u_2)$ so that $u_i$ is sampled uniformly in $\{0,1\}^n$ for $i=1,2$ is the same as sampling $u$ uniformly in $\{0,1\}^{2n}$. At the same time, since $z$ is sampled pointwise independently, we can sample in the form $z=(z_1, z_2)$ where $z_i\in\{0,1\}^n$ is sampled as in the single game for $i=1,2$. Then, the winning probability of the game can be written as $$n^2\mathbb E_{u_1,u_2}\mathbb E_{z_1,z_2} [A(u_1, u_2)B((u_1,u_2)\oplus (z_1, z_2))]=n^2\mathbb E_{u}\mathbb E_{z} [A(u)B(u\oplus z)].$$
Then, doing the same computations as in the proof of \cite[Theorem 4.1]{BuhrmanRSW12} we obtain the bound $n^2\big(\frac{1}{n^2}\big)^{1/(1-\eta)}\leq C/n^2$. This concludes the proof. \end{proof}
\begin{corollary}\label{thm optimal} The KV game leads to a tripartite game $\hat{G}$ with $(2^n/n)^2$ inputs and $n^2$ outputs per player, such that \begin{align}\label{Eq. estimate optimal} \frac{\omega_{\mathcal{Q}^3}(\hat{G})}{\omega_{\mathcal{BL}^3}(\hat{G})}\geq C\frac{n^2}{\ln^6 n}, \end{align}and the quantum lower bound in the previous equation is attained with a quantum state of local dimension $n^2$.
Moreover, this estimate is essentially optimal in the number of outputs and in the local dimension of the Hilbert space. \end{corollary}
First, this shows that tripartite quantum probability distributions can lead to unbounded violations with respect to bilocal ones. As we have seen in the previous section, this is in hard contrast with the case of correlations. But, second, this also proves optimality in the following sense. Once we have an example in which there exists unbounded violations, a natural question is how far our example is from the best possible construction. That is, does Corollary \ref{thm optimal} provide the best possible violation as a function of the number of inputs and outputs? In fact, when comparing quantum distributions with local (or bilocal) distributions, the amount of violations must be seen as a function of three parameters: number of inputs $N$, number of outputs $K$ and the local dimension of the Hilbert space $d$ used in the quantum distribution. As an example of this, in the bipartite scenario, it can be seen (\cite{JungeP11low, PV-Survey}) that \begin{align}\label{upper bounds for Bell inequalities} LV(\mathcal{Q}^2, \mathcal{L}^2)\leq C\min \{N, K,d\}, \end{align} where $C$ is a universal constant. Interestingly, the KV game provides an example which is essentially optimal (up to logarithmic factors) in the number of outputs and in the dimension $d$, since the estimate in Eq. (\ref{estimate KV}) for $\omega_{\mathcal{Q}^2}(G_{KV})$ is attained by using the maximally entangled state in dimension $n$. It is not known if the upper bound $O(N)$ can be attained, being $\sqrt{N}$ the best lower bound as a function of the number of inputs (see \cite{JungeP11low} for the corresponding game).
Corollary \ref{thm optimal} shows optimality in terms of the number of outputs and dimension of the Hilbert space. It follows from the following two lemmas:
\begin{lemma}\label{lemma d} Given a tripartite game $G$. If we denote by $\omega_{\mathcal{Q}_d^3}(G)$ the quantum value of $G$ when at least one of the player is restricted to local dimension $d$, then $$\omega_{\mathcal{Q}_d^3}(G)\leq d\, \omega_{\mathcal{BL}_{\mathcal {G}}^3}(G).$$ \end{lemma}
\begin{lemma}\label{lemma k} Given a tripartite game $G$ with $K$ outputs per player. Then, $$\omega_{\mathcal{Q}^3}(G)\leq K\omega_{\mathcal{BL}_{\mathcal G}^3}(G).$$ \end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma d}]
The proof can be obtained from a slight modification of the comments below \cite[Proposition 5.2]{PV-Survey}. Indeed, let us fix a quantum distribution $P$ which is defined by a quantum state $|\psi\rangle\in \mathbb{C}^d\otimes \mathbb{C}^n\otimes \mathbb{C}^m$ and some POVMs $\{\Pi_x^a\}_a$, $\{\Lambda_y^b\}_b$ and $\{\Upsilon_z^c\}_c$ acting on $\mathbb{C}^d$, $\mathbb{C}^n$ and $\mathbb{C}^m$ respectively, for every $x,y,z$. Then, from the Schmidt decomposition, we deduce that the state $|\psi\rangle$ can be written as $$|\psi\rangle=\sum_{i=1}^d\lambda_i |f_i\rangle |g_i\rangle,$$where $\sum_i|\lambda_i|^2=1$, and $|f_i\rangle$ and $|g_i\rangle$ are orthonormal systems in the unit ball of $\mathbb{C}^d$ and $\mathbb{C}^{nm}$ respectively. Then, we have \begin{align*}
|\langle G,P\rangle|&=\Big|\sum_{x,y,z,a,b,c}G_{xyz}^{abc}\langle \psi|\Pi_x^a\otimes \Lambda_y^b\otimes \Upsilon_z^c|\psi\rangle\Big|\\&\leq \sum_{i,j}|\lambda_i||\lambda_{j}|\sum_{x,y,z,a,b,c}G_{xyz}^{abc}|\langle f_i|\Pi_x^a|f_{j}\rangle||\langle g_i |\Lambda_y^b\otimes \Upsilon_z^c|g_{j}\rangle|\\&\leq d \max_{i,j}\sum_{x,y,z,a,b,c}G_{xyz}^{abc}|\langle f_i|\Pi_x^a|f_{j}\rangle||\langle g_i |\Lambda_y^b\otimes \Upsilon_z^c|g_{j}\rangle|,
\end{align*}where we have used the well known fact $\sum_{i=1}^d|\lambda_i|\leq \sqrt{d}\Big(\sum_{i=1}^d|\lambda_i|^2\Big)^{\frac{1}{2}}$.
Now, as it is shown in the comments below \cite[Proposition 5.2]{PV-Survey}, Cauchy-Schwarz inequality implies that for every $i$ and $j$, and for every $x$, $y$ and $z$, we have \begin{align*}
\sum_a|\langle f_i|\Pi_x^a|f_{j}\rangle|\leq 1 \hspace{0.2 cm} \text{ and } \hspace{0.2 cm} \sum_{b,c}|\langle g_i |\Lambda_y^b\otimes \Upsilon_z^c|g_{j}\rangle|\leq 1. \end{align*}Hence, we deduce that $\langle G,P\rangle\leq d\, \omega_{\mathcal{BL}_{\mathcal {G}}^3}(G)$, which concludes the proof. \end{proof}
\begin{proof}[Proof of Lemma \ref{lemma k}]
Notice that given a tripartite game $G=(G_{xyz}^{abc})_{x,y,z,a,b,c}$, we can define three different bipartite games $G_1$, $G_2$ and $G_3$ in which one party receives two inputs and gives two outputs. Put differently, we join Alice and Bob for $G_1$, Bob and Charlie for $G_2$ and Charlie and Alice for $G_3$. In case that the maximum value for $\omega_{\mathcal{BL}_{\mathcal {G}}^3}(G)$ is attained for a bilocal probability distribution of the form $(P(a,b|x,y)P(c|z))_{xyz}^{abc}$, then this probability distribution can be seen as a local bipartite probability distribution in the scenario where Alice and Bob are joint, and it will give the maximum value for the bipartite game $G_1$. As the other cases are similar we can say that: \begin{align}\label{sub functionals} \omega_{\mathcal{BL}_{\mathcal {G}}^3}(G)=\max\{\omega_{\mathcal{L}^2}(G_1),\omega_{\mathcal{L}^2}(G_2),\omega_{\mathcal{L}^2}(G_3)\}. \end{align}
Then, the upper bound we want to show follows from the known estimate for bipartite games (see comments below \cite[Proposition 4.5]{PV-Survey}) $$\omega_{\mathcal{Q}^2}(G)\leq\min\{K_1,K_2\}\omega_{\mathcal{L}^2}(G).$$ Indeed, according to our hypothesis, the functionals $G_1$, $G_2$ and $G_3$ from $G$ have $K$ outputs for one player and $K^2$ outputs for the other player. Then, for every $i=1,2,3$, we have $$\omega_{\mathcal{Q}^3}(G)\leq \omega_{\mathcal{Q}^2}(G_i)\leq\min\{K,K^2\}\omega_{\mathcal{L}^2}(G_i)=K\omega_{\mathcal{L}^2}(G_i),$$ which, according to Eq. (\ref{sub functionals}) gives the desired upper bound. \end{proof}
\section{Conclusions}
In this work we have extended the study of relative Bell violations of quantum resources over local and fully local ones to the genuinely multipartite scenario by comparing the power of quantum strategies over bilocal models. We have considered first the correlation scenario, where we have found that, as in the bipartite case, the ratio of Bell violation of quantum behaviours over bilocal ones is upper bounded by Grothendieck's constant for any number of inputs and, hence, there cannot be unbounded Bell violations. Since not all bilocal correlations are reproducible by quantum models, we have also investigated the relative power of the former over the latter. We have shown that this ratio is upper bounded by $O(\sqrt{N})$ and that this order is optimal. Next, we have considered the case of general conditional probability distributions. Contrary to the previous case, we have obtained here that quantum strategies lead to unbounded Bell violations over general bilocal behaviours. In order to do so, we have proved that if one considers tensor products of bipartite games the ratio of the quantum value over the general bilocal value is related to the ratio of the quantum and local values for the bipartite game. This has allowed us, by considering explicit games such as the Khot-Vishnoi game, to establish that there exist games for which the ratio of the quantum and general bilocal values grows unboundedly with the number of inputs and outputs. We moreover have proven that for a particular choice of games the given estimate of the asymptotic behaviour of this ratio is essentially optimal in the number of outputs and in the dimension of the Hilbert space.
It might be worth mentioning that the above games require a large number of inputs, i.e.\ exponential in the obtained violation. Random constructions of Bell functionals (see \cite{Pala2015, PV-Survey} for some surveys on this topic) could be used to show that there exist Bell inequalities in which the number of inputs and outputs grow polynomially with the amount of violation. This, however, is a non-constructive procedure and would come at the expense of not identifying an explicit Bell functional for this task.
It should be noticed that the two results about the ratio of Bell violations of quantum behaviours over bilocal ones -- boundedness in the correlation setting and unboundedness in the general case -- hold irrespectively of whether we consider general bilocal or non-signalling bilocal models. In the first case, this holds because both sets of models happen to coincide in the correlation scenario, as discussed in Sec.\ \ref{Sec: Bi-local correlations}. In the second case, unboudedness with respect to general bilocal behaviours automatically implies the same with respect to non-signalling bilocal behaviours due to the fact that this latter set is included in the former. Thus, our result can also be understood as showing an unlimited advantage of GMNL quantum behaviours irrespective of the underlying definition of bilocality. As mentioned in the introduction, the correlations contained in general bilocal models might be so strong that lead to undesirable unphysical effects in certain scenarios and this has motivated to consider more constrained hybrid models. Despite this fact, not only general bilocal models are unable to simulate all quantum behaviours as proven by Svetlichny in \cite{sve}, but our results show that quantum-mechanical resources can be, in a certain sense, unboundedly better than this strongest form of bilocality.
\section*{acknowledgment} This research was funded by the Spanish MINECO through Grant No. MTM2017-88385-P, MTM2017-84098-P, MTM2014-54240-P and by the Comunidad de Madrid through grant QUITEMAD-CM P2018/TCS4342. We also acknowledge funding from SEV-2015-0554-16-3.
\end{document} | arXiv |
Dynamics of a diffusive Leslie-Gower predator-prey model in spatially heterogeneous environment
Analyzing plasmid segregation: Existence and stability of the eigensolution in a non-compact case
November 2020, 25(11): 4165-4188. doi: 10.3934/dcdsb.2020092
Asymptotic behavior of solutions of a nonlinear degenerate chemotaxis model
Georges Chamoun 1, , Moustafa Ibrahim 2,, , Mazen Saad 3, and Raafat Talhouk 4,
Lebanese University, Faculty of Science Ⅳ. Laboratory of mathematics-EDST, Hadath, Lebanon
College of Engineering and Technology, American University of the Middle East, Kuwait
École Centrale de Nantes. UMR 6629 CNRS, laboratoire de mathématiques Jean Leray, F-44321, Nantes, France
Lebanese University, Faculty of Science Ⅰ. Laboratory of mathematics-EDST, Hadath, Lebanon
* Corresponding author: Moustafa Ibrahim
Received June 2019 Revised November 2019 Published April 2020
Figure(12)
Pattern formation in various biological systems has been attributed to Turing instabilities in systems of reaction-diffusion equations. In this paper, a rigorous mathematical description for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis is presented. We identify a generalized nonlinear degenerate chemotaxis model where a destabilization mechanism may lead to spatially non homogeneous solutions. Given any general perturbation of the solution nearby an homogenous steady state, we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along the finite number of fastest growing modes. The theoretical results are tested against two different numerical results in two dimensions showing an excellent qualitative agreement.
Keywords: Degenerate parabolic equation, Destabilization mechanism, Pattern formation, Growing modes, Nonlinear instability.
Mathematics Subject Classification: Primary: 35B40, 35K57, 35K55, 35K65, 65M08; Secondary: 70K50, 92C15.
Citation: Georges Chamoun, Moustafa Ibrahim, Mazen Saad, Raafat Talhouk. Asymptotic behavior of solutions of a nonlinear degenerate chemotaxis model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4165-4188. doi: 10.3934/dcdsb.2020092
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Figure 1. Unstructured triangular mesh for the space domain $ {\Omega} = {\left({0,1}\right)}\times{\left({0,1}\right)} $ with 14336 acute angle triangles
Figure 2. Plot of $ h({\left\|{q}\right\|}^{2}) $ as a function of $ {\left\|{q}\right\|}^{2} $defined by equation (13). When the chemosensitivity strength $ \zeta $ increases beyond the critical value $ \zeta_{c} $, $ h({\left\|{q}\right\|}^{2}) $ becomes negative for a finite range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $ marked with rhombi
Figure 3. To the top: Distribution of positive eigenvalues $ \lambda_{q}^{+} $ with respect to the range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $. To the bottom: Distribution of negative eigenvalues $ \lambda_{q}^{-} $ with respect to the range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $
Figure 4. Initial condition of the function $ u{\left({ {\mathbf{x}},t}\right)} $ given by equation (25) with a small perturbation around zero. 2D view of the function $ u{\left({ {\mathbf{x}},t}\right)} $ (to the left) and a 3D view of its magnitude (to the right)
Figure 5. First row from left to right: Nonlinear evolution of the function $ u{\left({ {\mathbf{x}},t}\right)} $ at $ t = 2.5 $, $ t = 325 $, and $ t = 997.5 $. Second row from left to right: Evolution of the heterogeneous stationary solutions at the same moments as for the evolution of $ u{\left({ {\mathbf{x}},t}\right)} $
Figure 6. Similarities of patterns between the nonlinear evolution $ u{\left({ {\mathbf{x}},t}\right)} $ (to the left) and the heterogeneous state (to the right)
Figure 7. Time evolution of the difference in $ {L^{2}} $ between $ u{\left({ {\mathbf{x}},t}\right)} $ and the heterogeneous solution
Figure 8. Plot of $ h({\left\|{q}\right\|}^{2}) $ as a function of $ {\left\|{q}\right\|}^{2} $defined by equation (13). When the death rate $ \beta $ decreases below the critical value $ \beta_{c} $, $ h({\left\|{q}\right\|}^{2}) $ becomes negative for a finite range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $ marked with rhombi, and pattern formation can be expected
Figure 10. Initial condition of the function $ u{\left({ {\mathbf{x}},t}\right)} $ given by equation (25) with a small perturbation around zero.2D view of the function $ u{\left({ {\mathbf{x}},t}\right)} $ (to the left) and a 3D view of its magnitude (to the right)
Figure 11. First row from left to right. Nonlinear evolution of the function $ u{\left({ {\mathbf{x}},t}\right)} $ at $ t = 10 $, $ t = 70 $, and $ t = 750 $. Second row from left to right. Evolution of the heterogeneous stationary solutions at the same moments as for $ u{\left({ {\mathbf{x}},t}\right)} $
Figure 12. Similarities of patterns between the nonlinear evolution $ u({\left({ {\mathbf{x}},t}\right)} $ (to the left) and the heterogeneous state (to the right)
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Georges Chamoun Moustafa Ibrahim Mazen Saad Raafat Talhouk | CommonCrawl |
Battering Rams: Estimating the energy of the 7.2 magnitude earthquake that destroyed Bohol's churches
"Little Boy" atomic bomb
(Disclaimer: Since I am not a geologist but only a physicist, I would appreciate if somebody can correct my assumptions and computations. Computations and opinions are solely mine and may not necessarily reflect those of the Department of Physics of Ateneo de Manila University and of the Manila Observatory.)
According to Philippine Star, Dr. Renato Solidum of Phivolcs reported that "a magnitude 7 earthquake has an energy equivalent to around 32 Hiroshima atomic bombs". I verified his estimate in Earthquake Energy Calculator and constructed the following table:
Table 1: Energy equivalent of earthquake magnitudes in Richter scale
Magnitude (R) Seismic Moment Energy (J) Seismic Radiated Energy (J) Hiroshima Bombs
7.0 $3.899420\times 10^{19}$ $1.995262\times 10^{15}$ 31.8
So a 7.0 quake is equivalent to 32 Hiroshima-type bombs, as Dr. Solidum said. But a 7.2 quake is 64 nuclear bombs!
Little Boy A-Bomb Drop Hiroshima Ruins 11x14 Silver Halide Photo Print
Numbers always have assumptions behind them. These earthquake energies do not mean that 32 "Little Boy" (Hiroshima-type) nuclear bombs were dropped from a plane and destroyed all the churches upon impact. There is still another parameter that the media failed to ask: the depth of the quake epicenter. According to USGS, the epicenter's depth is about 20 km and not 0 km (ground level). This means that you make a well 20 km deep, put all the 32 Hiroshima-type bombs there, one on top of each other, cover the well with soil and stones, then detonate the bombs. BOOM! The island of Bohol would then shake and the churches would be leveled to a heap of ruins. Actually, a similar procedure was done by France when they detonated nuclear bombs at the Moruroa Atoll, but the depth of the well is less than 1 km:
France abandoned nuclear testing in the atmosphere in 1974 and moved testing underground in the midst of intense world pressure which was sparked by the New Zealand Government of the time, which sent two frigates, HMNZS Canterbury and Otago, to the atoll in protest for a nuclear free Pacific. Shafts were drilled deep into the volcanic rocks underlying the atolls where nuclear devices were detonated. This practice created much controversy as cracking of the atolls was discovered, resulting in fears that the radioactive material trapped under the atolls would eventually escape and contaminate the surrounding ocean and neighboring atolls. A major accident occurred on 25 July 1979 when a test was conducted at half the usual depth because the nuclear device got stuck halfway down the 800 metre shaft.[2] It was detonated and caused a large submarine landslide on the southwest rim of the atoll, causing a significant chunk of the outer slope of the atoll to break loose and causing a tsunami affecting Mururoa and injuring workers.[2] The blast caused a 2 kilometre long and 40 cm wide crack to appear on the atoll.[2] (Wikipedia: Moruroa)
Richter's Scale: Measure of an Earthquake, Measure of a Man
A. Richter Scale and Radiated Energy
In order to relate the energy $E$ of an earthquake to the Richter scale $R$, we use the formula
\begin{equation}
E = 10^{1.5 R + 4.8}
\end{equation}
Let's try for $R=7.0$:
E = 10^{1.5(7.0)+4.8} = 10^{15.3} = 10^{0.3}10^{15} = 1.995262\times 10^{15} J,
where $J$ stands for Joules, a unit of energy. Notice that energy value corresponds not to the seismic moment energy but to the seismic radiated energy. Since Phivolcs says that the magnitude is 7.2 while USGS says it is 7.1, we will just stick to the value of 7.2, which gives
E = 10^{1.5(7.2)+4.8} = 10^{15.6}= 3.981\times 10^{15} \approx 4\times 10^{15} J.
Seismic Wave Propagation and Scattering in the Heterogeneous Earth : Second Edition
B. Earthquake Power and Intensity as Function of Distance from Epicenter
For simplicity, let us assume that the 7.2 magnitude quake happened in 1 second. This means that the power at the source is
P = \frac{E}{\Delta t} = \frac{4\times 10^{15} J}{1 s} = 4\times 10^{15} W,
where the $W$ stands for Watts, a unit of power. If the seismic wave spreads out equally in all directions, then the average intensity at the distance $r$ from the quake epicenter is
I_r = \frac{P}{4\pi r^2},
(c.f. Young and Freedman 2004, p. 567). This simply means that the farther you are from the epicenter, the quake intensity decreases. One way to imagine this is to take a balloon. Blow some air into it, then draw some equally spaced dots on the surface, about 1 cm apart. If you blow more air into the balloon, the balloon expands. The number of dots remain the same, but the number of dots per square centimeter decreases. In the same way, for earthquakes, the power remains the same, but the power per unit area decreases as you go farther from the epicenter.
Intensity of a spherical wave as a function of distance from source (Source: Hyperphysics)
Since the earthquake epicenter is 20 km deep, we set $r=20\times 10^3 m$ and solve for the intensity $I_r$:
I_r = \frac{4\times 10^{15} W}{4\times 3.14\times 400\times 10^6\ m^2} = 7.96\times 10^5 W/m^2 \approx 8\times 10^5 W/m^2.
Since we assumed that the duration of the quake is $1s$, then the energy $E_r$ absorbed per square meter by the church buildings per square meter is
I_r\Delta t = 8\times 10^5 J/m^2 = \frac{E_r}{A},
where $A= 1 m^2$. Hence,
E_r = 8\times 10^5 J.
Joint Scientific Papers of James Prescott Joule (Cambridge Library Collection - Physical Sciences) (Volume 2)
Energy Equivalent of a Falling Mass
How does one imagine an energy $E_r = 8\times 10^5 J$? Gravitational potential energy is $E = mgy_0$, where $m$ is the mass of the object, $g= 9.8 m/s^2$ is the gravitational acceleration, and $y_0$ is the initial height from the reference (e.g. ground). For simplicity, let us set $y_0 = 1 m$, so that the corresponding weight of the object must be
mg = 8\times 10^5 N,
where $N$ is Newtons, the unit of force. Dividing both sides by the gravitational acceleration $g=9.8 m/s^2$, we get the mass
m = 0.8\times 10^5 kg = 80,000 kg.
ANTIQUE PRINT -ROMAN- BATTERING RAM SYSTEM- 1729
Battering Ram!
How does one imagine a mass $m = 80,000 kg$?. Steel has a density of
\rho = \frac{m}{V} = 7,850 kg/m^3.
For a mass of $80,000 kg$, the corresponding volume of steel is
V = \frac{80,000 kg}{7,850 kg/m^3} = 10 m^3.
If the steel is is a box with base area of $1 m^2$, then the height of the box must be 10 m or about three building floors. So we have a pillar of steel: $1 m\times 1 m\times 10 m$. This is already a battering ram that can destroy the gates of medieval fortresses!
Old Historical Baroque Filipino Church of Loon - 52"W x 35"H - Peel and Stick Wall Decal by Wallmonkeys
Thus, Bohol's churches were destroyed because the energy of the quake is equivalent to a battering ram of $1 m\times 1 m\times 10 m$ that is rammed against every square meter of churches foundations with a speed equal to that as if the ram is lifted 1 m from the the point of impact and and then released. The speed of the impact can be computed from the Law of Conservation of Energy:
mgy_0 = \frac{1}{2}mv^2.
Solving for the final velocity $v$, we get
v = \sqrt{2gy_0} = \sqrt{2\times (9.8 m/s^2)\times (1 m)} = 4.43 m/s.
This speed is equivalent to
v = 4.43 m/s = \frac{4.43 m}{s}\times\frac{3600 s}{hr}\times\frac{1 km}{1000 m} = 15.9 = 16 kph.
Thus, the earthquake of magnitude 7.2 is equivalent to a battering rams of sizes $1 m\times 1 m \times 10 m$ lifted $1 m$ and then released. The battering rams swing like a pendulum and hit each square meter of the walls at a speed of $4.43 m/s = 16 kph$. Hence, the pile of ruins. (Note: Technically, you still have to compute the moment of inertia of the battering ram as it swings like a pendulum. But for simplicity, we approximated the ram into a ball of the same mass. This is not really important; what is more important is the speed of the ram just before impact, and we set this speed to be 4.43 m/s, the same speed if the ram touches the ground as it fell freely from a height 1 m above the ground.)
Medieval Battering Ram
Quake Magnitude and Length of Battering Ram
We can repeat the same procedure for 7.0 and 7.1 magnitude quakes. But I think it is best to derive a formula instead that relates the quake magnitude and the length of the battering ram.
We know that the formula for the quake energy $E$ as function of the Richter magnitude $R$ is
E = 10^{1.5 R + 4.8}.
Dividing this by the time interval $\Delta t$ yields the power
P = \frac{E}{\Delta t}.
The intensity $I_r$ of the quake at a distance $r$ from the epicenter is
In terms of energy $E$ and the Richter magnitude $R$, this is
I_r =\frac{E}{4\pi r^2\Delta t} = \frac{10^{1.5 R + 4.8}}{4\pi r^2\Delta t}.
This means that the energy $E_r$ per unit area $A$ at a distance $r$ from the epicenter is
\frac{E_r}{A} = \frac{E}{4\pi r^2} = \frac{10^{1.5 R + 4.8}}{4\pi r^2}.
Replica battering ram at Château des Baux, France
We define the energy $E_r$ in terms of a mass $m$ falling from a height $h$:
E_r = mgy_0.
For a steel of density $\rho$ which is shaped as solid box of base area $A$ and height $h$, the mass of the steel is
m = \rho V = \rho Ah.
Thus, the energy $E_r$ is
E_r = \rho Ahgy_0,
Using these results, we have
\frac{E_r}{A} = \rho hgy_0 = \frac{10^{1.5 R + 4.8}}{4\pi r^2}.
Solving for the height $h$ of the steel with $1 m^2$ base, we get
h = \frac{1}{\rho g y_0} \frac{10^{1.5 R + 4.8}}{4\pi r^2}.
This height $h$ is the length of the battering ram with $1 m^2$ base area of density $\rho$ raised at a distance $y_0$ from its starting point. This battering ram is the equivalent of the earthquake of Richter magnitude $R$ originating at a depth of $r$ below the surface.
Illustration of Roman Soldiers Forming Testulas and Using Siege Towers to Attack Fortified Wall - 42"W x 37"H - Peel and Stick Wall Decal by Wallmonkeys
Steel Ram vs Wooden Ram
Let us now set the values for a steel ram: $\rho = 7,850 kg/m^3$, $g = 9.8 m/s^2$, $y_0=1m$, $r=20\times 10^3 m$. These give
h = h(R) = \frac{10^{1.5 R + 4.8}}{3.867\times 10^{14}}.
On the other hand, had we set the values for a wooden ram with $1 m^2$ cross-section, we simply change the density to that of wood $\rho=1\times 10^3 kg/m^3$ (e.g. logwood $\approx 0.9\times 10^3 kg/m^3$). This would result to
As before, the ramming speed is assumed to be $4.43 m/s$ upon impact.
Let us tabulate our results. Note that this assumes that the structure is immediately above the quake epicenter. Notice that if the quake epicenter is 20 km below the ground, a 6.5 magnitude quake is approximately equivalent to a wrecking ball less slightly less than 1 m in radius; while a 6.9 magnitude quake is equivalent to 28.7 m (94 ft) long wooden ram hitting the building's foundations, such as that used by the Roman army in their siege:
Roman Siege Weapons - The Aries or Battering Ram. The aries, or battering-ram, consisted of a large beam made of the trunk of a tree, frequently one hundred feet in length, to one end of which was fastened a mace of iron or bronze resembling in form the head of a ram; it was often suspended by ropes from a beam fixed transversely over it, so that the soldiers were relieved from supporting its weight, and were able to give it a rapid and forcible swinging motion backward and forward. (Tribunes and Triumphs)
Games Workshop The Siege of Gondor Lord of the Rings Supplement
This ram is also similar to the one used by Sauron's armies during the siege of Gondor:
The drums rolled louder. Fires leaped up. Great engines crawled across the field; and in the midst was a huge ram, great as a forest-tree a hundred feet in length, swinging on mighty chains. Long had it been forging in the dark smithies of Mordor, and its hideous head founded of black steel, was shaped in the likeness of a ravening wolf; on it spells of ruin lay. Grond they named it, in memory of the Hammer of the Underworld of old. Great beasts drew it, orcs surrounded it, and behind walked mountain-trolls to wield it. (The Siege of Gondor, Lord of the Rings, p. 828)
This is only for a 6.9 quake, 20 km deep. How much more for quakes of magnitudes 7.0, 7.1, or 7.2?
Table 2: Equivalent battering ram lengths for earthquakes of different magnitudes with epicenter depth of 20 km
Earthquake (R) * Steel Ram (m)** Wooden Ram (m)**
6.0 0.16 1.3
6.5 0.92 (wrecking ball) 7.2
6.6 1.3 10.2
6.9 3.7 28.7 (Grond in LotR)
7.3 14.5 114.2
* Richter magnitude scale
** Cross-sectional area of 1 square meter and moving at a speed of $4.43 m/s$. This speed is equivalent to the speed of an object as it touches the ground after it was released from a height of 1 m above the ground.
Posted by Monk's Hobbit at 2:23 PM No comments:
Labels: Churches, Disasters, Earthquakes, Monk's Nation
Location: Bohol, Philippines
On the miraculous survival of Virgin Mary's picture amidst the destruction of Bohol's churches: A reply to Filipino Freethinkers
Image of the Blessed Virgin Mary amidst the destruction of a Bohol Church (photo from Megan Young's Facebook page, dated Oct. 15, 2013)
Sonnet: On the destruction of Bohol's Churches
I stand before the broken stones and glass
And gaze at roof beams lying now in dust
An ancient church once stood here tall and strong
Which brown hands built with sand and soft limestone
The brass bells tolled to mark the hours of day
And people stopped to cross themselves and pray
The brass bells tolled to mark the days of feasts
Of harvests, Sundays, Lents, and Christmas Eves
The organ played the old Gregorian chants
As Spanish friars said the Holy Mass
With zeal for souls, with love for Church and Christ
They preached, they teached, they blessed, and they baptized
And through their labors built the Christian lands
Though storms and quakes have passed our Faith still stands
--Monk's Hobbit, 2013.10.18
Freethinkers: A History of American Secularism
COMMENT BY FILIPINO FREETHINKERS
The Filipino Freethinkers made a video questioning the sudden focus on the miraculous survival of the image of the Blessed Virgin Mary during the 7.2 magnitude earthquake the destroyed Bohol's ancient churches. Because I am afraid of misquoting them, I tried my best to transcribe (though not perfectly) their thoughts based on their video entitled, "FF Podcast 018: Iglesia ni Cristo's Medical Mission and the Bohol Earthquake":
12:50 Just a particular thing that has been making the rounds on social media. It was about that one image. it was in that instagram photo that was regrammed by Julius Babao. It was an image of the Virgin Mary and it somehow it managed to survive one of the strong quakes that leveled the church that it was made. You are probably looking at that image right in this very second...What is the meaning of this? What is the significance of sharing and resharing this single image?
I think people are scrambling for something comforting at least to them in this period of devastation--the tragedy that happened in Bohol. So many people died and many buildings were destroyed. I am sure lots of infrastructure was lost. It is just them thinking that oh there must be reason why this small thing was not destroyed. This is a clear example of selection bias: that you decide to ignore the other bad stuff because it is overwhelming sometimes and you just look it is a good thing.
As a religious person, what is your take on this?
Cultures of Disaster: Society and Natural Hazard in the Philippines
14:24 I just find it really sad that suddenly because one easily replaceable picture gets saved vs all of the actual number of people who died. There are some people who need to latch on to that. Again I have been a lot through really terrible things. But at the same time i think that when something bad happens, you need to focus on the people who were hurt and not so much--symbols are important as well--but the tendency can be to conflate the importance. Oh, this one church was saved. Oh, this one picture was saved. This must mean that god doesn't hate us or isn't random or doesn't exist. The immediate response, I think, still needs to be a human response. There are people who are hurt. i think it is obscene to point to one object that may have been saved and how that it is a grand thing and it means that all of this are for a reason, whether there are dead bodies that are never gonna be be replaced or not.
15:35 People just grasp anything to use for hope to keep them from totally sinking in these hard times. It must be terrible what they are going through. But at the same time, I am not saying it is them directly. Perhaps not even the original person said, but a lot of the context in which that picture is being shared has that triumphalist veneer which I find rather disgusting. Maybe it is not even the people who actually went to the tragedy, but it is the people who are perhaps sitting comfortably in their chairs in Metro Manila and they point to this: "Hey, that tragedy happened in Bohol. we were saved. There must be some grand meaning. Checkmate atheists. Checkmate Filipino Freethinkers." That is how petty some people can be. But they will latch on to this in the face of horrible human suffering. Just for one argument. they will Ignore all of that. They will hoist this one symbol up. It can be that picture. It can be that tsunami in Japan because God is mad at them. All of these things. But that is really the way certain Fundamentalist seem to think.
Catholic Prophecy: The Coming Chastisement
16:58 I do understand there are people who need this kind of help whenever tragedies like this strike, but the skeptic in me still has to ask those questions like what is the meaning of that? Is there even meaning in it? How does it even work? is that image powerful in itself it some how deflected force that was falling on it. Or was there a targeting system from heaven that protected it from everything else. more and more questions could come out of all of this. like why wasn't that targeting system used on the people. why wasn't that force field installed on every human being that was hurt. Is there even a God? This question of course would be very hard to answer. And I guess we will just have to accept that these tragedies are a part of life, a part of nature. More and more earthquakes will happen. There is nothing special in it. There will be End Times prophets saying the end is near or they hear the voice of God. I like especially that the RH is not a hot topic nowadays, because if it is then you would expect people to relate this to the RH issue. It might as well be the INC mission.
I shall focus on five main comments by Filipino Freethinkers:
It is obscene to point to one object that may have been saved and how that it is a grand thing and it means that all of this are for a reason, whether there are dead bodies that are never gonna be be replaced or not.
People just grasp anything to use for hope to keep them from totally sinking in these hard times.
Why was the image of Mary saved from destruction, but not the hundreds who died?
Is there even a God? This question of course would be very hard to answer.
And I guess we will just have to accept that these tragedies are a part of life, a part of nature. More and more earthquakes will happen. There is nothing special in it.
Color Persuasion: The Science of using Color to Persuade and Influence Purchasing Decisions (Dynamic Media Series)
Comment 1: It is obscene to point to one object that may have been saved and how that it is a grand thing and it means that all of this are for a reason, whether there are dead bodies that are never gonna be be replaced or not.
In ordinary life in Bohol, people live and and go to mass during Sundays. So when all of a sudden people die and the churches destroyed, we focus on the destruction and it comes out in the news, in the same way as we focus on the black dot at the center of the white paper. Now, having expected destruction of old churches everywhere in Bohol and people dying in hundreds, when we see an image of the Blessed Virgin Mary that still stands despite the destruction of the Church that contains it, then we focus more on the image and not on the church or on the number of the dead, in the same way as we focus more on the white dot on a black paper. This is just a natural human reaction rooted in human vision. There is nothing obscene.
Is the survival of Mary's image a miracle? Maybe. The word miracle is rooted in the word "wonder":
miracle (n.) mid-12c., "a wondrous work of God," from Old French miracle (11c.) "miracle, story of a miracle, miracle play," from Latin miraculum "object of wonder" (in Church Latin, "marvelous event caused by God"), from mirari "to wonder at, marvel, be astonished," figuratively "to regard, esteem," from mirus "wonderful, astonishing, amazing," earlier *smeiros, from PIE *smei- "to smile, laugh" (cf. Sanskrit smerah "smiling," Greek meidan "to smile," Old Church Slavonic smejo "to laugh;" see smile (v.)). (Online Etymologyical Dictionary)
Since the image of Our Lady aroused wonder among many people, making them praise God for such a wonderful sight, then we can say that the survival of Mary's image is a miracle. For a more detailed discussion of miracles, please read the Catholic Encyclopedia.
Saved in Hope: Spe Salvi
Comment 2: People just grasp anything to use for hope to keep them from totally sinking in these hard times.
So what do you wish people do? Lose hope and kill themselves, just like the many who committed suicide in concentration camps? In order to survive the most inhuman conditions, man has to search for meaning and it is only those who found meaning in their life who will survive the ordeal. And this is what Victor Frankl found in his book, Man's Search For Meaning :
Man's Search for Meaning is a 1946 book by Viktor Frankl chronicling his experiences as an Auschwitz concentration camp inmate during World War II, and describing his psychotherapeutic method, which involved identifying a purpose in life to feel positively about, and then immersively imagining that outcome. According to Frankl, the way a prisoner imagined the future affected his longevity. The book intends to answer the question "How was everyday life in a concentration camp reflected in the mind of the average prisoner?" Part One constitutes Frankl's analysis of his experiences in the concentration camps, while Part Two introduces his ideas of meaning and his theory called logotherapy. According to a survey conducted by the Book-of-the-Month Club and the Library of Congress, Man's Search For Meaning belongs to a list of "the ten most influential books in the United States."[1] At the time of the author's death in 1997, the book had sold over 10 million copies and had been translated into 24 languages. (Wikipedia)
If one believes in Darwinian Law of Natural Selection which "the survival of the fittest and the removal of the unfit", then hope is necessary for the survival of the human species, so that we can rephrase Darwin's law as follows: "the survival of the hopeful and the death of the hopeless."
Bartolome Esteban Murillo Immaculate Conception - 18" x 27" Framed Premium Canvas Print
Comment 3: Why was the image of Mary saved from destruction, but not the hundreds who died?
I don't know. If God exists and we shall face Him during Judgment Day, we can ask Him this question. Well, we can't even determine the precise reasons why our friends betrayed us or why our loved ones abandoned us. Why? O, why? We asked ourselves. So how much more if we demand from God the reasons for His actions or inactions?
Job, the righteous person, also suffered in a single blow and the persons who survived are not his children but his lowly servants who reported to him:
One day, while his sons and daughters were eating and drinking wine in the house of their eldest brother, 14 a messenger came to Job and said, "The oxen were plowing and the donkeys grazing beside them, 15 and the Sabeans* carried them off in a raid. They put the servants to the sword, and I alone have escaped to tell you." 16 He was still speaking when another came and said, "God's fire has fallen from heaven and struck the sheep and the servants and consumed them; I alone have escaped to tell you."17 He was still speaking when another came and said, "The Chaldeans* formed three columns, seized the camels, carried them off, and put the servants to the sword; I alone have escaped to tell you." 18 He was still speaking when another came and said, "Your sons and daughters were eating and drinking wine in the house of their eldest brother,19 and suddenly a great wind came from across the desert and smashed the four corners of the house. It fell upon the young people and they are dead; I alone have escaped to tell you." (Job 1:13-19)
(13x19) Albrecht Durer Job Mocked by His Wife Art Print Poster
What was Job's response?
"Naked I came forth from my mother's womb,f and naked shall I go back there.* The LORD gave and the LORD has taken away; blessed be the name of the LORD!" (Job 1:21)
And even if the affliction is already on his own body, he still never cursed God:
He took a potsherd to scrape himself, as he sat among the ashes.9 Then his wife said to him,d "Are you still holding to your innocence? Curse God and die!"*10 But he said to her, "You speak as foolish women do. We accept good things from God; should we not accept evil?" Through all this, Job did not sin in what he said. (Job 2:8-10)
But like our friends in the Filipino Freethinkers, Job also would like to ask God why he suffered despite his righteousness:
I will give myself up to complaint; I will speak from the bitterness of my soul. 2 I will say to God: Do not put me in the wrong! Let me know why you oppose me. (Job 10:1-2)
What are my faults and my sins? My misdeed, my sin make known to me!24 Why do you hide your face and consider me your enemy? (Job 13:23-24)
And God appeared Job. God did not answer Job's question; rather, God only showed Job his wisdom and majesty. Then the Lord answered Job out of the storm and said:
Who is this who darkens counsel with words of ignorance?3 Gird up your loins* now, like a man; I will question you, and you tell me the answers!a 4 Where were you when I founded the earth? Tell me, if you have understanding. (Job 38:1-4)
Hidden Treasures in the Book of Job: How the Oldest Book in the Bible Answers Today's Scientific Questions (Reasons to Believe)
And Job replied:
I know that you can do all things, and that no purpose of yours can be hindered.3"Who is this who obscures counsel with ignorance?" I have spoken but did not understand; things too marvelous for me, which I did not know. "Listen, and I will speak; I will question you, and you tell me the answers."5 By hearsay I had heard of you, but now my eye has seen you.* 6 Therefore I disown what I have said, and repent in dust and ashes. (Job 42:2-3)
In the end God restored Job everything that he had and so much more::
Thus the LORD blessed the later days of Job more than his earlier ones. Now he had fourteen thousand sheep, six thousand camels, a thousand yoke of oxen, and a thousand she-donkeys.13 He also had seven sons and three daughters: 14 the first daughter he called Jemimah, the second Keziah, and the third Keren-happuch.*15 In all the land no other women were as beautiful as the daughters of Job; and their father gave them an inheritance* among their brothers. 16 After this, Job lived a hundred and forty years; and he saw his children, his grandchildren, and even his great-grandchildren.c 17 Then Job died, old and full of years. (Job 42:12-17)
And like Job we also pray that God may restore our churches not only in Bohol but in whole Philippine archipelago as well, with greater splendor than before. And we also pray that may God increase the number of the Church's faithful sons and daughter who shall proclaim the Gospel unto the ends of the earth.
New Proofs for the Existence of God: Contributions of Contemporary Physics and Philosophy
Comment 4: Is there even a God? This question of course would be very hard to answer.
Doubt is different from denial of God's existence. Here is what Cardinal Newman says:
Ten thousand difficulties do not make one doubt, as I understand the subject; difficulty and doubt are incommensurate. There of course may be difficulties in the evidence; but I am speaking of difficulties intrinsic to the doctrines themselves, or to their relations with each other. A man may be annoyed that he cannot work out a mathematical problem, of which the answer is or is not given to him, without doubting that it admits of an answer, or that a certain particular answer is the true one. Of all points of faith, the being of a God is, to my own apprehension, encompassed with most difficulty, and yet borne in upon our minds with most power.
The easy way out, of course, is just to pretend that the problem of the existence of God does not exist and live on life. But this question continue to haunt atheists and agnostics, in the same way as a woman who procured abortion would either deny the existence of the baby by calling it a piece of tissue or blood or pretend that nothing happened by trying to live life as normally as possible. In psychology, there is an explanation for this phenomenon. It is called cognitive dissonance:
The Brain Sell: When Science Meets Shopping; How the new mind sciences and the persuasion industry are reading our thoughts, influencing our emotions, and stimulating us to shop
Cognitive dissonance describes the feeling of discomfort we experience when we attempt to hold two conflicting beliefs simultaneously. Imagine you are a smoker who wants to quit but finds it hard to break the habit. Being exposed to health messages and dire warnings about the medical consequences of failing to give up while continuing to smoke will create cognitive dissonance. In order to free themselves from these discomforting feelings, smokers have one of two choices: to stop smoking or to rationalize away the risk. Lifelong smokers will often say: "Of course I understand it's a risky habit, but so is driving a car, riding a bike, or even crossing the street." Alternatively, they may point that their Uncle Charlie smoked 100 cigarettes a day and lived to be a healthy and sprightly 90-year-old. Both of these approaches will remove the dissonance, although the first actually safeguards health. (Dr. David Lewis, The Brain Sell: When Science Meets Shopping , p. 29)
Here is what St. Aquinas proposes as proofs for God's existence, for he does not shy away from the rational arguments extolled by our atheist and agnostic friends. These may be difficult to comprehend in the same way as calculus is difficult to comprehend, but not impossible:
The Existence Of God by St. Thomas Aquinas
I answer that, The existence of God can be proved in five ways.
The first and more manifest way is the argument from motion. It is certain, and evident to our senses, that in the world some things are in motion. Now whatever is in motion is put in motion by another, for nothing can be in motion except it is in potentiality to that towards which it is in motion; whereas a thing moves inasmuch as it is in act. For motion is nothing else than the reduction of something from potentiality to actuality. But nothing can be reduced from potentiality to actuality, except by something in a state of actuality. Thus that which is actually hot, as fire, makes wood, which is potentially hot, to be actually hot, and thereby moves and changes it. Now it is not possible that the same thing should be at once in actuality and potentiality in the same respect, but only in different respects. For what is actually hot cannot simultaneously be potentially hot; but it is simultaneously potentially cold. It is therefore impossible that in the same respect and in the same way a thing should be both mover and moved, i.e. that it should move itself. Therefore, whatever is in motion must be put in motion by another. If that by which it is put in motion be itself put in motion, then this also must needs be put in motion by another, and that by another again. But this cannot go on to infinity, because then there would be no first mover, and, consequently, no other mover; seeing that subsequent movers move only inasmuch as they are put in motion by the first mover; as the staff moves only because it is put in motion by the hand. Therefore it is necessary to arrive at a first mover, put in motion by no other; and this everyone understands to be God.
St. Anselm's Argument for the Existence of God: PROSLOGION The Nature of God Vol. I and II
The second way is from the nature of the efficient cause. In the world of sense we find there is an order of efficient causes. There is no case known (neither is it, indeed, possible) in which a thing is found to be the efficient cause of itself; for so it would be prior to itself, which is impossible. Now in efficient causes it is not possible to go on to infinity, because in all efficient causes following in order, the first is the cause of the intermediate cause, and the intermediate is the cause of the ultimate cause, whether the intermediate cause be several, or only one. Now to take away the cause is to take away the effect. Therefore, if there be no first cause among efficient causes, there will be no ultimate, nor any intermediate cause. But if in efficient causes it is possible to go on to infinity, there will be no first efficient cause, neither will there be an ultimate effect, nor any intermediate efficient causes; all of which is plainly false. Therefore it is necessary to admit a first efficient cause, to which everyone gives the name of God.
The third way is taken from possibility and necessity, and runs thus. We find in nature things that are possible to be and not to be, since they are found to be generated, and to corrupt, and consequently, they are possible to be and not to be. But it is impossible for these always to exist, for that which is possible not to be at some time is not. Therefore, if everything is possible not to be, then at one time there could have been nothing in existence. Now if this were true, even now there would be nothing in existence, because that which does not exist only begins to exist by something already existing. Therefore, if at one time nothing was in existence, it would have been impossible for anything to have begun to exist; and thus even now nothing would be in existence — which is absurd. Therefore, not all beings are merely possible, but there must exist something the existence of which is necessary. But every necessary thing either has its necessity caused by another, or not. Now it is impossible to go on to infinity in necessary things which have their necessity caused by another, as has been already proved in regard to efficient causes. Therefore we cannot but postulate the existence of some being having of itself its own necessity, and not receiving it from another, but rather causing in others their necessity. This all men speak of as God.
Metaphysical Demonstration of the Existence of God
The fourth way is taken from the gradation to be found in things. Among beings there are some more and some less good, true, noble and the like. But "more" and "less" are predicated of different things, according as they resemble in their different ways something which is the maximum, as a thing is said to be hotter according as it more nearly resembles that which is hottest; so that there is something which is truest, something best, something noblest and, consequently, something which is uttermost being; for those things that are greatest in truth are greatest in being, as it is written in Metaph. ii. Now the maximum in any genus is the cause of all in that genus; as fire, which is the maximum heat, is the cause of all hot things. Therefore there must also be something which is to all beings the cause of their being, goodness, and every other perfection; and this we call God.
The fifth way is taken from the governance of the world. We see that things which lack intelligence, such as natural bodies, act for an end, and this is evident from their acting always, or nearly always, in the same way, so as to obtain the best result. Hence it is plain that not fortuitously, but designedly, do they achieve their end. Now whatever lacks intelligence cannot move towards an end, unless it be directed by some being endowed with knowledge and intelligence; as the arrow is shot to its mark by the archer. Therefore some intelligent being exists by whom all natural things are directed to their end; and this being we call God. (Summa Theologiae, Ques. 2, Art. 3)
Adam and Eve After the Pill: Paradoxes of the Sexual Revolution
Comment 5. And I guess we will just have to accept that these tragedies are a part of life, a part of nature. More and more earthquakes will happen. There is nothing special in it.
So in the end, atheists and agnostics cannot provide a meaning to suffering and death, but simply lump these as part of the natural world. So where their naturalist explanation ends, we propose the good news and the logic of the Catholic Faith:
Man suffered and died because of the disobedience of our First Parents Adam and Eve. They were given a simple rule not to eat a fruit in the middle of garden, and all the preternatural gifts given to them would be theirs and to their descendants. But since they disobeyed, Adam has to toil to make the ground bear fruit unlike before, and the pains of childbearing of his wife, Eve, increased. The children of Adam and Eve and all their descendants bore the punishment of suffering and death due to the sin of Adam and Eve. This is called Original Sin. And this is why we all suffer and die.
Amazing Love: The Story of Hosea
The gravity of the sin of Adam and Eve separated them from God. In order to get a glimpse of the suffering and pain that God suffered because of the rejection of his love by the human race, we turn to the book of Hosea:
Again the LORD said to me: Go, love a woman who is loved by her spouse but commits adultery; just as the LORD loves the Israelites, though they turn to other gods and love raisin cakes. So I acquired her for myself for fifteen pieces of silver and a homer and a lethech of barley. 3Then I said to her: "You will wait for me for many days; you will not prostitute yourself or belong to any man; I in turn will wait for you." (Hos 3:1)
Here, God's love is likened to a man who married a prostitute (Israel) and made a marriage covenant with her to be his only and no other man's. But the woman continued in her prostitution (Israel worships other Gods). The whole Bible is a love story: a love story between God and Man. Since Man spurned God, God took the initiative and became man, subject to suffering and death:
The Crucifixion of Jesus, Completely Revised and Expanded: A Forensic Inquiry
And the Word became flesh*
and made his dwelling among us,
the glory as of the Father's only Son,
full of grace and truth. (Jn 1:14)
In doing so, God showed Man the way to salvation: not to reject suffering and death, but rather to embrace these and offer these to God, as Christ willingly embraced the Cross and offered to God his spirit. Thus, the path to salvation is the path of suffering and death, in the same way as Aragorn has to take the Paths of the Dead in order to save Gondor from destruction. As Christ said:
Whoever wishes to come after me must deny himself,* take up his cross, and follow me.25r For whoever wishes to save his life will lose it, but whoever loses his life for my sake will find it. (Mt 16:24:25)
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Moving sofa problem
In mathematics, the moving sofa problem or sofa problem is a two-dimensional idealisation of real-life furniture-moving problems and asks for the rigid two-dimensional shape of largest area that can be maneuvered through an L-shaped planar region with legs of unit width.[1] The area thus obtained is referred to as the sofa constant. The exact value of the sofa constant is an open problem. The currently leading solution, by Joseph L. Gerver, has a value of approximately 2.2195 and is thought to be close to the optimal, based upon subsequent study and theoretical bounds.
Unsolved problem in mathematics:
What is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?
(more unsolved problems in mathematics)
History
The first formal publication was by the Austrian-Canadian mathematician Leo Moser in 1966,[2] although there had been many informal mentions before that date.[1]
Bounds
Work has been done on proving that the sofa constant (A) cannot be below or above certain values (lower bounds and upper bounds).
Lower
Lower bounds can be proven by finding a specific shape of high area and a path for moving it through the corner. An obvious lower bound is $A\geq \pi /2\approx 1.57$. This comes from a sofa that is a half-disk of unit radius, which can slide up one passage into the corner, rotate within the corner around the center of the disk, and then slide out the other passage.
In 1968, John Hammersley stated a lower bound of $A\geq \pi /2+2/\pi \approx 2.2074$.[3] This can be achieved using a shape resembling a telephone handset, consisting of two quarter-disks of radius 1 on either side of a 1 by $4/\pi $ rectangle from which a half-disk of radius $2/\pi $ has been removed.[4][5]
In 1992, Joseph L. Gerver of Rutgers University described a sofa specified by 18 curve sections each taking a smooth analytic form. This further increased the lower bound for the sofa constant to approximately 2.2195.[6][7]
Upper
Hammersley stated an upper bound on the sofa constant of at most $2{\sqrt {2}}\approx 2.8284$.[3][1][8] Yoav Kallus and Dan Romik published a new upper bound in 2018, capping the sofa constant at $2.37$. Their approach involves rotating the corridor (rather than the sofa) through a finite sequence of distinct angles (rather than continuously), and using a computer search to find translations for each rotated copy so that the intersection of all of the copies has a connected component with as large an area as possible. As they show, this provides a valid upper bound for the optimal sofa, which can be made more accurate by using more rotation angles. A set of five carefully-chosen rotation angles leads to the stated upper bound.[9]
Ambidextrous sofa
A variant of the sofa problem asks the shape of largest area that can go round both left and right 90 degree corners in a corridor of unit width (where the left and right corners are spaced sufficiently far apart that one is fully negotiated before the other is encountered). A lower bound of area approximately 1.64495521 has been described by Dan Romik. His sofa is also described by 18 curve sections.[10][11]
See also
• Dirk Gently's Holistic Detective Agency – novel by Douglas Adams, a subplot of which revolves around such a problem.
• Moser's worm problem
• Square packing in a square
• "The One with the Cop" - an episode of the American TV series Friends with a subplot revolving around such a problem.
References
1. Wagner, Neal R. (1976). "The Sofa Problem" (PDF). The American Mathematical Monthly. 83 (3): 188–189. doi:10.2307/2977022. JSTOR 2977022. Archived from the original (PDF) on 2015-04-20. Retrieved 2009-07-25.
2. Moser, Leo (July 1966). "Problem 66-11, Moving furniture through a hallway". SIAM Review. 8 (3): 381. JSTOR 2028218.
3. J. M. Hammersley (1968). "On the enfeeblement of mathematical skills by 'Modern Mathematics' and by similar soft intellectual trash in schools and universities". Bulletin of the Institute of Mathematics and its Applications. 4: 66–85. See Appendix IV, Problems, Problem 8, p. 84.
4. Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1994). Halmos, Paul R. (ed.). Unsolved Problems in Geometry. Problem Books in Mathematics; Unsolved Problems in Intuitive Mathematics. Vol. II. Springer-Verlag. ISBN 978-0-387-97506-1. Retrieved 24 April 2013.
5. Finch, Steven, Moving Sofa Constant, Mathcad Library (includes a diagram of Gerver's sofa).
6. Gerver, Joseph L. (1992). "On Moving a Sofa Around a Corner". Geometriae Dedicata. 42 (3): 267–283. doi:10.1007/BF02414066. ISSN 0046-5755. S2CID 119520847.
7. Weisstein, Eric W. "Moving sofa problem". MathWorld.
8. Stewart, Ian (January 2004). Another Fine Math You've Got Me Into... Mineola, N.Y.: Dover Publications. ISBN 0486431819. Retrieved 24 April 2013.
9. Kallus, Yoav; Romik, Dan (December 2018). "Improved upper bounds in the moving sofa problem". Advances in Mathematics. 340: 960–982. arXiv:1706.06630. doi:10.1016/j.aim.2018.10.022. ISSN 0001-8708. S2CID 5844665.
10. Romik, Dan (2017). "Differential equations and exact solutions in the moving sofa problem". Experimental Mathematics. 26 (2): 316–330. arXiv:1606.08111. doi:10.1080/10586458.2016.1270858. S2CID 15169264.
11. Romik, Dan. "The moving sofa problem - Dan Romik's home page". UCDavis. Retrieved 26 March 2017.
External links
• Romik, Dan (March 23, 2017). "The Moving Sofa Problem" (video). YouTube. Brady Haran. Archived from the original on 2021-12-21. Retrieved 24 March 2017.
• SofaBounds - Program to calculate bounds on the sofa moving problem.
• A 3D model of Romik's ambidextrous sofa
| Wikipedia |
\begin{document}
\title{Focused Proof-search in the Logic of Bunched Implications hanks{This work has been partially supported by the UK’s EPSRC through research grant EP/S013008/1.}
\begin{abstract}
The logic of Bunched Implications (BI) freely combines additive and multiplicative connectives, including implications; however, despite its well-studied proof theory, proof-search in BI has always been a difficult problem. The focusing principle is a restriction of the proof-search space that can capture various goal-directed proof-search procedures. In this paper we show that focused proof-search is complete for BI by first reformulating the traditional bunched sequent calculus using the simpler data-structure of nested sequents, following with a polarised and focused variant that we show is sound and complete via a cut-elimination argument. This establishes an operational semantics for focused proof-search in the logic of Bunched Implications.
\keywords{Logic \and Proof-search \and Focusing \and Bunched Implications.} \end{abstract}
\section{Introduction} The \emph{Logic of Bunched Implications} (BI)~\cite{Hearn99} is well-known for its applications in systems modelling \cite{Pym19}, especially a particular theory (of a variant of BI) called \emph{Separation Logic}~\cite{Reynolds02,Ishtiaq2011} which has found industrial use in program verification.
In this work, we study an aspect of proof search in BI, relying on its well-developed and well-studied proof theory~\cite{Pym02}. We show that a goal-directed proof-search procedure known as \emph{focused proof-search} is complete; that is, if there is a proof then there is a focused one. Focused proofs are both interesting in the abstract, giving insight into the proof theory of the logic, and have (for other logics) been a useful modelling technology in applied settings. For example, focused proof-search forms an operational semantics of the DPLL SAT-solvers~\cite{Farooque13}, logic programming~\cite{Miller91,Andreoli92,Dyckhoff06,Chaudhuri06}, automated theorem provers \cite{Mclaughlin08}, and has been successful in providing a meta-theoretic framework in intuitionistic, substructural, and modal logics~\cite{Marin16,Miller13,Liang09}.
Syntactically BI combines additive and multiplicative connectives, but unlike related logics such as Linear Logic (LL) \cite{Girard87}, BI takes all the connectives as primitive. Indeed, it arose from a proof-theoretic investigation on the relationship between conjunction and implication. As a result, sequents in BI have a more complicated structure: each implication comes with an associated context-former. Therefore, in BI contexts are not lists, nor multisets, but instead are \emph{bunches}: binary trees whose leaves are formulas and internal nodes context-formers. Additive composition $(\Gamma;\Delta)$ admits the structural rules of weakening and contraction, whereas multiplicative composition $(\Gamma, \Delta)$ denies them. The principal technical challenges when studying proof-search in BI arise from the interaction between the additive and multiplicative fragments. We overcome these challenges by restricting the application of structural rules in the sequent calculus $\lbi$ as well as working with a representation of bunches as nested multisets.
Throughout we use the term \emph{sequent calculus} in a strict sense; that is, meaning a label-free internal sequent calculus, formed in the case of BI by a context (a bunch) and a consequent (a formula). The term \emph{proof-search} is consistently understood to be read as backward reduction within such a system. Although there is an extensive body of research on systems and procedures for semantics-based calculi in BI \cite{Galmiche2001,Galmiche2002,Galmiche2003,Galmiche05,Galmiche2019}, there has been comparatively little formal study on proof-search in the strict sense. One exception is the completeness result for (unit-simple) uniform proofs \cite{Armelin02} which is partially subsumed by the results herein.
The \emph{focusing principle} was introduced for Linear Logic \cite{Andreoli92} and is characterised by alternating \emph{focused} and \emph{unfocused} phases of goal-directed proof-search. The unfocused phase comprises rules which are safe to apply (i.e. rules where provability is invariant); conversely, the focused phase contains the reduction of a formula and its sub-formulas where potentially invalid sequents may arise, and backtracking may be required. During focused proof-search the unfocused phases are performed eagerly, followed by controlled goal-directed focused phases, until safe reductions are available again. We say that the focusing principle holds when every provable sequent has a focused proof. This alternation can be enforced by a mechanism based on a partition of the set of formulas into two classes, \emph{positive} and \emph{negative}, which correspond to safe behaviour on the left and right respectively; that is, for negative formulas provability is invariant with respect to the application of a right rule, and for positive formulas, of a left rule, but in the other cases the application may result in invalid sequents.
The original proof of the focusing principle in Linear Logic was via long and tedious permutations of rules \cite{Andreoli92}. In this paper, we use for BI a different methodology, originally presented in \cite{Laurent04}, which has since been implemented in a variety of logics \cite{Liang09,Chaudhuri16b,Chaudhuri16a} and proof systems \cite{Dyckhoff06}. The method is as follows: given a sequent calculus, first one polarises the syntax according to the positive/negative behaviours; second, one gives a focused variation of the sequent calculus where the control flow of proof-search is managed by polarisation; third, one shows that this system admits cut (the only non-analytic rule); and, finally, one shows that in the presence of cut the original sequent calculus may be simulated in the focused one. When the polarised system is complete, the focusing principle holds.
In $\lbi$ certain rules (the structural rules) have no natural placement in either the focused or the unfocused phases of proof-search. Thus, a design choice must be made: to eliminate/constrain these rules, or to permit them without restriction. The first gives a stricter control proof-search regime, but the latter typically achieves a more well-behaved proof theoretic meta-theory. In this paper, we choose the former as our motivation is to study computational behaviour of proof-search in BI, the latter being recovered by familiar admissibility results. The only case where confinement is not possible is the \emph{exchange} rule. In standard sequent calculi the exchange rule is made implicit by working with a more convenient data-structure such as multisets as opposed to lists; however, the specific structure of bunches in BI means that a more complex alternative is required. The solution presented is to use nested multisets of two types (additive and multiplicative) corresponding to the two different context-formers/conjunctions.
In Section~\ref{sec:BI} we present the logic of Bunched Implications; in particular, Section~\ref{sec:BI_trad} and Section~\ref{sec:BI_calc} contain the background on BI (the syntax and sequent calculus respectfully); meanwhile, Section~\ref{sec:BI_nests} gives representation of bunches as nested multisets. Section~\ref{sec:foc} contains the focused system: first, in Section~\ref{sec:foc_pol} we introduce the polarised syntax; second, in Section~\ref{sec:foc_calc} we introduce the focused sequents calculus and some metatheory, most importantly the $\cut$-admissibility result; finally, in Section~\ref{sec:foc_comp} we give the completeness theorem, from which the validity of the focusing principle follows as a corollary. We conclude in Section~\ref{sec:conclusion} with some further discussion and future directions.
\section{Re-presentations of BI} \label{sec:BI}
\subsection{Traditional Syntax} \label{sec:BI_trad}
The logic BI has a well-studied metatheory admitting familiar categorical, algebraic, and truth-functional semantics which have the expected dualities~\cite{Pym2004resource,Galmiche05,Pym02,Docherty19,Pym19} .
In practice, it is the free combination (or, more precisely, the fibration \cite{Gabbay1998,Pym02}) of intuitionistic logic (IL) and the multiplicative fragment of intuitionistic linear logic (MILL), which imposes the presence of two distinct context-formers in its sequent presentation. That is to say, the two conjunctions $\land$ and $*$ are represented at the meta-level by context-formers $;$ and $,$ in place of the usual commas for IL and MILL respectively.
\begin{definition}[Formula]
Let \emph{\textsf{P}} be a denumerable set of propositional letters. The \emph{formulas} of BI, denoted by small Greek letters ($\varphi, \psi, \chi, \ldots$), are defined by the following grammar, where $A \in \emph{\textsf{P}}$,
$$\varphi ::= \top \mid \bot \mid \top^* \mid A \mid (\varphi \land \varphi) \mid ( \varphi \lor \varphi ) \mid (\varphi \to \varphi) \mid (\varphi * \varphi) \mid (\varphi \wand \varphi) $$
If $\circ \in \{ \land, \lor, \to, \top \}$ then it is an additive connective and if $\circ \in \{ *, \wand, \top^* \}$ then it is a multiplicative connective. The set of all formulas is denoted $\mathbb{F}$.
\end{definition}
\begin{definition}[Bunch]
A \emph{bunch} is constructed from the following grammar, where $\varphi \in \mathbb{F}$,
$$\Delta ::= \varphi \mid \varnothing_+ \mid \varnothing_\times \mid (\Delta ; \Delta) \mid (\Delta , \Delta)$$
The symbols $\varnothing_+$ and $\varnothing_\times$ are the additive and multiplicative units respectively, and the symbols $;$ and $,$ are the additive and multiplicative context-formers respectively. A bunch is \emph{basic} if it is a formula, $\varnothing_+$, or $\varnothing_{\times}$ and \emph{complex} otherwise. The set of all bunches is denoted $\mathbb{B}$, the set of complex bunches with additive root context-former by $\mathbb{B}^+$, and the set of complex bunches with multiplicative root context-former by $\mathbb{B}^\times$.
\end{definition}
For two bunches $\Delta, \Delta' \in \mathbb{B}$ if $\Delta'$ is a sub-tree of $\Delta$, it is called a \emph{sub-bunch}.
We may use the standard notation $\Delta(\Delta')$ (despite its slight inpracticality) to denote that $\Delta'$ is a sub-bunch of $\Delta$, in which case $\Delta(\Delta'')$ is the result of replacing the occurrence of $\Delta'$ by $\Delta''$.
If $\delta$ is a sub-bunch of $\Delta$, then the context-former $\circ$ is said to be its principal context-former in $\Delta(\Delta' \circ \delta)$ (and $\Delta(\delta \circ \Delta')$).
\begin{example}\label{ex:abunch} Let $\varphi$, $\psi$ and $\chi$ be formulas, and let $\Delta = (\varphi,(\chi;\varnothing_+));(\psi;(\psi;\varnothing_\times))$. The bunch may be written for example as $\Delta(\varphi,(\chi;\varnothing_+))$ which means that we can have $\Delta(\varphi;\varphi)=(\varphi;\varphi);(\psi;(\psi;\varnothing_\times))$.
\end{example}
\begin{definition}[Bunched Sequent]
A bunched sequent is a pair of a bunch $\Delta$, called the context, and a formula $\varphi$, denoted $\Delta \Rightarrow \varphi$.
\end{definition}
Bunches are intended to be considered up-to \emph{coherent equivalence} $(\equiv)$.
It is the least relation satisfying:
\begin{itemize}
\item Commutative monoid equations for $;$ with unit $\varnothing_+$,
\item Commutative monoid equations for $,$ with unit $\varnothing_{\times}$,
\item Congruence: if $\Delta'\equiv\Delta'' $ then $\Delta(\Delta') \equiv \Delta(\Delta'')$.
\end{itemize}
It will be useful to have a measure on sub-bunches which can identify their distance from the root node.
\begin{definition}[Rank]
If $\Delta'$ is a sub-bunch of $\Delta$, then $\rho(\Delta')$ is the number of alternations of additive and multiplicative context-formers between the principal context-former of $\Delta'$, and the root context-former of $\Delta$.
\end{definition}
Let $\Delta$ be a complex bunch, we use $\Delta' \in \Delta$ to denote that $\Delta'$ is a (proper) top-most sub-bunch; that is, $\Delta$ is a sub-bunch satisfying $\Delta \neq \Delta'$ but $\rho(\Delta') = 0$. \begin{example} Let $\Delta$ be as in Example \ref{ex:abunch}, then $\rho(\varnothing_+)=2$ whereas $\rho(\varnothing_\times)=0$;
hence, $\psi$, $\varnothing_{\times}$ and $(\varphi,(\chi, \varnothing_{\times})) \in\Delta$. Consider the parse-tree of $\Delta$: \[ \xymatrix@R=2mm@C=2mm{
& & & ; \ar@{-}[dll] \ar@{-}[dr] & & &\\
& ,\ar@{-}[dl] \ar@{-}[dr] & & & ; \ar@{-}[dl] \ar@{-}[dr] & &\\
\varphi & & ; \ar@{-}[dl] \ar@{-}[dr] & \psi & & ; \ar@{-}[dl] \ar@{-}[dr]&\\
&\chi & & \varnothing_+ & \psi & & \varnothing_{\times} } \]
Reading upward from $\varnothing_+$ one encounters first $;$ which changes into $,$ and then back to $;$ so the rank is $2$; whereas counting up from $\varnothing_\times$ one only encounters $;$ so the rank is $0$. \end{example}
\subsection{Sequent Calculus} \label{sec:BI_calc}
The proof theory of BI is well-developed including familiar Hilbert, natural deduction, sequent calculi, tableaux systems, and display calculi ~\cite{Pym02,Galmiche05,Brotherston10a}. In the foregoing we restrict attention to the sequent calculus as it more amenable to studying proof-search as computation, having local correctness while enjoying the completeness of analytic proofs.
\begin{definition}[System $\lbi$]
The bunched sequent calculus $\lbi$ is composed of the rules in Figure \ref{fig:lbi}.
\end{definition}
The classification of $\land$ as additive may seem dubious upon reading the $\rrn\land$ rule, but the designation arises from the use of the structural rules; that is, the $\rrn\land$ and $\rrn\to$ rules may be replaced by \emph{additive} variants without loss of generality. The presentation in Figure \ref{fig:lbi} is as in \cite{Pym02} and simply highlights the nature of the additive and multiplicative context-formers. Nonetheless, the choice of rule does affect proof-search behaviours, and the consequences are discussed in more detailed in Section \ref{sec:foc_pol}.
\LBI
\begin{lemma}[Cut-elimination]\label{lem:LBI-cutelim}
If $\varphi$ has a $\lbi$-proof, then it has a $\cut$-free $\lbi$-proof, i.e., a proof with no occurence of the $\cut$ rule.
\end{lemma}
Throughout, unless specified otherwise, we take proof to mean $\cut$-free proof. Moreover, if $\mathsf{L}$ is a sequent calculus we use $\vdash_\mathsf{L }\Delta \Rightarrow \varphi$ to denote that there is an $\mathsf{L}$-proof of $\Delta \Rightarrow \varphi$. Further, if $\mathsf{R}$ is a rule, then we may denote $\mathsf{L+R}$ to denote the sequent calculus combining the rules of $\mathsf{L}$ with $\mathsf{R}$.
The following result, that a generalised version of the axiom is derivable in $\lbi$, will allow for such sequents to be used in proof-construction later on.
\begin{lemma}\label{lem:formulaAx}
For any formula $\varphi$, $\vdash_{\lbi} \varphi \Rightarrow \varphi$.
\end{lemma}
\begin{proof}
Follows from induction on size of $\varphi$.
\qed \end{proof}
The remainder of this section is the meta-theory required to control the structural rules, which pose the main issue to the study of proof-search in BI.
\begin{lemma}\label{lem:weakelim}
The following rules are derivable in $\lbi$, and replacing $\weak$ with them does not affect the completeness of the system.
$$
\infer[\rn{Ax'}]{\Delta;A \Rightarrow A}{}
\quad
\infer[\rrn{{\top^*}'}]{\Delta;\varnothing_\times \Rightarrow \top^*}{}
\quad
\infer[\rrn{\top'}]{\Delta;\varnothing_+ \Rightarrow \top}{}
$$
$$
\infer[\rrn{\ast'}]{(\Delta,\Delta');\Delta'' \Rightarrow \varphi*\psi }{\Delta \Rightarrow \varphi & \Delta' \Rightarrow \psi}
\quad
\infer[\lrn{\wand'}]{\Delta(\Delta',\Delta'',(\Delta''';\varphi \wand \psi)) \Rightarrow \chi }{\Delta' \Rightarrow \varphi & \Delta(\Delta'', \psi) \Rightarrow \chi}
$$
\end{lemma} \begin{proof}
We can construct in $\lbi$ derivations with the same premisses and conclusion as these rules by use of the structural rules. Let $\lbi'$ be $\lbi$ without $\weak$ but with these new rules (retaining also $\rrn\ast,\lrn\wand,\rrn\top^*,\rrn\top,$ and $\rn{Ax}$), then $\weak$ is admissible in $\lbi'$ using standard permutation argument.\qed \end{proof}
One may regard the above modification to $\lbi$ as forming a new calculus, but since all the new rules are derivable it is really a restriction of the calculus, in the sense that all proofs in the new system have equivalent proofs in $\lbi$ differing only by explicitly including instances of weakening.
\subsection{Nested Calculus} \label{sec:BI_nests} Originally, sequents in the calculi for classical and intuitionistic logics (\textsf{LK} and \textsf{LJ}, respectively) were introduced as lists, and a formal \emph{exchange} rule was required to permute elements when needed for a logical rule to be applied~\cite{Gentzen1969}. However, in practice, the exchange rule is often suppressed, and contexts are simply presented as multisets of formulas. This reduces the number of steps/choices being made during proof-search without increasing the complexity of the underlying data structure. Bunches have considerably more structure than lists, but a quotient with respect to coherent equivalence can be made resulting in two-sorted nested multisets; this was first suggested in \cite{Donnelly05}, though never formally realised.
\begin{definition}[Two-sorted Nest]
Nests $(\Gamma)$ are formulas or multisets, ascribed either additive $(\Sigma)$, or multiplicative $(\Pi)$ kind, containing nests of the opposite kind:
\begin{align*}
\Gamma := \Sigma \mid \Pi \qquad \Sigma := \varphi \mid \{\Pi_1,...,\Pi_n\}_+ \qquad \Pi := \varphi \mid \{\Sigma_1,...,\Sigma_n\}_\times
\end{align*}
The constructors are multiset constructors which may be empty in which case the nests are denoted $\varnothing_+$ and $\varnothing_\times$ respectively. No multiset is a singleton; and the set of all nests is denoted $\mathbb{B}\scriptstyle{/\equiv}\textstyle$.
\end{definition}
Given nests $\Lambda$ and $\Gamma$, we write $\Lambda \in \Gamma$ to denote either that $\Lambda =\Gamma$, if $\Gamma$ is a formula, or that $\Lambda$ is an element of the multiset $\Gamma$ otherwise. Furthermore, we write $\Lambda \subseteq \Gamma$ to denote $\forall \gamma \in \mathbb{B}\scriptstyle{/\equiv}\textstyle$ if $\gamma \in \Lambda$ then $\gamma \in \Gamma$.
We will depart from the standard, yet impractical subbunch notation, and adopt a context notation for nests instead.
We write $\Gamma\{\cdot\}_+$ (resp. $\Gamma\{\cdot\}_\times$) for a nest with a hole within one of its additive (resp. multiplicative) multisets.
The notation $\Gamma\{\Lambda\}_+$ (resp. $\Gamma\{\Lambda\}_\times$), denotes that $\Lambda$ is a sub-nest of $\Gamma$ of additive (resp. multiplicative) kind; we may use $\Gamma\{\Lambda\}$ when the kind is not specified. In either case $\Gamma\{\Lambda'\}$ denotes the substitution of $\Lambda$ for $\Lambda'$. A promotion in the syntax tree may be required after a substitution either to handle a singleton or an improper alternation of constructor types.
\begin{example}\label{ex:nests} The following inclusions are valid,
\[
\{\varphi \,, \chi \, \}_\times \in \Big \{ \, \{\varphi \,, \chi \, \}_\times , \psi \Big \}_+ \subseteq \Big\{ \, \{\varphi \,, \chi \, \}_\times , \psi \,, \psi \,, \varnothing_\times \, \Big\}_+ = \Gamma\{\{\varphi \,, \chi \, \}_\times\}_+
\]
It follow that $\Gamma\{\{\varphi\,,\varphi\}_+\}_+ = \{ \, \varphi \,, \varphi \,, \psi \,, \psi \,, \varnothing_\times \, \}_+$. Note the absence of the $\{ \cdot \}_+$ constructor after substitution, this is due to a promotion in the syntax tree to avoid having two nested additive constructors. Similarly, since $\varnothing_\times$ denotes the empty multiset of multiplicative kind, substituting $\chi$ with it gives $\{ \varphi , \psi \,, \psi \,, \varnothing_\times \, \}_+ $; that is, first the improper $\{\varphi, \varnothing_\times\}_\times$ becomes $\{\varphi\}_\times$; then, the resulting singleton $\{\varphi\}_\times$ is promoted to $\varphi$.
\end{example}
Typically we will only be interested in fragments of sub-nests so we have the following abuse of notation, where $\circ \in \{+,\times\}$: \[\Gamma\{\{\Pi_1,...,\Pi_i\}_\circ, \Pi_{i+1},..,\Pi_n\}_\circ := \Gamma\{\Pi_1,...,\Pi_n\}_\circ \]
The notion of rank has a natural analogue in this setting.
\begin{definition}[Depth, Rank]
Let $\circ \in \{+\,,\times\}$ be a nest, we define the depth on $\mathbb{B}$ as follows:
$$
\delta(\varphi) := 0 \qquad \delta(\{\Gamma_1,...,\Gamma_n\}_\circ) := \max\{\delta(\Gamma_1),...,\delta(\Gamma_n)\}+1 $$
\end{definition}
The equivalence of the two presentations, bunches and nests, follows from a moral (in the sense that bunches are intended to be considered modulo congruence) inverse between a \emph{nestifying} function $\eta$ and a \emph{bunching} function $\beta$. The transformation $\beta$ is simply going from a tree with arbitrary branching to a binary one, and $\eta$ is the reverse.
\begin{definition}[Canonical Translation]
The canonical translation $\eta:\mathbb{B} \to \mathbb{B}\scriptstyle{/\equiv}\textstyle$ is defined recursively as follows,
$$
\eta(\Delta) :=
\begin{cases}
\Delta & \text{if } \Delta \in \mathbb{F} \cup \{\varnothing_+,\varnothing_\times\} \\
\{\eta(\Delta') \in \mathbb{B}\scriptstyle{/\equiv}\textstyle \mid \rho(\Delta')=1 \text{ and } \Delta' \in \mathbb{B}^\times \}_+ & \text{if } \Delta \in \mathbb{B}^+ \\
\{\eta(\Delta') \in \mathbb{B}\scriptstyle{/\equiv}\textstyle \mid \rho(\Delta')=1 \text{ and } \Delta' \in \mathbb{B}^+ \}_\times & \text{if } \Delta \in \mathbb{B}^\times
\end{cases}
$$
The canonical translation $\beta:\mathbb{B}\scriptstyle{/\equiv}\textstyle \to \mathbb{B}$ is defined recursively as follows,
$$
\beta(\Gamma) :=
\begin{cases}
\Gamma & \text{ if } \Gamma \in \mathbb{F} \cup \{\varnothing_+, \varnothing_\times\} \\
\beta(\Pi_1);(\beta(\Pi_2);...) & \text{ if } \Gamma = \{\Pi_1, \Pi_2,...\}_+ \\
\beta(\Sigma_1),(\beta(\Sigma_2),...) & \text{ if } \Gamma = \{\Sigma_1, \Sigma_2,...\}_\times
\end{cases}
$$
\end{definition}
\begin{example} Applying $\eta$ to the bunch in Example \ref{ex:abunch} gives the nest in Example~\ref{ex:nests}:
$$
\xymatrix@R=2mm@C=2mm{
& & & + \ar@{-}[dll] \ar@{-}[dl] \ar@{-}[dr] \ar@{-}[drr] & & \\
&\times \ar@{-}[dl] \ar@{-}[dr] &\psi& & \psi& \varnothing_\times \\
\psi& &\chi & & &
}
$$
\end{example}
\begin{lemma} \label{lem:transinverse}
The translations are inverses up-to congruence; that is,
\begin{enumerate}
\item if $\Delta \in \mathbb{B}$ then $(\beta \circ \eta)(\Delta) \equiv \Delta$;
\item if $\Gamma \in \mathbb{B}\scriptstyle{/\equiv}\textstyle$ then $(\eta \circ \beta)(\Gamma) \equiv \Gamma$;
\item let $\Delta, \Delta' \in \mathbb{B}$, then $\Delta \equiv \Delta'$ if and only if $\eta(\Delta) = \eta(\Delta')$.
\end{enumerate}
\end{lemma} \begin{proof}
The first two statements follow by induction on the depth (either for bunches or nests), where one must take care to consider the case of a context consisting entirely of units. The third statement employs the first in the forward direction, and proceeds by induction on depth in the reverse direction. \qed \end{proof} \begin{definition}[System $\eta\lbi$]
The nested sequent calculus $\eta\lbi$ is composed of the rules in Figure~\ref{fig:taulbi}, where the metavariables denote possibly empty nests. \end{definition} Observe the use of metavariable $\Gamma'$ instead of $\Pi$ (resp. $\Sigma$) as sub-contexts in Figure~\ref{fig:taulbi}. This allows classes of inferences such as \[ \infer[\rrn\ast]{\{\Sigma_0,...,\Sigma_n \}_\times \Rightarrow \varphi *\psi}{
\{\Sigma_0,...,\Sigma_i\}_\times \Rightarrow \varphi
&
\{\Sigma_{i+1},...,\Sigma_n \}_\times \Rightarrow \varphi
} \]
to be captured by a single figure. In practice it implements the abuse of notation given above: $$\{\{\Sigma_0,...,\Sigma_i\}_\times, \{\Sigma_{i+1},...,\Sigma_n\}_\times \}_\times \Rightarrow \varphi *\psi$$
\etalbi
This system is a new and very convenient presentation of $\lbi$, not \emph{per se} a development of the proof theory for the logic. \begin{lemma}[Soundness and Completeness of $\eta \lbi$] \label{lem:etalbi}
Systems $\lbi$ and $\eta\lbi$ are equivalent:
\begin{enumerate}
\item[] Soundness: If $\vdash_{\eta\lbi} \Gamma \Rightarrow \varphi$ then $\vdash_{\lbi} \beta(\Gamma) \Rightarrow \varphi$;
\item[] Completeness: If $\vdash_{\lbi} \Delta \Rightarrow \varphi$ then $\vdash_{\eta\lbi} \eta(\Delta) \Rightarrow \varphi$.
\end{enumerate} \end{lemma} \begin{proof} Each claim follows by induction on the context, appealing to Lemma \ref{lem:transinverse} to organise the data structure for the induction hypothesis, without loss of generality. \end{proof} \begin{example}\label{ex:nestedproof}
The following is a proof in $\eta \lbi$.
\[\scalebox{.9}{$
\infer[\rrn\wand]{\varnothing_\times \Rightarrow (A * (B \land C)) \wand ((A * B)\land (A * C))}{
\infer[\lrn\ast]{A * (B \land C) \Rightarrow (A * B)\land (A * C)}{
\infer[\rrn\land]{\{A , (B \land C) \}_\times \Rightarrow (A * B)\land (A * C)}{
\infer[\lrn\land]{\{A , (B \land C) \}_\times \Rightarrow A * B}{
\infer[\rrn\ast]{\{A , \{B , C\}_+ \}_\times \Rightarrow A * B}{
\infer[\rn{Ax}]{A \Rightarrow A}{}
&
\infer[\rn{Ax}]{\{B, C\}_+ \Rightarrow B }{}
}
}
&
\infer[\rrn\ast]{\{A , (B \land C) \}_\times \Rightarrow A * C}{
\infer[\rn{Ax}]{A \Rightarrow A}{}
&
\infer[\lrn\land]{B\land C \Rightarrow C}{
\infer[\rn{Ax}]{\{B, C\}_+ \Rightarrow C}{}
}
}
}
}
}
$}\] \end{example} We expect no obvious difficulty in studying focused proof-search with bunches instead of nested multisets; the design choice is simply to reduce the complexity of the argument by pushing all uses of exchange (\rn{E}) to Lemma~\ref{lem:etalbi}, rather than tackle it at the same time as focusing itself. In particular, working without the nested system would mean working with a weaker notion of focusing since the exchange rule must then be permissible during both focused and unfocused phases of reduction.
\section{A Focused System} \label{sec:foc} At no point in this section will we refer to bunches, thus the variable $\Delta$, so far reserved for elements of $\mathbb{B}$, is re-appropriated as an alternative to $\Gamma$.
\subsection{Polarisation} \label{sec:foc_pol}
Polarity in the focusing principle is determined by the invariance of provability under application of a rule, that is, by the proof rules themselves. One way the distinction between positive and negative connectives is apparent is when their rule behave either \emph{synchronously} or \emph{asynchronously}. For example, the $\rrn\ast$ and $\lrn\wand$ highlight the {synchronous} behaviour of the multiplicative connectives since the structure of the context affects the applicability of the rule. Displaying such a synchronous behaviour on the left makes $\wand$ a negative connective, while having it on the right makes $\ast$ a positive connective.
Another way to characterise the polarity of a connective is the study of the inveribility properties of the corresponding rules.
For example, consider the inverses of the $\lrn\lor$ rule,
\[\scalebox{.9}{$
\infer[\text{$\lrn[1]{\lor^{inv}}$}]{\Gamma\{\varphi\} \Rightarrow \chi}{\Gamma\{\varphi\lor\psi\} \Rightarrow \chi }
\qquad\qquad
\infer[\text{$\lrn[2]{\lor^{inv}}$}]{\Gamma\{\psi\} \Rightarrow \chi}{\Gamma\{\varphi\lor\psi\} \Rightarrow \chi}
$}\]
They are derivable in $\lbi$ with $\cut$ (below -- the left branch being closed using Lemma~\ref{lem:formulaAx}) and therefore admissible in $\lbi$ without $\cut$ (by Lemma~\ref{lem:LBI-cutelim}).
\[\scalebox{.9}{$
\infer[\cut]{\Gamma\{\varphi\} \Rightarrow \chi}{
\infer[\rrn\lor]{\varphi \Rightarrow \varphi\lor\psi}{
\varphi \Rightarrow \varphi
}
&
\Gamma\{\varphi\lor\psi\} \Rightarrow \chi
}
\qquad
\infer[\cut]{\Gamma\{\psi\} \Rightarrow \chi}{
\infer[\rrn\lor]{\psi \Rightarrow \varphi\lor\psi}{
\psi \Rightarrow \psi
}
&
\Gamma\{\varphi\lor\psi\} \Rightarrow \chi
}
$}\]
This means that provability is invariant in general upon application of $\lrn\lor$ since it can always be reverted if needed, as follows
\[\scalebox{.9}{$
\infer[\lrn\lor]{\Gamma\{\varphi\lor\psi\} \Rightarrow \chi}{
\infer[\text{$\lrn[1]{\lor^{inv}}$}]{\Gamma\{\varphi\} \Rightarrow \chi}{
\Gamma\{\varphi\lor\psi\} \Rightarrow \chi
}
&
\infer[\text{$\lrn[2]{\lor^{inv}}$}]{\Gamma\{\psi\} \Rightarrow \chi}{
\Gamma\{\varphi\lor\psi\} \Rightarrow \chi
}
}
$}
\]
Note however that dual connectives do not necessarily have dual behaviours in terms of provability invariance, on the left and on the right. For example, consider all the possible rules for $\land$, of which some qualify as positive and others as positive.
\[\scalebox{.9}{$
\begin{array}{c@{\quad}c}
\infer[\text{$\lrn[1]{\land^-}$}]{\Gamma\{\varphi\land \psi\} \Rightarrow \chi}{\Gamma\{\varphi\} \Rightarrow \chi}
\quad
\infer[\text{$\lrn[2]{\land^-}$}]{\Gamma\{\varphi\land \psi\} \Rightarrow \chi}{\Gamma\{\psi\} \Rightarrow \chi}
&
\infer[\rrn{\land^- }]{\Gamma \Rightarrow \varphi \land \psi}{\Gamma \Rightarrow \varphi &\Gamma \Rightarrow \psi }
\\[2ex]
\infer[\lrn{\land^+}]{\Gamma\{\varphi\land\psi\} \Rightarrow\chi}{\Gamma\{\{\varphi,\psi\}_+\} \Rightarrow\chi}
&
\infer[\rrn{\land^+}]{\{\Gamma, \Delta \}_+ \Rightarrow \varphi \land \psi}{\Gamma \Rightarrow \varphi & \Gamma\{\{\varphi\land\psi\}_+\} \Rightarrow\chi}
\end{array}
$}\]
All of these rules are sound, and replacing the conjunction rules in $\lbi$ with any pair of a left and right rule will result in a sound and complete system. Indeed, the rules are inter-derivable when the structural rules are present, but otherwise they can be paired to form two sets of rules which have essentially different proof-search behaviours. That is, the rules in the top-row make $\land$ negative while the bottom row make $\land$ positive. Each conjunction also comes with an associated unit, that is, $\top^-$ for negative conjunctio and $\top^+$ for positive conjunction.
We choose to add all of them to our system in order to have access to those different proof search behaviours at will.
Finally, the polarity of the propositional letters can be assigned arbitrarily as long as only once for each.
\begin{definition}[Polarised Syntax]
Let $\mathsf{P}^+ \sqcup \mathsf{P}^-$ be a partition of $\mathsf{P}$, and let $A^+ \in \mathsf{P}^+$ and $A^- \in \mathsf{P}^-$, then the polarised formulas are defined by the following grammar,
\begin{align*}
P,Q &::= L \mid P \lor Q \, \, \mid \, P * Q \, \, \, \, \mid P \land^+ Q \, \mid \top^+ \mid \top^* \mid \bot &L ::= \downshift N \mid A^+ \\
N,M &::= R \mid P \to N \mid P \wand N \mid N \land^- M \mid \top^- &R ::= \upshift P \mid A^-
\end{align*}
The set of positive formulas $P$ is denoted $\mathbb{F}^+$; the set of negative formulas $N$ is denoted $\mathbb{F}^-$; and the set of all polarised formulas is denoted $\mathbb{F}^\pm$. The sub-classifications $L$ and $R$ are left-neutral and right-neutral formulas respectfully. \end{definition} The shift operators have no logical meaning; they simply mediate the exchange of polarity, and thus the \emph{shifting} into a new phase of proof-search. Consequently, to reduces cases in subsequent proofs, we will consider formulas of the form $\upshift \downshift N$ and $\downshift \upshift P$, but not $ \downshift \upshift \downshift N$, $\downshift \upshift \downshift \upshift P$, etc.
\begin{definition}[Depolarisation]
Let $\circ \in \{\lor\,, *\,, \to\,, \wand\}$, and let $A^+ \in \mathsf{P}^+$ and $A^- \in \mathsf{P}^-$, then the depolarisation function $\floor{\cdot}:\mathbb{F}^\pm \to \mathbb{F}$ is defined as follows:
\[\begin{array}{l}
\floor{A^+} := \floor{A^-} := A
\quad
\floor{\upshift \varphi} := \floor{\downshift \varphi} := \floor{\varphi}
\quad
\floor{\bot} := \bot
\quad\floor{\top^*} := \top^*
\\
\floor{\top^+} := \floor{\top^-} := \top
\quad
\floor{\varphi \circ \psi} := \floor{\varphi} \circ \floor{\psi}
\quad
\floor{\varphi \land^+ \psi} := \floor{\varphi \land^- \psi} := \floor{\varphi} \land \floor{\psi}
\end{array}\] \end{definition} Since proof-search is controlled by polarity, the construction of sequents in the focused system must be handled carefully to avoid ambiguity. \begin{definition}[Polarised Sequents]
\emph{Positive} and \emph{neutral} nests, denoted by $\Gamma$ and $\vec{\Gamma}$ resp., are defined according to the following grammars \[\begin{array}{l@{\ :=\ }l@{\ \mid\ }l@{\qquad}l@{\ :=\ }l@{\qquad}l@{\ :=\ }l@{\quad}}
\Gamma& \Sigma & \Pi
&
\Sigma & P \mid \{\Pi_1,...,\Pi_n\}_+
&
\Pi& P \mid \{\Sigma_1,...,\Sigma_n\}_\times
\\
\vec{\Gamma}&\vec{\Sigma} & \vec{\Pi}
&
\vec{\Sigma} & L \mid \{\vec{\Pi}_1,...,\vec{\Pi}_n\}_+
&
\vec{\Pi}& L \mid \{\vec{\Sigma}_1,...,\vec{\Sigma}_n\}_\times
\end{array}\]
A pair of a polarised nest and a polarised formula is a \emph{polarised sequent} if it falls into one of the following cases
$$
\Gamma \Rightarrow N \quad \mid \quad \vec{\Gamma} \Rightarrow \foc{P} \quad \mid \quad \vec{\Gamma}\{\foc{N}\} \Rightarrow R
$$ \end{definition} The decoration $\foc{\varphi}$ indicates that the formula is in focus; that is, it is a positive formula on the right, or a negative formula on the left. Of the three possible cases for well-formed polarised sequents, the first may be called \emph{unfocused}, with the particular case of being \emph{neutral} when of the form $\vec{\Gamma} \Rightarrow R$; and the latter two may be called \emph{focused}.
\begin{definition}[Depolarised Nest]
The depolarisation map extends to polarised nests $\floor{\cdot}:\mathbb{B}\scriptstyle{/\equiv}\textstyle^\pm \to \mathbb{B}\scriptstyle{/\equiv}\textstyle$ as follows:
$$\floor{\{\Pi_1,...,\Pi_n\}_+} = \{\floor{\Pi_1},...,\floor{\Pi_n} \}_+ \qquad \floor{\{\Sigma_1,...,\Sigma_n\}_\times} = \{\floor{\Sigma_1},...,\floor{\Sigma_n} \}_\times
$$
\end{definition}
\subsection{Focused Calculus} \label{sec:foc_calc} We may now give the focused system. That is, the operational semantics for focused proof-search in $\lbi$. All the rules, with the exception of $\rn P$ and $\rn N$, are polarised versions of the rules from $\eta\lbi$.
\focbi
\begin{definition}[System $\fbi$]
The focused system $\fbi$ is composed of the rules on Figure \ref{fig:fbi}. \end{definition}
Note the absence of a $\cut$-rule, this is because the above system is intended to encapsulate precisely \emph{focused} proof-search. Below we show that a $\cut$-rule is indeed admissible, but proofs in $\fbi+\cut$ are not necessarily focused themselves. Here the distinction between the methodologies for establishing the focusing principle becomes present since one may show completeness without leaving $\fbi$ by a permutation argument instead of a $\cut$-elimination one.
The $\rn P$ and $\rn N$ rules will allow us to move a formula from one side to another during the proof of the completeness of $\fbi+\cut$ (Lemma~\ref{lem:compFBIcut}).
The depolarised version are not directly present in $\lbi$, but are derivable in $\lbi$ (Lemma \ref{lem:formulaAx}). However, the way they are focused renders them not provable in $\fbi$ because it forces one to begin with a potentially \emph{bad} choice; for example, $A \lor B \Rightarrow A \lor B$ has no proof beginning with $\rrn\lor$. In practice, they are a feature rather than a bug since they allow one to terminate proof-search early, without unnecessary further expansion of the axiom.
In related works, such as~\cite{Chaudhuri16a,Chaudhuri16b}, the analogous rules are eliminated by initially working with a weaker notion of focused proof-search, and it is reasonable to suppose that the same may be true for BI. We leave this to future investigation.
Note also that, although it is perhaps proof-theoretically displeasing to incorporate weakening into the operational rules as in $\lrn{\wand'}$ and $\rrn{\ast'}$, it has good computational behaviour during focused proof-search since the reduction of $\varphi \wand \psi$ can only arise out of an explicit choice made earlier in the computation.
Soundness follows immediately from the depolarisation map; that is, the interpretation of polarised sequents as nested sequents, and hence proofs in $\fbi$ actually are focused proofs in $\eta\lbi$.
\begin{theorem}[Soundness of $\fbi$]
Let $\Gamma$ be a polarised nest and $N$ a negative formula.
If $\vdash_{\fbi} \Gamma \Rightarrow N$ then $\vdash_{\eta\lbi} \floor{\Gamma} \Rightarrow \floor{N}$ \end{theorem}
\begin{proof}
Every rule in $\fbi$ except the shift rules, as well as the $\rn P$ and $\rn N$ axioms, become a rule in $\eta \bi$ when the antecedent(s) and consequent are depolarised. Instance of the shift rule can be ignored since the depolarised versions of the consequent and antecedents are the same. Finally, the depolarised versions of $\rn P$ and $\rn N$ follow from Lemma \ref{lem:formulaAx} with the use of some weakening.\qed \end{proof}
\begin{example}
Consider the following proof in $\fbi$, we suppose here that propositional letters $A$ and $C$ are negative, but $B$ is positive.
\[\scalebox{.9}{$
\infer[\rrn\wand]{\varnothing_\times \Rightarrow (\downshift A * \downshift(\upshift B\land^- C)) \wand (\upshift (\downshift A * B)\land^- \upshift (\downshift A * \downshift C))}{
\infer[\lrn\ast]{\downshift A * \downshift(\upshift B\land^- C) \Rightarrow \upshift (\downshift A * B)\land^- \upshift (\downshift A * \downshift C)}{
\infer[\rrn{\land^-}]{\{ \downshift A , \downshift(\upshift B\land^- C) \}_\times \Rightarrow \upshift (\downshift A * B)\land^- \upshift (\downshift A * \downshift C)}{
\infer[\lrn\upshift\ {\color{blue}(1)}]{\{ \downshift A , \downshift(\upshift B\land^- C) \}_\times \Rightarrow \upshift (\downshift A * B)}{
\infer[\text{$\lrn[1]{\land^-}$}]{\{ \downshift A , \foc{\upshift B\land^- C} \}_\times \Rightarrow \upshift (\downshift A * B)}{
\infer[\lrn\upshift]{\{ \downshift A , \foc{\upshift B}\}_\times \Rightarrow \upshift (\downshift A * B)}{
\infer[\rrn\upshift]{\{ \downshift A , \ B\}_\times \Rightarrow \upshift (\downshift A * B) }{
\infer[\rrn\ast]{\{ \downshift A , B\}_\times \Rightarrow \foc{\downshift A * B} }{
\infer[\rrn\downshift]{\downshift A \Rightarrow \foc{\downshift A }}{
\infer[\lrn\downshift]{\downshift A \Rightarrow A }{
\infer[\rn{Ax^-}]{\foc{A} \Rightarrow A }{}
}
}
&
\infer[\rn{Ax^+}]{B \Rightarrow \foc{B}}{}
}
}
}
}
}
&
\infer[\rrn\upshift\ {\color{blue}(2)}]{\{ \downshift A , \downshift(\upshift B\land^- C) \}_\times \Rightarrow \upshift (\downshift A * \downshift C)}{
\infer[\rrn\ast]{\{ \downshift A , \downshift(\upshift B\land^- C) \}_\times \Rightarrow \foc{\downshift A * \downshift C}}{
\infer[\rrn\downshift]{\downshift A \Rightarrow \foc{\downshift A }}{
\infer[\lrn\downshift]{\downshift A \Rightarrow A }{
\infer[\rn{Ax^-}]{\foc{A} \Rightarrow A }{}
}
}
&
\infer[\rrn\downshift]{\downshift(\upshift B\land^- C) \}_\times \Rightarrow \foc{\downshift C }}{
\infer[\lrn\downshift]{\downshift(\upshift B\land^- C) \Rightarrow C}{
\infer[\text{$\lrn[2]{\land^-}$}]{ \foc{\upshift B\land^- C} \Rightarrow C}{
\infer[\rn{Ax^-}]{ \foc{C} \Rightarrow C }{}
}
}
}
}
}
}
}
}
$}\]
It is a focused version of the proof given in Example~\ref{ex:nestedproof}. Observe that the only non-deterministic choices are which formula to focus on, such as in steps (1) and (2), where different choices have been made for the sake of demonstration. The point of focusing is that \emph{only} at such points do choices that affect termination occur. The assignment of polarity to the propositional letters is what forced the shape of the proof; for example, if $B$ had been negative the above would not have been well-formed. This phenomenon is standarly observed in focused systems (e.g.~\cite{Chaudhuri06}). \end{example}
We now introduce the tool which will allow us to show that if there is a proof of a sequent (\emph{a priori} unstructured), then there is necessarily a focused one. \begin{definition}
All instances of the following rule where the sequents are well-formed are instances of $\cut$, where $\vec{\varphi}$ denotes that $\varphi$ is possibly prenexed with an additional shift
\[
\infer[\cut]{\Gamma\{\Delta\} \Rightarrow \chi}{\Delta \Rightarrow \varphi & \Gamma \{\vec{\varphi}\} \Rightarrow \chi}
\]
\end{definition} Admissibility follows from the usual argument, but within the focused system; that is, through the upward permutation of cuts until they are eliminated in the axioms or are reduced in some other measure. \begin{definition}[Good and Bad Cuts]
Let $\mathcal{D}$ be a $\fbi+\cut$ proof, a cut is a quadruple $\langle \mathcal{L}, \mathcal{R}, \mathcal{C}, \varphi \rangle$ where $\mathcal{L}$ and $\mathcal{R}$ are the premises to a $\cut$ rule, concluding $\mathcal{C}$ in $\D$, and $\varphi$ is the $\cut$-formula. They are classified as follows:
\begin{enumerate}
\item[] \textbf{Good} - If $\varphi$ is principal in both $\mathcal{L}$ and $\mathcal{R}$.
\item[] \textbf{Bad} - If $\varphi$ is not principal in one of $\mathcal{L}$ and $\mathcal{R}$.
\begin{enumerate}
\item[] Type 1: If $\varphi$ is not principal in $\mathcal{L}$.
\item[] Type 2: If $\varphi$ is not principal in $\mathcal{R}$.
\end{enumerate}
\end{enumerate} \end{definition} \begin{definition}[Cut Ordering]
The $\cut$-rank of a cut $\langle \mathcal{L}, \mathcal{R}, \mathcal{C}, \varphi \rangle$ in a proof is the triple $\langle \cut$-complexity, $\cut$-duplicity, $\cut$-level$\rangle$, where the $\cut$-complexity is the size of $\varphi$, the $\cut$-duplicity is the number of contraction instances above the cut, the $\cut$-level is the sum of the heights of the sub-proofs concluding $\mathcal{L}$ and $\mathcal{R}$.
Let $\D$ and $\D'$ be two $\fbi+\cut$ proofs, let $\sigma$ and $\sigma'$ denote their multiset of cuts respectively. Proofs are ordered by $\D \prec\D' \iff \sigma < \sigma'$, where $<$ is the multiset ordering derived from the lexicographic ordering on $\cut$-rank. \end{definition} It follows from a result in \cite{Dershowitz79} that the ordering on proofs is a well-order, since the ordering on cuts is a well-order. \begin{lemma}[Good Cuts Elimination]\label{lem:goodcutelim}
Let $\D$ be a $\fbi+\cut$ proof of $S$; there is a $\fbi+\cut$ proof $\D'$ of $S$ containing no good cuts such that $\D' \preceq \D$. \end{lemma} \begin{proof}
Let $\D$ be as in hypothesis, if it contains no good cuts then $\D =\D'$ gives the desired proof. Otherwise, there is at least one good cut $\langle \mathcal{L}, \mathcal{R}, \mathcal{C}, \varphi \rangle$. Let $\partial$ be the sub-proof in $\D$ concluding $\mathcal{C}$, then there is a transformation $\partial \mapsto \partial'$ where $\partial'$ is a $\fbi+\cut$ proof of $S$ with $\partial' \prec \partial$ such that the multiset of good cuts in $\partial'$ is smaller (with respect to $\prec$) than the multiset of good cuts in $\partial$. Since $\prec$ is a well-order indefinitely replacing $\partial$ with $\partial'$ in $\D$ for various cuts yields the desired $\D'$.
The key step is that a cut of a certain $\cut$-complexity is replaced by cuts of lower $\cut$-complexity, possibly increasing the $\cut$-duplicity or $\cut$-level of other cuts in the proof, but not modifying their complexity.
\scalebox{.85}{$
\infer[\cut]{\vec{\Gamma}\{\{\vec{\Gamma}' , A^+\}_+\} \Rightarrow \foc{A^+}}{
\infer[\rn{Ax^+}]{ \{\vec{\Gamma}', A^+\}_+ \Rightarrow \foc{A^+} }{}
&
\vec{\Gamma}\{A^+\} \Rightarrow \foc{A^+}
}
$}
$\quad\mapsto\quad$
\scalebox{.85}{$
\infer=[\weak]{\vec{\Gamma}\{\{\vec{\Gamma}' , A^+\}_+\} \Rightarrow \foc{A^+}}{\vec{\Gamma}\{A^+\} \Rightarrow \foc{A^+} }
$}\\
\scalebox{.85}{$
\infer[\cut]{\vec\Gamma\{\vec{\Delta},\vec{\Delta}',\{\vec{\Delta}'',\vec{\Delta}'''\}_+\}_\times \Rightarrow R}{
\infer[\rrn\wand]{\vec{\Delta}''' \Rightarrow P\wand N}{
\{\vec{\Delta}''', P\}_\times \Rightarrow N
}
&
\infer[\lrn\wand]{\vec\Gamma\{\vec{\Delta},\vec{\Delta}',\{\vec{\Delta}'',\foc{P\wand N}\}_+\}_\times \Rightarrow R}{
\vec{\Delta} \Rightarrow \foc{P}
&
\Gamma\{\vec{\Delta}', \foc{N}\}_\times \Rightarrow R
}
}
$}
\hspace*{2.5cm}
$\mapsto\quad$
\scalebox{.85}{$
\infer=[\weak]{\vec\Gamma\{\vec{\Delta},\vec{\Delta}',\{\vec{\Delta}'',\vec{\Delta}'''\}_+\}_\times \Rightarrow R}{
\infer[\cut]{\vec\Gamma\{\vec{\Delta},\vec{\Delta}',\vec{\Delta}''\}_\times \Rightarrow R}{
\vec{\Delta} \Rightarrow \foc{P}
&
\infer[\cut]{\vec\Gamma\{\vec{\Delta},\vec{\Delta}'',P\}_\times \Rightarrow R}{
\{\vec{\Delta}'', P\}_\times \Rightarrow N
&
\vec\Gamma\{\vec{\Delta}', \foc{N}\}_\times \Rightarrow R
}
}
}
$}\\
We denote by a double-line the fact that we do not actually use a weakening, but only the fact that it is admissible in $\fbi$ by construction (Lemma~\ref{lem:weakelim}). \qed
\end{proof}
\begin{lemma}[Bad Cuts Elimination]\label{lem:badcutelim}
Let $\D$ be a $\fbi+\cut$ proof of $S$ that contains only one cut which is bad, then there is a $\fbi+\cut$ proof $\D'$ of $S$ such that $\D' \prec \D$. \end{lemma} \begin{proof}
Without loss of generality suppose the cut is the last inference in the proof, then it may be replaced by other cuts whose $\cut$-level or $\cut$-duplicity is smaller, but with same $\cut$-complexity.
First we consider bad cuts when $\mathcal{L}$ and $\mathcal{R}$ are both axioms. There are no Type 1 bad cuts on axioms as the formula is always principal, meanwhile the Type $2$ bad cuts can trivially be permuted upwards or ignored; for example,\\
\scalebox{.85}{
$
\infer[\cut]{\vec\Gamma\{\vec\Delta,\vec\Delta',\{\vec\Delta'', \vec\Delta''', A_+,\foc{P\wand N}\}_+\}_\times \Rightarrow R}{
\infer[\rn{Ax^+}]{\{\vec\Delta''',A_+\}_+ \Rightarrow \foc{A_+}}{}
&
\infer[\lrn\wand]{\vec\Gamma\{\vec\Delta,\vec\Delta',\{\vec\Delta'', A_+,\foc{P\wand N}\}_+\}_\times \Rightarrow R}{
\vec\Delta \Rightarrow \foc P
&
\vec\Gamma\{\vec\Delta', \foc N\}_\times \Rightarrow R
}
}
$
}\\
\hspace*{2.5cm}
$\mapsto\quad$
\scalebox{.85}{
$
\infer=[\weak]{\vec\Gamma\{\vec\Delta,\vec\Delta',\{\vec\Delta'', \vec\Delta''', A_+,\foc{P\wand N}\}_+\}_\times \Rightarrow R}{
\infer[\lrn\wand]{\vec\Gamma\{\vec\Delta,\vec\Delta',\{\vec\Delta'', A_+,\foc{P\wand N}\}_+\}_\times \Rightarrow R}{
\vec\Delta \Rightarrow \foc P
&
\vec\Gamma\{\vec\Delta', \foc N\}_\times \Rightarrow R
} }
$
}
Here again we are using an appropriate version of Lemma~\ref{lem:weakelim}.
For the remaining cases the cuts are commutative in the sense that they may be permuted upward thereby reducing the $\cut$-level. An example is given below.\\
\scalebox{.85}{
$
\infer[\cut]{\vec\Gamma\{\vec\Delta\{\foc{N_1 \land^- N_2}\}\} \Rightarrow R}{
\infer[\text{$\lrn[1]{\land^-}$}]{\vec\Delta\{\foc{N_1 \land^- N_2}\} \Rightarrow M}{
\vec\Delta\{\foc{N_1}\} \Rightarrow M
}
&
\vec\Gamma\{ M\} \Rightarrow R
}
$
}
$\mapsto\quad$
\scalebox{.85}{
$
\infer[\text{$\lrn[1]{\land^-}$}]{\Gamma\{\Delta\{\foc{N_1 \land^- N_2}\}\} \Rightarrow R}{
\infer[\cut]{\Gamma\{\Delta\{\foc{N_1}\}\} \Rightarrow R}{
\Delta\{\foc{N_1}\} \Rightarrow M
&
\Gamma\{ M\} \Rightarrow R
}
}
$
}
The exceptional case is the interaction with contraction where the cut is replaced by cuts of possibly equal $\cut$-level, but $\cut$-duplicity decreases.\\
\scalebox{.85}{
$
\infer[\cut]{\vec\Gamma\{\vec{\Delta}\{\vec\Delta'\}\} \Rightarrow R}{
\vec\Delta' \Rightarrow \foc{L}
&
\infer[\cont]{\vec\Gamma\{\vec{\Delta}\{L\}\} \Rightarrow R}{
\vec\Gamma\{\{\vec{\Delta}\{L\}, \vec{\Delta}\{L\}\}_+\} \Rightarrow R
}
}
$
}
\hspace*{2.5cm}
$\quad\mapsto\quad$
\scalebox{.85}{
$
\infer[\cont]{\vec\Gamma\{\vec{\Delta}\{\vec\Delta'\}\} \Rightarrow R}{
\infer[\cut]{\vec\Gamma\{\{\vec{\Delta}\{\vec\Delta'\}, \vec{\Delta}\{\vec\Delta'\}\}_+\} \Rightarrow R }{
\vec\Delta' \Rightarrow \foc L
&
\infer[\cut]{\vec\Gamma\{\{\vec{\Delta}\{\vec\Delta'\}, \vec{\Delta}\{L\}\}_+\} \Rightarrow R}{
\vec\Delta' \Rightarrow \foc L
&
\vec\Gamma\{\{\vec{\Delta}\{L\}, \vec{\Delta}\{L\}\}_+\} \Rightarrow R
}
}
}
$
}
\qed\end{proof}
\begin{theorem}[Cut-elimination in $\fbi$]\label{lem:fbicutadmi}
Let $\Gamma$ be a positive nest and $N$ a negative formula. Then,
$\vdash_{\fbi} \Gamma \Rightarrow N$ if and only if $ \vdash_{\fbi+\cut} \Gamma \Rightarrow N$. \end{theorem} \begin{proof}
($\Rightarrow$) Trivial as any $\fbi$-proof is a $\fbi+\cut$-proof. ($\Leftarrow$) Let $\D$ be a $\fbi+\cut$-proof of $\Gamma \Rightarrow N$, if it has no cuts then it is a $\fbi$-proof so we are done. Otherwise, there is at least one $\cut$, and we proceed by well-founded induction on the ordering of proofs and sub-proofs of $\D$ with respect to $\prec$.
\textbf{Base Case.} Assume $\D$ is minimal with respect to $\prec$ with at least one cut; without loss of generality, by Lemma \ref{lem:goodcutelim}, assume the cut is bad. It follows from Lemma \ref{lem:badcutelim} that there is a proof strictly smaller in $\prec$-ordering, but this proof must be $\cut$-free as $\D$ is minimal.
\textbf{Inductive Step.} Let $\D$ be as in the hypothesis, then by Lemma \ref{lem:goodcutelim} there is a proof $\partial$ of $\Gamma \Rightarrow N$ containing no good cuts such that $\D' \preceq \D$. Either $\D'$ is $\cut$-free and we are done, or it contains bad cuts. Consider the topmost cut, and denote the sub-proof by $\partial$, it follows from Lemma \ref{lem:badcutelim} that there is a proof $\partial'$ of the same sequent such that $\partial' \prec \partial$. Hence, by inductive hypothesis, there is a $\cut$-free proof the sequent and replacing $\partial$ by this proof in $\D$ gives a proof of $\Gamma \Rightarrow \varphi$ strictly smaller in $\prec$-ordering, thus by inductive hypothesis there is a $\cut$-free proof as required.
\qed \end{proof}
\subsection{Completeness of $\fbi$} \label{sec:foc_comp} The completeness theorem of the focused system, the operational semantics, is with respect to an interpretation (i.e. a polarisation). Indeed, any polarisation may be considered; for example, both $(\downshift A^-*B^+) \land^+ \downshift A^-$ and $\downshift (A^+*\downshift B^-) \land^+ A^+$ are correct polarised versions of the formulas $(A*B) \land A$. Taking arbitrary $\varphi$ the process is as follows: first, fix a polarised syntax (i.e. a partition of the propositional letters into positive and negative sets), then assign a polarity to $\varphi$ with the following steps: \begin{itemize}
\item If $\varphi$ is a propositional atom, it must be polarised by default;
\item If $\varphi = \top$, then \emph{choose} polarisation $\top^+$ or $\top^-$;
\item If $\varphi = \psi_1 \land \psi_2$, first polarise $\psi_1$ and $\psi_2$, then \emph{choose} an additive conjunction and combine accordingly, using shifts to ensure the formula is well-formed;
\item If $\varphi = \psi_1 \circ \psi_2$ where $\circ \in\{*, \wand, \to, \lor\}$, then polarise $\psi_1$ and $\psi_2$ and combine with $\circ$ accordingly, using shifts where necessary. \end{itemize} \begin{example}
Suppose $A$ is negative and $B$ is positive, then $(A*B) \land A$ may be polarised by choosing the additive conjunction to be positive resulting in $(\downshift A*B) \land^+ \downshift A$ (when $\downshift (A*\downshift B) \land^+ A)$ would not be well-formed). Choosing to shift one can ascribe a negative polarisation $\upshift ((\downshift A*B) \land^+ \downshift A)$. \end{example} The above generates the set of all such polarised formulas when all possible choices are explored. The free assignment of polarity to formulas means several distinct focusing procedures are captured by the completeness theorem.
\begin{lemma}[Completeness of $\fbi+\cut$] \label{lem:compFBIcut}
For any unfocused sequent $\Gamma \Rightarrow N$, if $\vdash_{\eta \lbi}\floor{\Gamma \Rightarrow N}$ then $\vdash_{\fbi+\cut} \Gamma \Rightarrow N$. \end{lemma} \begin{proof}
We show that every rule in $\eta\lbi$ is derivable in $\fbi+\cut$, consequently every proof in $\eta\lbi$ may be simulated; hence, every provable sequent has a focused proof. For unfocused rules $\rrn\to, \rrn\wand, \rrn{\land^-}, \lrn{\land^+}, \lrn\lor, \lrn\ast, \lrn\bot, \rrn{\top^-}, \lrn{\top^+}, \lrn\top^*$, this is immediate; as well as for $\rn{Ax}$ and $\cont$. Below we give an example on how to simulate a focused rule.
Where it does not matter (e.g. in the case of inactive nests), we do not distinguish the polarised and unpolarised versions; each of the simulations can be closed thanks to the presence of the $\rn P$ and $\rn N$ rules in $\fbi$.\\
\noindent\scalebox{.9}{
$
\infer[\rrn\ast]{\{\{\Gamma,\Delta\}_\times,\Delta'\}_+ \Rightarrow \varphi * \psi}{\Gamma \Rightarrow \varphi & \Delta \Rightarrow \psi}
$
}
in $\eta\lbi$ is simulated in $\fbi+\cut$ by
\scalebox{.9}{
$
\infer[\cut]{\{\{ \Gamma, \Delta \}_\times,\Delta'\}_+ \Rightarrow \upshift(\varphi^+*\psi^+)}{
\infer[\rrn\ast]{\{ \Gamma,\Delta \}_\times \Rightarrow \foc{\downshift\upshift\varphi^+*\downshift\upshift\psi^+} }{
\infer[\rrn\downshift]{\Gamma \Rightarrow \foc{\downshift\upshift\varphi^+}}{
\Gamma \Rightarrow \upshift\varphi^+
}
&
\infer[\rrn\downshift]{\Delta \Rightarrow \foc{\downshift\upshift\psi^+}}{\Delta \Rightarrow \upshift\psi^+}
}
&
\infer[\lrn\ast]{\{ \downshift\upshift\varphi^+*\downshift\upshift\psi^+,\Delta'\}_+ \Rightarrow \upshift(\varphi^+*\psi^+)}{
\infer[\rrn\upshift]{\{\{ \downshift\upshift\varphi^+,\downshift\upshift\psi^+ \}_\times,\Delta'\}_+ \Rightarrow \upshift(\varphi^+*\psi^+)}{
\infer[\rrn\ast]{ \{\{ \downshift\upshift\varphi^+,\downshift\upshift\psi^+ \}_\times,\Delta'\}_+ \Rightarrow \foc{\varphi^+*\psi^+}}{
\infer[\rn P]{\downshift\upshift\varphi^+ \Rightarrow \foc{\varphi^+}}{}
&
\infer[\rn P]{\downshift\upshift\psi^+ \Rightarrow \foc{\psi^+}}{}
}
}
}
}
$
}
\qed \end{proof} \begin{theorem}[Completeness of $\fbi$]
For any unfocused $\Gamma \Rightarrow N$, if $\vdash_{\eta\lbi} \floor{\Gamma \Rightarrow N}$ then $\vdash_{\fbi} \Gamma \Rightarrow N$. \end{theorem} \begin{proof}
It follows from Lemma \ref{lem:compFBIcut} that there is a proof of $\Gamma \Rightarrow N$ in $\fbi+\cut$, and then it follows from Lemma \ref{lem:fbicutadmi} that there is a proof of $\Gamma \Rightarrow N$ in $\fbi$. \qed \end{proof} Given an arbitrary sequent the above theorem guarantees the existence of a focused proof, thus the focusing principle holds for $\eta \lbi$ and therefore for $\lbi$.
\section{Conclusion}\label{sec:conclusion} By proving the completeness of a focused sequent calculus for the logic of Bunched Implications, we have demonstrated that it satisfies the focusing principle; that is, any polarisation of a BI-provable sequent can be proved following a focused search procedure. This required a careful analysis of how to restrict the usage of structural rules. In particular, we had to fully develop the congruence-invariant representation of bunches as nested multisets (originally proposed in \cite{Donnelly05}) to treat the exchange rule within bunched structures.
Proof-theoretically the completeness of the focused systems suggests a syntactic orderliness of $\lbi$, though the $\rn P$ and $\rn N$ rules leave something to be desired. Computationally, these axioms are unproblematic as during search it makes sense to terminate a branch as soon as possible; however, unless they may be eliminated it means that the focusing principle holds in BI only up to a point.
In related works (c.f. \cite{Chaudhuri16a}) the analogous problem is overcome by first considering a \emph{weak} focused system; that is, one where the structural rules are not controlled and unfocused rules may be performed inside focused phases if desired. Completeness of (strong) focusing is achieved by appealing to a \emph{synthetic} system. It seems reasonable to suppose the same can be done for BI, resulting in a more proof-theoretically satisfactory focused calculus, exploring this possibility is a natural extension of the work on $\fbi$.
The methodology employed for proving the focusing principle can be interpreted as soundness and completeness of an operational semantics for goal-directed search. The robustness of this technique is demonstrated by its efficacy in modal \cite{Chaudhuri16a,Chaudhuri16b} and substructural logics \cite{Lincoln92}, including now bunched ones. Although BI may be the most employed bunched logic, there are a number of others, such as the family of relevant logics \cite{Read88}, and the family of bunched logics \cite{Docherty19}, for which the focusing principle should be studied. However, without the presence of a $\cut$-free sequent calculus goal-directed search becomes unclear, and currently such calculi do not exist for the two main variants of BI: Boolean BI \cite{Pym02} and Classical BI \cite{Brotherston10}. On the other hand, large families of bunched and substructural logics have been given hypersequent calculi \cite{Ciabattoni12,Ciabattoni17}. Effective proof-search procedures have been established for the hypersequent calculi in the substructural case \cite{Ramanayake20}, but not the bunched one, and focused proof-search for neither. There is a technical challenge in focusing these systems as one must not only decide which formula to reduce, but also which sequent.
In the future it will be especially interesting to see how focused search, when combined with the expressiveness of BI, increases its modelling capabilities. Indeed, the dynamics of proof-search can be used to represent models of computation within (propositional) logics; for example, the undecidability of Linear Logic involves simulating two-counter machines \cite{Lincoln92}. One particularly interesting direction is to see how focused proof-search in BI may prove valuable within the context of Separation Logic. Focused systems in particular have been used to emulate proofs for other logics \cite{Marin16}; and to give structural operational semantics for systems used in industry, such as algorithms for solving constraint satisfaction problems \cite{Farooque13}. A more immediate possibility though is the formulation of a theorem prover; we leave providing specific implementation or benchmarks to future research.
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\end{document} | arXiv |
\begin{document}
\title{ Hyperbolic groups have flat-rank at most~$1$}
\author{Udo Baumgartner, R{\"o}gnvaldur G. M{\"o}ller\inst{2} \and George A. Willis\inst{1}
\thanks{\textbf{The first and last author were supported by A.R.C. Grants DP0208137 and LX0349209}} }
\institute{School of Mathematical and Physical Sciences\\
University of Newcastle\\
University Drive\\
Callaghan, NSW 2308\\
Australia\\
\email {[email protected]}
\and
Science Institute\\
University of Iceland\\
107 Reykjavik\\
Iceland\\
\email{[email protected]}}
\date{}
\authorrunning{Baumgartner, M\"oller and Willis} \maketitle
\begin{abstract} The flat-rank\, of a totally disconnected, locally compact group $G$ is an integer, which is an invariant of $G$ as a topological group.
We generalize the concept of hyperbolic groups to the topological context and show that a totally disconnected, locally compact, hyperbolic group has flat-rank\, at most~$1$.
It follows that the simple totally disconnected locally compact groups constructed by Paulin and Haglund have flat-rank\, at most~$1$. \keywords {totally disconnected group, hyperbolic group, flat-rank\,\!, automorphism group, scale function}
\subclass{ 22D05, 22D45 (primary)
} \end{abstract}
\section{Introduction}\label{sec:intro}
\label{section-Intro}
The concept of a hyperbolic group can be generalized to the realm of compactly generated, topological groups by a straightforward adaption of the definition in the discrete case (see Definition~\ref{def:hyp-topG} on page~\pageref{def:hyp-topG}).
Such a generalization is an instance of \textit{`geometric group theory for topological groups'\/}, which is a line of investigation proposed in~\cite{analog(CayleyGphs(topGs))}.
This geometric approach is a natural one in the case of totally disconnected, locally compact groups, the subject of this paper, and has been pursued previously in \cite{struc(tdlcG-graphs+permutations)}, \cite{Gphs+Perm+topGs}, \cite{tdlcGs-geomObjs} and \cite{geo-char(flatG(auts))}. We take a different line to these papers, however, by studying hyperbolicity and relating it to a structural invariant for totally disconnected, locally compact groups, namely, the \textbf{flat-rank}.
The flat-rank\,\! of a totally disconnected, locally compact group (see Definition~\ref{defn:flat-rank}) is a non-negative integer that is analogous to the $k$-rank of a semisimple algebraic group over a local field~$k$. Indeed the flat-rank\,\! and $k$-rank coincide when $G$ is such a group, by Corollary~19 in~\cite{flatrk(AutGs(buildings))}.
Just as the $k$-rank of a simple algebraic group determines many properties of the group, the flat-rank\,\! can be expected to convey important information about general simple totally disconnected groups.
An indication of this is seen with the computation of the flat-rank\,\! of automorphism groups of buildings made
in~\cite{flatrk(AutGs(buildings))}, where it is shown, in conjunction with results from~\cite{geoFlats<CAT0-real(CoxGs+Tits-buildings)}, that if the group action is sufficiently transitive then the flat-rank\,\! of the group equals the rank of its building.
The following theorem, our main result, further demonstrates the relationship between the flat-rank\,\! and geometric properties of the group.
\begin{theorem}\label{MainThm-variant} The flat-rank\,\! of a totally disconnected, locally compact, hyperbolic group is at most~$1$. \end{theorem}
The major part of this paper is devoted to the proof of Theorem~\ref{MainThm-variant}. It is clear that the converse to this theorem does not hold, because discrete groups have flat-rank\,\!~0 and need not be hyperbolic. However, it may hold in the presence of further hypotheses that exclude discrete groups or non-discrete counterexamples based on them.
The properties of algebraic groups of $k$-rank $1$ differ notably from the properties of groups of higher $k$-rank. In the expectation that the same will be true of the flat-rank\,\!, a secondary aim of this paper is to seek further geometric criteria for a totally disconnected, locally compact group to have flat-rank\,\! at most~$1$. We establish two such criteria.
One is based on the action of the group on the space of compact open subgroups of the group.
The criterion and its proof are in the spirit of the papers~\cite{flatrk(AutGs(buildings))} and~\cite{geo-char(flatG(auts))}; the proof is contained in Section~\ref{sec:hyperbolic orbits}.
\begin{theorem}\label{thm:altMainThm} Let $\mathcal{A}$ be a group of automorphisms of the totally disconnected locally compact group $G$. Suppose that $\mathcal{A}$ has a hyperbolic orbit in the space of compact open subgroups of $G$. Then the flat-rank\,\! of $\mathcal{A}$ is at most $1$. \end{theorem}
The last criterion follows from results in~\cite{direction(aut(tdlcG))}, where the space of directions of a totally disconnected, locally compact group is defined; see page~\pageref{pref:pf(thm:flatrk-via-directions)} for its proof.
\begin{theorem}\label{thm:flatrk-via-directions} Let~$G$ be a totally disconnected, locally compact group whose space of directions is discrete.
Then the flat-rank\,\! of~$G$ is at most~$1$. If the space of directions is not empty, then the flat-rank\,\! is exactly~1. \end{theorem}
We do not know of a hyperbolic group whose space of directions is not discrete. In view of Theorem~\ref{thm:flatrk-via-directions}, it would be a strengthening of Theorem~\ref{MainThm-variant} to show that all hyperbolic groups have discrete spaces of directions.
\section{Basic concepts}\label{sec:basics}
\begin{definition}[hyperbolic group {[topological version]}] \label{def:hyp-topG} A topological group is called \textbf{hyperbolic} if and only if
it is compactly generated and its Cayley graph with respect to some (hence any) compact generating set is Gromov-hyperbolic. \end{definition}
The definition makes sense, because all Cayley graphs with respect to compact generating sets are quasi-isometric by part~(\romannumeral1) of Lemma~4.6 in~\cite{Gphs+Perm+topGs}.
The same definition is used in recent work by Yves Cornulier and Romain Tessera \cite{contrAut+L^p-Cohom<deg1}, where they characterize certain classes of non-discrete Gromov-hyperbolic groups.
In this paper we consider compactly generated, totally disconnected, locally compact topological groups only.
These groups admit a locally finite, connected graph with a vertex-transitive action by the group such that vertex-stabilizers are compact and open.
Such a graph with an action by the group is an instance of a so-called
{rough Cayley graph}, a concept introduced in~\cite{analog(CayleyGphs(topGs))}.
We now define this concept.
\begin{definition}\label{def:rough-Gayley-Gph} Let~$G$ be a topological group.
A connected graph~$X$ is said to be \textbf{a rough Cayley graph} of~$G$, if $G$ acts transitively on the vertex set of~$X$ and the stabilizers of vertices are compact open subgroups of~$G$. \end{definition}
The proof of our main result relies on the existence of a rough Cayley graph for the groups under consideration. The relevant result is Theorem~2.2+ in~\cite{analog(CayleyGphs(topGs))}, or Corollary~1 in~\cite{FC-ele<tdGs&auts(infGphs)}, which we restate for ease of reference.
In the formulation, $VX$ denotes the vertex set of~$X$.
\begin{theorem}[Existence of a locally finite, rough Cayley graph] \label{thm:existence(lf-roughCayleyGph)} Let~$G$ be a totally disconnected, compactly generated, locally compact group.
Then there is a locally finite, connected graph~$X$ such that: \begin{itemize} \item[(\romannumeral1)] $G$ acts as a group of automorphisms on~$X$ and is transitive on~$VX$; \item[(\romannumeral2)]
for every vertex~$v$ in~$X$ the subgroup~$G_v$ is compact and open in~$G$; \item[(\romannumeral3)]
if~$\Aut X_.$ is equipped with the permutation topology, then the homomorphism $\pi\colon G\to \Aut X_.$ given by the action of~$G$ on~$X$ is continuous, the kernel of this homomorphism is compact and the image of~$\pi$ is closed in~$\Aut X_.$. \end{itemize} Conversely, if~$G$ acts as a group of automorphisms on a locally finite, connected graph~$X$ such that~$G$ is transitive on the vertex set of~$X$ and the stabilizers of the vertices in~$X$ are compact and open, then~$G$ is compactly generated. \end{theorem}
For a totally disconnected, locally compact group hyperbolicity can be formulated in terms of any of its rough Cayley graphs as follows.
\begin{proposition}[hyperbolicity in terms of the rough Cayley graph] \label{prop:hyperbolicity(roughCayley-graph)} A totally disconnected, locally compact group is hyperbolic if and only if some (hence any) of its rough Cayley graphs is hyperbolic. \end{proposition}
\begin{proof} The claim is implied by the quasi-isometry of rough Cayley graphs; see Theorem~4.5 in~\cite{Gphs+Perm+topGs} or Theorem~2.7 in~\cite{analog(CayleyGphs(topGs))}. \hspace*{\fill}$\square$ \end{proof}
The flat-rank\,\! of a group $\mathcal{A}$ of automorphisms of a totally disconnected locally compact group $G$ was introduced in \cite{tidy<:commAut(tdlcG)}, although it was not given that name there. Some auxiliary definitions and results will be required for its definition and in later sections. \begin{definition} \label{defn:minimizing;flat} Let $G$ be a totally disconnected, locally compact group. \begin{itemize} \item[(\romannumeral1)] The \textbf{scale} of the automorphism, $\alpha$, of $G$ is the positive integer \begin{equation} \label{eq:scaledefn} s_G(\alpha) :=
\min\left\{ |\alpha(O)\colon \alpha(O)\cap O| \colon O\leqslant G\text{ compact and open}\right\}. \end{equation} \item[(\romannumeral2)] The compact, open subgroup $O$ is \textbf{minimizing} for $\alpha$ if the minimum index in {\rm(\ref{eq:scaledefn})} is attained at $O$. \item[(\romannumeral3)] The group, ${\mathcal A}$ of automorphisms of $G$ is \textbf{flat} if there is a compact open subgroup $O\leqslant G$ that is minimizing for every $\alpha\in {\mathcal A}$. \end{itemize} \end{definition}
\begin{theorem}[\cite{tidy<:commAut(tdlcG)}, Corollary 6.15] \label{thm:flat} Let ${\mathcal A}$ be a flat group of automorphisms of $G$ and $O$ be minimizing for ${\mathcal A}$. Then ${\mathcal A}_1 := \left\{\alpha\in{\mathcal A} \colon \alpha(O) = O\right\}$ is a normal subgroup of ${\mathcal A}$, and ${\mathcal A}/{\mathcal A}_1$ is a free abelian group. \end{theorem} The group ${\mathcal A}_1$ is independent of the minimizing subgroup used to define it. \begin{definition} \label{defn:flat-rank}
Let $G$ be a totally disconnected, locally compact group. \begin{itemize} \item[(\romannumeral1)] The \textbf{rank} of the flat group, ${\mathcal A}$ of automorphisms of $G$ is the rank of the free abelian group ${\mathcal A}/{\mathcal A}_1$. \item[(\romannumeral2)] The \textbf{flat-rank\,} of a group ${\mathcal A}$ of automorphisms of $G$ is the supremum of the ranks of all the flat subgroups of ${\mathcal A}$. \item[({\romannumeral3})] The \textbf{flat-rank\,} of $G$ is the flat-rank\, of the group of inner automorphisms. \end{itemize} \end{definition}
\section{Constructing hyperbolic topological groups}\label{sec:supplement}
The following proposition provides a method to construct totally disconnected, locally compact, hyperbolic groups.
For example one might take for~$X$ the Cayley graph of any discrete hyperbolic group and let~$G$ be its full automorphism group with the permutation topology (equivalently, the compact-open topology).
Further applications of this result will be provided in Section~\ref{sec:examples}.
\begin{proposition} \label{prop:hyperbolicity(cocp-<Gs(Aut(lf-Gromov-hyp-complexes))} Let $G$ be a totally disconnected, locally compact group acting cocompactly and with compact, open point stabilizers on a locally finite, connected Gromov-hyperbolic complex, $X$ say.
Then the $1$-skeleton of~$X$ is quasi-isometric to a locally finite, connected, rough Cayley graph for~$G$ which is also Gromov-hyperbolic, and $G$ is a hyperbolic group. \end{proposition}
\begin{proof} Since the group~$G$ acts cocompactly on~$X$, there is a finite subcomplex, $F$ say, of~$X$ whose $G$-translates cover~$X$.
The group~$G$ is then generated by its subset $\{x\in G\colon x.F\cap F\neq\emptyset\}$; this subset is compact, hence~$G$ is compactly generated.
Denote the $1$-skeleton of~$X$ by~$\Gamma$.
The graph~$\Gamma$ is a locally finite, connected $G$-set with compact, open point stabilizers. We use a standard argument, see \cite[Theorem~4.9]{Gphs+Perm+topGs} for instance, to complete the proof. Since the $G$-translates of the 1-skeleton of the finite complex, $F$, of the previous paragraph cover $\Gamma$, there are finitely many $G$-orbits in $\Gamma$ and there is a constant $k$ (for example, the diameter of $F$) such that every vertex is within distance $k$ of each orbit. Fix a vertex $x$ of $\Gamma$ and define a graph structure on the orbit $G.x$ by drawing an edge between vertices $g.x$ and $h.x$ if they are within distance $2k+1$. Then the graph $G.x$ is connected, locally finite and quasi-isometric to $\Gamma$. Hence $G.x$ is Gromov-hyperbolic. Furthermore, $G$ acts transitively with compact vertex stabilizers on this graph. Therefore the graph $G.x$ is a rough Cayley graph for $G$ and $G$ is a hyperbolic group. \hspace*{\fill}$\square$ \end{proof}
While the action of a group on its Cayley graph is always faithful, the action of a compactly generated topological group on its rough Cayley graph need not be.
The following result explains what happens when passing to the quotient of the group by the kernel of this action.
The proof is straightforward, and is left to the reader.
\begin{proposition}\label{prop:quot-by-kernel-of-action-on-rough-Cayley-graph} Let~$G$ be a compactly generated topological group that contains a compact, open subgroup and let~$\Gamma$ be any of its locally finite, connected, vertex-transitive rough Cayley graphs with compact, open vertex-stabilizers.
Let~$\widehat{G}$ be the quotient of~$G$ by the compact kernel of its action on~$\Gamma$.
Then~$\Gamma$ with its induced action by~$\widehat{G}$ is again a rough Cayley graph for~$\widehat{G}$ with the same properties.
In particular, if the group~$G$ is hyperbolic, then so is the group~$\widehat{G}$. \hspace*{\fill}$\square$ \end{proposition}
We also have the following transfer result for flat subgroups under continuous, open, surjective homomorphism with compact kernel.
\begin{proposition}\label{prop:flat<Gs&proper-maps} Let~$\pi\colon G\to \widehat{G}$ be a continuous, open, surjective homomorphism with compact kernel between totally disconnected, locally compact groups.
Further, let~$H$ be a flat subgroup of~$G$ and~$\widehat{H}$ a flat subgroup of~$\widehat{G}$.
Then~$\pi(H)$ is a flat subgroup of~$\widehat{G}$ of the same rank as~$H$ while $\pi^{-1}(\widehat{H})$ is a flat subgroup of~$G$ of the same rank as~$\widehat{H}$.
The groups~$G$ and~$\widehat{G}$ have the same flat rank.
\end{proposition}
\begin{proof}
Consider $h\in G$ and suppose that $O$ is minimizing for $h$, that is, is minimizing for the inner automorphism $\alpha_h : x\mapsto hxh^{-1}$. The index $|hOh^{-1}\colon hOh^{-1}\cap O|$ is unchanged if $O$ is replaced by $O\ker(\pi)$ and so it may be assumed that $\ker(\pi)\subseteq O\cap hOh^{-1}$.
The subgroup~$\pi(O)$ of~$\widehat{G}$, which is also compact and open, then satisfies \begin{equation}\label{eq:tidyness_under-proper-maps}
|hOh^{-1}\colon hOh^{-1}\cap O|=
|\pi(h)\pi(O)\pi(h)^{-1}\colon \pi(h)\pi(O)\pi(h)^{-1}\cap \pi(O)|\,, \end{equation} from which it follows that~$s_{\widehat G}(\pi(h))\leq s_G(h)$. On the other hand, if ${\widehat O} \leqslant {\widehat G}$ is minimizing for ${\hat h}\in {\widehat G}$, then $\pi^{-1}({\widehat O})$ is compact and open in $G$ and \begin{equation}\label{eq:tidyness_under-proper-maps2}
|h\pi^{-1}({\widehat O})h^{-1}\colon h\pi^{-1}({\widehat O})h^{-1}\cap \pi^{-1}({\widehat O})|=
|{\hat h}{\widehat O}{\hat h}^{-1}\colon {\hat h}{\widehat O}{\hat h}^{-1}\cap {\widehat O}|\,, \end{equation} for any $h\in G$ with $\pi(h) = {\hat h}$ and it follows that~$s_G(h) \leq s_{\widehat G}(\pi(h))$. Therefore the scales are equal and $O$ is minimizing for $h$ if and only if $\pi(O)$ is minimizing for $\pi(h)$.
Letting $H$ be a flat subgroup of $G$ and ${\widehat H}$ be a flat subgroup of ${\widehat G}$, it follows that $\pi(H)$ and $\pi^{-1}({\widehat H})$ are flat subgroups of $G$ and ${\widehat G}$ respectively as claimed. Moreover, $h\in H_1$ if and only if $\pi(h_1)\in \pi(H)_1$ and so the map $\pi : H\to \pi(H)$ induces an isomorphism $H/H_1\to \pi(H)/\pi(H)_1$. Hence the ranks of $H$ and $\pi(H)$ are equal as claimed. That the ranks of ${\widehat H}$ and $\pi^{-1}({\widehat H})$ also agree may be seen similarly. Therefore $G$ and ${\widehat G}$ have the same flat-rank\,\!. \hspace*{\fill}$\square$ \end{proof}
\section{Method of proof}\label{sec:method(proof)}
The proof of the main result uses the classification of group actions by isometries on Gromov-hyperbolic spaces in terms of fixed point properties versus existence of free subgroups. This is combined with topological properties of elliptic, parabolic and hyperbolic isometries and an analysis of the dynamics of actions of flat subgroups on the boundary of the hyperbolic space. For the geometric ideas we follow the approach in~\cite{woess}.
We begin by extending the necessary concepts of hyperbolic geometry to encompass topological groups.
\begin{definition}[boundary of a hyperbolic group] Let~$G$ be a hyperbolic topological group.
The \textbf{hyperbolic boundary of~$G$} is the Gromov-boundary of the Cayley graph of~$G$ with respect to some compact generating set of~$G$. \end{definition}
The usual properties of the hyperbolic boundary carry over from the discrete case.
That it is independent of the rough Cayley graph chosen will be important in subsequent arguments.
\begin{proposition}\label{prop:prop(hyp-boundary(G))} The hyperbolic boundary of a hyperbolic topological group, $G$, is independent of the choice of compact generating set used for its definition.
It is a metric space
which admits an action of $G$ by bi-Lipschitz maps.
If $G$ admits a compact, open subgroup, then metrics can be chosen such that its hyperbolic boundary is equivariantly isometric to the Gromov-boundary of any of its rough Cayley graphs; in particular, in that case the hyperbolic boundary of $G$ is compact. \end{proposition}
\begin{proof} Using standard results about Gromov-hyperbolic spaces, the above statements follow from Theorem~4.5 and Lemma~4.6 in~\cite{Gphs+Perm+topGs}. \hspace*{\fill}$\square$ \end{proof}
The classification of isometries of hyperbolic spaces also plays a central role in what follows.
Since the action of a topological group on a rough Cayley graph need not be faithful, we extend the usual definitions as follows.
\begin{definition}[elliptic, parabolic and hyperbolic elements] \label{def:elliptic,parabolic,hyperbolic} Let~$G$ be a group, $X$ be a Gromov-hyperbolic space and $\alpha\colon G\to \Aut X_.$ be an action of~$G$ on~$X$ by isometries.
An element~$g$ in~$G$ is called \begin{enumerate}
\item \textbf{$\alpha$-elliptic}
if there is a point of~$X$ whose $\alpha(g)$-orbit is bounded; in that case every other point of~$X$ has the same property;
\item \textbf{$\alpha$-parabolic}
if it is not $\alpha$-elliptic and~$\alpha(g)$ fixes a unique boundary point;
\item \textbf{$\alpha$-hyperbolic}
if it is not $\alpha$-elliptic and~$\alpha(g)$ fixes precisely two boundary points,
which, for arbitrary~$x\in X$ are then of the form
$\lim_{n\to \infty} \alpha^n(g).x$, called \textbf{attracting for~$g$}, and
$\lim_{n\to \infty} \alpha^{-n}(g).x$, called \textbf{repelling for~$g$}. \end{enumerate}
An element of a hyperbolic topological group is called \textbf{elliptic, parabolic or hyperbolic} respectively, if it is $\alpha$-elliptic, $\alpha$-parabolic or $\alpha$-hyperbolic respectively for $\alpha$ equal to the natural action of the group on its Cayley graph with respect to a compact generating set.
\end{definition}
Since Cayley graphs with respect to compact generating sets are quasi-isometric, the notions elliptic, parabolic or hyperbolic do not depend on the particular Cayley graph chosen and reference to the homomorphism~$\alpha$ is usually omitted in the following.
The elliptic elements of a locally compact, hyperbolic group can be characterized by an intrinsic topological property which generalizes the corresponding characterization in the discrete case.
\begin{proposition}\label{prop:top-char(elliptics)} An element of a locally compact, hyperbolic, topological group is elliptic if and only if it is topologically periodic. \end{proposition}
\begin{proof} By definition, an element, $g$ say, of the given group~$G$ is elliptic if and only if its orbits in the Cayley graph, $\Gamma$ say, of~$G$ with respect to a compact set of generators is bounded in the graph metric.
The property of being bounded is independent of the orbit chosen.
Hence $g$ is elliptic if and only if its orbit $\langle g\rangle.e= \langle g\rangle$ is bounded.
Abels' result~2.3 in~\cite{Specker-cp(lctG)} (Heine-Borel-Eigenschaft) implies that a subset of~$\Gamma$ is bounded in the graph-metric if and only if it is a relatively compact subset of~$G$.
The latter condition is satisfied by the set $\langle g\rangle$ if and only $g$ is topologically periodic.
\hspace*{\fill}$\square$ \end{proof}
\section{Properties of elliptic, parabolic and hyperbolic elements}\label{sec:types of elements}
\subsection{The scale of an elliptic element is~$1$}\label{subsec:scale(elliptic)=1}
An elliptic element in a totally disconnected, locally compact group is topologically periodic, by Proposition~\ref{prop:top-char(elliptics)}. Hence the set of conjugates of an open subgroup by powers of such an element is finite and the intersection of these conjugates is an open subgroup normalized by the element.
\begin{proposition}\label{prop:elliptic->scale1} The scale of every elliptic element in a totally disconnected, locally compact, hyperbolic group is~$1$. \hspace*{\fill}$\square$ \end{proposition}
\subsection{Totally disconnected,
hyperbolic groups contain no parabolics}\label{subsec:hypGs-not>parabolics}
Discrete hyperbolic groups do not contain parabolic elements.
This is proved in each one of the following sources: \cite[Corollary~8.1.D]{hypGs} (together with the obvious observation that torsion elements are elliptic), \cite[Chapitre~9, Th\'eor\`eme~3.4]{LNM1441} and \cite[Chapitre~8, Th\'eor\`eme~29]{surGhyp-d'apresGromov}.
Theorem~\ref{thm:Aut(hyperbolic-complex)-has-no-parabolics} below extends this result to totally disconnected, locally compact, hyperbolic groups,
the proof of which is modelled on the argument from~\cite{LNM1441}.
The following property of non-hyperbolic elements is of central importance in the proof.
\begin{lemma}\label{lem:parabolics-bounded-at-infty} Let $X$ be a geodesic $\delta$-hyperbolic space.
Then there is a constant~$k$, depending only on~$\delta$, such that: given a non-hyperbolic isometry, $g$, of~$X$ that fixes a boundary point~$\omega$,
and a geodesic ray, $x$, ending in~$\omega$, all points on $x$ which are sufficiently far out are moved by a distance of at most~$k$ by~$g$. \end{lemma}
\begin{proof} Choose any point~$p$ of~$X$ and denote the midpoint of a chosen geodesic segment connecting $p$ to $g.p$ by~$m$.
By Lemme~9.3.1 in~\cite{LNM1441}
\[ d(g.m,m)\le 6\delta\,. \] Applying Lemme~9.3.6 in~\cite{LNM1441} to the entities $x$ and $m$ chosen, we obtain that there is a number~$t_0\ge 0$ such that for each~$t\ge t_0$ \[ d(g.x(t),x(t))\le 72\delta +d(g.m,m)\le 72\delta+6\delta=78\delta\,, \] and so we may take $k$ to be $78\delta$. \hspace*{\fill}$\square$ \end{proof}
The following theorem is the main result of this subsection.
\begin{theorem}\label{thm:Aut(hyperbolic-complex)-has-no-parabolics} Suppose that a group~$G$ acts cocompactly and by automorphisms on a connected, locally finite, metric, Gromov-hyperbolic complex. Then~$G$ does not contain parabolic elements.
In particular, a hyperbolic topological group with a compact, open subgroup
does not contain parabolic elements. \end{theorem} \begin{proof} We argue by contradiction, and assume that some element, $g$ say, of~$G$ acts by a parabolic isometry.
All positive powers of $g$ are again parabolic and have the same unique fixed point on the boundary, $\omega$ say.
The diameter of cells in our space is bounded from above, say by the positive number~$D$, because the group~$G$ is assumed to act cocompactly.
Denote by~$n$ a natural number larger than the maximal number of vertices of the space that are contained in any closed ball whose radius is~$k+2D$, where~$k$ is the number introduced in the statement of Lemma~\ref{lem:parabolics-bounded-at-infty}.
Such a number~$n$ exists, because~$G$ acts cocompactly.
Choose a geodesic ray, $x$ say, that ends in~$\omega$.
Lemma~\ref{lem:parabolics-bounded-at-infty} applied to the elements $g, g^2,\ldots, g^n$ and this~$x$ implies that there is a number~$T$ such that for $t\ge T$ and~$i=1,\ldots, n$ we have $ d(g^i.x(t),x(t))\le k $.
By the definition of~$D$, there is a vertex, $v$ say, at distance at most~$D$ from $x(t)$.
Then the above bound on the displacement of the point~$x(t)$, implies that~$v$ is moved a distance at most $k+2D$ by each of the elements $g$, $g^2$, \ldots $g^n$.
By our choice of~$n$, there are exponents $i<j$ such that $g^i.v=g^j.v$.
But then $g^{j-i}$ fixes~$v$, and hence is elliptic.
Hence the element~$g$ is also elliptic, in contradiction to the assumption that $g$ is parabolic.
This contradiction shows that there is no parabolic element, finishing the proof. \hspace*{\fill}$\square$ \end{proof}
\section{Proof of the Main Result}\label{sec:Pf(MainResult)}
We will use the classification of actions on hyperbolic spaces established in~\cite{woess}, as already mentioned.
The bound on the rank of a flat subgroup is proved on a case-by-case basis according to this classification.
The following two lemmas prepare Proposition~\ref{prop:F2<flatG=>rk=0}, which provides this bound if the flat group contains a non-abelian free group consisting of hyperbolic elements.
\begin{lemma}\label{lem:Gv notfix rep.pt(elt.scale>1)} Let $G$ be a totally disconnected, locally compact group acting cocompactly and with compact, open point stabilizers on a locally finite, connected $\delta$-hyperbolic complex.
If~$h$ is an element of~$G$ of non-trivial scale (which is thus necessarily hyperbolic), then the orbit of~$\omega_h$, the repelling boundary point of~$h$, under every open subgroup is infinite. In particular, no open subgroup of $G$
fixes~$\omega_h$. \end{lemma}
\begin{proof} It follows from Proposition~\ref{prop:elliptic->scale1} and Theorem~\ref{thm:Aut(hyperbolic-complex)-has-no-parabolics} that an element~$h$ in~$G$ of non-trivial scale must indeed be hyperbolic as stated.
Proposition~\ref{prop:hyperbolicity(cocp-<Gs(Aut(lf-Gromov-hyp-complexes))} implies that~$h$ also acts as a hyperbolic automorphism of the given complex.
We now begin the proof proper; we will prove the contraposition.
Assume then that $V$ is an open subgroup of~$G$ such that the orbit of $\omega_h$ under~$V$ is finite.
Then a closed subgroup of finite index in~$V$ fixes~$\omega_h$ and we can assume that $V$ fixes~$\omega_h$.
Intersecting the group~$V$ with the stabilizer of a vertex, $v$ say, we may assume that $V$ fixes a given vertex~$v$ also.
Applying Theorem~7.7 in~\cite{struc(tdlcG-graphs+permutations)} with~$V$ equal to the group of the same name, and~$x$ equal to~$h$, we see that the scale of~$h$ is given by the limit \[
\lim_{n\to\infty} |V.(h^{-n}.v)|^{1/n}\,. \]
We will use our assumptions to show that there is a bound on the diameter of the orbits~$V.(h^{-n}.v)$ that is uniform in~$n\in\mathbb{N}$.
Because $G$ acts cocompactly, this implies that there is a uniform bound on the number of the vertices in these orbits.
The displayed formula above will then show that the scale of~$h$ is~$1$, and establish our claim.
The map~$f$ that sends an integer~$n$ to the vertex~$h^{-n}.v$ is a quasi-geodesic ray
that converges to~$\omega_h$.
By part~$(i)$ of Th\`eor\'eme~5.25 in~\cite{surGhyp-d'apresGromov}, there is a geodesic ray, $r$ say, that starts at~$v$ and is at Hausdorff-distance at most~$H$ from~$f$;
the ray~$r$ therefore ends in~$\omega_h$ also. Then part~$(i)$ of Corollaire~7.3 in~\cite{surGhyp-d'apresGromov} implies that $d(r(t),g.r(t))\le 8\delta$ for all $g\in V$ and all $t\ge 0$.
Let~$n$ be an integer.
On the geodesic ray~$r$ choose a point, $r(t_n)$ say, such that the distance between~$h^{-n}.v$ and~$r(t_n)$ is at most~$H$.
For every~$g$ in~$V$ the distance between~$g.(h^{-n}.v)$ and~$g.r(t_n)$ is then at most~$H$ also.
We conclude that for all~$g$ in~$V$ the distance between $h^{-n}.v$ and~$g.(h^{-n}.v)$ is at most~$2H+8\delta$; hence the diameter of the orbits~$V.(h^{-n}.v)$ is indeed uniformly bounded, and we are done. \hspace*{\fill}$\square$ \end{proof}
The claim of the following Lemma is false for flat groups of rank~$0$.
\begin{lemma}\label{lem:flatG(rk>0)=>no-hyperb<unisc<G} Let $G$ be a totally disconnected, locally compact, hyperbolic group and $H$ a flat subgroup of~$G$ of rank at least~$1$.
Then the group~$H_1$ does not contain hyperbolic elements. \end{lemma}
\begin{proof} We will derive a contradiction to Lemma~\ref{lem:Gv notfix rep.pt(elt.scale>1)} from the assumption that there is a hyperbolic element in~$H_1$ and an (automatically hyperbolic) element in $H\smallsetminus H_1$, thus establishing the claim.
Let $O$ be a minimizing subgroup
for~$H$.
The group~$O$ is the stabilizer in~$G$ of the point~$O$ in any rough Cayley graph of~$G$ constructed from $O$.
The group $H_1$ normalizes the subgroup~$O$.
Hence the group~$O$ fixes the whole orbit $H_1.O$ pointwise.
All boundary points fixed by hyperbolic elements in~$H_1$ are limit points of this orbit, hence the set of these points, $L_{hyp}(H_1)$ say, is fixed by the group~$O$ also.
Choose a hyperbolic element~$k$ in $H_1$.
Denote the repelling boundary point of~$k$
by $\omega_k\in L_{hyp}(H_1)$.
Further choose an element~$h$ in $H\smallsetminus H_1$;
which is possible because the rank of~$H$ is at least~$1$.
Replacing, if necessary, $h$ by its inverse, we may assume that $h$ has non-trivial scale.
Proposition~\ref{prop:elliptic->scale1} and Theorem~\ref{thm:Aut(hyperbolic-complex)-has-no-parabolics} imply that the element~$h$ is hyperbolic.
Denote the repelling boundary point of~$h$ by~$\omega_h$.
Since
the subgroup~$O$ fixes $\omega_k\in L_{hyp}(H_1)$ and $h$ has non-trivial scale, $\omega_k$ is different from $\omega_h$, for otherwise we would have a contradiction to Lemma~\ref{lem:Gv notfix rep.pt(elt.scale>1)}.
Because $H_1$ is
normal in~$H$, the group~$H$ leaves the set $L_{hyp}(H_1)$ invariant.
In particular, the sequence $\bigl(h^{-n}(\omega_k)\bigr)_{n\in\mathbb{N}}$, which converges to $\omega_h$ since $\omega_k\neq \omega_h$, is contained in~$L_{hyp}(H_1)$.
By continuity of the action of $G$ on the hyperbolic compactification, $\omega_h$ is contained in~$L_{hyp}(H_1)$ and hence is fixed by~$O$.
This is the anticipated contradiction to Lemma~\ref{lem:Gv notfix rep.pt(elt.scale>1)}. \hspace*{\fill}$\square$ \end{proof}
We are now ready to treat the first case in the classification.
\begin{proposition}\label{prop:F2<flatG=>rk=0} Let $G$ be a totally disconnected, locally compact, hyperbolic group and $H$ a flat subgroup of $G$ that contains a non-abelian free group consisting of hyperbolic elements.
Then the rank of~$H$ is~$0$. \end{proposition}
\begin{proof} Let $F$ be a non-abelian free subgroup of~$H$ consisting of hyperbolic elements.
Assume by way of contradiction that the rank of~$H$ is at least~$1$.
Using Lemma~\ref{lem:flatG(rk>0)=>no-hyperb<unisc<G}, we then conclude that the subgroup $H_1$ contains no hyperbolic elements.
Then the restriction of the canonical map $H\to H/H_1$ to the subgroup~$F$ of~$H$ has trivial kernel.
It follows that the abelian group~$H/H_1$ contains a non-abelian free group, which is absurd.
Therefore the rank of~$H$ is~$0$ as claimed. \hspace*{\fill}$\square$ \end{proof}
The second case of the classification is easy.
\begin{proposition}\label{prop:flatG-stab-nonempty-cpS=>rk=0} Let $G$ be a totally disconnected, locally compact, hyperbolic group and $H$ a flat subgroup of~$G$ that stabilizes a non-empty, compact subset of some rough Cayley graph of~$G$.
Then the rank of~$H$ is~$0$. \end{proposition}
\begin{proof} The condition satisfied by~$H$ implies that all elements of~$H$ are elliptic.
By Proposition~\ref{prop:elliptic->scale1}, $H$ is contained in its subgroup~$H_1$ and the flat-rank\,\! of~$H$ is~$0$ as claimed. \hspace*{\fill}$\square$ \end{proof}
The next lemma proves that a quasi-geodesic ray converging to a boundary point, $\omega$ say, is uniformly close to any geodesic ray converging to~$\omega$.
This lemma is used in the last cases in the classification, Proposition~\ref{prop:flatG-fix-1OR2boundary-pts=>rk le1} below.
\begin{lemma}\label{lem:q-geodesic rays hit close near their boundary point} Given real numbers~$\delta\ge 0$, $\lambda\ge 1$ and~$c\ge0$ there is a constant~$R$ (depending on~$\delta$, $\lambda$ and~$c$ only) such that for any proper $\delta$-hyperbolic space, $X$: given a geodesic ray, $r$, and a $(\lambda, c)$-quasi-geodesic ray, $f$, in $X$ that converge to the same boundary point~$\omega$, the image of $f$ intersects the ball of radius~$R$ centred on any point sufficiently far out on $r$. \end{lemma}
\begin{proof} By part~$(i)$ of Th\`eor\'eme~5.25 in~\cite{surGhyp-d'apresGromov} there is a geodesic ray~$g$ at Hausdorff distance at most~$H$ from~$f$, where~$H$ depends on~$\delta$, $\lambda$ and~$c$ only.
The geodesic ray~$g$ also converges to the boundary point~$\omega$.
Hence, according to Proposition~7.2 in~\cite{surGhyp-d'apresGromov} appropriate subrays $r'$ and $g'$ of the respective rays~$r$ and~$g$ have Hausdorff distance at most~$16\delta$. Then for each point on $r'$, the ball with radius~$16\delta$ centred on that point intersects $g'$, and for each point on $g'$ the ball with radius~$H$ centred on that point intersects $f$. Hence the claim holds with $R = H + 16\delta$. \hspace*{\fill}$\square$ \end{proof}
Finally, we cover the last two cases of the classification.
\begin{proposition}\label{prop:flatG-fix-1OR2boundary-pts=>rk le1} Let $G$ be a totally disconnected, locally compact, hyperbolic group and $H$ a flat subgroup of~$G$ that fixes a boundary point or a pair of boundary points (not necessarily pointwise).
Then the rank of~$H$ is at most~$1$. \end{proposition}
\begin{proof} We first reduce to the case where the flat subgroup~$H$ fixes a boundary point.
The other case is where~$H$ fixes a pair of distinct boundary points without fixing the points.
Then the subgroup of~$H$ that fixes both points is also flat and has index~$2$ in~$H$. Hence this subgroup has the same rank as~$H$ and it suffices to prove the claim for it.
Next we show that the images of any two hyperbolic elements, $g$ and $h$ say, that both fix a boundary point satisfy a nontrivial relation
in~$H/H_1$, thus showing that $H$ can not contain two elements mapping to linearly independent elements
in the quotient and thereby finishing the proof.
Inverting one of $g$, $h$ if necessary, we may assume that $g$ and $h$ have the same attracting boundary point, $\omega$ say.
Choose a vertex, $v$, in a rough Cayley graph, $\Gamma$, for~$G$.
The map $f_h\colon \mathbb{N}\to \Gamma$ defined by $f_h(n):=h^n.v$ is a quasi-geodesic ray, with quasi-isometry-constants $(\lambda, c)$ say.
Then all the maps $g^i.f_h$ with $i\geq0$ are $(\lambda, c)$-quasi-geodesic rays and converge to~$\omega$.
Choose a geodesic ray, $r$ say, ending in~$\omega$.
Denote by~$n$ a natural number larger than the maximal number of vertices of~$\Gamma$ that are contained in any closed ball whose radius is the constant~$R$ provided by Lemma~\ref{lem:q-geodesic rays hit close near their boundary point}.
Such a number~$n$ exists, because $G$ acts cocompactly on its rough Cayley graph~$\Gamma$.
According to Lemma~\ref{lem:q-geodesic rays hit close near their boundary point}, we may choose a point sufficiently far out on the ray~$r$ such that all the quasi--geodesic rays $g.f_h,\ldots, g^n.f_h$ intersect the ball~$B$ of radius~$R$ around it.
All points of intersection of $g.f_h,\ldots, g^n.f_h$ with~$B$ are vertices and so, by our choice of~$n$, there are integers~$i$ and~$j$ with $0<i<j$ such that $g^i(h^p.v)=g^j(h^q.v)$ for some integers~$p$ and~$q$.
The element~$h^{-q}g^{i-j}h^p$ fixes~$v$, hence is elliptic and has scale~$1$.
The relation $(p-q)\,hH_1+(i-j)\,gH_1=0$ therefore holds in~$H/H_1$ and,
since $j-i\neq0$, $hH_1$ and $gH_1$ are linearly dependent. \hspace*{\fill}$\square$ \end{proof}
The main result of the paper is now obtained by combining these cases.
\begin{theorem} The flat-rank\,\! of a totally disconnected, locally compact, hyperbolic group is at most~$1$. \end{theorem}
\begin{proof} Let $G$ be a totally disconnected, locally compact, hyperbolic group and $H$ a flat subgroup of~$G$.
Choose a connected, locally finite, rough Cayley graph, say~$\Gamma$, for~$G$.
The graph~$\Gamma$ is Gromov-hyperbolic and \S4C of~\cite{woess} explains how the results of that paper apply to $\Gamma$ and its hyperbolic compactification. In particular, \cite[Theorem~3]{woess} lists the possible types of actions for~$H$ on $\Gamma$.
Each of these possible types is covered by either Proposition~\ref{prop:F2<flatG=>rk=0} (type~$(a)$), Proposition~\ref{prop:flatG-stab-nonempty-cpS=>rk=0} (type~$(b)$) or Proposition~\ref{prop:flatG-fix-1OR2boundary-pts=>rk le1} (types~$(c)$ and~$(d)$) and the rank of~$H$ is seen to be at most~$1$ in all cases. \hspace*{\fill}$\square$ \end{proof}
\section{Flat subgroups, space of directions and hyperbolic boundary}\label{sec:furtherProp(flatGs)}
In this section we present the proof of Theorem~\ref{thm:flatrk-via-directions}, which shows that discreteness of the space of directions (defined in \cite{direction(aut(tdlcG))}) also imposes a bound of~1 on the flat-rank\,\!. The proof is followed by two conjectures that propose further links between flat subgroups of a hyperbolic, totally disconnected, locally compact group, its space of directions and hyperbolic boundary.
\begin{proof}[of {Theorem~\ref{thm:flatrk-via-directions}}] \label{pref:pf(thm:flatrk-via-directions)} The space of directions of a totally disconnected, locally compact group of flat-rank\,\!~$k$ contains a $k$-cell by Proposition~23 in~\cite{direction(aut(tdlcG))}.
Hence a group with a discrete space of directions can have flat-rank\,\! at most~$1$.
Furthermore, since a group has flat-rank\,\!~$0$ if and only if its space of directions is empty, we even have that a group with a non-empty, discrete space of directions has flat-rank\,\! equal to~$1$. \hspace*{\fill}$\square$ \end{proof}
The argument in the above proof in fact shows that the flat-rank\,\! of a group is bounded by the dimension of the space of directions. This pseudo-metric space need not be finite dimensional manifold however, as the example (a group having flat-rank\,\!~1) in~\cite[5.2.3]{direction(aut(tdlcG))} illustrates.
Theorem~\ref{thm:flatrk-via-directions} applies to closed subgroup of the automorphism group of a locally finite tree, by Proposition~36(2) in~\cite{direction(aut(tdlcG))}.
In fact, the bound on the flat-rank of such groups also follows from Theorem~\ref{MainThm-variant} in case they are compactly generated, because they are then hyperbolic.\\ \begin{proposition} \label{prop:tree-->hyperbolic} Let $G$ be a compactly generated topological group acting minimally on a
tree, $X$, such that the stabilizers of vertices are open subgroups of $G$.
Then $G$ acts with finitely many orbits on the vertices. \end{proposition} \begin{proof} The argument follows that of \cite[Proposition~7.9(b)]{Bass} which establishes the claim for discrete $G$. Let $\mathbf{A}$ denote the graph of groups arising from the
action of $G$ on $X$. Let $G_{\mathbf{B}}$ denote the fundamental group of a subgraph of groups $\mathbf{B}$ of $\mathbf{A}$. Note that $G_{\mathbf{B}}$ is always an open subgroup of~$G$. Clearly the groups $G_{\mathbf{B}}$, where $G_{\mathbf{B}}$ ranges over all finite subgraphs of groups form an open covering of $G$. Because $G$ has a a compact generating set we see that finitely many of the groups $G_{\mathbf{B}}$, with $\mathbf{B}$ a finite subgraph of groups, cover the generating set. Indeed, one sees from this that there is a finite subgraph of groups, $\mathbf{A}'$,
such that the fundemental group of $\mathbf{A}'$ contains the generating set and thus
the fundamental group of $\mathbf{A}'$ is equal to $G$. By \cite[Proposition~7.12]{Bass} we can now conclude that because the action is minimal that $\mathbf{A}' =\mathbf{A}$. The graph of groups $\mathbf{A}'$ is finite and thus $\mathbf{A}$ is also finite and hence the group $G$ has only finitely many orbits on both the vertices and edges of $X$. \hspace*{\fill}$\square$ \end{proof} \begin{corollary} \label{cor:tree-->hyperbolic} Let $G$ be a compactly generated group acting on a tree, $X$, such that the stabilizers of vertices
are compact open subgroups of $G$. Then $G$ is hyperbolic. \end{corollary} \begin{proof} If $G$ consists only of elliptic elements, then, since $G$ is compactly generated, \cite[Proposition~7.2]{Bass} implies that it is compact. Hence $G$ is in this case trivially hyperbolic.
Otherwise, $G$ contains a hyperbolic element and the union of all axes of all hyperbolic elements is a minimal $G$-invariant subtree of $X$. Replacing $X$ by this subtree and $G$ by its quotient by the (compact) stabilizer of this subtree, it may be assumed that the action is minimal. By Proposition~\ref{prop:tree-->hyperbolic}, $G$ acts with only finitely many orbits on the edges of $X$. Hence $X$ is locally finite and is quasi-isometric to a rough Cayley graph of $G$, which must therefore be hyperbolic. \hspace*{\fill}$\square$ \end{proof}
The following conjecture asks for a common extension of our Theorem~\ref{MainThm-variant} and Proposition~36(2) in~\cite{direction(aut(tdlcG))}.
\begin{conjecture} \label{conj:dir(hypGs)} Let $G$ be a hyperbolic, totally disconnected, locally compact group.
Then the following holds. \begin{enumerate} \item The map which assigns each element of non-trivial scale to its attracting boundary point defines an injection of the set of directions of~$G$ into the hyperbolic boundary. \item Elements of~$G$ with distinct directions have pseudo-distance 2. \end{enumerate} \end{conjecture}
The next conjecture asks whether the relationship between flat subgroups and the geometry of the rough Cayley graph that may be observed in automorphism groups of trees or in the setting of~\cite{flatrk(AutGs(buildings))} holds for hyperbolic totally disconnected, locally compact groups in general.
\begin{conjecture} \label{conj:struc(Gs(flatrk1))} Suppose that~$G$ is a hyperbolic, totally disconnected, locally compact group which does not fix a point of its hyperbolic boundary.
Then every flat subgroup of flat-rank\,\!~$1$, $H$ say, of~$G$ has a limit set that contains $2$ elements, which are both fixed by $H$.
The group~$H_1$ is relatively compact and is equal to the set of elliptic elements of~$H$. \end{conjecture}
The necessity of the hypothesis that the group should not fix a point on the hyperbolic boundary is shown by the following example.
\begin{example} \label{ex:flatrk1-fixes-single-boundary-point} Let $G$ be the semidirect product ${\mathbb Z}\ltimes \mathbb{F}_q((t))$, where $\mathbb{F}_q((t))$ is the ring of formal Laurent series over the finite field $\mathbb{F}_q$ and ${\mathbb Z}$ acts by multiplication by $t$. This group is isomorphic to the group of matrices of the form $\left(\begin{array}{cc}t^n &f \\0 & 1\end{array}\right)$ where $n\in {\mathbb Z}$ and $f\in \mathbb{F}_q((t))$. Then $G$ acts faithfully and co-compactly on the Bruhat-Tits-tree of $SL_2(\mathbb{F}_q((t)))$, a homogeneous tree where every vertex has valency $q+1$, and so is a hyperbolic group by Proposition~\ref{prop:hyperbolicity(cocp-<Gs(Aut(lf-Gromov-hyp-complexes))}. (Although $G$ is not a subgroup of $SL_2(\mathbb{F}_q((t)))$, it does act on this group by conjugation and this action induces an action on the tree.)
Put $O$ equal to $\mathbb{F}_q((t))$, a compact open subgroup of $G$. Direct calculation shows that, if $g=(n,f)\in G$, then $|gOg^{-1} : gOg^{-1}\cap O|$ is equal to~1 if $n\geq0$ and to $q^{-n}$ if $n<0$ and that these are the minimum possible. Hence $O$ is minimizing for $G$ and $G$ is flat.
However $G$ does not satisfy the last hypothesis of the conjecture because it fixes a point on the hyperbolic boundary, in this case the set of ends of the tree. It does not satisfy the conclusions because there is no other end of the tree fixed by $G$. Furthermore, as the above calculation shows, $G_1 = \mathbb{F}_q((t))$ which is the set of elliptic elements in $G$, and this group
is not compact. \end{example}
\section{Examples of simple groups of flat rank at most~$1$}\label{sec:examples}
Here we list examples of \textit{simple}, totally disconnected, locally compact groups whose flat rank is at most~$1$.
We expect none of the listed groups to have flat-rank\,\!~$0$.
Indeed, in any given case it is usually easy to exhibit a hyperbolic element of non-trivial scale.
In~\cite{arbre}, Tits showed that many closed subgroups of automorphism groups of locally finite trees are simple and provided concrete constructions of examples in terms of $a$-coverings.
As seen in the previous section, these groups have flat-rank\,\! at most~$1$.
Haglund and Paulin, in~\cite{simpl(G-aut(courb-))}, adapted Tits' methods to automorphism groups of negatively curved complexes, thus providing many more totally disconnected, locally compact, non-discrete, non-linear, simple groups.
In order to be able to formulate an analogue of Tits' property~(P), a central assumption in~\cite{arbre}, they introduced an axiomatic framework, namely, spaces with walls, which allowed them to generalize Tits' result to groups acting on hyperbolic spaces with walls; see Th\'eor\`eme~6.1 in their paper.
The groups studied by Haglund and Paulin are non-discrete under fairly general conditions; compare their Lemme~3.6.
We suspect that all non-discrete groups~$G^+$, where~$G$ satisfies the conditions of~\cite[Th\'eor\`eme~6.1]{simpl(G-aut(courb-))}, act cocompactly on the hyperbolic graph associated to the space with walls in the statement of that theorem.
If so, such a group~$G^+$ is a totally disconnected, locally compact, non-discrete, simple, hyperbolic group as a consequence of Proposition~\ref{prop:hyperbolicity(cocp-<Gs(Aut(lf-Gromov-hyp-complexes))} and thus has flat-rank\,\! at most~$1$ by Theorem~\ref{MainThm-variant}.
While there is some uncertainty as to whether the group~$G^+$ associated to a general group satisfying the conditions of Th\'eor\`eme~6.1 in Haglund and Paulin's paper acts cocompactly, all concrete examples given in that paper do act cocompactly.
These examples are \begin{enumerate}
\item the group of type-preserving automorphisms of a Bourdon building (Th\'eo\-r\`eme~1.1);
\item a subgroup of finite index in the automorphism group of a Benakli-Haglund building (Th\'eo\-r\`eme~1.2);
\item a subgroup of finite index in the automorphism group of the Cayley graph of a hyperbolic, non-rigid Weyl group with respect to its standard system of generators (Th\'eor\`eme~1.3);
\item subgroups of finite index in the automorphism groups of certain even polyhedral complexes (Th\'eor\`eme~1.4). \end{enumerate}
That these concrete examples do act cocompactly is no accident.
It is difficult to construct complexes with uncountable automorphism groups; most constructions of such complexes start from a discrete group acting cocompactly on some complex.
A notable exception is the horocyclic product of two locally finite trees with different valencies; while the automorphism group of this complex acts cocompactly, there is no discrete subgroup doing the same as shown in~\cite{qisos+rig(solvGs)}.
\section{Hyperbolic orbits for groups of automorphisms} \label{sec:hyperbolic orbits}
The idea of the proof of Theorem~\ref{thm:altMainThm} is straightforward. If $\mathcal{A}$ contains a flat subgroup whose rank is~$2$, then there is an orbit of $\mathcal{A}$ which contains a subset which `looks like' $\mathbb{Z}^2$; this is inconsistent with the assumption that $\mathcal{A}$ has a hyperbolic orbit.
Since we are now dealing with spaces which are not geodesic, we must use the general definition of $\delta$-hyperbolic space in terms of the Gromov-product; recall that this states that a metric space~$X$ is $\delta$-hyperbolic if and only if \[ (x\cdot y)_w\ge \min\{(x\cdot z)_w,(y\cdot z)_w\}-\delta \] for all $w$, $x$, $y$, $z\in X$ \cite[III.H.1.20]{GMW319}.
For our argument, we first need to be able to choose the orbit at our convenience, as the extent to which a flat subgroup of flat-rank\, $2$ `looks like' $\mathbb{Z}^2$ depends on the orbit.
The following two results take care of that problem.
\begin{lemma}\label{orbits_quasi-iso} Let a group $\mathcal{A}$ act by isometries on a metric space $\mathbf{B}$. Then any two orbits of $\mathcal{A}$ in $\mathbf{B}$ are $(1,\epsilon)$-quasi-isometric, with $\epsilon$ only depending on
the pair of orbits. \end{lemma}
\begin{proof} Fix two points, $M$ and $N$ say, in $\mathbf{B}$.
For every $P\in \mathcal{A}.M$ choose an element $\alpha_P\in\mathcal{A}$ such that $P=\alpha_P.M$. Once this choice is made, define a map $\tau_{M,N}\colon \mathcal{A}.M\to \mathcal{A}.N$ which maps $P$ to $\alpha_P.N$. Then, for $A$ and $B$ in $\mathcal{A}.M$ we have \begin{align*} d(A,B) &\leq d(\alpha_A.M,\alpha_A.N)+d(\alpha_A.N,\alpha_B.N)+d(\alpha_B.N,\alpha_B.M)\\ &=d(\alpha_A.N,\alpha_B.N)+2 d(M,N)\,, \end{align*} and \begin{align*} d(\alpha_A.N,\alpha_B.N) &\leq d(\alpha_A.N,\alpha_A.M)+d(\alpha_A.M,\alpha_B.M)+d(\alpha_B.M,\alpha_B.N)\\ &= d(A,B) + 2 d(M,N) \end{align*} hence
$|d(\tau_{M,N}(A),\tau_{M,N}(B))-d(A,B)|\leq 2 d(M,N)$, which shows that $\tau_{M,N}$ is a $(1,2 d(M,N))$-quasi-isometric embedding. Further, we have \begin{align*} d(\tau_{M,N}&(\alpha.M),\alpha.N) = d(\alpha_{\alpha.M}.N,\alpha.N)\\ &\leq d(\alpha_{\alpha.M}.N,\alpha_{\alpha.M}.M)+ d(\alpha_{\alpha.M}.M,\alpha.M)+ d(\alpha.M,\alpha.N)\\ &= 2d(M,N)\,, \end{align*} showing that $\tau_{M,N}$ is $2 d(M,N)$-dense. \hspace*{\fill}$\square$ \end{proof}
\begin{corollary}\label{some-orbit_hyp=all-orbits_hyp} If one orbit of $\mathcal{A}$ in $\bs G.$ is hyperbolic, then all are. \hspace*{\fill}$\square$ \end{corollary}
\begin{lemma}\label{(1,-)-qi-embed preserves_hyp} Let $f\colon \mathbf{B}_1\to \mathbf{B}_2$ be a $(1,\epsilon)$-quasi-isometric embedding. If $\mathbf{B}_2$ is $\delta$-hyperbolic, then $\mathbf{B}_1$ is $(\delta+3\epsilon)$-hyperbolic. \end{lemma}
\begin{proof} From our assumption $
|d(x,y) - d(f(x),f(y))|\leq \epsilon $ for any pair, $x$ and $y$, of points in $\mathbf{B}_1$, we infer that \[
|(x\cdot y)_o - (f(x)\cdot f(y))_{f(o)}|\leq 3\epsilon/2\,. \] We conclude that $(x\cdot z)_o-\min\{(x\cdot y)_o,(y\cdot z)_o\}$ differs from $$(f(x)\cdot f(z))_{f(o)}-\min\{(f(x)\cdot f(y))_{f(o)},(f(y)\cdot f(z))_{f(o)}\} $$ by at most $3\epsilon$. \hspace*{\fill}$\square$ \end{proof}
Taking $f$ in Lemma~\ref{(1,-)-qi-embed preserves_hyp} to be the inclusion of a subset we also obtain the following corollary.
\begin{corollary}\label{subspace(hyp)=hyp} Every subspace of a hyperbolic space is hyperbolic.
\hspace*{\fill}$\square$ \end{corollary}
Corollaries~\ref{some-orbit_hyp=all-orbits_hyp} and~\ref{subspace(hyp)=hyp} will prove Theorem~\ref{thm:altMainThm}, once we show that the presence of a flat group of rank~$r\ge 2$ implies the existence of a subspace of~$\bs G.$ that is an integer lattice in a normed space of dimension~$r$.
For completeness, the proof that such a lattice is not hyperbolic is outlined after the next result.
\begin{lemma}\label{norm>flat-gp} Let $\mathcal{H}$ be a flat group of automorphisms of the totally disconnected locally compact group $G$ with rank~$r$. Let $O$ be minimizing for $\mathcal{H}$. Then $\left(\mathcal{H}.O,d\right)$ is isometric to a lattice in in a normed real linear space of dimension~$r$. \end{lemma}
\begin{proof} Since $O$ is minimizing for $\mathcal{H}$, an automorphism $\alpha\in\mathcal{H}$ satisfies $\alpha.O = O$ if and only if $\alpha\in \mathcal{H}_1$ and the map $\alpha\mapsto \alpha.O$ induces a bijection $$ \mathcal{H}.O \to \mathcal{H}/\mathcal{H}_1. $$ Composition with the isomorphism $\mathcal{H}/\mathcal{H}_1\to \mathbb{Z}^r$ then produces a bijection $\mathcal{H}.O \to \mathbb{Z}^r$. The metric on $\mathcal{H}.O$ pushes forward to $\mathbb{Z}^r$ via this bijection and the resulting metric on $\mathbb{Z}^r$ is translation-invariant because $\mathcal{H}/\mathcal{H}_1$ and $\mathbb{Z}^r$ are isomorphic as groups.
An explicit formula may be given for the distance $d(O,\alpha.O)$ for $\alpha\in \mathcal{H}$.
There are: a finite set $\Phi=\Phi(\mathcal{H},G)$ of surjective homomorphisms $\rho\colon {\mathcal H} \to {\mathbb Z}$ such that the intersection of the kernels of elements in $\Phi$ equals $\mathcal{H}_1$; and a set $\left\{t_\rho\mid \rho\in\Phi\right\}$ of integers greater than one such that \[ s_G(\alpha) = \prod_{\rho\in\Phi,\,\rho(\alpha)>0} t_\rho^{\rho(\alpha)}\,,\ (\alpha\in\mathcal{H}), \] see~\cite[Theorems 6.12 and 6.14]{tidy<:commAut(tdlcG)}.
Since $O$ is minimizing for $\mathcal{H}$, we further have that $d(\alpha(O),O)=\log\bigl(s_G(\alpha)\cdot s_G(\alpha^{-1})\bigr)$, whence \begin{equation*} \label{eq:normform}
d(\alpha(O),O) = \sum_{\rho\in\Phi} \log(t_\rho)\,|\rho(\alpha)|\,, \ (\alpha\in\mathcal{H}). \end{equation*} Composition of each $\rho\in\Phi$ with the isomorphism $\mathcal{H}/\mathcal{H}_1\to \mathbb{Z}^r$ yields homomorphisms $\tilde\rho : \mathbb{Z}^r \to \mathbb{Z}$ and there are $\mathbf{z}_\rho\in \mathbb{Z}^r$ such that $\tilde\rho(\mathbf{z}) = \mathbf{z}_\rho.\mathbf{z}$ for each $\rho\in\Phi$. Hence the bijection $\mathcal{H}.O \to \mathbb{Z}^r$ becomes an isometry if $\mathbb{Z}^r$ is equipped with the translation-invariant metric \begin{equation} \label{eq:normform2}
\tilde d(\mathbf{w},\mathbf{z}) := \sum_{\rho\in\Phi} \log(t_\rho)\,| \mathbf{z}_\rho.(\mathbf{w}-\mathbf{z})|\,, \ (\mathbf{w},\,\mathbf{z}\in\mathbb{Z}^r). \end{equation} This metric extends to $\mathbb{R}^r$ by the same formula. \hspace*{\fill}$\square$ \end{proof}
We conclude with a sketch of the argument that a lattice~$X$ in a normed space of dimension~$r\ge 2$ is not hyperbolic.
Given $4$ points ${x}$, ${y}$, ${z}$ and ${w}$ in~$X$ let $\delta({x},{y},{z})_{w}$ denote the quantity $ \min\{ ({y}\cdot {x})_{w}, ({x}\cdot {z})_{w} \} -({y}\cdot {z})_{w} $. The number $\delta({x},{y},{z})_{w}$ is a lower bound for any $\delta$ such that $X$ could be $\delta$-hyperbolic.
But $\delta( \lambda{x},\lambda{y},\lambda{z} )_{\lambda{w}}=
|\lambda|\delta({x},{y},{z})_{w}$ for any $\lambda\in\mathbb{Z}$, showing that no such $\delta$ can exist, if we can find a quadruple $({x}, {y}, {z}, {w})$ such that $\delta({x},{y},{z})_{w}$ is positive.
If~$x$ and~$y$ are vectors for which
$\|x+y\|+\|x-y\|>\|x\|+\|y\|$ then $\delta({x},{y},{z})_{\mathbf{0}}$ is positive.
Such vectors exist for any normed linear space of dimension at least~$2$; vectors in the lattice can be found by rational approximation followed by scaling by integers.
\end{document} | arXiv |
Augmented triangular prism
In geometry, the augmented triangular prism is one of the Johnson solids (J49). As the name suggests, it can be constructed by augmenting a triangular prism by attaching a square pyramid (J1) to one of its equatorial faces. The resulting solid bears a superficial resemblance to the gyrobifastigium (J26), the difference being that the latter is constructed by attaching a second triangular prism, rather than a square pyramid.
Augmented triangular prism
TypeJohnson
J48 – J49 – J50
Faces3x2 triangles
2 squares
Edges13
Vertices7
Vertex configuration2(3.42)
1(34)
4(33.4)
Symmetry groupC2v
Dual polyhedronmonolaterotruncated triangular bipyramid
Propertiesconvex
Net
It is also the vertex figure of the nonuniform 2-p duoantiprism (if p ≥ 2). Despite the fact that p = 3 would yield a geometrically identical equivalent to the Johnson solid, it lacks a circumscribed sphere that touches all vertices.
Its dual, a triangular bipyramid with one of its 4-valence vertices truncated, can be found as cells of the 2-p duoantitegums (duals of the 2-p duoantiprisms).
A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]
External links
• Weisstein, Eric W. "Johnson Solid". MathWorld.
• Weisstein, Eric W. "Augmented triangular prism". MathWorld.
Johnson solids
Pyramids, cupolae and rotundae
• square pyramid
• pentagonal pyramid
• triangular cupola
• square cupola
• pentagonal cupola
• pentagonal rotunda
Modified pyramids
• elongated triangular pyramid
• elongated square pyramid
• elongated pentagonal pyramid
• gyroelongated square pyramid
• gyroelongated pentagonal pyramid
• triangular bipyramid
• pentagonal bipyramid
• elongated triangular bipyramid
• elongated square bipyramid
• elongated pentagonal bipyramid
• gyroelongated square bipyramid
Modified cupolae and rotundae
• elongated triangular cupola
• elongated square cupola
• elongated pentagonal cupola
• elongated pentagonal rotunda
• gyroelongated triangular cupola
• gyroelongated square cupola
• gyroelongated pentagonal cupola
• gyroelongated pentagonal rotunda
• gyrobifastigium
• triangular orthobicupola
• square orthobicupola
• square gyrobicupola
• pentagonal orthobicupola
• pentagonal gyrobicupola
• pentagonal orthocupolarotunda
• pentagonal gyrocupolarotunda
• pentagonal orthobirotunda
• elongated triangular orthobicupola
• elongated triangular gyrobicupola
• elongated square gyrobicupola
• elongated pentagonal orthobicupola
• elongated pentagonal gyrobicupola
• elongated pentagonal orthocupolarotunda
• elongated pentagonal gyrocupolarotunda
• elongated pentagonal orthobirotunda
• elongated pentagonal gyrobirotunda
• gyroelongated triangular bicupola
• gyroelongated square bicupola
• gyroelongated pentagonal bicupola
• gyroelongated pentagonal cupolarotunda
• gyroelongated pentagonal birotunda
Augmented prisms
• augmented triangular prism
• biaugmented triangular prism
• triaugmented triangular prism
• augmented pentagonal prism
• biaugmented pentagonal prism
• augmented hexagonal prism
• parabiaugmented hexagonal prism
• metabiaugmented hexagonal prism
• triaugmented hexagonal prism
Modified Platonic solids
• augmented dodecahedron
• parabiaugmented dodecahedron
• metabiaugmented dodecahedron
• triaugmented dodecahedron
• metabidiminished icosahedron
• tridiminished icosahedron
• augmented tridiminished icosahedron
Modified Archimedean solids
• augmented truncated tetrahedron
• augmented truncated cube
• biaugmented truncated cube
• augmented truncated dodecahedron
• parabiaugmented truncated dodecahedron
• metabiaugmented truncated dodecahedron
• triaugmented truncated dodecahedron
• gyrate rhombicosidodecahedron
• parabigyrate rhombicosidodecahedron
• metabigyrate rhombicosidodecahedron
• trigyrate rhombicosidodecahedron
• diminished rhombicosidodecahedron
• paragyrate diminished rhombicosidodecahedron
• metagyrate diminished rhombicosidodecahedron
• bigyrate diminished rhombicosidodecahedron
• parabidiminished rhombicosidodecahedron
• metabidiminished rhombicosidodecahedron
• gyrate bidiminished rhombicosidodecahedron
• tridiminished rhombicosidodecahedron
Elementary solids
• snub disphenoid
• snub square antiprism
• sphenocorona
• augmented sphenocorona
• sphenomegacorona
• hebesphenomegacorona
• disphenocingulum
• bilunabirotunda
• triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)
1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
| Wikipedia |
# Lecture 12
March $30^{\text {th }}, 2004$
Remark. Since many of our results rely on the regularity of the Newtonian Potential, and hence use Proposition 2 of Lecture 9, we will assume througout that the Hölder constant $\alpha$ ranges in the open interval $(0,1)$.
## Review from last time
Regularity Theorem. $\quad B \subseteq \mathbb{R}^{n}$, a ball, $u \in C^{2}(B) \cap C^{0}(\bar{B}), f \in C^{\alpha}(\bar{B})$, with $0<\alpha<1$. Suppose $u$ solves Laplace's Equation: $\Delta u=f$ on $B, u=0$ on $\partial B$. Then $u \in \mathcal{C}^{2, \alpha}(\bar{B})$.
In the interior of $B$, just use estimates on the Newtonian Potential (NP) and on harmonic functions. On the boundary of $B$ use translation \& inversion maps to map ball to upper half plane with flat boundary. Then note that the estimates on the NP work upto the boundary and an inversion map is smooth away from the origin.
Corollary. $\varphi \in \mathcal{C}^{2, \alpha}(\bar{B}), f \in \mathcal{C}^{\alpha}(\bar{B})$, with $0<\alpha<1$.. Then Poisson's Equation: $\Delta u=f$ on $B, u=\varphi$ on $\partial B$, has a unique solution $u \in \mathcal{C}^{2, \alpha}(\bar{B})$.
By the above if we can solve for $v$ such that $\Delta v=f-\Delta \varphi$ on $B, v=0$ on $\partial B$, then $v \in \mathcal{C}^{2, \alpha}(\bar{B})$. Let $u:=v+\varphi \in \mathcal{C}^{2, \alpha}(\bar{B})$. This $u$ solves our original equation! So we just need to be able to solve uniquely the above homogeneous equation with a $\mathcal{C}^{2}(B) \cap \mathcal{C}^{0}(\bar{B})$ solution. Then the Theorem will guarantee it is actually $\mathcal{C}^{2, \alpha}(\bar{B})$.
In order to do that, set $w:=\mathrm{NP}(g)$, where $g:=f-\Delta \varphi \in \mathcal{C}^{\alpha}$ (as $f \in \mathcal{C}^{\alpha}, \varphi \in \mathcal{C}^{2, \alpha}$ ). Indeed $w \in \mathcal{C}^{2}(B) \cap \mathcal{C}^{0}(\bar{B})$ from the elementary properties of the Newtonian Potential. Furthermore $\Delta w=g$. If we could make sure somehow the boundary values would be 0 we would be done as all assumptions of the Theorem would hold. In order to do that, we need to find a function $h \in \mathcal{C}^{2}(B) \cap \mathcal{C}^{0}(\bar{B})$ solving $\left\{\begin{array}{ll}\Delta h=0 & \text { on } B, \\ h=-w & \text { on } \partial B\end{array}\right.$. And indeed by Poisson's Integral Formula we can do this. Letting $v:=w+h \in \mathcal{C}^{2}(B) \cap \mathcal{C}^{0}(\bar{B})$ we have indeed a the required solution for the homogeneous problem.
## Solving Poisson's/Laplace's equation with regularity upto the boundary on gen- eral domains
Suppose we are given an (open) domain $\Omega \subseteq \mathbb{R}^{n}$, different than a ball, or equivalently some open subset in a Riemannian manifold $(M, g)$, and that we would like to develope a similar theory for the Poisson and Laplace equations on these domains. In other words prove a priori estimates upto the boundary for these domains.
Localizing to a neighborhood in $\mathbb{R}^{n}$ of a point on the boundary $\partial \Omega$ intersected with $\bar{\Omega}$, we could map it to a neighborhood of $\overline{\mathbb{H}}:=\left\{x=\left(x_{1}, \ldots, x_{n}\right) \in \mathbb{R}^{n} \mid x_{n} \geq 0\right\}$. This localization is tantamount to working with the manifolds local coordinates, and then we must work with $\Delta_{g}$, the Riemannian Laplacian.
We see that indeed we will be able to extend our theory to these generalized domains once we show our boundary estimates hold for general elliptic operators.
## Constant coefficients operators
Let $L_{0} u(x)=A^{i j} \mathrm{D}_{i j} u(x)=f(x)$ with $A^{i j}$ a constant matrix satisfying $0<\lambda|v|^{2} \leq A^{i j} v_{i} v_{j} \leq$ $\Lambda|v|^{2}, \forall 0 \neq v \in \mathbb{R}^{n}$. This two-sided inequality will be referred to as uniform ellipticity.
Theorem. Let $u$ be as above and $0<\alpha<1$.
I. If $u \in \mathcal{C}^{2}(\Omega), f \in \mathcal{C}^{\alpha}(\Omega)$, then $\forall \Omega^{\prime} \Subset \Omega$ (i.e " $\Omega^{\prime}$ precompact in $\Omega$ ") there exists $C=$ $C\left(\lambda, \Lambda, \Omega^{\prime}, \Omega, n\right)$ such that
$$
\|u\|_{\mathcal{C}^{2, \alpha}\left(\Omega^{\prime}\right)} \leq C \cdot\left(\|u\|_{\mathcal{C}^{0}(\Omega)}+\|f\|_{\mathcal{C}^{\alpha}(\Omega)}\right)
$$
II. If $u \in \mathcal{C}^{2}(\Omega) \cap \mathcal{C}^{0}(\Omega \cup T), f \in \mathcal{C}^{\alpha}(\Omega \cup T)$ and $u=0$ on $T$, then $\forall \Omega^{\prime} \Subset \Omega$ there exists $C=C\left(\lambda, \Lambda, \Omega^{\prime}, \Omega, n\right)$ such that
$$
\|u\|_{\mathcal{C}^{2, \alpha}\left(\Omega^{\prime} \cup T^{\prime}\right)} \leq C \cdot\left(\|u\|_{\mathcal{C}^{0}(\Omega \cup T)}+\|f\|_{\mathcal{C}^{\alpha}(\Omega \cup T)}\right)
$$
where $T^{\prime}:=\Omega^{\prime} \cap T$. We assume that $T$ is a flat boundary portion (portion of a hyperplane in $\mathbb{R}^{n}$ ) contained in $\partial \Omega$.
Setup: Let $H$ be an invertible linear transformation represented by multiplication by a constant matrix $H_{k l}$, and let $H^{-1}$ denote its inverse. Being linear, by rotating if necessary, we may assume it maps the upper half space to itself, and that the flat boundary portion remains flat. Put $\tilde{u}:=u \circ H^{-1}$ and $y=H x$. Then $\tilde{u}: \Omega \longrightarrow \mathbb{R}$
$$
\mathrm{D}_{i} \tilde{u}(y)=\mathrm{D}_{l} u\left(H^{-1} y\right) \cdot H_{l i}^{-1}(\text { summation) }
$$
from D applied to $u\left(\left[\begin{array}{c}y_{1} \\ \vdots \\ y_{n}\end{array}\right] \cdot\left[H^{-1}\right]\right)=u\left(\left[y_{l} H_{l 1}^{-1}, \ldots, H_{l n}^{-1} y_{n}\right]\right)$. Then
$$
\begin{aligned}
& \mathrm{D}_{i} \mathrm{D}_{j} \tilde{u}(y)=\mathrm{D}_{k} \mathrm{D}_{l} u\left(H^{-1} y\right) H_{k j}^{-1} H_{l i}^{-1}=\left(H^{-1}\right)^{T} \cdot \mathrm{D}^{2} u\left(H^{-1} y\right) \cdot H^{-1}, \\
& \Rightarrow H^{T} \cdot \mathrm{D}^{2} \tilde{u}(y) \cdot H=\mathrm{D}^{2} u(x) .
\end{aligned}
$$
Plugging this into our elliptic equation we get $A^{l k} H_{i l} \mathrm{D}_{i} \mathrm{D}_{j} \tilde{u} H_{j k}=A^{l k} \mathrm{D}_{l k} u(x)=f(x)$, or $H_{i l} A^{l k} H_{j k} \mathrm{D}^{2} \tilde{u}=\left(H A H^{T}\right) \mathrm{D}^{2} \tilde{u}(y)=f(x)=f\left(H^{-1} y\right)=: \tilde{f}(y)$.
Choosing appropriate $H$ can diagonalize $A: H A H^{T}=\operatorname{diag}\left(\lambda_{1}, \ldots, \lambda_{n}\right)$. Set
$P:=H \operatorname{diag}\left(\lambda_{1}^{-\frac{1}{2}}, \ldots, \lambda_{n}^{-\frac{1}{2}}\right)$. Then $P A P^{T}=I$, and in the domain $H(\Omega)$, which has flat boundary, we get the simple Poisson equation $\Delta \tilde{u}=\tilde{f} \in \mathcal{C}^{\alpha}$. By the theory we developed earlier in the course for this equation on such domains, $\forall \tilde{\Omega}^{\prime} \Subset H(\Omega)$ we have the interior estimates
$$
\|\tilde{u}\|_{\mathcal{C}^{2, \alpha}\left(\tilde{\Omega}^{\prime}\right)} \leq C \cdot\left(\|\tilde{u}\|_{\mathcal{C}^{0}(H(\Omega))}+\|\tilde{f}\|_{\mathcal{C}^{\alpha}(H(\Omega))}\right)
$$
Now $\|u\|_{\mathcal{C}^{2, \alpha}\left(\Omega^{\prime}\right)} \leq C \cdot\|\tilde{u}\|_{\mathcal{C}^{2, \alpha}\left(H\left(\Omega^{\prime}\right)\right)},\|u\|_{\mathcal{C}^{\alpha}\left(\Omega^{\prime}\right)} \leq C \cdot\|\tilde{u}\|_{\mathcal{C}^{\alpha}\left(H\left(\Omega^{\prime}\right)\right)}$, where we have used for the last two the identities $\|\tilde{g}\|_{\mathcal{C}^{0}(H(\Omega))}=\sup _{y \in H(\Omega)}|\tilde{g}(y)|=\sup _{x \in \Omega}|g(x)|=|| g \|_{\mathcal{C}^{0}(\Omega)}$ and
$$
\begin{gathered}
\left.|| \tilde{g}\right|_{\mathcal{C}^{\alpha}(H(\Omega))}=\sup _{y_{1} \neq y_{2} \in H(\Omega)} \frac{\left|\tilde{g}\left(y_{1}\right)-\tilde{g}\left(y_{2}\right)\right|}{\left|y_{1}-y_{2}\right|^{\alpha}}=\sup _{x_{1} \neq x_{2} \in \Omega} \frac{\left|g\left(x_{1}\right)-g\left(x_{2}\right)\right|}{\left|H x_{1}-H x_{2}\right|^{\alpha}} \\
=\sup _{x_{1} \neq x_{2} \in \Omega} \frac{\left|g\left(x_{1}\right)-g\left(x_{2}\right)\right|}{\left|x_{1}-x_{2}\right|^{\alpha}}\left(\frac{\left|x_{1}-x_{2}\right|}{\left|H x_{1}-H x_{2}\right|}\right)^{\alpha} \leq\left.|| g\right|_{\mathcal{C}^{\alpha}(\Omega)} \cdot(\text { smallest eigenvalue of } H)^{-1} .
\end{gathered}
$$
Here we use $H$ is a diffeomorphism. The $\mathcal{C}^{2, \alpha}$ inequality follows similarly using $H^{T} \mathrm{D}^{2} \tilde{u}(y) H=$ $\mathrm{D}^{2} u(x)$. Note that since $H, H^{-1}$ are both strictly positive, the above inequalities can be shown to hold in both directions (with different constants). That is to say all norms of $\tilde{u}$ are equivalent to those of $u$. This observation combined with the above interior estimates for $\tilde{u}$ gives us interior estimates for $u$ in $\Omega^{\prime}$.
As for boundary estimates (part II of the Theorem): we have seen that we can assume WLOG that $H$ maps the upper half plane to itself. Then our above inequalities for equivalence of the norms extend to the boundary of course, and since our theory (Lecture 11) gives boundary estimates for $\tilde{u}$ we are done.
## Interpolation
Theorem. Let $\Omega^{\prime} \Subset \Omega, u \in \mathcal{C}^{2, \alpha}(\Omega)$, with $0<\alpha<1$. For any $\epsilon>0, \exists C(\epsilon)$ such that
$$
|u|_{C^{k, \beta}\left(\Omega^{\prime}\right)} \leq C(\epsilon) \cdot|u|_{C^{o}(\Omega)}+\epsilon \cdot|u|_{C^{2, \alpha}\left(\Omega^{\prime}\right)}
$$
Note these are the semi-norms not the full norms! Also, as $\epsilon \rightarrow 0, C(\epsilon) \rightarrow \infty$.
The case $k=1, \beta=0$. Let $\bar{x}, x^{\prime} \in \Omega^{\prime}, x^{\prime \prime} \in \Omega \backslash \Omega^{\prime}$, such that all three points lie on a single line segment parallel to the $x^{i}$ axis and such that $\mathrm{D}_{i} u(\bar{x})=\frac{u\left(x^{\prime \prime}\right)-u\left(x^{\prime}\right)}{2 \epsilon} \leq \frac{2|u|_{C^{0}(\Omega)}}{2 \epsilon}$. From the fact that $x^{\prime \prime}$ is not in $\Omega^{\prime}$ we will get a global $\mathcal{C}^{0}$ norm involved (i.e norm over all $\Omega$ instead of just over $\Omega^{\prime}$ ). Now let $x \in \Omega^{\prime}$,
$$
\int_{x}^{\bar{x}} \mathrm{D}_{i i} u=\mathrm{D}_{i}(\bar{x})-\mathrm{D}_{i}(x)
$$
from which follows
$$
\left|\mathrm{D}_{i}(x)\right| \leq\left|\mathrm{D}_{i}(\bar{x})\right|+\left|\int_{x}^{\bar{x}} \mathrm{D}_{i i} u\right| \leq \frac{1}{\epsilon}|u|_{C^{0}(\Omega)}+\max _{\substack{x \in \text { segment } \overline{x \bar{x}} \\ \text { in } x^{i} \text { direction }}} \mathrm{D}_{i i} u \cdot|x-\bar{x}| \leq \frac{1}{\epsilon}|u|_{C^{0}(\Omega)}+\epsilon \cdot|u|_{C^{2}\left(\Omega^{\prime}\right)} .
$$
The case $k=2, \beta=0$. Fix $i$ and look at $\mathrm{D}_{i} u$. Again choose points on a segment in the $x^{l}$ direction such that $\mathrm{D}_{l i} u(\bar{x})=\frac{\mathrm{D}_{i} u\left(x^{\prime \prime}\right)-\mathrm{D}_{i} u\left(x^{\prime}\right)}{2 \epsilon} \leq \frac{2|\mathrm{D} u|_{C^{0}(\text { segment })}}{2 \epsilon}$. Now
$$
\left|\mathrm{D}_{l i} u(x)\right| \leq\left|\mathrm{D}_{l i} u(\bar{x})\right|+\left|\mathrm{D}_{l i} u(x)-\mathrm{D}_{l i} u(\bar{x})\right| \leq \frac{1}{\epsilon}|\mathrm{D} u|_{C^{0}(\text { segment })}+\left|\mathrm{D}^{2} u\right|_{\mathcal{C}^{2}(\Omega)} \cdot|x-\bar{x}|^{\alpha},
$$
which by the first case is
$$
\leq \frac{1}{\epsilon} \cdot\left(\frac{1}{\epsilon^{\prime}}|u|_{C^{0}(\Omega)}+\epsilon^{\prime} \cdot|u|_{C^{2}(\Omega)}\right)+\epsilon^{a} \cdot|u|_{C^{2, \alpha}(\Omega)}
$$
hence
$$
|u|_{C^{2}\left(\Omega^{\prime}\right)} \leq C \cdot|u|_{C^{0}(\Omega)}+C^{\prime} \cdot \epsilon^{a} \cdot|u|_{C^{2, \alpha}\left(\Omega^{\prime}\right)}
$$
For the cases to follow let $x, y \in \Omega^{\prime}$ and denote by $\bar{x}$ a point (to be chosen later) on the line segment $\overline{x y}$.
The case $k=0, \beta \in(0,1]$. We note that we can bound $|u|_{C^{0, \beta}\left(\Omega^{\prime}\right)}$ in a simple manner since
$$
\frac{u(x)-u(y)}{|x-y|^{\beta}} \leq\left\{\begin{array}{l}
\frac{\mathrm{D} u(\bar{x}) \cdot|x-y|}{|x-y|^{\beta}} \leq|u|_{C^{1}} \cdot|x-y|^{1-\beta} \leq|u|_{C^{1}} \cdot \epsilon^{1-\beta}, \text { if }|x-y| \leq \epsilon, \\
\frac{2|u|_{C^{0}}}{\epsilon^{\beta}}, \text { if }|x-y|>\epsilon .
\end{array}\right.
$$
or in other words
$$
|u|_{C^{0, \beta}\left(\Omega^{\prime}\right)} \leq \frac{2|u|_{C^{0}}}{\epsilon^{\beta}} \cdot|u|_{C^{0}(\Omega)}+\epsilon^{1-\beta} \cdot|u|_{C^{1}\left(\Omega^{\prime}\right)} .
$$
The case $k=1, \beta \in(0,1]$. We again note the dichotomy ( $\bar{x}$ as above)
$$
\frac{\mathrm{D}_{i} u(x)-\mathrm{D}_{i} u(y)}{|x-y|^{\beta}} \leq\left\{\begin{array}{l}
\frac{\mathrm{DD}_{i} u(\bar{x}) \cdot|x-y|}{|x-y|^{\beta}} \leq|u|_{C^{2}} \cdot|x-y|^{1-\beta} \leq|u|_{C^{2}} \cdot \epsilon^{1-\beta}, \text { if }|x-y| \leq \epsilon, \\
\frac{2|u|_{C^{1}}}{\epsilon^{\beta}}, \text { if }|x-y|>\epsilon .
\end{array}\right.
$$
or
$$
|u|_{C^{1, \beta}\left(\Omega^{\prime}\right)} \leq \frac{2|u|_{C^{1}}}{\epsilon^{\beta}} \cdot|u|_{C^{1}(\Omega)}+\epsilon^{1-\beta} \cdot|u|_{C^{2}(\Omega)} \cdot \leq C \cdot|u|_{C^{0}(\Omega)}+C^{\prime} \cdot|u|_{C^{2}\left(\Omega^{\prime}\right)}
$$
where in the last inequality we used one of the previous cases.
The case $k=2, \beta \in(0, \alpha)$. Once again
$$
\frac{\mathrm{D}_{i j} u(x)-\mathrm{D}_{i j} u(y)}{|x-y|^{\beta}}=\frac{\mathrm{D}_{i j} u(x)-\mathrm{D}_{i j} u(y)}{|x-y|^{\alpha}} \cdot|x-y|^{\alpha-\beta} \leq\left\{\begin{array}{l}
|u|_{C^{2, \alpha}\left(\Omega^{\prime}\right)} \epsilon^{\alpha-\beta}, \text { if }|x-y| \leq \epsilon \\
\frac{2|u|_{C^{2}(\Omega)}}{\epsilon^{\beta}}, \text { if }|x-y|>\epsilon
\end{array}\right.
$$
or
$$
|u|_{C^{2, \beta}\left(\Omega^{\prime}\right)} \leq \frac{2}{\epsilon^{\beta}} \cdot|u|_{C^{2}\left(\Omega^{\prime}\right)}+\epsilon^{\alpha-\beta} \cdot|u|_{C^{2, \alpha}\left(\Omega^{\prime}\right)} \cdot \leq C \cdot|u|_{C^{0}(\Omega)}+C^{\prime} \cdot|u|_{C^{2}(\alpha)} \Omega^{\prime}
$$
where in the last inequality we used one of the previous cases.
Remark. The Interpolation technique works also for $\Omega^{\prime} \Subset \Omega$ with flat boundary involved: we get inequalities with the flat boundary portion included, by the theory developed in Lecture 11 .
## Lecture 13
## April $1^{\text {st }}, 2004$
Now we would like to extend our estimates to general domains.
Proposition. Let $\Omega$ be $a \mathcal{C}^{2, \alpha}$ domain and $u \in \mathcal{C}^{2, \alpha}(\bar{\Omega})$, with $0<\alpha<1$. Given $\epsilon>0, \exists c=c(\epsilon, \Omega)$ s.t. for $k=0,1, \beta \in(0,1)$ and $k=2, \beta<\alpha$
$$
\|u\|_{C^{k, \beta}(\Omega)} \leq c \cdot\|u\|_{C^{0}(\Omega)}+\epsilon \cdot|u|_{C^{2, \alpha}(\Omega)}
$$
Note this is a global estimate, i.e. upto the boundary.
We will use the remark from last time concerning domains with a portion of a hyperplane on the boundary which will provide us with a needed estimate. We choose a $\mathcal{C}^{2, \alpha}$ injective function $\Psi$ which maps $B(x, r) \cap \Omega(x \in \partial \Omega)$ in such a manner as to map $B(x, r) \cap \partial \Omega$ onto a portion of a hyperplane. Its inverse $S:=\Psi^{-1}$ is also $\mathcal{C}^{2, \alpha}$. Set $\tilde{u}:=u \circ \Psi^{-1}$, the pulled-back function, and $T^{\prime}:=\Psi\left(B_{r / 2}(x) \cap \partial \Omega\right)$. For the domain in the image we have estimates as just mentioned
$$
|\tilde{u}|_{C^{k, \beta}\left(\Psi\left(B_{r / 2}(x)\right) \cap \tilde{T}\right)} \leq c(\epsilon) \cdot|\tilde{u}|_{C^{0}\left(\Psi\left(B_{r}(x)\right)\right)}+\epsilon \cdot|\tilde{u}|_{C^{2, \alpha}\left(\Psi\left(B_{r / 2}(x)\right) \cap \tilde{T}\right)} .
$$
Let $S=\left(S_{(1)}, \ldots, S_{(n)}\right)$. Now calculate
$$
\begin{aligned}
\mathrm{D}_{i} \tilde{u} & =\mathrm{D}_{l} u \cdot\left(S_{(l)}\right)_{i} \equiv \mathrm{D}_{l} u \cdot \mathrm{D}_{i}\left(S_{(l)}\right) \text { summation over } l \text { understood } \\
\mathrm{D}_{j} \mathrm{D}_{i} \tilde{u} & =\mathrm{D}_{k} \mathrm{D}_{l} u\left(S_{(k)}\right)_{j}\left(S_{(l)}\right)_{i}+\mathrm{D}_{l} u \mathrm{D}_{j}\left(S_{(l)}\right)_{i} \\
|u|_{C^{0}\left(B_{r}(x)\right)} & =|\tilde{u}|_{C^{0}\left(\Psi\left(B_{r}(x)\right)\right)} .
\end{aligned}
$$
Now
$$
\begin{aligned}
|\tilde{u}|_{C^{\alpha}\left(\Psi\left(B_{r}(x)\right)\right)} & =\sup _{y_{1} \neq y_{2}} \frac{\left|\tilde{u}\left(y_{1}\right)-\tilde{u}\left(y_{2}\right)\right|}{\left|y_{1}-y_{2}\right|^{\alpha}}=\sup _{x_{1} \neq x_{2}} \frac{\left|u\left(x_{1}\right)-u\left(x_{2}\right)\right|}{\left|\Psi\left(x_{1}\right)-\Psi\left(x_{2}\right)\right|^{\alpha}} \\
& \leq \sup _{x_{1} \neq x_{2}}|u|_{C^{\alpha}\left(B_{r}(x)\right)} \frac{\left|x_{1}-x_{2}\right|^{\alpha}}{\left|\Psi\left(x_{1}\right)-\Psi\left(x_{2}\right)\right|^{\alpha}} .
\end{aligned}
$$
Now since $\Psi, \Psi^{-1}$ are $\mathcal{C}^{2, \alpha}$ they are in particular Lipschitz $\left(\mathcal{C}^{0,1}\right)$, i.e $\exists K>0$ s.t.
$$
\left|\Psi\left(x_{1}\right)-\Psi\left(x_{2}\right)\right| \leq K\left|x_{1}-x_{2}\right|, \quad\left|\Psi^{-1}\left(y_{1}\right)-\Psi^{-1}\left(y_{2}\right)\right| \leq K\left|y_{1}-y_{2}\right|,
$$
or
$$
1 / K \cdot\left|x_{1}-x_{2}\right| \leq\left|\Psi\left(x_{1}\right)-\Psi\left(x_{2}\right)\right| \leq K \cdot\left|x_{1}-x_{2}\right|
$$
When plugged-in to our previous computation this yields
$$
\leq|u|_{C^{\alpha}\left(B_{r}(x)\right)} \cdot K^{-\alpha}
$$
Analogously we get
$$
|u|_{C^{\alpha}\left(B_{r}(x)\right)} \leq|\tilde{u}|_{C^{\alpha}\left(B_{r}(x)\right)} \cdot K^{\alpha}
$$
We also have analogously for the first derivatives
$$
|D \tilde{u}|_{C^{0}} \leq|D u|_{C^{0}} \cdot K_{1}
$$
and
$$
|D u|_{C^{0}} \leq|D \tilde{u}|_{C^{0}} \cdot K_{1}^{-1}
$$
where $K_{1}$ depends on $|\Psi|_{C^{1}}$.
And for the second derivatives $\exists K_{2}$ depending on $|\Psi|_{C^{2}}$ (i.e depending on the domain $\Omega$ !) with
$$
\left|D^{2} \tilde{u}\right| \leq K_{1} \cdot\left|D^{2} u\right|+K_{2}|D u| \leq K^{\prime}\|u\|_{C^{2}}
$$
Similarly
$$
\left|D^{2} u\right| \leq K^{\prime \prime}|| \tilde{u} \|_{C^{2}}
$$
Finally for $\mathcal{C}^{2, \alpha}$ norms, can again show norms on both sides are equivalent using $K, K^{\prime}, K^{\prime \prime}$. So we conclude - for each point $x \in \partial \Omega$ we have a ball and the estimate (1) holds there. The above long discussion yields that we have furthermore the desired estimate concerning the norms therein:
$$
\|u\|_{C^{k, \beta}\left(B_{r / 2}(x) \cap \Omega\right)} \leq c \cdot\|u\|_{C^{0}\left(B_{r}(x) \cap \Omega\right)}+\epsilon \cdot|u|_{C^{2, \alpha}\left(B_{r / 2}(x) \cap \Omega\right)} .
$$
Since $\bar{\Omega}$ is compact we can cover $\partial \Omega$ by finitely many balls. On the interior of each we have this (desired) estimate. To complete the proof we therefore just need to make sure this estimate also holds in the interior of $\Omega$. We take a set $\Omega^{\prime} \Subset \Omega$ and a number $\mathrm{D}>0$ s.t.
A) if $x, y \in \Omega$ and $d(x, y) \leq \mathrm{D}$ then either
i) both $x, y$ are both contained in one of the small ball covering the boundary of $\Omega$.
There we have the desired estimate already.
or
ii) both $x, y$ are in $\Omega^{\prime}$. Then we have the desired estimate as well from our interior estimate from the previous lecture for the semi-norms, which extends to give the desired estimate for the norms.
B) if $d(x, y)>\mathrm{D}$ we have $\frac{\left|D^{2} u(x)-D^{2} u(y)\right|}{|x-y|^{\alpha}} \leq 1 / \mathrm{D}^{\alpha} \cdot 2 \cdot\left|D^{2} u\right|_{C^{0}(\Omega)}$. And so also in this case we get the desired estimate.
We now try to extend our results to the case of non-constant coefficients, which was in fact our original goal.
Theorem. Let $u \in \mathcal{C}^{2, \alpha}(\Omega), L=a^{i j}(x) D_{i j}+b^{i}(x) D_{i}+c(x)$ (summation understood over double indices), and suppose $L u=f \in \mathcal{C}^{\alpha}(\Omega)$. Assume furthermore that $L$ has coefficients in $\mathcal{C}^{\alpha}(\Omega)$ and is uniformly elliptic, i.e
- $\frac{1}{\Lambda} \cdot \delta^{i j} \leq a^{i j}(x) \leq \Lambda \cdot \delta^{i j}$
- $\quad\left\|a^{i j}(x)\right\|_{C^{\alpha}(\Omega)},\left\|b^{i}(x)\right\|_{C^{\alpha}(\Omega)},\|c(x)\|_{C^{\alpha}(\Omega)}<\Lambda$.
Then $\forall \Omega^{\prime} \Subset \Omega, \exists c=c\left(\Lambda, n, \Omega^{\prime}, \Omega\right)$ such that
$$
\|u\|_{C^{2, \alpha}\left(\Omega^{\prime}\right)} \leq c\left(\|u\|_{C^{0}(\Omega)}+\|f\|_{C^{\alpha}(\Omega)}\right) .
$$
Thanks to the interpolation proposition we really only need to bound the $\mathcal{C}^{2, \alpha}$ semi-norm with the above RHS since all the other semi-norms contained in the $\mathcal{C}^{2, \alpha}$ norm are bounded above by it together with the $\mathcal{C}^{0}$ semi-norm which is part of the RHS already. We will try to make use of the Hölder constant of the coefficients to relate our situation to the constant coefficients case. The idea is that locally the coefficients are almost constant, and the degree to which this almost true is good enough for us (continuous wouldn't be good enough).
In this spirit, rewrite $L u=f$ as
$$
\left(a^{i j}(x)-a^{i j}\left(x_{0}\right)+a^{i j}\left(x_{0}\right)\right) u_{i j}(x)+b^{i}(x) u_{i}(x)+c(x) u(x)=f,
$$
or
$$
a^{i j}\left(x_{0}\right) u_{i j}(x)=-\left(a^{i j}(x)-a^{i j}\left(x_{0}\right)\right) u_{i j}(x)-b^{i}(x) u_{i}(x)-c(x) u(x)+f .
$$
Calling the LHS $L_{0} u$ and the RHS $F(x)$ we thus define a constant coefficient uniformly elliptic operator and a function.
Let $\Omega^{\prime} \Subset \Omega$ with $x_{0} \in \Omega^{\prime}$. Take $B\left(x_{0}, \mu \cdot D\right) \in \Omega^{\prime}, B\left(x_{0}, D\right) \in \Omega$ with $\mu>0$ small and denote $d:=\mu \cdot D$. As mentioned above we only need to prove an estimate for the $\left|\mathrm{D}^{2} u\right|_{C^{\alpha}\left(\Omega^{\prime}\right)}$. We observe that for any $y_{0} \in \Omega^{\prime}$, and for $d$ small enough
$$
\frac{\mathrm{D}^{2} u\left(x_{0}\right)-\mathrm{D}^{2} u\left(y_{0}\right)}{\left|x_{0}-y_{0}\right|^{\alpha}} \leq\left\{\begin{array}{l}
c \cdot\left(\|u\|_{C^{0}\left(B\left(x_{0}, \frac{d}{2}\right)\right)}+|| F||_{C^{\alpha}\left(B\left(x_{0}, \frac{d}{2}\right)\right)}\right), \text { if } y_{0} \in B\left(x_{0}, \frac{d}{2}\right) \\
\left.\frac{2|u|_{C^{2}\left(\Omega^{\prime}\right)}}{d^{\alpha}}+|| F \|_{C^{0}\left(\Omega^{\prime}\right)}\right), \text { if }\left|x_{0}-y_{0}\right| \geq \frac{d}{2}
\end{array}\right.
$$
since when $d$ is small enough, we can think of having a a uniform elliptic equation with almost constant coefficients in $B\left(x_{0}, \frac{d}{2}\right)$ and then apply our previous results. We therefore get
$$
\left|\mathrm{D}^{2} u\right|_{C^{\alpha}\left(\Omega^{\prime}\right)} \leq c^{\prime} \cdot\left(\|u\|_{C^{0}\left(\Omega^{\prime}\right)}+\frac{1}{d^{\alpha}}|u|_{C^{2}\left(\Omega^{\prime}\right)}+\sup _{x_{0} \in \Omega^{\prime}}\|F\|_{C^{\alpha}\left(B\left(x_{0}, \frac{d}{2}\right)\right)}\right) .
$$
We first estimate the last term which is of a local nature. Observe that for two Hölder functions $k, l \in \mathcal{C}^{\alpha}(\Omega), \quad|k \cdot l|_{C^{\alpha}(\Omega)} \leq|k|_{C^{\alpha}(\Omega)} \cdot|l|_{C^{0}(\Omega)}+|k|_{C^{0}(\Omega)} \cdot|l|_{C^{\alpha}(\Omega)} \leq C(\Omega) \cdot|k|_{C^{\alpha}(\Omega)}|l|_{C^{\alpha}(\Omega)}$, while for the norms themselves we see from the first inequality that $\|k \cdot l\|_{C^{\alpha}(\Omega)} \leq\|k\|_{C^{\alpha}(\Omega)} \cdot\|l\|_{C^{\alpha}(\Omega)}$. So
$$
\begin{aligned}
& \left.\|F\|_{C^{\alpha}\left(B\left(x_{0}, \frac{d}{2}\right)\right)}\right) \leq\left\|a^{i j}(x)-a^{i j}\left(x_{0}\right)\right\|_{C^{\alpha}\left(B\left(x_{0}, \frac{d}{2}\right)\right)} \cdot\left\|u_{i j}\right\|_{C^{\alpha}\left(B\left(x_{0}, \frac{d}{2}\right)\right)}+ \\
& \quad+\left\|b^{i}\right\|_{C^{\alpha}\left(B\left(x_{0}, \frac{d}{2}\right)\right)} \cdot\left\|u_{i}\right\|_{C^{\alpha}\left(B\left(x_{0}, \frac{d}{2}\right)\right)}+\|c\|_{C^{\alpha}\left(B\left(x_{0}, \frac{d}{2}\right)\right)}\|u\|_{C^{\alpha}\left(B\left(x_{0}, \frac{d}{2}\right)\right)}+\|f\|_{C^{\alpha}\left(B\left(x_{0}, \frac{d}{2}\right)\right)} .
\end{aligned}
$$
Remember that the $\mathcal{C}^{\alpha}$ norm includes the $\mathcal{C}^{0}$ and $\mathcal{C}^{\alpha}$ semi-norms. We have
$\left\|a^{i j}(x)-\left.a^{i j}\left(x_{0}\right)\right|_{C^{\alpha}\left(B\left(x_{0}, \frac{d}{2}\right)\right)}=\sup \left|a^{i j}(x)-a^{i j}\left(x_{0}\right)\right|+\left|a^{i j}(x)-a^{i j}\left(x_{0}\right)\right|_{\mathcal{C}^{\alpha}} \leq c \cdot|| a^{i j}\right\|_{C^{\alpha}\left(\Omega^{\prime}\right)}\left|x-x_{0}\right|^{\alpha}$ $\Longrightarrow$
$\|F\|_{C^{\alpha}\left(B\left(x_{0}, \frac{d}{2}\right)\right)} \leq c \cdot \Lambda d^{\alpha}\left(\left\|\mathrm{D}^{2} u\right\|_{C^{\alpha}\left(B\left(x_{0}, \frac{d}{2}\right)\right)}+\|\mathrm{D} u\|_{C^{\alpha}\left(B\left(x_{0}, \frac{d}{2}\right)\right)}+\|u\|_{C^{\alpha}\left(B\left(x_{0}, \frac{d}{2}\right)\right)}\right)+\|f\|_{C^{\alpha}\left(B\left(x_{0}, \frac{d}{2}\right)\right)}$ and using interpolation for the $2^{\text {nd }}$ and $3^{\text {rd }}$ terms we find
$\leq c \cdot \Lambda d^{\alpha}\left(|u|_{C^{2, \alpha}\left(B\left(x_{0}, \frac{d}{2}\right)\right)}+|u|_{C^{0}\left(\Omega^{\prime}\right)}\right)+\|f\|_{C^{\alpha}\left(B\left(x_{0}, \frac{d}{2}\right)\right)} \cdot$
We now come back to the $2^{\text {nd }}$ term of $(2)$.
We now let $x_{0}$ range over all points of $\Omega^{\prime}$. For each $x_{0}$ we will find different $d, D$ such that $(2)$ above holds.
It so happens that this inequality is useless unless we can not control from below the term involving $d^{-\alpha}$. The problem is that as $x_{0} \rightarrow \partial \Omega, d=d\left(x_{0}\right) \rightarrow 0$. To overcome this we assume we chose $\Omega^{\prime}$ such that $\Omega^{\prime} \Subset \Omega^{\prime \prime} \Subset \Omega$ and such that $\operatorname{dist}\left(\partial \Omega^{\prime}, \partial \Omega^{\prime \prime}\right) \geq \delta_{0}>0$. Then for problematic points $x_{0} \rightarrow \partial \Omega$ and $y_{0} \notin B\left(x_{0}, \frac{d}{2}\right)$ we can replace the term $\frac{2|u|_{C^{2}\left(\Omega^{\prime}\right)}}{d^{\alpha}}$ by $\frac{2|u|_{C^{2}\left(\Omega^{\prime}\right)}}{\left(d+\delta_{0}\right)^{\alpha}}$. Therefore we have overcome the problem by means of introducting a constant $c^{\prime \prime}$ depending on $\Omega^{\prime}$ (and $\Omega$ ) and we can replace (2) by
$$
\left|\mathrm{D}^{2} u\right|_{C^{\alpha}\left(\Omega^{\prime}\right)} \leq c^{\prime} \cdot\left(\|u\|_{C^{0}\left(\Omega^{\prime}\right)}+c^{\prime \prime}|u|_{C^{2}\left(\Omega^{\prime \prime}\right)}+\sup _{x_{0} \in \Omega^{\prime}}\|F\|_{C^{\alpha}\left(B\left(x_{0}, \frac{d}{2}\right)\right)}\right)
$$
The $2^{\text {nd }}$ term can be bounded above using our interpolation theory:
$$
|u|_{C^{2}\left(\Omega^{\prime \prime}\right)} \leq c\left(\epsilon_{1}\right) \cdot\|u\|_{C^{0}(\Omega)}+\epsilon_{1} \cdot|u|_{C^{2, \alpha}\left(\Omega^{\prime \prime}\right)}
$$
with $\epsilon_{1}>0$ of our choice. And so we have all in all
$|u|_{C^{2, \alpha}\left(\Omega^{\prime}\right)} \leq c^{\prime} \cdot\left(\|u\|_{C^{0}(\Omega)}+c\left(\epsilon_{1}\right) \cdot\|u\|_{C^{0}(\Omega)}+\epsilon_{1} \cdot|u|_{C^{2, \alpha}\left(\Omega^{\prime \prime}\right)}+c \Lambda d^{\alpha}\left(|u|_{C^{2, \alpha}\left(\Omega^{\prime}\right)}+|u|_{C^{0}\left(\Omega^{\prime}\right)}\right)+\|f\|_{C^{\alpha}\left(\Omega^{\prime}\right)}\right)$.
Now choose $d$ small enough such that $c \Lambda d^{\alpha}<\frac{1}{4}$ and $\epsilon_{1}<\frac{1}{4} \cdot \frac{|u|_{C^{2, \alpha}\left(\Omega^{\prime}\right)}}{|u|_{C^{2, \alpha}\left(\Omega^{\prime \prime}\right)}}$. Remark. This Theorem gives still just a Schauder type interior estimate. Next time we will try to extend it to the boundary.
## Lecture 14
April 6, th 2004
## Extending interior Schauder estimates to flat boundary part
Theorem. $u \in \mathcal{C}^{2, \alpha}(\Omega \cap T), L u=f, u=0$ on $T$, with $0<\alpha<1$. Assume coefficients are bounded in $\mathcal{C}^{2, \alpha}(\Omega \cap T)$ as well as uniformly elliptic. Then $\forall \Omega^{\prime} \cap T^{\prime} \Subset \Omega \cap T, \exists c=c\left(\Lambda, n, \Omega^{\prime}, \Omega, T^{\prime}, T\right)$ such that
$$
\|u\|_{C^{2, \alpha}\left(\Omega^{\prime} \cap T^{\prime}\right)} \leq c\left(\|u\|_{C^{0}(\Omega \cap T)}+\|f\|_{C^{\alpha}(\Omega \cap T)}\right) .
$$
Proof. As in the last remark we see that our proof consisted of perturbing the equation at any $x_{0} \in \Omega^{\prime}$ and relying on our constant coefficients estimates and interpolation methods. Both of these hold upto the flat boundary from our previous work.
## Global Schauder estimates
Theorem. Let $\Omega$ be a $\mathcal{C}^{2, \alpha}$ domain and $u \in \mathcal{C}^{2, \alpha}(\bar{\Omega})^{\star}$ with $0<\alpha<1$. Let $L$ be uniformly elliptic with $\mathcal{C}^{\alpha}(\bar{\Omega})$ bounds on coefficients . Let
$$
\begin{array}{cc}
L u=f, & f \in \mathcal{C}^{\alpha}(\bar{\Omega}), \\
u=\varphi & \text { on } \partial \Omega .
\end{array}
$$
Then $\exists c=c(\Omega, \Lambda, n)$ such that
$$
\|u\|_{C^{2, \alpha}(\Omega)} \leq c\left(\|u\|_{C^{0}(\Omega)}+\|f\|_{C^{\alpha}(\Omega)}+\|\varphi\|_{C^{2, \alpha}(\partial \Omega)}\right) .
$$
* We note that Gilbarg-Trudinger intend by this notation locally Hölder while we will take it henceforth to mean globally Hölder in the sense that we assume $\sup _{x_{0} \neq y_{0} \in \bar{\Omega}} \frac{\mathrm{D}^{2} u\left(x_{0}\right)-\mathrm{D}^{2} u\left(y_{0}\right)}{\left|x_{0}-y_{0}\right|^{\alpha}}$ is finite. Here we let $\|\varphi\|_{C^{2, \alpha}(\partial \Omega)}:=\inf _{\tilde{\varphi}: \Omega \rightarrow \mathbb{R}}\|\tilde{\varphi}\|_{C^{2, \alpha}(\Omega)}$
Proof. It is enough to prove for the case of zero boundary values: if we can solve the Dirichlet problem
$$
\begin{array}{ccccc}
L v & = & f-L \varphi=: f^{\prime} \in \mathcal{C}^{\alpha} & \text { on } & \bar{\Omega}, \\
v & = & 0 & \text { on } & \partial \Omega .
\end{array}
$$
we can also solve our original one by setting $v+\varphi$ solves the original equation. And if we have the above announced estimates for $v$ then by the triangle inequality (for the relevant norms) and the uniform ellipticity (which gives $\|L \varphi\|_{C^{\alpha}(\Omega)} \leq c \cdot\|\varphi\|_{C^{2, \alpha}(\Omega)}$ ) the same estimates will hold for $u$, possibly with a different constant.
So indeed we may assume $\varphi=0$.
By definition of a $\mathcal{C}^{2, \alpha}$ domain $\exists \Psi, \Psi^{-1} \in \mathcal{C}^{2, \alpha}\left(\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\right)$ mapping each small portion of the boundary of $\Omega$, say $B\left(x_{0}, R\right) \cap \partial \Omega$ for $x_{0} \in \partial \Omega$ to flat boundary. We set as in computations in the past $\tilde{u}:=u \circ \Psi^{-1}$ and then $\mathrm{D} \tilde{u}=\mathrm{D} u \circ \Psi^{-1}, \mathrm{D}^{2} \tilde{u}=\mathrm{D}^{2} u \cdot \Psi^{-1} \prime+\mathrm{D} u \cdot \mathrm{D}^{2} \Psi^{-1}$. These computations convince us once more that the relevant norms on $a, b, c$ and $\tilde{a}, \tilde{b}, \tilde{c}$ are equivalent using $\Psi, \Psi^{-1} \in \mathcal{C}^{2, \alpha}$ (e.g we find $\|\tilde{b}\|_{C^{\alpha}(\Omega)} \leq \|\left. b\right|_{C^{\alpha}(\Omega)}\left(|\Psi|_{C^{1, \alpha}(+)}|\Psi|_{C^{2, \alpha}(\Omega)} \leq C \cdot \Lambda\right.$ ).
We have for the flat boundary
$$
\|\tilde{u}\|_{C^{2, \alpha}\left(\Psi\left(B\left(x_{0}, \frac{1}{2} R\right) \cap \bar{\Omega}\right)\right)} \leq c\left(\|\tilde{u}\|_{C^{0}\left(\Psi\left(B\left(x_{0}, R\right) \cap \bar{\Omega}\right)\right)}+\|\tilde{f}\|_{C^{\alpha}\left(\Psi\left(B\left(x_{0}, R\right) \cap \bar{\Omega}\right)\right)}\right) .
$$
Now by our above work we know this holds also for $u$ in $B\left(x_{0}, R\right) \cap \bar{\Omega}$
$$
\|u\|_{C^{2, \alpha}\left(B\left(x_{0}, \frac{1}{2} R\right) \cap \bar{\Omega}\right)} \leq c\left(\|u\|_{C^{0}\left(B\left(x_{0}, R\right) \cap \bar{\Omega}\right)}+\|f\|_{C^{\alpha}\left(B\left(x_{0}, R\right) \cap \bar{\Omega}\right)}\right)
$$
Now we patch up the estimates over a countable cover of $\partial \Omega$ by small balls $\left\{B\left(x_{i}, \frac{1}{2} R_{i}\right)\right\}$. $\partial \Omega$ being compact we may choose a finite subcover say after relabeling $\left\{B\left(x_{i}, \frac{1}{2} R_{i}\right)\right\}_{i=1}^{N}$. Finally we adjoin to these estimates an interior estimate for some $\Omega^{\prime}$ such that $\Omega \backslash \cup_{i=1}^{N} B\left(x_{i}, \frac{1}{2} R_{i}\right) \Subset \Omega^{\prime} \Subset \Omega$. And having this we are done by analysing the different cases that might arise in a similar fashion to previous proofs.
## Banach Spaces
$\mathcal{L}$ et $V$ be a vector space equipped with a norm $\|\cdot\|: V \rightarrow \mathbb{R}$ i.e $i)\|x\| \geq 0$ with equality $\Leftrightarrow x=0$; ii) $\|\alpha x\|=|\alpha|\|x\|$; iii) $\Delta$ - inequality. With a norm we have a metric $d(x, y):=\|x-y\|$ and we can talk about topology induced from it, convergence etc.
Cauchy sequence: $\left\{x_{i}\right\}$ such that $d\left(x_{n}, x_{m}\right) \stackrel{N \rightarrow \infty}{\longrightarrow} 0, \forall m, n \geq N$.
Banach space: a normed space complete WRT the norm metric $\Leftrightarrow$ every Cauchy sequence converges (WRT the norm metric) in $V$ (limit in $V$ ).
We mention in passing a few examples.
- The Bolzano-Weierstrass theorem showing $\left(\mathbb{R}^{n},|\cdot|\right)$ is complete carries over to show finite dimensional normed spaces are Banach.
- $\quad\left(\mathcal{C}^{0}(\Omega),\|\cdot\|_{L^{1}}\right)$ is incomplete, so is not Banach;
- $\quad$ On the other handwhile $\left(\mathcal{C}^{0}(\Omega),\|\cdot\|_{C^{0}(\Omega)}\right)$ and in general $\left(\mathcal{C}^{k, \alpha}(\Omega),\|\cdot\|_{C k, \alpha}\right)$ are Banach, as can be demonstrated using the Arzelà-Ascoli theorem [cf. Peterson, Riemannian Geometry, Chapter 10].
- Sobolev spaces are yet another example.
Contraction Mapping Theorem. Let $\mathcal{B}$ a Banach space and $T: \mathcal{B} \rightarrow \mathcal{B}$ a contraction mapping (wRT to the norm metric). Then $T$ has a unique fixed point.
Proof. Here the assumption translates into $\|T x-T y\| \leq \theta \cdot\|x-y\|$ for $\theta \in[0,1)$. The idea is to look at the sequence $\left\{x_{n}:=T^{n} x_{0}\right\}$ and show it is Cauchy using the $\Delta$-inquality . Let $x \in V$ be its limit; we see that
$$
T x=T \lim x_{n}=\lim T x_{n}(\text { by continuity of } \mathrm{T} !)=\lim x_{n+1}=x .
$$
As for uniqueness, if $x, y$ are two fixed points,
$$
\|x-y\|=\|T x-T y\| \leq \theta\|x-y\| \Rightarrow\|x-y\|=0
$$
and by the norm properties $x=y$.
## Lecture 15
April $8^{\text {th }}, 2004$
## The Continuity Method
Let $T: \mathcal{B}_{1} \rightarrow \mathcal{B}_{2}$ be linear between two Banach spaces. $\mathrm{T}$ is bounded if
$$
\|T\|=\sup _{x \in \mathcal{B}_{1}} \frac{\|T x\|_{\mathcal{B}_{2}}}{\|x\|_{\mathcal{B}_{1}}}<\infty \Leftrightarrow\|T x\|_{\mathcal{B}_{2}} \leq c \cdot\|x\|_{\mathcal{B}_{1}} \text { for some } c>0 .
$$
Continuity Method Theorem. Let $\mathcal{B}$ be a Banach space, $V$ a normed space, $L_{0}, L_{1}: \mathcal{B} \rightarrow V$ bounded linear operators. Assume $\exists c$ such that $L_{t}:=(1-t) L_{0}+t L_{1}$ satisfies
$$
\|x\|_{\mathcal{B}} \leq c \cdot\left\|L_{t} x\right\|_{V}, \quad \forall t \in[0,1] .
$$
Then $-L_{0}$ is onto $\Leftrightarrow \quad L_{1}$ is.
Proof. Assume $L_{s}$ is onto for some $s \in[0,1]$; by $(*) L_{s}$ is also 1 -to- $1 \Rightarrow L_{s}^{-1}$ exists. For $t \in[0,1], y \in$ $V$ solving $L_{t} x=y$ is equivalent to solving $L_{s}(x)=y+\left(L_{s}-L_{t}\right) x=y+(t-s) L_{0} x+(t-s) L_{1} x$. By linearity now $x=L_{s}^{-1} y+(t-s) L_{s}{ }^{-1} \circ\left(L_{0}-L_{1}\right) x$.
Define a linear map $T: \mathcal{B} \rightarrow \mathcal{B}, T x=L_{s}^{-1} y+(t-s) L_{s}{ }^{-1} \circ\left(L_{0}-L_{1}\right) x$. One has $\| T x_{1}-$ $T x_{2}\left\|_{\mathcal{B}}=\right\|(t-s) L_{s}{ }^{-1} \circ\left(L_{0}-L_{1}\right)\left(x_{1}-x_{2}\right) \| .(*)$ now gives us a bound on $L_{s}{ }^{-1}$ : since $L_{s}$ is onto $\forall x \in \mathcal{B}, \exists y \in \mathcal{B}$ such that $L_{s} y=x$ and so
$$
\begin{gathered}
\left\|L_{s}{ }^{-1} x\right\|_{\mathcal{B}} \leq c \cdot\left\|L_{s} \circ L_{s}{ }^{-1} x\right\|_{V} \\
\left\|L_{s}{ }^{-1} x\right\|_{\mathcal{B}} \leq c \cdot\|x\|_{V} \quad \Rightarrow \quad\left\|L_{s}{ }^{-1}\right\| \leq c .
\end{gathered}
$$
As an application we see that
$$
\left\|T x_{1}-T x_{2}\right\|_{\mathcal{B}} \leq(t-s) c \cdot\left(\left\|L_{0}\right\|+\left\|L_{1}\right\|\right)\left\|x_{1}-x_{2}\right\|,
$$
and for $t$ close enough to $s$ (precisely for $t \in\left[s-\frac{1}{c\left(\left\|L_{0}\right\|+\left\|L_{1}\right\|\right)}, s+\frac{1}{c\left(\left\|L_{0}\right\|+\left\|L_{1}\right\|\right)}\right]$ ) we therefore have a contraction mapping! Therefore $T$ has a fixed point by the previous theorem which essentially means that we can solve $L_{t} x=y$ for any fixed $y$ or that $L_{t}$ is onto. Repeating this $c\left(\left\|L_{0}\right\|+\left\|L_{1}\right\|\right)$ many times we cover all $t \in[0,1]$.
Remark. Note as in the beginning of the proof that once such operators are onto they are in fact invertible as long as $(*)$ holds.
## Elliptic uniqueness
Let us summarize the properties we have establised for uniformly elliptic equations. Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$. Let $L=a^{i j}(x) \mathrm{D}_{i j}+b^{i}(x) \mathrm{D}_{i}+c(x)$ be uniformly elliptic, i.e
$$
\frac{1}{\Lambda} \cdot \delta^{i j} \leq a^{i j}(x) \leq \Lambda \cdot \delta^{i j}
$$
and assume $c(x) \leq 0$.
Let $u \in \mathcal{C}^{2}(\Omega) \cap \mathcal{C}^{0}(\bar{\Omega})$ be a solution of $L u=f \in \mathcal{C}^{\alpha}(\Omega)$ with $0<\alpha<1$. Then we have the following a priori estimates -
A. $\sup _{\Omega}|u| \leq c(\gamma, \Lambda, \Omega, n) \cdot\left(\sup _{\partial \Omega}|u|+\sup _{\Omega}|f|\right)$.
B. Under the additional assumptions
- in the case $L$ has $\alpha$ - Hölder continuous coefficients with Hölder constant $\Lambda$,
- $\Omega$ has $\mathcal{C}^{2, \alpha}$ boundary
- $u \in \mathcal{C}^{2, \alpha}(\bar{\Omega}), f \in \mathcal{C}^{\alpha}(\bar{\Omega})$,
we had the global Schauder estimate
$$
\|u\|_{C^{2, \alpha}(\bar{\Omega})} \leq c(\gamma, \Lambda, \Omega, n)\left(\|u\|_{C^{0}(\Omega)}+\|f\|_{C^{\alpha}(\Omega)}\right)
$$
C. Under the assumptions of $\mathrm{B}$, when $c(x) \leq 0$
$$
\|u\|_{C^{2, \alpha}(\bar{\Omega})} \leq c\left(\sup _{\partial \Omega}|u|+\sup _{\Omega}|f|\right)
$$
D. The above applies to the Dirichlet problem
$$
L u=f \text { on } \bar{\Omega}, \quad u=\varphi \text { on } \partial \Omega
$$
and in particular when $\varphi=0$ we get very simply
$$
\|u\|_{C^{2, \alpha}(\bar{\Omega})} \leq c \cdot\|L u\|_{C^{\alpha}(\bar{\Omega})} .
$$
Theorem. Let $\Omega$ be a $\mathcal{C}^{2, \alpha}$ domain, $L$ uniformly elliptic with $\mathcal{C}^{\alpha}(\bar{\Omega})$ coefficients and $(x) \leq 0$. Look at all $u \in \mathcal{C}^{2, \alpha}(\bar{\Omega})$ and assume $f \in \mathcal{C}^{\alpha}(\bar{\Omega})$. Then the Dirichlet problem Lu=f on $\bar{\Omega}, \quad u=$ $\varphi$ on $\partial \Omega$ has a unique solution $u \in \mathcal{C}^{2, \alpha}(\bar{\Omega})$ provided that the Dirichlet problem for $\Delta$ is solvable $\forall f \in \mathcal{C}^{\alpha}(\bar{\Omega}), \forall \varphi \in \mathcal{C}^{2, \alpha}(\bar{\Omega}) !$
Proof. Connect $L$ and $\Delta$ via a segment: $[0,1] \rightarrow L_{t}:=(1-t) L+t \Delta$. Since those operators are all linear it is enough to prove for $\varphi=0$ as we have seen previously. $\mathcal{C}^{2, \alpha}(\bar{\Omega})$ is a Banach space (Lecture 14), and so is its subspace $\mathcal{B}(\Omega):=\left\{u \in \mathcal{C}^{2, \alpha}(\bar{\Omega}), u=0\right.$ on $\left.\partial \Omega\right\}$. As a matter of fact $L_{t}$ is a bounded operator $\mathcal{B}(\Omega) \rightarrow \mathcal{C}^{\alpha}(\bar{\Omega})$ by the assumptions on the coefficients of $L$. And, by uniformly elliptic we see from D above
$$
\|u\|_{C^{2, \alpha}(\bar{\Omega})}=\|u\|_{C^{2, \alpha}(\mathcal{B}(\Omega))} \leq c \cdot\left\|L_{t} u\right\|_{C^{\alpha}(\bar{\Omega})}
$$
with $c$ independent of $t$ (depends just on $L$ ). Note $\mathcal{C}^{\alpha}(\bar{\Omega})$ is a Banach space and in particular a vector space. The Continuity Method thus applies.
Strangely enough, we are now back to solving Dirichlet's problem for $\Delta$ in domains.
Our methods so far were good for providing solution in balls, spherically symmetric domains. In other words we were able to solve (in $\left.\mathcal{C}^{2, \alpha}(\overline{B(0, R)}) !\right) \quad \Delta u=f \in \mathcal{C}^{\alpha}(\bar{\Omega}) \quad$ on $B(0, R), \quad u=$ $\varphi$ on $\partial B(0, R)$ using the Poisson Integral Formula and estimates for the Newtonian Potential. We used conformal mappings (inversion) to get indeed $\mathcal{C}^{2, \alpha}$ upto the boundary. We conclude therefore that Corollary. We can solve the Dirichlet Problem for any $L$ satisfying the assumptions of the Theorem in balls.
Perron's Method gives a solution in quite general domains but we will not go into its details as later on our regularity theory (weak solutions, Sobolev spaces etc.) will give us those answers.
## Elliptic $\mathcal{C}^{2, \alpha}$ regularity
Let $B:=$ ball, $T:=$ some connected boundary portion.
Theorem. Let $L$ be uniformly elliptic with $\mathcal{C}^{\alpha}$ coefficients and assume $c(x) \leq 0$. Let $u \in$ $\mathcal{C}^{2}(\Omega) \cap \mathcal{C}^{0}(\bar{\Omega})$ be a solution of the Dirichlet problem $L u=f \in \mathcal{C}^{\alpha}(\bar{B})$ in $B, \quad u=\varphi \in \mathcal{C}^{0}(\partial B) \cap$ $\mathcal{C}^{2, \alpha}(T)$ on $\partial B$ has a unique solution $u \in \mathcal{C}^{2, \alpha}(B \cup T) \cap \mathcal{C}^{0}(\bar{B})$.
We know by the previous theorem that if $\varphi \in \mathcal{C}^{2, \alpha}(\partial B)$ (and not just on $T$ ) then unique solvability would be equivalent to the unique solvability of $\Delta$ on $B$ which we have! Therefore this Theorem is a slight generalization.
Proof. As was just outlined the crucial problem lies in the (possible) absence of regularity of $\varphi$ on part of the boundary. So we approximate $\varphi$ by a sequence $\left\{\varphi_{k}\right\} \subset \mathcal{C}^{3}(\bar{B})$ such that both $\left\|\varphi_{k}-\varphi\right\|_{C^{0}(\bar{B})} \longrightarrow 0$ and $\left\|\varphi_{k}-\varphi\right\|_{C^{2, \alpha}(\bar{B})} \longrightarrow 0$. Solve $L u_{k}=f$, in $B, \quad u_{k}=\varphi_{k}$ on $\partial B$.
Now $L\left(u_{i}-u_{j}\right)=0, \quad$ in $B, \quad u_{i}-u_{j}=\varphi_{i}-\varphi_{j}$ on $\partial B$. And by A above $($ as $c(x) \leq 0)$ $\left\|u_{i}-u_{j}\right\|_{C^{0}(B)} \leq C \sup _{\partial B}\left|\varphi_{i}-\varphi_{j}\right|$. So we conclude our solutions $\left\{u_{k}\right\}$ form a Cauchy sequence
(not just subconvergence!) and furthermore this $u$ satisfies $u=\varphi$ on $p B$.
Now we shift our look to the $\mathcal{C}^{2, \alpha}$ situation; by our interior estimates we have for any $B^{\prime} \Subset B$ $\left\|u_{i}-u_{j}\right\|_{C^{2, \alpha}\left(B^{\prime}\right)} \leq c\left(\left\|u_{i}-u_{j}\right\|_{C^{0}(B)}+\|0\|_{C^{\alpha}(B)}\right)$.. That is our sequence is also a Cauchy sequence in the Banach space $\mathcal{C}^{2, \alpha}\left(B^{\prime}\right) \Rightarrow$ converges in $\mathcal{C}^{2, \alpha}\left(B^{\prime}\right)$ (in particular limit is $\mathcal{C}^{2, \alpha}$ regular). This limit must equal the limit $\left.u\right|_{B} ^{\prime}$ we obtained through the $\mathcal{C}^{0}$ norm. We do this for any $B^{\prime} \Subset B \Rightarrow$ get convergence in $\mathcal{C}^{2, \alpha}(B) \Rightarrow u$ satisfies $L u=f \underline{\text { on } B}$ and has the desired $\mathcal{C}^{2, \alpha}$ regularity on $B$. We now turn to the boundary portion: $\forall x_{0} \in T$ and $\rho>0$ such that $B\left(x_{0}, \rho\right) \cap \partial B \subseteq T$ we have the usual boundary Schauder estimates (for smooth enough functions) which give us $\left\|u_{i}-u_{j}\right\|_{C^{2, \alpha}\left(B\left(x_{0}, \rho\right) \cap \bar{B}\right)} \leq c \cdot\left(\left\|u_{i}-u_{j}\right\|_{C^{0}(B)}+\left\|\varphi_{i}-\varphi_{j}\right\|_{C^{2, \alpha}\left(B\left(x_{0}, \rho\right) \cap \bar{B}\right)}\right)$. This means that in fact $u_{i} \stackrel{\mathcal{C}^{2, \alpha}\left(B\left(x_{0}, \rho\right) \cap \bar{B}\right)}{\longrightarrow} u$ and in particular $u \in \mathcal{C}^{2, \alpha}$ at $x_{0} . \forall x_{0} \in T$.
## Lecture 16
April $13^{\text {th }}, 2004$
## Elliptic regularity
$\mathcal{H}$ itherto we have always assumed our solutions already lie in the appropriate $\mathcal{C}^{k, \alpha}$ space and then showed estimates on their norms in those spaces. Now we will avoid this a priori assumption and show that they do hold a posteriori. This is important for the consistency of our discussion. Precisely what we would like to show is -
$\mathcal{A}$ priori regularity. $\quad$ Let $u \in \mathcal{C}^{2}(\Omega)$ be a solution of $L u=f$ and assume $0<\alpha<1$. We do not assume $c(x) \leq 0$ but we do assume all the other assumptions on $L$ in the previous Theorem hold. If $f \in \mathcal{C}^{\alpha}(\Omega)$ then $u \in \mathcal{C}^{2, \alpha}(\Omega)$
- Here we mean the $\mathcal{C}^{\alpha}$ norm is locally bounded, i.e for every point exists a neighborhood where the $\mathcal{C}^{\alpha}$-norm is bounded. Had we written $\mathcal{C}^{\alpha}(\bar{\Omega})$ we would mean a global bound on $\sup _{x, y} \frac{|f(x)-f(y)|}{|x-y|^{\alpha}}$ (as in the footnote if Lecture 14).
- This result will allow us to assume in previous theorems only $\mathcal{C}^{2}$ regularity on (candidate) solutions instead of assuming $\mathcal{C}^{2, \alpha}$ regularity.
Proof. Let $u$ be a solution as above. Since the Theorem is local in nature we take any point in $\Omega$ and look at a ball $B$ centered there contained in $\Omega$. We then consider the Dirichlet problem
$$
\begin{array}{cccc}
L_{0} v & =f^{\prime} & \text { on } & B, \\
v & =u & \text { on } & \partial B .
\end{array}
$$
where $L_{0}:=L-c(x)$ and $f^{\prime}(x):=f(x)-c(x) \cdot u(x)$. This Dirichlet problem is on a ball, with $" c \leq 0 "$, uniform elliptic and with coefficients in $\mathcal{C}^{\alpha}$. Therefore we have uniqueness and existence of a solution $v$ in $\mathcal{C}^{2, \alpha}(B) \cap \mathcal{C}^{0}(\bar{B})$. But $u$ satisfies $L u=f$ or equivalently $L_{0} u=f^{\prime}$ on all of $\Omega$ so in particular on $\bar{B}$. By uniqueness on $B$ therefore we have $\left.u\right|_{\bar{B}}=v$, and so $u$ is $\mathcal{C}^{2, \alpha}$ smooth there. As this is for any point and all balls we have $u \in \mathcal{C}^{2, \alpha}(\Omega)$.
It is insightful to note at this point that these results are optimal under the above assumptions. Indeed need $\mathcal{C}^{2}$ smoothness (or atleast $\mathcal{C}^{1,1}$ ) in order to define $2^{\text {nd }}$ derivatives wRT $L$ ! If one takes $u$ in a larger function space, i.e weaker regularity of $u$, and defines $L u=f$ in a weak sense then need more regularity on coefficients of $L$ ! Under the assumption of $\mathcal{C}^{\alpha}$ continuity on the coefficients indeed we are in an optimal situation.
Higher a priori regularity. $\quad$ Let $u \in \mathcal{C}^{2}(\Omega)$ be a solution of $L u=f$ and $0<\alpha<1$. We do not assume $c(x) \leq 0$ but we assume uniformly elliptic and that all coefficients are in $\mathcal{C}^{k, \alpha}$. If $f \in \mathcal{C}^{k, \alpha}$ then $u \in \mathcal{C}^{k+2, \alpha}$. If $f \in \mathcal{C}^{\infty}$ then $u \in \mathcal{C}^{\infty}$.
Proof. $k=0$ was the previous Theorem.
The case $k=1$. The proof relies in an elegant way on our previous results with the combination of the new idea of using difference quotients. We would like to differentiate the $u$ three times and prove we get a $\mathcal{C}^{\alpha}$ function. Differentiating the equation $L u=f$ once would serve our purpose but it can not be done naïvely as it would involve 3 derivatives of $u$ and we only know that $u$ has two. To circumvent this hurdle we will take two derivatives of the difference quotients of $u$, which we define by (let $\mathbf{e}_{\mathbf{1}}, \ldots, \mathbf{e}_{\mathbf{n}}$ denote the unit vectors in $\mathbb{R}^{n}$ )
$$
\Delta^{h} u:=\frac{u\left(x+h \cdot \mathbf{e}_{\mathbf{l}}\right)-u(x)}{h}=: \cdot \frac{u^{h}(x)-u(x)}{h} .
$$
Namely we look at
$$
\Delta^{h} L u=\frac{L u\left(x+h \cdot \mathbf{e}_{\mathbf{l}}\right)-L u(x)}{h}=\frac{f\left(x+h \cdot \mathbf{e}_{\mathbf{l}}\right)-f(x)}{h}=\Delta^{h} u f .
$$
Note $\Delta^{h} v(x) \stackrel{h \rightarrow 0}{\longrightarrow} \mathrm{D}_{l} v(x)$ if $v \in \mathcal{C}^{1}$ (which we don't know a priori in our case yet).
Expanding our equation in full gives
$$
\begin{gathered}
\frac{1}{h}\left[\left(a^{i j}\left(x+h \cdot \mathbf{e}_{\mathbf{l}}\right)-a^{i j}(x)+a^{i j}(x)\right) \mathrm{D}_{i j} u^{h}-a^{i j}(x) \mathrm{D}_{i j} u(x)\right. \\
\left.+b^{i}\left(x+h \cdot \mathbf{e}_{\mathbf{l}}\right) \mathrm{D}_{i} u\left(x+h \cdot \mathbf{e}_{\mathbf{l}}\right)-b^{i}(x) \mathrm{D}_{i} u(x)+c\left(x+h \cdot \mathbf{e}_{\mathbf{l}}\right) u\left(x+h \cdot \mathbf{e}_{\mathbf{l}}\right)-c(x) u(x)\right] \\
=\Delta^{h} a^{i j} \mathrm{D}_{i j} u^{h}-a^{i j} \mathrm{D}_{i j} \Delta^{h} u+\Delta^{h} b^{i} \mathrm{D}_{i} u^{h}+b^{i} \mathrm{D}_{i} \Delta^{h} u+\Delta^{h} c \cdot u^{h}+c \cdot \Delta^{h} u=\Delta^{h} f .
\end{gathered}
$$
or succintly
$$
L \Delta^{h} u=f^{\prime}:=\Delta^{h} f-\Delta^{h} a^{i j} \cdot \mathrm{D}_{i j} u^{h}-\Delta^{h} b_{i} \cdot \mathrm{D}_{i} u^{h}-\Delta^{h} c \cdot u^{h}
$$
where $u^{h}:=u\left(x+h \cdot \mathbf{e}_{\mathbf{1}}\right)$.
We now analyse the regularity of the terms. $f \in \mathcal{C}^{1, \alpha}$ so so is $\Delta^{h} f$, but not (bounded) uniformly WRT $h$ (i.e $\mathcal{C}^{1, \alpha}$ norm of $\Delta^{h} f$ may go to $\infty$ as $h$ decreases). On the otherhand $\Delta^{h} f \in \mathcal{C}^{\alpha}(\Omega)$ uniformly WRT $h(\forall h>0): \Delta^{h} u f=\frac{f\left(x+h \cdot \mathbf{e}_{1}\right)-f(x)}{h}=\mathrm{D}_{l} f(\bar{x})$ for some $\bar{x}$ in the interval, and RHS has a uniform $\mathcal{C}^{\alpha}$ bound as $f \in \mathcal{C}^{1, \alpha}$ on all $\Omega$ ! (needed as $\bar{x}$ can be arbitrary).
For the same reason $\Delta^{h} a^{i j}, \Delta^{h} b_{i}, \Delta^{h} c \in \mathcal{C}^{\alpha}(\Omega)$. By the $k=0$ case we know $u \in \mathcal{C}^{2, \alpha}(\Omega)$ and not just in $\mathcal{C}^{2}(\Omega) . \Leftrightarrow \mathrm{D}_{i j} u^{h} \in \mathcal{C}^{\alpha}(\Omega)$ uniformly.
Remark. We take a moment to describe what we mean by uniformity. We say a function $g_{h}=g(h, \cdot): \Omega \rightarrow \mathbb{R}$ is uniformly bounded in $\mathcal{C}^{\alpha}$ WRT $h$ when $\forall \Omega^{\prime} \Subset \Omega$ exists $c(\Omega)$ such that $\left|g_{h}\right|_{C^{\alpha}\left(\Omega^{\prime}\right)} \leq c(\Omega)$. Note this definition goes along with our local definition of a function being in $\mathcal{C}^{\alpha}(\Omega)\left(\right.$ and not in $\left.\mathcal{C}^{\alpha}(\bar{\Omega}) !\right)$.
Putting the above facts together we now see that both sides of the equation $L \Delta^{h} u=f^{\prime}$ are in $\mathcal{C}^{\alpha}(\Omega)$. And they are also in $\mathcal{C}^{\alpha}\left(\Omega^{\prime}\right)$ with RHS uniformly so with constant $c\left(\Omega^{\prime}\right)$.
By the interior Schauder estimate, $\forall \Omega^{\prime \prime} \Subset \Omega^{\prime}$ and for each $h$
$$
\left\|\Delta^{h} u\right\|_{C^{2, \alpha}\left(\Omega^{\prime \prime}\right)} \leq c\left(\gamma, \Lambda, \Omega^{\prime \prime}\right) \cdot\left(\left\|\Delta^{h} u\right\|_{C^{0, \Omega^{\prime}(+)}}\left\|f^{\prime}\right\|_{C^{\alpha, \Omega^{\prime}}()}\right) \leq \tilde{c}\left(\gamma, \Lambda, \Omega^{\prime \prime}, \Omega^{\prime}, \Omega,\|u\|_{C^{1}(\Omega)},\right.
$$
which is independent of $h$ ! If we assume the Claim below taking the limit $h \rightarrow 0$ we get $\mathrm{D}_{l} u \in$ $\mathcal{C}^{2, \alpha}\left(\Omega^{\prime \prime}\right), \forall l=1, \ldots, n \quad u \in \mathcal{C}^{3, \alpha}\left(\Omega^{\prime \prime}\right) . \forall \Omega^{\prime \prime} \Subset \Omega^{\prime} \Subset \Omega \quad \Leftrightarrow u \in \mathcal{C}^{3, \alpha}(\Omega)$. Claim. $\left\|\Delta^{h} g\right\|_{C^{\alpha}(A)} \leq c$ independently of $h \quad \Leftrightarrow \quad D_{l} g \in \mathcal{C}^{\alpha}(A)$.
First we we show $g \in \mathcal{C}^{0,1}(A)$. This is tantamount to the existence of $\lim _{h \rightarrow 0} \Delta^{h} g(x)$ (since if it exists it equals $\mathrm{D}_{l} u \gamma(x)$ - that's how we define the first $l$-directional derivative at $\left.x\right)$. Now $\left\{\Delta^{h} g\right\}_{h>0}$ is family of uniformly bounded (in $\mathcal{C}^{0}(A)$ ) and equicontinuous functions (from the uniform Hölder constant). So by the Arzelà-Ascoli Theorem exists a sequence $\left\{\Delta^{h_{i}} g\right\}_{i=1}^{\infty}$ converging to some $\tilde{w} \in \mathcal{C}^{\alpha}(A)$ in the $\mathcal{C}^{\beta}(A)$ norm for any $\beta<\alpha$. But as we remarked above $\tilde{w}$ necessarily equals $\mathrm{D}_{l} g$ by definition.
Second, we show $g \in \mathcal{C}^{1}(A)$ (i.e such that derivative is continuous not just bounded) and actually $\in \mathcal{C}^{1, \alpha}(A)$
$$
c \geq\left\|\Delta^{h} g\right\|_{C^{\alpha}(A)} \geq \lim _{h \rightarrow 0} \frac{\Delta^{h} g(x)-\Delta^{h} g(y)}{|x-y|^{\alpha}}=\frac{\mathrm{D}_{l} g(x)-\mathrm{D}_{l} g(y)}{|x-y|^{\alpha}}=\left|\mathrm{D}_{l} g\right|_{C^{\alpha}(A)}
$$
where we used that $c$ is independent of $h$.
The case $k \geq 2$. Let $k=2$. By the $k=1$ case we can legitimately take 3 derivatives as $u \in \mathcal{C}^{3, \alpha}(\Omega)$. One has
$$
L\left(\mathrm{D}_{l} u\right)=f^{\prime}:=\mathrm{D}_{l} f-\mathrm{D}_{l} a^{i j} \cdot \mathrm{D}_{i j} u-\mathrm{D}_{l} b_{i} \cdot \mathrm{D}_{i} u-\mathrm{D}_{l} c \cdot u
$$
with $\mathrm{D}_{l} u, f^{\prime} \in \mathcal{C}^{1, \alpha}(\Omega)$. So again by the $k=1$ case we have now $\mathrm{D}_{l} u \in \mathcal{C}^{3, \alpha}(\Omega)$, hence $u \in \mathcal{C}^{4, \alpha}(\Omega)$. The instances $k \geq 3$ are in the same spirit.
## Boundary regularity
Let $\Omega$ be a $\mathcal{C}^{2, \alpha}$ domain, i.e whose boundary is locally the graph of a $\mathcal{C}^{2, \alpha}$ function. Let $L$ be uniformly elliptic with $\mathcal{C}^{\alpha}$ coefficients and $c \leq 0$.
with $0<\alpha<1$. Then $u \in \mathcal{C}^{2, \alpha}(\bar{\Omega})$. Proof. Our previous results give $u \in \mathcal{C}^{2, \alpha}(\Omega)$ and we seek to extend it to those points in $\partial \Omega$. Note that even though $u=\varphi$ on $\partial \Omega$ and $\varphi$ is $\mathcal{C}^{2, \alpha}$ there this does not give the same property for $u$. It just gives that $u$ is $\mathcal{C}^{2, \alpha}$ in directions tangent to $\partial \Omega$, but not in directions leading to the boundary. The question is local: restrict attention to $B\left(x_{0}, R\right) \cap \bar{\Omega}$ for each $x_{0} \in \partial \Omega$. We choose a $\mathcal{C}^{2, \alpha}$ homeomorphism $\Psi_{1}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ sending $B\left(x_{0}, R\right) \cap \partial \Omega$ to a portion of a (flat) hyperplane and $\partial B\left(x_{0}, R\right) \cap \Omega$ to the boundary of half a disc. We then choose another $\mathcal{C}^{2, \alpha}$ homeomorphism $\Psi_{2}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ sending the whole half disc into a disc (= a ball). Therefore $\Psi_{2} \circ \Psi_{1}$ maps our original boundary portion into a portion of the boundary of a ball.
Similarly to previous computations of this sort we define the induced operator $\tilde{L}$ on the induced domain $\Psi_{2} \circ \Psi_{1}\left(B\left(x_{0}, R\right) \cap \Omega\right)$ and define the induced functions $\tilde{u}, \tilde{\varphi}, \tilde{f}$ and we get a new Dirichlet problem with all norms of our original objects equivalent to those of our induced ones. Note that still $\tilde{c}:=c \circ \Psi_{1}^{-1} \circ \Psi_{2}^{-1} \leq 0$, therefore by our theory exists a unique solution $v \in \mathcal{C}^{2, \alpha}\left(\Psi_{2} \circ\right.$ $\left.\Psi_{1}\left(B\left(x_{0}, R\right) \cap \Omega\right) \cup \Psi_{2} \circ \Psi_{1}\left(B\left(x_{0}, R\right) \cap \partial \Omega\right)\right) \cap \mathcal{C}^{0}\left(\Psi_{2} \circ \Psi_{1}\left(\overline{B\left(x_{0}, R\right)} \cap \bar{\Omega}\right)\right)$ for the induced Dirichlet problem. Now our $\tilde{u}$ also solves it. So by uniqueness $\tilde{u}=v$ and $\tilde{u}$ has $\mathcal{C}^{2, \alpha}$ regularity as the induced boundary portion, and by pulling back through $\mathcal{C}^{2, \alpha}$ diffeomorphisms we get that so does $u$.
Remark. The assumption $c \leq 0$ is not necessary although modifying the proof is non-trivial without this assumption (exercise). We needed it in order to be able to use our existence result. But since we already assume a solution exists we may use some of our previous results which do not need $c \leq 0$ and which secure $\mathcal{C}^{2, \alpha}$ regularity upto the boundary.
## Lecture 17
April $15^{\text {th }}, 2004$
## Higher boundary regularity
$\mathcal{W}$ extend our results to include the boundary.
Higher a priori regularity upto the boundary. Let $u \in \mathcal{C}^{2}(\Omega) \cap \mathcal{C}^{0}(\bar{\Omega})$ be a solution of
$$
\begin{array}{rrrr}
L u & =f & \text { on } & \Omega, \\
u & =\varphi & \text { on } & \partial \Omega .
\end{array}
$$
Assume uniformly elliptic and that all coefficients are in $\mathcal{C}^{k, \alpha}(\bar{\Omega})$ with $0<\alpha<1$ and that $\Omega$ is a $\mathcal{C}^{k+2, \alpha}$ domain. If $f \in \mathcal{C}^{k, \alpha}(\bar{\Omega})$ and $\varphi \in \mathcal{C}^{k+2, \alpha}(\partial \Omega)$ then $u \in \mathcal{C}^{k+2, \alpha}$.
Proof. For $k=0$ our previous results apply unchanged (the case $c \not \leq 0$ can be handled if one believes the Remark above).
$k=1$, the crucial case, we use once again difference quotients. As usual, localize to $B^{+}:=$ $B\left(x_{0}, R\right) \cap \bar{\Omega}, x_{0} \in \partial \Omega$. Then flatten the boundary with the help of a $\mathcal{C}^{3, \alpha}$ diffeomorphism $\Psi$. Assume the flat portion lies on the $x_{n}=0$ hyperplane. We get
$$
\tilde{L} \Delta^{h} \tilde{u}=\Delta^{h} \tilde{f}-\Delta^{h} \tilde{a}^{i j} \cdot \mathrm{D}_{i j} \tilde{u}^{h}-\Delta^{h} \tilde{b}_{i} \cdot \mathrm{D}_{i} \tilde{u}^{h}-\Delta^{h} \tilde{c} \cdot \tilde{u}^{h}
$$
We know the RHS is uniformly $\mathcal{C}^{\alpha}\left(\Psi\left(B^{+}\right)\right)$bounded, while the LHS is only so for the directions $l=1, \ldots, n-1$, the tangent directions on $\Psi\left(\partial B^{+}\right)$, since the equation $u=\varphi$ holds there and may be differentiated in those directions (and $\varphi$ has 3 derivatives).
We therefore use Schauder estimates for $\Delta^{h} \tilde{u}$ which give it is uniformly bounded in $\mathcal{C}^{2, \alpha}\left(\Psi\left(B^{+} \prime\right)\right)$, $\forall B^{+} / \Subset B^{+}$similarly to the higher regularity Theorem for the interior. This is so since the estimates used there hold, in fact, upto the boundary. We get therefore $\mathrm{D}_{l} \tilde{u} \in \mathcal{C}^{2, \alpha}\left(\Psi\left(B^{+}\right)\right), l=1, \ldots, n-1$. We treat the remaining derivative. We have $\mathrm{D}_{i} \mathrm{D}_{l} \tilde{u} \mathcal{C}^{1, \alpha}\left(\tilde{B}^{+}\right), i=1, \ldots, n, l=1, \ldots, n-1 \Leftrightarrow$ $\mathrm{D}_{n l} \tilde{u}=\mathrm{D}_{l}\left(\mathrm{D}_{n} \tilde{u}\right)$ (mixed derivatives commute as $\left.\tilde{u} \in \mathcal{C}^{2} !\right)$. So all we have to show now is $\mathrm{D}_{n n} \tilde{u} \in$ $\mathcal{C}^{1, \alpha}\left(\tilde{B}^{+}\right)$.
From $\tilde{L} \tilde{u}=\tilde{f}$ we find
$$
\mathrm{D}_{n n} \tilde{u}=\frac{1}{\tilde{a}^{n n}}\left(\tilde{f}-\left(\tilde{L}-\tilde{a}^{n n}\right) \tilde{u}\right)
$$
From previous calculations of the form of $\tilde{a}$ we see that choosing $\Psi$ appropriately we may diagonalize it. Then uniformly elliptic gives $\frac{1}{\tilde{a}^{n n}}>\gamma>0$. The RHS is $\mathcal{C}^{1, \alpha}\left(\tilde{B}^{+}\right) \Leftrightarrow$ so is LHS $\Leftrightarrow \mathrm{D} \tilde{u} \in$ $\mathcal{C}^{2, \alpha}\left(\Psi\left(B^{+}\right)\right) \Leftrightarrow u$ is $\mathcal{C}^{3, \alpha}$ near $x_{0}$.
The cases $k \geq 2$ are handled as in the interior Theorem.
This wraps up our discussion on Hölder spaces/norms.
## Hilbert spaces
Let $V$ be a vector space over the field $\mathbb{R}$. Let $(\cdot, \cdot)$ be a map $V \times V \rightarrow \mathbb{R}$ such that
$$
\begin{aligned}
& \text { i) }(x, y)=(y, x) \\
& \text { ii) }\left(\alpha_{1} x_{1}+\alpha_{2} x, y\right)=\alpha_{1}\left(x_{1}, y\right)+\alpha_{2}\left(x_{2}, y\right), \quad \forall \alpha_{i} \in \mathbb{R} \\
& \text { iii) }(x, x)>0, \quad \forall x \neq 0
\end{aligned}
$$
Let $\|x\|:=(x, x)^{\frac{1}{2}}$. One can then demonstrate
$$
\begin{aligned}
& \|(x, y)\| \leq\|x\| \cdot\|y\| \quad \text { Schwarz inequality } \\
& \|x+y\| \leq\|x\|+\|y\| \quad \text { triangle inequality }
\end{aligned}
$$
The $2^{\text {nd }}$ affirms that $\|\cdot\|$ defines a norm.
If $\|\cdot\|$ is complete $(V,(\cdot, \cdot))$ is a Hilbert space. Let $F: V \rightarrow \mathbb{R}$ be linear, i.e a linear functional on $V$. If $\sup _{0 \neq x \in V} \frac{|F(x)|}{\|x\|}=:\|F\|_{V^{\star}}<\infty, F$ is bounded. Here $V^{\star}=\{$ bounded linear functional on $V\}$. Similary for a Hilbert space $\mathcal{H}$ define similarly $\mathcal{H}^{\star}$.
We give the statement of the main theorem regarding Hilbert spaces . Like the Continuity Method it will serve us as a strong tool for us to attack abstract questions, a tool from Functional Analysis.
Riesz Representation Theorem. Let $\mathcal{H}$ be a Hilbert space, $F \in \mathcal{H}^{\star}$. Then $\exists ! f \in \mathcal{H}$ such that
$$
\begin{aligned}
& \text { i) } F(x)=(f, x), \quad \forall x \in \mathcal{H} \\
& \text { ii) }\|F\|_{\mathcal{H}^{\star}}=\|f\|_{\mathcal{H}}
\end{aligned}
$$
In particular $\Leftrightarrow \mathcal{H}=\mathcal{H}^{\star}$.
## Sobolev Spaces
## Motivation
$$
\begin{aligned}
& \text { If } \Delta u=f, u \in \mathcal{C}^{2}(\Omega) \text { then } \forall \varphi \in \mathcal{C}_{0}^{1}(\Omega) \quad \varphi \Delta u=\varphi f \text { and } \\
& -\int_{\Omega} \nabla \varphi \nabla u=\int_{\Omega} \Delta u \cdot \varphi=\int_{\Omega} f \cdot \varphi .
\end{aligned}
$$
This observation lies at the heart of weak formulations of the Laplace equation.
Define an inner product on $\mathcal{C}_{0}^{1}(\Omega):=$ compactly supported functions in $\mathcal{C}^{1}(\Omega)$
$$
\left(\varphi_{1}, \varphi_{2}\right):=\int_{\Omega} \nabla \varphi_{1} \nabla \varphi_{2}
$$
$\left(\mathcal{C}_{0}^{1}(\Omega),(\cdot, \cdot)\right)$ is not complete: a sequence of functions may degenerate to a function which is not everywhere differentiable though continuous. Denote by $W_{0}^{1,2}(\Omega)$ the completion of $\mathcal{C}_{0}^{1}(\Omega)$ WRT this norm. It is nice to note that $(\cdot, \cdot)$ extends to an inner product on $W_{0}^{1,2}(\Omega)$ : represent any two elements there as limits of sequences of elements in $\mathcal{C}_{0}^{1}(\Omega)$ and take the limit of the inner products of those, which are well defined. Hence $W_{0}^{1,2}(\Omega)$ is a Hilbert space.
At this stage we do not yet know how $W_{0}^{1,2}(\Omega)$ looks like. Maye its elements are not even functions.
We continue with the motivation for defining those spaces. Let $F(\varphi):=-\int_{\Omega} f \cdot \varphi, \quad \forall \varphi \in \mathcal{C}_{0}^{1}(\Omega)$. $F=F(f, \Omega)$ extends to a linear functional on $W_{0}^{1,2}(\Omega)$.
Claim. $F$ is bounded.
since $\mathcal{C}_{0}^{1}(\Omega)$ is dense in its completion $W_{0}^{1,2}(\Omega)$.
$$
\frac{|F(\varphi)|}{\|\varphi\|_{W_{0}^{1,2}}}=\frac{\left|\int_{\Omega} \varphi \cdot f\right|}{\left(\int_{\Omega}|\nabla \varphi|^{2}\right)^{\frac{1}{2}}} \leq \frac{\left(\int_{\Omega} \varphi^{2}\right)^{\frac{1}{2}} \cdot\left(\int_{\Omega} f^{2}\right)^{\frac{1}{2}}}{\left(\int_{\Omega}|\nabla \varphi|^{2}\right)^{\frac{1}{2}}}
$$
Using the Poincaré inequality $\int_{\Omega} \varphi^{2} \leq c(\Omega) \cdot \int_{\Omega}|\nabla \varphi|^{2}$ we find a bound depending on $\Omega, f$ but not on $\varphi$.
Hence by the Riesz Representation Theorem exists $u \in W_{0}^{1,2}(\Omega)$, though we do not know it is a function or even if so whether it has any regularity, such that
$$
\begin{gathered}
F(\varphi)=(u, \varphi) \\
/ / \int_{\Omega} \int_{\Omega} \nabla u \nabla \varphi .
\end{gathered}
$$
We do not know if $u \in \mathcal{C}_{0}^{1}(\Omega)$, just that $u \in W_{0}^{1,2}(\Omega)$. We have a weak formulation of
$$
\begin{array}{rrrrr}
\Delta u & =f & \text { on } & \Omega, \\
u & =0 & \text { on } & \partial \Omega .
\end{array}
$$
for any $f \in L^{2}(\Omega)$ ! Our plan is now: if $f$ has certain regularity, $u$ has that regularity +2 . The philosophy is instead of classically solving the $\Delta$-equation with an exact explicit solution like Poisson's Integral Formula etc. we just enlarge our function spaces. Then the existence of a solution in the enlarged space becomes trivial (following Riesz). The work comes down to showing that the solution actually lies back in our original space of functions! That is regularity theory in a nutshell. We will focus on that in the sequel.
## Weak derivatives
For $u, v_{i} \in L_{\mathrm{loc}}^{1}(\Omega)$ say $" v_{i}=\mathrm{D}_{i} u "$ if
$$
\int_{\Omega} v \varphi=-\int_{\Omega} u \cdot \mathrm{D}_{i} \varphi, \quad \forall \varphi \in \mathcal{C}_{0}^{1}(\Omega)
$$
If such $v$ exists $\forall i=1, \ldots, n$ then $u$ is weakly differentiable in $\Omega$ with $\nabla u={ }^{\text {weak }}\left(v_{1}, \ldots, v_{n}\right)$.
If each $\mathrm{D}_{j} u$ satisfies the above conditions we say $u$ is twice weakly differentiable. We will omit the quotations marks in what follows.
The derivative does not exist pointwise in general. But by the Lesbegues Theorem it does exist pointwise almost everywhere (a.e).
## Definition
We are now in a position to define Sobolev spaces. Let $\|u\|_{L^{p}(\Omega)}:=\left(\int_{\Omega}|u|^{p}\right)^{\frac{1}{p}}$. Define
$$
L^{p}(\Omega):=\left\{\text { equivalence classes of measurable functions such that }\|\cdot\|_{L^{p}(\Omega)}<\infty\right\}
$$
where $f \sim g$ if $f=g$ a.e. Define
$$
W^{k}(\Omega):=\{k \text {-times weakly differentiable functions }\} \cap L_{\mathrm{loc}}^{1}(\Omega) \subseteq L_{\mathrm{loc}}^{1}(\Omega),
$$
Similarly define the Sobolev spaces
$$
W^{k, p}(\Omega) \equiv L^{k, p}(\Omega)=\left\{u \in W^{k}(\Omega), \mathrm{D}^{\alpha} u \in L^{p}(\Omega) \quad \text { all multi- } \quad \text { indices } \alpha,|\alpha| \leq k \subseteq L_{\mathrm{loc}}^{1}(\Omega)\right\}
$$
equipped with the norm
$$
\|\cdot\|_{W^{k}(p)} \Omega:=\left\{\sum_{|\alpha| \leq k} \int_{\Omega}\left|D^{\alpha} \cdot\right|^{p}\right\}^{\frac{1}{p}}
$$
(still need to prove it is a norm!). An equivalent norm is given by
$$
\sum_{|\alpha| \leq k} \int_{\Omega}\left\|D^{\alpha} \cdot\right\|_{L^{0}(\Omega)}
$$
$L^{p}(\Omega)$ is a Banach space! (Riesz-Fischer Theorem). Also $W^{k, p}(\Omega)=L^{k, p}(\Omega)$ are.
Claim. $\mathcal{C}^{\infty}(\Omega) \cap W^{k, p}(\Omega)$ is dense in $W^{k, p}(\Omega)$. i.e. we could have defined $W^{k, p}(\Omega)$ as the completion of $\mathcal{C}^{\infty}(\Omega)$ WRT $\|\cdot\|_{W^{k}(p)} \Omega$.
Given $u \in W^{k, p}(\Omega)$ mollify it to
$$
u_{h}(x):=\int_{\mathbb{R}^{n}} \frac{1}{h^{n}} \rho\left(\frac{|x-y|}{h}\right) u(y) d y
$$
with $\rho$ a smooth bump function on $\mathbb{R}$ with mass 1 and support in $\left[-\frac{1}{2}, \frac{1}{2}\right]$. Now $u \in \mathcal{C}^{\infty}(\Omega) \cap W^{k, p}(\Omega)$ and $u_{h} \rightarrow u$ in the $W^{k, p}(\Omega)$ norm.
We now define Sobolev spaces of compactly supported objects
$$
W_{0}^{k, p}(\Omega):=\text { completion of } \mathcal{C}_{0}^{k}(\Omega) \text { WRT }\|\cdot\|_{W^{k, p}(\Omega)}
$$
Note functions in $\mathcal{C}_{0}^{k}(\Omega)$ vanish on $\partial \Omega$ so in a sense $W_{0}^{1, p}(\Omega)$ (respectively $W_{0}^{k, p}(\Omega)$ ) can be thought of as (weak) functions which vanish on $\partial \Omega$ (whose first $k-1$ derivatives vanish on $\partial \Omega$ ).
## Equivalence of norms
For $\varphi \in W_{0}^{1,2}(\Omega)$ we defined two norms. One using the inner product $\int_{\Omega} \nabla \varphi_{1} \cdot \nabla \varphi_{2}$ on $\mathcal{C}_{0}^{1}(\Omega)$ which gave us the norm
$$
\|\varphi\|=\left\{\int_{\Omega}|\nabla \varphi|^{2}\right\}^{\frac{1}{2}}
$$
and another norm
$$
\|\varphi\|^{\prime}=\left\{\sum_{|\alpha| \leq 1} \int_{\Omega}\left|\mathrm{D}^{\alpha} \varphi\right|^{2}\right\}^{\frac{1}{2}}=\left\{\int_{\Omega}|\varphi|^{2}+\sum_{i=1}^{n}\left|\mathrm{D}^{i} \varphi\right|^{2}\right\}^{\frac{1}{2}} \leq\|\varphi\|_{L^{2}(\Omega)}+\|\nabla \varphi\|_{L^{2}(\Omega)}
$$
These norms are indeed equivalent since we are assuming compact support! The Poincaré inequality shows $\|\cdot\|^{\prime} \leq(1+c(\Omega)) \cdot\|\cdot\|$. This inequality fails grossly for non-compactly supported functions, e.g the constant function. Since $\|\cdot\| \leq\|\cdot\|^{\prime}$ the norms are equivalent.
Remark. in both of the above norms we define first the norms of functions which are also in $\mathcal{C}_{0}^{1}(\Omega)$ and then we extend the norm to the completion by means of norms of limits of sequences whose elements are all in $\mathcal{C}_{0}^{1}(\Omega)$ (those are dense).
## Lecture 18
April $22^{\text {nd }}, 2004$
## Embedding Theorems for Sobolev spaces
Sobolev Embedding Theorem. Let $\Omega$ a bounded domain in $\mathbb{R}^{n}$, and $1 \leq p<\infty$.
$$
W_{0}^{1, p}(\Omega) \subseteq\left\{\begin{array}{l}
L^{\frac{n p}{n-p}}(\Omega), \quad p<n \\
\mathcal{C}^{0, \alpha}(\Omega), \alpha=1-\frac{n}{p}, \quad p>n, \\
\text { i.e in particular } \subseteq \mathcal{C}^{0}(\Omega) .
\end{array}\right.
$$
Furthermore, those embeddings are continuous in the following sense: there exists $C(n, p, \Omega)$ such that for $u \in W_{0}^{1, p}(\Omega)$
$$
\begin{aligned}
&\|u\|_{L^{\frac{n p}{n-p}}(\Omega)} \leq C \cdot\|\nabla u\|_{L^{p}(\Omega)}, \quad \forall p<n \\
& \sup _{\Omega}|u| \leq C^{\prime} \cdot \operatorname{Vol}(\Omega)^{\frac{1}{n}-\frac{1}{p}} \cdot\|D u\|_{L^{p}(\Omega)}, \quad \forall p>n .
\end{aligned}
$$
We start with a function whose derivative and itself belong to $L^{p}$. The above theorem gives us more regularity for the function - it belongs to $L^{p \cdot \frac{n}{n-p}}$ - based on its regular derivative.
Proof. $\mathcal{C}_{0}^{1}(\Omega)$ is dense in $W_{0}^{1, p}(\Omega)$. We prove first for $u \in \mathcal{C}_{0}^{1}(\Omega)$ and will later justify why the proof actually extends to the larger space.
Case $p=1$. fix an index $i \in\{1, \ldots, n\}$ and observe
$$
u(x)=\int_{-\infty}^{x_{i}} \mathrm{D}_{i} u\left(x_{1}, \ldots, t, \ldots, x_{n}\right) d t .
$$
From which
$$
\begin{aligned}
|u(x)| & \leq \int_{-\infty}^{x_{i}}\left|\mathrm{D}_{i} u\right|\left(x_{1}, \ldots, t, \ldots, x_{n}\right) d t \\
& \leq \int_{-\infty}^{\infty}\left|\mathrm{D}_{i} u\right|\left(x_{1}, \ldots, t, \ldots, x_{n}\right) d t .
\end{aligned}
$$
Write this down for each $i$, take a product of the terms and take the $n-1^{\text {th }}$ root of the result to yield altogether
$$
|u(x)|^{\frac{n}{n-1}} \leq \prod_{i=1}^{n}\left(\int_{-\infty}^{\infty}\left|\mathrm{D}_{i} u\right| d x_{i}\right)^{\frac{1}{n-1}} .
$$
Quick Reminder. Hölder's inequality (HI) tells us
$$
\frac{1}{p}+\frac{1}{q}=1 \Rightarrow \int u \cdot v \leq\left(\int u^{p}\right)^{\frac{1}{p}} \cdot\left(\int v^{p}\right)^{\frac{1}{q}}
$$
or more generally
$$
\frac{1}{p_{1}}+\ldots+\frac{1}{p_{k}}=1 \Rightarrow \int u_{1} \cdots u_{k} \leq\left(\int u_{1}^{p_{1}}\right)^{\frac{1}{p_{1}}} \cdots\left(\int u_{k}^{p_{k}}\right)^{\frac{1}{p_{k}}}
$$
Coming back to our inequality, we integrate over the $x_{1}$ axis and subsequently apply the Hölder inequality with $k=n-1, p_{i}=n-1-$
$$
\begin{aligned}
\int_{-\infty}^{\infty}|u(x)|^{\frac{n}{n-1}} d x_{1} & \leq \int_{-\infty}^{\infty} \prod_{i=1}^{n}\left(\int_{-\infty}^{\infty}\left|\mathrm{D}_{i} u\right| d x_{i}\right)^{\frac{1}{n-1}} d x_{1} . \\
& =\left(\int_{-\infty}^{\infty}\left|\mathrm{D}_{1} u\right| d x_{1}\right)^{\frac{1}{n-1}} \cdot \int_{-\infty}^{\infty} \prod_{i=2}^{n}\left(\int_{-\infty}^{\infty}\left|\mathrm{D}_{i} u\right| d x_{i}\right)^{\frac{1}{n-1}} d x_{1} . \\
& \left.\left.=\left(\int_{-\infty}^{\infty}\left|\mathrm{D}_{1} u\right| d x_{1}\right)^{\frac{1}{n-1}} \cdot\left(\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\left|\mathrm{D}_{2} u\right| d x_{2} d x_{1}\right)^{\frac{1}{n-1}} \cdot \prod_{i=2}^{\infty} u \mid d x_{1}\right)^{\frac{1}{n-1}} \cdot \prod_{-\infty}^{n}\left[\int_{-\infty}^{\infty}\left|\mathrm{D}_{i} u\right| d x_{i}\right]^{\frac{n-1}{n-1}} d x_{1}\right)^{\frac{1}{n-1}} . \\
& \left.=\int_{-\infty}^{\infty}\left(\int_{-\infty}^{\infty}\left|\mathrm{D}_{i} u\right| d x_{i}\right) d x_{1}\right]^{\frac{1}{n-1}} .
\end{aligned}
$$
Now courageously continuing with this confusing calculation, we integrate over the $x_{2}$ axis. This is the reason we singled out the second terms from the $n-2$ others ones; if we would have integrated now over the $x_{j}$ axis we would have choosen a term involving integration over that axis. And indeed now the middle term is a constant WRT this operation, that is only the other two terms appear in this integral, hence -
$$
\begin{aligned}
& \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}|u(x)|^{\frac{n}{n-1}} d x_{1} d x_{2} \\
& \leq\left(\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\left|\mathrm{D}_{2} u\right| d x_{2} d x_{1}\right)^{\frac{1}{n-1}} \cdot\left(\int_{-\infty}^{\infty}\left\{\left(\int_{-\infty}^{\infty}\left|\mathrm{D}_{1} u\right| d x_{1}\right)^{\frac{1}{n-1}} \cdot \prod_{i=3}^{n}\left[\int_{-\infty}^{\infty}\left|\mathrm{D}_{i} u\right| d x_{i}\right] d x_{2}\right\}\right)^{\frac{1}{n-1}} .
\end{aligned}
$$
and using the Hölder Inequality the second term transforms, and we have
$$
\begin{aligned}
=\left(\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\left|\mathrm{D}_{2} u\right| d x_{2} d x_{1}\right)^{\frac{1}{n-1}} \cdot & \left(\int_{-\infty}^{\infty}\left[\int_{-\infty}^{\infty}\left|\mathrm{D}_{1} u\right| d x_{1}\right]^{\frac{n-1}{n-1}} d x_{2}\right)^{\frac{1}{n-1}} \\
& \cdot\left(\prod_{i=3}^{n} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\left|\mathrm{D}_{i} u\right| d x_{i} d x_{1} d x_{2}\right)^{\frac{1}{n-1}} .
\end{aligned}
$$
In the same vein, we now isolate among the $n$ terms the only term involving integration over the $x_{3}$ axis, integrate over that axis and then once again apply the Hölder Inequality for the remaining $n-1$ terms (at each stage we always have $n-1$ terms except from the isolated one; the Hölder Inequality allows us to lift the $\frac{1}{n-1}$ exponent and let another new $d x_{i}$ come in to the integral of those $n-1$ terms).
Finally, therefore, we will arrive at
$$
\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty}|u(x)|^{\frac{n}{n-1}} d x_{1} \cdots d x_{n} \leq \prod_{j=1}^{n}\left(\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty}\left|\mathrm{D}_{j} u\right| d x_{1} \cdots d x_{n}\right)^{\frac{1}{n-1}}
$$
In other words if we restrict to $\Omega$
$$
\left(\left.|| u\right|_{L^{\frac{n}{n-1}}(\Omega)}\right)^{\frac{n}{n-1}} \leq \prod_{j=1}^{n}\left(\int_{\Omega}\left|\mathrm{D}_{j} u\right| d \mathbf{x}\right)^{\frac{1}{n-1}}
$$
or still
$$
\begin{aligned}
\|u\|_{L^{\frac{n}{n-1}}(\Omega)} \leq \prod_{j=1}^{n}\left(\int_{\Omega}\left|\mathrm{D}_{j} u\right| d \mathbf{x}\right)^{\frac{1}{n}} \leq \frac{1}{n} \cdot \sum_{j=1}^{n} \cdot \int_{\Omega}\left|\mathrm{D}_{j} u\right| d \mathbf{x} \leq \frac{1}{n} \cdot \sum_{j=1}^{n} \cdot \int_{\Omega}|\mathrm{D} u| d \mathbf{x} & =\int_{\Omega}|\mathrm{D} u| d \mathbf{x} \\
& =\|\nabla u\|_{L^{1}(\Omega)} .
\end{aligned}
$$
This concludes the $p=1<n$ case. Let us remark that of course we neglected at the last steps to seek the best possible Sobolev constant and contented ourselves with the constant 1:
$$
\|u\|_{L^{\frac{n \cdot 1}{n-1}(\Omega)}} \leq 1 \cdot\|\nabla u\|_{L^{1}(\Omega)} .
$$
In fact the best possible Sobolev constant $c$ is achieved for $\Omega=B(0, r), u=\mathbb{I}_{B(0, r)}\left(\mathbb{I}_{A}\right.$ is the characteristic function on the set $A$, evaluating to 1 on $A$ and 0 otherwise); believing that, we compute
$$
\operatorname{Vol}(B(0, r))^{\frac{n}{n-1}}=c \cdot \int_{B(0, r)}\left|D \mathbb{I}_{B(0, r)}\right| d \mathbf{x}=c \cdot \int_{B(0, r)}\left|\delta_{\partial B(0, r)}\right| d \mathbf{x}=c \cdot \operatorname{Area}(S(r)),
$$
i.e
$$
\left(\omega_{n} r^{n}\right)^{\frac{n}{n-1}}=c \cdot n \omega_{n} r^{n-1} \quad \Rightarrow \quad c=\frac{1}{n \sqrt[n]{\omega_{n}}}
$$
Case $1<p<\infty$. A little trick will make our previous work apply to this case as well. Let $\gamma>1$ be a constant to be specified. We have by our previous case
$$
\|\left.|u|^{\gamma}\right|_{L^{\frac{n}{n-1}(\Omega)}} \leq\left.\left.\int_{\Omega}|\mathrm{D}| u\right|^{\gamma}\left|d \mathbf{x} \leq \gamma \int_{\Omega}\right| u\right|^{\gamma-1} \cdot|\mathrm{D} u| d \mathbf{x}
$$
Let $q$ be such that $\frac{1}{p}+\frac{1}{q}=1$. One has using the Hölder Inequality
$$
\left(\int_{\Omega}|u|^{\gamma \cdot \frac{n}{n-1}} d \mathbf{x}\right)^{\frac{n}{n-1}} \leq\left(\int_{\Omega}|u|^{(\gamma-1) q} d \mathbf{x}\right)^{\frac{1}{q}} \cdot\left(\int_{\Omega}|\mathrm{D} u|^{p} d \mathbf{x}\right)^{\frac{1}{p}}
$$
We have $q=\frac{p}{p-1}$. Choose $\gamma=\frac{n-1}{n-p} \cdot p$ in order to have $(\gamma-1) q=\frac{n}{n-1} \cdot \gamma$. Hence
$$
\left(\int_{\Omega}|u|^{\left(\frac{n-1}{n-p} \cdot p\right) \cdot \frac{n}{n-1}} d \mathbf{x}\right)^{\frac{n}{n-1}} \leq\left(\int_{\Omega}|u|^{\left.\left(\frac{n-1}{n-p} \cdot p\right)-1\right) \cdot\left(\frac{p}{p-1}\right)} d \mathbf{x}\right)^{\frac{p-1}{q}} \cdot\left(\int_{\Omega}|\mathrm{D} u|^{p} d \mathbf{x}\right)^{\frac{1}{p}}
$$
or succintly
$$
\|u\|_{L^{\frac{n p}{n-p}}(\Omega)}=\left\{\int_{\Omega}|u|^{\frac{n p}{n-p}}\right\}^{\frac{n-1}{n}-\frac{p-1}{p}} \leq \frac{n-1}{n-p} \cdot p\|\nabla u\|_{L^{p}(\Omega)} .
$$
This deals with the case $p<n$ indeed. We remark that characteristic functions no longer give the best Sobolev constants in the case $1<p<n$.
Remark. The above proof holds and is valid for $u \in \mathcal{C}_{0}^{1}(\Omega)$ ! We did not prove for distributional coefficient. If $u$ is only in $W_{0}^{1, p}(\Omega)$, take a sequence $\left\{u_{m}\right\} \subseteq \mathcal{C}_{0}^{1}(\Omega)$ such that $u_{m} \rightarrow u$ in the $W_{0}^{1, p}(\Omega)$-norm. This means that also
$$
\left\|u_{i}-u_{j}\right\|_{L^{\frac{n p}{n-p}(\Omega)}} \leq c \cdot\left\|\mathrm{D} u_{i}-d u_{j}\right\|_{L^{p}(\Omega)} \rightarrow 0
$$
$\left\{u_{m}\right\}$ is thus a Cauchy sequence in $L^{\frac{n p}{n-p}}(\Omega) . L^{\frac{n p}{n-p}}(\Omega)$ is a Banach space d'aprês Riesz-Fischer, i.e $u^{\prime}:=\lim \left\{u_{m}\right\} \in L^{\frac{n p}{n-p}}(\Omega) ; \quad \Leftrightarrow \quad u^{\prime}=u$ is in that space too. Now
$$
\begin{gathered}
\left\|u_{m}\right\|_{L^{\frac{n p}{n-p}}(\Omega)} \leq c \cdot\left\|\mathrm{D} u_{m}\right\|_{L^{p}(\Omega)} \rightarrow 0 . \\
\downarrow \\
\downarrow \\
\|u\|_{L^{\frac{n p}{n-p}}(\Omega)} \leq c \cdot\|\mathrm{D} u\|_{L^{p}(\Omega)} \rightarrow 0 .
\end{gathered}
$$
So we our Theorem applies equally well to functions in the larger space. In this last line we needed also mention that $\mathrm{D} u_{m} \rightarrow \mathrm{D} u$, but this is true since $\left\{\mathrm{D} u_{m}\right\}$ is a Cauchy sequence from same computation as above. Its limit lies in $L^{p}$ again as this is a Banach space and so indeed $\mathrm{D} u_{m} \rightarrow \mathrm{D} u$ in $L^{p}$ and hence also $\left\|\mathrm{D} u_{m}\right\|_{L^{p}(\Omega)} \rightarrow\|\mathrm{D} u\|_{L^{p}(\Omega)}$.
Case $p>n$. We postpone the proof of this case to state a Corollary. Corollary. By iterating, $\forall k \geq 2$ holds
$$
W_{0}^{k, p}(\Omega) \subseteq\left\{\begin{array}{l}
L^{\frac{n p}{n-k \cdot p}}(\Omega), \quad k p<n \\
\mathcal{C}^{m}, \quad 0 \leq m \leq k-\frac{n}{p}
\end{array}\right.
$$
Proof. For instance, if $k=2, u \in W^{2, p} \Rightarrow u, \mathrm{D} u \in W^{1, p}$. By the $k=1$ case above we have $u, \mathrm{D} u \in L^{\frac{n p}{n-p}}$. That means $\mathrm{D} u \in W^{1, \frac{n p}{n-p}} \Rightarrow \quad$ (by $k=1$ case once again) $u \in W^{1, p^{\prime}}$ where
$$
p^{\prime}=\frac{n \cdot\left(\frac{n p}{n-p}\right)}{n-\left(\frac{n p}{n-p}\right)}=\frac{n^{2} p}{n^{2}-n p-n p}=\frac{n p}{n-2 p}
$$
This proof repeated carries over $\forall k \in \mathbb{N}$.
Now for the second inclusion, the promised postponed. We will need the following lemma en passant.
Lemma. Let $\Omega$ be a bounded domain, $B:=$ Ball $\subseteq \Omega, u \in W^{1,1}$. Then for all $x \in \Omega$
$$
\left|u(x)-\frac{1}{\operatorname{Vol}(B)} \int_{B} u d \mathbf{x}\right| \equiv\left|u(x)-\int_{B} u d \mathbf{x}\right| \leq c \cdot \int_{B} \frac{|D u(y)|}{|x-y|^{n-1}} d \mathbf{y}
$$
Proof. By our density theorem $\mathcal{C}_{0}^{1}(\Omega)$ is dense in $W_{0}^{1, p}(\Omega)$ and thus work with $u$ in the former. Take $x, y \in \Omega$. Let $\omega:=\frac{y-x}{|y-x|}$,
$$
u(x)-u(y)=\int_{0}^{|x-y|} \mathrm{D}_{r} u(x+r \omega) d r
$$
Integrating over some ball $B$
$$
\operatorname{Vol}(B) \cdot u(x)-\int_{B} u(y)=\int_{B}\left(\int_{0}^{|x-y|} \mathrm{D}_{r} u(x+r \omega) d r\right) d \mathbf{y}
$$
Put
$$
v(x)= \begin{cases}\mathrm{D}_{r} u(x), & x \in \Omega \\ 0, & x \notin \Omega\end{cases}
$$
Take now a particular ball $B(x, R) \subseteq \Omega$ to get
$$
\left|u(x)-\int_{B} u(y) d \mathbf{y}\right| \leq \frac{1}{\operatorname{Vol}(B)} \int_{|x-y|<2 R}\left(\int_{0}^{\infty}|v(x+r \omega)| d r\right) d \mathbf{y} .
$$
Switch order of integration, and change coordinates to spherical ones
$$
=\frac{1}{\operatorname{Vol}(B)} \int_{0}^{\infty}\left(\int_{0}^{2 R}\left(\int_{S^{n-1}(1)}|v(x+r \omega)| \rho^{n-1} d \omega_{S^{n-1}(1)}\right) d \rho\right) d r
$$
after rescaling, where $(\rho, \omega)$ are the spherical coordinates, i.e $\omega$ are coordinates on the unit sphere. Now
$$
\begin{aligned}
& =\frac{(2 R)^{n}}{n \operatorname{Vol}(B)} \int_{0}^{\infty}\left(\int_{S^{n-1}(1)}|v(x+r \omega)| d \omega_{S^{n-1}(1)}\right) d r \\
& =\frac{(2 R)^{n}}{n \operatorname{Vol}(B)} \int_{0}^{\infty}\left(\int_{S^{n-1}(1)} \frac{|v(x+r \omega)|}{r^{n-1}} r^{n-1} d \omega_{S^{n-1}(1)}\right) d r
\end{aligned}
$$
Set $z:=x+r \omega, \rightarrow r=|r \omega|=|x-z|, r^{n-1} d \omega_{S^{n-1}(1)} d r=d \mathbf{z}$,
$$
=\frac{(2 R)^{n}}{n \operatorname{Vol}(B)} \int_{B} \frac{|v(z)|}{|x-z|^{n-1}} d \mathbf{z}
$$
and as $B(x, R) \subseteq \Omega \quad \Rightarrow$
$$
\leq \frac{(2 R)^{n}}{n \operatorname{Vol}(B)} \int_{\Omega} \frac{\left|\mathrm{D}_{r} u(z)\right|}{|x-z|^{n-1}} d \mathbf{z}
$$
## Claim.
$$
\int_{B_{R}}|x-y|^{1-n}|\mathrm{D} u(y)| d \mathbf{y} \leq C R^{1-\frac{n}{p}}|| \mathrm{D} u \|_{L^{p}\left(B_{R}\right)}, \quad \forall p>n .
$$
for $B_{R}:=B\left(x_{0}, R\right) \subseteq \mathbb{R}^{n}$ Proof. By the Hölder inequality, $\forall q$ such that $\frac{1}{q}+\frac{1}{p}=1$
$$
\begin{aligned}
\int_{B_{R}}|x-y|^{1-n}|\mathrm{D} u(y)| d \mathbf{y} & \leq\left\{\int_{B_{R}}|x-y|^{(1-n) q} d \mathbf{y}\right\}^{\frac{1}{q}} \cdot\|\mathrm{D} u\|_{L^{p}\left(B_{R}\right)} \\
& \leq \sup _{x \in \Omega}\left\{\int_{B_{R}}|x-y|^{(1-n) q} d \mathbf{y}\right\}^{\frac{1}{q}} \cdot\|\mathrm{D} u\|_{L^{p}\left(B_{R}\right)} \\
& =c \cdot\left\{\int_{B_{R}}\left|x_{0}-y\right|^{(1-n) q} d \mathbf{y}\right\}^{\frac{1}{q}} \cdot\|\mathrm{D} u\|_{L^{p}\left(B_{R}\right)} \\
& =c \cdot\left\{\int_{0}^{R} r^{(1-n) q} r^{n-1} d r\right\}^{\frac{1}{q}} \cdot\|\mathrm{D} u\|_{L^{p}\left(B_{R}\right)} \\
& =c \cdot\left\{\int_{0}^{R} r^{\frac{n-1}{1-p}} d r\right\}^{\frac{1}{q}} \cdot\|\mathrm{D} u\|_{L^{p}\left(B_{R}\right)} \\
& =c \cdot\left(\frac{n-1}{1-p}+1\right) R^{\left(\frac{n-1}{1-p}+1\right) / q} \cdot\|\mathrm{D} u\|_{L^{p}\left(B_{R}\right)} \\
& =C(n, p) R^{\frac{p-n}{p}} \cdot\|\mathrm{D} u\|_{L^{p}\left(B_{R}\right)}
\end{aligned}
$$
as $\left(\frac{n-1}{1-p}+1\right) \cdot \frac{1}{q}=\left(\frac{n-1}{1-p}+1\right) \cdot \frac{p}{1-p}=-\frac{n-1}{p}+\frac{p-1}{p}=\frac{p-n}{p}$.
Now we can finally, combining those last two results conclude the second inclusion in our Theorem as well as the estimate therein. First, using the triangle inequality together with the first lemma we have
$$
|u(x)-u(y)| \leq\left|u(x)-\int_{B} u d \mathbf{x}\right|+\int_{B} u d \mathbf{y}-u(y) \mid \leq 2 c \cdot \int_{B} \frac{|D u(y)|}{|x-y|^{n-1}} d \mathbf{y}
$$
which in turn is
$$
\leq c(n, p)|x-y|^{1-\frac{n}{p}}\|\mathrm{D} u\|_{L^{p}(B)}
$$
once we choose a ball $B=B(x,|x-y|)$ and apply the Claim. Since this is for any $x, y \in \Omega$, and $u \in W^{1, p}(\Omega)$ then $u \in \mathcal{C}^{1-\frac{n}{p}}(\Omega)$, if $p>n$.
Second and finally, we have as well
$$
\begin{aligned}
|u(x)| \leq\left|u(x)-\int_{B} u d \mathbf{x}\right| \leq 2 c \cdot \int_{\Omega} \frac{|D u(y)|}{|x-y|^{n-1}} d \mathbf{y} & \leq c(n, p) \cdot \operatorname{diam}(\Omega)^{1-\frac{n}{p}}\|\mathrm{D} u\|_{L^{p}(\Omega)} \\
& =c^{\prime}(n, p) \cdot \operatorname{Vol}(\Omega)^{\frac{1}{n}-\frac{1}{p}}\|\mathrm{D} u\|_{L^{p}(\Omega)}
\end{aligned}
$$
which gives the desired sup norm. Indeed for $k \geq 2$ the Corollary follows by iterating: we get first Hölder regularity of $u$, then we have $\mathrm{D} u$ is $W_{0}^{1, p}(\Omega)$ so we apply the first Theorem to it and get $\mathrm{D} u$ is Hölder and so on.
## Lecture 19
## April $27^{\text {th }}, 2004$
We give a slightly different proof of
Theorem. Let $\Omega$ a bounded domain in $\mathbb{R}^{n}$, and $1 \leq p<\infty$.
$$
W_{0}^{1, p}(\Omega) \subseteq \mathcal{C}^{0, \alpha}(\Omega), \quad \alpha=1-\frac{n}{p}, \quad p>n
$$
and $\exists C(n, p, \Omega)$ such that for $u \in W_{0}^{1, p}(\Omega)$
$$
\|u\|_{C^{0, \alpha}(\Omega)} \leq C \cdot\|u\|_{W^{1, p}(\Omega)}, \quad \forall p>n
$$
in other words
$$
\sup _{\Omega}|u|+|u|_{C^{0, \alpha}(\Omega)} \leq C \cdot\left\{\|u\|_{L^{p}(\Omega)}+\|\nabla u\|_{L^{p}(\Omega)}\right\}, \quad \forall p>n .
$$
Note the inequality is stronger than the one we stated in the previous lecture.
Proof. We take $u \in \mathcal{C}_{0}^{1}(\Omega)$ as before, WLOG (density argument). Extend $u$ to $\mathbb{R}^{n}$ trivially, i.e set $u=0$ on $\mathbb{R}^{n} \backslash \Omega$. Let $x, y \in \Omega$ and $\sigma=|x-y|$ and let $p$ be the point $\frac{x+y}{2}$. Put $B=B(p, \sigma)$ and take $z \in B$. By the Fundamental Theorem of Calculus
$$
\begin{aligned}
u(x)-u(z) & =\int_{0}^{1} \frac{d}{d t} u(x+(1-t) z) d t \\
& =\int_{0}^{1} \nabla u(x+t(z-x)) \cdot(z-x) d t .
\end{aligned}
$$
Integrating over $z \in B$
$$
\begin{aligned}
\left|\int_{B} u(z) d \mathbf{z}-\operatorname{Vol}(B) u(x)\right| & \leq \int_{B} \int_{0}^{1}|\nabla u(x+t(z-x))| \cdot|z-x| d t d \mathbf{z} \\
& \leq 2 \sigma \int_{B} \int_{0}^{1}|\nabla u(x+t(z-x))| d t d \mathbf{z} \\
& =2 \sigma \int_{0}^{1}\left(\int_{B}|\nabla u(x+t(z-x))| d \mathbf{z}\right) d t
\end{aligned}
$$
Change variables to
$$
\bar{z}:=x+t(z-x), \quad \rightarrow \quad d \overline{\mathbf{z}}=t^{n} d \mathbf{z}
$$
For $z \in B(x, \sigma) \Rightarrow \bar{z} \in B(x, t \sigma)=: \bar{B}$. In the new variable we have now
$$
\left|\int_{B} u d \mathbf{z}-\operatorname{Vol}(B) u(x)\right| \leq 2 \sigma \int_{0}^{1} t^{-n}\left(\int_{\bar{B}}|\nabla u(\bar{z})| d \overline{\mathbf{z}}\right) d t
$$
By the Hölder Inequality for $q$ such that $\frac{1}{p}+\frac{1}{q}=1$
$$
\begin{array}{ll}
\left.\int_{\bar{B}}|\nabla u(\bar{z})| d \overline{\mathbf{z}}\right) d t \leq\left\{\int_{\bar{B}} 1^{q}\right\}^{\frac{1}{q}} \cdot\left\{\int_{\bar{B}}|\nabla u(w)|^{p} d \mathbf{w}\right\}^{\frac{1}{p}} & \\
& =\operatorname{Vol}(B(t \sigma))^{\frac{1}{q}}\|\nabla u\|_{L^{p}(\bar{B})} \\
& \leq \operatorname{Vol}(B(t \sigma))^{\frac{1}{q}}\|\nabla u\|_{L^{p}(\Omega)} \\
& =\omega_{n}^{\frac{1}{q}} t^{\frac{n}{q}} \sigma^{\frac{n}{q}}\|\nabla u\|_{L^{p}(\Omega)}
\end{array} \quad \Rightarrow
$$
Divide now throughout by $\operatorname{Vol}(B)=\omega_{n} \sigma^{n}$
$$
\begin{aligned}
\left|\int_{B} u(z) d \mathbf{z}-u(x)\right| & \leq \sigma^{1+\frac{n}{q}-n} \omega_{n}^{\frac{1}{q}-1}\left(\int_{0}^{1} t^{-n\left(1-\frac{1}{q}\right)} d t\right)\|\nabla u\|_{L^{p}(\Omega)} \\
& =\sigma^{1-\frac{n}{p}} \omega_{n}^{-\frac{1}{p}}\left(\int_{0}^{1} t^{-n\left(\frac{1}{p}\right)} d t\right)\|\nabla u\|_{L^{p}(\Omega)}
\end{aligned}
$$
and the integral evaluates to $\left.\left[\frac{t^{-\frac{n}{p}+1}}{1-\frac{n}{p}}\right]\right|_{0} ^{1}$ which is finite iff $p>n$. We thus conclude
$$
\left|\int_{B} u(z) d \mathbf{z}-u(x)\right| \leq c(n, p) \cdot \sigma^{1-\frac{n}{p}}\|\nabla u\|_{L^{p}(\Omega)}
$$
We repeat the above computation with $x$ replaced by $y$ and use the triangle inequality, which gives us
$$
|u(x)-u(y)| \leq\left. 2 c(n, p) \cdot|x-y|^{1-\frac{n}{p}}|| \nabla u\right|_{L^{p}(\Omega)}
$$
and subsequently
$$
\frac{|u(x)-u(y)|}{|x-y|^{1-\frac{n}{p}}} \leq 2 c(n, p) \cdot\|\nabla u\|_{L^{p}(\Omega)}
$$
And concluding
$$
\|u\|_{C^{\alpha}(\bar{\Omega})}=\|u\|_{L^{\infty}(\Omega)}+\sup _{x \neq y \in \Omega} \frac{|u(x)-u(y)|}{|x-y|^{1-\frac{n}{p}}} \leq C(n, p, \Omega) \cdot\|\nabla u\|_{L^{p}(\Omega)} .
$$
since both $\mathcal{C}^{0}$ and $L^{\infty}$ norms coincide, being just $\sup _{\Omega}$, and finally because by our above computations we can also bound the $L^{\infty}$ norm in terms of the $L^{p}$ norm of $\mathrm{D} u$
$$
|u(x)| \leq 2 c(n, p, \operatorname{diam}(\Omega)) \cdot\|\nabla u\|_{L^{p}(\Omega)}
$$
so $\|u\|_{L^{\infty}(\Omega)}$ is bounded by the same RHS .
## Compactness Theorems
Lemma. Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$, and $1 \leq p<\infty$. Let $S$ be a bounded set in $L^{p}(\Omega)$. In other words,
$$
\forall u \in S, \quad\|u\|_{L^{p}(\Omega)} \leq M_{S} .
$$
Suppose $\forall \epsilon>0, \quad \exists \delta>0$ such that
$$
\forall u \in S, \forall|z|<\delta \quad \int_{\Omega}|u(y+z)-u(y)|^{p} d \mathbf{y}<\epsilon
$$
Then $S$ is precompact in $L^{p}(\Omega)$ (denoted $S \Subset L^{p}(\Omega)$ ), i.e every sequence of functions in $S$ has convergent subsequence ("subconverges"), or equivalently $\bar{S}$ is compact.
This is an Arzelà-Ascoli type theorem: bounded equicontinuous family is precompact. We just have to show somehow that the integral equicontinuity-type condition implies equicontinuity. Proof. Mollify $u$ as done previously in the course
$$
u_{h}=\int_{\mathbb{R}^{n}} \rho_{h}(x-y) u(y) d \mathbf{y}, \quad \rho_{h}(z)=\frac{1}{h^{n}} \rho\left(\frac{|z|}{h}\right) .
$$
Set $S_{h}:=\left\{u_{h}, u \in S\right\}$.
We compute
$$
\begin{aligned}
u_{h}=\int_{\mathbb{R}^{n}} \rho_{h}(x-y) u(y) d \mathbf{y} & =\int_{\mathbb{R}^{n}} \rho_{h}(x-y)|u(y)| d \mathbf{y} \\
& =\int_{\mathbb{R}^{n}} \rho_{h}^{\frac{1}{q}} \rho_{h}^{\frac{1}{p}}|u(y)| d \mathbf{y} \\
& \leq\left\{\int_{\mathbb{R}^{n}} \rho_{h}\right\}^{\frac{1}{q}} \cdot\left\{\rho_{h}|u(y)|^{\frac{1}{p}} d \mathbf{y}\right. \\
& \leq\|u\|_{L^{p}(\Omega)} .
\end{aligned}
$$
Now
$$
\begin{aligned}
u_{h}(x+z)-u_{h}(x) & =\int_{\mathbb{R}^{n}}\left[\rho_{h}(x+z-y)-\rho_{h}(x-y)\right] u(y) d \mathbf{y} \\
& =\int_{\mathbb{R}^{n}}\left[\rho_{h}(x-y)[u(y+z)-u(y)] d \mathbf{y}\right.
\end{aligned}
$$
and the same estimate as above yields
$$
u_{h}(x+z)-u_{h}(x) \leq 1 \cdot\left\{\int_{\Omega}|u(y+z)-u(y)|^{p}\right\}^{\frac{1}{p}} \leq \epsilon^{\frac{1}{p}}
$$
Now by our assumption for $\delta>0$ small enough and $|z|<\delta$ we will attain any desired $\epsilon$ on the RHS. Note $\int_{\mathbb{R}^{n}} \rho_{h}=1$ is fixed for all $h$ by our choice of $\rho$. Hence by definition we see that $S_{h}$ is an equicontinuous family, and bounded wrT the $L^{p}(\Omega)$ norm as inside $S$, hence by the Arzelà-Ascoli theorem $S_{h}$ is precompact in the space $L^{p}(\Omega)$.
Now $\lim _{h \rightarrow 0} S_{h} \rightarrow S$ as we have seen in previous lectures. So as the above estimates are independent of $h, S$ is precompact itself in $L^{p}(\Omega)$.
Theorem (Kodrachov) Let $\Omega$ be bounded in $\mathbb{R}^{n}$.
$$
\begin{aligned}
& \text { (I) } p<n: \quad W_{0}^{1, p}(\Omega) \subseteq L^{q}(\Omega) \quad \forall 1 \leq q<\frac{n p}{n-p} . \\
& \text { (II) } p>n: \quad W_{0}^{1, p}(\Omega) \subseteq \mathcal{C}^{0, \alpha}(\bar{\Omega}) \quad \forall 0<\alpha<1-\frac{n}{p} .
\end{aligned}
$$
and moreover $W_{0}^{1, p}(\Omega)$ is compactly embedded in each of the RHSs.
We have then a curious situation- $W_{0}^{1, p}(\Omega) \subseteq L^{\frac{n p}{n-p}} \subseteq L^{q} \quad$ for $1 \leq q<\frac{n p}{n-p}$ but the first inclusion is only continuous! Only for $q$ stricly smaller than $\frac{n p}{n-p}$ is it compact... And similarly for the case $p>n$.
For the sake of clarity: we say $B_{1} \subseteq B_{2}$ is compactly embedded if for every bounded set $S$ in $B_{1}$, $i(S) \subseteq B_{2}$ is precompact, where $i: B_{1} \rightarrow B_{2}$ is the inclusion map.
Proof. Case $q=1$. By the density argument we mentioned repeatedly we assume wLOG $S \subseteq \mathcal{C}_{0}^{1}(\Omega)$ and that $M_{S}=1$. Let $u \in S$. Then $\|u\|_{L^{p}(\Omega)} \leq 1,\|\mathrm{D} u\|_{L^{p}(\Omega)} \leq 1$. Hence $\|u\|_{L^{1}(\Omega)}=\int_{\Omega}|u(x)| \leq$ $\left\{\int_{\Omega} 1\right\}^{\frac{1}{q}}\left\{\int_{\Omega}|u|^{p}\right\}^{\frac{1}{p}} \leq \operatorname{Vol}(\Omega)^{\frac{1}{q}} \cdot 1$, in other words $S$ is also bounded in $L^{1}$. Once we show the condition of the Lemma holds then we will have precompactness in $L^{1}(\Omega)$. And indeed
$$
\begin{aligned}
u(y+z)-u(y)=\int_{0}^{1} \frac{d u}{d t}(y+t z) d t=\int_{0}^{1} \nabla u(y+t z) \cdot z d t \quad \Rightarrow \\
\int_{\Omega}|u(y+z)-u(y)| d \mathbf{y} \leq|z| \operatorname{Vol}(\Omega)^{\frac{1}{q}}|| \nabla u \|_{L^{p}(\Omega)} \leq c|z| .
\end{aligned}
$$
Case $1<q<\frac{n p}{n-p}$. We try to find some estimates for the $L^{q}(\Omega)$ norm using the indispensible Hölder Inequality. Naturally we will be able to take care of boundedness of all such $q$ together if we allude to the fact that the $\Lambda^{\frac{n p}{n-p}}(\Omega)$ is bounded, indeed the $L^{p}$ norms are increasing in $p$ - first choose $\lambda$ such that $q \lambda+q(1-\lambda) \frac{n-p}{n p}=1$
$$
\begin{aligned}
\left\{\int|u|^{q}\right\}= & \left\{\int|u|^{q \lambda} \cdot|u|^{q(1-\lambda)}\right\} \leq\left\{\int\left(|u|^{q \lambda}\right)^{\frac{1}{q \lambda}}\right\}^{q \lambda} \cdot\left\{\int\left(|u|^{q(1-\lambda)}\right)^{\frac{n p}{n-p} \frac{1}{q(\lambda-1)}}\right\}^{q(1-\lambda)\left(\frac{n-p}{n p}\right)} \Rightarrow \\
& \|u\|_{L^{q}(\Omega)} \leq\|u\|_{L^{1}(\Omega)}^{\lambda} \cdot\|u\|_{L^{\frac{n p}{n-p}}(\Omega)}^{1-\lambda} \\
& \leq\|u\|_{L^{1}(\Omega)}^{\lambda} \cdot c \cdot\|\nabla u\|_{L^{p}(\Omega)}^{1-\lambda} \\
& \leq\|u\|_{L^{1}(\Omega)}^{\lambda} \cdot c \cdot 1 \\
& \leq c(n, p, \operatorname{Vol}(\Omega))
\end{aligned}
$$
where we applied our Theorem from the previous lecture. Now note that we are done using the $q=1$ case: $S$ is bounded in $L^{q}(\Omega)$ and hence a subsequence converges in $L^{q}(\Omega)$, but then by the above inequality it will also converge in $L^{q}(\Omega)$ !
Case $p>n$. By the Theorem of the previous lecture $W_{0}^{1, p}(\Omega) \subseteq \mathcal{C}^{0, \alpha}(\bar{\Omega})$ continuously. But now $\mathcal{C}^{0, \alpha}(\bar{\Omega}) \subseteq \mathcal{C}^{0, \beta}(\bar{\Omega})$ compactly for any $0 \leq \beta<\alpha$ as mentioned in one of the previous lectures.
Remark. Replacing $W_{0}^{1, p}(\Omega)$ by $W^{1, p}(\Omega)$ (the completion of $\mathcal{C}^{1}(\Omega)$ WRT the $W^{1, p}$ norm) in the above embedding theorems require that the domain be Lipschitz, i.e $\partial \Omega$ is of class $\mathcal{C}^{0,1}$ (this is a local requirement).
## Lecture 20
April 29th, 2004
## Difference Quotients and Sobolev spaces
Define
$$
\Delta_{i}^{h} u:=\frac{u\left(x+h \cdot \mathbf{e}_{\mathbf{l}}\right)-u(x)}{h}, \quad h \neq 0 .
$$
Lemma. $\quad$ Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$, and $u \in W^{1, p}(\Omega)$, for some $1 \leq p<\infty$. Then for any $\Omega^{\prime} \Subset \Omega$ such that $\operatorname{dist}\left(\Omega^{\prime}, \partial \Omega\right)>h$ holds
$$
\left\|\Delta_{i}^{h} u\right\|_{L^{p}\left(\Omega^{\prime}\right)} \leq\left\|D_{i} u\right\|_{L^{p}(\Omega)} .
$$
Proof.
$$
\begin{aligned}
\left|\Delta_{i}^{h} u\right|=\left|\frac{u\left(x+h \cdot \mathbf{e}_{\mathbf{l}}\right)-u(x)}{h}\right| & \leq \frac{1}{h} \int_{0}^{h}\left|\mathrm{D}_{i} u\left(x_{1}, \ldots, x_{i}+\zeta, \ldots, x_{n}\right)\right| d \zeta \\
& \leq \frac{1}{h}\left\{\int_{0}^{h} 1^{q}\right\}^{\frac{1 q}{q}}\left\{\int_{0}^{h}\left|\mathrm{D}_{i} u\left(x_{1}, \ldots, x_{i}+\zeta, \ldots, x_{n}\right)\right|^{p} d \zeta\right\}^{\frac{1}{p}} \Rightarrow \\
\left|\Delta_{i}^{h} u\right|^{p} & \leq h^{\frac{p}{q}-p} \cdot \int_{0}^{h}\left|\mathrm{D}_{i} u\left(x_{1}, \ldots, x_{i}+\zeta, \ldots, x_{n}\right)\right|^{p} d \zeta \\
& =\frac{1}{h} \cdot \int_{0}^{h}\left|\mathrm{D}_{i} u\left(x_{1}, \ldots, x_{i}+\zeta, \ldots, x_{n}\right)\right|^{p} d \zeta \quad \Rightarrow \\
\int_{\Omega^{\prime}}\left|\Delta_{i}^{h} u\right|^{p} & \leq \frac{1}{h} \cdot \int_{\Omega^{\prime}} \int_{0}^{h}\left|\mathrm{D}_{i}\right|^{p} d \zeta d \mathbf{x}=\frac{1}{h} \cdot \int_{0}^{h} \int_{\Omega}^{\prime}\left|\mathrm{D}_{i}\right|^{p} d \mathbf{x} d \zeta \\
& =\frac{1}{h} \int_{0}^{h}\left\|\mathrm{D}_{i} u\right\|_{L^{p}\left(\Omega^{\prime}\right)}=|| \mathrm{D}_{i} u\left\|_{L^{p}\left(\Omega^{\prime}\right)} \leq|| \mathrm{D}_{i} u\right\|_{L^{p}(\Omega),}
\end{aligned}
$$
where we applied Fubini's Theorem in order to switch order of integration.
Conversely we have
Lemma. $\quad$ Let $u \in L^{p}(\Omega)$ for some $1 \leq p<\infty$ and suppose $\Delta_{i}^{h} u \in L^{p}\left(\Omega^{\prime}\right)$ with $\left\|\Delta_{i}^{h} u\right\|_{L^{p}\left(\Omega^{\prime}\right)} \leq K$ for all $\Omega^{\prime} \Subset \Omega$ and $0<|h|<\operatorname{dist}\left(\Omega^{\prime}, \Omega\right)$. Then the weak derivative satisfies $\left\|D_{i} u\right\|_{L^{p}(\Omega)} \leq K$. Consequently if this holds for all $i=1, \ldots, n$ then $u \in W^{1, p}(\Omega)$.
Proof. We will make use of
Alouglou's Theorem. A bounded sequence in a separable, reflexive Banach space has a weakly convergent subsequence.
A topological space is called separable if it contains a countable dense set.
A Banach space is called reflexive if $\left(B^{\star}\right)^{\star}=B$.
A sequence $\left\{x_{n}\right\}$ in a Banach space is said to converge weakly to $x$ when $\lim _{n \rightarrow \infty} F\left(x_{n}\right) \rightarrow F(x)$ for all linear functionals $F \in B^{\star}$. This is sometimes denoted $\lim _{n \rightarrow \infty} x_{n} \rightarrow x$.
Example: Let $\ell^{2}:=\left\{\left(a_{1}, a_{2}, \ldots\right): \sum_{i=1}^{\infty} a_{i}^{2}<\infty\right\}$. Consider the sequence $\left\{x_{i}:=(0, \ldots, 0,1,0, \ldots)\right\}$ $\subseteq \ell^{2}$. Any bounded linear functional on $\ell^{2}$ will be some linear combination of the linear functionals $F_{j}$, defined by $F_{j}\left(a_{1}, \ldots\right)=a_{j}$ (each such linear combination corresponds exactly to a point in $\ell^{2}$. That makes sense, indeed by the Riesz Representation Theorem $\left(\ell^{2}\right)^{\star}=\ell^{2}$ (note $\ell^{2}$ is a Hilbert space not just a Banach space as it has an inner product structure).). For any such $F=\left(a_{1}, \ldots\right)$, $\lim _{i \rightarrow \infty} F\left(x_{i}\right)=\lim _{i \rightarrow \infty} a_{i}=0$. So $x_{i}$ converges to the 0 vector weakly, though certainly not strongly: by Fourier Theory each point in $\ell^{2}$ corresponds to a periodic function on $[0,1]$, i.e an element of $L^{2}\left(S^{1}\right)$, and of course $\lim _{n \rightarrow \infty} \exp (n 2 \pi \sqrt{-1} z) \not 0(z)$. We come back to the proof. For the Banach space $B=L^{p}(\Omega), B^{\star}=L^{q}(\Omega)$ with $\frac{1}{p}+\frac{1}{q}=1$. This can be seen directly: If $F \in\left(L^{p}(\Omega)\right)^{\star}$, then exists $f$ such that $F(g)=\int_{\Omega} g \cdot f, \quad \forall g \in L^{p}(\Omega)$, and this will be bounded iff $f \in L^{q}(\Omega)$. So we get an identification $F \in\left(L^{p}(\Omega)\right)^{\star} \cong L^{q}(\Omega)$.
By Alouglou's Theorem there exists a sequence $\left\{h_{m}\right\} \rightarrow 0$ with $\Delta_{i}^{h_{m}} u \rightarrow v \in L^{p}(\Omega)$. In other words
$$
\int_{\Omega} \psi \cdot \Delta_{i}^{h_{m}} u \rightarrow \int_{\Omega} \psi \cdot v \in L^{p}(\Omega), \quad \forall \psi \in L^{q}(\Omega)
$$
And in particular for any $\psi \in \mathcal{C}_{0}^{1}(\Omega)$ (which is dense in $L^{q}(\Omega)$ so will suffice to look at such $\psi$ as will become clear ahead)
$$
\begin{aligned}
\int_{\Omega} \psi \Delta_{i}^{h_{m}} u & =\int_{\Omega} \psi \frac{1}{h}\left(u\left(x+h \cdot \mathbf{e}_{\mathbf{l}}\right)-u(x)\right) d \mathbf{x} \\
& =\frac{1}{h} \int_{\Omega} \psi\left(x-h \mathbf{e}_{\mathbf{i}}\right) u(x) d \mathbf{x}-\frac{1}{h} \int_{\Omega} \psi(x) u(x) d \mathbf{x} \\
& =\int_{\Omega} \frac{1}{h}\left(\psi\left(x-h \mathbf{e}_{\mathbf{i}}\right)-\psi(x)\right) u(x) d \mathbf{x} \\
& =\int_{\Omega}-\Delta_{i}^{h} \psi(x) u(x) d \mathbf{x} \stackrel{h \rightarrow 0}{\longrightarrow} \int_{\Omega}-\mathrm{D}_{i} \psi(x) u(x) d \mathbf{x}
\end{aligned}
$$
since $\psi$ is continuously differetiable. Altogether
$$
\int_{\Omega} \psi \cdot v \in L^{p}(\Omega)=\int_{\Omega}-\mathrm{D}_{i} \psi(x) u(x) d \mathbf{x}
$$
which by definition means $v$ is the weak derivative of $u$ in the direction of the $x_{i}$ axis, or simply the undistinctive notation $v=\mathrm{D}_{i} u$.
We also get the desired estimate, using the Fatou Lemma $\int \lim \inf \leq \lim \inf \int$
$$
\int_{\Omega}\left|\mathrm{D}_{i} u\right|^{p} d \mathbf{x}=\int_{\Omega} \liminf \left|\Delta_{i}^{h} u\right|^{p} d \mathbf{x} \leq \liminf \int_{\Omega}\left|\Delta_{i}^{h} u\right|^{p} d \mathbf{x} \leq K^{p}
$$
i.e $\left\|\mathrm{D}_{i} u\right\|_{L^{p}(\Omega)} \leq K$.
## $L^{2}$ Theory
Consider the second order equation in divergence form
$$
L u \equiv L(u):=\mathrm{D}_{i}\left(a^{i j} \mathrm{D}_{j} u\right)+b^{i} \mathrm{D}_{i} u+c \cdot u=f,
$$
with $a^{i j}, b^{i}, c \in L^{1}(\Omega)$ (integrable coefficients).
We call $u \in W^{1,2}(\Omega)$ a weak solution of the equation if
$$
\forall v \in \mathcal{C}_{0}^{1}(\Omega) \quad-\int_{\Omega} a^{i j} \mathrm{D}_{j} u \mathrm{D}_{i} v+\int_{\Omega}\left(b^{i} \mathrm{D}_{i} u+c u\right) v=\int_{\Omega} f v .
$$
## Elliptic Regularity
Let $u \in W^{1,2}(\Omega)$ be a weak solution of $L u=f$ in $\Omega$, and assume
- $\quad$ L strictly elliptic with $\left(a^{i j}\right)>\gamma \cdot I, \quad \gamma>0$
- $\quad a^{i j} \in \mathcal{C}^{0,1}(\Omega)$
- $\quad b^{i}, c \in L^{\infty}(\Omega)$
- $f \in L^{2}(\Omega)$
Then for any $\Omega^{\prime} \Subset \Omega, \quad u \in W^{2,2}\left(\Omega^{\prime}\right)$ and
$$
\|u\|_{W^{2,2}\left(\Omega^{\prime}\right)} \leq C\left(\left\|a^{i j}\right\|_{C^{0,1}(\Omega)},\|b\|_{C^{0}(\Omega)},\|c\|_{C^{0}(\Omega)}, \lambda, \Omega^{\prime}, \Omega, n\right) \cdot\left(\|u\|_{W^{1,2}(\Omega)}+\|f\|_{L^{2}(\Omega)}\right) .
$$
Note $L^{\infty}(\Omega)$ stands for bounded functions on $\Omega$ while $\mathcal{C}^{0}(\Omega)$ are functions that are also continuous ( $\Omega$ being bounded).
Proof. Start with the definition of $u$ being a solution in the weak sense, $\forall v \in \mathcal{C}_{0}^{1}(\Omega)$ :
$$
\int_{\Omega} a^{i j} \mathrm{D}_{j} u \mathrm{D}_{i} v=\int_{\Omega}\left(b^{i} \mathrm{D}_{i} u+c-f\right) v
$$
and take difference quotients, that is replace $v$ with $\Delta^{-h} v$.
$$
\int_{\Omega} a^{i j} \mathrm{D}_{j} u \mathrm{D}_{i}\left(\Delta^{-h} v\right)=\int_{\Omega}\left(b^{i} \mathrm{D}_{i} u+c-f\right)\left(\Delta^{-h} v\right)
$$
Taking $-h$ is a technicality that will unravel its reason later on, and we really mean $\Delta_{k}^{-h} v$ for some $k \in\{1, \ldots, n\}$ and then eventually repeat the computation for all $k$ in that range. This will become clear as well. Finally our goal will be to use the Chain Rule and move the difference quotient operator onto $u$ under the integral sign and get uniform bounds on $\Delta^{h} \mathrm{D} u$ and in this way get a priori $W^{2,2}(\Omega)$ estimates.
The Chain Rule gives
$$
\begin{aligned}
& \Delta^{h}\left(a^{i j} \mathrm{D}_{j} u\right)= \\
& \frac{1}{h}\left(a^{i j} u\left(x+h \cdot \mathbf{e}_{\mathbf{k}}\right) \mathrm{D}_{j} u\left(x+h \cdot \mathbf{e}_{\mathbf{k}}\right)-\left\{a^{i j}(x)-a^{i j}\left(x+h \cdot \mathbf{e}_{\mathbf{k}}\right)+a^{i j}\left(x+h \cdot \mathbf{e}_{\mathbf{k}}\right)\right\} \mathrm{D}_{j} u(x)\right) \\
& =a^{i j} u\left(x+h \cdot \mathbf{e}_{\mathbf{k}}\right) \Delta^{h} \mathrm{D}_{j} u-\Delta^{h} a^{i j} \mathrm{D}_{j} u .
\end{aligned}
$$
And applied to our previous equation, a short calculation verifies that we can 'integrate by part' WRT $\Delta^{h}-$
$$
\begin{gathered}
\int_{\Omega} a^{i j} \mathrm{D}_{j} u \mathrm{D}_{i}\left(\Delta^{-h} v\right)=\int_{\Omega} \Delta^{h}\left(a^{i j} \mathrm{D}_{j} u\right) \mathrm{D}_{i} v \Rightarrow \\
\int_{\Omega} a^{i j} u\left(x+h \cdot \mathbf{e}_{\mathbf{k}}\right) \Delta^{h} \mathrm{D}_{j} u \mathrm{D}_{i} v=\int_{\Omega}-\Delta^{h} a^{i j} \mathrm{D}_{j} u \mathrm{D}_{i} v+\int_{\Omega}\left(b^{i} \mathrm{D}_{i} u+c-f\right)\left(\Delta^{-h} v\right) \Rightarrow \\
\left|\int_{\Omega} a^{i j} u\left(x+h \cdot \mathbf{e}_{\mathbf{k}}\right) \Delta^{h} \mathrm{D}_{j} u \mathrm{D}_{i} v\right| \leq\left\|\Delta^{h} a^{i j} \mathrm{D}_{j} u\right\|_{L^{2}(\Omega)}\left\|\mathrm{D}_{i} v\right\|_{L^{2}(\Omega)}+ \\
+\left\|b^{i} \mathrm{D}_{i} u+c u-f\right\|_{L^{2}(\Omega)}\left\|\Delta^{-h} v\right\|_{L^{2}(\Omega)},
\end{gathered}
$$
where we have used the Hölder Inequality for $p=q=2$. This in turn can be bounded by
$$
\begin{aligned}
& \leq\left\|a^{i j}\right\|_{C^{0,1}(\Omega)} \mid\|\mathrm{D} u\|_{L^{2}(\Omega)}\|\mathrm{D} v\|_{L^{2}(\Omega)}+ \\
& \quad+\left(\left\|b^{i}\right\|_{L^{\infty}(\Omega)}\|\mathrm{D} u\|_{L^{2}(\Omega)}+\|c\|_{L^{\infty}(\Omega)}\|u\|_{L^{2}(\Omega)}+\|f\|_{L^{2}(\Omega)}\right)\|\mathrm{D} v\|_{L^{2}(\Omega)} \\
& \leq C\left(\|u\|_{W^{1,2}(\Omega)}+\|f\|_{L^{2}(\Omega)}\right) \cdot\|\mathrm{D} v\|_{L^{2}(\Omega)} .
\end{aligned}
$$
where we have used the Hölder Inequality for $p=1, q=\infty$, i.e a simple bounded integration argument $\left(\right.$ e.g $\left.\|c u\|_{L^{2}(\Omega)}=\left(\int c^{2} \cdot|u|^{2}\right)^{\frac{1}{2}} \leq\left(\sup |c|^{2} \int_{O}|u|^{2}\right)^{\frac{1}{2}}\right)$, and $\Delta^{h} a^{i j} \rightarrow \mathrm{D}_{k} a^{i j}$ as $a^{i j} \mathcal{C}^{0,1}(\Omega)$.
Take a cut-off function $\eta \in \mathcal{C}_{0}^{1}(\Omega), 0 \leq|\eta| \leq 1,\left.\eta\right|_{\Omega^{\prime}}=1$. We now choose a special $v: v:=\eta^{2} \Delta^{h} u$. From uniform ellipticity $\left(a^{i j} \zeta_{i} \zeta_{j} \geq \lambda|\zeta|^{2}\right)$
$$
\lambda \int_{\Omega}\left|\eta \mathrm{D} \Delta^{h} u\right|^{2} \leq \int_{\Omega} \eta^{2} a^{i j}\left(x+h \cdot \mathbf{e}_{\mathbf{k}}\right) \mathrm{D}_{i} \Delta^{h} u \mathrm{D}_{j} \Delta^{h} u
$$
Now
$$
\mathrm{D}_{i} v=2 \eta \mathrm{D}_{i} \eta \Delta^{h} u+\eta^{2} \mathrm{D}_{i} \Delta^{h} u
$$
which we substitute into our previous inequality,
$$
\begin{aligned}
\int_{\Omega} \eta^{2} a^{i j}\left(x+h \cdot \mathbf{e}_{\mathbf{k}}\right) \mathrm{D}_{j} \Delta^{h} u \mathrm{D}_{j} \Delta^{h} u \leq & \int_{\Omega} a^{i j}\left(x+h \cdot \mathbf{e}_{\mathbf{k}}\right) \mathrm{D}_{j} \Delta^{h} u \cdot\left(\mathrm{D}_{i} v-2 \eta \mathrm{D}_{i} \eta \Delta^{h} u\right) \\
\leq & C\left(\|u\|_{W^{1,2}(\Omega)}+\|f\|_{L^{2}(\Omega)}\right)\|\mathrm{D} v\|_{L^{2}(\Omega)}+ \\
& +C^{\prime}\left\|\left(\mathrm{D} \Delta^{h} u\right) \eta\right\|_{L^{2}(\Omega)}\left\|\mathrm{D} \eta \Delta^{h} u\right\|_{L^{2}(\Omega)}
\end{aligned}
$$
again by the Hölder Inequality. Now since $\eta \leq 1$
$$
\left\|\mathrm{D}_{i} v\right\|_{L^{2}(\Omega)} \leq C^{\prime \prime}\left(\left\|\mathrm{D}_{i} \eta \Delta^{h} u\right\|_{L^{2}(\Omega)}+\left\|\mathrm{D} \Delta^{h} u\right\|_{L^{2}(\Omega)}\right) .
$$
Combining all the above and again using $\eta \leq 1$,
$$
\begin{aligned}
\lambda \int_{\Omega}^{\prime}\left|\eta \mathrm{D} \Delta^{h} u\right|^{2} \leq & C\left(\|u\|_{W^{1,2}(\Omega)}+\|f\|_{L^{2}(\Omega)}\right) \cdot C^{\prime \prime}\left(\left\|\mathrm{D} \eta \Delta^{h} u\right\|_{L^{2}\left(\Omega^{\prime}\right)}+\left\|\mathrm{D} \Delta^{h} u\right\|_{L^{2}\left(\Omega^{\prime}\right)}\right) \\
& +C^{\prime}\left\|\left(\mathrm{D} \Delta^{h} u\right)\right\|_{L^{2}\left(\Omega^{\prime}\right)}\left\|\mathrm{D} \eta \Delta^{h} u\right\|_{L^{2}\left(\Omega^{\prime}\right)} \\
\leq & c\left(\|u\|_{W^{1,2}(\Omega)}+\|f\|_{L^{2}(\Omega)}+\left\|\mathrm{D} \eta \Delta^{h} u\right\|_{L^{2}\left(\Omega^{\prime}\right)}\right) \cdot\left\|\left(\mathrm{D} \Delta^{h} u\right)\right\|_{L^{2}\left(\Omega^{\prime}\right)} \\
& +c\left(\|u\|_{W^{1,2}(\Omega)}+\|f\|_{L^{2}(\Omega)}\right) \cdot\left\|\mathrm{D} \eta \Delta^{h} u\right\|_{L^{2}\left(\Omega^{\prime}\right)} .
\end{aligned}
$$
Using the AM-GM Inequality $a b=\sqrt{\frac{1}{\epsilon} a^{2} \cdot \epsilon b^{2}} \leq \frac{1}{2}\left(\frac{1}{\epsilon} a^{2}+\epsilon b^{2}\right)$ for the first term and the inequality $(a+b) c \leq \frac{1}{2}(a+b+c)^{2}$ for the second
$$
\begin{aligned}
\lambda \int_{\Omega^{\prime}}\left|\eta \mathrm{D} \Delta^{h} u\right|^{2} \leq & \frac{1}{\epsilon} c^{2}\left(\|u\|_{W^{1,2}(\Omega)}+\|f\|_{L^{2}(\Omega)}+\left\|\mathrm{D} \eta \Delta^{h} u\right\|_{L^{2}\left(\Omega^{\prime}\right)}\right)^{2}+\epsilon\left\|\left(\mathrm{D} \Delta^{h} u\right)\right\|_{L^{2}\left(\Omega^{\prime}\right)}^{2} \\
& +c\left(\|u\|_{W^{1,2}(\Omega)}+\|f\|_{L^{2}(\Omega)}+\left\|\mathrm{D} \eta \Delta^{h} u\right\|_{L^{2}\left(\Omega^{\prime}\right)}\right)^{2}
\end{aligned}
$$
Choose any $0<\epsilon<\lambda / 2$. Then subtract the second term on the first line of the RHS from the LHS to get
$$
\begin{aligned}
\left\|\eta \mathrm{D} \Delta^{h} u\right\|_{L^{2}\left(\Omega^{\prime}\right)}^{2} & \leq c\left(\|u\|_{W^{1,2}(\Omega)}+\|f\|_{L^{2}(\Omega)}+\left\|\mathrm{D} \eta \Delta^{h} u\right\|_{L^{2}\left(\Omega^{\prime}\right)}\right)^{2} \Rightarrow \\
\left\|\eta \mathrm{D} \Delta^{h} u\right\|_{L^{2}\left(\Omega^{\prime}\right)} & \leq c\left(\|u\|_{W^{1,2}(\Omega)}+\|f\|_{L^{2}(\Omega)}+\left\|\mathrm{D} \eta \Delta^{h} u\right\|_{L^{2}\left(\Omega^{\prime}\right)}\right) \\
& \leq c\left(\|u\|_{W^{1,2}(\Omega)}+\|f\|_{L^{2}(\Omega)}+\sup _{\Omega}|\mathrm{D} \eta| \cdot\left\|\Delta^{h} u\right\|_{L^{2}\left(\Omega^{\prime}\right)}\right) \\
& \leq c\left(\|u\|_{W^{1,2}(\Omega)}+\|f\|_{L^{2}(\Omega)}\right) \cdot\left(1+\sup _{\Omega}|\mathrm{D} \eta|\right),
\end{aligned}
$$
since $\left\|\Delta^{h} u\right\|_{L^{2}(\Omega)} \leq\|\mathrm{D} u\|_{L^{2}(\Omega)} \leq\|u\|_{W^{1,2}(\Omega)} \leq\|u\|_{W^{1,2}(\Omega)}+\|f\|_{L^{2}(\Omega)}$ where we have applied the first Lemma to $u \in W^{1,2}(\Omega)$. Now we are done as we can choose $\eta$ such that first $\left.\eta\right|_{\Omega^{\prime}}=1$ (for the LHS !) and second $|\mathrm{D} \eta| \leq \operatorname{dist}\left(\Omega^{\prime}, \partial \Omega\right.$ ) (for the RHS ) and so
$$
\left\|1 \cdot \mathrm{D} \Delta^{h} u\right\|_{L^{2}\left(\Omega^{\prime}\right)}^{2} \leq c\left(\|u\|_{W^{1,2}(\Omega)}+\|f\|_{L^{2}(\Omega)}\right)
$$
independently of $h$. So by our second Lemma the uniform boundedness of the difference quotients of $\mathrm{D} u$ in $L^{2}\left(\Omega^{\prime}\right)$ implies $\mathrm{D} u \in W^{1,2}\left(\Omega^{\prime}\right) \quad \Rightarrow \quad u \in W^{2,2}\left(\Omega^{\prime}\right)$ and we have the desired estimate for its $W^{2,2}\left(\Omega^{\prime}\right)$ norm by the above inequality combined with the Lemma.
Now that $u \in W^{2,2}\left(\Omega^{\prime}\right)$ then the our original equation holds in the usual sense
$$
L u=a^{i j} \mathrm{D}_{i} j u+\mathrm{D}_{i} a^{i j} \mathrm{D}_{j} u+b^{i} \mathrm{D}_{i} u+c \cdot u=f,
$$
a.e!
## Lecture 21
May $4^{\text {th }}, 2004$
## Higher Elliptic Regularity
We may obtain using our previous result yet higher regularity. This is a key point in regularity theory, and is quite beautiful. The previous theorem guaranteed that under certain conditions on the coefficients any $W^{1,2}(\Omega)$ solution is in fact $W^{2,2}\left(\Omega^{\prime}\right)$ for any $\Omega^{\prime} \Subset \Omega$. Naturally we would expect that if we had assumed more regularity on the coefficients we would get that any $W^{2,2}(\Omega)$ solution is in fact $W^{3,2}\left(\Omega^{\prime}\right)$. If this could work then we could thus start from a $W^{1,2}$ solution, a weak solution, and get arbitrary regularity if we are willing to shrink more and more the domain. Alternatively we could get just enough regularity so that our embedding theorems ensure us the solution is Hölder , then $\mathcal{C}^{1, \alpha}, \mathcal{C}^{2, \alpha}$ and by repeating this process any desired regularity! (Here we use the Corollary of Lecture 18) This is really something. We started from a weak solution which need not be a function and using our theory we are able to show it behaves well and is in fact smooth! We will make this discussion precise in the sequel.
Let us see how this interpolation process works.
We assume as before $u \in W^{1,2}(\Omega)$ is a weak solution of $L u=f$ (though we know this implies more regularity in the interior, we won't use it now). Then
$$
\forall v \in \mathcal{C}_{0}^{1}(\Omega) \quad-\int_{\Omega} a^{i j} \mathrm{D}_{j} u \mathrm{D}_{i} v+\int_{\Omega}\left(b^{i} \mathrm{D}_{i} u+c u\right) v=\int_{\Omega} f v
$$
and the idea is now to look at a smaller space of test functions $v$ and see what that gives. In a sense the extra regularity we find will come for free. Take in particular $v \in \mathcal{C}_{0}^{2}(\Omega)$. That means $v=\mathrm{D}_{k} w$ where $w \in \mathcal{C}_{0}^{1}(\Omega)$, and
$$
\int_{\Omega} a^{i j} \mathrm{D}_{j} u \mathrm{D}_{i} \mathrm{D}_{k} w=\int_{\Omega}\left(-b^{i} \mathrm{D}_{i} u-c+f\right) \mathrm{D}_{k} w .
$$
Since $w$ is twice continuously differentiable we may interchange derivatives above and integration by parts yields
$$
-\int_{\Omega} \mathrm{D}_{k}\left(a^{i j} \mathrm{D}_{j} u\right) \mathrm{D}_{i} w=\int_{\Omega}\left(-b^{i} \mathrm{D}_{i} u-c+f\right) \mathrm{D}_{k} w
$$
and
$$
-\int_{\Omega} a^{i j} \mathrm{D}_{k} \mathrm{D}_{j} u \mathrm{D}_{i} w=\int_{\Omega} \mathrm{D}_{k} a^{i j} \mathrm{D}_{j} u \mathrm{D}_{i} w+\int_{\Omega}\left(-b^{i} \mathrm{D}_{i} u-c+f\right) \mathrm{D}_{k} w
$$
and further
$$
\int_{\Omega} a^{i j} \mathrm{D}_{j}\left(\mathrm{D}_{k} u\right) \mathrm{D}_{i} w=\int_{\Omega}\left[-\mathrm{D}_{i}\left(\mathrm{D}_{k} a^{i j} \mathrm{D}_{j} u\right)+\mathrm{D}_{k}\left(b^{i} \mathrm{D}_{i} u-c+f\right)\right] \cdot w=: \int_{\Omega} g \cdot w
$$
which gives that $\mathrm{D}_{k} u$ is a weak solution of the second order equation
$$
\tilde{L}\left(\mathrm{D}_{k} u\right)=g
$$
since this holds $\forall w \in \mathcal{C}_{0}^{1}(\Omega)$.
Now we note that $\tilde{L}=a^{i j}$ is strictly elliptic, and that if
- $\quad a^{i j} \in \mathcal{C}^{1,1}(\Omega)$
- $\quad b^{i}, c \in \mathcal{C}^{0,1}(\Omega)$
- $u \in W^{2,2}(\Omega)$
then we will have $g \in L^{2}(\Omega)$, and the Theorem of the previous lecture applies and we have $\mathrm{D}_{k} u \in$ $W^{2,2}\left(\Omega^{\prime}\right) \quad \forall \Omega^{\prime} \Subset \Omega$, i.e $u \in W^{3,2}\left(\Omega^{\prime}\right)$ as we wished to show. Indeed we get this extra regularity seemingly for free and we may continue this for higher derivatives.
Let us see what kind of a priori estimates we get on the higher norms. From the Theorem we have
$$
\begin{aligned}
\left\|\mathrm{D}_{k} u\right\|_{W^{2,2}\left(\Omega^{\prime}\right)} & \leq c\left(\left\|\mathrm{D}_{k} u\right\|_{W^{1,2}(\Omega)}+\|g\|_{L^{2}(\Omega)}\right) \\
& \leq c\left(\|u\|_{W^{2,2}(\Omega)}+\|u\|_{W^{1,2}(\Omega)}+\|u\|_{W^{2,2}(\Omega)}+\|f\|_{L^{2}(\Omega)}\right)
\end{aligned}
$$
where the last three terms come from the definition of $g$. We now shrink from $\Omega$ to $\Omega^{\prime}$ and from $\Omega^{\prime}$ to $\Omega^{\prime \prime}$ so that terms on the LHS are evaluated on $\Omega^{\prime \prime}$ and the ones on the RHS on $\Omega^{\prime}$. But then those terms on the RHS can be evaluated by $\Omega$ terms using our theorem once again! We get altogether then
$$
\left\|\mathrm{D}_{k} u\right\|_{W^{2,2}\left(\Omega^{\prime \prime}\right)} \leq c\left(\|u\|_{W^{1,2}(\Omega)}+\|f\|_{L^{2}(\Omega)}\right)
$$
We state this as the following
Theorem. Let $u \in W_{0}^{1,2}(\Omega)$ be a weak solution of $L u=f$ in $\Omega$, and assume
- $\quad$ s strictly elliptic with $\left(a^{i j}\right)>\gamma \cdot I, \quad \gamma>0$
- $\quad a^{i j} \in \mathcal{C}^{k, 1}(\bar{\Omega})$
- $\quad b^{i}, c \in \mathcal{C}^{k-1,1}(\bar{\Omega})$
- $f \in W^{k, 2}(\Omega)$.
Then for any precompact set $\Omega^{\prime} \Subset \Omega, \quad u \in W^{k+2,2}\left(\Omega^{\prime}\right)$ and
$\|u\|_{W^{2,2}\left(\Omega^{\prime}\right)} \leq C\left(\left\|a^{i j}\right\|_{C^{k, 1}(\Omega)},\|b\|_{C^{k-1,1}(\Omega)},\|c\|_{C^{k-1,1}(\Omega)}, \lambda, \Omega^{\prime}, \Omega, k, n\right) \cdot\left(\|u\|_{W^{1,2}(\Omega)}+\|f\|_{W^{k, 2}(\Omega)}\right)$.
What we just did is the analogue in regularity theory of the technique we used in the Hölder part of the course. As there, we want to differentiate the original equation but the lack of regularity hinders us from doing do directly. We then take difference quotients and get bounds which now allow us to differentiate and get all higher estimates. This shows us how special solutions of such partial differetial equations are among general functions in those Sobolev Spaces.
Corollary. Let $u \in W_{0}^{1,2}(\Omega)$ be a weak solution of $L u=f$ in $\Omega$, and assume
- $\quad L$ strictly elliptic with $\left(a^{i j}\right)>\gamma \cdot I, \quad \gamma>0$
- $\quad f, a^{i j}, b^{i}, c \in \mathcal{C}^{\infty}(\bar{\Omega})$
Then on the whole domain, $u \in \mathcal{C}^{\infty}(\Omega)$. Proof. For all $k \in \mathbb{N}, f \in W^{k, 2}(\Omega) \quad \Rightarrow \quad \Omega^{\prime} \Subset \Omega u \in W^{k+2,2}\left(\Omega^{\prime}\right)$. By the Sobolev Embedding (Corollary to Lecture 18) then $u \in \mathcal{C}^{m}\left(\Omega^{\prime}\right), m<k+2-\frac{n}{2}$, hence $u \in \mathcal{C}^{\infty}\left(\Omega^{\prime}\right)$. Apply this reasoning for $\Omega^{\prime}$ a ball around each point $p \in \Omega$ ! to get $u \in \mathcal{C}^{\infty}(\Omega)$. Smoothness is indeed a notion defined pointwise.
## Global Regularity (upto the boundary)
$\mathcal{U} \mathrm{p}$ until now our regularity results were for the space $W_{0}^{1,2}(\Omega)$, i.e functions which vanish on $\partial \Omega$. We now study $W^{1,2}(\Omega)$.
Theorem. $\quad$ Suppose $u \in W^{1,2}(\Omega)$ is a (weak) solution of $L u=f$ and assume
- $\quad L$ strictly elliptic with $\left(a^{i j}\right)>\gamma \cdot I, \quad \gamma>0$
- $\quad a^{i j} \in \mathcal{C}^{0,1}(\bar{\Omega})$
- $\quad b^{i}, c \in L^{\infty}(\bar{\Omega})$
- $f \in L^{2}(\Omega)$
- $\Omega$ has $\mathcal{C}^{2}$ boundary.
- $\quad \exists \varphi \in W^{2,2}(\Omega)$ such that $u-\varphi \in W_{0}^{1,2}(\Omega)$.
Then $u \in W^{2,2}(\Omega)=W^{2,2}(\bar{\Omega})$ on all of $\Omega$ with
$$
\begin{array}{r}
\|u\|_{W^{2,2}\left(\Omega^{\prime}\right)} \leq C\left(\left\|a^{i j}\right\|_{C^{0,1}(\Omega)},\|b\|_{C^{0}(\Omega)},\|c\|_{C^{0}(\Omega)}, \lambda, \Omega^{\prime}, \Omega, \partial \Omega, n\right) . \\
\cdot\left(\|u\|_{W^{1,2}(\Omega)}+\|f\|_{L^{2}(\Omega)}+\|\varphi\|_{W^{2,2}(\Omega)}\right) .
\end{array}
$$
Note that even for the 0 boundary values case this theorem gives a stronger conclusion: regularity upto the boundary with a uniform estimate on all of $\Omega$. The price is the assumption that the boundary is regular enough. Proof. We reduce to the zero boundary case in the usual manner. Suppose we could prove the Theorem for all zero boundary Dirichlet Problems. Then for the problem $L(u-\varphi)=f^{\prime}:=f-L \varphi$ on $\Omega, u-\varphi=0$ on $\partial \Omega$ (this is precisely the assumption $u-\varphi \in W_{0}^{1,2}(\Omega)$ ) we would have the desired estimates
$$
\begin{array}{r}
\|u-\varphi\|_{W^{2,2}\left(\Omega^{\prime}\right)} \leq C \cdot\left(\|u-\varphi\|_{W^{1,2}(\Omega)}+\|f-L \varphi\|_{L^{2}(\Omega)}\right) \quad \Rightarrow \\
\|u\|_{W^{2,2}\left(\Omega^{\prime}\right)} \leq C \cdot\left(\|u\|_{W^{1,2}(\Omega)}+\|f\|_{L^{2}(\Omega)}+\|\varphi\|_{W^{2,2}(\Omega)}\right),
\end{array}
$$
since $\varphi \in W^{2,2}(\Omega)$ and $L$ is of second order.
So we assume indeed $u \in W_{0}^{1,2}(\Omega)$. We now take a neighborhood containing a boundary portion and map it through a $\mathcal{C}^{2}$ diffeomorphism $\psi$ (i.e $\psi^{-1}$ exists and is $\mathcal{C}^{2}$ ) onto $\mathbb{R}^{n}$ with the boundary portion mapping into the hyperplane $\left\{x_{n}=0\right\}$.
We pull back everything onto the flat boundary situation using $\left(\psi^{-1}\right)^{\star}$ - the original equation is
$$
\forall v \in \mathcal{C}_{0}^{1}(\Omega) \quad-\int_{\Omega} a^{i j} \mathrm{D}_{j} u \mathrm{D}_{i} v+\int_{\Omega}\left(b^{i} \mathrm{D}_{i} u+c u\right) v=\int_{\Omega} f v
$$
and the pulled-backed one
$$
\begin{aligned}
\forall v \circ \psi^{-1} \in \mathcal{C}_{0}^{1}(\psi(\Omega)) \quad & -\int_{\psi(\Omega)} \operatorname{Jac}\left(\psi^{-1}\right) a^{i j} \circ \psi^{-1}\left(\psi^{-1}\right)^{\star} \mathrm{D}_{j} u\left(\psi^{-1}\right)^{\star} \mathrm{D}_{i} v \\
& +\int_{\psi(\Omega)} \operatorname{Jac}\left(\psi^{-1}\right)\left(b^{i} \circ \psi^{-1}\left(\psi^{-1}\right)^{\star} \mathrm{D}_{i} u+c \circ \psi^{-1}\right) v \circ \psi^{-1}=\int_{\Omega} f v
\end{aligned}
$$
As we can assume $\psi^{-1}$ preserves the given orientation of $\mathbb{R}^{n}$ and it is a diffeomorphism then $\operatorname{Jac}\left(\psi^{-1}\right)>0$ and therefore we still have a strictly elliptic equation $\tilde{L} \tilde{u}=\tilde{f}$ and the $\mathcal{C}^{2}$ of $\psi^{-1}$ guarantees the $\tilde{f} \in L^{2}(\Omega)$ and that $\tilde{a}^{i j}$ is still Lipschitz, and $b^{i}$ and $c$ still bounded (e.g. $\left(\psi^{-1}\right)^{\star}\left(b^{i} \mathrm{D}_{i} u\right)=$ $b_{i} \circ \psi^{-1} \cdot\left(\psi^{-1}\right)^{\star} \mathrm{D}_{i} u=b_{i} \circ \psi^{-1} \cdot \mathrm{D}_{i}\left(\psi^{-1}\right)^{\star} u=b_{i} \circ \psi^{-1} \cdot \mathrm{D}_{i}\left(u \circ \psi^{-1}\right)=b_{i} \circ \psi^{-1} \cdot \mathrm{D}_{k} u \cdot \mathrm{D}_{i}\left(\psi^{-1}\right)_{k}$ (summation over $k)$ ).
Now we note that the difference quotients proof from last time still works for $\Delta_{l}^{h} \tilde{u}$ for each of the directions $l=1, \ldots, n-1$ tangent to the boundary. So by applying that Theorem we get $\mathrm{D}_{l} \tilde{u} \in W^{1,2}\left(\psi\left(\Omega^{\prime}\right)\right)$ and hence $\mathrm{D}_{i j} \tilde{u} \in L^{2}\left(\psi\left(\Omega^{\prime}\right)\right)$ except for $i=j=n$. Since $\psi$ is a $\mathcal{C}^{2}$ diffeomorphism, the same holds for $u$.
So in order to finish the proof we go back to the proof. We have $W^{2,2}$ except possibly in the boundary, so may write the equation
$$
L u=a^{i j} \mathrm{D}_{i} j u+\mathrm{D}_{i} a^{i j} \mathrm{D}_{j} u+b^{i} \mathrm{D}_{i} u+c \cdot u=f,
$$
a.e. All terms are in $L^{2}$ except $a^{n n} \mathrm{D}_{n n} u$. But then isolating it on one side of the equation we see it must be as well, so it can not blow up at the boundary.
So now indeed we are done: we cover all of $\bar{\Omega}$ with a finite number of small ball cover the boundary portion and another $\Omega^{\prime}$ covering the rest of the interior and we have the desired estimate on each of those domains.
We now have higher regularity upto the boundary:
Corollary. $\quad$ Let $u \in W^{1,2}(\Omega)$ be a weak solution of $L u=f$ in $\Omega$, and $u=\varphi$ on $\partial \Omega$ (i.e $\left.u-\varphi \in W_{0}^{1,2}(\Omega)\right)$ and assume
- $\quad$ L strictly elliptic with $\left(a^{i j}\right)>\gamma \cdot I, \quad \gamma>0$
- $\quad a^{i j} \in \mathcal{C}^{k, 1}(\bar{\Omega})$
- $\quad b^{i}, c \in \mathcal{C}^{k-1,1}(\bar{\Omega})$
- $\quad f \in W^{k, 2}(\Omega)$.
- $\partial \Omega$ is $\mathcal{C}^{k+2}$.
Then $u \in W^{k+2,2}(\Omega)$ uniformly on the whole domain and
$\|u\|_{W^{k+2,2}(\Omega)}$
$\leq C\left(\left\|a^{i j}\right\|_{C^{k, 1}(\Omega)},\|b\|_{C^{k-1,1}(\Omega)},\|c\|_{C^{k-1,1}(\Omega)}, \lambda, \partial \Omega, \Omega^{\prime}, \Omega, k, n\right) \cdot\left(\|u\|_{W^{1,2}(\Omega)}+\|f\|_{W^{k, 2}(\Omega)}+\|\varphi\|_{W^{k+2,2}(\Omega)}\right)$.
If $k=\infty$ then $u \in \mathcal{C}^{\infty}(\bar{\Omega})$. The only difference from the compactly supported case is that we need now to have at our disposal a modified Sobolev Embedding: $W^{k+2,2} \subseteq \mathcal{C}^{m}(\bar{\Omega})$ instead of the one we proved with $W_{0}^{k+2,2} \subseteq \mathcal{C}^{m}(\bar{\Omega})$. This is indeed the case as one can show by modifying the latter's proof under the assumption of smooth enough boundary.
## Improvement of our estimate
Assume $u \in W_{0}^{1,2}(\Omega), \quad L u=f \in L^{2}(\Omega), \quad a^{i j}, b^{i}, c \in L^{\infty}(\Omega)$. Then
$$
\|u\|_{W^{2,2}(\Omega)} \leq c \cdot\left(\|u\|_{L^{2}(\Omega)}+\|f\|_{L^{2}(\Omega)}\right)
$$
Proof. During the proof which involved the $W^{1,2}(\Omega)$ norm on the RHS we arrived at the inequality
$$
\lambda\|\mathrm{D} u\|_{L^{2}(\Omega)} \leq \int_{\Omega} a^{i j} \mathrm{D}_{i} u \mathrm{D}_{j} \varphi=\int_{\Omega}\left(-b^{i} \mathrm{D}_{i} u-c u+f\right) \varphi
$$
for all test functions $\varphi \in \mathcal{C}_{0}^{1}(\Omega)$ but in fact also for all test functions in the completion $W_{0}^{1,2}(\Omega)$ ! In particular we can take $\varphi=u$ ! But actually for $\varphi=u$ we can get this directly just from the strict ellipticity without having to go through the difference quotients process (just true for this special choice of $v$ !). In particular we needn't assume more than $L^{\infty}(\Omega)$ regularity on the $a^{i j}$ now! We continue then and get
$$
\begin{aligned}
\lambda\|\mathrm{D} u\|_{L^{2}(\Omega)} \leq \int_{\Omega} a^{i j} \mathrm{D}_{i} u \mathrm{D}_{j} u & =\int_{\Omega}\left(-b^{i} \mathrm{D}_{i} u-c u+f\right) u \\
& =-\int_{\Omega}\left(-b^{i} u\left(\mathrm{D}_{i} u\right)+\int_{\Omega}\left(-c u^{2}+f u\right)\right. \\
& \leq \frac{1}{2} \epsilon \int_{\Omega}|\mathrm{D} u|^{2}+\sum_{i=1}^{n} \frac{1}{2} \frac{1}{\epsilon} \int_{\Omega}\left|b^{i} u\right|^{2}+\int_{\Omega}\left(-c u^{2}+f u\right) .
\end{aligned}
$$
For $\epsilon<\lambda$ one can move the first term to the LHS to conclude that (thanks to strict ellipticity we can now divide out by $\lambda$ and get a uniform bound!)
$$
\|\mathrm{D} u\|_{L^{2}(\Omega)} \leq \frac{1}{2}\|u\|_{L^{2}(\Omega)}+\|f\|_{L^{2}(\Omega)}
$$
Now we can plug this in to our original estimate to get the desired improvement
$$
\begin{aligned}
\|u\|_{W^{2,2}(\Omega)} & \leq c \cdot\left(\|u\|_{W^{2,1}(\Omega)}+\|f\|_{L^{2}(\Omega)}\right) \\
& =c \cdot\left(\|u\|_{L^{2}(\Omega)}+\|\mathrm{D} u\|_{L^{2}(\Omega)}+\|f\|_{L^{2}(\Omega)}\right) \\
& \leq c^{\prime} \cdot\left(\|u\|_{L^{2}(\Omega)}+\|f\|_{L^{2}(\Omega)}\right)
\end{aligned}
$$
Similarly this improvement applies to $u \in W^{1,2}(\Omega)$ though it will not apply up to the boundary; we will have
$$
\|u\|_{W^{1,2}\left(\Omega^{\prime}\right)} \leq c\left(\|u\|_{L^{2}(\Omega)}+\|f\|_{L^{2}(\Omega)}\right)
$$
by taking $\varphi=\eta \cdot u$ with $\eta=1$ on $\Omega^{\prime}$ and applying the above argument. If we want an estimate on all of $\Omega$ we need to add a term $\|\varphi\|_{W^{2,2}(\Omega)}$ to the RHS by applying the above result to $u-\varphi \in W_{0}^{1,2}(\Omega)$ for $f^{\prime}:=f-L \varphi$.
## Lecture 22
May $6^{\text {th }}, 2004$
Define $u^{+}:=\max \{u, 0\}, \quad u^{-}:=\min \{u, 0\}$. For a generalized function $u \in W^{1,2}(\Omega)$ we say $u \leq 0$ on $\partial \Omega$ if $u^{+} \in W_{0}^{1,2}(\Omega)$. Similarly we say $u \leq v$ on $\partial \Omega$ if $u-v \leq 0$ on $\partial \Omega$. Finally define $\sup _{\partial \Omega} u:=\inf \{c: u \leq c$ on $\partial \Omega\}$
## Weak $\mathrm{L}^{2}$ Maximum Principle
$\mathcal{W}$ e consider the divergence form equation
$$
L u:=\mathrm{D}_{i}\left(a^{i j} \mathrm{D}_{j} u\right)+b^{i} \mathrm{D}_{i} u+c u=f,
$$
with $c \leq 0$.
Theorem. $\quad$ Suppose $u \in W^{1,2}(\Omega)$. Assume
$$
\begin{aligned}
& \text { - } c \leq 0 \\
& \text { - } \quad L \text { strictly elliptic with }\left(a^{i j}\right)>\gamma \cdot I, \quad \gamma>0 \\
& \text { - } \quad\left\|b^{i}\right\|_{C^{0}(\Omega)} \leq \Lambda \\
& \text { - } f \in W^{k, 2}(\Omega) \\
& \text { Then }\left\{\begin{array}{l}
\text { If } L u \geq 0 \text { then } \sup _{\Omega} u \leq \sup _{\partial \Omega} u^{+} . \\
\text {If } L u \leq 0 \text { then } \inf _{\Omega} u \geq \inf _{\partial \Omega} u^{-} . \\
\text {If } c=0 \text { then the above holds with }|u| \text { instead of } u .
\end{array}\right.
\end{aligned}
$$
The last conclusion follows from the first two since in that case $u$ and $-u$ each satisfy one inequality. Proof. From the statement we have that $u$ satisfies an inequality in the weak sense, the integral inequality
$$
\begin{array}{rlrl}
\forall v \in W_{0}^{1,2}(\Omega) & & -\int_{\Omega} a^{i j} \mathrm{D}_{j} u \mathrm{D}_{i} v+\int_{\Omega}\left(b^{i} \mathrm{D}_{i} u+c u\right) v \geq 0 \\
\text { or } & \int_{\Omega} a^{i j} \mathrm{D}_{j} u \mathrm{D}_{i} v \leq \int_{\Omega} b^{i} \mathrm{D}_{i} u v+\int_{\Omega} c u v .
\end{array}
$$
Now restrict to $v$ such that $u \cdot v \geq 0$. Since $c \leq 0$
$$
\int_{\Omega} a^{i j} \mathrm{D}_{j} u \mathrm{D}_{i} v \leq \int_{\Omega} b^{i} \mathrm{D}_{i} u v \leq \Lambda \int_{\Omega} v|\mathrm{D} u|
$$
If $\sup _{\Omega} u>\sup _{\partial \Omega} u^{+}$then choose $k \in \mathbb{R}$ such that $\sup _{\partial \Omega} u^{+} \leq k<\sup _{\Omega} u$. Now pick a specific $v$, $v:=(u-k)^{+}$. This $v$ is 0 everywhere except where $u$ exceed $k$, and in particular where it exceeds the supremum of the boundary values. Indeed we have $v \in W_{0}^{1,2}(\Omega)$ as well as
$$
\mathrm{D} v=\left\{\begin{array}{lll}
\mathrm{D} u & \text { for } u>k & (\text { there } v>0) \\
0 & \text { for } u \leq k & (\text { there } v=0)
\end{array} .\right.
$$
And so
$$
\int_{\Omega} a^{i j} \mathrm{D}_{j} v \mathrm{D}_{i} v \leq \Lambda \int_{\Gamma} v|\mathrm{D} v|
$$
where $\Gamma:=\operatorname{suppD} v \subseteq \operatorname{supp} v$. Now by strict ellipticity the LHS majorizes $\lambda \int_{\Omega}|\mathrm{D} v|^{2}$ hence
$$
\lambda\|\mathrm{D} v\|_{L^{2}(\Omega)}^{2}=\lambda \int_{\Omega}|\mathrm{D} v|^{2} \leq \Lambda \int_{\Gamma} v|\mathrm{D} v| \leq \Lambda\|v\|_{L^{2}(\Gamma)} \mid\|\mathrm{D} v\|_{L^{2}(\Omega)}
$$
by the Hölder Inequality (HI) (for $p=q=2$ ) and therefore
$$
\begin{aligned}
\|\mathrm{D} v\|_{L^{2}(\Omega)} \leq c(\lambda, \Lambda) \cdot\|v\|_{L^{2}(\Gamma)}=c \cdot\left(\int_{\Gamma} v^{2}\right)^{\frac{1}{2}} & \leq c \cdot\left(\left\{\int_{\Gamma}\left(v^{2}\right)^{\frac{n}{n-2}}\right\}^{\frac{n-2}{n}}\left\{\int_{\Gamma} 1^{\frac{n}{2}}\right\}^{\frac{2}{n}}\right)^{\frac{1}{2}} \\
& =c \cdot \operatorname{Vol}(\Gamma)^{\frac{1}{n}}\|v\|_{L^{\frac{2 n}{n-2}}(\Gamma)}
\end{aligned}
$$
once again by the HI for $p=\frac{n}{n-2}, q=\frac{n}{2}$. On the other hand by the Sobolev Embedding $\|v\|_{L^{\frac{2 n}{n-2}}(\Omega)} \leq C|| \mathrm{D} v \|_{L^{2}(\Omega)}$ and so over all
$$
\|v\|_{L^{\frac{2 n}{n-2}}(\Omega)} \leq C\|\mathrm{D} v\|_{L^{2}(\Omega)} \leq C\|v\|_{L^{2}(\Omega)} c \cdot \operatorname{Vol}(\Gamma)^{\frac{1}{n}}\|v\|_{L^{\frac{2 n}{n-2}}(\Omega)}
$$
and therefore $\operatorname{Vol}(\Gamma)^{\frac{1}{n}} \geq \tilde{C}$ where the constant is independent of $k !$ (note $v \in L^{2}(\Omega)$ ). Let therefore $k \rightarrow \sup _{\Omega} u$. Then we see $u$ must still attain its maximum on a set of positive measure! But then $\mathrm{D} v=\mathrm{D} u=0$ there! Which in turn contradicts this previous bound on the volume of $\Gamma=\operatorname{supp}(\mathrm{D} v)$. So we conclude that there exists no $k \in\left[\sup _{\partial \Omega} u^{+}, \sup _{\Omega} u\right)$, in other words $\sup _{\partial \Omega} u^{+} \geq \sup _{\Omega} u$. The second case of the Theorem follows now since if $L u \leq 0$ then $L(-u) \geq 0$ and the first case applies.
Corollary. $\quad$ Let $L$ be strictly elliptic with $c \leq 0$. Assume $u \in W_{0}^{1,2}(\Omega)$ satisfies Lu $=0$ on $\Omega$. Then $u=0$ on $\Omega$.
## An a priori Estimate
We improve slightly on the aesthetics of the higher regularity proved in the previous lecture for the case $c \leq 0$.
Theorem. Let $u \in W_{0}^{1,2}(\Omega) \cap W^{k+2,2}(\Omega)$ be a weak solution of $L u=f$ in $\Omega$, and assume
- $\quad L$ strictly elliptic with $\left(a^{i j}\right)>\gamma \cdot I, \quad \gamma>0$
- $\quad a^{i j} \in \mathcal{C}^{k, 1}(\bar{\Omega})$
- $\quad b^{i}, c \in \mathcal{C}^{k-1,1}(\bar{\Omega}) \quad\left(\right.$ for $\left.k=0, \mathcal{C}^{-1,1}:=\mathcal{C}^{0}=L^{\infty}\right)$
- $f \in W^{k, 2}(\Omega)$
- $\partial \Omega$ is $\mathcal{C}^{k+2}$ Then
$$
\|u\|_{W^{k+2,2}(\Omega)} \leq c \cdot\|L u\|_{W^{k, 2}(\Omega)} .
$$
Note that the assumption $u \in W^{k+2,2}(\Omega)$ is superfluous once $u \in W_{0}^{1,2}(\Omega)$ in light of our previous results.
Also note that this is exactly analogous to what we did in our Hölder theory study; there we proved $L u=f \in \mathcal{C}^{k, \alpha}(\Omega), c \leq 0$ implies $\|u\|_{C^{k+2, \alpha}(\Omega)} \leq c\|f\|_{C^{k, \alpha}(\Omega)}$.
Proof. Case $k=0$. We want to prove $\|u\|_{W^{2,2}(\Omega)} \leq c \cdot\|L u\|_{W^{2,2}(\Omega)}$. and we already know that
$$
\|u\|_{W^{2,2}(\Omega)} \leq c \cdot\left(\|u\|_{L^{2}(\Omega)}+\|L u\|_{W^{2,2}(\Omega)}\right)
$$
so we now try to demonstrate $\|u\|_{L^{2}(\Omega)} \leq c\|L u\|_{W^{2,2}(\Omega)}$ for all $u \in W^{2,2}(\Omega) \cap W_{0}^{1,2}(\Omega)$. If not, pick a sequence $\left\{u_{m}\right\} \subseteq W^{2,2}(\Omega) \cap W_{0}^{1,2}(\Omega)$ with $\left\|u_{m}\right\|_{L^{2}(\Omega)}=1, \quad\left\|L u_{m}\right\|_{W^{2,2}(\Omega)} \stackrel{m \rightarrow \infty}{\longrightarrow} 0$ and hence by what we know
$$
\left\|u_{m}\right\|_{W^{2,2}(\Omega)} \leq c
$$
Since $W^{2,2}(\Omega)$ is a Hilbert space exists a subsequence which converges weakly to $u \in W^{2,2}(\Omega)$ (note Alouglou's Theorem applies as we have separability and every Hilbert space is a reflexive Banach space). Since $W^{2,2}(\Omega) \hookrightarrow L^{2}(\Omega)$ is a compact embedding we actually have $u_{m} \rightarrow u \in L^{2}(\Omega)$ (i.e strongly). But now $\left\|L u_{m}\right\|_{L^{2}(\Omega)} \rightarrow 0$, hence $L u=0$ weakly. Since $c \leq 0$ this implies by our previous work $u=0$ ! In contradiction with $\left\|u_{m}\right\|_{L^{2}(\Omega)}=1$ as $u_{m} \rightarrow u$ in $L^{2}(\Omega)$ so $\|u\|_{L^{2}(\Omega)}=1$ allora ...
Corollary. Let $\Omega \subseteq \mathbb{R}^{n}$ be a bounded domain with $\mathcal{C}^{k+2}$ boundary. Then the map
$$
\Delta: W^{k+2,2}(\Omega) \cap W_{0}^{1,2}(\Omega) \longrightarrow W^{k, 2}(\Omega)
$$
is an isomorphism. Proof. Injective: By the previous Corollary if $L\left(u_{1}-u_{2}\right)=0$ on $\Omega$ and $u_{1}-u_{2} \in W_{0}^{1,2}(\Omega)$ then $u_{1}-u_{2}=0$. This actually applies also to any two such functions in $W^{1,2}(\Omega)$ with equal boundary values.
Surjective: Let $f \in W^{k, 2}(\Omega)$. We can find a solution $L u=f$ with $u$ in $W_{0}^{2,2}(\Omega)$ by Riesz Representation Theorem and our regularity theory. So $\Delta^{-1}$ exists and by our above Theorem satisfies
$$
\left\|\Delta^{-1} f\right\|_{W^{k+2,2}(\Omega)} \leq C \cdot\|f\|_{W^{k, 2}(\Omega)}
$$
So $\Delta^{-1}$ is continuous. From the definition of $\Delta$ we see that
$$
\|\Delta u\|_{W^{k, 2}(\Omega)} \leq\|u\|_{W^{k+2,2}(\Omega)}
$$
(note no constant on RHS ) we see also $\Delta$ itself is a continuous map between those spaces (WRT to their topologies).
Corollary. For appropriate $L$ (see above Theorems) with $c \leq 0$
$$
L: W^{k+2,2}(\Omega) \cap W_{0}^{1,2}(\Omega) \longrightarrow W^{k, 2}(\Omega)
$$
is an isomorphism.
Proof. Injective: Exactly as above.
Surjective: We employ the Continuity Method (CM) which will work out exactly as in the Schauder case. Consider the family of equations
$$
L_{t} u:=(1-t) \mathrm{D} u+t L u=f .
$$
Recall that the CM will provide for the surjectivity of $L$ based on the surjectivity of $\Delta$ (proved above) once we can prove
$$
\|u\|_{W^{k+2,2}(\Omega)} \leq c \cdot\left\|L_{t} u\right\|_{W^{k, 2}(\Omega)}
$$
with $c$ independent of $t$. And this is indeed the case since each of the $L_{t}$ satisfies the assumptions of the previous Theorem.
## Negative Sobolev Spaces
What happens for the $k=-1$ case? Where does $\Delta$ map to? $\Delta u$ is not defined as a function, though it is as a distribution: given $v \in W_{0}^{1,2}(\Omega)$ one can define
$$
\Delta u(v):=-\int_{\Omega} \nabla u \cdot \nabla v
$$
which realizes $\Delta u$ as a linear functional on $W_{0}^{1,2}(\Omega)$, in other words
$$
\Delta: W_{0}^{1,2}(\Omega) \longrightarrow\left(W_{0}^{1,2}(\Omega)\right)^{\star}
$$
The motivation for this definition lies in the fact that when we look at the equation $-\int_{\Omega} \nabla u \cdot \nabla v=$ $\int_{\Omega} \Delta u v$ we actually mean $\int_{\Omega} v \cdot(\Delta u d \mathbf{x})$ and $\Delta u d \mathbf{x}$ gives a distribution under the identification of distributions with measures.
Recall the inner product as we defined it in $W_{0}^{1,2}(\Omega)$ is
$$
(u, v)=+\int_{\Omega} \nabla u \cdot \nabla v
$$
By the Riesz Representation Theorem given any element $F \in\left(W_{0}^{1,2}(\Omega)\right)^{\star}$ there exists a unique $u \in W_{0}^{1,2}(\Omega)$ such that $F(v)=(u, v)$, so
$$
F(v)=(u, v)=+\int_{\Omega} \nabla u \cdot \nabla v=(-\Delta u)(v)
$$
as distributions. Therefore $\Delta$ is surjective. Injectivity follows from the definition of $\Delta$. Continuity of the inverse is also provided for by the Riesz Representation Theorem
$$
\|u\|_{W_{0}^{1,2}(\Omega)}=\|-\Delta u\|_{\left(W_{0}^{1,2}(\Omega)\right)^{\star}} .
$$
We conclude from this short discussion that $\Delta: W_{0}^{1,2}(\Omega) \longrightarrow\left(W_{0}^{1,2}(\Omega)\right)^{\star}=: W^{-1,2}(\Omega)$ is an isomorphism of Hilbert Spaces. This is a natural extension to our previous results, and adopting this notation they all extend now to the case $k=-1$.
## Lecture 23
May $11^{\text {th }}, 2004$
## $\mathrm{L}^{\mathbf{p}}$ Theory
Take $f$ any measurable function on a domain $\Omega \subseteq \mathbb{R}^{n}$ and define the distribution function of $f$ $\mu_{f}(t):=|\{x \in \Omega:|f(x)|>t\}|$. We use alternatively $|\cdot|$ and $\lambda(\cdot)$ to denote the Lebesgue measure.
Proposition. Assume $f \in L^{p}(\Omega)$ for some $p>0$.
$$
\begin{aligned}
& \text { I) } \mu_{f}(t) \leq t^{-p} \int_{\Omega}|f|^{p} d \mathbf{x} . \\
& \text { II) } \int_{\Omega}|f|^{p} d \mathbf{x}=p \int_{0}^{\infty} t^{p-1} \mu_{f}(t) d t .
\end{aligned}
$$
In order for the second equation to make sense we need the distribution function to be measurable and indeed it is as $f$ itself is.
Proof. First
$$
\int_{\Omega}|f|^{p} d \mathbf{x} \geq \int_{\{f>t\}}|f|^{p} d \mathbf{x} \geq t^{p} \lambda(\{x: f(x)>t\})=t^{p} \mu_{f}(t) .
$$
Second, assume first $p=1$. By Fubini's Theorem one can interchange order of integration in
$$
\int_{\Omega}|f|=\int_{\Omega} \int_{0}^{|f(x)|} d t d \mathbf{x}=\int_{0}^{\infty} \int_{\Omega} \mathbb{I}_{\{x \in \Omega: f(x)>t\}} d \mathbf{x} d t=\int_{0}^{\infty} \mu_{f}(t) d t
$$
For general $p$
$$
\mu_{f^{p}}(t)=\left|\left\{x: f^{p}(x)>t\right\}\right|=|\{x: f(x)>\sqrt[p]{t}\}|=\mu_{f}(\sqrt[p]{t})=
$$
and so
$$
p \int_{0}^{\infty} t^{p-1} \mu_{f}(t) d t=\int_{0}^{\infty} \mu_{f^{p}}\left(t^{p}\right) d\left(t^{p}\right)=\int_{\Omega}|f|^{p} d \mathbf{x} .
$$
Marcinkiewicz Interpolation Theorem. $\quad$ Let $1 \leq q<r<\infty$ and let $T: L^{q}(\Omega) \cap L^{r}(\Omega) \longrightarrow$ $L^{q}(\Omega) \cap L^{r}(\Omega)$ be a linear map. Suppose there exist constants $T_{1}, T_{2}$ such that
$$
\forall f \in L^{q}(\Omega) \cap L^{r}(\Omega) \quad \mu_{T f}(t) \leq\left(\frac{T_{1}\|f\|_{L^{q}(\Omega)}}{t}\right)^{q}, \quad \mu_{T f}(t) \leq\left(\frac{T_{2}\|f\|_{L^{r}(\Omega)}}{t}\right)^{r}, \quad \forall t>0 .
$$
Then for any exponent in between $q<p<r, T$ can be extended to a map $L^{p}(\Omega) \longrightarrow L^{p}(\Omega)$ for all $f \in L^{q}(\Omega) \cap L^{p}(\Omega)$. And moreover
$$
\|T f\|_{L^{p}(\Omega)} \leq\left[\frac{p}{q-p}\left(2 T_{1}\right)^{q}+\frac{p}{r-p}\left(2 T_{2}\right)^{r}\right]^{\frac{1}{p}}\|f\|_{L^{p}(\Omega)} .
$$
Otherwise stated: weak $(q, q) \&$ weak $(r, r) \Longrightarrow$ strong $(p, p) p \in(q, r)$, though not for the endpoints, the constants blow-up there (we say an operator is strong $\left(p_{1}, p_{2}\right)$ if it maps functions in $L^{p_{1}}$ to functions in $L^{p_{2}}$. We say it is weak $\left(p_{1}, p_{2}\right)$ if its domain is in $L^{p_{1}}$ and its distribution function satisfies the first inequality in the assumptions above with $q$ replaced by $p_{2}$ ).
Proof. Take $f \in L^{q}(\Omega) \cap L^{r}(\Omega)$, and let $s>0$. Let
$$
\begin{aligned}
& f_{1}:= \begin{cases}f(x) & |f(x)|>s \\
0 & |f(x)| \leq s\end{cases} \\
& f_{2}:= \begin{cases}0 & |f(x)|>s \\
f(x) & |f(x)| \leq s\end{cases}
\end{aligned}
$$
indeed one notices that $f=f_{1}+f_{2}$. The trick will be to let this splitting of $f$ vary by letting $s$ itself vary. So $|T f| \leq\left|T f_{1}\right|+\left|T f_{2}\right|$. If $T f(x)>t$ at some point $x \in \Omega$ then either $T f_{1}>t / 2$ or $T f_{2}>t / 2$. This translates into
$$
\begin{aligned}
\mu_{T f}(t) & \leq \mu_{T f_{1}}(t / 2)+\mu_{T f_{2}}(t / 2) \\
& \leq\left(\frac{T_{1}}{t / 2}\right)^{q} \int_{\Omega}\left|f_{1}\right|^{q}+\left(\frac{T_{2}}{t / 2}\right)^{r} \int_{\Omega}\left|f_{2}\right|^{r} .
\end{aligned}
$$
We choose the smaller exponent $q$ for the terms where $f$ is large $\left(f_{1}\right)$ and larger one $r$ for where $f$ is small $\left(f_{2}\right)$, intuitively. This will make sense in a moment when it will be clear how this guarantees that our two integrals - with different integration domains - are finite. By the Proposition we have
$$
\int_{\Omega}|T f|^{p} d \mathbf{x}=p \int_{0}^{\infty} t^{p-1} \mu_{T f}(t) d t
$$
and once we substitute in the above inequality we get
$$
\begin{aligned}
\int_{\Omega}|T f|^{p} d \mathbf{x} & \leq p \int_{0}^{\infty} t^{p-1}\left[\left(\frac{T_{1}}{t / 2}\right)^{q} \int_{\Omega}\left|f_{1}\right|^{q}+\left(\frac{T_{2}}{t / 2}\right)^{r} \int_{\Omega}\left|f_{2}\right|^{r}\right] d t \\
& =p\left(2 T_{1}\right)^{q} \int_{0}^{\infty}\left(\int_{\{|f|>s\}}\left|f_{1}\right|^{q}\right) t^{p-1-q} d t+p\left(2 T_{2}\right)^{r} \int_{0}^{\infty}\left(\int_{\{|f| \leq s\}}\left|f_{2}\right|^{r}\right) t^{p-1-r} d t .
\end{aligned}
$$
We chose $s>0$ arbitrary in the above construction of $f_{i}$. In particular we may let it vary. This is a neat trick. We set $s=t$ to get
$$
\begin{aligned}
& p\left(2 T_{1}\right)^{q} \int_{0}^{\infty}\left(\int_{\{|f|>s\}}|f|^{q}\right) s^{p-1-q} d s+p\left(2 T_{2}\right)^{r} \int_{0}^{\infty}\left(\int_{\{|f| \leq s\}}|f|^{r}\right) s^{p-1-r} d s \\
& =p\left(2 T_{1}\right)^{q} \int_{\Omega}|f|^{q} d \mathbf{x} \int_{0}^{|f|} s^{p-1-q} d s+p\left(2 T_{2}\right)^{r} \int_{\Omega}|f|^{r} d \mathbf{x} \int_{|f|}^{\infty} s^{p-1-r} d s \\
& =\left(2 T_{1}\right)^{q} \frac{p}{q-p} \int_{\Omega}|f|^{p}+\left(2 T_{2}\right)^{r} \frac{p}{r-p} \int_{\Omega}|f|^{p} .
\end{aligned}
$$
Altogether
$$
\int_{\Omega}|T f|^{p} d \mathbf{x} \leq\left[\frac{p}{q-p}\left(2 T_{1}\right)^{q}+\frac{p}{r-p}\left(2 T_{2}\right)^{r}\right] \cdot \int_{\Omega}|f|^{p}
$$
Remark. In Gilbarg-Trudinger, p.229, a different constant is achieved which is slightly stronger than ours (as can be seen using the AM-GM Inequality). This is done by introducing an additional constant $A$, letting $t=A s$ and later choosing $A$ appropriately.
## Back to the Newtonian Potential
We defined the Newtonian Potential of $f$
$$
\omega \equiv N f:=\int_{\Omega} \Gamma(x-y) f(y) d \mathbf{y}=\frac{1}{n(2-n) \omega_{n}} \int_{\Omega} \frac{1}{|x-y|^{n-2}} d \mathbf{y}
$$
Claim. (Young's Inequality) $N: L^{p}(\Omega) \longrightarrow L^{p}(\Omega)$. Moreover continuously so- $\exists C$ such that $\|N f\|_{L^{p}(\Omega)} \leq C\|f\|_{L^{p}(\Omega)}$.
Remark. For $p=2$ we proved in the past much more: $\Delta(N f)=f \in L^{2}(\Omega) \Rightarrow N f \in W^{2,2}(\Omega)$. Also our previous estimates on the Newtonian Potential can actually be made to extend our Claim to $W^{1, p}(\Omega)$ regularity. These estimates can not give though $W^{2, p}(\Omega)$ estimates (see the beginning of the next Lecture).
Proof.
$$
\begin{aligned}
\omega: & =\Gamma \star f=\int_{\Omega} \Gamma(x-y) f(y) d \mathbf{y} \\
& =\int_{\Omega} f(y) \Gamma(x-y)^{\frac{1}{p}} \Gamma(x-y)^{1-\frac{1}{p}} d \mathbf{y} \\
& \leq\left\{\int_{\Omega}\left|f(y)^{p} \Gamma(x-y)\right| d \mathbf{y}\right\}^{\frac{1}{p}}\left\{\int_{\Omega}|\Gamma(x-y)| d \mathbf{y}\right\}^{1-\frac{1}{p}} \\
& \leq C \cdot\left\{\int_{\Omega}\left|f(y)^{p} \Gamma(x-y)\right| d \mathbf{y}\right\}^{\frac{1}{p}} .
\end{aligned}
$$
since $\Gamma(x-y) \sim \frac{1}{|x-y|^{n-1}}$ and therefore is integrable over $\mathbb{R}^{n}$. Therefore we have an upper bound on $\omega^{p}$ which we can integrate
$$
\begin{aligned}
\int_{\Omega} \omega^{p} d \mathbf{x} & \leq \int_{\Omega} C^{p}\left\{\int_{\Omega}\left|f(y)^{p} \Gamma(x-y)\right| d \mathbf{y}\right\} d \mathbf{x} \\
& =C^{p} \int_{\Omega} \int_{\Omega}|f(y)|^{p}|\Gamma(x-y)| d \mathbf{x} d \mathbf{y} \\
& =C^{p} \int_{\Omega}|f(y)|^{p}\left(\int_{\Omega}|\Gamma(x-y)| d \mathbf{x}\right) d \mathbf{y} \\
& \leq \tilde{C} \int_{\Omega}|f(y)|^{p} d \mathbf{y} .
\end{aligned}
$$
where we applied Fubini's Theorem.
Theorem. Let $f \in L^{p}(\Omega)$ for some $1<p<\infty$ and let $\omega=N f$ be the Newtonian Potential of f. Then $\omega \in W^{2, p}(\Omega)$ and $\Delta w=f$ a.e. and
$$
\left\|D^{2} w\right\|_{L^{p}(\Omega)} \leq c(n, p, \Omega) \cdot\|f\|_{L^{p}(\Omega)}
$$
For $p=2$ we have even
$$
\int_{\mathbb{R}^{n}}\left|D^{2} \omega\right|^{2}=\int_{\Omega} f^{2}
$$
Proof. We prove just for $p=2$, leaving the hard work for the next and last lecture. First we assume $f \in \mathcal{C}_{0}^{\infty}\left(\mathbb{R}^{n}\right)$. From long time ago: $f \in \mathcal{C}_{0}^{\infty}\left(\mathbb{R}^{n}\right) \Rightarrow \omega \in \mathcal{C}^{\infty}\left(\mathbb{R}^{n}\right)$ and $\Delta \omega=f$ (Hölder Theory for the Newtonian Potential).
Let $B:=B_{R}$ a ball containing $\operatorname{supp} f \Rightarrow$
$$
\int_{B_{R}}(\mathrm{D} \omega)^{2}=\int_{B_{R}} f^{2}=\int_{\Omega} f^{2}
$$
We embark now on our main computation
$$
\begin{aligned}
\int_{B_{R}}\left|\mathrm{D}^{2} \omega\right|^{2}=\int_{B_{R}} \mathrm{D}_{i j} \omega \mathrm{D}_{i j} \omega \text { (summation) } & =-\int_{B_{R}} \mathrm{D}_{j}\left(\mathrm{D}_{i j} \omega\right) \mathrm{D}_{i} \omega+\int_{\partial B_{R}} \mathrm{D}_{i j} \omega \mathrm{D}_{i} \omega \nu_{j} d \theta \\
& =-\int_{B_{R}} \mathrm{D}_{i}\left(\mathrm{D}_{j j} \omega\right) \mathrm{D}_{i} \omega+\int_{\partial B_{R}} \mathrm{D}_{i j} \omega \mathrm{D}_{i} \omega \nu_{j} d \theta \\
& =-\int_{B_{R}} \mathrm{D}_{i}(\Delta \omega) \mathrm{D}_{i} \omega+\int_{\partial B_{R}} \frac{\partial}{\partial \nu} \mathrm{D} \omega \mathrm{D} \omega d \theta \\
& =\int_{B_{R}}(\Delta \omega)^{2}-\int_{\partial B_{R}} \Delta \omega \cdot \frac{\partial}{\partial \nu} \omega d \theta+\int_{\partial B_{R}} \frac{\partial}{\partial \nu} \mathrm{D} \omega \cdot \mathrm{D} \omega d \theta \\
& =\int_{B_{R}}(\Delta \omega)^{2}+\int_{\partial B_{R}} \frac{\partial}{\partial \nu} \mathrm{D} \omega \cdot \mathrm{D} \omega d \theta
\end{aligned}
$$
The last equality results from our assumption that $f$ vanishes on $\partial B$, i.e has compact support inside $\Omega$. Now since $f$ is smooth
$$
\begin{aligned}
\mathrm{D}_{i} \omega(x) & =\int_{\Omega} \mathrm{D}_{i} \Gamma(x-y) f(y) d \mathbf{y} \leq \frac{C}{R^{n-1}}, \\
\mathrm{D}_{i j} \omega(x) & =\int_{\Omega} \mathrm{D}_{i j} \Gamma(x-y) f(y) d \mathbf{y} \leq \frac{C}{R^{n}} .
\end{aligned}
$$
Therefore as we let $R \longrightarrow \infty$, the second term - which is integrated only over the sphere of radius $R$ in $\mathbb{R}^{n}$ - tends to 0 . Then we have in the limit the desired result (after substituting (1) for the RHS).
Now if $f \in L^{2}(\Omega)$, approximate it by functions $f_{m} \in \mathcal{C}_{0}^{\infty}\left(\mathbb{R}^{n}\right)$ (possible by the density argument used in the past: $\left.\overline{\mathcal{C}_{0}^{\infty}(\Omega)}=L^{2}(\Omega)\right)$ such that $f_{m} \stackrel{L^{2}(\Omega)}{\longrightarrow} f$. From the Claim above $\|N f\|_{L^{p}(\Omega)} \leq$ $C|| f \|_{L^{p}(\Omega)}$, hence $\left\|N\left(f_{i}-f_{j}\right)\right\|_{L^{p}(\Omega)} \leq C\left\|f_{i}-f_{j}\right\|_{L^{p}(\Omega)}$, from which $\omega_{m} \equiv N f_{m} \stackrel{L^{2}(\Omega)}{\longrightarrow} N f \equiv \omega$. Now $\Delta \omega_{j}=f_{j}$ and by the $\mathcal{C}_{0}^{\infty}(\Omega)$ case applied to the Dirichlet Problem $\Delta\left(\omega_{i}-\omega_{j}\right)=f_{i}-f_{j}$
$$
\int_{\mathbb{R}^{n}}\left|\mathrm{D}^{2}\left(\omega_{i}-\omega_{j}\right)\right|^{2}=\int_{\Omega}\left|f_{i}-f_{j}\right|^{2}
$$
As the RHS tends to 0 for $i, j$ large we have that $\left\{\mathrm{D}^{2} \omega_{m}\right\}$ converges in $L^{2}(\Omega)$, i.e $\left\{\omega_{m}\right\}$ converges in $W^{2,2}(\Omega)$. Since we already know its limit is $\omega \in L^{2}(\Omega)$ we conclude that in fact $\omega \in W^{2,2}(\Omega)$ !
## Lecture 24
May $13^{\text {th }}, 2004$
Our motivation for this last lecture in the course is to show a result using our regularity theory which is otherwise unprovable using classical techniques. This is the previous Theorem, and in particular the case $p \neq 2$ (which we haven't yet done) namely $N$ is a continuous map $L^{p}(\Omega)$ to $W^{2, p}(\Omega)$. Classical methods give at best $W^{1, p}(\Omega)$. For that we introduce the
## Calderon-Zygmund Decomposition Technique: Cube decomposition
Let $K_{0}$ be an $n$-dimensional cube in $\mathbb{R}^{n}, f \geq 0$ integrable and finally fix $t>0$ such that
$$
\int_{K_{0}} f \leq t\left|K_{0}\right| \equiv t \operatorname{Vol}\left(K_{0}\right), \quad \text { that is } \int_{K_{0}} f \leq t
$$
Next bisect $K_{0}$ into $2^{n}$ equal (in volume) subcubes. Let $\mathcal{S}$ be the collection of those subcubes
$K$ for which $\int_{K} f>t$. I.e the subcubes where $f$ is highly concentrated. On each of the remaining subcubes (those not in $\mathcal{S}$ ) repeat the same procedure, i.e bisect each one into $2^{n}$ sub-subcubes and add those where $f$ is highly concentrated to $\mathcal{S}$, bisect the rest et ceterà... Now for any $K \in \mathcal{S}$ denote by $\tilde{K}$ its immediate predecessor. Since $K \in \mathcal{S}$ while $\tilde{K} \notin \mathcal{S}$
$$
t<\frac{1}{\operatorname{Vol}(K)} \int_{K} f<\frac{1}{\operatorname{Vol}(K)} \int_{\tilde{K}} f=\frac{\operatorname{Vol}(\tilde{K})}{\operatorname{Vol}(K)} \cdot \frac{1}{\operatorname{Vol}(\tilde{K})} \int_{\tilde{K}} f \leq 2^{n} t
$$
In summary $\forall K \in \mathcal{S} \quad t<\int_{K} f \leq 2^{n} t$. Denote $F:=\bigcup_{K \in \mathcal{S}} K, G:=K_{0} \backslash F \equiv F^{C}=\bigcap_{K \in \mathcal{S}} K^{C}$. We see each point in $G$ lies in infinitely many nested cubes with bounded concentration of $f$ with diameters converging to 0: $\int_{K_{i}} f \leq t$ with $\operatorname{Vol}\left(K_{i}\right) \longrightarrow 0$. By Lebesgue's Theorem on differentiation the LHS $\longrightarrow f \lambda$-a.e $\left(\lambda\right.$ denotes Lebesgue's measure on $\left.\mathbb{R}^{n}\right)$, i.e $f \leq t$ a.e $(\equiv \lambda$-a.e) on $G$. In summary, on $F$ we have an average estimate, on $G$ we have a pointwise estimate.
## The promised proof
$\mathcal{W}$ e restate the desired result whose proof we gave for $p=2$.
Theorem. Let $f \in L^{p}(\Omega)$ for some $1<p<\infty$ and let $\omega=N f$ be the Newtonian Potential of f. Then $\omega \in W^{2, p}(\Omega)$ and $\Delta w=f$ a.e. and
$$
\left\|D^{2} w\right\|_{L^{p}(\Omega)} \leq c(n, p, \Omega) \cdot\|f\|_{L^{p}(\Omega)}
$$
For $p=2$ we have even
$$
\int_{\mathbb{R}^{n}}\left|D^{2} \omega\right|^{2}=\int_{\Omega} f^{2}
$$
Proof. Define an operator $T: L^{2}(\Omega) \longrightarrow L^{2}(\Omega), \quad T f=\mathrm{D}_{i j} N f$. Last time we showed $\left\|\mathrm{D}_{i j} N f\right\|_{L^{2}(\Omega)}=\|T f\|_{L^{2}(\Omega)}=\|f\|_{L^{2}(\Omega)}$. In other words $T$ is strong $(2,2)$ and therefore automatically weak $(2,2)$ i.e
$$
\mu_{T f} \leq\left(\frac{\|f\|_{L^{2}(\Omega)}}{t}\right)^{2}
$$
by the Proposition in the previous lecture. If we will now be able to bound it with $\frac{\|f\|_{L^{1}(O)}}{t}$, the Interpolation Theorem will then provide the desired bound on $\mathrm{D}^{2} \omega$ for all $1<p<2$. By duality $2<p<\infty$ will then be taken care of as well (to be made precise). So we
Claim. $\quad T$ is weak $(1,1)$ i.e
$$
\forall f \in L^{2}(\Omega) \cap L^{1}(\Omega) \quad \mu_{T f}(t) \leq C \frac{\|f\|_{L^{1}(\Omega)}}{t}, \quad \forall t>0 .
$$
Proof. Extend $f$ trivially outside $\Omega$ (i.e so the extension vanishes on $\mathbb{R}^{n} \backslash \Omega$ ), and given any fixed $t>0$ take a large enough cube $K_{0}$ containing $\Omega$ such that
$$
\int_{K_{0}}|f|=\frac{1}{\operatorname{Vol}\left(K_{0}\right)} \int_{K_{0}}|f| \leq t
$$
The Cube Decomposition furnishes a countable number of cubes $\left\{K_{l}\right\}$ such that on each $t<\int_{K_{l}}|f| \leq 2^{n} t$ and in addition $|f| \leq t$ a.e on $G:=K_{0} \backslash \bigcup_{l} K_{l}$. Split $f=b+g$ into bad, good parts by letting $g(x):=\left\{\begin{array}{ll}f(x) & \text { on } G \\ \int_{K_{l}} f & \text { on } K_{l}\end{array}\right.$ i.e $f$ could be oscillating on $K_{l}$, instead we just replace it there by its average value therein. Then let $b:=f-g$, the bad or highly oscillating part. Note: $|g| \leq 2^{n} t$ a.e, $b(x)=0$ on $G$ and $\int_{K_{l}} b=0$.
We have now $T f=T b+T g$. And as in the Interpolation Theorem of the previous lecture
$$
\mu_{T f}(t) \leq \mu_{T b}(t / 2)+\mu_{T g}(t / 2) .
$$
We would like to bound this with the $L^{1}(\Omega)$ norm of $f$. We divide the computation into 3 parts.
$L^{1}(\Omega)$ estimate for $\mu_{T g}(t / 2)$. Using (1) on the good part we have
$$
\begin{aligned}
\mu_{T g}(t / 2) & \leq\left(\frac{\|g\|_{L^{2}(\Omega)}}{t / 2}\right)^{2} \\
& \leq \frac{\int_{K_{0}} g^{2}}{(t / 2)^{2}}
\end{aligned}
$$
and since $g /\left(2^{n} t\right) \leq 1,\left(g /\left(2^{n} t\right)\right)^{2} \leq|g| /\left(2^{n} t\right)$ or $(g / t)^{2} \leq 2^{n}|g| / t$ from which
$$
\begin{aligned}
& \leq \frac{2^{n+2}}{t} \int_{K_{0}}|g| \\
& =\frac{2^{n+2}}{t} \int_{G}+\int_{\cup_{l} K_{l}}|g| \\
& =\frac{2^{n+2}}{t} \int_{G}|f|+\int_{\cup_{l} K_{l}}\left(\int_{K_{l}}|f|\right)
\end{aligned}
$$
$$
=\frac{2^{n+2}}{t} \int_{\Omega}|f|=\frac{2^{n+2}}{t}\|f\|_{L^{1}(\Omega)}
$$
We have not used so far any properties of $T$. On the bad part we will, and we will work with the kernel of the Newtonian Potential, in just a moment.
$L^{1}\left(K_{0} \backslash \bigcup_{l} B_{l}\right)$ estimate for $T b$. Let $\bar{y}$ be the center of the subcube $K_{l}$. Let $B_{l}:=B(\bar{y}, \delta)$ which strictly contains $K_{l}$. The diameter of $K_{l}$ is $\delta:=\operatorname{diam}(\Omega) \frac{\sqrt{n}}{2^{r}}$ if it belongs to the $r^{\text {th }}$ subdivision.
We content ourselves with bounding only the $L^{1}$ norm of $T b$ on $K_{0} \backslash \bigcup_{l} B_{l}$ since by part I) of the Proposition of Lecture 23 (with $p=1$ ) that will bound the distribution function $\mu_{T b}$ itself.
Write $b_{l}:=\mathbb{I}_{K_{l}}$, the characteristic function defined in Lecture 18. $\quad b=\sum_{l=1}^{\infty} b_{l}$. The advantage of this splitting is that each term is compactly supported unlike $b$ itself. Fix some $l \in \mathbb{N}$ and approximate $b_{l}$ by smooth functions $\left\{b_{l}^{(m)}\right\}_{m=0}^{\infty} \subseteq \mathcal{C}_{0}^{\infty}\left(K_{l}\right)$. By varying each with a constant one can make sure for each $n \in \mathbb{N} \int_{K_{l}} b_{l}^{(m)}=\int_{K_{l}} b_{l}=0$.
If $x \in K_{l}$,
$$
\begin{aligned}
T\left(b_{l}^{(m)}\right)(x) & =\int_{K_{l}} \mathrm{D}_{i j} \Gamma(x-y) b_{l}^{(m)}(y) d \mathbf{y} \\
& =\int_{K_{l}}\left[\mathrm{D}_{i j} \Gamma(x-y)-\mathrm{D}_{i j} \Gamma(x-\bar{y})\right] b_{l}^{(m)}(y) d \mathbf{y}
\end{aligned}
$$
by the zero average $b_{l}^{(m)}$.
## Computation.
$$
\left|T b_{l}^{(m)}(x)\right| \leq c \cdot \delta \cdot \frac{1}{\left[\operatorname{dist}\left(x, K_{l}\right)\right]^{n+1}} \int_{K_{l}}\left|b_{l}^{(m)}(y)\right| d \mathbf{y}
$$
Proof. Using the above equation in conjunction with the Mean Value Theorem of Calculus there exists $y_{0} \in K_{l}$ (and $\left.\left|y-y_{0}\right| \leq \delta \quad \forall y \in K_{l}\right)$ such that
$$
\begin{aligned}
\left|T b_{l}^{(m)}(x)\right| & =\left|\int_{K_{l}} \mathrm{DD}_{i j} \Gamma\left(x-y_{0}\right) \cdot\left(y-y_{0}\right) b_{l}^{(m)}(y) d \mathbf{y}\right| \\
& \leq \int_{K_{l}}\left|\mathrm{DD}_{i j} \Gamma\left(x-y_{0}\right)\right| \cdot\left|y-y_{0}\right|\left|b_{l}^{(m)}(y)\right| d \mathbf{y} \\
& \leq c \delta \int_{K_{l}} \frac{1}{\left|x-y_{0}\right|^{n+1}}\left|b_{l}^{(m)}(y)\right| d \mathbf{y} \\
& \leq c \delta \frac{1}{\left[\operatorname{dist}\left(x, K_{l}\right)\right]^{n+1}} \int_{K_{l}}\left|b_{l}^{(m)}(y)\right| d \mathbf{y} .
\end{aligned}
$$
This now helps us evaluate the $L^{1}$ norm
$$
\int_{K_{0} \backslash B_{l}}\left|T b_{l}^{(m)}\right| \leq c \cdot \delta \int_{|x-\bar{y}| \geq \delta} \frac{1}{\left[\operatorname{dist}\left(x, K_{l}\right)\right]^{n+1}} d \mathbf{x} \cdot\left(\int_{K_{l}}\left|b_{l}^{(m)}(y)\right| d \mathbf{y}\right) .
$$
Note there is some $\tilde{y}$ with $\operatorname{dist}\left(x, K_{l}\right)=|x-\tilde{y}|$ and then $|x-\bar{y}| \leq|x-\tilde{y}|+\left|\tilde{y}-y_{0}\right| \leq 2 \operatorname{dist}\left(x, K_{l}\right)$
$$
\begin{aligned}
& \leq 2^{n+1} c \cdot \delta \int_{|x-\bar{y}| \geq \delta} \frac{1}{|x-\bar{y}|^{n+1}} d \mathbf{x} \cdot\left(\int_{K_{l}}\left|b_{l}^{(m)}(y)\right| d \mathbf{y}\right) \\
& =c^{\prime} \int_{K_{l}}\left|b_{l}^{(m)}(y)\right| d \mathbf{y} .
\end{aligned}
$$
Let $m \rightarrow \infty$ in the above to get
$$
\int_{K_{0} \backslash B_{l}}\left|T b_{l}\right| \leq c^{\prime} \int_{K_{l}}\left|b_{l}(y)\right| d \mathbf{y}
$$
i.e we have taken care of things (have $L^{1}(\Omega)$ estimates there) on $K_{0} \backslash \bigcup_{l} B_{l}$, as can be seen by summing (the $b_{l}$ 's have disjoint supports so $|b|=\sum_{l}\left|b_{l}\right|$ )
$$
\begin{aligned}
\|T b\|_{L^{1}\left(K_{0} \backslash \bigcup_{l} B_{l}\right)}=\int_{K_{0} \backslash \bigcup_{l} B_{l}}|T b| & \leq \sum_{l=1}^{\infty} \int_{K_{0} \backslash B_{l}}\left|T b_{l}\right| \\
& \leq \sum_{l=1}^{\infty} c^{\prime} \int_{K_{l}}\left|b_{l}\right| \\
& \leq c^{\prime} \int_{\bigcup_{l} B_{l}}|b|=c^{\prime} \int_{\bigcup_{l} B_{l}}|f|=c^{\prime}|| f \|_{L^{1}(\Omega)} .
\end{aligned}
$$
$L^{1}\left(\bigcup_{l} B_{l}\right)$ estimates for $\mu_{T b}(t / 2)$.
$$
\mu_{T b}(t / 2)=|\{x \in \Omega: T b(x)>t / 2\}| \leq\left|\left\{\alpha \in K_{0} \backslash \bigcup_{l} B_{l}:|T b|>t / 2\right\}\right|+\left|\bigcup_{l} B_{l}\right| .
$$
The first term is taken care of (by applying part I of the Proposition in Lecture 23 with $p=1$ to the estimate above for $\left.\|T b\|_{L^{1}\left(K_{0} \backslash \bigcup_{l} B_{l}\right)}\right)$. For the second, there exists some constant $c$ such that $\left|\bigcup_{l} B_{l}\right| \leq c\left|\bigcup_{l} K_{l}\right|$ by the geometry of cubes and balls. Now the $K_{l}$ were chosen with
$$
t<\int_{K_{l}}|f|
$$
hence
$$
\operatorname{Vol}\left(K_{l}\right)<\frac{1}{t}\|f\|_{L^{1}\left(K_{l}\right)}
$$
Altogether
$$
\mu_{T b}(t / 2) \leq \frac{c}{t}\|f\|_{L^{1}(\Omega)}
$$
Combining the above 3 parts
$$
\forall f \in L^{2}(\Omega) \quad \mu_{T f}(t) \leq \mu_{T b}(t / 2)+\mu_{T g}(t / 2) . \leq \frac{c}{t}\|f\|_{L^{1}(\Omega)}+\frac{2^{n+1}}{t}\|f\|_{L^{1}(\Omega)}, \quad \forall t>0 .
$$
That is $T$ is weak $(1,1)$ proving the Claim. Thus by the Marcinkiewicz Interpolation Theorem (MIT) exists $c$ depending on the above constants, i.e on $n, p, \operatorname{diam}(\Omega)$, satisfying
$$
\forall f \in L^{2}(\Omega) \quad\|T f\|_{L^{p}(\Omega)} \leq c\|f\|_{L^{p}(\Omega)}, \quad 1<p<2 !
$$
From the proof of the MIT $c$ blows up as $p$ approaches either of the endpoints. We mention without proof that as stronger version of the MIT states that if $T$ is strong $(r, r)$ and/or strong $(q, q)$ then the constant does not blow-up at $r$ and/or $q$. Therefore we have in fact $1<p \leq 2$ in (2). As a matter of fact we do not even need to invoke this stronger Theorem since we have done the case $p=2$ independently (with constant $=1$ !) in the previous lecture.
Yet another idea would be to prove (2) for some one value $p$ greater than 2 , and apply the MIT to get (2) for $p=2$ as an intermediate value in the interval $(1, p)$ ! This will also conclude the proof of our Theorem as $p$ will be arbitrary.
To that end we use the so called Duality Method. Let $p>2$ be arbitrary. $\left(L^{p}(\Omega)\right)^{\star}=L^{q}(\Omega)$ with $1=\frac{1}{q}+\frac{1}{p}$. By the definition of the dual space (p. 3 of Lecture 17)
$$
\begin{aligned}
& \|T f\|_{L^{p}(\Omega)}=\sup _{\substack{g \in L^{q}(\Omega) \\
\|g\|_{L^{q}(O)}=1}} \int_{\Omega} T f \cdot g=\sup _{\substack{g \in L^{q}(\Omega) \\
\|g\|_{L^{q}(O)}=1}} \int_{\Omega} \mathrm{D}_{i j} \omega \cdot g \\
& =\sup _{\substack{g \in L^{q}(\Omega) \\
\mid g \|_{L} q(O)}} \int_{\Omega} \omega \cdot \mathrm{D}_{i j} g \\
& =\sup _{\substack{g \in L^{q}(\Omega) \\
\|g\|^{q}(O)=1}} \int_{\Omega}\left(\int_{\Omega} \Gamma(x-y) f(y) d \mathbf{y}\right) \mathrm{D}_{i j} g(x) d \mathbf{x} \\
& =\sup _{\substack{g \in L^{q}(\Omega) \\
\|g\|^{q}(O)=1}} \int_{\Omega}\left(\int_{\Omega} \Gamma(x-y) \mathrm{D}_{i j} g(x) f(y) d \mathbf{x}\right) d \mathbf{y} \\
& =\sup _{\substack{g \in L^{q}(\Omega) \\
\|g\|_{L^{q}(O)}=1}} \int_{\Omega}\left(\int_{\Omega} \mathrm{D}_{i j} \Gamma(x-y) g(x) d \mathbf{x}\right) f(y) d \mathbf{y} \\
& =\sup _{\substack{g \in L^{q}(\Omega) \\
\|g\|_{L^{q}(O)}=1}} \int_{\Omega} T g \cdot f
\end{aligned}
$$
$$
\begin{aligned}
& \underset{\text { Hölder's Ineq. }}{\leq} \sup _{\substack{g \in L^{q}(\Omega) \\
\|g\|_{L^{q}(O)}=1}}\|f\|_{L^{p}(\Omega)} \cdot\|T g\|_{L^{q}(\Omega)} \\
& \leq C \sup _{\substack{g \in L^{q}(\Omega) \\
\|g\|_{L^{q}(O)}=1}}\|f\|_{L^{p}(\Omega)} \cdot\|g\|_{L^{q}(\Omega)} \\
& =C\|f\|_{L^{p}(\Omega)} \cdot 1 .
\end{aligned}
$$
As we wished: $T$ is strong $(p, p)$. In the last inequality we simply used the fact that $1<q<2$ is in the range we have already taken care of.
In summary we have shown: If $f \in \mathcal{C}_{0}^{\infty}(\Omega), \omega:=N f$ then $\Delta \omega=f$ and $\left\|\mathrm{D}^{2} \omega\right\|_{L^{p}(\Omega)} \leq c\|f\|_{L^{p}(\Omega)}$ for any $1<p<\infty$. Now identically to how we finished the proof of the last Theorem in the previous lecture we extend this to all functions in $L^{p}(\Omega)$ by approximating and subsequently taking limits and making use of Young's Inequality (Lecture 23).
Our work can be rephrased
Corollary. $\quad$ Let $\Omega \subseteq \mathbb{R}^{n}$ be a bounded domain and assume $u \in W_{0}^{2, p}(\Omega)$ for some $1<p<\infty$. Then
$$
\left\|D^{2} u\right\|_{L^{p}(\Omega)} \leq c(n, p, \Omega) \cdot\|\Delta u\|_{L^{p}(\Omega)}
$$
For $p=2$
$$
\left\|D^{2} u\right\|_{L^{2}(\Omega)}=\|\Delta u\|_{L^{p}(\Omega)}
$$
Proof. $u-N(\Delta u)$ satisfies Laplace's equation $\Delta(u-N(\Delta u))=0$ a.e. In other words $u-N(\Delta u)$ is a harmonic function with compact support in $\mathbb{R}^{n}$ hence vanishes identically. Hence $u=N \Delta u$ and renaming $f:=\Delta u, \omega:=u$ gives the inequality from our above Theorem.
This is quite remarkable as it tells us that the whole Hessian, $\left(\begin{array}{l}n \\ 2\end{array}\right)$ functions, can be bounded simply in terms of the sum of its $n$ diagonal terms — its trace. Theorem. Let $L:=a^{i j} D_{i j}+b^{i} D_{i}+c$ and let $\Omega \subseteq \mathbb{R}^{n}$ be a bounded domain. Assume $u \in W^{2, p}(\Omega)$ for some $1<p<\infty$ satisfies Lu $=f$ a.e. Assume
- $\quad L$ strictly elliptic with $\left(a^{i j}\right)>\gamma \cdot I, \quad \gamma>0$
- $\quad a^{i j} \in \mathcal{C}^{0}(\Omega)$
- $\quad b^{i}, c \in L^{\infty}(\Omega)$
- $f \in L^{p}(\Omega)$.
Then $\forall \Omega^{\prime} \Subset \Omega$ holds
$$
\|u\|_{W^{2, p}\left(\Omega^{\prime}\right)} \leq c \cdot\left(\|u\|_{L^{p}(\Omega)}+\|f\|_{L^{p}(\Omega)}\right)
$$
Proof. We know this for $L=\Delta$, and therefore for any constant coefficients operator satisfying the above by Lecture 12. Then perturbing the coefficients and proceeding just like in the Schauder case works, as in Lecture 13, works.
Assuming $\mathcal{C}^{1,1}$ boundary, these estimates can be extended to hold globally on all of $\Omega$, as in done in Lecture 14.
| Textbooks |
\begin{definition}[Definition:Partial Derivative/Value at Point]
Let $\map f {x_1, x_2, \ldots, x_n}$ be a real function of $n$ variables
Let $f_i = \dfrac {\partial f} {\partial x_i}$ be the partial derivative of $f$ {{WRT|Differentiation}} $x_i$.
Then the value of $f_i$ at $x = \tuple {a_1, a_2, \ldots, a_n}$ can be denoted:
:$\valueat {\dfrac {\partial f} {\partial x_i} } {x_1 \mathop = a_1, x_2 \mathop = a_2, \mathop \ldots, x_n \mathop = a_n}$
or:
:$\valueat {\dfrac {\partial f} {\partial x_i} } {a_1, a_2, \mathop \ldots, a_n}$
or:
:$\map {f_i} {a_1, a_2, \mathop \ldots, a_n}$
and so on.
Hence we can express:
:$\map {f_i} {a_1, a_2, \mathop \ldots, a_n} = \valueat {\dfrac \partial {\partial x_i} \map f {a_1, a_2, \mathop \ldots, a_{i - 1}, x_i, a_{i + i}, \mathop \ldots, a_n} } {x_i \mathop = a_i}$
according to what may be needed as appropriate.
\end{definition} | ProofWiki |
HomeProgramming Languagehybridization – Why can there not be more than one sigma bond in a set of bonds?
hybridization – Why can there not be more than one sigma bond in a set of bonds?
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A question on an exam asked why there is exactly one sigma bond in double and triple covalent bonds. I looked in my text and online after the exam, but couldn't find an anawer to the question.
Why can there not be more than one sigma bond in a set of covalent bonds?
What atomic orbitals overlap to form a sigma bond?
@LordStryker, the s and p orbitals overlap to form the orbitals involved in sigma bonds, if that's what you mean.
"Why can there not be more than one sigma bond in a set of covalent bonds?" Actually that is quite a profound question. Your thinking is far ahead of your test. The answer to your question is that while introductory texts often display double and triple bonds as one sigma bond and the rest pi bonds, they can also be equivalently described as 2 or 3 "bent" sigma bonds. So double bonds and triple bonds can be described using only sigma bonds or as a mixture of pi and sigma bonds.
Strictly speaking, you can get two $\sigma$ bonds between the same two atoms, though it is rare. One example is in the gaseous dimolybdenum molecule $\ce{Mo2}$ with its sextuple bond, which has both a $s\! -\! s$ $\sigma$ sigma bond and a $d\! -\! d$ $\sigma$ sigma bond, as well as two $d\! -\! d$ $\pi$ pi bonds and two $d\! -\! d$ $\delta$ delta bonds. I've never heard of a double sigma bond for $\ce{C=C}$, though in some interpretations dicarbon $\ce{C2}$ has two $\pi$ bonds with no $\sigma$ bond.
This is where chemistry really gets interesting — when you try to pin a concept down and, after some wild times down the rabbit hole, find out that things are far more subtle and more weird than you ever thought possible. Nearly every concept is initially taught at a (relatively) comprehensible, modest-to-extreme level of approximation. As learning proceeds, the approximations are gradually stripped away until one finally starts to bump up against the limits of human knowledge.
Why can there not be more than one sigma bond in a set of bonds?
There can be, even in simple carbon compounds. Bent bonds, tau bonds or banana bonds; whatever you might like to call them were proposed by Linus Pauling; Erich Hückel proposed the alternative $\sigma – \pi$ bonding formalism. Hückel's description is the one commonly seen in introductory texts, but both methods produce equivalent descriptions of the electron distribution in a molecule.
In order to better understand the bent bond model let's first consider its application to cyclopropane and then move to ethylene.
In cyclopropane it has been found that significant electron density lies off the internuclear axis, rather than along the axis.
(image source)
Further, the $\ce{H-C-H}$ angle in cyclopropane has been measured and found to be 114 degrees. From this, and using Coulson's Theorem $$\ce{1+\lambda^2cos(114)=0}$$ where $\ce{\lambda^2}$ represents the hybridization index of the bond, the $\ce{C-H}$ bonds in cyclopropane can be deduced to be $\ce{sp^{2.46}}$ hybridized. Now, using the equation $$\ce{\frac{2}{1+\lambda^_{C-H}^2}+\frac{2}{1+\lambda_{C-C}^2}=1}$$ (which says that summing the "s" character in all bonds at a given carbon must total to 1) we find that $\ce{\lambda_{c-c}^2~=~}$$\mathrm{3.74}$, or the $\ce{C-C}$ bond is $\ce{sp^{3.74}}$ hybridized.
Pictorially, the bonds look as follows. They are bent (hence the strain in cyclopropane) and concentrate their electron density off of the internuclear axis as experimentally observed.
We can apply these same concepts to the description of ethylene. Using the known $\ce{H-C-H}$ bond angle of 117 degrees, Coulson's theorem and assuming that we have one p-orbital on each carbon (Hückel's $\sigma – \pi$ formalism), we would conclude that the carbon orbitals involved in the $\ce{C-H}$ bond are $\ce{sp^{2.2}}$ hybridized and the one carbon orbital involved in the $\ce{C-C}$ sigma bond is $\ce{sp^{1.7}}$ hybridized (plus there is the unhybridized p-orbital). This is the "$\ce{sp^2}$" description we see in most textbooks.
Alternately, if we only change our assumption of one pi bond and one sigma bond between the two carbon atoms to two equivalent sigma bonds (Pauling's bent bond formalism), we would find that the carbon orbitals involved in the $\ce{C-H}$ bond are again $\ce{sp^{2.2}}$ hybridized, but the two equivalent $\ce{C-C}$ sigma bonds are $\ce{sp^{4.3}}$ hybridized. We have constructed a two-membered ring cycloalkane analogue!
Hybridization is just a mathematical construct, another model to help us understand and describe molecular bonding. As shown in the ethylene example above, we can mix carbon's atomic s- and p-orbitals in different ways to describe ethylene. However, the two different ways we mixed these orbitals
one s-orbital plus two p-orbitals plus one unmixed p-orbital (Hückel's $\sigma – \pi$), or
two s-orbitals plus 2 p-orbitals (Pauling's bent bond)
must lead to equivalent electronic descriptions of ethylene. As noted in the Wikipedia article (first link at the beginning of this answer),
There is still some debate as to which of the two representations is
better,[10] although both models are mathematically equivalent. In a
1996 review, Kenneth B. Wiberg concluded that "although a conclusive
statement cannot be made on the basis of the currently available
information, it seems likely that we can continue to consider the σ/π
and bent-bond descriptions of ethylene to be equivalent.2 Ian
Fleming goes further in a 2010 textbook, noting that "the overall
distribution of electrons […] is exactly the same" in the two
models.[11]
The bent bond model has the advantage of explaining both cyclopropane (and other strained molecules) as well as olefins. The strain in cyclopropane and ethylene (e.g. heats of hydrogenation) also make intuitive sense with the bent bond model where the term "bent bond" conjures up an image of strain.
So, bent bonds are a mix of both sigma and pi properties and multiple bent bonds can be used to describe the bonding between adjacent atoms.
I'm not sure if this answering attempt is correct in the light of Mithoron's and ron's comments on your question, but this is the way I learnt it, so if this is wrong I will at least learn something, too.
We all know what s-, p- and d-orbitals look like, but what is the significance, and why do these orbitals preferentially form $\sigma$, $\pi$ and $\delta$ bonds, respectively?
Mathematically spoken, orbitals are functions of the hydrogen atom that solve the Schrödinger equation. The model in question is a non-rigid rotor,* i.e. the rotor's axis is not fixed in any spatial direction (the electron can rotate freely around the nucleus). For solving this equation, it is helpful to use polar coordinates $(r, \varphi, \theta)$, mainly because the solution can be split into a radial factor (dependent only on $r$) and angular factors (dependent on $\varphi$ and $\theta$).
$$\Psi (r, \varphi, \theta) = R(r) \cdot Y(\varphi, \theta)$$
$R(r)$ can be thought of giving an orbital its extension into space while $Y(\varphi, \theta)$ gives it its shape. Both functions depend heavily on quantum numbers: $R(r)$ does so for $n$ and $l$ while $Y(\varphi, \theta)$ depends on $l$ and $m_l$. For the simplest case ($l = 0; m_l = 0$, s-orbital), $Y (\varphi, \theta)$ degenerates to a simple constant, meaning that the orbital will have a totally symmetrical spherical shape. $l = 1$, (the p-orbital) while loosing spherical symmetry, still keeps total symmetry with respect to one axis, i.e. every slice you take through that orbital perpendicular to the axis of symmetry will be a circle. Higher quantum numbers lose more symmetry but it's not always as easily visualised, so I'll stick with these.
But you were talking about bonds, where do they come into play? Well, bonds also have a symmetry, but they also have an axis instead of a nucleus, so their symmetry will be reduced per se. The simplest symmetry along a bond axis is total rotational symmetry around the bonds axis. I hope you see the similarity between the s-orbital (total symmetry around a central point) and a $\sigma$ bond (total symmetry around the bond's central axis). Similarly, a $\pi$ bond will always have one degree of symmetry less, which turns out to mean 'having a plane of symmetry that includes the bond axis'. And a $\delta$ bond will have two planes of symmetry — yet another degree of symmetry less.
According to this definition, an orbital that can take part in a $\sigma$ bond needs to have full rotational symmetry along the bond's axis. That means, that there is only one, at most two orbitals that fulfil the criterion (but if there are two, one is going to be an unmodified s-orbital and likely not take part in bonding at all). Therefore, only one $\sigma$ bond would be possible between two atoms.
Writing this up, I remembered the 'banana bonds' that were introduced to us to explain the extremely small ($60°$) bond angles in $\ce{P4}$. I would need to go back, recheck and rethink what I would think of those and if I would treat them as exceptions of this 'rule' or simply as special cases that need additional information to be discussed. They certainly deserve consideration, as they are, de facto $\sigma$ bonds from the way they look, even though they do bend.
An interesting comment was left on the question pointing to sextuple bonds. I didn't know bonds of that order existed; my knowledge was stuck at 4. For a quadruple bond, possible between certain transition metals such as in $\ce{[Re2Cl8]^2-}$, four of the five d-orbitals form a bond to the other metal; one being $\sigma$, two being $\pi$ and a fourth one of $\delta$ type (to planes of symmetry). Extending that to a quintuple bond by adding a second $\delta$ layer with the last remaining pair of d-orbitals isn't hard.
The sextuple bond – eg $\ce{Mo2}$ — derives from an additional $\sigma$ bond between the s-orbitals of the higher shell. You thereby solve a problem you would otherwise have: The $4\mathrm{d}_{z^2}$ orbital can take part in $\sigma$ bonding along the $z$-axis; and the higher $5\mathrm{s}$ orbital is more diffuse, extends further into space and therefore is still able to form a contact to the neighbouring atom's counterpart. Because it is more or less a sphere, it can only form $\sigma$ bonds.
* I don't think this is the model's correct name. In my German quantum chemistry class, the rigid rotor was a raumstarrer Rotator and thus the model here was a raumfreier Rotator. Somebody who might know the proper name please comment (or edit).
The main thing that determines the shape of orbitals is that they must have zero net overlap with all other orbitals (i.e. that they are orthogonal). It's quite hard to construct two orthogonal $\sigma$-bonding MOs connecting the same two atoms.
$\sigma$-bonding MOs are typically $sp^x$ hybridized orbitals with maximum overlap along the line connecting two atomic centers
It's hard to construct another bonding MO with maximum overlap along the same line that is orthogonal to this MO. To stay orthogonal we have introduce a node and the next $\sigma$ MO becomes anti-bonding
The usual way to get another, orthogonal, bonding MO is a $\pi$-bond.
If you have valence $d$-electrons available then it is possible to construct another, orthogonal, $\sigma$-orbital
However, it is still rare to have two $\sigma$ bonding MOs because transition metals tend to loose the $s$-electrons needed for one of the $\sigma$-MOs very readily. So double $\sigma$-bonding tend to be only observed in special cases such as gas phase $\ce{Mo2}$.
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# Understanding the concept of optimization
Optimization is the process of finding the best solution to a problem. In many real-world scenarios, we are faced with the task of maximizing or minimizing a certain objective while satisfying a set of constraints. Optimization algorithms help us find the optimal values of the variables that will achieve this objective.
The concept of optimization can be applied to various fields such as engineering, economics, machine learning, and many others. It allows us to make informed decisions and improve the performance of systems.
To understand optimization better, let's consider a simple example. Imagine you are a delivery driver and you want to find the shortest route to deliver packages to multiple locations. The objective is to minimize the total distance traveled. The variables in this case would be the order in which you visit each location.
By using optimization techniques, you can find the optimal order of locations that will minimize your travel distance. This is just one example of how optimization can be applied to solve real-world problems.
Let's consider another example in the field of finance. Suppose you have a certain amount of money that you want to invest in different stocks. Your objective is to maximize your return on investment while considering the risk associated with each stock. The variables in this case would be the allocation of your money to each stock.
Using optimization algorithms, you can find the optimal allocation of your money that will maximize your return while managing the risk. This can help you make informed investment decisions.
## Exercise
Think of a real-world scenario where optimization can be applied. Describe the objective, variables, and constraints involved in that scenario.
### Solution
One possible scenario is workforce scheduling in a call center. The objective is to minimize the waiting time for customers while ensuring that there are enough agents available to handle the calls. The variables would be the shift schedules of the agents, and the constraints would include the number of agents available at each time slot and the maximum working hours for each agent.
# Types of optimization problems
Optimization problems can be classified into different types based on their characteristics and constraints. Understanding these types can help us choose the appropriate optimization algorithm for a given problem.
The main types of optimization problems include:
1. Linear Programming: In linear programming, the objective function and constraints are linear. The variables are continuous and the feasible region forms a convex polyhedron. Linear programming is widely used in various fields such as operations research, economics, and engineering.
2. Nonlinear Programming: Nonlinear programming deals with optimization problems where the objective function or constraints are nonlinear. The variables can be continuous or discrete, and the feasible region can be nonconvex. Nonlinear programming is used in fields such as engineering design, finance, and data analysis.
3. Integer Programming: Integer programming involves optimization problems where some or all of the variables are restricted to integer values. This adds an additional level of complexity to the problem, as the feasible region becomes discrete. Integer programming is used in various applications such as production planning, network optimization, and scheduling.
4. Quadratic Programming: Quadratic programming deals with optimization problems where the objective function is quadratic and the constraints can be linear or quadratic. Quadratic programming is commonly used in fields such as machine learning, portfolio optimization, and control systems.
5. Convex Optimization: Convex optimization involves optimization problems where the objective function and constraints are convex. Convex optimization has the advantage of having efficient algorithms and global optimality guarantees. It is used in various applications such as signal processing, image reconstruction, and machine learning.
6. Combinatorial Optimization: Combinatorial optimization deals with optimization problems where the feasible solutions are discrete and can be represented as combinations or permutations. Combinatorial optimization is used in fields such as network optimization, scheduling, and logistics.
Each type of optimization problem has its own characteristics and requires specific algorithms and techniques for solving. It is important to understand the nature of the problem and choose the appropriate optimization approach accordingly.
## Exercise
Match each type of optimization problem with its description:
- Linear Programming
- Nonlinear Programming
- Integer Programming
- Quadratic Programming
- Convex Optimization
- Combinatorial Optimization
Descriptions:
1. Optimization problems with linear objective function and constraints.
2. Optimization problems with discrete feasible solutions.
3. Optimization problems with quadratic objective function and linear or quadratic constraints.
4. Optimization problems with convex objective function and constraints.
5. Optimization problems with nonlinear objective function or constraints.
6. Optimization problems with integer variables.
### Solution
- Linear Programming: Optimization problems with linear objective function and constraints.
- Nonlinear Programming: Optimization problems with nonlinear objective function or constraints.
- Integer Programming: Optimization problems with integer variables.
- Quadratic Programming: Optimization problems with quadratic objective function and linear or quadratic constraints.
- Convex Optimization: Optimization problems with convex objective function and constraints.
- Combinatorial Optimization: Optimization problems with discrete feasible solutions.
# Overview of gradient descent algorithm
The gradient descent algorithm is a widely used optimization technique that is particularly effective for solving problems with large amounts of data and complex models. It is an iterative algorithm that aims to find the minimum of a function by iteratively updating the parameters in the direction of the negative gradient.
The basic idea behind gradient descent is to start with an initial set of parameter values and then update these values in small steps, guided by the gradient of the function. The gradient is a vector that points in the direction of the steepest ascent of the function. By taking steps in the opposite direction of the gradient, we can gradually move towards the minimum of the function.
The update rule for gradient descent is as follows:
$$
\theta_{i+1} = \theta_i - \alpha \nabla f(\theta_i)
$$
where $\theta_i$ is the current set of parameter values, $\alpha$ is the learning rate (which determines the size of the steps taken), and $\nabla f(\theta_i)$ is the gradient of the function evaluated at $\theta_i$.
The learning rate is an important parameter in gradient descent. If the learning rate is too large, the algorithm may overshoot the minimum and fail to converge. If the learning rate is too small, the algorithm may take a long time to converge. Finding the right learning rate is often a matter of trial and error.
Gradient descent can be used for both convex and non-convex optimization problems. In convex problems, gradient descent is guaranteed to converge to the global minimum. In non-convex problems, gradient descent may converge to a local minimum, depending on the starting point and the shape of the function.
Let's consider a simple example to illustrate how gradient descent works. Suppose we have a function $f(x) = x^2$. Our goal is to find the minimum of this function using gradient descent.
We start with an initial value of $x = 3$. The gradient of the function at this point is $\nabla f(x) = 2x$. Using the update rule, we can calculate the next value of $x$ as follows:
$$
x_{i+1} = x_i - \alpha \nabla f(x_i) = 3 - \alpha (2 \cdot 3) = 3 - 6\alpha
$$
We repeat this process for a certain number of iterations or until convergence is achieved.
## Exercise
Consider the function $f(x) = x^3 - 2x^2 + 3x - 1$. Use the gradient descent algorithm to find the minimum of this function. Start with an initial value of $x = 2$ and a learning rate of $\alpha = 0.1$. Perform 5 iterations.
### Solution
Iteration 1:
$$
x_1 = 2 - 0.1 \cdot (3 \cdot 2^2 - 2 \cdot 2 + 3) = 1.6
$$
Iteration 2:
$$
x_2 = 1.6 - 0.1 \cdot (3 \cdot 1.6^2 - 2 \cdot 1.6 + 3) = 1.408
$$
Iteration 3:
$$
x_3 = 1.408 - 0.1 \cdot (3 \cdot 1.408^2 - 2 \cdot 1.408 + 3) = 1.32704
$$
Iteration 4:
$$
x_4 = 1.32704 - 0.1 \cdot (3 \cdot 1.32704^2 - 2 \cdot 1.32704 + 3) = 1.3032192
$$
Iteration 5:
$$
x_5 = 1.3032192 - 0.1 \cdot (3 \cdot 1.3032192^2 - 2 \cdot 1.3032192 + 3) = 1.297955328
$$
# Calculating gradients and updating parameters
To perform gradient descent, we need to calculate the gradient of the function with respect to the parameters. The gradient is a vector that contains the partial derivatives of the function with respect to each parameter.
For example, let's say we have a function $f(x, y) = x^2 + 2y$. To calculate the gradient, we need to take the partial derivatives of $f$ with respect to $x$ and $y$. The partial derivative of $f$ with respect to $x$ is denoted as $\frac{\partial f}{\partial x}$, and the partial derivative of $f$ with respect to $y$ is denoted as $\frac{\partial f}{\partial y}$.
In this case, the partial derivatives are:
$$
\frac{\partial f}{\partial x} = 2x
$$
$$
\frac{\partial f}{\partial y} = 2
$$
Once we have the gradient, we can update the parameters using the update rule of gradient descent:
$$
\theta_{i+1} = \theta_i - \alpha \nabla f(\theta_i)
$$
where $\theta_i$ is the current set of parameter values, $\alpha$ is the learning rate, and $\nabla f(\theta_i)$ is the gradient of the function evaluated at $\theta_i$.
Let's continue with the example of the function $f(x, y) = x^2 + 2y$. Suppose we start with an initial set of parameter values $\theta_0 = (1, 2)$ and a learning rate of $\alpha = 0.1$. We can calculate the gradient at $\theta_0$ as follows:
$$
\nabla f(\theta_0) = \left(\frac{\partial f}{\partial x}(\theta_0), \frac{\partial f}{\partial y}(\theta_0)\right) = (2 \cdot 1, 2) = (2, 2)
$$
Using the update rule, we can calculate the next set of parameter values $\theta_1$ as follows:
$$
\theta_1 = \theta_0 - \alpha \nabla f(\theta_0) = (1, 2) - 0.1 \cdot (2, 2) = (0.8, 1.8)
$$
We repeat this process for a certain number of iterations or until convergence is achieved.
## Exercise
Consider the function $f(x, y) = x^3 + 3xy + y^2$. Calculate the gradient of $f$ with respect to $x$ and $y$. Use the gradient descent algorithm to update the parameters for 3 iterations, starting with an initial set of parameter values $\theta_0 = (1, 2)$ and a learning rate of $\alpha = 0.01$.
### Solution
The partial derivatives of $f$ with respect to $x$ and $y$ are:
$$
\frac{\partial f}{\partial x} = 3x^2 + 3y
$$
$$
\frac{\partial f}{\partial y} = 3x + 2y
$$
Using the update rule, we can calculate the next set of parameter values $\theta_1$ as follows:
$$
\theta_1 = \theta_0 - \alpha \nabla f(\theta_0)
$$
$$
\theta_2 = \theta_1 - \alpha \nabla f(\theta_1)
$$
$$
\theta_3 = \theta_2 - \alpha \nabla f(\theta_2)
$$
# Convergence criteria for gradient descent
Convergence criteria are used to determine when to stop the gradient descent algorithm. The algorithm continues to update the parameters until a certain condition is met. There are several common convergence criteria for gradient descent:
1. **Maximum number of iterations**: The algorithm stops after a specified maximum number of iterations. This is useful when we want to limit the computation time or prevent the algorithm from running indefinitely.
2. **Minimum change in parameters**: The algorithm stops when the change in the parameters between iterations falls below a certain threshold. This indicates that the algorithm has converged to a stable solution.
3. **Minimum change in the cost function**: The algorithm stops when the change in the cost function between iterations falls below a certain threshold. This indicates that the algorithm has reached a minimum of the cost function.
4. **Target value of the cost function**: The algorithm stops when the value of the cost function falls below a specified target value. This indicates that the algorithm has reached a satisfactory solution.
It's important to note that the convergence criteria may vary depending on the specific problem and the desired level of accuracy. It's also common to use a combination of criteria to ensure convergence.
Let's consider the example of optimizing a linear regression model using gradient descent. We can use the mean squared error (MSE) as the cost function. The convergence criteria can be set as follows:
1. **Maximum number of iterations**: Set a maximum number of iterations, such as 1000, to limit the computation time.
2. **Minimum change in parameters**: Set a threshold, such as 0.001, for the change in the parameters between iterations. If the change falls below this threshold, the algorithm stops.
3. **Minimum change in the cost function**: Set a threshold, such as 0.01, for the change in the cost function between iterations. If the change falls below this threshold, the algorithm stops.
4. **Target value of the cost function**: Set a target value, such as 0.1, for the cost function. If the value falls below this target, the algorithm stops.
By using a combination of these convergence criteria, we can ensure that the gradient descent algorithm stops when it has reached a satisfactory solution.
## Exercise
Consider the following convergence criteria for gradient descent:
1. Maximum number of iterations: 100
2. Minimum change in parameters: 0.001
3. Minimum change in the cost function: 0.01
4. Target value of the cost function: 0.1
For each convergence criterion, explain when it would be appropriate to use it.
### Solution
1. Maximum number of iterations: This criterion is useful when we want to limit the computation time or prevent the algorithm from running indefinitely. It ensures that the algorithm stops after a specified maximum number of iterations, regardless of the convergence.
2. Minimum change in parameters: This criterion is useful when we want to stop the algorithm when the change in the parameters between iterations falls below a certain threshold. It indicates that the algorithm has converged to a stable solution.
3. Minimum change in the cost function: This criterion is useful when we want to stop the algorithm when the change in the cost function between iterations falls below a certain threshold. It indicates that the algorithm has reached a minimum of the cost function.
4. Target value of the cost function: This criterion is useful when we want to stop the algorithm when the value of the cost function falls below a specified target value. It indicates that the algorithm has reached a satisfactory solution.
# Understanding the trade-off between speed and accuracy
When using optimization algorithms, it's important to understand the trade-off between speed and accuracy. In general, increasing the accuracy of the optimization algorithm requires more computational resources and time. On the other hand, increasing the speed of the algorithm may result in sacrificing some accuracy.
The trade-off between speed and accuracy depends on the specific problem and the desired level of optimization. For some problems, a fast but less accurate solution may be sufficient. For example, in real-time applications where decisions need to be made quickly, a fast algorithm that provides a reasonably good solution may be preferred.
However, in other cases, such as scientific research or critical applications, a more accurate solution may be necessary, even if it takes longer to compute. In these situations, it may be worth investing more computational resources to ensure a higher level of accuracy.
It's important to consider the constraints and requirements of the problem at hand when deciding on the trade-off between speed and accuracy. By understanding this trade-off, you can make informed decisions about the optimization algorithm to use and the level of accuracy to aim for.
Let's consider the example of training a machine learning model. In this case, accuracy is typically a key consideration, as the goal is to minimize the difference between the predicted and actual values. However, training a model with high accuracy often requires more computational resources and time.
If the model is being trained for a real-time application, such as autonomous driving, a fast but reasonably accurate model may be sufficient. On the other hand, if the model is being used for scientific research or medical diagnosis, a higher level of accuracy may be necessary, even if it takes longer to train the model.
Understanding the trade-off between speed and accuracy allows us to make informed decisions about the resources and time to allocate for optimization algorithms.
## Exercise
Consider a scenario where you need to optimize a manufacturing process. The process involves multiple parameters that can be adjusted to improve the quality of the final product. You have limited computational resources and time.
What factors would you consider when deciding on the trade-off between speed and accuracy in this scenario?
### Solution
When deciding on the trade-off between speed and accuracy in this scenario, the following factors should be considered:
1. Importance of accuracy: Consider the impact of the optimization on the quality of the final product. If a high level of accuracy is critical for the success of the manufacturing process, it may be necessary to allocate more computational resources and time to achieve the desired level of accuracy.
2. Time constraints: Evaluate the time available for the optimization process. If there are strict deadlines or time-sensitive requirements, it may be necessary to prioritize speed over accuracy to meet the timeline.
3. Computational resources: Assess the available computational resources. If there are limitations on the computing power or budget, it may be necessary to find a balance between speed and accuracy that can be achieved within the available resources.
4. Iterative improvements: Consider the potential for iterative improvements. If the optimization process can be performed in multiple iterations, it may be possible to start with a faster algorithm to achieve an initial level of optimization and then refine the results with a more accurate algorithm in subsequent iterations.
By considering these factors, you can make an informed decision on the trade-off between speed and accuracy in the manufacturing process optimization.
# Stochastic gradient descent and its applications
Stochastic gradient descent (SGD) is a variation of the gradient descent algorithm that is commonly used in machine learning and optimization problems. While the standard gradient descent algorithm updates the model parameters using the average of the gradients computed over the entire dataset, SGD updates the parameters using the gradient computed on a single randomly selected data point or a small batch of data points.
The main advantage of SGD is its computational efficiency. By using a random subset of the data, SGD can perform updates more frequently and converge faster than the standard gradient descent algorithm. This makes SGD particularly useful when working with large datasets or in real-time applications where computational resources are limited.
In addition to its computational efficiency, SGD also has the advantage of escaping from local minima. The random selection of data points introduces noise into the optimization process, which can help the algorithm explore different regions of the parameter space and avoid getting stuck in local optima.
SGD is widely used in various machine learning algorithms, including linear regression, logistic regression, and neural networks. It has also been applied to optimization problems in other domains, such as image processing and natural language processing.
Let's consider the example of training a neural network for image classification using SGD. In this case, the neural network model has millions of parameters that need to be optimized. Instead of computing the gradient over the entire dataset, which can be computationally expensive, SGD randomly selects a small batch of images and computes the gradient based on the predictions and the corresponding labels of those images.
The model parameters are then updated using the computed gradient, and this process is repeated for multiple iterations until the model converges. By using SGD, the training process can be significantly faster compared to using the standard gradient descent algorithm.
## Exercise
Consider a scenario where you are training a logistic regression model using SGD. The dataset contains 1000 samples and 10 features. You decide to use a batch size of 100 for SGD.
1. How many iterations will it take to process the entire dataset once?
2. How many updates to the model parameters will be performed in total?
### Solution
1. It will take 10 iterations to process the entire dataset once. Since the batch size is 100, each iteration processes 100 samples, and there are 1000 samples in total.
2. There will be 10 updates to the model parameters in total. Each iteration performs one update, and there are 10 iterations in total.
# Regularization techniques for better optimization
Regularization is a technique used in optimization to prevent overfitting and improve the generalization performance of a model. Overfitting occurs when a model becomes too complex and starts to fit the noise in the training data, resulting in poor performance on unseen data.
There are several regularization techniques that can be used to control the complexity of a model and prevent overfitting. Two commonly used regularization techniques are L1 regularization and L2 regularization.
L1 regularization, also known as Lasso regularization, adds a penalty term to the loss function that encourages the model to have sparse weights. This means that some of the weights will be set to zero, effectively selecting a subset of the features that are most relevant for the task at hand. L1 regularization can be particularly useful for feature selection and reducing the dimensionality of the input space.
L2 regularization, also known as Ridge regularization, adds a penalty term to the loss function that encourages the model to have small weights. This helps to prevent the model from becoming too sensitive to the training data and improves its generalization performance. L2 regularization can be seen as a form of smoothing that reduces the impact of individual data points on the model's parameters.
Both L1 and L2 regularization can be used together, resulting in a technique called elastic net regularization. Elastic net regularization combines the sparsity-inducing property of L1 regularization with the smoothing property of L2 regularization, providing a balance between feature selection and parameter shrinkage.
Regularization techniques can be applied to various optimization algorithms, including gradient descent. By adding a regularization term to the loss function, the optimization process is guided towards finding a solution that not only fits the training data well but also has good generalization performance on unseen data.
Let's consider the example of training a linear regression model using gradient descent with L2 regularization. The goal is to predict the price of a house based on its features, such as the number of bedrooms, the size of the living area, and the location. By adding an L2 regularization term to the loss function, the model is encouraged to have small weights, which helps to prevent overfitting and improve its ability to generalize to new houses.
The regularization term is controlled by a hyperparameter called the regularization parameter, which determines the strength of the regularization. A higher value of the regularization parameter results in stronger regularization, while a lower value allows the model to have larger weights.
## Exercise
Consider a scenario where you are training a logistic regression model for binary classification. You want to prevent overfitting and improve the generalization performance of the model.
1. Which regularization technique would you use: L1 regularization, L2 regularization, or elastic net regularization?
2. What effect does increasing the regularization parameter have on the model's weights?
### Solution
1. You can use either L1 regularization, L2 regularization, or elastic net regularization to prevent overfitting and improve the generalization performance of the model. The choice depends on the specific requirements of the problem and the desired trade-off between feature selection and parameter shrinkage.
2. Increasing the regularization parameter has the effect of shrinking the model's weights. This means that the weights become smaller, reducing the impact of individual features on the model's predictions. As a result, the model becomes less sensitive to the training data and more likely to generalize well to unseen data. However, increasing the regularization parameter too much can lead to underfitting, where the model is too simple and fails to capture the underlying patterns in the data.
# Optimization techniques for non-convex problems
So far, we have discussed optimization techniques for convex problems, where the objective function and constraints are convex. However, many real-world optimization problems are non-convex, meaning that the objective function or constraints are not convex.
Optimizing non-convex problems is more challenging because there may be multiple local optima, and it is difficult to guarantee finding the global optimum. Nevertheless, there are several techniques that can be used to tackle non-convex optimization problems.
One approach is to use heuristic algorithms, such as genetic algorithms or simulated annealing. These algorithms are inspired by natural processes and can explore the search space more extensively, potentially finding better solutions. However, they do not provide guarantees of finding the global optimum.
Another approach is to use gradient-based optimization algorithms, such as gradient descent, even though they are designed for convex problems. In practice, gradient descent can still be effective for non-convex problems, especially if combined with random restarts or other techniques to escape local optima.
Let's consider the example of training a neural network for image classification. The objective is to minimize the loss function, which measures the difference between the predicted and actual labels. The neural network has multiple layers with non-linear activation functions, making the optimization problem non-convex.
Despite the non-convexity of the problem, gradient descent can still be used to train the neural network. The gradients are computed using backpropagation, and the weights are updated iteratively to minimize the loss function. Although gradient descent may converge to a local optimum, it can still find good solutions in practice.
## Exercise
Consider a scenario where you are optimizing a non-convex function using gradient descent. You have initialized the algorithm with a random starting point and are updating the parameters iteratively.
1. What is the risk of getting stuck in a local optimum?
2. How can you mitigate the risk of getting stuck in a local optimum?
### Solution
1. The risk of getting stuck in a local optimum is high in non-convex optimization problems. This is because the objective function may have multiple local optima, and gradient descent can converge to one of them instead of the global optimum.
2. To mitigate the risk of getting stuck in a local optimum, you can use random restarts. This involves running gradient descent multiple times with different initializations and selecting the solution with the lowest objective function value. Random restarts increase the chances of finding a better solution by exploring different regions of the search space. Another approach is to use more advanced optimization algorithms, such as genetic algorithms or simulated annealing, which are designed to handle non-convex problems. These algorithms can explore the search space more extensively and potentially find better solutions.
# Parallelization and distributed computing for faster optimization
Optimization problems can often be computationally intensive, especially when dealing with large datasets or complex models. In such cases, it can be beneficial to leverage parallelization and distributed computing techniques to speed up the optimization process.
Parallelization involves dividing the computation into smaller tasks that can be executed simultaneously on multiple processors or cores. This can significantly reduce the overall computation time, especially for algorithms that involve repetitive calculations, such as gradient descent.
Distributed computing takes parallelization a step further by distributing the computation across multiple machines or nodes in a network. This allows for even greater scalability and can handle larger datasets or more complex models.
Let's consider the example of training a machine learning model on a large dataset. The training process involves iteratively updating the model parameters using gradient descent. By parallelizing the computation, we can divide the dataset into smaller subsets and assign each subset to a different processor or core. Each processor can then independently calculate the gradients for its subset and update the model parameters. This can significantly speed up the training process, especially for models with millions or billions of data points.
Distributed computing can further enhance the performance by distributing the computation across multiple machines. Each machine can process a subset of the data and communicate with other machines to exchange information and synchronize the model updates. This allows for even faster training times and the ability to handle even larger datasets.
## Exercise
Consider a scenario where you are training a deep neural network on a large dataset using gradient descent. The training process is computationally intensive and takes a long time to complete. You want to speed up the process by leveraging parallelization and distributed computing techniques.
1. How can you parallelize the computation to speed up the training process?
2. How can you further enhance the performance by using distributed computing?
### Solution
1. To parallelize the computation, you can divide the dataset into smaller subsets and assign each subset to a different processor or core. Each processor can independently calculate the gradients for its subset and update the model parameters. This can significantly reduce the overall computation time and speed up the training process.
2. To further enhance the performance, you can use distributed computing techniques. This involves distributing the computation across multiple machines or nodes in a network. Each machine can process a subset of the data and communicate with other machines to exchange information and synchronize the model updates. This allows for even faster training times and the ability to handle larger datasets.
# Real-world applications of gradient descent
1. Machine Learning: Gradient descent is widely used in machine learning for training models. It is used to minimize the loss function by iteratively updating the model parameters. This allows the model to learn from the data and make accurate predictions. Gradient descent is used in popular machine learning algorithms such as linear regression, logistic regression, and neural networks.
2. Image and Signal Processing: Gradient descent is used in image and signal processing applications such as image denoising, image reconstruction, and audio signal processing. It is used to optimize the parameters of algorithms that enhance the quality of images and signals. For example, in image denoising, gradient descent can be used to minimize the difference between the noisy image and the denoised image.
3. Natural Language Processing: Gradient descent is used in natural language processing tasks such as language modeling, machine translation, and sentiment analysis. It is used to optimize the parameters of models that generate or classify text. For example, in machine translation, gradient descent can be used to minimize the difference between the predicted translation and the reference translation.
4. Recommender Systems: Gradient descent is used in recommender systems to optimize the parameters of models that predict user preferences. It is used to minimize the difference between the predicted ratings and the actual ratings given by users. This allows the system to make personalized recommendations to users based on their preferences.
5. Financial Modeling: Gradient descent is used in financial modeling to optimize the parameters of models that predict stock prices, asset returns, and other financial variables. It is used to minimize the difference between the predicted values and the actual values. This allows financial analysts and traders to make informed decisions based on the predictions.
6. Robotics: Gradient descent is used in robotics for motion planning and control. It is used to optimize the parameters of control policies that allow robots to perform tasks such as grasping objects, navigating obstacles, and manipulating objects. Gradient descent is used to minimize the difference between the desired robot behavior and the actual robot behavior.
These are just a few examples of the many real-world applications of gradient descent. The algorithm's ability to optimize complex models and find optimal solutions makes it a valuable tool in various fields. By understanding the concepts and techniques of gradient descent, you can apply it to solve optimization problems in your own domain.
## Exercise
Think of a real-world application in your field or area of interest where gradient descent can be used to solve an optimization problem. Describe the problem and how gradient descent can be applied to find an optimal solution.
### Solution
In the field of renewable energy, gradient descent can be used to optimize the placement and configuration of wind turbines in a wind farm. The goal is to maximize the energy output of the wind farm by finding the optimal positions and orientations of the turbines.
To solve this optimization problem, we can define a cost function that represents the energy output of the wind farm as a function of the positions and orientations of the turbines. The cost function takes into account factors such as wind speed, wind direction, and the wake effects of neighboring turbines.
By applying gradient descent, we can iteratively update the positions and orientations of the turbines to minimize the cost function. This involves calculating the gradients of the cost function with respect to the turbine parameters and updating the parameters in the direction of steepest descent.
The optimization process continues until a convergence criterion is met, such as reaching a certain level of energy output or a maximum number of iterations. The result is an optimized configuration of wind turbines that maximizes the energy output of the wind farm.
By using gradient descent in this way, we can design more efficient and cost-effective wind farms that harness the power of the wind to generate renewable energy. | Textbooks |
Seasonal energy exchange in sea ice retreat regions contributes to differences in projected Arctic warming
Robyn C. Boeke1 &
Patrick C. Taylor2
Nature Communications volume 9, Article number: 5017 (2018) Cite this article
238 Altmetric
Projection and prediction
Rapid and, in many cases, unprecedented Arctic climate changes are having far-reaching impacts on natural and human systems. Despite state-of-the-art climate models capturing the rapid nature of Arctic climate change, termed Arctic amplification, they significantly disagree on its magnitude. Using a regional, process-oriented surface energy budget analysis, we argue that differences in seasonal energy exchanges in sea ice retreat regions via increased absorption and storage of sunlight in summer and increased upward surface turbulent fluxes in fall/winter contribute to the inter-model spread. Models able to more widely disperse energy drawn from the surface in sea ice retreat regions warm more, suggesting that differences in the local Arctic atmospheric circulation response contribute to the inter-model spread. We find that the principle mechanisms driving the inter-model spread in Arctic amplification operate locally on regional scales, requiring an improved understanding of atmosphere-ocean-sea ice interactions in sea ice retreat regions to reduce the spread.
The Arctic has warmed 2–3 times faster than globally-averaged warming, a phenomenon known as Arctic amplification (AA)1,2. AA is evident in surface temperature observations over the last century3,4, in model projections made starting in the 1970's5,6 and suggested by Arrhenius7 more than 100 years ago. Arctic climate change has global consequences by influencing glacial melt and sea level rise, permafrost thaw and the carbon cycle, atmospheric and oceanic circulations, and potentially extreme mid-latitude weather8,9. Because of the significant implications for the physical climate and human systems, accurate projections of AA are needed. Unfortunately, climate models project a wide range of possible futures for the Arctic—a larger inter-model spread than any other region. The 2 °C globally-averaged warming specified in the Paris Climate Agreement equates to an Arctic warming between 3 and 7 °C, according to Coupled Model Intercomparison 5 (CMIP510) models. Narrowing the inter-model spread requires an understanding of how Arctic feedback processes and their interactions shape the temperature response.
A collection of interacting processes support AA: sea ice loss and surface albedo feedback2,5,7,11,12,13,14,15,16, changes in longwave and/or temperature feedbacks17,18, cloud changes6,19,20,21, intraseasonal cycling of heat22,23, and poleward energy transport24,25,26,27. While the surface albedo feedback (SAF) has often been cited as the leading contributor to AA28, idealized climate simulations show that AA can occur in its absence29,30. Recent studies argue for a remote forcing of observed AA, whereby atmospheric heat transport into the Arctic from lower latitudes drives warming and thinner sea ice31,32, whereas others attribute observed AA to local mechanisms such as the surface albedo and evaporation feedbacks11. Our understanding of AA mechanisms has evolved significantly over the last decade, yet the relative importance of each feedback and its contributions to the inter-model spread in Arctic warming projections remains under debate.
This study offers a seasonal, process-oriented surface energy budget decomposition using the multi-model CMIP5 archive and methods introduced in Lu and Cai17. CMIP5 models show an increased inter-model spread in surface temperature and sea ice compared to CMIP333. Pithan and Mauritsen18 argue that temperature feedbacks explain these inter-model differences, however our analysis and interpretation differ. Previous studies have analyzed AA in CMIP5 models, yet generally lack the regional perspective required to isolate the energy exchanges that appear to regulate AA in observations34. Moreover, CMIP5 models exhibit large differences in the seasonal cycles of radiative fluxes, clouds, and turbulent fluxes15,35,36 and a seasonality in the inter-model spread in projected warming.
Our results outline the primary drivers of inter-model spread in AA found in CMIP5 models using a surface energy budget perspective highlighting the important contribution of seasonal energy exchange in sea ice retreat regions facilitated by ocean heat storage to the inter-model spread in Arctic warming. We argue that the atmospheric and ocean processes that modulate the seasonal energy exchange in sea ice retreat regions drive model differences in projected Arctic warming. The models that more effectively disperse energy drawn from the surface in sea ice retreat regions warm more. Therefore, reconciling the differences in AA projections requires constraining the representation of atmosphere-ocean-sea ice interactions in sea ice retreat regions.
Model projections of Arctic amplification
The current generation of CMIP5 climate models unanimously simulate AA in response to increasing CO2 (Fig. 1). Figure 1 illustrates AA using the normalized temperature change—hereafter, amplification factor—defined as the ratio between the 1° zonally-averaged temperature change to global temperature change. All models simulate surface-based warming—at least 1.5 times global average warming—extending into the lower troposphere poleward of 75°N (Fig. 1b). The solid black line in Fig. 1a represents the ensemble mean amplification factor, which exceeds 2.5 at the pole. CMIP5 models also simulate a similar seasonality of AA (Fig. 1a, inset) with minimum warming (amplification factor < 1) in summer and maximum warming in fall and winter (amplification factor >2). Despite unanimous agreement in the existence of AA, models disagree on the magnitude and spatial characteristics.
Nature of projected Arctic amplification. Surface temperature amplification for CMIP5 RCP8.5 models. Figure 1a shows the zonally-averaged temperature change normalized to global temperature change for each CMIP5 model (hereafter amplification factor). The solid black line in a represents the ensemble mean amplification factor; the black dashed line represents the zonally-averaged inter-model standard deviation normalized by the global inter-model standard deviation. The boxed region in a represents the Arctic domain 60–90°N used in this study. The inset in a shows the seasonal cycle of the amplification factor for the Arctic domain. b shows the vertical profile of ensemble mean temperature change by latitude
Inter-model differences in projected Arctic warming exceed those for any other latitude. The Arctic domain (defined as 60°–90°N) shows increasing model differences at more northerly latitudes, approaching an amplification factor spread of 1.6–3.6 (Fig. 1a). The dashed black line in Fig. 1a represents the ratio between the 1° zonally-averaged inter-model standard deviation in temperature change to the global inter-model standard deviation—a measure of inter-model spread. This ratio approaches four times the global average moving poleward. The seasonal cycle of AA (Fig. 1a, inset) shows the largest model spread in winter (amplification factor between 2 and 4) and smallest in summer (amplification factor between 0.5 and 1.25).
Arctic warming projections display stark regional contrasts (Fig. 2) where sea ice retreat regions exhibit the greatest warming and possess the largest model disagreement. Figure 2 shows the annual and seasonal ensemble mean temperature and sea ice concentration (SIC) changes and the corresponding standard deviations across the ensemble. The Barents/Kara and Beaufort/Chukchi Seas regions (see map, Supplementary Figure 1) exhibit the largest projected warming and sea ice loss; wintertime temperature projections exceed +20 K in both regions. The central Arctic Ocean shows the largest seasonality of the temperature response, warming by 15–20 K in fall/winter and less than 5 K in spring/summer. During all seasons, the smallest temperature changes between 1–4 K occur in the sea ice-free ocean regions of the North Atlantic, Norwegian Sea, and Davis Strait due to small surface energy budget changes consistent with reductions in ocean heat transport37.
Ensemble mean and standard deviation of Arctic climate change by 2100. CMIP5 RCP8.5 projected surface temperature change by 2100 for (a) annual mean, (b) winter (January–February), (c) sunlit season (March through September), and (d) autumn (October, November, and December) with the corresponding ensemble standard deviations (e–h). i–l show the ensemble mean projected changes in sea ice concentration (ΔSIC) for the seasons above, and m–p are the corresponding ensemble standard deviations in ΔSIC
The spatial warming pattern varies significantly between models; some show larger temperature increases over the central Arctic Ocean and others show the greatest warming in regions of the largest sea ice loss. In general, the inter-model spread is greatest in the sea ice retreat regions (Fig. 2e–h), particularly the Barents/Kara and Beaufort/Chukchi Seas.
Individual contributions to Arctic amplification
The presence of surface ice and the massive amounts of energy sequestered and released during water phase change indicates that the surface energy budget (SEB) is more relevant to the Arctic surface temperature than the top-of-atmosphere (TOA) energy budget. Previous work demonstrates that the surface and TOA perspectives can show opposite signs for the individual feedback contributions to warming, as Taylor et al.38 illustrated for clouds. Our SEB decomposition approach analyzes the energy flux changes and how they contribute to AA in each model (see Methods) by linearly decomposing the total surface temperature change into partial temperature contributions (PTCs). These terms include SAF, cloud radiative effect (CRE), changes in shortwave clear-sky radiation unrelated to SAF (SWCS), longwave clear-sky radiation (LWCS), ocean heat storage and transport (HSTOR), and surface turbulent fluxes (HFLUX, positive from atmosphere to ocean). In this decomposition, the LWCS term includes the effects of CO2, air temperature, and water vapor changes from both local and remote sources. We do not separate these effects as in previous work15,18 because biases due to the decomposition approach interfere with the assessment of inter-model differences. The HSTOR term represents the surface energy imbalance, including surface heat storage and ocean heat transport, computed as a residual (see Methods). Since the heat storage capacity of land is small compared to ocean, this term represents ocean heat content. Ocean heat transport, while potentially important to the inter-model spread, could not be assessed separately because few CMIP5 models archived the necessary output. This limits our ability to consider ocean heat storage and transport separately; however, previous work demonstrated that ocean heat storage in the mixed layer dominates changes in ocean heat transport in CMIP5 models over the 21st century15; thus, we consider HSTOR changes to be from ocean heat storage.
Annual mean PTCs and formulae are listed in Table 1; PTCs are additive, each representing the individual feedback contributions to the total ensemble mean Arctic temperature change of 7.38 K by 2100 in RCP8.5. At least three times larger than any other feedback, the strongest annual surface warming contributions are from LWCS changes, 7.27 K. The SAF feedback exhibits the second strongest ensemble average annual warming contribution (1.82 K) and the cooling influence of HFLUX (−1.67 K) the third largest contributor.
Table 1 Annual mean partial temperature contributions for the Arctic domain (60°–90°N) from CMIP5 RCP8.5
The strong seasonality of the Arctic SEB renders the annual mean picture incomplete. Important factors influencing AA—HSTOR, CRE, and HFLUX—exhibit strong seasonal variations. Applying the decomposition methodology to monthly SEB changes enables the assessment of seasonal energy exchanges (Fig. 3).
Seasonality of partial temperature contributions. Seasonal cycles of (a) surface albedo feedback, (b) cloud radiative effect, (c) ocean heat storage, and (d) surface turbulent flux partial temperature contributions averaged over the Arctic domain (60°–90° N). The gray shaded region denotes the ensemble mean (solid black line) +/− one standard deviation
The seasonality of the PTCs from CMIP5 remains unchanged from CMIP3 model analysis and is consistent with reanalysis17,34,39. Figure 3 shows a strong seasonality in SAF, CRE, HSTOR, and HFLUX. The SAF exhibits significant warming contributions peaking in June/July that mirror the increased HSTOR (Fig. 3a, c). While the SAF is partially offset by negative CRE PTCs (Fig. 3b), summer SAF increases the energy deposited into the ocean14,40,41. The negative HSTOR PTC (April–September) represents the accumulation of solar energy during spring and summer (more energy into the surface than out). Evidenced by the approximate equivalence of the combined SAF + CRE to HSTOR, changes in SAF and CRE primarily determine the summer HSTOR PTC. In fall/winter, HSTOR and HFLUX are approximately equivalent indicating that model HFLUX determines the fall/winter surface cooling rate change. From a process perspective, HFLUX is controlled by the air–sea temperature contrast, strongly influenced by both the presence of sea ice and atmospheric advection16,42. The negative fall/winter HFLUX PTCs in Fig. 4d are more than offset by positive fall/winter LWCS PTCs (not shown).
Spatial variablity of partial temperature contributions. Ensemble mean annually-averaged partial temperature contributions shown for (a) surface albedo feedback, (b) cloud radiative effect, (c) longwave clear-sky, (d) shortwave clear-sky, (e) ocean heat storage, and (f) surface turbulent flux. The corresponding inter-model standard deviation for each partial temperature contribution is shown for (g) surface albedo feedback, (h) cloud radiative effect, (i) longwave clear-sky, (j) shortwave clear-sky, (k) ocean heat storage, and (l) surface turbulent flux
Inter-model differences
The model spread between the contributions of individual SEB terms is represented by the inter-model standard deviation in Table 1. LWCS and HSTOR exhibit the largest annual mean, domain-averaged standard deviation, 1.4 K and 1.2 K, respectively. Considering percent differences however, the HSTOR inter-model spread is 400%. Clouds represent the second largest percentage spread at ~125%. Different from other feedbacks, the ±1 standard deviation bounds for these two feedbacks includes zero indicating that models disagree on the annual mean magnitude and sign of these contributions.
The strong seasonality of the PTCs renders the annual mean picture potentially misleading. Despite a modest annual mean inter-model spread (~40%), the SAF shows the widest inter-model range in any month, between +2 to +13 K in July (Fig. 3a). A strong correlation (R = −0.89) is found between annual mean albedo changes (peaking in summer) and AA (peaking in winter) even though the SAF seasonal cycle is out-of-phase with the maximum warming. The inter-model range in HSTOR (from −2 to −11 K) also maximizes in summer, aligning with the SAF inter-model spread. The inter-model spread in the CRE PTCs is the largest in fall (from +1 to +7 K) and smallest during summer. Models with the largest summer SAF do not exhibit the largest negative CRE PTC and therefore do not compensate for sea ice loss by simulating more reflective clouds. HFLUX exhibits its largest inter-model spread in winter (from −1 to −7 K) and smallest spread in summer (from −1 to +0.5 K). Seasonal cycles of the PTCs exhibit greater amplitude over ocean than land as in Laîné et al.15 (not shown), with HSTOR and HFLUX over the ocean having the largest inter-model range in winter of ~11 K between models. The SAF and HSTOR over the ocean show the largest inter-model range in summer of 18 K and 15 K, respectively.
Spatial variability of process contributions
Regional differences are important because spatial patterns of sea ice loss affect Arctic climate variability and the warming response43,44,45. For instance, the pattern of sea ice loss modulates the position of regional baroclinic zones that are favored regions of cyclogenesis46,47. Moreover, Overland et al.8 suggest that the spatial pattern of warming and sea ice loss alters the mid-latitude circulation. Several recent studies indicate that local feedbacks in sea ice-retreat areas accelerate warming16,48. Moreover, future Arctic climate change is likely to occur in response to episodic energy fluxes, via surface turbulent fluxes and poleward heat transport42,49,50. Since these factors act on a regional level, it is important to assess model feedbacks and differences spatially.
Figure 4 shows the ensemble mean annually-averaged PTCs (a–f) and the corresponding spatial inter-model standard deviation for each PTC (g–l). The largest model differences occur in seasonal sea ice regions where small differences in sea ice extent correspond to large differences in HSTOR (Fig. 4k) and HFLUX (Fig. 4l). The most striking feature is the contrast between the spatial variability of PTCs and their associated inter-model spread for radiative and non-radiative feedbacks, revealing in almost every case that the spatial variation of non-radiative feedbacks (HSTOR and HFLUX) is stronger than for radiative feedbacks (SAF, CRE, SWCS, and LWCS). The spatial contours in Fig. 4 show that non-radiative feedbacks vary strongly across models, in sea ice-retreat regions the HSTOR standard deviation exceeds 10.0 in the annual mean, more than seven times the domain average, and also approaches 10.0 in the Barents/Kara Seas, more than five times the domain average. Thus, the inter-model spread in Arctic warming is strongly influenced by the non-radiative feedbacks (HSTOR and HFLUX) in sea ice-retreat regions.
Sorting regional PTCs by the local warming amount (Fig. 5) indicates the most important terms driving the inter-model spread. Evident from Fig. 5, regions with the most warming exhibit the largest inter-model spread between the PTCs; the only exception being summer SAF (Fig. 5d). Finding the largest inter-model spread in regions that warm the most may not seem surprising; however, this result is not guaranteed. This result supports our conclusion that the mechanisms driving the inter-model spread in AA operate regionally.
Relationship between regional warming pattern and individual partial temperature contributions. Select partial temperature contributions are plotted against projected temperature change by year 2100 using individual grid boxes for (a) winter longwave clear-sky, (b) winter ocean heat storage, (c) winter surface turbulent flux, (d) summer surface albedo feedback, and (e) autumn cloud radiative effect. The bracketed ranges show the ensemble mean temperature change (black filled circle) and the ±1 standard deviation for various Arctic regions; temperature changes for some individual models exceed this range
Individual SEB terms contribute differently to the regional warming pattern. LWCS (Fig. 5a) and CRE (Fig. 5e) PTCs indicate a direct relationship with the regional warming pattern, where larger PTCs are found in regions of larger warming. Models overwhelmingly agree on the relationship between the regional warming and LWCS PTCs. The correspondence between regional characteristics of LWCS and warming indicates a direct local relationship exhibiting very little inter-model spread and supported by the small spatial variation in the LWCS PTC ensemble standard deviation (Fig. 4i). The CRE PTCs exhibit a larger inter-model spread and smaller PTCs than LWCS. Regions that warm the most show a positive year-round CRE PTC.
The SAF PTCs suggest a different relationship with regional warming. The summer SAF PTCs' dependence on regional warming (Fig. 5d) resembles a 'U-shape'—large PTCs in regions of small and large warming. Since regions that warm most in summer also warm most in fall/winter (Fig. 2), this suggests that the regional warming pattern itself is partially independent of the SAF, also supported by Kim et al.34.
HSTOR (Fig. 5b) and HFLUX (Fig. 5c) show monotonic relationships between PTCs and regional warming. HFLUX is negative (surface cooling) in regions that warm most and positive (surface warming) in regions that warm least, the opposite holds for HSTOR. Further, HFLUX and HSTOR PTCs are very close to zero in regions of modest warming, primarily land regions. This behavior indicates that changes in HFLUX control HSTOR and the regional surface heating/cooling rates. The largest inter-model spread in HSTOR and HFLUX PTCs are found in the regions of greatest warming (sea ice retreat regions) exhibiting an inter-model range exceeding 40 K. Overall, these results suggest that the regions that warm the most exhibit a strong summer SAF and fall/winter HFLUX and HSTOR PTCs.
The process of Arctic amplification
Synthesis of our results and previous work paints a picture of AA whereby increased downwelling LW radiation (LWDN) dominates Arctic warming17,18,29,51. Observational evidence corroborates this model behavior demonstrating a significant contribution to recent Arctic warming and fall sea ice variability from LWDN20,31,32,51,52. Understanding the drivers of AA then reduces to quantifying the processes driving changes in LWDN.
Figure 6 illustrates two primary process loops contributing to increased LWDN: remote and local mechanisms. In the remote mechanism, changes in the non-polar (tropical and mid-latitude) circulation increase atmospheric poleward heat transport (APHT) into the Arctic and warm, moisten, and produce a cloudier Arctic atmosphere, increasing LWDN. Proposed processes facilitating the increased APHT include mid-latitude circulation changes such as increased moisture intrusions50 and teleconnections with the tropical climate53,54. Atmospheric energy convergence from midlatitude moisture fluxes into the Arctic has increased LWDN and contributed to observed Arctic temperature trends between 1989 and 200948,51.
Local and remote Arctic amplification processes. The local and remote mechanisms represent two process loops that contribute to Arctic amplification. In the remote mechanism, changes in the non-polar circulation increase atmospheric poleward heat transport warming, moistening, and producing a cloudier Arctic, increasing longwave downward radiation. The local mechanism represents the combined surface albedo (radiative) and ice-insulation (non-radiative) feedbacks whereby a less sea ice covered Arctic stores more sunlight in the ocean in summer supporting increased surface turbulent fluxes in fall/winter, which warms, moistens, and produces cloudier conditions and increases longwave downward radiation. The common influence of both the local and remote mechanisms on longwave downward radiation links the two mechanisms facilitating constructive interference and makes them challenging to separate. A line connecting warmer Arctic temperatures with non-polar circulation changes suggests a potential feedback, however the dashed line is used to indicate the current lack of consensus on the magnitude and influence of Arctic temperature changes on the mid-latitude circulation
The local mechanism represents the combined surface albedo (radiative) and ice-insulation (non-radiative) feedbacks whereby a warmer Arctic with less sea ice stores more energy in the ocean in summer via the SAF supporting increased HFLUX in fall/winter. The increased absorbed radiation is not immediately radiated away but stored and transferred from summer to fall/winter delaying fall sea ice freeze-up and providing an energy source to the atmosphere11,55. Increased HFLUX subsequently warms, moistens, and increases clouds, contributing to increased LWDN16,34,39. Increased LWDN and warmer Arctic temperatures in fall/winter promote thinner sea ice and further sea ice loss48, reinforcing the feedback loop by increasing the potential for ocean heat storage55,56.
Ocean mixed layer processes and heat transport modulate the local and remote mechanisms. The summer SAF warms the upper ocean making it less dense and more stably stratified. Heat entering the ocean is trapped near the surface, where it melts and thins sea ice [Graham et al. 2013]. Sea ice melt freshens the mixed layer and increases stratification encouraging the development of a near-surface temperature maximum55,57,58 —a layer of warm water below the shallow summer ocean mixed layer (OML). In fall, the OML cools, deepens, and transfers heat to the atmosphere via HFLUX and longwave radiation59. More ice-free ocean supports greater upper ocean mixing by atmospheric winds, which can entrain warm water from the near-surface temperature maximum layer and delay fall sea ice freeze-up55,58. However, this process is likely poorly represented across CMIP5 models due to insufficient upper ocean vertical mixing60.
Ocean heat transport into the Arctic influences temperature and sea ice, passes energy to the atmosphere via HFLUX, and modifies the Arctic circulation. Reconstructions of long-term records27 and model output point to ocean heat transport as a potential source of AA and inter-model spread6,61. Oceanic heat transport into the Barents Sea influences Arctic climate variability and the North Atlantic Oscillation62,63 by reducing sea ice, enhancing HFLUX, and lowering the local atmospheric pressure. However, ocean heat transport in CMIP5 models contributes little to projected warming over 21st century15. These interactions suggest that changes in the position and intensity of the jet stream and frequency of synoptic cyclones can modulate the local and remote mechanisms by influencing sea ice, HFLUX, and the OML, representing a process link between the two mechanisms.
The fact that the remote and local mechanisms contribute to AA via increased LWDN (Fig. 6) also links these two mechanisms29,64. It is clear that the remote forcing mechanism can accelerate the local mechanism by increasing LWDN and influencing sea ice52. It is an open question if changes in the Arctic drive changes in the mid-latitude circulation (dashed line in Fig. 6)65. Zappa et al.66 suggest that sea ice loss in CMIP5 models influences the position of the midlatitude westerly jet, promoting an equatorward shift and consistent with previous work67,68. These links suggest the potential for constructive and destructive interference between these mechanisms, however it is unclear how this interference influences the inter-model spread and cannot be assessed from the current methodology. We hypothesize that quantifying the strength of interactions between the local and remote mechanisms in observations is key to determining if the largest projected AA is likely. This is an important area for future work.
Drivers of inter-model spread
The largest inter-model differences in the warming response (Fig. 2) and feedback contributions (Fig. 4) are found in sea ice retreat regions. These regions exhibit the largest summer SAF and fall/winter HFLUX/HSTOR PTCs, as well as the largest LWCS increases (Fig. 5). The HFLUX and HSTOR PTCs also show the largest inter-model differences in sea ice retreat regions. These factors suggest that the local mechanism drives the inter-model spread.
The local mechanism links the SAF, HSTOR, and HFLUX terms through seasonal energy transfer. To test our hypothesis that differences in the local mechanism explain the AA inter-model spread, we identify a metric based upon the amplitude of seasonal energy exchanges—the seasonal ocean heat flux (OHFSEASONAL). OHFSEASONAL is defined as the difference between the month of minimum and maximum HSTOR and ΔOHFSEASONAL represents its change by 2100. Figure 7a shows a statistically significant relationship between the ΔOHFSEASONAL and AA (r = 0.89), supporting our hypothesis. Using ΔOHFSEASONAL as an observational constraint on projected AA may lack practical relevance if the signal emergence does not occur before warming. Conversely, the remote forcing mechanism, represented by ΔAPHT (see Methods), anticorrelates with AA suggesting that it dampens the inter-model spread (Fig. 7b). Thus, the seasonal exchanges of energy related to the local feedback mechanism widen the inter-model spread in AA.
Energy transfer and Arctic amplification. a Correlation between the changes in seasonal ocean heat flux amplitude (ΔOHFseasonal) and Arctic amplification. b Correlation between annual change in atmospheric poleward heat transport and correlation between model summer surface albedo feedback partial temperature contribution and fall/winter surface turbulent flux partial temperature contribution for (c) Arctic domain, (d) Barents Sea, and (e) Beaufort/Chukchi Sea
Models increase ΔOHFSEASONAL through a stronger summer SAF and a larger fall/winter HFLUX (Fig. 7c). This seasonal energy transfer is amplified in sea ice retreat regions, such as the Barents/Kara and Beaufort/Chukchi Seas regions (Fig. 7d, e) and does not operate in ice-free ocean areas (not shown). Therefore, models accomplish a stronger seasonal transfer of energy through a larger summer SAF, storing heat in the ocean and enhancing fall/winter HFLUX in sea ice retreat regions.
Figure 8 illustrates the series of relationships that contribute to the inter-model spread due to the local mechanism in sea ice retreat regions. First, the SAF and increased summer heat storage cannot directly increase HFLUX, rather increased summer heat storage supports a delayed fall sea ice freeze-up and a positive fall/winter air–sea temperature gradient (Ts − Ta). Figure 8a illustrates that models with less fall/winter sea ice produce larger HFLUX. Increased HFLUX is supported by stronger air–sea temperature gradients (Fig. 8b) that are maintained by the atmospheric circulation through advection.
Dispersal of energy from sea ice retreat regions to the broader Arctic. a Regression slope between annual mean change in sea ice concentration and fall surface turbulent fluxes. b Ensemble mean change in the air–sea temperature (Ts − Ta) gradient. c Regression slope between the fall surface turbulent flux change partial temperature contribution in the Barents/Kara Seas (boxed region) and the fall longwave clear-sky partial temperature contribution across the ensemble and d as is c for the Beaufort/Chukchi Seas. Regions where the regression slope is significant at the 90% level are bounded by the dotted line. The statistically significant regression slopes suggest that models with larger changes in surface turbulent fluxes in the Barents/Kara and Beaufort/Chukchi Seas also produce larger increases in the downward longwave clear-sky flux across the broader Arctic
Inter-model differences in sea ice retreat regions have non-local effects through the atmospheric circulation. Increased HFLUX warms, moistens, and produces clouds locally, increasing LWDN, deepening the OML, and contributing to a delayed fall sea ice freeze-up. The influence of local HFLUX changes on the Arctic-wide LWCS requires the atmospheric circulation to disperse this energy. Figure 8c, d shows a statistically significant linear regression slope between model fall/winter HFLUX PTC in the Barents/Kara and Beaufort/Chukchi Seas and the spatial pattern of ΔLWCS; models with a larger magnitude fall/winter HFLUX PTC in sea ice retreat regions generate a larger Arctic-wide LWCS PTC. Comparing Fig. 8c, d indicates a larger Arctic-wide LWCS PTC due to ΔHFLUX in the Beaufort-Chukchi Seas, suggesting that the atmospheric circulation is more sensitive to a perturbation in this region. The anti-correlation between ΔOHFSEASONAL and ΔAPHT (Fig. 7) suggests that the inter-model differences in how this energy is dispersed are not from the large-scale circulation, but due to the local circulation. We infer that local atmospheric circulation differences contribute to the inter-model spread in AA by influencing how energy drawn from the surface in sea ice retreat regions is dispersed. Burt et al.16 hypothesize a possible mechanism whereby reduced fall/winter sea ice induces a thermal contrast between the Arctic Ocean and the colder Arctic continents driving a circulation response, termed a shallow winter monsoon, promoting stronger HFLUX.
OML depth modulates the seasonal energy transfer and can impact the inter-model spread in AA. Eight CMIP5 models that archived OML depth show very large inter-model differences in the relationships between OML depth, SIC, and HFLUX in Barents/Kara and Beaufort/Chukchi Seas regions. A deeper OML contains a higher heat capacity supporting larger HFLUX and less sea ice (Fig. 9). Moreover, the average OML depth in the Barents/Kara Seas region ranges from 13 to 95 m in fall. Motivated by this inter-model spread and the relationship in Fig. 7, we hypothesized that the OML depth influences the inter-model spread in AA by modulating the seasonal energy transfer. However, we found no correlation between the mean state OML depth or its change with AA. The lack of correlation may indicate a bias in the model representation of OML dynamics, such as the known bias in upper ocean vertical mixing60. Despite the lack of correlation, this range of OML depth significantly influences the ability of the ocean to store energy and modulate surface turbulent fluxes, sea ice variability, and the atmospheric circulation. Therefore, an improved understanding of the causes and consequences of the inter-model spread in OML depth and its relationship with sea ice and surface turbulent fluxes is needed.
Implications of changes in ocean mixed layer depth. Relationships between ocean mixed layer depth, sea ice concentration, and surface turbulent flux in sea ice retreat regions, showing (a) Beaufort/Chukchi Seas fall mixed layer depth and fall sea ice concentration, (b) Barents/Kara Seas fall mixed layer depth and fall sea ice concentration, (c) Beaufort/Chukchi Seas fall mixed layer depth and fall surface turbulent flux, and (d) Barents/Kara Sea fall mixed layer depth and fall surface turbulent flux. A linear regression fit is drawn for each model (solid lines). The results indicate significant inter-model differences in the present-day and future, changes in ocean mixed layer depth as well as the relationship with sea ice concentration and surface turbulent fluxes
Our results indicate that the local AA mechanism—the combined sea ice albedo and ice insulation feedbacks—significantly contributes to the inter-model spread in AA. The remote AA mechanism is shown to dampen the inter-model spread.
We conclude that models that transfer more energy from summer to fall produce a larger AA. This seasonal energy transfer is primarily accomplished by increased absorption and storage of solar insolation in summer and increased surface turbulent fluxes in fall/winter in sea ice retreat regions. Models simulating greater reductions in summer/fall sea ice and larger fall/winter surface turbulent fluxes produce more AA. Our results suggest that the local Arctic circulation and its response contribute to the inter-model spread in AA by dispersing the energy drawn from the surface in sea ice retreat regions Arctic-wide and reinforcing the local air–sea temperature gradients. Models that more widely disperse the energy drawn from the surface in sea ice retreat regions warm more. We found significant inter-model differences in ocean mixed layer depths and its relationships with sea ice concentration and surface turbulent fluxes that modulate seasonal energy transfer, yet do not correspond to the inter-model spread in AA. This lack of correlation suggests a bias in the model representation of the ocean mixed layer with the potential to significantly impact projected AA.
This explanation contains familiar arguments for the processes that drive the AA inter-model spread; however, the regional picture presented should not be overlooked. The local mechanism does not occur throughout the Arctic but is focused in regions of sea ice retreat. Neglecting the regional variation in AA mechanisms paints an incomplete picture, as the inter-model spread in warming possesses a regional structure. Only after considering the regional perspective does the importance of the local atmospheric circulation to AA become clear. Our results suggest that an atmospheric circulation change local to the Arctic may be an important factor in the inter-model spread. While much of the scientific literature attempts to explain inter-model differences in AA using atmospheric-only mechanisms, we contend that a complete theory of AA and its inter-model spread must consider the atmosphere-ocean-sea ice system. Our results indicate that the principle mechanisms driving the inter-model spread in AA operate on regional not Arctic-wide scales and that reductions in the inter-model spread in projected Arctic warming require an improved process representation of atmosphere-ocean-sea ice interactions in sea ice retreat regions.
Surface energy budget decomposition
Data analyzed during this study comes from 21 CMIP5 models, listed in Supplementary Table 1, which archived the required data to perform the surface energy budget decomposition10. All data are available from the Earth System Grid Federation Peer-to-Peer enterprise system at https://esgf-node.llnl.gov/projects/esgf-llnl/. The model simulations were forced with RCP8.5, a high-emission scenario from 2006 to 2100. Present (future) climatology was determined from the average of the first (last) 20 years of the simulation. Using available data, surface energy budget can be computed as
$$Q = \left( {1 - \alpha } \right)S_{{\mathrm{dn}}} + F_{{\mathrm{dn}}} - \varepsilon \sigma T_s^4 - \left( {S + L} \right),$$
where Q represents the storage of heat for all surface types as well as oceanic transport, α is the surface albedo defined by the ratio of upward to downward shortwave clear-sky fluxes, Sdn is incident solar radiation, Fdn is the downwelling longwave radiation, εσTs4 is the longwave emission from the surface at temperature Ts (where the emissivity, ε, is assumed to be equal to 1), and (S + L) are the sensible and latent heat fluxes (defined as positive upward). All variables are available from the CMIP5 data portal except for Q, which is obtained as a residual. After solving for T, a perturbation form of (1) yields
$$4\sigma T_{\mathrm{s}}^3\Delta T = \Delta \left[ {\left( {1 - \alpha } \right)S_{{\mathrm{dn}}}} \right] + \Delta F_{{\mathrm{dn}}} - \Delta Q - \Delta \left( {S + L} \right),$$
where Δ-values represent the change in a variable between present-day and future.
Following Lu and Cai17, the change in the cloud radiative effect (CRE) is calculated as
$$\Delta{\mathrm{ CRE}} = \left( {1 - \overline{ \alpha} } \right)\Delta S_{{\mathrm{dn}},{\mathrm{cld}}} + \Delta F_{{\mathrm{dn}},{\mathrm{cld}}},$$
where \(\Delta S_{{\mathrm{dn}},{\mathrm{cld}}}\,{\mathrm{and}}\,\Delta F_{{\mathrm{dn}},{\mathrm{cld}}}\) are computed as all-sky minus clear-sky radiative fluxes and \({{\bar \alpha }}\) represents mean state surface albedo. This formulation of ΔCRE controls for the influence of surface albedo on CRE. Substitution of (3) into (2) and dividing by 4σTs3 yields
$${ \Delta T = \frac{{ - \left( {\Delta \alpha } \right)\left( {\overline {S_{{\mathrm{dn}}}} + \Delta S_{{\mathrm{dn}}}} \right) + \Delta {{CRE}} + \left( {1 - \overline{\alpha} } \right)\Delta S_{{\mathrm{dn}},{\mathrm{clr}}} + \Delta F_{{\mathrm{dn}},{\mathrm{clr}}} - \Delta Q - \Delta \left( {S + L} \right)}}{{4\sigma T_s^3}}}.$$
Each term on the right-hand side of (4) represents a partial temperature contribution for SAF, CRE, changes in shortwave clear-sky radiation unrelated to SAF, changes in longwave clear-sky radiation, changes in heat storage, and changes in surface turbulent fluxes, respectively (Table 1). Each PTC represents the temperature contribution of the corresponding feedback to the total Arctic temperature change by 2100.
Calculation of atmospheric poleward heat transport (APHT)
The annual mean net radiation balance for latitudes poleward of ~40° is negative, meaning it emits more energy than it receives, requiring poleward heat transport. The energy balance of the Arctic domain can be written as in (5)
$$\frac{{\partial E}}{{\partial t}} = R_{{\mathrm{TOA}}} - F_{{\mathrm{ao}}}$$
where \(\frac{{\partial E}}{{\partial t}}\) is the time rate of change in energy content of the Arctic, RTOA is the net incoming radiation at TOA, and Fao is the horizontal flux divergence for atmosphere and ocean (poleward heat transport term). Averaging over timescales greater than one year makes the energy storage term \(\frac{{\partial E}}{{\partial t}}\) negligible and then the implied poleward heat transport (PHT) is calculated by requiring a balance between RTOA and Fao. RTOA is calculated using CMIP5 model outputs at TOA for downwelling shortwave radiation (STOA,dn), upwelling shortwave radiation (STOA,up), and upwelling longwave radiation (FTOA,up). The equation for the combined atmospheric and oceanic poleward heat transport then becomes
$$F_{{\mathrm{ao}}} = R_{{\mathrm{TOA}}} = - \left( {S_{{\mathrm{TOA}},{\mathrm{dn}}} - S_{{\mathrm{TOA}},{\mathrm{up}}} - F_{{\mathrm{TOA}},{\mathrm{up}}}} \right).$$
The atmospheric component of poleward heat transport (APHT) is determined by taking the difference between RTOA from the surface energy imbalance. After averaging over timescales greater than one year and neglecting \(\frac{{\partial E}}{{\partial t}}\), Eq. (5) becomes
$$APHT = - \left( {R_{{\mathrm{TOA}}} - R_{{\mathrm{SFC}}}} \right)$$
$$R_{{\mathrm{SFC}}} = F_{{\mathrm{net}}} + S_{{\mathrm{net}}} - \left( {S + L} \right).$$
Code availability
Computer code used for the analysis was written in IDL and is available from the authors upon request.
The CMIP5 model data analyzed and support the finding of this study are deposited in the Earth System Grid Federation Peer-to-Peer enterprise system and available at https://esgf-node.llnl.gov/projects/esgf-llnl/.
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This work is funded by the NASA Interdisciplinary Studies Program grant NNH12ZDA001N-IDS. We acknowledge the World Climate Research Program's Working Group on Coupled Modeling, which is responsible for CMIP, and we thank the climate modeling groups for producing and making available their model output. For CMIP the U.S. Department of Energy's Program for Climate Model Diagnosis and Intercomparison provides coordinating support and leads development of software infrastructure in partnership with the Global Organization for Earth System Science Portals.
Science Systems Applications Inc, Hampton, VA, 23666, USA
Robyn C. Boeke
NASA Langley Research Center, Climate Science Branch, Hampton, VA, 23681-2199, USA
Patrick C. Taylor
R.C.B. downloaded the model output and performed the calculations. P.C.T. and R.C.B. both formulated the study, analyzed the calculations, and jointly wrote the manuscript.
Correspondence to Patrick C. Taylor.
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Boeke, R.C., Taylor, P.C. Seasonal energy exchange in sea ice retreat regions contributes to differences in projected Arctic warming. Nat Commun 9, 5017 (2018). https://doi.org/10.1038/s41467-018-07061-9
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\begin{document}
\vskip1.5cm
\centerline {\bf \chuto On Properties of Geodesic Semilocal E-Preinvex Functions
}
\vskip.2cm
\centerline {\bf \chuto }
\vskip.8cm \centerline {\kamy Adem Kili\c{c}man $ ^{a} $ and Wedad Saleh $^b,$\footnote{{\tt Corresponding author email: wed\_10\[email protected] ( Wedad Saleh)}}}
\vskip.5cm
\centerline {$^a$ Department of Mathematics, Putra University of Malaysia (UPM),Serdang, Malaysia }
\centerline {$^{b}$ Department of Mathematics, Taibah University, Al- Medina, Saudi Arabia }
\vskip.5cm \hskip-.5cm{\small{\bf Abstract :} The authors define
a class of functions on Riemannian manifolds, which is called geodesic semilocal E-preinvex functions, as a generalization of geodesic semilocal E-convex and geodesic semi E-preinvex functions and some of its properties are established. Furthermore, a nonlinear fractional multiobjective programming is considered, where the functions involved are geodesic E-$ \eta $-semidifferentiability, sufficient optimality conditions are obtained. A dual is formulated and duality results are proved using concepts of geodesic semilocal E-preinvex functions, geodesic pseudo-semilocal E-prinvex functions and geodesic quasi-semilocal E-preinvex functions.
\vskip0.3cm\noindent {\bf Keywords :} Generalized convexity, Riemannian geometry, Duality
\noindent{\bf 2000 Mathematics Subject Classification : 52A41; 53B20; 90C46 }
\hrulefill
\section{Introduction} \hskip0.6cm
Convexity and generalized covexity play a significant role in many fields, for example, in biological system, economy, optimization, and so on \cite{kS,GrinalattLinnainmaa2011,RuelAyres1999}.
Generalized convex functions, labelled as semilocal convex functions were introduced by Ewing \cite{Ewing} by using more general semilocal perinvexity and $ \eta- $ semidifferentiability. After that optimality conditions for weak vector minima was given \cite{Preda1997}. Also, optimality conditions and duality results for a nonlinear fractioal involving $ \eta- $ semidifferentiability were established \cite{Preda2003}.
Furthermore,some optimality conditions and duality results for semilocal E-convex programming were established \cite{Hu2007}. E-convexity was extedned to E-preinvexity \cite{FulgaPreda}. Recently, semilocal E-prenivexity (SLEP) and some of its applications were introdued \cite{Jiao2011,Jiao2012,Jiao2013}.
Generalized convex functions in manifolds such as Riemannian manifolds were studied by many authors; see \cite{Agarwal,BG,Ferrara,Mordukhovch2011}. Udrist \cite{Udriste1994} and Rapcsak \cite{Rapcsak1997} considered a generalization of convexity called geodesic convexity. In 2012, geodesic E-convex (GEC) sets and geodesic E-convex (GEC)functions on Riemannian manifolds were studied \cite{IAA2012}. Moreover, geodesic semi E-convex (GsEC) functions were introduced \cite{Iqbal}. Recently, geodesic strongly E-convex (GSEC) functions were introduced and discussed some of their properties \cite{AW}.
\section{Geodesic Semilocal E-Preinvexity}
\hskip0.6cm
\begin{definition}\label{df}
A nonempty set $ B \subset \aleph $ is said to be
\begin{enumerate}
\item geodesic E-invex (GEI) with respect to $ \eta $ if there is exactly one geodesic $ \gamma_{E(\kappa_{1}), E(\kappa_{2})}:\left[0,1 \right]\longrightarrow \aleph $ such that
\begin{eqnarray*}
\gamma_{E(\kappa_{1}), E(\kappa_{2})}(0)=E(\kappa_{2}),\acute{\gamma}_{E(\kappa_{1}), E(\kappa_{2})}=\eta(E(\kappa_{1}),E(\kappa_{2}), \gamma_{E(\kappa_{1}), E(\kappa_{2})}(t)\in B,
\end{eqnarray*}
$ \forall \kappa_{1},\kappa_{2}\in B$ and $t\in[0,1]. $
\item a geodesic local E-invex (GLEI) respect to $ \eta $, if there is $ u(\kappa_{1},\kappa_{2})\in\left(\left.0,1 \right] \right. $ such that $ \forall t\in [0,u (\kappa_{1},\kappa_{2})] $,
\begin{equation}\label{eq2}
\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t)\in B \ \ \forall\kappa_{1},\kappa_{2}\in B.
\end{equation}
\item
a geodesic local starshaped E-convex, if there is a map $ E $ such that corresponding to each pair of points $ \kappa_{1},\kappa_{2}\in A $, there is a maximal positive number $ u(\kappa_{1},\kappa_{2})\leq 1 $ such as
\begin{equation}\label{eq1}
\gamma_{E(\kappa_{1}),E(\kappa_{2})}\in A, \ \ \forall t\in [0, u(\kappa_{1},\kappa_{2})]
\end{equation}
\end{enumerate}
\end{definition}
\begin{definition}
A function $ f: A\subset \aleph\longrightarrow \mathbb{R} $ is said to be
\begin{enumerate}
\item Geodesic E-preinvex (GEP) on $ A\subset \aleph$ with repect to $ \eta $ if $A$ is a GEI set and $$f\left(\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t)\right)\leq t f(E(\kappa_{1}))+(1-t)f(E(\kappa_{2})) , \ \ \forall \kappa_{1},\kappa_{2}\in A, t\in[0,1]; $$
\item Geodesic semi E-preinvex (GSEP) on $ A $ with respect to $ \eta $ if if $A$ is a GEI set and
$$f\left(\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t)\right)\leq t f(\kappa_{1})+(1-t)f(\kappa_{2}) , \ \ \forall \kappa_{1},\kappa_{2}\in A, t\in[0,1]. $$
\item Geodesic Local E-preinvex (GLEP) on $ A\subset \aleph $ with respect to $ \eta $, if for any $ \kappa_{1},\kappa_{2}\in A $ there exists $ 0<v(\kappa_{1},\kappa_{2})\leq u(\kappa_{1},\kappa_{2}) $ such that $ A $ is a GLEI set and
$$f\left(\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t)\right)\leq t f(E(\kappa_{1}))+(1-t)f(E(\kappa_{2})) , \ \ \forall t\in[0,v(\kappa_{1},\kappa_{2})].$$
\end{enumerate} \end{definition} \begin{definition}
A function $ f:\aleph\longrightarrow \mathbb{R} $ is a geodesic semilocal E-convex ( GSLEC) on a geodesic local starshaped E-convex set $ B\subset \aleph $ if for each pair of $ \kappa_{1},\kappa_{2}\in B $ ( with a maximal positive number $ u(\kappa_{1},\kappa_{2})\leq1 $ satisfying \ref{eq1}), there exists a positive number $ v(\kappa_{1},\kappa_{2})\leq u(\kappa_{1},\kappa_{2}) $ satisfying $$f\left(\gamma_{E(\kappa_{1}), E(\kappa_{2})}(t)\right)\leq t f(\kappa_{1})+(1-t)f(\kappa_{2}) , \ \ \forall t\in[0,v(\kappa_{1},\kappa_{2})].$$ \end{definition} \begin{remark}
Every GEI set with respect to $ \eta $ is a GLEI set with respect to $ \eta $, where $ u(\kappa_{1},\kappa_{2})=1, \forall \kappa_{1},\kappa_{2}\in \aleph $. On the other hand, their coverses are not recessarily true and we can see that in the next example. \end{remark}
\begin{example}
Put $ A=\left[ \left. -4,-1\right) \right. \cup[1,4] $, \begin{eqnarray*}
E(\kappa) &=& \begin{cases}
\kappa^{2} \ \ if\ \ \left|\kappa \right| \leq 2,\\ -1\ \ if \ \ \left|\kappa \right| > 2;
\end{cases}
\end{eqnarray*}
\begin{eqnarray*}
\eta (\kappa,\iota) &=& \begin{cases}
\kappa-\iota \ \ if\ \ \kappa\geqslant 0, \iota\geqslant 0\ \ or\ \ \kappa\leq 0, \iota\leq0 ,\\ -1-\iota \ \ if \ \ \kappa>0, \iota\leq 0 \ \ or \ \ \kappa\geqslant0 ,\iota< 0, \\ 1-\iota \ \ if \ \ \kappa<0, \iota\geqslant 0 \ \ or\ \ \kappa\leq 0, \iota>0;
\end{cases}
\end{eqnarray*}
\begin{eqnarray*}
\gamma_{\kappa,\iota}(t) &=& \begin{cases}
\iota+t(\kappa-l) \ \ if \ \ \kappa\geqslant 0, \iota\geqslant 0 \ \ or \ \ \kappa\leq 0, \iota\leq0 ,\\ \iota+t(-1-\iota) \ \ if \ \ \kappa>0, \iota\leq 0 \ \ or\ \ \kappa\geqslant0 ,\iota< 0, \\ \iota+t(1-\iota) \ \ if \ \ \kappa<0, \iota\geqslant 0 \ \ or \ \ \kappa\leq 0, \iota>0.
\end{cases}
\end{eqnarray*}
Hence $ A $ is a GLEI set with respect to $ \eta $.
But, when $ \kappa=3, \iota=0 $, there is a $ t_{1}\in[0,1] $ such that $ \gamma_{E(\kappa),E(\iota)}(t_{1})=-t_{1} $, then if $ t_{1}=1 $, we obtain $\gamma_{E(\kappa),E(\iota)}(t_{1})\notin A $. \end{example}
\begin{definition}\label{de1}
A function $ f: \aleph\longrightarrow \mathbb{R} $ is GSLEP on $ B\subset \aleph $ with respect to $ \eta $ if for any $ \kappa_{1},\kappa_{2}\in B $, there is $ 0<v(\kappa_{1},\kappa_{2})\leq u(\kappa_{1},\kappa_{2})\leq1 $ such that $ B $ is a GLEI set and
\begin{equation}\label{eq3}
f\left(\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t)\right)\leq t f(\kappa_{1})+(1-t)f(\kappa_{2}) , \ \ \forall t\in[0,v(\kappa_{1},\kappa_{2})].
\end{equation}
If $$f\left(\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t)\right)\geqslant t f(\kappa_{1})+(1-t)f(\kappa_{2}) , \ \ \forall t\in[0,v(\kappa_{1},\kappa_{2})],$$ then $ f $ is GSLEP on $ B $.
\end{definition} \begin{remark}
Any GSLEC function is a GSLEP function. Also, any GSEP function with respect to $ \eta $ is a GSLEP function. On the other hand, their converses are not necessarily true. \end{remark} The next example shows SLGEP, which is neither a GSLEC function nor a GSEP function. \begin{example}
Assume that $ E: \mathbb{R}\longrightarrow \mathbb{R} $ is given as
\begin{eqnarray*}
E(m) &=& \begin{cases}
0 \ \ if\ \ m<0,\\ 1 \ \ if\ \ 1<m\leq 2,\\ m \ \ if\ \ 0\leq m\leq1 \ \ or\ \ m>2
\end{cases}
\end{eqnarray*} and the map $ \eta: \mathbb{R}\times \mathbb{R}\longrightarrow \mathbb{R} $ is defined as
\begin{eqnarray*}
\eta (m,n) &=& \begin{cases}
0 \ \ if\ \ m= n,\\ 1-m \ \ if\ \ m\neq n ;
\end{cases}
\end{eqnarray*} also,
\begin{eqnarray*}
\gamma_{m,n}(t) &=& \begin{cases}
n \ \ if\ \ m= n,\\ n+t(1-m) \ \ if\ \ m\neq n.
\end{cases}
\end{eqnarray*}
Since $ \mathbb{R} $ is a geodesic local starshaped E-convex set and a geodesic local E-invex set with respect to $ \eta $. Assume that $ h: \mathbb{R}\longrightarrow \mathbb{R} $, where
\begin{eqnarray*}
h(m) &=& \begin{cases}
0 \ \ if\ \ 1<m\leq 2,\\ 1 \ \ if\ \ m>2, \\ -m+1 \ \ if\ \ 0\leq m\leq1,\\ -m+2 \ \ if\ \ m<0.
\end{cases}
\end{eqnarray*} Then $ h $ is a GSLEP on $ \mathbb{R} $ with respect to $ \eta $. However, when $ m_{0}=2, n_{0}=3 $ and for any $ v\in\left(\left.0,1 \right] \right. $, there is a sufficiently small $ t_{0}\in\left(\left.0,v \right] \right. $ such as $$h\left(\gamma_{E(m_{0}),E(n_{0})}(t_{0}) \right)=1>(1-t_{0})=t_{0}h(m_{0})+(1-t_{0})h(n_{0}) .$$ Then $ h(m) $ is not a GSLEC function on $ \mathbb{R} $.
Similarly, taking $ m_{1}=1, n_{1}=4 $, we have $$h\left(\gamma_{E(m_{1}),E(n_{1})}(t_{1}) \right)=1>(1-t_{1})=t_{1}h(m_{1})+(1-t_{1})h(n_{1}) .$$ for some $ t_{1}\in[0,1] $.\\
Hence $ h(m) $ is not a GSEP function on $ \mathbb{R} $ with respect to $ \eta $. \end{example} \begin{definition}\label{de2} A function $ h:S\subset \aleph\longrightarrow \mathbb{R} $ , where $ S $ a GLEI set, is said to be a geodesic quasi-semilocal E-preinvex (GqSLEP) (with respect to $ \eta $) if for all $ \kappa_{1},\kappa_{2}\in S $ satisfying $ h(\kappa_{1})\leq h(\kappa_{2}) $, there is a positive number $ v(\kappa_{1},\kappa_{2})\leq u(\kappa_{1},\kappa_{2}) $ such that $$h\left(\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t)\right) \leq h(\kappa_{2}), \forall t\in[0,v(\kappa_{1},\kappa_{2})]. $$ \end{definition} \begin{definition}\label{de3} A function $ h:S\subset \aleph\longrightarrow \mathbb{R} $ , where $ S $ a GLEI set, is said to be a geodesic pseudo-semilocal E-preinvex ( GpSLEP) (with respect to $ \eta $) if for all $ \kappa_{1},\kappa_{2}\in S $ satisfying $ h(\kappa_{1})<h(\kappa_{2}) $, there are positive numbers $ v(\kappa_{1},\kappa_{2})\leq u(\kappa_{1},\kappa_{2}) $ and $ w(\kappa_{1},\kappa_{2}) $ such that
$$h\left(\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t)\right) \leq h(\kappa_{2})-t w(\kappa_{1},\kappa_{2}), \forall t\in[0,v(\kappa_{1},\kappa_{2})]. $$ \end{definition} \begin{remark}
Every GSLEP on a GLEI set with respect to $ \eta $ is both a GqELEP function and a GpSLEP function. \end{remark} \begin{definition}\label{de4} A function $ h:S\longrightarrow \mathbb{R} $
is called geodesic E-$ \eta $- semidifferentiable at $ \kappa^{*}\in S $ where $ S\subset \aleph $ is a GLEI set with respect to $ \eta $ if $ E(\kappa^{*})=\kappa^{*} $ and \begin{equation*} h'_{+}\left(\gamma_{\kappa^{*},E(\kappa)}(t) \right)= \lim_{t\longrightarrow 0^{+}} \frac{1}{t}\left[h\left(\gamma_{\kappa^{*},E(\kappa)}(t)\right) -h(\kappa^{*}) \right], \end{equation*} exists for every $ \kappa\in S. $. \end{definition} \begin{remark}
\begin{enumerate}
\item Let $ \aleph=\mathbb{R}^{n} $, then the geodesic E-$ \eta $- semidifferentiable is E-$ \eta $-semidifferentiable \cite{Jiao2011}.
\item If $ \aleph=\mathbb{R}^{n} $ and $ E=I $, then the geodesic E-$ \eta $-semidifferentiable is the $ \eta $-semidifferentiablitiy \cite{Niculescu2007Optimality} .
\item If $ \aleph=\mathbb{R}^{n} $ , $ E=I $ and $ \eta(\kappa,\kappa^{*})=\kappa-\kappa^{*} $, then geodesic E-$ \eta $-semidifferentiable is the semidifferentiability \cite{Jiao2011}.
\end{enumerate} \end{remark} \begin{lemma}\label{lemma2}
\begin{enumerate}
\item Assume that $ h $ is a GSLEP (E-preconcave) and geodesic E-$ \eta $-semidifferentiable at $ \kappa^{*}\in S\subset \aleph $, where $ S $ is a GLEI set with respect to $ \eta $. Then
$$h(\kappa)-h(\kappa^{*})\geqslant (\leq) h'_{+}(\gamma_{\kappa^{*},E(\kappa)}(t)), \forall \kappa\in S.$$
\item Let $ h $ be GqSLEP (GpSLEP) and geodesic E-$ \eta $-semidifferentiable at $ \kappa^{*}\in S\subset \aleph $, where $ S $ is a LGEI set with respect to $ \eta $. Hence $$h(\kappa)\leq(<) h(\kappa^{*})\Rightarrow h'_{+}(\gamma_{\kappa^{*},E(\kappa)}(t))\leq (<)0, \forall \kappa\in S.$$
\end{enumerate} \end{lemma} The above lemma is directly by using definitions (\ref{de1},\ref{de2},\ref{de3} and \ref{de4}). \begin{theorem}
Let $ f: S\subset \aleph\longrightarrow \mathbb{R} $ be a GLEP function on a GLEI set $ S $ with respect to $ \eta $, then $ f $ is a GSLEP function iff $ f(E(\kappa))\leq f(\kappa),\forall \kappa\in S $. \end{theorem} \begin{proof}
Assume that $ f $ is a GSLEP function on set $ S $ with respect to $ \eta $, then $\forall \kappa_{1},\kappa_{2}\in S $, there is a positive number $ v(\kappa_{1},\kappa_{2})\leq u(\kappa_{1},\kappa_{2}) $ where
$$f(\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t))\leq tf(\kappa_{2})+(1-t)f(\kappa_{1}), t\in[0,v(\kappa_{1},\kappa_{2})].$$
By letting $ t=0 $, then $ f(E(\kappa_{1}))\leq f(\kappa_{1}),\forall \kappa_{1}\in S $.\\
Conversely, consider $ f $ is a GLEP function on a GLEI set $ S $, then for any $ \kappa_{1},\kappa_{2}\in S $, there exist $ u(\kappa_{1},\kappa_{2}) \in \left(\left.0,1 \right] \right. $ (\ref{eq2}) and $ v(\kappa_{1},\kappa_{2}) \in \left(\left.0,u(\kappa_{1},\kappa_{2}) \right] \right. $
such that
$$f(\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t))\leq tf(E(\kappa_{1}))+(1-t)f(E(\kappa_{2})), t\in[0,v(\kappa_{1},\kappa_{2})].$$
Since $ f(E(\kappa_{1})) \leq f(\kappa_{1}), \forall \kappa_{1}\in S$, then
$$f(\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t))\leq tf(\kappa_{1})+(1-t)f(\kappa_{2}), t\in[0,v(\kappa_{1},\kappa_{2})].$$
\end{proof} \begin{definition}
The set $ \omega=\left\lbrace(\kappa,\alpha):\kappa\in B\subset \aleph, \alpha\in \mathbb{R} \right\rbrace $ is said to be a GLEI set with respect to $ \eta $ corresponding to $ \aleph $ if there are two maps $ \eta, E $ and a maximal positive number $ u((\kappa_{1},\alpha_{1}), (\kappa_{2},\alpha_{2}))\leq 1 $, for each $ (\kappa_{1},\alpha_{1}), (\kappa_{2},\alpha_{2})\in \omega $ such that
$$ \left(\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t),t\alpha_{1}+(1-t)\alpha_{2} \right)\in \omega, \forall t\in\left[0,u((\kappa_{1},\alpha_{1}), (\kappa_{2},\alpha_{2})) \right]. $$ \end{definition} \begin{theorem}\label{th1}
Let $ B\subset \aleph$ be a GLEI set with respect to $ \eta $. Then $ f $ is a GSLEP function on $ B $ with respect to $ \eta $ iff its epigraph $$ \omega_{f}=\left\lbrace (\kappa_{1},\alpha):\kappa_{1}\in B, f(\kappa_{1})\leq\alpha, \alpha\in \mathbb{R} \right\rbrace $$ is a GLEI set with respect to $ \eta $ corresponding to $ \aleph $. \end{theorem} \begin{proof}
Suppose that $ f $ is a GSLEP on $ B $ with respect to $ \eta $ and $ (\kappa_{1},\alpha_{1}), (\kappa_{2},\alpha_{2})\in \omega_{f} $, then $ \kappa_{1},\kappa_{2}\in B, f(\kappa_{1})\leq \alpha_{1}, f(\kappa_{2})\leq \alpha_{2} $. By applying definition \ref{df}, we obtain
$ \gamma_{E(\kappa_{1}),E(\kappa_{2})}(t)\in B, \forall t\in\left[0, u(\kappa_{1},\kappa_{2}) \right]. $\\
Moreover,
there is a positive number $ v(\kappa_{1},\kappa_{2})\leq u(\kappa_{1},\kappa_{2}) $ such that
$$f\left(\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t), t\alpha_{1}+(1-t)\alpha_{2} \right)\in \omega_{f}, \forall t\in[0,v(\kappa_{1},\kappa_{2})]. $$
Conversely, if $ \omega_{f} $ is a GLEI set with respect to $ \eta $ corresponding to $ \aleph $ ,then for any points $ (\kappa_{1},f(\kappa_{1})) , (\kappa_{2},f(\kappa_{2}))\in \omega_{f}$, there is a maximal positive number $ u((\kappa_{1},f(\kappa_{1})), (\kappa_{2},f(\kappa_{2}))\leq 1 $ such that $$\left( \gamma_{E(\kappa_{1}),E(\kappa_{2})}(t), tf(\kappa_{1}) +(1-t)f(\kappa_{2})\right) \in \omega_{f},\forall t\in\left[0, u((\kappa_{1},f(\kappa_{1})),(\kappa_{2},f(\kappa_{2}))) \right].$$
That is, $\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t) \in B, $ $$f\left(\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t) \right)\leq tf(\kappa_{1}) +(1-t)f(\kappa_{2}), \ \ t\in\left[0,u((\kappa_{1},f(\kappa_{1})),(\kappa_{2},f(\kappa_{2}))) \right]. $$ Thus, $ B $ is a GLEI set and $ f $ is a GSLEP function on $ B $. \end{proof} \begin{theorem}
If $ f $ is a GSLEP function on a GLEI set $ B\subset \aleph $ with respect to $ \eta $ , then the level $ K_{\alpha}=\left\lbrace \kappa_{1}\in B: f(\kappa_{1})\leq \alpha \right\rbrace $ is a GLEI set for any $ \alpha\in \mathbb{R} $. \end{theorem} \begin{proof}
For any $ \alpha\in \mathbb{R}$ $ $ and $ \kappa_{1},\kappa_{2}\in K_{\alpha} $, then $ \kappa_{1},\kappa_{2}\in B $ and $ f(\kappa_{1})\leq\alpha, f(\kappa_{2})\leq\alpha $. Since $ B $ is a GLEI set, then there is a maximal positive number $ u(\kappa_{1},\kappa_{2})\leq1 $ such that $$ \gamma_{E(\kappa_{1}),E(\kappa_{2})}(t)\in B, \ \ \forall t\in\left[0,u(\kappa_{1},\kappa_{2}) \right] .$$
In addition, since $ f $ is GSLEP, there is a positive number $ v(\kappa_{1},\kappa_{2})\leq u(y_{1},y_{2}) $ such that
\begin{eqnarray}
f\left(\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t) \right)&\leq& t f(\kappa_{1}) +(1-t)f(\kappa_{2})\nonumber\\&\leq& t\alpha+(1-t)\alpha\nonumber\\ &=& \alpha, \ \ \forall t\in\left[0,v(\kappa_{1},\kappa_{2}) \right].\nonumber\end{eqnarray}
That is , $ \gamma_{E(\kappa_{1}),E(\kappa_{2})}(t)\in K_{\alpha}, \ \ \forall t\in\left[0,v(\kappa_{1},\kappa_{2}) \right] $.
Therefore, $ K_{\alpha} $ is a GLEI set with respect to $ \eta $ for any $ \alpha \in \mathbb{R} $. \end{proof} \begin{theorem}
Let $ f:B\subset \aleph\longrightarrow \mathbb{R} $ where $ B $ is a GLEI. Then $ f $ is a GSLEP function with respect to $ \eta $ iff for each pair of points $ \kappa_{1},\kappa_{2}\in B $, there is a positive number $ v(\kappa_{1},\kappa_{2})\leq u(\kappa_{1},\kappa_{2})\leq 1 $ such that
$$f\left(\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t) \right) \leq t \alpha +(1-t)\beta , \ \ \forall t\in\left[0,v(\kappa_{1},\kappa_{2}) \right].$$
\begin{proof}
Let $ \kappa_{1},\kappa_{2}\in B $ and $ \alpha,\beta\in \mathbb{R} $ such that $ f(\kappa_{1})<\alpha $ and $ f(\kappa_{2})<\beta $. Since $ B $ is GLEI, there is a maximal positive number $ u(\kappa_{1},\kappa_{2})\leq 1 $ such that
$$\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t) \in B , \ \ \forall t\in\left[0,u(\kappa_{1},\kappa_{2}) \right].$$
In addition, there is a positive number $ v(\kappa_{1},\kappa_{2})\leq u(\kappa_{1},\kappa_{2}) $ where
$$f\left(\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t) \right) \leq t \alpha +(1-t)\beta , \ \ \forall t\in\left[0,v(\kappa_{1},\kappa_{2}) \right].$$
Conversely, let $ (\kappa_{1},\alpha) \in \omega_{f} $ and $ (\kappa_{2},\beta) \in \omega_{f} $, then $ \kappa_{1},\kappa_{2}\in B $, $ f(\kappa_{1})<\alpha $ and $ f(\kappa_{2})<\beta $. Hence, $ f(\kappa_{1})<\alpha+\varepsilon $ and $ f(\kappa_{2})<\beta+\varepsilon $ hold for any $ \varepsilon>0 $. According to the hypothesis for $ \kappa_{1},\kappa_{2}\in B $, there is a positive number $ v(\kappa_{1},\kappa_{2})\leq u(\kappa_{1},\kappa_{2})\leq 1 $ such that
$$f\left(\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t) \right) \leq t \alpha +(1-t)\beta+\varepsilon , \ \ \forall t\in\left[0,v(\kappa_{1},\kappa_{2}) \right].$$
Let $ \varepsilon\longrightarrow0^{+} $, then
$$f\left(\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t)\right) \leq t \alpha +(1-t)\beta , \ \ \forall t\in\left[0,v(\kappa_{1},\kappa_{2}) \right].$$
That is
$\left(\gamma_{E(\kappa_{1}),E(\kappa_{2})}(t) , t \alpha +(1-t)\beta\right) \in \omega_{f} , \ \ \forall t\in\left[0,v(\kappa_{1},\kappa_{2}) \right].$\\
Therefore, $ \omega_{f} $ is a GLEI set corresponding to $ \aleph $.
From Theorem\ref{th1}, it follows that $ f $ is a GSLEP on $ B $ with respect to $ \eta $.
\end{proof} \end{theorem} \section{Optimality Criteria} \hskip0.6cm
In this section, let us consider the nonlinear fractional multiobjective programming problem such as :\\
\begin{eqnarray*} (VFP) \begin{cases}
minimize \frac{f(\kappa)}{g(\kappa)}=\left(\frac{f_{1}(\kappa)}{g_{1}(\kappa)},\cdots,\frac{f_{p}(\kappa)}{g_{p}(\kappa)} \right),\\ subject \ \ to\ \ h_{j}(\kappa)\leq0, j\in Q={1,2,\cdots q} \\ \kappa\in K_{0}
\end{cases} \end{eqnarray*} where $ K_{0}\subset \aleph $ is a GLEI set and $ g_{i}(\kappa)>0, \forall \kappa\in K_{0} , i\in P={1,2,\cdots, p} $.
Let $ f=(f_{1},f_{2},\cdots, f_{p}), g=(g_{1},g_{2},\cdots,g_{p}) $ and $ h=(h_{1},h_{2},\cdots,h_{q}) $\\ and denote $ K=\left\lbrace \kappa:h_{j}(\kappa)\leq 0, j\in Q, \kappa\in K_{0}\right\rbrace $, the feasible set of problem ($ VFP $).\\ For $ \kappa^{*}\in K $, we put $ Q(\kappa^{*})=\left\lbrace j:h_{j}(\kappa^{*})= 0, j\in Q \right\rbrace $, $ L(\kappa^{*})=\frac{Q}{Q(\kappa^{*})} $.
We also formulate the nonlinear multiobjective programming problem as follows: \begin{eqnarray*}
(VFP_{\lambda}) \begin{cases}
minimize \left( f_{1}(\kappa)-\lambda_{1}g_{1}(\kappa),\cdots f_{p}(\kappa)-\lambda_{p}g_{p}(\kappa) \right),\\ subject \ \ to\ \ h_{j}(\kappa)\leq0, j\in Q={1,2,\cdots q} \\ \kappa\in K_{0}
\end{cases}
\end{eqnarray*} where $ \lambda=(\lambda_{1},\lambda_{2},\cdots ,\lambda_{p})\in \mathbb{R}^{p} $.
The followinng lemma connects the weak efficient solutions for ($ VFP $) and ($ VFP_{\lambda} $). \begin{lemma}\label{Lemma1}
A point $ \kappa^{*} $ is a weak efficient solution for ($ VFP_{\lambda} $) iff $ \kappa^{*} $ is a weak efficient solution for ($ VFP^{*}_{\lambda} $), where $ \lambda^{*}=(\lambda^{*}_{1},\cdots,\lambda^{*}_{p} )=\left(\frac{f_{1}(\kappa^{*})}{g_{1}(\kappa^{*})},\cdots,\frac{f_{p}(\kappa^{*})}{g_{p}(\kappa^{*})} \right) $. \end{lemma} \begin{proof}
Assume that there is a feasible point $ \kappa\in K $, where
$$f_{i}(\kappa)-\lambda^{*}_{i}g_{i}(\kappa)<f_{i}(\kappa^{*})-\lambda^{*}_{i}g_{i}(\kappa^{*}),\forall i\in Q $$
$ \Longrightarrow $$$f_{i}(\kappa)<\frac{f_{i}(\kappa^{*})}{g_{i}(\kappa^{*})g_{i}(\kappa)}$$
$ \Longrightarrow $ $$\frac{f_{i}(\kappa)}{g_{i}(\kappa)}<\frac{f_{i}(\kappa^{*})}{g_{i}(\kappa^{*})},$$
which is a contradiction the weak efficiency of $ \kappa^{*} $ for ($ VFP $).
Now let us take $ \kappa\in K $ as a feasible point such that
$$\frac{f_{i}(\kappa)}{g_{i}(\kappa)}<\frac{f_{i}(\kappa^{*})}{g_{i}(\kappa^{*})}= \lambda^{*}_{i},$$ then $ f_{i}(\kappa)-\lambda^{*}_{i}g_{i}(\kappa)<0=f_{i}(\kappa^{*})-\lambda^{*}_{i}g_{i}(\kappa^{*}), \forall i\in Q $, which is agian contradiction to the weak efficiency of $ \kappa^{*} $ for ($ VFP^{*}_{\lambda} $). \end{proof} Next, some sufficient optimality conditions for the problem ($ VFP $) are established. \begin{theorem}\label{th2} Let $ \bar{\kappa}\in K, E(\bar{\kappa})=\bar{\kappa} $ and $ f,h $ be GSLEP and $ g $ be a geodesic semilocal E-preincave, and they are all geodesic E-$ \eta $- semidifferentiable at $ \bar{\kappa} $. Further, assume that there are $ \zeta^{o}=\left(\zeta^{o}_{i}, i=1,\cdots,p \right)\in\mathbb{R}^{p} $ and $ \xi^{o}=\left(\xi^{o}_{j}, j=1,\cdots,m \right)\in\mathbb{R}^{m} $ such that \begin{equation}\label{eq4} \zeta^{o}_{i}f'_{i+}\left(\gamma_{\bar{\kappa},E(\widehat{\kappa})}(t) \right)+\xi^{o}_{j} h'_{j+}\left(\gamma_{\bar{\kappa},E(\widehat{\kappa})}(t) \right)\geqslant 0\forall \kappa\in K, t\in[0,1], \end{equation} \begin{equation}\label{eq5} g'_{i+}\left(\gamma_{\bar{\kappa},E(\kappa)}(t) \right)\leq 0, \forall \kappa\in K, i\in P, \end{equation} \begin{equation}\label{eq6} \xi^{o}h(\bar{\kappa})=0 \end{equation} \begin{equation}\label{eq7} \zeta^{o}\geqslant 0 , \xi^{o}\geqslant 0. \end{equation} Then $ \bar{\kappa} $ is a weak efficient solution for ($ VFP $). \end{theorem} \begin{proof}
By contradiction, let $ \bar{\kappa} $ be not a weak efficient solution for ($ VFP $), then there exist a point $ \widehat{\kappa}\in K $ such that
\begin{equation}\label{eq8}
\frac{f_{i}(\widehat{\kappa})}{g_{i}(\widehat{\kappa})}<\frac{f_{i}(\bar{\kappa})}{g_{i}(\bar{\kappa})}, i\in P.
\end{equation}
By the above hypotheses and Lemma \ref{Lemma1}, we have
\begin{equation}\label{eq9}
f_{i}(\widehat{\kappa})-f_{i}(\bar{\kappa})\geqslant f'_{i+}\left(\gamma_{\bar{\kappa},E(\widehat{\kappa})}(t)\right) , i\in P
\end{equation}
\begin{equation}\label{eq10}
g_{i}(\widehat{\kappa})-g_{i}(\bar{\kappa})\leq g'_{i+}\left(\gamma_{\bar{\kappa},E(\widehat{\kappa})}(t)\right) , i\in P
\end{equation}
\begin{equation}\label{eq11}
h_{i}(\widehat{\kappa})-h_{i}(\bar{\kappa})\geqslant h'_{j+}\left(\gamma_{\bar{\kappa},E(\widehat{\kappa})}(t)\right) , j\in Q.
\end{equation}
Multiplying (\ref{eq9}) by $ \zeta^{o}_{i} $ and (\ref{eq11}) by $ \xi^{o}_{j} $, then we get
\begin{eqnarray}\label{eq12}&&
\sum_{i=1}^{p} \zeta^{o}_{i} \left(f_{i}(\widehat{\kappa})-f_{i}(\bar{\kappa}) \right) + \sum_{j=1}^{m} \xi^{o}_{j} \left(h_{j}(\widehat{\kappa})-h_{j}(\bar{\kappa}) \right)\nonumber\\&&\hspace{0.5in} \geqslant \zeta^{o}_{i} f'_{i+}\left(\gamma_{\bar{\kappa},E(\widehat{\kappa})}(t)\right) +\xi^{o}_{j} h'_{j+}\left(\gamma_{\bar{\kappa},E(\widehat{\kappa})}(t)\right) \geqslant 0.
\end{eqnarray}
Since $ \widehat{\kappa}\in K, \xi^{o}\geqslant 0 $ by (\ref{eq6}) and (\ref{eq12}), we have \begin{equation}\label{eq13} \sum_{i=1}^{p} \zeta^{o}_{i} \left(f_{i}(\widehat{\kappa})-f_{i}(\bar{\kappa}) \right)\geqslant 0. \end{equation} Utilizing (\ref{eq7}) and (\ref{eq13}),then there is at least an $ i_{0} $ ($ 1\leq i_{0}\leq p $) such that \begin{equation}\label{eq14} f_{i_{0}}(\widehat{\kappa})\geqslant f_{i_{0}}(\bar{\kappa}). \end{equation} On the other hand, (\ref{eq5}) and (\ref{eq10}) imply \begin{equation}\label{eq15} g_{i}(\widehat{\kappa})\leq g_{i}(\bar{\kappa}), i\in P. \end{equation} By using (\ref{eq14}), (\ref{eq15}) and $ g>0 $, we have \begin{equation}\label{eq16} \frac{f_{i_{0}}(\widehat{\kappa})}{g_{i_{0}}(\widehat{\kappa})}\geqslant\frac{f_{i_{0}}(\bar{\kappa})}{g_{i_{0}}(\bar{\kappa})}, \end{equation} which is a contradition with \ref{eq8}, then the proof of throrem is completed. \end{proof} Similarly we can prove the next theorem:
\begin{theorem}
Consider that $ \bar{\kappa}\in B, E(\bar{\kappa})=\bar{\kappa} $ and $ f,h $ are geodesic E-$ \eta $- semidifferentiable at $ \bar{\kappa} $. If there exist $ \zeta^{o}\in \mathbb{R}^{n} $ and $ \xi^{o}\in \mathbb{R}^{m} $ such that condition (\ref{eq4})-(\ref{eq7}) hold and $ \zeta^{o}f(x)+\xi^{o}h(x) $ is a GSLEP function, then $ \bar{\kappa} $ is a weak efficient solution for ($ VFP $). \end{theorem} \begin{theorem}
Consider $ \bar{\kappa}\in B, E(\bar{\kappa})=\bar{\kappa} $ and $ \lambda_{i}^{o}=\frac{f_{i}(\bar{\kappa})}{g_{i}(\bar{\kappa})}(i\in P) $ are all pSLGEP functions and $ h_{j}(\kappa)(j\in \aleph(\bar{\kappa})) $ are all GqSLEP functions and $ f,g,h $ are all geodesic E-$ \eta $-semidifferentiable at $ \bar{\kappa} $. If there is $ \zeta^{o}\in \mathbb{R}^{p} $ and $ \xi^{o}\in \mathbb{R}^{m} $ such that
\begin{eqnarray}\label{eq17}
\sum_{i=1}^{p}\zeta_{i}^{o}\left(f'_{i+}\left( \gamma_{\bar{\kappa},E(\kappa)}(t)\right) -\lambda_{i}^{o}g'_{i+}\left( \gamma_{\bar{\kappa},E(\kappa)}(t)\right) \right) +\xi^{o}h'_{i+}\left( \gamma_{\bar{\kappa},E(\kappa)}(t)\right) \geqslant 0
\end{eqnarray}
\begin{eqnarray}\label{eq18}
\xi^{o}h(\bar{\kappa})=0,
\end{eqnarray}
\begin{equation}\label{eq19}
\zeta^{o}\geqslant 0, \xi^{o}\geqslant 0,
\end{equation}
then $ \bar{\kappa} $ is a weak efficient solution for ($ VFP $). \end{theorem} \begin{proof}
Assume that $ \bar{\kappa} $ is not a weak efficient solution for ($ VFP $). Therefore, there exists $ \kappa^{*}\in B $, yields
$$\frac{f_{i}(\kappa^{*})}{g_{i}(\kappa^{*})}<\frac{f_{i}(\bar{\kappa})}{g_{i}(\bar{\kappa})}.$$ Then $$f_{i}(\kappa^{*})-\lambda_{i}^{o}g_{i}(\kappa^{*})<0,\ \ \ i\in P,$$ which means that $$f_{i}(\kappa^{*})-\lambda_{i}^{o}g_{i}(\kappa^{*})< f_{i}(\bar{\kappa})-\lambda_{i}^{o}g_{i}(\bar{\kappa})<0,\ \ \ i\in P. $$
By the pSLGEP of $ \left( f_{i}(\kappa)-\lambda_{i}^{o}g_{i}(\kappa)\right) (i\in P) $ and Lemma \ref{lemma2}, we have
$$ f'_{i+}\left( \gamma_{\bar{\kappa},E(\kappa)}(t)\right) -\lambda_{i}^{o}g'_{i+}\left( \gamma_{\bar{\kappa},E(\kappa)}(t)\right) ,\ \ \ i\in P. $$
Utilizing $\zeta^{o}\geqslant 0 $, then
\begin{equation}\label{eq20}
\sum_{i=1}^{p}\zeta_{i}^{o}\left(f'_{i+}\left( \gamma_{\bar{\kappa},E(\kappa)}(t)\right) -\lambda_{i}^{o}g'_{i+}\left( \gamma_{\bar{\kappa},E(\kappa)}(t)\right) \right)< 0.
\end{equation}
For $ h(\kappa^{*})\leq 0 $ and $ h_{j}(\bar{\kappa})= 0,\ \ \ j\in \aleph(\bar{\kappa}) $ , we have $ h_{j}(\kappa^{*})\leq h_{j}(\bar{\kappa}),\ \ \ \forall j\in \aleph(\bar{\kappa}). $
By the GqSLEP of $ h_{j} $ and Lemma \ref{lemma2}, we have
$$h_{j+}\left( \gamma_{\bar{\kappa},E(\kappa)}(t)\right) \leq 0,\ \ \ \forall j\in \aleph(\bar{\kappa}). $$
Considering $ \xi^{o}\geqslant 0 $ and $ \xi_{j}^{o}= 0,\ \ \ j\in \aleph(\bar{\kappa}), $ then
\begin{equation}\label{eq21}
\sum_{j=1}^{m}\xi_{j}^{o}h'_{j+}\left( \gamma_{\bar{\kappa},E(\kappa^{*})}(t)\right) \leq 0.
\end{equation}
Hence, by (\ref{eq20}) and (\ref{eq21}), we have
\begin{eqnarray}
\sum_{i=1}^{p}\zeta_{i}^{o}\left(f'_{i+}\left( \gamma_{\bar{\kappa},E(\kappa^{*})}(t)\right) -\lambda_{i}^{o}g'_{i+}\left( \gamma_{\bar{\kappa},E(\kappa^{*})}(t)\right) \right) +\xi^{o}h'_{i+}\left( \gamma_{\bar{\kappa},E(\kappa^{*})}(t)\right) < 0,\nonumber\\
\end{eqnarray}
which is contradiction with relation (\ref{eq17}) at $ \kappa^{*}\in B $.
Therefore, $ \bar{\kappa} $ is a weak efficient solution for ($ VFP $). \end{proof}
\begin{theorem} Consider $ \bar{\kappa}\in B, E(\bar{\kappa})=\bar{\kappa} $ and $ \lambda_{i}^{o}=\frac{f_{i}(\bar{\kappa})}{g_{i}(\bar{\kappa})}(i\in P) $. Also, assume that $ f,g,h $ are geodesic E-$ \eta $-semidifferentiable at $ \bar{\kappa} $. If there is $ \zeta^{o}\in \mathbb{R}^{p} $ and $ \xi^{o}\in \mathbb{R}^{m} $ such that the conditions (\ref{eq17})-(\ref{eq19}) hold and $ \sum_{i=1}^{p}\zeta^{o}_{i}\left(f_{i}(\kappa)-\lambda^{o}_{i}g_{i}(\kappa) \right)+\xi^{o}_{\aleph(\bar{\kappa})}h_{\aleph(\bar{\kappa})}(\kappa) $ is a GpSLEP function, then $ \bar{\kappa} $ is a weak efficient soluion for ($ VFP $). \end{theorem} \begin{corollary} Let $ \bar{\kappa}\in B, E(\bar{\kappa})=\bar{\kappa} $ and $ \lambda_{i}^{o}=\frac{f_{i}(\bar{\kappa})}{g_{i}(\bar{\kappa})}(i\in P) $. Futher let $ f, h_{\aleph(\bar{\kappa})} $ be all GSLEP function, $ g $ be a geodesic semilocal E-preincave function and $ f,g,h $ be all geodesic E-$ \eta $- semidifferentiable at $ \bar{\kappa} $. If there exist $ \zeta^{o}\in \mathbb{R}^{p} $ and $ \xi^{o}\in \mathbb{R}^{m} $ such that the conditions (\ref{eq17})-(\ref{eq19}) hold, then $ \bar{\kappa} $ is a weak efficient soluion for ($ VFP $). \end{corollary}
The dual problem for ($ VFP $) is formulated as follows \begin{eqnarray*}
(VFD) \begin{cases}
minimize \left(\zeta_{i}, i=1,2,\cdots, p \right) ,\\ subject \ \ to\ \ \sum_{i=1}^{p}\alpha_{i}\left(f'_{i+}\left( \gamma_{\lambda,E(\kappa)}(t)\right) -\zeta_{i}g'_{i+}\left( \gamma_{\lambda,E(\kappa)}(t)\right) \right)\\\hspace{1.5in} +\sum_{j=1}^{m}\beta_{j}h'_{j+}\left( \gamma_{\lambda,E(\kappa)}(t)\right) \geqslant 0 \\ \kappa\in K_{0}, t\in[0,1],\\
f_{i}(\lambda)-\zeta_{i}g_{i}(\lambda)\geqslant0,\ \ \ i\in P,
\beta_{j}h_{j}(\lambda)\geqslant 0,\ \ \ j\in \aleph,\\
\end{cases} \end{eqnarray*} where $\zeta =(\zeta_{i}, i=1,2,\cdots, p)\geqslant 0$, $\alpha =(\alpha_{i}, i=1,2,\cdots, p)> 0$,\\ $\beta =(\beta_{i}, i=1,2,\cdots, m)\geqslant 0$, $\lambda\in K_{0}. $
Denote the feasible set problem ($ VFD $) by $ K^{,} $. \begin{theorem}[General Weak Duality] Let $ \kappa\in K $, $ (\alpha,\beta,\lambda,\zeta)\in K^{,} $ and $ E(\lambda)=\lambda $. If $ \sum_{i=1}^{p}\alpha_{i}(f_{i}-\zeta_{i}g_{i}) $ is a GpSLEP function and $ \sum_{j=1}^{m}\beta_{j}h_{j} $ is a GqSLEP function and they are all geodesic E-$ \eta $-semidifferentiable at $ \lambda $, then $ \frac{f(\kappa)}{g(\kappa)}\nleq \zeta $. \end{theorem} \begin{proof} From $ \alpha>0 $ and $ (\alpha, \beta,\lambda,\zeta)\in K^{,} $, we have $$\sum_{i=1}^{p}\alpha_{i}(f_{i}(\kappa)-\zeta_{i}g_{i}(\kappa))<0\leq\sum_{i=1}^{p}\alpha_{i}(f_{i}(\lambda)-\zeta_{i}g_{i}(\lambda)). $$ By the GpSLEP of $ \sum_{i=1}^{p}\alpha_{i}(f_{i}-\zeta_{i}g_{i}) $ and Lemma \ref{lemma2}, we obtain $$\left( \sum_{i=1}^{p}\alpha_{i}(f_{i}-\zeta_{i}g_{i}) \right)'_{+}\left(\gamma_{\lambda,E(\kappa)}(t) \right) <0, $$ that is, $$\sum_{i=1}^{p}\alpha_{i}\left(f'_{i+}\left( \gamma_{\lambda,E(\kappa)}(t)\right) -\zeta_{i}g'_{i+}\left( \gamma_{\lambda,E(\kappa)}(t)\right] \right)<0. $$ Also, from $ \beta\geqslant 0$ and $ \kappa\in K $, then $$\sum_{j=1}^{m}\beta_{j}h_{j}(\kappa)\leq 0 \leq\sum_{j=1}^{m}\beta_{j}h_{j}(\lambda). $$ Using the GqSLEP of $ \sum_{j=1}^{m}\beta_{j}h_{j} $ and Lemma \ref{lemma2}, one has $$\left( \sum_{j=1}^{m}\beta_{j}h_{j} \right)'_{+}\left(\gamma_{\lambda,E(\kappa)}(t) \right) \leq 0. $$ Then $$ \sum_{j=1}^{m}\beta_{j}h'_{j+} \left(\gamma_{\lambda,E(\kappa)}(t) \right) \leq 0. $$ Therefore $$\sum_{i=1}^{p}\alpha_{i}\left(f'_{i+}\left( \gamma_{\lambda,E(\kappa)}(t)\right) -\zeta_{i}g'_{i+}\left( \gamma_{\lambda,E(\kappa)}(t)\right) \right) +\sum_{j=1}^{m}\beta_{j}h'_{j+}\left( \gamma_{\lambda,E(\kappa)}(t)\right) < 0, $$ This is a contradiction with $ (\alpha,\beta,\lambda,\zeta)\in K^{,} $. \end{proof}
\begin{theorem}
Consider that $ \kappa\in K $, $ (\alpha, \beta,\lambda,\zeta)\in K^{,} $ and $ E(\lambda)=\lambda $. If $ \sum_{i=1}^{p}\alpha_{i}(f_{i}-\zeta_{i}g_{i})+\sum_{j=1}^{m}\beta_{j}h_{j} $ is a GpSLEP function and geodesic E-$ \eta $-semidifferentiable at $ \lambda $, then $ \frac{f(\kappa)}{g(\kappa)}\nleq \zeta $. \end{theorem} \begin{theorem}[General Converse Duality] Let $ \bar{\kappa}\in K $ and $ (\kappa^{*},\alpha^{*}, \beta^{*},\zeta^{*})\in K^{,} $,$ E(\kappa^{*})=\kappa^{*} $, where $\zeta^{*}= \frac{f(\kappa^{*})}{g(\kappa^{*})}=\frac{f(\bar{\kappa})}{g(\bar{\kappa})}=(\zeta^{*}_{i}, \ \ \ i=1,2,\cdots, p) $. If $ f_{i}-\zeta_{i}^{*}g_{i} (i\in P), h_{j}(j\in \aleph)$ are all GSLEP functions and all geodesic E-$ \eta $-semidifferentiable at $ \kappa^{*} $, then $ \bar{\kappa} $ is a weak efficient solution for ($ VFP $). \end{theorem} \begin{proof}
By using the hypotheses and Lemma \ref{lemma2}, for any $ \kappa\in K $, we obtain $$\left( f_{i}(\kappa)-\zeta_{i}^{*}g_{i}(\kappa)\right) -\left(f_{i}(\kappa^{*})-\zeta_{i}^{*}g_{i}(\kappa^{*}) \right)\geqslant f'_{i+}\left( \gamma_{\kappa^{*},E(\kappa)}(t)\right) -\zeta_{i}g'_{i+}\left( \gamma_{\kappa^{*},E(\kappa)}(t)\right) $$
$$h_{j}(y)-h_{j}(\kappa^{*})\geqslant h'_{j+}\left( \gamma_{\kappa^{*},E(\kappa)}(t)\right). $$
Utilizing the fiest constraint condition for ($ VFD $), $ \alpha^{*}>0,\beta^{*}\geqslant 0, \zeta^{*}\geqslant 0 $ and the two inequalilities above, hence
\begin{eqnarray}&&
\sum_{i=1}^{p}\alpha^{*}_{i}\left(\left( f_{i}(\kappa)-\zeta_{i}^{*}g_{i}(\kappa)\right) -\left(f_{i}(\kappa^{*})-\zeta_{i}^{*}g_{i}(\kappa^{*}) \right) \right) + \sum_{j=1}^{m}\beta^{*}_{j}\left(h_{j}(\kappa)-h_{j}(\kappa^{*}) \right) \nonumber\\\hspace{0.5in} &\geqslant& \sum_{i=1}^{p}\left(f'_{i+}\left( \gamma_{\kappa^{*},E(\kappa)}(t)\right) -\zeta_{i}g'_{i+}\left( \gamma_{\kappa^{*},E(\kappa)}(t)\right) \right)\nonumber\\\hspace{0.5in}&+&\sum_{j=1}^{m}\beta^{*}_{j} h'_{j+}\left( \gamma_{\kappa^{*},E(\kappa)}(t)\right)\nonumber\\\hspace{0.5in} &\geqslant& 0.
\end{eqnarray}
In view of $ h_{j}(\kappa)\leq 0, \beta^{*}_{j}\geqslant 0, \beta^{*}_{j}h_{j}(\kappa^{*})\geqslant (j\in \aleph) $ and $ \zeta^{*}_{i}= \frac{f_{i}(\kappa^{*})}{g_{i}(\kappa^{*})}\ \ \ (i\in P) $, then
\begin{equation}\label{eq22}
\sum_{i=1}^{p}\alpha^{*}_{i}\left( f_{i}(\kappa)-\zeta_{i}^{*}g_{i}(\kappa)\right)\geqslant 0 \ \ \ \forall y\in Y .
\end{equation}
Consider that $ \bar{\kappa} $ is not a weak efficient solution for ($ VFP $). From $ \zeta^{*}_{i}= \frac{f_{i}(\bar{\kappa})}{g_{i}(\bar{\kappa})}\ \ \ (i\in P) $ and Lemma \ref{Lemma1}, it follows that $ \bar{\kappa} $ is not a weak efficient solution for ($ VFP_{\zeta^{*}} $). Hence, $ \tilde{\kappa}\in K $ such that
$$ f_{i}(\tilde{\kappa})-\zeta_{i}^{*}g_{i}(\tilde{\kappa}) <f_{i}(\bar{\kappa})-\zeta_{i}^{*}g_{i}(\bar{\kappa}) = 0,\ \ \ i\in P, $$
hence
$ \sum_{i=1}^{p}\alpha^{*}_{i}\left(f_{i}(\tilde{\kappa})-\zeta_{i}^{*}g_{i}(\tilde{\kappa}) \right)<0 $.
This is a contradiction to the inequality (\ref{eq22}). The proof of theorem is completed. \end{proof}
\end{document} | arXiv |
RSA Timing Attack on "Extra" Montgomery Reduction
In "A practical implementation of the timing attack", the authors take advantage of a timing difference that stems from "extra reductions" that occur when multiplying numbers in the Montgomery form. After implementing a toy example of this attack, I thought I understood it. I created two sets of messages, one set containing messages $m$ where $m^3$ did not require an extra reduction ($M1$) and another set where $m^3$ did require an extra reduction ($M2$). I then augmented a square-and-multiply algorithm to take advantage of Montgomery multiplication while tracking the number of extra reductions that took place over the course of the exponentiation.
This quick experiment did appear to show that the set in which $m^3$ did not require an extra reduction ($M1$) on average would have less extra reductions than the other set. However, the difference in the two averages would be much larger than one. Most of the paper seems to suggest the timing attack they exploited relied on the timing difference a single extra reduction (e.g. section 7.1).
I believe I see an attack here, but its not the exact situation that the authors describe. It sounds like the authors are suggesting that a square-and-multiply exponentiation algorithm that relied on Montgomery multiplication would use $n$ extra reductions for messages in the set $M1$ and $n+1$ extra reductions for messages in the set $M2$ which does not make sense to me intuitively. Am I misunderstanding this attack?
Edit: After comparing my attack code with others, it turns out I understood the attack just fine and my experiment was different. The difference between my results and the results from the original paper was due to my selection of the Montgomery parameters. The value of $R$ used to put numbers in Montgomery form can have a major effect on the number of reductions required per multiplication.
rsa side-channel-attack timing-attack montgomery-multiplication
kklkkl
Montgomery multiplication
Theorem (Montgomery, 1985). For any odd integer $N$ and any integer $0 \le T < N2^k$, one has: $$T 2^{-k} \equiv \frac{T + UN}{2^k} \pmod N$$ where $U = T N' \bmod 2^k$ and $N' = -N^{-1} \bmod 2^k$. Further, one has: $$\begin{cases} T+UN \propto 2^k\\ 0\le \frac{T + UN}{2^k} < 2N\end{cases}$$
The last 2 properties show that $R:=T2^{-k} \bmod{N}$ can be computed in two steps as:
$R \gets \frac{T+UN}{2^k}$
If $R \ge N$ then $R \gets R - N\qquad$ (extra subtraction)
Montgomery arithmetic represents an integer $0\le x <N$ as $\tilde{x} = x2^k \bmod N$. It defines the Montgomery multiplication of two representatives $\tilde{x}$ and $\tilde{y}$ as $$\tilde{x} \otimes \tilde{y} := (\tilde{x}\cdot\tilde{y})2^{-k} \bmod N\tag{*}$$ Observe that letting $z= x\cdot y \bmod N$, one has $\tilde{z} \equiv (x\cdot y)2^k \equiv (\tilde{x}\cdot \tilde{y})2^{-k} \equiv \tilde{x} \otimes \tilde{y}\pmod N$.
Timing attack
Suppose that each Montgomery multiplication, cf. Eq. ($^*$), is done using the above two-step process. We see that depending on $\tilde{x}$ and $\tilde{y}$, there may be an extra subtraction to get the result $\tilde{z}$ in the range $[0, N)$.
To fix ideas, suppose that the computation of $S = m^d \bmod N$ is carried out with the square-and-multiply algorithm. The attack can easily be adapted to other exponentiation algorithm. The algorithm takes as input a message $m$ and private exponent $d = (d_{\ell-1}, \dots, d_0)_2$.
$R_0 \gets \tilde{1}$; $R_1 \gets \tilde{m}$
for $i=\ell-1$ downto $0$ do
$\quad R_0 \gets R_0 \otimes R_0$
$\quad$if $d_i = 1$ then $R_0 \gets R_0 \otimes R_1$
return $\tilde{R_0}$
The goal of the attacker is to recover the value of $d$. Observe that the square-and-multiply algorithm processes the bits of $d$ from the left to the right.
Let $d = (d_{\ell-1}, \dots, d_0)_2$
At step $j$, the attacker
already knows bits $d_{\ell-1}, d_{\ell-2}, \dots, d_{j+1}$
guesses that next bit is $d_j=1$
chooses $T$ random messages $m_1, \dots, m_T$ and computes $$X_t = m_t^{(d_{\ell-1}, d_{\ell-2}, \dots, d_{j+1},1)_2} \bmod N\quad\text{for $1\le t \le T$}$$
prepares two sets $$\mathcal{S}_0 = \{m_t \mid \text{subtraction}\}\quad\text{and}\quad \mathcal{S}_0 = \{m_t \mid \text{no subtraction}\}$$
plays all messages in set $\mathcal{S}_0$ and obtains the average running time $\tau(\mathcal{S}_0)$ for an exponentiation; does the same for $\mathcal{S}_1$ and obtains $\tau(\mathcal{S}_1)$
if $\tau(\mathcal{S}_0) - \tau(\mathcal{S}_1) \not\approx 0$ then $d_j=1$ (the guess was correct); if $\tau(\mathcal{S}_0) - \tau(\mathcal{S}_1) \approx 0$ then $d_j=0$ (the guess was incorrect)
Iterate the attack to find $d_{j-1},\dots$
In the above description, set $\mathcal{S}_0$ denotes the set of messages such that there is an extra subtraction at iteration $i=j$ in Step 4 of the square-and-multiply algorithm (i.e., Montgomery multiplication $R_0 \otimes R_1$) for the computation of $X_t$; set $\mathcal{S}_0$ denotes the set of messages such that there is no extra subtraction at iteration $i=j$ for the computation of $X_t$.
Note that the attacker can evaluate the exponentiation $X_t$ by herself as she knows $d_{\ell-1}, d_{\ell-2}, \dots, d_{j+1}$. Further, since $d_j$ is assumed to be $1$, the last operation of this exponentiation $X_t$ will be a Montgomery multiplication $R_0 \otimes R_1$ (i.e., Step 4 is executed at iteration $i=j$).
If $d_j=1$ then the average time for an exponentiation for messages in set $\mathcal{S}_0$ will be greater than the average time for an exponentiation for messages in set $\mathcal{S}_1$ because there is always an extra subtraction for $\mathcal{S}_0$.
If $d_j=0$ then the average time for an exponentiation for messages in set $\mathcal{S}_0$ or in set $\mathcal{S}_1$ will be (roughly) the same because the sorting between the two sets looks like random. Remember that the attacker made the guess $d_j = 1$.
What a coincidence, I implemented this attack yesterday!
I'm executing it right now and I can tell you that the difference at each step is around 1 reduction (as the paper suggests). See for example my logs for bits 6 to 8 of a 96-bit key:
bit 6: avg M2 (#111)= 1.2072072072072073, avg M1 (#99889)= 0.1684369650311846
[1, 1, 1, 1, 1, 1, 1]
bit 7: avg M2 (#99)= 0.18181818181818182, avg M1 (#99901)= 0.16957788210328226
[1, 1, 1, 1, 1, 1, 1, 0]
bit 8: avg M2 (#98)= 1.1938775510204083, avg M1 (#99902)= 0.16858521350923905
As you can see, in bits 6 and 8, which are set to 1, the average of M2 and M1 differs in approximately 1 reduction (my proof-of-concept measures reductions for the moment, not time), while in bit 5, which is set to 0, there is almost no difference.
As far as I understand, the idea of the attack is that on each step (i.e., on each bit guess) the attacker partitions the samples in those that needed a reduction on that particular step (M2), and those that didn't (M1). You seem to think that this implies that messages in M2 would tend to have more reductions apart from the one that is isolated in the current step. However, from the numbers I got, that doesn't seem to be the case: at each step, only the 0.1% of all the samples needed a reduction (e.g., in bit 6, only 111 out of 100000 samples), which indicates that reductions are actually very rare, so most samples will have 0, 1 or perhaps 2 reductions for the whole exponentiation.
The trick is that if you are capable of identifying the moments where reductions occur (which in the attack is achieved by computing all the square-and-multiply operations for each sample, for each step), then you have a criterion for telling the samples apart. As a result, the most probable result is that the difference between M2 and M1 is due only to the presence of a reduction for the current step.
cygnusvcygnusv
$\begingroup$ "... which indicates that reductions are actually very rare, so most samples will have 0, 1 or perhaps 2 reductions for the whole exponentiation..." I think our implementations are very different, as I get far more reductions than that! My code seems to be functionally correct, as it returns the correct modexp results though! Hm. I may try rewriting my experiment and see where I may have done differently. Thanks! $\endgroup$ – kkl Oct 16 '17 at 20:35
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•https://doi.org/10.1364/OE.426510
Ultrafast and low power all-optical switching in the mid-infrared region based on nonlinear highly doped semiconductor hyperbolic metamaterials
Ebrahim Azmoudeh and Saeed Farazi
Ebrahim Azmoudeh and Saeed Farazi*
Materials and Energy Research Center, Dezful Branch, Islamic Azad University, Dezful, Iran
*Corresponding author: [email protected]
Saeed Farazi https://orcid.org/0000-0001-5460-1280
E Azmoudeh
S Farazi
Ebrahim Azmoudeh and Saeed Farazi, "Ultrafast and low power all-optical switching in the mid-infrared region based on nonlinear highly doped semiconductor hyperbolic metamaterials," Opt. Express 29, 13504-13517 (2021)
All-optical switching of nonlinear hyperbolic metamaterials in visible and near-infrared regions
Maziar Shoaei, et al.
Structural parameters of hyperbolic metamaterials controlling high-k mode resonant wavelengths
Patrick Sohr, et al.
Single-material semiconductor hyperbolic metamaterials
D. Wei, et al.
Nanophotonics, Metamaterials, and Photonic Crystals
Extinction ratios
Guided waves
Negative refractive index
Spontaneous emission
Surface plasmon polaritons
Surface plasmons
Original Manuscript: March 29, 2021
Manuscript Accepted: April 9, 2021
Article Outline
Hyperbolic metamaterials based on highly doped semiconductors
Nonlinear response and bistability in the semiconductor-based hyperbolic metamaterials
High-performance switch based on nonlinear semiconductor hyperbolic metamaterials
Figures (10)
Guided wave modes in the uniaxial anisotropic hyperbolic metamaterials (HMMs) based on highly doped semiconductor instead of metal in the mid-infrared region are investigated theoretically. The heavily doped semiconductor is used to overcome the restrictions of the conventional metal-based structures caused by the lake of tunability and high metal loss at mid-infrared wavelengths. The unit cells of our proposed metamaterial are composed of alternating layers of undoped InAs as a dielectric layer and highly doped InAs as a metal layer. We numerically study the linear and nonlinear behavior of such multilayer metamaterials, for different arrangements of layers in the parallel (vertical HMM) and perpendicular (horizontal HMM) to the input wave vector. The effect of doping concentration, metal to dielectric thickness ratio in the unit cell (fill-fraction), and the total thickness of structure on the guided modes and transmission/reflection spectra of the metamaterials are studied. Moreover, the charge redistribution due to band-bending in the alternating doped and undoped layers of InAs is considered in our simulations. We demonstrate that the guided modes of the proposed hyperbolic metamaterial can change by increasing the intensity of the incident lightwave and entering the nonlinear regime. Therefore, the transition from linear to the nonlinear region leads to high-performance optical bistability. Furthermore, the switching performance in the vertical and horizontal HMMs are inspected and an ultrafast, low power, and high extinction ratio all-optical switch is presented based on a vertical structure of nonlinear highly doped semiconductor hyperbolic metamaterials.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Confinement and control of the electromagnetic energy in the nanoscale are required to achieve ultrafast and energy-efficient all-optical nanophotonic devices [1–3]. The use of plasmonic nanostructures and metamaterials are two main approaches, to addressing these requirements [4,5]. Surface plasmon polaritons are the most promising solutions for tight light confinement in deep sub-wavelength scales [6,7]. Plasmonic nanostructures can overcome the diffraction limit of light and increase localized electromagnetic fields [8]. Therefore, the emergence of plasmonics has revolutionized the light-based technologies and enabled low power, compact, and ultra-fast applications of linear and nonlinear optical phenomena [9–11]. Plasmonic nanostructures have been extensively studied and developed in the last decade [12]. Nowadays, the optimization of plasmonic waveguides and the use of new materials have attracted significant interest for improving tunability, reducing losses, and increasing the efficiency of optical devices [13,14]. In addition to developing plasmonics knowledge, metamaterials have also been investigated to overcome the diffraction limit of light and implement high-performance devices [15–18]. Due to the unique and unusual behavior of metamaterials such as extraordinary electromagnetic properties that are not available or not easily obtained in nature, it is expected that the combination of metamaterials and plasmonic nanostructures can lead to the production of novel devices with high efficiency and unprecedented functionalities [19]. One type of plasmonic metamaterials is a uniaxial anisotropic multilayer structure with an isofrequency contour (IFC) in the form of an open hyperboloid. Such metamaterials can be implemented using alternating subwavelength layers of isotropic dielectric and metal [20]. The open hyperboloidal isofrequency contour of these metamaterials arises from the opposite signs of the principal components of their magnetic or electric tensor [21–23]. This unusual IFC profile of HMMs and their ability to manipulate and control the light at sub-wavelength scales can be utilized in high-performance nanophotonics devices [24–27]. For example, the hyperbolic metamaterials have applications in negative refraction [28], hyperlens [29], enhancement of the spontaneous emission [30], and sensing [31]. Furthermore, the plasmonic hyperbolic metamaterials can strongly enhance the nonlinear optical effects due to the highly localized fields. As a result, the plasmonic hyperbolic metamaterials with strong nonlinear Kerr effect and flexible control of electromagnetic waves by tuning the hyperbolic shape of isofrequency surface can be utilized in optical bistability and design the high-performance optical nanoswitches. However, the plasmonic hyperbolic waveguides composed of metal and dielectric layers deal with some limitations such as high metal loss and lack of tunability [32,33]. One approach to overcome these problems is to utilize graphene instead of metal in these structures. The use of graphene in plasmonic hyperbolic metamaterials leads to high tunability and reduction of propagation losses [34–36]. However, the surface plasmons frequencies of graphene typically fall in the terahertz range, and to change the plasma frequency, an external voltage must be applied [37]. In the mid-infrared range, which is widely used in optical communication systems, due to the large imaginary part of the noble metal permittivity, ohmic losses are more critical. Highly doped semiconductors are an alternative to metals that offer lower loss and better tunability of the plasma frequency in the Mid-infrared region [38–40].
In this context, the various guided wave modes in a plasmonic hyperbolic metamaterial based on a highly doped semiconductor in the mid-infrared region are investigated. The linear and nonlinear behavior of our proposed structure considering the effects of important parameters such as doping concentration, fill-fraction of the unit cells, and the total thickness of multilayer structure in two different arrangements of layers (vertical and horizontal HMMs) are studied. We show that by increasing the intensity of the input TM lightwave and the transition from linear to the nonlinear regime, the high-performances switching can be realized in our proposed nanostructure. Furthermore, a low power and high extinction ratio all-optical nanoswitch in the mid-infrared region based on nonlinear highly doped semiconductor hyperbolic metamaterials is designed. The finite difference time domain (FDTD) method has been adopted to simulate our proposed nanostructures.
2. Hyperbolic metamaterials based on highly doped semiconductors
The usual HMMs structures, which consists of metal and dielectric layers, must be modified to implement the high efficiency plasmonic hyperbolic metamaterials in the Mid-infrared region. Here, considering the high metal losses at the mid-infrared wavelength range, a doped semiconductor has been used instead of metal to improve tunability and reduce the ohmic losses. However, the use of alternating doped and undoped layers of InAs in the implementation of structures would cause the band-bending to become serious. The band-bending can have a significant effect on the performance of semiconductor-based metamaterials. Therefore, to simulate such structures by the FDTD method, we used the constituent materials and first calculated the charge distribution and the width of depletion regions by electrical analysis of the multilayer structures and solving the drift-diffusion and Poisson equations. Next, we entered the charge redistribution profile due to the band-bending into our FDTD code. Figure 1 shows the charge distribution profile for a multilayer structure consists of altering doped and undoped layers of InAs. The doping concentration of doped layers is $1 \times {10^{19}}({c{m^{ - 3}}} )$ and the thickness of layers is 100 nm. Three regions of A, B, and C in Fig. 1 show the undoped, graded, and constant doped regions, respectively. Region A has a constant permittivity and the Drude model can be used to calculate the permittivity of constant carrier density in region C. For region B, the plasma wavelength is a function of depth and is calculated using the depth-dependent charge distribution and considering the effects of the non-parabolic effective mass.
Fig. 1. Conduction band diagram (black dashed line) and charge distribution (red line) in the doped/undoped InAs superlattice calculated using Poisson solver. Region A is undoped InAs with a constant permittivity, region C is doped InAs with a constant carrier density, and region B is the graded region with depth-dependent carrier concentration.
Download Full Size | PPT Slide | PDF
Figure 2 shows two different arrangements of the proposed multilayer structure consisting of altering layers of doped InAs acting as metal and undoped InAs acting as the dielectric with the thickness of ${d_m}$ and ${d_d}$, respectively. The total thickness of structures in the direction of wave propagation is indicated by LV and LH. Figure 2(a) illustrates the schematic in which the direction of TM wave propagation is parallel to the layers (vertical structure) and Fig. 2(b) describes the structure with layers perpendicular to the direction of wave vector (horizontal structure).
Fig. 2. Schematics of the multilayer structures. (a) The direction of TM wave propagation is parallel to layers (vertical structure). (b) The layers are perpendicular to the direction of the incident wave vector (horizontal structure). ${d_m}$ and ${d_d}$ are the thickness of doped InAs as a metal layer and undoped InAs as a dielectric layer, respectively and L is the thickness of structures.
The thickness of layers is much smaller than the wavelength and an effective medium approximation (EMA) can be applied to investigate the dispersion behavior of the proposed structures [41,42] and a better understanding of the multilayer structures guided modes. But because the EMA model does not take into account the effect of band-bending, it may not be accurate enough compared to the method of the constituent materials. Our simulations, confirming the results of previous experimental works [42–44], show that with increasing layer thickness and doping density, the effect of band-bending on the metamaterial behavior is significantly reduced and the results of the EMA model will be more consistent with the results of the constituent structure. For this reason, we have chosen layer thicknesses and doping in such a way that the EMA model helps to explain the behavior of our structures with a good approximation.
In the following, the transverse and longitudinal components of permittivity are shown as pairs of $({{\varepsilon_ \bot },{\varepsilon_\textrm{||}}} )$ and $({\varepsilon_ \bot^{\prime},\; \varepsilon_\textrm{||}^{\prime}} )$ for the vertical and horizontal structures, respectively. The dielectric components in the parallel $({{\varepsilon_\textrm{||}}} )$ and perpendicular $({{\varepsilon_ \bot }} )$ directions to incident wave vector can be written as:
(1)$${\varepsilon _ \bot } = \varepsilon _{||}^{\prime} = \frac{{{\varepsilon _m}{\varepsilon _d}}}{{{f_d}{\varepsilon _m} + {f_m}{\varepsilon _d}}}$$
(2)$${\varepsilon _{||}} = \varepsilon _ \bot ^{\prime} = {f_m}{\varepsilon _m} + {f_d}{\varepsilon _d}$$
where ${\varepsilon _m}$ and ${\varepsilon _d}$ are the metal (or highly doped semiconductor) and dielectric permittivity, respectively. The permittivity of doped semiconductor ${\varepsilon _m}$ for different doping concentrations can be calculated from Drude model as follows [45].
(3)$${\varepsilon _m} = {\varepsilon _\infty } - \frac{{\omega _p^2}}{{\omega (\omega + j\gamma )}}$$
Here, ${\varepsilon _\infty }$ is the high frequency permittivity, $\gamma $ is the damping term, and ${\omega _p}$ is the plasma angular frequency given by
(4)$$\omega _p^2 = \frac{{{N_d}{q^2}}}{{{\varepsilon _0}{m^\ast }}}$$
${N_d}$, $q$, and ${m^\ast }$ are the doping concentration, the electron charge, and the electron effective mass, respectively, and ${\varepsilon _0}$ is the permittivity of free space. The effective mass of electrons in the doped InAs depends on the doping concentration. Therefore, the effective mass is calculated from Eq. (5) by empirical model as follows [46,47]:
(5)$$\Delta E = \left( {\frac{{{h^2}}}{{2{m^\ast }({N_d})}}} \right){\left( {\frac{{3{N_d}}}{{8\pi }}} \right)^{\frac{2}{3}}}$$
where $\varDelta E$ is the band-gap narrowing and ${m^\ast }({{N_d}} )$ is the effective mass as function of the doping concentration. The band-gap narrowing ($\varDelta E$) for doped InAs is calculated by the equation given in Ref. [46]. The required parameters for our studied doping concentrations are calculated in the Table 1.
Table 1. Effective mass, scattering time, and plasma wavelength for studied doping concentrations.
The coefficients ${f_m}$ and ${f_d}$ are the metal and dielectric thickness ratios to the total thickness of a unit cell and are defined as:
(6)$${f_m} = \frac{{{d_m}}}{{{d_m} + {d_d}}}$$
(7)$${f_d} = \frac{{{d_d}}}{{{d_m} + {d_d}}}$$
In such multilayer structures, the effective permittivity is anisotropic and the signs of its longitudinal and transverse components are the function of wavelength. Therefore, by increasing the wavelength to values greater than the resonance wavelength $({{\lambda_p}} )$, and solving the Maxwell equations for an incident TM wave, four different operating modes can be obtained [48]. For both vertical and horizontal structures, the different modes are Effective dielectric $({\varepsilon _ \bot },\varepsilon _ \bot ^{\prime} > 0,\; {\varepsilon _\parallel },\varepsilon _\textrm{||}^{\prime} > 0)$, effective metal $({\varepsilon _ \bot },\varepsilon _ \bot ^{\prime} < 0,\; {\varepsilon _\parallel },\varepsilon _\textrm{||}^{\prime} < 0)$, Type Ι HMM$({\varepsilon _ \bot },\varepsilon _ \bot ^{\prime} > 0,\; {\varepsilon _\parallel },\varepsilon _\textrm{||}^{\prime} < 0)$, and Type ΙΙ HMM $\left( {{\varepsilon_ \bot },\varepsilon_ \bot^{\prime}\left\langle {0,\; {\varepsilon_\parallel },\varepsilon_\textrm{||}^{\prime}} \right\rangle 0} \right)$. In the effective dielectric mode, the incident light can propagate through the structure, while in the effective metal mode, no real k vector satisfies the dispersion relation and the incident light is totally reflected. On the other hand, the signs of the longitudinal and transverse components of permittivity are opposite in both Type Ι HMM and Type ΙΙ HMM, and these modes have hyperboloidal dispersion curves in two orthogonal directions. Type Ι HMM exhibits a negative refractive index and incident light can propagate through the structure, but in Type ΙΙ HMM, such as effective metal mode the incident lightwave is reflected.
Guided modes of metamaterial are affected by some parameters of structure. The fill-fraction (FF) is a parameter that plays an important role in the optical behavior of hyperbolic metamaterials. The fill-fraction represents the thickness ratio of the metal or highly doped semiconductor in a unit cell. According to Eqs. (1) and (2), the permittivity components are the function of the fill-fraction, and as a result, it is possible to modify the guided modes by changing the fill-fraction. Here, the effect of fill-fraction on the metamaterial guided modes is studied for three different fill-fractions in a horizontal structure consisting of InAs/doped InAs layers with a doping concentration of ${N_d} = 1 \times {10^{19}}({c{m^{ - 3}}} )$ and a unit cell thickness of 200 nm. Figure 3 shows the longitudinal $({\varepsilon_\textrm{||}^{\prime}} )$ and transverse $({\varepsilon_ \bot^{\prime}} )$ components of permittivity at various wavelengths for three different fill-fractions. As displayed in Fig. 3, with a fill-fraction of 0.5, depending on the signs of parallel and perpendicular components of permittivity, two modes can occur. For wavelengths higher than the plasma wavelength$({{\lambda_p}} )$, the structure first enters in Type Ι HMM, and with further increasing of the wavelength, the Type ΙΙ HMM has appeared. However, for a fill-fraction smaller than 0.5, the effective dielectric mode has occurred before the emergence of the Type ΙΙ HMM, and for a fill-fraction larger than 0.5, an effective metal mode is observed in a wavelength window between the Type Ι HMM and Type ΙΙ HMM. Therefore, the fill-fraction modification can be applied to control the metamaterial operating modes. Moreover, Figs. 4(a)–4(b) show the transmission and reflection spectra of the mentioned horizontal metamaterial with a thickness of 1.2 μm for the various fill-fractions. In the multilayer structure with a fill-fraction of 0.6, the reflection rate has increased in the wavelength range that effective metal mode has appeared. However, in the structure with the fill-fraction of 0.4, the transmission rate has increased in the effective dielectric mode region.
Fig. 3. Longitudinal and transverse components of permittivity at different wavelengths for various fill-fractions in a horizontal structure of InAs/doped InAs with doping concentration of $1 \times {10^{19}}({c{m^{ - 3}}} )$. The effective metal and dielectric modes appear by increasing and decreasing the FF around 0.5, respectively.
Fig. 4. (a) Transmission and (b) reflection spectra in the horizontal InAs/doped InAs multilayer structure with a total thickness of 1.2 μm and the doping concentration of $1 \times {10^{19}}({c{m^{ - 3}}} )$. The transmission rate has increased in the effective dielectric region for the fill-fraction of 0.4 and the reflection rate has increased in the effective metal mode region when the fill-fraction is 0.6. The effective dielectric and effective metal regions are shown in Fig. 3.
Due to the use of doped semiconductor instead of metal in our structure, the optical properties of the metamaterial can be modified by changing the doping concentration. For vertical and horizontal metamaterials with a fill-fraction of 0.5, the transverse $({\varepsilon _ \bot },\varepsilon _ \bot ^{\prime})$ and longitudinal $({{\varepsilon_\textrm{||}},\varepsilon_\textrm{||}^{\prime}} )$ components of permittivity and related guided modes at various wavelengths for different doping concentrations (different plasma wavelengths) are sketched in Fig. 5. Increasing the resonance wavelength $({{\lambda_p}} )$ at lower doping concentrations allows for negative permittivity and hyperbolic dispersion at higher wavelengths. As shown in Fig. 5, for a horizontal structure with fill-fraction of 0.5 and total thickness of 400 nm, by increasing the wavelength of input TM wave to values slightly greater than the resonance wavelength (${\lambda _p})$, the permittivity components can be calculated as $\varepsilon _ \bot ^{\prime} > 0\; \; $ and $\varepsilon _\textrm{||}^{\prime} < 0$, thus the multilayer structure acts as the Type Ι HMM. In this mode, the metamaterial exhibits a negative refractive index, and as a result, the incident light propagates through the structure with little reflection. If the wavelength increases to higher values, the signs of parallel and perpendicular components of the permittivity change $(\varepsilon _ \bot ^{\prime} < 0\; \; $ and $\varepsilon _\textrm{||}^{\prime} > 0)$, and structure acts as Type ΙΙ HMM. In this mode the propagation of incident light is prohibited and the light is reflected. The transmission and reflection spectra of proposed horizontal metamaterial are shown in Figs. 6(a)–6(b). Also, the effects of doping concentration on the behavior of a vertical structure are considered. The Fill-fraction of vertical metamaterial is set to 0.5 and its thickness to 200 nm. In this structure due to displacement of the transverse and longitudinal components of permittivity in comparison to the horizontal structure, by increasing the wavelength to values higher than the resonance wavelength$({{\lambda_p}} )$, Type ΙΙ HMM (reflection mode) appears first, and then with further increase in the wavelength the structure enters in Type Ι HMM (transmission mode). Figures 6(c)–6(d) illustrate the transmission and reflection spectra of the vertical structure.
Fig. 5. The transverse $({\varepsilon _ \bot },\varepsilon _ \bot ^{\prime})$ and longitudinal $({{\varepsilon_\textrm{||}},\varepsilon_\textrm{||}^{\prime}} )$ components of permittivity and related guided modes at various wavelengths for different doping concentrations (different plasma wavelengths$({\lambda _p})$). The fill-fractions of the vertical and horizontal metamaterials are 0.5.
Fig. 6. (a)Transmission and (b) reflection spectra for different doping concentrations in InAs/doped InAs multilayer for a horizontal metamaterial with the total thickness of 400 nm. (c) Transmission and (d) reflection spectra for vertical metamaterial with the total thickness of 200 nm. The fill-fraction for both structures is 0.5.
The thickness of the structure (L) in the direction of the wave vector is another parameter affecting the transmission and reflection spectra of the multilayer hyperbolic metamaterials. Figures 7(a)–7(b) illustrate the transmission and reflection spectra of a horizontal HMM for various thicknesses of 200, 400, and 600 nm. The thickness of each unit cell is 200nm, the fill-fraction is 0.5, and the doping concentration is$7.5 \times {10^{19}}({c{m^{ - 3}}} )$. As expected, increasing the number of unit cells resulting in a higher thickness of structure reduces the transmission rate and increases the reflection rate in different wavelengths. Figure 7(c) shows the sum of the normalized values of transmission and reflection rates, which can be a criterion for measuring losses in the nanostructure. According to Fig. 7(c), in the wavelength range where metamaterial is in the Type Ι HMM (transmission mode), for a structure with higher thickness, due to the increasing losses in the doped semiconductor, the sum of transmission and reflection rates is reduced. Although, a structures with higher thickness in the guided mode of Type ΙΙ HMM (reflection mode) exhibit lower losses because of the higher reflection. Furthermore, the effect of higher doping concentration on losses is explained in Fig. 7(d). The higher doping concentrations result in the larger imaginary part of the permittivity and the losses during transmission are increased.
Fig. 7. The effect of the total thickness of InAs/doped InAs multilayer metamaterial on optical behavior of the horizontal structure for the doping concentration of $7.5 \times {10^{19}}({c{m^{ - 3}}} )$, (a) transmission, (b) reflection, and (c) losses at different wavelengths, respectively. (d) The losses in different doping concentrations for a horizontal structure with a total thickness of 600 nm and the fill-fraction of 0.5.
3. Nonlinear response and bistability in the semiconductor-based hyperbolic metamaterials
The strong field confinement achieved by plasmonic hyperbolic metamaterials enhances the optical nonlinear effects which are utilized in many different applications. Here, the effect of nonlinear coefficients on the permittivity of highly doped InAs layers of metamaterial are inspected. Moreover, the optical bistability and switching performance in such epsilon-near-zero (ENZ) metamaterials are investigated. The nonlinear susceptibilities that appear with increasing amplitude of the input electric field, lead to changing the permittivity components. Therefore, transition from linear to nonlinear regime can modify the guided modes of the metamaterial. Furthermore, the optical bistability and high-performance switching can be implemented by moving from a reflection mode to a transmission mode or vice versa. Our simulations demonstrate the nonlinear Kerr effect is responsible for the metamaterial behavior in the nonlinear region. The Kerr effect modifies the corresponding permittivity of InAs nanolayers as [49]
(8)$${\varepsilon _{NL}} = \varepsilon + {\chi ^{(3)}}{|{{E_0}} |^2}$$
where ${\chi ^{(3 )}}$ is the third-order nonlinear optical susceptibility of InAs, which is dependent on doping concentration, and ${E_0}$ is the field intensity. Therefore, according to Eqs. (1) and (2), the transverse and longitudinal components of permittivity of metamaterial are modified by increasing the field intensity in the nonlinear regime.
In this section, the nonlinear behavior of both vertical and horizontal HMMs at the same wavelength (12 μm) and doping concentration of $1 \times {10^{19}}c{m^{ - 3}}$ are studied. The thickness of the vertical structure in the direction of the input wave propagation is 150 nm, and the fill-fraction is 0.5. In these conditions, for low input intensities (linear regime) the metamaterial is biased in the Type ΙΙ HMM (reflection mode). By increasing the intensity of the input electric field and entering the nonlinear regime, the permittivity of doped InAs becomes positive and as a result, the operating mode changes to dielectric mode. In dielectric mode the incident TM wave propagates through the multilayer structure. Considering the nonlinear coefficients in the permittivity of layers, Fig. 8(a) shows the transverse and longitudinal components of permittivity at different wavelengths, for the linear regime with low electric field amplitude $({1\; V/m} )$ and nonlinear regime with high electric field amplitude$({{{10}^6}\; V/m} )$. Figure 8(b) illustrates the transmission and reflection spectra of the vertical structure. As seen, in the nonlinear region, in contrast to the linear regime, the predominant phenomenon is the transmission.
Fig. 8. The transverse and longitudinal components of permittivity at different wavelengths and transmission/reflection spectra at 12 μm in the linear (L) and nonlinear (NL) regimes. (a)–(b) for vertical structure with the thickness of 150 nm and fill-fraction of 0.5, and (c)–(d) for horizontal structure with the thickness of 800 nm and fill-fraction of 0.8. The doping concentration for two structures is $1 \times {10^{19}}({c{m^{ - 3}}} ).$
The second structure to be studied is an InAs/doped InAs multilayer metamaterial in which the layers are arranged horizontally (horizontal structure) in the direction of the incident TM wave. This structure is composed of 4 unit cells with a fill-fraction of 0.8 and a total thickness of 800 nm. The fill-fraction is selected so that for low input intensity in the linear regime a reflection mode can be realized at the wavelength of 12 μm. At this wavelength, the only reflection mode that can be achieved in this structure is the effective metal mode. An analysis similar to that performed for the vertical structure was repeated for this structure and results are reported in Figs. 8(c)–8(d). According to Fig. 8(c), in the nonlinear regime, with an electric field of order${10^6}({V/m} )$, the operating mode of structure changes from effective metal (reflection mode) to dielectric mode (transmission mode) and as stated in Fig. 8(d), the incident light propagates through the structure. The transmission and reflection spectra in Fig. 8(d) reveal the high losses in the horizontal multilayer structure due to high thickness of structure and high metal loss in the transition from effective metal mode to dielectric mode.
Figure 9 shows the operating modes of vertical and horizontal structures for different fill-fraction at some wavelengths in the mid-infrared region. The doping concentration for two structures is$1 \times {10^{19}}c{m^{ - 3}}$. According to Fig. 8, the effective dielectric mode, the effective metal mode, and the dielectric region (where the permittivity of highly doped semiconductor is positive) are the same for both structures, but Type Ι and Type ΙΙ HMM, which are the transmission and reflection modes respectively, have replaced. Therefore, to design an optical device such as a switch at a given wavelength using these structures, the initial mode can be determined by selecting the structure type and the fill-fraction ratio. In this study, the transition is assumed from a reflection mode in linear regime to a transmission mode in nonlinear regime, so the arrows show the transition path from reflection mode to transmission mode in Fig. 9.
Fig. 9. The guided modes of vertical and horizontal structures for different fill-fraction at some wavelengths in the mid-infrared region. The doping concentration for two structures is$1 \times {10^{19}}c{m^{ - 3}}$. Arrows indicate the transition path from reflection mode to transmission mode by changing the operating regime from linear to nonlinear.
However, it should be noted that the horizontal HMM structure that needs negative longitudinal component of permittivity ($\varepsilon _\textrm{||}^{\prime}$) for transition to Type Ι HMM, utilizes resonance to realize negative $\varepsilon _\textrm{||}^{\prime}$ and suffers high losses from it. On the other hand, high speed mode transition is required to achieve a very fast switch. Our simulations reveal that the transition from reflection mode to transmission mode due to increasing the input intensity at a given wavelength in vertical structure is faster than horizontal structure because the vertical metamaterial is a non-resonant structure that can be achieved with smaller thickness than horizontal structure. A structure with the desired modes can be selected using Fig. 9 to design a switch at a given wavelength in the mid-infrared range.
4. High-performance switch based on nonlinear semiconductor hyperbolic metamaterials
All-optical switching is an essential function to realize optical signal processing and optical communication systems. Small footprint, high-speed operation, and low power consumption are the features of an ideal optical switch [50]. In this section, a high-performance all-optical nanoswitch based on a multilayer InAs/doped InAs hyperbolic metamaterial is presented. According to the results extracted from the previous sections, an optimal structure is considered to design our proposed ultrafast and low power all-optical switch. As shown in Fig. 10(a) a vertical multilayer hyperbolic structure consists of 3 unit cells with 200 nm thickness of each cell is utilized to design the proposed switch at 13 μm. The thickness (L) of metamaterial in the direction of incident TM wave propagation is 150 nm and the doping concentration of the doped InAs layer is $1 \times {10^{19}}({{ c}{{ m}^{ - 3}}} )$. The charge distribution due to band-bending in the multilayer structure is calculated by electrical analysis and then the FDTD method is used to simulate the proposed switch. Our structure switches between the reflection mode (Off) in the linear regime and the transmission mode (On) in the nonlinear regime. In the linear regime, to set the initial mode in Type ΙΙ HMM (reflection mode), the fill-fraction is assigned to (FF=0.5). In the nonlinear regime, when the electric field amplitude increases to about ${10^6}{ \; }({{ V}/{ m}} )$, the permittivity of the doped InAs becomes positive and the operating mode changes to dielectric mode (transmission mode). This transition from Type ΙΙ HMM to dielectric mode provides a high-performance optical switch. To accurately simulate the performance of the proposed switch and prevent the field scattering, we assume that the metal contacts extend as a waveguide to the sides of the metamaterial. The electric field intensity profiles in linear and nonlinear regimes are illustrated in Figs. 10(b) and 10(c), respectively. Considering the electrical analysis and solving the drift-diffusion and Poisson equations, the effect of charge redistribution on the electric field profile has been calculated. The R and T factors indicate the normalized reflected and transmitted power (the ratio of reflection and transmission power to the input power). The results show our proposed structure is a low power and high extinction ratio all-optical switch. In terms of practical implementation, the maximum amplitude of the electric field in the nonlinear regime is notably less than the breakdown threshold of all components. Our proposed semiconductor-based metamaterial is compatible with the fabrication of semiconductor devices. Moreover, because the layers are of the same material in the multilayer structure, no issues are arising from the lattice mismatch in the deposition process. The structure consisted of alternating layers of doped and undoped InAs that can be grown by molecular beam epitaxy (MBE) [43,51]. The sputtering methods and electron-beam deposition have been reported for creating metal contacts on the semiconductor layers [44,52].
Fig. 10. (a) The schematic of our proposed switch consists of a vertical hyperbolic metamaterial. (b) The electric field intensity profile in the linear regime for field amplitude of $1\; ({V/m} )$. The structure is in the Type ΙΙ HMM (reflection mode). (c) The electric field intensity profile in the nonlinear regime for field amplitude about$\; {10^6}({V/m} )$. The structure is in the dielectric region (transmission mode). T and R show the normalized transmission and reflection, respectively.
We numerically investigate the linear and nonlinear behavior of multilayer hyperbolic metamaterials based on heavily doped semiconductors instead of metal to reduce the high ohmic losses in the mid-infrared range. The effects of fill-fraction, the doping concentration, and the thickness of multilayer on the guided modes and transmission/reflection spectra of our proposed semiconductor-based metamaterial in the vertical and horizontal structures are studied. We demonstrate that tuning the operating wavelength and guided modes are attained by changing the fill-fraction and the doping concentration. Also, the transmission and reflection rates in various wavelengths are affected by the thickness of the structure, fill-fraction, and the doping concentration. Furthermore, we show by increasing the field intensity and changing the operating regime from linear to nonlinear, the components of permittivity are modified due to the Kerr effect. Consequently, by changing the guided modes of metamaterial, optical bistability and high-performance switching are achievable. We also find that switching from reflection to transmission mode at a given wavelength needs more thickness in the horizontal structure than the vertical to realize acceptable transmission/reflection spectra and high extinction ratio. Besides, the propagation losses in horizontal structures are higher in comparison to vertical structures due to the greater thickness, higher metal loss and the need for resonance mode to obtain negative refraction. Finally, relying on the results of this research, we presented an ultrafast, low power, and high extinction ratio all-optical switch based on a vertical hyperbolic metamaterial in the mid-infrared range at the wavelength of 13 μm.
The authors declare no conflicts of interest.
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Article Order
M. Ono, M. Hata, M. Tsunekawa, K. Nozaki, H. Sumikura, H. Chiba, and M. Notomi, "Ultrafast and energy-efficient all-optical switching with graphene-loaded deep-subwavelength plasmonic waveguides," Nat. Photonics 14(1), 37–43 (2020).
[Crossref]
X. Sun, M. Z. Alam, M. Mojahedi, and J. S. Aitchison, "Confinement and Integration Density of Plasmonic Waveguides," IEEE J. Sel. Top. Quantum Electron. 21(4), 268–275 (2015).
H. Lu, Y. Gong, D. Mao, X. Gan, and J. Zhao, "Strong plasmonic confinement and optical force in phosphorene pairs," Opt. Express 25(5), 5255–5263 (2017).
A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, "Hyperbolic metamaterials," Nat. Photonics 7(12), 948–957 (2013).
A. Ciattoni, C. Rizza, A. Marini, A. Di Falco, D. Faccio, and M. Scalora, "Enhanced nonlinear effects in pulse propagation through epsilon-nearzero media," Laser Photonics Rev. 10(3), 517–525 (2016).
J. A. Wang, "Review of recent progress in plasmon-assisted nanophotonic devices," Front. Optoelectron. 7(3), 320–337 (2014).
L. Chen, T. Zhang, X. Li, and W. Huang, "Novel hybrid plasmonic waveguide consisting of two identical dielectric nanowires symmetrically placed on each side of a thin metal film," Opt. Express 20(18), 20535–20544 (2012).
W. L. Barnes, A. Dereus, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424(6950), 824–830 (2003).
M. Kauranen and A. V. Zayats, "Nonlinear plasmonics," Nat. Photonics 6(11), 737–748 (2012).
C. Argyropoulos, P. Y. Chen, G. D. Aguano, N. Engheta, and A. Alu, "Boosting optical nonlinearities in epsilon-near-zero plasmonic channels," Phys. Rev. B 85(4), 045129 (2012).
H. M. G. Wassel, D. Dai, M. Tiwari, J. Valamehr, L. Theogarajan, J. Dionne, F. Chong, and T Sherwood, "Opportunities and Challenges of Using Plasmonic Components in Nanophotonic Architectures," IEEE J. Emerg. Sel. Topics Circuits Syst. 2(2), 154–168 (2012).
J. Abed, N. S. Rajput, A. E. Moutaouakil, and M. Jouiad, "Recent Advances in the Design of Plasmonic Au/TiO2 Nanostructures for Enhanced Photocatalytic Water Splitting," Nanomaterials 10(11), 2260 (2020).
Q. Hong, F. Xiong, W. Xu, Z. Zhu, K. Liu, X. Yuan, J. Zhang, and S. Qin, "Towards high performance hybrid two-dimensional material plasmonic devices: strong and highly anisotropic plasmonic resonances in nanostructured graphene-black phosphorus bilayer," Opt. Express 26(17), 22528–22535 (2018).
A. V. Krasavin and A. V. Zayats, "Active Nanophotonic Circuitry Based on Dielectric-loaded Plasmonic Waveguides," Adv. Optical Mater. 3(12), 1662–1690 (2015).
M. Tanaka, T. Amemiya, H. Kagami, N. Nishiyama, and S. Arai, "Control of slow-light effect in a metamaterial-loaded Si waveguide," Opt. Express 28(16), 23198–23208 (2020).
X. Fang, M. L. Tseng, J. Y. Ou, K. F. MacDonald, D. P. Tsai, and N. I. Zheludev, "Ultrafast all-optical switching via coherent modulation of metamaterial absorption," Appl. Phys. Lett. 104(14), 141102 (2014).
T. Amemiya, T. Kanazawa, S. Yamasaki, and S. Arai, "Metamaterial Waveguide Devices for Integrated Optics," Materials 10(9), 1037 (2017).
E. L. Lyashko and A. I. Maimistov, "Guided waves in asymmetric hyperbolic slab waveguides: the TM mode case," J. Opt. Soc. Am. B 33(11), 2320–2330 (2016).
P.R. West, S. Ishii, G.V. Naik, N.K. Emani, V.M. Shalaev, and A. Boltasseva, "Searching for better plasmonic materials," Laser Photonics Rev. 4(6), 795–808 (2010).
P. Shekhar, J. Atkinson, and Z. Jacob, "Hyperbolic metamaterials: fundamentals and applications," Nano Convergence 1(1), 14 (2014).
V. Popov, A. V. Lavrinenko, and A. Novitsky, "Surface waves on multilayer hyperbolic metamaterials: Operator approach to effective medium approximation," Phys. Rev. B 97(12), 125428 (2018).
G. T. Papadaki, D. Fleischma, A. Davoyan, P. Yeh, and H. A. Atwater, "Optical magnetism in planar metamaterial heterostructures," Nat. Commun. 9(1), 296 (2018).
Y. H. Yang, P. F. Qin, B. Zheng, L. Shen, H. P. Wang, Z. J. Wang, E. P. Li, R. Singh, and H. S. Chen, "Magnetic Hyperbolic Metasurface," Adv. Sci. 5(12), 1801495 (2018).
Z. Guo, H. Jiang, and H. Chen, "Hyperbolic metamaterials: From dispersion manipulation to applications," Appl. Phys. 127(7), 071101 (2020).
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\begin{document}
\maketitle \begin{abstract}
Shitov recently gave a counterexample to Comon's conjecture that the symmetric tensor rank and tensor rank of a symmetric tensor are the same. In this paper we show that an analog of Comon's conjecture for the $G$-stable rank introduced by Derksen is true: the symmetric $G$-stable rank and $G$-stable rank of a symmetric tensor are the same. We also show that the log-canonical threshold of a singularity is bounded by the $G$-stable rank of the defining ideal. \end{abstract}
\section{Introduction}
An order $d$ tensor is a vector in a tensor product of $d$ vector spaces. The are several generalizations of the rank of a matrix to tensors of order $\geq 3$, for example the tensor rank, border rank, sub-rank, slice rank and $G$-stable rank. A simple tensor in $V_1\otimes V_2\otimes\cdots \otimes V_d$ is a tensor of the form $v_1\otimes v_2\otimes \cdots\otimes v_d$, where $v_i\in V_i$. The tensor rank of $T\in V_1\otimes V_2\otimes \cdots \otimes V_d$ is the smallest number of simple tensors that sum up to $T$. The $G$-stable rank of a tensor was introduced by Derksen in \cite{G-stable rank}. The slice rank and $G$-stable rank have been used to find bounds for the cap set problem (see~\cite{G-stable rank}, \cite{EllenbergGijswijt}, \cite{Jiang}, \cite{Tao}).
If $V_1=V_2=\cdots=V_d=V$ then there is a natural action of the symmetric group $S_d$ on $V^{\otimes d}=V\otimes V\otimes \cdots\otimes V$. A tensor invariant under this action is called a symmetric tensor of order $d$. The Waring rank or symmetric rank of a symmetric tensor $T\in V^{\otimes d}$ is the smallest number $d$ such that $T$ can be written as a sum of $d$ tensors of the form $v^{\otimes d}=v\otimes v\otimes \cdots \otimes v$. It is clear that the tensor rank is less than or equal to the symmetric rank. It was conjectured by Comon \cite{Comon} that the symmetric rank and tensor rank of a symmetric tensor are equal. Recently, Shitov gave a counterexample \cite{Shitov}. In this paper, we study the notion of $G$-stable rank of a tensor. The $G$-stable rank of a tensor is defined in terms of geometric invariant theory and the notion of stability for algebraic group actions on tensors. It is also natural to define a symmetric $G$-stable rank for a symmetric tensor. One main result of this paper is that the symmetric $G$-stable rank and $G$-stable rank of a symmetric tensor are the same.
In algebraic geometry and singularity theory, the log canonical threshold is an important invariant of singularities. We will show that the symmetric $G$-stable rank and the log canonical threshold are closely related. We extend the notion of $G$-stable rank to ideals in a coordinate ring of a smooth irreducible affine variety. In this context, we show
that the log canonical threshold is less than or equal to the $G$-stable rank. In the case of monomial ideals in the polynomial ring we show equality.
\subsection{Stability of tensors} Let $K$ be a perfect field, and $V$ a finite dimensional vector space over $K$, we consider the action of the group of product of special linear groups $\operatorname{SL}(V)^d = \operatorname{SL}(V)\times \operatorname{SL}(V) \times \cdots \times \operatorname{SL}(V)$ on the tensor product space $V^{\otimes d}=V\otimes V\otimes\cdots\otimes V$. A 1-parameter subgroup of an algebraic group $G$ is a homomorphism of algebraic groups $\lambda:{\mathbb G}_m\to G$, where $\mathbb G_m$ is the multiplicative group. For any integer $m$, we define the multiple of $\lambda$ by $m$, denoted by $m\cdot\lambda$, which is also a 1-parameter subgroup with $(m\cdot\lambda)(t) = (\lambda(t))^m$. A 1-parameter subgroup is {\em indivisible} if it is not a multiple of any other 1-parameter subgroup with factor $m\geq 2$. We say a tensor $v\in V^{\otimes d}$ is $\operatorname{SL}(V)^d$-{\em unstable} if there is a 1-parameter subgroup $\lambda: \mathbb G_m\to \operatorname{SL}(V)^d$, such that \[ \lim_{t\to 0}\lambda(t)\cdot v = 0. \]
If no such 1-parameter subgroup exists, then $v$ is called $\operatorname{SL}(V)^d$-{\em semistable}. Let $G$ be a reductive algebraic group over $K$. By a $G$-scheme $X$ we mean a separated, finite type scheme $X$ over $K$ as well as a morphism $G\times X\to X$ mapping $(g,x)$ to $g\cdot x$, such that $g\cdot(h\cdot x) = (gh)\cdot x$, for all $g,h\in G$ and for all $x\in X$. A morphism $f: X\to Y$ between two $G$-schemes $X$ and $Y$ is $G$-equivariant if for all $g\in G$ and $x\in X$, we have $f(g\cdot x) = g\cdot f(x)$. A subscheme $S$ of a $G$-scheme $X$ is called a $G$-subscheme if $S$ is a $G$-scheme and the immersion $S\hookrightarrow X$ is $G$-equivariant.
Throughout this paper, we will work over a perfect field $K$. In \cite{Kempf}, Kempf proved a $K$-rational version of the \textit{Hilbert-Mumford} criterion:
\begin{theorem}[\cite{Kempf}, Corollary 4.3] \label{HM}
Let $G$ be a reductive algebraic group. Suppose that $X$ is a $G$-scheme and $x\in X$ is a $K$-point. Assume $S$ is a closed $G$-subscheme of $X$ which does not contain $x$ and $S$ meets the closure of the orbit $G\cdot x$. Then there exits a $K$-rational 1-parameter subgroup $\lambda :\mathbb G_m\to G$, such that \[ \lim_{t\to 0}\lambda(t)\cdot x\in S. \] \end{theorem}
If $G = \operatorname{SL}(V)^d$, $X = V^{\otimes d}$ , then by Theorem \ref{HM}, $v$ is unstable if and only if $0$ is in the closure of the orbit of $v$, i.e. $0\in \overline{\operatorname{SL}(V)^d\cdot v}$.
A tensor $T\in V^{\otimes d}$ is called symmetric if it is invariant under the action of symmetric group $S_d$. Let $D^d V\subseteq V^{\otimes d}$ be the space of symmetric tensors. As a representation of $\operatorname{GL}(V)$, this is the space of divided powers, which is isomorphic to the $d$-th symmetric power $S^dV$ if the characteristic of $K$ is $0$ or $>d$.
It is interesting to look at the diagonal action of $\operatorname{SL}(V)$ on ${D}^dV$ via the diagonal embedding: \begin{equation} \Delta: \operatorname{SL}(V)\hookrightarrow \operatorname{SL}(V)^d. \end{equation}
\begin{definition}\label{ss} A symmetric tensor $v\in {D}^dV$ is $\operatorname{SL}(V)$-\textit{unstable} if there is a 1-parameter subgroup $\lambda: \mathbb G_m\to \operatorname{SL}(V)$, such that \[ \lim_{t\to 0}\lambda(t)\cdot v = 0. \] Otherwise we say $v$ is $\operatorname{SL}(V)$-{\em semistable}. \end{definition}
\subsection{G-stable rank for tensors}
In \cite{G-stable rank}, Derksen introduced $G$-stable rank for tensors. Suppose the base field $K$ is perfect. If $\lambda:\mathbb{G}_m\to \operatorname{GL}_n$ is a $1$-parameter subgroup, then we can view $\lambda(t)$ as an invertible $n\times n$ matrix whose entries lie in the ring $K[t,t^{-1}]$ of Laurent polynomials. We say that $\lambda(t)$ is a polynomial $1$-parameter subgroup of $\operatorname{GL}_n$ if all these entries lie in the polynomial ring $K[t]$.
Consider the action of the group $G = \operatorname{GL}(V_1)\times \operatorname{GL}(V_2)\times\cdots\times \operatorname{GL}(V_d)$ on the tensor product space $W = V_1\otimes V_2\otimes\cdots\otimes V_d$. A 1-parameter subgroup $\lambda:\mathbb{G}_m\to G$ can be written as \[ \lambda(t) = (\lambda_1(t), \cdots, \lambda_d(t)), \] where $\lambda_i(t)$ is a $1$-parameter subgroup of $\operatorname{GL}(V_i)$ for all $i$. We say that $\lambda(t)$ is polynomial if and only if $\lambda_i(t)$ is a polynomial $1$-parameter subgroup for all $i$.
The $t$-valuation $\operatorname{val}(a(t))$ of a polynomial $a(t)\in K[t]$ is the biggest integer $n$ such that $a(t) = t^nb(n)$ for some $b(t)\in K[t]$. For $a(t), b(t)\in K[t]$, the $t$-valuation $\operatorname{val}\big(\frac{a(t)}{b(t)}\big)$ of the rational function $\frac{a(t)}{b(t)}\in K(t)$ is $\operatorname{val}\big(\frac{a(t)}{b(t)}\big) = \operatorname{val}(a(t)) - \operatorname{val}(b(t))$. For a tuple $u(t) = (u_1(t),u_2(t),\cdots, u_d(t))\in K(t)^d$, we define the $t$-valuation of $u(t)$ as \begin{equation}\label{fval}
\operatorname{val}(u(t)) = \min_i\{\operatorname{val}(u_i(t))| 1\le i\le d\}. \end{equation}
If $\lambda$ is a $1$-parameter subgroup of $G$ and $v\in W$ is a tensor, then we have $\lambda(t)\cdot v\in K(t)\otimes W$. We view $K(t)\otimes W$ as a vector space over $K(t)$ and define the $t$-valuation $\operatorname{val}(\lambda(t)\cdot v)$ as in \eqref{fval}. Assume $\operatorname{val}(\lambda(t)\cdot v) > 0$, then for for any $\alpha = (\alpha_1,\alpha_2,\cdots,\alpha_d)\in \mathbb R^d_{>0}$, we define the slope \begin{equation}
\mu_\alpha(\lambda(t), v) = \frac{\sum_{i = 1}^d\alpha_i \operatorname{val}(\det(\lambda_i(t)))}{\operatorname{val}(\lambda(t)\cdot v)}. \end{equation}
The $G$-stable rank for $v\in W$ is the infimum of the slope with respect to all such 1-parameter subgroups. More precisely:
\begin{definition}[\cite{G-stable rank}, Theorem 2.4] If $\alpha\in \mathbb R^d_{>0}$, then the $G$-stable rank $\operatorname{rk}^G_\alpha(v)$ is the infimum of $\mu_\alpha(\lambda(t), v)$ where $\lambda(t)$ is a polynomial 1-parameter subgroup of $G = \operatorname{GL}(V_1)\times\cdots\times \operatorname{GL}(V_n)$ and $\operatorname{val}(\lambda(t)\cdot v) > 0$. If $\alpha = (1,1,\cdots, 1)$, we simply write $\operatorname{rk}^G(v)$. \end{definition}
Let $V$ be a finitely dimensional vector space over $K$. Let ${D}^dV\subset V^{\otimes d}$ be the space of all symmetric tensors. Assume the group $\operatorname{GL}(V)$ acts on ${D}^dV$ via the diagonal embedding: $\operatorname{GL}(V) \ \hookrightarrow \operatorname{GL}(V)^d$. \begin{definition} Let $v\in {D}^dV$ be a symmetric tensor, the symmetric $G$-stable rank $\operatorname{symmrk}^G(v)$ of $v$ is the infimum of $\mu(\lambda(t),v) = d\frac{\operatorname{val}(\det(\lambda(t)))}{\operatorname{val}(\lambda(t)\cdot v)}$, where $\lambda(t)$ is a polynomial 1-parameter subgroup of $\operatorname{GL}(V)$ and $\operatorname{val}(\lambda(t)\cdot v) > 0$. \end{definition} Since any 1-parameter subgroup of $\operatorname{GL}(V)$ is also a 1-parameter subgroup of $\operatorname{GL}(V)^d$ via the diagonal embedding, we have $\operatorname{symmrk}^G(v)\ge \operatorname{rk}^G(v)$ for any $v\in {D}^d V$. It turns out that the other inequality is also true, \begin{theorem}\label{second result} Let $v\in {D}^d V$ be a symmetric tensor, then we have \[ \operatorname{symmrk}^G(v) = \operatorname{rk}^G(v). \] \end{theorem}
\begin{example} Suppose that $V=K^2$, and $v = e_2\otimes e_1\otimes e_1+e_1\otimes e_2\otimes e_1+e_1\otimes e_1\otimes e_2\in V^{\otimes 3}$, where $\{e_1,e_2\}$ is the standard basis of $V=K^2$. Let $\lambda(t) = \begin{pmatrix} t&0\\ 0 &1 \end{pmatrix}$ be a polynomial 1-parameter subgroup of $\operatorname{GL}(K^2)$. Then $\lambda(t)\cdot v = t^2 v$, $\det(\lambda(t)) = t$, the slope is \[ \mu(\lambda(t), v) = 3\frac{\operatorname{val}(\det(\lambda(t)))}{\operatorname{val}(\lambda(t)\cdot v)}=\frac{3}{2}. \]
Therefore we have $\operatorname{symmrk}^G(v)\le \frac{3}{2}$. It was proved in \cite{G-stable rank} that $\operatorname{rk}^G(v)=\frac{3}{2}$. Hence by the fact $\operatorname{symmrk}^G(v)\ge \operatorname{rk}^G(v)$ we have $\operatorname{symmrk}^G(v) = \frac{3}{2}$.
\end{example}
A 1-parameter subgroup of $\operatorname{SL}(V)$ is also a 1-parameter subgroup of $\operatorname{SL}(V)^d$ via the diagonal embedding. It follows that if a symmetric tensor $v\in {D}^d V$ is $\operatorname{SL}(V)$-unstable, then $v$ is also $\operatorname{SL}(V)^d$-unstable. Equivalently, if $v$ is $\operatorname{SL}(V)^d$-semistable, then $v$ is also $\operatorname{SL}(V)$-semistable. It follows from Theorem \ref{second result} that the converse direction is also true:
\begin{corollary}\label{first result}
Let $v\in {D}^d V$ be a symmetric tensor, then $v$ is $\operatorname{SL}(V)^d$-semistable if and only if it is $\operatorname{SL}(V)$-semistable.
\end{corollary}
\subsection{G-stable rank for ideals and log canonical threshold}
Let $V$ be an $n$-dimensional vector space over a perfect field $K$. By choosing a basis of $V$ and a dual basis $\{x_1,x_2,\dots,x_n\}$ of $V^\star$, we have an isomorphism of algebras $SV^\star\cong K[x_1,\cdots, x_n]$, where $SV^\star$ is the symmetric algebra on the vector space $V^\star$.
We have defined the symmetric $G$-stable rank for symmetric tensors, it is natural to extend this idea to polynomials and more generally to ideals in the polynomial ring $K[x_1,\cdots,x_n]$. Furthermore, let $X$ be a smooth irreducible affine variety with coordinate ring $K[X]$, and let $\mathfrak a\subset K[X]$ be an ideal. We can define the $G$-stable rank $\operatorname{rk}^G(P, \mathfrak a) $ for the ideal $\mathfrak a$ at a point $P\in V(\mathfrak a)$. We postpone the precise definition of $G$-stable rank for ideals to Section~\ref{ideal rank}. It turns out that the $G$-stable rank $\operatorname{rk}^G(P, \mathfrak a)$ of an ideal $\mathfrak a$ at $P$ is closely related to the {\em log canonical threshold} $\text{\rm lct}_P(\mathfrak a)$ of the ideal $\mathfrak a$ at the point $P\in V(\mathfrak a)$.
Log canonical threshold is an invariant of singularities in algebraic geometry, \cite{Mus} gives a comprehensive introduction to this subject. Let $K = \mathbb C$ be the complex field. Let $H\subset \mathbb C^n$ be a hypersurface defined by a polynomial $f\in \mathbb C[x_1,\cdots,x_n]$, and let $P\in H$ be a closed point. The log canonical threshold $\text{\rm lct}_P(f)$ of $f$ at the point $P$ tells us how singular $f$ is at the point $P$. More precisely, $\text{\rm lct}_P(f)$ is a rational number bounded above by 1, and equal to 1 if $P$ is a smooth point of $H$.
There are several equivalent ways to define the log canonical threshold, here we give an analytic definition, which we will use later. \begin{definition}\label{lct} Let $X$ be a smooth irreducible affine variety. Let $\mathfrak a = (f_1,\cdots,f_r) \subset \mathbb C[X]$ be an ideal, and $P\in V(\mathfrak a)$ is a closed point. The {\em log canonical threshold} $\text{\rm lct}_P(\mathfrak a)$ of the ideal $\mathfrak a$ at $P$ is \begin{equation}
\text{\rm lct}_P(\mathfrak a) = \sup\Big\{s>0\ \Big|\ \frac{1}{(\sum_{i = 0}^r|f_i|^2)^s} \
\text{is}\ \text{integrable}\ \text{around}\ P\Big\}. \end{equation} \end{definition}
The log canonical threshold $\text{\rm lct}_P(f)$ of a polynomial $f\in \mathbb C[X]$ is the log canonical threshold of the principle ideal $\mathfrak a = (f)$. We have the following relation between the log canonical threshold and the $G$-stable rank:
\begin{theorem}\label{relation} In the situation of Definition \ref{lct}, the log canonical threshold of $\mathfrak a$ is less than or equal to the $G$-stable rank of $\mathfrak a$ at $P$: \begin{equation} \text{\rm lct}_P(\mathfrak a) \le \operatorname{rk}^G(P,\mathfrak a). \end{equation} \end{theorem}
When $\mathfrak a$ is a monomial ideal, i.e. $\mathfrak a$ is generated by monomials, the equality holds.
\begin{theorem}\label{equal} Suppose $\mathfrak a\subset \mathbb C[x_1,\cdots,x_n]$ is a proper nonzero ideal generated by monomials and $P=(0,\cdots,0)$ is the origin. Then we have \begin{equation} \text{\rm lct}_P(\mathfrak a) = \operatorname{rk}^G(P, \mathfrak a). \end{equation} \end{theorem}
\section{ Kempf's theory of optimal subgroups}\label{Kempf Results} Let $G$ be a reductive algebraic group over a perfect field $K$. In our case, $G$ is one of $\operatorname{SL}(V)^d, \operatorname{SL}(V), \operatorname{GL}(V)^d$ or $\operatorname{GL}(V)$, depend on the situation. Let $\Gamma(G)$ denote the set of all 1-parameter subgroups of $G$. In \cite{Kempf}, Kempf provided a way to approach the boundary of an orbit. Following~\cite{Kempf}, we have the definition:
\begin{definition}
Let $X$ be a $G$-scheme over a perfect field $K$ and let $x\in X$ be a $K$-point. We define $ |X,x|$ to be the set of all 1-parameter subgroups of $G$ such that $\lim_{t\to 0}\lambda(t)\cdot x$ exists in $X$.
Assume $S$ is a $G$-invariant closed sub-scheme of X not containing $x$, we define a subset $|X,x|_S\subset |X,x|$ by \begin{equation}
|X,x|_S =\{\lambda\in |X,x|\mid\lim_{t\to 0}\lambda(t)\cdot x \in S\}. \end{equation}
\end{definition}
\begin{remark}
If $S\cap \overline{G\cdot x}\ne \varnothing$, then by Theorem \ref{HM}, there exists a 1-parameter subgroup $\lambda(t)\in \Gamma(G)$, such that $\lim_{t\to 0}\lambda(t)\cdot x\in S$. Hence $|X,x|_S\ne \varnothing$. If $X = V^{\otimes d}$, $S = 0$ and $v\in V^{\otimes d}$ is $\operatorname{SL}(V)^d$-unstable, then $|V^{\otimes d}, v|_{\{0\}}\ne \varnothing$. \end{remark}
Let $\lambda\in |X,x|$ be a 1-parameter subgroup of $G$, we get a morphism $\phi_\lambda:\mathbb A^1\to X$ by $\phi_\lambda(t) = \lambda(t)\cdot x$ if $t\neq 0$ and $\phi_\lambda(0)=\lim_{t\to 0} \lambda(t)\cdot x$. Assume $S$ is a $G$-invariant closed sub-scheme of X not containing $x$, the inverse image $\phi_\lambda^{-1}(S)$ is an effective divisor supported inside $t=0$. Let $a_{S,x}(\lambda)$ denote the degree of the divisor $\phi_\lambda^{-1}(S)$ for $\lambda\in |X,x|$. Note that we have a natural conjugate action of $G$ on the set of 1-parameter subgroups $\Gamma(G)$ by $(g\cdot \lambda)(t) = g\lambda(t)g^{-1}$, where $g\in G$, $\lambda\in \Gamma(G)$.
\begin{definition}\label{def:length}
A {\it length} function $\|\cdot\|$ is a non-negative real-valued function on $\Gamma(G)$ such that \begin{enumerate}
\item $\|g\cdot\lambda\| = \|\lambda\|$ for any $\lambda\in\Gamma(G)$ and $g\in G$.
\item For any maximal torus $T \subseteq G$, we have $\Gamma(T)\subseteq \Gamma(G)$, the restriction of $\|\cdot\|$ on $\Gamma(T)$ is integral valued and extends to a {\it norm} on the vector space $\Gamma(T)\otimes_\mathbb Z \mathbb R$.
\end{enumerate} \end{definition}
\begin{remark}\label{len_exists} Such a length function exists. Let $T$ be a maximal torus of $G$. Let $N$ be the normalizer of $T$. Then the Weyl group with respect to $T$ is defined by $W = N/T$. By the fact that $\Gamma(G)/G\cong \Gamma(T)/W$, it suffices to define a $W$-invariant norm on $\Gamma(T)\otimes_\mathbb Z \mathbb R$. Since $W$ is a finite group, any norm on $\Gamma(T)\otimes_\mathbb Z \mathbb R$ and then average over $W$ will work. \end{remark}
\begin{remark}
In the original paper \cite{Kempf}, Kempf defined a length function $\|\cdot\|$ that satisfies a different condition (2): for any maximal torus $T$ of $G$, there is a positive definite integral-valued bilinear form $(\ ,\ )$ on $\Gamma(T)$, such that $(\lambda,\lambda) = \|\lambda\|^2$ for any $\lambda$ in $\Gamma(T)$. But the proof in \cite{Kempf} of the theorem below is also valid for our slightly weaker definition of length function.
\end{remark}
\begin{theorem}[Kempf \cite{Kempf}]\label{main theorem}
Let $X$ be an affine $G$-scheme over a perfect field $K$. Let $x\in X$ be a $K$-point. Assume $S$ is an $G$-invariant closed sub-scheme not containing $x$ such that $S\cap \overline{G\cdot x} \ne \varnothing$. Fix a length function $\|\cdot\|$ on $\Gamma(G)$, then we have \begin{enumerate}
\item The function $\frac{a_{S,x}(\lambda)}{\|\lambda\|}$ has a maximum positive value $B_{S,x}$ on the set of non-trivial 1-parameter subgroups in $|X,x|$.
\item Let $\Lambda_{S,x}$ be the set of indivisible 1-parameter subgroups $\lambda\in |X,x|$ such that $a_{S,x}(\lambda)=B_{S,x}\cdot \|\lambda\|$, then we have
\begin{enumerate}
\item $\Lambda_{S,x}\ne\varnothing$.
\item For $\lambda\in \Lambda_{S,x}$, Let $P(\lambda)=\{g\in G|\lim_{t\to 0} \lambda(t)\cdot g\cdot\lambda(t)^{-1} \ \mbox{exists}\}$, then $P(\lambda)$ is a parabolic subgroup and independent of $\lambda$. We denote it by $P_{S,x}$.
\item Any maximal torus of $P_{S,x}$ contains a unique member of $\Lambda_{S,x}$.
\end{enumerate} \end{enumerate} \end{theorem}
\begin{comment} \section{Stability of symmetric tensors}\label{stabilitysymmetric}
We now proceed to the proof of Theorem~\ref{first result}, which says that a symmetric tensor $v\in {D}^dV$ is $\operatorname{SL}(V)^d$-unstable if and only if it is $\operatorname{SL}(V)$-unstable. Let $G=\operatorname{SL}(V)^d$, $X= V^{\otimes d}$, $S=\{0\}$ and let $v\in {D}^dV\subset V^{\otimes d}$ be a symmetric tensor. Recall the definition of $t$-valuation in equation (\ref{val}), we have the following result:
\begin{lemma} Let $v\in {D}^dV\subset V^{\otimes d}$ be a symmetric tensor, and $G = \operatorname{SL}(V)^d$ acts on $V^{\otimes d} $ in the usual way, then \begin{enumerate}
\item $|X,v| = \{\lambda\in \Gamma(G)\mid \operatorname{val}(\lambda(t)\cdot v)\ge 0\}$.
\item $|X,v|_{\{0\}} = \{\lambda\in \Gamma(G)\mid \operatorname{val}(\lambda(t)\cdot v)> 0\}$.
\item $a_{\{0\},v}(\lambda) = \operatorname{val}(\lambda(t)\cdot v)$ for $\lambda\in |X,v|$. \end{enumerate} \end{lemma}
Let $S_d$ be the symmetric group on the set of $d$ elements, then there is a natural action of $S_d$ on $V^{\otimes d}$ by permuting the components. The symmetric group $S_d$ also acts on $G=\operatorname{SL}(V)^d$ and $\Gamma(G)$ by permutation. We will show that the function $\frac{a_{S,x}(\lambda)}{\|\lambda\|}$ behaves well under the action of $S_d$.
\begin{lemma}\ \begin{enumerate}
\item There exists a length function $\|\cdot\|: \Gamma(G)\to \mathbb R$ which is invariant under the action of $S_d$.
\item $a_{\{0\},v}(\lambda)=a_{\{0\},v}(\sigma(\lambda))$ for any $\lambda\in |X,v|$ and $\sigma\in S_d$. \end{enumerate} \end{lemma}
\begin{proof}\ \begin{enumerate}
\item Since $S_d$ is a finite group, we take any length function and then averaging over $S_d$ gives us the desired length function.
\item Since $v$ is a symmetric tensor, $\sigma(v) = v$,
\[
\operatorname{val}(\sigma(\lambda(t))\cdot v) = \operatorname{val}(\sigma(\lambda(t))\cdot \sigma(v))=\operatorname{val}(\sigma(\lambda(t)\cdot v)) = \operatorname{val}(\lambda(t)\cdot v)
\]
Therefore, we get $a_{\{0\},v}(\lambda)=a_{\{0\},v}(\sigma(\lambda))$. \end{enumerate} \end{proof}
Recall that $\Lambda_{\{0\},v}$ is the set of indivisible one parameter subgroups $\lambda\in |V^{\otimes d},v|$ such that $\frac{a_{S,x}(\lambda)}{\|\lambda\|}$ attains the maximal value. We have the following immediate consequence:
\begin{corollary}
For a $S_d$-invariant length function $\|\quad \|$ we have \begin{enumerate}
\item $\Lambda_{\{0\},v}$ is invariant under $S_d$.
\item Since $G = \operatorname{SL}(V)^d$, the parabolic subgroup $P_{\{0\},v}$ is also a product of parabolic subgroups of $\operatorname{SL}(V)$, and $P_{\{0\},v}$ is $S_d$ invariant. In other words, $P_{\{0\},v}=P^d\subset \operatorname{SL}(V)^d$ for some parabolic subgroup $P$ of $\operatorname{SL}(V)$. \end{enumerate} \end{corollary}
\begin{proof}\ \begin{enumerate}
\item Since both $a_{\{0\},v}(\lambda)$ and the length function are $S_d$ invariant, we have
$$\frac{a_{\{0\},v}(\lambda)}{\|\lambda\|}=\frac{a_{\{0\},v}(\sigma(\lambda))}{\|\sigma(\lambda)\|}$$
for all $\sigma\in S_d$ and $\lambda\in \Lambda_{\{0\},v}$. Therefore if $\lambda$ lies in $\Lambda_{\{0\},v}$, so does $\sigma(\lambda)$.
\item By definition, $S_d$ acts on $P_{\{0\},v}$ and $\lambda = (\lambda_1,\cdots,\lambda_d)$ both by permuting the components. Therefore for any $\sigma\in S_d$, we have
\[
\sigma(P_{\{0\},v}) = P(\sigma(\lambda)) = P_{\{0\},v}
\]
We used the fact that $P_{\{0\},v}=P(\lambda)$ is independent of $\lambda\in\Lambda_{\{0\},v}$. \end{enumerate} \end{proof}
Corollary~\ref{first result} is equivalent to the following corollary. \begin{corollary} For a symmetry tensor $v\in {D}^dV\subset V^{\otimes d}$, $v$ is $\operatorname{SL}(V)^d$-unstable if and only if it is $\operatorname{SL}(V)$-unstable. \end{corollary} \begin{proof}
If $v$ is $\operatorname{SL}(V)$-unstable, then it is clear $v$ is $\operatorname{SL}(V)^d$-unstable. Assume $v$ is $\operatorname{SL}(V)^d$-unstable, then there exists a one parameter subgroup $\lambda$ of $\operatorname{SL}(V)^d$, such that $\lim_{t\to 0}\lambda(t)\cdot v = 0$, hence $|V^{\otimes d}, v|\ne \varnothing$. By Theorem \ref{main theorem} and previous corollary, we have $\Lambda_{\{0\},v}\ne \varnothing$, and the parabolic subgroup $P_{\{0\},v}$ is of the form $P^d$ for some parabolic subgroup $P$ of $\operatorname{SL}(V)$. Let $T$ be a maximal torus of $P$, then $T^d$ is a maximal torus of $P_{\{0\},v}$. Then by Theorem \ref{main theorem}, $T^d$ contains a unique member of $\Lambda_{\{0\},v}$, say $\lambda=(\lambda_1,\cdots,\lambda_d)\in T^d$, then $\gamma = \Pi_{i=1}^d\lambda_i$ is a one parameter subgroup of $\operatorname{SL}(V)$, and we have $\lim_{t\to 0}\gamma(t)\cdot v = 0$ by the diagonal embedding, which means that $v$ is also $\operatorname{SL}(V)$-unstable. This completes the proof. \end{proof}
\end{comment}
\section{$G$-stable rank and symmetric $G$-stable rank}\label{equal ranks} Let $V$ be an $n$ dimensional vector space over $K$, fix a maximal torus $T$ of $\operatorname{GL}(V)$, we have an isomorphism $\Gamma(T)\cong \mathbb Z^n$. Any 1-parameter subgroup $\lambda$ of the maximal torus $T$ is given by a tuple of $n$ integers $(\nu_1,\cdots,\nu_n)$, we define a function on $\Gamma(T) \cong \mathbb Z^n$ by \begin{equation}\label{len1}
\|\lambda\| = \sum_{i = 1}^n|\nu_i|. \end{equation} This function extends linearly to a norm on the vector space $\Gamma(T)\otimes_{\mathbb Z} \mathbb R$. The Weyl group of $\operatorname{GL}(V)$ with respect to $T$ is the symmetric group $S_n$. It is clear that the function is invariant under the action of $S_n$ by permutation, therefore by Remark \ref{len_exists}, it defines a length function on $\Gamma(\operatorname{GL}(V))$. Let $G = \operatorname{GL}(V)^d$, fix a maximal torus $T_i\subset\operatorname{GL}(V)$ for each component of $\operatorname{GL}(V)^d$, then $T = T_1\times\cdots\times T_d$ is a maximal torus of $G$. We have $\Gamma(T) \cong (\mathbb Z^n)^d$, Let $\lambda = (\lambda_1,\cdots,\lambda_d)$ be a 1-parameter subgroup of $T$, where \[ \lambda_i=(\lambda_{i,1},\cdots,\lambda_{i, n}), \ \lambda_{i, j} \in \mathbb{Z}\ \text{for\ all}\ j \] is a tuple of $n$ integers.
We define a function on $\Gamma(T)$ by
\begin{equation}\label{len2}
\|\lambda\| = \sum_{i = 1}^d \|\lambda_i\|,
\end{equation}
where $\|\lambda_i\| = \sum_{j = 1}^n |\lambda_{i, j}|$. This extends to a length function on $\Gamma(G) = \Gamma(\operatorname{GL}(V)^d)$.
Let $G=\operatorname{GL}(V)^d$, $X= V^{\otimes d}$ and $S=\{0\}$. Recall the definition of $t$-valuation in equation~(\ref{fval}).
\begin{lemma} Let $v\in {D}^dV\subset V^{\otimes d}$ be a symmetric tensor, and $G = \operatorname{GL}(V)^d$ acts on $V^{\otimes d} $ in the usual way. If $\lambda(t)$ is a 1-parameter subgroup of $G$, then \begin{enumerate}
\item $|X,v| = \{\lambda\in \Gamma(G)\mid \operatorname{val}(\lambda(t)\cdot v)\ge 0\}$.
\item $|X,v|_{\{0\}} = \{\lambda\in \Gamma(G)\mid \operatorname{val}(\lambda(t)\cdot v)> 0\}$.
\item $a_{\{0\},v}(\lambda) = \operatorname{val}(\lambda(t)\cdot v)$ for $\lambda\in |X,v|$. \end{enumerate} \end{lemma}
\begin{proof}
This follows immediately from the definition.
\end{proof} \begin{lemma}\label{>=0}
Let $v\in {D}^dV\subset V^{\otimes d}$ be a symmetric tensor, then the function $\frac{\operatorname{val}(\lambda(t)\cdot v)}{\|\lambda\|}: \Gamma(G)\to \mathbb{R} $ attains its maximal value at some 1-parameter subgroup $\lambda\in \Gamma(\operatorname{GL}(V)^d)$. There exists a maximal torus $T\subset \operatorname{GL}(V)^d$, such that $\lambda\in \Gamma(T)$ and under the isomorphism $\Gamma(T^d)\cong(\mathbb Z^n)^d$, we can write $\lambda = (\lambda_1,\cdots,\lambda_d)$, where $\lambda_i =(\lambda_{i,1},\cdots,\lambda_{i,n})$ such that $\lambda_{i,j}\in \mathbb{Z}$ and $\lambda_{i, j} \ge 0$ for all $1\le i\le d, 1\le j\le n$, in other words, $\lambda$ is a polynomial 1-parameter subgroup. \end{lemma}
\begin{proof}
By Theorem \ref{main theorem}, the maximal value of $\frac{\operatorname{val}(\lambda(t)\cdot v)}{\|\lambda\|}$ exists. Let $T\subset G$ be a maximal torus and $\lambda\in \Gamma(T)$ such that the function attains its maximum at $\lambda$. Assume $\lambda$ is of the form in the lemma and $\lambda_{i, j} < 0$ for some $i$ and $j$. If we replace $\lambda_{i, j}$ by $-\lambda_{i, j}$, $\operatorname{val}(\lambda(t)\cdot v)$ never decreases and $\|\lambda\|$ does not change. Therefore by the maximality of $\frac{\operatorname{val}(\lambda(t)\cdot v
)}{\|\lambda\|}$, the value $\frac{\operatorname{val}(\lambda(t)\cdot v)}{\|\lambda\|}$ does not change after the replacement. Hence without loss of generality we can assume all $\lambda_{i, j} \ge 0$. \end{proof}
Recall that for a tensor $v\in V^{\otimes d}$ and a polynomial 1-parameter subgroup $\lambda$ of $G = \operatorname{GL}(V)^d$ such that $\operatorname{val}(\lambda(t)\cdot v)>0$, we have the slope function \begin{equation}\label{slope} \mu(\lambda(t), v) = \frac{\sum_{i = 1}^d \operatorname{val}(\det(\lambda_i(t)))}{\operatorname{val}(\lambda(t)\cdot v)}. \end{equation}
Let $\lambda = (\lambda_1,\cdots, \lambda_d)\in \Gamma(G)$ be a polynomial 1-parameter subgroup of $G=\operatorname{GL}(V)^d$, then by Lemma \ref{>=0}, $\sum_{i=1}^d\operatorname{val}(\det(\lambda_i(t)))$ is the restriction of the length function defined by equation (\ref{len2}). Let $S_d$ be the symmetric group acting on $G=\operatorname{GL}(V)^d$ by permuting the $d$ components. Then the length function defined by equation (\ref{len2}) is invariant under the action of $S_d$. From now on, fix this length function on $\Gamma(G)$. We have a corollary following from Theorem \ref{main theorem}:
\begin{corollary}\label{Pd}
Let $v\in {D}^d V\subset V^{\otimes d}$ be a symmetric tensor. Let $\Lambda_{\{0\},v}$ be the set of indivisible 1-parameter subgroups $\lambda\in |V^{\otimes d},v|$ such that $\frac{\operatorname{val}(\lambda(t)\cdot v)}{\|\lambda\|}$ attains the maximum value. Then we have \begin{enumerate}
\item $\Lambda_{\{0\},v}$ is invariant under $S_d$.
\item $P_{\{0\},v}$ is $S_d$ invariant. In other words, $P_{\{0\},v} = P^d\subset \operatorname{GL}(V)^d$ for some parabolic subgroup $P\subset\operatorname{GL}(V)$.
\end{enumerate} \end{corollary}
\begin{proof}\ \begin{enumerate}
\item It is clear that $\frac{\operatorname{val}(\lambda(t)\cdot v)}{\|\lambda\|}$ is $S_d$ invariant. Indeed, Let $\sigma\in S_d$, since $v\in {D}^d V$ is a symmetric tensor and $\|\cdot\|$ is $S_d$ invariant, we have
\[
\frac{\operatorname{val}((\sigma \lambda)(t)\cdot v)}{\|\sigma \lambda\|} = \frac{\operatorname{val}((\sigma \lambda)(t)\cdot (\sigma v))}{\|\sigma \lambda\|} =
\frac{\operatorname{val}(\sigma (\lambda(t)\cdot v))}{\|\sigma \lambda\|} = \frac{\operatorname{val}(\lambda(t)\cdot v)}{\|\lambda\|}.
\]
Therefore if $\lambda\in \Lambda_{\{0\},v}$, so is $\sigma(\lambda)$.
\item
Since $G = \operatorname{GL}(V)^d$, the parabolic subgroup $P_{\{0\},v}$ is a product of parabolic subgroups of $\operatorname{GL}(V)$, the symmetric group $S_d$ acts on $P_{\{0\},v}$ by permuting the components. Let $\lambda\in \Lambda_{\{0\},v}$, for any $\sigma\in S_d$, we have
\[
\sigma(P_{\{0\},v}) = P(\sigma(\lambda)) = P_{\{0\},v}.
\]
We used the fact that $P_{\{0\},v}=P(\lambda)$ is independent of $\lambda\in\Lambda_{\{0\},v}$ and $\sigma(\lambda)\in \Lambda_{\{0\},v}$. So $P_{\{0\},v}$ is $S_d$ invariant and we can find a parabolic subgroup $P\subset \operatorname{GL}(V)$ such that $P_{\{0\},v} = P^d$. \end{enumerate} \end{proof}
Let $T\subset P\subset \operatorname{GL}(V)$ be a maximal torus, then $T^d$ is a maximal torus of $P^d=P_{\{0\}, v}$. By (2.c) in Theorem \ref{main theorem} and Lemma \ref{>=0}, there is a polynomial 1-parameter subgroup $\lambda = (\lambda_1,\cdots, \lambda_n)$ of $T^d\subset \operatorname{GL}(V)^d$, such that the slope function \[
\mu(\lambda(t),v)=\frac{\sum_{i = 1}^d \operatorname{val}(\det(\lambda_i(t)))}{\operatorname{val}(\lambda(t)\cdot v)} = \frac{\|\lambda\|}{\operatorname{val}(\lambda(t)\cdot v)} \] has a minimum value at $\lambda$. The minimal value of $\mu(\lambda(t), v)$ is by definition the $G$-stable rank $\operatorname{rk}^G(v)$ of $v$. In rest of the section, we fix such a maximal torus $T\subset \operatorname{GL}(V)$.
Let $\lambda=(\lambda_1,\cdots,\lambda_d)$ be a polynomial 1-parameter subgroup of $T^d\subset \operatorname{GL}(V)^d$, then $\gamma = \Pi_{i=1}^d\lambda_i$ is a polynomial 1-parameter subgroup of $\operatorname{GL}(V)$. Furthermore, $\gamma$ acts on $v\in {D}^dV$ via the diagonal embedding $\operatorname{GL}(V)\hookrightarrow \operatorname{GL}(V)^d$. We have the following lemma:
\begin{lemma} For any symmetric tensor $v\in {D}^d V$, we have $\operatorname{val}(\gamma(t)\cdot v) \ge d\cdot \operatorname{val}(\lambda(t)\cdot v)$. \end{lemma}
\begin{proof} Let $C = \operatorname{val}(\lambda(t)\cdot v)$, we define a subspace $W$ of $V^{\otimes}$ as following \[
W = \{w\in V^{\otimes d}|\operatorname{val}(\sigma(\lambda(t))\cdot w)\ge C, \forall \sigma\in S_d\}. \] Since $\operatorname{val}(\sigma(\lambda(t))\cdot v) = \operatorname{val}(\sigma(\lambda(t)\cdot v)) = \operatorname{val}(\lambda(t)\cdot v) = C$, we have $v\in W$. For any $\sigma\in S_d$, it is clear that $\sigma(\lambda(t))\cdot W\subset t^CK[t]\cdot W$. We can write \begin{align*} \gamma(t)\cdot v &=(\Pi_{i=1}^d\lambda_i,\cdots,\Pi_{i=1}^d\lambda_i)\cdot v \\ &=(\lambda_1,\lambda_2,\cdots,\lambda_d)(\lambda_2,\lambda_3,\cdots,\lambda_d,\lambda_1)\cdots(\lambda_d,\lambda_1,\cdots,\lambda_{d-1})\cdot v \\ &=(\lambda_1,\lambda_2,\cdots,\lambda_d)\sigma(\lambda_1,\lambda_2,\cdots,\cdots,\lambda_d)\cdots\sigma^{d- 1}(\lambda_1,\lambda_2,\cdots,\lambda_{d})\cdot v \\ &=\lambda\sigma(\lambda)\cdots\sigma^{d-1}(\lambda)\cdot v, \end{align*} where $\sigma\in S_d$ satisfies $\sigma(1) = 2, \sigma(2) = 3,\cdots,\sigma(d) = 1$. Therefore $\gamma(t)\cdot v \in t^{dC}K[t]\cdot W$, hence we get $\operatorname{val}(\gamma(t)\cdot v) \ge dC = d\cdot \operatorname{val}(\lambda(t)\cdot v)$. \end{proof}
\begin{comment} \begin{proof} Since $v\in {D}^d V$ is a symmetric tensor, for any element $\sigma$ in the symmetric group $\sigma\in S_d$, we have $\sigma(\lambda(t))\cdot v = \lambda(t)\cdot v$. Then we have
\begin{align*} \gamma(t)\cdot v &=(\Pi_{i=1}^d\lambda_i,\cdots,\Pi_{i=1}^d\lambda_i)\cdot v \\ &=(\lambda_1,\lambda_2,\cdots,\lambda_d)(\lambda_2,\lambda_3,\cdots,\lambda_d,\lambda_1)\cdots(\lambda_d,\lambda_1,\cdots,\lambda_{d-1})\cdot v \\ &=(\lambda_1,\cdots,\lambda_d)^d\cdot v \\ &=\lambda^d\cdot v. \\ \end{align*} Therefore we get $\operatorname{val}(\gamma(t)\cdot v) = \operatorname{val}(\lambda(t)^d\cdot v)=d\cdot \operatorname{val}(\lambda(t)\cdot v)$. \end{proof} \end{comment}
Next we prove that the symmetric $G$-stable rank is the same as the $G$-stable rank for symmetric tensors.
\begin{proof}[Proof of Theorem \ref{second result}] Let $T$ be the chosen maximal torus of $\operatorname{GL}(V)$ as above. Let $\lambda = (\lambda_1,\cdots,\lambda_d)$ be a polynomial 1-parameter subgroup of $T^d\subset \operatorname{GL}(V)^d$ such that the slope function $\mu(\lambda(t),v)$ attains its minimum value. In other words, $\lambda$ computes the $G$-stable rank $\operatorname{rk}^G(v)$ of $v$: \[ \operatorname{rk}^G(v) = \frac{\sum_{i=1}^d \operatorname{val}(\det(\lambda_i(t)))}{\operatorname{val}(\lambda(t)\cdot v)}. \] Let $\gamma = \Pi_{i=1}^d\lambda_i$ as above, then we have \[ \operatorname{symmrk}^G(v)\le \frac{\sum_{i=1}^d \operatorname{val}(\det(\gamma(t)))}{\operatorname{val}(\gamma(t)\cdot v)} \le \frac{d\sum_{i=1}^d \operatorname{val}(\det(\lambda_i(t)))}{d\cdot \operatorname{val}(\lambda(t)\cdot v)} = \operatorname{rk}^G(v). \] On the other hand, it is clear that $\operatorname{symmrk}^G(v)\ge \operatorname{rk}^G(v)$. Therefore $\operatorname{symmrk}^G(v) = \operatorname{rk}^G(v)$, this completes the proof. \end{proof}
\section{Stability of symmetric tensors}\label{stability}
As a result of Theorem \ref{second result}, we prove Corollary \ref{first result}, which says that for a symmetric tensor $v\in {D}^d V$, $v$ is $\operatorname{SL}(V)^d$-semistable if and only if $v$ is $\operatorname{SL}(V)$-semistable. It is clear that $\operatorname{SL}(V)^d$-semistability implies $\operatorname{SL}(V)$-semistability. To prove the other direction, we will use a result which relates semistability with $G$-stable rank.
\begin{prop}[\cite{G-stable rank}, Proposition 2.6] \label{2.6} Suppose that $\alpha=(\frac{1}{n_1},\cdots\frac{1}{n_d})$ where $n_i =\dim V_i$. For $v\in V_1\otimes V_2\otimes \cdots\otimes V_d$ we have $\operatorname{rk}^G_\alpha(v) \le 1$. Moreover, $\operatorname{rk}^G_\alpha(v) = 1$ if and only if $v$ is semistable with respect to the group $H = \operatorname{SL}(V_1)\times\operatorname{SL}(V_2)\times\cdots\times\operatorname{SL}(V_d)$. \end{prop}
If $v\in {D}^d V$ is a symmetric tensor, $\alpha = (1,1,\cdots,1)$ and $n = \dim V$, then by the above proposition, $\operatorname{rk}^G(v) = n$ if and only if $v$ is $\operatorname{SL}(V)^d$-semistable. We have a similar result for symmetric $G$-stable rank.
\begin{prop}\label{2.66} For a symmetric tensor $v\in {D}^d V$, we have $\operatorname{symmrk}^G(v) \le n$, where $n = \dim V$. Moreover, $\operatorname{symmrk}^G(v) = n$ if and only if $v$ is $\operatorname{SL}(V)$-semistable. \end{prop} \begin{proof} The first statement is clear from Theorem \ref{second result}. If $\operatorname{symmrk}^G(v) = n$, by Proposition \ref{2.6}, we have $\operatorname{rk}^G(v) = n$ and $v$ is $\operatorname{SL}(V)^d$-semistable, hence $v$ is $\operatorname{SL}(V)$-semistable. On the other hand, assume $v$ is $\operatorname{SL}(V)$-semistable. Let $\lambda$ be a polynomial 1-parameter subgroup of $\operatorname{GL}(V)$ such that $\lim_{t\to 0}\lambda(t)\cdot v=0$. Then we can define another 1-parameter subgroup $\lambda'(t) = \lambda(t)^n t^{-e}$, where $\det(\lambda(t)) = t^e$, such that $\det(\lambda')=1$ and $\lambda'\in \operatorname{SL}(V)$. Since $v$ is $\operatorname{SL}(V)$-semistable, we have $\operatorname{val}(\lambda'(t)\cdot v) \le 0$. It follows that \[ \operatorname{val}(\lambda'(t)\cdot v) = \operatorname{val}(\lambda(t)^d t^{-e}\cdot v) = n\operatorname{val}(\lambda(t)\cdot v) - ed\le 0. \] The slope function \[ \mu(\lambda(t), v) = \frac{d\operatorname{val}(\det(\lambda(t)))}{\operatorname{val}(\lambda(t)\cdot v)} = \frac{de}{\operatorname{val}(\lambda(t)\cdot v)}\ge n. \] We get $\operatorname{symmrk}^G(v) = n$. \end{proof}
\begin{proof}[Proof of Corollary \ref{first result}] It suffices to prove that if $v$ is $\operatorname{SL}(V)$-semistable, then $v$ is $\operatorname{SL}(V)^d$-semistable. Let us assume $v$ is $\operatorname{SL}(V)$-semistable, then by Proposition \ref{2.66} and Theorem \ref{second result}, \[ \operatorname{rk}^G(v) = \operatorname{symmrk}^G(v) = n. \] It follows from Proposition \ref{2.6} that $v$ is $\operatorname{SL}(V)^d$-semistable. \end{proof}
\section{G-stable rank for ideals and log canonical threshold} \label{ideal rank}
\subsection{G-stable rank for ideals} Let $X = \text{Spec}(R)$ be a nonsingular irreducible complex affine algebraic variety of dimension $n$, and $\mathfrak a\subset R$ be a nonzero ideal, and let $P\in V(\mathfrak a)$ be a closed point, $\mathcal O_P$ be the local ring at $P$ and $\mathfrak m_P$ be the maximal ideal corresponding to $P$.
\begin{definition}[\cite{Sha}] Functions $x_1,\cdots,x_n\in \mathcal O_P$ are {\em a system of local parameters} at $P$ if each $x_i\in \mathfrak m_P$, and the images of $x_1,\cdots,x_n$ form a basis of the vector space $\mathfrak m_P/\mathfrak m_P^2$. \end{definition}
Let $T = \{x_1, x_2,\cdots, x_n\}$ be a system of local parameters at $P$. Let $\mathbb{C}\{x_1,x_2,\dots,x_n\}$ be the ring of convergent power series in $x_1,x_2,\dots,x_n$. The ring $\mathcal O_P$ is contained in $\mathbb{C}\{x_1,x_2,\dots,x_n\}$. If $y_1,y_2,\dots,y_n$ is any system of local parameters, then $\mathbb{C}\{x_1,x_2,\dots,x_n\}=\mathbb{C}\{y_1,y_2,\dots,y_n\}$. For any $\lambda = (\lambda_1,\lambda_2,\cdots,\lambda_n)\in \mathbb Z^n_{\ge 0}$, we have a natural action of $\mathbb C^*$ on $\mathbb{C}\{x_1,x_2,\cdots,x_n\}$ by $t\cdot x_i = t^{\lambda_i}x_i$ for any $t\in\mathbb C^*$.
\begin{definition}\label{val&ord} Let $T = \{x_1,x_2,\cdots,x_n\}$ be a system of local parameters at $P$ and $\lambda = (\lambda_1,\lambda_2,\cdots,\lambda_n)\in \mathbb Z^n_{\ge 0}$ a tuple of non-negative integers. Let $f\in \mathbb{C}\{x_1,x_2,\cdots,x_n\}$ be a convergent power series. We define the {\em valuation} of $f$ with respect to $T$ by \begin{equation}\label{val}
\operatorname{val}^T_\lambda(f) = \max\{k|f(t^{\lambda_1}x_1,t^{\lambda_2}x_2,\cdots,t^{\lambda_n}x_n) = t^kg(x_1,\cdots,x_n,t), \text{for}\ \text{some}\ g\in \mathbb C \{x_1,\cdots,x_n,t\}\}. \end{equation}
Let $\mathfrak a\subset R$ be a nonzero ideal as before, the {\em order} of $\mathfrak a$ with respect to this system of local parameters $T$ and $\lambda\in \mathbb Z^n_{\ge 0}$ is defined as \begin{equation}\label{ord}
\operatorname{ord}^T_\lambda(\mathfrak a) = \min\{\operatorname{val}^T_{\lambda}(f)|f\in \mathfrak a\}. \end{equation}
\begin{remark}\label{gen} If $\mathfrak a$ is generated by $f_1,\cdots,f_r$, then \[
\operatorname{ord}^T_\lambda(\mathfrak a) = \min\{\operatorname{val}^T_\lambda(f_i)| i = 1,\cdots,r\}. \]
Indeed, it is clear that $\min\{\operatorname{val}^T_{\lambda}(f)|f\in \mathfrak a\}\le \min\{\operatorname{val}^T_\lambda(f_i)| i = 1,\cdots,r\}$. On the other hand, if $f\in \mathfrak a$ computes $\operatorname{ord}^T_\lambda(\mathfrak a)$, then we can write $f = \sum_ia_if_i$, for some $a_i\in R$, we have $\operatorname{val}^T_\lambda(f) = \operatorname{val}^T_\lambda(\sum_ia_if_i) \ge \min\{\operatorname{val}^T_\lambda(a_if_i)|i=1,\cdots,r\}\ge \min\{\operatorname{val}^T_\lambda(f_i)|i = 1,\cdots,r\}$. \end{remark}.
\end{definition}
\begin{definition}\label{Gdef} Assume $T = \{x_1,\cdots,x_n\}$ is a system of local parameters at $P$ and $\lambda=\{\lambda_1,\cdots,\lambda_n\}\in \mathbb Z^n_{\ge 0}$, we define the {\em slope} function $\mu_P(\lambda,\mathfrak a)$ at $P$ as \begin{equation} \mu_P(\lambda,\mathfrak a) = \frac{\sum_{i = 1}^n \lambda_i}{\operatorname{ord}^T_\lambda(\mathfrak a)}. \end{equation} The $T$-{\em stable} {\em rank} of $\mathfrak a$ at $P$ is the infimum of the slope function $\mu_P(\lambda,\mathfrak a)$ with respect to the tuple $\lambda= (\lambda_1,\lambda_2,\cdots,\lambda_n)\in \mathbb Z^n_{\ge 0}$, \begin{equation}
\operatorname{rk}^T(P, \mathfrak a) = \inf_\lambda\mu_P(\lambda,\mathfrak a) = \inf_\lambda\frac{\sum_{i = 1}^n \lambda_i}{\operatorname{ord}^T_\lambda(\mathfrak a)}. \end{equation} The $G$-{\em stable} {\em rank} of $\mathfrak a$ is defined by taking the infimum of $T$-stable rank with respect to all system of local parameters $T$ at $P$, \begin{equation} \operatorname{rk}^G(P, \mathfrak a) = \inf_T(\operatorname{rk}^T(P, \mathfrak a)). \end{equation}
\end{definition}
If $P\notin V(\mathfrak a)$, we define $\operatorname{rk}^G(P,\mathfrak a) = \infty$, we write $\operatorname{rk}^G(\mathfrak a)$ and $\operatorname{rk}^T(\mathfrak a)$ if $P$ is known in the context. In the following example, we see that an ideal $\mathfrak a$ can have different $T$-stable rank with respect to different system of local parameters $T$ at a point $P$.
\begin{example} Let $R = \mathbb C[x, y]$, $T = \{x, y\}$ and assume $\mathfrak a = (x^2+2xy+y^2)$ is a principle ideal generated by a polynomial $f(x,y) = x^2+2xy+y^2$, $P=(0,0)$ is the origin. Let $\lambda = (\lambda_1,\lambda_2)\in \mathbb Z^2_{\ge 0}$, then we have \[ \operatorname{rk}^T(f) = \inf_{\lambda}\frac{\lambda_1+\lambda_2}{\min(2\lambda_1, \lambda_1+\lambda_2,2\lambda_2)} = 1 \] Let us choose a different system of local parameters $T' = \{u = x + y, v = x - y\}$, then $\mathfrak a = (u^2)$, and $f(u,v) = u^2$, then \[ \operatorname{rk}^{T'}(f) = \inf_\lambda\frac{\lambda_1+\lambda_2}{2\lambda_1} = \frac{1}{2} \] In fact, $\text{\rm lct}_P(u^2) = \frac{1}{2}$, and by Theorem \ref{relation}, we have $\text{\rm lct}_P(\mathfrak a) \le \operatorname{rk}^G(\mathfrak a)$, therefore we get $\operatorname{rk}^G(f) = \frac{1}{2}$. \end{example}
\begin{example} Let $\mathfrak a = (x^2y,y^2z,z^2x) \subset \mathbb C[x,y,z]$, $T = \{x, y, x\}$, $P=(0,0, 0)$, then we get \[ \operatorname{rk}^T(\mathfrak a) = \inf_{\lambda}\frac{\lambda_1 + \lambda_2 + \lambda_3}{\min(2\lambda_1+\lambda_2,2\lambda_2+\lambda_3,2\lambda_3+\lambda_1)} = 1 \] The ideal $\mathfrak a = (x^2y, y^2z, z^2x)$ is a monomial ideal and we will see later that for a monomial ideal $\mathfrak a$, we have $\operatorname{rk}^G(\mathfrak a) = \text{\rm lct}_P(\mathfrak a)$. Using the fact that $\text{\rm lct}_P(\mathfrak a) = 1$, we obtain $\operatorname{rk}^G(\mathfrak a) = 1$. \end{example}
\begin{remark}\label{ses} We have a short exact sequence \begin{equation} 1\to K\to \text{\rm Aut}(\mathbb C\{x_1,\cdots,x_n\})\to \operatorname{GL}(n)\to 1 , \end{equation} where $K$ is a normal subgroup of the group $\text{\rm Aut}(\mathbb C\{x_1,\cdots,x_n\})$ of local holomorphic automorphisms. The morphism $\text{\rm Aut}(\mathbb C\{x_1,\cdots,x_n\})\to \operatorname{GL}(n)$ is given by computing the Jacobian matrix at $(0, \cdots, 0)$. Furthermore, this sequence splits, we have $\text{\rm Aut}(\mathbb C\{x_1\cdots,x_n\}) = K\rtimes \operatorname{GL}(n)$. \end{remark}
\begin{comment} First we define a group homomorphism $\pi: \text{\rm Aut}(\mathbb C[[x_1,\cdots,x_n]]) \to \operatorname{GL}(n)$. Let $g\in \text{\rm Aut}(\mathbb C[[x_1,\cdots,x_n]])$, then we can write \[ g(x_i) = \sum_{j = 1}^n x_ja_{ji} + p_i(x_1,\cdots, x_n), \] where $a_{ji}\in \mathbb C$ and $p_i(x_1,\cdots, x_n) = 0$ or has degree greater than 1. Let $h = g^{-1}\in \text{\rm Aut}(\mathbb C[[x_1,\cdots,x_n]])$. Let $h$ act on $x_i$, similarly we can write \[ h(x_i) = \sum_{j = 1}^n x_jb_{ji} + q_i(x_1,\cdots, x_n), \] where $b_{ji}\in \mathbb C$ and $q_i(x_1,\cdots, x_n) = 0$ or has degree greater than 1. By the above two equations, we get \[ h(g(x_i)) =h(\sum_{j=1}^nx_ja_{ji} + p_i(x_1,\cdots, x_n)) \] \[ =\sum_{j,k=1}^nx_kb_{kj}a_{ji} + r_i(x_1,\cdots, x_n), \] here $r_i(x_1,\cdots, x_n) = 0$ or has degree greater that 1. By the fact that $h(g(x_i)) = x_i$, we obtain $b_{kj}a_{ji} = \delta_{ki}$, and $r_i(x_1,\cdots, x_n) = 0$. Therefore $A = (a_{ji})$ is an element in $\operatorname{GL}(n)$, we define $\pi: \text{\rm Aut}(\mathbb C[[x_1,\cdots,x_n]]) \to \operatorname{GL}(n)$ by $\pi(g) = A \in \operatorname{GL}(n)$. It is clear that this defines a group homomorphism. Let $K$ be the kernel of $\pi$, we prove the first claim.
To prove that the sequence splits, we define a map $\eta: \operatorname{GL}(n)\to\text{\rm Aut}(\mathbb C[[x_1,\cdots,x_n]])$. Given $A = (a_{ij})\in \operatorname{GL}(n)$, let $\eta(A)(x_i) = \sum_{j = 1}^nx_ja_{ji} $. It is clear that $\pi\circ\eta = \text{id}$, it remains to show that this is a group homomorphism. Let $A=(a_{ij}), B=(b_{ij})\in \operatorname{GL}(n)$, then we have \[ \eta(AB)(x_i)=\sum_lx_l(AB)_{li}=\sum_{l,j}x_la_{lj}b_{ji} =\sum_{j}\eta(A)(x_j)b_{ji} = \eta(A)(\sum_j x_jb_{ji}) \] \[ = \eta(A)\eta(B)(x_i). \] \end{comment}
By Remark \ref{ses}, there is an action of $\operatorname{GL}(n)$ on the set of system of local parameters. Assume $T=\{x_1,\cdots,x_n\}$ is a system of local parameters at $P$. For $g\in \operatorname{GL}(n)$, $g\cdot T$ is another
system of local parameters.
We say a system of local parameters $T=\{x_1,\cdots,x_n\}$ is {\em good} for $\mathfrak a$ if $\operatorname{rk}^G(P,\mathfrak a) = \operatorname{rk}^{g\cdot T}(P, \mathfrak a)$ for some $g\in \operatorname{GL}(n)$. In other words, to compute the $G$-stable rank of $\mathfrak a$, it is enough to consider all systems of local parameters obtained from $T$ by actions of $\operatorname{GL}(n)$.
\begin{example} Let $f(x,y) = x + y^2 \in \mathbb C[x,y]$, $P = (0, 0)$, we take $T = \{x,y\}$. It can be shown that $\operatorname{rk}^{T}(f) = \frac{3}{2}$. However, if we choose another system of local parameters $T' = \{u = x + y^2, v = y\}$, then $f(u, v) = u$, and $\operatorname{rk}^{T'}(f) = 1$. Indeed, this system of local parameters is optimal, in other words, we can compute the $G$-stable rank in this system of local parameters, and we have $\operatorname{rk}^G(f) = 1$. \end{example}
\begin{prop}\label{homog} If $\mathfrak a$ is homogeneous in local parameters $T=\{x_1,\cdots,x_n\}$, then $T$ is good for $\mathfrak a$. \end{prop}
\begin{proof} Since $\mathfrak a$ is a homogeneous ideal, we can find a set of generators which are homogeneous polynomials. By Remark \ref{gen}, it is enough to assume that $\mathfrak a$ is generated by a single homogeneous polynomial $f$. Let $g\in K\subseteq \text{\rm Aut}(\mathbb C\{x_1,\cdots,x_n\})$, we can write the action of $g$ on $T$ as following \[ g(x_i) = x_i + p_i(x_1,\cdots,x_n), \] where $p_i\in \mathbb C\{x_1\cdots,x_n\}$ with no constant and degree 1 terms. Since $f$ is a homogeneous polynomial, we have \[ \operatorname{val}^T_\lambda(f(g(x_1),\cdots,g(x_n)) \le \operatorname{val}^T_\lambda(f(x_1,\cdots,x_n)). \] Let $T'$ be the system of local parameters obtained from $T$ by the action of $g$, then we have \begin{equation}\label{comp slopes} \frac{\sum_{i=1}^n\lambda_i}{\operatorname{ord}^{T'}_\lambda(f)} \ge \frac{\sum_{i=1}^n\lambda_i}{\operatorname{ord}^{T}_\lambda(f)} . \end{equation} By Lemma \ref{ses}, given any $h\in \text{\rm Aut}(\mathbb C\{x_1,\cdots,x_n\})$, we can decompose the action of $h$ into an action of $K$ following by an action of $\operatorname{GL}(n)$. By inequality (\ref{comp slopes}), in the system of local parameters obtained by the action of $K$ from $T$, we have larger slope than the slope computed in $T$, hence to compute the $G$-stable rank of $f$, it suffices to consider the action of $\operatorname{GL}(n)$. This shows that $T$ is good for $\mathfrak a$. \end{proof}
\begin{corollary} If $f\in \mathbb C[x_1,\cdots,x_n]$ is a homogeneous polynomial of degree $d \ge 2$ and $f$ has isolated singularity at $P = (0,\cdots,0)$. Then $\operatorname{rk}^G(f) = \frac{n}{d}$. \end{corollary}
\begin{proof} By Corollary \ref{homog}, the system of local parameters $T = \{x_1,\cdots,x_n\}$ is good for $f$, therefore we only need to consider the group action of $\operatorname{GL}(n)$.
We claim that $f$ is $\operatorname{SL}(n)$-semistable in the sense of Definition \ref{ss}. Indeed, since $f$ has an isolated singularity at origin, $\frac{\partial f}{\partial x_1},\cdots, \frac{\partial f}{\partial x_n}$ only have a common zero at origin. By \cite{GKZ}, chapter 13, their resultant ${\rm Res}(\frac{\partial f}{\partial x_1},\cdots,\frac{\partial f}{\partial x_n})$ is nonzero and invariant under the action of $\operatorname{SL}(n)$. Now assume there is a one parameter subgroup $\lambda: \mathbb G_m\to \operatorname{SL}(n)$, such that \[ \lim_{t\to 0} \lambda(t)\cdot v = 0. \] Then the resultant ${\rm Res}(\frac{\partial f}{\partial x_1},\cdots,\frac{\partial f}{\partial x_n})$ is 0, which is impossible. This proves the claim.
By the claim that $f$ is $\operatorname{SL}(n)$-semistable, the corollary follows immediately from Proposition 2.6 in \cite{G-stable rank} \end{proof}
\subsection{Relation to log canonical threshold} Let $X = \text{Spec}(R)$ be a nonsingular irreducible complex affine variety, $\mathfrak a\subset R$ is an ideal, $P\in V(\mathfrak a)$. If $\mathfrak a = (f_1,\cdots,f_r)\subseteq R$ is a nonzero ideal, recall the Definition \ref{lct}, the log canonical threshold $\text{\rm lct}_P(\mathfrak a)$ of the ideal $\mathfrak a$ at point $P$ is
\begin{equation}
\text{\rm lct}_P(\mathfrak a) = \sup\{s>0|\frac{1}{(\sum_{i = 0}^r|f_i|^2)^s} \
\text{is}\ \text{integrable}\ \text{around}\ P\}. \end{equation}
Theorem \ref{relation} says that the log canonical threshold of an ideal is less than or equal to the $G$-stable rank of that ideal \begin{equation} \text{\rm lct}_P(\mathfrak a) \le \operatorname{rk}^G(P,\mathfrak a). \end{equation}
\begin{proof}[Proof of Theorem \ref{relation}]
Let $s>0$ be such that $\frac{1}{(\sum_{i = 0}^r|f_i|^2)^s}$ is integrable around $P$, then there is a neighborhood $U_P$ of $P$, such that
\[
\int_{U_P} \frac{dV}{(\sum_{i = 0}^r|f_i|^2)^s} < C < \infty, \]
for some constant $C$. Choose a system of local parameters $T = \{x_1,\cdots,x_n\}$ at $P$, let $\lambda = (\lambda_1,\cdots,\lambda_n)\in\mathbb Z^n_{\ge0}$, then let $t\in \mathbb C^*$ act on the coordinates by $x_i\to t^{\lambda_i}x_i$. We denote $t\cdot U_P$ for the image of $U_P$ under this action. If $|t|<1$, we have $t\cdot U_P\subset U_P$, therefore \begin{equation}
\int_{t\cdot U_P} \frac{dV}{(\sum_{i = 0}^r|f_i|^2)^s} < \int_{U_P} \frac{dV}{(\sum_{i = 0}^r|f_i|^2)^s} <C \end{equation}
Let $y_i = t^{-\lambda_i} x_i$ for $i = 1,\cdots, n$, then we have \begin{equation}\label{int}
\int_{t\cdot U_P} \frac{dx_1d\bar{x}_1\cdots dx_nd\bar x_n}{(\sum_{i = 0}^r|f_i(x_1,\cdots,x_n)|^2)^s}
= \int_{U_P}\frac{|t|^{2\sum_{i=1}^n \lambda_i}dy_1d\bar y_1\cdots dy_nd\bar y_n}{(\sum_{i=1}^r|f_i(t^{\lambda_1}y_1,\cdots,t^{\lambda_n}y_n)|^2)^s} < C \end{equation}
Recall the Definition \ref{val&ord} for the valuation (\ref{val}) and order of an ideal (\ref{ord}), we can write \[
\sum_{i=0}^r|f_i(t^{\lambda_1}y_1,\cdots,t^{\lambda_n}y_n)|^2= |t|^{2\min_i(\operatorname{val}^T_\lambda(f_i))}(\sum_{i=1}^r|\tilde f_i(y_1,\cdots,y_n, t)|^2) \]
\[= |t|^{2\operatorname{ord}_\lambda^T(\mathfrak a)}(\sum_{i=1}^r|\tilde f_i(y_1,\cdots, y_n, t)|^2), \]
for some $\tilde f_i(y_1,\cdots, y_n, t) \in \mathbb C\{y_1,y_2,\cdots,y_n,t\}$. In particular, we know that $\sum_{i=1}^r|\tilde f_i(y_1,\cdots, y_n, 0)|^2$ is not constantly zero in $U_P$. So we can find a point $Q\in U_P$ such that $ 0<\sum_{i=1}^r|\tilde f_i(Q, 0)|^2 < B$ for some constant $B > 0$. By the continuity, there is a neighborhood $U_Q$ such that $Q\in U_Q \subset U_P$ and some $\epsilon > 0$, such that $ 0<\sum_{i=1}^r|\tilde f_i(y_1,\cdots,y_n, t)|^2 < B$ for any $(y_1,\cdots,y_n) \in U_Q$ and $0 \le t < \epsilon$. We can write integral (\ref{int}) as
\begin{equation}\label{new_int}
\int_{U_P}\frac{|t|^{2\sum_{i=1}^n \lambda_i}dy_1d\bar y_1\cdots dy_nd\bar y_n}{|t|^{2\operatorname{ord}_\lambda^T(\mathfrak a)}(\sum_{i=1}^r|\tilde f_i(y_1,\cdots, y_n, t)|^2)^s} = \int_{U_P}\frac{|t|^{2(\sum_{i=1}^n \lambda_i - s\operatorname{ord}_\lambda^T(\mathfrak a))}dV}{(\sum_{i=1}^r|\tilde f_i(y_1,\cdots, y_n, t)|^2)^s}< C. \end{equation}
Therefore we get \[
|t|^{2(\sum_{i=1}^n \lambda_i - s\operatorname{ord}_\lambda^T(\mathfrak a))}\int_{U_Q}\frac{dV}{B^s}
=\int_{U_Q}\frac{|t|^{2(\sum_{i=1}^n \lambda_i - s\operatorname{ord}_\lambda^T(\mathfrak a))}dV}{B^s}
<\int_{U_P}\frac{|t|^{2(\sum_{i=1}^n \lambda_i - s\operatorname{ord}_\lambda^T(\mathfrak a))}dV}{(\sum_{i=1}^r|\tilde f_i(y_1,\cdots, y_n, t)|^2)^s} < C \] for all $t\in [0,\epsilon)$, where $\epsilon > 0$. Since $\int_{U_Q}\frac{dV}{B^s} > 0$, we must have
\[ \sum_{i=1}^n\lambda_i-s\operatorname{ord}^T_{\lambda}(\mathfrak a) \ge 0. \] Hence we get $s\le \frac{\sum_{i=1}^n\lambda_i}{\operatorname{ord}^T_{\lambda}(\mathfrak a)}$. This holds for any system of local coordinates and $\lambda = (\lambda_1,\cdots, \lambda_n)\in\mathbb Z^n_{\ge0}$, therefore \[ \text{\rm lct}_P(\mathfrak a) \le \operatorname{rk}^G(P, \mathfrak a). \]
This completes the proof of Theorem \ref{relation}. \end{proof} \begin{example} Let $f=x_1^{u_1}+x_2^{u_2}+\cdots +x_n^{u_n}\in \mathbb C[x_1,\cdots,x_n]$, $P = (0,\cdots,0)$ be the origin. It was shown in \cite{Mus} that $\text{\rm lct}_P(f) = \min(1,\sum_{i=1}^n\frac{1}{u_i})$. However, we will show that $\operatorname{rk}^G(f) =\sum_{i=1}^n \frac{1}{u_i}$. So we get $\text{\rm lct}_P(f)\le \operatorname{rk}^G(f)$. If $n = 3$, $u_1 = u_2 = u_3 = 2$, then we have $f = x_1^2+x_2^2+x_3^2$ and $\text{\rm lct}_P(f) = 1 < \operatorname{rk}^G(f) = \frac{3}{2}$. \end{example}
Suppose $\mathfrak a = (m_1,\cdots,m_r) \subset \mathbb C[x_1,\cdots,x_n]$ is a proper nonzero ideal generated by monomials $\{m_1,\cdots, m_r\}$ and let $P = (0,\cdots, 0)$ be the origin. Given $u = (u_1,\cdots, u_n)\in \mathbb Z^n_{\ge 0}$, we write $x^u = x_1^{u_1}\cdots x_n^{u_n}$. The $Newton$ $Polyhedron$ of $\mathfrak a$ is \begin{equation}
P(\mathfrak a) =\text{convex}\ \text{hull}\ (\{u\in\mathbb Z^n_{\ge 0} |x^u\in \mathfrak a\}). \end{equation}
It was shown in \cite{Mus} that \begin{equation}
\text{\rm lct}_P(\mathfrak a) = \max\{\nu\in\mathbb R_{\ge 0}|(1,1,\cdots,1)\in \nu\cdot P(\mathfrak a)\}. \end{equation} In other words, $\text{\rm lct}_P(\mathfrak a)$ is equal to the largest $\nu$ such that $\sum_{i=1}^n\lambda_i \ge \nu\cdot \min_{u\in P(\mathfrak a)}\langle u,\lambda\rangle$ for any $\lambda = (\lambda_1,\cdots,\lambda_n)\in \mathbb Z^n_{\ge 0}$, where we use the standard inner product $\langle u,\lambda\rangle = \sum_{i=1}^n u_i\lambda_i$.
Theorem \ref{equal} says that the log canonical threshold is equal to the $G$-stable rank for monomial ideals. More precisely, suppose $\mathfrak a\subset \mathbb C[x_1,\cdots,x_n]$ is a proper nonzero ideal generated by monomials and $P=(0,\cdots,0)$ is the origin. Then \begin{equation} \text{\rm lct}_P(\mathfrak a) = \operatorname{rk}^G(\mathfrak a). \end{equation}
\begin{proof}[Proof of Theorem \ref{equal}] Let $\mathfrak a = (m_1,\cdots,m_r)$ and $\{m_i = x_1^{l_{i1}}x_2^{l_{i2}}\cdots x_n^{l_{in}}\}_{i=1,\cdots,r}$ be a set of generators, where $l_i = (l_{i1},\cdots,l_{in})\in \mathbb Z^n_{\ge 0}$, we have \[
\text{\rm lct}_P(\mathfrak a) = \max\{\nu\in\mathbb R_{\ge 0}|(1,\cdots,1)\in \nu \cdot P(\mathfrak a)\} \] \[
=\max\{\nu\in \mathbb R_{\ge 0}| \sum_j\lambda_j \ge \nu\cdot \min_{u\in P(\mathfrak a)} \langle u,\lambda\rangle, \forall \lambda\in \mathbb Z^n_{\ge 0}\} \] \[
=\max\{\nu\in \mathbb R_{\ge 0}| \sum_j\lambda_j \ge \nu\cdot \min_{i} \langle l_i,\lambda\rangle, \forall \lambda\in \mathbb Z^n_{\ge 0}\} \] \[
=\max\{\nu\in \mathbb R_{\ge 0}| \nu \le \frac{\sum_j\lambda_j}{\min_i(\sum_j l_{ij}\lambda_j)}, \forall \lambda\in \mathbb Z^n_{\ge 0}\}. \] In this system of local parameters $T = \{x_1,\cdots,x_n\}$, we have $\operatorname{ord}^T_\lambda(\mathfrak a) = \min_i(\sum_j l_{ij}\lambda_j)$, therefore we get \[
\text{\rm lct}_P(\mathfrak a) = \max\{\nu\in \mathbb R_{\ge 0}| \nu \le \frac{\sum_i\lambda_i}{\operatorname{ord}^T_\lambda(\mathfrak a)}, \forall \lambda\in \mathbb Z^n_{\ge 0}\} \] \[
= \max\{\nu\in \mathbb R_{\ge 0}| \nu \le \operatorname{rk}^T(\mathfrak a)\} = \operatorname{rk}^T(\mathfrak a). \] Since log canonical threshold does not depend on the local coordinates, hence we have \[ \text{\rm lct}_P(\mathfrak a) = \operatorname{rk}^G(\mathfrak a). \] This completes the proof of Theorem \ref{equal}. \end{proof} \begin{example} Suppose $\mathfrak a = (x_1^{u_1},\cdots,x_n^{u_n})$ and $P = (0,\cdots,0) $, then $\text{\rm lct}_P(\mathfrak a) = \sum_{i=1}^n\frac{1}{u_i}$. Therefore we also have $\operatorname{rk}^G(\mathfrak a) = \sum_{i=1}^n\frac{1}{u_i}$. \end{example}
\subsection{Some properties of G-stable rank for ideals}
Some results for log canonical threshold can be found in~\cite{Mus}. Here, we also prove similar results for the $G$-stable rank of ideals.
\begin{prop} If $\mathfrak a\subseteq \mathfrak b$ are nonzero ideals on $X$, then we have $\text{\rm lct}_P(\mathfrak a)\le \text{\rm lct}_P(\mathfrak b)$ and $\operatorname{rk}^G(P,\mathfrak a)\le \operatorname{rk}^G(P,\mathfrak b)$. \end{prop} \begin{proof} The first inequality was shown in \cite{Mus}. If $P\notin V(\mathfrak b)$, it is trivial. Assume $P\in V(\mathfrak b)$, let $T$ be a system of local parameters at $P$ and $\lambda=(\lambda_1,\cdots,\lambda_n)\in \mathbb Z^n_{\ge 0}$. Since $\mathfrak a\subseteq \mathfrak b$, we have $\operatorname{ord}^T_\lambda(\mathfrak b) \le \operatorname{ord}^T_\lambda(\mathfrak a)$, therefore $\mu_P(\lambda,\mathfrak a) \le \mu_P(\lambda,\mathfrak b)$, it follows immediately that $\operatorname{rk}^G(P,\mathfrak a)\le \operatorname{rk}^G(P,\mathfrak b)$. \end{proof}
\begin{prop} We have $\text{\rm lct}_P(\mathfrak a^r) = \frac{\text{\rm lct}_P(\mathfrak a)}{r}$ and $\operatorname{rk}^G(P,\mathfrak a^r) = \frac{\operatorname{rk}^G(P,\mathfrak a)}{r}$ for every $r\ge 1$. \end{prop} \begin{proof} The first claim was shown in \cite{Mus} and the second claim follows from the fact that $\operatorname{ord}^T_\lambda(\mathfrak a^r) = r\cdot\operatorname{ord}^T_\lambda(\mathfrak a)$ and Definition \ref{Gdef}. \end{proof}
\begin{prop} If $\mathfrak a$ and $\mathfrak b$ are ideals of $X$, then \[ \frac{1}{\text{\rm lct}_P(\mathfrak a\cdot\mathfrak b)} \le \frac{1}{\text{\rm lct}_P(\mathfrak a)} + \frac{1}{\text{\rm lct}_P(\mathfrak b)}, \quad \frac{1}{\operatorname{rk}^G(P,\mathfrak a\cdot\mathfrak b)}\le \frac{1}{\operatorname{rk}^G(P,\mathfrak a)}+\frac{1}{\operatorname{rk}^G(P,\mathfrak b)}. \] \end{prop} \begin{proof} The first inequality was shown in \cite{Mus}, we show the second inequality. Let $T$ be a system of local parameters at $P$ and $\lambda = (\lambda_1,\cdots,\lambda_n)\in \mathbb Z^n_{\ge 0}$, then \[ \operatorname{ord}_\lambda^T(\mathfrak a\cdot\mathfrak b) = \operatorname{ord}_\lambda^T(\mathfrak a) +\operatorname{ord}_\lambda^T(\mathfrak b). \] Therefore, we have \[ \sup_{T,\lambda}\frac{\operatorname{ord}_\lambda^T(\mathfrak a\cdot\mathfrak b)}{\sum\lambda_i}= \sup_{T,\lambda}\left(\frac{\operatorname{ord}_\lambda^T(\mathfrak a)}{\sum\lambda_i}+\frac{\operatorname{ord}_\lambda^T(\mathfrak b)}{\sum\lambda_i}\right) \] \[ \le \sup_{T,\lambda}\frac{\operatorname{ord}_\lambda^T(\mathfrak a)}{\sum\lambda_i}+\sup_{T,\lambda}\frac{\operatorname{ord}_\lambda^T(\mathfrak b)}{\sum\lambda_i}. \] \end{proof}
The following two propositions are from \cite{Mus}. A lot of evidence suggests the same results for $G$-stable rank of ideals, we give them as conjectures. \begin{prop} If $H\subset X$ is a nonsingular hypersurface such that $\mathfrak a\cdot \mathcal O_H$ is nonzero, then $\text{\rm lct}_P(\mathfrak a\cdot \mathcal O_H)\le \text{\rm lct}_P(\mathfrak a)$. \end{prop}
\begin{prop} If $\mathfrak a$ and $\mathfrak b$ are ideals on $X$, then \[ \text{\rm lct}_P(\mathfrak a + \mathfrak b) \le \text{\rm lct}_P(\mathfrak a) + \text{\rm lct}_P(\mathfrak b) \] for every $P\in X$. \end{prop}
\begin{conjecture} If $H\subset X$ is a nonsingular hypersurface such that $\mathfrak a\cdot \mathcal O_H$ is nonzero, then $\operatorname{rk}^G(P,\mathfrak a\cdot\mathcal O_H)\le \operatorname{rk}^G(P,\mathfrak a)$ for any $P\in H$. \end{conjecture}
\begin{conjecture} Let $\mathfrak a$ and $\mathfrak b$ be two nonzero proper ideals of $X$, then for any point $P\in X$, we have \[
\operatorname{rk}^G(P, \mathfrak a+\mathfrak b) \le \ \operatorname{rk}^G(P, \mathfrak a) + \operatorname{rk}^G(P, \mathfrak b). \] \end{conjecture}
\end{document} | arXiv |
G/M/1 queue
In queueing theory, a discipline within the mathematical theory of probability, the G/M/1 queue represents the queue length in a system where interarrival times have a general (meaning arbitrary) distribution and service times for each job have an exponential distribution.[1] The system is described in Kendall's notation where the G denotes a general distribution, M the exponential distribution for service times and the 1 that the model has a single server.
The arrivals of a G/M/1 queue are given by a renewal process. It is an extension of an M/M/1 queue, where this renewal process must specifically be a Poisson process (so that interarrival times have exponential distribution).
Models of this type can be solved by considering one of two M/G/1 queue dual systems, one proposed by Ramaswami and one by Bright.[2]
Queue size at arrival times
Let $(X_{t},t\geq 0)$ be a $G/M(\mu )/1$ queue with arrival times $(A_{n},n\in \mathbb {N} )$ that have interarrival distribution A. Define the size of the queue immediately before the nth arrival by the process $U_{n}=X_{A_{n}-}$. This is a discrete-time Markov chain with stochastic matrix:
$P={\begin{pmatrix}1-a_{0}&a_{0}&0&0&0&\cdots \\1-(a_{0}+a_{1})&a_{1}&a_{0}&0&0&\cdots \\1-(a_{0}+a_{1}+a_{2})&a_{2}&a_{1}&a_{0}&0&\cdots \\1-(a_{0}+a_{1}+a_{2}+a_{3})&a_{3}&a_{2}&a_{1}&a_{0}&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \end{pmatrix}}$
where $a_{v}=\mathbb {E} \left({\frac {(\mu X)^{v}e^{-\mu A}}{v!}}\right)$.[3]: 427–428
The Markov chain $U_{n}$ has a stationary distribution if and only if the traffic intensity $\rho =(\mu \mathbb {E} (A))^{-1}$ is less than 1, in which case the unique such distribution is the geometric distribution with probability $\eta $ of failure, where $\eta $ is the smallest root of the equation $\mathbb {E} (\exp(\mu (\eta -1)A))$.[3]: 428
In this case, under the assumption that the queue is first-in first-out (FIFO), a customer's waiting time W is distributed by:[3]: 430
$\mathbb {P} (W\leq x)=1-\eta \exp(-\mu (1-\eta )x)~{\text{ for }}x\geq 0$
Busy period
The busy period can be computed by using a duality between the G/M/1 model and M/G/1 queue generated by the Christmas tree transformation.[4]
Response time
The response time is the amount of time a job spends in the system from the instant of arrival to the time they leave the system. A consistent and asymptotically normal estimator for the mean response time, can be computed as the fixed point of an empirical Laplace transform.[5]
References
1. Adan, I.; Boxma, O.; Perry, D. (2005). "The G/M/1 queue revisited" (PDF). Mathematical Methods of Operations Research. 62 (3): 437. doi:10.1007/s00186-005-0032-6.
2. Taylor, P. G.; Van Houdt, B. (2010). "On the dual relationship between Markov chains of GI/M/1 and M/G/1 type" (PDF). Advances in Applied Probability. 42: 210. doi:10.1239/aap/1269611150.
3. Grimmett, G. R.; Stirzaker, D. R. (1992). Probability and Random Processes (second ed.). Oxford University Press. ISBN 0198572220.
4. Perry, D.; Stadje, W.; Zacks, S. (2000). "Busy period analysis for M/G/1 and G/M/1 type queues with restricted accessibility". Operations Research Letters. 27 (4): 163. doi:10.1016/S0167-6377(00)00043-2.
5. Chu, Y. K.; Ke, J. C. (2007). "Interval estimation of mean response time for a G/M/1 queueing system: Empirical Laplace function approach". Mathematical Methods in the Applied Sciences. 30 (6): 707. doi:10.1002/mma.806.
Queueing theory
Single queueing nodes
• D/M/1 queue
• M/D/1 queue
• M/D/c queue
• M/M/1 queue
• Burke's theorem
• M/M/c queue
• M/M/∞ queue
• M/G/1 queue
• Pollaczek–Khinchine formula
• Matrix analytic method
• M/G/k queue
• G/M/1 queue
• G/G/1 queue
• Kingman's formula
• Lindley equation
• Fork–join queue
• Bulk queue
Arrival processes
• Poisson point process
• Markovian arrival process
• Rational arrival process
Queueing networks
• Jackson network
• Traffic equations
• Gordon–Newell theorem
• Mean value analysis
• Buzen's algorithm
• Kelly network
• G-network
• BCMP network
Service policies
• FIFO
• LIFO
• Processor sharing
• Round-robin
• Shortest job next
• Shortest remaining time
Key concepts
• Continuous-time Markov chain
• Kendall's notation
• Little's law
• Product-form solution
• Balance equation
• Quasireversibility
• Flow-equivalent server method
• Arrival theorem
• Decomposition method
• Beneš method
Limit theorems
• Fluid limit
• Mean-field theory
• Heavy traffic approximation
• Reflected Brownian motion
Extensions
• Fluid queue
• Layered queueing network
• Polling system
• Adversarial queueing network
• Loss network
• Retrial queue
Information systems
• Data buffer
• Erlang (unit)
• Erlang distribution
• Flow control (data)
• Message queue
• Network congestion
• Network scheduler
• Pipeline (software)
• Quality of service
• Scheduling (computing)
• Teletraffic engineering
Category
| Wikipedia |
\begin{document}
\title{Continuous-time quantum walks in the presence of a quadratic perturbation} \author{Alessandro Candeloro} \email{[email protected]} \affiliation{Quantum Technology Lab, Dipartimento di Fisica {\em Aldo Pontremoli}, Universit\`{a} degli Studi di Milano, I-20133 Milano, Italy}
\author{Luca Razzoli} \email{[email protected]} \affiliation{Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Universit\`{a} di Modena e Reggio Emilia, I-41125 Modena, Italy}
\author{Simone Cavazzoni} \email{[email protected]} \affiliation{Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Universit\`{a} di Modena e Reggio Emilia, I-41125 Modena, Italy}
\author{Paolo Bordone} \email{[email protected]} \affiliation{Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Universit\`{a} di Modena e Reggio Emilia, I-41125 Modena, Italy} \affiliation{Centro S3, CNR-Istituto di Nanoscienze, I-41125 Modena, Italy}
\author{Matteo G. A. Paris} \email{[email protected]} \affiliation{Quantum Technology Lab, Dipartimento di Fisica {\em Aldo Pontremoli}, Universit\`{a} degli Studi di Milano, I-20133 Milano, Italy} \affiliation{INFN, Sezione di Milano, I-20133 Milano, Italy}
\begin{abstract} We address the properties of continuous-time quantum walks with Hamiltonians of the form $\mathcal{H}= L + \lambda L^2$, being $L$ the Laplacian matrix of the underlying graph and being the perturbation $\lambda L^2$ motivated by its potential use to introduce next-nearest-neighbor hopping. We consider cycle, complete, and star graphs because paradigmatic models with low/high connectivity and/or symmetry. First, we investigate the dynamics of an initially localized walker. Then, we devote attention to estimating the perturbation parameter $\lambda$ using only a snapshot of the walker dynamics. Our analysis shows that a walker on a cycle graph is spreading ballistically independently of the perturbation, whereas on complete and star graphs one observes perturbation-dependent revivals and strong localization phenomena. Concerning the estimation of the perturbation, we determine the walker preparations and the simple graphs that maximize the Quantum Fisher Information. We also assess the performance of position measurement, which turns out to be optimal, or nearly optimal, in several situations of interest. Besides fundamental interest, our study may find applications in designing enhanced algorithms on graphs. \end{abstract}
\date{\today} \maketitle
\section{Introduction} \label{sec:intro} A continuous-time quantum walk (CTQW) describes the dynamics of a quantum particle confined to discrete spatial locations, i.e. to the vertices of a graph \cite{farhi1998quantum,childs2002example,wong2016laplacian}. In these systems, the graph Laplacian $L$ (also referred to as the {\em Kirchhoff} matrix of the graph) plays the role of the free Hamiltonian, i.e. it corresponds to kinetic energy of the particle. Perturbations to ideal CTQW have been investigated earlier \cite{Rao_2011,Caruso_2014,Siloi_2017,Rossi_2017,Cattaneo_2018,Rossi_2018,Herrman_2019,De_Santis_2019,Melnikov_2020}, however with the main focus being about the decoherence effects of stochastic noise, rather than the quantum effects induced by a perturbing Hamiltonian. A notable exception exists, though, given by quantum spatial search, where the perturbation induced by the so-called {\em oracle Hamiltonian} has been largely investigated as a tool to induce localization on a desired site \cite{childs2004spatial,kendon_2006,Wong_2015,Wong_2016,Philipp_2016,Delvecchio_2020}. As a matter of fact, quantum walks have found several applications ranging from universal quantum computation \cite{childs09} to quantum algorithms \cite{kendon06,ambainis07,farhi08,coppersmith10,Tamascelli_2014,PhysRevE.97.013301}, and to the study of excitation transport on networks \cite{mulken11,alvir16,tamas16}, and biological systems \cite{mohseni08,hoyer10}. As such, and due to the diversity of the physical platforms on which quantum walks have been implemented \cite{preiss2015strongly,peruzzo2010quantum}, a precise characterization of the quantum-walk Hamiltonian is desired.
In the present work, we investigate the dynamics of an initially localized quantum walker propagating on cycle, complete, and star graphs (see Fig. \ref{fig:graphs}) under perturbed Hamiltonians of the form $\mathcal{H}= L + \lambda L^2$. Characterizing these Hamiltonians amounts to determining the value of the coupling parameter $\lambda$, which quantifies the effects of the quadratic term. For this purpose, we investigate whether, and to which extent, a snapshot of the walker dynamics at a given time suffices to estimate the value of $\lambda$.
Besides the fundamental interest, there are few reasons to address these particular systems. The topologies of these graphs describe paradigmatic situations with low (cycle and star) or high (complete) connectivity, as well as low (star) and high (cycle and complete) symmetry. At the same time, CTQW Hamiltonians with quadratic perturbation of the form $\lambda L^2$ are of interest, e.g., because they represent a physically motivated and convenient way to introduce next-nearest-neighbor hopping in one-dimensional lattices, or intrinsic spin-orbit coupling in two-dimensional ones. Moreover, considering such perturbations is the first step towards the description of dephasing and dechoerence processes, which result from making the parameter $\lambda$ a stochastic process.
\begin{figure}
\caption{The three types of graphs considered in the present work: (a) cycle, (b) complete, and (c) star graphs. Examples for $N=5$ vertices.}
\label{fig:graphs}
\end{figure}
To analyze both semi-classical and genuinely quantum features of the dynamics, we employ a set of different quantifiers, including site distribution, mixing, inverse participation ratio, and coherence. In this framework, mixing has been studied for CTQWs on some circulant graphs \cite{ahmadi2003mixing}, e.g. the cycle and the complete graph, and also employed together with the temporal standard deviation to study the dynamics of CTQW on the cycle graph \cite{inui2005evolution}. Moreover, a spectral method has been introduced to investigate CTQW on graphs \cite{jafarizadeh2007investigation,salimi2009continuous}. Coherent transport has been analytically analyzed for CTQW on star graphs \cite{xu2009exact}, showing the occurrence of perfect revivals and strong localization on the initial node.
The rest of the paper is organized as follows. In Sec. \ref{sec:dynamics} we address the dynamics of an initially localized walker in the different graphs. In Sec. \ref{sec:estimation}, we focus on the estimation of the parameter of the perturbation by evaluating the Quantum Fisher Information (QFI). We consider initially localized states as well as the states maximizing the QFI, and we compare the QFI to the Fisher Information (FI) of position measurement. Moreover, we determine the simple graphs that allow to obtain the maximum QFI. In Sec. \ref{sec:conclusions}, we summarize and discuss our results and findings. Then, in Appendix \ref{app:an_res_ccsg} we provide further analytical details about the dynamics of the CTQWs over the different graphs. In Appendix \ref{app:q_fi_scg}, we prove the results concerning the (Q)FI.
\section{Dynamics} \label{sec:dynamics} A graph is a pair $G=(V, E)$, where $V$ denotes the non-empty set of vertices and $E$ the set of edges. In a graph, the kinetic energy term ($\hbar=1$) ${T}=-\nabla^2/2m$ is replaced by ${T}=\gamma L$, where $\gamma \in \mathbb{R}^+$ is the hopping amplitude of the walk and $L=D-A$ the graph Laplacian, with $A$ the adjacency matrix ($A_{jk}=1$ if the vertices $j$ and $k$ are connected, $0$ otherwise) and $D$ the diagonal degree matrix ($D_{jj}=\operatorname{deg}(j)$). The hopping amplitude $\gamma$ plays the role of a time-scaling factor, thus the time dependence of the results is significant when expressed in terms of the dimensionless time $\gamma t$. Please notice that in the following we set $\gamma = \hbar= 1$, and, as a consequence, hereafter time and energy will be dimensionless. We consider finite graphs of order $\abs{V}=N$, i.e. graphs with $N$ vertices which we index from $0$ to $N-1$, and we focus on the dynamics of a walker whose initial state $\ket{\psi(0)}$ is a vertex of the graph, i.e. the walker is initially localized.
We consider the Hamiltonian \begin{equation} {\mathcal{H}}={\mathcal{H}}_0+\lambda{\mathcal{H}}_1=L+\lambda L^2\,, \label{eq:H_ctqwp4} \end{equation} where $\lambda$ is a dimensionless perturbation parameter. Because of this choice, the eigenproblem of $\mathcal{H}$ is basically the eigenproblem of $L$. The laplacian eigenvalue $\varepsilon = 0$ is common to all simple graphs, it is not degenerate for connected graphs, like cycle, complete, and star graph, and the corresponding eigenvector is $(1,\ldots,1)/\sqrt{N}$. The time evolution of the system is coherent and ruled by the unitary time-evolution operator \begin{equation} \mathcal{U}_\lambda(t)=e^{-i{\mathcal{H}}t}=\sum_{n=0}^{N-1} e^{-i(\varepsilon_n+\lambda\varepsilon_n^2) t} \dyad{e_n}\,, \label{eq:t_evol_op} \end{equation} where the second equality follows from the spectral decomposition of $L$. To study the dynamics of the walker, we consider the following quantities, which basically arise from the density matrix $\rho(t)=\dyad{\psi(t)}$.
The (\textit{instantaneous}) probability of finding the walker in the vertex $k$ at time $t$ is \begin{equation}
P(k,t\vert \lambda)=\abs{\langle k \vert \mathcal{U}_\lambda(t) \vert \psi(0)\rangle}^2\,, \end{equation} whereas the \textit{average} probability is \begin{equation}
\bar{P}(k\vert \lambda)=\lim_{T\to+\infty}\frac{1}{T}\int_0^T P(k,t\vert \lambda) dt\,.
\label{eq:avg_prob} \end{equation} There are two main notions of mixing in quantum walks \cite{aharonov2001quantum,moore2002quantum,ahmadi2003mixing}. A graph has the \textit{instantaneous exactly uniform mixing} property if there are times when the probability distribution $P(t)$ of the walker is exactly uniform; it has the \textit{average uniform mixing} property if the average probability distribution $\bar{P}$ is uniform.
In addition, we consider the inverse participation ratio (IPR) \cite{thouless1974electrons,kramer1993localization,Rossi_2017} \begin{equation}
\mathcal{I}(t) = \sum_{k=0}^{N-1} \langle k \vert \rho (t) \vert k \rangle ^2 = \sum_{k=0}^{N-1} P^2(k,t\vert \lambda)\,,
\label{eq:ipr_local} \end{equation} which allows us to assess the amount of localization in position space of the walker. Indeed, the IPR is bounded from below by $1/N$ (complete delocalization) and from above by $1$ (localization on a single vertex). In this sense, the IPR is an alternative quantity to study the instantaneous exactly uniform mixing. The inverse of the IPR indicates the number of vertices over which the walker is distributed \cite{ingold2002delocalization}.
Finally, to further analyze the quantum features of the dynamics, we consider the quantum coherence. A proper measure is provided by the $l_1$ norm of coherence \cite{baumgratz2014quantifying} \begin{equation} \mathcal{C}(t)=\sum_{\substack{j,k=0,\\j\neq k}}^{N-1} \abs{\rho_{j,k}(t)}=\sum_{j,k=0}^{N-1} \abs{\rho_{j,k}(t)}-1\,. \label{eq:coherence_def} \end{equation}
Please refer to Appendix \ref{app:an_res_ccsg} for the details about the analytical derivation of the results shown in the following.
\subsection{Cycle graph} \label{subsec:cyclegraph} In the cycle graph each vertex is adjacent to $2$ other vertices, so its degree is $2$. Hence, the graph Laplacian is \begin{equation} L = 2{I}-\sideset{}{'}\sum_{k=0}^{N-1} \left( \dyad{k-1}{k}+\dyad{k+1}{k}\right)\,. \label{eq:H0_cycle_matrix} \end{equation} The primed summation symbol means that we look at the cycle graph as a path graph provided with periodic boundary conditions, thus the terms $\dyad{-1}{0}$ and $\dyad{N}{N-1}$ are $\dyad{N-1}{0}$ and $\dyad{0}{N-1}$, respectively. The matrix representation of this Laplacian is symmetric and circulant (a special case of Toeplitz matrix), and the related eigenproblem is analytically solved in Ref. \cite{gray2006toeplitz} and reported in Table \ref{tab:eig_pbm_cycle}.
\begin{table}[tb]
\centering
\begin{ruledtabular}
\begin{tabular}{lcc}
$\ket{e_n}$ & $\varepsilon_n$ & $\mu_n$\\\hline
$\ket{e_n}=\frac{1}{\sqrt{N}}\sum\limits_{k=0}^{N-1}e^{-i\frac{2\pi n}{N}k}\ket{k}$ & $ 2\left[1-\cos\left( \frac{2\pi n}{N} \right) \right]$ & $\ast$\\
\text{with $n=0,\ldots,N-1$}\\
\end{tabular}
\end{ruledtabular}
\caption{Eigenvectors $\ket{e_n}$ and eigenvalues $\varepsilon_n$ of the graph Laplacian in the cycle graph. ($\ast$) The multiplicity of the eigenvalues depends on the parity of $N$. In particular, the ground state $n=0$ is always unique, whereas the highest energy level is unique for even $N$ and doubly degenerate for odd $N$. Independently of the parity of $N$, the remaining eigenvalues have multiplicity $2$, since $\varepsilon_n = \varepsilon_{N-n}$.}
\label{tab:eig_pbm_cycle} \end{table}
The ground state ($n=0$) is unique and equal to \begin{align} \varepsilon_{min}&= 0 \label{eq:eval_cyc_min}\,,\\ \ket{e_{min}}&=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1} \ket{k} \label{eq:evec_cyc_min}\,. \end{align} Instead, the highest energy level depends on the parity of $N$ and \begin{enumerate}[(i)] \item is unique for even $N$ ($n=N/2$): \begin{align} \varepsilon_{max} &=4 \,,\label{eq:eval_cyc_max_even}\\ \ket{e_{max}}&= \frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}(-1)^k\ket{k}\,,\label{eq:evec_cyc_max_even} \end{align} \item has degeneracy $2$ for odd $N$ ($n=(N\pm1)/2$): \begin{align} \varepsilon_{max} &= 2\left[1+\cos\left (\frac{\pi}{N}\right )\right]\,,\label{eq:eval_cyc_max_odd}\\ \ket{e_{max}}&= \frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}(-1)^k e^{\pm i \frac{\pi}{N}k}\ket{k}\,,\label{eq:evec_cyc_max_pm} \end{align} where the phase factors are all either with the $+$ sign or with the $-$ sign. \end{enumerate}
Since for odd $N$ the highest energy level is doubly degenerate, we may be interested in finding the corresponding orthonormal eigenstates having real components \footnote{The further reason is that some numerical routines solving the eigenproblem for real symmetric matrices may return orthonormal eigenvectors with real components.}. Therefore we define the following states by linearly combining the two eigenstates in Eq. \eqref{eq:evec_cyc_max_pm} in one case with the plus sign and with the minus sign in the other \footnote{The linear combination leading to Eq. \eqref{eq:evec_cyc_max_sin} introduces also an imaginary unit. However, this is a global phase factor, and, as such, we neglect it.}, respectively: \begin{align} \ket{e_{max}^+} &= \sqrt{\frac{2}{N}} \sum_{k=0}^{N-1} (-1)^k \cos \left(\frac{\pi}{N}k\right)\ket{k} \,,\label{eq:evec_cyc_max_cos}\\ \ket{e_{max}^-} &= \sqrt{\frac{2}{N}} \sum_{k=0}^{N-1}(-1)^k \sin \left(\frac{\pi}{N}k\right)\ket{k} \,.\label{eq:evec_cyc_max_sin} \end{align}
The perturbation involves \begin{equation} L^2 =6{I}+ \sideset{}{'}\sum_{k=0}^{N-1} (\dyad{k-2}{k}-4\dyad{k-1}{k}+H.c.)\,, \label{eq:L2_cyc} \end{equation} where the Hermitian conjugate of $\dyad{k-n}{k}$ is a $+n$-vertices hopping term, and, as such, has to be intended as $\dyad{k+n}{k}$. Hence, the perturbed Hamiltonian \eqref{eq:H_ctqwp4} reads as follows: \begin{align} {\mathcal{H}}&=(2+6\lambda){I}+\sideset{}{'}\sum_{k=0}^{N-1}\left[ \lambda\dyad{k-2}{k}\right.\nonumber\\ &\quad \left.-(1+4\lambda)\dyad{k-1}{k} +H.c. \right]\,. \label{eq:totH_cyc} \end{align} The perturbation $\lambda L^2$, thus, introduces the next-nearest neighbor hopping, affects the nearest-neighbor one, and also the on-site energies $\propto {I}$.
In a cycle graph all the vertices are equivalent, so an initially localized walker will show the same time evolution independently of the starting vertex chosen. We denote the initial state by $\ket{j}$. The probability of finding the walker in the vertex $k$ at time $t$ for a given value of $\lambda$ is (Fig. \ref{fig:prob_loc_cycle}) \begin{align}
P_j&(k,t\vert\lambda) =\frac{1}{N} + \frac{2}{N^2} \nonumber\\
& \times \sum_{\substack{n=0, \\ m>n}}^{N-1} \cos\left[(E^\lambda_n - E^\lambda_m)t- \frac{2\pi}{N}(n-m)(j-k)\right]\,,
\label{eq:pjk_cyc_cos} \end{align} which is symmetric with respect to the starting vertex $j$, i.e. $P_j(j+k,t\vert\lambda)=P_j(j-k,t\vert\lambda)$ (proof in Appendix \ref{subapp:an_res_cyc}). The average probability distribution is the same as the one reported in \cite{inui2005evolution}, which is basically our unperturbed CTQW \footnote{The CTQW Hamiltonian in \cite{inui2005evolution} is $\mathcal{H}=A/d$, instead of being the Laplacian. In regular graphs $d$, the degree of the vertex, is the same for all the vertices. The diagonal degree matrix $D$ is thus proportional to the identity, and this introduces an irrelevant phase factor in the time evolution of the quantum state. The time scale of the evolutions under the Hamiltonians $A$ and $A/d$ are clearly different, but the resulting time-averaged probability distribution is the same.}.
\begin{figure}
\caption{Probability distribution $P_j(k,t\vert\lambda)$ of the walker as a function of time in the cycle graph. The walker is initially localized in the vertex $\ket{j=2}$. The probability distribution is symmetric with respect to the starting vertex, i.e. $P_j(j+k,t\vert\lambda)=P_j(j-k,t\vert\lambda)$. Numerical results suggest that revivals in the starting vertex are most likely not exact. Indeed, to be exact, the periods of the cosine functions entering the definition of the probability \eqref{eq:pjk_cyc_cos} have to be commensurable and such periods strongly depend on the choice of $N$ and $\lambda$. Results for $N=5$ and $\lambda=0.2$.}
\label{fig:prob_loc_cycle}
\end{figure}
The solution of the time-dependent Schr\"{odinger} equation of the unperturbed system ($\lambda=0$) can be expressed in terms of Bessel functions \cite{ahmadi2003mixing}. This allowed to analytically prove the ballistic spreading in a one-dimensional infinite lattice \cite{endo2009ballistic}, i.e. that the variance of the position is $\sigma^2(t) = \langle \hat {x}(t)^2 \rangle - \langle \hat {x}(t) \rangle^2 \propto t^2$. We expect the same ballistic spreading to characterize the CTQW on a finite cycle at short times, i.e. as long as the walker does not feel the topology of the cycle graph.
We can find a simple expression describing the variance of the position for $\lambda \neq 0$ at short times. The variance is meaningful if we consider sufficiently large $N$, and the assumption $t\ll 1$ ensures that the wavefunction does not reach the vertices $\ket{0}$ and $\ket{N-1}$. Indeed, the position on the graph is the corresponding vertex, but the topology of the cycle graph allows the walker to jump from $\ket{0}$ to $\ket{N-1}$ and \textit{vice versa}. This, in turn, affects the computation of the variance. To ensure the maximum distance from the extreme vertices, we consider a walker initially localized in the central vertex. We assume even $N$, so the starting vertex is $\ket{j=N/2}$. Under these assumptions, we have that \begin{equation} \sigma^2 (t) \approx \left[40(\lambda-\lambda_0)^2+\frac{2}{5}\right ]t^2\,, \label{eq:xVar_cyc} \end{equation} with $\lambda_0=-1/5$ (see Appendix \ref{subapp:an_res_cyc}). The spreading of the walker is ballistic in spite of the perturbation. Nevertheless, increasing $\abs{\lambda-\lambda_0}$ makes the walker spread faster by affecting the factor in front of $t^2$. Such factor, indeed, is related to the square of the parameter characterizing the speed of the walker \cite{endo2009ballistic}. The lowest variance is for $\lambda=\lambda_0$, which is the value for which the nearest-neighbor hopping $-(1+4\lambda)$ equals the next-nearest-neighbor one $\lambda$ (see Eq. \eqref{eq:totH_cyc}). Numerical simulations of the CTQW provide evidences that the same behavior in Eq. \eqref{eq:xVar_cyc} characterizes also the CTQW on the cycle with odd $N$ or when the starting vertex is not the central one, again assuming that the wavefunction does not reach the extreme vertices.
For completeness, we report in Fig. \ref{fig:fpd_loc_cycle} the numerical results for the probability distribution \eqref{eq:pjk_cyc_cos} at a given time and at varying $\lambda$. The pattern of the probability distribution is not symmetric with respect to $\lambda_0$. Nevertheless, at short times the resulting variance of the position \eqref{eq:xVar_cyc} turns out to be symmetric with respect to $\lambda_0$.
\begin{figure}
\caption{Map of the probability distribution \eqref{eq:pjk_cyc_cos} as a function of the position (vertex) and $\lambda$ at $t=4$. The walker is initially localized in the center of the cycle graph ($N=100$). The horizontal dashed white line highlights $\lambda_0=-1/5$, the value at which the variance of the position is minimum. For clarity, here vertices are indexed from $1$ to $N$.}
\label{fig:fpd_loc_cycle}
\end{figure}
Next, we numerically evaluate the IPR \eqref{eq:ipr_local} for the probability distribution in Eq. \eqref{eq:pjk_cyc_cos}, and the results are shown in Fig. \ref{fig:cycle_IPR_multi}. As expected from the previous results about the probability distribution (see also Fig. \ref{fig:prob_loc_cycle}), the IPR does not show a clear periodicity, it strongly fluctuates, and there are instants of time when it gets closer to $1$, meaning that the walker is more localized. The numerical results also suggest that the instantaneous exactly uniform mixing is achievable for $N\leq 4$, while there is no exact delocalization for $N>4$, as already conjectured \cite{ahmadi2003mixing}. However, for large $N$ the probability distribution \eqref{eq:pjk_cyc_cos} approaches the uniform one, and so the IPR approaches $1/N$. \begin{figure}
\caption{Inverse participation ratio (IPR) for a walker initially localized in the cycle graph. Numerical results suggest that for $t>0$ the IPR reaches neither the lower bound $1/N$ (green dashdotted line), i.e. the delocalization, nor the upper bound $1$ (orange dashed line), i.e. the localization. The fact that the (de)localization is achievable or not is most likely related to the choice of $N$ and $\lambda$. This choice, in turn, might result in the commensurability or incommensurability of the periods of the cosine functions entering the definition of the probability \eqref{eq:pjk_cyc_cos}. For large $N$ the IPR approaches $1/N$, since the probability distribution approaches the uniform one. Results for $\lambda=0.2$.}
\label{fig:cycle_IPR_multi}
\end{figure}
Finally, we focus on the time dependence of the coherence \eqref{eq:coherence_def} for an initially localized walker. The exact numerical results are shown in Fig. \ref{fig:coher_cyc_loc}. Under the assumption $t \ll 1$, we can find a simple expression. We Taylor expand the time-evolution operator up to the first order, so the density matrix is approximated as ${\rho}(t) = {\rho}(0)-it\left[\mathcal{H},\rho(0)\right]+\mathcal{O}(t^2)$. Then, being the Hamiltonian \eqref{eq:totH_cyc}, the behavior characterizing the earlier steps of the time evolution of the coherence is \begin{equation}
\mathcal{C}(t,\lambda) \approx 4(\abs{\lambda}+\abs{1+4\lambda} )t\,,
\label{eq:coher_smallt_cycle} \end{equation} consistently with the results shown in Fig. \ref{fig:coher_cyc_loc}. Hence, at short times the coherence is minimum for $\lambda = -1/4$. For such value the nearest-neighbor hopping $-(1+4\lambda)$ is null, while the next-nearest-neighbor hopping $\lambda$ is nonzero (see Eq. \eqref{eq:totH_cyc}).
\begin{figure}
\caption{Coherence for a walker initially localized in the cycle graph with $N=5$. For $t\ll 1$ the minimum is for $\lambda =-1/4$, as expected from the linear approximation in Eq. \eqref{eq:coher_smallt_cycle}.}
\label{fig:coher_cyc_loc}
\end{figure}
\subsection{Complete graph} \label{subsec:completegraph} In the complete graph each vertex is adjacent to all the others, so its degree is $N-1$. Hence, the graph Laplacian is \begin{equation}
L = (N-1){I}-\sum_{\substack{j,k=0,\\j\neq k}}^{N-1} \dyad{j}{k}\,,
\label{eq:H0_complete_matrix} \end{equation} and has the following property \begin{equation} L^n =N^{n-1}L\,. \label{eq:Ln_proptoL} \end{equation}
\begin{table}[tb]
\centering
\begin{ruledtabular}
\begin{tabular}{clcc}
$n$ & $\ket{e_n}$ & $\varepsilon_n$ & $\mu_n$\\\hline
$0$ & $\ket{e_0}=\frac{1}{\sqrt{N}}\sum\limits_{k=0}^{N-1}\ket{k}$ & $0$ & $1$ \\
$1$ & $\ket{e_1^l}=\frac{1}{\sqrt{l(l+1)}}\left(\sum\limits_{k=0}^{l-1}\ket{k}-l\ket{l}\right)$ & $N$ & $N-1$\\
&\text{with $l=1,\ldots,N-1$}\\
\end{tabular}
\end{ruledtabular}
\caption{Eigenvectors $\ket{e_n}$, eigenvalues $\varepsilon_n$ with multiplicity $\mu_n$ of the graph Laplacian in the complete graph.}
\label{tab:eig_pbm_complete} \end{table}
The eigenproblem related to Eq. \eqref{eq:H0_complete_matrix} is solved in Table \ref{tab:eig_pbm_complete}. The graph Laplacian has two energy levels: \begin{enumerate}[(i)]
\item the level $\varepsilon_0 = 0$, having eigenstate $\ket{e_0}$;
\item the $(N-1)$-degenerate level $\varepsilon_1 = N$, having orthonormal eigenstates $\ket{e_1^l}$, with $l=1,\ldots,N-1$. \end{enumerate}
The perturbed Hamiltonian is therefore \begin{equation}
\mathcal{H}=(1+N\lambda)L\,,
\label{eq:totH_comp} \end{equation} i.e. it is basically the CTQW Hamiltonian of the complete graph multiplied by a constant which linearly depends on $\lambda$. We observe that the value $\lambda^\ast=-1/N$ makes the Hamiltonian null, and so it makes this case trivial. The perturbation affects the energy scale of the unperturbed system, and so its time scale. Therefore, we can directly compare the next results with the well-known ones concerning the unperturbed system \cite{ahmadi2003mixing}.
The time evolution operator \eqref{eq:t_evol_op} is \begin{equation} e^{-i \mathcal{H} t}=I+\frac{1}{N}\left [e^{-i2\omega_N(\lambda)t}-1\right ]L\,. \end{equation} where we have Taylor expanded the exponential, used Eq. \eqref{eq:Ln_proptoL}, and defined the angular frequency \begin{equation}
\omega_N(\lambda)=\frac{N}{2}(1+\lambda N)\,,
\label{eq:ang_freq} \end{equation} which depends on $\lambda$. For large $N$, the time evolution basically results in adding a phase to the initial state, since $\lim_{N\to+\infty} \mathcal{U}_\lambda(t)=\exp\left[-i2\omega_N(\lambda)t\right] I$.
In a complete graph all the vertices are equivalent, so an initially localized walker will show the same time evolution independently of the starting vertex chosen. We denote the initial state by $\ket{0}$. The probabilities of finding the walker in $\ket{0}$ or elsewhere, $\ket{1\leq i \leq N-1}$, at time $t$ for a given value of $\lambda$ are periodic (Fig. \ref{fig:prob_loc_complete}) \begin{align}
P_0(0,t\vert\lambda) & = 1 - \frac{4(N-1)}{N^2} \sin^2\left(\omega_N(\lambda)t\right), \label{eq:p0_cg}\\
P_0(i,t\vert \lambda) & = \frac{4}{N^2}\sin^2\left(\omega_N(\lambda)t\right)\,. \label{eq:pi_cg} \end{align} Hence, the walker comes back periodically to the starting vertex and can be found in it with certainty. This occurs for $t_k= 2 k \pi /(N+\lambda N^2)$, with $k\in \mathbb{N}$. Increasing the order of the graph makes the angular frequency higher, and $\lim_{N\to+\infty}P_0(0,t\vert\lambda) = 1$, while $\lim_{N\to+\infty}P_0(i,t\vert\lambda) = 0$. The perturbation only affects the periodicity of the probabilities. The probability distribution is symmetric with respect to $\lambda^\ast$, since $\omega_N(\lambda^\ast \pm\lambda)=\pm \lambda N^2 / 2$ and $\sin^2\left( \lambda N^2 / 2 \right)=\sin^2\left(-\lambda N^2 / 2 \right)$. As expected, for $\lambda^\ast$ the walker remains in the starting vertex all the time, since $\omega_N(\lambda^\ast)=0$ and so $P_0(0,t\vert\lambda^\ast) = 1\,\forall\,t$. The average probability distribution is the same as the one reported in \cite{ahmadi2003mixing}, which is basically our unperturbed CTQW \footnote{The CTQW Hamiltonian in \cite{ahmadi2003mixing} is $\mathcal{H}=A/d$. See also footnote \cite{Note3}.}.
\begin{figure}
\caption{Probability of finding the walker in the starting vertex $P_0(0,t\vert\lambda)$ (red solid line) or in any other vertex $P_0(i,t\vert\lambda)$ (blue dashed line) as a function of time in the complete graph. The walker is initially localized in the vertex $\ket{0}$. Results for $N=5$ and $\lambda=0.2$.}
\label{fig:prob_loc_complete}
\end{figure}
Next, the IPR \eqref{eq:ipr_local} for the probability distribution in Eqs. \eqref{eq:p0_cg}--\eqref{eq:pi_cg} reads as \begin{align}
\mathcal{I}(t) = & 1 - \frac{8(N-1)}{N^2}\sin^2(\omega_N(\lambda)t)\nonumber \\
& + \frac{16(N-1)}{N^3}\sin^4(\omega_N(\lambda)t)\,. \end{align} The IPR has the same properties of the probability distribution: it is periodic, reaches the upper bound $1$ (localization of the walker) for $t_k$ such that $P_0(0,t_k\vert \lambda)=1$, and $\lim_{N \to + \infty}\mathcal{I}=1$, since for large $N$ the walker tends to be localized in the starting vertex (Fig. \ref{fig:cg_IPR_multi}). The lower bound $\mathcal{I}_{m}:=\min_t \mathcal{I}$ actually depends on $N$: \begin{equation}
\mathcal{I}_{m} =\mathcal{I}(t_l)= \begin{dcases}
\frac{1}{N} & \text{for $N\leq 4$}\,,\\
1-\frac{8}{N}+\frac{24}{N^2}-\frac{16}{N^3} & \text{for $N>4$}\,,
\end{dcases}
\label{eq:low_bound_cg_IPR} \end{equation} where
\begin{equation}
t_l =\begin{dcases}
\frac{2 [\pm\arcsin{(\sqrt{N}/2)}+\pi l]}{N+\lambda N^2} & \text{for $N\leq 4$}\,,\\
\frac{2\pi(1/2 + l)}{N+\lambda N^2} & \text{for $N>4$}\,,
\end{dcases} \end{equation} with $l\in \mathbb{N}$. Please notice that the two definitions of $\mathcal{I}_m$ match in $N=4$. For $N\leq 4$ there are instants of time when the walker is delocalized ($\mathcal{I}_m=1/N$) and there is instantaneous exactly uniform mixing. Instead, for $N>4$ the walker is never delocalized, since $\mathcal{I}_m>1/N$.
\begin{figure}
\caption{Inverse participation ratio (IPR) for a walker initially localized in the complete graph. The IPR periodically reaches the upper bound $1$ (orange dashed line), i.e. the localization, but for $N>4$ does not reach the value $1/N$ (green dashdotted line), i.e. the delocalization. The lower bound of the IPR is defined in Eq. \eqref{eq:low_bound_cg_IPR}. For $N\to+\infty$ the IPR approaches 1, since the probability of finding the walker in the starting vertex approaches $1$ (see Eqs. \eqref{eq:p0_cg}--\eqref{eq:pi_cg}). Results for $\lambda=0.2$.}
\label{fig:cg_IPR_multi}
\end{figure}
Finally, we focus on the time dependence of the coherence, which we derive in Appendix \ref{subapp:an_res_cg} and is shown in Fig. \ref{fig:coher_complete_loc}. The modulus of the off-diagonal elements of the density matrix can be expressed in terms of the square root of probabilities (see Appendix \ref{app:an_res_ccsg}), thus the coherence is periodic and it is symmetric with respect to $\lambda^\ast$, as well as the probability distribution. As expected, the dependence on the perturbation is encoded only in the angular frequency $\omega_n(\lambda)$, and the coherence is identically null, thus minimum, for $\lambda^\ast$. For $\lambda \neq \lambda^\ast$, the coherence periodically reaches the following extrema \begin{align}
\max \mathcal{C}&=\frac{8(N-1)(N-2)}{N^2} \quad \text{for } t_k = \frac{(2k+1)\pi}{N+\lambda N^2}\,,\\
\min \mathcal{C}&=0\quad \text{for } t_k = \frac{2k\pi}{N+\lambda N^2}\,, \end{align} with $k\in\mathbb{N}$, and assuming $N\geq 2$.
\begin{figure}
\caption{Coherence for a walker initially localized in the complete graph with $N=5$. The coherence is null, thus minimum, for $\lambda^\ast =-1/N$, and it is symmetric with respect to $\lambda^\star$, so only the data for $\lambda\geq\lambda^\ast$ are shown.}
\label{fig:coher_complete_loc}
\end{figure}
\subsection{Star graph} \label{subsec:stargraph} In the star graph, the central vertex is adjacent to all the others, so its degree is $N-1$. On the other hand, the other vertices are only connected to the central one, thus their degree is $1$. Hence, the graph Laplacian is \begin{equation} L=I+(N-2)\dyad{0}-\sum_{k=1}^{N-1}(\dyad{k}{0}+\dyad{0}{k})\,, \label{eq:H0_star_matrix} \end{equation} where $\ket{0}$ denotes the central vertex.
\begin{table}[tb]
\centering
\begin{ruledtabular}
\begin{tabular}{clcc}
$n$ & $\ket{e_n}$ & $\varepsilon_n$ & $\mu_n$\\\hline
$0$ & $\ket{e_0}=\frac{1}{\sqrt{N}}\sum\limits_{k=0}^{N-1}\ket{k}$ & $0$ & $1$ \\
$1$ & $\ket{e_1^l}=\frac{1}{\sqrt{l(l+1)}}\left(\sum\limits_{k=1}^{l}\ket{k}-l\ket{l+1}\right)$ & $1$ & $N-2$\\
&\text{with $l=1,\ldots,N-2$}\\
$2$ & $\ket{e_2}=\frac{1}{\sqrt{N(N-1)}}\left[(N-1)\ket{0}-\sum\limits_{k=1}^{N-1}\ket{k}\right]$ & $N$ & $1$\\
\end{tabular}
\end{ruledtabular}
\caption{Eigenvectors $\ket{e_n}$, eigenvalues $\varepsilon_n$ with multiplicity $\mu_n$ of the graph Laplacian in the star graph.}
\label{tab:eig_pbm_star} \end{table}
The eigenproblem related to Eq. \eqref{eq:H0_star_matrix} is solved in Table \ref{tab:eig_pbm_star}. The graph Laplacian has three energy levels: \begin{enumerate}[(i)]
\item the level $\varepsilon_0 = 0$, having eigenstate $\ket{e_0}$;
\item the $(N-2)$-degenerate level $\varepsilon_1 = 1$, having orthonormal eigenstates $\ket{e_1^l}$, with $l=1,\ldots,N-2$;
\item the level $\varepsilon_1 = N$, having eigenstate $\ket{e_2}$. \end{enumerate}
The perturbation involves \begin{align} L^2&=2I+(N^2-N-2)\dyad{0}\nonumber\\ &-N\sum_{k=1}^{N-1}(\dyad{k}{0}+\dyad{0}{k})+\sum_{\substack{j,k=1,\\j\neq k}}^{N-1}\dyad{j}{k}\,, \label{eq:L2_star} \end{align} so the perturbed Hamiltonian \eqref{eq:H_ctqwp4} reads as follows: \begin{align} {\mathcal{H}}&=(1+2\lambda)I+[N-2+\lambda(N^2-N-2)]\dyad{0}\nonumber\\ &-(1+\lambda N)\sum_{k=1}^{N-1}(\dyad{k}{0}+\dyad{0}{k})+\lambda\sum_{\substack{j,k=1,\\j\neq k}}^{N-1}\dyad{j}{k}\,. \label{eq:totH_star} \end{align} The perturbation $\lambda L^2$, thus, introduces the hopping among all the outer vertices (next-nearest neighbors), affects the hopping to and from the central vertex, i.e. the nearest-neighbor hopping, and also the on-site energies $\propto {I}$.
For an initially localized state, there are two different time evolutions. If at $t=0$ the walker is in the central vertex $\vert 0 \rangle$, then the time evolution is equal to the corresponding one in the complete graph of the same size. Therefore, also the resulting probability distribution, the IPR, and the coherence are equal between star and complete graph. Instead, if at $t=0$ the walker is localized in any of the outer vertices, then we have a different time evolution. All the outer vertices $\ket{1 \leq i \leq N -1}$ are equivalent and differ from the central vertex $\ket{0}$, so, if we keep the central vertex as $\ket{0}$, we can always relabel the outer vertices in such a way that the starting vertex is denoted by $\ket{1}$.
The probabilities of finding the walker in the central vertex $\ket{0}$, in the starting vertex $\ket{1}$ or in any other outer vertex $\ket{2\leq i\leq N-1}$ at time $t$ for a given value of $\lambda$ are respectively (Fig. \ref{fig:prob_loc_star}) \begin{align}
P_1(0,t\vert \lambda) & = \frac{4}{N^2}\sin^2(\omega_N(\lambda)t)\,, \label{eq:p_10_sg}\\
P_1(1,t\vert \lambda) & = 1 - \frac{4}{N(N-1)} \Big[(N-2)\sin^2(\omega_1(\lambda)t) \nonumber\\
& + \frac{N-2}{N-1} \sin^2[(\omega_{N}(\lambda )-\omega_1(\lambda))t] \nonumber \\
& + \frac{1}{N} \sin^2(\omega_N(\lambda)t)\Big]\,,\label{eq:p_11_sg}\\
P_1(i,t\vert \lambda) & = \frac{4}{N(N-1)} \Big[ \sin^2(\omega_1(\lambda)t) \nonumber \\
& + \frac{1}{N-1} \sin^2[(\omega_{N}(\lambda )-\omega_1(\lambda))t] \nonumber \\
& -\frac{1}{N} \sin^2(\omega_N(\lambda)t)\Big]\label{eq:p_1i_sg}\,, \end{align} where the angular frequency is defined in Eq. \eqref{eq:ang_freq}. In particular, $P_1(0,t\vert \lambda)$ is periodic with period $T_N:=\pi / \omega_N(\lambda)$, it is symmetric with respect to $\lambda^\ast=-1/N$, and $P_1(0,t\vert \lambda^\ast)=0$, which means that the walker lives only in the outer vertices of the star graph. Indeed, $\lambda^\ast$ makes the hopping terms to and from the central vertex $\ket{0}$ null (see Eq. \eqref{eq:totH_star}). Instead, $P_1(1,t\vert \lambda)$ and $P_1(i,t\vert \lambda)$ are periodic if and only if the periods $T_1$, $T_N$, and $\pi/(\omega_N(\lambda)-\omega_1(\lambda))$ of the summands are commensurable. When this happens, then the overall probability distribution is periodic. This happens also for the particular values $\lambda=-1,-1/N,-1/(N+1)$, which make null $\omega_1$, $\omega_N$, and $\omega_N-\omega_1$, respectively. Indeed, when $\omega_1$ ($\omega_N$) is null, the probabilities \eqref{eq:p_10_sg}--\eqref{eq:p_1i_sg} only involve sine functions with $\omega_N$ ($\omega_1$). When $\omega_N-\omega_1=0$, i.e. $\omega_N=\omega_1$, all the sine functions have the same angular frequency. We address in details the periodicity of the probability distribution in Appendix \ref{subapp:an_res_sg}. For $P_1(1,t\vert \lambda)$ and $P_1(i,t\vert \lambda)$ results suggest that there is no symmetry with respect to $\lambda$. Increasing the order of the graph makes the angular frequency higher, and $\lim_{N\to+\infty}P_1(1,t\vert\lambda) = 1$, while $\lim_{N\to+\infty}P_1(0,t\vert\lambda)=\lim_{N\to+\infty}P_1(i,t\vert\lambda) = 0$. Again, the perturbation affects the probabilities only through the angular frequency. The average probability distribution is the same as the one reported in \cite{xu2009exact}, which is exactly our unperturbed CTQW.
\begin{figure}
\caption{Probability of finding the walker in the central vertex $P_1(0,t\vert\lambda)$ (green dotted line), in the starting vertex $P_1(1,t\vert\lambda)$ (red dashed line) or in any other vertex $P_1(i,t\vert\lambda)$ (blue solid line) as a function of time in the star graph. The walker is initially localized in the vertex $\ket{1}$. Results for $N=5$ and $\lambda=0.2$.}
\label{fig:prob_loc_star}
\end{figure}
Next, we numerically evaluate the IPR \eqref{eq:ipr_local} for the probability distribution in Eqs. \eqref{eq:p_10_sg}--\eqref{eq:p_1i_sg}, and the results are shown in Fig. \ref{fig:sg_IPR_multi}. The IPR oscillates between $1$ and its minimum value, which grows with $N$, similarly to what happens in the complete graph. Indeed, for $N\to+\infty$ the IPR approaches 1 (localization), since the probability of finding the walker in the starting vertex approaches $1$. The periodicity of the IPR relies upon that of the probability distribution. When the latter is periodic, the IPR periodically reaches $1$, since the walker is initially localized in a vertex, and periodically comes back to it. By considering $P_1(0,t\vert \lambda)=1/N$, we notice that the instantaneous exactly uniform mixing is never achievable for $N>4$, and so the IPR is never close to $1/N$, independently of $\lambda$. Instead, for $N \leq 4$ the mixing properties strongly depend on the choice of $N$ and $\lambda$, e.g. it is achievable for $\lambda=-1/(N+1)$ and for $N=2 \wedge \lambda=-1$. The instantaneous exactly uniform mixing is never achievable for $\lambda^\ast$, since $P_1(0,t\vert \lambda^\ast)=0 \,\forall\, t$.
\begin{figure}
\caption{Inverse participation ratio (IPR) for a walker initially localized in $\ket{1}$ in the star graph. Results suggest that for $t>0$ there are instants of time when the IPR is close to the upper bound $1$ (orange dashed line), i.e. the localization. In particular, the IPR periodically reaches $1$ when the probability distribution is periodic. For $N>4$ the IPR does not reach the value $1/N$ (green dashdotted line), i.e. the delocalization. For $N\to+\infty$ the IPR approaches 1, since the probability of finding the walker in the starting vertex approaches $1$ (see Eqs. \eqref{eq:p_10_sg}--\eqref{eq:p_1i_sg}). Results for $\lambda=0.2$.}
\label{fig:sg_IPR_multi}
\end{figure}
Finally, we focus on the time dependence of the coherence of a walker initially localized in $\ket{1}$, which we derive in Appendix \ref{subapp:an_res_sg} and it is shown in Fig. \ref{fig:coher_star_loc}. The coherence shows a complex structure of local maxima and minima. However, it is smoother and periodic for the values of $\lambda$ which make the overall probability distribution periodic (see Appendix \ref{subapp:an_res_sg}), e.g. $\lambda=-1,-1/N,-1/(N+1)$.
\begin{figure}
\caption{Coherence for a walker initially localized in $\ket{1}$ in the star graph with $N=5$. The coherence is smooth and periodic for $\lambda^\ast =-1/N$.}
\label{fig:coher_star_loc}
\end{figure}
\section{Characterization} \label{sec:estimation} In this section we address the characterization of the CTQW Hamiltonian \eqref{eq:H_ctqwp4}, i.e. the estimation of the parameter $\lambda$ that quantifies the amplitude of the perturbation $\mathcal{H}_1=L^2$. Our aim is to assess whether, and to which extent, we may determine the value of $\lambda$ using only a snapshot of the walker dynamics, i.e. by performing measurements at a given time $t$. Hence, we briefly review some useful concepts from classical and quantum estimation theory \cite{paris2009quantum}.
Purpose of classical estimation theory is to find an {\em estimator}, i.e a function $\hat{\lambda}$ that, taking as input $n$ experimental data $\{x_i\}_{i=1,\dots,n}$ whose probabilistic distribution $P(x_i \vert \lambda)$ depends on $\lambda$, gives the most precise estimate of the parameter. A particular class of $\hat{\lambda}$ are the {\em unbiased estimators}, for which the expectation value is the actual value of the parameter $\lambda$, i.e. $E_\lambda [\hat{\lambda}] = \int dx P(x\vert \lambda) \hat{\lambda}(x) \equiv \lambda$. The main result regarding the precision of an estimator $\hat{\lambda}$ is given by the Cram\'{e}r-Rao Bound, which sets a lower bound on the variance of any unbiased estimator $\hat{\lambda}$, provided that the family of distribution $P(x\vert \lambda)$ realizes a so-called {\em regular statistical model}. In this case, the variance of any unbiased estimator $\hat{\lambda}$ satisfies the inequality \begin{equation}
\sigma^2(\hat{\lambda}) \geq \frac{1}{n \mathcal{F}_c(\lambda)}\,,
\label{eq:classcrb} \end{equation} where $\mathcal{F}_c(\lambda)$ is the Fisher Information (FI) of the probability distribution $P(x\vert\lambda)$ \begin{equation}
\mathcal{F}_c(\lambda) = \int dx \frac{(\partial_\lambda P(x\vert \lambda))^2}{P(x\vert \lambda)}\,.
\label{eq:fi_def_cont} \end{equation} Regular models are those with a constant support, i.e. the region in which $P(x\vert \lambda) \neq 0$ does not depend on the parameter $\lambda$, and with non-singular FI. If these hypotheses are not satisfied, estimators with vanishing variance may be easily found. Optimal estimators are those saturating the inequality \eqref{eq:classcrb}, and it can be proved that for $n\to+\infty$ maximum likelihood estimators attain the lower bound \cite{newey1994large}.
In a quantum scenario, the parameter must be encoded in the density matrix of the system In turn, a {\em quantum statistical model} is defined as a family of quantum states $\{\rho_\lambda\}$ parametrized by the value of $\lambda$. In order to extract information from the system, we need to perform measurements, i.e. positive operator-valued measure (POVM) $\{\mathcal{E}_m\}$, where $m$ is a continuous or discrete index labeling the outcomes. According to the Born rule, a conditional distribution $P(m\vert \lambda) = \Tr{\rho_\lambda \mathcal{E}_m}$ naturally arises. Unlike the classical regime, the probability depends both on the state and on the measurement, so we can suitably choose them to get better estimates. In particular, given a family of quantum states $\left\{\rho_\lambda\right\}$, we can find a POVM which maximizes the FI, i.e. \begin{equation}
\mathcal{F}_c (\lambda) \leq \mathcal{F}_q(\lambda) = \Tr{\rho_\lambda \Lambda_\lambda^2}\,,
\label{eq:qfi_def} \end{equation} where $\mathcal{F}_q(\lambda)$ is the Quantum Fisher Information (QFI) and $\Lambda_\lambda$ is the Symmetric Logarithmic Derivative (SLD), which is implicitly defined as \begin{equation}
\frac{\rho_\lambda \Lambda_\lambda + \Lambda_\lambda \rho_\lambda}{2} = \partial_\lambda \rho_\lambda\,. \end{equation} The optimal POVM saturating the inequality \eqref{eq:qfi_def} is given by the projectors on the eigenspaces of the SLD. Since $\mathcal{F}_q(\lambda) = \max_{\mathcal{E}_m}\{\mathcal{F}_c(\lambda)\}$, we have a more precise bound on $\sigma^2(\hat{\lambda})$ which goes by the name of Quantum Cram\'{e}r-Rao (QCR) inequality \begin{equation}
\sigma^2(\hat{\lambda}) \geq \frac{1}{n\mathcal{F}_q(\lambda)}\,.
\label{eq:qcrb} \end{equation} This establishes the ultimate lower bound of the precision in estimating a parameter $\lambda$ encoded in a quantum state. Notice that the QCR is valid for {\em regular quantum statistical model}, i.e. families of quantum states made of density matrices with constant rank (i.e. the rank does not depend on the parameter) and leading to a nonsingular QFI \cite{bycr,bcr,vrank}. \par In the present work we focus on pure states subjected to the unitary evolution in Eq. \eqref{eq:t_evol_op}, i.e. $\ket{\psi_\lambda(t)}={\mathcal{U}}_\lambda(t)\ket{\psi(0)}$. For such states the QFI reads as \begin{equation}
\mathcal{F}_q(t,\lambda) = 4 \left[\langle \partial_\lambda \psi_\lambda(t) \vert \partial_\lambda \psi_\lambda(t) \rangle- \abs{\langle \psi_\lambda(t) \vert \partial_\lambda \psi_\lambda(t) \rangle}^2 \right]\,.
\label{eq:qfi_def_braket} \end{equation} When dealing with CTQWs on a graph, a reasonable and significant measurement is the position one. For such a measurement the FI reads as \begin{equation}
\mathcal{F}_c(t,\lambda) = \sum_{k=0}^{N-1} \frac{\left(\partial_\lambda P(k,t \vert \lambda)\right)^2}{P(k,t\vert\lambda)}\,,
\label{eq:fi_def} \end{equation} where $P(k,t\vert\lambda)$ is the conditional probability of finding the walker in the $k$-th vertex at time $t$ when the value of the parameter is $\lambda$.
When the perturbation $\mathcal{H}_1$ commutes with the unperturbed Hamiltonian $\mathcal{H}_0$ (which is our case, see Eq. \eqref{eq:H_ctqwp4}), the unitary time evolution simplifies to \begin{equation}
\mathcal{U}_\lambda(t) = e^{-it\mathcal{H}_0}e^{-i t\lambda \mathcal{H}_1}\,. \end{equation} Then, the QFI has a simple representation in terms of the perturbation and of time. Indeed, if our probe $\vert \psi \rangle$ at time $t=0$ does not depend on $\lambda$ and undergoes the evolution $\mathcal{U}_\lambda(t)$, at a later time $t>0$ we can write \begin{align}
\mathcal{F}_q(t) &= 4 t^2 \big[ \langle \psi \vert \mathcal{H}_1^2 \vert \psi \rangle - \langle \psi \vert \mathcal{H}_1 \vert \psi \rangle ^2 \big]\nonumber\\
&=4t^2 \langle (\Delta \mathcal{H}_1)^2\rangle\,.
\label{eq:qfi_quadratic_time} \end{align} since $\vert \partial_\lambda \psi_\lambda (t) \rangle = -it\mathcal{H}_1 \vert \psi_\lambda (t)\rangle$ when $[\mathcal{H}_0,\mathcal{H}_1] = 0$. We emphasize that the QFI does not depend on the parameter $\lambda$ to be estimated. This is due to the unitary evolution, and to the fact that at $t=0$ the probe $\ket{\psi}$ does not depend on $\lambda$.
In the following, we evaluate the QFI of localized states, whose dynamics is addressed in Sec. \ref{sec:dynamics}, and we determine the states maximizing the QFI for cycle, complete, and star graph. We compare the QFI with the FI for a position measurement to assess whether it is an optimal measurement or not. Moreover, we find the simple graphs allowing the maximum QFI. Please refer to Appendix \ref{app:q_fi_scg} for the details about the analytical derivation of the results shown in the following.
\subsection{Localized states}
\subsubsection{Cycle graph} The QFI of an initially localized state in the cycle graph is \begin{equation}
\mathcal{F}_q (t)=136 t^2\,,
\label{eq:qfi_loc_cyc} \end{equation} and it is independent of $N$. We numerically evaluate the FI \eqref{eq:fi_def} for the probability distribution in Eq. \eqref{eq:pjk_cyc_cos}. The results are shown in Fig. \ref{fig:qfi_cyc_loc} and suggest that the FI never reaches the QFI. Specific behaviors of the FI strongly depend on the choice of $N$ and $\lambda$.
\begin{figure}
\caption{Quantum (black solid line) and classical Fisher Information (colored non-solid lines) of position measurement for an initially localized state in the cycle graph. Results for $N=5$.}
\label{fig:qfi_cyc_loc}
\end{figure}
\subsubsection{Complete Graph} The QFI of an initially localized state in the complete graph is \begin{equation}
\mathcal{F}_q(N,t) = 4 N^2(N-1)t^2 \,.
\label{eq:qfi_loc_cg} \end{equation} The FI is \begin{equation}
\mathcal{F}_c(N,t,\lambda) = \frac{4N^4(N-1) t^2\cos^2\left(\omega_N(\lambda) t \right)}{N^2-4(N-1) \sin^2\left(\omega_N(\lambda) t\right)}\,,
\label{eq:fi_loc_cg} \end{equation} with $\omega_N(\lambda)$ defined in Eq. \eqref{eq:ang_freq}. Due to the symmetry of the graph, both the QFI and the FI do not depend on the starting vertex, i.e. the estimation is completely indifferent to the choice of the initially localized state. Unlike the QFI, the FI does depend on $\lambda$ and is symmetric with respect to $\lambda^\ast = -1/N$, as well as the probability distribution in Eqs. \eqref{eq:p0_cg}--\eqref{eq:pi_cg}. In particular, $\mathcal{F}_c(t,\lambda^\ast)=\mathcal{F}_q(t)$. {However, we recall that $P_0(0,t\vert\lambda^\ast) = 1$ and $P_0(i,t\vert\lambda^\ast) = 0$, i.e. the walker is in the starting vertex all the time. In this case the hypotheses leading to the Cram\'{e}r-Rao Bound \eqref{eq:classcrb} do not hold, since the model is
not regular, and the bound may be easily surpassed. Indeed, if we perform
the measurement described by the POVM $\{\vert 0 \rangle \langle 0 \vert , \mathbbm{1}-\vert 0 \rangle \langle 0 \vert\}$, the variance of the estimator is identically zero, outperforming both classical and quantum bounds.}
For $\lambda \neq \lambda^\ast$, the periodicity of the probabilities in Eqs. \eqref{eq:p0_cg}--\eqref{eq:pi_cg} results in a dependence of the FI on $\lambda$ and an analogous oscillating behavior (Fig. \ref{fig:qfi_cg_loc}). The FI reaches periodically its local maxima when the numerator is maximum and the denominator is mininum, and these maxima saturate the Quantum Cram\'{e}r-Rao Bound \begin{equation}
\mathcal{F}_c(t_k,\lambda) = \mathcal{F}_q(t_k,\lambda)\,. \end{equation} This occurs for $t_k= 2 k \pi /(N+\lambda N^2)$, with $k\in\mathbb{N}$, i.e. when the walker is completely localized and we definitely find it in the starting vertex. Indeed, in the probability distribution the parameter $\lambda$ is encoded only in the angular frequency, thus knowing when the walker is certainly in the starting vertex means knowing exactly its period, and thus the parameter $\lambda$. However, to perform such a measurement one needs some \textit{a priori} knowledge of the value of the parameter. In fact, the POVM saturating the Quantum Cram\'{e}r-Rao Bound \eqref{eq:qcrb} strongly depends on the parameter $\lambda$.
\begin{figure}
\caption{Quantum (black solid line) and classical Fisher Information (colored non-solid lines) of position measurement for an initially localized state in the complete graph. The same results are obtained for a walker initially localized in the central vertex $\ket{0}$ of the star graph of the same size.
Results for $N=5$.}
\label{fig:qfi_cg_loc}
\end{figure}
\subsubsection{Star Graph} The time evolution of the state localized in the center of the star graph is equivalent to that of a localized state in the complete graph, as already pointed out in Sec. \ref{subsec:stargraph}. Thus, for this state the QFI and FI are provided in Eq. \eqref{eq:qfi_loc_cg} and Eq. \eqref{eq:fi_loc_cg}, respectively (see also Fig. \ref{fig:qfi_cg_loc}).
Things change when we consider a walker initially localized in one of the outer vertices of the star graph. In this case the QFI is \begin{equation}
\mathcal{F}_q(N,t) = 4 (N^2+N-2) t^2\,.
\label{eq:qfi_loc_sg} \end{equation}
We numerically evaluate the FI \eqref{eq:fi_def} for the probability distribution in Eqs. \eqref{eq:p_10_sg}--\eqref{eq:p_1i_sg} and the results are shown in Fig. \ref{fig:qfi_sg_loc}. Unlike the complete graph, for the star graph there is no saturation of the Quantum Cram\'{e}r-Rao Bound. Notice, however, that for $\lambda^\ast=-1/N$ the walker cannot reach the central site and, in principle, one may exploit this feature to build a non regular model, as we discussed in the previous section.
\begin{figure}
\caption{Quantum (black solid line) and classical Fisher Information (colored non-solid lines) of position measurement for a walker initially localized in an outer vertex of the star graph. Results for $N=5$.}
\label{fig:qfi_sg_loc}
\end{figure}
\subsection{States maximizing the QFI} In the previous section, we have studied how localized states behave as quantum probes for estimating the parameter $\lambda$ of the perturbation. However, we might be interested in finding the best estimate for such parameter by searching for the state $\rho_\lambda$ maximizing the QFI, hence minimizing the variance $\sigma^2(\hat{\lambda})$. For this purpose, it is worth introducing an alternative formula for QFI. When there is only one parameter to be estimated and the state is pure, the QFI reads as \begin{equation} \mathcal{F}_q(\lambda,t)=\lim_{\delta\lambda\to0}\frac{8\left( 1-\abs{\braket{\psi_\lambda(t)}{\psi_{\lambda+\delta\lambda}(t)}}\right )}{\delta\lambda^2}\,. \label{eq:qfi_def_dyn} \end{equation} This expression involves the modulus of the following scalar product: \begin{equation} \braket{\psi_\lambda(t)}{\psi_{\lambda+\delta\lambda}(t)}=\matrixel{\psi(0)}{{U}_{\delta\lambda}(t)}{\psi(0)}\,, \label{eq:scpr} \end{equation} where \begin{align} {U}_{\delta\lambda}(t)&:=e^{+i({\mathcal{H}}_0+\lambda{\mathcal{H}}_1)t}e^{-i[{\mathcal{H}}_0+(\lambda+\delta\lambda){\mathcal{H}}_1]t}\nonumber\\ &=e^{-i\delta\lambda{\mathcal{H}}_1t} \label{eq:unitary_op} \end{align} is a unitary operator given by the product of two unitary operators \eqref{eq:t_evol_op} related to the time evolutions for $\lambda$ and $\lambda+\delta\lambda$, and the last equality holds since $[{\mathcal{H}}_0,{\mathcal{H}}_1]=0$ (see Eq. \eqref{eq:H_ctqwp4}).
The QFI strongly depends on the quantum state considered. To maximize the QFI, we recall the following lemma \cite{parthasarathy2001consistency}. \begin{lemma}[K. R. Parthasarathy] Let $W$ be any unitary operator in the finite dimensional complex Hilbert space $\mathscr{H}$ with spectral resolution $\sum_{j=1}^k e^{i\theta_j}P_j$ where $e^{i\theta_1},\ldots,e^{i\theta_k}$ are the distinct eigenvalues of $W$ with respective eigenprojections $P_1,\ldots,P_k$. Define \begin{equation} m(W)=\min_{\norm{\psi}=1}\abs{\matrixel{\psi}{W}{\psi}}^2\,. \end{equation} Then the following hold: \begin{enumerate}[(i)] \item If there exists a unit vector $\ket{\psi_0}$ such that $\matrixel{\psi_0}{W}{\psi_0}=0$, then $m(W)=0$. \item If $\matrixel{\psi}{W}{\psi}>0$ for every unit vector $\ket{\psi}$, then \begin{equation} m(W)=\min_{i\neq j}\cos^2\left ( \frac{\theta_i-\theta_j}{2}\right )\,. \label{eq:mW_def} \end{equation} Furthermore, when the right-hand side is equal to $\cos^2\left ( \frac{\theta_{i_0}-\theta_{j_0}}{2}\right )$, \begin{equation} m(W)=\abs{\matrixel{\psi_0}{W}{\psi_0}}^2 \end{equation} where \begin{equation} \ket{\psi_0} = \frac{1}{\sqrt{2}}\left (\ket{e_{i_0}}+\ket{e_{j_0}}\right ) \label{eq:psi_0} \end{equation} and $\ket{e_{i_0}}$ and $\ket{e_{j_0}}$ are arbitrary unit vectors in the range of $P_{i_0}$ and $P_{j_0}$ respectively. \end{enumerate} \label{lemma:parthasarathy} \end{lemma}
The idea is to exploit the Lemma to compute the QFI. We consider $\ket{\psi_0}$ as initial state and we identify $W$ with ${U}_{\delta\lambda}(t)$, since $\braket{\psi_\lambda(t)}{\psi_{\lambda+\delta\lambda}(t)}=\langle \psi_0 \vert {U}_{\delta\lambda}(t)\vert \psi_0\rangle$, so that \begin{equation} \mathcal{F}_q(\lambda,t)=\lim_{\delta\lambda\to 0}\frac{8\left[ 1-\sqrt{m({U}_{\delta\lambda}(t))}\right]}{\delta\lambda^2}\,. \label{eq:qfi_mw} \end{equation} Indeed, the state $\ket{\psi_0}$ in Eq. \eqref{eq:psi_0} maximizes the QFI by minimizing the modulus of the scalar product \eqref{eq:scpr}. The unit vectors involved by $\ket{\psi_0}$ are eigenvectors of the unitary operator \eqref{eq:unitary_op} and so, ultimately, of ${\mathcal{H}}_1$. In particular, such states are those whose eigenvalues minimize Eq. \eqref{eq:mW_def}. The eigenvalues of the unitary operator \eqref{eq:unitary_op} are $e^{i\theta_j}=e^{-i\delta\lambda t \varepsilon_j^2}$, with $\{\varepsilon_j^2\}$ eigenvalues of ${\mathcal{H}}_1=\mathcal{H}_0^2$, being $\{\varepsilon_j\}$ those of ${\mathcal{H}}_0=L$. Thus, we can identify $\theta_j=-\delta\lambda t \varepsilon_j^2$. Because of this relation, we may assume $\ket{e_{i_0}}$ and $\ket{e_{j_0}}$ to be the eigenstates corresponding to the lowest and the highest energy eigenvalue. Indeed, in the limit for $\delta\lambda t \to 0$ the cosine in Eq. \eqref{eq:mW_def} is minimized by maximizing the difference $\theta_i-\theta_j$. Then, the QFI reads as follows: \begin{equation} \mathcal{F}_q(t) = t^2(\varepsilon_{max}^2-\varepsilon_{min}^2)^2=t^2 \varepsilon_{max}^4\,. \label{eq:qfi_delta_eigenval} \end{equation} Because of the choice of the state $\ket{\psi_0}$, which involves the lowest and the highest energy eigenstates, the first equality follows from Eq. \eqref{eq:qfi_quadratic_time}, whereas the second equality holds since $\varepsilon_{min}=0$ for simple graphs. An eventual phase difference between the two eigenstates in Eq. \eqref{eq:psi_0} would result in the same QFI, but a different FI, as shown in Appendix \ref{subapp:phi_maxQFI_states}.
\subsubsection{General Graph} \label{subsubsec:general_graph_maxQFI} We prove that for a specific class of graphs the maximum QFI is always equal to $N^4t^2$, provided that the probe of the system is the state \eqref{eq:psi_0}. Indeed, according to Lemma \ref{lemma:parthasarathy}, in order to find quantum probes maximizing the QFI, we need to search for systems whose eigenvalues separation is maximum. For a graph of $N$ vertices with no loops, the row sums and the column sums of the graph Laplacian $L_N$ are all equal to $0$, and the vector $(1,\ldots,1)$ is always an eigenvector of $L$ with eigenvalue $0$. It follows that any Laplacian spectrum contains the zero eigenvalue and to maximize the QFI we need to find graphs having the largest maximum eigenvalue.
Following Ref. \cite{brouwer2011spectra}, the Laplacian spectrum of a graph $G(V,E)$ is the set of the eigenvalues of $L_N$ \begin{equation}
S_L(G) =\{\mu_1 = 0,\mu_2 ,\dots,\mu_N\} \,, \end{equation} where the eigenvalues $\mu_i$ are sorted in ascending order. To study the maximum eigenvalue $\mu_N$ we introduce the complementary graph $\bar{G}$ of $G$. The complementary graph $\bar{G}$ is defined on the same vertices of $G$ and two distinct vertices are adjacent in $\bar{G}$ if and only if they are not adjacent in $G$. So, the adjacency matrix $\bar{A}$ can be easily obtained from $A$ by replacing all the off-diagonals $0$s with $1$s and all the $1$s with $0$s. In other words \begin{equation}
\bar{A}_N= \mathbb{J}_N - \mathbbm{1}_N - A_N\,, \end{equation} where $\mathbb{J}_N$ denotes the $N\times N$ all-ones matrix and $\mathbbm{1}_N$ the $N\times N$ identity matrix. A vertex in $G$ can be at most adjacent to $N-1$ vertices, since no loops are allowed. Then, the degree $\bar{d}_j$ of a vertex in $\bar{G}$ is $N-1-d_j$, i.e. the complement to $N-1$ of the degree of the same vertex in $G$. The diagonal degree matrix is therefore \begin{equation}
\bar{D}_N = (N-1)\mathbbm{1}_N - D_N\,. \end{equation} In conclusion, the Laplacian matrix $\bar{L}_N$ associated to the complementary graph $\bar{G}$ is \begin{equation} \label{barlapl}
\bar{L}_N = \bar{D}_N-\bar{A}_N = N \mathbbm{1}_N - \mathbb{J}_N- L_N\,. \end{equation}
\begin{lemma} Any eigenvector $\vec{n}$ of $L_N$ is an eigenvector of $\bar{L}_N$. If the eigenvalue of $\vec{n}$ for $L_N$ is $0$, then it is $0$ also for $\bar{L}_N$. If the eigenvalue of $\vec{n}$ for $L_N$ is $\mu_i$, then the eigenvalue for $\bar{L}_N$ is $N - \mu_i$. Thus, the spectrum of $\bar{L}_N$ is given by \begin{equation}
S_{\bar{L}}(\bar{G}) = \{0,N-\mu_N,\ldots,N-\mu_2\}\,,
\label{eq:s_barL} \end{equation} where the eigenvalues are still sorted in ascending order. \end{lemma}
Any $L_N$ is positive-semidefinite, i.e. $\mu_i \geq 0 \ \forall \ i$, so this holds for $\bar{L}_N$ too. According to these remarks and to Eq. \eqref{eq:s_barL}, we then observe that $\mu_N \leq N$, i.e. the largest eigenvalue is bounded from above by the number of vertices $N$. Moreover, the second-smallest eigenvalue $\mu_2$ of $L_N$ is the algebraic connectivity of $G$: it is greater than 0 if and only if $G$ is a connected graph. Indeed, the algebraic multiplicity of the eigenvalue 0 is the number of connected components of the graph \cite{fiedler1973algebraic,mohar1991laplacian,marsden2013eigenvalues}. So, if $\bar{G}$ has at least two distinct components, then the second-smallest eigenvalue of $\bar{L}_N$ is $N-\mu_N=0$, from which $\mu_N=N$.
\begin{lemma}
Given a graph $G$ and its Laplacian spectrum $S_{L}(G) = \{0,\mu_2,\ldots,\mu_N\}$, the largest Laplacian eigenvalue $\mu_N$ is bounded from above by $\mu_N \leq N$, and the equality is saturated only if the complementary graph $\bar{G}$ is disconnected. \end{lemma}
This result in spectral graph theory has a direct impact on our estimation problem. Since our perturbation is the square of the graph Laplacian, the maximum QFI is given by Eq. \eqref{eq:qfi_delta_eigenval} and involves the lowest and the largest eigenvalue of the Laplacian spectrum. \begin{lemma}
The simple graphs $G$ whose complementary graph $\bar{G}$ is disconnected are the only ones providing the maximum QFI for the estimate of the parameter $\lambda$ in Eq. \eqref{eq:H_ctqwp4}. For such graphs, the largest eigenvalue of the graph Laplacian is $N$ and the lowest is $0$. This results in the following maximum QFI
\begin{equation}
\label{eq:maxQFIgraph}
\mathcal{F}^{max}_q(N,t) = N^4 t^2 \,.
\end{equation}
\label{lemma:mleg_maxqfi} \end{lemma}
This lemma allows us to predict whether or not a graph provides the maximum QFI and its value, with no need to diagonalize the graph Laplacian. Some graphs satisfying Lemma \ref{lemma:mleg_maxqfi} are the complete, the star, the wheel, and the complete bipartite graph. The cycle graph allows the maximum QFI only for $N\leq 4$: for $N=2,3$ it is just a complete graph; for $N=4$ the complementary graph has two disconnected components, and for $N>4$ it is connected.
\subsubsection{Cycle graph} \label{subsubsec:cycle_max_qfi} The cycle graph satisfies Lemma \ref{lemma:mleg_maxqfi} only for $N\leq 4$. For $N>4$ the maximum QFI is lower than $N^4t^2$, and it depends on $N$. Indeed, the energy spectrum of the cycle graph is sensitive to the parity of $N$, and the state maximizing the QFI at $t=0$ is therefore \begin{equation} \vert\psi_0^{(\pm)}\rangle= \frac{1}{\sqrt{2}}(\ket{e_{min}}+\vert e_{max}^{(\pm)}\rangle)\,, \label{eq:psi0_Nevenodd} \end{equation} where $\vert e_{min} \rangle$ is the ground state, while $\vert e_{max}^{(\pm)} \rangle$ is the eigenstate corresponding to the highest energy level and it depends on the parity of $N$. For even $N$ it is unique, whereas for odd $N$ the highest energy level is doubly degenerate, which is the reason for the $\pm$ sign (see Eqs. \eqref{eq:evec_cyc_max_cos}--\eqref{eq:evec_cyc_max_sin} and Table \ref{tab:eig_pbm_cycle}).
The resulting QFI is \begin{align} \mathcal{F}_q(t)= \begin{cases} 256 t^2 & \text{if $N$ is even}\,,\\ 16 \left[1+\cos\left (\frac{\pi}{N}\right )\right]^4 t^2 & \text{if $N$ is odd}\,. \end{cases} \label{eq:qfi_cycle_evenodd} \end{align} The QFI for odd $N$ depends on $N$, and for large $N$ it approaches the QFI for even $N$, which, instead, does not depend on $N$. Even the FI discriminates between even and odd $N$, because of the ambiguity in choosing the highest energy eigenstate for odd $N$ (see Appendix \ref{subapp:q_fi_cyc}). For even $N$ the position measurement is optimal, i.e. $\mathcal{F}_c(t)=\mathcal{F}_q(t)$. For odd $N$, both the eigenstates for $n=(N\pm1)/2$ in Table \ref{tab:eig_pbm_cycle} lead to $\mathcal{F}_c(t)=\mathcal{F}_q(t)$. Instead, if we choose the linear combinations of them in Eqs. \eqref{eq:evec_cyc_max_cos}--\eqref{eq:evec_cyc_max_sin}, the FI of position measurement is no longer optimal, as shown in Fig. \ref{fig:qfi_cyc_max}.
\begin{figure}
\caption{Quantum (black solid line) and classical Fisher Information (colored non-solid lines) of position measurement for the states maximizing the QFI in the cycle graph for odd $N$: (a) $\ket{\psi_0^+}$, where the highest energy state is Eq. \eqref{eq:evec_cyc_max_cos}; and (b) $\ket{\psi_0^-}$, where the highest energy state is Eq. \eqref{eq:evec_cyc_max_sin}. For odd $N$, indeed, the highest energy level is doubly degenerate. While the QFI does not depend on the choice of the corresponding eigenstate, the FI does.
Results for $N=5$.}
\label{fig:qfi_cyc_max}
\end{figure}
\subsubsection{Complete graph} \label{subsubsec:complete_max_qfi} The complementary graph of the complete graph has $N$ disconnected components, so it satisfies the Lemma \ref{lemma:mleg_maxqfi}. A possible choice of the state maximizing the QFI (at $t=0$) is the following \begin{equation}
\vert \psi_0^l \rangle = \frac{1}{\sqrt{2}} ( \vert e_0 \rangle + \vert e^l_1 \rangle)\,,
\label{eq:max_qfi_states_cg} \end{equation} where $\vert e_0 \rangle$ is the ground state, while $\vert e_1^l \rangle$, with $l=1,\ldots,N-1$, is the eigenstate corresponding to the highest energy level $\varepsilon_1 = N$, which is $(N-1)$-degenerate (see Table \ref{tab:eig_pbm_complete}). Then, we are free to choose any eigenstate from the eigenspace $\{\vert e^l_1 \rangle\}$ (or even a superposition of them) and the QFI is always given by Eq. \eqref{eq:maxQFIgraph}. On the other hand, the FI does depend on the choice of $\vert e^l_1 \rangle$.
As an example, let us consider the two states \begin{align}
\vert \psi_0^1 \rangle &= \frac{1}{\sqrt{2}}(\vert e_0 \rangle + \vert e^1_1 \rangle )\,, \label{eq:max_qfi_states_cg_1}\\
\vert \psi_0^{N-1} \rangle &= \frac{1}{\sqrt{2}}(\vert e_0 \rangle + \vert e^{N-1}_1 \rangle\,. \label{eq:max_qfi_states_cg_2} \end{align} These states are equivalent for the QFI (both maximize it), but they are not for the FI (see Fig. \ref{fig:qfi_cg_max}), which reads as follows \begin{align}
&\mathcal{F}_c(\vert \psi_0^1\rangle;N,t,\lambda) = \frac{4 N^4 (N+2)t^2\sin^2(2t \omega_N(\lambda))}{(N+2)^2-8N \cos^2(2t\omega_N(\lambda))}\,, \label{eq:fi_max_qfi_states_cg_1}\\
&\mathcal{F}_c(\vert \psi_0^{N-1} \rangle; N,t,\lambda) = \frac{4 N^4 (N-1) t^2 \sin ^2(2t\omega_N(\lambda))}{N^2-4 (N-1) \cos ^2(2t\omega_N(\lambda))}\,. \label{eq:fi_max_qfi_states_cg_2} \end{align} In both cases the FI is symmetric with respect to $\lambda^\ast=-1/N$, and for such value it vanishes. The local maxima occur for $t_k = \pi(k+1/2)/(N+\lambda N^2)$, with $k\in\mathbb{N}$, and are the following \begin{align}
\mathcal{F}^{max}_c(\vert \psi_0^1\rangle;N,t_k,\lambda) &= \frac{4 N^4}{N+2} t_k^2\,, \\
\mathcal{F}^{max}_c(\vert \psi_0^{N-1} \rangle; N,t_k,\lambda)&= 4(N-1)N^2t_k^2\,. \label{eq:qfi_local_max_qfi_states_cg} \end{align} For these states the FI never reaches the value of the QFI \eqref{eq:maxQFIgraph}, so the position measurement on $\vert \psi^l_0 \rangle$ is not optimal.
\begin{figure}\label{fig:qfi_cg_max}
\end{figure}
\subsubsection{Star graph} \label{subsubsec:star_max_qfi} The complementary graph of the star graph has two disconnected components, so it satisfies the Lemma \ref{lemma:mleg_maxqfi}. The state maximizing the QFI (at $t=0$) is \begin{equation}
\vert \psi_0 \rangle = \frac{1}{\sqrt{2}} \left(\vert e_0 \rangle + \vert e_2 \rangle\right)\,,
\label{eq:max_qfi_states_sg} \end{equation} where $\vert e_0 \rangle$ is the ground state, while $\vert e_2 \rangle$ is the eigenstate corresponding to the highest energy level $\varepsilon_2 = N$ (see Table \ref{tab:eig_pbm_star}). The resulting QFI is given by Eq. \eqref{eq:maxQFIgraph}. Since the highest energy level is not degenerate, there is no ambiguity in the state maximizing the QFI. For such state the FI reads as Eq. \eqref{eq:fi_max_qfi_states_cg_2}, so please also refer to Fig. \ref{fig:qfi_cg_max}(b).
\begin{table*}[tb]
\centering
\begin{ruledtabular}
\begin{tabular}{rcccccc}
& \multicolumn{3}{c}{\textbf{QFI}} & \multicolumn{3}{c}{\textbf{FI}}\\ \cline{2-4}\cline{5-7}
& \textit{cycle} & \textit{complete} & \textit{star} & \textit{cycle} & \textit{complete} & \textit{star}\\ \hline
\textit{Localized states} & $\mathcal{O}(1)$ & $\mathcal{O}(N^3)$ & $\mathcal{O}(N^2)$ & $\mathcal{O}(1)$ & $\mathcal{O}(N^3)$ & $\mathcal{O}(N^2)$\\
\textit{Maximum QFI states} & $\mathcal{O}(1)$ & $\mathcal{O}(N^4)$ & $\mathcal{O}(N^4)$ & $\mathcal{O}(1)$ & $\mathcal{O}(N^3)$ & $\mathcal{O}(N^3)$\\
\end{tabular}
\end{ruledtabular}
\caption{Asymptotic behavior of the quantum Fisher information and of the classical Fisher information for large order $N$ of the cycle, complete, and star graphs, for localized and maximum QFI states.}
\label{tab:big-o_q_fi} \end{table*}
\begin{table*}[tb]
\centering
\begin{ruledtabular}
\begin{tabular}{rccc}
& \multicolumn{3}{c}{\textbf{FI}}\\ \cline{2-4}
& \textit{cycle} & \textit{complete} & \textit{star}\\ \hline
\textit{Localized states} & $\mathcal{O}(t^2)$ & $\mathcal{O}(t^2)$ & $\mathcal{O}(t^2)$\\
\textit{Maximum QFI states} &
\multicolumn{1}{l}{$\mathcal{O}(t^2)$ for energy eigenstates in Table \ref{tab:eig_pbm_cycle}}& $\mathcal{O}(t^4)$ & $\mathcal{O}(t^4)$\\
&\multicolumn{1}{l}{$\mathcal{O}(t^4)$ for odd $N$ and highest energy eigenstate \eqref{eq:evec_cyc_max_cos} or \eqref{eq:evec_cyc_max_sin}}
\end{tabular}
\end{ruledtabular}
\caption{Behavior at short times $t$ of the classical Fisher information of the cycle, complete, and star graphs, for localized and maximum QFI states. The maximum QFI state is the superposition of the ground state and the highest energy eigenstate. The QFI is always $\mathcal{O}(t^2)$, even at short and long times (see Eq. \eqref{eq:qfi_quadratic_time}), since the perturbation $\mathcal{H}_1$ is time-independent.}
\label{tab:smallt_q_fi} \end{table*}
\section{Discussion and Conclusions} \label{sec:conclusions} In this paper we have investigated the dynamics and the characterization of continuous-time quantum walks (CTQW) with Hamiltonians of the form $\mathcal{H}= L + \lambda L^2$, being $L$ the Laplacian (Kirchhoff) matrix of the underlying graph. We have considered cycle, complete, and star graphs, as they describe paradigmatic models with low/high connectivity and/or symmetry. The perturbation $\lambda L^2$ to the CTQW Hamiltonian $L$ introduces next-nearest-neighbor hopping. This strongly affects the CTQW in the cycle and in the star graph, whereas it is negligible in the complete graph, since each of its vertices is adjacent to all the others and $L^2=NL$. Clearly $[L,\lambda L^2]=0$, so the commutator between the unperturbed Hamiltonian and the perturbation is not indicative of how much the system is perturbed. Therefore, we consider how different is $L^2$ from $L$ by assessing the Frobenius norm of the operator $\Delta=L - L^2/N$, i.e. $\norm{\Delta}_F=\sqrt{\Tr{\Delta^\dagger \Delta}}$ \cite{horn2012matrix}. This turns out to be null for the complete graph, equal to $\sqrt{6N-40+70/N}$ for the cycle, and to $\sqrt{N-4+5/N-2/N^2}$ for the star graph. According to this, the cycle graph is the most perturbed, and the complete the least one.
Our results indicate the general quantum features of CTQWs on graphs, e.g. revivals, interference, and creation of coherence, are still present in their perturbed versions. On the other hand, novel interesting effects emerge, such as the appearance of symmetries in the behavior of the probability distribution and of the coherence. In the cycle graph (for $t\ll 1$), the perturbation affects the speed of the walker, while preserving the ballistic spreading. The variance is symmetric with respect to $\lambda _0$, despite the fact that the probability distribution is not. The value $\lambda_0$ makes the next-nearest-neighbor hopping equal to the nearest-neighbor hopping. The physical interpretation of this behavior is still an open question, which deserves to be further investigated. In the complete graph the perturbation does not affect the dynamics, since $L^2=N L$, so the resulting perturbed Hamiltonian is proportional to $L$. In the star graph, the perturbation affects the periodicity of the system. We have determined the values of $\lambda$ allowing the system to be periodic, thus to have exact revivals. In particular, the value $\lambda^\ast=-1/N$ makes the walker live only in the outer vertices, provided it starts in one of them.
Characterizing the perturbed Hamiltonian amounts to estimating the parameter $\lambda$ of the perturbation. We have addressed the optimal estimation of $\lambda$ by means of the quantum Fisher information (QFI) and using only a snapshot of the walker dynamics. The states maximizing the QFI turn out to be the equally-weighted linear combination of the eigenstates corresponding to the lowest and highest energy level. In addition, we have found that the simple graphs whose complementary graph is disconnected, e.g. the complete and the star graph, are the only ones providing the maximum QFI $N^4 t^2$. Moreover, we have evaluated the Fisher information (FI) of position measurements to assess whether it is optimal. We sum up the asymptotic behavior of the (Q)FI for large $N$ in Table \ref{tab:big-o_q_fi} and for $t\ll1$ in Table \ref{tab:smallt_q_fi}. When the probe is a localized state, the QFI in the cycle graph is independent of the order $N$ of the graph. In the complete graph, the local maxima of the FI equal the QFI and occur when the walker is localized in the starting vertex with probability $1$, and this happens periodically. However, to perform such a measurement one needs some \textit{a priori} knowledge of the value of $\lambda$. When the probe is the maximum QFI state, the QFI in the cycle graph depends on $N$, and FI is optimal for even $N$. In general, when the highest energy level is degenerate, the QFI does not depend on the choice of the corresponding eigenstates when defining the optimal state, instead the FI does.
Besides fundamental interest, our study may find applications in designing enhanced algorithms on graphs, e.g. spatial search, and as a necessary ingredient to study dephasing and decoherence.
\acknowledgments This study has been partially supported by SERB through the VAJRA scheme (grant VJR/2017/000011). P.B. and M.G.A.P. are members of GNFM-INdAM.
A.C. and L.R. contributed equally to this work.
\appendix
\section{Analytical derivation of the results for the dynamics} \label{app:an_res_ccsg} The dynamics of the system is essentially encoded in the time evolution of the density matrix. For an initially localized state $\ket{i}$, the density matrix is given by $\rho(t) = \vert i(t) \rangle \langle i(t) \vert$, whose generic element in the position basis is \begin{align}
\rho_{j,k}(t) & = \langle j \vert \mathcal{U}_\lambda(t) \vert i \rangle \langle i\vert \mathcal{U}^\dag_\lambda(t) \vert k \rangle\,,
\label{eq:rho_mn} \end{align} where the time-evolution operator $\mathcal{U}_\lambda(t)$ is defined in Eq. \eqref{eq:t_evol_op}. The probability distribution is given by the diagonal elements of the density matrix \begin{equation}
P_i(j,t\vert \lambda) = \vert \langle j \vert i(t) \rangle \vert^2 = \langle j \vert i(t) \rangle \langle i(t) \vert j \rangle = \rho_{j,j}(t)\,.
\label{eq:prob_from_state} \end{equation} On the other hand, the modulus of the off-diagonal elements of the density matrix entering the definition of coherence in Eq. \eqref{eq:coherence_def} can also be expressed in terms of probabilities: \begin{align}
\abs{\rho_{j,k}(t)} &=\vert \langle j \vert \mathcal{U}_\lambda(t) \vert i \rangle \vert \vert \langle k \vert \mathcal{U}_\lambda(t) \vert i \rangle \vert \nonumber \\ &=\sqrt{P_i(j,t\vert\lambda)P_i(k,t\vert\lambda)}\,.
\label{eq:abs_rho_mn} \end{align}
\subsection{Cycle Graph} \label{subapp:an_res_cyc} According to the time-evolution operator and to the spectral decomposition in Table \ref{tab:eig_pbm_cycle}, in a cycle graph an initially localized state $\ket{j}$ evolves in time as follows \begin{equation}
\ket{j(t)} =\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}e^{-iE_n^\lambda t}e^{i\frac{2\pi }{N}jn}\ket{e_n}\,,
\label{eq:jt_cyc} \end{equation} where $E_n^\lambda:=\varepsilon_n+\lambda \varepsilon_n^2$, and $\exp\{ i\frac{2\pi }{N}jn \} /\sqrt{N}=\braket{e_n}{j}$. Then, the probability of finding the walker in the vertex $k$ at time $t$ is \begin{equation}
P_j(k,t\vert\lambda) =\frac{1}{N^2}\sum_{n,m=0}^{N-1}e^{-i(E_n^\lambda-E_m^\lambda)t}e^{i\frac{2\pi }{N}(n-m)(j-k)}\,. \label{eq:pjk_cyc} \end{equation} This expression leads to Eq. \eqref{eq:pjk_cyc_cos} as follows. Let $p_{nm}$ be the summand, excluding $1/N^2$. The summation over $m$ can be split in three different summations: one over $m=n$ (providing $\sum_{n}p_{nn}=N$), one over $m>n$, and one over $m<n$. Since $p_{nm}=p_{mn}^\ast$, then $\sum_{m<n}p_{nm}=\sum_{m>n}p_{mn}^\ast$, so $\sum_{m>n}(p_{nm}+p_{mn}^\ast)=2\sum_{m>n}\Re\{p_{nm}\}$, with $\Re\{p_{nm}\}=\cos[\operatorname{arg} (p_{nm})]$.
To prove that the probability distribution is symmetric with respect to the starting vertex $j$, i.e. that $P_j(j+k,t\vert \lambda)=P_j(j-k,t\vert \lambda)$, we consider Eq. \eqref{eq:pjk_cyc}. The left-hand side is \begin{equation}
P_j(j+k,t\vert \lambda) = \frac{1}{N^2}\sum_{n,m=0}^{N-1}e^{-i(E_n^\lambda-E_m^\lambda)t}e^{i\frac{2\pi }{N}(n-m)(-k)}\,. \end{equation} Now, letting $l = N-n$ ($q=N-m$) be the new summation indices, and since $\varepsilon_{N-l} = \varepsilon_l$ ($\varepsilon_{N-q} = \varepsilon_q$), we have that \begin{align}
P_j(j+k,t\vert \lambda) &= \frac{1}{N^2} \sum_{l,q=1}^{N}e^{-i(E_{l}^\lambda-E_{q}^\lambda)t}e^{i\frac{2\pi }{N}(q-l)(-k)} \nonumber \\
&= \frac{1}{N^2} \sum_{l,q=0}^{N-1}e^{-i(E_{l}^\lambda-E_{q}^\lambda)t}e^{i\frac{2\pi }{N}(l-q)[j-(j-k)]} \nonumber \\
&= P_j(j-k, t \vert \lambda)\,. \end{align} The second equality holds since the summand of index $l=N$ ($q=N$) is equal to that of index $l=0$ ($q=0$). Indeed, according to Table \ref{tab:eig_pbm_cycle}, the virtual $E_N^\lambda$ is equal to $E_0^\lambda$ (the actual energies have index running from $0$ to $N-1$). In addition, $\exp\{i\frac{2\pi }{N}(q-l)k\}$ returns the same value if evaluated in $l=N$ ($q=N$) or $l=0$ ($q=0$).
Finally, we justify the expression of the variance of the position in Eq. \eqref{eq:xVar_cyc}. We assume even $N$, $\ket{j=N/2}$ as initial state, and $t\ll 1$. The variance requires the expectation values of $\hat{x}$ and $\hat{x}^2$, and the vertex states are eigenstates of the position operator. The probability distribution \eqref{eq:pjk_cyc_cos} is symmetric about the starting vertex, thus $\langle \hat{x} \rangle=N/2$, and it involves summands of the form \begin{align} \cos(\alpha t +\beta)&=\cos(\alpha t) \cos \beta -\sin( \alpha t) \sin\beta\nonumber\\ &= (1-\frac{\alpha^2}{2}t^2)\cos \beta-\alpha t \sin \beta+\mathcal{O}(t^3)\,, \end{align} since $t\ll 1$. Hence, letting $\alpha_{nm}=E_n^\lambda-E_m^\lambda$ and $\beta_{nm}^{kj}=\frac{2\pi}{N}(n-m)(k-j)$, we can write \begin{align}
\langle \hat{x}^2(t) \rangle&\approx\frac{1}{N}\sum_{k=0}^{N-1} k^2+\frac{2}{N^2}\sum_{\substack{n=0, \\ m>n}}^{N-1}\sum_{k=0}^{N-1}\left[\vphantom{\frac{t^2}{2}} k^2 \cos{\beta_{nm}^{kj}}\right.\nonumber\\
&\quad\left.-\frac{t^2}{2} k^2 \alpha_{nm}^2 \cos{\beta_{nm}^{kj}}-t k^2 \alpha_{nm} \sin{\beta_{nm}^{kj}}\right]\nonumber\\
&=\frac{1}{6}(N-1)(2N-1)-\frac{1}{12}[N(N-6)+2]\nonumber\\
&\quad+2t^2 (20\lambda^2+8\lambda+1)\nonumber\\
&=\frac{N^2}{4}+\Big[ 40\Big(\lambda+\frac{1}{5}\Big)^2+\frac{2}{5}\Big]t^2\,. \end{align} Then, the variance \eqref{eq:xVar_cyc} follows, and is symmetric with respect to $\lambda_0 = -1/5$.
\subsection{Complete Graph} \label{subapp:an_res_cg} The complete graph has two energy levels (see Table \ref{tab:eig_pbm_complete}), thus the unitary time-evolution operator has the following spectral decomposition: \begin{equation}
\mathcal{U}_\lambda(t) = \vert e_0\rangle \langle e_0 \vert + e^{-2i\omega_N(\lambda)t} \sum_{l=1}^{N-1} \vert e^l_1\rangle \langle e^l_1 \vert \,, \end{equation} being $\omega_N(\lambda)$ defined in Eq. \eqref{eq:ang_freq}. Hence, a localized state $\ket{0}$ evolves in time according to \begin{align}
\vert 0(t) \rangle = & \frac{1}{N}\left[1+(N-1)e^{-i2\omega_N(\lambda)t}\right] \vert 0 \rangle \nonumber\\
&+\frac{1}{N} \left(1-e^{-i2\omega_N(\lambda)t}\right)\sum_{k=1}^{N-1} \vert k \rangle\,.
\label{eq:jt_cg} \end{align}
Then, the density matrix $\rho(t) = \vert 0(t) \rangle \langle 0(t) \vert$ in the position basis is \begin{align}
\rho(t)= &[1-(N-1)A]\dyad{0}+(A+B)\sum_{k=1}^{N-1}\dyad{0}{k}\nonumber\\
&+(A+B^\ast)\sum_{k=1}^{N-1}\dyad{k}{0}+A\sum_{j,k=1}^{N-1}\dyad{j}{k}\,,
\end{align} where \begin{align}
A &= \frac{4}{N^2}\sin^2(\omega_N(\lambda)t) \,,\\
B &= \frac{1}{N}\left(e^{-2it\omega_N(\lambda)}-1\right)\,. \end{align}
The diagonal elements of $\rho(t)$ provide the probability distribution in Eqs. \eqref{eq:p0_cg}--\eqref{eq:pi_cg}. Instead, the off-diagonal elements allow us to compute the coherence according to Eq. \eqref{eq:coherence_def}, and it reads as \begin{equation}
\mathcal{C}(t) = 2 (N-1) \vert A+ B \vert + (N-1)(N-2) \vert A \vert \,. \end{equation}
\subsection{Star graph} \label{subapp:an_res_sg} The star graph has three energy levels (see Table \ref{tab:eig_pbm_star}), thus the unitary time-evolution operator has the following spectral decomposition: \begin{align}
\mathcal{U}_\lambda(t) = &\vert e_0 \rangle \langle e_0 \vert + e^{-2it\omega_{1}} \sum_{l=1}^{N-2} \vert e^l_1 \rangle\langle e^l_1 \vert \nonumber\\
& + e^{-2it\omega_{N}} \vert e_2 \rangle \langle e_2 \vert\,, \end{align} being $\omega_N(\lambda)$ defined in Eq. \eqref{eq:ang_freq}. Hence, a localized state $\ket{1}$, i.e. an outer vertex, evolves in time according to \begin{align}
\vert 1(t) \rangle =&\frac{1}{N}(1-e^{-i2\omega_N(\lambda)t})\ket{0}\nonumber\\
&+\left(\frac{1}{N}+\frac{N-2}{N-1}e^{-i2\omega_1(\lambda)t}+\frac{e^{-i2\omega_N(\lambda)t}}{N(N-1)} \right)\ket{1}\nonumber\\
&+\left(\frac{1}{N}-\frac{e^{-i2\omega_1(\lambda)t}}{N-1} +\frac{e^{-i2\omega_N(\lambda)t}}{N(N-1)} \right)\sum_{k=2}^{N-1} \ket{k} \,.
\label{eq:1t_sg} \end{align} Instead, if the initial state is the central vertex $\ket{0}$, we recover the time evolution of a localized state in the complete graph of the same size (see Eq. \eqref{eq:jt_cg}), and so the same results. Then, the density matrix $\rho(t) = \vert 1(t) \rangle \langle 1(t) \vert$ in the position basis is \begin{align}
\rho(t)=&\abs{A}^2\dyad{0}+\abs{B}^2 \dyad{1}+\abs{C}^2 \sum_{k=2}^{N-1}\dyad{k} +\nonumber\\
&+\Bigg[\abs{C}^2 \sum_{\substack{j,k=2,\\k>j}}^{N-1}\dyad{j}{k}+AB^\ast \dyad{0}{1}\nonumber\\
&+\sum_{k=2}^{N-1}(AC^\ast\dyad{0}{k}+BC^\ast \dyad{1}{k})+H.c.\Bigg]\,,
\label{eq:rho_loc_sg} \end{align} where $H.c.$ denotes the Hermitian conjugate of the off-diagonal terms only, and \begin{align}
A &=\frac{1}{N}(1-e^{-i2\omega_N(\lambda)t})\,,\\
B &= \frac{1}{N}+\frac{N-2}{N-1}e^{-i2\omega_1(\lambda)t}+\frac{e^{-i2\omega_N(\lambda)t}}{N(N-1)}\,,\\
C & =\frac{1}{N}-\frac{e^{-i2\omega_1(\lambda)t}}{N-1} +\frac{e^{-i2\omega_N(\lambda)t}}{N(N-1)} \,, \end{align} are the coefficients of $\ket{1(t)}$ in the position basis (see Eq. \eqref{eq:1t_sg}). The diagonal elements of $\rho(t)$ provide the probability distribution in Eqs. \eqref{eq:p_10_sg}--\eqref{eq:p_1i_sg}. Instead, the off-diagonal elements allow us to compute the coherence according to Eq. \eqref{eq:coherence_def}. Given the counting of the different matrix elements, and since $\abs{\rho_{j,k}(t)}=\abs{\rho_{k,j}(t)}$, the coherence reads as \begin{align}
\mathcal{C} (t) =& 2 \abs{AB^\ast}+ 2(N-2)(\abs{AC^\ast}+\abs{BC^\ast})\nonumber\\
&+(N-2)(N-3)\abs{C}^2 \,. \end{align}
It is still pending the issue about the periodicity of the probability distribution. Ultimately, the overall probability distribution is periodic if and only if the periods of the sine functions involved by the probabilities \eqref{eq:p_10_sg}--\eqref{eq:p_1i_sg} are commensurable. Since such sine functions are squared, the periods are: \begin{align}
&T_1(\lambda):=\frac{\pi}{\omega_1(\lambda)}=\frac{2\pi}{1+\lambda}\,,\\
&T_N(\lambda):=\frac{\pi}{\omega_N(\lambda)}=\frac{2\pi}{N+\lambda N^2}\,,\\
&T_{N,1}(\lambda):=\frac{\pi}{\omega_{N}(\lambda )-\omega_1(\lambda)}=\frac{2\pi}{(N-1)[1+\lambda(N+1)]}\,. \end{align} Two non-zero real numbers are commensurable if their ratio is a rational number. The idea is therefore to express both $T_1(\lambda)$ and $T_{N,1}(\lambda)$ as multiple integers of $T_N(\lambda)$. From the ratio $T_1(\lambda)/T_N(\lambda)$ we get \begin{equation}
T_1(\lambda)=\frac{N(1+\lambda N)}{1+\lambda}T_N(\lambda)=:p_N^\lambda T_N(\lambda)\,,
\label{eq:t1_over_tn_sg} \end{equation} with $\lambda \neq -1 \wedge \lambda \neq -1/N$, and from $T_{N,1}(\lambda)/T_N(\lambda)$ \begin{equation}
T_{N,1}(\lambda)=\frac{N(1+\lambda N)}{(N-1)[1+\lambda(N+1)]}T_N(\lambda)=:q_N^\lambda T_N(\lambda)\,,
\label{eq:tn1_over_tn_sg} \end{equation} with $\lambda \neq -1/N \wedge \lambda \neq -1/(N+1)$. Then, we need to find the value of $\lambda$ such that $p_N^\lambda, q_N^\lambda \in \mathbb{N}$ at the same time. Combining the definition of $p_N^\lambda$ and $q_N^\lambda$ in Eqs. \eqref{eq:t1_over_tn_sg}--\eqref{eq:tn1_over_tn_sg} we find that they are related to $\lambda$ and $N$ by \begin{equation}
\lambda = \frac{p_N^\lambda-q_N^\lambda(N-1)}{q_N^\lambda(N^2-1)-p_N^\lambda}\,.
\label{eq:lambda_sg_pq} \end{equation}
Please notice that Eq. \eqref{eq:lambda_sg_pq} is to be understood together with Eqs. \eqref{eq:t1_over_tn_sg}--\eqref{eq:tn1_over_tn_sg}. As an example, for $p_N^\lambda=q_N^\lambda$ we get $\lambda=(2-N)/(N^2-2)$ from Eq. \eqref{eq:lambda_sg_pq}. However, the period is unique, so we can not choose any $p_N^\lambda=q_N^\lambda$. Indeed, for such value of $\lambda$ we get $p_N^\lambda=q_N^\lambda=2$ from Eqs. \eqref{eq:t1_over_tn_sg}--\eqref{eq:tn1_over_tn_sg}. In the end, by considering the least common multiple of the latter two integers, the total period of the probability distribution is \begin{equation}
T=\operatorname{lcm}(p_N^\lambda, q_N^\lambda) T_N(\lambda)\,.
\label{eq:period_sg} \end{equation}
The above ratios \eqref{eq:t1_over_tn_sg}--\eqref{eq:tn1_over_tn_sg} between the different periods are properly defined unless $\lambda=-1,-1/N,-1/(N+1)$. Nevertheless, for such values of $\lambda$ the overall probability distribution is actually periodic. If we let $p,q\in\mathbb{Z}$, we recover them from Eq. \eqref{eq:lambda_sg_pq} for $q=0$, $q=-p$, and $p=0$, respectively. These values of $\lambda$ make $\omega_1$, $\omega_N$, and $\omega_N-\omega_1$ to vanish, respectively. When $\omega_1=0$ ($\omega_N=0$), the probabilities only involve sine functions with $\omega_N$ ($\omega_1$). When $\omega_N=\omega_1$, all the sine functions have the same angular frequency.
\section{Fisher Information and Quantum Fisher Information for localized states and states maximizing the QFI} \label{app:q_fi_scg}
In this appendix we prove the analytical results about the Quantum Fisher Information (QFI) in Eq. \eqref{eq:qfi_def_braket} and the Fisher Information (FI) in Eq. \eqref{eq:fi_def} in the different graphs. We provide the FI for a local position measurement whose POVM is given by $\{\dyad{0}, \dyad{1}, \dots, \dyad{N-1}\}$, i.e. by the projectors on the vertex states. The Parthasarathy's Lemma \ref{lemma:parthasarathy} leads to the QFI in Eq. \eqref{eq:qfi_delta_eigenval}, because the state maximizing the QFI involves the ground state and the highest energy eigenstate. The highest energy level might be degenerate, but choosing any eigenstate of such level results in the same QFI. Instead, the FI does depend on such choice.
\subsection{Cycle graph} \label{subapp:q_fi_cyc} \paragraph*{Localized state} The QFI in Eq. \eqref{eq:qfi_quadratic_time} requires the expectation values of $L^2$ \eqref{eq:L2_cyc} and of \begin{align} L^4 =&70{I}+ \sideset{}{'}\sum_{k=0}^{N-1} ( \dyad{k-4}{k}-8\dyad{k-3}{k}\nonumber\\ &+28\dyad{k-2}{k}-56\dyad{k-1}{k}+H.c.)\,, \end{align} on the initial state $\ket{j}$. These are $\langle L^2 \rangle=6$ and $\langle L^4 \rangle=70$, from which Eq. \eqref{eq:qfi_loc_cyc} follows.
\paragraph*{States maximizing the QFI} In the cycle graph the ground state is unique, whereas the degeneracy of the highest energy level depends on the parity of $N$ (see Table \ref{tab:eig_pbm_cycle}). We define $E_\lambda:=\varepsilon_{max}+\lambda \varepsilon_{max}^2$, where $\varepsilon_{max}$ is the highest energy eigenvalue of $L$ (see Eq. \eqref{eq:eval_cyc_max_even} and Eq. \eqref{eq:eval_cyc_max_odd} for even and odd $N$, respectively).
For even $N$, according to Eq. \eqref{eq:psi0_Nevenodd} the state maximizing the QFI is \begin{equation}
\ket{\psi_0(t)} = \frac{1}{\sqrt{2N}}\sum_{k=0}^{N-1}\left[1+(-1)^k e^{-i E_\lambda t} \right]\ket{k}\,, \end{equation} and the maximum QFI \eqref{eq:qfi_cycle_evenodd} follows from Eq. \eqref{eq:qfi_delta_eigenval}. Then, the probability distribution associated to a position measurement is \begin{align}
P_M(k, t \vert \lambda) = \frac{1}{N}\left[1+(-1)^k \cos(E_\lambda t)\right]\,. \end{align} Hence, observing that the dependence on the vertex is encoded only into an alternating sign, the FI is \begin{align}
&\mathcal{F}_c(N,t,\lambda) =\nonumber\\
&=\sum_{k=0}^{N/2-1}\left[ \frac{(\partial_\lambda P_M(2k,t\vert \lambda))^2}{P_M(2k,t\vert \lambda)}+\frac{(\partial_\lambda P_M(2k+1,t\vert \lambda))^2}{P_M(2k+1,t\vert \lambda)} \right] \nonumber \\
&=\frac{\varepsilon_{max}^4 t^2 \sin^2(E_\lambda t)}{N^2}\sum_{k=0}^{N/2-1}\left[ \frac{N}{1+\cos(E_\lambda t)}+\frac{N}{1-\cos(E_\lambda t)}\right]\nonumber\\
&=\frac{\varepsilon_{max}^4 t^2 \sin^2(E_\lambda t)}{N^2}\frac{N}{2} \frac{2N}{\sin^2(E_\lambda t)}= \varepsilon_{max}^4 t^2 = \mathcal{F}_q(t)\,. \end{align}
In other words, the position measurement for the state maximizing the QFI in a cycle graph having an even number of vertices is optimal, since the corresponding FI equals the QFI.
For odd $N$, the situation is trickier: the state maximizing the QFI is not unique, because of the degeneracy of the highest energy level. We may consider the two corresponding eigenstates according to Table \ref{tab:eig_pbm_cycle}, which lead to the following states maximizing the QFI \begin{equation}
\ket{\varphi_0^\pm(t)} = \frac{1}{\sqrt{2N}}\sum_{k=0}^{N-1}\left[1+(-1)^k e^{\pm i \theta_k}e^{-i E_\lambda t} \right]\ket{k}\,,
\label{eq:psi0_cyc_odd_pf} \end{equation}
where $\theta_k=\pi k / N$. On the other hand, we may also consider the linear combinations of such eigenstates (see Eqs. \eqref{eq:evec_cyc_max_cos}--\eqref{eq:evec_cyc_max_sin}), which lead to the following states maximizing the QFI \begin{align}
\ket{\psi_0^+(t)} &= \frac{1}{\sqrt{2N}}\sum_{k=0}^{N-1}\left[1+\sqrt{2}(-1)^k \cos\theta_k e^{-i E_\lambda t} \right]\ket{k}\label{eq:psi0_cyc_odd_cos}\,,\\
\ket{\psi_0^-(t)} &=\frac{1}{\sqrt{2N}}\sum_{k=0}^{N-1}\left[1+\sqrt{2}(-1)^k \sin\theta_k e^{-i E_\lambda t} \right]\ket{k}\label{eq:psi0_cyc_odd_sin}\,. \end{align} Under the assumption of odd $N$, and according to the following results \begin{align}
&\sum_{k=0}^{N-1}\cos^2\theta_k=\sum_{k=0}^{N-1}\sin^2\theta_k=\frac{N}{2}\,,\\
& \sum_{k=0}^{N-1}(-1)^k e^{\pm i \theta_k}=\frac{1+(-1)^N}{1+e^{\pm i \frac{\pi}{N}}}\stackrel{odd\, N}{=}0\,, \end{align} the maximum QFI \eqref{eq:qfi_cycle_evenodd} follows from Eq. \eqref{eq:qfi_delta_eigenval} and does not depend on the choice of these states. Instead, we prove that the FI does depend on them. Again, there is an alternating sign which depends on the vertex. In the following, we will split the sum over even and odd indices, and for odd $N$ it reads as follows: \begin{equation}
\sum_{k=0}^{N-1}a_k=\sum_{k=0}^{(N-1)/2}a_{2k}+\sum_{k=0}^{(N-1)/2-1}a_{2k+1}\,. \end{equation}
We first consider the states $\ket{\varphi_0^\pm (t)}$ in Eq. \eqref{eq:psi0_cyc_odd_pf}. The probability distribution associated to a position measurement is \begin{equation}
P_M^\pm(k, t \vert \lambda) = \frac{1}{N}\left[1+(-1)^k \cos(E_\lambda t\mp \theta_k)\right]\,. \end{equation} Hence the FI is \begin{align}
\mathcal{F}_c^\pm&(\ket{\varphi_0^\pm};N,t,\lambda)=\nonumber\\
&=\frac{\varepsilon_{max}^4 t^2}{N}\left[\sum_{k=0}^{(N-1)/2} \frac{\sin^2(E_\lambda t\mp \theta_{2k})}{1+\cos(E_\lambda t\mp \theta_{2k})}\right.\nonumber\\
&\quad\left.+\sum_{k=0}^{(N-1)/2-1} \frac{\sin^2(E_\lambda t\mp \theta_{2k+1})}{1-\cos(E_\lambda t\mp \theta_{2k+1})} \right]\nonumber\\
& = \frac{\varepsilon_{max}^4 t^2}{N}\left[\sum_{k=0}^{(N-1)/2} (1-\cos(E_\lambda t\mp \theta_{2k}))\right.\nonumber\\
&\quad\left.+\sum_{k=0}^{(N-1)/2-1} (1+\cos(E_\lambda t\mp \theta_{2k+1})) \right]\nonumber\\
&= \frac{\varepsilon_{max}^4 t^2}{N}\left[\frac{N-1}{2}+1+\frac{N-1}{2}-1+1 \right]\nonumber\\
&=\varepsilon_{max}^4 t^2=\mathcal{F}_q(t)\,. \end{align} Indeed, for odd $N$ \begin{align}
&\sum_{k=0}^{(N-1)/2} \cos(\theta_{2k}+\phi)-\sum_{k=0}^{(N-1)/2-1} \cos(\theta_{2k+1}+\phi)\nonumber\\
&=\sum_{k=0}^{N-1} (-1)^k\cos(\theta_{k}+\phi)=0\,. \end{align}
Now, we focus on the state $\ket{\psi_0^+(t)}$ in Eq. \eqref{eq:psi0_cyc_odd_cos}. The probability distribution associated to a position measurement is \begin{align}
P_M^+(k, t \vert \lambda) &= \frac{1}{2N}\left[1+2\sqrt{2}(-1)^k \cos\theta_k \cos(E_\lambda t)\right.\nonumber\\
&\quad\left.\vphantom{\sqrt{2}(-1)^k}+2 \cos^2\theta_k )\right]\,. \end{align} Hence the FI is \begin{align}
&\mathcal{F}_c^+(\ket{\psi_0^+};N,t,\lambda) = \frac{4 \varepsilon_{max}^4 t^2 \sin^2(E_\lambda t)}{N}\nonumber\\
&\times\left[\sum_{k=0}^{(N-1)/2} \frac{\cos^2\theta_{2k}}{1+2\sqrt{2} \cos\theta_{2k} \cos(E_\lambda t)+2 \cos^2\theta_{2k} }\right.\nonumber\\
&\quad\left.+\sum_{k=0}^{(N-1)/2-1} \frac{\cos^2\theta_{2k+1}}{1-2\sqrt{2} \cos\theta_{2k+1} \cos(E_\lambda t)+2 \cos^2\theta_{2k+1} }\right]\,. \end{align} Analogously for $\ket{\psi_0^-(t)}$ in Eq. \eqref{eq:psi0_cyc_odd_sin}, we find \begin{align}
&\mathcal{F}_c^-(\ket{\psi_0^-};N,t,\lambda) = \frac{4 \varepsilon_{max}^4 t^2 \sin^2(E_\lambda t)}{N}\nonumber\\
&\times\left[\sum_{k=0}^{(N-1)/2} \frac{\sin^2\theta_{2k}}{1+2\sqrt{2} \sin\theta_{2k} \cos(E_\lambda t)+2 \sin^2\theta_{2k} }\right.\nonumber\\
&\quad\left.+\sum_{k=0}^{(N-1)/2-1} \frac{\sin^2\theta_{2k+1}}{1-2\sqrt{2} \sin\theta_{2k+1} \cos(E_\lambda t)+2 \sin^2\theta_{2k+1} }\right]\,. \end{align} Numerical results suggest that $\mathcal{F}_c^\pm(\ket{\psi_0^\pm};N,t,\lambda)<\mathcal{F}_q(t)$. Notice that $\mathcal{F}_c^\pm(\ket{\psi_0^\pm};N,t,\lambda=-1/\varepsilon_{max}) = 0 \,\forall\, t$. Indeed, for such value of $\lambda$ we have that $E_\lambda=0$.
\subsection{Complete graph} \label{subapp:q_fi_cg} \paragraph*{Localized state} The QFI in Eq. \eqref{eq:qfi_quadratic_time} requires the expectation values of $L^2$ and of $L^4$ on the initial state $\ket{0}$. Because of Eqs. \eqref{eq:H0_complete_matrix} and \eqref{eq:Ln_proptoL}, we only need $\langle L \rangle = N-1$, from which Eq. \eqref{eq:qfi_loc_cg} follows.
\paragraph*{States maximizing the QFI} The complete graph has two energy levels: the ground state is unique, but the highest energy level is $(N-1)$-degenerate (see Table \ref{tab:eig_pbm_complete}). The QFI does not depend on the choice of the eigenstate of the highest energy level, but the FI does. As an example, we consider two different states maximizing the QFI $\vert \psi_0^1\rangle$ and $\vert \psi_0^{N-1} \rangle$, i.e. the states in Eq. \eqref{eq:max_qfi_states_cg} for $l=1$ and $l=N-1$, respectively.
The first state is \begin{equation}
\vert \psi^0_1(t)\rangle = \frac{1}{\sqrt{2}} \left[\vert e_0 \rangle + \frac{1}{\sqrt{2}}e^{-2it\omega_N(\lambda)}\left(\vert 0 \rangle - \vert 1 \rangle\right)\right]\,. \end{equation} The probability distribution associated to a position measurement is \begin{align}
P_M^0(0,t\vert \lambda) &= \frac{1}{4} + \frac{1}{2N} + \frac{\cos\left(2t \omega_N(\lambda)\right)}{\sqrt{2N}}\,, \\
P_M^0(1,t\vert \lambda) &= \frac{1}{4} + \frac{1}{2N} - \frac{\cos\left(2t \omega_N(\lambda)\right)}{\sqrt{2N}}\,, \\
P_M^0(k,t\vert \lambda) &= \frac{1}{2N}\,, \text{with $2\leq k \leq N-1$.} \end{align} Then, being null the $(N-2)$ contributions from the vertices $2\leq k \leq N-1$, since $\partial_\lambda (P_M^0(k,t\vert \lambda))= 0$, only the probabilities associated to the vertices $\ket{0}$ and $\ket{1}$ contribute to the FI \eqref{eq:fi_def}, which results in Eq. \eqref{eq:fi_max_qfi_states_cg_1}.
Similarly, the second state is \begin{align}
\vert \psi^{N-1}_0 (t) \rangle = &\frac{1}{\sqrt{2}} \left\lbrace\vert e_0 \rangle + \frac{1}{\sqrt{N^2-N}}e^{-2it\omega_N(\lambda)}\left[\vert 0 \rangle+\dots\right.\right. \nonumber \\
& \left.\vphantom{\frac{}{}}\left.+ \vert N-2 \rangle - (N-1) \vert N-1 \rangle \right]\right\rbrace\,. \end{align} The probability distribution associated to a position measurement is \begin{equation}
P^{N-1}_M (k,t\vert \lambda) =\frac{1}{2(N-1)} + \frac{1}{N\sqrt{N-1}}\cos\left(2t\omega_N(\lambda)\right)
\end{equation} with $0\leq k\leq N-2$, and \begin{equation}
P^{N-1}_M(N-1,t\vert \lambda) = \frac{1}{2}-\frac{\sqrt{N-1}}{N}\cos(2t\omega_N(\lambda))\,. \end{equation} Then, having $(N-1)$ equal contributions from the vertices $0\leq k \leq N-2$ and a particular one from $N-1$, the FI \eqref{eq:fi_def} results in Eq. \eqref{eq:fi_max_qfi_states_cg_2}.
\subsection{Star graph} \label{subapp:q_fi_sg} \paragraph*{Localized state} Considering the central vertex $\ket{0}$ as the initial state provides the same results observed in the complete graph of the same size. Thus, we consider as initial state $\ket{1}$, i.e. one of the outer vertices The QFI in Eq. \eqref{eq:qfi_quadratic_time} requires the expectation values of $L^2$ \eqref{eq:L2_star} and of \begin{align} L^4=&(N^2+N+2)I-N^3 \sum_{k=1}^{N-1}(\dyad{k}{0}+\dyad{0}{k})\nonumber\\ &+(N-2)(N^3+N^2+N+1)\dyad{0}\nonumber\\ &+(N^2+N+1)\sum_{\substack{j,k=1,\\j\neq k}}^{N-1}\dyad{j}{k} \end{align} on the initial state $\ket{1}$. These are $\langle L^2 \rangle=2$ and $\langle L^4 \rangle=N^2+N+2$, from which Eq. \eqref{eq:qfi_loc_sg} follows.
\paragraph*{States maximizing the QFI} In the star graph the state maximizing the QFI, according to Eq. \eqref{eq:max_qfi_states_sg}, is \begin{equation}
\vert \psi_0 (t) \rangle = \frac{1}{\sqrt{2}} \left(\vert e_0 \rangle + e^{-2it\omega_N(\lambda)}\vert e_2 \rangle\right)\,, \end{equation} being both the ground and the highest energy levels not degenerate (see Table \ref{tab:eig_pbm_star}). Then, the probability distribution associated to a position measurement is \begin{align}
P_M(0, t \vert \lambda) &= \frac{1}{2} + \frac{\sqrt{N-1}}{N} \cos(2 t \omega_{N}(\lambda))\,, \\
P_M(k, t\vert \lambda) &= \frac{1}{2(N-1)} - \frac{1}{N\sqrt{N-1}}\cos(2t\omega_N(\lambda))\,, \end{align} with $1 \leq k \leq N-1$. Then, the FI follows from Eq. \eqref{eq:fi_def}.
\subsection{Maximum QFI states: the role of the phase factor in the superposition of energy eigenstates} \label{subapp:phi_maxQFI_states} So far we have studied the states maximizing the QFI without bothering to consider a different linear combination of the ground state and the highest energy state. According to the Parthasarathy's lemma \ref{lemma:parthasarathy}, the two eigenstates defining the state in Eq. \eqref{eq:psi_0} are equally weighted. However, we may suppose the second one to have a phase factor, i.e \begin{equation}
\vert \psi_0 \rangle = \frac{1}{\sqrt{2}}(\vert e_0 \rangle + e^{i\phi} \vert e_1 \rangle)\,. \end{equation} In this section, we study how the phase $\phi$ affects the FI and QFI.
The states $\ket{e_0}$ and $\ket{e_1}$ denote the eigenstates of minimum and maximum energy eigenvalue, i.e. $\varepsilon_{min}$ and $\varepsilon_{max}$ respectively, and we know that for simple graphs $\ket{e_0}=(1,\ldots,1)/\sqrt{N}$ and $\varepsilon_{min}=0$. Moreover, since the Laplacian matrix is real and symmetric, we can always deal with real eigenstates. Because of Eq. \eqref{eq:qfi_quadratic_time}, we already know that the QFI is \eqref{eq:qfi_delta_eigenval} and therefore it is independent of a phase shift. On the other hand, the FI reads as follows: \begin{align}
&\mathcal{F}_c(t, \lambda) = 2 t^2 \varepsilon_{max}^4 \sin^2(E_\lambda t - \phi)\nonumber\\
&\times \sum_{i=0}^{N-1}\frac{\langle i \vert e_1 \rangle^2}{N\langle i \vert e_1 \rangle^2 + 2\sqrt{N}\langle i \vert e_1 \rangle \cos(E_\lambda t - \phi) +1}\,, \end{align} where $E_\lambda:=\varepsilon_{max}+\lambda \varepsilon_{max}^2$, and $\langle i \vert e_1 \rangle\in \mathbb{R}$, since the vectors involved are real. Hence, the phase is encoded as a phase shift in all the sine and cosine functions. However, this does not result in a global time shift, because the quadratic term in $t$ is not affected by $\phi$.
\end{document} | arXiv |
\begin{document}
\maketitle
\begin{abstract}
Triggered by the fact that, in the hydrodynamic limit, several different kinetic equations of physical interest all lead to the same Navier-Stokes-Fourier system, we develop in the paper an abstract framework which allows to explain this phenomenon. The method we develop can be seen as a significant improvement of known approaches for which we fully exploit some structural assumptions on the linear and nonlinear collision operators as well as a good knowledge of the Cauchy theory for the limiting equation. We adopt a perturbative framework in a Hilbert space setting and first develop a general and fine spectral analysis of the linearized operator and its associated semigroup. Then, we introduce a splitting adapted to the various regimes (kinetic, acoustic, hydrodynamic) present in the kinetic equation which allows, by a fixed point argument, to construct a solution to the kinetic equation and prove the convergence towards suitable solutions to the Navier-Stokes-Fourier system. Our approach is robust enough to treat, in the same formalism, the case of the Boltzmann equation with hard and moderately soft potentials, with and without cut-off assumptions, as well as the Landau equation for hard and moderately soft potentials in presence of a spectral gap. New well-posedness and strong convergence results are obtained within this framework. In particular, for initial data with algebraic decay with respect to the velocity variable, our approach provides the first result concerning the strong Navier-Stokes limit from Boltzmann equation without Grad cut-off assumption or Landau equation. The method developed in the paper is also robust enough to apply, at least at the linear level, to quantum kinetic equations for Fermi-Dirac or Bose-Einstein particles.\end{abstract}
\tableofcontents
\section{Introduction}
\subsection{From nonlinear collisional model to Navier-Stokes-Fourier system}
The connection between the Navier-Stokes and Boltzmann equations originates seemingly from the work \cite{H1912} regarding the mathematical treatment of the axioms of physics. Since this original idea, the derivation of suitable hydrodynamic equations from nonlinear kinetic equations has attracted a lot of attention in the recent years. We will review later in this introduction several of the main contributions in the field, illustrating in particular the large variety of models considered in the literature, but we wish to focus here on some striking universal features shared by several binary collisional models in the diffusive scaling.
Namely, for kinetic equations in adimensional form given by the evolution of a {particles number} density $f^{{\varepsilon}}(x,v,t)$ {(with $x \in \mathbb R^d$ denoting position, $v \in \mathbb R^d$ the velocity, $t \geqslant 0$ the time and ${\varepsilon} > 0$ the mean free path between particles collisions)}
\begin{equation}\label{eq:Kin-Intro}
\partial_{t}f^{{\varepsilon}}+\frac{1}{{\varepsilon}}v\cdot \nabla_{x}f^{{\varepsilon}} =\frac{1}{{\varepsilon}^{2}}\mathcal L f^{{\varepsilon}} + \frac{1}{{\varepsilon}}\mathcal Q(f^{{\varepsilon}},f^{{\varepsilon}}), \qquad f^{{\varepsilon}}(x,v,0)=f_{\textnormal{in}}(x,v)\end{equation}
for some suitable \emph{linear operator} $\mathcal L$ and \emph{quadratic operator} $\mathcal Q$, it has been shown in various contexts that, in the limit ${\varepsilon} \to 0$, the solution $f^{{\varepsilon}}$ converges (in some sense to determine) towards a ``macroscopic'' distribution $f_{\textnormal{NS}}(x,v,t)$ of the form
\begin{equation}\label{eq:11}
f_{\textnormal{NS}}(x,v,t)= \left(\varrho(t, x) + u(t, x) \cdot v + C_{0}\theta(t, x)\left( |v|^2 - E \right)\right) \mu(v)\end{equation}
where $C_{0} >0,E >0$ are depending {only on the universal distribution $\mu$ (independent of $f_\textnormal{in}$)}. More surprisingly, it is also known that the triple of functions
$$(\varrho(t,x),u(t,x),\theta(t,x)) \in \mathbb R \times \mathbb R^{d}\times \mathbb R$$
{associated to} the macroscopic mass, mean velocity and temperature of the gas
are suitable solutions to the Navier-Stokes-Fourier system
\begin{equation}\label{eq:NSFint}
\begin{cases}
\partial_{t}u-\kappa_{\textnormal{inc}}\,\Delta_{x}u + \vartheta_{\textnormal{inc}}\,u\cdot \nabla_{x}\,u = \nabla_{x}p \,,\\[6pt]
\partial_{t}\,\theta-\kappa_{\textnormal{Bou}}\,\Delta_{x}\theta +\vartheta_{\textnormal{Bou}}\,u\cdot \nabla_{x}\theta=0,\\[8pt]
\nabla_{x}\cdot u=0\,, \qquad \nabla_x\left(\varrho + \theta\right)= 0\,,
\end{cases}
\end{equation}
where the third line describe respectively the \emph{incompressibility} condition of the fluid and the \emph{Boussinesq relation} between mass and temperature. The pressure of the fluid $p$ is here above obtained implicitly as a Lagrange multiplier associated to the incompressibility constraint $\nabla_{x}\cdot u=0.$
The striking phenomena we wish to discuss in this paper is the fact that a large variety of kinetic models described by \eqref{eq:Kin-Intro} provide in the hydrodynamic limit the \emph{same Navier-Stokes-Fourier system} \eqref{eq:NSFint}, making that system a \emph{universal hydrodynamic limit} for \eqref{eq:Kin-Intro}. The only memory of the original equation \eqref{eq:Kin-Intro} kept in the system \eqref{eq:NSFint} is encapsulated in the various coefficients:
$$\kappa_{\textnormal{inc}} >0, \quad \kappa_{\textnormal{Bou}} >0,$$
which represent the viscosity and thermal conductivity, as well as and $\vartheta_\textnormal{inc},\vartheta_\textnormal{Bou}$ , all of being defined explicitly in terms of the operators $\mathcal L$ and $\mathcal Q$ that encode the collision process. We refer to Section \ref{sec:detail} for more details on those coefficients.
Recall that, in the kinetic equation \eqref{eq:Kin-Intro}, the unknown $f^{\varepsilon}(x,v,t)$ denotes typically the density of particles having position $x \in \mathbb R^{d}$ and velocity $v \in \mathbb R^{d}$ at time $t\geq0$ while the parameter ${\varepsilon}$ represents the \emph{Knudsen number} which is proportional to the mean free path between collisions. Typically, small values of ${\varepsilon}$ correspond to a case in which particles suffer a very large number of collisions. The hydrodynamic limit ${\varepsilon} \to 0$ consists in assuming that the mean free path is negligible when compared to the typical physical scale length. We refer to \cite{Cercignani,Sone} for details on the kinetic description of gases.
That kinetic equation \eqref{eq:Kin-Intro} leads to \eqref{eq:NSFint} in the limit ${\varepsilon} \to 0^{+}$ is a well-understood fact that have been proven, for several type of solutions and various mode of convergence, in the case of the classical Boltzmann equation for which
\begin{equation}\label{eq:QQBoltz}
\mathcal Q(f,f)(v)=\mathcal Q_{\mathrm{Boltz}}(f,f)=\int_{\mathbb R^{d}\times\mathbb S^{d-1}}B(|v-v_{*}|,\sigma)\left[f(v')f(v_{*}')-f(v)f(v_{*})\right]\mathrm{d} v_{*}\mathrm{d}\sigma\end{equation}
where
$$v'=\frac{v+v_*}{2}+\frac{|v-v_*|}{2}\sigma, \qquad v_{*}'=\frac{v+v_*}{2}-\frac{|v-v_*|}{2}\sigma, \qquad \sigma \in \mathbb S^{d-1}$$
and the collision kernel $B(|v-v_{*}|,\sigma)$ is given by
$$B(|v-v_{*}|,\sigma)=|v-v_{*}|^{\gamma}\,b(\cos\theta), \qquad \cos \theta=\sigma \cdot \frac{v-v_{*}}{|v-v_{*}|}.$$
The method developed in the paper allows to consider \emph{all kinds of collision kernel} of physical interest, covering the cases of hard {and Maxwell} potentials $(\gamma \geqslant 0)$ with and without cut-off assumptions as well as that of moderately soft potentials {(without cut-off assumption) for which $b(\cos \theta) \approx \theta^{-(d-1) - 2 s}$ and $\gamma + 2 s \geqslant 0$}. We refer to Appendix \ref{sec:Landau-Boltz} for details.
Besides this Boltzmann model, our approach is also robust enough to treat in the same formalism the case of the Landau equation
\begin{equation*}
\begin{split}
\mathcal Q(f,f)&=\mathcal Q_{\mathrm{Landau}}(f,f)\\
&=\nabla_{v}\cdot \int_{\mathbb R^{d}} |v-v_{*}|^{\gamma+2} \, \Pi_{v-v_{*}}
\Big\{f(t,v_{*}) \nabla_{v} f(t,v) - f(t,v) {\nabla_{v_{*}} f}(t,v_{*}) \Big\}
\, \mathrm{d} v_{*}\end{split}\end{equation*}
where {$\gamma \geqslant -d$} and
$$\Pi_{z}=\mathrm{Id}-\frac{z \otimes z}{|z|^{2}}, \qquad z \in \mathbb R^{d} \setminus \{0\}$$
denotes the projection in the direction orthogonal to $z \in \mathbb R^{d},z\neq 0$. As before, our results cover the two cases of hard {or Maxwell $(\gamma \geqslant 0)$ and moderately soft potentials $(\gamma + 2 \geqslant 0)$}.
For both these models, the solutions to \eqref{eq:Kin-Intro} converges to a solution $f$ given by \eqref{eq:11} where
$$\mu(v)=(2\pi)^{-\frac{d}{2}}\exp\left(-\frac{|v|^{2}}{2}\right)$$
is a Maxwellian distribution with {unit mass, unit energy and mean zero velocity}, which is an equilibrium state of the collision operator $\mathcal Q$, i.e.
$$\mathcal Q(\mu,\mu)=0$$
whereas $\mathcal L$ is the linearized operator around that equilibrium, i.e.
\begin{equation}\label{eq:linearized}
\mathcal L f=\mathcal Q(\mu,f)+\mathcal Q(f,\mu)\end{equation}
for any suitable $f$ for which this makes sense.
In this paper, we introduce an abstract framework allowing to recover the above universal behaviour, as well as the well-posedness of \eqref{eq:Kin-Intro} in a perturbative framework. Even tough the Boltzmann and Landau equations are the two main models we have in mind as field of applications of our method, we wish again to point out that we are able to prove the convergence towards \eqref{eq:NSFint} for much more general models than those ones. In particular, we can handle general linear operator $\mathcal L$ and do not ask for the rest of the analysis that $\mathcal L$ and $\mathcal Q$ are related through \eqref{eq:linearized}.
The abstract framework developed in the paper is very general and robust and rely only on core assumptions about the linear part $\mathcal L$ and the quadratic part $\mathcal Q$. In particular, our approach can also be adapted to handle the case of the Boltzmann equation with \emph{relativistic velocities} and it is flexible enough to also encompass, at the price of some modifications, the case of quantum kinetic model (for which the collision operator is actually trilinear). Work is in progress in that direction in order to prove the strong convergence of solutions to the Boltzmann-Fermi-Dirac equation towards the above NSF system \eqref{eq:NSFint}, see \cite{GL2023}.
\subsection{Literature review}
As said, the derivation of hydrodynamic limits from linear and nonlinear equation is an important problem which received a lot of attention since the pioneering work of \cite{H1912} and \cite{E1917}. We do not review here the vast literature on the problem of diffusion approximation for transport processes, just referring to the classical references \cite{BLP1979,BSS} and the more recent contributions \cite{GW,BM} and the references therein.
For nonlinear collisional models, we refer the reader to {\cite{SR,golse}} for a more exhaustive description of the mathematically relevant results in the field regarding the Boltzmann equation. Depending on the limiting equation and the type of convergence one is interested with, there are mainly three different approaches for the derivation of hydrodynamical limit from the Boltzmann equation: a first approach consists in justifying rigorously suitable (truncated) asymptotic expansions of the solution to the kinetic equation around some hydrodynamic solution
\begin{equation*}\label{eq:C-E}
f_{{\varepsilon}}(t,x,v)=f_{0}(t,x,v)\left(1+\sum_{n}{\varepsilon}^{n}F_{n}(t,x,v)\right)\end{equation*}
where, typically $f_{0}(t,x,v)$ is a local Maxwelllian whose macroscopic fields are required to satisfy the limiting fluid model. With such an approach, the works \cite{caflisch} and \cite{demasi} obtained respectively the first rigorous justification of the compressible Euler limit up to the first singular time for the solution of the Euler system and a justification of the incompressible Navier-Stokes limit from Boltzmann equation. The work \cite{G2006} is another important reference on this line of research and we point out that, with such an approach, one is mainly interested with strong solutions for both the kinetic and fluid equations.
Regarding now \emph{weak solutions} at both the kinetic and fluid models, a very important program has been introduced in {\cite{BaGoLe1,BaGoLe2}} whose goal was to prove the convergence of the renormalized solutions to the Boltzmann equation towards weak solutions to the compressible Euler system or to the incompressible Navier-Stokes equations. This program has been continued exhaustively and the convergence have been obtained in several important results (see {\cite{golseSR,golseSR1,jiang-masm,lever,lions-masm1,lions-masm2}} to mention just a few).
The present contribution belongs to the third line of research which investigates \emph{strong solutions close to equilibrium} and exploits a careful spectral analysis of the linearized kinetic equation. Strong solutions to the Boltzmann equation close to equilibrium have been obtained in a weighted $L^{2}$-framework in the work \cite{U1974} and the \emph{local-in-time} convergence of these solutions towards solution to the compressible Euler equations have been derived in {\cite{nishida}}. For the limiting incompressible Navier-Stokes solution, a similar result have been carried out in {\cite{BU1991}} for smooth global solutions in $\mathbb R^{3}$ with a small {initial datum}. The recent work \cite{GT2020} recently removed this smallness assumption, allowing to treat also non global in time solutions to the Navier-Stokes equation. A recent extension to less restrictive integrability conditions has been obtained in \cite{G2023}. Our work is falling into this framework and is closer in spirit to the work \cite{GT2020} than to \cite{BU1991} since it fully exploits the Cauchy theory of the limiting NSF system. This line of research, complemented for instance with {\cite{B2015,BMM2019, CC2023}}, exploits a very careful description of the spectrum of the linearized Boltzmann equation derived in {\cite{EP1975}}. We notice that they are framed in the space $L^{2}(\mu^{-1})$ where the linearized Boltzmann operator is self-adjoint and coercive. The fact that the analysis of \cite{EP1975} has been extended recently in \cite{G2021} to larger functional spaces of the type $L^{2}_{v}(\langle \cdot\rangle^{q})$ opens the gate to some refinements of several of the aforementioned results. We also mention here the work {\cite{zhao}} which deals with an energy method in $L^{2}(\mu^{-1})$ spaces (see also {\cite{guo,guo2}} and~\cite{R2021}) in order to prove the \emph{strong convergence} of the solutions to the Boltzmann or Landau equation towards the incompressible Navier-Stokes equation without resorting to the work of {\cite{EP1975}}.
Besides the above lines of research and contributions which are dealing mainly with Boltzmann or Landau equation, we wish to point out that other kinetic and fluid models have been considered in the literature. Exhaustive list of contributions to the field is out of reach and we just mention some recent works spanning from high friction regimes for kinetic models of swarwing (see e.g. \cite{karper,figalli} for the Cucker-Smale model) to the reaction-diffusion limit for Fitzhugh-Nagumo kinetic equations \cite{crevat}. For fluid-kinetic systems, the literature is even more important, we mention simply here the works \cite{goudona,goudonb} dealing with light or fine particles regimes for the Vlasov-Navier-Stokes system and refer to \cite{daniel} for the more recent advances on the subject. We also mention the challenging study of gas of charged particles submitted to electro-magnetic forces (Vlasov-Maxwell-Boltzmann system) for which several incompressible fluid limits have been derived recently in the monograph \cite{arsenio}.
\subsection{Objectives of the paper}
The main scope of the paper is threefold:
\begin{enumerate}
\item[\textbf{(I)}] First, we provide a unified framework which allows to capture a large variety of quadratic models and explain the emergence of the universal NSF system \eqref{eq:NSFint} in the hydrodynamic limit. To do so, we provide a general though seemingly minimal set of Assumptions under which the NSF would emerge. Those are structural assumptions on the collision operator $\mathcal Q$ as well as the linear operator $\mathcal L$. They are related to physical properties of the kinetic equation: we assume in particular the rotational symmetry of $\mathcal L$ and $\mathcal Q$ due to the isotropy of the collision process as well as usual \emph{local conservation laws} related to mass, bulk velocity and energy.
\item[\textbf{(II)}] Second, within the abstract framework considered here, we aim to provide a very fine spectral analysis of the linearized operator $\mathcal L-v\cdot \nabla_{x}$ as well as a thorough description of the decay {and regularization} properties of the associated semigroup. As in previous contributions to the field, such an analysis is performed in a Fourier-based formalism under which the linearized operator of peculiar interest becomes
$$\mathcal L_{\xi}:=\mathcal L-i(\xi\cdot v)$$
where the transport term has been transformed in the more tractable multiplication operator by $i(v\cdot \xi)$ in Fourier variable (see Section \ref{sec:detail} for details). The advantage of working in this Fourier-based formalism is that it encompass the various scales of frequencies according to
$$|\xi| \simeq {\varepsilon}, \quad |\xi| \ll {\varepsilon} \quad \text{ or } {\varepsilon} \ll |\xi|$$
which let emerge the various (kinetic, hydrodynamic, dispersive) regimes of description at the linearized level. Under the structural assumptions on the linear part $\mathcal L$, we give a full description of the spectrum of $\mathcal L_{\xi}$, including the asymptotic expansion of both its leading eigenvalues and associated spectral projectors in the regime of small frequencies, $|\xi| \simeq 0$. Such a spectral description yields to result similar to those obtained in the seminal work \cite{EP1975} but we provide here a \textbf{\textit{completely new and more direct approach to this question}} in the unified and abstract framework. Our new approach is based upon a combination of Kato's perturbation theory \cite{K1995} and enlargment and factorization techniques from \cite{GMM2017}.
\item[\textbf{(III)}] Finally, we provide a \emph{strong convergence} result from solutions $f^{\varepsilon}$ to \eqref{eq:Kin-Intro} towards the solution $f$ given in \eqref{eq:11} associated to \eqref{eq:NSFint}. Moreover, the strong convergence result is in essence \emph{quantitative} since we carefully estimate the difference between the solution $f^{{\varepsilon}}$ and the solution $f=f_{\textnormal{NS}}$ by introducing a suitable splitting of $f^{{\varepsilon}}$ which, roughly speaking, can be given as
$$f^{{\varepsilon}}=f_{\textnormal{NS}} + h^{{\varepsilon}}_{\textnormal{err}}$$
where $h_{\textnormal{err}}^{{\varepsilon}}$ is an error term that that we aim to estimate as
$$\sup_{t\geqslant t_{*}}\|h_{\textnormal{err}}^{{\varepsilon}}(t)\| \leqslant \beta({\varepsilon}), \qquad \lim_{{\varepsilon}\to0}\beta({\varepsilon})=0$$
for any $t_{*} >0$ and some quantified error estimate $\beta({\varepsilon})$. Here above, the norm $\|\cdot\|$ is quite involved and takes into account several phenomena that produce different convergence rates (e.g, acoustic waves, dissipation of entropy). The restriction $t \geqslant t_{*}$ stems from the difficult task of estimating the initial layer and can be removed in the case of \emph{well-prepared} initial datum (see Theorem \ref{thm:hydrodynamic_limit}
for a precise statement and a complete description of the difference $f^{{\varepsilon}}-f_{\textnormal{NS}}$). \end{enumerate}
As a by-product of our third objective \textbf{(III)} here above, we show, for this variety of model, a close-to-equilibrium Cauchy theory for the kinetic equation \eqref{eq:Kin-Intro} for suitably small value of ${\varepsilon}$.
One of the main feature of our approach is that, inspired by the work \cite{GT2020}, our methodology is ‘‘top-down" from the limit equation to the kinetic equation rather than ‘‘bottom-up" as usually done. This means that, as far as possible, we adapt our approach to the existing Cauchy theory for the limiting system \eqref{eq:NSFint} and deduce the Cauchy theory for the kinetic equation \eqref{eq:Kin-Intro} by comparing it to the limiting equation \eqref{eq:NSFint} for small values of ${\varepsilon}.$ This is achieved through a suitable fixed-point argument involving fluctuations around the solution $f_{\textnormal{NS}}$. The fixed-point argument is based upon a simple use of Banach fixed point theorem or, for the more general case considered in the paper, by the convergence of a suitable scheme mimicking Picard iteration. Such an approach allows in particular to obtain well-posedness results \emph{without} any smallness assumption on the initial datum $f_{\textnormal{in}}$ but only under some smallness assumption on the scaling parameter ${\varepsilon}$ yielding several improvements of known results in the field.
Among the novelty of the paper, as just said, we adapt our approach to the existing Cauchy theory for the limiting system \eqref{eq:NSFint}. A lot of efforts in the present paper are given to adapt several tools used in the estimates of the Navier-Stokes system and, in particular, we resort to several Fourier analysis tools as developed in \cite{BCD2011} to treat nonlinear terms. We in particular adapt the paraproduct estimates described in \cite{BCD2011} to handle $x$-estimates of products of the form $\mathcal Q(f,g)$ (see Appendix \ref{scn:littlewood-paley} for more details). The case $d=2$ needs in particular a peculiar treatment for which we face several technical difficulties to handle nonlinear estimates.
Regarding the method used to achieve the above objectives, as in previous contributions to the field, we start by studying \eqref{eq:Kin-Intro} without its non-linear part and in Fourier variables:
\begin{equation}\label{eq:FoKin}
\partial_{t}\widehat{f}^{{\varepsilon}}(\xi,v,t)+\frac{i}{{\varepsilon}}(\xi \cdot v)\widehat{f}(\xi,v,t)=\frac{1}{{\varepsilon}^{2}}\mathcal L \widehat{f}^{{\varepsilon}}(\xi,v,t)
\end{equation}
where
$$\widehat{f}^{{\varepsilon}}(\xi,v,t)=\int_{\mathbb R^{d}}e^{-i \xi \cdot x}f^{{\varepsilon}}(x,v,t)\mathrm{d} x$$
is the Fourier transform with respect to the position variable $x\in \mathbb R^{d}$ and we exploited the fact that $\mathcal L$ is local in $x.$ In this framework, the linear operator of peculiar interest becomes
$$\mathcal L_{\xi}:=\mathcal L-i(\xi\cdot v)$$
where the transport term has been transformed in the more tractable multiplication operator by $i(v\cdot \xi)$ in Fourier variable. The idea of studying \eqref{eq:FoKin} originates from the seminal work \cite{EP1975} where a careful spectral analysis of the linearized Boltzmann operator was performed. It enforces somehow the study of both \eqref{eq:Kin-Intro} and \eqref{eq:NSFint} in $L^{2}_{x}$-functional spaces. Here, we push forward this idea and try to extract from it minimal assumptions and optimal estimates for $\mathcal L$ and $\mathcal Q$.
\color{black}
\subsection{Notations} In all the sequel, given a closed densely defined linear operator on a Banach space $Y$ of functions $f\::v \in \mathbb R^{d} \mapsto f(v)\in \mathbb C$,
$$L\::\:\mathscr{D}(L) \subset Y \to Y$$
we denote, for any $\xi \in \mathbb R^{d}$, the operator
$L_{\xi}\::\:\mathscr{D}(L_{\xi}) \subset Y \to Y$
by
$$\mathscr{D}(L_{\xi})=\{f \in \mathscr{D}(L)\;;\; v f \in Y\}\, \qquad L_{\xi}f=f-i(v\cdot \xi)f, \qquad f \in \mathscr{D}(L_{\xi}).$$
The spectrum of $L$ is denoted $\mathfrak{S}(L)$ (or $\mathfrak{S}_X(L)$ if it appears necessary to explicit the underlying Banach space) and, for $z \in \mathbb C \setminus \mathfrak{S}(L)$, the resolvent of $L$ at $z$ is denoted by
$$\mathcal R(z,L)=(z-L)^{-1} \in \mathscr B(Y)$$
where $\mathscr B(Y)$ is the space of all bounded linear operators on $Y$ (with its usual norm $\|\cdot\|_{\mathscr B(Y)}$).
We introduce, for any $a \in \mathbb R$ the right-half plane of the complex field $\mathbb C$ as
$$\Delta_{a}:=\left\{z \in \mathbb C\,;\;\mathrm{Re} z >a\right\}.$$
To handle now functions depending on the position variable $x\in \mathbb R^d$, we define the {inhomogeneous} Sobolev spaces of order $s \in \mathbb R$,
$$\mathbb{H}^{s}_{x}(\mathbb R^{d})=\left\{f \in L^{2}(\mathbb R^{d})\;;\;\|f\|_{\mathbb{H}_{x}^{s}}^{2}=\int_{\mathbb R^{d}}\langle \xi\rangle^{2s}|\widehat{f}(\xi)|^{2}\mathrm{d}\xi <\infty\right\}$$
and the homogeneous Sobolev space
$$\dot{\mathbb{H}}^{s}_{x}(\mathbb R^{d})=\left\{f \in \mathscr{S}'_x(\mathbb R^{d})\;;\;\|f\|_{\dot{\mathbb{H}}_{x}^{s}}^{2}=\int_{\mathbb R^{d}}|\xi|^{2s}|\widehat{f}(\xi)|^{2}\mathrm{d}\xi <\infty\right\}$$
where $\mathscr{S}'_x(\mathbb R^d)$ denotes the space of tempered distributions over $\mathbb R^d$. One can identify $\mathbb{H}^{s}_{x}(\mathbb R^{d})$ as the space of tempered distribution $f \in \mathscr{S}'_{x}(\mathbb R^{d})$ such that $\left(\mathrm{Id}-\Delta_{x}\right)^{s/2}f \in L^{2}_{x}(\mathbb R^{d})$ whereas $\dot{\mathbb{H}}_{x}^{s}(\mathbb R^{d})$ is the space of mappings $f \in \mathscr{S}'_{x}(\mathbb R^{d})$ such that $(-\Delta_{x})^{s/2}f = |\nabla_x |^s f \in L^{2}_{x}(\mathbb R^{d})$.
We also introduce the homogeneous Besov spaces for $p, q \in [1, \infty]$ and $s \in \mathbb R$
$$\dot{\mathbb{B}}^s_{p, q}\left( \mathbb R^d \right) = \left\{ f \in \mathscr{S}'_x \left(\mathbb R^d\right) \, ; \, \| f \|_{ \dot{\mathbb{B}}^s_{p, q} }^q = \sum_{n \in \mathbb Z} \left( 2^{ n s } \| \dot{\Delta}_{n} f \|_{ L^p_x } \right)^q < \infty\right \}$$
where the homogeneous dyadic projector $\dot{\Delta}_n$ from Littlewood-Paley theory is recalled in Appendix \ref{scn:littlewood-paley}.
For a Banach space $(Y_{v},\|\cdot\|{Y_{v}})$ of mappings depending on the variable $v$, the space
$\mathbb{H}^{s}_{x}\left(Y_{v}\right)$
denotes the space of functions $f\::\;(x,v) \mapsto f(x,v)$ such that $$\|f\|_{\mathbb{H}^{s}_{x}\left(Y_{v}\right)}=\left\|\,\|f(x,\cdot)\|_{Y_{v}}\,\right\|_{\mathbb{H}^{s}_{x}} < \infty.$$
Equivalently, one has
\begin{equation}\label{eq:normHsx}
\|f\|_{\mathbb{H}^s_x\left(Y_v\right)}^2=\int_{\mathbb R^d}\,\langle\xi\rangle^{2s}\,\left\|\widehat{f}(\xi)\right\|_{Y}^2\,\mathrm{d}\xi.
\end{equation} A similar definition applies to Besov spaces.
\subsection{Assumptions}
We work in a general setting of a perturbed kinetic equation of the form \eqref{eq:Kin-Intro} which, for ${\varepsilon}=1$, reads
\begin{gather*}
(\partial_t + v \cdot \nabla_x)f = \mathcal L f + \mathcal Q(f, f),
\end{gather*}
where $\mathcal L$ and $\mathcal Q$ are local in $x$, that is to say, they act on functions depending only on $v$. Their actions on functions $f=f(x, v)$ depending on both $x$ and $v$ are naturally defined as
$$[\mathcal L f](x, v) = \big[ \mathcal L f(x, \cdot) \big](v), \qquad \mathcal Q(f, f)(x, v) = \Big( \mathcal Q\big( f(x, \cdot), f(x, \cdot) \big) \Big)(v).$$
At the linear level, we make the following assumptions on the linearized operator $\mathcal L$ in the space
$$H=L^{2}\left(\mu^{-1}(v)\mathrm{d} v\right),$$
of functions depending only on the velocity variable where $\mu \, : \, \mathbb R^d \to [0,\infty)$ is some measurable weight function.
\begin{hypL}\label{AsL1} The linear operator $\mathcal L\::\mathscr{D}(\mathcal L) \subset H \to H$ satisfies the following.
\begin{enumerate}[label=\hypst{L\arabic*}]
\item \label{L1} The operator $\mathcal L$ is self-adjoint in $H$ and commutes with orthogonal matrices:
\begin{gather*}
\langle \mathcal L f, g \rangle_{H} = \langle f, \mathcal L g \rangle_{H} = \langle \mathcal L ({\Theta}f), \Theta g \rangle_H,
\end{gather*}
for any $f, g \in \mathscr{D}(\mathcal L)$ and orthogonal matrix $\Theta \in \mathscr M_{d \times d}(\mathbb R)$, where $[\Theta f](v):=f(\Theta v)$.
\item \label{L2} The weight function $\mu$ is nonnegative, normalized, radial, and such that:
\begin{gather*}
\mu=\mu(|v|) \geqslant 0, \qquad \int_{\mathbb R^d} \mu(v) \mathrm{d} v = 1,\\
E = \int_{\mathbb R^d} |v|^2 \mu(v) \mathrm{d} v < \infty, \qquad K = \frac{1}{E^2} \int_{\mathbb R^d} | v |^4 \mu(v) \mathrm{d} v < \infty.
\end{gather*}
\item \label{L3} The null-space of $\mathcal L$ is given by
\begin{equation*}
\nul\left( \mathcal L \right) =\mathrm{Span}\left\{ \mu, v_1 \mu, v_2 \mu, \dots, v_d \mu, |v|^2 \mu \right\}\end{equation*}
and there exists a Hilbert space $H^{\bullet}$ such that
\begin{equation*}
\mathscr{D}(\mathcal L) \subset H^{\bullet} \subset H, \qquad \| \cdot \|_{H} \leqslant \| \cdot \|_{H^{\bullet}},
\end{equation*}
and such that there holds
\begin{equation*}
\langle \mathcal L f, f \rangle_{H} \leqslant - \lambda_\mathcal L \| f \|^2_{H^{\bullet}} \qquad \text{ for any } f \in \mathscr{D}(\mathcal L) \cap \nul\left( \mathcal L \right)^\perp\,.
\end{equation*}
\item \label{L4} The operator $\mathcal L$ can be decomposed as
$$\mathcal L = \mathcal B+ \mathcal A, \qquad \mathscr{D}(\mathcal B)=\mathscr{D}(\mathcal L), \qquad \mathcal A \in \mathscr B(H),$$
where the splitting is compatible with a hierarchy of Hilbert spaces $(H_j)_{j=0}^{2}$ such that
\begin{enumerate}
\item \label{assumption_hierarchy} the spaces $H_j$ continuously and densely embed into one another:
$$H_{2} \hookrightarrow H_1 \hookrightarrow H_0 = H,$$
\item \label{assumption_multi-v} the multiplication by $v$ is bounded from $H_{j+1}$ to $H_{j}$, i.e.
\begin{equation*}
\|v f\|_{H_{j}} \lesssim \|f\|_{H_{j+1}} \qquad f \in H_{j+1}, \quad j=0,1,
\end{equation*}
\item \label{assumption_bounded_A} the operator $\mathcal A : H_j \rightarrow H_{j+1}$ is bounded:
$$\mathcal A \in \mathscr B(H_j,H_{j+1}), \qquad j=0,1,$$
\item \label{assumption_dissipative} the part $\mathcal B_{\xi}$ is hypo-dissipative on each space $H_j$ uniformly in $\xi \in \mathbb R^d$, that is to say there exists $\lambda_\mathcal B \geqslant \lambda_{\mathcal L}$ such that, for $j=0,1,2$
$$\mathfrak{S}_{H_{j}}(\mathcal B_{\xi}) \cap \Delta_{-\lambda_\mathcal B} = \varnothing,$$
and
$$\sup_{\xi\in \mathbb R^{d}}
\left\| \mathcal R( z,\mathcal B_{\xi})\right\|_{\mathscr B(H_j)} \lesssim | \mathrm{Re}\, z + \lambda_{\mathcal B}|^{-1}, \qquad \forall z \in \Delta_{-\lambda_{\mathcal B}}.$$
\end{enumerate}
\end{enumerate}
\end{hypL}
\begin{rem}
Note that $K > 1$ by a simple use of Jensen's inequality applied to the probability measure $\mu(v)\mathrm{d} v$. Moreover, according to \ref{L1}, Assumption \ref{L3} can be formulated as follows:
\begin{equation*}
\forall f \in \mathscr{D}(\mathcal L), \quad \langle \mathcal L f, \mu \rangle_{H} = \langle \mathcal L f, v \mu \rangle_{H} = \left\langle \mathcal L f, |v|^2 \mu \right\rangle_{H} = 0,
\end{equation*}
that is to say $\nul(\mathcal L)^{\perp}=\range(\mathcal L)$, and the operator $\mathcal L$ has a spectral gap in $H$:
\begin{equation*}
\mathfrak{S}_{H}(\mathcal L) \cap \Delta_{-\lambda_\mathcal L} = \{0\}\,.
\end{equation*}
\end{rem}
\begin{rem}\label{rem:KernelH2} Notice also that $\nul(\mathcal L) \subset H_2$. Indeed, given $f \in \nul(\mathcal L)$, with the spitting given in \ref{L4}, $\mathcal L f=0$ implies
$$f=\mathcal R(0,\mathcal B)\mathcal A f$$
and thanks to \ref{assumption_bounded_A}, $\mathcal A f \in H_1$ which, with now \ref{assumption_dissipative}, yields $\mathcal R(0,\mathcal B)\mathcal A f\in H_1.$ Thus $f \in H_1$ and one can repeat the argument to deduce that $f \in H_2$.
\end{rem}
\begin{rem}\label{rem:L1B1} We show in Appendix \ref{sec:Landau-Boltz} that the various Assumptions \ref{L1}--\ref{L4} hold for several models of physical interest, expliciting for each of those models the precise definition of the various spaces $H^{\bullet}$ and $H_{j}$ as well as the splitting $\mathcal L=\mathcal A+\mathcal B$. {Typically, Assumptions \ref{L1}--\ref{L4} apply to the Boltzmann equation with hard potentials with or without Grad's cut-off assumptions or to Landau equation \emph{in spaces with Gaussian weights}.} To clarify right away the role of this set of Assumptions in our analysis, we illustrate here the form of the spaces $H^{\bullet},H_{j}$ in the case of Boltzmann equation with hard-spheres interactions. This corresponds to \eqref{eq:QQBoltz} with the choice
$$B(|v-v_{*}|,\sigma)=|v-v_{*}|, \qquad v,v_{*} \in \mathbb R^{d}\times\mathbb R^{d}, \quad \sigma \in \mathbb S^{d-1}.$$
In such a case, as said, $\mu$ is a Maxwellian distribution:
$$\mu(v) := (2 \pi)^{-d/2} \exp\left( - \frac{|v|^2}{2} \right), \qquad E = d, \quad K = 1 + \frac{2}{d} \, ,$$
and the usual linearized operator given by \eqref{eq:linearized}
is known to satisfy \ref{L1}--\ref{L2} with $\mathscr{D}(\mathcal L)=L^2( \langle v\rangle^2\mu^{-1}(v)\mathrm{d} v)$. Moreover, assumption \ref{L3} is met with the choice
$$H^{\bullet}=L^2\left( \langle v\rangle \mu^{-1}(v)\mathrm{d} v\right)$$
Regarding assumption \ref{L4}, one can chose the hierarchy of spaces $H_{j}$ as
$$H_j := L^2\left( \langle v \rangle^{2 j} \mu^{-1}(v) \mathrm{d} v \right),$$
for $j=0,1,2$. The splitting is taken to be Grad's splitting:
$$(\mathcal B f)(v) = - f(v) \int_{\mathbb R^d} |v-v_*| \mu(v_*) \mathrm{d} v_*, \qquad \mathcal A f = \mathcal L - \mathcal B.$$
Details are given in Appendix \ref{sec:Landau-Boltz}. We point out that, in full generality, $H^{\bullet}$ maybe much more complicated than the above one and this is what motivated the introduction of the abstract framework \ref{L1}--\ref{L4}.\end{rem}
\begin{defi}
\label{defi:Hhstar}
Under Assumption \ref{L2}, we define the ``dual'' space $H^{\circ}$ of the dissipation Hilbert space $H^{\bullet}$ as the completion of~$H$ for the norm
$$\| f \|_{H^{\circ}} := \sup_{ \| \varphi \|_{H^{\bullet}} \leqslant 1 } \langle f, \varphi \rangle_{H}.$$
\end{defi}
\begin{rem}
\label{rem:rangHh}
Since $\|\cdot\|_{H} \leqslant \|\cdot\|_{H^{\bullet}}$, for any $f \inH$ one has from Cauchy-Schwarz inequality
$$\|f\|_{H^{\circ}} \leqslant \sup_{\|\varphi\|_{H^{\bullet}}\leqslant 1} \|f\|_{H}\|\varphi\|_{H} \leqslant \|f\|_{H}$$
we thus have the following comparison:
$$H^{\bullet} \hookrightarrow H \hookrightarrow H^{\circ}, \qquad \|\cdot\|_{H^{\circ}} \leqslant \|\cdot\|_{H} \leqslant \|\cdot\|_{H^{\bullet}}.$$
\end{rem}
At the nonlinear level, we make the following assumptions on $\mathcal Q$.
\begin{hypQ}\label{AsB1} The nonlinear operator $\mathcal Q$ is satisfying the following assumptions:
\begin{enumerate}[label=\hypst{B\arabic*}]
\item \label{Bortho} The bilinear operator is $H$-orthogonal to the null-space of $\mathcal L$:
$$\langle \mathcal Q(f, g), \mu \rangle_{H} = \langle \mathcal Q(f, g), v \mu \rangle_{H} = \langle \mathcal Q(f, g), |v|^2 \mu \rangle_{H} = 0,$$
or, equivalently, in terms of integrals:
$$\int_{\mathbb R^d} \mathcal Q(f, g)(v) \mathrm{d} v = \int_{\mathbb R^d} v \mathcal Q(f, g)(v) \mathrm{d} v = \int_{\mathbb R^d} |v|^2 \mathcal Q(f, g)(v) \mathrm{d} v = 0.$$
\item \label{Bisotrop} The bilinear operator commutes with orthogonal matrices:
$$\langle \mathcal Q(f, g), h \rangle_{H} = \langle \mathcal Q\left( \Theta f, \Theta g \right), \Theta h \rangle_{H},$$
for any orthogonal matrix $\Theta \in \mathscr M_{d \times d}(\mathbb R).$
\item \label{Bbound} The bilinear operator satisfies the following dual estimate
$$\| \mathcal Q(f, g) \|_{H^{\circ}} \lesssim \| f \|_{H} \| g \|_{H^{\bullet}} + \| f \|_{H^{\bullet}} \| g \|_{H}\,,$$
or, in other words, there holds
$$\left\langle \mathcal Q(f, g) , h \right\rangle_{H} \lesssim \| h \|_{H^{\bullet}} \left( \| f \|_{H} \| g \|_{H^{\bullet}} + \| f \|_{H^{\bullet}} \| g \|_{H} \right)\,.$$
\end{enumerate}
\end{hypQ}
\subsection{Main results -- first version} Under the above structural assumptions \ref{L1}--\ref{L4}, the full description of the spectrum of $\mathcal L_{\xi}$ and the decay {and regularization properties} of the associated semigroup are made explicit in the following.
\begin{theo}[\textit{\textbf{Main spectral theorem}}]
\label{thm:spectral_study}
Assume \ref{L1}--\ref{L4}, there exist {explicitly computable constants} $C, \alpha_{0}, \lambda, \gamma, \sigma_0 > 0$ such that the following spectral and dynamical properties hold.\\
\noindent \textit{\textbf{(1)}} \textit{\textbf{Localization of the spectrum.}}
The spectrum of $\mathcal L_\xi$ is localized as follows.
\begin{itemize}
\item If $|\xi| \geqslant \alpha_{0}$, the spectrum is at a positive distance from $\{ \mathrm{Re}\, z \geqslant 0 \}$:
$$ \mathfrak{S}_{H}\left(\mathcal L_{\xi}\right) \cap \Delta_{-\gamma} = \varnothing.$$
\item If $|\xi| \leqslant \alpha_{0}$, the spectrum is at a positive distance from $\{ \mathrm{Re}\, z \geqslant 0 \}$, except for a finite number of small eigenvalues:
$$\mathfrak{S}_{H}(\mathcal L_{\xi}) \cap \Delta_{-\lambda} = \big\{ \lambda_\textnormal{inc}(\xi), \, \lambda_\textnormal{Bou}(\xi), \, \lambda_{- \textnormal{wave}}(\xi), \,\lambda_{+\textnormal{wave}}(\xi) \big\},$$
and these eigenvalues $\lambda_\star(\xi)$ expand for $\xi \to 0$ as
\begin{subequations}
\label{eq:lambda_star}
\begin{gather}
\lambda_{\pm \textnormal{wave}}(\xi) = \pm i c | \xi | - \kappa_\textnormal{wave} | \xi |^2 + \mathcal O\left( | \xi |^3 \right),\\
\lambda_{\star}(\xi) = - \kappa_\star | \xi |^2 + \mathcal O\left( | \xi |^3 \right), \quad \star = \textnormal{Bou}, \textnormal{inc},
\end{gather}
\end{subequations}
where the speed of sound is defined as
\begin{equation}
\label{eq:speed_sound}
c:=\sqrt{\frac{KE}{d}},
\end{equation}
and the diffusion coefficients {$\kappa_\star \in (0, \infty)$} are given by
\begin{equation}\label{eq:kappa}
\begin{split}
\kappa_\textnormal{inc} := - \dfrac{1}{(d-1)(d+1) } &\left \langle \mathcal L^{-1} {\mathbf{A}} ,{\mathbf{A}} \right \rangle_{H},
\qquad \kappa_\textnormal{Bou} := - \dfrac{1}{d} \left \langle \mathcal L^{-1} {\mathbf{B}} , {\mathbf{B}} \right \rangle_{H},\\
\kappa_\textnormal{wave} &:= \frac{d-1}{2 d} \kappa_\textnormal{inc} + \frac{E^2(K-1)}{2} \kappa_\textnormal{Bou},
\end{split}
\end{equation}
where the Burnett functions ${\mathbf{A}} $ and ${\mathbf{B}} $ are defined as
\begin{equation}\label{eq:burnett}\begin{split}\begin{cases}
{\mathbf{A}} (v) & := \displaystyle \sqrt{ \frac{d}{E} } \left( v \otimes v - \frac{|v|^2}{d} \mathrm{Id} \right) \mu(v),\\
\\
{\mathbf{B}} (v) & := \displaystyle \frac{1}{\sqrt{K (K-1)}} v \left( K - \frac{|v|^2}{E} \right) \mu(v).
\end{cases}\end{split}\end{equation}
\end{itemize}
\noindent \textit{\textbf{(2)}} \textit{\textbf{Asymptotic behavior of the spectral projectors.}}
For any non-zero $| \xi | \leqslant \alpha_0$, the spectral projectors associated with these eigenvalues expand in $\mathscr B\left( H^{\circ} ; H^{\bullet} \right)$ as
\begin{equation}
\label{eq:PPstar}
\mathsf P_\star(\xi) = \mathsf P^{(0)}_\star\left( \frac{\xi}{|\xi|} \right) + i \xi \cdot \mathsf P^{(1)}_\star\left( \frac{\xi}{|\xi|} \right) + S_{\star}(\xi), \qquad \star=\textnormal{inc},\textnormal{Bou},\pm\textnormal{wave},
\end{equation}
where $S_{\star}(\xi) \in \mathscr B(H^{\circ};H^{\bullet})$ with $\|S_{\star}(\xi)\|_{\mathscr B(H^{\circ};H^{\bullet})} \lesssim | \xi |^2$.
The zeroth order coefficients are defined for any $\omega \in \mathbb S^{d-1}$ as
\begin{gather*}
\mathsf P_\textnormal{inc}^{(0)}\left( \omega \right) f(v)=\dfrac{d}{E} \big( \Pi_\omega \langle f, v \mu \rangle_{H}\big) \cdot v \mu(v),\\
\mathsf P_\star^{(0)}(\omega)f = \langle f, \psi_\star(\omega) \rangle_{H} \, \psi_\star(\omega), \qquad \star = \textnormal{Bou}, \pm \textnormal{wave},
\end{gather*}
where we denoted $\Pi_\omega=\mathrm{Id} - \omega \otimes \omega$ the orthogonal projection onto $( \mathbb R \omega )^\perp$, and the first order terms write explicitly for any $f \in \ker(\mathcal L)^\perp$ as
\begin{gather*}
\mathsf P^{(1)}_\textnormal{inc}(\omega) f(v) = \sqrt{\frac{d}{E}} \left\langle f , \mathcal L^{-1}{\mathbf{A}} \right\rangle_{H} \Pi_\omega v \mu,\\
\mathsf P^{(1)}_{\pm \textnormal{wave}}(\omega) f = \left(\pm \frac{1}{\sqrt{2} } \left\langle f , \mathcal L^{-1}{\mathbf{A}} \omega \right\rangle_{H} + E \sqrt{\frac{K-1}{2}} \left\langle f, \mathcal L^{-1} {\mathbf{B}} \right\rangle_{H} \right) \psi_{\pm \textnormal{wave}}(\omega),
\end{gather*}
and
\begin{equation}
\label{eq:P_1_Bou}
\mathsf P^{(1)}_{\textnormal{Bou}}(\omega) f = \langle f, \mathcal L^{-1} {\mathbf{B}} \rangle_{H} \psi_\textnormal{Bou},
\end{equation}
{where the zeroth order eigenfunctions $\psi_{\pm \textnormal{wave}}$ and $\psi_\textnormal{Bou}$ are defined as}
\begin{gather}
\label{eq:def_psi_wave}
\psi_{\pm \textnormal{wave}}(\omega, v) := \frac{1}{\sqrt{2 K}} \left( 1 \pm \sqrt{\frac{d K}{E}} \omega \cdot v + \frac{1}{E}\left(|v|^2 - E\right) \right) \mu(v),\\
\label{eq:def_psi_Bou}
\psi_{\textnormal{Bou}}(v) := \frac{1}{\sqrt{K (K - 1) } } \left(K - \frac{|v|^2}{E} \right) \mu(v).
\end{gather}
\color{black}
Notice, in particular, that
\begin{equation}
\label{eq:alternative_P_1_inc}
\omega \cdot \mathsf P^{(1)}_\textnormal{inc}(\omega) f = \sqrt{\frac{d}{E}} \Big( \Pi_\omega \left\langle f , \mathcal L^{-1}{\mathbf{A}} \right\rangle_{H} \omega \Big) \cdot v \mu.
\end{equation}
\noindent \textit{\textbf{(3)}} \textit{\textbf{Resolvent bounds and decay estimates.}} Setting
\begin{equation}\label{eq:splitPP}
\mathsf P(\xi)= \mathbf{1}_{|\xi| \leqslant \alpha_0} \bigg(\mathsf P_{\textnormal{Bou}}(\xi)+\mathsf P_{\textnormal{inc}}(\xi)+\mathsf P_{+\textnormal{wave}}(\xi)+\mathsf P_{-\textnormal{wave}}(\xi)\bigg)\end{equation}
the spectral projector associated to the part of the spectrum from point \textbf{(1)}, the following resolvent bound holds
\begin{equation}
\label{eq:ResolI-PP}
\sup_{z \in \Delta_{-\sigma_{0}}}\bigg\|\mathcal R(z,\mathcal L_{\xi})\left(\mathrm{Id}-\mathsf P(\xi)\right)\bigg\|_{\mathscr B(H)} \leqslant C, \qquad \forall \xi\in \mathbb R^d ,
\end{equation}
where $\sigma_0 := \min\{\lambda, \gamma\}$. Finally, the $C^{0}$-semigroup $\left(U_{\xi}(t)\right)_{t\geq0}$ generated by $(\mathcal L_{\xi},\mathscr{D}(\mathcal L_{\xi}))$ satisfies for any $\sigma \in (0, \sigma_0)$, any $\xi \in \mathbb R^{d}$ and any $f \in H$
\begin{subequations}\label{decay-semigroup}
\begin{equation}
\label{eq:decay}
\begin{split} \sup_{t \geqslant 0} \, & e^{2 \sigma_0 t} \left\| U_{\xi}(t)\ {\left( \mathrm{Id} - \mathsf P(\xi) \right)} f \right\|^2_{H} \\
&+ \int_0^\infty e^{2 \sigma t} \left\| U_{\xi}(t) \left( \mathrm{Id} - \mathsf P(\xi) \right)f \right\|_{H^{\bullet}}^2 \, \mathrm{d} t \leqslant C_\sigma \| {\left( \mathrm{Id} - \mathsf P(\xi) \right)}f \|_{H}^2,
\end{split}\end{equation}
whereas, for any $f \in H^{\circ}$,
\begin{equation}
\label{eq:decay_Hh'}
\int_0^\infty e^{2 \sigma t} \left\| U_{\xi}(t) {\left( \mathrm{Id} - \mathsf P(\xi) \right)} f \right\|_{H}^2 \, \mathrm{d} t \leqslant C_\sigma \| {\left( \mathrm{Id} - \mathsf P(\xi) \right)}f \|_{H^{\circ}}^2\,.
\end{equation}
\end{subequations}
\end{theo}
\begin{figure}
\caption{Localization of the spectrum of $\mathcal L - i (v \cdot \xi) $ for $| \xi | \leqslant \alpha_0 $ and for~$| \xi | \geqslant \alpha_0$ }
\end{figure}
\begin{rem}
Recall from Remark \ref{rem:KernelH2} that \ref{L4} implies that $|\cdot|^2\mu \in H_2$. Using then \ref{assumption_multi-v} twice, we deduce that the mapping $| \cdot |^4\mu$ belongs to $H$. Thus,
$$\int_{\mathbb R^d}|v|^8\mu(v)\mathrm{d} v < \infty.$$
Consequently, ${\mathbf{A}} , {\mathbf{B}} \in H$ and $\mathcal L^{-1}{\mathbf{A}} ,\mathcal L^{-1}{\mathbf{B}} \in H$, and thus
$$\kappa_\star < \infty, \qquad \star=\textnormal{Bou},\textnormal{inc},\textnormal{wave}.$$
We point out that, in a sense, we only assume (almost) enough integrability for the diffusion coefficients $\kappa_\star$ to be finite. This is to be contrasted with the work \cite{Mellet} in which they prove that if $\kappa_\star = \infty$, then, under some appropriate scaling, one observes fractional diffusion in the limit ${\varepsilon} \to 0$. In this framework, a corresponding version of Theorem \ref{thm:spectral_study} was proved in \cite{Puel}. We also refer to \cite{BM} for a unified spectral approach to the (fractional) diffusion limit for a large variety of linear collisional kinetic equations with a single conservation law. Finally, we point out that contrary to previous proofs of Theorem \ref{thm:spectral_study} for specific models, we do not assume that the weight $\mu$ decays like a gaussian.
\end{rem}
\begin{rem}
Notice that $\mathcal R(z,\mathcal L_{\xi})\left(\mathrm{Id}-\mathsf P(\xi)\right)=\mathcal R\Big(z,\mathcal L_{\xi}(\mathrm{Id}-\mathsf P(\xi))\Big)$ and, by virtue of \eqref{eq:PPstar} the above resolvent bound \eqref{eq:ResolI-PP} can be rewritten as
$$\sup_{z \in \Delta_{-\lambda}}\bigg\|\mathcal R(z,\mathcal L_{\xi})-\sum_{\star=\textnormal{inc},\pm\textnormal{wave},\textnormal{Bou}}\left(z-\lambda_{\star}(\xi)\right)^{-1}\mathsf P_{\star}(\xi)\bigg\|_{\mathscr B(H)} \leqslant C.$$
Note also that as $\mathcal L$ is self adjoint in {$H$}, the dual semigroup
$$\Big(U_{\xi}(t) \left(\mathrm{Id} - \mathsf P(\xi)\right) \Big)^\star = U_{-\xi}(t) \left(\mathrm{Id} - \mathsf P(-\xi)\right) \,,\, \qquad t \geqslant 0,$$ automatically satisfies the same estimates \eqref{decay-semigroup}.
\end{rem}
\begin{rem}
\label{rem:macro_representation_spectral}
The zeroth order terms in the expansions of the projectors are macroscopic in the sense that
$$\mathsf P \mathsf P^{(0)}_\star = \mathsf P^{(0)}_\star \mathsf P = \mathsf P^{(0)}_\star, \quad \star = \textnormal{inc}, \textnormal{Bou}, \pm \textnormal{wave}.$$
As a consequence, they can be characterized in terms of the macroscopic components $\varrho[\cdot], u[\cdot]$ and $\theta[\cdot]$ where
\begin{equation}
\label{eq:fluctuat}
\begin{cases}
\varrho_f &= \varrho[f] := \langle f, \mu \rangle_H, \\
\\
u_f & = u[f] := \dfrac{d}{E} \langle f, v \mu \rangle_H ,\\
\\
\theta_f &= \theta[f] := \dfrac{1}{E} \left\langle f , (|v|^2 - E) \mu \right\rangle_H
\end{cases}
\end{equation}
for any $f\in H.$ We refer to Proposition \ref{prop:macro_representation_spectral} for a precise statement.
\end{rem}
For the Boltzmann equation with hard potential interaction, the above {theorem} has been proven first in the seminal work \cite{EP1975} whose method has been {adapted} subsequently to encompass much more general models in the recent work \cite{YY} (see also \cite{LY2016, LY2017,YY23}). The method in these contributions is based upon some compactness argument and a study of the eigenvalue problem through the use of the Implicit Function Theorem.
The approach we perform in the present paper appears much more direct and simpler. Any explicit computation relies solely on the isotropy of the operator $\mathcal L$. To be more precise, we adapt here the perturbation theory of eigenvalues introduced in \cite{K1995} and exploit the structural assumption \ref{L4} to prove the regularity and expansion of the eigen-projectors. Notice that, except for some peculiar cases (including the Boltzmann equation for hard-spheres interactions), our perturbative approach does not directly fall into the realm of the classical perturbation theory of unbounded operators developed in \cite{K1995} since the multiplication operator $i(v\cdot\xi)$ is \emph{not} $\mathcal L$-bounded in general. This induces some technical complications and requires to adapt the method of \cite{K1995} to the general situation we are dealing with here. This is done, borrowing and pushing further some ideas of \cite{T2016}, by fully exploiting the splitting of $\mathcal L$ as
$$\mathcal L=\mathcal A+\mathcal B$$
where $\mathcal B$ enjoys dissipative properties whereas $\mathcal A$ is a regularizing operator which compensates the unboundedness of the multiplication by $v$ (see \ref{L4}). Moreover, in contrast with existing results based upon \cite{EP1975}, our method takes into account the role of the dissipation space $H^{\bullet}$ and its dual $H^{\circ}$. This allows us to emphasise and exploit regularizing effects of $\mathcal L$ in the scale of spaces $H^{\bullet} \hookrightarrow H \hookrightarrow H^{\circ}$. A more detailed description of the approach we follow will be given in Section \ref{sec:newspec}. We point out already that we strongly use here the fact that all functional spaces considered here are Hilbert spaces: this allows to use a suitable ``diagonalization'' of the transport operator thanks to Fourier transform and also permits to deduce spectral properties of the semigroup $\left(U_{\xi}(t)\right)_{t\ge0}$ through some of its generator $\mathcal L_{\xi}$ thanks to Gearhart-Pruss theorem.
We strongly believe that the new method we propose here to the fine spectral analysis of kinetic models is robust enough to be adapted to various contexts and can become a valid alternative to the technical approach of \cite{EP1975}. In our opinion, it replaces in an efficient way the compactness arguments introduced in \cite{EP1975} for the localization of the spectrum by a much modern and quantitative approach, combining enlargement techniques from \cite{GMM2017} to describe small frequencies $\xi\simeq 0$ with hypocoercivity methods from \cite{D2011} for frequencies $|\xi| \gtrsim 1.$ Moreover, since it is based on the isotropic nature of $\mathcal L$ and $\mathcal Q$, it can be directly adapted to more general equations of the type
$$\partial_t f+ \mathfrak{a}(|v|) v \cdot \nabla_x f=\mathcal L f+ \mathcal Q(f,f)$$
where $\mathfrak{a}(\cdot)$ is a suitable smooth radial mapping and
$$\nul \mathcal L=\mathrm{Span}\left\{\mu,v_1\mu,\ldots,v_d\mu,\mathfrak{b}(|v|)\mu\right\}$$
for a suitable radial mapping $\mathfrak{b}(\cdot)$ such that
$$\int_{\mathbb R^d}\mathcal Q(f,f)\mathfrak{b}(|v|)\mathrm{d} v=0.$$
The relativistic Boltzmann and Landau equations both fall within the above framework with
$$\mathfrak{a}(|v|)=\frac{1}{\mathfrak{b}(|v|)}, \qquad \mathfrak{b}(|v|)=\sqrt{1+c_0^{-2}|v|^2}, \qquad \mu(v)=Z^{-1}e^{-\mathfrak{b}(|v|)}, \qquad c_0 > 0,$$
where $Z >0$ is a normalization constant so that the Juttner distribution $\mu$ satisfies \ref{L1}. Our structural assumptions \ref{L1}--\ref{L4} can then easily be modified to cover such a case. For instance, Assumption \ref{assumption_multi-v} should read now
$$\left\|\mathfrak{a}(|v|)vf\right\|_{H_j} \lesssim \|f\|_{H_{j+1}}.$$
We point out that several moments of the Juttner distribution $\mu$ involving powers of $\mathfrak{a}(|v|)v$ and $\mathfrak{b}(|v|)$ would have to be considered in Assumption \ref{L2}. In particular the expressions of $\psi_\star$ and $\kappa_\star$ would be much more intricate. Thus, we do not pursue further this line of research since the present contribution is already quite technical and lengthy.
Besides the thorough description of the spectrum of $\mathcal L_\xi$ and the relevant eigen-projectors, Theorem \ref{thm:spectral_study} also describe the long-time behaviour of the associated linearized semigroup $\left(U_\xi(t)\right)_{t\ge0}$. Our approach uses, as said enlargement techniques from \cite{GMM2017} as well as an abstract hypocoercivity result from \cite{D2011}. The decay of the linearized semigroup $\left(U_{\xi}(t)\right)_{t\geq0}$ in \eqref{decay-semigroup} is one of the fundamental brick on which it is possible to build the Cauchy theory for \eqref{eq:Kin-Intro} whereas a comparison of $\left(U_\xi(t)\right)_{t \geq0}$ with the linearized semigroup associated to \eqref{eq:NSFint} is the main tool for the study of the hydrodynamic limit. This yields, in the Hilbert space setting $\mathbb{H}^s_x(H_v)$
to our main result as far as the above objective \textbf{(III)} is concerned:
\begin{theo}[\textit{\textbf{Hydrodynamic limit theorem}}]
\label{thm:hydrodynamic_limit}
Let $s > \frac{d}{2}$ be given as well as some initial data
$$f_\textnormal{in} \in\mathbb{H}^s_x \left( H_v \right),$$
satisfying additionally, if $d=2$, $f_\textnormal{in} \in \dot{\mathbb{H}}^{-\alpha}_x \left( H_v \right)$ for some $0 < \alpha < \frac{1}{2}.$
Consider the solution of the Navier-Stokes-Fourier system (see Theorem \ref{thm:spectral_study} and Proposition \ref{prop:equivalence_kinetic_hydrodynamic_INSF} for the definitions of the coefficients, {and Theorem \ref{thm:cauchy_NSF} for the existence of this solution})
\begin{equation}\label{eq:main-NSF}
\begin{cases}
\partial_{t}u-\kappa_{\textnormal{inc}}\,\Delta_{x}u + \vartheta_{\textnormal{inc}}\,u\cdot \nabla_{x}\,u= \nabla_{x}p\,,\\[6pt]
\partial_{t}\,\theta-\kappa_{\textnormal{Bou}}\,\Delta_{x}\theta +\vartheta_{\textnormal{Bou}}\,u\cdot \nabla_{x}\theta=0,\\[8pt]
\nabla_{x}\cdot u=0\,, \qquad \nabla_{x}\left(\varrho + \theta\right)= 0\,,
\end{cases}
\end{equation}
spanned by the initial conditions
$$u(0, x) = \mathbb{P} u[f_\textnormal{in}](x), \qquad \theta(0, x) = \frac{1}{K(K-1)}\left( ( K-1 ) \varrho[f_\textnormal{in}](x) - \theta[f_\textnormal{in}](x) \right),$$
and which satisfies for some $T \in (0, \infty]$
\begin{gather*}
(\varrho, u, \theta) \in \mathcal C_b\left( [0, T) ;\mathbb{H}^s_x \right), \qquad (\nabla_x \varrho, \nabla_x u, \nabla_x \theta) \in L^2\left( [0, T) ;\mathbb{H}^s_x \right).
\end{gather*}
Introducing, for any $(x,v) \in \mathbb R^{d}\times\mathbb R^{d}$ and $t \in [0,T)$,
$$ f_\textnormal{NS}(t, x, v) = \left(\varrho(t, x) + u(t, x) \cdot v + \frac{\theta(t, x)}{E(K-1)} \left( |v|^2 - E \right)\right) \mu(v),$$
the following holds.
\begin{enumerate}
\item[\textit{\textbf{(1)}}]\textit{\textbf{Existence of a unique solution.}}
There exists some small $c_0 > 0$ and ${\varepsilon}_0 > 0$ such that the equation
$$\partial_t f^{\varepsilon} = \frac{1}{{\varepsilon}^2} \left( \mathcal L - {\varepsilon} v \cdot \nabla_x \right) f^{\varepsilon} + \frac{1}{{\varepsilon}} \mathcal Q\left( f^{\varepsilon}, f^{\varepsilon} \right), \quad f^{\varepsilon}(0, x, v) = f_\textnormal{in}(x, v)$$
admits for any ${\varepsilon} \in (0, {\varepsilon}_0]$ a unique solution
$$f^{\varepsilon} \in L^2_{\rm{loc}} \left( [0, T) ; \mathbb{H}^s_x \left( H^{\bullet}_v\right) \right) \cap \mathcal C\left([0,T);\mathbb{H}^s_x(H_v)\right)$$
such that
$$\sup_{0 \leqslant t < T} \| f^{\varepsilon}(t) \|_{ \mathbb{H}^s_x \left( H_v \right) } \leqslant \frac{c_0}{{\varepsilon}},$$
which satisfies furthermore the following uniform estimate:
$$\sup_{0 \leqslant t < T} \| f^{\varepsilon}(t) \|_{ \mathbb{H}^s_x \left( H_v \right) }^2 + \int_0^T \| | \nabla_x |^{ 1-\alpha} f^{\varepsilon}(t) \|_{ \mathbb{H}^{s - (1-\alpha) }_x \left( H^{\bullet}_v \right) }^2 \mathrm{d} t \, \lesssim 1,$$
where we recall that $0 < \alpha < \frac{1}{2}$ if $d=2$ and $\alpha=0$ if $d\geq3.$
\item[\textit{\textbf{(2)}}]\textit{\textbf{Decomposition and convergence of the solution.}}
The solution $f^{\varepsilon}$ splits as the sum of some limiting part $f_\textnormal{NS}$, some initial layers $(f^{\varepsilon}_\textnormal{disp}, f^{\varepsilon}_\textnormal{kin})$, and a vanishing part~$f^{\varepsilon}_\textnormal{err}$:
\begin{equation}\label{eq:decomp}
f^{\varepsilon} = f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp} + f^{\varepsilon}_\textnormal{kin} + f^{\varepsilon}_\textnormal{err}\end{equation}
where {each part belongs to $L^\infty\left( [0, T) ; \mathbb{H}_x^s \left( H_v \right) \right)$ uniformly in ${\varepsilon}$ and}
\begin{itemize}
\item The dispersive part $f^{\varepsilon}_\textnormal{disp}$ vanishes in an averaged sense:
\begin{equation*}
\lim_{{\varepsilon} \to 0}\int_0^{t_*} \|f^{\varepsilon}_\textnormal{disp}(\tau)\|^p_{L^\infty_x(H_v)}\mathrm{d}\tau=0, \qquad \forall 0 < t_\star < T, \quad p >\frac{2}{d-1}
\end{equation*}
and uniformly away from $t=0$:
\begin{equation*}
\lim_{{\varepsilon} \to 0}\sup_{t_\star \leqslant t < T} \| f^{\varepsilon}_\textnormal{disp}(t) \|_{ L^\infty_x \left( H_v \right) }=0, \qquad \forall 0 < t_\star < T.
\end{equation*}
\item The kinetic part $f^{\varepsilon}_\textnormal{kin}$ satisfies for some universal $\sigma > 0$
\begin{equation}\label{eq:fKIN}
\sup_{0 \leqslant t < T} e^{2 \sigma t / {\varepsilon}^2} \| f^{\varepsilon}_\textnormal{kin}(t) \|_{ \mathbb{H}^s_x \left( H_v \right) }^2 + \frac{1}{{\varepsilon}^2} \int_0^T e^{2 \sigma t / {\varepsilon}^2} \| f^{\varepsilon}_\textnormal{kin}(t) \|_{ \mathbb{H}^s_x \left( H^{\bullet}_v \right) }^2 \mathrm{d} t \, {\lesssim 1}.
\end{equation}
\item The error term $f^{\varepsilon}_\textnormal{err}$ vanishes uniformly:
\begin{equation*}
\lim_{{\varepsilon} \to 0} \sup_{0 \leqslant t < T} \| f^{\varepsilon}_\textnormal{err}(t) \|_{ \mathbb{H}^s_x \left( H_v \right) }=0.
\end{equation*}
\end{itemize}
\end{enumerate}
\end{theo}
\begin{rem}
The rate of convergence of $f^{\varepsilon}_\textnormal{disp}$ and $f^{\varepsilon}_\textnormal{err}$ can be made explicit. Namely, the dispersive part $f^{\varepsilon}_\textnormal{disp}$ satisfies:
$$ \| f^{\varepsilon}_\textnormal{disp} \|_{L^\infty_x \left( H_v \right) } \lesssim 1 \land \left( \frac{{\varepsilon}}{t} \right)^{ \frac{d-1}{2} } \left( {\| \mathsf P_\textnormal{disp} f_\textnormal{in}} \|_{ \dot{\mathbb{B}}^{\frac{d+1}{2}}_{1,1} \left( H_v \right) } + \| \mathsf P_\textnormal{disp} f_\textnormal{in} \|_{ \mathbb{H}^s_x (H_v) } \right),$$
whereas the error term $f^{\varepsilon}_\textnormal{err}$ is such that
$$\| f^{\varepsilon}_\textnormal{err} \|_{ L^\infty_t \mathbb{H}^s_x H_v } \lesssim {\beta_{\textnormal{disp}}}(f_\textnormal{in}, {\varepsilon}) + \beta_\textnormal{NS}(f_\textnormal{NS}, f_\textnormal{in}, {\varepsilon}),$$
where ${\mathsf P_\textnormal{disp}}$ is defined in Definition \ref{def:hydrodynamic_projectors}, and ${\beta_{\textnormal{disp}}}$ and $\beta_\textnormal{NS}$ are rates of convergence to zero described in Proposition \ref{prop:source_term}.
\end{rem}
To study both the kinetic equation \eqref{eq:Kin-Intro} and the Navier-Stokes-Fourier system \eqref{eq:NSFint}, we adopt a mild formulation which consists in writing the equations in Duhamel form
\begin{equation*}\begin{split}
f^{\varepsilon}(t)&=U^{\varepsilon}(t)f_\textnormal{in}+\frac{1}{{\varepsilon}}\int_0^t U^{\varepsilon}(t-\tau) \mathcal Q\left(f^{\varepsilon}(\tau),f^{\varepsilon}(\tau)\right)\mathrm{d}\tau\\
&=:U^{\varepsilon}(t)f_\textnormal{in}+\Psi^{\varepsilon}[f^{\varepsilon},f^{\varepsilon}](t)\end{split}\end{equation*}
where $\left(U^{\varepsilon}(t)\right)_{t\ge0}$ is the semigroup generated by $-{{\varepsilon}^{-2}}\mathcal L - {{\varepsilon}}^{-1}v\cdot \nabla_x$.
{Of course, the most obvious difficulty in establishing the hydrodynamic limit ${\varepsilon} \to 0$ lies in the control of the stiff term
$\frac{1}{{\varepsilon}}\mathcal Q\left(f^{\varepsilon},f^{\varepsilon}\right)$, however one first needs to construct a solution for any ${\varepsilon}$.} In \cite{BU1991, GT2020} in which the cutoff Boltzmann equation is considered, the authors only need to consider uniform in time estimates for the semigroup $(U^{\varepsilon}(t))_{t\ge0}$ to prove that $\Psi^{\varepsilon}[f,f]$ is well-defined. This means that, in such a case, $\Psi^{\varepsilon}$ is bounded in a space of the type $L^\infty_t\mathbb{H}^s_x H_v$. The study of the Boltzmann equation under cut-off assumption makes this approach possible since then, in the splitting
$$\mathcal L=\mathcal A+\mathcal B$$
the dissipative part $\mathcal B$ is simply the multiplication by the collision frequency.
In \cite{CRT2022} in which the Landau equation is considered, the authors prove uniform in time regularization estimates for the semigroup and the kinetic solution which subsequently yield the boundedness of $\Psi^{\varepsilon}[f,f].$ The short-time regularization effects are of course due to the elliptic-like nature of the Landau collision operator.
The abstract framework we are considering in this paper covers both cases, but the assumptions \ref{L1}--\ref{L4} or \ref{LE} are not strong enough to directly deduce regularization effects and boundededness of $\Psi^{\varepsilon}$. To overcome this, we draw inspiration from suitable energy methods used to construct close-to-equilibrium solutions for Boltzmann and Landau that leads us to a study of the solution in spaces of the form $$L^\infty_t H \cap L^2_t H^{\bullet}$$
as expressed in \eqref{eq:fKIN}. See Section \ref{sec:detail} for more details of the functional setting and the mathematical difficulties encoutered in the proof of Theorem \ref{thm:hydrodynamic_limit}.
We wish to point out also that the nonlinear effects induced by the collision and the lack of control for the $L^2_{x}H_v$ norm requires a refined inequality of quantity like $\| f g \|_{\mathbb{H}^s_x}$ of the type
$$\| f g \|_{\mathbb{H}^s_x} \lesssim \| | \nabla_x |^{r} f \|_{\mathbb{H}^{s - r}} \| g \|_{\mathbb{H}^s_x}, \quad {0 \leqslant r < \frac{d}{2} < s}.$$
In the case of $d \geqslant 3$, one can take $r = 1$ so that the dissipation of the $L^2_x$-norm and the energy control the nonlinearity. In the case $d = 2$, we cannot take $r= 1$ so we consider $r = 1-\alpha$ where $\alpha > 0$. This explains
why we require, in this specific case,
$$f_\textnormal{in} \in \dot{\mathbb{H}}^{-\alpha}$$
which is related to the control of the heat semigroup of the type
$$\| e^{t \Delta_x} | \nabla_x|^{1-\alpha}_x f \|_{L^2_{t, x}} \lesssim \| f \|_{ \dot{\mathbb{H}}^{-\alpha} }.$$
Note that in \cite{GT2020, G2023}, the stronger assumption $f_\textnormal{in} \in L^1_x \left( H_v \right)$ was made.
More details about the proof of Theorem \ref{thm:hydrodynamic_limit} will be given in the next Section \ref{sec:detail}. However, let us already anticipate that the main relevant facts of our approach lie in the following
\begin{enumerate}
\item[(i)] We insist here on the fact that, in the broad generality we are dealing with here, Theorem \ref{thm:hydrodynamic_limit} is new even if results of similar flavors do exist in the literature for the Boltzmann or Landau equations. In particular, as already said, we do not assume here any special link between $\mathcal L$ and $\mathcal Q$ apart from the structure of $\nul(\mathcal L)$ and compatible nonlinear estimates. Precisely, \emph{we do not require here} that $\mathcal L$ is the linearized version of $\mathcal Q$ around the equilibrium $\mu$.
\item[(ii)] In dimension $d=2$, one knows that solutions to the NSF system exist globally in time. When working in dimension $d\geqslant 3$ one can prove that the solutions to the NSF systems are global assuming
$$\left\|(\varrho_\textnormal{in},u_\textnormal{in},\theta_\textnormal{in})\right\|_{\mathbb{H}^{\frac{d}{2}-1}_x} \ll 1$$
(see Appendix \ref{sec:N-S} for details), or equivalently, provided therefore that {the corresponding parts of the initial datum $f_\textnormal{in}$ are small in $\mathbb{H}^{\frac{d}{2}-1}_{x}(H_v)$ norms.} In both cases, solutions to \eqref{eq:Kin-Intro} constructed in Theorem \ref{thm:hydrodynamic_limit} are also global. This was already the case in the work \cite{GT2020} and this is an important contrast with respect to the result in \cite{BU1991} which assumed small $f_{\textnormal{in}}$ to generate global solutions.
\item[(iii)] In the same spirit, contrary to \cite{BU1991, CRT2022, CC2023}, we do not work with small $f_\textnormal{in}$, but as in \cite{GT2020, G2023}, we consider the \emph{a priori} limit $f_{\textnormal{NS}}$ (which exists at least locally in time) and construct the kinetic solution in its neighborhood with the same lifespan. The smallness assumption we impose is transferred to the physical parameter ${\varepsilon}$, i.e. assuming that a large number of collisions are experienced by the gas. This for instance extends the results of \cite{CRT2022,CC2023} to a larger class of initial data. Notice also that, since solutions to the NSF system \eqref{eq:NSFint} can be global depending on the properties of the initial data (such as {symmetry, etc.}), the kinetic solution to \eqref{eq:Kin-Intro} we construct are also global.
\item[(iv)] We also point out that our analysis is performed in the whole space $\mathbb R^{d}_{x}$. The strategy we adopt in the paper can be easily adapted to treat the case of a spatial torus $\mathbb T^{d}_{x}$. Furthermore, in such a case, assuming the initial datum $\mathsf P f_\textnormal{in}$ to be mean-free in space, i.e.
$$\int_{\mathbb T^d} \mathsf P f_\textnormal{in}(x) \mathrm{d} x = 0,$$ one can show the exponential trend to equilibrium for solutions to the kinetic equation \eqref{eq:Kin-Intro}. This is an easy consequence of Theorem \ref{thm:spectral_study} and this can be seen in the case of the Boltzmann equation (see \cite{GT2020, G2023, GMM2017, BMM2019}). The situation is much more delicate in the case of the whole space $\mathbb R^{d}_{x}$ and the trend to equilibrium for solutions to \eqref{eq:Kin-Intro} is not addressed in this paper.
\end{enumerate}
Moreover, we wish to emphasize that, for \emph{well-prepared} initial datum, i.e. in the case in which $f_{\textnormal{in}}$ is such that
$$\nabla_x\cdot u[f_\textnormal{in}]=0, \quad \nabla_x\left(\varrho[f_\textnormal{in}]+\theta[f_\textnormal{in}]\right)=0$$
then {no acoustic waves are produced:}
$$f^{\varepsilon}_\textnormal{disp}(t)=0 \qquad t \in [0,T]$$
and, with the notations of Proposition \ref{prop:source_term}, there holds $\beta_\textnormal{wave}(f_\textnormal{in},{\varepsilon})=0$. In particular, for a smooth initial datum $f_\textnormal{in} \in \mathbb{H}^{s+1}_x(H_v)$, this yields to the convergence rate {$\beta_{\textnormal{NS}} = \mathcal{O}({\varepsilon})$} which is optimal (see \cite{G2006}). Let us state this clearly in the following corollary.
\begin{cor}[\thttl{Optimal convergence rate}]
If the initial datum is smooth and well-prepared, in the sense that
$$f_\textnormal{in} \in \mathbb{H}^{s+1}_x \left( H_v \right), \quad \nabla_x\cdot u[f_\textnormal{in}]=0, \quad \nabla_x\left(\varrho[f_\textnormal{in}]+\theta[f_\textnormal{in}]\right)=0,$$
then the conclusion of Theorem \ref{thm:hydrodynamic_limit} holds with the decomposition
$$f^{\varepsilon} = f_\textnormal{NS} + f^{\varepsilon}_\textnormal{kin} + f^{\varepsilon}_\textnormal{err},$$
where, in this case, the error term is such that
\begin{gather*}
\sup_{0 \leqslant t < T} \| f^{\varepsilon}_\textnormal{err}(t) \|_{ \mathbb{H}^s_x \left( H_v \right) } \lesssim {\varepsilon},
\end{gather*}
and, in particular, away from $t=0$
$$\sup_{t_* \leqslant t < T} \| f^{\varepsilon}(t) - f_\textnormal{NS}(t) \|_{ \mathbb{H}^s_x \left( H_v \right) } \lesssim {\varepsilon}, \qquad \forall 0 < t_* < T.$$
\end{cor}
\color{black}
\subsection{Main results -- second version in larger functional spaces} We improve also the two main results here above by showing that the same conclusion still holds in a larger functional space $X$ such that
$$H \hookrightarrow X.$$
To do so, our analysis requir{es a new set of Assumption which complement \ref{L1}--\ref{L4}:
\begin{hypLE} Besides Assumptions \ref{L1}--\ref{L4}, one assumes that $\mathcal L$ satisfies
\begin{enumerate}[label=\hypst{LE}]
\item \label{LE} Besides the splitting provided in \ref{L3}, the operator $\mathcal L$ can be decomposed as
$$\mathcal L = \mathcal B^{(0)}+ \mathcal A^{(0)}, \qquad \mathscr{D}\left(\mathcal B^{(0)}\right)=\mathscr{D}(\mathcal L), \qquad \mathcal A^{(0)} \in \mathscr B(X,H)$$
where the splitting is
compatible with a hierarchy of Hilbert spaces $\left(X_j\right)_{j=0}^{2}$ such that
\begin{enumerate}
\item the spaces $X_j$ continuously and densely embed into one another:
$$X_{2} \hookrightarrow X_{1} \hookrightarrow X_{0} =X, \quad H \hookrightarrow X,$$
\item \label{assumption_large_multi-v} the multiplication by $v$ and its adjoint are bounded from $X_{j+1}$ to $X_{j}$:
\begin{equation*}
\|v f\|_{X_{j}} \lesssim \|f\|_{X_{j+1}}, \qquad
\|v^{\star} f\|_{X_{j}} \lesssim \|f\|_{X_{j+1}}, \quad j=0,1,
\end{equation*}
\item the part $\mathcal B^{(0)}_{\xi} = \mathcal B^{(0)} - i v \cdot \xi$ is dissipative on each space $X_j$ and $H$ uniformly in $\xi \in \mathbb R^d$, that is to say, for $Y = X_0, X_1, X_2, H$
$$
\mathfrak{S}_Y\left(\mathcal B_{\xi}^{(0)} \right) \cap \Delta_{-\lambda_\mathcal B} = \varnothing$$
and $$\sup_{\xi \in \mathbb R^d}\left\|\mathcal R\left(z,\mathcal B^{(0)}_{\xi} \right)\right\|_{\mathscr B(Y)} \lesssim | \mathrm{Re}\, z + \lambda_\mathcal B|^{-1}, \qquad \forall z \in \Delta_{-\lambda_\mathcal B}.$$
Specifically, in the space $X$, there holds
$$\Re \left\langle \mathcal B^{(0)}_{\xi} f , f \right\rangle_{X} \leqslant - \lambda_\mathcal B \| f \|_{X^{\bullet}}^2, \qquad \forall f \in X^{\bullet}$$
for some dissipation Hilbert space $X^{\bullet}$ satisfying
$$H^{\bullet} \hookrightarrow X^{\bullet} \hookrightarrow X, \qquad \| \cdot \|_{X} \leqslant \| \cdot \|_{X^{\bullet}},$$
\item \label{assumption_large_bounded_A} the operator $\mathcal A^{(0)}$ and its adjoint $(\mathcal A^{(0)})^{\star}$ are bounded in the following spaces
$$\mathcal A^{(0)} \in \mathscr B(X;H) \cap \mathscr B(X_{j};X_{j+1}), \qquad (\mathcal A^{(0)}) ^{\star} \in \mathscr B(X_{j};X_{j+1}), \quad j=0,1\,.$$
\end{enumerate}
\end{enumerate}
\begin{enumerate}[label=\hypst{BE}]
\item \label{BE} The corresponding nonlinear assumption is then the following:
$$\left\langle \mathcal Q(f, g) , h \right\rangle_{X} \lesssim \| h \|_{X^{\bullet}} \left( \| f \|_{X} \| g \|_{X^{\bullet}} + \| f \|_{X^{\bullet}} \| g \|_{X} \right), \qquad f,g,h \in X^{\bullet}\,.$$
\end{enumerate}
\end{hypLE}
\begin{rem}
As shown in \cite{GMM2017, BMM2019, G2023}, the operator $\mathcal L$ and $\mathcal Q$ satisfy \ref{LE} and \ref{BE} in the spaces
$$X_j = L^2\left( \langle v \rangle^{k + 2j} \mathrm{d} v \right), \qquad X^{\bullet} = L^2\left( \langle v \rangle^{k+1} \mathrm{d} v \right), \qquad \mathscr{D}\left( \mathcal L \right) = L^2\left( \langle v \rangle^{k+2} \mathrm{d} v \right)$$
for some $k > 0$.
\end{rem}
\color{black}
As in Definition \ref{defi:Hhstar}, one can define the dual space $X^{\circ}$ of $X^{\bullet}$ as the completion of $X$ for the norm
$$\| f \|_{X^{\circ}} := \sup_{ \| \varphi \|_{X^{\bullet}} \leqslant 1 } \langle f, \varphi \rangle_{X}.$$
In that space $X$, combining suitable enlargement techniques introduced in \cite{GMM2017} with a bootstrap argument, we derive the following improvement of the spectral Theorem \ref{thm:spectral_study}.
\begin{theo}[\textit{\textbf{Enlarged spectral result}}]
\label{thm:enlarged_thm}
Assume \ref{L1}--\ref{L4} as well as \ref{LE}. Then the results of Theorem \ref{thm:spectral_study} hold with $(H, H^{\bullet}, H^{\circ})$ replaced by $(X, X^{\bullet}, X^{\circ})$. Furthermore the spectral projectors are \emph{regularizing} in the sense that, in the decomposition
\begin{equation*}
\mathsf P_\star(\xi) = \mathsf P^{(0)}_\star\left( \widetilde{\xi} \right) + i \xi \cdot \mathsf P_\star^{(1)}\left( \widetilde{\xi} \right) + S_{\star}(\xi),
\end{equation*}
each term belongs to $\mathscr B\left(X^{\circ} ; H^{\bullet} \right)$ uniformly in $|\xi| \leqslant \alpha_{0}$, and $\| S_\star(\xi) \|_{\mathscr B( X^{\circ} ; H^{\bullet} )} \lesssim | \xi |^2$. Finally, the decay estimate \eqref{decay-semigroup}
extends to $X$ as follows: for any $\xi \in \mathbb R^d$, any $\sigma \in (0,\sigma_0)$ and any $f \in X$
\begin{subequations}
\label{decay-semigroup-EE}
\begin{align}
\notag
\sup_{t \geqslant 0} \, e^{2 \sigma_0 t} \big\| & U_{\xi}(t) \left( \mathrm{Id} - \mathsf P(\xi) \right) f \big\|^2_{X} \\
\label{eq:decay-EE}
& + \int_0^\infty e^{2 \sigma t} \big\| U_{\xi}(t)\left( \mathrm{Id} - \mathsf P(\xi) \right)f \big\|_{X^{\bullet}}^2 \, \mathrm{d} t \leqslant C_\sigma \|\left( \mathrm{Id} - \mathsf P(\xi) \right)f \|_{X}^2,
\end{align}
whereas for any $f \in X^{\circ}$
\begin{equation}
\label{eq:decay_Ee'}
\int_0^\infty e^{2\sigma t} \left\| U_{\xi}(t)\left( \mathrm{Id} - \mathsf P(\xi) \right) f \right\|_{X}^2 \, \mathrm{d} t \leqslant C_\sigma \| \left( \mathrm{Id} - \mathsf P(\xi) \right)f \|_{X^{\circ}}^2\,.
\end{equation}
\end{subequations}
\end{theo}
{The extension of Theorem \ref{thm:spectral_study} to a larger Hilbert space $X$ is done using the enlargement procedure developed in \cite{GMM2017}, which inspired the subtle bootstrap argument leading to the regularity properties of $\mathsf P_\star$}. Again, we refer to Section \ref{sec:newspec} for a description of the proof. {The aforementioned regularity of $\mathsf P_\star$ can be improved in the presence of yet another suitable splitting of $\mathcal L$, this time in Banach spaces instead of just Hilbert spaces}. This is done in Theorem \ref{thm:regularized_thm}, but, since such a result is not necessary for the derivation of the NSF system \eqref{eq:NSFint}, we do not give the statement in this introduction.
We assume for this next theorem \ref{L1}--\ref{L4} and \ref{Bortho}--\ref{Bisotrop}, as well as \ref{LE} and \ref{BE}.
\begin{theo}[\textit{\textbf{Enlarged hydrodynamic limit theorem}}]
\label{thm:hydrodynamic_limit-gen_symmetric}
Under the assumptions of Theorem \ref{thm:hydrodynamic_limit} {on $f_\textnormal{in} \in \mathbb{H}^s_x \left( X_v \right)$} and on the solution to the Navier-Stokes-Fourier system, {the conclusion of Theorem \ref{thm:hydrodynamic_limit} still holds with the following differences:}
\begin{enumerate}
\item[\textit{\textbf{(1)}}]\textit{\textbf{Existence of a unique solution.}}
There exists some small $c_0 > 0$ and ${\varepsilon}_0 > 0$ such that the equation
$$\partial_t f^{\varepsilon} = \frac{1}{{\varepsilon}^2} \left( \mathcal L - {\varepsilon} v \cdot \nabla_x \right) f^{\varepsilon} + \frac{1}{{\varepsilon}} \mathcal Q\left( f^{\varepsilon}, f^{\varepsilon} \right), \quad f^{\varepsilon}(0, x, v) = f_\textnormal{in}(x, v)$$
admits for any ${\varepsilon} \in (0, {\varepsilon}_0]$ a unique solution among those satisfying
$$\sup_{0 \leqslant t < T} \| f^{\varepsilon}(t) \|_{ \mathbb{H}^s_x \left( X_v \right) } \leqslant \frac{c_0}{{\varepsilon}}, \qquad f^{\varepsilon} \in L^2_{\rm{loc}} \left( [0, T) ; \mathbb{H}^s_x \left( X^{\bullet}_v\right) \right),$$
and it satisfies furthermore the following uniform estimate:
$$\sup_{0 \leqslant t < T} \| f^{\varepsilon}(t) \|_{ \mathbb{H}^s_x \left( H_v \right) }^2 + \int_0^T \| | \nabla_x |^{1-\alpha } f^{\varepsilon}(t) \|_{ \mathbb{H}^{s-1+\alpha}_x \left( H^{\bullet}_v \right) }^2 \mathrm{d} t \, \lesssim 1.$$\color{black}
Moreover,
$$f^{\varepsilon} \in\mathcal C\left( [0, T) ; \mathbb{H}^s_x \left( X_v \right) \right)\,.$$
\item[\textit{\textbf{(2)}}]\textit{\textbf{Decomposition and convergence of the solution.}}
The solution $f^{\varepsilon}$ splits as the sum of some limiting part $f_\textnormal{NS}$, some initial layers $(f^{\varepsilon}_\textnormal{disp}, f^{\varepsilon}_\textnormal{kin})$, and a vanishing part~$f^{\varepsilon}_\textnormal{err}$:
$$f^{\varepsilon} = f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp} + f^{\varepsilon}_\textnormal{kin} + f^{\varepsilon}_\textnormal{err}$$
where $f^{\varepsilon}_\textnormal{disp}$ and $f^{\varepsilon}_\textnormal{err}$ satisfy the same estimates, and this time the kinetic part $f^{\varepsilon}_\textnormal{kin}$ satisfies
$$\sup_{0 \leqslant t < T} e^{2 \sigma t / {\varepsilon}^2} \| f^{\varepsilon}_\textnormal{kin}(t) \|_{ \mathbb{H}^s_x \left( X_v \right) }^2 + \frac{1}{{\varepsilon}^2} \int_0^T e^{2 \sigma t / {\varepsilon}^2} \| f^{\varepsilon}_\textnormal{kin}(t) \|_{ \mathbb{H}^s_x \left( X^{\bullet}_v \right) }^2 \mathrm{d} t \, {\lesssim 1}.$$
\color{black}
\end{enumerate}
\end{theo}
Finally, drawing inspiration from works such as \cite{CM2017, CTW2016, HTT2020, CG2022} which dealt with the Boltzmann equation without cutoff or the Landau equation, we present an alternative to the nonlinear assumption \ref{BE} in which the two arguments of $\mathcal Q$ do not play symmetric roles.
\begin{hypQE}
Besides Assumptions \ref{L1}--\ref{L4}, we assume the following:
\begin{enumerate}[label=\hypst{BED}]
\item \label{BED} We consider a hierarchy of spaces $(\bm{X}_j)_{j=-2-s}^1$ for some $s \geqslant 0$ such that
$$H \hookrightarrow \bm{X}_1 \hookrightarrow X = \bm{X}_{0} \hookrightarrow \dots \hookrightarrow \bm{X}_{-2-s},$$
\end{enumerate}
whose dissipation spaces are embedded in the same way:
$$H^{\bullet} \hookrightarrow \bm{X}^{\bullet}_1 \hookrightarrow X^{\bullet} = \bm{X}^{\bullet}_{0} \hookrightarrow \dots \hookrightarrow \bm{X}^{\bullet}_{-2-s},$$
and such that the following conditions hold:
\begin{enumerate}
\item the assumption \ref{LE} is satisfied in the larger spaces $\bm{X}_{-2-s}, \dots, \bm{X}_{-2}$ (but not necessarily in $\bm{X}_1$) and with a splitting of $\mathcal L$ that may be different in each space $\bm{X}_j$,
\item the nonlinear operator satisfies the following \emph{non-closed} dual estimate:
\begin{equation}\label{eq:BEDNon}
\| \mathcal Q(f, g) \|_{\bm{X}^{\circ}_{j} }
\lesssim
\| f \|_{\bm{X}_{j}} \| g \|_{\bm{X}^{\bullet}_{j+1}}
+ \| f \|_{\bm{X}^{\bullet}_{j}} \| g \|_{\bm{X}_{j+1}}
, \quad j=-1-s, \dots, 0,
\end{equation}
\item the nonlinear operator satisfies the following \emph{closed} dual estimates:
\begin{equation}\label{eq:BEDclosed}
\begin{split}
\langle \mathcal Q(f, g), g \rangle_{ \bm{X}_{j} }
\lesssim \sum_{ \{ a, b, c \} = \{j, j, -1-s \} } & \| f \|_{\bm{X}_{ a }} \| g \|_{\bm{X}^{\bullet}_{b}} \| g \|_{\bm{X}^{\bullet}_{c}} \\
& + \| f \|_{\bm{X}^{\bullet}_{a}} \| g \|_{\bm{X}_{b}} \| g \|_{\bm{X}^{\bullet}_{c}}, \quad j = -1-s, \dots, 0,
\end{split}
\end{equation}
\item $\mathcal A$ sends $\bm{X}_{j-1}$ to $\bm{X}_{j}$ at the dual level, in the sense that
$$\langle \mathcal A f, f \rangle_{\bm{X}_{j}} \lesssim \| f \|^2_{ \bm{X}_{j-1} }, \quad j=-1-s, \dots, 0.$$
\end{enumerate}
\end{hypQE}
\begin{theo}[\textit{\textbf{Enlarged hydrodynamic limit theorem under \ref{BED}}}]
\label{thm:hydrodynamic_limit-gen_degenerate}
Consider $s \in \mathbb N$ such that $s \geqslant \max\{3, d/2 + 1\}$ and denote for $j = -1, 0$ the spaces $\rSSh{s}$ and $\rSShp{s}$ defined by the norms
$$\|f\|_{\rSSh{s}_j} = \| f \|_{ L^2_x \left( \bm{X}_j \right) } + \| \nabla_x^s f \|_{ L^2_x(\bm{X}_{j-s}) },
\qquad \|f\|_{\rSShp{s}_j} = \| f \|_{ L^2_x \left( \bm{X}^{\bullet}_j \right) } + \| \nabla_x^s f \|_{ L^2_x (\bm{X}^{\bullet}_{j-s}) }.$$
For any $f_\textnormal{in} \in \rSSh{s}_0$, and under the assumptions of Theorem \ref{thm:hydrodynamic_limit-gen_symmetric} on the solution to the Navier-Stokes-Fourier system, the conclusion of Theorem \ref{thm:hydrodynamic_limit-gen_symmetric} still holds with the following difference.
There exists some small $c_0 > 0$ and ${\varepsilon}_0 > 0$ such that the equation
$$\partial_t f^{\varepsilon} = \frac{1}{{\varepsilon}^2} \left( \mathcal L - {\varepsilon} v \cdot \nabla_x \right) f^{\varepsilon} + \frac{1}{{\varepsilon}} \mathcal Q\left( f^{\varepsilon}, f^{\varepsilon} \right), \quad f^{\varepsilon}(0, x, v) = f_\textnormal{in}(x, v)$$
admits for any ${\varepsilon} \in (0, {\varepsilon}_0]$ a unique solution among those satisfying
$$\sup_{0 \leqslant t < T} \| f^{\varepsilon}(t) \|_{ \rSSh{s}_0 } \leqslant \frac{c_0}{{\varepsilon}}, \qquad f^{\varepsilon} \in L^2_{\rm{loc}} \left( [0, T) ; \rSShp{s}_0 \right).$$
Moreover, it satisfies the following uniform estimate:
$$\sup_{0 \leqslant t < T} \| f^{\varepsilon}(t) \|_{ \rSSh{s}_0 }^2 + \int_0^T \| | \nabla_x |^{ \bm \alpha } f^{\varepsilon}(t) \|_{ \left( \rSShp{s-\boldsymbol{\alpha}}_0 \right) }^2 \mathrm{d} t \, \lesssim 1$$
and is continuous in the larger space $\rSSh{s}_{-1}$:
$$f^{\varepsilon} \in \mathcal C\left( [0, T) ; \mathbb{H}^s_x \left( \rSSh{s}_{-1} \right) \right).$$
Finally, the kinetic initial layer is such that
$$\sup_{0 \leqslant t < T} e^{2 \sigma t / {\varepsilon}^2} \| f^{\varepsilon}_\textnormal{kin}(t) \|_{ \rSSh{s}_0 }^2 + \frac{1}{{\varepsilon}^2} \int_0^T e^{2 \sigma t / {\varepsilon}^2} \| f^{\varepsilon}_\textnormal{kin}(t) \|_{ \rSShp{s}_0 }^2 \, \mathrm{d} t \, {\lesssim 1}.$$
\end{theo}
A few comments are in order regarding Theorem \ref{thm:hydrodynamic_limit-gen_symmetric} and \ref{thm:hydrodynamic_limit-gen_degenerate}:
\begin{itemize}
\item Only the kinetic part is now living in the smaller space $\mathbb{H}^s_x(X_v)$, whereas the dispersive parts $f_\textnormal{disp}^{\varepsilon}$ and $f^{\varepsilon}_\textnormal{err}$ satisfy the same properties as in Theorem \ref{thm:hydrodynamic_limit}.
\item The method of proof of Theorem \ref{thm:hydrodynamic_limit-gen_symmetric} is the same as for \ref{thm:hydrodynamic_limit}; the existence of solution to the kinetic equation can be deduced from the application of Banach fixed point theorem {(and the proof is led independently of whether we assume \ref{BE} or not)}. Concerning Theorem \ref{thm:hydrodynamic_limit-gen_degenerate}, the approach is more involved and a careful study of an approximating scheme (a variation of Picard iterations) has to be done in order to overcome the difficulty induced by the lack of symmetry of $\mathcal Q.$
\item Notice that, as for Theorem \ref{thm:hydrodynamic_limit-gen_symmetric}, we \emph{do not require} here any smallness assumptions on $f_{\rm in}$ and the smallness is totally transferred in the parameter ${\varepsilon}.$ Recalling that assumptions \ref{BED} are suited for the study of the Boltzmann equation without cut-off assumption and for the Landau equation, this is an important improvement with respect to the previous results in the field which all require some additional restriction on the size of the initial datum to derive the hydrodynamical limit (see \cite{CRT2022} for the Landau equation and \cite{CC2023} for the Boltzmann equation).
\item {Finally, we emphasize that this provides, up to our knowledge, the first result concerning the strong Navier-Stokes limit for initial data with algebraic decay with respect to the velocity variable in the case of Boltzmann equation without cut-off assumption or Landau equation (see Appendix \ref{sec:Landau-Boltz}).}
\end{itemize}
We will comment with more details the conclusion of Theorems \ref{thm:hydrodynamic_limit}, \ref{thm:hydrodynamic_limit-gen_symmetric} and \ref{thm:hydrodynamic_limit-gen_degenerate} in Section \ref{sec:detail} where a detailed description of the proof and the role of the various assumptions will be illustrated.
\subsection{Outline of the paper} In the next Section \ref{sec:detail}, we introduce the main ideas underlying the proofs of {the hydrodynamic limit Theorems} \ref{thm:hydrodynamic_limit}, \ref{thm:hydrodynamic_limit-gen_symmetric} and \ref{thm:hydrodynamic_limit-gen_degenerate}. Notations and mathematical objects that are used in the rest of the analysis are also introduced in Section \ref{sec:detail}.
Section \ref{scn:spectral_study} gives the full proof of both the spectral theorems \ref{thm:spectral_study} and \ref{thm:enlarged_thm}. A detailed description of approach is given in Section \ref{sec:newspec} and the proof of Theorem \ref{thm:spectral_study} is then derived in various steps, together with its ‘‘regularized version‘‘ Theorem \ref{thm:regularized_thm}.
Section \ref{scn:study_physical_space} establishes the main consequences of Theorem \ref{thm:spectral_study} on the semigroup $(U^{\varepsilon}(t))_{t\geq0}$ generated by the linear part ${\varepsilon}^{-2} \left( \mathcal L - {\varepsilon} v \cdot \nabla_x \right)$ of \eqref{eq:Kin-Intro} in the various regimes/time scales relevant for the hydrodynamic limit. In particular, the comparison between the linearized semigroups associated to \eqref{eq:Kin-Intro} and \eqref{eq:NSFint} is given in Section \ref{scn:study_physical_space}.
The main bilinear estimates are then established in Section \ref{sec:Bilin} as well as the main tools used for the hydrodynamic limit (and in particular the mild formulation of \eqref{eq:NSFint}). The proof of Theorem \ref{thm:hydrodynamic_limit} and \ref{thm:hydrodynamic_limit-gen_symmetric} under assumptions \ref{BE}, is then given in Section \ref{scn:proof_hydrodynamic_limit_symmetrizable}, whereas the proof of Theorem \ref{thm:hydrodynamic_limit-gen_degenerate} under Assumption \ref{BED} is given in Section \ref{scn:hydrodynamic_limit_BED}.
To make the paper self-contained, we end it with three different Appendices. In Appendix \ref{sec:Landau-Boltz}, we discuss the general assumptions \ref{L1}--\ref{L4}, \ref{Bortho}--\ref{Bisotrop}, \ref{LE}--\ref{BE} and \ref{BED} for an extensive list of physical models including, as said, the classical Boltzmann and Landau equations as well as their quantum counterpart covering, at the linearized level, both the Fermi-Dirac and Bose-Einstein descriptions. Appendix \ref{sec:toolbox} gives the functional toolbox with particular emphasis to Littlewood-Paley theory (Section \ref{scn:littlewood-paley} and other results relevant for our analysis). We also present in Section \ref{scn:boostrap_projectors} the bootstrap argument for projection operators which is a cornerstone of Theorem \ref{thm:enlarged_thm}. The final Appendix \ref{sec:N-S} recalls the main properties of the Navier-Stokes-Fourier system \eqref{eq:NSFint} that are needed for our analysis {and proves some results necessary for our framework}.
\subsection*{Acknowledgments} Both the author gratefully acknowledge the financial support from the Italian Ministry of Education, University and Research (MIUR), ``Dipartimenti di Eccellenza'' grant 2022-2027. They also thank Isabelle Gallagher and Isabelle Tristani for insightful discussions about hydrodynamic limits.
\section{Detailed description of our proofs}\label{sec:detail}
We give here a precise description of the main steps of our approach to prove the above two Theorem \ref{thm:hydrodynamic_limit} and \ref{thm:hydrodynamic_limit-gen_symmetric}. We use repeatedly the spectral properties of $\mathcal L_\xi$ and the properties of the associated semigroup $\left(U_\xi(t)\right)_{t\ge0}$ as established in Theorems \ref{thm:enlarged_thm} and \ref{thm:enlarged_thm}. Notations are those introduced in those two results.
\subsection{The functional setting} The conclusion of the Theorem \ref{thm:hydrodynamic_limit} and the splitting of $f^{\varepsilon}$ in \eqref{eq:decomp} suggest to introduce the following definitions of position-velocity spaces and time-position-velocity spaces suited to the different regimes (kinetic, diffusive and mixed) we will consider in this work.
\begin{defi}\label{defi:NORMSPACES} Let $s \in \mathbb R$ be given.
\begin{enumerate}
\item For $Y=H$ or $Y=X$, we define the position-velocity spaces
$$\VV = \VV^s = \mathbb{H}^{s}_{x}(Y_v),$$
which we endow with their natural norms $\|\cdot\|_{\VV}$ defined in \eqref{eq:normHsx}.
We define in the same way the spaces
$$\rSSgp{s}=\mathbb{H}^s_x(Y^{\bullet}_v), \qquad \rSSgm{s}=\mathbb{H}^s_x(Y^{\circ}_v)$$
with $Y^{\bullet}_v=H^{\bullet}$ or $X^{\bullet}$ and $Y^{\circ}_v=H^{\circ}$ or $X^{\circ}.$
\item Given $T \in (0,\infty]$ and $\sigma \in (0,\sigma_0)$ (with $\sigma_0$ defined in Theorem \ref{thm:spectral_study} or \ref{thm:enlarged_thm}), we introduce here the kinetic-type time-position-velocity space
$$\GGG=\rSSSl{s}(T, \sigma, {\varepsilon}):= \big\{ f \in \mathcal C_b\left( [0, T) ; \rSSl{s} \right) \, ; \, |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{\rSSSl{s}} < \infty \big\},$$
where the norm $|\hskip-0.04cm|\hskip-0.04cm| \cdot |\hskip-0.04cm|\hskip-0.04cm|_{\rSSSl{s}}$ is given by
$$ |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{ \rSSSl{s}}^2 := \sup_{0 \leqslant t < T} \, e^{2 \sigma t / {\varepsilon}^2 } \| f(t) \|_{\rSSl{s}}^2 + \frac{1}{{\varepsilon}^2} \int_0^T e^{2 \sigma t / {\varepsilon}^2} \| f(t) \|_{{\rSSlp{s}}}^2 \mathrm{d} t.$$
\item Given $T \in (0,\infty]$, $0 < \eta \ll 1$ and some mapping $\phi$ such that
\begin{equation}\label{eq:nablaPhi}
\left|\nabla_x\right|^{1-\alpha}\phi \in L^2\left( [0, T) ; \rSSsp{s} \right)
\end{equation}
where $\alpha \in \left(0, \frac{1}{2}\right)$ if $d = 2$ and $\alpha = 0$ if $d \geqslant 3$, we introduce the parabolic-type time-position-velocity space
$$\HHH=\rSSSs{s}(T, \phi, \eta):= \left\{ f \in \mathcal C_b\left( [0, T) ; \rSSs{s} \right) \, ; \, |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{\rSSSs{s}} < \infty \right\}$$
endowed with the norm
$$
|\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{\rSSSs{s}}^2 := \sup_{0 \leqslant t < T} \left\{ w_{\phi, \eta}(t)^2 \| f(t) \|_{\rSSsp{s}}^2 + w_{\phi, \eta}(t)^2 \int_0^t \left\| |\nabla_x|^{1-\alpha} f( \tau ) \right\|_{\rSSsp{s}}^2 \mathrm{d} \tau \right\},
$$
where we set
\begin{equation}\label{eq:wphieta}
w_{\phi, \eta}(t) = \exp\left( \frac{1}{2 \eta^2} \int_0^t\| |\nabla_x|^{1-\alpha} \phi(\tau) \|_{\rSSsp{s}}^2 \mathrm{d} \tau \right)\,.\end{equation}
\item Finally, with the notation of the previous points, we introduce the mixed-type time-position-velocity space
$$\FFF = \rSSSm{s}(T, \phi, \eta, {\varepsilon}):= \left\{ \mathcal C_b\left( [0, T) ; \rSSs{s} \right) \, ; \, |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{\rSSSm{s}} < \infty \right\}$$
endowed with the norm
$$|\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{\rSSSm{s}}^2 := \sup_{0 \leqslant t < T } \left\{ w_{\phi, \eta}(t)^2 \| f(t) \|_{\rSSs{s}}^2 + \frac{1}{{\varepsilon}^2} w_{\phi, \eta}(t)^2 \int_0^t \| f(\tau) \|_{\rSSsp{s}}^2 \mathrm{d} \tau \right\}.$$
\end{enumerate}\end{defi}
\begin{rem}
{Drawing inspiration from \cite{GT2020} or more precisely \cite{G2023},} the above norm $|\hskip-0.04cm|\hskip-0.04cm|\cdot|\hskip-0.04cm|\hskip-0.04cm|_{\HHH}$ and $|\hskip-0.04cm|\hskip-0.04cm| \cdot |\hskip-0.04cm|\hskip-0.04cm|_\FFF$, {and more specifically the time weight $w_{\phi, \eta}$,} are designed so that, no matter how big $\phi$ is, there holds
\begin{equation}
\label{eq:NS_exponential_weight}
\forall t \in [0, T), \quad w_{\phi, \eta}(t) \left\| w_{\phi, \eta}^{-1} | \nabla_{x} |^{1-\alpha} \phi \right\|_{L^2 \left( [0, t] ; {\rSSsp{s}} \right) } \leqslant \eta
\end{equation}
as can be seen by a simple computation (see Proposition \ref{prop:special_bilinear_hydrodynamic}). Notice also that $w_{\phi, \eta}$ is decreasing and satisfies the bounds $0 < w_{\phi, \eta}(T) \leqslant w_{\phi, \eta} \leqslant 1$, consequently
\begin{equation}
\label{eq:roughNt}
|\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{\rSSSs{s}}^2 \leqslant \sup_{0 \leqslant t \leqslant T}\|f(t)\|_{\rSSsp{s}}^2+\int_0^T\left\|\,|\nabla_x|^{1-\alpha}f(\tau)\right\|_{\rSSsp{s}}^2\mathrm{d} \tau \leqslant w_{\phi, \eta}(T)^{-2} |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{\rSSSs{s}}^2,
\end{equation}
and
\begin{equation*}
|\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{\rSSSm{s}}^2 \leqslant \sup_{0 \leqslant t \leqslant T}\|f(t)\|_{\rSSs{s}}^2 + \frac{1}{{\varepsilon}^2} \int_0^T\left\|\,f(\tau)\right\|_{\rSSsp{s}}^2\mathrm{d} \tau \leqslant w_{\phi, \eta}(T)^{-2} |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{\rSSSm{s}}^2.
\end{equation*}
We point out already that we will work in the proof of Theorems \ref{thm:hydrodynamic_limit}, \ref{thm:hydrodynamic_limit-gen_symmetric} and \ref{thm:hydrodynamic_limit-gen_degenerate} with the choice $\phi=f_\textnormal{NS}$ in \eqref{eq:nablaPhi}.
\end{rem}
\begin{rem}
Clearly, Theorems \ref{thm:hydrodynamic_limit-gen_symmetric} and \ref{thm:hydrodynamic_limit-gen_degenerate} suggest that, in the decomposition of the solution $f^{\varepsilon}$ in \eqref{eq:decomp}, we will look for the kinetic part $f^{\varepsilon}_\textnormal{kin}$ in the space $\GGG$ in the sense that
$$|\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_\textnormal{kin}|\hskip-0.04cm|\hskip-0.04cm|_{\rSSSl{s}} < \infty, \qquad s >\frac{d}{2},$$
and, although its is not as obvious, the error term $f^{\varepsilon}_\textnormal{err}$ will be constructed as the sum of a part in $\HHH$ and one in $\FFF$, which explains the global energy estimate satisfied by the solution.
The first two regimes $\GGG$ and $\HHH$ are preserved by the two corresponding parts $U^{\varepsilon}_\textnormal{kin}(\cdot)$ and $U^{\varepsilon}_\textnormal{hydro}(\cdot)$ of the linearized flow of the equation \eqref{eq:Kin-Intro} (see Sections \ref{scn:reduction_problem}--\ref{scn:pseudo_hydro_kinetic} for their definitions), and the mixed regime $\FFF$ is introduced to describe the interactions between the two different regimes (see Lemmas \ref{lem:decay_semigroups_hydro} and \ref{lem:decay_regularization_convolution_kinetic_semigroup}).
\color{black}
\end{rem}
\subsection{Reduction of the problem}
\label{scn:reduction_problem}
We present in this section how we frame the problem of hydrodynamic limits. We start from the integral formulation
\begin{equation}\label{eq:Kin-Mild}
f^{\varepsilon}(t) = U^{\varepsilon}(t) f_\textnormal{in} + \Psi^{\varepsilon}\left[ f^{\varepsilon}, f^{\varepsilon} \right](t),
\end{equation}
where, denoting $2 \mathcal Q^\mathrm{sym}(f, g) := \mathcal Q(f, g) + \mathcal Q(g, f)$
\begin{gather*}
\Psi^{\varepsilon}[f,g](t) := \frac{1}{{\varepsilon}} \int_0^t U^{\varepsilon}(t - \tau) \mathcal Q^\mathrm{sym} (f(\tau), g(\tau)) \mathrm{d} \tau
\end{gather*}
and $(U^{\varepsilon}(t))_{t\geq0}$ is the $C^0$-semigroup in $\VV^s$ generated by the full linearized operator (in original variables):
$$\mathcal{G}_{\varepsilon} f:=-\frac{1}{{\varepsilon}} v\cdot \nabla_x f +\frac{1}{{\varepsilon}^2}\mathcal L f, \qquad \mathscr{D}(\mathcal{G}_{\varepsilon})=\left\{f \in \VV\,;\,\mathcal L f \in \VV\right\}$$
where $Y=H$ or $Y=X.$ Notice that the semigroup $\left(U^{\varepsilon}(t)\right)_{t\geq0}$ is related to the semigroup $\left(U_{{\varepsilon}\,\xi}(t)\right)_{t \ge0}$; for $g \in \VV$, setting
$$\widehat{g}(\xi)=\widehat{g}(\xi,\cdot)=\mathcal F_{x}\left[g(x,\cdot)\right](\xi)=\int_{\mathbb R^{d}}e^{i\xi\cdot x}g(x,\cdot)\mathrm{d} x \in Y$$
one has
$$\mathcal F_x\left[\mathcal{G}_{\varepsilon} g\right](\xi) = {\varepsilon}^{-2}\mathcal L_{{\varepsilon}\,\xi}\widehat{g}(\xi,\cdot)$$
so that
$$\mathcal F_x U^{\varepsilon}(t)\mathcal F_x^{-1}=U_{{\varepsilon}\,\xi}\left(\frac{t}{{\varepsilon}^2}\right), \qquad t\ge0.$$
We define now the linearized semigroup $(U_\textnormal{NS}(t))_{t\ge0}$, adopting again a Fourier-based description which involves the projectors $\mathsf P^{(0)}_\textnormal{inc}$ and $\mathsf P_\textnormal{Bou}^{(0)}$ as defined in Theorem \ref{thm:spectral_study}.
\begin{defi}
\label{def:U_NS_V_NS}
We define the \emph{diffusive} Navier-Stokes semigroup $\left(U_\textnormal{NS}(t)\right)_{t\geq0}$ through its Fourier transform for any $g \in \VV^{\circ}=\HH^{\circ}$ or $\VV^{\circ}=\GG^{\circ}$ and $t \geqslant 0$ as
\begin{align*}
U_\textnormal{NS}(t) g = \mathcal F_x^{-1} \Big[ & \exp( - t \kappa_\textnormal{inc} | \xi |^2 ) \mathsf P^{(0)}_\textnormal{inc} \left( \widetilde{\xi} \right) \widehat{g}(\xi) \\
& + \exp( - t \kappa_\textnormal{Bou} | \xi |^2 ) \mathsf P^{(0)}_\textnormal{Bou} \left( \widetilde{\xi} \right) \widehat{g}(\xi) \Big], \quad \widetilde{\xi} := \frac{\xi}{|\xi|}.
\end{align*}
We also define the one parameter family $\left(V_\textnormal{NS}(t)\right)_{t\geqslant 0}$ as
\begin{align}\label{eq:defi_V_ns}
V_\textnormal{NS}(t) g := \mathcal F^{-1}_x\Big[ \exp( - t \kappa_\textnormal{inc} | \xi |^2 ) \mathsf P^{(1)}_\textnormal{inc}\left( \widetilde{\xi} \right) \widehat{g}(\xi) + \exp( - t \kappa_\textnormal{Bou} | \xi |^2 ) \mathsf P^{(1)}_\textnormal{Bou}\left( \widetilde{\xi} \right) \widehat{g}(\xi) \Big].
\end{align}
\end{defi}
The link between the above objects and solutions to the NSF system \eqref{eq:NSFint} is given by the following Proposition whose proof is postponed to Appendix \ref{sec:N-S}.
\begin{prop}
\label{prop:equivalence_kinetic_hydrodynamic_INSF}
Consider $T_0 > 0$ and $(\varrho, u, \theta) \in L^\infty\left( [0, T_0] ; \mathbb{H}^s_x \right) \cap L^2\left( [0, T_0] ; H^{s+1}_x \right)$,
and define the corresponding macroscopic distribution $f$ as
\begin{equation}\label{eq:f-macro}
f(t, x, v) = \varrho(t, x) \mu(v) + u(t, x) \cdot v \mu(v) + \frac{1}{E(K-1)} \theta(t, x) \left( |v|^2 - E \right) \mu(v).\end{equation}
The macroscopic distribution $f$ satisfies the integral equation
\begin{equation}\begin{split}
\label{eq:NS-IntVNS}
f(t) & = U_\textnormal{NS}(t) f_\textnormal{in} + \Psi_\textnormal{NS}\left[ f, f \right](t) \\
& = U_\textnormal{NS}(t) f_\textnormal{in} + \int_0^t \nabla_x \cdot V_\textnormal{NS}(t-\tau) \mathcal Q^\mathrm{sym} (f(\tau), f(\tau) )(t).
\end{split}\end{equation}
if and only if the coefficients $(\varrho, u, \theta)$ satisfy the incompressible Navier-Stokes-Fourier equations:
$$
\begin{cases}
\partial_t u + \vartheta_\textnormal{inc} u \cdot \nabla_x u = \kappa_\textnormal{inc} \Delta_x u - \nabla_x p, & \nabla_x \cdot u = 0, \\
\\
\partial_t \theta + \vartheta_\textnormal{Bou} u \cdot \nabla_x \theta = \kappa_\textnormal{Bou} \Delta_x \theta, & \nabla_x (\varrho + \theta) = 0,
\end{cases}
$$
where we denoted
\begin{gather*}
\vartheta_\textnormal{inc} = - \frac{\vartheta_1}{2} \left( \frac{d}{E} \right)^{\frac{3}{2}}, \qquad
\vartheta_\textnormal{Bou} = -\frac{1}{ K \sqrt{K(K-1)} } \left( 2 \vartheta_2 + \frac{2 \vartheta_3}{E (K-1)} \right)
\end{gather*}
with $\vartheta_1,\vartheta_2$ and $\vartheta_3$ defined in Lemma \ref{lem:Q_A}.
\end{prop}
With this at hands, one sees that \eqref{eq:NS-IntVNS} provides a \emph{kinetic formulation} of the NSF system \eqref{eq:NSFint}. As said in the Introduction, our approach is ‘‘top-down" so we start with solutions $(\varrho,u,\theta)$ to the Navier-Stokes-Fourier system \eqref{eq:NSFint} to recover information about the kinetic equation \eqref{eq:Kin-Intro} in its mild formulation \eqref{eq:Kin-Mild}. This allows in particular to define the ‘‘kinetic formulation'' to the NSF system
\begin{equation}
\label{eq:reduction_NS_kin}
f_\textnormal{NS}(t)=U_\textnormal{NS}(t)f_\textnormal{in} + \Psi_\textnormal{NS}[f_\textnormal{NS},f_\textnormal{NS}](t).
\end{equation}
Then, on the basis of the above Proposition, the hydrodynamic limit problem consists in proving
$$\lim_{{\varepsilon} \to0}\Big(U^{\varepsilon}(t) f_\textnormal{in} + \Psi^{\varepsilon}\left[ f^{\varepsilon}, f^{\varepsilon} \right](t) \Big)= U_\textnormal{NS}(t) f_\textnormal{in} + \Psi[f_\textnormal{NS}, f_\textnormal{NS}](t) = f_\textnormal{NS}(t)$$
in some precise sense. We point out already that using the representation \eqref{eq:f-macro}, the solution $f_\textnormal{NS}$ to \eqref{eq:reduction_NS_kin} actually belongs to $\HHH$ (see Lemma \ref{lem:NS_parabolic_space}).\color{black}
The key point will be therefore to split suitably $U^{\varepsilon}(\cdot)$ (and $\Psi^{\varepsilon}$ accordingly) in order to prove the convergence. The splitting will be based upon the different parts of the spectrum identified in Theorem \ref{thm:spectral_study}:
$$U^{\varepsilon}(t)=U^{\varepsilon}_\textnormal{NS}(t)+U^{\varepsilon}_\textnormal{wave}(t)+U^{\varepsilon}_\textnormal{kin}(t).$$
Here $U^{\varepsilon}_\textnormal{NS}(t)$ is the leading order term which is expected to converge, as ${\varepsilon}\to0$ towards the linearized Navier-Stokes semigroup $U_\textnormal{NS}(t)$ whereas $U^{\varepsilon}_\textnormal{wave}(t)$ contains the acoustic waves {responsible for dispersive effects (which are absent if the initial data is well-prepared)}, and the combination of these two semigroups can be seen as a \emph{pseudo-hydrodynamic semigroup} encapsulating the macroscopic behavior of the solution $f^{\varepsilon}$. The part $(U^{\varepsilon}_\textnormal{kin}(t))_{t\ge0}$ keeps track of the {pseudo-}kinetic (microscopic) behavior of the solution which is exponentially small in $\frac{t}{{\varepsilon}^2}$ due to the dissipation of entropy, enhanced by the numerous collisions in this hydrodynamic scaling. Since
$$\Psi^{\varepsilon}[f,f](t)=\frac{1}{{\varepsilon}}\int_0^t U^{\varepsilon}(t-\tau)\mathcal Q^{\mathrm{sym}}(f(\tau),f(\tau))\mathrm{d}\tau$$
the above splitting of $U^{\varepsilon}(\cdot)$ induces a similar splitting of the nonlinear term as
$$\Psi^{\varepsilon}[f,f]=\Psi^{\varepsilon}_\textnormal{NS}[f,f]+\Psi^{\varepsilon}_\textnormal{wave}[f,f]+\Psi^{\varepsilon}_\textnormal{kin}[f,f].$$
Precise definitions of these objects are given in the next sections. Before this, we
briefly describe the main difficulties faced in the proof of Theorem \ref{thm:hydrodynamic_limit}:
\begin{enumerate}
\item As said, the major difficulty in establishing the hydrodynamic limit ${\varepsilon} \to 0$ lies in the control of the stiff term $\frac{1}{{\varepsilon}}\mathcal Q\left(f^{\varepsilon},f^{\varepsilon}\right)$. This requires a precise understanding of the asymptotic behavior when ${\varepsilon} \to 0$ of both $U^{\varepsilon}(t)f$ and convolutions of the type
$$\frac{1}{{\varepsilon}} \int_0^t U^{\varepsilon}(t-\tau)\varphi(\tau)\mathrm{d}\tau$$
in various norms, having in mind that $\varphi=\mathcal Q(f,f)$. {Furthermore,} the nonlinear operator $\mathcal Q$ induces a loss of regularity in the sense that $\mathcal Q(f, f) \in H^{\circ}$ when $f \in H^{\bullet}$, where we recall
$$ {H^{\bullet} \subset H \subset H^{\circ} = ( H^{\bullet} )'}.$$
One of the difficulty is therefore to show that convolution by $\left( U^{\varepsilon}(t) \right)_{t \geqslant 0}$ is able to compensate this loss of regularity.
\item As explained in the introduction, in the abstract framework considered here, our minimal assumptions on $\mathcal L$ and $\mathcal Q$ are not sufficient to deduce in a direct way regularization estimates or direct boundedness of $\Psi^{\varepsilon}$ as it is the case for the Boltzmann equation under cut-off assumptions in \cite{BU1991,GT2020} or for the Landau equation \cite{CRT2022}. In a more explicit way, our splitting
$$\mathcal L=\mathcal A+\mathcal B$$
does not induce, in full generality, regularization estimates of the form
$$t \mapsto \| \exp(t \mathcal B) \|_{ \mathscr B\left( H^{\circ} ; H \right) } + \| \exp(t \mathcal B) \|_{ \mathscr B\left( H ; H^{\bullet} \right) } \in L^1_{\text{loc}}((0,T))$$
which would allow to compensate the unboundedness of $\mathcal Q$ in the Duhamel nonlinear term $\Psi^{\varepsilon}$. Inspired by known energy methods introduced for instance in \cite{G2004} and which rely on a suitable dissipation in $L^2$-norm, the abstract functional setting which is adapted to our framework is the one involving spaces of the type
$$L^\infty_t H \cap L^2_t H^{\bullet}.$$
{Such spaces correspond to the above defined space $\GGG, \HHH$ and $\FFF$.}
\end{enumerate}
\subsection{The pseudo-hydrodynamic and pseudo-kinetic projectors}
\label{scn:pseudo_hydro_kinetic}
In this section, we denote by $(Y, \VV, \VV^{\bullet}, \VV^{\circ})$ the spaces $(H, \HH, \HH^{\bullet}, \HH^{\circ})$ under assumption \ref{L1}--\ref{L4}, as well as $(X, \GG, \GG^{\bullet}, \GG^{\circ})$ under the extra assumption \ref{LE}.
We introduce the \textit{pseudo-hydrodynamic projector}, denoted $\mathsf P_\textnormal{hydro}^{{\varepsilon}}$, corresponding to the small eigenvalues identified in Theorem \ref{thm:spectral_study}, and defined as a Fourier multiplier:
\begin{gather*}
\mathsf P_\textnormal{hydro}^{\varepsilon} g := \mathcal F^{-1}_\xi \Big[ \mathsf P({\varepsilon} \xi) \widehat{g}(\xi)\Big].
\end{gather*}
Using the splitting of $\mathsf P$ in \eqref{eq:splitPP}, one sees that it is made up of two parts; one corresponding to the acoustic modes, denoted $\mathsf P^{\varepsilon}_\textnormal{wave}$, and another one corresponding to the Navier-Stokes-Fourier modes, denoted $\mathsf P^{\varepsilon}_\textnormal{NS}$:
$$\mathsf P_{\textnormal{hydro}}^{{\varepsilon}}=\mathsf P^{{\varepsilon}}_{\textnormal{wave}} + \mathsf P^{{\varepsilon}}_{\textnormal{NS}}$$
defined in the following.
\begin{defi}
\label{def:hydrodynamic_projectors}
The projectors $\mathsf P_{\textnormal{wave}}^{{\varepsilon}}$ and $\mathsf P_{\textnormal{NS}}^{{\varepsilon}}$ are defined through their Fourier transform, namely for any $g \in \VV^{\circ}$
\begin{gather*}
\mathsf P_\textnormal{wave}^{\varepsilon} g := \mathcal F^{-1}_{\xi} \Big[\mathsf P_{+\textnormal{wave}} ({\varepsilon} \xi) \widehat{g}(\xi) + \mathsf P_{-\textnormal{wave}} ({\varepsilon} \xi) \widehat{g}(\xi)\Big],\\
\mathsf P_\textnormal{NS}^{\varepsilon} g := \mathcal F^{-1}_{\xi} \Big[\mathsf P_{\textnormal{inc}} ({\varepsilon} \xi) \widehat{g}(\xi) + \mathsf P_{\textnormal{Bou}} ({\varepsilon} \xi) \widehat{g}(\xi)\Big].
\end{gather*}
We also define the limit of the first one as ${\varepsilon} \to 0$ provided by the expansions of the projectors in Theorem \ref{thm:spectral_study}:
\begin{equation*}
\mathsf P_\textnormal{disp} g := \mathcal F^{-1}_{\xi} \Big[\mathsf P_{+\textnormal{wave}}^{(0)} \left( \widetilde{\xi} \right) \widehat{g}(\xi) + \mathsf P_{-\textnormal{wave}}^{(0)} \left( \widetilde{\xi} \right) \widehat{g}(\xi)\Big], \quad \widetilde{\xi} := \frac{\xi}{|\xi|}.
\end{equation*}
\end{defi}
Using these projectors, we define the corresponding partial semigroups:
\begin{gather*}
U^{\varepsilon}_\star(\cdot):= \mathsf P^{\varepsilon}_\star U^{\varepsilon}(\cdot) = U^{\varepsilon} (\cdot)\mathsf P^{\varepsilon}_\star, \quad \star = \textnormal{hydro}, \textnormal{NS}, \textnormal{wave},
\end{gather*}
which gives
\begin{equation*}
U^{\varepsilon}_\textnormal{hydro}(\cdot) := U^{\varepsilon}_\textnormal{NS}(\cdot) + U^{\varepsilon}_\textnormal{wave}(\cdot).
\end{equation*}
More precisely, the above semigroups are defined as follows.
\begin{defi}
\label{def:hydro_semigroups}
The pseudo-Navier-Stokes (diffusive) part $\left(U^{\varepsilon}_\textnormal{NS}(t)\right)_{t\geqslant 0}$ is defined through its Fourier transform for any $g \in \VV^{\circ}$ and $t \geqslant 0$ as
\begin{align}
\label{eq:Unst}
U^{\varepsilon}_\textnormal{NS}(t) g = \mathcal F_\xi^{-1} \Big[& \exp\left( - {\varepsilon}^{-2} t \lambda_\textnormal{Bou}({\varepsilon} \xi) \right) \mathsf P_\textnormal{Bou}({\varepsilon} \xi)\widehat{g}(\xi) \\
\notag
& + \exp\left( - {\varepsilon}^{-2} t \lambda_\textnormal{inc}({\varepsilon} \xi) \right) \mathsf P_\textnormal{inc}({\varepsilon} \xi)\widehat{g}(\xi)\Big] ,
\end{align}
whereas the pseudo-acoustic (dispersive) part $\left(U^{\varepsilon}_\textnormal{wave}(t)\right)_{t\geqslant 0}$ is defined as
\begin{align*}
\label{eq:UWave}
U^{\varepsilon}_\textnormal{wave}(t) g = \mathcal F_\xi^{-1} \Big[ & \exp\left( {\varepsilon}^{-2} t \lambda_{+ \textnormal{wave}}({\varepsilon} \xi) \right) \mathsf P_{+ \textnormal{wave}}({\varepsilon} \xi)\widehat{g}(\xi)\\
\notag
& + \exp\left( {\varepsilon}^{-2} t \lambda_{- \textnormal{wave}}({\varepsilon} \xi) \right) \mathsf P_{- \textnormal{wave}}({\varepsilon} \xi)\widehat{g}(\xi)\Big].
\end{align*}
\end{defi}
Because $\mathsf P_\textnormal{wave}^{\varepsilon} \to \mathsf P_\textnormal{disp}$ as ${\varepsilon} \to 0$, the leading order terms of $U^{\varepsilon}_\textnormal{wave}(t)$ denoted respectively $U^{\varepsilon}_\textnormal{disp}(t)$ will play also a crucial roles in the study of hydrodynamic limits:
\begin{defi}
The dispersive semigroup $\left(U^{\varepsilon}_\textnormal{disp}(t)\right)_{t\geqslant 0}$ is defined as
\begin{align*}
U^{\varepsilon}_\textnormal{disp}(t)g = \mathcal F_\xi^{-1} \Big[ & \exp\left( i c {\varepsilon}^{-1} t|\xi| - t \kappa_\textnormal{wave} | \xi |^2 \right) \mathsf P^{(0)}_{+ \textnormal{wave}}\left( \widetilde{\xi} \right) \widehat{g}(\xi) \\
& + \exp\left( - i c {\varepsilon}^{-1} t|\xi| - t \kappa_\textnormal{wave} | \xi |^2 \right) \mathsf P^{(0)}_{- \textnormal{wave}}\left( \widetilde{\xi} \right) \widehat{g}(\xi) \Big].
\end{align*}
\end{defi}
\color{black}
\begin{rem}
Recall that $U_\textnormal{NS}(t)$ and $V_\textnormal{NS}(t)$ were introduced in Definition \ref{def:U_NS_V_NS}. Observe that $U_\textnormal{NS}(t)$ is the leading order term in the expansion of $U^{\varepsilon}_\textnormal{NS}(t)$ while $\nabla_x \cdot V_{\textnormal{NS}}(t)$ is the leading order term of $U^{\varepsilon}_\textnormal{NS}(t)$ on $\nul(\mathcal L)^\perp$, that is to say
$${U^{\varepsilon}_\textnormal{NS}(t) \approx U_\textnormal{NS}(t), \qquad \left(U^{\varepsilon}_\textnormal{NS}(t)\right)_{| \nul(\mathcal L)^\perp } \approx {\varepsilon} \nabla_{x} \cdot V_\textnormal{NS}(t)}$$
as will be {exploited} in Lemma \ref{lem:asymptotic_equiv_NS_semigroup}.
\end{rem}
Note that the projectors $\mathsf P_{\textnormal{inc}}^{(0)}$, $\mathsf P_{\textnormal{Bou}}^{(0)}$ and $\mathsf P_{\pm \textnormal{wave}}^{(0)}$ are macroscopic in the sense that they take values in
$$\nul(\mathcal L) = \left\{ \left(\varrho + u \cdot v + \theta \left( |v|^2 - E \right) \right) \mu \, ; \, \varrho, u, \theta \in L^2\left( \mathbb R^d \right) \right \}$$
and vanish on its orthogonal, thus they can be characterized using the macroscopic components $\varrho, u$ and $\theta$ (see Remark \ref{rem:macro_representation_spectral}). Similarly, the first order projectors $\mathsf P^{(1)}_\textnormal{inc}$ and $\mathsf P^{(1)}_\textnormal{Bou}$ restricted to $\nul(\mathcal L)^\perp$ can be characterized in such a way, which will be useful for describing $V_\textnormal{NS}$.
\begin{prop}
\label{prop:macro_representation_spectral}
The zeroth order projector related to the Navier-Stokes (incompressible) mode is characterized for $f=f(x,v) \in L^2_x(H_v)$ by
$$\varrho\left[ \mathsf P^{(0)}_\textnormal{inc} f \right] = \theta\left[ \mathsf P^{(0)}_\textnormal{inc} f \right] = 0, \qquad u \left[ \mathsf P^{(0)}_\textnormal{inc} f \right] = \frac{E}{d} \mathbb{P} u[f],$$
the one related to the Fourier (Boussineq) mode for $f=f(x, v)$ by
$$u\left[ \mathsf P^{(0)}_\textnormal{Bou} f \right] =\varrho\left[ \mathsf P^{(0)}_\textnormal{Bou} f \right] + \theta\left[ \mathsf P^{(0)}_\textnormal{Bou} f \right] = 0,$$
$$\sqrt{K(K-1)} \big( (K-1) \varrho \left[ \mathsf P^{(0)}_\textnormal{Bou} f \right] - \theta \left[ \mathsf P^{(0)}_\textnormal{Bou} f \right] \big) = (K-1) \varrho[f] - \theta[f] \,,$$
and the ones related to the acoustic modes (recall that $( \mathrm{Id} - \mathbb{P}) \nabla_x = \nabla_x $) for $f=f(x, v)$ by
$$(K-1) \varrho\left[ \mathsf P^{(0)}_{\pm \textnormal{wave}} f \right] - \theta\left[ \mathsf P^{(0)}_{\pm \textnormal{wave}} f \right] = 0,$$
$$\sqrt{2K} u\left[ \mathsf P^{(0)}_{\pm \textnormal{wave}} f \right] = (- \Delta_x)^{-\frac{1}{2}} \nabla_x \left( \varrho[f] + \theta[f]\right) \pm c \left( \mathrm{Id} - \mathbb{P} \right) u[f],$$
$$\sqrt{2K} \left( 1 - \frac{1}{K} \right)\left( \varrho \left[ \mathsf P^{(0)}_{\pm \textnormal{wave}} f \right] + \theta \left[ \mathsf P^{(0)}_{\pm \textnormal{wave}} f \right] \right) = ( \varrho[f] + \theta[f]) \pm c (- \Delta_x)^{-\frac{1}{2}} \nabla_x \cdot u[f].$$
The first order projectors related to the Navier-Stokes (incompressible) mode satisfy the identities for $f(x, \cdot) \perp \nul(\mathcal L)$
$$\varrho\left[ \mathsf P^{(1)}_\textnormal{inc} f \right] = \theta\left[ \mathsf P^{(1)}_\textnormal{inc} f \right] = 0,$$
$$\left( \frac{E}{d} \right)^{\frac{3}{2}} u \left[ \nabla_x \cdot \mathsf P^{(1)}_\textnormal{inc} f \right] = \mathbb{P} \left( \nabla_x \cdot \langle f, \mathcal L^{-1}{\mathbf{A}} \rangle_{H} \right),$$
and the first order coefficient related to the Fourier (Boussinesq) mode for $f(x, \cdot) \perp \nul(\mathcal L)$
$$u\left[ \mathsf P^{(1)}_\textnormal{Bou} f \right] = \varrho\left[ \mathsf P^{(1)}_\textnormal{Bou} f \right] + \theta\left[ \mathsf P^{(1)}_\textnormal{Bou} f \right] = 0,$$
$$ (K-1) \varrho\left[ \mathsf P^{(1)}_\textnormal{Bou} f \right] - \theta\left[ \mathsf P^{(1)}_\textnormal{Bou} f \right] = \frac{1}{\sqrt{K(K-1)}}\langle f, \mathcal L^{-1}{\mathbf{B}} \rangle_{H}.$$
\end{prop}
We end this section by defining, in a similar way, the pseudo-kinetic part of the whole linearized semigroup
\begin{defi}\label{def:kinetic_semigroups}
We define the \textit{pseudo-kinetic projector} $\mathsf P_\textnormal{kin}^{\varepsilon}$ through its Fourier transform for any $g \in \VV$
\begin{equation*}
\mathsf P_\textnormal{kin}^{\varepsilon} g := \mathcal F_\xi^{-1} \Big[ \left(\mathrm{Id} - \mathsf P({\varepsilon} \xi)\right) \widehat{g}(\xi) \Big] = \left(\mathrm{Id} - \mathsf P^{\varepsilon}_\textnormal{hydro}\right) g,
\end{equation*}
as well as the corresponding semigroup $\left(U^{\varepsilon}_\textnormal{kin}(t)\right)_{t\ge0}$
\begin{align*}
U^{\varepsilon}_\textnormal{kin}(t)g := \mathcal F^{-1}_\xi \Big[ U_{\xi}({\varepsilon}^{-2}t)\left( \mathrm{Id} - \mathsf P({\varepsilon} \xi) \right) \widehat{g}(\xi) \Big]=\mathcal F^{-1}_\xi \Big[ \left( \mathrm{Id} - \mathsf P({\varepsilon} \xi) \right)U_{\xi}({\varepsilon}^{-2}t)\widehat{g}(\xi) \Big].
\end{align*}
\end{defi}
\color{black}
\subsection{Decomposition of the solution}
With the above definitions, we obtain the following decomposition of the semigroup $U^{\varepsilon}(t)$ as
\begin{equation}\label{eq:decompUeps}
U^{\varepsilon}(t) = U^{\varepsilon}_\textnormal{hydro}(t) + U^{\varepsilon}_\textnormal{kin}(t)
=U^{\varepsilon}_\textnormal{NS}(t)+U^{\varepsilon}_\textnormal{wave}(t) + U^{\varepsilon}_\textnormal{kin}(t)\qquad \quad t\ge0\end{equation}
and we split the nonlinear integral operator $\Psi^{\varepsilon}$ accordingly, that is to say as a hydrodynamic part and a kinetic part:
$$ \Psi^{\varepsilon}[f,g](t) = \Psi^{\varepsilon}_\textnormal{hydro} [f, g](t) + \Psi^{\varepsilon}_\textnormal{kin} [f, g](t),$$
with
$$ \Psi^{\varepsilon}_\star [f, g](t) := \mathsf P^{\varepsilon}_\star \Psi^{\varepsilon}[f,g](t) = \frac{1}{{\varepsilon}} \int_0^t U^{\varepsilon}_\star(t - \tau) \mathcal Q^\mathrm{sym} (f(\tau), g(\tau)) \mathrm{d} \tau.$$
The main idea behind the proof of Theorems \ref{thm:hydrodynamic_limit} or \ref{thm:hydrodynamic_limit-gen_symmetric} or \ref{thm:hydrodynamic_limit-gen_degenerate} is to consider an \textit{a priori} decomposition of the unknown $f^{\varepsilon}$ in $\GGG + \FFF + \HHH$:
\begin{equation}\label{eq:decompfeps}
f^{\varepsilon} = f^{\varepsilon}_\textnormal{kin} + f^{\varepsilon}_\textnormal{mix} + f^{\varepsilon}_\textnormal{hydro}, \qquad f^{\varepsilon}_\textnormal{kin} \in \GGG, ~ f^{\varepsilon}_\textnormal{mix} \in \FFF, ~ f^{\varepsilon}_\textnormal{hydro} \in \HHH\end{equation}
which will enable us to reduce the construction of a solution of \eqref{eq:Kin-Mild} to that of a solution to some appropriate system of equations for all the new unknowns
$$(f^{\varepsilon}_\textnormal{kin},f^{\varepsilon}_\textnormal{mix},f^{\varepsilon}_\textnormal{hydro}) \in \GGG\times \FFF\times \HHH.$$
The term $f^{\varepsilon}_\textnormal{mix}$ is a coupling term between the purely kinetic $f^{\varepsilon}_\textnormal{kin}$ and macroscopic $f^{\varepsilon}_\textnormal{hydro}$ parts and which need to be studied separately.
Let us dive more deeply in such a strategy, aiming to determine the system solved by $(f^{\varepsilon}_\textnormal{kin},f^{\varepsilon}_\textnormal{mix},f^{\varepsilon}_\textnormal{hydro})$. The splitting \eqref{eq:decompfeps} induces the \textit{a priori} decomposition of the kinetic part of the non-linear term:
$$\Psi^{\varepsilon}_\textnormal{kin}[f^{\varepsilon}, f^{\varepsilon}] = \Psi^{\varepsilon}_\textnormal{kin} \left[ f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{kin} \right] + 2 \Psi^{\varepsilon}_\textnormal{kin} \left[f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{hydro} + f^{\varepsilon}_\textnormal{mix}\right] + A_\textnormal{mix} $$
where we expect the first two terms to belong to $\GGG$ and the third one
$$A_\textnormal{mix}:=\Psi^{\varepsilon}_\textnormal{kin} \left[f^{\varepsilon}_\textnormal{mix} + f^{\varepsilon}_\textnormal{hydro} , f^{\varepsilon}_\textnormal{mix} + f^{\varepsilon}_\textnormal{hydro} \right] \in \FFF.$$
In the same way, we introduce the following \textit{a priori} decomposition of the hydrodynamic part of the non-linearity, which will only be used to make the following presentation more compact:
$$\Psi^{\varepsilon}_\textnormal{hydro}\left[f^{\varepsilon}, f^{\varepsilon}\right]=\Psi^{\varepsilon}_\textnormal{hydro} \left[f^{\varepsilon}_\textnormal{hydro}, f^{\varepsilon}_\textnormal{hydro}\right] + A_\textnormal{hydro}\left[f^{\varepsilon}_\textnormal{hydro}\right] + B_\textnormal{hydro}$$
where we denoted
$$A_\textnormal{hydro}\left[f^{\varepsilon}_\textnormal{hydro}\right]=2 \Psi^{\varepsilon}_\textnormal{hydro}\left[f^{\varepsilon}_\textnormal{hydro} , f^{\varepsilon}_\textnormal{mix} + f^{\varepsilon}_\textnormal{kin}\right], \qquad B_\textnormal{hydro}:=\Psi^{\varepsilon}_\textnormal{hydro} \left[f^{\varepsilon}_\textnormal{mix} + f^{\varepsilon}_\textnormal{kin} , f^{\varepsilon}_\textnormal{mix} + f^{\varepsilon}_\textnormal{kin} \right]\,.$$
We consider an \emph{arbitrary} system of equations for each part, where $A_\textnormal{mix}$ is assigned to the equation for $f^{\varepsilon}_\textnormal{mix}$:
\begin{equation*}\label{eq:systemE}
\begin{cases}
f^{\varepsilon}_\textnormal{kin}(t) = U^{\varepsilon}_\textnormal{kin}(t) f_\textnormal{in} + \Psi^{\varepsilon}_\textnormal{kin} \left[ f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{kin} \right](t) + 2 \Psi^{\varepsilon}_\textnormal{kin} \left[f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{hydro} + f^{\varepsilon}_\textnormal{mix}\right],\\
\\
f^{\varepsilon}_\textnormal{mix}(t) = A_\textnormal{mix}(t),\\
\\
f^{\varepsilon}_\textnormal{hydro}(t) = U^{\varepsilon}_\textnormal{hydro}(t) f_\textnormal{in} + \Psi^{\varepsilon}_\textnormal{hydro}\left[f^{\varepsilon}_\textnormal{hydro}, f^{\varepsilon}_\textnormal{hydro}\right](t) + A_\textnormal{hydro}\left[f^{\varepsilon}_\textnormal{hydro}\right](t) + B_\textnormal{hydro}(t).
\end{cases}
\end{equation*}
We see that, under the \emph{ansatz} \eqref{eq:decompfeps}, solving \eqref{eq:Kin-Mild} amounts to solve the above system for
$$(f^{\varepsilon}_\textnormal{kin},f^{\varepsilon}_\textnormal{mix},f^{\varepsilon}_\textnormal{hydro}) \in \GGG\times \FFF\times \HHH,$$
{as well as proving the uniqueness of solutions to the original equation \eqref{eq:Kin-Intro} since our system is arbitrary.}
In the hydrodynamic limit, we moreover expect $f^{\varepsilon}_\textnormal{hydro}$ to be the leading term of $f^{\varepsilon}$ converging to $f_\textnormal{NS}$, the other two terms being expected to converge to zero. Notice that we look for the solution $f^{\varepsilon}_\textnormal{hydro} \in \HHH$ and, as observed already, this is the functional space to which $f_\textnormal{NS}$ actually belongs. In other words, we expect
$$\lim_{{\varepsilon}\to0}|\hskip-0.04cm|\hskip-0.04cm| f_\textnormal{hydro}^{\varepsilon}-f_\textnormal{NS} |\hskip-0.04cm|\hskip-0.04cm|_{\HHH}=0.$$
Moreover, we need to prove that, in the above system, all terms are well-defined and belong to the desired spaces. This is one the main technical difficulties of the work and, as explained in the introduction, will follow from a careful study of the behaviour of the various semigroups $U^{\varepsilon}_\star(\cdot)$ as well as their action on convolutions. See Section \ref{scn:study_physical_space} for full proofs.
\subsection{Removing the acoustic initial layer and the hydrodynamic limit}
To justify the convergence of $f^{\varepsilon}_\textnormal{hydro}$ towards $f_\textnormal{NS}$, we actually will need to rewrite the equation for $f^{\varepsilon}_\textnormal{hydro}(t)$ by removing its leading order terms, namely a Navier-Stokes part and an acoustic part . The construction of those leading order terms rely on already existing theory for the Navier-Stokes equations and the wave equation.
More precisely, we split the hydrodynamic part $f^{\varepsilon}_\textnormal{hydro}$ into an oscillating one $f^{\varepsilon}_\textnormal{disp}(t)$ (which is explicit) and another one $f^{\varepsilon}_\textnormal{NS}(t)$ that will be shown to be an approximation of the hydrodynamic limit $f_\textnormal{NS}(t)$:
\begin{equation}\label{eq:fepshyd}
f^{\varepsilon}_\textnormal{hydro}(t) = f^{\varepsilon}_\textnormal{disp}(t) + f^{\varepsilon}_\textnormal{NS}(t), \qquad
f^{\varepsilon}_\textnormal{disp}(t) := U^{\varepsilon}_\textnormal{disp}(t) f_{\textnormal{in}}\,.
\end{equation}
Inserting this into the equation solved by $f^{\varepsilon}_\textnormal{hydro}$, we see that $f^{\varepsilon}_\textnormal{NS}(t)$ satisfies
\begin{multline*}
f^{\varepsilon}_\textnormal{NS}(t) = \left(U^{\varepsilon}_\textnormal{hydro}(t)f_\textnormal{in} - U^{\varepsilon}_\textnormal{disp}(t)f_\textnormal{in}\right) + \Psi^{\varepsilon}_\textnormal{hydro}\left[ f^{\varepsilon}_\textnormal{NS}, f^{\varepsilon}_\textnormal{NS} \right](t)\\+ 2 \Psi^{\varepsilon}_\textnormal{hydro}\left[f^{\varepsilon}_\textnormal{disp}, f^{\varepsilon}_\textnormal{NS}\right](t) + A_\textnormal{hydro}\left[f^{\varepsilon}_\textnormal{NS}\right](t)\\
+ A_\textnormal{hydro}\left[f^{\varepsilon}_\textnormal{disp}\right](t) + B_\textnormal{hydro}(t) + \Psi^{\varepsilon}_\textnormal{hydro}\left[ f_\textnormal{disp}^{\varepsilon}, f_\textnormal{disp}^{\varepsilon} \right](t).
\end{multline*}
We further split $f^{\varepsilon}_\textnormal{NS}$ into its \emph{a priori} limit $f_\textnormal{NS}(t)$ and an error term $g^{\varepsilon}(t)$:
$$f^{\varepsilon}_\textnormal{NS}(t) = f_\textnormal{NS}(t) + g^{\varepsilon}(t),$$
so that, using \eqref{eq:reduction_NS_kin}, the part $g^{\varepsilon}(t)$ satisfies the equation
\begin{align*}
g^{\varepsilon}(t) &= \left(U^{\varepsilon}_\textnormal{hydro}(t)f_{\textnormal{in}} - U_\textnormal{NS}(t)f_{\textnormal{in}} - U^{\varepsilon}_\textnormal{disp}(t)f_\textnormal{in}\right) + \left(\Psi^{\varepsilon}_\textnormal{hydro}\left[ f_\textnormal{NS} , f_\textnormal{NS} \right](t) - \Psi_\textnormal{NS}\left[ f_\textnormal{NS} , f_\textnormal{NS} \right](t)\right ) \\
&\phantom{+++} + 2 \Psi^{\varepsilon}_\textnormal{hydro}\left[ f_\textnormal{NS}, g^{\varepsilon} \right](t) + 2 \Psi^{\varepsilon}_\textnormal{hydro}\left[f^{\varepsilon}_\textnormal{disp}, g^{\varepsilon} \right](t) + A_\textnormal{hydro}\left[g^{\varepsilon}\right](t) + \Psi^{\varepsilon}_\textnormal{hydro}\left[g^{\varepsilon}, g^{\varepsilon}\right](t) \\
&\phantom{+++} + 2 \Psi^{\varepsilon}_\textnormal{hydro}\left[ f^{\varepsilon}_\textnormal{disp}, f_\textnormal{NS} \right](t) + \Psi^{\varepsilon}_\textnormal{hydro}\left[ f^{\varepsilon}_\textnormal{disp}, f^{\varepsilon}_\textnormal{disp}\right](t) \\
&\phantom{+++} + A_\textnormal{hydro}\left[f_\textnormal{NS}\right](t) + A_\textnormal{hydro}\left[ f^{\varepsilon}_\textnormal{disp}\right](t) + B_\textnormal{hydro}(t) \\
&=2 \Psi^{\varepsilon}_\textnormal{hydro}\left[ g^{\varepsilon}, f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp} + f^{\varepsilon}_\textnormal{mix} + f^{\varepsilon}_\textnormal{kin}\right](t) + \Psi^{\varepsilon}_\textnormal{hydro}\left[g^{\varepsilon}, g^{\varepsilon}\right](t) + \mathcal S^{\varepsilon}(t)
\end{align*}
Here, we have denoted the vanishing non-linear source term (which depends on $f^{\varepsilon}_\textnormal{kin}$ and $f^{\varepsilon}_\textnormal{mix}$ but not on $g^{\varepsilon}$), as
\begin{subequations}\label{eq:source}
\begin{equation}\label{eq:sourceSS}
\mathcal S^{\varepsilon}(t) =\mathcal S^{\varepsilon}_1(t) + \mathcal S^{\varepsilon}_2(t) + \mathcal S^{\varepsilon}_3[f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{mix}](t)
\end{equation} where {the first two parts depend only on $f^{\varepsilon}_\textnormal{disp}$ through $f_\textnormal{in}$ and $f_\textnormal{NS}$, which are considered given}
\begin{equation}\label{eq:sourceS1S2} \begin{split}
\mathcal S^{\varepsilon}_1(t)&=\left(U^{\varepsilon}_\textnormal{hydro}(t)f_{\textnormal{in}} - U_\textnormal{NS}(t)f_{\textnormal{in}} - U^{\varepsilon}_\textnormal{disp}(t)f_{\textnormal{in}} \right) + \left(\Psi^{\varepsilon}_\textnormal{hydro}\left[ f_\textnormal{NS} , f_\textnormal{NS} \right](t) - \Psi_\textnormal{NS}\left[ f_\textnormal{NS} , f_\textnormal{NS} \right](t)\right) \\
\mathcal S^{\varepsilon}_2(t) &= \Psi^{\varepsilon}_\textnormal{hydro}\left[ f^{\varepsilon}_\textnormal{disp}, 2 f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp} \right](t)\,,\end{split}\end{equation}
and {the third one also depends on the partial solutions $f^{\varepsilon}_\textnormal{kin}$ and $f^{\varepsilon}_\textnormal{mix}$}
\begin{equation}\label{eq:sourceS3} \begin{split}
\mathcal S^{\varepsilon}_3[f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{mix}](t) &= A_\textnormal{hydro}\left[f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp}\right](t) + B_\textnormal{hydro}(t)\\
&= \Psi^{\varepsilon}_\textnormal{hydro}\left[ f^{\varepsilon}_\textnormal{kin} + f^{\varepsilon}_\textnormal{mix} , f^{\varepsilon}_\textnormal{kin} + f^{\varepsilon}_\textnormal{mix} + f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp} \right](t).
\end{split}\end{equation}
\end{subequations}
\subsection{Summary of the proof}\label{sec:detail-sum} The above technical splitting allows us to consider the solution to \eqref{eq:Kin-Mild} we aim to construct in the form
\begin{equation*}\begin{split}
f^{\varepsilon}(t)&=f^{\varepsilon}_\textnormal{kin}(t)+f^{\varepsilon}_\textnormal{mix}(t)+f^{\varepsilon}_\textnormal{hydro}(t)\\
&=f^{\varepsilon}_\textnormal{kin}(t)+f^{\varepsilon}_\textnormal{mix}(t)+f^{\varepsilon}_\textnormal{disp}(t)+ f_\textnormal{NS}(t)+g^{\varepsilon}(t)
\end{split}
\end{equation*}
where $f_\textnormal{NS}(\cdot)$ as well as $f_\textnormal{in}$ (and thus $f^{\varepsilon}_\textnormal{disp}(t) = U^{\varepsilon}_\textnormal{disp}(t) f_\textnormal{in}$) are functions to be considered as fixed parameters since they depend only on the initial datum $f_\textnormal{in}$ (and ${\varepsilon}$).
According to the analysis performed in Section \ref{scn:study_physical_space} (see Lemma \ref{lem:decay_regularization_kinetic_semigroup}, Lemma \ref{lem:decay_semigroups_hydro} and Lemma \ref{lem:NS_parabolic_space} respectively) that
$$|\hskip-0.04cm|\hskip-0.04cm| U^{\varepsilon}_\textnormal{kin}(\cdot) f_\textnormal{in} |\hskip-0.04cm|\hskip-0.04cm|_{\GGG} \lesssim 1, \qquad |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_\textnormal{disp} |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} \lesssim 1, \qquad |\hskip-0.04cm|\hskip-0.04cm| f_\textnormal{NS} |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} \lesssim 1.$$
Since $f_\textnormal{NS}$ is entirely determined by the solutions $(\varrho,u,\theta)$ to the NSF system \eqref{eq:main-NSF}, we point out that, in the definition of the space $\HHH$, we choose the function $\phi$ to be \emph{exactly} the solution $f_\textnormal{NS}$. This corresponds, in Eq. \eqref{eq:wphieta}, to the choice of the weight function
$$w_{f_\textnormal{NS},\eta}(t)=\exp\left( \frac{1}{2 \eta^2} \int_0^t\| |\nabla_x|^{1-\alpha} f_\textnormal{NS}(\tau) \|_{\rSSsp{s}}^2 \mathrm{d} \tau \right) \qquad t \ge0,$$
with $\eta >0$ is a parameter which is still to free to be chosen {suitably small for the upcoming fixed point argument to work}. The above system considered in Section \ref{scn:reduction_problem} writes now:
\begin{equation}\label{eq:systemKinMixG}
\begin{cases}
f^{\varepsilon}_\textnormal{kin}(t) &= U^{\varepsilon}_\textnormal{kin}(t) f_\textnormal{in} + \Psi^{\varepsilon}_\textnormal{kin} \left[ f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{kin} \right](t) + 2 \Psi^{\varepsilon}_\textnormal{kin}\left[ f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{hydro} + f^{\varepsilon}_\textnormal{mix}\right](t),\\
\\
f^{\varepsilon}_\textnormal{mix}(t) &=\Psi^{\varepsilon}_\textnormal{kin}\left[\left(f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp}\right)+ f_\textnormal{mix}^{\varepsilon}+g^{\varepsilon}\, , \,\left(f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp}\right)+ f_\textnormal{mix}^{\varepsilon}+g^{\varepsilon} \right](t),\\
\\
g^{\varepsilon}(t) &= \Phi^{\varepsilon}[ f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{mix} ](t) g^{\varepsilon}(t)+
\Psi^{\varepsilon}_\textnormal{hydro}\left[ g^{\varepsilon}, g^{\varepsilon}\right] + \mathcal S^{\varepsilon}(t).
\end{cases}
\end{equation}
We will construct a solution $\left( f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{mix}, g^{\varepsilon} \right)$ of this system in the space
$ \GGG \times \FFF \times \HHH $ and more specifically, in a product of suitable balls in such spaces. This is achieves through a suitable use of Banach fixed point Theorem in the case of Assumptions \ref{BE} whereas, under Assumptions \ref{BED}, the situation is much more involved and we adapt a Picard-like scheme to construct our solution $(f^{\varepsilon}_\textnormal{kin},f^{\varepsilon}_\textnormal{mix},g^{\varepsilon})$. \\
As said already, in order to study the system \eqref{eq:systemKinMixG}, we first need to prove that all the various terms make sense under the \emph{ansatz} \eqref{eq:decompfeps}, that is we need to show that the various bilinear terms $\Psi^{\varepsilon}_\textnormal{kin} \left[ f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{kin} \right]$, $\Psi^{\varepsilon}_\textnormal{kin}\left[ f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{hydro} + f^{\varepsilon}_\textnormal{mix}\right]$ are defined and belong to $\GGG$ if $\left( f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{mix}, g^{\varepsilon} \right) \in \GGG \times \FFF \times \HHH$, that the bilinear term appearing in the equation for $f_\textnormal{mix}^{\varepsilon}$ is well defined and belong to $\FFF$ while the bilinear terms involved in the equation for $g^{\varepsilon}$ are well-defined and belong to $\HHH.$ This is done in Section \ref{sec:Bilin} which is based on the thorough analysis led in Section \ref{scn:study_physical_space} of the various semigroups $U^{\varepsilon}_\textnormal{kin}(\cdot)$, $U^{\varepsilon}_\textnormal{hydro}(\cdot)$ and convolutions of the type
$$\frac{1}{{\varepsilon}} U^{\varepsilon}_\textnormal{kin}(\cdot) \ast \varphi \quad \text{ and } \quad \frac{1}{{\varepsilon}}U^{\varepsilon}_\textnormal{hydro}(\cdot) \ast \varphi.$$
\section{Spectral analysis of the linearized operator}
\label{scn:spectral_study}
This section is mainly devoted to the proof of the main spectral result Theorem \ref{thm:spectral_study} in the Introduction about the linearized operator $\mathcal L_{\xi} = \mathcal L - i (v \cdot \xi).$
\subsection{Description of the novel spectral approach} \label{sec:newspec}
We give here a precise description of the main steps of our approach to prove the above two main results.
In order to prove our main spectral result, we adopt a ``direct method'' which appears much simpler than the original method of \cite{EP1975}.
First, to study the spectrum of $\mathcal L_{\xi}$, we use the fact that, for $\xi=0,$ the spectrum of $\mathcal L_{0}=\mathcal L$ is explicit thanks to Assumption \ref{L3} and show that, for $|\xi|$ small enough, the structure of $\mathfrak{S}(\mathcal L_{\xi})$ is similar to that of $\mathfrak{S}(\mathcal L),$ i.e. there exists some explicit value $\alpha_{0}$ and $\gamma >0$ such that
$$\mathfrak{S}(\mathcal L_{\xi}) \cap \left\{z \in \mathbb{C}_{+}\;;\;\mathrm{Re}\, z >-\gamma\right\}, \qquad \forall |\xi | \leqslant \alpha_0$$
consists in a finite number of eigenvalues. Such a localization of the spectrum is obtained here \emph{without resorting to any compactness argument}. This is the main contrast with respect to the original work \cite{EP1975} whose approach disseminates in the literature.
The localization of the spectrum is not enough for the purpose of the paper and we also need to compute explicitly {the asymptotic spectrum $\mathfrak{S}(\mathcal L_{\xi}) \cap \left\{z \in \mathbb{C}_{+}\;;\;\mathrm{Re}\, z >-\gamma\right\}$ and the associated spectral projector}. This is done in a quantitative way, using Kato's perturbation theory as developed in \cite{K1995}.
Typically, one observes that, since for any $\xi \in \mathbb R^{d}$, $\mathcal L_{\xi}$ is a perturbation of $\mathcal L$ by the multiplication operator with $-i(v\cdot \xi)$, for any $z \notin \mathfrak{S}(\mathcal L_{\xi}) \cup \mathfrak{S}(\mathcal L)$, the following expansion formulae are valid for $N \geqslant 0$:
\begin{equation}
\label{eq:factorization_L_ivxi_left}
\mathcal R(z,\mathcal L_{\xi}) = \sum_{n = 0}^{N-1} \mathcal R(z,\mathcal L) \Big((- i v \cdot \xi) \mathcal R(z,\mathcal L)\Big)^n + \mathcal R(z,\mathcal L_{\xi}) \Big((- i v \cdot \xi) \mathcal R(z,\mathcal L)\Big)^N,\end{equation}
as well as
\begin{equation} \label{eq:factorization_L_ivxi_right}
\mathcal R(z,\mathcal L_{\xi}) = \sum_{n = 0}^{N-1} \mathcal R(z,\mathcal L) \Big((- i v \cdot \xi) \mathcal R(z,\mathcal L)\Big)^n + \mathcal R(z,\mathcal L_{\xi}) \Big((- i v \cdot \xi) \mathcal R(z,\mathcal L)\Big)^N.
\end{equation}
Various choice of the parameter $N\geq1$ would allow us to recover estimates on $\mathcal R(z,\mathcal L_{\xi})$ from known result on $\mathcal R(z,\mathcal L)$ and provide the asymptotic expansion of the eigenvalue and eigen-projectors. In a more specific way, the proof of Theorem \ref{thm:spectral_study} is done according to the following roadmap:
\begin{itemize}
\item In Lemma \ref{lem:localization_spectrum}, we show that the spectrum of $\mathcal L_\xi := \mathcal L - i v \cdot \xi$ contained in some right half plane is confined in a ball of radius of order $\xi$ centered around the origin, and establish some bounds on the resolvent.
\item In Lemma \ref{lem:expansion_projection}, we prove that the spectral projector associated with this part of the spectrum has a first order expansion as $\xi \to 0$. {To study the aforementionned part of the spectrum, we then introduce $\mathbb{L}_{ \xi}$ which is a matrix conjugated to the restriction of $\mathcal L_\xi$ to the corresponding stable subspace (sum of eigenspaces)}, thus allowing to rely on perturbation theory in finite dimension.
\item We establish in Lemma \ref{lem:rectified_operator} some invariance \textit{(isotropy)} properties satisfied by $\mathbb{L}_{ \xi}$ and give its first order expansion.
\item A block matrix representation of $\mathbb{L}_{ \xi}$ is presented in Lemma \ref{lem:rectified_block_matrix}, thus identifying its only multiple eigenvalue, and isolating it from the three simple remaining ones as is shown in Lemma \ref{lem:expansion_rectified_matrix}.
\item From that point on, we use finite dimensional perturbation theory and show that $\mathbb{L}_{ \xi}$ is diagonalizable and establish a second order expansion of its eigenvalues as well as a first order expansion of its spectral projectors in Lemma \ref{lem:diagonalization_rectified}, from which we deduce the spectral decomposition of the original operator $\mathcal L_\xi$, as well as expansions of the projectors in Lemma \ref{lem:expansion_projectors}.
\end{itemize}
Finally, we combine the resolvent bounds for $| \xi | \ll 1$ from the previous lemmas, and use a hypocoercivity theorem from \cite{D2011} for $ | \xi | \gtrsim$ to obtain an uniform exponential decay in $H$ of the semigroup generated by $\mathcal L_\xi$ on the stable subspace associated with the rest of the spectrum. We then improve this uniform decay estimate in $H$ as an {integral regularization and decay in $H-H^{\bullet}$ and $H^{\circ}-H$} by combining it with an energy method.
\subsection{The spatially homogeneous setting}
Before undertaking the program described here above, it is important to recall the spectral picture in the spatially homogeneous setting corresponding to $\xi=0$. Assumptions \ref{L1}--\ref{L4} directly give the localization of the spectrum and the fact that $0$ is a semi-simple eigenvalue of $\mathcal L$ with $d+2$-dimensional (geometric) multiplicity. Associated to such an eigenvalue, the spectral projection
$$\mathsf P:=\frac{1}{2i\pi}\oint_{|z|=r}\mathcal R(z,\mathcal L)\mathrm{d} z$$
has the following properties:
Let us present the properties of the orthogonal projection on the null space of $\mathcal L$.
\begin{prop}[\textit{\textbf{Representation formulae involving $\mathsf P$}}]
\label{prop:representation_P}
We recall the macroscopic (fluctuations of) mass~$\varrho \in \mathbb R$, velocity $u \in \mathbb R^d$ and temperature~$\theta \in \mathbb R$ as defined in \eqref{eq:fluctuat}. Under Assumptions \ref{L1}-\ref{L2}, the spectral projector $\mathsf P$ on the null-space $\nul(\mathcal L)$ is $H$-orthogonal and has the following explicit representation: \begin{equation}\label{eq:representation_P}
\mathsf P f(v) = \left(\varrho_f + u_f \cdot v + \frac{\theta_f}{E(K-1)} \left( |v|^2 - E \right)\right) \mu(v),
\end{equation}
as well as, denoting $\Pi_\omega=\mathrm{Id}-\omega\otimes\omega$ the orthogonal projection on $\omega^\perp= \left\{ u \in \mathbb R^d ~| ~ u \perp \omega \right\}$ for any $\omega \in \mathbb S^{d-1}$:
\begin{align*}
\mathsf P f(v) = \left( \Pi_\omega u_f \right) \cdot v \mu(v) & + \frac{1}{K(K-1)} \Big( (K-1) \varrho_f - \theta_f \Big) \left( K - \frac{|v|^2}{E} \right) \mu(v) \\
& + \frac{1}{d c^2} (\varrho_f + \theta_f) |v|^2 \mu(v) + \big( \left( \mathrm{Id} - \Pi_\omega \right) u_f \big) \cdot v \mu(v)
\end{align*}
where we introduced the speed of sound $c$ in \eqref{eq:speed_sound}. More compactly, {in terms of the eigenfunctions $\psi_{\pm \textnormal{wave}}$ and $\psi_\textnormal{Bou}$ defined in \eqref{eq:def_psi_wave}--\eqref{eq:def_psi_Bou}}
\begin{equation}
\label{eq:representation_hydrodynamic_modes}
\begin{aligned}
\mathsf P f(v)= \frac{d}{E} \Pi_\omega \langle f, v\mu \rangle_H v \mu &+ \langle f, \psi_{- \textnormal{wave}}(\omega) \rangle_H \psi_{- \textnormal{wave}}(\omega) \\
& + \langle f, \psi_{+ \textnormal{wave}}(\omega) \rangle_H \psi_{+ \textnormal{wave}}(\omega) + \langle f, \psi_{\textnormal{Bou}} \rangle_H \psi_{\textnormal{Bou}}.
\end{aligned}
\end{equation}
We finally notice that the Burnett functions introduced in \eqref{eq:burnett} are related to $v \mu \in \nul(\mathcal L)$ and $\psi_\textnormal{Bou} \in \nul(\mathcal L)$ through
$${\mathbf{A}} (v)=(\mathrm{Id} - \mathsf P) [ v \otimes v \mu ] \quad \text{ and } \quad {\mathbf{B}} (v)=( \mathrm{Id} - \mathsf P ) [ v \psi_\textnormal{Bou} ].$$
\end{prop}
\begin{rem}
Note that, if a given function $\mu$ satisfies \ref{L2} , and if one defines $\mathsf P$ as the $H$-orthogonal projection on $\Span \{\mu, v_1, \dots, v_d \mu, |v|^2 \mu\}$, i.e. as \eqref{eq:representation_P}, then Proposition \ref{prop:representation_P} still holds.
\end{rem}
\subsection{Proof of Theorem \ref{thm:spectral_study}}
We are now ready to attack the full proof of Theorem \ref{thm:spectral_study}. We begin with the localization of the spectrum.
\begin{lem}[\textit{\textbf{Localization of the spectrum}}]
\label{lem:localization_spectrum}
For any gap size
$$0 < \lambda < \lambda_\mathcal L,$$
there exists some $C_0=C_0(\lambda) > 0$ and $\alpha_{0} = \alpha_{0}(\lambda) > 0$ that can be assumed small, such that the spectrum is localized as follows:
$$\mathfrak{S}(\mathcal L_{\xi}) \cap \Delta_{-\lambda} \subset \big\{ | z | \leqslant C_0 | \xi | \big\}, \qquad \forall | \xi | \leqslant \alpha_{0}\,.$$
Moreover, there exist $C_1=C_1(\alpha_{0}, \lambda) > 0$ such that, for any $|\xi| \leqslant \alpha_{0}$
\begin{equation}
\label{eq:bound_L_xi}
\sup_{ | z | = r } \| \mathcal R(z,\mathcal L_{\xi}) \|_{\mathscr B(H;H^{\bullet}) } + \sup_{|z| = r} \| \mathcal R(z, \mathcal L_\xi ) \|_{ \mathscr B( H^{\circ} ; H ) } + \sup_{z\in\Omega} \| \mathcal R(z,\mathcal L_{\xi}) \|_{\mathscr B(H)} \leqslant C_1
\end{equation}
where $r=C_{0}\alpha_{0} >0$ and $\Omega := \Delta_{-\lambda} \cap \{ | z | \geqslant r \}$.
\end{lem}
\begin{proof}
In all the proof, we assume $0 < \lambda < \lambda_\mathcal L$ to be \emph{fixed}.
\step{1}{Resolvent properties in the spaces $H_j$} We first observe that, $\mathcal B_{\xi} - \lambda$ being dissipative in the spaces $H_j$ by hypothesis \ref{assumption_dissipative}, it holds for any $j = 0, 1, 2$
\begin{equation}
\label{eq:ResBB_xi}
\|\mathcal R(z,\mathcal B_{\xi})\|_{\mathscr B(H_{j})} \lesssim 1, \qquad \forall z \in \Delta_{-\lambda},
\end{equation}
uniformly in $\xi \in \mathbb R^{d}$.
In particular, the above is true for $\xi=0$, that is to say for $\mathcal R(z,\mathcal B)$. Using that $\mathfrak{S}_{H}\left(\mathcal L\right) \cap \Delta_{-\lambda} = \{0\}$ from \ref{L3}, we have the factorization formula for any $N \geqslant 0$:
\begin{gather}
\label{eq:shrinkage_A_B}
\mathcal R(z,\mathcal L) = \sum_{n = 0}^{N-1} \mathcal R(z,\mathcal B) \Big( \mathcal A \mathcal R(z,\mathcal B) \Big)^n + \Big(\mathcal R(z,\mathcal B) \mathcal A \Big)^N \mathcal R(z,\mathcal L).
\end{gather}
holds for any $z \in \Delta_{-\lambda} \setminus\{0\}.$
Furthermore, since $\mathcal L$ is self-adjoint in $H$ by hypothesis \ref{L1}, it is well-known that the zero eigenvalue is semi-simple so that
\begin{equation}
\label{eq:RRLLH}\|\mathcal R(z,\mathcal L)\|_{\mathscr B(H)} \lesssim \frac{1}{|z|} \qquad \qquad
\forall z \in \Delta_{-\lambda}\setminus\{0\}.
\end{equation}
Thus, using the factorization \eqref{eq:shrinkage_A_B} with $N=1$, we deduce from a repeated use of \eqref{eq:ResBB_xi}
that, for any $f \in H_{1}$ and any $z \in \Delta_{-\lambda}\setminus\{0\}$,
\begin{align*}
\|\mathcal R(z,\mathcal L)f\|_{H_{1}} \leqslant \|\mathcal R(z,\mathcal B)f\|_{H_{1}} + \|\mathcal R(z,\mathcal B)\mathcal A\mathcal R(z,\mathcal L)f\|_{H_{1}}
\lesssim \|f\|_{H_{1}} +\|\mathcal A\mathcal R(z,\mathcal L)f\|_{H_{1}},
\end{align*}
and using the boundedness of $\mathcal A : H_j \to H_{j+1}$ from \ref{assumption_bounded_A} as well as \eqref{eq:RRLLH}
\begin{align*}
\|\mathcal R(z,\mathcal L)f\|_{H_{1}} & \lesssim \|f\|_{H_{1}} + \|\mathcal A\|_{\mathscr B(H_{1},H)}\|\mathcal R(z,\mathcal L)f\|_{H}\\
&\lesssim \|f\|_{H_{1}}+ |z|^{-1}\|f\|_{H} \lesssim \left(1+\frac{1}{|z|}\right)\|f\|_{H_{1}}
\end{align*}
where we used $H_{1} \hookrightarrow H$ in the last inequality. Using this estimate and proceeding in the same way for $j=2$, we deduce that
\begin{equation}
\label{eq:bound_resolvent_L_X_j}
\| \mathcal R(z,\mathcal L) \|_{\mathscr B(H_{j})} \lesssim 1 + \frac{1}{|z|}, \qquad \forall z \in \Delta_{-\lambda} \setminus \{0\}, \quad j=0,1,2.
\end{equation}
We conclude that, in each space $H_j$, the eigenvalue $0$ is semi-simple (i.e. a simple pole of $\mathcal R(\cdot,\mathcal L)$) and the resolvent writes in $\mathscr B(H_j)$ as the sum of a singular part and a holomorphic part (see \cite[Chapter 3, Section 6.5]{K1995}):
\begin{equation}
\label{eq:pseudo_Laurent_expansion} \mathcal R(z,\mathcal L) = z^{-1}\mathsf P + \mathcal R^\perp(z),
\qquad \forall z \in \Delta_{-\lambda} \setminus \{0\}\,.
\end{equation}
with the regular part defined as
\begin{equation}
\label{eq:pseudo_Laurent_expansion1}
\mathcal R^{\perp}(z)=\sum_{n=1}^{\infty}z^{n}\mathsf{R_{0}}^{n+1}, \qquad \mathsf{R}_{0}:=\mathcal R(0,\mathcal L)\left(\mathrm{Id}-\mathsf P\right)\,,
\end{equation}
where we notice that $\mathsf{R}_{0} =-\mathcal L^{-1} \left(\mathrm{Id}-\mathsf P\right) \in \mathscr B(H_j)$.
\step{2}{Localization of the spectrum and resolvent bound in $\mathscr B(H)$}
We draw inspiration from the proof of \cite[Lemma 2.16]{T2016}. Let us start with the following factorization formula permitted by the dissipativity hypothesis from \ref{assumption_dissipative} for some large enough $a > 0$:
$$\mathcal R(z,\mathcal L_{\xi}) = \mathcal R(z,\mathcal B_{\xi}) + \mathcal R(z,\mathcal L_{\xi}) \mathcal A \mathcal R(z,\mathcal B_{\xi}) \qquad \forall z \in \Delta_{a}$$
and, expanding the term $\mathcal R(z,\mathcal L_{\xi})$ using \eqref{eq:factorization_L_ivxi_left} with $N = 1$, we deduce now
\begin{align}
\label{eq:factorization_R_xi_localization}
\mathcal R(z,\mathcal L_{\xi}) = & \mathcal R(z,\mathcal B_{\xi}) + \mathcal R(z,\mathcal L) \mathcal A \mathcal R(z,\mathcal B_{\xi}) + \mathcal R(z,\mathcal L_{\xi}) (- i v \cdot \xi) \mathcal R(z,\mathcal L) \mathcal A \mathcal R(z,\mathcal B_{\xi}).
\end{align}
This allows to localize the spectrum using the bounds \eqref{eq:bound_resolvent_L_X_j} and the regularization hypothesis for $\mathcal A$ coming from \ref{assumption_bounded_A}. Indeed, according to \ref{assumption_multi-v}
$$\left\|v \mathcal R(z,\mathcal L) \mathcal A \mathcal R(z,\mathcal B_{\xi}) \right\|_{\mathscr B(H)} \lesssim \left\|\mathcal R(z,\mathcal L)\mathcal A\mathcal R(z,\mathcal B_{\xi})\right\|_{\mathscr B(H,H_{1})}$$
and, using now \eqref{eq:bound_resolvent_L_X_j} and the fact that $\mathcal A \in \mathscr B(H,H_{1})$ from \ref{assumption_bounded_A}, we deduce that
$$\left\|v \mathcal R(z,\mathcal L) \mathcal A \mathcal R(z,\mathcal B_{\xi}) \right\|_{\mathscr B(H)} \lesssim \left(1+\frac{1}{|z|}\right)\|\mathcal R(z,\mathcal B_{\xi})\|_{\mathscr B(H)} \lesssim 1+\frac{1}{|z|},$$
thanks to \eqref{eq:ResBB_xi}. In particular, for any $c_0 >0,$
$$ \left\|\left(\xi \cdot v\right)\mathcal R(z,\mathcal L) \mathcal A \mathcal R(z,\mathcal B_{\xi}) \right\|_{\mathscr B(H)} \lesssim |\xi|+c_0, \qquad \forall | z | > \frac{| \xi |}{c_0}\,,$$
thus, considering $c_0, \alpha_{0} > 0$ small enough, we deduce that
$$\| \xi \cdot v \mathcal R(z,\mathcal L) \mathcal A \mathcal R(z,\mathcal B_{\xi}) \|_{\mathscr B(H)} \leqslant \frac{1}{2}, \qquad \forall |z| > \frac{|\xi|}{c_0}, \quad | \xi | \leqslant \alpha_0\, ,$$
and in particular, for such a choice of $(\xi, z)$, the operator $\mathrm{Id} + (i v \cdot \xi) \mathcal R(z,\mathcal L) \mathcal A\mathcal R(z,\mathcal B_{\xi})$ is invertible in $\mathscr B(H)$ with
\begin{equation}
\label{eq:inverseId} \left\|\Big( \mathrm{Id} + (i v \cdot \xi) \mathcal R(z,\mathcal L) \mathcal A\mathcal R(z,\mathcal B_{\xi}) \Big)^{-1}\right\|_{\mathscr B(H)} \leqslant 2,\qquad \text{ for any } |z| > \frac{|\xi|}{c_0}, ~ |\xi | \leqslant \alpha_0\, ,
\end{equation}
thus it follows from \eqref{eq:factorization_R_xi_localization} that
\begin{align*}
\mathcal R(z,\mathcal L_{\xi}) =\Big( \mathcal R(z,\mathcal B_{\xi}) + \mathcal R(z,\mathcal L) \mathcal A \mathcal R(z,\mathcal B_{\xi}) \Big)\Big( \mathrm{Id} + (i v \cdot \xi) \mathcal R(z,\mathcal L) \mathcal A\mathcal R(z,\mathcal B_{\xi}) \Big)^{-1} .
\end{align*}
Each term on the right hand side belongs to $\mathscr B(H)$ for $z \in \Delta_{-\lambda} \cap \Big\{ |z| > c_0^{-1} | \xi | \Big\}$, thus the following localization of the spectrum holds:
\begin{equation*} \mathfrak{S}_{H}(\mathcal L_{\xi}) \cap \Delta_{-\lambda} \subset \Big\{z \in \mathbb C \; ; \; | z | \leqslant c_0^{-1} | \xi | \Big\}, \qquad \forall |\xi| \leqslant \alpha_{0}.
\end{equation*}
More precisely, using the bounds on $\mathcal R(z,\mathcal B_{\xi})$ from \ref{L4} and \eqref{eq:bound_resolvent_L_X_j} together with \eqref{eq:inverseId} we have
\begin{equation}
\label{eq:general_bound_L_xi} \| \mathcal R(z,\mathcal L_{\xi}) \|_{\mathscr B(H)} \lesssim 1 + \frac{1}{|z|}
\end{equation}
for any $z \in \Delta_{-\lambda}, ~ | z | > \frac{|\xi|}{c_0},$ with $|\xi|\leqslant \alpha_{0}$.
This proves the $\mathscr B(H)$-bound in \eqref{eq:bound_L_xi}. This concludes this step.
\step{3}{Resolvent bound in $\mathscr B(H; H^{\bullet}) \cap \mathscr B(H^{\circ} ; H)$}
First of all, note that the following identity for bounded operator $T\::\:H \to H^{\bullet}$ is proved in Appendix \ref{scn:duality} (where the adjoint $T^{\star}$ is considered for the inner product of $H$):
\begin{equation*}
\| T^\star \|_{\mathscr B(H^{\circ};H)} = \| T \|_{ \mathscr B(H;H^{\bullet}) },
\end{equation*}
furthermore, using that $\mathcal L_\xi^\star = \mathcal L_{-\xi}$ since $\mathcal L$ is self adjoint, one has
$$ \mathcal R(z,\mathcal L_{\xi})^{\star}=\mathcal R(\overline{z},\mathcal L_{-\xi})\qquad \forall \xi \in \mathbb R^d, ~ z \in \mathfrak{S}(\mathcal L_\xi),$$
thus it is enough to prove the bound in $\mathscr B(H; H^{\bullet})$.
Using the dissipativity estimate for $\mathcal L$ from \ref{L3} and the fact that the multiplication by~$i v \cdot \xi$ is skew-adjoint, we have for any $z > 0$
\begin{align*}
\left\langle (\mathcal L_\xi - z) f, f \right\rangle_{H} & \leqslant - \lambda_\mathcal L \| (\mathrm{Id} - \mathsf P) f \|_{H^{\bullet}}^2 - z \| f \|^2_{H}.
\end{align*}
Furthermore, using that $\mathsf P$ is $H$-orthogonal as well as the fact that $\mathsf P^2 = \mathsf P$ and $\| \mathsf P \|_{\mathscr B(H;H^{\bullet})} \leqslant M$ for some $M > 0$
\begin{align*}
\left\langle (\mathcal L_\xi - z) f, f \right\rangle_{H} & \leqslant - \lambda_\mathcal L \| (\mathrm{Id} - \mathsf P) f \|_{H^{\bullet}}^2 - z \| \mathsf P f \|^2_{H} \\
& \leqslant - \lambda_\mathcal L \| (\mathrm{Id} - \mathsf P) f \|_{H^{\bullet}}^2 - z {M^{-2}} \| \mathsf P f \|^2_{H^{\bullet}}.
\end{align*}
The term $\| (\mathrm{Id} - \mathsf P) f \|_{H^{\bullet}}^2$ can be estimated using the polar identity and Young's inequality (note that $\mathsf P$ may not be $H^{\bullet}$-orthogonal):
\begin{align}
\notag
\| (\mathrm{Id} - \mathsf P) f \|_{H^{\bullet}}^2 & = \frac{1}{2} \| (\mathrm{Id} - \mathsf P) f \|_{H^{\bullet}}^2 + \frac{1}{2} \Big(\| f \|^2_{H^{\bullet}} - \| \mathsf P f \|_{H^{\bullet}}^2 - 2 \left\langle (\mathrm{Id} - \mathsf P) f, \mathsf P f \right\rangle_{H^{\bullet}}\Big) \\
\label{eq:degenerate_pythagora_X_star}
& \geqslant \frac{1}{2} \| f \|^2_{H^{\bullet}} - \| \mathsf P f \|_{H^{\bullet}}^2,
\end{align}
therefore we have
\begin{align*}
\left\langle (\mathcal L_\xi - z) f, f \right\rangle_{H} \leqslant - \frac{\lambda_\mathcal L}{2} \| f \|_{H^{\bullet}}^2 - \left(z {M^{-2}} - \lambda_\mathcal L \right) \| \mathsf P f \|^2_{H^{\bullet}}.
\end{align*}
We conclude that for $z_0 = {\lambda_\mathcal L \,M^{2}}$, we have
$$\forall f \in \mathscr{D}(\mathcal L_{\xi}), \quad \left\langle (\mathcal L_\xi - z_0) f, f \right\rangle_{H} \leqslant - \frac{\lambda_\mathcal L}{2} \| f \|_{H^{\bullet}}^2.$$ This, together with the comparison $\|\cdot\|_{H} \leqslant \|\cdot\|_{H^{\bullet}}$, implies the resolvent bound
\begin{equation}
\label{eq:bound_L_xi_X_X_star_z_0}
\| \mathcal R(z_{0},\mathcal L_{\xi})\|_{\mathscr B(H;H^{\bullet})} \leqslant \frac{2}{\lambda_\mathcal L}.
\end{equation}
Using the resolvent identity
\begin{equation*}
\mathcal R(z,\mathcal L_{\xi}) = \mathcal R(z_{0},\mathcal L_{\xi})+ (z_0 - z) \mathcal R(z_{0},\mathcal L_{\xi})\mathcal R(z,\mathcal L_{\xi}),
\end{equation*}
we can combine \eqref{eq:general_bound_L_xi} and \eqref{eq:bound_L_xi_X_X_star_z_0} so as to obtain, for $|\xi| \leqslant \alpha_{0}$,
\begin{equation}
\| \mathcal R(z,\mathcal L_{\xi}) \|_{\mathscr B(H;H^{\bullet})}
\lesssim 1 + |z| + \frac{1}{|z|}, \qquad \forall z \in \Delta_{-\lambda}, ~ |z| > c_0^{-1}|\xi|,
\end{equation}
from which we deduce the $\mathscr B(H;H^{\bullet})$-bound of \eqref{eq:bound_L_xi}. This concludes the proof.
\end{proof}
\begin{figure}
\caption{Localization of the spectrum from Lemma \ref{lem:localization_spectrum}. The hatched blue part contains the ‘‘pseudo hydrodynamic part'' of the spectrum of $\mathcal L_\xi$ (i.e. the perturbation of the macroscopic eigenvalue of $\mathcal L$, that is to say $0$). The hatched red part contains the ‘‘pseudo kinetic part'' of the spectrum of $\mathcal L_\xi$ (i.e. the perturbation of the microscopic part of the spectrum of $\mathcal L$, that is to say $\mathfrak{S}(\mathcal L) \setminus \{0\}$).}
\end{figure}
\begin{lem}[\textit{\textbf{Expansion of the total projector}}]
\label{lem:expansion_projection}
With the notations of Lemma \ref{lem:localization_spectrum}, considering some $0 < \lambda < \lambda_\mathcal L$, for any $|\xi| \leqslant \alpha_{0}$, the spectral projector $\mathsf P(\xi)$ associated with the $0$-group of Lemma \ref{lem:localization_spectrum} and defined as
$$\mathsf P(\xi) = \frac{1}{2 i \pi} \oint_{|z| = r } \mathcal R(z,\mathcal L_{\xi}) \mathrm{d} z, \qquad \forall | \xi | \leqslant \alpha_0,$$
where the integration along the circle $\{ |z| = r \}$ is counterclockwise, has the following first order expansion in $\mathscr B(H^{\circ} ; H^{\bullet} )$:
\begin{equation*}
\mathsf P(\xi) = \mathsf P + i \xi \cdot \Big( \mathsf P v\mathsf{R}_{0} +\mathsf{R}_{0} v \mathsf P \Big) + \mathsf{S}(\xi), \qquad \mathsf{R}_{0}:=\mathcal R(0,\mathcal L)\left(\mathrm{Id}-\mathsf P\right)
\end{equation*}
where $\mathsf{S}(\xi) \in \mathscr B(H^{\circ};H^{\bullet})$ with $\|\mathsf{S}(\xi)\|_{\mathscr B(H^{\circ};H^{\bullet})} \lesssim | \xi |^2$.
\end{lem}
\begin{proof} For a fixed $0 < \lambda < \lambda_{\mathcal L}$, let $\alpha_{0}=\alpha_{0}(\lambda)$ be provided by Lemma \ref{lem:localization_spectrum}. Since
$$\mathsf P(\xi)=\frac{1}{2i\pi}\oint_{|z|=r}\mathcal R(z,\mathcal L_{\xi}) \mathrm{d} z,$$
we deduce directly from the bound \eqref{eq:bound_L_xi} that
$$\| \mathsf P(\xi) \|_{\mathscr B(H;H^{\bullet})} + \| \mathsf P(\xi) \|_{\mathscr B(H^{\circ};H)} \lesssim 1,$$
and using that $\mathsf P(\xi)^2 = \mathsf P(\xi)$, we actually deduce
\begin{equation}
\label{eq:reg_P_xi_order_0}
\| \mathsf P(\xi) \|_{\mathscr B(H^{\circ};H^{\bullet})} \lesssim 1, \quad \mathsf P(0) = \mathsf P \in \mathscr B(H^{\circ} ; H^{\bullet}).
\end{equation}
We will refine this information and prove a second order Taylor expansion using the bootstrap formulae from Appendix \ref{scn:boostrap_projectors}. Before doing so, we observe that the following factorization holds true
\begin{equation*}
\mathcal R(z,\mathcal L_{\xi}) =\mathcal R(z,\mathcal B_{\xi})+ \big( \mathcal R(z,\mathcal B_{\xi}) \mathcal A \big)^2 \mathcal R(z,\mathcal B_{\xi}) + \big(\mathcal R(z,\mathcal B_{\xi}) \mathcal A \big)^2 \mathcal R(z,\mathcal L_{\xi}),
\end{equation*}
where the first two terms are actually $\mathscr B(H)$-valued holomorphic in $z \in \Delta_{-\lambda}$. Therefore, $\mathsf P(\xi)$ can be written equivalently as
$$\mathsf P(\xi)=\frac{1}{2i\pi}\oint_{|z|=r} \big(\mathcal R(z,\mathcal B_{\xi})\mathcal A\big)^{2}\mathcal R(z,\mathcal L_{\xi})\mathrm{d} z.$$
Since, according to the regularization properties \ref{assumption_bounded_A} of $\mathcal A$ together with the resolvent bounds \eqref{eq:bound_L_xi} and \eqref{eq:ResBB_xi} for $\mathcal L_\xi$ and $\mathcal B_\xi$ respectively, it holds, uniformly in $|z|=r$ and $| \xi | \leqslant \alpha_0$
\begin{align*}
\| \mathcal R(z, \mathcal B_\xi) \mathcal A \|_{ \mathscr B(H ; H_1 ) } + \| \mathcal R(z, \mathcal B_\xi) \mathcal A \|_{ \mathscr B(H_1 ; H_2 ) } + \| \mathcal R(z, \mathcal L_\xi) \|_{ \mathscr B(H^{\circ} ; H ) } \lesssim 1,
\end{align*}
and thus, using $H_{2} \hookrightarrow H_1$
\begin{equation*}
\left\| \big(\mathcal R(z,\mathcal B_{\xi})\mathcal A\big)^{2}\mathcal R(z,\mathcal L_{\xi}) \right\|_{ \mathscr B(H^{\circ} ; H_2) } \lesssim 1,
\end{equation*}
which once integrated along $| z | = r$, gives
\begin{equation}
\label{eq:regularization_P_xi} \| \mathsf P(\xi) \|_{\mathscr B(H^{\circ};H_{j})} \lesssim 1, \qquad j=0,1,2, \qquad | \xi | \leqslant \alpha_0\,.
\end{equation}
\step{1}{First order expansion}
A representation of the first order Taylor expansion is given by integrating \eqref{eq:factorization_L_ivxi_left} or \eqref{eq:factorization_L_ivxi_right} with $N = 1$ yielding
$$ \mathsf P(\xi) = \mathsf P + \xi \cdot \mathsf P^{(1)}(\xi), \qquad | \xi | \leqslant \alpha_0,$$
where $\mathsf P^{(1)}(\xi)$ has the integral representation
$$ \mathsf P^{(1)}(\xi) := \frac{1}{2 i \pi} \oint_{|z| = r} \mathcal R(z,\mathcal L_{\xi}) (- i v) \mathcal R(z,\mathcal L) \mathrm{d} z = \frac{1}{2 i \pi} \oint_{|z| = r} \mathcal R(z,\mathcal L) (- i v) \mathcal R(z,\mathcal L_{\xi}) \mathrm{d} z.$$
We know that $\mathsf P^{(1)}(\xi) \in \mathscr B(H^{\circ} ; H^{\bullet})$ (since $\xi\cdot \mathsf P^{(1)}(\xi)=\mathsf P(\xi)-\mathsf P$) let us prove that this holds uniformly in $\xi$. We begin with the following estimate in $\mathscr B(H_{1};H^{\bullet})$ which is made possible since the multiplication by $v$ is bounded from $H_{1}$ to $H$. Namely, thanks to \ref{assumption_multi-v} and the resolvent bounds \eqref{eq:bound_L_xi} and \eqref{eq:bound_resolvent_L_X_j}, we have uniformly in $|\xi| \leqslant \alpha_0$
\begin{equation}
\label{eq:bound_PP1_H1X*}
\begin{split}
\| \mathsf P^{(1)}(\xi) \|_{\mathscr B(H_{1};H^{\bullet})} & \leqslant \frac{1}{2 \pi} \oint_{|z|=r} \| \mathcal R(z,\mathcal L_{\xi}) v \mathcal R(z,\mathcal L) \|_{\mathscr B(H_{1};H^{\bullet}) } \mathrm{d} |z| \\
& \lesssim \oint_{|z|=r} \| \mathcal R(z,\mathcal L_{\xi}) \|_{\mathscr B(H;H^{\bullet}) } \| \mathcal R(z,\mathcal L) \|_{\mathscr B(H_{1})} \mathrm{d} |z|\lesssim 1.
\end{split}
\end{equation}
Let us now explain how to extend such an estimate to $\mathscr B(H^{\circ};H)$. Using the formula \eqref{eq:bootstrap_P_1_A}, we have, for $|\xi| \leqslant \alpha_0$
$$\mathsf P^{(1)}(\xi) = \mathsf P(\xi) \mathsf P^{(1)}(\xi) + \mathsf P^{(1)}(\xi) \mathsf P,$$
and therefore
\begin{align*}
\| \mathsf P^{(1)}(\xi) \|_{\mathscr B(H^{\circ};H^{\bullet})} & \leqslant \| \mathsf P(\xi) \mathsf P^{(1)}(\xi) \|_{\mathscr B(H^{\circ};H^{\bullet})} + \| \mathsf P^{(1)}(\xi) \mathsf P \|_{\mathscr B(H^{\circ};H^{\bullet})}\\
& \leqslant \| \mathsf P^{(1)}(\xi)^{\star} \mathsf P(\xi)^{\star} \|_{\mathscr B(H^{\circ};H^{\bullet})} + \| \mathsf P^{(1)}(\xi) \mathsf P \|_{\mathscr B(H^{\circ};H^{\bullet})}
\end{align*}
where we used the adjoint identity \eqref{eq:twisted_adjoint_identity} (where the adjoint is considered for $\langle \cdot , \cdot \rangle_{H}$). Since we have
$ \mathsf P(\xi)^\star = \mathsf P(-\xi)$ and $\mathsf P^\star = \mathsf P,$
one has $\mathsf P^{(1)}(\xi)^{\star}=-\mathsf P^{(1)}(-\xi)$. Therefore
\begin{align*}
\| \mathsf P^{(1)}(\xi) \|_{\mathscr B(H^{\circ};H^{\bullet})} & \leqslant \| \mathsf P^{(1)}(-\xi) \mathsf P(-\xi) \|_{\mathscr B(H^{\circ};H^{\bullet})} + \| \mathsf P^{(1)}(\xi) \mathsf P \|_{\mathscr B(H^{\circ};H^{\bullet})} \\
& \leqslant \| \mathsf P^{(1)}(-\xi) \|_{\mathscr B(H_{1};H^{\bullet})} \| \mathsf P(-\xi) \|_{\mathscr B(H^{\circ};H_{1}) } + \| \mathsf P^{(1)}(\xi) \|_{\mathscr B(H_{1};H^{\bullet})} \| \mathsf P \|_{\mathscr B(H^{\circ};H_{1}) } \\
& \lesssim \| \mathsf P^{(1)}(-\xi) \|_{\mathscr B(H_{1};H^{\bullet})} + \| \mathsf P^{(1)}(\xi) \|_{\mathscr B(H_{1};H^{\bullet})},
\end{align*}
where we used the regularizing property \eqref{eq:regularization_P_xi} of $\mathsf P(\xi)$ in the last line (recall that $\mathsf P=\mathsf P(0)$). Using the above estimates \eqref{eq:bound_PP1_H1X*}, we deduce that
$$\| \mathsf P^{(1)}(\xi) \|_{\mathscr B(H^{\circ};H^{\bullet})} \lesssim 1, \qquad |\xi| \leqslant \alpha_0,$$
which concludes this step. Notice that a clear implication is that, for any $|\xi| \leqslant \alpha_0$
$$\|\mathsf P(\xi)-\mathsf P\|_{\mathscr B(H^{\circ};H^{\bullet})} \lesssim |\xi|.$$
\step{2}{Second order expansion} In a similar fashion, we obtain a second order expansion integrating once again \eqref{eq:factorization_L_ivxi_left} or \eqref{eq:factorization_L_ivxi_right} with~$N = 2$:
\begin{gather*}
\mathsf P(\xi) = \mathsf P + i \xi \cdot \mathsf P_{1} + \xi \otimes \xi : \mathsf P^{(2)}(\xi), \qquad | \xi | \leqslant \alpha_0,
\end{gather*}
where the first and second order terms are defined as
\begin{gather*}
\mathsf P_{1} = - \frac{1}{2 i \pi} \oint_{|z| = r} \mathcal R(z,\mathcal L) v \mathcal R(z,\mathcal L) \mathrm{d} z,\\
\mathsf P^{(2)}(\xi) := - \frac{1}{2 i \pi} \oint_{ | z | = r } \Big( \mathcal R(z,\mathcal L) v \Big)^{\otimes 2} \mathcal R(z,\mathcal L_{\xi}) \mathrm{d} z = - \frac{1}{2 i \pi} \oint_{ | z | = r } \mathcal R(z,\mathcal L_{\xi}) \Big(v \mathcal R(z,\mathcal L)\Big)^{\otimes 2} \mathrm{d} z.
\end{gather*}
The first order term is explicitly computable using the Laurent series expansion \eqref{eq:pseudo_Laurent_expansion}:
\begin{equation*}
\mathsf P_{1} = - \frac{1}{2 i \pi} \oint_{|z| = r} \mathcal R(z,\mathcal L) v \mathcal R(z,\mathcal L) \mathrm{d} z = \mathsf{R}_{0} v \mathsf P + \mathsf P v \mathsf{R}_{0},
\end{equation*}
so in particular, one checks directly that
$$ \mathsf P_{1} = \mathsf P \mathsf P_{1} + \mathsf P_{1} \mathsf P.$$
We use now the formula \eqref{eq:bootstrap_P_2_A}, which gives
\begin{align*}
\| \mathsf P^{(2)}(\xi) \|_{\mathscr B(H^{\circ};H^{\bullet})} \leqslant \| \mathsf P(\xi) \mathsf P^{(2)}(\xi) \|_{\mathscr B(H^{\circ};H^{\bullet})} + \| \mathsf P^{(1)}(\xi) \otimes \mathsf P_{1} \|_{\mathscr B(H^{\circ};H^{\bullet})} + \| \mathsf P^{(2)}(\xi) \mathsf P \|_{\mathscr B(H^{\circ};H^{\bullet})}.
\end{align*}
We use the bound from \textit{Step 1} for the second term, and, after using duality as in \textit{Step 1} (for the first term), we use the regularization property \eqref{eq:regularization_P_xi} of $\mathsf P(\xi)$:
\begin{align*}
\| \mathsf P^{(2)}(\xi) \|_{\mathscr B(H^{\circ};H^{\bullet})} & \lesssim 1 + \| \mathsf P^{(2)}(-\xi) \mathsf P(-\xi) \|_{\mathscr B(H^{\circ};H^{\bullet})}
+ \| \mathsf P^{(2)}(\xi) \mathsf P \|_{\mathscr B(H^{\circ};H^{\bullet})} \\
& \lesssim 1 + \| \mathsf P^{(2)}(-\xi) \|_{\mathscr B(H_{2},H^{\bullet})}
+ \| \mathsf P^{(2)}(\xi) \|_{\mathscr B(H_{2},H^{\bullet})}.
\end{align*}
As previously, using the fact that the multiplication by $v$ is bounded from $H_{j}$ to $H_{j+1}$ for $j=0,1$, we show using the resolvent bounds \eqref{eq:bound_L_xi} for $\mathcal L_\xi$ and \eqref{eq:bound_resolvent_L_X_j} for $\mathcal L$ that
\begin{align*}
\| \mathsf P^{(1)}(\xi) \|_{\mathscr B(H_{2};H^{\bullet})} & \leqslant \frac{1}{2 \pi} \oint_{|z|=r} \left\| \mathcal R(z,\mathcal L_{\xi}) \Big(v \mathcal R(z,\mathcal L)\Big)^{\otimes 2} \right\|_{\mathscr B(H_{2};H^{\bullet}) } \mathrm{d}|z |\\
& \lesssim \oint_{|z|=r} \| \mathcal R(z,\mathcal L_{\xi}) \|_{\mathscr B(H;H^{\bullet}) } \| \mathcal R(z,\mathcal L) \|_{\mathscr B(H_{1})} \| \mathcal R(z,\mathcal L) \|_{\mathscr B(H_{2})} \mathrm{d}| z|\lesssim 1.
\end{align*}
This concludes the proof, setting $\mathsf{S}(\xi) := \xi \otimes \xi : \mathsf P^{(2)}(\xi)$.
\end{proof}
Following Kato's reduction process from \cite[Section I-4.6, pp. 32--34]{K1995}, we introduce for any $| \xi | \leqslant \alpha_0$ a ``rectified'' version of $\mathcal L_\xi$ in which we cut off any spectral points that is not a small eigenvalue. Precisely, following \cite[Section I-4.6]{K1995} and since
$$\|\mathsf P(\xi)-\mathsf P\|_{\mathscr B(H^{\circ},H^{\bullet})} \lesssim |\xi| \qquad \forall | \xi | \leqslant \alpha_0, \quad $$
according to the previous Lemma and the injection $H^{\bullet} \hookrightarrow H \hookrightarrow H^{\circ}$, we deduce that for $\alpha_0$ small enough
$$ \|\left(\mathsf P(\xi)-\mathsf P\right)^{2}\|_{\mathscr B(H)} < 1, \qquad \forall | \xi | \leqslant \alpha_0,$$
where we recall that $H^{\bullet} \hookrightarrow H\hookrightarrow H^{\circ}.$
In particular (see \cite[Eqs. (4.36)--(4.39), p. 33]{K1995}), we can define
\begin{align*}
\mathcal U_\xi &= \Big( \mathsf P(\xi) \mathsf P + (\mathrm{Id} - \mathsf P(\xi))(\mathrm{Id} - \mathsf P) \Big) \Big(\mathrm{Id} -(\mathsf P(\xi) - \mathsf P)^2 \Big)^{-\frac{1}{2}} \\
&= \Big(\mathrm{Id} -(\mathsf P(\xi) - \mathsf P)^2 \Big)^{-\frac{1}{2}} \Big( \mathsf P(\xi) \mathsf P + (\mathrm{Id} - \mathsf P(\xi))(\mathrm{Id} - \mathsf P) \Big), \qquad |\xi| \leqslant \alpha_0
\end{align*}
where we used the definition $(\mathrm{Id} - T)^{-\frac{1}{2}} = \sum_{n= 0}^\infty \binom{-\frac{1}{2}}{n} (-T)^n$ for any $T \in \mathscr B(H)$. With such a definition, $\mathcal U_{\xi}$ mapping isomorphically the null-space of $\mathcal L$ onto the eigenspaces corresponding to the small eigenvalues of $\mathcal L_\xi$:
\begin{equation*}
\nul(\mathcal L) = \range(\mathsf P) \xrightarrow{\mathcal U_{\xi}} \range\big(\mathsf P(\xi) \big),
\end{equation*}
we define the \emph{finite dimensional} rectified operator on $\nul(\mathcal L)$:
\begin{equation*} \quad \mathbb{L}_{\xi} := \Big( \mathcal U_{\xi}^{-1} \, \mathcal L_{\xi} \, \mathcal U_{\xi} \Big)_{| \nul(\mathcal L) } \in \mathscr B\left( \nul(\mathcal L) \right), \qquad \forall | \xi | \leqslant \alpha_0, \quad
\end{equation*}
whose spectrum is related to that of $\mathcal L_{\xi}$ by
$ \mathfrak{S}(\mathcal L_{\xi}) \cap \Delta_{-\lambda} = \mathfrak{S}\left(\mathbb{L}_{\xi}\right).$
In the rest of this section, we will study the structure of $\mathbb{L}_{ \xi}$ so as to reduce its diagonalization to the perturbation of a diagonalizable matrix with simple eigenvalues. Using (Kato's) classical perturbation theory for matrices, this will provide expansion of the eigenvalues and eigenfunctions of $\mathbb{L}_\xi$, and in turn those of $\mathcal L_{\xi}$.
\begin{lem}[\textit{\textbf{Properties of the rectified operator}}]
\label{lem:rectified_operator}
The rectified operator $\mathbb{L}_{ \xi}$ has the following properties:
\begin{enumerate}
\item $\mathbb{L}_{ \xi}$ is compatible with any orthogonal $d \times d$ matrix:
\begin{equation*}
\Theta \mathbb{L}_{\Theta \xi} = \mathbb{L}_{\xi} \Theta,
\end{equation*}
in particular, $\mathbb{L}_{\xi}$ commutes with orthogonal matrices that fix $\xi$, and it preserves eveness and oddity in directions orthogonal to $\xi$:
\begin{gather*}
\Theta \xi = \xi \Longrightarrow \Theta \mathbb{L}_{\xi} = \mathbb{L}_\xi \Theta,\\
\Big( u \perp \xi \text{ and } \varphi(-u) = \pm \varphi(u) \Big) \Longrightarrow \Big( \mathbb{L}_{\xi} \varphi \Big)(-u) = \pm \Big( \mathbb{L}_{\xi} \varphi \Big)(u).
\end{gather*}
\item $\mathbb{L}_{\xi}$ has the following second order expansion in $\mathscr B\left( \nul(\mathcal L) \right)$
\begin{equation*}
\mathbb{L}_{\xi} = -i \mathsf P ( v \cdot \xi) + \mathsf P (v \cdot \xi) \mathsf{R}_{0} (v \cdot \xi) + \mathbf{r}_0
(\xi) \end{equation*}
where $\mathbf{r}_0(\xi) \in \mathscr B(\nul(\mathcal L))$ is such that $\|\mathbf{r}_0(\xi)\|_{\mathscr B( \nul(\mathcal L) ) } \lesssim |\xi|^3.$
\end{enumerate}
\end{lem}
\begin{proof}
As $\Theta \mathcal L_{\Theta \xi} = \mathcal L_\xi \Theta$ holds for any orthogonal matrix $\Theta$, it is clear that the resolvent satisfies the same relation $\Theta \mathcal R(z, \mathcal L_{\Theta \xi})= \mathcal R(z,\mathcal L_\xi)\Theta$, and thus $\Theta \mathsf P(\Theta \xi) = \mathsf P(\xi) \Theta$. Using the above definition of $\mathcal U_{\xi}$, this implies $\Theta \mathcal U_{\Theta \xi} = \mathcal U_{\xi} \Theta$ and thus the first point of this lemma. We only need to check the expansion for $\mathbb{L}_{ \xi}$. According to Lemma \ref{lem:expansion_projection}, there holds
\begin{gather*}
\mathsf P(\xi) \mathsf P = \mathsf P +i \mathsf{R}_{0} (v \cdot \xi) \mathsf P + r_{1}(\xi), \\
(\mathrm{Id} - \mathsf P(\xi))(\mathrm{Id} - \mathsf P) = \mathrm{Id} - \mathsf P - i \mathsf P (v \cdot \xi) \mathsf{R}_{0} + r_{2}(\xi),
\end{gather*}
and
$$ (\mathsf P(\xi) - \mathsf P)^2=r_{3}(\xi),
$$
where the \emph{remainder operators} $r_{k}$ are such that
$$\left\|r_{k}(\xi)\right\|_{\mathscr B(H^{\circ};H^{\bullet})} \lesssim |\xi|^{2}, \qquad k=1,2,3,$$
for $|\xi| \leqslant \alpha_0$ with $\alpha_0$ small enough. Inserting this in the definition of $\mathcal U_{\xi}$, there exist additional remainder operators $r_{k}(\xi) \in \mathscr B(H^{\circ},H^{\bullet})$ \footnote{with for instance $r_{4}(\xi)=r_{1}(\xi)+r_{2}(\xi)$ and $r_{5}(\xi)=\sum_{n=1}^{\infty}{-\frac{1}{2} \choose n}(-r_{3}(\xi))^{n}.$} such that
\begin{equation*}
\label{eq:expansion_Kato_isomorphism}\begin{split}
\mathcal U_\xi & = \Big( \mathrm{Id} - i \mathsf P (v \cdot \xi)\mathsf{R}_{0} + i\mathsf{R}_{0} (v \cdot \xi) \mathsf P + r_{4}(\xi) \Big) \Big( \mathrm{Id} + r_{5}(\xi) \Big) \\
& = \mathrm{Id} - i \mathsf P (v \cdot \xi)\mathsf{R}_{0} + i\mathsf{R}_{0} (v \cdot \xi) \mathsf P + r_{6}(\xi),
\end{split}
\end{equation*}
where $\|r_{k}(\xi)\|_{\mathscr B(H^{\circ},H^{\bullet})} \lesssim |\xi|^{2}$ for $k=1,\ldots,6.$ Note that there holds as well
\begin{equation*}
\label{eq:expansion_Kato_isomorphism_inverse}
\mathcal U^{-1}_\xi = \mathrm{Id} - i\mathsf{R}_{0} (v \cdot \xi) \mathsf P + i \mathsf P (v \cdot \xi)\mathsf{R}_{0} + r_{7}(\xi),
\end{equation*}
with $\|r_{7}(\xi)\|_{\mathscr B(H^{\circ};H^{\bullet})} \lesssim |\xi|^{2}.$ Notice that in the above decompositions of $\mathcal U_{\xi}$ and $\mathcal U_{\xi}^{-1}$, all terms (except $\mathrm{Id}$) are in $\mathscr B\left( H^{\circ} ; H^{\bullet} \right)$. We thus compute the second order expansion
$$\mathbb{L}_\xi =\mathsf P \mathbb{L}_\xi \mathsf P=\mathsf P \Big(\mathcal U^{-1}_\xi \mathcal L_\xi \mathcal U_\xi\Big) \mathsf P $$
as follows. Observe that $\mathsf P \mathsf{R}_{0}=\mathsf{R}_{0}\mathsf P=0$ so that
\begin{equation}
\label{eq:expansion_Kato_isomorphismPP}
\mathsf P\mathcal U^{-1}_\xi=\mathsf P + i \mathsf P (v \cdot \xi)\mathsf{R}_{0} + \mathsf P\, r_{7}(\xi), \qquad \mathcal U_{\xi}\mathsf P=\mathsf P +
i\mathsf{R}_{0}(v\cdot\xi)\mathsf P+r_{6}(\xi)\mathsf P\,,
\end{equation}
and that $\range(r_{6}(\xi)\mathsf P) \subset \mathscr{D}(\mathcal L)$ so that $r_{8}(\xi):=\mathcal L\,r_{6}(\xi)\mathsf P$ is well-defined and in the end we have $\|r_{8}(\xi) \|_{\mathscr B(H^{\circ},H^{\bullet})} \lesssim |\xi|^{2}$. Therefore, using also that $\mathcal L \mathsf P=0$ while $\mathcal L\mathsf{R}_{0}=\mathrm{Id}-\mathsf P$, we deduce that
\begin{equation*}\begin{split}
\mathbb{L}_\xi & = \Big( \mathsf P + i \mathsf P (v \cdot \xi)\mathsf{R}_{0} + \mathsf P r_{7}(\xi) \Big) (\mathcal L - i v \cdot \xi) \Big( \mathsf P + i\mathsf{R}_{0} (v \cdot \xi) \mathsf P + r_{6}(\xi)\mathsf P\Big) \\
&= \Big( \mathsf P + i \mathsf P (v \cdot \xi)\mathsf{R}_{0} + \mathsf P r_{7}(\xi) \Big) \Big(- i \mathsf P (v \cdot \xi) \mathsf P + (v \cdot\xi) \mathsf{R}_{0} (v \cdot \xi) \mathsf P + r_{8}(\xi) + \tilde{r}_{1}(\xi)\Big) \\
&= - i \mathsf P (v \cdot \xi) \mathsf P + \mathsf P ( v \cdot \xi)\mathsf{R}_{0} (v \cdot \xi) \mathsf P + \tilde{r}_{2}(\xi),
\end{split}\end{equation*}
where $\|\tilde{r}_{k}(\xi)\|_{\mathscr B(H^{\circ},H^{\bullet})} \lesssim |\xi|^{3}$ for $k=1,2$ because the second order remainder term $\mathsf P r_{8}(\xi)$ vanishes since $\mathsf P\mathcal L=0$. The lemma is proved.
\end{proof}
\begin{rem}
Note that in order to compute a second order expansion of the rectified operator, only a first order expansion of the spectral projector is needed.
\end{rem}
\begin{lem}[\textit{\textbf{Block matrix representation of the rectified operator}}]
\label{lem:rectified_block_matrix}
For any non-zero $|\xi| \leqslant \alpha_0$, the rectified operator $\mathbb{L}_\xi$ writes, along the $H$-orthogonal decomposition
\begin{equation*}
\nul(\mathcal L) = \Big\{ u \cdot v \mu ~ ; ~ u \perp \xi \Big \} \oplus \Span\left( \psi_{\textnormal{Bou}}, \psi_{+\textnormal{wave}}\left( \widetilde{\xi} \right), \psi_{-\textnormal{wave}}\left( \widetilde{\xi} \right) \right), \quad \widetilde{\xi}=\frac{\xi}{|\xi|},
\end{equation*}
as a block matrix:
\begin{equation}\label{eq:LLXIMXI}\mathbb{L}_{\xi} = \left(
\begin{matrix}
\lambda_\textnormal{inc}(\xi) \, \mathrm{Id}_{(d-1) \times (d-1)} & 0_{(d-1) \times 3} \\
0_{3 \times (d-1)} & \mathbb{M}(\xi) \\
\end{matrix}
\right), \qquad \mathbb{M}(\xi) \in \mathscr{M}_{3\times3}(\mathbb R)\end{equation}
where the ``incompressible'' eigenvalue $\lambda_\textnormal{inc}(\xi)$ can be expressed for any normalized $u \perp \xi$ as
\begin{equation*}
\lambda_\textnormal{inc}(\xi) = \frac{d}{E} \left \langle \mathbb{L}_\xi (u \cdot v \mu), u \cdot v \mu \right \rangle_{H}.
\end{equation*}
\end{lem}
\begin{proof}
Fix some non-zero $| \xi | \leqslant \alpha_0$ and consider some $u \perp \xi$, using the first point of Lemma \ref{lem:rectified_operator}, the functions $\mathbb{L}_{\xi} ( u \cdot v \mu)$ and~$\left(\mu, \xi \cdot v \mu, |v|^2 \mu\right)$ are respectively odd and even in the direction $u$, and thus $H$-orthogonal. Similarly, $\mathbb{L}_\xi \left(\mu, \xi \cdot v \mu, |v|^2 \mu\right)$ and $u \cdot v \mu$ are also~$H$-orthogonal. Thus, one may write
$$\mathbb{L}_{\xi} = \left(
\begin{matrix}
\mathbb{M}'(\xi) & 0_{(d-1) \times 3} \\
0_{3 \times (d-1)} & \mathbb{M}(\xi) \\
\end{matrix}
\right),$$
for some $(d-1) \times (d-1)$ matrix $\mathbb{M}'(\xi)$, because $\left\{\left(a + b \xi + c | v |^2\right)\mu ~ | ~ a, b, c \in \mathbb R \right\}$ is spanned by $\psi_{\textnormal{Bou}}, \psi_{- \textnormal{wave}}\left( \widetilde{\xi} \right)$ and $\psi_{+\textnormal{wave}}\left( \widetilde{\xi} \right)$.
In the case $d \geqslant 3$, we still have to show that $\mathbb{M}'(\xi)$ is a multiplication matrix. To do so, consider a pair of $H$-orthogonal functions $\varphi = u \cdot v \mu$ and $\varphi' = u' \cdot v \mu$ for some vectors~$u, u' \in \mathbb R^d$ such that~$(u, u', \xi)$ is an orthogonal triple. From the first point of Lemma~\ref{lem:rectified_operator}, the function $\mathbb{L}_{\xi} \varphi$ is odd in the direction $u$ and even in the direction $u'$, thus
\begin{gather*}
\left\langle \mathbb{L}_\xi (u \cdot v \mu), u' \cdot v \mu \right\rangle_{H} = \left\langle \mathbb{L}_\xi \left(u' \cdot v \mu\right), u \cdot v \mu \right\rangle_{H} = 0.
\end{gather*}
Furthermore, choosing an orthogonal matrix $\Theta$ mapping $(\xi, u, u')$ onto $(\xi, u', u)$, we have
\begin{align*}
\langle \mathbb{L}_\xi (u \cdot v \mu), u \cdot v \mu \rangle_{H} = \langle \mathbb{L}_\xi (\Theta u) \cdot v \mu, (\Theta u) \cdot v \mu \rangle_{H} = \left \langle \mathbb{L}_\xi \left(u' \cdot \mu\right), u' \cdot v \mu \right \rangle_{H}.
\end{align*}
We conclude by applying these two relations to any orthogonal basis of $\Big\{ u \cdot v \mu ~| ~ u \perp \xi \Big \}$.
\end{proof}
\begin{lem}[\textit{\textbf{Expansion in matrix form}}]
\label{lem:expansion_rectified_matrix}
Recall that $\kappa_\star$ and $c$ are defined in Theorem \ref{thm:spectral_study}. With the notations of Lemma \ref{lem:rectified_block_matrix},
the ``incompressible'' eigenvalue expands~as
\begin{equation*}
\lambda_\textnormal{inc}(\xi) = - \kappa_\textnormal{inc} | \xi |^2 + \mathcal O\left( | \xi |^3 \right).
\end{equation*}
Furthermore, the matrix $\mathbb{M}(\xi)$ in \eqref{eq:LLXIMXI} can be written in the basis $\psi_{\textnormal{Bou}}, \psi_{- \textnormal{wave}}\left( \widetilde{\xi} \right), \psi_{ + \textnormal{wave}}\left( \widetilde{\xi} \right)$ as
\begin{equation}
\label{eq:expansion_rectified_matrix}
\mathbb{M}(\xi) = i | \xi |
\left(
\begin{matrix}
0 & 0 & 0 \\
0 & c & 0 \\
0 & 0 & - c
\end{matrix}
\right) + | \xi |^2 \left(
\begin{matrix}
- \kappa_\textnormal{Bou} & * & * \\
* & - \kappa_{\textnormal{wave}} & * \\
* & * & - \kappa_{\textnormal{wave}}
\end{matrix}
\right) + \mathcal O\left( | \xi |^3 \right).
\end{equation}
\end{lem}
\begin{proof}
Straightforward calculations show that, in the basis $\psi_{\textnormal{Bou}}, \psi_{- \textnormal{wave}}\left( \widetilde{\xi} \right), \psi_{+\textnormal{wave}}\left( \widetilde{\xi} \right)$, the first order coefficient is diagonal:
\begin{equation*}
\mathbb{M}(\xi) = -i \mathsf P (v \cdot \xi) + \mathcal O\left( | \xi |^2 \right) = i | \xi |
\left(
\begin{matrix}
0 & 0 & 0 \\
0 & c & 0 \\
0 & 0 & - c
\end{matrix}
\right) + \mathcal O\left( | \xi |^2 \right).
\end{equation*}
We then turn to the second order coefficient $ \left( v \cdot \widetilde{\xi} \right)\mathsf{R}_{0} \left( v \cdot \widetilde{\xi} \right)$.
\step{1}{The diffusion coefficient $\kappa_\textnormal{inc}$}
In this step, we will use the following identity:
\begin{equation}
\label{eq:invariance_L_inv_A}
\left\langle \mathcal L^{-1}{\mathbf{A}} u \cdot u', {\mathbf{A}} u \cdot u' \right\rangle_{H} = \left\langle \mathcal L^{-1}{\mathbf{A}} (\Theta u) \cdot (\Theta u'),{\mathbf{A}} (\Theta u) \cdot (\Theta u') \right\rangle_{H},
\end{equation}
which holds for any $u, u' \in \mathbb R^d$ and any orthogonal matrix $\Theta$, and is a consequence of the identities (where we recall $\mathsf{R}_0 = -\mathcal L^{-1} (\mathrm{Id} - \mathsf P)$ )
$$\Theta\mathsf{R}_{0} =\mathsf{R}_{0} \Theta, \qquad {\mathbf{A}} (v) u \cdot u' = {\mathbf{A}} (\Theta v) (\Theta u) \cdot (\Theta u').$$
The diffusion coefficient $\kappa_\textnormal{inc}$ is given for any $\sigma \in \mathbb S^{d-1}$ orthogonal to $\widetilde{\xi}$ by
$$
\kappa_\textnormal{inc} = \frac{ d}{E} \left\langle\mathsf{R}_{0} \left( v \cdot \widetilde{\xi} \right) (v \cdot \sigma) \mu, \left( v \cdot \widetilde{\xi} \right) \left(v \cdot \sigma\right) \mu \right\rangle_{H} = \left\langle \mathcal L^{-1}{\mathbf{A}} \widetilde{\xi} \cdot \sigma, {\mathbf{A}} \widetilde{\xi} \cdot \sigma \right\rangle_{H},
$$ which, for any orthogonal pair $\omega, \sigma \in \mathbb S^{d-1}$, rewrites using \eqref{eq:invariance_L_inv_A} as
\begin{equation}
\label{eq:rel_non_diag}
\kappa_\textnormal{inc} = - \left\langle \mathcal L^{-1}{\mathbf{A}} \omega \cdot \sigma, {\mathbf{A}} \omega \cdot \sigma \right\rangle_{H}.
\end{equation}
Choosing in particular $\omega = \frac{1}{\sqrt{2}}(u - u')$ and $\sigma = \frac{1}{\sqrt{2}}(u + u')$, where $u, u' \in \mathbb S^{d-1}$ are orthogonal, we have
$$\kappa_{\textnormal{inc}}=- \frac{1}{2} \left\langle \mathcal L^{-1} \big({\mathbf{A}} u \cdot u - {\mathbf{A}} u' \cdot u'\big), {\mathbf{A}} u \cdot u - {\mathbf{A}} u' \cdot u' \right\rangle_{H}.$$
where we used the fact that $A$ is symmetric. Consequently, we have by \eqref{eq:invariance_L_inv_A}
\begin{align*}
\kappa_\textnormal{inc} = & - \left\langle \mathcal L^{-1} {\mathbf{A}} u \cdot u, {\mathbf{A}} u \cdot u \right\rangle_{H} + \left\langle \mathcal L^{-1} {\mathbf{A}} u \cdot u, {\mathbf{A}} u' \cdot u' \right\rangle_{H},
\end{align*}
and since ${\mathbf{A}}$ is trace-free, one can get rid of the term involving $u'$ by averaging over some orthonormal family~$u' = u_1', \dots, u_{d-1}' \in \left( \mathbb R u \right)^\perp$:
\begin{align*}
\kappa_\textnormal{inc} = & - \left\langle \mathcal L^{-1} {\mathbf{A}} u \cdot u, {\mathbf{A}} u \cdot u \right\rangle_{H} + \frac{1}{d-1} \left\langle \mathcal L^{-1} {\mathbf{A}} u \cdot u , \sum_{i=1}^{d-1} {\mathbf{A}} u'_i \cdot u'_i \right\rangle_{H} \\
= & - \left\langle \mathcal L^{-1} {\mathbf{A}} u \cdot u, {\mathbf{A}} u \cdot u \right\rangle_{H} - \frac{1}{d-1} \left\langle \mathcal L^{-1} {\mathbf{A}} u \cdot u , {\mathbf{A}} u \cdot u \right\rangle_{H},
\end{align*}
and therefore
\begin{equation}
\label{eq:rel_diag}
\kappa_\textnormal{inc} = - \frac{d}{d-1} \left\langle \mathcal L^{-1} {\mathbf{A}} u \cdot u, {\mathbf{A}} u \cdot u \right\rangle_{H}.
\end{equation}
Summing \eqref{eq:rel_non_diag} and \eqref{eq:rel_diag} over all pairs of vectors in the canonical basis of $\mathbb R^d$, we rewrite the coefficient $\kappa_\textnormal{inc}$ as a Hilbert-Schmidt norm of matrices:
\begin{equation*}
\kappa_\textnormal{inc} = -\frac{1}{(d-1)(d+1)} \left\langle \mathcal L^{-1}{\mathbf{A}} , {\mathbf{A}} \right\rangle_{H}.
\end{equation*}
\step{2}{The diffusion coefficient $\kappa_\textnormal{Bou}$}
The coefficient $\kappa_\textnormal{Bou}$ writes
\begin{align*}
\kappa_\textnormal{Bou} & = - \left \langle\mathsf{R}_{0} \left( v \cdot \widetilde{\xi} \right) \psi_{\textnormal{Bou}} , \left( v \cdot \widetilde{\xi} \right) \psi_{\textnormal{Bou}} \right\rangle_{H}= - \left \langle \mathcal L^{-1}{\mathbf{B}} \cdot \widetilde{\xi}, {\mathbf{B}} \cdot \widetilde{\xi} \right \rangle_{H} = - \frac{1}{d} \left \langle \mathcal L^{-1} {\mathbf{B}} , {\mathbf{B}} \right \rangle_{H},
\end{align*}
where we used the invariance of $\mathcal L^{-1}$ again, as well as the identity ${\mathbf{B}} (v) \cdot u = {\mathbf{B}} (\Theta v) \cdot (\Theta u)$, allowing to sum over $\widetilde{\xi}$ taken in the canonical basis of $\mathbb R^d$.
\step{3}{The diffusion coefficient $\kappa_{\textnormal{wave}}$}
The coefficient $\kappa_{\textnormal{wave}}$ writes
\begin{align*}
\kappa_\textnormal{wave} & = - \left \langle\mathsf{R}_{0} \left( v \cdot \widetilde{\xi} \right) \psi_{\pm \textnormal{wave}} \left(\widetilde{\xi}\right), \left( v \cdot \widetilde{\xi} \right) \psi_{\pm \textnormal{wave}}\left(\widetilde{\xi}\right) \right\rangle_{H} \\
& = - \frac{1}{2} \left \langle \mathcal L^{-1}{\mathbf{A}} \widetilde{\xi} \cdot \widetilde{\xi}, {\mathbf{A}} \widetilde{\xi} \cdot \widetilde{\xi} \right \rangle_{H} - \frac{E^2(K-1)}{2} \left \langle \mathcal L^{-1}{\mathbf{B}} \cdot \widetilde{\xi}, {\mathbf{B}} \cdot \widetilde{\xi} \right \rangle_{H},
\end{align*}
where we have used the fact that ${\mathbf{A}} \widetilde{\xi} \cdot \widetilde{\xi}$, and thus $\mathcal L^{-1}{\mathbf{A}} \widetilde{\xi} \cdot \widetilde{\xi}$ again by the invariance of~$\mathcal L^{-1}$, is even in the direction $\widetilde{\xi}$, whereas $B \cdot \widetilde{\xi}$ is odd in this direction, and therefore these functions are $H$-orthogonal. Using the results of the previous steps, we actually have
$$
\kappa_\textnormal{wave} = \frac{d-1}{2 d} \kappa_\textnormal{inc} + \frac{E^2(K-1)}{2} \kappa_\textnormal{Bou}.$$
The lemma is proved.
\end{proof}
\newcommand{\mathbb{P}}{\mathbb{P}}
\begin{lem}[\textit{\textbf{Second order diagonalization and decomposition of the rectified operator}}]
\label{lem:diagonalization_rectified} With the notations of Lemma \ref{lem:rectified_block_matrix},
the rectified operator $\mathbb{L}_\xi$ has four distinct eigenvalues which expand as
\begin{gather*}
\lambda_\textnormal{inc}(\xi) = - \kappa_\textnormal{inc} | \xi |^2 + \mathcal O\left( |\xi|^3 \right), \\
\lambda_\textnormal{Bou}(\xi) = - \kappa_\textnormal{Bou} | \xi |^2 + \mathcal O\left( |\xi|^3 \right), \quad
\lambda_{\pm \textnormal{wave}}(\xi) = \pm i c | \xi| - \kappa_\textnormal{wave} |\xi|^2 + \mathcal O\left( |\xi|^3 \right)\,.
\end{gather*}
Furthermore, the ``incompressible'' eigenvalue is associated with the following spectral projector:
\begin{equation*}
\mathbb{P}_\textnormal{inc}\left( \xi \right) \left[ a \mu + u \cdot v \mu + c |v|^2 \mu \right] = {\left(\Pi_{\widetilde{\xi}}u\cdot v\right)} \mu,
\end{equation*}
and the spectral projectors associated with the ``Boussinesq'' and ``waves'' eigenvalues expand in the basis $\psi_{\textnormal{Bou}}, \psi_{- \textnormal{wave}}\left( \widetilde{\xi} \right), \psi_{+ \textnormal{wave}}\left( \widetilde{\xi} \right)$ as
\begin{equation}\label{eq:textbook_expansion_projector}\mathbb{P}_{\star}(\xi)=\mathbb{P}_{\star}^{(0)} + |\xi|\mathbb{P}^{(1)}_{\star}+\mathcal O\left(|\xi|^{2}\right), \qquad \star=\textnormal{Bou},\pm\textnormal{wave}\end{equation}
where
\begin{equation}\label{eq:textbook_rectP0}
\mathbb{P}_{\textnormal{Bou}}^{(0)}= \left(\begin{matrix}
1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0
\end{matrix}\right), \quad \mathbb{P}_{-\textnormal{wave}}^{(0)} = \left(\begin{matrix}
0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0
\end{matrix}\right), \quad \mathbb{P}_{\textnormal{wave}}^{(0))} = \left(\begin{matrix}
0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1
\end{matrix}\right)\end{equation}
and $\mathbb{P}_{\star}^{(1)}$ is a constant $3\times 3$-matrix for $\star=\textnormal{Bou},\pm\textnormal{wave}$.
\end{lem}
\begin{proof}
Because the operator $\mathbb{M}(\xi)$ has the asymptotic expansion
$$\mathbb{M}(\xi) = |\xi| \Big( \mathbb{M}^{(0)}\left( \widetilde{\xi} \right) + | \xi | \mathbb{M}^{(1)}\left( \widetilde{\xi} \right) + \mathcal O\left( |\xi|^2 \right) \Big)$$
where $\mathbb{M}^{(0)}\left( \widetilde{\xi} \right)$ has three distinct eigenvalues by Lemma \ref{lem:expansion_rectified_matrix}, we know from \cite[Chapter 2, Theorem 5.4]{K1995} that $\mathbb{M}(\xi)$ has three distinct eigenvalues for $\xi$ small, and thus admits the following spectral decomposition:
$$\mathbb{M}(\xi) = \lambda_\textnormal{Bou}(\xi) \mathbb{P}_\textnormal{Bou}(\xi) + \lambda_{+ \textnormal{wave}}(\xi) \mathbb{P}_{+\textnormal{wave}}(\xi) + \lambda_{- \textnormal{wave}}(\xi) \mathbb{P}_{-\textnormal{wave}}(\xi).$$
The expansions of the eigenvalues is given by \cite[Chapter II-(5.12)]{K1995} applied to \eqref{eq:expansion_rectified_matrix}:
\begin{gather*}
\lambda_\textnormal{inc}(\xi) = - \kappa_\textnormal{inc} | \xi |^2 + \mathcal O\left( |\xi|^3 \right), \quad \lambda_\textnormal{Bou}(\xi) = - \kappa_\textnormal{Bou} | \xi |^2 + \mathcal O\left( | \xi |^2 \right),\\
\lambda_{\pm \textnormal{wave}}(\xi) = \pm i c | \xi| - \kappa_{\textnormal{wave}} | \xi |^2 + \mathcal O \left( | \xi |^3 \right),
\end{gather*}
and the expansions of the spectral projectors are given by \cite[Chapter II-(5.9)]{K1995} applied to~\eqref{eq:expansion_rectified_matrix} yielding
\eqref{eq:textbook_expansion_projector} and \eqref{eq:textbook_rectP0} and we point out that the first order coefficients $\mathbb{P}^{(1)}_{\star}$ in matrix form has coefficients independent of $\widetilde{\xi}$ and can be explicitly computed, although their expression will not be needed.
\end{proof}
\begin{figure}
\caption{Localization of the spectrum of $\mathcal L - i (v \cdot \xi) $ for $| \xi | \leqslant \alpha_0 $.}
\end{figure}
\begin{lem}[\textit{\textbf{Expansion of the spectral projectors}}]
\label{lem:expansion_projectors}
For $\alpha_{0} >0$ small enough, the following spectral decomposition holds for any $| \xi | \leqslant \alpha_{0}$:
\begin{equation*}\begin{split}
\mathcal L_\xi \mathsf P(\xi) &= \mathsf P(\xi) \mathcal L_\xi = \lambda_{\textnormal{Bou}}(\xi)\mathsf P_{\textnormal{Bou}}(\xi)+\lambda_{\textnormal{inc}}(\xi)\mathsf P_{\textnormal{inc}}(\xi)\\
&\phantom{++++} +\lambda_{+\textnormal{wave}}(\xi)\mathsf P_{+\textnormal{wave}}(\xi)+\lambda_{-\textnormal{wave}}(\xi)\mathsf P_{-\textnormal{wave}}(\xi)
\end{split}\end{equation*}
where the projector operators $\mathsf P_\star(\xi)$ $(\star = \textnormal{Bou}, \textnormal{inc}, \pm\textnormal{wave})$ expand in $\mathscr B(H^{\circ} ; H^{\bullet})$ as \eqref{eq:PPstar}
with the zeroth order coefficients being defined in Theorem \ref{thm:spectral_study}.
\end{lem}
\begin{proof}
Recall that $\mathbb{L}_\xi$ and $\mathcal L_\xi$ are related through
$\mathbb{L}_\xi = \left(\mathcal U_\xi^{-1} \mathcal L_{\xi} \mathcal U_\xi\right)_{| \nul(\mathcal L)},$
thus, using the fact that $\mathcal U_\xi \mathsf P= \mathsf P(\xi) \mathcal U_{\xi}$, we deduce that
\begin{equation*}
\mathsf P(\xi) \mathcal L_{\xi} = \mathcal L_\xi \mathsf P(\xi)= \mathcal U_\xi \mathbb{L}_{\xi} \mathcal U_\xi^{-1}.
\end{equation*}
This lemma is therefore a lifted version of Lemma \ref{lem:diagonalization_rectified}, and the corresponding projectors are related through
\begin{equation*}
\mathsf P_\star(\xi) = \mathcal U_{\xi} \mathbb{P}_\star(\xi) \mathcal U_\xi^{-1} = \big(\mathcal U_{\xi} \mathsf P\big) \mathbb{P}_\star(\xi) \big(\mathsf P \mathcal U_\xi^{-1}\big).
\end{equation*}
We then deduce the expansion of each $\mathsf P_\star(\xi)$ from those of $\mathbb{P}_\star(\xi)$ in $\mathscr B\left( \nul(\mathcal L) \right)$ established in Lemma~\ref{lem:diagonalization_rectified} and those of $\mathcal U_\xi \mathsf P$ and $\mathsf P \mathcal U_\xi^{-1}$ in $\mathscr B(H^{\circ} ; H^{\bullet})$ from \eqref{eq:expansion_Kato_isomorphismPP}:
$$
\mathsf P_\star(\xi) = \Big( \mathsf P + i \xi\cdot \mathsf{R}_{0} v \mathsf P + \mathsf P r_{7}(\xi) \Big)
\Big( \mathbb{P}_\star^{(0)}\left( \widetilde{\xi} \right) + |\xi| \mathbb{P}^{(1)} \left( \widetilde{\xi} \right) + \mathbb{S}(\xi)\Big)\Big( \mathsf P + i \xi \cdot\mathsf P v\mathsf{R}_{0} + r_{6}(\xi)\mathsf P \Big)
$$
where we recall that $\|r_{j}(\xi)\|_{\mathscr B(H^{\circ};H^{\bullet})} \lesssim |\xi|^{2}$ while $\mathbb{S}(\xi) \in \mathscr M_{(d+2)\times (d+2)}(\mathbb R)$ with norm of order~$\mathcal O\left( |\xi|^2 \right).$ We can expand further to deduce
$$\mathsf P_\star(\xi) =: \mathsf P_\star^{(0)}\left(\widetilde{\xi}\right) + i \xi \cdot \mathsf P^{(1)}_\star\left(\widetilde{\xi}\right) + \mathsf{S}_{\star}(\xi),$$
where we have denoted \footnote{Of course, $\mathsf P\mathbb{P}^{(0)}_{\star}(\cdot)\mathsf P$ can be identified with $\mathbb{P}^{(0)}_{\star}(\cdot)$ but we make here the (slight) distinction between operators defined on the finite dimensional space and the associated matrices.}
\begin{gather*}
\mathsf P^{(0)}_\star\left( \widetilde{\xi} \right) := \mathsf P\mathbb{P}^{(0)}_\star\left( \widetilde{\xi} \right)\mathsf P,
\qquad
\mathsf P_\star^{(1)}\left( \widetilde{\xi} \right) := \mathsf P^{(0)}_\star(\widetilde{\xi}) v \mathsf{R}_{0} +\mathsf{R}_{0}v \mathsf P^{(0)}_{\star}\left(\widetilde{\xi}\right)-i\widetilde{\xi}\mathsf P\,\mathbb{P}^{(1)}_{\star}(\widetilde{\xi})\mathsf P\,.
\end{gather*}
We notice that both $\mathsf{R}_{0}v\mathsf P^{(0)}_{\star}(\widetilde{\xi})$ and $\mathsf P\,\mathbb{P}^{(1)}_{\star}(\widetilde{\xi})\mathsf P$ vanish on $\nul(\mathcal L)^\perp$:
$$\varphi \in \nul(\mathcal L)^\perp \Rightarrow \mathsf P^{(1)} \left( \widetilde{\xi} \right) \varphi = \mathsf P^{(0)} \left( \widetilde{\xi} \right) v \mathsf{R_{0}} \varphi = \mathsf P^{(0)} \left( \widetilde{\xi} \right) v \varphi.$$
Therefore, the first order term of the projector associated with the ``incompressible'' eigenvalue writes explicitly for any~$\varphi\in \nul(\mathcal L)^{\perp}$, and $\omega \in \mathbb S^{d-1}$
\begin{align*}
\mathsf P^{(1)}_\textnormal{inc}(\omega) \varphi
& = \frac{d}{E} \Bigg( \Big[ \left(\mathrm{Id} - \omega \otimes \omega \right) \left\langle v_j \mathsf{R}_0 \varphi , v \mu \right\rangle_{H} \Big] \cdot v \mu \Bigg)_{j=1}^d \\
&= \frac{d}{E} \Bigg( \left\langle v_j \mathsf{R}_0 \varphi , v \mu \right\rangle_{H} \cdot \Big[ \left(\mathrm{Id} - \omega \otimes \omega \right) v \Big] \mu \Bigg)_{j=1}^d \\
& = \sqrt{\frac{d}{E}} \left\langle \varphi , \mathcal L^{-1}{\mathbf{A}} \right\rangle_{H} \Big[ \left(\mathrm{Id} - \omega \otimes \omega \right) v \Big] \mu,
\end{align*}
and, in particular,
\begin{align*}
\omega \cdot \mathsf P^{(1)}_\textnormal{inc}(\omega) \varphi
& = \frac{d}{E} \Big[ \left(\mathrm{Id} - \omega \otimes \omega \right) \left\langle v \cdot \omega \mathsf{R}_0 \varphi , v \mu \right\rangle_{H} \Big] \cdot v \mu \\
& = \sqrt{\frac{d}{E}} \Big[ \left(\mathrm{Id} - \omega \otimes \omega \right) \left\langle \varphi , \mathcal L^{-1}{\mathbf{A}} \omega \right\rangle_{H} \Big] \cdot v \mu,
\end{align*}
the one associated with the ``Boussinesq'' eigenvalue writes for any $\varphi \in \nul(\mathcal L)^\perp$
$$
\mathsf P^{(1)}_\textnormal{Bou}(\omega) \varphi = \left \langle \mathsf{R}_{0} \varphi, v \psi_\textnormal{Bou} \right \rangle_{H} \psi_\textnormal{Bou}= \langle \varphi, \mathcal L^{-1} {\mathbf{B}} \rangle_{H} \psi_\textnormal{Bou},
$$
and the ones associated with the ``waves'' eigenvalues write, for $\varphi\in \nul(\mathcal L)^{\perp}$,
\begin{align*}
\mathsf P^{(1)}_{\pm \textnormal{wave}}(\omega) \varphi&= \left \langle v \mathsf{R}_{0} \varphi, \psi_{\pm \textnormal{wave}}(\omega) \right \rangle_{H} \psi_{\pm \textnormal{wave}}(\omega)\\
&= \left(\pm \frac{1}{\sqrt{2} } \left\langle \varphi , \mathcal L^{-1} A(v) \omega \right\rangle_{H} + E \sqrt{\frac{K-1}{2}} \left\langle \varphi, \mathcal L^{-1} B(v) \right\rangle_{H} \right) \psi_{\pm \textnormal{wave}}(\omega).
\end{align*}
This concludes the proof.
\end{proof}
We recall now the following \emph{hypocoercivity} result extracted from \cite{D2011}:
\begin{lem}[\textit{\textbf{Hypocoercivity \cite[Lemma 4.1]{D2011}}}]\label{lem:lemhypocoercivity}
Assume that $\mathcal L\::\:\mathscr{D}(\mathcal L) \subset H \to H$ satisfies Assumptions \ref{L1}--\ref{L4}. Then, for any $\xi \in\mathbb R^{d}$, there exists some~$\xi$-dependent bilinear symmetric form $\Phi_\xi[\cdot, \cdot]\::\:H\times H \to \mathbb R^{d}$ defined through
$$\Phi_\xi\left[f, f\right] = \frac{\xi}{1 + | \xi |^2} \cdot \Big\langle \mathsf P f, \, \mathcal T_1 \mathsf P f + \mathcal T_2 \mathsf P v \left( \mathrm{Id} - \mathsf P \right) f \Big\rangle_{H},$$
where $\mathcal T_1 \in \mathscr B\left( \nul(\mathcal L) ; \mathbb R^d \right)$ and $\mathcal T_2 \in \mathscr B\Big( \big(\nul(\mathcal L)\big)^d ; \mathbb R^d \Big)$,
such that there holds for some $c > 0$
\begin{equation}
\label{eq:hypocoercivity}
\Phi_\xi\Big[\mathcal L_{\xi} f, f\Big] \leqslant - \frac{ c | \xi |^2 }{1 + | \xi |^2 } \| \mathsf P f \|^2_{H} + \frac{1}{c} \| (\mathrm{Id} - \mathsf P) f \|_{H}^2,
\end{equation}
uniformly in $\xi \in \mathbb R^d$ and $f \in \mathscr{D}(\mathcal L_\xi)$.
\end{lem}
With this at hands, we may turn to the proof of the decay estimates of Theorem \ref{thm:spectral_study}.
\begin{lem}[\textit{\textbf{Resolvent bounds and decay estimates of the semigroup}}]
\label{lem:spectral_decay} With the notations of Lemma \ref{lem:localization_spectrum}, let $0 < \lambda < \lambda_\mathcal L$.
There exist some constants $C, \gamma > 0$ such that, for any $|\xi| \geqslant \alpha_{0}$, the spectrum is localized as follows:
$$\mathfrak{S}_{H}(\mathcal L_\xi) \cap \Delta_{-\gamma} = \varnothing,$$
and the resolvent satisfies
$$ \sup_{z \in \Delta_{-\gamma}} \| \mathcal R(z,\mathcal L_{\xi}) \|_{\mathscr B(H)} \leqslant C.$$
Furthermore, for any $0 < \sigma < \sigma_0 := \min\{\lambda, \gamma\}$, the decay estimates
$$
\sup_{t \geqslant 0} \,e^{2 \sigma_0 t} \left\| U_{\xi}(t) {\left( \mathrm{Id} - \mathsf P(\xi) \right)} f \right\|^2_{H} + \int_0^\infty e^{2 \sigma t} \left\| U_{\xi}(t) {\left( \mathrm{Id} - \mathsf P(\xi) \right)}f \right\|_{H^{\bullet}}^2 \, \mathrm{d} t \leqslant C_{\sigma}\| \left( \mathrm{Id} - \mathsf P(\xi) \right)f \|^2_{H}$$
and
$$\int_{0}^{\infty} e^{2\sigma t} \left\|U_{\xi}(t) {\left(\mathrm{Id}-\mathsf P(\xi)\right)}f\right\|_{H}^{2}\mathrm{d} t \leqslant C_{\sigma}\|{\left(\mathrm{Id}-\mathsf P(\xi)\right)}f\|_{H^{\circ}}^{2}$$
hold uniformly in $\xi \in \mathbb R^{d}$ and $f \in {H}$, where $\left(U_{\xi}(t)\right)_{t\geq0}$ denotes the $C^{0}$-semigroup in $H$ generated by $(\mathcal L_{\xi},\mathscr{D}(\mathcal L_{\xi}))$.
\end{lem}
\begin{proof}
In a first step, we prove resolvent bounds using the above hypocoercivity result as well as the uniform decay estimate in $H$. In a second step, we prove the $H^{\bullet}-\mathcal S$ integral decay estimate using an energy method, from which we deduce the $H-H^{\circ}$ one in a third step. Let us fix $0<\lambda < \lambda_\mathcal L$.
\step{1}{Resolvent bounds and uniform decay estimate} Using the above Lemma \ref{lem:lemhypocoercivity}, we define, for any $|\xi| \geqslant \alpha_{0}$ the equivalent inner product on $H$
$$( \cdot , \cdot )_{H, \xi} := \langle \cdot, \cdot \rangle_{H} + \eta \Phi_\xi[\cdot, \cdot]$$
with some small $\eta > 0$. By combining the control of $(\mathrm{Id} - \mathsf P)f$ from \ref{L3} and the control of $\mathsf P f$ from \eqref{eq:hypocoercivity}, we have for any $| \xi | \geqslant \alpha_0$
\begin{align*}
\left( \mathcal L_\xi f, f \right)_{H, \xi} & \leqslant \left(\frac{\eta}{c} - \lambda_\mathcal L\right) \| (\id - \mathsf P) f \|_{H}^2 - \frac{c |\xi|^2}{1 + |\xi| ^2} \| \mathsf P f \|^2_{H} \\
& \leqslant -\min\left\{ \frac{\lambda_\mathcal L}{2}, \frac{c \alpha_{0} ^2 }{1 + \alpha_{0} ^2} \right\} \| f \|^2_{H},
\end{align*}
where we chose $\eta \leqslant \frac{c}{2} \lambda_\mathcal L$. Assuming $\eta$ small enough so that the norm induced by~$( \cdot, \cdot )_{H, \xi}$ is equivalent to $\| \cdot \|_{H}$ uniformly in $\xi \in \mathbb R^d$, we have for some $\gamma > 0$
\begin{equation*}
( \mathcal L_\xi f, f )_{H, \xi} \leqslant -\gamma \| f \|_{H}^2, \qquad
\forall | \xi | \geqslant \alpha_0, ~ f \in \mathscr{D}(\mathcal L_{\xi})\,. \end{equation*}
We thus deduce that for $| \xi | \geqslant \alpha_{0}$
\begin{equation*}
\| U_{\xi}(t) \|_{\mathscr B(H)} \lesssim e^{-\gamma t},
\end{equation*}
as well as (up to a reduction of $\gamma$ for the resolvent bound)
\begin{equation*}
\mathfrak{S}(\mathcal L_{\xi}) \cap \Delta_{-\gamma} = \varnothing, \quad \sup_{z \in \Delta_{-\gamma}} \| \mathcal R(z,\mathcal L_{\xi}) \|_{\mathscr B(H)} \lesssim 1.
\end{equation*}
On the other hand, for $|\xi| \leqslant \alpha_{0}$, the resolvent of $(\mathrm{Id} - \mathsf P(\xi)) \mathcal L_\xi$ is given for $z \in \Delta_{-\lambda}$ by
$$\mathcal R(z,\mathcal L_{\xi}(\mathrm{Id}-\mathsf P(\xi)))=\mathcal R(z,\mathcal L_{\xi}) - \sum_{\star = \textnormal{inc}, \textnormal{Bou}, \pm \textnormal{wave}}(z-\lambda_{\star}(\xi))^{-1} \mathsf P_\star(\xi) $$
which is therefore holomorphic in $z \in \Delta_{-\lambda}$ and thus can be explicitly bounded using the bound \eqref{eq:bound_L_xi} and the maximum principle. We deduce that the semigroup it generates is bounded by the Gearhart-Pruss theorem \cite[Theorem V.1.10]{engel}, i.e.
\begin{equation*}
\left\| U_{\xi}(t)\left(\mathrm{Id}-\mathsf P(\xi)\right) f \right\|_{\mathscr B(H)} \lesssim e^{-\lambda t} \| f \|_{H} \qquad \quad
\forall | \xi | \leqslant \alpha_0, ~ f \in H\,.
\end{equation*}
To sum up, putting together both decay estimates and denoting $\sigma_0 := \min\{ \lambda, \gamma\}$, there holds
\begin{equation*}
\sup_{\xi \in \mathbb R^d}\left\| U_{\xi}(t)\left(\mathrm{Id}-\mathsf P(\xi)\right)f \right\|_{\mathscr B(H)} \lesssim e^{-\sigma_0 t} \| f \|_{H}, \qquad \forall f \in H\,,
\end{equation*}
where we recall that $\mathsf P(\xi) = 0$ for $|\xi| > \alpha_0$. This concludes this step.
\step{2}{Integral decay estimates}
Let us prove both integral decay estimates.
\step{2a}{The $H^{\bullet}-H$-integral decay estimate}
Let $f \in \range (\mathrm{Id} - \mathsf P(\xi)) {\cap \mathscr{D}(\mathcal L_{\xi})}$ and denote~$f(t) =U_{\xi}(t)f$ the unique solution to
$$
\begin{cases}
\partial_{t} f(t)=\mathcal L_{\xi}f(t)=\mathcal L_{\xi}\left(\mathrm{Id}-\mathsf P(\xi)\right)f(t),\\
f(0) = f.
\end{cases}
$$
Using that $\mathcal L$ is self-adjoint in $H$ and that the multiplication by $i v \cdot \xi$ is skew-adjoint, we have the energy estimate
$$\frac{\mathrm{d}}{\mathrm{d} t} \| f(t) \|^2_{H} = 2 \Re \left\langle \mathcal L_\xi f(t) , f(t) \right\rangle_{H} = 2 \left\langle \mathcal L f(t) , f(t) \right\rangle_{H}.$$
Furthermore, using the dissipativity estimate of $\mathcal L$ from \ref{L3}, we get
$$\frac{\mathrm{d}}{\mathrm{d} t} \| f(t) \|^2_{H} + 2 \lambda_\mathcal L \| (\mathrm{Id} - \mathsf P) f(t) \|^2_{H^{\bullet}} \leqslant 0,$$
which we complete using \eqref{eq:degenerate_pythagora_X_star} and $\mathsf P \in \mathscr B(H; H^{\bullet})$ as
\begin{equation}
\label{eq:EnergyHH_Hh}
\frac{\mathrm{d}}{\mathrm{d} t} \| f(t) \|^2_{H} + \lambda_\mathcal L \| f(t) \|^2_{H^{\bullet}} \lesssim \| \mathsf P f(t) \|_{H^{\bullet}}^2 \lesssim \| f(t) \|_{H}^2 \lesssim e^{-2 \sigma_0 t} \| f \|_{H}^2,
\end{equation}
where we also used the decay estimate established in the previous step. Multiplying this by~$e^{2 \sigma t}$ and integrating, one easily deduces
\begin{equation*}
\sup_{t \geqslant 0} e^{2 \sigma t} \| f(t) \|_{H}^2 + \lambda_\mathcal L \int_0^\infty e^{2 \sigma t} \| f(t) \|_{H^{\bullet}}^2 \mathrm{d} t \lesssim \| f \|^2_{H}.
\end{equation*}
This identity holds for any $f \in \left(\mathrm{Id}-\mathsf P(\xi)\right) \cap \mathscr{D}(\mathcal L_{\xi})$ and we conclude the proof using that~$\mathscr{D}(\mathcal L_{\xi})$ is dense in $H$.
\step{2b}{The $H-H^{\circ}$-integral decay estimate} As before, we assume now $f \in \left(\mathrm{Id}-\mathsf P(\xi)\right) \cap H^{\circ}$ and set $f(t)=U_{\xi}(t)f$ for any $t\geq0.$
We use a duality argument together with a density argument and prove that
$$\langle e^{\sigma t} f(t) , \phi \rangle_{L^2_t(H)} \lesssim \| f\|_{H^{\circ}} \| \phi \|_{L^2_t(H)}.$$
To perform the duality argument, we need to point out that $\mathcal L$ is self-adjoint, thus
$$\left(U_{\xi}(t) \left( \mathrm{Id} - \mathsf P(\xi) \right) \right)^\star = U_{-\xi}(t) \left( \mathrm{Id} - \mathsf P(-\xi) \right).$$
Since the step functions span a dense subspace of $L^2_t(H)$, it is enough to check that the dual estimate holds for such a following function:
\begin{gather*}
\phi(t) = \begin{cases}
\phi_0 \in H, & t \in [t_1 , t_2] ,\\
0, & t \notin [t_1, t_2],
\end{cases}, \quad \| \phi \|_{L^2_t(H)} = \sqrt{t_2 - t_1} \| \phi_0 \|_{H}.
\end{gather*}
For such a step function, the inner product writes explicitly as
\begin{align*}
\langle e^{\sigma t} f(t) , \phi \rangle_{L^2_t(H)} = \int_0^\infty \big\langle e^{\sigma t} f(t) , \phi(t) \big\rangle_{H} \mathrm{d} t= \int_{t_1 }^{ t_2 } \left\langle e^{\sigma t} U_{\xi}(t)\left( \mathrm{Id} - \mathsf P(\xi) \right) f , \phi_0 \right\rangle_{H} \mathrm{d} t.
\end{align*}
We then get by duality and using Cauchy-Schwarz's inequality
\begin{equation*}\begin{split}
\langle e^{\sigma t} f(t) , \phi \rangle_{L^2_t(H)} & = \int_{t_1 }^{ t_2 } \big\langle f , e^{\sigma t} U_{-\xi}(t) \left( \mathrm{Id} - \mathsf P(-\xi) \right) \phi_0 \big\rangle_{H} \mathrm{d} t \\
& \leqslant \| f \|_{H^{\circ}} \, \int_{ t_1 }^{ t_2 } e^{\sigma t} \| U_{-\xi}(t) \left( \mathrm{Id} - \mathsf P(-\xi) \right) \phi_0 \|_{H^{\bullet}} \mathrm{d} t \\
& \leqslant \| f \|_{H^{\circ}} \, \sqrt{t_2-t_1} \left(\int_0^\infty e^{2 \sigma t} \| U_{-\xi}(t) \left( \mathrm{Id} - \mathsf P(-\xi) \right) \phi_0 \|_{H^{\bullet}}^2 \mathrm{d} t\right)^{\frac{1}{2}}
\end{split}\end{equation*}
thus, using the $H^{\bullet}-H$ estimate from \textit{Step 2a}
\begin{align*}
\langle e^{\sigma t} f(t) , \phi \rangle_{L^2_t(H)} & \lesssim \| f \|_{H^{\circ}} \sqrt{t_2-t_1} \| \phi_0 \|_{H} \lesssim \| f \|_{H^{\circ}} \| \phi \|_{L^2_t(H)}.
\end{align*}
\color{black}
This concludes the proof.\end{proof}
\begin{figure}
\caption{Localization of the spectrum of $\mathcal L_\xi $ provided by the hypocoercivity lemma \ref{lem:lemhypocoercivity} for $\xi \in \mathbb R^d$, compared with the localization of the hydrodynamic eigenvalues defined for $|\xi| \leqslant \alpha_0$. }
\end{figure}
\subsection{Proof of Theorem \ref{thm:enlarged_thm}}
We present here the full proof a the ‘‘enlarged'' version of Theorem \ref{thm:spectral_study} as provided in Theorem \ref{thm:enlarged_thm}. We present in a first step how to extend the resolvent bounds, in a second step how to extend the decay estimate. In a third step, we extend the projector bounds from Lemma \ref{lem:expansion_projection} as this is enough to deduce the same bounds on the expansion of $\mathcal U_\xi$ and in turn of $\mathsf P_\star(\xi)$.
\step{1}{Resolvent bounds}
Using the factorization formulae
\begin{equation}
\label{eq:shrinkage_Y}
\begin{split}
\mathcal R(z,\mathcal L_{\xi}) & = \mathcal R(z,\mathcal B^{(0)}_{\xi}) + \mathcal R(z,\mathcal L_{\xi}) \mathcal A^{(0)} \mathcal R(z,\mathcal B^{(0)}_{\xi}) \\ & =\mathcal R(z,\mathcal B^{(0)}_{\xi})+ \mathcal R(z,\mathcal B^{(0)}_{\xi}) \mathcal A^{(0)} \mathcal R(z,\mathcal L_{\xi})
\end{split}
\end{equation}
and the fact that the function $z \in \Delta_{-\lambda} \mapsto \mathcal R(z,\mathcal B^{(0)}_{\xi}) \in \mathscr B(H) \cap \mathscr B(X)$ is holomorphic, as well as $H \hookrightarrow X$ and $\mathcal A^{(0)}\in \mathscr B(X;H)$,
we deduce that, for any $z \in \Delta_{-\lambda}$
$$ \mathcal R(z,\mathcal L_{\xi}) \in \mathscr B\left( X \right) \iff \mathcal R(z,\mathcal L_{\xi}) \in \mathscr B(H)\,,$$
or in other words, the spectrum of $\mathcal L_\xi$ in $\Delta_{-\lambda}$ does not depend on the space $H$ or $X$:
$$\mathfrak{S}_{H}(\mathcal L_{\xi}) \cap \Delta_{-\lambda} = \mathfrak{S}_{X}(\mathcal L_\xi) \cap \Delta_{-\lambda}.$$
More precisely, since $\mathcal R\left( z, \mathcal B^{(0)}_\xi \right) \in \mathscr B(H) \cap \mathscr B(X)$ uniformly in $\xi \in \mathbb R^d$ and $z \in \Delta_{-\lambda}$, there holds for any $z \in \Delta_{-\lambda} \setminus \mathfrak{S}(\mathcal L_\xi)$
$$1 + \|\mathcal R(z,\mathcal L_{\xi})\|_{\mathscr B(H)} \lesssim 1 + \|\mathcal R(z,\mathcal L_{\xi})\|_{\mathscr B(X)} \lesssim 1 + \|\mathcal R(z,\mathcal L_{\xi})\|_{\mathscr B(H)}.$$
This implies that the bounds in $\mathscr B(H)$ on the resolvent from Theorem \ref{thm:spectral_study} also hold in~$\mathscr B\left( X \right)$. Actually, also the bounds \eqref{eq:bound_L_xi} in Lemma \ref{lem:localization_spectrum} can be refined for $|\xi| \leqslant \alpha_{0}$ small enough as
\begin{equation}
\label{eq:bound_L_xi_Y}
\sup_{ | z | = r } \| \mathcal R(z,\mathcal L_{\xi}) \|_{\mathscr B(X;X^{\bullet})} + \sup_{ | z | = r } \| \mathcal R(z,\mathcal L_{\xi}) \|_{\mathscr B(X^{\circ};X)}
+ \sup_{z\in\Omega} \| \mathcal R(z,\mathcal L_{\xi}) \|_{\mathscr B(X)} \leqslant C_1,
\end{equation}
where $r$ is small enough and $\Omega := \Delta_{-\lambda} \cap \{ | z | \geqslant r \}$. Indeed, starting from the dissipativity estimate involving $X^{\bullet}$ and $\mathcal A^{(0)} \in \mathscr B(X)$ from \ref{LE}:
\begin{align*}
\Re \left \langle (\mathcal L_\xi - z) f, f \right\rangle_{X} & = \Re \left \langle \mathcal B^{(0)} f, f \right\rangle_{X} + \Re \left \langle \mathcal A^{(0)} f, f \right\rangle_{X} - z \| f \|^2_{X} \\
& \leqslant - \lambda_\mathcal B \| f \|^2_{X^{\bullet}} + \left(\| \mathcal A^{(0)}\|_{\mathscr B(X)} - z \right)\| f \|_{X}^2
\end{align*}
which gives for some $z_0 \geqslant \| \mathcal A^{(0)} \|_{\mathscr B(X)}$
$$\| \mathcal R(z_{0},\mathcal L_{\xi})\|_{\mathscr B(X,X^{\bullet})} \leqslant \lambda_\mathcal B^{-1}.$$
Performing the same computations with the decomposition
$$\mathcal L^\star_{\xi} = (\mathcal B^{(0)})^{\star} + i v \cdot \xi + (\mathcal A^{(0)})^{\star},$$
where $(\mathcal B^{(0)})^{\star}$ satisfies the same dissipativity as $\mathcal B^{(0)}$, we get $\|\mathcal R(z_{0},\mathcal L_{\xi})^{\star} \|_{\mathscr B(X;X^{\bullet})} \leqslant \lambda_\mathcal B^{-1}$ and thus by the adjoint identity \eqref{eq:twisted_adjoint_identity_one_sided}
$$\| \mathcal R(z_{0},\mathcal L_{\xi})\|_{\mathscr B(X^{\circ};X)} \leqslant \lambda_\mathcal B^{-1}.$$ Using the resolvent identity as in the proof of Lemma \ref{lem:localization_spectrum}, we deduce \eqref{eq:bound_L_xi_Y}.
\step{2}{Decay estimates}
We first prove the uniform estimates, and then the integral ones.
\step{2a}{The uniform decay estimate}
To improve the decay estimate of \eqref{eq:decay} to \eqref{eq:decay-EE}, we apply the Duhamel formula to the decomposition $\mathcal L_\xi = \mathcal B_{\xi}^{(0)}+ \mathcal A^{(0)}$
$$U_{\xi}(t) = V_{\xi}^{(0)}(t) + \int_0^t U_{\xi}(t-\tau)\mathcal A^{(0)} V_{\xi}^{(0)}(\tau)\mathrm{d} \tau,$$
where $\left(V_{\xi}^{(0)}(t)\right)_{t\geq0}$ denotes the $C^{0}$-semigroup in $X$ generated by $(\mathcal B^{(0)}_{\xi},\mathscr{D}(\mathcal L_{\xi}))$. After
composing with $\mathrm{Id} - \mathsf P(\xi)$ from the left, we get
\begin{align*}
\left( \mathrm{Id} - \mathsf P(\xi) \right)U_{\xi}(t) = & \left( \mathrm{Id} - \mathsf P(\xi) \right)V_{\xi}^{(0)}(t) + \int_0^t \left( \mathrm{Id} - \mathsf P(\xi) \right) U_{\xi}(t-\tau)\mathcal A^{(0)} V_{\xi}^{(0)}(\tau) \mathrm{d} \tau.
\end{align*}
Since $\| \mathsf P(\xi) \|_{\mathscr B(X)} \lesssim 1$ from \eqref{eq:bound_L_xi_Y}, and using $\mathcal A^{(0)} \in \mathscr B( X ; H )$, we have
\begin{align*}
\big\| ( \mathrm{Id} - \mathsf P(\xi) ) U_{\xi}(t) \big\|_{\mathscr B(X)}
\lesssim \left\|V_{\xi}^{(0)}(t)\right\|_{\mathscr B(X)}
+ \int_0^t \left\| \left( \mathrm{Id} - \mathsf P(\xi) \right) U_{\xi}(t - \tau) \right\|_{\mathscr B(H)} \left\| V_{\xi}^{(0)}(\tau)\right\|_{\mathscr B(X)} \mathrm{d} \tau,
\end{align*}
and using the decay estimate of $\left( \mathrm{Id} - \mathsf P(\xi) \right)U_{\xi}(t)$ in $\mathscr B(H)$ from Theorem \ref{thm:spectral_study}, as well as the dissipativity hypothesis for $\mathcal B^{(0)}_{\xi}$ from \ref{LE}, we then get (recall that $\sigma_0 \leqslant \lambda$)
\begin{equation*}
\left\| \left( \mathrm{Id} - \mathsf P(\xi) \right) U_{\xi}(t)\right\|_{\mathscr B(X)} \lesssim e^{-\lambda\,t}
+ \int_0^t e^{-\sigma_0 (t - \tau)} e^{-\lambda \tau} \mathrm{d} \tau \lesssim e^{-\sigma_0 t}.
\end{equation*}
This proves the uniform in time decay.
\step{2b}{The integral $X^{\bullet}-X$ and $X-X^{\circ}$ decay estimates}
From then on, the proof of the integral decay estimate follows the same strategy as the one adopted for the proof of Lemma \ref{lem:spectral_decay}, starting from the decomposition $\mathcal L = \mathcal B^{(0)} + \mathcal A^{(0)}$ and, resuming the computations of Lemma \ref{lem:spectral_decay}. Typically, estimate \eqref{eq:EnergyHH_Hh} can be adapted to give now
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d} t} \| f(t) \|^2_{X} + 2 \lambda_\mathcal B \| f(t) \|^2_{X^{\bullet}} & \lesssim \| \mathcal A^{(0)} \|_{\mathscr B(X)} \| f(t) \|_{X}^2 \lesssim \| \mathcal A^{(0)} \|_{\mathscr B(X)} e^{-2 \sigma_0 t} \| f \|^2_{X}.
\end{align*}
After integration, one obtains easily \eqref{eq:decay-EE} as in Lemma \ref{lem:spectral_decay}, as well as the corresponding estimate for $\Big(U_\xi(t) (\mathrm{Id} - \mathsf P(\xi)) \Big)^\star$, from which we deduce \eqref{eq:decay_Ee'} by a similar duality argument.
\step{3}{Expansion of the projectors}
To establish the uniform bounds in $\mathscr B(X^{\circ} ;H^{\bullet})$ on the expansion of the spectral projectors
$$\mathsf P_\star(\xi) = \mathsf P^{(0)}_\star\left( \widetilde{\xi} \right) + i \xi \cdot \mathsf P^{(1)}_\star\left( \widetilde{\xi} \right) + S_{\star}\left( \xi \right),$$
we use a similar bootstrap strategy as in the proofs of Lemma \ref{lem:expansion_projection} and Theorem~\ref{thm:regularized_thm}. More precisely, in each step, we will prove uniform bounds in $\mathscr B\left(X^{\circ} ; X \right)$ and then combine with those in $\mathscr B\left( H ; H^{\bullet}\right)$ from Lemma \ref{lem:expansion_projection} to conclude. Similarly, we will need the following regularization properties for $\mathsf P(\xi)$:
\begin{equation}
\label{eq:regularization_P_Y}
\| \mathsf P(\xi) \|_{ \mathscr B(X^{\circ} ; X_2 ) } + \| \mathsf P(\xi) \|_{ \mathscr B(X^{\circ} ; H^{\bullet} ) } \lesssim 1,
\end{equation}
which, as in the original proof, comes from combining the identity $\mathsf P(\xi)^2 = \mathsf P(\xi)$ with the resolvent bound \eqref{eq:bound_L_xi_Y} (for the $X^{\circ}-X$ bound), with Lemma \ref{lem:expansion_projection} (for the $H-H^{\bullet}$ bound), and with the regularization hypothesis \ref{assumption_large_bounded_A} for $\mathcal A^{(0)}$ (for the $X-X_2$ and $X-H$ bounds) applied to the representation
\begin{gather*}
\mathsf P(\xi) = \frac{1}{2 i \pi} \oint_{ | z | = r } \left( \mathcal R\left( z, \mathcal B^{(0)}_\xi \right) \mathcal A^{(0)} \right)^2 \mathcal R\left( z , \mathcal L_\xi \right) \mathrm{d} z.
\end{gather*}
Similarly, we will need the regularization properties
\begin{equation}
\label{eq:regularization_P_Y_adjoint}
\| \mathsf P(\xi)^\star \|_{ \mathscr B(X^{\circ} ; X_j ) } \lesssim 1, \quad j = 1, 2,
\end{equation}
which comes from the regularization hypothesis \ref{assumption_large_bounded_A} for $\left( \mathcal A^{(0)} \right)^\star$ applied to the representation
$$\mathsf P(\xi)^\star = \frac{1}{2 i \pi} \oint_{ | z | = r } \left[ \mathcal R\left( z, \mathcal B^{(0)}_\xi \right)^\star \left(\mathcal A^{(0)}\right)^\star \right]^j \left(\mathcal R\left( z , \mathcal L_\xi \right)\right)^\star \mathrm{d} z.$$
Finally, we will the need the resolvent bounds
\begin{equation}
\label{eq:bound_resolvent_Y_j} \| \mathcal R(z,\mathcal L) \|_{\mathscr B(X_{j})} \lesssim 1 + \frac{1}{|z|}, \quad j = 0,1,2, \qquad z \in \Delta_{-\lambda} \setminus \{0\},
\end{equation}
which also come from a factorization strategy as in the original proof.
\step{3a}{First order expansion of $\mathsf P(\xi)$}
We use the the bootstrap formula \eqref{eq:bootstrap_P_1_A}:
$$\mathsf P^{(1)}(\xi) = \mathsf P(\xi) \mathsf P^{(1)}(\xi) + \mathsf P^{(1)}(\xi) \mathsf P,$$
and the adjoint identity $\| T \|_{\mathscr B(X^{\circ};X)} = \| T^\star \|_{\mathscr B(X;X^{\bullet})}$ from \eqref{eq:twisted_adjoint_identity_one_sided} to establish the bound
\begin{align*}
\| \mathsf P^{(1)}(\xi) \|_{\mathscr B(X^{\circ};X)}
&\leqslant \| \mathsf P^{(1)}(\xi)^\star \mathsf P(\xi)^\star \|_{\mathscr B(X;X^{\bullet})} + \| \mathsf P^{(1)}(\xi) \mathsf P \|_{\mathscr B(X^{\circ};X)} \\
&\leqslant \| \mathsf P^{(1)}(\xi)^\star \|_{\mathscr B(X_1;X^{\bullet})} \| \mathsf P(\xi)^\star \|_{\mathscr B(X;X_{1})} + \| \mathsf P^{(1)}(\xi) \|_{\mathscr B(X_1;X)} \| \mathsf P \|_{\mathscr B(X^{\circ};X_1)}\\
&\lesssim \| \mathsf P^{(1)}(\xi)^\star \|_{\mathscr B(X_1;X^{\bullet})} + \| \mathsf P^{(1)}(\xi) \|_{\mathscr B(X_1;X)}
\end{align*}
where we used the the regularization property \eqref{eq:regularization_P_Y} for $\mathsf P=\mathsf P(0)$ and \eqref{eq:regularization_P_Y_adjoint} for $\mathsf P^\star(\xi)$ in the last estimate. We now turn to the first term $\| \mathsf P^{(1)}(\xi)^\star \|_{\mathscr B(X_{1};X^{\bullet})}$:
\begin{align*}
\| \mathsf P^{(1)}(\xi)^\star \|_{\mathscr B(X_{1};X^{\bullet})} & \leqslant \frac{1}{2 \pi} \oint_{|z|=r} \left\| \Big(\mathcal R(z,\mathcal L) v \mathcal R(z,\mathcal L_{\xi})\Big)^\star \right\|_{\mathscr B(X_{1};X^{\bullet}) } \mathrm{d} |z| \\
&\leqslant\frac{1}{2 \pi} \oint_{|z|=r} \left\| \mathcal R(z,\mathcal L_{\xi})^\star v^{\star}\mathcal R(z,\mathcal L)^\star \right\|_{\mathscr B(X_{1};X^{\bullet}) } \mathrm{d} |z| \\
& \lesssim \oint_{|z|=r} \| \mathcal R(z,\mathcal L_{\xi})^\star \|_{\mathscr B(X;X^{\bullet})} \| \mathcal R(z,\mathcal L)^\star \|_{\mathscr B(X_{1}) } \mathrm{d} |z|,
\end{align*}
where we used the fact that the adjoint of the multiplication by $v$ is in $\mathscr B(X_1 ; X)$ according to \ref{assumption_large_multi-v}. We can rewrite this estimate without the adjoints and estimate it using the resolvent bounds \eqref{eq:bound_L_xi_Y} and \eqref{eq:bound_resolvent_Y_j}:
\begin{align*}
\| \mathsf P^{(1)}(\xi)^\star \|_{\mathscr B(X_{1};X^{\bullet})} \lesssim \oint_{|z|=r} \| \mathcal R(z,\mathcal L_{\xi}) \|_{\mathscr B(X^{\circ};X) } \| \mathcal R(z,\mathcal L) \|_{\mathscr B(X_{1})} \mathrm{d} |z|
\lesssim 1.
\end{align*}
The second term $\| \mathsf P^{(1)}(\xi)\|_{\mathscr B(X_1;X)} \lesssim 1$ is estimated in the same way, thus we obtain
$$\|\mathsf P^{(1)}(\xi)\|_{\mathscr B(X^{\circ};X)} \lesssim 1.$$
Integrating the formula \eqref{eq:shrinkage_Y} and combining with the $H-H^{\bullet}$ resolvent bound \eqref{eq:bound_L_xi}, one proves the estimate
$$\|\mathsf P(\xi)\|_{\mathscr B(X;H^{\bullet})} \lesssim 1,$$
which then allows to perform another simpler (duality-free) bootstrap argument by combining the above estimates with the bounds of Lemma \ref{lem:expansion_projection}:
\begin{align*}
\| \mathsf P^{(1)}(\xi) \|_{ \mathscr B(X^{\circ} ; H^{\bullet} ) } & \leqslant \| \mathsf P(\xi) \mathsf P^{(1)}(\xi) \|_{ \mathscr B(X^{\circ} ; H^{\bullet} ) } + \| \mathsf P^{(1)}(\xi) \mathsf P \|_{ \mathscr B(X^{\circ} ; H^{\bullet} ) } \\
& \leqslant \| \mathsf P(\xi) \|_{ \mathscr B(X ; H^{\bullet} ) } \| \mathsf P^{(1)}(\xi) \|_{ \mathscr B(X^{\circ} ; X ) } + \| \mathsf P^{(1)}(\xi) \|_{ \mathscr B(X ; H^{\bullet} ) } \| \mathsf P \|_{ \mathscr B(X^{\circ} ; X ) } \\
& \lesssim 1.
\end{align*}
This concludes this step.
\step{3b}{Second order expansion}
We use this time the bootstrap formula \eqref{eq:bootstrap_P_2_A} and the first order estimates, together with the duality identity \eqref{eq:twisted_adjoint_identity_one_sided}
\begin{align*}
\| \mathsf P^{(2)}(\xi) \|_{\mathscr B(X^{\circ};X)} & \lesssim 1
+ \| \mathsf P \mathsf P^{(2)}(\xi) \|_{\mathscr B(X^{\circ};X) } + \| \mathsf P^{(2)}(\xi) \mathsf P(\xi) \|_{\mathscr B(X^{\circ};X)} \\
& \lesssim 1
+ \| \mathsf P^{(2)}(\xi)^\star \mathsf P^\star \|_{\mathscr B(X;X^{\bullet}) } + \| \mathsf P^{(2)}(\xi) \mathsf P(\xi) \|_{\mathscr B(X^{\circ};X)}.
\end{align*}
Using the regularization estimate \eqref{eq:regularization_P_Y} on $\mathsf P$, we obtain
\begin{align*}
\| \mathsf P^{(2)}(\xi) \|_{\mathscr B(X^{\circ};X)} \lesssim 1 & + \| \mathsf P^{(2)}(\xi)^\star \|_{\mathscr B(X_2;X^{\bullet})} \| \mathsf P^\star \|_{\mathscr B(X;X_{2})} + \| \mathsf P^{(2)}(\xi) \|_{\mathscr B(X_{2};X) } \| \mathsf P \|_{\mathscr B(X^{\circ};X_2) } \\
\lesssim 1 & + \| \mathsf P^{(2)}(\xi)^\star \|_{\mathscr B(X_2;X^{\bullet})} + \| \mathsf P^{(2)}(\xi) \|_{\mathscr B(X_{2};X) }.
\end{align*}
We conclude as in the previous step that $\| \mathsf P^{(2)}(\xi) \|_{\mathscr B(X^{\circ};X)} \lesssim 1$, and then perform a second bootstrap to deduce $\| \mathsf P^{(2)}(\xi) \|_{\mathscr B(X^{\circ};H^{\bullet})} \lesssim 1$ from the estimates of Lemma \ref{lem:expansion_projection}. This concludes the proof.
\begin{rem}
Notice that, since
$$\left\|\left[\left(\mathrm{Id}-\mathsf P(\xi)\right)U_{\xi}(t)\right]^{\star}\right\|_{\mathscr B(X)}=\left\|\left(\mathrm{Id}-\mathsf P(\xi)\right)U_{\xi}(t)\right\|_{\mathscr B(X)}$$
the decay estimates \eqref{decay-semigroup-EE} extends easily to the adjoint $ U_{\xi}(t)^{\star} {\left( \mathrm{Id} - \mathsf P(\xi) \right)^{\star}}$.
\end{rem}
\subsection{Regularized version of the spectral result}
\label{sec:LR}
We present here yet another improved version of Theorem \ref{thm:spectral_study}, taking now advantage of possible alternative splittings of the linearized operator $\mathcal L$. In order to prove a ``regularized'' version of our main result, we will need the following extra assumption.
\begin{enumerate}[label=\hypst{LR}]
\item \label{LR}
Besides Assumptions \ref{L1}--\ref{L4}, assume that the operator can be decomposed in a way $\mathcal L = \mathcal B^{(1)} + \mathcal A^{(1)}$ compatible with a hierarchy of \textbf{Banach} spaces $\left(W_j\right)_{j=-\ell}^{2}$, where $\ell \geqslant 0$, such that
\begin{enumerate}
\item the spaces $W_j$ embed into one another and the regular space embeds into the original space:
$$W_2 \hookrightarrow W_1 \hookrightarrow W = W_{0} \hookrightarrow W_{-1} \hookrightarrow \dots \hookrightarrow W_{-\ell}, \quad W \hookrightarrow H,$$
\item the multiplication by $v$ is bounded from $W_{j}$ to $W_{j-1}$ for some $j$:
$$\|vf\|_{W_{j}} \lesssim \|f\|_{W_{j+1}}, \quad \quad j=0,1,$$
\item \label{ass:reg_A_reg} the operator $\mathcal A^{(1)}$ is bounded from $W_{j}$ to $W_{j+1}$ and from $H$ to $W_{-\ell}$:
$$\mathcal A^{(1)} \in \mathscr B(W_{j};W_{j+1}) \cap \mathscr B(H;W_{-\ell}), \qquad j=1,\ldots,-\ell,$$
\item \label{ass:reg_diss} the part $\mathcal B^{(1)}_{\xi}$ is dissipative on $Y = W_{-\ell}, \dots, W_2, H$ in the sense that
$$\mathfrak{S}_{Y}(\mathcal B^{(1)}_{\xi}) \cap \Delta_{-\lambda_{\mathcal B}} = \varnothing, \qquad \sup_{\xi \in \mathbb R^d}\left\|\mathcal R(z,\mathcal B^{(1)}_{\xi})\right\|_{\mathscr B(Y)} \lesssim | \mathrm{Re}\, z + \lambda_\mathcal B|^{-1},$$
uniformly in $z \in \Delta_{-\lambda_{\mathcal B}}$.
\end{enumerate}
\end{enumerate}
Under this new set of Assumptions, we derive the following version of Theorem \ref{thm:spectral_study}:
\begin{theo}[\textit{\textbf{Regularized result}}]
\label{thm:regularized_thm}
If Assumptions \ref{LR} are in force, then the spectral projectors from Theorem \ref{thm:spectral_study} are regularizing in the sense that in the decomposition \eqref{eq:PPstar}
\begin{equation*}
\mathsf P_\star(\xi) = \mathsf P^{(0)}_\star\left( \widetilde{\xi} \right) + i \xi \cdot \mathsf P_\star^{(1)}\left( \widetilde{\xi} \right) + \mathsf{S}_{\star}(\xi),
\end{equation*}
each term belongs to $\mathscr B\left(H^{\circ} ; W \right)$ uniformly in $|\xi| \leqslant \alpha_{0}$, and $\| \mathsf{S}_\star(\xi) \|_{ \mathscr B(H^{\circ} ; W) } \lesssim | \xi |^2$.
\end{theo}
\begin{rem}
Once again, we illustrate this set of assumptions in the case of the Boltzmann equation for hard spheres. In this context the hierarchy of spaces can be taken to be
$$W_j = L^\infty\left( \mu^{-1/2} \langle v \rangle^{j + \ell} \mathrm{d} v \right), \qquad j=2,\ldots,-\ell\,,$$
for some integer $\ell > \frac{d}{2}$, and the splitting is also Grad's splitting (see Remark \ref{rem:L1B1}).
\end{rem}
\begin{proof}[Proof of Theorem \ref{thm:regularized_thm}]
Let us prove that the coefficients of the expansion
$$\mathsf P(\xi) = \mathsf P + \xi \cdot \mathsf P^{(1)} + \xi \otimes \xi : \mathsf P^{(2)}(\xi)$$
belong to $\mathscr B(H^{\circ};W)$ uniformly in $\xi$ small enough. As pointed out in the proof of Theorem \ref{thm:enlarged_thm}, this will be enough to deduce it also holds for $\mathsf P_\star(\xi)$.
\step{1}{Estimate for the resolvent in the regular space $W$}
Starting from the factorization formula
\begin{equation}
\label{eq:factorization_L_xi_reg}
\mathcal R(z,\mathcal L_{\xi}) = \sum_{n=0}^{\ell-1} \left( \mathcal R\left(z,\mathcal B^{(1)}_{\xi} \right) \mathcal A^{(1)} \right)^n \mathcal R\left(z,\mathcal B_{\xi}^{(1)}\right) + \left(\mathcal R\left(z,\mathcal B^{(1)}_{\xi} \right) \mathcal A^{(1)} \right)^{\ell} \mathcal R(z,\mathcal L_{\xi}),
\end{equation}
one gets from \eqref{eq:bound_L_xi}, the embedding $W \hookrightarrow H$, as well as the bounds \ref{ass:reg_diss} on $\mathcal R\left(z,\mathcal B_{\xi}^{(1)}\right)$ and the regularization hypothesis \ref{ass:reg_A_reg} for $\mathcal A^{{(1)}}$ that, for any $0 < \lambda < \lambda_\mathcal L$, there are some $\alpha_{0},r >0$ small enough such that
\begin{equation}
\label{eq:bound_L_xi_reg}
\sup_{ z \in \Omega } \| \mathcal R(z,\mathcal L_{\xi}) \|_{\mathscr B(W)} \leqslant C, \qquad |\xi| \leqslant \alpha_0
\end{equation}
with, as before, $\Omega=\Delta_{-\lambda}\cap \{ |z|\geqslant r\}$.
\step{2}{Behavior of the spectral projector as $\xi \to 0$}
We use a similar bootstrap strategy. It is simpler because no duality argument is involved, in exchange we replace the use of adjoint operators by estimates in $W_{-1}$ and $W_{-2}$.
The first step is to extend the bound \eqref{eq:bound_L_xi_reg} from $W$ to $W_j$ for $j=-2, \dots, 2$ using similar factorization arguments as in the previous proofs.
The second step is then to deduce, using the bounds \ref{ass:reg_A_reg}--\ref{ass:reg_diss} and the representation formula
\begin{equation*}
\mathsf P(\xi) = \frac{1}{2 i \pi} \int_{ | z | = r } \left( \mathcal R\left(z,\mathcal B^{(1)}_{\xi} \right) \mathcal A^{(1)} \right)^{2} \mathcal R(z,\mathcal L_{\xi}) \mathrm{d} z
\end{equation*}
the regularization bounds
$$\| \mathsf P(\xi) \|_{ \mathscr B\left( W ; W_2 \right) } + \| \mathsf P(\xi) \|_{ \mathscr B\left( W_{-2} ; W \right) } \lesssim 1, \quad | \xi | \leqslant \alpha_0.$$
From then, we follow a simplified version of the bootstrap procedures used in the proofs of Lemma \ref{lem:expansion_projection} and Theorem \ref{thm:enlarged_thm}. This concludes this proof.
\end{proof}
\section{Properties of the linearized semigroup in the physical space}
\label{scn:study_physical_space}
If one assumes only \ref{L1}--\ref{L4}, we denote in this section $X = H$. Under the extra assumption \ref{LE}, $X$ is the space from \ref{LE}.
In this section, we exploit the spectral description of the previous Section to study the main properties of the semigroup $(U^{\varepsilon}(t))_{t\geq0}$.
We adopt the notations and definitions introduced in Section \ref{sec:detail}. We only recall that
$$U^{\varepsilon}(t)=U^{\varepsilon}_\textnormal{kin}(t)+ U^{\varepsilon}_\textnormal{hydro}(t), \qquad U^{\varepsilon}_\textnormal{hydro}(t)=U^{\varepsilon}_\textnormal{NS}(t) + U^{\varepsilon}_\textnormal{wave}(t)$$
where the various semigroups are defined in Definitions \ref{def:hydro_semigroups} and \ref{def:kinetic_semigroups}. As explained in the Introduction and in Section \ref{sec:detail}, it is important for the definition of the various stiff terms $\Psi^{\varepsilon}_\textnormal{hydro}[f,g]$ and $\Psi^{\varepsilon}_\textnormal{kin}[f,g]$ to study suitable bounds on the semigroups $U^{\varepsilon}_\textnormal{kin}(t) $ and $U^{\varepsilon}_\textnormal{hydro}(t) $ as well as their convolution with suitable time-dependent functions.
We begin with the following uniform estimates on $U^{{\varepsilon}}_{\textnormal{hydro}}(\cdot) $
\begin{lem}[\textit{\textbf{Bounds for the hydrodynamic semigroup}}]
\label{lem:decay_semigroups_hydro}
For $\star=\textnormal{NS},\textnormal{wave},\textnormal{disp}$, the hydrodynamic semigroups~$U^{\varepsilon}_\star(\cdot)$ are bounded from $\GG$ to $\HHH$:
\begin{equation}
\label{eq:decay_semigroups_hydro}
|\hskip-0.04cm|\hskip-0.04cm| U^{\varepsilon}_\star(\cdot) g |\hskip-0.04cm|\hskip-0.04cm|_\HHH \lesssim \| g \|_{\GG} +\|g\|_{\dot{\mathbb{H}}_x^{-\alpha}(X^{\circ})}.
\end{equation}
Furthermore, if $\varphi \in L^2\left( [0, T) ; \GG^{\circ} \right)$ where $T \in (0, \infty]$ is such that $\mathsf P \varphi(t) = 0$, then for~$\star=\textnormal{NS}, \textnormal{wave}$ and thus $\star=\textnormal{hydro}$
\begin{subequations}
\label{eq:decay_semigroups_hydro_orthogonal}
\begin{equation}
\label{eq:decay_semigroups_hydro_orthogonal_negative}
\frac{1}{{\varepsilon}} |\hskip-0.04cm|\hskip-0.04cm| U^{\varepsilon}_\star(\cdot) * \varphi |\hskip-0.04cm|\hskip-0.04cm|_\HHH \lesssim \sup_{0 \leqslant t < T} \left\{w_{\phi, \eta}(t) \left(\int_0^t \| \varphi(\tau) \|_{ \GG^{\circ} \cap \dot{\mathbb{H}}^{-\alpha}_x (X^{\circ}_v) }^2 \mathrm{d} \tau \right)^{\frac{1}{2}} \right\},
\end{equation}
and
\begin{equation}
\label{eq:decay_semigroups_hydro_orthogonal_positive}
\frac{1}{{\varepsilon}} |\hskip-0.04cm|\hskip-0.04cm| U^{\varepsilon}_\star(\cdot) * \varphi |\hskip-0.04cm|\hskip-0.04cm|_\HHH \lesssim \sup_{0 \leqslant t < T} \left\{w_{\phi, \eta}(t) \left(\int_0^t \| \varphi(\tau) \|_{ \GG^{\circ} }^{ \frac{2}{1+\alpha} } \mathrm{d} \tau \right)^{ \frac{1+\alpha}{2} } \right\},
\end{equation}
\end{subequations}
\end{lem}
\begin{rem} The above estimates still hold true for $\star=\textnormal{hydro}$ since $$U^{\varepsilon}_\textnormal{hydro}(t)=U^{\varepsilon}_\textnormal{wave}(t)+U^{\varepsilon}_\textnormal{NS}(t).$$
Notice also that \eqref{eq:decay_semigroups_hydro_orthogonal_positive} shows that, with respect to \eqref{eq:decay_semigroups_hydro_orthogonal_negative}, no use of Sobolev space of negative order is required under the stronger integrability assumption $\varphi \in L^{\frac{2}{1+\alpha}}\left( [0, T) ; \GG^{\circ} \right).$\end{rem}
\begin{proof}
Let us fix $\star=\textnormal{Bou},\textnormal{inc},\pm\textnormal{wave}$. First of all, notice that for ${\varepsilon} | \xi | \leqslant \alpha_{0}$, where we recall from \eqref{eq:lambda_star} that $\alpha_{0}$ can be taken small enough
$$\mathrm{Re}\, \left({\varepsilon}^{-2} \lambda_\star({\varepsilon} \xi) \right) = - \kappa_\star | \xi |^2 + \mathcal O\left( {\varepsilon} | \xi |^3 \right) \leqslant - \frac{\kappa_\star}{2} | \xi |^2,$$
and thus, the following estimate holds:
$$\exp\left({\varepsilon}^{-2} t \lambda_\star({\varepsilon} \xi) \right) \leqslant \exp(- t \kappa_\star \frac{|\xi|^2}{2}).$$
Let us prove in the first step \eqref{eq:decay_semigroups_hydro} and in the second step \eqref{eq:decay_semigroups_hydro_orthogonal}.
\step{1}{Proof of \eqref{eq:decay_semigroups_hydro}}
Using the Fourier representation from Definition \ref{def:hydro_semigroups} of $U^{\varepsilon}_\star f$, together with the boundedness $\| \mathsf P_\star({\varepsilon} \xi) \|_{\mathscr B( X^{\circ} ; H^{\bullet} )}$ of Theorems \ref{thm:spectral_study} and \ref{thm:enlarged_thm}, we easily have for $d \geqslant 2$
\begin{equation*}\begin{split}
\| U^{\varepsilon}_\star(t) g \|_{\HH^{\bullet}}^2 &=\int_{\mathbb R^d}\left\|\mathcal F_x\left[U^{{\varepsilon}}_\star(t)g\right](\xi)\right\|_{H^{\bullet}}^2\,\langle \xi\rangle^{2s}\mathrm{d} \xi\\
&\lesssim \int_{\mathbb R^{d}} e^{- t \kappa_\star | \xi |^2 } \| \widehat{g}(\xi) \|_{X^{\circ}}^2 \langle \xi \rangle^{2s} \mathrm{d} \xi \lesssim \| g \|_{\GG^{\circ}}^2,
\end{split}
\end{equation*}
as well as
\begin{equation*}\begin{split}
\int_0^T \left\| |\nabla_x|^{1-\alpha} U^{\varepsilon}_\star(t) g \right\|_{\HH^{\bullet}}^2 \mathrm{d} t &=\int_0
^T\mathrm{d} t \int_{\mathbb R^d}|\xi|^{2-2\alpha}\left\|\mathcal F_x\left[U^{\varepsilon}_\star(t) g\right](\xi)\right\|^2_{H^{\bullet}}\langle\xi\rangle^{2s}\mathrm{d} \xi\\
&\lesssim \int_0^T \int_{\mathbb R^{d}} | \xi |^{2-2\alpha} e^{- t \kappa_\star | \xi |^2 } \| \widehat{g}(\xi) \|_{X^{\circ}}^2 \langle \xi \rangle^{2s} \mathrm{d} \xi \mathrm{d} t \\
& \lesssim \int_{\mathbb R^{d}} \| \widehat{g}(\xi) \|_{X^{\circ}}^2 \langle \xi \rangle^{2s} \left(\int_0^T | \xi |^{2 - 2\alpha} e^{- t \kappa_\star \frac{|\xi|^2}{2} } \mathrm{d} t\right) \mathrm{d} \xi\,.
\end{split}\end{equation*}
Consequently,
\begin{equation}\label{eq:graU_*}
\int_0^T \left\| |\nabla_x|^{1-\alpha} U^{\varepsilon}_\star(t) g \right\|_{\HH^{\bullet}}^2 \mathrm{d} t\lesssim \int_{\mathbb R^{d}} \| \widehat{g}(\xi) \|_{X^{\circ}}^2 | \xi |^{-2\alpha} \langle \xi \rangle^{2s} \left(\int_0^T | \xi |^{2} e^{- t \kappa_\star \frac{|\xi|^2}{2} } \mathrm{d} t\right) \mathrm{d} \xi.\end{equation}
Using that
\begin{equation}\label{eq:XIALPHA}
|\xi|^{-2\alpha}\langle \xi\rangle^{2s} \lesssim \langle \xi\rangle^{2s}\mathbf{1}_{|\xi|\geqslant 1} +|\xi|^{-2\alpha}\mathbf{1}_{|\xi|\leqslant 1} \qquad \text{ and } \quad \int_0^T | \xi |^{2} e^{- t \kappa_\star \frac{|\xi|^2}{2} } \mathrm{d} t \lesssim 1\end{equation}
we deduce that
$$\int_0^T \left\| |\nabla_x|^{1-\alpha} U^{\varepsilon}_\star(t) g \right\|_{\HH^{\bullet}}^2 \mathrm{d} t \lesssim \|g\|_{\GG^{\circ}}^{2}+\| g \|_{\dot{\mathbb{H}}^{-\alpha}_x \left(X^{\circ}_v\right)}^2.$$
This concludes this step thanks to \eqref{eq:roughNt}.
\step{2}{Proof of \eqref{eq:decay_semigroups_hydro_orthogonal}}
Recall from the expansion \eqref{eq:PPstar} of $\mathsf P_\star$ that
\begin{equation}
\label{eq:overPP_star}
\begin{split}
\mathsf P_\star({\varepsilon} \xi) &= \mathsf P_\star^{(0)}\left( \widetilde{\xi} \right) +i {\varepsilon} \xi \cdot \mathsf P_\star^{(1)}( \widetilde{\xi} ) + S_{\star}({\varepsilon} \xi)\\
&=:\mathsf P_{\star}^{(0)}\left(\widetilde{\xi}\right) + {\varepsilon} \xi \cdot \overline{\mathsf P}_{\star}^{(1)}({\varepsilon} \xi),
\end{split}
\end{equation}
where the remainder $\overline{\mathsf P}_{\star}^{(1)}({\varepsilon} \xi)$ satisfies
$$\sup_{{\varepsilon}|\xi| \leqslant \alpha_0}\|\overline{\mathsf P}_{\star}^{(1)}({\varepsilon} \xi)\|_{\mathscr B(X^{\circ};H^{\bullet})} \lesssim 1$$ by virtue of Theorems \ref{thm:spectral_study} and \ref{thm:enlarged_thm} whereas $\mathsf P_{\star}^{(0)}\left( \widetilde{\xi} \right)$ is an $H$-orthogonal projection on a subspace of $\nul(\mathcal L)$. In particular, we have
\begin{equation}
\label{eq:KernelP}
\mathsf P_\star({\varepsilon} \xi) \widehat{\varphi}(t, \xi) = {\varepsilon} \xi \cdot \overline{\mathsf P}_\star^{(1)}({\varepsilon} \xi) \widehat{\varphi}(t, \xi)\end{equation}
and thus there holds, for any $t\geqslant 0,\tau \geqslant 0$
\begin{align*}
\left\| \exp\left( {\varepsilon}^{-2} t \lambda_\star({\varepsilon} \xi) \right) \mathsf P_\star({\varepsilon} \xi) \, \widehat{\varphi}(\tau, \xi) \right\|_{ H^{\bullet} }
& \leqslant {\varepsilon} |\xi| e^{- t \kappa_\star \frac{|\xi|^2}{2}} \| \overline{\mathsf P}^{(1)}_\star({\varepsilon} \xi) \, \widehat{\varphi}(\tau,\xi) \|_{H^{\bullet} } \\
& \lesssim {\varepsilon} |\xi| e^{- t \kappa_\star \frac{|\xi|^2}{2}} \| \widehat{\varphi}(\tau,\xi) \|_{X^{\circ} }.
\end{align*}
Therefore,
\begin{equation}\label{eq:convVarphi}\begin{split}
\frac{1}{{\varepsilon}^2} \| U^{\varepsilon}_\star(\cdot) * \varphi(t) \|_{\HH^{\bullet}}^2 &=\frac{1}{{\varepsilon}^2}\int_{\mathbb R^d}\langle \xi\rangle^{2s}\left\|\int_0^t \mathcal F_x\left[U^{\varepsilon}_\star(t-\tau)\varphi(\tau)\right](\xi)\mathrm{d} \tau\right\|_{H^{\bullet}}^2\mathrm{d} \xi\\
& \leqslant \frac{1}{{\varepsilon}^2}\int_{\mathbb R^d}\langle \xi\rangle^{2s}\left(\int_0^t {\varepsilon}|\xi|e^{- (t-\tau) \kappa_\star \frac{|\xi|^2}{2}} \| \widehat{\varphi}(\tau,\xi)\|_{X^{\circ} }\mathrm{d} \tau\right)^2\mathrm{d} \xi.
\end{split}\end{equation}
Using Cauchy-Schwarz's inequality to estimate the integral over $[0,t]$, we have
\begin{align*}
\frac{1}{{\varepsilon}^2} \| U^{\varepsilon}_\star(\cdot) * \varphi(t) \|_{\HH^{\bullet}}^2 & \lesssim \int_{\mathbb R^{d}} \langle \xi \rangle^{2 s} \left(\int_0^t \left[| \xi | e^{- (t-\tau) \kappa_\star \frac{|\xi|^2}{2} } \right]^2\mathrm{d}\tau\right)\left(\int_0^t \| \widehat{\varphi}(\tau, \xi) \|_{X^{\circ}}^2 \mathrm{d} \tau\right) \mathrm{d} \xi \\
& \lesssim \int_{\mathbb R^{d}} \langle \xi \rangle^{2 s} \mathrm{d}\xi \int_0^t \| \widehat{\varphi}(\tau, \xi) \|_{X^{\circ}}^2 \mathrm{d} \tau \lesssim \int_0^t \| \varphi(\tau) \|_{\GG^{\circ}}^2 \mathrm{d} \tau.
\end{align*}
In the same way,
\begin{multline}\label{eq:younIne}
\frac{1}{{\varepsilon}^2} \int_0^T \| |\nabla_x|^{1-\alpha} U^{\varepsilon}_\star(\cdot) * \varphi(t) \|_{\HH^{\bullet}}^2 \mathrm{d} t
\\
\lesssim \frac{1}{{\varepsilon}^2}\int_0^T\mathrm{d} t\int_{\mathbb R^d}\langle \xi\rangle^{2s}|\xi|^{2-2\alpha}
\left(\int_0^t {\varepsilon}|\xi|e^{-(t-\tau)\kappa_{\star}\frac{|\xi|^2}{2}}\|\widehat{\varphi}(\tau,\xi)\|_{X^{\circ}}\mathrm{d}\tau\right)^2\mathrm{d}\xi\\
\lesssim \int_{\mathbb R^d}\langle \xi\rangle^{2s}|\xi|^{-2\alpha}\mathrm{d} \xi\int_0^T \left(\int_0^t |\xi|^2e^{-(t-\tau)\kappa_{\star}\frac{|\xi|^2}{2}}\|\widehat{\varphi}(\tau,\xi)\|_{X^{\circ}}\mathrm{d}\tau\right)^2\mathrm{d} t.
\end{multline}
Recalling \eqref{eq:XIALPHA} and using Young's convolution inequality in the form $L^1\left( [0, T] \right) \ast L^2\left([0, T]\right) \hookrightarrow L^2\left([0, T]\right)$ we deduce that
\begin{align*}
\frac{1}{{\varepsilon}^2} \int_0^T \| |\nabla_x|^{1-\alpha} U^{\varepsilon}_\star(t) * \varphi \|_{\HH^{\bullet}}^2 \mathrm{d} t & \lesssim \int_{\mathbb R^{d}} \langle \xi \rangle^{2 s} | \xi |^{-2\alpha} \int_0^T \| \widehat{\varphi}(t, \xi) \|_{X^{\circ}}^2 \mathrm{d} t \mathrm{d} \xi \\
& \lesssim \int_0^T \| \varphi(t) \|_{\GG^{\circ}}^2 \mathrm{d} t + \int_0^T \| \varphi(t) \|_{ \dot{\mathbb{H}}^{-\alpha}_x \left(X^{\circ}_v\right) }^2 \mathrm{d} t\,
\end{align*}
which easily prove \eqref{eq:decay_semigroups_hydro_orthogonal_negative}. To prove \eqref{eq:decay_semigroups_hydro_orthogonal_positive}, we rewrite \eqref{eq:younIne} as
\begin{align*}
\frac{1}{{\varepsilon}^2} \int_0^T \| |\nabla_x|^{1-\alpha} & U^{\varepsilon}_\star(\cdot) * \varphi(t) \|_{\HH^{\bullet}}^2 \mathrm{d} t
\\
& \lesssim \int_{\mathbb R^d}\langle \xi\rangle^{2s}\mathrm{d} \xi\int_0^T \left(\int_0^t |\xi|^{2-\alpha}e^{-(t-\tau)\kappa_{\star}\frac{|\xi|^2}{2}}\|\widehat{\varphi}(\tau,\xi)\|_{X^{\circ}}\mathrm{d}\tau\right)^2\mathrm{d} t.
\end{align*}
Using now Young's convolution inequality in the form $L^{\frac{2}{2-\alpha}}\left( [0, T] \right) \ast L^{\frac{2}{1+\alpha}}\left([0, T]\right) \hookrightarrow L^2\left( [0, T] \right)$ we deduce that
\begin{align*}
\frac{1}{{\varepsilon}^2} \int_0^T \| |\nabla_x|^{1-\alpha} U^{\varepsilon}_\star(\cdot) * \varphi(t) \|_{\HH^{\bullet}}^2 \mathrm{d} t
& \lesssim \int_{\mathbb R^{d}} \langle \xi \rangle^{2 s} \left(\int_0^T \| \widehat{\varphi}(t, \xi) \|_{X^{\circ}}^{\frac{2}{1+\alpha}} \mathrm{d} t\right)^{1+\alpha} \mathrm{d} \xi \\
& \lesssim \left(\int_0^T \| \varphi(t) \|_{\GG^{\circ}}^{\frac{2}{1+\alpha}} \mathrm{d} t\right)^{1+\alpha},
\end{align*}
where we used Minkowski's integral inequality for the last estimate. Since the estimates established are uniform in $T$, this concludes the proof.
\end{proof}
We now make precise the asymptotic equivalence between the semigroup $\left(U^{{\varepsilon}}_{\textnormal{wave}}(t)\right)_{t\geqslant 0}$ and its leading order $\left(U^{{\varepsilon}}_{\textnormal{disp}}(t)\right)_{t\geq0}$.
\begin{lem}[\textit{\textbf{Asymptotic equivalence of the oscillating semigroups}}]
\label{lem:asymptotic_equiv_oscillating_semigroup} Given $s \ge0$ and some regularity parameter $r \in (s, s+1]$, it holds
\begin{equation}
\label{eq:asymptotic_equiv_oscillating_reg}
|\hskip-0.04cm|\hskip-0.04cm| U^{\varepsilon}_\textnormal{wave}(\cdot) f - U^{\varepsilon}_\textnormal{disp}(\cdot) f |\hskip-0.04cm|\hskip-0.04cm|_{\rSSSs{s}} \lesssim {\varepsilon}^{r-s} \left(\| f \|_{\rSSlm{r}} + \| f \|_{\dot{\mathbb{H}}^{-\alpha}_x (X^{\circ}_v) } \right),
\end{equation}
for any $f \in \rSSlm{r} \cap \dot{\mathbb{H}}_x^{-\alpha}(X^{\circ}_v)$ while there holds
\begin{equation}
\label{eq:asymptotic_equiv_oscillating_general}
\lim_{{\varepsilon} \to 0}|\hskip-0.04cm|\hskip-0.04cm| U^{\varepsilon}_\textnormal{wave}(\cdot) f - U^{\varepsilon}_\textnormal{disp}(\cdot) f |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} =0
\end{equation}
for any $f \in \GG^{\circ} \cap \dot{\mathbb{H}}^{-\alpha}_x (X^{\circ}_v),$ i.e. whenever $r=s$.
\end{lem}
\begin{proof}
We start by expanding the symbol of $U^{\varepsilon}_{\pm \textnormal{wave}}(t)$ using the decomposition of $\mathsf P_{\pm \textnormal{wave}}({\varepsilon} \xi)$ from \emph{Step 2} of the proof of Lemma \ref{lem:decay_semigroups_hydro}, we obtain
\begin{equation*}\begin{split}
\exp\left( {\varepsilon}^{-2} t \lambda_{\pm \textnormal{wave}}({\varepsilon} \xi) \right)&\mathsf P_{\pm \textnormal{wave}}({\varepsilon} \xi) = \exp\left( {\varepsilon}^{-2} t \lambda_{\pm \textnormal{wave}}({\varepsilon} \xi) \right)\left( \mathsf P_{\pm \textnormal{wave}}^{(0)}\left( \widetilde{\xi} \right) + i{\varepsilon} \xi \cdot \overline{\mathsf P}^{(1)}_{\pm \textnormal{wave}}({\varepsilon} \xi) \right)\\
= & \exp\left( \pm i c {\varepsilon}^{-1}t| \xi | - t \kappa_\textnormal{wave} | \xi |^2 \right) \mathsf P_{\star}^{(0)}\left( \widetilde{\xi} \right) \\
& + \Bigg[ \exp\left( {\varepsilon}^{-2} t \lambda_{\star}({\varepsilon} \xi) \right) - \exp\left( \pm i c {\varepsilon}^{-1}t| \xi | - t \kappa_\textnormal{wave} | \xi |^2 \right) \Bigg] \mathsf P_{\pm \textnormal{wave}}^{(0)}\left( \widetilde{\xi} \right) \\
& + {\varepsilon} \exp\left( {\varepsilon}^{-2} t \lambda_{\star}({\varepsilon} \xi) \right) \xi \cdot \overline{\mathsf P}^{(1)}_{\pm \textnormal{wave}}({\varepsilon} \xi).
\end{split}\end{equation*}
Thus, the symbol of the difference $U^{\varepsilon}_\textnormal{wave}(t) - U^{\varepsilon}_\textnormal{disp}(t)$ writes, for ${\varepsilon} |\xi| \leqslant \alpha_{0},$ as the sum of the two terms (corresponding to $\star=\pm\textnormal{wave}$):
\begin{align*}
\Big[\exp\left( {\varepsilon}^{-2} t \lambda_{\star}({\varepsilon} \xi) \right) - \exp\left( \pm i c {\varepsilon}^{-1}t | \xi | - t \kappa_\textnormal{wave} | \xi |^2 \right) \Big]& \mathsf P_{\star}^{(0)}\left( \widetilde{\xi} \right)\\
+ {\varepsilon} \exp\left( {\varepsilon}^{-2} t \lambda_{\star}({\varepsilon} \xi) \right) \xi \cdot & \overline{\mathsf P}^{(1)}_{\star}({\varepsilon} \xi).
\end{align*}
On the one hand, when ${\varepsilon} |\xi| > \alpha_{0}$, since $\overline{\mathsf P}^{(1)}_{\star}({\varepsilon} \xi)$ is supported in $\{ {\varepsilon} |\xi | \leqslant \alpha_{0} \}$, the symbol reduces to
$$- \exp\left( i c {\varepsilon}^{-1}t | \xi | - t \kappa_\textnormal{wave} | \xi |^2 \right) \mathsf P_\textnormal{wave}^{(0)}\left( \widetilde{\xi} \right)-\exp\left( - i c {\varepsilon}^{-1}t | \xi | - t \kappa_\textnormal{wave} | \xi |^2 \right) \mathsf P_{-\textnormal{wave}}^{(0)}\left( \widetilde{\xi} \right).$$
On the other hand, when ${\varepsilon} | \xi | \leqslant \alpha_{0}$, we estimate the difference of exponentials using the inequality $\left|1 - e^a\right| \leqslant a e^{|a|}$ as well as the expansion \eqref{eq:lambda_star} of
$\lambda_{\pm\textnormal{wave}}(\xi)$:
\begin{align*}
\Big| \exp\left( {\varepsilon}^{-2} t \lambda_{\star}({\varepsilon} \xi) \right) & - \exp\left( \pm i c {\varepsilon}^{-1} |\xi| - t \kappa_\textnormal{wave} | \xi |^2 \right) \Big| \\
& = \left| \exp\left( \pm i c t {\varepsilon}^{-1}|\xi| - t \kappa_\textnormal{wave} | \xi |^2 \right) \right| \Big| \exp \left( \mathcal O\left( t {\varepsilon} | \xi |^3 \right) \right) -1 \Big| \\
& \lesssim \left({\varepsilon} | \xi|\right) \left(t | \xi |^2\right) \exp\left( - t \kappa_\textnormal{wave} | \xi |^2 \right) \exp\Big( \mathcal O\left( t {\varepsilon} | \xi |^3 \right) \Big),
\end{align*}
thus, using $re^{-r} \lesssim e^{-\frac{1}{2}r}$ and assuming $\alpha_{0}$ small enough so that $\mathcal O\left( {\varepsilon} | \xi |^3 \right) \leqslant \frac{1}{4}\kappa_\textnormal{wave} | \xi |^2$, we obtain
\begin{align*}
\Big| \exp\left( {\varepsilon}^{-2} t \lambda_{\star}({\varepsilon} \xi) \right) & - \exp\left( \pm i c t {\varepsilon}^{-1}|\xi| - t \kappa_\textnormal{wave} | \xi |^2 \right) \Big|\\
& \lesssim {\varepsilon} | \xi | \exp\left( - \frac{t}{2} \kappa_\textnormal{wave} | \xi |^2 \right) \exp\Big( \mathcal O\left( t {\varepsilon} | \xi |^3 \right) \Big) \\
& \lesssim {\varepsilon} | \xi | \exp\left( - \frac{t}{4} \kappa_\textnormal{wave} | \xi |^2 \right).
\end{align*}
Putting together the previous estimates, we then bound the operator norm in $\mathscr B(X^{\circ} ;H^{\bullet} )$ of the symbol of the difference~$U^{\varepsilon}_\textnormal{wave}(t) - U^{\varepsilon}_\textnormal{disp}(t)$. It is controlled by
\begin{align*}
\mathbf{1} _{{\varepsilon} | \xi | \leqslant \alpha_0} {\varepsilon} | \xi | \exp\left( - \frac{t}{4} \kappa_\textnormal{wave} | \xi |^2 \right) & + \mathbf{1} _{{\varepsilon} | \xi | > \alpha_0} \exp\left( - \frac{t}{4} \kappa_\textnormal{wave} | \xi |^2 \right) \\
& \lesssim \left({\varepsilon} | \xi |\right)^{r - s} \exp\left( - \frac{t}{4} \kappa_\textnormal{wave} | \xi |^2 \right),
\end{align*}
where we used the comparison $u\mathbf{1}_{u\leqslant \alpha_{0}} + \mathbf{1}_{u >\alpha_{0}} \lesssim u^{r-s}$ for any $u \geq0$ since $r-s \in [0, 1]$.
As in the proof of Lemma \ref{lem:decay_semigroups_hydro}, such an estimate on the symbol of $U^{\varepsilon}_\textnormal{wave}(t) - U^{\varepsilon}_\textnormal{disp}(t)$ yields the controls \eqref{eq:asymptotic_equiv_oscillating_reg}, from which we deduce \eqref{eq:asymptotic_equiv_oscillating_general} by density.
\end{proof}
A similar result holds for the difference between $U^{{\varepsilon}}_{\textnormal{NS}}(t)-U_{\textnormal{NS}}(t)$.
\begin{lem}[\textit{\textbf{Asymptotic equivalence of the Navier-Stokes semigroup}}]
\label{lem:asymptotic_equiv_NS_semigroup}
Given $s \ge0$ and consider some regularity parameter $r \in (s, s+1]$, the part $U^{\varepsilon}_\textnormal{NS}(\cdot)$ of the hydrodynamic semigroup is such that
\begin{equation}
\label{eq:asymptotic_equiv_NS_reg}
|\hskip-0.04cm|\hskip-0.04cm| U^{\varepsilon}_\textnormal{NS}(\cdot) f - U_\textnormal{NS}(\cdot) f |\hskip-0.04cm|\hskip-0.04cm|_{\rSSSs{s}} \lesssim {\varepsilon}^{r - s} \left(\| f \|_{\rSSlm{r}} + \| f \|_{ \dot{\mathbb{H}}^{-\alpha}_x (X^{\circ}_v) } \right),
\end{equation}
for any $f \in \rSSlm{r} \cap \dot{\mathbb{H}}^{-\alpha}_x(X^{\circ}_v)$, while, for $f \in \GG^{\circ} \cap \dot{\mathbb{H}}^{-\alpha}_x (X^{\circ}_v)$ (i.e. $r=s$), there holds
\begin{equation}
\label{eq:asymptotic_equiv_NS_general}
\lim_{{\varepsilon} \to 0}|\hskip-0.04cm|\hskip-0.04cm| U^{\varepsilon}_\textnormal{NS}(\cdot) f - U_\textnormal{NS}(\cdot) f |\hskip-0.04cm|\hskip-0.04cm|_\HHH = 0.
\end{equation}
Furthermore, if $\varphi \in L^2\left( [0, T) ; \GG^{\circ} \right)$ where $T \in (0, \infty]$ is such that $\mathsf P \varphi(t) = 0$, then
\begin{equation}
\label{eq:asymptotic_equiv_NS_orthogonal_reg}
\frac{1}{{\varepsilon}} |\hskip-0.04cm|\hskip-0.04cm| U^{\varepsilon}_\textnormal{NS}(\cdot) * \varphi - {\varepsilon} \nabla_x \cdot V_\textnormal{NS}(\cdot) * \varphi |\hskip-0.04cm|\hskip-0.04cm|_{\rSSSs{s}} \lesssim {\varepsilon}^{r - s} \sup_{0 \leqslant t < T} \left\{w_{\phi, \eta}(t) \left(\int_0^t \| \varphi(\tau) \|_{ \rSSlm{r} \cap \dot{\mathbb{H}}^{-\alpha}_x (X^{\circ}_v) }^2 \mathrm{d} \tau \right)^{\frac{1}{2}} \right\}.
\end{equation}
\end{lem}
\begin{proof}
Let us fix $\star=\textnormal{Bou},\textnormal{inc}$. As in the previous proof, we start by expanding the symbol of $U^{\varepsilon}_\textnormal{NS}(t)$ so as to compare it with those of $U_\textnormal{NS}(t)$ and $V_\textnormal{NS}(t)$. We first prove \eqref{eq:asymptotic_equiv_NS_reg} and \eqref{eq:asymptotic_equiv_NS_general}, and then \eqref{eq:asymptotic_equiv_NS_orthogonal_reg}.
\step{1}{Proof of \eqref{eq:asymptotic_equiv_NS_reg} and \eqref{eq:asymptotic_equiv_NS_general}}
For ${\varepsilon} | \xi | \leqslant \alpha_0$, using the decomposition \eqref{eq:overPP_star}, there holds
\begin{align*}
\exp\left( {\varepsilon}^{-2} t \lambda_\star({\varepsilon} \xi) \right) \mathsf P_\star({\varepsilon} \xi)
= & \exp\left( - t \kappa_\star | \xi |^2 \right) \mathsf P_\star^{(0)}\left( \widetilde{\xi} \right) + {\varepsilon} \exp\left( {\varepsilon}^{-2} t \lambda_\star({\varepsilon} \xi) \right)\xi \cdot \overline{\mathsf P}_\star^{(1)}({\varepsilon} \xi) \\
& + \Big[\exp\left( {\varepsilon}^{-2} t \lambda_\star({\varepsilon} \xi) \right) - \exp\left( - t \kappa_\star | \xi |^2 \right) \Big] \mathsf P^{(0)}_\star\left( \widetilde{\xi} \right)
\end{align*}
whereas, for ${\varepsilon} | \xi| > \xi_0$, since $\mathsf P_\star({\varepsilon} \xi)$ vanishes, the symbol of the difference $U^{\varepsilon}_\textnormal{NS}(t) - U_\textnormal{NS}(t)$ reduces to that of $-U_\textnormal{NS}(t)$ given by
$$-\exp\left( - t \kappa_\textnormal{Bou} | \xi |^2 \right) \mathsf P_\textnormal{Bou}^{(0)}\left( \widetilde{\xi} \right) - \exp\left(-t\kappa_{\textnormal{inc}}|\xi|^{2}\right)\mathsf P_{\textnormal{inc}}^{(0)}\left(\widetilde{\xi}\right).$$
To sum up, the symbol of the difference $U_\textnormal{NS}^{\varepsilon}(t) - U_\textnormal{NS} (t)$ writes as the sum over $\star=\textnormal{Bou},\textnormal{inc}$ of the symbols
\begin{align*}
\mathbf{1}_{{\varepsilon} | \xi | \leqslant \alpha_0} \Big( {\varepsilon} \exp\left( {\varepsilon}^{-2} t \lambda_\star({\varepsilon} \xi) \right)\xi \cdot \overline{\mathsf P}_\star^{(1)}({\varepsilon} \xi) & + \Big[\exp\left( {\varepsilon}^{-2} t \lambda_\star({\varepsilon} \xi) \right) - \exp\left( - t \kappa_\star | \xi |^2 \right) \Big] \mathsf P^{(0)}_\star\left( \widetilde{\xi} \right) \Big) \\
& - \mathbf{1}_{{\varepsilon} | \xi | > \alpha_0} \exp\left( - t \kappa_\star | \xi |^2 \right) \mathsf P_\star^{(0)}\left( \widetilde{\xi} \right),
\end{align*}
and its operator norm in $\mathscr B( X^{\circ} ; H^{\bullet} )$ is controlled as in the proof of Lemma \ref{lem:decay_semigroups_hydro} by
$${\varepsilon} | \xi | \exp\left( - t \kappa_\star \frac{| \xi |^2}{4} \right).$$
We then deduce \eqref{eq:asymptotic_equiv_NS_reg} as well as \eqref{eq:asymptotic_equiv_NS_general} by density as in the proof of Lemma \ref{lem:asymptotic_equiv_oscillating_semigroup}.
\step{2}{Proof of \eqref{eq:asymptotic_equiv_NS_orthogonal_reg}}
In the case $\mathsf P \varphi(t)=0$, the projector $\mathsf P^{(0)}_\star \varphi(t)$ vanishes, and we use the second order expansion of $\mathsf P_{\star}({\varepsilon} \xi)$ provided in \eqref{eq:PPstar} in Theorem \ref{thm:spectral_study}:
$$\mathsf P_{\star}({\varepsilon} \xi) = i{\varepsilon} \xi \cdot \mathsf P^{(1)}\left( \widetilde{\xi} \right) + S_{\star}({\varepsilon} \xi),$$
where we recall that $\| S_\star({\varepsilon} \xi) \|_{ \mathscr B(X^{\circ} ; H^{\bullet}) } \lesssim {\varepsilon}^2 | \xi|^2$ uniformly in ${\varepsilon} | \xi | \leqslant \alpha_0$.
Similarly, the symbol of $U^{\varepsilon}_\textnormal{NS}(t) - {\varepsilon} \nabla_x \cdot V_\textnormal{NS}(t)$ restricted to $\nul(\mathsf P)$ then writes
\begin{align*}
& \mathbf{1}_{{\varepsilon} | \xi | \leqslant \alpha_0} \Big( {\varepsilon}^2 \exp\left( {\varepsilon}^{-2} t \lambda_\star({\varepsilon} \xi) \right) |\xi|^{2} S_\star({\varepsilon} \xi) + i{\varepsilon} \Big[\exp\left( {\varepsilon}^{-2} t \lambda_\star({\varepsilon} \xi) \right) - \exp\left( - t \kappa_\star | \xi |^2 \right) \Big] \xi \cdot \mathsf P^{(1)}_\star\left( \widetilde{\xi} \right) \Big) \\
& \phantom{++++} - \mathbf{1}_{{\varepsilon} | \xi | > \alpha_0} i{\varepsilon} \exp\left( - t \kappa_\star | \xi |^2 \right) \xi \cdot \mathsf P_\star^{(1)}\left( \widetilde{\xi} \right),
\end{align*}
which is similarly controlled by
$${\varepsilon} | \xi | ({\varepsilon} | \xi |)^{r-s} \exp\left( - t \kappa_\star \frac{| \xi |^2}{4} \right).$$
These representations allow to proceed as in the proofs of Lemmas \ref{lem:decay_semigroups_hydro} and \ref{lem:asymptotic_equiv_oscillating_semigroup} to get the desired conclusion.
\end{proof}
We present now a dispersive estimate for the semigroup $\left(U_{\textnormal{disp}}^{{\varepsilon}}(t)\right)_{t\geq0}$ which is deduced from a general result about the decay rate for solutions to the wave equation.
\begin{lem}[\textit{\textbf{Dispersive estimate}}]
The part $U^{\varepsilon}_\textnormal{disp}$ of the hydrodynamic semigroup satisfies the dispersive estimate
\begin{gather}
\label{eq:dispersive}
\left\| U^{\varepsilon}_\textnormal{disp}(t) g \right\|_{ W^{s, \infty}_x \left( H^{\bullet}_v \right) } \lesssim \left(\frac{{\varepsilon}}{t}\right)^{\frac{d-1}{2}} \| g \|_{ \dot{\mathbb{B}}^{ \frac{d+1}{2}+ s }_{1,1} \left( X^{\circ}_v \right) }.
\end{gather}
\end{lem}
\begin{proof}
In virtue of the macroscopic representation of $U^{\varepsilon}_\textnormal{disp}$ from Proposition \ref{prop:macro_representation_spectral} and the continuity of the heat semigroup on $L^1$, we can deduce \eqref{eq:dispersive} directly from Lemma \ref{lem:wave-equation}.\end{proof}
\begin{lem}[\textit{\textbf{Vanishing estimate for the convoluted oscillating semigroup}}]
\label{lem:convolution_wave}
Suppose $\varphi \in L^{\infty}([0,T)\,;\,\GG^{\circ})$ is such that $|\nabla_{x}|^{1-\alpha}\varphi \in L^{2}([0,T)\;;\;\GG^{\circ})$ and
$\mathsf P \varphi(t) = 0$ for any $t\ge0$ together with
$$\partial_t\varphi \in L^2 \cap L^{ \frac{2}{1+\alpha} } \left( [0, T) ; \rSSlm{s-1}\right) \bigcap L^{\frac{4}{3}} \cap L^{\frac{4}{3 + 2 \alpha}} \left( [0, T) ; \dot{\mathbb{H}}^{-\frac{1}{2}}_x \left(X^{\circ}_v\right) \right).$$
Then, there holds
\begin{equation*}
\begin{aligned}
\frac{1}{{\varepsilon}^2} |\hskip-0.04cm|\hskip-0.04cm| U^{\varepsilon}_\textnormal{wave}(\cdot) * \varphi |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} \lesssim \| \varphi(0) \|_{ \dot{\mathbb{H}}^{-\alpha}_x \left( X^{\circ}_v \right) } & + \| \varphi \|_{ L^\infty \left( [0, T) ; \GG^{\circ} \right) } + \| | \nabla_x|^{1 - \alpha} \varphi \|_{ L^2 \left( [0, T) ; \GG^{\circ} \right) } \\
& + \| \partial_t \varphi \|_{ \left(L^2 \cap L^{ \frac{2}{1+\alpha} }\right) \left( [0, T) ; \rSSlm{s-1} \right) } \\
& + \| \partial_t \varphi \|_{ \left(L^{ \frac{4}{3} } \cap L^{\frac{4}{3 + 2 \alpha}}\right) \left( [0, T) ; \dot{\mathbb{H}}^{-\frac{1}{2}}_x \left(X^{\circ}_v\right) \right) }.
\end{aligned}
\end{equation*}
\end{lem}
\begin{rem}
Note that if $T < \infty$, we have $L^{\frac{4}{3}} \cap L^{\frac{4}{3+2\alpha}}=L^{\frac{4}{3}}$ and $L^{\frac{2}{1+\alpha}}\cap L^2=L^2$.
\end{rem}
\begin{proof}
In the first step, we establish a preparatory estimate for any $\xi \in \mathbb R^d$ satisfying ${\varepsilon} | \xi | \leqslant \alpha_0$, which we will use in the following step to prove the lemma. Since $w_{\phi, \eta} \leqslant 1$, we neglect it for the estimates on $\partial_t \varphi$.
\step{1}{Preparatory estimate}
Recall that $\| U^{{\varepsilon}}_{\textnormal{wave}}(\cdot) * \varphi(t)\|_{\HH^{\bullet}}^{2}$ is given by \eqref{eq:convVarphi} which allows us to
work, as in the proof of Lemma \ref{lem:asymptotic_equiv_oscillating_semigroup}, on the two parts of the symbol of $U^{{\varepsilon}}_{\textnormal{wave}}(t).$ Recalling \eqref{eq:KernelP}, for any fixed $t \ge0$ and any $\tau \in [0,t]$ one has
\begin{equation*}\begin{split}
\exp\left( {\varepsilon}^{-2} \tau \lambda_{\pm \textnormal{wave}}({\varepsilon} \xi) \right) & \mathsf P_{\pm \textnormal{wave}}({\varepsilon} \xi) \widehat{\varphi}(t-\tau, \xi) \\
& = {\varepsilon}\exp\left( {\varepsilon}^{-2} \tau \lambda_{\pm \textnormal{wave}}({\varepsilon} \xi) \right) \xi \cdot \overline{\mathsf P}^{(1)}_{\pm \textnormal{wave}}({\varepsilon} \xi) \widehat{\varphi}(t-\tau, \xi) \\
& = {\varepsilon}\exp\left( {\varepsilon}^{-2} \tau \lambda_{\pm \textnormal{wave}}({\varepsilon} \xi) \right) \xi \cdot {\phi^{\pm}
}(\tau, \xi)
\end{split}\end{equation*}
where we denoted ${\phi^{\pm}}(\tau, \xi) := \overline{\mathsf P}^{(1)}_{\pm \textnormal{wave}}({\varepsilon} \xi) \widehat{\varphi}(t-\tau, \xi)$. We now integrate with respect to $\tau \in [0, t]$ using integration by parts:
\begin{align*}
\int_0^t \exp\left( {\varepsilon}^{-2} \tau \lambda_{\pm \textnormal{wave}}({\varepsilon} \xi) \right) & \mathsf P_{\pm \textnormal{wave}}({\varepsilon} \xi) \widehat{\varphi}(t-\tau, \xi) \mathrm{d} \tau\\
= & {\varepsilon} \xi \cdot \int_0^t \exp\left( {\varepsilon}^{-2} \tau \lambda_{\pm \textnormal{wave}}({\varepsilon} \xi) \right) \phi^{\pm}(\tau, \xi) \mathrm{d} \tau \\
= & - \frac{{\varepsilon}^3 \xi }{\lambda_{\pm \textnormal{wave}}({\varepsilon} \xi)} \cdot \int_0^t \exp\left( {\varepsilon}^{-2} \tau\lambda_{\pm \textnormal{wave}}({\varepsilon} \xi) \right) \partial_\tau {\phi}^{\pm}(\tau, \xi) \mathrm{d} \tau \\
& + \frac{{\varepsilon}^3 \xi }{\lambda_{\pm \textnormal{wave}}({\varepsilon} \xi)} \cdot \left[ {\phi}^{\pm}(t, \xi)\exp\left( {\varepsilon}^{-2} t \lambda_{\pm \textnormal{wave}}({\varepsilon} \xi ) \right)
- {\phi}^{\pm}(0, \xi)\right].
\end{align*}
As in Lemma \ref{lem:asymptotic_equiv_oscillating_semigroup}, we can choose $\alpha_0$ small enough so that
$$|\lambda_{\pm \textnormal{wave}}({\varepsilon} \xi)| \approx {\varepsilon} | \xi |, \qquad \mathrm{Re}\,\left( {\varepsilon}^{-2} \lambda_{\pm \textnormal{wave}} ({\varepsilon} \xi) \right) \leqslant - \frac{1}{2}\kappa_{ \textnormal{wave}} | \xi |^2,$$
uniformly in $|\xi| \leqslant \alpha_{0}$, and one notices
$$
\|\phi^{\pm}(t,\xi)\|_{H^{\bullet}}=\|\overline{\mathsf P}^{(1)}_{\pm \textnormal{wave}}({\varepsilon} \xi) \widehat{\varphi}(0, \xi)\|_{H^{\bullet}} \lesssim \|\widehat{\varphi}(0,\xi)\|_{X^{\circ}},$$
while, in the same way,
$$\|\phi^{\pm}(0,\xi)\|_{H^{\bullet}}\lesssim \|\widehat{\varphi}(t,\xi)\|_{X^{\circ}},\qquad
\|\partial_{\tau}\phi^{\pm}(\tau,\xi)\|_{H^{\bullet}}
\lesssim \|\partial_{\tau}\widehat{\varphi}(t-\tau,\xi)\|_{X^{\circ}}\,.$$
Those considerations lead to
\begin{equation*}\begin{split}
\frac{1}{{\varepsilon}^2} \Big\|\int_0^t & \exp\left( {\varepsilon}^{-2} \tau \lambda_{\pm \textnormal{wave}}({\varepsilon} \xi) \right) \mathsf P_{\pm \textnormal{wave}}({\varepsilon} \xi) \widehat{\varphi}(t-\tau, \xi) \mathrm{d} \tau\Big\|_{H^{\bullet}}\\
\leqslant & ~ \int_{0}^{t} \exp\left(- \frac{\tau}{2} \kappa_\textnormal{wave} |\xi |^2\right) \|\partial_{\tau}\widehat{\varphi}(t-\tau,\xi)\|_{X^{\circ}}\mathrm{d} \tau + \|\widehat{\varphi}(t,\xi)\|_{X^{\circ}}\\
&\phantom{+++++} + \|\widehat{\varphi}(0,\xi)\|_{X^{\circ}}\exp\left(-\frac{t}{2}\kappa_{\textnormal{wave}}|\xi|^{2}\right).
\end{split}\end{equation*}
In other words, we have shown that
\begin{equation}\label{eq:FxUwave}\begin{split}
\frac{1}{{\varepsilon}^2} \Big\| \mathcal F_x \big[ & U^{\varepsilon}_\textnormal{wave}(\cdot) * \varphi \big](t, \xi) \Big\|_{H^{\bullet}} \leqslant ~ \int_{0}^{t} \exp\left(- \frac{\tau}{2} \kappa_\textnormal{wave} |\xi |^2 \right) \|\partial_{\tau}\widehat{\varphi}(t-\tau,\xi)\|_{X^{\circ}}\mathrm{d} \tau
\\ &+ \|\widehat{\varphi}(t,\xi)\|_{X^{\circ}}+ \|\widehat{\varphi}(0,\xi)\|_{X^{\circ}}\exp\left(-\frac{t}{2}\kappa_{\textnormal{wave}}|\xi|^{2}\right).
\end{split}\end{equation}
\step{2}{Completion of the proof} We first deduce from \eqref{eq:FxUwave} that
$$\frac{1}{{\varepsilon}^2} \| U^{\varepsilon}_\textnormal{wave}(\cdot) * \varphi(t) \|_{\HH^{\bullet}} \lesssim \sup_{0 \leqslant \tau \leqslant t} \| \varphi(\tau) \|_{ \GG^{\circ} } + \left\| \langle \xi \rangle^{s} \int_0^t e^{- \tau \kappa_{\textnormal{wave}} \frac{|\xi|^2}{2}} \| \partial_\tau \widehat{\varphi}(t-\tau, \xi) \|_{X^{\circ}} \mathrm{d} \tau \right\|_{ L^2_\xi }.$$
For notations simplicity, we call $J=J(t,\varphi)$ the above $L^2_\xi$-norm and split it according to $|\xi| \leqslant 1$ or $|\xi| >1$, i.e. $J=J_1+J_2$
where
$$J_1^2=\int_{|\xi| \leqslant 1}\langle \xi\rangle^{2s}\left(\int_0^t e^{- \tau \kappa_{\textnormal{wave}} \frac{|\xi|^2}{2}} \| \partial_\tau \widehat{\varphi}(t-\tau, \xi) \|_{X^{\circ}} \mathrm{d} \tau \right)^2 \mathrm{d}\xi$$
and
\begin{multline*}
J_2^2=\int_{|\xi| \geqslant 1}\langle \xi\rangle^{2s}\left(\int_0^t e^{- \tau \kappa_{\textnormal{wave}} \frac{|\xi|^2}{2}} \| \partial_\tau \widehat{\varphi}(t-\tau, \xi) \|_{X^{\circ}} \mathrm{d} \tau \right)^2 \mathrm{d}\xi\\
\leqslant \int_{|\xi| \geqslant 1}\langle \xi\rangle^{2s-2}\left(\int_0^t \left[|\xi|\,e^{- \tau \kappa_{\textnormal{wave}} \frac{|\xi|^2}{2}}\right] \| \partial_\tau \widehat{\varphi}(t-\tau, \xi) \|_{X^{\circ}} \mathrm{d} \tau \right)^2 \mathrm{d}\xi. \end{multline*}
On the one hand, using Cauchy-Schwarz inequality and the second estimate in \eqref{eq:XIALPHA}, one has
$$J_2^2 \lesssim \int_{\mathbb R^d}\langle\xi\rangle^{2s-2}\mathrm{d}\xi\int_0^t\|\partial_\tau\widehat{\varphi}(\tau,\xi)\|_{X^{\circ}}^2\mathrm{d}\tau=\|\partial_t \varphi\|_{L^2_t H^{s-1}_x(X^{\circ}_v)}^2.$$
On the other hand,
invoking H\"older's inequality (with exponents $p=4,q=\frac{4}{3}$) to estimate the time integral, we deduce that
\begin{equation*}\begin{split}
J_1^2 &\lesssim \int_{|\xi| \leqslant 1} \left(\int_0^t\left[e^{- \tau \kappa_{\textnormal{wave}} \frac{|\xi|^2}{2}}\right]^4\mathrm{d} \tau\right)^{\frac{1}{2}}\,\left(\int_0^t\| \partial_\tau \widehat{\varphi}(t-\tau, \xi) \|_{X^{\circ}}^{\frac{4}{3}}\mathrm{d}\tau\right)^{\frac{3}{2}}\mathrm{d}\xi \\
&\lesssim \int_{\mathbb R^d} |\xi|^{-1}\left(\int_0^t\| \partial_\tau \widehat{\varphi}(\tau, \xi) \|_{X^{\circ}}^{\frac{4}{3}}\mathrm{d}\tau\right)^{\frac{3}{2}}\mathrm{d}\xi
\end{split}\end{equation*}
which, thanks to Minkowski's integral inequality, yields
$$J_1^{\frac{4}{3}}\lesssim \int_{0}^t\left(\int_{\mathbb R^d} |\xi|^{-1}\| \partial_\tau \widehat{\varphi}(\tau, \xi) \|_{X^{\circ}}^{2}\mathrm{d}\xi\right)^{\frac{2}{3}}\mathrm{d}\tau=\int_0^t \|\partial_t \varphi(t)\|_{\dot{\mathbb{H}}^{-\frac{1}{2}}_x(X^{\circ}_v)}^{\frac{4}{3}}\mathrm{d}\tau.$$
Therefore,
$$J \lesssim \|\partial_t \varphi\|_{L^2_t H^{s-1}_x(X^{\circ}_v)} + \| \partial_t \varphi \|_{ L^{\frac{4}{3}}_t \dot{\mathbb{H}}^{-\frac{1}{2}}_x X^{\circ}_v } $$
i.e.
$$\frac{1}{{\varepsilon}^2} \| U^{\varepsilon}_\textnormal{wave}(\cdot) * \varphi(t) \|_{\HH^{\bullet}} \lesssim \sup_{0 \leqslant \tau \leqslant t} \| \varphi(\tau) \|_{ \GG^{\circ} } + \| \partial_t \varphi \|_{ L^{ \frac{4}{3} }_t \dot{\mathbb{H}}^{-\frac{1}{2}}_x X^{\circ}_v } + \| \partial_t \varphi \|_{ L^2_t H^{s-1}_x X^{\circ}_v }.$$
Furthermore, coming back to \eqref{eq:FxUwave},
\begin{multline*}
\frac{1}{{\varepsilon}^4} \int_0^T | \xi |^{2-2\alpha} \Big\| \mathcal F_x \big[ U^{\varepsilon}_\textnormal{wave}(\cdot) * \varphi \big](t, \xi) \Big\|_{ H^{\bullet} }^2 \mathrm{d} t \\
\lesssim \int_0^T \left(\int_{0}^{t} \left[ | \xi |^{1-\alpha} e^{-\frac{\tau}{2} \kappa_\textnormal{wave} | \xi |^2} \right] \|\partial_{\tau}\widehat{\varphi}(t-\tau,\xi)\|_{X^{\circ}} \mathrm{d} \tau\right)^2 \mathrm{d} t\\
+ \int_0^T | \xi |^{2-2\alpha} \big\| \widehat{\varphi}(t, \xi) \big\|_{\GG^{\circ}}^2 \mathrm{d} t + \| \widehat{\varphi}(0, \xi) \|_{X^{\circ}}^2 \left(\int_0^T |\xi|^{2-2\alpha} e^{-t \kappa_{\textnormal{wave}}|\xi|^{2}} \mathrm{d} t\right)
\end{multline*}
Using now Young's convolution inequality in the form $L^{\frac{2p}{3 p - 2}}\left([0, T]\right) \ast L^p\left( [0, T] \right) \hookrightarrow L^2\left([0, T]\right)$ ($p \in [1, 2]$) in the first time integral,
we deduce that
\begin{multline*}
\frac{1}{{\varepsilon}^4} \int_0^T | \xi |^{2-2\alpha} \Big\| \mathcal F_x \big[ U^{\varepsilon}_\textnormal{wave}(\cdot) * \varphi \big](t, \xi) \Big\|_{ H^{\bullet} }^2 \mathrm{d} t \lesssim \int_0^T | \xi |^{2-2\alpha} \big\| \widehat{\varphi}(t, \xi) \big\|_{X^{\circ}}^2 \mathrm{d} t + | \xi |^{-2\alpha} \| \widehat{\varphi}(0, \xi) \|_{X^{\circ}}^2 \\
+ \left( \int_0^T \left(| \xi |^{-\left(2 + \alpha - \frac{2}{p}\right)} \|\partial_{\tau}\widehat{\varphi}(\tau,\xi)\|_{X^{\circ}}\right)^p \mathrm{d} \tau\right)^{\frac{2}{p}}.
\end{multline*}
We integrate this inequality against $\langle \xi \rangle^{2s}$ with the choice $p=\frac{2}{1+\alpha} \in \left[\frac{4}{3}, 2\right]$ on the region $|\xi| \geqslant 1$ and with~$p=\frac{4}{3 + 2\alpha } \in \left[1, \frac{4}{3} \right]$ on the region $|\xi| \leqslant 1$, to obtain, after a simple use of Minkowski's integral inequality,
\begin{align*}
\frac{1}{{\varepsilon}^2} \Bigg(\int_0^T \| |\nabla_x|^{1-\alpha} U^{\varepsilon}_\textnormal{wave}(\cdot) * \varphi(t) \|_{\HH^{\bullet}}^2 \mathrm{d} t\Bigg)^{\frac{1}{2}}
\lesssim &
\left\| | \nabla_x|^{1-\alpha} \varphi \right\|_{ L^2\left( [0, T) ; \GG^{\circ} \right) } + \| \varphi(0) \|_{ \dot{\mathbb{H}}^{-\alpha}_x \left( X^{\circ}_v \right) } \\
& + \| \partial_t \varphi \|_{ L^{\frac{4}{3 + 2 \alpha}}_t \dot{\mathbb{H}}^{-\frac{1}{2}}_x X^{\circ}_v }
+ \| \partial_t \varphi \|_{ L^{ \frac{2}{1+\alpha} }_t H^{s-1}_x X^{\circ}_v }.
\end{align*}
Since the estimates established are uniform in $T$ and $w_{\phi, \eta} \leqslant 1$, this concludes the proof.
\end{proof}
The decay and regularization estimates for $\left(U^{\varepsilon}_\textnormal{kin}(t)\right)_{t\ge0}$ are given by scaling the estimates from Theorem \ref{thm:spectral_study}, or under the enlargement assumptions \ref{LE}, Theorem \ref{thm:enlarged_thm}.
\begin{lem}[\textit{\textbf{Decay and regularization of the kinetic semigroup}}]
\label{lem:decay_regularization_kinetic_semigroup}
For any fixed decay rate~$\sigma \in (0, \sigma_0)$, the kinetic part~$\left(U^{\varepsilon}_\textnormal{kin}(t)\right)_{t\ge0}$ of the semigroup satisfies the decay and regularization estimates
$$ \sup_{t \geqslant 0} \, e^{2 \sigma_0 t / {\varepsilon}^2} \| U^{\varepsilon}_\textnormal{kin}(t) f \|_{\GG}^2 + \frac{1}{{\varepsilon}^2} \int_0^\infty e^{2 \sigma t / {\varepsilon}^2} \| U^{\varepsilon}_\textnormal{kin}(t) f \|_{\GG^{\bullet}}^2 \, \mathrm{d} t \lesssim \| f \|_{\GG}^2$$
as well as
$$ \frac{1}{{\varepsilon}^2} \int_0^\infty e^{2 \sigma t / {\varepsilon}^2} \| U^{\varepsilon}_\textnormal{kin}(t) f \|_{\GG}^2 \, \mathrm{d} t \lesssim \| f \|^2_{\GG^{\circ}},$$
with exactly the same estimate satisfied by the adjoint $\left((U^{{\varepsilon}}_{\textnormal{kin}}(t))^{\star}\right)_{t\geq0}$.
\end{lem}
As for the hydrodynamic semigroup $U^{\varepsilon}_\textnormal{hydro}(\cdot)$, we establish now suitable convolution estimates:
\begin{lem}[\textit{\textbf{Decay and regularization of the convoluted kinetic semigroup}}]
\label{lem:decay_regularization_convolution_kinetic_semigroup}
Consider $T \in (0, \infty]$. For any $\varphi \in L^2\left( [0, T) ; \HH^{\circ} \right)$, there holds uniformly in ${\varepsilon}$
\begin{subequations}
\begin{equation}
\label{eq:decay_convolution_semigroup_no_exp}
\begin{aligned}
\frac{1}{{\varepsilon}} |\hskip-0.04cm|\hskip-0.04cm| U^{\varepsilon}_\textnormal{kin}(\cdot) * \varphi |\hskip-0.04cm|\hskip-0.04cm|_{\FFF(T, \phi, \eta, {\varepsilon})} \lesssim \sup_{0 \leqslant t < T} \left\{w_{\phi, \eta}(t) \left(\int_0^t \| \varphi(\tau) \|_{ \HH^{\circ} }^2 \mathrm{d} \tau \right)^{\frac{1}{2}} \right\}.
\end{aligned}
\end{equation}
Furthermore, consider $\sigma \in [0, \sigma_0)$, there holds uniformly in ${\varepsilon}$
\begin{equation}
\label{eq:decay_convolution_semigroup_exp}
\begin{aligned}
\frac{1}{{\varepsilon}} |\hskip-0.04cm|\hskip-0.04cm| U^{\varepsilon}_\textnormal{kin}(\cdot) * \varphi |\hskip-0.04cm|\hskip-0.04cm|_{\GGG(\sigma, {\varepsilon})} \lesssim \left(\int_0^T e^{2 \sigma t / {\varepsilon}^2 } \| \varphi(t) \|_{\GG^{\circ} }^2 \mathrm{d} t\right)^{\frac{1}{2}}
\end{aligned}
\end{equation}
for any $\varphi$ for which the right-hand-side is finite.
\end{subequations}
\end{lem}
\begin{proof}
Denote by $(Y, \VV, \VV^{\bullet}, \VV^{\circ})$ either $(H, \HH, \HH^{\bullet}, \HH^{\circ})$ or $(X, \GG, \GG^{\bullet}, \GG^{\circ})$.
In a first step, we use a duality argument to prove the $\VV-\VV^{\circ}$-integral decay:
\begin{equation*}
\frac{1}{{\varepsilon}^2} \int_0^T e^{2 \sigma t / {\varepsilon}^2} \| U^{\varepsilon}_\textnormal{kin}(\cdot) * \varphi (t) \|_{\VV^{\bullet}}^2 \, \mathrm{d} t \lesssim {\varepsilon}^2 \int_0^T \| \varphi(t) \|_{ \VV^{\circ} }^2 \, \mathrm{d} t.
\end{equation*}
and we deduce from it the $\VV-\VV^{\circ}$-uniform decay together with the stronger $\VV^{\bullet}-\VV^{\circ}$-integral decay using an energy method in a second~step:
\begin{equation*}
\sup_{0 \leqslant t < T} e^{2 \sigma t / {\varepsilon}^2} \| U^{\varepsilon}_\textnormal{kin}(\cdot) * \varphi (t) \|_{\VV}^2 + \frac{1}{{\varepsilon}^2} \int_0^T e^{2 \sigma t / {\varepsilon}^2} \| U^{\varepsilon}_\textnormal{kin}(\cdot) * \varphi (t) \|_{\VV^{\bullet}}^2 \lesssim {\varepsilon}^2 \int_0^T \| \varphi(t) \|_{ \VV^{\circ} }^2 \, \mathrm{d} t.
\end{equation*}
Note that this proves \eqref{eq:decay_convolution_semigroup_exp}, and it is enough to prove \eqref{eq:decay_convolution_semigroup_no_exp} as it follows from the particular case~$\sigma = 0$ and $Y = H$.
\step{1}{Integral decay in $\VV-\VV^{\circ}$}
We will prove the following estimate uniformly in $T \in (0, \infty]$ and~$\phi \in L^2\left( [0, T) ; \VV\right)$:
$$\left\langle e^{\sigma t / {\varepsilon}^2 } (U^{\varepsilon}_\textnormal{kin}(\cdot) * \varphi), \phi \right\rangle_{L^2\left( [0, T) ; \VV\right) } \lesssim {\varepsilon}^2 \| \phi \|_{L^2\left( [0, T) ; \VV\right) } \left(\int_0^T e^{2\sigma \tau / {\varepsilon}^2 } \| \varphi(\tau) \|_{\VV^{\circ}}^2 \mathrm{d} \tau\right)^{\frac{1}{2}},$$
and as in the proof of Lemma \ref{lem:spectral_decay}, it is enough to check that it holds for $\phi$ of the form
\begin{gather*}
\phi(t) = \begin{cases}
\phi_0 \in \VV, & t \in [t_1 , t_2]\\
0, & t \notin [t_1, t_2]
\end{cases}, \qquad \| \phi \|_{ L^2\left( [0, T) ; \VV \right) } = \sqrt{t_2 - t_1} \| \phi_0 \|_{\VV}.
\end{gather*}
By duality, we have
\begin{equation*}\begin{split}
\big\langle e^{\sigma t / {\varepsilon}^2} & (U^{\varepsilon}_\textnormal{kin}(\cdot) * \varphi), \phi \big\rangle_{ L^2\left( [0, T) ; \VV \right) } \\
& = \int_{t_1}^{t_2} \int_0^t \left\langle e^{\sigma t / {\varepsilon}^2} U^{\varepsilon}_\textnormal{kin}(t-\tau) \varphi(\tau) , \phi_0 \right\rangle_{\VV} \mathrm{d} \tau \, \mathrm{d} t \\
& = \int_{t_1}^{t_2} \int_0^t \left\langle e^{\sigma \tau / {\varepsilon}^2} \varphi(\tau) , e^{\sigma (t-\tau) / {\varepsilon}^2 } U^{\varepsilon}_\textnormal{kin}(t-\tau)^\star \, \phi_0 \right\rangle_{\VV} \mathrm{d} \tau \, \mathrm{d} t \\
& \leqslant \int_{t_1}^{t_2} \int_0^t \left\| e^{\sigma \tau / {\varepsilon}^2 } \varphi(\tau) \right\|_{\VV^{\circ} } \left\| e^{\sigma (t-\tau) / {\varepsilon}^2 } U^{\varepsilon}_\textnormal{kin}(t-\tau)^\star \, \phi_0 \right\|_{\VV^{\bullet}} \mathrm{d} \tau \, \mathrm{d} t,
\end{split}\end{equation*}
and so, using first Cauchy-Schwarz's inequality and then Young's convolution inequality in the form~$L^2\left( [0, T] \right) \ast L^1\left([0, T]\right) \hookrightarrow L^2\left([0, T]\right)$, there holds
\begin{equation*}\begin{split}
\big\langle e^{\sigma t / {\varepsilon}^2} & (U^{\varepsilon}_\textnormal{kin}(\cdot) * \varphi), \phi \big\rangle_{ L^2\left( [0, T) ; \VV \right) } \\
& \leqslant \sqrt{t_1-t_1} \left( \int_{0}^{T} \left( \int_0^t \left\| e^{\sigma \tau / {\varepsilon}^2} \varphi(\tau) \right\|_{\VV^{\circ} } \left\| e^{\sigma (t-\tau) / {\varepsilon}^2 } U^{\varepsilon}_\textnormal{kin}(t-\tau)^\star \phi_0 \right\|_{\VV^{\bullet}} \mathrm{d} \tau \right)^2 \mathrm{d} \tau \right)^{\frac{1}{2}} \\
& \leqslant \sqrt{t_1-t_1} \left(\int_0^T \left\| e^{\sigma t / {\varepsilon}^2 } \varphi(t) \right\|_{\VV^{\circ} }^2 \mathrm{d} t\right)^{\frac{1}{2}} \int_0^T \left\| e^{\sigma t / {\varepsilon}^2 } U^{\varepsilon}_\textnormal{kin}(t)^\star \phi_0 \right\|_{\VV^{\bullet}} \mathrm{d} t.
\end{split}\end{equation*}
Furthermore, using Cauchy-Schwarz's inequality, we deduce for some $\sigma' \in (\sigma, \sigma_0)$
\begin{equation*}\begin{split}
\langle e^{\sigma t / {\varepsilon}^2} & (U^{\varepsilon}_\textnormal{kin}(\cdot) * \varphi), \phi \rangle_{ L^2\left( [0, T) ; \VV \right) } \\
& \lesssim {\varepsilon} \sqrt{t_1-t_1} \left(\int_0^T \| e^{\sigma t / {\varepsilon}^2 } \varphi(t) \|_{\VV^{\circ} }^2 \mathrm{d} t\right)^{\frac{1}{2}} \left(\int_0^T \| e^{\sigma' t / {\varepsilon}^2 } U^{\varepsilon}_\textnormal{kin}(t)^\star \phi_0 \|_{\VV^{\bullet}}^2 \mathrm{d} t\right)^{\frac{1}{2}}.
\end{split}\end{equation*}
from which, using the $\VV^{\bullet}-\VV$-integral estimate for $\left((U^{\varepsilon}_\textnormal{kin}(t))^\star\right)_{t\geq0}$ obtained in Lemma \ref{lem:decay_regularization_kinetic_semigroup}, we obtain
\begin{equation*}\begin{split}
\big\langle e^{\sigma t / {\varepsilon}^2} & (U^{\varepsilon}_\textnormal{kin}(\cdot) * \varphi), \phi \big\rangle_{ L^2\left( [0, T) ; \VV \right) } \\
& \lesssim {\varepsilon}^2 \sqrt{t_1-t_1} \left(\int_0^T \| e^{\sigma t / {\varepsilon}^2 } \varphi(t) \|_{\VV^{\circ} }^2 \mathrm{d} t\right)^{\frac{1}{2}} \| \phi_0 \|_{\VV}.
\end{split}\end{equation*}
This concludes this step.
\step{2}{Regularized uniform and integral decay}
Denote $u(t) := U^{\varepsilon}_\textnormal{kin} * \varphi(t) = U^{\varepsilon} * \mathsf P^{\varepsilon}_\textnormal{kin} \varphi(t)$, it satisfies the evolution equation
$$\partial_t u = \frac{1}{{\varepsilon}^2} \left(\mathcal L - {\varepsilon} v \cdot \nabla_x \right) u + \mathsf P^{\varepsilon}_\textnormal{kin} \varphi, \quad u(0) = 0.$$
Note that, considering the decomposition $\mathcal L = \left(\mathcal L - \mathsf P\right) + \mathsf P$ in the case $Y = H$ (from Assumption \ref{L1}--\ref{L4}), or $\mathcal L = \mathcal B+ \mathcal A$ in the case $Y = X$ ( from Assumption \ref{LE}), the following degenerate dissipativity estimate holds for some $\lambda > 0$:
$$\Re \langle \mathcal L f, f \rangle_\VV + \lambda \| f \|_{\VV^{\bullet}}^2 \lesssim \| f \|^2_\VV.$$
We now write an energy estimate, using the skew-adjointness of $v \cdot \nabla_{x}$:
$$\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d} t} \| u \|^2_{\VV} + \frac{\lambda}{{\varepsilon}^2} \| u \|_{\VV^{\bullet}}^2 \lesssim \| u \|^2_{\VV} + \left| \langle \mathsf P^{\varepsilon}_\textnormal{kin} \varphi, u \rangle_\VV \right|.$$
Recall that $\mathsf P^{\varepsilon}_\textnormal{kin} = \mathrm{Id} - \mathsf P^{\varepsilon}_\textnormal{hydro}$, where we know from the spectral analysis performed in Theorems \ref{thm:spectral_study} or \ref{thm:enlarged_thm} that $\mathsf P^{\varepsilon}_\textnormal{hydro} \in \mathscr B(\VV^{\circ} ; \VV^{\bullet}) \subset \mathscr B(\VV^{\circ})$ uniformly in ${\varepsilon}$ from the embedding $\VV^{\bullet} \hookrightarrow \VV \hookrightarrow \VV^{\circ}$. Thus, we have $\mathsf P^{\varepsilon}_\textnormal{kin} \in \mathscr B(\VV^{\circ})$ uniformly in ${\varepsilon}$, from which we deduce
$$\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d} t} \| u \|^2_{\VV} + \frac{\lambda}{{\varepsilon}^2} \| u \|_{\VV^{\bullet}}^2 \lesssim \| u \|^2_{\VV} + \| \mathsf P^{\varepsilon}_\textnormal{kin} \varphi \|_{\VV^{\circ}} \| u \|_{\VV^{\bullet}} \lesssim \| u \|^2_{\VV} + \| \varphi \|_{\VV^{\circ}} \| u \|_{\VV^{\bullet}}.$$
Therefore, multiplying by $e^{2\sigma t / {\varepsilon}^2}$, we obtain
\begin{align*}
\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d} t} \left(e^{2 \sigma t / {\varepsilon}^2} \| u \|^2_{\VV}\right) + & \frac{\lambda}{{\varepsilon}^2} e^{2 \sigma t / {\varepsilon}^2} \| u \|_{\VV^{\bullet}}^2 \\
& \lesssim \frac{1}{{\varepsilon}^2} e^{2 \sigma t / {\varepsilon}^2} \| u \|^2_{\VV} + {\varepsilon} \left(e^{\sigma t / {\varepsilon}^2} \| \varphi \|_{\VV^{\circ}}\right) \left( \frac{1}{{\varepsilon}} e^{\sigma t / {\varepsilon}^2} \| u \|_{\VV^{\bullet}}\right),
\end{align*}
or more simply by Young's inequality
\begin{align*}
\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d} t} \left(e^{2 \sigma t / {\varepsilon}^2} \| u \|^2_{\VV}\right) + & \frac{\lambda}{2 {\varepsilon}^2} e^{2 \sigma t / {\varepsilon}^2} \| u \|_{\VV^{\bullet}}^2
\lesssim \frac{1}{{\varepsilon}^2} e^{2 \sigma t / {\varepsilon}^2} \| u \|^2_{\VV} + {\varepsilon}^2 e^{2 \sigma t / {\varepsilon}^2} \| \varphi \|_{\VV^{\circ}}^2.
\end{align*}
Integrating in time, we finally deduce from the previous step
$$|\hskip-0.04cm|\hskip-0.04cm| u |\hskip-0.04cm|\hskip-0.04cm|^2_{\GGG(\sigma, {\varepsilon})} \lesssim {\varepsilon}^2 \int_0^T e^{2 \sigma t / {\varepsilon}^2} \| \varphi(t) \|_{\VV^{\circ}}^2\mathrm{d} t .$$
This concludes the proof.
\end{proof}
\section{Bilinear theory}\label{sec:Bilin}
We come now to the main nonlinear estimates involbing the various stiff terms
$$\Psi^{\varepsilon}[f,g]=\frac{1}{{\varepsilon}^2}U^{\varepsilon} \ast \mathcal Q^\mathrm{sym}(f,g).$$
We will exploit the decomposition of $ U^{\varepsilon}(t) $ given in \eqref{eq:decompUeps} and the associated nonlinear decomposition
$$ \Psi^{\varepsilon}[f,g](t) = \Psi^{\varepsilon}_\textnormal{hydro} [f, g](t) + \Psi^{\varepsilon}_\textnormal{kin} [f, g](t),$$
with
$$ \Psi^{\varepsilon}_\star [f, g](t) := \mathsf P^{\varepsilon}_\star \Psi^{\varepsilon}[f,g](t) = \frac{1}{{\varepsilon}} \int_0^t U^{\varepsilon}_\star(t - \tau) \mathcal Q^\mathrm{sym} (f(\tau), g(\tau)) \mathrm{d} \tau.$$
We first need the following spatially inhomogeneous nonlinear estimates of $\mathcal Q$.
\begin{lem}[\textit{\textbf{Nonlinear Sobolev estimates for $\mathcal Q$}}]
\label{lem:Q_sobolev}
Denote $Y = H$ under assumption \ref{Bbound}, or $Y = X$ under assumption~\ref{BE}.
Consider $s > \frac{d}{2}$ and recall that~$\alpha \in \left(0, \frac{1}{2}\right)$ if $d = 2$, or $\alpha = 0$ if $d \geqslant 3$. There holds
\begin{subequations}
\label{eq:Q_refined_sobolev_algebra}
\begin{gather}
\label{eq:Q_refined_sobolev_negative_algebra_inequality}
\begin{aligned}
\| \mathcal Q(f, g) \|_{ \dot{\mathbb{H}}^{-\alpha}_x \left( Y^{\circ}_v \right) } & + \| \mathcal Q(f, g) \|_{\rSSgm{s} } \\
\lesssim& \| f \|_{ \rSSg{s} } \| | \nabla_x |^{1-\alpha} g \|_{ \rSSgp{s-(1-\alpha)} }
+ \| | \nabla_x |^{1-\alpha} f \|_{ \rSSgp{s-(1-\alpha)} } \| g \|_{ \rSSg{s} },
\end{aligned} \\
\label{eq:Q_refined_sobolev_negative_algebra_inequality_same}
\begin{aligned}
\| \mathcal Q(f, g) \|_{ \dot{\mathbb{H}}^{-\alpha}_x \left( Y^{\circ}_v \right) } & + \| \mathcal Q(f, g) \|_{\rSSgm{s} } \\
\lesssim & \| f \|_{ \rSSg{s} } \| | \nabla_x |^{1-\alpha} g \|_{ \rSSgp{s-(1-\alpha)} }
+ \| f \|_{ \rSSgp{s} } \| | \nabla_x |^{1-\alpha} g \|_{ \rSSg{s- (1-\alpha) } }.
\end{aligned}
\end{gather}
\end{subequations}
Furthermore, we have the following control when $g \in W^{s', \infty}_x \left( Y^{\bullet}_v \right)$ for $s' > s$:
\begin{equation}
\label{eq:Q_sobolev_algebra_Holder}
\| \mathcal Q(f, g) \|_{\rSSgm{s} } \lesssim \| f \|_{ \rSSg{s} } \| g \|_{ W^{s', \infty}_x \left( Y^{\bullet}_v \right) } + \| f \|_{ \rSSgp{s} } \| g \|_{ W^{s', \infty}_x \left( Y_v \right) }.
\end{equation}
\end{lem}
\begin{proof}
Due to the locality in $x$ of the bilinear operator $\mathcal Q$, one can adapt classical results from paradifferential calculus, replacing the multiplication $(u, v) \mapsto uv$ (resp. the modulus $|\cdot|$) by the collision operator $(u, v) \mapsto \mathcal Q(u, v)$ (resp. the $Y^{\circ}$-norm). In particular, we redefine the homogeneous paraproduct and remainder (see Appendix \ref{scn:littlewood-paley}) as
$$\dot{T}_u v= \sum_{j} \mathcal Q\left( \dot{S}_{j-1} u , \dot{\Delta}_j v \right), \qquad \dot{R}(u, v) = \sum_{|j - k| \leqslant 1} \mathcal Q\left( \dot{\Delta}_k u , \dot{\Delta}_j v \right),$$
which satisfy, under the assumption \ref{Bbound} or \ref{BE}, the estimates
$$ \left\| \mathcal Q\left( \dot{S}_{j-1} u , \dot{\Delta}_j v \right) \right\|_{ L^p_x \left( Y^{\circ}_v \right) } \lesssim
\left\| \| \dot{S}_{j-1} u \|_{ Y^{\bullet}_v } \| \dot{\Delta}_j v \|_{ Y_v } \right\|_{ L^p_x } +
\left\| \| \dot{S}_{j-1} u \|_{ Y_v } \| \dot{\Delta}_j v \|_{ Y^{\bullet}_v } \right\|_{ L^p_x },$$
$$
\left\| \mathcal Q\left( \dot{\Delta}_k u , \dot{\Delta}_j v \right) \right\|_{ L^p_x \left( Y^{\circ}_v \right) } \lesssim
\left\| \| \dot{\Delta}_k u \|_{ Y^{\bullet}_v } \| \dot{\Delta}_j v \|_{ Y_v } \right\|_{ L^p_x } +
\left\| \| \dot{\Delta}_k u \|_{ Y_v } \| \dot{\Delta}_j v \|_{ Y^{\bullet}_v } \right\|_{ L^p_x }.
$$
One then checks that \eqref{eq:Q_sobolev_algebra_Holder} is the $\mathcal Q$-version of Proposition \ref{prop:product_sobolev_holder}. Furthermore, denoting for compactness $\boldsymbol{\alpha} = 1-\alpha \in \left(0, \frac{d}{2}\right)$, one gets from the $\mathcal Q$-version of Proposition~\ref{prop:product_homogeneous_sobolev} with $s_1 = \frac{d}{2} -1 \in \big[ 0, \frac{d}{2} \big)$ and $s_2 = \boldsymbol{\alpha}$, so that $s_1 + s_2 - \frac{d}{2} = - \alpha$:
\begin{equation*}\begin{split}
\| \mathcal Q(f, g) \|_{\dot{\mathbb{H}}^{-\alpha}_x \left( Y^{\circ}_v \right) }
&\lesssim \| f \|_{ \dot{\mathbb{H}}_x^{\frac{d}{2} - 1} \left( Y_v \right) } \left\| | \nabla_{x} |^{\boldsymbol{\alpha}} g \right\|_{ L^2_x \left( Y^{\bullet}_v \right) }
+ \| | \nabla_x |^{\boldsymbol{\alpha}} f \|_{ L^2_x\left( Y^{\bullet}_v \right) } \| g \|_{ \dot{\mathbb{H}}^{\frac{d}{2} - 1}_x \left( Y_v \right) } \\
&\lesssim \| f \|_{ \mathbb{H}^s_x \left( Y_v \right) } \left\| | \nabla_{x} |^{\boldsymbol{\alpha}} g \right\|_{ L^2_x \left( Y^{\bullet}_v \right) }
+ \left\| | \nabla_{x} |^{\boldsymbol{\alpha}} f \right\|_{ L^2_x \left( Y^{\bullet}_v \right) } \| g \|_{ \mathbb{H}^s_x \left( Y_v \right) }.
\end{split}\end{equation*}
We now turn to the estimate in $\rSSgm{s} = \mathbb{H}^s_x \left( Y^{\circ}_v \right)$. Using \ref{Bbound} or \ref{BE}, we have
\begin{align*}
\| \mathcal Q(f, g) \|_{ \mathbb{H}^s_x \left( Y^{\circ}_v \right) } = & \left\| \langle \xi \rangle^s \int_{\mathbb R^{d}} \mathcal Q\left( \widehat{f}(\xi - \zeta) , \widehat{g}(\zeta) \right) \mathrm{d} \zeta \right\|_{ L^2_\xi \left( Y^{\circ}_v \right) } \\
\lesssim & \left\| \langle \xi \rangle^s \int_{\mathbb R^{d}} \| \widehat{f}(\xi - \zeta) \|_{ Y^{\bullet}_v } \| \widehat{g}(\zeta) \|_{ Y_v } \mathrm{d} \zeta \right\|_{ L^2_\xi } \\
& + \left\| \langle \xi \rangle^s \int_{\mathbb R^{d}} \| \widehat{f}(\xi - \zeta) \|_{ Y_v } \| \widehat{g}(\zeta) \|_{ Y^{\bullet}_v } \mathrm{d} \zeta \right\|_{ L^2_\xi } =: I_1 + I_2.
\end{align*}
We split the frequency weight as
$$\langle \xi \rangle^s \approx 1 + | \xi |^s \lesssim 1 + | \xi - z |^s + | \zeta |^s \lesssim | \xi - z|^{\boldsymbol{\alpha}} \langle \xi - z \rangle^{s - \boldsymbol{\alpha}} + \langle z\rangle^s,$$
since $s-\boldsymbol{\alpha} >0,$
which allows to control the term $I_1$ as follows:
\begin{equation*}\begin{split}
I_1 &\lesssim \left\| \int_{\mathbb R^{d}} \Big[ | \xi - z |^{\boldsymbol{\alpha}} \langle \xi - z \rangle^{s - \boldsymbol{\alpha}} \| \widehat{f}(\xi -z) \|_{ Y^{\bullet}_v } \Big] \| \widehat{g}(z) \|_{ Y_v } \mathrm{d} z \right\|_{ L^2_\xi } \\
&\phantom{+++++*} + \left\| \int_{\mathbb R^{d}} \| \widehat{f}(\xi -z) \|_{ Y^{\bullet}_v } \Big[\langle \zeta \rangle^s \| \widehat{g}(z) \|_{ Y_v }\Big] \mathrm{d} z \right\|_{ L^2_\xi }.
\end{split}\end{equation*}
Using Young's convolution inequality $L^2_\xi \ast L^1_\xi \hookrightarrow L^2_\xi$
we deduce that
\begin{equation*}\begin{split}
I_1 &\lesssim \left\|\,|\xi|^{\boldsymbol{\alpha}}\langle\xi\rangle^{\boldsymbol{\alpha}}\,
\|\widehat{f}\|_{Y^{\bullet}_v}\,\right\|_{L^2_\xi}
\left\|\,\|\widehat{g}(\xi)\|_{Y_v}\right\|_{L^1_\xi}+
\left\|\,\|\widehat{f}(\xi)\|_{Y^{\bullet}_v}\right\|_{L^1_\xi}\,\left\|\,\langle\xi\rangle^{s}\,\|\widehat{g}\|_{Y_v}\right\|_{L^2_\xi}\\
&\lesssim \left\|\,|\nabla_x|^{\boldsymbol{\alpha}}\,f\right\|_{H^{s-\boldsymbol{\alpha}}_x(Y^{\bullet}_v)}\,\|\widehat{g}\|_{L^1_\xi(Y_v)}+\left\|\widehat{f}\right\|_{L^1_\xi(Y^{\bullet}_v)}\,\left\|g\right\|_{\mathbb{H}^s_x(Y_v)}.
\end{split}\end{equation*}
Using the fact that~$\langle \xi \rangle^{ -s } \in L^2_\xi$ (resp. $| \xi |^{ -\boldsymbol{\alpha} } \langle \xi \rangle^{-(s - \boldsymbol{\alpha}) } \in L^2_\xi$), a simple use of Cauchy-Schwarz's inequality allows to estimate $L^1_\xi$-norms with weighted $L^2_\xi$-norms resulting in
$$ I_1 \lesssim \| | \nabla_x |^{\boldsymbol{\alpha}} f \|_{ H^{ s-\boldsymbol{\alpha} }_x \left( Y^{\bullet}_v \right) } \| g \|_{ \mathbb{H}^s_x \left( Y_v \right) }.$$
We prove in the exact same way that
$$I_2 \lesssim \| f \|_{ \mathbb{H}^s_x \left( Y_v \right) } \| | \nabla_x |^{\boldsymbol{\alpha}} g \|_{ H^{ s-\boldsymbol{\alpha} }_x \left( Y^{\bullet}_v \right) },$$
thus \eqref{eq:Q_refined_sobolev_negative_algebra_inequality} is proved, and the proof of \eqref{eq:Q_refined_sobolev_negative_algebra_inequality_same} is similar. The proofs of \eqref{eq:Q_sobolev_algebra_degenerate} and \eqref{eq:Q_sobolev_algebra_degenerate_closed} are also similar. This concludes the proof.
\end{proof}
\subsection{Bilinear and linear hydrodynamic estimates}
We have all in hands to estimate the bilinear ‘‘hydrodynamic'' operator $\Psi^{\varepsilon}_\textnormal{hydro}(f,g)=\frac{1}{{\varepsilon}}U_\textnormal{hydro}^{\varepsilon}(\cdot) \ast \mathcal Q^\mathrm{sym}(f,g)$. \textbf{The results of this section hold under any assumption \ref{Bbound}, \ref{BE} or \ref{BED}}. They are based upon the above properties of $\mathcal Q$ as well as the results of Section \ref{scn:study_physical_space} on the various semigroups involved:
\begin{prop}[\textit{\textbf{General bilinear hydrodynamic estimates}}]
\label{prop:bilinear_hydrodynamic}
The bilinear operator~$\Psi^{\varepsilon}_\textnormal{hydro}$ satisfies the following continuity estimates in $\HHH$ when at least one argument is in $\HHH$:
\begin{subequations}
\label{eq:bilinear_hyd_hyd}
\begin{gather}
\label{eq:bilinear_hyd_hyd_hyd}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_\HHH \lesssim w_{\phi, \eta}(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_\HHH |\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_\HHH,\\
\label{eq:bilinear_hyd_hyd_mix}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_\HHH \lesssim {\varepsilon} w_{\phi, \eta}(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_\HHH |\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_\FFF,\\
\label{eq:bilinear_hyd_hyd_kin}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_\HHH \lesssim {\varepsilon} |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_\HHH |\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_\GGG,
\end{gather}
\end{subequations}
as well as as the following ones when at least one argument is in $\FFF$:
\begin{subequations}
\label{eq:bilinear_hyd_mix}
\begin{gather}
\label{eq:bilinear_hyd_mix_mix}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_\HHH \lesssim {\varepsilon} w_{\phi, \eta}(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_\FFF |\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_\FFF,\\
\label{eq:bilinear_hyd_mix_kin}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_\HHH \lesssim {\varepsilon} |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_\FFF |\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_\GGG,
\end{gather}
\end{subequations}
and the following one when both arguments are in $\GGG$:
\begin{equation}
\label{eq:bilinear_hyd_kin_kin}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_\HHH \lesssim {\varepsilon} |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_\GGG |\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_\GGG.
\end{equation}
Furthermore, it is strongly continuous at $t=0$:
$$\lim_{t \to 0}\| \Psi_\textnormal{hydro}^{\varepsilon}(f, g)(t) \|_{\HH}=0$$
in all cases considered above.
\end{prop}
\begin{proof}
We recall the definition of $\Psi^{\varepsilon}_\textnormal{hydro}$ and the orthogonality property of $\mathcal Q$:
$$\Psi^{\varepsilon}_\textnormal{hydro} [f, g](t) = \frac{1}{{\varepsilon}} \int_0^t U^{\varepsilon}_\textnormal{hydro}(t-\tau) \mathcal Q(f(\tau), g(\tau)) \mathrm{d} \tau, \qquad \mathsf P \mathcal Q = 0,$$
thus, denoting for compactness $w = w_{\phi, \eta}$, the convolution estimate \eqref{eq:decay_semigroups_hydro_orthogonal} gives
\begin{equation}
\label{eq:pre_estimate_hyd}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_{\rSSSs{s}}^2 \lesssim \sup_{0 \leqslant t < T} \left\{w(t)^2 \int_0^t \| \mathcal Q(f(\tau), g(\tau) ) \|_{ \rSSgm{s} \cap \dot{\mathbb{H}}^{-\alpha}_x \left(Y^{\circ}_v \right) }^2 \mathrm{d} \tau\right\},
\end{equation}
where $(\VV, Y) = (\HH, H), (\GG, X)$ or $(\GG_{-1}, X_{-1})$, and we recall that $w$ is non-increasing and bounded from above and below:
\begin{equation}
\label{eq:bound_w}
\forall 0 \leqslant t_1 \leqslant t_2 < T, \quad 0 < w(T) \leqslant w(t_2) \leqslant w(t_1) \leqslant 1.
\end{equation}
The continuity at $t = 0$ will be an easy consequence of the estimate \eqref{eq:pre_estimate_hyd}.
\step{1}{Proof of \eqref{eq:bilinear_hyd_hyd} for $f \in \HHH$}
When $g \in \HHH$, we combine \eqref{eq:pre_estimate_hyd} with the bilinear estimate~\eqref{eq:Q_refined_sobolev_negative_algebra_inequality} for $\mathcal Q$, to deduce the following, where $\boldsymbol{\alpha} := 1 - \alpha$
\begin{align*}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_\HHH^2
\lesssim & \sup_{ 0 \leqslant t < T }\left\{w(t)^2\int_0^t \Big[ \|f(\tau) \|_{ \rSSs{s} }^2 \left\| |\nabla_x|^{\boldsymbol{\alpha}} g(\tau) \right\|_{\rSSsp{s-{\boldsymbol{\alpha}}}}^2 \right.\\
& \phantom{+++++} \left.+ \left\| |\nabla_x|^{\boldsymbol{\alpha}} f(\tau) \right\|_{\rSSsp{s-{\boldsymbol{\alpha}}}}^2 \| g(\tau) \|_{ \rSSs{s} }^2 \Big] \mathrm{d} \tau\right\}\\
\lesssim & \sup_{ 0 \leqslant t < T }\int_0^t \Big[w^2(\tau)\|f(\tau) \|_{ \rSSs{s} }^2 \left\| |\nabla_x|^{\boldsymbol{\alpha}} g(\tau) \right\|_{\rSSsp{s-{\boldsymbol{\alpha}}}}^2 \\
&\phantom{+++++} +\left\| |\nabla_x|^{\boldsymbol{\alpha}} f(\tau) \right\|_{\rSSsp{s-{\boldsymbol{\alpha}}}}^2 w(\tau)^2\| g(\tau) \|_{ \rSSs{s} }^2 \Big] \mathrm{d} \tau
\end{align*}
where we used \eqref{eq:bound_w} in the second inequality. Recalling that
$$|\hskip-0.04cm|\hskip-0.04cm| h |\hskip-0.04cm|\hskip-0.04cm|_{\rSSSs{s}}^2 := \sup_{0 \leqslant t < T} \left\{ w(t)^2 \| h(t) \|_{\rSSsp{s}}^2 + w(t)^2 \int_0^t \left\| |\nabla_x|^{1-\alpha} h( \tau ) \right\|_{\rSSsp{s}}^2 \mathrm{d} \tau \right\},$$
and using $\HH^{\bullet} \hookrightarrow \HH$ and \eqref{eq:bound_w}, we deduce
\begin{equation*}\begin{split}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro}(f,g) |\hskip-0.04cm|\hskip-0.04cm|_\HHH^2 &\lesssim
|\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH }^2 \int_0^T \left\| |\nabla_x|^{\boldsymbol{\alpha}} g(t) \right\|_{\rSSsp{s-{\boldsymbol{\alpha}}}}^2 \mathrm{d} t
+ |\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH }^2 \int_0^T \left\| |\nabla_x|^{\boldsymbol{\alpha}} f(t) \right\|_{\rSSsp{s-{\boldsymbol{\alpha}}}}^2 \mathrm{d} t\\
& \lesssim w(T)^{-2} |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_\HHH^2 |\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_\HHH^2
\end{split}\end{equation*}
which is exactly ~\eqref{eq:bilinear_hyd_hyd_hyd}.
When $g \in \FFF$, using furthermore $\HH^{\bullet} \hookrightarrow \HH$, we similarly have \eqref{eq:bilinear_hyd_hyd_mix}:
\begin{equation*}\begin{split}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_\HHH^2 &\lesssim {\varepsilon}^2 \int_0^T \left( w(\tau)\|f(\tau) \|_{ \HH^{\bullet} } \right)^2 \left(\frac{1}{{\varepsilon}} \| g(\tau) \|_{\HH^{\bullet}}\right)^2 \mathrm{d} \tau \\
&\lesssim {\varepsilon}^2 |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_\HHH^2 \int_0^T \left(\frac{1}{{\varepsilon}} \| g(\tau) \|_{\HH^{\bullet}}\right)^2 \mathrm{d} \tau\lesssim {\varepsilon}^2 w(T)^{-2} |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_\HHH^2 |\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_\FFF^2,
\end{split}\end{equation*}
which gives now \eqref{eq:bilinear_hyd_hyd_mix}. In the same way, using also $\HH^{\bullet} \hookrightarrow \GG^{\bullet}$, we have \eqref{eq:bilinear_hyd_hyd_kin}.
\step{2}{Proof of \eqref{eq:bilinear_hyd_mix} and \eqref{eq:bilinear_hyd_kin_kin}}
When $f \in \FFF$ and $g \in \GGG$, we combine~\eqref{eq:pre_estimate_hyd} with the bilinear estimate \eqref{eq:Q_refined_sobolev_negative_algebra_inequality} for $\mathcal Q$. Using that $\HH^{\bullet} \hookrightarrow \GG^{\bullet}$ and the property \eqref{eq:bound_w} of~$w$, we have:
\begin{equation*}\begin{split}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro} (f, g) |\hskip-0.04cm|\hskip-0.04cm|_\HHH^2
\lesssim & {\varepsilon}^2 \sup_{0 \leqslant t < T} \Bigg\{w(t)^2 \int_0^t \left[ \| f(\tau) \|_{ \HH }^2 \left( \frac{1}{{\varepsilon}} \| g(\tau) \|_{\GG^{\bullet}}\right)^2 \right. \\
& \left.\phantom{+++++++++} + \left( \frac{1}{{\varepsilon}} \| f(\tau) \|_{\HH^{\bullet}} \right)^2 \| g(\tau) \|_{ \GG }^2 \, \right] \mathrm{d} \tau \Bigg\} \\
\lesssim & {\varepsilon}^2 \int_0^T w(\tau)^2 \|f(\tau) \|_{ \HH }^2 \left( \frac{1}{{\varepsilon}} \| g(\tau) \|_{\GG^{\bullet}}\right)^2 \mathrm{d} t\\
& + {\varepsilon}^2 \sup_{0 \leqslant t < T} \Bigg\{w(t)^2 \int_0^t \left( \frac{1}{{\varepsilon}} \| f(\tau) \|_{\HH^{\bullet}} \right)^2 \| g(\tau) \|_{ \GG }^2 \mathrm{d} \tau \Bigg\} \\
\lesssim & {\varepsilon}^2 |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_\FFF^2 |\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_\GGG^2.
\end{split}\end{equation*}
This proves ~\eqref{eq:bilinear_hyd_mix_kin}
The proofs of \eqref{eq:bilinear_hyd_mix_mix} and \eqref{eq:bilinear_hyd_kin_kin} are similar and omitted.
\end{proof}
\begin{prop}[\textit{\textbf{Special bilinear hydrodynamic estimates}}]
\label{prop:special_bilinear_hydrodynamic}
When $f \in \HHH$ and $\phi$ is the parameter defining the $\HHH$-norm
\begin{equation}
\label{eq:bilinear_hyd_phi}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro}[f, \phi] |\hskip-0.04cm|\hskip-0.04cm|_\HHH \lesssim \eta |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_\HHH\,.
\end{equation}
Furthermore, when $g^{\varepsilon}_\textnormal{disp} = U^{\varepsilon}_\textnormal{disp}(\cdot) g$ where $g = \mathsf P_{\textnormal{wave}} g \in \rSSs{s} \cap \dot{\mathbb{H}}^{-\alpha}_x \left( H_v \right)$, there holds
\begin{equation}
\label{eq:bilinear_hyd_disp}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro}[ f , g^{\varepsilon}_\textnormal{disp}] |\hskip-0.04cm|\hskip-0.04cm|_\HHH \lesssim {\beta_{\textnormal{disp}}}(g, {\varepsilon}) |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH }, \qquad \lim_{{\varepsilon} \to 0} {\beta_{\textnormal{disp}}}(g, {\varepsilon})=0,
\end{equation}
and in the case $d \geqslant 3$, the rate of convergence is explicit assuming $g \in \dot{\mathbb{B}}^{s' + (d+1)/2}_{1, 1} \left( H_v \right) \cap \rSSs{s}$ for some $s'>s$:
\begin{equation*}
{\beta_{\textnormal{disp}}}(g, {\varepsilon}) \lesssim \sqrt{{\varepsilon}}
\left( \| g \|_{ \rSSs{s} } + \| g \|_{ \dot{\mathbb{B}}^{s' + (d+1)/2 }_{1, 1} \left( H_v \right) }\right).
\end{equation*}
\end{prop}
\begin{proof}
We start by proving \eqref{eq:bilinear_hyd_phi} and then prove \eqref{eq:bilinear_hyd_disp}. We use again the shorthand notation~$w = w_{\phi, \eta}$ and recall that, besides \eqref{eq:pre_estimate_hyd}, the convolution estimate
\eqref{eq:decay_semigroups_hydro_orthogonal} also leads to
\begin{equation}
\label{eq:pre_estimate_2_pos}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_{\rSSSs{s}}^2 \lesssim \sup_{0 \leqslant t < T} \left\{w(t)^2 \left(\int_0^t \| \mathcal Q(f(\tau), g(\tau) ) \|_{ \HH^{\circ} }^{ \frac{2}{1+\alpha} } \mathrm{d} \tau \right)^{ 1+\alpha } \right\},
\end{equation}
\step{1}{Proof of \eqref{eq:bilinear_hyd_phi}}
We combine the convolution estimate \eqref{eq:pre_estimate_hyd} with the nonlinear bound \eqref{eq:Q_refined_sobolev_negative_algebra_inequality_same}, and use $\HH^{\bullet} \hookrightarrow \HH$ to obtain (where we denote for compactness ${\boldsymbol{\alpha}} = 1-\alpha$)
\begin{equation*}\begin{split}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro} [f, \phi] |\hskip-0.04cm|\hskip-0.04cm|_\HHH^2
&\lesssim \sup_{0 \leqslant t < T} \left\{ w(t)^2 \int_0^t \Big[w(\tau)\| f(\tau) \|_{ \rSSsp{s} }\Big]^2 \Big[w(\tau)^{-1} \| |\nabla_x|^{\boldsymbol{\alpha}} \phi(\tau) \|_{\rSSsp{s-{\boldsymbol{\alpha}}}}\Big]^2 \mathrm{d} \tau \right\}\\
& \lesssim |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_\HHH^2 \sup_{0 \leqslant t < T} \Bigg\{ w(t)^2 \int_0^t \Big[w(\tau)^{-1} \| |\nabla_x|^{\boldsymbol{\alpha}} \phi(\tau) \|_{\rSSsp{s-\boldsymbol{\alpha}}}\Big]^2 \mathrm{d} \tau \Bigg\}\,
\end{split}\end{equation*}
and then, using \eqref{eq:NS_exponential_weight}, we finally get
$
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro} (f, \phi) |\hskip-0.04cm|\hskip-0.04cm|_\HHH^2
\lesssim \eta^2 |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_\HHH^2.
$
which is exactly \eqref{eq:bilinear_hyd_phi}.
\step{2}{Proof of \eqref{eq:bilinear_hyd_disp}}
As in the previous step, but using the nonlinear bound~\eqref{eq:Q_refined_sobolev_negative_algebra_inequality_same} for $\mathcal Q$, one has
\begin{align*}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro}[ f , g^{\varepsilon}_\textnormal{disp}] |\hskip-0.04cm|\hskip-0.04cm|^2_\HHH & \lesssim |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH}^2 \int_0^T \left\| | \nabla_{x} |^{\boldsymbol{\alpha}} g^{\varepsilon}_\textnormal{disp}(t) \right\|^2_{ \rSSsp{s-{\boldsymbol{\alpha}}} } \mathrm{d} t,
\end{align*}
which, according to \eqref{eq:graU_*} and \eqref{eq:XIALPHA}, satisfies for some universal $\kappa > 0$
\begin{equation}
\label{eq:continuity_hyd_disp}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro}[ f , g^{\varepsilon}_\textnormal{disp}] |\hskip-0.04cm|\hskip-0.04cm|^2_\HHH \lesssim |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH}^2 \left( \| g \|_{ \rSSs{s} } + \| g \|_{ \dot{\mathbb{H}}^{-\alpha}_x \left( H_v \right) } \right)^2 \sup_{\xi \in \supp \| \widehat{g} \|_{H^{\circ}_v} } \int_0^T | \xi |^2 e^{-t \kappa | \xi |^2 } \mathrm{d} t.
\end{equation}
Since the supremum term is bounded uniformly in $T \in (0, \infty)$ and $g \in \rSSsm{s} \cap \dot{\mathbb{H}}^{-\alpha}_x \left( H_v \right)$, it is enough to prove \eqref{eq:bilinear_hyd_disp} in the case where $g \in \rSSsm{s} \cap \dot{\mathbb{B}}^{s' + (d+1)/2 }_{1, 1} \left( H^{\circ}_v \right)$ for some $s' > s$ and~$\xi \mapsto \| \widehat{g}(\xi) \|_{ H^{\circ}_v }$ is supported away from $0$, as it will allow to conclude by a density argument.
We assume therefore there is some $\nu > 0$ such that
$$\forall | \xi | \leqslant \nu, \quad \widehat{g}(\xi) = 0,$$
and we point out that when $T < \infty$, using $e^{-r} \lesssim r^{-1}$, for any $R > 0$
$$\sup_{|\xi | \geqslant \nu} \int_R^T | \xi|^2 e^{-\kappa t | \xi |^2} \mathrm{d} t \lesssim \int_{ R }^{ T } \frac{\mathrm{d} t}{t} = \log(T) - \log(R),$$
and when $T = \infty$
$$\sup_{|\xi | \geqslant \nu} \int_R^\infty | \xi|^2 e^{-\kappa t | \xi |^2} \mathrm{d} t = \sup_{|\xi | \geqslant \nu } \int_{ R | \xi |^2 }^\infty e^{- \kappa t } \mathrm{d} t \lesssim \exp\left( - R \nu^2 \right),$$
so that, in both cases, we have
$$C_\nu(R, T) := \sup_{|\xi | \geqslant \nu } \int_R^T | \xi|^2 e^{-\kappa t | \xi |^2} \mathrm{d} t \xrightarrow[]{R \to T} 0.$$
We split for some $0 < R < T$ the nonlinear term:
$$\mathcal Q(g^{\varepsilon}_\textnormal{disp}, f) = \mathbf{1}_{0 \leqslant t \leqslant R} \mathcal Q(g^{\varepsilon}_\textnormal{disp}, f) + \mathbf{1}_{R \leqslant t < T} \mathcal Q(g^{\varepsilon}_\textnormal{disp}, f) =: \varphi^-(t) + \varphi^+(t),$$
so that, using an estimate analogous to \eqref{eq:continuity_hyd_disp} we have
\begin{align*}
\left(\frac{1}{{\varepsilon}} |\hskip-0.04cm|\hskip-0.04cm| U^{\varepsilon}_\textnormal{hydro}(\cdot) * \varphi^+ |\hskip-0.04cm|\hskip-0.04cm|_\HHH\right)^2 \lesssim &
|\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH}^2 \left( \| g \|_{ \rSSs{s} } + \| g \|_{ \dot{\mathbb{H}}^{-\alpha}_x \left( H_v \right) } \right)^2 C_\nu(R, T).
\end{align*}
Furthermore, combining this time \eqref{eq:pre_estimate_2_pos} with the boundedness estimate \eqref{eq:decay_semigroups_hydro} (resp. the dispersive estimate \eqref{eq:dispersive}) $U^{\varepsilon}_\textnormal{disp}(\cdot)$, together with the corresponding nonlinear bound \eqref{eq:Q_refined_sobolev_algebra} (resp.~\eqref{eq:Q_sobolev_algebra_Holder}) for $\mathcal Q$, we have for $\varphi^-(t)$
\begin{equation*}
\label{eq:continuity_hyd_disp_vanish}
\left(\frac{1}{{\varepsilon}} |\hskip-0.04cm|\hskip-0.04cm| U^{\varepsilon}_\textnormal{hydro}(\cdot) * \varphi^- |\hskip-0.04cm|\hskip-0.04cm|_\HHH \right)^2\lesssim |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{\HHH}^2 \left( \| g \|_{ \rSSsm{s} }^2 + \| g \|_{ \dot{\mathbb{B}}^{s + (d+1)/2 }_{1, 1} \left( H^{\circ}_v \right) }^2 \right) \left(\int_0^R 1 \land \left(\frac{{\varepsilon}}{t}\right)^{\frac{d-1}{1+\alpha}} \mathrm{d} t\right)^{1+\alpha}.
\end{equation*}
Put together, the two previous controls yield uniformly in $R \in (0, T)$
\begin{align*}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro}[ f , g^{\varepsilon}_\textnormal{disp}] |\hskip-0.04cm|\hskip-0.04cm|_\HHH^2 \lesssim &
|\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{\HHH}^2 \left( \| g \|_{ \rSSsm{s} }^2 + \| g \|_{ \dot{\mathbb{B}}^{s + (d+1)/2 }_{1, 1} \left( H^{\circ}_v \right) }^2 \right) \left(\int_0^{R} 1 \land \left(\frac{{\varepsilon}}{t}\right)^{\frac{d-1}{1+\alpha}} \mathrm{d} t\right)^{1+\alpha} \\
& + |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH}^2 \left( \| g \|_{ \rSSs{s} } + \| g \|_{ \dot{\mathbb{H}}^{-\alpha}_x \left( H_v \right) } \right)^2 C_\nu(R, T).
\end{align*}
Letting ${\varepsilon} \to 0$ and then $R \to T$, one deduces \eqref{eq:bilinear_hyd_disp} for $g \in \rSSsm{s} \cap \dot{\mathbb{B}}^{s' + (d+1)/2 }_{1, 1} \left( H^{\circ}_v \right)$ with~$s' > s$ and whose Fourier transform is supported away from $0$. We conclude to the general case of~$g \in \rSSsm{s}$ by density thanks to \eqref{eq:continuity_hyd_disp}. Note that in dimension $d \geqslant 3$, there holds $\alpha = 0$, thus one has
$$\int_0^T 1 \land \left(\frac{{\varepsilon}}{t}\right)^{d-1} \mathrm{d} t =\int_0^{\varepsilon} \mathrm{d} t+ {\varepsilon}^{d-1}\int_{{\varepsilon}}^T t^{1-d}\mathrm{d} t \lesssim {\varepsilon}.$$
This concludes the proof.
\end{proof}
\subsection{Bilinear kinetic and mixed estimates}
The results of this section hold assuming \ref{Bbound}. We point out that only one estimate, namely \eqref{eq:bilinear_kin_kin_kin}, holds assuming \ref{Bbound} or \ref{BE} \textbf{but not \ref{BED}}, which is why it has to be treated using the alternative strategy of Section \ref{scn:hydrodynamic_limit_BED}.
We have the analogue of Proposition \ref{prop:bilinear_hydrodynamic} for the kinetic bilinear operator $\Psi^{\varepsilon}_\textnormal{kin}(f, g) = \frac{1}{{\varepsilon}} U^{\varepsilon}_\textnormal{kin} \ast \mathcal Q^\mathrm{sym}(f, g)$.
\begin{prop}[\textit{\textbf{General bilinear kinetic and mixed estimates}}]
\label{prop:bilinear_kinetic}
The bilinear operator $\Psi^{\varepsilon}_\textnormal{kin}$ satisfies the following continuity estimates in the mixed space $\FFF$ when at least one argument is in $\FFF$:
\begin{gather}
\label{eq:bilinear_mix_mix_mix}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_\FFF \lesssim {\varepsilon} w_{\phi, \eta}(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_\FFF \min\left\{|\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_\FFF\,,\,|\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_\HHH\right\}\,,
\end{gather}
and the following one when $f, g \in \HHH$ (note the absence of a factor ${\varepsilon}$):
\begin{equation}
\label{eq:bilinear_mix_hyd_hyd}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_\FFF \lesssim w_{\phi, \eta}(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_\HHH |\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_\HHH.
\end{equation}
Furthermore, considering $X = H$ in the definition of $\GGG$ under Assumption \ref{Bbound}, or considering $X$ to be the space from \ref{BE} under this assumption, there holds in the kinetic space $\GGG$ when at least one argument is in $\GGG$
\begin{subequations}
\label{eq:bilinear_kin}
\begin{gather}
\label{eq:bilinear_kin_kin_kin}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_\GGG \lesssim {\varepsilon} |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_\GGG |\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_\GGG,\\
\label{eq:bilinear_kin_kin_mix}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_\GGG \lesssim {\varepsilon} w_{\phi, \eta}(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_\GGG \min\left\{ |\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_\FFF\,,\,|\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_\HHH\right\}.
\end{gather}
\end{subequations}
Finally, it is strongly continuous at $t = 0$ in the sense that in the corresponding cases
$$\lim_{t\to 0}\| \Psi^{\varepsilon}_\textnormal{kin} [f, g](t) \|_\GG= 0, \qquad \lim_{t\to 0}\| \Psi^{\varepsilon}_\textnormal{kin} [f, g](t) \|_\HH = 0.$$
\end{prop}
\begin{proof}
Recall the definition of $\Psi^{\varepsilon}_\textnormal{kin}$:
$$\Psi^{\varepsilon}_\textnormal{kin} [f, g](t) = \frac{1}{{\varepsilon}} \int_0^t U^{\varepsilon}_\textnormal{kin}(t-\tau) \mathcal Q(f(\tau), g(\tau)) \mathrm{d} \tau,$$
thus denoting for compactness $w_{\phi, \eta}(t) = w(t)$, the convolution estimates \eqref{eq:decay_convolution_semigroup_exp} and \eqref{eq:decay_convolution_semigroup_no_exp} give respectively
\begin{equation}
\label{eq:pre_estimate_kin}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG }^2 \lesssim \int_0^T e^{2\sigma t / {\varepsilon}^2} \left\| \mathcal Q( f(t), g(t) ) \right\|^2_{ \GG^{\circ} } \mathrm{d} t,
\end{equation}
and
\begin{equation} \label{eq:pre_estimate_mix}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF }^2 \lesssim \sup_{0 \leqslant t < T} \left\{ w(t)^2 \int_0^T \left\| \mathcal Q( f(\tau), g(\tau) ) \right\|^2_{ \HH^{\circ} } \mathrm{d} t \right\}.
\end{equation}
We also recall the bound \eqref{eq:bound_w} for $w$. The continuity at $t = 0$ will be immediate from the estimates below by letting $T \to 0$. For the reader convenience, we also recall the definitions of $|\hskip-0.04cm|\hskip-0.04cm|\cdot|\hskip-0.04cm|\hskip-0.04cm|_{\GGG}$ and $|\hskip-0.04cm|\hskip-0.04cm| \cdot |\hskip-0.04cm|\hskip-0.04cm|_{\FFF}$:
$$ |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG}^2 := \sup_{0 \leqslant t < T} \, e^{2 \sigma t / {\varepsilon}^2 } \| f(t) \|_{\GG}^2 + \frac{1}{{\varepsilon}^2} \int_0^T e^{2 \sigma t / {\varepsilon}^2} \| f(t) \|_{\GG^{\bullet}}^2 \mathrm{d} t,$$
$$|\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_{\FFF}^2 := \sup_{0 \leqslant t < T } \left\{ w(t)^2 \| g(t) \|_{\HH}^2 + \frac{w(t)^2}{{\varepsilon}^2} \int_0^t \| g(\tau) \|_{\HH^{\bullet}}^2 \mathrm{d} \tau \right\}.$$
\step{1}{Proof of \eqref{eq:bilinear_kin} for $f \in \GGG$}
On the one hand, if $g \in \GGG$, combining the estimate \eqref{eq:pre_estimate_kin} with the bilinear estimate \eqref{eq:Q_refined_sobolev_negative_algebra_inequality} for $\mathcal Q$, one has:
\begin{equation*}\begin{split}
|\hskip-0.04cm|\hskip-0.04cm| \Psi_\textnormal{kin}^{\varepsilon} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_\GGG^2
& \lesssim {\varepsilon}^2 \int_0^T \left\{ \left( \frac{1}{{\varepsilon}} e^{\sigma t / {\varepsilon}^2} \| f(t) \|_{\GG^{\bullet}}\right)^2 \| g(t) \|_{ \GG }^2 + \| f(t) \|_{ \GG }^2 \left( \frac{1}{{\varepsilon}} e^{\sigma t / {\varepsilon}^2} \| g(t) \|_{\GG^{\bullet}}\right)^2 \right\} \mathrm{d} t \\
& \lesssim {\varepsilon}^2 |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{\GGG}^2 |\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_{\GGG}^2,
\end{split}\end{equation*}
which is ~\eqref{eq:bilinear_kin_kin_kin}. Similarly, when $g \in \FFF$, we have
\begin{equation*}\begin{split}
|\hskip-0.04cm|\hskip-0.04cm| \Psi_\textnormal{kin}^{\varepsilon} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_\GGG^2
&\lesssim {\varepsilon}^2 \int_0^T \left\{ \left( \frac{1}{{\varepsilon}} e^{\sigma t / {\varepsilon}^2} \| f(t) \|_{\GG^{\bullet}}\right)^2 \| g(t) \|_{ \HH }^2 + \left( e^{\sigma t / {\varepsilon}^2} \| f(t) \|_{ \GG } \right)^2 \left( \frac{1}{{\varepsilon}} \| g(t) \|_{\HH^{\bullet}}\right)^2 \right\} \mathrm{d} t \\
&\lesssim {\varepsilon}^2 w(T)^{-2} |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{\GGG}^2 |\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_{\HHH}^2\,,
\end{split}\end{equation*}
where we used \eqref{eq:bound_w}. This proves \eqref{eq:bilinear_kin_kin_mix} for $g \in \HH^{\bullet}$. On the other hand, if $g \in \HHH$, using furthermore $\GG^{\bullet} \hookrightarrow \GG$ and $\HH^{\bullet} \hookrightarrow \GG^{\bullet}$, we have
$$
|\hskip-0.04cm|\hskip-0.04cm| \Psi_\textnormal{kin}^{\varepsilon} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_\GGG^2
\lesssim {\varepsilon}^2\int_0^T \left( \frac{1}{{\varepsilon}} e^{\sigma t / {\varepsilon}^2} \| f(t) \|_{\GG^{\bullet}}\right)^2 \| g(t) \|_{ \HH^{\bullet} }^2 \mathrm{d} t \lesssim {\varepsilon}^2 w(T)^{-2} |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{\GGG}^2 |\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_{\HHH}^2
$$ which gives \eqref{eq:bilinear_kin_kin_mix}.
\step{2}{Proof of \eqref{eq:bilinear_mix_mix_mix} for $f \in \FFF$}
In the case $f, g \in \FFF$, combining the estimate \eqref{eq:pre_estimate_mix} with the bilinear estimate \eqref{eq:Q_refined_sobolev_negative_algebra_inequality}, and using the bound \eqref{eq:bound_w} for $w$, we have
\begin{align*}
|\hskip-0.04cm|\hskip-0.04cm| & \Psi_\textnormal{kin}^{\varepsilon} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_\FFF^2 \\
& \lesssim {\varepsilon}^2 \int_0^T \left\{ \left(\frac{1}{{\varepsilon}} \| f(t) \|_{\HH^{\bullet}}\right)^2 \Big[ w(t) \|g(t) \|_{ \HH }\Big]^2 + \Big[w(t)\| f(t) \|_{ \HH }\Big]^2 \left(\frac{1}{{\varepsilon}} \| g(t) \|_{\HH^{\bullet}}\right)^2 \right\} \mathrm{d} t
\end{align*}
which readily gives \eqref{eq:bilinear_mix_mix_mix} for $f,g\in \FFF.$
In the case $f \in \FFF$ and $g \in \HHH$, using furthermore $\HH^{\bullet} \hookrightarrow \HH$, we have
$$
|\hskip-0.04cm|\hskip-0.04cm| \Psi_\textnormal{kin}^{\varepsilon} [f, g] |\hskip-0.04cm|\hskip-0.04cm|_\FFF^2 \lesssim {\varepsilon}^2 \int_0^T \left(\frac{1}{{\varepsilon}} \| f(t) \|_{\HH^{\bullet}}\right)^2 \| w(t) g(t) \|_{ \HH^{\bullet} }^2 \mathrm{d} t
\lesssim {\varepsilon}^2 w(T)^{-2} |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{\FFF}^2 |\hskip-0.04cm|\hskip-0.04cm| g |\hskip-0.04cm|\hskip-0.04cm|_{\HHH}^2.$$
This shows \eqref{eq:bilinear_mix_mix_mix}.
Similarly, using the nonlinear estimate~\eqref{eq:Q_refined_sobolev_negative_algebra_inequality_same} for $\mathcal Q$, denoting for compactness ${\boldsymbol{\alpha}} = {1-\alpha}$, we have
$$
|\hskip-0.04cm|\hskip-0.04cm| \Psi_\textnormal{kin}^{\varepsilon} [f, g]|\hskip-0.04cm|\hskip-0.04cm|_\FFF^2
\lesssim \int_0^T \| w(t) f(t) \|_{\HH^{\bullet}}^2 \left\| |\nabla_x|^{\boldsymbol{\alpha}} g(t) \right\|_{ \rSSsp{s-{\boldsymbol{\alpha}}} }^2 \mathrm{d} t \\
\lesssim w(T)^{-2} \| f \|_{\HHH}^2 \| g \|_{\HHH}^2.
$$
This proves \eqref{eq:bilinear_mix_hyd_hyd} and concludes the proof.
\end{proof}
This next proposition is proved as Proposition \ref{prop:special_bilinear_hydrodynamic} and its proof is omitted.
\begin{prop}[\textit{\textbf{Special bilinear mixed estimates}}]
\label{prop:special_bilinear_mix}
When $f \in \HHH$ and $\phi$ is the parameter defining the $\HHH$-norm
\begin{equation}
\label{eq:bilinear_mix_phi}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin} [f, \phi] |\hskip-0.04cm|\hskip-0.04cm|_\FFF \lesssim \eta |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_\HHH,
\end{equation}
furthermore, when $g^{\varepsilon}_\textnormal{disp} = U^{\varepsilon}_\textnormal{disp}(\cdot) g$ where $g = \mathsf P_{\textnormal{disp}} g \in \rSSs{s} \cap \dot{\mathbb{H}}^{-\alpha}_x \left( H_v \right)$, there holds
\begin{equation}
\label{eq:bilinear_mix_disp}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin} [f, g^{\varepsilon}_\textnormal{disp}] |\hskip-0.04cm|\hskip-0.04cm|_\FFF \lesssim {\beta_{\textnormal{disp}}}(g , {\varepsilon}) |\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH }, \qquad \lim_{{\varepsilon}\to0} {\beta_{\textnormal{disp}}}(g, {\varepsilon})=0,
\end{equation}
and in the case $d \geqslant 3$, the rate of convergence is explicit assuming $g \in \dot{\mathbb{B}}^{s' + (d+1)/2}_{1, 1} \left( H^{\circ}_v \right) \cap \rSSsm{s}$ for some $s'>s$:
\begin{equation*}
{\beta_{\textnormal{disp}}}(g, {\varepsilon}) \lesssim \sqrt{{\varepsilon}} \left( \| g \|_{ \rSSs{s} } + \| g \|_{ \dot{\mathbb{B}}^{s' + (d+1)/2 }_{1, 1} \left( H_v \right) }\right).
\end{equation*}
\end{prop}
\section{Proof of Theorems \ref{thm:hydrodynamic_limit} and \ref{thm:hydrodynamic_limit-gen_symmetric}}
\label{scn:proof_hydrodynamic_limit_symmetrizable}
In this section, we denote $X = H$ under the sole assumptions \ref{L1}--\ref{L4} and \ref{Bortho}--\ref{Bbound}, and $X$ is the space from assumptions \ref{LE} and \ref{BE} under these extra assumptions.
In this section, we construct a solution of the perturbed equation and then show it must be unique. We follow the approach described in Section \ref{sec:detail}. We refer more specifically to Section \ref{sec:detail-sum} that we briefly resume here. Recall that we look for a solution of the form
\begin{equation}\label{eq:ansatz}\begin{split}
f^{\varepsilon}(t)&=f^{\varepsilon}_\textnormal{kin}(t)+f^{\varepsilon}_\textnormal{mix}(t)+f^{\varepsilon}_\textnormal{hydro}(t)\\
&=f^{\varepsilon}_\textnormal{kin}(t)+f^{\varepsilon}_\textnormal{mix}(t)+f^{\varepsilon}_\textnormal{disp}(t)+ f_\textnormal{NS}(t)+g^{\varepsilon}(t)
\end{split}
\end{equation}
where $f_\textnormal{NS}(\cdot)$ as well as $f_\textnormal{in}$ (and thus $f^{\varepsilon}_\textnormal{disp}(t) = U^{\varepsilon}_\textnormal{disp}(t) f_\textnormal{in}$) are functions to be considered as fixed parameters since they depend only on the initial datum $f_\textnormal{in}$ (and ${\varepsilon}$). We point out that by Lemma \ref{lem:decay_regularization_kinetic_semigroup}, Lemma \ref{lem:decay_semigroups_hydro} and Lemma \ref{lem:NS_parabolic_space} respectively
$$|\hskip-0.04cm|\hskip-0.04cm| U^{\varepsilon}_\textnormal{kin}(\cdot) f_\textnormal{in} |\hskip-0.04cm|\hskip-0.04cm|_{\GGG} \lesssim 1, \qquad |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_\textnormal{disp} |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} \lesssim 1, \qquad |\hskip-0.04cm|\hskip-0.04cm| f_\textnormal{NS} |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} \lesssim 1,$$
without those quantities being necessarily small. The smallness necessary to our fixed-point argument is coming from Proposition \ref{prop:special_bilinear_hydrodynamic} which yields
$$\| \Psi^{\varepsilon}_\textnormal{hydro}(f_\textnormal{NS}, \cdot) \|_{ \mathscr B\left( \HHH \right) } + \| \Psi^{\varepsilon}_\textnormal{kin}(f_\textnormal{NS}, \cdot) \|_{ \mathscr B\left( \HHH ; \FFF \right) } \lesssim \eta,$$
and from Proposition \ref{prop:special_bilinear_mix} which yields
$$\lim_{{\varepsilon} \to 0} \left( \| \Psi^{\varepsilon}_\textnormal{hydro}(f_\textnormal{disp}^{\varepsilon}, \cdot) \|_{ \mathscr B\left( \HHH \right) } + \| \Psi^{\varepsilon}_\textnormal{kin}(f_\textnormal{disp}^{\varepsilon}, \cdot) \|_{ \mathscr B\left( \HHH ; \FFF \right) } \right) = 0.$$
From now on, we work with the space $\FFF$ and $\HHH$ associated to the solution $f_\textnormal{NS}$ which corresponds, in Eq. \eqref{eq:wphieta}, to the choice of the weight function
$$w_{f_\textnormal{NS},\eta}(t)=\exp\left( \frac{1}{2 \eta^2} \int_0^t\| |\nabla_x|^{1-\alpha} f_\textnormal{NS}(\tau) \|_{\rSSsp{s}}^2 \mathrm{d} \tau \right) \qquad t \ge0,$$
with $\eta >0$ still to be chosen.
We recall that we showed in Section \ref{sec:detail-sum} that solving equation
$$f^{\varepsilon}(t)=U^{\varepsilon}(t)f_\textnormal{in}+\Psi^{\varepsilon}[f^{\varepsilon},f^{\varepsilon}](t)$$
can be reformulated, under the above \emph{ansatz}, into the system of coupled nonlinear equations
\begin{equation}\label{eq:systemKinMixG-1}
\begin{cases}
f^{\varepsilon}_\textnormal{kin}(t) &= U^{\varepsilon}_\textnormal{kin}(t) f_\textnormal{in} + \Psi^{\varepsilon}_\textnormal{kin} \left[ f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{kin} \right](t) + 2 \Psi^{\varepsilon}_\textnormal{kin}\left[ f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{hydro} + f^{\varepsilon}_\textnormal{mix}\right](t),\\
\\
f^{\varepsilon}_\textnormal{mix}(t) &=\Psi^{\varepsilon}_\textnormal{kin}\left[\left(f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp}\right)+ f_\textnormal{mix}^{\varepsilon}+g^{\varepsilon}\,;\,\left(f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp}\right)+ f_\textnormal{mix}^{\varepsilon}+g^{\varepsilon} \right](t),\\
\\
g^{\varepsilon}(t) &= \Phi^{\varepsilon}[ f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{mix} ](t) g^{\varepsilon}(t)+
\Psi^{\varepsilon}_\textnormal{hydro}\left[ g^{\varepsilon}, g^{\varepsilon}\right] + \mathcal S^{\varepsilon}(t),
\end{cases}
\end{equation}
where the source term $\mathcal S^{\varepsilon}(t)$ is defined through \eqref{eq:source}. We construct a solution $\left( f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{mix}, g^{\varepsilon} \right)$ of this system in the space
$ \GGG \times \FFF \times \HHH $ and more specifically, in a product of the following balls for some small radii $ c_2, c_3 >0$:
\begin{equation*}\begin{split}
B_1 := \bigg\{ U^{\varepsilon}_\textnormal{kin}(\cdot) f_\textnormal{in} + \phi \, : \, &|\hskip-0.04cm|\hskip-0.04cm| \phi |\hskip-0.04cm|\hskip-0.04cm|_{\GGG} \leqslant 1 \bigg\}, \qquad
B_2 := \bigg\{ \varphi \in \FFF \, : \, \| \varphi |\hskip-0.04cm|\hskip-0.04cm|_{\FFF} \leqslant c_2 \bigg\}, \\
B_3 &:= \bigg\{ \psi \in \HHH \, : \, \| \psi |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} \leqslant c_3 \bigg\},
\end{split}\end{equation*}
where $\HHH=\HHH(T, f_\textnormal{NS}, \eta)$ with $T$ being the lifespan of $f_\textnormal{NS}$. To do so, we reformulate the system \eqref{eq:systemKinMixG-1} as a fixed point problem of the type
$$(f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{mix}, g^{\varepsilon})=\bm{\Xi}\left[f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{mix}, g^{\varepsilon}\right]$$
where the mapping $\bm{\Xi}=\left( \Xi_1, \Xi_2, \Xi_3 \right)$
\begin{equation*}
\bm{\Xi} \, : \, B_1\times B_2 \times B_3 \longrightarrow B_1\times B_2\times B_3
\end{equation*}
is defined through its components:
\begin{equation}\label{eq:systemXIi}\begin{cases}
\Xi_1\left[\phi, \varphi, \psi \right] &= U^{\varepsilon}_\textnormal{kin}(\cdot) f_\textnormal{in} + \Psi^{\varepsilon}_\textnormal{kin}\left[\phi, \phi\right] +2 \Psi^{\varepsilon}_\textnormal{kin}\left[\phi, f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp}\right]+ 2 \Psi^{\varepsilon}_\textnormal{kin}\left[\phi, \psi + \varphi\right],\\
\\
\Xi_2\left[\phi, \varphi, \psi\right] &= \Psi^{\varepsilon}_\textnormal{kin}\left[\left(f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp}\right) + \varphi + \psi , \left(f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp}\right) + \varphi + \psi \right] ,\\
\\
\Xi_3\left[\phi, \varphi, \psi\right] &= \Phi^{\varepsilon}[\phi,\varphi]\psi + \Psi^{\varepsilon}_\textnormal{hydro}\left[\psi , \psi\right] + \mathcal S^{\varepsilon}[\phi, \varphi]\,,
\end{cases}\end{equation}
for any
$$(\phi,\varphi,\psi) \in \bm{B} := B_1\times B_2 \times B_3 \subset \GGG \times \FFF \times \HHH , $$
where we recall that $f_\textnormal{NS}$ and $f_\textnormal{disp}^{\varepsilon}$ are \emph{fixed} parameters for this problem and thus, with a slight abuse of notation, the source term $\mathcal S^{\varepsilon}[\phi,\varphi](t)$ writes
$$\mathcal S^{\varepsilon}[\phi,\varphi](t)=\mathcal S_1^{\varepsilon}(t)+\mathcal S^{\varepsilon}_2(t)+\mathcal S^{\varepsilon}_3[\phi,\varphi](t)$$
where we recall that $\mathcal S_1^{\varepsilon}(t)$ depends only on $f_\textnormal{NS}(t)$ and $f_\textnormal{in}$ while $\mathcal S_2^{\varepsilon}(t)$ depends only on $f_\textnormal{disp}^{\varepsilon}$ and $f_\textnormal{NS}$. We also defined, for $(\phi,\varphi) \in \GGG \times \FFF$, the linear operator $\Phi^{\varepsilon}[\phi,\varphi]$ on $\HHH$ as
$$\Phi^{\varepsilon}\left[\phi,\varphi\right]h=2\Psi_\textnormal{hydro}^{\varepsilon}\left[h,\left(f_\textnormal{NS}+f_\textnormal{disp}^{\varepsilon}\right)+\varphi+\phi\right], \qquad \quad \forall h\in \HHH.$$
\subsection{Linear estimates and source terms estimates}
The source term $\mathcal S^{\varepsilon}$ and the linear terms involved in the above system \eqref{eq:systemKinMixG-1} can be estimated with a simple use of the results of Section \ref{sec:Bilin}. In particular, we recall that the norm $\HHH$ depends on a parameter $\eta >0$ which can be chosen freely.
\begin{prop}[\textit{\textbf{Linear hydrodynamic estimate}}]
\label{prop:linear_hydrodynamic}
With the notation ${\beta_{\textnormal{disp}}}$ of Proposition~\ref{prop:special_bilinear_hydrodynamic} and assuming that $f_\textnormal{in} \in \GG$ and $f^{\varepsilon}_\textnormal{disp} = U^{\varepsilon}_\textnormal{disp} f_\textnormal{in}$ are given, the following continuity estimate holds in $\HHH=\HHH(T, f_\textnormal{NS}, \eta)$:
\begin{align*}
\big\| \Psi^{\varepsilon}_\textnormal{hydro}\big[ h, & f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp} + g_\textnormal{mix} + g_\textnormal{kin}\big] \big\|_{ \HHH } \\
& \lesssim \left(\eta + {\beta_{\textnormal{disp}}}(\mathsf P_\textnormal{disp} f_\textnormal{in}, {\varepsilon}) + {\varepsilon} w_{f_\textnormal{NS}, \eta}(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| g_\textnormal{mix} |\hskip-0.04cm|\hskip-0.04cm|_{\FFF} + {\varepsilon} |\hskip-0.04cm|\hskip-0.04cm| g_\textnormal{kin} |\hskip-0.04cm|\hskip-0.04cm|_\GGG\right) |\hskip-0.04cm|\hskip-0.04cm| h |\hskip-0.04cm|\hskip-0.04cm|_\HHH ,
\end{align*}
as well as the following stability estimate:
\begin{align*}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro}[ h, f_\textnormal{NS} & + f^{\varepsilon}_\textnormal{disp} + g _\textnormal{mix} + g _\textnormal{kin}] - \Psi^{\varepsilon}_\textnormal{hydro}\left[ h', f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp} + g_\textnormal{mix} '+ g_\textnormal{kin}'\right] |\hskip-0.04cm|\hskip-0.04cm|_\HHH \\
\lesssim & |\hskip-0.04cm|\hskip-0.04cm| h - h' |\hskip-0.04cm|\hskip-0.04cm|_\HHH \left( \eta + \beta_{g}({\varepsilon}) + {\varepsilon} w_{f_\textnormal{NS}, \eta}(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| g_\textnormal{mix} |\hskip-0.04cm|\hskip-0.04cm|_{\FFF} + {\varepsilon} |\hskip-0.04cm|\hskip-0.04cm| g_\textnormal{kin} |\hskip-0.04cm|\hskip-0.04cm|_\GGG \right) \\
& + |\hskip-0.04cm|\hskip-0.04cm| h |\hskip-0.04cm|\hskip-0.04cm|_\HHH \left( {\varepsilon} w_{f_\textnormal{NS}, \eta}(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| g_\textnormal{mix} - g_\textnormal{mix}' |\hskip-0.04cm|\hskip-0.04cm|_\FFF + {\varepsilon} |\hskip-0.04cm|\hskip-0.04cm| g_\textnormal{kin} - g_\textnormal{kin}' |\hskip-0.04cm|\hskip-0.04cm|_\GGG \right).
\end{align*}
\end{prop}
\begin{proof}
The two estimates are direct consequences of Propositions \ref{prop:bilinear_hydrodynamic} and \ref{prop:special_bilinear_hydrodynamic} since \begin{align*}
\Psi^{\varepsilon}_\textnormal{hydro}\left[ h, f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp} + g _\textnormal{mix} + g _\textnormal{kin}\right] & - \Psi^{\varepsilon}_\textnormal{hydro}\left[ h', f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp} + g_\textnormal{mix} '+ g_\textnormal{kin}'\right]\\
= & \Psi^{\varepsilon}_\textnormal{hydro}\left[h- h', f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp} + g_\textnormal{mix} + g_\textnormal{kin}\right] \\
& + 2 \Psi^{\varepsilon}_\textnormal{hydro}\left[h, g_\textnormal{mix}-g_\textnormal{mix}'\right] + 2 \Psi^{\varepsilon}_\textnormal{hydro}\left[h, g_\textnormal{kin}-g_\textnormal{kin}'\right]\,.
\end{align*}
This proves the result.
\end{proof}
The first part $\mathcal S_1^{\varepsilon}$ of the source term $\mathcal S^{\varepsilon}$ which depends only on the initial data $f_\textnormal{in}$ and the Navier-Stokes solution $f_\textnormal{NS}$ (but not on the partial solutions $f^{\varepsilon}_\textnormal{kin}$, $f^{\varepsilon}_\textnormal{mix}$ or $g^{\varepsilon}$) is estimated in this next lemma.
\begin{lem}[\textit{\textbf{Estimate of the first source term $\mathcal S^{\varepsilon}_1$}}]\label{lem:SourceS1}
Consider some $f_\textnormal{in} \in \GG$. The source term $\mathcal S^{\varepsilon}_1$ satisfies
$$|\hskip-0.04cm|\hskip-0.04cm| \mathcal S^{\varepsilon}_1 |\hskip-0.04cm|\hskip-0.04cm|_\HHH \leqslant \beta_\textnormal{NS}(f_\textnormal{NS}, f_\textnormal{in}, {\varepsilon}), \qquad \lim_{{\varepsilon} \to 0}\beta_\textnormal{NS}(f_\textnormal{NS}, f_\textnormal{in}, {\varepsilon})=0.$$
If we assume additionally that the initial data $f_\textnormal{in}$ lies in $\rSSl{s+\delta} \cap \dot{\mathbb{H}}^{-\alpha}_x \left( X_v \right)$ for some $\delta \in (0, 1]$, then the rate of convergence can be made explicit as
\begin{align*}
\beta_\textnormal{NS}(f_\textnormal{NS}, f_\textnormal{in}, {\varepsilon}) \lesssim {\varepsilon}^{\delta} \Big( & 1 + \| f_\textnormal{in} \|_{\rSSl{s+\delta}} + \| f_\textnormal{in} \|_{ \dot{\mathbb{H}}^{-\alpha}_x \left( X_v \right) } \\
& + \| f_\textnormal{NS} \|_{ L^\infty\left( [0, T) ; \rSSs{s+\delta} \right) } + \| \nabla_x f_\textnormal{NS} \|_{ L^2\left( [0, T) ; \rSSs{s+\delta} \right) } \Big)^3.
\end{align*}
\end{lem}
\begin{proof} Recalling that $U_\textnormal{hydro}^{{\varepsilon}}(t)=U^{{\varepsilon}}_{\textnormal{NS}}(t)+U_{\textnormal{wave}}^{{\varepsilon}}(t)$, we write the source term $\mathcal S^{\varepsilon}_1(t)$ as
\begin{align*}
\mathcal S^{\varepsilon}_1(t) = & \left(U^{\varepsilon}_\textnormal{wave}(t)f_{\textnormal{in}} - U^{\varepsilon}_\textnormal{disp}(t)f_{\textnormal{in}}\right) + \left(U^{\varepsilon}_\textnormal{NS}(t)f_{\textnormal{in}} - U_\textnormal{NS}(t)f_{\textnormal{in}}\right) \\
& + \Psi^{\varepsilon}_\textnormal{wave}\left[f_\textnormal{NS}, f_\textnormal{NS}\right](t) + \left(\Psi^{\varepsilon}_\textnormal{NS}\left[f_\textnormal{NS}, f_\textnormal{NS}\right](t)- \Psi_\textnormal{NS}\left[f_\textnormal{NS}, f_\textnormal{NS}\right](t)\right) .
\end{align*}
Using Lemmas \ref{lem:asymptotic_equiv_oscillating_semigroup} and \ref{lem:asymptotic_equiv_NS_semigroup}, we have for a smooth initial data $f_\textnormal{in}$
\begin{equation*}\begin{split}
|\hskip-0.04cm|\hskip-0.04cm| U^{\varepsilon}_\textnormal{hydro}(\cdot)f_\textnormal{in} &- U_\textnormal{NS}(\cdot)f_\textnormal{in} - U^{\varepsilon}_\textnormal{disp}(\cdot) f_\textnormal{in} |\hskip-0.04cm|\hskip-0.04cm|_\HHH \\
& \leqslant |\hskip-0.04cm|\hskip-0.04cm| U^{\varepsilon}_\textnormal{wave}(\cdot) f_\textnormal{in} - U^{\varepsilon}_\textnormal{disp} f_\textnormal{in} |\hskip-0.04cm|\hskip-0.04cm|_\HHH + |\hskip-0.04cm|\hskip-0.04cm| U^{\varepsilon}_\textnormal{NS}(\cdot) f_\textnormal{in} - U_\textnormal{NS}(\cdot) f_\textnormal{in} |\hskip-0.04cm|\hskip-0.04cm|_\HHH \\
& \lesssim {\varepsilon}^{\delta} \left( \| f_\textnormal{in} \|_{\rSSlm{s+\delta}} + \| f_\textnormal{in} \|_{ \dot{\mathbb{H}}^{-\alpha}_x \left( X^{\circ}_v \right) } \right),
\end{split}\end{equation*}
and in general, by a limiting argument
$$\lim_{{\varepsilon}\to0}|\hskip-0.04cm|\hskip-0.04cm| U^{\varepsilon}_\textnormal{hydro}(\cdot)f_\textnormal{in} - U_\textnormal{NS}(\cdot)f_\textnormal{in} - U^{\varepsilon}_\textnormal{disp}(\cdot) f_\textnormal{in} |\hskip-0.04cm|\hskip-0.04cm|_\HHH=0.$$
Furthermore, using the estimate of Lemma \ref{lem:convolution_wave} with $\varphi = \mathcal Q(f_\textnormal{NS}, f_\textnormal{NS})$, and where we point out that, for $d=2$, we have $0 < \alpha < \frac{1}{2}$ and thus
$$ \frac{4}{3 + 2 \alpha} \in \left(1, \frac{4}{3}\right) , \qquad \frac{2}{1+\alpha} \in \left(\frac{4}{3}, 2\right),$$
whereas, for $d \geq3$, we have $\alpha=0$ and thus
$$\frac{4}{3 + 2 \alpha} = \frac{4}{3}, \qquad \frac{2}{1+\alpha} = 2,$$
we estimate $ \| \varphi(0) \|_{ \dot{\mathbb{H}}_x^{-\alpha} \left( H_v \right) }$ thanks to \eqref{eq:Q_refined_sobolev_negative_algebra_inequality}, and the other ones using using Lemmas \ref{lem:estimates_derivative_Q_navier_stokes}--\ref{lem:NS_parabolic_space} to deduce that
\begin{equation*}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{wave}\left[f_\textnormal{NS}, f_\textnormal{NS}\right] |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} \lesssim {\varepsilon} \Big( 1 + \| f_\textnormal{NS}(0) \|_{ \dot{\mathbb{H}}^{-\alpha}_x \left( H_v \right) } + \| f_\textnormal{NS} \|_{ L^\infty\left( [0, T) ;\rSSs{s} \right) } + \| \nabla_x f_\textnormal{NS} \|_{ L^2\left( [0, T) ; \rSSs{s} \right) }\Big)^3.
\end{equation*}
Finally, one proves as for \eqref{eq:bilinear_hyd_hyd} using this time \eqref{eq:asymptotic_equiv_NS_orthogonal_reg}, that for any $\delta \in [0, 1]$
$$|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{NS}\left[f_\textnormal{NS}, f_\textnormal{NS}\right] - \Psi_\textnormal{NS}\left[f_\textnormal{NS}, f_\textnormal{NS}\right] |\hskip-0.04cm|\hskip-0.04cm|_{ \rSSSs{s} } \lesssim {\varepsilon}^{\delta} |\hskip-0.04cm|\hskip-0.04cm| f_\textnormal{NS} |\hskip-0.04cm|\hskip-0.04cm|_{\rSSSs{s+\delta}}^2,$$
which, on the one hand, implies by Lemma \ref{lem:NS_parabolic_space}
\begin{align*}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{NS} & \left[f_\textnormal{NS}, f_\textnormal{NS}\right] - \Psi_\textnormal{NS}\left[f_\textnormal{NS}, f_\textnormal{NS}\right] |\hskip-0.04cm|\hskip-0.04cm|_{ \rSSSs{s} } \\
& \lesssim {\varepsilon}^{\delta} \Big( \| f_\textnormal{NS}(0) \|_{ \dot{\mathbb{H}}^{-\alpha}_x \left( H_v \right) } + \| f_\textnormal{NS} \|_{ L^\infty\left( [0, T) ;\rSSs{s+\delta} \right) } + \| \nabla_x f_\textnormal{NS} \|_{ L^2\left( [0, T) ; \rSSs{s+\delta} \right) }\Big)^2,
\end{align*}
and on the other hand, since $f_\textnormal{NS}$ can be approximated by elements of $\rSSSs{s+1}$ by Lemma \ref{lem:NS_parabolic_space}, we have in general
$$\lim_{{\varepsilon}\to0}|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{NS}\left[f_\textnormal{NS}, f_\textnormal{NS}\right] - \Psi_\textnormal{NS}\left[f_\textnormal{NS}, f_\textnormal{NS}\right] |\hskip-0.04cm|\hskip-0.04cm|_{ \rSSSs{s} }=0.$$
This concludes the proof.
\end{proof}
\begin{prop}[\textit{\textbf{Estimate for the source term $\mathcal S^{\varepsilon}$}}]
\label{prop:source_term}
Consider some $f_\textnormal{in} \in \GG$ and denote $f^{\varepsilon}_\textnormal{disp} = U^{\varepsilon}_\textnormal{disp}(\cdot) f_\textnormal{in}$, the source term~$\mathcal S^{\varepsilon}$ satisfies in $\rSSSs{s} = \rSSSs{s}(T, f_\textnormal{NS}, \eta)$
\begin{equation*}\begin{split}
|\hskip-0.04cm|\hskip-0.04cm| \mathcal S^{\varepsilon}[ g_\textnormal{kin}, g_\textnormal{mix}] |\hskip-0.04cm|\hskip-0.04cm|_\HHH \lesssim & {\beta_{\textnormal{disp}}}(\mathsf P_\textnormal{disp} f_\textnormal{in}, {\varepsilon}) \left(|\hskip-0.04cm|\hskip-0.04cm| f_\textnormal{NS} |\hskip-0.04cm|\hskip-0.04cm|_\HHH + \| g \|_{\HH}\right) + \beta_\textnormal{NS}(f_\textnormal{NS}, f_\textnormal{in}, {\varepsilon}) \\
& + {\varepsilon} w_{f_\textnormal{NS}, \eta}(T)^{-1} \left( |\hskip-0.04cm|\hskip-0.04cm| g_\textnormal{kin} |\hskip-0.04cm|\hskip-0.04cm|_\GGG + |\hskip-0.04cm|\hskip-0.04cm| g_\textnormal{mix} |\hskip-0.04cm|\hskip-0.04cm|_\FFF \right) \\
& \quad \times \left( |\hskip-0.04cm|\hskip-0.04cm| g_\textnormal{kin} |\hskip-0.04cm|\hskip-0.04cm|_\GGG + |\hskip-0.04cm|\hskip-0.04cm| g_\textnormal{mix} |\hskip-0.04cm|\hskip-0.04cm|_\FFF + |\hskip-0.04cm|\hskip-0.04cm| f_\textnormal{NS} |\hskip-0.04cm|\hskip-0.04cm|_\HHH + \| \mathsf P_\textnormal{disp} f_\textnormal{in} \|_{ \HH } + \| \mathsf P_\textnormal{disp} f_\textnormal{in} \|_{ \dot{\mathbb{H}}^{-\alpha}_x \left( H_v \right) }\right),
\end{split}\end{equation*}
where there holds
$$\displaystyle \lim_{{\varepsilon}\to0} {\beta_{\textnormal{disp}}}( \mathsf P_\textnormal{disp} f_\textnormal{in}, {\varepsilon}) =\lim_{{\varepsilon}\to0}\beta_\textnormal{NS}(f_\textnormal{NS}, f_\textnormal{in}, {\varepsilon})=0.$$
The rate of convergence of the term~$\beta_\textnormal{NS}(f_\textnormal{NS}, f_\textnormal{in}, {\varepsilon})$ can be made explicit if the initial data $f_\textnormal{in}$ lies in $\rSSl{s+\delta} \cap \dot{\mathbb{H}}^{-\alpha}_x \left( X_v \right)$ for some $\delta \in (0, 1]$:
\begin{align*}
\beta_\textnormal{NS}(f_\textnormal{NS}, f_\textnormal{in}, {\varepsilon}) \lesssim {\varepsilon}^{\delta} \Big( & 1 + \| f_\textnormal{in} \|_{\rSSl{s+\delta}} + \| f_\textnormal{in} \|_{ \dot{\mathbb{H}}^{-\alpha}_x \left( X_v \right) } \\
& + \| f_\textnormal{NS} \|_{ L^\infty\left( [0, T) ; \rSSs{s+\delta} \right) } + \| \nabla_x f_\textnormal{NS} \|_{ L^2\left( [0, T) ; \rSSs{s+\delta} \right) } \Big)^3.
\end{align*}
and if $d \geqslant 3$, the rate of convergence of $ {\beta_{\textnormal{disp}}}(f_\textnormal{in}, {\varepsilon})$ is explicit if $\mathsf P_\textnormal{disp} f_\textnormal{in} \in \dot{\mathbb{B}}^{s' + (d+1)/2}_{1, 1} \left( H_v \right) \cap \rSSs{s}$ for some $s'>s$:
\begin{equation*}
{\beta_{\textnormal{disp}}}(f_\textnormal{in}, {\varepsilon}) \lesssim \sqrt{{\varepsilon}} \left( \| \mathsf P_\textnormal{disp} f_\textnormal{in} \|_{ \rSSs{s} } + \| \mathsf P_\textnormal{disp} f_\textnormal{in} \|_{ \dot{\mathbb{B}}^{s' + (d+1)/2 }_{1, 1} \left( H_v \right) }\right).
\end{equation*}
Furthermore, the source term $\mathcal S^{\varepsilon}$ satisfies the stability estimate
\begin{align*}
|\hskip-0.04cm|\hskip-0.04cm| \mathcal S^{\varepsilon}_3[g_\textnormal{kin}, & g_\textnormal{mix}] - \mathcal S^{\varepsilon}_3[g_\textnormal{kin}', g_\textnormal{mix}'] |\hskip-0.04cm|\hskip-0.04cm|_\HHH \\
\lesssim {\varepsilon} & w_{f_\textnormal{NS}, \eta}(T)^{-1} \left( |\hskip-0.04cm|\hskip-0.04cm| g_\textnormal{kin} - g_\textnormal{kin}' |\hskip-0.04cm|\hskip-0.04cm|_\GGG + |\hskip-0.04cm|\hskip-0.04cm| g_\textnormal{mix} - g_\textnormal{mix}' |\hskip-0.04cm|\hskip-0.04cm|_\FFF \right) \\
& \times \left( |\hskip-0.04cm|\hskip-0.04cm| g_\textnormal{kin} + g_\textnormal{kin}' |\hskip-0.04cm|\hskip-0.04cm|_\GGG + |\hskip-0.04cm|\hskip-0.04cm| g_\textnormal{mix} + g_\textnormal{mix}' |\hskip-0.04cm|\hskip-0.04cm|_\FFF + |\hskip-0.04cm|\hskip-0.04cm| f_\textnormal{NS} |\hskip-0.04cm|\hskip-0.04cm|_\HHH + \| \mathsf P_\textnormal{disp} f_\textnormal{in} \|_{\HH} + \| \mathsf P_\textnormal{disp} f_\textnormal{in} \|_{ \dot{\mathbb{H}}^{-\alpha}_x \left( H_v \right) } \right)\,.
\end{align*}
\end{prop}
\begin{rem}
Note that the terms $\beta_{g}$ and $\beta_\textnormal{NS}$ depend respectively on $g$, which stands for~$\mathsf P_\textnormal{disp} f_\textnormal{in}$, and $f_\textnormal{NS}$ which are to be considered as fixed data of the problem, thus the lack of uniform estimate for their convergence is not an issue for the iterative scheme.
\end{rem}
\begin{proof}
Recalling the definition of $\mathcal S^{\varepsilon}_2$:
$$
\mathcal S^{\varepsilon}_2 = \Psi^{\varepsilon}_\textnormal{hydro}\left[ g^{\varepsilon}_\textnormal{disp}, 2 f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp} \right]
$$
we easily have thanks to Eq. \eqref{eq:bilinear_hyd_disp} in Proposition \ref{prop:special_bilinear_hydrodynamic} and Lemma \ref{lem:decay_semigroups_hydro}
$$|\hskip-0.04cm|\hskip-0.04cm| \mathcal S^{\varepsilon}_2 |\hskip-0.04cm|\hskip-0.04cm|_\HHH \lesssim \beta_\textnormal{disp}( \mathsf P_\textnormal{disp} f_\textnormal{in}, {\varepsilon}) |\hskip-0.04cm|\hskip-0.04cm| 2f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp}|\hskip-0.04cm|\hskip-0.04cm|_\HHH \lesssim \beta_\textnormal{disp}( \mathsf P_\textnormal{disp} f_\textnormal{in}, {\varepsilon}) \left(|\hskip-0.04cm|\hskip-0.04cm| f_\textnormal{NS} |\hskip-0.04cm|\hskip-0.04cm|_\HHH + \| g \|_{\HH}\right).$$
Furthermore, recalling the definition of $\mathcal S^{\varepsilon}_3$:
$$
\mathcal S^{\varepsilon}_3[g_\textnormal{kin}, g_\textnormal{mix}] = \Psi^{\varepsilon}_\textnormal{hydro}\left[ g_\textnormal{kin} + g_\textnormal{mix} , g_\textnormal{kin} + g_\textnormal{mix} + f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp} \right],
$$
we easily have thanks to the various estimates of Proposition \ref{prop:bilinear_hydrodynamic} and Lemma \ref{lem:decay_semigroups_hydro} together with the bilinearity of $\Psi^{\varepsilon}_\textnormal{hydro}$
\begin{align*}
|\hskip-0.04cm|\hskip-0.04cm| \mathcal S^{\varepsilon}_3[g_\textnormal{kin}, g_\textnormal{mix}] |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH}
\lesssim {\varepsilon} & w_{\phi, \eta}(T)^{-1} \left( |\hskip-0.04cm|\hskip-0.04cm| g_\textnormal{kin} |\hskip-0.04cm|\hskip-0.04cm|_\GGG + |\hskip-0.04cm|\hskip-0.04cm| g_\textnormal{mix} |\hskip-0.04cm|\hskip-0.04cm|_\FFF \right) \\
& \times \left( |\hskip-0.04cm|\hskip-0.04cm| g_\textnormal{kin} |\hskip-0.04cm|\hskip-0.04cm|_\GGG + |\hskip-0.04cm|\hskip-0.04cm| g_\textnormal{mix} |\hskip-0.04cm|\hskip-0.04cm|_\FFF + |\hskip-0.04cm|\hskip-0.04cm| f_\textnormal{NS} |\hskip-0.04cm|\hskip-0.04cm|_\HHH + \| \mathsf P_\textnormal{disp} f_\textnormal{in} \|_{ \HH } + \| \mathsf P_\textnormal{disp} f_\textnormal{in} \|_{ \dot{\mathbb{H}}^{-\alpha}_x \left( H_v \right) } \right).
\end{align*}
The stability estimate comes from the identity
\begin{align*}
\mathcal S^{\varepsilon}_3[g_\textnormal{kin}, g_\textnormal{mix}] - & \mathcal S^{\varepsilon}_3[g_\textnormal{kin}', g_\textnormal{mix}'] \\
= & \Psi^{\varepsilon}_\textnormal{hydro}\left[ g_\textnormal{kin} - g_\textnormal{kin}' + g_\textnormal{mix} - g_\textnormal{mix}' , g_\textnormal{kin} + g_\textnormal{kin}' + g_\textnormal{mix} + g_\textnormal{mix}' \right] \\
& + \Psi^{\varepsilon}_\textnormal{hydro}\left[ g_\textnormal{kin} - g_\textnormal{kin}' + g_\textnormal{mix} - g_\textnormal{mix}' , f_\textnormal{NS} + f_\textnormal{disp}^{\varepsilon} \right]
\end{align*}
which we control using the same estimates. This concludes the proof.
\end{proof}
\subsection{The mapping is a contraction}
\label{scn:mapping_contraction}
In what follows, we will simplify some estimates by using the fact that
$$c_2, c_3, \eta, {\varepsilon} \lesssim 1, \qquad 1 \leqslant w_{f_\textnormal{NS}, \eta}(T)^{-1}$$
and that $\beta({\varepsilon}) = {\beta_{\textnormal{disp}}}(\mathsf P_\textnormal{disp} f_\textnormal{in}, {\varepsilon}) + \beta_{\textnormal{NS}}(f_\textnormal{NS}, f_\textnormal{in}, {\varepsilon})$ (see Proposition \ref{prop:source_term}) can be assumed to vanish at a slower rate that ${\varepsilon}$:
$${\varepsilon} \lesssim \beta({\varepsilon}).$$
To prove the existence and uniqueness of a fixed point for $\bm{\Xi}$, we need to check that $\bm{\Xi}$ is a a contraction on $\bm{B}$. We begin by showing that $\bm{B}$ is stable under the action of $\bm{\Xi}$ under suitable smallness assumption on ${\varepsilon},c_3,\eta,c_2$:
\begin{lem} For a suitable choice of
$${\varepsilon} \ll c_3 \ll \eta \ll c_2 \ll 1$$
the mapping $\bm{\Xi}$ is well-defined on $\bm{B}$ and
$\bm{\Xi}(\bm{B}) \subset \bm{B}.$
\end{lem}
\begin{proof} Let us check that the first component $\Xi_1$ is well defined and take values in $B_1$. We assume $(\phi,\varphi,\psi) \in \bm{B}$
\begin{align*}
|\hskip-0.04cm|\hskip-0.04cm| \Xi_1\left[\phi, \varphi, \psi\right] - U^{\varepsilon}_\textnormal{kin}(\cdot) f_\textnormal{in} |\hskip-0.04cm|\hskip-0.04cm|_{\GGG} \leqslant & |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[\phi, \phi\right] |\hskip-0.04cm|\hskip-0.04cm|_{\GGG} + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[\phi, f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp} \right] |\hskip-0.04cm|\hskip-0.04cm|_{\GGG} \\
& + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[\phi, \psi\right] |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG } + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[\phi, \varphi\right] |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG }.
\end{align*}
Using \eqref{eq:bilinear_kin}, we have the estimates
$$|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[ \phi, \phi\right] |\hskip-0.04cm|\hskip-0.04cm|_{\GGG} \lesssim {\varepsilon} |\hskip-0.04cm|\hskip-0.04cm| \phi|\hskip-0.04cm|\hskip-0.04cm|_{\GGG}^2 \lesssim {\varepsilon},$$
\begin{align*}
2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin} \left[ \phi, \varphi\right] |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG } & \lesssim {\varepsilon} w_{f_\textnormal{NS}, \eta }(T)^{-1}|\hskip-0.04cm|\hskip-0.04cm| \phi|\hskip-0.04cm|\hskip-0.04cm|_{\GGG}\,|\hskip-0.04cm|\hskip-0.04cm| \psi|\hskip-0.04cm|\hskip-0.04cm|_{\FFF} \lesssim {\varepsilon} w_{f_\textnormal{NS}, \eta }(T)^{-1},
\end{align*}
as well as
\begin{equation*} \begin{split}
2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[\phi, f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp} \right] |\hskip-0.04cm|\hskip-0.04cm|_{\GGG} &+ 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[\phi, \psi\right] |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG }\\
&\lesssim {\varepsilon} w_{f_\textnormal{NS},\eta}(T)^{-1}|\hskip-0.04cm|\hskip-0.04cm| \phi|\hskip-0.04cm|\hskip-0.04cm|_{\GGG}\left(|\hskip-0.04cm|\hskip-0.04cm| f_\textnormal{NS} +f^{\varepsilon}_\textnormal{disp} |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} + |\hskip-0.04cm|\hskip-0.04cm| \psi |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} \right)
\\
&\lesssim {\varepsilon} w_{f_\textnormal{NS}, \eta}(T)^{-1}\,.
\end{split}\end{equation*}
Consequently
$$|\hskip-0.04cm|\hskip-0.04cm| \Xi_1\left[\phi, \varphi, \psi\right] - U^{\varepsilon}_\textnormal{kin}(\cdot) f_\textnormal{in} |\hskip-0.04cm|\hskip-0.04cm|_{\GGG} \lesssim {\varepsilon} w_{f_\textnormal{NS}, \eta}(T)^{-1},$$
thus, considering ${\varepsilon} \ll \eta$, we conclude that $\Xi_1[\phi,\varphi,\psi] \in B_1$.
The second component $\Xi_2$ is also well-defined. We have
\begin{equation*}\begin{split}
|\hskip-0.04cm|\hskip-0.04cm| \Xi_2\big[\phi, &\varphi, \psi\big] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } \\
&\leqslant |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[ \varphi , \varphi \right] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } + |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[ f_\textnormal{NS} , f_\textnormal{NS} \right] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } + |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[f^{\varepsilon}_\textnormal{disp} , f^{\varepsilon}_\textnormal{disp}\right] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } + |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[ \psi , \psi \right] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } \\
& + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[ \varphi , f_\textnormal{NS} \right] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[ \varphi , f_\textnormal{disp}^{\varepsilon} \right] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[ \varphi , \psi \right] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } \\
& + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[ f_\textnormal{NS} , f^{\varepsilon}_\textnormal{disp} \right] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[ f_\textnormal{NS} , \psi \right] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[ f^{\varepsilon}_\textnormal{disp} , \psi \right] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF }\,.
\end{split}\end{equation*}
Using \eqref{eq:bilinear_mix_mix_mix}, \eqref{eq:bilinear_mix_phi}, \eqref{eq:bilinear_mix_disp} and \eqref{eq:bilinear_mix_hyd_hyd} respectively, we have
\begin{equation*}\begin{split}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[ \varphi , \varphi \right] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF }& + |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[ f_\textnormal{NS} , f_\textnormal{NS} \right] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } + |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[f^{\varepsilon}_\textnormal{disp} , f^{\varepsilon}_\textnormal{disp}\right] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } + |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[ \psi , \psi \right] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } \\
&\lesssim {\varepsilon} w_{f_\textnormal{NS}, \eta}(T)^{-1}|\hskip-0.04cm|\hskip-0.04cm| \varphi|\hskip-0.04cm|\hskip-0.04cm|_{\FFF}^2 + \eta |\hskip-0.04cm|\hskip-0.04cm| f_\textnormal{NS} |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} +\beta({\varepsilon})|\hskip-0.04cm|\hskip-0.04cm| f_\textnormal{disp}^{\varepsilon}|\hskip-0.04cm|\hskip-0.04cm|_{\HHH}+w_{f_\textnormal{NS}, \eta}(T)^{-1}|\hskip-0.04cm|\hskip-0.04cm| \psi|\hskip-0.04cm|\hskip-0.04cm|_{\HHH}^2\\
& \lesssim {\varepsilon} c_2^2 w_{f_\textnormal{NS}, \eta}(T)^{-1} + \eta + \beta({\varepsilon}) + c_3^2 w_{f_\textnormal{NS}, \eta}(T)^{-1} \\
&\lesssim \left(\beta({\varepsilon}) + c_3\right) w_{f_\textnormal{NS}, \eta}(T)^{-1} + \eta\,.
\end{split}\end{equation*}
Furthermore, using \eqref{eq:bilinear_mix_mix_mix}, we have
\begin{equation*}\begin{split}
2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[ \varphi , f_\textnormal{NS}\right] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } &+ 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[\varphi , f_\textnormal{disp}^{\varepsilon}\right] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[ \varphi , \psi\right]|\hskip-0.04cm|\hskip-0.04cm|_{ \FFF }\\
&\lesssim {\varepsilon} w_{f_\textnormal{NS}, \eta}(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| \varphi |\hskip-0.04cm|\hskip-0.04cm|_{\FFF}\bigg(|\hskip-0.04cm|\hskip-0.04cm| f_\textnormal{NS}|\hskip-0.04cm|\hskip-0.04cm|_{\HHH}
+ |\hskip-0.04cm|\hskip-0.04cm| f_\textnormal{disp}^{\varepsilon} |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} + |\hskip-0.04cm|\hskip-0.04cm| \psi |\hskip-0.04cm|\hskip-0.04cm|_{\HHH}\bigg)\\
&\lesssim {\varepsilon} c_2 ( 1 + c_3) w_{f_\textnormal{NS}, \eta}(T)^{-1} \lesssim {\varepsilon} w_{f_\textnormal{NS}, \eta}(T)^{-1},
\end{split}\end{equation*}
whereas, \eqref{eq:bilinear_mix_phi} and \eqref{eq:bilinear_mix_disp} give respectively
\begin{align*}
2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[ f_\textnormal{NS} , f^{\varepsilon}_\textnormal{disp} \right] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[ f_\textnormal{NS} , \psi \right] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } & \lesssim \eta\left(|\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_\textnormal{disp}|\hskip-0.04cm|\hskip-0.04cm|_{\HHH} + |\hskip-0.04cm|\hskip-0.04cm| \psi |\hskip-0.04cm|\hskip-0.04cm|_{\HHH}\right) \\
&\lesssim\eta (1+c_3) \lesssim \eta,
\end{align*}
and
\begin{align*}
2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}\left[ f^{\varepsilon}_\textnormal{disp} , \psi \right] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } \lesssim \beta({\varepsilon})|\hskip-0.04cm|\hskip-0.04cm| \psi|\hskip-0.04cm|\hskip-0.04cm|_{\HHH}
\lesssim \beta({\varepsilon}) c_3 \lesssim \beta({\varepsilon}).
\end{align*}
Gathering these estimates yield
$$|\hskip-0.04cm|\hskip-0.04cm| \Xi_2\left[\phi, \varphi, \psi\right] |\hskip-0.04cm|\hskip-0.04cm|_{\GGG} \lesssim \left(\beta({\varepsilon}) + c_3\right) w_{f_\textnormal{NS}, \eta}(T)^{-1} + \eta.$$
We deduce for $\max\{ {\varepsilon} , c_3 \} \ll \eta \ll c_2$ that $\Xi_2$ takes value in $B_2$.
Finally, the third component $\Xi_3$ is well defined. We have
$$|\hskip-0.04cm|\hskip-0.04cm| \Xi_3\left[\phi, \varphi, \psi\right]|\hskip-0.04cm|\hskip-0.04cm|_{\HHH} \leqslant |\hskip-0.04cm|\hskip-0.04cm| \Phi^{\varepsilon}[ \phi , \varphi ] \psi |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} + |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro} [\psi , \psi] |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} + |\hskip-0.04cm|\hskip-0.04cm| \mathcal S^{\varepsilon}[\phi, \varphi] |\hskip-0.04cm|\hskip-0.04cm|_{\HHH}.$$
By Proposition \ref{prop:linear_hydrodynamic}
\begin{align*}
|\hskip-0.04cm|\hskip-0.04cm| \Phi^{\varepsilon}[ \phi , \varphi ] \psi |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} & \lesssim \left(\eta + \beta({\varepsilon}) + {\varepsilon} c_2 w_{f_\textnormal{NS}, \eta}(T)^{-1} + {\varepsilon} \right) c_3 \lesssim \beta({\varepsilon}) w_{f_\textnormal{NS}, \eta}(T)^{-1} + \eta c_3,
\end{align*}
while, from \eqref{eq:bilinear_hyd_hyd} and Proposition \ref{prop:source_term}
$$
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro} [\psi, \psi] |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} \lesssim c_3^2 w_{f_\textnormal{NS}, \eta}(T)^{-1}, \qquad |\hskip-0.04cm|\hskip-0.04cm| \mathcal S^{\varepsilon}[\phi, \varphi] |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH } \lesssim \beta({\varepsilon}).$$
All these estimates gathered together give
\begin{align*}
|\hskip-0.04cm|\hskip-0.04cm| \Xi_3[\phi, \varphi, \psi] |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} \lesssim \beta({\varepsilon}) w_{ f_\textnormal{NS}, \eta }(T)^{-1} + \left( \eta + c_3 w_{ f_\textnormal{NS}, \eta }(T)^{-1} \right) c_3
\end{align*}
thus $\Xi_3$ takes value in $B_3$ by taking ${\varepsilon} \ll c_3 \ll \eta \ll 1$.
This completes the proof.\end{proof}
We show now that, up to reducing further the parameters ${\varepsilon},c_2,c_3,\eta$, the mapping $\bm{\Xi}$ is a contraction on $\bm{B}$.
\begin{prop} Under the smallness assumption
$$\max\{ {\varepsilon} , c_3 \} \ll \eta \ll 1\,,$$
the mapping $\bm{\Xi}\::\:\bm{B} \to \bm{B} \subset \GGG\times \FFF \times \HHH$ is a contraction.
\end{prop}
\begin{proof} Let us fix $(\phi,\varphi,\psi) \in \bm{B},$ $(\phi',\varphi',\psi') \in \bm{B}$. We prove that each component of $\bm{\Xi}$ is contractive. We have
\begin{equation*}\begin{split}
|\hskip-0.04cm|\hskip-0.04cm| \Xi_1[\phi, \varphi, \psi] & - \Xi_1[\phi', \varphi', \psi' ] |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG } \\
\leqslant & |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}[\phi - \phi', \phi + \phi'] |\hskip-0.04cm|\hskip-0.04cm|_{\GGG} + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}[\phi - \phi', f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp}] |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG } \\
& + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}[\phi - \phi', \psi] |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG } + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}[\phi', \psi - \psi'] |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG } \\
& + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}[\phi - \phi', \varphi] |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG } + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}[\phi', \varphi - \varphi']|\hskip-0.04cm|\hskip-0.04cm|_{ \GGG }
\end{split}\end{equation*}
As in the previous proof, using \eqref{eq:bilinear_kin} we have
\begin{equation*}\begin{split}
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}[\phi - \phi', \phi + \phi'] |\hskip-0.04cm|\hskip-0.04cm|_{\GGG} & + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}[\phi - \phi', f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp}] |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG } \\
& \lesssim {\varepsilon} \left(1 + w_{f_\textnormal{NS}, \eta}(T)^{-1} \right) |\hskip-0.04cm|\hskip-0.04cm| \phi - \phi' |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG } \\
& \lesssim {\varepsilon} w_{f_\textnormal{NS}, \eta}(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| \phi - \phi' |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG },
\end{split}\end{equation*}
and
\begin{equation*}\begin{split}
2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}[\phi - \phi', \psi] |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG } &+ 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}[\phi', \psi - \psi'] |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG } \\
&\lesssim {\varepsilon} c_3 w_{f_\textnormal{NS}, \eta}(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| \phi - \phi' |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG }
+ {\varepsilon} w_{f_\textnormal{NS}, \eta}(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| \psi - \psi' |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH } \\
&\lesssim {\varepsilon} w_{ f_\textnormal{NS}, \eta }(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| (\phi, \psi) - (\phi', \psi') |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG \times \HHH },\end{split}\end{equation*}
as well as
\begin{align*}
2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}[\phi - \phi', \varphi] |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG } & + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}[\phi', \varphi - \varphi'] |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG } \\
\lesssim & {\varepsilon} c_2 w_{ f_\textnormal{NS}, \eta }(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| \phi - \phi' |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG } + {\varepsilon} w_{ f_\textnormal{NS}, \eta }(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| \varphi - \varphi' |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } \\
\lesssim & {\varepsilon} w_{ f_\textnormal{NS}, \eta }(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| (\phi, \varphi) - (\phi', \varphi') |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG \times \FFF }.
\end{align*}
This shows that
$$|\hskip-0.04cm|\hskip-0.04cm| \Xi_1[\phi, \varphi, \psi] - \Xi_1[\phi', \varphi', \psi' ] |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG } \lesssim {\varepsilon} w_{ f_\textnormal{NS}, \eta }(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| (\phi, \varphi, \psi) - (\phi', \varphi', \psi') |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG \times \FFF \times \HHH }.$$
Thus, taking ${\varepsilon} \ll \eta$, the first component $\Xi_1$ is indeed a contraction. We argue in the same way for the second component. It holds
\begin{equation*}\begin{split}
|\hskip-0.04cm|\hskip-0.04cm| \Xi_2[\phi, \varphi, \psi] - \Xi_2[\phi', \varphi', \psi' ] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF }
&\leqslant |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}[\varphi - \varphi' , \varphi + \varphi ] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } \\
& + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin} [\varphi - \varphi' , f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp}] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } \\
& + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}[ \varphi - \varphi', \psi ] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } + |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin} [ \varphi' , \psi - \psi' ] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } \\
& + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin} [f_\textnormal{NS} , \psi - \psi' ] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}[ f^{\varepsilon}_\textnormal{disp} , \psi - \psi' ] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF }.
\end{split}\end{equation*}
As in the previous proof, resorting to \eqref{eq:bilinear_mix_mix_mix}, one deduces that
$$
|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin} [ \varphi - \varphi' , \varphi + \varphi ] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } \lesssim {\varepsilon} c_2 w_{f_\textnormal{NS}, \eta}(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| \varphi - \varphi' |\hskip-0.04cm|\hskip-0.04cm|_{\FFF} \\
\lesssim {\varepsilon} w_{f_\textnormal{NS}, \eta}(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| \varphi - \varphi' |\hskip-0.04cm|\hskip-0.04cm|_{\FFF}$$
and
$$2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin} [\varphi - \varphi' , f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp}] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } \lesssim {\varepsilon} w_{f_\textnormal{NS}, \eta}(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| \varphi - \varphi' |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF },$$
whereas \begin{equation*}\begin{split}
2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin} [ \varphi - & \varphi', \psi ] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } + |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin} [ \varphi' , \psi - \psi' ] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } \\
& \lesssim {\varepsilon} c_3 w_{f_\textnormal{NS}, \eta}(T)^{-1} \| \varphi - \varphi' \|_{ \FFF } + {\varepsilon} c_2 w_{f_\textnormal{NS}, \eta}(T)^{-1} \| \psi - \psi' \|_{ \HHH } \\
& \lesssim {\varepsilon} w_{f_\textnormal{NS}, \eta}(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| (\varphi, \psi) - (\varphi', \psi' ) |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF \times \HHH }\,.
\end{split}\end{equation*}
Finally, using \eqref{eq:bilinear_mix_phi} and \eqref{eq:bilinear_mix_disp}, one has as previously
$$2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin} [ f_\textnormal{NS} , \psi - \psi' ] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } + 2 |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{kin}[ f^{\varepsilon}_\textnormal{disp} , \psi - \psi' ] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } \lesssim \left(\eta + \beta({\varepsilon})\right) |\hskip-0.04cm|\hskip-0.04cm| \psi - \psi' |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH }\,$$
which, together with the previous estimates, yields
\begin{align*}
|\hskip-0.04cm|\hskip-0.04cm| \Xi_2[\phi, & \varphi, \psi] - \Xi_2[\phi', \varphi', \psi' ] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } \\
& \lesssim \left[{\varepsilon} w_{ f_\textnormal{NS}, \eta }(T)^{-1} +\eta + \beta({\varepsilon})\right]|\hskip-0.04cm|\hskip-0.04cm| (\phi, \varphi, \psi) - (\phi', \varphi', \psi') |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG \times \FFF \times \HHH },
\end{align*}
thus, taking ${\varepsilon} \ll \eta$, the second component $\Xi_2$ is also a contraction. As far as the third component is concerned, one has
\begin{equation*}\begin{split}
|\hskip-0.04cm|\hskip-0.04cm| \Xi_3[\phi, \varphi, \psi] & - \Xi_3[\phi', \varphi', \psi'] |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH } \\
& \leqslant |\hskip-0.04cm|\hskip-0.04cm| \Phi^{\varepsilon}[\phi, \varphi] \psi - \Phi^{\varepsilon}[\phi', \varphi'] \psi' |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH } + |\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro}[\psi - \psi', \psi + \psi'] |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH }\\
&\phantom{++++} + |\hskip-0.04cm|\hskip-0.04cm| \mathcal{S}^{\varepsilon}[\phi-\phi',\varphi]|\hskip-0.04cm|\hskip-0.04cm|_{\HHH} + |\hskip-0.04cm|\hskip-0.04cm| \mathcal{S}^{\varepsilon}[\phi',\varphi-\varphi']|\hskip-0.04cm|\hskip-0.04cm|_{\HHH}.
\end{split}\end{equation*}
Now, using Proposition \ref{prop:linear_hydrodynamic},
\begin{equation*}\begin{split}
|\hskip-0.04cm|\hskip-0.04cm| \Phi^{\varepsilon}[\phi, \varphi] \psi & - \Phi^{\varepsilon}[\phi', \varphi'] \psi' |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH } \\
\lesssim & \left( \eta + \beta({\varepsilon}) + {\varepsilon} c_2 w_{ f_\textnormal{NS}, \eta }(T)^{-1} + {\varepsilon}\right) |\hskip-0.04cm|\hskip-0.04cm| \psi - \psi' |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH } \\
& + c_3 {\varepsilon} w_{ f_\textnormal{NS}, \eta }(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| \varphi - \varphi' |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF }
+ c_3 {\varepsilon} |\hskip-0.04cm|\hskip-0.04cm| \phi - \phi' |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG } \\
\lesssim & \left( \eta + \beta({\varepsilon}) + {\varepsilon} c_2 w_{ f_\textnormal{NS}, \eta }(T)^{-1} + {\varepsilon} \right) |\hskip-0.04cm|\hskip-0.04cm| (\phi , \varphi, \psi) - (\phi' , \varphi', \psi') |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG \times \FFF \times \HHH } \\
\lesssim & \left( \eta + \beta({\varepsilon}) w_{ f_\textnormal{NS}, \eta }(T)^{-1} \right) |\hskip-0.04cm|\hskip-0.04cm| (\phi , \varphi, \psi) - (\phi' , \varphi', \psi') |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG \times \FFF \times \HHH },
\end{split}\end{equation*}
while \eqref{eq:bilinear_hyd_hyd_hyd} yields
$$|\hskip-0.04cm|\hskip-0.04cm| \Psi^{\varepsilon}_\textnormal{hydro}[\psi - \psi' , \psi + \psi' ] |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH } \lesssim w_{f_\textnormal{NS}, \eta}(T)^{-1} c_3 |\hskip-0.04cm|\hskip-0.04cm| \psi - \psi' |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH }.$$
Finally, Proposition \ref{prop:source_term} easily gives
\begin{align*}
|\hskip-0.04cm|\hskip-0.04cm| \mathcal{S}^{\varepsilon}[\phi,\varphi] -\mathcal{S}^{\varepsilon}[\phi',\varphi']|\hskip-0.04cm|\hskip-0.04cm|_{\HHH}
& \leqslant |\hskip-0.04cm|\hskip-0.04cm| \mathcal{S}_3^{\varepsilon}[\phi-\phi',\varphi]|\hskip-0.04cm|\hskip-0.04cm|_{\HHH} + |\hskip-0.04cm|\hskip-0.04cm| \mathcal{S}_3^{\varepsilon}[\phi',\varphi-\varphi']|\hskip-0.04cm|\hskip-0.04cm|_{\HHH} \\
& \lesssim {\varepsilon}\,\bigg(|\hskip-0.04cm|\hskip-0.04cm| \phi-\phi'|\hskip-0.04cm|\hskip-0.04cm|_{\GGG} + |\hskip-0.04cm|\hskip-0.04cm| \varphi-\varphi'|\hskip-0.04cm|\hskip-0.04cm|_{\FFF}\bigg)\,.
\end{align*}
All these estimates yield
\begin{align*}
|\hskip-0.04cm|\hskip-0.04cm| \Xi_3[\phi, \varphi, \psi] & - \Xi_3[\phi', \varphi', \psi'] |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH } \\
& \lesssim \Big[{{\varepsilon} +} \eta + \left( \beta({\varepsilon}) + c_3 \right) w_{ f_\textnormal{NS}, \eta }(T)^{-1} \Big] |\hskip-0.04cm|\hskip-0.04cm| (\phi , \varphi, \psi) - (\phi' , \varphi', \psi') |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG \times \FFF \times \HHH },
\end{align*}
thus, taking $\max\{ {\varepsilon}, c_3 \} \ll \eta \ll 1$, the third component $\Xi_3$ is indeed a contraction.
\end{proof}
\subsection{Proof of Theorem \ref{thm:hydrodynamic_limit}: existence and convergence}
We have established that $\bm{\Xi}$ is a well-defined contraction on $\bm{B}=B_1 \times B_2 \times B_3$ under the smallness assumption
$${\varepsilon} \ll c_3 \ll \eta \ll c_2 \ll 1,$$
thus it admits a unique fixed point denoted $(f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{mix}, g^{\varepsilon})$. The part $g^{\varepsilon}$ satisfies (for some sufficiently small $c > 0$)
\begin{equation*}
|\hskip-0.04cm|\hskip-0.04cm| g^{\varepsilon} |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} \lesssim c |\hskip-0.04cm|\hskip-0.04cm| g^{\varepsilon} |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH } + |\hskip-0.04cm|\hskip-0.04cm| \mathcal S^{\varepsilon} |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH },
\end{equation*}
and therefore, considering $c$ small enough, we have:
\begin{equation*}
|\hskip-0.04cm|\hskip-0.04cm| g^{\varepsilon} |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} \lesssim |\hskip-0.04cm|\hskip-0.04cm| \mathcal S^{\varepsilon} |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} \lesssim \beta({\varepsilon}).
\end{equation*}
The part $f^{\varepsilon}_\textnormal{mix}$ satisfies the equation
\begin{equation*}\begin{split}
f^{\varepsilon}_\textnormal{mix} - \Psi^{\varepsilon}_\textnormal{kin}[ f_\textnormal{NS} , f_\textnormal{NS}]
&= \Psi^{\varepsilon}_\textnormal{kin}[f^{\varepsilon}_\textnormal{mix} , f^{\varepsilon}_\textnormal{mix}] + \Psi^{\varepsilon}_\textnormal{kin}[ g^{\varepsilon} , g^{\varepsilon} ] + \Psi^{\varepsilon}_\textnormal{kin}[ f^{\varepsilon}_\textnormal{disp} , f^{\varepsilon}_\textnormal{disp} ] \\
& + 2 \Psi^{\varepsilon}_\textnormal{kin}[ f^{\varepsilon}_\textnormal{mix} , f_\textnormal{NS}] + 2 \Psi^{\varepsilon}_\textnormal{kin}[ f^{\varepsilon}_\textnormal{mix} , f_\textnormal{disp}^{\varepsilon} ] + 2 \Psi^{\varepsilon}_\textnormal{kin}[ f^{\varepsilon}_\textnormal{mix} , g^{\varepsilon} ] \\
& + 2 \Psi^{\varepsilon}_\textnormal{kin}[ f_\textnormal{NS} , f^{\varepsilon}_\textnormal{disp} ] + 2 \Psi^{\varepsilon}_\textnormal{kin}[ f_\textnormal{NS} , g^{\varepsilon} ] + 2 \Psi^{\varepsilon}_\textnormal{kin}[ f^{\varepsilon}_\textnormal{disp} , g^{\varepsilon} ],
\end{split}\end{equation*}
therefore, from the computations of Section \ref{scn:mapping_contraction}, we have
$$|\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_\textnormal{mix} - \Psi^{\varepsilon}_\textnormal{kin}[f_\textnormal{NS} , f_\textnormal{NS} ] |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } \lesssim |\hskip-0.04cm|\hskip-0.04cm| g^{\varepsilon} |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH } + \beta({\varepsilon}) \lesssim \beta({\varepsilon}).$$
Furthermore, by a duality argument similar to the one from the proof of \eqref{eq:decay_convolution_semigroup_exp}
\begin{align*}
\int_{t_1}^{t_2} \big\langle \Psi^{\varepsilon}_\textnormal{kin}[f_\textnormal{NS}, f_\textnormal{NS}](t) , \phi_0 \big\rangle_{\HH} \mathrm{d} t
& = \frac{1}{{\varepsilon}} \int_{t_1}^{t_2} \int_0^t \left\langle \mathcal Q\left( f_\textnormal{NS}(\tau) , f_\textnormal{NS}(\tau) \right) , U^{\varepsilon}_\textnormal{kin}(t-\tau)^\star \phi_0 \right\rangle_{\HH} \mathrm{d} \tau \, \mathrm{d} t \\
& \lesssim \frac{1}{{\varepsilon}} \| f_\textnormal{NS} \|_{ L^\infty\left( [0, T) ; \HH^{\bullet} \right) } (t_2 - t_1) \int_0^\infty \left\| U^{\varepsilon}_\textnormal{kin}(t)^\star \phi_0 \right\|_{ \HH^{\bullet} } \mathrm{d} t \\
& \lesssim (t_2 - t_1) \left(\int_0^\infty e^{2 \sigma t / {\varepsilon}^2} \left\| U^{\varepsilon}_\textnormal{kin}(t)^\star \phi_0 \right\|_{ \HH^{\bullet} }^2 \mathrm{d} t\right)^{\frac{1}{2}} \lesssim {\varepsilon} (t_2 - t_1) \| \phi_0 \|_{\HH},
\end{align*}
where we used that $\| f_\textnormal{NS} \|_{ L^\infty\left( [0, T) ; \HH^{\bullet} \right) } \lesssim 1$. Thus, we deduce
$$\left\| \Psi^{\varepsilon}_\textnormal{kin}[f_\textnormal{NS}, f_\textnormal{NS}] \right\|_{ L^\infty\left( [0, T) ; \HH \right) } \lesssim {\varepsilon},$$
from which we conclude that
$\left\| f^{\varepsilon}_\textnormal{mix} + g^{\varepsilon} \right\|_{ L^\infty\left( [0, T) ; \HH \right) } \lesssim \beta({\varepsilon}).$
We conclude to Theorem \ref{thm:hydrodynamic_limit} by letting
\begin{gather*}
f^{\varepsilon}_\textnormal{err} := g^{\varepsilon} + f^{\varepsilon}_\textnormal{mix}.
\end{gather*}
This concludes the proof.
\subsection{Proof of Theorem \ref{thm:hydrodynamic_limit}: uniqueness}
\label{scn:uniqueness_simmetrizable}
Consider another solution associated with the same initial data $f_\textnormal{in}$:
$$\overline{f}^{\varepsilon} \in L^\infty\left( [0, T) ; \GG \right) \cap L^2_{\text{loc}}\left( [0, T) ; \GG^{\bullet} \right)$$
satisfying for some universal small $c > 0$ the bound (note that the same bound holds for $f^{\varepsilon}$ since $\| f^{\varepsilon} \|_{L^\infty_t (\GG)} \lesssim 1$)
$$\| \overline{f}^{\varepsilon} \|_{ L^\infty \left( [0, T) ; \GG \right) } \leqslant \frac{c}{{\varepsilon}}.$$
Define the difference of solutions
$$h^{\varepsilon} = f^{\varepsilon} - \overline{f}^{\varepsilon}$$
and observe it satisfies the equation
$$\partial_t h^{\varepsilon} = \frac{1}{{\varepsilon}^2} \left( \mathcal L - {\varepsilon} v \cdot \nabla_x \right) h^{\varepsilon} + \frac{1}{{\varepsilon}} \mathcal Q \left( h^{\varepsilon}, f^{\varepsilon} + \overline{f}^{\varepsilon} \right), \quad h^{\varepsilon}(0) = 0.$$
We write an energy estimate for $h^{\varepsilon}$ (see \textit{Step 2} of the proof of Lemma \ref{lem:decay_regularization_convolution_kinetic_semigroup} for the dissipative part):
$$\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d} t} \| h^{\varepsilon} \|^2_{ \GG } + \frac{\lambda}{{\varepsilon}^2} \| h^{\varepsilon} \|^2_{ \GG^{\bullet} } \lesssim \frac{1}{{\varepsilon}^2} \| h^{\varepsilon} \|^2_{ \GG } + \frac{1}{{\varepsilon}} \| h^{\varepsilon} \|_{ \GG^{\bullet} }^2 \| f^{\varepsilon} + \overline{f}^{\varepsilon} \|_{ \GG } + \frac{1}{{\varepsilon}} \| h^{\varepsilon} \|_{ \GG } \| h^{\varepsilon} \|_{ \GG^{\bullet} } \| f^{\varepsilon} + \overline{f}^{\varepsilon} \|_{ \GG^{\bullet} }, $$
which gives after integrating on $[0, t]$ in the space $\GGG(\sigma=0, t, {\varepsilon})$:
\begin{align*}
|\hskip-0.04cm|\hskip-0.04cm| h^{\varepsilon} |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG }^2 \lesssim \frac{t}{{\varepsilon}^2} |\hskip-0.04cm|\hskip-0.04cm| h^{\varepsilon} |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG }^2 + c |\hskip-0.04cm|\hskip-0.04cm| h^{\varepsilon} |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG }^2 + |\hskip-0.04cm|\hskip-0.04cm| h^{\varepsilon} |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG }^2 \left(\int_0^t \| f^{\varepsilon}(\tau) + \overline{f}^{\varepsilon}(\tau) \|^2_{\GG^{\bullet}} \mathrm{d} \tau\right)^{1/2}.
\end{align*}
Thus, since $c$ is supposed to be small, taking $t$ close enough to $0$ yields (for instance)
$$|\hskip-0.04cm|\hskip-0.04cm| h^{\varepsilon} |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG } \leqslant \frac{1}{2} |\hskip-0.04cm|\hskip-0.04cm| h^{\varepsilon} |\hskip-0.04cm|\hskip-0.04cm|_{ \GGG },$$
which in turn implies $h(\tau) = 0$, or equivalently $f^{\varepsilon}(\tau) = \overline{f}^{\varepsilon}(\tau)$ for any $\tau \in [0, t]$. Repeating this argument yields the uniqueness of the solution.
\section{Proof of Theorem \ref{thm:hydrodynamic_limit-gen_degenerate}}
\label{scn:hydrodynamic_limit_BED}
We prove here Theorem~\ref{thm:hydrodynamic_limit-gen_degenerate} under the assumption \ref{BED}. Note that the following strategy can also be seen as an alternative proof under assumption \ref{BE}.
\subsection{Modification of the strategy}
Under the assumption \ref{BED}, the arguments of $\mathcal Q(f, g)$ in $X$ no longer play symmetric roles, so \textbf{we no longer consider $\mathcal Q$ in its symmetrized form}. This does not induce any change for the parts $f^{\varepsilon}_\textnormal{hydro}$ and $f^{\varepsilon}_\textnormal{mix}$ of the solution since they are constructed using assumption \ref{Bbound} and not \ref{BED}, for which both arguments play symmetric roles. The only modification is therefore the need to adjust the iterative scheme constructing $f^{\varepsilon}_\textnormal{kin}$ as well as its space-velocity functional space so as to take into account the assumptions \ref{BED} following the strategy adopted in \cite{GMM2017, CTW2016, CM2017, HTT2020, CG2022}. We detail below this new strategy.
\begin{itemize}
\item We no longer consider the equation on $f^{\varepsilon}_\textnormal{kin}$ in integral form
$$f^{\varepsilon}_\textnormal{kin}(t) = U^{\varepsilon}_\textnormal{kin}(t) f_\textnormal{in} + \Psi^{\varepsilon}_\textnormal{kin} \left[ f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{kin} \right](t)+ 2 \Psi^{\varepsilon}_\textnormal{kin}\left[f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{hydro} + f^{\varepsilon}_\textnormal{mix}\right](t),$$
but we study the evolution of $f_\textnormal{kin}^{\varepsilon}$ it in its differential form
\begin{equation}\label{eq:diffFkin}\begin{split}
\partial_t f^{\varepsilon}_\textnormal{kin} = \frac{1}{{\varepsilon}^2} \left(\mathcal L - {\varepsilon} v \cdot \nabla_x\right) f^{\varepsilon}_\textnormal{kin} &+ \frac{1}{{\varepsilon}} \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q(f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{kin}) \\ \phantom{+++}&+ \frac{2}{{\varepsilon}} \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q^\mathrm{sym}(f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{hydro} + f^{\varepsilon}_\textnormal{mix}).
\end{split}\end{equation}
This equation can be studied through a suitable energy method so as to be able to use the ‘‘closing estimate'' of \ref{BED} (which does not translate in integral form).
\item Since the roles played by both arguments of $\mathcal Q(f, g)$ in $X$ under the assumption \ref{BED} are different, we do not construct $f^{\varepsilon}_\textnormal{kin}$ using Banach's theorem, which, as far as \eqref{eq:diffFkin} is concerned, would correspond to the convergence an iterative scheme of the form
\begin{equation*}\begin{split} \partial_t f^{\varepsilon}_{\textnormal{kin}, N} = \frac{1}{{\varepsilon}^2} \left( \mathcal L - {\varepsilon} v \cdot \nabla_x\right) f^{\varepsilon}_{\textnormal{kin}, N} &+ \frac{1}{{\varepsilon}} \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q(f^{\varepsilon}_{\textnormal{kin}, N-1}, f^{\varepsilon}_{\textnormal{kin}, N-1}) \\ \phantom{+++}&+ \frac{2}{{\varepsilon}} \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q^\mathrm{sym}(f^{\varepsilon}_{\textnormal{kin}, N-1} f^{\varepsilon}_{\textnormal{hydro}, N-1} + f^{\varepsilon}_{\textnormal{mix},N-1})\end{split}\end{equation*}
but using a variation of such a scheme which allows to use the the "closing estimate'' of \ref{BED}. Namely, we prove the stability of the scheme \eqref{eq:scheme} hereafter.
\item We define a new hierarchy of spaces $(\bm{\GG}_j)_{j=-2-s}^1$ of the form
$$\bm{\GG}_j = L^2_x \left( \bm{X}_j \right) \cap \dot{\mathbb{H}}^s_x \left( \bm{X}_{j-s} \right)$$
which allows to prove spatially inhomogeneous counterparts of the estimates of \ref{BED}. Notice here that we assume our ``regularity parameter'' $s$ to be integer $s \in \mathbb N$ and it is now assumed an additional role in the hierarchy of spaces $\bm{\GG}_{-2-s},\ldots,\bm{\GG}_{1}.$
\item The operator $\mathcal L-{\varepsilon} v \cdot \nabla_x$ is not dissipative for the inner product of $\bm{X}$, but it is \emph{hypo-dissipative} on $\range\left( \mathsf P^{\varepsilon}_\textnormal{kin} \right)$, so, we introduce an equivalent inner product of the form
\begin{equation*}
\dlla f, g \drra_{\bm{\GG}_j, {\varepsilon}} := \delta \langle f, g \rangle_{\bm{\GG}_j} + \frac{1}{{\varepsilon}^2} \int_0^\infty \left\langle U^{\varepsilon}_\textnormal{kin}(t) f, U^{\varepsilon}_\textnormal{kin}(t) g \right\rangle_{\bm{\GG}_{j-1}} \mathrm{d} t,
\end{equation*}
for which $\mathcal L-{\varepsilon} v \cdot \nabla_x$ is dissipative and $\mathcal Q$ satisfies the same estimates as \ref{BED}.
\end{itemize}
To summarize, our approach will be aimed at constructing a solution $(f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{mix}, g^{\varepsilon})$ as the limit of a sequence of approximate solutions $\left\{( f^{\varepsilon}_{\textnormal{kin}, N}, f^{\varepsilon}_{\textnormal{mix}, N}, g^{\varepsilon}_N )\right\}_{N \geqslant 0}$, where the first component $f_{\textnormal{kin},N}^{\varepsilon}$ is constructed inductively by solving the following differential equation:
\begin{equation}\label{eq:scheme}
\begin{cases}
\displaystyle
\begin{aligned}
\partial_t f^{\varepsilon}_{\textnormal{kin}, N} = \frac{1}{{\varepsilon}^2} \left(\mathcal L - {\varepsilon} v \cdot \nabla_x\right) f^{\varepsilon}_{\textnormal{kin}, N} & + \frac{1}{{\varepsilon}} \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q(f^{\varepsilon}_{\textnormal{kin}, N-1}, f^{\varepsilon}_{\textnormal{kin}, N}) \\
& + \frac{2}{{\varepsilon}} \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q^\mathrm{sym}(f^{\varepsilon}_{\textnormal{kin}, N}, f^{\varepsilon}_{\textnormal{hydro}, N-1} + f^{\varepsilon}_{\textnormal{mix}, N-1})
\end{aligned}\\
f^{\varepsilon}_{\textnormal{kin}, N}(0) = \mathsf P^{\varepsilon}_\textnormal{kin} f_\textnormal{in},\\
f^{\varepsilon}_{\textnormal{kin}, 0} = 0,
\end{cases}
\end{equation}
where we naturally denoted $f^{\varepsilon}_{\textnormal{hydro}, N} = f_\textnormal{NS} + f^{\varepsilon}_\textnormal{disp} + g^{\varepsilon}_N$, and the other parts are still constructed as in Section \ref{scn:proof_hydrodynamic_limit_symmetrizable}:
\begin{equation*}
\begin{cases}
g^{\varepsilon}_N = \Phi^{\varepsilon}[f^{\varepsilon}_{\textnormal{kin}, N-1} , f^{\varepsilon}_{\textnormal{mix}, N-1} ] g^{\varepsilon}_{N-1} + \Psi^{\varepsilon}_\textnormal{hydro}[g^{\varepsilon}_{N-1}, g^{\varepsilon}_{N-1}]
+ \mathcal S^{\varepsilon}[f^{\varepsilon}_{\textnormal{kin}, N-1} , f^{\varepsilon}_{\textnormal{mix}, N-1}], \\
\\
f^{\varepsilon}_{\textnormal{mix}, N} = \Psi^{\varepsilon}_\textnormal{kin}\left[ f^{\varepsilon}_{\textnormal{mix},N-1} + f^{\varepsilon}_{\textnormal{kin}, N-1} , f^{\varepsilon}_{\textnormal{mix},N-1} + f^{\varepsilon}_{\textnormal{kin}, N-1} \right], \\
\\
g^{\varepsilon}_0 = 0, \qquad f^{\varepsilon}_{\textnormal{mix},0} = 0.
\end{cases}
\end{equation*}
Let us define the new functional space we will use in this section.
\begin{defi}
Suppose $s \in \mathbb N$ and satisfies $s \geqslant 3$ if $d = 2$ or $s \geqslant \frac{d}{2} + 1$ if $d \geqslant 3$, we define
\begin{equation*}
\| f \|_{ \bm{\GG}_j^s } := \| f \|_{ L^2_x \left( \bm{X}_j \right) } + \| \nabla_x^s f \|_{ L^2_x \left( \bm{X}_{j-s} \right) },
\end{equation*}
and note that, since $\bm{X}_k \hookrightarrow \bm{X}_j$ as soon as $j \leqslant k$, the following equivalence of norms holds:
\begin{equation*}
\| f \|_{ \bm{\GG}_j^s } \approx \sum_{k = 0}^{s} \| f \|_{ \mathbb{H}^k_x \left( \bm{X}_{j-k} \right) } \approx \sum_{k = 0}^{s} \| f \|_{ \bm{\GG}_j^k }.
\end{equation*}
In particular, this hierarchy of spaces is decreasing in both indexes:
\begin{equation}
\label{eq:hierarchy_degenerate_spaces}
s_1 \leqslant s_2 ~ \text{ and } ~ j_1 \leqslant j_2 \Longrightarrow \bm{\GG}_{j_2}^{s_2} \hookrightarrow \bm{\GG}_{j_1}^{s_1}.
\end{equation}
\end{defi}
\begin{lem}
The bilinear operator $\mathcal Q$ satisfies in $\bm{\GG}_j$ the estimates
\begin{gather}
\label{eq:Q_sobolev_algebra_degenerate}
\left\| \mathcal Q(f, g) \right\|_{ \bm{\GG}^{\circ}_j } \lesssim \| f \|_{ \bm{\GG}_j } \| g \|_{ \bm{\GG}^{\bullet}_{j+1} } + \| f \|_{ \bm{\GG}^{\bullet}_j } \| g \|_{ \bm{\GG}_{j+1} }, \\
\label{eq:Q_sobolev_algebra_degenerate_closed}
\langle \mathcal Q(f, g), g \rangle_{ \bm{\GG}_j } \lesssim \| f \|_{ \bm{\GG}_j } \| g \|_{ \bm{\GG}^{\bullet}_j }^2 + \| f \|_{ \bm{\GG}^{\bullet}_j } \| g \|_{ \bm{\GG}_j } \| g \|_{ \bm{\GG}^{\bullet}_j }.
\end{gather}
\end{lem}
\begin{proof}
The general non-closed control \eqref{eq:Q_sobolev_algebra_degenerate} is easily obtained from its spatially homogeneous counterpart of \ref{BED} so we only prove the closed control \eqref{eq:Q_sobolev_algebra_degenerate_closed}.
The inner product writes according to Leibniz's formula and the locality of $\mathcal Q$ as
\begin{equation*}\begin{split}
\left\langle \mathcal Q(f, g), h \right\rangle_{ \bm{\GG}_j } = & \left\langle \mathcal Q(f, g), h \right\rangle_{ L^2_x \left( \bm{X}_{j} \right) } + \sum_{| \beta | = s } \left\langle \partial_x^\beta \mathcal Q(f, g), \partial_x^\beta h \right\rangle_{ L^2_x \left( \bm{X}_{j-s} \right) } \\ = & \left\langle \mathcal Q(f, g), h \right\rangle_{ L^2_x \left( \bm{X}_{j} \right) } + {\sum_{|\gamma|+|\beta-\gamma|=|\beta|=s}} \left\langle \mathcal Q\left( \partial_x^\gamma f, \partial_x^{\beta-\gamma} g \right), \partial_x^{\beta} h \right\rangle_{ L^2_x \left( \bm{X}_{j-s} \right) }.
\end{split}\end{equation*}
We first look at the terms which can be controlled using the closed estimate \eqref{eq:BEDclosed} of \ref{BED}. Using H\"older's inequality in $L^\infty_x \times L^2_x \times L^2_x$ (or some appropriate permutation) together with the embeddings $\mathbb{H}^s_x \hookrightarrow L^\infty_x$ and $\bm{X}_{j-s} \hookrightarrow \bm{X}_{-1-s}$ and therefore $\mathbb{H}^s_x \left( \bm{X}_{j-s} \right) \hookrightarrow L^\infty_x \left( \bm{X}_{-1-s} \right)$, we immediately have the estimate
\begin{equation*}
\begin{split}
\langle \mathcal Q(f, g), g \rangle_{ L^2_x \left( \bm{X}_{j} \right) }
\lesssim & \sum_{ \{ a, b, c \} = \{j, j, -1-s \} } \left( \int_{\mathbb R^d} \| f \|_{\bm{X}_{ a }} \| g \|_{\bm{X}^{\bullet}_{b}} \| g \|_{\bm{X}^{\bullet}_{c}} \mathrm{d} x + \int_{\mathbb R^{d}}\| f \|_{\bm{X}^{\bullet}_{a}} \| g \|_{\bm{X}_{b}} \| g \|_{\bm{X}^{\bullet}_{c}} \mathrm{d} x \right)\\
\lesssim & \| f \|_{ \bm{\GG}_j } \| g \|_{ \bm{\GG}^{\bullet}_j }^2 + \| f \|_{ \bm{\GG}^{\bullet}_j } \| g \|_{ \bm{\GG}_j } \| g \|_{ \bm{\GG}^{\bullet}_j }
\end{split}
\end{equation*}
\color{black}
as well as the terms associated with $| \beta | = s$ and $|\gamma| = 0$:
$$\left\langle \mathcal Q\left(f, \partial_x^\beta g\right), \partial_x^\beta g \right\rangle_{ L^2_x \left( \bm{X}_{j-s} \right) } \lesssim \| f \|_{ \bm{\GG}_j } \| g \|_{ \bm{\GG}^{\bullet}_j }^2 + \| f \|_{ \bm{\GG}^{\bullet}_j } \| g \|_{ \bm{\GG}_j } \| g \|_{ \bm{\GG}^{\bullet}_j }.$$
We are thus left with the terms associated with $|\beta|=s$ and $|\gamma| \geqslant 1$ which have to be controlled starting from the non-closed estimate \eqref{eq:BEDNon} of \ref{BED}:
\begin{equation}\label{eq:QQgammabeta}
\begin{split}
\Big\langle \mathcal Q\big( \partial_x^\gamma f, & \partial_x^{\beta-\gamma} g\big), \partial_x^\beta g \Big\rangle_{ L^2_x \left( \bm{X}_{j-s} \right) } \\
\lesssim & \int_{\mathbb R^{d}} \| \partial_x^\gamma f(x) \|_{ \bm{X}_{j-s} } \| \partial_x^{\beta-\gamma} g(x) \|_{ \bm{X}^{\bullet}_{j - (s-1) } } \| \partial_x^\beta g(x) \|_{ \bm{X}^{\bullet}_{j - s } } \mathrm{d} x \\
& + \int_{\mathbb R^{d}} \| \partial_x^\gamma f(x) \|_{ \bm{X}^{\bullet}_{j-s} } \| \partial_x^{\beta-\gamma} g(x) \|_{ \bm{X}_{j - (s-1) } } \| \partial_x^\beta g(x) \|_{ \bm{X}^{\bullet}_{j - s } } \mathrm{d} x \, ,
\end{split}
\end{equation}
and we will show in each case that \begin{equation}
\label{eq:degenerate_bilinear_goal}
\begin{split}
\left\|\| \partial_x^\gamma f(x) \|_{ \bm{X}_{j-s} } \| \partial_x^{\beta-\gamma} g(x) \|_{ \bm{X}^{\bullet}_{j - (s-1) } } \right\|_{L^{2}_{x}} &\lesssim \|f\|_{\bm{\GG}_{j}}\|g\|_{\bm{\GG}^{\bullet}_{j}}\\
\left\|\| \partial_x^\gamma f(x) \|_{ \bm{X}^{\bullet}_{j-s} } \| \partial_x^{\beta-\gamma} g(x) \|_{ \bm{X}_{j - (s-1) } } \right\|_{L^{2}_{x}} &\lesssim \|f\|_{\bm{\GG}^{\bullet}_{j}}\|g\|_{\bm{\GG}_{j}}\,.\end{split} \end{equation}
\step{1}{The case $d = 2$}
When $| \gamma | = s$, we have $|\beta - \gamma| = 0$, thus, using the fact that $\mathbb{H}^{s-1}_x \hookrightarrow L^\infty_x$ since $s \geqslant 3$, followed by \eqref{eq:hierarchy_degenerate_spaces}
$$\| \partial_x^{\beta-\gamma} g \|_{ L^\infty_x \left( \bm{X}_{j- (s-1) } \right) } \lesssim \| g \|_{ \mathbb{H}^{s-1}_x \left( \bm{X}_{j- (s-1) } \right) } \lesssim \| g \|_{ \bm{\GG}_j }, $$
thus, \eqref{eq:degenerate_bilinear_goal} holds. When $| \gamma | = s-1$, using the injection $\mathbb{H}^1_x \hookrightarrow L^4_x$, we have
$$\| \partial_x^\gamma f \|_{ L^4_x \left( \bm{X}_{j-s} \right) } \lesssim \| f \|_{ \mathbb{H}^{ s }_x \left( \bm{X}_{j-s} \right) } \lesssim \| f \|_{ \bm{\GG}_j },$$
similarly, since $| \beta - \gamma | = 1 \leqslant s - 2$, we also have
$$\| \partial_x^{\beta-\gamma} g \|_{ L^4_x \left( \bm{X}_{j - (s-1) } \right) } \lesssim \| f \|_{ \mathbb{H}^{ s-1 }_x \left( \bm{X}_{j-(s-1) } \right) } \lesssim \| f \|_{ \bm{\GG}_j },$$
thus, \eqref{eq:degenerate_bilinear_goal} holds. When $| \gamma | \leqslant s-2$, we have
$$\| \partial_x^\gamma f \|_{ L^\infty_x \left( \bm{X}_{j-s} \right) } \lesssim \| f \|_{ \mathbb{H}^s_x \left( \bm{X}_{j-s} \right) }, \lesssim \| f \|_{ \bm{\GG}_j }$$
and similarly, since $| \beta - \gamma | \leqslant s-1$ (recall that $|\beta|=s$ and $| \gamma \geqslant 1$)
$$\| \partial_x^{\beta-\gamma} g \|_{ L^2_x \left( \bm{X}_{j-(s-1)} \right) } \lesssim \| f \|_{ \mathbb{H}^{s-1}_x \left( \bm{X}_{j-(s-1)} \right) } \lesssim \| f \|_{ \bm{\GG}_j },$$
thus \eqref{eq:degenerate_bilinear_goal} holds. This concludes this step.
\step{2}{The case $d = 3$}
First, a simple use of Cauchy-Schwarz inequality yields
$$\big\| \| \partial_x^\gamma f \|_{ \bm{X}_{j-s} } \| \partial_x^{\beta-\gamma} g \|_{ \bm{X}^{\bullet}_{j- (s-1) } } \big \|_{L^1_x} \leqslant \| \partial_x^\gamma f \|_{ L^2_x \left( \bm{X}_{j-s} \right) } \| \partial_x^{\beta-\gamma}g \|_{ L^2_x \left( \bm{X}^{\bullet}_{j- (s-1) } \right) }$$
and, since $|\beta-\gamma|+|\gamma| \leqslant s$, we deduce that \begin{equation}\label{productL1}
\big\| \| \partial_x^\gamma f \|_{ \bm{X}_{j-s} } \| \partial_x^{\beta-\gamma} g \|_{ \bm{X}^{\bullet}_{j- (s-1) } } \big \|_{L^1_x} \lesssim \|f \|_{\bm{\GG}_j}\,\|g \|_{\bm{\GG}^{\bullet}_j}.\end{equation} Now, using Sobolev embeddings, we have
$$\| \partial_x^\gamma f \|_{ L^p_x \left( \bm{X}_{j-s} \right) } \lesssim \| f \|_{ \mathbb{H}^s_x \left( \bm{X}_{j-s} \right) } \lesssim \|f \|_{\bm{\GG}_j}, \qquad \frac{1}{p} = \frac{1}{2} - \frac{s - | \gamma| }{d}$$
as well as
$$\| \partial_x^{\beta-\gamma}g \|_{ L^q_x \left( \bm{X}^{\bullet}_{j- (s-1) } \right) } \lesssim \| g \|_{ \mathbb{H}^{s-1}_x \left( \bm{X}^{\bullet}_{j-(s-1) } \right) } \lesssim \|g \|_{\bm{\GG}^{\bullet}_j}, \qquad \frac{1}{q} = \frac{1}{2} - \frac{ (s-1) - | \beta - \gamma| }{d}.$$
Since $| \beta | = s$ and $s \geqslant \frac{d}{2} + 1$, H\"older's inequality implies
$$\big\| \| \partial_x^\gamma f \|_{ \bm{X}_{j-s} } \| \partial_x^{\beta-\gamma} g \|_{ \bm{X}^{\bullet}_{j- (s-1) } } \big \|_{L^r_x} \lesssim \| f \|_{\bm{\GG}_j} \| g \|_{ \bm{\GG}^{\bullet}_j }, \qquad \frac{1}{r} = \frac{1}{p} + \frac{1}{q} = 1 - \frac{s - 1}{d} \leqslant \frac{1}{2}\,.$$
Using \eqref{productL1} and simple
interpolation, we deduce that
$$\big\| \| \partial_x^\gamma f \|_{ \bm{X}_{j-s} } \| \partial_x^{\beta-\gamma} g \|_{ \bm{X}_{j- (s-1) } } \big \|_{ L^2_x} \lesssim \| f \|_{\bm{\GG}_j} \| g \|_{ \bm{\GG}^{\bullet}_j },$$ this proves the first estimate in \eqref{eq:degenerate_bilinear_goal} and the second one is done in the same way.
Combining then \eqref{eq:degenerate_bilinear_goal} with \eqref{eq:QQgammabeta} and Cauchy-Schwarz inequality yields
\begin{equation*}
\Big\langle \mathcal Q\big( \partial_x^\gamma f, \partial_x^{\beta-\gamma} g\big), \partial_x^\beta g \Big\rangle_{ L^2_x \left( \bm{X}_{j-s} \right) } \lesssim \| f \|_{ \bm{\GG}_j } \| g \|_{ \bm{\GG}^{\bullet}_j }^2 + \| f \|_{ \bm{\GG}^{\bullet}_j } \| g \|_{ \bm{\GG}_j } \| g \|_{ \bm{\GG}^{\bullet}_j }.
\end{equation*}
This concludes this step and the proof.
\end{proof}
As stated above, we will study the equation \eqref{eq:scheme} using an equivalent inner product. The next proposition defines it and presents its properties.
\begin{prop}[\textit{\textbf{Kinetic dissipative inner product}}]
\label{prop:dissipative_kinetic_inner_product}
Let $j = -1, 0$ and $\sigma \in (0, \sigma_0)$. For some $\delta > 0$ small enough, the inner product defined for any $f, g \in \range\left( \mathsf P^{\varepsilon}_\textnormal{kin} \right) \cap \bm{\GG}_j$ as
\begin{equation*}
\dlla f, g \drra_{\bm{\GG}_{j}, {\varepsilon}} := \delta \langle f, g \rangle_{\bm{\GG}_{j}} + \frac{1}{{\varepsilon}^2} \int_0^\infty e^{2 \sigma t / {\varepsilon}^2} \left\langle U^{\varepsilon}_\textnormal{kin}(t) f, U^{\varepsilon}_\textnormal{kin}(t) g \right\rangle_{ \bm{\GG}_{j-1} } \mathrm{d} t,
\end{equation*}
induces a norm equivalent to that of $\bm{\GG}_{j}$ (uniformly in ${\varepsilon}$), i.e. there exists $C >0$ independent of ${\varepsilon}$ such that
\begin{equation}
\label{eq:equivE}
\frac{1}{C} \| f \|_{\bm{\GG}_{j}} \leqslant \| f \|_{\bm{\GG}_{j}, {\varepsilon}} \leqslant C \| f \|_{\bm{\GG}_{j}}.
\end{equation}
Moreover, there is $\mu >0$ such that
$$\mathrm{Re}\, \dlla (\mathcal L - {\varepsilon} v \cdot \nabla_x) f, f \drra_{\bm{\GG}_{j}, {\varepsilon} } \leqslant - \sigma \| f \|_{\bm{\GG}_{j}, {\varepsilon}}^2 - \mu \| f \|^2_{\bm{\GG}^{\bullet}_{j}},$$
and the nonlinear estimates for $\mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q$ are the same as those from \eqref{eq:Q_sobolev_algebra_degenerate} and \eqref{eq:Q_sobolev_algebra_degenerate_closed}:
\begin{align*}
\dlla \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q(f, g) , h \drra_{\bm{\GG}_{j} , {\varepsilon} } \lesssim & \| h \|_{\bm{\GG}^{\bullet}_{j}} \left( \| f \|_{\bm{\GG}^{\bullet}_{j}} \| g \|_{\bm{\GG}_{j+1}} + \| f \|_{\bm{\GG}_{j}} \| g \|_{\bm{\GG}^{\bullet}_{j+1}} \right),
\end{align*}
and
\begin{equation*}
\dlla \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q(f, g) , g \drra_{\bm{\GG}_{j} , {\varepsilon} } \lesssim \| f \|_{\bm{\GG}_{j}} \| g \|_{\bm{\GG}^{\bullet}_{j}}^2 + \| g \|_{ \bm{\GG}_{j} } \| g \|_{\bm{\GG}^{\bullet}_{j}} \| f \|_{ \bm{\GG}^{\bullet}_j }.
\end{equation*}
\end{prop}
\begin{proof}
\step{1}{Proof of the equivalence of norms}
The norm $\| \cdot \|_{\bm{\GG}_{j}, {\varepsilon} }$ writes
$$\| f \|_{ \bm{\GG}_{j} , {\varepsilon} }^2 = \delta \| f \|^2_{ \bm{\GG}_j} + \frac{1}{{\varepsilon}^2} \int_0^\infty e^{2 \sigma t / {\varepsilon}^2 } \left\| U^{\varepsilon}_\textnormal{kin}(t) f \right\|^2_{\bm{\GG}_{j-1}} \mathrm{d} t,$$
and thus, using $\bm{\GG}_j \hookrightarrow \bm{\GG}_{j-1}$ and the decay estimate for the semigroup $U^{\varepsilon}_\textnormal{kin}(t)$ from Lemma~\ref{lem:decay_regularization_kinetic_semigroup}, we have for some $C > 0$
$$\delta \| f \|_{\bm{\GG}_j }^2 \leqslant \| f \|_{\bm{\GG}_j , {\varepsilon} }^2 \leqslant (\delta + C) \| f \|^2_{ \bm{\GG}_j}.$$
This proves \eqref{eq:equivE}.
\step{2}{Proof of the dissipative estimate}
Using the decomposition $\mathcal L = \mathcal B + \mathcal A$ coming from \ref{LE} in the space $\bm{X}_{j}$ (recall that, from \ref{BED}, the dissipativity estimate of \ref{LE} is valid in $\bm{X}_{j}$) together with the estimate for $\mathcal A$ from \ref{BED}, we have for some~$K > 0$
\begin{equation*}\begin{split}
\mathrm{Re}\, & \dlla (\mathcal L - {\varepsilon} v \cdot \nabla_x) f, f \drra_{\bm{\GG}_j , {\varepsilon} } \\
&= \delta \mathrm{Re}\, \left\langle (\mathcal L - {\varepsilon} v \cdot \nabla_x) f , f \right\rangle_{\bm{\GG}_j} \\
& + \mathrm{Re}\, \int_0^\infty e^{2 \sigma t / {\varepsilon}^2} \left\langle U^{\varepsilon}_\textnormal{kin}(t) \left\{ \frac{1}{{\varepsilon}^2} \left( \mathcal L - {\varepsilon} v \cdot \nabla_x\right) f \right\} , U^{\varepsilon}_\textnormal{kin}(t) f \right\rangle_{\bm{\GG}_{j-1}} \mathrm{d} t \\
&\leqslant - \delta \lambda_\mathcal B \| f \|_{ \bm{\GG}_j}^2 + \delta K \| f \|_{ \bm{\GG}_{j-1}}^2 \\
& -\frac{\sigma}{{\varepsilon}^2} \int_0^\infty e^{2 \sigma t / {\varepsilon}^2} \left\| U^{\varepsilon}_\textnormal{kin}(t) f \right\|_{\bm{\GG}_{j-1}}^2 \mathrm{d} t + \frac{1}{2} \int_0^\infty \frac{\mathrm{d}}{\mathrm{d} t} \left[ e^{2 \sigma t / {\varepsilon}^2} \| U^{\varepsilon}_\textnormal{kin}(t) f \|^2_{\bm{\GG}_{j-1} } \right] \, \mathrm{d} t,
\end{split}\end{equation*}
and thus, since $\lim_{t\to\infty}\| U^{\varepsilon}(t) f \|_{\bm{\GG}_{j-1}}=0$ and $\| \cdot \|_{ \bm{\GG}^{\bullet}_{j}} \geqslant \| \cdot \|_{\bm{\GG}_j}$, we have
\begin{equation*}\begin{split}
\mathrm{Re}\, \dlla (\mathcal L - {\varepsilon} v \cdot \nabla_x) f, f \drra_{\bm{\GG}_j, {\varepsilon} }
\leqslant& - \delta \lambda_\mathcal B \| f \|_{ \bm{\GG}^{\bullet}_{j} }^2 + \delta K \| f \|_{ \bm{\GG}_{j-1}}^2 \\
&-\frac{\sigma}{{\varepsilon}^2} \int_0^\infty e^{2 \sigma t / {\varepsilon}^2} \left\| U^{\varepsilon}_\textnormal{kin}(t) f \right\|_{\bm{\GG}_{j-1}}^2 \mathrm{d} t - \frac{1}{2} \| f \|^2_{ \bm{\GG}_{j-1}} \\
\leqslant&
- \sigma \left( \delta \| f \|^2_{\bm{\GG}_j} + \frac{1}{{\varepsilon}^2} \int_0^\infty e^{2 \sigma t / {\varepsilon}^2} \left\| U^{\varepsilon}_\textnormal{kin}(t) f \right\|_{\bm{\GG}_{j-1}}^2 \mathrm{d} t \right) \\
& - \delta (\lambda_\mathcal B - \sigma ) \| f \|^2_{ \bm{\GG}^{\bullet}_{j} } - \left( \frac{1}{2} - \delta K \right) \| f \|^2_{ \mathbb{H}^s_x(\bm{X}^{\bullet}_{j-1}) }.
\end{split}\end{equation*}
We finally deduce, considering $\delta \leqslant (2K)^{-1}$ and letting $\mu = \delta(\lambda_\mathcal B - \sigma) > 0$
$$\mathrm{Re}\, \dlla (\mathcal L - {\varepsilon} v \cdot \nabla_x) f, f \drra_{\bm{\GG}_j , {\varepsilon} } \leqslant - \sigma \| f \|^2_{ \bm{\GG}_j, {\varepsilon} } - \mu \| f \|_{\bm{\GG}^{\bullet}_{j}}^2.$$
This concludes this step.
\step{3}{Proof of the nonlinear estimates}
Using the definition $\mathsf P^{\varepsilon}_\textnormal{kin} = \mathrm{Id} - \mathsf P^{\varepsilon}_\textnormal{hydro}$, we have
$$
\dlla \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q(f, g) , h \drra_{\bm{\GG}_j, {\varepsilon}} = \langle \mathcal Q(f, g), h \rangle_{\bm{\GG}_j} + R(f, g, h),$$
where $$
R(f, g, h) := - \left\langle \mathsf P^{\varepsilon}_\textnormal{hydro} \mathcal Q(f, g), h \right\rangle_{\bm{\GG}_j } + \frac{1}{{\varepsilon}^2} \int_0^\infty e^{2 \sigma t / {\varepsilon}^2} \left\langle U^{\varepsilon}_\textnormal{kin}(t) \mathcal Q(f, g), U^{\varepsilon}_\textnormal{kin}(t) h \right\rangle_{ \bm{\GG}_{j-1}} \, \mathrm{d} t.$$
On the one hand, the boundedness $\mathsf P^{\varepsilon}_\textnormal{hydro} \in \mathscr B\left( \bm{\GG}^{\circ}_{j-1} ; \HH \right) \subset \mathscr B\left( \bm{\GG}^{\circ}_{j-1} ;\bm{\GG}^{\circ}_{j} \right)$ implies
\begin{equation*}\begin{split} \left|\, \langle \mathsf P^{\varepsilon}_\textnormal{hydro} \mathcal Q(f, g), h \rangle_{\bm{\GG}_j} \right| & \lesssim \| h \|_{\bm{\GG}^{\bullet}_{j} } \| \mathsf P^{\varepsilon}_\textnormal{hydro} \mathcal Q(f, g) \|_{ \bm{\GG}^{\circ}_{j}} \\
& \lesssim \| h \|_{ \bm{\GG}^{\bullet}_{j}} \| \mathcal Q(f, g) \|_{ \bm{\GG}^{\circ}_{j-1}},
\end{split}\end{equation*}
and on the other hand, using Cauchy-Schwarz inequality and the integral estimates of Lemma~\ref{lem:decay_regularization_kinetic_semigroup}:
\begin{equation*}\begin{split}
\frac{1}{{\varepsilon}^2} \int_0^\infty e^{2 \sigma t / {\varepsilon}^2 } & \langle U^{\varepsilon}_\textnormal{kin}(t) \mathcal Q(f, g), U^{\varepsilon}_\textnormal{kin}(t) h \rangle_{ \bm{\GG}_{j-1}} \, \mathrm{d} t \\
& \leqslant \frac{1}{{\varepsilon}^2} \int_0^\infty \left( e^{\sigma t / {\varepsilon}^2} \| U^{\varepsilon}_\textnormal{kin}(t) \mathcal Q(f, g) \|_{ \bm{\GG}_{j-1}} \right) \left( e^{\sigma t / {\varepsilon}^2} \| U^{\varepsilon}_\textnormal{kin}(t) h \|_{ \bm{\GG}_{j-1}} \right) \mathrm{d} t \\
& \lesssim \| \mathcal Q(f, g) \|_{ \bm{\GG}^{\circ}_{j-1}} \| h \|_{ \bm{\GG}_{j-1} } \lesssim \| \mathcal Q(f, g) \|_{ \bm{\GG}^{\circ}_{j-1} } \| h \|_{ \bm{\GG}^{\bullet}_{j-1} }.
\end{split}\end{equation*}
Combining these estimates and using the injection $ \bm{\GG}^{\bullet}_{j}\hookrightarrow \bm{\GG}_{j-1}$, we deduce that
$$R(f, g, h) \lesssim \| h \|_{ \bm{\GG}^{\bullet}_{j} } \| \mathcal Q(f, g) \|_{ \bm{\GG}^{\circ}_{j-1} }.$$
Recalling that $ \bm{\GG}_{j-1}$ is defined so that the general non-closed estimate of $\mathcal Q$ in $\langle \cdot , \cdot \rangle_{\bm{\GG}^{\bullet}_{j-1} }$ only involves the norms of the space $ \bm{X}_{j} \hookrightarrow \bm{X}_{j-1}$ and $\bm{X}^{\bullet}_{j} \hookrightarrow \bm{X}_{j-1}$ (but not~$\bm{X}_{j+1}$ nor $\bm{X}^{\bullet}_{j+1}$), that is to say,
$$ \| \mathcal Q(f, g) \|_{ \bm{\GG}^{\circ}_{j-1} } \lesssim \| f \|_{ \bm{\GG}^{\bullet}_{j} } \| g \|_{ \bm{\GG}_j} + \| f \|_{ \bm{\GG}_j} \| g \|_{ \bm{\GG}^{\bullet}_{j}},$$
we finally end up with
$$
R(f, g, h) \lesssim \| h \|_{ \bm{\GG}^{\circ}_{j} } \left( \| f \|_{ \bm{\GG}^{\bullet}_{j} } \| g \|_{ \bm{\GG}_j} + \| f \|_{ \bm{\GG}_j } \| g \|_{ \bm{\GG}^{\bullet}_{j} } \right).
$$ This allows to conclude this step and the proof.
\end{proof}
\subsection{Stability estimates}
We study here the scheme \eqref{eq:scheme} in the kinetic-type time-position-velocity space $\pmb{\GGG}_j$ which corresponds to the space $\GGG$ introduced in Definition \ref{defi:NORMSPACES} with $E$ replaced with $\bm{X}_{j}$, i.e $\pmb{\GGG}_j$ is characterized by the norm:
\begin{equation}
\label{eq:defiTildeSSSl}
|\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{ \pmb{\GGG}_j}^2 := \sup_{0 \leqslant t < T} \, e^{2 \sigma t / {\varepsilon}^2 } \| f(t) \|_{\bm{\GG}_j}^2 + \frac{1}{{\varepsilon}^2} \int_0^T e^{2 \sigma t / {\varepsilon}^2} \| f(t) \|_{\bm{\GG}^{\bullet}_{j}}^2 \mathrm{d} t.
\end{equation}
We have the following estimates, valid for any $N\geq0$:
\begin{lem}\label{lem:stable-scheme} With the notations of Section \ref{scn:proof_hydrodynamic_limit_symmetrizable}, one can choose
$${\varepsilon} \ll c_3 \ll \eta \ll c_2 \ll 1, \qquad C_1 \approx 1$$
such that, for any $N \geqslant 0$ and $j = -1, 0$:
\begin{equation}
\label{eq:inductive_stability}
|\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{kin}, N} |\hskip-0.04cm|\hskip-0.04cm|_{\pmb{\GGG}_j} \leqslant C_1 \| \mathsf P^{\varepsilon}_\textnormal{kin} f_\textnormal{in} \|_{ \bm{\GG}_j} , \qquad |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{mix},N} |\hskip-0.04cm|\hskip-0.04cm|_{\FFF} \leqslant c_2, \qquad |\hskip-0.04cm|\hskip-0.04cm| g^{\varepsilon}_N |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} \leqslant c_3\,.
\end{equation}
\end{lem}
\begin{proof} The proof of the Lemma is made by induction over $N$.
Thanks to \eqref{eq:equivE}, it will turn useful to estimate the various norms rather with $\|\cdot\|_{ \bm{\GG}_j, {\varepsilon}}$.
The estimates \eqref{eq:inductive_stability} are satisfied for $N = 0$. Let us assume they hold at rank $N-1$ for some~$N \geqslant 1$ and deduce them at rank $N$.
We recall that $f^{\varepsilon}_{\textnormal{kin},N}$ is a solution to \eqref{eq:scheme} where, thanks to Proposition \ref{prop:dissipative_kinetic_inner_product}, it holds
\begin{gather*}
\frac{1}{{\varepsilon}^2} \dlla (\mathcal L - {\varepsilon} v \cdot \nabla_x) f^{\varepsilon}_{\textnormal{kin}, N}, f^{\varepsilon}_{\textnormal{kin}, N} \drra_{ \bm{\GG}_j, {\varepsilon}} \leqslant - \frac{\mu}{{\varepsilon}^2} \| f^{\varepsilon}_{\textnormal{kin}, N} \|^2_{ \bm{\GG}^{\bullet}_{j}} - \frac{\sigma}{{\varepsilon}^2} \| f^{\varepsilon}_{\textnormal{kin}, N} \|^2_{ \bm{\GG}_j, {\varepsilon}}\,,
\end{gather*}
as well as
\begin{equation*}\begin{split}
\dlla \mathsf P^{\varepsilon}_\textnormal{kin} &\mathcal Q( f^{\varepsilon}_{\textnormal{kin}, N-1} , f^{\varepsilon}_{\textnormal{kin}, N}), f^{\varepsilon}_{\textnormal{kin}, N} \drra_{ \bm{\GG}_j, {\varepsilon}} \\
& \lesssim \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{ \bm{\GG}^{\bullet}_{j}} \left( \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{ \bm{\GG}^{\bullet}_{j}} \| f^{\varepsilon}_{\textnormal{kin}, N-1} \|_{ \bm{\GG}_{j-1}} + \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{ \bm{\GG}_j} \| f^{\varepsilon}_{\textnormal{kin}, N-1} \|_{ \bm{\GG}^{\bullet}_{j}} \right).
\end{split}\end{equation*}
Still using Proposition \ref{prop:dissipative_kinetic_inner_product}, the coupling term is estimated thanks to the non-closed estimate combined with the injections $\HH^{\bullet} \hookrightarrow \bm{\GG}^{\bullet}_{j+1}$ and $\HH \hookrightarrow \bm{\GG}_{j+1}$, and the closed estimate respectively:
\begin{equation*}\begin{split}
\dlla \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q( & f^{\varepsilon}_{\textnormal{kin}, N} , f^{\varepsilon}_{\textnormal{hydro}, N-1} + f^{\varepsilon}_{\textnormal{mix}, N-1}), f^{\varepsilon}_{\textnormal{kin}, N} \drra_{ \bm{\GG}_j, {\varepsilon}} \\
&\lesssim \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{ \bm{\GG}^{\bullet}_{j}}^2 \left( \| f^{\varepsilon}_{\textnormal{hydro}, N-1} \|_{\HH} + \| f^{\varepsilon}_{\textnormal{mix}, N-1} \|_{\HH} \right) \\
& + \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{ \bm{\GG}^{\bullet}_{j}} \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{ \bm{\GG}_j} \left( \| f^{\varepsilon}_{\textnormal{hydro}, N-1} \|_{\HH^{\bullet}} + \| f^{\varepsilon}_{\textnormal{mix}, N-1} \|_{\HH^{\bullet}} \right),
\end{split}\end{equation*}
and
\begin{equation*}\begin{split}
\dlla \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q( & f^{\varepsilon}_{\textnormal{hydro}, N-1} + f^{\varepsilon}_{\textnormal{mix}, N-1} , f^{\varepsilon}_{\textnormal{kin}, N}), f^{\varepsilon}_{\textnormal{kin}, N} \drra_{ \bm{\GG}_j, {\varepsilon}} \\
\lesssim & \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{ \bm{\GG}^{\bullet}_{j}}^2 \left( \| f^{\varepsilon}_{\textnormal{hydro}, N-1} \|_{\HH} + \| f^{\varepsilon}_{\textnormal{mix}, N-1} \|_{\HH} \right) \\
& + \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{ \bm{\GG}_j} \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{ \bm{\GG}^{\bullet}_{j}} \left( \| f^{\varepsilon}_{\textnormal{hydro}, N-1} \|_{\HH^{\bullet}} + \| f^{\varepsilon}_{\textnormal{mix}, N-1} \|_{\HH^{\bullet}} \right).
\end{split}\end{equation*}
Note that the second part of both previous estimates coincide. Put together, and multiplying by $e^{2 \sigma t / {\varepsilon}^2}$, we have the energy estimate
\begin{equation}\begin{split}\label{eq:I_1--I_4}
\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d} t} \left( e^{2 \sigma t / {\varepsilon}^2} \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{ \bm{\GG}_j, {\varepsilon}}^2 \right) + \frac{\mu}{{\varepsilon}^2} e^{2 \sigma t / {\varepsilon}^2} \| f^{\varepsilon}_{\textnormal{kin}, N} \|^2_{ \bm{\GG}^{\bullet}_{j}}\lesssim \mathscr{I}_{1}(t)+\mathscr{I}_2(t)+\mathscr{I}_3(t)+\mathscr{I}_4(t)\end{split}\end{equation}
where we introduced
\begin{equation*}\begin{split}
\mathscr{I}_1&=\frac{1}{{\varepsilon}} e^{2 \sigma t / {\varepsilon}^2} \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{ \bm{\GG}^{\bullet}_{j}}\bigg( \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{ \bm{\GG}^{\bullet}_{j}} \| f^{\varepsilon}_{\textnormal{kin}, N-1} \|_{\bm{\GG}_j} + \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}_j} \| f^{\varepsilon}_{\textnormal{kin}, N-1} \|_{\bm{\GG}^{\bullet}_{j}} \bigg) \\
\mathscr{I}_2&= \frac{1}{{\varepsilon}} e^{2 \sigma t / {\varepsilon}^2} \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}^{\bullet}_{j}} \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}_j} \left( \| f^{\varepsilon}_{\textnormal{hydro}, N-1} \|_{\HH^{\bullet}} + \| f^{\varepsilon}_{\textnormal{mix}, N-1} \|_{\HH^{\bullet}} \right) \\
\mathscr{I}_3&=\frac{1}{{\varepsilon}} e^{2 \sigma t / {\varepsilon}^2} \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}^{\bullet}_{j}}^2 \left( \| f^{\varepsilon}_{\textnormal{hydro}, N-1} \|_{\HH} + \| f^{\varepsilon}_{\textnormal{mix}, N-1} \|_{\HH} \right) \\
\mathscr{I}_4&=\frac{1}{{\varepsilon}} e^{2 \sigma t / {\varepsilon}^2} \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}_j} \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}^{\bullet}_{j}} \left( \| f^{\varepsilon}_{\textnormal{hydro}, N-1} \|_{\HH^{\bullet}} + \| f^{\varepsilon}_{\textnormal{mix}, N-1} \|_{\HH^{\bullet}} \right).
\end{split}\end{equation*}
One easily sees that
$$\int_0^T\mathscr{I}_1(t) \mathrm{d} t \lesssim {\varepsilon} |\hskip-0.04cm|\hskip-0.04cm| f_{\textnormal{kin},N}^{\varepsilon}|\hskip-0.04cm|\hskip-0.04cm|_{\pmb{\GGG}_j}^2\,|\hskip-0.04cm|\hskip-0.04cm| f_{\textnormal{kin},N-1}^{\varepsilon}|\hskip-0.04cm|\hskip-0.04cm|_{\pmb{\GGG}_j}$$
where we recall the definition \eqref{eq:defiTildeSSSl} and the fact that $\|\cdot\|_{\bm{\GG}_j} \lesssim \|\cdot\|_{\bm{\GG}^{\bullet}_{j}}$.
We write the time integral of the second term as follows
\begin{multline*}
\int_0^T \mathscr{I}_2(t)\mathrm{d} t={\varepsilon} \int_0^T \left[ \frac{1}{{\varepsilon}} e^{\sigma t / {\varepsilon}^2} \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}^{\bullet}_{j}} \right] \left[ \frac{1}{{\varepsilon}} e^{\sigma t / {\varepsilon}^2 } \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}_j} \right] \| f^{\varepsilon}_{\textnormal{hydro}, N-1} \|_{\HH^{\bullet}} \mathrm{d} t\\
+{\varepsilon} \int_0^T \left[ \frac{1}{{\varepsilon}} e^{\sigma t / {\varepsilon}^2} \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}^{\bullet}_{j}} \right]\left[ \frac{1}{{\varepsilon}} \| f^{\varepsilon}_{\textnormal{mix}, N-1} \|_{\HH^{\bullet}} \right] \left[ e^{\sigma t / {\varepsilon}^2 } \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}_j} \right] \mathrm{d} t
\end{multline*}
where, in each integral, the first two terms in brackets belong to $L^2(0,T)$ whereas the third one belongs to $L^\infty(0,T)$. Then, it is easy to deduce that
$$\int_0^T \mathscr{I}_2(t) \mathrm{d} t\lesssim {\varepsilon} w_{ f_\textnormal{NS}, \eta }(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{kin}, N} |\hskip-0.04cm|\hskip-0.04cm|^2_{\pmb{\GGG}_j} \left( |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{hydro}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_\HHH + |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{mix}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_\FFF \right)\,
$$
where we used again that $\|\cdot\|_{\bm{\GG}_j} \lesssim \|\cdot\|_{\bm{\GG}^{\bullet}_{j}}$. One also sees that
\begin{multline*}
\int_0^T\mathscr{I}_3(t)\mathrm{d} t \lesssim
{\varepsilon} \int_0^T \left[ \frac{1}{{\varepsilon}} e^{\sigma t / {\varepsilon}^2} \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}^{\bullet}_{j}} \right]^2 \left( \| f^{\varepsilon}_{\textnormal{hydro}, N-1} \|_{\HH} + \| f^{\varepsilon}_{\textnormal{mix}, N-1} \|_{\HH} \right) \mathrm{d} t \\
\lesssim {\varepsilon} w_{ f_\textnormal{NS}, \eta }(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{kin}, N} |\hskip-0.04cm|\hskip-0.04cm|^2_{\pmb{\GGG}_j} \left( |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{hydro}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} + |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{mix}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{\FFF} \right)\,.\end{multline*}
Finally, the term involving $\mathscr{I}_4$ is dealt with as the one involving $\mathscr{I}_2$ writing
\begin{multline*}
\int_0^T \mathscr{I}_4(t)\mathrm{d} t = {\varepsilon} \int_0^T \left[ \frac{1}{{\varepsilon}} e^{\sigma t / {\varepsilon}^2} \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}_j} \right] \left[ \frac{1}{{\varepsilon}} e^{\sigma t / {\varepsilon}^2} \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}^{\bullet}_{j}} \right] \| f^{\varepsilon}_{\textnormal{hydro}, N-1} \|_{\HH^{\bullet}} \mathrm{d} t \\
+ {\varepsilon} \int_0^T \left[ \frac{1}{{\varepsilon}} e^{\sigma t / {\varepsilon}^2} \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}^{\bullet}_{j}} \right] \left[ \frac{1}{{\varepsilon}} \| f^{\varepsilon}_{\textnormal{mix}, N-1} \|_{\HH^{\bullet}} \right] \left[ e^{\sigma t / {\varepsilon}^2} \| f^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}_j} \right]\mathrm{d} t
\end{multline*}
where for both integrals, the first two terms in brackets belong to $L^2(0,T)$ whereas the third one belongs to $L^\infty(0,T)$. This gives easily
$$\int_0^T \mathscr{I}_4(t)\mathrm{d} t \lesssim {\varepsilon} w_{ f_\textnormal{NS}, \eta }(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{kin}, N} |\hskip-0.04cm|\hskip-0.04cm|^2_{\pmb{\GGG}_j} \left( |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{hydro}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} + |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{mix}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{\FFF} \right).$$
Coming back to \eqref{eq:I_1--I_4}, we finally deduce the recursive estimate
\begin{equation*}\begin{split}
|\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{kin}, N} |\hskip-0.04cm|\hskip-0.04cm|^2_{\pmb{\GGG}_j} \lesssim {\varepsilon} & |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{kin}, N} |\hskip-0.04cm|\hskip-0.04cm|^2_{ \pmb{\GGG}_j} |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{kin}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{ \pmb{\GGG}_j} + \| \mathsf P^{\varepsilon}_\textnormal{kin} f_\textnormal{in} \|^2_{\mathbb{H}^s_x(\bm{X}_j)}\\
&+ {\varepsilon} w_{ f_\textnormal{NS}, \eta }(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{kin}, N} |\hskip-0.04cm|\hskip-0.04cm|^2_{\pmb{\GGG}_j} \left( |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{hydro}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} + |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{mix}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{\FFF} \right),
\end{split}\end{equation*}
and using the inductive hypothesis \eqref{eq:inductive_stability}
\begin{equation*}
|\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{kin}, N} |\hskip-0.04cm|\hskip-0.04cm|^2_{\pmb{\GGG}_j} \lesssim {\varepsilon} \Big[ C_1 + w_{ f_\textnormal{NS}, \eta }(T)^{-1} \left( 1 + c_2 + c_3 \right) \Big] |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{kin}, N} |\hskip-0.04cm|\hskip-0.04cm|^2_{ \pmb{\GGG}_j} + \| \mathsf P^{\varepsilon}_\textnormal{kin} f_\textnormal{in} \|^2_{\tilde{\GG}_j},
\end{equation*}
so that, for ${\varepsilon}$ small enough, there holds
$$
|\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{kin}, N} |\hskip-0.04cm|\hskip-0.04cm|_{\pmb{\GGG}_j} \lesssim \| \mathsf P^{\varepsilon}_\textnormal{kin} f_\textnormal{in} \|_{\pmb{\GGG}_j},
$$
which allows to deduce the stability estimate at rank $N$ for $f^{\varepsilon}_{\textnormal{kin}, N}$.
The proof of the stability estimates for $f^{\varepsilon}_{\textnormal{mix}, N}$ and $g^{\varepsilon}_N$ is the same as in Section \ref{scn:proof_hydrodynamic_limit_symmetrizable}. This concludes the proof.\end{proof}
\subsection{Convergence of the scheme}
The convergence will be proved in the larger space $\pmb{\GGG}_{-1}$ using, of course, the stability estimate in $\pmb{\GGG}_{-1} $, but also those in $\pmb{\GGG}_0$ because of the non-closed estimates. To do so, we denote the difference and sum of successive approximate solutions as
\begin{gather*}
d^{\varepsilon}_{\textnormal{kin}, N} := f^{\varepsilon}_{\textnormal{kin}, N} - f^{\varepsilon}_{\textnormal{kin}, N-1}, \qquad d^{\varepsilon}_{\textnormal{mix}, N} := f^{\varepsilon}_{\textnormal{mix}, N} - f^{\varepsilon}_{\textnormal{mix}, N-1}\\
d^{\varepsilon}_{\textnormal{hydro}, N} := f^{\varepsilon}_{\textnormal{hydro}, N} - f^{\varepsilon}_{\textnormal{hydro}, N-1} = g^{\varepsilon}_N - g^{\varepsilon}_{N-1}\,.
\end{gather*}
One has the following recursive estimate.
\begin{prop} With the notations of Lemma \ref{lem:stable-scheme}, up to reducing again
$${\varepsilon} \ll c_3 \ll \eta \ll c_2 \ll 1,$$
the following estimate
\begin{equation*}\begin{split}
|\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{kin}, N} |\hskip-0.04cm|\hskip-0.04cm|_{ \pmb{\GGG}_{-1}} & + |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{hydro}, N} |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH } + |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{mix}, N} |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } \\
&\leqslant \frac{1}{2} \Bigg( |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{kin}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{\pmb{\GGG}_{-1}} + |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{hydro}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} + |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{mix}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{\FFF}\Bigg),
\end{split}\end{equation*}
holds for any $N\ge0.$
\end{prop}
\begin{proof} As for the proof of Lemma \ref{lem:stable-scheme}, the difficulty lies in estimating $d^{\varepsilon}_{\textnormal{kin},N}$ which solves
\begin{equation*}
\begin{cases}
\displaystyle
\begin{aligned}
\partial_t d^{\varepsilon}_{\textnormal{kin}, N} = & \frac{1}{{\varepsilon}^2} \big(\mathcal L - {\varepsilon} v \cdot \nabla_x\big) d^{\varepsilon}_{\textnormal{kin}, N} \\
& + \frac{1}{{\varepsilon}} \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q(f^{\varepsilon}_{\textnormal{kin}, N-1}, d^{\varepsilon}_{\textnormal{kin}, N}) + \frac{1}{{\varepsilon}} \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q(d^{\varepsilon}_{\textnormal{kin}, N-1}, f^{\varepsilon}_{\textnormal{kin}, N-1}) \\
& + \frac{2}{{\varepsilon}} \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q^\mathrm{sym}(f^{\varepsilon}_{\textnormal{hydro}, N-1} + f^{\varepsilon}_{\textnormal{mix}, N-1} , d^{\varepsilon}_{\textnormal{kin}, N}) \\
&+ \frac{2}{{\varepsilon}} \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q^\mathrm{sym}(f^{\varepsilon}_{\textnormal{kin}, N-1}, d^{\varepsilon}_{\textnormal{hydro}, N-1} + d^{\varepsilon}_{\textnormal{mix}, N-1} ),
\end{aligned}\\
d^{\varepsilon}_{\textnormal{kin}, N}(0) = 0.
\end{cases}
\end{equation*}
We use as previously the equivalent norms $\|\cdot\|_{\bm{\GG}_{-1}, {\varepsilon}}$ which allows the use of dissipativity:
\begin{gather*}
\frac{1}{{\varepsilon}^2} \dlla (\mathcal L - {\varepsilon} v \cdot \nabla_x) d^{\varepsilon}_{\textnormal{kin}, N}, d^{\varepsilon}_{\textnormal{kin}, N} \drra_{\bm{\GG}_{-1}, {\varepsilon}} \leqslant - \frac{\sigma}{{\varepsilon}^2} \| d^{\varepsilon}_{\textnormal{kin}, N} \|^2_{\bm{\GG}_{-1}, {\varepsilon}} - \frac{\mu}{{\varepsilon}^2} \| d^{\varepsilon}_{\textnormal{kin}, N} \|^2_{\bm{\GG}^{\bullet}_{-1}},
\end{gather*}
as well as the closed estimate:
\begin{equation*}\begin{split}
\dlla \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q( f^{\varepsilon}_{\textnormal{kin}, N-1} , d^{\varepsilon}_{\textnormal{kin}, N}), d^{\varepsilon}_{\textnormal{kin}, N} \drra_{\bm{\GG}_{-1}, {\varepsilon}}
&\lesssim \| d^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}^{\bullet}_{-1}}^2 \| f^{\varepsilon}_{\textnormal{kin}, N-1} \|_{\bm{\GG}_{-1}} \\
&\phantom{++++} + \| d^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}^{\bullet}_{-1}} \| d^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}_{-1}} \| f^{\varepsilon}_{\textnormal{kin}, N-1} \|_{\bm{\GG}^{\bullet}_{-1}},
\end{split}\end{equation*}
and the non-closed estimate involving the $\bm{\GG}_{0} $ and $\bm{\GG}^{\bullet}_{0}$--norms of $f^{\varepsilon}_{\textnormal{kin}, N-1}$:
\begin{equation*}\begin{split}
\dlla \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q( d^{\varepsilon}_{\textnormal{kin}, N-1} &, f^{\varepsilon}_{\textnormal{kin}, N-1}), d^{\varepsilon}_{\textnormal{kin}, N} \drra_{\bm{\GG}_{-1}, {\varepsilon}}\\
&\lesssim \| d^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}^{\bullet}_{-1}} \| d^{\varepsilon}_{\textnormal{kin}, N-1} \|_{\bm{\GG}_{-1}} \| f^{\varepsilon}_{\textnormal{kin}, N-1} \|_{\bm{\GG}^{\bullet}_{0}} \\
& \phantom{++++} + \| d^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}^{\bullet}_{-1}} \| d^{\varepsilon}_{\textnormal{kin}, N-1} \|_{\bm{\GG}^{\bullet}_{-1}} \| f^{\varepsilon}_{\textnormal{kin}, N-1} \|_{\bm{\GG}_{0}}.
\end{split}\end{equation*}
The coupling term involving $d^{\varepsilon}_{\textnormal{kin}, N}$ is estimated using both the closed and non-closed estimates, together with the injections $\HH \hookrightarrow \bm{\GG}_{0}$ and $\HH^{\bullet} \hookrightarrow \bm{\GG}^{\bullet}_{0}$:
\begin{equation*}\begin{split}
\dlla \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q^\mathrm{sym}&\left( f^{\varepsilon}_{\textnormal{hydro}, N-1} + f^{\varepsilon}_{\textnormal{mix}, N-1} , d^{\varepsilon}_{\textnormal{kin}, N} \right), d^{\varepsilon}_{\textnormal{kin}, N} \drra_{\bm{\GG}_{-1}, {\varepsilon}} \\
&\lesssim \| d^{\varepsilon}_{\textnormal{kin}, N} \|_{ \bm{\GG}^{\bullet}_{-1}}^2 \left( \| f^{\varepsilon}_{\textnormal{hydro}, N-1} \|_{ \HH } + \| f^{\varepsilon}_{\textnormal{mix}, N-1} \|_{ \HH } \right) \\
& + \| d^{\varepsilon}_{\textnormal{kin}, N} \|_{ \bm{\GG}_{-1} } \| d^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}_{-1} } \left( \| f^{\varepsilon}_{\textnormal{hydro}, N-1} \|_{ \HH^{\bullet} } + \| f^{\varepsilon}_{\textnormal{mix}, N-1} \|_{ \HH^{\bullet} } \right),
\end{split}\end{equation*}
and the coupling term involving $f^{\varepsilon}_{\textnormal{kin}, N-1}$ using the non-closed estimate, together with the injections $\HH \hookrightarrow \bm{\GG}_{0} \hookrightarrow \bm{\GG}_{-1}$ and $\HH^{\bullet} \hookrightarrow \bm{\GG}^{\bullet}_{0} \hookrightarrow \bm{\GG}^{\bullet}_{-1}$:
\begin{equation*}\begin{split}
\dlla \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q^\mathrm{sym}&\left( d^{\varepsilon}_{\textnormal{hydro}, N-1} + d^{\varepsilon}_{\textnormal{mix}, N-1} , f^{\varepsilon}_{\textnormal{kin}, N-1} \right), d^{\varepsilon}_{\textnormal{kin}, N} \drra_{\bm{\GG}_{-1}, {\varepsilon}} \\
&\lesssim \| d^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}^{\bullet}_{-1} } \| f^{\varepsilon}_{\textnormal{kin}, N-1} \|_{ \bm{\GG}^{\bullet}_{0} } \left( \| d^{\varepsilon}_{\textnormal{hydro}, N-1} \|_{ \HH } + \| d^{\varepsilon}_{\textnormal{mix}, N-1} \|_{ \HH } \right) \\
&\phantom{+++} + \| d^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}^{\bullet}_{-1} } \| f^{\varepsilon}_{\textnormal{kin}, N-1} \|_{\bm{\GG}_{0} } \left( \| d^{\varepsilon}_{\textnormal{hydro}, N-1} \|_{ \HH^{\bullet} } + \| d^{\varepsilon}_{\textnormal{mix}, N-1} \|_{ \HH^{\bullet} } \right)\,.
\end{split}\end{equation*}
Multiplying these estimates by $e^{2 \sigma t / {\varepsilon}^2}$, we get the energy estimate
\begin{equation*}\begin{split}
\frac{1}{2} \frac{\mathrm{d} }{\mathrm{d} t} \Big( e^{2 \sigma t / {\varepsilon}^2} & \| d^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}_{-1}, {\varepsilon}}^2 \Big) + \frac{\mu}{{\varepsilon}^2} e^{2 \sigma t / {\varepsilon}^2} \| d^{\varepsilon}_{\textnormal{kin}, N} \|^2_{\bm{\GG}^{\bullet}_{-1}} \\
\lesssim & \frac{1}{{\varepsilon}} e^{2 \sigma t / {\varepsilon}^2} \| d^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}^{\bullet}_{-1}}^2 \| f^{\varepsilon}_{\textnormal{kin}, N-1} \|_{\bm{\GG}_{-1}} \\
&+ \frac{1}{{\varepsilon}} e^{2 \sigma t / {\varepsilon}^2} \| d^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}^{\bullet}_{-1}} \Bigg(\| d^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}_{0}} \| f^{\varepsilon}_{\textnormal{kin}, N-1} \|_{\bm{\GG}_{-1}} \\
&\phantom{++++} + \| d^{\varepsilon}_{\textnormal{kin}, N-1} \|_{\bm{\GG}_{-1}} \| f^{\varepsilon}_{\textnormal{kin}, N-1} \|_{\bm{\GG}^{\bullet}_{0}}
+ \| d^{\varepsilon}_{\textnormal{kin}, N-1} \|_{\bm{\GG}^{\bullet}_{-1}} \| f^{\varepsilon}_{\textnormal{kin}, N-1} \|_{\bm{\GG}^{\bullet}_{1}} \Bigg)\\
& + \frac{1}{{\varepsilon}} e^{2 \sigma t / {\varepsilon}^2} \| d^{\varepsilon}_{\textnormal{kin}, N} \|_{ \bm{\GG}^{\bullet}_{-1} }^2 \left( \| f^{\varepsilon}_{\textnormal{hydro}, N-1} \|_{ \HH } + \| f^{\varepsilon}_{\textnormal{mix}, N-1} \|_{ \HH } \right) \\
& + \frac{1}{{\varepsilon}} e^{2 \sigma t / {\varepsilon}^2} \| d^{\varepsilon}_{\textnormal{kin}, N} \|_{ \bm{\GG}_{-1} } \| d^{\varepsilon}_{\textnormal{kin}, N} \|_{ \bm{\GG}_{-1} } \left( \| f^{\varepsilon}_{\textnormal{hydro}, N-1} \|_{ \HH^{\bullet} } + \| f^{\varepsilon}_{\textnormal{mix}, N-1} \|_{ \HH^{\bullet} } \right) \\
& + \frac{1}{{\varepsilon}} e^{2 \sigma t / {\varepsilon}^2} \| d^{\varepsilon}_{\textnormal{kin}, N} \|_{\bm{\GG}^{\bullet}_{-1} } \| f^{\varepsilon}_{\textnormal{kin}, N-1} \|_{ \bm{\GG}^{\bullet}_{0} } \left( \| d^{\varepsilon}_{\textnormal{hydro}, N-1} \|_{ \HH } + \| d^{\varepsilon}_{\textnormal{mix}, N-1} \|_{ \HH } \right) \\
& + \frac{1}{{\varepsilon}} e^{2 \sigma t / {\varepsilon}^2} \| d^{\varepsilon}_{\textnormal{kin}, N} \|_{ \bm{\GG}^{\bullet}_{-1} } \| f^{\varepsilon}_{\textnormal{kin}, N-1} \|_{ \bm{\GG}_{0} } \left( \| d^{\varepsilon}_{\textnormal{hydro}, N-1} \|_{ \HH^{\bullet} } + \| d^{\varepsilon}_{\textnormal{mix}, N-1} \|_{ \HH^{\bullet} } \right).
\end{split}\end{equation*}
Arguing as in the proof of Lemma \ref{lem:stable-scheme}, we then obtain
\begin{align*}
|\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{kin}, N} |\hskip-0.04cm|\hskip-0.04cm|^2_{ \pmb{\GGG}_{-1} } &\lesssim {\varepsilon} |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{kin}, N} |\hskip-0.04cm|\hskip-0.04cm|_{\pmb{\GGG}_{-1}}^2 |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{kin}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{\pmb{\GGG}_{-1} } + {\varepsilon} |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{kin}, N} |\hskip-0.04cm|\hskip-0.04cm|_{\pmb{\GGG}_{-1}} |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{kin}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{\pmb{\GGG}_{-1}} |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{kin}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{ \pmb{\GGG}_{0} } \\
& + {\varepsilon} w_{ f_\textnormal{NS}, \eta }(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{kin}, N} |\hskip-0.04cm|\hskip-0.04cm|_{\pmb{\GGG}_{-1}}^2 \left( |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{hydro}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_\HHH + |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{mix}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_\FFF \right) \\
& + {\varepsilon} w_{ f_\textnormal{NS}, \eta }(T)^{-1} |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{kin}, N} |\hskip-0.04cm|\hskip-0.04cm|_{\pmb{\GGG}_{-1}} |\hskip-0.04cm|\hskip-0.04cm| f^{\varepsilon}_{\textnormal{kin}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{\pmb{\GGG}_{0}} \left( |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{hydro}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_\HHH + |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{mix}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_\FFF \right),
\end{align*}
and thus using the stability estimates \eqref{eq:inductive_stability}
$$|\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{kin}, N} |\hskip-0.04cm|\hskip-0.04cm|_{ \pmb{\GGG}_{-1} } \lesssim {\varepsilon} w_{ f_\textnormal{NS}, \eta }(T)^{-1} \left( |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{kin}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{ \pmb{\GGG}_{-1}} + |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{hydro}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} + |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{mix}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{\FFF} \right)$$
i.e., and assuming ${\varepsilon}$ small enough,
$$|\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{kin}, N} |\hskip-0.04cm|\hskip-0.04cm|_{ \pmb{\GGG}_{-1} } \leqslant \frac{1}{4} \left( |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{kin}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{ \pmb{\GGG}_{-1}} + |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{hydro}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} + |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{mix}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{\FFF} \right).$$
Arguing in the very same way as in Section \ref{scn:mapping_contraction}, one also shows
\begin{align*}
|\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{hydro}, N} |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH } + |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{mix}, N} |\hskip-0.04cm|\hskip-0.04cm|_{ \FFF } \leqslant & \frac{1}{4} \bigg( |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{kin}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{\pmb{\GGG}_{-1}} + |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{hydro}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{\HHH} + |\hskip-0.04cm|\hskip-0.04cm| d^{\varepsilon}_{\textnormal{mix}, N-1} |\hskip-0.04cm|\hskip-0.04cm|_{\FFF} \bigg)\,.
\end{align*}
Summing up the last two estimates yields the result.
\end{proof}
The above result allows to prove in a standard way the convergence of the approximate solutions $\left\{( f^{\varepsilon}_{\textnormal{kin}, N} , f^{\varepsilon}_{\textnormal{mix}, N} , g^{\varepsilon}_N )\right\}_N$ to some limit $( f^{\varepsilon}_{\textnormal{kin}} , f^{\varepsilon}_{\textnormal{mix}} , g^{\varepsilon} )$ solving the system
\begin{equation*}
\begin{cases}
\displaystyle
\begin{aligned}
\partial_t f^{\varepsilon}_\textnormal{kin} = \frac{1}{{\varepsilon}^2} \left(\mathcal L - {\varepsilon} v \cdot \nabla_x\right) f^{\varepsilon}_\textnormal{kin} & + \frac{1}{{\varepsilon}} \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q(f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{kin}) \\
& + \frac{2}{{\varepsilon}} \mathsf P^{\varepsilon}_\textnormal{kin} \mathcal Q^\mathrm{sym}(f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{hydro} + f^{\varepsilon}_\textnormal{mix}), \quad f^{\varepsilon}_\textnormal{kin}(0) = \mathsf P^{\varepsilon}_\textnormal{kin} f_\textnormal{in},
\end{aligned} \\
f^{\varepsilon}_\textnormal{mix} = \Psi^{\varepsilon}_\textnormal{kin}\left[f^{\varepsilon}_\textnormal{mix} + f^{\varepsilon}_\textnormal{hydro} , f^{\varepsilon}_\textnormal{mix} + f^{\varepsilon}_\textnormal{hydro} \right],\\
g^{\varepsilon} = \Phi^{\varepsilon}[ f^{\varepsilon}_\textnormal{kin}, f^{\varepsilon}_\textnormal{mix} ] g^{\varepsilon} + \Psi^{\varepsilon}_\textnormal{hydro}(g^{\varepsilon}, g^{\varepsilon}) + \mathcal S^{\varepsilon}.
\end{cases}
\end{equation*}
A similar argument as the one from Section \ref{scn:uniqueness_simmetrizable} performed in $\pmb{\GGG}_{-1}$ yields the uniqueness, which implies the uniqueness in $\pmb{\GGG}_{0} = \GGG$ and achieves the proof of Theorem \ref{thm:hydrodynamic_limit} under Assumptions \ref{BED}.
\appendix
\section{About Assumptions \ref{L1}--\ref{L4} and \ref{Bortho}-\ref{Bbound}}
\label{sec:Landau-Boltz}
The various assumptions \ref{L1}--\ref{L4} and \ref{Bortho}--\ref{Bisotrop}, as well as the ``enlargement ones'' \ref{LE}, \ref{BE} and \ref{BED} were identified as being the properties shared by the Boltzmann and Landau equations in a close to equilibrium setting. We specify in this section that the aforementioned assumptions are proven in the literature, with precise references.
\subsection{The case of the classical Boltzmann equation}
Let us recall that, in the case of the classical Boltzmann equations, the distribution $\mu$ can be taken assumed to be the centered maxwellian:
$$\mu(v) := (2 \pi)^{-d/2} \exp\left( - \frac{|v|^2}{2} \right), \qquad E = d, \quad K = 1 + \frac{2}{d} \, .$$
The linear operator $\mathcal L$ is the linearized nonlinear operator $\mathcal Q$ around $\mu$:
$$\mathcal L f := \mathcal Q(\mu, f) + \mathcal Q(\mu, f)$$
where $\mathcal Q$ is defined as
$$\mathcal Q(f, f) := \int_{\mathbb R^3_{v_*} \times \mathbb S^{d-1}_\sigma } | v - v_* |^\gamma b(\cos \theta) \Big( f(v') f(v'_*) - f(v) f(v_*) \Big) \mathrm{d} \sigma \mathrm{d} v_* \, ,$$
for some parameter $\gamma \in (-3, 1)$ and some positive function $b$ smooth on $(0, 1]$, and the pre-collisional velocities $v', v_*'$ as well as the deviation angle are defined as
$$v' = \frac{v+v_*}{2} + \sigma \frac{|v - v_* |}{2}, \qquad v' = \frac{v+v_*}{2} - \sigma \frac{|v - v_* |}{2}, \qquad \cos \theta := \sigma \cdot \frac{v-v_*}{ | v - v_* | }$$
and illustrated in Figure \ref{fig:collisional geometry}.
\begin{figure}
\caption{On the left, the representation of a binary collision. On the right, an illustration of the so-called \emph{collisional geometry}, representing the distribution of the pre-collisional velocities $(v', v'_*)$ and the post-collisional ones $(v, v_*)$ on the circle centered about (the conserved mean velocity) $\frac{v + v_*}{2}$ of radius (the conserved relative velocity) $|v-v_*|$ in the plane $\Span(v-v_*, \sigma)$. This representation allows to visualize the deviation angle $\theta$.}
\label{fig:collisional geometry}
\end{figure}
\subsubsection{The cutoff case}
One talks about an \emph{angular cutoff assumption} when the function $b$ is assumed to be well-behaved enough close to $\theta = 0$ to satisfy some integrability property. Concerning our basics assumptions \ref{L1}--\ref{L4} and \ref{Bortho}--\ref{Bisotrop}, one considers hard or Maxwell potentials under a mild cutoff assumption:
$$\gamma \geqslant 0, \qquad \sin(\theta) b(\cos \theta) \in L^1_\theta\left( [-1, 1] \right) .$$
The hierarchy of spaces $H_j$ and the dissipation space $H^{\bullet}$ are defined as
$$H_j := L^2\left( \mu^{-1} \langle v \rangle^{2j} \mathrm{d} v \right), \quad H^{\bullet} := L^2\left( \mu^{-1} \langle v \rangle^{\gamma/2} \mathrm{d} v\right),$$
and the dissipation estimate is given by the interpolation of the following two estimates coming respectively from Hilbert-Grad's splitting (\cite{H1912, G1963}) and the spectral gap estimate \cite[Theorem 1.1]{BM2005}:
$$\langle \mathcal L g, g \rangle_H \leqslant - c \| g \|^2_{H^{\bullet}} + C \| g \|_{H}^2, \qquad \langle \mathcal L g, g \rangle_H \leqslant - c \| g \|^2_{H}.$$
The splitting we consider is
$$\mathcal B g := \mathcal L g- M \mathbf{1}_{| v | \leqslant M} g, \qquad \mathcal A g := M \mathbf{1}_{| v | \leqslant M} g,$$
for a large enough $M > 0$, and it satisfies \ref{L4} thanks to the following weighted estimates coming from \cite[Proof of Lemma 3.3 for $\beta = 0$]{G2006}:
$$\langle \mathcal L g, g \rangle_{H_j} \leqslant - c \| \langle v \rangle^{\gamma/2} g \|^2_{H_j} + C \| \langle v \rangle^{\gamma / 2} f \|_{H_0}^2,$$
for some $c, C > 0$. The nonlinear estimate \ref{Bbound} is given by \cite[Lemma 4.1]{G2004}.
The ``enlarged'' assumptions \ref{LE} and \ref{BE} are satisfied for some restricted versions of the cutoff model. The linear assumptions \ref{LE} are proved under the strong cutoff assumption:
$$b =b(\cos \theta) \in \mathcal C^1\left( [-1, 1] \right)$$
in the larger energy space $X$ and dissipation space $X^{\bullet}$ defined as
$$X := L^2\left( \langle v \rangle^k \mathrm{d} v \right), \quad X^{\bullet} := L^2\left( \langle v \rangle^{k+\gamma/2} \mathrm{d} v\right), \quad k > 2.$$
The splitting considered is of the form
\begin{gather*}
\mathcal L = (\mathcal L - \mathcal A) + \mathcal A =: \mathcal B+ \mathcal A,\\
\mathcal A g(v) := \int_{\mathbb R^d} a(v, v_*) g(v_*) \mathrm{d} v_*, \quad a \in \mathcal C^\infty_c\left( \mathbb R^d_v \times \mathbb R^d_{v_*} \right),
\end{gather*}
and the dissipation estimate is given in \cite[Lemma 3.4]{BMM2019}, whereas the regularization estimate follows from the form of $\mathcal A$. The nonlinear estimate \ref{BE} is given by \cite[Lemma 4.4]{BMM2019}.
\subsubsection{The non cutoff case}
One talks about a \emph{non cutoff case} when the function $b$ has a non-integrable singularity close to $\theta = 0$, and more precisely when it behaves as follows:
$$\sin (\theta)^{d-2} b(\cos \theta) \approx \theta^{-2 s}, \quad s \in (0, 1).$$
In this situation, the angular singularity improves the dissipation in the sense that $\mathcal L$ presents a spectral gap even for some negative values of $\gamma$, at least in the suitable space $H$. The basics assumptions \ref{L1}--\ref{L4} are satisfied for Maxwel, hard and moderately soft potentials:
$$0 < s < 1, \qquad \gamma + 2 s \geqslant 0.$$
The energy space is defined as in the cutoff case, whereas this time, the dissipation space $H^{\bullet}$ is defined as (see \cite{AMUXY2011, AHL2019}, we also mention \cite{GS2011})
\begin{align*}
\| g \|^2_{H^{\bullet}} & \approx \| \langle v \rangle^{\gamma/2 + s} g \|_{H}^2 + \int_{\mathbb R^6_{v, v_*} \times \mathbb S^{d-1}_\sigma } \langle v_* \rangle^{\gamma} \mu_* \left( g(v') - g(v) \right)^2 \mathrm{d} \sigma \mathrm{d} v_* \mathrm{d} v \\
& \approx \| \langle v \rangle^{\gamma/2 + s} g \|_{H}^2 + \left\| \langle v \rangle^{\gamma/2} | v \land \nabla_v |^s g \right\|^2_{H},
\end{align*}
where $| v \land \nabla_v |^s$ is defined as a pseudo-differential operator. The dissipativity estimate \ref{L4} is given by \cite[Lemma 2.6]{AMUXY2011}, and the nonlinear one \ref{Bbound} is given by~\cite[Theorem 2.1]{GS2011}. The intermediate spaces $H_j$ are defined as in the cutoff case, but this time the associated splitting is
$$\mathcal B g := \mathcal L g- M \chi\left( \frac{|v|}{M} \right) g, \qquad \mathcal A g := M \chi\left( \frac{|v|}{M} \right) g,$$
where $\chi$ is some smooth bump function and $M$ is large enough. This splitting satisfies \ref{L4} thanks to the following weighted estimates from \cite[Lemmas 2.4-2.5]{GS2011}:
$$\langle \mathcal L g, g \rangle_{H_j} \leqslant - c \| g \|^2_{H_j} + C \| \mathbf{1}_{|v| \leqslant C} f \|_{H_j}^2.$$
The enlargement assumptions are known to hold only for hard and Maxwell potentials ($\gamma \geqslant 0$) and the enlarged spaces are defined as
$$X_j := L^2\left( \langle v \rangle^{2k+2j} \mathrm{d} v \right), \quad k > 3 + \gamma/2 + 2 s,$$
and the dissipation space $X^{\bullet}$ is defined in a similar fashion as in the Gaussian case. The decomposition from \ref{LE} is defined as in the Gaussian case and the dissipativity estimates follow from the weighted estimates
\begin{gather*}
\langle \mathcal L g, g \rangle_{X_j} \leqslant - c \| g \|^2_{X_j} + C \| f \|_{L^2}^2,
\quad \langle \mathcal L g, g \rangle_{X} \leqslant - c \| g \|^2_{X^{\bullet}} + C \| f \|_{L^2}^2,
\end{gather*}
which are proved in \cite[Lemma 4.2]{HTT2020}. The nonlinear estimates are proved in \cite[Lemma 2.12]{CDL2022} (see also \cite{HTT2020, AMSY2021}).
\subsection{The case of the classical Landau equation}
The Landau equation corresponds in some sense to the non-cutoff Boltzmann equation for $s = 1$. This time, the nonlinear operator $\mathcal Q$ is defined as
\begin{equation*}
\mathcal Q(f, f) = \nabla_v \cdot \int_{\mathbb R^3_{v_*} \times \mathbb S^{d-1}_\sigma } | v - v_* |^{\gamma+2} \Pi(v-v_*) \Big( f(v_*) \nabla_v f(v) - \nabla_{v_*} f(v_*) f(v) \Big) \mathrm{d} v_*,
\end{equation*}
where $\Pi(z) = \mathrm{Id} - |z|^{-2} z \otimes z$ is the orthogonal projection onto $z^\perp$. The basics assumptions \ref{L1}--\ref{L4} and \ref{Bortho}--\ref{Bisotrop} are satisfied for hard, Maxwell and moderately soft potentials ($\gamma + 2 \geqslant 0$) and the energy space remains the same as for the Boltzmann equation, but the dissipation space is defined as
\begin{align*}
\| g \|^2_{H^{\bullet}} & \approx \| \langle v \rangle^{\gamma/2 + 1} g \|_H^2 + \| \langle v \rangle^{\gamma/2} (v \land \nabla_v) g \|^2_H \\
& \approx \| \langle v \rangle^{\gamma/2 + 1} g \|_H^2 + \| \langle v \rangle^{\gamma/2} \nabla_v g \|^2_H + \| \langle v \rangle^{\gamma / 2 + 1} \Pi(v) \nabla_v g \|_H^2.
\end{align*}
The dissipativity estimate \ref{L4} is given by \cite[(24)]{G2002}, and the nonlinear one \ref{Bbound} is given by~\cite[Lemma 2.2]{R2021}. The intermediate spaces $H_j$ and splitting are the same as for the non-cutoff Boltzmann equation, which satisfies \ref{L4} using this time the weighted estimates which can be proved as \cite[Lemma 6]{G2002}:
$$\langle \mathcal L g, g \rangle_{H_j} \leqslant - c \| g \|^2_{H_j} + C \| \mu f \|_{H}^2.$$
The enlarged assumptions \ref{LE} and \ref{BED} hold for Maxwell and hard potentials ($\gamma \geqslant 0$) and the enlarged spaces are defined as
$$X_j = L^2\left( \langle v \rangle^{2k + 2 j} \mathrm{d} v \right), \quad k > \gamma + 17/2,$$
and the dissipation space is defined as
\begin{align*}
\| g \|^2_{X^{\bullet}} = \| \langle v \rangle^{\gamma/2 + 1} g \|_X^2 + \| \langle v \rangle^{\gamma/2} \nabla_v g \|^2_X + \| \langle v \rangle^{\gamma / 2 + 1} \Pi(v) \nabla_v g \|_X^2.
\end{align*}
The decomposition from \ref{LE} is defined as in the gaussian case and the dissipativity is proved in \cite[(2.22)-(2.23)]{CTW2016}. The nonlinear bounds \ref{BED} are proved in \cite[Lemma 3.5]{CTW2016}.
\subsection{The quantum Boltzmann and Landau equations}
The quantum Boltzmann and Landau equations are variations which can be written as
$$\partial_t F + v \cdot \nabla_x F = \mathcal Q(F, F) + \delta \mathcal T(F, F, F),$$
where $\delta > 0$ for Fermi-Dirac models while $\delta < 0$ for Bose-Einstein models. The parameter $\delta$ is a small parameter related to the Planck constant, and $\mathcal T$ is defined for the quantum Boltzmann equation
$$\mathcal T(F, G, H) = \int_{\mathbb R^d_{v_*} \times \mathbb S^{d-1}_\sigma } |v - v_* |^\gamma b(\cos \theta) \Big( F' F'_* (F + F_*) - F F_* \left( F' + F'_* \right) \Big) \mathrm{d} v_* \mathrm{d} \sigma,$$
and for the quantum Landau equation
$$\mathcal T(F, F, F) = \nabla_v \cdot \int_{\mathbb R^3_{v_*} \times \mathbb S^{d-1}_\sigma } | v - v_* |^{\gamma+2} \Pi(v-v_*) \Big( F(v_*)^2 \, \nabla_v F(v) - F(v)^2 \, \nabla_{v_*}F(v_*) \Big) \mathrm{d} v_*.$$
At the linear level, the quantum Boltzmann and Landau equations fit within our framework. The various assumptions \ref{L1}--\ref{L4} have been shown to hold for the Boltzmann-Bose-Einstein equation $(\delta < 0)$ in \cite{YZ22, Z22} in the case of very soft potentials ($\gamma + 2 s < 0$) for which there does not hold $\| \cdot \|_{H^{\bullet}} \geqslant \| \cdot \|_H$, but is more intricate than the one for which \ref{L4} would be satisfied ($\gamma + 2 s \geqslant 0$).
For Fermi-Dirac statistics, a spectral gap estimate can be found in \cite{zhao} in the case $\gamma = b(\cos \theta) = 1$. Concerning the Landau equation in the case $\delta > 0$, the linear assumptions \ref{L1}--\ref{L4} as well as \ref{LE} have been partially checked in \cite{ABL}. We refer the reader to the work in preparation \cite{GL2023} for more details on the quantum Boltzmann equation.
\section{Technical toolbox}\label{sec:toolbox}
\subsection{Littlewood-Paley theory}
\label{scn:littlewood-paley}
For some appropriate $\varphi \in \mathcal C^\infty\left( \mathbb R^d \right)$ supported in an annulus centered about $0$ and $\chi \in \mathcal C^\infty\left( \mathbb R^d \right)$ supported in a ball centered about $0$ such that
$$0 \leqslant \varphi, \chi \leqslant 1, \qquad \chi(\xi) + \sum_{j=0}^\infty \varphi\left( 2^{-j} \xi \right) = \sum_{j=-\infty}^\infty \varphi\left( 2^{-j} \xi \right) = 1,$$
one defines the \textit{homogeneous Littlewood-Paley projectors} for any $j \in \mathbb Z$ (see \cite[Section 2.2]{BCD2011}):
\begin{gather*}
\dot{\Delta}_j u := \mathcal F^{-1}_\xi \left[ \varphi\left( 2^{-j} \xi \right) \widehat{u}(\xi) \right], \qquad \dot{S}_j u := \mathcal F^{-1}_\xi \left[ \chi\left( 2^{-j} \xi \right) \widehat{u}(\xi) \right],
\end{gather*}
as well as \textit{Bony's homogeneous decomposition} (see \cite[Section 2.6.1]{BCD2011}):
$$u v = \dot{T}_u v + \dot{T}_v u + \dot{R}(u, v),$$
where the homogeneous paraproduct $\dot{T}_f g$ and the homogeneous remainder $\dot{R}(f, g)$ are defined as
$$\dot{T}_f g := \sum_{j=-\infty}^\infty \dot{S}_{j-1} f \dot{\Delta}_j g, \qquad \dot{R}(f, g) := \sum_{|j - k| \leqslant 1} \dot{\Delta}_j f \dot{\Delta}_k g.$$
This decomposition allows to prove the following product rule, which follows from the combination of \cite[Corollary 2.55]{BCD2011} and the embedding $\dot{\mathbb{B}}^s_{2, 1} \hookrightarrow \dot{\mathbb{B}}^s_{2, 2} = \dot{\mathbb{H}}^s$.
\begin{prop}
\label{prop:product_homogeneous_sobolev}
For any $s_1, s_2 \in \left( -\frac{d}{2} , \frac{d}{2} \right)$ such that $s_1 + s_2 > 0$, there holds
$$\| u v \|_{ \dot{\mathbb{H}}^{-\frac{d}{2} + s_1 + s_2} } \lesssim \| u \|_{ \dot{\mathbb{H}}^{s_1} } \| v \|_{ \dot{\mathbb{H}}^{s_2} }.$$
\end{prop}
One also defines the \textit{inhomogeneous Littlewood-Paley projectors} for $j \geqslant -1$ (see \cite[Section 2.2]{BCD2011}):
$$\Delta_j u := \begin{cases}
\mathcal F^{-1}_\xi \left[ \chi(\xi) \widehat{u}(\xi) \right], & j = -1, \\
\\
\dot{\Delta}_j u, & j \geqslant 0,
\end{cases} \qquad S_j := \sum_{k = -1}^{j-1} \Delta_k u,$$
as well as Bony's inhomogeneous decomposition (see \cite[Section 2.8.1]{BCD2011}):
$$uv = T_u v + T_v u + R(u, v),$$
where the inhomogeneous paraproduct $T_f g$ and remainder $R(f, g)$ are defined as
$$T_f g := \sum_{j=-1}^\infty S_{j-1} f \Delta_j g, \qquad R(f, g) := \sum_{|j - k| \leqslant 1} \Delta_j f \Delta_k g.$$
This decomposition allows to prove the following product rule, which follows from the combination of \cite[Corollary 2.86]{BCD2011} and the embedding $H^{\sigma} = B^{\sigma}_{2, 2} \hookrightarrow L^\infty$ whenever $\sigma > d/2$.
\begin{prop}
For any $s > 0$ and $\sigma > d/2$, there holds
$$\| u v \|_{H^s} \lesssim \| u \|_{H^s} \| v \|_{H^\sigma} + \| u \|_{H^\sigma} \| v \|_{H^s}.$$
\end{prop}
We also present the following Sobolev-H\"older product rule.
\begin{prop}
\label{prop:product_sobolev_holder}
For any $s' > s > 0$, there holds
$$\| u v \|_{H^s} \lesssim \| u \|_{H^s} \| v \|_{W^{s', \infty}}.$$
\end{prop}
\begin{proof}
When $s \in \mathbb N$, the estimate follows from the combination of Leibniz's formula and H\"older's inequality in $L^2 \times L^\infty$: for any $\sigma \leqslant s$
\begin{align*}
\| \nabla^\sigma (u v) \|_{L^2} \lesssim \sum_{r = 0}^\sigma \| \nabla^r u \nabla^{\sigma - r} v \|_{L^2}
\lesssim \sum_{r = 0}^\sigma \| \nabla^r u \|_{L^2} \| \nabla^{\sigma - r} v \|_{L^\infty} \lesssim \| u \|_{H^\sigma} \| v \|_{W^{\sigma, \infty}},
\end{align*}
and thus $\| uv \|_{H^s} \lesssim \| u \|_{H^s} \| v \|_{W^{s, \infty}}$.
When $s \notin \mathbb N$, then $W^{s, \infty} = B^{s}_{\infty, \infty}$, and we rely on Bony's inhomogeneous decomposition. On the one hand, \cite[Theorem 2.82]{BCD2011} and \cite[Theorem 2.85]{BCD2011} yield
$$\| T_v u \|_{H^s} + \| R(u, v) \|_{H^s} \lesssim \| u \|_{H^s} \| v \|_{W^{s, \infty}},$$
so we are left with estimating $T_u v$. Following the proof of \cite[Theorem 2.82]{BCD2011} (or more precisely \cite[Theorem 2.82]{BCD2011}), there holds
\begin{equation*}\begin{split}
\| T_u v \|_{H^s}^2 & \lesssim \sum_{j = -1}^\infty 2^{2 j s} \| S_{j-1} u \Delta_j v \|_{L^2}^2 \\
& \lesssim \sum_{j = -1}^\infty 2^{-2 j (s'-s)} \left( 2^{ j s'} \| \Delta_j v \|_{L^\infty} \right)^2 \| S_{j-1} u \|_{L^2}^2 \\
& \lesssim \| u \|_{W^{s', \infty}}^2 \| u \|_{L^2}^2 ,
\end{split}\end{equation*}
where we used the definition of the norm $B^{s}_{\infty, \infty}=W^{s, \infty}$ and the fact that $\| S_{j-1} u \|_{L^2} \lesssim \| u \|_{L^2}$.
\end{proof}
\subsection{About the wave equation} On the basis of the above Littlewood-Paley decomposition, we also establish the following decay in time of the wave semigroup. Such a result is very likely to be part of the folklore knowledge for dispersive equations and we give a full proof here:
\begin{lem}\label{lem:wave-equation} Given $u \in \dot{\mathbb{B}}_{1,1}^{\frac{d+1}{2}}(\mathbb R^d)$ one has
$$\|e^{it|D_x|}u\|_{L^\infty} \lesssim t^{-\frac{d-1}{2}}\|u\|_{\dot{\mathbb{B}}_{1,1}^{\frac{d+1}{2}}}, \qquad t >0$$
whereas, for $s \geqslant 0$ and $u \in \dot{\mathbb{B}}_{1,1}^{\frac{d+1}{2}+s}$,
$$\|e^{it|D_x|}u\|_{W^{s,\infty}}\lesssim t^{-\frac{d-1}{2}}\|u\|_{\dot{\mathbb{B}}^{\frac{d+1}{2}+s}_{1,1}} \qquad t >0.$$
\end{lem}
\begin{proof}
For any $j \in \mathbb Z$, we denote $u_j(x) = u\left( 2^j x \right)$ and notice
$$e^{i t | D_x | } \dot{\Delta}_j u = \left(e^{i 2^j t | D_x | } \dot{\Delta}_0 u_{-j} \right)_j, \qquad \dot{\Delta}_0 u_{-j} = \left(\dot{\Delta}_j u \right)_{-j}.$$
Using the scaling properties of the $L^1$ and $L^\infty$-norms, we deduce
$$\| e^{i t | D_x | } \dot{\Delta}_j u \|_{L^\infty} = \| e^{i 2^j t | D_x | } \dot{\Delta}_0 u_{-j} \|_{L^\infty}, \qquad \| \dot{\Delta}_0 u_{-j} \|_{L^1_x} = 2^{j d } \| \dot{\Delta}_j u \|_{L^1_x}, $$
and thus, using the dispersive estimate for functions whose frequencies are localized in an annulus (see \cite[Proposition 8.15]{BCD2011})
\begin{equation*}\begin{split}
\| e^{i t | D_x | } u \|_{L^\infty_x} & \leqslant \sum_{j \in \mathbb Z} \| e^{i t | D_x | } \dot{\Delta}_j u \|_{L^\infty_x} \lesssim \sum_{j \in \mathbb Z} \left(2^j t\right)^{-\frac{d-1}{2}} \| \dot{\Delta_0} u_{-j} \|_{L^1_x} \\
& \lesssim t^{-\frac{d-1}{2}} \sum_{j \in \mathbb Z} \left(2^j \right)^{\frac{d+1}{2}} \| \Delta_j u \|_{L^1_x} = t^{-\frac{d-1}{2}} \| u \|_{ \dot{\mathbb{B}}^{(d+1)/2}_{1,1} }.
\end{split}\end{equation*}
Furthermore, there holds
\begin{align*}
\| e^{i t | D_x | } u \|_{W^{s, \infty}_x} & \approx \| e^{i t | D_x | } u \|_{L^\infty_x} + \sup_{j \geqslant 0} 2^{j s} \| e^{i t | D_x | } \dot{\Delta}_j u \|_{L^\infty_x} \\
& \lesssim t^{-\frac{d-1}{2}} \left( \| u \|_{ \dot{\mathbb{B}}^{(d+1)/2}_{1,1} } + \sup_{j \geqslant 0} 2^{j \left(s+\frac{d+1}{2}\right) } \| \dot{\Delta}_j u \|_{L^1_x}\right) \\
& \lesssim t^{-\frac{d-1}{2}} \| u \|_{ \dot{\mathbb{B}}^{(d+1)/2 + s }_{1,1} }.
\end{align*}
This concludes the proof.
\end{proof}
\subsection{Duality}
\label{scn:duality}
Consider some surjective isometry $\Lambda : Y^{\bullet} \to Y$, that is to say
$$\| f \|_{Y^{\bullet}} = \| \Lambda f \|_{Y}.$$
Its adjoint $\Lambda^\star : Y \to Y^{\circ}$ for the inner product of $Y$ is then an isometry as well:
\begin{equation*}
\| \Lambda^\star f \|_{Y^{\circ}} = \sup_{\| g \|_{Y^{\bullet}} = 1} \langle f, \Lambda g \rangle_{Y} = \sup_{\| g' \|_{Y} = 1} \langle f, g' \rangle_{Y} = \| f \|_{Y},
\end{equation*}
and it extends naturally to a surjective isometry since $Y^{\circ}$ was defined as the completion of $Y$.
This allows to write the~$\mathscr B( Y^{\circ} ; Y^{\bullet} )$-norm in term of the $\mathscr B(Y)$-norm as well as the isometries $\Lambda$ and $\Lambda^\star$:
$$\| T \|_{ Y^{\circ} \to Y^{\bullet} } = \| \Lambda T \Lambda^\star \|_{ Y \to Y},$$
and thus deduce the identity
\begin{equation}
\label{eq:twisted_adjoint_identity}
\| T \|_{ Y^{\circ} \to Y^{\bullet} } = \| T^\star \|_{ Y^{\circ} \to Y^{\bullet} },
\end{equation}
from the classical one $\| S \|_{Y \to Y} = \| S^\star \|_{Y \to Y}$ and $(\Lambda^\star)^\star=\Lambda$. Similarly, we have
\begin{equation}
\label{eq:twisted_adjoint_identity_one_sided}
\| T \|_{Y \to Y^{\bullet} } = \| T^\star \|_{ Y^{\circ} \to Y },
\end{equation}
\subsection{Bootstrap formula for projectors}
\label{scn:boostrap_projectors}
We present some formulas relating the remainder of Taylor expansions for projectors with the lower order terms and remainders. This will allow to prove inductively regularizing properties on each term of said expansion.
\begin{lem}
Consider a projector $P(r) \in \mathscr B(E)$ depending on a parameter $r \in [0, 1]$ and its Taylor expansion at order $N \geqslant 0$:
\begin{gather*}
P(r) = \sum_{n=0}^{N-1} r^n P^{(n)} + r^N P^{(N)}(r),
\end{gather*}
whose (constant) coefficients belong to $\mathscr B(E)$ and satisfy the identities
\begin{equation}
\label{eq:convolution_polynom_projector}
\sum_{n=0}^{M} P^{(n)} P^{(M-n)} =P^{(M)}, \quad 0 \leqslant M \leqslant N-1,
\end{equation}
then the remainder satisfies a similar one:
\begin{align}
\label{eq:bootstrap_P_N_A}
P^{(N)}(r) & = P(r) P^{(N)}(r) + \sum_{n=1}^{N} P^{(n)} (r) P^{(N-n)} \\
\label{eq:bootstrap_P_N_B}
& = P^{(N)}(r) P(r) + \sum_{n=1}^{N} P^{(N-n)} P^{(n)} (r).
\end{align}
\end{lem}
\begin{proof}
We only take care of \eqref{eq:bootstrap_P_N_A}. Note that when $N=0$, this reduces to $P(r) = P(r) P(r)$, which is true since $P(r)$ is a projector. We prove the case $N \geqslant 1$ by induction. We start from the induction hypothesis at order $N$:
$$P^{(N)}(r) = P(r) P^{(N)}(r) + \sum_{n=1}^{N} P^{(n)} (r) P^{(N-n)},$$
and inject the expansions $P^{(N)}(r) = P^{(N)} + r P^{(N+1)}(r)$ and $P^{(n)}(r) = P^{(n)} + r P^{(n+1)}(r)$:
\begin{equation*}\begin{split}
P^{(N)}(r) = & P(r) \Big( P^{(N)} + r P^{(N+1)}(r) \Big) + \sum_{n=1}^{N} \Big(P^{(n)} + r P^{(n+1)}(r)\Big) P^{(N-n)} \\
=& P(r) P^{(N)} + \sum_{n=1}^{N} P^{(n)} P^{(N-n)} + r \Big( P(r) P^{(N+1)}(r) + \sum_{n=1}^{N} P^{(n+1)}(r) P^{(N-n)} \Big).
\end{split}\end{equation*}
Next, we expand $P(r) = P^{(0)} + r P^{(1)}(r)$ in the first line:
\begin{align*}
P^{(N)}(r) =& \Big( P^{(0)} + r P^{(1)}(r) \Big) P^{(N)} + \sum_{n=1}^{N} P^{(n)} P^{(N-n)} \\
&+ r \Big( P(r) P^{(N+1)}(r) + \sum_{n=1}^{N} P^{(n+1)}(r) P^{(N-n)} \Big) \\
= & \sum_{n=0}^{N} P^{(n)} P^{(N-n)} + r \Big( P(r) P^{(N+1)}(r) + \sum_{n=0}^{N} P^{(n+1)}(r) P^{(N-n)} \Big).
\end{align*}
Since we assumed \eqref{eq:convolution_polynom_projector}, we replace the first term and thus have
$$P^{(N)}(r) = P^{(N)} + r \Big( P(r) P^{(N+1)}(r) + \sum_{n=0}^{N} P^{(n+1)}(r) P^{(N-n)} \Big),$$
from which we conclude using $P^{(N)}(r) = P^{(N)} + r P^{(N+1)}(r)$.
\end{proof}
In the case of the spectral projector from Lemma \ref{lem:expansion_projection}, we use the following corollary.
\begin{cor}
The following identities hold for $N=1$:
\begin{align}
\label{eq:bootstrap_P_1_A}
\mathsf P^{(1)}(\xi) & = \mathsf P(\xi) \mathsf P^{(1)}(\xi) + \mathsf P^{(1)}(\xi) \mathsf P,\\
\label{eq:bootstrap_P_1_B}
& = \mathsf P^{(1)}(\xi) \mathsf P(\xi) + \mathsf P \mathsf P^{(1)}(\xi),
\end{align}
and, assuming $\mathsf P^{(1)} = \mathsf P \mathsf P^{(1)} + \mathsf P^{(1)} \mathsf P$, for $N=2$:
\begin{align}
\label{eq:bootstrap_P_2_A}
\mathsf P^{(2)}(\xi) & = \mathsf P(\xi) \mathsf P^{(2)}(\xi) + \mathsf P^{(1)}(\xi) \otimes \mathsf P^{(1)} + \mathsf P^{(2)}(\xi) \mathsf P,\\
\label{eq:bootstrap_P_2_B}
& = \mathsf P^{(2)}(\xi) \mathsf P(\xi) + \mathsf P^{(1)} \otimes \mathsf P^{(1)}(\xi) + \mathsf P \mathsf P^{(2)}(\xi).
\end{align}
\end{cor}
\section{Properties of the Navier-Stokes equations}
\label{sec:N-S}
We recall here some classical results on the Navier-Stokes equations and refer to \cite{R2016} and references therein.
\begin{theo}[\textit{\textbf{Cauchy theory for Navier-Stokes}}]
\label{thm:cauchy_NSF}
Let $s \geqslant \frac{d}{2} - 1$ and consider a triple of initial conditions $(\varrho_\textnormal{in}, u_\textnormal{in}, \theta_\textnormal{in}) \in \mathbb{H}^s_x$ satisfying
$$\nabla_x (\varrho_\textnormal{in} + \theta_\textnormal{in}) = 0, \qquad \nabla_x \cdot u_\textnormal{in} = 0.$$
There exists a unique maximal lifespan $T_* \in (0, \infty]$ such that, for any $T < T_*$, the initial data~$(\varrho_\textnormal{in}, u_\textnormal{in}, \theta_\textnormal{in})$ generates a unique solution
$$(\varrho, u, \theta) \in \mathcal C\left( [0, T] ; \mathbb{H}^s_x\right) \cap L^2\left( [0, T] ; H^{s+1}_x \right)$$
to the incompressible Navier-Stokes-Fourier system
\begin{equation}
\label{eq:NSF}
\begin{cases}
\partial_t u + u \cdot \nabla_x u = \kappa_\textnormal{inc} \Delta_x u - \nabla_x p, & \nabla_x \cdot u = 0,\\
\partial_t \theta + u \cdot \nabla_x \theta = \kappa_\textnormal{Bou} \Delta_x \theta, & \nabla_x (\varrho + \theta) = 0,
\end{cases}
\end{equation}
and it satisfies for some universal constant $C >0$
\begin{align*}
\| (\varrho, u, \theta) \|_{ L^\infty \left( [0, T] ; H^s \right) } + \| \nabla_x & (\varrho, u, \theta) \|_{ L^2 \left( [0, T] ; H^s \right) } \\
& \leqslant C \| (\varrho_\textnormal{in}, u_\textnormal{in}, \theta_\textnormal{in}) \|_{\mathbb{H}^s_x } \exp\left( C \| \nabla_x u \|_{ L^2 \left( [0, T] ; H^{\frac{d}{2}-1}_x \right) } \right).
\end{align*}
If the solution is global (i.e. $T_* = \infty$), the solution vanishes for large times:
$$\lim_{t \infty} \left\| (\rho(t), u(t), \theta(t)) \right\|_{\mathbb{H}^s} = 0,$$
this is the case if $d = 2$, or if $d \geqslant 3$ and $\| u \|_{ \mathbb{H}^{\frac{d}{2} - 1}_x }$ is small.
\end{theo}
Note that, on the one hand, $\varrho(t), \theta(t) \in L^2_x$ thus the Boussinesq condition $\nabla_x (\varrho + \theta) = 0$ is equivalent to $\varrho + \theta = 0$, and on the other hand, since $u$ is incompressible, the pressure (which is to be interpreted as a Lagrange multiplier) can be eliminated using Leray's projector $\mathbb{P}$ on incompressible fields:
\begin{equation}
\label{eq:NSF_1}
\begin{cases}
\partial_t u + \mathbb{P} \left( u \cdot \nabla_x u \right) = \kappa_\textnormal{inc} \Delta_x u,\\
\partial_t \theta = \kappa_\textnormal{Bou} \Delta_x \theta + u \cdot \nabla_x \theta, \\
\varrho =- \theta,
\end{cases}
\end{equation}
or, equivalently,
\begin{equation}
\label{eq:NSF_2}
\begin{cases}
\partial_t u + \mathbb{P} \big[ \nabla_x \cdot (u \otimes u) \big] = \kappa_\textnormal{inc} \Delta_x u,\\
\partial_t \theta = \kappa_\textnormal{Bou} \Delta_x \theta + \nabla_x \cdot (u \theta) , \\
\varrho = -\theta.
\end{cases}
\end{equation}
The next two results detail in what sense the Navier-Stokes-Fourier system is equivalent to \eqref{eq:reduction_NS_kin}, proving Proposition \ref{prop:equivalence_kinetic_hydrodynamic_INSF}.
\begin{lem}
The following identities hold.
\label{lem:Q_A}
\begin{enumerate}
\item For the Burnett function ${\mathbf{A}} $, one has
\begin{gather*}
\left\langle \mathcal Q^\mathrm{sym}( v_i \mu, v_j \mu), \mathcal L^{-1}{\mathbf{A}} \right\rangle_{H} = \frac{\vartheta_1}{2} \left( E_{i, j} + E_{j, i} - \frac{2}{d} \delta_{i, j} \mathrm{Id} \right),
\end{gather*}
where $(E_{i, j})_{i, j = 1}^d$ is the canonical basis of $\mathscr{M}_{d\times d}$, and the coefficient $\vartheta_1$ is defined as
\begin{equation*}
\vartheta_1 := - d \sqrt{ \dfrac{d}{E} } \left\langle \mathcal Q^\mathrm{sym}(v_1 \mu, v_1 \mu), \mathcal L^{-1} \left( \mathrm{Id} - \mathsf P \right) v_2^2 \mu \right\rangle_{H}.
\end{equation*}
Moreover, there holds for $\varphi = \mu, (|v|^2-E) \mu$
\begin{equation*}
\left\langle \mathcal Q^\mathrm{sym}( \varphi , v \mu ) , \mathcal L^{-1}{\mathbf{A}} \right\rangle_{H} = \left\langle \mathcal Q^\mathrm{sym}( \varphi , \varphi ) , \mathcal L^{-1}{\mathbf{A}} \right\rangle_{H} = 0.
\end{equation*}
\item Regarding the Burnett function ${\mathbf{B}} $, one has
\begin{gather*}
\left\langle \mathcal Q^\mathrm{sym}( v_i \mu, \mu ), \mathcal L^{-1}{\mathbf{B}} \right\rangle_{H} = \vartheta_2 \, \mathbf{e}_i, \\
\left\langle \mathcal Q^\mathrm{sym}( v_i \mu, \left( |v|^2 - E \right) \mu ), \mathcal L^{-1}{\mathbf{B}} \right\rangle_{H} = \vartheta_3 \, \mathbf{e}_i,
\end{gather*}
where $(\bm{e}_i)_{i=1}^d$ is the canonical basis of $\mathbb R^d$ and the coefficients $\vartheta_i$ are defined as
\begin{equation*}\begin{cases}
\vartheta_2 &:= - \dfrac{1}{E \sqrt{K (K-1)}} \left\langle \mathcal Q^\mathrm{sym}(v_1 \mu, \mu), \mathcal L^{-1} \left( \mathrm{Id} - \mathsf P \right) v_1 |v|^2 \mu \right\rangle_{H},\\
\\
\vartheta_3 &:= - \dfrac{1}{E \sqrt{K (K-1)}} \left\langle \mathcal Q^\mathrm{sym}(v_1 \mu, \left( |v|^2 - E \right) \mu), \mathcal L^{-1} \left( \mathrm{Id} - \mathsf P \right) v_1 |v|^2 \mu \right\rangle_{H}\,.
\end{cases}\end{equation*}
Furthermore, for $\varphi, \psi = \mu, \, v \mu, \, \left( |v|^2 - E \right) \mu$, one has
\begin{equation*}
\left\langle \mathcal Q^\mathrm{sym}( \varphi , \psi ), \mathcal L^{-1}{\mathbf{B}} \right\rangle_{H} = 0 \qquad \text{ and } \quad
\left\langle \mathcal Q^\mathrm{sym}( v_i \mu , v_j \mu ), \mathcal L^{-1}{\mathbf{B}} \right\rangle_{H} = 0.
\end{equation*}
\end{enumerate}
\end{lem}
\begin{proof}
We recall the notation from Section \ref{scn:spectral_study}
$$\mathsf{R}_0 := \mathcal L^{-1} \left( \mathrm{Id} - \mathsf P\right) \quad \text{ as well as the identity } \quad \mathcal L^{-1}{\mathbf{A}} = \sqrt{ \frac{d}{E} } \mathsf{R}_0 \left[ v \otimes v \mu \right].$$
We also recall that both $\mathcal L$ and $\mathcal Q^\mathrm{sym}$ commute with orthogonal matrices, and in particular preserve the evenness/oddity.
In this proof, we only prove the first identity which is the most intricate, that is we compute for all $i, j, k, \ell$
$$\left\langle \mathcal Q^\mathrm{sym}(v_i \mu, v_j \mu ), \mathsf{R}_0 \left[ v_k v_\ell \mu \right] \right\rangle.$$
The other identities are proved in a similar yet simpler manner.
\step{1}{The case $i \neq j$}
If $\{ i, j \} \neq \{ k, \ell \}$, then $\mathcal Q^\mathrm{sym}(v_i \mu, v_j \mu )$ is odd in the variables $v_i$ and $v_j$, however~$\mathsf{R}_0 \big[ v_k v_\ell \mu \big]$ is even in at least one of these variables, thus
$$\left\langle \mathcal Q^\mathrm{sym}(v_i \mu, v_j \mu) , \mathsf{R}_0 \big[ v_k v_\ell \mu \big] \right\rangle = 0.$$
If $\{ i, j \} = \{ k, \ell \}$, we use the isometric change of variables $(v_i, v_j) \rightarrow \left(\frac{v_1 + v_2}{\sqrt{2} }, \frac{v_1 - v_2}{\sqrt{2} } \right)$, which is compatible with the invariance of $\mathcal L$ and $\mathcal Q^\mathrm{sym}$:
\begin{align*}
\big\langle \mathcal Q^\mathrm{sym}(v_i \mu, v_j \mu) , \mathsf{R}_0 \big[ v_i v_j \mu \big] \big\rangle
&= \frac{1}{4} \left\langle \mathcal Q^\mathrm{sym}( (v_1 + v_2) \mu, (v_1 - v_2) \mu) , \mathsf{R}_0 \big[ (v_1-v_2) (v_1 + v_2) \mu \big] \right\rangle \\
&= \frac{1}{4} \left\langle \mathcal Q^\mathrm{sym}( v_1 \mu, v_1 \mu) - \mathcal Q^\mathrm{sym}( v_2 \mu, v_2 \mu) , \mathsf{R}_0 \big[ v_1^2 \mu \big] - \mathsf{R}_0 \big[ v_2^2 \mu \big] \right\rangle \\
&= \frac{1}{2} \left( \left\langle \mathcal Q^\mathrm{sym}( v_1 \mu, v_1 \mu) , \mathsf{R}_0 \big[ v_1^2 \mu \big] \right\rangle - \left\langle \mathcal Q^\mathrm{sym}( v_1 \mu, v_1 \mu) , \mathsf{R}_0 \big[ v_2^2 \mu \big] \right\rangle\right)
\end{align*}
where we used the change of variables $(v_1, v_2) \leftrightarrow (v_2, v_1)$ in the last identity. Using that
$$\mathsf{R}_0 \big[ v_1^2 \mu \big] + \sum_{j = 2}^d \mathsf{R}_0 \big[ v_j^2 \mu \big] = \mathsf{R}_0 \left[ |v|^2 \mu \right] = 0$$
together with the change of variables $v_j \rightarrow v_2$, we can rewrite the previous identity as
\begin{equation*}\begin{split}
\left\langle \mathcal Q^\mathrm{sym}(v_i \mu, v_j \mu) , \mathsf{R}_0 \big[ v_i v_j \mu \big] \right\rangle & = \frac{1}{2} \Bigg( - \sum_{j = 2}^d \left\langle \mathcal Q^\mathrm{sym}( v_1 \mu, v_1 \mu) , \mathsf{R}_0 \big[ v_j^2 \mu \big] \right\rangle \\
&\phantom{+++} - \left\langle \mathcal Q^\mathrm{sym}( v_1 \mu, v_1 \mu) , \mathsf{R}_0 \big[ v_2^2 \mu \big] \right\rangle\Bigg) \\
& = - \frac{d}{2} \left\langle \mathcal Q^\mathrm{sym}( v_1 \mu, v_1 \mu) , \mathsf{R}_0 \big[ v_2^2 \mu \big] \right\rangle.
\end{split}\end{equation*}
To sum up, if $i \neq j$, we have
$\displaystyle \left\langle \mathcal Q^\mathrm{sym}(v_i \mu, v_j \mu) , \mathcal L^{-1}{\mathbf{A}} \right\rangle = \frac{ \vartheta_1 }{2} \left( E_{i, j} + E_{j, i} \right).$
\step{2}{The case $i = j$}
If $k \neq \ell$, then $\mathsf{R}_0 \big[ v_ k v_\ell \mu \big]$ is odd in both $v_k$ and $v_\ell$, whereas $\mathcal Q^\mathrm{sym}(v_i \mu, v_i \mu)$ is even in all directions, thus
$$\left\langle \mathcal Q^\mathrm{sym}(v_i \mu, v_i \mu ) , \mathsf{R}_0\big[ v_k v_\ell \mu \big] \right\rangle = 0.$$
When $k = \ell$, arguing as in \textit{Step 1}, we have
$$\left\langle \mathcal Q^\mathrm{sym}(v_i \mu, v_i \mu), \mathsf{R}_0 \big[ v_k^2 \mu \big] \right\rangle =
\begin{cases}
-(d-1) \left\langle \mathcal Q^\mathrm{sym}(v_1 \mu, v_1 \mu), \mathsf{R}_0 \big[ v_1^2 \mu \big] \right\rangle, & k = i, \\
\left\langle \mathcal Q^\mathrm{sym}(v_1 \mu, v_1 \mu), \mathsf{R}_0 \big[ v_2^2 \mu \big] \right\rangle, & k \neq i.
\end{cases}$$
To sum up, when $i = j$, we have
$\displaystyle \left\langle \mathcal Q^\mathrm{sym}(v_i \mu, v_i \mu), \mathcal L^{-1}{\mathbf{A}} \right\rangle = \vartheta_1 \left( E_{i, i} - \frac{1}{d} \mathrm{Id} \right)$ and this proves the result.
\step{3}{Comments on the other coefficients}
Regarding the Burnett function ${\mathbf{A}} $, one proves similarly for $\varphi = \mu, (|v|^2 - E) \mu$
$$\left\langle \mathcal Q^\mathrm{sym}( \varphi, \varphi ) , \mathcal L^{-1}{\mathbf{A}} \right\rangle_{H} = \sqrt{ \dfrac{d}{E} } \left\langle \mathcal Q^\mathrm{sym}(\varphi, \varphi), \mathcal L^{-1} \left( \mathrm{Id} - \mathsf P \right) v_1^2 \mu \right\rangle_{H} \mathrm{Id},$$
and observing that the $\mathcal Q(\varphi, \varphi)$ is radial and that $|v|^2 \mu = \sum_{i=1}^d v_i^2 \mu \in \nul(\mathcal L)$, we deduce using the change of variable $v_1 \rightarrow v_i$ that these coefficients vanish.
Regarding the Burnett function ${\mathbf{B}} $, one proves similarly for $\varphi = \mu, (|v|^2 - E) \mu$
$$\left\langle \mathcal Q^\mathrm{sym}( v_i \mu, \varphi ), \mathcal L^{-1}{\mathbf{B}} \right\rangle_{H} = - \dfrac{1}{E \sqrt{K (K-1)}} \left\langle \mathcal Q^\mathrm{sym}(v_1 \mu, \varphi), \mathcal L^{-1} \left( \mathrm{Id} - \mathsf P \right) v_1 |v|^2 \mu \right\rangle_{H},$$
and for $\varphi, \psi = \mu, (|v|^2 - E) \mu$, we have that
$$\left\langle \mathcal Q^\mathrm{sym}( \varphi, \psi ), \mathcal L^{-1}{\mathbf{B}} \right\rangle_{H} = - \dfrac{1}{E \sqrt{K (K-1)}} \left\langle \mathcal Q^\mathrm{sym}( \varphi, \psi ), \mathcal L^{-1} \left( \mathrm{Id} - \mathsf P \right) v_1 |v|^2 \mu \right\rangle_{H} = 0,$$
where we used that $\mathcal Q(\varphi, \psi)$ is radial and $\mathcal L^{-1} \left( \mathrm{Id} - \mathsf P \right) v_1 |v|^2 \mu$ is odd in $v_1$. Finally, there holds
$$\langle \mathcal Q^\mathrm{sym}(v_i \mu, v_j \mu) , \mathcal L^{-1}{\mathbf{B}} \rangle_H = 0$$
because $\mathcal Q^\mathrm{sym}(v_i \mu, v_j \mu)$ is odd in both $v_i$ and $v_j$ if $i \neq j$, or even in $v_i = v_j$ otherwise, and~$\mathcal L^{-1} B$ is odd. This concludes the proof.
\end{proof}
\begin{rem}
Note that in the case of the classical Boltzmann and Landau equations, the operator $\mathcal L$ is related to $\mathcal Q$ through a linearization procedure, and one can show the identity
$$\forall f \in \nul(\mathcal L), \quad \mathcal Q(f, f) = - \frac{1}{2} \mathcal L \left(f^2 \mu^{-1} \right),$$
which implies that $\vartheta_2 = 0$ and the coefficients $\vartheta_1$ and $\vartheta_3$ can be computed explicitly.
\end{rem}
Thanks to the above result, we are in position to prove the Proposition \ref{prop:equivalence_kinetic_hydrodynamic_INSF}.
\begin{proof}[Proof of Proposition \ref{prop:equivalence_kinetic_hydrodynamic_INSF}]
We recall the integral formulation of the incompressible Navier-Stokes system:
$$
\begin{cases}
\displaystyle u(t) = e^{ \kappa_\textnormal{inc} t \Delta_x } \mathbb{P} u_\textnormal{in} - \vartheta_\textnormal{inc} \mathbb{P} \int_0^t e^{(t-\tau) \kappa_\textnormal{inc} \Delta_x} \, \nabla_x \cdot \left( u \otimes u \right) (\tau) \, \mathrm{d} \tau, \\
\displaystyle \theta(t) = e^{ t \kappa_\textnormal{Bou} \Delta_x} \theta_\textnormal{in} - \vartheta_\textnormal{Bou} \int_0^t e^{(t-\tau) \kappa_\textnormal{Bou} \Delta_x } \nabla_x \cdot (u \theta)(\tau) \mathrm{d} \tau, \\
\varrho = - \theta,
\end{cases}
$$
where we recall that $\mathbb{P}$ is Leray's projector on incompressible fields, and we point out the equivalence between $\nabla_x(\varrho +\theta) = 0$ and $\varrho + \theta = 0$ since $\varrho(t), \theta(t) \in L^2_x$.
Regarding the kinetic integral equation, we recall the definitions of $U_\textnormal{NS}$ and $V_\textnormal{NS}$:
$$U_\textnormal{NS}(t) = e^{t \kappa_\textnormal{inc} \Delta_x} \mathsf P^{(0)}_{\textnormal{inc}} + e^{t \kappa_\textnormal{Bou} \Delta_x} \mathsf P^{(0)}_{\textnormal{Bou}}, \qquad V_\textnormal{NS}(t) = e^{t \kappa_\textnormal{inc} \Delta_x} \mathsf P^{(1)}_{\textnormal{inc}} + e^{t \kappa_\textnormal{Bou} \Delta_x} \mathsf P^{(1)}_{\textnormal{Bou}},$$
and point out the equivalence coming from Proposition \ref{prop:macro_representation_spectral}:
\begin{equation}
\label{eq:inc_bou_conditions}
\left( \mathrm{Id} - \mathsf P^{(0)}_\textnormal{inc} \right) f = \left( \mathrm{Id} - \mathsf P_\textnormal{Bou}^{(0)} \right) f = 0 \iff \nabla_x (\varrho + \theta) = 0 ~\text{ and }~ \nabla_x \cdot u = 0,
\end{equation}
and since \ref{Bortho} assumes $\mathcal Q(f, f) \perp \nul(\mathcal L)$, we have
$$\mathsf P^{(0)}_\textnormal{inc} \mathsf P^{(1)}_\textnormal{inc} \mathcal Q(f, f) = \mathsf P^{(1)}_\textnormal{inc} \mathcal Q(f, f), \qquad\mathsf P^{(0)}_\textnormal{Bou} \mathsf P^{(1)}_\textnormal{Bou} \mathcal Q(f, f) = \mathsf P^{(1)}_\textnormal{Bou} \mathcal Q(f, f),$$
thus \textit{we assume \eqref{eq:inc_bou_conditions} from now on}.
Since $U_\textnormal{NS}$ and $V_\textnormal{NS}$ both take values in macroscopic distributions, it is enough to consider their macroscopic components.
\step{1}{Description of $\mathsf P^{(1)}_\textnormal{inc} \mathcal Q(f, f)$ and $\mathsf P^{(1)}_\textnormal{Bou} \mathcal Q(f, f)$}
Plugging the expression \eqref{eq:f-macro} of $f$ into the nonlinearity $\mathcal Q(f, f)$, we have
\begin{align*}
\mathcal Q(f, f) = & \varrho^2 \mathcal Q( \mu, \mu ) + \mathcal Q( u \cdot v \mu, u \cdot v \mu ) + \frac{\theta^2}{E^2(K-1)^2} \mathcal Q\left( \left( |v|^2-E \right) \mu , \left( |v|^2-E \right) \mu \right) \\
& + 2 \varrho u \cdot \mathcal Q(\mu, v \mu) + \frac{2\varrho \theta}{E(K-1)} \mathcal Q\left( \mu, (|v|^2 - E) \mu \right)
+ \frac{2\theta u}{E(K-1)} \cdot \mathcal Q\left( v \mu, (|v|^2 - E) \mu \right).
\end{align*}
On the one hand, Lemma \ref{lem:Q_A} yields
$$\left\langle \mathcal Q( f , f ) , \mathcal L^{-1}{\mathbf{A}} \right\rangle_{H} = \frac{\vartheta_1}{2} \left( u \otimes u - \frac{2}{d} |u|^2 \mathrm{Id} \right),$$
and, since for any $g : \mathbb R^d \rightarrow \mathbb R$, there holds $\nabla_x \cdot (g \mathrm{Id}) = \nabla_x g$ and thus $\mathbb{P} \left( \nabla_x \cdot (g \mathrm{Id}) \right) = 0$, we deduce from Proposition \ref{prop:macro_representation_spectral} that
\begin{gather*}
u\left[ \nabla_x \cdot \mathsf P_\textnormal{inc}^{(1)} \mathcal Q( f , f ) \right] = \left( \frac{d}{E} \right)^{\frac{3}{2}} \mathbb{P} \left\{ \nabla_x \cdot \left\langle \mathcal Q( f , f ) , \mathcal L^{-1}{\mathbf{A}} \right\rangle_{H} \right\}
= \left( \frac{d}{E} \right)^{\frac{3}{2}} \frac{\vartheta_1}{2} \mathbb{P} \left\{ \nabla_x \cdot \left( u \otimes u \right) \right\}.
\end{gather*}
On the other hand, Lemma \ref{lem:Q_A} and \eqref{eq:inc_bou_conditions} yield
\begin{equation*}
\langle \mathcal Q(f, f) , \mathcal L^{-1}{\mathbf{B}} \rangle_{H} = 2 \vartheta_2 u \varrho + \frac{2 \vartheta_3}{E (K-1)} u \theta
= \left( 2 \vartheta_2 + \frac{2 \vartheta_3}{E (K-1)} \right) u \theta,
\end{equation*}
and thus, according to Proposition \ref{prop:macro_representation_spectral},
\begin{align*}
\theta\left[ \nabla_x \cdot \mathsf P^{(1)}_{\textnormal{Bou}} \mathcal Q(f, f) \right] = \frac{1}{K \sqrt{K(K-1)}} \left( 2 \vartheta_2 + \frac{2 \vartheta_3}{E (K-1)} \right) \nabla_x \cdot (u \theta).
\end{align*}
\step{2}{The integral formulation in macroscopic variables}
We are only left with checking that the $u$-part of the kinetic integral system satisfies the Navier-Stokes equations, and that the $\theta$-part satisfies the Fourier equation.
Indeed, by Proposition \ref{prop:macro_representation_spectral} and the previous step, there holds
\begin{align*}
u(t) = u\left[ f(t) \right] & = u\left[ U_\textnormal{NS}(t) f_\textnormal{in} \right] + u\left[ \int_0^t V_\textnormal{NS}(t-\tau) \mathcal Q(f(\tau), f(\tau) ) \right] \\
& = e^{t \kappa_\textnormal{inc} \Delta_x} u_\textnormal{in} - \vartheta_\textnormal{inc} \mathbb{P} \int_0^t e^{(t-\tau) \kappa_\textnormal{inc} \Delta_x} \, \nabla_x \cdot \left( u \otimes u \right) (\tau) \, \mathrm{d} \tau,
\end{align*}
as well as
\begin{align*}
\theta(t) = \theta\left[ f(t) \right] & = \theta\left[ U_\textnormal{NS}(t) f_\textnormal{in} \right] + \theta\left[ \int_0^t V_\textnormal{NS}(t-\tau) \mathcal Q(f(\tau), f(\tau) ) \right] \\
& = e^{t \kappa_\textnormal{inc} \Delta_x} \theta_\textnormal{in} - \vartheta_\textnormal{Bou} \int_0^t e^{(t-\tau) \kappa_\textnormal{Bou} \Delta_x } \nabla_x \cdot (u \theta)(\tau) \mathrm{d} \tau.
\end{align*}
This concludes the proof.
\end{proof}
The next two lemmas provide estimates related to the kinetic version of the Navier-Stokes-Fourier solution, and the first one is essentially a quantitative version of those proved in \cite{GT2020}.
\color{black}
\begin{lem}[\textit{\textbf{Estimates for Navier-Stokes squared}}]
\label{lem:estimates_derivative_Q_navier_stokes}
Suppose $s > \frac{d}{2}$ and denote the bilinear term $\varphi = \mathcal Q(f, f)$ where~$f$ is defined as
$$
f(t, x, v) = \varrho(t, x) \mu(v) + u(t, x) \cdot v \mu(v) + \frac{1}{E(K-1)} \theta(t, x) \left( |v|^2 - E \right) \mu(v),$$
and the coefficients $(\varrho, u, \theta)$ are a solution to the Navier-Stokes-Fourier equations given by Theorem \ref{thm:cauchy_NSF}. Then $\varphi$ satisfies
\begin{equation*}\begin{split}
\| \varphi & \|_{ L^\infty \left( [0, T) ; \mathbb{H}^s_x \left( H^{\circ}_v \right) \right) } + \| | \nabla_x |^{1-\alpha} \varphi \|_{ L^2 \left( [0, T) ; \mathbb{H}^s_x \left( H^{\circ}_v \right) \right) } \\
& \lesssim \Big(1 + \| (\varrho_\textnormal{in}, u_\textnormal{in}, \theta_\textnormal{in}) \|_{ \dot{\mathbb{H}}^{-\alpha} } + \| (\varrho, u, \theta) \|_{ L^\infty\left( [0, T) ; \mathbb{H}^{s}_x \right) } + \| \nabla_x (\varrho, u, \theta) \|_{ L^2\left( [0, T) ; \mathbb{H}^{s}_x \right) }\Big)^2,
\end{split}\end{equation*}
and for any $p \in (1, 2]$, where $p = 1$ is allowed for $d \geqslant 3$, its derivative $\partial_{t} \varphi$ satisfies
\begin{equation*}\begin{split}
\| \partial_t \varphi \|_{ L^p\left( [0, T) ; \mathbb{H}^{s-1}_x(H^{\circ}_v) \right) } & + \| \partial_t \varphi \|_{ L^p\left( [0, T) ; \dot{\mathbb{H}}^{-\frac{1}{2}}_x (H^{\circ}_v)\right) } \\
& \lesssim \Big(1 + \| (\varrho, u, \theta) \|_{ L^\infty\left( [0, T) ; \mathbb{H}^{s}_x \right) } + \| \nabla_x (\varrho, u, \theta) \|_{ L^2\left( [0, T) ; \mathbb{H}^{s}_x \right) }\Big)^3.
\end{split}\end{equation*}
\end{lem}
\begin{proof}
The function $\varphi(t, x) \in H^{\circ}$ writes for some $\varphi_j \in H^{\circ}$ as (note that $\varrho = - \theta$)
$$\varphi(t, x) = u(t, x) \otimes u(t, x) : \varphi_1 + u(t, x) \varrho(t, x) \cdot \varphi_2 + \varrho^2(t, x) \varphi_3,$$
and its derivative writes
\begin{align*}
\partial_t \varphi = & 2 \left(\partial_t u\right) \otimes u : \varphi_1
+ \big[\left(\partial_t u\right) \varrho + u \left( \partial_t \varrho \right)\big] \cdot \varphi_2
+ 2 \varrho \left(\partial_t \varrho\right) \varphi_3
\end{align*}
We only prove that the term $\varrho \left(\partial_t u\right)$ satisfies the estimates of the lemma, the other ones being treated the same way. Since $(\varrho, u, \theta)$ is a solution of the Navier-Stokes-Fourier system, using the formulation \eqref{eq:NSF_1}, the term writes (omitting constants)
$$\varrho \left(\partial_t u\right) = \varrho \Big( \mathbb{P} \left(u \cdot \nabla_x u\right) \Big) + \varrho \Delta_x u.$$
We will require the product rules recalled in Appendix \ref{scn:littlewood-paley}:
\begin{gather}
\label{eq:homogeneous_product}
\forall s_1, s_2 \in \left(-\frac{d}{2}, \frac{d}{2}\right), \quad s_1 + s_2 > 0, \quad \| g h \|_{ \dot{\mathbb{H}}^{s_1 + s_2 - \frac{d}{2}} } \lesssim \| g \|_{\dot{\mathbb{H}}^{s_1} } \| h \|_{\dot{\mathbb{H}}^{s_2} },\\
\label{eq:inhomogeneous_product}
\forall s_1 > 0, ~\forall s_2 > \frac{d}{2}, \quad \| g h \|_{ H^s_1 } \lesssim \| g \|_{\dot{\mathbb{H}}^{s_1} } \| h \|_{\dot{\mathbb{H}}^{s_2} } + \| g \|_{\dot{\mathbb{H}}^{s_2} } \| h \|_{\dot{\mathbb{H}}^{s_1} },
\end{gather}
and this last one which can be proved as \eqref{eq:Q_refined_sobolev_negative_algebra_inequality}:
\begin{equation}
\label{eq:inhomogeneous_product_2}
\forall s > \frac{d}{2}, ~\forall r \in \left[0, \frac{d}{2}\right), \quad \| g h \|_{ H^s } \lesssim \| | \nabla_x |^r g \|_{ H^{s-r}_x } \| h \|_{H^{s} } .
\end{equation}
\step{1}{The estimates for $\varrho \Big( \mathbb{P} \left(u \cdot \nabla_x u\right) \Big)$}
Using the algebra structure of $H^s$ and $\mathbb{P} \in \mathscr B\left( H_x^s \right)$:
$$\left\| \varrho \Big( \mathbb{P} \left(u \cdot \nabla_x u\right) \Big) \right\|_{H^{s-1}_x} \leqslant \left\| \varrho \Big( \mathbb{P} \left(u \cdot \nabla_x u\right) \Big) \right\|_{H^{s}_x} \lesssim \| \varrho \|_{\mathbb{H}^s_x} \| u \|_{\mathbb{H}^s_x} \| \nabla_x u \|_{\mathbb{H}^s_x} \in L^2_t.$$
Applying the product rule \eqref{eq:homogeneous_product} a first time with the parameters $(s_1, s_2)= \left( \frac{1}{2}, \frac{d-1}{2} \right)$ and using the boundedness $\mathbb{P} \in \mathscr B\left( \dot{\mathbb{H}}^{(d-1)/2} \right)$
\begin{align*}
\left\| \varrho \Big( \mathbb{P} \left(u \cdot \nabla_x u\right) \Big) \right\|_{ \dot{\mathbb{H}}^{-\frac{1}{2}} } & \lesssim \| \varrho \|_{L^2_x} \left\| \mathbb{P} \left(u \cdot \nabla_x u\right) \right\|_{ \dot{\mathbb{H}}^{(d-1)/2}_x } \lesssim \| \varrho \|_{L^2_x} \left\| u \cdot \nabla_x u \right\|_{ \dot{\mathbb{H}}^{(d-1)/2}_x },
\end{align*}
and then using the product rule \eqref{eq:homogeneous_product} a second time with the parameters $(s_1, s_2) = \left( \nu , d - \frac{1}{2} - \nu \right)$ for some $\nu \in \left( \frac{d-1}{2} , \frac{d}{2} \right)$ so that $s_2 \in \left( \frac{d-1}{2} , \frac{d}{2} \right)$
$$
\left\| \varrho \Big( \mathbb{P} \left(u \cdot \nabla_x u\right) \Big) \right\|_{ \dot{\mathbb{H}}^{-\frac{1}{2}} } \lesssim \| \varrho \|_{L^2_x} \| u \|_{ \dot{\mathbb{H}}^\nu_x } \left\| \nabla_x u \right\|_{ \dot{\mathbb{H}}^{ d - \nu - \frac{1}{2} }_x }.$$
When $d \geqslant 3$, we have $\nu \in \left( 1, s \right)$ and $d - \nu - \frac{1}{2} < s$, and thus
$$\| \varrho u \cdot \nabla_x u \|_{ \dot{\mathbb{H}}^{-\frac{1}{2}} } \lesssim \| \varrho \|_{ \mathbb{H}^s_x } \| \nabla_{x} u \|_{ H^{s-1}_x } \| \nabla_{x} u \|_{ H^{s}_x } \in L^1_t \cap L^2_t,$$
and when $d = 2$, since $\nu \in (\frac{1}{2}, 1)$, we have by interpolation
$$\| \varrho u \cdot \nabla_x u \|_{ \dot{\mathbb{H}}^{-\frac{1}{2}} } \lesssim \| \varrho \|_{ \mathbb{H}^s_x } \| u \|_{\mathbb{H}^s_x}^{1-\nu} \| \nabla_{x} u \|_{\mathbb{H}^s_x}^{1+\nu} \in L^2_t \cap L^{\frac{2}{1+\nu}}_t,$$
thus, taking $\nu$ arbitrarily small to $1$ yields the result.
\step{2}{The estimates for $\varrho \Delta_x u$}
We rewrite this term as
$$\varrho \Delta_x u = \nabla_x \cdot \left( \varrho \nabla_x u \right) - \nabla_x \varrho \cdot \nabla_x u,$$
from which we deduce
$$\| \varrho \Delta_x u \|_{H^{s-1} } \lesssim \| \varrho \nabla_x u \|_{\mathbb{H}^s_x} + \| \nabla_x \varrho \cdot \nabla_x u \|_{H^{s-1} }.$$
Using for the first term the product rule \eqref{eq:inhomogeneous_product_2} with $\nu \in (0, 1)$ when $d = 2$ or $\nu = 1$ when $d \geqslant 3$, and \eqref{eq:inhomogeneous_product} for the second term, we have
\begin{equation*}\begin{split}
\| \varrho \Delta_x u \|_{H^{s-1} } \lesssim &
\| | \nabla_x |^\nu \varrho \|_{H^{s-\nu}_x} \| \nabla_x u \|_{\mathbb{H}^s_x}
+ \| \nabla_x \varrho \|_{H^{s-1}_x} \| \nabla_x u \|_{\mathbb{H}^s_x} \\
& + \| \nabla_x \varrho \|_{\mathbb{H}^s_x} \| \nabla_x u \|_{H^{s-1}_x} \in L^2_t \cap L^{ \frac{2}{1 + \nu } }.
\end{split}\end{equation*}
Furthermore, using the product rule \eqref{eq:homogeneous_product} with the parameters $(s_1, s_2) = \left( \nu, \frac{ d + 1 }{2} - \nu\right)$ for some~$\nu \in \left( \frac{1}{2} , \frac{d}{2} \right)$, and with $(s_1, s_2) = \left( \frac{d-1}{2}, 0 \right)$, we have
\begin{equation*}\begin{split}
\| \varrho \Delta_x u \|_{ \dot{\mathbb{H}}^{-\frac{1}{2}}_x } & \lesssim \| \varrho \nabla_x u \|_{ \dot{\mathbb{H}}^{\frac{1}{2}}_x } + \| \nabla_x \varrho \cdot \nabla_x u \|_{ \dot{\mathbb{H}}^{-\frac{1}{2}}_x } \\
& \lesssim \| \varrho \|_{ \dot{\mathbb{H}}^{\nu}_x } \| \nabla_x u \|_{ \dot{\mathbb{H}}^{(d+1)/2 - \nu}_x } + \| \nabla_x \varrho \|_{ \dot{\mathbb{H}}^{(d-1)/2 }_x } \| \nabla_x u \|_{ L^2_x } \\
& \lesssim \| \nabla_x u \|_{\mathbb{H}^s_x} \left( \| \varrho \|_{ \dot{\mathbb{H}}^\nu_x } + \| \nabla_x \varrho \|_{ H^{s-1}_x } \right).
\end{split}\end{equation*}
In the case $d \geqslant 3$, we deduce taking $\nu = 1$
$$\| \varrho \Delta_x u \|_{ \dot{\mathbb{H}}^{-\frac{1}{2}}_x } \lesssim \| \nabla_x u \|_{\mathbb{H}^s_x} \| \nabla_x \varrho \|_{ H^{s-1}_x } \in L^1_t \cap L^2_t,$$
and when $d = 2$, by interpolation,
$$\| \varrho \Delta_x u \|_{ \dot{\mathbb{H}}^{-\frac{1}{2}}_x } \lesssim \| \nabla_x u \|_{\mathbb{H}^s_x} \| \nabla_x \varrho \|_{ H^{s-1} }^\nu \| \varrho \|_{ \mathbb{H}^s_x }^{1-\nu} \in L^{\frac{2}{1+\nu}}_t \cap L^2_t,$$
from which we conclude the the result by taking $\nu$ arbitrarily close to $1$. This concludes the proof of the estimates for $\partial_t \varphi$.
For the estimates of $\varphi$, one proves similarly
$$\| \varrho u \|_{ \mathbb{H}^s_x } \lesssim \| \varrho \|_{ \mathbb{H}^s_x } \| u \|_{ \mathbb{H}^s_x }$$
and
$$\| | \nabla_x |^{1 - \alpha} (\varrho u) \|_{ \mathbb{H}^s_x } \lesssim \left( \| | \nabla_x |^{1 - \alpha} \varrho \|_{ \mathbb{H}^s_x } + \| \nabla_x \varrho \|_{ \mathbb{H}^s_x } \right) \| u \|_{ \mathbb{H}^s_x } + \| \varrho \|_{ \mathbb{H}^s_x } \| \nabla_x u \|_{ \mathbb{H}^s_x }$$
which allows to conclude using Lemma \ref{lem:NS_parabolic_space}. This concludes the proof.
\end{proof}
\begin{lem}[\thttl{The Navier-Stokes-Fourier solution and the space $\HHH$}]
\label{lem:NS_parabolic_space}
The Navier-Stokes solution in its kinetic form belongs to the space $\rSSSs{s}$ (where the parameter $\alpha$ defines this space):
\begin{align*}
|\hskip-0.04cm|\hskip-0.04cm| f |\hskip-0.04cm|\hskip-0.04cm|_{ \HHH } \lesssim \| (\varrho_\textnormal{in}, u_\textnormal{in}, \theta_\textnormal{in}) \|_{ \dot{\mathbb{H}}^{-\alpha}_x } + \| (\varrho, u, \theta) \|_{ L^\infty \left( [0, T) ; \mathbb{H}^s_x \right) } + \| \nabla_x (\varrho, u, \theta) \|_{ L^2 \left( [0, T) ; \mathbb{H}^s_x \right) }\,.
\end{align*}
Furthermore, it can be approximated by a smoother sequence; there exists $\left(f_{\varepsilon}\right)_{ {\varepsilon} \in (0, 1] } \in \rSSSs{s+1}$ such that
$$\lim_{{\varepsilon} \to 0} |\hskip-0.04cm|\hskip-0.04cm| f_{\varepsilon} - f |\hskip-0.04cm|\hskip-0.04cm|_{\rSSSs{s}}=0.$$
\end{lem}
\begin{proof}
\step{1}{Bound in $\HHH$}
We only need to consider the case $d = 2$ and only prove the estimate for $u$. We start by applying Duhamel's principle to \eqref{eq:NSF_2}:
$$u(t) = e^{t \kappa_\textnormal{inc} \Delta_x} u_\textnormal{in} - \int_0^t e^{(t-\tau) \Delta_x} \varphi(\tau) \mathrm{d} \tau, \qquad \varphi := \mathbb{P} \big[ \nabla_x \cdot \left( u \otimes u \right) \big],$$
from which we obtain
\begin{align*}
\int_0^T \| |\nabla_x|^{ 1 - \alpha } u(t) \|^2_{ L^2_x } \mathrm{d} t \lesssim & \int_0^T \left\| |\nabla_x|^{ 1 - \alpha } e^{ t \kappa_\textnormal{inc} \Delta_x} u_\textnormal{in} \right\|_{L^2_x}^2 \mathrm{d} t \\
& + \int_0^T \left\| |\nabla_x|^{ 1 - \alpha } \int_0^t e^{(t-\tau) \Delta_x} \varphi(\tau) \mathrm{d} \tau \right\|_{L^2_x}^2 \mathrm{d} t,
\end{align*}
or, equivalently, in Fourier variables:
\begin{equation*}\begin{split}
\int_0^T \| |\nabla_x|^{ 1 - \alpha } u(t) \|^2_{ L^2_x } \mathrm{d} t \lesssim & \int_{\mathbb R^{d}} | \xi |^{-2 \alpha} | \widehat{u}_\textnormal{in}(\xi) |^2 \int_0^T | \xi |^2 e^{ - 2 \kappa_\textnormal{inc} t |\xi|^2} \mathrm{d} t \, \mathrm{d} \xi \\
& + \int_{\mathbb R^{d}} | \xi|^{ - 2 \alpha } \int_0^T \left( \int_0^t | \xi | e^{(t-\tau) |\xi|^2} \widehat{\varphi}(\tau, \xi) \mathrm{d} \tau \right)^2 \mathrm{d} t \, \mathrm{d} \xi.
\end{split}\end{equation*}
Using Young's convolution inequality in the form $L^2\left( [0, T] \right) \ast L^1\left( [0, T] \right) \hookrightarrow L^2\left( [0, T] \right)$ followed by Minkowski's integral inequality for the second term, we thus get
\begin{equation*}\begin{split}
\int_0^T \| |\nabla_x|^{ 1 - \alpha } u(t) \|^2_{ L^2_x } \mathrm{d} t &\lesssim \int_{\mathbb R^{d}} | \xi |^{-2 \alpha} | \widehat{u}_\textnormal{in}(\xi) |^2 \mathrm{d} \xi
+ \int_{\mathbb R^{d}} | \xi|^{ - 2 \alpha } \left(\int_0^T \left| \widehat{\varphi}(t, \xi) \right| \mathrm{d} t\right)^2 \, \mathrm{d} \xi \\
&\lesssim \| u_\textnormal{in} \|^2_{ \dot{\mathbb{H}}^{-\alpha}_x } + \left(\int_0^T \| \varphi(t) \|_{ \dot{\mathbb{H}}^{-\alpha}_x } \mathrm{d} t\right)^2.
\end{split}\end{equation*}
Since $\mathbb{P} \in \mathscr B\left( \dot{\mathbb{H}}_x^{-\alpha} \right)$, we get using the product rule \eqref{eq:homogeneous_product} (using that $\frac{d}{2} = 1$) and then by interpolation
$$
\| \varphi \|_{ \dot{\mathbb{H}}_x^{-\alpha} } \lesssim \| u \otimes u \|_{\dot{\mathbb{H}}_x^{1-\alpha}} \lesssim \| u \|_{\dot{\mathbb{H}}_x^{1-\frac{\alpha}{2}}}^2 \lesssim \| \nabla_x u \|_{L^2_x} \| | \nabla_x |^{1-\alpha} u \|_{L^2_x}$$
from which we conclude using Cauchy-Schwarz
$$
\int_0^T \| |\nabla_x|^{ 1 - \alpha } u(t) \|^2_{ L^2_x } \mathrm{d} t \lesssim \| u_\textnormal{in} \|^2_{ \dot{\mathbb{H}}^{-\alpha}_x } + \left(\int_0^T \| |\nabla_x|^{ 1 - \alpha } u(t) \|^2_{ L^2_x } \mathrm{d} t\right)^{\frac{1}{2}} \left(\int_0^T \| \nabla_x u(t) \|^2_{ L^2_x } \mathrm{d} t\right)^{\frac{1}{2}},
$$
and thus, by Young's inequality
$$ \int_0^T \| |\nabla_x|^{ 1 - \alpha } u(t) \|^2_{ L^2_x } \mathrm{d} t \lesssim \| u_\textnormal{in} \|^2_{ \dot{\mathbb{H}}^{-\alpha}_x } +\int_0^T \| \nabla_x u(t) \|^2_{ L^2_x } \mathrm{d} t.
$$
This concludes this step.
\step{2}{Approximation by functions in $\rSSSs{s+1}$}
Since the solution is instantly regularized in the sense that
$$(\nabla_x \varrho, \nabla_x u, \nabla_x \theta) \in L^2 \left( [0, T) ; H^s_x \right),$$
and in virtue of the control from Theorem \ref{thm:cauchy_NSF}, there holds for any $\delta \in (0, T)$
\begin{equation*}\begin{split}
\| (\varrho, u, \theta) \|_{ L^\infty \left( [\delta, T) ; H^{s+1} \right) } + \| \nabla_x & (\varrho, u, \theta) \|_{ L^2 \left( [\delta, T) ; H^{s+1} \right) } \\
& \leqslant C \left\| \left(\varrho(\delta), u(\delta), \theta(\delta) \right) \right\|_{\mathbb{H}^{s+1}_x } \exp\left( C \| \nabla_x u \|_{ L^2 \left( [0, T) ; H^{\frac{d}{2}-1}_x \right) } \right),
\end{split}\end{equation*}
and thus from \textit{Step 1}, we have that
$f( \delta + \cdot ) \in \rSSSs{s+1}$. From this observation, considering (by the continuity of $f$ and the density of $H^{s+1}_x$) for any ${\varepsilon} > 0$ some $\delta_{\varepsilon} > 0$ and $f_{\textnormal{in}, {\varepsilon}} \in H^{s+1}_x \left( H_v \right)$ such that
$$\sup_{0 \leqslant t \leqslant \delta_{\varepsilon}} \| f(t) - f_{\textnormal{in}, {\varepsilon}} \|_{ H^s_x \left( H_v \right) } \leqslant {\varepsilon}, \qquad \int_0^{\delta_{\varepsilon}} \| | \nabla_x |^{1-\alpha} f(t) \|_{ H^s_x \left( H_v \right) }^2 \mathrm{d} t \leqslant {\varepsilon},$$
and up to a reduction of $\delta_{\varepsilon}$
$$\| | \nabla_x |^{1 - \alpha} f_{\textnormal{in}, {\varepsilon}} \|_{ H^s_x \left( H_v \right) } \leqslant \frac{{\varepsilon}}{\delta_{\varepsilon}}.$$
We can therefore define
$$f_{\varepsilon}(t) =
\begin{cases}
f_{\textnormal{in}, {\varepsilon}}, & t \in [0, \delta_{\varepsilon}), \\
f(t), & t \in [\delta_{\varepsilon}, T),
\end{cases}
$$
so as to have $f_{\varepsilon} \in \rSSSs{s+1}$ and
$$|\hskip-0.04cm|\hskip-0.04cm| f_{\varepsilon} - f |\hskip-0.04cm|\hskip-0.04cm|_{\rSSSs{s}}^2 \lesssim \sup_{0 \leqslant t \leqslant \delta_{\varepsilon}} \| f(t) - f_{\textnormal{in}, {\varepsilon}} \|_{ H^s_x \left( H_v \right) }^2 + \int_0^{\delta_{\varepsilon}} \left\| | \nabla_x |^{1-\alpha} f(t) - | \nabla_x |^{1-\alpha} f_{\textnormal{in}, {\varepsilon}} \right\|_{ H^s_x \left( H_v \right) }^2 \mathrm{d} t \lesssim {\varepsilon}^2,$$ which concludes the proof.
\end{proof}
\end{document} | arXiv |
Superreal number
In abstract algebra, the superreal numbers are a class of extensions of the real numbers, introduced by H. Garth Dales and W. Hugh Woodin as a generalization of the hyperreal numbers and primarily of interest in non-standard analysis, model theory, and the study of Banach algebras. The field of superreals is itself a subfield of the surreal numbers.
Dales and Woodin's superreals are distinct from the super-real numbers of David O. Tall, which are lexicographically ordered fractions of formal power series over the reals.[1]
Formal definition
Suppose X is a Tychonoff space and C(X) is the algebra of continuous real-valued functions on X. Suppose P is a prime ideal in C(X). Then the factor algebra A = C(X)/P is by definition an integral domain that is a real algebra and that can be seen to be totally ordered. The field of fractions F of A is a superreal field if F strictly contains the real numbers $\mathbb {R} $, so that F is not order isomorphic to $\mathbb {R} $.
If the prime ideal P is a maximal ideal, then F is a field of hyperreal numbers (Robinson's hyperreals being a very special case).
References
1. Tall, David (March 1980), "Looking at graphs through infinitesimal microscopes, windows and telescopes" (PDF), Mathematical Gazette, 64 (427): 22–49, CiteSeerX 10.1.1.377.4224, doi:10.2307/3615886, JSTOR 3615886, S2CID 115821551
Bibliography
• Dales, H. Garth; Woodin, W. Hugh (1996), Super-real fields, London Mathematical Society Monographs. New Series, vol. 14, The Clarendon Press Oxford University Press, ISBN 978-0-19-853991-9, MR 1420859
• Gillman, L.; Jerison, M. (1960), Rings of Continuous Functions, Van Nostrand, ISBN 978-0442026912
Number systems
Sets of definable numbers
• Natural numbers ($\mathbb {N} $)
• Integers ($\mathbb {Z} $)
• Rational numbers ($\mathbb {Q} $)
• Constructible numbers
• Algebraic numbers ($\mathbb {A} $)
• Closed-form numbers
• Periods
• Computable numbers
• Arithmetical numbers
• Set-theoretically definable numbers
• Gaussian integers
Composition algebras
• Division algebras: Real numbers ($\mathbb {R} $)
• Complex numbers ($\mathbb {C} $)
• Quaternions ($\mathbb {H} $)
• Octonions ($\mathbb {O} $)
Split
types
• Over $\mathbb {R} $:
• Split-complex numbers
• Split-quaternions
• Split-octonions
Over $\mathbb {C} $:
• Bicomplex numbers
• Biquaternions
• Bioctonions
Other hypercomplex
• Dual numbers
• Dual quaternions
• Dual-complex numbers
• Hyperbolic quaternions
• Sedenions ($\mathbb {S} $)
• Split-biquaternions
• Multicomplex numbers
• Geometric algebra/Clifford algebra
• Algebra of physical space
• Spacetime algebra
Other types
• Cardinal numbers
• Extended natural numbers
• Irrational numbers
• Fuzzy numbers
• Hyperreal numbers
• Levi-Civita field
• Surreal numbers
• Transcendental numbers
• Ordinal numbers
• p-adic numbers (p-adic solenoids)
• Supernatural numbers
• Profinite integers
• Superreal numbers
• Normal numbers
• Classification
• List
| Wikipedia |
Alice underground: the door, the quaternion and the relativity
By Gianluigi Filippelli on Saturday, May 28, 2016
Alice underground is the first version of Aline in the wonderland by Lewis Carroll. The original manuscript, illustrated by Carroll himself, was given to the little Alice Liddell for Christmas in 1864 and picked up the story that he had told to Alice and her sisters Lorina and Edith during a summer's afternoon, precisely on July the 4th, 1862. This first version of the carrollian fantasy novel is, ultimately, a restricted version of Alice, where various characters and episodes completely absent in Underground are added, such as the Duchess or the team composed by the Mad Hatter, the March Hare and the Dormouse.
The intial, interesting considerations about underground is about the importance of the trees and the doors: following the suggestion by Adele Cammarata(3), we can assume that the tree and the door that Alice cross to enter the garden of the Queen of Hearts, completely absent in Wonderland, is linked with the Celtic tradition. Indeed the oak is one of the sacred trees of the druids, symbolizing a link between heaven and earth(1). In this way the oak, which in Celtic was called duir, is a real door that connects people with the gods, but also ourselves with our inner part. So, from an etymological point of view, a carved door in a tree trunk is a Celtic symbol used to identify the Alice's passage towards a more stable phase after the size's changes of the previous scenes.
These changes in size, alluding both to the transition to adulthood, in perfect connection with the Druidic symbolism, and with the more classic homothetic transformations, i.e. the transformations which, without changing the proportions of a geometric figure, change its size. All these changes remain unchanged in the transition to the second version, including the meeting with the Caterpillar, who continues to ask Alice:
Quaternions
One piece, absent in Underground, is the Mad Hatter's tea party, that it is not only a particular scene in which mention the time (in this case stopped for the protagonists of the party), and even a way to mention the cyclical partitions, but it could be a way to introduce in a fairy tale quaternions, discovered by William Rowan Hamilton.
Hamilton began to work on an extension of complex numbers, constituted by two parts, a real and an imaginary (where for imaginary number is defined as a multiple of the square root of $-1 = i^2$). Earlier this extension had only three numbers, and Hamilton could only get rotations in the plane; then he added the fourth number, allowing him to get also the spatial rotations. In order to give an interpretation to his results, Hamilton wrote in 1853 on Lectures on Quaternions:
It seemed (and still seems) to me natural to connect this extra-spatial unit with the conception of time.
Starting from these elements, we can intend the party in this way: the Hatter, the March Hare and the Dormouse are three elements of a quaternion orphans of the fourth element, and this forces them to constantly go around the table, always drenching the same biscuits in the same cups without ever being able to change their condition. The fact that Alice is not able to change their condition suggests that she is like a space coordinate.
Another passage that seems to fit with the mathematics of quaternions is the Hatter's response to Alice at the end of this exchange:
'You should learn not to make personal remarks,' Alice said with some severity; 'it's very rude.'
The Hatter opened his eyes very wide on hearing this; but all he said was, 'Why is a raven like a writing-desk?'
'Come, we shall have some fun now!' thought Alice. 'I'm glad they've begun asking riddles.—I believe I can guess that,' she added aloud.
'Do you mean that you think you can find out the answer to it?' said the March Hare.
'Exactly so,' said Alice.
'Then you should say what you mean,' the March Hare went on.
'I do,' Alice hastily replied; 'at least—at least I mean what I say—that's the same thing, you know.'
'Not the same thing a bit!' said the Hatter. 'You might just as well say that "I see what I eat" is the same thing as "I eat what I see"!'
that we can intend as an allusion to the non-commutativity of quaternions.
Shortly, a quaternion is a kind of 4-vector written (or defined) as follows: \[q = a + bi + cj + dk\] where $a$, $b$, $c$, $d$ are real numbers, while $i$, $j$, $k$ are the coordinates into a 3d imaginary space (in some sense the equivalent of $x$, $y$, $z$ in the real space). It's not a case if $a$ is called the scalar part, while $bi + cj + dk$ is the vectoral part of the quaternion. The three imaginary vector directions are then related by the following equation: \[i^2 = j^2 = k^2 = ijk = -1\] But when you start playing a little with quaternions, some interesting properties check out: for example, you can build your own group of rotations starting from quaternionic units (to every quaternion we can associate a rotation in space) or even the so-called quaternion group, non-commutative, and which we can provide a representation using both matrices $2 \times 2$ with complex values, both $4 \times 4$ with real values(2).
Th last observation: the group theory comes from the studies by Evariste Galois and Niels Abel about the solutions of the 5th grade, and higher polynomials; in the same way, as shown in 1981 by Richard Dean, quaternion group comes from the following polynomial: \[x^8 - 72 x^6 + 180 x^4 - 144 x^2 + 36\]
Alice at theatre
Carroll's novel has had in its history several theater transpositions. Of all, I saw one of the Milan theater "Elfo Puccini", in 2013, based on Alice Underground: absurd and unusual scenes in carrollian style with some scientific ideas. For example the presence of Mr. Time and Mr. Space that opened the piece talking to each other while Alice spleeping her adventures in the Wonderland. In the meanwhile on the white wall behind the actors images and drawings are projected: clocks, numbers, math symbols and equations like the famous Einstein's relativity field equation: \[R_{\mu \nu} - {1 \over 2} g_{\mu \nu} R + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}\] Despite the absence of the Queen of Hearts (only as a projected image and voiceover), appreciable the attempt to make the logic as one of the subject of the piece: there is not only the Mad Hatter's tea party, but also the scene with the Duchess, the encounter with the twins Tweedledee and Tweedledum and, above all, the famous race of the Red Queen. In particular this last scene is beautiful thanks to the actress: as Alice runs to enter into the garden of the Queen of Hearts, or to go into the world beyond the mirror (through a metaphorical hall of mirrors), the actress runs on site, making so a run which leaves the runner at the starting point!
Interesting the closing considerations about dream and dreamer: what could happen to the dream if the dreamer wake up? This theme was explored by a lot of science fiction, for example in Tonight the Sky Will Fall by Daniel F. Galouye or in the Joe Lansdale's Drive-In saga. And, in one of the various interpretation of carrollian works, it could be also a Lewis Carroll's quest.
(1) Many Celtic gods are represented with a face set in a tree trunk. I also remember the tradition of the tree of life, apparently symbolic in the Jewish tradition, but, if we think of its Celtic equivalent, present also in The Silmarillion and in the Tolkien's epic.
(2) Read also Alice adventures in Algebra: Wonderland solved (3rd part)
(3) Adele Cammarata (2002). Introduction to Alice underground, Stampa Alternativa, Vietrbo
Labels: albert einstein, alice in wonderland, dream, general relativity, lewis carroll, mathematics, quaternions, review, time, william rowan hamilton
Just the kind of intro I had been looking for, aiming to hook female high schoolers on the pursuit of math, following the example of Katherine Johnson (of "Hidden Figures" fame).
Brandon Wiers
Alice underground: the door, the quaternion and th...
Doubt like a salmon
Transit of Mercury
How to become a superhero | CommonCrawl |
\begin{document}
\title{Imprinting Light Phase on Matter Wave Gratings in Superradiance Scattering }
\author{Xiaoji Zhou} \email[E-mail: ]{[email protected]} \author{Fan Yang} \author{Xuguang Yue} \author{T. Vogt} \author{Xuzong Chen} \email[E-mail: ]{[email protected]}
\affiliation{ School of Electronics Engineering $\&$ Computer Science, Peking University, Beijing 100871, P. R. China}
\date{\today}
\begin{abstract}
Superradiance scattering from a Bose-Einstein condensate is studied with a two-frequency pumping beam. We demonstrate the possibility of fully tuning the backward mode population as a function of the locked initial relative phase between the two frequency components of the pumping beam. This result comes from an imprinting of this initial relative phase on two matter wave gratings, formed by the forward mode or backward mode condensate plus the condensate at rest, so that cooperative scattering is affected. A numerical simulation using a semiclassical model agrees with our observations. \end{abstract}
\pacs{03.75.Kk, 42.50.Gy, 42.50.Ct, 32.80.Qk}.
\maketitle
Superradiance from a Bose-Einstein Condensation (BEC) offers the possibility of studying a novel physics associated with cooperative scattering of light in ultracold atomic systems. A series of experiments~\cite{Inouye1999science, Schneble2003scince, 1999, Yutaka} and related theories~\cite{Moore1999prl, Zobay2006pra, Pu2003prl} have sparked new interests in matter wave amplification~\cite{Inouye1999science, 1999}, holographic storage~\cite{Yutaka}, scattering spectroscopy in optical lattice~\cite{xu}, coherent imaging~\cite{sadler}, coherent atomic recoil lasing~\cite{Courteille1, Courteille2,zhou}, and quantum states storage and retrieval~\cite{Lukin,Matsuk}.
In a typical BEC superradiance experiment, an elongated condensate is illuminated by an off-resonant pumping laser pulse along its short axis. Due to the phase-matching condition and mode competition, highly directional light is emitted along the long axis of the condensate, in the so-called end-fire modes. Consequently, the recoiled atoms acquire a well-defined momentum at $\pm 45^{\circ}$ angles with respect to the pumping laser direction. These atomic modes are referred to as forward modes. This forward scattering is interpreted as optical diffraction from a matter wave grating~\cite{Inouye1999science, Moore1999prl}. Meanwhile, atoms in the condensate may scatter photons in the end-fire modes back into the pumping mode and recoil at $\pm135^{\circ}$ angles, forming the so-called backward modes~\cite{Schneble2003scince, Pu2003prl}, when the pumping pulse is short and intense. This pattern was interpreted as a result of diffraction of atoms off a light grating. A four wave mixing interpretation was proposed, involving two optical fields--the pumping laser field and an end-fire mode, and two matter wave modes--the condensate and a mode of momentum~\cite{ Pu2003prl}. \begin{figure}
\caption{(Color online) Schematic
diagram of our experiment. The two-frequency pumping beam is incident along the short $x$ direction, with a linear polarization along the $y$ direction. The atomic side modes are denoted in momentum space, each labeled with a pair of integers which describe the order in the $x$ and $z$ directions, respectively. Within this notation, atoms of the condensate at rest are in mode $(0,0)$. A forward scattering event transfers an atom from mode $(n,m)$ to mode $(n+1,m\pm 1)$, and a backward event transfers one to mode $(n-1,m\pm 1)$.The end fire mode $e_{\pm}$ is along the long axis of the condensate in $z$ direction.}
\label{setup1}
\end{figure}
There is an energy mismatch of four times the recoil frequency for this backward scattering, due to the increased kinetic energy of recoiled atoms~\cite{Schneble2003scince, Zobay2006pra}. Recently, a two-frequency-pumping scheme has been implemented~\cite{KMRvdStam2007arxiv, Bar-Gill2007arxiv, yang}, where the pumping beam consisted of two frequency components and the frequency difference was controlled to compensate for the energy mismatch and excite the backward scattering on a long time scale with a weak pump intensity. The presence of the backward mode in the spectroscopic response~\cite{Bar-Gill2007arxiv} and the enhancement of the diagonal sequential scattering~\cite{yang} have been reported. In those experiments the relative phase between the two pumping frequency components is maintained constant. Although phase is a very important factor for understanding interference and coherence phenomena, phase effects on matter wave gratings and cooperative scattering have not been reported so far.
We present here our new experimental results that show a possibility to fully control the backward mode population as a function of the locked initial relative phase between the two pumping frequencies. A theoretical analysis of the cooperative process is used to confirm and explain this phase dependence. It requires to consider the relationship between matter wave gratings and optical waves beating for the first order scattering. The initial relative phase of the pumping beam is imprinted into two matter wave gratings, one formed by the condensate with a forward mode, the other formed by the condensate with a backward mode.
\begin{figure}
\caption{Two-frequency superradiance scattering experimental setup. The pump laser beam is split into two beams with orthogonal polarizations using a PBS. With two AOMs, each beam frequency is shifted about 90 MHz, with a little difference of $\Delta\omega$. In order to keep the coherence of the two beams after frequency shifting, the phase-locked-loops of the radio frequency (rf) sources for the two AOMs are locked to the same crystal oscillator. A phase detection circuit is used to detect the initial relative phase of beating signal of the two frequency beams after they are combined. Note that the beat signal of the pump pulse is opposite in phase with the signal measured by the photodiode.}
\label{twofrequency}
\end{figure}
In our experiment, a nearly pure Bose-Einstein condensate of about $2\times10^{5}$ $^{87}$Rb atoms in the $|F=2, m_{F}=2\rangle$ hyperfine ground state is generated in a quadrupole-Ioffe-configuration magnetic trap, with Thomas-Fermi radii of $50 \mathrm{\mu m}$ and $5 \mathrm{\mu m}$ along the axial and radial directions~\cite{yang}. The pumping pulse with two-frequency components is incident along the short axis of the condensate, with its polarization perpendicular to the long axis, as shown in Fig.~\ref{setup1}. This arrangement of polarization induces Rayleigh superradiance where all side modes possess the same atomic internal state~\cite{Inouye1999science}. To get the two-frequency pumping beam, a laser beam from an external cavity diode laser is split into two equal-intensity beams. Their frequencies are shifted individually by acousto-optical modulators (AOM1 and AOM2) which are driven by phase-locked radio frequency signals. Therefore the frequency difference $\Delta\omega$ between the two beams can be controlled precisely, as shown in Fig.~\ref{twofrequency}. After that, they are recombined to form our linear-polarized two-frequency pump beam, which is red detuned by $2\pi \times 1.5 \mathrm{GHz}$ from the $|F=2, m_{F}=2\rangle$ to $|F'=3, m_{F}'=3\rangle$ transition.
The frequency difference $\Delta\omega$ is chosen to be $2\pi\times15$kHz (the corresponding period $T$ is then $66.67 \mu$s), i.e. four times the recoil frequency ($4\omega_{r} = 2 \hbar k_{l}^{2}/M$) so that we reach the two-photon resonance condition taking into account the recoil energies deficit for the backward scattering. The magnetic trap is shut off immediately after the pumping pulse, and the distribution of atomic side modes is measured by absorption imaging after $21\mathrm{ms}$ of ballistic expansion, as shown in Fig.~\ref{intphase}.
\begin{figure}
\caption{Superradiance with different initial relative phases and time control sequence.
(a) and (b): Initial relative phase of $\pi/2$ and $1.4\pi$ corresponding to a generation time of 50$\mu$s and 20$\mu$s respectively.
(c) and (d): Superradiance patterns corresponding to (a) and
(b), respectively.
(e) Control sequence for generating the pump pulse. Note that there
is a phase difference of $\pi$ between the phase shown in figures (a) and (b) and that of the real pumping pulse. }
\label{intphase}
\end{figure}
The experiment is repeated with different initial relative phases between the two frequency components. Monitoring the beating signal $\cos(\Delta \omega t+\phi)$ on a photodiode, a light pulse of duration $T$ with a definite initial relative phase $\phi_0=\Delta \omega t_0+\phi$ can be generated. Different initial relative phases $\phi_0$ varying between $0$ and $2\pi$, are obtained by switching on the pulse at different time $t_0$, which we call ``generation time", as shown in Fig. 3(e). The generation time $t_0$ and the switch off time $t_0+T$ can be controlled precisely with an AOM. Figure~3(a) and (b) demonstrate the initial relative phases of the two frequency pumping beam being $\pi/2$ and $1.4\pi$, the corresponding generation time $t_0$ being $50\mu$s and $20\mu$s, respectively. This phase can be read out related to the reference signal corresponding to $\phi_0=0$.
In Fig.~\ref{intphase} (c), the backward mode is obvious for a relative initial phase $\phi_{0}=\pi/2$ while in Fig.~\ref{intphase} (d), when the phase is $\phi_{0}=1.4\pi$, backward scattering is almost suppressed. The ratio between the backward population and the total atom number is plotted in Fig.~\ref{experdata} versus different phases $\phi_0$, with a pulse duration equal to $66.67 \mu$s or one cycle, a time step between two points given by $\Delta t_{0}=10\mu$s and each point being the average on four experimental data. The error bar is their standard deviation. For each experiment, the atom number in the BEC, the initial quantum noise and the temperature are not exactly the same. These fluctuations are the main reason for the experimental uncertainty. Nevertheless, the number of backward scattered atoms shows a high sensitivity to the initial relative phase between the two frequency components and it is possible to obtain an almost complete cancellation of the backward scattering for $\phi_{0}=3\pi/2$.
\begin{figure}
\caption{(Color online) Experimental results and theoretical analysis about
the ratio of backward scattered atoms to the total atom number vs the initial
phase $\phi_{0}$ between the two pump beams. Each point is an average on four
experimental data. The dashed line is a numerical simulation using a
semi-classical theory depicted in the text with one initial seed and c
oupling factor $g=1.5\times 10^{6}$. The dotted line is drawn using
Eq.(\ref{results}) with $\Delta\varphi_{0,1}=\pi/2$.}
\label{experdata}
\end{figure}
Such a result was not expected considering in particular the interpretations given in previous experiments. Especially, the exponential growth model of scattered atoms used in previous articles \cite{Inouye1999science} would not accurately account for the phase dependence. Even considering the depletion of the original mode of the condensate it writes ${\mathrm{d} N_s(t)}/{\mathrm{d} t}= G(t) \left( N_0 - N_s(t) \right) N_s(t)$, where $N_S$ is the number of scattered atoms and $N_0$ the initial number of atoms in the condensate~\cite{Inouye1999science}. The gain $G(t)$ is proportional to the pump intensity $I(t)$. This logistic differential equation can be solved analytically as $ N_s(t) =[N_0 e^{N_0 \int_0^t G(t) \mathrm{d}t}]/{[N_0 -1 + e^{N_0 \int_0^t G(t) \mathrm{d}t}]}$. Within this model, the number of scattered atoms is determined by the integration of the pump intensity $I(t)$, thus proportional to $\int_{0}^{t}I(t)dt = I_0 \int_{0}^{t} [1+\cos(\Delta \omega t +\phi_0)]dt\geq0$. When the pumping pulse duration equals to an exact cycle, the integration of the pump intensity is always the same, regardless of the relative initial phase between the two frequency components. This means, within this nonlinear growth model, the number of scattered atoms is independent of the initial relative phase.
The above model is not valid since more than one scattered mode should be taken into account. Actually, the different modes form matter wave gratings and affect the scattering behavior. To give a physical explanation, we need to clarify the relationship between the population of the different modes and the initial relative phase $\phi_0$. Here we restrain our analysis to a set of diagonal modes : $(-1, -1)$, $(0, 0)$, and $(1, 1)$. Results are similar for the other diagonal modes ($(1, -1)$, $(0, 0)$, and $(-1, 1)$) because of symmetry. Using the slowly varying envelope approximation (SVEA)~\cite{Zobay2006pra}, the population of mode $(-1, -1)$ is given by its wave function $\psi_{-1,-1}(\xi,\tau)$ : \begin{equation}\label{number} \frac{\partial N_{-1,-1}(\tau)}{\partial t}=\int_{-\infty}^{\infty}d\xi \left(\frac{\partial\psi_{-1,-1}} {\partial t}\psi_{-1,-1}^{*} +\frac{\partial\psi_{-1,-1}^{*}} {\partial t}\psi_{-1,-1}\right). \end{equation} We use the semiclassical Maxwell-Schr\"{o}dinger equations to describe the coupled dynamics of matter-wave and optical fields. Although this theory can not be used to analyze the initial quantum spontaneous process, the pulse in the experiment is about several tens of $\mu s$ and far beyond its initial quantum characteristic. So the effect of the pump beam phase on the initial seeds can be omitted to analyze the experimental data~\cite{Zobay2006pra,Bar-Gill2007arxiv,yang}. Considering the trapped BEC is tightly constrained at its short axis degrees and the Fresnel index number of the optical field is around 1, we consider a one-dimensional semiclassical model including spatial propagation effects. Then the envelope function of the end-fire mode optical fields with a pump laser consisting of two components having same intensity and polarization but different frequencies $\omega_{l}= c k_{l}$ and $\omega_{l}-\Delta\omega$, is given by~\cite{Zobay2006pra,yang} \begin{eqnarray}\label{e}
e_{-}(\xi,\tau) &=& -i \kappa(\phi_{0}) \sqrt{\frac{L}{k_{l}}}
\int_{\xi}^{\infty}\mathrm{d}\xi'
(\psi_{0,0} \psi_{1,1}^{*}
\nonumber\\&+&
\psi_{-1,-1} \psi_{0,0}^{*}e^{-i2\tau}), \end{eqnarray} where $\tau = 2\omega_{r}t$ and $\xi = k_{l}z$. $\kappa(\phi_{0})=g \left(1+ e^{i (2 \tau + \phi_{0})}\right)$ is connected with the initial relative phase $\phi_{0}$ and the coupling factor between light and atom $g=\sqrt{3 \pi c R / (2 \omega_{l}^{2} A L)}$. $R$ is the Rayleigh scattering rate of the pump component and $L$ the BEC length. It indicates that the end-fire mode field $e_{-}$ is due to the coherence between different modes, such as modes $(0,0)$ and $(1,1)$, $(-1,-1)$ and $(0,0)$, and spatial overlap between these modes is needed. There exists $4\omega_{r}$ frequency difference between adjacent modes.
The coupled evolution equations of atomic side modes is given by \begin{equation}\label{eq1} \frac{\partial\psi_{-1,-1}(\xi,\tau)}{\partial \tau}=- i \kappa^{*}(\phi_{0}) e_{-} \psi_{0,0} e^{i 2\tau} \end{equation} This equation describes the atom exchange between modes $(-1,-1)$ and $(0,0)$ through the pump laser and end-fire mode field. An atom in $(-1,-1)$ mode may absorb a laser photon and emit it into an end-fire mode $e_{-}$, and the accompanying recoil drives the atom into the $(0,0)$ mode. The atom may absorb an endfire mode photon and deposit it into the laser mode, to form the backward scattering mode $(-2,-2)$ which process is very weak and omitted. Here we have omitted the dispersion term, spatial translation, photon exchange between modes and non-diagonal modes connected with $e_{+}$ such as mode $(-2,0)$ and $(0,-2)$~\cite{Zobay2006pra,yang}.
Inserting Eq.~(\ref{e}) and (\ref{eq1}) into Eq.~(\ref{number}), we can get the evolution equation of $N_{-1,-1}$ \begin{eqnarray}{\label{a}}
\frac{\partial N_{-1,-1}(\tau)}{\partial \tau} &=& -2\frac{g^2Lc}
{\omega_rc}[1+\cos(2\tau +\phi_0)]
\nonumber\\
&\times&\left[C_{0,1}e^{i2\tau}+C_{-1,0}\right]+c.c., \end{eqnarray}
where the envelope function of each side mode $\psi_{m,m}$ is written as $\psi_{m,m}=|\psi_{m,m}|e^{-i\varphi_{m,m}}$ with the phase $\varphi_{m,m}$ assumed to be space independent. Here we define $C_{m,n}(\tau)=\int_{-\infty}^{\infty}d\xi
\tilde{C}_{m,n}(\xi,\tau)$=$|C_{m,n}(\tau)|e^{-i\Delta\varphi_{m,n}}$ with the phase difference between the matter waves $\Delta\varphi_{m,n}=\varphi_{m,m}-\varphi_{n,n}+\varphi_{0,0}-\varphi_{-1,-1}$, and the interference grating $\tilde{C}_{m,n}(\xi,\tau)=\psi_{0,0}\psi^*_{-1,-1}\int_\xi^\infty d\xi^\prime\psi_{m,m}\psi^*_{n,n}$. Then we have $\Delta\varphi_{0,1} = 2\varphi_{0,0}-\varphi_{1,1}-\varphi_{-1,-1}$, and $\Delta\varphi_{-1,0} = 0$.
In this case the above Eq.~(\ref{a}) can be expanded using trigonometric formula \begin{eqnarray}
\frac{\partial N_{-1,-1}(\tau)}{\partial \tau}&=&-\frac{g^2L}{\omega_rc}
\left\{\left[|C_{0,1}|2\cos(2\tau-\Delta\varphi_{0,1})\right.\right.\nonumber\\
&+&\cos(\phi_0+\Delta\varphi_{0,1})+\cos(4\tau+\phi_0-\Delta\varphi_{0,1})]\nonumber\\
&+&\left.2|C_{-1,0}|[1+\cos(2\tau+\phi_0)]\right\} \end{eqnarray}
In a first approximation, we replace $|C_{m,n}|$ by its time average
$\overline{|C_{m,n}|}$. Equation~(5) can then be integrated from $0$ to one cycle, and the cosine functions of time vanish, leaving only the $\cos(\phi_0 +\Delta\varphi_{0,1})$ term. A simpler expression for $N_{-1,-1}$ at the end of the pumping pulse($\Delta\omega t=2\pi$ or $\tau=\pi$) is obtained: \begin{equation}{\label{results}}
N_{-1,-1}(\pi)=-2\pi\frac{g^2L}{\omega_rc}\overline{|C_{0,1}|}\cos(\phi_0
+\Delta\varphi_{0,1})+\alpha, \end{equation} The first term of Eq.~(\ref{results}) indicates that the population of side mode $(-1,-1)$ is connected with the module of four waves
$|C_{0,1}|= |\psi_{0,0}\psi^*_{-1,-1}|\int_\xi^\infty d\xi^\prime|\psi_{0,0}\psi^*_{1,1}|$, the relative phase $\Delta
\varphi_{0,1}$ and the optical initial relative phase $\phi_{0}$, the coefficient of the cosine function determining the amplitude of the oscillation. The second part $\alpha$ approximated to be a constant in the above expression should actually include other terms due to the residual time dependent part of $|C_{m,n}|$ and may also slightly depend on $\phi_0$ . This dependence may be neglected in a first approximation as the time-dependent terms in Eq. (5) have different time-dependent signatures and may cancel each other after integration. The main purpose of Eq.~(\ref{results}) is to show the dependence of the backward scattering on the initial relative phase $\phi_0$, although it is hard for us to prove its positivity due to the complexity of $\alpha$. However, numerical calculations demonstrate that this backward scattered atom number is always positive.
To give a physical explanation to this initial relative phase dependence and because of the assumption that the phase $\varphi_{m,m}$ of each side mode does not depend on space, we can focus our attention on the terms $\tilde{C}_{m,n}$ since the spatial integral from $-\infty$ to $\infty$ in the definition of $C_{m,n}$ does not change the relative phase of the matter waves. The term $\tilde{C}_{0,1}=\psi_{0,0}\psi^*_{-1,-1}\int_\xi^\infty d\xi^\prime \psi_{0,0}\psi^*_{1,1}$ indicates the presence of two sequential diffractions by two gratings, which we analyze as follows. With the spatial coordinates described in Fig.~1, the incident laser light traveling in the $x$ direction is diffracted by the grating formed by modes $(0,0)$ and $(1,1)$, which is along the diagonal of the $x-z$ plane, and can be expressed as
$\int_{z}^\infty dz |\psi_{0,0}\psi_{1,1}^*
|\cos(kx+kz+\varphi_{1,1} -\varphi_{0,0})$. The scattered light (end-fire mode light) propagating along the long axis of the condensate in $-z$ direction is then diffracted again by the grating formed by modes $(0,0)$ and $(-1,-1)$, which is also along the diagonal of the $x-z$ plane, and can be described as
$|\psi_{0,0}\psi_{-1,-1}^*| \cos(kx+kz-\varphi_{-1,-1} +\varphi_{0,0})$, resulting in a scattered light traveling back into $x$ direction which is highly correlated to the backward scattered atom number. The spatial integral in the former grating describes the propagating effect. Here a phase shift of $\pi/2$ takes place as the integral can be approximated to be
$-|\psi_{0,0}\psi_{1,1}^*|\sin(kx+kz+\varphi_{1,1}
-\varphi_{0,0})=|\psi_{0,0}\psi_{1,1}^*|\cos(kx+kz+\varphi_{1,1}-\varphi_{0,0} +\pi/2)$. Based on light diffraction theory, when a beam is incident at an angle of $45^\circ$ onto a grating, which is the combination of two gratings, the maximum diffraction light along the incident direction occurs when these two gratings are in phase. That is, $-\varphi_{-1,-1} + \varphi_{0,0} =\varphi_{1,1} -\varphi_{0,0}+\pi/2$, or \begin{equation}\label{phMat} (\varphi_{1,1}-\varphi_{0,0})+\frac{\pi}{2}+(\varphi_{-1,-1}-\varphi_{0,0})=0. \end{equation} That means $\Delta\varphi_{0,1}=\pi/2$. According to our simulation we find that the relative phase of the matter waves $\Delta \varphi_{0,1}$ is nearly $\pi/2$, and there is a little deviation for different initial relative phases $\phi_0$ of pump beam. The dependence of $\Delta \varphi_{0,1}$ on $ \phi_0$ reflects the intrinsic nature of nonlinearity in the superradiance process since it should not depend on the initial relative phase between the two frequency components when the pumping pulse duration equals to an exact cycle for a linear process. The initial relative phase of the pumping beams slightly alters the phase matching condition, which maybe results in the divergence between our simple model and the numerical result. However, in our discussed model, the result giving $\Delta\varphi_{0,1}=\pi/2$ is reasonable and meaningful for understanding the scattering picture.
According to Eq.~({\ref{results}) and ({\ref{phMat}}) and neglecting $\alpha$, the ratio of backward scattered atoms to the total atom number versus the initial phase $\phi_{0}$ is simply a sinusoidal function and is plotted as the dotted line in Fig.~\ref{experdata}, with its amplitude adjusted to the experimental data. Backward scattering is enhanced when the relative phase is nearly $\phi_{0}=\pi/2$, while suppressed for $\phi_{0}$ close to $3\pi/2$. This simple model agrees well with our experimental data except for a small shift of the minimum.
Furthermore, the ratio of the backward scattering atom number to the total atom number is simulated numerically using coupled Maxwell-Schr\"{o}dinger equations~\cite{Zobay2006pra,yang} with an initial seed of one atom in modes $(1,1)$ and $(-1,-1)$, as shown by the dashed line in Fig.~\ref{experdata} with the coupling factor $g=1.5\times 10^{6}$. This simulation is in good qualitative agreement with our data but not completely satisfactory. The difference may be due to the necessity to take into account higher order modes in the scattering process and to go beyond 1D approximation. The matching condition of matter waves may also not be exactly satisfied. Describing this mechanism under full quantum methods~\cite{guo} and considering wave mixing for the nonlinear high-order scattering modes is our future work.
\begin{figure}
\caption{The simulation for the ratio of backward scattered atoms to the total atom number vs the initial phase $\phi_{0}$ for different weak coupling factors $g$: $1.0\times10^{6}$ (dashed line, its value is amplified by 10 times to show); $1.5\times 10^{6}$(solid line); $1.6\times 10^{6}$(dotted line) and $1.7\times 10^{6}$(dashed-dotted line). }
\label{gas}
\end{figure}
To show the features of the position of the maximum and minimum of the scattering ratio vs the relative phase $\phi_{0}$, we simulate it for different pumping factors $g$ in the week coupling regime, as shown in Fig.~\ref{gas}. The backward scattering ratio increases with the coupling factor, which depends on the system parameters like the pump beam intensity and detuning, the total number of atoms or the BEC shape. The simulation clearly shows that the position of the maximum and minimum of the scattering very weakly depends on the experimental parameters, this is a generic feature. However, the value of the maximum increases with the coupling factor.
\begin{figure}
\caption{The ratio of backward scattered atoms to the total atom number vs the initial phase
$\phi_{0}$ between the two pump beams for different seeds with $g=1.5\times 10^{6}$.}
\label{seed}
\end{figure}
The superradiant scattering is initiated by quantum mechanical noise, i.e., spontaneous Rayleigh scattering from individual condensate atoms. Subsequent stimulated scattering and bosonic enhancement lead to rapid growth of the side-mode populations. The ratio of backward scattered atoms to the total atom number vs the initial phase for different initial seeds in modes $(1,1)$ and $(-1,-1)$ are shown in Fig.~\ref{seed}. We can see that the positions of maximum and minimum do not vary with different seeds, although the amplitude of the scattering events changes obviously. The smaller the seeds, the lower the backward scattering amplitude. This shows that these features caused by the initial relative phase are not related to quantum fluctuation effects.
The above discussion is for the condition $\Delta\omega t=2\pi$, which means identical coupling factor between light and atoms for the different initial relative phase. If the condition $\Delta\omega t=2\pi$ is violated, the integration of the pump intensity is different for different relative phases, so that the coupling factor is different. The ratio of backward scattered atoms to the total atom number vs the initial phase $\phi_{0}$ is shown in Fig.7 for the same frequency difference $15 \mathrm{KHz}$ and different pump times $t$. From Fig.~\ref{violate}, we know that the position of the extrema will change at different $t$, while the value of the maximum will increase with $t$.
\begin{figure}
\caption{The ratio of backward scattered atoms to the total atom number vs the initial phase $\phi_{0}$ for the same frequency difference $15$KHz but different pump times $t$: 22.22 $\mu$s (dashed line); 44.44 $\mu$s (dotted line); 66.67 $\mu$s (solid line) and 88.89 $\mu$s (dashed-dotted line). $g=1.5\times 10^{6}$.}
\label{violate}
\end{figure}
In summary, we have studied superradiant scattering from a Bose-Einstein condensate with a two-frequency pump beam. We have shown that this phenomenon depends much on the initial relative phase between the two optical components. Two matter gratings, one formed by the condensate at rest plus the side mode with positive momentum, the other by the condensate at rest plus the side mode with negative momentum, scatter endfire modes photons, which in the end affects the atom scattering process. We found that the relative optical phase is imprinted on matter wave gratings and can enhance or annihilate scattering in backward modes. Adjusting this phase provides a powerful tool for controlling superradiant scattering. It is already very beneficial for understanding the interaction between matter waves and optical waves in cooperative scattering.
We thank the anonymous referee for useful and detailed suggestions. This work is partially supported by the state Key Development Program for Basic Research of China (No.2005CB724503, 2006CB921401, 2006CB921402),NSFC (No.10874008, 10934010 and 60490280) and the Project-sponsored by SRF for ROCS, SEM.
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\end{document} | arXiv |
\begin{document}
\newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{question}[theorem]{Question} \newtheorem{problem}[theorem]{Problem} \newtheorem*{claim}{Claim} \newtheorem*{criterion}{Criterion}
\theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{construction}[theorem]{Construction} \newtheorem{notation}[theorem]{Notation} \newtheorem{object}[theorem]{Object} \newtheorem{operation}[theorem]{Operation}
\theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example}
\numberwithin{equation}{subsection}
\newcommand\id{\textnormal{id}}
\newcommand\Z{\mathbb Z} \newcommand\R{\mathbb R} \newcommand\C{\mathbb C} \newcommand\CC{\mathbf C} \newcommand\BB{\mathbf B} \newcommand\TT{\mathbf T} \newcommand\PP{\mathcal P} \newcommand\W{\mathcal W} \newcommand\RR{\mathcal R} \newcommand\Aut{\textnormal{Aut}}
\title{Wiggle Island}
\author{Danny Calegari} \address{University of Chicago \\ Chicago, Ill 60637 USA} \email{[email protected]} \date{\today}
\begin{abstract} A {\em wiggle} is an embedded curve in the plane that is the attractor of an iterated function system associated to a complex parameter $z$. We show the space of wiggles is disconnected --- i.e.\/ there is a {\em wiggle island}. \end{abstract}
\maketitle \setcounter{tocdepth}{1}
\section{Squiggles and wiggles}
A {\em squiggle} is a continuous map $w_z:I \to \C$ where $I:=[0,1]$, depending on a complex parameter $z$ with $|z|<1$ and $|1-z|<1$ in the following way.
If we define the two maps $$f_z: x \to -zx+z, \quad g_z: x \to (z-1)x+1$$ and inductively define a sequence of maps $w^i_z:I \to \C$ by \begin{enumerate} \item{$w^0_z:[0,1] \to \C$ is the identity map; and} \item{$w^j_z$ is obtained by concatenating the two maps $(g_z w^{j-1}(z))$ and $(f_z w^{j-1}(z))$, and reversing the orientation (i.e.\/ precomposing with $t \to 1-t$);} \end{enumerate}
then $w_z$ is the limit of $w^i_z$ as $i \to \infty$. This limit exists because when $|z|<1$ and $|1-z|<1$ both $f_z$ and $g_z$ are uniformly contracting. See Figure~\ref{wiggle_iterates} illustrating $w^i_z$ for $z=0.4+0.4i$ and $i=0,1,2,3$.
\begin{figure}
\caption{$w^i_z$ for $z=0.4+0.4i$ and $i=0,1,2,3$}
\label{wiggle_iterates}
\end{figure}
A {\em wiggle} is a squiggle which is an embedding. The set of wiggles $w_z$ is parameterized by an open subset $\W:= \lbrace z \text{ such that } w_z \text{ is a wiggle}\rbrace$ of $\C$. A wiggle with parameter $z$ has Hausdorff dimension $d$
where $|z|^d + |1-z|^d = 1$; thus $\W$ is a subset of the open disk of radius $1/2$ centered at $1/2$ (because a squiggle might intersect itself its Hausdorff dimension only satisfies the inequality
$|z|^d + |1-z|^d \ge 1$). Denote the complement of $\W$ in this disk by $\RR$. Figure~\ref{W} depicts $\W$ (in white) as a subset of this disk (note how $\W$ very nearly fills the entire disk!) \begin{figure}
\caption{$\W$ (in white) as a subset of $\lbrace z: |z-1/2|<1/2\rbrace$.}
\label{W}
\end{figure}
The set $\W$ contains one {\em big component}, the connected component of $1/2$; this big component contains the real interval $(0,1)$ and the imaginary interval $(-0.5i,0.5i)$. However, the big component is not all of $\W$:
\begin{theorem}[Wiggle Island]\label{theorem:wiggle_island} $\W$ is not connected. \end{theorem}
Figure~\ref{wiggle_curve} depicts a wiggle $w_z$ for $z$ in an `island' component of $\W$ centered at approximately $z=0.3409+0.43486i$. The island is invisible at the resolution of Figure~\ref{W}; a zoomed in image of the island is Figure~\ref{W_island_rigorous}.
The wiggle $w_z$ for $z$ as above {\em may not} be deformed through wiggles to $w_{1/2}$ (i.e.\/ the unit interval).
\begin{figure}
\caption{An approximation to $w_z$ for $z=0.3409+0.43486i$; this is an embedded arc.}
\label{wiggle_curve}
\end{figure}
\begin{remark}[The Carpenter's Rule Problem] The {\em Carpenter's Rule Problem}, first posed by Stephen Schanuel and George Bergman in the early 1970's, asks whether every embedded planar polygonal arc may be {\em straightened} (i.e.\/ moved through embeddings to a straight arc by changing the angles but not the lengths of the segments). Connelly, Demaine and Rote showed in 2000 \cite{Connelly_Demaine_Rote} that the answer to the Carpenter's Rule Problem is {\em yes}.
The {\em Wiggle Problem} asks analogously whether every wiggle may be straightened through a family of wiggles; Theorem~\ref{theorem:wiggle_island} says that the answer to the Wiggle Problem is {\em no}. \end{remark}
\section{Certifying in and out}
In this section we give two {\em stable numerical} criteria to certify that $z\in \W$ resp. $z\in \RR$.
\subsection{Certifying $z\in \W$}
Fix $z$ and write $\gamma:=w_z(I)$ and abbreviate $f_z$ and $g_z$ by $f$ and $g$. Observe that $\gamma = f\gamma \cup g\gamma$, and by induction $z\in \W$ if and only if $f\gamma \cap g\gamma = \lbrace z \rbrace$. Let $S_n$ denote the set of words of length $n$ in the alphabet $\lbrace f,g\rbrace$ and let $S=\cup_n S_n$. By abuse of notation we think of $u \in S_n$ as a map, obtained by composing $f$ or $g$ according to the letters of $u$. Let $fS_{n-1}$ resp. $gS_{n-1}$ represent words of length $n$ beginning with $f$ and $g$ respectively.
\begin{lemma} Define
$$R = \max\left( \frac {|z-1|}{2|1-|z|}, \frac {|z|} {2|1-|1-z||} \right)$$ and let $B$ be the ball of radius $R$ about $1/2$. Then $\gamma$ is contained in $B$. \end{lemma} \begin{proof} Both $f$ and $g$ take $B$ inside itself. \end{proof}
Thus to show $z\in \W$ it would suffice to show, for any $n$, that $uB \cap vB = \emptyset$ for all $u\in fS_n$ and $v\in gS_n$ except for one specific pair for which $uB \cap vB = \lbrace z \rbrace$. Unfortunately this is impossible; necessarily $z$ itself is in the interior of some $uB \cap vB$
The key observation is that $ffgf^{-1} = ggfg^{-1}$; in particular, $ffg\gamma \cup ggf \gamma$ is a dilated copy of $\gamma$ itself. Thus if $ffg\gamma \cap ggf \gamma$ contains a point other than $z$, then $f\gamma \cap g\gamma$ contains a point other than $z$ which is not in $ffg\gamma \cap ggf \gamma$. We therefore obtain the following algorithm which, if it terminates, certifies that $z$ is in the interior of $\W$:
\begin{algorithm} Initialize $L$ to the set of pairs $(u,v) \in fS_2 \times gS_2 - (ffg,ggf)$
\While{$L\ne \emptyset$}{ \ForAll{$(u,v) \in L$}{
\eIf{$uB \cap vB = \emptyset$}{ remove $(u,v)$ from $L$ }{ replace $(u,v)$ with $\lbrace (uf,vf),(uf,vg),(ug,vf),(ug,vg) \rbrace$ } } } \end{algorithm}
\subsection{Certifying $z\in \RR$}
We now show how to modify the algorithm from the previous subsection to certify (numerically) that $z\in \RR$. Actually our modified algorithm certifies that $z$ is contained in the {\em interior} of $\RR$, and therefore fails for $z$ in the frontier of $\RR$. However one consequence of the nature of the algorithm is that it implies that the interior of $\RR$ is dense.
The idea is a modification of the method of {\em traps}, introduced in \cite{Calegari_Koch_Walker} to prove Bandt's Conjecture on interior points in $\mathcal{M}$, the connectivity locus for another 1-parameter family of complex 1-dimensional IFSs (and the analog to $\RR$ for this family)
The idea is very simple. Let $u\in fS$ and $v\in gS$ and choose some finite $n$. Let $uS_n B$ denote the union of translates of $B$ for all elements of $uS_n$ and likewise $vS_n B$.
\begin{definition}[Stable Crossing] The triple $(u,v,n)$ is a {\em stable crossing} if there are {\em disjoint proper rays} $r^\pm$, $s^\pm$ from points $p^\pm$ and $q^\pm$ in $u\gamma$, $v\gamma$ to infinity so that $r^\pm$ are disjoint from $vS_nB$, so that $s^\pm$ are disjoint from $uS_nB$, and so that $r^\pm$ link $s^\pm$ at infinity. \end{definition}
The existence of a stable crossing is, in fact, stable in $z$; and given $(u,v,n)$ stable for $z$ one may easily estimate a lower bound on the radius of a ball around $z$ for which this triple continues to be stable.
Furthermore, the existence of a stable crossing certifies $z\in \RR$ for elementary topological reasons.
\begin{lemma} If $(u,v,n)$ is a stable crossing for $z$ then $z\in \RR$. \end{lemma} \begin{proof} Let $p^\pm \in u\gamma$ and $q^\pm \in v\gamma$ be the finite endpoints of $r^\pm$ and $s^\pm$ respectively. Since $u\gamma$ and $v\gamma$ are path connected, there are arcs $\alpha \subset u\gamma$ and $\beta \subset v\gamma$ joining $p^\pm$ and $q^\pm$. Then $r^+\cup \alpha \cup r^-$ and $s^+\cup \beta\cup s^-$ are properly immersed lines in $\C$ that are embedded and disjoint at infinity where their endpoints are linked. Thus their algebraic intersection number (rel. their ends) is odd, so they must intersect. But by hypothesis the only place they might intersect is $\alpha$ with $\beta$. \end{proof}
To search for stable crossings we enumerate pairs $(u,v)$ by the algorithm from the previous section, then for each we compute $uS_nB$ and $vS_nB$ for some finite $n$ and look for rays $r^\pm$, $s^\pm$ and points $p^\pm$, $q^\pm$ forming a stable crossing. If we find one we certify $z$ (and some ball of computable radius about it) as lying in $\RR$.
Let's say more generally that two compact path-connected subsets $K,L \subset \C$ have a stable crossing if there is some $\epsilon$ and disjoint rays $r^\pm$, $s^\pm$ from points $p^\pm$, $q^\pm$ in $K$ and $L$ so that $r^\pm$ is disjoint from the $\epsilon$-neighborhood of $L$ and $s^\pm$ is disjoint from the epsilon-neighborhood of $K$, and $r^\pm$, $s^\pm$ link at infinity. Thus $u\gamma,v\gamma$ have a stable crossing if and only if $(u,v,n)$ is a stable crossing for some $n$. See Figure~\ref{Gamma_crossings} for some relevant examples.
As an application of stable crossings we prove that the interior of $\RR$ is dense in $\RR$. Both the statement and the proof are very similar to Theorem~7.2.7 from \cite{Calegari_Koch_Walker}.
\begin{theorem}[Interior is Dense]\label{interior_is_dense} The interior of $\RR$ is dense in $\RR$. \end{theorem} \begin{proof}
It suffices to find a stable crossing arbitrarily close to any $z_0\in \RR$. Let $z_0\in \RR$ so that $u\gamma$ intersects $v\gamma$ for some finite words $u,v$ which do not start with the prefix $ffg,ggf$. Without loss of generality, we may assume $u,v$ both have length $n\gg 1$. There are complex numbers $\alpha,\beta$ so that $\gamma = \alpha v^{-1}u\gamma+\beta$, and by padding $u$ or $v$ or both with additional letters $f$ or $g$ if necessary we may arrange for $1/C<|\alpha|<C$ for some $C$ depending on $z$ while still having $u\gamma$ intersect $v\gamma$.
As we vary $z$ we get $\gamma(z) = \alpha(z) v^{-1}u\gamma(z) + \beta(z)$ for suitable analytic functions $\alpha$ and $\beta$. Now, $\beta(z)$ is not constant; one way to see this is to observe that it is much much bigger at $z=1/2$ than at $z_0$. If $n$ is big enough, $u\gamma$ and $v\gamma$ have diameter extremely small compared to the first nonvanishing derivative of $\beta$ at $z_0$. Thus for any $\epsilon$, there is an $n$ so that if we choose $n$ as above, and $U$ is the disk of radius $\epsilon$ about $z_0$, then $\lbrace \beta(z), z\in U\rbrace$ contains the disk of radius $1/\epsilon$ about zero, while $\alpha(z)$ stays essentially constant. Thus we would be done if we could show that there is some $\mu \le 1/\epsilon$ so that $\gamma$ and $\alpha \gamma+\mu$ have a stable crossing.
Actually, with more work we may eliminate the factor of $\alpha$. Note that $\alpha(z)$ is just a product of integer powers of $-z$ and $(z-1)$ according to how many $f$s and $g$s are in $u$ and $v$. Thus by multiplying $u$ and $v$ on the right by some power of $u$ and $v$ we may arrange for $\gamma(z) = v^{-1}u\gamma(z) + \beta(z)$ for some new $\beta(z)$, where again we can arrange for the image of $\beta(z)$ to contain the disk of radius $1/\epsilon$ about zero. It follows that to prove the theorem we just need to show that {\em whatever} shape $\gamma(z_0)$ is, there is some constant $\mu\in \C$ so that $\gamma(z_0)$ and $\gamma(z_0)+\mu$ have a stable crossing. This is proved in Lemma~\ref{translation_lemma} and Lemma~\ref{gamma_not_convex}. \end{proof}
Thus Theorem~\ref{interior_is_dense} is reduced to the following two lemmas:
\begin{lemma}[Translation Lemma]\label{translation_lemma} Let $K$ be a compact full subset of $\C$ and suppose $K$ is not convex. There is some complex number $\mu$ of order $\text{diam}(K)$ so that $K$ and $K + \mu$ have a stable crossing. \end{lemma}
This is Lemma~7.2.2 from \cite{Calegari_Koch_Walker}. Finally we need to understand the set of $z$ for which $\gamma(z)$ is convex.
\begin{lemma}[$\gamma$ not convex]\label{gamma_not_convex}
For $|z-1/2|<1/2$ and $z$ not real, $\gamma(z)$ is not convex. \end{lemma}
Of course if $|z-1/2|<1/2$ and $z$ {\em is} real then $\gamma(z)=[0,1]$ and $z\in \W$. \begin{proof} If $\gamma(z)$ is convex but not real it encloses a full subset $X\subset \C$ with nonempty interior (in particular $X$ has Hausdorff dimension $2$). We claim that actually $\gamma(z)=X$. To see this, let's let $p\in X$ be a point which realizes the maximal distance $\epsilon$ to $\gamma(z)$. Because $X$ is convex, and $f(X)$, $g(X)$ intersect, it follows that $X=f(X)\cup g(X)$ (without convexity there might be some omitted region `trapped' between $f(X)$ and $g(X)$). But then $p\in f(X)$ or $g(X)$; thus the distance from $p$ to
$\gamma(z)$ is at most $\max(|z|\epsilon,|1-z|\epsilon)$. Since $|z|$ and
$|1-z|$ are both strictly less than $1$ it follows that $\epsilon=0$ and therefore $\gamma(z)=X$ and has Hausdorff dimension $d=2$. But then
$|z|^2 +|1-z|^2\ge 1$ so $|z-1/2|\ge 1/2$ and we are done. \end{proof}
Compare to Lemma~7.2.3 from \cite{Calegari_Koch_Walker}. This completes the proof of Theorem~\ref{interior_is_dense}. We end this subsection with a conjecture:
\begin{conjecture} Every interior point of $\RR$ is certified by some stable crossing $(u,v,n)$. \end{conjecture}
\subsection{Proof of Theorem~\ref{theorem:wiggle_island}}
Using the techniques of the previous two sections, we may numerically certify some $z\in \W$ and numerically check that a polygonal loop separating $z$ from $0$ is contained in $\RR$. To make this computationally feasible we pursue the following strategy: \begin{enumerate} \item{find an apparent island by some experimentation, in a small region $U\subset \C$;} \item{compute a region $O\subset \C^*\times \C$ so that $\gamma(z)$ and $\alpha \gamma(z)+\beta$ have a stable crossing for all $(\alpha,\beta)\in O$ and all $z\in U$;} \item{run the algorithm on a narrowly spaced grid of $z\in U$ to generate crossing pairs $(u,v)$ and for each compute $(\alpha(z),\beta(z))$ such that $\gamma(z) = \alpha(z) v^{-1}u\gamma(z) + \beta(z)$;} \item{if $(\alpha(z),\beta(z))$ is in $O$, compute an $\epsilon$ so that $(\alpha(z'),\beta(z'))\in O$ when
$|z'-z|<\epsilon$; and finally} \item{when we have generated enough $\epsilon$-balls centered at $z$ in a fine enough grid to surround a point in $\W$, we have proved the theorem.} \end{enumerate}
We elaborate on these points.
\begin{figure}
\caption{The disk $\Gamma$ encloses $\gamma(z)$ for all $z\in U$. The vertices $0$ and $1$ of $\gamma(z)$ are distinguished.}
\label{Gamma}
\end{figure}
The region $U$ is an open square centered at $0.3409+0.43486i$ with width $0.00005$. We may then readily determine the vertices of a polyhedral disk $\Gamma$ that contains a thin neighborhood of $\gamma(z)$ for all $z\in U$; see Figure~\ref{Gamma}.
\begin{figure}
\caption{Stable crossings of $\Gamma$ and $\alpha\Gamma+\beta$.}
\label{Gamma_crossings}
\end{figure}
Next we compute some open polydisks in $\C^*\times \C$ parameterizing $(\alpha,\beta)$ for which $\Gamma$ and $\alpha \Gamma + \beta$ have a suitable stable crossing (with rays landing at $0,1$ and $\beta,\alpha+\beta$ respectively, which are always in $\gamma(z)$ and $\alpha\gamma(z)+\beta$); see Figure~\ref{Gamma_crossings}.
The program {\tt wiggle} implements the algorithm from the previous section, and rigorously finds $\epsilon$-balls (actually polygons) centered at points $z$ near $\W$ with a stable crossing for some fixed $(u,v,n)$. The result of the output is Figure~\ref{W_island_rigorous}. The blocks of solid color in the figure correspond to specific $(u,v,n)$. These blocks surround Wiggle Island (in the north of the figure), completing the proof of Theorem~\ref{theorem:wiggle_island}. The isthmus to the southeast is part of the big component of $\W$.
\begin{figure}
\caption{Stable crossings in $\RR$ surround Wiggle Island.}
\label{W_island_rigorous}
\end{figure}
Intermediate between Wiggle Island and the mainland there is apparent in the Figure a smaller island, and a speck (which on magnification turns out too be an island too). It seems likely that there are infinitely many islands, arranged in an asymptotically geometric spiral converging to an algebraic point $z_c\sim 0.340922 + 0.43481i$. One could give a rigorous numerical proof of this by a modification of the argument proving Theorem~\ref{theorem:wiggle_island}, applied to the {\em tangent cone} to $\W$ centered at $z_c$, though we have not pursued this. An exactly analogous argument for $\mathcal{M}$ certifying the existence of a spiral of islands is proved in \S~9 of \cite{Calegari_Koch_Walker} and we refer the interested reader to that paper for details.
We end with a conjecture, parallel to Conjecture~9.2.7 from \cite{Calegari_Koch_Walker}:
\begin{conjecture} Every point in the frontier of $\RR$ is the limit of islands in $\W$ of diameter going to zero. \end{conjecture}
\begin{remark} In fact the existence of Wiggle Island has been well-known to Europeans as least since the early 17th century, and is well-documented in the literature \cite{Wiggles}. \end{remark}
\end{document} | arXiv |
\begin{document}
\title{\textbf{Involution matchings, the semigroup of orientation-preserving and orientation-reversing mappings, and inverse covers of the full transformation semigroup}}
\author{Peter M. Higgins, University of Essex, U.K.} \maketitle \begin{abstract} We continue the study of permutations of a finite regular semigroup that map each element to one of its inverses, providing a complete description in the case of semigroups whose idempotent generated subsemigroup is a union of groups. We show, in two ways, how to construct an involution matching on the semigroup of all transformations which either preserve or reverse orientation of a finite cycle. Finally, by way of application, we prove that when the base set has more than three members, a finite full transformation semigroup has no cover by inverse subsemigroups which is closed under intersection. \end{abstract}
\section{Introduction and General Results}
\subsection{Background}
In {[}6{]} the author introduced the study of \emph{permutation matchings, }which are permutations on a finite regular semigroup $S$ that map each element to one of its inverses. It follows from Hall's Marriage Lemma that $S$ will possess a permutation matching if and only if
$S$ satisfies the condition that $|A|\leq|V(A)|$ for all subsets $A$ of $S$ with set of inverses $V(A)$. Although not all finite regular semigroups have a permutation matching, there are positive results for many important classes. In {[}7{]} the author characterised some classes of finite regular semigroups by the nature of their permutation matchings and determined, in terms of Green's relations on principal factors, when a finite orthodox semigroup $S$ has a permutation matching. In this case a permutation matching implies the existence of an involution matching. In Section 1.3 we show how this result may be extended to semigroups whose idempotent-generated subsemigroup is a union of groups.
It is not known whether the semigroup ${\cal O}_{n}$ of all order-preserving mappings on a finite $n$-chain has a permutation matching of any kind. It was shown in {[}6{]} however that ${\cal OP}_{n}$, the semigroup of all orientation-preserving mappings on an $n$-cycle, has a natural involution matching. In Section 2.1 we summarise relevant properties of this semigroup and of ${\cal P}_{n}$, the semigroup of all orientation-preserving and orientation-reversing mappings on an $n$-cycle. This latter semigroup, which was introduced in {[}1{]} and independently by McAlister in {[}9{]}, has an intricate structure, which is manifested in the context of the problem of this paper. In Section 3 we construct a dual pair of involution matchings of ${\cal P}_{n}$.
There are no known examples of a finite regular semigroup $S$ that has a permutation matching but no involution matching. It was proved in {[}6{]} by graph theoretic techniques that ${\cal T}_{n}$, the full transformation semigroup on an $n$-set, has a permutation matching but it is not known if ${\cal T}_{n}$ has an involution matching. However in Section 4 we show that ${\cal T}_{n}$ $(n\geq4)$ has no involution matching through so-called strong inverses, which allows us to show that ${\cal T}_{n}$ $(n\geq4)$ has no cover by inverse semigroups that is closed under intersection.
Following the texts {[}8{]} and {[}5{]}, we denote the set of idempotents of a semigroup $S$ by $E(S)$. We shall write $(a,b)\in V(S)$ if $a$ and $b$ are mutual inverses in $S$ and denote this as $b\in V(a)$ so that $V(a)$ is the set of inverses of $a\in S$. We extend the notation for inverses to sets $A$: $V(A)=\bigcup_{a\in A}V(a)$. Standard results on Green's relations, particularly those stemming from Green's Lemma, will be assumed (Chapter 2 of {[}5{]}, specifically Lemma 2.2.1) and fundamental facts and definitions concerning semigroups that are taken for granted in what follows are all to be found in {[}5{]}. We shall sometimes write ${\cal G}$ to stand for either of the Green's relations ${\cal L}$ or ${\cal R}$.
We say that a semigroup $S$ is \emph{combinatorial }(or \emph{aperiodic}) if Green's ${\cal H}$-relation on $S$ is trivial. A completely $0$-simple combinatorial semigroup is known as a \emph{$0$-rectangular band}. The full transformation semigroup on a base set $X$ is denoted by ${\cal T}_{X}$ or by ${\cal T}_{n}$ when $X=X_{n}=\{0,1,2,\cdots,n-1\}$.
Let $C=\{A_{i}\}_{i\in I}$ be any finite family of finite sets (perhaps with repetition of sets). A set $\tau\subseteq\bigcup A_{i}$ is a \emph{transversal }of $C$ if there exists a bijection $\phi:\tau\rightarrow C$
such that $t\in\phi(t)$ for all $t\in\tau$. We assume Hall's Marriage Lemma in the form that $C$ has a transversal if and only if \emph{Hall's Condition }is satisfied, which says that for all $1\leq k\leq|I|$, the union of any $k$ sets from $C$ has at least $k$ members.
\subsection{Permutation matchings}
\textbf{Definitions 1.2.1 }Let $S$ be any semigroup and let $F=\{f\in T_{S}:f(a)\in V(a)\,\forall a\in\mbox{dom\,\ \ensuremath{f}\}}.$ We call $F$ the set of \emph{inverse matchings }of $S$. We call $f\in F$ a \emph{permutation matching }if $f$ is a permutation of $S$; more particularly $f$ is an \emph{involution matching} if $f^{2}=\varepsilon$, the identity mapping on $S$.
In the remainder of the paper we shall assume that $S$ is regular and finite unless otherwise indicated. We shall often denote a matching simply by $'$, so that the image of $a$ is $a'$. We use the shorthand $a''$ as an abbreviation for $(a')'$. We shall work with the family of subsets of $S$ given by $V=\{V(a)\}_{a\in S}$. The members of $V$ may have repeated elements\textemdash for example $S$ is a rectangular band if and only if $V(a)=S$ for all $a\in S$. However, we consider the members of $V$ to be marked by the letter $a$, so that $V(a)$ is an unambiguous member of $V$ (strictly, we are using the pairs $\{a,V(a)\},$ $(a\in S)$). We summarise some results of {[}5{]}.
\textbf{Theorem 1.2.2 }{[}6{]} For a finite regular semigroup $S$ the following are equivalent:
(i) $S$ has a permutation matching;
(ii) $S$ is a transversal of $V=\{V(a)\}_{a\in A}$;
(iii) $|A|\leq|V(A)|$ for all $A\subseteq S$;
(iv) $S$ has a permutation matching that preserves the ${\cal H}$-relation; (meaning that $a{\cal H}b\Rightarrow a'{\cal H}b'$);
(v) each principal factor $D_{a}\cup\{0\}$ $(a\in S)$ has a permutation matching;
(vi) each $0$-rectangular band $B=(D_{a}\cup\{0\})/{\cal H}$ has a permutation matching.
In {[}7, Remark 1.5{]} it was shown that we may replace `permutation matching' by `involution matching' in Theorem 1.2.2 as regards the implications ((i) $\Leftrightarrow$ (v)) $\Leftarrow$ ((iv) $\Leftrightarrow$ (vi)) although the missing forward implication has not been resolved.
\subsection{Permutation matchings for an E-solid semigroup}
\textbf{Definition 1.3.1 }A regular semigroup $S$ is defined to be $E$-\emph{solid} if $S$ satisfies the condition that for all idempotents $e,f,g\in E(S)$ \[ e{\cal L}f{\cal R}g\rightarrow\exists h\in E(S):e{\cal R}h{\cal L}f. \] An alternative characterisation of an $E$-solid semigroup is that of a regular semigroup $S$ for which the idempotent-generated subsemigroup $\langle E(S)\rangle$ is a union of groups {[}3, Theorem 3{]}.
We prove our result on $E$-solid semigroups via the corresponding result for orthodox semigroups. The proof of this latter result involved reducing the general problem to the case of $0$-rectangular bands and then showing that the corresponding ${\cal D}$-class may be diagonalised in that the ${\cal R}$- and ${\cal L}$-classes may be ordered so that all idempotents are contained in rectangular blocks (which then form the maximal rectangular subbands of $D$); $S$ then has a permutation matching if and only if, within each ${\cal D}$-class of $S$, these blocks are similar in the following sense.
\textbf{Definition 1.3.2 }Let $U_{1}$ and $U_{2}$ be finite rectangular bands, let $m_{i}$ and $n_{i}$ denote the respective number of ${\cal R}$-classes and ${\cal L}$-classes of $U_{i}$ $(i=1,2)$. We say that $U_{1}$ and $U_{2}$ are \emph{similar }if $\frac{m_{1}}{n_{1}}=\frac{m_{2}}{n_{2}}$.
\textbf{Theorem 1.3.3 }{[}7, Theorems 3.7 and 3.1{]} Let $S$ be a finite orthodox semigroup. Then $S$ has a permutation matching if and only if for each $0$-rectangular band $B=(D_{a}\cup\{0\})/{\cal H}$ $(a\in S)$ the maximal rectangular subbands of $B$ are pairwise similar. In that case the permutation matching of $S$ may be chosen to be an involution matching.
\textbf{Proposition 1.3.4 }Each $0$-rectangular band $B=(D_{a}\cup\{0\})/{\cal H}$ $(a\in S)$ of a finite $E$-solid semigroup $S$ is orthodox.
For completeness, we record a proof of the Proposition but note that the class of all (not necessarily finite) $E$-solid semigroups is a so-called \emph{e}-\emph{variety}, meaning that the class is\emph{ }closed under the taking of homomorphic images, of direct products, and regular subsemigroups {[}4{]}. Also in {[}4{]} is shown that a semigroup is orthodox if and only if the same is true of each of its principal factors: (also see {[}5, Ex. 1.4.13(iv){]}).
\textbf{Proof} From the definition of $E$-solidity we see that each principal factor $D_{a}\cup\{0\}$ of $S$ is itself $E$-solid, and $B$ certainly is regular. Next we note that $B$ is $E$-solid through two observations: $H\in E(B)$ if and only if $H=H_{e}$ for some $e\in E(S)$, and $H_{a}{\cal G}H_{b}$ in $B$ if and only if $a{\cal G}b$ in $S$. Hence if $B$ contain three idempotents $H_{e},H_{f},$ and $H_{g}$ with $e,f,g\in E(S)$, and they are such that $H_{e}{\cal L}H_{f}{\cal R}H_{g}$ in $B$, then $e{\cal L}f{\cal R}g$ in $S$ and by the $E$-solid condition on $S$ we have $e{\cal R}h{\cal L}g$ for some $h\in E(S)$. We now have $H_{e}{\cal R}H_{h}{\cal L}H_{f}$ in $B$ and $H_{h}\in E(B)$. Therefore $B$ is $E$-solid.
To show that $B$ is indeed orthodox, first note that by our second observation, $B$ is combinatorial. Then for any two idempotents of $B$, which we now write as $e,f$, we have either that $ef=0\in E(B)$, or otherwise $e{\cal L}g{\cal R}f$ for some $g\in E(B)$ whence, since $B$ is $E$-solid, it follows that $h=ef\in E(B)$. Therefore $E^{2}(B)=E(B)$, as required. $\quad\vrule height4.17pt width4.17pt depth0pt$
\textbf{Theorem 1.3.5 }Let $S$ be a finite $E$-solid semigroup. Then $S$ has a permutation matching if and only if the maximal rectangular subbands of each of the $0$-rectangular bands $(D_{a}\cup\{0\})/{\cal H}$ are pairwise similar. Moreover if $S$ has a permutation matching then $S$ has an involution matching.
\textbf{Proof} By Theorem 1.2.2, $S$ has a permutation matching if and only if the same can be said for all $B=(D_{a}\cup\{0\})/{\cal H}$ $(a\in S)$. By Proposition 1.3.4, each such $B$ is a finite orthodox $0$-rectangular band. By Theorem 1.3.3, each such $B$ then has a permutation matching if and only if the maximal rectangular subbands of $B$ are pairwise similar, giving the first statement of Theorem 1.3.5. In this case, again by Theorem 1.3.3, each permutation matching of each $B$ may be chosen to be an involution matching of $B$. Then by (vi) implies (i) in Theorem 1.2.2 as it applies to involutions, we conclude that $S$ itself has an involution matching, thus completing the proof. $\quad\vrule height4.17pt width4.17pt depth0pt$
\section{Matchings for ${\cal OP}_{n}$ and ${\cal P}_{n}$}
\subsection{The semigroups ${\cal OP}_{n}$ and ${\cal P}_{n}$}
We recap some of the important properties of the semigroups ${\cal OP}_{n}$ and ${\cal P}_{n}$. We also augment these results in order to build a type of calculus for these semigroups. All semigroups under consideration will be subsemigroups of ${\cal T}_{n}$. Basic properties of the representation of $\alpha\in{\cal T}_{n}$ as a digraph $G(\alpha)$ can be found in the text {[}5, Section 1.5{]}. Each component $C$ of $G(\alpha)$ is \emph{functional}, meaning that each vertex has out-degree $1$ so in consequence $C$ consists of a unique cycle $Z(\alpha)$ with a number of directed trees rooted around the vertices of $Z(\alpha)$. The set of cycle points of $G(\alpha)$ are exactly the points in the \emph{stable range }of $\alpha$, denoted by stran$(\alpha)$, which are the points of $X_{n}$ contained in the range of all powers of $\alpha$. Pictures of these digraphs are helpful in seeing what is going on and the reader is invited to draw them where relevant, especially in the examples of Section 4 where they are a natural aid to understanding.
For $\alpha\in{\cal T}_{n}$ we write $R=R(\alpha)$ for its range
$X\alpha$, while $t=|R(\alpha)|$ will stand for the \emph{rank} of $\alpha$. The kernel relation of $\alpha$ on $X$ will be denoted as ker$(\alpha)$ with the corresponding partition of $X_{n}$ written as Ker$(\alpha)$. The set of fixed points of $\alpha$ will be denoted by $F(\alpha)$. Facts from the source paper {[}1{]} are listed using the term Result.
\textbf{Definitions 2.1.1 }(i) the \emph{cyclic interval }$[i,i+t]$ $(0\leq t\leq n-1)$ is the set $\{i,i+1,\cdots,i+t\}$ if $i+t\leq n-1$ and otherwise is the set \newline $\{i,i+1,\cdots,n-1,0,1,\cdots,(i+t)\,\mbox{(mod\,\ensuremath{n)\}}}$.
(ii) A finite sequence $A=(a_{0},a_{1},\cdots,a_{t})$ from $[n]$ is \emph{cyclic }if there exists no more than one subscript $i$ such that $a_{i}>a_{i+1}$ (taking $t+1=0$). We say that $A$ is \emph{anti-cyclic }if the reverse sequence $A^{r}=(a_{t},a_{t-1},\cdots,a_{0})$ is cyclic.
\textbf{Remarks 2.1.2 }To say that $A$ is cyclic as in (ii) is equivalent to saying that for some subscript $i$, $a_{i+1}\leq\cdots\leq a_{t}\leq a_{0}\leq\cdots\leq a_{i}$ and the subscript $i$ with this property is unique unless $A$ is constant. On the other hand $A$ is anti-cyclic means $A^{r}$ is cyclic so that $A$ is anti-cyclic if and only if for some subscript $i$ we have $a_{i+1}\geq\cdots\geq a_{t}\geq a_{0}\geq\cdots\geq a_{i}$ (and $i$ is unique if $A$ is not constant). The properties of cyclicity and anti-cyclicity are inherited by subsequences and by sequences obtained by cyclic re-ordering.
\textbf{Definition 2.1.3 }A mapping $\alpha\in{\cal T}_{n}$ is \emph{orientation-preserving }if its list of images, $(0\alpha,1\alpha,\cdots,(n-1)\alpha)$, is cyclic. The collection of all such mappings is denoted by ${\cal OP}_{n}$.\textbf{ }We say that $\alpha\in{\cal T}_{n}$ is \emph{orientation-reversing }if $(0\alpha,1\alpha,\cdots,(n-1)\alpha)$ is anti-cyclic and the collection of all orientation-reversing mappings is denoted by ${\cal OR}_{n}$.
\textbf{Result 2.1.4} ${\cal OP}_{n}$ is a regular submonoid of ${\cal T}_{n}$. Each kernel class of $\alpha\in{\cal OP}_{n}$ is a cyclic interval of $[n]$ and the maximal cycles of the components of the digraph $G(\alpha)$ have the same number of vertices, denoted by $c(\alpha)$.
\textbf{Definition 2.1.5} Let $\alpha\in{\cal OP}_{n}$ be of rank $t\geq2$. We index the members of Ker$(\alpha)$ as $K_{i}$ $(0\leq i\leq t-1)$ in such a way that the set of initial points $a_{i}$ of the cyclic intervals $K_{i}$ satisfy $a_{0}<a_{1}<\cdots<a_{t-1}$, denoting this ordered set by $K(\alpha)$. The list $\{K_{0},K_{1},\cdots,K_{t-1}\}$ is called the \emph{canonical listing} of the kernel classes of $\alpha$. For $r_{i}\in R(\alpha)$ where $r_{0}<r_{1}<\cdots<r_{t-1}$ we denote the cyclic interval $[r_{i},r_{i}+1,\cdots,r_{i+1}-1]$ by $R_{i}$.
\textbf{Result 2.1.6 }({[}1{]}, Theorem 3.3) For $t\geq2$ there is a one-to-one correspondence $\Phi_{0}$ between the set of triples $(K,R,i)$ where $K$ and $R$ are ordered $t$-sets of $X_{n}$ $(0\leq i\leq t-1)$
and $\{\alpha\in{\cal OP}_{n}:|X\alpha|=t\}$ whereby $(K,R,i)\mapsto\alpha$, where each $a_{j}\in K$ is an initial point of a kernel class of
$\alpha$ and $a_{j}\alpha=r_{i+j}$$\,(0\leq i\leq t-1)$, subscripts calculated modulo $t$. Moreover $H_{\alpha}=\{\Phi_{0}(K,R,i):\,i=0,1,\cdots,t-1\}$, and so $|H_{\alpha}|=t$.
\textbf{Result 2.1.7 }(i) The collection ${\cal P}_{n}={\cal OP}_{n}\cup{\cal OR}_{n}$ is a regular submonoid of ${\cal T}_{n}$; ${\cal R}$- ${\cal L}$- and ${\cal D}$-classes are determined by equality or kernels, of images, and of ranks respectively (as in ${\cal T}_{n}$ and ${\cal OP}_{n}$).
(ii) the \emph{reflection mapping} $\gamma:[n]\rightarrow[n]$, whereby $i\mapsto n-i-1$ $(i\in[n])$ is orientation-reversing and ${\cal P}_{n}=\langle a,e,\gamma\rangle$, where $a$ is the $n$-cycle $(0\,1\,\cdots\,n-1)$ and $e$ is any idempotent in ${\cal OP}_{n}$ of rank $n-1$; $\langle a,e\rangle={\cal OP}_{n}$.
(iii) ${\cal OP}_{n}\cap{\cal OR}_{n}=\{\alpha\in{\cal OP}_{n}:\,\mbox{rank\ensuremath{(\alpha)\leq2\}}. }$
(iv) $({\cal OR}_{n})^{2}={\cal OP}_{n}$, ${\cal OP}_{n}\cdot{\cal OR}_{n}={\cal OR}_{n}\cdot{\cal OP}_{n}={\cal OR}_{n}$.
It is also proved in {[}1{]} and in {[}9{]} that the respective maximal subgroups of rank $t$ of ${\cal OP}_{n}$ and of ${\cal P}_{n}$ are cyclic groups of order $t$ and dihedral groups of order $2t$. Also every non-constant member $\alpha\in{\cal OP}_{n}$ factorizes uniquely as $\alpha=a^{r}\phi$ where $a$ is the $n$-cycle as above and $\phi\in{\cal O}_{n}$. The constant mappings on $[n]$ comprise $D_{1}$, the lowest ${\cal D}$-class of ${\cal P}_{n}$. Any permutation of $D_{1}$ is a permutation matching of $D_{1}$ and for that reason $D_{1}$ will not need to feature in our subsequent discussion.
\textbf{Definition 2.1.8 }For $\alpha\in{\cal P}_{n}$ we shall write $\rho(\alpha)=(K,R)$, where $K$ and $R$ are the respective sets $K(\alpha)$ of initial points of kernel classes and $R(\alpha)$.
Note that for any $\alpha,\beta\in S={\cal OP}_{n}$ or ${\cal P}_{n}$, $\alpha{\cal H}\beta$ if and only if $\rho(\alpha)=\rho(\beta)$. We now extend Result 2.1.6 to ${\cal P}_{n}$.
\textbf{Theorem 2.1.9 }For $t\geq2$ there is a one-to-one correspondence $\Phi$ between the set of quadruples $(K,R,i,k)$ where $K$ and $R$ are ordered $t$-sets of $[n]$, $0\leq i\leq t-1$, $k=\pm1$
and $\{\alpha\in{\cal P}_{n}:|X\alpha|=t\}$. The correspondence is given by $(K,R,i,k)\mapsto\alpha$, where each $a_{j}\in K$ is an initial point of a kernel class of $\alpha$ and $a_{j}\alpha=r_{i+kj}$$\,(0\leq j\leq t-1)$, subscripts calculated modulo $t$.
\textbf{Proof} The first statement of Result 2.1.6 implies that $\Phi|_{k=1}$
maps bijectively onto the set of non-constant mappings in ${\cal OP}_{n}$. We show that $\Phi|_{k=-1}$ maps bijectively onto the set of non-constant mappings in ${\cal OR}_{n}$. The equation $K_{j}\alpha=r_{i-j}$ certainly specifies a unique mapping $\alpha=\Phi(K,R,i,-1)\in{\cal T}_{n}$, and distinct quadruples yield distinct mappings. We need to check that $\alpha\in{\cal OR}_{n}$. We have however the following equality of two lists: \begin{equation} a_{i+1}\alpha=r_{t-1}>a_{i+2}\alpha=r_{t-2}>\cdots>a_{i}\alpha=r_{0} \end{equation}
It follows from (1) and Remarks 2.1.2 that the image of the cyclic list $K$ under $\alpha$ is anti-cyclic and so $\alpha=\Phi(K,R,i,-1)\in{\cal OR}_{n}$; hence $\Phi|_{k=-1}$ is a one-to-one mapping into the set of mappings of ${\cal OR}_{n}$ of rank at least $2$.
Conversely, let $\alpha\in{\cal OR}_{n}$ be of rank $t\geq2$. Since multiplication on the right by $\gamma$ defines a bijection of ${\cal OP}_{n}$ onto ${\cal OR}_{n}$, it follows that the kernel classes of $\alpha$ are cyclic intervals and so $H_{\alpha}$ is determined by a pair of ordered $t$-sets $(K,R)$. Take $i$ such that $a_{i}\alpha=r_{0}$. Then since $a_{i+1},a_{i+2},\cdots,a_{i}$ is cyclic and $\alpha\in{\cal OR}_{n}$, it follows that $a_{i+1}\alpha,a_{i+2}\alpha,\cdots,a_{i}\alpha$
is anti-cyclic. However, since $a_{i}\alpha=r_{0}=$ min$R$, it follows by Remarks 2.1.2 that (1) holds for $\alpha$ and so $\alpha=\Phi(K,R,i,-1)$. Therefore $\Phi|_{|k=-1}$ is a bijection onto the set of non-constant mappings of ${\cal OR}_{n}$. Finally note that for $k=\pm1$, the rank of $\alpha=\Phi(K,R,i,k)$ is indeed $t=|R|=|K|$. $\quad\vrule height4.17pt width4.17pt depth0pt$
\textbf{Corollary 2.1.10 }For $t\geq3$, each ${\cal H}$-class $H$ of ${\cal P}_{n}$ contained in $D(t)$ is a disjoint union $H=(H\cap{\cal OP}_{n})\cup(H\cap{\cal OR}_{n})$ with each set in the union of cardinal $t$.
\textbf{Proof} Let $H=\{\alpha\in{\cal P}_{n}:\rho(\alpha)=(K,R)\}$. Then by Theorem 2.1.9, $H\cap{\cal OP}_{n}=\{\Phi(K,R,i,1):0\leq i\leq t-1\}$ and $H\cap{\cal OR}_{n}=\{\Phi(K,R,i,-1):0\leq i\leq t-1\}$; these two sets each have $t$ members and are disjoint by Result 2.1.7(iii). $\quad\vrule height4.17pt width4.17pt depth0pt$
We shall refer to the coding of each $\alpha\in{\cal P}_{n}$ in the form $(K,R,i,k)$ as the \emph{KRik-coordinates }of $\alpha$, noting that $(K,R,i,k)=\Phi^{-1}(\alpha)$. We call $i$ and $k$ respectively the \emph{shift }and the \emph{parity }of $\alpha$.
\textbf{Lemma 2.1.11 }Let $\alpha\in{\cal P}_{n}$ with $\rho(\alpha)=(K,R)$. Then
(i) $\rho(\alpha\gamma)=(K,n-1-R)$; ~~~(ii) $\rho(\gamma\alpha)=(n-K,R)$;
(iii) $\rho(\gamma\alpha\gamma)=(n-K,n-1-R)$.
\textbf{Proof }(i) is immediate from definition as is the fact that $R(\gamma\alpha)=R(\alpha)$ in (ii). Continuing in (ii), suppose that $\alpha=\Phi(K,R,i,k)$. Then for $r_{i+kj}\in R$ $(0\leq j\leq t-1)$, we obtain: \[ r_{i+kj}(\gamma\alpha)^{-1}=r_{i+kj}\alpha^{-1}\gamma^{-1}=K_{j}\gamma \] \[ =(n-1)-\{a_{j},a_{j}+1,\cdots,a_{j+1}-1\}=\{n-a_{j+1},n-a_{j+1}+1,\cdots,n-a_{j}-1\}, \] where $j+1$ is calculated modulo $t$. It follows that $K(\gamma\alpha)=n-K(\alpha)$, thereby establishing (ii). Applying (i) and then (ii) now gives (iii) as follows: \[ \rho(\gamma\alpha\cdot\gamma)=(K(\gamma\alpha),n-1-R(\gamma\alpha))=(n-K,n-1-R)\,\,\quad\vrule height4.17pt width4.17pt depth0pt \]
\textbf{Proposition 2.1.12} Let $\alpha=\Phi(K,R,i,k)\in{\cal P}_{n}$. Then
(i) $\alpha\gamma=\Phi(K,n-1-R,-(i+1),-k)$;
(ii) if $0\not\not\in K$ then $\gamma\alpha=\Phi(n-K,R,i-2k,-k)$;
(iii) if $0\in K$ then $\gamma\alpha=\Phi(n-K,R,i-k,-k)$;
(iv) if $0\not\in K$ then $\gamma\alpha\gamma=\Phi(n-K,n-1-R,2k-(i+1),k);$
(v) if $0\in K$ then $\gamma\alpha\gamma=\Phi(n-K,n-1-R,k-(i+1),k)$.
\textbf{Proof} (i) We are working throughout modulo $t$ on subscripts. By Lemma 2.1.11(i) we have $\rho(\alpha\gamma)=(K,n-1-R)$. Now \[ n-1-R=\{n-1-r_{t-1}<n-1-r_{t-2}<\cdots<n-1-r_{0}\}. \] Let us denote $n-1-r_{-(j+1)}$ by $s_{j}$ $(0\leq j\leq t-1)$ so that $R(\alpha\gamma)=\{s_{0}<s_{1}<\cdots<s_{t-1}\}$. Hence $a_{j}\alpha\gamma=r_{i+kj}\gamma=n-1-r_{i+kj}$; now \[ i+kj=-(-i-kj),\,\text{so that\,\ensuremath{a_{j}\alpha\gamma}}=s_{-(i+1)-kj}, \] which establishes equation (i).
(ii) By Lemma 2.1.11(ii) we have $\rho(\gamma\alpha)=(n-K,R)$. Since $1\leq a_{0}$ \[ n-K=n-a_{t-1}<n-a_{t-2}<\cdots<n-a_{0}. \] Let us denote $n-a_{-(j+1)}$ by $b_{j}$ $(0\leq j\leq t-1)$ so that $K(\gamma\alpha)=\{b_{0}<b_{1}<\cdots<b_{t-1}\}$. Hence: \[ b_{j}\gamma\alpha=(n-1-(n-a_{-(j+1)}))\alpha=(a_{-(j+1)}-1)\alpha=a_{-(j+2)}\alpha=r_{i-k(j+2)}=r_{(i-2k)-kj}, \] which establishes equation (ii).
(iii) Now since $a_{0}=0$ we have $n-a_{0}=n\equiv0$ (mod $n)$ and so: \[ n-K=n-a_{0}<n-a_{t-1}<n-a_{t-2}<\cdots<n-a_{1}. \] Let us denote $n-a_{-j}$ by $b_{j}$ $(0\leq j\leq t-1)$ so that $K(\gamma\alpha)=\{b_{0}<b_{1}<\cdots<b_{t-1}\}$. Hence \[ b_{j}\gamma\alpha=(n-1-(n-a_{t-j}))\alpha=(a_{-j}-1)\alpha=a_{-(j+1)}\alpha=r_{i-k(j+1)}=r_{(i-k)-kj}, \] which establishes equation (iii).
(iv) By Lemma 2.1.11(iii) we have $\rho(\gamma\alpha\gamma)=(n-K,n-1-R)$. Now using (ii) we obtain \[ b_{j}\gamma\alpha\gamma=r_{(i-2k)-kj}\gamma=n-1-r_{-(2k-i+kj)}=s_{(2k-(i+1))+kj}, \] which establishes equation (iv).
(v) By Lemma 2.1.11(iii) we have $\rho(\gamma\alpha\gamma)=(n-K,n-1-R)$. Now using (iii) we obtain \[ b_{j}\gamma\alpha\gamma=r_{i-k-kj}\gamma=n-1-r_{-(k-i+kj)}=s_{(k-(i+1))+kj}, \] which establishes equation (v). $\quad\vrule height4.17pt width4.17pt depth0pt$
\textbf{Example 2.1.13 }As an example we find $\gamma\alpha\gamma$ for $\alpha\in{\cal OR}_{10}$ given by: \[ \alpha=\begin{pmatrix}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ 3 & 2 & 2 & 8 & 8 & 6 & 6 & 4 & 3 & 3 \end{pmatrix}; \] so that $n=10$, $t=5$, $K=\{1,3,5,7,8\}$, $R=\{2,3,4,6,8\}$ and $\alpha=\Phi(K,R,0,-1)$. Since $0\not\in K$, according to Proposition 2.12(iv), we should find that $\gamma\alpha\gamma=\Phi(10-K,9-R,2,-1)$ as $i(\gamma\alpha\gamma)=2(-1)-(0+1)=-3\equiv2$ (mod $5$). Now $n-K=\{2,3,5,7,9\}$, and $n-1-R=\{1,3,5,6,7\}$. This accords with the direct calculation of $\gamma\alpha\gamma$, which corresponds to reversing the image line of $\alpha$ (to get $\gamma\alpha)$ and subtracting the images from $n-1=9$. \[ \gamma\alpha\gamma=\begin{pmatrix}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ 6 & 6 & 5 & 3 & 3 & 1 & 1 & 7 & 7 & 6 \end{pmatrix}. \]
\section{Permutation matchings for ${\cal OP}_{n}$ and ${\cal P}_{n}$ }
\subsection{An approach via subset involutions}
\textbf{Lemma 3.1.1 }Let $A\subseteq S$ and let $(\cdot')$ denote an ${\cal H}$-class-preserving involution matching on the set $A$. Then $(\cdot')$ may be uniquely extended to an involution matching on $A_{{\cal H}}=\cup_{a\in A}H_{a}$. In particular, if $A$ meets every ${\cal H}$-class of $S$ then $(\cdot')$ extends uniquely to an ${\cal H}$-class-preserving involution of $S$, which we shall call the \emph{induced involution matching }on $S$.
\textbf{Proof }Since $(\cdot')$ is ${\cal H}$-class-preserving it induces an involution on $A_{{\cal H}}$ by $H_{a}\mapsto H_{a'}$. We then have an involution matching on $A_{{\cal H}}$ defined by $b\mapsto b'$ where $b\in H_{a}$ say and $b'$ is the unique inverse of $b$ in $H_{a'}$. $\quad\vrule height4.17pt width4.17pt depth0pt$
We now construct what we shall refer to as the \emph{natural matching involution }for ${\cal P}_{n}$, which is the induced involution matching on ${\cal P}_{n}$ extending the involution of ${\cal OP}_{n}$ recorded in {[}6{]}.
\textbf{Theorem 3.1.2} The semigroup ${\cal P}_{n}$ has an ${\cal H}$-class-preserving involution matching $(\cdot')$ defined by the rule: \begin{equation} (\alpha=\Phi(K,R,i,k))\Rightarrow(\alpha'=\Phi(R,K,-ki,k)) \end{equation} \textbf{Proof }From Theorem 2.1.9 and its proof it follows that (2) defines an ${\cal H}$-class-preserving involution that maps each of ${\cal OP}_{n}$ and ${\cal OR}_{n}$ onto ${\cal OP}_{n}$ and ${\cal OR}_{n}$ respectively: certainly $\alpha''=\alpha$ as $(-k)(-ki)=k^{2}i=i.$ It remains only to check that $(\alpha,\alpha')\in V({\cal P}_{n})$ and by symmetry it is enough to verify that $\alpha=\alpha\alpha'\alpha$. To this end take $x\in[n]$ with $x\in K_{j}$ say where the kernel classes of $\alpha$ are labelled by subscripts in the canonical order. Then since $-ki+k(i+kj)=-ki+ki+k^{2}j=j$ we obtain: \[ x\alpha\alpha'\alpha=a_{j}\alpha\alpha'\alpha=r_{i+kj}\alpha'\alpha=a_{-ki+k(i+kj)}\alpha=a_{j}\alpha=x\alpha, \] and so $\alpha=\alpha\alpha'\alpha$, as required. $\quad\vrule height4.17pt width4.17pt depth0pt$
Note that for $t=2$ we have ${\cal OP}_{n}\cap D_{2}={\cal OR}_{n}\cap D_{2}=D_{2}$ and in this case $k=-k$ (mod $t)$. This collapse in the fourth entry of the \emph{KRik-}coordinates leads to the involution matching taking on the simpler form $\alpha=\Phi(K,R,i)\mapsto\alpha'=\Phi(R,K,i)$.
\textbf{Examples 3.1.3 }Let $n=8$, $t=4$, $K=\{0,2,4,6\}$, $R=\{1,3,5,7\}$ and $\alpha\in{\cal OR}_{8}$ defined by $\alpha=\Phi(K,R,3,-1)$ so that $\alpha'=\Phi(R,K,3,-1)$. \[ \alpha=\begin{pmatrix}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 7 & 7 & 5 & 5 & 3 & 3 & 1 & 1 \end{pmatrix}\,\,\alpha'=\begin{pmatrix}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 0 & 6 & 6 & 4 & 4 & 2 & 2 & 0 \end{pmatrix}. \] For an example in ${\cal OP}_{n}$ let us take $n=10,$ $t=6$, $K=\{0,2,4,7,8,9\},$ $R=\{0,1,2,5,6,7\}$ and $\text{\ensuremath{\alpha}}=\Phi(K,R,4,1)$. Since $-i=-4=2$ (mod $6)$ we obtain $\alpha'=\Phi(R,K,2,1)$: \[ \alpha=\begin{pmatrix}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ 6 & 6 & 7 & 7 & 0 & 0 & 0 & 1 & 2 & 5 \end{pmatrix}\,\,\alpha'=\begin{pmatrix}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ 4 & 7 & 8 & 8 & 8 & 9 & 0 & 2 & 2 & 2 \end{pmatrix}. \]
\textbf{Example 3.1.4 }The natural involution inverse of a group element is not necessarily a group element and nor does the natural involution map ${\cal O}_{n}$ into itself. Both features are seen in the following example. Take $n=3$, $t=2$, $K=\{0,2\}$, $R=\{1,2\}$, and put $\alpha=\Phi(K,R,0,1)$ so that $\alpha'=\Phi(R,K,0,1)$: \[ \alpha=\begin{pmatrix}0 & 1 & 2\\ 1 & 1 & 2 \end{pmatrix}\,\,\alpha'=\begin{pmatrix}0 & 1 & 2\\ 2 & 0 & 2 \end{pmatrix}; \] we see that $\alpha\in E({\cal O}_{3})\subseteq E({\cal OP}_{3})$ and so is an order-preserving group element while $\alpha'^{2}$ is the constant mapping with range $\{2\}$ and so $\alpha'$ is not contained in any subgroup of ${\cal OP}_{3}$ and nor is $\alpha'$ order-preserving. In the next example $\alpha\in E({\cal O}_{2n})$, but $\alpha'$ has no fixed points.
\textbf{Example 3.1.5 }Take $\alpha\in E({\cal O}_{2n})$ so that $R(\alpha)=F(\alpha)$, putting $R(\alpha)$ as the set of odd members of $[2n]$ and for each even integer $i\in[2n]$ we put $i\alpha=i+1$. This yields an order-preserving idempotent $\alpha$ on $[2n]$ of rank $n$ for which \[ (0\alpha,1\alpha,\cdots,(2n-2)\alpha,(2n-1)\alpha)=(1,1,3,3,\cdots,2n-1,2n-1). \] Now $\alpha=\Phi(K,R,0,1)$ where $K=\{0,2,4,\cdots,2n-2\}$ and \newline $R=\{1,3,5,\cdots,2n-1\}$. Hence $\alpha'=\Phi(R,K,0,1)$. The kernel classes of $\alpha'$ have the form $(i,i+1),\,(i=1,3,\cdots,2n-1)$. We see that $\alpha'$ contains the $n$-cycle: \[ \sigma=(2n-2\,2n-4\,2n-6\,\cdots\,2\,0), \] and for all odd $i$ we have $i\alpha'=i-1$. (The digraph $G(\alpha)$ has exactly one component consisting of the $n$-cycle $\sigma$ along with $n$ endpoints, one for each point on $\sigma$.) In particular $c(\alpha)=1$ but $c(\alpha')=n$.
\subsection{The dual matching involution of ${\cal P}_{n}$}
The mapping on ${\cal P}_{n}$ defined by $\alpha\mapsto\alpha\gamma$ (resp. $\alpha\mapsto\gamma\alpha)$ is an involution on ${\cal P}_{n}$ that maps ${\cal OP}_{n}$ onto ${\cal OR}_{n}$ and maps ${\cal OR}_{n}$ onto ${\cal OP}_{n}$. Additionally, for any $\alpha\gamma\in{\cal OR}_{n}$ (resp. $\gamma\alpha\in{\cal OR}_{n}$) and $\alpha'\in V(\alpha)$ in ${\cal OP}_{n}$, $\gamma\alpha'\in V(\alpha\gamma)$ (resp. $\alpha'\gamma\in V(\gamma\alpha)$) which lies in ${\cal OR}_{n}$. The upshot of this is that any permutation matching $(\cdot')$ on ${\cal OP}_{n}$ may be extended to one on ${\cal P}_{n}$ by defining $(\alpha\gamma)'=\gamma\alpha'$ (or dually, $(\gamma\alpha)'=\alpha'\gamma)$. However, if $(\cdot')$ is an involution matching, the same is not generally true of either of these extensions, even in the case of the natural inverse matching. Lemma 2.1.11 supplies enough information to make this point.
Let $\beta=\Phi(K,R,i,-1)\in{\cal OR}_{n}$ so that $\rho(\beta)=(K,R)$. Writing $\beta=\alpha\gamma$ so that $\alpha=\beta\gamma$ we get $\rho(\alpha)=(K,n-1-R)$ and so $\rho(\alpha')=(n-1-R,K)$. Then for $\overline{\beta}=\gamma\alpha'$ we have $\rho(\overline{\beta})=(R+1,K)$. Factorizing $\overline{\beta}$ as $(\gamma\alpha'\gamma)\gamma$ we obtain $\rho(\gamma\alpha'\gamma)=(R+1,n-K-1)$ so that $\rho((\gamma\alpha'\gamma)')=(n-K-1,R+1)$. Finally, $\overline{\overline{\beta}}=\gamma(\gamma\alpha'\gamma)'$ for which we have $\rho(\overline{\overline{\beta}})=(K+1,R+1)$. In particular we see that in general $\overline{\overline{\beta}}\neq\beta$, as $K+1=K$ if and only if $\beta$ is a member of the group of units of ${\cal P}_{n}$. However, by replacing the standard linear ordering by the reverse, or as we shall call it the \emph{dual ordering} of $[n]$, we automatically obtain a dual involution matching on ${\cal P}_{n}$, which we shall denote by $(\overline{\cdot})$. This generates a distinct matching involution of ${\cal P}_{n}$ to that of Theorem 3.1.2 and we now seek to express $(\overline{\cdot})$ in \emph{KRik-}coordinates.
Let $\alpha=\Phi(K,R,i,k)$. Under $(\overline{\cdot})$, each $r\in R$ is mapped to the initial point of $r\alpha^{-1}$ in the dual ordering, which is the terminal point $r\alpha^{-1}$ in the standard ordering. It follows that $X\overline{\alpha}=K-1$. Similarly, under $(\overline{\cdot})$, $R$ becomes the set of initial points of kernel classes in the dual ordering, which is then the set of terminal points of those same classes when expressed in the standard ordering, and so $K(\overline{\alpha})=R+1$. Therefore $\rho(\overline{\alpha})=(R+1,K-1).$ Since the choice of ordering does not affect whether or not $\alpha\in{\cal P}_{n}$ preserves or reverses orientation, we may write $\overline{\alpha}=\Phi(R+1,K-1,i(\overline{\alpha}),k),$ where it only remains to determine $i(\overline{\alpha})$.
\textbf{Theorem 3.2.2 }The dual matching involution $\overline{(\cdot)}:{\cal P}_{n}\rightarrow{\cal P}_{n}$ acts by $\alpha=\Phi(K,R,i,k)\mapsto\Phi(R+1,K-1),i(\overline{\alpha}),k)$ where:
Case (1): $0\not\in K$ and $n-1\not\in R$: $i(\overline{\alpha})=1+k(1-i)$;
Case (2): $0\not\in K$ and $n-1\in R$: $i(\overline{\alpha})=1-ki$;
Case (3): $0\in K$ and $n-1\not\in R$: $i(\overline{\alpha})=k(1-i)$;
Case (4): $0\in K$ and $n-1\in R$: $i(\overline{\alpha})=-ki.$
Moreover, $\alpha$ is in Case (1/4) if and only if $\overline{\alpha}$ is in Case (1/4) and $\alpha$ is in Case (2/3) if and only if $\overline{\alpha}$ is in Case (3/2).
\textbf{Proof }Since $\rho(\overline{\alpha})=(R+1,K-1)$ it follows that $\rho(\alpha\overline{\alpha})=(K,K-1)$. Observe that the unique member of $K-1$ in the kernel class $K_{j}$ is $a_{j+1}-1$ and since $\alpha\overline{\alpha}\in E({\cal P}_{n})$ we have: \[ K_{j}\alpha\overline{\alpha}=a_{j+1}-1 \] \[ \Rightarrow r_{i+kj}\overline{\alpha}=a_{j+1}-1\Rightarrow r_{i+j}\overline{\alpha}=a_{kj+1}-1 \] \[ \Rightarrow r_{j}\overline{\alpha}=a_{k(j-i)+1}-1=a_{(1-ki)+kj}-1 \] \begin{equation} \therefore(r_{j}+1)\overline{\alpha}=r_{j+1}\overline{\alpha}=a_{(1+k(1-i))+kj}-1 \end{equation} The value of $\overline{i}$ now falls into four cases.
Case (1): $0\not\in K$ and $n-1\not\in R$. Since $1\leq a_{0}$ and $r_{t-1}\le n-2$ we have, in ascending order: \begin{equation} K-1=\{a_{0}-1,a_{1}-1,\cdots,a_{t-1}-1\},\,R+1=\{r_{0}+1,r_{1}+1,\cdots,r_{t-1}+1\} \end{equation} Then putting $j=0$ in (3) yields $(r_{0}+1)\overline{\alpha}=a_{1+k(1-i)}-1$ and so $\overline{\alpha}=\Phi(R+1,K-1,1+k(1-i),k)$.
Case (2): $0\not\in K$ but $n-1\in R$ so that the ordered set $R+1=\{r_{t-1}+1=0,r_{0}+1,\cdots,r_{t-2}+1\}$. Then since $r_{t-1}+1$ is the initial entry of $R+1$ we substitute $j=t-1\equiv-1$ (mod $t$) into (3) to recover that $\overline{i}$ is $1-ki$ and so $\overline{\alpha}=\Phi(R+1,K-1,1-ki,k)$.
Case (3): $0\in K$ but $n-1\not\in R$ so that the ordered set $K-1=\{a_{1}-1,a_{2}-1,\cdots,a_{t-1}-1,a_{0}-1=n-1\}$. As in Case (1) we put $j=0$ in (3) to get $a_{1+k(1-i)}-1$ but since each entry is now listed one place earlier in the ordered set $K-1$ compared to Case (1), we subtract $1$ from the outcome in Case (1) to obtain $\overline{\alpha}=\Phi(R+1,K-1,k(1-i),k)$.
Case (4): $0\in K$ and $n-1\in R$ so the ordered set $R+1$ is as in Case (2) and $K-1$ is as in Case (3). Hence $r_{-1}+1$ is the first entry of $R+1$, which, by (3), is mapped to $a_{1-ki}$, which is the entry at position $-ki$ in the list of $K-1.$ Hence we obtain $\overline{\alpha}=\Phi(R+1,K-1,-ki,k)$.
Also note that $0\in K\Leftrightarrow n-1\in K-1$ and $n-1\in R\Leftrightarrow0\in R+1$. It follows that $\alpha$ is in Case (1/4) if and only if $\overline{\alpha}$ is in Case (1/4) and that $\alpha$ is in Case (2/3) if and only if $\overline{\alpha}$ is in Case (3/2). $\quad\vrule height4.17pt width4.17pt depth0pt$
\textbf{Remarks 3.2.3 }If we take the union of the natural involution matching $(\cdot)'$ on ${\cal OP}_{n}$ with the dual involution matching $\overline{(\cdot)}$ on its complement in ${\cal P}_{n}$, we have another involution matching on ${\cal P}_{n}$. Since the natural involution matching on ${\cal P}_{n}$ is the unique involution matching that extends $(\cdot)'$ to ${\cal P}_{n}$ while preserving ${\cal H}$-classes, it follows that this alternative matching is an example of an involution matching of ${\cal P}_{n}$ that does not preserve ${\cal H}.$
We may check directly that $\overline{(\cdot)}$ defines an involution: the only non-obvious feature is that the formulas for the shift co-ordinates are self-inverse in Cases (1) and (4) and mutally inverse for Cases (2) and (3) but these are readily checked: for example in Case (1): $1+k(1-(1+k(1-i)))=i.$
An approach by `half duals' leads to permutation matchings that are however not involutions. For instance we may look to inverses that map to terminal points of kernel classes while keeping $R$ as the set of initial points of kernel classes of the inverse. In detail, for $\alpha\in{\cal P}_{n}$ such that $\rho(\alpha)=(K,R)$ the ${\cal H}$-classes defined by the kernel-range pairs $(K,K-1)$ and $(R,R)$ are groups and so there exists a unique inverse $\dot{\alpha}$ of $\alpha$ such that $\rho(\dot{\alpha})=(R,K-1)$. The mapping $\alpha\mapsto\dot{\alpha}$ is then a permutation matching of ${\cal P}_{n}$ but not an involution as $\rho(\ddot{\alpha})=(K-1,R-1)\neq\rho(\alpha)$ (unless $\alpha$ lies in the group of units of ${\cal P}_{n}$). A dual comment applies to the other half dual where the inverse of $\alpha$ lies in the ${\cal H}$-class defined by the pair $(R+1,K)$.
\textbf{Examples 3.2.4 }We take $n=8$, $t=4$, $K=\{0,2,4,6\}$, $R=\{1,3,5,7\}$ and put $\alpha=\Phi(K,R,0,1)$. We have $R+1=\{0,2,4,6\}=K,\,\,K-1=\{1,3,5,7\}=R.$ Here we have $0\in K$ and $n-1=7\in R$ so that we are in Case (4). By Theorem 3.2.1 we obtain $\overline{\alpha}=\Phi(R+1,K-1,0,1)=\alpha$, and indeed $\alpha$ is an idempotent. In contrast, the natural inverse $\alpha'=\Phi(R,K,0,1)$: \[ \alpha=\alpha^{2}=\begin{pmatrix}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 1 & 1 & 3 & 3 & 5 & 5 & 7 & 7 \end{pmatrix}=\overline{\alpha},\,\,\alpha'=\begin{pmatrix}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 6 & 0 & 0 & 2 & 2 & 4 & 4 & 6 \end{pmatrix}. \] Next we revisit the first of Examples 3.1.3: $\alpha=\Phi(K,R,3,-1){\cal \in OR}_{8}$, where $K=\{0,2,4,6\}$\textbf{ }and $R=\{1,3,5,7\}$. Since $0\in K$ and $n-1=7\in R$ we are in Case (4) and so $\overline{\alpha}=\Phi(R+1,K-1,3,-1)$ and so $R+1=\{0,2,4,6\}=K$ and $K-1=\{1,3,5,7\}=R$. Hence: \[ \alpha=\begin{pmatrix}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 7 & 7 & 5 & 5 & 3 & 3 & 1 & 1 \end{pmatrix}=\overline{\alpha}. \] Since $\overline{\alpha}=\alpha$, therefore $\overline{\overline{\alpha}}=\alpha$ also, and $\alpha^{3}=\alpha.$ In particular, $\overline{\alpha}\neq\alpha'$ . In general, ${\cal OR}_{n}$ contains no idempotents of rank greater than $2$, so that $\alpha=\alpha^{2}$ is impossible for $\alpha\in{\cal OR}_{n}\setminus{\cal OP}_{n}$.
As a third example let $n=10$, $t=5$, $K=\{1,3,5,7,8\},$ $R=\{2,3,4,6,8\}$ with $\alpha=\Phi(K,R,4,-1)$. Here $0\not\in K$ and $n-1=9\not\in R$ and so we are in Case (1). Note that since $k=-1$ we have for all $i$ that $\overline{i}=1-(1-i)=i$, so in particular $i(\overline{\alpha})=4$ and so $\overline{\alpha}=\Phi(R+1,K-1,4,-1)$. We see that $R+1=\{3,4,5,7,9\}$ and $K-1=\{0,2,4,6,7\}$: \[ \alpha=\begin{pmatrix}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ 2 & 8 & 8 & 6 & 6 & 4 & 4 & 3 & 2 & 2 \end{pmatrix}\,\,\overline{\alpha}=\begin{pmatrix}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ 0 & 0 & 0 & 7 & 6 & 4 & 4 & 2 & 2 & 0 \end{pmatrix}. \] Beginning with $\overline{\alpha}=\Phi(R+1,K-1,4,-1)$ we have $0\not\in R+1$ and $9\not\in K-1$ so we are (necessarily) again in Case (1). As before $i(\overline{\overline{\alpha}})=4$ and we obtain as expected: \[ \overline{\overline{\alpha}}=\Phi((K-1)+1,(R+1)-1,4,-1)=\Phi(K,R,4,-1)=\alpha. \] In contrast the natural inverse of $\alpha$ is $\alpha'=\Phi(R,K,1,-1).$
As an example illustrating Cases (2/3) let $n=10,$ $t=6$, $K=\{1,2,5,7,8,9\}$, $R=\{0,4,5,6,7,9\}$ with $\alpha=\Phi(K,R,4,1)$. Here $0\not\in K$ but $n-1=9\in R$, putting $\alpha$ in Case (2). Theorem 3.2.1 gives $i(\overline{\alpha})=1-ki(\alpha)=1-4=-3\equiv3$ (mod $6$). Hence $\overline{\alpha}=\Phi(R+1,K-1,3,1)$, where $0\in R+1=\{0,1,5,6,7,8\}$ and $9\not\in K-1=\{0,1,4,6,7,8\}$, placing $\overline{\alpha}$ in Case (3). Therefore \[ \alpha=\begin{pmatrix}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ 6 & 7 & 9 & 9 & 9 & 0 & 0 & 4 & 5 & 6 \end{pmatrix}\,\,\overline{\alpha}=\begin{pmatrix}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ 6 & 7 & 7 & 7 & 7 & 8 & 0 & 1 & 4 & 4 \end{pmatrix}, \] and $\overline{\overline{\alpha}}=\alpha$: $i(\overline{\overline{\alpha}})=k(1-i(\overline{\alpha}))=1(1-3)=-2\equiv4$ (mod $6)=i(\alpha)$.
\section{Inverse covers and involution matchings for ${\cal T}_{n}$}
\subsection{Inverse covers for ${\cal T}_{n}$ }
In 1971 it was shown by Schein {[}10{]} that every finite full transformation semigroup ${\cal T}_{n}$ is covered by its inverse subsemigroups, a result that does not extend to the case of an infinite base set, {[}5, Ex. 6.2.8{]}. If there existed a cover ${\cal A}=\{A_{i}\}_{1\leq i\leq m}$ of inverse subsemigroups of ${\cal T}_{n}$ with the additional property that the intersection of any pair of semigroups of ${\cal A}$ was also an inverse subsemigroup of ${\cal T}_{n}$ then we could deduce (as explained below) that ${\cal T}_{n}$ had an involution matching. (The semigroups ${\cal OP}_{n}$ and ${\cal P}_{n}$ of the previous sections have an inverse cover only if $n\leq3$, {[}2{]}).
It is convenient in what follows to consider the empty set also to be an inverse semigroup. Suppose that ${\cal A}=\{A_{i}\}_{1\leq i\leq m}$ is an \emph{inverse cover} of ${\cal T}_{n}$ meaning that each $A_{i}$ is an inverse subsemigroup of ${\cal T}_{n}$ and that $\cup_{i=1}^{m}A_{i}={\cal T}_{n}$. It follows that, for all $1\leq i,j\leq m$, the subsemigroup $A_{i,j}=:A_{i}\cap A_{j}$ of ${\cal T}_{n}$ has commuting idempotents. Indeed it follows easily from this that Reg$(A_{i,j})$, the set of regular elements of $A_{i,j}$, forms an inverse subsemigroup of $A_{i,j}$. However it does not automatically follow that $A_{i,j}=$ Reg$(A_{i,j})$.
Let $S$ be an arbitrary semigroup and $a\in S$. We say that $b\in V(a)$ is a \emph{strong inverse }of $a$ if the subsemigroup $\langle a,b\rangle$ of $S$ is an inverse semigroup. We denote the set of strong inverses of $a$ by $S(a)$. We next observe that $S$ has an inverse cover if and only if every element of $S$ has a strong inverse for, on the one hand, if every element $a$ has a strong inverse then $S$ is covered by its inverse subsemigroups $\langle a,b\rangle$ where $b\in S(a)$. On the other hand suppose that $S$ has an inverse cover. Take $a\in S$ and choose an inverse subsemigroup $A_{a}$ of $S$ containing $a$ and let $b$ be the (unique) inverse of $a$ in $A_{a}$. Then $A=\langle a,b\rangle$ is a subsemigroup of $A_{a}$ with commuting idempotents and every element of $A$ is regular as for any product $p=c_{1}c_{2}\cdots c_{k}\in A$ $(c_{i}\in\{a,b\}$) we see that $p'=c_{k}'c_{k-1}'\cdots c_{1}'$ is an inverse of $p$ in $A$, where we take $a'=b$ and $b'=a$, because both products take place within the inverse semigroup $A_{a}$. It follows that to prove that a given semigroup $S$ has an inverse cover is equivalent to showing that $S(a)$ is non-empty for every $a\in S$.
The following general observation applies to any inverse cover ${\cal A}=\{A_{i}\}_{i\in I}$ of an arbitrary semigroup $S$: if the pairwise intersection of any two members of ${\cal A}$ is an inverse subsemigroup of $S$ then the same is true of arbitrary intersections. To see this let $J\subseteq I$ and consider $A=\cap_{j\in J}A_{j}$. Either $A$ is the empty inverse subsemigroup or we may choose $a\in A$ and consider an arbitrary $A_{j}$ $(j\in J$). Then $a$ has a unique inverse $a^{-1}$ in $A_{j}$. Now let $k\in J$. By hypothesis, $A_{j}\cap A_{k}$ is an inverse subsemigroup of $S$ that contains $a$. Since $A_{j}\cap A_{k}$ is an inverse subsemigroup of the inverse semigroup $A_{j}$, it follows that the unique inverse of $a$ in $A_{j}\cap A_{k}$ is $a^{-1}$. Since $k\in J$ was arbitrary, it follows that $a^{-1}\in A$ and so $A$ is indeed an inverse subsemigroup of $S$. We shall say that $S$ has a \emph{closed inverse cover} if $S$ has a cover by inverse subsemigroups for which all pairwise intersections of its members are themselves inverse semigroups.
\textbf{Theorem 4.1.1} For a finite semigroup $S$:
(i) if $S$ has a closed inverse cover then $S$ has an involution matching by strong inverses.
(ii) If every element $a\in S$ has a unique strong inverse $b$ then $S$ has a closed inverse cover \[ {\cal C}=\{\cap_{i=1}^{k}U_{i},\,U_{i}=\langle a,b\rangle,\,a\in S,\,k\geq1\}. \]
(iii) If $a$ is a group element of $S$ then $a^{-1},$ the group inverse of $a$ in $S$, is the unique strong inverse of $a$.
\textbf{Proof }(i) Suppose there exists a closed inverse cover ${\cal A}=\{A_{0}\}_{0\leq i\leq m}$ of $S$ where, without loss, we include $\emptyset$ as $A_{0}$. The collection ${\cal A}$ is partially ordered by inclusion. Since every partial order may be extended to a total order, we may order the members of ${\cal A}$ in such a way that if $A_{i}\subset A_{j}$, then $A_{i}$ appears before $A_{j}$ in the list. This is assumed in the following argument.
We now show how ${\cal A}$ could be used to build an involution matching $(\cdot')$ of $S$ for which $a'\in S(a)$. First $A_{0}$ has an involution matching $(\cdot')$ in the empty function. Next let $U=\cup_{i=0}^{k}A_{i}$ $(k\geq1)$. Suppose inductively that we have extended the involution $(\cdot')$ to $V=\cup_{i=0}^{k-1}A_{i}$ and that for each $a\in A_{j}$, $(0\leq j\leq k-1)$ $a'\in A_{j}$ (so that $a'\in S(a))$. Let $a\in A_{k}$. Suppose first that $a\in V$ so that $a\in A_{j}$ for some $0\leq j\leq k-1$. Then $a'$ is already defined and by the nature of the linear order we have imposed on ${\cal A}$, $A_{i}=A_{j}\cap A_{k}\subseteq V$, with $i\leq j\leq k-1$. Therefore by induction we have $a'\in A_{j}\cap A_{k}$ and so the induction continues in the case where $a\in A_{k}\cap V$.
Otherwise $a\not\in V$. Then there exists a unique strong inverse $a'\in S(a)\cap A_{k}$. What is more $a'\not\in V$ for if to the contrary $a'\in A_{j}$ say $(0\leq j\leq k-1)$ then $a'$ is again a member of the inverse semigroup $A_{i}=A_{j}\cap A_{k}\subseteq V$, where $i\leq k-1$. In this event, $(a')'$ is already defined and would be an inverse of $a'$ in $A_{j}\cap A_{k}$, whence $(a')'=a$. But then $a\in A_{j}$, contrary to our choice of $a$. It follows that $a'\not\in V$ and so we may extend the involution by strong inverses $(\cdot')$ to $U=V\cup A_{k}$ by setting $a'$ as the unique inverse of $a$ in $A_{k}\setminus V$ for all $a\in A_{k}\setminus V$. Therefore we see that in both cases the induction continues. Since ${\cal A}$ covers $S$, the process terminates when $k=m$, yielding an involution matching by strong inverses $(\cdot')$ of $S$.
(ii) Let $U,V\in{\cal A}$ and suppose that $U\cap V\neq\emptyset$. For any $a\in U\cap V$ let $b$ be the unique strong inverse of $a$. Let $u\in V(a)$ in the inverse semigroup $U$. Then $u\in S(a)$, whence $u=b$. We may draw the corresponding conclusion for the inverse $v\in V(a)\cap V$, so that $u=b=v$. In particular $b\in U\cap V$, whence it follows that $U\cap V$ is an inverse subsemigroup. Therefore by adjoining all intersections $U_{1}\cap U_{2}\cap\cdots\cap U_{k}$ $(k\geq2)$ of members $U_{i}\in{\cal A}$ to the inverse cover ${\cal A}$ we generate a closed inverse cover for $S$.
(iii) Clearly $a^{-1}\in S(a)$. Consider an arbitrary $b\in S(a)$ and let $e=ab,$ $f=ba$. Then we have $e{\cal R}a{\cal L}f$ in $S$. Then since $H_{a}$ is a group we have $fe{\cal H}b$. Since $b\in S(a)$ it follows that $ef=fe\in E(S)$. But then $a{\cal H}ef=fe{\cal H}b$ and so $b,a^{-1}\in V(a)$ with $b{\cal H}a^{-1}$, whence $b=a^{-1}$ by uniqueness of inverses within an ${\cal H}$-class. $\quad\vrule height4.17pt width4.17pt depth0pt$
\textbf{Remark 4.1.2} As explained prior to Theorem 4.1.1, for any semigroup with a closed inverse cover, the intersection of any collection of members of ${\cal A}$ is also an inverse subsemigroup of $S$ and so we may assume further that ${\cal A}$ is closed under the taking of arbitrary intersections of its members. This allows the argument of the previous proof to be extended to arbitrary semigroups through the Axiom of choice and transfinite induction. Part (ii) of Theorem 4.1.1 is a partial converse of part (i). It remains open as to whether or not the full converse holds.
In the next section, we shall prove that in general ${\cal T}_{n}$ has no involution matching by strong inverses, from which it follows from the contrapositive of Theorem 4.1.1(i) that ${\cal T}_{n}$ has no closed inverse cover.
\subsection{Closed inverse covers for ${\cal T}_{n}$ do not exist}
In this section our context throughout will be ${\cal T}_{n}$. The account here of the construction of strong inverses in ${\cal T}_{n}$ follows {[}5, Section 6.2{]}.
Let $\alpha\in{\cal T}_{n}$. For $x\in X_{n}$ the \emph{depth }of $x$, denoted by $d(x)$, is the length of the longest dipath in $G(\alpha)$ ending at $x$; if $x\in$ stran$(\alpha)$ we conventionally define $d(x)=\infty$. Note that $d(x)=k<\infty$ if and only if $x\in X\alpha^{k}\setminus X\alpha^{k+1}$. The \emph{height }of $x$, denoted by $h(x)$ is the least positive integer $k$ such that $d(x\alpha^{k})\geq d(x)+k+1$; again we take $h(x)=\infty$ if $x\in$ stran$(\alpha)$. The height of $x$ is the length of the dipath which begins at $x$ and terminates at the first point $u$ which is also the terminal point of some dipath that is longer than the dipath from $x$ to $u$. A necessary condition for membership of $S(a)$ is the following.
\textbf{Lemma 4.2.1} Let $\beta\in S(\alpha)$. Then for all $x\in X\alpha$, $x\beta$ is a member of $x\alpha^{-1}$ of maximal depth.
When constructing strong inverses, the correct treatment of the endpoints of $G(\alpha),$ which are those $x\in X_{n}$ for which $d(x)=0$, is more complicated. The next parameter is defined on the vertices of $G(\alpha)$ in terms of some fixed but arbitrary $\beta\in{\cal T}_{n}$, but is only significant when $\beta\in V(\alpha)$. For each $x\in X_{n}$ the \emph{grasp }$g(x)$ of $\alpha$ is the greatest non-negative integer $k$ such that $x\alpha^{k}\beta^{k}=x$.
\textbf{Lemma 4.2.2 }Let $\beta\in S(\alpha)$. If $d(x)=0$, and $h(x)=h$ then $x\beta=y$ satisfies $g(y)\geq g(x)+1$ and $y\alpha^{h+1}=x\alpha^{h}$.
Lemmas 4.2.1 and 4.2.2 are all we require here. However, if $\beta\in{\cal T}_{n}$ satisfies these conditions together with the equality $x\beta\alpha\cdot\alpha^{g(x)+1}\beta^{g(x)+1}=x\alpha^{g(x)+1}\beta^{g(x)+1}\cdot\beta\alpha$, it may then be proved that that $\beta\in S(\alpha)$. We may show from this point that ${\cal T}_{n}$ has an inverse cover for the lemmas represent the two stages in the construction of a particular type of strong inverse $\beta\in S(\alpha)$: Lemma 4.2.1 applies to points of positive depth in $G(\alpha)$, while for each endpoint $x$ we may follow the dipath (of length $k$ say) from $x$ until we meet a point $u$ of depth exceeding $k$. Then $u\beta^{t}$ has already been defined for all $0\leq t\leq k+1$ and we then put $x\beta=x\alpha^{k}\beta^{k+1}$. This stage can always be carried out and indeed this $\beta\in S(\alpha)$ is uniquely determined by the choices made in determining $X\alpha\beta$. The outcome of this is a particular strong inverse $\beta\in S(\alpha)$ for such a $\beta$ will also satisfy the additional condition and indeed the set of all idempotents $\alpha^{t}\beta^{t},$$\beta^{s}\alpha^{s}$ then commute with each other, as is required for $\langle\alpha,\beta\rangle$ to be an inverse semigroup.
The main result of this section is the following.
\textbf{Theorem 4.2.3 }The full transformation semigroup ${\cal T}_{n}$ has a closed inverse cover if and only if $n\leq3$.
\textbf{Lemma 4.2.4 }For $n\leq3$, ${\cal T}_{n}$ has a closed inverse cover.
\textbf{Proof }For $n=1,2$ we note that ${\cal T}_{n}$ is a union of groups so the claim follows from Theorem 4.1.1. Although ${\cal T}_{3}$ is not a union of groups, we may verify that each $\alpha\in{\cal T}_{3}$ has a unique strong inverse as follows.
In general, an element $\alpha\in{\cal T}_{n}$ is a group element if and only if $X\alpha=X\alpha^{2}$. It follows that all members of ${\cal T}_{3}$ of ranks 1 or 3 are group elements. There are $3^{3}-3-3!=18$
mappings in ${\cal T}_{3}$ of rank $2$. All of those with two components are idempotent (these number $3\times2=6$). Those with one component for which $|X\alpha^{2}|=2$ are group elements (these also number
$6$). This leaves $18-6-6=6$ mappings of rank $2$ with a single component and for which $|X\alpha^{2}|=1$. These are evidently the $6$ mappings $\alpha$ of the form $a\mapsto b\mapsto c\mapsto c$ where $\{a,b,c\}=\{1,2,3\}$, which we denote for our current purposes by $(a\,b\,c)$. Observe that each such $\alpha$ has a unique strong inverse, which is $\alpha'=(b\,a\,c)$. The result now follows by Theorem 4.1.1(ii). $\quad\vrule height4.17pt width4.17pt depth0pt$
\textbf{Examples 4.2.5 }For $\alpha=(1\,2\,3)$ we have $S(\alpha)=\alpha'=(2\,1\,3)$, $\alpha\alpha':\text{\ensuremath{1\mapsto1,\,\,2,3\mapsto3}}$, $\alpha'\alpha:1,3\mapsto3,\,2\mapsto2$. The inverse subsemigroup $U_{3}=\langle\alpha,\alpha'\rangle$ is a 5-element combinatorial Brandt semigroup with zero element given by $\alpha^{2}=\alpha'^{2}$, which is the constant mapping with range $\{3\}$. The subsemigroups $U_{1},U_{2},U_{3}$ are pairwise disjoint. However not all intersections of distinct members of the inverse cover ${\cal C=}\{\langle a,b\rangle:b\in S(a)\}$ are empty: for example consider the mapping $\gamma:1\mapsto3,\,2,3\mapsto1$, which is its own strong inverse. Since $\gamma^{2}=\alpha\alpha'$ we obtain $\langle\gamma\rangle\cap\langle\alpha,\alpha'\rangle=\{\alpha\alpha'\}$.
By way of contrast, let us examine a subsemigroup $\langle\alpha,\alpha_{1}\rangle$, where $\alpha_{1}\in V(\alpha)\setminus S(\alpha)$: we take $\alpha_{1}=(3\,2\,1)$. Then $e=\alpha\alpha_{1}\in E({\cal T}_{3})$ and has fixed point set of $\{1,2\}$ with $3e=2$. Also $f=\alpha^{2}\in E({\cal T}_{3})$ is the constant mapping with range $\{3\}$, so that $ef=f$. However $fe$ is the constant mapping onto $2$ and so idempotents do not commute in $U=\langle\alpha,\alpha_{1}\rangle$. In fact $U$ is a 7-element regular subsemigroup of $S$, containing all three constant mappings, as $\alpha_{1}^{2}$ is the constant with range $\{1\}$.
Given Lemma 4.2.4 we now need to prove that ${\cal T}_{n}$ does not have a closed inverse cover for $n\geq4$. The remaining substantial task is to show that ${\cal T}_{4}$ has no involution matching through strong inverses for that implies that ${\cal T}_{4}$ has no closed inverse cover. For $n\geq5$ we then consider the copy of ${\cal T}_{4}$ embedded in ${\cal T}_{n}$ defined by $T=\{\alpha\in{\cal T}_{n}:\,k\alpha=k\,\forall\,k\geq5\}$. Suppose that ${\cal C}$ were a closed inverse cover for ${\cal T}_{n}$. If $\alpha\in T$ then for any $\beta\in S(\alpha)$ we have $\beta\in T$. It follows that ${\cal C}_{T}=\{A\cap T:A\in{\cal C}\}$ would be a closed inverse cover for $T$, which is isomorphic to ${\cal T}_{4}$, which would then also have such a cover. Therefore, to complete the proof of our theorem, it remains only to show that ${\cal T}_{4}$ does not have an involution matching by strong inverses.
First we identify every member of ${\cal T}_{4}$ that possesses a unique strong inverse, a collection that includes all mappings of ranks 1 or $4$ as these are group elements. Indeed for any rank we only need consider non-group elements, which are the mappings $\alpha$ such that $X\alpha^{2}$ is proper subset of $X\alpha$.
Consider mappings of rank $3$. It follows that $|X\alpha^{2}|\leq2$. If $\alpha$ has two components, since $X\alpha^{2}\neq X\alpha,$ it follows that $\alpha$ has an isolated fixed point $d$ say and a second component of the form $a\mapsto b\mapsto c$, which has a unique strong inverse $b\mapsto a\mapsto c$, $d\mapsto d$. If $\alpha$
has just one component with $|$stran$(\alpha)|=2$ then $\alpha$ necessarily now has the form $a\mapsto b\mapsto c\mapsto d\mapsto c$, which has a unique strong inverse, which is the mapping $b\mapsto a\mapsto d\mapsto c\mapsto d$. We conclude that all mappings of ranks 1, 3, or 4 have a unique strong inverse.
Finally consider mappings of rank $2$. Since we may assume there exists a point $x\in X\alpha\setminus$stran$(\alpha)$ (as otherwise $\alpha$ is a group element) it follows that we are restricted to mappings $\alpha$ with a single component and that component has a fixed point. The two remaining cases are:
A: the form of a `Y': $\alpha=\begin{pmatrix}a & b & c & d\\ c & c & d & d \end{pmatrix}$ or B: the form $\beta=\begin{pmatrix}a & b & c & d\\ b & d & d & d \end{pmatrix}$.
We next act the strong inverse operator $S(\cdot)$ for mappings of these two types. We will see that this generates a set of four $9$-cycles, with each mapping within a cycle sharing the same fixed point. Consequently these cycles are pairwise disjoint.
The given mapping $\alpha$ of type A has exactly two strong inverses, both of which are of type B: \begin{equation} \beta_{1}:\,\begin{pmatrix}a & b & c & d\\ d & d & a & d \end{pmatrix}\,\beta_{2}=\begin{pmatrix}a & b & c & d\\ d & d & b & d \end{pmatrix}\,(B) \end{equation} The mapping $\beta$ of type B also has exactly two strong inverses, the first of type A, the second of type B: \begin{equation} \alpha_{1}:\,\begin{pmatrix}a & b & c & d\\ d & a & a & d \end{pmatrix}\,(A)\,\,\beta{}_{2}:\begin{pmatrix}a & b & c & d\\ d & a & d & d \end{pmatrix}\,(B) \end{equation} Consider the collection $C$ of all mappings of rank $2$ with two strong inverses and a common fixed point, $d$. There are $3$ mappings of type A and $6$ of type B, so that $C$ has $9$ members. We use the symbols $\alpha$ and $\beta$, with appropriate subscripts, to denote mappings of types A and of B respectively.
The strong inverse operator $S(\cdot)$ acting on a point in $C$ outputs exactly two distinct mappings, which are also members of $C$, in accord with the rules (5) and (6). Let us write $\alpha_{1}=\alpha$ for the type A mapping above. We write $\beta_{1}\rightarrow\alpha_{1}\rightarrow\beta_{2}$ with the arrow indicating the first map is a strong inverse of the second (so that the reverse arrow is equally valid). We now act the operator $S(\cdot)$ on the rightmost member of our sequence, which will produce as outputs the previous member and a new sequence member. Bearing in mind rules (5) and (6) our sequence $C$ will thus take on the form: \begin{equation} C:\,\beta_{1}\rightarrow\alpha_{1}\rightarrow\beta_{2}\rightarrow\beta_{3}\rightarrow\alpha_{2}\rightarrow\beta_{4}\rightarrow\beta_{5}\rightarrow\alpha_{3}\rightarrow\beta_{6}\cdots. \end{equation} The output of $S(\gamma)$ when acting on $\gamma\in C$ comprises two distinct mappings, neither of which is $\gamma$, and one of which is the predecessor of $\gamma$ in the sequence. Eventually the output
$S(\gamma)$ will produce a repeated member of $C$ (in addition to the predecessor of $\gamma$), which must appear at least two steps before $\gamma$. However, all such members of $C$, apart from $\beta_{1}$, have already had their two strong inverses appear in $C$, and so cannot have $\gamma$ as a third strong inverse. Therefore the repeated sequence member is necessarily $\beta_{1}$. Hence $C$ is a cycle of length $l$ say with $l\geq2$ and $l|9$, and so $l=3$ or $l=9$. However $l=3$ would imply that $\beta_{1}$ and $\beta_{2}$ were mutual inverses, which is not the case. Therefore $l=9$ and and the cycle $C$ is completed by $\beta_{6}\rightarrow\beta_{1}$.
Suppose now that ${\cal T}_{4}$ possessed an involution $(\cdot')$ by strong inverses. Any mapping $\alpha$ with a unique strong inverse $\beta$ is necessarily paired with $\beta$ under $'$. This includes all mappings in ${\cal T}_{4}$ except for the mappings which are the vertices of the four disjoint $9$-cycles we have just identified. Each member of such a $9$-cycle $C$ is then paired with an adjacent partner in that cycle, but since $9$ is odd, this is not possible and so we have a contradiction. Therefore ${\cal T}_{4}$ has no involution by strong inverses, which implies by Theorem 4.1.1(i) that ${\cal T}_{4}$ has no inverse cover closed under the taking of intersections. $\quad\vrule height4.17pt width4.17pt depth0pt$
\textbf{Remarks 4.2.6 }We may explicitly calculate the $9$-cycle $C$ that contains the mapping $\alpha$ above, denoted here as $\alpha_{1}$, through repeated use of rules (5) and (6) as follows. We write $S(\alpha_{1})=\{\beta_{1},\beta_{2}\}$, with the $\beta_{i}$ as given in (5). Then following (7) the subsequent members of $C$ are $\beta_{2}\rightarrow\beta_{3}=\begin{pmatrix}a & b & c & d\\ d & c & d & d \end{pmatrix}\rightarrow$ $\alpha_{2}=\begin{pmatrix}a & b & c & d\\ b & d & b & d \end{pmatrix}\rightarrow$ $\beta_{4}=\begin{pmatrix}a & b & c & d\\ d & a & d & d \end{pmatrix}\rightarrow$ $\beta_{5}=\begin{pmatrix}a & b & c & d\\ b & d & d & d \end{pmatrix}\rightarrow$$\alpha_{3}=\begin{pmatrix}a & b & c & d\\ d & a & a & d \end{pmatrix}\rightarrow$ $\beta_{6}=\begin{pmatrix}a & b & c & d\\ c & d & d & d \end{pmatrix}\rightarrow$ $\beta_{7}=\beta_{1}=\begin{pmatrix}a & b & c & d\\ d & d & a & d \end{pmatrix}$, giving the anticipated $9$-cycle $C$.
Running down the ranks from $4$ to $1$, elementary combinatorial considerations give that: \[
|E({\cal T}_{4})|=1+2\binom{4}{2}+\Big(3\binom{4}{1}+(2)(2)\binom{4}{2}\Big)+4=1+12+36+4=53. \] In a similar fashion, bracketing term sum contributions from a common rank, the number of non-idempotent self-inverse elements is given by: \[ \Big(\binom{4}{2}+\frac{1}{2}\binom{4}{2}\Big)+(2)(3)\binom{4}{2}+2\binom{4}{2}=9+36+12=57. \] The number of mappings with a distinct unique strong inverse is given by: \[ (3!+2!\binom{4}{3}\big)+\Big((3)(2!)\binom{4}{3}+(2)(2)\binom{4}{2})+4!)+(2)(2)\binom{4}{2}\Big)=14+72+24=110. \]
The number of mappings with exactly two strong inverses is $4\times9=36$, giving the total of $(53+57)+110+36=256=4^{4}=|{\cal T}_{4}|$. The graph of strong inverses of ${\cal T}_{4}$ then consists of $110$ singletons, $55$ pairs, and four $9$-cycles. In particular the above analysis shows that ${\cal T}_{4}$ does possess a permutation matching by strong inverses. We may use one of these permutations to construct an involution matching of ${\cal T}_{4}$. (There are $2^{4}=16$ such permutations, determined by the $2$ choices of orientation of the $4$ cycles). First consider the $9$-cycle explicitly calculated above in which all mappings fix a point $d$. The mapping $\alpha=\alpha_{1}$ has an idempotent inverse $\varepsilon_{d}=\begin{pmatrix}a & b & c & d\\ b & b & b & d \end{pmatrix}$. We then remove the pair $(\varepsilon_{d},\varepsilon_{d})$ from our permutation, replacing it by $(\varepsilon_{d},\alpha)$ and pair up the remaining $8$ members of the associated $9$-cycle in neighbouring pairs. We repeat this procedure with the other three cycles, noting that there is no repetition of idempotents used in our pairings. This then yields an involution matching for ${\cal T}_{4}$.
We close with an example showing however that in general ${\cal T}_{n}$ does not possess a permutation matching by strong inverses.
\textbf{Example 4.2.7 }Consider the following pair of members of ${\cal T}_{8}$: \[ \alpha_{1}=\begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ 2 & 3 & 4 & 5 & 5 & 3 & 8 & 4 \end{pmatrix}\,\,\alpha_{2}=\begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ 2 & 3 & 4 & 5 & 6 & 8 & 3 & 4 \end{pmatrix}. \] The two mappings are identical except for the interchange of the images of $6$ and $7$, and so their digraphs are isomorphic. They share a common range: $X\alpha_{1}=X\alpha_{2}=\{2,3,4,5,8\}$. Moreover for each $x\in X\alpha_{i}$ $(i=1,2)$ there is a unique member of $y\in x\alpha_{i}^{-1}$ of maximal depth and so by Lemma 4.2.1 we see that any strong inverse $\beta_{i}\in S(\alpha_{i})$ has the following form: \[ \beta_{1}=\begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ - & 1 & 2 & 3 & 5 & - & - & 7 \end{pmatrix}\,\,\beta_{2}=\begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ - & 1 & 2 & 3 & 5 & - & - & 6 \end{pmatrix}. \] In each case the points of depth zero are $1,6,$ and $7$. For both mappings and for any strong inverses $\beta_{i}$ we see that $g(1)=3$ so that for any choice of $y=1\beta_{i}$ we have by Lemma 4.2.2 that $g(y)\geq4$, which implies that $y=1\beta_{i}=5$. To determine $6\beta_{1}$ we note that $6\alpha_{1}=3$ and so $d(6\alpha_{1})=2>0+1=d(6)+1$; hence $h(6)=1$ and $g(6)=0$. Writing $y=6\beta_{1}$ we have by Lemma 2.2 that $g(y)\geq1$ and $y\alpha_{1}^{2}=6\alpha_{1}=3$ so that $y=6\beta_{1}=1$. By the same argument with $6$ replaced by $7$ we obtain $7\beta_{2}=1$.
Finally consider $7\beta_{1}$. We have $7\alpha_{1}=8$ so we see that $d(7\alpha_{1})=1$ and $g(7)=1$, $h(7)=2$ as $d(7\alpha_{1}^{2})=d(4)=3>2+0=2+d(7)$ while $d(7\alpha)=d(8)=1\not>1+0$. Hence we have by Lemma 4.2.2 that $y=7\beta_{1}$ must satisfy $g(y)\geq2$ and $y\alpha^{3}=7\alpha^{2}=4$ so that $7\beta_{1}=1$. By symmetry we also obtain $6\beta_{2}=1$. Therefore each of the $\alpha_{i}$ has a unique strong inverse $\beta_{i}$: \[ \beta_{1}=\begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ 5 & 1 & 2 & 3 & 5 & 1 & 1 & 7 \end{pmatrix}\,\,\beta_{2}=\begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ 5 & 1 & 2 & 3 & 5 & 1 & 1 & 6 \end{pmatrix}. \] We now consider a third mapping $\beta\in{\cal T}_{8}$ and a putative strong inverse $\beta'\in S(\beta)$. As before we have $X\beta=\{2,3,4,5,8\}$ and again Lemma 4.2.2 gives the following unique partial definition of $\beta'$: \[ \beta=\begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ 2 & 3 & 4 & 5 & 5 & 8 & 8 & 4 \end{pmatrix}\,\,\beta'=\begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ - & 1 & 2 & 3 & 5 & - & - & - \end{pmatrix}. \] We see that $8\beta'\in\{6,7\}$; for the moment let us make the choice $8\beta'=7$ and henceforth denote $\beta'$ by $\beta_{1}'$. The points of zero depth are again $1,6,$ and 7 and the same analysis that applied to the $\alpha_{i}$ again yields $1\beta_{1}'=5$. Next we note that $g(6)=0$ and $h(6)=2$ as $d(6\alpha^{2})=d(4)=3>2+0$ but $d(6\alpha)=d(8)=1\not>1+0$. Hence $y=6\beta_{1}'$ must satisfy $g(y)\geq1$ and $y\beta^{3}=6\beta^{2}=4$ so that $y=6\beta_{1}'=1$. Finally we have $g(7)=1$ and $h(7)=2$ as for $h(6)$. Hence $y=7\beta'$ must satisfy $g(y)\geq2$ and $y\beta^{3}=6\beta^{2}=4$ so that $y=7\beta'=1$ also. We have then identified one strong inverse of $\beta_{1}'\in S(\beta)$. Similarly there exists a second strong inverse $\beta_{2}'\in S(\beta)$ determined by the alternative choice $8\beta_{2}'=6$. Exchanging the roles of the symbols $6$ and $7$ makes no difference to the images of the other domain points in that we again obtain that $1\beta_{2}'=5,$ $6\beta_{2}'=7\beta_{2}'=1$. Therefore we find that \[ \beta'_{1}=\beta_{1}=\begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ 5 & 1 & 2 & 3 & 5 & 1 & 1 & 7 \end{pmatrix}\,\,\beta'_{2}=\beta_{2}=\begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ 5 & 1 & 2 & 3 & 5 & 1 & 1 & 6 \end{pmatrix}. \] The upshot of all this is that we have a set of three members of ${\cal T}_{8}$ in $U=\{\alpha_{1},\alpha_{2},\beta\}$ such that the set $S(U)$ of all strong inverses of elements of $U$ is the two-element set $S(U)=\{\beta_{1},\beta_{2}\}$. It follows that there is no permutation matching $(\cdot)'$ on ${\cal T}_{8}$ that maps the set $U$ into the set $S(U)$, thereby giving us the result mentioned earlier, which we now formally state.
\textbf{Corollary 4.2.8 }There is no permutation matching $(\cdot')$ of ${\cal T}_{n}$ $(n\geq8)$ such that $a'$ is a strong inverse of $a$ for all $a\in{\cal T}_{n}$.
\textbf{Proof }Example 4.2.7 shows the corollary is true for $n=8$. For $n\geq9$ we may extend the above example with each of the the mappings $\alpha_{1},\alpha_{2},\beta$ acting identically on all integers exceeding $8$. Since any strong inverse preserves components we again obtain the conclusion that $S(\alpha_{1})=\{\beta_{1}\}$, $S(\alpha_{2})=\{\beta_{2}\}$ and $S(\beta)=\{\beta_{1},\beta_{2}\}$, which implies the result. $\quad\vrule height4.17pt width4.17pt depth0pt$
\end{document} | arXiv |
\begin{document}
\maketitle
\begin{abstract}
In this paper we consider the 2D Ericksen-Leslie equations which describes the hydrodynamics of nematic Liquid crystal with external body forces and anisotropic energy modeling the energy of applied external control such as magnetic or electric field. Under general assumptions on the initial data, the external data and the anisotropic energy, we prove the existence and uniqueness of global weak solutions with finitely many singular times. If the initial data and the external forces are sufficiently small, then we establish that the global weak solution does not have any singular times and is regular as long as the data are regular.
\end{abstract}
\tableofcontents
\section{Introduction}
We consider a hydrodynamical system modeling the flow of liquid crystal materials with anisotropic energy in a 2D bounded domain. More precisely, let $T>0$ and $\Omega\subset \mathbb{R}^2$ be a bounded domain with a smooth boundary $\partial \Omega$ and let us consider
\begin{subequations} \label{1a}
\begin{align}
&\partial_t v+ v\cdot \nabla v-\Delta v +\nabla \mathrm{p}=-\Div~ (\nabla d\odot\nabla d) + f ,\text{in $ [0,T)\times \Omega $}\\ & \partial_t d + v\cdot \nabla d=-d\times(d\times(\Delta d-\phi^\prime(d))) + d\times g,~\text{in~$[0,T)\times \Omega $},\\
& \Div v=0,~~\text{in~$[0,T)\times \Omega $},\\ & v = {\frac{\partial d}{\partial \nu}}=0,\text{ on~$[0,T)\times \partial \Omega $},\\ & \lvert d \lvert=1,\text{in~$[0,T)\times \Omega $},\\ & (v(0),d(0))=(v_{0},d_{0}),\text{in~$ \Omega $},
\end{align}
\end{subequations}
where $v:[0,T)\times \Omega \to\mathbb{R}^{2}$, $d:[0,T)\times \Omega \to\mathbb{S}^{2}$, where $\mathbb{S}^2$ is the unit sphere in $\mathbb{R}^{3}$, and $P:[0,T)\times \Omega \to\mathbb{R}$ represent the velocity field of the flow, the macroscopic molecular orientation of the liquid crystal material and the pressure function, respectively. In the system (\ref{1a}), the function $\phi:\mathbb{R}^{3}\to\mathbb{R}^{+}$ is a given map, $f:[0,T)\times \Omega \to \mathbb{R}^2 $ and $g:[0,T)\times \Omega \to \mathbb{R}^3$ are given external forces. The symbol $\nu(x)$ is a unit outward normal vector at each point $x\in \partial \Omega$. The matrix $\nabla d\odot\nabla d$ is defined by
\begin{equation*}
[\nabla d\odot\nabla d]_{ij}= \sum_{k=1}^3 \partial_i d_k \partial_j d_k~ \text{for}~ 1\le i,j\le 3.
\end{equation*}
Using the identities \begin{align*}
-\Div (\nabla d \odot \nabla d) = -\nabla d \Delta d + \frac12 \nabla \lvert \nabla d \rvert^2, \\
a\times (b\times c)=(a\cdot c)b-(a\cdot b)c \text{ for } a, b, c\in\mathbb{R}^{3},
\end{align*}
we can rewrite system (\ref{1a}) as follows
\begin{subequations}\label{1b-0}
\begin{align}
& \partial_t v+v\cdot \nabla v-\Delta v+\nabla \tilde{\mathrm{p}}=-\nabla d \Delta d + f,\\
& \partial_t d+v\cdot \nabla d=\Delta d+ |\nabla d|^{2}d-\phi^\prime(d)+(\phi^\prime(d)\cdot d)d+d\times g,\\ & \Div v=0,\\
& v_{\lvert_{\partial\Omega}} = {\frac{\partial d}{\partial \nu}}{\Big\lvert_{\partial\Omega}}=0, \label{Eq:BC-DirNeu}\\
& \lvert d \rvert=1,\\ & (v(0),d(0))=(v_{0},d_{0}),
\end{align} \end{subequations} where \begin{equation*} \tilde{\mathrm{p}}= \mathrm{p} + \frac12 \lvert \nabla d \rvert^2 . \end{equation*}
While we focus our mathematical analysis on the system \eqref{1b-0} with the Dirichlet and the Neumann boundary conditions \eqref{Eq:BC-DirNeu}, our results remain valid for the case of the periodic boundary conditions. That is, our results remain true in the case that $\Omega$ is a 2D torus $\mathbb{T}^2$ and \eqref{Eq:BC-DirNeu} is replaced by
\begin{equation*}
\int_\Omega v(t, x) \;dx=0, \; \forall t \in (0,T].
\end{equation*}
The model \eqref{1b-0} is an oversimplification of a Ericksen-Leslie model of nematic liquid crystal with a simplified energy density
\begin{equation*}
\frac12 \lvert \nabla d \rvert^2 + \phi(d).
\end{equation*}
The term $\frac12 \lvert \nabla d \rvert^2 $ represents the one-constant simplification of the Frank-Oseen energy density and $\phi(d)$ represents an anisotropic energy density. One example of such anisotropic energy density is the magnetic energy density
$$ \phi(n) = (n\cdot H)^2,$$
when the nematic liquid crystal is subjected to the action of a constant magnetic field $H \in \mathbb{R}^3$. We also give different examples of mathematical models of anisotropic energy density later on. For more details on physical modeling of liquid crystal under the action of external control such as magnetic or electric field we refer to the books \cite{Gennes} and \cite{Stewart} and the papers \cite{Ericksen} and \cite{Leslie}.
We should note that since \eqref{1b-0} was obtained by neglecting several terms such as the viscous Leslie stress tensor in the equation for $v$(see for instance \cite{Lin-Liu,Lin-Liu2}), the stretching and rotational effects for $d$, one does not known whether it is thermodynamically stable or consistent with the laws of thermodynamics. However, this model still retains many mathematical and essential features of the hydrodynamic equations for nematic liquid crystals. In recent years, several liquid crystal models which are thermodynamically consistent and stable have been recently developed and analyzed, see for instance the \cite{Feireisl-2019}, \cite{Feireisl-2012}, \cite{Feiresil-2011}, \cite{MH+JP-2017}, \cite{MH+JP-2018}, \cite{MH et al-2014}, \cite{Sun+Liu}, \cite{Lin+Wang-2014} and references therein.
In the absence of external forcings $f$, $g$ and the anisotropic energy potential $\phi(d)$, the system \eqref{1b-0} has extensively studied and several important results have been obtained. In addition to the papers we cited above we refer, among others, to \cite{Hong,Hong12,LLW,Lin-Liu,
Lin-Liu2,Lin-Wang,WZZ} for results obtained prior to 2013, and to \cite{YC+SK+YY-2018,Hong14,MCH+YM-2019,Huang14,Huang16,JL+ET+ZX-2016,Lin+Wang16,Wang2014,Wang2016,WZZ15} for results obtained after 2014. For detailed reviews of the literature about the mathematical theory of nematic liquid crystals and other related models, we recommend the review articles \cite{Lin+Wang-2014,MH+JP-2018,Climent} and the recent papers \cite{MCH+YM-2019,Lin+Wang16}.
Let us now outline the contributions of our manuscript. \begin{trivlist} \item[\;\;\;(1)] In Section \ref{Sec:ExistRegularSol}, we prove by using Banach fixed point theorem that if $(v_0,d_0)\in D(\mathrm{A} ^\frac12)\times (D(\hat{\mathrm{A}} ) \cap \mathcal{M} )$ and $(f,g)\in L^2(0,T; \mathrm{H} \times D(\hat{\mathrm{A}} ))$, then there exists a unique local regular solution $(v,d): [0,T_0] \to D(\mathrm{A} ^\frac12)\times D(\hat{\mathrm{A}} ) \cap \mathcal{M}$ such that $C([0,T_0]; D(\mathrm{A} ^\frac12) \times (D(\hat{\mathrm{A}} )\cap \mathcal{M})) \cap L^2(0,T; D(\mathrm{A} )\times D(\hat{\mathrm{A}} ^\frac32)$, see Theorem \ref{th}. Here \[ \mathcal{M}=\{ d: \Omega \to \mathbb{R}{^3}: \lvert d(x) \rvert=1 \;\;\mathrm{Leb}\text{-a.e.} \}, \] $\mathrm{A} $ and $\hat{\mathrm{A}} $ are respectively the Stokes operator and the Neumann Laplacian, see Section \ref{Sec:NotaPrelim-Sec} for the definitions of these operators and the space $\mathrm{H} $.
\item[\;\;\;(2)] We exploit this result and the local energy method developed in \cite{Struwe}, \cite{LLW} and \cite{Hong} to show in Section \ref{Sec:LocRegSolSmallENERG} that there exists universal constants $\varepsilon\,_0>0$ and $r_0$ such that the following statement hold.
\textit{ If $(f,g)\in L^2(0,T; \mathrm{H}\times D(\hat{\mathrm{A}} ^{1/2}))$ and $(v_0, d_0)$ belongs to $\mathrm{V}\times (D(\hat{\mathrm{A}} )\cap \mathcal{M}) $ with small energy, i.e., there exists $R_0\in (0,r_0]$ such that \begin{equation*} \sup_{x\in \Omega}\int_{B(x, R_0)} [\lvert v_0(x) \rvert^2+ \lvert \nabla d_0(x) \rvert^2 + \phi(d_0(x))]<\varepsilon\,_0^2. \end{equation*} Then, there exists a time $T_0>0$ a unique $(v,d): [0,T_0] \to D(\mathrm{A} ^\frac12)\times (D(\hat{\mathrm{A}} ) \cap \mathcal{M})$ such that $C([0,T_0]; D(\mathrm{A} ^\frac12) \times (D(\hat{\mathrm{A}} )\cap \mathcal{M})) \cap L^2(0<T; D(\mathrm{A} )\times D(\hat{\mathrm{A}} ^\frac32)$ such that \begin{equation*} \frac12 \sup_{0\le t\le T_0 } \sup_{x\in \Omega}\int_{B(x,R_0)} \left(\lvert v(t,y)\rvert^2 + \lvert \nabla d(t,y) \rvert^2 + 2 \phi(d(t,y)) \right) dy\le 2\varepsilon\,_1^2. \end{equation*} } We refer to Proposition \ref{Prop:LocalSolwithSmallEnergy} and its proof for more details.
\item[\;\;\;(3)] The two results above are exploited in Section \ref{Sec:ExistMaxLocStrongSol} in order to prove the global existence of our problem. This is our main result. It holds under weaker assumptions than those listed in (1) and (2) above, and is presented in Theorem \ref{thm-main}. It can be summarized as follows.
\begin{Thm}\label{thm-main-intro}
Let $(v_0,d_0)\in \mathrm{H}\times (\mathrm{H} ^1\cap \mathcal{M})$. Then, there exist constants $\varrho_0>0 $ and $\varepsilon\,_0>0 $ such that the following hold.
If $(f,g)\in L^2(0,T; \mathrm{H} ^{-1} \times L^2)$, then \begin{enumerate}[(i)]
\item a number $L\in \mathbb{N}$, depending only on the norms of $(v_0,d_0)\in \mathrm{H} \times \mathrm{H} ^1$ and $(f,g)\in L^2(0,T; \mathrm{H} ^{-1}\times L^2)$,
a finite sequence $0=T_0<T_1<\cdots<T_L\leq T$ and,
\item a function $(\mathbf{u},\mathbf{d})\in C_{w}([0, T]; \mathrm{H} \times \mathrm{H}^1) \cap L^2(0, T; \mathrm{V}\times D(\hat{\mathrm{A}} ) )$
such that {for all $t\in [0,T], \mathbf{d}(t)\in \mathcal{M} $} and
\begin{enumerate}
\item for every $i\in \{1, \ldots, L\}$, $(\mathbf{u},\mathbf{d})_{\lvert_{[T_{i-1}, T_i)}}\in C([T_{i-1}, T_i); \mathrm{H} \times \mathrm{H}^1)$ with the left-limit at $T_i$, which satisfies the variational form of problem \eqref{1b-0} on the interval $[T_{i-1}, T_i)$ with initial data $(v(T_{i-1}), d(T_{i-1}) )$. \item If $T_L<T$, then $(\mathbf{u},\mathbf{d})_{\lvert_{[T_{L}, T]}}$ belongs to $ C([T_{L}, T]; \mathrm{H} \times \mathrm{H}^1)$ and satisfies the variational form problem \eqref{1b-0} on the interval $[T_{L}, T]$ with initial data $(v(T_{L}), d(T_{L}) )$.
\item {There exists $\varepsilon\,_1\in (0,\varepsilon\,_0)$ such that for all $i\in \{1, \ldots, L\}$ and all $R\in (0,\varrho_0]$}
\begin{equation*}
\lim_{t\nearrow T_i} \sup_{x\in \Omega} \int_{B(x, R)}\left( \frac12 \lvert \mathbf{u}(t,y)\rvert^2 + \frac12\lvert \nabla \mathbf{d}(t,y) \rvert^2 + \phi(d(t,y)) \right) dy\ge \varepsilon\,_1^2.
\end{equation*}
\item At every time $T_i$, $i\in \{1, \ldots, L\}$, there is a loss of energy at least $\varepsilon\,_1^2\in (0, \varepsilon\,_0^2)$, \textit{i.e.},
\begin{equation*}
\begin{split}
& \hspace{1truecm} \int_{\Omega }\left( \frac12 \lvert \mathbf{u}(T_i,y)\rvert^2 + \frac12 \lvert \nabla \mathbf{d}(T_i,y) \rvert^2 + \phi(d(T_i,y)) \right) dy \\
& \hspace{1truecm} \le \int_{\Omega }\left( \frac12 \lvert \mathbf{u}(T_{i-1},y)\rvert^2 + \frac12 \lvert \nabla \mathbf{d}(T_{i-1},y) \rvert^2 + \phi(d(T_{i-1},y)) \right) dy
+\frac12 \int_{T_{i-1}}^{T_1} \left[\lvert f\rvert_{\mathrm{H} ^{-1}}^2 + \lvert g\rvert^2_{L^2}\right] dt -\varepsilon\,_1^2 .
\end{split}
\end{equation*}
\end{enumerate} \end{enumerate} The numbers $T_1,\cdots, T_L$ are called the \textbf{singular times} of the solution $(\mathbf{u},\mathbf{d})$. \end{Thm} Because of the presence of the anisotropic energy and the external forces, this result is a generalization of the global existence of weak solution obtained in \cite{Hong} and \cite{LLW}. \item[(4)] Finally, in Section \ref{Sec:RegCompactSmallData} we prove that the set of singular times is empty when the data $(v_0,d_0)\in \mathrm{H}\times \mathrm{H} ^1$ and $(f,g)\in L^2(0,T; \mathrm{H} ^{-1}\times L^2)$ are sufficiently small. We also show that if the data are sufficiently regular and small, \textit{i.e.} $(v_0,d_0)\in D(\mathrm{A} ^\frac12 )\times D(\hat{\mathrm{A}} )$ and $(f,g)\in L^2(0,T; \mathrm{H} \times \mathrm{H} ^1)$, then the weak solution becomes regular for all time. Moreover, for all $t \in [0,T)$ $(u(t),d(t))$ lies in a compact set of $\mathrm{H} \times \mathrm{H} ^1$. We refer the reader to Theorems \ref{Thm:NoSingSmall} and \ref{Thm:PrecompactInHH1} for more detail about these results. \end{trivlist}
We close this introduction with the presentation of the layout of the present paper. In Section \ref{Sec:NotaPrelim-Sec} we fix the frequently used notation in the manuscript. We also state and prove some auxiliary results which are essential to our analysis. Section \ref{Sec:ExistRegularSol} is devoted to the existence and uniqueness of of a regular solution to Problem \eqref{1b}. In Section \ref{Sec:LocRegSolSmallENERG} we prove that one can find a small number $R_0>0$ and a unique maximal local regular solution $((v,d);T_0)$ such that its energy at any time $t\in [0,T_0)$ does not exceed twice the supremum of all energies on $B(x,2R_0)$, $x\in \Omega$ of the initial data. This result will play a pivotal role in the proof of the existence of a maximal local strong solution to Problem \eqref{1b} in Section \ref{Sec:ExistMaxLocStrongSol}.
\section{Notation and preliminaries}\label{Sec:NotaPrelim-Sec}
Let $\Omega\subset\mathbb{R}^{2}$ be an open and bounded set. We denote by $\Gamma=\partial\Omega$ the boundary of $\Omega$. We assume that the closure $\overline{\Omega}$ of the set $\Omega$ is a manifold with $C^\infty$ boundary $\Gamma:=\partial\Omega$ which is a
$1$-dimensional infinitely differentiable manifold being
locally on one side of $\Omega$.
Throughout this paper $L^p(\Omega; \mathbb{R}^\ell)$, $\mathrm{W}^{p,k}(\Omega; \mathbb{R}^\ell)$( $\mathrm{H} ^k(\Omega; \mathbb{R}^\ell)= \mathrm{W}^{2,p}(\Omega; \mathbb{R}^\ell)$), $p\in[1,\infty], k \in \mathbb{N}$, and $\mathrm{H} ^\alpha(\Omega; \mathbb{R}^\ell)$, $\alpha \in (0,\infty)$, denote the Lebesgue and Sobolev spaces whose elements take values in $\mathbb{R}^\ell$, $\ell=2,3$. To shorten the notation we will just write $L^p$, $\mathrm{W}^{p,k}$, $\mathrm{H} ^k$ and $\mathrm{H} ^\alpha$ irrespectively if the elements of these spaces take values in $\mathbb{R}^2$ or $\mathbb{R}^3$.
\subsection{Notations for the velocity field $v$}\label{subsect-Dirichlet}
The following is an abridged version of notations and preliminary of the paper \cite{Brz+Cer+F_2015}. The facts we enumerate here can be found in \cite[Section 2]{Brz+Cer+F_2015} and references therein.
Let $\mathcal{D}(\Omega)$ (resp. $\mathcal{D
}(\overline{\Omega})$) be the set of all $C^\infty$ class vector fields $u:\mathbb{R}^2\to \mathbb{R}^2$ with compact support contained in the set $\Omega$ (resp. $\overline{\Omega}$). Then, let us define
\begin{eqnarray*}
E(\Omega) &=& \{ u\in L^2(\Omega,\mathbb{R}^2):\mathrm{div}\, u \in\,L^2(\Omega,\mathbb{R}^2)\},\\
\mathcal{V}&=&\big\{ u\in C_0(\Omega,\mathbb{R}^2): \mathrm{div}\, u=0\big\} ,\\
\mathrm{H} &=& \mbox{the closure of $\mathcal{V}$ in } L^2(\Omega,\mathbb{R}^2),\\
\mathrm{H} _0^1(\Omega,\mathbb{R}^2)&=& \mbox{the closure of $\mathcal{D}(\Omega,\mathbb{R}^2)
$ in } \mathrm{H} ^1(\Omega,\mathbb{R}^2),\\
\mathrm{V}&=& \mbox{the closure of $\mathcal{V}$ in } \mathrm{H} ^1_0(\Omega,\mathbb{R}^2).
\end{eqnarray*}
The inner products in all $L^2$ spaces will be denoted by $\langle \cdot,\cdot\rangle$. The space $E(\Omega)$ is a Hilbert space with a scalar product \begin{equation}\label{Temam_1.13}
\langle u,v\rangle_{E(\Omega)}:=\langle u,v\rangle +\langle \mathrm{div}\, u, \mathrm{div}\, v\rangle.
\end{equation}
We endow the set $\mathrm{H} $ with the inner product $\rangle \cdot,\cdot\rangle$ and the norm $\left\vert \cdot\right\vert_{\mathrm{H} } $ induced by $L^2$.
The space $\mathrm{H} $ can also be characterized in the following way, see
\cite[Theorem I.1.4]{Temam_2001},
\[
\mathrm{H} =\{ u \in E(\Omega): \mathrm{div}\, u=0 \mbox{ and } u\cdot \nu_{\lvert_{\partial \Omega}} =0\}.
\]
Let us denote by $\Pi :L^2(\Omega,\mathbb{R}^2) \rightarrow \mathrm{H}$
the orthogonal projection called usually the Leray-Helmholtz projection. It is known,
see \cite[Remark I.1.6]{Temam_2001} that $\Pi$ maps continuously the Sobolev space $\mathrm{H} ^{1}$ into itself.
Observe that $\Omega$ is a Poincar\'e
domain, \textit{i.e.} there exists a constant $\lambda_1>0$ such that the following Poincar\'{e} inequality is satisfied
\begin{equation}
\lambda_1\int_{\Omega}\varphi^{2}(x)\,\;dx\leq\int_{\Omega}|\nabla
\varphi(x)|^{2}\,\;dx,\ \ \ \varphi\in \mathrm{H} ^1_0({\Omega}).
\label{ineq-Poincare}
\end{equation}
Because of this the norms on the space $\mathrm{V} $ induced by $\lvert \cdot \rvert_\mathrm{H} ^1= \lvert \cdot \rvert_{L^2} +\lvert \nabla \cdot \rvert_{L^2}$ and by $\lvert \nabla \cdot \rvert_{L^2}$ are equivalent.
Since the space $\mathrm{V} $ is densely and continuously
embedded into $\mathrm{H}$, by identifying $\mathrm{H}$
with its dual $\mathrm{H}^\prime$, we have the following embedding
\begin{equation}
\label{eqn:Gelfanf}
\mathrm{V} \hookrightarrow \mathrm{H}\cong\mathrm{H}^\prime \hookrightarrow
\mathrm{V}^\prime.
\end{equation}
Let us observe here that, in particular, the spaces $\mathrm{V}$, $\mathrm{H}$ and
$\mathrm{V}^\prime$ form a Gelfand triple.
We will denote by $| \cdot |_{\mathrm{V}^\ast}$ and $\left\langle \cdot,\cdot\right\rangle $ the norm in
$\mathrm{V}^\ast$ and the
duality pairing between $\mathrm{V} $ and $\mathrm{V}^\ast$, respectively.
Now, define the bilinear
form $a:\mathrm{V}\times \mathrm{V} \to \mathbb{R}$ by setting
\begin{equation}
\label{form-a}
a(u,v):=\langle \nabla u,\nabla v\rangle, \quad u,v \in \mathrm{V}.
\end{equation}
It is well-known that this bilinear map is
$\mathrm{V}$-continuous and $\mathrm{V}$-coercive, i.e. there exist some $C_0, C_1>0$ such that
\[ C_0 \lvert u \rvert^2_{\mathrm{V} }\le \vert a(u,u) \vert
\leq C \vert u \vert_\mathrm{V}^2,\ \ \ \ u \in\,\mathrm{V}.\]
Hence, by the Riesz Lemma and Lax-Milgram theorem, see for instance
Temam \cite[Theorem II.2.1]{Temam_2001},
there exists a unique isomorphism
$\mathcal{A}:\mathrm{V} \to \mathrm{V}^\prime$, such that
$a(u,v)=\langle \mathcal{A}u,v\rangle$, for $u, v \in \mathrm{V}$.
Next we define an unbounded linear operator
$\mathrm{A}$ in $\mathrm{H}$ as follows
\begin{equation}
\label{def-A} \left\{
\begin{array}{ll}
D(\mathrm{A}) &= \{u \in \mathrm{V}: \mathcal{A}u \in
\mathrm{H}\},\\
&
\\
\mathrm{A}u&=\mathcal{A}u, \, u \in
D(\mathrm{A}).
\end{array}
\right.
\end{equation}
Under our assumption on $\Omega$, $\mathrm{A} $ and $D(\mathrm{A})$ can be characterized as follows
\begin{equation}
\label{eqn:4.3} \left\{
\begin{array}{l}
D(\mathrm{A}) = \mathrm{V} \cap \mathrm{H} ^2=\mathrm{H} \cap \mathrm{H} ^1_0\cap \mathrm{H} ^2,\\
\mathrm{A}u=-\mathrm{P}\Delta u, \quad u\in D(\mathrm{A}).
\end{array}
\right.
\end{equation}
It is also well-known, see \cite[Section]{Brz+Cer+F_2015} and references therein, that $\mathrm{A}$ is a positive self adjoint operator in
$\mathrm{H}$ and
\[D(\mathrm{A} ^{\alpha/2})=[\mathrm{H} ,D(\mathrm{A} )]_{\frac{\alpha}{2}},\] where
$[\cdot,\cdot]_\frac{\alpha}{2}$ is the complex interpolation functor of
order $\frac{\alpha}{2}$. Furthermore, for $\alpha \in (0, \frac12)$
\begin{equation}\label{eqn-domains}
D(\mathrm{A} ^{\alpha/2})= \mathrm{H} \cap \mathrm{H} ^{\alpha}(\Omega,\mathbb{R}^2).
\end{equation}
In particular, $\mathrm{V}=D(\mathrm{A}^{1/2})$ and $\lvert \mathrm{A} ^\frac12 u \rvert^2= \lvert \nabla u \rvert^2=: \lVert u \rVert^2$ for $u \in \mathrm{V}$.
The equality \eqref{eqn-domains} leads to the following result which was proved in \cite[Proposition 2.1]{Brz+Cer+F_2015}.
\begin{proposition}\label{prop-Leray-fractional} Assume that $\alpha \in (0, \frac12)$.
Then the Leray-Helmholtz projection $\Pi$ is a well defined and continuous map from
$\mathrm{H} ^{\alpha}$ into $ D(\mathrm{A} ^{\alpha/2})$.
\end{proposition}
Let us finally recall that the projection $\Pi$ extends to a
bounded linear projection in the space $L^q$, for any $q \in\,(1,\infty)$.
Now, consider the trilinear form $b$ on $V\times V\times V$ given by
\[
b(u,v,w)=\sum_{i,j=1}^{2}\int_{\Omega}u_{i}{\frac{\partial v_{j}
}{\partial x_{i}}}w_{j}\,\,{d}x,\quad u,v,w\in \mathrm{V} .
\]
Indeed, $b$ is a continuous trilinear form such that
\begin{equation}
\label{eqn:b01}
b(u,v,w)=-b(u,w,v),
\quad \, u\in \mathrm{V}, v, w\in
\mathrm{H} _0^{1}(\Omega,\mathbb{R}^2),
\end{equation}
or a proof see for instance \cite[Lemma 1.3, p.163]{Temam_2001} .
Define next the bilinear map $B:\mathrm{V} \times \mathrm{V} \rightarrow \mathrm{V} ^{\ast}$ by setting
\[
{}_{\mathrm{V}^\ast}\left\langle B(u,v),w\right\rangle_{\mathrm{V}}=b(u,v,w),\quad u,v,w\in \mathrm{V} ,
\]
and the homogenous polynomial of second degree $B:\mathrm{V} \rightarrow \mathrm{V}^{\ast}$ by
\[
B(u)=B(u,u),\; u\in \mathrm{V} .
\]
Let us observe that if $v \in\,D(\mathrm{A} )$, then $B(u,v) \in\,H$ and the following identity is a direct consequence of \eqref{eqn:b01}.
\begin{equation}
\label{eqn-B02}
{}_{\mathrm{V}^\ast}\langle \mathrm{B}(u,v),v \rangle_{\mathrm{V}} =0,\;\; u,v\in \mathrm{V} .
\end{equation}
The restriction of the map $\mathrm{B}$ to the space $D(\mathrm{A} )\times D(\mathrm{A} )$ has also the following representation
\begin{equation}
\label{eqn-B-using-LH}
\mathrm{B}(u,v)= \Pi( u\cdot \nabla v), \;\; u,v\in D(\mathrm{A} ),
\end{equation}
where $\Pi$ is the Leray-Helmholtz projection operator and $u\nabla v=\sum_{j=1}^2 u^jD_jv \in L^2(\Omega,\mathbb{R}^2)$.
\subsection{The Laplacian for the director field $d$}
Throughout this section we still denote $L^2(\Omega; \mathbb{R}^3)$ and $\mathrm{H} ^k(\Omega; \mathbb{R}^3)$, $k \in \mathbb{N},$ by $L^2$ and $\mathrm{H} ^1$, respectively.
We aim in this subsection to introduce the Laplacian for the director $d:\Omega \to \mathbb{R}^3$ with the Neumann boundary conditions. We can do this by mimicking the way we define the Stokes operator $\mathrm{A} $. We define the bilinear map $\hat{a}: \mathrm{H} ^1\times \mathrm{H} ^1 \to \mathbb{R}$ by
\begin{equation*}
\hat{a}(d,n) = \int_\Omega (\nabla\; d \nabla n ) \;dx=\sum_{i=1}^3\sum_{j=1}^{3}\int_{\Omega}(\partial_i d_j \partial_in_j)\,dx, \; d, n \in \mathrm{H} ^1.
\end{equation*}
It is clear that $\hat{a}$ is continuous, and hemce, by Riesz representation lemma, there exists a unique bounded linear operator $\hat{\mathcal{A}}: \mathrm{H} ^1 \to (\mathrm{H} ^1)^\ast$ such that ${}_{(\mathrm{H} ^1)^\ast} \langle\hat{\mathcal{A}} d, n\rangle_{\mathrm{H} ^1} =\hat{a}(d,n) $, for $d,n\in \mathrm{H} ^1$. Next, we define an unbounded linear operator
$\hat{\mathrm{A}} $ in $L^2$ as follows
\begin{equation}
\label{def-ANeum} \begin{cases}
D(\hat{\mathrm{A}} ) = \{d \in \mathrm{H} ^1: \hat{\mathrm{A}} d \in L^2 \}\\
\hat{\mathrm{A}} =\hat{\mathcal{A}}d, \, d \in
D(\hat{\mathrm{A}} ).
\end{cases}
\end{equation}
Under our assumption on $\Omega$, it is known, see for instance \cite[Section2, p. 65]{Temam_1997} that $\hat{\mathrm{A}} $ and $D(\hat{\mathrm{A}} )$ can be characterized by
\begin{equation}\label{eqn-def-A}
\begin{cases}
D(\hat{\mathrm{A}} )&:=\{d\in \mathbb{H}^{2}: \frac{\partial d}{\partial \nu }=0 \;\text{ on }\; \partial \Omega \},\\
\hat{\mathrm{A}} d&:=-\Delta d,\quad d\in D(\hat{\mathrm{A}} ),
\end{cases}
\end{equation}
where $\nu=(\nu_1,\nu_2,\nu_3)$ is the unit outward normal vector field on $\partial \Omega $ and $\frac{\partial d}{\partial \nu}$
is the directional derivative of $d$ in the direction $\nu$.
Let us recall that the operator $\hat{\mathrm{A}} $ is self-adjoint and nonnegative and
$D\left(\hat{\mathrm{A}} ^{1/2}\right)$ when endowed with the
graph norm coincides with $\mathrm{H} ^1$. Moreover, the operator $(I+\hat{\mathrm{A}} )^{-1}$ is
compact. Furthermore, if we denote
\[
\hat{\mathrm{V}} := D(\hat{\mathrm{A}} ^{1/2}),
\]
the $(\hat{\mathrm{V}} ,L^2, \mathrm{V}^\ast)$ is a Gelfand triple and
\[
\fourIdx{}{\hat{\mathrm{V}} ^\ast}{}{\hat{\mathrm{V}} }{\langle d_1,d_2 \rangle}= \int_{\Omega} d_1(x) d_2(x)\, \;dx
\]
\subsection{An abstract formulation of problem \eqref{1a}}
With the notations we have introduced above, we can now rewrite problem \eqref{1b-0} as an abstract equations. In fact by projecting the first equation in \eqref{1b} onto $\mathrm{H} $ we obtain the following system
\begin{equation}
\begin{cases}
\dot{v}+\mathrm{A} v =-B(v,v)-\Pi \left(\Div [ \nabla d\odot \nabla d]\right)+ \Pi f,\\
\partial_t d+\hat{\mathrm{A}} d = |\nabla d|^{2}d-v\cdot \nabla d-\phi^\prime(d)+(\phi^\prime(d),d)d+d\times g,\\
\lvert d \rvert=1\\
(v,d)(0)=(v_{0},d_{0}). \label{1b}
\end{cases}
\end{equation}
\section{The existence and the uniqueness of a regular solution to Problem \eqref{1b}} \label{Sec:ExistRegularSol}
Throughout the whole section, we fix a map $\phi:\mathbb{R}^3\to \mathbb{R}^3$ satisfying the following set of conditions.
\begin{assume}\label{Assum:EnergyPotential}
The map $\phi:\mathbb{R}^3 \to \mathbb{R}^3$ is of class $C^2$ and there exist constants $M_0>0$, $M_1>0$ and $M_2>0$ such that for all $n, d \in \mathbb{R}^3$
\begin{align}
\lvert \phi^\prime(n) \rvert \le & M_0(1+ \lvert n\rvert),\label{Eq:LInearGrowthPhiprime}\\
\lvert \phi^{\prime \prime} (n)-\phi^{\prime \prime} (d)\rvert \le & M_1\lvert n-d\rvert,
\label{Eq:LipSchitzPhibis}\\
\lvert \phi^{\prime \prime} (n)\rvert\le & M_2.\label{Eq:BoundednessPhibis}
\end{align}
\end{assume} \begin{example}
Let $H\in \mathbb{R}^3$ be a constant vector. Then the anisotropy energy potential $\phi$ due to the action of a magnetic or electric is defined by
$$ \phi(d)= \frac12[ \lvert H\rvert^2- (d\cdot H)^2], d\in \mathbb{R}^3.$$
This potential $\phi$ satisfies the Assumption \ref{Assum:EnergyPotential}. In this case $H$ represents a constant magnetic or electric field applied to the nematic liquid crystal.
\noindent Another mathematical example which satisfies Assumption \ref{Assum:EnergyPotential} is the potential defined by
$$ \phi(d)= \frac12 \lvert d -\xi \rvert^2, d\in \mathbb{R}^3,$$
where $\xi \in \mathbb{R}^3$ is a fixed constant vector. \end{example}
Next, we consider the problem \eqref{1b} on a finite time horizon $[0,T]$. Throughout the paper we put \[ \mathcal{M}=\{ d: \Omega \to \mathbb{R}{^3}: \lvert d(x) \rvert=1 \;\;\mathrm{Leb}\text{-a.e.} \}, \]
For this section, we impose the following set of conditions on the data.
\begin{assume}
We assume that $({f},g)\in L^2(0,T; L^2 \times D(\hat{\mathrm{A}} ^{1/2}))$ and $(v_0, d_0)\in \mathrm{V}\times (D(\hat{\mathrm{A}} )\cap \mathcal{M})$.
\end{assume}
Under this assumption we will prove in this section that Problem \eqref{1b} has a unique local regular solution.
Before stating and proving this result
we define what we mean by a maximal local regular solution.
\begin{Def}\label{Def:Local-Regular-Sol}
Let $T_0\in (0, T]$. A function $(v,d): [0,T_0] \to \mathrm{V}\times D(\hat{\mathrm{A}} )$ is called a local regular solution to Problem \eqref{1b} with initial data $(v_0,d_0)$ iff
\begin{enumerate}[(1)]
\item $(v,d)\in C([0,T_0];\mathrm{V}\times D(\hat{\mathrm{A}} ))\cap L^2(0,T_0; D(\mathrm{A} )\times D(\hat{\mathrm{A}} ^{3/2}))$,
\item for all $t \in [0,T_0]$ the integral equations
\begin{align}
v(t) =& v_0 +\int_0^t[-\mathrm{A} v(s) -B(v(s),v(s)) - \Pi( \Div[\nabla d(s)\odot \nabla d(s)] ) ] ds + \int_0^t f(s) ds,\label{eq:LocVelo}\\
d(t)=& d_0 +\int_0^t[-\hat{\mathrm{A}} d(s)+\lvert \nabla d(s)\rvert^2 d(s) -v(s)\cdot \nabla d(s) -\phi^\prime(d(s)) +(\phi^\prime(d(s) )\cdot d(s) ) d(s) ] ds \nonumber \\
&\qquad \qquad + \int_0^t (d(s)\times g(s)) ds,\label{eq:LocDir}
\end{align}
hold in $\mathrm{H} $ and $D(\hat{\mathrm{A}} ^{1/2})$, respectively.
\item For all $t\in [0,T_0]$ $d(t)\in \mathcal{M}$.
\item \label{Item4-DefRegularSol} $(\partial_tv, \partial_td)\in L^2(0,T_0; \mathrm{H} \times D(\hat{\mathrm{A}} ^{1/2}))$.
\end{enumerate}
Throughout this paper we will denote by $((v,d);T_0)$ a local regular solution defined on $[0,T_0]$.
\end{Def}
We also need the definition of a maximal local regular solution.
\begin{Def}\label{Def:Maximal-Sol}
A pair $((v,d);T_0)$ with $T_0\in (0,T)$ and $(v,d):[0,T_0) \to \mathrm{V} \times D(\hat{\mathrm{A}} )$ is called a maximal local regular solution to \eqref{1b} with initial data $(v(0), d(0))=(v_0,d_0)$ if
\begin{enumerate}
\item $((v,d);T_0)$ defined on $[0,T_0)$ is a local regular solution to \eqref{1b},
\item for any other local regular solution $((\tilde{v}, \tilde{d}); \tilde{T}_0)$ we have
\begin{equation}
\tilde{T}_0 \le T_0 \text{ and } (v,d)_{\lvert_{[0,\tilde{T}_0)}}= (\tilde{v},\tilde{d}).\notag
\end{equation}
\end{enumerate}
\end{Def}
We state the following important remark.
\begin{Rem}
Let
\begin{align}
F(v,d):=& -B(v, v) -\Pi(\Div [\nabla d \odot \nabla d ]),\label{Eq:NonlinVelo} \\
\tilde{G}(v,d)=& \lvert \nabla d \rvert^2 d -v\cdot \nabla d -\phi^\prime(d) + (\phi^\prime(d) \cdot d) d, \\
G(v,d):=&\lvert \nabla d \rvert^2 d -v\cdot \nabla d -\phi^\prime(d) + (\phi^\prime(d) \cdot d) d + d\times g.\label{Eq:NonlinDir}
\end{align}
Then, the condition \eqref{Item4-DefRegularSol} of Definition \ref{Def:Local-Regular-Sol} is equivalent to the following
\begin{equation}
(F(v,d), G(v,d)) \in L^2(0,T_0; \mathrm{H} \times D(\hat{\mathrm{A}} ^{1/2}) ).\notag
\end{equation}
We will see in Lemma \ref{Lem:RighthandSideinL2VxH1} that if $(v,d)\in C([0,T_0];\mathrm{V}\times D(\hat{\mathrm{A}} ))\cap L^2(0,T_0; D(A)\times D(\hat{\mathrm{A}} ^{3/2}))$, then $(F(v,d), G(v,d)) \in L^2(0,T_0; H\times D(\hat{\mathrm{A}} ^{1/2})) $.
\end{Rem}
With the definitions and remark in mind we are now ready to formulate our first result.
\begin{Thm}\label{th}
Let $R_1>0$, $R_2>0$ and $g\in L^2(0,T; D(\hat{\mathrm{A}} ^{1/2}))$. Then, there exist $T_1(g)$ and $T_2(R_1,R_2)>0$ such that the following holds.
\noindent If $(v_0, d_0)\in \mathrm{V}\times (D(\hat{\mathrm{A}} )\cap \mathcal{M})$ and $(f,g)\in L^2(0,T; L^2\times D(\hat{\mathrm{A}} ^{1/2}))$ are such that
\begin{equation} \label{Eq:DatainBoundedSet}
\left[\lvert v_0\rvert^2_{V} + \lvert d_0\rvert^2_{D(\hat{\mathrm{A}} )}\right]^\frac12 \le R_1 \text{ and }
\left[\int_0^T \left( \lvert f(s) \rvert^2_{L^2} + \lvert g(s) \rvert^2_{D(\hat{\mathrm{A}} ^{1/2})} \right)\;ds \right]^\frac12 \le R_2,
\end{equation}
then the problem \eqref{1b} has a local regular solution $((v,d);T_0)$ with $T_0=T_1(g)\wedge T_2(R_1,R_2)$
Moreover, if $((u,n);T_0)$ is another local regular solution, then
$$(u(t),n(t))=(v(t),d(t)) \text{ for all } t\in [0,T_0] . $$
\end{Thm}
In order to prove this theorem we shall introduce the following spaces
\begin{align}
\mathbf{X}^1_T=& C([0,T]; \mathrm{V} ) \cap L^2(0,T; D(\mathrm{A} ) ),\notag\\
\mathbf{X}^2_T=& C([0,T]; D(\hat{\mathrm{A}} ) ) \cap L^2(0,T; D(\hat{\mathrm{A}} ^\frac32) ),\textbf{}\notag\\
\mathbf{Y}^1_T= & L^2(0,T; \mathrm{H} ) \notag\\
\mathbf{Y}^2_T=& L^2(0,T; D(\hat{\mathrm{A}} ^\frac12))\notag
\end{align}
We also set
\begin{align}
\mathbf{X}_T=& \mathbf{X}^1_{T} \times \mathbf{X}^2_{T},\notag\\
\mathbf{Y}_T=&\mathbf{Y}^1_T \times \mathbf{Y}^2_T.\notag
\end{align}
Let $(v,n)\in \mathbf{X}_T$ and consider the following decoupled linear problem
\begin{equation}\notag
\begin{pmatrix}
\partial_t {u}\\
\partial_t {d}
\end{pmatrix}
+ \begin{pmatrix}
\mathrm{A} u \\
\hat{\mathrm{A}} d
\end{pmatrix}
=
\begin{pmatrix}
F(v,n)+f\\
G(v,n)+d\times g
\end{pmatrix}.
\end{equation}
Before continuing further, let us recall the following result.
\begin{Lem}\label{Lem:ExistenceStokes+Heat}
If $(u_0,d_0)\in \mathrm{V} \times D(\hat{\mathrm{A}} )$, $(\mathfrak{f}, \mathfrak{g})\in L^2(0,T; \mathrm{H} \times D(\hat{\mathrm{A}} ^\frac12) )$ and $T>0$, then the problem
\begin{equation}\label{Eq:SystemStokesheat}
\begin{pmatrix}
\partial_t {u}\\
\partial_t{d}
\end{pmatrix}
+ \begin{pmatrix}
\mathrm{A} u \\
\hat{\mathrm{A}} d
\end{pmatrix}
=
\begin{pmatrix}
\mathfrak{f}\\
\mathfrak{g}
\end{pmatrix}
\end{equation}
has a unique strong solution $(u,d)\in \mathbf{X}_T$. Moreover, there exists a constant $C>0$, independent of $T$, such that
\begin{equation}\notag
\lvert (u,d) \rvert^2_{\mathbf{X}_T} \le C \lvert (u_0,d_0) \rvert^2_{\mathrm{V}\times D(\hat{\mathrm{A}} )} + C \lvert (\mathfrak{f}, \mathfrak{g}) \rvert^2_{L^2(0,T; \mathrm{H} \times D(\hat{\mathrm{A}} ^\frac12))}.
\end{equation}
\end{Lem}
Now we state the following lemma whose proof will be given in the appendix.
\begin{Lem}\label{Lem:RighthandSideinL2VxH1}
There exists a constant $C_0>0$, independent of $T$, such that for all $v_i\in \mathbf{X}^1_T, d_i\in \mathbf{X}^2_T$, $i=1,2$, the following inequalities hold
\begin{align}
\lvert F(v_1,n_1)-F(v_2, n_2) \rvert^2_{L^2(0,T; \mathrm{H} )} \le & C_0 T^\frac12 \lvert (v_1,n_1) - (v_2,n_2)\rvert^2_{\mathbf{X}_T} [\lvert (v_1,n_1) \rvert^2_{\mathbf{X}_T}+ \lvert (v_2,n_2)\rvert^2_{\mathbf{X}_T}]\label{Eq:FPT-ForcingVelo} \\
\lvert \tilde{G}(v_1, n_1) - \tilde{G}(v_2,n_2) \rvert^2_{L^2(0,T; D(\hat{\mathrm{A}} ^\frac12))}\le & C_0 (T\vee T^\frac12) \lvert (v_1,n_1) - (v_2,n_2)\rvert^2_{\mathbf{X}_T} \Bigl[ 1+\sum_{ i=1}^{2} \lvert (v_i, n_i) \rvert^{6} _{\mathbf{X}_T} \Bigr]. \label{Eq:FPT-ForcingOptDir-1}\\
\lvert n_1\times g - n_2 \times g \rvert^2_{L^2(0,T; \mathrm{H} ^1)}\le & C_0 \lvert n_1-n_2 \rvert^2_{\mathbf{X}_T^2} \lvert g \rvert^2_{L^2(0,T; \mathrm{H} ^1)}. \label{Eq:FPT-ForcingOptDir-2}
\end{align}
\end{Lem}
Now, we will give the proof of Theorem \ref{th}.
\begin{proof}[Proof of Theorem \ref{th}]
Let $\Psi:\mathbf{X}_T \to \mathbf{X}_T$ be the map defined as follows.
If $(v,n)\in \mathbf{X}_T $, then $\Psi(v,n)=(u,d)$ iff $(u,d)$ is the unique regular solution to \eqref{Eq:SystemStokesheat} with right hand side of the form
\begin{equation}\label{Eq:nonlinear-linearized}
\begin{pmatrix}
\mathfrak{f}\\
\mathfrak{g}
\end{pmatrix}
=\begin{pmatrix}
F(v,n) + f \\
\tilde{G}(v,n) +n\times g
\end{pmatrix}.
\end{equation}
Let us observe that by Lemma \ref{Lem:RighthandSideinL2VxH1}
the term $(\mathfrak{f}, \mathfrak{g}) $ defined in \eqref{Eq:nonlinear-linearized} belongs to $L^2(0,T; \mathrm{H} \times D(\hat{\mathrm{A}} ^\frac12))$. Hence, by Lemma \ref{Lem:ExistenceStokes+Heat} $(u,d)\in \mathbf{X}_T$, and so the map $\Psi$ is well-defined.
Now, let $R_1>0$, $R_2>0$, $(f,g)\in L^2(0,T; H\times D(\hat{\mathrm{A}} ^{1/2}))$ and $(v_0, d_0)\in \mathrm{V}\times (D(\hat{\mathrm{A}} )\cap \mathcal{M})$.
Let
\begin{equation}
\mathbf{K}_{R_1,R_2}= \Bigl\{ (v,n)\in \mathbf{X}_T: \lvert (v, n) \rvert^2_{\mathbf{X}_T}\le R_1^2 + R_2^2 \Bigr\}.
\end{equation}
Let $(v_i,n_i)\in \mathbf{K}_{R_1,R_2}$ and $(u_i,d_i)=\Psi(v_i,n_i)$, $i=1,2$. Put $(u,d)=(u_1-u_2, d_1-d_2)$. Then, it is easy to check that $(u,d)$ solves the following problem
\begin{equation}\label{Eq:SystemStokesheat-1}
\begin{pmatrix}
\partial_t {u}\\
\partial_t{d}
\end{pmatrix}
+ \begin{pmatrix}
\mathrm{A} u \\
\hat{\mathrm{A}} d
\end{pmatrix}
=
\begin{pmatrix}
F(v_1,n_1)-F(v_2,n_2)\\
\tilde{G}(v_1,n_1)-\tilde{G}(v_1,n_2)+ (n_1-n_2)\times g
\end{pmatrix}
.
\end{equation}
Hence, by Lemma \ref{Lem:ExistenceStokes+Heat} there exists a constant $C>0$, independent of $T$, such that
\begin{equation}\notag
\begin{split}
\lvert (u,d)\rvert^2_{\mathbf{X}_T}\le & C \bigl(\lvert F(v_1,n_1)-F(v_2,n_2)\rvert^2_{L^2(0,T;\mathrm{H} )} + \lvert \tilde{G}(v_1,n_1)-\tilde{G}(v_2,n_2)\rvert^2_{L^2(0,T;\mathrm{H} ^1)} \\& + \lvert (n_1-n_2)\times g\rvert^2_{L^2(0,T;\mathrm{H} ^1)} \bigr).
\end{split}
\end{equation}
Then, by plugging \eqref{Eq:FPT-ForcingVelo}, \eqref{Eq:FPT-ForcingOptDir-1} and \eqref{Eq:FPT-ForcingOptDir-2} in the above inequality and performing elementary calculations imply that there exists a constant $C_1>0$, independent of $T$, such that for all $(v_i,n_i)\in \mathbf{K}_{R_1,R_2}$, $i=1,2$,
\begin{equation}\notag
\begin{split}
\lvert \Psi(v_1,n_1)-\Psi(v_2,n_2) \rvert^2_{\mathbf{X}_T}\le C_1 \lvert (v_1,n_1)-(v_2,n_2) \rvert^2_{\mathbf{X}_T}\Biggl( \Bigl[1+R_1^{6} +R_2^{6} \Bigr](T\vee T^\frac12) + \int_0^T \lvert g(s)\rvert^2_{\mathrm{H} ^1} ds \Biggr).
\end{split}
\end{equation}
Since $g\in L^2(0,T; \mathrm{H} ^1) $, for any $\varepsilon\,>0$ there exists $T_1\in (0,T)$ such that
\begin{equation}\notag
C_1 \int_0^{T_1} \lvert g(s)\rvert^2_{\mathrm{H} ^1}\le \varepsilon\,.
\end{equation}
Next, we choose a number $T_2$ such that
\begin{equation}\notag
C_1 \Bigl[1+R_1^{6} +R_2^{6} \Bigr] (T_2 \vee T_2^\frac12) \le \frac14.
\end{equation}
Hence, by choosing $\varepsilon\,=\frac14$ and setting $T_0=T_1\wedge T_2$ we infer that
for all $(v_i,n_i)\in \mathbf{K}_{R_1,R_2}$, $i=1,2$,
\begin{equation}\notag
\begin{split}
\lvert \Psi(v_1,n_1)-\Psi(v_2,n_2) \rvert^2_{\mathbf{X}_T}\le \frac12.
\end{split}
\end{equation}
Hence, $\Psi$ has a unique fixed point $(u,d)\in \mathbf{X}_{T_0}$ satisfying
\begin{equation}\label{Eq:SystemStokesheat-A}
\begin{pmatrix}
\partial_t{u}\\
\partial_t {d}
\end{pmatrix}
+ \begin{pmatrix}
\mathrm{A} u \\
\hat{\mathrm{A}} d
\end{pmatrix}
=
\begin{pmatrix}
F(u,d) + f \\
G(u,d) +d\times g
\end{pmatrix}.
\end{equation}
Thus, in order to prove Theorem \ref{th} it remains to prove that
\begin{equation}\label{Eq:tu5}
d(t) \in \mathcal{M} \text{ for all } t\in [0,T_0].
\end{equation}
For this purpose, let
$$z(t) =\lvert d(t) \rvert^2-1,\; t\in [0,T_0].$$
We recall that there exists a constant $C>0$ such that for all $n\in \mathrm{H} ^3$
\begin{equation}\label{Eq:InterpolationofH52}
\lvert n \rvert_{\mathrm{H}^{\frac52}}\le C \lvert n \vert^\frac12_{\mathrm{H}^2} \lvert n \rvert^\frac12_{\mathrm{H} ^3}.
\end{equation} Hence, since $d\in \mathbf{X}^2_{T_0}:=C([0,T_0]; D(\hat{\mathrm{A}} )) \cap L^2(0,T; D(\hat{\mathrm{A}} ^\frac32))$, $D(\hat{\mathrm{A}} ^{\theta})\subset \mathrm{H}^{2\theta}$ and $\mathrm{H} ^{2\theta}, \theta>\frac12$ is an algebra, by using the interpolation inequality \eqref{Eq:InterpolationofH52} we easily show that \begin{equation}\label{Eq:Regularity-of-z-1} z\in C([0,T_0]; \mathrm{H}^2) \cap L^2(0,T_0; \mathrm{H}^{\frac52}). \end{equation} Also, since $(u,d)\in X_{T_0}$ we infer from Lemma \ref{Lem:RighthandSideinL2VxH1} that \begin{equation}\label{Eq:TimeDrivative-of-d} \partial_t d= -\hat{\mathrm{A}} d + \tilde{G}(u,d)+ d\times g \in L^2(0,T_0;\mathrm{H} ^1). \end{equation} Using this and $d\in \mathbf{X}^2_{T_0}$ we easily prove that \begin{equation}\label{Eq:Regularity-of-z-2} \partial_tz \in L^2(0,T_0; \mathrm{H} ^1). \end{equation} Now we will claim that $z$ satisfies the weak form of the following problem \begin{equation}\label{Eq:ViscousTransport-z} \begin{cases}
\partial_{t}z-\Delta z +u\cdot \nabla z=2|\nabla d|^{2}z-2(\phi^\prime(d).d)z,\\ {\frac{\partial z}{\partial \nu}}{\Big\lvert_{\partial\Omega}}=0,\\ z(0)=0. \end{cases} \end{equation} Towards this end let $\varphi\in H^{1}(\Omega;\mathbb{R})$ and fix $t\in [0,T_0]$.
Since $d\in C([0,T_0]; D(\hat{\mathrm{A}} )) \cap L^2(0,T; D(\hat{\mathrm{A}} ^\frac32))$ and $D(\hat{\mathrm{A}} )\subset L^\infty$ we easily prove that $\varphi d \in C([0,T_0]; \mathrm{H} ^1)\subset L^2(0,T_0; \mathrm{H} ^1)$.
Also, since $(u,d)\in X_{T_0}$ we infer from Lemma \ref{Lem:RighthandSideinL2VxH1} that
$$\partial_t d= -\hat{\mathrm{A}} d + \tilde{G}(u,d)+ d\times g \in L^2(0,T_0;\mathrm{H} ^1).$$
Hence, in view of the Lions-Magenes lemma, see \cite[Lemma III.1.2]{Temam_2001}, we have
\begin{align}
\frac{1}{2}\frac{d}{dt}\int_{\Omega}\varphi(x)|d(t,x))|^{2} \;dx= & \langle \partial_t d(t), \varphi d(t)\rangle \nonumber \\
=& -\int_{\Omega}\varphi(x)\mathrm{A} d(t,x)\cdot d(t,x) \;dx-\int_{\Omega}\varphi(x) [u(t,x)\cdot \nabla d(t,x)]\cdot d(t,x) \;dx \nonumber \\& + \int_{\Omega}\varphi(x) |\nabla d(t,x)|^{2}|d(t,x)|^{2}\;dx -\int_{\Omega}\varphi(|d(t,x)|^{2}-1)(\phi^\prime(d(t,x))\cdot d(t,x))\;dx, \label{Eq:Weakformz-1}
\end{align}
where we used the fact that $\varphi d \perp_{\mathbb{R}^3} d\times g$.
Since $d(t)\in D(\hat{\mathrm{A}} )$ and $\varphi d(t)\in \mathrm{H} ^1$ for all $t \in [0,T_0]$, by using \cite[Equation (2.6)]{ZB+BG+TJ} we infer that
\[
-\int_\Omega \hat{\mathrm{A}} d(t,x)\cdot \varphi(x) d(t,x) \;dx= - \int_\Omega (\nabla d(t,x) ) (\nabla[\varphi(x)d(t,x)]) \;dx.
\]
Thus, straightforward calculation yields
\[
-\int_\Omega \hat{\mathrm{A}} d(t,x)\cdot \varphi(x) d(t,x) \;dx= - \int_\Omega \varphi(x) \lvert \nabla d(t,x) \rvert^2 \;dx- \frac12\int_\Omega \varphi(x) \nabla \lvert d(t,x) \rvert^2 \;dx.
\]
Hence, recalling the definition of $z$ and using the last identity in \eqref{Eq:Weakformz-1} implies
\begin{equation}\label{Eq:ProofSphere-1}
\begin{split}
\frac{1}{2}\int_{\Omega}\partial_{t}z(t,x) \varphi(x) \;dx+\frac{1}{2}\int_{\Omega}\nabla z(t,x)\nabla\varphi(x) \;dx+\frac{1}{2}\int_{\Omega} u(t,x)\nabla z(t,x) \varphi(x)\; \;dx\\
=\int_{\Omega}|\nabla d(t,x)|^{2}z(t,x) \varphi(x) \;dx-\int_{\Omega} z(t,x)(\phi^\prime(d(t,x))\cdot d(t,x))\varphi(x)\;\;dx.
\end{split}
\end{equation}
This is exactly the weak form of \eqref{Eq:ViscousTransport-z}.
By Proposition \ref{tu4} $z$ is the unique solution of \eqref{Eq:ViscousTransport-z} and satisfies
\begin{equation}\notag
\sup_{0\le t\le T}|z(t)|^{2}_{L^{2}}+\int_{0}^{T}|\nabla z(t)|^{2}dt\le
|z(0)|^{2}_{L^{2}}e^{c\int_{0}^{T}\left[|\nabla d|^{4}_{L^{4}}+(1+|d|^{2}_{H^{2}})\right]dt}.
\end{equation}
Since $z(0)=0$, we infer that
\begin{equation}
\sup_{0\le t\le T}|z(t)|^{2}_{L^{2}}=\sup_{0\le t \le T}\int_{\Omega}(|d|^{2}-1)^{2}\;dx=0,\notag
\end{equation}
which implies that $d(t) \in \mathcal{M}$ for all $t\in [0,T_0]$.
This completes the proof of \eqref{Eq:tu5}. This also completes the proof of Theorem \ref{th}.
\end{proof}
\section{The existence and the uniqueness of a maximal local regular solution to \eqref{1b} } \label{Sec:LocRegSolSmallENERG}\
The aim of this section is to prove that Problem \eqref{1b} has a unique maximal local regular solution when the initial data has small energy. The main result of the section is Proposition \ref{Prop:LocalSolwithSmallEnergy} and it is a generalization of
\cite[Lemma 5.2]{LLW}. Before proceeding to a precise statement and a detailed proof of the result let us introduce few notations. For $R>0$ and $(u,n)\in \mathrm{H} \times \mathrm{H}^1$ we set
\begin{equation}\label{eqn-energy-loc}
\mathcal{E}_{R}(u,n):= \frac12\sup_{x\in \Omega} \int_{B(x, 2R)} \left(\lvert u(y)\rvert^2 + \lvert \nabla n(y) \rvert^2 + 2 \phi(n(y)) \right) dy,
\end{equation}
and
\begin{equation}\label{eqn-energy-glob}
\mathcal{E}(u,n):=\frac12 (\lvert u\rvert^2_{\mathrm{H}} + \lvert \nabla n\rvert^2_{L^2} ) + \int_\Omega \phi(n(x)) \;dx.
\end{equation}
We also recall the following important lemma, see \cite[Lemma 3.1\& 3.2]{Struwe}.
\begin{Lem}\label{Lem:Struwe}
There exist $c_1>0$ and $r_0>0$ such that for every $h\in L^\infty(0,T; L^2)\cap L^2(0,T; \mathrm{H} ^1)$ we have
\begin{equation}\label{Eq:Ladyzhenskaya-Struwe}
\int_0^T \lvert h(t) \rvert^4_{L^4}dt \le c_1 \left(\sup_{(t,x) \in [0,T]\times \Omega } \int_{B(x,r_0)} \lvert h(t,y) \rvert^2 dy \right) \left(\int_0^T \lvert \nabla h(t)\rvert^2_{L^2} dt + \frac{1}{r_0^2} \int_0^T \lvert h(t)\rvert^2_{L^2} dt \right).
\end{equation}
\end{Lem}
\begin{Rem}
Let $r_0>0$ be as in Lemma \ref{Lem:Struwe}.
In view of \ref{Eq:Ladyzhenskaya-Struwe} and \cite[Theorem 3.4]{Simader}, we infer that there exists $c_2>0$ such that for all $h\in L^\infty(0,T;\mathrm{H} ^1)\cap L^2(0,T; D(\hat{\mathrm{A}} ))$ we have
\begin{equation}\label{Eq:Ladyzhenskaya-Struwe-2}
\int_0^T \lvert \nabla h(t) \rvert^4_{L^4} \le c_2 \left(\sup_{(t,x) \in [0,T]\times \Omega } \int_{B(x,r_0)} \lvert \nabla h(t,y) \rvert^2 dy \right) \left(\int_0^T \lvert \Delta h(t)\rvert^2_{L^2} dt + \frac{1}{r_0^2} \int_0^T \lvert \nabla h(t)\rvert^2_{L^2} dt \right).
\end{equation}
\end{Rem}
We state and prove the following important result.
\begin{Prop}\label{Prop:LocalSolwithSmallEnergy}
There exist a constant $\varepsilon_0>0$ and a function
\[\theta_0: (0,\varepsilon\,_0)\times (0,\infty) \to (0,\infty),,
\]
which is non-increasing w.r.t. the second variable and nondecreasing w.r.t. the first one,
such that the following holds: \\
Let $r_0>0$ be as in Lemma \ref{Lem:Struwe}.
Let $(f,g)\in L^2(0,T; \mathrm{H}\times D(\hat{\mathrm{A}} ^{1/2}))$, $(v_0, d_0)\in\mathrm{V}\times D(\hat{\mathrm{A}} ) $ and $R_0\in (0, r_0]$ are such that
\begin{equation}\label{Eq:AssumSmallInitialNRJ}
\mathcal{E}_{2R_0}(v_0,d_0)< \varepsilon\,_0^2.
\end{equation}
Then, there exists a unique maximal local regular solution $((v,d);T_0)$ to problem \eqref{1b} satisfying
\begin{align}\label{eqn-t_o}
T_0 \ge \frac{R^2_0}{(R_0^\frac12 + 1)^4} \theta_0(\varepsilon\,_1,E_0),\\
\sup_{0\le t\le T_0 } \mathcal{E}_{R_0}(v(t),d(t)) \le 2\varepsilon\,_1^2,\label{eqn-small estimates}
\end{align}
where $E_0:= \mathcal{E}(v_0,d_0)$ and $\varepsilon\,_1^2=\mathcal{E}_{2R_0}(v_0,d_0)$.
\end{Prop}
\begin{Rem}
In this theorem, the length $T_0$ is not the length of the existence interval but the length of the existence interval as long as the condition \eqref{eqn-small estimates} is satisfied.
Note also that \eqref{eqn-small estimates} is equivalent to
\begin{equation*}\label{eqn-small estimates-B}
\frac12 \sup_{0\le t\le T_0 } \sup_{x\in \Omega}\int_{B(x,R_0)} \left(\lvert v(t,y)\rvert^2 + \lvert \nabla d(t,y) \rvert^2 + 2 \phi(d(t,y)) \right) dy\le 2\varepsilon\,_1^2.
\end{equation*}
\end{Rem}
In order to prove the above proposition we need several results. For $n\in \mathbb{R}^3$
\begin{equation*}
\alpha(n)= \phi^\prime(n)\cdot n.
\end{equation*}
We state and prove the following elementary results.
\begin{Clm}\label{Claim:perp}
Let $u\in \mathrm{H} $, $n\in D(\hat{\mathrm{A}} )\cap \mathcal{M}$ and $m\in C([0,T_\ast); (D(\hat{\mathrm{A}} )\cap \mathcal{M}))$ such that $\partial_t m \in L^2(0,T_\ast; L^2)$. Then,
\begin{align}
&\langle u\cdot \nabla n, \lvert \nabla n\rvert^2 n -\phi^\prime(n) +\alpha(n)n\rangle=0,\notag\\
& \langle \partial_t m, \lvert \nabla m \rvert^2 m -\alpha(m) m \rangle=0.\notag
\end{align}
\end{Clm}
\begin{proof}
Let us fix $u\in \mathrm{V}$, $n\in D(\hat{\mathrm{A}} )$ and $m: [0,T_\ast)\to D(\hat{\mathrm{A}} )\cap \mathcal{M}$ satisfying the assumptions of Claim \ref{Claim:perp}. Then, since $\Div u=0$ and the fact $n(x) \in \mathbb{S}^2$ $x$-a.e. we get
\begin{equation*}
\begin{split}
\langle u\cdot \nabla n, \lvert \nabla n\rvert^2 n -\phi^\prime(n) +\alpha(n)n\rangle = \frac12 \int_\Omega u(x)\cdot \nabla \lvert n(x) \rvert^2_{\mathbb{R}^3}( \lvert \nabla n(x) \rvert^2 + \alpha(n(x)) ) \;dx \\- \int_\Omega u(x)\cdot \nabla \phi(n(x)) \;dx =0.
\end{split}
\end{equation*}
Since $ m\in C([0,T_\ast); D(\hat{\mathrm{A}} )\cap \mathcal{M}) $ we infer that for all $t\in [0,T_\ast)$
\begin{align*}
\langle \partial_t m(t), \lvert \nabla m(t) \rvert^2 m(t) -\alpha(m(t)) m(t) \rangle= \frac12 \int_\Omega \partial_t \lvert m (t,x)\rvert^2_{\mathbb{R}^3} ( \lvert \nabla m(t,x) \rvert^2 - \alpha(m(t,x)) ) \;dx =0,
\end{align*}
which completes the proof of the claim.
\end{proof}
We also recall the following result, see \cite[Eq. (2.10)]{ZB+EH+PR-SPDE_2019}.
\begin{Clm}\label{Claim:Dissip}
For any $u\in \mathrm{H} \cap L^4$ and $n \in D(\hat{\mathrm{A}} )$
\begin{equation}\label{Eq:Dissip}
-\langle\Div (\nabla n \odot \nabla n ), u\rangle + \langle u\cdot \nabla n, \Delta n \rangle =0.
\end{equation}
\end{Clm}
By Theorem \ref{th}, Problem \eqref{1b} has a unique maximal local regular solution $(v,d) \in X_{T_\ast} $ provided that $(v_0,d_0)\in \mathrm{V}\times D(\hat{\mathrm{A}} )$ and $(f,g)\in L^2(0,T; L^2 \times D(\hat{\mathrm{A}} ^{1/2}))$.
Hereafter, we fix such a maximal local regular solution and we set,
\begin{equation}\notag
\mathcal{E}(v(t),d(t)) = \frac12\left(\lvert v(t) \rvert^2_{L^2} + \lvert \nabla d(t) \rvert^2_{L^2} \right) + \int_\Omega \phi(d(t,x)) \;dx,\text{ for $t \in [0,T_\ast]$,}.
\end{equation}
We then prove the following important a' priori estimates.
\begin{Lem}
Let $(v_0,d_0)\in \mathrm{V} \times D(\hat{\mathrm{A}} )$, $(f,g)\in L^2(0,T;L^2\times \mathrm{H} ^1)$ and $(v,d) \in X_{T_\ast} $ be a maximal local regular solution to the problem \eqref{1b}.
Then for all $s,t\in [0,T_\ast] $ with $s\le t$, the following inequality holds
\begin{equation}\label{Eq:NRJ-Inequality}
\mathcal{E}(v(t), d(t) ) +\frac12 \int_s^t\left( \lvert \nabla v(r) \rvert^2_{L^2} + \lvert R(d(r) ) \rvert^2_{L^2} \right) dr \le \mathcal{E}(v(s), d(s)) + \frac12 \int_s^t \left(\lvert f(r) \rvert^2_{\mathrm{H} ^{-1}}+ \lvert g(r) \rvert^2_{L^2} \right) dr,
\end{equation}
where, for $ n \in D(\hat{\mathrm{A}} )$, we put
\begin{align*}
&\alpha(n) = \phi^\prime(n)\cdot n,\\
&R(n)= \Delta n + \lvert \nabla n \rvert^2 n -\phi^\prime(n) + \alpha(n).
\end{align*}
\end{Lem}
\begin{proof}
Since the maximal smooth solution $(v,d)$ satisfies part (1) of Definition \ref{Def:Local-Regular-Sol}, it is not difficult to show that $( v(t),R(d(t)))\in L^2(0,T_\ast; D(\mathrm{A} ) \times D(\hat{\mathrm{A}} ^\frac12))$. We also have $(\partial_t v,\partial_t d) \in L^2(0,T_\ast; \mathrm{H} \times D(\hat{\mathrm{A}} ^\frac12))$ because $(v,d)$ satisfies. Hence, by applying the Lions-Magenes Lemma (\cite[Lemma III.1.2]{Temam_2001}) and the Claims \ref{Claim:perp} and \ref{Claim:Dissip} we infer that for all $t\in [0,T_\ast)$
\begin{align*}
-\langle \partial_t d(t), R(d(t)) \rangle - \langle \partial_t v(t), \Delta v\rangle =& \frac{d}{dt} \mathcal{E}(v(t),d(t)) + \lvert \nabla v(t) \rvert^2_{L^2} + \lvert R(d(t) ) \rvert^2_{L^2} \\
= & \langle f(t), v(t) \rangle + \langle d(t)\times g(t) , R(d(t) ) \rangle.
\end{align*}
From the Cauchy-Schwarz and the Young inequalities and, the fact $d(t)\in \mathcal{M}, t\in [0, T_\ast)$, which is part (4) of Definition \ref{Def:Local-Regular-Sol}, we infer that there exists a constant $C>0$ such that
\begin{align*}
\frac{d}{dt} \mathcal{E}(v(t),d(t) ) + \lvert \nabla v(t) \rvert^2_{L^2} + \lvert R(d(t) ) \rvert^2_{L^2} \le C \vert f(t) \rvert_{\mathrm{H} ^{-1}} \lvert \nabla v(t) \rvert_{L^2} + C \vert d(t)\times g(t) \rvert_{L^2}
\lvert R(d(t) ) \rvert_{L^2}\\
\le \frac12 \left(\lvert \nabla v(t) \rvert^2_{L^2} + \lvert R(d(t) ) \rvert^2_{L^2}\right) + \frac12 \left( \lvert f(t) \rvert^2_{\mathrm{H} ^{-1}}+ \lvert g(t) \rvert^2_{L^2}\right).
\end{align*}
Absorbing the first term on the right hand side of the last inequality into the left hand side and integrating over $[s,t] \subset [0,T_\ast)$ completes the proof of the lemma.
\end{proof}
For any $\varepsilon\,>0$ and $R>0$ we define the time
\begin{equation}\label{Eq:DefStoppingTime}
T(\varepsilon\,, R)= \inf\left\{t\in [0,T_\ast): \mathcal{E}_{R}(v(t),d(t)) > 2 \varepsilon\,^2 \right \}\wedge T_\ast.
\end{equation}
\begin{Rem}
Let $\varepsilon\,>0$ and $R>0$. Then, for any $t\in [0,T(\varepsilon\,, R)]$
\begin{equation}\label{Eq:BelowStoppingTime}
\mathcal{E}_{R}(v(t),d(t)) \le 2 \varepsilon\,^2.
\end{equation}
\end{Rem} We state and prove the following lemma.
\begin{Lem}\label{Lem:EstimateofDeltaD+L4norms}
Let $(v_0,d_0)\in \mathrm{V} \times D(\hat{\mathrm{A}} )$ and $(f,g)\in L^2(0,T;L^2\times \mathrm{H} ^1)$.
There exist $\varepsilon\,_0>0$ and $K_0>0$ such that if $(v,d) \in X_{T_\ast} $ is a maximal local regular solution to the problem \eqref{1b}, then for all $\varepsilon\,\in (0, \varepsilon\,_0)$, $R\in (0,r_0]$, where $r_0>0$ is the constant from Lemma \ref{Lem:Struwe}, and for all $t\in [0,T(\varepsilon\,, R)]$
\begin{align}
\int_0^{t} \lvert \Delta d(r) \rvert^2_{L^2} dr \le K_0\left[ E_0 + \frac12 \int_0^{t}\left(\lvert f (s) \rvert^2_{\mathrm{H} ^{-1}}+ \lvert g(s) \rvert^2_{L^2}\right) ds + \left(1+\frac{2\varepsilon\,^2}{R^2}\right)t \right],\label{Eq:EstDeltad}\\
\int_0^{t } \left(\lvert v(s) \rvert^4_{L^4} + \lvert \nabla d (s) \rvert^4_{L^4} \right) ds \le K_0\varepsilon\,^2\left[ E_0 + \frac12 \int_0^{t}\left(\lvert f (s) \rvert^2_{\mathrm{H} ^{-1}}+ \lvert g(s) \rvert^2_{L^2}\right) ds + \left(1+\frac{2\varepsilon\,^2}{R^2}\right)t \right].\label{Eq:EstL4normsofU+NablaD}
\end{align}
\end{Lem}
\begin{proof}
Let $r_0>0$ be the constant from Lemma \ref{Lem:Struwe}, $R\in [0,r_0]$ and $\varepsilon\,\in (0,\varepsilon\,_0)$ where $\varepsilon\,_0$ is number to be chosen later.
We set
$$E_0= \mathcal{E}(v_0,d_0).$$
Since $\phi$ is twice continuously differentiable and the 2-sphere $\mathbb{S}^2$ is compact, we can and will assume throughout that for some constant $M>0$
\begin{equation}
2 \lvert \phi^\prime(n) - \alpha(n) n \rvert^2 \le M, \; n \in \mathbb{S}^2.\notag
\end{equation}
From this observation we infer that for all $t\in [0,T(\varepsilon\,, R)]$
\begin{align*}
\int_0^{t} \lvert \Delta d \rvert^2_{L^2} ds \le 2 \int_0^{t} \lvert \Delta d - \phi^\prime(d) + \alpha(d) d \rvert^2_{L^2} ds + M t\\
\le 4 \int_0^t \lvert R(d) \rvert^2_{L^2} ds + 4 \int_0^{t} \lvert \nabla d \rvert^4_{L^4} ds + M t.
\end{align*}
The last line of the above inequalities, \eqref{Eq:Ladyzhenskaya-Struwe-2}, \eqref{Eq:NRJ-Inequality} and \eqref{Eq:BelowStoppingTime} imply that
\begin{align}
\int_0^{t} \lvert \Delta d \rvert^2_{L^2} ds\le & 4 c_2 \left(\sup_{(s,x) \in [0,t]\times \Omega } \int_{B(x,R)} \lvert \nabla d(s,y) \rvert^2 dy \right) \left(\int_0^t \lvert \Delta d(t)\rvert^2_{L^2} dt + \frac{1}{R^2} \int_0^t \lvert \nabla d(s)\rvert^2_{L^2} ds \right)\nonumber \\
& + 4 \int_0^t \lvert R(d) \rvert^2_{L^2} ds \nonumber \\
\le & 4 \left[E_0 + \frac12 \int_0^{t}\left(\lvert f (s) \rvert^2_{\mathrm{H} ^{-1}}+ \lvert g(s) \rvert^2_{L^2}\right) ds \right] + 8 c_2 \varepsilon\,^2 \left(\int_0^{t} \lvert \Delta d \rvert^2_{L^2}ds + \frac{2 \varepsilon\,^2 }{R^2} t \right)+M t.\notag
\end{align}
Now choosing $\varepsilon\,_0>0$ so that $1-8c_2\varepsilon\,_0^2\ge \frac12$, we infer that $1-8c_2\varepsilon\,^2>\frac12$ and for all $t\in [0,T(\varepsilon\,, R)]$
\begin{equation}
\begin{split}
\int_0^{t} \lvert \Delta d \rvert^2_{L^2} ds \le 8 \left[E_0 + \frac12 \int_0^{t}\left(\lvert f (s) \rvert^2_{\mathrm{H} ^{-1}}+ \lvert g(s) \rvert^2_{L^2}\right) ds \right] \\
+ 16 c_2 \left(\frac{2 \varepsilon\,^2 }{R^2} +M\right)t,\notag
\end{split}
\end{equation}
which completes the proof of \eqref{Eq:EstDeltad}.
We now proceed to the proof of \eqref{Eq:EstL4normsofU+NablaD}. For this we observe that by Lemma \ref{Lem:Struwe} and \eqref{Eq:EstDeltad} we infer that for all $t\in [0,T(\varepsilon\,, R)]$
\begin{align}
\int_0^{t} \lvert \nabla d \rvert^4_{L^4} ds \le 4 c_2\varepsilon\,^2 \left( \int_0^{t} \lvert \Delta d \rvert^2_{L^2} ds + \frac{1}{R^2} \int_0^{t} \lvert \nabla d \rvert^2_{L^2} ds \right)\nonumber \\
\le 4 c_2\varepsilon\,^2 \left(K_0\left[ E_0 + \frac12 \int_0^{t}\left(\lvert f (s) \rvert^2_{\mathrm{H} ^{-1}}+ \lvert g(s) \rvert^2_{L^2}\right) ds + \left(1+\frac{2\varepsilon\,^2}{R^2}\right)t \right] + \frac{2\varepsilon\,^2}{R^2} t \right).\label{Est:L4NablaD}
\end{align}
In a similar way, we prove that for all $t\in [0,T(\varepsilon\,, R)]$
\begin{align}
\int_0^{t} \lvert v \rvert^4_{L^4} ds \le 4 c_2\varepsilon\,^2 \left( \int_0^{t} \lvert \nabla v \rvert^2_{L^2} ds + \frac{1}{r_0^2} \int_0^{t} \lvert v \rvert^2_{L^2} ds \right)\nonumber \\
\le 4 c_2\varepsilon\,^2 \left(K_0\left[ E_0 + \frac12 \int_0^{t}\left(\lvert f (s) \rvert^2_{\mathrm{H} ^{-1}}+ \lvert g(s) \rvert^2_{L^2}\right) ds \right] + \frac{2\varepsilon\,^2}{R^2} t \right),\label{Eq:EstL4u}
\end{align}
which altogether with \eqref{Est:L4NablaD} imply \eqref{Eq:EstL4normsofU+NablaD}.
\end{proof}
We will need the following estimates which will be proved in the Appendix \ref{App:EstHighOrder}.
\begin{Clm}\label{Clm:EstimateHigherorder}
There exists a constant $K_1>0$ such that for all $v\in D(A)$ and $n \in D(\hat{\mathrm{A}} ^{3/2})$ we have
\begin{align}
& \lvert \langle A v, -\Pi[\Div(\nabla n \odot \nabla n) ] \rangle \lvert \le \frac1{12}\left(\lvert Av \rvert^2_{L^2}+ \lvert \nabla \Delta n \rvert^2_{L^2}\right) + K_1 \lvert \nabla n \rvert^4_{L^4} \lvert \Delta n \rvert^2_{L^2},\label{Eq:EstHighOrd-1}\\ & \lvert {} _{(\mathrm{H} ^1)^\ast}\langle \hat{\mathrm{A}} ^2 n, (v\cdot \nabla n )\rangle_{\mathrm{H} ^1} \lvert \le \frac1{12} \left(\lvert \nabla \Delta n\rvert^2_{L^2} +\lvert \mathrm{A} v\rvert^2_{L^2}\right)+ K_1 \lvert \nabla v\rvert^2_{L^2 } \rvert(\nabla n) \rvert^2_{L^4}+K_1[\lvert v \rvert^2_{L^4}+\lvert v\rvert^4_{L^4} ]\lvert \Delta n\rvert_{L^2},\label{Eq:EstHighOrd-2}\\ & \lvert {}_{(\mathrm{H} ^1)^\ast}\langle \hat{\mathrm{A}} ^2 n, (\lvert \nabla n \rvert^2 n ) \rangle_{\mathrm{H} ^1} \lvert \le \frac1{12} \lvert \nabla \Delta n \rvert^2_{L^2} + K_1 \Big( [\lvert \nabla n \rvert^4_{L^4} +\lvert \nabla n\rvert^2_{L^4} ]\lvert \Delta n \rvert^2_{L^2}+ \lvert \nabla n \rvert^4_{L^4}(\lvert \nabla n\rvert^2_{L^2} + \lvert \Delta n \rvert^2_{L^2})\Big).\label{Eq:EstHighOrd-3}
\end{align}
\end{Clm}
We will also need the following results.
\begin{Clm}\label{Clm:EstimateDeltaAnisotropy}
There exists a constant $K_2>0$ such that for all $n \in D(\hat{\mathrm{A}} ^{3/2})$ we have
\begin{align}
\lvert {}_{ (\mathrm{H} ^1)^\ast }\langle \hat{\mathrm{A}} ^2 n, \alpha(n)n-\phi^\prime(n) \rangle_{\mathrm{H} ^1} \lvert \le \frac1{12} \lvert \nabla \Delta n \rvert^2_{L^2}+ K_2 \lvert \nabla n \rvert^2_{L^2}.\notag
\end{align}
\end{Clm}
\begin{proof}
Using the Cauchy-Schwarz inequality and the fact that there exists a constant $M>0$ such that
\begin{equation*}
\lvert \phi^\prime(n) \rvert + \lvert \phi^{\prime\prime}(n) \rvert \le M, n \in \mathbb{S}^2,
\end{equation*}
we infer that there exists a constant $C>0$ such that
\begin{align*}
\lvert {}_{(\mathrm{H} ^1)^\ast}\langle \hat{\mathrm{A}} ^2 n, \alpha(n)n-\phi^\prime(n) \rangle_{\mathrm{H} ^1} \lvert\le \lvert \nabla \Delta n \rvert_{L^2}\left( \lvert \nabla (\alpha(n)n) \rvert_{L^2}+ \lvert \phi^{\prime\prime}(n)\nabla n \rvert_{L^2} \right)\\
\le C \lvert \nabla \Delta n \rvert_{L^2}\left( \lvert \lvert \nabla n\rvert \; \rvert\phi^\prime(n)\rvert\; \lvert n\rvert\; \rvert_{L^2}+\lvert \lvert \nabla n\rvert \; \lvert \phi^{\prime\prime}(n) \rvert\; \lvert n \rvert^2 \rvert_{L^2} + \lvert \phi^{\prime\prime}(n)\nabla n \rvert_{L^2} \right)\\
\le C \lvert \nabla \Delta n \rvert_{L^2}\lvert \nabla n \rvert_{L^2}.
\end{align*}
We now complete the proof using the Young inequality in the last line.
\end{proof}
Hereafter, we put for all $s,t \in [0,T]$ with $s\le T$ \begin{align}
\Psi(s,t)=&\frac12 \int_s^t \left(\lvert f \rvert^2_{\mathrm{H} ^{-1}}+ \lvert g \rvert^2_{L^2} \right)dr \text{ and } \Psi(t)=\Psi(0,t), \label{Eq:DefPsi}\\
\Xi(s,t)=&\frac12 \int_s^t \left(\lvert f \rvert^2_{L^2} + \lvert g \rvert^2_{\mathrm{H} ^1} \right)dr \text{ and } \Xi(t)=\Xi(0,t) \;\; \forall s\le t\in [0,T]. \label{Eq:DefXi} \end{align} \begin{Lem}\label{Lem:EstinHigherorderNorms}
Let $(v_0,d_0)\in \mathrm{V} \times D(\hat{\mathrm{A}} )$, $(f,g)\in L^2(0,T;L^2\times \mathrm{H} ^1)$ and $(v,d) \in X_{T_\ast} $ be a maximal local regular solution to the problem \eqref{1b}.
Let us put
\begin{equation}\label{Eq:DefSigma}
\Sigma_0 =[E_0 + \lvert \Delta d_0\rvert^2_{L^2} + \lvert \nabla v_0\rvert^2_{L^2}+ \Psi(T)+ \Xi(T)+ (E_0+1)\Xi(T)]<\infty.
\end{equation}
Then, there exist constants $K_3>0$ and $K_4>0$ such that for all $\tau\in [0,T_\ast]$ we have
\begin{equation}\label{Eq:EstinHigherorderNorms}
\begin{split}
\sup_{0\le s\le \tau} \left(\lvert \nabla v(s) \rvert^2_{L^2} + \lvert \Delta d(s) \rvert^2_{L^2} \right) + 2 \int_0^{\tau} \left( \lvert \nabla \Delta d(s) \rvert^2_{L^2} + \lvert A v(s) \rvert^2_{L^2}\right)ds \\
\le K_3\Sigma_0 e^{K_4\Xi(T)} e^{K_4 \int_0^{\tau}[\lvert \nabla d(r) \rvert^4_{L^4}+ \lvert v(r) \rvert^4_{L^4} +\lvert \nabla d(r) \rvert^2_{L^4}+ \lvert v(r)\rvert^2_{L^4} ]dr }.
\end{split}
\end{equation} \end{Lem} \begin{proof}
Throughout this proof $C>0$ will denote an universal constant which may change from one term to the other. Let $(v_0,d_0)\in \mathrm{H} \times D(\hat{\mathrm{A}} )$ and $((v,d); T_\ast) $ be a local regular solution to the problem \eqref{1b}.
By part (1) and (4) of Definition \ref{Def:Local-Regular-Sol} we have $(v,d)\in L^2(0,T_\ast; D(\mathrm{A} ) \times D(\hat{\mathrm{A}} ^\frac32))$ and $(\partial_t v, \partial d) \in L^2(0,T_\ast; \mathrm{H} \times D(\hat{\mathrm{A}} ^\frac12)) \subset L^2(0,T_\ast; \mathrm{V}^\ast \times D(\hat{\mathrm{A}} ))$. Hence, by the Lions-Magenes Lemma (\cite[Lemma III.1.2]{Temam_2001}) we infer that
\begin{align*}
\frac12 \frac{d}{dt}\left( \lvert \Delta d(t) \rvert^2_{L^2} + \lvert \nabla v(t) \rvert^2_{L^2} \right)
&= \langle \partial_t \Delta d(t), \Delta d(t) \rangle+ \langle \partial_t \nabla v(t), \nabla v(t)\rangle\\
= & {}_{\mathrm{H} ^{1}}\langle \partial_t d(t), \hat{\mathrm{A}} ^2 d(t)\rangle_{(\mathrm{H} ^{1})^\ast} + (\partial_tv(t) , A v(t) ).
\end{align*}
Hence,
\begin{align}\label{Eq:Derivativehigheroerdernorms}
\frac12 \frac{d}{dt}\left( \lvert \Delta d(t) \rvert^2_{L^2} + \lvert \nabla v(t) \rvert^2_{L^2} \right) = & -\lvert \nabla \Delta d(t) \rvert^2_{L^2} - \lvert \mathrm{A} v(t) \rvert^2_{L^2} + \langle f(t), Av(t) \rangle-\langle \nabla (g(t)\times d(t)), \nabla \Delta d(t)\rangle \nonumber \\
&+ {}_{(\mathrm{H} ^{1})^\ast}\langle \hat{\mathrm{A}} ^2 d(t),\lvert \nabla d(t) \rvert^2 d(t)+\alpha(d(t))d(t)
-\phi^\prime(d(t))-v(t)\cdot \nabla d(t) \rangle_{\mathrm{H} ^{1}}\nonumber \\
&- \langle \Div(\nabla d(t) \odot \nabla d(t))+ v(t)\cdot \nabla v(t), A v(t)\rangle.
\end{align}
By using the Cauchy-Schwarz, the Young inequalities and the Ladyzhenskaya inequality (\cite[Lemma III.3.3]{Temam_2001})
we obtain
\begin{align*}
\langle v\cdot \nabla v, \mathrm{A} v \rangle \le C \lvert Av \rvert_{L^2} \lvert v \rvert_{L^4} \lvert \nabla v \rvert_{L^4} \\
\le C \lvert \mathrm{A} v \rvert^\frac32 \lvert v \rvert_{L^4} \lvert \nabla v \rvert^\frac12_{L^2}\\
\le \frac14 \lvert \mathrm{A} v \rvert^2+ C \lvert v \rvert^4_{L^4} \lvert \nabla v \rvert^2_{L^2}.
\end{align*}
\noindent Using the H\"older and Young inequalities, ,and the Sobolev embedding $\mathrm{H} ^1 \hookrightarrow L^4$ we also have
\begin{align*}
{}_{(\mathrm{H} ^{1})^\ast}\langle \hat{\mathrm{A}} ^2 d, g\times d \rangle_{\mathrm{H} ^{1}} = & ( \nabla \Delta d , \nabla (g\times d) )\\
\le & C \lvert \nabla \Delta d \rvert_{L^2} \left(\lvert \nabla g \rvert_{L^2}+ \lvert g \rvert_{L^4} \lvert \nabla d \rvert_{L^4} \right)\\
\le & C \lvert \nabla \Delta d \rvert_{L^2} \left(\lvert \nabla g \rvert_{L^2}+ \lvert g \rvert_{L^4} \lvert \nabla d \rvert_{L^2}^\frac12 \lvert \Delta d \rvert_{L^2}^\frac12 \right)\\
\le & \frac{1}{2} \lvert \nabla \Delta d \rvert^2_{L^2} + \frac12 \lvert g \rvert^2_{\mathrm{H} ^1} + C \lvert g\rvert^2_{\mathrm{H} ^1}\left(\lvert \nabla d \rvert^2_{L^2}+\lvert \Delta d \rvert^2_{L^2}\right).
\end{align*}
Plugging these estimates and the ones in Claims \ref{Clm:EstimateHigherorder}-\ref{Clm:EstimateDeltaAnisotropy} into \eqref{Eq:Derivativehigheroerdernorms} yield
\begin{equation}\label{Eq:Derivativehigheroerdernorms-Fin}
\begin{split}
\frac12 \frac{d}{dt}\left( \lvert \Delta d\rvert^2_{L^2} + \lvert \nabla v \rvert^2_{L^2} \right) + \frac12 \left(\lvert \mathrm{A} v \rvert^2_{L^2} + \lvert \nabla \Delta d \rvert^2_{L^2}\right) - C \lvert v \rvert^4_{L^4} \lvert \nabla v \rvert^2_{L^2} \\
\le C \left(1+
\lvert \nabla d \rvert^4_{L^4} + \lvert g \rvert^2_{\mathrm{H} ^1} \right) \lvert \Delta d \rvert^2_{L^2}
+ \frac12 \left(\lvert f \rvert^2_{L^2}+ \lvert g \rvert^2_{\mathrm{H} ^1}+ C \lvert g \rvert^2_{\mathrm{H} ^1} \lvert \nabla d \rvert^2_{L^2} \right)\\
+ C \lvert \nabla d \rvert^2 \lvert \nabla d \rvert^4_{L^4}.
\end{split}
\end{equation}
Hence,
\begin{equation}\label{Eq:Derivativehigheroerdernorms-Fin-A}
\begin{split}
\frac12 \frac{d}{dt}\left( \lvert \Delta d(t) \rvert^2_{L^2} + \lvert \nabla v(t) \rvert^2_{L^2} \right) \le C \left(1+
\lvert \nabla d(t)\rvert^4_{L^4} + \lvert g(t)\rvert^2_{\mathrm{H} ^1} + \lvert v(t)\rvert^4_{L^4} \right) \left( \lvert \Delta d(t) \rvert^2_{L^2} + \lvert \nabla v(t) \rvert^2_{L^2} \right) \\ + \frac12 \left(\lvert f(t) \rvert^2_{L^2}+ \lvert g(t) \rvert^2_{\mathrm{H} ^1}+ C \lvert g(t) \rvert^2_{\mathrm{H} ^1} \lvert \nabla d(t)\rvert^2_{L^2} \right)+ C \lvert \nabla d(t) \rvert^2 \lvert \nabla d(t)\rvert^4_{L^4}.
\end{split}
\end{equation}
Let us put
\begin{equation*}
\Theta(t):=e^{2 C \left( \int_0^t[
\lvert \nabla d(r)\rvert^4_{L^4} + \lvert \nabla d(r)\rvert^4_{L^4} + \lvert g(r) \rvert^2_{\mathrm{H} ^1} + \lvert v(r) \rvert^2_{L^4} + \lvert v(r) \rvert^4_{L^4}]\,dr \right) }ds,\; t\in [0,T_\ast).
\end{equation*}
Thus, by the Gronwall Lemma, we obtain
\begin{equation}\label{Eq:Derivativehigheroerdernorms-Fin-B}
\begin{split}
& \lvert \Delta d (t) \rvert^2_{L^2} + \lvert \nabla u (t) \rvert^2_{L^2} - \left( \lvert \Delta d_0 \rvert^2_{L^2} + \lvert \nabla v_0 \rvert^2_{L^2} \right) \Theta(t) \\
&\le \left(\int_0^t \frac12 \lvert f(s) \rvert^2_{L^2} ds+ \left(C \sup_{s\in [0,\tau]} \lvert \nabla d (s) \rvert^2_{L^2} +\frac12\right) \int_0^t [\lvert g (s) \rvert^2_{\mathrm{H} ^1} +\lvert \nabla d(s) \rvert^4_{L^4}] ds \right) \Theta(t),
\end{split}
\end{equation}
which along with the inequality \eqref{Eq:NRJ-Inequality} the fact $\theta \le e^\theta,\; \theta\ge 0$ implies that
\begin{equation}\label{Eq:EstinHigherorderNorms-0}
\begin{split}
\sup_{0\le s\le \tau} \left(\lvert \nabla v(s) \rvert^2_{L^2} + \lvert \Delta d(s) \rvert^2_{L^2} \right)
\le C \Sigma_0 e^{C \Xi(T)} e^{C \int_0^{\tau}[\lvert \nabla d(r) \rvert^4_{L^4}+ \lvert v(r) \rvert^4_{L^4}+\lvert \nabla d(r) \rvert^2_{L^4}+ \lvert v(r) \rvert^2_{L^4} ]dr }.
\end{split}
\end{equation}
Integrating \eqref{Eq:Derivativehigheroerdernorms-Fin}, using \eqref{Eq:EstinHigherorderNorms-0} and the fact $\theta \le e^\theta,\; \theta\ge 0$ yield
\begin{align}
\int_0^{\tau} \left(\lvert \mathrm{A} v(s) \rvert^2_{L^2} + \lvert \nabla \Delta d(r) \rvert^2 \right) ds\le& \sup_{s\in [0,\tau]} \lvert \Delta d(s) \rvert^2_{L^2} C \int_0^{\tau}
\lvert \nabla d(s)\rvert^4_{L^4} + \lvert g(s) \rvert^2_{\mathrm{H} ^1} \,ds \nonumber \\
& + \int_0^\tau \left(\lvert f(s) \rvert^2_{L^2}+ \lvert g(s) \rvert^2_{\mathrm{H} ^1}\right) ds +\lvert \nabla v_0\rvert^2_{L^2} + \lvert \Delta d_0\rvert^2_{L^2} \nonumber \\
& + C \sup_{s\in [0,\tau)} \lvert \nabla d(s) \rvert^2_{L^2} \int_0^\tau \lvert g(s)\rvert^2_{\mathrm{H} ^1} ds \nonumber \\
\le & C \Sigma_0 e^{C \Xi(T)} e^{C \int_0^{\tau}[\lvert \nabla d(r) \rvert^4_{L^4}+ \lvert v(r) \rvert^4_{L^4} +\lvert \nabla d(r) \rvert^2_{L^4}+ \lvert v(r) \rvert^2_{L^4}]dr } \label{Eq:EstinHigherorderNorms-1}
\end{align}
We easily infer from \eqref{Eq:EstinHigherorderNorms-0} and \eqref{Eq:EstinHigherorderNorms-1} that \eqref{Eq:EstinHigherorderNorms} holds. This completes the proof of the lemma. \end{proof} We have the following consequence of the above lemma. \begin{Cc}\label{Cor:EstinHigherorderNorms-CC}
Let $(v_0,d_0)\in \mathrm{V} \times D(\hat{\mathrm{A}} )$, $(f,g)\in L^2(0,T;L^2\times \mathrm{H} ^1)$, $r_0$ and $\varepsilon\,_0$ be as in Lemma \ref{Lem:Struwe} and Lemma \ref{Lem:EstimateofDeltaD+L4norms}.
Let $(v,d) \in X_{T_\ast} $ be a maximal local regular solution to the problem \eqref{1b}.
Let $\Sigma_0$ be defined as in \eqref{Eq:DefSigma}.
Then, there exists a constant $K_3>0$ such that for any $R\in (0, r_0]$, $\varepsilon\,\in (0, \varepsilon\,_0)$ and $t\in [0,T(\varepsilon\,,R)]$,
\begin{equation}\label{Eq:EstinHigherorderNorms-CC}
\begin{split}
\left(\lvert \nabla v(t) \rvert^2_{L^2} + \lvert \Delta d(t) \rvert^2_{L^2} \right) + 2 \int_0^{t } \left( \lvert \nabla \Delta d \rvert^2_{L^2} + \lvert \mathrm{A} v \rvert^2_{L^2}\right)ds
\le K_3 \Sigma_0 e^{K_3 [\Xi(T)+\varepsilon\,^2 (E_0 + \Psi(T)+(1+\frac{2\varepsilon\,^2}{R^2}) t) ]}.
\end{split}
\end{equation} \end{Cc} \begin{proof}
Let $(v,d) \in X_{T_\ast} $ be a maximal local regular solution to the problem \eqref{1b} with initial data $(v_0,d_0)\in \mathrm{V}\times D(\hat{\mathrm{A}} )$.
Let us fix $R\in (0,r_0]$ and $\varepsilon\,\in (0,\varepsilon\,_0)$. Since $(v,d)\in X_{T(\varepsilon\,, R)}$ then we can apply Lemma \ref{Lem:EstinHigherorderNorms} and infer that \eqref{Eq:EstinHigherorderNorms} holds for $\tau=T(\varepsilon\,, R)$. Hence, in order to complete the proof of the corollary we need to estimate the exponential term in the right-hand side of \eqref{Eq:EstinHigherorderNorms}. For this purpose, we use \eqref{Eq:EstL4normsofU+NablaD} and infer that there exists a universal constant $K>0$ such that for any $R\in (0,r_0]$, $\varepsilon\,(0,\varepsilon\,_0)$ and $t\in [0,T(\varepsilon\,,R)]$
\begin{equation}\label{Eq:EstExpo}
e^{ K_4\int_0^t\left(
\lvert \nabla d(s)\rvert^4_{L^4} + \lvert v(s) \rvert^4_{L^4} +\lvert \nabla d(r) \rvert^2_{L^4}+ \lvert v(r) \rvert^2_{L^4} \right)ds }\le K e^{K[\Xi(T)+\varepsilon\,^2 (E_0 + \Psi(T)+(1+\frac{2\varepsilon\,^2}{R^2}) t) ] }.
\end{equation}
This completes the proof of the corollary.
\end{proof}
\begin{Cc}\label{Cor:SmallNRJBeforemaxTime}
Let $(v_0,d_0)\in \mathrm{V} \times D(\hat{\mathrm{A}} )$, $(f,g)\in L^2(0,T;L^2\times \mathrm{H} ^1)$ and $(v,d) \in X_{T_\ast} $ be a maximal local regular solution to the problem \eqref{1b}.
Let $r_0>0$ and $\varepsilon\,_0>0$ be as in Lemma \ref{Lem:Struwe} and Lemma \ref{Lem:EstimateofDeltaD+L4norms}, respectively. Then, for all $\varepsilon\,\in (0,\varepsilon\,_0)$ and $R\in (0,r_0]$ we have
$$ T(\varepsilon\,, R) < T_\ast .$$ \end{Cc} \begin{proof}
We argue by contradiction. Assume that there exists $\varepsilon\,\in (0,\varepsilon\,_0)$ and $R\in (0,r_0]$ such that $T(\varepsilon\,, R) = T_\ast$. Let us put
$$ R_2=\int_0^{T_\ast} (\lvert f(r) \rvert^2_{L^2}+\lvert g(r) \rvert^2_{\mathrm{H} ^1}) dr.$$ By Corollary \ref{Cor:EstinHigherorderNorms-CC} and \eqref{Eq:NRJ-Inequality} we infer that there exists a constant $\tilde{K}_3>0$ such that for all $t\in [0,T_\ast)$
\begin{equation*}
\lvert (v(t), d(t) \rvert^2_{\mathrm{V}\times D(\hat{\mathrm{A}} ) }\le \tilde{K}_3.
\end{equation*}
Let $T_0= T_1(\tilde{K}_3, R_2) \wedge T_2(g)\wedge T_\ast >0$ be the time given by Theorem \ref{th}. Let $T_1= \frac{T_0}{2}$. By Theorem \ref{th} the problem \eqref{1b} with initial data $(v(T_1), d(T_1))$ has a unique local regular solution $(\tilde{v}(t), \tilde{d}(t) )$ defined on $[T_\ast- T_1, T_\ast-T_1 + T_0 ]$.
We then define $(\bar{v}, \bar{d}):[0,T_\ast+ T_1]\to \mathrm{V}\times D(\hat{\mathrm{A}} )$ by
\begin{equation*}
(\bar{v}(t),\bar{d}(t))=
\begin{cases}
(v(t), d(t) ) \text{ if } t\in [0,T_\ast-T_1]\\
(\tilde{v}(t), \tilde{d}(t) ) \text{ if } t\in [T_\ast-T_1, T_\ast + T_1].
\end{cases}
\end{equation*}
It is easily seen that $((\bar{v}, \bar{d});T_1)$ is a local regular solution to \eqref{1b} with initial data $(v_0, d_0)$ and time of existence $T_\ast+T_1>T_\ast$.
This contradicts the fact that $((v,d);T_\ast)$, with $T_\ast=T(\varepsilon\,, R)$, is a maximal smooth solution to \eqref{1b}. This completes the proof of the corollary. \end{proof}
We now state and prove a local energy inequality which will play an important role in the proof of Proposition \ref{Prop:LocalSolwithSmallEnergy}.
\begin{Lem}\label{Lem:EstPressure}
Let $(v_0,d_0)\in \mathrm{V} \times D(\hat{\mathrm{A}} )$, $(f,g)\in L^2(0,T;L^2\times \mathrm{H} ^1)$, $r_0$ and $\varepsilon\,_0$ be as in Lemma \ref{Lem:EstimateofDeltaD+L4norms}. Also, let $R\in (0, r_0]$, $\varepsilon\,\in (0, \varepsilon\,_0)$ and $(v,d) \in X_{T_\ast} $ be a local regular solution to the problem \eqref{1b}.
Then, there exists a function $\mathrm{p}:[0,T_\ast )\to L^1$ such that $\mathrm{p}\in L^\frac43(0,T_\ast; L^4)$, $\nabla \mathrm{p} \in L^\frac43(0,T_\ast ; L^\frac43 ) $ and
\begin{equation}\label{Eq:pressure}
v(t) + \int_0^t [v(s) \cdot \nabla v(s) + \nabla \mathrm{p}(s)]\;ds=v_0 + \int_0^t [\Delta v(s) - \nabla d(s)\Delta d(s) +f(s)]\; ds, t\in [0,T_\ast].
\end{equation}
Moreover, there exists a constant $K_5>0$, which may depend on the norms of $(v_0,d_0)\in \mathrm{V} \times D(\hat{\mathrm{A}} )$ and $(f,g)\in L^2(0,T;L^2\times \mathrm{H} ^1)$, such that for any $t\in [0,T_\ast)$
\begin{equation}
\lvert \nabla \mathrm{p} \rvert_{L^\frac43(0,t; L^\frac43 )} \le K_5 \varepsilon\,^\frac12_1 \left[E_0 + \Psi(t) + (1+ \frac{2\varepsilon\,^2}{R^2}) t\right]^\frac{3}{4}.\notag
\end{equation}
\end{Lem}
\begin{proof}
Let us fix $\varepsilon\,\in (0,\varepsilon\,_0)$ and $R\in (0,r_0]$. We fix $t\in [0,T_\ast)$. From \eqref{Eq:NRJ-Inequality} and \eqref{Eq:EstDeltad} we have $(v,d)\in C([0,t]; \mathrm{H}\times \mathrm{H} ^1)\cap L^2(0,t; \mathrm{V}\times D(\hat{\mathrm{A}} ))$. Hence, one can apply \cite[Lemma 4.4]{LLW} and infer that there exists function $\mathrm{p}:[0,T_\ast)\to L^1$ such that $\mathrm{p}\in L^\frac43(0,T_\ast; L^4)$, $\nabla \mathrm{p} \in L^\frac43(0,T_\ast ; L^\frac43 ) $ and the identity \eqref{Eq:pressure} holds. Moreover,
\begin{align}\label{Eq:EstPressure-0}
\lvert \nabla \mathrm{p} \rvert_{L^\frac43(0,t; L^\frac43 )} \le \lvert f\rvert_{L^\frac43(0,t; L^\frac43 )} + \lvert v\cdot \nabla v\rvert_{L^\frac43(0,t; L^\frac43 )} + \lvert \nabla d\Delta d \rvert_{L^\frac43(0,t; L^\frac43 )}.
\end{align}
From the H\"older inequality, \eqref{Eq:NRJ-Inequality} and \eqref{Eq:EstL4normsofU+NablaD} we infer that
\begin{align}
\lvert v\cdot \nabla v\rvert_{L^\frac43(0,t; L^\frac43 )} \le \left(\int_0^{t} \lvert v(r) \rvert^4_{L^4} dr \right)^\frac14 \left(\int_0^{t}\lvert \nabla v(r) \rvert^2_{L^2} dr \right)^\frac12\nonumber \\
\le C \varepsilon\,^\frac12 \left[E_0 + \Psi(t) + (1+ \frac{2\varepsilon\,^2}{R^2}) t\right]^\frac{3}{4}.\label{Eq:EstPressure-1}
\end{align}
In a similar way,
\begin{align}
\lvert \nabla d \Delta d \rvert_{L^\frac43(0,t; L^\frac43) } \le \left(\int_0^{t} \lvert \nabla d(r) \rvert^4_{L^4} dr \right)^\frac14 \left(\int_0^{t}\lvert \Delta d(r) \rvert^2_{L^2} dr \right)^\frac12\nonumber \\
\le C \varepsilon\,^\frac12 \left[E_0 + \Psi(t) + (1+ \frac{2\varepsilon\,^2}{R^2}) t\right]^\frac{3}{4}.\label{Eq:EstPressure-2}
\end{align}
Plugging \eqref{Eq:EstPressure-1} and \eqref{Eq:EstPressure-2} into \eqref{Eq:EstPressure-0} completes the proof of Lemma \ref{Lem:EstPressure}.
\end{proof} We now continue with some estimates of local energy. Hereafter, in order to save space we simply write $ \int_\mathcal{O} \Xi\, d\mu $ instead of $ \int_\mathcal{O} \Xi(y)\, d\mu(y) $ for an integrable function $\Xi$ defined on a measure space $(\mathcal{O}, \mathcal{A}, \mu)$.
\begin{Lem}
Let $(v_0,d_0)\in \mathrm{V} \times D(\hat{\mathrm{A}} )$, $(f,g)\in L^2(0,T;L^2\times \mathrm{H} ^1)$ and $(v,d) \in X_{T_\ast} $ be a maximal local regular solution to the problem \eqref{1b}.
Let $\varphi \in C^\infty_c(\Omega; \mathbb{R})$ and put
\begin{equation}
\mathcal{E}_\varphi(v(t),d(t))=\frac12 \int_\Omega \varphi(x) \left( \lvert v(t,x) \rvert^2 + \lvert \nabla d(t,x) \rvert^2 + 2 \phi(d(t,x)) \right)\;dx, \;\; t\in [0,T_\ast) .\notag
\end{equation}
We also set
$$ \mathrm{p}_\Omega(t)= \frac1{\lvert \Omega\rvert} \int_\Omega \mathrm{p}(t,x) \;dx, \; t\in [0,T_\ast).$$
Then, for any $s,t\in [0,T_\ast)$ with $s\le t$ we have
\begin{equation}\label{Eq:LocalNRJ-DU}
\begin{split}
\mathcal{E}_\varphi(v(t), d(t)) - \mathcal{E}_\varphi(v(s), d(s)) +\int_s^t \int_\Omega \varphi \left(\lvert \nabla v \rvert^2 + \lvert R(d) \rvert^2 \right) \;dxdr \\
\le
\int_s^t \int_\Omega \lvert \nabla \varphi \rvert \Bigl( \frac12 \lvert \nabla d \rvert^2 \lvert v \rvert + \lvert \partial_t \rvert \lvert \nabla d \rvert+ \phi(d) \lvert v \rvert + \lvert v \rvert^3 \Bigr) \;dxdr\\
+ \int_s^t \int_\Omega \lvert \nabla \varphi\rvert \Bigl(\lvert \nabla v \rvert \lvert v \rvert + \lvert \mathrm{p}- \mathrm{p}_\Omega \rvert \lvert v \rvert + \lvert g \rvert \lvert \nabla d\rvert \Bigr) \;dxdr\\
+ \int_s^t \int_\Omega \lvert \varphi \rvert \Bigl(\lvert d \rvert \lvert \nabla g \rvert \lvert \nabla d \rvert + \lvert d \rvert \lvert g \rvert \lvert \phi^\prime (d) \rvert + \lvert f \rvert \lvert v \rvert \Bigr) \;dxdr.
\end{split}
\end{equation}
\end{Lem}
\begin{proof}
We fix $\varphi \in C_c^\infty(\Omega;\mathbb{R})$ and $s\le t\in [0,T_\ast)$. We firstly observe that because of the fact $ d(t)\in \mathcal{M}$ for all $t\in [0,T_\ast)$ we have
\begin{align}\label{Eq:pointwisePerp}
(\partial_t d + v\cdot \nabla d)\cdot(\lvert \nabla d \rvert^2 d + \alpha(d) d)=0,
\end{align}
for all $t\in [0,T_\ast)$ and a.e. $x\in \Omega$. Multiplying $\partial_t d + v\cdot \nabla d$ by $-\varphi R(d)$ in $L^2$ and using the pointwise orthogonality yields
\begin{equation}\label{Eq:LocalNRJD-0}
\begin{split}
A+B:= & -\int_\Omega \varphi (\partial_td + v\cdot \nabla d)\cdot \Delta d \;dx + \int_\Omega (\partial_t d + v\cdot \nabla d)\cdot \phi^\prime(d) \;dx\\
= & -\int_\Omega \varphi \lvert R(d) \rvert^2 \;dx - \int_\Omega \varphi(d\times g)\cdot R(d) \;dx .
\end{split}
\end{equation}
Using integration by parts and \cite[Eq. (4.16)]{LLW} we obtain
\begin{align}
A=& \frac12 \frac{d}{dt}\int_\Omega \lvert \nabla d \rvert^2 \varphi \;dx + \int_\Omega \partial_t d\cdot (\nabla d \nabla \varphi) \;dx -\int_\Omega \varphi (v\cdot \nabla d)\cdot \Delta d \;dx\nonumber \\
=& \frac12 \frac{d}{dt}\int_\Omega \lvert \nabla d \rvert^2 \varphi \;dx + \int_\Omega \partial_t d\cdot (\nabla d \nabla \varphi) \;dx - \int_\Omega \frac12 \lvert \nabla d\rvert^2 v\cdot \nabla \varphi \;dx + \int_\Omega \varphi (\nabla d\odot \nabla d)\nabla u \;dx \nonumber \\
&\qquad -\int_\Omega (v\cdot \nabla d)\cdot (\nabla d \nabla \varphi) \;dx.\label{Eq:LocalNRJD-1}
\end{align}
For the term $B$ it is easy to show that
\begin{align}
B= & \int_\Omega (\partial_t d \cdot \phi^\prime(d) )\varphi \;dx + \int_\Omega \varphi (v\cdot \nabla d) \cdot \phi^\prime (d) \;dx\nonumber \\
=& \frac{d}{dt} \int_\Omega \phi(d) \varphi \;dx + \int_\Omega v\cdot \nabla \phi(d) \varphi \;dx\nonumber \\
=& \frac{d}{dt} \int_\Omega \phi(d) \varphi \;dx -\int_\Omega v\cdot \nabla \varphi \phi(d) \;dx.\label{Eq:LocalNRJD-2}
\end{align}
Note also that for all $t\in [0,T_\ast)$ and a.e. $x\in \Omega$.
\begin{equation}
(d\times g)\cdot (\lvert \nabla d\rvert^2 d + \alpha(d) d)=0.\notag
\end{equation}
Hence, using integration by parts and the Cauchy-Schwarz inequality we obtain
\begin{align}
-\int_\Omega \varphi (d\times g) \cdot R(d) \;dx= & -\int_\Omega \varphi (d\times g)\cdot (\Delta d- \phi^\prime(d) ) \;dx \nonumber \\
\le& \int_\Omega \lvert \varphi \rvert \left( \lvert d \rvert \lvert \nabla d\rvert \nabla g \rvert \right) \;dx + \int_\Omega \lvert g \rvert \nabla d \rvert \lvert \nabla \varphi \rvert \;dx + \int_\Omega \lvert g \rvert \lvert d \rvert \lvert \phi^\prime(d)\rvert \lvert \varphi \rvert \;dx . \label{Eq:LocalNRJD-3}
\end{align}
Plugging \eqref{Eq:LocalNRJD-1}, \eqref{Eq:LocalNRJD-2} and \eqref{Eq:LocalNRJD-3} into \eqref{Eq:LocalNRJD-0} yields
\begin{equation}
\begin{split}
& \frac12 \frac{d}{dt}\int_\Omega \left( \lvert \nabla d \rvert^2 + 2 \phi(d)\right)\varphi \;dx +\int_\Omega \varphi \lvert R(d) \rvert^2 \;dx \\
& \le \int_\Omega \frac12 \lvert \nabla d\rvert^2 v\cdot \nabla \varphi \;dx+ \int_\Omega (v\cdot \nabla d)\cdot (\nabla d \nabla \varphi) \;dx-\int_\Omega \partial_t d\cdot (\nabla d \nabla \varphi) \;dx\\
& \qquad - \int_\Omega \varphi (\nabla d\odot \nabla d)\nabla u \;dx
+ \int_\Omega \lvert \varphi \rvert \left( \lvert d \rvert \lvert \nabla d\rvert \nabla g \rvert \right) \;dx + \int_\Omega \lvert g \rvert \nabla d \rvert \lvert \nabla \varphi \rvert \;dx \\
&\qquad + \int_\Omega \lvert g \rvert \lvert d \rvert \lvert \phi^\prime(d)\rvert \lvert \varphi \rvert \;dx .
\end{split}
\end{equation}
We can follow the same calculation in \cite{LLW} to derive the following local inequality for the velocity $v$
\begin{equation}\notag
\begin{split}
\frac12 \frac{d}{dt} \int_\Omega \lvert v \rvert^2 \varphi \;dx + \int_\Omega \le \frac14 \int_\Omega \lvert v\rvert^2 v\cdot \nabla \varphi \;dx -\int_\Omega (\nabla u) v\cdot \nabla \varphi \;dx +
\int_\Omega (\mathrm{p}-\mathrm{p}_\Omega) v\cdot \nabla \varphi \;dx \\
+ \int_\Omega \varphi (\nabla d \odot \nabla d) \nabla u \;dx + \int_\Omega ( \nabla d) \cdot \nabla d\nabla \varphi \;dx + \int_\Omega \lvert f\rvert \lvert v \rvert \lvert \varphi \rvert \;dx.
\end{split}
\end{equation}
Adding up the last inequalities side by side and using the Cauchy-Schwarz inequality and integrating over $[s,t]$ yield the sought estimate \eqref{Eq:LocalNRJ-DU}.
\end{proof}
The following lemma is also important for our analysis.
\begin{Lem}\label{Lem:LocalNRJ-Ball}
Let $(v_0,d_0)\in \mathrm{V} \times D(\hat{\mathrm{A}} )$, $(f,g)\in L^2(0,T;L^2\times \mathrm{H} ^1)$, $r_0$ and $\varepsilon\,_0$ be as in Lemma \ref{Lem:EstimateofDeltaD+L4norms}. Also, let $(v,d) \in X_{T_\ast} $ be a maximal local regular solution to the problem \eqref{1b}.
Then, there exists a constant $K_4>0$ such that for all $\varepsilon\,\in (0,\varepsilon\,_0)$, $R\in (0,r_0]$ and $t\in [0,T(\varepsilon\,,R)] $
\begin{equation}\label{Eq:EstNRJonBalls}
\begin{split}
\frac12 \int_{B(x,R)} \left( \lvert v(t) \rvert^2 + \lvert \nabla d (t) \rvert^2 + 2 \phi(d(t))\right) dy - \frac12 \int_{B(x,2R)} \left( \lvert v_0 \rvert^2 + \lvert \nabla d_0 \rvert^2 + 2\phi(d_0) \right) dy
\\
\le K_4 t^\frac14 \left(1+ E_0 + \frac12 \int_0^t [\lvert f \rvert^2_{\mathrm{H} ^{-1}}+ \lvert g \rvert^2_{L^2}] ds +(1+ \frac{2\varepsilon\,^2}{R^2}) t \right)^{\frac{5}{4}}\left( R^{-\frac12} (\varepsilon\,^\frac32 + \varepsilon\,^\frac12)+ \varepsilon\,^\frac12 \right) .
\end{split}
\end{equation}
\end{Lem}
\begin{proof}
Let us fix $\varepsilon\,\in (0,\varepsilon\,_0)$, $R\in (0,r_0]$, $x\in \Omega$ and $t\in [0,T(\varepsilon\,,R)]$. We also fix $\varphi \in C^\infty_c(\Omega; [0,1])$ such that
\begin{equation}\label{Eq:Condvarphi}
\mathds{1}_{B(x,R)} \le \varphi \le \mathds{1}_{B(x,2R)} \text{ and } \lvert \nabla \varphi \rvert\le \frac{c_4}{R},
\end{equation}
for some constant $c_4>0$.
For such particular $\varphi$ we will estimate each term on the right hand side of \eqref{Eq:LocalNRJ-DU}. To start this quest we observe that
\begin{equation*}
\left(\int_0^t \int_{B(x,2R)} \lvert \nabla \varphi \rvert \;dx ds\right)^\frac14 \le \frac{c_4}{R} \lvert B(x,2R) \rvert^\frac14 t^\frac14\le c_5 \left(\frac{t}{R^2}\right)^\frac14.
\end{equation*}
We will use this inequality below without further notice.
Using the H\"older inequality and \eqref{Eq:EstL4normsofU+NablaD} we get
\begin{align}
\int_0^t \int_\Omega \frac{\lvert \nabla d\rvert^2}{2} \lvert v\rvert \lvert \nabla \varphi\rvert \;dx dr \le \left( \int_0^t \lvert \nabla d \rvert^4_{L^4}\right)^\frac14 \left( \int_0^t \lvert v \rvert^4_{L^4}\right)^\frac14 \left( \int_0^t \lvert \nabla \varphi \rvert^4_{L^4}\right)^\frac14\nonumber \\
\le C \varepsilon\,^\frac32 \left(\frac{t}{R^2}\right)^\frac14 \left[E_0 + \Psi(t) + (1+ \frac{2\varepsilon\,^2}{R^2}) t\right]^\frac{3}{4}.\label{Eq:LocalNRj-Start}
\end{align}
In a similar way we get
\begin{equation}\notag
\int_0^t \int_\Omega \lvert \nabla \varphi \rvert \phi(d) \lvert v \rvert \;dx dr \le C \varepsilon\,^\frac12 \left(\frac{t}{R^2}\right)^\frac14 \left[E_0 + \Psi(t) + (1+ \frac{2\varepsilon\,^2}{R^2}) t\right]^\frac{3}{4}.
\end{equation}
Here we used the boundedness of $\phi$ on $\mathbb{S}^2$.
Similarly,
\begin{align*}
\int_0^t \int_\Omega \lvert \nabla \varphi \rvert \lvert v \rvert^3 \;dx dr \le C \varepsilon\,^\frac32 \left(\frac{t}{R^2}\right)^\frac14 \left[E_0 + \Psi(t) + (1+ \frac{2\varepsilon\,^2}{R^2}) t\right]^\frac{3}{4}\\
\int_0\int_\Omega \lvert \nabla \varphi \rvert \lvert g \rvert \lvert \nabla d \rvert \;dx dr \le C \varepsilon\,^\frac12 \left(\frac{t}{R^2}\right)^\frac14 \left[E_0 + \Psi(t) + (1+ \frac{2\varepsilon\,^2}{R^2}) t\right]^\frac{3}{4}.
\end{align*}
Using the H\"older inequality, \eqref{Eq:EstL4normsofU+NablaD} and \eqref{Eq:NRJ-Inequality}
\begin{align*}
\int_0^t \int_\Omega \lvert \nabla \varphi \rvert \lvert \nabla v \rvert \lvert v \rvert \;dx dr \le & c_5 \left(\frac{t}{R^2}\right)^\frac14 \left(\int_0^t \lvert v \rvert^4_{L^4} dr \right)^\frac14 \left(\int_0^t \lvert \nabla v \rvert^2_{L^2} dr \right)^\frac12\nonumber \\
\le & C \varepsilon\,^\frac12 \left(\frac{t}{R^2}\right)^\frac14 \left[E_0 + \Psi(t) + (1+ \frac{2\varepsilon\,^2}{R^2}) t\right]^\frac{3}{4}.
\end{align*}
To deal with the term containing $\partial_t d$ we argue as follows
\begin{align*}
\int_t\int_\Omega \lvert \partial_t d \rvert \lvert \nabla d\rvert \lvert \nabla \varphi \rvert \;dx dr \le c_5 \left(\frac{t}{R^2}\right)^\frac14 \left(\int_0^t \lvert \nabla d \rvert^4_{L^4} dr \right)^\frac14
\left(\int_0^t \lvert \partial_t d \rvert^2_{L^2} dr \right)^\frac12\\
\le c_5 \left(\frac{t}{R^2}\right)^\frac14 \left(\int_0^t \lvert \nabla d \rvert^4_{L^4} dr \right)^\frac14
\left(\int_0^t \left[\lvert R(d) \rvert^2_{L^2} + \lvert g \rvert^2_{L^2}\right] dr \right)^\frac12.
\end{align*}
Now, using \eqref{Eq:EstL4normsofU+NablaD} and \eqref{Eq:NRJ-Inequality} we obtain
\begin{align*}
\int_t\int_\Omega \lvert \partial_t d\rvert \lvert \nabla d\rvert \lvert \nabla \varphi \rvert \;dx dr \le
C \varepsilon\,^\frac12 \left(\frac{t}{R^2}\right)^\frac14 \left[E_0 + \Psi(t) + (1+ \frac{2\varepsilon\,^2}{R^2}) t\right]^\frac{3}{4}.
\end{align*}
We now deal with term containing the pressure $\mathrm{p}$. First by the
Using the H\"older and Poincar\'e inequalities and the estimates \eqref{Eq:NRJ-Inequality} we obtain
\begin{align*}
\int_0^t \int_\Omega \lvert \mathrm{p}- \mathrm{p}_\Omega \rvert \lvert v \rvert \lvert \nabla \varphi \rvert \;dx dr \le \sup_{0\le t < T_\ast} \lvert v(t) \rvert_{L^2} \left(\int_0^t \lvert \mathrm{p}-\mathrm{p}_\Omega \rvert_{L^4}^\frac4{2} dr \right)^\frac34 \left( \int_0^t \lvert \nabla \varphi \rvert^4_{L^4} \right)^\frac14 \\
\le C \left(\frac{t}{R^2}\right)^\frac14 \left[E_0 + \Psi(t) \right]^\frac12 \left(\int_0^t \lvert \mathrm{p}-\mathrm{p}_\Omega \rvert_{L^4}^\frac4{3} dr \right)^\frac34 \\
\le C \left(\frac{t}{R^2}\right)^\frac14 \left[E_0 + \Psi(t) \right]^\frac12 \left(\int_0^t \lvert \nabla \mathrm{p} \rvert_{L^\frac43}^\frac4{3} dr \right)^\frac34.
\end{align*}
From the last line and Lemma \ref{Lem:EstPressure} we infer that
\begin{align*}
\int_0^t \int_\Omega \lvert \mathrm{p}- \mathrm{p}_\Omega \rvert \lvert v \rvert \lvert \nabla \varphi \rvert \;dx dr \le C \varepsilon\,^\frac12 \left(\frac{t}{R^2}\right)^\frac14 \left[E_0 + \Psi(t) + (1+ \frac{2\varepsilon\,^2}{R^2}) t\right]^\frac{5}{4}.
\end{align*}
We now deal with the terms containing $\lvert \varphi\rvert$. Applying the H\"older inequality and \eqref{Eq:EstL4normsofU+NablaD} yields
\begin{align*}
\int_0^t \int_\Omega \lvert \varphi \rvert \lvert \nabla g \rvert \lvert \nabla d \rvert \;dx dr \le &C \left(\int_0^t \int_\Omega \varphi^4 \;dx dr \right)^\frac14 \left(\int_0^t \int_\Omega \lvert \nabla g \rvert^2 \;dx dr \right)^\frac12 \left(\int_0^t \int_\Omega \lvert \nabla d \rvert^4 \;dx dr \right)^\frac14\nonumber \\
\le& C t^\frac14 \varepsilon\,^\frac12 \left[E_0 + \Psi(t) + (1+ \frac{2\varepsilon\,^2}{R^2}) t\right]^\frac{3}{4}.
\end{align*}
In a similar way,
\begin{align*}
\int_0^t \int_\Omega \lvert \varphi \rvert \lvert g \rvert \lvert \phi^\prime (d) \rvert \;dx dr \le C \left(\int_0^t \int_\Omega \varphi^4 \;dx dr \right)^\frac14 \left(\int_0^t \int_\Omega \lvert g \rvert^2 \;dx dr \right)^\frac12 \left(\int_0^t \int_\Omega \lvert d \rvert^4 \;dx dr \right)^\frac14\nonumber\\
\le C \left(\int_0^t \int_\Omega \varphi^4 \;dx dr \right)^\frac14 \left(\int_0^t \int_\Omega \lvert g \rvert^2 \;dx dr \right)^\frac12 \left(1+ \sup_{r\in [0, t]} \lvert \nabla d (r)\rvert_{L^2} \right)\nonumber \\
\le C t^\frac14 \varepsilon\,^\frac12 \left[1+ E_0 + \Psi(t) + (1+ \frac{2\varepsilon\,^2}{R^2}) t\right].
\end{align*}
We also have
\begin{equation}
\int_0^t \int_\Omega \lvert f \rvert \lvert v \rvert \lvert \varphi \rvert \;dx dr \le C t^\frac14 \varepsilon\,^\frac12 \left[ E_0 + \Psi(t) + (1+ \frac{2\varepsilon\,^2}{R^2}) t\right]^\frac34 .\label{Eq:LocalNRJ-Final}
\end{equation}
Taking $s=0$, dropping out the positive term $\int_0^t \int_\Omega \varphi \lvert R(d) \rvert^2 \;dx dr $, plugging the inequalities \eqref{Eq:LocalNRj-Start}-\eqref{Eq:LocalNRJ-Final} and using the first fact in \eqref{Eq:Condvarphi} in \eqref{Eq:LocalNRJ-DU} completes the proof of the Lemma \ref{Lem:LocalNRJ-Ball}
\end{proof}
We are now ready to give the proof of Proposition \ref{Prop:LocalSolwithSmallEnergy}.
\begin{proof}[Proof of Proposition \ref{Prop:LocalSolwithSmallEnergy}] We recall that under the assumption of Proposition \ref{Prop:LocalSolwithSmallEnergy} there exists a unique solution $(v,d) \in X_{T_\ast} $ to the problem \eqref{1b}, see Theorem \ref{th}. We can assume that $T_\ast>0$ is the maximal time of existence of $(v,d)$. Let $r_0>0$ and $\varepsilon\,_0>0$ be as in Lemma \ref{Lem:Struwe} and Lemma \ref{Lem:EstimateofDeltaD+L4norms}, respectively. Let $R_0 \in (0,r_0)$ be chosen such that \[
\varepsilon\,_1^2= \mathcal{E}_{2R_0}(v_0,d_0)=\frac12 \sup_{x\in\Omega}\int_{\Omega\cap B_{2{R_{0}}}}\left(|v_{0}|^{2}+|\nabla d_{0}|^{2}+\phi(d_{0})\right)\;dx< \varepsilon\,_0^2. \] Let us observe that since $\mathcal{E}(v_0,d_0)<\infty$ and
$\mu(A)=\int_{\Omega\cap A}\left[|v_{0}|^{2}+|\nabla d_{0}|^{2}+\phi(d_{0})\right]\;dx$ is absolutely continuous, then it is possible to choose such $R_0$. We also observe that $\varepsilon\,^2_1< E_0.$
Now, let
\[\theta_0(\varepsilon\,_1, E_0):= \min\left\{\frac{K_4^{-4} \varepsilon\,_1^6(1+r_0+\varepsilon\,_1)^{-4}} {[1+ 2E_0 +\Psi(T) +T]^5}, \frac{3}{4}\right\}.
\]
and $T_0=T(\varepsilon\,_1, R_0)$. By Corollary \ref{Cor:SmallNRJBeforemaxTime} $T_0< T_\ast$.
We will now distinguish two cases.
\begin{itemize}
\item
If $T_0>R_0^2$, then because $\theta_0(\varepsilon\,_1, E_0) \in (0, \frac34]$, \[ T_0\ge \theta_0(\varepsilon\,_1, E_0)R_0^2,\].
\item
If $T_0\le R_0^2$, then by Corollary \ref{Cor:SmallNRJBeforemaxTime}, the definition of $T_0$ and the continuity of $(v,d)$ at $t=T_0$ we infer that
\begin{align}
\mathcal{E}_{R_0}(v(T_0), d(T_0)) -\mathcal{E}_{2R_0}(v_0,d_0)= 2\varepsilon\,_1^2 - \varepsilon\,_1^2=\varepsilon\,_1^2\nonumber
\end{align}
Hence, by using the inequality \eqref{Eq:EstNRJonBalls} with $t=T_0$ and the fact $\varepsilon\,^2_1< E_0$, we infer the existence of a universal constant $K_7>0$ such that
\begin{align*}
\varepsilon\,_1^2 \le & K_7 T_0^\frac14 \left(1+ E_0 + \Psi(T) + T+ \frac{T_0E_0}{R_0^2} \right)^{\frac{5}{4}}\left( R_0^{-\frac12} (\varepsilon\,_1^\frac32 + \varepsilon\,_1^\frac12)+ \varepsilon\,_1^\frac12 \right) \\
\le & K_7 \varepsilon\,_1^\frac12 (1+R_0^{-\frac12}[1+\varepsilon\,_1]) T_0^\frac14 \left[1+2E_0 + \Psi(T) + T \right]^\frac{5}{4},
\end{align*}
where $\Psi$ is defined in \eqref{Eq:DefPsi}. Since $R_0\le r_0$ we have $$ R_0^\frac12 +\varepsilon\,_1+1\le 1+r_0^\frac12+\varepsilon\,_1. $$
Hence,
\begin{align*}
\varepsilon\,_1^2 \le K_4 \varepsilon\,_1^\frac12 R_0^{-\frac12} (1+r_0^\frac12+\varepsilon\,_1) T_0^\frac14 \left[1+2E_0 + \Psi(T) + T \right]^\frac{5}{4},
\end{align*}
from which we deduce that
\begin{equation*}
T_0\ge \frac{K_4^{-4} \varepsilon\,_1^6(1+r_0^\frac12+\varepsilon\,_1)^{-4} } {[1+ 2 E_0 +\Psi(T) +T]^5} R_0^2 =\theta_0(\varepsilon\,_1, E_0) R_0^2 .
\end{equation*}
\end{itemize}
By the definition of $T_0=T(\varepsilon\,_1, R_0)$, see \eqref{Eq:DefStoppingTime}, and the fact $T_0<T_\ast$ we automatically obtain \eqref{eqn-small estimates-B}.
Thus, the proof of Proposition \ref{Prop:LocalSolwithSmallEnergy} is complete.
\end{proof}
\section{The existence and the uniqueness of a global weak solution} \label{Sec:ExistMaxLocStrongSol}
In this section we will prove global existence of a weak solutions to problem \eqref{1b}. Before we state and prove this result let us define the concept of a weak solution.
\begin{Def}
A global weak solution to \eqref{1b} is a pair of functions $(v,d):[0,T)\to \mathrm{H} \times \mathrm{H} ^1$ such that
\begin{enumerate}
\item $(v,d) \in L^\infty(0,T; \mathrm{H} \times \mathrm{H} ^1)$
\item for all $t \in [0,T)$ the following integral equations
\begin{align}
v(t) =& v_0 +\int_0^t[Av(s) -B(v(s)) - \Pi(\Div [\nabla d(s)\odot \nabla d(s)] ) ] ds + \int_0^t \Pi f(s) ds,\nonumber\\
d(t)=& d_0 +\int_0^t[\Delta d(s)+\lvert \nabla d(s)\rvert^2 d(s) -v(s)\cdot \nabla d(s) -\phi^\prime(d(s)) +(\phi^\prime(d(s) )\cdot d(s) ) d(s) ] ds \nonumber \\
&\qquad \qquad + \int_0^t (d(s)\times g(s)) ds,\nonumber
\end{align}
hold in $D(A^{-\frac32})$ and $\mathrm{H} ^{-2}$, respectively.
\item For all $t\in [0,T)$ $d(t) \in \mathcal{M}$,
\item and $(\partial_tv, \partial_td)\in L^2(0,T_0; D(A^{-\frac32})\times \mathrm{H} ^{-2})$.
\end{enumerate}
\end{Def} We also introduce the notion of local strong solution which will be needed to prove the existence of a global weak solution to our problem.
\begin{Def}\label{Def:Strong-Sol}
Let $T_0\in (0, T]$.
A function $(v,d):[0,T_0]\to H\times \mathrm{H} ^1 $ is a local strong solution to \eqref{1b} with initial data $(v(0), d(0))=(v_0,d_0)$ iff
\begin{enumerate}
\item $(v,d)\in C([0,T_0]; \mathrm{H} \times \mathrm{H} ^1) \cap L^2(0,T_0; \mathrm{V}\times D(\hat{\mathrm{A}} ))$,
\item for all $t \in [0,T_0]$ the equations \eqref{eq:LocVelo} and \eqref{eq:LocDir} hold in $\mathrm{V}^\ast$ and $L^2$, respectively.
\item For all $t\in [0,T_0]$\; $d(t)\in \mathcal{M}$,
\item and $(\partial_tv, \partial_td)\in L^2(0,T_0; \mathrm{V}^\ast\times L^2)$.
\end{enumerate}
As usual we denote by $((v,d);T_0)$ a local strong solution defined on $[0,T_0]$.
Similarly to Definition \ref{Def:Maximal-Sol}, one we can also define the notion of a maximal local strong solution.
\end{Def}
We state the following important remark.
\begin{Rem}
From the definition it is clear that a maximal local solution $(v,d)$ defined on $[0,T_0)$ is a local solution on the open interval $[0,T_0)$.
In the definitions above, the condition $(\partial_tv, \partial_t d)\in L^2(0,T_0; D(A^{-\frac{j+1}{2}}) \times \mathrm{H} ^{j-2}) $, $j=0,2$, is equivalent to
\begin{align*}
F(v,d)=& -\Pi(v\cdot \nabla v) -\Pi(\Div[\nabla d \odot \nabla d]) \in L^2(0,T_0; D(A^{-\frac{j+1}{2}}) ),\; j=0,2,\\
G(v,d)=&\lvert \nabla d \rvert^2 d -v\cdot \nabla d -\phi^\prime(d) + (\phi^\prime(d) \cdot d) d \in L^2(0,T_0; H^{j-2}) , j=0,2.
\end{align*}
\end{Rem}
Let us now state the standing assumptions of this section.
\begin{assume}\label{AssumptionMain}
Let $T>0$ and assume that $(f,g)\in L^2(0,T; \mathrm{H} ^{-1}\times L^2)$.
We also assume that $(v_0,d_0)\in H\times \mathrm{H} ^1$ satisfies
\begin{align*}
& E_0=\mathcal{E}(v_0,d_0)=\int_\Omega (\lvert u_0\rvert^2 + \lvert \nabla d_0\rvert^2+ \phi(d_0)) dy <\infty,\\
& d_0 \in \mathcal{M}.
\end{align*}
\end{assume}
The first main result of this section is the following uniqueness result.
\begin{Prop}\label{Prop:Uniq}
Let $(v_i, d_i)\in C([0,T]; \mathrm{H}\times \mathrm{H} ^1 )\cap L^2(0,T; D(\mathrm{A} ^\frac12)\times D(\hat{\mathrm{A}} ^\frac32 )$, $i=1,2$, be two strong solutions to \eqref{1b} defined on $[0,T]$.
Then,
\begin{equation}
(v_1, d_1)= (v_2, d_2).\notag
\end{equation}
\end{Prop}
\begin{proof}
In order to prove this result we closely follow the approach of \cite{JL+ET+ZX-2016}.
Let $(v_i,d_i)$ be a two strong solutions to \eqref{1b}, $v=v_1-v_2$ and $d=d_1-d_2$. Hence, $v$ satisfies the equation
\begin{equation}
\frac{d v}{dt} + \mathrm{A} v + B(v, v_1)+B(v_2,v) = -\Pi\left(\Div[\nabla d \odot \nabla d_1 + \nabla d_2 \odot \nabla d ]\right).\notag
\end{equation}
Let $w=\mathrm{A} ^{-1}v$. It is not difficult to show that $w$ satisfies
\begin{equation}
\frac{dw }{dt} +\mathrm{A} w + A^{-1} \left(B(v, v_1 ) + B(v_2,v)\right) =-\mathrm{A} ^{-1} \Pi \left(\Div[\nabla d \odot \nabla d_1 + \nabla d_2 \odot \nabla d ]\right).\notag
\end{equation}
By parts (1) and (4) of Definition 5.2, we have $w \in L^2(0,T; D(\mathrm{A} ^\frac32)) $ and $\partial_tw= A^{-1}\partial_t v\in L^2(0,T; \mathrm{V})\subset L^2(0,T; D(\mathrm{A} )^\ast)$. Then, by applying the Lions-Magenes Lemma (\cite[Lemma III.1.2]{Temam_2001}) and using the facts that $\mathrm{A} $ is self-adjoint and $\Div w=0$ we infer that
\begin{align}
\frac12 \frac{d}{dt} \lvert \mathrm{A} ^\frac12 w \rvert^2_{L^2} + \lvert \mathrm{A} w\rvert^2_{L^2}=- \langle B(v, v_1) + B(v_1, v), w \rangle
-\langle \Pi \left(\Div[\nabla d \odot \nabla d_1 + \nabla d_2 \odot \nabla d ]\right), w\rangle \notag\\
= - \langle B(v, v_1) + B(v_1, v), w \rangle -\langle \nabla d \odot \nabla d_1 + \nabla d_2 \odot \nabla d, \nabla w \rangle.\notag
\end{align}
We also used the integration by parts to obtain the second line. Let us now estimate the terms on the right hand side of the last line of the chain of identities above.
Hereafter we fix $\varepsilon\,,\gamma>0$, the symbols $C_\varepsilon\,, C_{\varepsilon\,,\gamma}$ denote two positive constants depending only on $\varepsilon\,$ and $\gamma$.
Firstly, by using the H\"older, the Young inequalities and the Ladyzhenskaya inequality (\cite[Lemma III.3.3]{Temam_2001}) we infer that
\begin{align*}
-\langle B(v, v_1), w\rangle =& \langle B(v, w), v_1\rangle\\
\le & \lvert v \rvert_{L^2} \lvert \nabla w \rvert_{L^4} \lvert v_1\rvert_{L^4}\\
\le & \varepsilon\, \lvert v \rvert^2_{L^2} + C_{\varepsilon\,}\lvert \nabla w\rvert_{L^2} \lvert \nabla^2 w\rvert_{L^2} \lvert v_1\rvert^2_{L^4}\\
\le & \varepsilon\, \lvert v \rvert^2_{L^2}+C_{\varepsilon\,} \lvert \mathrm{A} ^\frac12 w\rvert_{L^2} \lvert \mathrm{A} w\rvert_{L^2} \lvert v_1\rvert^2_{L^4}\\
\le & \varepsilon\, \lvert v \rvert^2_{L^2}+\varepsilon\, \lvert \mathrm{A} w \rvert^2_{L^2} + C_\varepsilon\, \lvert \mathrm{A} ^\frac12 w \rvert^2_{L^2} \lvert v_1 \rvert^2_{L^4}.
\end{align*}
Observe that $\lvert v\rvert^2_{L^2} = \lvert \mathrm{A} w \rvert^2_{L^2}$. Thus,
\begin{align*}
-\langle B(v, v_1), w\rangle
\le 2 \varepsilon\, \lvert \mathrm{A} w \rvert^2_{L^2} + C_\varepsilon\, \lvert \mathrm{A} ^\frac12 w \rvert^2_{L^2} \lvert v_1 \rvert^4_{L^4}.
\end{align*}
In a similar way, we can prove that
\begin{align}
-\langle B(v_2, v), w\rangle
\le 2 \varepsilon\, \lvert \mathrm{A} w \rvert^2_{L^2} + {C_\varepsilon\,} \lvert \mathrm{A} ^\frac12 w \rvert^2_{L^2} \lvert v_2 \rvert^4_{L^4}.\notag
\end{align}
Secondly, making use of the Ladyzhenskaya inequality (\cite[Lemma III.3.3]{Temam_2001}), the H\"older and the Young inequalities we obtain
\begin{align*}
-\langle \nabla d \odot \nabla d_1 , \nabla w \rangle &\le \lvert \nabla d \rvert_{L^2} \lvert \nabla d_1 \rvert_{L^4} \lvert \nabla w \rvert_{L^4} \\
&\le \gamma \lvert \nabla d \rvert^2_{L^2} + C_\gamma \lvert \nabla d_1 \rvert^2_{L^4} \lvert \nabla w \rvert_{L^2} \lvert \nabla^2 w\rvert_{L^2}\\
&\le \gamma \lvert \nabla d \rvert^2_{L^2} + C_\gamma \lvert \nabla d_1 \rvert^2_{L^4} \lvert \mathrm{A} ^\frac12 w \rvert_{L^2} \lvert \mathrm{A} w\rvert_{L^2}\\
& \le \gamma \lvert \nabla d \rvert^2_{L^2} + \varepsilon\, \lvert \mathrm{A} w \rvert^2_{L^2} + C_{\gamma, \varepsilon\,} \lvert \mathrm{A} ^\frac12 w \rvert^2_{L^2} \lvert \nabla d_1 \rvert^4_{L^4}.
\end{align*}
Similarly,
\begin{align*}
-\langle \nabla d_2 \odot \nabla d , \nabla w \rangle
\le \gamma \lvert \nabla d \rvert^2_{L^2} + \varepsilon\, \lvert \mathrm{A} w \rvert^2_{L^2} + C_{\gamma, \varepsilon\,} \lvert \mathrm{A} ^\frac12 w \rvert^2_{L^2} \lvert \nabla d_2 \rvert^4_{L^4}.
\end{align*}
Collecting all these inequalities we obtain
\begin{align}\label{Eq:DifferenceVelo-Fin}
\frac12 \frac{d}{dt} \lvert \mathrm{A} ^\frac12 w \rvert^2_{L^2} + \lvert \mathrm{A} w\rvert^2_{L^2} &\le \varepsilon\, \lvert \mathrm{A} w \rvert^2_{L^2} + \gamma \lvert \nabla d \rvert^2_{L^2} \\ &+ C_{\varepsilon\,,\gamma} \lvert \mathrm{A} ^\frac12 w \rvert^2_{L^2}\left(\lvert v_1 \rvert^4_{L^4} + \lvert v_2 \rvert^4_{L^4} + \lvert \nabla d_1 \rvert^4_{L^4} + \lvert \nabla d_2 \rvert^4_{L^4} \right).
\nonumber \end{align}
Let us turn our attention to the function $d=d_1-d_2$. We notice that $d$ satisfies
\begin{equation*}
\begin{split}
\frac{d}{dt}d +\hat{\mathrm{A}} d+ v\cdot \nabla d_1 + v_2\cdot \nabla d =\left( \lvert \nabla d_1 \rvert^2- \lvert \nabla d_2\rvert^2\right)d_1+ \lvert \nabla d_2 \rvert^2 d -[\phi^\prime(d_1) - \phi^\prime(d_2)]
\\ + [\alpha(d_1)-\alpha(d_2)]d_1 + \alpha(d_2)d + d\times g\\
=\left( \nabla d_1 -\nabla d_2 : \nabla d_1 + \nabla d_2 \right)d_1 + \lvert \nabla d_2 \rvert d -[\phi^\prime(d_1) - \phi^\prime(d_2)]
\\ + [\alpha(d_1)-\alpha(d_2)]d_1 + \alpha(d_2)d + d\times g\\
=\left( \nabla d : \nabla d_1 + \nabla d_2 \right)d_1 + \lvert \nabla d_2 \rvert d -[\phi^\prime(d_1) - \phi^\prime(d_2)]
\\ + [\alpha(d_1)-\alpha(d_2)]d_1 + \alpha(d_2)d + d\times g.
\end{split}
\end{equation*}
Since $d_1,d_2\in L^2(0,T; D(\hat{\mathrm{A}} ))$, and $\partial_t d_1, \partial_t d_2 \in L^2(0,T; L^2)$, we infer by applying Lions-Magenes Lemma (\cite[Lemma III.1.2]{Temam_2001}), and the facts $\langle v_2 \cdot \nabla d, d\rangle=0$ and $ \langle d\times g, d\rangle =0$ (because $d\times g \perp_{\mathbb{R}^3} d$) that
\begin{align} \nonumber
\frac12 \frac{d}{dt} \lvert d \rvert^2_{L^2} + \lvert \nabla d \rvert^2_{L^2}&= -\langle v \cdot \nabla d_1+[\nabla d : \nabla(d_1+d_2)]d_1 + \lvert \nabla d_2 \rvert^2 d -[\phi^\prime(d_1) \phi^\prime(d_2)]_+\alpha(d_2) d, d \rangle \\
&+ \langle [\alpha(d_1)-\alpha(d_2)]d_1, d\rangle. \label{Eq:DifferenceDinL2} \end{align}
Let us estimate the terms in the right hand side of \eqref{Eq:DifferenceDinL2}. First, by using the H\"older, the Young inequalities and the Gagliardo-Nirenberg inequality (\cite[Section 9.8, Example C.3]{Brezis})
we show that
\begin{align*}
-\langle v \cdot \nabla d_1 , d \rangle \le &\lvert v \rvert_{L^2} \lvert \nabla d_1 \rvert_{L^4} \lvert d \rvert_{L^4} \\
\le &\varepsilon\, \lvert v \rvert^2_{L^2} + C_\varepsilon\, \lvert \nabla d_1\rvert^2_{L^4} \lvert d \rvert_{L^2}(\lvert d \rvert_{L^2} + \lvert \nabla d \rvert_{L^2} )\\
\le& \varepsilon\, \lvert v \rvert^2_{L^2} + \gamma \lvert \nabla d \rvert^2_{L^2}+ C_\varepsilon\, \lvert \nabla d_1 \rvert^2_{L^4} \lvert d\rvert^2 + \lvert d \rvert^2_{L^2} \lvert \nabla d_1\rvert^4_{L^4}\\
\le & \varepsilon\, \lvert \mathrm{A} w \rvert^2_{L^2} + \gamma \lvert \nabla d \rvert^2_{L^2}+ C_{\varepsilon\,, \gamma} \ \lvert d \rvert^2_{L^2}\left( 1+ \lvert \nabla d_1\rvert^4_{L^4}\right).
\end{align*}
With the same idea, we prove that
\begin{align*}
\langle [\nabla d : \nabla (d_1 +d_2) ]d_1, d \rangle \le & \lvert \nabla d \rvert_{L^2} \lvert d\rvert_{L^4} [\lvert \nabla d_1 \rvert_{L^4}+ \lvert \nabla d_2\rvert_{L^4} ] \lvert d_1 \rvert_{L^\infty}\\
\le& \gamma \lvert \nabla d \rvert^2_{L^2} +C_\gamma \lvert d \rvert_{L^2} [\lvert d\rvert_{L^2} + \lvert \nabla d \rvert_{L^2} ] [ \lvert \nabla d_1 \rvert^2_{L^4} + \lvert \nabla d_2 \rvert^2_{L^4} ]\lvert d_1 \rvert^2_{L^\infty}\\
\le & 2 \gamma \lvert \nabla d \rvert^2_{L^2} + C_\gamma \lvert d \rvert^2_{L^2} [ \lvert \nabla d_1 \rvert^2_{L^4} + \lvert \nabla d_2 \rvert^2_{L^4} + \lvert \nabla d_1 \rvert^4_{L^4} + \lvert \nabla d_2 \rvert^4_{L^4}]\\
\le & 2 \gamma \lvert \nabla d \rvert^2_{L^2} + C_\gamma \lvert d \rvert^2_{L^2} [ 1+ \lvert \nabla d_1 \rvert^4_{L^4} + \lvert \nabla d_2 \rvert^4_{L^4}]
\end{align*}
In the last line we used the fact that $\lvert d_1 \rvert_{L^\infty} \le 1$.
\noindent Utilizing the H\"older, the Young inequalities and the Gagliardo-Nirenberg inequality (\cite[Section 9.8, Example C.3]{Brezis}) we obtain
\begin{align*}
\langle \lvert \nabla d_2 \rvert^2 d, d \rangle \le & \lvert \nabla d_2 \rvert^2_{L^4} \lvert d \rvert^2_{L^4}\\
\le & \lvert \nabla d_2 \rvert^2_{L^4} \lvert d \rvert_{L^2}( \lvert d \rvert_{L^2} + \lvert \nabla d \rvert_{L^2})\\
\le & \gamma \lvert \nabla d \rvert^2_{L^2} +C_\gamma \lvert d \rvert^2_{L^2} \left( 1+ \lvert \nabla d_2 \rvert^4_{L^4} \right).
\end{align*}
Since $\lvert \phi^{\prime\prime}\rvert\le M $, the map $\phi^\prime: \mathbb{R}^3 \to \mathbb{R}^3$ is Lipschitz and
\begin{equation}
- \langle \phi^\prime(d_1) - \phi^\prime(d_2), d\rangle\le M \lvert d \rvert^2_{L^2}. \label{Eq:DiffPhiprime}
\end{equation}
Using the definition of $\alpha(d_2)=(\phi^\prime(d_2) \cdot d_2)$, the fact $\lvert d_2 \rvert=1$ and \eqref{Eq:LInearGrowthPhiprime} we have
\begin{align}
\langle \alpha(d_2) d, d \rangle = \langle (\phi^\prime(d_2) \cdot d_2) d, d\rangle
\le 2 M\lvert d\rvert^2_{L^2}. \label{Eq:PhiprimeD}
\end{align}
Using again the definition of $\alpha(d_1)$ and $\alpha(d_2)$ we obtain
\begin{align*}
\langle [\alpha(d_1)-\alpha(d_2)]d_1, d \rangle = &\langle [\phi^\prime(d_1)\cdot d_1 -\phi^\prime(d_2)\cdot d_2]d_1, d\rangle \\
=& \langle \left([\phi^\prime(d_1)-\phi^\prime(d_2) ]\cdot d_1 + \phi^\prime(d_2)\cdot d \right)d_1, d\rangle.
\end{align*}
Since $\phi^\prime$ is Lipschitz, $d_i(t)\in \mathcal{M}$ for all $t\in [0,T]$, we show with the same ideas as used in \eqref{Eq:DiffPhiprime} and \eqref{Eq:PhiprimeD} that
\begin{align*}
\langle [\alpha(d_1)-\alpha(d_2)]d_1, d \rangle \le 3 M \lvert d \rvert^2_{L^2}.
\end{align*}
Hence, collecting all these inequalities related to the terms in the right hand side of the equation \eqref{Eq:DifferenceDinL2} we obtain
\begin{equation}\label{Eq:DifferenceOptDir-fin}
\frac12 \frac{d}{dt} \lvert d \rvert^2_{L^2} + \lvert \nabla d \rvert^2_{L^2}\le \varepsilon\, \lvert \mathrm{A} w \rvert^2_{L^2}+ \gamma \lvert \nabla d \rvert^2_{L^2} + C_\gamma \lvert d \rvert^2_{L^2} \left(1 + \lvert \nabla d_1 \rvert^4_{L^4} + \lvert \nabla d_2 \rvert^4_{L^4}\right).
\end{equation}
Thus, summing \eqref{Eq:DifferenceVelo-Fin} and \eqref{Eq:DifferenceOptDir-fin} up, we have
\begin{align}\nonumber \label{Eq:DifferenceVeloOptDir}
\frac12 \frac{d}{dt} \left(\lvert d \rvert^2_{L^2} + \lvert \mathrm{A} ^\frac12 w\rvert^2_{L^2} \right)+\lvert \mathrm{A} w\rvert^2_{L^2} + \lvert \nabla d \rvert^2_{L^2} &\le \varepsilon\, \lvert \mathrm{A} w \rvert^2_{L^2}+ \gamma \lvert \nabla d \rvert^2_{L^2} + C_{\gamma, \varepsilon\,} \lvert d \rvert^2_{L^2} \left(1 + \lvert \nabla d_1 \rvert^4_{L^4} + \lvert \nabla d_2 \rvert^4_{L^4}\right) \\
&\hspace{-1truecm}+ C_{\varepsilon\,,\gamma}
\lvert \mathrm{A} ^\frac12 w\rvert^2_{L^2}\left( \lvert \nabla d_1 \rvert^4_{L^4} + \lvert \nabla d_2 \rvert^4_{L^4}+\lvert v_1 \rvert^4_{L^4} + \lvert v_2 \rvert^4_{L^4}\right).
\end{align}
Let choose $\varepsilon\,=\gamma=\frac12$ and put
\begin{align*}
y(t)= &\lvert d(t) \rvert^2_{L^2} + \lvert \mathrm{A} ^\frac 12 w(t)\rvert^2_{L^2}, t\in [0,T], \\
\Phi(t)=& 2C_{\frac12,\frac12} \left( 1+\lvert \nabla d_1(t) \rvert^4_{L^4} + \lvert \nabla d_2(t) \rvert^4_{L^4}+\lvert v_1(t) \rvert^4_{L^4} + \lvert v_2 (t) \rvert^4_{L^4} \right), t\in [0,T].
\end{align*}
Then, we see from \eqref{Eq:DifferenceVeloOptDir} that $y$ satisfies
\begin{equation}\label{Eq:NeedGornwall}
\dot{y}(t) \le \Phi(t) y(t), t\in [0,T].
\end{equation}
Observe that since $(v_i,d_i)\in C([0,T], \mathrm{H} \times \mathrm{H} )\cap L^2(0,T; D(\mathrm{A} ^\frac12) \times D(\hat{\mathrm{A}} ))$ we infer that
\begin{equation*}
\int_0^T \Phi(s) ds<\infty.
\end{equation*}
Thus, one can apply the Gronwall inequality to \eqref{Eq:NeedGornwall} and deduce that
\begin{equation*}
y(t) \le y(0)e^{\int_0^t \Phi(s) ds }\le y(0)e^{\int_0^T\Phi(s) ds}, \; t\in [0,T].
\end{equation*}
Since $$y(0)=\lvert d_1(0) -d_2(0)\rvert^2_{L^2} + \lvert \mathrm{A} ^{-1\frac12} v_1(0)-\mathrm{A} ^{-\frac12} v_2(0)\rvert^2_{L^2}= 0,$$
we have
\begin{equation*}
y(t)= \lvert d_1(t)-d_2(t)\rvert^2_{L^2} + \lvert v_1(t) -v_2(t) \rvert^2_{D(\mathrm{A} ^{-\frac12})}=0, \; t\in [0,T].
\end{equation*}
This completes the proof of the Proposition \ref{Prop:Uniq}.
\end{proof}
The second main result of this paper is the following theorem.
\begin{Thm}\label{thm-main}
Let $(v_0,d_0)\in \mathrm{H}\times \mathrm{H} ^1$, $r_0>0$ and $\varepsilon\,_0>0$ be the constants from Lemma \ref{Lem:Struwe} and \ref{Lem:EstimateofDeltaD+L4norms}, respectively. Then, there exist constants $\varrho_0\in (0, r_0]$ and $\varepsilon\,_1\in (0,\varepsilon\,_0)$ such that the following hold.
If Assumption \ref{AssumptionMain} holds, then there are a number $L\in \mathbb{N}$, depending only on the norms of $(v_0,d_0)\in H\times \mathrm{H} ^1$ and $(f,g)\in L^2(0,T; \mathrm{H} ^{-1} \times L^2)$, a collection of times $0<T_1<\cdots<T_L\le T$ and a global weak solution $(\mathbf{u},\mathbf{d})\in C_{w}([0, T]; \mathrm{H} \times \mathrm{H}^1) \cap L^2(0, T; \mathrm{V}\times D(\hat{\mathrm{A}} ) )$ to \eqref{1b-0} such that
\begin{enumerate}
\item for each $i\in \{1, \ldots, L\}$, $(\mathbf{u},\mathbf{d})_{\lvert_{[T_{i-1}, T_i)}}\in C([T_{i-1}, T_i); \mathrm{H} \times (\mathrm{H} ^1\cap \mathcal{M})) $ with the left-limit at $T_i$,
is a maximal local regular solution to \eqref{1b} with initial data $(v(T_{i-1}), d(T_{i-1}) )$. Here we understand that $T_0=0$. \item If $T_L<T$, then $(\mathbf{u},\mathbf{d})_{\lvert_{[T_{L}, T]}}$ belongs to $ C([T_{L}, T]; \mathrm{H} \times \mathrm{H}^1)$ and satisfies the variational form problem \eqref{1b-0} on the interval $[T_{L}, T]$ with initial data $(v(T_{L}), d(T_{L}) )$.
\item For each $i\in \{1, \ldots, L\}$
\begin{equation*}
\lim_{t\nearrow T_i} \mathcal{E}_{R}(\mathbf{u}(t), \mathbf{d}(t)) \ge \varepsilon\,_1^2,
\end{equation*}
for all $R\in (0,\varrho_0]$.
\item At each $T_i$ there is a loss of energy at least $\varepsilon\,_1^2\in (0, \varepsilon\,_0^2)$, \textit{i.e.},
\begin{equation*}
\mathcal{E}(\mathbf{u}(T_i), \mathbf{d}(T_i)) \le \mathcal{E}(\mathbf{u}(T_{i-1}), \mathbf{d}(T_{i-1})) +\frac12 \int_{T_{i-1}}^{T_1} \left[\lvert f\rvert_{\mathrm{H} ^{-1}}^2 + \lvert g\rvert^2_{L^2}\right] dt -\varepsilon\,_1^2.
\end{equation*}
\end{enumerate}
\end{Thm}
The proof of this theorem is established in several steps. The first of such steps is the proof of the existence of a maximal local strong solution for the Ericksen-Leslie system \eqref{1b} with data satisfying Assumption \ref{AssumptionMain}.
\begin{Thm}\label{Ma1}
Let $(u_0,d_0)\in \mathrm{H}\times(\mathrm{H} ^1\cap \mathcal{M})$, $r_0>0$ and $\varepsilon\,_0>0$ be the constants from Lemma \ref{Lem:Struwe} and \ref{Lem:EstimateofDeltaD+L4norms}, respectively.
Then, there exist $\varrho_0\in (0, r_0]$ such that
\begin{align}
\varepsilon\,_1^2=\mathcal{E}_{2\varrho_0}(u_0,d_0)< \varepsilon\,_0^2,\label{Eq:AssumSmallInitialNRJ-1-0},
\end{align}
and a maximal local strong solution $((v,d); T_\ast)$ satisfying
\begin{align*}
\limsup_{t\nearrow T_{\ast}}\mathcal{E}_{R}(v(t), d(t)) \ge\varepsilon^{2}_{1},
\end{align*}
for all $R\in (0, \varrho_0]$.
\end{Thm}
For the proof of this theorem, we first prove the existence of a local strong solution which is given by the following proposition.
\begin{Prop}\label{Ma2}
There exist $\varepsilon\,_0>0$ and a function
\[\theta_0: (0,\varepsilon\,_0)\times (0,\infty) \to (0,\frac34]
\]
which is non-increasing w.r.t. the second variable and nondecreasing w.r.t. the first one
such that the following holds. \\
Let $r_0>0$ be the constant from Lemma \ref{Lem:Struwe} and assume that the initial data $(v_0,d_0)$ and the forcing $(f,g)$ satisfies Assumption \ref{AssumptionMain}.
Then, there exists $\varrho_0\in (0, r_0]$ such that
\begin{equation}\label{Eq:AssumSmallInitialNRJ-1}
\varepsilon\,_1^2=2\mathcal{E}_{2\varrho_0}(u_0,d_0)< \varepsilon\,_0^2.
\end{equation}
Moreover, there exists a local strong solution $((v,d);T_0)$ satisfying
\begin{equation}\label{eqn-t_o-2}
\begin{split}
&T_0 \ge \frac{\varrho^2_0}{(\varrho_0^\frac12 + 1)^4} \theta_0(\varepsilon\,_1,E_0)\\
&\sup_{t\in [0,T_0]}\mathcal{E}_{\varrho_0}(v(t), d(t)) \le 2\varepsilon\,_1^2.
\end{split}
\end{equation}
Furthermore, there exists a constant $K_5>0$
\begin{align}
\sup_{t\in [0,T_0]} \mathcal{E}(v(t), d(t)) + \int_0^{T_0} \lvert \nabla v (r) \rvert^2_{L^2} dr\le E_0 +\frac12 \int_0^T \left( \lvert f(s) \rvert^2_{\mathrm{H} ^{-1} } + \lvert g(s\rvert^2_{L^2}\right) ds \label{Eq:UnifEstStrongSol-1}\\
\int_0^{T_0} \lvert \Delta d(t) \rvert^2_{L^2} dt \le K_5 \left[ E_0 + \frac12 \int_0^{T}\left(\lvert f (s) \rvert^2_{\mathrm{H} ^{-1}}+ \lvert g(s) \rvert^2_{L^2}\right) ds + \left(1+\frac{E_0}{R^2}\right)T \right], \;\; R \in (0,\varrho_0].\label{Eq:UnifEstStrongSol-2}
\end{align}
where $E_0:= \mathcal{E}(u_0,d_0)$.
\delb{and $(v,d)\in C([0,T_{0}];H\times \mathrm{H} ^{1}_{w})\cap L^{2}(0,T_{0}; V\times \mathrm{H} ^{2})$.
Also for all $t\in [0,T_0]$, a.e. $x\in \Omega$
$|d(t,x)|=1$.}
\end{Prop}
\begin{proof}[Proof of Proposition \ref{Ma2}]
There exists a sequence $\{d_{0}^{k}\}_{k=1}^{\infty}\subset C^{\infty}(\Omega;\mathbb{S}^{2})$ such that
\[
\lim_{k\to\infty}\|d_{0}^{k}-d_{0}\|_{H^{1}}=0,
\]
see \cite[Section 4]{Schoen-Uhlenbeck-83}.
By definition, $v_{0}\in \mathrm{H} $ can be approximated by a sequence $\{v_{0}^{k}\}_{k=1}^{\infty}\subset \mathcal{V}$ such that
\[
\lim_{k\to\infty}|v_{0}^{k}-v_{0}|_{L^{2}}=0.
\]
Since embedding $L^2 \hookrightarrow \mathrm{H} ^{-1}$ and $\mathrm{H} ^1\subset L^2$ are dense, then one can also approximate $(f,g)\in L^2(0,T; \mathrm{H}^{-1} \times L^2)$ by a sequence $((f_k, g_k))_{k \in \mathbb{N}} \subset L^2(0,T; L^2 \times \mathrm{H}^1)$ in the following sense
\begin{equation}
(f_k, g_k) \rightarrow (f,g) \text{ strong in } L^2(0,T; \mathrm{H}^{-1} \times L^2).
\end{equation}
Let $\varepsilon\,_0>0$ and $R_0\in (0,r_0]$ be the constants from Proposition \ref{Prop:LocalSolwithSmallEnergy}. Since $\mathcal{E}(v_0,d_0)<\infty$ and
$\mu(A)=\int_{\Omega\cap A}\left[|v_{0}|^{2}+|\nabla d_{0}|^{2}+\phi(d_{0})\right]\;dx$ is absolutely continuous, then there exists $\tilde{R_{0}}>0$ such that
\[
\varepsilon\,^2_1:= \sup_{x\in\Omega}\int_{\Omega\cap B_{2\tilde{R_{0}}}}\left(|v_{0}|^{2}+|\nabla d_{0}|^{2}+\phi(d_{0})\right)\;dx< \varepsilon\,_0^2.
\]
Choosing $\varrho_0=\tilde{R}_0 \wedge R_0$ yields \eqref{Eq:AssumSmallInitialNRJ-1}.
Let $\varepsilon\,>0$ be an arbitrary real number. Since $(v_{0}^{k}, d_{0}^{k})$ strongly converges to $(v_{0},d_{0})$ in $ L^{2}\times \mathrm{H} ^{1}$, there exists a number $k_0\in \mathbb{N}$ such that for all $k\ge k_0$
\begin{align}
\frac12\int_{\Omega\cap B_{2\varrho_0}(x)}|v_{0}^{k}|^{2}\;dx
&=\frac12 \int_{\Omega\cap B_{2\varrho_0}(x)}|v_{0}^{k}-v_{0}+v_{0}|^{2}\notag\\
&\le \int_{\Omega\cap B_{2\varrho_0(x)}}|v_{0}^{k}-v_{0}|^{2}\;dx+ \int_{\Omega\cap B_{2\varrho_0(x)}}|v_{0}|^{2}\;dx\notag\\
&\le \frac\eps3+ \int_{\Omega\cap B_{2\varrho_0(x)}}|v_{0}|^{2}\;dx.\notag
\end{align}
In a similar way one can prove that,
\[
\int_{\Omega\cap B_{2\varrho_0}(x)}|\nabla d_{0}^{k}|^{2}\;dx\le \frac\eps3+ \int_{\Omega\cap B_{2\varrho_0(x)}}|\nabla d_{0}|^{2}\;dx.
\]
Observe that
\begin{align}
\int_{\Omega\cap B_{2\varrho_0}(x)}\phi(d_{0}^{k})\;dx
&= \int_{\Omega\cap B_{2\varrho_0}(x)}\left(\phi(d_{0}^{k})-\phi(d_{0})+\phi(d_{0})\right)\;dx\notag\\
&\le \int_{\Omega\cap B_{2\varrho_0}(x)}|\phi(d_{0}^{k})-\phi(d_{0})|\;dx +\int_{\Omega\cap B_{2\varrho_0}(x)}\phi(d_{0})\;dx\notag\\
&\le \int_{\Omega\cap B_{2\varrho_0}(x)}|\phi(d_{0}^{k})-\phi(d_{0})|\;dx + \int_{\Omega\cap B_{2\varrho_0}(x)}\phi(d_{0})\;dx\notag\\
&\le \int_{\Omega\cap B_{2\varrho_0}(x)}|\phi(d_{0}^{k})-\phi(d_{0})|\;dx + \int_{\Omega\cap B_{2\varrho_0}(x)}\phi(d_{0})\;dx.\notag
\end{align}
Since $|d_{0}^{k}|=|d_{0}|=1$, $$ \sup_{n \in \mathbb{S}^2 } \lvert\phi^\prime(n) \rvert\le M $$ and $d_0^k \to d_0$ in $\mathrm{H} ^1$ we deduce that there exists $k_0\in \mathbb{N}$ such that for all $k\ge k_0$
\begin{align}
\int_{\Omega\cap B_{2\varrho_0}(x)}\phi(d_{0}^{k})\;dx
&\le M \int_{\Omega} \lvert d_0^k - d_0\rvert \;dx + \int_{\Omega\cap B_{2\varrho_0}(x)}\phi(d_{0})\;dx\nonumber\\
&\le \frac\eps3 + \int_{\Omega\cap B_{2\varrho_0}(x)}\phi(d_{0})\;dx.\notag
\end{align}
Hence, there exists a constant $k_0\in \mathbb{N}$ such that for all $k\ge k_0$
\begin{equation}
\mathcal{E}_{2\varrho_0}(v_0^k, d_0^k) \le \varepsilon\, + 2 \mathcal{E}_{2\varrho_0}(v_0, d_0).\notag
\end{equation}
Since $\varepsilon\,$ is arbitrary we infer that
\begin{equation}
\mathcal{E}_{2\varrho_0}(v_0^k, d_0^k) \le \varepsilon\,_1^2 < \varepsilon\,_0^2.\notag
\end{equation}
Without loss of generality, we will assume that for all $k\ge 1$
\[
\mathcal{E}_{2\varrho_0}(v_0^k, d_0^k)\le \varepsilon_{1}^{2}.
\]
By Proposition \ref{Prop:LocalSolwithSmallEnergy} there exist a function $\theta_0:(0,\varepsilon\,_0) \times [0, \infty)\to (0,\frac34]$ satisfying the properties stated in Proposition \ref{Ma2}, a sequence of time $T_0^k$ satisfying \eqref{eqn-t_o-2} and a sequence of regular solutions $(v^{k},d^{k}):[0,T_{0}^{k}]\to \mathrm{V} \times \mathrm{H} ^{2}$, with initial condition
\[
(v^{k}_{0}(0),d^{k}_{0}(0))=(v_{0}^{k},d_{0}^{k}).
\]
Moreover,
\begin{equation}\label{Eq:NRGSmall}
\sup_{t\in [0,T_{0}^{k}]}\mathcal{E}_{\varrho_0}(v^k(t), d^k(t))\le 2\varepsilon_{1}^{2}.
\end{equation}
We recall that for all $k\ge 1$ and $t\in [0,T_{0}^{k})$, we have
\begin{align}
&\sup_{t\le T^{k}_{0}}\mathcal{E}(v^{k}(t), d^{k}(t)) + \int_{0}^{T^{k}_{0}}\left[|\nabla v^{k}|^{2}_{L^{2}}+|\Delta d^{k}-\phi^\prime(d^{k})+\alpha(d^{k})d^{k}+|\nabla d^{k}|^{2}d^{k}|^{2}_{L^{2}}\right]dt\notag\\
&\le \mathcal{E}(v_{0}^{k},d^{k}_{0})+ \frac12 \int_0^{T^{k}_{0} } (\lvert f_k \rvert^2_{\mathrm{H} ^{-1}} + \lvert g_k\rvert^2_{L^2} ) \,dt\le E_{0} + \frac12 \int_0^{T^{k}_{0}} (\lvert f \rvert^2_{\mathrm{H} ^{-1}} + \lvert g\rvert^2_{L^2} ) \,dt .\notag
\end{align}
Hereafter, we put
\begin{equation}
\Xi_0=E_{0} + \frac12 \int_0^{T^{k}_{0}} (\lvert f \rvert^2_{\mathrm{H} ^{-1}} + \lvert g\rvert^2_{L^2} ) \,dt.\notag
\end{equation}
We also recall that there exists $C>0$ such that for all $k\in \mathbb{N}$ and $R\in (0,\varrho_0]$
\begin{align}
&\int_{0}^{T^{k}_{0}}|\Delta d^{k}|^{2}_{L^{2}}dt\le C \left[ E_0 + \frac12 \int_0^{T}\left(\lvert f (s) \rvert^2_{\mathrm{H} ^{-1}}+ \lvert g(s) \rvert^2_{L^2}\right) ds + \left(1+\frac{E_0}{R^2}\right)T \right],\label{Eq:UnifEstLaplace}\\
&\int_{0}^{T^{k}_{0}}\left(|v^{k}|^{4}_{L^{4}}+|\nabla d^{k}|^{4}_{L^{4}}\right)dt\le C \varepsilon\,_1^2\left[ E_0 + \frac12 \int_0^{T}\left(\lvert f (s) \rvert^2_{\mathrm{H} ^{-1}}+ \lvert g(s) \rvert^2_{L^2}\right) ds + \left(1+\frac{E_0}{R^2}\right)T \right],\label{Eq:UnifEstL4}\\
&\int_{0}^{T^{k}_{0}}|\nabla v^{k}|^{2}_{L^{2}}dt\le E_{0}+\frac12 \int_0^{T}\left(\lvert f (s) \rvert^2_{\mathrm{H} ^{-1}}+ \lvert g(s) \rvert^2_{L^2}\right) ds, \label{Eq:UnifEstL2}
\end{align}
and
\begin{equation}\label{Eq:UnifEstNRG}
\sup_{t\le T^{k}_{0}}\left[|v^{k}(t)|^{2}_{L^{2}}+|\nabla d^{k}(t)|^{2}_{L^{2}}+\int_{\Omega}\phi(d^{k}(t))\;dx\right]\le E_{0}+\frac12 \int_0^{T}\left(\lvert f (s) \rvert^2_{\mathrm{H} ^{-1}}+ \lvert g(s) \rvert^2_{L^2}\right) ds.
\end{equation}
We now estimate the time derivatives. Let us put
\begin{equation}
\mathcal{E}(d(s)) =\frac12\lvert \nabla d(s) \rvert^2_{L^2}+ \int_\Omega \phi(d(s,x)) \;dx.\notag
\end{equation}
Then, we deduce from Claim \ref{Claim:perp} that
\begin{align}
\int_{0}^{T^{k}_{0}}|\partial_sd^{k}(s)|^{2}_{L^{2}}ds =&\int_0^{T^k_0} \langle R(d(s)), \partial_s d(s)\rangle ds -\int_0^{T^k_0} \langle v^k(s)\cdot \nabla d^k(s), \partial_s d^k(s) \rangle ds\notag
\\
=&\int_{0}^{T^{k}_{0}}\frac{d}{ds}\mathcal{E}(d^{k}(s))ds-\int_{0}^{T^{k}_{0}}(v^{k}(s)\cdot \nabla d^{k}(s),\partial_sd^{k}(s) )ds.\notag
\end{align}
Hence
\begin{align}
\int_{0}^{T^{k}_{0}}|\partial_sd^{k}(s)|^{2}_{L^{2}} ds
&\le \sup_{t\le T^{k}_{0}}\mathcal{E}(d^{k}(t))-\mathcal{E}(d^{k}(0))
+\int_{0}^{T^{k}_{0}}(v^{k}(s)\cdot \nabla d^{k}(s),\partial_sd^{k}(s))ds\notag\\
&\le\mathcal{E}(v_{0}^{k},d_{0}^{k})-\mathcal{E}(d^{k}(0))+\int_{0}^{T^{k}_{0}}(v^{k}(s)\cdot \nabla d^{k}(s),\partial_sd^{k}(s))\,ds\notag\\
&\le \frac{1}{2}|v_{0}^{k}|^{2}_{L^{2}}+\int_{0}^{T^{k}_{0}}(v^{k}(s)\cdot \nabla d^{k}(s),\partial_sd^{k}(s))\,ds\notag\\
&\le E_{0}+\int_{0}^{T^{k}_{0}}(v^{k}\cdot \nabla d^{k},\partial_sd^{k})\,ds.\notag
\end{align}
Now we estimate the integral on the right hand side of the last line as follow:
\begin{align}
\int_{0}^{T^{k}_{0}}(v^{k}(s)\cdot \nabla d^{k}(s),\partial_{s}d^{k}(s)) ds
&\le \int_{0}^{T^{k}_{0}}|v^{k}(s)\cdot \nabla d^{k}(s)|_{L^{2}}|\partial_{s}d^{k}(s)|_{L^{2}} ds \notag\\
&\le\frac{1}{2}\int_{0}^{T^{k}_{0}}|\partial_{s}d^{k}(s)|^{2}_{L^{2}}\,ds+\frac{1}{2}\int_{0}^{T^{k}_{0}}|v^{k}(s)\cdot \nabla d^{k}(s)|^{2}_{L^{2}}\,ds\notag\\
&\le\frac{1}{2}\int_{0}^{T^{k}_{0}}|\partial_{s}d^{k}(s)|^{2}_{L^{2}}\,ds+
\frac{1}{2}\left(\int_{0}^{T^{k}_{0}}|v^{k}(s)|^{4}_{L^{4}}\,ds\right)^{\frac{1}{2}}\left(\int_{0}^{T^{k}_{0}}|\nabla d^{k}(s)|^{4}_{L^{4}}\,ds\right)^{\frac{1}{2}}.\notag\\
\end{align}
By employing \eqref{Eq:UnifEstL4} in the last line we get
\begin{equation}\notag
\int_{0}^{T^{k}_{0}}(v^{k}\cdot \nabla d^{k},\partial_{t}d^{k}) dt\le\frac{1}{2}\int_{0}^{T^{k}_{0}}|\partial_{t}d^{k}|^{2}_{L^{2}}+\frac{C}{2}\varepsilon^{2}_{1}\Xi_{0}.
\end{equation}
Summing up these discussion, we get
\begin{equation}
\sup_{k \in \mathbb{N}}\int_{0}^{T^{k}_{0}}|\partial_{t}d^{k}|^{2}_{L^{2}}dt \le E_{0}+\frac{1}{2}\sup_{k \in \mathbb{N}}\int_{0}^{T^{k}_{0}}|\partial_{t}d^{k}|^{2}_{L^{2}}\,dt+ C\varepsilon^{2}_{1}\Xi_{0}.\notag
\end{equation}
Thus, we obtain the following uniform estimate of $\partial_t d$
\begin{equation}\notag
\sup_{k \in \mathbb{N}}\int_{0}^{T^{k}_{0}}|\partial_{t}d^{k}|^{2}_{L^{2}} \,dt\le 2E_{0} + 2C\varepsilon_{1}^{2} \Xi_{0}.
\end{equation}
We now estimate the time derivative $\partial_{t}v^{k}$.
Let $\varphi\in$ V. We have
\begin{equation}\notag
(\partial_{t}v^{k},\varphi)=-(\nabla v^{k},\nabla\varphi)-\int_{\Omega}v^{k}\cdot \nabla v^{k}\varphi \;dx-\int_{\Omega}Div(\nabla d^{k}\odot\nabla d^{k})\varphi \;dx +\int_{D}f(t) \;dx.
\end{equation}
Hence
\begin{align}
|(\partial_{t}v^{k},\varphi)|
&\le |\nabla v^{k}|_{L^{2}}|\nabla\varphi|_{L^{2}}+\int_{\Omega}|v^{k}|^{2}|\nabla\varphi|\;dx+\int_{\Omega}|\nabla d^{k}|^{2}|\nabla\varphi|\;dx\notag\\
&\le |\nabla\varphi|_{L^{2}}\left[|\nabla v^{k}|_{L^{2}}+|v^{k}|^{2}_{L^{4}}+|\nabla d^{k}|^{2}_{L^{4}}+|f(t)|_{L^{2}}\right]\notag
\end{align}
This altogether with \eqref{Eq:UnifEstL4} and \eqref{Eq:UnifEstL2} imply that there exists $C>0$ such that for all $k\in \mathbb{N}$
\begin{align}
\sup_{k\in \mathbb{N}}\int_{0}^{T^{k}_{0}}|\partial_{t}v^{k}|^{2}_{\mathrm{V}^{\ast}}
&\le 4\left[\int_{0}^{T^{k}_{0}}|\nabla v^{k}|^{2}_{L^{2}} \,ds+\int_{0}^{T^{k}_{0}}|v^{k}|^{4}_{L^{4}}\,ds+\int_{0}^{T^{k}_{0}}|\nabla d^{k}|^{4}_{L^{4}}\,ds+\int_{0}^{T^{k}_{0}}|f(t)|_{L^{2}}^{2} dt\right]\notag\\
&\le C.\notag
\end{align}
Now, let us set $$T_{0}=\inf_{k\ge 1}T^{k}_{0}.$$ Since for all $k\ge 1$ $$T^{k}_{0}\ge \theta_{0}(\varepsilon\,_1, E_0) \frac{\varrho^2_0}{(\varrho_0^\frac12 + 1)^4} ,$$ then by the definition of $T_{0}$, we get $$T_{0}\ge\theta_{0}(\varepsilon\,_1, E_0) \frac{\varrho^2_0}{(\varrho_0^\frac12 + 1)^4} .$$
It then follows from the previous analysis that the sequence $\left((v^{k},d^{k})\right)_{k \in \mathbb{N}}$ is bounded in $C([0,T_{0}];\mathrm{H} \times \mathrm{H} ^{1})\cap L^{2}(0,T_{0};\mathrm{V} \times \mathrm{H} ^{2})$ and the sequence $\left((\partial_{t}v^{k},\partial_{t}d^{k})\right)_{k \in \mathbb{N}}$ is bounded in $L^{2}(0,T_{0};\mathrm{V}^{\ast}\times L^{2})$.
Hence by Aubin-Lions compactness lemma and Banach-Alaoglu theorem, one can extract a subsequence $\left((v^{k_{j}},d^{k_{j}})\right)_{j \in \mathbb{N}}$ from $\left((v^{k},d^{k})\right)_{k \in \mathbb{N}}$ and find $(v,d)$ such that as $j \to \infty$
\begin{equation}
\begin{split}
&(v^{k_{j}},d^{k_{j}})\to (v,d)~ \text{weakly in}~ L^{2}(0,T_{0};\mathrm{V}\times \mathrm{H} ^{2}),\\
&(v^{k_{j}},d^{k_{j}})\to (v,d)~ \text{weakly star}~ \text{in}~ L^{\infty}(0,T_{0}; \mathrm{H} \times \mathrm{H} ^{1}),\\
&(v^{k_{j}},d^{k_{j}})\to (v,d)~ \text{strongly in}~ L^{2}(0,T_{0};D(\mathrm{A} ^{\frac{\theta}{2}})\times \mathrm{H} ^{1+\theta})
~\text{for any}~ \theta\in [0,1)\label{fiu}.
\end{split}
\end{equation}
Let $t\in [0,T_0]$. Then, the sequence $(v^{k}(t), d^k(t))_{k \in \mathbb{N}}$ is bounded in $\mathrm{H} \times \mathrm{H} ^1$. Hence, thanks to the compact embedding $\mathrm{H} ^1\hookrightarrow L^2$ we can and we will assume that the subsequence $\left((v^{k_{j}},d^{k_{j}})\right)_{j \in \mathbb{N}}$ satisfies, for all $t\in [0,T_0]$
\begin{equation}
d^{k_j}(t) \to d(t) \text{ strongly in } L^2.\notag
\end{equation}
This and the fact $d^{k_j}(t)\in \mathcal{M}$ for all $j \in \mathbb{N}$, $t \in [0,T_0]$ implies that there exists a constant $C>0$
such that for all $j \in \mathbb{N}$, $t \in [0,T_0]$
\begin{align}
\int_\Omega \lvert 1-\rvert d(t,x) \rvert^2\rvert \;dx =& \int_\Omega \lvert \rvert d^{k_j}(t,x)\rvert^2 -\lvert d(t,x)\rvert^2\rvert \;dx\notag\\
\le & \lvert d^{k_j}(t,x) -d(t,x)\rvert_{L^2} \lvert d^{k_j}(t,x) + d(t,x)\rvert_{L^2}\notag\\
\le & C \lvert d^{k_j}(t,x) - d(t,x)\rvert_{L^2} .\notag
\end{align}
Passing to the limit as $j\to \infty$ yields
\begin{equation*}
\int_\Omega \lvert 1-\rvert d(t,x) \rvert^2\rvert \;dx=0, \text{ for all $t\in [0,T_0]$}.\notag
\end{equation*}
Hence, for all $t\in [0,T_0]$
\begin{equation}\label{Eq:SphereCondition}
d(t)\in \mathcal{M}.
\end{equation}
Our next step is to show that the limit $(v,d)$ satisfies the system \eqref{1b}. Hence, we need to pass to the limit in the nonlinear terms. In order to do this, we firstly observe that the
convergences
\begin{align}
&v^{k_{j}}\cdot \nabla v^{k_{j}}\rightharpoonup v\cdot \nabla v~ \text{in}~ L^{2}(0,T_{0};\mathrm{V}^{\ast})\notag\\
&-\Div(\nabla d^{k_{j}}\odot\nabla d^{k_{j}})\rightharpoonup -\Div(\nabla d\odot\nabla d)~ \text{in}~ L^{2}(0,T_{0};\mathrm{V}^{\ast})\notag\\
& v^{k_{j}}\cdot \nabla d^{k_{j}}\rightharpoonup v\cdot \nabla d ~\text{in}~ L^{2}(0,T_{0}; L^{2}).\notag
\end{align}
are now well-known, see, for instance, \cite{Temam_2001} or \cite{ZB+EH+PR-SPDE_2019}, hence we omit their proof.
Secondly, the most difficult point is the convergence
\begin{equation}
\lim_{k\to\infty}\int_{0}^{T_{0}}||\nabla d^{k}|^{2}d^{k}-|\nabla d|^{2}d|_{L^{2}}dt=0, \label{fiu1}
\end{equation}
and hence we prove it here. For this quest we notice that there exists a constant $C>0$ such that for all $k\ge 1$
\begin{align}
\int_{0}^{T_{0}}||\nabla d^{k}|^{2}d^{k}-|\nabla d|^{2}d|^{2}_{L^{2}}dt
&\le C\int_{0}^{T_0}|(\nabla d^{k}-\nabla d):(\nabla d^{k}+\nabla d)d^{k}|_{L^{2}} dt \notag\\
&+ C\int_{0}^{T_0}||\nabla d|(d^{k}-d)|_{L^{2}}dt.\notag
\end{align}
By the H\"older inequality, we have
\begin{align}
\int_{0}^{T_{0}}||\nabla d^{k}|^{2}d^{k}-|\nabla d|^{2}d|_{L^{2}}\,ds
&\le C\left(\int_{0}^{T_{0}}|\nabla d^{k}-\nabla d|^{2}_{L^{4}}\,ds\right)^{\frac{1}{2}}
\left(\int_{0}^{T_{0}}(|\nabla d^{k}|^{2}_{L^{4}}+|\nabla d|^{2}_{L^{4}})\,ds\right)^{\frac{1}{2}}\sup_{t\in [0,T_0]}|d^{k}(t)|_{L^{\infty}} \notag\\
&+ \left(\int_{0}^{T_{0}}|\nabla d|^{2}_{L^{4}}\,ds\right)^{\frac{1}{2}}\left(\int_{0}^{T_{0}}|d^{k}-d|^{2}_{L^{\infty}}\,ds\right)^{\frac{1}{2}}\notag
\end{align}
By the Ladyzhenskaya inequality (\cite[Lemma III.3.3]{Temam_2001}) and the Sobolev embedding $\mathrm{H} ^{1+\theta}\hookrightarrow L^{\infty} (\theta\in (0,1))$, we arrive at
\begin{align}
\int_{0}^{T_{0}}||\nabla d^{k}|^{2}d^{k}-|\nabla d|^{2}d|_{L^{2}}\,ds
&\le C\left(\int_{0}^{T_{0}}|\nabla d^{k}-\nabla d|_{L^{2}}|\Delta d^{k}-\Delta d|_{L^{2}}\,ds\right)^{\frac{1}{2}}\notag\\
&\qquad +C\left(\int_{0}^{T_{0}}|d^{k}-d|^{2}_{\mathrm{H} ^{1+\theta}}\,ds\right)^{\frac{1}{2}}\notag\\
&\le \left(\int_{0}^{T_{0}}|\nabla d^{k}-\nabla d|^{2}_{L^{2}}\,ds\right)^{\frac{1}{2}}\left(\int_{0}^{T_{0}}|\Delta d^{k}-\Delta d|^{2}_{L^{2}}\,ds\right)^{\frac{1}{2}} \notag \\
&\qquad + C\left(\int_{0}^{T_{0}}|d^{k}-d|^{2}_{\mathrm{H} ^{1+\theta}}\,ds\right)^{\frac{1}{2}}\notag
\end{align}
where we have also used the fact that
\[
\sup_{k}\int_{0}^{T}|\nabla d^{k}|^{2}_{L^{4}}\,ds<\infty~ \text{and}~ d_{k}(t)\in \mathcal{M},\, t\in [0,T_0).
\]
The strong convergence (\ref{fiu}) and the fact that
\[
\sup_{k}\left(\int_{0}^{T_{0}}|\Delta d^{k}-\Delta d|^{2}_{L^{2}}\,ds\right)^{\frac{1}{2}}<\infty
\]
completes the proof of (\ref{fiu1}).
\noindent We now study the convergence of the term $\phi^\prime (d^{k})$.
Since $d^{k_{j}}$ $\to$ d strongly in $L^{2}(0,T_{0};\mathrm{H} ^{1})$, we can assume that
$d^{k_{j}}\to$ $d$ ~a.e.~ $(t,x)\in [0,T_{0}]\times\Omega$. Thus,
by the continuity of $\phi^\prime(.)$, we get
\begin{equation}
\phi^\prime(d^{k_{j}})\to \phi^\prime(d)~ a.e.~ (t,x)\in [0,T_{0})\times\Omega.\notag
\end{equation}
Since $d^{k_{j}}(t)\in \mathcal{M}, t\in [0,T_0)$ and, by assumption, $|\phi^\prime(d^{k_{j}})|\le M$, the Lebesgue dominated convergence theorem implies that
\begin{equation}
\phi^\prime(d^{k_{j}})\to \phi^\prime(d)~ \text{in}~ L^{2}([0,T_{0})\times\Omega)=L^{2}(0,T_{0};L^{2}).\notag
\end{equation}
Since $d^{k_{j}}(t) \in \mathcal{M}, \,t\in [0,T_0)$, $|\phi^\prime(d^{k_{j}})|\le M$, we have
\begin{align}
&\int_{0}^{T_{0}}|(d^{k_{j}},\phi^\prime(d^{k_{j}}))d^{k_{j}}-(d,\phi^\prime(d))d|^{2}_{L^{2}}\notag\\
&\le \int_{0}^{T_{0}}|(d^{k_{j}}-d,\phi^\prime(d^{k_{j}}))d^{k_{j}}+ (d,\phi^\prime(d^{k_{j}})-\phi^\prime(d))d^{k_{j}}|^{2}_{L^{2}}\notag\\
&+\int_{0}^{T_{0}}|(d,\phi^\prime(d))(d^{k_{j}}-d)|^{2}_{L^{2}}\notag\\
&\le M\int_{0}^{T_{0}}|d^{k_{j}}-d|^{2}_{L^{2}}dt+\int_{0}^{T_{0}}|\phi^\prime(d^{k_{j}})-\phi^\prime(d)|^{2}_{L^{2}}dt\notag\\
& +\int_{0}^{T_{0}}|d^{k_{j}}-d|^{2}_{L^{2}}dt.\notag
\end{align}
By the convergence $\phi^\prime(d^{k_{j}})\to\phi^\prime(d)$ strongly in $L^{2}(0,T_{0}; L^{2})$ and $d^{k_{j}}\to d$ strongly in $L^{2}(0,T_{0};L^{\infty})$, we obtain
\begin{equation}
\alpha(d^{k_{j}})d^{k_{j}}\to \alpha(d)d ~\text{in}~ L^{2}(0,T_{0};L^{2}).\notag
\end{equation}
Since
\begin{align}
&\int_{0}^{T_{0}}|d^{k_{j}}\times g(t)-d\times g(t)|_{L^{2}}dt\notag\\
&=\int_{0}^{T_{0}}|(d^{k_{j}}-d)\times g(t)|_{L^{2}}dt\notag\\
&\le\int_{0}^{T_{0}}|d^{k_{j}}-d|_{L^{\infty}}|g(t)|_{L^{2}}dt\notag\\
&\le\int_{0}^{T_{0}}|d^{k^{j}}-d|^{2}_{L^{\infty}}dt\int_{0}^{T_{0}}|g(t)|^{2}_{L^{2}}dt,\notag
\end{align}
and $d^{k_{j}}\to d$ strongly in $L^{2}(0,T_{0};L^{\infty})$, we get
\begin{equation}
d^{k_{j}}\times g \to d\times g~ \text{in}~ L^{1}(0,T_{0};L^{\infty}).\notag
\end{equation}
We will now prove that $(v,d)$ satisfies the initial conditions and that $(v,d)\in C([0,T_0]; \mathrm{H} \times \mathrm{H} ^1)$. Towards these goals, we first observe that
since $(v,d)\in L^{\infty}(0,T_{0};\mathrm{H} \times \mathrm{H} ^{1})$ and $$(\partial_{t}v^{k_{j}},\partial_{t}d^{k_{j}}) \rightharpoonup (\partial_{t}v,\partial_{t}d)\text{ in } L^{2}(0,T_{0};\mathrm{V}^{\ast}\times L^{2}),$$ then by the Strauss theorem, see \cite[Lemma III.1.2]{Temam_2001}, we get
\[
(v,d)\in C([0,T_{0}];\mathrm{H} _{w}\times \mathrm{H} ^{1}_{w}),
\]
and $d\in C([0,T_{0}];L^{2})$. Hence,
\begin{align}
\lim_{t\to 0}(v(t),\varphi)=(v_{0},\varphi), \forall \varphi\in \mathrm{H} ,\label{Eq:Weak-ConvInitDatavelo}\\
\lim_{t\to 0}(\nabla d(t),\Psi)=(\nabla d_{0},\Psi), \forall \Psi\in L^{2},\label{Eq:Weak-ConvInitDataOptDir}\\
\lim_{t\to 0}\phi(d(t))=\phi(d_{0})~ \text{in}~ L^{2}.\label{Eq:StrongConve}
\end{align}
From all these passages to the limits we see that the limit $v$ and $d$ satisfy the equations \eqref{eq:LocVelo} and \eqref{eq:LocVelo} in $\mathrm{V}^\ast$ and $L^2$, respectively.
The estimates \eqref{Eq:UnifEstStrongSol-1} and \eqref{Eq:UnifEstStrongSol-2} are established by passing to the limit and using the weak lower semicontinuity of the norms in
the estimates \eqref{Eq:UnifEstNRG}, \eqref{Eq:UnifEstL2} and \eqref{Eq:UnifEstLaplace}.
What remains to prove is the continuity of $(v,d):[0,T_0] \to \mathrm{H} \times D(\hat{\mathrm{A}} ^\frac12)$.
For this we will firstly establish that
\begin{equation}\label{Eq:EstDerrivativeintime-1}
(\partial_tv, \partial d) \in L^2(0,T; \mathrm{V}^\ast \times L^2).
\end{equation}
Towards this aim we recall that it was proved in \cite[Proofs of Eqs (3.27), (3.28), (3.31) and (3.32)]{ZB+EH+PR-SPDE_2019} that there exists a constant $C>0$ such that for $(v,d)\in C([0,T_0);\mathrm{H} \times D(\hat{\mathrm{A}} ^\frac12) )\cap L^2(0,T_0; \mathrm{V}\times D(\hat{\mathrm{A}} ))$
\begin{align}
\lvert -\mathrm{A} v -B(v,v)-\Pi(\Div [\nabla d \odot \nabla d]) \rvert_{L^2(0,T_0; \mathrm{V}^\ast)} \le C,\notag\\
\lvert -\hat{\mathrm{A}} d -v \cdot \nabla d + d\times g \rvert_{L^2(0,T_0; L^2)} \le C.\notag
\end{align}
Note that the estimates in \cite[Eqs (3.27), (3.28), (3.31) and (3.32)]{ZB+EH+PR-SPDE_2019} are for $L^2(0,T_0; \mathrm{V}^\ast)$- and $L^2(0,T; L^2)$-valued random variables, but they remain valid for deterministic $L^2(0,T_0; \mathrm{V}^\ast)$ and $L^2(0,T; L^2)$ functions.
Since $d(t)\in \mathcal{M}$, $t\in [0,T_0)$, we easily infer that there exists a constant $C>0$ such that
\begin{align}
\lvert \lvert \nabla d \rvert^2 d \rvert^2_{L^2(0,T_0;L^2)} \le \lvert \nabla d \rvert^4_{L^4(0,T_0; L^4)} \le \sup_{t\in [0,T_0]} \lvert \nabla d(t) \rvert^2 \int_0^{T_0} \lvert d(s) \rvert^2_{\mathrm{H} ^2} ds< C.\notag
\end{align}
Using \eqref{Eq:LInearGrowthPhiprime} and the constraint $d(t)\in \mathcal{M}$, $t\in [0,T_0)$, easily implies that there exists a constant $C>0$ such that
\begin{equation}
\lvert -\phi^\prime(d) + (\phi^\prime(d) \cdot d) d \rvert_{L^2(0,T_0; L^2)}\le C.\notag
\end{equation}
These estimates completes the proof of \eqref{Eq:EstDerrivativeintime-1}.
Secondly, from \eqref{Eq:EstDerrivativeintime-1}, the fact $(v,d)\in L^2(0,T_0; \mathrm{V}\times D(\hat{\mathrm{A}} ))$ and \cite[Lemma 3.1.2]{Temam_2001} we infer that
$$ (v,d) \in C([0,T_0]; \mathrm{H} \times D(\hat{\mathrm{A}} ^\frac12).$$
This completes the proof of Proposition \ref{Ma2}.
\end{proof}
Now, we proceed to the proof of Theorem \ref{Ma1}.
\begin{proof}[Proof of Theorem \ref{Ma1}]
In order to prove the theorem, let us denote by $\Sigma$ the set of local solutions to problem \eqref{1b}. By Proposition \ref{Ma2}, the set $\Sigma$ is non empty and we can and will assume that the time of existence $T_0$ of any local solution $(v,d)\in \Sigma$ satisfies the property \eqref{eqn-t_o-2} stated in Proposition \ref{Ma2}.
Let us define the relation $\lesssim$ on $\Sigma$ by
$$(y_{1};\sigma_{1})\lesssim (y_{2};\sigma_{2})\text{ if } \sigma_{1}\le\sigma_{2} \text{ and } y_{2}=y_{1} \text{ on } [0,\sigma_{1}] , \text{ for all }
(y_{i};\sigma_{i}):=((u_{i},d_{i});\sigma_{i})\in\Sigma, i=1,2.$$
We briefly show below that the relation $\lesssim$ is reflexive, antisymmetric and transitive.
\noindent \textbf{Reflexivity}. Let $(y_{1};\sigma_{1})\in\Sigma$. Then $(y_{1};\sigma_{1})\lesssim (y_{1};\sigma_{1})$ by definition.
\noindent \textbf{Antisymmetry}. Let $(y_{i};\sigma_{i})\in\Sigma$, $i=1,2$. Suppose that $(y_{1};\sigma_{1})\lesssim (y_{2};\sigma_{2})$ and $(y_{2};\sigma_{2})\lesssim (y_{1};\sigma_{1})$. This implies that
$y_{2}=y_{1}$ on $[0,\sigma_{1}]$, $y_{1}=y_{2}$ on $[0,\sigma_{2}]$ and $\sigma_{1}=\sigma_{2}$. This proves the antisymmetric property of $\lesssim$.
\noindent \textbf{Transitivity}. Let $(y_{i},\sigma_{i})\in\Sigma$, $i=1,2,3$. Suppose that $(y_{1};\sigma_{1})\lesssim (y_{2};\sigma_{2})$ and $(y_{2};\sigma_{2})\lesssim (y_{3},\sigma_{3})$. We will prove that $(y_{1};\sigma_{1})\lesssim (y_{3};\sigma_{3})$. For this purpose, we observe that
\begin{align}
&\sigma_{1}\le\sigma_{2}~ \text{and} ~y_{2}=y_{1}~ \text{on}~ [0,\sigma_{1}],\notag\\
&\sigma_{2}\le\sigma_{3}~ \text{and}~ y_{3}=y_{2}~ \text{on}~ [0,\sigma_{2}].\notag
\end{align}
Hence
\begin{equation}
\sigma_{1}\le \sigma_{3},\label{Ma5}
\end{equation}
\begin{equation}
y_{3}=y_{2}=y_{1}~ \text{on}~ [0,\sigma_{1}].\label{Ma6}
\end{equation}
The facts (\ref{Ma5}) and (\ref{Ma6}) prove the transitivity property of $\lesssim$.
In order to prove the existence of a maximal element in $\Sigma$ we shall use the Kuratowski-Zorn Lemma. Hence, we need to prove that all increasing chain in $\Sigma$ has an upper bound. For this purpose, let $((v_k, d_k);\sigma_{k})_{k \in \mathbb{N}}$ be an increasing chain in $\Sigma$.
We will show that this sequence has an upper bound. In order to do that we set
$$ \sigma=\sup_{k \in \mathbb{N} } \sigma_k$$ and define $(v,d):[0,\sigma) \to \mathrm{H} \times \mathrm{H} ^1$ by
\begin{equation}
(v,d)_{\vert_{[0,\sigma_k)}}=(v_k,d_k).\notag
\end{equation}
From these definitions it is clear that for all $k$ $\sigma_k\le\sigma $ and $(v,d)=(v_k,d_k)$ on $[0,\sigma_k)$ for all $k \in \mathbb{N}$. Hence, $((v,d);\sigma)\in \Sigma$ and it is an upper bound of the increasing chain $((v_k, d_k);\sigma_{k})_{k \in \mathbb{N}}$.
Therefore, it now follows from the Kuratowski-Zorn lemma that $\Sigma$ has a maximal element which is a maximal local solution to \eqref{1b}. This completes the proof of Theorem \ref{Ma1}.
\end{proof}
The following result gives an important property of the energy of the maximal solution $((v,d); T_{\ast})$ near the point $T_\ast$.
\begin{Prop}
Let $\varepsilon\,_0>0$, $\varrho_0>0$ and $\theta_0$ be the constants and the function from Proposition \ref{Ma2} and
$$ \varepsilon\,_1^2= 2\mathcal{E}_{2\varrho_0}(v_0,d_0).$$ Let $((v,d);T_{\ast})$ be the maximal solution defined in Theorem \ref{Ma1}. Then,
\begin{equation}\label{Eq:LossNRJ}
\limsup_{t\nearrow T_{\ast}}\sup_{x\in\Omega}\int_{B_{R}(x)}\left[|v(t)|^{2}+|\nabla d(t)|^{2}+\phi(d(t))\right]dy\ge\varepsilon^{2}_{1},
\end{equation}
for all $R\in (0, \varrho_0]$.
\end{Prop}
\begin{proof}
We prove the proposition by contradiction. Suppose that there exists $R>0$ such that
\[
\limsup_{t\nearrow T_{\ast}}\sup_{x\in\Omega}\int_{B_{R}(x)}\left[|v(t)|^{2}+|\nabla d(t)|^{2}+\phi(d(t))\right]dy<\varepsilon^{2}_{1}.
\]
Thus, there exists an increasing sequence $(t_{n})_{n\in\mathbb{N}}$ such that
\begin{equation}
t_{n}\nearrow T_{\ast}~ \text{as}~ n\to\infty,\label{Ma10-0}
\end{equation}
and
\begin{equation}\label{Ma10}
\lim_{n\to\infty}\sup_{x\in\Omega}\int_{B_{R}(x)}\left[|v(t_{n})|^{2}+
|\nabla d(t_{n})|^{2}+\phi(d(t_{n}))\right]dy<\varepsilon^{2}_{1}.
\end{equation}
From (\ref{Ma10-0}) and \eqref{Ma10} we deduce that
\begin{itemize}
\item[(a)] $\forall$ $\delta>0$, there exists $m\in\mathbb{N}$ such that $0<T_{\ast}-t_{m}<\delta$,
\item[(b)] we can and will assume that for all $n\in\mathbb{N}$
\begin{equation}
\sup_{x\in\Omega}\int_{B_{R}(x)}\left[|v(t_{n})|^{2}+|\nabla d(t_{n})|^{2}+\phi(d(t_{n}))\right]dy<\varepsilon^{2}_{1}.\notag
\end{equation}
\end{itemize}
By the global energy inequality \eqref{Eq:NRJ-Inequality}, we get
\begin{equation}
E_{\ast}=\sup_{t<T_{\ast}}\mathcal{E}(v(t),d(t))<\infty.\label{Ma10-1}
\end{equation}
By (a), for $\delta=\frac{1}{2}\theta_{0}(\varepsilon^{2}_{1},E_{\ast})R^{2}$, there exists $m\in\mathbb{N}$ such that
\begin{equation}
0<T_{\ast}-t_{m}<\frac{1}{2}\theta_{0}(\varepsilon^{2}_{1},E_{\ast})R^{2}_{0}.\label{Ma11}
\end{equation}
By (b), (\ref{Ma10-1}) and \eqref{Ma10} we get
\begin{equation}
\mathcal{E}(v(t_{m}),d(t_{m}))\le E_{\ast},\notag
\end{equation}
and
\begin{equation}
\sup_{x\in\Omega}\int_{B_{R}(x)}\left[|v(t_{m})|^{2}+|\nabla d(t_{m})|^{2}+\phi(d(t_{m}))\right]dy<\varepsilon^{2}_{1}.\notag
\end{equation}
Hence, by Proposition \ref{Ma2}, there exists a solution $(\tilde{v},\tilde{d})$ defined on $[t_{m},\tau +t_{m}]$ with
$$\tau\ge\theta_{0}(\varepsilon^{2}_{1},\mathcal{E}(v(t_{m}),d(t_{m}))R^{2}_{0}.$$ But observe that $\theta_{0}(\varepsilon^{2}_{1},E_{0})$ is a non-increasing function of the initial energy $E_{0}$. Hence, by (\ref{Ma11})
\begin{equation}
\tau\ge \theta_{0}(\varepsilon^{2}_{1},\mathcal{E}(v(t_{m}),d(t_{m}))R^{2}_{0}\ge\theta_{0}(\varepsilon^{2}_{1},E_{\ast})R^{2}_{0}>2(T_{\ast}-t_{m}).\notag
\end{equation}
By doing elementary calculation we obtain
\begin{align}
t_{m}+\tau
&\ge 2(T_{\ast}-t_{m})+t_{m}\notag\\
&\ge T_{\ast}+T_{\ast}-t_{m} \notag\\
&>T_{\ast}~ (~\text{because}~ T_{\ast}-t_{m}>0, \text{ see \eqref{Ma11}}).\notag
\end{align}
Hence, we get the existence of a local solution $(\tilde{v},\tilde{d}):[0,\tau+t_{m})\to \mathrm{H} \times \mathrm{H} ^{1}$ with $\tau+t_{m}>T_{\ast}$ and
$(\tilde{v},\tilde{d})=(v,d)~ \text{on}~ [0,T_{\ast})$. This contradicts the fact that $((v,d);T_{\ast})$ is a maximal solution.
\end{proof}
We now give the promised proof of Theorem \ref{thm-main}.
\begin{proof}[Proof of Theorem \ref{thm-main}]
Let $((v,d),T_{\ast})$ be the maximal local strong solution to \eqref{1b} constructed from Theorem \ref{Ma1}. Firstly, we set $T_\ast=T_1$ and prove the following result.
\begin{Lem}\label{Lem:Continuity}
The maximal local strong solution $((v,d),T_1)$ satisfies
\begin{align}
(\partial_t v, \partial_td) \in L^2(0,T_1; D(\mathrm{A} ^{-\frac32}) \times D(\hat{\mathrm{A}} )^{\ast} ),\notag\\
(v,d)\in C([0,T_1 ];\mathrm{H} \times L^{2}(\Omega)).\notag
\end{align}
\end{Lem}
\begin{proof}[Proof of Lemma \ref{Lem:Continuity}]
We recall that
\begin{equation*}
\partial_{t}v+Av + \Pi(v\cdot\nabla v)=-\Pi(\mathrm{div}\, (\nabla d\odot\nabla d)) + f.\notag
\end{equation*}
Firstly, let us prove that $\partial_{t}v\in L^{2}(0,T_{1};D(\mathrm{A} ^{-\frac32}))$. For this aim, let $\varphi\in D(\mathrm{A} ^\frac32)$ be fixed. Then,
\begin{align*}
|\langle {\partial_t } v ,\varphi\rangle|
&=\left|\int_{\Omega}(\nabla v\cdot \nabla\varphi+v\cdot \nabla v.\varphi)\;dx-\int_{\Omega}\nabla d\odot\nabla d:\nabla\varphi \;dx +\int_{\Omega}f\varphi \;dx\right|\notag\\
&\le \|\nabla v\|_{L^{2}}\|\nabla\varphi\|_{L^{2}}+\|v\|_{L^{2}}\|\nabla v\|_{L^{2}}\|\varphi\|_{C^{0}(\Omega)}+\|\nabla d\|^{2}_{L^{2}}\|\nabla\varphi\|_{L^{2}}+\|f(t)\|_{\mathrm{H} ^{-1}}\|\nabla \varphi\|_{L^{2}}\notag\\
&\le\left(\|\nabla v\|_{L^{2}}+\|v\|_{L^{2}}\|\nabla v\|_{L^{2}}+\|\nabla d\|^{2}_{L^{2}}+\|f(t)\|_{\mathrm{H} ^{-1}}\right)\|\varphi\|_{\mathrm{H} ^{3}}\notag\\
&\le\left(\|\nabla v\|_{L^{2}}+\|v\|_{L^{2}}\|\nabla v\|_{L^{2}}+\|\nabla d\|^{2}_{L^{2}}+\|f(t)\|_{\mathrm{H} ^{-1}}\right)\|\varphi\|_{D(\mathrm{A} ^{\frac32})},\notag
\end{align*}
where we used the fact $\mathrm{H} ^{3}(\Omega)\subset C^{0}(\Omega)$ and $D(\mathrm{A} ^\frac32)\hookrightarrow \mathrm{H} ^3$.
Then, we deduce that
\begin{equation*}
|\partial_tv|_{D(\mathrm{A} ^{-\frac32})}\le \|\nabla v\|_{L^{2}}+\|v\|_{L^{2}}\|\nabla v\|_{L^{2}}+\|\nabla d\|^{2}_{L^{2}}+\|f \|_{\mathrm{H} ^{-1}}.
\end{equation*}
The last line, the Assumption \ref{AssumptionMain} and the facts $(v,d)\in L^\infty(0,T_1;\mathrm{H} \times \mathrm{H} ^1)$ and $v\in L^2(0,T_1; \mathrm{V})$ imply that $\partial_t v \in L^{2}(0,T_{1};D(\mathrm{A} ^{-\frac32}))$. Hence, since $v\in L^{2}(0,T_{1};\mathrm{V})$ we have $v\in C([0,T_{1}];L^{2})$.
Secondly, we estimate $\partial_{t}d$. For this purpose, let $\Psi$ $\in$ $D(\hat{\mathrm{A}} )$ be fixed. Recall that
\begin{equation}
\partial_{t}d=\Delta d + |\nabla d|^{2}d+v\cdot \nabla d-\phi^\prime(d) +\alpha(d)d+d\times g.\notag
\end{equation}
Then,
\begin{align}
&\left|\langle \Delta d+|\nabla d|^{2}d-v\cdot \nabla,\Psi\rangle\right|\notag\\
&=\left|\int_{\Omega}(\nabla d\cdot \nabla\Psi + v\cdot \nabla d.\Psi)\;dx-\int_{\Omega}|\nabla d|^{2}d.\Psi \;dx\right|\notag\\
&\le \left(\|\nabla d\|_{L^{2}}+\|v\|_{L^{2}}\|\nabla d\|_{L^{2}}+\|\nabla d\|^{2}_{L^{2}}\right)\|\Psi\|_{\mathrm{H} ^{2}}.\label{F1}
\end{align}
By the boundedness of $\phi^\prime(d)$ on $\mathbb{S}^{2}$, we have
\begin{equation}
\left|\langle -\phi^\prime(d)+\alpha(d)d, \Psi\rangle\right|\le C\|\Psi\|_{\mathrm{H} ^{2}}.\label{F2}
\end{equation}
We also get
\begin{equation}
\left|\langle d\times g,\Psi\rangle\right|\le C|g(t)|_{L^{2}}\|\Psi\|_{\mathrm{H} ^{2}}. \label{F3}
\end{equation}
The estimates (\ref{F1})-(\ref{F3}) imply that $\partial_{t}d\in L^{2}(0,T_{1};D(\hat{\mathrm{A}} )^\ast)$ because $d\in L^{\infty}(0,T_{1};\mathrm{H} ^{1})$ and $v\in L^{\infty}(0,T_{1};L^{2})$ and $D(\hat{\mathrm{A}} ) \hookrightarrow \mathrm{H} ^2$. The fact $\partial_{t}d\in L^{2}(0,T_{1};D(\hat{\mathrm{A}} )^\ast)$ combined with $d\in L^{2}(0,T_{1};\mathrm{H} ^{1})$ imply that $d\in C^{0}([0,T_{1}];L^{2}(\Omega))$, see \cite[Lemma III.1.4 ]{Temam_2001}. This ends the proof of the lemma.
\end{proof}
We can now continue with the proof of Theorem \ref{Ma1}. From Lemma \ref{Lem:Continuity}, we can define
\begin{equation}
(v(T_{1}),d(T_{1}))=\lim_{t\nearrow T_{1}}\left(v(t),d(t)\right)~ \text{in}~ L^{2}(\Omega)\times L^{2}(\Omega).\notag
\end{equation}
Since $\nabla d\in L^{\infty}(0,T_{1};L^{2}(\Omega))$, then $\nabla d(t)\rightharpoonup \nabla d(T_{1})$ weakly in $L^{2}(\Omega)$. This and Theorem \ref{Thm:StraussThm} implies that
$v(T_{1})\in \mathrm{H} $ and $d(T_{1})\in \mathrm{H} ^{1}$ and $(v,d)\in C_w([0,T_1]; \mathrm{H} \times \mathrm{H} ^1)$. Moreover,
thanks to the strong convergence $d(t)\to d(T_{1})$ in $L^{2}(\Omega)$, we show using the same idea as in the proof of \eqref{Eq:SphereCondition} that
\begin{equation}
d(T_{1})\in \mathcal{M}.\label{F5}
\end{equation}
Also, it is not difficult to prove that
\begin{align}
\mathcal{E}(v(T_{1}),d(T_{1}))
&\le \lim_{t\nearrow T_{1}}\mathcal{E}(v(t),d(t))\notag\\
&\le \mathcal{E}(v_{0},d_{0}) + C\left(|f|^{2}_{L^{2}(0,T;\mathrm{H} ^{-1})}+|g|^{2}_{L^{2}(0,T_{1};L^{2})}\right)<\infty. \label{F4}
\end{align}
We also prove that
\begin{equation}\label{F8}
\mathcal{E}(v(T_{1}),d(T_{1}))\le \mathcal{E}(v_{0},d_{0}) + \frac12 \left(\int_{0}^{T_{1}}|f(t)|^{2}_{\mathrm{H} ^{-1}}dt+\int_{0}^{T_{1}}|g(t)|^{2}_{L^2}dt\right)-\varepsilon\,_{1}^{2}.
\end{equation}
In fact, the inequality \eqref{Eq:LossOfNRJ} implies that there exists a sequence $t_{n}\nearrow T_{1}$ and $x_{0}\in {\Omega}$ such that
\begin{equation}
\limsup_{t_{n}\nearrow T_{1}}\int_{B_{R}(x_{0})}\left[|v(t_{n})|^{2}+|\nabla d(t_{n})|^{2}+\phi(d(t_{n}))\right]\;dx\ge \varepsilon\,^{2}_{1}, \forall R\in (0, \varrho_0].\notag
\end{equation}
Therefore,
\begin{align}
\mathcal{E}(v(T_{1}),d(T_{1}))
&=\lim_{R\searrow 0}\int_{\Omega \setminus B_{R}(x_{0})}\mathcal{E}(v(T_{1}),d(T_{1}))dy\notag\\
&\le \lim_{R\searrow 0}\liminf_{t_{n}\nearrow T_{1}}\int_{\Omega \setminus B_{R}(x_{0})}\mathcal{E}(v(t_{n}),d(t_{n}))dy\notag\\
&\le \lim_{R\searrow 0}\left[\liminf_{t_{n}\nearrow T_{1}}\int_{\Omega}\mathcal{E}(v(t_{n}),d(t_{n}))dy-\limsup_{t_{n}\nearrow T_{1}}\int_{B_{R}(x_{0})}\mathcal{E}(v(t_{n}),d(t_{n}))dy\right]\notag\\
&\le \liminf_{t_{n}\nearrow T_{1}}\mathcal{E}(v(t_{n}),d(t_{n}))-\varepsilon\,_{1}^{2}\notag\\
&\le \mathcal{E}(v_0,d_0)+ \frac12 \left(\int_{0}^{T_{1}}|f(t)|^{2}dt +\int_{0}^{T_{1}}|g(t)|^{2}dt\right)-\varepsilon\,_{1}^{2}.\notag
\end{align}
This completes the proof of (\ref{F8}).
With (\ref{F4}) and (\ref{F5}) at hand, one can apply Proposition \ref{Ma2} to construct a local strong solution $(\tilde{v},\tilde{d}):[T_{1}.T_{2}]\to \mathrm{H} \times \mathrm{H} ^{1}$ with the initial data $(\tilde{v}(T_{1}),\tilde{d}(T_{1}))=(v(T_{1}),d(T_{1}))\in \mathrm{H} \times \mathrm{H} ^{1}$.
Moreover, there are constants $\varrho_0>0$, $\varepsilon\,_2\in (0,\varepsilon\,_0)$ such that
\begin{equation}
\limsup_{t\nearrow T_{2}}\sup_{x\in{\Omega}}\int_{B_{R}(x)}\mathcal{E}(\tilde{v}(t),\tilde{d}(t))\ge \varepsilon\,^{2}_{2},~~ \forall R\in (0, \varrho_0].\label{F6}
\end{equation}
Furthermore,
\begin{equation}\label{F8-b}
\mathcal{E}(\tilde{v}(T_{2}),\tilde{d}(T_{2}))< \mathcal{E}(v(T_1),d(T_1)) + \frac12 \left(\int_{T_1}^{T_{2}}|f(t)|^{2}_{\mathrm{H} ^{-1}}dt+\int_{T_1}^{T_{2}}|g(t)|^{2}_{L^2}dt\right)-\varepsilon\,_{2}^{2}.
\end{equation}
Hence, we can construct a sequence of maximal local strong solutions $((v_i, d_i);T_i)_{i=1}^L$ satisfying: there are constants\footnote{In fact, $ \tilde{\varepsilon\,}_1=\min\{\varepsilon\,_i;\,1\le i\le L \}$} $\varrho_0>0$, $\tilde{\varepsilon\,}_1\in (0,\varepsilon\,_0)$ such that for all $1\le i\le L$
\begin{align}
\limsup_{t\nearrow T_{i}}\sup_{x\in{\Omega}}\int_{B_{R}(x)}\mathcal{E}(v_i(t),d_i(t))\ge \tilde{\varepsilon\,}^{2}_{1},~~ \forall R\in (0, \varrho_0].\label{F6-b}\\
\mathcal{E}(v_i(T_{i}),d_i(T_{i}))\le \mathcal(v_i(T_{i-1}),d_i(T_{i-1}))+ \frac12\left( \int_{T_{i-1}}^{T_{i}}|f(t)|^{2}_{\mathrm{H} ^{-1}}dt +\int_{T_{i-1}}^{T_{i}}|g(t)|^{2}_{L^2}dt\right)-\tilde{\varepsilon\,}_1^2.\label{Eq:LossOfNRJ-0}
\end{align}
In order to construct the global solution we consider a function $(\mathbf{v},\mathbf{d})$ defined
\begin{equation}\label{Eq:DefGlobalWeakSol}
(\mathbf{v},\mathbf{d})_{\lvert_{[T_{i-1}, T_i)} } =(v_i,d_i), \;\; 1\le i \le L,
\end{equation}
and prove that $L<\infty$. Towards this aim, we first deduce from \eqref{Eq:LossOfNRJ-0} that
\begin{equation}\label{Eq:LossOfNRJ}
\mathcal{E}(\mathbf{v}(T_{i}),\mathbf{d}(T_{i}))\le \mathcal(\mathbf{v}(T_{i-1}),\mathbf{d}(T_{i-1}))+ \frac12\left( \int_{T_{i-1}}^{T_{i}}|f(t)|^{2}_{\mathrm{H} ^{-1}}dt +\int_{T_{i-1}}^{T_{i}}|g(t)|^{2}_{L^2}dt\right)-\tilde{\varepsilon\,}_1^2.
\end{equation}
for $i=1,\ldots ,L$. Here $T_{0}=0$.
Iterating the estimate \eqref{Eq:LossOfNRJ} yields
\begin{equation}
\mathcal{E}(\mathbf{v}(T_{L}), \mathbf{d}(T_{L}))\le \mathcal{E}(v_{0},d_{0})+ C\left(\int_{0}^{T_{L}}(|f(t)|^{2}_{\mathrm{H} ^{-1}}+|g(t)|^{2}_{L^{2}})dt\right) -L\tilde{\varepsilon\,}^{2}_{1}.
\end{equation}
This implies that
\begin{equation}
L\le\frac{1}{\tilde{\varepsilon\,}_{1}^{2}}\left[\mathcal{E}(v_{0},d_{0})+ C\left(\int_{0}^{T}(|f(t)|^{2}_{\mathrm{H} ^{-1}}+|g(t)|^{2}_{L^{2}})dt\right)\right]<\infty.\notag
\end{equation}
In order to complete the proof we need to check that $(\mathbf{v},\mathbf{d})$ is indeed a global weak solution. But this follows from the definition \eqref{Eq:DefGlobalWeakSol},
the fact that each $((v_i,d_i);T_i)$ are maximal local strong solution defined on $[T_{i-1}, T_i)$ and satisfying
\begin{align}
& (v_i,d_i)\in C([T_{i-1}, T_i);\mathrm{H} \times \mathrm{H} ^1 ),\notag\\
& (\partial_tv_i, \partial_t d_i) \in L^2(T_{i-1}, T_i; D(\mathrm{A} ^{-\frac32})\times D(\hat{\mathrm{A}} )^\ast),\notag\\
& d_i(t) \in \mathcal{M} \text{ for all } t \in [ T_{i-1}, T_i).\notag
\end{align}
\end{proof}
\section{On the regularity and the set of singular times of the solutions when the data is small}\label{Sec:RegCompactSmallData}
In this section we prove that the set of singular time reduces to the final time horizon $T$ when the data are small enough.
Let us start with the following conditional regularity of a strong solution $((v,d);T_\ast)$.
\begin{Prop}\label{prop-reg-zb}
Let $(v_0,d_0)\in \mathrm{V}\times D(\hat{\mathrm{A}} )$, $T>0$, $f\in L^2(0,T; L^2)$, $g\in L^2(0,T; \mathrm{H} ^1)$ and $(v,d)$ be a strong solution to \eqref{1b} such that
\begin{equation}\notag
\begin{split}
v\in C([0,T];\mathrm{H} ) \cap L^2(0,T;\mathrm{V} )\\
d\in C([0,T];\mathrm{H} ^1) \cap L^2(0,T;D(\hat{\mathrm{A}} )).
\end{split}
\end{equation}
Then,
\begin{equation}\notag
\begin{split}
v\in C([0,T];\mathrm{V} ) \cap L^2(0,T;D(\mathrm{A} ))\\
d\in C([0,T]; D(\hat{\mathrm{A}} )) \cap L^2(0,T;D(\hat{\mathrm{A}} ^{3/2})).
\end{split}
\end{equation}
\end{Prop}
\begin{proof}[Proof of Proposition \ref{prop-reg-zb}]
We start the proof with the following claim.
\begin{Clm}\label{Clm:Regular-Sol}
There exist constants $K>0$ and $K_7>0$ depending only on the norms of $(v_0,d_0)\in \mathrm{V} \times D(\hat{\mathrm{A}} ^\frac12)$ and the norms of $(f,g)\in L^2([0,T]; \mathrm{H} \times \mathrm{H} ^1)$ such that the following holds.
If $(v,d)$ is a regular solution on
some interval $[0,T_\ast] \subset [0,T]$ such that
\begin{equation}\label{EQ:AssumL4Norms}
\int_0^T \lvert \nabla d(s)\rvert^4_{L^4} ds \le K_7 \text{ and } \int_0^T \lvert v(s)\rvert^4_{L^4} ds \le K_7,
\end{equation}
then,
\begin{equation}\label{Eq:Derivativehigheroerdernorms-Fin-04}
\begin{split}
\lvert \Delta d (t) \rvert^2_{L^2} + \lvert \nabla v(t) \rvert^2_{L^2} \le K,\;\; t \in [0,T_\ast],\\
\int_0^{T_\ast}\left(\lvert \mathrm{A} v (r) \rvert^2_{L^2} + \lvert \nabla \Delta d(r) \rvert^2_{L^2}\right)dr \le K.
\end{split}
\end{equation}
\end{Clm}
\begin{proof}[Proof of Claim \ref{Clm:Regular-Sol}]
Let $(v,d)$ be a regular solution to \eqref{1b} on $[0,T_\ast]\subset [0,T]$. Then, we infer from Lemma \ref{Lem:EstinHigherorderNorms} that there exists a constant $K_3>0$ which depends on the norms of $(v_0,d_0)\in \mathrm{V}\times \mathrm{H} ^1$ and $(f,g)\in L^2(0,T; H\times \mathrm{H} ^1)$ such that
\begin{equation}\label{Eq:EstinHigherorderNorms-6}
\begin{split}
\sup_{0\le s\le \tau} \left(\lvert \nabla v(s) \rvert^2_{L^2} + \lvert \Delta d(s) \rvert^2_{L^2} \right) + 2 \int_0^{\tau} \left( \lvert \nabla \Delta d(s) \rvert^2_{L^2} + \lvert A v(s) \rvert^2_{L^2}\right)ds \\
\le K_3e^{K_3 \int_0^{\tau}[\lvert \nabla d(r) \rvert^4_{L^4}+ \lvert v(r) \rvert^4_{L^4}+ \lvert \nabla d(r) \rvert^2_{L^4}+ \lvert v(r) \rvert^2_{L^4}]dr }.
\end{split}
\end{equation}
This and the assumption \eqref{EQ:AssumL4Norms} implies the desired inequality \eqref{Eq:Derivativehigheroerdernorms-Fin-04}. This proves the claim.
\end{proof}
Now we give the proof of Proposition.
Since $(v_0,d_0)\in \mathrm{V}\times D(\hat{\mathrm{A}} )$, $f\in L^2(0,T; L^2)$, $g\in L^2(0,T; \mathrm{H} ^1)$, we can apply Theorem \ref{th} to assert that there exists a time $T_0 \leq T$ which depends on $K$ and
the norms of $(f,g)$ (on [0,T]) and a unique solution $(\tilde{v},\tilde{d}):[0,T_0] \to \mathrm{V}\times D(\hat{\mathrm{A}} )$ to \eqref{1b} such that
\begin{equation}\notag
\begin{split}
\tilde v\in C([0,T_0],\mathrm{V} ) \cap L^2(0,T_0;D(\mathrm{A} ))\\
\tilde d\in C([0,T_0],D(\hat{\mathrm{A}} )) \cap L^2(0,T_0;D(\hat{\mathrm{A}} ^{3/2})).
\end{split}
\end{equation}
Hence, by Proposition \ref{Prop:Uniq} we infer that
$$ (\tilde{v}, \tilde{d})=(v,d) \text{ on } [0,T_\ast].$$
Thus, since $(v,d)$ is a strong solution on $[0,T_0]$, it follows from the estimates \eqref{Eq:UnifEstStrongSol-1} and \eqref{Eq:UnifEstStrongSol-2} that the regular solution $(v,d)$ on $[0,T_0]$ satisfies \eqref{EQ:AssumL4Norms} on $[0,T_0]$. Hence, the above claim implies that
\begin{equation}\notag
(v,d)\in C([0,T_0]; \mathrm{V} \times D(\hat{\mathrm{A} } )) \cap L^2(0,T_0; D(\mathrm{A} ) \times D(\hat{\mathrm{A} }^\frac32).
\end{equation}
Thanks to the claim again, we can repeat the above procedure finitely many times, say, on $[T_0, T_1]$, \ldots $[T_N, T_\ast]$ to assert that
\begin{equation}\notag
(v,d)\in C([T_i,T_{i+1}]; \mathrm{V} \times D(\hat{\mathrm{A} } )) \cap L^2(0,T_0; D(\mathrm{A} ) \times D(\hat{\mathrm{A} }^\frac32),\;\; i\in \{0, \ldots, N \}.
\end{equation}
With this we conclude the proof of the proposition.
\end{proof}
\begin{Thm}\label{Thm:NoSingSmall}
Let $\varepsilon\,_0>0$ and $\tilde{\varepsilon\,}_1\in (0,\varepsilon\,_0)$ be the constants from Proposition \ref{Ma2} and Theorem \ref{thm-main}, respectively. If the data $(v_0,d_0)\in \mathrm{H} \times \mathrm{H} ^1$ and $(f,g)\in L^2(0,T; L^2\times \mathrm{H} ^1)$ satisfy
\begin{equation}\label{Eq:AsummSmallData}
\mathcal{E}(v_0,d_0) +\frac12\int_0^T\left(\lvert f \rvert^2_{\mathrm{H} ^{-1}} + \lvert g \rvert^2_{L^2}\right)ds\le 2 \tilde{\varepsilon\,}_1^2,
\end{equation}
then \eqref{1b} has a unique global strong solution $(v,d):[0,T) \to \mathrm{H} \times \mathrm{H} ^1$ satisfying
\begin{equation}\label{Eq:MaxRegSmallData}
(v,d)\in C([0,T), \mathrm{H} \times (\mathrm{H} ^1\cap\mathcal{M}))\cap C([0,T]; \mathrm{H} \times L^2) \cap L^2(0,T; \mathrm{V}\times D(\hat{\mathrm{A}} )).
\end{equation}
That is, the global strong solution does not have any singular times.
Moreover, if $(v_0, d_0)\in V\times D(\hat{\mathrm{A} })$ and $(f,g)\in L^2(0, T; \mathrm{H} \times \mathrm{H} ^1)$, then
\begin{equation}\label{Eq:MaxRegSmallData-2}
(v,d)\in C([0,T), \mathrm{V}\times (D(\hat{\mathrm{A} })\cap\mathcal{M}) )\cap L^2(0,T; D{\mathrm{A} }\times D(\hat{\mathrm{A}} ^\frac32)).
\end{equation}
\end{Thm}
\begin{proof}
Let $(v_0,d_0)\in \mathrm{H} \times \mathrm{H} ^1$ and $(f,g)\in L^2(0,T; \mathrm{H} ^{-1} \times \mathrm{H} ^1)$. Then, by Theorem \ref{thm-main} there exist a finite set $S=\{0< T_1< \ldots<T_L< T\}$ and a global weak solution $(v,d)$ to \eqref{1b} satisfying the properties (1)-(3) of Theorem \ref{thm-main}. We will show that $T_1=T$ if the smallness condition \eqref{Eq:AsummSmallData} holds. For this purpose, we argue by contradiction. Assume that $T_1<T$ and that $T_2=T$. Then, by part (2) of Theorem \ref{thm-main}
\begin{equation}\notag
\mathcal{E}(v(T_2), d(T_2)) \le \mathcal{E}(v_0,d_0) + \frac12\int_{0}^{T} \left(\lvert f\rvert^2_{\mathrm{V}^\ast}+ \lvert g\rvert^2_{L^2} \right) ds -2\tilde{\varepsilon\,}_1^2,
\end{equation}
which altogether with the assumption \eqref{Eq:AsummSmallData} yields
\begin{equation}\notag
\limsup_{t\to T_{2}}\sup_{x\in\Omega}\int_{B_{R}(x)}\left[|v(t,y)|^{2}+|\nabla d(t,y)|^{2}+\phi(d(t,y))\right]dy\le \mathcal{E}(v(T_2), d(T_2)) \le 0.
\end{equation}
This clearly contradicts the fact that
\begin{equation}\notag
\limsup_{t\to T_{2}}\sup_{x\in\Omega}\int_{B_{R}(x)}\left[|v(t,y)|^{2}+|\nabla d(t,y)|^{2}+\phi(d(t,y))\right]dy\ge\tilde{\varepsilon}^{2}_{1}>0,
\end{equation}
for all $R\in (0, \varrho_0]$. This completes the proof of the first part of the theorem.
The second part of the theorem follows easily from the Proposition \ref{prop-reg-zb}. Hence, the proof of the theorem is completed.
\end{proof}
The last, but not the least, result of this section is about the precompactness of the orbit $(v(t), d(t))$, $t\in [0,T)$, in $\mathrm{H} \times D(\hat{\mathrm{A}} )$. This result will require the following set of conditions on the map $\phi$.
\begin{assume}\label{Assum:PhiSpecial}
Let $\xi\in \mathbb{R}^3$ be a constant vector and $\phi: \mathbb{R}^3 \to [0,\infty)$ be the map defined by
\begin{equation}\notag
\phi(d)= \frac12 \lvert d -\xi \rvert^2, d\in \mathbb{R}^3.
\end{equation}
\end{assume}
It is clear that if $\phi$ satisfies this assumption, the it also satisfies Assumption \ref{Assum:EnergyPotential}.
\begin{Thm}\label{Thm:PrecompactInHH1}
Let $\varepsilon\,_0>0$ and $\varepsilon\,_1\in (0,\varepsilon\,_0)$ be the constants from Proposition \ref{Ma2} and Theorem \ref{thm-main}, respectively.
Then, there exists a constant $\kappa_1\in (0,\varepsilon\,_1]$ such that the following hold.
If $(v_0,d_0)\in \mathrm{H} \times \mathrm{H} ^1$ and $(f,g)\in L^2(0,T; \mathrm{V}^\ast \times \mathrm{H} ^1)$ satisfy
\begin{equation}\label{Eq:SmallAssumSmooth}
\mathcal{E}(v_0,d_0) +\frac12\int_0^T\left(\lvert f \rvert^2_{\mathrm{H} ^{-1}}+ \lvert g \rvert^2_{L^2}\right)ds\le 2 \kappa_1^2,
\end{equation}
then \eqref{1b} has a unique global strong solution $(v,d):[0,T) \to \mathrm{H} \times \mathrm{H} ^1$ satisfying:
\begin{enumerate}
\item $(v,d)$ does not have any singular point,
\begin{equation}\label{Eq:MaxRegSmallData-2-B}
(v,d)\in C([0,T), \mathrm{V}\times D(\hat{\mathrm{A} }) )\cap L^2(0,T; D({\mathrm{A} })\times D(\hat{\mathrm{A}} ^\frac32)).
\end{equation}
\item There exists a constant $K_1>0$ such that for all $T>0$
\begin{equation}\label{Eq:EstVeloOptDirInVH2-0}
\sup_{t\in [0,T)} (\lvert v(t) \rvert^2_{\mathrm{V}} + \lvert d(t)\rvert^2_{D(\hat{\mathrm{A}} )} ) + \int_0^T \left( \lvert \mathrm{A} v (s) \rvert^2_{L^2} +\lvert \nabla \Delta d(s) \rvert^2_{L^2} \right) ds \le K_1.
\end{equation}
In particular, the orbit of $(v,d)$ is precompact in $\mathrm{H} \times D(\hat{\mathrm{A}} ^\frac12)$.
\end{enumerate}
\end{Thm}
\begin{proof}
Let $(v_0,d_0)\in \mathrm{V}\times D(\hat{\mathrm{A}} )$, $f\in L^2(0,T; L^2)$, $g\in L^2(0,T; \mathrm{H} ^1)$.
Let $\varepsilon\,_0>0$ and ${\varepsilon\,}_1\in (0,\varepsilon\,_0)$ be the constants from Proposition \ref{Ma2} and Theorem \ref{thm-main}, respectively. Let $\kappa_0\in (0,\varepsilon\,_1]$ be a number to be chosen in \eqref{Eq:ChoiceOfKappa} (see the proof of Proposition \ref{Thm:PrecompactinVastL2}) such that \eqref{Eq:SmallAssumSmooth} holds. Then, by
Theorem \ref{Thm:NoSingSmall} we deduce that the problem \eqref{1b} has a solution $(v,d)$ which does not have singular times and satisfying
\eqref{Eq:MaxRegSmallData-2-B}. The proof of \eqref{Eq:EstVeloOptDirInVH2-0} will be proved in Proposition \ref{Prop:EstVeloOptDirInVH2} below.
\end{proof}
\begin{Prop} \label{Thm:PrecompactinVastL2}
Let $\varepsilon\,_0>0$ and ${\varepsilon\,}_1\in (0,\varepsilon\,_0)$ be the constants from Proposition \ref{Ma2} and Theorem \ref{thm-main}, respectively. Then, there exists $\kappa_1\in (0, \varepsilon\,_0)$ and $K_4, K_0>0$ such that the following hold.
If $(v_0,d_0)\in \mathrm{V}\times D(\hat{\mathrm{A}} )$, $f\in L^2(0,T; L^2)$, $g\in L^2(0,T; \mathrm{H} ^1)$ satisfy
\begin{equation}\notag
\mathbf{E}_0= \mathcal{E}(v_0,d_0)+\frac12 \int_0^\infty \left[\lvert f \rvert^2_{\mathrm{H} ^{-1}}+ \lvert g \rvert^2_{L^2}\right] dt<2 \kappa_1^2,
\end{equation}
then any regular solution $(v,d)$ to \eqref{1b} defined on $[0,T)$ satisfies
\begin{equation}\label{Eq:EstVeloOptDirInHH1}
\begin{split}
\frac12\sup_{t\in [0,T)} \left( \lvert v(t) \rvert^2_{L^2} + \lvert \nabla d(t) \rvert^2 \right) + \int_0^{T}\left( \lvert \nabla v (r)\rvert^2_{L^2} + K_0 \lvert \nabla d(r)\rvert^2 + K_0 \lvert \Delta d (r)\rvert^2_{L^2} \right)dr
\le \mathbf{E}_0.
\end{split}
\end{equation}
Furthermore,
\begin{equation}\label{Eq:EstOptDirInL4}
\int_0^{T} [\lvert \nabla d\rvert^4_{L^4} + \lvert \nabla d\rvert^2_{L^4} ]dt \le \frac{K_4}{K_0}(1+\mathbf{E}_0^2)
\end{equation}
\end{Prop}
\begin{proof}
Let $(v_0,d_0)\in \mathrm{V}\times D(\hat{\mathrm{A}} )$, $f\in L^2(0,T; L^2)$, $g\in L^2(0,T; \mathrm{H} ^1)$. Let $\varepsilon\,_0>0$ and ${\varepsilon\,}_1\in (0,\varepsilon\,_0)$ be the constants from Proposition \ref{Ma2} and Theorem \ref{thm-main}, respectively. Let $\kappa_0\in (0,{\varepsilon\,}_1]$ be a number to be chosen later such that \eqref{Eq:SmallAssumSmooth} holds.
Let $(v,d)$ be a regular solution to \eqref{1b} defined on $[0,T)$.
Then, multiplying the velocity equation by $v$ and using the Cauchy-Schwarz and Young inequalities imply
\begin{equation}\label{Eq:EstVeloinL2-1}
\begin{split}
\frac12 \frac{d}{dt} \lvert v\rvert^2_{L^2} + \frac12 \lvert \nabla v\rvert^2_{L^2} \le -\langle \Pi\Div(\nabla d \odot \nabla d), u \rangle + \frac12 \lvert f\rvert^2_{\mathrm{H}^{-1}}.
\end{split}
\end{equation}
We multiply the optical director equation by $-\Delta d$, then use Cauchy-Schwarz and Young inequalities and the constraint $d(t)\in \mathcal{M}$, $t\in [0,T]$, and obtain
\begin{equation*}
\begin{split}
\frac12 \frac{d}{dt} \lvert \nabla d \rvert^2_{L^2} + \frac12 \lvert \Delta d \rvert^2_{L^2} \le \langle v \cdot \nabla d, \Delta d \rangle - \langle \lvert \nabla d \rvert^2 d -\phi^\prime(d) +(\phi^\prime(d) \cdot d)d, \Delta d\rangle +\frac12 \lvert g \rvert^2_{L^2}.
\end{split}
\end{equation*}
Using the fact $\phi^\prime(d)= d-\xi$ and the integration-by-parts on the torus we find
\begin{equation}\label{Eq:EstOptDirInH1-1}
\begin{split}
\frac12 \frac{d}{dt} \lvert \nabla d \rvert^2_{L^2} + \frac12 \lvert \Delta d \rvert^2_{L^2} + \lvert\nabla d \rvert^2_{L^2} \le \langle v \cdot \nabla d, \Delta d \rangle +\frac12 \lvert g \rvert^2_{L^2}- \langle \lvert \nabla d \rvert^2 d +(\phi^\prime(d) \cdot d)d, \Delta d\rangle.
\end{split}
\end{equation}
Before proceeding further, let us estimate the last term of the above inequality. Toward this end we divide the task into two parts. Firstly, we use fact that $\Delta d\cdot d = -\lvert \nabla d \rvert^2$, the H\"older, the Gagliardo-Nirenberg (\cite[Section 9.8, Example C.3]{Brezis}) , the Young inequalities and \cite[Theorem 3.4]{Simader} to get the following chain of inequalities
\begin{align}
\lvert \langle (\phi^\prime(d)\cdot d)d, \Delta d \rangle \rvert & = -\int_{\Omega} (\phi^\prime(d(x)) \cdot d(x) ) \lvert \nabla d(x)\rvert^2 \;dx \notag\\
& \le \lvert \phi^\prime(d) \rvert_{L^2} \lvert d \rvert_{L^\infty} \lvert \nabla d \rvert^2_{L^4}\notag\\
&\le C_0\lvert d- \xi \rvert_{L^2} \left(\lvert \nabla d \rvert_{L^2} \lvert \nabla^2 d \rvert_{L^2}+ \lvert \nabla d \rvert^2_{L^2} \right)\notag\\
& \le C_0 \lvert d-\xi \rvert_{L^2} \left(\lvert \nabla d \rvert_{L^2} \lvert \Delta d \rvert_{L^2}+ \lvert \nabla d \rvert^2_{L^2} \right)\notag\\
&\le \frac12 \lvert \nabla d \rvert^2_{L^2} + 2C^2_0 \frac12\lvert d-\xi \rvert^2_{L^2} \left(\lvert \Delta d \rvert^2_{L^2} + \lvert \nabla d \rvert^2_{L^2}\right).
\label{Eq:EstAnisopNRJ}
\end{align}
Secondly, using the Gagliardo-Nirenberg inequality (\cite[Section 9.8, Example C.3]{Brezis}) and \cite[Theorem 3.4]{Simader} we obtain
\begin{align}
-\langle \lvert \nabla d\rvert^2 d, \Delta d \rangle&= \lvert \nabla d \rvert^4_{L^4}\notag\\
&\le C_1 \lvert \nabla d \rvert^2_{L^2} (\lvert \Delta d \rvert^2_{L^2}+ \lvert \nabla d \rvert^2_{L^2} ).\label{Eq:EstTransportTerm-1}
\end{align}
Plugging \eqref{Eq:EstAnisopNRJ} and \eqref{Eq:EstTransportTerm-1} into the inequality \eqref{Eq:EstOptDirInH1-1} yields
\begin{equation}\label{Eq:EstOptDirInH1-2}
\begin{split}
\frac12 \frac{d}{dt} \lvert \nabla d \rvert^2_{L^2} + \lvert\nabla d \rvert^2_{L^2} \le \langle v \cdot \nabla d, \Delta d \rangle +\frac12 \lvert g \rvert^2_{L^2}-\left(\frac12-2C_0^2 \frac12 \lvert d-\xi \rvert^2_{L^2}-C_1 \lvert \nabla d\rvert^2_{L^2} \right)\lvert \Delta d\rvert^2_{L^2}\\
-\left(\frac12 -2C_0^2 \frac12 \lvert d-\xi \rvert^2_{L^2}-C_1 \lvert \nabla d\rvert^2_{L^2} \right)\lvert \nabla d \rvert^2_{L^2}.
\end{split}
\end{equation}
Now adding the inequalities \eqref{Eq:EstVeloinL2-1} and \eqref{Eq:EstOptDirInH1-2} side by side, and using \eqref{Eq:Dissip} imply
\begin{equation}\notag
\begin{split}
\frac12 \frac{d}{dt} \left(\lvert \nabla d \rvert^2_{L^2} + \lvert v \rvert^2_{L^2}\right) \frac12 \lvert \nabla v \rvert^2_{L^2} \le-\left(\frac12-2C_0^2 \frac12 \lvert d-\xi \rvert^2_{L^2}-C_1 \lvert \nabla d\rvert^2_{L^2} \right)\lvert \Delta d\rvert^2_{L^2}\\
-\left(\frac12 -2C_0^2 \frac12 \lvert d-\xi \rvert^2_{L^2}-C_1 \lvert \nabla d\rvert^2_{L^2} \right)\lvert \nabla d \rvert^2_{L^2}\\
+ \frac12 \left(\lvert g \rvert^2_{L^2}+\lvert f \rvert_{\mathrm{H} ^{-1}}\right).
\end{split}
\end{equation}
Thanks to the energy inequality \eqref{Eq:NRJ-Inequality} we obtain
\begin{equation}\label{Eq:EstVeloOptDirInL2H1}
\begin{split}
\frac12 \frac{d}{dt} \left(\lvert \nabla d \rvert^2_{L^2} + \lvert v \rvert^2_{L^2}\right) + \lvert\nabla d \rvert^2_{L^2}+ \frac12 \lvert \nabla v \rvert^2_{L^2} \le-\left(\frac12-2C_0^2 \mathbf{E}_0-C_1 \mathbf{E}_0 \right)\lvert \Delta d\rvert^2_{L^2}\\
-\left(\frac12 -2C_0^2 \mathbf{E}_0-C_1\mathbf{E}_0 \right)\lvert \nabla d \rvert^2_{L^2}
+ \frac12 \left(\lvert g \rvert^2_{L^2}+\lvert f \rvert_{\mathrm{H} ^{-1}}\right).
\end{split}
\end{equation}
We now easily conclude the proof of \eqref{Eq:EstVeloOptDirInHH1} in the proposition by integrating \eqref{Eq:EstVeloOptDirInL2H1}, taking the supremum over $t\in [0,T_0]$ and choosing
\begin{equation}\label{Eq:ChoiceOfKappa}
\kappa_1^2= \min\{\varepsilon\,_1^2, \frac{1}{4(C_0^2 \lvert \Omega \rvert+ C_1)} \} \text{ and } K_0= \frac12- \mathbf{E}_0\left(C_0^2 \lvert \Omega \rvert +C_1\right).
\end{equation}
In order to prove the second estimate, we use the Gagliardo-Nirenberg inequality (see \cite[Section 9.8, Example C.3]{Brezis}) and \eqref{Eq:EstVeloOptDirInHH1} to obtain
\begin{align*}
\int_0^T( \lvert \nabla d \rvert^4_{L^4} +\lvert \lvert \nabla d\rvert^2_{L^4} )dt &\le C \sup_{t\in [0,T]} \lvert \nabla d(t)\rvert^2_{L^2} \int_0^T \lvert \Delta d(t) \rvert^2 dt +C \int_0^{T} \lvert \nabla d (s) \rvert^2_{L^2} ds \\
& + C \int_0^T (\lvert \nabla d(s) \rvert^2+ \lvert \Delta d(s) \rvert^2_{L^2} ds ) \\
&\le \frac{C}{K_0}[\mathbf{E}_0^2+\mathbf{E}_0 ].
\end{align*}
This completes the proof of the proposition.
\end{proof}
The next result is crucial for the proof of Theorem \ref{Thm:PrecompactInHH1}.
\begin{Prop}\label{Prop:EstVeloOptDirInVH2}
Let $\varepsilon\,_0$ be as in Proposition \ref{Ma2} and $(v_0,d_0)\in \mathrm{V}\times D(\hat{\mathrm{A}} )$, $f\in L^2(0,T; L^2)$, $g\in L^2(0,T; \mathrm{H} ^1)$. Assume that
\begin{equation}\notag
\mathbf{E}_0= \mathcal{E}(v_0,d_0)+\frac12 \int_0^\infty \left[\lvert f \rvert^2_{\mathrm{H} ^{-1}}+ \lvert g \rvert^2_{L^2}\right] dt< \kappa^2_1,
\end{equation}
where $\kappa_1\in (0,\varepsilon\,_0)$ is defined in Proposition \ref{Thm:PrecompactinVastL2}. Then, there exist a constant $K_2>0$, which depends only on the norms of $(v_0,d_0)\in \mathrm{V}\times D(\hat{\mathrm{A}} )$ and $(f,g)\in L^2(0,T; \mathrm{H}\times \mathrm{H} ^1)$, such that for a regular solution $(v,d)$ to \eqref{1b} we have
\begin{equation}\label{Eq:EstVeloOptDirInVH2}
\begin{split}
\frac12\sup_{t\in [0,T)} \left( \lvert \mathrm{A} ^\frac12 v(t) \rvert^2_{L^2} + \lvert \Delta d(t) \rvert^2 \right) + \int_0^{T_\ast}\left( \lvert \mathrm{A} v (r)\rvert^2_{L^2} + \lvert \Delta d(r)\rvert^2 + \lvert \Delta d (r)\rvert^2_{L^2} \right)dr
\le K_2.
\end{split}
\end{equation}
\end{Prop}
\begin{proof}
The proof of the estimate \eqref{Eq:EstVeloOptDirInVH2} is very similar to the proof of \eqref{Eq:EstinHigherorderNorms}, hence we only sketch the proof.
We start with the same idea as in the proof of \eqref{Eq:EstinHigherorderNorms}, i.e., we multiply the velocity and optical director equations by $Av$ and $\mathrm{A} ^2 d$, respectively. This procedure implies the equation \eqref{Eq:Derivativehigheroerdernorms}. In the present proof, we estimate the term ${}_{\mathrm{H} ^1}\langle\phi^\prime(d), -\hat{\mathrm{A}} ^2 d\rangle_{(\mathrm{H} ^1)^\ast} $ as follows:
\begin{align}\notag
\;{}_{\mathrm{H} ^1}\langle\phi^\prime(d), -\hat{\mathrm{A}} ^2 d\rangle_{(\mathrm{H} ^1)^\ast} =-\langle \Delta\phi^\prime(d), \Delta d \rangle = -\lvert \Delta d \rvert^2_{L^2}.
\end{align}
This gives raise to the term $\int_0^T \lvert \Delta d(r)\rvert^2 dr$ in the left hand side of \eqref{Eq:EstVeloOptDirInVH2}.
Now, the remaining terms are estimated with exactly the same way as in the proof of \eqref{Eq:Derivativehigheroerdernorms}. In particular, we infer that
there exist constants $K_3>0$, which depends on the norms of $(u_0,d_0)\in \mathrm{V}\times D(\hat{\mathrm{A}} )$ and $(f,g)\in L^2(0,T; \mathrm{H} \times D(\hat{\mathrm{A}} ^{1/2}))$, and $K_4>0$ such that for all $\tau\in [0,T)$ we have
\begin{equation}\label{Eq:EstinHigherorderNorms-A}
\begin{split}
\sup_{0\le s\le \tau} \left(\lvert \nabla
v(s) \rvert^2_{L^2} + \lvert \Delta d(s) \rvert^2_{L^2} \right) + 2 \int_0^{\tau} \left( \lvert \nabla \Delta d(s) \rvert^2_{L^2} +\lvert \Delta d(s)\rvert^2_{L^2} +\lvert A v(s) \rvert^2_{L^2}\right)ds \\
\le K_3e^{K_4 \int_0^{\tau}[\lvert \nabla d(r) \rvert^4_{L^4}+ \lvert v(r) \rvert^4_{L^4}+\lvert \nabla d(r) \rvert^2_{L^4}+ \lvert v(r) \rvert^2_{L^4} ]dr }.
\end{split}
\end{equation}
Since $L^4\cap \mathrm{H} \subset \mathrm{V} $, by using \eqref{Eq:EstVeloOptDirInHH1} and the Ladyzhenskaya inequality (\cite[Lemma III.3.3]{Temam_2001}) we infer that there exists $C>0$ such that
\begin{equation}\notag
\int_0^T[ \lvert v(s) \rvert^4_{L^4} +\lvert v(s) \rvert^2_{L^4}] ds\le \int_0^T[ \lvert v(s) \rvert^4_{L^4} +\lvert \nabla v(s) \rvert^2] ds \le C (1+\mathbf{E}_0^2).
\end{equation}
Thus, plugging the latter estimate and \eqref{Eq:EstOptDirInL4} into \eqref{Eq:EstinHigherorderNorms-A} imply \eqref{Eq:EstVeloOptDirInVH2}. This completes the proof of the proposition.
\end{proof}
\appendix
\section{Proof of Lemma \ref{Lem:RighthandSideinL2VxH1}} In this section we will prove Lemma \ref{Lem:RighthandSideinL2VxH1}. \begin{proof}[Proof of Lemma \ref{Lem:RighthandSideinL2VxH1}]
Firstly, we infer from
\cite[Lemma 2.5]{ZB+EH+PR-RIMS}
\footnote{ Note that in
\cite[Lemma 2.5]{ZB+EH+PR-RIMS}
the authors used $B_1(v_i,v_i)$ and $M(n_i,n_i)$ in places of $B(v_i,v_i)$ and $\Pi\left(\nabla n_i \Delta n_i\right)$, respectively. } that there exists a constant $C>0$ such that for all $(v_i,d_i)\in D(\mathrm{A} )\times D(\hat{\mathrm{A}} ^\frac32)$
\begin{equation*}
\begin{split}
\lvert F(v_1,n_1)-F(v_2, n_2) \rvert_{\mathrm{H} } \le C \lvert\mathrm{A} ^\frac12[ v_1 -v_2]\rvert^\frac12_{L^2} \left(\lvert \mathrm{A} [v_1 -v_2] \rvert^\frac12_{L^2} \lvert \mathrm{A} ^\frac12 u_2 \rvert_{L^2} + \lvert \mathrm{A} ^\frac12[v_1-v_2] \rvert^\frac12_{L^2} \lvert \mathrm{A} ^\frac12 u_1 \rvert^\frac12_{L^2} \lvert \mathrm{A} u_1 \rvert^\frac12_{L^2} \right)\\
+ \lvert n_1 -n_2 \rvert_{\mathrm{H} ^2} \lvert n_1 \rvert^\frac12_{\mathrm{H} ^2} \lvert n_1\rvert^\frac12_{\mathrm{H} ^3} + \lvert n_1-n_2 \rvert^\frac12_{\mathrm{H} ^2} \lvert n_1-n_2 \rvert^\frac12_{\mathrm{H} ^3} \lvert n_2 \rvert^\frac12_{\mathrm{H} ^2}
\end{split}
\end{equation*}
This clearly implies that there exists a constant $C>0$ such that for all $(v_i,d_i)\in \mathbf{X}_T$, $i=1,2$,
\begin{equation*}
\begin{split}
\lvert F(v_1,n_1)-F(v_2, n_2) \rvert^2_{L^2(0,T; \mathrm{H} )} \le & C T^\frac12 \lvert v_1 -v_2\rvert_{C([0,T];\mathrm{V})} \lvert u_2 \rvert_{C([0,T];\mathrm{V})} \lvert \mathrm{A} [v_1 -v_2] \rvert_{L^2(0,T;D(\mathrm{A} ))} \\
&+ C T^\frac12 \lvert \mathrm{A} ^\frac12[v_1-v_2] \rvert^2_{C([0,T]; \mathrm{V})} \lvert u_1 \rvert_{C([0,T];\mathrm{V})} \lvert \mathrm{A} u_1 \rvert_{L^2(0,T;D(\mathrm{A} ))} \\
&+ C T^\frac12 \lvert n_1 -n_2 \rvert^2_{C([0,T];\mathrm{H} ^2)} \lvert n_1 \rvert_{C([0,T];\mathrm{H} ^2)} \lvert n_1\rvert_{L^2(0,T;\mathrm{H} ^3)} \\
&+ CT^\frac12\lvert n_1-n_2 \rvert_{C([0,T]; \mathrm{H} ^2)} \lvert n_1-n_2 \rvert_{L^2(0,T;\mathrm{H} ^3)} \lvert n_2 \rvert_{C([0,T];\mathrm{H} ^2)}.
\end{split}
\end{equation*}
Hence, we infer that there exists a constant $C>0$ such that for all $(v_i,d_i)\in \mathbf{X}_T$, $i=1,2$,
\begin{equation*}
\begin{split}
\lvert F(v_1,n_1)-F(v_2, n_2) \rvert^2_{L^2(0,T; \mathrm{H} )} \le C T^\frac12 \lvert v_1-v_2\rvert^2_{\mathbf{X}^1_T}[\lvert u_1\rvert^2_{\mathbf{X}^1_T}+\lvert u_2\rvert^2_{\mathbf{X}^1_T} ]\\
C T^\frac12 \lvert n_1-n_2\rvert^2_{\mathbf{X}^1_T}[\lvert d_1\rvert^2_{\mathbf{X}^2_T}+\lvert d_2\rvert^2_{\mathbf{X}^2_T} ],
\end{split}
\end{equation*}
from which we easily deduce the inequality \eqref{Eq:FPT-ForcingVelo}.
We will now proceed to the proof of \eqref{Eq:FPT-ForcingOptDir-1} which will be divided into four steps. Firstly, because of the assumption \eqref{Eq:BoundednessPhibis} the map $\phi^\prime:\mathbb{R}^3 \to \mathbb{R}^3$ is Lipschitz continuous. Hence, for all $n_i\in \mathrm{H} ^2$, $i=1,2$,
\begin{align*}
\lvert \phi^\prime(n_1)- \phi^\prime(n_2) \rvert_{\mathrm{H} ^1}=& \lvert \phi^\prime(n_1)- \phi^\prime(n_2) \rvert_{L^2}+\lvert \nabla \phi^\prime(n_1)- \nabla \phi^\prime(n_2) \rvert_{L^2}\\
\le & M_2 \lvert n_1 -n_2\lvert_{L^2} + \lvert \phi^{\prime\prime}(n_1)\nabla n_1-\phi^{\prime\prime}(n_2) \nabla n_2\rvert_{L^2}\\
\le & M_2 \lvert n_1 -n_2\lvert_{L^2} + \lvert [\phi^{\prime\prime}(n_1)-\phi^{\prime\prime}(n_2)]\nabla n_1+\phi^{\prime\prime}(n_2) \nabla (n_1-n_2)\lvert_{L^2}.
\end{align*}
Using \eqref{Eq:LipSchitzPhibis}, \eqref{Eq:BoundednessPhibis} and the Sobolev emebedding $\mathrm{H} ^1\subset L^{4}$ we obtain
\begin{align*}
\lvert \phi^\prime(n_1)- \phi^\prime(n_2) \rvert_{\mathrm{H} ^1} \le & M_2\lvert n_1 -n_2\lvert_{L^2}+ \lvert \lvert n_1 -n_2\lvert \lvert \nabla n_1\rvert\rvert_{L^2}+ M_2 \lvert \nabla (n_1-n_2) \rvert_{L^2}\\
\le & M_2 \lvert n_1 -n_2 \rvert_{\mathrm{H} ^1}+ \lvert n_1 -n_2 \rvert_{L^4} \lvert \nabla n_1 \rvert_{L^4}\\
\le & M_2 \lvert n_1- n_2 \rvert_{\mathrm{H} ^1} + \lvert n_1 - n_2 \rvert_{\mathrm{H} ^1} \lvert n_1 \rvert_{\mathrm{H} ^2}.
\end{align*}
Hence, there exists a constant $C>0$ such that for all $n_i\in \mathbf{X}^2_T$, $i=1,2$
\begin{equation} \label{Eq:EstimateL2H1PhiPrime}
\lvert \phi^\prime(n_1)- \phi^\prime(n_2) \rvert_{L^2(0,T;\mathrm{H} ^1)}^2 \le M_2 T \lvert n_1 -n_2\rvert^2_{\mathbf{X}^2_T} [1 + \lvert n_1 \rvert^2_{\mathbf{X}^2_T} ].
\end{equation}
Secondly, we estimate the term involving $(\lvert (\phi^\prime(n_i) \cdot n_i)n_i)$, $i=1,2$, as follows. Let $n_i\in \mathrm{H} ^2$, $i=1,2$. Then, we have
\begin{align*}
\lvert (\phi^\prime(n_1) \cdot n_1)n_1 -(\phi^\prime(n_2) \cdot n_2)n_2\rvert_{\mathrm{H} ^1} =& \lvert ([\phi^\prime(n_1) -\phi^\prime(n_2)] \cdot n_1)n_1+ (\phi^\prime(n_2)\cdot n_1)n_1-(\phi^\prime(n_2) \cdot n_2)n_2 \rvert_{\mathrm{H} ^1}\\
\le & \lvert ([\phi^\prime(n_1) -\phi^\prime(n_2)] \cdot n_1)n_1\rvert_{\mathrm{H} ^1} +\lvert (\phi^\prime(n_2)\cdot [n_1-n_2])n_1\rvert_{\mathrm{H} ^1} \\
&\qquad + \lvert (\phi^\prime(n_2)\cdot n_2)[n_1-n_2] \rvert_{\mathrm{H} ^1}.
\end{align*}
{We recall the following classical fact whose proof is easy and omitted. }
\begin{equation}\label{Eq:ProductH1Linfty}
\lvert fg\rvert_{\mathrm{H} ^1}\le 2 \left( \lvert f\rvert_{L^\infty} \lvert g \rvert_{\mathrm{H} ^1} + \lvert f\rvert_{\mathrm{H} ^1}\lvert g \rvert_{L^\infty}\right),\;\; \text{ for all } f,g\in \mathrm{H}^1\cap L^\infty.
\end{equation}
Since, $n_i\in \mathrm{H}^2$ and $\mathrm{H} ^2\subset L^\infty$, it follows from \eqref{Eq:LInearGrowthPhiprime} and \eqref{Eq:BoundednessPhibis} that $\phi^\prime(n_i)\in \mathrm{H}^1\cap L^\infty$. Hence, we can repeatedly use \eqref{Eq:ProductH1Linfty}, the Lipschitz property of $\phi^\prime$ to infer that for all $n_i\in \mathrm{H} ^2$, $i=1,2$,
\begin{align*}
\lvert \left(\left[\phi^\prime(n_1)-\phi^\prime(n_2)\right]\cdot n_1\right)n_1 \rvert_{\mathrm{H} ^1}\le C \lvert \phi^\prime(n_1) -\phi^\prime(n_2)\rvert_{L^\infty} \lvert n_1\rvert_{L^\infty} \lvert n_1\rvert_{\mathrm{H} ^1} + \lvert \phi^\prime(n_1)-\phi^\prime(n_2) \rvert_{\mathrm{H} ^1} \lvert n_1\rvert_{L^\infty}^2.\\
\le \lvert n_1 -n_2\rvert_{L^\infty} \lvert n_1 \rvert^2_{\mathrm{H} ^2} + \lvert \phi^\prime(n_1) -\phi^\prime(n_2)\rvert_{\mathrm{H} ^1} \lvert n_1\rvert^2_{\mathrm{H} ^2}\\
\le \lvert n_1-n_2\rvert_{\mathrm{H} ^2} \lvert n_1\rvert^2_{\mathrm{H} ^2}+ \lvert \phi^\prime(n_1) -\phi^\prime(n_2)\rvert_{\mathrm{H} ^1} \lvert n_1\rvert^2_{\mathrm{H} ^2}.
\end{align*}
Thanks to the last line and \eqref{Eq:EstimateL2H1PhiPrime} we deduce that there exists $C>0$ such that for all $n_1,n_2\in \mathbf{X}^2_T$
\begin{align*}
\lvert \left(\left[\phi^\prime(n_1)-\phi^\prime(n_2)\right]\cdot n_1\right)n_1 \rvert^2_{L^2(0,T;\mathrm{H} ^1)}\le CT \lvert n_1-n_2\rvert^2_{\mathbf{X}^2_T } \lvert n_1\rvert^4_{\mathbf{X}^2_T}\left[ 1 + \lvert n_1 \rvert^2_{\mathbf{X}^2_T} \right],
\end{align*}
\noindent Using \eqref{Eq:ProductH1Linfty} we infer that there exists a constant $C>0$ such that for all $a,b,c\in \mathrm{H} ^2$
\begin{align*}
\lvert (\phi^\prime(a)\cdot b)c\rvert_{\mathrm{H} ^1} \le & C \lvert\phi^\prime(a) \rvert_{L^\infty}[ \lvert b \rvert_{L^\infty} \lvert c \rvert_{\mathrm{H} ^1}+\lvert b \rvert_{\mathrm{H} ^1} \lvert c \rvert_{L^\infty} ]+ \lvert \phi^\prime(a)\rvert_{\mathrm{H} ^1} \lvert b \rvert_{L^\infty} \lvert a \rvert_{L^\infty}.\\
\end{align*}
This altogether with \eqref{Eq:LInearGrowthPhiprime}, \eqref{Eq:BoundednessPhibis} and the continuous Sobolev embeddings $\mathrm{H} ^2\hookrightarrow L^\infty$and $\mathrm{H} ^2\subset \mathrm{H} ^1$ implies that for all $a,b,c\in \mathrm{H} ^2$
\begin{align*}
\lvert (\phi^\prime(a)\cdot b)c\rvert_{\mathrm{H} ^1} \le & C (1+\lvert a \rvert_{\mathrm{H} ^2}) \lvert b \rvert_{\mathrm{H} ^2} \lvert c \rvert_{\mathrm{H} ^2},
\end{align*}
Thus, there exists a constant $C>0$ such that $n_1,n_2\in \mathrm{H} ^2$
\begin{align*}
\lvert (\phi^\prime(n_2)\cdot [n_1-n_2])n_1\rvert_{\mathrm{H} ^1} \le C (1+\lvert n_2 \rvert_{\mathrm{H} ^2}) \lvert n_1-n_2 \rvert_{\mathrm{H} ^2} \lvert n_1 \rvert_{\mathrm{H} ^2}\\
\lvert (\phi^\prime(n_2)\cdot n_2)[n_1-n_2] \rvert_{\mathrm{H} ^1}\le C (1+\lvert n_2 \rvert_{\mathrm{H} ^2}) \lvert n_2 \rvert_{\mathrm{H} ^2} \lvert n_1-n_2 \rvert_{\mathrm{H} ^2}.
\end{align*}
Thus, we deduce that there exists constant $C>0$ such that for all
$n_1,n_2\in \mathbf{X}^2_T$
\begin{align}
\lvert (\phi^\prime(n_2)\cdot [n_1-n_2])n_1\rvert^2_{L^2(0,T;\mathrm{H} ^1)} \le C T (1+\lvert n_2 \rvert_{\mathbf{X}^2_T}^2) \lvert n_1-n_2 \rvert^2_{\mathbf{X}^2_T} \lvert n_1 \rvert^2_{\mathbf{X}^2_T}\label{Eq:alphaphiprime-1}\\
\lvert (\phi^\prime(n_2)\cdot n_2)[n_1-n_2] \rvert^2_{L^2(0,T;\mathrm{H} ^1)}\le C T (1+\lvert n_2 \rvert^2_{\mathbf{X}^2_T}) \lvert n_2 \rvert_{\mathbf{X}^2_T}^2 \lvert n_1-n_2 \rvert^2_{\mathbf{X}^2_T}.\label{Eq:alphaphiprime-2}
\end{align}
Therefore, there exists a constant $C>0$ such that for all $n_i\in \mathbf{X}^2_T$ , $i=1,2$,
\begin{align} \label{Eq:alphaphiprime-3}
\lvert \left(\left[\phi^\prime(n_1)-\phi^\prime(n_2)\right]\cdot n_1\right)n_1 \rvert_{L^2(0,T; \mathrm{H} ^1)} \le C T \lvert n_1 -n_2\rvert^2_{\mathbf{X}^2_T} \left(1 + \lvert n_1 \rvert^4_{\mathbf{X}^2_T} + \lvert n_1\rvert^6_{\mathbf{X}^2_T} + \lvert n_2 \rvert^2_{\mathbf{X}^2_T} \right).
\end{align}
From \eqref{Eq:alphaphiprime-1}-\eqref{Eq:alphaphiprime-3} we infer that there exists a constant $C>0$ such that for all $n_i\in \mathbf{X}^2_T$, $i=1,2$,
\begin{equation*}
\lvert (\phi^\prime(n_1)\cdot n_1)n_1-(\phi^\prime(n_2)\cdot n_2)n_2 \rvert_{L^2(0,T; \mathrm{H} ^1)} \le C T \lvert n_1 -n_2\rvert^2_{\mathbf{X}^2_T} \left(1 + \sum_{i=1}^2 \lvert n_i \rvert^6_{\mathbf{X}^2_T}\right).
\end{equation*}
\noindent Thirdly, it was proved in \cite[Lemma 2.6]{ZB+EH+PR-RIMS}, where the notation $B_2(u_i,n_i)$ was used in place of $u_i\cdot \nabla n_i$, that there exists a constant $C>0$ such that for all $u_i\in \mathrm{V} $ and $n_i\in D(\hat{\mathrm{A}} ^\frac32)$, $i=1,2$, we have
\begin{align*}
\lvert u_1\cdot \nabla n_1 - u_2\cdot\nabla n_2\rvert_{\mathrm{H} ^1}\le C
\biggl(&\lvert \nabla(u_1-u_2)\rvert_{L^2} \lvert
n_1\rvert_{\mathrm{H} ^2}^{\frac12}\lvert n_1\rvert_{\mathrm{H} ^3}^\frac12\\
&+ \lvert n_1-n_2\lvert^{\frac12}_{\mathrm{H} ^2} \lvert n_1-n_2\lvert^{\frac12}_{\mathrm{H} ^3}\lvert
\nabla u_2\lvert_{L^2}\biggr).
\end{align*}
From this inequality we easily infer that there exists a constant $C>0$ such that for all {$(u_i,n_i)\in \mathbf{X}^1_T\times \mathbf{X}^2_T $}, $i=1,2$
\begin{align*}
\lvert u_1\cdot \nabla n_1 - u_2\cdot\nabla n_2\rvert^2_{L^2(0,T; \mathrm{H} ^1)}\le C
\biggl(& \lvert u_1-u_2\rvert_{C([0,T]; \mathrm{V})}^2 \lvert
n_1\rvert_{C([0,T];D(\hat{\mathrm{A}} ))}\int_0^T \lvert n_1(s)\rvert_{\mathrm{H} ^3} ds \\
&+ \lvert
\nabla u_2\lvert^2_{C([0,T];\mathrm{V})} \lvert n_1-n_2\lvert_{C([0,T];D(\hat{\mathrm{A}} ))}\int_0^T \lvert n_1(s)-n_2(s)\lvert_{\mathrm{H} ^3}ds \biggr)\\
\le & C T^\frac12
\biggl(\lvert u_1-u_2\rvert_{\mathbf{X}^1_T}^2 \lvert
n_1\rvert_{\mathbf{X}^2_T}^2+ \lvert u_2\lvert^2_{\mathbf{X}^1_T} \lvert n_1-n_2\lvert_{\mathbf{X}^2_T}^2 \biggr)
\end{align*}
Fourthly, we prove that there exists a constant $C>0$ such that for all $n_i\in \mathbf{X}^2_T$, $i=1,2$,
\begin{equation}\label{Eq:GradientNonlinearity-Lipschitz}
\lvert \lvert \nabla n_1\rvert^2n_1-\lvert \nabla n_2 \rvert^2n_2 \rvert^2_{L^2(0,T; \mathrm{H} ^1)} \le C (T\vee T^\frac12) \lvert n_1-n_2\rvert^2_{\mathbf{X}^2_T} \left(\lvert n_1\rvert^4_{\mathbf{X}^2_T}+\lvert n_2\rvert^4_{\mathbf{X}^2_T} \right).
\end{equation}
We need the following claim to prove this.
\begin{Clm}\label{Claim:Projection-Of-Laplacian}
There exists $C>0$ such that for $a,b\in D(\hat{\mathrm{A}} ^\frac32)$ and $c\in D(\hat{\mathrm{A}} )$ the following holds
\begin{equation*}
\lvert [\nabla a : \nabla b ] c \rvert_{\mathrm{H} ^1}^2 \le \lvert a \rvert_{\mathrm{H} ^2}^2 \lvert b\rvert_{\mathrm{H} ^2}^2\lvert c \rvert_{\mathrm{H} ^2}^2+ \lvert c \rvert_{\mathrm{H} ^2}^2[ \lvert a \rvert_{\mathrm{H} ^2}\lvert a \rvert_{\mathrm{H} ^3}\lvert b \rvert^2_{\mathrm{H} ^2}+\lvert a \rvert^2_{\mathrm{H} ^2}\lvert b \rvert_{\mathrm{H} ^2}\lvert b \rvert_{\mathrm{H} ^3} ]
\end{equation*}
\end{Clm}
\begin{proof}[Proof of the Claim \ref{Claim:Projection-Of-Laplacian}]
We infer from the Cauchy-Schwarz inequality and the Gagliardo-Nirenberg inequality (\cite[Section 9.8, Example C.3]{Brezis}) that there exists a constant $C>0$ such that for all $a,b\in D(\hat{\mathrm{A}} ^\frac32)$ and $c\in D(\hat{\mathrm{A}} )$ we have
\begin{align*}
\lvert [\nabla a : \nabla b]c \rvert_{\mathrm{H} ^1}^2 =& \lvert [\nabla a : \nabla b]c \rvert_{L^2}^2 + \sum_{i=1}^2 \lvert \partial_i \left( [\nabla a : \nabla b]c \right) \lvert^2_{L^2}\\
\le & C \lvert\nabla a \rvert_{L^4}^2 \lvert \nabla b \rvert_{L^4}^2 \lvert c \rvert_{L^\infty}^2 +C \lvert c \rvert^2_{L^\infty} \sum_{i=1}^2\left( \lvert \nabla \partial_i a \rvert^2_{L^4} \lvert \nabla b \rvert^2_{L^4} + \lvert \nabla a \rvert^2_{L^4} \lvert \nabla \partial_i b \rvert^2_{L^4}\right) \\
& \qquad \qquad +C \sum_{i=1}^2 \lvert \nabla a\rvert^2_{L^8} \lvert \nabla b \rvert^2_{L^8 } \lvert \partial_i c \rvert^2_{L^4} \\
\le & C \lvert a \rvert_{\mathrm{H} ^2}^2 \lvert b \rvert_{\mathrm{H} ^2}^2 \lvert c \rvert_{\mathrm{H} ^2}^2+C \lvert c \rvert^2_{\mathrm{H} ^2} \sum_{i=1}^2\left( \lvert \nabla \partial_i a \rvert_{L^2} \lvert \nabla \partial_i a \rvert_{\mathrm{H} ^1 }\lvert b \rvert^2_{\mathrm{H} ^2} + \lvert \nabla a \rvert^2_{\mathrm{H} ^2} \lvert \nabla \partial_i b \rvert_{L^2} \lvert \nabla \partial_i b \rvert_{\mathrm{H} ^1} \right) \\
& \qquad \qquad +C \sum_{i=1}^2 \lvert \nabla a\rvert^2_{\mathrm{H} ^1} \lvert \nabla b \rvert^2_{\mathrm{H} ^1} \lvert \partial_i c \rvert^2_{L^4} \\
\le & C \lvert a \rvert_{\mathrm{H} ^2}^2 \lvert b \rvert_{\mathrm{H} ^2}^2 \lvert c \rvert_{\mathrm{H} ^2}^2 + \lvert c \rvert^2_{\mathrm{H} ^2}\left(\lvert a \rvert_{\mathrm{H} ^2} \lvert a \rvert_{\mathrm{H} ^3} \lvert b \rvert^2_{\mathrm{H} ^2} + \lvert a \rvert^2_{\mathrm{H} ^2} \lvert b \rvert_{\mathrm{H} ^2} \lvert b\rvert_{\mathrm{H} ^3} \right).
\end{align*}
Thus, the proof of Claim \ref{Claim:Projection-Of-Laplacian} is complete.
\end{proof}
Let us resume the proof of \eqref{Eq:GradientNonlinearity-Lipschitz}. Applying the claim and integrating over $[0,T]$ yields
\begin{align*}
\lvert [ \nabla n_2: \nabla n_2 ] (n_1-n_2)\rvert^2_{L^2(0,T;\mathrm{H} ^1)} \le C T \lvert n_2\rvert^4_{C([0,T;D(\hat{\mathrm{A}} )])} \lvert n_1-n_2\rvert^2_{C([0,T;D(\hat{\mathrm{A}} )])} \\
+ C \lvert n_1-n_2\rvert^2_{C([0,T;D(\hat{\mathrm{A}} )])}\Bigl(\lvert n_2\rvert^3_{C([0,T;D(\hat{\mathrm{A}} )])}\int_0^T \lvert n_2(s) \rvert_{\mathrm{H} ^3} ds \Bigr)\\
\le
C T^\frac12 \lvert n_1-n_2\rvert^2_{\mathbf{X}^2_T} \Bigl(\lvert n_2\rvert^2_{\mathbf{X}^2_T} \lvert n_2 \rvert_{C([0,T;D(\hat{\mathrm{A}} )])}\left(\int_0^T \lvert n_2(s) \rvert^2_{\mathrm{H} ^3} ds\right)^\frac12 \Bigr)\\
+ C T \lvert n_2\rvert^4_{\mathbf{X}^2_T } \lvert n_1-n_2\rvert^2_{\mathbf{X}^2_T},
\end{align*}
for some constant $C>0$ and for all $n_i\in \mathbf{X}^2_T$, $i=1,2$.
The last line of the above chain of inequalities yields that there exists a constant $C>0$ such that for all $n_i\in \mathbf{X}^2_T$, $i=1,2$,
\begin{align}\label{Eq:GradientNonlinearity-1}
\lvert [ \nabla n_2: \nabla n_2 ] (n_1-n_2)\rvert^2_{L^2(0,T;\mathrm{H} ^1)} \le C( T \vee T^\frac12) \lvert n_2\rvert^4_{\mathbf{X}^2_T}\lvert n_1-n_2\rvert^2_{\mathbf{X}^2_T}.
\end{align}
In a similar way, one can show that there exists a constant $C>0$ such that for all $n_i\in \mathbf{X}^2_T$, $i=1,2$,
\begin{align}\label{Eq:GradientNonlinearity-2}
\lvert [ \nabla (n_1-n_2): \nabla(n_1+n_2)] n_1\rvert_{L^2(0,T; \mathrm{H} ^1)}^2 \le C(T\vee T^\frac12 )\lvert n_1-n_2\rvert^2_{\mathbf{X}^2_T}\left( \lvert n_2\rvert^4_{\mathbf{X}^2_T}+ \lvert n_1\rvert^4_{\mathbf{X}^2_T}\right).
\end{align}
We easily complete the proof of \eqref{Eq:GradientNonlinearity-Lipschitz} by using \eqref{Eq:GradientNonlinearity-1} and \eqref{Eq:GradientNonlinearity-2} in the following inequality
\begin{align*}
\lvert \lvert \nabla n_1\rvert^2n_1 - \lvert \nabla n_2\rvert^2n_2\rvert^2_{L^2(0,T; \mathrm{H} ^1)} \le &
2 \lvert [ \nabla (n_1-n_2): \nabla(n_1+n_2)] n_1\rvert_{L^2(0,T; \mathrm{H} ^1)} ^2 \\
&+ 2 \lvert [\nabla n_2 : \nabla n_2] (n_2-n_1 ) \rvert^2_{L^2(0,T; \mathrm{H} ^1)} .
\end{align*}
In order to complete the proof of Lemma \ref{Lem:RighthandSideinL2VxH1} we need to establish the inequality \eqref{Eq:FPT-ForcingOptDir-2}. For this purpose, we firstly observe that it is not difficult to prove that there exists a constant $C>0$ such that for all $n_i$, $i=1,2$,
\begin{align*}
\lvert (n_1-n_2) \times g \rvert^2_{L^2(0,T; \mathrm{H} ^1)} \le & \lvert n_1- n_2 \rvert^2_{C([0,T]; D(\hat{\mathrm{A}} ))} \lvert g \rvert^2_{L^2(0,T; \mathrm{H}^1)} + \lvert \nabla (n_1-n_2) \rvert^2_{C([0,T]; L^4)} \int_0^T\lvert g(s)\rvert^2_{L^4}ds.
\end{align*}
Hence, there exists a constant $C>0$ such that for all $n_i\in \mathbf{X}^2_T$, $i=1,2$, we have
\begin{equation*}
\lvert (n_1-n_2) \times g \rvert^2_{L^2(0,T; \mathrm{H} ^1)} \le \lvert n_1- n_2 \rvert^2_{\mathbf{X}^2_T} \lvert g \rvert^2_{L^2(0,T;\mathrm{H} ^1)},
\end{equation*}
which is the inequality \eqref{Eq:FPT-ForcingOptDir-2}. Hence, the proof of Lemma \ref{Lem:RighthandSideinL2VxH1} is complete. \end{proof}
\section{Weak solution of a modified viscous transport equation} Let $d\in C([0,T];\mathrm{H} ^{1})\cap L^{2}(0,T;\mathrm{H} ^{2})$ and $u \in C([0,T]; \mathrm{H} )\cap L^2(0,T;\mathrm{V} )$. Consider the following problem \begin{equation} \begin{cases}
\partial_{t}z-\Delta z= 2|\nabla d|^{2}z-2(\phi^\prime(d)\cdot d)z-u\cdot \nabla z\\
{\frac{\partial z}{\partial \nu}}{\Big\lvert_{\partial\Omega}}=0,\\ z(0)=z_{0}.\label{tu1} \end{cases} \end{equation} We introduce the definition of weak solution of problem \ref{tu1}. \begin{Def}\label{Def:WeakSol-ViscousTransport}
Let $z_0\in L^2$. A function $z: [0,T] \to L^2$ is a weak solution to (\ref{tu1}) iff
\begin{enumerate}
\item \label{Item:DefWeakSol-1} $z\in C([0,T]; L^2)\cap L^2(0,T; \mathrm{H} ^1)$;
\item for all $t\in [0,T]$ and $\varphi \in \mathrm{H} ^1$
\begin{equation*}
(z(t) -z_0, \varphi)+ \int_0^t (\nabla z(s), \nabla \varphi) ds = \int_0^t \left(2 |\nabla d(s)|^{2}z(s)-2 (\phi^\prime(d(s))\cdot d(s) )z(s)-u(s)\cdot \nabla z(s), \varphi\right) ds
\end{equation*}
\end{enumerate} \end{Def} \begin{Rem}
Let $d\in C([0,T];\mathrm{H} ^{1})\cap L^{2}(0,T;\mathrm{H} ^{2})$, $u \in C([0,T]; \mathrm{H} )\cap L^2(0,T;\mathrm{V} )$ and $z\in C([0,T]; L^2)\cap L^2(0,T; \mathrm{H} ^1)$ be a weak solution to \eqref{Eq:ViscousTransport-z}. Then, by using the H\"older inequality and Sobolev embeddings such as $\mathrm{H} ^1\hookrightarrow L^4$ and $\mathrm{H} ^2\hookrightarrow\infty$ we can easily shows that
\begin{align*}
\lvert \partial_t z\rvert_{(\mathrm{H} ^1)^\ast}=&\sup_{\varphi \in \mathrm{H} ^1 : \lvert \varphi \rvert_{\mathrm{H} ^1}=1 } \lvert\langle \partial_t z, \varphi \rangle \rvert\\
\le & \lvert \nabla z \rvert_{L^2}+ \lvert \nabla d \rvert^2_{L^4} \lvert z\rvert_{L^4}+ \lvert u \rvert_{L^4} \lvert z\rvert_{L^4}+ (1+ \lvert d \rvert_{\mathrm{H} ^2})\lvert d \rvert_{L^4} \lvert z \rvert_{L^2}.
\end{align*}
Hence, since $u, \nabla d ,z\in C([0,T];L^2)\cap L^2(0,T; \mathrm{H} ^1)\subset L^4(0,T; L^4)$, we can show by using the H\"older inequality and \eqref{tu3} that
\begin{equation}\label{Item:DefWeakSol-3}
\lvert \partial_t z\rvert_{L^2(0,T; (\mathrm{H} ^1)^\ast)}\le c,
\end{equation}
for a universal constant which depends only on $\Omega$ and $T$. \end{Rem} The following result gives the existence and uniqueness of problem (\ref{tu1}). \begin{Prop}\label{tu4}
Let $d\in C([0,T];\mathrm{H} ^{1})\cap L^{2}(0,T;\mathrm{H} ^{2})$, $u \in C([0,T]; \mathrm{H} )\cap L^2(0,T;\mathrm{V} )$ and $z_{0}\in L^{2}$. Let also $\phi:\mathbb{R}^{3}\to \mathbb{R}^{+}$ be of class $C^{1}$ such that
\begin{equation*}
|\phi^\prime(d)|_{\mathbb{R}^{3}}\le c(1+|d|).
\end{equation*}
Then problem (\ref{tu1}) has a unique weak solution $z$. Moreover, there exists a constant $c>0$ such that
\begin{equation}
\sup_{0\le t\le T}|z(t)|^{2}_{L^{2}}+\int_{0}^{T}|\nabla z(t)|^{2}dt\le
|z(0)\rvert^{2}_{L^{2}}e^{c\int_{0}^{T}\left[|\nabla d|^{4}_{L^{4}}+(1+|d|^{2}_{\mathrm{H} ^{2}})\right]dt}\label{tu3}
\end{equation} \end{Prop} \begin{proof}
Throughout this proof $c>0$ will denote an universal constant which depends only on $\Omega$ and may change from one term to the other. For the sake of simplicity we will omit the dependence on the space variable inside any integral over $\Omega$.
Let $d\in C([0,T];\mathrm{H} ^{1})\cap L^{2}(0,T;\mathrm{H} ^{2})$, $u \in C([0,T]; \mathrm{H} )\cap L^2(0,T;\mathrm{V} )$ and $z_{0}\in L^{2}$.
Since the problem \eqref{tu1} is linear, the proofs of the existence, which can be done via Galerkin and compactness methods, and the uniqueness are easy and omitted.
So we only prove \eqref{tu3}.
For this purpose, $z\in C([0,T]; L^2)\cap L^2(0,T; \mathrm{H} ^1)$ be a weak solution to \eqref{Eq:ViscousTransport-z}. We firstly observe that since $\mathrm{H} ^2 \subset L^\infty$
\begin{align*}
\int_{\Omega}(\phi^\prime(d)\cdot d)|z|^{2}\;dx \le &
c|d|_{L^{\infty}}(1+|d|_{L^\infty})\int_{\Omega}|z|^{2}\;dx\notag\\
&\le c(1+|d|^{2}_{\mathrm{H} ^{2}})|z|^{2}_{L^{2}}.
\end{align*}
Also, since $u\in \mathrm{H} ^1_0$ and $\Div u =0$ we can prove that
\begin{equation*}
\langle u \cdot\nabla z, z \rangle =\frac12 \int_\Omega u\cdot \nabla \lvert z\rvert^2 dx =0.
\end{equation*}
Hence, by the Gagliardo-Nirenberg inequality (\cite[Section 9.8, Example C.3]{Brezis}) and the Lions-Magenes lemma (\cite[Lemma III.1.2]{Temam_2001}), which is applicable because of Definition \ref{Def:WeakSol-ViscousTransport}\eqref{Item:DefWeakSol-1} and \eqref{Item:DefWeakSol-3}, we have the following inequalities
\begin{align*}
\langle \partial_t z, z\rangle=& \frac12 \frac{d}{dt}\lvert z\rvert^2_{L^2}\\
=& - \frac{1}{2}|\nabla z|^{2}_{L^{2}}+ 2\int_\Omega \lvert \nabla d \rvert^2 \lvert z\rvert^2\; dx -2 \int_\Omega (\phi^\prime(d).d)|z|^{2}\;dx \\
&\le - \frac{1}{2}|\nabla z|^{2}_{L^{2}}+ \int_{\Omega}|\nabla d|^{2}|z|^{2}\;dx + C(1+|d|^{2}_{\mathrm{H} ^{2}})|z|^{2}_{L^{2}}\notag\\
&\le - \frac{1}{2}|\nabla z|^{2}_{L^{2}}+ c|\nabla d|^{2}_{L^{4}}|z|^{2}_{L^{4}}+ c(1+|d|^{2}_{\mathrm{H} ^{2}})|z|^{2}_{L^{2}}\notag\\
&\le - \frac{1}{2}|\nabla z|^{2}_{L^{2}}+ c |\nabla d|^{2}_{L^{4}}|z|_{L^{2}}|\nabla z|_{L^{2}}+ \lvert \nabla d \rvert^2_{L^4} \lvert z\rvert^2_{L^2} + c(1+ |d|^{2}_{\mathrm{H} ^{2}})|z|^{2}_{L^{2}}\notag\\
&\le- \frac{1}{2}|\nabla z|^{2}_{L^{2}}+\frac{1}{4}|\nabla z|^{2}_{L^{2}}+ c|\nabla d|^{4}_{L^{4}}|z|^{2}_{L^{2}}+ c(1+|d|^{2}_{\mathrm{H} ^{2}})|z|^{2}_{L^{2}}.
\end{align*}
Thus,
\begin{align*}
\frac12 \frac{d}{dt}\lvert z\rvert^2_{L^2}+\frac14 |\nabla z|^{2}_{L^{2}}\le c|\nabla d|^{4}_{L^{4}}|z|^{2}_{L^{2}}+ c(1+|d|^{2}_{\mathrm{H} ^{2}})|z|^{2}_{L^{2}},
\end{align*}
By applying the Gronwall lemma we infer (\ref{tu3}).
This also completes the proof of the Proposition \ref{tu4}. \end{proof}
\section{Proof of Claim \ref{Clm:EstimateHigherorder}}\label{App:EstHighOrder} In this section we will the estimates \eqref{Eq:EstHighOrd-1}-\eqref{Eq:EstHighOrd-3} in Claim \ref{Clm:EstimateHigherorder}. Throughout this section $C>0$ will denote an universal constant which may change from one term to the other.
\begin{proof}[Proof of inequality \eqref{Eq:EstHighOrd-1}]
Let us choose and fix $v \in D(\mathrm{A} )$ and $n \in D(\hat{\mathrm{A}} ^\frac32)$.
Since $D(\hat{\mathrm{A}} ^\frac32)\subset \mathrm{H} ^3$ and $\mathrm{H} ^2$ is an algebra, it is not difficult to show that $\Div (\nabla n\odot \nabla n)\in L^2$. Thus, using the fact $\Pi:L^2 \to \mathrm{H} $ is bounded, the H\"older inequality and Gagliardo-Nirenberg inequality (\cite[Section 9.8, Example C.3]{Brezis}) we infer that
\begin{align*}
\lvert \langle \mathrm{A} v, \Pi [\Div(\nabla n \odot \nabla n) ] \rangle \rvert=&\lvert \langle \mathrm{A} v, \Pi[\nabla n)^{\mathrm{T}} \Delta n ]\rangle \rvert\\
& \le C \lvert \mathrm{A} v\rvert_{L^2} \lvert \nabla n \rvert_{L^4} \lvert \Delta n \rvert_{L^4} \\
& \le C \lvert \mathrm{A} v \rvert_{L^2} \lvert \nabla n \rvert_{L^4} \lvert \Delta n \rvert_{L^2}^\frac12 \lvert \nabla \Delta n \rvert^\frac12_{L^2}.
\end{align*}
Now, applying the Young inequality twice implies
\begin{align*}
\lvert \langle \mathrm{A} v, \Pi_L [\Div(\nabla n \odot \nabla n) ] \rangle \rvert
\le \frac12( \lvert \mathrm{A} v \rvert_{L^2}^2+ \lvert \nabla \Delta n \rvert^2_{L^2} )+ \lvert \nabla n \rvert_{L^4}^4\lvert \Delta n \rvert_{L^2}^2.
\end{align*}
This complete the proof of the inequality \eqref{Eq:EstHighOrd-1}. \end{proof}
\begin{proof}[Proof of the inequality \eqref{Eq:EstHighOrd-2}]
Let $v\in D(\mathrm{A} )$ and $n \in D(\hat{\mathrm{A}} ^\frac32)$ be fixed.
Because $D(\mathrm{A} )\subset \mathrm{H} ^2$, $D(\hat{\mathrm{A}} ^\frac32)\subset \mathrm{H} ^3$ and $\mathrm{H} ^\theta$, $\theta>1$, is an algebra, it is not difficult to show that $v\cdot \nabla n \in \mathrm{H} ^2\subset \mathrm{H} ^1$. Hence, by using the Cauchy-Schwarz and Young inequalities we infer
\begin{align*}
{}_{(\mathrm{H} ^1)^\ast}\langle \mathrm{A} ^2 n, v\cdot \nabla n \rangle_{\mathrm{H} ^1} = &\langle \nabla \Delta n, \nabla (v\cdot \nabla n) \rangle \\
&\le \frac1{24} \lvert \nabla \Delta n\rvert^2_{L^2} + C \lvert \nabla (v\cdot \nabla n) \rvert^2_{L^2}.
\end{align*}
Using the H\"older inequality in the last line we infer that
\begin{align*}
{}_{(\mathrm{H} ^1)^\ast}\langle \mathrm{A} ^2 n, v\cdot \nabla n \rangle_{\mathrm{H} ^1} \le \frac1{24} \lvert \nabla \Delta n\rvert^2_{L^2} + C \lvert (\nabla v) (\nabla n) \rvert^2_{L^2}+C \lvert v\cdot \nabla(\nabla n)\rvert^2_{L^2}\\
\le \frac1{24} \lvert \nabla \Delta n\rvert^2_{L^2} + C \lvert (\nabla v)\rvert^2_{L^4 } \rvert(\nabla n) \rvert^2_{L^4}+C \lvert v\rvert^2_{L^4} \lvert \nabla^2 n\rvert^2_{L^4}
\end{align*}
Now, by using \cite[Theorem 3.4]{Simader} to estimate $\lvert \nabla^2 n \rvert_{L^4}$ by $\lvert \Delta n \rvert_{L^4}$, then by applying the Gagliardo-Nirenberg (\cite[Section 9.8, Example C.3]{Brezis}) , the Young inequalities we infer that
\begin{align}
{}_{(\mathrm{H} ^1)^\ast}\langle \mathrm{A} ^2 n, v\cdot \nabla n \rangle_{\mathrm{H} ^1}
\le \frac1{24} \lvert \nabla \Delta n\rvert^2_{L^2} + C \lvert \nabla v\rvert_{L^2 }\lvert \mathrm{A} v \rvert_{L^2} \rvert(\nabla n) \rvert^2_{L^4}+C \lvert v\rvert^2_{L^4} \lvert \Delta n\rvert^2_{L^4}\nonumber\\
\le \frac1{24} \lvert \nabla \Delta n\rvert^2_{L^2} +\frac1{12} \lvert \mathrm{A} v\rvert^2_{L^2}+ C \lvert \nabla v\rvert^2_{L^2 } \rvert(\nabla n) \rvert^4_{L^4}+C \lvert v\rvert^2_{L^4} [\lvert \Delta n\rvert_{L^2} \lvert \nabla \Delta n \rvert_{L^2} +\lvert \Delta n \rvert^2_{L^2} ]\nonumber \\
\le \frac1{12} \lvert \nabla \Delta n\rvert^2_{L^2} +\frac1{12} \lvert \mathrm{A} v\rvert^2_{L^2}+ C \lvert \nabla v\rvert^2_{L^2 } \rvert(\nabla n) \rvert^2_{L^4}+C [ \lvert v \rvert^2_{L^4}+\lvert v\rvert^4_{L^4} ]\lvert \Delta n\rvert_{L^2}.\label{Eq:EstHighOrd-1b}
\end{align}
This completes the proof of the inequality \eqref{Eq:EstHighOrd-2}. \end{proof} \begin{proof}[Proof of the inequality \eqref{Eq:EstHighOrd-3}]
Let $v \in D(\mathrm{A} )$ and $n \in D(\hat{\mathrm{A}} ^\frac32)$. As in the proof of \eqref{Eq:EstHighOrd-1}, we can show that $\lvert \nabla n \rvert^2n \in \mathrm{H} ^1$. Hence, using the Cauchy-Schwarz, the Young and the Gagliardo-Nirenberg inequalities (see \cite[Section 9.8, Example C.3]{Brezis}) we infer that
\begin{align*}
{}_{(\mathrm{H} ^1)^\ast}\langle \hat{\mathrm{A}} ^2 n, \lvert \nabla n\rvert^2n \rangle_{\mathrm{H} ^1} = & \langle \nabla \Delta n, \nabla (\lvert \nabla n \rvert^2 n) \rangle \\
&\le \frac1{24} \lvert \nabla \Delta n \rvert^2_{L^2} + C \lvert \nabla (\lvert \nabla n \rvert^2 n) \rvert^2_{L^2}\\
& \le \frac1{24} \lvert \nabla \Delta n \rvert^2_{L^2} + C \lvert \nabla (\lvert \nabla n \rvert^2 n) \rvert^2_{L^2}\\
&\le \frac{1}{24}\lvert \nabla \Delta n \rvert^2_{L^2} +C \lvert (\nabla n)(\nabla^2 n ) \rvert^2_{L^2}+ C \lvert \lvert \nabla n \rvert^2 \nabla n \rvert^2_{L^2}\\
&\le \frac{1}{24}\lvert \nabla \Delta n \rvert^2_{L^2} +C \lvert (\nabla n)\rvert^2_{L^4} \lvert (\nabla^2 n ) \rvert^2_{L^4}+ C \lvert \nabla n \rvert^6_{L^6}\\
&\le \frac{1}{24}\lvert \nabla \Delta n \rvert^2_{L^2} +C \lvert (\nabla n)\rvert^2_{L^4} \lvert (\nabla^2 n ) \rvert^2_{L^4}+ C \lvert \nabla n \rvert^4_{L^4}(\lvert \nabla n\rvert^2_{L^2} + \lvert \nabla^2 n \rvert^2_{L^2}).
\end{align*}
Proceeding as in the proof of \eqref{Eq:EstHighOrd-1b}, we show that
\begin{align*}
{}_{(\mathrm{H} ^1)^\ast}\langle \hat{\mathrm{A}} ^2 n, \lvert \nabla n\rvert^2n \rangle_{\mathrm{H} ^1}
&\le \frac{1}{12}\lvert \nabla \Delta n \rvert^2_{L^2} +C [\lvert (\nabla n)\rvert^4_{L^4}+ \lvert (\nabla n)\rvert^2_{L^4}]\lvert (\Delta n ) \rvert^2_{L^2}+ C \lvert \nabla n \rvert^4_{L^4}(\lvert \nabla n\rvert^2_{L^2} + \lvert \Delta n \rvert^2_{L^2}).
\end{align*}
This completes the proof of \eqref{Eq:EstHighOrd-3}.
\end{proof}
\section{A weak continuity of Banach space valued functions}
In this section we will state and prove of continuity theorem for Banach-valued map similar to \cite{Strauss}. This theorem was initially proven in the work in progress \cite{ZB+AP+UM}, but for the sake of completeness we repeat the proof here.
\begin{Thm}\label{Thm:StraussThm}
Let $X$ and $Y$ be two Banach spaces such that $X$ is reflexive, $X\subset Y$ and the canonical injection $i:X\to Y$ is dense and continuous.
Let $T>0$ be fixed and $u \in L^\infty([0,T); X)$. Let also $b \in Y$ and $v:[0,T] \to Y$, defined by
\begin{equation*}
v(t)=
\begin{cases}
i(u(t)) \text{ if } t\in [0,T),\\
b \text{ if } t=T,
\end{cases}
\end{equation*}
be weakly continuous.
Then, $b\in X$ and the map $\tilde{u}: [0,T] \to X$ defined by
\begin{equation}\label{Eq:ModifiedMap}
\tilde{u}(t)=
\begin{cases}
u(t) \text{ if } t\in [0,T),\\
b \text{ if } t=T,
\end{cases}
\end{equation}
is weakly continuous.
\end{Thm} \begin{proof}
Let $X$, $Y$, $b\in Y$ , $T>0$, $u\in L^\infty([0,T);X)$, $v:[0,T]\to Y$ be as in the statement of the theorem. Let also $(t_n)_{n\in \mathbb{N}}\subset [0,T)$ be a sequence such that $t_n \nearrow T$.
Let us prove the first part of the theorem, \text{i.e.}, that $b\in X$. For this aim we observe that by assumption there exists $M>0$ such that
\begin{equation*}
\lvert u(t) \rvert_X \le M \text{ for all } t\in [0,T).
\end{equation*}
Hence, by the Banach-Alaoglu theorem we can extract from $(t_n)_{n \in \mathbb{N}}$ a subsequence, which is still denoted by $(t_n)_{n\in \mathbb{N}}$, such that $t_n\nearrow T$ and
\begin{equation*}
u(t_n) \to x \text{ weakly in } X.
\end{equation*}
Since, by assumption, $X\subset Y$ we infer that $v(t) = i(u(t)) =u(t)$ for all $t\in [0,T]$. Hence, by the weak continuity of $v$ we infer that
\begin{equation}\notag
v(t_n)=u(t_n) \to b \text{ weakly in } Y.
\end{equation}
By the uniqueness of weak limit we infer that $x=b \in X$.
It remains to prove the second part of the theorem, \textit{i.e.}, we shall show that the map $\tilde{u}:[0,T] \to X$ defined in \eqref{Eq:ModifiedMap} is weakly continuous. For this purpose, we will closely follow the proof of \cite[Lemma III.1.4 ]{Temam_2001}. We will divide the task into two cases.
\begin{trivlist}
\item[\textbf{Case 1}] Let $t, t_0 \in [0,T)$. Let $X^\ast$ and $Y^\ast$ be the dual spaces of $X$ and $Y$, respectively.
Recall that since the embedding $X\subset Y$ is dense and continuous, the embedding $Y^\ast\subset X^\ast$ is dense and continuous, too. Let us choose and fix $\varepsilon\,>0$ and $\eta\in X^\ast$. Then, there exists $\eta_\varepsilon\,\in Y^\ast$ such that
\begin{equation}\notag
\lvert \eta -\eta_\varepsilon\, \rvert_{X^\ast}<\varepsilon\,.
\end{equation} Thus, using the boundedness of $u=\tilde{u}\Big\vert_{[0,T)}$ we infer that
\begin{align}
\lvert {}_{X}\langle \tilde{u}(t) -\tilde{u}(t_0), \eta \rangle_{X^\ast}\rvert\le &\lvert {}_{X}\langle \tilde{u}(t) -\tilde{u}(t_0), \eta-\eta_\varepsilon\, \rangle_{ X^\ast}\rvert+ \lvert {}_{Y}\langle \tilde{u}(t) -\tilde{u}(t_0), \eta \rangle_{Y^\ast}\rvert\notag\\
\le & 2 M \lvert \eta -\eta_\varepsilon\,\rvert_{X^\ast} + \lvert {}_{Y}\langle \tilde{u}(t) -\tilde{u}(t_0), \eta \rangle_{Y^\ast}\rvert.\notag
\end{align}
\end{trivlist}
By the weak continuity of $v\Big\vert_{[0,T)}=u=\tilde{u}$ we have
\begin{align}
\lim_{t\to t_0} \lvert {}_{X}\langle \tilde{u}(t) -\tilde{u}(t_0), \eta \rangle_{X^\ast}\rvert\le 2M \lvert \eta- \eta_\varepsilon\,\rvert_{X^\ast} + \lim_{t\to t_0} \lvert {}_{Y}\langle v(t) -v(t_0), \eta \rangle_{Y^\ast} \rvert
\le 2M \varepsilon\,.
\end{align}
Since $\varepsilon\,>0$ is arbitrary we infer that
\begin{equation}
\lim_{t\to t_0} \lvert {}_{X}\langle \tilde{u}(t) -\tilde{u}(t_0), \eta \rangle_{X^\ast}\rvert=0.\notag
\end{equation}
Hence, $\tilde{u}\Big\vert_{[0,T)}$ is weakly continuous.
\item[\textbf{Case 2}.] We will prove that $\tilde{u}$ is weakly continuous at $t=T$. Towards this aims let $\eta\in X^\ast$, $\varepsilon\,>0$ be fixed.
From the proof of the first part of the theorem we infer that there exists $\delta>0$ such that for $t>0$ with $0<T-t<\delta$
\begin{equation*}
\lvert{}_{X}\langle u(t) -b, \eta \rangle_{X^\ast} \rvert= \lvert{}_{X}\langle u(t) -\tilde{u}(T), \eta \rangle_{X^\ast} \rvert<\varepsilon\,.
\end{equation*}
But for $t\in (0,T)$ we have $u(t)=\tilde{u}(t)$, hence for $t>0$ with $0<T-t<\delta$
\begin{equation*}
\lvert{}_{X}\langle \tilde{u}(t) -\tilde{u}(T), \eta \rangle_{X^\ast} \rvert \le \lvert{}_{X}\langle \tilde{u}(t) -u(t), \eta \rangle_{X^\ast} \rvert+ \lvert{}_{X}\langle u(t) -\tilde{u}(T), \eta \rangle_{ X^\ast} \rvert
< \varepsilon\,.
\end{equation*}
Thus, $\tilde{u}$ is weakly continuous at $t=T$. This completes the proof of Case 2, the second part of the theorem and hence the whole theorem. \end{proof}
\end{document} | arXiv |
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\title{\mytitle\thanks{The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement no. 340506. This work was partially supported by DFG grants SA 933/10-2. }}
\author{Sebastian Lamm\thanks{Institute for Theoretical Informatics, Karlsruhe Institute of Technology, Karlsruhe, Germany. Parts of this research have been done while the author was at the University of Vienna.}, Christian Schulz\thanks{Faculty of Computer Science, University of Vienna, Vienna, Austria.}, Darren Strash\thanks{Department of Computer Science, Hamilton College, Clinton, NY, USA.}, Robert Williger\thanks{Institute for Theoretical Informatics, Karlsruhe Institute of Technology, Karlsruhe, Germany.}, Huashuo Zhang\thanks{Department of Computer Science, Colgate University, Hamilton, NY, USA.}}
\date{} \maketitle
\begin{abstract} One powerful technique to solve NP-hard optimization problems in practice is branch-and-reduce search---which is branch-and-bound that intermixes branching with reductions to decrease the input size. While this technique is known to be very effective in practice for unweighted problems, very little is known for weighted problems, in part due to a lack of known effective reductions. In this work, we develop a full suite of new reductions for the maximum weight independent set problem and provide extensive experiments to show their effectiveness in practice on real-world graphs of up to millions of vertices and edges.
Our experiments indicate that our approach is able to outperform existing state-of-the-art algorithms, solving many instances that were previously infeasible. In particular, we show that branch-and-reduce is able to solve a large number of instances up to two orders of magnitude faster than existing (inexact) local search algorithms---and is able to solve the majority of instances within 15 minutes. For those instances remaining infeasible, we show that combining kernelization with local search produces higher-quality solutions than local search alone.
\end{abstract}
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\section{Introduction} The maximum weight independent set problem is an NP-hard problem that has attracted much attention in the combinatorial optimization community, due to its difficulty and its importance in many fields. Given a graph $G=(V,E,w)$ with weight function $w:V\rightarrow \mathbb{R}^+$, the goal of the maximum weight independent set problem is to compute a set of vertices $\mathcal{I}\subseteq V$ with maximum total weight, such that no vertices in $\mathcal{I}$ are adjacent to one another. Such a set is called a \emph{maximum weight independent set} (MWIS). The maximum weight independent set problem has applications spanning many disciplines, including signal transmission, information retrieval, and computer vision~\cite{balas1986finding}. As a concrete example, weighted independent sets are vital in labeling strategies for maps~\cite{gemsa2014dynamiclabel,barth-2016}, where the objective is to maximize the number of visible non-overlapping labels on a map. Here, the maximum weight independent set problem is solved in the label conflict graph, where any two overlapping labels are connected by an edge and vertices have a weight proportional to the city's population.
Similar to their unweighted counterparts, a maximum weight independent set $\mathcal{I}\subseteq V$ in $G$ is a maximum weight clique in $\overline{G}$ (the complement of $G$), and $V \setminus \mathcal{I}$ is a minimum vertex cover of $G$~\cite{xu2016new,cai-dynwvc}. Since all of these problems are NP-hard~\cite{garey1974}, heuristic algorithms are often used in practice to efficiently compute solutions of high quality on \emph{large} graphs~\cite{pullan-2009,hybrid-ils-2018,li2017efficient,cai-dynwvc}.
Small graphs with hundreds to thousands of vertices may often be solved in practice with traditional branch-and-bound methods~\cite{balas1986finding,babel1994fast,warren2006combinatorial,butenko-trukhanov}. However, even for medium-sized synthetic instances, the maximum weight independent set problem becomes infeasible. Further complicating the matter is the lack of availability of large real-world test instances --- instead, the standard practice is to either systematically or randomly assign weights to vertices in an unweighted graph. Therefore, the performance of exact algorithms on real-world data sets is virtually unknown.
In stark contrast, the unweighted variants can be quickly solved on \emph{large} real-world instances---even with millions of vertices---in practice, by using \emph{kernelization}~\cite{strash-power-2016,chang2017,hespe2018scalable} or the \emph{branch-and-reduce} paradigm~\cite{akiba-tcs-2016}. For those instances that can't be solved exactly, high-quality (and often exact) solutions can be found by combining kernelization with either local search~\cite{dahlum2016,chang2017} or evolutionary algorithms~\cite{redumis-2017}.
These algorithms first remove (or fold) whole subgraphs from the input graph while still maintaining the ability to compute an optimal solution from the resulting smaller instance. This so-called \emph{kernel} is then solved by an exact or heuristic algorithm. While these techniques are well understood, and are effective in practice for the unweighted variants of these problems, very little is known about the weighted~problems.
While the unweighted maximum independent set problem has many known reductions, we are only aware of \emph{one} explicitly known reduction for the maximum weight independent set problem: the weighted critical independent set reduction by Butenko and Trukhanov~\cite{butenko-trukhanov}, which has only been tested on small synthetic instances with unit weight (unweighted case). However, it remains to be examined how their weighted reduction performs in practice.
There is only one other reduction-like procedure of which we are aware, although it is neither called so directly nor is it explicitly implemented as a reduction. Nogueria~et~al.~\cite{hybrid-ils-2018} introduced the notion of a ``$(\omega,1)$-swap'' in their local search algorithm, which swaps a vertex into a solution if its neighbors in the current solution have smaller total weight. This swap is not guaranteed to select a vertex in a true MWIS; however, we show how to transform it into a reduction that does.
\myparagraph{Our Results.} In this work, we develop a full suite of new reductions for the maximum weight independent set problem and provide extensive experiments to show their effectiveness in practice on real-world graphs of up to millions of vertices and edges. While existing exact algorithms are only able to solve graphs with hundreds of vertices, our experiments show that our approach is able to exactly solve real-world label conflict graphs with thousands of vertices, and other larger networks with synthetically generated vertex weights---all of which are infeasible for state-of-the-art solvers. Further, our branch-and-reduce algorithm is able to solve a large number of instances up to two orders of magnitude faster than existing \emph{inexact} local search algorithms---solving the majority of instances within 15 minutes. For those instances remaining infeasible, we show that combining kernelization with local search produces higher-quality solutions than local search alone.
Finally, we develop new \emph{meta} reductions, which are general rules that subsume traditional reductions. We show that weighted variants of popular unweighted reductions can be explained by two general (and intuitive) rules---which use MWIS search as a subroutine. This yields a simple framework covering many~reductions.
\ifFull \myparagraph{Organization.} The rest of the paper is organized as follows. We begin in Section~\ref{sec:related_work} by highlighting important related work for the maximum weight independent set problem. This includes exact and heuristic algorithms, as well as hybrid approaches. Afterwards, we present basic concepts used in our algorithm in Section~\ref{sec:prelim}. This section also contains reductions for that are used for the unweighted maximum independent set in practice in order to make the weighted reductions that we define later more accessible. We describe the overall structure of our branch-and-reduce framework in Section~\ref{sec:noveltechniques}. The full set of reductions for the weighted maximum independent set problem that are employed by our algorithm are described in Section~\ref{sec:labelofchapterforoutlineofpaper_reductions}.
An extensive experimental evaluation of our method is presented in Section~\ref{sec:experiments}. Finally, we present conclusions in Section~\ref{sec:conclusion}. \fi{}
\section{Related Work} \label{sec:related_work} We now present important related work on finding high-quality weighted independent sets. This includes exact branch-and-bound algorithms, reduction based approaches, as well as inexact heuristics, e.g.\ local search algorithms. We then highlight some recent approaches that combine both exact and inexact~algorithms.
\subsection{Exact Algorithms.} Much research has been devoted to improve exact branch-and-bound algorithms for the MWIS and its complementary problems. These improvements include different pruning methods and sophisticated branching schemes~\cite{ostergaard2002fast,balas1986finding,babel1994fast,warren2006combinatorial}.
Warren and Hicks~\cite{warren2006combinatorial} proposed three combinatorial branch-and-bound algorithms that are able to quickly solve DIMACS and weighted random graphs. These algorithms use weighted clique covers to generate upper bounds that reduce the search space via pruning. Furthermore, they all use a branching scheme proposed by Balas and Yu~\cite{balas1986finding}. In particular, their first algorithm is an extension and improvement of a method by Babel~\cite{babel1994fast}. Their second one uses a modified version of the algorithm by Balas and Yu that uses clique covers that borrow structural features from the ones by Babel~\cite{babel1994fast}. Finally, their third approach is a hybrid of both previous algorithms. Overall, their algorithms are able to quickly solve instances with hundreds of vertices.
An important technique to reduce the base of the exponent for exact branch-and-bound algorithms are so-called \emph{reduction rules}. Reduction rules are able to reduce the input graph to an irreducible \emph{kernel} by removing well-defined subgraphs. This is done by selecting certain vertices that are provably part of some maximum(-weight) independent set, thus maintaining optimality. We can then extend a solution on the kernel to a solution on the input graph by undoing the previously applied reductions. There exist several well-known reduction rules for the unweighted vertex cover problem (and in turn for the unweighted MIS problem)~\cite{akiba-tcs-2016}. However, there are only a few reductions known for the MWIS problem.
One of these was proposed by Butenko and Trukhanov~\cite{butenko-trukhanov}. In particular, they show that every critical weighted independent set is part of a maximum weight independent set. A critical weighted set is a subset of vertices such that the difference between its weight and the weight of its neighboring vertices is maximal for all such sets. They can be found in polynomial time via minimum cuts. Their neighborhood is recursively removed from the graph until the critical set is~empty.
As noted by Larson~\cite{larson-2007}, it is possible that in the unweighted case the initial critical set found by Butenko and Trukhanov might be empty. To prevent this case, Larson~\cite{larson-2007} proposed an algorithm that finds a \emph{maximum} (unweighted) critical independent set. \longRW{His algorithm accumulates vertices that are in some critical set and removes their neighborhood. Additionally, he provides a method to quickly check if a given vertex is part of some critical set.} Later Iwata~\cite{iwata-2014} has shown how to remove a large collection of vertices from a maximum matching all at once; however, it is not known if these reductions are equivalent.
For the maximum weight clique problem, Cai and Lin~\cite{cai2016fast} give an exact branch-and-bound algorithm that interleaves between clique construction and reductions. \longRW{In particular, their algorithm picks different starting vertices to form a clique and then maintains a candidate set to iteratively extend this clique. In each iteration, the vertex to be added is selected using a benefit estimation function and a dynamic best from multiple selection heuristic~\cite{cai2015balance}. Once the candidate set is empty, the new solution is compared to the best solution found so far. If an improvement is found, their algorithm then tries to apply reductions and reduce the graph size. To be more specific, they use two reduction rules that are able to remove a vertex $v$ by computing upper bounds related to the weight of different neighborhoods of $v$.} We briefly note that their algorithm and reductions are targeted at sparse graphs, and therefore their reductions would likely work well for the MWIS problem on \emph{dense} graphs---but not on the sparse graphs we consider here.
\subsection{Heuristic Algorithms.} Heuristic algorithms such as local search work by maintaining a single solution that is gradually improved through a series of vertex deletions, insertion and swaps. Additionally, \emph{plateau search} allows these algorithms to explore the search space by performing node swaps that do not change the value of the objective function. We now cover state-of-the-art heuristics for both the unweighted and weighted maximum independent set problem.
For the unweighted case, the iterated local search algorithm by Andrade~et~al.~\cite{andrade-2012} (ARW) is one of the most successful approaches in practice. Their algorithm is based on finding improvements using so-called $(1,2)$-swaps that can be found in linear time. Such a swap removes a single vertex from the current solution and inserts two new vertices instead.
Their algorithm is able to find (near-)optimal solutions for small to medium-sized instances in milliseconds, but struggles on massive instances with millions of vertices and edges~\cite{dahlum2016}. \longRW{Their algorithm was improved significantly by applying (2,1)- and ($k$,1)-swaps as perturbation steps~\cite{jin-hao-swap-2015}, omitting high-degree vertices~\cite{dahlum2016} and using reduction rules~\cite{dahlum2016,redumis-2017}.}
\longRW{As most other local search algorithm PLS interleaves sequences of iterative improvements and plateau search. Improvements are found using different sub-algorithms that select vertices either at random, by using their degree or by using a penalty strategy that is dynamically adjusted during the main algorithm. Additionally, these sub-algorithms differ in their perturbation mechanisms. PLS achieved state-of-the-art quality on various benchmark sets including the popular DIMACS benchmark.}
Several local search algorithms have been proposed for the maximum weight independent set problem. Most of these approaches interleave a sequence of iterative improvements and plateau search. Further strategies developed for these algorithms include the usage of sub-algorithms for vertex selection~\cite{pullan-2006,pullan-2009}, tabu mechanisms using randomized restarts~\cite{wu2012multi}, and adaptive perturbation strategies~\cite{benlic2013breakout}. Local search approaches are often able to obtain high-quality solutions on medium to large instances that are not solvable using exact algorithms. Next, we cover some of the most recent state-of-the-art local search algorithms in greater detail.
The hybrid iterated local search (HILS) by Nogueria~et~al.~\cite{hybrid-ils-2018} extends ARW to the weighted case. It uses two efficient neighborhood structures: $(\omega, 1)$-swaps and weighted $(1,2)$-swaps. Both of these structures are explored using a variable neighborhood descent procedure. \longRW{Additionally, they introduce a dynamic perturbation mechanism that regulates intensification and diversification.} Their algorithm outperforms state-of-the-art algorithms on well-known benchmarks and is able to find known optimal solution in milliseconds.
Recently, Cai~et~al.~\cite{cai-dynwvc} proposed a heuristic algorithm for the weighted vertex cover problem that was able to derive high-quality solution for a variety of large real-world instances. Their algorithm is based on a local search algorithm by Li~et~al.~\cite{li2017efficient} and uses iterative removal and maximization of a valid vertex cover. \longRW{Vertices are removed and inserted using two scoring function: a \emph{gain} function and a \emph{loss} function. In particular, the baseline algorithm removes two vertices in each iteration. The first vertex is chosen using the minimal loss value, the second one is selected by the best from multiple selection heuristic~\cite{cai2015balance} (w.r.t. the minimal loss value). They then introduce two dynamic approaches for selecting between different scoring functions for the gain and loss values of a vertex. To be more precise, their first approach selects between two different scoring functions by counting the number of non-improving steps. Their second approach, dynamically adjust the number of vertices which are removed in a single iteration based on their total degree.}
\subsection{Hybrid Algorithms.} In order to overcome the shortcomings of both exact and inexact methods, new approaches that combine reduction rules with heuristic local search algorithms were proposed recently~\cite{dahlum2016,redumis-2017}. A very successful approach using this paradigm is the reducing-peeling framework proposed by Chang~et~al.~\cite{chang2017} which is based on the techniques proposed by Lamm~et~al.~\cite{redumis-2017}. Their algorithm works by computing a kernel using practically efficient reduction rules in linear and near-linear time. Additionally, they provide an extension of their reduction rules that is able to compute good initial solutions for the kernel. In particular, they greedily select vertices that are unlikely to be in a large independent set, thereby opening up the reduction space again. Thus, they are able to significantly improve the performance of the ARW local search algorithm that is applied on the kernelized graph. To speed-up kernelization, Hespe~et~al.~\cite{hespe2018scalable} proposed a shared-memory algorithm using partitioning and parallel bipartite matching. \longRW{Additionally, they propose two pruning techniques to provide additional performance gains over the algorithm by Akiba and Iwata~\cite{akiba-tcs-2016}. These techniques are called dependency checking and reduction tracking. Dependency checking allows them to prune reductions when they will provably not succeed, therefore significantly reducing the number of failed reductions. Reduction tracking enables them to stop local reductions when they are not effectively reducing the global graph sizes. However, using reduction tracking does not produce a full kernel.}
\section{Preliminaries} \label{sec:prelim}
Let $G=(V=\{0,\ldots, n-1\},E, w)$ be an undirected graph with $n = |V|$ nodes and $m = |E|$ edges.
$w:V \rightarrow \mathbb{R}^+$ is the real-valued vertex weighting function such that $w(v) \in \mathbb{R}^+$ for all $v \in V$. Furthermore, for a non-empty set $S \subseteq V$ we use $w(S) = \sum_{v\in S} w(v)$ and $|S|$ to denote the \emph{weight} and \emph{size} of $S$. The set $N(v) = \setGilt{u}{\set{v,u}\in E}$ denotes the neighbors of $v$. We further define the neighborhood of a set of nodes $U \subseteq V$ to be $N(U) = \cup_{v\in U} N(v)\setminus U$, $N[v] = N(v) \cup \{v\}$, and $N[U] = N(U) \cup U$. A graph $H=(V_H, E_H)$ is said to be a \emph{subgraph} of $G=(V, E)$ if $V_H \subseteq V$ and $E_H \subseteq E$. We call $H$ an \emph{induced} subgraph when $E_H = \setGilt{\{u,v\} \in E}{u,v\in V_H}$. For a set of nodes $U\subseteq V$, $G[U]$ denotes the subgraph induced by $U$. The \emph{complement} of a graph is defined as $\overline{G} = (V,\overline{E})$, where $\overline{E}$ is the set of edges not present in $G$. An \emph{independent set} is a set $\mathcal{I} \subseteq V$, such that all nodes in $\mathcal{I}$ are pairwise non-adjacent. An independent set is \emph{maximal} if it is not a subset of any larger independent set. Furthermore, an independent set $\I$ has \emph{maximum weight} if there is no heavier independent set, i.e.\ ~there exists no independent set $I'$ such that $w(I) < w(I')$.
The weight of a maximum independent set of $G$ is denoted by $\alpha_w(G)$. The \emph{maximum weight independent set problem} (MWIS) is that of finding the independent set of largest weight among all possible independent sets. A \emph{vertex cover} is a subset of nodes $C \subseteq V$, such that every edge $e \in E$ is incident to at least one node in $C$. The \emph{minimum-weight vertex cover problem} asks for the vertex cover with the minimum total weight. Note that the vertex cover problem is complementary to the independent set problem, since the complement of a vertex cover $V \setminus C$ is an independent set. Thus, if $C$ is a minimum vertex cover, then $V \setminus C$ is a maximum independent set. A \emph{clique} is a subset of the nodes $Q \subseteq V$ such that all nodes in $Q$ are pairwise adjacent. An independent set is a clique in the complement graph.
\subsection{Unweighted Reductions.} \label{subsec:reductions} In this section, we describe reduction rules for the \emph{unweighted} maximum independent set problem. These reductions perform exceptionally well in practice and form the basis of our weighted reductions described~in~Section~\ref{sec:new-reductions}.
\myparagraph{Vertex Folding~\cite{chen1999}.} Vertex folding was first introduced by Chen et al.~\cite{chen1999} to reduce the theoretical running time of exact branch-and-bound algorithms for the maximum independent set problem. This reduction is applied whenever there is a vertex $v$ with degree two and non-adjacent neighbors $u$ and $w$. Chen et al.~\cite{chen1999} then showed that either $v$ or both $u$ and $w$ are in some maximum independent set. Thus, we can contract $u$, $v$, and $w$ into a single vertex $v'$ (called a \emph{fold}), forming a new graph $G'$. Then $\alpha(G) = \alpha(G') + 1$ and after finding a MIS $\I'$ of $G'$, if $v'\notin \I'$ then $\I = \I'\cup\{v\}$ is an MIS of $G$, otherwise $\I = (\I'\setminus\{v'\})\cup \{u,w\}$ is.
\myparagraph{Isolated Vertex Removal~\cite{butenko-correcting-codes-2009}.} An \emph{isolated} vertex, also called a \emph{simplicial} vertex, is a vertex $v$ whose neighborhood forms a clique. That is, there is a clique $C$ such that $V(C) \cap N[v] = N[v]$; this clique is called an \emph{isolated clique}. Since $v$ has no neighbors outside of the clique, by a cut-and-paste argument, it must be in \emph{some} maximum independent set. Therefore, we can add $v$ to the maximum independent set we are computing, and remove $v$ and $C$ from the graph. Isolated vertex removal was shown by Butenko~et~al.~\cite{butenko-correcting-codes-2009} to be highly effective in finding exact maximum independent sets on graphs derived from error-correcting codes. In order to work efficiently in practice, this reduction is typically limited to cliques with size at most 2 or 3~\cite{chang2017,dahlum2016}.
Although Chang~et~al.~\cite{chang2017} showed that the domination reduction (described below) captures the isolated vertex removal reduction, that reduction must be applied several times: once per neighbor in the clique.
\myparagraph{Twin.} Two non-adjacent vertices $u$ and $v$ are called \emph{twins} if $N(u) = N(v)$.
Note that either both $u$ and $v$ are in some MIS, or some subset of $N(u)$ is in some MIS. If $|N(u)| = |N(v)| = 3$, then either $u$ and $v$ are together or \emph{at least} two vertices of $N(u)$ must be in an MIS. The following case of the reduction is relevant to our result: If $N(u)$ is independent, then we can fold $u$, $v$, and $N(v)$ into a single vertex $v'$ and $\alpha(G) = \alpha(G') + 2$.
\myparagraph{Domination~\cite{fomin-2009}.} Given two vertices $u$ and $v$, $u$ is said to \emph{dominate} $v$ if and only if $N[u] \supseteq N[v]$. In this case there is an MIS in $G$ that excludes $u$ and therefore, $u$ can be removed from the graph.
\myparagraph{Critical Independent Set.}
A subset $U_c \subseteq V$ is called a \emph{critical set} if $|U_c| - |N(U_c)| = \max\{|U| - |N(U)| : U \subseteq V\}$.
Likewise, an independent set $I_c \subseteq V$ is called a \emph{critical independent set} if $|I_c| - |N(I_c)| = \max\{|I| - |N(I)| : I \text{ is an independent set of } G\}$. Butenko and Trukhanov~\cite{butenko-trukhanov} show that any critical independent set is a subset of a maximum independent set. They then continue to develop a reduction that uses critical independent sets which can be computed in polynomial time. In particular, they start by finding a critical set in $G$ by using a reduction to the maximum matching problem in a bipartite graph~\cite{ageev1994finding} . In turn, this problem can then be solved in $\mathcal{O}(|V|\sqrt{|E|})$ time using the Hopcroft-Karp algorithm. They then obtain a critical independent set by setting $I_c = U_c \setminus N(U_c)$. Finally, they can remove $I_c$ and $N(I_c)$ from $G$.
\myparagraph{Linear Programming (LP) Relaxation.} The LP-based reduction rule by Nemhauser and Trotter~\cite{nemhauser-1975}, is based on an LP relaxation of the vertex cover problem: \begin{align*}
\text{minimize } & \sum_{v \in V} x_v & \\
\text{s.t. } & x_u + x_v \geq 1 & \text{ for } (u,v) \in E, \\
& x_v \geq 0 & \text{ for } v \in V. \end{align*}
Nemhauser and Trotter~\cite{nemhauser-1975} showed that there exists an optimal half-integral solution for this problem. Additionally, they prove that if a variable $x_v$ takes an integer value in an optimal solution, then there exists an optimal integer solution where $x_v$ has the same value. Just as in the critical set reduction, they use a reduction to the maximum bipartite matching problem to compute a half-integral solution. To develop a reduction rule for the vertex cover problem, they afterwards fix the integral part of their solution and output the remaining graph. Their approach was successively improved by Iwata~et~al.~\cite{iwata-2014} and was shown to be effective in practice by Akiba and Iwata~\cite{akiba-tcs-2016}.
\subsection{Critical Weighted Independent Set Reduction.} We now briefly describe the critical weighted independent set reduction, which is the \emph{only} reduction that has appeared in the literature for the \emph{weighted} maximum independent set problem. Similar to the unweighted case, a subset $U_c \subseteq V$ is called a \emph{critical weighted set} if $w(U_c) - w(N(U_c)) = \max\{w(U) - w(N(U)) : U \subseteq V\}$. A weighted independent set $I_c \subseteq V$ is called a \emph{critical weighted independent set} (CWIS) if $w(I_c) - w(N(I_c)) = \max\{w(I) - w(N(I)) : I \text{ is an independent set of } G\}$. Butenko and Trukhanov~\cite{butenko-trukhanov} show that any CWIS is a subset of a maximum weight independent set. Additionally, they propose a weighted critical set reduction which works similar to its unweighted counterpart. However, instead of computing a maximum matching in a bipartite graph, a critical weighted set is obtained by solving the selection problem~\cite{ageev1994finding}. The problem is equivalent to finding a minimum cut in a bipartite graph. For a proof of correctness, see the paper by Butenko and Trukhanov~\cite{butenko-trukhanov}.
\begin{reduction}[CWIS Reduction] Let $U\subseteq V$ be a critical weighted independent set of $G$. Then $U$ is in some MWIS of $G$. We set $G' = G[V\setminus N[U]]$ and $\alpha_w(G) = \alpha_w(G') + w(U)$. \end{reduction}
\section{Efficient Branch-and-Reduce} \label{sec:noveltechniques} We now describe our branch-and-reduce framework in full detail. This includes the pruning and branching techniques that we use, as well as other algorithm details. An overview of our algorithm can be found in Algorithm~\ref{branchreducelabel}. To keep the description simple, the pseudocode describes the algorithm such that it outputs the weight of a maximum weight independent set in the graph. However, our algorithm is implemented to actually output the maximum weight independent set. Throughout the algorithm we maintain the current solution weight as well as the best solution weight. Our algorithm applies a set of reduction rules before branching on a node. We describe these reductions in the following section. Initially, we run a local search algorithm on the reduced graph to compute a lower bound on the solution weight, which later helps pruning the search space. We then prune the search by excluding unnecessary parts of the branch-and-bound tree to be explored. If the graph is not connected, we separately solve each connected component. If the graph is connected, we branch into two cases by applying a branching rule. If our algorithm does not finish with a certain time limit, we use the currently best solution and improve it using a greedy algorithm. More precisely, our algorithm sorts the vertices in decreasing order of their weight and adds vertices in that order if feasible. We give a detailed description of the subroutines of our~algorithm~below.
\subsection{Incremental Reductions.} Our algorithm starts by running all reductions that are described in the following section. Following the lead of previous works~\cite{strash-power-2016,chang2017,hespe2018scalable}, we apply our reductions \emph{incrementally}. For each reduction rule, we check if it is applicable to any vertex of the graph. After the checks for the current reduction are completed, we continue with the next reduction if the current reduction has not changed the graph. If the graph was changed, we go back to the first reduction rule and repeat. Most of the reductions we introduce in the following section are \emph{local}: if a vertex changes, then we do not need to check the entire graph to apply the reduction again, we only need to consider the vertices whose neighborhood has changed since the reduction was last applied. The critical weighted independent set reduction defined above is the only \emph{global} reduction that we use; it always considers all vertices in~the~graph.
For each of the local reductions there is a queue of changed vertices associated. Every time a node or its neighborhood is changed it is added to the queues of all reductions. When a reduction is applied only the vertices in its associated queue have to be checked for applicability. After the checks are finished for a particular reduction its queue is cleared. Initially, the queues of all reductions are filled with every vertex~in~the~graph.
\begin{algorithm}[t] \SetAlgoLined \begin{algorithmic}
\STATE \textbf{input} graph $G=(V,E)$, current solution weight $c$ (initially zero), best solution weight $\mathcal{W}$ (initially zero)
\vspace*{-.45cm}
\STATE \textbf{procedure} Solve($G$, $c$, $\mathcal{W}$)
\STATE \quad $(G,c) \leftarrow$ Reduce$(G, c)$
\STATE \quad \textbf{if} $\mathcal{W} = 0$ \textbf{then} $\mathcal{W} \leftarrow$ $c+\mathrm{ILS}(G)$
\STATE \quad \textbf{if} $c$ + UpperBound($G$) $\leq \mathcal{W}$ \textbf{then} \textbf{return} $\mathcal{W}$
\STATE \quad \textbf{if} $G$ is empty \textbf{then} \textbf{return} $\max\{\mathcal{W}, c\}$
\STATE \quad \textbf{if} $G$ is not connected \textbf{then}
\STATE \quad \quad \textbf{for all} $G_i \in $ Components($G$) \textbf{do}
\STATE \quad \quad \quad $c \leftarrow c + \text{Solve}$($G_i$, 0, 0)
\STATE \quad \quad \textbf{return} $\max(\mathcal{W},c)$
\STATE \quad $(G_1, c_1), (G_2, c_2) \leftarrow $ Branch$(G, c)$
\STATE \quad \COMMENT{Run 1st case, update currently best solution}
\STATE \quad $\mathcal{W} \leftarrow $ Solve$(G_1, c_1, \mathcal{W})$
\STATE \quad \COMMENT{Use updated $\mathcal{W}$ to shrink the search space}
\STATE \quad $\mathcal{W}\leftarrow $ Solve$(G_2, c_2, \mathcal{W})$
\STATE \textbf{return} $\mathcal{W}$ \end{algorithmic}
\caption{Branch-and-Reduce Algorithm for MWIS} \label{branchreducelabel} \end{algorithm}
\subsection{Pruning.} Exact branch-and-bound algorithms for the MWIS problem often use weighted clique covers to compute an upper bound for the optimal solution~\cite{warren2006combinatorial}. A weighted clique cover of $G$ is a collection of (possibly overlapping) cliques $C_1, \ldots, C_k \subseteq V$, with associated weights $W_1, \ldots, W_k$ such that $C_1 \cup C_2 \cup \cdots \cup C_k = V$, and for every vertex $v \in V$, $\sum_{i\,:\,v\in C_i} W_i \geq w(v)$. The weight of a clique cover is defined as $\sum_{i=1}^k W_i$ and provides an upper bound on $\alpha_w(G)$. This holds because the intersection of a clique and any IS of $G$ is either a single vertex or empty. The objective then is to find a clique cover of small weight. This can be done using an algorithm similar to the coloring method of Brelaz~\cite{brelazcoloring}. However, this method can become computationally expensive since its running time is dependent on the maximum weight of the graph~\cite{warren2006combinatorial}. Thus, we use a faster method to compute a weighted clique cover which is similar to the one used in Akiba and Iwata~\cite{akiba-tcs-2016}.
We begin by sorting the vertices in descending order of their weight (ties are broken by selecting the vertex with higher degree). Next, we initiate an empty set of cliques $\mathcal{C}$. We then iterate over the sorted vertices and search for the clique with maximum weight which it can be added to. If there are no candidates for insertion, we insert a new single vertex clique to $\mathcal{C}$ and assign it the weight of the vertex. Afterwards the vertex is marked as processed and we continue with the next one. Computing a weighted clique cover using this algorithm has a linear running time independent of the maximum weight. Thus, we are able to obtain a bound much faster. However, this algorithm produces a higher weight clique cover than the method of Brelaz~\cite{brelazcoloring,akiba-tcs-2016}.
In addition to computing an upper bound, we also add an additional lower bound using a heuristic approach. In particular, we run a modified version of the ILS by Andrade~et~al.~\cite{andrade-2012} that is able to handle vertex weights for a fixed fraction of our total running time. This lower bound is computed once after we apply our reductions initially and then again when splitting the search space on connected components.
\subsection{Connected Components.} Solving the maximum weight independent set problem for a graph $G$ is equal to solving the problem for all $c$ connected components $G_1, \dots, G_{c}$ of $G$ and then combining the solution sets $\mathcal{I}_1, \dots, \mathcal{I}_{c}$ to form a solution $\mathcal{I}$ for $G$: $\mathcal{I} = \bigcup_{i=1}^{c} \mathcal{I}_{i}$. We leverage this property by checking the connectivity of $G$ after each completed round of reduction applications. If the graph disconnects due to branching or reductions then we apply our branch-and-reduce algorithm recursively to each of the connected components and combine their solutions afterwards. This technique can reduce the size of the branch-and-bound tree significantly on some instances.
\subsection{Branching.} Our algorithm has to pick a branching order for the remaining vertices in the graph. Initially, vertices are sorted in non-decreasing order by degree, with ties broken by weight. Throughout the algorithm, the next vertex to be chosen is the highest vertex in the ordering. This way our algorithm quickly eliminates the largest neighborhoods and makes the problem ``simpler''.
\section{Weighted Reduction Rules} \label{sec:labelofchapterforoutlineofpaper_reductions} \label{sec:new-reductions} We now develop a comprehensive set of reduction rules for the maximum weight independent set problem. We first introduce two \emph{meta} reductions, which we then use to instantiate many efficient reductions similar to already-known unweighted reductions.
\begin{figure*}
\caption{Neighbor removal (left) and neighborhood folding (right)}
\label{fig:general_folding}
\label{fig:neighbor_removal}
\end{figure*}
\subsection{Meta Reductions.} \label{sec:general-reductions} There are two operations that are commonly used in reductions: vertex removal and vertex folding. In the following reductions, we show general ways to detect when these operations can be applied in the neighborhood of a vertex.
\myparagraph{Neighbor Removal.} In our first meta reduction, we show how to determine if a neighbor can be outright removed from the graph. We call this reduction the \emph{neighbor removal} reduction. (See Figure~\ref{fig:neighbor_removal}.)
\begin{reduction}[Neighbor Removal] Let $v \in V$. For any $u\in N(v)$, if $\alpha_w(G[N(v) \setminus N[u]]) + w(u) \leq w(v)$, then $u$ can be removed from $G$, as there is some MWIS of $G$ that excludes $u$, and $\alpha_w(G) = \alpha_w(G[V\setminus\{u\}])$. \end{reduction} \begin{proof} Let $\I$ be an MWIS of $G$. We show by a cut-and-paste argument that if $u\in \I$ then there is another MWIS $\I'$ that contains $v$ instead. Let $u\in N(v)$, and suppose that $\alpha_w(G[N(v) \setminus N[u]]) + w(u) \leq w(v)$. There are two cases, if $u$ is not in $\I$ then it is safe to remove. Otherwise, suppose $u\in \I$. Then $v\in N(u)$ is not in $\I$, and $w(\I\cap N(v)) = w(\I \cap (N(v)\setminus N[u])\cup\{u\}) = \alpha_w(G[N(v) \setminus N[u]]) + w(u) = w(v)$; otherwise we can swap $\I\cap N(v)$ for $v$ in $\I$ obtaining an independent set of larger weight. Thus $\I' = (\I \setminus N(v)) \cup \{v\}$ is an MWIS of $G$ excluding $u$ and $\alpha_w(G) = \alpha_w(G[V\setminus\{u\}])$. \end{proof}
\myparagraph{Neighborhood Folding.} For our next meta reduction, we show a general condition for folding a vertex with its neighborhood. We first briefly describe the intuition behind the reduction. Consider $v$ and its neighborhood $N(v)$. If $N(v)$ has a unique independent set $\I_{N(v)}$ with weight larger than $w(v)$, then we only need to consider two independent sets: independent sets that contain $v$ or $\I_{N(v)}$. Otherwise, any other independent set in $N(v)$ can be swapped for $v$ and achieve higher overall weight. By folding $v$ with $\I_{N(v)}$, we can solve the remaining graph and then decide which of the two options will give an MWIS of the graph. (See Figure~\ref{fig:general_folding}.) \begin{reduction}[Neighborhood Folding] Let $v \in V$, and suppose that $N(v)$ is independent. If $w(N(v)) > w(v)$, but $w(N(v)) - \min_{u\in N(v)}\{w(u)\} < w(v)$, then fold $v$ and $N(v)$ into a new vertex $v'$ with weight $w(v') = w(N(v)) - w(v)$. Let $\I'$ be an MWIS of $G'$, then we construct an MWIS $\I$ of $G$ as follows: If $v'\in \I'$ then $\I = (\I'\setminus\{v'\}) \cup N(v)$, otherwise if $v\in \I'$ then $\I = \I' \cup \{v\}$. Furthermore, $\alpha_w(G) = \alpha_w(G') + w(v)$. \end{reduction} \begin{proof} Proof can be found in Appendix~\ref{ommittedproofs}. \end{proof}
However, these reductions require solving the MWIS problem on the neighborhood of a vertex, and therefore may be as expensive as computing an MWIS on the input graph. We next show how to use these meta reductions to develop efficient reductions.
\subsection{Efficient Weighted Reductions.} We now construct new efficient reductions using the just defined meta reductions. \myparagraph{Neighborhood Removal.} In their HILS local search algorithm, Nogueria~et~al.~\cite{hybrid-ils-2018} introduced the notion of a ``$(\omega,1)$-swap'', which swaps a vertex $v$ into a solution if its neighbors in the current solution $I$ have weight $w(N(v)\cap I) < w(v)$. This can be transformed into what we call the \emph{neighborhood removal reduction}.
\begin{reduction}[Neighorhood Removal] For any $v\in V$, if $w(v)\geq w(N(v))$ then $v$ is in some MWIS of $G$. Let $G' = G[V\setminus N[v]]$ and $\alpha_w(G) = \alpha_w(G') + w(v)$. \end{reduction} \begin{proof}
\iffalse We give a cut-and-paste argument. Suppose $w(v) \geq w(N(v))$, and let $\mathcal{I}$ be an MWIS of $G$. Either $v \in \mathcal{I}$, or not. If it is, then we are done. Otherwise, it must be that $w(N(v)) = w(v)$, otherwise, the independent set $\mathcal{I}'=(\mathcal{I}\setminus N(v)) \cup \{v\}$ has weight higher than $\mathcal{I}$. $\mathcal{I}'$ has the same weight as $\mathcal{I}$ and is an MWIS containing $v$. \else Since $w(N(v)) \leq w(v)$, $\forall u\in N(v)$ we have that \[\alpha_w(G[N(v)\cap N(u)]) + w(u) \leq w(N(v)) \leq w(v).\] Then we can remove all $u\in N(v)$ and are left with $v$ in its own component. Calling this graph $G'$, we have that $v$ is in some MWIS and $\alpha_w(G) = \alpha_w(G') + w(v)$. \fi{} \end{proof} For the remaining reductions, we assume that the neighborhood removal reduction has already been applied. Thus, $\forall v\in V$, $w(v) < w(N(v))$.
\myparagraph{Weighted Isolated Vertex Removal.} Similar to the (unweighted) isolated vertex removal reduction, we now argue that an isolated vertex is in some MWIS---if it has highest weight in its clique. \begin{reduction}[Isolated Vertex Removal.] Let $v\in V$ be isolated and $w(v)\geq \max_{u\in N(v)}w(u)$. Then $v$ is in some MWIS of $G$. Let $G' = G[V\setminus N[v]]$ and $\alpha_w(G) = \alpha_w(G') + w(v)$. \end{reduction} \begin{proof} \iffalse By a cut-and-paste argument, let $\mathcal{I}\subseteq V$ be an MWIS of $G$. Then either $v$ is in $\mathcal{I}$ and we are done, or not. If $v\notin\mathcal{I}$, then there must be a neighbor $u\in \mathcal{I}\cap N(v)$ such that $w(u) = w(v)$. Otherwise the independent set $\mathcal{I}' = (\mathcal{I} \setminus N(v)) \cup \{v\}$ has higher weight than $\mathcal{I}$. $\mathcal{I}'$ is an MWIS containing $v$. \else Since $N(v)$ is a clique, $\forall u\in N(v)$ we have that \[\alpha_w(G[N(v)\cap N(u)]) \leq \alpha_w(N(v)) = \max_{u\in N(v)}\{w(u)\} \leq w(v).\] Similar to neighborhood removal, remove all $u\in N(v)$ producing $G'$ and $\alpha_w(G) = \alpha_w(G') + w(v)$. \fi \end{proof}
\vspace*{-.25cm} \myparagraph{Isolated Weight Transfer.} Given its weight restriction, the weighted isolated vertex removal reduction may be ineffective. We therefore introduce a reduction that supports more liberal vertex removal. \begin{figure}
\caption{Isolated weight transfer}
\label{fig:weighted_clique}
\end{figure}
\begin{reduction}[Isolated weight transfer] Let $v\in V$ be isolated, and suppose that the set of isolated vertices $S(v)\subseteq N(v)$ is such that $\forall u\in S(v)$, $w(v) \geq w(u)$. We \begin{enumerate}[(i)] \item remove all $u\in N(v)$ such that $w(u)\leq w(v)$, and let the remaining neighbors be denoted by $N'(v)$, \item remove $v$ and $\forall x\in N'(v)$ set its new weight to $w'(x) = w(x) - w(v)$, and \end{enumerate} let the resulting graph be denoted by $G'$. Then $\alpha_w(G) = w(v) + \alpha_w(G')$ and an MWIS $\I$ of $G$ can be constructed from an MWIS $\I'$ of $G'$ as follows: if $\I' \cap N'(v) = \emptyset$ then $\I = \I'\cup\{v\}$, otherwise $\I = \I'$. \end{reduction} \begin{proof} Proof can be found in Appendix~\ref{ommittedproofs}. \end{proof}
\vspace*{-.25cm} \myparagraph{Weighted Vertex Folding.} Similar to the unweighted vertex folding reduction, we show that we can fold vertices with two non-adjacent neighbors---however, not all weight configurations permit this. \begin{figure}
\caption{Weighted vertex folding}
\label{fig:weighted_folding}
\end{figure} \begin{reduction}[Vertex Folding] Let $v\in V$ have $d(v) = 2$, such that $v$'s neighbors $u$, $x$ are not adjacent. If $w(v) < w(u) + w(x)$ but $w(v)\geq \max\{w(u),w(x)\}$, then we fold $v,u,x$ into vertex $v'$ with weight $w(v') = w(u) + w(x) - w(v)$ forming a new graph $G'$. Then $\alpha_w(G) = \alpha_w(G') + w(v)$. Let $\mathcal{I}'$ be an MWIS of $G'$. If $v'\in \mathcal{I}'$ then $\I = (\I'\setminus\{v'\})\cup\{u,x\}$ is an MWIS of $G$. Otherwise, $\I = \I' \cup \{v\}$ is an MWIS of $G$. \end{reduction} \begin{proof} \iffalse
First note that after folding, the following graphs are identical: $G[V\setminus N[\{u,x\}]] = G'[V'\setminus N_{G'}[\{v'\}]]$ and $G'[V'\setminus \{v'\}] = G[V\setminus N[v]]$. Let $\I'$ be an MWIS of $G'$. Then we have two cases.
\noindent \emph{Case 1 ($v'\in\I'$):} Suppose that $v'\in\I'$. We show that $w(u) + w(x) + \alpha_w(G[V\setminus N[\{u,x\}]]) \geq w(v) + \alpha_w(G[V\setminus N[v]])$, which shows that $u$ and $x$ are together in some MWIS of $G$.
\noindent Since $v'\in\I'$, we have that
\begin{align*} w(v) + \alpha_w(G') &= w(v) + w(v') + \alpha_w(G'[V'\setminus N_{G'}[v']])\\
&= w(v) + w(u) + w(x) - w(v) \\
&\phantom{= w(v) + w(u)\text{ }} + \alpha_w(G'[V'\setminus N_{G'}[v']])\\
&= w(u) + w(x) + \alpha_w(G[V\setminus N[\{u,x\}]]). \end{align*}
\noindent But since $\I'$ is an MWIS of $G'$, we have that \begin{align*} w(v) + \alpha_w(G') &\geq w(v) + \alpha_w(G'[V'\setminus \{v'\}]) \\
&= w(v) + \alpha_w(G[V\setminus\{u,x\}]) \\
&= w(v) + \alpha_w(G[V\setminus N[v]]). \end{align*}
\noindent Thus, $w(u) + w(x) + \alpha_w(G[V\setminus N[\{u,x\}]]) \geq w(v) + \alpha_w(G[V\setminus N[v]])$ and $u$ and $x$ are together in some MWIS of $G$. Furthermore, we have that
\begin{align*} \alpha_w(G) &= w(u) + w(x) + \alpha_w(G[V\setminus N[\{u,x\}]]) \\ &= \alpha_w(G') + w(v). \end{align*}
\noindent \emph{Case 2: ($v'\notin\I'$):} Suppose that $v'\notin\I'$. We show that $w(v) + \alpha_w(G[V\setminus N[v]]) \geq w(u) + w(x) + \alpha_w(G[V\setminus N[\{u,x\}]])$, which shows that $v$ is in some MWIS of $G$.
\noindent Since $v'\notin\I'$, we have that
\begin{align*} w(v) + \alpha_w(G') &= w(v) + \alpha_w(G'[V'\setminus \{v'\}])\\
&= w(v) + \alpha_w(G[V\setminus N[v]]) \end{align*}
\noindent But since $\I'$ is an MWIS of $G'$, we have that \begin{align*} w(v) + \alpha_w(G') &\geq w(v) + w(v') + \alpha_w(G'[V'\setminus N_{G'}[v']]) \\
&= w(v) + w(u) + w(x) - w(v) \\
&\phantom{= w(v) + w(u)\text{ } }+ \alpha_w(G[V\setminus N[\{u,x\}]]) \\
&= w(u) + w(x) + \alpha_w(G[V\setminus N[\{u,x\}]]). \end{align*}
\noindent Thus, $w(v) + \alpha_w(G[V\setminus N[v]]) \geq w(u) + w(x) + \alpha_w(G[V\setminus N[\{u,x\}]])$ and $v$ is in some MWIS of $G$. Lastly,
\begin{align*} \alpha_w(G) &= w(v) + \alpha_w(G[V\setminus N[v]]) \\ &= \alpha_w(G') + w(v). \end{align*}
\else Apply neighborhood folding to $v$. \fi{} \end{proof}
\vspace*{-.25cm} \myparagraph{Weighted Twin.} \begin{figure}
\caption{Illustrating proof of weighted twin reduction}
\label{fig:weighted_twin}
\end{figure} The twin reduction, as described by Akiba and Iwata~\cite{akiba-tcs-2016} for the unweighted case, works for twins with $3$ common neighbors. We describe our variant in the same terms, but note that the reduction supports an arbitrary number of common neighbors.
\begin{reduction}[Twin] Let vertices $u$ and $v$ \ifFull be twins with \else have \fi{} independent neighborhoods $N(u) = N(v) = \{p,q,r\}$. We have two cases: \begin{enumerate}[(i)] \item If $w(\{u,v\}) \geq w(\{p,q,r\})$, then $u$ and $v$ are in some MWIS of $G$. Let $G' = G[V\setminus N[\{u,v\}]]$. \item If $w(\{u,v\}) < w(\{p,q,r\})$, but $w(\{u,v\}) > w(\{p,q,r\}) - \min_{x\in\{p,q,r\}} w(x)$, then we can fold $u, v, p, q, r$ into a new vertex $v'$ with weight $w(v') = w(\{p,q,r\}) - w(\{u,v\})$ and call this graph $G'$. Let $I'$ be an MWIS of $G'$. Then we construct an MWIS $\I$ of $G$ as follows: if $v'\in \I'$ then $\I = (\I'\setminus \{v'\})\cup \{p,q,r\}$, if $v'\notin \I'$ then $\I = \I' \cup \{u,v\}$. \end{enumerate} Furthermore, $\alpha_w(G) = \alpha_w(G') + w(\{u,v\})$. \end{reduction} \begin{proof} Just as in the unweighted case, either $u$ and $v$ are simultaneously in an MWIS or some subset of $p$, $q$, $r$ is in. First fold $u$ and $v$ into a new vertex $\{u,v\}$ with weight $w(\{u,v\})$. To show (i), apply the neighborhood reduction to vertex $\{u,v\}$. For (ii), since $N(\{u,v\})$ is independent, we apply the neighborhood folding reduction to $\{u,v\}$, giving the claimed result. \end{proof}
If $p,q,r$ are not independent, further reductions are possible; however, introducing a comprehensive list is not illuminating. Instead, we can simply let meta reductions reduce as appropriate.
\myparagraph{Weighted Domination.} Lastly, we give a weighted variant of the domination reduction. \begin{figure}
\caption{Weighted domination}
\label{fig:weighted_domination}
\end{figure}
\begin{reduction}[Domination] Let $u,v\in V$ be vertices such that $N[u]\supseteq N[v]$ (i.e., $u$ dominates $v$). If $w(u)\leq w(v)$, there is an MWIS in $G$ that excludes $u$ and $\alpha_w(G) = \alpha_w(G[V\setminus \{u\}])$. Therefore, $u$ can be removed from the graph. \end{reduction} \begin{proof} We show by a cut-and-paste argument that there is an MWIS of $G$ excluding $u$. Let $\I$ be an MWIS of $G$. If $u$ is not in $\I$ then we are done. Otherwise, suppose $u\in \I$. Then it must be the case that $w(v) = w(u)$, otherwise $\I' = (\I\setminus\{u\}) \cup \{v\}$ is an independent set with weight larger than $\I$. Thus, $\I'$ is an MWIS of $G$ excluding $u$, and $\alpha_w(G) = \alpha_w(G[V\setminus \{u\}])$. \end{proof}
\section{Experimental Evaluation} \label{sec:experiments} We now compare the performance of our branch-and-reduce algorithm to existing state-of-the-art algorithms on large real-world graphs. Furthermore, we examine how our reduction rules can drastically improve the quality of existing heuristic approaches.
\vspace*{-.25cm} \subsection{Methodology and Setup.} All of our experiments were run on a machine with four Octa-Core Intel Xeon E5-4640 processors running at $2.4$ GHz, $512$ GB of main memory, $420$ MB L3-Cache and $48256$ KB L2-Cache. The machine runs Ubuntu 14.04.3 and Linux kernel version 3.13.0-77. All algorithms were implemented in \Cminus11 and compiled with \GG~version 4.8.4 with optimization flag \texttt{-O3}. Each algorithm was run sequentially for a total of $1000$ seconds\footnote{Results with more than 1000 seconds are due to initial kernelization taking longer than the time limit.}.
We present two kinds of data: (1) the best solution found by each algorithm and the time (in seconds) required to obtain it, (2) \emph{convergence plots}, which show how the solution quality changes over time. In particular, whenever an algorithm finds a new best independent set $S$ at time $t$, it reports a tuple ($t$, $|S|$)\footnote{For the convergence plots of the heuristic algorithms we use the maximum values over five runs with varying random seeds}. \begin{table*}[ht] \scriptsize \centering \setlength{\tabcolsep}{0.5ex}
\begin{tabular}{l|r|r r|r r|r r}
Graph & $|V|$ & $t_\text{max}$ & $w_\text{max}$ & $t_\text{max}$ & $w_\text{max}$ & $t_\text{max}$ & $w_\text{max}$ \\ \hline
\rule{0pt}{3ex}OSM networks & & \multicolumn{2}{c|}{DynWVC1} & \multicolumn{2}{c|}{HILS} & \multicolumn{2}{c}{B~\&~R\textsubscript{\text{dense}}} \\ \hline \Id{\detokenize{alabama-AM3}} & \numprint{3504} & \numprint{464.02} & \numprint{185527} & \numprint{0.73} & \textbf{\numprint{185744}} & \numprint{15.79} & \numprint{185707} \\ \rowcolor{lightergray} \Id{\detokenize{florida-AM2}} & \numprint{1254} & \numprint{1.14} & \textbf{\numprint{230595}} & \numprint{0.04} & \textbf{\numprint{230595}} & \numprint{0.03} & \textbf{\numprint{230595}} \\ \Id{\detokenize{georgia-AM3}} & \numprint{1680} & \numprint{0.88} & \textbf{\numprint{222652}} & \numprint{0.05} & \textbf{\numprint{222652}} & \numprint{4.88} & \numprint{214918} \\ \Id{\detokenize{kansas-AM3}} & \numprint{2732} & \numprint{46.87} & \textbf{\numprint{87976}} & \numprint{0.84} & \textbf{\numprint{87976}} & \numprint{11.35} & \numprint{87925} \\ \rowcolor{lightergray} \Id{\detokenize{maryland-AM3}} & \numprint{1018} & \numprint{1.34} & \textbf{\numprint{45496}} & \numprint{0.02} & \textbf{\numprint{45496}} & \numprint{3.34} & \textbf{\numprint{45496}} \\ \Id{\detokenize{massachusetts-AM3}} & \numprint{3703} & \numprint{435.31} & \numprint{145863} & \numprint{2.73} & \textbf{\numprint{145866}} & \numprint{12.87} & \numprint{145617} \\ \rowcolor{lightergray} \Id{\detokenize{utah-AM3}} & \numprint{1339} & \numprint{136.15} & \numprint{98802} & \numprint{0.08} & \textbf{\numprint{98847}} & \numprint{64.04} & \textbf{\numprint{98847}} \\ \Id{\detokenize{vermont-AM3}} & \numprint{3436} & \numprint{119.63} & \numprint{63234} & \numprint{0.95} & \textbf{\numprint{63302}} & \numprint{95.81} & \numprint{55584} \\ \hline
\rule{0pt}{3ex}Solved instances & & \multicolumn{2}{r|}{} & \multicolumn{2}{r|}{} & \multicolumn{2}{r}{44.12\% (15/34) } \\
Optimal weight & & \multicolumn{2}{r|}{60.00\% (9/15) } & \multicolumn{2}{r|}{100.00\% (15/15) } & \multicolumn{2}{r}{} \\
\rule{0pt}{4ex}SNAP networks & & \multicolumn{2}{c|}{DynWVC2} & \multicolumn{2}{c|}{HILS} & \multicolumn{2}{c}{B~\&~R\textsubscript{\text{full}}} \\ \hline \Id{\detokenize{as-skitter}} & \numprint{1696415} & \numprint{576.93} & \numprint{123105765} & \numprint{998.75} & \numprint{122539706} & \numprint{746.93} & \textbf{\numprint{123904741}} \\ \rowcolor{lightergray} \Id{\detokenize{ca-AstroPh}} & \numprint{18772} & \numprint{108.35} & \numprint{796535} & \numprint{46.76} & \textbf{\numprint{796556}} & \numprint{0.03} & \textbf{\numprint{796556}} \\ \rowcolor{lightergray} \Id{\detokenize{email-EuAll}} & \numprint{265214} & \numprint{179.26} & \textbf{\numprint{25330331}} & \numprint{501.09} & \textbf{\numprint{25330331}} & \numprint{0.19} & \textbf{\numprint{25330331}} \\ \rowcolor{lightergray} \Id{\detokenize{p2p-Gnutella08}} & \numprint{6301} & \numprint{0.19} & \textbf{\numprint{435893}} & \numprint{0.25} & \textbf{\numprint{435893}} & \numprint{0.01} & \textbf{\numprint{435893}} \\ \rowcolor{lightergray} \Id{\detokenize{roadNet-TX}} & \numprint{1379917} & \numprint{1000.78} & \numprint{77525099} & \numprint{1697.13} & \numprint{76366577} & \numprint{33.49} & \textbf{\numprint{78606965}} \\ \Id{\detokenize{soc-LiveJournal1}} & \numprint{4847571} & \numprint{1001.23} & \numprint{277824322} & \numprint{12437.50} & \numprint{280559036} & \numprint{270.96} & \textbf{\numprint{283948671}} \\ \rowcolor{lightergray} \Id{\detokenize{web-Google}} & \numprint{875713} & \numprint{683.63} & \numprint{56190870} & \numprint{994.58} & \numprint{55954155} & \numprint{3.16} & \textbf{\numprint{56313384}} \\ \rowcolor{lightergray} \Id{\detokenize{wiki-Talk}} & \numprint{2394385} & \numprint{991.31} & \numprint{235874419} & \numprint{996.02} & \numprint{235852509} & \numprint{3.36} & \textbf{\numprint{235875181}} \\ \hline
\rule{0pt}{3ex}Solved instances & & \multicolumn{2}{r|}{} & \multicolumn{2}{r|}{} & \multicolumn{2}{r}{80.65\% (25/31) } \\
Optimal weight & & \multicolumn{2}{r|}{28.00\% (7/25) } & \multicolumn{2}{r|}{68.00\% (17/25) } & \multicolumn{2}{r}{} \\ \end{tabular}
\caption{Best solution found by each algorithm and time (in seconds) required to compute it. The global best solution is highlighted in \textbf{bold}. Rows are highlighted in gray if B~\&~R is able to find an exact solution. } \label{tab:local_rt} \vspace*{-.5cm} \end{table*}
\myparagraph{Algorithms Compared.} We use two different variants of our branch-and-reduce algorithm. The first variant, called B~\&~R\textsubscript{\text{full}}, uses our \emph{full set} of reductions each time we branch. The second variant, called B~\&~R\textsubscript{\text{dense}}, omits the more costly reductions and also terminates the execution of the remaining reductions faster than B~\&~R\textsubscript{\text{full}}. In particular, this configuration completely omits the weighted critical set reductions from both the initialization and recursion. Additionally, we also omit the weighted clique reduction from the first reduction call and use a faster version that only considers triangles during recursion. Finally, we do not use the generalized neighborhood folding during recursion. This configuration find solutions more quickly on dense graphs.
We also include the state-of-the-art heuristics \emph{HILS} by Nogueria~et~al.~\cite{hybrid-ils-2018} and both versions of \emph{DynWVC} by Cai~et~al.~\cite{cai-dynwvc} (see Section~\ref{sec:related_work} for a short explanation of these algorithms). Finally, we do not include any other exact algorithms (e.g.\ ~\cite{butenko-trukhanov,warren2006combinatorial}) as their code is not available. Also note that these exact algorithms are either not tested in the weighted case~\cite{butenko-trukhanov} or the largest instances reported consist of a few hundred vertices~\cite{warren2006combinatorial}.
To further evaluate the impact of reductions on existing algorithms, we also propose combinations of the heuristic approaches with reductions (\emph{Red~+~HILS} and \emph{Red~+~DynWVC}). We do so by first computing a kernel graph using our set of reductions and then run the existing algorithms on the resulting graph.
\myparagraph{Instances.} We test all algorithms on a large corpus of sparse data sets. For this purpose, we include a set of real-world conflict graphs obtained from OpenStreetMap \cite{OSMWEB} files of North America, according to the method described by Barth~et~al.~\cite{barth-2016}. More specifically, these graphs are generated by identifying map labels with vertices that have a weight corresponding to their importance. Edges are then inserted between vertices if their labels overlap each other. Conflict graphs can also be used in a dynamic setting by associating vertices with intervals that correspond to the time they are displayed. Furthermore, solving the MWIS problem on these graphs eliminates label conflicts and maximizes the importance of displayed labels. Finally, different activity models (AM1, AM2 and AM3) are used to generate different conflict graphs. The instances we use for our experiments are the same ones used by Cai~et~al.~\cite{cai-dynwvc}. We omit all instances with less than $1000$ vertices from our experiments, as these are easy to solve and our focus is on large scale networks~\cite{cai-dynwvc}.
In addition to the OSM networks, we also include collaboration networks, communication networks, additional road networks, social networks, peer-to-peer networks, and Web crawl graphs from the Stanford Large Network Dataset Repository~\cite{snapnets} (SNAP).
These networks are popular benchmark instances commonly used for the maximum independent set problem~\cite{akiba-tcs-2016,dahlum2016,redumis-2017}. However, all SNAP instances are unweighted and comparable weighted instances are very scarce. Therefore, a common approach in literature is to assign vertex weights uniformly at random from a fixed size interval~\cite{cai-dynwvc,li2017efficient}. To keep our results in line with existing work, we thus decided to select vertex weights uniformly at random from $[1,200]$. Basic properties of our benchmark instances can be found in Table~\ref{tab:props}.
\subsection{Comparison with State-of-the-Art.} \label{sec:sota} A representative sample of our experimental results for the OSM and SNAP networks is presented in Table~\ref{tab:local_rt}. For a full overview of all instances, we refer to Table~\ref{tab:osm_local_rt} (OSM) and Table~\ref{tab:snap_local_rt} (SNAP) respectively. For each instance, we list the best solution computed by each algorithm $w_\text{Algo}$ and the time in seconds required to find it $t_\text{Algo}$. For each data set, we highlight the best solution found across all algorithms in \textbf{bold}. Additionally, if any version of our algorithm is able to find an exact solution, the corresponding row is highlighted in gray. Finally, recall that our algorithm computes a solution on unsolved instances once the time-limit is reached by additionally running a greedy algorithm as post-processing.
Examining the OSM graphs, B~\&~R is able to solve 15 out of the 34 instances we tested. However, HILS is also able to compute a solution with the same weight on all of these instance. Furthermore, HILS obtains a higher or similar quality solution than both versions of DynWVC and B~\&~R for all remaining unsolved instances. Overall, HILS is able to find the best solution on all OSM instances that we tested. Additionally, on most of these instances it does so significantly faster than all of its competitors. Note though, that neither HILS nor DynWVC provide any optimality guarantees (in contrast to B~\&~R).
Looking at both versions of DynWVC, we see that DynWVC1 performs better than DynWVC2, which is also reported by Cai~et~al.~\cite{cai-dynwvc}. Comparing both variants of our branch-and-reduce algorithm, we see that they are able to solve the same instances. Nonetheless, B~\&~R\textsubscript{\text{dense}}~is able to compute better solutions on roughly half of the remaining instances. Additionally, it almost always requires significantly less time to achieve its maximum compared to B~\&~R\textsubscript{\text{full}}.
\begin{figure*}
\caption{Solution quality over time for two OSM instances (left) and two SNAP instances (right).}
\label{fig:convergence}
\end{figure*}
\begin{table*}[ht] \scriptsize \centering \setlength{\tabcolsep}{0.5ex}
\begin{tabular}{l|r r|r r r|r r|r r r|r r} Graph & $t_\text{max}$ & $w_\text{max}$ & $t_\text{max}$ & $w_\text{max}$ & $S_\text{base}$ & $t_\text{max}$ & $w_\text{max}$ & $t_\text{max}$ & $w_\text{max}$ & $S_\text{base}$ & $t_\text{max}$ & $w_\text{max}$ \\ \hline
\rule{0pt}{4ex}OSM instances & \multicolumn{2}{c|}{DynWVC1} & \multicolumn{3}{c|}{Red+DynWVC1} & \multicolumn{2}{c|}{HILS} & \multicolumn{3}{c|}{Red + HILS} & \multicolumn{2}{c}{B~\&~R\textsubscript{\text{dense}}} \\ \hline \Id{\detokenize{alabama-AM3}} & \numprint{464.02} & \numprint{185527} & \numprint{370.80} & \numprint{185727} & \textcolor{darkgreen}{\numprint{1.25}} & \numprint{0.73} & \textbf{\numprint{185744}} & \numprint{4.05} & \textbf{\numprint{185744}} & \textcolor{darkred}{\numprint{0.18}} & \numprint{15.79} & \numprint{185707} \\ \rowcolor{lightergray} \Id{\detokenize{florida-AM2}} & \numprint{1.14} & \textbf{\numprint{230595}} & \numprint{0.03} & \textbf{\numprint{230595}} & \textcolor{darkgreen}{\numprint{44.19}} & \numprint{0.04} & \textbf{\numprint{230595}} & \numprint{0.03} & \textbf{\numprint{230595}} & \textcolor{darkgreen}{\numprint{1.75}} & \numprint{0.03} & \textbf{\numprint{230595}} \\ \Id{\detokenize{georgia-AM3}} & \numprint{0.88} & \textbf{\numprint{222652}} & \numprint{2.64} & \textbf{\numprint{222652}} & \textcolor{darkred}{\numprint{0.33}} & \numprint{0.05} & \textbf{\numprint{222652}} & \numprint{2.43} & \textbf{\numprint{222652}} & \textcolor{darkred}{\numprint{0.02}} & \numprint{4.88} & \numprint{214918} \\ \Id{\detokenize{kansas-AM3}} & \numprint{46.87} & \textbf{\numprint{87976}} & \numprint{13.59} & \textbf{\numprint{87976}} & \textcolor{darkgreen}{\numprint{3.45}} & \numprint{0.84} & \textbf{\numprint{87976}} & \numprint{2.06} & \textbf{\numprint{87976}} & \textcolor{darkred}{\numprint{0.41}} & \numprint{11.35} & \numprint{87925} \\ \rowcolor{lightergray} \Id{\detokenize{maryland-AM3}} & \numprint{1.34} & \textbf{\numprint{45496}} & \numprint{2.07} & \textbf{\numprint{45496}} & \textcolor{darkred}{\numprint{0.65}} & \numprint{0.02} & \textbf{\numprint{45496}} & \numprint{2.07} & \textbf{\numprint{45496}} & \textcolor{darkred}{\numprint{0.01}} & \numprint{3.34} & \textbf{\numprint{45496}} \\ \Id{\detokenize{massachusetts-AM3}} & \numprint{435.31} & \numprint{145863} & \numprint{10.68} & \textbf{\numprint{145866}} & \textcolor{darkgreen}{\numprint{40.75}} & \numprint{2.73} & \textbf{\numprint{145866}} & \numprint{2.92} & \textbf{\numprint{145866}} & \textcolor{darkred}{\numprint{0.93}} & \numprint{12.87} & \numprint{145617} \\ \rowcolor{lightergray} \Id{\detokenize{utah-AM3}} & \numprint{136.15} & \numprint{98802} & \numprint{168.07} & \textbf{\numprint{98847}} & \textcolor{darkred}{\numprint{0.81}} & \numprint{0.08} & \textbf{\numprint{98847}} & \numprint{2.10} & \textbf{\numprint{98847}} & \textcolor{darkred}{\numprint{0.04}} & \numprint{64.04} & \textbf{\numprint{98847}} \\ \Id{\detokenize{vermont-AM3}} & \numprint{119.63} & \numprint{63234} & \numprint{62.85} & \numprint{63280} & \textcolor{darkgreen}{\numprint{1.90}} & \numprint{0.95} & \numprint{63302} & \numprint{2.95} & \textbf{\numprint{63312}} & \textcolor{darkred}{\numprint{0.32}} & \numprint{95.81} & \numprint{55584} \\ \hline
\rule{0pt}{3ex}Solved instances & \multicolumn{2}{r|}{} & \multicolumn{3}{r|}{} & \multicolumn{2}{r|}{} & \multicolumn{3}{r|}{} & \multicolumn{2}{r}{44.12\% (15/34) } \\
Optimal weight & \multicolumn{2}{r|}{60.00\% (9/15) } & \multicolumn{3}{r|}{93.33\% (14/15) } & \multicolumn{2}{r|}{100.00\% (15/15) } & \multicolumn{3}{r|}{100.00\% (15/15) } & \multicolumn{2}{r}{} \\
\rule{0pt}{4ex}SNAP instances & \multicolumn{2}{c|}{DynWVC2} & \multicolumn{3}{c|}{Red+DynWVC2} & \multicolumn{2}{c|}{HILS} & \multicolumn{3}{c|}{Red + HILS} & \multicolumn{2}{c}{B~\&~R\textsubscript{\text{dense}}} \\ \hline \Id{\detokenize{as-skitter}} & \numprint{576.93} & \numprint{123105765} & \numprint{85.60} & \numprint{123995808} & \textcolor{darkgreen}{\numprint{6.74}} & \numprint{998.75} & \numprint{122539706} & \numprint{845.70} & \textbf{\numprint{123996322}} & \textcolor{darkgreen}{\numprint{1.18}} & \numprint{746.93} & \numprint{123904741} \\ \rowcolor{lightergray} \Id{\detokenize{ca-AstroPh}} & \numprint{108.35} & \numprint{796535} & \numprint{0.02} & \textbf{\numprint{796556}} & \textcolor{darkgreen}{\numprint{4962.17}} & \numprint{46.76} & \textbf{\numprint{796556}} & \numprint{0.02} & \textbf{\numprint{796556}} & \textcolor{darkgreen}{\numprint{2142.48}} & \numprint{0.03} & \textbf{\numprint{796556}} \\ \rowcolor{lightergray} \Id{\detokenize{email-EuAll}} & \numprint{179.26} & \textbf{\numprint{25330331}} & \numprint{0.12} & \textbf{\numprint{25330331}} & \textcolor{darkgreen}{\numprint{1548.08}} & \numprint{501.09} & \textbf{\numprint{25330331}} & \numprint{0.12} & \textbf{\numprint{25330331}} & \textcolor{darkgreen}{\numprint{4327.82}} & \numprint{0.19} & \textbf{\numprint{25330331}} \\ \rowcolor{lightergray} \Id{\detokenize{p2p-Gnutella08}} & \numprint{0.19} & \textbf{\numprint{435893}} & \numprint{0.00} & \textbf{\numprint{435893}} & \textcolor{darkgreen}{\numprint{46.98}} & \numprint{0.25} & \textbf{\numprint{435893}} & \numprint{0.00} & \textbf{\numprint{435893}} & \textcolor{darkgreen}{\numprint{63.80}} & \numprint{0.01} & \textbf{\numprint{435893}} \\ \rowcolor{lightergray} \Id{\detokenize{roadNet-TX}} & \numprint{1000.78} & \numprint{77525099} & \numprint{771.05} & \numprint{78601813} & \textcolor{darkgreen}{\numprint{1.30}} & \numprint{1697.13} & \numprint{76366577} & \numprint{946.32} & \numprint{78602984} & \textcolor{darkgreen}{\numprint{1.79}} & \numprint{33.49} & \textbf{\numprint{78606965}} \\ \Id{\detokenize{soc-LiveJournal1}} & \numprint{1001.23} & \numprint{277824322} & \numprint{996.68} & \numprint{283973997} & \textcolor{darkgreen}{\numprint{1.00}} & \numprint{12437.50} & \numprint{280559036} & \numprint{761.51} & \textbf{\numprint{283975036}} & \textcolor{darkgreen}{\numprint{16.33}} & \numprint{270.96} & \numprint{283948671} \\ \rowcolor{lightergray} \Id{\detokenize{web-Google}} & \numprint{683.63} & \numprint{56190870} & \numprint{3.30} & \numprint{56313349} & \textcolor{darkgreen}{\numprint{207.26}} & \numprint{994.58} & \numprint{55954155} & \numprint{3.01} & \textbf{\numprint{56313384}} & \textcolor{darkgreen}{\numprint{330.28}} & \numprint{3.16} & \textbf{\numprint{56313384}} \\ \rowcolor{lightergray} \Id{\detokenize{wiki-Talk}} & \numprint{991.31} & \numprint{235874419} & \numprint{2.30} & \textbf{\numprint{235875181}} & \textcolor{darkgreen}{\numprint{430.22}} & \numprint{996.02} & \numprint{235852509} & \numprint{2.30} & \textbf{\numprint{235875181}} & \textcolor{darkgreen}{\numprint{432.26}} & \numprint{3.36} & \textbf{\numprint{235875181}} \\ \hline
\rule{0pt}{3ex}Solved instances & \multicolumn{2}{r|}{} & \multicolumn{3}{r|}{} & \multicolumn{2}{r|}{} & \multicolumn{3}{r|}{} & \multicolumn{2}{r}{80.65\% (25/31) } \\
Optimal weight & \multicolumn{2}{r|}{28.00\% (7/25) } & \multicolumn{3}{r|}{84.00\% (21/25) } & \multicolumn{2}{r|}{68.00\% (17/25) } & \multicolumn{3}{r|}{88.00\% (22/25) } & \multicolumn{2}{r}{} \\ \end{tabular}
\caption{Best solution found by each algorithm and time (in seconds) required to compute it. $S_\text{base} = \frac{t_\text{base}}{t_\text{modified}}$ denotes the speedup between the modified and base versions of each local search. The global best solution is highlighted in \textbf{bold}. Rows are highlighted in gray if B~\&~R is able to find an exact solution. } \label{tab:reduce_rt} \vspace*{-.25cm} \end{table*}
For the SNAP networks, we see that B~\&~R solves $25$ of the $31$ instances we tested~\footnote{Using a longer time limit of $48$ hours we are able to solve 27 our of 31 instances.}. Most notable, on seven of these instances where either HILS or DynWVC1 also find a solution with optimal weight, it does so up to two orders of magnitude faster. This difference in performance compared to the OSM networks can be explained by the significantly lower graph density and less uniform degree distribution of the SNAP networks.
These structural differences seem to allow for our reduction rules to be applicable more often, resulting in a significantly smaller kernel (as seen in Table~\ref{tab:props}). This is similar to the behavior of unweighted branch-and-reduce~\cite{akiba-tcs-2016}. Therefore, except for a single instance, our algorithm is able to find the best solution on \emph{all} graphs tested.
Comparing the heuristic approaches, both versions of DynWVC perform better than HILS on most instances, with DynWVC2 often finding better solution than DynWVC1. Nonetheless, HILS finds higher weight solutions than DynWVC1 and DynWVC2.
\subsection{The Power of Weighted Reductions.} \label{sec:sota_reductions} We now examine the effect of using reductions to improve existing heuristic algorithms. For this purpose, we compare the combined approaches Red + HILS and Red + DynWVC with their base versions as well as our branch-and-reduce algorithm. Our sample of results for the OSM and SNAP networks is given in Table~\ref{tab:reduce_rt}. In addition to the data used in our state-of-the-art comparison, we now also report speedups between the modified and base versions of each local search. Additionally, we give the percentage of instances solved by B~\&~R, as well as the percentage of solutions with optimal weight found be the inexact algorithms compared to B~\&~R. For a full overview of all instances, we refer to Table~\ref{tab:osm_reduce_rt} and Table~\ref{tab:snap_reduce_rt} respectively.
When looking at the speedups for the SNAP graphs, we can see that using reductions allows local search to find optimal solutions orders of magnitude faster. Additionally, they are now able to find an optimal solution more often than without reductions. DynWVC2 in particular achieves an increase of $56\%$ of optimal solutions when using reductions. Overall, we achieve a speedup of up to three orders of magnitude for the SNAP instances. Thus, the additional costs for computing the kernel can be neglected for these instances. However, for the OSM instances our reduction rules are less applicable and reducing the kernel comes at a significant cost compared to the unmodified local searches.
To further examine the influence of using reductions, Figure~\ref{fig:convergence} shows the solution quality over time for all algorithms and four instances. For additional convergence plots, we refer to Figure~\ref{fig:apdxconvergence}. For the OSM instances, we can see that initially DynWVC and HILS are able find good quality solutions much faster compared to their combined approaches. However, once the kernel has been computed, regular DynWVC and HILS are quickly outperformed by the hybrid algorithms.
A more drastic change can be seen for the SNAP instances. Instances were both DynWVC and HILS examine poor performance, Red~+~DynWVC and Red~+~HILS now rival our branch-and-reduce algorithm and give near-optimal solutions in less time. Thus, using reductions for instances that are too large for traditional heuristic approaches allows for a~drastic~improvement.
\section{Conclusion and Future Work} \label{sec:conclusion} In this paper, we engineered a new branch-and-reduce algorithm as well as a combination of kernelization with local search for the maximum weight independent set problem. The core of our algorithms are a full suite of new reductions for the maximum weight independent set problem. We performed extensive experiments to show the effectiveness of our algorithms in practice on real-world graphs of up to millions of vertices and edges. Our experimental evaluation shows that our branch-and-reduce algorithm can solve many large real-world instances quickly in practice, and that kernelization has a highly positive effect on local search~algorithms.
As HILS often finds optimal solutions in practice, important future works include using this algorithm for the lower bound computation within our branch-and-reduce algorithm. Furthermore, we would like to extend our discussion on the effectiveness of novel reduction rules. In particular, we want to evaluate how much quality we gain from applying each individual rule and how the order we apply them in changes the resulting kernel size.
\renewcommand\bibsection{\section*{\refname}}
\begin{appendix} \section{Omitted Proofs} \setcounter{reduction}{2} \begin{reduction}[Neighborhood Folding] Let $v \in V$, and suppose that $N(v)$ is independent. If $w(N(v)) > w(v)$, but $w(N(v)) - \min_{u\in N(v)}\{w(u)\} < w(v)$, then fold $v$ and $N(v)$ into a new vertex $v'$ with weight $w(v') = w(N(v)) - w(v)$. Let $\I'$ be an MWIS of $G'$, then we construct an MWIS $\I$ of $G$ as follows: If $v'\in \I'$ then $\I = (\I'\setminus\{v'\}) \cup N(v)$, otherwise if $v\in \I'$ then $\I = \I' \cup \{v\}$. Furthermore, $\alpha_w(G) = \alpha_w(G') + w(v)$. \end{reduction} \begin{proof} \label{ommittedproofs} First note that after folding, the following graphs are identical: $G'[V'\setminus N_{G'}[v']] = G[V\setminus N[N[v]]$ and $G'[V'\setminus \{v'\}] = G[V\setminus N[v]]$. Let $\I'$ be an MWIS of $G'$. We have two cases.
\noindent \emph{Case 1 ($v'\in\I'$):} Suppose that $v'\in\I'$. We show that $w(N(v)) + \alpha_w(G[V\setminus N[N[v]]]) \geq w(v) + \alpha_w(G[V\setminus N[v]])$, which shows that the vertices of $N(v)$ are together in some MWIS of $G$.
Since $v'\in\I'$, we have that
\begin{align*} w(v) + \alpha_w(G') &= w(v) + w(v') + \alpha_w(G'[V'\setminus N_{G'}[v']])\\
&= w(v) + w(N(v)) - w(v) \\
&\phantom{= w(v) + w(N(v))\text{ }} + \alpha_w(G'[V'\setminus N_{G'}[v']])\\
&= w(N(v)) + \alpha_w(G[V\setminus N[N[v]]]). \end{align*}
\noindent But since $\I'$ is an MWIS of $G'$, we have that \begin{align*} w(v) + \alpha_w(G') &\geq w(v) + \alpha_w(G'[V'\setminus \{v'\}]) \\
&= w(v) + \alpha_w(G[V\setminus N[v]]). \end{align*}
\noindent Thus, $w(N(v)) + \alpha_w(G[V\setminus N[N[v]]]) \geq w(v) + \alpha_w(G[V\setminus N[v]])$ and the vertices of $N(v)$ are together in some MWIS of $G$. Furthermore, we have that
\begin{align*} \alpha_w(G) &= w(N(v)) + \alpha_w(G[V\setminus N[N[v]]]) \\ &= \alpha_w(G') + w(v). \end{align*}
\noindent \emph{Case 2: ($v'\notin\I'$):} Suppose that $v'\notin\I'$. We show that $w(v) + \alpha_w(G[V\setminus N[v]]) \geq w(N(v)) + \alpha_w(G[V\setminus N[N[v]]])$, which shows that $v$ is in some MWIS of $G$. Since $v'\notin\I'$, we have that
\begin{align*} w(v) + \alpha_w(G') &= w(v) + \alpha_w(G'[V'\setminus \{v'\}])\\
&= w(v) + \alpha_w(G[V\setminus N[v]]) \end{align*}
\noindent But since $\I'$ is an MWIS of $G'$, we have that \begin{align*} w(v) + \alpha_w(G') &\geq w(v) + w(v') + \alpha_w(G'[V'\setminus N_{G'}[v']]) \\
&= w(v) + w(N(v)) - w(v) \\
&\phantom{= w(v) + w(N(v))\text{ } }+ \alpha_w(G[V\setminus N[N[v]]]) \\
&= w(N(v)) + \alpha_w(G[V\setminus N[N[v]]]). \end{align*}
\noindent Thus, $w(v) + \alpha_w(G[V\setminus N[v]]) \geq w(u) + w(x) + \alpha_w(G[V\setminus N[\{u,x\}]])$ and $v$ is in some MWIS of $G$. Lastly,
\begin{align*} \alpha_w(G) &= w(v) + \alpha_w(G[V\setminus N[v]]) \\ &= \alpha_w(G') + w(v). \end{align*}
\end{proof}
\setcounter{reduction}{5} \begin{reduction}[Isolated weight transfer] Let $v\in V$ be isolated, and suppose that the set of isolated vertices $S(v)\subseteq N(v)$ is such that $\forall u\in S(v)$, $w(v) \geq w(u)$. We \begin{enumerate}[(i)] \item remove all $u\in N(v)$ such that $w(u)\leq w(v)$, and let the remaining neighbors be denoted by $N'(v)$, \item remove $v$ and $\forall x\in N'(v)$ set its new weight to $w'(x) = w(x) - w(v)$, and \end{enumerate} let the resulting graph be denoted by $G'$. Then $\alpha_w(G) = w(v) + \alpha_w(G')$ and an MWIS $\I$ of $G$ can be constructed from an MWIS $\I'$ of $G'$ as follows: if $\I' \cap N'(v) = \emptyset$ then $\I = \I'\cup\{v\}$, otherwise $\I = \I'$. \end{reduction} \begin{proof} For (i), note that it is safe to remove all $u\in N(v)$ such that $w(u)\leq w(v)$ since these vertices meet the criteria for the neighbor removal reduction. All vertices $x\in N'(v)$ that remain have weight $w(x) > w(v)$ and are not isolated.
\emph{Case 1 ($\I'\cap N'(v) = \emptyset$):} Let $\I'$ be an MWIS of $G'$, we show that if $\I'\cap N'(v) = \emptyset$ then $I = I'\cup\{v\}$. To show this, we show that $w(v) + \alpha_w(G[V\setminus N[v]]) \geq \alpha_w(G[V\setminus\{v\}])$.
Let $x\in N'(v)$. Since $x\notin I'$, we have that \begin{align*} w(v) + \alpha_w(G') &= w(v) + \alpha_w(G'[V'\setminus N'(v)]) \\
&= w(v) + \alpha_w(G[V\setminus N[v]]) \end{align*} and \begin{align*} w(v) + \alpha_w(G') &\geq w(v) + w'(x) + \alpha_w(G'[V'\setminus N[x]]) \\
&= w(v) + w(x) - w(v) + \alpha_w(G'[V'\setminus N[x]])\\
&= w(x) + \alpha_w(G[V\setminus N[x]]). \end{align*}
Thus, for any $x\in N'(v)$, we have that \begin{align*} w(v) + \alpha_w(G') &= w(v) + \alpha_w(G[V\setminus N[v]]) \\
&\geq w(x) + \alpha_w(G[V\setminus N[x]]) \end{align*} and therefore the heaviest independent set containing $v$ is at least the weight of the heaviest independent containing \emph{any} neighbor of $v$. Concluding, we have that \[w(v) + \alpha_w(G[V\setminus N[v]]) \geq \alpha_w(G[V\setminus\{v\}])\] and therefore $\I = \I'\cup\{v\}$ is an MWIS of $G$.
\emph{Case 2 ($\I'\cap N'(v) \neq \emptyset$):} Let $\I'$ be an MWIS of $G'$, we show that if $\I' \cap N'(v) \neq \emptyset$ then $\I = \I'$. To show this, let $\{x\} = \I'\cap N'(v)$. Define $G''$ as the graph resulting from increasing the weight of $N'(v)$ by $w(v)$, i.e.\ ~$\forall~u~\in~N'(v)$ we set $w''(u) = w'(u) + w(v) = w(u)$. We first show that $\I'' = \I'$ is an MWIS of $G'$. Therefore, assume that $\I^*$ is an MWIS of $G''$ with $w(\I^*) > w(\I')$ that does not contain $x$. However, then $\I^*$ is also a better MWIS on $G'$ which contradicts our initial assumption. Finally, we have that $w(\I'') = w(\I') + w(v)$, since exactly one node in $N'(v)$ is in $\I'$.
Next, we define $G'''$ as the graph resulting from adding back $v$ to $G'$' and show that $\I''' = \I''$ is a MWIS of $G'''$. For this purpose, we assume that $\I^*$ is a MWIS of $G'''$ with $w(\I^*) > w(\I''')$. Then, $v \in \I^*$ since we only added this node to $G''$. Likewise, $x \not\in \I^*$ since its a neighbor of $v$.
Since $w(\I^*) > w(\I''')$, we have that: \begin{align*}
w(\I^*\setminus\{v\}) &= w(\I^*) - w(v)\\
&> w(\I'') - w(v)\\
&= w(\I') + w(v) - w(v)\\
&= w(\I'). \end{align*} However, since $\I^*\setminus\{v\}$ does neither include v nor any neighbor of $v$ it is also an IS of $G'$ that is larger than $\I'$. This contradicts our initial assumption and thus $\I''' = \I'' = \I'$. Furthermore, since $G''' = G$, we have that $\I''' = \I = \I'$. \end{proof}
\pagebreak
\section{Graph Properties, Kernel Sizes, Full Tables, Convergence Plots} \begin{table}[h!] \scriptsize \centering \vspace*{-.5cm}
\begin{tabular}{l|r r r r}
Graph & $|V|$ & $|E|$ & $\mathcal{K}_{\text{dense}}$ & $\mathcal{K}_{\text{full}}$ \\ \hline \Id{\detokenize{alabama-AM2}} & \numprint{1164} & \numprint{38772} & \numprint{173} & \numprint{173} \\ \Id{\detokenize{alabama-AM3}} & \numprint{3504} & \numprint{619328} & \numprint{1614} & \numprint{1614} \\ \Id{\detokenize{district-of-columbia-AM1}} & \numprint{2500} & \numprint{49302} & \numprint{800} & \numprint{800} \\ \Id{\detokenize{district-of-columbia-AM2}} & \numprint{13597} & \numprint{3219590} & \numprint{6360} & \numprint{6360} \\ \Id{\detokenize{district-of-columbia-AM3}} & \numprint{46221} & \numprint{55458274} & \numprint{33367} & \numprint{33367} \\ \Id{\detokenize{florida-AM2}} & \numprint{1254} & \numprint{33872} & \numprint{41} & \numprint{41} \\ \Id{\detokenize{florida-AM3}} & \numprint{2985} & \numprint{308086} & \numprint{1069} & \numprint{1069} \\ \Id{\detokenize{georgia-AM3}} & \numprint{1680} & \numprint{148252} & \numprint{861} & \numprint{861} \\ \Id{\detokenize{greenland-AM3}} & \numprint{4986} & \numprint{7304722} & \numprint{3942} & \numprint{3942} \\ \Id{\detokenize{hawaii-AM2}} & \numprint{2875} & \numprint{530316} & \numprint{428} & \numprint{428} \\ \Id{\detokenize{hawaii-AM3}} & \numprint{28006} & \numprint{98889842} & \numprint{24436} & \numprint{24436} \\ \Id{\detokenize{idaho-AM3}} & \numprint{4064} & \numprint{7848160} & \numprint{3208} & \numprint{3208} \\ \Id{\detokenize{kansas-AM3}} & \numprint{2732} & \numprint{1613824} & \numprint{1605} & \numprint{1605} \\ \Id{\detokenize{kentucky-AM2}} & \numprint{2453} & \numprint{1286856} & \numprint{442} & \numprint{442} \\ \Id{\detokenize{kentucky-AM3}} & \numprint{19095} & \numprint{119067260} & \numprint{16871} & \numprint{16871} \\ \Id{\detokenize{louisiana-AM3}} & \numprint{1162} & \numprint{74154} & \numprint{382} & \numprint{382} \\ \Id{\detokenize{maryland-AM3}} & \numprint{1018} & \numprint{190830} & \numprint{187} & \numprint{187} \\ \Id{\detokenize{massachusetts-AM2}} & \numprint{1339} & \numprint{70898} & \numprint{196} & \numprint{196} \\ \Id{\detokenize{massachusetts-AM3}} & \numprint{3703} & \numprint{1102982} & \numprint{2008} & \numprint{2008} \\ \Id{\detokenize{mexico-AM3}} & \numprint{1096} & \numprint{94262} & \numprint{620} & \numprint{620} \\ \Id{\detokenize{new-hampshire-AM3}} & \numprint{1107} & \numprint{36042} & \numprint{247} & \numprint{247} \\ \Id{\detokenize{north-carolina-AM3}} & \numprint{1557} & \numprint{473478} & \numprint{1178} & \numprint{1178} \\ \Id{\detokenize{oregon-AM2}} & \numprint{1325} & \numprint{115034} & \numprint{35} & \numprint{35} \\ \Id{\detokenize{oregon-AM3}} & \numprint{5588} & \numprint{5825402} & \numprint{3670} & \numprint{3670} \\ \Id{\detokenize{pennsylvania-AM3}} & \numprint{1148} & \numprint{52928} & \numprint{315} & \numprint{315} \\ \Id{\detokenize{rhode-island-AM2}} & \numprint{2866} & \numprint{590976} & \numprint{1103} & \numprint{1103} \\ \Id{\detokenize{rhode-island-AM3}} & \numprint{15124} & \numprint{25244438} & \numprint{13031} & \numprint{13031} \\ \Id{\detokenize{utah-AM3}} & \numprint{1339} & \numprint{85744} & \numprint{568} & \numprint{568} \\ \Id{\detokenize{vermont-AM3}} & \numprint{3436} & \numprint{2272328} & \numprint{2630} & \numprint{2630} \\ \Id{\detokenize{virginia-AM2}} & \numprint{2279} & \numprint{120080} & \numprint{237} & \numprint{237} \\ \Id{\detokenize{virginia-AM3}} & \numprint{6185} & \numprint{1331806} & \numprint{3867} & \numprint{3867} \\ \Id{\detokenize{washington-AM2}} & \numprint{3025} & \numprint{304898} & \numprint{382} & \numprint{382} \\ \Id{\detokenize{washington-AM3}} & \numprint{10022} & \numprint{4692426} & \numprint{8030} & \numprint{8030} \\ \Id{\detokenize{west-virginia-AM3}} & \numprint{1185} & \numprint{251240} & \numprint{991} & \numprint{991} \\ \end{tabular}
\vspace*{.5cm}
\begin{tabular}{l|r r r r}
Graph & $|V|$ & $|E|$ & $\mathcal{K}_{\text{dense}}$ & $\mathcal{K}_{\text{full}}$ \\ \hline \Id{\detokenize{as-skitter}} & \numprint{1696415} & \numprint{22190596} & \numprint{27318} & \numprint{9180} \\ \Id{\detokenize{ca-AstroPh}} & \numprint{18772} & \numprint{396100} & \numprint{0} & \numprint{0} \\ \Id{\detokenize{ca-CondMat}} & \numprint{23133} & \numprint{186878} & \numprint{0} & \numprint{0} \\ \Id{\detokenize{ca-GrQc}} & \numprint{5242} & \numprint{28968} & \numprint{0} & \numprint{0} \\ \Id{\detokenize{ca-HepPh}} & \numprint{12008} & \numprint{236978} & \numprint{0} & \numprint{0} \\ \Id{\detokenize{ca-HepTh}} & \numprint{9877} & \numprint{51946} & \numprint{0} & \numprint{0} \\ \Id{\detokenize{email-Enron}} & \numprint{36692} & \numprint{367662} & \numprint{0} & \numprint{0} \\ \Id{\detokenize{email-EuAll}} & \numprint{265214} & \numprint{728962} & \numprint{0} & \numprint{0} \\ \Id{\detokenize{p2p-Gnutella04}} & \numprint{10876} & \numprint{79988} & \numprint{0} & \numprint{0} \\ \Id{\detokenize{p2p-Gnutella05}} & \numprint{8846} & \numprint{63678} & \numprint{0} & \numprint{0} \\ \Id{\detokenize{p2p-Gnutella06}} & \numprint{8717} & \numprint{63050} & \numprint{0} & \numprint{0} \\ \Id{\detokenize{p2p-Gnutella08}} & \numprint{6301} & \numprint{41554} & \numprint{0} & \numprint{0} \\ \Id{\detokenize{p2p-Gnutella09}} & \numprint{8114} & \numprint{52026} & \numprint{0} & \numprint{0} \\ \Id{\detokenize{p2p-Gnutella24}} & \numprint{26518} & \numprint{130738} & \numprint{0} & \numprint{0} \\ \Id{\detokenize{p2p-Gnutella25}} & \numprint{22687} & \numprint{109410} & \numprint{11} & \numprint{0} \\ \Id{\detokenize{p2p-Gnutella30}} & \numprint{36682} & \numprint{176656} & \numprint{10} & \numprint{0} \\ \Id{\detokenize{p2p-Gnutella31}} & \numprint{62586} & \numprint{295784} & \numprint{0} & \numprint{0} \\ \Id{\detokenize{roadNet-CA}} & \numprint{1965206} & \numprint{5533214} & \numprint{233083} & \numprint{63926} \\ \Id{\detokenize{roadNet-PA}} & \numprint{1088092} & \numprint{3083796} & \numprint{135536} & \numprint{38080} \\ \Id{\detokenize{roadNet-TX}} & \numprint{1379917} & \numprint{3843320} & \numprint{151570} & \numprint{39433} \\ \Id{\detokenize{soc-Epinions1}} & \numprint{75879} & \numprint{811480} & \numprint{6} & \numprint{0} \\ \Id{\detokenize{soc-LiveJournal1}} & \numprint{4847571} & \numprint{85702474} & \numprint{61690} & \numprint{29779} \\ \Id{\detokenize{soc-Slashdot0811}} & \numprint{77360} & \numprint{938360} & \numprint{0} & \numprint{0} \\ \Id{\detokenize{soc-Slashdot0902}} & \numprint{82168} & \numprint{1008460} & \numprint{20} & \numprint{0} \\ \Id{\detokenize{soc-pokec-relationships}} & \numprint{1632803} & \numprint{44603928} & \numprint{927214} & \numprint{902748} \\ \Id{\detokenize{web-BerkStan}} & \numprint{685230} & \numprint{13298940} & \numprint{37004} & \numprint{17482} \\ \Id{\detokenize{web-Google}} & \numprint{875713} & \numprint{8644102} & \numprint{2892} & \numprint{1178} \\ \Id{\detokenize{web-NotreDame}} & \numprint{325729} & \numprint{2180216} & \numprint{14038} & \numprint{6760} \\ \Id{\detokenize{web-Stanford}} & \numprint{281903} & \numprint{3985272} & \numprint{14280} & \numprint{2640} \\ \Id{\detokenize{wiki-Talk}} & \numprint{2394385} & \numprint{9319130} & \numprint{0} & \numprint{0} \\ \Id{\detokenize{wiki-Vote}} & \numprint{7115} & \numprint{201524} & \numprint{246} & \numprint{237} \\ \end{tabular}
\caption{Basic properties as well as kernel sizes computed by both variants of our branch-and-reduce algorithm for the OSM networks (top) and SNAP networks (bottom). } \label{tab:props} \end{table}
\begin{table*}[ht] \scriptsize \centering \setlength{\tabcolsep}{0.5ex}
\begin{tabular}{l|r r|r r|r r|r r|r r}
& \multicolumn{2}{c|}{DynWVC1} & \multicolumn{2}{c|}{DynWVC2} & \multicolumn{2}{c|}{HILS} & \multicolumn{2}{c|}{B~\&~R\textsubscript{\text{dense}}} & \multicolumn{2}{c}{B~\&~R\textsubscript{\text{full}}} \\ \hline Graph & $t_\text{max}$ & $w_\text{max}$ & $t_\text{max}$ & $w_\text{max}$ & $t_\text{max}$ & $w_\text{max}$ & $t_\text{max}$ & $w_\text{max}$ & $t_\text{max}$ & $w_\text{max}$ \\ \hline \rowcolor{lightergray} \Id{\detokenize{alabama-AM2}} & \numprint{0.62} & \numprint{174241} & \numprint{26.83} & \numprint{174297} & \numprint{0.04} & \textbf{\numprint{174309}} & \numprint{0.40} & \textbf{\numprint{174309}} & \numprint{0.79} & \textbf{\numprint{174309}} \\ \Id{\detokenize{alabama-AM3}} & \numprint{464.02} & \numprint{185527} & \numprint{887.55} & \numprint{185652} & \numprint{0.73} & \textbf{\numprint{185744}} & \numprint{15.79} & \numprint{185707} & \numprint{80.78} & \numprint{185707} \\ \Id{\detokenize{district-of-columbia-AM1}} & \numprint{12.64} & \textbf{\numprint{196475}} & \numprint{11.40} & \textbf{\numprint{196475}} & \numprint{0.26} & \textbf{\numprint{196475}} & \numprint{1.97} & \textbf{\numprint{196475}} & \numprint{4.13} & \textbf{\numprint{196475}} \\ \Id{\detokenize{district-of-columbia-AM2}} & \numprint{272.37} & \numprint{208942} & \numprint{596.62} & \numprint{208954} & \numprint{717.75} & \textbf{\numprint{209132}} & \numprint{20.03} & \numprint{147450} & \numprint{233.70} & \numprint{147450} \\ \Id{\detokenize{district-of-columbia-AM3}} & \numprint{949.96} & \numprint{224289} & \numprint{782.62} & \numprint{223780} & \numprint{989.68} & \textbf{\numprint{227598}} & \numprint{553.84} & \numprint{92784} & \numprint{918.07} & \numprint{92714} \\ \rowcolor{lightergray} \Id{\detokenize{florida-AM2}} & \numprint{1.14} & \textbf{\numprint{230595}} & \numprint{0.72} & \textbf{\numprint{230595}} & \numprint{0.04} & \textbf{\numprint{230595}} & \numprint{0.03} & \textbf{\numprint{230595}} & \numprint{0.02} & \textbf{\numprint{230595}} \\ \rowcolor{lightergray} \Id{\detokenize{florida-AM3}} & \numprint{553.56} & \numprint{237127} & \numprint{181.58} & \numprint{237081} & \numprint{2.76} & \textbf{\numprint{237333}} & \numprint{20.52} & \textbf{\numprint{237333}} & \numprint{324.38} & \numprint{226767} \\ \Id{\detokenize{georgia-AM3}} & \numprint{0.88} & \textbf{\numprint{222652}} & \numprint{1.29} & \textbf{\numprint{222652}} & \numprint{0.05} & \textbf{\numprint{222652}} & \numprint{4.88} & \numprint{214918} & \numprint{14.35} & \numprint{214918} \\ \Id{\detokenize{greenland-AM3}} & \numprint{73.16} & \textbf{\numprint{14011}} & \numprint{51.09} & \numprint{14008} & \numprint{1.72} & \textbf{\numprint{14011}} & \numprint{14.52} & \numprint{13152} & \numprint{47.25} & \numprint{13069} \\ \rowcolor{lightergray} \Id{\detokenize{hawaii-AM2}} & \numprint{4.85} & \numprint{125273} & \numprint{3.20} & \numprint{125276} & \numprint{0.33} & \textbf{\numprint{125284}} & \numprint{3.59} & \textbf{\numprint{125284}} & \numprint{10.89} & \textbf{\numprint{125284}} \\ \Id{\detokenize{hawaii-AM3}} & \numprint{898.64} & \numprint{140596} & \numprint{904.15} & \numprint{140486} & \numprint{332.32} & \textbf{\numprint{141035}} & \numprint{288.58} & \numprint{106251} & \numprint{1177.95} & \numprint{129812} \\ \Id{\detokenize{idaho-AM3}} & \numprint{76.55} & \textbf{\numprint{77145}} & \numprint{85.35} & \textbf{\numprint{77145}} & \numprint{1.49} & \textbf{\numprint{77145}} & \numprint{866.90} & \numprint{77010} & \numprint{61.26} & \numprint{76831} \\ \Id{\detokenize{kansas-AM3}} & \numprint{46.87} & \textbf{\numprint{87976}} & \numprint{44.26} & \textbf{\numprint{87976}} & \numprint{0.84} & \textbf{\numprint{87976}} & \numprint{11.35} & \numprint{87925} & \numprint{18.99} & \numprint{87925} \\ \rowcolor{lightergray} \Id{\detokenize{kentucky-AM2}} & \numprint{5.12} & \textbf{\numprint{97397}} & \numprint{7.39} & \textbf{\numprint{97397}} & \numprint{0.47} & \textbf{\numprint{97397}} & \numprint{11.35} & \textbf{\numprint{97397}} & \numprint{42.05} & \textbf{\numprint{97397}} \\ \Id{\detokenize{kentucky-AM3}} & \numprint{932.32} & \numprint{100463} & \numprint{722.69} & \numprint{100430} & \numprint{802.03} & \textbf{\numprint{100507}} & \numprint{172.30} & \numprint{91864} & \numprint{3346.94} & \numprint{96634} \\ \rowcolor{lightergray} \Id{\detokenize{louisiana-AM3}} & \numprint{0.32} & \numprint{60005} & \numprint{0.27} & \numprint{60002} & \numprint{0.03} & \textbf{\numprint{60024}} & \numprint{3.38} & \textbf{\numprint{60024}} & \numprint{20.17} & \textbf{\numprint{60024}} \\ \rowcolor{lightergray} \Id{\detokenize{maryland-AM3}} & \numprint{1.34} & \textbf{\numprint{45496}} & \numprint{0.87} & \textbf{\numprint{45496}} & \numprint{0.02} & \textbf{\numprint{45496}} & \numprint{3.34} & \textbf{\numprint{45496}} & \numprint{11.08} & \textbf{\numprint{45496}} \\ \rowcolor{lightergray} \Id{\detokenize{massachusetts-AM2}} & \numprint{0.37} & \textbf{\numprint{140095}} & \numprint{0.09} & \textbf{\numprint{140095}} & \numprint{0.02} & \textbf{\numprint{140095}} & \numprint{0.46} & \textbf{\numprint{140095}} & \numprint{0.48} & \textbf{\numprint{140095}} \\ \Id{\detokenize{massachusetts-AM3}} & \numprint{435.31} & \numprint{145863} & \numprint{154.61} & \numprint{145863} & \numprint{2.73} & \textbf{\numprint{145866}} & \numprint{12.87} & \numprint{145617} & \numprint{23.97} & \numprint{145631} \\ \rowcolor{lightergray} \Id{\detokenize{mexico-AM3}} & \numprint{0.14} & \textbf{\numprint{97663}} & \numprint{46.86} & \textbf{\numprint{97663}} & \numprint{0.04} & \textbf{\numprint{97663}} & \numprint{14.25} & \textbf{\numprint{97663}} & \numprint{289.14} & \textbf{\numprint{97663}} \\ \rowcolor{lightergray} \Id{\detokenize{new-hampshire-AM3}} & \numprint{0.22} & \textbf{\numprint{116060}} & \numprint{0.42} & \textbf{\numprint{116060}} & \numprint{0.03} & \textbf{\numprint{116060}} & \numprint{3.25} & \textbf{\numprint{116060}} & \numprint{8.75} & \textbf{\numprint{116060}} \\ \Id{\detokenize{north-carolina-AM3}} & \numprint{796.26} & \numprint{49716} & \numprint{285.91} & \textbf{\numprint{49720}} & \numprint{0.08} & \textbf{\numprint{49720}} & \numprint{10.45} & \numprint{49562} & \numprint{11.55} & \numprint{49562} \\ \rowcolor{lightergray} \Id{\detokenize{oregon-AM2}} & \numprint{0.22} & \textbf{\numprint{165047}} & \numprint{0.25} & \textbf{\numprint{165047}} & \numprint{0.04} & \textbf{\numprint{165047}} & \numprint{0.04} & \textbf{\numprint{165047}} & \numprint{0.09} & \textbf{\numprint{165047}} \\ \Id{\detokenize{oregon-AM3}} & \numprint{393.23} & \numprint{175046} & \numprint{126.97} & \numprint{175060} & \numprint{3.36} & \textbf{\numprint{175078}} & \numprint{351.99} & \numprint{174334} & \numprint{474.15} & \numprint{164941} \\ \rowcolor{lightergray} \Id{\detokenize{pennsylvania-AM3}} & \numprint{0.09} & \textbf{\numprint{143870}} & \numprint{0.15} & \textbf{\numprint{143870}} & \numprint{0.04} & \textbf{\numprint{143870}} & \numprint{9.98} & \textbf{\numprint{143870}} & \numprint{38.76} & \textbf{\numprint{143870}} \\ \Id{\detokenize{rhode-island-AM2}} & \numprint{6.66} & \numprint{184562} & \numprint{24.74} & \numprint{184576} & \numprint{0.40} & \textbf{\numprint{184596}} & \numprint{10.70} & \numprint{184543} & \numprint{16.79} & \numprint{184543} \\ \Id{\detokenize{rhode-island-AM3}} & \numprint{54.99} & \numprint{201553} & \numprint{609.14} & \numprint{201344} & \numprint{43.34} & \textbf{\numprint{201758}} & \numprint{399.33} & \numprint{162639} & \numprint{931.05} & \numprint{163080} \\ \rowcolor{lightergray} \Id{\detokenize{utah-AM3}} & \numprint{136.15} & \numprint{98802} & \numprint{233.52} & \textbf{\numprint{98847}} & \numprint{0.08} & \textbf{\numprint{98847}} & \numprint{64.04} & \textbf{\numprint{98847}} & \numprint{285.22} & \textbf{\numprint{98847}} \\ \Id{\detokenize{vermont-AM3}} & \numprint{119.63} & \numprint{63234} & \numprint{88.35} & \numprint{63238} & \numprint{0.95} & \textbf{\numprint{63302}} & \numprint{95.81} & \numprint{55584} & \numprint{443.88} & \numprint{55577} \\ \rowcolor{lightergray} \Id{\detokenize{virginia-AM2}} & \numprint{0.89} & \numprint{295794} & \numprint{1.32} & \numprint{295668} & \numprint{0.12} & \textbf{\numprint{295867}} & \numprint{0.93} & \textbf{\numprint{295867}} & \numprint{0.77} & \textbf{\numprint{295867}} \\ \Id{\detokenize{virginia-AM3}} & \numprint{289.23} & \numprint{307867} & \numprint{883.75} & \numprint{307845} & \numprint{3.75} & \textbf{\numprint{308305}} & \numprint{109.20} & \numprint{306985} & \numprint{786.05} & \numprint{233572} \\ \rowcolor{lightergray} \Id{\detokenize{washington-AM2}} & \numprint{2.00} & \textbf{\numprint{305619}} & \numprint{15.60} & \textbf{\numprint{305619}} & \numprint{0.62} & \textbf{\numprint{305619}} & \numprint{2.44} & \textbf{\numprint{305619}} & \numprint{2.20} & \textbf{\numprint{305619}} \\ \Id{\detokenize{washington-AM3}} & \numprint{79.77} & \numprint{313808} & \numprint{401.59} & \numprint{313827} & \numprint{13.88} & \textbf{\numprint{314288}} & \numprint{248.77} & \numprint{271747} & \numprint{532.25} & \numprint{271747} \\ \Id{\detokenize{west-virginia-AM3}} & \numprint{1.10} & \textbf{\numprint{47927}} & \numprint{0.87} & \textbf{\numprint{47927}} & \numprint{0.08} & \textbf{\numprint{47927}} & \numprint{14.38} & \textbf{\numprint{47927}} & \numprint{854.73} & \textbf{\numprint{47927}} \\ \end{tabular}
\caption{Best solution found by each algorithm and time (in seconds) required to compute it. The global best solution is highlighted in \textbf{bold}. Rows are highlighted in gray if B~\&~R is able to find an exact solution. } \label{tab:osm_local_rt} \end{table*}
\begin{table*}[ht] \scriptsize \centering \setlength{\tabcolsep}{0.5ex}
\begin{tabular}{l|r r|r r|r r|r r|r r}
& \multicolumn{2}{c|}{DynWVC1} & \multicolumn{2}{c|}{DynWVC2} & \multicolumn{2}{c|}{HILS} & \multicolumn{2}{c|}{B~\&~R\textsubscript{\text{dense}}} & \multicolumn{2}{c}{B~\&~R\textsubscript{\text{full}}} \\ \hline Graph & $t_\text{max}$ & $w_\text{max}$ & $t_\text{max}$ & $w_\text{max}$ & $t_\text{max}$ & $w_\text{max}$ & $t_\text{max}$ & $w_\text{max}$ & $t_\text{max}$ & $w_\text{max}$ \\ \hline \Id{\detokenize{as-skitter}} & \numprint{997.39} & \numprint{123412428} & \numprint{576.93} & \numprint{123105765} & \numprint{998.75} & \numprint{122539706} & \numprint{641.38} & \numprint{123172824} & \numprint{746.93} & \textbf{\numprint{123904741}} \\ \rowcolor{lightergray} \Id{\detokenize{ca-AstroPh}} & \numprint{207.99} & \numprint{796467} & \numprint{108.35} & \numprint{796535} & \numprint{46.76} & \textbf{\numprint{796556}} & \numprint{0.03} & \textbf{\numprint{796556}} & \numprint{0.03} & \textbf{\numprint{796556}} \\ \rowcolor{lightergray} \Id{\detokenize{ca-CondMat}} & \numprint{71.54} & \numprint{1143431} & \numprint{222.30} & \numprint{1143471} & \numprint{45.07} & \textbf{\numprint{1143480}} & \numprint{0.02} & \textbf{\numprint{1143480}} & \numprint{0.02} & \textbf{\numprint{1143480}} \\ \rowcolor{lightergray} \Id{\detokenize{ca-GrQc}} & \numprint{1.75} & \textbf{\numprint{289481}} & \numprint{0.82} & \textbf{\numprint{289481}} & \numprint{0.60} & \textbf{\numprint{289481}} & \numprint{0.00} & \textbf{\numprint{289481}} & \numprint{0.00} & \textbf{\numprint{289481}} \\ \rowcolor{lightergray} \Id{\detokenize{ca-HepPh}} & \numprint{26.36} & \numprint{579624} & \numprint{17.31} & \numprint{579662} & \numprint{11.44} & \textbf{\numprint{579675}} & \numprint{0.02} & \textbf{\numprint{579675}} & \numprint{0.02} & \textbf{\numprint{579675}} \\ \rowcolor{lightergray} \Id{\detokenize{ca-HepTh}} & \numprint{9.87} & \numprint{560630} & \numprint{12.64} & \numprint{560642} & \numprint{94.19} & \textbf{\numprint{560662}} & \numprint{0.01} & \textbf{\numprint{560662}} & \numprint{0.01} & \textbf{\numprint{560662}} \\ \rowcolor{lightergray} \Id{\detokenize{email-Enron}} & \numprint{295.02} & \numprint{2457460} & \numprint{910.50} & \numprint{2457505} & \numprint{79.40} & \textbf{\numprint{2457547}} & \numprint{0.04} & \textbf{\numprint{2457547}} & \numprint{0.03} & \textbf{\numprint{2457547}} \\ \rowcolor{lightergray} \Id{\detokenize{email-EuAll}} & \numprint{180.92} & \textbf{\numprint{25330331}} & \numprint{179.26} & \textbf{\numprint{25330331}} & \numprint{501.09} & \textbf{\numprint{25330331}} & \numprint{0.13} & \textbf{\numprint{25330331}} & \numprint{0.19} & \textbf{\numprint{25330331}} \\ \rowcolor{lightergray} \Id{\detokenize{p2p-Gnutella04}} & \numprint{2.46} & \numprint{667496} & \numprint{866.88} & \numprint{667503} & \numprint{2.64} & \textbf{\numprint{667539}} & \numprint{0.01} & \textbf{\numprint{667539}} & \numprint{0.01} & \textbf{\numprint{667539}} \\ \rowcolor{lightergray} \Id{\detokenize{p2p-Gnutella05}} & \numprint{24.23} & \textbf{\numprint{556559}} & \numprint{3.54} & \textbf{\numprint{556559}} & \numprint{0.60} & \textbf{\numprint{556559}} & \numprint{0.01} & \textbf{\numprint{556559}} & \numprint{0.01} & \textbf{\numprint{556559}} \\ \rowcolor{lightergray} \Id{\detokenize{p2p-Gnutella06}} & \numprint{532.67} & \numprint{547585} & \numprint{1.38} & \numprint{547586} & \numprint{1.47} & \textbf{\numprint{547591}} & \numprint{0.01} & \textbf{\numprint{547591}} & \numprint{0.01} & \textbf{\numprint{547591}} \\ \rowcolor{lightergray} \Id{\detokenize{p2p-Gnutella08}} & \numprint{0.21} & \textbf{\numprint{435893}} & \numprint{0.19} & \textbf{\numprint{435893}} & \numprint{0.25} & \textbf{\numprint{435893}} & \numprint{0.00} & \textbf{\numprint{435893}} & \numprint{0.01} & \textbf{\numprint{435893}} \\ \rowcolor{lightergray} \Id{\detokenize{p2p-Gnutella09}} & \numprint{0.23} & \textbf{\numprint{568472}} & \numprint{0.22} & \textbf{\numprint{568472}} & \numprint{0.15} & \textbf{\numprint{568472}} & \numprint{0.01} & \textbf{\numprint{568472}} & \numprint{0.01} & \textbf{\numprint{568472}} \\ \rowcolor{lightergray} \Id{\detokenize{p2p-Gnutella24}} & \numprint{10.83} & \numprint{1970325} & \numprint{9.81} & \numprint{1970325} & \numprint{4.06} & \textbf{\numprint{1970329}} & \numprint{0.02} & \textbf{\numprint{1970329}} & \numprint{0.02} & \textbf{\numprint{1970329}} \\ \rowcolor{lightergray} \Id{\detokenize{p2p-Gnutella25}} & \numprint{2.22} & \textbf{\numprint{1697310}} & \numprint{6.33} & \textbf{\numprint{1697310}} & \numprint{1.64} & \textbf{\numprint{1697310}} & \numprint{0.01} & \textbf{\numprint{1697310}} & \numprint{0.02} & \textbf{\numprint{1697310}} \\ \rowcolor{lightergray} \Id{\detokenize{p2p-Gnutella30}} & \numprint{10.06} & \numprint{2785926} & \numprint{22.66} & \numprint{2785922} & \numprint{7.36} & \textbf{\numprint{2785957}} & \numprint{0.02} & \textbf{\numprint{2785957}} & \numprint{0.03} & \textbf{\numprint{2785957}} \\ \rowcolor{lightergray} \Id{\detokenize{p2p-Gnutella31}} & \numprint{169.03} & \numprint{4750622} & \numprint{43.15} & \numprint{4750632} & \numprint{34.33} & \textbf{\numprint{4750671}} & \numprint{0.13} & \textbf{\numprint{4750671}} & \numprint{0.04} & \textbf{\numprint{4750671}} \\ \rowcolor{lightergray} \Id{\detokenize{roadNet-CA}} & \numprint{1001.61} & \numprint{109028140} & \numprint{1000.88} & \numprint{109023976} & \numprint{3312.19} & \numprint{108167310} & \numprint{931.36} & \numprint{106500027} & \numprint{774.56} & \textbf{\numprint{111408830}} \\ \rowcolor{lightergray} \Id{\detokenize{roadNet-PA}} & \numprint{720.57} & \numprint{60940033} & \numprint{787.59} & \numprint{60940033} & \numprint{998.56} & \numprint{59915775} & \numprint{988.62} & \numprint{58927755} & \numprint{32.06} & \textbf{\numprint{61686106}} \\ \rowcolor{lightergray} \Id{\detokenize{roadNet-TX}} & \numprint{1001.45} & \numprint{77498612} & \numprint{1000.78} & \numprint{77525099} & \numprint{1697.13} & \numprint{76366577} & \numprint{870.62} & \numprint{75843903} & \numprint{33.49} & \textbf{\numprint{78606965}} \\ \rowcolor{lightergray} \Id{\detokenize{soc-Epinions1}} & \numprint{617.40} & \numprint{5668054} & \numprint{625.89} & \numprint{5668180} & \numprint{694.51} & \numprint{5668382} & \numprint{0.07} & \textbf{\numprint{5668401}} & \numprint{0.11} & \textbf{\numprint{5668401}} \\ \Id{\detokenize{soc-LiveJournal1}} & \numprint{1001.31} & \numprint{277850684} & \numprint{1001.23} & \numprint{277824322} & \numprint{12437.50} & \numprint{280559036} & \numprint{86.66} & \numprint{283869420} & \numprint{270.96} & \textbf{\numprint{283948671}} \\ \rowcolor{lightergray} \Id{\detokenize{soc-Slashdot0811}} & \numprint{809.97} & \numprint{5650118} & \numprint{477.14} & \numprint{5650303} & \numprint{767.51} & \numprint{5650644} & \numprint{0.10} & \textbf{\numprint{5650791}} & \numprint{0.18} & \textbf{\numprint{5650791}} \\ \rowcolor{lightergray} \Id{\detokenize{soc-Slashdot0902}} & \numprint{783.10} & \numprint{5953052} & \numprint{272.11} & \numprint{5953235} & \numprint{786.70} & \numprint{5953436} & \numprint{0.13} & \textbf{\numprint{5953582}} & \numprint{0.21} & \textbf{\numprint{5953582}} \\ \Id{\detokenize{soc-pokec-relationships}} & \numprint{999.99} & \numprint{82522272} & \numprint{1001.42} & \textbf{\numprint{82640035}} & \numprint{2482.18} & \numprint{82381583} & \numprint{287.40} & \numprint{82595492} & \numprint{1404.57} & \numprint{75717984} \\ \Id{\detokenize{web-BerkStan}} & \numprint{347.17} & \numprint{43595139} & \numprint{372.33} & \numprint{43593142} & \numprint{994.73} & \numprint{43319988} & \numprint{22.58} & \numprint{43138612} & \numprint{831.75} & \textbf{\numprint{43766431}} \\ \rowcolor{lightergray} \Id{\detokenize{web-Google}} & \numprint{759.75} & \numprint{56193138} & \numprint{683.63} & \numprint{56190870} & \numprint{994.58} & \numprint{55954155} & \numprint{2.08} & \textbf{\numprint{56313384}} & \numprint{3.16} & \textbf{\numprint{56313384}} \\ \Id{\detokenize{web-NotreDame}} & \numprint{963.44} & \textbf{\numprint{25975765}} & \numprint{875.22} & \numprint{25968209} & \numprint{998.79} & \numprint{25970368} & \numprint{354.79} & \numprint{25947936} & \numprint{28.11} & \numprint{25957800} \\ \Id{\detokenize{web-Stanford}} & \numprint{999.97} & \numprint{17731195} & \numprint{997.98} & \numprint{17735700} & \numprint{999.91} & \numprint{17679156} & \numprint{47.62} & \numprint{17634819} & \numprint{4.69} & \textbf{\numprint{17799469}} \\ \rowcolor{lightergray} \Id{\detokenize{wiki-Talk}} & \numprint{961.05} & \numprint{235874406} & \numprint{991.31} & \numprint{235874419} & \numprint{996.02} & \numprint{235852509} & \numprint{3.85} & \textbf{\numprint{235875181}} & \numprint{3.36} & \textbf{\numprint{235875181}} \\ \rowcolor{lightergray} \Id{\detokenize{wiki-Vote}} & \numprint{0.74} & \textbf{\numprint{500436}} & \numprint{0.75} & \textbf{\numprint{500436}} & \numprint{23.96} & \textbf{\numprint{500436}} & \numprint{0.05} & \textbf{\numprint{500436}} & \numprint{0.06} & \textbf{\numprint{500436}} \\ \end{tabular}
\caption{Best solution found by each algorithm and time (in seconds) required to compute it. The global best solution is highlighted in \textbf{bold}. Rows are highlighted in gray if B~\&~R is able to find an exact solution. } \label{tab:snap_local_rt} \end{table*}
\begin{table*}[ht] \scriptsize \centering \setlength{\tabcolsep}{0.5ex}
\begin{tabular}{l|r r|r r|r r|r r|r r}
& \multicolumn{2}{c|}{Red+DynWVC1} & \multicolumn{2}{c|}{Red+DynWVC2} & \multicolumn{2}{c|}{Red+HILS} & \multicolumn{2}{c|}{B~\&~R\textsubscript{\text{dense}}} & \multicolumn{2}{c}{B~\&~R\textsubscript{\text{full}}} \\ \hline Graph & $t_\text{max}$ & $w_\text{max}$ & $t_\text{max}$ & $w_\text{max}$ & $t_\text{max}$ & $w_\text{max}$ & $t_\text{max}$ & $w_\text{max}$ & $t_\text{max}$ & $w_\text{max}$ \\ \hline \rowcolor{lightergray} \Id{\detokenize{alabama-AM2}} & \numprint{0.11} & \textbf{\numprint{174309}} & \numprint{0.11} & \textbf{\numprint{174309}} & \numprint{0.10} & \textbf{\numprint{174309}} & \numprint{0.40} & \textbf{\numprint{174309}} & \numprint{0.79} & \textbf{\numprint{174309}} \\ \Id{\detokenize{alabama-AM3}} & \numprint{370.80} & \numprint{185727} & \numprint{295.20} & \numprint{185729} & \numprint{4.05} & \textbf{\numprint{185744}} & \numprint{15.79} & \numprint{185707} & \numprint{80.78} & \numprint{185707} \\ \Id{\detokenize{district-of-columbia-AM1}} & \numprint{0.92} & \textbf{\numprint{196475}} & \numprint{0.92} & \textbf{\numprint{196475}} & \numprint{0.37} & \textbf{\numprint{196475}} & \numprint{1.97} & \textbf{\numprint{196475}} & \numprint{4.13} & \textbf{\numprint{196475}} \\ \Id{\detokenize{district-of-columbia-AM2}} & \numprint{334.12} & \numprint{209125} & \numprint{982.91} & \numprint{209056} & \numprint{220.82} & \textbf{\numprint{209132}} & \numprint{20.03} & \numprint{147450} & \numprint{233.70} & \numprint{147450} \\ \Id{\detokenize{district-of-columbia-AM3}} & \numprint{879.25} & \numprint{225535} & \numprint{789.47} & \numprint{225031} & \numprint{320.06} & \textbf{\numprint{227534}} & \numprint{553.84} & \numprint{92784} & \numprint{918.07} & \numprint{92714} \\ \rowcolor{lightergray} \Id{\detokenize{florida-AM2}} & \numprint{0.03} & \textbf{\numprint{230595}} & \numprint{0.03} & \textbf{\numprint{230595}} & \numprint{0.03} & \textbf{\numprint{230595}} & \numprint{0.03} & \textbf{\numprint{230595}} & \numprint{0.02} & \textbf{\numprint{230595}} \\ \rowcolor{lightergray} \Id{\detokenize{florida-AM3}} & \numprint{8.66} & \numprint{237331} & \numprint{8.57} & \numprint{237331} & \numprint{8.01} & \textbf{\numprint{237333}} & \numprint{20.52} & \textbf{\numprint{237333}} & \numprint{324.38} & \numprint{226767} \\ \Id{\detokenize{georgia-AM3}} & \numprint{2.64} & \textbf{\numprint{222652}} & \numprint{2.62} & \textbf{\numprint{222652}} & \numprint{2.43} & \textbf{\numprint{222652}} & \numprint{4.88} & \numprint{214918} & \numprint{14.35} & \numprint{214918} \\ \Id{\detokenize{greenland-AM3}} & \numprint{712.63} & \numprint{14007} & \numprint{462.23} & \numprint{14006} & \numprint{10.34} & \textbf{\numprint{14011}} & \numprint{14.52} & \numprint{13152} & \numprint{47.25} & \numprint{13069} \\ \rowcolor{lightergray} \Id{\detokenize{hawaii-AM2}} & \numprint{0.96} & \textbf{\numprint{125284}} & \numprint{0.96} & \textbf{\numprint{125284}} & \numprint{0.93} & \textbf{\numprint{125284}} & \numprint{3.59} & \textbf{\numprint{125284}} & \numprint{10.89} & \textbf{\numprint{125284}} \\ \Id{\detokenize{hawaii-AM3}} & \numprint{405.34} & \numprint{140714} & \numprint{957.61} & \numprint{140709} & \numprint{329.20} & \textbf{\numprint{141011}} & \numprint{288.58} & \numprint{106251} & \numprint{1177.95} & \numprint{129812} \\ \Id{\detokenize{idaho-AM3}} & \numprint{40.38} & \textbf{\numprint{77145}} & \numprint{20.79} & \textbf{\numprint{77145}} & \numprint{203.76} & \textbf{\numprint{77145}} & \numprint{866.90} & \numprint{77010} & \numprint{61.26} & \numprint{76831} \\ \Id{\detokenize{kansas-AM3}} & \numprint{13.59} & \textbf{\numprint{87976}} & \numprint{18.43} & \textbf{\numprint{87976}} & \numprint{2.06} & \textbf{\numprint{87976}} & \numprint{11.35} & \numprint{87925} & \numprint{18.99} & \numprint{87925} \\ \rowcolor{lightergray} \Id{\detokenize{kentucky-AM2}} & \numprint{1.13} & \textbf{\numprint{97397}} & \numprint{1.13} & \textbf{\numprint{97397}} & \numprint{1.07} & \textbf{\numprint{97397}} & \numprint{11.35} & \textbf{\numprint{97397}} & \numprint{42.05} & \textbf{\numprint{97397}} \\ \Id{\detokenize{kentucky-AM3}} & \numprint{766.39} & \numprint{100479} & \numprint{759.20} & \numprint{100480} & \numprint{973.22} & \textbf{\numprint{100486}} & \numprint{172.30} & \numprint{91864} & \numprint{3346.94} & \numprint{96634} \\ \rowcolor{lightergray} \Id{\detokenize{louisiana-AM3}} & \numprint{1.35} & \textbf{\numprint{60024}} & \numprint{1.35} & \textbf{\numprint{60024}} & \numprint{1.33} & \textbf{\numprint{60024}} & \numprint{3.38} & \textbf{\numprint{60024}} & \numprint{20.17} & \textbf{\numprint{60024}} \\ \rowcolor{lightergray} \Id{\detokenize{maryland-AM3}} & \numprint{2.07} & \textbf{\numprint{45496}} & \numprint{2.07} & \textbf{\numprint{45496}} & \numprint{2.07} & \textbf{\numprint{45496}} & \numprint{3.34} & \textbf{\numprint{45496}} & \numprint{11.08} & \textbf{\numprint{45496}} \\ \rowcolor{lightergray} \Id{\detokenize{massachusetts-AM2}} & \numprint{0.04} & \textbf{\numprint{140095}} & \numprint{0.04} & \textbf{\numprint{140095}} & \numprint{0.04} & \textbf{\numprint{140095}} & \numprint{0.46} & \textbf{\numprint{140095}} & \numprint{0.48} & \textbf{\numprint{140095}} \\ \Id{\detokenize{massachusetts-AM3}} & \numprint{10.68} & \textbf{\numprint{145866}} & \numprint{8.38} & \textbf{\numprint{145866}} & \numprint{2.92} & \textbf{\numprint{145866}} & \numprint{12.87} & \numprint{145617} & \numprint{23.97} & \numprint{145631} \\ \rowcolor{lightergray} \Id{\detokenize{mexico-AM3}} & \numprint{5.39} & \textbf{\numprint{97663}} & \numprint{5.34} & \textbf{\numprint{97663}} & \numprint{5.28} & \textbf{\numprint{97663}} & \numprint{14.25} & \textbf{\numprint{97663}} & \numprint{289.14} & \textbf{\numprint{97663}} \\ \rowcolor{lightergray} \Id{\detokenize{new-hampshire-AM3}} & \numprint{1.51} & \textbf{\numprint{116060}} & \numprint{1.51} & \textbf{\numprint{116060}} & \numprint{1.50} & \textbf{\numprint{116060}} & \numprint{3.25} & \textbf{\numprint{116060}} & \numprint{8.75} & \textbf{\numprint{116060}} \\ \Id{\detokenize{north-carolina-AM3}} & \numprint{1.76} & \textbf{\numprint{49720}} & \numprint{0.79} & \textbf{\numprint{49720}} & \numprint{0.48} & \textbf{\numprint{49720}} & \numprint{10.45} & \numprint{49562} & \numprint{11.55} & \numprint{49562} \\ \rowcolor{lightergray} \Id{\detokenize{oregon-AM2}} & \numprint{0.04} & \textbf{\numprint{165047}} & \numprint{0.04} & \textbf{\numprint{165047}} & \numprint{0.04} & \textbf{\numprint{165047}} & \numprint{0.04} & \textbf{\numprint{165047}} & \numprint{0.09} & \textbf{\numprint{165047}} \\ \Id{\detokenize{oregon-AM3}} & \numprint{135.72} & \numprint{175073} & \numprint{167.56} & \numprint{175075} & \numprint{5.18} & \textbf{\numprint{175078}} & \numprint{351.99} & \numprint{174334} & \numprint{474.15} & \numprint{164941} \\ \rowcolor{lightergray} \Id{\detokenize{pennsylvania-AM3}} & \numprint{4.35} & \textbf{\numprint{143870}} & \numprint{4.34} & \textbf{\numprint{143870}} & \numprint{4.33} & \textbf{\numprint{143870}} & \numprint{9.98} & \textbf{\numprint{143870}} & \numprint{38.76} & \textbf{\numprint{143870}} \\ \Id{\detokenize{rhode-island-AM2}} & \numprint{1.03} & \textbf{\numprint{184596}} & \numprint{2.40} & \textbf{\numprint{184596}} & \numprint{0.43} & \textbf{\numprint{184596}} & \numprint{10.70} & \numprint{184543} & \numprint{16.79} & \numprint{184543} \\ \Id{\detokenize{rhode-island-AM3}} & \numprint{993.86} & \numprint{201667} & \numprint{255.71} & \numprint{201668} & \numprint{605.61} & \textbf{\numprint{201734}} & \numprint{399.33} & \numprint{162639} & \numprint{931.05} & \numprint{163080} \\ \rowcolor{lightergray} \Id{\detokenize{utah-AM3}} & \numprint{168.07} & \textbf{\numprint{98847}} & \numprint{2.36} & \textbf{\numprint{98847}} & \numprint{2.10} & \textbf{\numprint{98847}} & \numprint{64.04} & \textbf{\numprint{98847}} & \numprint{285.22} & \textbf{\numprint{98847}} \\ \Id{\detokenize{vermont-AM3}} & \numprint{62.85} & \numprint{63280} & \numprint{690.58} & \numprint{63256} & \numprint{2.95} & \textbf{\numprint{63312}} & \numprint{95.81} & \numprint{55584} & \numprint{443.88} & \numprint{55577} \\ \rowcolor{lightergray} \Id{\detokenize{virginia-AM2}} & \numprint{0.25} & \textbf{\numprint{295867}} & \numprint{0.25} & \textbf{\numprint{295867}} & \numprint{0.23} & \textbf{\numprint{295867}} & \numprint{0.93} & \textbf{\numprint{295867}} & \numprint{0.77} & \textbf{\numprint{295867}} \\ \Id{\detokenize{virginia-AM3}} & \numprint{708.66} & \numprint{308052} & \numprint{790.21} & \numprint{308090} & \numprint{19.34} & \textbf{\numprint{308305}} & \numprint{109.20} & \numprint{306985} & \numprint{786.05} & \numprint{233572} \\ \rowcolor{lightergray} \Id{\detokenize{washington-AM2}} & \numprint{0.24} & \textbf{\numprint{305619}} & \numprint{0.24} & \textbf{\numprint{305619}} & \numprint{0.23} & \textbf{\numprint{305619}} & \numprint{2.44} & \textbf{\numprint{305619}} & \numprint{2.20} & \textbf{\numprint{305619}} \\ \Id{\detokenize{washington-AM3}} & \numprint{59.08} & \numprint{314097} & \numprint{505.58} & \numprint{314079} & \numprint{863.47} & \textbf{\numprint{314288}} & \numprint{248.77} & \numprint{271747} & \numprint{532.25} & \numprint{271747} \\ \Id{\detokenize{west-virginia-AM3}} & \numprint{3.06} & \textbf{\numprint{47927}} & \numprint{3.77} & \textbf{\numprint{47927}} & \numprint{2.54} & \textbf{\numprint{47927}} & \numprint{14.38} & \textbf{\numprint{47927}} & \numprint{854.73} & \textbf{\numprint{47927}} \\ \end{tabular}
\caption{Best solution found by each algorithm and time (in seconds) required to compute it. The global best solution is highlighted in \textbf{bold}. Rows are highlighted in gray if B~\&~R is able to find an exact solution. } \label{tab:osm_reduce_rt} \end{table*}
\begin{table*}[ht] \scriptsize \centering \setlength{\tabcolsep}{0.5ex}
\begin{tabular}{l|r r|r r|r r|r r|r r}
& \multicolumn{2}{c|}{Red+DynWVC1} & \multicolumn{2}{c|}{Red+DynWVC2} & \multicolumn{2}{c|}{Red+HILS} & \multicolumn{2}{c|}{B~\&~R\textsubscript{\text{dense}}} & \multicolumn{2}{c}{B~\&~R\textsubscript{\text{full}}} \\ \hline Graph & $t_\text{max}$ & $w_\text{max}$ & $t_\text{max}$ & $w_\text{max}$ & $t_\text{max}$ & $w_\text{max}$ & $t_\text{max}$ & $w_\text{max}$ & $t_\text{max}$ & $w_\text{max}$ \\ \hline \Id{\detokenize{as-skitter}} & \numprint{64.52} & \numprint{123995654} & \numprint{85.60} & \numprint{123995808} & \numprint{845.70} & \textbf{\numprint{123996322}} & \numprint{641.38} & \numprint{123172824} & \numprint{746.93} & \numprint{123904741} \\ \rowcolor{lightergray} \Id{\detokenize{ca-AstroPh}} & \numprint{0.02} & \textbf{\numprint{796556}} & \numprint{0.02} & \textbf{\numprint{796556}} & \numprint{0.02} & \textbf{\numprint{796556}} & \numprint{0.03} & \textbf{\numprint{796556}} & \numprint{0.03} & \textbf{\numprint{796556}} \\ \rowcolor{lightergray} \Id{\detokenize{ca-CondMat}} & \numprint{0.01} & \textbf{\numprint{1143480}} & \numprint{0.01} & \textbf{\numprint{1143480}} & \numprint{0.01} & \textbf{\numprint{1143480}} & \numprint{0.02} & \textbf{\numprint{1143480}} & \numprint{0.02} & \textbf{\numprint{1143480}} \\ \rowcolor{lightergray} \Id{\detokenize{ca-GrQc}} & \numprint{0.00} & \textbf{\numprint{289481}} & \numprint{0.00} & \textbf{\numprint{289481}} & \numprint{0.00} & \textbf{\numprint{289481}} & \numprint{0.00} & \textbf{\numprint{289481}} & \numprint{0.00} & \textbf{\numprint{289481}} \\ \rowcolor{lightergray} \Id{\detokenize{ca-HepPh}} & \numprint{0.02} & \textbf{\numprint{579675}} & \numprint{0.02} & \textbf{\numprint{579675}} & \numprint{0.02} & \textbf{\numprint{579675}} & \numprint{0.02} & \textbf{\numprint{579675}} & \numprint{0.02} & \textbf{\numprint{579675}} \\ \rowcolor{lightergray} \Id{\detokenize{ca-HepTh}} & \numprint{0.00} & \textbf{\numprint{560662}} & \numprint{0.00} & \textbf{\numprint{560662}} & \numprint{0.00} & \textbf{\numprint{560662}} & \numprint{0.01} & \textbf{\numprint{560662}} & \numprint{0.01} & \textbf{\numprint{560662}} \\ \rowcolor{lightergray} \Id{\detokenize{email-Enron}} & \numprint{0.03} & \textbf{\numprint{2457547}} & \numprint{0.03} & \textbf{\numprint{2457547}} & \numprint{0.03} & \textbf{\numprint{2457547}} & \numprint{0.04} & \textbf{\numprint{2457547}} & \numprint{0.03} & \textbf{\numprint{2457547}} \\ \rowcolor{lightergray} \Id{\detokenize{email-EuAll}} & \numprint{0.12} & \textbf{\numprint{25330331}} & \numprint{0.12} & \textbf{\numprint{25330331}} & \numprint{0.12} & \textbf{\numprint{25330331}} & \numprint{0.13} & \textbf{\numprint{25330331}} & \numprint{0.19} & \textbf{\numprint{25330331}} \\ \rowcolor{lightergray} \Id{\detokenize{p2p-Gnutella04}} & \numprint{0.01} & \textbf{\numprint{667539}} & \numprint{0.01} & \textbf{\numprint{667539}} & \numprint{0.01} & \textbf{\numprint{667539}} & \numprint{0.01} & \textbf{\numprint{667539}} & \numprint{0.01} & \textbf{\numprint{667539}} \\ \rowcolor{lightergray} \Id{\detokenize{p2p-Gnutella05}} & \numprint{0.01} & \textbf{\numprint{556559}} & \numprint{0.01} & \textbf{\numprint{556559}} & \numprint{0.01} & \textbf{\numprint{556559}} & \numprint{0.01} & \textbf{\numprint{556559}} & \numprint{0.01} & \textbf{\numprint{556559}} \\ \rowcolor{lightergray} \Id{\detokenize{p2p-Gnutella06}} & \numprint{0.01} & \textbf{\numprint{547591}} & \numprint{0.01} & \textbf{\numprint{547591}} & \numprint{0.01} & \textbf{\numprint{547591}} & \numprint{0.01} & \textbf{\numprint{547591}} & \numprint{0.01} & \textbf{\numprint{547591}} \\ \rowcolor{lightergray} \Id{\detokenize{p2p-Gnutella08}} & \numprint{0.00} & \textbf{\numprint{435893}} & \numprint{0.00} & \textbf{\numprint{435893}} & \numprint{0.00} & \textbf{\numprint{435893}} & \numprint{0.00} & \textbf{\numprint{435893}} & \numprint{0.01} & \textbf{\numprint{435893}} \\ \rowcolor{lightergray} \Id{\detokenize{p2p-Gnutella09}} & \numprint{0.01} & \textbf{\numprint{568472}} & \numprint{0.01} & \textbf{\numprint{568472}} & \numprint{0.01} & \textbf{\numprint{568472}} & \numprint{0.01} & \textbf{\numprint{568472}} & \numprint{0.01} & \textbf{\numprint{568472}} \\ \rowcolor{lightergray} \Id{\detokenize{p2p-Gnutella24}} & \numprint{0.01} & \textbf{\numprint{1970329}} & \numprint{0.01} & \textbf{\numprint{1970329}} & \numprint{0.01} & \textbf{\numprint{1970329}} & \numprint{0.02} & \textbf{\numprint{1970329}} & \numprint{0.02} & \textbf{\numprint{1970329}} \\ \rowcolor{lightergray} \Id{\detokenize{p2p-Gnutella25}} & \numprint{0.01} & \textbf{\numprint{1697310}} & \numprint{0.01} & \textbf{\numprint{1697310}} & \numprint{0.01} & \textbf{\numprint{1697310}} & \numprint{0.01} & \textbf{\numprint{1697310}} & \numprint{0.02} & \textbf{\numprint{1697310}} \\ \rowcolor{lightergray} \Id{\detokenize{p2p-Gnutella30}} & \numprint{0.01} & \textbf{\numprint{2785957}} & \numprint{0.01} & \textbf{\numprint{2785957}} & \numprint{0.01} & \textbf{\numprint{2785957}} & \numprint{0.02} & \textbf{\numprint{2785957}} & \numprint{0.03} & \textbf{\numprint{2785957}} \\ \rowcolor{lightergray} \Id{\detokenize{p2p-Gnutella31}} & \numprint{0.02} & \textbf{\numprint{4750671}} & \numprint{0.02} & \textbf{\numprint{4750671}} & \numprint{0.02} & \textbf{\numprint{4750671}} & \numprint{0.13} & \textbf{\numprint{4750671}} & \numprint{0.04} & \textbf{\numprint{4750671}} \\ \rowcolor{lightergray} \Id{\detokenize{roadNet-CA}} & \numprint{918.32} & \numprint{111398659} & \numprint{866.70} & \numprint{111398243} & \numprint{994.57} & \numprint{111402080} & \numprint{931.36} & \numprint{106500027} & \numprint{774.56} & \textbf{\numprint{111408830}} \\ \rowcolor{lightergray} \Id{\detokenize{roadNet-PA}} & \numprint{733.57} & \numprint{61680822} & \numprint{639.56} & \numprint{61680822} & \numprint{947.93} & \numprint{61682180} & \numprint{988.62} & \numprint{58927755} & \numprint{32.06} & \textbf{\numprint{61686106}} \\ \rowcolor{lightergray} \Id{\detokenize{roadNet-TX}} & \numprint{952.53} & \numprint{78601859} & \numprint{771.05} & \numprint{78601813} & \numprint{946.32} & \numprint{78602984} & \numprint{870.62} & \numprint{75843903} & \numprint{33.49} & \textbf{\numprint{78606965}} \\ \rowcolor{lightergray} \Id{\detokenize{soc-Epinions1}} & \numprint{0.08} & \textbf{\numprint{5668401}} & \numprint{0.08} & \textbf{\numprint{5668401}} & \numprint{0.08} & \textbf{\numprint{5668401}} & \numprint{0.07} & \textbf{\numprint{5668401}} & \numprint{0.11} & \textbf{\numprint{5668401}} \\ \Id{\detokenize{soc-LiveJournal1}} & \numprint{916.65} & \numprint{283973802} & \numprint{996.68} & \numprint{283973997} & \numprint{761.51} & \textbf{\numprint{283975036}} & \numprint{86.66} & \numprint{283869420} & \numprint{270.96} & \numprint{283948671} \\ \rowcolor{lightergray} \Id{\detokenize{soc-Slashdot0811}} & \numprint{0.14} & \textbf{\numprint{5650791}} & \numprint{0.14} & \textbf{\numprint{5650791}} & \numprint{0.14} & \textbf{\numprint{5650791}} & \numprint{0.10} & \textbf{\numprint{5650791}} & \numprint{0.18} & \textbf{\numprint{5650791}} \\ \rowcolor{lightergray} \Id{\detokenize{soc-Slashdot0902}} & \numprint{0.17} & \textbf{\numprint{5953582}} & \numprint{0.17} & \textbf{\numprint{5953582}} & \numprint{0.17} & \textbf{\numprint{5953582}} & \numprint{0.13} & \textbf{\numprint{5953582}} & \numprint{0.21} & \textbf{\numprint{5953582}} \\ \Id{\detokenize{soc-pokec-relationships}} & \numprint{1400.47} & \numprint{43734005} & \numprint{1400.47} & \numprint{43734005} & \numprint{2400.00} & \textbf{\numprint{82845330}} & \numprint{287.40} & \numprint{82595492} & \numprint{1404.57} & \numprint{75717984} \\ \Id{\detokenize{web-BerkStan}} & \numprint{373.58} & \numprint{43877439} & \numprint{612.64} & \numprint{43877349} & \numprint{859.76} & \textbf{\numprint{43877507}} & \numprint{22.58} & \numprint{43138612} & \numprint{831.75} & \numprint{43766431} \\ \rowcolor{lightergray} \Id{\detokenize{web-Google}} & \numprint{3.20} & \numprint{56313343} & \numprint{3.30} & \numprint{56313349} & \numprint{3.01} & \textbf{\numprint{56313384}} & \numprint{2.08} & \textbf{\numprint{56313384}} & \numprint{3.16} & \textbf{\numprint{56313384}} \\ \Id{\detokenize{web-NotreDame}} & \numprint{147.60} & \numprint{25995575} & \numprint{850.00} & \numprint{25995615} & \numprint{173.50} & \textbf{\numprint{25995648}} & \numprint{354.79} & \numprint{25947936} & \numprint{28.11} & \numprint{25957800} \\ \Id{\detokenize{web-Stanford}} & \numprint{5.08} & \numprint{17799379} & \numprint{5.19} & \numprint{17799405} & \numprint{131.24} & \textbf{\numprint{17799556}} & \numprint{47.62} & \numprint{17634819} & \numprint{4.69} & \numprint{17799469} \\ \rowcolor{lightergray} \Id{\detokenize{wiki-Talk}} & \numprint{2.30} & \textbf{\numprint{235875181}} & \numprint{2.30} & \textbf{\numprint{235875181}} & \numprint{2.30} & \textbf{\numprint{235875181}} & \numprint{3.85} & \textbf{\numprint{235875181}} & \numprint{3.36} & \textbf{\numprint{235875181}} \\ \rowcolor{lightergray} \Id{\detokenize{wiki-Vote}} & \numprint{0.04} & \textbf{\numprint{500436}} & \numprint{0.04} & \textbf{\numprint{500436}} & \numprint{0.03} & \textbf{\numprint{500436}} & \numprint{0.05} & \textbf{\numprint{500436}} & \numprint{0.06} & \textbf{\numprint{500436}} \\ \end{tabular}
\caption{Best solution found by each algorithm and time (in seconds) required to compute it. The global best solution is highlighted in \textbf{bold}. Rows are highlighted in gray if B~\&~R is able to find an exact solution. } \label{tab:snap_reduce_rt} \end{table*}
\begin{figure*}
\caption{Solution quality over time for our sample of eight OSM instances (upper two rows) and eight SNAP instances (lower two rows) as given in Section~\ref{sec:sota}.}
\label{fig:apdxconvergence}
\end{figure*}
\end{appendix}
\end{document} | arXiv |
\begin{document}
\title{\bf{Single-world theory of the extended Wigner's friend experiment}}
\author{Anthony Sudbery$^1$\centerdot[10pt] \small Department of Mathematics, University of York, \centerdot[-2pt] \small Heslington, York, England YO10 5DD\centerdot \small $^1$ [email protected]}
\date{14 November 2016}
\maketitle
\begin{abstract}
Frauchiger and Renner have recently claimed to prove that ``Single-world interpretations of quantum theory cannot be self-consistent". This is contradicted by a construction due to Bell, inspired by Bohmian mechanics, which shows that any quantum system can be modelled in such a way that there is only one ``world" at any time, but the predictions of quantum theory are reproduced. This Bell-Bohmian theory is applied to the experiment proposed by Frauchiger and Renner, and their argument is critically examined. It is concluded that it is their version of ``standard quantum theory", incorporating state vector collapse upon measurement, that is not self-consistent.
\end{abstract}
\section{Introduction}
In 1984 John Bell \cite{Bell:beables} proposed an interpretation of quantum field theory in which certain field variables always have definite values. This can be generalised to any quantum system \cite{QMPN}, giving a theory in which any chosen set of commuting observables --- the \emph{beables} of the theory --- always have definite values, and yet the results of measurements are always distributed as predicted by quantum mechanics. Bell's theory was an extension of Bohmian quantum mechanics.
Recently Frauchiger and Renner have declared that this is impossible \cite{FrauchigerRenner}. They describe an experiment, the ``extended Wigner's friend experiment", in which, they claim, the predictions of quantum mechanics and the assumption that each measurement in the experiment has a unique result, together lead to a contradiction. In this paper we examine the Bell-Bohmian description of the extended Wigner's friend experiment in an attempt to identify the source of this contradiction.
The contents of the paper are as follows. Section 2 is an outline of the Bell-Bohmian theory. Section 3 contains a description of the extended Wigner's friend experiment and a summary of the argument of Frauchiger and Renner. In Section 4 we analyse the experiment in terms of Bell-Bohmian theory and show how it avoids the contradiction found by Frauchiger and Renner. Section 5 contains discussion, leading to the conclusion that the source of the contradiction is the use of the projection postulate for measurements by different agents.
\section{Bell-Bohmian theory}\label{BellBohm}
This interpretation was inspired by Bohm's interpretation of non-relativistic many-particle quantum mechanics (see e.g. \cite{Bohmian} p. 145), according to which particles always have definite positions. The motion of the particles is governed deterministically by the wave function, which thus has the role of a force acting on the system rather than a description of the state of the system. To emphasise this role, Bell \cite{Bell:pilot} calls the wave function a ``pilot wave". This evolves according to the time-dependent Schr\"odinger equation.
In Bell's generalisation of this interpretation (\cite{Bell:beables}, \cite{QMPN} p. 215), the many-particle system can be replaced by any quantum system S, with states described by vectors in a Hilbert space $\mathcal{S}$, and the positions of the particles replaced by any set of commuting variables, which are known as \emph{beables}. These are taken to have definite values, so the actual real state of the system is described by a state vector in one of the simultaneous eigenspaces $\centerdot_i$ of the beables (which are also known \cite{QMPN} as \emph{viable} subspaces). The evolution of this state is governed by another time-dependent vector, the \emph{pilot vector} $|\Psi\centerdot\in \centerdot$, which satisfies the time-dependent Schr\"odinger equation with the Hamiltonian $H$ determined by the physics of the system. This pilot vector can be decomposed into its components in the viable subspaces $\centerdot_i$: \centerdot
|\Psi(t)\centerdot = \sum_i |\psi_i(t)\centerdot \qquad \text{with} \quad |\psi_i(t)\centerdot \in \centerdot_i, \centerdot
and the real state at time $t$ is taken to be one of the components $|\psi_i(t)\centerdot$.
The real state changes in time, not deterministically as in the original Bohmian mechanics, but stochastically: it makes transitions between the preferred subspaces $S_i$ with transition probabilities given by
{\bf Bell's postulate}: The real state of the system is one of the components of the pilot state vector in one of the viable subspaces $\centerdot_i$. If, at time $t$, the real state is the component $|\psi_i(t)\centerdot\in\centerdot_i$, then the probability that at time $t + \delta t$ the real state is $|\psi_j(t + \delta t)\centerdot\in\centerdot_j$ is $w_{ij}\delta t$ where the transition probability $w_{ij}$ is given by \begin{equation}\label{Bell}
w_{ij} = \begin{cases} \frac{2\text{Re}[(i\hbar)^{-1}\langle\psi_j(t)|H|\psi_i(t)\centerdot]}{\langle\psi_i(t)|\psi_i(t)\centerdot} &\text{if this is } \ge 0\centerdot
0 &\text{if it is negative}
\end{cases}
\end{equation} It follows from this \cite{QMPN}\footnote{The proof in \cite{QMPN} refers to a slightly different, and less satisfactory, form of Bell's postulate, but it is easily adapted so as to apply to the form given here.} that the probability $p_i(t) $ that the real state of the system at time $t$ is $|\psi_i(t)\centerdot$ is given by the Born rule ($p_i(t) = \langle\psi_i(t)|\psi_i(t)\centerdot$) at all positive times $t$, if the probabilities are so given at the initial time $t = 0$.
This framework can be generalised still further \cite{BacciaDickson, verdammte} to allow for the possibility that the viable subspaces $\centerdot_i$ vary with time; it then includes the modal interpretation of quantum mechanics.
Although this theory is indeterministic, it can be shown \cite{determlimit, Vink} that Bohm's deterministic theory can be obtained as a continuum limit of Bell's original theory of the above form, in which he took the points of space to be a discrete lattice.
\section{The extended Wigner's friend experiment}
This section contains a description of the experiment designed by Frauchiger and Renner \cite{FrauchigerRenner} to demonstrate that any theory which is compliant with quantum theory and describes a single world cannot be self-consistent. After describing the experiment, we will outline the argument of Frauchiger and Renner for this conclusion.
The experiment contains two experimenters $F_1$ and $F_2$ (Wigner's friends), who perform experiments on two two-state quantum systems, a coin $C$ with orthonormal basis states $|\text{head}\centerdot_C$ and $|\text{tail}\centerdot_C$, and an electron $S$ with spin states $|\uparrow\centerdot_S$ and $|\downarrow\centerdot_S$; it also contains Wigner $W$ and his assistant $A$, who can perform measurements on $F_1$ and $F_2$ as well as the coin $C$ and the electron spin $S$. Irrelevant degrees of freedom of the four experimenters are suppressed, so each of them is regarded as having just two independent states, which record the results of their measurements.
Before the experiment starts the coin is prepared in the state
$\sqrt{\third}|\text{head}\centerdot + \sqrt{\tfrac{2}{3}}|\text{tail}\centerdot$.
At time $t = 0$ experimenter $F_1$ observes the coin and records the result $r =$ ``head" or ``tail", thereby being put into a memory state $|r\centerdot_{F_1}$.
At time $t = 1$, $F_1$ prepares the electron as follows: if the result of the measurement at $t = 0$ was $r =$ ``head", $F_1$ prepares the electron in spin state $|\downarrow\centerdot_S$; if $r = $ ``tail", they prepare it in spin state $|\rightarrow\centerdot_S = \tfrac{1}{\sqrt{2}}\big(|\uparrow\centerdot_S + |\downarrow\centerdot_S$.
At time $t = 2$ experimenter $F_2$ measures the spin $z\half\hbar$ of the electron $(z = \pm)$ in the basis $\centerdot|\uparrow\centerdot, |\downarrow\centerdot\centerdot$ and records the result, thereby being put into a memory state $|z\centerdot_{F_2}$.
At time $t = 3$ Wigner's assistant $A$ measures $F_1$, together with the coin, in the basis \begin{align*}
|\text{ok}\centerdot_{F_1C} &= \tfrac{1}{\sqrt{2}}\big(|\text{head}\centerdot_{F_1}|\text{head}\centerdot_C - |\text{tail}\centerdot_{F_1}|\text{tail}\centerdot_C\big)\centerdot
|\text{fail}\centerdot_{F_1C} &= \tfrac{1}{\sqrt{2}}\big(|\text{head}\centerdot_{F_1}|\text{head}\centerdot_C + |\text{tail}\centerdot_{F_1}|\text{tail}\centerdot_C\big), \end{align*} and records the result $x =$ ``ok" or ``fail".
At time $t = 4$ Wigner measures $F_2$, together with the electron, in the basis \begin{align*}
|\text{ok}\centerdot_{F_2S} &= \tfrac{1}{\sqrt{2}}\big(|-\centerdot_{F_2}|\downarrow\centerdot_S - |+\centerdot_{F_2}|\uparrow\centerdot_S\big)\centerdot
|\text{fail}\centerdot_{F_2S} &= \tfrac{1}{\sqrt{2}}\big(|-\centerdot_{F_2}|\downarrow\centerdot_S + |+\centerdot_{F_1}|\uparrow\centerdot_S\big), \end{align*} and records the result $w =$ ``ok" or ``fail".
At the end of the experiment Wigner and his assistant compare the results of their measurements. They repeat the experiment again and again, stopping when they find $x = w = ``\text{ok}"$. The question is whether it is possible for the procedure to stop.
Frauchiger and Renner argue as follows. Let us assume that the experiment is described by a theory $T$ with the following three properties:
{\bf QT} \emph{Compliance with quantum theory}: $T$ forbids all experimental results that are forbidden by standard quantum theory.
{\bf SW} \emph{Single world}: $T$ rules out the occurrence of more than one single outcome if an experimenter measures a system once.
{\bf SC} \emph{Self-consistency}: $T$'s statements about measurement outcomes are logically consistent (even if they are obtained by considering the perspectives of different experimenters).\footnote{This is the formulation of Frauchiger and Renner. Elsewhere they state that this property ``demands that the laws of a theory $T$ do not contradict each other". These are not the same. If the laws of a theory $T$ contradicted each other, then $T$ simply would not exist as a theory. But as stated here, {\bf SC} is not a very interesting requirement: there is no logical reason why statements existing in different perspectives should be consistent (think of statements about the order of events in different frames of reference, in special relativity). However, we show in this paper that even in this form there is no contradiction between {\bf QT}, {\bf SW} and {\bf SC}.}
Then we have the following implications:
{\bf 1.} Suppose that $F_1$, in the measurement at $t = 1$, gets the result $r =$ ``tail". Then $F_1$ prepares the electron spin $S$ in the state $|\rightarrow\centerdot_S$. When $F_2$ measures $S$ at $t = 2$, $F_2$ and $S$ are put into the entangled state $|\text{fail}\centerdot_{F_2S}$. This is not affected by $A$'s measurement of $F_1C$ at $t = 3$, so $W$, on measuring $F_1C$ at $t = 4$, will, by {\bf QT}, get the result $w =$ ``fail". Thus \begin{equation}\label{r implies w} r(1) = \text{tail}\centerdot \Longrightarrow \centerdot w(4) = \text{fail} \end{equation} (this allows for the possibility that the values of $r$ and $w$ might vary with time). Since, by {\bf SW}, the value of $r(1)$ must be either ``head" or ``tail", and the value of $w(4)$ must be either ``ok" or ``fail", it follows that \begin{equation}\label{w implies r} w(4) = \text{ok} \centerdot \Longrightarrow \centerdot r(1) = \text{head}. \end{equation}
{\bf 2.} Suppose, on the other hand, that $F_1$ gets the result $r =$ ``head" at $t = 1$. Then the state of the electron spin after this measurement must be $|\downarrow\centerdot$. Hence, by {\bf QT}, $F_2$, in the measurement at $t = 2$, must get the result $z = -$. Thus \begin{equation}\label{r implies z} r(1) = \text{head} \centerdot \Longrightarrow \centerdot z(2) = -. \end{equation}
{\bf 3.} Now consider $F_2$'s measurement of $z$ at $t = 2$. After $F_1$'s preparation of the electron spin, the state of $F_1$, the coin and the electron is \centerdot
\sqrt{\third}|\text{head}\centerdot_{F_1C}|\downarrow\centerdot_S + \sqrt{\tfrac{2}{3}}|\text{tail}\centerdot_{F_1C}|\rightarrow\centerdot_S = \sqrt{\third}|\text{tail}\centerdot_{F_1C}|\uparrow\centerdot_S + \sqrt{\tfrac{2}{3}}|\text{fail}\centerdot_{F_1C}|\downarrow\centerdot_S. \centerdot Hence if the result of $F_2$'s measurement of $S$ is $z = -$, then the result of $A$'s measurement of $F_1C$ at $t = 3$ must be $x =$ ``fail": \begin{equation}\label{z implies x} z(2) = - \centerdot \Longrightarrow \centerdot x(3) = \text{fail}. \end{equation}
{\bf 4.} After $F_2$'s measurement of the electron spin, the state of $F_1$ and $F_2$ (and their laboratories) is \begin{multline*}
\sqrt{\third}|\text{tail}\centerdot_{F_1}|\text{tail}\centerdot_C|+\centerdot_{F_2}|\uparrow\centerdot_S + \sqrt{\tfrac{2}{3}}|\text{fail}\centerdot_{F_1C}|-\centerdot_{F_2}|\downarrow\centerdot_S\centerdot
= \tfrac{1}{2\sqrt{3}}\big(|\text{ok}\centerdot_{F_1C}|\text{ok}\centerdot_{F_2S} - |\text{ok}\centerdot_{F_1C}|\text{fail}\centerdot_{F_2S} + |\text{fail}\centerdot|\text{ok}\centerdot_{F_2S}\big) + \tfrac{\sqrt{3}}{2}|\text{fail}\centerdot_{F_1C}|\text{fail}\centerdot_{F_2S}. \end{multline*}
This has non-zero coefficient of $|\text{ok}\centerdot_{F_1C}|\text{ok}\centerdot_{F_2S}$, so \centerdot x(4) = w(4) = \text{ok} \text{ is possible.} \centerdot But $W$'s measurement of $F_2S$ does not affect the state of $A$, so $x(4) = x(3)$. Thus in the measurements of $A$ and $W$ at $t = 3$ and $4$, \begin{equation}\label{x and w} x(3) = w(4) = \text{ok} \text{ is possible.} \end{equation}
Now we have \begin{align*} w(4) = \text{ok} \centerdot &\Longrightarrow \centerdot r(1) = \text{head} \qquad \text{by \eqref{w implies r}}\centerdot
&\Longrightarrow \centerdot z(2) = - \qquad\qquad \text{by \eqref{r implies z}}\centerdot
&\Longrightarrow \centerdot x(3) = \text{fail} \quad \text{by \eqref{z implies x}} \end{align*} which contradicts \eqref{x and w}. Frauchiger and Renner conclude that no theory can have all three properties {\bf SW}, {\bf QT} and $\bf SC$. \section{Bell-Bohmian theory of the experiment}\label{Bell-Bohm}
Bell-Bohmian theory assumes a pilot vector in the Hilbert space of the whole experiment, evolving purely according to the unitary operator describing the dynamics (i.e. with no application of the projection postulate after measurements). In this it resembles Everettian quantum mechanics, but the metaphysical interpretation is different, as described in \secref{BellBohm}. The Hilbert space in question is \centerdot \centerdot_{F_1}\otimes\centerdot_{F_2}\otimes\centerdot_A\otimes\centerdot_W\otimes\centerdot_C\otimes\centerdot_S \centerdot
where $\centerdot_C$ and $\centerdot_S$ are two-dimensional, with orthonormal bases $\centerdot|\text{head}\centerdot,|\text{tail}\centerdot\centerdot$ and $\centerdot|\uparrow\centerdot, |\downarrow\centerdot\centerdot$ respectively; and $\centerdot_{F_1}, \centerdot_{F_2}, \centerdot_A$ and $\centerdot_W$ are all 3-dimensional, with bases labelled by $r, z, x$ and $w$, each taking the two values described in Section 2 and also a third value $0$ to describe the ``ready" state of the observer before making any measurement. We take $r, z, x$ and $w$ to be the beables of the system, which always have definite values. Thus the real state vector of the system always lies in one of the 81 viable subspaces \centerdot
|r\centerdot_{F_1}|z\centerdot_{F_2}|x\centerdot_A|w\centerdot_W\otimes\centerdot_C\otimes\centerdot_S. \centerdot and is one of the projections of the pilot vector onto these subspaces.
In order to analyse the experiment, we need to be more precise about the way in which $F_1$ prepares the spin state after the coin toss at $t = 0$. I will assume that before the coin toss, the electron spin is prepared in some known initial state $|0\centerdot_S\in\centerdot_S$; after the coin toss, $F_1$ applies to the electron either a unitary operator which takes $|0\centerdot$ to $|\downarrow\centerdot$ or one which takes $|0\centerdot$ to $|\rightarrow\centerdot$, according to the result of the toss. Then the real state vector before the experiment starts is the same as the pilot state, namely \centerdot
|0\centerdot_{F_1}|0\centerdot_{F_2}|0\centerdot_A|0\centerdot_W\centerdot\sqrt{\third}|\text{head}\centerdot_C + \sqrt{\tfrac{2}{3}}|\text{tail}\centerdot_C\centerdot|0\centerdot_S. \centerdot
At $t=0$, after $F_1$'s measurement of the coin, the pilot vector becomes \centerdot
|\Psi(0)\centerdot = \left(\sqrt{\third}|\text{head}\centerdot_{F_1C} + \sqrt{\tfrac{2}{3}}|\text{tail}\centerdot_{F_1C}\right)|0\centerdot_S|0\centerdot_{F_2}|0\centerdot_A|0\centerdot_W, \centerdot
where $|\text{head}\centerdot_{F_1C} = |\text{head}\centerdot_{F_1}|\text{head}\centerdot_C$ and similarly for ``tail", but the real state vector is one of the two summands in this. We will consider \centerdot
|\Phi(0)\centerdot = |\text{tail}\centerdot_{F_1C}|0\centerdot_S|0\centerdot_{F_2}|0\centerdot_A|0\centerdot_W. \centerdot At $t =1$, after $F_1$ has prepared the electron spin, the pilot state is \centerdot
|\Psi(1)\centerdot = \left(\sqrt{\third}|\text{head}\centerdot_{F_1C}|\downarrow\centerdot_S + \sqrt{\tfrac{2}{3}}|\text{tail}\centerdot_{F_1C}|\rightarrow\centerdot_S\right)|0\centerdot_{F_2}|0\centerdot_A|0\centerdot_W \centerdot
The real state is one of the two summands in $|\Psi(1)\centerdot$; we take \centerdot
|\Phi(1) = \sqrt{\tfrac{2}{3}}|\text{tail}\centerdot_{F_1C}|\rightarrow\centerdot_S|0\centerdot_{F_2}|0\centerdot_A|0\centerdot_W. \centerdot After $F_2$'s measurement of $S$ at $t = 2$, the pilot state becomes \centerdot
|\Psi(2)\centerdot = \sqrt{\tfrac{1}{3}}\bigg(|\text{head}\centerdot_{F_1C}|-\centerdot_{F_2S} + |\text{tail}\centerdot_{F_1C}|+\centerdot_{F_2S} +
|\text{tail}\centerdot_{F_1C}|-\centerdot_{F_2S}\bigg)|0\centerdot_A|0\centerdot_W, \centerdot which has three components with definite values of $r, z, x$ and $w$ (\emph{viable} components), one of which is \centerdot
|\Phi(2)\centerdot = \sqrt{\tfrac{1}{3}}|\text{tail}\centerdot_{F_1C}|+\centerdot_{F_2S}|0\centerdot_A|0\centerdot_W. \centerdot After $A$'s measurement of $F_1$ and $C$ at $t = 3$, the pilot vector becomes \begin{multline*}
|\Psi(3)\centerdot = \bigg(\sqrt{\tfrac{1}{6}}\Big( - |\text{ok}\centerdot_{F_1C}|\text{ok}\centerdot_A + |\text{fail}\centerdot_{F_1C}|\text{fail}\centerdot_A\Big)|+\centerdot_{F_2S}\centerdot\centerdot
+ \sqrt{\tfrac{2}{3}}|\text{fail}\centerdot_{F_1C}|\text{fail}\centerdot_A|-\centerdot_{F_2S}\bigg)|0\centerdot_W \end{multline*} which has six viable components, one of which is \centerdot
|\Phi(3)\centerdot = \sqrt{\tfrac{1}{12}}|\text{tail}\centerdot_{F_1C}|+\centerdot_{F_2S}|\text{ok}\centerdot_A|0\centerdot_W. \centerdot After $W$'s measurement of $F_2$ and $S$ at $t = 4$, the pilot vector becomes \begin{multline*}
|\Psi(4)\centerdot = \sqrt{\tfrac{1}{12}}\Big(|\text{ok}\centerdot_{F_1C}|\text{ok}\centerdot_A + |\text{fail}\centerdot_{F_1C}|\text{fail}\centerdot_A\Big)|\text{ok}\centerdot_{F_2S}|\text{ok}\centerdot_W\centerdot
+ \sqrt{\tfrac{1}{12}}\Big(- |\text{ok}\centerdot_{F_1C}|\text{ok}\centerdot_A + 3|\text{fail}\centerdot_{F_1C}|\text{fail}\centerdot_A\Big)|\text{fail}\centerdot_{F_2S}|\text{fail}\centerdot_W \end{multline*} which has sixteen viable components, one of which is \centerdot
|\Phi(4)\centerdot = -\sqrt{\tfrac{1}{24}}|\text{tail}\centerdot_{F_1C}|-\centerdot_{F_2S}|\text{ok}\centerdot_A|\text{ok}\centerdot_W. \centerdot
According to Bell-Bohmian theory, at all times Wigner, his assistant and his two friends are in a single world with definite values of $r, z, x$ and $w$, the results of their measurements. But Frauchiger and Renner argue that this leads to the contradictory implications \eqref{r implies w}, \eqref{r implies z}, \eqref{z implies x} and \eqref{x and w}. We will show, on the contrary, that in Bell-Bohmian theory it is possible that the real state undergoes the transitions \centerdot
|\Phi(0)\centerdot \longrightarrow |\Phi(1)\centerdot \longrightarrow |\Phi(2)\centerdot \longrightarrow |\Phi(3)\centerdot \longrightarrow\centerdot |\Phi(4)\centerdot. \centerdot It follows that in this theory the implication \eqref{r implies w} ($r(1) = \text{tail}\centerdot \Longrightarrow \centerdot w(4) = \text{fail}$) does not hold: it is possible for $F_1$ to get the result $r = $ ``tail" (and, incidentally, to remain in a state registering this result) while $W$ gets the result $w = $``ok".
To establish this, we will need to see what transitions between viable states are allowed by Bell's postulate, and for this we need a model of the processes by which the measurements are made. The following is a general theory of such a process. We consider an experimenter $E$ measuring an observable $X$ on a system $S$, whose basis of eigenstates of $X$ is $\centerdot|1\centerdot_S,|2\centerdot_S\centerdot$, and suppose that the process takes place as follows. The relevant states of the experimenter are taken to be $|0\centerdot_E,|1\centerdot_E,|2\centerdot_E$, where $|0\centerdot_E$ is the state of the experimenter before the measurement, and $|1\centerdot_E$ and $|2\centerdot_E$ are the states of the experimenter registering the results $X = 1$ and $X = 2$. In the course of the measurement the joint state $|1\centerdot_S|0\centerdot_E$ evolves to $|1\centerdot_S|1\centerdot_E$ and the joint state $|2\centerdot_S|0\centerdot_E$ evolves to $|2\centerdot_S|2\centerdot_E$. We assume that each of these evolutions is a simple rotation in the joint state space $\centerdot_E\otimes\centerdot_S$, lasting for a time $\tau$: \centerdot
|k\centerdot_S|0\centerdot_E \centerdot \longrightarrow \centerdot |\Psi_k(t)\centerdot = \cos\lambda t|k\centerdot_S|0\centerdot_E + \sin\lambda t|k\centerdot_S|k\centerdot_E \centerdot $(k = 1,2; \centerdot 0\le t\le \tau)$ where $\lambda = \pi/2\tau$. At times outside the interval $[0,\tau]$, the joint state of the system and the experimenter is assumed to be stationary (with zero energy). This time development is produced by the Hamiltonian \centerdot
H = i\hbar\lambda\bigg(|1\centerdot\<1|_S\otimes\big[|1\centerdot\<0| - |0\centerdot\<1|\big]_E +
|2\centerdot\<2|_S\otimes\big[|2\centerdot\<0| - |0\centerdot\<2|\big]_E\bigg), \centerdot which is switched on at $t = 0$ and off at $t = \tau$.
Suppose the system has just one beable $M$, the observation of the experimenter, with values $(0,1,2)$, and suppose the initial state of the joint system is $\big(a|1\centerdot_S + b|2\centerdot_S\big)|0\centerdot_E$. This has the definite value 0 for the beable $M$, so it is both the real state vector for the joint system and the pilot vector at $t = 0$. Then in the time interval $[0,\tau]$ during which the measurement is proceeding, the pilot state is \begin{align*}
|\Psi(t)\centerdot &= a|\Psi_1(t)\centerdot + b|\Psi_2(t)\centerdot\centerdot
&= \cos\lambda t\big(a|1\centerdot+ b|2\centerdot\big)_S|0\centerdot_E + \sin\lambda t\big(a|1\centerdot_S|1\centerdot_E + b|2\centerdot_S|1\centerdot_E\big) \end{align*}
and the real state of the joint system at any time in this interval is one of the three states $|\Psi(0)\centerdot = (a|1\centerdot_S + b|2\centerdot_S)|0\centerdot_E$, $|1\centerdot_S|1\centerdot_E$ or $|2\centerdot_S|2\centerdot_E$. It can make a transition from $|\Psi(0)\centerdot$ to $|1\centerdot_S|1\centerdot_E$ or to $|2\centerdot_S|2\centerdot_E$ because the (real) matrix elements $(i\hbar)^{-1}\big(\<k|_S\<k|_E\big) H \big(|k\centerdot_S|0\centerdot_E\big)$ ($k = 1,2$) are both positive. It cannot make the reverse transitions because the
matrix elements $(i\hbar)^{-1}\big(\<k|_S\<0|_E\big) H \big(|k\centerdot_S|k\centerdot_E\big)$ are negative, and it cannot make transitions between $|1\centerdot_S|1\centerdot_E$ and $|2\centerdot_S|2\centerdot_E$ because the relevant matrix elements of $H$ are zero. Thus at time $t = 0$ the real state vector and the pilot vector coincide; between $t = 0$ and $t = \tau$ the pilot vector $|\Psi(t)\centerdot$ changes smoothly but the real state vector remains at its initial value $|k\centerdot_S|0\centerdot_E$ until some undetermined intermediate time at which it changes discontinuously to either $|1\centerdot_S|1\centerdot_E$ or $|2\centerdot_S|2\centerdot_E$ and remains at that value until $t = \tau$. A calculation of the final probabilities from the transition probabilities as given by Bell yields the expected values $|a|^2$ and $|b|^2$.
To examine the implication \eqref{r implies w}, we will apply this theory to the measurements in the extended Wigner's friend experiment. We will assume that each of the measurements has duration $\tau < 1$ before the time assigned to it (e.g.\ A's measurement ``at time $t = 3$" occupies the interval $[3 - \tau, 3])$, and that each measurement consists of a simple rotation as described above.
If the result of $F_1$'s measurement at $t = 0$ is $r = $``tail", then the component of $|\Psi(0)\centerdot$ describing the actual world must be $|\Phi(0)\centerdot$.
The pilot vector is still $|\Psi(0)\centerdot$. $F_1$'s preparation of the electron spin at $t=1$ is accomplished by a unitary operator acting only on $F_1$ and $S$, such that there are no matrix elements of the Hamiltonian between states with different values of the beables $r,x,z,w$; therefore the real state at $t=1$ is $|\Phi(1)\centerdot$. The next measurement, by $F_2$ at $t = 2$, is driven by the Hamiltonian $\mathbf{1}_{F_1C}\otimes (H_2)_{F_2S}\otimes\mathbf{1}_A\otimes \mathbf{1}_W$ where \begin{multline}
H_2 = i\hbar\lambda\Big(|-\centerdot_{F_2S}\big(\langle\downarrow|_S\<0|_{F_2}\big) - \big(|\downarrow\centerdot_S|0\centerdot_{F_2}\big)\langle-|_{F_2S}\centerdot
+ |+\centerdot_{F_2S}\big(\langle\uparrow|_S\<0|_{F_2}\big) - \big(|\uparrow\centerdot_S|0\centerdot_{F_2}\big)\langle+|_{F_2S}\Big). \end{multline}
The pilot state during the measurement is $\cos\lambda t|\Psi(1)\centerdot + \sin\lambda t|\Psi(2)\centerdot$; the real state must therefore be one of the viable components of $|\Psi(1)\centerdot$ or $|\Psi(2)\centerdot$. Since this Hamiltonian has no matrix elements betweeen states containing $|\text{head}\centerdot_{F_1C}$ and states containing $|\text{tail}\centerdot_{F_1C}$, the only possible transitions from $|\Phi(1)\centerdot$ are to the second or third term in $|\Psi(2)\centerdot$, followed by transitions back to $|\Phi(1)\centerdot$ or to other components of $|\Psi(2)\centerdot$. But the Hamiltonian also has no matrix elements between different viable components of $|\Psi(2)\centerdot$, and the only positive matrix elements of $H/i\hbar$ are those corresponding to transitions in the forward direction, so once a transition has been made to one of the three terms in $|\Psi(2)\centerdot$, there will be no further transitions during this measurement. Thus if the real state after $F_1$'s measurement has $r = $ ``tail", this will still be the case after $F_2$'s measurement and the real state will be the second or third term of $|\Psi(2)\centerdot$, and both of these are possible. Thus there is a non-zero probability that the real state evolves as $|\Psi(0)\centerdot \rightarrow |\Phi(1)\centerdot \rightarrow |\Phi(2)\centerdot$.
$A$'s measurement of $F_1$ and $C$ at $t = 3$ is driven by the Hamiltonian
$H_3\otimes\mathbf{1}_{F_2S}\otimes\mathbf{1}_W$ where $H_3$, acting in $\centerdot_{F_1C}\otimes\centerdot_A$, rotates $|\text{fail}\centerdot_{F_1C}|0\centerdot_A$ to $|\text{fail}\centerdot_{F_1C}|\text{fail}\centerdot_A$ and $|\text{ok}\centerdot_{F_1C}|0\centerdot_A$ to $|\text{ok}\centerdot_{F_1C}|\text{ok}\centerdot_A$. In terms of the viable states, this is \begin{align*}
H_3 &= \half i\hbar\lambda\big(|\text{head}\centerdot + |\text{tail}\centerdot\big)\big(\langle\text{head}| + \langle\text{tail}|\big)_{F_1C}\otimes\big(|\text{fail}\centerdot\<0| - |0\centerdot\langle\text{fail}|\big)_A\centerdot
&+ \half i\hbar\lambda\big(|\text{head}\centerdot - |\text{tail}\centerdot\big)\big(\langle\text{head}| - \langle\text{tail}|\big)_{F_1C}\otimes\big(|\text{ok}\centerdot\<0| - |0\centerdot\langle\text{ok}|\big)_A. \end{align*} This Hamiltonian $H$ has \centerdot
\langle\Phi(3)|\frac{H}{i\hbar}|\Phi(2)\centerdot > 0, \centerdot
and there are no positive matrix elements $\langle\phi|\tfrac{H}{i\hbar}|\Phi(3)\centerdot$ for viable states $|\phi\centerdot$, so the transition $|\Phi(2)\centerdot \rightarrow |\Phi(3)\centerdot$ is possible, and if it occurs the system remains in the state $|\Phi(3)\centerdot$ until the next measurement.
$W$'s measurement of $F_2$ and $S$ at $t = 4$ is driven by the Hamiltonian $\mathbf{1}_{F_1C}\otimes\mathbf{1}_A\otimes H_4$ where $H_4$ is the following operator on $\centerdot_{F_2S}\otimes\centerdot_W$: \begin{align*}
H_4 = i&\hbar\lambda|\text{ok}\centerdot\langle\text{ok}|_{F_2S}\big(|\text{ok}\centerdot\<0|-|0\centerdot\langle\text{ok}|\big)_W\centerdot
+i&\hbar\lambda|\text{fail}\centerdot\langle\text{fail}|_{F_2S}\big(\text{fail}\centerdot\<0| - |0\centerdot\langle\text{fail}|\big)_W. \end{align*} This has \centerdot
\langle\Phi(3)|\frac{H}{i\hbar}|\Phi(4)\centerdot > 0, \centerdot
and there are no positive matrix elements $\langle\phi|\tfrac{H}{i\hbar}|\Phi(4)\centerdot$ for viable states $|\phi\centerdot$, so the transition $|\Phi(3)\centerdot \rightarrow |\Phi(4)\centerdot$ is possible during $W$'s measurement, and if it occurs the system remains in the state $|\Phi(4)\centerdot$.
Thus it is possible that $W$ and $A$ both get the result ``ok" for their measurements, and this happens even though $F_1$ records the result $r =$ ``tail". This contradicts the theorem of Frauchiger and Renner.
\section{Discussion}
The purpose of this paper has been to show that there is a counter-example to the theorem that Frauchiger and Renner claim to prove. There is a theory which is self-consistent, in which any experiment has only one result, and which reproduces the predictions of quantum mechanics. It is not the purpose of the paper to advocate this theory as a true description of the experiment, but simply to show that it exists. This disproves the theorem. But what is wrong with Frauchiger and Renner's proof?
Let us examine the implication \eqref{r implies w}: if the result of $F_1$'s measurement at $t = 1$ is $r = $ ``tail", then $F_1$ acts on this information and calculates the future development of the whole system by means of the Schr\"odinger equation, with the measurement result ``tail" as initial condition. This is to follow the instructions of the quantum mechanics textbooks, so Frauchiger and Renner describe it as ``compliance with quantum theory". It incorporates a collapse of the state vector on measurement, otherwise known as the collapse postulate. In Bell-Bohmian theory, on the other hand, although the result of measurement determines the real state, the Schr\"odinger equation is applied with a different initial condition, namely the pilot vector. This includes a term corresponding to the result of measurement which did not actually occur.
Naturally, these two procedures give different results. They are both presented as ``compliant with quantum theory", but this cannot be true if ``quantum theory" has a well-defined meaning. This does not seem to be so. The contradiction between the Frauchiger-Renner claim that ``It is impossible for any theory to obey ({\bf QT}), ({\bf SW}) and ({\bf SW})" and the claim of this paper that ``Bell-Bohmian theory obeys ({\bf QT}), ({\bf SW}) and ({\bf SW})" is due to different meanings of ({\bf QT}) in the two claims.
The version of quantum theory assumed by Frauchiger and Renner \ seems appropriate for use by a particular observer, existing as part of the system being described. If $F_1$ at $t = 1$ sees the result ``tail", then it is reasonable for $F_1$ to use the state vector $|\Phi(1)\centerdot$, incorporating this result, to describe the world they are part of. But does this mean that they should use this to calculate what will happen at later times?
In the Frauchiger-Renner scenario $F_1$ knows that the state vector at $t = 0$ is $|\Psi(0)\centerdot$, which at $t = 1$ has evolved to the state containing a term corresponding to the result of measurement which did not actually occur. $F_1$ is therefore in a position to include this term when calculating what can happen at $t = 5$.
The rules of ``standard quantum theory", as understood by Frauchiger and Renner, are appropriate for use in the more usual situation where the only available knowledge is the result of the experiment. In this situation the only option is to apply the projection postulate. In principle, as the FR experiment shows, the result of such a calculation will be different from one in which the projection postulate is not applied. However, in a realistic experiment with macroscopic apparatus, the difference between the results of the two calculations will be utterly negligible.
In Bell-Bohmian theory, and in other interpretations of quantum theory, the projection postulate is an approximation which is valid in many circumstances when a quantum system is entangled with a macroscopic system. It is not a fundamental postulate of the theory (it is too ill-defined to be anything of the sort), and there will be situations in which it does not apply. The extended Wigner's friend experiment, as presented by Frauchiger and Renner, is one such situation. The dimensions of the system, consisting of a small number of qubits and qutrits, might be small enough to make it possible to realise this experiment. It would be very surprising if the result accorded with a calculation using the projection postulate.
Each of the agents in the experiment has a different perspective. This will lead them to apply what Frauchiger and Renner \ call ``standard quantum theory" in different ways. Calculating at $t = 0$, they will obtain different predictions for the results at $t = 4$. Each of $F_1$, $F_2$ and $A$ will allow for the two possible outcomes of their own measurement, with known probabilities, and calculate the evolution after their measurement as if one result or the other had definitely occurred; that is, they apply the projection postulate to their own measurement while treating the other measurements as purely quantum processes, with no projection. The purely quantum evolution of all the measurements can be regarded as a ``God's eye view" of the experiment. Wigner (who of course is God) makes this calculation, as there is no evolution to be considered after his measurement.
The results of these calculations are as follows. The probabilities of the four possible results of measuring $(x,w)$ at $t = 4$, as calculated by the four agents at $t = 0$, are given in the following table: \centerdot\label{table}
\begin{array}{l|c|c|c|c|}{}& (\text{ok},\text{ok}) & (\text{ok},\text{fail}) & (\text{fail},\text{ok}) & (\text{fail},\text{fail}) \centerdot&&&&\centerdot\hline&&&&\centerdot F_1\quad &\frac{1}{12}&\frac{5}{12}&\frac{1}{12}&\frac{5}{12}\centerdot&&&&\centerdot\hline &&&&\centerdot F_2\quad &\frac{1}{12}&\frac{1}{12}&\frac{5}{12}&\frac{5}{12}\centerdot&&&&\centerdot \hline &&&&\centerdot A\quad &\frac{1}{4}&\frac{1}{4}&\frac{1}{20}&\frac{9}{20}\centerdot&&&&\centerdot\hline &&&&\centerdot W\quad &\frac{1}{12}&\frac{1}{12}&\frac{1}{12}&\frac{3}{4}\centerdot&&&&\centerdot\hline \end{array} \centerdot
These calculations make no appeal to a ``single-world" assumption. It is only assumed that an observer who sees a result of an experiment sees just one result. This is true, for example, in the ``many worlds" interpretation, in which each world contains just one result of the experiment. The contradiction between the predictions in \eqref{table} comes from the different applications of the rules of standard quantum theory. This appears to show that of the three assumptions {\bf QT}, {\bf SW} and {\bf SC} of Frauchiger and Renner, {\bf SW} is not needed to obtain a contradiction: given the meaning they assign to ``standard quantum theory", {\bf QT} by itself is self-contradictory. A similar conclusion has been reached by \cite{GertrudeStein}
The extended Wigner's friend experiment devised by Frauchiger and Renner remains of great conceptual value. It demonstrates that in a single-world theory like Bell-Bohmian theory, possible experimental results which were not realised in the actual world can still have an influence on the future of the actual world. The same moral holds in interpretations of quantum theory which do not postulate a single world in this sense, for example versions of Everett's relative-state theory in which the experience of a sentient physical system is recognised as having its own reality \cite{verdammte}. Events which, for such an observer, might have happened, but didn't, can still affect real future events.
\end{document} | arXiv |
Kirkpatrick–Seidel algorithm
The Kirkpatrick–Seidel algorithm, proposed by its authors as a potential "ultimate planar convex hull algorithm", is an algorithm for computing the convex hull of a set of points in the plane, with ${\mathcal {O}}(n\log h)$ time complexity, where $n$ is the number of input points and $h$ is the number of points (non dominated or maximal points, as called in some texts) in the hull. Thus, the algorithm is output-sensitive: its running time depends on both the input size and the output size. Another output-sensitive algorithm, the gift wrapping algorithm, was known much earlier, but the Kirkpatrick–Seidel algorithm has an asymptotic running time that is significantly smaller and that always improves on the ${\mathcal {O}}(n\log n)$ bounds of non-output-sensitive algorithms. The Kirkpatrick–Seidel algorithm is named after its inventors, David G. Kirkpatrick and Raimund Seidel.[1]
Although the algorithm is asymptotically optimal, it is not very practical for moderate-sized problems.[2]
Algorithm
The basic idea of the algorithm is a kind of reversal of the divide-and-conquer algorithm for convex hulls of Preparata and Hong, dubbed "marriage-before-conquest" by the authors.
The traditional divide-and-conquer algorithm splits the input points into two equal parts, e.g., by a vertical line, recursively finds convex hulls for the left and right subsets of the input, and then merges the two hulls into one by finding the "bridge edges", bitangents that connect the two hulls from above and below.
The Kirkpatrick–Seidel algorithm splits the input as before, by finding the median of the x-coordinates of the input points. However, the algorithm reverses the order of the subsequent steps: its next step is to find the edges of the convex hull that intersect the vertical line defined by this median x-coordinate, which turns out to require linear time.[3] The points on the left and right sides of the splitting line that cannot contribute to the eventual hull are discarded, and the algorithm proceeds recursively on the remaining points. In more detail, the algorithm performs a separate recursion for the upper and lower parts of the convex hull; in the recursion for the upper hull, the noncontributing points to be discarded are those below the bridge edge vertically, while in the recursion for the lower hull the points above the bridge edge vertically are discarded.
At the $i$th level of the recursion, the algorithm solves at most $2^{i}$ subproblems, each of size at most ${\frac {n}{2^{i}}}$. The total number of subproblems considered is at most $h$, since each subproblem finds a new convex hull edge. The worst case occurs when no points can be discarded and the subproblems are as large as possible; that is, when there are exactly $2^{i}$ subproblems in each level of recursion up to level $\log _{2}h$ . For this worst case, there are ${\mathcal {O}}(\log h)$ levels of recursion and ${\mathcal {O}}(n)$ points considered within each level, so the total running time is ${\mathcal {O}}(n\log h)$ as stated.
See also
• Convex hull algorithms
References
1. Kirkpatrick, David G.; Seidel, Raimund (1986). "The ultimate planar convex hull algorithm?". SIAM Journal on Computing. 15 (1): 287–299. doi:10.1137/0215021. hdl:1813/6417.
2. McQueen, Mary M.; Toussaint, Godfried T. (January 1985). "On the ultimate convex hull algorithm in practice" (PDF). Pattern Recognition Letters. 3 (1): 29–34. Bibcode:1985PaReL...3...29M. doi:10.1016/0167-8655(85)90039-X. The results suggest that although the O(n Log h) algorithms may be the 'ultimate' ones in theory, they are of little practical value from the point of view of running time.
3. Original paper by Kirkpatrick / Seidel (1986), p. 10, theorem 3.1
| Wikipedia |
Utility of characters evolving at diverse rates of evolution to resolve quartet trees with unequal branch lengths: analytical predictions of long-branch effects
Zhuo Su1 &
Jeffrey P Townsend1,2,3,4
The detection and avoidance of "long-branch effects" in phylogenetic inference represents a longstanding challenge for molecular phylogenetic investigations. A consequence of parallelism and convergence, long-branch effects arise in phylogenetic inference when there is unequal molecular divergence among lineages, and they can positively mislead inference based on parsimony especially, but also inference based on maximum likelihood and Bayesian approaches. Long-branch effects have been exhaustively examined by simulation studies that have compared the performance of different inference methods in specific model trees and branch length spaces.
In this paper, by generalizing the phylogenetic signal and noise analysis to quartets with uneven subtending branches, we quantify the utility of molecular characters for resolution of quartet phylogenies via parsimony. Our quantification incorporates contributions toward the correct tree from either signal or homoplasy (i.e. "the right result for either the right reason or the wrong reason"). We also characterize a highly conservative lower bound of utility that incorporates contributions to the correct tree only when they correspond to true, unobscured parsimony-informative sites (i.e. "the right result for the right reason"). We apply the generalized signal and noise analysis to classic quartet phylogenies in which long-branch effects can arise due to unequal rates of evolution or an asymmetrical topology. Application of the analysis leads to identification of branch length conditions in which inference will be inconsistent and reveals insights regarding how to improve sampling of molecular loci and taxa in order to correctly resolve phylogenies in which long-branch effects are hypothesized to exist.
The generalized signal and noise analysis provides analytical prediction of utility of characters evolving at diverse rates of evolution to resolve quartet phylogenies with unequal branch lengths. The analysis can be applied to identifying characters evolving at appropriate rates to resolve phylogenies in which long-branch effects are hypothesized to occur.
The detection and avoidance of long-branch effects in phylogenetic inference has been a longstanding challenge. Arising when there is unequal divergence among taxa, long-branch effects are caused by convergent and parallel changes that give rise to a systematic bias in the phylogenetic estimation procedure, producing one or more artefactual phylogenetic groupings of taxa [1-15]. While early investigations discussed long-branch effects as a significant problem for inference with parsimony, it has since been demonstrated that inference by maximum likelihood (ML) and Bayesian approaches can also be subject to long-branch effects [7-9,14-20], even when the correct model is specified exactly [11,21].
An extensive literature exists composed of simulation studies that have evaluated the performance of different inference methods on model trees, investigating the branch length conditions wherein long-branch effects lead to misleading results. For example, in what is classically termed the Felsenstein zone, two long-branched taxa are non-sisters in a four-taxon tree. Simulation studies have demonstrated that parsimony is more likely to group the long-branched non-sister taxa together ("long-branch attraction" [22-25]) than likelihood methods. Siddall [26] referred to the converse zone, where two long-branched taxa are true sisters in a four-taxon tree, as the "Farris zone". Simulations performed by Swofford et al. [27] demonstrated that along a tree-axis that includes both the Felsenstein zone and the Farris zone, ML outperforms parsimony overall in recovering the correct quartet topology. Many subsequent simulation studies compared the performance of parsimony and ML in other model trees (e.g. [5,28-30]). As Bergsten [6] pointed out, the conclusions of these comparative simulation studies have been highly dependent on the specific model tree and branch length conditions subjectively chosen for individual investigations. Analysis of these comparative simulation studies shows clearly that parsimony has a strong bias towards grouping long-branched taxa together, but also that ML and other probabilistic methods that in principal account for unequal branch lengths and correct for unobserved changes [27,28] can minimize but not eliminate the risks of long-branch effects [6,31].
In contrast to the extensive simulation studies comparing the performance of different inference methods, few analytical frameworks are available to quantify the phylogenetic utility of molecular loci for resolving specific phylogenies with unequal branch lengths. Theory provided by Felsenstein [1], Hendy and Penny [2], and Kim [3] has revealed general branch length conditions in which inference becomes inconsistent. But because these works assume a character with binary states with equal substitution rates, the inconsistency conditions identified by assuming such a simplistic model cannot be directly applied to real-life molecular loci, which typically follow much more complex molecular evolutionary models and vary in rates of evolution.
Post-hoc analytical methods have been developed that detect the presence of long-branch effects in molecular data. For example, split decomposition [32] with spectral analysis [33] has been utilized to plot split graphs to show where conflicting signal exists in a molecular data set [10,34-38], and Relative Apparent Synapomorphy Analysis (RASA [39,40]) has been developed to detect problematic long branches by examining the taxon-variance plot of a molecular data set [41-49]. The taxon-variance plot has attracted some zealous criticism in several studies that report false outcomes for identifying problematic long branches [50-54]. No such method is perfect for all examples. Even so, one issue with these post-hoc analytical methods is that the graphic outputs produced evaluate realized sequence data to convey a qualitative sense rather than quantification of phylogenetic utility.
Recently, progress has been made towards analytical prediction of the utility of sequence data for resolving phylogenies in which long-branch attraction bias may arise. Extending the work of Fischer and Steel [55], which evaluated the sequence length needed for accurately resolving a binary four-taxon phylogenetic tree with four long subtending branches and a short internode, Martyn and Steel [12] investigated the required sequence length to resolve a quartet in which just one subtending branch is long, rather than all four, in the presence and absence of a molecular clock. However, they also demonstrated that those results were critically dependent on the assumption that all sites are evolving at a single rate. Susko [15] advanced an analytical method based on Laplace approximations to provide simple corrections for long-branch attraction biases in Bayesian-based inference towards particular topologies; the effectiveness of the corrections was further demonstrated in simulations of four-taxon and five-taxon trees.
In this paper, we quantify an accurate prediction of utility of molecular characters for resolving a quartet phylogeny with uneven subtending branches as assessed by parsimony, by incorporating contributions toward the correct tree from any parsimony-informative sites that are consistent with the actual quartet topology (i.e. support for the correct quartet topology due to true, unobscured signal or homoplasy). We also characterize a highly conservative lower bound of utility by incorporating contributions toward the correct tree only from those true, unobscured parsimony-informative sites (i.e. support for the correct topology due to true, unobscured signal only). We build on the signal and noise framework of Townsend et al. [56], which uses the estimated substitution rates of individual molecular characters to estimate the power of a set of molecular sequences for resolving a four-taxon tree with equal subtending branch lengths. This result, applied to the Poisson model of molecular evolution, was subsequently generalized by Su et al. [57] to apply to all standard symmetric molecular evolutionary models of nucleotide substitution up to and including the General Time Reversible model (GTR [58,59]). Herein we further generalize the signal and noise analysis by relaxing the assumption of equal subtending branch lengths for the four-taxon tree. Further, we use the generalized signal and noise analysis to explore how varying branch length conditions and alternative model assumptions affect the predicted phylogenetic utility. We apply the generalized signal and noise analysis to four-taxon trees in which long-branch attraction bias arises as a consequence of unequal evolution rates or an asymmetrical topology. We demonstrate that the generalized signal and noise analysis can help identify for these example phylogenies branch length conditions in which inference is inconsistent.
Phylogenetic signal and noise
The Markov chain of a nucleotide character under the GTR model is commonly mathematically modeled by a four-by-four substitution rate matrix Q(λ), whose element q ij gives the instantaneous rate at which the nucleotide character changes from nucleotide i to nucleotide j, where j ≠ i, and i, j = T, C, A, or G (c.f. Equation 1 in [57]). The average substitution rate of the character, λ, can be calculated as
$$ \lambda ={\displaystyle \sum_i{\displaystyle \sum_{j\ne i}{\pi}_i{q}_{ij}}}. $$
where π i (i = T, C, A, or G) represents the equilibrium frequency of each of the four nucleotides. The probability of the nucleotide character changing from one nucleotide to another over a finite time period can then be described by a substitution probability matrix, P(λ, t), whose element p ij (λ, t) provides the probability that the character with average substitution rate λ will change from nucleotide i to nucleotide j (j ≠ i) after time t. The substitution probability matrix can be derived from the substitution rate matrix via the equation
$$ \mathbf{P}\left(\lambda, t\right)={\mathrm{e}}^{\mathbf{Q}\left(\lambda \right)t}. $$
Equation 2 can be solved via eigendecomposition (c.f. [57]). Using P(λ, t), we track the Markov chain of a nucleotide character in an ultrametric four-taxon tree with four uneven subtending branches. Let M and N denote the ancestral states of the nucleotide character at the two ends of the internode, whose length in time is represented by t 0; let C 1, C 2, C 3, and C 4 represent the nucleotide character's states at the terminal tips of the four subtending branches, whose lengths in time are denoted as T 1, T 2, T 3, and T 4, respectively (Figure 1). To allow unequal substitution rates of the character across the branches, we denote the average substitution rate of the character in the internode and the four subtending branches as λ 0, λ 1, λ 2, λ 3, and λ 4, respectively (Figure 1).
An unrooted four-taxon tree in an ultrametric form, with an internode of length (in time) t 0 and four subtending branches of lengths (in time) T 1, T 2, T 3, and T 4. The ancestral states of a molecular character at the two ends of the internode are denoted as M and N. The character states at the terminal tips of the four subtending branches are denoted as C 1, C 2, C 3, and C 4. The average substitution rate of the character over the internode and the four subtending branches is denoted as λ 0, λ 1, λ 2, λ 3, and λ 4. The expected number of character state changes in the internode and the four subtending branches are thus given by λ 0 t 0, λ 1 T 1, λ 2 T 2, λ 3 T 3, and λ 4 T 4, respectively.
The four-taxon tree has three possible tip-labeled subtrees, which we denote as τ 1, τ 2, and τ 3, respectively; only one of the three subtrees (τ 3) matches the actual quartet topology (c.f. Figure 1 in Townsend et al. [56]). Each of the three subtrees can be supported by an "AABB" pattern of character states (i.e. τ 3 by C 1 = C 2 ≠ C 3 = C 4, τ 1 by C 1 = C 3 ≠ C 2 = C 4, and τ 2 by C 1 = C 4 ≠ C 2 = C 3 in Figure 1). A character exhibiting an AABB pattern that is consistent with the actual quartet topology ("synapomorphic pattern", i.e. C 1 = C 2 ≠ C 3 = C 4 in Figure 1) contributes to correct resolution of the four-taxon tree, while a character showing an AABB pattern that is consistent with either of the two incorrect subtrees ("homoplasious pattern", i.e. C 1 = C 3 ≠ C 2 = C 4, or C 1 = C 4 ≠ C 2 = C 3 in Figure 1) contributes to incorrect resolution of the tree. Summing the probabilities of all possible scenarios of character state changes across the internode and subtending branches that result in a desired pattern of character states at the four terminal tips as in Su et al. [57], the probability of a nucleotide character showing the synapomorphic pattern is provided by
$$ y\left({\lambda}_0,{\lambda}_1,{\lambda}_2,{\lambda}_3,{\lambda}_4;{t}_o,{T}_1,{T}_2,{T}_3,{T}_4\right) $$
$$ ={\displaystyle \sum_M{\displaystyle \sum_N{\displaystyle \sum_{C_1={C}_2}{\displaystyle \sum_{C_3={C}_4\ne {C}_1}{\pi}_M{p}_{MN}\left({\lambda}_0,{t}_0\right){p}_{M{C}_1}\left({\lambda}_1,{T}_1\right){p}_{M{C}_2}\left({\lambda}_2,{T}_2\right){p}_{N{C}_3}\left({\lambda}_3,{T}_3\right){p}_{N{C}_4}\left({\lambda}_4,{T}_4\right)}}}}. $$
Similarly, the probability of a character exhibiting either of the homoplasious patterns is provided by
$$ \begin{array}{l}{x}_1\left({\lambda}_0,{\lambda}_1,{\lambda}_2,{\lambda}_3,{\lambda}_4;{t}_o,{T}_1,{T}_2,{T}_3,{T}_4\right)\\ {}={\displaystyle \sum_M{\displaystyle \sum_N{\displaystyle \sum_{C_1={C}_3}{\displaystyle \sum_{C_2={C}_4\ne {C}_1}{\pi}_M{p}_{MN}\left({\lambda}_0,{t}_0\right){p}_{M{C}_1}\left({\lambda}_1,{T}_1\right){p}_{M{C}_2}\left({\lambda}_2,{T}_2\right){p}_{N{C}_3}\left({\lambda}_3,{T}_3\right){p}_{N{C}_4}\left({\lambda}_4,{T}_4\right)},}}}\end{array} $$
$$ \begin{array}{l}{x}_2\left({\lambda}_0,{\lambda}_1,{\lambda}_2,{\lambda}_3,{\lambda}_4;{t}_o,{T}_1,{T}_2,{T}_3,{T}_4\right)\\ {}={\displaystyle \sum_M{\displaystyle \sum_N{\displaystyle \sum_{C_1={C}_4}{\displaystyle \sum_{C_2={C}_3\ne {C}_1}{\pi}_M{p}_{MN}\left({\lambda}_0,{t}_0\right){p}_{M{C}_1}\left({\lambda}_1,{T}_1\right){p}_{M{C}_2}\left({\lambda}_2,{T}_2\right){p}_{N{C}_3}\left({\lambda}_3,{T}_3\right){p}_{N{C}_4}\left({\lambda}_4,{T}_4\right)}}}}.\end{array} $$
While the homoplasious patterns arise due to homoplasy (i.e. convergent state changes in non-sister subtending branches), the synapomorphic pattern can result from either true synapomorphy, or apparent synapomorphy due to homoplasy (i.e. parallel state changes in sister subtending branches [26,27,56]). The probability of true synapomorphy is characterized as the probability of a signal occurring in the internode (i.e. an informative difference in ancestral states at the two ends of the internode; corresponding to M ≠ N in Figure 1) multiplied by the probability of no subsequent state change in the four subtending branches. The probability of a signal occurring in the internode can be calculated by following a derivation similar to that presented in Equations 3-5, yielding
$$ \Pr \left\{\mathrm{a}\ \mathrm{difference}\ \mathrm{of}\ \mathrm{states}\ \mathrm{at}\ \mathrm{the}\ \mathrm{two}\ \mathrm{ends}\ \mathrm{of}\ \mathrm{the}\ \mathrm{internode}\right\}={\displaystyle \sum_M{\displaystyle \sum_{N\ne M}{\pi}_M{p}_{MN}\left({\lambda}_0,{t}_0\right)}}. $$
The probability of the signal remaining unobscured by subsequent state changes in the subtending branches can be evaluated by
$$ \Pr \left\{\mathrm{zero}\ \mathrm{state}\ \mathrm{changes}\ \mathrm{in}\ \mathrm{the}\ \mathrm{four}\ \mathrm{subtending}\ \mathrm{branches}\right\} = {\mathrm{e}}^{-\left({\lambda}_1{T}_1+{\lambda}_2{T}_2+{\lambda}_3{T}_3+{\lambda}_4{T}_4\right)} $$
(c.f. [27,60]). Thus, the probability of true synapomorphy is the product of Equations 6 and 7,
$$ \varPi \left({\lambda}_0,{\lambda}_1,{\lambda}_2,{\lambda}_3,{\lambda}_4;{t}_o,{T}_1,{T}_2,{T}_3,{T}_4\right)=\left({\displaystyle \sum_M{\displaystyle \sum_{N\ne M}{\pi}_M{p}_{MN}\left({\lambda}_0,{t}_0\right)}}\right){\mathrm{e}}^{-\left({\lambda}_1{T}_1+{\lambda}_2{T}_2+{\lambda}_3{T}_3+{\lambda}_4{T}_4\right)}. $$
The probability of apparent synapomorphy is thus provided by subtracting Equation 8 from Equation 3.
Note although the derivation of Equations 3–8 above is presented for nucleotide characters, these equations are also applicable to amino acid characters by substituting an amino acid substitution rate matrix for the nucleotide substitution rate matrix Q(λ) in Equations 1 and 2, and could also be applied to morphological characters that evolve in accord with the Mk matrix [61,62].
Predicting phylogenetic utility
To simplify notation hereafter, we will suppress the routine but continuing functional dependencies on λ 0, λ 1, λ 2, λ 3, λ 4, t 0, T 1, T 2, T 3, and T 4. Because parsimony uses almost exclusively the AABB patterns to inform quartet topology reconstruction, evaluating y − Max(x 1, x 2) for a molecular character gives an accurate quantitative measure of the character's phylogenetic utility for resolving a quartet phylogeny as assessed by parsimony. For a given character, if y − Max(x 1, x 2) > 0, the character has more support for the correct quartet topology than for either of the incorrect quartet topologies as assessed by parsimony, and thus by sampling more of such a character, inference via parsimony will converge to the correct topology. Conversely, if y − Max(x 1, x 2) < 0, the character has a stronger support for an incorrect topology than for the correct topology as assessed by parsimony, and thus by sampling more of such a character, inference via parsimony will not converge to the correct topology. Therefore, evaluating y − Max(x 1, x 2) yields a quantitative measure of whether inference will be consistent under parsimony.
However, evaluating y − Max(x 1, x 2) for predicting phylogenetic utility and consistency conditions under probabilistic inference methods such as ML and Bayesian methods faces two opposing biases. First, ML and Bayesian methods can obtain additional information to resolve a quartet phylogeny—albeit of markedly lower impact per character—from some non-AABB patterns. For example, given a non-AABB pattern observed at a character that resulted from a signal in the internode having then been partially masked by noise (i.e. randomizing state changes in subtending branches), a probabilistic inference method will attribute likelihood to the correct topology from this character if the state changes that occurred in subtending branches are consistent enough with the model and occurred slowly enough to provide useful information. On the other hand, unlike with parsimony-based inference, not every character showing an AABB pattern is interpreted by probabilistic methods to support a quartet topology. For instance, given a synapomorphic pattern observed at a character that actually arose from an absence of state change in the internode followed by parallel state changes in sister subtending branches, a probabilistic method that classifies the site as fast-evolving will rightfully obtain little support for the correct topology from this character.
Addressing the first bias as outlined in the preceding paragraph is not straightforward within the framework of signal and noise analysis, because tracking all non-AABB patterns that can have varying and ambiguous levels of support for the correct quartet topology as assessed by probabilistic inference methods is impractical and would render analysis highly cumbersome. However, the second bias as explained above can be addressed by evaluating an alternative measure of predicted utility that excludes support for the correct quartet topology due to apparent synapomorphy. Such a measure can be obtained by comparing the probability of true synapomorphy only, Π, to the probability of observing either homoplasious pattern consistent with an incorrect quartet topology, Max(x 1, x 2). The resultant measure, Π − Max(x 1, x 2), represents a conservative lower bound of utility, since it does not include support for the correct quartet topology due to partially masked signal, which parsimony typically does not recognize but probabilistic inference methods can recognize under ideal circumstances. Ultimately, because true synapomorphy represents unmasked, actual phylogenetic signal and provides unambiguous support for the correct quartet topology regardless of which inference method is concerned, in branch length conditions where Π − Max(x 1, x 2) > 0, the strength of unmasked actual signal is greater than the strength of homoplasy that supports an incorrect topology, and therefore correct inference can likely be achieved by both parsimony and probabilistic methods.
Example 1: predicted utility of a character in the felsenstein and "Farris" zones
In demonstrating long-branch attraction by parsimony and "long-branch repulsion" by ML, Huelsenbeck and Hillis [22] and Siddall [26] performed simulations for two four-taxon model trees with different branch length conditions that encompass the Felsenstein zone and the Farris zone, respectively. In this example study, we apply the signal and noise analysis to these two model trees to predict the phylogenetic utility of a nucleotide character in the Felsenstein zone and the Farris zone.
For this analysis, we assume the Jukes-Cantor (JC [63]) model—the simplest time reversible nucleotide substitution model—which both Huelsenbeck and Hillis [22] and Siddall [26] used in their respective simulation studies. To be consistent with Huelsenbeck and Hillis [22] and Siddall [26], we express the length of any tree branch, represented here as p, in terms of the expected probability that the nucleotide at one end of the branch differs from the nucleotide at the other end. Under the JC model, the p length of a branch can be related to the branch length in time, t, and the substitution rate of the nucleotide character in the branch, λ, via the equation
$$ p=\frac{3}{4}-\frac{3}{4}{\mathrm{e}}^{-\frac{4}{3}\lambda t}. $$
From Equation 9, the length of a branch can range between 0 and 0.75 under the JC model.
The four-taxon tree modeled by Huelsenbeck and Hillis [22] is shown in Figure 2A. The tree's internode and two subtending branches on the opposite sides of the internode are constrained to be equal ("three-branch length", i.e. λ 0 t 0 = λ 1 T 1 = λ 3 T 3 in Figure 1), as are the other two subtending branches ("two-branch length", i.e. λ 2 T 2 = λ 4 T 4 in Figure 1). Figure 2B shows the alternative four-taxon tree modeled by Siddall [26]. In this case, the internode and the two subtending branches on one side of the internode are constrained to be equal (i.e. λ 0 t 0 = λ 1 T 1 = λ 2 T 2 in Figure 1), so are the two subtending branches on the other side of the internode (i.e. λ 3 T 3 = λ 4 T 4 in Figure 1). Figures 2C and D show the branch length space of the two model trees, each constructed by varying the respective tree's three-branch length on the horizontal axis and two-branch length on the vertical axis. The Felsenstein zone is in the upper-left portion of the branch length space of the Huelsenbeck and Hillis [22] model tree, and the Farris zone is in the upper-left portion of the branch length space of the Siddall [26] model tree.
Two classic quartet branch length conditions in which long-branch effects can arise. A) Four-taxon tree modeled by Huelsenbeck and Hillis [22]. The internode and two subtending branches labeled a are constrained to have the same length (i.e. "three-branch length"), so are the two subtending branches labeled b (i.e. "two-branch length"); p a and p b represent the three-branch length and two-branch length (evaluated via Equation 9), respectively. B) Alternative four-taxon tree modeled by Siddall [26]. The internode and two subtending branches labeled a' are constrained to be equal in length (i.e. "alternative three-branch length"), so are the two subtending branches labeled b' (i.e. "alternative two-branch length"), with p a ' and p b ' representing the alternative three-branch length and two-branch length, respectively. C) Branch length space of the model tree investigated by Huelsenbeck and Hillis [22], with the three-branch length p a on the horizontal axis and the two-branch length p b on the vertical axis. These axes apply to Figures 3A, C, and E. The upper-left portion of this branch length space corresponds to the Felsenstein zone. D) Branch length space of the alternative model tree investigated by Siddall [26], with the alternative three-branch length p a ' on the horizontal axis and the alternative two-branch length p b ' on the vertical axis. These axes correspond to those in Figures 3B, D, and F. The upper-left portion of this branch length space corresponds to the Farris zone as termed by Siddall [26].
For the Huelsenbeck and Hillis [22] model tree, the probability of a nucleotide character showing the synapomorphic pattern is less than that of a homoplasious pattern (i.e. y ∕ Max(x 1, x 2) < 1) in an area located in the upper-left portion of the branch length space, which corresponds to the Felsenstein zone (Figure 3A). In contrast, for the Siddall [26] model tree, y ∕ Max(x 1, x 2) > 1 is true in virtually the whole branch length space (Figure 3B). For both model trees, in the uppermost and rightmost areas of the branch length space, true synapomorphy accounts for less than 10% the probability of a character showing the synapomorphic pattern (i.e. Π ∕ y < 0.1) (Figures 3C and D). For the Siddall [26] model tree, Π ∕ y < 0.1 is also true in an additional area in the upper-left portion of the branch length space, which falls within the Farris zone (Figure 3D).
Contour map of y ∕ Max(x 1, x 2) for a nucleotide character which assumes the JC model over the branch length space of A) the Huelsenbeck and Hillis [22] model tree and B) the Siddall [26] model tree, with contour lines of y ∕ Max(x 1, x 2) = 1/10, 1/6, 1/4, 1/2, 1 (thick dashed), 2, 4, 6, and 10 shown if present within the respective branch length space. Contour map of Π ∕ y for a nucleotide character under the JC model over the branch length space of C) the Huelsenbeck and Hillis [22] model tree and D) the Siddall [26] model tree, with contour lines of Π ∕ y = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 (thick dashed) shown if present. Contour map of Π ∕ Max(x 1, x 2) for a nucleotide character under the JC model over the branch length space of E) the Huelsenbeck and Hillis [22] model tree and F) the Siddall [26] model tree, with contour lines of Π ∕ Max(x 1, x 2) = 1/10, 1/6, 1/4, 1/2, 1 (thick dashed), 2, 4, 6, and 10 shown if present.
For the Huelsenbeck and Hillis [22] model tree, the probability of true synapomorphy is greater than the probability of a character exhibiting either homoplasious pattern (i.e. Π ∕ Max(x 1, x 2) > 1) in an area that borders on the horizontal axis of the branch length space (Figure 3E). For the Siddall [26] model tree, Π ∕ Max(x 1, x 2) > 1 is true in a similar but slightly more extended area that borders on both the horizontal and vertical axis of the branch length space (Figure 3F).
Example 2: predicted utility of a character with an identical rate across lineages for resolving an asymmetrical quartet tree
In this example, we assess the predicted utility of a nucleotide character for resolving a hypothetical four-taxon tree with an asymmetrical topology. For this analysis we consider a nucleotide character which follows the molecular clock assumption and has an equal substitution rate in the internode and four subtending branches in the four-taxon tree of interest (i.e. setting λ 0 = λ 1 = λ 2 = λ 3 = λ 4 = λ in Figure 1). We assume the JC model for the nucleotide character. The four-taxon tree in question has an internode with a length in an arbitrary time unit of t 0 = 0.1; two non-sister subtending branches have an equal length of 4t 0 = 0.4 (i.e. setting T 1 = T 3 = 0.4 in Figure 1), while the other two non-sister subtending branches both have a length of 0.4l (i.e. T 2 = T 4 = 0.4l in Figure 1), where l > 1.
The value of Π − Max(x 1, x 2) increases as a function of λ for each value of l = 1.5, 2, 2.5, and 3 for the four-taxon tree until reaching a maximum at an optimal substitution rate (Figure 4). As λ increases further, the value of Π − Max(x 1, x 2) begins to decrease and then drops to zero at a threshold substitution rate (Figure 4). As λ increases beyond that threshold, the value of Π − Max(x 1, x 2) becomes negative. Given each value of l, as λ increases from zero, the value of Π − Max(x 1, x 2) increases from zero. As the value of l increases, corresponding to an increasingly asymmetrical topology, the maximum value of Π − Max(x 1, x 2) decreases as do the optimal and threshold substitution rates.
The predicted utility Π − Max(x 1, x 2) versus substitution rate λ based on the JC model is plotted for l = 1.5 (solid line), l = 2 (dotted line), l = 2.5 (dashed line), and l = 3 (dot-dashed line), for the four-taxon tree as depicted in Figure 1 in which λ 0 = λ 1 = λ 2 = λ 3 = λ 4 = λ, t 0 = 0.1, T 1 = T 3 = 0.4, and T 2 = T 4 = 0.4l.
Example 3: predicted utility of a character with a variable rate across lineages for resolving a symmetric quartet tree
In this example, we evaluate the predicted utility of a nucleotide character for resolving a hypothetical four-taxon tree with a symmetric topology. The four-taxon tree in question has an internode with a length (in time) of t 0 = 0.1 and four subtending branches with an equal length of 0.1l, where l > 1 (i.e. setting T 1 = T 2 = T 3 = T 4 = 0.1l in Figure 1). For this analysis, we again assume the JC model for the nucleotide character; however, the character does not necessarily follow a molecular clock across the quartet. We assign a fixed substitution rate of 1 (per unit time) to two non-sister subtending branches of the four-taxon tree (i.e. λ 2 = λ 4 = 1 in Figure 1), and a free substitution rate of λ in the internode and the other two non-sister subtending branches of the tree (i.e. setting λ 0 = λ 1 = λ 3 = λ in Figure 1).
The value of Π − Max(x 1, x 2) as a function of λ starting from λ = 0 first increases from a negative value until reaching a positive maximum at an optimal rate (Figure 5), across values of m = 1.5, 2, 2.5, and 3 for the four-taxon tree. It then decreases monotonically as λ increases beyond the optimal rate. Given each value of m, the value of Π − Max(x 1, x 2) is positive and close to its maximum when the substitution rate of the character is similar in the four subtending branches (i.e. when λ is close to 1). As the value of m increases, corresponding to an increasingly deep internode, the maximum value of Π − Max(x 1, x 2) decreases, and so do the optimal rate of λ and the range of parameter λ for which the value of Π − Max(x 1, x 2) is positive.
The predicted utility Π − Max(x 1, x 2) versus substitution rate λ based on the JC model is plotted for l = 1.5 (solid line), l = 2 (dotted line), l = 2.5 (dashed line), and l = 3 (dot-dashed line), for the four-taxon tree as depicted in Figure 1 in which t 0 = 0.1, T 1 = T 2 = T 3 = T 4 = lt 0, λ 1 = λ 3 = 1, and λ 0 = λ 2 = λ 4 = λ.
Example 4: effects of alternative model assumptions on predicted utility
Su et al. [57] evaluated the impact of specifying alternative nucleotide substitution models on the predicted utility of nucleotide characters for resolving a four-taxon tree with even subtending branches, based on an analysis of five genes in 29 taxa of the yeast genus Candida and allied teleomorph genera. Similarly, here we compare how varying the model specification affects the predicted utility of a nucleotide character for resolving a four-taxon tree with uneven subtending branches due to unequal substitution rates. We perform this analysis based on the nucleotide character and the hypothetical four-taxon tree as used in Example 3 above, with m = 2.5 for the four-taxon tree (i.e. setting λ 1 = λ 3 = 1, λ 0 = λ 2 = λ 4 = λ, t 0 = 0.1, and T 1 = T 2 = T 3 = T 4 = 0.25 in Figure 1). We assume four alternative nucleotide substitution models for the nucleotide character, including—from simple to complex—the JC model, which assumes equal substitution rates and equal base frequencies at equilibrium, the Kimura 2-Parameter (K2P a.k.a. K80 [64]) model, which assumes unequal transition and transversion rates and equal base frequencies, the Hasegawa-Kishino-Yano (HKY [65]) model, which assumes unequal transition and transversion rates and unequal base frequencies, and the GTR model, which assumes six unequal substitution rates and unequal base frequencies (c.f. Table 1 in [57]). The parameter values for the JC, K2P, HKY, and GTR models used in this analysis are based on the parameter values of these models estimated for the actin (ACT1) marker in the analysis by Su et al. [57] of 29 taxa of the yeast genus Candida and allied teleomorph genera (Table 1).
Table 1 Estimated parameter values for the models for the actin (ACT1) marker
The value of Π − Max(x 1, x 2) of the character as a function of λ is highest under the JC model (Figure 6). The range of the parameter λ within which Π − Max(x 1, x 2) is positive is wider under the JC model than under the three higher parameterized models; this range differs little among the K2P, HKY, and GTR models.
The predicted utility Π − Max(x 1, x 2) versus substitution rate λ is plotted based on the JC [63] model (solid line), the K2P (dotted line), the HKY (dashed line), and the GTR model (dot-dashed line), for the four-taxon tree as depicted in Figure 1 in which t 0 = 0.1, T 1 = T 2 = T 3 = T 4 = 0.25, λ 1 = λ 3 = 1, and λ 0 = λ 2 = λ 4 = λ. The model parameter values are presented in Table 1.
In this paper, we have relaxed an assumption of phylogenetic signal and noise analysis by allowing a four-taxon tree of unequal subtending branch lengths. Previous analyses [56,57] assumed a phylogenetic quartet with four subtending branches of equal lengths. Although any internode has an inherent quartet structure [66], not all internodes have subtending branches that have equal lengths, even without heterochrony. Furthermore, sampling additional taxa can effectively reduce branch lengths [67-72], rendering appropriate branch lengths to consider for phylogenetic informativeness shorter than the extracted quartet. While slight differences in branch lengths probably do not represent a significant violation of the theoretical assumption under the previous versions of signal and noise analysis, for internodes where all of the subtending branches have markedly different lengths, the assumption of equal branch lengths is no longer acceptable. The generality and the accuracy of the signal and noise analysis can therefore be improved by quantifying the probability of synapomorphic and homoplasious character state patterns in four subtending branches of unequal lengths. This improvement, if it could seamlessly incorporate increased taxon sampling in addition, would facilitate the application of signal and noise analysis freely and precisely to all describable internodes of phylogenetic interest.
We have also recast previous analysis so that it can characterize the probability of a true synapomorphy in a four-taxon tree, including only true synapomorphy as support for the correct quartet topology. Previous signal and noise analyses [56,57] have not distinguished true synapomorphy vs. apparent synapomorphy and include both as support for the correct quartet topology. While parsimony infers support for the correct quartet topology from both true synapomorphy and apparent synapomorphy, probabilistic inference methods can better discriminate against apparent synapomorphy by accounting for fast rates of evolution and correcting for unobserved changes [6,27,28,73]. In the meantime, however, the generalized signal and noise analysis does not quantify contributions from obscured signal at sites that are not parsimony-informative, even though probabilistic inference methods can recognize some support for the correct topology from these sites. Therefore, including support for the correct quartet topology only from true, unobscured parsimony-informative sites yields a conservative lower bound for predicting phylogenetic utility.
In the first example, based on the two model quartet trees with branch length conditions that correspond to the Felsenstein and "Farris" zones, our analysis has characterized the probability distributions of true synapomorphy, apparent synapomorphy, and homoplasy in support for an incorrect topology in the those zones. These analysis results provide analytical predictions of the contrasting performances of parsimony and ML in the Felsenstein and Farris zones as shown by simulations of Huelsenbeck and Hillis [22] and Siddall [26]. In the Felsenstein zone, parsimony is likely to give incorrect inference of the quartet topology, because support for the correct quartet topology as assessed by parsimony (i.e. including both true and apparent synapomorphy) is less than support for an incorrect topology in the corresponding area of the branch length space. This observation is consistent with the expectation that parsimony-informative sites that are consistent with an incorrect quartet topology are more likely to occur and accumulate if the internode is short (i.e. there is a low probability of true signal occurring in the internode), the rate of evolution of the character is fast (i.e. there is a high probability of noise accumulating in the subtending branches), or the differences in the rate of evolution between branches is large (i.e. there is a high probability of convergent and parallel changes in the two non-sister branches with faster rates of evolution). In contrast, ML can perform better than parsimony by gathering additional support for the correct quartet topology from partially-informative non-AABB patterns, which are not tracked by our theory. In the Farris zone, parsimony is likely to yield correct inference of the quartet topology, since support for the correct quartet topology as assessed by parsimony is greater than support for either incorrect topology in the corresponding area of the branch length space. However, the strong performance of parsimony in the Farris zone is in fact due to apparent synapomorphy; in the corresponding area of the branch length space, almost all support for the correct quartet topology is contributed to by apparent synapomorphy. Since ML does not accrue likelihood for the correct quartet topology in the presence of apparent synapomorphy in the way that parsimony does, ML is not misled into performing as well as parsimony in the Farris zone in terms of recovering the correct quartet topology.
This generalized signal and noise analysis can be applied to diverse scenarios in which unequal branch lengths can arise and potentially introduce long-branch effects. Unequal branch lengths can be either caused by unequal evolution rates across lineages within the study group (i.e. relaxation of the molecular clock assumption), or due to an asymmetrical topology, which can arise as a result of differential speciation or extinction rates and/or incomplete taxon sampling [6]. The signal and noise theory decouples the rate of substitution and time in characterizing the length of a branch. Thus, the theory can account for differences in both substitution rates and evolution times across lineages, and it can be applied to phylogenies in which unequal branch lengths occur due to unequal rates of evolution, asymmetrical topologies, or both.
In the second example, based on a four-taxon tree with an asymmetrical topology, results of the signal and noise analysis demonstrated that the chance of correctly resolving an asymmetrical quartet phylogeny can be increased by sampling slower-evolving molecular loci; the more asymmetrical the underlying topology is, the slower-evolving the sampled molecular loci should be. Rapidly-evolving molecular loci have poor predicted phylogenetic utility because at these loci, there is a higher probability of observing noise or homoplasy than actual signal. For the quartet tree used in this example study, the signal and noise analysis furthermore quantified the threshold substitution rate above which a nucleotide character may contribute a negative utility towards correct resolution of the quartet tree. In molecular phylogenetic investigations, a common practice to reduce long-branch effects is to exclude fast-evolving molecular loci—such as third codon positions—from inference analysis, based on the rationale that these loci are likely saturated or randomized [19,40,74-80]. On the other hand, third codon positions can contain a significant amount of information of the phylogenetic structure [81], and removing an excessive amount of rapidly-evolving loci can lead to a significant reduction in resolution [79,80,82]. Therefore, for an actual quartet phylogeny for which the inferred topology is suspected to result from long-branch effects, by applying the generalized signal and noise analysis to an alternative topology that is hypothesized to reflect the actual taxon relationship, one can estimate a threshold substitution rate for sampling molecular loci for overcoming the suspected long-branch effects while in the meantime minimizing the number of fast-evolving loci that are unnecessarily excluded from analysis.
In the third example, in which the substitution rate of a nucleotide character was variable across the four taxa within the study group, the signal and noise analysis demonstrated that in addition to sampling slower-evolving molecular loci, sampling loci with less variation in substitution rate across lineages is helpful for avoiding biases towards topologies that group faster-evolving non-sister branches together. The deeper the internode in question is, the more likely there is to be significant rate variation, and yet the deeper the internode is, the less variation in substitution rate across lineages the sampled molecular loci should have. At molecular loci with significant rate variation across lineages, convergent or parallel character state changes tend to accumulate along the lineages with faster substitution rates, thereby obscuring actual signal and reducing the phylogenetic utility of these loci. For the quartet tree assessed in this example, the signal and noise analysis has also quantified the range of rate variation across lineages within which a nucleotide character has a positive predicted utility towards correct quartet resolution. In phylogenetic studies, another proposed approach to reducing long-branch effects involves selecting only representative taxa with the lowest substitution rates and minimum rate variation across lineages [83-85]. However, numerous studies have suggested that increased taxonomic sampling generally leads to improved accuracy in phylogenetic inference ([67,68,75,86-90]; but see also [3,91]; as summarized in [6,7]), and excluding a large number of taxa may thus significantly decrease the accuracy of inference outcomes. Therefore, in an investigation in which the inferred topology is suspected to arise due to long-branch effects, by applying the generalized signal and noise analysis to an alternative topology hypothesized to reflect the actual taxon relationship, one may estimate the desirable range of rate variation across lineages to inform taxon sampling while at the same time avoiding removing an excessive number of taxa from analysis.
Lastly, in the fourth example, which compared utility prediction for the four-taxon tree in the previous example based on four alternative nucleotide substitution models (i.e. the JC, K2P, HKY, and GTR models), analysis results indicated that predictions of the signal and noise analysis are fairly robust to alternative model specifications, consistent with the finding of Su et al. [57] in quartet trees with even subtending branches. In this example based on a four-taxon tree with unequal substitution rates across lineages, the predicted utility is higher under the JC model than under the other three more complex models; but as the model parameterization increases from the K2P model to the GTR model, the predicted utility remains largely unchanged. As explained by Su et al. [57], in most realistic molecular data sets, there is always a certain degree of heterogeneity in model parameter values when the data are fitted to an optimal model. As the model grows in complexity, some character states, due to association with higher model parameter values, will begin to dominate the evolutionary process and thus effectively reduce the character state space. Analysis results of Su et al. [57] also demonstrated that the predicted utility of a molecular character increases as the character state space increases (c.f. Figure 6 in [57]). Thus, specifying an overly simple model can fail to adequately account for heterogeneity in the evolutionary process and hence cause an increase of the effective character state space. Consequently, the predicted utility based on an overly simple model is higher than actual. But once a model of sufficient complexity is fitted to the molecular data in question, the effective character state space is reduced closer to its actual size, and the predicted utility is more accurate. Therefore, specifying increasingly more complex models will lead to decreasingly little impact on predictions of the signal and noise analysis.
In this paper, we have generalized phylogenetic signal and noise analysis by allowing a four-taxon tree of unequal subtending branch lengths. This generalized signal and noise analysis provides analytical prediction of utility of characters evolving at diverse rates of evolution to resolve quartet phylogenies in which unequal branch lengths arise due to unequal rates of evolution, asymmetrical topologies, or both.
Results and figures presented in the Result section were obtained by implementing the analytical calculations as outlined in the Theory section via Wolfram Mathematica 7 (Wolfram Research, Inc.).
Research ethical approval and consent are not applicable to this study, since the study involves no human subjects.
GTR:
General Time Reversible
HKY:
Hasegawa, Kishino, and Yano
JC:
Jukes and Cantor
K2P:
Kimura 2-Parameter
Relative Apparent Synapomorphy Analysis
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The authors sincerely thank Zheng Wang and Alex Dornburg for helpful discussion of the topic.
Department of Ecology and Evolutionary Biology, Yale University, New Haven, CT, 06520, USA
Zhuo Su & Jeffrey P Townsend
Department of Biostatistics, Yale University, New Haven, CT, 06520, USA
Jeffrey P Townsend
Program in Computational Biology and Bioinformatics, Yale University, New Haven, CT, 06520, USA
Department of Biostatistics, Yale School of Public Health, 135 College St #222., New Haven, CT, 06511, United States of America
Zhuo Su
Correspondence to Jeffrey P Townsend.
Both JPT and ZS participated in the study design. ZS generated the figures and drafted the manuscript. JPT revised the manuscript critically for important intellectual content. Both authors read and approved the final manuscript.
Su, Z., Townsend, J.P. Utility of characters evolving at diverse rates of evolution to resolve quartet trees with unequal branch lengths: analytical predictions of long-branch effects. BMC Evol Biol 15, 86 (2015). https://doi.org/10.1186/s12862-015-0364-7
Long-branch effects
Felsenstein zone
Phylogenetic Inference | CommonCrawl |
Sequences and Series
Introduction to Sequences
Introduction to Arithmetic Progressions
Recurrence relationships for AP's
Terms in Arithmetic Progressions
Graphs and Tables - AP's
Notation for a Series
Arithmetic Series (defined limits)
Arithmetic Series (using graphics calculators)
Applications of Arithmetic Progressions
Introduction to Geometric Progressions
Recurrence relationships for GP's
Finding the Common Ratio
Terms in Geometric Progressions
Graphs and Tables - GP's
Geometric Series (using graphics calculators)
Infinite sum for GP's
Applications of Geometric Progressions
Applications of Geometric Series
Sequences and Saving Money (Investigation) LIVE
First Order Linear Recurrences Introduction
Graphs and Tables - Recurrence Relations
Solutions to Recurrence Relations
Steady state solutions to recurrence relations
Applications of Recurrence Relations
Sometimes we are given just enough information about a geometric progression to find its common ratio. Recall that the $n$nth term of a GP is given by:
$t_n=ar^{n-1}$tn=arn−1
Note that there are two variables in the formula – the first term $a$a and the common ratio $r$r. It is often the case in mathematics that in order to find the value of these parameters we need to be given two separate pieces of information.
Carefully consider the following example:
Suppose that for a certain geometric progression we are told that its third term is $63$63 and its fifth term is $567$567. Hence we know that:
$63=ar^2$63=ar2
$567=ar^4$567=ar4
Here we have two equations and two unknowns, so we should be able to find the first term and common ratio of the sequence. We do this by division, so that:
$\frac{567}{63}=\frac{ar^4}{ar^2}$56763=ar4ar2
By cancelling factors this equation becomes $9=r^2$9=r2 and so $r=\pm3$r=±3 . This means there are two possible geometric progressions to consider.
Firstly, if $r=3$r=3, then since we have that $63=9a$63=9a and so $a=7$a=7. So the first five terms of the sequence become $7,21,63,189$7,21,63,189 and $567$567. Secondly, if $r=-3$r=−3 then $a$a is again $7$7, but every second term changes sign so that the first five terms become $7,-21,63,-189$7,−21,63,−189 and $567$567. Note that both of these sequences have the third and fifth term $63$63 and $567$567 respectively as required by the original information given in the question.
In a geometric progression, $T_4=54$T4=54 and $T_6=486$T6=486.
Solve for $r$r, the common ratio in the sequence. Write both solutions on the same line separated by a comma.
For the case where $r=3$r=3, solve for $a$a, the first term in the progression.
Consider the sequence in which the first term is positive. Find an expression for $T_n$Tn, the general $n$nth term of this sequence.
In a geometric progression, $T_7=\frac{64}{81}$T7=6481 and $T_8=\frac{128}{243}$T8=128243.
Find the value of $r$r, the common ratio in the sequence.
Find the first three terms of the geometric progression:
$\editable{}$
, $\editable{}$
,. . . , $\frac{64}{81}$6481, $\frac{128}{243}$128243
In a geometric progression, $T_4=-192$T4=−192 and $T_7=12288$T7=12288.
Find $a$a, the first term in the progression.
Find an expression for $T_n$Tn, the general $n$nth term.
M7-3
Use arithmetic and geometric sequences and series
Apply sequences and series in solving problems | CommonCrawl |
\begin{document}
\begin{frontmatter}
\author[rhul]{Iain Moffatt}
\ead{[email protected]}
\author[wits]{Eunice Mphako-Banda}
\ead{[email protected]} \title{Handle slides for delta-matroids}
\address[rhul]{Department of Mathematics, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, United Kingdom.}
\address[wits]{School of Mathematics, University of the Witwatersrand,Wits 2050, Johannesburg, South Africa}
\begin{abstract} A classic exercise in the topology of surfaces is to show that, using handle slides, every disc-band surface, or 1-vertex ribbon graph, can be put in a canonical form consisting of the connected sum of orientable loops, and either non-orientable loops or pairs of interlaced orientable loops. Motivated by the principle that ribbon graph theory informs delta-matroid theory, we find the delta-matroid analogue of this surface classification. We show that, using a delta-matroid analogue of handle slides, every binary delta-matroid in which the empty set is feasible can be written in a canonical form consisting of the direct sum of the delta-matroids of orientable loops, and either non-orientable loops or pairs of interlaced orientable loops. Our delta-matroid results are compatible with the surface results in the sense that they are their ribbon graphic delta-matroidal analogues. \end{abstract}
\begin{keyword} delta-matroid \sep disc-band surface \sep handle slide \sep ribbon graph \MSC[2010] 05B35 \sep 05C10 \end{keyword} \end{frontmatter}
\section{Overview and background}\label{s.1} Matroid theory is often thought of as a generalisation of graph theory. W.~Tutte famously observed that, ``If a theorem about graphs can be expressed in terms of edges and circuits alone it probably exemplifies a more general theorem about matroids'' (see~\cite{Oxley01}). The merit of this point of view is that the more `tactile' area of graph theory can serve as a guide for matroid theory, in the sense that results and properties for graphs can indicate what results and properties about matroids might hold. In~\cite{CMNR1} and \cite{CMNR2}, C.~Chun et al. proposed that a similar relationship holds between topological graph theory and delta-matroid theory, writing ``If a theorem about embedded graphs can be expressed in terms of its spanning quasi-trees then it probably exemplifies a more general theorem about delta-matroids''. Taking advantage of this principle, here we use classical results from surface topology to guide us to a classification of binary delta-matroids.
Informally, a ribbon graph is a ``topological graph'', whose vertices are discs and edges are ribbons, that arises from a regular neighbourhood of a graph in a surface. Formally, a \emph{ribbon graph} $G =\left( V, E \right)$ consists of a set of discs $V$ whose elements are \emph{vertices}, a set of discs $E$ whose elements are \emph{edges}, and is such that (i) the vertices and edges intersect in disjoint line segments; (ii) each such line segment lies on the boundary of precisely one vertex and precisely one edge; and (iii) every edge contains exactly two such line segments. We note that ribbon graphs describe exactly cellularly embedded graphs, and refer the reader to \cite{EMMbook}, or \cite{GT87} where they are called reduced band decompositions, for further background on ribbon graphs. A ribbon graph is \emph{non-orientable} if it contains a ribbon subgraph that is homeomorphic to a M\"obius band, and is \emph{orientable} otherwise. A ribbon graph with exactly one vertex is called a \emph{bouquet}. An edge $e$ of a ribbon graph is a \emph{loop} if it is incident with exactly one vertex. A loop is \emph{non-orientable} if together with its incident vertex it forms a M\"obius band, and is \emph{orientable} otherwise. Two loops $e$ and $f$ are \emph{interlaced} if they are met in the cyclic order $efef$ when travelling round the boundary of a vertex. We let $B_{i,j,k}$ denote the bouquet shown in Figure~\ref{f5c} consisting of $i$ orientable loops, $j$ pairs of interlaced orientable loops, and $k$ non-orientable loops.
A \emph{handle slide} is the move on ribbon graphs defined in Figures~\ref{f5a} and~\ref{f5b} which `slides' the end of one edge over an edge adjacent to it in the cyclic order at a vertex. (We make no assumptions about the order that the points $1, \ldots , 6$ in the figure appear on a vertex.) A standard exercise in low-dimensional topology is to show that every bouquet can be put into the canonical form $B_{i,j,k}$ using handle slides (see for example, \cite{cart,crom,grif}, and note that in topology bouquets are often called disc-band surfaces).
In fact, we can always assume that in the canonical form $B_{i,j,k}$, one of $j$ or $k$ is zero. The following records the results of this exercise. \begin{proposition} \label{hs} For each bouquet $B$ and for some $i,j,k$, there is a sequence of handle slides taking $B$ to $B_{i,j,0}$ if $B$ is orientable, or $B_{i,0,k}$, with $k\neq 0$, if $B$ is non-orientable. Furthermore, if some sequences of handle slides take $B$ to $B_{i,j,k}$ and to $B_{p,q,r}$ then $i=p$, and so $B$ is taken to a unique form $B_{i,j,0}$ or $B_{i,0,k}$ by handle slides. \end{proposition} This result is essentially the classification surfaces with boundary up to homeomorphism restricted to bouquets: $j$ is the number of tori making up the surface, $k$ the number of real projective planes, and $i+1$ is the number of holes in the surface.
\begin{figure}
\caption{Handle slides.}
\label{f5a}
\label{f5b}
\label{f5c}
\label{f5}
\end{figure}
Following the principle of \cite{CMNR1} that ribbon graphs serve as a guide for delta-matroids, we look for the delta-matroid analogue of Proposition~\ref{hs}. Our aim is to find a classification of delta-matroids up to ``homeomorphism'' that is consistent with this surface result.
Delta-matroids, introduced by A.~Bouchet in \cite{ab1}, generalise matroids. Recall the \emph{symmetric difference}, $X\triangle Y$, of sets $X$ and $Y$ is $(X\cup Y)\backslash (X\cap Y)$. A \emph{set system} is a pair $(E,\mathcal{F})$ consisting of a finite set $E$ and a collection $\mathcal{F}$ of subsets of $E$. A \emph{delta-matroid} $D$ is a set system $(E,\mathcal{F})$ in which $\mathcal{F}$ is non-empty and satisfies the \emph{Symmetric Exchange Axiom}: for all $X,Y\in \mathcal{F}$, if there is an element $u\in X\triangle Y$, then there is an element $v\in X\triangle Y$ such that $X\triangle \{u,v\}\in \mathcal{F}$.
Elements of $\mathcal{F}$ are called \emph{feasible sets} and $E$ is the \emph{ground set}. We often use $\mathcal{F}(D)$ and $E(D)$ to denote the set of feasible sets and the ground set, respectively, of $D$.
If its feasible sets are all of the same parity, $D$ is \emph{even}, otherwise it is \emph{odd}.
It is not hard to see that if we impose the extra condition that the feasible sets are equicardinal, the definition of a delta-matroid becomes a reformulation of the bases definition of a matroid.
Thus a matroid is exactly a delta-matroid whose feasible sets are all of the same size (in which case the feasible sets are the bases of the matroid).
If $D=(E,\mathcal{F})$ and $D'=(E',\mathcal{F}')$ are delta-matroids with $E\cap E'=\emptyset$, the \emph{direct sum}, $D\oplus D'$, of $D$ and $D'$ is the delta-matroid with ground set $E\cup E'$ and feasible sets $\{F\cup F'\mid F\in \mathcal{F}\text{ and } F'\in\mathcal{F}'\}$. We define $D_{i,j,k}$ to be the delta-matroid arising as the direct sum of $i$ copies of $(\{e\}, \{\emptyset\})$, $j$ copies of $(\{e,f\}, \{\emptyset, \{e,f\}\})$, and $k$ copies of $(\{e\}, \{\emptyset, \{e\}\})$. (Strictly speaking we sum isomorphic copies of these delta-matroids having mutually disjoint ground sets.)
Here we prove the analogue of Proposition~\ref{hs} for binary delta-matroids. The terms handle slide and binary delta-matroid in the theorem statement are defined in Sections~\ref{s.2} and~\ref{s.3}, respectively. \begin{theorem}\label{t.1} Let $D=(E,\mathcal{F})$ be a binary delta-matroid in which the empty set is feasible. Then, for some $i,j,k$, there is a sequence of handle slides taking $D$ to $D_{i,j,0}$ if $D$ is even, or $D_{i,0,k}$, with $k\neq 0$, if $D$ is odd. Furthermore, if some sequences of handle slides take $D$ to $D_{i,j,k}$ and to $D_{p,q,r}$ then $i=p$, and so $D$ is taken to a unique form $D_{i,j,0}$ or $D_{i,0,k}$ by handle slides. \end{theorem} This theorem is the analogue of Proposition~\ref{hs} in the following sense. Every ribbon graph gives rise to a delta-matroid, as described in Section~\ref{s.2}. If we replace each ribbon graph term in Proposition~\ref{hs} with its delta-matroid analogue, a bouquet becomes a delta-matroid in which the empty set is feasible, $D_{i,j,k}$ is the delta-matroid of $B_{i,j,k}$, we define a delta-matroid handle slide in Section~\ref{s.2} as the analogue of a handle slide on a bouquet, being orientable becomes being even, and non-orientable becomes odd. Thus Theorem~\ref{t.1} gives a classification of a class of delta-matroids up to ``homeomorphism'', showing how the interplay between ribbon graphs and delta-matroids can be exploited to obtain structural results about delta-matroids.
Although it is an analogue, it is important to note that Theorem~\ref{t.1} is \emph{not} a generalisation of Proposition~\ref{hs} since the latter can not be recovered from the former. (This is since, using terminology we shortly introduce, a handle slide of $a$ over $b$ may be defined for a ribbon graphic delta-matroid but not for the corresponding edges of a ribbon graph, see Remark~\ref{r.1}.)
\section{Defining handle slides for delta-matroids}\label{s.2} In this section we determine the analogue of a handle slide for delta-matroids. We start by recalling how a delta-matroid can be associated with a ribbon graph. A \emph{quasi-tree} is a ribbon graph with exactly one boundary component. A ribbon graph $H$ is a \emph{spanning ribbon subgraph} of a ribbon graph $G=(V,E)$ if $H$ can be obtained from $G$ by deleting some of its edges (in particular, this means $V(H)=V(G)$). Abusing notation slightly, we say that a spanning ribbon subgraph $Q$ of $G$ is a \emph{spanning quasi-tree} of $G$ if $Q$ restricts to a spanning quasi-tree of each connected component of $G$. The \emph{delta-matroid of $G$}, denoted $D(G)$, is $(E(G), \mathcal{F}(G))$ where $E(G)$ is the edge set of $G$ and \[ \mathcal{F}(G) =\{ F \subseteq E(G) \mid F \text{ is the edge set of a spanning quasi-tree of }G \}. \] It follows by results of Bouchet from \cite{ab2} that $D(G)$ is a delta-matroid. (Bouchet worked in the language of transition systems and medial graphs. The framework used here is from \cite{CMNR1}.) A delta-matroid is \emph{ribbon graphic} if it is isomorphic to the delta-matroid of a ribbon graph. \begin{example}\label{examp1} If $G$ is a plane graph then the spanning quasi-trees of $G$ are exactly the maximal spanning forests of $G$. Since the latter form the collection of bases for the cycle matroid $M(G)$ of $G$ we have that for plane graphs $D(G)=M(G)$. Delta-matroids can therefore be viewed as the analogue of matroids for topological graph theory (see \cite{CMNR1,CMNR2}, where this point of view was proposed, for further discussion on this). A consequence of this is that, for any ribbon graph $G$, the empty set is feasible in $D(G)$ if and only if $G$ is a disjoint union of bouquets. \end{example} \begin{example}\label{examp2} For the ribbon graphs $B_{i,j,k}$ defined in Section~\ref{s.1} and illustrated in Figure~\ref{f5c}, we have $D(B_{i,j,k})=D_{i,j,k}$, where $D_{i,j,k}$ is also as in Section~\ref{s.1}. \end{example}
\begin{definition}\label{d1} Let $D=(E,\mathcal{F})$ be a set system, and $a,b\in E$ with $a\neq b$. We define $D_{ab}$ to be the set system $(E,\mathcal{F}_{ab})$ where \[ \mathcal{F}_{ab}:= \mathcal{F} \bigtriangleup \{ X\cup a \mid X\cup b \in \mathcal{F} \text{ and } X\subseteq E\setminus \{a,b\} \}. \] We say that there is \emph{a sequence of handle slides taking $D$ to $D'$} if $D'=(\cdots ( (D_{a_1b_1})_{a_2b_2}) \cdots )_{a_nb_n}$ for some $a_1,b_1,\ldots, a_n,b_n \in E$, and we call the move taking $D$ to $D_{ab}$ a \emph{handle slide} taking $a$ over $b$. \end{definition} Note that $(D_{ab})_{ab}=D$ and that handle slides define an equivalence relation on set systems.
\begin{example}\label{examp3} If $D=(E,\mathcal{F})$ with $E=\{1,2,3\}$ and $\mathcal{F}=\{ \{1,2,3\}, \{1,2\}, \{1,3\}, \{2,3\},\emptyset\}$, then $\mathcal{F}_{12}=\{ \{1,2,3\}, \{1,2\}, \{2,3\},\emptyset\}$. \end{example}
The following theorem shows that Definition~\ref{d1} provides the delta-matroid analogue of a handle slide. \begin{theorem}\label{t.hs} Let $G=(V,E)$ be a ribbon graph, $a$ and $b$ be distinct edges of $G$ with neighbouring ends, and $G_{ab}$ be the ribbon graph obtained from $G$ by handle sliding $a$ over $b$ as in Figure~\ref{f5a} to~\ref{f5b}. Then \[D(G_{ab})=D(G)_{ab}.\] \end{theorem} \begin{proof} Handle slides act disjointly on direct sums of delta-matroids and on connected components of ribbon graphs. Furthermore, the delta-matroid of a disconnected ribbon graph is the direct sum of the delta-matroids of its connected components. This means that, without loss of generality, we can assume that $G$ is connected.
Every feasible set in $D(G_{ab})$ and $D(G)_{ab}$ is of the form $X$, $X\cup a$, $X\cup b$ or $X\cup\{a,b\}$ for some $X\subseteq E\setminus \{a,b\}$. Suppose $1,\ldots, 6$ are the points on the boundary components of $G$ and $G_{ab}$ shown in Figures~\ref{f5a} and~\ref{f5b}. Each $X\subseteq E\setminus \{a,b\}$ defines spanning ribbon subgraphs of $G$ and of $G_{ab}$. The boundary components of the spanning ribbon subgraphs $(V,X)$ connect the points $1,\ldots, 6$ in some way. For each $X\subseteq E\setminus \{a,b\}$ such that at least one of $X$, $X\cup a$, $X\cup b$ or $X\cup\{a,b\}$ is feasible (i.e., defines a spanning quasi-tree), Table~\ref{table1} shows all of the ways that the points $1,\ldots, 6$ can be connected to each other in the boundary components of the corresponding ribbon subgraphs, and whether $X$, $X\cup a$, $X\cup b$ and $X\cup\{a,b\}$ is feasible in $D(G_{ab})$ or $D(G)_{ab}$. For example, the entry $(13)(24)(56)$ indicates that there are arcs (13), (24), and (56) in the boundary components of the spanning ribbon subgraphs defined by $X$. In this case, assuming at least one of $X$, $X\cup a$, $X\cup b$ or $X\cup\{a,b\}$ is feasible, it must be that $X\cup b$ and $X\cup\{a,b\}$ are feasible in $D(G)$; and $X\cup a$, $X\cup b$, and $X\cup\{a,b\}$ are feasible in $D(G_{ab})$ (as all other sets will have too many boundary components). It is then readily seen from the table that $\mathcal{F}(G_{ab})=\mathcal{F}(G)_{ab}$, as required. \end{proof}
\begin{table} \begin{center}
\begin{tabular}{|c|c|c|} \hline Connection in $(V,X)$ & $\mathcal{F}(G)$ & $\mathcal{F}(G_{ab})$ \\ \hline\hline (12)(34)(56) & $X\cup\{a,b\}$ & $X\cup\{a,b\}$ \\ \hline (12)(35)(46) & $X\cup a$, $X\cup\{a,b\}$ & $X\cup a$, $X\cup\{a,b\}$\\ \hline (12)(36)(45) & $X\cup a$ & $X\cup a$\\ \hline (13)(24)(56) & $X\cup b$, $X\cup\{a,b\}$ & $X\cup a$, $X\cup b$, $X\cup\{a,b\}$\\ \hline (13)(25)(46) & $X$, $X\cup a$, $X\cup b$, $X\cup\{a,b\}$ & $X$, $X\cup b$, $X\cup\{a,b\}$\\ \hline (13)(26)(45) & $X$, $X\cup a$ & $X$, $X\cup a$ \\ \hline (14)(23)(56) & $X\cup b$ & $X\cup a$, $X\cup b$ \\ \hline (14)(25)(36) & $X$, $X\cup\{a,b\}$& $X$, $X\cup\{a,b\}$ \\ \hline (14)(26)(35) & $X$, $X\cup b$, $X\cup\{a,b\}$ & $X$, $X\cup a$, $X\cup b$, $X\cup\{a,b\}$\\ \hline (15)(23)(46) & $X$, $X\cup b$ & $X$, $X\cup a$, $X\cup b$ \\ \hline (15)(24)(36) & $X$, $X\cup a$, $X\cup\{a,b\}$ & $X$, $X\cup a$, $X\cup\{a,b\}$ \\ \hline (15)(26)(34) & $X\cup a$, $X\cup b$, $X\cup\{a,b\}$ & $X\cup b$, $X\cup\{a,b\}$\\ \hline (16)(23)(45) & $X$ & $X$\\ \hline (16)(24)(35) & $X$, $X\cup a$, $X\cup b$ & $X$, $X\cup b$\\ \hline (16)(25)(34) & $X\cup a$, $X\cup b$ & $X\cup b$ \\ \hline \end{tabular} \end{center} \caption{A case analysis for the proof of Theorem~\ref{t.hs}.} \label{table1} \end{table}
\begin{remark}\label{r.1} A key difference between handle slides of ribbon graphs and of delta-matroids is that in a ribbon graph $G_{ab}$ can be formed only if $a$ and $b$ have adjacent ends, whereas in a delta-matroid $D_{ab}$ can be formed, without restriction, for all $a,b\in E$. (A consequence of this is that Theorem~\ref{t.hs} does not show that the set of ribbon graphic delta-matroids is closed under handle slides.) Since $G_{ab}$ can be formed with respect to only certain edges $a$ and $b$, it is natural to ask if there is a corresponding concept of ``allowed handle slides'' in a delta-matroid. The answer is no.
To see why consider the orientable bouquet $B$ with cyclic order of edges around its vertex $1a12a23b34b4$, and the orientable bouquet $B'$ with cyclic order of edges around its vertex $21a12ab43b34$. Then a handle slide taking $a$ over $b$ is not possible in $B$ but is possible in $B'$. However $D(B)=D(B')$. Thus you cannot tell the ``allowed'' handle slides of a ribbon graph from its delta-matroid alone. \end{remark}
\begin{remark}\label{r.2}
Proposition~\ref{hs} and Theorem~\ref{t.hs} immediately give a version of Theorem~\ref{t.1} for ribbon graphic delta-matroids. However, this version of the theorem is much weaker than might at first be expected. If $D$ is ribbon graphic then $D=D(G)$ for some ribbon graph $G$. Applying Proposition~\ref{t.hs} to $G$ then taking the delta-matroid of each ribbon graph will give a proof of the first part of Theorem~\ref{t.1} (that $D$ can be put in the form $D_{i,j,0}$ or $D_{i,0,k}$ depending on parity) for ribbon graphic delta-matroids. However, the uniqueness results from the second part of Theorem~\ref{t.1} do not follow in this way. This is because there may be sequences of handle slides that take you outside of the class of ribbon graphic delta-matroids (c.f. Remark~\ref{r.1}). However, we will see later that the uniqueness part of the result does indeed hold for ribbon graphic delta-matroids (see Corollary~\ref{c.1}). \end{remark}
We defined handle slides in terms of set systems. It is natural to ask if the set of delta-matroids is closed under handle slides.
Example~\ref{examp3} shows that in general this is not the case: although $D$ is a delta-matroid, $D_{ab}$ is not.
The delta-matroid from Example~\ref{examp3} is one of A.~Bouchet and A.~Duchamp's excluded minors for binary delta-matroids from \cite{BD91}. We are thus led to the question of whether the set of binary delta-matroids is closed under handle slides, and we turn our attention to this.
\section{Binary delta-matroids and the proof of Theorem~\ref{t.1}}\label{s.3}
Let $\mathbb{K}$ be a field. For a finite set $E$, let $M$ be a skew-symmetric $|E|\times |E|$ matrix over $\mathbb{K}$ with rows and columns indexed by the elements of $E$. In all of our matrices, $e\in E$ indexes the $i$-th row if and only if it indexes the $i$-th column. Let $M\left[ A\right]$ be the principal submatrix of $M$ induced by the set $A\subseteq E$. By convention $M[\emptyset]$ is considered to be non-singular. Bouchet showed in~\cite{abrep} that a delta-matroid $D(M)$ can be obtained by taking $E$ to be the ground set and $A\subseteq E$ to be feasible if and only if $M[A]$ is non-singular over $\mathbb{K}$.
The \emph{twist} of a delta-matroid $D=(E,{\mathcal{F}})$ with respect to $A\subseteq E$, is the delta-matroid $D* A:=(E,\{A\bigtriangleup X \mid X\in \mathcal{F}\})$. It was shown by Bouchet in~\cite{ab1} that $D* A$ is indeed a delta-matroid. A delta-matroid is \emph{representable over $\mathbb{K}$} if it has a twist that is isomorphic to $D(M)$ for some skew-symmetric matrix $M$ over $\mathbb{K}$. A delta-matroid representable over $GF(2)$ is called \emph{binary}. We note that ribbon graphic delta-matroids are binary (see \cite{abrep}), and also record the following result. \begin{lemma}[Bouchet \cite{abrep}]\label{l.rep}
Let $E$ be a finite set, $A\subseteq E$, and $M$ be a skew-symmetric $|E|\times |E|$ matrix over a field $\mathbb{K}$ with rows and columns indexed by $E$. Then if $D=D(M)$ and $\emptyset \in \mathcal{F}(D\ast A)$, we have $D\ast A=D(N)$ for some skew-symmetric $|E|\times |E|$ matrix $N$ over $\mathbb{K}$. \end{lemma}
We now describe handle slides in terms of matrices. \begin{definition}\label{d.2}
Let $E$ be a finite set and $M$ be a symmetric $|E|\times|E|$ matrix over $GF(2)$ with rows and columns indexed by the elements of $E$, and let $a,b\in E$ with $a\neq b$. We define $M_{ab}$ to be the matrix obtained from $M$ by adding the column of $b$ to the column of $a$, then, in the resulting matrix, adding the row of $b$ to the row of $a$. We say that $M_{ab}$ is obtained by a \emph{handle slide}, or by \emph{handle sliding $a$ over $b$}. \end{definition} Note that in Definition~\ref{d.2} adding the row of $b$ to the row of $a$ then, in the resulting matrix, the column of $b$ to the column of $a$ also results in the matrix $M_{ab}$. Definition~\ref{d.2} by no means describes a new operation on matrices. For example the operation was considered by R.~Kirby in the context of handle slides and 3-manifolds in \cite{MR0467753}.
The following theorem shows that all the concepts of handle slides defined here agree. \begin{theorem}\label{con2} Let $M$ be a symmetric matrix over $GF(2)$. Then \[ D(M_{ab})=D(M)_{ab}. \] \end{theorem} \begin{proof} We need to show that $D(M_{ab})$ and $D(M)_{ab}$ have the same feasible sets. In view of the definition of handle slides, Definition~\ref{d1}, it suffices to prove that, for $Y\subseteq E$, \begin{enumerate} \item \label{con2.a} $ \det\left( M_{ab}[Y] \right) = \det\left( M[Y] \right) $ if $a\notin Y$, \item \label{con2.b} $ \det\left( M_{ab}[Y] \right) = \det\left( M[Y] \right) $ if $a,b\in Y$, and \item \label{con2.c} $\det(M_{ab}[Y])=\det(M[Y])+\det(M[Y \bigtriangleup \{a, b\}])$ if $a\in Y$ and $b\notin Y$. \end{enumerate}
The first item is trivial since $M[Y]=M_{ab}[Y]$ when $a\notin Y$.
For the second item, suppose that $a,b\in Y$. Observe that in this case applying the construction in Definition~\ref{d.2} to $M[Y]$ results in $M_{ab}[Y]$ (i.e., $(M[Y])_{ab}=M_{ab}[Y]$). Since adding one row or column of a matrix to another row or column does not change the determinant, $\det(M[Y])=\det(M_{ab}[Y])$.
For the third item, suppose that $a\in Y$ and $b\notin Y$. Set $X=Y\setminus a$, so that $Y=X\cup a$ and $Y \bigtriangleup \{a, b\} = X\cup b$. We then need to show \begin{equation}\label{e.con2a} \det(M_{ab}[X\cup a])=\det(M[X\cup a])+\det(M[X\cup b]). \end{equation}
(We will work in terms of the set $X$, rather than $Y$, as it simplifies the exposition.) Suppose that $M[X\cup \{a,b\}]=[a_{i,j}]_{1\leq i,j\leq n}$. Without loss of generality, we assume that $a$ indexes the first row and column of $M[X\cup \{a,b\}]$, and $b$ indexes the second. Then $M[X] = [a_{i,j}]_{3\leq i,j\leq n} $; $M[X\cup a]$ is the $(n-1)\times(n-1)$ matrix obtained from $M[X\cup \{a,b\}]$ by deleting its second row and column; $M[X\cup b]$ is the $(n-1)\times(n-1)$ matrix obtained from $M[X\cup \{a,b\}]$ by deleting its first row and column; and $M_{ab}[X\cup a]$ is the $(n-1)\times(n-1)$ matrix whose first row is $ \begin{bmatrix} a_{1,1}+a_{2,2} & a_{1,3}+a_{2,3} & \cdots & a_{1,n}+a_{2,n}\end{bmatrix}$, first column is $\begin{bmatrix} a_{1,1}+a_{2,2} & a_{3,1}+a_{3,2} & \cdots & a_{n,1}+a_{n,2}\end{bmatrix}^T$, with the rest of the matrix given by $M[X]$.
Letting $M[X]_{i,j}$ denote the matrix obtained by deleting the $i$-th row and $j$-th column of $M[X]$, using the Laplace (cofactor) expansion of the determinant, expanding down the first row and column, gives \begin{multline}\label{e.con2b} \det(M_{ab}[X\cup a]) = \left( a_{1,1}\det (M[X])+ \sum_{3\leq i,j\leq n} a_{1,i}a_{j,1}\det (M[X]_{i,j}) \right) \\ + \left( a_{2,2}\det (M[X])+ \sum_{3\leq i,j\leq n} a_{2,i}a_{j,2}\det (M[X]_{i,j}) \right) \\ + \left( \sum_{3\leq i,j\leq n} a_{1,i}a_{j,2}\det (M[X]_{i,j}) \right) +\left( \sum_{3\leq i,j\leq n} a_{2,i}a_{j,1}\det (M[X]_{i,j}) \right). \end{multline} By expanding down the first row and column of $M[X\cup a]$ and of $M[X\cup b]$, we see the first and second bracketed terms on the right-hand side of \eqref{e.con2b} equal $\det(M[X\cup a])$ and $\det(M[X\cup b])$, respectively. The remaining two sums in \eqref{e.con2b} are also determinants. Let $N$ be the $(n-1)\times(n-1)$ matrix whose first row is $ \begin{bmatrix} 0 & a_{1,3} & a_{1,4} & \cdots & a_{1,n}\end{bmatrix}$, first column is $\begin{bmatrix} 0 & a_{3,2} & a_{4,2} & \cdots & a_{n,2}\end{bmatrix}^T$, with the rest of the matrix given by $M[X]$. Let $P$ be the $(n-1)\times(n-1)$ matrix whose first row is $ \begin{bmatrix} 0 & a_{2,3} & a_{2,4} & \cdots & a_{2,n}\end{bmatrix}$, first column is $\begin{bmatrix} 0 & a_{3,1} & a_{4,1} & \cdots & a_{n,1}\end{bmatrix}^T$, with the rest of the matrix given by $M[X]$. By expanding down the first row and column of the $N$ and $P$, we see the third and fourth bracketed terms on the right-hand side of \eqref{e.con2b} equal $\det(N)$ and $\det(P)$, respectively. However, since $M[X\cup \{a,b\}]$ is symmetric, we see $N=P^T$, and so $\det(N)=\det(P)$. Since we are working over $GF(2)$, Equation~\eqref{e.con2a}, and so the theorem, holds. \end{proof}
The following observation is an immediate consequence of Lemma~\ref{l.rep} and Theorem~\ref{t.hs}. It should be contrasted with the observations made in Remark~\ref{r.1}. We note that Corollary~\ref{c.3} is generalised by Theorem~\ref{th.cl} where the assumption that the empty set is feasible is removed. \begin{corollary}\label{c.3} The set of binary delta-matroids in which the empty set is feasible is closed under handle slides. \end{corollary}
\begin{remark} A proof of Theorem~\ref{t.hs} in the special case where $G$ is a bouquet can be obtained from Theorem~\ref{con2}. The interlacement between, and the orientability of, edges of a bouquet $B$ can be used to obtain a matrix $M$ such that $D(B)=D(M)$ (see \cite{CMNR1} for a description of how). By examining how interlacement and orientability changes under a handle slide, it can be shown that $D(B_{ab})= D(M_{ab})$. Theorem~\ref{con2} then gives $D(M_{ab}) = D(M)_{ab}=D(B)_{ab}$. \end{remark}
Our starting point was the observation that handle slides can be used to put any bouquet into the form $B_{i,j,k}$. The following says that this result holds on the level of binary delta-matroids. \begin{lemma}\label{t.3} Let $D$ be a binary delta-matroid such that the empty set is feasible. Then there is a sequence of handle slides taking $D$ to $D_{i,j,k}$, for some $i,j,k$. \end{lemma} \begin{proof} Since the empty set is feasible, by Lemma~\ref{l.rep}, $D=D(M)$ for some symmetric matrix $M$ over $GF(2)$. We need to use handle slides and reordering of rows and columns to put $M$ in a block diagonal form in which each block is one of $\begin{bmatrix} 0\end{bmatrix}$, $\begin{bmatrix} 1 \end{bmatrix}$, or $\begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}$. (It is clear that the delta-matroid of such a matrix equals $D_{i,j,k}$, for some $i,j,k$.) To do this first observe that once we have a block of a matrix then a handle slide $M_{ab}$ preserves that block as long as $a$ does not index a row or column of it. Thus, by induction, it is enough to show that we can always use handle slides to construct a block of the required form in the matrix $M$.
If $M$ has a diagonal entry $m_{e,e}=1$. Then for each $f$ with $m_{f,e}=m_{e,f}=1$ handle slide $f$ over $e$. In the resulting matrix, all other entries of the $e$-th row and $e$-th column are zero, giving a block $\begin{bmatrix} 1 \end{bmatrix}$.
Now suppose all diagonal entries of $M$ are zero. If there is some $e$ such that all entries of the $e$-th row and $e$-th column are zero, then we have a block $\begin{bmatrix} 0\end{bmatrix}$.
Otherwise there is some $f$ with $m_{f,e}=m_{e,f}=1$. For convenience, and without loss of generality, we can reorder the rows and columns so that $e$ labels the first row and column, and $f$ labels the second. So we have the submatrix $\begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}$ in the top left corner of $M$. We need to use handle slides to make all other entries in the first two rows and columns zero. This can be done as follows. If $m_{i,e}=m_{e,i}=1$ and $m_{i,f}=m_{f,i}=0$ sliding $i$ over $f$ makes the $(i,e)$ and $(e,i)$ entries zero. If $m_{i,e}=m_{e,i}=0$ and $m_{i,f}=m_{f,i}=1$ sliding $i$ over $e$ makes the $(i,f)$ and $(f,i)$ entries zero. If $m_{i,e}=m_{e,i}=1$ and $m_{i,f}=m_{f,i}=1$ sliding $i$ over $f$, then $i$ over $e$ makes the four entries zero. Thus we can obtain a block $\begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}$, as required. This completes the proof of the lemma. \end{proof}
We can now prove Theorem~\ref{t.1}. \begin{proof}[Proof of Theorem~\ref{t.1}] By Lemma~\ref{t.3}, there is a sequence of handle slides taking $D$ to $D_{i,j,k}$, for some $i,j,k$. We have that $D_{i,j,k} = D(M_{i,j,k})$ where $M_{i,j,k}$ consist of $i$ blocks of $\begin{bmatrix} 0\end{bmatrix}$, $j$ blocks of $\begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}$, and $k$ blocks of the matrix $\begin{bmatrix} 1 \end{bmatrix}$.
It is readily seen from Definition~\ref{d1} that handle slides of delta-matroids preserve parity, so $D$ is odd if and only if $D_{i,j,k}$ is. A delta-matroid $D(M)$, where $M$ is a symmetric matrix over $GF(2)$, is odd if and only if there is a 1 on the diagonal of $M$ (this follows from the fact that a symmetric matrix of odd size over $GF(2)$ with zeros on the diagonal must be singular). Thus $D$ is even if and only if $D_{i,j,k}$ has $k= 0$, and the even case of the theorem follows.
Now suppose that $D$ is odd. Then handle slides can be used to put it in the form $D_{i,j,k}$ with $k>0$. It remains to put this $D_{i,j,k}$ in the form $D_{i,0,p}$ for some $p\in \mathbb{N}$. If $j=0$ we are done, otherwise, possibly after reordering rows and columns, there is a block $\begin{bmatrix} 0 & 1 &0\\ 1 & 0 &0\\ 0&0&1\end{bmatrix}$ whose rows and columns are labelled by $a,b,c$, say, in that order. The sequence of handle slides $a$ over $c$, $c$ over $b$, and $b$ over $a$ transforms this into the $3\times 3$ identity matrix. It follows that if $D_{i,j,k}$ has $k\neq 0$, then there is a sequence of handle slides taking $M$ to $D_{i,0,k+2j}$, completing the proof of the first part of the theorem.
For the second claim, suppose that there are sequences of handle slides take $D$ to $D_{i,j,k}$ and to $D_{p,q,r}$. Then there is a sequence of handle slides taking $D_{i,j,k}$ to $D_{p,q,r}$. Since a determinant of a block diagonal matrix is the product of the determinants of its blocks, the size of the largest feasible set in $D_{i,j,k}$ is $|E|-i$, and in $D_{p,q,r}$ is $|E|-p$. Upon observing from Definition~\ref{d1} that handle slides preserve the size of the largest feasible sets, we have that $i=p$, as required. \end{proof}
It is worth emphasising that we have shown that if $D$ can be taken to $D_{i,j,k}$, then $2j+k$ is the size of the largest feasible set in $D$, and $i$ is the size of the ground set minus this number.
\begin{corollary}\label{c.2} Let $D=(E,\mathcal{F})$ be a binary delta-matroid in which the empty set is feasible and such that there is a sequence of handle slides taking $D$ to $D_{i,j,k}$. \begin{enumerate} \item Suppose $D$ is even. There is a sequence of handle slides taking $D$ to $D_{p,q,r}$ if and only if $p=i$, $q=j$, and $r=k=0$.
\item Suppose $D$ is odd. There is a sequence of handle slides taking $D$ to $D_{p,q,r}$ if and only if $p=i$, $q=\ell$, and $r=|E|-i-2\ell$, for some $0\leq \ell\leq \lfloor \frac{|E|-i}{2} \rfloor$. \end{enumerate} \end{corollary} \begin{proof} The first item follows from Theorem~\ref{t.1} upon noting that handle slides preserve parity.
For the second item, suppose $D$ is odd. By Theorem~\ref{t.1}, $D_{i,\ell,|E|-i-2\ell}$ can be taken to $D_{i,0,|E|-i}$ using handle slides, and $D_{i,j,k}$ can be taken to $D_{i,0,|E|-i}$, thus $D$ can be taken to $D_{i,\ell,|E|-i-2\ell}$ by handle slides. Conversely, by Theorem~\ref{t.1}, $D_{i,j,k}$ and $D_{p,q,r}$ can both be taken to $D_{i,0,|E|-i}$ by handle slides, and so $i=p$ and $2q+r=|E|-i$, and result follows. \end{proof}
Theorem~\ref{t.hs} can be used to show that ribbon graphic delta-matroids are not closed under handle slides. Choose a binary delta-matroid $D$ with empty set feasible that is not ribbon graphic. There is a sequence of handle slides taking $D$ to a ribbon graphic delta matroid $D_{i,j,k}$. Thus there must be a handle slide between a graphic and non-graphic delta-matroid. Despite this, the following result says that we can always work with handle slides within the class of ribbon graphic delta-matroids. \begin{corollary}\label{c.1} Let $D=(E,\mathcal{F})$ be a ribbon graphic delta-matroid in which the empty set is feasible. If there is a sequence of handle slides taking $D$ to $D_{i,j,k}$, then there is a sequence of handle slides in which every delta-matroid is ribbon graphic that takes $D$ to $D_{i,j,k}$. \end{corollary} \begin{proof} First suppose that $D$ is even. Then, by Theorem~\ref{t.1}, $D_{i,j,k}= D_{i,j,0}$. Since $D$ is ribbon graphic $D=D(B)$ for some bouquet $B$. By Proposition~\ref{hs} there is a sequence of (ribbon graph) handle slides taking $B$ to $B_{p,q,0}$. Taking the delta-matroids of the ribbon graphs that appear in this sequence and applying Theorem~\ref{t.hs} gives a sequence of (delta-matroid) handle slides, in which every delta-matroid is ribbon graphic, that takes $D$ to $D_{p,q,0}$. The result then follows by Corollary~\ref{c.2}.
Now suppose that $D$ is odd. Then, by Corollary~\ref{c.2}, $ D_{i,j,k}= D_{i,\ell,|E|-i-2\ell}$ for some $0\leq \ell\leq \lfloor \frac{|E|-i}{2} \rfloor$.
Since $D$ is ribbon graphic $D=D(B)$ for some bouquet $B$. By Proposition~\ref{hs} there is a sequence of (ribbon graph) handle slides taking $B$ to $B_{p,0,r}$. For a bouquet $H$ consisting of three non-interlaced non-orientable loops $a$, $b$, and $c$ whose ends appear in the order $aabbcc$ when travelling round the vertex, observe that $((H_{cb})_{ba})_{ac}$ consists of a pair of interlaced orientable loops $b$ and $c$, and a non-interlace non-orientable loop $a$. It follows that there is a sequence of (ribbon graph) handle slides taking $B_{p,0,r}$, and hence $B$, to $B_{p,m,r-2m}$ for each $0\leq m \leq \lfloor \frac{|E|-p}{2} \rfloor$. Taking the delta-matroids of the ribbon graphs that appear in this sequence from $B$, and applying Theorem~\ref{t.hs}, gives a sequence of (delta-matroid) handle slides in which every delta-matroid is ribbon graphic that takes $D$ to $D_{p,m,r-2m}$. By Corollary~\ref{c.2}, for some $m$, $D_{p,m,r-2m}=D_{i,\ell,|E|-i-2\ell}=D_{i,j,k}$, and the result follows. \end{proof}
\begin{remark} It is natural to ask if the binary condition in Theorem~\ref{t.1} can be dropped. That is, can every delta-matroid in which the empty set is feasible be taken to a canonical form $D_{i,j,k}$ by a sequence of handle slides? The answer is no. For example, the delta-matroid over $E=\{1,2,3\}$ with feasible sets $\mathcal{F}=\{ \{1,2,3\}, \{1,2\}, \{1,3\}, \{2,3\},\emptyset\}$ cannot be. (Alternatively, that the answer is no follows from Theorem~\ref{th.cl} below since the $D_{i,j,k}$ are binary.) However, there should be a version of Theorem~\ref{t.1} that includes non-binary delta-matroids or set systems. The key problem is determining the canonical forms (i.e., the analogues of the $D_{i,j,k}$, which may not be delta-matroids) for other classes of delta-matroids. \end{remark}
\section{Closure under handle slides}\label{s.4} Although handle slides are defined for all delta-matroids, because of our motivation from the classification of bouquets we have so far focused on delta-matroids in which the empty set is feasible. We now examine what happens when it is not.
\begin{theorem}\label{th.cl} The set of binary delta-matroids is closed under handle slides. \end{theorem} \begin{proof} For any delta-matroid $D$, $A\subseteq E(D)$, and $a,b\in E(D)$ with $a\neq b$. If $a, b\notin A$, \begin{equation}\label{e.cl1}
D_{ab}\ast A=(D\ast A)_{ab}, \end{equation} and if $a, b\in A$, \begin{equation}\label{e.cl4}
D_{ab}\ast A = (D\ast A)_{ba} . \end{equation} Equation~\eqref{e.cl1} follows easily from the observation that, since $a, b\notin A$, for any $F\subseteq E(D)$, either of $a$ or $b$ is in $F$ if and only if it is in $F\bigtriangleup A$. Equation~\eqref{e.cl4} follows by direct computation. Start by writing \[ \mathcal{F} (D\ast A) = \left\{ X_i, Y_j\cup a, Z_k\cup b, W_l\cup a, W_l\cup b , T_m\cup\{a,b\}\right\}_{i\in \mathcal{I},j\in \mathcal{J},k\in \mathcal{K},l\in \mathcal{L}, m\in \mathcal{M} } , \] where $X_i, Y_j,Z_k,W_l \subseteq E\setminus \{a,b\}$, none of the $Y_j,Z_k,W_l$ are equal, and where the $\mathcal{I}$, $\mathcal{J}$, $\mathcal{K}$, $\mathcal{L}$, and $\mathcal{M}$ are indexing sets. From this it is easy to compute the feasible sets of $(D\ast A)_{ba}$, $D$, $D_{ab}$, and $D_{ab}\ast A$, upon which it is seen that $\mathcal{F} (D\ast A) = \mathcal{F}( (D\ast A)_{ba})$, and Equation~\eqref{e.cl4} follows.
Now suppose that $D=(E,\mathcal{F})$ is a binary delta-matroid and $a,b\in E$ with $a\neq b$. Then there is some $A\subseteq E$ and some symmetric matrix $M$ over $GF(2)$ such that $D\ast A=D(M)$. We need to show that $D_{ab}$ is binary. That is, we need to show that $D_{ab}\ast B=D(N)$ for some $B\subseteq E$ and some symmetric matrix $N$ over $GF(2)$. We will consider four cases given by the membership of $a$ and $b$ in $A$.
\noindent \underline{Case 1:} Suppose that $a,b\notin A$. Then, by Equation~\eqref{e.cl1} and Theorem~\ref{con2}, \begin{equation}\label{e.cl2}
D_{ab}\ast A = (D\ast A)_{ab} = D(M)_{ab}= D(M_{ab}), \end{equation} and so $D_{ab}$ is binary.
\noindent \underline{Case 2:} Suppose that $a\in A$ and $b\in A$. Then, by Equation~\eqref{e.cl4} and Theorem~\ref{con2}, \begin{equation}\label{e.cl5}
D_{ab}\ast A = (D\ast A)_{ba} = D(M)_{ba}= D(M_{ba}), \end{equation} and so $D_{ab}$ is binary.
\noindent \underline{Case 3:} Suppose that $a\in A$ and $b\notin A$. If there is some $F\in \mathcal{F}(D\ast A)$ with $a\in F$ and $b\notin F$, then by Lemma~\ref{l.rep}, we see that $D\ast(A\bigtriangleup F)= D(N)$, for some symmetric matrix $N$ over $GF(2)$. Since $ a,b\notin A\bigtriangleup F$, Case 1 applies and so $D_{ab}$ is binary.
Similarly, if there is some $F\in \mathcal{F}(D\ast A)$ with $a\notin F$ and $b\in F$, then by Lemma~\ref{l.rep}, $D\ast(A\bigtriangleup F)= D(N)$, for some symmetric matrix $N$ over $GF(2)$. Since $ a,b\in A\bigtriangleup F$, Case 2 now applies and so $D_{ab}$ is binary.
Otherwise every feasible set of $D\ast A$ contains both $a$ and $b$, or neither of $a$ or $b$. Suppose this is the case. There is either some $F\in \mathcal{F}(D\ast A)$ containing both $a$ and $b$ or there is not.
First suppose that there is, and let $F\in \mathcal{F}(D\ast A)$ with $a,b\in F$. Let $X \in \mathcal{F}(D\ast A)$ be such that $a,b\notin X$ (we know such a set exists, since the empty set is feasible). Then $a\in X\bigtriangleup F$, and by the Symmetric Exchange Axiom, $X\bigtriangleup \{a,u\}\in \mathcal{F}$ for some $u\in X\bigtriangleup F$. Since, by hypothesis, $a$ or $b$ cannot appear in a feasible set without the other, we must have $u=b$, and so $X\cup \{a,b\}\in \mathcal{F}$. Similarly, the Symmetric Exchange Axiom gives that $F\bigtriangleup \{a,u\}\in \mathcal{F}$ for some $u\in X\bigtriangleup F$. Again we must have that $b=u$ and so $F\setminus \{a,b\}\in \mathcal{F}$. These two observations together give that we can partition the feasible sets of $D\ast A$ to get
$\mathcal{F} (D\ast A) = \left\{ X_i, X_i\cup\{a,b\}\right\}_{i\in \mathcal{I}} $, where $X_i\subseteq E\setminus \{a,b\}$ and $\mathcal{I}$ is an indexing set. From this we see that
$\mathcal{F} (D) = \left\{ \hat{X}_i\cup a, \hat{X}_i\cup b \right\}_{i\in \mathcal{I}} $, where for each set $X_i$, $\hat{X_i}$ denotes $X_i\bigtriangleup (A\setminus\{a,b\})$, and that
$\mathcal{F} (D_{ab}) = \left\{ \hat{X}_i\cup b \right\}_{i\in \mathcal{I}}$. We then see that $D_{ab} = D\setminus a$. Since $D$ is binary, and the set of binary delta-matroids is minor-closed, it follows that $D_{ab} $ is binary.
All that remains is the case where no feasible set of $D\ast A$ contains $a$ or $b$ (so $a$ and $b$ are loops). In this case each feasible set of $D$ contains $a$ but not $b$, and it follows that $D=D_{ab}$. Since $D$ is binary, so is $D_{ab}$.
\noindent \underline{Case 4:} Suppose that $a\notin A$ and $b\in A$. If there is some $F\in \mathcal{F}(D\ast A)$ with $a\notin F$ and $b\in F$, then, by Lemma~\ref{l.rep}, $D\ast(A\bigtriangleup F)= D(N)$, for some symmetric matrix $N$ over $GF(2)$. Since $ a,b\notin A\bigtriangleup F$, Case 1 now applies and so $D_{ab}$ is binary.
If there is some $F\in \mathcal{F}(D\ast A)$ with $a\in F$ and $b\notin F$, then $D\ast(A\bigtriangleup F)= D(N)$. Since $ a,b \in A\bigtriangleup F$, Case 2 now applies and so $D_{ab}$ is binary.
If there is some $F\in \mathcal{F}(D\ast A)$ with $a\in F$ and $b\in F$, then $D\ast(A\bigtriangleup F)= D(N)$. Since $ a\in A\bigtriangleup F$ and $b\notin A\bigtriangleup F$, Case 3 now applies and so $D_{ab}$ is binary.
All that remains is the case in which no feasible set of $D\ast A$ contains $a$ or $b$ (so $a$ and $b$ are loops). In this case we can write $\mathcal{F} (D\ast A) = \left\{ X_i\right\}_{i\in \mathcal{I}} $,
where $X_i\subseteq E\setminus \{a,b\}$ and $\mathcal{I}$ is an indexing set. From this we see that $\mathcal{F} (D_{ab} \ast A) = \left\{ \hat{X}_i\cup \{a, b\}, \hat{X}_i \right\}_{i\in \mathcal{I}} $, and that $D_{ab} \ast A = (D\ast A) \oplus D_{0,1,0} $. Since both $D\ast A$ and $D_{0,1,0}$ are binary, it follows that $D_{ab}$ is. This completes the proof of the theorem. \end{proof}
Since handle slides preserve the maximum (and minimum) sizes of a feasible set in a delta-matroid, we have the following. \begin{corollary} The set of binary matroids is closed under slides. \end{corollary}
The problem of extending Theorem~\ref{t.1} to all binary delta-matroids now arises. One way to try to extend the Theorem is to augment the set of canonical forms to include delta-matroids $D_{i,j,k,l}$ consisting of the direct sum of $D_{i,j,k}$ with $l$ copies of delta-matroids isomorphic to $(\{e\}, \{\{e\}\})$. We conjecture that a version of Theorem~\ref{t.1} holds for all binary delta-matroids with these terminal forms. \begin{conjecture}\label{conj} For each binary delta-matroid $D$, there is a sequence of handle slides taking $D$ to some $D_{i,j,k,l}$ where $i$ is the size of the ground set minus the size of a largest feasible set, $l$ is the size of a smallest feasible set, $2j+k$ is difference in the sizes of a largest and a smallest feasible set. Moreover, $k=0$ if and only if $D$ is even, and if $D$ is odd then every value of $j$ from 0 to $ \lfloor \frac{w}{2} \rfloor$, where $w$ is the difference between the sizes of a largest and a smallest feasible set, can be attained. \end{conjecture} Conjecture~\ref{conj} is true for ribbon graphic delta-matroids. This can be proven by induction on the size of a smallest feasible set. The base case is Theorem~\ref{t.1}. For the inductive step take a ribbon graph $G$ such that $D=D(G)$; choose any non-loop edge $e=(u,v)$ of $G$; handle slide each edge, other than $e$, that is incident with $u$ over $e$ so that $u$ becomes a degree 1 vertex; the corresponding handle slides in $D(G)$ transform it into a ribbon graphic delta-matroid with a direct summand $(\{e\}, \{\{e\}\})$. The main step in this argument is that any non-loop element $e$ of a ribbon graphic delta-matroid can be transformed into a coloop using only handle slides over $e$. This result does not hold for delta-matroids in general. For example, it is readily checked that in the uniform matroid $U_{2,4}$ no element $e$ can be transformed into a coloop using only handle slides over $e$. In fact, with a little more work, it can be checked that no sequence of handle slides applied to $U_{2,4}$ will create a coloop. Of course the (delta-)matroid $U_{2,4}$ is not binary and so this example says nothing about the validity of Conjecture~\ref{conj}. However it does indicate that any approach to isolating $(\{e\}, \{\{e\}\})$ for the conjecture will be intimately tied to the binary structure of the delta-matroid.
It is perhaps also worth commenting on the alternative approach of considering sequences of twists, and handle slides that can only act on delta-matroids in which the empty set is feasible, rather than just sequences of handle slides. Such an extension results in non-unique terminal forms. For example if $D=(\{e,f\}, \{ \emptyset, \{e\}, \{e,f\} \})$ then there is a sequence of handle slides taking $D$ to $D_{0,0,2}$, but also there is a sequence of handle slides taking $D\ast e$ to $D_{1,0,1}$.
In fact, ribbon graph theory indicates that this approach should fail. The ribbon graph analogue is to consider ribbon graphs up to partial duals (see \cite{Ch09,CMNR1}), and handle slides that can only act on bouquets. But partial duality changes the topology of a surface, and so our choice of terminal form will need to reflect this. Nevertheless, this relation on binary delta-matroids will result in some set of terminal forms. What are they?
We conclude with one final open question. We have seen that binary delta-matroids are closed under handle slides, but that delta-matroids, in general, are not. What classes of delta-matroids are closed under handle slides?
\end{document} | arXiv |
3.2 Calculating x and y Force Components in Truss Members
The equilibrium equations in \eqref{eq:TrussEquil} define forces in terms of x- and y- directions.
\begin{equation}\label{eq:TrussEquil} \tag{1} \sum_{i=1}^{n}{F_{xi}} = 0; \sum_{i=1}^{p}{F_{yi}} = 0; \end{equation}
Truss members are often inclined relative to the horizontal as shown in Figure 3.2.
Figure 3.2: Free Body Diagram of an Angled Truss Member with Corresponding Joint Forces and Force Components
This figure shows a free body diagram of a truss member in tension with the forces at either end of the member parallel to the member itself and pointing away from it. Using our sign convention, this represents a positive axial force. The truss member axial forces ($F$ in the figure) are typically the unknown forces that we are trying to calculate in a truss analysis. As previously discussed in Section 1.6, although there is a sign convention for the internal axial force itself (positive for tension, negative for compression), in our solution, we will have to consider the actual direction of force arrows (as external forces) in a free body diagram of the member or the connected joints. For the axial force $F$ shown, the force on the left of the member points down and to the left, while the force on the right points up and to the right. These forces in opposite directions together create the internal tension axial force in the truss member.
The free body diagram of the truss member shown on the left side of Figure 3.2 is cut at the location of the hinges at either end; however, at either end of the member, it is connected to a joint, which may be represented as a single point as shown. Note that the joints shown in the figure are not free body diagrams because they are clearly not in equilibrium (unless $F=0$). The force on the joint caused by the truss member is in the opposite direction of the force on the truss member caused by the joint. This follows from Newton's third law and was described previously in Section 1.6, where it was shown that the forces on either side of a cut in a structure must be equal in magnitude and opposite in direction. In a truss analysis, it is important to keep careful track of which way the forces are pointing. In general, a tension force will point away from a truss member end or a joint and a compression force will point towards a member end or joint.
The $x$- and $y$-direction force components of the total force in a truss member, may be found using either the slope of the member or the angle relative to the horizontal (shown as $\theta$ in Figure 3.2).
To use the slope, you need to know the relative lengths $L$, $L_x$ and $L_y$, all of which may be easily found using the geometry of the truss. The lengths $L_x$ and $L_y$ are the projected lengths of the member total length $L$ onto the $x$ or $y$ axes, respectively. The total force and force components are relative to their lengths, so:
\begin{align} \tag{2} F_x = \left( \frac{L_x}{L} \right) F \\ \tag{3} F_y = \left( \frac{L_y}{L} \right) F \end{align}
Often, you know the projected lengths $L_x$ and $L_y$ and can use these to find the total length $L$ using the Pythagorean Theorem:
\begin{equation} \tag{4} L = \sqrt{L_x^2 + L_y^2} \end{equation}
Alternately, the force component may be found using trigonometry and the brace angle $\theta$. Using trigonometry, the force components are simply equal to:
\begin{align} \tag{5} F_x = F \cos \theta \\ \tag{6} F_y = F \sin \theta \end{align}
The angle $\theta$ may be found using trigonometry if it is not given. If the projected lengths $L_x$ and $L_y$ are known, then:
\begin{equation} \theta = \arctan \left( \frac{L_y}{L_x} \right) \label{eq:incl-angle}\tag{7} \end{equation}
In the end, both of these methods are identical, the only difference being that they use different starting information. The most efficient method to use will depend on what information is known at the start of a truss analysis problem. Both methods will work though, regardless of the problem.
Book traversal links for 3.2 Calculating x and y Force Components in Truss Members
3.3 Identifying Zero Force Members
3.4 Using Global Equilibrium to Calculate Reactions
3.5 The Method of Joints
3.6 The Method of Sections
3.7 Practice Problems | CommonCrawl |
Dobiński's formula
In combinatorial mathematics, Dobiński's formula[1] states that the n-th Bell number Bn (i.e., the number of partitions of a set of size n) equals
$B_{n}={1 \over e}\sum _{k=0}^{\infty }{\frac {k^{n}}{k!}},$
where $e$ denotes Euler's number. The formula is named after G. Dobiński, who published it in 1877.
Probabilistic content
In the setting of probability theory, Dobiński's formula represents the nth moment of the Poisson distribution with mean 1. Sometimes Dobiński's formula is stated as saying that the number of partitions of a set of size n equals the nth moment of that distribution.
Reduced formula
The computation of the sum of Dobiński's series can be reduced to a finite sum of $n+o(n)$ terms, taking into account the information that $B_{n}$ is an integer. Precisely one has, for any integer $K>1$
$B_{n}=\left\lceil {1 \over e}\sum _{k=0}^{K-1}{\frac {k^{n}}{k!}}\right\rceil $
provided ${\frac {K^{n}}{K!}}\leq 1$ (a condition that of course implies $K>n$, but that is satisfied by some $K$ of size $n+o(n)$). Indeed, since $K>n$, one has
${\Big (}{\frac {K+j}{K}}{\Big )}^{n}\leq {\Big (}{\frac {K+j}{K}}{\Big )}^{K}={\Big (}1+{\frac {j}{K}}{\Big )}^{K}\leq {\Big (}1+{\frac {j}{1}}{\Big )}{\Big (}1+{\frac {j}{2}}{\Big )}\dots {\Big (}1+{\frac {j}{K}}{\Big )}={\frac {1+j}{1}}{\frac {2+j}{2}}\dots {\frac {K+j}{K}}={\frac {(K+j)!}{K!j!}}.$
Therefore ${\frac {(K+j)^{n}}{(K+j)!}}\leq {\frac {K^{n}}{K!}}{\frac {1}{j!}}\leq {\frac {1}{j!}}$ for all $j\geq 0$ so that the tail $\sum _{k\geq K}{\frac {k^{n}}{k!}}=\sum _{j\geq 0}{\frac {(K+j)^{n}}{(K+j)!}}$ is dominated by the series $\sum _{j\geq 0}{\frac {1}{j!}}=e$, which implies $0<B_{n}-{\frac {1}{e}}\sum _{k=0}^{K-1}{\frac {k^{n}}{k!}}<1$, whence the reduced formula.
Generalization
Dobiński's formula can be seen as a particular case, for $x=0$, of the more general relation:
${1 \over e}\sum _{k=x}^{\infty }{\frac {k^{n}}{(k-x)!}}=\sum _{k=0}^{n}{\binom {n}{k}}B_{k}x^{n-k}$
and for $x=1$ in this formula for Touchard polynomials
$T_{n}(x)=e^{-x}\sum _{k=0}^{\infty }{\frac {x^{k}k^{n}}{k!}}$
Proof
One proof[2] relies on a formula for the generating function for Bell numbers,
$e^{e^{x}-1}=\sum _{n=0}^{\infty }{\frac {B_{n}}{n!}}x^{n}.$
The power series for the exponential gives
$e^{e^{x}}=\sum _{k=0}^{\infty }{\frac {e^{kx}}{k!}}=\sum _{k=0}^{\infty }{\frac {1}{k!}}\sum _{n=0}^{\infty }{\frac {(kx)^{n}}{n!}}$
so
$e^{e^{x}-1}={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {1}{k!}}\sum _{n=0}^{\infty }{\frac {(kx)^{n}}{n!}}$
The coefficient of $x^{n}$ in this power series must be $B_{n}/n!$, so
$B_{n}={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {k^{n}}{k!}}.$
Another style of proof was given by Rota.[3] Recall that if x and n are nonnegative integers then the number of one-to-one functions that map a size-n set into a size-x set is the falling factorial
$(x)_{n}=x(x-1)(x-2)\cdots (x-n+1)$
Let ƒ be any function from a size-n set A into a size-x set B. For any b ∈ B, let ƒ −1(b) = {a ∈ A : ƒ(a) = b}. Then {ƒ −1(b) : b ∈ B} is a partition of A. Rota calls this partition the "kernel" of the function ƒ. Any function from A into B factors into
• one function that maps a member of A to the element of the kernel to which it belongs, and
• another function, which is necessarily one-to-one, that maps the kernel into B.
The first of these two factors is completely determined by the partition π that is the kernel. The number of one-to-one functions from π into B is (x)|π|, where |π| is the number of parts in the partition π. Thus the total number of functions from a size-n set A into a size-x set B is
$\sum _{\pi }(x)_{|\pi |},$
the index π running through the set of all partitions of A. On the other hand, the number of functions from A into B is clearly xn. Therefore, we have
$x^{n}=\sum _{\pi }(x)_{|\pi |}.$
Rota continues the proof using linear algebra, but it is enlightening to introduce a Poisson-distributed random variable X with mean 1. The equation above implies that the nth moment of this random variable is
$E(X^{n})=\sum _{\pi }E((X)_{|\pi |})$
where E stands for expected value. But we shall show that all the quantities E((X)k) equal 1. It follows that
$E(X^{n})=\sum _{\pi }1,$
and this is just the number of partitions of the set A.
The quantity E((X)k) is called the kth factorial moment of the random variable X. To show that this equals 1 for all k when X is a Poisson-distributed random variable with mean 1, recall that this random variable assumes each value integer value $j\geq 0$ with probability $1/(ej!)$. Thus
$E((X)_{k})=\sum _{j=0}^{\infty }{\frac {(j)_{k}}{ej!}}={\frac {1}{e}}\sum _{j=0}^{\infty }{\frac {j(j-1)\cdots (j-k+1)}{j(j-1)\cdots 1}}={\frac {1}{e}}\sum _{j=i}^{\infty }{\frac {1}{(j-i)!}}=1.$
Notes and references
1. Dobiński, G. (1877). "Summirung der Reihe $\textstyle \sum {\frac {n^{m}}{n!}}$ für m = 1, 2, 3, 4, 5, …". Grunert's Archiv (in German). 61: 333–336.
2. Bender, Edward A.; Williamson, S. Gill (2006). "Theorem 11.3, Dobiński's formula". Foundations of Combinatorics with Applications (PDF). Dover. pp. 319–320. ISBN 0-486-44603-4.
3. Rota, Gian-Carlo (1964), "The number of partitions of a set" (PDF), American Mathematical Monthly, 71 (5): 498–504, doi:10.2307/2312585, JSTOR 2312585, MR 0161805.
| Wikipedia |
Modeling of non-Gaussian colored noise and application in CR multi-sensor networks
Zheng Dou1,
Chengzhuo Shi1,
Yun Lin1 &
Wenwen Li1
EURASIP Journal on Wireless Communications and Networking volume 2017, Article number: 192 (2017) Cite this article
Motivated by the practical and accurate demand of intelligent cognitive radio (CR) sensor networks, a new modeling method of practical background noise and a novel sensing scheme are presented, where the noise model is the non-Gaussian colored noise based on α stable process and the sensing method is improved fractional low-order moment (FLOM) detection algorithm with balance parameter. First, we establish the non-Gaussian colored noise model through combining α-distribution with a linear system represented by a matrix. And a fitting curve of practical noise data is given to verify the validity of the proposed model. Then we present a parameter estimation method with low complexity to obtain the balance parameter, which is an important part of the detection algorithm. The balance parameter-based FLOM (BP-FLOM) detector does not require any a priori knowledge about the primary user signal and channels. Monte Carlo simulations clearly demonstrate the performance of the proposed method versus the generalized signal-to-noise ratio, the characteristic exponent α, and the number of detectors in sensing networks.
The cognitive radio (CR) nodes with sensing and adaptive abilities have been recognized as a promising solution [1] to realize the next-generation intelligent sensing networks; the key ideas behind detector nodes lie in sensing spectrum information accurately under the practical noise background. Gaussian white noise are typically used to model practical noise processes that affect digital sensing systems [1], such as the multi-radar system and underwater acoustic detection system. In practice, however, Gaussian models reveal difficulties in fitting data that often have distinct spiky and impulsive characteristics leading them deviate from Gaussian distributions which is known as non-Gaussian [2]. Such non-Gaussian makes the common Gaussian assumption not valid for traditional spectrum sensing [3]. One of the most important challenges in sensor networks is to detect as quickly and reliably as possible the absence or presence of the signal in complex radio environments such as those characterized by non-Gaussian noises. Thus, the effective model of practical noise and the realization of accurate detection are the main problems to be solved.
Non-Gaussian noise impairments may result from human factors and the natural factors, such as man-made impulse noise, electromagnetic equipment, atmospheric storms, and out of band spectral leakage [4, 5]. The non-Gaussian noise model should not only take into account its exact description of the nature for the noise, but also the simplicity of the calculation. Large measured data show that the probability density distribution of the impulse noise process is similar to the Gaussian process: symmetrical, smooth, and bell-shaped, but its tail is heavier than the Gaussian distribution [6]. The Gaussian mixture density (GMD) [7], centered generalized Gaussian density (GDD) [8], and the symmetric α-stable (S αS) density distribution [9] are most common models in recent years. The S αS distribution has proved to be the most promising model to fit many impulsive noise processes in communications channels, and, in fact, includes the Gaussian density as a special case [10]. Due to its good performance, the α-stable distribution is used to fit the noise and interference in cognitive radio multi-sensor networks [11, 12], but the colored noise is not considered.
Many spectrum sensing schemes for non-Gaussian noise has been presented in many literatures. The performances of Cauchy detector and global optimal detector are not ideal in the non-Gaussian noise [13, 14]. Polarity-Coincidence-Array (PCA)-based spectrum sensing is proposed in [15], a significant performance enhancement is achieved by the PCA detector, but the prior knowledge such as the variance of the noise and the PU signal cyclic frequency are also needed in the algorithm, which is difficult in practical system. The Lp-Norm Spectrum Sensing method for cognitive radio networks is presented in [16]. It does not require any prior knowledge about the non-Gaussian noise, but the condition that the noise do not have second-order statistics and high-order statistics such as α-stable distribution are not taken into account, there are limitations when applied. The authors proposed a novel FLOM-based detector for the detection of a primary user in the S αS noises that can significantly enhance the detection performance compared to other detector algorithms [4, 17]. But it is same to other detection algorithms, it is applicable only in the background non-Gaussian white noise. In addition, the algorithm relies on the characteristic exponent α of the S αS distribution, but it is difficult to obtain in the actual detection process.
Although non-Gaussian noise in sensing networks are given a variety of modeling and spectrum detection algorithm [18], most of them remain in the simulation and limit to white noise. We sample the practical noise data in laboratory environment with CR equipment USRP X-300 and analyze the power spectrum density. The result shows that it is not flat, which means the CR sensing system is working in the background of colored noise. That is why the performance of the algorithm mentioned above is declined when applied to practice, as pointed in [19, 20]. Therefore, we propose a novel model to describe the non-Gaussian colored noise and present a new detection method to sensing signals. What is more, at special values, Gaussian white noise or non-Gaussian white noise can be included, which is more widely used in the practical system.
In this paper, we first fit the curve of practice noise data to study its characteristics and give a novel model to present the colored non-Gaussian noise through combining symmetric α-distribution with a linear system represented by a matrix. Then we give an improved method to estimate the parameters (characteristic exponent α and dispersion γ same to α-stable density distribution) of the new distribution from a time series [21]. According to the estimation result, we propose a new sensing method of balance parameter-based fractional lower order moment (BP-FLOM), referred to as BP-FLOM detector. No prior knowledge is needed and the calculation is simplified. We also investigate the detection performance with different characteristic exponent α and the performance at different signal-to-noise ratios. In addition, multi-sensor performance is also simulated to verify the validity.
The remainder of this paper is organized as follows: In Section 2, we present the analysis of practical noise and establish the model of non-Gaussian noise. The estimation method for the new model is proposed in Section 3, and we give a new BP-FLOM algorithm based on the estimation result. Simulations results and analysis are presented in Section 4 and we conclude the paper in Section 5.
Observation models and problem description
In this section, a brief description of the most commonly models used in survival literature for the CR sensing node is provided. They include the system model of CR networks [22] and the symmetric α-stable density distribution [17]. And the new model is presented based on the analysis of practical noise data.
Assume that the CR comprised of one Primer User (PU) and M Secondary Users (SU). The received observation vector at the multi-sensors CR form the PU at time n under each hypothesis (PU absent/present) is given by
$$ \begin{aligned} &\mathcal{H}_{0}: Y_{m}[n] = \xi_{m}[n], \quad\qquad\,\,\, n = 0, 1, \dots, N - 1.\\ &\mathcal{H}_{1}: Y_{m}[n] = S_{m}[n] + \xi_{m}[n], n = 0, 1, \dots, N - 1. \end{aligned} $$
where Y m [n]=[y 1(n),y 2(n),…,y m (n)]T is the sample of received signal by mth detector at the time n, N is the length of the sample sequence. ξ m [n]=[ξ 1(n),ξ 2(n),…,ξ m (n)]T is an additive background noise and S m [n] is primary signal to be detected. The primary signal is assumed to be random sequence of Gaussian distributions, and symmetric α-stable distribution has proved to be a good way to describe the non-Gaussian noise [4].
The probability density function (PDF) of an α-stable random variable cannot be given in closed form, but the characteristic function can always be given as followed [23]
$$ \phi(t) = \text{exp} \{-\gamma\lvert t \rvert^{\alpha} + i\delta t\} $$
where α∈(0,2] is the characteristic exponent, and it describes the tail of the distribution. The values α=2 and α=1 correspond to the Gaussian distribution and Cauchy distribution. The other two parameters are γ>0 for dispersion scale and δ∈R for location. Let X ∼(α,β,γ,δ), the symmetric α stable (S αS) distribution is given by X ∼(α,0,1,0) and only the S αS is considered in this paper.
Noise model
The power spectral density of the practical noise is shown in Fig. 1. It is obvious that the power spectral density of practical noise is not flat. So the S αS distribution can not describe the practical noise perfectly as it is colored. To describe the practical noise accurately, we analyze the characteristics of S αS distribution as follows.
Power spectral density of practical noise. The practical noise data are sampled in laboratory environment with the CR equipment USRP X-300
if X(n),∀n∈{1,2,…,N} are independent and identically distributed (I.I.D) copies of X and
$$ \sum_{n=1}^{N}a_{n}X^{(n)} \triangleq X_{new} $$
where N∈Z +and a n ,c∈R, then X new is the S αS distribution with same parameter α[17]. According to the nature of colored noise that the noise in the sequence is correlated at each moment, we consider the colored noise to be a linear output driven by a white noise sequence. We propose a novel model for non-Gaussian colored noise as
$$\begin{array}{@{}rcl@{}} \begin{aligned} \xi(n) &= X^{T}(n)\cdot A \\ &= \sum_{i = 1}^{n}{X_{i}\sim(\alpha, 0, a_{i}\gamma, 0)}\\ &= X_{nov}\sim(\alpha, 0, \gamma_{nov}, 0) \end{aligned} \end{array} $$
where X[n]=[x(n),x(n−1)],…,x(0)]T, is the sequence of S αS distribution. A=[a 1,a 2,…,a N ] is the linear transformation matrix. ξ(n)=X nov ∼(α,0,γ nov ,0) is the colored noise sequence and has the same characteristic exponent to X(n) according to [17]. In particular, when special parameter is taken as A=[a 1,0,…,0], the proposed model is the S αS distribution. The PDF of the new model based on S αS and different characteristic exponent α are plotted in Fig. 2. As expected, the heavy tail of non-Gaussian colored noise is consistent with the characteristics of the α distribution.
PDF tails of non-Gaussian colored noise with different characteristic exponent α
The power spectral density of the novel model is shown in Fig. 3, it can be concluded that it is very close to the practical noise that proves the validity of the model. As the S αS distribution has only the fractional lower moments, and its variance does not exist, the conventional signal-to-noise ratio is meaningless. So the signal-to-noise ratio for non-Gaussian colored noise is defined by mixed signal-to-noise, we call it general signal-to-noise ratio (GSNR)
$$ {GSNR}_{dB} = 10{log}_{10}\left(\frac{\sigma_{s}^{2}}{\gamma_{nov}}\right) $$
Power spectral density of simulation colored noise. The data length is 10e4 and the Fs = 10e4 MHz, α=1.8
where \(\sigma _{s}^{2}\) is the variance of Gaussian signal, and γ nov is the dispersion scale of Gaussian colored noise. The GSNR will be applied to the subsequent simulation analysis.
BP-FLOM-based spectrum sensing
In this section, we propose a new spectrum sensing scheme, namely balance parameter-based fractional low-order moment (BP-FLOM) detector, for the non-Gaussian colored noise background. Adopted parameter estimation method associated with the proposed BP-FLOM method along with the potential of employing BP-FLOM detector in cooperative sensing is presented.
Estimation of characteristic exponent α and γ nov
As introduced in Section 2, we assume that ξ=a ξ 1+b ξ 2, ξ 1 and ξ 2 are I.I.D sequence, the non-Gaussian colored noise ξ has the same characteristic exponent α with S αS, and a different dispersion scale γ nov . To simplify the calculation, we do not consider the specific value of γ nov , and assume it is a known parameter. Then we improve the estimation method in [21], so it has a finite factional lower order moment for a ξ 1+b ξ 2
$$ E(\lvert a\xi_{1} + b\xi_{2}\rvert^{p}) = C_{1}(p,\alpha)\gamma_{nov}^{p/\alpha}, \quad for\quad 0 < p < \alpha $$
where \(C_{1}(p,\alpha) = \frac {2^{p}\Gamma (\frac {p+1}{2})\Gamma (1-p/\alpha)}{\sqrt {\pi }\Gamma (1-p/2)}\), 0<α≤2, γ is the dispersion scale and Γ(·) is the gamma function. Then we obtained from (6)
$$ \log{\gamma_{nov}} = \frac{\alpha}{p}\log{\frac{E(\mid a\xi_{1} + b\xi_{2}\mid^{p})}{ C_{1}(p,\alpha)}} $$
If pth order of ξ satisfies (6), we can write E(|a ξ 1+b ξ 2|p)) as \(E(e^{p\log \lvert a\xi _{1} + b\xi _{2}\rvert })\) and define a new variable Z= log|a ξ 1+b ξ 2|. Then
$$ E(\lvert \xi\rvert^{p}) = E\left(e^{p\log \lvert a\xi_{1} + b\xi_{2} \rvert}\right) = E\left(e^{pZ}\right),\quad 0<p<\alpha $$
where E(e pZ) is the moment-generating function of Z. The power series can be expressed by
$$ E(e^{pZ})=\sum_{k=0}^{\infty}E(Z^{k})\frac{p^{k}}{k!} $$
It is obvious that the moment of Z is finite in any order, together with (6) we have
$$ E(Z^{k}) = \frac{d^{k}}{dp^{k}}(C_{1}(p,\alpha))\gamma_{nov}^{p/\alpha}, \quad p = 0 $$
Simplify the above equation according to [21]
$$ E\left(\log \lvert a\xi_{1} + b\xi_{2}\rvert\right) = C_{e}\left(\frac{1}{\alpha}-1\right)+\frac{1}{\alpha}\log\gamma_{nov} $$
where C e =0.57721566… is the Euler constant, α is the characteristic exponent, then combined (6) and (11) we have
$$ \begin{aligned} E\left\{\left(\log \lvert a\xi_{1}\vphantom{1^{2^{2}}_{2}}\right.\right. & \left.\left.\vphantom{1^{2^{2}}_{2}}+ b\xi_{2}\rvert-E\left[\log \lvert a\xi_{1} + b\xi_{2}\rvert\right]\right)^{2}\right\}\\[-1pt] &= \frac{\pi^{2}}{6}\left(\frac{1}{\alpha^{2}}+\frac{1}{2}\right) \end{aligned} $$
Equation (12) can be used to estimate α in this way without calculating the value of γ nov , and it is easily obtained with (11).
Simulation result is given in Table 1. As can be seen from the table, we can estimate the value of α exactly with a small relative error when the sampling of noise sequence is enough. The verification of the estimation method will be discussed in Section 4.
Table 1 Estimation of α. The estimation result of α with different sample size, where real value is 1.5 and 1.2
Spectrum sensing based on balance parameter
We have known that the fractional low-order moment detection (FLOM) has a good performance under the S αS distribution noise that is presented in [4], and it is more suitable than Cauchy detector. But when applied to the non-Gaussian colored noise background, the performance is declined. Based on the analysis in Section 2, we give the hypothesis A: The practical non-Gaussian colored noise is obtained by the linear transformation of the non-Gaussian white noise sequence obeying the S αS distribution.
Since the impulse response of linear transformation is difficult to compute without prior knowledge, we propose a simpler and more accurate approximation algorithm.
Derivation of sensing threshold
It is easy for us to obtain a S αS distribution sequence based on the parameter α e , X k ∼(α e ,0,1,0), α e is estimated by the estimation algorithm presented in Section 2, which is same to the practical noise sequence. Then, the sensing threshold of detector for the constructed sequence X k ∼(α e ,0,1,0), which is non-Gaussian white noise will be derived according to [4]. The detection statistic obtained in multi-user detection is :
$$ \begin{aligned} T_{F} = \frac{1}{MN}\sum_{m=1}^{M}\sum_{n=1}^{N}\lvert X_{m}(n)\rvert^{p_{e}} \quad \end{aligned} $$
where M is the number of detectors, and N is the observation signal of each detector. 0<p e <α e /2 is the order of the fractional moment, and it is the only parameter to be determined. Then make a comparison by statistics T F and threshold η, if T F >η, it is considered that the primary signal is present, otherwise it is absent. The detection signal does not require any priori information such as the channel gain and primary signal. It is practical and provides a scheme for the detection of non-Gaussian colored noise. The expressions for the probability of false alarm and the detection under the hypothesis \(\mathcal {H}_{0}\) and \(\mathcal {H}_{1}\) are derived, including the multi-detectors.
Under hypothesis \(\mathcal {H}_{0}\), the mean of the T F is calculated by
$$ \mu_{0} = E[T_{F} \mid \mathcal{H}_{0}] = \frac{1}{MN}E\left[\sum_{m=1}^{M}\sum_{n=1}^{N}\lvert X_{m}(n) \rvert^{p_{e}}\right] $$
According to the properties of α stable distribution, the fractional lower order moments of any S αS random variable S can be represented by its characteristic index α and the dispersion scale γ [24]
$$ E(\lvert S\rvert^{p}) = C(p,\alpha)\gamma^{p/\alpha},\quad \text{for} \quad p < \alpha $$
$$ C(p,\alpha)= \frac{2^{p+1}\Gamma(\frac{p+1}{2})\Gamma(-p/\alpha)}{\alpha\sqrt{\pi}\Gamma(-p/2)} $$
Here, \(\Gamma (\sigma) = \int _{0}^{\infty }x^{\sigma -1}e^{-x}dx\). Applying (15) and (16) to Eq. (14), it can be rewritten as
$$ \mu_{0} = \frac{1}{M}\sum_{m=1}^{M}C(p_{e},\alpha_{e})\gamma^{p_{e}/\alpha_{e}} $$
And the variance of statistic under \(\mathcal H_{0}\)
$$ \begin{aligned} \sigma_{0}^{2} &= E\left[\left(T_{F}\mid\mathcal H_{0}\right)^{2}-E^{2}\left[\left(T_{F}\mid\mathcal H_{0}\right)\right]\right]\\ &=\frac{1}{M^{2}N}\left\{\sum_{m=1}^{M}E\left[\lvert V_{m}(n)\rvert^{2p_{e}}\right]\right.\\ &\quad-\left.\sum_{m=1}^{M}E^{2}\left[\lvert V_{m}(n)\rvert^{p_{e}}\right]\right\} \end{aligned} $$
According to the central limit theorem, T F is a Gaussian random variable when N is large enough. With the result μ 0 in (17)and \(\sigma _{0}^{2}\) in (18), the probability of false alarm is obtained
$$ \begin{aligned} P_{fa1} = \{T_{F} > \eta_{t}\mid \mathcal H_{0}\}= Q\left(\frac{\eta_{t}-\mu_{0}}{\sqrt{\sigma_{0}^{2}}}\right) \end{aligned} $$
Thus, the sensing threshold for constructed sequence X k ∼(α e ,0,1,0) is then
$$ \eta_{t} = \sqrt{\sigma_{0}^{2}}Q^{-1}(P_{fa1})+\mu_{0},\quad P_{fa1} \leq \bar{P}_{fa1} $$
Balance parameter-based FLOM detector
The simulation curve of η t and the P f a1 for the X k ∼(α e ,0,1,0) constructed in the previous section can be fitted easily. When the P f a1 is determined, η t can be obtained by (21). But if we use this value as standard sensing threshold, detection is invalid. So, another statistics T p is needed to calculate the balance parameter.
According to hypothesis A, we assume that ξ(n) is the non-Gaussian noise sequence under \(\mathcal H_{0}\), a different statistics of ξ(n) can be obtained.
$$ T_{P} = \frac{1}{MN}\sum_{m=1}^{M}\sum_{n=1}^{N}\lvert \xi_{m}(n)\rvert^{p_{\bar{m}}} $$
When the sampling sequence N is large, according to the central limit theorem, the statistical values of the two sequences tend to be stable value. Then we have the equation
$$ \quad\quad\eta_{p} = \eta_{t}\cdot\Delta $$
where Δ=T F /T P , we call it the balance parameter. η p is the sensing threshold for practical sensing node. In particular, when vector A in Section 2.2 is [1,0,…,0], Δ=1 and η t =η p . To obtain the false alarm probability and detection probability, the \(\sigma _{pi}^{2}\) and μ pi is needed, i=0,1.
Substituting (13) to (19), and assume X m (n)=a ξ 1(n)+b ξ 2(n), combined (15), we obtain
$$ \begin{aligned} \sigma_{p0}^{2} &= E\left[\left(\frac{1}{MN}\sum_{m=1}^{M}\sum_{n=1}^{N}\left[\lvert X_{m}(n)\rvert^{p_{e}}\right]\right)^{2}\right]\\ &\quad-\left\{\frac{1}{MN}\sum_{m=1}^{M} \sum_{n=1}^{N} E\left[\lvert X_{m}(n)\rvert^{p_{e}}\right]\right\}^{2}\\ &=\frac{1}{M^{2}N^{2}}\left\{NE\left[\sum_{m=1}^{M}\lvert X_{m}(n)\rvert^{2p_{e}}\right]\right.\\ &\quad+\sum_{Case:a}^{M}\sum_{n,j =1}^{N}E\left[\lvert X_{m}(n)\rvert^{p_{e}}\lvert X_{i}(j)\rvert^{p_{j}}\right]\\ &\quad-N\sum_{m=1}^{M}E^{2}\left[\lvert x_{m}(n)\rvert^{p}_{m}\right]\\ &\quad-\sum_{Case:b}^{M}\sum_{n,j =1}^{N}E\left[\lvert X_{m}(n)\rvert^{p_{e}}\lvert X_{i}(j)\rvert^{p_{j}}\right]\\ &=\frac{1}{M^{2}N}\left\{\sum_{m=1}^{M}E\left[\lvert \xi_{1}(n)+b\xi_{2}(n)\rvert^{2p_{e}}\right]\right.\\ &\quad-\left.\sum_{m=1}^{M}E^{2}\left[\lvert \xi_{1}(n)+b\xi_{2}(n)\rvert^{p_{e}}\right]\right\} \end{aligned} $$
Noting that both c a s e a and c a s e b are m,i=1,m≠i or n≠j. As p e <α e /2, using (15) to (24) is then
$$ \begin{aligned} \sigma_{P0}^{2} &=\frac{1}{M^{2}N}\sum_{m=1}^{M}\left[C(2p_{e},\alpha_{e})\gamma_{e}^{2p_{e}/\alpha_{e}}\vphantom{\left(C(p_{e},\alpha_{e})\gamma_{e}^{p_{e}/\alpha_{e}}\right)^{2}}\right.\\ &\quad-\left.\left(C(p_{e},\alpha_{e})\gamma_{e}^{p_{e}/\alpha_{e}}\right)^{2}\right] \end{aligned} $$
where γ e =γ nov and
$$ \mu_{p0} = \frac{1}{M}\sum_{m=1}^{M}C(p_{e},\alpha_{e})\gamma_{e}^{p_{e}/\alpha_{e}} $$
Similarly, the mean μ 1 and variance \(\sigma _{1}^{2}\) of T F under \(\mathcal H_{1}\) are derived then
$$ \mu_{p1} = E[T_{F}\mid\mathcal H_{1}] = \mu_{0} + \sum_{m=1}^{M}\phi_{m,0} $$
$$ \phi_{m,0} = \frac{\sigma_{s}^{2}p_{e}(p_{e}-1)C(p_{e}-2),\alpha_{e}}{2M}\gamma_{e}^{(p_{e}-2)/\alpha_{e}} $$
And the variance under \(\mathcal H_{1}\)
$$ \begin{aligned} \sigma_{p1}^{2} &= E\left[\left(T_{F}\mid\mathcal H_{1}\right)^{2}-E^{2}\left[\left(T_{F}\mid\mathcal H_{1}\right)\right]\right]\\ &=\sigma_{p0}^{2}+\frac{1}{N}\left(\frac{\sigma_{s}}{M}\right)^{2}\sum_{m=1}^{M}\phi_{m,1} \end{aligned} $$
$$ \begin{aligned} &\phi_{m,1}=p_{e}(2p_{e}-1)C(2p_{e}-2,\alpha_{e})\gamma_{e}^{2(p_{e}-1)/\alpha_{e}}\\ &-p_{e}(p_{e}-1)C(p_{e},\alpha_{e}\gamma_{e}^{p_{e}/\alpha_{e}})(2p_{e}-2,\alpha_{e})\gamma_{e}^{(p_{e}-2)/\alpha_{e}} \end{aligned} $$
According to the central limit theorem, T F is a Gaussian random variable when N is large enough. With the result μ p0 in (24)and \(\sigma _{p0}^{2}\) in (25), the probability of false alarm is obtained
$$ P_{fa} = \{T_{P} > \eta_{P}\mid \mathcal H_{0}\}= Q\left(\frac{\eta_{t}\Delta-\mu_{p0}}{\sqrt{\sigma_{p0}^{2}}}\right) $$
and detection probability with μ p1 and \(\sigma _{p1}^{2}\)
$$ P_{d} = \{T_{P} > \eta_{P}\mid \mathcal H_{0}\}= Q\left(\frac{\eta_{t}\Delta-\mu_{p1}}{\sqrt{\sigma_{p1}^{2}}}\right) $$
where μ pi and \(\sigma _{i}^{2}\) (i=1,2) are the mean and variance of practical noise sequence. The expression of probability detection can be deduced by the combination of (28) and (29)
$$ P_{d} = Q\left(\frac{\sqrt{\left(\sigma_{0}^{2}\right)}Q^{-1}(P_{fa})+\mu_{p0}-\mu_{p1}}{\sqrt{\sigma_{p1}^{2}}}\right) $$
Substituting (26−28) into (31)
$$ \begin{aligned} P_{d} = Q\left(\frac{\sqrt{\left(\sigma_{0}^{2}\right)}Q^{-1}(P_{fa})-\sum_{m=1}^{M}{\phi_{m,0}}}{\sqrt{ \sigma_{0}^{2}+\frac{\sigma_{s}^{2}}{M^{2}N}\sum_{m=1}^{M}\phi_{m,1}}}\right) \end{aligned} $$
The optimal P d can be obtained by searching for the order vector \(p_{\bar {m}}\) because \(\sigma _{0}^{2}\), ϕ m,0 and ϕ m,1 are all related to \(p_{\bar {m}}\). Then numerical computation of (31) will be conducted in Section 4 as well as Monte Carlo simulations to validate our algorithm.
In this section, the simulation result will be discussed to evaluate the performance of the novel model for non-Gaussian colored noise as well as the BP-FLOM detector.
Effectiveness of noise models
First, we investigate the effectiveness of the non-Gaussian colored noise model as well as the parameter estimated method. The comparison for power spectral density is presented in Section 2, the proposed non-Gaussian colored noise model has the same statistical properties to the practical noise.
Then we construct a new non-Gaussian colored noise sequence ξ(n), the characteristic exponent α = 1.25, the linear transformation matrix A = [0.9 −1.25 0.7 0], sample size is set to L = 100,000. As shown in Fig. 4, the distribution of the data follow the shape of the probability density distribution.
The PDF and fit curve of non-Gaussian colored sequence. The blue part are the data distribution, the red line is the fit-curve with estimated parameters
Next we estimate the parameters of the sequence to draw a new fit curve. It is obvious that the fitted curve is consistent with the data distribution. The estimation result is shown in Table 2. The result of parameter estimation is very close to the parameter α of the constructed sequence, and the estimation relative error is very small.
Table 2 Estimation of result. The estimation result of α with different sample size, where real value is 1.5
In conclusion, the proposed model has the same statistical characteristics, the practical noise is estimated, and the reconstructed data are consistent with the actual ones. all the analysis above show that non-Gaussian colored noise model we proposed can describe the real noise accurately.
The performance of BP-FLOM detector
We assume that the primary signal is Gaussian with 0 mean, variance \(\sigma _{s}^{2}\), the noise background is the non-Gaussian colored noise based on I.I.D S αS with dispersion scale γ=1. We set the sample size N = 1000, and the simulation results are achieved by 20,000 Monte Carlo simulations.
Figure 5 shows the curves of the detection threshold versus the probability of false alarm, and different values for the single-node detection parameters. Curves labeled with "theory" are calculated according to expression (28), curves labeled with "simulation" are obtained statistic and detection threshold. In the case of cooperative detection, we set α 1=2,p 1=0.7 for the first detector and α 2=0.8,p 2=0.33. Figure 6 shows M=2 for cooperative detection, the comparison is also displayed. It proves that the simulation is consistent with the theory.
Probability of false alarm versus detection threshold of single detector for different values of α, p. The comparison of theory and simulation result are displayed
Probability of false alarm versus detection threshold of a multi-detector. The number of detector is 2, and the comparison of theory and simulation result are displayed
Figure 7 shows the comparison of the three sets of simulation results, and the range of GSNR is − 15∼5 dB. Three different characteristic exponents and corresponding fractional lower order are presented to make a comparison between FLOM detector and the BP-FLOM we proposed. When the characteristic component α=2, it is the Gaussian colored noise and the performance of the two algorithm is close to each other. For α=1.5 and α=0.8, our proposed detector has a better detection performance than the FLOM detector under all levels of GSNR. For instance, when GSNR = −4dB, P f a=0.1 and α=1.5, the probability of detection of our detector is 71.4%, but that of the FLOM detector is 42% only, which fails to detect the primary signal. But the detection is effective only under a large GSNR when α=0.8. In summary, the performance of BP-FLOM detector is better than FLOM detector in the background of non-Gaussian colored noise.
Probability of false alarm versus detection threshold of BP-FLOM detector for different values of α, p, and M
Figure 8 shows the performance of the BP-FLOM detector under single detection and multi-detection; it is obvious that the performance of multi-detection is much better than that of single detection. For example, the probability of detection of multi-detection is 75% while the single detection is 42%, under the same condition P f s=0.1.
ROC of BP-FLOM detector with different number of detectors. M=2 and M=1 represent different number of detectors for sensing
Figure 9 shows the ROC curves of the BP-FLOM detector, both the simulation and theory curves. The simulation result are obtained with different values of p=0.33 and p=0.7 under two values of GSNR (−5 dB and −10 dB). The theory curves are obtained with (31) under the same conditions. As shown in the figure, our proposed detector performs better for smaller values of p. For example, the probability of detection is 70% with p=0.33 while 37% with p=0.7, under the same GSNR = −10 dB.
ROC of BP-FLOM detector with multi-detectors. Performance in different conditions when GSNR = − 10 dB or − 5 dB, p=0.33 or 0.7, the theory and simulation are all displayed
This paper propose a novel non-Gaussian noise model based on the analysis of practical noise, and present a balance parameter-based fractional low-order moment (BP-FLOM) detector. It was shown that although the BP-FLOM scheme exhibits an approximately identical detection performance with the FLOM detector when it is Gaussian colored noise, performance is better when the background is the non-Gaussian colored noise, and the multi-detection performance is much better than FLOM detector and single detector. Simulation results, as well as the presented analysis, conform to the superior performance of the proposed balance parameter-based sensing scheme relative to competing solutions, in particular, the balance parameter is self adjusted when the noise sequence changed, which is practical for the system.
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This work was supported by the National Nature Science Foundation of China (no. 61301095), National Nature Science Foundation of China (no. 61671167).
School of Electronics and Information Engineering, Harbin Engineering University, Nantong Street No.145, Nangang District, Harbin, 150001, China
Zheng Dou
, Chengzhuo Shi
, Yun Lin
& Wenwen Li
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CS proposed the non-Gaussian colored noise based on S αS distribution and presented a new detection method of balance parameter-based FLOM detector. CS completed all the computer simulation and wrote the whole paper as well as the proofreading. All authors read and approved the final manuscript.
Correspondence to Chengzhuo Shi.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Dou, Z., Shi, C., Lin, Y. et al. Modeling of non-Gaussian colored noise and application in CR multi-sensor networks. J Wireless Com Network 2017, 192 (2017) doi:10.1186/s13638-017-0983-3
Non-Gaussian colored noise model
BP-FLOM
Practical noise data
CR multi-sensor networks
Radar and Sonar Networks | CommonCrawl |
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Florian Spieckermann ORCID: orcid.org/0000-0003-4836-32841,
Daniel Şopu ORCID: orcid.org/0000-0001-5531-462X2,3,
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Michael B. Kerber4,
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Erhard Schafler4 &
Jürgen Eckert1,2
Nature Communications volume 13, Article number: 127 (2022) Cite this article
Structure of solids and liquids
The atomistic mechanisms occurring during the processes of aging and rejuvenation in glassy materials involve very small structural rearrangements that are extremely difficult to capture experimentally. Here we use in-situ X-ray diffraction to investigate the structural rearrangements during annealing from 77 K up to the crystallization temperature in Cu44Zr44Al8Hf2Co2 bulk metallic glass rejuvenated by high pressure torsion performed at cryogenic temperatures and at room temperature. Using a measure of the configurational entropy calculated from the X-ray pair correlation function, the structural footprint of the deformation-induced rejuvenation in bulk metallic glass is revealed. With synchrotron radiation, temperature and time resolutions comparable to calorimetric experiments are possible. This opens hitherto unavailable experimental possibilities allowing to unambiguously correlate changes in atomic configuration and structure to calorimetrically observed signals and can attribute those to changes of the dynamic and vibrational relaxations (α-, β- and γ-transition) in glassy materials. The results suggest that the structural footprint of the β-transition is related to entropic relaxation with characteristics of a first-order transition. Dynamic mechanical analysis data shows that in the range of the β-transition, non-reversible structural rearrangements are preferentially activated. The low-temperature γ-transition is mostly triggering reversible deformations and shows a change of slope in the entropic footprint suggesting second-order characteristics.
The atomistic mechanisms underlying the aging and rejuvenation of bulk metallic glasses (BMGs) still remain unclear to a great extent. The first studies on aging due to the mechanical degradation of glassy polymers emerging in the 1950s and the fact that aging and rejuvenation occur (as discussed by Kovacs1), culminated in a lively discussion about the existence of rejuvenation by Struik and McKenna in the late 1990s and the early years of this millennium2,3. As aging leads to enhanced brittleness, it was found to be detrimental for many potential applications of BMGs. Many efforts have been undertaken to tune the aging and rejuvenation. The degree of rejuvenation and, hence, the amount of the stored energy as well as the free volume in a BMG can be controlled by different methods such as deformation4,5, high-pressure torsion (HPT)6, ion irradiation7, flash annealing8, or even cooling to cryogenic temperatures9,10. All these studies reveal promising enhancement of mechanical properties without fully resolving their atomistic origins.
Recent developments in the understanding of the atomic-scale mechanisms of rejuvenation from computer simulations have developed very insightful knowledge regarding the structural and thermodynamic origin of aging and rejuvenation due to either thermal processing or mechanical (or cyclic) loadings11,12,13. The kinetic aspects, as well as the experimental proof of these theoretical findings, which relate the dynamic relaxation modes to the (structural) atomic-scale reorganization processes during ageing and rejuvenition in metallic glasses are however still sparse.
Different studies have tried to correlate the stress-driven processes, such as activation of shear transformation zones (STZs) and shear-banding, with thermally activated dynamic relaxations, i.e., β- and α-relaxation modes14,15,16,17. The α-relaxation is typically a low-frequency mode (10−2 Hz) commonly associated with the glass transition, i.e., large cooperative rearrangements. The β-relaxation is a higher frequency mode (103 Hz) related to structural rearrangements on a smaller scale in the glassy state. The coupling of α- and β-modes sometimes results in the observation of an excess wing in the loss modulus. More recent studies suggested a third dynamic relaxation mechanism, termed γ- or \(\beta ^{\prime}\)-relaxation, activated at low temperatures for low-frequency actuation. The formation and relaxation of stress inhomogeneities at cryogenic temperatures might be correlated to this relaxation mechanism18,19; however, little is known about the structural origin of this relaxation due to its recent discovery and the experimental challenges associated with cryogenic cooling.
The present paper aims to give deeper insight into the interplay of structural reorganization of the material and the dynamic relaxations. The understanding and experimental validation of the mechanisms occurring on the atomistic scale in glassy materials and particularly in bulk metallic glasses during aging and rejuvenation are crucial to improve and understand the origins of their limited ductility. Many structural characterization methods, however, fail to catch the very small changes related to aging and rejuvenation in metallic glasses. In this work, we use in situ synchrotron X-ray diffraction to study the structural rearrangements during annealing from 77 K up to the crystallization temperature of Cu44Zr44Al8Hf2Co2 BMGs. This is done by determining small configurational changes in topological ordering with high time and temperature resolution. We propose here to use an equivalent of a configurational entropy of the experimentally determined X-ray pair distribution function (PDF) to make the subtle changes occurring during annealing visible. The samples were rejuvenated by high-pressure torsion (HPT) performed at cryogenic and room temperatures prior to the in situ annealing experiments. The as-deformed state of the samples was preserved by cryogenic storage prior to the in situ annealing experiments (please refer "Methods" section for details). Structural changes reflected in the X-ray-derived equivalent configurational entropy are correlated with dynamic mechanical analysis (DMA) as well as with differential scanning calorimetry (DSC) to determine dynamic relaxations and crystallization. The DMA measurements provide a clear picture of the relaxation process and are able to identify and distinguish between the well-known β- and α-relaxation modes and also reveal the presence of the fast γ-relaxation mechanism in the glassy material.
Structural characteristics
Figure 1 shows selected reduced pair distribution functions (PDFs) G(r) as determined by X-ray diffraction at 303.15 K while heating in situ. It can be seen that the differences are large for the as-cast state where no shoulder (see inset in Fig. 1c) is discernible in the first peak, whereas for the other two samples (deformation at RT and at 77 K) that were immediately stored at 77 K after HPT a clear shoulder is discernible. More intriguing is the fact, that the sample deformed by HPT at room temperature and relaxed for 7 days does not exhibit the shoulder and also the second peak (above 5 Å medium-range order (MRO) is considered to start) approximates the as-cast state and the two present shoulders smear out as well. Upon further heating, the shoulders also start to disappear for the cryogenically stored samples (Supplementary Movies 1 and 2). The shoulder is an indication for HPT-induced short-range ordering. This very local structure is related to the elevated pressure during HPT and is not stable at room temperature and ambient pressure.
Fig. 1: 2-D diffraction and pair distribution function.
a Diffraction pattern of an as-cast amorphous sample, (b) diffraction pattern of the crystallized state. c Comparison of the reduced pair distribution function G(r) at room temperature for the as-cast, and three different HPT-deformed states. If no cryogenic storage is ensured after HPT, the sample relaxes within 1 week and the shoulders in the first and second peaks of G(r) smear out indicating a certain degree of disordering/aging.
The development of a shoulder in the first diffraction peak indicates that clusters are affected by the HPT deformation process in the short-range order (SRO) regime. BMGs have a high degree of SRO, and the clusters in their structure have a preference to develop fivefold symmetry (close-packed)20. The hydrostatic stress during the HPT process rejuvenates the glassy structure by increasing the free volume. At the same time, the amount of local ordering and the fraction of favored fivefold motifs21,22 are increased and stabilized by hydrostatic stress5. This effect is not present in the sample stored at room temperature after HPT. Here, local reconfiguration via thermally induced rearrangement can relax the structure.
The entropy is a useful measure of the energetic state of glass concerning aging and rejuvenation23. Multibody entropies have been derived since the early stages of statistical physics. Baranyai and Evans showed that two-body contributions24, the pair correlation, dominate the configurational entropy of a liquid. With low temperature resolution, the configurational entropy was used to study liquid-to-liquid phase transitions in In20Sn8025. Here, we use a measure of the configurational entropy as calculated from the pair correlation function for further analysis. The pair correlation function g(r) can be calculated from the experimentally determined reduced pair distribution (correlation) function G(r) by
$$g(r)=\frac{G(r)}{4\pi {\rho }_{0}r}+1$$
with ρ0 the mean atomic number density of the alloy determined from the slope of G(r) in the range between 0 and 2 Å and r the radius.
The analysis of the measured G(r) suggests that the correlational equivalent of the configurational entropy Seq— derived from the two-body correlation—could be used as a measure of the state of aging or rejuvenation of the metallic glass, comparable to the approach applied by Piaggi et al.26 on pair correlation data derived from molecular dynamics simulations. The configurational entropy Seq can then be calculated with the equation derived by Nettleton and Green27,28
$${S}_{{{{{{{{\rm{eq}}}}}}}}}=-2\pi {\rho }_{0}{k}_{{{{{{{{\rm{B}}}}}}}}}\int \left(g(r){{{{{{\mathrm{ln}}}}}}}\,g(r)-g(r)+1\right){r}^{2}dr,$$
with kB the Boltzmann constant. It has been derived to determine the two-body contribution to the excess configurational entropy of a single-component liquid. The definition used here refers to the excess entropy with respect to the gas state29 (other definitions sometimes used in glass physics refer to the excess with respect to the relaxed glass or to the crystalline state23,30).
We apply this formula to the experimentally derived XRD pair correlation function. As such, we treat the five-component metallic glass in a first approximation similar to a monoatomic liquid, which is the condition for which the present formula has been derived and tested. As the formula is nonlinear, the derived entropy can only be used to visualize structural differences in the same material but not to directly derive quantitative results. Such results would require the determination of partial PDF's, which is planned for future work. For small changes in the topological ordering which occur before and during glass transition, the evaluation does not become unstable. As crystallization involves also substantial chemical reordering, any derived entropies from the above formula should, however, be treated with great care in order to avoid unphysical conclusions.
For the calculation of the equivalent configurational entropy, it is assumed that the uncertainty introduced by the ad-hoc variational corrections applied by PDFgetX331 is small enough to assume a linear dependence.
We decided here not to derive the partial PDF data for the following reasons. The high number of components would make the derivation of the partial PDFs unreliable. We also aim at proposing an a priori model-free approach to assess structural disordering. In order to underline the difference of the quantity derived here from a correlational entropy (which would be the sum of all partial entropies), we will denote it as "equivalent entropy Seq". Since contributions to the configurational entropy other than the two-body correlation (i.e., chemical) are not considered, the equivalent entropy is smaller than the total entropy.
Figure 2 depicts the resultant equivalent entropies as derived from the in situ diffraction experiments. The configurational state reflects the rejuvenation of the cryogenically stored samples with respect to the as-cast state. The HPT sample stored at room temperature relaxed into a state where Seq was closer to the as-cast state. Crystallization was characterized by a rapid reduction of Seq upon heating. After reaching crystallization, the samples were cooled to room temperature and a curve for the crystalline state was recorded. The glass transition Tg can be seen at 709 K (in good agreement with the literature32).
Fig. 2: Change in equivalent configurational entropy as determined by Eq. (2).
Each point is derived from a reduced PDF calculated from an X-ray diffraction pattern. The curves have been shifted by an additive constant on the ordinate axis (Seq-axis) in order to assure overlapping in the liquid state (unshifted data in Supplementary Fig. 1). The lower x axis is normalized to the glass transition temperature Tg = 709 K.
In the glassy state, characteristic changes of the slope of Seq occur which are correlated in Fig. 3a with the dynamic transitions. In thermodynamic equilibrium, the entropy can be used to calculate an equivalent configurational heat flow Δϕeq for a given heating rate βh with the equation
$${{\Delta }}{\phi }_{{{{{{{{\rm{eq}}}}}}}}}=\frac{T\cdot d{S}_{{{{{{{{\rm{eq}}}}}}}}}}{dT}\cdot {\beta }_{{{{{{{{\rm{h}}}}}}}}}={{\Delta }}{c}_{{{{{{{{\rm{eq,p}}}}}}}}}\cdot {\beta }_{{{{{{{{\rm{h}}}}}}}}},$$
where T is the temperature and Δceq,p the change in equivalent configurational heat capacity at constant pressure. By numerical differentiation (after smoothing) and application of Eq. (3), the equivalent configurational heat flow Δϕeq as depicted in Fig. 3b can be correlated with the DMA-derived dynamical relaxations (Fig. 3c). Especially in the region of the β-relaxation and the excess wing, structural rearrangements are triggered that lead to a change of slope in Seq. Figure 4 shows an excellent correlation between the DSC and the XRD-derived heat flows for the as-cast state. Interestingly, an enthalpic peak at low temperatures is observable in both evaluations that might originate from the relaxation of casting-induced stresses, triggered for instance by the mobilization due to the γ-transition. After the glass transition (at T/Tg = 1), the calorimetric heat flow shows a clear endothermic enthalpic peak that is not visible in the equivalent configurational heat flow ϕeq. ϕeq also increases when entering the undercooled liquid until crystallization occurs. Oxidation leads to a slight curvature of the calorimetric signal at elevated temperatures.
Fig. 3: Structure dynamics relationship.
Derivation of the equivalent configurational heat flow (shown in (b)) from the equivalent configurational entropy (shown in (a)). (dashed line) HPT at 77 K, (full line) HPT at RT, (dash-dotted line) as-cast. A change of slope at the γ-transition is indicated by solid gray lines in (a). Considerable enthalpic relaxation (the peaks are indicated by dashed gray lines in (b)) occurs when entering the β-relaxation region and between β and α in the excess wing region. c Displays the loss tangent tan(δ) as determined from torsion geometry DMA of the as-cast material.
Fig. 4: Calorimetric (dashed line) and equivalent configurational heat flow (solid line) for the as-cast state.
At low temperatures, a very clear peak is observable—in contrast to the HPT-deformed state.
Relaxation kinetics
The kinetics of the dynamic mechanical relaxations have been studied using ex situ dynamic mechanical analysis. The results of DMA experiments performed on the as-cast and HPT-deformed material are represented in Fig. 5a–c with a focus on the β and the α-transition. In accordance with the earlier figures, the transition intervals are shown in colors derived from the torsion DMA experiment (represented reference in Fig. 5a). Further experiments focusing on the frequency dependence of the γ-transition studied at low temperatures are presented in Fig. 6. Figure 5a shows tan δ of the as-cast state and the HPT-deformed state from cryogenic temperature up to the crystallization temperature. Both states show evidence of the γ and the β-transition. In the undeformed case, the β-transition appears as a well-separated peak, while in the deformed case a pronounced excess wing with a slight shoulder is formed. The excess wing is very pronounced for the HPT-deformed case and less for the as-cast material (Fig. 5a–c). This small wing is not suppressed when heating below or into the glass transition (Fig. 5a–c). The excess wing has been interpreted to be related to the coupling of β- and α-relaxation modes33. Such a coupling could explain the strong structural aging to a lower entropy state before the glass transition, as shown in Fig. 3, as it allows a transition from local atomic rearrangements (STZ activation) to larger collective rearrangements. Only the full activation of the α-modes and their percolation would then allow the glass transition and therefore the occurrence of homogeneous viscous flow. In the present experiments, the appearance of the β-transition peak in \(\tan \delta\) can be suppressed by heating just over this transition followed by subsequent cooling (Fig. 5b). When heating deep into the supercooled liquid (Fig. 5c), the β peak is also undetectable, but the \(\tan \delta\) is slightly increased. The β-transition is considered to be related to the local plastic deformation (STZ activation) of metallic glasses.
Fig. 5: Kinetics of α-, β-, and γ-transitions.
a DMA Experiments on as-cast and HPT-deformed material. b β-transition upon heating just below Tg and (c) upon heating deep into the undercooled liquid. In accordance with the previous figures, the transition intervals (shown in color) are derived from the torsion curve represented in (a).
Fig. 6: DMA experiments probing the γ-relaxation.
a Representing the storage modulus and (b) representing the loss modulus. Storage modulus curves for frequencies f = 0.05, 0.5, 7, and 15 Hz are shifted on the ordinate axis from the 3 Hz data for clarity. Cole–Cole fits of the data indicated in (b) by red lines are used for the determination of the activation energy.
For glasses rejuvenated by HPT or other methods (i.e., by affine cryogenic deformation, elastostatic loading, cold rolling, shot peening, uniaxial compression, triaxial compression) a fair amount of enthalpy relaxation is observed4,6,9,34,35 in the temperature range characteristic for the β-transition. In the range of the β-transition recent literature reported that some glasses tend to stiffen in DMA16,36 due to annihilation of free volume. It can hence be assumed, that rejuvenation and the β-transition are related.
In the present work, we observe that the occurrence of the β-transition peak is easily canceled by thermal treatment, indicating that the mobility of the underlying mobile species—i.e., deformation carriers or flow units—can be erased for our alloy. Considering that the local atomic mobility in the β-transition region is correlated with structural heterogeneities of enhanced free volume, this result can be considered as a further strong evidence for the viscoelastic and non-affine nature of the deformation mechanism associated with the β-transition.
Figure 6 shows DMA experiments carried out on an as-cast sample with different frequencies. The sample was cooled to 123.15 K and heated with 5 K/min and dynamic deformations with frequencies from 0.05 to 15 Hz have been applied on the same sample successively. Using Cole–Cole fitting, the transition temperatures were derived. Figure 7 shows the resulting Arrhenius evaluation that is used to determine the activation energy of the γ-relaxation (Arrhenius evaluations for the α- and β-relaxations are provided in Supplementary Fig. 2).
Fig. 7: Activation energy of the γ-relaxation.
Arrhenius plot of the frequency (f) dependence of the γ-transition yielding activation energy on the order of 27 ± 2 kJ mol−1 = 0.27 ± 0.02 eV.
The γ-transition can be reproduced during each heating-cooling cycle. This confirms that the structural origin that allows this transition peak to be detectable is not destroyed when the experiment is performed without heating into the β-transition (which would allow annihilation of free volume, Fig. 6). These data suggest that the deformations active during the γ-transition are affine in nature but can be used to activate stress-driven non-affine relaxation/annihilation processes such as those reported by Ketov et al.9. The DMA-derived activation energies determined for the α, β and γ-relaxations are represented in Table 1.
Table 1 Activation energies for the α-, β-, and γ-relaxation derived from the DMA experiments.
All relaxation modes (α, β, and γ) are the origin of aging (or rejuvenation) at different length- and timescales. This is reflected in the rather fast transition of the HPT-deformed sample stored at room temperature to an aged state featuring an entropy loss and a change in the peak shape of the first PDF maximum (SRO) and the second PDF maximum (MRO) in accordance with the results of Bian et al.37 and Sarac et al.38,39.
The DMA experiments which involve annealing in the β-relaxation regime and the supercooled liquid regime suppress the β-relaxation peak for subsequent experiments. Therefore, we interpret the β-relaxation and hence also the excess wing, to be of diffusional nature involving collective atomic movements/deformations (permanent). This also suggests that the β-relaxation is predominantly involved local, non-affine deformation. The excess wing may be interpreted as the activating mechanism allowing for the percolation of mobile species, hence confirming the nature of the wing as an overlap of α- and β-modes. In contrast, the γ-relaxation is reproducible over many heating-cooling cycles (Fig. 6). This shows that the involved deformation is recoverable, suggesting predominantly affine recoverable structural rearrangements40. Although macroscopically BMGs are rather isotropic in nature, they are highly heterogeneous structures on the microscopic and atomic scales. Kinetic factors may furthermore allow a certain degree of super- or undercooling and, very recently, heterogeneity in time has been shown for glassy relaxations by means of X-ray photon correlation spectroscopy41,42. This temporal and structural heterogeneity is also reflected by broadened relaxation peaks due to deformation by HPT (Fig. 5a).
In the entropy curve, one may distinguish between a change of slope, corresponding to a first-order phase transition, and a non-monotonous step for second-order transitions. Crystallization clearly involves a sharp step and can hence be interpreted as a first-order transition (Fig. 2). If the transition smears out over a larger temperature regime for instance due to structural effects (e.g., melting of crystals with different sizes), the separation becomes more difficult. Such behavior would result in a broadened peak in the heat flow. The percolated cooperative motion in the β-transition region can be seen as a thermal annealing process responsible for the annihilation of free volume. This is reflected by several overlapping peaks in the equivalent configurational heat flow. The peak in the equivalent configurational heat flow may be an indication of overlapping processes with characteristics of a first-order phase transition. The excess wing as a result of coupled α- and β-modes seems to be related to a substantial peak of the equivalent configurational heat flow shown in Fig. 3 for the HPT-deformed state.
To our understanding, the HPT process increases the number of more mobile species (softer heterogeneities) in the glassy state. This is characterized by a transition to a higher entropy amorphous state of glassy nature that allows activating β-type string-like43,44 and vortex-like motion45,46,47, in contrast to liquid-like complete cooperative flow for the supercooled liquid state. This leads to the well-known phenomenon of strain localization in shear bands48 during deformation processes. During the in situ heating process performed in this work, the glass will tend to relax to a lower entropy state using the thermodynamically and kinetically favorable β-relaxation modes (Fig. 8).
Fig. 8: Potential energy landscape scheme.
Schematic representation of the energetic scales of the relevant dynamic relaxation modes at a constant temperature T < Tg.
This could imply that a brittle-to-ductile transition involves the β-transition while the material is still in the solid state. The activation of mobile species through the β-transition can therefore be considered as a local bond-breaking mechanism with release of latent heat leading to an apparent character of a first-order phase transition in the equivalent configurational heat flow. The percolation of such localized events is necessary to achieve plastic flow in the metallic glass, schematically represented in the potential energy landscape in Fig. 8. In colloidal glasses, it has been reported that the failure transition may be associated with a first-order phase transition49,50. Evidently, the XRD-derived equivalent configurational entropy has high potential for giving new insights into the phenomena related to the glass-to-liquid transition17,51,52,53.
This prompts the question whether the local bond-breaking mechanisms or structural rearrangements related to the β-transition in metallic glasses may be attributed to a first-order phase transition. Recent results reported in the literature interpret the β-transition as a mobilizing mechanism allowing for a polyamorphic, nucleation-controlled transition from the rejuvenated state to an aged state48. Over the last few years, the random first-order transition (RFOT) theory is one of the models for the glassy behavior that is able to well explain a number of phenomena observed in glassy materials with respect to their mechanical properties54. Recently, also the first-order characteristics have been shown for the glass transition of ultra-fragile glasses55. The observations made in the framework of this work are not conclusive in this respect; however, the equivalent configurational heat flow, as well as calorimetric heat flow, are associated with enthalpic peaks related to the β-transition that can be interpreted as showing first-order characteristics.
In summary, the pair distribution function was continuously measured to calculate the entropy of the pair correlation (excess equivalent configurational entropy Seq) that allows us to derive an equivalent configurational heat flow of the changes in pair correlation with unprecedented high-temperature resolution. We present here evidence that Seq changes in a characteristic manner when moving through the temperature regimes of the dynamic relaxations (γ, β, excess wing, and α). The structural rearrangements are occurring at the same temperatures as the dynamic γ- and β-relaxations, the excess wing as well as the α-relaxation determined by DMA. The β-transition and the excess wing have stronger first-order characteristics, as indicated by clear peaks in the equivalent configurational heat flow. These peaks may be related to local bond-breaking mechanisms leading to symmetry changes on the level of clusters. Seq as derived from the pair correlation shows that the β-transition, the excess wing, and the α-transition have very clear structural footprints. Our work also shows some indication that the γ-transition might be related to a change of slope of Seq, characteristic for a second-order phase transition similar to the α-relaxation.
The present work suggests to use Seq as a measure for the state of rejuvenation of metallic glasses. With improving synchrotron sources, the possibility to derive configurational entropy data with high time or spacial resolution may spark new research to better understand the nature and structural origins of different relaxation mechanisms occurring in glassy alloys. In future works, the derivation of partial PDFs may even allow to quantitatively determine the configurational entropy as a material property during in situ diffraction experiments, potentially unveiling to date unknown phenomena in metastable (amorphous) solids and liquids.
The bulk metallic glass Cu44Zr44Al8Hf2Co2, was chosen based on the well-studied bulk metallic glass system Cu44Zr44Al8 with minor additions of Hf to further increase the glass-forming ability56 and the addition of Co in order to maximize the ability of the material to rejuvenate by moderate atomic distance shortening57.
Samples of Cu44Zr44Al8Hf2Co2 were obtained by the suction casting of rods (3 mm diameter) and plates (1 mm thickness) in an Edmund Bühler arc melter under vacuum after multiple purging with Ar gas and purification by Ti getter. For the deformation experiments, discs with diameter d = 8 mm and height h = 1 mm were prepared by grinding and fine polishing from the as-cast plates.
HPT deformation was chosen in the present paper as it allows inducing a high degree of rejuvenation6. HPT was performed up to 20 revolutions at a pressure of 6 GPa on an HPT press (type WAK-01 Mark 1) machine with martensitic chromium steel anvils. Cooling during deformation using liquid nitrogen was applied to selected samples in order to study the effect of cryogenic deformation. For room temperature deformed samples, the samples were cooled to liquid nitrogen (LN2) temperature (77 K) by submerging the whole anvil setup in LN2 in order to suppress relaxation. Only then, was the pressure removed and the sample transferred at 77 K to a transportation dewar allowing to "freeze-in" the state post-HPT before unloading the pressure. The "frozen" samples were then transported to the synchrotron and transferred with the cryogenically arrested status (post-HPT) to the cold (77 K) heating stage. The different sample states are reproduced in Table 2. No intermediate heating before the start of the synchrotron experiment was allowed by this procedure for the states labeled as HPT (\({{{{{{{{\rm{LN}}}}}}}}}_{2}\)) + stored (\({{{{{{{{\rm{LN}}}}}}}}}_{2}\)) and HPT (RT) + stored (\({{{{{{{{\rm{LN}}}}}}}}}_{2}\)).
Table 2 Sample matrix showing the number of HPT rotations, deformation temperature (Tdef.) and storage temperature for all sample treatments Tstor..
In situ X-ray diffraction was performed on the P02.1 Powder Diffraction and Total Scattering Beamline of PETRA III using a Perkin Elmer XRD1621 (200 μm × 200 μm pixel size) detector with a photon energy of 60 keV in transmission setup. The cold sample, still at liquid nitrogen temperature, was mounted to the sample stage of a LINKAM THS 600 temperature controller using a custom-built sample environment58,59. To prohibit a temperature gradient between the sample and the instrument, the sample stage was pre-cooled to a temperature of 93.15 ± 1 K. Wide-Angle X-ray diffraction (WAXD) patterns of 12 s were then recorded while heating with 10 K/min from 93.15 K up to 873.15 K. After crystallization, the samples were again cooled to room temperature with 50 K/min. The diffraction patterns were carefully calibrated using a CeO2 reference (NIST 674b) and the pyFAI software60, and background subtraction was performed for the sample container. Pair distribution functions were determined using the software PDFgetX361. The mean atomic number density was determined from G(r) on the interval 0−2 Å, where G(r) = −4πrρ0. The nominal mean atomic number density of the alloy as calculated from the atomic composition amounts to ρ0 = 5.38345 ⋅ 1028 m−3.
Dynamic mechanical analyses were performed on a TA Instruments Discovery Hybrid Rheometer DHR 3 in torsional and three-point bending (TPB) mode. The experiments were performed by heating with constant heating rates from 123.15 to 873.15 K with heating rates varying from 2 to 10 K/min. For the determination of the activation energies, the frequencies were varied from 0.05 to 15 Hz in TPB mode and up to 60 Hz in a torsional mode. The loss tangent was fitted with the Cole–Cole equation in order to derive the frequency dependence of the transition temperatures.
Differential scanning calorimetry was performed on a Mettler Toledo DSC 3+ using platinum crucibles in a temperature range from 153.15 to 793.15 K with a heating rate of 20 K/min. The second heating signal was not subtracted.
The raw/processed data required to reproduce these findings can be provided by the corresponding author upon reasonable request.
The GnuOctave code required to calculate the equivalent entropies can be provided by the corresponding author upon reasonable request.
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We thank DESY (Hamburg, Germany), a member of the Helmholtz Association (HGF), for providing experimental facilities. Parts of this research were carried out at PETRA III using the Powder Diffraction and Total Scattering beamline P02.1. The research leading to our findings took place in the framework of project CALIPSOplus under the Grant Agreement 730872 of the EU Framework Programme for Research and Innovation HORIZON 2020 (F.S., B.S., and E.S.). This work was funded by the European Research Council under the ERC Advanced Grant INTELHYB (grant ERC-2013-ADG-340025, J.E., B.S., and A.R.), the ERC Proof of Concept Grant TriboMetGlass (grant ERC-2019-PoC-862485, J.E.), and by the Austrian Science Fund (FWF) under project grant I3937-N36 (B.S. and A.R.). We thank Dr. Christoph Gammer and Dr. Ivan Kaban for fruitful discussions. Cameron Quick, MChem. is thanked for English language editing.
Department of Materials Science, Chair of Materials Physics, Montanuniversität Leoben, Jahnstraße 12, 8700, Leoben, Austria
Florian Spieckermann & Jürgen Eckert
Erich Schmid Institute of Materials Science of the Austrian Academy of Sciences, Jahnstraße 12, 8700, Leoben, Austria
Daniel Şopu, Viktor Soprunyuk, Amir Rezvan, Sergey Ketov, Baran Sarac & Jürgen Eckert
Institut für Materialwissenschaft, Fachgebiet Materialmodellierung, Technische Universität Darmstadt, Otto-Berndt-Strasse 3, Darmstadt, D-64287, Germany
Daniel Şopu
Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090, Vienna, Austria
Viktor Soprunyuk, Michael B. Kerber & Erhard Schafler
Deutsches Elektronen Synchrotron (DESY), Notkestraße 85, 22607, Hamburg, Germany
Jozef Bednarčík & Alexander Schökel
P. J. Šafarik University in Košice, Faculty of Science, Institute of Physics, Park Angelinum 9, 041 54, Košice, Slovakia
Jozef Bednarčík
Florian Spieckermann
Viktor Soprunyuk
Michael B. Kerber
Alexander Schökel
Amir Rezvan
Sergey Ketov
Baran Sarac
Erhard Schafler
Jürgen Eckert
F.S., E.S., B.S., and J.E. designed the research. F.S., E.S., B.S., M.K., J.B., and A.S. performed the synchrotron experiments. F.S. and D.S. evaluated and cured the data. V.S. performed the DMA experiments. F.S. performed the DSC experiments. S.K., B.S., and A.R. produced the alloy. F.S. wrote the original draft. J.E. and B.S. provided funding. All authors contributed to the interpretation of the data and revised the manuscript.
Correspondence to Florian Spieckermann.
Nature Communications thanks Yun-Jiang Wang and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Description of Additional Supplementary Files
Supplementary Movie 1
Spieckermann, F., Şopu, D., Soprunyuk, V. et al. Structure-dynamics relationships in cryogenically deformed bulk metallic glass. Nat Commun 13, 127 (2022). https://doi.org/10.1038/s41467-021-27661-2 | CommonCrawl |
\begin{document}
\title{\textbf{Hermitian $K$-theory and $2$-regularity for totally real number fields}} \author{A.\thinspace J.~Berrick, M.~Karoubi, P.\thinspace A.~{\O }{}stv{\ae }{}r} \date{\today} \maketitle
\begin{abstract} We completely determine the $2$-primary torsion subgroups of the hermitian $K$-groups of rings of $2$-integers in totally real $2$-regular number fields. The result is almost periodic with period $8$. Moreover, the $2$-regular case is precisely the class of totally real number fields that have homotopy cartesian \textquotedblleft B\"{o}kstedt square", relating the $K$-theory of the $2$-integers to that of the fields of real and complex numbers and finite fields. We also identify the homotopy fibers of the forgetful and hyperbolic maps relating hermitian and algebraic $K$-theory. The result is then exactly periodic of period $8$ in the orthogonal case. In both the orthogonal and symplectic cases, we prove a $2$-primary hermitian homotopy limit conjecture for these rings.
\end{abstract}
\section{Introduction and statement of results}
\label{section:mainresults}
Let $F$ be a real number field with $r$ real embeddings, ring of integers $\mathcal{O}_{F}$ and ring of $2$-integers $R_{F}=\mathcal{O}_{F}[1/2]$. A key ingredient in the study of the $K$-theory of these rings is the commuting square of $2$-completed connective\textsf{ }$K$-theory spectra \begin{equation} \begin{array} [c]{ccc} \mathcal{K}(R_{F})_{\#} & \rightarrow & {\displaystyle\bigvee\limits^{r}} {}\mathcal{K}(\mathbb{R})_{\#}^{c}\\ \downarrow & & \downarrow\\ \mathcal{K}(\mathbb{F}_{q})_{\#} & \rightarrow & {\displaystyle\bigvee \limits^{r}}\mathcal{K}(\mathbb{C})_{\#}^{c} \end{array} \label{algebraic K Bokstedt square} \end{equation} introduced by B\"{o}kstedt for the rational case ($r=1$) in \cite{Bok}, and in \cite{HO}, \cite{Mitchell} for the general case; see also Appendix A. Here, $\mathbb{F}_{q}$ is a finite residue field described later in this Introduction. The $K$-theory spectrum $\mathcal{K}(\Lambda)$ refers to the spectrum defined by $\mathcal{K}(\Lambda)_{n}=\Omega{B\mathrm{GL}(S} ^{n+1}{\Lambda)}^{+}$ for $n\geq0$, where $S^{m}\Lambda$ denotes the $m$-iterated suspension of $\Lambda$ (when $\Lambda$ is a discrete ring) or Calkin algebra ($\Lambda$ a topological ring), as defined in \cite[Appendix A]{BK}. Throughout this paper, we use the notation \thinspace$\mathcal{E}^{c}$ to denote the connective covering of a spectrum $\mathcal{E}$, and $_{\#}$ to denote $2$-adic completion of connective spectra or groups. The often-used notation $\mathcal{E}_{\#}^{c}$ means $(\mathcal{E}^{c})_{\#}.$ For instance, \[ \mathcal{K}(\mathbb{R})_{\#}^{c}=(\mathcal{K}(\mathbb{R})^{c})_{\#} \quad\text{and\quad}\mathcal{K}(R_{F})_{\#}^{c}=(\mathcal{K}(R_{F})^{c} )_{\#}=\mathcal{K}(R_{F})_{\#} \] (see also the discussion after Lemma \ref{Connective.Spectra}). The bottom horizontal map is the Brauer lift, corresponding to the fibring of Adams' map $\psi^{q}-1$ on the $2$-completed connective complex topological $K$-theory spectrum $\mathcal{K}(\mathbb{C})_{\#}^{c}$. The remaining maps are induced from the obvious ring homomorphisms via Suslin's identification of the $2$-completed algebraic $K$-theory spectra of the real and complex numbers with $\mathcal{K}(\mathbb{R})_{\#}^{c}$ and $\mathcal{K}(\mathbb{C})_{\#}^{c} $, respectively, in \cite{Suslin :local fields}. If preferred, one can think of the above in terms of spaces and maps; however, the spaces also have an infinite loop space structure that is preserved by the maps.
In the rational case, the Dwyer-Friedlander formulation of the Quillen-Lichtenbaum conjecture for $\mathbb{Z}$ at the prime $2$ is that the above square is homotopy cartesian, see \cite[Conjecture 1.3, Proposition 4.2]{DF}. This has been affirmed in work of Rognes and Weibel \cite{RW} (see \cite[Corollary 8]{W:CR}), as a consequence of \cite{Bok}, Voevodsky's solution of the Milnor Conjecture \cite{V} and his subsequent joint work with Suslin \cite{SV}.
In the general number field case, Rognes and Weibel \cite{RW} determined the groups $K_{n}(R_{F})_{\#}$ up to extensions. It turns out that in the case of $2$-regular real number rings, discussed below, these extension problems disappear \cite{RO}. This leads to the above square being homotopy cartesian in that case too. Many of the foregoing results having been developed for spaces, we present the transition to spectra in Appendix A.
These developments raise the question of for which class of real number fields $F$ the square (\ref{algebraic K Bokstedt square}) is homotopy cartesian.
We turn now to the hermitian analog of the above. For the definition of hermitian $K$-theory we refer to \cite{K:AnnM112hgo} and \cite[Introduction and Appendix A]{BK}. Briefly, for a ring $\Lambda$ with involution ${}_{\varepsilon}{KQ}_{0}{(\Lambda)}$ denotes the Grothendieck group of isomorphism classes of finitely generated projective $\Lambda$-modules with nondegenerate $\varepsilon$-hermitian form, where we let $\varepsilon=\pm1$ according to whether orthogonal ($\varepsilon=+1$) or symplectic ($\varepsilon=-1$) actions on the ring $\Lambda$ are involved. If $\Lambda$ is discrete, $B{{}_{\varepsilon}O(\Lambda)}^{+}$ represents the plus-construction of the classifying space of the limit ${{}_{\varepsilon}O(\Lambda)}$ of the $\varepsilon$\emph{-}orthogonal\emph{ }groups ${}_{\varepsilon}O_{n,n} (\Lambda)$. This last group is the group of automorphisms of the $\varepsilon $-hyperbolic module ${}_{\varepsilon}H(\Lambda^{n})$, whose elements can be described as $2\times2$ matrices written in $n$-blocks \[ M=\left[ \begin{array} [c]{cc} a & b\\ c & d \end{array} \right] \] such that $M^{\ast}M=MM^{\ast}=I$, where the \textquotedblleft$\varepsilon $-hyperbolic adjoint\textquotedblright\ $M^{\ast}$ is defined as \[ M^{\ast}=\left[ \begin{array} [c]{cc} ^{\mathrm{t}}\check{d} & \varepsilon\,^{\mathrm{t}}\check{b}\\ \check{\varepsilon}\,^{\mathrm{t}}\check{c} & ^{\mathrm{t}}\check{a} \end{array} \right] \text{.} \] For a discrete ring $A$, the $\varepsilon$-hermitian $K$-theory spectrum ${}_{\varepsilon}\mathcal{KQ}(A)$ refers to the spectrum defined by ${}_{\varepsilon}\mathcal{KQ}(A)_{n}=\Omega{B{}_{\varepsilon}O(S}^{n+1} {A)}^{+}$ for $n\geq0$, where $S^{m}A$ denotes the $m$-iterated suspension of $A$, as defined in \cite[Appendix A]{BK}. We note that ${}_{\varepsilon }\mathcal{KQ}(A)_{0}$ has non-naturally the homotopy type of ${}_{\varepsilon }KQ_{0}(A)\times B{}_{\varepsilon}O(A)^{+}$, where ${}_{\varepsilon}KQ_{0}(A)$ is endowed with the discrete topology. The same definition applies for the $K$-theory spectrum $\mathcal{K}(A)$ on replacing the orthogonal group by the general linear group. There is however a significant difference between the two theories, at least for regular noetherian rings like fields or Dedekind rings. For such rings $A$, $\mathcal{K}(A)=\mathcal{K}(A)^{c}$ is connective \textsl{i.e.} $K_{n}(A)=0$ for $n<0$, whereas ${}_{\varepsilon}\mathcal{KQ} (A)$ is not connective in general.
On the other hand, the spectra of topological hermitian $K$-theory (with trivial involutions on $\mathbb{R}$ and $\mathbb{C}$) ${}_{\varepsilon }\mathcal{KQ}(\mathbb{R})$ and ${}_{\varepsilon}\mathcal{KQ}(\mathbb{C})$ have been defined in \cite[Appendix A]{BK}. A geometric description of these spectra appears in Appendix B below. In order to avoid a potential confusion between hermitian $K$-theory and surgery theory, we are writing ${} _{\varepsilon}\mathcal{KQ}$\ for the hermitian $K$-theory spectrum, and ${}_{\varepsilon}{KQ}_{n}$\ for the corresponding homotopy groups. (These are denoted by ${}_{\varepsilon}\mathcal{L}$\ and ${}_{\varepsilon}{L}_{n} $\ respectively in \cite{BK}.)
In \cite{BK}, the first two authors constructed a Brauer lift in hermitian $K$-theory and considered the hermitian analogue of the B\"{o}kstedt square for the rational numbers $\mathbb{Q}$, \textsl{i.e.} for $r=1$ and $R_{F}=\mathbb{Z}[1/2]$. For general number fields, the commuting B\"{o}kstedt square for hermitian $K$-theory takes the form: \begin{equation} \begin{array} [c]{ccc} {}_{\varepsilon}\mathcal{KQ}(R_{F})_{\#}^{c} & \rightarrow & {\displaystyle \bigvee\limits^{r}}{}_{\varepsilon}\mathcal{KQ}(\mathbb{R} )_{\#}^{c}\\ \downarrow & & \downarrow\\ {}_{\varepsilon}\mathcal{KQ}(\mathbb{F}_{q})_{\#}^{c} & \rightarrow & {\displaystyle\bigvee\limits^{r}}{}_{\varepsilon}\mathcal{KQ}(\mathbb{C} )_{\#}^{c} \end{array} \label{hermitian K Bokstedt square} \end{equation}
It was shown in \cite{BK} that when $F=\mathbb{Q}$ the square (\ref{hermitian K Bokstedt square}) too is homotopy cartesian; the result leads to another version of the homotopy limit problem related to the Quillen-Lichtenbaum conjecture, expressed as the $2$-adic homotopy equivalence of \emph{ }the fixed point set and the homotopy fixed point set of the ${}_{\varepsilon}\mathbb{Z}/2$ action on $\mathcal{K}(\mathbb{Z}[1/2])$. Thus again, one is led to ask \textit{for which class of totally real number fields the square (\ref{hermitian K Bokstedt square}) is homotopy cartesian, and for which the homotopy equivalence generalizes.}
These are the principal questions addressed in the present work.
Tackling these questions leads to a focus on a particular class of number fields $F$ with the associated rings of integers $\mathcal{O}_{F}$, as follows. From a theorem of Tate \cite[Theorem 6.2]{Tate}, one knows that the $2$-primary part of the finite abelian group $K_{2}(\mathcal{O}_{F})$ has order at least $2^{r}$, where $r$ is the number of real embeddings. We call $F$ (and $\mathcal{O}_{F}$, $R_{F}$) $2$\emph{-regular }when this order is exactly $2^{r}$. See Proposition \ref{2+-regular characterization} below for alternative characterizations. In the totally real case, which is our concern here, $r=[F:\mathbb{Q}]$. The simplest examples are the rational numbers $\mathbb{Q}$ and the following fields recorded in \cite[\S 4]{RO}.
\begin{enumerate} \item Let $b\geq2$. The maximal real subfield $F=\mathbb{Q}(\zeta_{2^{b}} +\bar{\zeta}_{2^{b}})$ of $\mathbb{Q}(\zeta_{2^{b}})$ is a totally real $2$-regular number field with $r=2^{b-2}$.
\item Let $m$ be an odd prime power such that $2$ is a primitive root modulo $m$. Then $F=\mathbb{Q}(\zeta_{m}+\bar{\zeta}_{m})$ is a totally real $2$-regular number field when Euler's $\phi$-function $\phi(m)\leq66$ (except for $m=29$), and also for Sophie Germain primes ($m$ and $(m-1)/2$ both prime) with $m\not \equiv 7\;(\mathrm{mod}\ 8)$ (the first few instances are $m=5,11,59,83,107$ and $179$). The number $r$ of real embeddings is $\phi(m)/2$.
\item Let $F=\mathbb{Q}(\sqrt{d})$ be a quadratic number field with $d>0$ square free. Then $F$ is $2$-regular if and only if $d=2$, $d=p$ or $d=2p$ with $p\equiv\pm3\;(\mathrm{mod}\ 8)$ prime \cite{BS}. Here, $r=2$. \end{enumerate}
The residue field $\mathbb{F}_{q}$ of $R_{F}$ referred to above is now chosen in the following manner. The number $q$ is a prime number with this property: the elements corresponding to the Adams operations $\psi^{q}$ and $\psi^{-1}$ in the ring of operations of the periodic complex topological $K$-theory spectrum generate the Galois group of $F(\mu_{2^{\infty}}(\mathbb{C}))/F$ obtained by adjoining all $2$-primary roots of unity $\mu_{2^{\infty} }(\mathbb{C})\subset\mathbb{C}$ to $F$ \cite[\S 1]{Mitchell}. The Cebotarev density theorem guarantees the existence of infinitely many such prime powers. By Dirichlet's theorem on arithmetic progressions we may assume that $q$ is a prime number, an hypothesis we assume throughout all the paper.
According to \cite{Mitchell}, if $a_{F}:=(|\mu_{2^{\infty}}(F(\sqrt
{-1}))|)_{2}$ is the $2$-adic valuation, then $q$ is $\equiv\pm 1\;(\mathrm{mod}\ 2^{a})$ but not $(\mathrm{mod}\ 2^{a+1})$. In the examples above: when $F=\mathbb{Q}(\zeta_{2^{b}}+\bar{\zeta}_{2^{b}})$, then $a_{F}=b$; when $F=\mathbb{Q}(\zeta_{m}+\bar{\zeta}_{m})$ or $\mathbb{Q}(\sqrt{d})$ with $d>2$, $a_{F}=2$; and finally, when $F=\mathbb{Q}(\sqrt{2})=\mathbb{Q} (\zeta_{8}+\bar{\zeta}_{8})$, we have $a_{F}=3$.
We are now ready to state our main results.
\begin{theorem} \label{theorem1} For every totally real $2$-regular number field $F$, and for any $q$ as discussed above, the square (\ref{hermitian K Bokstedt square}) is homotopy cartesian for $\varepsilon=\pm1$. \end{theorem}
Our proof of this theorem is based on the techniques employed in the case of the rational numbers \cite{BK} and the analogous algebraic $K$-theoretic result established in \cite{HO}, \cite{Mitchell} and \cite{RO} (see Appendix A for an overview).
The next result crystallizes the special role of $2$-regular fields in this setting.
\begin{theorem} \label{converse to theorem 1}Let $q$ be as above. Then, for every totally real number field $F$, the following are equivalent.
\begin{enumerate} \item[(i)] $F$ is $2$-regular.
\item[(ii)] The square (\ref{hermitian K Bokstedt square}) is homotopy cartesian for $F$ when $\varepsilon=1$.
\item[(iii)] The square (\ref{algebraic K Bokstedt square}) is homotopy cartesian for $F$. \end{enumerate} \end{theorem}
Since the Quillen-Lichtenbaum conjecture has been established for every real number field by the third author \cite{Ostvar}, one consequence of this theorem is that, in contrast to the rational case, for general real number fields the Quillen-Lichtenbaum conjecture fails to imply that the square (\ref{algebraic K Bokstedt square}) is homotopy cartesian.
Theorem \ref{theorem1} allows us to compute explicitly the $2$-primary torsion of the nonnegative hermitian $K$-groups ${}_{\varepsilon}KQ_{n}(R_{F})$ for $F$ as above. We tabulate and compare these groups with the corresponding algebraic $K$-groups $K_{n}(R_{F})$ computed in \cite{HO} and \cite{RO}.
\begin{theorem} \label{theorem2}Let $F$ be a totally real $2$-regular number field. Up to finite groups of odd order, the groups ${}_{\varepsilon}KQ_{n}(R_{F})$ are given in the following table. (If $m$ is even, let $w_{m}=2^{a_{F}+\nu_{2} (m)}$; also, $\delta_{n0}$ denotes the Kronecker symbol.) \begin{table}[tbh] \begin{center} \begin{tabular}
[c]{p{0.4in}|p{1.5in}|p{1.1in}|p{1.5in}|}\hline $n\geq0$ & ${}_{-1}KQ_{n}(R_{F})$ & ${}_{1}KQ_{n}(R_{F})$ & $K_{n}(R_{F} )$\\\hline $8k$ & $\delta_{n0}\mathbb{Z}$ & $\delta_{n0}\mathbb{Z}\oplus\mathbb{Z} ^{r}\oplus\mathbb{Z}/2$ & $\delta_{n0}\mathbb{Z}$\\ $8k+1$ & $0$ & $(\mathbb{Z}/2)^{r+2}$ & $\mathbb{Z}^{r}\oplus\mathbb{Z}/2$\\ $8k+2$ & $\mathbb{Z}^{r}$ & $(\mathbb{Z}/2)^{r+1}$ & $(\mathbb{Z}/2)^{r}$\\ $8k+3$ & $(\mathbb{Z}/2)^{r-1}\oplus\mathbb{Z}/2w_{4k+2}$ & $\mathbb{Z} /w_{4k+2}$ & $(\mathbb{Z}/2)^{r-1}\oplus\mathbb{Z}/2w_{4k+2}$\\ $8k+4$ & $(\mathbb{Z}/2)^{r}$ & $\mathbb{Z}^{r}$ & $0$\\ $8k+5$ & $\mathbb{Z}/2$ & $0$ & $\mathbb{Z}^{r}$\\ $8k+6$ & $\mathbb{Z}^{r}$ & $0$ & $0$\\ $8k+7$ & $\mathbb{Z}/w_{4k+4}$ & $\mathbb{Z}/w_{4k+4}$ & $\mathbb{Z}/w_{4k+4} $\\\hline \end{tabular} \end{center} \end{table} \end{theorem}
The proof of Theorem \ref{theorem2} makes use of a splitting result for ${}_{\varepsilon}\mathcal{KQ}(R_{F})^{c}$ shown in \S \ref{section:splittingresults} and an explicit computation carried out in \S \ref{section:proofoftheorem2}.
We recall that the forgetful and hyperbolic functors induce the two homotopy fiber sequences \[ {}_{\varepsilon}\mathcal{V}(R_{F})\longrightarrow{}_{\varepsilon} \mathcal{KQ}(R_{F})\longrightarrow\mathcal{K}(R_{F})\text{ and } {}_{\varepsilon}\mathcal{U}(R_{F})\longrightarrow\mathcal{K}(R_{F} )\longrightarrow{}_{\varepsilon}\mathcal{KQ}(R_{F})\text{.} \] The fundamental theorem in hermitian $K$-theory \cite{K:AnnM112fun} states that there is a natural homotopy equivalence \[ {}_{\varepsilon}\mathcal{V}(R_{F})\simeq\Omega{}_{-\varepsilon}\mathcal{U} (R_{F})\text{.} \]
Our next result gives an explicit computation of the homotopy groups of these spectra.
\begin{theorem} \label{theorem4} For any totally real $2$-regular number field $F$, the groups \[ {}_{\varepsilon}V_{n}(R_{F}):=\pi_{n}({}_{\varepsilon}\mathcal{V}(R_{F} ))\cong\pi_{n}(\Omega{}_{-\varepsilon}\mathcal{U}(R_{F}))=:{}_{-\varepsilon }U_{n+1}(R_{F}) \] are given, up to finite groups of odd order, as below.\begin{table}[tbh] \begin{center} \begin{tabular}
[c]{p{0.4in}|p{1.0in}|p{1.2in}|}\hline $n\geq0$ & ${}_{-1}V_{n}(R_{F})$ & ${}_{1}V_{n}(R_{F})$\\\hline $8k$ & $\mathbb{Z}^{r}\oplus\mathbb{Z}/2$ & $\mathbb{Z}^{2r}$\\ $8k+1$ & $0$ & $(\mathbb{Z}/2)^{2r}$\\ $8k+2$ & $\mathbb{Z}^{r}$ & $(\mathbb{Z}/2)^{2r}$\\ $8k+3$ & $0$ & $0$\\ $8k+4$ & $\mathbb{Z}^{r}$ & $\mathbb{Z}^{2r}$\\ $8k+5$ & $\mathbb{Z}/2$ & $0$\\ $8k+6$ & $\mathbb{Z}^{r}\oplus\mathbb{Z}/2$ & $0$\\ $8k+7$ & $\mathbb{Z}/2$ & $0$\\\hline \end{tabular} \end{center} \end{table}
More precisely, the spectrum ${}_{1}\mathcal{V}(R_{F})_{\#}^{c}$ has the homotopy type of \[ \bigvee^{2r}\mathcal{K}(\mathbb{R})_{\#}^{c}\simeq{}_{1}\mathcal{V} (R_{\mathbb{Q}})_{\#}^{c}\vee\bigvee^{2(r-1)}\mathcal{K}(\mathbb{R})_{\#} ^{c}\,\text{,} \] The cup-product with a generator of $K_{8}(\mathbb{R})$ induces a periodicity homotopy equivalence \[ {}_{1}\mathcal{V}(R_{F})_{\#}^{c}\simeq(\Omega^{8}{}_{1}\mathcal{V} (R_{F}))_{\#}^{c}\,\text{.} \]
\end{theorem}
It would be interesting to have more information about the class of number fields for which the intriguing periodicity result in this theorem holds. Note also that for $\varepsilon=-1$, the homotopy type of ${}_{\varepsilon }\mathcal{V}(R_{F})_{\#}^{c}$ is more complicated to determine explicitly, although we know that its homotopy groups are periodic (in Section \ref{V-computation} and Appendix D we show that an analogous splitting of the spectrum ${}_{-1}\mathcal{V}(R_{F})_{\#}^{c}$ as the product of classical topological spectra does not hold).
Recall from \cite[\S 7]{BK} that the standard ${}_{\varepsilon}\mathbb{Z} /2$-action on $\mathcal{K}(R_{F})$ defined via conjugation of the involute transpose of a matrix by \[ {}_{\varepsilon}J_{n}=\left[ \begin{array} [c]{cc} 0 & \varepsilon I_{n}\\ I_{n} & 0 \end{array} \right] \] induces an isomorphism between the fixed point spectrum $\mathcal{K} (R_{F})^{{}_{\varepsilon}\mathbb{Z}/2}$ and ${}_{\varepsilon}\mathcal{KQ} (R_{F})$.
Our next result solves in the affirmative a $2$-primary homotopy limit problem for ${}_{\varepsilon}\mathbb{Z}/2$-homotopy fixed point $K$-theory spectra, in the special case of totally real $2$-regular number fields. (In \cite{Ostvar} the corresponding problem in algebraic $K$-theory was solved in the affirmative for every real number field.) Recall the homotopy fixed point spectrum \[ \mathcal{K}(R_{F}){}^{h({}_{\varepsilon}\mathbb{Z}/2)}:=\mathrm{map} _{{}_{\varepsilon}\mathbb{Z}/2}(\Sigma^{\infty}E(\mathbb{Z}/2)_{+} ,\,\mathcal{K}(R_{F}))\text{.} \] Here $\mathrm{map}_{{}_{\varepsilon}\mathbb{Z}/2}$ denotes the function spectrum of ${}_{\varepsilon}\mathbb{Z}/2$-equivariant maps and $E(\mathbb{Z} /2)$ is a free contractible $\mathbb{Z}/2$-space, such as the CW-complex $S^{\infty}$ with antipodal action. Our hermitian analogue is now the following:
\begin{theorem} \label{theorem3} For every totally real $2$-regular number field $F$ there is a natural homotopy equivalence of $2$-completed spectra \[ {}_{\varepsilon}\mathcal{KQ}(R_{F})_{\#}^{c}\simeq(\mathcal{K}(R_{F}){} _{\#}^{h({}_{\varepsilon}\mathbb{Z}/2)})^{c}. \]
\end{theorem}
The proof of Theorem \ref{theorem3} follows from Theorem \ref{theorem1} and results established in \cite{BK}. Earlier results in this direction inspired a much more general conjecture formulated by B.~Williams in \cite[3.4.2] {Williams}. However, in Appendix C below, we provide a counterexample to that conjecture, in the form of a ring with vanishing $K$-theory but nontrivial $KQ$-theory.
\begin{remark} Most of the theorems in this introduction are also true if we replace the $2$-completions of the spectra involved by their $2$-localizations. As we shall see through the paper, the proofs of most of our main theorems work as well in this context. There is an important exception however, namely our Theorem \ref{theorem3}, which is only true for completions.
\end{remark}
In our work in progress \cite{BKO:bottperiodicity} the results shown in this paper are used to give the first algebraic examples of Bott periodicity isomorphisms for hermitian $K$-groups. Another project uses these results to provide homological information.
\section{Preliminaries}
\label{section:preliminaryresults}We begin with a list of alternative characterizations of the class of real number fields $F$ of interest. To fix terminology, recall that the $r$ real embeddings of $F$ define the \emph{signature} map $F^{\times}/(F^{\times})^{2}\rightarrow(\mathbb{Z} /2)^{r}$ where a unit is mapped to the signs of its images under the real embeddings. One says that $R_{F}=\mathcal{O}_{F}[\frac{1}{2}]$, the ring of $2$-integers in $F$, has \emph{units of independent signs} if the signature map remains surjective when restricted to the square classes $R_{F}^{\times }/(R_{F}^{\times})^{2}$ of $R_{F}^{\times}$. A \emph{dyadic prime} in $F$ is a prime ideal in the ring of integers $\mathcal{O}_{F}$ lying over the rational prime ideal $(2)$. The \emph{narrow Picard group} \textrm{Pic}$_{+}(R_{F})$ consists of fractional $R_{F}$-ideals modulo totally positive principal ideals, defined as in \cite[V\S 1]{FrohlichTaylor}.
The \emph{Witt ring} $W(A)$ of a commutative unital ring $A$ with involution is defined in terms of Witt classes of symmetric nondegenerate bilinear forms \cite[I (7.1)]{MH}; in $K$-theoretic terms, as an abelian group it coincides with the cokernel ${}_{1}W_{0}(A)$ of the hyperbolic map $K_{0}(A)\rightarrow {}_{1}KQ_{0}(A)$ if $2$ is invertible in $A$ \cite{K:AnnM112fun}. Symmetrically, we define the \emph{coWitt group} $W^{\prime}(A)={}_{1} W_{0}^{\prime}(A)$ as the kernel of the forgetful rank map ${}_{1} KQ_{0}(A)\rightarrow{}K_{0}(A)$. In this section especially, we often use the simplified notations $W(A)$ and $W^{\prime}(A)$ instead of ${}_{1}W_{0}(A)$ and $_{1}W_{0}^{\prime}(A)$.
In the statement below, for a finite abelian group $G$, we write $G\{2\}$ for its $2$-Sylow subgroup.
\begin{proposition} \label{2+-regular characterization} Let $F$ be a number field with $r$ real embeddings and $c$ pairs of complex embeddings. Then the following are equivalent.
\begin{enumerate} \item $F$ is $2$-regular; that is, the $2$-Sylow subgroup of the finite abelian group $K_{2}(\mathcal{O}_{F})$ has order $2^{r}$.
\item The real embeddings of $F$ induce isomorphisms on $2$-Sylow subgroups \[ K_{2}(\mathcal{O}_{F})\{2\}\overset{\cong}{\longrightarrow}K_{2} (R_{F})\{2\}\overset{\cong}{\longrightarrow}\oplus^{r}K_{2}(\mathbb{R} )\cong(\mathbb{Z}/2)^{r}. \]
\item The finite abelian group $K_{2}(\mathcal{O}_{F(\sqrt{-1})})$ has odd order.
\item There is a unique dyadic prime in $F$ and the narrow Picard group \textrm{Pic}$_{+}(R_{F})$ has odd order.
\item There is a unique dyadic prime in $F$, the Picard group \textrm{Pic} $(R_{F})$ has odd order, and $R_{F}$ has units with independent signs.
\item The nilradical of the Witt ring $W(R_{F})$ is a finite abelian group of order $2^{c+1}$. In particular, if $F$ is a totally real number field, so that $c=0$, then the nilradical of $W(R_{F})$ has order $2$.
\item The Witt ring $W(R_{F})$ is a finitely generated abelian group of rank $r$ with torsion subgroup of order $2^{c+1}$. In particular, if $F$ is a totally real number field, then $W(R_{F})$ is isomorphic to $\mathbb{Z} ^{r}\oplus\mathbb{Z}/2$. \end{enumerate}
If $F$ is a totally real number field and satisfies any of the equivalent conditions above, then the free part of $W(R_{F})$ is generated by elements $\left\langle 1\right\rangle ,\left\langle u_{1}\right\rangle ,\dots ,\left\langle u_{r-1}\right\rangle $ where $u_{i}\in R_{F}^{\times}$ is negative at the $i$\thinspace th embedding and positive elsewhere. \end{proposition}
\noindent\textbf{Proof.} Obviously, (2) implies (1). For the converse, we apply Tate's $2$-rank formula for $K_{2}$ \cite[Theorem 6.2]{Tate}, which shows that the group $K_{2}(\mathcal{O}_{F})$ maps, via square power norm residue symbols, onto the subgroup of the direct sum (over all archimedean places and places over $2$) of copies of $\mu_{2}(F)=\mathbb{Z}/2$ that consists of the elements $z=(z_{v})$ such that $\sum z_{v}=0$. This forces $r$ to be a lower bound for the $2$-rank of $K_{2}(\mathcal{O}_{F})$. Of course, the unique group of order $2^{r}$ and $2$-rank at least $r$ is $(\mathbb{Z} /2)^{r}$; so, the converse follows.
See \cite[Proposition 2.2]{RO} for the equivalence between (2) and (4). By \cite[(4.1),(4.6)]{ConnerHurrelbrink}, again using \cite[Theorem 6.2]{Tate}, (2), (3) and (5) are equivalent. The equivalences between (4), (6), and (7) are immediate from \cite[Corollary 3.6, Theorem 4.7]{Czogala}. Finally, the claim concerning the generators of the free part of $W(R_{F})$ follows as in \cite[IV (4.3)]{MH}.
$\Box$
\begin{remark} 1. Further equivalent conditions, in terms of \'{e}tale cohomology, appear in \cite[Proposition 2.2]{RO}.
2. Berger \cite{Berger} calls a totally real number field $F$ satisfying (5) above $2^{+}$\emph{-regular}, and has shown in \cite{Berger} that each totally real $2$-regular number field has infinitely many totally real quadratic field extensions satisfying the equivalent conditions in Proposition \ref{2+-regular characterization}. In particular, there exist totally real $2$-regular number fields of arbitrarily high degree.
3. In the other direction, by \cite[(4.1)]{ConnerHurrelbrink} every subfield of a totally real $2$-regular number field also enjoys this property.
4. In \cite[pg.~95]{MH}, it is shown that for totally real $F$ there is also an equivalence between amended conditions (4)--(7) above, in which $R_{F}$ is replaced by $\mathcal{O}_{F}$ and $2^{c+1}$ by $2^{c}$. However, in view of the surjection \textrm{Pic}$_{+}(\mathcal{O}_{F})\twoheadrightarrow $\textrm{Pic}$_{+}(R_{F})$, these amended conditions define a strict subclass of those considered here. After Gauss, one knows that for a real quadratic number field $F=\mathbb{Q}(\sqrt{d})$ the narrow Picard group \textrm{Pic} $_{+}(\mathcal{O}_{F})$ is an odd torsion group if and only if $F$ has prime-power discriminant; that is, $d=2$ or $d=p$ with $p\equiv 1\;(\mathrm{mod}\ 4)$ a prime number. Thus $\mathbb{Q}(\sqrt{p})$ with $p\equiv3\;(\mathrm{mod}\ 4)$ a prime number and $\mathbb{Q}(\sqrt{2p})$ with $p\equiv\pm3\;(\mathrm{mod}\ 8)$ a prime number fail to lie in the subclass, cf.~\cite{CO}. \end{remark}
\textit{Suppose henceforth} that $2$ is a unit in a domain $A$ of dimension at most $1$, \textsl{e.g}.~a subring of some number field $F$. Let $k_{0}(A)$ denote the $0$\thinspace th Tate cohomology group $\widehat{H}^{0} (\mathbb{Z}/2;\,K_{0}(A))$ of $\mathbb{Z}/2$ acting on $K_{0}(A)$ by duality as in \cite[pg.~278]{K:AnnM112fun}, and $k_{0}^{\prime}(A)$ denote the $1$\thinspace st Tate cohomology group $\widehat{H}^{1}(\mathbb{Z} /2;\,K_{0}(A))$.
There is a well defined induced \emph{rank map }$\rho\colon W(A)\rightarrow k_{0}(A)$, whose image always has a $\mathbb{Z}/2$ summand. From the 12-term exact sequence established in \cite[pg.~278]{K:AnnM112fun} there is an exact sequence \[ k_{0}^{\prime}(A)\longrightarrow W^{\prime}(A)\overset{\varphi} {\longrightarrow}W(A)\overset{\rho}{\longrightarrow}k_{0}(A), \] where $\varphi$ is simply the composition $W^{\prime}(A)\hookrightarrow{} _{1}KQ_{0}(A)\twoheadrightarrow W(A)$.
The next result is almost immediate, but worth recording. Part (c) uses the observation that $\tilde{K}_{0}(A)\cong\mathrm{Pic}(A)$ when $A$ is of dimension at most $1$. If further \textrm{Pic}$(A)$ is an odd torsion group, then $k_{0}^{\prime}(A)=0$ and $k_{0}(A)\cong\mathbb{Z}/2$. Also, (d) refers to the fact that, when $A$ is a field, its \emph{fundamental ideal} is the unique ideal $I$ of $W(A)$ with $W(A)/I\cong\mathbb{F}_{2}$ \cite[III (3.3)]{MH}.
\begin{lemma} \label{proposition:dedekindandoddtorsionpicardgroup}(a) In general, $\varphi$ maps the coWitt group of $A$ onto the kernel of the rank map $\rho$.
(b) When $k_{0}^{\prime}(A)=0$, then $\varphi$ is an isomorphism between $W^{\prime}(A)$ and $\mathrm{Ker}\rho$.
(c) When $A$ has dimension at most $1$ and $\mathrm{Pic}(A)$ is an odd torsion group, then there is a natural short exact sequence \[ 0\rightarrow W^{\prime}(A)\overset{\varphi}{\longrightarrow}W(A)\overset{\rho }{\longrightarrow}\mathbb{Z}/2\rightarrow0\text{.} \]
(d) When $A$ is a field, then $\varphi$ induces an isomorphism between the coWitt group of $A$ and the fundamental ideal of the Witt ring.$
\Box$ \end{lemma}
We can now give a further characterization of the class of $2$-regular totally real number fields, in terms of the coWitt group.
\begin{lemma} \label{2-regular coWitt}A totally real number field $F$ with $r$ real embeddings is $2$-regular if and only if both
\begin{enumerate} \item[(i)] the Picard group $\mathrm{Pic}(R_{F})$ has odd order, and
\item[(ii)] the coWitt group $W^{\prime}(R_{F})$ is isomorphic to $\mathbb{Z}^{r}\oplus\mathbb{Z}/2$. \end{enumerate} \end{lemma}
\noindent\textbf{Proof.} Here we use the fact that, as for both $K_{0}$ and $KQ_{0}$, the Witt group $W(\mathbb{R})=W_{0}(\mathbb{R})$ and the coWitt group $W^{\prime}(\mathbb{R})=W_{0}^{\prime}(\mathbb{R})$ do not depend on the topology of $\mathbb{R}$.
In both directions of the statement of the lemma (one way uses Proposition \ref{2+-regular characterization}(5)), we have from Lemma \ref{proposition:dedekindandoddtorsionpicardgroup}(c) the map of short exact sequences (where the middle vertical map is surjective): \[ \begin{array} [c]{ccccccc} 0\rightarrow & W^{\prime}(R_{F}) & \overset{\varphi}{\longrightarrow} & W(R_{F}) & \overset{\rho}{\longrightarrow} & \mathbb{Z}/2 & \rightarrow0\\ & \downarrow & & \downarrow & & \downarrow\mathrm{id} & \\ 0\rightarrow & W^{\prime}(\mathbb{R}) & \overset{\varphi}{\longrightarrow} & W(\mathbb{R}) & \overset{\rho}{\longrightarrow} & \mathbb{Z}/2 & \rightarrow0 \end{array} \] Since $W(\mathbb{R})\cong\mathbb{Z}$ \cite[III (2.7)]{MH}, in the upper sequence $\rho$ must be trivial on torsion elements. By combining with Proposition \ref{2+-regular characterization}(7), we obtain the result.
$\Box$
When $F$ has $r$ real embeddings, for $A=R_{F},F$, the map $W(A)\rightarrow W(\mathbb{R})^{r}\cong\mathbb{Z}^{r}$ is the \emph{total signature }$\sigma$ \cite[III (2.9)]{MH}.
To define further invariants, we recall from \cite{K:AnnM112fun} generalizations of some of the definitions above. For $\varepsilon=\pm1$, and $n\geq1$, we set \begin{align*} {}_{\epsilon}W_{n}(A) & =\mathrm{Coker}[K_{n}(A)\rightarrow{}_{\epsilon }KQ_{n}(A)],\\ {}_{\epsilon}W_{n}^{\prime}(A) & =\mathrm{Ker}[{}_{\epsilon}KQ_{n} (A)\rightarrow K_{n}(A)],\\ k_{n}(A) & =\widehat{H}^{0}(\mathbb{Z}/2;\,K_{n}(A)),\\ k_{n}^{\prime}(A) & =\widehat{H}^{1}(\mathbb{Z}/2;\,K_{n}(A)). \end{align*}
The next invariant we shall employ is the \emph{discriminant map } ${}_{\epsilon}W_{0}^{\prime}(A)\rightarrow k_{1}^{\prime}(A)$. To recall the definition, suppose that $M$ and $N$ are quadratic modules with isomorphic underlying $A$-modules. The elements of ${}_{\epsilon}W_{0}^{\prime}(A)$ are of the form $M-N$ with $M$ and $N$ isomorphic modules. An isomorphism $\alpha\colon M\rightarrow N$ induces an automorphism $\alpha^{\ast}\alpha$ of $M$ that is antisymmetric for the $\mathbb{Z}/2$-action. Its class in $k_{1}^{\prime}(A)$, which is independent of $\alpha$, defines the desired invariant. For a generalization of the above we refer to \cite{K:AnnM112fun}.
If $SK_{1}(A)=0$ (\textsl{e.g. }$A$ a ring of $S$-integers in a number field \cite[Corollary 4.3]{BassMilnorSerre} or $A$ a field), then $k_{1}^{\prime }(A)$ is isomorphic to the group of square classes $A^{\times}/(A^{\times })^{2}$ of units in $A$ and $k_{1}(A)=\{\pm1\}$. This affords the following description.
\begin{lemma} \label{SK_1(A) = 0}Assume that $SK_{1}(A)=0$\textsf{.}
Then the rank/determinant map ${}_{1}W_{1}(A)\rightarrow k_{1}(A)=\{\pm1\}$ is surjective, and there is a short exact sequence \[ 0\rightarrow{}_{-1}W_{2}(A)\longrightarrow{}_{1}W_{0}^{\prime}(A)\overset {\mathrm{disc}}{\longrightarrow}k_{1}^{\prime}(A)=A^{\times}/(A^{\times} )^{2}\rightarrow0\text{.} \]
If $A$ is a field, we may identify $W^{\prime}(A)={}_{1}W_{0}^{\prime}(A)$ (resp.~ ${}_{-1}W_{2}(A)$) with the fundamental ideal in the Witt group $W(A)$ (resp.~ its square), so that the exact sequence above reduces to \[ 0\rightarrow I^{2}\longrightarrow I\longrightarrow I/I^{2}\rightarrow0\text{.} \] The result is still true for the field $\mathbb{R}$ and $\mathbb{C}$ with their usual topology. \end{lemma}
\noindent\textbf{Proof.} Since $SK_{1}(A)=0$, we have $K_{1}(A)=A^{\times}$. Therefore, $k_{1}(A)$ is reduced to $\pm1$ which is a $1\times1$ unitary matrix. This explains the surjectivity. By the exact sequence \[ {}_{1}W_{1}(A)\longrightarrow k_{1}(A)\longrightarrow{}_{-1}W_{2} (A)\longrightarrow{}_{1}W_{0}^{\prime}(A)\longrightarrow k_{1}^{\prime}(A) \] from \cite{K:AnnM112fun}, ${}_{-1}W_{2}(A)$ now identifies with the kernel of the induced discriminant map $W^{\prime}(A)\rightarrow k_{1}^{\prime}(A)\cong A^{\times}/(A^{\times})^{2}$.
Moreover, taking $\varepsilon=1$, $W^{\prime}(A)$ surjects onto $A^{\times }/(A^{\times})^{2}$ since the discriminant maps $\left\langle u\right\rangle -\left\langle 1\right\rangle $ to $u\in A^{\times}/(A^{\times})^{2}$.
When $A$ is a field, by Lemma \ref{proposition:dedekindandoddtorsionpicardgroup}\thinspace(d) above the coWitt group identifies with the fundamental ideal $I$, while by \cite[III (5.2)]{MH}\textsf{ }the discriminant induces an isomorphism $I/I^{2}\cong A^{\times}/(A^{\times})^{2}$. Finally, if $A=\mathbb{R}$ or $\mathbb{C}$ with the usual topology, it is easy to see that the group${}_{-1}W_{2}(A)$ coincides with the same group when we regard $\mathbb{R}$ and $\mathbb{C}$ with the discrete topology, thanks to the fundamental theorem in topological hermitian $K$-theory proved in \cite{Karoubi LNM343}.
$\Box$
\begin{examples} \label{examples for A/A^{2}}In the case of $A=\mathbb{R}$, from \cite[III (2.7)]{MH} the discriminant map above is the surjection $\mathbb{Z} \rightarrow\mathbb{Z}/2$.
For $A=\mathbb{F}_{q}$ by \cite[III (5.2), (5.9)]{MH} it is the isomorphism $A^{\times}/(A^{\times})^{2}\cong\mathbb{Z}/2$.
For $A=R_{F}$, with $F$ totally real, the Dirichlet $S$-unit theorem for $R_{F}$ \cite[Ch. IV Theorem 9]{Weil}\textsf{ }gives $R_{F}^{\times} \cong\mathbb{Z}^{r+d-1}\times\mu(F)$, where $d$ is the number of dyadic places, and the group of roots of unity $\mu(F)$ has order $2$. In the $2$-regular case, $d=1$ by Proposition \ref{2+-regular characterization}(4), so $R_{F}^{\times}/(R_{F}^{\times})^{2}\cong(\mathbb{Z}/2)^{r+1}$. \end{examples}
What follows is a key ingredient in the proof of Theorem \ref{theorem1}.
\begin{proposition} \label{proposition:keyinput} Suppose that $F$ is a totally real $2$-regular number field with $r$ real embeddings. Then the residue field map $R_{F}\rightarrow\mathbb{F}_{q}$ and the real embeddings of $F$ induce an isomorphism between coWitt groups \[ \begin{array} [c]{c} W^{\prime}(R_{F})\overset{\cong}{\longrightarrow}\bigoplus\limits^{r} W^{\prime}(\mathbb{R})\oplus W^{\prime}(\mathbb{F}_{q})\cong\mathbb{Z} ^{r}\oplus\mathbb{Z}/2\text{.} \end{array} \]
\end{proposition}
\noindent\textbf{Proof.} Recalling that $SK_{1}$ is trivial for $R_{F}$ and $F$ \cite[Corollary 4.3]{BassMilnorSerre}, we obtain from the last lemma a map of short exact sequences \[ \begin{array} [c]{ccccccc} 0\rightarrow & {}_{-1}W_{2}(R_{F}) & \rightarrow & W^{\prime}(R_{F}) & \rightarrow & R_{F}^{\times}/(R_{F}^{\times})^{2} & \rightarrow0\\ & \downarrow & & \downarrow & & \downarrow & \\ 0\rightarrow & \bigoplus\limits^{r}{}_{-1}W_{2}(\mathbb{R})\oplus{}_{-1} W_{2}(\mathbb{F}_{q}) & \rightarrow & \bigoplus\limits^{r}W^{\prime }(\mathbb{R})\oplus W^{\prime}(\mathbb{F}_{q}) & \rightarrow & \bigoplus \limits^{r}\mathbb{R}^{\times}/(\mathbb{R}^{\times})^{2}\oplus\mathbb{F} _{q}^{\times}/(\mathbb{F}_{q}^{\times})^{2} & \rightarrow0 \end{array} \] which by Lemma \ref{2-regular coWitt} and Example (\ref{examples for A/A^{2}}) above\textsf{ }takes the form: \begin{equation} \begin{array} [c]{ccccccc} 0\rightarrow & \mathbb{Z}^{r} & \rightarrow & \mathbb{Z}^{r}\oplus \mathbb{Z}/2 & \rightarrow & (\mathbb{Z}/2)^{r+1} & \rightarrow0\\ & \downarrow & & \downarrow & & \downarrow & \\ 0\rightarrow & \bigoplus\limits^{r}\mathbb{Z}\oplus0 & \rightarrow & \bigoplus\limits^{r}\mathbb{Z}\oplus\mathbb{Z}/2 & \rightarrow & \bigoplus\limits^{r}\mathbb{Z}/2\oplus\mathbb{Z}/2 & \rightarrow0 \end{array} \label{diagram: computed discriminant} \end{equation}
From the homotopy cartesian square (\ref{algebraic K Bokstedt square}) of $2$-completed spectra, established in \cite{HO} and \cite{Mitchell} -- see Appendix A, we deduce the short exact sequence of $2$-completed groups \begin{equation} \begin{array} [c]{c} 0\rightarrow\bigoplus^{r}K_{2}(\mathbb{C})\rightarrow K_{1}(R_{F} )\rightarrow\bigoplus^{r}K_{1}(\mathbb{R})\oplus K_{1}(\mathbb{F} _{q})\rightarrow0\text{.} \end{array} \end{equation} (The left hand $0$ is justified by the facts that $K_{2}(\mathbb{F}_{q})=0$ and $K_{2}(\mathbb{R})=\mathbb{Z}/2$, while $K_{2}(\mathbb{C})$ is torsion-free\textsf{.) }Therefore, the final vertical map in (\ref{diagram: computed discriminant}) is an isomorphism. Observe that commutativity of the right-hand square implies that the middle vertical map in (\ref{diagram: computed discriminant}) is a monomorphism on the\textsf{ }$\mathbb{Z}/2$ summand.\textsf{ }Thus, to show that the middle vertical map is an isomorphism, because this is equivalent to its being an epimorphism,\textsf{ }it suffices to show that either of the homomorphisms $W^{\prime}(R_{F})\rightarrow\bigoplus\limits^{r}W^{\prime}(\mathbb{R})$ or ${}_{-1}W_{2}(R_{F})\rightarrow\bigoplus\limits^{r}{}_{-1}W_{2}(\mathbb{R})$ is surjective.
For the former homomorphism, we use the map of short exact sequences afforded by Lemma \ref{proposition:dedekindandoddtorsionpicardgroup}(c): \[ \begin{array} [c]{ccccccc} 0\rightarrow & W^{\prime}(R_{F}) & \longrightarrow & W(R_{F}) & \longrightarrow & \mathbb{Z}/2 & \rightarrow0\\ & \downarrow\sigma^{\prime} & & \downarrow\sigma & & \downarrow\Delta & \\ 0\rightarrow & \bigoplus\limits^{r}W^{\prime}(\mathbb{R}) & \longrightarrow & \bigoplus\limits^{r}W(\mathbb{R}) & \longrightarrow & \bigoplus\limits^{r} \mathbb{Z}/2 & \rightarrow0 \end{array} \] Since the diagonal map $\Delta$ is injective, the snake lemma gives the exact sequence \[ 0\longrightarrow\mathrm{Coker}\sigma^{\prime}\longrightarrow\mathrm{Coker} \sigma\longrightarrow(\mathbb{Z}/2)^{r-1}\longrightarrow0\text{.} \] According to \cite[Corollary 4.8]{Czogala} (applicable because by Proposition \ref{2+-regular characterization}(4) \textrm{Pic}$_{+}(R_{F})$ has odd order), $\sigma:W(R_{F})\rightarrow\bigoplus\limits^{r}W(\mathbb{R})$ has the same cokernel as $\sigma:W(F)\rightarrow\bigoplus\limits^{r}W(\mathbb{R})$. Now, by \cite[pp.~64-65]{MH} (applicable because by Proposition \ref{2+-regular characterization}(5) $R_{F}$ has units with independent signs), $\mathrm{Coker}\sigma$ is $(\mathbb{Z}/2)^{r-1}$. Hence, $\mathrm{Coker}\sigma^{\prime}=0$, as desired.
The second approach, showing that ${}_{-1}W_{2}(R_{F})\rightarrow \bigoplus\limits^{r}{}_{-1}W_{2}(\mathbb{R})$ is surjective, instead of \cite{Czogala} uses the \textquotedblleft classical\textquotedblright \ treatment of \cite{MH}, by invoking the generalized Hasse-Witt invariant for quadratic forms.
Recall the map ${}_{-1}W_{2}(A)\longrightarrow k_{2}(A)$ employed in the definition of the 12-term exact sequence in \cite{K:AnnM112fun}. If $A$ is a field, then by Lemma \ref{SK_1(A) = 0} above we may identify ${}_{-1}W_{2}(A)$ with $I^{2}(A)$; according to \cite{Guin}, this map from $I^{2}(A)$ to $k_{2}(A)=K_{2}(A)/2$ gives an equivalent definition of the classical \emph{Hasse-Witt invariant}. Moreover, the Hasse-Witt invariants for $R_{F}$ and $F$ induce a commutative diagram with exact rows: \[ \begin{array} [c]{ccccc} {}_{-1}W_{2}(R_{F}) & \rightarrow & k_{2}(R_{F}) & \rightarrow & 0\\ \downarrow & & \downarrow & & \\ {}_{-1}W_{2}(F) & \rightarrow & k_{2}(F) & \rightarrow & 0 \end{array} \] Now, from \cite{HO} and \cite{Mitchell}, the real embeddings of $F$ induce a surjective map from $R_{F}^{\times}$ to $\oplus^{r}K_{1}(\mathbb{R})$. Hence, ${}_{-1}W_{2}(R_{F})$ surjects onto $k_{2}(R_{F})\cong(\mathbb{Z}/2)^{r}$ because the tensor product $(\left\langle u\right\rangle -\left\langle 1\right\rangle )\otimes(\left\langle v\right\rangle -\left\langle 1\right\rangle )$ maps to the symbol $\{u,v\}$ in $k_{2}(R_{F})$. By reference to \cite[III (5.9)]{MH}, where the arguments can be extended to any ring $D$ of $S$-integers in a number field, the kernels of the Hasse-Witt surjections for $R_{F}$ and $F$ are isomorphic to $I^{3}(F)\cong\bigoplus^{r} I^{3}(\mathbb{R})\cong8\mathbb{Z}^{r}$ via the signature map, and hence there is a map of short exact sequences: \[ \begin{array} [c]{ccccccc} 0\rightarrow & 8\mathbb{Z}^{r} & \longrightarrow & {}_{-1}W_{2}(R_{F}) & \longrightarrow & (\mathbb{Z}/2)^{r} & \rightarrow0\\ & \downarrow\cong & & \downarrow & & \downarrow\cong & \\ 0\rightarrow & 8\mathbb{Z}^{r} & \longrightarrow & \oplus^{r}{}_{-1} W_{2}(\mathbb{R}) & \overset{\chi}{\longrightarrow} & (\mathbb{Z}/2)^{r} & \rightarrow0 \end{array} \] where the notation $8\mathbb{Z}^{r}$ means the image in $\oplus^{r} W(\mathbb{R})=\mathbb{Z}^{r}$ of the kernel of $\chi$ by the total signature map. The $5$-lemma now shows that ${}_{-1}W_{2}(R_{F})$ is isomorphic to $\oplus^{r}{}_{-1}W_{2}(\mathbb{R})$ by the middle vertical map.
$\Box$
We finish this section by relating the numbers $w_{m}$ in the formulation of Theorem \ref{theorem2} to $t_{n}$, the $2$-adic valuation $(q^{(n+1)/2} -1)_{2}$ of $q^{(n+1)/2}-1$ for $n$ odd.
\begin{lemma} Suppose that $q\equiv1\;(\mathrm{mod}\ 4)$, and write $(-)_{2}$ for the $2$-adic valuation. Then $(q^{m}-1)_{2}=(q-1)_{2}(m)_{2}$. \end{lemma}
\noindent\textbf{Proof.} With $q=4r+1$ and $t=(m)_{2}$, note that $q^{t}-1$ is divisible by $4rt$ but not by $8rt$, due to the binomial identity. Set $s=8t(r)_{2}$ and $u=m/t$. Since $(\mathbb{Z}/2^{s})^{\times}$ has even order and $u$ is odd, $q^{m}-1=(q^{t})^{u}-1$ is divisible by $s/2$ but not by $s$.
$\Box$
\begin{lemma} Let $q$ be an odd number. If $m$ is odd, then $(q^{m}-1)_{2}=(q-1)_{2}$. If $m=2m^{\prime}$ is even, then $(q^{m}-1)_{2}=(q^{2}-1)_{2}(m^{\prime})_{2}$. \end{lemma}
\noindent\textbf{Proof.} If $m$ is odd, writing $q^{m}-1=(q-1)(q^{m-1} +\cdots+1)$ shows that $(q^{m}-1)_{2}=(q-1)_{2}$ since $(q^{m-1}+\cdots+1)$ is odd. If $m=2m^{\prime}$ is even, write $q^{m}-1$ as $(q^{2})^{m^{\prime}}-1$ where $q^{2}\equiv1\;(\mathrm{mod}\ 4)$, and apply the previous lemma.
$\Box$
Returning now to the setting of this paper, we have a prime $q$ defined, as in the Introduction, in terms of the number field $F$ such that $q$ is $\equiv \pm1\;(\mathrm{mod}\ 2^{a_{F}})$ but not $(\mathrm{mod}\ 2^{a_{F}+1})$. Thus, for even $m$, the last lemma above gives \[ (q^{m}-1)_{2}=2^{a_{F}}(m)_{2}=:w_{m}\text{,} \] where the number $w_{m}$ appears in Theorem \ref{theorem2}. Hence, we have the following description of $t_{n}:=(q^{(n+1)/2}-1)_{2}$ in terms of $w_{m}$.
\begin{lemma} \label{lemma: t and w}Let $F$ be a totally real number field for which $q$ is chosen as in the Introduction. Then, for $n\equiv3\;(\mathrm{mod}\ 4)$, \[ t_{n}=w_{(n+1)/2}\text{.}
\]
$\Box
$ \end{lemma}
\section{Proof of Theorem \ref{theorem1}}
\label{section:proofoftheorem1} Before commencing consideration of Theorem \ref{theorem1}, we make a few preparatory remarks.
The following result, of independent interest, will be used below.
\begin{proposition} \label{Wodd}Let $R$ be any ring of $S$-integers in a number field $F$ with $1/2\in R$. Then the odd torsion of the generalized Witt groups \[ {}_{\varepsilon}W_{n}(R)=\mathrm{Coker}\left[ K_{n}(R)\longrightarrow {}_{\varepsilon}KQ_{n}(R)\right] \] and the coWitt groups \[ {}_{\varepsilon}W_{n}^{^{\prime}}(R)=\mathrm{Ker}\left[ {}_{\varepsilon }KQ_{n}(R)\longrightarrow K_{n}(R)\right] \] is trivial for all values of $n\in
\mathbb{Z}
$. \end{proposition}
\noindent\textbf{Proof.} We first remark that the higher Witt groups and coWitt groups have the same odd torsion because of the $12$ term exact sequence proved in \cite{K:AnnM112fun}. The same exact sequence shows that we have an isomorphism modulo odd torsion between ${}_{\varepsilon}W_{n}(R)$ and ${}_{-\varepsilon}W_{n+2}(R)$. Therefore, we have only to compute the four cases $\varepsilon=\pm1$ and $n=0,1$.
For $\varepsilon=-1$, recall that ${}_{-1}KQ_{1}(R)=\mathrm{Sp}(R)\left/ \left[ \mathrm{Sp}(R),\,\mathrm{Sp}(R)\right] \right. =0$ according to \cite[Corollary $4.3$]{BassMilnorSerre}, and therefore its quotient ${} _{-1}W_{1}(R)=0$. By \cite[I (3.5)]{MH} there are isomorphisms ${}_{-1} KQ_{0}(R)\cong\mathbb{Z}$ detected by the (even) rank of the free symplectic $R$-inner product space. Therefore, ${}_{-1}W_{0}(R)=\mathrm{Coker}\left[ K_{0}(R)\longrightarrow{}_{-1}KQ_{0}(R)\right] =0.$
For $\varepsilon=1,$ it is well known (see for instance \cite[Corollary IV.3.3]{MH}) that the Witt group ${}_{1}W(R)$ injects in ${}_{1}W(F)$ and that the torsion of ${}_{1}W(F)$ is $2$-primary according to the same book \cite[Theorem III.3.10]{MH}. Finally, we consider the exact sequence \[ K_{1}(R)\longrightarrow{}_{1}KQ_{1}(R)\longrightarrow{}_{2}\mathrm{Pic} (R)\oplus Z_{2}(R)\longrightarrow0 \] proved in \cite[(4.7.6)]{Bass} with different notations. Here, ${} _{2}\mathrm{Pic}(R)$ is the $2$-torsion of the Picard group and $Z_{2}(R)$ is the group of locally constant maps from $\mathrm{Spec}(R)$ to $
\mathbb{Z}
/2$. Moreover, $\mathrm{Spec}(R)$ is connected, so that $Z_{2}(R)=
\mathbb{Z}
/2$, and the previous exact sequence yields the isomorphism \begin{equation} {}_{1}W(R)\cong{}_{2}\mathrm{Pic}(R)\oplus
\mathbb{Z}
/2\text{.}
\label{W1} \end{equation}
$\Box
$
The next follows by comparing the splittings of the hermitian $K$-theory spectrum and the $K$-theory spectrum according to their canonical involutions \cite[pg.~253]{K:AnnM112fun}. We record it for the sake of completeness: by the previous result, it may be applied to any ring of $2$-integers in a number field.
\begin{proposition} Let $A$ be any ring, with $n$ an integer such that the odd torsion ${}_{\varepsilon}W_{n}(A)$ vanishes. Then the odd torsion subgroup of ${}_{\varepsilon}KQ_{n}(A)$ is the invariant part of the odd torsion subgroup of $K_{n}(A)$ induced by the involution $M\mapsto{}^{t}M^{-1}$ on $\mathrm{GL}(A)$.
$\Box
$ \end{proposition}
We now turn to consideration of spectra. The following simple exercise in stable homotopy theory ((iv), (v) are from \cite[Chapter VI (5.1),(5.2)] {BK304}, suitably adopted to spectra in \cite[Proposition 2.5]{Bousfieldloc}) will be used implicitly at various times. Here and subsequently, we write $\Omega\mathcal{E}$ to indicate the shifted spectrum whose space at level $n$ is $\mathcal{E}_{n-1}\simeq\Omega(\mathcal{E}_{n})$; similarly for other shifts. In (iv), $\mathbb{Z}_{2^{\infty}}$ denotes the quasicyclic $2$-group $\underrightarrow{\mathrm{lim}}\,\mathbb{Z}/2^{n}$.
\begin{lemma} \label{Connective.Spectra} For any spectrum $\mathcal{E}$ and fiber sequence $\mathcal{F}\rightarrow\mathcal{E}\rightarrow\mathcal{B}$ of spectra, the connective covers and $2$-completions have the following properties.
\begin{enumerate} \item[\emph{(i)}] $\mathcal{F}^{c}\rightarrow\mathcal{E}^{c}\rightarrow \mathcal{B}^{c}$ is a fiber sequence, provided that $\pi_{0}(\mathcal{E} )\rightarrow\pi_{0}(\mathcal{B})$ is an epimorphism.
\item[\emph{(ii)}] $(\Omega\mathcal{E})^{c}\simeq\Omega(\mathcal{E}^{c})$, provided that $\pi_{0}(\mathcal{E})=0$.
\item[\emph{(iii)}] $(\Omega^{-1}\mathcal{E})^{c}\simeq\Omega^{-1} (\mathcal{E}^{c})$, provided that $\pi_{-1}(\mathcal{E})=0$.
\item[\emph{(iv)}] $(\mathcal{E}_{\#})^{c}\simeq(\mathcal{E}^{c})_{\#} $\thinspace, provided that $\mathrm{Hom}(\mathbb{Z}_{2^{\infty}},$ \thinspace$\pi_{-1}(\mathcal{E}))=0$, as for example when $\pi_{-1} (\mathcal{E})$ is finitely generated.
\item[\emph{(v)}] When all homotopy groups of $\mathcal{E}$ are finitely generated, then for all $i\in\mathbb{Z}$ the $2$-completed spectrum $\mathcal{E}_{\#}$ has \[ \pi_{i}(\mathcal{E}_{\#})=\pi_{i}(\mathcal{E})\otimes\mathbb{Z}_{2} \text{.}
\]
$\Box
$ \end{enumerate} \end{lemma}
Even though (iv) above applies to our spectra whose homotopy groups are finitely generated, we clarify our convention by defining $\mathcal{E} _{\#}^{c}$ as $(\mathcal{E}^{c})_{\#}$. This spectrum $\mathcal{E}_{\#}^{c}$ is named the $2$-completed connective spectrum associated to $\mathcal{E}$. For instance, \[ \mathcal{K}(R_{F})_{\#}^{c}=(\mathcal{K}(R_{F})^{c})_{\#}=\mathcal{K} (R_{F})_{\#}\,\text{,} \] and for $n\geq0$ \[ \pi_{n}(\mathcal{K}(R_{F})_{\#}^{c})=K_{n}(R_{F})\otimes\mathbb{Z} _{2}\,\text{;} \] while \[ {}_{\varepsilon}\mathcal{KQ}(R_{F})_{\#}^{c}=({}_{\varepsilon}\mathcal{KQ} (R_{F})^{c})_{\#}\,\text{,} \] and for $n\geq0$ \[ \pi_{n}({}_{\varepsilon}\mathcal{KQ}(R_{F})_{\#}^{c})={}_{\varepsilon} KQ_{n}(R_{F})\otimes\mathbb{Z}_{2}\,\text{.} \] This uses the result of \cite[(3.6), pg. 795.]{BK} that the groups ${}_{\varepsilon}KQ_{n}(R_{F})$ are finitely generated. For completeness' sake, we give an alternative proof in Proposition \ref{Finiteness} at the end of this section.
The following lemma is used later on in our argument.
\begin{lemma} \label{Lemma modulo m} Let $\mathcal{E}$ and $\mathcal{F}$ be two spectra with finitely generated homotopy groups in each degree. Let \[ f:\mathcal{E\longrightarrow F} \] be a morphism that induces an isomorphism between homotopy groups $\pi _{i}(-;\,\mathbb{Z}/2^{r})$ for $i\geq n$ and sufficiently large $r$.
\emph{(a) }Then $f$ induces an isomorphism between the $2$-primary torsion of $\mathcal{E}$ and $\mathcal{F}$ in all degrees $\geq n$.
\emph{(b) }Moreover, if also $f$ induces an isomorphism between rational homotopy groups for degrees $\geq n$, then it induces an isomorphism between $\mathcal{E}$ and $\mathcal{F}$ in these degrees after tensoring with $2$-local integers $\mathbb{Z}_{(2)}$, and hence after tensoring with $2$-adic integers $\mathbb{Z}_{2}$. \end{lemma}
\noindent\textbf{Proof.} \textbf{(a) }We consider the map of Bockstein short exact sequences \[ \begin{array} [c]{ccccccc} 0\rightarrow & \pi_{i}(\mathcal{E})/2^{r} & \longrightarrow & \pi _{i}(\mathcal{E};\,\mathbb{Z}/2^{r}) & \longrightarrow & {}_{2^{r}}\pi _{i-1}(\mathcal{E}) & \rightarrow0\\ & \downarrow & & \downarrow^{\cong} & & \downarrow & \\ 0\rightarrow & \pi_{i}(\mathcal{F})/2^{r} & \longrightarrow & \pi _{i}(\mathcal{F};\,\mathbb{Z}/2^{r}) & \longrightarrow & {}_{2^{r}}\pi _{i-1}(\mathcal{F}) & \rightarrow0 \end{array} \] Evidently, this implies that for all $r$ the map ${}_{2^{r}}\pi_{i-1} (\mathcal{E})\rightarrow{}_{2^{r}}\pi_{i-1}(\mathcal{F})$ between $2^{r} $-torsion subgroups is surjective whenever $i\geq n$. If now $r$ is sufficiently large, then injectivity of the map $\pi_{i}(\mathcal{E} )/2^{r}\rightarrow\pi_{i}(\mathcal{F})/2^{r}$ yields an isomorphism of the $2$-primary torsion subgroups, again whenever $i\geq n$.
\noindent\textbf{(b) }Here, following \cite[pg.32]{Sullivan}, we first note that $\pi_{i}$ commutes with direct limits of coefficient groups, giving an isomorphism on $\pi_{i}(-;\,\mathbb{Z}_{2^{\infty}})$. Then, from the exact sequence of homotopy groups associated to the short exact sequence of coefficients \[ 0\rightarrow\mathbb{Z}_{(2)}\longrightarrow\mathbb{Q}\longrightarrow \mathbb{Z}_{2^{\infty}}\rightarrow0\text{,} \] we obtain an isomorphism on $\pi_{i}(-;\,\mathbb{Z}_{(2)})$. However, since $\mathbb{Z}_{(2)}$ is a flat $\mathbb{Z}$-module, the $\mathrm{Tor}$ term vanishes in the universal coefficient sequence \[ 0\rightarrow\pi_{i}(\mathcal{E})\otimes_{\mathbb{Z}}\mathbb{Z}_{(2)} \longrightarrow\pi_{i}(\mathcal{E};\,\mathbb{Z}_{(2)})\longrightarrow \mathrm{Tor}_{\mathbb{Z}}(\pi_{i-1}(\mathcal{E}),\,\mathbb{Z}_{(2)} )\rightarrow0\text{.} \] Finally, of course, we use the fact that $\mathbb{Z}_{(2)}\otimes_{\mathbb{Z} }\mathbb{Z}_{2}\cong\mathbb{Z}_{2}$.
$\Box$
Following the exposition in \cite[\S 5]{BK}, we start out the proof of Theorem \ref{theorem1} by choosing an embedding of the field of $q$-adic numbers $\mathbb{Q}_{q}$ into the complex numbers $\mathbb{C}$ such that the induced composite map \[ {}_{\varepsilon}\mathcal{KQ}(\mathbb{Z}_{q})^{c}\rightarrow{}_{\varepsilon }\mathcal{KQ}(\mathbb{Q}_{q})^{c}\rightarrow{}_{\varepsilon}\mathcal{KQ} (\mathbb{C})^{c} \] agrees with Friedlander's Brauer lift ${}_{\varepsilon}\mathcal{KQ} (\mathbb{F}_{q})^{c}\rightarrow{}_{\varepsilon}\mathcal{KQ}(\mathbb{C})^{c}$ from \cite{Friedlander} under the rigidity equivalence\textsf{ }between ${}_{\varepsilon}\mathcal{KQ}(\mathbb{Z}_{q})^{c}$ and ${}_{\varepsilon }\mathcal{KQ}(\mathbb{F}_{q})^{c}$, \textsl{cf}.~\cite[Lemma 5.3]{BK}. This idea is the hermitian analogue of a widely used construction in algebraic $K$-theory originating in the works of B{\"{o}}kstedt \cite{Bok}, Dwyer-Friedlander \cite{DF}, and Friedlander \cite{Friedlander}.
The ring maps relating $R_{F}$ to $\mathbb{F}_{q}$, $\mathbb{R}$ and $\mathbb{C}$ induce the commuting B{\"{o}}kstedt square for hermitian $K$-theory spectra (\ref{hermitian K Bokstedt square}) in the Introduction.
The same ring maps induce the analogous commuting B{\"{o}}kstedt square for algebraic $K$-theory spectra (\ref{algebraic K Bokstedt square}) in the Introduction comprising ${}_{\varepsilon}\mathbb{Z}/2$-equivariant maps.
Denote by ${}_{\varepsilon}\overline{\mathcal{KQ}}(R_{F})$ the homotopy cartesian product of ${}_{\varepsilon}\mathcal{KQ}(\mathbb{F}_{q})^{c}$ and $\vee^{r}{}_{\varepsilon}\mathcal{KQ}(\mathbb{R})^{c}$ over $\vee^{r} {}_{\varepsilon}\mathcal{KQ}(\mathbb{C})^{c}$, afforded by the B{\"{o}}kstedt square of hermitian $K$-theory spectra. (${}_{\varepsilon}\overline {\mathcal{KQ}}(R_{F})$ is thereby connective because of the epimorphism ${}_{\varepsilon}KQ_{0}(\mathbb{R})\twoheadrightarrow{}_{\varepsilon} KQ_{0}(\mathbb{C})$. Moreover, since the spectra ${}_{\varepsilon} \mathcal{KQ}(\mathbb{F}_{q})^{c}$ and $\vee^{r}{}_{\varepsilon}\mathcal{KQ} (\mathbb{R})^{c}$ have finitely generated homotopy groups, so too does ${}_{\varepsilon}\overline{\mathcal{KQ}}(R_{F})$.)
Thus, Theorem \ref{theorem1} becomes the assertion that there is a homotopy equivalence between the $2$-completed connective spectra associated to the map \begin{equation} {}_{\varepsilon}\mathcal{KQ}(R_{F})\rightarrow{}_{\varepsilon}\overline {\mathcal{KQ}}(R_{F})\text{.} \label{weakequivalence} \end{equation} We write ${}_{\varepsilon}\overline{KQ}_{n}(R_{F})$ for the homotopy groups of the target spectrum above.
We prove Theorem \ref{theorem1} by means of the following strategy, previously adopted in \cite[\S 5]{BK} in the case of the rational field. The low-dimensional computations in Theorem \ref{lowdimensionalcomputations} below show that (\ref{weakequivalence}) induces an isomorphism modulo odd torsion, on integral homotopy groups ${}_{\varepsilon}\pi_{n}$ for $n=0,1$. Moreover, these homotopy groups are trivial in degree $n=-1$. When this is combined with the fact that the corresponding algebraic $K$-theory square (\ref{algebraic K Bokstedt square}) is homotopy cartesian (which is shown in both \cite{HO} and \cite{Mitchell}, see also Appendix \ref{section:K-theorybackground}), and the induction methods for hermitian $K$-groups in \cite[\S 3]{BK}, we deduce that the morphism of spectra (\ref{weakequivalence}) induces an isomorphism of all homotopy groups both rationally and with finite $2$-group coefficients. Since both spectra have their integral homotopy groups finitely generated in all dimensions, we may apply Lemma \ref{Lemma modulo m} to obtain an isomorphism of $2$-completions of the integral homotopy groups of these spectra. Then Lemma \ref{Connective.Spectra}\thinspace(v) finishes the proof.
We must therefore establish the following key computational result which extends the low-dimensional calculations for the rational integers in \cite[\S 4]{BK} to every totally real $2$-regular number field.
\begin{theorem} \label{lowdimensionalcomputations} Let $n=-1,0$ or $1$ and $\varepsilon=\pm1$. Then the map of homotopy groups \[ {}_{\varepsilon}\pi_{n}\colon{}_{\varepsilon}KQ_{n}(R_{F})\rightarrow {}_{\varepsilon}\overline{KQ}_{n}(R_{F}) \] is an isomorphism, except for $\varepsilon=1$ and $n=0$ where it is an isomorphism modulo odd torsion. \end{theorem}
\noindent\textbf{Proof.} The theorem will be proved after many preliminary lemmas listed below. According to the definition of the spectrum ${}_{\varepsilon}\overline{\mathcal{KQ}}(R_{F})$ as the homotopy cartesian product of ${}_{\varepsilon}\mathcal{KQ}(\mathbb{F}_{q})^{c}$ and $\vee^{r} {}_{\varepsilon}\mathcal{KQ}(\mathbb{R})^{c}$ over $\vee^{r}{}_{\varepsilon }\mathcal{KQ}(\mathbb{C})^{c}$, there is a naturally induced long exact sequence \begin{equation} \begin{array} [c]{ccc} \cdots\rightarrow\overset{r}{\bigoplus}{}_{\varepsilon}KQ_{n+1}(\mathbb{C} )\rightarrow{}_{\varepsilon}\overline{KQ}_{n}(R_{F})\rightarrow{} _{\varepsilon}KQ_{n}(\mathbb{F}_{q})\oplus\overset{r}{\bigoplus} {}_{\varepsilon}KQ_{n}(\mathbb{R})\rightarrow\cdots. & & \end{array} \label{longexactsequence1} \end{equation} We shall deal in turn with each of the six cases $\varepsilon=\pm1$, $n=-1,0,1$, in the remaining lemmas of this section.
\begin{lemma} \label{lemma:firstcase} The map ${}_{-1}\pi_{1}\colon{}_{-1}KQ_{1} (R_{F})\rightarrow{}_{-1}\overline{KQ}_{1}(R_{F})$ is an isomorphism between trivial groups. \end{lemma}
\noindent\textbf{Proof.} \label{epsilon=-1, n=1} Recall that ${}_{-1} KQ_{1}(A)=0$ when $A=R_{F},\mathbb{F}_{q},\mathbb{R}$ by for example \cite[Th\'eor\`eme 2.13]{K:AnnScENS} (using \cite[Corollary 12.5] {BassMilnorSerre} ). Thus (\ref{longexactsequence1}) shows that it suffices to note that ${}_{-1}KQ_{2}(\mathbb{C})=\pi_{1}(\mathrm{Sp})=0$ for the infinite symplectic group $\mathrm{Sp}$.
$\Box$
\begin{lemma} \label{lemma:secondcase} The map ${}_{-1}\pi_{0}\colon{}_{-1}KQ_{0} (R_{F})\rightarrow{}_{-1}\overline{KQ}_{0}(R_{F})\cong\mathbb{Z}$ is an isomorphism. \end{lemma}
\noindent\textbf{Proof.} \label{epsilon=-1, n=0} By \cite[I (3.5)]{MH} again, there are isomorphisms ${}_{-1}KQ_{0}(A)\cong\mathbb{Z}$ when $A=R_{F} ,\mathbb{F}_{q},\mathbb{R}$, and $\mathbb{C}$ detected by the (even) rank of the free symplectic $A$-inner product space. Now since every ring map preserves the rank, we obtain a cartesian square: \begin{equation} \begin{array} [c]{ccc} {}_{-1}KQ_{0}(R_{F}) & \rightarrow & \overset{r}{\bigoplus}{}_{-1} KQ_{0}(\mathbb{R})\\ \downarrow & & \downarrow\\ {}_{-1}KQ_{0}(\mathbb{F}_{q}) & \rightarrow & \overset{r}{\bigoplus}{} _{-1}KQ_{0}(\mathbb{C}) \end{array} \label{cartesiandiagram} \end{equation} Combining (\ref{longexactsequence1}), (\ref{cartesiandiagram}), and the fact that ${}_{-1}KQ_{1}(\mathbb{C})$ is the trivial group, it follows that ${}_{-1}\pi_{0}$ is an isomorphism.
$\Box$
\begin{lemma} \label{lemma:thirdcase} The map ${}_{1}\pi_{1}\colon{}_{1}KQ_{1} (R_{F})\rightarrow{}_{1}\overline{KQ}_{1}(R_{F})$ is an isomorphism. \end{lemma}
\noindent\textbf{Proof.} \label{epsilon=1, n=1} We first show that the determinant and the spinor norm of $R_{F}$ induce an isomorphism \begin{equation} {}_{1}KQ_{1}(R_{F})\overset{\cong}{\longrightarrow}R_{F}^{\times} /(R_{F}^{\times})^{2}\oplus\mathbb{Z}/2. \label{Bassisomorphism} \end{equation} (Moreover, from (\ref{examples for A/A^{2}}) above we note that the right-hand side reduces to $(\mathbb{Z}/2)^{r+2}$.) To see this, consider the exact sequence of units, discriminant modules, and Picard groups in \cite[(2.1)] {Bass} \begin{equation} 0\rightarrow\mu_{2}(F)\rightarrow R_{F}^{\times}\overset{(\ )^{2}} {\rightarrow}R_{F}^{\times}\rightarrow\mathrm{Discr}(R_{F})\rightarrow \mathrm{Pic}(R_{F})\overset{2\cdot}{\rightarrow}\mathrm{Pic}(R_{F})\text{.} \label{exactsequencebass} \end{equation} Combining (\ref{exactsequencebass}) with the $2$-regular assumption on $F$ and the vanishing of $SK_{1}(R_{F})$ \cite[Corollary 4.3]{BassMilnorSerre}, the isomorphism in (\ref{Bassisomorphism}) follows from \cite[(4.7.6)]{Bass}. As a corollary, we deduce that the hyperbolic map $R_{F}^{\times}\cong K_{1}(F)\rightarrow{}_{1}KQ_{1}(F)\cong R_{F}^{\times}/(R_{F}^{\times} )^{2}\oplus\mathbb{Z}/2$ may be identified with the composition of the quotient map $R_{F}^{\times}\twoheadrightarrow R_{F}^{\times}/(R_{F}^{\times })^{2}$ by the inclusion of this last group in $R_{F}^{\times}/(R_{F}^{\times })^{2}\oplus\mathbb{Z}/2$, and so that ${}_{1}W_{1}(R_{F})\cong\mathbb{Z}/2$. An analogous result holds if we replace $R_{F}$ by $\mathbb{F}_{q},\mathbb{R}$ or $\mathbb{C}$.
The group ${}_{1}KQ_{2}(\mathbb{R})\cong\mathbb{Z}/2\oplus\mathbb{Z}/2$ maps split surjectively onto ${}_{1}KQ_{2}(\mathbb{C})\cong\mathbb{Z}/2$, as shown in the first two lemmas of Appendix B. Hence by (\ref{longexactsequence1}) there is a short split exact sequence \begin{equation} 0\rightarrow{}_{1}\overline{KQ}_{1}(R_{F})\overset{\theta}{\rightarrow}{} _{1}KQ_{1}(\mathbb{F}_{q})\oplus\overset{r}{\bigoplus}{}_{1}KQ_{1} (\mathbb{R})\overset{\varphi}{\rightarrow}\overset{r}{\bigoplus}{}_{1} KQ_{1}(\mathbb{C})\rightarrow0 \label{exactsequence2} \end{equation} where we may choose a splitting of $\varphi$ whose image lies in a direct summand of $\overset{r}{\bigoplus}{}_{1}KQ_{1}(\mathbb{R})\cong(\mathbb{Z} /2)^{2r}$, while $\overset{r}{\bigoplus}{}_{1}KQ_{1}(\mathbb{C})\cong (\mathbb{Z}/2)^{r}$. Therefore, we have an induced isomorphism \[ {}_{1}\overline{KQ}_{1}(R_{F})\cong{}_{1}KQ_{1}(\mathbb{F}_{q})\oplus (\mathbb{Z}/2)^{r}\text{.} \] Thus, since ${}_{1}KQ_{1}(\mathbb{F}_{q})\cong\mathbb{Z}/2\oplus\mathbb{Z}/2$, using (\ref{Bassisomorphism}) and (\ref{exactsequence2}), we deduce that ${}_{1}\overline{KQ}_{1}(R_{F})$ and ${}_{1}KQ_{1}(R_{F})$ are both abstractly isomorphic to direct sums of $r+2$ copies of $\mathbb{Z}/2$. Therefore, to finish the proof of the lemma, it suffices to show that ${}_{1}\pi_{1}:{} _{1}KQ_{1}(R_{F})\rightarrow{}_{1}\overline{KQ}_{1}(R_{F})$ is surjective. The argument is now broken into three steps.
\noindent\textbf{Step 1. }For $A=R_{F},\mathbb{F}_{q},\mathbb{R}$ and $\mathbb{C}$, by Lemma \ref{SK_1(A) = 0}, we have surjections \[ {}_{1}KQ_{1}(A)\twoheadrightarrow{}_{1}W_{1}(A)\overset{\cong} {\twoheadrightarrow}k_{1}(A)=\{\pm1\}\text{.} \] Let $SKQ_{1}(A)$ denote the kernel of this determinant map ${}_{1} KQ_{1}(A)\rightarrow\{\pm1\}$ which is obviously split surjective, so that ${}_{1}KQ_{1}(A)\cong S{}KQ_{1}(A)\oplus\mathbb{Z}/2$. Naturality of the determinant map implies that the maps between the various ${}_{1}KQ_{1}(A)$ restrict to maps between the corresponding subgroups $SKQ_{1}(A)$. Moreover, because ${}_{1}W_{1}(A)$ is the cokernel of the hyperbolic map, the map $K_{1}(A)\rightarrow{}_{1}KQ_{1}(A)$ factors through $SKQ_{1}(A)$.
The computations above now imply that in the commutative diagram \[ \begin{array} [c]{ccc} R_{F}^{\times}\cong K_{1}(R_{F}) & \longrightarrow & K_{1}(\mathbb{F} _{q})\oplus\overset{r}{\bigoplus}K_{1}(\mathbb{R})\\ \downarrow & & \downarrow\\ R_{F}^{\times}/(R_{F}^{\times})^{2}\cong SKQ_{1}(R_{F}) & \longrightarrow & SKQ_{1}(\mathbb{F}_{q})\oplus\overset{r}{\bigoplus}SKQ_{1}(\mathbb{R}) \end{array} \] both the vertical and top horizontal arrows are surjective. Therefore, the lower horizontal map is also surjective.
\noindent\textbf{Step 2. }In a symmetric way, we introduce a subgroup $S\overline{KQ}_{1}(R_{F})$ of ${}_{1}\overline{KQ}_{1}(R_{F})$ as the kernel of the composition ${}_{1}\overline{KQ}_{1}(R_{F})\cong{}_{1}KQ_{1} (\mathbb{F}_{q})\oplus(\mathbb{Z}/2)^{r}\longrightarrow{}_{1}KQ_{1} (\mathbb{F}_{q})\overset{\mathrm{det}}{\longrightarrow}\{\pm1\}$ which is a split surjection. Therefore, we have a decomposition of both ${}_{1} KQ_{1}(R_{F})$ and ${}_{1}\overline{KQ}_{1}(R_{F})$ as compatible direct sums: \[ {}_{1}KQ_{1}(R_{F})\cong SKQ_{1}(R_{F})\oplus\mathbb{Z}/2\text{\quad and\quad }{}_{1}\overline{KQ}_{1}(R_{F})\cong S{}\overline{KQ}_{1}(R_{F})\oplus \mathbb{Z}/2\text{,} \] and it is enough to prove the surjectivity of the induced map $\sigma:{} _{1}SKQ_{1}(R_{F})\rightarrow{}_{1}\overline{SKQ}_{1}(R_{F})$.
\noindent\textbf{Step 3. }Finally, consider the commuting diagram: \[ \begin{array} [c]{ccccc} SKQ_{1}(R_{F}) & \overset{\sigma}{\longrightarrow} & S\overline{KQ}_{1} (R_{F}) & \overset{\gamma}{\longrightarrow} & SKQ_{1}(\mathbb{F}_{q} )\oplus\overset{r}{\bigoplus}SKQ_{1}(\mathbb{R})\\ \downarrow & & \downarrow & & \downarrow\\ {}_{1}KQ_{1}(R_{F}) & \overset{{}_{1}\pi_{1}}{\longrightarrow} & {} _{1}\overline{KQ}_{1}(R_{F}) & \overset{\theta}{\longrightarrow} & {} _{1}KQ_{1}(\mathbb{F}_{q})\oplus\overset{r}{\bigoplus}{}_{1}KQ_{1}(\mathbb{R}) \end{array} \] From (\ref{exactsequence2}), $\theta$ is a monomorphism; so, its restriction $\gamma$ is also injective. Meanwhile, from Step 1, the composite $\gamma \circ\sigma$ is surjective. It follows that $\sigma$ is surjective too, as we sought.
$\Box$
\begin{lemma} \label{lemma:fourthcase} The map ${}_{1}\pi_{0}\colon{}_{1}KQ_{0} (R_{F})\rightarrow{}_{1}\overline{KQ}_{0}(R_{F})$ is an isomorphism modulo odd torsion. \end{lemma}
\noindent\textbf{Proof.} Since $\mathrm{Pic}(A)$ is an odd torsion group we obtain for $A=R_{F},\mathbb{F}_{q},\mathbb{R}$, and $\mathbb{C}$ the short exact sequence (modulo odd torsion) \begin{equation} 0\rightarrow W^{\prime}(A)\rightarrow{}_{1}KQ_{0}(A)\rightarrow K_{0} (A)\rightarrow0.\label{ses W' -> KQ -> K} \end{equation} From the last three of these four cases we obtain the vertical map of short exact sequences: \[ \begin{array} [c]{ccccccc} 0\rightarrow & W^{\prime}(\mathbb{F}_{q})\oplus\bigoplus\limits^{r}W^{\prime }(\mathbb{R}) & \rightarrow & {}_{1}KQ_{0}(\mathbb{F}_{q})\oplus \bigoplus\limits^{r}{}_{1}KQ_{0}(\mathbb{R}) & \rightarrow & K_{0} (\mathbb{F}_{q})\oplus\bigoplus\limits^{r}K_{0}(\mathbb{R}) & \rightarrow0\\ & \downarrow & & \downarrow & & \downarrow & \\ 0\rightarrow & \bigoplus\limits^{r}W^{\prime}(\mathbb{C}) & \rightarrow & \bigoplus\limits^{r}{}_{1}KQ_{0}(\mathbb{C}) & \rightarrow & \bigoplus \limits^{r}K_{0}(\mathbb{C}) & \rightarrow0 \end{array} \] By \cite{HO} and \cite{Mitchell}, $K_{0}(R_{F})$ is the kernel of the rightmost vertical map. Further, the bottom right horizontal epimorphism maps between two copies of $\mathbb{Z}^{r}$, making the coWitt group $W^{\prime }(\mathbb{C})$ trivial. As already noted, ${}_{1}KQ_{1}(\mathbb{R})$ maps onto ${}_{1}KQ_{1}(\mathbb{C})$, making ${}_{1}\overline{KQ}_{0}(R_{F})$ the kernel of the center vertical map. Therefore, the desired isomorphism between ${} _{1}KQ_{0}(R_{F})$ and ${}_{1}\overline{KQ}_{0}(R_{F})$ follows by combining the short exact sequence (\ref{ses W' -> KQ -> K}) for $A=R_{F}$ and Proposition \ref{proposition:keyinput}.$
\Box
$
Before dealing with the remaining cases $n=-1$ with $\varepsilon=\pm1$, we need a proposition which is interesting by itself.
\begin{proposition} \label{Dedekind forgetful}Let $A$ be a Dedekind ring and $\varepsilon=\pm1$ and let us consider the forgetful map \[ {}_{\varepsilon}KQ_{0}(A)\overset{\varphi}{\longrightarrow}K_{0} (A)=\mathbb{Z}\oplus\mathrm{Pic}(A)\text{.} \] Its image on the first summand is $\mathbb{Z}$ if $\varepsilon=1$ and $2\mathbb{Z}$ if $\varepsilon=-1$. Its image on the second summand is a group of order at most $2$. Therefore, if $A$ is a $2$-regular ring of integers, the image of $\varphi$ lies in the first summand. \end{proposition}
\noindent\textbf{Proof.} The projection to the $\mathbb{Z}$ summand is given by the rank. On the other hand, if $E$ is equipped with a symmetric or antisymmetric form, it is isomorphic to its dual. The duality on $\mathrm{Pic}(A)$ is given by $I\longmapsto I^{\ast}=I^{-1}$. This implies that the image of $E$ in $\mathrm{Pic}(A)$ is at most of order $2$. In particular, when $\mathrm{Pic}(A)$ is odd torsion, the image of $\varphi$ is included in the $\mathbb{Z}$ summand.
$\Box$
For the cases $n=-1$ with $\varepsilon=\pm1$, we are going to show that ${}_{\varepsilon}KQ_{-1}(R_{F})$ is trivial, as is ${}_{\varepsilon} \overline{KQ}_{-1}(R_{F})$ because the spectrum ${}_{\varepsilon} \overline{\mathcal{KQ}}(R_{F})$ is connective. As in \cite{BK}, we exploit the two exact sequences \[ K_{1}(R_{F})\longrightarrow{}_{-\varepsilon}KQ_{1}(R_{F})\longrightarrow {}_{-\varepsilon}U_{0}(R_{F})\longrightarrow K_{0}(R_{F})\longrightarrow {}_{-\varepsilon}KQ_{0}(R_{F}), \] and (via the fundamental theorem proved in \cite{K:AnnM112fun}) \[ {}_{\varepsilon}KQ_{0}(R_{F})\overset{\varphi}{\longrightarrow}K_{0} (R_{F})\longrightarrow{}_{-\varepsilon}U_{0}(R_{F})\longrightarrow {}_{\varepsilon}KQ_{-1}(R_{F})\longrightarrow K_{-1}(R_{F})=0\text{.} \]
\begin{lemma} \label{lemma:epsilon=1, n=-1} The map ${}_{1}\pi_{-1}\colon{}_{1}KQ_{-1} (R_{F})\rightarrow{}_{1}\overline{KQ}_{-1}(R_{F})$ is an isomorphism between trivial groups. \end{lemma}
\noindent\textbf{Proof.} For $\varepsilon=1$, ${}_{-\varepsilon}KQ_{1} (R_{F})=0$ by a well-known theorem of Bass, Milnor and Serre \cite[Corollary 12.5]{BassMilnorSerre}; and the map $K_{0}(R_{F})\rightarrow{}_{-\varepsilon }KQ_{0}(R_{F})$ has its kernel identified with $\mathrm{Pic}(R_{F})$ since ${}_{-1}KQ_{0}(R_{F})=\mathbb{Z}$. Hence ${}_{-\varepsilon}U_{0}(R_{F} )\cong\mathrm{Pic}(R_{F})$. On the other hand, according to the previous proposition, the cokernel of $\varphi$ is also identified with $\mathrm{Pic} (R_{F})$. The second exact sequence therefore provides an injective map from $\mathrm{Pic}(R_{F})$ to itself. Since $\mathrm{Pic}(R_{F})$ is finite, this map is bijective, and it follows that ${}_{1}KQ_{-1}(R_{F})$ is the trivial group.
$\Box$
\begin{lemma} \label{lemma:epsilon=-1, n=-1} The map ${}_{-1}\pi_{-1}\colon{}_{-1} KQ_{-1}(R_{F})\rightarrow{}_{-1}\overline{KQ}_{-1}(R_{F})$ is an isomorphism between trivial groups\textsf{.} \end{lemma}
\noindent\textbf{Proof.} In order to compute ${}_{-1}KQ_{-1}(R_{F}),$ let us first work modulo odd torsion, which makes the map $K_{0}(R_{F} )\longrightarrow{}_{-\varepsilon}KQ_{0}(R_{F})$ injective. The first exact sequence above written for $\varepsilon=-1$ shows that ${}_{1}U_{0} (R_{F})=\mathbb{Z}/2$, because ${}_{1}W_{1}(R_{F})=\mathbb{Z}/2$ by (\ref{W1}). On the other hand, the cokernel of the map $\varphi$ in the second exact sequence is also $\mathbb{Z}/2$ by Proposition \ref{Dedekind forgetful}. It follows that $_{-1}KQ_{-1}(R_{F})=0$, and so the map ${}_{-1}KQ_{-1} (R_{F})\rightarrow{}_{-1}\overline{KQ}_{-1}(R_{F})$ is a homomorphism between odd order finite groups.
Finally, according to Proposition \ref{Wodd}, the odd torsion of the Witt groups in degree $-1$ is trivial. Since $K_{-1}(R_{F})=0$, it follows that the odd torsion of ${}_{-1}KQ_{-1}(R_{F})$ is also trivial.
$\Box
$
These various lemmas conclude the proof of Theorem \ref{lowdimensionalcomputations}, and hence of Theorem \ref{theorem1} with the exception of the following proposition which was announced at the beginning:
\begin{proposition} \label{Finiteness}Let $F$ be a number field and let $R$ be a ring of $S$-integers in $F$, where $S$ is a finite set of primes containing the dyadic ones. Then the groups ${}_{\varepsilon}KQ_{n}(R)$ are finitely generated for $\varepsilon=\pm1$ and $n\in\mathbb{Z}$. \end{proposition}
\noindent\textbf{Proof.} The same arguments used in the proofs of Lemmas \ref{lemma:firstcase} and \ref{lemma:secondcase} show that ${}_{-1} KQ_{0}(R)\equiv\mathbb{Z}$ and $_{-1}KQ_{1}(R)=0$. According to \cite[(2.1)] {Bass}, $_{1}KQ_{1}(R)$ ($=KO_{1}(R)$ in Bass's notation) is inserted in an exact sequence between two finitely generated groups $K\mathrm{SL}_{1}(R)$ and $\mathrm{Disc}(R)\oplus\mathbb{Z}/2$, where $\mathrm{Disc}(R)$ is described in \cite[pg. 156.]{Bass}. Therefore, $_{1}KQ_{1}(R)$ and {}$_{1}W_{1}(R)$ are finitely generated.
On the other hand, it is well known (see \textsl{e.g.} \cite[ Corollary 3.3, pg. 93]{MH}), that the canonical map between classical Witt groups $W(R)\longrightarrow W(F)$ is injective. Moreover, any element of $W(F)$ is determined by the classical invariants which are the rank, signature, discriminant and Hasse-Witt invariant. Since $S$ is finite, the discriminant and the Hasse-Witt invariant computed on the elements of $W(R)$, considered as a subgroup of $W(F)$, can take only a finite set of values. Moreover, the signature takes integral values defined by the various real embeddings$.$ Therefore, the group $W(R)$ is finitely generated.
In order to deal with the higher Witt groups ${}_{\varepsilon}W_{n}(R)$ and coWitt groups ${}_{\varepsilon}W_{n}^{^{\prime}}(R)$ for $n\in\mathbb{Z}$, we use the following two basic tools:
1) According to Quillen \cite{Quillenfinite} the groups $K_{n}(R)$ are finitely generated. Therefore, the Tate cohomology groups $k_{n}(R)$ and $k_{n}^{^{\prime}}(R)$ are also finitely generated.
2) There is a $12$-term exact sequence detailed in \cite[pg. 278.] {K:AnnM112fun} which shows by double induction from the cases $\varepsilon =\pm1$ and $n=0,1$, that the groups ${}_{\varepsilon}W_{n}(R)$ and ${}_{\varepsilon}W_{n}^{^{\prime}}(R)$ are finitely generated for all values of $n\in\mathbb{Z}$ and $\varepsilon=\pm1$.
Finally, from 1) and 2), we deduce that ${}_{\varepsilon}KQ_{n}(R)$, which lies in an exact sequence between $K_{n}(R)$ and ${}_{\varepsilon}W_{n}(R)$, is finitely generated.\textbf{
}$\Box$
\begin{remark} \label{localization}1. By means of Lemma \ref{Lemma modulo m}\thinspace(b), the proof of Theorem \ref{theorem1} also works when dealing with $2$-localizations instead of $2$-completions. Likewise, Theorems \ref{theorem2} and \ref{theorem4} are also true in the framework of $2$-localizations. However, Theorem \ref{theorem3} is only true for completions.
2. In this paper, we work with the spectra ${}\mathcal{K}(\mathbb{R})_{\#} ^{c}$ and ${}\mathcal{K}(\mathbb{C})_{\#}^{c}$ and their hermitian analogs. The Suslin equivalence \cite{Suslin :local fields} with ${}\mathcal{K} (\mathbb{R}^{\delta})_{\#}$ and ${}\mathcal{K}(\mathbb{C}^{\delta})_{\#}$ respectively, where $\delta$ means the discrete topology, enables these spectra to be replaced by those of the discrete rings when considering completions. However, this method fails if we deal with localizations instead of completions. \end{remark}
\section{Splitting results}
\label{section:splittingresults} The purpose of this section is to prove some generic splitting results employed in the proofs of Theorems \ref{theorem2} and \ref{theorem4}.
To start with, fix some residue field $\mathbb{F}_{q}$ of $R_{F}$ as in paragraph prior to the statement of Theorem \ref{theorem1} in the Introduction, and define the spectrum ${}_{\varepsilon}\mathcal{KQ} (\overline{R}_{F})$ (this is just a convenient notation, which we shall use below for $K$-theory and $V$-theory too) by the homotopy cartesian square: \[
\xymatrix{ {}_\varepsilon\mathcal{KQ}(\overline{R}_{F})\ar@{}[d]|-{\hbox
{\large{$\downarrow$}}} \ar@{}[r]|-{\hbox{\large{$\rightarrow$}}}&
{}_\varepsilon\mathcal{KQ}(\mathbb{R})^c\ar@{}[d]|-{\hbox{\large{$\downarrow
$}}}\\ {}_\varepsilon\mathcal{KQ}(\mathbb{F}_q)^c\ar@{}[r]|-{\hbox {\large{$\rightarrow$}}} & {}_\varepsilon\mathcal{KQ}(\mathbb{C})^c }\label{RFoverbar} \] It is connective because of the epimorphism ${}_{\varepsilon}KQ_{0}(
\mathbb{R}
)\rightarrow{}_{\varepsilon}KQ_{0}(
\mathbb{C}
)$. Similarly, define $\mathcal{K}(\overline{R}_{F})$ by the same type of homotopy cartesian square by replacing ${}_{\varepsilon}\mathcal{KQ}$ by $\mathcal{K}.$ It is also connective because of the epimorphism $K_{0}(
\mathbb{R}
){}\rightarrow K_{0}(
\mathbb{C}
)$. Finally, we define ${}_{\varepsilon}\mathcal{V}(\overline{R}_{F})$ by the similar homotopy cartesian square replacing ${}_{\varepsilon}\mathcal{KQ}$ by ${}_{\varepsilon}\mathcal{V}$. It too is connective, because of the epimorphism ${}_{\varepsilon}V_{0}(
\mathbb{R}
)\rightarrow{}_{\varepsilon}V_{0}(
\mathbb{C}
)$ proved in Lemmas \ref{_1VR}, \ref{_1VC} and \ref{_-1VRC} of Appendix B.
The first homotopy cartesian square can be recast as a homotopy fiber sequence${}$ \begin{equation} _{\varepsilon}\mathcal{KQ}(\overline{R}_{F})\longrightarrow{}_{\varepsilon }\mathcal{KQ}(
\mathbb{R}
)^{c}{}\vee{}_{\varepsilon}\mathcal{KQ}(\mathbb{F}_{q})^{c}\longrightarrow {}_{\varepsilon}\mathcal{KQ}(
\mathbb{C}
)^{c}\text{,} \label{_epsilonKQ(R^-_F) fibration} \end{equation} and similarly for the others.
In the next step we form the naturally induced diagram with horizontal homotopy fiber sequences: \begin{equation} \begin{array} [c]{ccccc} _{\varepsilon}\mathcal{KQ}(\overline{R}_{F}) & \longrightarrow & {}_{\varepsilon}\mathcal{KQ}(
\mathbb{R}
)^{c}{}\vee{}_{\varepsilon}\mathcal{KQ}(\mathbb{F}_{q})^{c} & \longrightarrow & _{\varepsilon}\mathcal{KQ}(
\mathbb{C}
)^{c}\\ \downarrow & & \downarrow^{\nabla\vee\mathrm{id}} & & \downarrow^{\nabla}\\ _{\varepsilon}\overline{\mathcal{KQ}}(R_{F}) & \longrightarrow &
{\displaystyle\bigvee\nolimits^{r}}
{}_{\varepsilon}\mathcal{KQ}(
\mathbb{R}
)^{c}{}\vee{}_{\varepsilon}\mathcal{KQ}(\mathbb{F}_{q})^{c} & \longrightarrow &
{\displaystyle\bigvee\nolimits_{\varepsilon}^{r}}
\mathcal{KQ}(
\mathbb{C}
)^{c}\\ \downarrow & & \downarrow & & \downarrow\\%
{\displaystyle\bigvee\nolimits^{r-1}}
{}_{\varepsilon}\mathcal{F} & \longrightarrow &
{\displaystyle\bigvee\nolimits^{r-1}}
{}_{\varepsilon}\mathcal{KQ}(
\mathbb{R}
)^{c}{} & \longrightarrow &
{\displaystyle\bigvee\nolimits_{\varepsilon}^{r-1}}
\mathcal{KQ}(
\mathbb{C}
)^{c} \end{array} \label{3 x 3} \end{equation} Note that ${}_{\varepsilon}\mathcal{F}$ is connective, again from the epimorphism ${}_{\varepsilon}KQ_{0}(
\mathbb{R}
)\rightarrow{}_{\varepsilon}KQ_{0}(
\mathbb{C}
)$. Since the two right-hand columns are also homotopy fiber sequences, the same holds for the left-hand column, \textsl{cf.}~\cite[Lemma 2.1]{CMN}. Using the compatibility of the two evident splittings of the right-hand columns, we obtain a splitting of ${}_{\varepsilon}\mathcal{KQ}(\overline{R} _{F})\rightarrow{}_{\varepsilon}\overline{\mathcal{KQ}}(R_{F})$. In other words, \[ {}_{\varepsilon}\overline{\mathcal{KQ}}(R_{F})\simeq{}_{\varepsilon }\mathcal{KQ}(\overline{R}_{F})\vee\bigvee^{r-1}{}_{\varepsilon} \mathcal{F}\text{.} \]
Let us define the connective spectrum $\overline{\mathcal{K}}(R_{F})$ as the homotopy pull-back of the diagram \[ \begin{array} [c]{ccc} \overline{\mathcal{K}}(R_{F}) & \rightarrow & {\displaystyle\bigvee \limits^{r}}{}\mathcal{K}(\mathbb{R})^{c}\\ \downarrow & & \downarrow\\ \mathcal{K}(\mathbb{F}_{q}) & \rightarrow & {\displaystyle\bigvee\limits^{r} }\mathcal{K}(\mathbb{C})^{c} \end{array} \] It is connective because the map $K_{0}(
\mathbb{R}
)\longrightarrow K_{0}(
\mathbb{C}
)$ is surjective. By an argument similar to before, we can split $\overline{\mathcal{K}}(R_{F})$ as $\overline{\mathcal{K}}(R_{F} )\simeq\mathcal{K}(\overline{R}_{F})\vee\bigvee^{r-1}\Omega^{-1} \mathcal{K}(\mathbb{R})^{c}$, where $\Omega^{-1}\mathcal{K}(\mathbb{R} )^{c}=(\Omega^{-1}\mathcal{K}(\mathbb{R}))^{c}$ (since $K_{-1}(
\mathbb{R}
)=0$) is the homotopy fiber of the map $\mathcal{K}(\mathbb{R})^{c} \rightarrow\mathcal{K}(\mathbb{C})^{c}$ by a direct consequence of a well-known result due to Bott: see for instance \cite[Section III.5] {K:ktheorybook}. This splitting will be used incidentally in the computation of one $V$-group later on. It also implies the explicit computation of the groups $K_{n}(R_{F})$ listed in Theorem $1.3.$
In ${}_{1}\mathcal{KQ}$-theory, we make use of the following lemma, which is a consequence of (\ref{3 x 3}) and the homotopy equivalences $_{1}\mathcal{KQ}(
\mathbb{R}
)\simeq\mathcal{K}(
\mathbb{R}
)\vee\mathcal{K}(
\mathbb{R}
)$ and $_{1}\mathcal{KQ}(
\mathbb{C}
)\simeq\mathcal{K}(
\mathbb{R}
)$ proved in Appendix B which give the homotopy type of $_{1}\mathcal{F}$ as $\mathcal{K}(\mathbb{R})^{c}$.
\begin{lemma} \label{_1KQ(R_F) splits}There is a homotopy equivalence \[ {}_{1}\overline{\mathcal{KQ}}(R_{F})\simeq{}_{1}\mathcal{KQ}(\overline{R} _{F})\vee
{\displaystyle\bigvee\nolimits^{r-1}}
\mathcal{K}(\mathbb{R})^{c}.
\]
$\Box
$ \end{lemma}
Next, we consider the fiber ${}_{-1}\mathcal{F}$ of \[ {}_{-1}\mathcal{KQ}(\mathbb{R})^{c}=\mathcal{K}(\mathbb{C})^{c}\longrightarrow {}_{-1}\mathcal{KQ}(\mathbb{C})^{c}=\mathcal{K}(\mathbb{H})^{c} \] (where $\mathbb{H}$ refers to the quaternions with the usual topology). If we write the commutative diagram due to Bott \[ \begin{array} [c]{ccc} \mathcal{K}(\mathbb{C})^{c} & \longrightarrow & \mathcal{K}(\mathbb{H})^{c}\\ \downarrow^{\simeq} & & \downarrow^{\simeq}\\ (\Omega^{4}(\mathcal{K}(\mathbb{C})))^{c} & \longrightarrow & (\Omega ^{4}(\mathcal{K}(\mathbb{R})))^{c} \end{array} \] we see that the homotopy fiber of the map $_{-1}\mathcal{KQ}(\mathbb{R} )^{c}\rightarrow{}_{-1}\mathcal{KQ}(\mathbb{C})^{c}$ may be identified with the homotopy fiber $(\Omega^{6}\mathcal{K}(\mathbb{R}))^{c}$ of $(\Omega ^{4}(\mathcal{K}(\mathbb{C})))^{c}\rightarrow(\Omega^{4}(\mathcal{K} (\mathbb{R})))^{c}$.
\begin{lemma} \label{_-1KQ(R_F) splits}There is a homotopy equivalence \[ {}_{-1}\overline{\mathcal{KQ}}(R_{F})\simeq{}_{-1}\mathcal{KQ}(\overline {R}_{F})\vee\bigvee^{r-1}(\Omega^{6}\mathcal{K}(\mathbb{R}))^{c} .
\]
$\Box
$ \end{lemma}
The next proposition is a consequence of the two previous lemmas and Theorem \ref{theorem1}.
\begin{proposition} There are homotopy equivalences of $2$-completed connective spectra \[ {}_{1}\mathcal{KQ}(R_{F})_{\#}^{c}\simeq{}_{1}\mathcal{KQ}(\overline{R} _{F})_{\#}\vee
{\displaystyle\bigvee\nolimits^{r-1}}
\mathcal{K}(\mathbb{R})_{\#}^{c} \] and \[ {}_{-1}\mathcal{KQ}(R_{F})_{\#}^{c}\simeq{}_{-1}\mathcal{KQ}(\overline{R} _{F})_{\#}\vee\bigvee^{r-1}(\Omega^{6}\mathcal{K}(\mathbb{R}))_{\#} ^{c}.
\] $
\Box
$ \end{proposition}
In order to obtain the diagram in ${}_{\varepsilon}\mathcal{V}$-theory corresponding to (\ref{3 x 3}), we start with the diagram \begin{equation} \begin{array} [c]{ccccc} {}_{\varepsilon}\mathcal{V}(R_{F})^{c} & \longrightarrow & \bigvee^{r} {}_{\varepsilon}\mathcal{V}(\mathbb{R})^{c}\vee{}_{\varepsilon}\mathcal{V} (\mathbb{F}_{q})^{c} & \longrightarrow & \bigvee^{r}{}_{\varepsilon }\mathcal{V}(\mathbb{C})^{c}\\ \downarrow & & \downarrow & & \downarrow\\ {}_{\varepsilon}\mathcal{KQ}(R_{F})^{c} & \longrightarrow & \bigvee^{r} {}_{\varepsilon}\mathcal{KQ}(\mathbb{R})^{c}\vee{}_{\varepsilon} \mathcal{KQ}(\mathbb{F}_{q})^{c} & \longrightarrow & \bigvee^{r} {}_{\varepsilon}\mathcal{KQ}(\mathbb{C})^{c}\\ \downarrow & & \downarrow & & \downarrow\\ \mathcal{K}(R_{F}) & \longrightarrow & \bigvee^{r}\mathcal{K}(\mathbb{R} )^{c}\vee\mathcal{K}(\mathbb{F}_{q}) & \longrightarrow & \bigvee ^{r}\mathcal{K}(\mathbb{C})^{c} \end{array} \label{3 x 3 V} \end{equation} in which by definition all three columns are fiber sequences; by Theorem \ref{theorem1} and its counterpart in algebraic $K$-theory (\ref{algebraic K Bokstedt square}). The lower two rows are fiber sequences, and hence the top row is as well \cite[Lemma 2.1]{CMN}. A similar argument reveals that ${}_{\varepsilon}\mathcal{V}(\overline{R}_{F})$ is the homotopy fiber of the map ${}_{\varepsilon}\mathcal{KQ}(\overline{R}_{F})\rightarrow \mathcal{K}(\overline{R}_{F})$.
Let us define the connective spectrum ${}_{\varepsilon}\overline{\mathcal{V} }(R_{F})$ as the homotopy pull-back of the diagram \[ \begin{array} [c]{ccc} {}_{\varepsilon}\overline{\mathcal{V}}(R_{F}) & \rightarrow & {\displaystyle \bigvee\limits^{r}}{}{}_{\varepsilon}\mathcal{V}(\mathbb{R} )^{c}\\ \downarrow & & \downarrow\\ {}_{\varepsilon}\mathcal{V}(\mathbb{F}_{q}) & \rightarrow & {\displaystyle \bigvee\limits^{r}}{}_{\varepsilon}\mathcal{V}(\mathbb{C})^{c} \end{array} \] This spectrum is connective because the map ${}_{\varepsilon}V_{0}(
\mathbb{R}
)\longrightarrow$ ${}_{\varepsilon}V_{0}(
\mathbb{C}
)=0$ is surjective according to Lemmas \ref{_1VC} and \ref{_-1VRC} of Appendix B.
By considering the homotopy fibers of the map from Diagram \ref{3 x 3} to its $\mathcal{K}$-counterpart, we now obtain the commuting diagram \begin{equation} \begin{array} [c]{ccccc} _{\varepsilon}\mathcal{V}(\overline{R}_{F}) & \longrightarrow & {} _{\varepsilon}\mathcal{V}(
\mathbb{R}
)^{c}{}\vee{}_{\varepsilon}\mathcal{V}(\mathbb{F}_{q})^{c} & \longrightarrow & _{\varepsilon}\mathcal{V}(
\mathbb{C}
)^{c}\\ \downarrow & & \downarrow^{\nabla\vee\mathrm{id}} & & \downarrow^{\nabla}\\ _{\varepsilon}\overline{\mathcal{V}}(R_{F}) & \longrightarrow &
{\displaystyle\bigvee\nolimits^{r}}
{}_{\varepsilon}\mathcal{V}(
\mathbb{R}
)^{c}{}\vee{}_{\varepsilon}\mathcal{V}(\mathbb{F}_{q})^{c} & \longrightarrow &
{\displaystyle\bigvee\nolimits_{\varepsilon}^{r}}
\mathcal{V}(
\mathbb{C}
)^{c}\\ \downarrow & & \downarrow & & \downarrow\\%
{\displaystyle\bigvee\nolimits^{r-1}}
{}_{\varepsilon}\mathcal{G} & \longrightarrow &
{\displaystyle\bigvee\nolimits^{r-1}}
{}_{\varepsilon}\mathcal{V}(
\mathbb{R}
)^{c}{} & \longrightarrow &
{\displaystyle\bigvee\nolimits_{\varepsilon}^{r-1}}
\mathcal{V}(
\mathbb{C}
)^{c} \end{array} \label{3 x 3 V 2nd} \end{equation} wherein we conclude that the first column is a homotopy fiber sequence, since all other columns and rows are. As for (\ref{3 x 3}), we deduce that the first column is split. Thus, a complete determination of ${}_{\varepsilon} \overline{\mathcal{V}}(R_{F})$ in terms of ${}_{\varepsilon}\mathcal{V} (\overline{R}_{F})$ requires only a computation of the summand {} $_{\varepsilon}\mathcal{G}$ which is (by definition) the homotopy fiber of ${}_{\varepsilon}\sigma:{}_{\varepsilon}\mathcal{V}(\mathbb{R})^{c} \rightarrow{}_{\varepsilon}\mathcal{V}(\mathbb{C})^{c}$. The following is included in Appendix B, as Lemmas \ref{_1VR -> C} and \ref{_-1V(R) -> _-1V(C)}.
\begin{lemma} \label{Fiber V(R) to V(C)}The homotopy fiber {}$_{\varepsilon}\mathcal{G}$ of ${}_{\varepsilon}\sigma:{}_{\varepsilon}\mathcal{V}(\mathbb{R})^{c} \rightarrow{}_{\varepsilon}\mathcal{V}(\mathbb{C})^{c}$ is \[ \left\{ \begin{array} [c]{lll} \mathcal{K}(\mathbb{R})^{c}\vee\mathcal{K}(\mathbb{R})^{c} & \quad & \varepsilon=1,\\ \mathcal{K}(\mathbb{C})^{c} & & \varepsilon=-1. \end{array} \right.
\]
$\Box
$ \end{lemma}
The above computations therefore imply the following two lemmas.
\begin{lemma} \label{_1V(R_F) splits} There is a homotopy equivalence
\[ {}_{1}\overline{\mathcal{V}}(R_{F})\simeq{}_{1}\mathcal{V}(\overline{R} _{F})\vee\bigvee^{r-1}\mathcal{K}(\mathbb{R})^{c}\vee\bigvee^{r-1} \mathcal{K}(\mathbb{R})^{c}.
\]
$\Box
$ \end{lemma}
\begin{lemma} \label{_-1V(R_F) splits} There is a homotopy equivalence
\[ {}_{-1}\overline{\mathcal{V}}(R_{F})\simeq{}_{-1}\mathcal{V}(\overline{R} _{F})\vee\bigvee^{r-1}\mathcal{K}(\mathbb{C})^{c}\text{.}
\]
$\Box
$ \end{lemma}
Let us now consider the following diagram of fibrations \[ \begin{array} [c]{ccccc} {}_{\varepsilon}\mathcal{V}(R_{F})_{\#}^{c} & \longrightarrow & {} _{\varepsilon}\mathcal{KQ}(R_{F})_{\#}^{c} & \longrightarrow & \mathcal{K} (R_{F})_{\#}^{c}\\ \downarrow^{\alpha} & & \downarrow^{\beta} & & \downarrow^{\gamma}\\ {}_{\varepsilon}\overline{\mathcal{V}}(R_{F})_{\#} & \longrightarrow & {}_{\varepsilon}\overline{\mathcal{KQ}}(R_{F})_{\#} & \longrightarrow & \overline{\mathcal{K}}(R_{F})_{\#} \end{array} \] Since $\beta$ and $\gamma$ are homotopy equivalences, $\alpha$ is a homotopy equivalence. To finish the computations of the $KQ$ and $V$-groups, it remains to compute explicitly the groups ${}_{\varepsilon}KQ_{n}(\overline{R}_{F} )=\pi_{n}({}_{\varepsilon}\mathcal{KQ}(\overline{R}_{F}))$ and ${} _{\varepsilon}V_{n}(\overline{R}_{F})=\pi_{n}({}_{\varepsilon}\mathcal{V} (\overline{R}_{F}))$ in the next two sections.
\section{Proof of Theorem \ref{theorem2}}
\label{section:proofoftheorem2} The splitting results Lemma \ref{_1KQ(R_F) splits} and Lemma \ref{_-1KQ(R_F) splits} in \S \ref{section:splittingresults} show that in order to prove Theorem \ref{theorem2}, it suffices to compute the groups ${}_{\varepsilon} KQ_{n}(\overline{R}_{F})=\pi_{n}(_{\varepsilon}\mathcal{KQ}(\overline{R} _{F}))$, and then sum with $r-1$ copies of the well-known $K$-groups of $\mathbb{R}$. We formulate the computation in terms of the numbers $t_{n}=(q^{(n+1)/2}-1)_{2}$ introduced at the end of \S \ref{section:preliminaryresults}, where we recall from Lemma \ref{lemma: t and w} that they are related to the numbers $w_{m}$ of Theorem \ref{theorem2} by the formulae: \[ t_{8k+3}=w_{4k+2}\qquad\text{and}\qquad t_{8k+7}=w_{4k+4}\text{.} \]
\begin{theorem} \label{theorem5} Up to finite groups of odd order, the groups ${} _{\varepsilon}KQ_{n}(\overline{R}_{F})$ are given in the following table. (Recall that $\delta_{n0}$ denotes the Kronecker symbol.)
\begin{table}[tbh] \begin{center} \begin{tabular}
[c]{p{0.4in}|p{1.5in}|p{1.5in}|}\hline $n\geq0$ & ${}_{-1}KQ_{n}(\overline{R}_{F})$ & ${}_{1}KQ_{n}(\overline{R} _{F})$\\\hline $8k$ & $\delta_{n0}\mathbb{Z}$ & $\delta_{n0}\mathbb{Z}\oplus\mathbb{Z} \oplus\mathbb{Z}/2$\\ $8k+1$ & $0$ & $(\mathbb{Z}/2)^{3}$\\ $8k+2$ & $\mathbb{Z}$ & $(\mathbb{Z}/2)^{2}$\\ $8k+3$ & $\mathbb{Z}/2t_{8k+3}$ & $\mathbb{Z}/t_{8k+3}$\\ $8k+4$ & $\mathbb{Z}/2$ & $\mathbb{Z}$\\ $8k+5$ & $\mathbb{Z}/2$ & $0$\\ $8k+6$ & $\mathbb{Z}$ & $0$\\ $8k+7$ & $\mathbb{Z}/t_{8k+7}$ & $\mathbb{Z}/t_{8k+7}$\\\hline \end{tabular} \end{center} \end{table} \end{theorem}
\noindent\textbf{Proof.} Throughout the proof, we exploit Friedlander's computation of ${}_{\varepsilon}KQ_{n}(\mathbb{F}_{q})$ given in \cite[Theorem 1.7]{Friedlander}, and work modulo odd torsion.\newline\noindent\textit{First case}: $\varepsilon=1$.\ Applying Lemma \ref{_1KQ(R) splits} of Appendix B to the homotopy fiber sequence (\ref{_epsilonKQ(R^{-}_F) fibration}) gives for each $n$ a split short exact sequence \[ 0\rightarrow{}_{1}KQ_{n}(\overline{R}_{F})\longrightarrow K_{n}(\mathbb{R} )\oplus K_{n}(\mathbb{R})\oplus{}_{1}KQ_{n}(\mathbb{F}_{q})\longrightarrow K_{n}(\mathbb{R})\rightarrow0\text{.} \] From this splitting we deduce an isomorphism \[ {}_{1}KQ_{n}(\overline{R}_{F})\cong K_{n}(\mathbb{R})\oplus{}_{1} KQ_{n}(\mathbb{F}_{q}). \] \noindent\textit{Second case}: $\varepsilon=-1$.\ For the computation of ${}_{-1}\mathcal{KQ}(\overline{R}_{F})$ we need to make several case distinctions arising from the $2$-completed homotopy cartesian square: \[ \begin{array} [c]{ccc} _{-1}\mathcal{KQ}(\overline{R}_{F})_{\#}^{c} & \longrightarrow & _{-1}\mathcal{KQ}(\mathbb{R})_{\#}^{c}\simeq\mathcal{K}(\mathbb{C})_{\#} ^{c}\simeq(\Omega^{4}(\mathcal{K}(\mathbb{C})))_{\#}^{c}\\ \downarrow & & \downarrow\\ _{-1}\mathcal{KQ}(\mathbb{F}_{q})_{\#}^{c} & \longrightarrow & _{-1} \mathcal{KQ}(\mathbb{C})_{\#}^{c}\simeq\mathcal{K}(\mathbb{H})_{\#}^{c} \simeq(\Omega^{4}(\mathcal{K}(\mathbb{R})))_{\#}^{c} \end{array} \] Note that the vertical homotopy fiber is $(\Omega^{6}\mathcal{K} (\mathbb{R}))_{\#}^{c}$ by Lemma \ref{Connective.Spectra} since the map $K_{4}(\mathbb{C)}\longrightarrow K_{4}(\mathbb{R)}$ is onto. The horizontal homotopy fiber is $(\Omega^{5}\mathcal{K}(\mathbb{R}))_{\#}^{c}$ according to \cite[Theorem 1.7]{Friedlander}. In particular, there is the \textquotedblleft vertical\textquotedblright\ exact sequence \begin{equation} \cdots\rightarrow{}_{-1}KQ_{n+1}(\mathbb{F}_{q})\rightarrow K_{n+6} (\mathbb{R})\rightarrow{}_{-1}KQ_{n}(\overline{R}_{F})\rightarrow{}_{-1} KQ_{n}(\mathbb{F}_{q})\rightarrow K_{n+5}(\mathbb{R})\rightarrow\cdots, \label{equation:vertical} \end{equation}
and the \textquotedblleft horizontal\textquotedblright\ exact sequence \begin{equation} \cdots\rightarrow K_{n+5}(\mathbb{C})\rightarrow K_{n+5}(\mathbb{R} )\rightarrow{}_{-1}KQ_{n}(\overline{R}_{F})\rightarrow K_{n+4}(\mathbb{C} )\rightarrow K_{n+4}(\mathbb{R})\rightarrow\cdots. \label{equation:horizontal} \end{equation}
If $n\equiv0,1\;(\mathrm{mod}\ 8)$ is nonzero, then (\ref{equation:vertical}) implies ${}_{-1}{KQ}_{n}(\overline{R}_{F})=0$. Likewise, when $n\equiv 2\;(\mathrm{mod}\ 8)$, (\ref{equation:vertical}) shows that ${}_{-1}{KQ} _{n}(\overline{R}_{F})\cong K_{8}(\mathbb{R})\cong\mathbb{Z}$.
For $n\equiv4\;(\mathrm{mod}\ 8)$, we use the segment \[
\xymatrix{ K_{n+5}(\mathbb{C})\ar@{}[r]|-{\hbox{\large{$\rightarrow$}}}& K_{n+5}(\mathbb{R})
\ar@{}[r]|-{\hbox{\large{$\rightarrow$}}}&{}_{-1}{KQ}_n(\overline{R}_{F})\ar@{}[r]|-{\hbox{\large{$\rightarrow$}}}& K_{n+4}(\mathbb{C}) } \] of (\ref{equation:horizontal}). By analyzing (\ref{equation:vertical}), it follows that ${}_{-1}{KQ}_{n}(\overline{R}_{F})$ is finite. Thus ${}_{-1} {KQ}_{n}(\overline{R}_{F})$ has order $2$.
For $n\equiv5\;(\mathrm{mod}\ 8)$, (\ref{equation:horizontal}) implies that ${}_{-1}{KQ}_{n}(\overline{R}_{F})$ is cyclic, whence by (\ref{equation:vertical}) there is an isomorphism ${}_{-1}{KQ}_{n} (\overline{R}_{F})\cong\mathbb{Z}/2$.
For $n\equiv3\;(\mathrm{mod}\ 8)$, we use the exact sequence (\ref{equation:vertical}) from ${}_{-1}{KQ}_{n+2}(\overline{R}_{F})$ to $K_{n+5}(\mathbb{R})$. In view of the two previous results, this takes the form \[ \mathbb{Z}/2\rightarrow\mathbb{Z}/2\oplus\mathbb{Z}/2\rightarrow \mathbb{Z}/2\rightarrow\mathbb{Z}/2\rightarrow\mathbb{Z}/2\rightarrow \mathbb{Z}/2\rightarrow{}_{-1}{KQ}_{n}(\overline{R}_{F})\rightarrow \mathbb{Z}/t_{n}\rightarrow\mathbb{Z}\text{.} \] Chasing this sequence from the left reveals that ${}_{-1}{KQ}_{n}(\overline {R}_{F})$ is a finite group of order $2t_{n}$. Meanwhile, the exact sequence (\ref{equation:horizontal}) obliges this group to be cyclic.
For $n\equiv6\;(\mathrm{mod}\ 8)$, (\ref{equation:vertical}) implies that ${}_{-1}{KQ}_{n}(\overline{R}_{F})\cong\mathbb{Z}$ or $\mathbb{Z} \oplus\mathbb{Z}/2$. On the other hand, the exact sequence (\ref{equation:horizontal}) shows that ${}_{-1}{KQ}_{n}(\overline{R}_{F})$ is a subgroup of $\mathbb{Z}$.
Finally, if $n\equiv7\;(\mathrm{mod}\ 8)$ then (\ref{equation:vertical}) produces an isomorphism between ${}_{-1}{KQ}_{n}(\overline{R}_{F})$ and ${}_{-1}{KQ}_{n}(\mathbb{F}_{q})\cong\mathbb{Z}/t_{n}$.
$\Box
$
We can now prove Theorem \ref{converse to theorem 1}.
\begin{theorem} For every totally real number field $F$, the following are equivalent.
\begin{enumerate} \item[(i)] $F$ is $2$-regular.
\item[(ii)] The square (\ref{hermitian K Bokstedt square}) is homotopy cartesian for $F$ when $\varepsilon=1$.
\item[(iii)] The square (\ref{algebraic K Bokstedt square}) is homotopy cartesian for $F$. \end{enumerate} \end{theorem}
\noindent\textbf{Proof.} Theorem \ref{theorem1} shows
that (i) implies (ii).
In the other direction, the preceding proof shows that (ii) leads to the second column of the table of Theorem \ref{theorem5}. Since $W(R_{F})$ injects into $W(F)$ \cite[IV (3.3)]{MH}, which has no odd-order torsion \cite[III (3.10)]{MH}, we may work modulo odd torsion. From the epimorphisms ($i=0,1$) ${}_{1}{KQ}_{i}(\mathbb{R})\twoheadrightarrow{}_{1}{KQ}_{i}(\mathbb{C})$, we obtain, from the Mayer-Vietoris sequence following from Theorem \ref{theorem1} , a short exact sequence \[ 0\rightarrow{}_{1}{KQ}_{0}(R_{F})\longrightarrow{}_{1}{KQ}_{0}(\mathbb{F} _{q})\oplus\bigoplus^{r}{}_{1}{KQ}_{0}(\mathbb{R})\longrightarrow\bigoplus ^{r}{}_{1}{KQ}_{0}(\mathbb{C})\rightarrow0\text{,} \] which splits because the final group is free abelian. It follows that \[ {}_{1}{KQ}_{0}(R_{F})\cong\mathbb{Z}\oplus\mathbb{Z}^{r}\oplus\mathbb{Z} /2\text{,} \] such that its $\mathbb{Z}/2$ summand maps nontrivially in the commuting square \[ \begin{array} [c]{ccccc} \mathbb{Z}\oplus\mathbb{Z}^{r}\oplus\mathbb{Z}/2\cong & {}_{1}{KQ}_{0} (R_{F}) & \longrightarrow & {}_{1}{KQ}_{0}(\mathbb{F}_{q}) & \cong \mathbb{Z}\oplus\mathbb{Z}/2\\ & \downarrow & & \downarrow & \\ & {}_{1}W_{0}(R_{F}) & \longrightarrow & {}_{1}W_{0}(\mathbb{F}_{q}) & \cong\mathbb{Z}/2\oplus\mathbb{Z}/2 \end{array} \] whose vertical maps are, by definition, surjective. Thus, in the exact sequence \[ K_{0}(R_{F})\overset{H}{\longrightarrow}{}_{1}{KQ}_{0}(R_{F})\longrightarrow {}_{1}W_{0}(R_{F})\rightarrow0 \] the hyperbolic homomorphism $H$ must map the finite summand \textrm{Pic} $(R_{F})$ of $K_{0}(R_{F})$ trivially, and so have its cokernel $W(R_{F})$ isomorphic to $\mathbb{Z}^{r}\oplus\mathbb{Z}/2$. Then by Proposition \ref{2+-regular characterization}(1), (7), it follows that $F$ is $2$-regular.
Similarly, the $K$-theoretic theorem of \cite{HO} and \cite{Mitchell} in case (i) asserts that (i) implies (iii).
Again, the computation $K_{2}(R_{F})\{2\}\cong(\mathbb{Z}/2)^{r}$ follows from (iii). Here, Proposition \ref{2+-regular characterization}(1), (2) yield that $F$ is $2$- regular.
$\Box$
\section{Proof of Theorem \ref{theorem4}\label{V-computation}}
As for Theorem \ref{theorem2}, the splitting results proven in Lemmas \ref{_1V(R_F) splits} and \ref{_-1V(R_F) splits} show that in order to prove Theorem \ref{theorem4}, it suffices to compute the groups ${}_{\varepsilon }V_{n}(\overline{R}_{F})$ introduced in the same section.
\begin{theorem} \label{theorem6} Up to finite groups of odd order, the groups \[ {}_{\varepsilon}V_{n}(\overline{R}_{F}):=\pi_{n}({}_{\varepsilon} \mathcal{V}(\overline{R}_{F})) \] are as follows. \begin{table}[tbh] \begin{center} \begin{tabular}
[c]{p{0.4in}|p{1.0in}|p{1.2in}|}\hline $n\geq0$ & ${}_{-1}V_{n}(\overline{R}_{F})$ & ${}_{1}V_{n}(\overline{R}_{F} )$\\\hline $8k$ & $\mathbb{Z}\oplus\mathbb{Z}/2$ & $\mathbb{Z}^{2}$\\ $8k+1$ & $0$ & $(\mathbb{Z}/2)^{2}$\\ $8k+2$ & $\mathbb{Z}$ & $(\mathbb{Z}/2)^{2}$\\ $8k+3$ & $0$ & $0$\\ $8k+4$ & $\mathbb{Z}$ & $\mathbb{Z}^{2}$\\ $8k+5$ & $\mathbb{Z}/2$ & $0$\\ $8k+6$ & $\mathbb{Z}\oplus\mathbb{Z}/2$ & $0$\\ $8k+7$ & $\mathbb{Z}/2$ & $0$\\\hline \end{tabular} \end{center} \end{table}
More precisely, for $\varepsilon=1$, the spectrum ${}_{1}\mathcal{V} (\overline{R}_{F})_{\#}^{c}$ (resp.~ ${}_{1}\mathcal{V}(R_{F})_{\#}^{c}$) has the homotopy type of $2$ copies (resp.~ $2r$ copies) of the spectrum $\mathcal{K}(\mathbb{R})_{\#}^{c}$. \end{theorem}
\noindent\textbf{Proof.} Throughout the proof, we again work modulo odd torsion.\newline\textit{First case}: $\varepsilon=1$.\ In the homotopy cartesian square \[ \begin{array} [c]{ccc} {}_{1}\mathcal{V}(\overline{R}_{F})_{\#}^{c} & \longrightarrow & {} _{1}\mathcal{V}(\mathbb{R})_{\#}^{c}\simeq\mathcal{K}(\mathbb{R})_{\#}^{c}\\ \downarrow & & \downarrow^{{}_{1}\sigma}\\ {}_{1}\mathcal{V}(\mathbb{F}_{q})_{\#}^{c} & \overset{\chi}{\longrightarrow} & {}_{1}\mathcal{V}(\mathbb{C})_{\#}^{c}\simeq\Omega^{-1}\mathcal{K} (\mathbb{R})_{\#}^{c} \end{array} \] as noted in Lemma \ref{_1VR -> C} of Appendix B, the map ${}_{1}\sigma$ is nullhomotopic. If we replace $\chi$ by a Serre fibration, then ${} _{1}\mathcal{V}(\overline{R}_{F})_{\#}^{c}$ has the homotopy type of its pullback over the nullhomotopic map ${}_{1}\sigma$. This means that ${} _{1}\mathcal{V}(\overline{R}_{F})_{\#}^{c}$ has the homotopy type of a fiber homotopy trivial fibration with base ${}_{1}\mathcal{V}(\mathbb{R})_{\#} ^{c}=\mathcal{K}(\mathbb{R})_{\#}^{c}$, and fiber of the homotopy type of the fiber of the lower horizontal map, which has been\textsf{ }identified in \cite[Corollary 1.6]{Friedlander} as $\mathcal{K}(\mathbb{R})_{\#}^{c}$. By Lemmas \ref{_1V(R_F) splits} and \ref{_-1V(R_F) splits}, a similar argument holds for $_{1}\mathcal{V}(R_{F})_{\#}^{c}$. Hence, the spectrum ${} _{1}\mathcal{V(}\overline{R}_{F})_{\#}^{c}$ (resp${}$ $_{1}\mathcal{V} (R_{F})_{\#}^{c}$) has the homotopy type of $\vee^{2}\mathcal{K} (\mathbb{R})_{\#}^{c}$ (resp.~ $\vee^{2r}\mathcal{K}(\mathbb{R})_{\#}^{c}$).
\textit{Second case}: $\varepsilon=-1$.\ Combining the results of Friedlander \cite[Corollary 1.6]{Friedlander} and Quillen \cite{Quillen} via the forgetful map relating algebraic and hermitian $K$-theory gives a homotopy commutative diagram \[ \begin{array} [c]{ccccc} {}_{-1}\mathcal{V}(\mathbb{F}_{q})_{\#}^{c} & \longrightarrow & {} _{-1}\mathcal{V}(\mathbb{C})_{\#}^{c} & \longrightarrow & {}_{-1} \mathcal{V}(\mathbb{C})_{\#}^{c}\\ \downarrow & & \downarrow & & \downarrow\\ {}_{-1}\mathcal{KQ}(\mathbb{F}_{q})_{\#}^{c} & \longrightarrow & {} _{-1}\mathcal{KQ}(\mathbb{C})_{\#}^{c} & \overset{\psi^{q}-1}{\longrightarrow} & {}_{-1}\mathcal{KQ}(\mathbb{C})_{\#}^{c}\\ \downarrow & & \downarrow & & \downarrow\\ \mathcal{K}(\mathbb{F}_{q})_{\#}^{c} & \longrightarrow & \mathcal{K} (\mathbb{C})_{\#}^{c} & \overset{\psi^{q}-1}{\longrightarrow} & \mathcal{K} (\mathbb{C})_{\#}^{c} \end{array} \] in which all colums and both lower rows are homotopy fibrations/cofibrations. It follows from \cite[Lemma 2.1]{CMN} once more that the top row is also a fibration/cofibration, with the cofiber map induced from $\psi^{q}-1$. It follows that the defining homotopy cartesian square for ${}_{-1} \mathcal{V}(\overline{R}_{F})_{\#}^{c}$ from Section \ref{section:splittingresults}, by means of substitutions according to Lemma \ref{_-1VRC} in Appendix B, gives rise to a homotopy commutative diagram where the horizontal maps are homotopy fibrations (note that $\pi_{0}(\Omega ^{3}\mathcal{K}(\mathbb{R}))=0$): \begin{equation} \begin{array} [c]{ccccc} {}_{-1}\mathcal{V}(\overline{R}_{F})_{\#}^{c} & \longrightarrow & {} _{-1}\mathcal{V}(\mathbb{R})_{\#}^{c}\simeq(\Omega^{2}\mathcal{K} (\mathbb{R}))_{\#}^{c} & \overset{\tau}{\longrightarrow} & (\Omega ^{3}\mathcal{K}(\mathbb{R}))_{\#}^{c}\\ \downarrow & & \downarrow^{{}_{-1}\sigma} & & \downarrow^{\mathrm{id}}\\ {}_{-1}\mathcal{V}(\mathbb{F}_{q})_{\#}^{c} & \longrightarrow & {} _{-1}\mathcal{V}(\mathbb{C})_{\#}^{c}\simeq(\Omega^{3}\mathcal{K} (\mathbb{R}))_{\#}^{c} & \overset{\psi^{q}-1}{\longrightarrow} & (\Omega ^{3}\mathcal{K}(\mathbb{R}))_{\#}^{c} \end{array} \label{-1VDiagram} \end{equation}
From the proof of Lemma \ref{_-1V(R) -> _-1V(C)} in Appendix B again, the map ${}_{-1}\sigma$ corresponds to the cup-product with the generator of $K_{1}(\mathbb{R})$. Now, the image of $_{-1}\sigma$ is a torsion element in $K_{\ast}(\mathbb{R})$; and it is easy to see by a direct checking that, for odd $q$, any torsion element of $K_{\ast}(\mathbb{R})$ is killed by $\psi ^{q}-1$. Therefore, the homotopy exact sequence for the upper horizontal maps includes the exact sequence \[
\xymatrix{ K_{n+4}(\mathbb{R}) \ar@{}[r]|-{\hbox{\large{$\rightarrow$}}}&
{}_{-1}V_{n}(\overline{R}_{F}) \ar@{}[r]|-{\hbox{\large{$\rightarrow$}}}&
K_{n+2}(\mathbb{R}) \ar@{}[r]|-{\hbox{\large{$\rightarrow$}}}& K_{n+3}(\mathbb{R}), } \] in which the last map is trivial because it is given by the cup-product with the generator of $K_{1}(\mathbb{R})$ composed with $\psi^{q}-1$. The groups ${}_{-1}V_{n}(\overline{R}_{F})$ are therefore included in the short exact sequences \[ 0\longrightarrow K_{n+4}(\mathbb{R})\longrightarrow{}_{-1}V_{n}(\overline {R}_{F})\longrightarrow K_{n+2}(\mathbb{R})\longrightarrow0 \] which determine them, except for $n\equiv0\;(\mathrm{mod}\ 8)$, where we can say only that ${}_{-1}V_{n}(\overline{R}_{F})=\mathbb{Z}$ or $\mathbb{Z} /2\oplus\mathbb{Z}$. One way to resolve this ambiguity is to write the exact sequence \[ 0={}_{-1}KQ_{n+1}(\overline{R}_{F})\longrightarrow K_{n+1}(\overline{R} _{F})\longrightarrow{}_{-1}V_{n}(\overline{R}_{F})\longrightarrow{}_{-1} KQ_{n}(\overline{R}_{F})=0 \] which implies that $K_{n+1}(\overline{R}_{F})\cong{}_{-1}V_{n}(\overline {R}_{F})$.
In general, the computation of the groups $K_{n}(\overline{R}_{F})$ for $n>0$ follows from the analog of Diagram ($\ref{RFoverbar}$) for $\mathcal{K} (\overline{R}_{F})$. They are the following for $n\equiv k\;(\mathrm{mod} \ 8)$, starting from $k=0$: \[ 0,\ \mathbb{Z}/2\oplus\mathbb{Z},\ \mathbb{Z}/2,\ \mathbb{Z}/2w_{4k+2} ,\ 0,\ \mathbb{Z},\ 0,\ \mathbb{Z}/w_{4k+1}\text{.} \]
This computation is straightforward, except for $n\equiv1\;(\mathrm{mod}\ 8)$, where we have to use two exact sequences extracted from the analog of the previous square for the spectrum \ $\mathcal{K}(\overline{R}_{F})$. The first one \[ 0\longrightarrow K_{n+1}(\mathbb{C})\longrightarrow K_{n}(\overline{R} _{F})\longrightarrow K_{n}(\mathbb{R})\longrightarrow K_{n}(\mathbb{C})=0 \] shows as expected that $K_{n}(\overline{R}_{F})=$ $\mathbb{Z}$ or $\mathbb{Z}/2\oplus\mathbb{Z}$. In the second one, we write the Mayer-Vietoris exact sequence associated to the same previous square: \[ 0\longrightarrow K_{n+1}(\mathbb{C})=\mathbb{Z}\longrightarrow K_{n} (\overline{R}_{F})\longrightarrow K_{n}(\mathbb{R})\oplus K_{n}(\mathbb{F} _{q})\longrightarrow K_{n}(\mathbb{C})=0\text{.} \] It shows that $K_{n}(\overline{R}_{F})=\pi_{n}(\mathcal{K}(\overline{R}_{F}))$ cannot be isomorphic to $\mathbb{Z}$, since $K_{n}(\mathbb{R})\oplus K_{n}(\mathbb{F}_{q})$ is a direct sum of two nontrivial cyclic groups. The computation of the groups ${}_{-1}V_{n}(\overline{R}_{F})$ is therefore accomplished for all values of $n$.$
\Box
$
\begin{remark} On the level of spectra the composition $\tau$ in (\ref{-1VDiagram}), \[ {}_{-1}\mathcal{V}(\mathbb{R})_{\#}^{c}\sim(\Omega^{2}\mathcal{K} (\mathbb{R}))_{\#}^{c}\overset{\sigma}{\longrightarrow}(\Omega^{3} \mathcal{K}(\mathbb{R}))_{\#}^{c}\overset{\Omega^{3}(\psi^{q}-1)} {\longrightarrow}(\Omega^{3}\mathcal{K}(\mathbb{R}))_{\#}^{c} \] where $\sigma$ is induced by the cup-product with the generator of $K_{1}(\mathcal{\mathbb{R}}),$ is NOT nullhomotopic. This fact is proved in Appendix D. \end{remark}
We can now use Theorems \ref{theorem2} and \ref{theorem4} to determine the composition \[ {}_{\varepsilon}{KQ}_{n}(R_{F})\overset{F}{\longrightarrow}{K}_{n} (R_{F})\overset{H}{\longrightarrow}{}_{\varepsilon}{KQ}_{n}(R_{F}) \] of the homomorphisms induced by the forgetful and hyperbolic functors. From their respective induced homotopy fiber sequences ${}_{\varepsilon} \mathcal{V}(R_{F})\longrightarrow{}_{\varepsilon}\mathcal{KQ}(R_{F} )\longrightarrow\mathcal{K}(R_{F})$, ${}_{\varepsilon}\mathcal{U} (R_{F})\longrightarrow\mathcal{K}(R_{F})\longrightarrow{}_{\varepsilon }\mathcal{KQ}(R_{F})$, and the natural homotopy equivalence ${}_{\varepsilon }\mathcal{V}(R_{F})\simeq\Omega{}_{-\varepsilon}\mathcal{U}(R_{F})$ of \cite{K:AnnM112fun}, we have the exact sequences \[ \cdots\rightarrow{}_{\varepsilon}V_{n}(R_{F})\longrightarrow{}_{\varepsilon }KQ_{n}(R_{F})\overset{F}{\longrightarrow}K_{n}(R_{F})\longrightarrow {}_{\varepsilon}V_{n-1}(R_{F})\rightarrow\cdots, \] and \[ \cdots\rightarrow{}_{-\varepsilon}V_{n-1}(R_{F})\longrightarrow K_{n} (R_{F})\overset{H}{\longrightarrow}{}_{\varepsilon}KQ_{n}(R_{F} )\longrightarrow{}_{-\varepsilon}V_{n-2}(R_{F})\rightarrow\cdots\text{.} \] Since all terms are now known (and many are zero), a routine computation gives the following.
\begin{corollary} \label{HF computation}For $n\geq1$, the endomorphism $HF$ of the group ${}_{\varepsilon}{KQ}_{n}(R_{F})$ modulo odd torsion
\begin{enumerate} \item[(i)] is multiplication by $2$, when $n\equiv3\;(\mathrm{mod}\ 4)$ ($\varepsilon=\pm1$),
\item[(ii)] has image of order $2$, when both $n\equiv1,2\;(\mathrm{mod}\ 8)$ and $\varepsilon=1$, and
\item[(iii)] is zero otherwise.$
\Box
$ \end{enumerate} \end{corollary}
A similar computation affords the corresponding result for the other composition of $F$ and $H$. From \cite[pg.~230]{K:AnnM112hgo} we note that this endomorphism of ${K}_{n}(R_{F})$ is the sum of the identity and the involution induced by the duality functor. Since this involution is independent of $\varepsilon$, we need consider only the simpler case $\varepsilon=-1$.
\begin{corollary} For $n\geq1$ and $\varepsilon=\pm1$, the endomorphism $FH$ of ${K}_{n}(R_{F})$ modulo odd torsion
\begin{enumerate} \item[(i)] is multiplication by $2$, when $n\equiv3\;(\mathrm{mod}\ 4)$, and
\item[(ii)] is zero, otherwise.$
\Box
$ \end{enumerate} \end{corollary}
\begin{corollary} The canonical involution on $K_{n}(R_{F})$ modulo odd torsion
\begin{enumerate} \item[(i)] is the identity, for $n=0$ and $n\equiv3\;(\mathrm{mod}\ 4)$, and
\item[(ii)] is the opposite of the identity, otherwise.$
\Box
$ \end{enumerate} \end{corollary}
\begin{remark} Concerning the odd torsion, in general the functors $F$ and $H$ induce bijections between the symmetric parts of the $K$- and $KQ$-groups of a ring $A$. Here the involution on the $K$-groups is induced by the duality functor. Clearly the composition $FH$ is the multiplication by $2$ map on the symmetric part. The same result holds for $HF$ on the symmetric part, while it is trivial on the antisymmetric part. We note that it remains to compute the odd torsion part of ${}_{\varepsilon}KQ_{n}(R_{F})$, even for $F=\mathbb{Q}$. However, as we have seen more generally in Proposition \ref{Wodd}, the odd torsion of the higher Witt groups and coWitt groups of $R_{F}$ is trivial. \end{remark}
\section{Proof of Theorem \ref{theorem3}}
\label{section:proofoftheorem3}
In the terminology at the end of the Introduction, consider the naturally induced map between $2$-completed connective spectra induced by the forgetful functor and the homotopy fixed point functor for the $_{\varepsilon} \mathbb{Z}/2$-action: \begin{equation} \begin{array} [c]{ccc} {}_{\varepsilon}\mathcal{KQ}(R_{F})_{\#}^{c} & \rightarrow & {\displaystyle \bigvee\limits^{r}}{}_{\varepsilon}\mathcal{KQ}(\mathbb{R} )_{\#}^{c}\\ \downarrow & & \downarrow\\ {}_{\varepsilon}\mathcal{KQ}(\mathbb{F}_{q})_{\#}^{c} & \rightarrow & {\displaystyle\bigvee\limits^{r}}{}_{\varepsilon}\mathcal{KQ}(\mathbb{C} )_{\#}^{c} \end{array} \quad\rightarrow\quad \begin{array} [c]{ccc} \mathcal{K}(R_{F})_{\#}{}^{h({}_{\varepsilon}\mathbb{Z}/2)} & \rightarrow & {\displaystyle\bigvee\limits^{r}}\mathcal{K}(\mathbb{R})_{\#}^{c}{} ^{h({}_{\varepsilon}\mathbb{Z}/2)}\\ \downarrow & & \downarrow\\ \mathcal{K}(\mathbb{F}_{q})_{\#}{}^{h({}_{\varepsilon}\mathbb{Z}/2)} & \rightarrow & {\displaystyle\bigvee\limits^{r}}\mathcal{K}(\mathbb{C} )_{\#}^{c}{}^{h({}_{\varepsilon}\mathbb{Z}/2)} \end{array} \label{homotopycartesiandiagrams1} \end{equation} As noted in the beginning of Section \ref{section:proofoftheorem1}, the spectrum maps in the B{\"{o}}kstedt square are $_{\varepsilon}\mathbb{Z} /2$-equivariant, being induced by ring maps. Theorem \ref{theorem1} and the main results in \cite{HO}, \cite{Mitchell} (\textsl{cf.}~Appendix \ref{section:K-theorybackground} for more details) show that both the hermitian and the algebraic $K$-theory squares are homotopy cartesian squares (since the homotopy fixed point functor is a homotopy functor).
By \cite[Lemmas 7.3-7.5]{BK} the map \[ {}_{\varepsilon}\mathcal{KQ}(A)_{\#}^{c}\rightarrow(\mathcal{K(A)}_{\#} {}^{h({}_{\varepsilon}\mathbb{Z}/2)})^{c} \] in (\ref{homotopycartesiandiagrams1}) is a homotopy equivalence for $A=\mathbb{F}_{q},\mathbb{R},\mathbb{C}$. It is worth mentioning that the most delicate case is when $A=\mathbb{R},$ where the machinery of Fredholm operators in an infinite dimensional real Hilbert space is used. It follows that the induced map of homotopy pullbacks is also a homotopy equivalence.
$\Box
$
\appendix
\section{$K$-theory background}
\label{section:K-theorybackground}
In this appendix we deduce the homotopy cartesian square of $K$-theory spectra (\ref{algebraic K Bokstedt square}) using the space level results given in \cite{HO}. The examples of \'{e}tale $K$-theory spectra of real number fields at the prime $2$ in \cite[\S 5]{Mitchell} provide an alternate proof on account of the solution of the Quillen-Lichtenbaum conjecture in \cite{Ostvar}. Throughout we retain the assumptions and notations employed in the main body of the text. Recall from the Introduction that $q$ is a prime number.
Recall the construction of the square (\ref{algebraic K Bokstedt square}) from the beginning of Section \ref{section:proofoftheorem1}: one starts out by choosing an embedding of the field of $q$-adic numbers $\mathbb{Q}_{q}$ into the complex numbers $\mathbb{C}$ such that the induced composite map \[ \mathcal{K}(\mathbb{Z}_{q})_{\#}\rightarrow\mathcal{K}(\mathbb{Q}_{q} )_{\#}\rightarrow\mathcal{K}(\mathbb{C})_{\#}^{c} \] agrees with Quillen's Brauer lift $\mathcal{K}(\mathbb{F}_{q})_{\#} \rightarrow\mathcal{K}(\mathbb{C})_{\#}^{c}$ from \cite{Quillen} under the rigidity equivalence between $\mathcal{K}(\mathbb{Z}_{q})_{\#}$ and $\mathcal{K}(\mathbb{F}_{q})_{\#}$ \cite{Gabber}. The ring maps relating $R_{F}$ to $\mathbb{F}_{q}$, $\mathbb{R}$ and $\mathbb{C}$ induce the commuting B{\"{o}}kstedt square (\ref{algebraic K Bokstedt square}) via Suslin's identifications of the $2$-completed algebraic $K$-theory spectra of the real numbers with $\mathcal{K}(\mathbb{R})_{\#}^{c}$ and likewise for the complex numbers and $\mathcal{K}(\mathbb{C})_{\#}^{c}$ \cite{Suslin :local fields}.
\begin{theorem} The B{\"{o}}kstedt square (\ref{algebraic K Bokstedt square}) is a commuting homotopy cartesian square of $2$-completed spectra: \end{theorem}
\noindent\textbf{Proof.} Let $JK(q)$ denote the fiber of the composite map \[ BO_{\#}\overset{c}{\longrightarrow}BU_{\#}\overset{\psi^{q}-1}{\longrightarrow }BU_{\#} \] where $c$ denotes the complexification map and $\psi^{q}$ the $q$\thinspace th Adams operation on the $2$-completion of the classifying space $BU$. As usual, $U$ and $O$ are the stable unitary and orthogonal groups. When $q\equiv \pm3\;(\mathrm{mod}\ 8)$, $JK(q)$ is a space level model for $\mathcal{K} (R_{\mathbb{Q}})_{\#}$ \cite{HO}. By the main result in \cite{Quillen}, the fiber of $\psi^{q}-1$ identifies with the $2$-completed algebraic $K$-theory space of $\mathbb{F}_{q}$. Moreover, the product decomposition \[ JK(q)\times\prod^{r-1}U_{\#}/O_{\#} \] of the $2$-completed algebraic $K$-theory space of $R_{F}$ established in \cite[Theorem 1.1]{HO} shows that the space level analogue of (\ref{algebraic K Bokstedt square}) is homotopy cartesian. On the other hand, the Quillen-Lichtenbaum conjecture for totally real number fields \cite{Ostvar} implies that $\mathcal{K}(R_{F})_{\#}$ is homotopy equivalent to the connective cover of its $K(1)$-localization $L_{K(1)}\mathcal{K} (R_{F})_{\#}$, and likewise for $\mathbb{F}_{q}$, $\mathbb{R}$ and $\mathbb{C}$. Here $K(1)$ is the first Morava $K$-theory spectrum at the prime $2$. In order to conclude we incorporate \cite{Bousfield}, which reduces questions about $K(1)$-local spectra to space level questions. That is, applying Bousfield's homotopy functor $T$ from spaces to spectra yields the desired conclusion since by \textsl{loc.~cit.}~$L_{K(1)}\mathcal{K} (R_{F})_{\#}$ identifies with $T\Omega^{\infty}\mathcal{K}(R_{F})_{\#}$.
$\Box$
We refer the reader to \cite{MitchellSurvey} for an extensive background on the stable homotopy-theoretic interpretation of the Quillen-Lichtenbaum conjecture.
\begin{remark} As we already mentioned in the Introduction (see Remark \ref{localization}), we also have a \textquotedblleft B{\"{o}}kstedt square\textquotedblright\ if we decide to consider $2$-localizations instead of $2$-completions, provided we follow the convention of our paper that the fields $\mathbb{R}$ and $\mathbb{C}$ are considered with their usual topology. \end{remark}
\section{Homology module maps}
Most theories in this paper are modules over the graded ring ${}_{\varepsilon }{KQ}_{\ast}(R_{F})$ in the case $F=\mathbb{Q}$, in other words, ${}_{\varepsilon}{KQ}_{\ast}(\mathbb{Z}[1/2])$. The framework for such considerations is laid out in \cite[\S 3]{K:asterisque149}, using the description of algebraic $K$-theory in terms of flat \textquotedblleft virtual\textquotedblright\ bundles.
In the topological case when $A=\mathbb{R}$, $\mathbb{C}$, the module structures on $\mathcal{K}(A)$, ${}_{\varepsilon}\mathcal{KQ}(A)$, ${}_{\varepsilon}\mathcal{U}(A)$ and ${}_{\varepsilon}\mathcal{V}(A)$ are much simpler to define. For clarity, we discuss the examples of ${}_{\varepsilon }\mathcal{V}(\mathbb{R})$ and ${}_{\varepsilon}\mathcal{V}(\mathbb{C})$, leading to a determination of the homotopy fiber of the map \[ {}_{\varepsilon}\mathcal{V}(\mathbb{R})\longrightarrow{}_{\varepsilon }\mathcal{V}(\mathbb{C}) \] for both cases $\varepsilon=\pm1$.
We start with a geometric viewpoint: the cohomology theory associated to the spectrum ${}_{1}\mathcal{KQ}(\mathbb{R})$ is constructed as the $K$-theory of real vector bundles equipped with nondegenerate quadratic forms. As shown in \cite[Exercise 9.22]{K:ktheorybook}, such a vector bundle $E$ splits as a Whitney sum \[ E=E^{+}\oplus E^{-}, \] where the quadratic form is positive on $E^{+}$ and negative on $E^{-}$. A bundle version of Sylvester's theorem tells us that the isomorphism classes of $E^{+}$ and $E^{-}$ are independent of the sum decomposition (see also the remarks below). Hence, we have the following assertion.
\begin{lemma} \label{_1KQ(R) splits}There are splittings \[ {}_{1}\mathcal{KQ}(\mathbb{R})\simeq\mathcal{K}(\mathbb{R})\vee\mathcal{K} (\mathbb{R})\text{\quad and\quad}_{1}\mathcal{KQ}(\mathbb{R})^{c} \simeq\mathcal{K}(\mathbb{R})^{c}\vee\mathcal{K}(\mathbb{R})^{c}\text{.} \] Moreover, the $K$-theory of the category of real vector bundles, on a compact space $X$ and provided with a nondegenerate quadratic form, is canonically isomorphic to the direct sum of two copies of the usual real $K$-theory of $X$.$
\Box
$ \end{lemma}
Another result of interest is also shown in \cite[Exercise 9.22] {K:ktheorybook}:
\begin{lemma} \label{_1KQ(C)}There are homotopy equivalences of spectra \[ {}_{1}\mathcal{KQ}(\mathbb{C})\simeq\mathcal{K}(\mathbb{R})\text{\quad and\quad}_{1}\mathcal{KQ}(\mathbb{C})^{c}\simeq\mathcal{K}(\mathbb{R} )^{c}\text{.} \] Moreover, the $K$-theory of the category of complex vector bundles, on a compact space $X$ and provided with a nondegenerate quadratic form, is canonically isomorphic to the usual real $K$-theory of $X$. \end{lemma}
\begin{remark} A slightly different proof of these lemmas is to use the following classical result: a Lie group has the homotopy type of its compact form \cite[pp. 218--219]{Helgason}. For instance, the Lie group $O(p,q)$ has the homotopy type of $O(p)\times O(q)$. This implies that the homotopy theory of real vector bundles provided with a quadratic form of type $(p,q)$ is equivalent to the homotopy theory of couples of real vector bundles $(E^{+},E^{-})$, of dimensions $p$ and $q$ respectively. A similar example of interest is the Lie group $O(n,\mathbb{C})$ which has the homotopy type of the usual compact Lie group $O(n)$. This implies that the homotopy theory of complex vector bundles of dimension $n$ provided with a nondegenerate quadratic form is equivalent to the homotopy theory of real vector bundles of rank $n$. These results of course imply the previous lemmas. Moreover, since all theories involved are $8$-periodic according to Bott, the homotopy equivalences on the level of the $0$-space imply the homotopy equivalence of spectra. \end{remark}
In the considerations that follow, we prefer to take the bundle viewpoint which is easier to handle than its homotopy counterpart, especially for module or ring structures which are simply given by the tensor product of vector bundles in the appropriate categories.
We illustrate this philosophy by a concrete description of the spectrum ${}_{1}\mathcal{V}(\mathbb{R})$ which is the homotopy fiber of the forgetful map ${}_{1}\mathcal{KQ}(\mathbb{R})\overset{F}{\rightarrow}\mathcal{K} (\mathbb{R})$. Strickly speaking, one should describe the full spectrum. However, by classical Bott periodicity, it is enough to describe the $0$-part of the spectrum. Since ${}_{1}\mathcal{KQ}(\mathbb{R})$ splits as $\mathcal{K}(\mathbb{R})\vee\mathcal{K}(\mathbb{R})$, the map $F$ being induced by the direct sum, the homotopy fiber should be $\mathcal{K} (\mathbb{R})$. We want to be more precise in terms of module structures and consider the \textquotedblleft relative\textquotedblright\ cohomology theory associated to this homotopy fiber ${}_{1}\mathcal{V}(\mathbb{R})$. It can be described by a well-known scheme going back to Atiyah-Hirzebruch \cite{Atiyah-Hirzebruch}, reproduced in \cite[pp. 59--63]{K:ktheorybook} (in a slightly different context) and also in \cite[pg $269$]{K:AnnM112fun}. One considers homotopy classes of triples $\tau=(E,F,\alpha)$, where $E$ and $F$ are real vector bundles equipped with nondegenerate quadratic forms, and $\alpha$ is an isomorphism between the underlying real vector bundles. If $G$ is another real vector bundle with a nondegenerate quadratic form, then its cup-product with $\tau$ is given as the triple \[ (G\otimes E,\,G\otimes F,\,\mathrm{id}\otimes\alpha)\text{.} \] This defines a ${}_{1}\mathcal{KQ}(\mathbb{R})$-module structure on ${} _{1}\mathcal{V}(\mathbb{R})$. By associating to every real vector bundle a metric,\textsl{ i.e}.~a positive quadratic form, we obtain a well-defined map up to homotopy $\mathcal{K}(\mathbb{R})\rightarrow{}_{1}\mathcal{KQ} (\mathbb{R})$, which is a right inverse to the forgetful map. Therefore, every ${}_{1}\mathcal{KQ}(\mathbb{R})$-module acquires a naturally induced $\mathcal{K}(\mathbb{R})$-module structure.
\begin{lemma} \label{_1VR}The spectrum ${}_{1}\mathcal{V}(\mathbb{R})$ is homotopy equivalent to the real topological $K$-theory $\mathcal{K}(\mathbb{R})$ as a $\mathcal{K}(\mathbb{R})$-module spectrum, and hence $_{1}\mathcal{V} (\mathbb{R})^{c}\simeq\mathcal{K}(\mathbb{R})^{c}$ as $\mathcal{K} (\mathbb{R})^{c}$-module spectra. \end{lemma}
\noindent\textbf{Proof.} We can identify ${}_{1}\mathcal{V}(\mathbb{R})$ with $\mathcal{K}(\mathbb{R})$ as modules over $\mathcal{K}(\mathbb{R})$ as follows: if $E$ is a real vector bundle there is an associated triple $(E_{+},E_{-},\mathrm{id})$ where $E_{+}$ is the bundle $E$ equipped with a positive quadratic form, and likewise for $E_{-}$ but with a negative quadratic form. This correspondence has an inverse defined by associating to a triple $\tau=(E,F,\alpha)$ as above the formal difference $E_{+}-F_{+}$ of the respective positive-form summands.
$\Box$
\begin{lemma} \label{_1VC}The spectrum ${}_{1}\mathcal{V}(\mathbb{
\mathbb{C}
})$ is homotopy equivalent to the spectrum $\Omega^{-1}(\mathcal{K} (\mathbb{R}))$, and therefore ${}_{1}\mathcal{V}(\mathbb{
\mathbb{C}
})^{c}\simeq\Omega^{-1}(\mathcal{K}(\mathbb{R})^{c})$. \end{lemma}
\noindent\textbf{Proof.} The theory ${}_{1}\mathcal{V}(\mathbb{C})$ is the homotopy fiber of the map ${}_{1}\mathcal{KQ}(\mathbb{C})\overset {F}{\longrightarrow}\mathcal{K}(\mathbb{C})$, which arises from triples $(E_{1},E_{2},\alpha)$, where $E_{1}$ and $E_{2}$ are real vector bundles and $\alpha$ is an isomorphism between their corresponding complexified vector bundles. By a well known theorem of Bott (see for instance \cite[Section III.5]{K:ktheorybook}) , this homotopy fiber may be identified with $\Omega^{-1}(\mathcal{K}(\mathbb{R}))$. Since $K_{-1}(\mathbb{R})=0$, we also have ${}_{1}\mathcal{V}(\mathbb{
\mathbb{C}
})^{c}\simeq\Omega^{-1}(\mathcal{K}(\mathbb{R})^{c})$ according to Lemma \ref{Connective.Spectra}.
$\Box
$
\begin{lemma} \label{_1VR -> C}The map ${}_{1}\mathcal{V}(\mathbb{R})\longrightarrow{} _{1}\mathcal{V}(\mathbb{C})$ is nullhomotopic and its homotopy fiber has the homotopy type of $\mathcal{K}(\mathbb{R})\vee\mathcal{K}(\mathbb{R})$. In the same way, the map ${}_{1}\mathcal{V}(\mathbb{R})^{c}\longrightarrow{} _{1}\mathcal{V}(\mathbb{C})^{c}$ is nullhomotopic and its homotopy fiber has the homotopy type of $\mathcal{K}(\mathbb{R})^{c}\vee\mathcal{K} (\mathbb{R})^{c}$. \end{lemma}
\noindent\textbf{Proof.} By the above considerations, the two theories also have a $\mathcal{K}(\mathbb{R})$-module structure. Since ${}_{1} \mathcal{V}(\mathbb{R})$ is free of rank one as a $\mathcal{K}(\mathbb{R} )$-module, the map ${}_{1}\mathcal{V}(\mathbb{R})\longrightarrow{} _{1}\mathcal{V}(\mathbb{C})$ is determined up to homotopy equivalence by its effect on the zeroth homotopy groups. This means that the map of associated real topological $K$-theories \[ K_{\mathbb{R}}^{n}(X)\longrightarrow K_{\mathbb{R}}^{n+1}(X) \] is induced by the cup-product with an element of $K_{\mathbb{R}} ^{1}(\mathrm{point})=K_{-1}(\mathbb{R})=0$. This shows that our first map is nullhomotopic. Its fiber has the homotopy type of $\mathcal{K}(\mathbb{R} )\vee\mathcal{K}(\mathbb{R})$ since the fiber of this nullhomotopic fibration has the homotopy type of the product of the total space (see (\ref{_1VR})) with the loop space of the base (see (\ref{_1VC})). The same statements hold for connective covers since, from the previous lemma for example, the map ${}_{1}V_{0}(
\mathbb{R}
)\longrightarrow{}_{1}V_{0}(
\mathbb{C}
)=0$ is an epimorphism.
$\Box
$
The determination of the map ${}_{-1}\mathcal{V}(\mathbb{R})\rightarrow{} _{-1}\mathcal{V}(\mathbb{C})$ is more delicate but uses the same arguments. By definition, the spectrum $_{-1}\mathcal{V}(\mathbb{R})$ is the fiber of ${}_{-1}\mathcal{KQ}(\mathbb{R})=\mathcal{K}(\mathbb{C})\rightarrow \mathcal{K}(\mathbb{R})$ and so identifies with $\Omega^{2}(\mathcal{K} (\mathbb{R}))$, according to a classical result of Bott. For the same reasons, $_{-1}\mathcal{V}(\mathbb{R})$, which is the fiber of \[ _{-1}\mathcal{KQ}(\mathbb{C})\simeq\Omega^{4}(\mathcal{K}(\mathbb{R} ))\longrightarrow\mathcal{K}(\mathbb{C})\simeq\Omega^{4}(\mathcal{K} (\mathbb{C}))\text{,} \] is homotopically equivalent to $\Omega^{3}(\mathcal{K}(\mathbb{R}))$. Summarizing, we have proved the following lemma:
\begin{lemma} \label{_-1VRC}The spectrum ${}_{-1}\mathcal{V}(\mathbb{R})$ is homotopy equivalent to $\Omega^{2}(\mathcal{K}(\mathbb{R}))$, while $_{-1} \mathcal{V}(\mathbb{C})$ is homotopy equivalent to $\Omega^{3}(\mathcal{K} (\mathbb{R}))$.
$\Box
$ \end{lemma}
Since all maps are module maps, our geometric viewpoint shows us that the required morphism $\Omega^{2}(\mathcal{K}(\mathbb{R}))\longrightarrow \Omega^{3}(\mathcal{K}(\mathbb{R}))$ is induced by the cup-product with an element in the group $K_{1}(\mathbb{R})=\mathbb{Z}/2$. The following lemma resolves this ambiguity and describes the homotopy fiber of the morphism.
\begin{lemma} \label{_-1V(R) -> _-1V(C)}The map \[ {}_{-1}\mathcal{V}(\mathbb{R})\simeq\Omega^{2}(\mathcal{K}(\mathbb{R} ))\longrightarrow{}_{-1}\mathcal{V}(\mathbb{C})\simeq\Omega^{3}(\mathcal{K} (\mathbb{R})) \] is induced by the cup-product with the generator of $K_{1}(\mathbb{R})$ and its fiber has the homotopy type of $\mathcal{K}(\mathbb{C}).$ Therefore, the homotopy fiber ${}_{-1}\mathcal{G}$ of the induced map on connective covers \[ {}_{-1}\mathcal{V}(\mathbb{R})^{c}\simeq(\Omega^{2}(\mathcal{K}(\mathbb{R} ))^{c}\longrightarrow{}_{-1}\mathcal{V}(\mathbb{C})^{c}\simeq(\Omega ^{3}(\mathcal{K}(\mathbb{R})))^{c} \] has the homotopy type of $\mathcal{K}(\mathbb{C})^{c}.$ \end{lemma}
\noindent\textbf{Proof.} To decide which element of $K_{1}(\mathbb{R})$ is involved, one may use the fundamental theorem of hermitian $K$-theory in a topological context (which is equivalent to Bott periodicity; see \cite{Karoubi LNM343}). In other words, we can work in ${}_{1}U$-theory instead of ${}_{-1}V$-theory. More precisely, if we show that the map ${} _{1}U_{0}(\mathbb{R})\longrightarrow{}_{1}U_{0}(\mathbb{C})$ is nontrivial, this implies that our original map ${}_{-1}\mathcal{V}(\mathbb{R} )\rightarrow{}_{-1}\mathcal{V}(\mathbb{C})$ is \textit{not} nullhomotopic and is therefore defined by the cup-product with the nontrivial element in $K_{1}(\mathbb{R})$. For this purpose, we form the diagram \[ \begin{array} [c]{ccc} K_{1}(\mathbb{R}) & \rightarrow{}_{1}KQ_{1}(\mathbb{R}) & \rightarrow{} _{1}U_{0}(\mathbb{R})\\ \downarrow & \downarrow & \downarrow\\ K_{1}(\mathbb{C}) & \rightarrow{}_{1}KQ_{1}(\mathbb{C}) & \rightarrow{} _{1}U_{0}(\mathbb{C}) \end{array} \] with exact rows. With the aid of Lemmas \ref{_1KQ(R) splits} and \ref{_1KQ(C)}, this can be rewritten as \[ \begin{array} [c]{cccc} \mathbb{Z}/2 & \rightarrow & \mathbb{Z}/2\oplus\mathbb{Z}/2 & \rightarrow {}_{1}U_{0}(\mathbb{R})\\ \downarrow & & \downarrow & \downarrow\\ 0 & \rightarrow & \mathbb{Z}/2 & \rightarrow{}_{1}U_{0}(\mathbb{C}) \end{array} \] It is important to notice that the map \[ {}_{1}\mathcal{KQ}(\mathbb{R})\simeq\mathcal{K}(\mathbb{R})\times \mathcal{K}(\mathbb{R})\longrightarrow{}_{1}\mathcal{KQ}(\mathbb{C} )\simeq\mathcal{K}(\mathbb{R}) \] is the sum map. Therefore, the map $\mathbb{Z}/2\oplus\mathbb{Z} /2\rightarrow\mathbb{Z}/2$ in the diagram above is surjective. Since by exactness ${}_{1}KQ_{1}(\mathbb{C})\rightarrow{}_{1}U_{0}(\mathbb{C})$ is injective, it follows that {}$_{1}KQ_{1}(\mathbb{R})\rightarrow{}_{1} U_{0}(\mathbb{C})$ is nontrivial, and thus {}$_{1}U_{0}(\mathbb{R} )\rightarrow{}_{1}U_{0}(\mathbb{C})$ is nontrivial too.
Thus we obtain a nontrivial fiber sequence ${}_{-1}\mathcal{V}(\mathbb{R} )\longrightarrow{}_{-1}\mathcal{V}(\mathbb{C})$ with homotopy fiber $\Omega^{2}(\mathcal{K}(\mathbb{C}))\simeq\mathcal{K}(\mathbb{C})$ by taking a double loop of the Bott fibration \[ \mathcal{K}(\mathbb{C})\longrightarrow\mathcal{K}(\mathbb{R})\longrightarrow \Omega(\mathcal{K}(\mathbb{R})) \] whose last map is defined by the cup-product with the nontrivial element in $K_{1}(\mathbb{R}).$ Therefore, we also have a fibration of connective covers \[ \mathcal{K}(\mathbb{C})^{c}\longrightarrow{}_{-1}\mathcal{V}(\mathbb{R} )^{c}\longrightarrow{}_{-1}\mathcal{V}(\mathbb{C})^{c} \] since ${}_{-1}V_{0}(\mathbb{C})=0$ by the previous lemma.
$\Box
$
\section{A ring with trivial $K$-theory and nontrivial $KQ$-theory}
Our first purpose here is to construct, from any ring $A$ in which $2$ is invertible, a ring $R_{\infty}=R_{\infty}(A)$ with the following properties:
\begin{enumerate} \item[(i)] \emph{For all }$n\in\mathbb{Z}$\emph{, }$K_{n}(R_{\infty} )=0$\emph{; but }
\item[(ii)] \emph{not all groups }${}_{\varepsilon}KQ_{n}(R_{\infty})$\emph{ need be trivial. }\newline\emph{In particular, for }$A$\emph{ a field of characteristic }$\neq2$ \emph{and }$\varepsilon=1$\emph{,} $\emph{we}$ \emph{have }${}_{\varepsilon}KQ_{0}(R_{\infty})\cong W(A)$\emph{, the Witt group of }$A$\emph{.} \end{enumerate}
The existence of such a ring $R_{\infty}$ is used below to provide a counterexample to a conjecture of \cite[3.4.2]{Williams}.
First recall that the suspension $S\Lambda$ of a discrete ring $\Lambda$ is defined to be the quotient ring $C\Lambda/\tilde{\Lambda}$, where the cone $C\Lambda$ of $\Lambda$ is the ring of infinite matrices (indexed by $\mathbb{N}$) over $\Lambda$ for which there exists a natural number that bounds:
\begin{enumerate} \item[(i)] the number of nonzero entries in each row and column; and
\item[(ii)] the number of distinct entries in the entire matrix. \end{enumerate}
\noindent The ideal $\tilde{\Lambda}$ of $C\Lambda$ comprises matrices with only finitely many nonzero entries.
Writing $\mathbb{Z}^{\prime}=\mathbb{Z}[1/2]$, for a $\mathbb{Z}^{\prime} $-algebra $A$ denote by $\mathcal{P}(A)$ the category of its finitely generated projective right modules. The tensor product of modules over $\mathbb{Z}^{\prime}$ then defines a biadditive functor \[ \mathcal{P}(A)\times\mathcal{P}(S^{2}\mathbb{Z}^{\prime})\longrightarrow \mathcal{P}(S^{2}A)\text{,} \] where $S^{2}$ refers to the double suspension.
Now provide $M_{2}(S^{2}\mathbb{Z}^{\prime})$ and $M_{2}(S^{2}A)$ with the involution \[ \left[ \begin{tabular} [c]{cc} $a$ & $b$\\ $c$ & $d$ \end{tabular} \right] \longmapsto\left[ \begin{tabular} [c]{cc} $\overline{d}$ & $-\overline{b}$\\ $-\overline{c}$ & $\overline{a}$ \end{tabular} \right] \] and choose a self-adjoint projection operator $p$ in $M_{2}(S^{2} \mathbb{Z}^{\prime})$ whose image defines a class in $_{1}KQ_{0}(M_{2} (S^{2}\mathbb{Z}^{\prime}))\cong\mathsf{{}}_{-1}KQ_{-2}(\mathbb{Z}^{\prime })\cong\mathbb{Z}\oplus\mathbb{Z}/2$ that is a generator of the free summand \cite{K:AnnM112hgo}. The tensor product with $p$ induces a nonunital map between unital rings \[ \phi:A\longrightarrow M_{2}(S^{2}A) \] defined by $a\mapsto a\otimes p$. It is easy to see that $\phi$ induces on $K_{0}$ and $KQ_{0}$ the cup-product with the class in $_{-1}KQ_{-2} (\mathbb{Z}^{\prime})$ mentioned above.
In order to deal with the technical problem that $\phi(1)\neq1$, let us replace the ring $A$ by the suspension $SB$ of a ring $B$. We can view $A$ as a bimodule over the cone $CB$ of this ring $B$. This allows us to \textquotedblleft add a unit\textquotedblright\ to the ring $A$ by defining $R$ as $CB\times A$ with the multiplication rule \[ (\lambda,u)(\mu,v)=(\lambda\mu,\,\lambda v+u\mu+uv)\text{.} \] We obtain an exact sequence of rings and nonunital ring homomorphisms \[ 0\longrightarrow A\longrightarrow R\longrightarrow CB\longrightarrow0 \] which shows that in negative degrees the $K$-theories of $R$ and $A$ coincide, as do their $KQ$-theories. We then change $\phi$ into a map between unital rings \[ \Phi:R\longrightarrow R\otimes_{\mathbb{Z}^{\prime}}S^{2}(\mathbb{Z}^{\prime })=S^{2}(R)=R_{1} \] by the formula $\Phi(\lambda,u)=(\lambda\otimes1,\,u\otimes p)$. This map restricts to $\phi$ on the ideal $A$ and we can safely use $\Phi$ as a substitute for $\phi$. Thus, we are able to define a direct system of unital rings with involution $R_{t}$ by the inductive formula $R_{t+1}=(R_{t})_{1}$.
By a well-known theorem of Wagoner \cite{Wagoner} (see also \cite{K:AnnM112fun}), we have canonical isomorphisms $K_{n+1}(SD)\cong K_{n}(D)$ and ${}_{\varepsilon}KQ_{n+1}(SD)\cong{}_{\varepsilon}KQ_{n}(D)$. Therefore, if we define $R_{\infty}$ as the direct limit of the $R_{t}$, we have \[ K_{n}(R_{\infty})\cong\underrightarrow{\lim}{}K_{n}(R_{t})\cong \underrightarrow{\lim}{}K_{n-2t}(A)=0 \] since $K_{-2}(\mathbb{Z}^{\prime})=0$.\textsf{ }
On the other hand ${}_{\varepsilon}KQ_{n}(R_{\infty})\cong\underrightarrow {\lim}{}{}_{\varepsilon}KQ_{n-4s}(A)$ is the stabilized Witt group of $A$ \cite{Karoubi stab.Witt}, which is not trivial in general. For instance, if $A$ is a commutative regular noetherian ring and $\varepsilon=1$, this is the classical Witt group of $A$. Hence, our construction of the ring $R_{\infty}$ is complete.
This construction provides many counterexamples to a conjecture of B. Williams \cite{Williams}. More specifically, we have for instance the following theorem.
\begin{theorem} Let $A$ be a commutative regular noetherian ring with finitely generated $\varepsilon$-Witt groups in degrees $0$ and $1$, and let $R_{\infty}$ be the associated ring defined above. Then the canonical map \[ {}_{1}\mathcal{KQ}(R_{\infty})_{\#}^{c}\longrightarrow(\mathcal{K}(R_{\infty })_{\#}{}^{h({}_{1}\mathbb{Z}/2)})^{c} \] is NOT a homotopy equivalence. \end{theorem}
\noindent\textbf{Proof.} According to the computation before (using Lemma \ref{Connective.Spectra}\thinspace(v)), $\pi_{0}({}_{1}\mathcal{KQ}(R_{\infty })_{\#}^{c})$ is the $2$-completed Witt group $W(A)_{\#}$ because $W(R_{\infty})=W(A)$ in this case \cite{Karoubi stab.Witt}. The group $W(A)_{\#}$ is not trivial since the rank map induces a surjection from this group to $\mathbb{Z}/2$. On the other hand, since $\mathcal{K}(R)_{\#}$ has trivial homotopy groups, it is contractible, which implies that the group $\pi_{0}$ of the right hand side is reduced to $0$.
$\Box
$
\section{Adams operations on the real $K$-theory spectrum}
Here we consider the composite \[ \Omega^{2}\mathcal{K}(\mathbb{R})\overset{\sigma}{\longrightarrow}\Omega ^{3}\mathcal{K}(\mathbb{R})\overset{\Omega^{3}(\psi^{q}-1)}{\longrightarrow }\Omega^{3}\mathcal{K}(\mathbb{R}) \] where $q$ is odd and $\sigma$ is induced by the cup-product with the generator $H-1$ of $K_{1}(\mathbb{R})=\mathbb{Z}/2$, and the canonical real line bundle $H$ over $S^{1}$ has $H^{2}=1$.
\begin{proposition} The composite \[ \Omega^{2}\mathcal{K}(\mathbb{R})_{\#}^{c}\overset{\sigma}{\longrightarrow }(\Omega^{3}\mathcal{K}(\mathbb{R}))_{\#}^{c}\overset{\Omega^{3}(\psi^{q} -1)}{\longrightarrow}(\Omega^{3}\mathcal{K}(\mathbb{R}))_{\#}^{c} \] is NOT nullhomotopic. \end{proposition}
\noindent\textbf{Proof.\ }We first show that for any space $X$, the composition of homotopy groups (where $\left[ X,\mathcal{E}\right] $ refers to pointed homotopy classes from $X$ to $\mathcal{E}_{0}$) \[ \left[ X,\,\mathcal{K}(\mathbb{R})\right] \overset{\sigma_{\ast} }{\longrightarrow}\left[ X,\,\Omega\mathcal{K}(\mathbb{R})\right] \overset{\Omega(\psi^{q}-1)_{\ast}}{\longrightarrow}\left[ X,\,\Omega \mathcal{K}(\mathbb{R})\right] \] is equal to the composite in reverse order \[ \left[ X,\,\mathcal{K}(\mathbb{R})\right] \overset{(\psi^{q}-1)_{\ast} }{\longrightarrow}\left[ X,\,\mathcal{K}(\mathbb{R})\right] \overset {\sigma_{\ast}}{\longrightarrow}\left[ X,\,\Omega\mathcal{K}(\mathbb{R} )\right] \] (in other words, $\sigma$ commutes with $\psi^{q}$). This follows from the following straightforward computation: \[ \psi^{q}(\sigma(x))=\psi^{q}((H-1)\cdot x)=\psi^{q}(H-1)\cdot\psi ^{q}(x)=(H^{q}-1)\cdot\psi^{q}(x)=(H-1)\cdot\psi^{q}(x)=\sigma(\psi^{q}(x)), \] where $x\in K_{\mathbb{R}}(X).$
Therefore, on passing to $2$-completions of connective covers, our claim will follow from the nontriviality of the composition \[ (\Omega^{2}\mathcal{K}(\mathbb{R}))_{\#}^{c}\overset{\Omega^{2}(\psi^{q} -1)}{\longrightarrow}(\Omega^{2}\mathcal{K}(\mathbb{R}))_{\#}^{c} \overset{\sigma}{\longrightarrow}(\Omega^{3}\mathcal{K}(\mathbb{R}))_{\#}^{c} \] or \begin{equation} (\Omega^{8}\mathcal{K}(\mathbb{R}))_{\#}^{c}\overset{\Omega^{8}(\psi^{q} -1)}{\longrightarrow}(\Omega^{8}\mathcal{K}(\mathbb{R}))_{\#}^{c} \overset{\sigma}{\longrightarrow}\Omega^{9}\mathcal{K}(\mathbb{R})_{\#}^{c}. \label{Adams} \end{equation}
By a theorem of Bott (see for instance \cite[Section III.5]{K:ktheorybook}), there is a fiber sequence \[ \mathcal{K}(\mathbb{C})\overset{r}{\longrightarrow}\mathcal{K}(\mathbb{R} )\overset{\sigma}{\longrightarrow}\Omega\mathcal{K}(\mathbb{R}) \] where the homotopy fiber of $\sigma$ is $\mathcal{K}(\mathbb{C})$, the classifying space of complex topological $K$-theory, and $r$ is the realification map. Therefore, if the sequence (\ref{Adams}) were trivial, one would have a factorization \[ \Omega^{8}(\psi^{q}-1):(\Omega^{8}\mathcal{K}(\mathbb{R}))_{\#}^{c} \longrightarrow(\Omega^{8}\mathcal{K}(\mathbb{C}))_{\#}^{c}\overset {r}{\longrightarrow}(\Omega^{8}\mathcal{K}(\mathbb{R))}_{\#}^{c} \] or equally, by Bott periodicity, \[ q^{4}\psi^{q}-1:\mathcal{K}(\mathbb{R})_{\#}^{c}\longrightarrow\mathcal{K} (\mathbb{C})_{\#}^{c}\overset{r}{\longrightarrow}\mathcal{K}(\mathbb{R} )_{\#}^{c}. \]
A way to prove the impossibility of such a factorization is to find a test space $X$ and map it into the three spaces involved. For such a space we choose the classifying space $X=$ $BG$, where $G$ is the connected Lie group $SO(3)$. We thereby transform our problem into an algebraic one: by Atiyah and Hirzebruch \cite[Theorem 4.8]{Atiyah-Hirzebruch}, we know that $K_{\mathbb{C} }(BG)=\widehat{R(G)}$, the complex representation ring of $G$ completed at its augmentation ideal, while by Anderson \cite{Anderson}, we have $K_{\mathbb{R} }(BG)=\widehat{RO(G)}$, the completed real representation ring of $G$.
It is well-known (for example, \cite{Adams}) that $RO(G)$ is the polynomial algebra in one variable $\mathbb{Z}\left[ \lambda^{1}\right] $, where $\lambda^{1}$ is the standard representation of $SO(3)$ in $\mathbb{R}^{3}$. The complexification $c$ induces an isomorphism $RO(G)\overset{\cong }{\rightarrow}R(G)$, while the realification from $R(G)$ to $RO(G)\cong R(G)$ is identified with the multiplication by $2$ (any complex representation of $G$ is isomorphic to its conjugate). If we choose $SO(2)$ as a maximal compact torus in $SO(3)$ embedded as \[ \left[ \begin{array} [c]{ccc} \cos\theta & -\sin\theta & 0\\ \sin\theta & \cos\theta & 0\\ 0 & 0 & 1 \end{array} \right] \] we may view $R(G)$ as the ring of polynomials on the variable $\lambda ^{1}=t+t^{-1}+1$, where $t$ represents the standard one-dimensional representation $\theta\mapsto e^{i\theta}$ of $SO(2)=S^{1}$.
Now, since $G$ is a compact connected Lie group, $R(G)$ injects into its completion. If we put $t=1-u$, we may identify the completed representation ring $\widehat{R(G)}$ as a subring of the ring of formal power series $\mathbb{Z}\left[ \left[ u\right] \right] $. However, we are considering homotopy classes from $X=BG$ not just to $\mathcal{K}(\mathbb{R})_{0}$ or $\mathcal{K}(\mathbb{C})_{0}$ but to its $2$-adic completion $\mathcal{K} (\mathbb{R})_{0\#}$ or $\mathcal{K}(\mathbb{C})_{0\#}$ \cite[(v), pg.~205]{Adamsbluebook}. From the algebraic point of view, this means that we have to compute in the power series ring $\mathbb{Z}_{2}\left[ \left[ u\right] \right] $ instead of $\mathbb{Z}\left[ \left[ u\right] \right] $.
Since $t$ is one-dimensional and $\psi^{q}$ commutes with the complexification isomorphism $c$ between $R(G)$ and $RO(G)$, we can write, in $\mathbb{Z} _{2}\left[ \left[ u\right] \right] $: \begin{align*} c(q^{4}\psi^{q}-1)(\lambda^{1}) & =q^{4}t^{q}+q^{4}t^{-q}+q^{4}-t-t^{-1}-1\\ & =(1-u)^{-q}[q^{4}(1-u)^{2q}-(1-u)^{q+1}+(q^{4}-1)(1-u)^{q}-(1-u)^{q-1} +q^{4}]\text{.} \end{align*} Because multiplication by $(1-u)^{q}$ leaves an odd coefficient of $u^{2q}$, the power series is not divisible by $2$. Therefore $(q^{4}\psi^{q} -1)(\lambda^{1})$ cannot be in the image of the realification map $r$ (which furnishes elements divisible by $2$ in $\widehat{RO(G)}\overset{\underset {\cong}{c}}{\longrightarrow}\widehat{R(G)}$). This contradiction concludes the proof of the proposition.$
\Box
$
\begin{center} A.\thinspace Jon Berrick \\[0pt]Department of Mathematics, National University of Singapore, Singapore. \\[0pt]e-mail: [email protected]
Max Karoubi \\[0pt]UFR de Math{\'{e}}matiques, Universit{\'{e}} Paris 7, France. \\[0pt]e-mail: [email protected]
Paul Arne {\O }stv{\ae }r \\[0pt]Department of Mathematics, University of Oslo, Norway. \\[0pt]e-mail: [email protected] \end{center}
\end{document} | arXiv |
\begin{document}
\title[Heisenberg-Virasoro vertex operator algebras]
{Simple restricted modules over the Heisenberg-Virasoro algebra as VOA modules}
\author{Haijun Tan}
\address{Tan: School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, 130024, P. R. China.} \email{[email protected]}
\author{Yufeng Yao$^\dagger$}
\address{Yao: Department of Mathematics, Shanghai Maritime University,
Shanghai, 201306, China.}\email{[email protected]}
\author{Kaiming Zhao}
\address{Zhao: Department of Mathematics, Wilfrid
Laurier University, Waterloo, ON, Canada N2L 3C5, and School of
Mathematical Science, Hebei Normal (Teachers) University, Shijiazhuang, Hebei, 050024 P. R. China.}
\email{[email protected]}
\thanks{$^\dagger$the corresponding author}
\begin{abstract} In this paper, we determine all simple restricted modules over the mirror Heisenberg-Virasoro algebra ${\mathfrak{D}}$, and the twisted Heisenberg-Virasoro algebra $\bar \mathfrak{D}$ with nonzero level. As applications, we characterize simple Whittaker modules and simple highest weight modules over ${\mathfrak{D}}$. A vertex-algebraic interpretation of our result is the classification of simple weak twisted and untwisted modules over the Heisenberg-Virasoro vertex operator algebras $\mathcal V^{c} \cong V_{Vir} ^{c} \otimes M(1)$. We also present a few examples of simple restricted ${\mathfrak{D}}$-modules and $\bar \mathfrak{D}$-modules induced from simple modules over finite dimensional solvable Lie algebras, that are not tensor product modules of Virasoro modules and Heisenberg modules. This is very different from the case of simple highest weight modules over $\mathfrak{D}$ and $\bar \mathfrak{D}$ which are always tensor products of simple Virasoro modules and simple Heisenberg modules.
\end{abstract}
\subjclass[2010]{17B10, 17B65, 17B66, 17B68, 17B69, 17B81}
\keywords{The Virasoro algebra, the mirror Heisenberg-Virasoro algebra, restricted module, Heisenberg-Virasoro vertex operator algebra, tensor product module}
\maketitle
\setcounter{tocdepth}{1}\tableofcontents \begin{center} \end{center}
\section{Introduction}
Throughout the paper we denote by ${\mathbb Z}, {\mathbb Z}^*,{\mathbb N},{\mathbb Z}_+, {\mathbb Z}_{\leq 0}, {\mathbb R}, {\mathbb C},$ and ${\mathbb C}^*$ the sets of integers, nonzero integers, non-negative integers, positive integers, non-positive integers, real numbers, complex numbers, and nonzero complex numbers, respectively. All vector spaces and Lie algebras are assumed to be over ${\mathbb C}$. For a Lie algebra $\mathcal{G}$, the universal algebra of $\mathcal{G}$ is denoted by $\mathcal{U}(\mathcal{G})$.
The Virasoro algebra $ \mathfrak{Vir}$ and the Heisenberg algebra $\mathcal{H}$ are infinite-dimensional Lie algebras with bases $\{{\bf c}, d_n: n\in{\mathbb Z}\}$ and $\{{\bf l}, h_n: n\in{\mathbb Z}\}$, respectively. Their Lie brackets are given by $$[\mathfrak{Vir},{\bf c}]=0,\,[d_m,d_n]=(m-n)d_{m+n}+\frac{m^3-m}{12}\delta_{m+n,0}{\bf c},\,m,n\in{\mathbb Z},$$ and $$[\mathcal{H},{\bf l}]=0,\,\,\,[h_m,h_n]=m\delta_{m+n,0}{\bf l},\,\,\,m,n\in{\mathbb Z},$$ respectively. The twisted Heisenberg-Virasoro algebra $ \bar \mathfrak{D}$ is the universal central extension of the Lie algebra $$\Big\{ f(t)\frac{d}{dt}+g(t) : f,g\in{\mathbb C}[t,t^{-1}]\Big\}$$ of differential operators of order at most one on the Laurent polynomial algebra ${\mathbb C} [t,t^{-t}]$. Since the Lie algebra $\bar \mathfrak{D}$ contains the Virasoro algebra $\mathfrak{Vir}$ and the Heisenberg algebra $\mathcal{H}$ as subalgebras (but not the semi-direct product of the two subalgebras), many properties of $\bar \mathfrak{D}$ are closely related to the algebras $\mathfrak{Vir}$ and $\mathcal{H}$.
The Virasoro algebra $\mathfrak{Vir}$, the Heisenberg algebra $\mathcal{H}$ and the twisted Heisenberg-Virasoro algebra $ \bar \mathfrak{D}$ are very important infinite-dimensional Lie algebras in mathematics and in mathematical physics because of its beautiful representation theory (see \cite{IK, KRR}), and its widespread applications to vertex operator algebras (see \cite{DMZ,FZ}), quantum physics (see \cite{GO}), conformal field theory (see \cite{DMS}), and so on. Many other interesting and important algebras contain the Virasoro algebra as a subalgebra, such as the Schr{\"o}dinger-Virasoro algebra (see \cite{H,H1}), the mirror Heisenberg-Virasoro algebra $\mathfrak{D}$ (see \cite{B,GZ,LPXZ}) and so on. These Lie algebras have nice structures and perfect theory on simple Harish-Chandra modules. The mirror Heisenberg-Virasoro algebra $\mathfrak{D}$ is the even part of the mirror $N=2$ superconformal algebra (see \cite{B}), and
is the semi-direct product of the Virasoro algebra and the twisted Heisenberg algebra (see Definition \ref{def.1}).
\subsection{Connection with representation theory of Lie algebras} Representation theory of Lie algebras has attracted a lot of attentions of mathematicians and physicists. For a Lie algebra $\mathcal{G}$ with a triangular decomposition $\mathcal{G}=\mathcal{G}_+\oplus \mathfrak{h}\oplus \mathcal{G}_-$ in the sense of \cite{MP}, one can study its weight and non-weight representation theory. For weight representation approach, to some extent, Harish-Chandra modules are well understood for many infinite-dimensional Lie algebras, for example, the affine Kac-Moody algebras in \cite{CP, MP}, the Virasoro algebra in \cite{FF,KRR,M}, the twisted Heisenberg-Virasoro algebra in \cite{ACKP, LZ1}, the Schr{\"o}dinger-Virasoro algebra (partial results) in \cite{H, H1, LS}, and the mirror Heisenberg-Virasoro algebra in \cite{LPXZ}. There are also some researches about weight modules with infinite-dimensional weight spaces (see \cite{BBFK,CGZ,LZ3}).
Recently, non-weight module theory over Lie algebras $\mathcal{G}$ attracts more attentions from mathematicians. In particular, $\mathcal{U}(\mathfrak{h})$-free $\mathcal{G}$-modules, Whittaker modules, and restricted modules have been widely studied for many Lie algebras.
The notation of $\mathcal{U}(\mathfrak{h})$-free modules was first introduced by Nilsson \cite{N} for the simple Lie algebra $\mathfrak{sl}_{n+1}$. At the same time these modules were introduced in a very different approach in the paper \cite{TZ}. Later, $\mathcal{U}(\mathfrak{h})$-free modules for many infinite-dimensional Lie algebras are determined, for example, the Kac-Moody algebras in \cite{C,CTZ, GZ1}, the Virasoro algebra in \cite{LGZ,LZ2,MZ2}, the Witt algebra in \cite{TZ}, the twisted Heisenberg-Virasoro algebra and $W(2,2)$ algebra in \cite{CHSY,CG, LZ3}, and so on.
Whittaker modules for $\mathfrak{sl}_2({\mathbb C})$ were first constructed by Arnal and Pinzcon (see \cite{AP}). Whittaker modules for arbitrary finite-dimensional complex semisimple Lie algebra $\mathfrak{L}$ were introduced and systematically studied by Kostant in \cite{Ko}, where he proved that these modules with a fixed regular Whittaker function (Lie homomorphism) on a nilpotent radical are (up to isomorphism) in bijective correspondence with central characters of $\mathcal{U}(\mathfrak{L})$. In recent years, Whittaker modules for many other Lie algebras have been investigated (see \cite{AHPY,ALZ,BM,BO,C,MD1,MD2}).
\subsection{Restricted modules} The restricted modules for a ${\mathbb Z}$-graded Lie algebra are the modules in which any vector can be annihilated by sufficiently large positive part of the Lie algebra. Whittaker modules and highest weight modules are restricted modules, and, in some sense, restricted modules can be seen as generalization of Whittaker modules and highest weight modules. Understanding restricted modules for an infinite-dimensional Lie algebra with a ${\mathbb Z}$-gradation is one of core topics in Lie theory, for this class of modules are closely connected with the modules for corresponding vertex operator algebras. The first step of studying restricted modules is to classify all restricted modules for a Lie algebra. But this is a difficult challenge. Up to now all simple restricted modules for the Virasoro algebra are classified in \cite{MZ2}. There are some partial results of simple restricted modules for other Lie algebras. Some simple restricted modules for twisted Heisenberg-Virasoro algebra and mirror Heisenberg-Virasoro algebra with level $0$ were constructed in \cite{CG1, G, LPXZ}. Rudakov investigated a class of simple modules over Lie algebras of Cartan type $W, S, H$ in \cite{Ru1,Ru2}, and these modules are restricted modules over the Cartan type Lie algebras of the formal power series.
\subsection{Vertex algebraic approach} For many infinite-dimensional $\mathbb{Z}$-graded Lie algebras and superalgebras $\mathcal G$, one can construct the associated (unversal) vertex algebra $\mathcal V_{\mathcal G}$ with the property: \begin{itemize} \item Any restricted $\mathcal G$-module is a weak $\mathcal V_{\mathcal G}$-module; \item Any weak module for the vertex algebra $\mathcal V_{\mathcal G}$ has the structure of a restricted $\mathcal G$-module. \end{itemize} This approach is very prominent for the following cases: \begin{itemize} \item Affine Kac-Moody algebra of type $X_n ^{(1)}$, when the associated vertex algebra is the universal affine vertex algebra $V^k(\mathfrak{g})$ for certain simple Lie algebra $\mathfrak{g}$. This approach was used in \cite{ALZ} for studying Whittaker modules. \item Virasoro Lie algebra, when the associated vertex algebra is the universal Virasoro vertex algebra $V_{Vir} ^c$ (cf. \cite{LL}) \item Heisenberg vertex algebra, when the associated vertex algebra is $M(1)$ (cf. \cite{LL}). \item Heisenberg-Virasoro algebra; super conformal algebras, etc. \end{itemize}
From the vertex-algebraic point of view, the twisted Heisenberg-Virasoro algebra and its untwisted modules were investigated in \cite{AR, GW}.
The restricted representations of nonzero level for the twisted Heisenberg-Virasoro algebra corresponds to representations of the Heisenberg-Virasoro vertex operator algebra $\mathcal V^{c}=V_{Vir} ^c \otimes M(1)$ ,
where $V_{Vir} ^c$ is the universal Virasoro vertex operator algebra of central charge $c= \ell_1-1$, and
$M(1)$
is the Heisenberg vertex operator algebra of level $1$. (Since $M(\ell_2) \cong M(1)$ (cf. \cite{LL}), we usually assume that the level $\ell_2=1$.)
Moreover, the restricted representations of the mirror Heisenberg-Virasoro algebra $\mathfrak{D}$ can be treated as twisted modules for the Heisenberg-Virasoro vertex operator algebra $\mathcal V^{c}=V_{Vir} ^c \otimes M(1)$.
We summarize:
\begin{itemize}
\item The category of restricted $\bar{\mathfrak{D}}$-modules of level $1$ is equivalent to the category of weak (untwisted) modules for the vertex operator algebra $\mathcal V^{c}$;
\item The category of restricted $ {\mathfrak{D}}$-modules of level $1$ is equivalent to the category of weak twisted modules for the vertex operator algebra $\mathcal V^{c}$.
\end{itemize}
\subsection{Main results} In this paper, our main goal is to classify all simple restricted modules for mirror Heisenberg-Virasoro algebra $\mathfrak{D}$, and classify simple restricted modules with non-zero level for the twisted Heisenberg-Virasoro algebra $\bar \mathfrak{D}$. As applications, we describe the simple untwisted and twisted modules for Heisenberg-Virasoro vertex operator algebras $\mathcal V^{c}$. The main results are the following theorems:
\noindent {\bf Main theorem A} (Theorem \ref{mainthm}) {\it Let $S$ be a simple restricted module over the mirror Heisenberg-Virasoro algebra $\mathfrak{D}$ with level $\ell \ne 0$. Then
{\rm(i)} $S\cong H^{\mathfrak{D}}$ where $H$ is a simple restricted module over the Heisenberg algebra $\mathcal{H}$, or
{\rm(ii)} $S$ is an induced $\mathfrak{D}$-module from a simple restricted $\mathfrak{D}^{(0,-n)}$-module, or
{\rm(iii)} $S\cong U^{\mathfrak{D}}\otimes H^{\mathfrak{D}}$ where $U$ is a simple restricted $\mathfrak{Vir}$-module, and $H$ is a simple restricted module over the Heisenberg algebra $\mathcal{H}$. }
\noindent {\bf Main theorem B } (Theorem \ref{mainthm'}) {\it Let $M$ be a simple restricted module over the twisted Heisenberg-Virasoro algebra $\bar \mathfrak{D}$ with level $\ell \ne 0$. Then
{\rm(i)} $M\cong K(z)^{\bar \mathfrak{D}}$ where $K$ is a simple restricted $\bar \mathcal{H}$-module and $z\in{\mathbb C}$, or
{\rm(ii)} $M$ is an induced $\bar \mathfrak{D}$-module from a simple restricted $\bar \mathfrak{D}^{(0,-n)}$-module for some $n\in{\mathbb Z}_+$, or
{\rm(iii)} $M\cong K(z)^{\bar \mathfrak{D}}\otimes U^{\bar \mathfrak{D}}$ where $z\in{\mathbb C}$, $K$ is a simple restricted $\bar \mathcal{H}$-module and $U$ is a simple restricted $\mathfrak{Vir}$-module. }
These simple restricted modules over the (mirror) Heisenberg-Virasoro algebra are actually all simple weak ($\theta$-twisted) modules over Heisenberg-Virasoro vertex operator algebras $\mathcal V^{c}$ (where the involution $\theta$ is defined in Section \ref{voasection}, see Theorem \ref{5.3-untw}, Theorem \ref{5.3}). As a consequence, we obtain the classification of twisted and untwisted simple modules for the Heisenberg-Virasoro vertex operator algebra $\mathcal V^{c}$, i.e., we obtain all weak simple $\mathcal V^{c}$-modules and all weak simple $\theta$-twisted $\mathcal V^{c}$-modules.
It is important to notice that certain weak modules induced from simple restricted $\mathfrak{D}^{(0,-n)}$, as a (twisted) modules for $V_{Vir} ^c \otimes M(\ell_2)$, do not have the form $M_1 \otimes M_2$ (see Section \ref{examples}). This is interesting, since in the category of ordinary (twisted) modules for the vertex operator alegbras, such modules don't exist (see \cite[Theorem 4.7.4]{FHL} and its twisted analogs).
\subsection{Organization of the paper} The present paper is organized as follows. In Section 2, we recall notations related to the algebras $\mathfrak{D}$ and $\bar\mathfrak{D}$, collect some known results, and establish a general result for a simple module to be a tensor product module over a class of Lie algebras (Theorem \ref{generalize}). In Section 3, we construct a class of induced simple $\mathfrak{D}$-modules (Theorem \ref{thmmain1.1}). In Section 4, we completely determine all simple restricted modules over the mirror Heisenberg-Virasoro algebra $\mathfrak{D}$ (Theorems \ref{mainthm}, \ref{MT}). In Section 5, we use a similar method as in Section 4 to classify the simple restricted modules of level nonzero over the twisted Heisenberg-Virasoro algebra $\bar \mathfrak{D}$ (see Theorem \ref{mainthm'}). In Section 6, we apply Theorem \ref{mainthm} to generalize the result in \cite{MZ1} to the algebra ${\mathfrak{D}}$, i.e., we give a new characterization of simple highest weight modules over ${\mathfrak{D}}$ (Theorem \ref{thmmain}). We also characterize simple Whittaker modules over ${\mathfrak{D}}$ (Theorem \ref{thmmain17}).
In Section 7, we present a few examples of simple restricted ${\mathfrak{D}}$-modules and $\bar \mathfrak{D}$-modules induced from simple modules over finite dimensional solvable Lie algebras, that are not tensor product modules of Virasoro modules and Heisenberg modules. This is very different from the case of simple highest weight modules over $\mathfrak{D}$ and $\bar \mathfrak{D}$ which are always tensor products of simple Virasoro modules and simple Heisenberg modules. In Appendix A, we apply Theorems \ref{mainthm} and \ref{mainthm'} to classify
simple weak twisted and untwisted modules over the Heisenberg-Virasoro vertex operator algebras $\mathcal V^{c} \cong V_{Vir} ^{c} \otimes M(1)$ (Theorems \ref{5.3-untw}, \ref{5.3}). The main method in this paper is the (twisted) weak module structure of the Heisenberg vertex operator algebra $M(1)$ on simple restricted $\mathfrak{D}$-modules.
\section{Notations and preliminaries}
In this section we recall some notations and known results related to the algebras $\mathfrak{D}$ and $\bar \mathfrak{D}$.
\begin{defi}\label{def.1}
The {\bf twisted Heisenberg-Virasoro algebra} ${\bar {\mathfrak{D}}}$ is a Lie algebra with a basis
$$\left\{d_{m},h_{r},\bar{\bf c}_1,\bar{\mathbf{c}}_2, \bar{{\bf c}}_3: m, r\in{\mathbb Z}\right\}$$ and subject to the commutation relations
\begin{equation}\label{thva}\begin{aligned}
&[d_m, d_n]=(m-n)d_{m+n}+\delta_{m+n,0}\frac{ m^3-m}{12}\bar{{\bf c}}_1,\\
&[d_m,h_{r}]=-rh_{m+r}+\delta_{m+r,0}(m^2+m)\bar{{\bf c}}_2,\\
&[h_{r},h_{s}]=r\delta_{r+s,0}\bar{{\bf c}}_3,\\
&[\bar{{\bf c}}_1,{\bar {\mathfrak{D}}}]=[\bar{{\bf c}}_2,{\bar {\mathfrak{D}}}]=[\bar{{\bf c}}_3,{\bar {\mathfrak{D}}}]=0,\end{aligned}
\end{equation}
for $m,n,r,s\in{\mathbb Z}$.
\end{defi}
It is clear that ${\bar {\mathfrak{D}}}$ contains a copy of the Virasoro subalgebra $\mathfrak{Vir}=\text{span}\{\bar{\bf c}_1, d_i: i\in{\mathbb Z}\}$ and the Heisenberg algebra $\bar {\mathcal{H}}=\bigoplus_{r\in{\mathbb Z}}{\mathbb C} h_r\oplus{\mathbb C} \bar{{\bf c}}_3$. So $\bar {\mathfrak{D}}$ has a quotient algebra that is isomorphic to a copy of {\bf Heisenberg-Virasoro algebra} $$\widetilde{\mathfrak{D}}=\text{span}_{{\mathbb C}}\left\{d_{m},h_{r},\bar{\bf c}_1,\bar{{\bf c}}_3: m, r\in{\mathbb Z}\right\}$$ whose relations are defined by (\ref{thva}) (but the second and fourth equalities are replaced by $[d_m,h_{r}]=-rh_{m+r}$ and $[\bar{{\bf c}}_1,\widetilde{\mathfrak{D}}]=[\bar{{\bf c}}_3,\widetilde{\mathfrak{D}}]=0$).
Note that ${\bar {\mathfrak{D}}}$ is ${\mathbb Z}$-graded and equipped with a triangular decomposition:
$
{{\bar {\mathfrak{D}}}}={\bar {\mathfrak{D}}}^{+}\oplus {\mathfrak{h}}\oplus {\bar {\mathfrak{D}}}^{-},
$
where
\begin{eqnarray*}
&&{\bar {\mathfrak{D}}}^{\pm}=\bigoplus_{n, r\in{\mathbb Z}_+}({\mathbb C} d_{\pm n}\oplus {\mathbb C} h_{\pm r}),\quad {\mathfrak{h}}={\mathbb C} d_0\oplus{\mathbb C} h_{0}\oplus{\mathbb C} \bar{{\bf c}}_1+{\mathbb C} \bar{{\bf c}}_2+ {\mathbb C} \bar{{\bf c}}_3.
\end{eqnarray*} Moreover, ${\bar {\mathfrak{D}}}=\oplus_{i\in{\mathbb Z}}{\bar {\mathfrak{D}}}_i$ is ${\mathbb Z}$-graded with ${\bar {\mathfrak{D}}}_i={\mathbb C} d_i\oplus{\mathbb C} h_{i}$ for $i\in{\mathbb Z}^*$, $\bar {\mathfrak{D}}_0={\mathfrak{h}}$.
\begin{defi}\label{def.2}
The {\bf mirror Heisenberg-Virasoro algebra} ${\mathfrak{D}}$ is a Lie algebra with a basis
$$\left\{d_{m},h_{r},{\bf c}_1,\mathbf{c}_2\mid m\in{\mathbb Z},r\in \frac{1}{2}+{\mathbb Z}\right\}$$ and subject to the commutation relations
\begin{eqnarray*}
&&[d_m, d_n]=(m-n)d_{m+n}+\delta_{m+n,0}\frac{ m^3-m}{12}{\bf c}_1,\\
&&[d_m,h_{r}]=-rh_{m+r},\\
&&[h_{r},h_{s}]=r\delta_{r+s,0}{\bf c}_2,\\
&&[{\bf c}_1,{\mathfrak{D}}]=[\mathbf{c}_2,{\mathfrak{D}}]=0,
\end{eqnarray*}
for $m,n\in{\mathbb Z}, r,s\in \frac{1}{2}+{\mathbb Z}$.
\end{defi} It is clear that ${\mathfrak{D}}$ is the semi-direct product of the Virasoro subalgebra $\mathfrak{Vir}=\text{span}\{{\bf c}_1, d_i\mid i\in{\mathbb Z}\}$ and the twisted Heisenberg algebra $\mathcal{H}=\bigoplus_{r\in\frac{1}{2}+{\mathbb Z}}{\mathbb C} h_r\oplus{\mathbb C} {\bf c}_2$. Note that ${\mathfrak{D}}$ is $\frac{1}{2}{\mathbb Z}$-graded and equipped with triangular decomposition:
$
{{\mathfrak{D}}}={\mathfrak{D}}^{+}\oplus {\mathfrak{D}}^{0}\oplus {\mathfrak{D}}^{-},
$
where
\begin{eqnarray*}
&&{\mathfrak{D}}^{\pm}=\bigoplus_{n\in{\mathbb Z}_+}{\mathbb C} d_{\pm n}\oplus \bigoplus_{r\in\frac{1}{2}+{\mathbb N}}{\mathbb C} h_{\pm r},\quad {\mathfrak{D}}^{0}={\mathbb C} d_0\oplus{\mathbb C} {\bf c}_1\oplus{\mathbb C} \mathbf{c}_2.
\end{eqnarray*} Moreover, ${\mathfrak{D}}=\oplus_{i\in{\mathbb Z}}{\mathfrak{D}}_i$ is ${\mathbb Z}$-graded with ${\mathfrak{D}}_i={\mathbb C} d_i\oplus{\mathbb C} h_{i+\frac{1}{2}}$ for $i\in{\mathbb Z}^*\setminus\{-1\}$, ${\mathfrak{D}}_0={\mathbb C} d_0\oplus{\mathbb C} h_{\frac{1}{2}}\oplus{\mathbb C} {\bf c}_1$ and ${\mathfrak{D}}_{-1}={\mathbb C} d_{-1}\oplus{\mathbb C} h_{-\frac{1}{2}}\oplus{\mathbb C} \mathbf{c}_2$.
\begin{defi} Let $\mathcal{G}=\oplus_{i\in{\mathbb Z}}{\mathcal{G}}_{i}$ be a ${\mathbb Z}$-graded Lie algebra. A $\mathcal{G}$-module $V$ is called the $\bf{restricted}$ module if for any $v\in V$ there exists $n\in{\mathbb N}$ such that $\mathcal{G}_iv=0$, for $i>n$. The category of restricted modules over $\mathcal{G}$ will be denoted as $\mathcal{R}_{\mathcal{G}}$. \end{defi}
\begin{defi} Let $\mathfrak{a}$ be a subalgebra of a Lie algebra $\mathcal{G}$, and $V$ be a $\mathcal{G}$-module. We denote $${\rm Ann}_V(\mathfrak{a})=\{v\in V:\mathfrak{a}v=0\}. $$ \end{defi}
\begin{defi} Let $\mathcal{G}$ be a Lie algebra and $V$ a $\mathcal{G}$-module and $x\in \mathcal{G}$. \begin{itemize} \item[\rm (1)] If for any $v\in V$ there exists $n\in\mathbb Z_+$ such that $x^nv=0$, then we say that the action of $x$ on $V$ is {\bf locally nilpotent}. \item[\rm (2)] If for any $v\in V$ we have $\mathrm{dim}(\sum_{n\in{\mathbb N}}\mathbb C x^nv)<+\infty$, then the action of $x$ on $V$ is said to be {\bf locally finite}. \item[\rm (3)] The action of $\mathcal{G}$ on $V$ is said to be {\bf locally nilpotent} if for any $v\in V$ there exists an $n\in\mathbb Z_+$ (depending on $v$) such that $x_1x_2\cdots x_nv=0$ for any $x_1,x_2,\cdots, x_n\in L$. \item[\rm (4)] The action of $\mathcal{G}$ on $V$ is said to be {\bf locally finite} if for any $v\in V$ there is a finite-dimensional $L$-submodule of $V$ containing $v$. \end{itemize} \end{defi}
\begin{defi}
If $W$ is a $\mathfrak D$-module ({\it resp.} $\bar {\mathfrak{D}}$-module) on which ${\bf c}_1$ ({\it resp.} $\bar{{\bf c}}_1$) acts as complex
scalar $c$, we say that $W$ is of {\bf central charge} $c$. If $W$ is a $\mathfrak D$-module ({\it resp.} $\bar {\mathfrak{D}}$-module) on which ${\bf c}_2$ ({\it resp.} $\bar{{\bf c}}_3$) acts as complex
scalar $\ell$, we say that $W$ is of {\bf level} $\ell$.
\end{defi}
Note that if $V$ is a $\mathfrak{Vir}$-module, then $V$ can be easily viewed as a $\mathfrak{D}$-module ({ resp.} $\bar {\mathfrak{D}}$-module) by defining $\mathcal{H} V=0$ ({ resp.} $(\bar {\mathcal{H}}+{\mathbb C} \bar{{\bf c}}_2)V=0$), the resulting module is denoted by $V^{\mathfrak{D}}$({ resp.} $V^{\bar {\mathfrak{D}}}$).
Thanks to \cite{LPXZ}, for any $H\in\mathcal{R}_{\mathcal{H}}$ with the action of ${\bf c}_2$ as a nonzero scalar $\ell$, we can give $H$ a $\mathfrak{D}$-module structure denoted by $H^{{\mathfrak{D}}}$ via the following map \begin{align} d_n&\mapsto L_n= \frac{1}{2\ell}\sum_{k\in{\mathbb Z}+\frac{1}{2}}h_{n-k}h_k,\quad\forall n\in{\mathbb Z}, n\neq0 ,\label{rep1}\\
d_0&\mapsto L_0=\frac{1}{2\ell}\sum_{k\in{\mathbb Z}+\frac{1}{2}}h_{-|k|}h_{|k|}+\frac{1}{16},\label{rep2}\\ h_r&\mapsto h_r,\quad\forall r\in\frac{1}{2}+{\mathbb Z}, \quad {\bf{c}_1} \mapsto 1, \quad {\bf{c}_2}\mapsto \ell.\label{rep3} \end{align}
\begin{rem} The vertex operator algebra interpretation of the formula (\ref{rep1})-(\ref{rep2}) will be given in Section \ref{voa-interp}. The operators $L_n$, $n \in {{\mathbb Z}}$, will be represented as components of the field $L_{tw} ^{Heis} (z)$ in (\ref{heis-tw-vir}). \end{rem}
According to (9.4.13) and (9.4.15) in \cite{FLM}, we know that for all $m,n\in{\mathbb Z}, r\in\frac{1}{2}+{\mathbb Z}$, we have \begin{equation}\label{rep4} \aligned {} [{L}_n,h_r]&=-rh_{n+r},\\ [{L}_m,{L}_n]&=(m-n){L}_{m+n}+\frac{m^3-m}{12}\delta_{m+n,0} .\endaligned \end{equation} Morover, since $$[d_m,h_{n-k}h_k]=[d_m,h_{n-k}]h_k+h_{n-k}[d_m,h_k]=[{L}_m,h_{n-k}]h_k+h_{n-k}[{L}_m,h_k]=[{L}_m,h_{n-k}h_k],$$ we see that \begin{equation}\label{rep4'}[d_m,{L}_n]=[{L}_m,{L}_n]\end{equation}
By \cite{LZ3}, for any $z\in{\mathbb C}$ and $H\in\mathcal{R}_{\bar {\mathcal{H}}}$ with the action of ${\bar{{\bf c}}}_3$ as a nonzero scalar $\ell$, we can give $H$ a $\bar {\mathfrak{D}}$-module structure (denoted by $H(z)^{\bar {\mathfrak{D}}}$) via the following map \begin{align} d_n&\mapsto \bar{L}_n= \frac{1}{2\ell}\sum_{k\in{\mathbb Z}}:h_{n-k}h_k:+\frac{(n+1)z}{\ell}h_n,\quad\forall n\in{\mathbb Z},\label{rep1-untw}\\ h_r&\mapsto h_r,\quad\forall r\in{\mathbb Z}, \quad {\bar{{\bf c}}_1} \mapsto 1-\frac{12z^2}{\ell}, \,\quad {\bar{{\bf c}}_2} \mapsto z, \quad {\bar{{\bf c}}_3}\mapsto \ell,\label{rep2-untw} \end{align} where the normal order is define as
$$\normOrd{h_{r}h_s}=\normOrd{h_{s}h_r}=h_{r}h_s,\text{ if } r\le s.$$
According to (8.7.9), (8.7.13) in \cite{FLM} and by some simple computation, we deduce that for all $m,n,r\in{\mathbb Z}$, \begin{equation}\label{rep3-untw} \aligned {} [\bar{L}_m,h_r]&=-rh_{m+r}+\delta_{m+r,0}(m^2+m)z, \\ [\bar{L}_m,\bar{L}_n]&=(m-n)\bar L_{m+n}+\frac{m^3-m}{12}\delta_{m+n,0}(1-\frac{12z^2}{\ell}) .\endaligned \end{equation} \begin{rem} The vertex operator algebra interpretation of the operators (\ref{rep1-untw}) will be given later in (\ref{rep1-voa}). Then relations (\ref{rep3-untw}) can be obtained using commutator formula, similarly as in \cite{AR}. \end{rem} Moreover, since $$[d_m,h_{n-k}h_k]=[d_m,h_{n-k}]h_k+h_{n-k}[d_m,h_k]=[\bar{L}_m,h_{n-k}]h_k+h_{n-k}[\bar{L}_m,h_k]=[\bar{L}_m,h_{n-k}h_k],$$ we see that \begin{equation}\label{rep3'}[d_m,\bar{L}_n]=[\bar{L}_m,\bar{L}_n].\end{equation}
For convenience, we define the following subalgebras of $\mathfrak{D}$. For any $m\in{\mathbb N}, n\in{\mathbb Z}$, set \begin{equation}\label{Natations} \begin{split} &\mathfrak{D}^{(m, n)}=\sum_{i\in{\mathbb N}}{\mathbb C} d_{m+i}\oplus {\mathbb C} h_{n+i+\frac12}\oplus{\mathbb C}{\bf c}_1\oplus {\mathbb C}{\bf c}_2, \\ &\mathfrak{D}^{(m,-\infty)}=\sum_{i\in{\mathbb N}}{\mathbb C} d_{m+i}\oplus \sum_{i\in{\mathbb Z}}{\mathbb C} h_{i+\frac12}+{\mathbb C}{\bf c}_1+{\mathbb C}{\bf c}_2,\\ &\mathfrak{Vir}^{(m)}= \sum_{i\in{\mathbb N}}{\mathbb C} d_{m+i}\oplus{\mathbb C}{\bf c}_1,\\ &\mathfrak{Vir}_{\geq m}= \sum_{i\in{\mathbb N}}{\mathbb C} d_{m+i},\\ &\mathfrak{Vir}_{\leq 0}= \sum_{i\in{\mathbb N}}{\mathbb C} d_{-i},\\
&\mathfrak{Vir}_+=\text{span}\{{\bf c}_1, d_i: i\ge 0\},\\
&\mathcal{H}^{(n)}=\sum_{i\in{\mathbb N}}{\mathbb C} h_{n+i+\frac12}\oplus{\mathbb C}{\bf c}_2,\\
&\mathcal{H}_{\ge n}=\sum_{i\in{\mathbb N}}{\mathbb C} h_{n+i+\frac12}. \end{split} \end{equation}
Similarly, we define the subalgebras of $\bar {\mathfrak{D}}$ as following: for $m\in{\mathbb N}, n\in{\mathbb Z}$, set \begin{equation}\label{Natations'} \begin{split} &\bar {\mathfrak{D}}^{(m, n)}=\sum_{i\in{\mathbb N}}{\mathbb C} d_{m+i}\oplus {\mathbb C} h_{n+i}\oplus{\mathbb C}{\bar{{\bf c}}}_1\oplus {\mathbb C}{\bar{{\bf c}}}_2+{\mathbb C}{\bar{{\bf c}}}_3, \\ &\bar {\mathfrak{D}}^{(m,-\infty)}=\sum_{i\in{\mathbb N}}{\mathbb C} d_{m+i}\oplus \sum_{i\in{\mathbb Z}}{\mathbb C} h_{i}+{\mathbb C}{\bar{{\bf c}}}_1\oplus {\mathbb C}{\bar{{\bf c}}}_2+{\mathbb C}{\bar{{\bf c}}}_3,\\ &\mathfrak{Vir}^{(m)}= \sum_{i\in{\mathbb N}}{\mathbb C} d_{m+i}\oplus{\mathbb C}{\bar{{\bf c}}}_1,\\ &\mathfrak{Vir}_{\geq m}= \sum_{i\in{\mathbb N}}{\mathbb C} d_{m+i},\\ &\mathfrak{Vir}_{\leq 0}= \sum_{i\in{\mathbb N}}{\mathbb C} d_{-i},\\
&\mathfrak{Vir}_+=\text{span}\{{\bar{{\bf c}}}_1, d_i: i\ge 0\},\\
&\bar {\mathcal{H}}^{(n)}=\sum_{i\in{\mathbb N}}{\mathbb C} h_{n+i}\oplus{\mathbb C}{\bar{{\bf c}}}_3,\\
&\bar {\mathcal{H}}_{\ge n}=\sum_{i\in{\mathbb N}}{\mathbb C} h_{n+i}. \end{split} \end{equation} Note that we use the same notations $\mathfrak{Vir}^{(m)}, \mathfrak{Vir}_{\geq m}, \mathfrak{Vir}_{\leq 0}, \mathfrak{Vir}_+$ to denote the subalgebras of $\mathfrak{D}$ and of $\bar {\mathfrak{D}}$ since there will be no ambiguities in later contexts.
Denote by $\mathbb{M}$ the set of all infinite vectors of the form $\mathbf{i}:=(\ldots, i_2, i_1)$ with entries in $\mathbb N$, satisfying the condition that the number of nonzero entries is finite. We can make
$(\mathbb{M}, +)$ a monoid by
$$(\ldots, i_2, i_1)+(\ldots, j_2, j_1)=(\ldots, i_2+j_2, i_1+j_1).$$
Let $\mathbf{0}$ denote the element $(\ldots, 0, 0)\in\mathbb{M}$ and for $i\in\mathbb Z_+$ let $\epsilon_i=(\ldots,0,1,0,\ldots,0)\in\mathbb{M}$, where $1$ is in the $i$'th position from right. For any $\mathbf{i}\in\mathbb{M}$ we define \begin{equation} {\rm w}(\mathbf{i})=\sum_{n\in\mathbb Z_+}i_n\cdot n,\,\,\,
\label{length}\end{equation}
Let $\prec$ be the {\bf reverse lexicographic} total order on $\mathbb{M}$, that is, for any ${\mathbf{i}},{\mathbf{j}}\in \mathbb{M}$, \begin{equation*} {\mathbf{j}} \prec {\mathbf{i}}\Longleftrightarrow {\rm\ there\ exists\ } r\in{\mathbb N} {\rm\ such\ that\ } j_r<i_r {\rm\ and\ } j_s=i_s,\ \forall\ 1\leq s<r. \end{equation*}
We extend the above total order on $\mathbb{M} \times \mathbb{M}$, that is, for all ${\mathbf{i}},{\mathbf{j}},{\mathbf{k}},{\mathbf{l}}\in \mathbb{M}$, $$({\mathbf{i}},{\mathbf{j}}) \prec ({\mathbf{k}},{\mathbf{l}})\iff \aligned ({\mathbf{j}},{\rm w}({\mathbf{j}}), {\rm w}({\mathbf{i}})+{\rm w}({\mathbf{j}}))&\prec({\mathbf{l}}, {\rm w}({\mathbf{l}}), {\rm w}({\mathbf{k}})+{\rm w}({\mathbf{l}})), \text{ or }\\ ({\mathbf{j}},{\rm w}({\mathbf{j}}), {\rm w}({\mathbf{i}})+{\rm w}({\mathbf{j}}))&\prec({\mathbf{l}}, {\rm w}({\mathbf{l}}), {\rm w}({\mathbf{k}})+{\rm w}({\mathbf{l}})), \text{ and } {\mathbf{i}}\prec{\mathbf{k}} .\endaligned$$
Now we define another total order $\prec'$ on $\mathbb{M} \times \mathbb{M}$: for all ${\mathbf{i}},{\mathbf{j}},{\mathbf{k}},{\mathbf{l}}\in \mathbb{M}$, $$({\mathbf{i}},{\mathbf{j}}) \prec' ({\mathbf{k}},{\mathbf{l}}) \iff ({\mathbf{j}},{\mathbf{i}}) \prec ({\mathbf{l}},{\mathbf{k}}) $$ The symbols $\preceq$ and $\preceq'$ have the obvious meanings.
It is not hard to verify that
$$(a, b) \preceq (c,d)\,\,\, \&\,\,\, (c',d')\prec(a',b') \implies (a-a', b-b') \prec (c-c',d-d'),$$ provided $(a, b) , (c,d)\,\,\, (c',d'),(a',b'), (a-a', b-b') , (c-c',d-d')\in \mathbb{M} \times \mathbb{M},$ where the difference is the corresponding entry difference.
For $n\in {\mathbb Z}$, let $V$ be an simple ${\mathfrak{D}}^{(0,-n)}$-module. According to the $\mathrm{PBW}$ Theorem, every nonzero element $v\in \mathrm{Ind}_{{\mathfrak{D}}^{(0,-n)}}^ {\mathfrak{D}}(V)$ can be uniquely written in the following form \begin{equation}\label{def2.1} v=\sum_{\mathbf{i}, \mathbf{k}\in\mathbb{M}} h^{\mathbf{i}}d^{\mathbf{k}} v_{\mathbf{i}, \mathbf{k}}, \end{equation} where $$ h^{\mathbf{i}}d^{\mathbf{k}}=\ldots h_{-2-n+\frac{1}{2}}^{i_2} h_{-1-n+\frac{1}{2}}^{i_1}\ldots d_{-2}^{k_2} d_{-1}^{k_1}\in U(\mathfrak D^{-}),v_{\mathbf{i}, \mathbf{k}}\in V,$$ and only finitely many $v_{\mathbf{i}, \mathbf{k}}$ are nonzero. For any nonzero $v\in\mathrm{Ind}(V)$ as in \eqref{def2.1}, we will use the following notations for later use:
(1) Denote by $\mathrm{supp}(v)$ the set of all $(\mathbf{i},\mathbf{k})\in \mathbb{M}\times \mathbb{M}$ such that $v_{\mathbf{i}, \mathbf{k}}\neq0$.
(2) Denote by $$\mathrm{w}(v) =\mathrm{max}\{{\rm w}(\mathbf{i})+{\rm w}({\bf k}): (\mathbf{i}, {\bf k})\in \mathrm{supp}(v)\}, $$ called the {\bf length} of $v$.
(3) Denote by $\deg(v)$ to be the largest element in $\mathrm{supp}(v)$ with respect to the total order $\prec$.
(4) Denote by $\deg'(v)$ to be the largest element in $\mathrm{supp}(v)$ with respect to the total order $\prec'$.
We first recall from \cite{MZ2} the classification for simple restricted $\mathfrak{Vir}$-modules.
\begin{theo} \label{Vir-modules} Any simple restricted $\mathfrak{Vir}$-module is a highest weight module, or isomorphic to $ {\rm Ind}_{\mathfrak{Vir}_+}^{\mathfrak{Vir}}V$ for an simple $\mathfrak{Vir}_+$-module $V$ such that for some $k\in \mathbb Z_+$, \begin{enumerate}[$($a$)$] \item $d_k$ acts injectively on $V$; \item$d_iV=0$ for all $i>k$. \end{enumerate} \end{theo}
Simple restricted $\mathfrak{D}$-modules with level $0$ are classified in \cite{LPXZ} by the following two theorems.
\begin{theo}\label{simple-theo}
Let $V$ be a simple ${\mathfrak{D}}^{(0,-n)}$-module for some $n\in \mathbb{Z}_+$ and $c\in {\mathbb C}$ such that ${\bf c}_1v=cv, {\bf c}_2v=0$ for any $v\in V$. Assume that there exists an integer $k\ge-n$ satisfying the following two conditions:
\begin{itemize}
\item[(a)] the action of $h_{k+\frac12}$ on $V$ is bijective;
\item[(b)] $h_{m+\frac{1}{2}}V=0=d_{m+n}V$ for all $m> k$.
\end{itemize}
Then the induced ${\mathfrak{D}}$-module $\mathrm{Ind}_{{\mathfrak{D}}^{(0,-n)}}^ {\mathfrak{D}} (V)$ is simple. \end{theo}
\begin{theo}\label{MT}
Every simple restricted ${\mathfrak{D}}$-module of level 0 is isomorphic to a restricted $\mathfrak{Vir}$-module with $\mathcal{H} S=0$, or $S\cong\mathrm{Ind}_{{\mathfrak{D}}^{(0,-n)}}^ {\mathfrak{D}} (V)$ for some $n\in \mathbb{N}$ and a simple ${\mathfrak{D}}^{(0,-n)}$-module $V$ as in Theorem \ref{simple-theo}. \end{theo}
Actually the simple ${\mathfrak{D}}^{(0,-n)}$-module $V$ can be considered as a simple module over a finite dimensional solvable Lie algebra
${\mathfrak{D}}^{(0,-n)}/{\mathfrak{D}}^{(t+n+1, t-n)}$ for some $ t\in{\mathbb Z}_+$ and injective action of $h_{t+\frac12}$ on $V$.
For simple restricted $\bar \mathfrak{D}$-modules with level $0$, we know the following results from \cite{CG1}.
\begin{theo} Let $n\in{\mathbb N}$ and $V$ be a simple module over $\bar \mathfrak{D}^{(0,-n)}$ or over $\bar \mathfrak{D}^{(0,-\infty)}$ with $\ell =0$, $h_0=\mu, \bar{\bf c}_2=z$. If there exists $k\in{\mathbb N}$ such that
(a)\begin{equation*} \begin{cases}h_k \,\text{acts\, injectively \,on\, V},& \text{if}\, k\ne 0,\\ \mu+(1-r)z\neq 0,\forall r\in{\mathbb Z}\setminus\{0\}, &\text{if}\,k=0; \end{cases} \end{equation*}
(b) $h_iV=d_jV=0$ for all $i>k$ and $j>k+n$.
\noindent then
(1) $\text{\rm Ind}(V)$ is a simple $\bar \mathfrak{D}$-module;
(2) $h_i, d_j$ act locally nilpotently on $\text{\rm Ind}(V)$ for all $i>k$ and $j>k+n$.
\end{theo}
Now we generalize Theorem 12 in \cite{LZ3} as follows.
Let ${\mathfrak g}={\mathfrak a}\ltimes{\mathfrak b}$ be a Lie algebra where ${\mathfrak a}$ is a Lie subalgebra of ${\mathfrak g}$ and ${\mathfrak b}$ is an ideal of ${\mathfrak g}$. Let $M$ be a ${\mathfrak g}$-module with a ${\mathfrak b}$-submodule $H$ so that the ${\mathfrak b}$-submodule structure on $H$ can be extended to a ${\mathfrak g}$-module structure on $H$. We denote this ${\mathfrak g}$-module by $H^{\mathfrak g}$. For any ${\mathfrak a}$-module $U$, we can make it into a ${\mathfrak g}$-module by ${\mathfrak b}U=0$. We denote this ${\mathfrak g}$-module by $U^{\mathfrak g}$.
\begin{theo}\label{generalize} Let ${\mathfrak g}={\mathfrak a}\ltimes{\mathfrak b}$ be a countable dimensional Lie algebra where ${\mathfrak a}$ is a Lie subalgebra of ${\mathfrak g}$ and ${\mathfrak b}$ is an ideal of ${\mathfrak g}$.
Let $M$ be a simple ${\mathfrak g}$-module with a simple ${\mathfrak b}$-submodule $H$ so that an $H^{\mathfrak g}$ exists. Then $M\cong H^{\mathfrak{g}}\otimes U^{\mathfrak{g}}$ as $\mathfrak{g}$-modules for some simple ${\mathfrak a}$-module $U$. \end{theo}
\begin{proof} Define the one-dimensional ${\mathfrak b}$-module ${\mathbb C} v_0$ by ${\mathfrak b} v_0=0$. Then $H\cong H\otimes {\mathbb C} v_0$ as ${\mathfrak b}$-modules. Now from Lemma 8 in \cite{LZ3}, we have $$\text{\rm Ind}_{{\mathfrak b}}^{\mathfrak{g}}H\cong \text{\rm Ind}_{{\mathfrak b}}^{\mathfrak{g}}\left(H\otimes {\mathbb C} v_0\right)\cong H^{\mathfrak{g}}\otimes \text{\rm Ind}_{{\mathfrak b}}^{\mathfrak{g}}{\mathbb C} v_0 .$$ Note that $\text{\rm Ind}_{{\mathfrak b}}^{\mathfrak{g}}{\mathbb C} v_0 \cong W^{\mathfrak{g}}$ for the universal ${\mathfrak a}$-module $W$. Since $M$ is a simple quotient of $\text{\rm Ind}_{{\mathfrak b}}^{\mathfrak{g}}H$, from Theorem 7 in \cite{LZ3} we know that there is a simple quotient ${\mathfrak a}$-module $U$ of $W$ such that $M\cong H^{\mathfrak{g}}\otimes U^{\mathfrak{g}}$ as $\mathfrak{g}$-modules. Now the theorem follows.\end{proof}
{\bf Remark.} This theorem has particular meaning for ${\mathfrak g}={\mathfrak a}\oplus {\mathfrak b}$ since $H^{\mathfrak g}$ automatically exists (see for example \cite{Li}). Also, Theorem \ref{generalize} holds for associative algebras.
Applying the above theorem to our mirror Heisenberg-Virasoro algebra $\mathfrak{D}=\mathfrak{Vir}\ltimes \mathcal{H}$ and twisted Heisenberg-Virasoro algebra $\bar {\mathfrak{D}}=\mathfrak{Vir}\ltimes (\bar {\mathcal{H}}+{\mathbb C} \bar{{\bf c}}_2)$, we have the following results.
\begin{cor}\label{tensor} Let $V$ be a simple ${\mathfrak{D}}$-module with nonzero action of ${\bf c}_2$. Then $V\cong H^{\mathfrak{D}}\otimes U^{\mathfrak{D}}$ as a $\mathfrak{D}$-module for some simple module $H\in{\mathcal R}_{\mathcal{H}}$ and some simple $\mathfrak{Vir}$-module $U$ if and only if $ V$ contains
a simple $\mathcal{H}$-submodule $H\in{\mathcal R}_{\mathcal{H}}$.
\end{cor} \begin{proof} The sufficiency follows from Theorem \ref{generalize}; and the necessity follows from that $H\otimes u$ is a simple $\mathcal{H}$-submodule of $H^{\mathfrak{D}}\otimes U^{\mathfrak{D}}$ for any nonzero $u\in U$. \end{proof}
\section{Induced modules over the mirror Heisenberg-Virasoro algebra $\mathfrak{D}$}
In this section, we construct some simple restricted $\mathfrak{D}$-modules induced from some simple ones over some subalgebras $\mathfrak{D}^{(0,-n)}$ for $n\in{\mathbb Z}_+$. For that, we need the following formulas in $U(\mathfrak{D})$ which can be shown by induction on $t$: let $i,j_s\in{\mathbb Z},1\le s\le t$ with $j_1\le j_2\le\cdots\le j_t$, \begin{equation}\label{bracket3.1} [h_{i-\frac{1}{2}},h_{j_1+\frac{1}{2}}h_{j_2+\frac{1}{2}}\cdots h_{j_t+\frac{1}{2}}]=\sum_{1\le s\le t}\delta_{i+j_s,0}(i-\frac{1}{2}) {\bf{c}}_2 h_{j_1+\frac{1}{2}}\cdots \hat{h}_{j_{s}+\frac{1}{2}}\cdots h_{j_t+\frac{1}{2}}, \end{equation} \begin{equation}\label{bracket3.2} \begin{aligned} &[d_{i},h_{j_1+\frac{1}{2}}h_{j_2+\frac{1}{2}}\cdots h_{j_t+\frac{1}{2}}]=\sum_{1\le s\le t}(-j_s-\frac{1}{2}) h_{j_1+\frac{1}{2}}\cdots\hat{h}_{j_s+\frac{1}{2}} \cdots h_{j_t+\frac{1}{2}}{h_{i+j_{s}+\frac{1}{2}}}\\ &\hskip 20pt +\sum_{1\le s_1<s_2\le t}(-j_{s_1}-\frac{1}{2})(i+j_{s_1}+\frac{1}{2})\delta_{i+j_{s_1}+j_{s_2}+1,0}{\bf{c}}_2 h_{j_1+\frac{1}{2}}\cdots \hat{h}_{j_{s_1}+\frac{1}{2}}\cdots \hat{h}_{j_{s_2}+\frac{1}{2}} \cdots h_{j_t+\frac{1}{2}}, \end{aligned} \end{equation} \begin{equation}\label{bracket3.3} \begin{aligned} &[h_{i-\frac{1}{2}},d_{j_1}d_{j_2}\cdots d_{j_t}]=\sum_{1\le s\le t}(i-\frac{1}{2})d_{j_1}\cdots \hat{d_{j_s}}\cdots d_{j_t}h_{i+j_s-\frac{1}{2}}\\ &\hskip 20pt +\sum_{1\le s_1<s_2\le t}a_{s_1,s_2}d_{j_1}\cdots \hat{d}_{j_{s_1}}\cdots\hat{d}_{j_{s_2}}\cdots d_{j_t}h_{i+j_{s_1}+j_{s_2}-\frac{1}{2}}+\cdots+ a_{1,2,\cdots,t}h_{i+j_1+j_2+\cdots+j_t-\frac{1}{2}},\end{aligned} \end{equation} \begin{equation}\label{bracket3.4} \begin{aligned} &[d_{i},d_{j_1}d_{j_2}\cdots d_{j_t}]=\sum_{1\le s\le t}(i-j_s)d_{j_1}\cdots \hat{d_{j_s}}\cdots d_{j_t}\tilde{d}_{i+j_s}\\ &\hskip 30pt +\sum_{1\le s_1<s_2\le t}b_{s_1,s_2}d_{j_1}\cdots \hat{d}_{j_{s_1}}\cdots\hat{d}_{j_{s_2}}\cdots d_{j_t}\tilde{d}_{i+j_{s_1}+j_{s_2}}+\cdots+ b_{1,2,\cdots,t}\tilde{d}_{i+j_1+j_2+\cdots+j_t},\end{aligned} \end{equation} where $\hat{h}_{j_{s}+\frac{1}{2}},\hat{d}_{j_s}$ mean that $h_{j_{s}+\frac{1}{2}}, d_{j_s}$ are deleted in the corresponding products, $a_{s_1,s_2},\cdots,$ $a_{1,2,\cdots,t},$ $b_{s_1,s_2},\cdots,b_{1,2,\cdots,t}\in {\mathbb C}$, and $\tilde{d}_{i+j_1+\cdots+j_s}=d_{i+j_1+\cdots+j_s}+\frac{j_{s}^2-1}{24}\delta_{i+j_1+\cdots+j_s,0}{\bf{c}}_1,1\leq s\leq t.$
We are now in the position to present the following main result in this section.
\begin{theo} \label{thmmain1.1} Let $k\in{\mathbb Z}_+$ and $ n\in{\mathbb Z}$ with $k\ge n$. Let $V$ be a simple $\mathfrak{D}^{(0,-n)}$-module with level $\ell \not=0$ such that
there exists $l\in{\mathbb N}$ satisfying both conditions: \begin{enumerate}[$(a)$] \item $h_{k-\frac{1}{2}}$ acts injectively on $V$; \item $h_{i -\frac{1}{2}} V=d_j V=0$ for all $i>k$ and $j>l$. \end{enumerate} Then $\text{\rm Ind}^{\mathfrak{D}}_{\mathfrak{D}^{(0,-n)}}(V)$ is a simple $\mathfrak{D}$-module if one of the following conditions holds: \begin{itemize} \item[\rm(1)] $k=n$, $l\geq 2n$, and $d_l$ acts injectively on $V$ ; \item[\rm(2)] $k>n$, $k+n\geq 2$, and $l=n+k-1$. \end{itemize} \end{theo}
Theorem \ref{thmmain1.1} follows from Lemmas \ref{main1'}-\ref{main4} directly.
\begin{lem}\label{main1'}Let $n\in{\mathbb Z}_+$ and $V$ be a $\mathfrak{D}^{(0,-n)}$-module such that
$h_{n-\frac{1}{2}}$ acts injectively on $V$, and $h_{i -\frac{1}{2}} V=0$ for all $i>n$. For any $v\in \text{\rm Ind}(V)\setminus V$, let $\deg(v)=({\mathbf{i}},{\mathbf{j}})$ .
If ${\mathbf{i}}\not=\bf0$, then $\deg(h_{p+n-\frac{1}{2}}v)=({\mathbf{i}}-\epsilon_p,{\mathbf{j}})$ where $p=\min\{s:i_s\neq 0\}$. \end{lem}
\begin{proof} Write $v$ in the form of \eqref{def2.1} and let $({\mathbf{k}},{\mathbf{l}})\in\mathrm{supp}(v)$.
Noticing that $h_{p+n-\frac{1}{2}}V=0$, we have $$h_{p+n-\frac{1}{2}}h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}=[h_{p+n-\frac{1}{2}},h^{{\mathbf{k}}}]d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}+h^{{\mathbf{k}}}[h_{p+n-\frac{1}{2}},d^{{\mathbf{l}}}]v_{{\mathbf{k}},{\mathbf{l}}}.$$
First we consider the term $[h_{p+n-\frac{1}{2}},h^{{\mathbf{k}}}]d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$ which is zero if $k_p=0$. In the case that $k_p>0$, since the level $\ell\ne 0$, it follows from (\ref{bracket3.1}) that $[h_{p+n-\frac{1}{2}},h^{{\mathbf{k}}}]=\lambda h^{{\mathbf{k}}-\epsilon_p}$ for some $\lambda\in{\mathbb C}^*$. So $$\text{deg}([h_{p+n-\frac{1}{2}},h^{{\mathbf{k}}}]d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}})=({\mathbf{k}}-\epsilon_p,{\mathbf{l}})\preceq ({\mathbf{i}}-\epsilon_p,{\mathbf{j}}),$$ where the equality holds if and only if $({\mathbf{k}},{\mathbf{l}})=({\mathbf{i}},{\mathbf{j}})$.
Now we consider the term $h^{{\mathbf{k}}}[h_{p +n-\frac{1}{2}},d^{{\mathbf{l}}}]v_{{\mathbf{k}},{\mathbf{l}}}$ which is by (\ref{bracket3.3}) a linear combination of some vectors in the form $h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}h_{p+n-\frac{1}{2}-j}v_{{\mathbf{k}},{\mathbf{l}}}$ with $j\in{\mathbb Z}_+$ and ${\rm{w}} ({\mathbf{l}}_j)={\rm{w}}({\mathbf{l}})-j$. If $h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}h_{p+n-\frac{1}{2}-j}v_{{\mathbf{k}},{\mathbf{l}}}\ne0$, we denote $\deg(h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}h_{p+n-\frac{1}{2}-j}v_{{\mathbf{k}},{\mathbf{l}}})=({\mathbf{k}}^*,{\mathbf{l}}^*)$. We will show that \begin{equation}\label{Eq3.5}({\mathbf{k}}^*,{\mathbf{l}}^*) \prec ({\mathbf{i}}-\epsilon_p,{\mathbf{j}}).\end{equation} We have four different cases to consider.
(a) $j<p$. Then $p+n-j>n$ and $h_{p+n-\frac{1}{2}-j}v_{{\mathbf{k}},{\mathbf{l}}}=0$. Hence $h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}h_{p+n-\frac{1}{2}-j}v_{{\mathbf{k}},{\mathbf{l}}}=0$.
(b) $j=p$. Noting that $h_{n-\frac{1}{2}}$ acts injectively on $V$, we see $({\mathbf{k}}^*,{\mathbf{l}}^*)=({\mathbf{k}}, {\mathbf{l}}_p)$ and ${\rm{w}}({\mathbf{k}}^*)+{\rm{w}({\mathbf{l}}^*)=({\mathbf{k}})+\rm{w}({\mathbf{l}})-}p$ with ${\rm{w}}({\mathbf{l}}_p)={\rm{w}({\mathbf{l}})}-p<\mathrm{w}({\mathbf{l}}).$
If ${\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})}<{\rm{w}({\mathbf{i}})+\rm{w}({\mathbf{j}})}$, then $({\mathbf{k}}^*,{\mathbf{l}}^*)\prec ({\mathbf{i}}-\epsilon_p,{\mathbf{j}}).$
If ${\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})}={\rm{w}({\mathbf{i}})+\rm{w}({\mathbf{j}})}$, then there is $\tau\in \mathbb{M}$ such that ${\rm w}(\tau)=p$ and ${\mathbf{l}}_p={\mathbf{l}}-\tau$. Since $ (\epsilon_p, \bf0)\prec (\bf0, \tau)$ and $({\mathbf{k}},{\mathbf{l}})\preceq ({\mathbf{i}},{\mathbf{j}})$, we see that $$({\mathbf{k}}^* ,{\mathbf{l}}^*)=({\mathbf{k}} ,{\mathbf{l}})-(\bf0, \tau) \prec ({\mathbf{i}} ,{\mathbf{j}})- (\epsilon_p,\bf0)=
({\mathbf{i}}-\epsilon_p,{\mathbf{j}}).$$
In both cases, (\ref{Eq3.5}) holds.
(c) $p<j<2n+p$. Then $h_{p+n-\frac{1}{2}-j}\in \mathfrak{D}^{(0,-n)}$ and $h_{p+n-\frac{1}{2}-j}v_{{\mathbf{k}},{\mathbf{l}}}\in V.$ So $$ {\rm{w}({\mathbf{k}}^*)+\rm{w}({\mathbf{l}}^*)}= {\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})}-j<{\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})}-p$$ and (\ref{Eq3.5}) holds.
(d) $j\ge 2n+p$. Then $p+n-\frac{1}{2}-j<-n+\frac{1}{2}$. Assume $p+n-\frac{1}{2}-j=-s-n+\frac{1}{2}$ for some $s\in {\mathbb Z}_+$, that is, $-j+s= -2n-p+1<-p$. Clearly, the corresponding vector $h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}h_{p+n-\frac{1}{2}-j}v_{{\mathbf{k}},{\mathbf{l}}}$ can be written in the form $$h^{{\mathbf{k}}}h_{-s-n+\frac{1}{2}}d^{{\mathbf{l}}_j}v_{{\mathbf{k}},{\mathbf{l}}}+\text{lower\, terms},$$ which means $$ {\rm{w}({\mathbf{k}}^*)+\rm{w}({\mathbf{l}}^*)}= {\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})}-j+s<{\rm{w}}({\mathbf{k}})+{\rm{w}}({\mathbf{l}})-p,$$ and hence (\ref{Eq3.5}) holds.
In conclusion, $\text{deg}(h_{p+n-\frac{1}{2}}h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}) \preceq ({\mathbf{i}}-\epsilon_p,{\mathbf{j}}),$
where the equality holds if and only if $({\mathbf{k}},{\mathbf{l}})=({\mathbf{i}},{\mathbf{j}})$, that is, $\deg(h_{p+n-\frac{1}{2}}v)=({\mathbf{i}}-\epsilon_p,{\mathbf{j}})$.\end{proof}
\begin{lem}\label{main1} Let $n\in{\mathbb Z}_+$ and $V$ be a $\mathfrak{D}^{(0,-n)}$-module satisfying Conditions (a), (b) and (1) in Theorem \ref{thmmain1.1}. If $v\in \text{\rm Ind}(V)\setminus V$ with $\deg(v)=(\bf0,{\mathbf{j}})$ , then $\deg(d_{q+l}v)=({\mathbf{0}},{\mathbf{j}}-\epsilon_q)$ where $q=\min\{s:j_s\neq 0\}$. \end{lem}
\begin{proof} Write $v$ in the form of \eqref{def2.1} and let $({\mathbf{k}},{\mathbf{l}})\in\mathrm{supp}(v)$.
Since $d_{q+l}V=0$, we have $$d_{q+l}h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}=[d_{q+l},h^{{\mathbf{k}}}]d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}+h^{{\mathbf{k}}}[d_{q+l},d^{{\mathbf{l}}}]v_{{\mathbf{k}},{\mathbf{l}}}.$$
We first consider the degree of $h^{{\mathbf{k}}}[d_{q+l},d^{{\mathbf{l}}}]v_{{\mathbf{k}},{\mathbf{l}}}$ with $d_{q+l}h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}\neq 0$. Clearly, by (\ref{bracket3.4}) we see that $h^{{\mathbf{k}}}[d_{q+l},d^{{\mathbf{l}}}]v_{{\mathbf{k}},{\mathbf{l}}}$ is is a linear combination of some vectors of the forms $h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}d_{q+l-j}v_{{\mathbf{k}},{\mathbf{l}}}$ ,$j\in{\mathbb Z}_+$ and $h^{{\mathbf{k}}}d^{{\mathbf{l}}_{q+l}}v_{{\mathbf{k}},{\mathbf{l}}}$ where ${\rm{w}}({\mathbf{l}}_j)={\rm{w}}({\mathbf{l}})-j$. Clearly, $\deg(h^{{\mathbf{k}}}d^{{\mathbf{l}}_{q+l}}v_{{\mathbf{k}},{\mathbf{l}}})=({\mathbf{k}},{\mathbf{l}}_{q+l})$ has weight $${\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})}-q-l<{\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})}-q\le {\rm{w}}({\mathbf{j}})-q,$$ so $\deg(h^{{\mathbf{k}}}d^{{\mathbf{l}}_{q+l}}v_{{\mathbf{k}},{\mathbf{l}}})\prec (\bf0,{\mathbf{j}}-\epsilon_q)$. Then we need only to consider $h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}d_{q+l-j}v_{{\mathbf{k}},{\mathbf{l}}}$. Denote $\deg(h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}d_{q+l-j}v_{{\mathbf{k}},{\mathbf{l}}}) $ by $({{\mathbf{k}}},{{\mathbf{l}}}^*)$. We will show that
\begin{equation}\label{Eq3.7}({{\mathbf{k}}},{{\mathbf{l}}}^*) \preceq (\bf0,{\mathbf{j}}-\epsilon_q),\end{equation} where the equality holds if and only if $({\mathbf{k}},{\mathbf{l}})=(\bf0,{\mathbf{j}})$. We have four different cases to consider.
(i) $j<q$. Then $q+l-j>l$ and $h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}d_{q+l-j}v_{{\mathbf{k}},{\mathbf{l}}}=0$.
(ii) $j=q$. Then $q+l-j=l$. Since $d_l$ acts injectively on $V$, we see $ ({{\mathbf{k}}},{{\mathbf{l}}}^*)=({\mathbf{k}},{\mathbf{l}}_q)$ and $\mathrm{w}({\mathbf{k}})+\mathrm{w}({\mathbf{l}}^*) =\mathrm{w}({\mathbf{k}})+\mathrm{w}({\mathbf{l}})-q$. If
${\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})}<{\rm{w}(\bf0)+\rm{w}({\mathbf{j}})}$, then $ ({{\mathbf{k}}},{{\mathbf{l}}}^*) \prec (\bf0,{\mathbf{j}}-\epsilon_q).$ If ${\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})}={\rm{w}(\bf0)+\rm{w}({\mathbf{j}})}$, there is $\tau\in \mathbb{M}$ such that ${\rm w}(\tau)=q$ and ${\mathbf{l}}_q={\mathbf{l}}-\tau$. Then $(\bf0,\epsilon_q)\preceq (\bf0, \tau)$. Since $({\mathbf{k}},{\mathbf{l}})\preceq (\bf0,{\mathbf{j}})$, we see that $$({\mathbf{k}} ,{\mathbf{l}}^*)=({\mathbf{k}} ,{\mathbf{l}})-(\bf0, \tau) \preceq (\bf0 ,{\mathbf{j}})- (\bf0,\epsilon_q )=
(\bf0,{\mathbf{j}}-\epsilon_q).$$ In both cases we have that $$({{\mathbf{k}}},{{\mathbf{l}}}^*) \preceq (\bf0,{\mathbf{j}}-\epsilon_q),$$ where the equality holds if and only if $({\mathbf{k}},{\mathbf{l}})=(\bf0,{\mathbf{j}})$.
(iii) $q+1\le j\le q+l$. Then $0\le q+l-j\le l-1$ and $d_{q+l-j}v_{{\mathbf{k}},{\mathbf{l}}}\in V$. So if $h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}d_{q+l-j}v_{{\mathbf{k}},{\mathbf{l}}}\neq 0$, then $\mathrm{w}({\mathbf{k}})+\mathrm{w}({\mathbf{l}}^*) =\mathrm{w}({\mathbf{k}})+\mathrm{w}({\mathbf{l}})-j<\mathrm{w}({\mathbf{k}})+\mathrm{w}({\mathbf{l}})-q$.
(iv) $j>q+l$. Then $ q+l-j<0$. Clearly, $\mathrm{w}({\mathbf{l}}^*) =\mathrm{w}({\mathbf{l}}_j)+(j-q-l)=\mathrm{w}({\mathbf{l}}) -q-l$, and hence $$\mathrm{w}({\mathbf{k}})+\mathrm{w}({\mathbf{l}}^*) =\mathrm{w}({\mathbf{k}})+\mathrm{w}({\mathbf{l}})-q-l<\mathrm{w}({\mathbf{k}})+\mathrm{w}({\mathbf{l}})-q.$$
Therefore, we conclude that (\ref{Eq3.7}) holds, i.e., $\sum_{({\mathbf{k}},{\mathbf{l}})}h^{{\mathbf{k}}}[d_{q+l},d^{{\mathbf{l}}}]v_{{\mathbf{k}},{\mathbf{l}}}$ has degree $({\bf0,{\mathbf{j}}}-\epsilon_q)$.
Next we consider the degree of the nonzero vector $[d_{q+l},h^{{\mathbf{k}}}]d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$. By (\ref{bracket3.2}) we can see that $[d_{q+l},h^{{\mathbf{k}}}]d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$ is a linear combination of some vectors of the forms
$h^{{\mathbf{k}}_s}h_{q+l-s-n+\frac{1}{2}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}, s\in{\mathbb Z}_+$ and $h^{{\mathbf{k}}_{q+l+1-2n}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$, where $\mathrm{w}({\mathbf{k}}_s)=\mathrm{w}({\mathbf{k}})-s$. Noting that $l\geq 2n$, the degree of $h^{{\mathbf{k}}_{q+l+1-2n}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$ has weight $$\mathrm{w}({\mathbf{k}})-(q+l+1-2n)+\mathrm{w}({\mathbf{l}})<\mathrm{w}({\mathbf{k}})+\mathrm{w}({\mathbf{l}})-q.$$ So $$\deg(h^{{\mathbf{k}}_{q+l+1-2n}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}) \prec (\bf0,{\mathbf{j}}-\epsilon_q).$$
Next we will show that \begin{equation}\label{Eq3.8}\deg(h^{{\mathbf{k}}_s}h_{q+l-s-n+\frac{1}{2}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}) \prec (\bf0,{\mathbf{j}}-\epsilon_q),\end{equation} We have two different cases to consider.
(a) $s>q+l$. The degree of $h^{{\mathbf{k}}_s}h_{q+l-s-n+\frac{1}{2}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$ has weight
$$\mathrm{w}({\mathbf{k}}_s)+(s-q-l)+\mathrm{w}({\mathbf{l}})=\mathrm{w}({\mathbf{k}})+\mathrm{w}({\mathbf{l}})-q-l<\mathrm{w}({\mathbf{k}})+\mathrm{w}({\mathbf{l}})-q.$$
So, (\ref{Eq3.8}) holds in this case.
(b) $1\le s\le q+l$. We have $$h^{{\mathbf{k}}_s}h_{q+l-s-n+\frac{1}{2}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}=h^{{\mathbf{k}}_s}[h_{q+l-s-n+\frac{1}{2}}, d^{{\mathbf{l}}}]v_{{\mathbf{k}},{\mathbf{l}}}+h^{{\mathbf{k}}_s}d^{{\mathbf{l}}}h_{q+l-s-n+\frac{1}{2}}v_{{\mathbf{k}},{\mathbf{l}}}.$$ Noting that $h_{q+l-s-n+\frac{1}{2}}v_{{\mathbf{k}},{\mathbf{l}}}\in V$ (in particular, $h_{q+l-s-n+\frac{1}{2}}v_{{\mathbf{k}},{\mathbf{l}}}=0$ for $1\le s\le q+l-2n$), we see that if $h^{{\mathbf{k}}_s}d^{{\mathbf{l}}}h_{q+l-s-n+\frac{1}{2}}v_{{\mathbf{k}},{\mathbf{l}}}\neq 0$ for $q+l-2n+1\le s\le q+l$, its degree has weight $$\mathrm{w}({\mathbf{k}}_s)+\mathrm{w}({\mathbf{l}})=\mathrm{w}({\mathbf{k}})+\mathrm{w}({\mathbf{l}})-s<\mathrm{w}({\mathbf{k}})+\mathrm{w}({\mathbf{l}})-q. $$ Now we consider $\deg(h^{{\mathbf{k}}_s}[h_{q+l-s-n+\frac{1}{2}}, d^{{\mathbf{l}}}]v_{{\mathbf{k}},{\mathbf{l}}})$ which is denoted by $(\tilde{{\mathbf{k}}},\tilde{{\mathbf{l}}})$.
(b1) $1\le s\le q$, that is, $q+l-s-n\geq n$. Then $q+l-s-n+\frac{1}{2}=n+p-\frac{1}{2}$ for some $p\in{\mathbb Z}_+$ and hence $s+p=q+l-2n+1\geq q+1$. Thus, by the same arguments in proof of Lemma 3.2, we see $$ \mathrm{w}(\tilde{{\mathbf{k}}})+\mathrm{w}(\tilde{{\mathbf{l}}}) \le \mathrm{w}({\mathbf{k}}_s)+\mathrm{w}({\mathbf{l}})-p=\mathrm{w}({\mathbf{k}})-s+\mathrm{w}({\mathbf{l}})-p \leq \mathrm{w}({\mathbf{k}})+\mathrm{w}({\mathbf{l}})-q-1<\mathrm{w}({\mathbf{k}})+\mathrm{w}({\mathbf{l}})-q.$$
So, (\ref{Eq3.8}) holds in this case.
(b2) $q+1\le s \le q+l$. Then by (\ref{bracket3.3}) and the same arguments in proof of Lemma 3.2, we see $$\mathrm{w}(\tilde{{\mathbf{k}}})+\mathrm{w}(\tilde{{\mathbf{l}}})\le \mathrm{w}({\mathbf{k}}_s)+\mathrm{w}({\mathbf{l}}) =\mathrm{w}({\mathbf{k}})+\mathrm{w}({\mathbf{l}})-s\le \mathrm{w}({\mathbf{k}})+\mathrm{w}({\mathbf{l}})-q-1<\mathrm{w}({\mathbf{k}})+\mathrm{w}({\mathbf{l}})-q.$$
So, (\ref{Eq3.8}) holds in this case as well.
Therefore, $ \deg\big(d_{q+l}v)=({\mathbf{0}},{\mathbf{j}}-\epsilon_q)$, as desired.
\end{proof}
\begin{lem}\label{main3} Let $k\in{\mathbb Z}_+, n\in{\mathbb Z}$ with $ k\ge n$ and $k+n\ge2$, and let $V$ be a $\mathfrak{D}^{(0,-n)}$-module such that $h_{k-\frac{1}{2}}$ acts injectively on $V$, and $h_{i -\frac{1}{2}} V=0$ for all $i>k$. If $v\in \text{\rm Ind}(V)\setminus V$ with $\deg'(v)=({\mathbf{i}},{\mathbf{j}})$ and ${\mathbf{j}}\not=\bf0$, then $\deg'(h_{p+k-\frac{1}{2}}v)=({\mathbf{i}},{\mathbf{j}}-\epsilon_p)$ where $p=\min\{s:j_s\neq 0\}$. \end{lem}
\begin{proof}As in (\ref{def2.1}), write $v=\sum_{({\mathbf{k}},{\mathbf{l}})}h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$. Consider $\text{deg}'(h_{p+k-\frac{1}{2}}h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}})$ if $h_{p+k-\frac{1}{2}}h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}\ne 0.$ Noting that $h_{p+k-\frac{1}{2}}V=0,$ we see $$h_{p+k-\frac{1}{2}}h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}=[h_{p+k-\frac{1}{2}},h^{{\mathbf{k}}}]d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}+h^{{\mathbf{k}}}[h_{p+k-\frac{1}{2}},d^{{\mathbf{l}}}]v_{{\mathbf{k}},{\mathbf{l}}}.$$
First we consider the term $[h_{p+k-\frac{1}{2}},h^{{\mathbf{k}}}]d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$ which is zero if $k_{p'}=0$ for $p':=p+k-n$. In the case that $k_{p'}>0$, since the level $\ell\ne 0$, it follows from (\ref{bracket3.1}) that $[h_{p+k-\frac{1}{2}},h^{{\mathbf{k}}}]=\lambda h^{{\mathbf{k}}-\epsilon_{p'}}$ for some $\lambda\in{\mathbb C}^*$. Note that $({\mathbf{k}},{\mathbf{l}})\preceq' ({\mathbf{i}},{\mathbf{j}}),
(\bf0, \epsilon_p)\prec'(\epsilon_{p'},\bf0).$ So $$\text{deg}'([h_{p+k-\frac{1}{2}},h^{{\mathbf{k}}}]d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}})=({\mathbf{k}}-\epsilon_{p'},{\mathbf{l}})=({\mathbf{k}},{\mathbf{l}})- (\epsilon_{p'},\bf0)\prec' ({\mathbf{i}},{\mathbf{j}})- (\bf0,\epsilon_p)=({\mathbf{i}},{\mathbf{j}}-\epsilon_p).$$
Now we consider the term $h^{{\mathbf{k}}}[h_{p +k-\frac{1}{2}},d^{{\mathbf{l}}}]v_{{\mathbf{k}},{\mathbf{l}}}$ which is by (\ref{bracket3.3}) a linear combination of some vectors in the form $h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}h_{p+k-\frac{1}{2}-j}v_{{\mathbf{k}},{\mathbf{l}}}$ with $j\in{\mathbb Z}_+$ and ${\rm{w}} ({\mathbf{l}}_j)={\rm{w}}({\mathbf{l}})-j$. We will show that \begin{equation}\label{Eq3.6}\deg'(h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}h_{p+k-\frac{1}{2}-j}v_{{\mathbf{k}},{\mathbf{l}}})=({\mathbf{k}}^*,{\mathbf{l}}^*) \preceq' ({\mathbf{i}},{\mathbf{j}}-\epsilon_p),\end{equation}
where the equality holds if and only if $({\mathbf{k}},{\mathbf{l}})=({\mathbf{i}},{\mathbf{j}})$.
We have four different cases to consider.
(a) $j<p$. Then $p+k-j>n$ and $h_{p+k-\frac{1}{2}-j}v_{{\mathbf{k}},{\mathbf{l}}}=0$. Hence $h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}h_{p+k-\frac{1}{2}-j}v_{{\mathbf{k}},{\mathbf{l}}}=0$.
(b) $j=p$. Noting that $h_{k-\frac{1}{2}}$ acts injectively on $V$, we see $({\mathbf{k}}^*,{\mathbf{l}}^*)=({\mathbf{k}}, {\mathbf{l}}_p)$ and ${\rm{w}}({\mathbf{k}}^*)+{\rm{w}({\mathbf{l}}^*)={\rm{w}}({\mathbf{k}})+\rm{w}({\mathbf{l}})-}p$.
If ${\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})}<{\rm{w}({\mathbf{i}})+\rm{w}({\mathbf{j}})}$, then $({\mathbf{k}}^*,{\mathbf{l}}^*)\preceq' ({\mathbf{i}},{\mathbf{j}}-\epsilon_p).$
If ${\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})}={\rm{w}({\mathbf{i}})+\rm{w}({\mathbf{j}})}$, then there is $\tau\in \mathbb{M}$ such that ${\rm w}(\tau)=p$ and ${\mathbf{l}}_p={\mathbf{l}}-\tau$. Since $ (\bf0, \epsilon_p)\preceq' (\bf0, \tau)$ and $({\mathbf{k}},{\mathbf{l}})\preceq' ({\mathbf{i}},{\mathbf{j}})$, we see that $$({\mathbf{k}}^* ,{\mathbf{l}}^*)=({\mathbf{k}} ,{\mathbf{l}})-({\bf}0, \tau) \preceq' ({\mathbf{i}} ,{\mathbf{j}})- ({\bf0},\epsilon_p)= ({\mathbf{i}},{\mathbf{j}}-\epsilon_p),$$
where the equality holds if and only if $({\mathbf{k}},{\mathbf{l}})=({\mathbf{i}},{\mathbf{j}})$.
(c) $p<j<n+k+p$. Then $h_{p+k-\frac{1}{2}-j}\in \mathfrak{D}^{(0,-n)}$ and $h_{p+k-\frac{1}{2}-j}v_{{\mathbf{k}},{\mathbf{l}}}\in V.$ So $$ {\rm{w}({\mathbf{k}}^*)+\rm{w}({\mathbf{l}}^*)}= {\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})}-j<{\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})}-p$$ and $({\mathbf{k}}^*,{\mathbf{l}}^*) \prec' ({\mathbf{i}},{\mathbf{j}}-\epsilon_p)$.
(d) $j\ge n+k+p$. Then $p+k-\frac{1}{2}-j<-n+\frac{1}{2}$. Assume $p+k-\frac{1}{2}-j=-s-n+\frac{1}{2}$ for some $s\in {\mathbb Z}_+$, that is, $-j+s= -n-k-p+1<-p$ since $k+n\ge2$. Since the corresponding vector $h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}h_{p+k-\frac{1}{2}-j}v_{{\mathbf{k}},{\mathbf{l}}}=h^{{\mathbf{k}}}h_{-s-n+\frac{1}{2}}d^{{\mathbf{l}}_j}v_{{\mathbf{k}},{\mathbf{l}}}-h^{{\mathbf{k}}}[h_{-s-n+\frac{1}{2}},d^{{\mathbf{l}}_j}]v_{{\mathbf{k}},{\mathbf{l}}}, $ by (\ref{bracket3.3}) and simple computations we see $h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}h_{p+k-\frac{1}{2}-j}v_{{\mathbf{k}},{\mathbf{l}}}$ can written as a linear combination of vectors in the form $h^{{\mathbf{k}}}h_{-s'-s-n+\frac{1}{2}}d^{{\mathbf{l}}_{s'+j}}v_{{\mathbf{k}},{\mathbf{l}}} $ where $s'\in{\mathbb N}$ and $\deg'(h^{{\mathbf{k}}}h_{-s'-s-n+\frac{1}{2}}d^{{\mathbf{l}}_{s'+j}}v_{{\mathbf{k}},{\mathbf{l}}}) $ has weight $${\rm{w}}({\mathbf{k}})+s'+s+{\rm{w}}({\mathbf{l}}_{s'+j})={\rm{w}}({\mathbf{k}})+{\rm{w}}({\mathbf{l}})+s-j. $$ So $$ {\rm{w}({\mathbf{k}}^*)+\rm{w}({\mathbf{l}}^*)}= {\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})}-j+s<{\rm{w}}({\mathbf{k}})+{\rm{w}}({\mathbf{l}})-p\le {\rm{w}}({\mathbf{i}})+{\rm{w}}({\mathbf{j}})-p,$$ and hence $({\mathbf{k}}^*,{\mathbf{l}}^*) \prec' ({\mathbf{i}},{\mathbf{j}}-\epsilon_p)$.
In conclusion, $\text{deg}'(h_{p+k-\frac{1}{2}}h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}) \preceq' ({\mathbf{i}},{\mathbf{j}}-\epsilon_p),$
where the equality holds if and only if $({\mathbf{k}},{\mathbf{l}})=({\mathbf{i}},{\mathbf{j}})$, that is, $\deg'(h_{p+k-\frac{1}{2}}v)=({\mathbf{i}},{\mathbf{j}}-\epsilon_p)$.\end{proof}
\begin{lem}\label{main4} Let $k\in{\mathbb Z}_+, n\in{\mathbb Z}$ such that $k>n$ and $k+n\geq 2$, and $V$ be a $\mathfrak{D}^{(0,-n)}$-module such that $h_{k-\frac{1}{2}}$ acts injectively on $V$, and $h_{i -\frac{1}{2}} V=d_jV=0$ for all $i>k$, $j>k+n-1$. Assume that $v=\sum_{({\mathbf{k}},{\mathbf{l}})}h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}\in \text{\rm Ind}(V)\setminus V$ with $\deg'(v)=({\mathbf{i}}, \bf0)$. Set $q=\min\{s: i_s\neq 0\}$. \begin{enumerate} \item[\rm(1)] If the sum $\sum_{({\mathbf{k}},{\mathbf{l}})}h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$ does not contain terms $h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$ satisfying \begin{equation}\label{weight} {\rm{w}({\mathbf{k}})}+{\rm{w}({\mathbf{l}})}={\rm{w}({\mathbf{i}})}, {\rm{w}({\mathbf{i}})}-q\le {\rm{w}({\mathbf{k}})}<{\rm{w}({\mathbf{i}})}, \end{equation} then $\deg'(d_{q+k+n-1}v)=({\mathbf{i}}-\epsilon_q, \bf0)$; \item[\rm(2)] Assume that the sum $\sum_{({\mathbf{k}},{\mathbf{l}})}h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$ contain terms $h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$ satisfying (\ref{weight}). Let $v'=v-\sum_{\rm{w}({\mathbf{k}})={\rm{w}({\mathbf{i}})}}h^{{\mathbf{k}}}v_{{\mathbf{k}},\bf0}$ and $\deg'(v')=({\mathbf{k}}^*, {\mathbf{l}}^*)$ with $t=min\{s: l^*_s\neq 0\}$. Then $\deg'(h_{k+t-\frac{1}{2}}v)=({\mathbf{k}}^*, {\mathbf{l}}^*-\epsilon_t)$. \end{enumerate}
\end{lem}
\begin{proof} Consider $\text{deg}'(d_{q+k+n-1}h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}})$ with $d_{q+k+n-1}h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}\ne 0.$ Noting that $d_{q+k+n-1}V=0,$ we see that $$d_{q+k+n-1}h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}=[d_{q+k+n-1},h^{{\mathbf{k}}}]d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}+h^{{\mathbf{k}}}[d_{q+k+n-1},d^{{\mathbf{l}}}]v_{{\mathbf{k}},{\mathbf{l}}}.$$
First we consider the term $[d_{q+k+n-1},h^{{\mathbf{k}}}]d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$. It follows from (\ref{bracket3.2}) that $[d_{q+k+n-1},h^{{\mathbf{k}}}]d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$ is a linear combination of vectors in the forms $h^{{\mathbf{k}}_j}h_{(q-j)+k-\frac{1}{2}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$ and $h^{\bf{s}}d^{{\mathbf{l}}}v_{{\mathbf{k}}, {\mathbf{l}}}$
where ${\mathbf{k}}_j={\mathbf{k}}-\epsilon_j$, ${\rm{w}}({\bf{s}}) ={\rm{w}}({\mathbf{k}})-(k+q-n)$. If ${\mathbf{l}}=0$, it is not hard to see that $\text{deg}'(d_{q+k+n-1}h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}})\preceq' ({\mathbf{i}}-\epsilon_q,\bf0)$ where the equality holds if and only if $({\mathbf{k}}, {\mathbf{l}})=({\mathbf{i}},\bf0)$.
Next we assume that ${\mathbf{l}}\ne\bf0$, and continue to consider the term $[d_{q+k+n-1},h^{{\mathbf{k}}}]d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$. We first consider the term $h^{{\mathbf{k}}_j}h_{(q-j)+k-\frac{1}{2}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$. We break the arguments into to four different cases next.
(a) $j<q$. In this case, we have $h^{{\mathbf{k}}_j}h_{(q-j)+k-\frac{1}{2}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}=h^{{\mathbf{k}}_j}[h_{(q-j)+k-\frac{1}{2}}, d^{{\mathbf{l}}}]v_{{\mathbf{k}},{\mathbf{l}}}$. Then it follows from (\ref{bracket3.3}) that $h^{{\mathbf{k}}_j}[h_{(q-j)+k-\frac{1}{2}}, d^{{\mathbf{l}}}]v_{{\mathbf{k}},{\mathbf{l}}}$ is a linear combination of vectors in the form $h^{{\mathbf{k}}_j} d^{{\mathbf{l}}_s}h_{(q-j-s)+k-\frac{1}{2}}v_{{\mathbf{k}},{\mathbf{l}}}$ where ${\rm{w}}({\mathbf{l}}_s)={\rm{w}}({\mathbf{l}})-s$.
(a1) If $s<q-j$, then $h^{{\mathbf{k}}_j} d^{{\mathbf{l}}_s}h_{(q-j-s)+k-\frac{1}{2}}v_{{\mathbf{k}},{\mathbf{l}}}=0.$
(a2) If $s=q-j$, then $\deg'(h^{{\mathbf{k}}_j} d^{{\mathbf{l}}_s}h_{k-\frac{1}{2}}v_{{\mathbf{k}},{\mathbf{l}}})$ has weight
$${\rm{w}}({\mathbf{k}}_j)+{\rm{w}}({\mathbf{l}}_s)={\rm{w}}({\mathbf{k}})+{\rm{w}}({\mathbf{l}})-j-s={\rm{w}}({\mathbf{k}})+{\rm{w}}({\mathbf{l}})-q.$$ If ${\rm{w}}({\mathbf{k}})+{\rm{w}}({\mathbf{l}})<{\rm{w}}({\mathbf{i}})$, or ${\rm{w}}({\mathbf{k}})+{\rm{w}}({\mathbf{l}})={\rm{w}}({\mathbf{i}})$ and ${\rm{w}}({\mathbf{k}})<{\rm{w}}({\mathbf{i}})-q$, then $\deg'(h^{{\mathbf{k}}_j} d^{{\mathbf{l}}_s}h_{k-\frac{1}{2}}v_{{\mathbf{k}},{\mathbf{l}}})\prec' ({\mathbf{i}}-\epsilon_q,\bf0).$ We will discuss the remaining cases that
$({\mathbf{k}}, {\mathbf{l}})$ satisfies (3.9) in Case (2) later.
(a3) If $q-j<s\le q+k+n-1-j$, then $ h_{(q-j-s)+k-\frac{1}{2}}v_{{\mathbf{k}},{\mathbf{l}}}\in V$ and $\deg'(h^{{\mathbf{k}}_j} d^{{\mathbf{l}}_s}h_{k-\frac{1}{2}}v_{{\mathbf{k}},{\mathbf{l}}})$ has weight
$${\rm{w}}({\mathbf{k}}_j)+{\rm{w}}({\mathbf{l}}_s)={\rm{w}}({\mathbf{k}})+{\rm{w}}({\mathbf{l}})-j-s<{\rm{w}}({\mathbf{k}})+{\rm{w}}({\mathbf{l}})-q\le {\rm{w}}({\mathbf{i}})-q.$$ So
$\deg'(h^{{\mathbf{k}}_j} d^{{\mathbf{l}}_s}h_{(q-j-s)+k-\frac{1}{2}}v_{{\mathbf{k}},{\mathbf{l}}})\prec' ({\mathbf{i}}-\epsilon_q,\bf0).$
(a4) If $s>q+k+n-1-j$, then $q-j-s+k-\frac{1}{2}=-s'-n+\frac{1}{2} $ for some $s'\in{\mathbb Z}_+$. It is easy to see
$h^{{\mathbf{k}}_j} d^{{\mathbf{l}}_s}h_{(q-j-s)+k-\frac{1}{2}}v_{{\mathbf{k}},{\mathbf{l}}}$ can be written as a linear combination of vectors of the form $h^{{\mathbf{k}}_j} h_{-s'-s''-n+\frac{1}{2}}d^{{\mathbf{l}}_{s+s''}}v_{{\mathbf{k}},{\mathbf{l}}}, 0\le s''\le {\rm{w}}({\mathbf{l}}_s)$. Note that both $\deg'(h^{{\mathbf{k}}_j} h_{-s'-s''-n+\frac{1}{2}}d^{{\mathbf{l}}_{s+s''}}v_{{\mathbf{k}},{\mathbf{l}}})$ and $\deg'(h^{{\mathbf{k}}_j} h_{-s'-n+\frac{1}{2}}d^{{\mathbf{l}}_{s}}v_{{\mathbf{k}},{\mathbf{l}}})$ have the same weight and $-j-s+s'=-q-k-n+1<-q$, we see
$\deg'(h^{{\mathbf{k}}_j} d^{{\mathbf{l}}_s}h_{(q-j-s)+k-\frac{1}{2}}v_{{\mathbf{k}},{\mathbf{l}}})$ has weight
$${\rm{w}}({\mathbf{k}}_j)+{\rm{w}}({\mathbf{l}}_s)+s'={\rm{w}}({\mathbf{k}})+{\rm{w}}({\mathbf{l}})-j-s+s'<{\rm{w}}({\mathbf{k}})+{\rm{w}}({\mathbf{l}})-q\le {\rm{w}}({\mathbf{i}})-q.$$So
$\deg'(h^{{\mathbf{k}}_j} d^{{\mathbf{l}}_s}h_{(q-j-s)+k-\frac{1}{2}}v_{{\mathbf{k}},{\mathbf{l}}})\prec' ({\mathbf{i}}-\epsilon_q,\bf0).$
(b) $j=q$. In this case, we have $h^{{\mathbf{k}}_q}h_{k-\frac{1}{2}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}=h^{{\mathbf{k}}_q}d^{{\mathbf{l}}}h_{k-\frac{1}{2}}v_{{\mathbf{k}},{\mathbf{l}}}+h^{{\mathbf{k}}_q}[h_{k-\frac{1}{2}}, d^{{\mathbf{l}}}]v_{{\mathbf{k}},{\mathbf{l}}}$. Clearly, $\text{deg}'(h^{{\mathbf{k}}_q}d^{{\mathbf{l}}}h_{k-\frac{1}{2}}v_{{\mathbf{k}},{\mathbf{l}}})=({\mathbf{k}}_q,{\mathbf{l}})\prec' ({\mathbf{i}}-\epsilon_q,\bf0)$ since ${\mathbf{l}}\ne\bf0$. By (\ref{bracket3.3}) and the similar arguments in Cases (a3) and (a4) we can deduce that $\text{deg}'(h^{{\mathbf{k}}_q}[h_{k-\frac{1}{2}},d^{{\mathbf{l}}}]v_{{\mathbf{k}},{\mathbf{l}}})\prec' ({\mathbf{i}}-\epsilon_q,\bf0)$. Hence $\text{deg}'(h^{{\mathbf{k}}_q}h_{k-\frac{1}{2}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}})\prec' ({\mathbf{i}}-\epsilon_q,\bf0)$.
(c) $q<j\leq q+k+n-1$. In this case, we have $h^{{\mathbf{k}}_j}h_{(q-j)+k-\frac{1}{2}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}=h^{{\mathbf{k}}_j}d^{{\mathbf{l}}}h_{(q-j)+k-\frac{1}{2}}v_{{\mathbf{k}},{\mathbf{l}}}+h^{{\mathbf{k}}_j}[h_{(q-j)+k-\frac{1}{2}}, d^{{\mathbf{l}}}]v_{{\mathbf{k}},{\mathbf{l}}}$. Clearly, $\deg'(h^{{\mathbf{k}}_j}d^{{\mathbf{l}}}h_{(q-j)+k-\frac{1}{2}}v_{{\mathbf{k}},{\mathbf{l}}})= {\rm{w}}({\mathbf{k}})+{\rm{w}}({\mathbf{l}})-j<{\rm{w}}({\mathbf{i}})-q.$ Then by (\ref{bracket3.3}) and the similar arguments in Cases (a3) and (a4) we can deduce that $\text{deg}'(h^{{\mathbf{k}}_q}[h_{(q-j)+k-\frac{1}{2}}, d^{{\mathbf{l}}}]v_{{\mathbf{k}},{\mathbf{l}}})\prec' ({\mathbf{i}}-\epsilon_q,\bf0)$. Hence, $\text{deg}'(h^{{\mathbf{k}}_j}h_{(q-j)+k-\frac{1}{2}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}})\prec' ({\mathbf{i}}-\epsilon_q,\bf0)$.
(d) $j>q+k+n-1$. In this case, we have $h^{{\mathbf{k}}_j}h_{(q-j)+k-\frac{1}{2}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}=h^{{\mathbf{k}}_j}h_{-(j-(q+k+n-1))-n+\frac{1}{2}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$. Then $\text{deg}'(h^{{\mathbf{k}}_j}h_{(q-j)+k-\frac{1}{2}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}})=({\mathbf{k}}^*,{\mathbf{l}})$ with weight ${\rm{w}}({\mathbf{k}}^*)+{\rm{w}}({\mathbf{l}})= {\rm{w}}({\mathbf{k}})+{\rm{w}}({\mathbf{l}}) -(q+k+n-1)<{\rm{w}}({\mathbf{i}})-q$. Hence, $\text{deg}'(h^{{\mathbf{k}}_j}h_{(q-j)+k-\frac{1}{2}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}})\prec' ({\mathbf{i}}-\epsilon_q,\bf0)$.
Next consider the term $h^{\bf{s}}d^{{\mathbf{l}}}v_{{\mathbf{k}}, {\mathbf{l}}}$. Since ${\rm{w}}(\text{deg}'(h^{\bf{s}}d^{{\mathbf{l}}}v_{{\mathbf{k}}, {\mathbf{l}}}))={\rm{w}}(\bf{s})+{\rm{w}}({\mathbf{l}})< {\rm{w}}(\textbf{k})+{{\rm{w}}(\textbf{l})}$-$q\leq {\rm{w}}(\textbf{i})$-$q$, it follows that $\text{deg}'(h^{\bf{s}}d^{{\mathbf{l}}}v_{{\mathbf{k}}, {\mathbf{l}}})\prec' ({\mathbf{i}}-\epsilon_q,\bf0)$.
Thus, if $h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$ does not satisfy (\ref{weight}) we have $$\deg'([d_{q+k+n-1},h^{{\mathbf{k}}}]d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}})\preceq' ({\mathbf{i}}-\epsilon_q,\bf0)$$ where the equality holds if and only if $({\mathbf{k}}, {\mathbf{l}})=({\mathbf{i}},\bf0)$.
Now, consider the term $h^{{\mathbf{k}}}[d_{q+k+n-1},d^{{\mathbf{l}}}]v_{{\mathbf{k}},{\mathbf{l}}}$ where we still assume that ${\mathbf{l}}\ne\bf0$. By (\ref{bracket3.4}) we see $h^{{\mathbf{k}}}[d_{q+k+n-1},d^{{\mathbf{l}}}]v_{{\mathbf{k}},{\mathbf{l}}}$ is a linear combination of vectors $h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}d_{q+k+n-1-j}v_{{\mathbf{k}},{\mathbf{l}}}$ and $h^{{\mathbf{k}}}d^{{\mathbf{l}}_{q+k+n-1}}v_{{\mathbf{k}},{\mathbf{l}}}$ where $\rm{w}({{\mathbf{l}}_j})=\rm{w}({{\mathbf{l}}})-j,j\in{\mathbb N}$. Since $\deg'(h^{{\mathbf{k}}}d^{{\mathbf{l}}_{q+k+n-1}}v_{{\mathbf{k}},{\mathbf{l}}})$ has weight $${\rm{w}({\mathbf{k}})}+{\rm{w}}({{\mathbf{l}}})-(q+k+n-1)<{\rm{w}({\mathbf{k}})}+{\rm{w}}({{\mathbf{l}}})-q\le {\rm{w}}({\mathbf{i}})-q,$$ we see $\deg'(h^{{\mathbf{k}}}d^{{\mathbf{l}}_{q+k+n-1}}v_{{\mathbf{k}},{\mathbf{l}}})\prec' ({\mathbf{i}}-\epsilon_q,\bf0)$. So we need only to consider the vectors $h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}d_{q+k+n-1-j}v_{{\mathbf{k}},{\mathbf{l}}}$. There are four different cases.
(i) $j<q$. Then $q+k+n-1-j>k+n-1$ and $h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}d_{q+k+n-1-j}v_{{\mathbf{k}},{\mathbf{l}}}=0$. In particular, for ${\rm{w}({\mathbf{l}})}<q$ we have $h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}d_{q+k+n-1-j}v_{{\mathbf{k}},{\mathbf{l}}}=0$.
(ii) $j=q$. Then $q+k+n-1-q=k+n-1$ and hence $\deg'(h^{{\mathbf{k}}}d^{{\mathbf{l}}_q}d_{k+n-1}v_{{\mathbf{k}},{\mathbf{l}}})=({\mathbf{k}},{\mathbf{l}}_q)$ ( in the case $d_{k+n-1}v_{{\mathbf{k}},{\mathbf{l}}}\ne 0$ ) with ${\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}}_q)=\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})}-q$.
If $\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})<\rm{w}({\mathbf{i}})$, or $\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})=\rm{w}({\mathbf{i}})$ and ${\rm{w}({\mathbf{k}})<\rm{w}({\mathbf{i}})}-q$, then $({\mathbf{k}},{\mathbf{l}}_q)\prec' ({\mathbf{i}}-\epsilon_q,\bf0).$ We will discuss the remaining cases that
$({\mathbf{k}}, {\mathbf{l}})$ satisfies (3.9) in Case (2) later.
(iii) $q<j\le q+k+n-1$. Then $d_{q+k+n-1-j}v_{{\mathbf{k}},{\mathbf{l}}}\in V$ and $h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}d_{q+k+n-1-j}v_{{\mathbf{k}},{\mathbf{l}}}=0$ or $\deg'(h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}d_{q+k+n-1-j}v_{{\mathbf{k}},{\mathbf{l}}})$ has weight $${\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}}_j)=\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})}-j<{\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})}-q\le {\rm{w}}({\mathbf{i}})-q,$$ so $\deg'(h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}d_{q+k+n-1-j}v_{{\mathbf{k}},{\mathbf{l}}})\prec' ({\mathbf{i}}-\epsilon_q,\bf0)$.
(iv) $j>q+k+n-1$. Then $q+k+n-1-j<0$. Assume $q+k+n-1-j=-j'$, $j'\in {\mathbb Z}_+$. Then $-j+j'=-(q+k+n-1)<-q$. So $\deg'(h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}d_{q+k+n-1-j}v_{{\mathbf{k}},{\mathbf{l}}})$ has weight $${\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}}_j)+j'=\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})}-j+j'={\rm{w}({\mathbf{k}})+\rm{w}({\mathbf{l}})}-(q+k+n-1) < {\rm{w}}({\mathbf{i}})-q,$$ which means $\deg'(h^{{\mathbf{k}}}d^{{\mathbf{l}}_j}d_{q+k+n-1-j}v_{{\mathbf{k}},{\mathbf{l}}}) \prec' ({\mathbf{i}}-\epsilon_q,\bf0).$
(1) If $v=\sum_{({\mathbf{k}},{\mathbf{l}})}h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$ does not contain a term $h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$ satisfying (\ref{weight}), then by the above arguments we see $\deg'(d_{q+k+n-1}v)=({\mathbf{i}}-\epsilon_q, \bf0)$.
(2) If $v=\sum_{({\mathbf{k}},{\mathbf{l}})}h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$ contain terms $h^{{\mathbf{k}}}d^{{\mathbf{l}}}v_{{\mathbf{k}},{\mathbf{l}}}$ satisfying (\ref{weight}), then we see $\deg'(v')=({\mathbf{k}}^*,{\mathbf{l}}^*)$ with $${\rm{w}({\mathbf{k}}^*)+\rm{w}({\mathbf{l}}^*)=\rm{w}({\mathbf{i}}), }\,
{\rm{w}({\mathbf{k}}^*)}\ge {\rm{w}({\mathbf{i}})}-q, \,1\le {\rm{w}}{({\mathbf{l}}^*)}\le q.$$
Then by Lemma \ref{main3} we see $\deg'(h_{t+k-\frac{1}{2}}v')=({\mathbf{k}}^*, {\mathbf{l}}^*-\epsilon_t)$.
Noticing that $k>n$, by (\ref{bracket3.1}) we see $h_{t+k-\frac{1}{2}}h^{{\mathbf{k}}}v_{{\mathbf{k}},\bf0}=\bf0$ or $\lambda h^{{\mathbf{k}}_{t'}}v_{{\mathbf{k}},\bf0},\lambda\in{\mathbb C}^*$ with $t'=t+k-n>t$ and ${\rm{w}({\mathbf{k}}_{t'})=\rm{w}({\mathbf{k}})}-t'$, so $\deg'(h_{t+k-\frac{1}{2}}(h^{{\mathbf{k}}}v_{{\mathbf{k}},\bf0} ))=({\mathbf{k}}_{t'},\bf0)$ has weight ${\rm{w}({\mathbf{k}}_{t'})=\rm{w}({\mathbf{k}})}-t'<{\rm{w}({\mathbf{k}}^*)+\rm{w}({\mathbf{l}}^*)}-t={\rm{w}({\mathbf{k}}^*)+\rm{w}}({\mathbf{l}}^*-\epsilon_t)$. Hence $$\deg'(h_{t+k-\frac{1}{2}}v)=\deg'\Big(h_{t+k-\frac{1}{2}}\Big(v-\sum_{\rm{w}({\mathbf{k}})={\rm{w}({\mathbf{i}})}}h^{{\mathbf{k}}}v_{{\mathbf{k}},\bf0}\Big)\Big)=({\mathbf{k}}^*, {\mathbf{l}}^*-\epsilon_t).$$ \end{proof}
\section{Simple restricted $\mathfrak{D}$-modules}\label{char}
In this section we will determine all simple restricted ${\mathfrak{D}}$-modules. Based on Theorem \ref{MT}, we only need to determine all simple restricted ${\mathfrak{D}}$-modules $S$ of level $\ell\ne0$.
For a given simple restricted $\mathfrak{D}$-module $S$ with level $\ell \not=0$, we define the following invariants of $S$ as follows: $$S(r)={\text{Ann}}_S(\mathcal{H}^{(r)}), n_S=\min\{r\in {\mathbb Z}:S(r)\ne0\}, W_0=S{(n_S)}, $$ and $$U(r)={\text{Ann}}_{W_0}(\mathfrak{Vir}^{(r)}), m_S=\min\{r\in {\mathbb Z}:U(r)\ne0\}, U_0=U(m_S).$$
\begin{lem}\label{lem4.2} Let $S$ be a simple restricted $\mathfrak{D}$-module with level $\ell \not=0$. \begin{enumerate} \item[\rm(i)] $h_{n_S-\frac12}$ acts injectively on $W_0$, $d_{m_S-1}$ acts injectively on $U_0$. \item[\rm(ii)] $n_S, m_S\in{\mathbb N}$. \item[\rm(iii)] $W_0$ is a nonzero $\mathfrak{D}^{(0,-n_S)}$-module, and is invariant under the action of the operators $L_n$ defined in (\ref{rep1})-(\ref{rep3}) for $n\in{\mathbb N}$. \item[\rm(iv)] If $m_S\ge 2n_S$, then $U_0$ is a nonzero $\mathfrak{D}^{(0,-n_S)}$-submodule of $W_0$, and is invariant under the action of the operators $L_n$ defined in (\ref{rep1})-(\ref{rep3}) for $n\in{\mathbb N}$. \end{enumerate} \end{lem}
\begin{proof} (i) follows from the definitions of $n_S$ and $m_S$.
(ii) Suppose $n_S<0$, take any nonzero $v\in W_0$, we then have $$h_{\frac12}v=0=h_{-\frac12}v.$$ This implies that $\frac12\ell v=[h_{\frac12},h_{-\frac12}]v=0$, a contradiction. Hence, $n_S\in{\mathbb N}$.
Suppose $m_S<0$. Take any nonzero $v\in U_0$, we then have $d_{-1}v=0=h_{n_S+\frac{1}{2}}v$. Then $$-(n_S+\frac{1}{2})h_{n_S-\frac{1}{2}}v=[d_{-1}, h_{n_S+\frac{1}{2}}]v=0,$$ a contradiction with (1). Hence, $m_S\in{\mathbb N}$.
(iii) It is obvious that $W_0\neq 0$ by definition. For any $w\in W_0$, $i, j, k\in{\mathbb N}$, we have $$h_{k+n_S+\frac{1}{2}}d_iw=d_ih_{k+n_S+\frac{1}{2}}w+(k+n_S+\frac{1}{2})h_{i+k+n_S+\frac{1}{2}}w=0,$$ and $$h_{k+n_S+\frac{1}{2}}h_{j-n_S+\frac{1}{2}}w=h_{j-n_S+\frac{1}{2}}h_{k+n_S+\frac{1}{2}}w=0.$$ Hence, $d_iu\in W_0$ and $h_{j-n_S+\frac{1}{2}}u\in W_0$, i.e., $W_0$ is a nonzero $\mathfrak{D}^{(0,-n_S)}$-module.
For $n\in{\mathbb N}, i\in{\mathbb N}$, $w\in W_0$, by (\ref{rep4}) we have \begin{eqnarray*} h_{i+n_S+\frac{1}{2}}L_nw&=\Big(L_nh_{i+n_S+\frac{1}{2}}-(i+n_S+\frac{1}{2})h_{n+i+n_S+\frac{1}{2}}\Big)w=0.
\end{eqnarray*}
This implies that $L_iw\in W_0$ for $i\in{\mathbb N}$, that is, $W_0$ is invariant under the action of the operators $L_i$ for $i\in{\mathbb N}$.
(iv) It is obvious that $0\neq U_0\subseteq W_0$. Suppose that $m_S\geq 2n_S$. For any $u\in U_0$, $i, j, k\in{\mathbb N}$, it follows from (iii) that $d_iu\in W_0$ and $h_{j-n_S+\frac{1}{2}}u\in W_0$. Furthermore, $$d_{k+m_S}d_iu=d_id_{k+m_S}u+(k-i-m_S)d_{k+i+m_S}u=0,$$ and $$d_{k+m_S}h_{j-n_S+\frac{1}{2}}u=h_{j-n_S+\frac{1}{2}}d_{k+m_S}u-(j-n_S+\frac{1}{2})h_{k+j+m_S-n_S+\frac{1}{2}}u=0.$$ Hence, $d_iu\in U_0$ and $h_{j-n_S+\frac{1}{2}}u\in U_0$, i.e., $U_0$ is a nonzero $\mathfrak{D}^{(0,-n_S)}$ submodule of $W_0$.
Furthermore, if in addition $m_S>0$, then for $n, i\in{\mathbb N}$, $u\in U_0$, it follows from (iii) that $L_nu\in W_0$. Moreover, for $n\in{\mathbb N}$, using (\ref{rep1}-\ref{rep4'}) we have \begin{eqnarray*} d_{i+m_S}L_nu=L_nd_{i+m_S} u+[d_{i+m_S},L_n]u =[d_{i+m_S},L_n]u =(n-i-m_S)L_{i+n+m_S}u=0. \end{eqnarray*} This implies that $L_iu\in U_0$ for $i\in{\mathbb N}$, that is, $U_0$ is invariant under the action of the operators $L_i$ for $i\in{\mathbb N}$. \end{proof}
\begin{pro}\label{prop4.3} Let $S$ be a simple restricted $\mathfrak{D}$-module with level $\ell \not=0$. \begin{enumerate} \item[\rm(i)] If $n_S=0$, then $S\cong H^{\mathfrak{D}}\otimes U^{\mathfrak{D}}$ as $\mathfrak{D}$-modules for some simple modules $H\in \mathcal{R}_{\mathcal{H}}$ and $U\in \mathcal{R}_{\mathfrak{Vir}}$. \item[\rm(ii)] If $m_S>2n_S>0$, then $S\cong \text{\rm Ind}^{\mathfrak{D}}_{\mathfrak{D}^{(0,-n_S)}}(U_0)$ and $U_0$ is a simple $\mathfrak{D}^{(0,-n_S)}$-module. \item[\rm(iii)] If $m_S< 2n_S$, then $U_0$ is a nonzero $\mathfrak{D}^{(0,-(m_S-n_S))}$-submodule of $W_0$. Moreover, \begin{enumerate} \item[\rm(iii-1)] If $m_S\geq 2$, then $S\cong \text{\rm Ind}^{\mathfrak{D}}_{\mathfrak{D}^{(0,-(m_S-n_S))}}(U_0)$ and $U_0$ is a simple $\mathfrak{D}^{(0,-(m_S-n_S))}$-module. \item[\rm(iii-2)] If $m_S=0$ or $1$, and $n_S>1$, then $U(2)$ is a simple $\mathfrak{D}^{(0,-(2-n_S))}$-module, and $S\cong \text{\rm Ind}^{\mathfrak{D}}_{\mathfrak{D}^{(0,-(2-n_S))}}(U(2))$. \end{enumerate} \end{enumerate} \end{pro}
\begin{proof} (i) Since $n_S=0$, we take any nonzero $v\in W_0$. Then ${\mathbb C} v$ is a trivial $\mathcal{H}^{(0)}$-module. Let $H=U(\mathcal{H})v$, the $\mathcal{H}$-submodule of $S$ generated by $v$. It follows from representation theory of Heisenberg algebras (or from the same arguments as in the proof of Lemma \ref{main1'}) that $\text{\rm Ind}^{\mathcal{H}}_{\mathcal{H}^{(0)}}({\mathbb C} v)$ is a simple $\mathcal{H}$-module. Consequently, the following surjective $\mathcal{H}$-module homomorphism \begin{eqnarray*} \varphi:\, \text{\rm Ind}^{\mathcal{H}}_{\mathcal{H}^{(0)}}({\mathbb C} v) &\longrightarrow & H\\ \sum_{\mathbf{i}\in\mathbb{M}}a_{\mathbf{i}} h^{\mathbf{i}}\otimes v&\mapsto & \sum_{\mathbf{i}\in\mathbb{M}} a_{\mathbf{i}} h^{\mathbf{i}} v \end{eqnarray*} is an isomorphism, that is, $H$ is a simple $\mathcal{H}$-module, which is certainly restricted. Then the desired assertion follows directly from Corollary \ref{tensor}.
(ii) By taking $V=U_0$, $k=n=n_S$ and $l=m_S-1$ in Theorem \ref{thmmain1.1}(1) we see that any nonzero $\mathfrak{D}$-submodule of $\text{\rm Ind}^{\mathfrak{D}}_{\mathfrak{D}^{(0,-n_S)}}(U_0)$ has a nonzero intersection with $U_0$. Consequently, the surjective $\mathfrak{D}$-module homomorphism \begin{eqnarray*} \varphi:\, \text{\rm Ind}^{\mathfrak{D}}_{\mathfrak{D}^{(0,-n_S)}}(U_0) &\longrightarrow & S\\ \sum_{\mathbf{i}, \mathbf{k}\in\mathbb{M}} h^{\mathbf{i}}d^{\mathbf{k}}\otimes v_{\mathbf{i}, \mathbf{k}}&\mapsto & \sum_{\mathbf{i}, \mathbf{k}\in\mathbb{M}} h^{\mathbf{i}}d^{\mathbf{k}} v_{\mathbf{i}, \mathbf{k}} \end{eqnarray*} is an isomorphism, i.e., $S\cong \text{\rm Ind}^{\mathfrak{D}}_{\mathfrak{D}^{(0,-n_S)}}(U_0)$. Since $S$ is simple, we see $U_0$ is a simple $\mathfrak{D}^{(0,-n_S)}$-module.
(iii) Suppose that $m_S< 2n_S$. For any $u\in U_0$, $i, j, k\in{\mathbb N}$, it follows from Lemma \ref{lem4.2} (iii) that $d_iu\in W_0$ and $h_{j-(m_S-n_S)+\frac{1}{2}}u\in W_0$. Furthermore, $$d_{k+m_S}d_iu=d_id_{k+m_S}u+(k-i+m_S)d_{k+i+m_S}u=0,$$ and $$d_{k+m_S}h_{j-(m_S-n_S)+\frac{1}{2}}u=h_{j-(m_S-n_S)+\frac{1}{2}}d_{k+m_S}u-(j-(m_S-n_S)+\frac{1}{2})h_{k+j+n_S+\frac{1}{2}}u=0.$$ Hence, $d_iu\in U_0$ and $h_{j-(m_S-n_S)+\frac{1}{2}}u\in U_0$, i.e., $U_0$ is a nonzero $\mathfrak{D}^{(0,-(m_S-n_S))}$ submodule of $W_0$.
Now suppose $m_S\geq 2$. Then it follows from Theorem \ref{thmmain1.1}(2) that
any nonzero $\mathfrak{D}$-submodule of $\text{\rm Ind}^{\mathfrak{D}}_{\mathfrak{D}^{(0,-(m_S-n_S))}}(U_0)$ has a nonzero intersection with $U_0$ by taking $k=n_S, n=m_S-n_S$ and $l=m_S-1$ therein. Consequently, $S\cong \text{\rm Ind}^{\mathfrak{D}}_{\mathfrak{D}^{(0,-(m_S-n_S))}}(U_0)$ by similar arguments as in (ii). Since $S$ is simple, we see $U_0$ is a simple $\mathfrak{D}^{(0,-n_S)}$-module.
Suppose that $m_S=0$ or $1$, and $n_S>1$. Then $\mathfrak{D}^{(0,-(2-n_S))}\subseteq \mathfrak{D}^{(0,-n_S)}$. Hence, $W_0$ is a $\mathfrak{D}^{(0,-(2-n_S))}$-module. Moreover, for any $u\in U(2)$, $i, j\in{\mathbb N}$, we have $$d_{j+2}d_{i}u=d_{i}d_{j+2}u=0,$$ and $$d_{j+2}h_{i-(2-n_S)+\frac{1}{2}}u=h_{i-(2-n_S)+\frac{1}{2}}d_{j+2}u+(2-n_S-i-\frac{1}{2})h_{i+j+n_S+\frac{1}{2}}u=0.$$ Therefore, $U(2)$ is a $\mathfrak{D}^{(0,-(2-n_S))}$-module. Then it follows from Theorem \ref{thmmain1.1}(2) that
any nonzero $\mathfrak{D}$-submodule of $\text{\rm Ind}^{\mathfrak{D}}_{\mathfrak{D}^{(0,-(2-n_S))}}(U(2))$ has a nonzero intersection with $U(2)$
by taking $V=U(2)$, $k=n_S$, $n=2-n_S$ and $l=1$ therein. Consequently, $S\cong \text{\rm Ind}^{\mathfrak{D}}_{\mathfrak{D}^{(0,-(2-n_S))}}(U(2))$ by similar arguments as in (ii). In particular, $U(2)$ is a simple $\mathfrak{D}^{(0,-(2-n_S))}$-module. \end{proof}
From Proposition \ref{prop4.3}, what remains to consider are the following two cases: (1) $m_S=2n_S>0$, (2) $m_S=0$ or $1$, and $n_S=1$.
Now we first consider Case (1): $m_S=2n_S>0$. For that, we define the operators $d_n'=d_n-L_n$ on $S$ for $n\in{\mathbb Z}$. Since $S$ is a restricted $\mathfrak{D}$-module, then $d_n'$ is well-defined for any $n\in{\mathbb Z}$. By (\ref{rep4}) and (\ref{rep4'}), we have \begin{equation}\label{vir-bracket} [d_m',{\bf c}_1]=0, [d_m',d_n']=(m-n)d_{m+n}'+\frac{m^3-m}{12}\delta_{m+n,0}(c-1), m,n\in{\mathbb Z}, \end{equation}where ${\bf c}'_1={\bf c}_1-\text{id}_S$ and $c$ is the central charge of $S$. So the operator algebra $$\mathfrak{Vir}'=\bigoplus_{n\in{\mathbb Z}}{\mathbb C} d_n'\oplus {\mathbb C}{\bf c}'_1$$
is isomorphic to the Virasoro algebra $\mathfrak{Vir}$. Since $[d_n,h_{k+\frac{1}{2}}]=[L_n,h_{k+\frac{1}{2}}]=-({k+\frac{1}{2}})h_{n+k+\frac{1}{2}},$ we have $[d'_n,h_{k+\frac{1}{2}}]=0, n,k\in{\mathbb Z}$ and hence $[\mathfrak{Vir}',\mathcal{H}]=0$. Clearly, the operator algebra
$\mathfrak{D}'=\mathfrak{Vir}'\oplus \mathcal{H}$ is a direct sum, and $S=\mathcal{U}(\mathfrak{D})v=\mathcal{U}(\mathfrak{D}')v, 0\ne v\in S$.
Similar to (\ref{Natations}) we can define its subalgebras, $\mathfrak{D}'^{(m,n)} $ and the likes.
Let $$Y_n=\bigcap_{p\ge n}{\rm Ann}_{U_0}(d_p'), r_S=\min\{n\in{\mathbb Z}:Y_n\ne0\} , K_0=Y_{r_S}. $$ If $Y_n\ne0$ for any $n\in{\mathbb Z}$, we define $r_S=-\infty$. Denote by $K=U(\mathcal{H})K_0$.
\begin{lem}\label{lem4.4} Let $S$ be a simple restricted $\mathfrak{D}$-module with level $\ell \not=0$. Assume that $m_S=2n_S>0$. Then the following statements hold. \begin{enumerate} \item[\rm(i)] $-1\le r_S\le m_S$ or $r_S=-\infty$. \item[\rm(ii)] $K_0$ is a $\mathfrak{D}^{(0,-n_S)}$-module and $h_{n_S-\frac{1}{2}}$ acts injectively on $K_0$. \item[\rm(iii)] $K$ is a $\mathfrak{D}^{(0,-\infty)}$-module and $K^{\mathfrak{D}}$ has a $\mathfrak{D}$-module structure by (\ref{rep1})-(\ref{rep3}). \item[\rm{(iv)}] $K_0$ and $K$ are invariant under the action of $d_n'$ for $n\in{\mathbb N}$. \item[\rm(v)] If $r_S\ne -\infty$, then $d'_{r_S-1}$ acts injectively on $K_0$ and $K$. \end{enumerate} \end{lem}
\begin{proof} (i) Since $m_S=2n_S>0$, the operators $d_m$ and $L_{m}=\frac{1}{2\ell}\sum_{k\in{\mathbb Z}+\frac{1}{2}}h_{m-k}h_k$ act trivially on $U_0$ for any $m\geq m_S$. This implies that $Y_{m_S}=U_0\neq 0$. Consequently, $r_S\leq m_S$ by the definition of $r_S$.
If $Y_{-2}\ne 0$, then $d'_{-2}K_0=d'_{-1}K_0=0$. We deduce that $\mathfrak{Vir}' K_0=0$ and hence $r_S=-\infty$.
If $Y_{-2}=0$, then $r_S\ge -1$ and hence $-1\le r_S\le m_S$.
(ii) For any $0\ne v\in K_0$ and $x\in {\mathfrak{D}^{(0,-n_S)}}$, it follows from Lemma \ref{lem4.2}(iv) that $xv\in U_0$. We need to show that $d'_pxv=0, p\ge r_S$. Indeed, $d_p'h_{k+\frac{1}{2}}v=h_{k+\frac{1}{2}}d_p'v=0$ by (\ref{rep4}) for any $k\geq -n_S$. Moreover, it follows from (\ref{rep4'}) and (\ref{vir-bracket}) that $$d_p'd_nv=d_nd_p'v+[d_p', d_n]v=(p-n)d_{p+n}'v=0.$$ Hence, $d'_pxv=0, p\ge r_S$, that is, $xv\in K_0$, as desired.
Since $0\ne K_0 \subseteq U_0\subseteq W_0$, we see that $h_{n_S-\frac{1}{2}}$ acts injectively on $K_0$ by Lemma \ref{lem4.2}(i).
(iii) follows from (ii).
(iv) It follows from Lemma \ref{lem4.2}(iv) that $U_0$ is invariant under the action of $d_n'$ for $n\in{\mathbb N}$, so is $K_0$ by (\ref{vir-bracket}). Moreover, since $[\mathfrak{Vir}',\mathcal{H}]=0$, $K$ is also is invariant under the action of $d_n'$ for $n\in{\mathbb N}$.
(v) follows directly from the definition of $r_S$ and $K$. \end{proof}
\begin{pro}\label{prop for -inf}Let $S$ be a simple restricted $\mathfrak{D}$-module with central charge $c$ and level $\ell \not=0$. Assume that $m_S=2n_S>0$. If $r_S=-\infty$, then $c=1$. Moreover, $S= K^{\mathfrak{D}}$ and $K$ is a simple $\mathcal{H}$-module. \end{pro} \begin{proof} Since $r_S=-\infty$, we see that $\mathfrak{Vir}' K_0=0$. This together with (\ref{vir-bracket}) implies that $c=1$. Noting that $[\mathfrak{Vir}',\mathcal{H}]=0$, we further obtain that $\mathfrak{Vir}' K=0$, that is, $d_nv=L_nv\in K$ for any $v\in K$ and $n\in{\mathbb Z}$. Hence $K^{\mathfrak{D}}$ is a $\mathfrak{D}$-submodule of $S$, yielding that $S= K^{\mathfrak{D}}$. In particular, $K$ is a simple $\mathcal{H}$-module. \end{proof}
\begin{pro}\label{prop4.6} Let $S$ be a simple restricted $\mathfrak{D}$-module with level $\ell \not=0$. If $r_S\ge 2$, then $K_0$ is a simple $\mathfrak{D}^{(0,-n_S)}$-module and $S\cong \text{\rm Ind}_{\mathfrak{D}^{(0,-n_S)}}^{\mathfrak{D}}K_0$. \end{pro}
\begin{proof}We first show that $\text{\rm Ind}_{\mathfrak{D}^{(0,-n_s)}}^{\mathfrak{D}^{(0,-\infty)}}K_0 \cong K$ as $\mathfrak{D}^{(0,-\infty)}$ modules. For that, let \begin{eqnarray*} \phi:\, \text{\rm Ind}_{\mathfrak{D}^{(0,-n_s)}}^{\mathfrak{D}^{(0,-\infty)}}K_0 &\longrightarrow &K\\ \sum_{{\mathbf{k}}\in\mathbb{M}} h^{{\mathbf{k}}}\otimes v_{{\mathbf{k}}}&\mapsto & \sum_{{\mathbf{k}}\in\mathbb{M}} h^{{\mathbf{k}}} v_{{\mathbf{k}}},
\end{eqnarray*} where $h^{{\mathbf{k}}}=\cdots h^{k_2}_{-2-n_S+\frac{1}{2}}h_{-1-n_S+\frac{1}{2}}^{k_1}$. Then $\phi$ is a $\mathfrak{D}^{(0,-\infty)}$-module epimorphism and $\phi|_{K_0}$ is one-to-one. By similar arguments in the proof of Lemma \ref{main1'} we see that any nonzero submodule of $\text{\rm Ind}_{\mathfrak{D}^{(0,-n_s)}}^{\mathfrak{D}^{(0,-\infty)}}K_0$ contains nonzero vectors of $K_0$, which forces that the kernel of $\phi$ must be zero and hence $\phi$ is an isomorphism.
By Lemma \ref{lem4.4}(v), we see that $d_{r_S-1}'$ acts injectively on $K$.
As $\mathfrak{D}$-modules, $$\text{\rm Ind}_{\mathfrak{D}^{(0,-n_S)}}^{\mathfrak{D}}K_0\cong\text{\rm Ind}_{\mathfrak{D}^{(0,-\infty)}}^{\mathfrak{D}}(\text{\rm Ind}_{\mathfrak{D}^{(0,-n_S)}}^{(0,-\infty)}K_0)\cong\text{\rm Ind}_{\mathfrak{D}^{(0,-\infty)}}^{\mathfrak{D}}K.$$ And we further have $\text{\rm Ind}_{\mathfrak{D}^{(0,-\infty)}}^{\mathfrak{D}}K\cong \text{\rm Ind}_{\mathfrak{Vir}'^{(0)}}^{\mathfrak{Vir}'}K$ as vector spaces. Moreover, we have the following $\mathfrak{D}$-module epimorphism \begin{eqnarray*} \pi: \text{\rm Ind}_{\mathfrak{D}^{(0,-\infty)}}^{\mathfrak{D}}K=\text{\rm Ind}_{\mathfrak{Vir}'^{(0)}}^{\mathfrak{Vir}'}K&\rightarrow& S,\cr \sum_{{\mathbf{l}}\in\mathbb{M}}d'^{{\mathbf{l}}}\otimes v_{{\mathbf{l}}}&\mapsto& \sum_{{\mathbf{l}}\in\mathbb{M}}d'^{{\mathbf{l}}} v_{{\mathbf{l}}}, \end{eqnarray*}
where $d'^{{\mathbf{l}}}=\cdots (d'_{-2})^{l_2}(d'_{-1})^{l_1}$. We see that $\pi$ is also a $\mathfrak{Vir}'$-module epimorphism. By the proof of Theorem 2.1 in \cite{MZ2} we know that any nonzero $\mathfrak{Vir}'$-submodule of $\text{\rm Ind}_{\mathfrak{Vir}'^{(0)}}^{\mathfrak{Vir}'}K$ contain nonzero vectors of $K$. Note that $\pi|_K$ is one-to-one, we see that the image of any nonzero $\mathfrak{D}$-submodule ( and hence $\mathfrak{Vir}'$-submodule ) of $\text{\rm Ind}_{\mathfrak{D}^{(0,-\infty)}}^{\mathfrak{D}}K$ must be a nonzero $\mathfrak{D}$-submodule of $S$ and hence be the whole module $S$, which forces that the kernel of $\pi$ must be $ 0$. Therefore, $\pi$ is an isomorphism. Since $S$ is simple, we see $K_0$ is a simple $\mathfrak{D}^{(0,-n_S)}$-module.\end{proof}
As a direct consequence of Proposition \ref{prop4.6}, we have \begin{cor}\label{case 2} Let $S$ be a simple restricted $\mathfrak{D}$-module with level $\ell \not=0$. If $m_S\leq 1$ and $n_S=1$, then $K_0$ is a simple $\mathfrak{D}^{(0,-1)}$-module and $S\cong \text{\rm Ind}_{\mathfrak{D}^{(0,-1)}}^{\mathfrak{D}}K_0$. \end{cor}
\begin{proof} For any nonzero $u\in U_0$, since $m_S\leq 1$ and $n_S=1$, it follows from the definitions of $m_S, n_S$ and Lemma \ref{lem4.2}(i) that $$d_1u=0,\, L_1u=\frac{1}{2\ell}\sum_{k\in{\mathbb Z}+\frac{1}{2}}h_{1-k}h_ku=\frac{1}{2\ell}(h_{\frac{1}{2}})^2u\neq 0.$$ This implies that $d_1^{\prime}u\neq 0$, i.e., $d_1^{\prime}$ acts injectively on $U_0$. Hence $r_S\geq 2$. More precisely, since $$d_{2+i}v=L_{2+i}v=0,\,\,\forall\,i\in{\mathbb N}, v\in U_0,$$ we see that $r_S=2$. Now the desired assertion follows directly from Proposition \ref{prop4.6}. \end{proof}
\begin{rem} From Corollary \ref{case 2}, we have dealt with the Case (2). \end{rem}
What remains to consider for Case (1) is that $ m_S=2n_S\ge2$ and $r_S\le 1$. In this case we will show that $K$ is a simple $\mathcal{H}$-module.
For the Verma module $M_{\mathfrak{Vir}}(c,h)$ over $\mathfrak{Vir}$, it is well-known from \cite{A, FF} that there exist two homogeneous elements $P_1, P_2\in \mathcal{U}(\mathfrak{Vir}^-)\mathfrak{Vir}^-$ such that $ \mathcal{U}(\mathfrak{Vir}^-)P_1w_1+ \mathcal{U}(\mathfrak{Vir}^-)P_2w_1$ is the unique maximal proper $\mathfrak{Vir}$-submodule of $M_{\mathfrak{Vir}}(c,h)$, where $P_1, P_2$ are allowed to be zero and $w_1$ is the highest weight vector in $M_{\mathfrak{Vir}}(c,h)$.
\begin{lem}\label{lem4.5'} Let $d=0,-1$. Suppose $M$ is a $\mathfrak{Vir}^{(d)}$-module on which $d_0$ acts as multiplication by a given scalar $\lambda$. Then there exists a unique maximal submodule $N$ of ${\rm Ind}^{\mathfrak{Vir}}_{\mathfrak{Vir}^{(d)}}M$ with $N\cap M=0$. More precisely, $N$ is generated by $P_1M$ and $P_2M$, i.e., $N= \mathcal{U}(\mathfrak{Vir}^-)(P_1M+P_2M)$. \end{lem}
\begin{proof} Note that $d_0$ acts diagonalizably on ${\rm Ind}^{\mathfrak{Vir}}_{\mathfrak{Vir}^{(d)}}M$ and its submodules, and $$M=\{u\in{\rm Ind}^{\mathfrak{Vir}}_{\mathfrak{Vir}^{(d)}}M\mid d_0u=\lambda u\},$$ i.e., $M$ is the highest weight space of ${\rm Ind}^{\mathfrak{Vir}}_{\mathfrak{Vir}^{(d)}}M$.
Let $N$ be the sum of all $\mathfrak{Vir}$-submodules of ${\rm Ind}^{\mathfrak{Vir}}_{\mathfrak{Vir}^{(d)}}M$ which intersect with $M$ trivially. Then $N$ is the desired unique maximal $\mathfrak{Vir}$-submodule of ${\rm Ind}^{\mathfrak{Vir}}_{\mathfrak{Vir}^{(d)}}M$ with $N\cap M=0$.
Let $N^{\prime}$ be the $\mathfrak{Vir}$-submodule generated by $P_1M$ and $P_2M$, i.e., $N^{\prime}= \mathcal{U}(\mathfrak{Vir}^-)(P_1M+P_2M)$. Then $N^{\prime}\cap M=0$. Hence, $N^{\prime}\subseteq N$. Suppose there is a proper submodule $U$ of ${\rm Ind}^{\mathfrak{Vir}}_{\mathfrak{Vir}^{(d)}}M$ that is not contained in $N^{\prime}$. There is a nonzero homogeneous $v=\sum _{i=1}^ru_iv_i\in U\setminus N^{\prime}$ where $u_i\in \mathcal{U}(\mathfrak{Vir}^-)$ and $v_1,...v_r\in M$ are linearly independent. Note that all $u_i$ have the same weight. Then some $u_iv_i\notin N^{\prime}$, say $u_1v_1\notin N^{\prime}$. There is a homogeneous $u\in \mathcal{U}(\mathfrak{Vir})$ such that $uu_1v_1=v_1$. Noting that all $uu_i$ has weight $0$, so $uu_iv_i\in {\mathbb C} v_i$. Thus $uv\in M\setminus\{0\}.$ This implies that $N\subseteq N^{\prime}$. Hence, $N=N^{\prime}$, as desired. \end{proof}
\begin{pro}\label{pro4.10} Let $S$ be a simple restricted $\mathfrak{D}$-module with level $\ell \not=0$. If $ m_S=2n_s\ge2$, and $r_S=0$ or $ -1$, then $K$ is a simple $\mathcal{H}$-module and $S\cong U^{\mathfrak{D}}\otimes K^{\mathfrak{D}}$ for some simple $U\in \mathcal{R}_{\mathfrak{Vir}}$. \end{pro}
\begin{proof} By Lemma \ref{lem4.4} (iii), we see that $K^{\mathfrak{D}}$ is a $\mathfrak{D}$-module, and hence $K^{\mathfrak{D}'}$ is a $\mathfrak{D}'$-module with $d_n' K=0$ for any $n\in{\mathbb Z}$. Let ${\mathbb C} v_0$ be a one-dimensional $\mathfrak{D}'^{(r_S,-\infty)}$-module with module structure defining by $d'_nv_0=h_{k+\frac{1}{2}}v_0={\bf c}_2v_0=0, n\ge r_S, k\in{\mathbb Z}, {\bf c}'_1v_0=(c-2)v_0.$ Then ${\mathbb C} v_0\otimes K^{\mathfrak{D}'} $ is a $\mathfrak{D}'^{(r_S,-\infty)}$-module with central charge $c-1$ and level $\ell$. It is easy to see that we have the following $\mathfrak{D}'^{(r_S,-\infty)}$-module homomorphism \begin{eqnarray*} \tau_{K}: {\mathbb C} v_0\otimes K^{\mathfrak{D}'}& \longrightarrow& S,\cr
v_0\otimes u&\mapsto& u, \forall u\in K. \end{eqnarray*} Clearly, $\tau_K$ is an injective map and can be extended to a $\mathfrak{D}'$-module epimorphism \begin{eqnarray*} \tau:\text{\rm Ind}_{{\mathfrak{D}'}^{(r_S,-\infty)}}^{\mathfrak{D}'}({\mathbb C} v_0\otimes K^{\mathfrak{D}'})&\longrightarrow& S,\cr
x(v_0\otimes u)&\mapsto& xu, x\in \mathcal{U}(\mathfrak{D}'), u\in K. \end{eqnarray*} By Lemma 8 in \cite{LZ3} we know that $$\text{\rm Ind}_{\mathfrak{D}'^{(r_S,-\infty)}}^{\mathfrak{D}'}({\mathbb C} v_0\otimes K^{\mathfrak{D}'})\cong (\text{\rm Ind}_{\mathfrak{D}'^{(r_S,-\infty)}}^{\mathfrak{D}'}{\mathbb C} v_0)\otimes K^{\mathfrak{D}'} =(\text{\rm Ind}_{\mathfrak{Vir}'^{(r_S)}}^{\mathfrak{Vir}'}{\mathbb C} v_0)^{\mathfrak{D}'}\otimes K^{\mathfrak{D}'}. $$Then we have the following $\mathfrak{D}'$-module epimorphism \begin{eqnarray*} \tau':(\text{\rm Ind}_{\mathfrak{Vir}'^{(r_S)}}^{\mathfrak{Vir}'}{\mathbb C} v_0)^{\mathfrak{D}'}\otimes K^{\mathfrak{D}'}&\longrightarrow& S,\cr
xv_0\otimes u&\mapsto& xu, x\in \mathcal{U}(\mathfrak{Vir}'), u\in K. \end{eqnarray*}
Note that $(\text{\rm Ind}_{\mathfrak{Vir}'^{(r_S)}}^{\mathfrak{Vir}'}{\mathbb C} v_0)^{\mathfrak{D}'}\otimes K^{\mathfrak{D}'}\cong\text{\rm Ind}_{\mathfrak{Vir}'^{(r_S)}}^{\mathfrak{Vir}'}({\mathbb C} v_0\otimes K^{\mathfrak{D}'})$ as $\mathfrak{Vir}'$-modules, and $\tau'$ is also a $\mathfrak{Vir}'$-module epimorphism, $\tau'|_{{\mathbb C} v_0\otimes K^{\mathfrak{D}'}}$ is one-to-one, and $(\text{\rm Ind}_{\mathfrak{Vir}'^{(r_S)}}^{\mathfrak{Vir}'}{\mathbb C} v_0)^{\mathfrak{D}'}\otimes K^{\mathfrak{D}'}$ is a highest weight $\mathfrak{Vir}'$-module.
Let $V=\text{\rm Ind}_{\mathfrak{Vir}'^{(r_S)}}^{\mathfrak{Vir}'}{\mathbb C} v_0$ and $\mathfrak{K}=\text{Ker}(\tau')$. It should be noted that $${\mathbb C} v_0\otimes K^{\mathfrak{D}'}=\{u\in V^{\mathfrak{D}'}\otimes K^{\mathfrak{D}'}\mid d_0^{\prime}u=0\}.$$ We see that $({\mathbb C} v_0\otimes K^{\mathfrak{D}'})\cap \mathfrak{K}=0$. Let $\mathfrak{K}^{\prime}$ be the sum of all $\mathfrak{Vir}'$-submodules $W$ of $V^{\mathfrak{D}'}\otimes K^{\mathfrak{D}'}$ with $({\mathbb C} v_0\otimes K^{\mathfrak{D}'})\cap W=0$, that is, the unique maximal $\mathfrak{Vir}'$-submodule of $V^{\mathfrak{D}'}\otimes K^{\mathfrak{D}'}$ with trivial intersection with $({\mathbb C} v_0\otimes K^{\mathfrak{D}'})$. It is obvious that $\mathfrak{K}\subseteq\mathfrak{K}'$. Next we further show that $\mathfrak{K}=\mathfrak{K}'$. For that, take any $\mathfrak{Vir}'$-submodule $W$ of $V^{\mathfrak{D}'}\otimes K^{\mathfrak{D}'}$ such that $({\mathbb C} v_0\otimes K^{\mathfrak{D}'})\cap W=0$. Then for any weight vector $w=\sum_{{\mathbf{l}}\in \mathbb{M}}d'^{{\mathbf{l}}}v_0\otimes u_{{\mathbf{l}}}\in W$, where $u_{{\mathbf{l}}}\in K^{\mathfrak{D}'}, d'^{{\mathbf{l}}}=\cdots (d'_{-2})^{l_2}(d'_{-1})^{l_1}$ if $r_S=0$, or $d'^{{\mathbf{l}}}=\cdots (d'_{-2})^{l_2}$ if $r_S=-1$, and all ${\rm{w}}({\mathbf{l}})\ge 1$ are equal. Note that $h_{k+\frac{1}{2}}w=\sum_{{\mathbf{l}}\in \mathbb{M}}d'^{{\mathbf{l}}}v_0\otimes h_{k+\frac{1}{2}}u_{{\mathbf{l}}}$ either equals to $0$ or has the same weight as $w$ under the action of $d_0^{\prime}$. So $U(\mathfrak{D}')\mathfrak{K}'\cap ({\mathbb C} v_0\otimes K^{\mathfrak{D}'})=0$. The maximality of $\mathfrak{K}'$ forces that $\mathfrak{K}'=U(\mathfrak{D}')\mathfrak{K}'$ is a proper $\mathfrak{D}'$-submodule of $V^{\mathfrak{D}'}\otimes K^{\mathfrak{D}'}$. Since $\mathfrak{K}$ is a maximal proper $\mathfrak{D}'$-submodule of $V^{\mathfrak{D}'}\otimes K^{\mathfrak{D}'}$, it follows that $\mathfrak{K}=\mathfrak{K}'$.
By Lemma \ref{lem4.5'} we know that $\mathfrak{K}$ is generated by $P_1( {\mathbb C} v_0\otimes K^{\mathfrak{D}'})={\mathbb C} P_1 v_0\otimes K^{\mathfrak{D}'}$ and $P_2 ({\mathbb C} v_0\otimes K^{\mathfrak{D}'})={\mathbb C} P_2 v_0\otimes K^{\mathfrak{D}'}$. Let $V'$ be the maximal submodule of $V$ generated by $P_1v_0$ and $P_2v_0$, then $\mathfrak{K}=V'^{\mathfrak{D}'}\otimes K^{\mathfrak{D}'}$. Therefore, $$S\cong (V^{\mathfrak{D}'}\otimes K^{\mathfrak{D}'})/(V'^{\mathfrak{D}'}\otimes K^{\mathfrak{D}'})\cong (V/V')^{\mathfrak{D}'}\otimes K^{\mathfrak{D}'},$$ which forces that $K^{\mathfrak{D}'}$ is a simple $\mathfrak{D}'$-module and hence a simple $\mathcal{H}$-module. So $S$ contains a simple $\mathcal{H}$-module $K$. By Corollary \ref{tensor} we know there exists a simple $\mathfrak{Vir}$-module $U\in {\mathcal{R}}_{\mathfrak{Vir}}$ such that $S\cong U^{\mathfrak{D}}\otimes K^{\mathfrak{D}}$, as desired. \end{proof}
\begin{lem}\label{lem4.11} Let $M$ be a $\mathfrak{Vir}^{(0)}$-module on which $\mathfrak{Vir}^{(1)}$ acts trivially. If any finitely generated ${\mathbb C}[d_0]$-submodule of $M$ is a free ${\mathbb C}[d_0]$-module, then any nonzero submodule of ${\rm Ind}^{\mathfrak{Vir}}_{\mathfrak{Vir}^{(0)}}M$ intersects with $M$ non-trivially. \end{lem}
\begin{proof} Let $V$ be a nonzero submodule of ${\rm Ind}^{\mathfrak{Vir}}_{\mathfrak{Vir}^{(0)}}M$. Take a nonzero $u\in V$. If $u\in M$, there is nothing to do. Now assume $u\in V\backslash M$. Write $u=\sum_{i=1}^n a_iu_i$ where $a_i\in \mathcal{U}(\mathfrak{Vir}_{\le 0})$, $u_i\in M$. Since $M_1=\sum_{1\le i\le n}{\mathbb C}[d_0]u_i$ ( a $\mathfrak{Vir}^{(0)}$-submodule of $M$ ) is a finitely generated ${\mathbb C}[d_0]$-module, we see $M_1$ is a free module over ${\mathbb C}[d_0]$ by the assumption. Without loss of generality, we may assume that $M_1=\oplus_{1\le i\le n}{\mathbb C} [d_0]u_i$ with basis $u_1,\cdots,u_n$ over ${\mathbb C}[d_0]$. Note that each $a_i$ can be expressed as a sum of eigenvalue subspaces of $\mbox{\rm ad}\, d_0$ for $1\leq i\leq n$. Assume that $a_1$ has a maximal eigenvalue among all $a_i$ for $1\leq i\leq n$. Then $a_1u_1\notin M$. For any $\lambda\in{\mathbb C}$, let $M_1(\lambda)$ be the ${\mathbb C}[d_0]$-submodule of $M_1$ generated by $u_2, u_3,\cdots, u_n, d_0u_1-\lambda u_1$. Then $M_1/M_1(\lambda)$ is a one-dimensional $\mathfrak{Vir}^{(0)}$-module with $d_0(u_1+M_1(\lambda))=\lambda u_1+M_1(\lambda)$. By the Verma module theory for Virasoro algebra we know that there exists some $0\ne \lambda_0\in {\mathbb C}$ such that the corresponding Verma module $\mathfrak{V}={\rm Ind}_{\mathfrak{Vir}^{(0)}}^{\mathfrak{Vir}}(M_1/M_1(\lambda_0))$ is irreducible. We know that $u=a_1u_1\ne0$ in $\mathfrak{V}$. Hence we can find a homogeneous $w\in \mathcal{U}(\mathfrak{Vir}^+)$ such that $wa_1u_1=f_1(d_0)u_1$ in ${\rm Ind}^{\mathfrak{Vir}}_{\mathfrak{Vir}^{(0)}}M$, where $0\neq f_1(d_0)\in{\mathbb C}[d_0]$. So $wu=\sum_{i=1}^n wa_iu_i=\sum_{i=1}^n f_i(d_0)u_i $ for $f_i(d_0)\in{\mathbb C}[d_0]$, $1\leq i\leq n$. Therefore, $0\ne wu\in V\cap M_1\subset V\cap M,$ as desired. \end{proof}
\begin{pro}\label{eigenvalue prop} Let $S$ be a simple restricted $\mathfrak{D}$-module with level $\ell \not=0$. If $m_S=2n_s\ge2$, $r_S=1$, then $d_0'$ has an eigenvector in $K$. \end{pro}
\begin{proof} Suppose first that any finitely generated ${\mathbb C}[d'_0]$-submodule of $K={\rm Ind}_{\mathcal{H}^{(-n_S)}}^{\mathcal{H}}K_0$ is a free ${\mathbb C}[d'_0]$-module. By Lemma \ref{lem4.11} we see that the following $\mathfrak{D}'$-module homomorphism \begin{eqnarray*} \tau:\text{\rm Ind}_{\mathfrak{D}'^{(0,-\infty)}}^{\mathfrak{D}'}K=\text{\rm Ind}_{\mathfrak{Vir}'^{(0)}}^{\mathfrak{Vir}'}K&\longrightarrow & S, \cr
x\otimes u&\mapsto &xu, x\in \mathcal{U}(\mathfrak{Vir}'), u\in K. \end{eqnarray*} is an isomorphism. So $S={\rm Ind}_{\mathfrak{Vir}'^{(0)}}^{\mathfrak{Vir}'}K$, and consequently, $K$ is an irreducible $\mathfrak{D}'^{(0,-\infty)}$-module. Since $\mathfrak{Vir}'^{(1)}K=0$, we consider $K$ as an irreducible module over the Lie algebra $\mathcal{H}\oplus {\mathbb C} d'_0$. Since $d'_0$ is the center of the Lie algebra $\mathcal{H}\oplus {\mathbb C} d'_0$, we see that the action of $d'_0$ on $K$ is a scalar, a contradiction. So this case does not occur.
Now there exists some finitely generated ${\mathbb C}[d'_0]$-submodule $M$ of $K$ that is not a free ${\mathbb C}[d'_0]$-module. Since ${\mathbb C}[d'_0]$ is a principal ideal domain, by the structure theorem of finitely generated modules over a principal ideal domain, there exists a monic polynomial $f(d'_0)\in{\mathbb C}[d'_0]$ with positive degree and nonzero element $u\in M$ such that $f(d'_0)u=0$. Furthermore, we can write $f(d'_0)=(d_0'-\lambda_1) (d_0'-\lambda_2)\cdots (d_0'-\lambda_p)$ for some $\lambda_1,\cdots,\lambda_p\in{\mathbb C}$. Then there exists some $s\leq p$ such that $w:=\prod_{i=s+1}^p(d_0'-\lambda_{j})u\neq 0$ and $d_0'w=\lambda_sw$, where we make convention that $w=u$ if $s=p$. Then $w$ is a desired eigenvector of $d_0'$. \end{proof}
\begin{pro} \label{pro4.13} Let $S$ be a simple restricted $\mathfrak{D}$-module with level $\ell \not=0$. If $m_S=2n_s\ge2$, $r_S=1$, then $K$ is
a simple $\mathcal{H}$-module and $S\cong U^{\mathfrak{D}}\otimes K^{\mathfrak{D}}$ for some simple $U\in \mathcal{R}_{\mathfrak{Vir}}$. \end{pro}
\begin{proof} We see that $S$ is a weight $\mathfrak{D}'$-module since $S$ is a simple $\mathfrak{D}'$-module and $d'_0$ has an eigenvector. From Lemma \ref{lem4.4}(iii), $K$ and $K_0$ are weight $\mathfrak{D}'$-modules as well. We can take some $0\ne u_0\in K$ such that $d_0'u_0=\lambda u_0$ for some $\lambda \ne 0$ by Proposition \ref{eigenvalue prop}. Set $K'=U(\mathcal{H})u_0$, which is an $\mathcal{H}$ submodule of $K$. Then we have the $\mathfrak{D}'$-module $K'^{\mathfrak{D}'}$, on which $\mathfrak{Vir}'$ acts trivially by definition for any $n\in{\mathbb Z}$. Let ${\mathbb C} v_0$ be the one-dimensional $\mathfrak{D}'^{(0,-\infty)}$-module defined by $d'_0v_0=\lambda v_0, d_n'v_0=h_{k+\frac{1}{2}}v_0={\bf c}_2v_0=0, n\in{\mathbb Z}_+,k\in{\mathbb Z}$, ${\bf c}_1'v_0=(c-2)v_0$. Then ${\mathbb C} v_0\otimes K'^{\mathfrak{D}'}$ is a $\mathfrak{D}'^{(0,-\infty)}$-module with central charge $c-1$ and level $\ell$. There is a $\mathfrak{D}'^{(0,-\infty)}$-module homomorphism \begin{eqnarray*} \tau_{K'}: {\mathbb C} v_0\otimes K'^{\mathfrak{D}'}& \longrightarrow & S,\cr
v_0\otimes u &\mapsto & u, \forall u\in K', \end{eqnarray*} which is injective and can be extended to be the following $\mathfrak{D}'$-module homomorphism \begin{eqnarray*} \tau:\text{\rm Ind}_{{\mathfrak{D}'}^{(0,-\infty)}}^{\mathfrak{D}'}({\mathbb C} v_0\otimes K'^{\mathfrak{D}'})&\longrightarrow & S, \cr
x(v_0\otimes u)&\mapsto &xu, x\in \mathcal{U}(\mathfrak{D}'), u\in K'. \end{eqnarray*} Since $S$ is a simple $\mathfrak{D}'$ module and $\tau\ne 0$, we see that $\tau$ is surjective. By similar arguments in the proof of Proposition \ref{pro4.10}, we can obtain that $K'$ is a simple $\mathcal{H}$-module. By Corollary \ref{tensor} we know there exists a simple $\mathfrak{Vir}$-module $U\in {\mathcal{R}}_{\mathfrak{Vir}}$ such that $S\cong U^{\mathfrak{D}}\otimes K'^{\mathfrak{D}}$, as desired. Now it is clear that $K=K'$. \end{proof}
We are now in a position to present the following main result on a classification of simple restricted $\mathfrak{D}$-modules with nonzero level.
\begin{theo}\label{mainthm} Let $S$ be a simple restricted $\mathfrak{D}$-module with level $\ell \not=0$. The invariants $m_S, n_S, r_S$ of $S$, $U_0, U(2), K_0, K$ are defined as before. Then one of the following cases occurs.
{\rm Case 1}: $n_S=0$.
In this case, $S\cong H^{\mathfrak{D}}\otimes U^{\mathfrak{D}}$ as $\mathfrak{D}$-modules for some simple modules $H\in \mathcal{R}_{\mathcal{H}}$ and $U\in \mathcal{R}_{\mathfrak{Vir}}$.
{\rm Case 2}: $n_S>0$.
In this case, we further have the following three subcases.
{\rm Subcase 2.1}: $m_S>2n_S$.
In this subcase, $S\cong \text{\rm Ind}^{\mathfrak{D}}_{\mathfrak{D}^{(0,-n_S)}}(U_0)$.
{\rm Subcase 2.2}: $m_S=2n_S$.
In this subcase, we have \begin{equation*} S\cong\begin{cases} K^{\mathfrak{D}}, &{\text{ if }}r_S=-\infty,\cr U^{\mathfrak{D}}\otimes K^{\mathfrak{D}}, &{\text{ if }}-1\leq r_S\leq 1, \cr \text{\rm Ind}_{\mathfrak{D}^{(0,-n_S)}}^{\mathfrak{D}}K_0, &{\text { otherwise, }}\end{cases} \end{equation*} where $U\in \mathcal{R}_{\mathfrak{Vir}}$.
{\rm Subcase 2.3}: $m_S<2n_S$.
In this subcase, we have \begin{equation*} S\cong\begin{cases} \text{\rm Ind}^{\mathfrak{D}}_{\mathfrak{D}^{(0,-(m_S-n_S))}}(U_0),
&{\text{ if }}m_S\geq 2,\cr \text{\rm Ind}^{\mathfrak{D}}_{\mathfrak{D}^{(0,-(2-n_S))}}(U(2)),
&{\text{ if }} m_S<2, n_S>1, \cr \text{\rm Ind}_{\mathfrak{D}^{(0,-1)}}^{\mathfrak{D}}K_0, &{\text { otherwise. }}\end{cases} \end{equation*} \end{theo}
\begin{proof} The assertion follows directly from Proposition \ref{prop4.3}, Proposition \ref{prop for -inf}, Proposition \ref{prop4.6}, Corollary \ref{case 2}, Proposition \ref{pro4.10} and Proposition \ref{pro4.13}. \end{proof}
\begin{rem}By Theorems \ref{MT} and \ref{mainthm}, we know that any simple restricted module $S$ is a highest weight $\mathfrak{Vir}$-module with trivial action of $\mathcal{H}$, or a tensor product of a simple restricted $\mathfrak{Vir}$-module and a simple restricted $\mathcal{H}$-module, or an induced module from some simple module $M$ over certain subalgebra of $\mathfrak{D}$. Moreover, $M$ can be viewed as a simple module over some finite-dimensional solvable Lie algebra. This reduces the study of such $\mathfrak{D}$-modules to the study of simple modules over the corresponding finite-dimensional solvable Lie algebras. \end{rem}
\section{Simple restricted $\bar {\mathfrak{D}}$-modules with nonzero level}\label{char}
In this section we will determine all simple restricted ${\bar {\mathfrak{D}}}$-modules $M$ of level $\ell\ne0$. The main method we will use is similar to the one used in Section 4.
For a given simple restricted $\bar {\mathfrak{D}}$-module $M$ with level $\ell \not=0$, we define the following invariants of $M$ as follows: $$M(r)={\text{Ann}}_M(\bar {\mathcal{H}}^{(r)}), n_M=\min\{r\in {\mathbb Z}:M(r)\ne0\}, M_0=M{(n_M)}.$$
\begin{lem}\label{lem4.2'} Let $M$ be an irreducible restricted $\bar \mathfrak{D}$-module with level $\ell \not=0$. \begin{enumerate} \item[\rm(i)] $n_M\in{\mathbb N}$, and $h_{n_M-1}$ acts injectively on $M_0$. \item[\rm(ii)] $M_0$ is a nonzero $\bar \mathfrak{D}^{(0,-(n_M-1))}$-module, and is invariant under the action of the operators $\bar{L}_n$ defined in (\ref{rep1}) for $n\in{\mathbb N}$. \end{enumerate} \end{lem}
\begin{proof}
(i) Assume that $n_M<0$. Take any nonzero $v\in M_0$, we then have $$h_{1}v=0=h_{-1}v.$$ This implies that $v=\frac{1}{\ell}[h_{1},h_{-1}]v=0$, a contradiction. Hence, $n_M\in{\mathbb N}$.
The definition of $n_M$ means that $h_{n_M-1}$ acts injectively on $M_0$.
(ii) It is obvious that $M_0\neq 0$ by definition. For any $w\in M_0$, $i, j, k\in{\mathbb N}$, we have $$h_{k+n_M}d_iw=d_ih_{k+n_M}w+(k+n_M)h_{i+k+n_M}w=0,$$ and $$h_{k+n_M}h_{j-n_M+1}w=h_{j-n_M+1}h_{k+n_M}w=0.$$ Hence, $d_iw, h_{j-n_M+1}w\in M_0$, i.e., $M_0$ is a nonzero $\mathfrak{D}^{(0,-(n_M-1))}$-module.
For $i, n\in{\mathbb N}$, $w\in M_0$, noticing $n_M\ge 0$ by (i), it follows from (\ref{rep3}) that \begin{eqnarray*} h_{i+n_M}\bar L_nw&=\Big(\bar L_nh_{i+n_M}+(i+n_M)h_{n+i+n_M}\Big)w=0.
\end{eqnarray*} This implies that $\bar L_nw\in M_0$ for $n\in{\mathbb N}$, that is, $M_0$ is invariant under the action of the operators $\bar L_n$ for $n\in{\mathbb N}$. \end{proof}
\begin{pro}\label{prop4.3'} Let $M$ be a simple restricted $\bar \mathfrak{D}$-module with level $\ell \not=0$.
If $n_M=0, 1$, then $M\cong H^{\bar \mathfrak{D}}\otimes U^{\bar \mathfrak{D}}$ as $\bar \mathfrak{D}$-modules for some simple modules $H\in \mathcal{R}_{\bar \mathcal{H}}$ and $U\in \mathcal{R}_{\mathfrak{Vir}}$. \end{pro} \begin{proof} Since $n_M=0, 1$, we take any nonzero $v\in M_0$. Then ${\mathbb C} v$ is a $\bar \mathcal{H}^{(0)}$-module. Let $H=\mathcal{U}(\bar \mathcal{H})v$, the $\bar \mathcal{H}$-submodule of $M$ generated by $v$. It follows from representation theory of Heisenberg algebras that $\text{\rm Ind}^{\bar \mathcal{H}}_{\bar \mathcal{H}^{(0)}}({\mathbb C} v)$ is a simple $\bar \mathcal{H}$-module. Consequently, the following surjective $\bar \mathcal{H}$-module homomorphism \begin{eqnarray*} \varphi:\, \text{\rm Ind}^{\bar \mathcal{H}}_{\bar \mathcal{H}^{(0)}}({\mathbb C} v) &\longrightarrow & H\\ \sum_{\mathbf{i}\in\mathbb{M}}a_{\mathbf{i}} h^{\mathbf{i}}\otimes v&\mapsto & \sum_{\mathbf{i}\in\mathbb{M}} a_{\mathbf{i}} h^{\mathbf{i}} v \end{eqnarray*} is an isomorphism, that is, $H$ is a simple $\bar \mathcal{H}$-module, which is certainly restricted. Then the desired assertion follows directly from \cite[Theorem 12]{LZ3}. \end{proof}
Next we assume that $n_M\ge 2$.
We define the operators $d_n'=d_n-\bar L_n$ on $M$ for $n\in{\mathbb Z}$. Since $M$ is a restricted $\bar \mathfrak{D}$-module, then $d_n'$ is well-defined for any $n\in{\mathbb Z}$. By (\ref{rep3}) and (\ref{rep3'}), we have \begin{equation}\label{vir-bracket'} [d_m',{\bar {\bf c}}'_1]=0, [d_m',d_n']=(m-n)d_{m+n}'+\frac{m^3-m}{12}\delta_{m+n,0}{\bar {\bf c}}'_1, m,n\in{\mathbb Z}, \end{equation}where ${\bar {\bf c}}'_1=c-(1-\frac{12z^2}{\ell})\text{id}_M$ and $c$ is the central charge of $M$. So the operator algebra $$\mathfrak{Vir}'=\bigoplus_{n\in{\mathbb Z}}{\mathbb C} d_n'\oplus {\mathbb C}{\bar {\bf c}}'_1$$
is isomorphic to the Virasoro algebra $\mathfrak{Vir}$. Since $[d_n,h_{k}]=[\bar L_n,h_{k}]=-kh_{n+k}+\delta_{n+k,0}(n^2+n){\bar {\bf c}}_2,$ we have
\begin{equation}\label{d'bracket}
[d'_n,h_{k}]=0, n, k\in{\mathbb Z}\end{equation}
and hence $[\mathfrak{Vir}',\bar \mathcal{H}+{\mathbb C} {\bar {\bf c}}_2]=0$. Clearly, the operator algebra
$\bar \mathfrak{D}'=\mathfrak{Vir}'\oplus (\bar \mathcal{H}+{\mathbb C} {\bar {\bf c}}_2)$ is a direct sum, and $M=\mathcal{U}(\bar \mathfrak{D})v=\mathcal{U}(\bar \mathfrak{D}')v$ for any $ v\in M\setminus\{0\}$. Let $$Y_n=\bigcap_{p\ge n}{\rm Ann}_{M_0}(d_p'), r_M=\min\{n\in{\mathbb Z}:Y_n\ne0\}, K_0=Y_{r_M}. $$ Noting that $M$ is a restricted $\bar \mathfrak{D}$-module, we know that $r_M<+\infty$. If $Y_n\ne0$ for any $n\in{\mathbb Z}$, we define $r_M=-\infty$. Denote by $K=\mathcal{U}(\bar \mathcal{H})K_0$.
\begin{lem}\label{lem4.4'} Let $M$ be a simple restricted $\bar \mathfrak{D}$-module with level $\ell \not=0$. Then the following statements hold. \begin{enumerate} \item[\rm(i)] $ r_M\ge -1$ or $r_M=-\infty$. \item[\rm(ii)] If $r_M\ge -1$, then $K_0$ is a $\bar \mathfrak{D}^{(0,-(n_M-1))}$-module and $h_{n_M-1}$ acts injectively on $K_0$. \item[\rm(iii)] $K$ is a $\bar \mathfrak{D}^{(0,-\infty)}$-module and $K(z)^{\bar \mathfrak{D}}$ has a $\bar \mathfrak{D}$-module structure by (\ref{rep1})-(\ref{rep2}). \item[\rm{(iv)}] $K_0$ and $K$ are invariant under the actions of $L_n$ and $d_n'$ for $n\in{\mathbb N}$. \item[\rm(v)] If $r_M\ne -\infty$, then $d'_{r_M-1}$ acts injectively on $K_0$ and $K$. \end{enumerate} \end{lem}
\begin{proof} (i) If $Y_{-2}\ne 0$, then $d'_{p}K_0=0, p\ge -2$. We deduce that $\mathfrak{Vir}' K_0=0$ and hence $r_M=-\infty$.
If $Y_{-2}=0$, then $r_M\ge -1$.
(ii) For any $0\ne v\in K_0$ and $x\in {\bar \mathfrak{D}^{(0,-(n_M-1))}}$, it follows from Lemma \ref{lem4.2'}(ii) that $xv\in M_0$. We need to show that $d'_pxv=0, p\ge r_M$. Indeed, $d_p'h_{k}v=h_{k}d_p'v=0$ by (\ref{d'bracket}) for any $k\geq -(n_M-1)$. Moreover, it follows from (\ref{rep3'}) and (\ref{vir-bracket'}) that $$d_p'd_nv=d_nd_p'v+[d_p', d_n]v=(n-p)d_{p+n}'v=0, \forall n\in{\mathbb N}.$$ Hence, $d'_pxv=0, p\ge r_M$, that is, $xv\in K_0$, as desired.
Since $0\ne K_0 \subseteq M_0$, we see that $h_{n_M-1}$ acts injectively on $K_0$ by Lemma \ref{lem4.2'}(i).
(iii) follows from (ii).
(iv) Note that if $n_M=0$, then $\bar L_nK_0=0$ for any $n\in{\mathbb N}$. For $n_M>0$ we compute that $$\bar L_n=\frac{1}{2\ell}\sum_{k\in{\mathbb Z}}:h_{n-k}h_k:+\frac{(n+1)z}{\ell}h_n=\frac{1}{2\ell}\sum_{-(n_M-1)\le k\le n_M-1}:h_{n-k}h_k:+\frac{(n+1)z}{\ell}h_n, n\in{\mathbb N}.$$ We see $\bar L_nK_0\subset K_0$ and $\bar L_nK\subset K$ by (ii), and hence $d'_nK_0\subset K_0$ and $d'_nK\subset K$.
(v) follows directly from the definitions of $r_M$ and $K$. \end{proof}
We first consider the case $r_M=-\infty$.
\begin{pro}\label{pro3.5'} Let $M$ be a simple restricted $\bar \mathfrak{D}$-module with central charge $c$ and level $\ell \not=0$. If $r_M=-\infty$, then $M= K(z)^{\bar \mathfrak{D}}$ for some $z\in{\mathbb C}$. Hence $c=1-\frac{12z^2}{\ell}$
and $K$ is a simple $\bar \mathcal{H}$-module. \end{pro} \begin{proof} Since $r_M=-\infty$, we see that $\mathfrak{Vir}' K_0=0$. This together with (\ref{vir-bracket'}) implies that $c=1-\frac{12z^2}{\ell}$. Noting that $[\mathfrak{Vir}',\bar \mathcal{H}+{\mathbb C}{ \bar {\bf c}}_2]=0$, we further obtain that $\mathfrak{Vir}' K=0$, that is, $d_nv=\bar L_nv\in K$ for any $v\in K$ and $n\in{\mathbb Z}$. Hence $K(z)^{\bar \mathfrak{D}}$ is a $\bar \mathfrak{D}$-submodule of $M$, yielding that $M= K(z)^{\bar \mathfrak{D}}$. In particular, $K$ is a simple $\bar \mathcal{H}$-module. \end{proof}
\begin{pro}\label{prop4.6'} Let $M$ be a simple restricted $\bar \mathfrak{D}$-module with level $\ell \not=0$. If $r_M\ge 2$ and $n_M\ge 2$, then $K_0$ is a simple $\bar \mathfrak{D}^{(0,-(n_M-1))}$-module and $M\cong \text{\rm Ind}_{\bar \mathfrak{D}^{(0,-(n_M-1))}}^{\bar \mathfrak{D}}K_0$. \end{pro}
\begin{proof}We first show that $\text{\rm Ind}_{\bar \mathfrak{D}^{(0,-(n_M-1))}}^{\bar \mathfrak{D}^{(0,-\infty)}}K_0 \cong K$ as $\bar \mathfrak{D}^{(0,-\infty)}$ modules. For that, let \begin{eqnarray*} \phi:\, \text{\rm Ind}_{\bar \mathfrak{D}^{(0,-(n_M-1))}}^{\bar \mathfrak{D}^{(0,-\infty)}}K_0 &\longrightarrow &K\\ \sum_{{\mathbf{k}}\in\mathbb{M}} h^{{\mathbf{k}}}\otimes v_{{\mathbf{k}}}&\mapsto & \sum_{{\mathbf{k}}\in\mathbb{M}} h^{{\mathbf{k}}} v_{{\mathbf{k}}},
\end{eqnarray*} where $h^{{\mathbf{k}}}=\cdots h^{k_2}_{-2-(n_M-1)}h_{-1-(n_M-1)}^{k_1}\in\mathcal{U}(\bar \mathcal{H})$. Then $\phi$ is a $\bar \mathfrak{D}^{(0,-\infty)}$-module epimorphism and $\phi|_{K_0}$ is one-to-one.
{\bf Claim}. Any nonzero submodule $V$ of $\text{\rm Ind}_{\bar \mathfrak{D}^{(0,-(n_M-1))}}^{\bar \mathfrak{D}^{(0,-\infty)}}K_0$ does not intersect with $K_0$ trivially.
Assume $V\cap K_0=0$. Let $v=\sum_{{\mathbf{k}}\in\mathbb{M}} h^{{\mathbf{k}}}\otimes v_{{\mathbf{k}}}\in V\backslash K_0 $ with minimal degree ${\mathbf{i}}$. Then $\bf 0\prec {\mathbf{i}}$.
Let $p=\text{min}\{s:i_s\ne 0\}$. Since $h_{p+n_M-1}v_{{\mathbf{k}}}=0$, we have $h_{p+n_M-1}h^{{\mathbf{k}}}v_{{\mathbf{k}}}=[h_{p+n_M-1},h^{{\mathbf{k}}}]v_{{\mathbf{k}}}$. The following equality
\begin{equation*} [h_{i},h_{j_1}h_{j_2}\cdots h_{j_t}]=\sum_{1\le s\le t}\delta_{i+j_s,0}i {\bar {\bf c}}_3 h_{j_1}\cdots \hat{h}_{j_{s}}\cdots h_{j_t}, \,i, j_1\le j_2\le\cdots\le j_t\in{\mathbb Z} \end{equation*} implies that if $k_p=0$ then $h_{p+n_M-1}h^{{\mathbf{k}}}v_{{\mathbf{k}}}=0$; and if $k_p\ne 0$, noticing the level $\ell\ne 0$, then $[h_{p+n},h^{{\mathbf{k}}}]=\lambda h^{{\mathbf{k}}-\epsilon_p}$ for some $\lambda\in{\mathbb C}^*$ and hence $$\text{deg}([h_{p+n_M-1},h^{{\mathbf{k}}}]v_{{\mathbf{k}}})={\mathbf{k}}-\epsilon_p\preceq {\mathbf{i}}-\epsilon_p,$$ where the equality holds if and only if ${\mathbf{k}}={\mathbf{i}}$. Hence $\deg(h_{p+n_M-1}v)={\mathbf{i}}-\epsilon_p\prec {\mathbf{i}}$ and $h_{p+n_M-1}v\in V$, contrary to the choice of $v$. Thus, the claim holds.
From the Claim we know that the kernel of $\phi$ must be zero and hence $\phi$ is an isomorphism.
By Lemma \ref{lem4.4'}(v), we see that $d_{r_M-1}'$ acts injectively on $K$.
As $\bar \mathfrak{D}$-modules, $$\text{\rm Ind}_{\bar \mathfrak{D}^{(0,-(n_M-1))}}^{\bar \mathfrak{D}}K_0\cong\text{\rm Ind}_{\bar \mathfrak{D}^{(0,-\infty)}}^{\bar \mathfrak{D}}(\text{\rm Ind}_{\bar \mathfrak{D}^{(0,-(n_M-1))}}^{\bar \mathfrak{D}^{(0,-\infty)}}K_0)\cong\text{\rm Ind}_{\bar \mathfrak{D}^{(0,-\infty)}}^{\bar \mathfrak{D}}K.$$ And we further have $\text{\rm Ind}_{\bar \mathfrak{D}^{(0,-\infty)}}^{\bar \mathfrak{D}}K\cong \text{\rm Ind}_{\mathfrak{Vir}'^{(0)}}^{\mathfrak{Vir}'}K$ as vector spaces. Moreover, we have the following $\bar \mathfrak{D}$-module epimorphism \begin{eqnarray*} \pi: \text{\rm Ind}_{\bar \mathfrak{D}^{(0,-\infty)}}^{\bar \mathfrak{D}}K=\text{\rm Ind}_{\mathfrak{Vir}'^{(0)}}^{\mathfrak{Vir}'}K&\rightarrow& M,\cr \sum_{{\mathbf{l}}\in\mathbb{M}}d'^{{\mathbf{l}}}\otimes v_{{\mathbf{l}}}&\mapsto& \sum_{{\mathbf{l}}\in\mathbb{M}}d'^{{\mathbf{l}}} v_{{\mathbf{l}}}, \end{eqnarray*}
where $d'^{{\mathbf{l}}}=\cdots (d'_{-2})^{l_2}(d'_{-1})^{l_1}$. We see that $\pi$ is also a $\mathfrak{Vir}'$-module epimorphism. By the proof of Theorem 2.1 in \cite{MZ2} we know that any nonzero $\mathfrak{Vir}'$-submodule of $\text{\rm Ind}_{\mathfrak{Vir}'^{(0)}}^{\mathfrak{Vir}'}K$ contain nonzero vectors of $K$. Note that $\pi|_K$ is one-to-one, we see that the image of any nonzero $\bar \mathfrak{D}$-submodule (and hence $\mathfrak{Vir}'$-submodule) of $\text{\rm Ind}_{\bar \mathfrak{D}^{(0,-\infty)}}^{\bar \mathfrak{D}}K$ must be a nonzero $\bar \mathfrak{D}$-submodule of $M$ and hence be the whole module $M$, which forces that the kernel of $\pi$ must be $ 0$. Therefore, $\pi$ is an isomorphism. Since $M$ is simple, we see $K_0$ is a simple $\bar \mathfrak{D}^{(0,-(n_M-1))}$-module.\end{proof}
\begin{pro}\label{eigenvalue prop'} Let $M$ be a simple restricted $\bar \mathfrak{D}$-module with level $\ell \not=0$. If $r_M=1$, then $d_0'$ has an eigenvector in $K$. \end{pro}
\begin{proof}Lemma \ref{lem4.4'} (iv) means that $K$ is a $\bar \mathfrak{D}'^{(0,-\infty)}$-module. Assume that any finitely generated ${\mathbb C}[d'_0]$-submodule of $K$ is a free ${\mathbb C}[d'_0]$-module. By Lemma \ref{lem4.11} we see that the following $\bar \mathfrak{D}'$-module homomorphism \begin{eqnarray*} \tau:\text{\rm Ind}_{\bar \mathfrak{D}'^{(0,-\infty)}}^{{\bar \mathfrak{D}}'}K=\text{\rm Ind}_{\mathfrak{Vir}'^{(0)}}^{\mathfrak{Vir}'}K&\longrightarrow & M, \cr
x\otimes u&\mapsto &xu, x\in \mathcal{U}(\mathfrak{Vir}'), u\in K. \end{eqnarray*} is an isomorphism. So $M={\rm Ind}_{\mathfrak{Vir}'^{(0)}}^{\mathfrak{Vir}'}K$, and consequently, $K$ is a simple $\bar \mathfrak{D}'^{(0,-\infty)}$-module. Since $r_M=1$ and $\mathfrak{Vir}'^{(1)}K=0$, $K$ can be seen as a simple module over the Lie algebra $\mathcal{H}\oplus {\mathbb C}{\bf c}_2\oplus{\mathbb C} d'_0$ where ${\mathbb C} d'_0$ lies in the center of the Lie algebra. Schur's lemma tells us that $d'_0$ acts as a scalar on $K$, a contradiction. So this case will not occur.
Therefore, there exists some finitely generated ${\mathbb C}[d'_0]$-submodule $W$ of $K$ that is not a free ${\mathbb C}[d'_0]$-module. Since ${\mathbb C}[d'_0]$ is a principal ideal domain, by the structure theorem of finitely generated modules over a principal ideal domain, there exists a monic polynomial $f(d'_0)\in{\mathbb C}[d'_0]$ with minimal positive degree and nonzero element $u\in W$ such that $f(d'_0)u=0$. Write $f(d'_0)=\Pi_{1\le i\le s}(d_0'-\lambda_i)$, $\lambda_1,\cdots,\lambda_s\in{\mathbb C}$. Denote $w:=\prod_{i=1}^{s-1}(d_0'-\lambda_{i})u\neq 0$, we see $(d_0'-\lambda_s)w=0$ where we make convention that $w=u$ if $s=1$. Then $w$ is a desired eigenvector of $d_0'$. \end{proof}
\begin{pro} \label{pro3.8} Let $M$ be a simple restricted $\bar \mathfrak{D}$-module with level $\ell \not=0$. If $r_M=0,\pm 1$, then $K$ is
a simple $\mathcal{H}$-module and $M \cong K(z)^{\bar \mathfrak{D}}\otimes U^{\bar \mathfrak{D}}$ for some simple module $U\in \mathcal{R}_{\mathfrak{Vir}}$ and some $z\in{\mathbb C}$. \end{pro}
\begin{proof} If $r_M=1$, then by Proposition \ref{eigenvalue prop'} we know that there exists $0\ne u\in K$ such that $d_0'u=\lambda u$ for some $\lambda \ne 0$; if $r_M=0,-1$, then $d_0'K=0$. In summary, for all the three cases, $d_0'$ has an eigenvector in $ K$. Since $M$ is a simple $\bar \mathfrak{D}'$-module, Schur's lemma implies that $h_0, {\bar {\bf c}}'_1, {\bar {\bf c}}_2, {\bar {\bf c}}_3$ act as scalars on $M$. So $M$ is a weight $\bar \mathfrak{D}'$-module, and $K$ is a weight module for $\bar \mathfrak{D}'^{(r_M-\delta_{r_{_M}, 1},-\infty)}$. Take a weight vector $u_0\in K$ with $d'_0u_0=\lambda_0u_0$ for some $\lambda_0\in{\mathbb C}$.
Set $K'=\mathcal{U}(\bar \mathcal{H})u_0$, which is an $\bar \mathcal{H}$ submodule of $K$. Now we define the $\bar \mathfrak{D}'$-module $K'^{{\bar \mathfrak{D}}'}$ with trivial action of $\mathfrak{Vir}'$. Let ${\mathbb C} v_0$ be the one-dimensional $\bar \mathfrak{D}'^{(r_M-\delta_{r_{_M}, 1},-\infty)}$-module defined by $${\bar {\bf c}}_1'v_0=(c-1+\frac{12z^2}{\ell})v_0,\,\,\, d'_0 v_0=\lambda_0v_0, \,\,\,d_n'v_0=h_{k}v_0={\bar{\bf c}}_2v_0={\bar{\bf c}}_3v_0=0, \,\,\, 0\ne n\ge r_M,k\in{\mathbb Z}.$$ Then ${\mathbb C} v_0\otimes K'^{{\bar \mathfrak{D}}'}$ is a $\bar \mathfrak{D}'^{(r_M-\delta_{r_{_M}, 1},-\infty)}$-module with central charge $c-1+\frac{12z^2}{\ell}$ and level $\ell$. There is a $\bar \mathfrak{D}'^{(r_M-\delta_{r_{_M}, 1},-\infty)}$-module homomorphism \begin{eqnarray*} \tau_{K'}: {\mathbb C} v_0\otimes K'^{{\bar \mathfrak{D}}'}& \longrightarrow & M,\cr
v_0\otimes u &\mapsto & u, \forall u\in K', \end{eqnarray*} which is injective and can be extended to be the following $\bar \mathfrak{D}'$-module epimomorphism \begin{eqnarray*} \tau:\text{\rm Ind}_{{\bar \mathfrak{D}}'^{(r_M-\delta_{r_{_M}, 1},-\infty)}}^{\bar \mathfrak{D}'}({\mathbb C} v_0\otimes K'^{{\bar \mathfrak{D}}'})&\longrightarrow & M, \cr
x(v_0\otimes u)&\mapsto &xu, x\in \mathcal{U}(\bar \mathfrak{D}'), u\in K'. \end{eqnarray*}
By Lemma 8 in \cite{LZ3} we know that $$\text{\rm Ind}_{\bar \mathfrak{D}'^{(r_M-\delta_{r_{_M}, 1},-\infty)}}^{{\bar \mathfrak{D}}'}({\mathbb C} v_0\otimes K'^{{\bar \mathfrak{D}}'})\cong (\text{\rm Ind}_{\bar \mathfrak{D}'^{(r_M-\delta_{r_{_M}, 1},-\infty)}}^{{\bar \mathfrak{D}}'}{\mathbb C} v_0)\otimes K'^{{\bar \mathfrak{D}}'} =(\text{\rm Ind}_{\mathfrak{Vir}'^{(r_M-\delta_{r_{_M}, 1})}}^{\mathfrak{Vir}'}{\mathbb C} v_0)^{{\bar \mathfrak{D}}'}\otimes K'^{{\bar \mathfrak{D}}'}. $$Then we have the following $\bar \mathfrak{D}'$-module epimorphism \begin{eqnarray*} \tau':(\text{\rm Ind}_{\mathfrak{Vir}'^{(r_M-\delta_{r_{_M}, 1})}}^{\mathfrak{Vir}'}{\mathbb C} v_0)^{{\bar \mathfrak{D}}'}\otimes K'^{{\bar \mathfrak{D}}'}&\longrightarrow& M,\cr
xv_0\otimes u&\mapsto& xu, x\in \mathcal{U}(\mathfrak{Vir}'), u\in K'. \end{eqnarray*}
Note that $(\text{\rm Ind}_{\mathfrak{Vir}'^{(r_M-\delta_{r_{_M}, 1})}}^{\mathfrak{Vir}'}{\mathbb C} v_0)^{{\bar \mathfrak{D}}'}\otimes K'^{{\bar \mathfrak{D}}'}\cong\text{\rm Ind}_{\mathfrak{Vir}'^{(r_M-\delta_{r_{_M}, 1})}}^{\mathfrak{Vir}'}({\mathbb C} v_0\otimes K'^{{\bar \mathfrak{D}}'})$ as $\mathfrak{Vir}'$-modules, and $\tau'$ is also a $\mathfrak{Vir}'$-module epimorphism, $\tau'|_{{\mathbb C} v_0\otimes K'^{{\bar \mathfrak{D}}'}}$ is one-to-one, and $(\text{\rm Ind}_{\mathfrak{Vir}'^{(r_M-\delta_{r_{_M}, 1})}}^{\mathfrak{Vir}'}{\mathbb C} v_0)^{{\bar \mathfrak{D}}'}\otimes K'^{{\bar \mathfrak{D}}'}$ is a highest weight $\mathfrak{Vir}'$-module. Let $V=\text{\rm Ind}_{\mathfrak{Vir}'^{(r_M-\delta_{r_{_M}, 1})}}^{\mathfrak{Vir}'}{\mathbb C} v_0$ and $\mathfrak{K}=\text{Ker}(\tau')$. It should be noted that $${\mathbb C} v_0\otimes K'^{{\bar \mathfrak{D}}'}=\{u\in V^{{\bar \mathfrak{D}}'}\otimes K'^{{\bar \mathfrak{D}}'}\mid d_0^{\prime}u=\lambda_0u\}.$$ We see that $({\mathbb C} v_0\otimes K'^{{\bar \mathfrak{D}}'})\cap \mathfrak{K}=0$. Let $\mathfrak{K}^{\prime}$ be the sum of all $\mathfrak{Vir}'$-submodules $W$ of $V^{{\bar \mathfrak{D}}'}\otimes K'^{{\bar \mathfrak{D}}'}$ with $({\mathbb C} v_0\otimes K'^{{\bar \mathfrak{D}}'})\cap W=0$, that is, the unique maximal (weight) $\mathfrak{Vir}'$-submodule of $V^{{\bar \mathfrak{D}}'}\otimes K'^{{\bar \mathfrak{D}}'}$ with trivial intersection with $({\mathbb C} v_0\otimes K'^{{\bar \mathfrak{D}}'})$. It is obvious that $\mathfrak{K}\subseteq\mathfrak{K}'$. Next we further show that $\mathfrak{K}=\mathfrak{K}'$. For that, take any $\mathfrak{Vir}'$- submodule $W$ of $V^{{\bar \mathfrak{D}}'}\otimes K'^{{\bar \mathfrak{D}}'}$ such that $({\mathbb C} v_0\otimes K'^{{\bar \mathfrak{D}}'})\cap W=0$. Then for any weight vector $w=\sum_{{\mathbf{l}}\in \mathbb{M}} d'^{{\mathbf{l}}}v_0\otimes u_{{\mathbf{l}}}\in W$, where $u_{{\mathbf{l}}}\in {K'}^{{\bar \mathfrak{D}}'}, d'^{{\mathbf{l}}}=\cdots (d'_{-2})^{l_2}(d'_{-1})^{l_1}$ if $r_M=1,0$, or $d'^{{\mathbf{l}}}=\cdots (d'_{-2})^{l_2}$ if $r_M=-1$, and all ${\rm{w}}({\mathbf{l}})\ge 1$ are equal. Note that $h_{k}w=\sum_{{\mathbf{l}}\in \mathbb{M}}d'^{{\mathbf{l}}}v_0\otimes h_{k}u_{{\mathbf{l}}}$ either equals to $0$ or has the same weight as $w$ under the action of $d_0^{\prime}$. So $\mathcal{U}(\bar \mathfrak{D}')W\cap ({\mathbb C} v_0\otimes K'^{{\bar \mathfrak{D}}'})=0$, i.e., $\mathcal{U}(\bar \mathfrak{D}')W\subset \mathfrak{K}'$. Hence $\mathcal{U}(\bar \mathfrak{D}')\mathfrak{K}'\cap ({\mathbb C} v_0\otimes K'^{{\bar \mathfrak{D}}'})=0$. The maximality of $\mathfrak{K}'$ forces that $\mathfrak{K}'=\mathcal{U}(\bar \mathfrak{D}')\mathfrak{K}'$ is a proper $\bar \mathfrak{D}'$-submodule of $V^{{\bar \mathfrak{D}}'}\otimes K'^{{\bar \mathfrak{D}}'}$. Since $\mathfrak{K}$ is a maximal proper $\bar \mathfrak{D}'$-submodule of $V^{{\bar \mathfrak{D}}'}\otimes K'^{{\bar \mathfrak{D}}'}$, it follows that $\mathfrak{K}=\mathfrak{K}'$.
By Lemma \ref{lem4.5'} we know that $\mathfrak{K}$ is generated by $P_1({\mathbb C} v_0\otimes K^{{\bar \mathfrak{D}}'})={\mathbb C} P_1 v_0\otimes K'^{{\bar \mathfrak{D}}'}$ and $P_2 ({\mathbb C} v_0\otimes K'^{{\bar \mathfrak{D}}'})={\mathbb C} P_2 v_0\otimes K'^{{\bar \mathfrak{D}}'}$. Let $V'$ be the maximal submodule of $V$ generated by $P_1v_0$ and $P_2v_0$, then $\mathfrak{K}=V'^{{\bar \mathfrak{D}}'}\otimes K'^{{\bar \mathfrak{D}}'}$. Therefore, \begin{equation}\label{cong} M\cong (V^{{\bar \mathfrak{D}}'}\otimes K'^{{\bar \mathfrak{D}}'})/(V'^{{\bar \mathfrak{D}}'}\otimes K'^{{\bar \mathfrak{D}}'})\cong (V/V')^{{\bar \mathfrak{D}}'}\otimes K'^{{\bar \mathfrak{D}}'},\end{equation} which forces that $K'^{{\bar \mathfrak{D}}'}$ is a simple $\bar \mathfrak{D}'$-module and hence a simple $\bar \mathcal{H}$-module. So $K'$ is a simple $\bar \mathcal{H}$-module. By \cite[Theorem 12]{LZ3} we know there exists a simple $\mathfrak{Vir}$-module $U\in {\mathcal{R}}_{\mathfrak{Vir}}$ such that $M\cong K'^{\bar \mathfrak{D}} \otimes U^{\bar \mathfrak{D}} $. From this isomorphism and some computations we see that $K_0\subseteq K'^{\bar \mathfrak{D}}\otimes v_0$ where $v_0$ is a highest weight vector. So $K=K'$.\end{proof}
We are now in a position to present the following main result on characterization of simple restricted $\bar \mathfrak{D}$-modules with nonzero level.
\begin{theo}\label{mainthm'} Let $M$ be a simple restricted $\bar \mathfrak{D}$-module with level $\ell \not=0$. The invariants $ n_M, r_M$ of $M$, $ K_0, K$ are defined as before. Then \begin{equation*} M\cong\begin{cases} K(z)^{\bar \mathfrak{D}}, &{\text{ if }}r_M=-\infty,\cr K(z)^{\bar \mathfrak{D}}\otimes U^{\bar \mathfrak{D}}, &{\text{ if }}-1\leq r_M\leq 1 \text{ or } n_M=0, 1, \cr \text{\rm Ind}_{\bar \mathfrak{D}^{(0,-(n_M-1))}}^{\bar \mathfrak{D}}K_0, &{\text { otherwise, }}\end{cases} \end{equation*} for some $U\in \mathcal{R}_{\mathfrak{Vir}}$ and some $z\in{\mathbb C}$. \end{theo}
\begin{proof} The assertion follows directly from Proposition \ref{prop4.3'}, Proposition \ref{pro3.5'}, Proposition \ref{prop4.6'}, Proposition \ref{pro3.8}. \end{proof}
The following result characterizes simple Whittaker modules over the twisted Heisenberg-Virasoro algebra $\bar \mathfrak{D}$.
\begin{theo}\label{thmmain3.10} Let $M$ be a $\bar \mathfrak{D}$-module (not necessarily weight) on which the algebra ${\bar \mathfrak{D}}^{+}$ acts locally finitely. Then the following statements hold. \begin{enumerate} \item[\rm(i)] The module $M$ contains a nonzero vector $v$ such that ${\bar \mathfrak{D}}^{+}\, v\subseteq{\mathbb C} v$. \item[\rm(ii)] If $M$ is simple, then $M$ is a Whittaker module or a highest weight module. \end{enumerate} \end{theo}
\begin{proof}
(i) Let $(M_1,\rho)$ be a finite dimensional ${\bar \mathfrak{D}}^{+}$-submodule of $M$. Then $M_1$ is also a finite dimensional $\mathfrak{Vir}_{\geq 1}$-module. Let $\mathfrak{a}:=\ker(\rho|_{\mathfrak{Vir}_{\geq 1}})$ be the kernel of the representation map of $\mathfrak{Vir}_{\geq 1}$ on $M_1$. Then $\mathfrak{a}$ is an ideal of $\mathfrak{Vir}_{\geq 1}$ of finite codimension. We claim that $d_n\in\mathfrak{a}$ for some $n\in{\mathbb Z}_+$. If this is not true, then there exists a minimal $m\in{\mathbb Z}_+$ such that $\mathfrak{a}$ contains an element of the form $a_{i_1}d_{i_1}+a_{i_2}d_{i_2}+\cdots+a_{i_{m+1}}d_{i_{m+1}}$ for positive integers $i_1<i_2<\cdots<i_{m+1}$ and nonzero complex numbers $a_{i_1}, a_{i_2},\cdots, a_{i_{m+1}}$. We further see that $\mathfrak{a}$ contains $$[d_{i_1},a_{i_1}d_{i_1}+a_{i_2}d_{i_2}+\cdots+a_{i_{m+1}}d_{i_{m+1}}]=a_{i_2}(i_2-i_1)d_{i_1+i_2}+a_{i_3}(i_3-i_1)d_{i_1+i_3}+ \cdots+a_{i_{m+1}}(i_{m+1}-i_1)d_{i_1+i_{m+1}},$$ which contradicts with the minimality of $m$. Hence the claim follows. Consequently, $$\widetilde{\mathfrak{Vir}}_{\geq n}:=\sum_{i\geq n,\, i\neq 2n}{\mathbb C} d_i={\mathbb C} d_n+ [d_n, \mathfrak{Vir}_{\geq 1}]\subseteq\mathfrak{a}.$$ Then $$\widetilde{\mathfrak{Vir}}_{\geq n}+ {\bar \mathcal{H}_{\geq n+1}}=\widetilde{\mathfrak{Vir}}_{\geq n}+[ {\bar \mathcal{H}_{\geq 1}},\widetilde{\mathfrak{Vir}}_{\geq n}]\subseteq\ker(\rho).$$ This implies that $M_1$ is a finite dimensional module over a finite dimensional solvable Lie algebra $\bar {\mathfrak{D}}^{+}/(\widetilde{\mathfrak{Vir}}_{\geq n}+ {\bar \mathcal{H}_{\geq n+1}})$. The desired assertion follows directly from Lie Theorem.
(ii) follows directly from (i) and \cite{MZ2}. \end{proof}
\begin{rem}From Theorem \ref{thmmain3.10} we know that if $M$ is a simple Whittaker module over $\bar \mathfrak{D}$ with nonzero level, and $\bar \mathfrak{D}^+ v\subset {\mathbb C} v$ for some nonzero vector $v\in M$, then $K=\mathcal{U}(\bar \mathcal{H})v=\mathcal{U}(\oplus_{r\in-{\mathbb Z}_+}{\mathbb C} h_r) v$ is a simple Whittaker module over $\bar \mathcal{H}$. Therefore, \cite[Theorem 12]{LZ3} implies that $M\cong U^{\bar \mathfrak{D}}\otimes K(z)^{\bar \mathfrak{D}}$ for some $U\in{\mathcal R}_{\mathfrak{Vir}}$. Clearly, $U$ is a simple Whittaker module or a simple highest weight module over $\mathfrak{Vir}$. \end{rem}
\section{Application one: characterization of simple highest weight modules and Whittaker modules over the mirror Heisenberg-Virasoro algebra}
Based on the results on structure of simple restricted modules over the mirror Heisenberg-Virasoro algebra $\mathfrak{D}$ given in Theorem \ref{MT} and Theorem \ref{mainthm}, we give characterization of simple highest weight $\mathfrak{D}$-modules and simple Whittaker $\mathfrak{D}$-modules in this section.
We first have the following result characterizing simple highest weight modules over the mirror Heisenberg-Virasoro algebra.
\begin{theo}\label{thmmain} Let $\mathfrak{D}$ be the mirror Heisenberg-Virasoro algebra with the triangular decomposition ${{\mathfrak{D}}}={\mathfrak{D}}^{+}\oplus {\mathfrak{D}}^{0}\oplus {\mathfrak{D}}^{-}$. Let $S$ be a $\mathfrak{D}$-module (not necessarily weight) on which every element in the algebra ${\mathfrak{D}}^{+}$ acts locally nilpotently. Then the following statements hold. \begin{enumerate} \item[\rm(i)] The module $S$ contains a nonzero vector $v$ such that ${\mathfrak{D}}^{+}\, v=0$. \item[\rm(ii)] If $S$ is simple, then $S$ is a highest weight module. \end{enumerate} \end{theo}
\begin{proof} (i) It follows from \cite[Theorem 1]{MZ1} that there exists a nonzero vector $v\in S$ such that $d_iv=0$ for any $i\in{\mathbb Z}_+$. If $h_{\frac{1}{2}}v=0$, then ${\mathfrak{D}}^{+}\, v=0$ as $d_1, d_2$ and $h_{\frac{1}{2}}$ generate ${\mathfrak{D}}^{+}$. Assume that $w:=h_{\frac{1}{2}}v\neq 0$. Then $$d_1w=d_1h_{\frac{1}{2}}v=h_{\frac{1}{2}}d_1v+[d_1, h_{\frac{1}{2}}]v=-\frac{1}{2}h_{\frac{3}{2}}v.$$ Similar arguments yield that the element $d_1^jw=\lambda h_{j+\frac{1}{2}}v$ for some $\lambda\in{\mathbb C}^*$ and $j\in{\mathbb Z}_+$. As $d_1$ acts locally nilpotently on $S$, it follow that there exists some $n\in{\mathbb Z}_+$ such that $h_{j+\frac{1}{2}}v=0$ for $j\geq n$.
We now show that for every $m\in{\mathbb N}$ there exists some nonzero element $u\in S$ such that $d_iu=h_{k+\frac{1}{2}}u=0$ for $i\in{\mathbb Z}_+$ and $k\geq m$ by a backward induction on $m$. The above arguments imply that the assertion is true for $m\geq n$. Assume that $0\neq u\in S$ satisfies that $d_iu=h_{k+\frac{1}{2}}u=0$ for $i\in{\mathbb Z}_+$ and $k\geq m>0$. If $h_{m-\frac{1}{2}}u=0$, then the induction step is proved. Otherwise, $h_{m-\frac{1}{2}}u\neq 0$, and there exists some $l\in{\mathbb N}$ such that $u^{\prime}:=h_{m-\frac{1}{2}}^lu\neq 0$ and $h_{m-\frac{1}{2}}u^{\prime}=h_{m-\frac{1}{2}}^{l+1}u=0$. Moreover, $d_iu^{\prime}=h_{k+\frac{1}{2}}u^{\prime}=0$ for $i\in{\mathbb Z}_+$ and $k\geq m-1$. The induction step follows.
(ii) By (i), we know that $S$ is a simple restricted $\mathfrak{D}$-module with $n_S=0$ and $ m_S\le 1$. From Theorem \ref{MT} and Case 1 of Theorem \ref{mainthm} we know that $S\cong H^{\mathfrak{D}}\otimes U^{\mathfrak{D}}$ as $\mathfrak{D}$-modules for some simple modules $H\in \mathcal{R}_{\mathcal{H}}$ and $U\in \mathcal{R}_{\mathfrak{Vir}}$. Moreover, $H=\text{\rm Ind}^{\mathcal{H}}_{\mathcal{H}^{(0)}}({\mathbb C} v)$ is a simple highest weight module over $\mathfrak{D}$. Note that every element in the algebra $\mathfrak{Vir}^{(1)}$ acts locally nilpotently on ${\mathbb C} v\otimes U$ by the assumption. This implies that the same property also holds on $U$. From \cite[Theorem 1]{MZ1} we know that $U$ is a simple highest weight $\mathfrak{Vir}$-module. This completes the proof. \end{proof}
As a direct consequence of Theorem \ref{thmmain}, we have \begin{cor} Let $S$ be an simple restricted $\mathfrak{D}$-module with $m_S\leq 1$ and $n_S=0$. Then $S$ is a highest weight module. \end{cor}
\begin{proof} The assumption that $m_S\leq 1$ and $n_S=0$ implies that there exists a nonzero vector $v\in M$ such that ${\mathfrak{D}}^{+}v=0$. Then $M=\mathcal{U}({\mathfrak{D}}^{-}+{\mathfrak{D}}^{0})v$. It follows that each element in ${\mathfrak{D}}^{+}$ acts locally nilpotently on $M$. Consequently, the desired assertion follows directly from Theorem \ref{thmmain}. \end{proof}
The following result characterizes simple Whittaker modules over the mirror Heisenberg-Virasoro algebra.
\begin{theo}\label{thmmain17} Let $M$ be a $\mathfrak{D}$-module (not necessarily weight) on which the algebra ${\mathfrak{D}}^{+}$ acts locally finitely. Then the following statements hold. \begin{enumerate} \item[\rm(i)] The module $M$ contains a nonzero vector $v$ such that ${\mathfrak{D}}^{+}\, v\subseteq{\mathbb C} v$. \item[\rm(ii)] If $M$ is simple, then $M$ is a Whittaker module or a highest weight module. \end{enumerate} \end{theo}
\begin{proof}
(i) Let $(M_1,\rho)$ be a finite dimensional ${\mathfrak{D}}^{+}$-submodule of $M$. Then $M_1$ is also a finite dimensional $\mathfrak{Vir}_{\geq 1}$-module. Let $\mathfrak{a}:=\ker(\rho|_{\mathfrak{Vir}_{\geq 1}})$ be the kernel of the representation map of $\mathfrak{Vir}_{\geq 1}$ on $M_1$. Then $\mathfrak{a}$ is an ideal of $\mathfrak{Vir}_{\geq 1}$ of finite codimension. We claim that $d_n\in\mathfrak{a}$ for some $n\in{\mathbb Z}_+$. If this is not true, then there exists a minimal $m\in{\mathbb Z}_+$ such that $\mathfrak{a}$ contains an element of the form $a_{i_1}d_{i_1}+a_{i_2}d_{i_2}+\cdots+a_{i_{m+1}}d_{i_{m+1}}$ for positive integers $i_1<i_2<\cdots<i_{m+1}$ and nonzero complex numbers $a_{i_1}, a_{i_2},\cdots, a_{i_{m+1}}$. We further see that $\mathfrak{a}$ contains $$[d_{i_1},a_{i_1}d_{i_1}+a_{i_2}d_{i_2}+\cdots+a_{i_{m+1}}d_{i_{m+1}}]=a_{i_2}(i_1-i_2)d_{i_1+i_2}+a_{i_3}(i_1-i_3)d_{i_1+i_3}+ \cdots+a_{i_{m+1}}(i_1-i_{m+1})d_{i_1+i_{m+1}},$$ which contradicts with the minimality of $m$. Hence the claim follows. Consequently, $$\widetilde{\mathfrak{Vir}}_{\geq n}:=\sum_{i\geq n,\, i\neq 2n}{\mathbb C} d_i={\mathbb C} d_n+ [d_n, \mathfrak{Vir}_{\geq 1}]\subseteq\mathfrak{a}.$$ Then $$\widetilde{\mathfrak{Vir}}_{\geq n}+\mathcal{H}_{\geq n}=\widetilde{\mathfrak{Vir}}_{\geq n}+[{\mathbb C} h_{\frac{1}{2}}+{\mathbb C} h_{\frac{3}{2}},\widetilde{\mathfrak{Vir}}_{\geq n}]\subseteq\ker(\rho).$$ This implies that $M_1$ is a finite dimensional module over a finite dimensional solvable Lie algebra ${\mathfrak{D}}^{+}/(\widetilde{\mathfrak{Vir}}_{\geq n}+\mathcal{H}_{\geq n})$. The desired assertion follows directly from Lie Theorem.
(ii) follows directly from (i). \end{proof}
\section{Examples} \label{examples}
In this section, we will give a few examples of simple restricted $\bar \mathfrak{D}$- and $\mathfrak{D}$-modules, which are also weak (simple) untwisted and twisted $\mathcal{V}^{c}$-modules.
\begin{exa} For any $n\in{\mathbb Z}_+$, let $\mathcal{W}_0={\mathbb C}[x_1,\cdots,x_n]$ be the polynomial algebra in indeterminates $x_1,\cdots,x_n$. Define the $\mathcal{H}^{(-n)}$-module structure on $\mathcal{W}_0$ by \begin{equation}\label{1stexample} \begin{aligned} &h_{i-\frac{1}{2}}\cdot f(x_1,\cdots,x_i,\cdots,x_n)=\lambda_if(x_1,\cdots,x_i-1,\cdots,x_n),\\ &h_{-i+\frac{1}{2}}\cdot f(x_1,\cdots,x_i,\cdots,x_n)=-\frac{\ell(i-\frac{1}{2})}{\lambda_i}(x_i+a_i)f(x_1,\cdots,x_i+1,\cdots,x_n),\\ &h_{n+j+\frac{1}{2}}\cdot f(x_1,\cdots,x_i,\cdots,x_n)=0, \\ &{\bf c}_2\cdot f(x_1,\cdots,x_n)=\ell f(x_1,\cdots,x_n)\end{aligned} \end{equation} where $\ell, \lambda_i\in{\mathbb C}^*, a_i\in{\mathbb C}, j\in{\mathbb N}, 1\le i\le n$. It is not hard to check that $\mathcal{W}_0$ is a simple $\mathcal{H}^{(-n)}$-module. Then the induced $\mathcal{H}$-module $K=\text{\rm Ind}_{\mathcal{H}^{(-n)}}^{\mathcal{H}}\mathcal{W}_0$ is a simple restricted $\mathcal{H}$-module. So $K^{\mathfrak{D}}$ is a simple restricted $\mathfrak{D}$-module with central charge $1$ and level $\ell$. We may denote $K^{\mathfrak{D}}=K^{\mathfrak{D}}(\ell, \Lambda_n,\mathfrak{a}_n)$ for any $\ell\in{\mathbb C}^*,$ $\Lambda_n=(\lambda_1,\cdots,\lambda_n)\in ({\mathbb C}^*)^n,$ $\mathfrak{a}_n=(a_1,\cdots,a_n)\in{\mathbb C}^n$.
Let $U$ be a simple restricted $\mathfrak{Vir}$-module (Theorem \ref{Vir-modules} classified all simple restricted $\mathfrak{Vir}$-modules). From Corollary \ref{tensor}, then $S=U^{\mathfrak{D}}\otimes K^{\mathfrak{D}}(\ell, \Lambda_n,\mathfrak{a}_n)$ is a simple restricted $\mathfrak{D}$-module.
If we replace (\ref{1stexample}) by \begin{equation*} \begin{aligned} &h_{i}\cdot f(x_1,\cdots,x_i,\cdots,x_n)=\lambda_if(x_1,\cdots,x_i-1,\cdots,x_n),\\ &h_{-i}\cdot f(x_1,\cdots,x_i,\cdots,x_n)=-\frac{\ell i}{\lambda_i}(x_i+a_i)f(x_1,\cdots,x_i+1,\cdots,x_n),\\ &h_{n+j+1}\cdot f(x_1,\cdots,x_i,\cdots,x_n)=0, \\ &\bar{\bf c}_3\cdot f(x_1,\cdots,x_n)=\ell f(x_1,\cdots,x_n)\end{aligned} \end{equation*} for $\ell, \lambda_i\in{\mathbb C}^*, a_i\in{\mathbb C}, j\in{\mathbb N}, 1\le i\le n$, then $\mathcal{W}_0$ is a simple $\bar \mathcal{H}^{(-n)}$-module, and the induced $\bar \mathcal{H}$-module $\bar K=\text{\rm Ind}_{\bar \mathcal{H}^{(-n)}}^{\bar \mathcal{H}}\mathcal{W}_0$ is a simple restricted $\bar \mathcal{H}$-module. Hence, for any $z\in{\mathbb C}$, we have the simple $\bar \mathfrak{D}$-module $\bar K(z)^{\bar \mathfrak{D}}=\bar K(z)^{\bar \mathfrak{D}}(\ell, \Lambda_n,\mathfrak{a}_n)$ for any $\ell\in{\mathbb C}^*,$ $\Lambda_n=(\lambda_1,\cdots,\lambda_n)\in ({\mathbb C}^*)^n,$ $\mathfrak{a}_n=(a_1,\cdots,a_n)\in{\mathbb C}^n$. For any simple $\mathfrak{Vir}$-module $U\in\mathcal{R}_{\mathfrak{Vir}}$, the tensor product $M=U^{\bar \mathfrak{D}}\otimes \bar K(z)^{\bar \mathfrak{D}}(\ell, \Lambda_n,\mathfrak{a}_n)$ is a simple restricted $\bar \mathfrak{D}$-module. \end{exa}
For characterizing simple induced restricted $\mathfrak{D}$-and $\bar {\mathfrak{D}}$-module which are not tensor product modules, we need the following
\begin{lem}\label{restricted submodule} Let $S=U^{\mathfrak{D}}\otimes V^{\mathfrak{D}}$ be a simple restricted $\mathfrak{D}$-module with $n_S>0$ and nonzero level, where $U\in{\mathcal R}_{\mathfrak{Vir}}$ and $V\in{\mathcal R}_{\mathcal{H}}$. Let $V_0=\text{Ann}_V(\mathcal{H}^{(n_S)})$ and $W_0=\text{Ann}_S(\mathcal{H}^{(n_S)})$. Then $V_0$ is a simple $\mathfrak{D}^{(0,-n_S)}$-module, and $W_0=U\otimes V_0$. Hence $W_0$ contains a simple $ \mathcal{H}^{(-n_S)}$ submodule. \end{lem} \begin{proof} This is clear. \end{proof} We also have the $\bar \mathfrak{D}$-module version of Lemma \ref{restricted submodule}: \begin{lem}\label{restricted submodule'} Let $M=H(z)^{\bar \mathfrak{D}}\otimes U^{\bar \mathfrak{D}}$ be a simple restricted $\bar \mathfrak{D}$-module with $n_M>1$ and nonzero level, where $z\in{\mathbb C}$, $H\in{\mathcal R}_{\bar \mathcal{H}}$ and $U\in{\mathcal R}_{\mathfrak{Vir}}$. Let $H_0=\text{Ann}_H(\bar \mathcal{H}^{(n_M)})$ and $M_0=\text{Ann}_M(\bar \mathcal{H}^{(n_M)})$. Then $H_0$ is a simple $\bar \mathfrak{D}^{(0,-n_M+1)}$-module, and $M_0=H_0\otimes U$. Hence $M_0$ contains a simple $\bar \mathcal{H}^{(-n_M+1)}$ submodule. \end{lem}
Lemma \ref{restricted submodule} (resp. Lemma \ref{restricted submodule'}) means that if $S\in{\mathcal R}_{\mathfrak{D}}$ (resp. $M\in{\mathcal R}_{\bar \mathfrak{D}}$) is not a tensor product module, then $W_0$ (resp. $M_0$) contains no simple $\mathcal{H}^{(-n_S)}$-submodule (resp. $\bar \mathcal{H}^{(-n_M+1)}$-submodules).
Here we will first consider the case $n_S=1$ (resp. $n_M=2$).
Let $\mathfrak{b}={\mathbb C} h+{\mathbb C} e$ be the 2-dimensional solvable Lie algebra with basis $h,e$ and subject to Lie bracket $[h,e]=e$.
The following concrete example using \cite[Example 13]{LMZ} tells us
how to construct induced restricted $\mathfrak{D}$-module (resp. $\bar \mathfrak{D}$-module) from a ${\mathbb C}[e]$-torsion-free simple $\mathfrak{b}$-module.
\begin{exa}[Simple induced restricted module, $n_S=1/n_M=2$] \label{ex-7.4} Let $c_1,c_2\in{\mathbb C}$ with $c_2\ne0$. Let $W'=(t-1)^{-1}{\mathbb C}[t,t^{-1}]$. From \cite[Example 13]{LMZ} we know that $W'$ is a simple $\mathfrak{b}$-module whose structure is given by \begin{equation*} h\cdot f(t)=t\frac{d}{dt}(f(t))+\frac{f(t)}{t^2(t-1)},\,\, e\cdot f(t)=tf(t),\forall f(t)\in W'.
\end{equation*} We can make $W'$ into a $\mathfrak{D}^{(0,0)}$-module by $$\aligned {\bf c}_1\cdot f(t)&=c_1 f(t),\,\,\, {\bf c}_2\cdot f(t)=c_2 f(t),\\ d_0\cdot f(t)&=-\frac{1}{2}h\cdot f(t), \,h_{\frac{1}{2}}\cdot f(t)=e\cdot f(t), \,d_i\cdot f(t)=h_{\frac{1}{2}+i}\cdot f(t)=0,\,\,\,i\in{\mathbb Z}_+.\endaligned $$ Then $W'$ is a simple $\mathfrak{D}^{(0,0)}$-module. Clearly, the action of $h_{\frac{1}{2}}$ on $W'$ implies that $W'$ contains no simple $\mathcal{H}^{(0)}$-module. Then $W_0=\text{\rm Ind}_{\mathfrak{D}^{(0,0)}}^{\mathfrak{D}^{(0,-1)}}W'$ is a simple $\mathfrak{D}^{(0,-1)}$-module and contains no simple $\mathcal{H}^{(-1)}$-module. So $W_0$ is not a tensor product $\mathfrak{D}^{(0,-1)}$-module. Let $S=\text{\rm Ind}_{\mathfrak{D}^{(0,-1)}}^{\mathfrak{D}}W_0$. It is easy to see $n_S=1,m_S=2=r_S$, and $W_0=U_0=K_0$. The proof of Proposition \ref{prop4.6} implies that $S$ is a simple restricted $\mathfrak{D}$-module. And Lemma \ref{restricted submodule} means that $S$ is not a tensor product $\mathfrak{D}$-module.
For $c,z,z'\in{\mathbb C}, \ell\in{\mathbb C}^*$, we also can make $W'$ into a $\bar \mathfrak{D}^{(0,0)}$-module by $$\begin{aligned}&d_0\cdot f(t)=h\cdot f(t), \,h_{1}\cdot f(t)=e\cdot f(t), \\ &h_0\cdot f(t)=z'f(t), h_{1+i}\cdot f(t)=d_i\cdot f(t)=0,\,\,\,i\in{\mathbb Z}_+,\\ & \bar{\bf c}_1\cdot f(t)=cf(t),\, \bar{\bf c}_2\cdot f(t)=zf(t),\, \bar{\bf c}_3\cdot f(t)=\ell f(t),\\ \end{aligned} $$where $f(t)\in W'$. Then $W'$ is a simple $\bar \mathfrak{D}^{(0,0)}$-module. Clearly, the action of $h_{1}$ on $W'$ implies that $W'$ contains no simple $\bar \mathcal{H}^{(0)}$-module. Then $M_0=\text{\rm Ind}_{\bar \mathfrak{D}^{(0,0)}}^{\bar \mathfrak{D}^{(0,-1)}}W'$ is a simple $\bar \mathfrak{D}^{(0,-1)}$-module and contains no simple $\bar \mathcal{H}^{(-1)}$-module. Let $M=\text{\rm Ind}_{\bar \mathfrak{D}^{(0,-1)}}^{\bar \mathfrak{D}}M_0$. It is easy to see $n_M=2,$$r_M= 3$. The proof of Proposition \ref{prop4.6'} implies that $M$ is a simple restricted $\bar \mathfrak{D}$-module. And Lemma \ref{restricted submodule'} means that $M$ is not a tensor product $\bar \mathfrak{D}$-module.
\end{exa}
\begin{exa}[Simple induced modules of semi-Whittaker type, $n_S\ge 2, n_M\ge 3$]\footnote{This example is a modified version of the one provided by Drazen Adamovic.} \label{ex-7.6} Take $p,q\in{\mathbb Z}_+, {\bf a} =(a_1, \dots, a_{q}) \in ({{\mathbb C}}^*)^{q}$, ${\bf b} = (b_1, \dots, b_p) \in ({{\mathbb C}}^*)^p$, $c,\ell\in{\mathbb C}$ with $\ell\ne0$. Define the $1$-dimensional $\mathfrak{D}^{(p,q)}$-module ${{\mathbb C}}_{{\bf a}, {\bf b}} = {{\mathbb C}} v_0$ with \begin{equation}\label{induced-modules} \aligned {\bf c}_1\cdot v_0&=c v_0,\,\,\, {\bf c}_2\cdot v_0=\ell v_0,\\
d_{p} v_0 &= a_1 v_0, \cdots, d_{p+q-1} v_0 = a_q v_0, \ d_i v_0 = 0 \ \mbox{for} \ i > p+q-1, \\
h_{q+\tfrac{1}{2}} v_0 &= b_1 v_0, \cdots h_{p+q- \tfrac{1}{2}} v_0 = b_p v_0, \ h_{i-\tfrac{1}{2}} v_0 = 0 \ \mbox{for} \ i > p+q.\endaligned\end{equation} It is not hard to show that $U({\bf a},{\bf b}) := \text{\rm Ind}_{\mathfrak{D}^{(p,q)}}^{\mathfrak{D}^{(0,-1)}}{{\mathbb C}}_{{\bf a},{\bf b}}$ is a simple $\mathfrak{D}^{(0,-1)}$-module. Then in Theorem \ref{thmmain1.1} (2) we have $V=U({\bf a},{\bf b}), n=1, k=p+q=l$, and so $S= \widehat U({\bf a},{\bf b}) := \text{\rm Ind}_{\mathfrak{D}^{(0,-1)}}^{\mathfrak{D} } U({\bf a},{\bf b})$ is a simple restricted $\mathfrak{D}$-module. In Lemma \ref{restricted submodule}, $n_S=p+q$, and $W_0=\text{\rm Ind}_{\mathcal{H}^{(q)}}^{\mathcal{H}^{(-(p+q))}}\left(\text{\rm Ind}_{\mathfrak{D}^{(p,q)}}^{\mathfrak{D}^{(0,q)}} {\mathbb C}_{{\bf a},{\bf b}}\right)$ does not contain any simple $\mathcal{H}^{(-(p+q))}$-module ( for $h_{\pm 1/2}$ acts freely on $W_0$). Hence, by Lemma \ref{restricted submodule}, $\widehat U({\bf a},{\bf b})$ is not a tensor product $\mathfrak{D}$-module.
If we, in the above example, replace (\ref{induced-modules}) by \begin{equation*}\label{induced module} \aligned & {\bar {\bf c}}_1\cdot v_0=c v_0,\,{\bar {\bf c}}_2\cdot v_0=z v_0,\,\bar{\bf c}_3 v_0=\ell v_0,\\ &d_{p} v_0 = a_1 v_0, \cdots, d_{p+q-1} v_0 = a_q v_0, \ d_i v_0 = 0 \ \mbox{for} \ i > p+q-1, \\ & h_{q+1} v_0 = b_1 v_0, \cdots h_{p+q} v_0 = b_p v_0, \ h_{i} v_0 = 0 \ \mbox{for} \ i > p+q, \endaligned\end{equation*}where $z\in{\mathbb C}$
and leave other parts invariant, then for any $z'\in{\mathbb C}$, the induced
$\bar \mathfrak{D}^{(0,-(p+q))}$-module $$\bar V=\text{\rm Ind}_{\bar \mathfrak{D}^{(p,q+1)}}^{\bar \mathfrak{D}^{(0,-(p+q))}} {{\mathbb C}}_{\bf a,\bf b}/\Big(\mathcal{U}( \bar \mathfrak{D}^{(0,-(p+q))})(h_0-z')(1\otimes v_0)\Big) $$
is a simple $\bar \mathfrak{D}^{(0,-(p+q))}$-module. Let $M=\text{\rm Ind}_{\bar \mathfrak{D}^{(0,-(p+q))}}^{\bar \mathfrak{D}}\bar V$. The proof of Theorem \ref{prop4.6'} implies that $M$ is a simple restricted $\bar \mathfrak{D}$-module where $n_M=p+q+1,r_M=2(p+q)+1$, and $K_0=\bar V=M_0$. Since $\bar V$ contains no simple $\bar \mathcal{H}^{(-n_M+1)}$-module, we see, by Lemma \ref{restricted submodule'}, that $M$ is not a tensor product $\bar \mathfrak{D}$-module.
\end{exa}
\begin{rem}From Theorem \ref{mainthm} (resp. Theorem \ref{prop4.3'}) we know that if $n_S=0$ (resp. $n_M=0, 1$), then simple restricted $\mathfrak{D}$-modules(resp. $\bar \mathfrak{D}$-modules) must be tensor product modules. And Examples \ref{ex-7.4}-\ref{ex-7.6} mean that for any $n_S>0$ (resp. $n_M>1$), there do exist simple restricted $\mathfrak{D}$-modules (resp. $\bar \mathfrak{D}$-modules) which are not tensor product modules. Clearly, the $\bar \mathfrak{D}$-modules here are simple restricted $\widetilde\mathfrak{D}$-modules for $z=0$. \end{rem}
\appendix
\section{Application two: simple modules for Heisenberg-Virasoro vertex operator algebras $\mathcal V^{c}$ (by Drazen Adamovic)}\label{voasection}
\label{voa-interp}
A connection between restricted modules over the Heisenberg-Virasoro algebra and VOA modules in untwisted cases was considered by Guo and Wang in \cite{GW}. In this appendix we extend this correspondence for restricted modules for the mirror Heisenberg-Virasoro algebra. Restricted modules of nonzero level for the mirror Heisenberg-Virasoro algebra can be treated as weak twisted modules for the Heisenberg-Virasoro vertex algebras, and restricted modules of nonzero level for the twisted Heisenberg-Virasoro algebra can be treated as weak modules for the Heisenberg-Virasoro vertex algebras. For convenience, in the Heisenberg-Virasoro algebra
$\widetilde \mathfrak{D}$, we denote ${\bf c}_1=\bar{\bf c}_1, {\bf c}_2=\bar {\bf c}_3$.
Let $\mathcal P$ be the subalgebra of $\widetilde {\mathfrak{D}}$ spanned by
$$ {\bf c}_1, {\bf c}_2, d_m, h_m, \quad (m \in {{\mathbb Z}}_{\ge 0}). $$
Let $(\ell_1, \ell_2) \in {{\mathbb C}} ^2$. Consider the $1$-dimensional $\mathcal P$--module ${{\mathbb C}} v_{\ell_1,\ell_2}$ such that
$$ {\bf c}_1 v_{\ell_1,\ell_2} = \ell_1 v_{\ell_1,\ell_2}, \ {\bf c}_2 v_{\ell_1,\ell_2} = \ell_2 v_{\ell_1,\ell_2}, \ d_m v_{\ell_1,\ell_2} = h_m v_{\ell_1,\ell_2} = 0 \quad (m \in {{\mathbb Z}}_{\ge 0}). $$
Let $\mathcal V^{\ell_1, \ell_2}$ be the following induced $\widetilde {\mathfrak{D}}$-module: $$\mathcal V^{\ell_1, \ell_2} = U( \widetilde {\mathfrak{D}}) \otimes_{ U(\mathcal P)} {{\mathbb C}} v_{\ell_1,\ell_2}. $$ Then $\mathcal V^{\ell_1, \ell_2}$ is a highest weight $\widetilde {\mathfrak{D}}$-module, with the highest weight vector ${\bf 1}_{\ell_1, \ell_2} =1 \otimes v_{\ell_1, \ell_2}$.
Define the following fields acting on $\mathcal V^{\ell_1, \ell_2}$:
$$ h(z) = \sum_{m \in {{\mathbb Z}}} h_m z^{-m-1}, \quad d(z) = \sum_{m \in {{\mathbb Z}}} d_m z^{-n-2}.$$
\begin{itemize}
\item Let $V_{Vir} ^c$ denotes the universal Virasoro vertex algebra of central charge $c$ generated by the Virasoro field $L^{(1)} (z) = \sum_{m \in {\mathbb Z}} L^{(1)}_n z^{-n-2}$ (cf. \cite{LL}, \cite{FZ}).
\item Let $ M(\ell) $ denotes the Heisenberg vertex algebra of level $\ell$, with the vertex operator $Y_{M(\ell)}( \cdot, z)$ which is uniquely generated with the Heisenberg field $$ Y_{M(\ell)}(h(-1) {\bf 1}, z) =h(z) = \sum_{n \in {{\mathcal Z}}} h_n z^{-n-1}. $$ If $\ell \ne 0$, then $M(\ell)$ is simple and always isomorphic to $M(1)$ (cf. \cite{LL}).
\item $M(\ell)$ contains a Virasoro vertex subalgebra $V_{Vir} ^{c=1}$ generated by the Virasoro field
$$ L^{Heis} (z) := \frac{1}{2\ell} :h(z) h(z): = \sum_{n \in {{\mathbb Z}}} L^{Heis} _n z^{-n-2}. $$
Moreover, we have
$$ L^{Heis}_n = \frac{1}{2\ell} \sum_{k \in {\mathbb Z}} :h_k h_{n-k}:$$
For details see \cite{FLM}, \cite{LL}.
\item $M(\ell)$ contains the Virasoro vector
$$\omega_{\lambda } = \frac{1}{2\ell} (h_{-1} ^2){\bf 1} - \frac{\lambda}{\ell} h_{-2} {\bf 1}$$
of the central charge $c = 1 - 12 {\lambda} ^2 / \ell$. The components of the vertex operator
\begin{align} Y_{M(\ell)} (\omega_{\lambda}) = \bar L(z) = L^{Heis}(z) - \frac{\lambda}{\ell} \partial_z h(z) = \sum _{n \in {{\mathbb Z}}} \bar L_n z^{-n-2} \label{rep1-voa}\end{align}
represent the operators $\bar L_n$ defined directly by (\ref{rep1}).
\item Note that the field $\bar L(z)$ and operators $\bar L_n$ acts on any weak $M(\ell)$-module, and in particular on any restricted module for the Heisenberg algebra.
\end{itemize}
Note that $M(\ell_2)$ is naturally a vertex subalgebra of $\mathcal V^{\ell_1, \ell_2}$.
\begin{pro} Assume that $\ell_2 \ne 0$. We have the following isomorphism of vertex algebras
$$ V_{Vir}^{ c } \otimes M(\ell_2) \cong \mathcal V^{\ell_1, \ell_2},$$
such that $c= \ell_1 -1$ and
$$ L^{(1)} (z) \mapsto d(z) - L^{Heis}(z), \quad h(z) \mapsto h(z). $$
In particular, $\mathcal V^{\ell_1, \ell_2} \cong V_{Vir}^{ c } \otimes M(1)$.
\end{pro}
Since $M(\ell) \cong M(1)$, without loss of generality we can assume that $\ell =1$.
In what follows we set $\mathcal V^{c} := V_{Vir}^{ c } \otimes M(1)$.
The following theorem relates restricted $\bar \mathfrak{D}$-modules as (untwisted) modules for the vertex operator algebra $\mathcal V^{c}$.
\begin{theo}\label{5.3-untw}(cf. \cite{GW}) The following statements holds. \item[\rm(1)] Assume that $W$ is a (simple) restricted $\bar \mathfrak{D}$-module of central charge $\ell_1$ and level one. Then $W$ has the unique structure of a weak (simple) $\mathcal V^{c=\ell_1-1}$-module generated by the fields: $$ h (z) = \sum_{r \in {{\mathbb Z}}} h_r z^{-r-1}, \quad L^{(1)}(z) = d(z) - L^{Heis}(z) = \sum_{n \in {{\mathbb Z}}} L^{(1)} _n z^{-n-2}. $$
\item[\rm(2)] Assume that $W$ is a weak (simple) $\mathcal V^{c}$-module. Then $W$ has the structure of a (simple) restricted $\bar \mathfrak{D}$-module of level one such that
$$ h_n \mapsto h_n, \quad d_n = L^{(1)}_n + L^{Heis} _n $$
\end{theo}
The vertex--algebraic interpretation of the restricted $\mathfrak{D}$--modules is via the twisted $\mathcal V^{c}$-modules.
There is an automorphism $\theta_1$ of order two of $M(1)$ such that $ \theta_{1} (h) = -h$ (cf. \cite{FLM}). We extend this automorphism to the automorphism $\theta = \mbox{Id} \otimes \theta_1$ of $\mathcal V^{c}$.
We have the following theorem.
\begin{theo}\label{5.3} The following statements holds. \item[\rm(1)] Assume that $W$ is a (simple) restricted $\mathfrak{D}$-module of central charge $\ell_1$ and level one. Then $W$ has the unique structure of a weak (simple) $\theta$-twisted $\mathcal V^{c=\ell_1-1}$-module generated by the fields: $$ h^{tw}(z) = \sum_{r \in \tfrac{1}{2} + {{\mathbb Z}}} h_r z^{-r-1}, L^{(1)} (z) = d(z) - L_{tw} ^{Heis} (z), $$ where
\begin{align} L_{tw} ^{Heis} (z):=\tfrac{1}{2} : h^{tw} (z) ^2 : + \frac{1}{16} z^{-2} = \sum _{n\in {{\mathbb Z}}} L _n z^{-n-2}. \label{heis-tw-vir} \end{align}
\item[\rm(2)] Assume that $W$ is a weak (simple) $\theta$-twisted $\mathcal V^{c}$-module. Then $W$ has the structure of a (simple) restricted $\mathfrak{D}$-module of level one such that
$$ h_r \mapsto h_r, \quad d_n = L^{(1)}_n + L_n. $$
\end{theo}
\begin{proof}
As in \cite{FLM} (see also \cite{Tanabe}, \cite{HY}) we see that the field $ h^{tw}(z)$ defines on $W$ the unique structure of a $\theta_1$-twisted $M(1)$-module with the Virasoro field (\ref{heis-tw-vir}).
Then we define $L^{(1)} (z) = d(z) - L _{tw}^{Heis}(z) = \sum_{n \in {{\mathbb Z}}} L^{(1)}_n z^{-n-2}$. The field $L^{(1)}(z)$ defines on $W$ the structure of a restricted module for the Virasoro algebra of central charge $c= \ell_1 -1$. Since
$$[L^{(1)} _n, h_r] = 0, \quad \forall n \in {{\mathbb Z}}, \ r \in \tfrac{1}{2} + {{\mathbb Z}}, $$
we have that the action of $L^{(1)}(z)$ commutes with $h^{tw}(z)$. Therefore
$W$ is a $\theta$-twisted $\mathcal V_{Vir}^{c}$-module.
Since all components of the vertex operators are obtained from the action of $d_n, h_r$, $n \in {{\mathbb Z}}, r \in \tfrac{1}{2} + {{\mathbb Z}}$, we see that $W$ is an irreducible $\mathfrak{D}$-module if and only if $W$ is an irreducible module for the vertex algebra $\mathcal V^{c}$. This proves the assertion (1).
Assume that $W$ is $\theta$-twisted $\mathcal V^{c}$-module with the vertex operator $Y^{tw} _W(v, z)$, $v \in \mathcal V^{c}$. Then the twisted Jacobi identity (cf. \cite{FLM}) shows that:
\begin{itemize} \item The components of the field $$ L^{(1)} (z) = Y_{W} ^{tw} (L^{(1)} _{-2}{\bf 1}, z)= \sum_{n \in {\mathbb Z} } L^{(1)} _n z^{-n-2}$$ define on $W$ structure of a restricted module for the Virasoro algebra with central charge $c$; \item The components of the field $$h^{tw}(z) = Y_{W} ^{tw} (h_{-1} {\bf 1}, z)= \sum_{r \in \tfrac{1}{2} + {\mathbb Z} } h_{r} z^{-r-1}$$ define on $W$ the structure of a restricted module for the Heisenberg algebra of level one.
\item The fields $h^{tw}(z)$ and $L^{(1)}(z)$ commute.
\item The field $$L_{tw}^{Heis} (z) =Y(\tfrac{1}{2} h_{-1} ^2 {\bf 1}, z) = \sum_{n \in {\mathbb Z} } L_n z^{-n-2}$$ is a Virasoro field commuting with $L^{(1)}(z)$ such that
$$[L_n, h_{r}] = - r h_{n+r}, \quad n \in {\mathbb Z} , r \in \tfrac{1}{2} + {\mathbb Z}. $$
Define $d(z) = L^{(1)}(z) + L_{tw}^{Heis}(z) = \sum_{n \in {\mathbb Z}} d_n z^{-n-2}$. Then the components of the fields $d(z)$ and $h^{tw}(z)$ define on $W$ the structure of a restricted $\mathfrak{D}$-module of central charge $\ell_1 = c+1$ and level one. \end{itemize}
Arguments for the irreducibility are the same as in (1). This proves the assertion (2). \end{proof}
\begin{rem} The simple modules in Examples \ref{ex-7.4}-\ref{ex-7.6} show that the vertex operator algebra $V_{Vir} ^{c} \otimes M(1)$ has simple weak (untwisted and twisted) modules which are not isomorphic to any tensor product $S_1 \otimes S_2$ for any simple weak $V_{Vir} ^{c}$-module $S_1$ and any simple weak (untwisted and twisted) $M(1)$-module $S_2$. It would be interesting to find analogs of these non-tensor product modules for other types of vertex operator algebras.\end{rem}
\begin{center}{\bf Acknowledgments} \end{center}
The authors would like to thank Drazen Adamovic to provide Appendix A and Example 7.5, and help modifying the introduction. The last two authors would like to thank H. Chen and D. Gao for helpful discussions at the early stage on the restricted modules over the twisted Heisenberg-Virasor algebra.
H.T. is partially supported by the Fundamental Research Funds for the Central Universities (135120008). Y.Y. is partially supported by the National Natural Science Foundation of China (11771279, 12071136, 11671138). K.Z. is partially supported by the National Natural Science Foundation of China (11871190) and NSERC (311907-2020).
\end{document} | arXiv |
# FUNDAMENTALS
## FUNDAMENTALS OF MATRIX ALGEBRA
Third Edition, Version 3.1110
Gregory Hartman, Ph.D.
Department of Mathematics and Computer Science
Virginia Military Institute Copyright (C) 2011 Gregory Hartman Licensed to the public under Creative Commons Attribution-Noncommercial 3.0 United States License
## THANKS
This text took a great deal of effort to accomplish and I owe a great many people thanks.
I owe Michelle (and Sydney and Alex) much for their support at home. Michelle puts up with much as I continually read $\mathrm{LT}_{\mathrm{E}}^{\mathrm{X}}$ manuals, sketch outlines of the text, write exercises, and draw illustrations.
My thanks to the Department of Mathematics and Computer Science at Virginia Military Institute for their support of this project. Lee Dewald and Troy Siemers, my department heads, deserve special thanks for their special encouragement and recognition that this effort has been a worthwhile endeavor.
My thanks to all who informed me of errors in the text or provided ideas for improvement. Special thanks to Michelle Feole and Dan Joseph who each caught a number of errors.
This whole project would have been impossible save for the efforts of the $\mathrm{AT}_{\mathrm{E}}^{\mathrm{X}} \mathrm{X}$ community. This text makes use of about 15 different packages, was compiled using MiKT $T_{E} X$, and edited using $T_{E} X n i c$ Center, all of which was provided free of charge. This generosity helped convince me that this text should be made freely available as well.
## SYSTEMS OF LINEAR EQUATIONS
You have probably encountered systems of linear equations before; you can probably remember solving systems of equations where you had three equations, three unknowns, and you tried to find the value of the unknowns. In this chapter we will uncover some of the fundamental principles guiding the solution to such problems.
Solving such systems was a bit time consuming, but not terribly difficult. So why bother? We bother because linear equations have many, many, many applications, from business to engineering to computer graphics to understanding more mathematics. And not only are there many applications of systems of linear equations, on most occasions where these systems arise we are using far more than three variables. (Engineering applications, for instance, often require thousands of variables.) So getting a good understanding of how to solve these systems effectively is important.
But don't worry; we'll start at the beginning.
### Introduction to Linear Equations
## AS YOU READ
1. What is one of the annoying habits of mathematicians?
2. What is the difference between constants and coefficients?
3. Can a coefficient in a linear equation be 0 ?
We'll begin this section by examining a problem you probably already know how to solve.
Example 1 Suppose a jar contains red, blue and green marbles. You are told that there are a total of 30 marbles in the jar; there are twice as many red marbles as green ones; the number of blue marbles is the same as the sum of the red and green marbles. How many marbles of each color are there?
Solution We could attempt to solve this with some trial and error, and we'd probably get the correct answer without too much work. However, this won't lend itself towards learning a good technique for solving larger problems, so let's be more mathematical about it.
Let's let $r$ represent the number of red marbles, and let $b$ and $g$ denote the number of blue and green marbles, respectively. We can use the given statements about the marbles in the jar to create some equations.
Since we know there are 30 marbles in the jar, we know that
$$
r+b+g=30 .
$$
Also, we are told that there are twice as many red marbles as green ones, so we know that
$$
r=2 g \text {. }
$$
Finally, we know that the number of blue marbles is the same as the sum of the red and green marbles, so we have
$$
b=r+g .
$$
From this stage, there isn't one "right" way of proceeding. Rather, there are many ways to use this information to find the solution. One way is to combine ideas from equations 1.2 and 1.3 ; in 1.3 replace $r$ with $2 g$. This gives us
$$
b=2 g+g=3 g .
$$
We can then combine equations $1.1,1.2$ and 1.4 by replacing $r$ in 1.1 with $2 g$ as we did before, and replacing $b$ with $3 g$ to get
$$
\begin{aligned}
r+b+g & =30 \\
2 g+3 g+g & =30 \\
6 g & =30 \\
g & =5
\end{aligned}
$$
We can now use equation 1.5 to find $r$ and $b$; we know from 1.2 that $r=2 g=10$ and then since $r+b+g=30$, we easily find that $b=15$.
Mathematicians often see solutions to given problems and then ask "What if. . .?" It's an annoying habit that we would do well to develop - we should learn to think like a mathematician. What are the right kinds of "what if" questions to ask? Here's another annoying habit of mathematicians: they often ask "wrong" questions. That is, they often ask questions and find that the answer isn't particularly interesting. But asking enough questions often leads to some good "right" questions. So don't be afraid of doing something "wrong;" we mathematicians do it all the time.
So what is a good question to ask after seeing Example 1? Here are two possible questions: 1. Did we really have to call the red balls " $r$ "? Could we call them " $q$ "?
2. What if we had 60 balls at the start instead of 30 ?
Let's look at the first question. Would the solution to our problem change if we called the red balls $q$ ? Of course not. At the end, we'd find that $q=10$, and we would know that this meant that we had 10 red balls.
Now let's look at the second question. Suppose we had 60 balls, but the other relationships stayed the same. How would the situation and solution change? Let's compare the "orginal" equations to the "new" equations.
$$
\begin{array}{c|c}
\text { Original } & \text { New } \\
\hline r+b+g=30 & r+b+g=60 \\
r=2 g & r=2 g \\
b=r+g & b=r+g
\end{array}
$$
By examining these equations, we see that nothing has changed except the first equation. It isn't too much of a stretch of the imagination to see that we would solve this new problem exactly the same way that we solved the original one, except that we'd have twice as many of each type of ball.
A conclusion from answering these two questions is this: it doesn't matter what we call our variables, and while changing constants in the equations changes the solution, they don't really change the method of how we solve these equations.
In fact, it is a great discovery to realize that all we care about are the constants and the coefficients of the equations. By systematically handling these, we can solve any set of linear equations in a very nice way. Before we go on, we must first define what a linear equation is.
## Definition 1
So in Example 1, when we answered "how many marbles of each color are there?," we were also answering "find a solution to a certain system of linear equations."
## Linear Equation
A linear equation is an equation that can be written in the form
$$
a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n}=c
$$
where the $x_{i}$ are variables (the unknowns), the $a_{i}$ are coefficients, and $c$ is a constant.
A system of linear equations is a set of linear equations that involve the same variables.
A solution to a system of linear equations is a set of values for the variables $x_{i}$ such that each equation in the system is satisfied. The following are examples of linear equations:
$$
\begin{aligned}
2 x+3 y-7 z & =29 \\
x_{1}+\frac{7}{2} x_{2}+x_{3}-x_{4}+17 x_{5} & =\sqrt[3]{-10} \\
y_{1}+14^{2} y_{4}+4 & =y_{2}+13-y_{1} \\
\sqrt{7} r+\pi s+\frac{3 t}{5} & =\cos \left(45^{\circ}\right)
\end{aligned}
$$
Notice that the coefficients and constants can be fractions and irrational numbers (like $\pi, \sqrt[3]{-10}$ and $\cos \left(45^{\circ}\right)$ ). The variables only come in the form of $a_{i} x_{i}$; that is, just one variable multiplied by a coefficient. (Note that $\frac{3 t}{5}=\frac{3}{5} t$, just a variable multiplied by a coefficient.) Also, it doesn't really matter what side of the equation we put the variables and the constants, although most of the time we write them with the variables on the left and the constants on the right.
We would not regard the above collection of equations to constitute a system of equations, since each equation uses differently named variables. An example of a system of linear equations is
$$
\begin{aligned}
x_{1}-x_{2}+x_{3}+x_{4} & =1 \\
2 x_{1}+3 x_{2}+x_{4} & =25 \\
x_{2}+x_{3} & =10
\end{aligned}
$$
It is important to notice that not all equations used all of the variables (it is more accurate to say that the coefficients can be 0 , so the last equation could have been written as $0 x_{1}+x_{2}+x_{3}+0 x_{4}=10$ ). Also, just because we have four unknowns does not mean we have to have four equations. We could have had fewer, even just one, and we could have had more.
To get a better feel for what a linear equation is, we point out some examples of what are not linear equations.
$$
\begin{aligned}
2 x y+z & =1 \\
5 x^{2}+2 y^{5} & =100 \\
\frac{1}{x}+\sqrt{y}+24 z & =3 \\
\sin ^{2} x_{1}+\cos ^{2} x_{2} & =29 \\
2^{x_{1}}+\ln x_{2} & =13
\end{aligned}
$$
The first example is not a linear equation since the variables $x$ and $y$ are multiplied together. The second is not a linear equation because the variables are raised to powers other than 1 ; that is also a problem in the third equation (remember that $1 / x=x^{-1}$ and $\sqrt{x}=x^{1 / 2}$ ). Our variables cannot be the argument of function like sin, cos or In, nor can our variables be raised as an exponent. At this stage, we have yet to discuss how to efficiently find a solution to a system of linear equations. That is a goal for the upcoming sections. Right now we focus on identifying linear equations. It is also useful to "limber" up by solving a few systems of equations using any method we have at hand to refresh our memory about the basic process.
## Exercises 1.1
In Exercises 1 - 10, state whether or not the given equation is linear.
1. $x+y+z=10$
2. $x y+y z+x z=1$
3. $-3 x+9=3 y-5 z+x-7$
4. $\sqrt{5} y+\pi x=-1$
5. $(x-1)(x+1)=0$
6. $\sqrt{x_{1}^{2}+x_{2}^{2}}=25$
7. $x_{1}+y+t=1$
8. $\frac{1}{x}+9=3 \cos (y)-5 z$
9. $\cos (15) y+\frac{x}{4}=-1$
10. $2^{x}+2^{y}=16$
In Exercises 11 - 14, solve the system of linear equations.
11. $\begin{aligned} x+y & =-1 \\ 2 x-3 y & =8\end{aligned}$ 12. $\begin{array}{r}2 x-3 y=3 \\ 3 x+6 y=8\end{array}$
$x-y+z=1$
13. $2 x+6 y-z=-4$
$4 x-5 y+2 z=0$
$x+y-z=1$
14. $2 x+y=2$
$y+2 z=0$
15. A farmer looks out his window at his chickens and pigs. He tells his daughter that he sees 62 heads and 190 legs. How many chickens and pigs does the farmer have?
16. A lady buys 20 trinkets at a yard sale. The cost of each trinket is either $\$ 0.30$ or $\$ 0.65$. If she spends $\$ 8.80$, how many of each type of trinket does she buy?
### Using Matrices To Solve Systems of Linear Equations
## AS YOU READ
1. What is remarkable about the definition of a matrix?
2. Vertical lines of numbers in a matrix are called what?
3. In a matrix $A$, the entry $a_{53}$ refers to which entry?
4. What is an augmented matrix?
In Section 1.1 we solved a linear system using familiar techniques. Later, we commented that in the linear equations we formed, the most important information was the coefficients and the constants; the names of the variables really didn't matter. In Example 1 we had the following three equations:
$$
\begin{aligned}
r+b+g & =30 \\
r & =2 g \\
b & =r+g
\end{aligned}
$$
Let's rewrite these equations so that all variables are on the left of the equal sign and all constants are on the right. Also, for a bit more consistency, let's list the variables in alphabetical order in each equation. Therefore we can write the equations as
$$
\begin{aligned}
b+g+r & =30 \\
-2 g+r & =0 \\
-b+g+r & =0
\end{aligned} .
$$
As we mentioned before, there isn't just one "right" way of finding the solution to this system of equations. Here is another way to do it, a way that is a bit different from our method in Section 1.1.
First, lets add the first and last equations together, and write the result as a new third equation. This gives us:
$$
\begin{aligned}
b+g+r & =30 \\
-2 g+r & =0 \\
2 g+2 r & =30
\end{aligned} .
$$
A nice feature of this is that the only equation with $a b$ in it is the first equation.
Now let's multiply the second equation by $-\frac{1}{2}$. This gives
$$
\begin{aligned}
b+g+r & =30 \\
g-1 / 2 r & =0 \\
2 g+2 r & =30
\end{aligned} .
$$
Let's now do two steps in a row; our goal is to get rid of the $g$ 's in the first and third equations. In order to remove the $g$ in the first equation, let's multiply the second equation by -1 and add that to the first equation, replacing the first equation with that sum. To remove the $g$ in the third equation, let's multiply the second equation by -2 and add that to the third equation, replacing the third equation. Our new system of equations now becomes
$$
\begin{aligned}
b+3 / 2 r & =30 \\
g-1 / 2 r & =0 \\
3 r & =30
\end{aligned} .
$$
Clearly we can multiply the third equation by $\frac{1}{3}$ and find that $r=10$; let's make this our new third equation, giving
$$
\begin{aligned}
b+3 / 2 r & =30 \\
g-1 / 2 r & =0 \\
r & =10
\end{aligned} .
$$
Now let's get rid of the $r$ 's in the first and second equation. To remove the $r$ in the first equation, let's multiply the third equation by $-\frac{3}{2}$ and add the result to the first equation, replacing the first equation with that sum. To remove the $r$ in the second equation, we can multiply the third equation by $\frac{1}{2}$ and add that to the second equation, replacing the second equation with that sum. This gives us:
$$
\begin{aligned}
& \text { b }=15 \\
& g=5 \text {. }
\end{aligned}
$$
Clearly we have discovered the same result as when we solved this problem in Section 1.1.
Now again revisit the idea that all that really matters are the coefficients and the constants. There is nothing special about the letters $b, g$ and $r$; we could have used $x$, $y$ and $z$ or $x_{1}, x_{2}$ and $x_{3}$. And even then, since we wrote our equations so carefully, we really didn't need to write the variable names at all as long as we put things "in the right place."
Let's look again at our system of equations in (1.6) and write the coefficients and the constants in a rectangular array. This time we won't ignore the zeros, but rather write them out.
$$
\begin{array}{r}
b+g+r=30 \\
-2 g+r=0 \\
-b+g+r=0
\end{array} \Leftrightarrow\left[\begin{array}{cccc}
1 & 1 & 1 & 30 \\
0 & -2 & 1 & 0 \\
-1 & 1 & 1 & 0
\end{array}\right]
$$
Notice how even the equal signs are gone; we don't need them, for we know that the last column contains the coefficients.
We have just created a matrix. The definition of matrix is remarkable only in how unremarkable it seems.
Definition 2
## Matrix
A matrix is a rectangular array of numbers.
The horizontal lines of numbers form rows and the vertical lines of numbers form columns. A matrix with $m$ rows and $n$ columns is said to be an $m \times n$ matrix ("an $m$ by $n$ matrix").
The entries of an $m \times n$ matrix are indexed as follows:
$$
\left[\begin{array}{ccccc}
a_{11} & a_{12} & a_{13} & \cdots & a_{1 n} \\
a_{21} & a_{22} & a_{23} & \cdots & a_{2 n} \\
a_{31} & a_{32} & a_{33} & \cdots & a_{3 n} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
a_{m 1} & a_{m 2} & a_{m 3} & \cdots & a_{m n}
\end{array}\right] .
$$
That is, $a_{32}$ means "the number in the third row and second column." In the future, we'll want to create matrices with just the coefficients of a system of linear equations and leave out the constants. Therefore, when we include the constants, we often refer to the resulting matrix as an augmented matrix.
We can use augmented matrices to find solutions to linear equations by using essentially the same steps we used above. Every time we used the word "equation" above, substitute the word "row," as we show below. The comments explain how we get from the current set of equations (or matrix) to the one on the next line.
We can use a shorthand to describe matrix operations; let $R_{1}, R_{2}$ represent "row 1 " and "row 2," respectively. We can write "add row 1 to row 3 , and replace row 3 with that sum" as " $R_{1}+R_{3} \rightarrow R_{3}$." The expression " $R_{1} \leftrightarrow R_{2}$ " means "interchange row 1 and row 2."
$$
\begin{aligned}
b+g+r & =30 \\
-2 g+r & =0 \\
-b+g+r & =0
\end{aligned}
$$
Replace equation 3 with the sum of equations 1 and 3
$$
\begin{aligned}
b+g+r & =30 \\
-2 g+r & =0 \\
2 g+2 r & =30
\end{aligned}
$$
Multiply equation 2 by $-\frac{1}{2}$
$$
\begin{aligned}
b+g+r & =30 \\
g+-1 / 2 r & =0 \\
2 g+2 r & =30
\end{aligned}
$$
Replace equation 1 with the sum of $(-1)$ times equation 2 plus equation 1;
Replace equation 3 with the sum of $(-2)$ times equation 2 plus equation 3
$$
\begin{aligned}
b+3 / 2 r & =30 \\
g-1 / 2 r & =0 \\
3 r & =30
\end{aligned}
$$
Multiply equation 3 by $\frac{1}{3}$
$$
\left[\begin{array}{cccc}
1 & 1 & 1 & 30 \\
0 & -2 & 1 & 0 \\
-1 & 1 & 1 & 0
\end{array}\right]
$$
Replace row 3 with the sum of rows 1 and 3 .
$$
\left(R_{1}+R_{3} \rightarrow R_{3}\right)
$$
$$
\left[\begin{array}{cccc}
1 & 1 & 1 & 30 \\
0 & -2 & 1 & 0 \\
0 & 2 & 2 & 30
\end{array}\right]
$$
$$
\text { Multiply row } 2 \text { by }-\frac{1}{2}
$$
$$
\left(-\frac{1}{2} R_{2} \rightarrow R_{2}\right)
$$$$
\left[\begin{array}{cccc}
1 & 1 & 1 & 30 \\
0 & 1 & -\frac{1}{2} & 0 \\
0 & 2 & 2 & 30
\end{array}\right]
$$
Replace row 1 with the sum of $(-1)$ times row 2 plus row 1
$$
\left(-R_{2}+R_{1} \rightarrow R_{1}\right) \text {; }
$$
Replace row 3 with the sum of $(-2)$ times row 2 plus row 3
$$
\left(-2 R_{2}+R_{3} \rightarrow R_{3}\right)
$$
$$
\left[\begin{array}{cccc}
1 & 0 & \frac{3}{2} & 30 \\
0 & 1 & -\frac{1}{2} & 0 \\
0 & 0 & 3 & 30
\end{array}\right]
$$
Multiply row 3 by $\frac{1}{3}$
$$
\left(\frac{1}{3} R_{3} \rightarrow R_{3}\right)
$$
$$
\begin{aligned}
b+3 / 2 r & =30 \\
g-1 / 2 r & =0 \\
r & =10
\end{aligned}
$$
Replace equation 2 with the sum of $\frac{1}{2}$ times equation 3 plus equation 2;
Replace equation 1 with the sum of $-\frac{3}{2}$ times equation 3 plus equation 1
$$
b \quad \begin{aligned}
& =15 \\
g & =5 \\
r & =10
\end{aligned}
$$
$$
\left[\begin{array}{cccc}
1 & 0 & \frac{3}{2} & 30 \\
0 & 1 & -\frac{1}{2} & 0 \\
0 & 0 & 1 & 10
\end{array}\right]
$$
Replace row 2 with the sum of $\frac{1}{2}$ times row 3 plus row 2
$$
\left(\frac{1}{2} R_{3}+R_{2} \rightarrow R_{2}\right)
$$
Replace row 1 with the sum of $-\frac{3}{2}$ times row 3 plus row 1
$$
\left(-\frac{3}{2} R_{3}+R_{1} \rightarrow R_{1}\right)
$$
$$
\left[\begin{array}{cccc}
1 & 0 & 0 & 15 \\
0 & 1 & 0 & 5 \\
0 & 0 & 1 & 10
\end{array}\right]
$$
The final matrix contains the same solution information as we have on the left in the form of equations. Recall that the first column of our matrices held the coefficients of the $b$ variable; the second and third columns held the coefficients of the $g$ and $r$ variables, respectively. Therefore, the first row of the matrix can be interpreted as " $b+0 g+0 r=15$," or more concisely, " $b=15$."
Let's practice this manipulation again.
Example 2 Find a solution to the following system of linear equations by simultaneously manipulating the equations and the corresponding augmented matrices.
$$
\begin{aligned}
x_{1}+x_{2}+x_{3} & =0 \\
2 x_{1}+2 x_{2}+x_{3} & =0 \\
-1 x_{1}+x_{2}-2 x_{3} & =2
\end{aligned}
$$
Solution We'll first convert this system of equations into a matrix, then we'll proceed by manipulating the system of equations (and hence the matrix) to find a solution. Again, there is not just one "right" way of proceeding; we'll choose a method that is pretty efficient, but other methods certainly exist (and may be "better"!). The method use here, though, is a good one, and it is the method that we will be learning in the future.
The given system and its corresponding augmented matrix are seen below.
Original system of equations
$$
\begin{gathered}
x_{1}+x_{2}+x_{3}=0 \\
2 x_{1}+2 x_{2}+x_{3}=0 \\
-1 x_{1}+x_{2}-2 x_{3}=2
\end{gathered}
$$
Corresponding matrix
$$
\left[\begin{array}{cccc}
1 & 1 & 1 & 0 \\
2 & 2 & 1 & 0 \\
-1 & 1 & -2 & 2
\end{array}\right]
$$
We'll proceed by trying to get the $x_{1}$ out of the second and third equation. Replace equation 2 with the sum of $(-2)$ times equation 1 plus equation 2;
Replace equation 3 with the sum of equation 1 and equation 3 Replace row 2 with the sum of $(-2)$ times row 1 plus row 2 $\left(-2 R_{1}+R_{2} \rightarrow R_{2}\right)$;
Replace row 3 with the sum of row 1 and row 3
$$
\left(R_{1}+R_{3} \rightarrow R_{3}\right)
$$
$$
\begin{aligned}
x_{1}+x_{2}+x_{3} & =0 \\
-x_{3} & =0 \\
2 x_{2}-x_{3} & =2
\end{aligned}
$$
Notice that the second equation no longer contains $x_{2}$. We'll exchange the order of the equations so that we can follow the convention of solving for the second variable in the second equation.
Interchange equations 2 and 3
$$
\begin{array}{r}
x_{1}+x_{2}+x_{3}=0 \\
2 x_{2}-x_{3}=2
\end{array}
$$
Multiply equation 2 by $\frac{1}{2}$
$$
\begin{aligned}
x_{1}+x_{2}+x_{3} & =0 \\
x_{2}-\frac{1}{2} x_{3} & =1 \\
-x_{3} & =0
\end{aligned}
$$
Multiply equation 3 by -1
$$
\begin{aligned}
x_{1}+x_{2}+x_{3} & =0 \\
x_{2}-\frac{1}{2} x_{3} & =1 \\
x_{3} & =0
\end{aligned}
$$
Interchange rows 2 and 3
$$
R_{2} \leftrightarrow R_{3}
$$
$\left[\begin{array}{cccc}1 & 1 & 1 & 0 \\ 0 & 2 & -1 & 2 \\ 0 & 0 & -1 & 0\end{array}\right]$
Multiply row 2 by $\frac{1}{2}$
$$
\left(\frac{1}{2} R_{2} \rightarrow R_{2}\right)
$$$$
\left[\begin{array}{cccc}
1 & 1 & 1 & 0 \\
0 & 1 & -\frac{1}{2} & 1 \\
0 & 0 & -1 & 0
\end{array}\right]
$$
Multiply row 3 by -1
$$
\left(-1 R_{3} \rightarrow R_{3}\right)
$$
$$
\left[\begin{array}{cccc}
1 & 1 & 1 & 0 \\
0 & 1 & -\frac{1}{2} & 1 \\
0 & 0 & 1 & 0
\end{array}\right]
$$
Notice that the last equation (and also the last row of the matrix) show that $x_{3}=0$. Knowing this would allow us to simply eliminate the $x_{3}$ from the first two equations. However, we will formally do this by manipulating the equations (and rows) as we have previously.
Replace equation 1 with the sum of $(-1)$ times equation 3 plus equation 1 ;
Replace equation 2 with the sum of $\frac{1}{2}$ times equation 3 plus equation 2 Replace row 1 with the sum of $(-1)$ times row 3 plus row 1
$$
\left(-R_{3}+R_{1} \rightarrow R_{1}\right) \text {; }
$$
Replace row 2 with the sum of $\frac{1}{2}$ times row 3 plus row 2
$$
\left(\frac{1}{2} R_{3}+R_{2} \rightarrow R_{2}\right)
$$
$$
\begin{aligned}
x_{1}+x_{2} & =0 \\
x_{2} & =1 \\
x_{3} & =0
\end{aligned}
$$
Notice how the second equation shows that $x_{2}=1$. All that remains to do is to solve for $x_{1}$.
Replace equation 1 with the sum of $(-1)$ times equation 2 plus equation 1
$$
\begin{aligned}
x_{1} & \\
x_{2} & =-1 \\
& =1 \\
x_{3} & =0
\end{aligned}
$$
Replace row 1 with the sum of $(-1)$ times row 2 plus row 1
$$
\left(-R_{2}+R_{1} \rightarrow R_{1}\right)
$$
$$
\left[\begin{array}{cccc}
1 & 0 & 0 & -1 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & 0
\end{array}\right]
$$
Obviously the equations on the left tell us that $x_{1}=-1, x_{2}=1$ and $x_{3}=0$, and notice how the matrix on the right tells us the same information.
## Exercises 1.2
In Exercises 1-4, convert the given system of linear equations into an augmented matrix.
$$
\begin{aligned}
& 3 x+4 y+5 z=7 \\
& \text { 1. }-x+y-3 z=1 \\
& 2 x-2 y+3 z=5 \\
& 2 x+5 y-6 z=2 \\
& \text { 2. } 9 x-8 z=10 \\
& -2 x+4 y+z=-7 \\
& x_{1}+3 x_{2}-4 x_{3}+5 x_{4}=17 \\
& \text { 3. }-x_{1}+4 x_{3}+8 x_{4}=1 \\
& 2 x_{1}+3 x_{2}+4 x_{3}+5 x_{4}=6 \\
& 3 x_{1}-2 x_{2}=4 \\
& \text { 4. } \begin{aligned}
2 x_{1} & =3 \\
-x_{1}+9 x_{2} & =8
\end{aligned} \\
& 5 x_{1}-7 x_{2}=13
\end{aligned}
$$
In Exercises 5 - 9, convert the given augmented matrix into a system of linear equations. Use the variables $x_{1}, x_{2}$, etc.
5. $\left[\begin{array}{ccc}1 & 2 & 3 \\ -1 & 3 & 9\end{array}\right]$
6. $\left[\begin{array}{ccc}-3 & 4 & 7 \\ 0 & 1 & -2\end{array}\right]$
7. $\left[\begin{array}{ccccc}1 & 1 & -1 & -1 & 2 \\ 2 & 1 & 3 & 5 & 7\end{array}\right]$
8. $\left[\begin{array}{ccccc}1 & 0 & 0 & 0 & 2 \\ 0 & 1 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 & 5 \\ 0 & 0 & 0 & 1 & 3\end{array}\right]$
$$
\text { 9. }\left[\begin{array}{llllll}
1 & 0 & 1 & 0 & 7 & 2 \\
0 & 1 & 3 & 2 & 0 & 5
\end{array}\right]
$$
In Exercises 10 - 15, perform the given row operations on $A$, where
$$
A=\left[\begin{array}{ccc}
2 & -1 & 7 \\
0 & 4 & -2 \\
5 & 0 & 3
\end{array}\right] .
$$
10. $-1 R_{1} \rightarrow R_{1}$
11. $R_{2} \leftrightarrow R_{3}$
12. $R_{1}+R_{2} \rightarrow R_{2}$
13. $2 R_{2}+R_{3} \rightarrow R_{3}$
14. $\frac{1}{2} R_{2} \rightarrow R_{2}$
15. $-\frac{5}{2} R_{1}+R_{3} \rightarrow R_{3}$
A matrix $A$ is given below. In Exercises 16 20 , a matrix $B$ is given. Give the row operation that transforms $A$ into $B$.
$$
A=\left[\begin{array}{lll}
1 & 1 & 1 \\
1 & 0 & 1 \\
1 & 2 & 3
\end{array}\right]
$$
16. $B=\left[\begin{array}{lll}1 & 1 & 1 \\ 2 & 0 & 2 \\ 1 & 2 & 3\end{array}\right]$
17. $B=\left[\begin{array}{lll}1 & 1 & 1 \\ 2 & 1 & 2 \\ 1 & 2 & 3\end{array}\right]$
18. $B=\left[\begin{array}{lll}3 & 5 & 7 \\ 1 & 0 & 1 \\ 1 & 2 & 3\end{array}\right]$ $\begin{aligned} \text { 19. } B & =\left[\begin{array}{lll}1 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 2 & 3\end{array}\right] \\ \text { 20. } B & =\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 0 & 1 \\ 0 & 2 & 2\end{array}\right]\end{aligned}$
In Exercises 21 - 26, rewrite the system of equations in matrix form. Find the solution to the linear system by simultaneously manipulating the equations and the matrix.
21. $\begin{aligned} x+y & =3 \\ 2 x-3 y & =1\end{aligned}$
22. $\begin{aligned} 2 x+4 y & =10 \\ -x+y & =4\end{aligned}$ 23. $\begin{array}{r}-2 x+3 y=2 \\ -x+y=1\end{array}$
23. $\begin{aligned} 2 x+3 y & =2 \\ -2 x+6 y & =1\end{aligned}$
24. $\begin{aligned}-5 x_{1}+2 x_{3} & =14 \\ -3 x_{1} x_{2}+x_{3} & =8\end{aligned}$
25. $\begin{array}{rlll}-5 x_{2}+2 x_{3} & = & -11 \\ +2 x_{3} & =15 \\ -3 x_{2}+x_{3} & = & -8\end{array}$
### Elementary Row Operations and Gaussian Elimina- tion
## AS YOU READ . .
1. Give two reasons why the Elementary Row Operations are called "Elementary."
2. T/F: Assuming a solution exists, all linear systems of equations can be solved using only elementary row operations.
3. Give one reason why one might not be interested in putting a matrix into reduced row echelon form.
4. Identify the leading $1 \mathrm{~s}$ in the following matrix:
$$
\left[\begin{array}{llll}
1 & 0 & 0 & 1 \\
0 & 1 & 1 & 0 \\
0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0
\end{array}\right]
$$
5. Using the "forward" and "backward" steps of Gaussian elimination creates lots of making computations easier.
In our examples thus far, we have essentially used just three types of manipulations in order to find solutions to our systems of equations. These three manipulations are:
1. Add a scalar multiple of one equation to a second equation, and replace the second equation with that sum 2. Multiply one equation by a nonzero scalar
## Swap the position of two equations in our list
We saw earlier how we could write all the information of a system of equations in a matrix, so it makes sense that we can perform similar operations on matrices (as we have done before). Again, simply replace the word "equation" above with the word "row."
We didn't justify our ability to manipulate our equations in the above three ways; it seems rather obvious that we should be able to do that. In that sense, these operations are "elementary." These operations are elementary in another sense; they are fundamental - they form the basis for much of what we will do in matrix algebra. Since these operations are so important, we list them again here in the context of matrices.
## Key Idea 1
## Elementary Row Operations
1. Add a scalar multiple of one row to another row, and replace the latter row with that sum
2. Multiply one row by a nonzero scalar
3. Swap the position of two rows
Given any system of linear equations, we can find a solution (if one exists) by using these three row operations. Elementary row operations give us a new linear system, but the solution to the new system is the same as the old. We can use these operations as much as we want and not change the solution. This brings to mind two good questions:
1. Since we can use these operations as much as we want, how do we know when to stop? (Where are we supposed to "go" with these operations?)
2. Is there an efficient way of using these operations? (How do we get "there" the fastest?)
We'll answer the first question first. Most of the time ${ }^{1}$ we will want to take our original matrix and, using the elementary row operations, put it into something called reduced row echelon form. ${ }^{2}$ This is our "destination," for this form allows us to readily identify whether or not a solution exists, and in the case that it does, what that solution is.
In the previous section, when we manipulated matrices to find solutions, we were unwittingly putting the matrix into reduced row echelon form. However, not all solutions come in such a simple manner as we've seen so far. Putting a matrix into reduced
${ }^{1}$ unless one prefers obfuscation to clarification
${ }^{2}$ Some texts use the term reduced echelon form instead. row echelon form helps us identify all types of solutions. We'll explore the topic of understanding what the reduced row echelon form of a matrix tells us in the following sections; in this section we focus on finding it.
## Definition 3
## Reduced Row Echelon Form
A matrix is in reduced row echelon form if its entries satisfy the following conditions.
1. The first nonzero entry in each row is a 1 (called a leading 1).
2. Each leading 1 comes in a column to the right of the leading $1 \mathrm{~s}$ in rows above it.
3. All rows of all Os come at the bottom of the matrix.
4. If a column contains a leading 1 , then all other entries in that column are 0 .
A matrix that satisfies the first three conditions is said to be in row echelon form.
Example 3 Which of the following matrices is in reduced row echelon form?
a) $\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
b) $\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 2\end{array}\right]$
c) $\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$
d) $\left[\begin{array}{lll}1 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
e) $\left[\begin{array}{llll}1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 3\end{array}\right]$
f) $\left[\begin{array}{llll}1 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 4\end{array}\right]$
g) $\left[\begin{array}{llllll}0 & 1 & 2 & 3 & 0 & 4 \\ 0 & 0 & 0 & 0 & 1 & 5 \\ 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]$
h) $\left[\begin{array}{lll}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
Solution The matrices in a), b), c), d) and g) are all in reduced row echelon form. Check to see that each satisfies the necessary conditions. If your instincts were wrong on some of these, correct your thinking accordingly.
The matrix in e) is not in reduced row echelon form since the row of all zeros is not at the bottom. The matrix in $f$ ) is not in reduced row echelon form since the first nonzero entries in rows 2 and 3 are not 1 . Finally, the matrix in $h$ ) is not in reduced row echelon form since the first entry in column 2 is not zero; the second 1 in column 2 is a leading one, hence all other entries in that column should be 0 .
We end this example with a preview of what we'll learn in the future. Consider the matrix in b). If this matrix came from the augmented matrix of a system of linear equations, then we can readily recognize that the solution of the system is $x_{1}=1$ and $x_{2}=2$. Again, in previous examples, when we found the solution to a linear system, we were unwittingly putting our matrices into reduced row echelon form.
We began this section discussing how we can manipulate the entries in a matrix with elementary row operations. This led to two questions, "Where do we go?" and "How do we get there quickly?" We've just answered the first question: most of the time we are "going to" reduced row echelon form. We now address the second question.
There is no one "right" way of using these operations to transform a matrix into reduced row echelon form. However, there is a general technique that works very well in that it is very efficient (so we don't waste time on unnecessary steps). This technique is called Gaussian elimination. It is named in honor of the great mathematician Karl Friedrich Gauss.
While this technique isn't very difficult to use, it is one of those things that is easier understood by watching it being used than explained as a series of steps. With this in mind, we will go through one more example highlighting important steps and then we'll explain the procedure in detail.
Example $4 \quad$ Put the augmented matrix of the following system of linear equations into reduced row echelon form.
$$
\begin{aligned}
-3 x_{1}-3 x_{2}+9 x_{3} & =12 \\
2 x_{1}+2 x_{2}-4 x_{3} & =-2 \\
-2 x_{2}-4 x_{3} & =-8
\end{aligned}
$$
Solution We start by converting the linear system into an augmented matrix.
$$
\left[\begin{array}{cccc}
-3 & -3 & 9 & 12 \\
2 & 2 & -4 & -2 \\
0 & -2 & -4 & -8
\end{array}\right]
$$
Our next step is to change the entry in the box to a 1 . To do this, let's multiply row 1 by $-\frac{1}{3}$.
$$
-\frac{1}{3} R_{1} \rightarrow R_{1} \quad\left[\begin{array}{cccc}
1 & 1 & -3 & -4 \\
2 & 2 & -4 & -2 \\
0 & -2 & -4 & -8
\end{array}\right]
$$
We have now created a leading 1 ; that is, the first entry in the first row is a 1. Our next step is to put zeros under this 1 . To do this, we'll use the elementary row operation given below.
$$
-2 R_{1}+R_{2} \rightarrow R_{2} \quad\left[\begin{array}{cccc}
1 & 1 & -3 & -4 \\
0 & 0 & 2 & 6 \\
0 & -2 & -4 & -8
\end{array}\right]
$$
Once this is accomplished, we shift our focus from the leading one down one row, and to the right one column, to the position that is boxed. We again want to put a 1 in this position. We can use any elementary row operations, but we need to restrict ourselves to using only the second row and any rows below it. Probably the simplest thing we can do is interchange rows 2 and 3 , and then scale the new second row so that there is a 1 in the desired position.
$$
\begin{aligned}
R_{2} \leftrightarrow R_{3} & {\left[\begin{array}{cccc}
1 & 1 & -3 & -4 \\
0 & -2 & -4 & -8 \\
0 & 0 & 2 & 6
\end{array}\right] } \\
-\frac{1}{2} R_{2} \rightarrow R_{2} \quad & {\left[\begin{array}{cccc}
1 & 1 & -3 & -4 \\
0 & 1 & 2 & 4 \\
0 & 0 & 2 & 6
\end{array}\right] }
\end{aligned}
$$
We have now created another leading 1 , this time in the second row. Our next desire is to put zeros underneath it, but this has already been accomplished by our previous steps. Therefore we again shift our attention to the right one column and down one row, to the next position put in the box. We want that to be a 1. A simple scaling will accomplish this.
$$
\frac{1}{2} R_{3} \rightarrow R_{3} \quad\left[\begin{array}{cccc}
1 & 1 & -3 & -4 \\
0 & 1 & 2 & 4 \\
0 & 0 & 1 & 3
\end{array}\right]
$$
This ends what we will refer to as the forward steps. Our next task is to use the elementary row operations and go back and put zeros above our leading $1 \mathrm{~s}$. This is referred to as the backward steps. These steps are given below.
$$
\begin{array}{ll}
\begin{array}{c}
3 R_{3}+R_{1} \rightarrow R_{1} \\
-2 R_{3}+R_{2} \rightarrow R_{2}
\end{array} & {\left[\begin{array}{cccc}
1 & 1 & 0 & 5 \\
0 & 1 & 0 & -2 \\
0 & 0 & 1 & 3
\end{array}\right]} \\
-R_{2}+R_{1} \rightarrow R_{1} & {\left[\begin{array}{cccc}
1 & 0 & 0 & 7 \\
0 & 1 & 0 & -2 \\
0 & 0 & 1 & 3
\end{array}\right]}
\end{array}
$$
It is now easy to read off the solution as $x_{1}=7, x_{2}=-2$ and $x_{3}=3$. We now formally explain the procedure used to find the solution above. As you read through the procedure, follow along with the example above so that the explanation makes more sense.
## Forward Steps
1. Working from left to right, consider the first column that isn't all zeros that hasn't already been worked on. Then working from top to bottom, consider the first row that hasn't been worked on.
2. If the entry in the row and column that we are considering is zero, interchange rows with a row below the current row so that that entry is nonzero. If all entries below are zero, we are done with this column; start again at step 1.
3. Multiply the current row by a scalar to make its first entry a 1 (a leading 1).
4. Repeatedly use Elementary Row Operation 1 to put zeros underneath the leading one.
5. Go back to step 1 and work on the new rows and columns until either all rows or columns have been worked on.
If the above steps have been followed properly, then the following should be true about the current state of the matrix:
1. The first nonzero entry in each row is a 1 (a leading 1 ).
2. Each leading 1 is in a column to the right of the leading 1 s above it.
3. All rows of all zeros come at the bottom of the matrix.
Note that this means we have just put a matrix into row echelon form. The next steps finish the conversion into reduced row echelon form. These next steps are referred to as the backward steps. These are much easier to state.
## Backward Steps
1. Starting from the right and working left, use Elementary Row Operation 1 repeatedly to put zeros above each leading 1.
The basic method of Gaussian elimination is this: create leading ones and then use elementary row operations to put zeros above and below these leading ones. We can do this in any order we please, but by following the "Forward Steps" and "Backward Steps," we make use of the presence of zeros to make the overall computations easier. This method is very efficient, so it gets its own name (which we've already been using). Definition 4
Let's practice some more.
## Example 5
elon form, whereUse Gaussian elimination to put the matrix $A$ into reduced row ech-
$$
A=\left[\begin{array}{ccccc}
-2 & -4 & -2 & -10 & 0 \\
2 & 4 & 1 & 9 & -2 \\
3 & 6 & 1 & 13 & -4
\end{array}\right]
$$
Solution We start by wanting to make the entry in the first column and first row a 1 (a leading 1). To do this we'll scale the first row by a factor of $-\frac{1}{2}$.
$$
-\frac{1}{2} R_{1} \rightarrow R_{1} \quad\left[\begin{array}{ccccc}
1 & 2 & 1 & 5 & 0 \\
2 & 4 & 1 & 9 & -2 \\
3 & 6 & 1 & 13 & -4
\end{array}\right]
$$
Next we need to put zeros in the column below this newly formed leading 1 .
$$
\begin{aligned}
& -2 R_{1}+R_{2} \rightarrow R_{2} \\
& -3 R_{1}+R_{3} \rightarrow R_{3}
\end{aligned}
$$
$$
\left[\begin{array}{ccccc}
1 & 2 & 1 & 5 & 0 \\
0 & 0 & -1 & -1 & -2 \\
0 & 0 & -2 & -2 & -4
\end{array}\right]
$$
Our attention now shifts to the right one column and down one row to the position indicated by the box. We want to put a 1 in that position. Our only options are to either scale the current row or to interchange rows with a row below it. However, in this case neither of these options will accomplish our goal. Therefore, we shift our attention to the right one more column.
We want to put a 1 where there is a -1 . A simple scaling will accomplish this; once done, we will put a 0 underneath this leading one.
$$
-R_{2} \rightarrow R_{2} \quad\left[\begin{array}{ccccc}
1 & 2 & 1 & 5 & 0 \\
0 & 0 & 1 & 1 & 2 \\
0 & 0 & -2 & -2 & -4
\end{array}\right]
$$
$$
2 R_{2}+R_{3} \rightarrow R_{3} \quad\left[\begin{array}{ccccc}
1 & 2 & 1 & 5 & 0 \\
0 & 0 & 1 & 1 & 2 \\
0 & 0 & 0 & 0 & 0
\end{array}\right]
$$
Our attention now shifts over one more column and down one row to the position indicated by the box; we wish to make this a 1 . Of course, there is no way to do this, so we are done with the forward steps.
Our next goal is to put a 0 above each of the leading 1s (in this case there is only one leading 1 to deal with).
$$
-R_{2}+R_{1} \rightarrow R_{1} \quad\left[\begin{array}{ccccc}
1 & 2 & 0 & 4 & -2 \\
0 & 0 & 1 & 1 & 2 \\
0 & 0 & 0 & 0 & 0
\end{array}\right]
$$
This final matrix is in reduced row echelon form.
Example $6 \quad$ Put the matrix
$$
\left[\begin{array}{llll}
1 & 2 & 1 & 3 \\
2 & 1 & 1 & 1 \\
3 & 3 & 2 & 1
\end{array}\right]
$$
into reduced row echelon form.
Solution Here we will show all steps without explaining each one.
$$
\begin{aligned}
& \begin{array}{l}
-2 R_{1}+R_{2} \rightarrow R_{2} \\
-3 R_{1}+R_{3} \rightarrow R_{3}
\end{array} \quad\left[\begin{array}{cccc}
1 & 2 & 1 & 3 \\
0 & -3 & -1 & -5 \\
0 & -3 & -1 & -8
\end{array}\right] \\
& -\frac{1}{3} R_{2} \rightarrow R_{2} \quad\left[\begin{array}{cccc}
1 & 2 & 1 & 3 \\
0 & 1 & 1 / 3 & 5 / 3 \\
0 & -3 & -1 & -8
\end{array}\right] \\
& 3 R_{2}+R_{3} \rightarrow R_{3} \quad\left[\begin{array}{cccc}
1 & 2 & 1 & 3 \\
0 & 1 & 1 / 3 & 5 / 3 \\
0 & 0 & 0 & -3
\end{array}\right] \\
& -\frac{1}{3} R_{3} \rightarrow R_{3} \quad\left[\begin{array}{cccc}
1 & 2 & 1 & 3 \\
0 & 1 & 1 / 3 & 5 / 3 \\
0 & 0 & 0 & 1
\end{array}\right] \\
& \begin{array}{l}
-3 R_{3}+R_{1} \rightarrow R_{1} \\
-\frac{5}{3} R_{3}+R_{2} \rightarrow R_{2}
\end{array} \quad\left[\begin{array}{cccc}
1 & 2 & 1 & 0 \\
0 & 1 & 1 / 3 & 0 \\
0 & 0 & 0 & 1
\end{array}\right] \\
& -2 R_{2}+R_{1} \rightarrow R_{1} \quad\left[\begin{array}{cccc}
1 & 0 & 1 / 3 & 0 \\
0 & 1 & 1 / 3 & 0 \\
0 & 0 & 0 & 1
\end{array}\right]
\end{aligned}
$$
The last matrix in the above example is in reduced row echelon form. If one thinks of the original matrix as representing the augmented matrix of a system of linear equations, this final result is interesting. What does it mean to have a leading one in the last column? We'll figure this out in the next section.
Example 7 Put the matrix $A$ into reduced row echelon form, where
$$
A=\left[\begin{array}{cccc}
2 & 1 & -1 & 4 \\
1 & -1 & 2 & 12 \\
2 & 2 & -1 & 9
\end{array}\right]
$$
Solution We'll again show the steps without explanation, although we will stop at the end of the forward steps and make a comment.
$$
\begin{array}{cccc}
\frac{1}{2} R_{1} \rightarrow R_{1} & {\left[\begin{array}{cccc}
1 & 1 / 2 & -1 / 2 & 2 \\
1 & -1 & 2 & 12 \\
2 & 2 & -1 & 9
\end{array}\right]} \\
\begin{array}{c}
-R_{1}+R_{2} \rightarrow R_{2} \\
-2 R_{1}+R_{3} \rightarrow R_{3}
\end{array} & {\left[\begin{array}{cccc}
1 & 1 / 2 & -1 / 2 & 2 \\
0 & -3 / 2 & 5 / 2 & 10 \\
0 & 1 & 0 & 5
\end{array}\right]} \\
-\frac{2}{3} R_{2} \rightarrow R_{2} & {\left[\begin{array}{cccc}
1 & 1 / 2 & -1 / 2 & 2 \\
0 & 1 & -5 / 3 & -20 / 3 \\
0 & 1 & 0 & 5
\end{array}\right]} \\
-R_{2}+R_{3} \rightarrow R_{3} & {\left[\begin{array}{cccc}
1 & 1 / 2 & -1 / 2 & 2 \\
0 & 1 & -5 / 3 & -20 / 3 \\
0 & 0 & 5 / 3 & 35 / 3
\end{array}\right]} \\
\frac{3}{5} R_{3} \rightarrow R_{3} & {\left[\begin{array}{cccc}
1 & 1 / 2 & -1 / 2 & 2 \\
0 & 1 & -5 / 3 & -20 / 3 \\
0 & 0 & 1 & 7
\end{array}\right]}
\end{array}
$$
Let's take a break here and think about the state of our linear system at this moment. Converting back to linear equations, we now know
$$
\begin{aligned}
x_{1}+1 / 2 x_{2}-1 / 2 x_{3} & =2 \\
x_{2}-5 / 3 x_{3} & =-20 / 3 . \\
x_{3} & =7
\end{aligned} .
$$
Since we know that $x_{3}=7$, the second equation turns into
$$
x_{2}-(5 / 3)(7)=-20 / 3 \text {, }
$$
telling us that $x_{2}=5$.
Finally, knowing values for $x_{2}$ and $x_{3}$ lets us substitute in the first equation and find
$$
x_{1}+(1 / 2)(5)-(1 / 2)(7)=2,
$$
so $x_{1}=3$.
This process of substituting known values back into other equations is called back substitution. This process is essentially what happens when we perform the backward steps of Gaussian elimination. We make note of this below as we finish out finding the reduced row echelon form of our matrix.
$$
\begin{array}{ccc}
\begin{array}{c}
\frac{5}{3} R_{3}+R_{2} \rightarrow R_{2} \\
\text { (knowing } x_{3}=7 \text { allows us } \\
\text { to find } \left.x_{2}=5\right)
\end{array} & {\left[\begin{array}{cccc}
1 & 1 / 2 & -1 / 2 & 2 \\
0 & 1 & 0 & 5 \\
0 & 0 & 1 & 7
\end{array}\right]} \\
\frac{1}{2} R_{3}+R_{1} \rightarrow R_{1} \\
-\frac{1}{2} R_{2}+R_{1} \rightarrow R_{1} \\
\begin{array}{c}
\text { (knowing } x_{2}=5 \text { and } x_{3}=7 \\
\text { allows us to find } \left.x_{1}=3\right)
\end{array} & {\left[\begin{array}{llll}
1 & 0 & 0 & 3 \\
0 & 1 & 0 & 5 \\
0 & 0 & 1 & 7
\end{array}\right]}
\end{array}
$$
We did our operations slightly "out of order" in that we didn't put the zeros above our leading 1 in the third column in the same step, highlighting how back substitution works.
In all of our practice, we've only encountered systems of linear equations with exactly one solution. Is this always going to be the case? Could we ever have systems with more than one solution? If so, how many solutions could there be? Could we have systems without a solution? These are some of the questions we'll address in the next section.
## Exercises 1.3
In Exercises 1-4, state whether or not the given matrices are in reduced row echelon form. If it is not, state why.
1. (a) $\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
(b) $\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$
2. (a) $\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]$
(b) $\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 1\end{array}\right]$
3. (a) $\left[\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right]$
(b) $\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{array}\right]$
(c) $\left[\begin{array}{lll}0 & 0 & 0 \\ 1 & 0 & 0\end{array}\right]$
(d) $\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]$
(c) $\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$
(d) $\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 2\end{array}\right]$
4. (a) $\left[\begin{array}{llll}2 & 0 & 0 & 2 \\ 0 & 2 & 0 & 2 \\ 0 & 0 & 2 & 2\end{array}\right]$
(b) $\left[\begin{array}{llll}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0\end{array}\right]$
(c) $\left[\begin{array}{cccc}0 & 0 & 1 & -5 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{array}\right]$
(d) $\left[\begin{array}{llllll}1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0\end{array}\right]$ In Exercises 5 - 22, use Gaussian Elimination to put the given matrix into reduced row echelon form.
5. $\left[\begin{array}{cc}1 & 2 \\ -3 & -5\end{array}\right]$
6. $\left[\begin{array}{ll}2 & -2 \\ 3 & -2\end{array}\right]$
7. $\left[\begin{array}{cc}4 & 12 \\ -2 & -6\end{array}\right]$
8. $\left[\begin{array}{cc}-5 & 7 \\ 10 & 14\end{array}\right]$
9. $\left[\begin{array}{lll}-1 & 1 & 4 \\ -2 & 1 & 1\end{array}\right]$
10. $\left[\begin{array}{lll}7 & 2 & 3 \\ 3 & 1 & 2\end{array}\right]$
11. $\left[\begin{array}{ccc}3 & -3 & 6 \\ -1 & 1 & -2\end{array}\right]$
12. $\left[\begin{array}{ccc}4 & 5 & -6 \\ -12 & -15 & 18\end{array}\right]$
13. $\left[\begin{array}{ccc}-2 & -4 & -8 \\ -2 & -3 & -5 \\ 2 & 3 & 6\end{array}\right]$
14. $\left[\begin{array}{lll}2 & 1 & 1 \\ 1 & 1 & 1 \\ 2 & 1 & 2\end{array}\right]$ 15. $\left[\begin{array}{ccc}1 & 2 & 1 \\ 1 & 3 & 1 \\ -1 & -3 & 0\end{array}\right]$
15. $\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 6 & 9\end{array}\right]$
16. $\left[\begin{array}{cccc}1 & 1 & 1 & 2 \\ 2 & -1 & -1 & 1 \\ -1 & 1 & 1 & 0\end{array}\right]$
17. $\left[\begin{array}{cccc}2 & -1 & 1 & 5 \\ 3 & 1 & 6 & -1 \\ 3 & 0 & 5 & 0\end{array}\right]$
18. $\left[\begin{array}{cccc}1 & 1 & -1 & 7 \\ 2 & 1 & 0 & 10 \\ 3 & 2 & -1 & 17\end{array}\right]$
19. $\left[\begin{array}{cccc}4 & 1 & 8 & 15 \\ 1 & 1 & 2 & 7 \\ 3 & 1 & 5 & 11\end{array}\right]$
20. $\left[\begin{array}{llllll}2 & 2 & 1 & 3 & 1 & 4 \\ 1 & 1 & 1 & 3 & 1 & 4\end{array}\right]$
21. $\left[\begin{array}{cccccc}1 & -1 & 3 & 1 & -2 & 9 \\ 2 & -2 & 6 & 1 & -2 & 13\end{array}\right]$
### Existence and Uniqueness of Solutions
## AS YOU READ
1. $\mathrm{T} / \mathrm{F}$ : It is possible for a linear system to have exactly 5 solutions.
2. T/F: A variable that corresponds to a leading 1 is "free."
3. How can one tell what kind of solution a linear system of equations has?
4. Give an example (different from those given in the text) of a 2 equation, 2 unknown linear system that is not consistent.
5. T/F: A particular solution for a linear system with infinite solutions can be found by arbitrarily picking values for the free variables. So far, whenever we have solved a system of linear equations, we have always found exactly one solution. This is not always the case; we will find in this section that some systems do not have a solution, and others have more than one.
We start with a very simple example. Consider the following linear system:
$$
x-y=0 .
$$
There are obviously infinite solutions to this system; as long as $x=y$, we have a solution. We can picture all of these solutions by thinking of the graph of the equation $y=x$ on the traditional $x, y$ coordinate plane.
Let's continue this visual aspect of considering solutions to linear systems. Consider the system
$$
\begin{aligned}
& x+y=2 \\
& x-y=0
\end{aligned}
$$
Each of these equations can be viewed as lines in the coordinate plane, and since their slopes are different, we know they will intersect somewhere (see Figure 1.1 (a)). In this example, they intersect at the point $(1,1)$ - that is, when $x=1$ and $y=1$, both equations are satisfied and we have a solution to our linear system. Since this is the only place the two lines intersect, this is the only solution.
Now consider the linear system
$$
\begin{array}{r}
x+y=1 \\
2 x+2 y=2
\end{array}
$$
It is clear that while we have two equations, they are essentially the same equation; the second is just a multiple of the first. Therefore, when we graph the two equations, we are graphing the same line twice (see Figure 1.1 (b); the thicker line is used to represent drawing the line twice). In this case, we have an infinite solution set, just as if we only had the one equation $x+y=1$. We often write the solution as $x=1-y$ to demonstrate that $y$ can be any real number, and $x$ is determined once we pick a value for $y$.
(a)
(b)
(c)
Figure 1.1: The three possibilities for two linear equations with two unknowns. Finally, consider the linear system
$$
\begin{aligned}
& x+y=1 \\
& x+y=2
\end{aligned}
$$
We should immediately spot a problem with this system; if the sum of $x$ and $y$ is 1 , how can it also be 2 ? There is no solution to such a problem; this linear system has no solution. We can visualize this situation in Figure 1.1 (c); the two lines are parallel and never intersect.
If we were to consider a linear system with three equations and two unknowns, we could visualize the solution by graphing the corresponding three lines. We can picture that perhaps all three lines would meet at one point, giving exactly 1 solution; perhaps all three equations describe the same line, giving an infinite number of solutions; perhaps we have different lines, but they do not all meet at the same point, giving no solution. We further visualize similar situations with, say, 20 equations with two variables.
While it becomes harder to visualize when we add variables, no matter how many equations and variables we have, solutions to linear equations always come in one of three forms: exactly one solution, infinite solutions, or no solution. This is a fact that we will not prove here, but it deserves to be stated.
Theorem 1
Solution Forms of Linear Systems
Every linear system of equations has exactly one solution, infinite solutions, or no solution.
This leads us to a definition. Here we don't differentiate between having one solution and infinite solutions, but rather just whether or not a solution exists.
Definition 5
## Consistent and Inconsistent Linear Systems
A system of linear equations is consistent if it has a solution (perhaps more than one). A linear system is inconsistent if it does not have a solution.
How can we tell what kind of solution (if one exists) a given system of linear equations has? The answer to this question lies with properly understanding the reduced row echelon form of a matrix. To discover what the solution is to a linear system, we first put the matrix into reduced row echelon form and then interpret that form properly.
Before we start with a simple example, let us make a note about finding the re- duced row echelon form of a matrix.
Technology Note: In the previous section, we learned how to find the reduced row echelon form of a matrix using Gaussian elimination - by hand. We need to know how to do this; understanding the process has benefits. However, actually executing the process by hand for every problem is not usually beneficial. In fact, with large systems, computing the reduced row echelon form by hand is effectively impossible. Our main concern is what "the rref" is, not what exact steps were used to arrive there. Therefore, the reader is encouraged to employ some form of technology to find the reduced row echelon form. Computer programs such as Mathematica, MATLAB, Maple, and Derive can be used; many handheld calculators (such as Texas Instruments calculators) will perform these calculations very quickly.
As a general rule, when we are learning a new technique, it is best to not use technology to aid us. This helps us learn not only the technique but some of its "inner workings." We can then use technology once we have mastered the technique and are now learning how to use it to solve problems.
From here on out, in our examples, when we need the reduced row echelon form of a matrix, we will not show the steps involved. Rather, we will give the initial matrix, then immediately give the reduced row echelon form of the matrix. We trust that the reader can verify the accuracy of this form by both performing the necessary steps by hand or utilizing some technology to do it for them.
Our first example explores officially a quick example used in the introduction of this section.
Example $8 \quad$ Find the solution to the linear system
$$
\begin{aligned}
x_{1}+x_{2} & =1 \\
2 x_{1}+2 x_{2} & =2
\end{aligned} .
$$
Solution Create the corresponding augmented matrix, and then put the matrix into reduced row echelon form.
$$
\left[\begin{array}{lll}
1 & 1 & 1 \\
2 & 2 & 2
\end{array}\right] \quad \overrightarrow{\operatorname{rref}} \quad\left[\begin{array}{lll}
1 & 1 & 1 \\
0 & 0 & 0
\end{array}\right]
$$
Now convert the reduced matrix back into equations. In this case, we only have one equation,
$$
x_{1}+x_{2}=1
$$
or, equivalently,
$$
\begin{aligned}
& x_{1}=1-x_{2} \\
& x_{2} \text { is free. }
\end{aligned}
$$
We have just introduced a new term, the word free. It is used to stress that idea that $x_{2}$ can take on any value; we are "free" to choose any value for $x_{2}$. Once this value is chosen, the value of $x_{1}$ is determined. We have infinite choices for the value of $x_{2}$, so therefore we have infinite solutions.
For example, if we set $x_{2}=0$, then $x_{1}=1$; if we set $x_{2}=5$, then $x_{1}=-4$.
Let's try another example, one that uses more variables.
Example $9 \quad$ Find the solution to the linear system
$$
\begin{aligned}
x_{2}-x_{3} & =3 \\
x_{1}-2 x_{3} & =2 \\
-3 x_{2}+3 x_{3} & =-9
\end{aligned} .
$$
Solution To find the solution, put the corresponding matrix into reduced row echelon form.
$$
\left[\begin{array}{cccc}
0 & 1 & -1 & 3 \\
1 & 0 & 2 & 2 \\
0 & -3 & 3 & -9
\end{array}\right] \quad \underset{\operatorname{rref}}{\longrightarrow} \quad\left[\begin{array}{cccc}
1 & 0 & 2 & 2 \\
0 & 1 & -1 & 3 \\
0 & 0 & 0 & 0
\end{array}\right]
$$
Now convert this reduced matrix back into equations. We have
$$
\begin{aligned}
x_{1}+2 x_{3} & =2 \\
x_{2}-x_{3} & =3
\end{aligned}
$$
or, equivalently,
$$
\begin{aligned}
& x_{1}=2-2 x_{3} \\
& x_{2}=3+x_{3} \\
& x_{3} \text { is free. }
\end{aligned}
$$
These two equations tell us that the values of $x_{1}$ and $x_{2}$ depend on what $x_{3}$ is. As we saw before, there is no restriction on what $x_{3}$ must be; it is "free" to take on the value of any real number. Once $x_{3}$ is chosen, we have a solution. Since we have infinite choices for the value of $x_{3}$, we have infinite solutions.
As examples, $x_{1}=2, x_{2}=3, x_{3}=0$ is one solution; $x_{1}=-2, x_{2}=5, x_{3}=2$ is another solution. Try plugging these values back into the original equations to verify that these indeed are solutions. (By the way, since infinite solutions exist, this system of equations is consistent.)
In the two previous examples we have used the word "free" to describe certain variables. What exactly is a free variable? How do we recognize which variables are free and which are not?
Look back to the reduced matrix in Example 8. Notice that there is only one leading 1 in that matrix, and that leading 1 corresponded to the $x_{1}$ variable. That told us that $x_{1}$ was not a free variable; since $x_{2}$ did not correspond to a leading 1 , it was a free variable. Look also at the reduced matrix in Example 9. There were two leading $1 \mathrm{~s}$ in that matrix; one corresponded to $x_{1}$ and the other to $x_{2}$. This meant that $x_{1}$ and $x_{2}$ were not free variables; since there was not a leading 1 that corresponded to $x_{3}$, it was a free variable.
We formally define this and a few other terms in this following definition.
## Definition 6
## Dependent and Independent Variables
Consider the reduced row echelon form of an augmented matrix of a linear system of equations. Then:
a variable that corresponds to a leading 1 is a basic, or dependent, variable, and
a variable that does not correspond to a leading 1 is a free, or independent, variable.
One can probably see that "free" and "independent" are relatively synonymous. It follows that if a variable is not independent, it must be dependent; the word "basic" comes from connections to other areas of mathematics that we won't explore here.
These definitions help us understand when a consistent system of linear equations will have infinite solutions. If there are no free variables, then there is exactly one solution; if there are any free variables, there are infinite solutions.
Key Idea 2
## Consistent Solution Types
A consistent linear system of equations will have exactly one solution if and only if there is a leading 1 for each variable in the system.
If a consistent linear system of equations has a free variable, it has infinite solutions.
If a consistent linear system has more variables than leading $1 \mathrm{~s}$, then the system will have infinite solutions.
A consistent linear system with more variables than equations will always have infinite solutions.
Note: Key Idea 2 applies only to consistent systems. If a system is inconsistent, then no solution exists and talking about free and basic variables is meaningless.
When a consistent system has only one solution, each equation that comes from the reduced row echelon form of the corresponding augmented matrix will contain exactly one variable. If the consistent system has infinite solutions, then there will be at least one equation coming from the reduced row echelon form that contains more than one variable. The "first" variable will be the basic (or dependent) variable; all others will be free variables.
We have now seen examples of consistent systems with exactly one solution and others with infinite solutions. How will we recognize that a system is inconsistent? Let's find out through an example.
Example 10 Find the solution to the linear system
$$
\begin{gathered}
x_{1}+x_{2}+x_{3}=1 \\
x_{1}+2 x_{2}+x_{3}=2 \\
2 x_{1}+3 x_{2}+2 x_{3}=0
\end{gathered}
$$
Solution We start by putting the corresponding matrix into reduced row echelon form.
$$
\left[\begin{array}{llll}
1 & 1 & 1 & 1 \\
1 & 2 & 1 & 2 \\
2 & 3 & 2 & 0
\end{array}\right] \quad \underset{\operatorname{rref}}{\longrightarrow} \quad\left[\begin{array}{llll}
1 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{array}\right]
$$
Now let us take the reduced matrix and write out the corresponding equations. The first two rows give us the equations
$$
\begin{aligned}
x_{1}+x_{3} & =0 \\
x_{2} & =0 .
\end{aligned}
$$
So far, so good. However the last row gives us the equation
$$
0 x_{1}+0 x_{2}+0 x_{3}=1
$$
or, more concisely, $0=1$. Obviously, this is not true; we have reached a contradiction. Therefore, no solution exists; this system is inconsistent.
In previous sections we have only encountered linear systems with unique solutions (exactly one solution). Now we have seen three more examples with different solution types. The first two examples in this section had infinite solutions, and the third had no solution. How can we tell if a system is inconsistent?
A linear system will be inconsistent only when it implies that 0 equals 1 . We can tell if a linear system implies this by putting its corresponding augmented matrix into reduced row echelon form. If we have any row where all entries are 0 except for the entry in the last column, then the system implies $0=1$. More succinctly, if we have a leading 1 in the last column of an augmented matrix, then the linear system has no solution.
## Key Idea 3
Inconsistent Systems of Linear Equations
A system of linear equations is inconsistent if the reduced row echelon form of its corresponding augmented matrix has a leading 1 in the last column.
Example 11 Confirm that the linear system
$$
\begin{gathered}
x+y=0 \\
2 x+2 y=4
\end{gathered}
$$
has no solution.
Solution We can verify that this system has no solution in two ways. First, let's just think about it. If $x+y=0$, then it stands to reason, by multiplying both sides of this equation by 2 , that $2 x+2 y=0$. However, the second equation of our system says that $2 x+2 y=4$. Since $0 \neq 4$, we have a contradiction and hence our system has no solution. (We cannot possibly pick values for $x$ and $y$ so that $2 x+2 y$ equals both 0 and 4.)
Now let us confirm this using the prescribed technique from above. The reduced row echelon form of the corresponding augmented matrix is
$$
\left[\begin{array}{lll}
1 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]
$$
We have a leading 1 in the last column, so therefore the system is inconsistent.
Let's summarize what we have learned up to this point. Consider the reduced row echelon form of the augmented matrix of a system of linear equations. ${ }^{3}$ If there is a leading 1 in the last column, the system has no solution. Otherwise, if there is a leading 1 for each variable, then there is exactly one solution; otherwise (i.e., there are free variables) there are infinite solutions.
Systems with exactly one solution or no solution are the easiest to deal with; systems with infinite solutions are a bit harder to deal with. Therefore, we'll do a little more practice. First, a definition: if there are infinite solutions, what do we call one of those infinite solutions?
${ }^{3}$ That sure seems like a mouthful in and of itself. However, it boils down to "look at the reduced form of the usual matrix." Definition 7
## Particular Solution
Consider a linear system of equations with infinite solutions. A particular solution is one solution out of the infinite set of possible solutions.
The easiest way to find a particular solution is to pick values for the free variables which then determines the values of the dependent variables. Again, more practice is called for.
Example 12 Give the solution to a linear system whose augmented matrix in reduced row echelon form is
$$
\left[\begin{array}{ccccc}
1 & -1 & 0 & 2 & 4 \\
0 & 0 & 1 & -3 & 7 \\
0 & 0 & 0 & 0 & 0
\end{array}\right]
$$
and give two particular solutions.
Solution We can essentially ignore the third row; it does not divulge any information about the solution. ${ }^{4}$ The first and second rows can be rewritten as the following equations:
$$
\begin{array}{r}
x_{1}-x_{2}+2 x_{4}=4 \\
x_{3}-3 x_{4}=7 .
\end{array}
$$
Notice how the variables $x_{1}$ and $x_{3}$ correspond to the leading 1 s of the given matrix. Therefore $x_{1}$ and $x_{3}$ are dependent variables; all other variables (in this case, $x_{2}$ and $x_{4}$ ) are free variables.
We generally write our solution with the dependent variables on the left and independent variables and constants on the right. It is also a good practice to acknowledge the fact that our free variables are, in fact, free. So our final solution would look something like
$$
\begin{aligned}
& x_{1}=4+x_{2}-2 x_{4} \\
& x_{2} \text { is free } \\
& x_{3}=7+3 x_{4} \\
& x_{4} \text { is free. }
\end{aligned}
$$
To find particular solutions, choose values for our free variables. There is no "right" way of doing this; we are "free" to choose whatever we wish.
${ }^{4}$ Then why include it? Rows of zeros sometimes appear "unexpectedly" in matrices after they have been put in reduced row echelon form. When this happens, we do learn something; it means that at least one equation was a combination of some of the others. By setting $x_{2}=0=x_{4}$, we have the solution $x_{1}=4, x_{2}=0, x_{3}=7, x_{4}=0$. By setting $x_{2}=1$ and $x_{4}=-5$, we have the solution $x_{1}=15, x_{2}=1, x_{3}=-8$, $x_{4}=-5$. It is easier to read this when are variables are listed vertically, so we repeat these solutions:
One particular solution is:
Another particular solution is:
$$
\begin{aligned}
& x_{1}=4 \\
& x_{2}=0 \\
& x_{3}=7 \\
& x_{4}=0 .
\end{aligned}
$$
$$
\begin{aligned}
& x_{1}=15 \\
& x_{2}=1 \\
& x_{3}=-8 \\
& x_{4}=-5 .
\end{aligned}
$$
Example $13 \quad$ Find the solution to a linear system whose augmented matrix in reduced row echelon form is
$$
\left[\begin{array}{lllll}
1 & 0 & 0 & 2 & 3 \\
0 & 1 & 0 & 4 & 5
\end{array}\right]
$$
and give two particular solutions.
Solution Converting the two rows into equations we have
$$
\begin{aligned}
& x_{1}+2 x_{4}=3 \\
& x_{2}+4 x_{4}=5 .
\end{aligned}
$$
We see that $x_{1}$ and $x_{2}$ are our dependent variables, for they correspond to the leading 1s. Therefore, $x_{3}$ and $x_{4}$ are independent variables. This situation feels a little unusual, ${ }^{5}$ for $x_{3}$ doesn't appear in any of the equations above, but cannot overlook it; it is still a free variable since there is not a leading 1 that corresponds to it. We write our solution as:
$$
\begin{aligned}
& x_{1}=3-2 x_{4} \\
& x_{2}=5-4 x_{4} \\
& x_{3} \text { is free } \\
& x_{4} \text { is free. }
\end{aligned}
$$
To find two particular solutions, we pick values for our free variables. Again, there is no "right" way of doing this (in fact, there are ... infinite ways of doing this) so we give only an example here.
${ }^{5}$ What kind of situation would lead to a column of all zeros? To have such a column, the original matrix needed to have a column of all zeros, meaning that while we acknowledged the existence of a certain variable, we never actually used it in any equation. In practical terms, we could respond by removing the corresponding column from the matrix and just keep in mind that that variable is free. In very large systems, it might be hard to determine whether or not a variable is actually used and one would not worry about it. When we learn about eigenvectors and eigenvalues, we will see that under certain circumstances this situation arises. In those cases we leave the variable in the system just to remind ourselves that it is there. Chapter 1 Systems of Linear Equations
One particular solution is:
$$
\begin{aligned}
& x_{1}=3 \\
& x_{2}=5 \\
& x_{3}=1000 \\
& x_{4}=0 .
\end{aligned}
$$
Another particular solution is:
$$
\begin{aligned}
& x_{1}=3-2 \pi \\
& x_{2}=5-4 \pi \\
& x_{3}=e^{2} \\
& x_{4}=\pi .
\end{aligned}
$$
(In the second particular solution we picked "unusual" values for $x_{3}$ and $x_{4}$ just to highlight the fact that we can.)
Example 14 Find the solution to the linear system
$$
\begin{aligned}
& x_{1}+x_{2}+x_{3}=5 \\
& x_{1}-x_{2}+x_{3}=3
\end{aligned}
$$
and give two particular solutions.
Solution The corresponding augmented matrix and its reduced row echelon form are given below.
Converting these two rows into equations, we have
$$
\begin{array}{r}
x_{1}+x_{3}=4 \\
x_{2}=1
\end{array}
$$
giving us the solution
$$
\begin{aligned}
& x_{1}=4-x_{3} \\
& x_{2}=1 \\
& x_{3} \text { is free. }
\end{aligned}
$$
Once again, we get a bit of an "unusual" solution; while $x_{2}$ is a dependent variable, it does not depend on any free variable; instead, it is always 1 . (We can think of it as depending on the value of 1.) By picking two values for $x_{3}$, we get two particular solutions.
One particular solution is:
Another particular solution is:
$$
\begin{array}{ll}
x_{1}=4 & x_{1}=3 \\
x_{2}=1 & x_{2}=1 \\
x_{3}=0 & x_{3}=1 .
\end{array}
$$
The constants and coefficients of a matrix work together to determine whether a given system of linear equations has one, infinite, or no solution. The concept will be fleshed out more in later chapters, but in short, the coefficients determine whether a matrix will have exactly one solution or not. In the "or not" case, the constants determine whether or not infinite solutions or no solution exists. (So if a given linear system has exactly one solution, it will always have exactly one solution even if the constants are changed.) Let's look at an example to get an idea of how the values of constants and coefficients work together to determine the solution type.
Example $15 \quad$ For what values of $k$ will the given system have exactly one solution, infinite solutions, or no solution?
$$
\begin{array}{r}
x_{1}+2 x_{2}=3 \\
3 x_{1}+k x_{2}=9
\end{array}
$$
Solution We answer this question by forming the augmented matrix and starting the process of putting it into reduced row echelon form. Below we see the augmented matrix and one elementary row operation that starts the Gaussian elimination process.
$$
\left[\begin{array}{lll}
1 & 2 & 3 \\
3 & k & 9
\end{array}\right] \quad \overrightarrow{-3 R_{1}+R_{2} \rightarrow R_{2}} \quad\left[\begin{array}{ccc}
1 & 2 & 3 \\
0 & k-9 & 0
\end{array}\right]
$$
This is as far as we need to go. In looking at the second row, we see that if $k=9$, then that row contains only zeros and $x_{2}$ is a free variable; we have infinite solutions. If $k \neq 9$, then our next step would be to make that second row, second column entry a leading one. We don't particularly care about the solution, only that we would have exactly one as both $x_{1}$ and $x_{2}$ would correspond to a leading one and hence be dependent variables.
Our final analysis is then this. If $k \neq 9$, there is exactly one solution; if $k=9$, there are infinite solutions. In this example, it is not possible to have no solutions.
As an extension of the previous example, consider the similar augmented matrix where the constant 9 is replaced with a 10 . Performing the same elementary row operation gives
$$
\left[\begin{array}{ccc}
1 & 2 & 3 \\
3 & k & 10
\end{array}\right] \quad \overrightarrow{-3 R_{1}+R_{2} \rightarrow R_{2}} \quad\left[\begin{array}{ccc}
1 & 2 & 3 \\
0 & k-9 & 1
\end{array}\right]
$$
As in the previous example, if $k \neq 9$, we can make the second row, second column entry a leading one and hence we have one solution. However, if $k=9$, then our last row is $\left[\begin{array}{lll}0 & 0 & 1\end{array}\right]$, meaning we have no solution.
We have been studying the solutions to linear systems mostly in an "academic" setting; we have been solving systems for the sake of solving systems. In the next section, we'll look at situations which create linear systems that need solving (i.e., "word problems").
## Exercises 1.4
In Exercises 1 - 14, find the solution to the given linear system. If the system has infinite solutions, give 2 particular solutions.
1. $\begin{aligned} 2 x_{1}+4 x_{2} & =2 \\ x_{1}+2 x_{2} & =1\end{aligned}$
2. $\begin{aligned}-x_{1}+5 x_{2} & =3 \\ 2 x_{1}-10 x_{2} & =-6\end{aligned}$
3. $\begin{aligned} x_{1}+x_{2} & =3 \\ 2 x_{1}+x_{2} & =4\end{aligned}$
4. $\begin{aligned}-3 x_{1}+7 x_{2} & =-7 \\ 2 x_{1}-8 x_{2} & =8\end{aligned}$
5. $2 x_{1}+3 x_{2}=1$
6. $-2 x_{1}-3 x_{2}=1$
7. $\begin{gathered}x_{1}+2 x_{2}=1 \\ -x_{1}-2 x_{2}=1\end{gathered}$
8. $\begin{gathered}-2 x_{1}+4 x_{2}+4 x_{3}=6 \\ x_{1}-3 x_{2}+2 x_{3}=1\end{gathered}$
9. $\begin{aligned}-x_{1}+2 x_{2}+2 x_{3} & =2 \\ 2 x_{1}+5 x_{2}+x_{3} & =2\end{aligned}$
10. $-x_{1}-x_{2}+x_{3}+x_{4}=0$
$-2 x_{1}-2 x_{2}+x_{3}=-1$
10. $x_{1}+x_{2}+6 x_{3}+9 x_{4}=0$
$-x_{1}-x_{3}-2 x_{4}=-3$ $2 x_{1}+x_{2}+2 x_{3}=0$
11. $x_{1}+x_{2}+3 x_{3}=1$
$$
3 x_{1}+2 x_{2}+5 x_{3}=3
$$
$x_{1}+3 x_{2}+3 x_{3}=1$
12. $2 x_{1}-x_{2}+2 x_{3}=-1$
$4 x_{1}+5 x_{2}+8 x_{3}=2$
$x_{1}+2 x_{2}+2 x_{3}=1$
13. $2 x_{1}+x_{2}+3 x_{3}=1$
$$
3 x_{1}+3 x_{2}+5 x_{3}=2
$$
$2 x_{1}+4 x_{2}+6 x_{3}=2$
14. $1 x_{1}+2 x_{2}+3 x_{3}=1$
$$
-3 x_{1}-6 x_{2}-9 x_{3}=-3
$$
In Exercises 15 - 18, state for which values of $k$ the given system will have exactly 1 solution, infinite solutions, or no solution.
15. $\begin{aligned} x_{1}+2 x_{2} & =1 \\ 2 x_{1}+4 x_{2} & =k\end{aligned}$
16. $\begin{aligned} x_{1}+2 x_{2} & =1 \\ x_{1}+k x_{2} & =1\end{aligned}$
17. $x_{1}+2 x_{2}=1$
18. $x_{1}+k x_{2}=2$
19. $\begin{aligned} x_{1}+2 x_{2} & =1 \\ x_{1}+3 x_{2} & =k\end{aligned}$
### Applications of Linear Systems
## AS YOU READ
1. How do most problems appear "in the real world?"
2. The unknowns in a problem are also called what?
3. How many points are needed to determine the coefficients of a $5^{\text {th }}$ degree polynomial?
We've started this chapter by addressing the issue of finding the solution to a system of linear equations. In subsequent sections, we defined matrices to store linear equation information; we described how we can manipulate matrices without changing the solutions; we described how to efficiently manipulate matrices so that a working solution can be easily found.
We shouldn't lose sight of the fact that our work in the previous sections was aimed at finding solutions to systems of linear equations. In this section, we'll learn how to apply what we've learned to actually solve some problems.
Many, many, many problems that are addressed by engineers, businesspeople, scientists and mathematicians can be solved by properly setting up systems of linear equations. In this section we highlight only a few of the wide variety of problems that matrix algebra can help us solve.
We start with a simple example.
Example $16 \quad \mathrm{~A}$ jar contains 100 blue, green, red and yellow marbles. There are twice as many yellow marbles as blue; there are 10 more blue marbles than red; the sum of the red and yellow marbles is the same as the sum of the blue and green. How many marbles of each color are there?
Solution Let's call the number of blue balls $b$, and the number of the other balls $g, r$ and $y$, each representing the obvious. Since we know that we have 100 marbles, we have the equation
$$
b+g+r+y=100
$$
The next sentence in our problem statement allows us to create three more equations.
We are told that there are twice as many yellow marbles as blue. One of the following two equations is correct, based on this statement; which one is it?
$$
2 y=b \quad \text { or } \quad 2 b=y
$$
The first equation says that if we take the number of yellow marbles, then double it, we'll have the number of blue marbles. That is not what we were told. The second equation states that if we take the number of blue marbles, then double it, we'll have the number of yellow marbles. This is what we were told.
The next statement of "there are 10 more blue marbles as red" can be written as either
$$
b=r+10 \quad \text { or } \quad r=b+10 .
$$
Which is it?
The first equation says that if we take the number of red marbles, then add 10, we'll have the number of blue marbles. This is what we were told. The next equation is wrong; it implies there are more red marbles than blue.
The final statement tells us that the sum of the red and yellow marbles is the same as the sum of the blue and green marbles, giving us the equation
$$
r+y=b+g .
$$
We have four equations; altogether, they are
$$
\begin{aligned}
b+g+r+y & =100 \\
2 b & =y \\
b & =r+10 \\
r+y & =b+g .
\end{aligned}
$$
We want to write these equations in a standard way, with all the unknowns on the left and the constants on the right. Let us also write them so that the variables appear in the same order in each equation (we'll use alphabetical order to make it simple). We now have
$$
\begin{aligned}
b+g+r+y & =100 \\
2 b-y & =0 \\
b-r & =10 \\
-b-g+r+y & =0
\end{aligned}
$$
To find the solution, let's form the appropriate augmented matrix and put it into reduced row echelon form. We do so here, without showing the steps.
$$
\left[\begin{array}{ccccc}
1 & 1 & 1 & 1 & 100 \\
2 & 0 & 0 & -1 & 0 \\
1 & 0 & -1 & 0 & 10 \\
-1 & -1 & 1 & 1 & 0
\end{array}\right] \quad \underset{\operatorname{rref}}{\longrightarrow} \quad\left[\begin{array}{lllll}
1 & 0 & 0 & 0 & 20 \\
0 & 1 & 0 & 0 & 30 \\
0 & 0 & 1 & 0 & 10 \\
0 & 0 & 0 & 1 & 40
\end{array}\right]
$$
We interpret from the reduced row echelon form of the matrix that we have 20 blue, 30 green, 10 red and 40 yellow marbles.
Even if you had a bit of difficulty with the previous example, in reality, this type of problem is pretty simple. The unknowns were easy to identify, the equations were pretty straightforward to write (maybe a bit tricky for some), and only the necessary information was given.
Most problems that we face in the world do not approach us in this way; most problems do not approach us in the form of "Here is an equation. Solve it." Rather, most problems come in the form of:
Here is a problem. I want the solution. To help, here is lots of information. It may be just enough; it may be too much; it may not be enough. You figure out what you need; just give me the solution.
Faced with this type of problem, how do we proceed? Like much of what we've done in the past, there isn't just one "right" way. However, there are a few steps that can guide us. You don't have to follow these steps, "step by step," but if you find that you are having difficulty solving a problem, working through these steps may help. (Note: while the principles outlined here will help one solve any type of problem, these steps are written specifically for solving problems that involve only linear equations.)
## Key Idea 4
Having identified some steps, let us put them into practice with some examples.
## Mathematical Problem Solving
1. Understand the problem. What exactly is being asked?
2. Identify the unknowns. What are you trying to find? What units are involved?
3. Give names to your unknowns (these are your variables).
4. Use the information given to write as many equations as you can that involve these variables.
5. Use the equations to form an augmented matrix; use Gaussian elimination to put the matrix into reduced row echelon form.
6. Interpret the reduced row echelon form of the matrix to identify the solution.
7. Ensure the solution makes sense in the context of the problem.
Example 17 A concert hall has seating arranged in three sections. As part of a special promotion, guests will recieve two of three prizes. Guests seated in the first and second sections will receive Prize $A$, guests seated in the second and third sections will receive Prize $B$, and guests seated in the first and third sections will receive Prize C. Concert promoters told the concert hall managers of their plans, and asked how many seats were in each section. (The promoters want to store prizes for each section separately for easier distribution.) The managers, thinking they were being helpful, told the promoters they would need 105 A prizes, $103 \mathrm{~B}$ prizes, and $88 \mathrm{C}$ prizes, and have since been unavailable for further help. How many seats are in each section?
Solution Before we rush in and start making equations, we should be clear about what is being asked. The final sentence asks: "How many seats are in each section?" This tells us what our unknowns should be: we should name our unknowns for the number of seats in each section. Let $x_{1}, x_{2}$ and $x_{3}$ denote the number of seats in the first, second and third sections, respectively. This covers the first two steps of our general problem solving technique. (It is tempting, perhaps, to name our variables for the number of prizes given away. However, when we think more about this, we realize that we already know this - that information is given to us. Rather, we should name our variables for the things we don't know.)
Having our unknowns identified and variables named, we now proceed to forming equations from the information given. Knowing that Prize $A$ goes to guests in the first and second sections and that we'll need 105 of these prizes tells us
$$
x_{1}+x_{2}=105
$$
Proceeding in a similar fashion, we get two more equations,
$$
x_{2}+x_{3}=103 \text { and } x_{1}+x_{3}=88 .
$$
Thus our linear system is
$$
\begin{aligned}
& x_{1}+x_{2}=105 \\
& x_{2}+x_{3}=103 \\
& x_{1}+x_{3}=88
\end{aligned}
$$
and the corresponding augmented matrix is
$$
\left[\begin{array}{cccc}
1 & 1 & 0 & 105 \\
0 & 1 & 1 & 103 \\
1 & 0 & 1 & 88
\end{array}\right]
$$
To solve our system, let's put this matrix into reduced row echelon form.
$$
\left[\begin{array}{cccc}
1 & 1 & 0 & 105 \\
0 & 1 & 1 & 103 \\
1 & 0 & 1 & 88
\end{array}\right] \quad \underset{\operatorname{rref}}{\longrightarrow} \quad\left[\begin{array}{cccc}
1 & 0 & 0 & 45 \\
0 & 1 & 0 & 60 \\
0 & 0 & 1 & 43
\end{array}\right]
$$
We can now read off our solution. The first section has 45 seats, the second has 60 seats, and the third has 43 seats.
Example 18 A lady takes a 2-mile motorized boat trip down the Highwater River, knowing the trip will take 30 minutes. She asks the boat pilot "How fast does this river flow?" He replies "I have no idea, lady. I just drive the boat."
She thinks for a moment, then asks "How long does the return trip take?" He replies "The same; half an hour." She follows up with the statement, "Since both legs take the same time, you must not drive the boat at the same speed."
"Naw," the pilot said. "While I really don't know exactly how fast I go, I do know that since we don't carry any tourists, I drive the boat twice as fast."
The lady walks away satisfied; she knows how fast the river flows.
(How fast does it flow?)
Solution This problem forces us to think about what information is given and how to use it to find what we want to know. In fact, to find the solution, we'll find out extra information that we weren't asked for! We are asked to find how fast the river is moving (step 1). To find this, we should recognize that, in some sense, there are three speeds at work in the boat trips: the speed of the river (which we want to find), the speed of the boat, and the speed that they actually travel at.
We know that each leg of the trip takes half an hour; if it takes half an hour to cover 2 miles, then they must be traveling at $4 \mathrm{mph}$, each way.
The other two speeds are unknowns, but they are related to the overall speeds. Let's call the speed of the river $r$ and the speed of the boat $b$. (And we should be careful. From the conversation, we know that the boat travels at two different speeds. So we'll say that $b$ represents the speed of the boat when it travels downstream, so $2 b$ represents the speed of the boat when it travels upstream.) Let's let our speed be measured in the units of miles/hour ( $\mathrm{mph}$ ) as we used above (steps 2 and 3 ).
What is the rate of the people on the boat? When they are travelling downstream, their rate is the sum of the water speed and the boat speed. Since their overall speed is $4 \mathrm{mph}$, we have the equation $r+b=4$.
When the boat returns going against the current, its overall speed is the rate of the boat minus the rate of the river (since the river is working against the boat). The overall trip is still taken at $4 \mathrm{mph}$, so we have the equation $2 b-r=4$. (Recall: the boat is traveling twice as fast as before.)
The corresponding augmented matrix is
$$
\left[\begin{array}{ccc}
1 & 1 & 4 \\
2 & -1 & 4
\end{array}\right]
$$
Note that we decided to let the first column hold the coefficients of $b$.
Putting this matrix in reduced row echelon form gives us:
$$
\left[\begin{array}{ccc}
1 & 1 & 4 \\
2 & -1 & 4
\end{array}\right] \quad \overrightarrow{\operatorname{rref}} \quad\left[\begin{array}{lll}
1 & 0 & 8 / 3 \\
0 & 1 & 4 / 3
\end{array}\right]
$$
We finish by interpreting this solution: the speed of the boat (going downstream) is $8 / 3 \mathrm{mph}$, or $2 . \overline{6} \mathrm{mph}$, and the speed of the river is $4 / 3 \mathrm{mph}$, or $1 . \overline{3} \mathrm{mph}$. All we really wanted to know was the speed of the river, at about $1.3 \mathrm{mph}$.
Example $19 \quad$ Find the equation of the quadratic function that goes through the points $(-1,6),(1,2)$ and $(2,3)$.
Solution This may not seem like a "linear" problem since we are talking about a quadratic function, but closer examination will show that it really is.
We normally write quadratic functions as $y=a x^{2}+b x+c$ where $a, b$ and $c$ are the coefficients; in this case, they are our unknowns. We have three points; consider the point $(-1,6)$. This tells us directly that if $x=-1$, then $y=6$. Therefore we know that $6=a(-1)^{2}+b(-1)+c$. Writing this in a more standard form, we have the linear equation
$$
a-b+c=6 \text {. }
$$
The second point tells us that $a(1)^{2}+b(1)+c=2$, which we can simplify as $a+b+c=2$, and the last point tells us $a(2)^{2}+b(2)+c=3$, or $4 a+2 b+c=3$. Thus our linear system is
$$
\begin{aligned}
a-b+c & =6 \\
a+b+c & =2 \\
4 a+2 b+c & =3 .
\end{aligned}
$$
Again, to solve our system, we find the reduced row echelon form of the corresponding augmented matrix. We don't show the steps here, just the final result.
$$
\left[\begin{array}{cccc}
1 & -1 & 1 & 6 \\
1 & 1 & 1 & 2 \\
4 & 2 & 1 & 3
\end{array}\right] \quad \overrightarrow{\operatorname{rref}} \quad\left[\begin{array}{cccc}
1 & 0 & 0 & 1 \\
0 & 1 & 0 & -2 \\
0 & 0 & 1 & 3
\end{array}\right]
$$
This tells us that $a=1, b=-2$ and $c=3$, giving us the quadratic function $y=x^{2}-2 x+3$.
One thing interesting about the previous example is that it confirms for us something that we may have known for a while (but didn't know why it was true). Why do we need two points to find the equation of the line? Because in the equation of the a line, we have two unknowns, and hence we'll need two equations to find values for these unknowns.
A quadratic has three unknowns (the coefficients of the $x^{2}$ term and the $x$ term, and the constant). Therefore we'll need three equations, and therefore we'll need three points.
What happens if we try to find the quadratic function that goes through 3 points that are all on the same line? The fast answer is that you'll get the equation of a line; there isn't a quadratic function that goes through 3 colinear points. Try it and see! (Pick easy points, like $(0,0),(1,1)$ and $(2,2)$. You'll find that the coefficient of the $x^{2}$ term is 0 .)
Of course, we can do the same type of thing to find polynomials that go through 4, 5 , etc., points. In general, if you are given $n+1$ points, a polynomial that goes through all $n+1$ points will have degree at most $n$.
Example $20 \quad$ A woman has $32 \$ 1, \$ 5$ and \\$10 bills in her purse, giving her a total of $\$ 100$. How many bills of each denomination does she have?
Solution Let's name our unknowns $x, y$ and $z$ for our ones, fives and tens, respectively (it is tempting to call them $o, f$ and $t$, but $o$ looks too much like 0 ). We know that there are a total of 32 bills, so we have the equation
$$
x+y+z=32 \text {. }
$$
We also know that we have $\$ 100$, so we have the equation
$$
x+5 y+10 z=100 .
$$
We have three unknowns but only two equations, so we know that we cannot expect a unique solution. Let's try to solve this system anyway and see what we get.
Putting the system into a matrix and then finding the reduced row echelon form, we have
$$
\left[\begin{array}{cccc}
1 & 1 & 1 & 32 \\
1 & 5 & 10 & 100
\end{array}\right] \quad \overrightarrow{\operatorname{rref}} \quad\left[\begin{array}{cccc}
1 & 0 & -\frac{5}{4} & 15 \\
0 & 1 & \frac{9}{4} & 17
\end{array}\right]
$$
Reading from our reduced matrix, we have the infinite solution set
$$
\begin{aligned}
& x=15+\frac{5}{4} z \\
& y=17-\frac{9}{4} z \\
& z \text { is free. }
\end{aligned}
$$
While we do have infinite solutions, most of these solutions really don't make sense in the context of this problem. (Setting $z=\frac{1}{2}$ doesn't make sense, for having half a ten dollar bill doesn't give us $\$ 5$. Likewise, having $z=8$ doesn't make sense, for then we'd have " -1 " $\$ 5$ bills.) So we must make sure that our choice of $z$ doesn't give us fractions of bills or negative amounts of bills.
To avoid fractions, $z$ must be a multiple of $4(-4,0,4,8, \ldots)$. Of course, $z \geq 0$ for a negative number wouldn't make sense. If $z=0$, then we have 15 one dollar bills and 17 five dollar bills, giving us $\$ 100$. If $z=4$, then we have $x=20$ and $y=8$. We already mentioned that $z=8$ doesn't make sense, nor does any value of $z$ where $z \geq 8$.
So it seems that we have two answers; one with $z=0$ and one with $z=4$. Of course, by the statement of the problem, we are led to believe that the lady has at least one $\$ 10$ bill, so probably the "best" answer is that we have $20 \$ 1$ bills, $8 \$ 5$ bills and $4 \$ 10$ bills. The real point of this example, though, is to address how infinite solutions may appear in a real world situation, and how suprising things may result.
Example 21 In a football game, teams can score points through touchdowns worth 6 points, extra points (that follow touchdowns) worth 1 point, two point conversions (that also follow touchdowns) worth 2 points and field goals, worth 3 points. You are told that in a football game, the two competing teams scored on 7 occasions, giving a total score of 24 points. Each touchdown was followed by either a successful extra point or two point conversion. In what ways were these points scored?
Solution The question asks how the points were scored; we can interpret this as asking how many touchdowns, extra points, two point conversions and field goals were scored. We'll need to assign variable names to our unknowns; let $t$ represent the number of touchdowns scored; let $x$ represent the number of extra points scored, let $w$ represent the number of two point conversions, and let $f$ represent the number of field goals scored.
Now we address the issue of writing equations with these variables using the given information. Since we have a total of 7 scoring occasions, we know that
$$
t+x+w+f=7
$$
The total points scored is 24; considering the value of each type of scoring opportunity, we can write the equation
$$
6 t+x+2 w+3 f=24 .
$$
Finally, we know that each touchdown was followed by a successful extra point or two point conversion. This is subtle, but it tells us that the number of touchdowns is equal to the sum of extra points and two point conversions. In other words,
$$
t=x+w .
$$
To solve our problem, we put these equations into a matrix and put the matrix into reduced row echelon form. Doing so, we find
$$
\left[\begin{array}{ccccc}
1 & 1 & 1 & 1 & 7 \\
6 & 1 & 2 & 3 & 24 \\
1 & -1 & -1 & 0 & 0
\end{array}\right] \quad \underset{\operatorname{rref}}{\longrightarrow} \quad\left[\begin{array}{ccccc}
1 & 0 & 0 & 0.5 & 3.5 \\
0 & 1 & 0 & 1 & 4 \\
0 & 0 & 1 & -0.5 & -0.5
\end{array}\right] .
$$
Therefore, we know that
$$
\begin{aligned}
t & =3.5-0.5 f \\
x & =4-f \\
w & =-0.5+0.5 f .
\end{aligned}
$$
We recognize that this means there are "infinite solutions," but of course most of these will not make sense in the context of a real football game. We must apply some logic to make sense of the situation.
Progressing in no particular order, consider the second equation, $x=4-f$. In order for us to have a positive number of extra points, we must have $f \leq 4$. (And of course, we need $f \geq 0$, too.) Therefore, right away we know we have a total of only 5 possibilities, where $f=0,1,2,3$ or 4 .
From the first and third equations, we see that if $f$ is an even number, then $t$ and $w$ will both be fractions (for instance, if $f=0$, then $t=3.5$ ) which does not make sense. Therefore, we are down to two possible solutions, $f=1$ and $f=3$.
If $f=1$, we have 3 touchdowns, 3 extra points, no two point conversions, and (of course), 1 field goal. (Check to make sure that gives 24 points!) If $f=3$, then we 2 touchdowns, 1 extra point, 1 two point conversion, and (of course) 3 field goals. Again, check to make sure this gives us 24 points. Also, we should check each solution to make sure that we have a total of 7 scoring occasions and that each touchdown could be followed by an extra point or a two point conversion.
We have seen a variety of applications of systems of linear equations. We would do well to remind ourselves of the ways in which solutions to linear systems come: there can be exactly one solution, infinite solutions, or no solutions. While we did see a few examples where it seemed like we had only 2 solutions, this was because we were restricting our solutions to "make sense" within a certain context.
We should also remind ourselves that linear equations are immensely important. The examples we considered here ask fundamentally simple questions like "How fast is the water moving?" or "What is the quadratic function that goes through these three points?" or "How were points in a football game scored?" The real "important" situations ask much more difficult questions that often require thousands of equations! (Gauss began the systematic study of solving systems of linear equations while trying to predict the next sighting of a comet; he needed to solve a system of linear equations that had 17 unknowns. Today, this a relatively easy situation to handle with the help of computers, but to do it by hand is a real pain.) Once we understand the fundamentals of solving systems of equations, we can move on to looking at solving bigger systems of equations; this text focuses on getting us to understand the fundamentals.
## Exercises 1.5
In Exercises 1 - 5, find the solution of the given problem by:
(a) creating an appropriate system of linear equations
(b) forming the augmented matrix that corresponds to this system
(c) putting the augmented matrix into reduced row echelon form
(d) interpreting the reduced row echelon form of the matrix as a solution
1. A farmer looks out his window at his chickens and pigs. He tells his daughter that he sees 62 heads and 190 legs. How many chickens and pigs does the farmer have?
2. A lady buys 20 trinkets at a yard sale. The cost of each trinket is either $\$ 0.30$ or $\$ 0.65$. If she spends $\$ 8.80$, how many of each type of trinket does she buy?
3. A carpenter can make two sizes of table, grande and venti. The grande table requires 4 table legs and 1 table top; the venti requires 6 table legs and 2 table tops. After doing work, he counts up spare parts in his warehouse and realizes that he has 86 table tops left over, and 300 legs. How many tables of each kind can he build and use up exactly all of his materials?
4. A jar contains 100 marbles. We know there are twice as many green marbles as red; that the number of blue and yellow marbles together is the same as the number of green; and that three times the number of yellow marbles together with the red marbles gives the same numbers as the blue marbles. How many of each color of marble are in the jar?
5. A rescue mission has 85 sandwiches, 65 bags of chips and 210 cookies. They know from experience that men will eat 2 sandwiches, 1 bag of chips and 4 cookies; women will eat 1 sandwich, a bag of chips and 2 cookies; kids will eat half a sandwhich, a bag of chips and 3 cookies. If they want to use all their food up, how many men, women and kids can they feed?
In Exercises 6 - 15, find the polynomial with the smallest degree that goes through the given points.
6. $(1,3)$ and $(3,15)$
7. $(-2,14)$ and $(3,4)$
8. $(1,5),(-1,3)$ and $(3,-1)$
9. $(-4,-3),(0,1)$ and $(1,4.5)$
10. $(-1,-8),(1,-2)$ and $(3,4)$
11. $(-3,3),(1,3)$ and $(2,3)$
12. $(-2,15),(-1,4),(1,0)$ and $(2,-5)$
13. $(-2,-7),(1,2),(2,9)$ and $(3,28)$
14. $(-3,10),(-1,2),(1,2)$ and $(2,5)$
15. $(0,1),(-3,-3.5),(-2,-2)$ and $(4,7)$
16. The general exponential function has the form $f(x)=a e^{b x}$, where $a$ and $b$ are constants and $e$ is Euler's constant $1 \approx$ 2.718). We want to find the equation of the exponential function that goes through the points $(1,2)$ and $(2,4)$. (a) Show why we cannot simply subsitute in values for $x$ and $y$ in $y=a e^{b x}$ and solve using the techniques we used for polynomials.
(b) Show how the equality $y=a e^{b x}$ leads us to the linear equation $\ln y=\ln a+b x$.
(c) Use the techniques we developed to solve for the unknowns In $a$ and $b$.
(d) Knowing In $a$, find $a$; find the exponential function $f(x)=a e^{b x}$ that goes through the points $(1,2)$ and $(2,4)$.
17. In a football game, 24 points are scored from 8 scoring occasions. The number of successful extra point kicks is equal to the number of successful two point conversions. Find all ways in which the points may have been scored in this game.
18. In a football game, 29 points are scored from 8 scoring occasions. There are 2 more successful extra point kicks than successful two point conversions. Find all ways in which the points may have been scored in this game. 19. In a basketball game, where points are scored either by a 3 point shot, a 2 point shot or a 1 point free throw, 80 points were scored from 30 successful shots. Find all ways in which the points may have been scored in this game.
19. In a basketball game, where points are scored either by a 3 point shot, a 2 point shot or a 1 point free throw, 110 points were scored from 70 successful shots. Find all ways in which the points may have been scored in this game.
20. Describe the equations of the linear functions that go through the point $(1,3)$. Give 2 examples.
21. Describe the equations of the linear functions that go through the point $(2,5)$. Give 2 examples.
22. Describe the equations of the quadratic functions that go through the points $(2,-1)$ and $(1,0)$. Give 2 examples.
23. Describe the equations of the quadratic functions that go through the points $(-1,3)$ and $(2,6)$. Give 2 examples.
## MATRIX ARITHMETIC
A fundamental topic of mathematics is arithmetic; adding, subtracting, multiplying and dividing numbers. After learning how to do this, most of us went on to learn how to add, subtract, multiply and divide " $x$ ". We are comfortable with expressions such as
$$
x+3 x-x \cdot x^{2}+x^{5} \cdot x^{-1}
$$
and know that we can "simplify" this to
$$
4 x-x^{3}+x^{4}
$$
This chapter deals with the idea of doing similar operations, but instead of an unknown number $x$, we will be using a matrix $A$. So what exactly does the expression
$$
A+3 A-A \cdot A^{2}+A^{5} \cdot A^{-1}
$$
mean? We are going to need to learn to define what matrix addition, scalar multiplication, matrix multiplication and matrix inversion are. We will learn just that, plus some more good stuff, in this chapter.
### Matrix Addition and Scalar Multiplication
## AS YOU READ . .
1. When are two matrices equal?
2. Write an explanation of how to add matrices as though writing to someone who knows what a matrix is but not much more.
3. T/F: There is only 1 zero matrix.
4. T/F: To multiply a matrix by 2 means to multiply each entry in the matrix by 2 . In the past, when we dealt with expressions that used " $x$," we didn't just add and multiply $x$ 's together for the fun of it, but rather because we were usually given some sort of equation that had $x$ in it and we had to "solve for $x$. .
This begs the question, "What does it mean to be equal?" Two numbers are equal, when, ..., uh, ..., nevermind. What does it mean for two matrices to be equal? We say that matrices $A$ and $B$ are equal when their corresponding entries are equal. This seems like a very simple definition, but it is rather important, so we give it a box.
## Definition 8
## Matrix Equality
Two $m \times n$ matrices $A$ and $B$ are equal if their corresponding entries are equal.
Notice that our more formal definition specifies that if matrices are equal, they have the same dimensions. This should make sense.
Now we move on to describing how to add two matrices together. To start off, take a wild stab: what do you think the following sum is equal to?
$$
\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]+\left[\begin{array}{cc}
2 & -1 \\
5 & 7
\end{array}\right]=\text { ? }
$$
If you guessed
$$
\left[\begin{array}{cc}
3 & 1 \\
8 & 11
\end{array}\right]
$$
you guessed correctly. That wasn't so hard, was it?
Let's keep going, hoping that we are starting to get on a roll. Make another wild guess: what do you think the following expression is equal to?
$$
3 \cdot\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]=?
$$
If you guessed
$$
\left[\begin{array}{cc}
3 & 6 \\
9 & 12
\end{array}\right]
$$
you guessed correctly!
Even if you guessed wrong both times, you probably have seen enough in these two examples to have a fair idea now what matrix addition and scalar multiplication are all about.
Before we formally define how to perform the above operations, let us first recall that if $A$ is an $m \times n$ matrix, then we can write $A$ as
$$
A=\left[\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1 n} \\
a_{21} & a_{22} & \cdots & a_{2 n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m 1} & a_{m 2} & \cdots & a_{m n}
\end{array}\right]
$$
Secondly, we should define what we mean by the word scalar. A scalar is any number that we multiply a matrix by. (In some sense, we use that number to scale the matrix.) We are now ready to define our first arithmetic operations.
Definition 9 Matrix Addition
Let $A$ and $B$ be $m \times n$ matrices. The sum of $A$ and $B$, denoted $A+B$, is
$$
\left[\begin{array}{cccc}
a_{11}+b_{11} & a_{12}+b_{12} & \cdots & a_{1 n}+b_{1 n} \\
a_{21}+b_{21} & a_{22}+b_{22} & \cdots & a_{2 n}+b_{2 n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m 1}+b_{m 1} & a_{m 2}+b_{m 2} & \cdots & a_{m n}+b_{m n}
\end{array}\right]
$$
Definition 10
## Scalar Multiplication
Let $A$ be an $m \times n$ matrix and let $k$ be a scalar. The scalar multiplication of $k$ and $A$, denoted $k A$, is
$$
\left[\begin{array}{cccc}
k a_{11} & k a_{12} & \cdots & k a_{1 n} \\
k a_{21} & k a_{22} & \cdots & k a_{2 n} \\
\vdots & \vdots & \ddots & \vdots \\
k a_{m 1} & k a_{m 2} & \cdots & k a_{m n}
\end{array}\right]
$$
We are now ready for an example.
Example 22 Let
$$
A=\left[\begin{array}{ccc}
1 & 2 & 3 \\
-1 & 2 & 1 \\
5 & 5 & 5
\end{array}\right], \quad B=\left[\begin{array}{ccc}
2 & 4 & 6 \\
1 & 2 & 2 \\
-1 & 0 & 4
\end{array}\right], \quad C=\left[\begin{array}{lll}
1 & 2 & 3 \\
9 & 8 & 7
\end{array}\right]
$$
Simplify the following matrix expressions.
1. $A+B$
2. $A-B$
3. $-3 A+2 B$
4. $5 A+5 B$
5. $B+A$
6. $A+C$
7. $A-A$
8. $5(A+B)$
## SOLUTION
1. $A+B=\left[\begin{array}{lll}3 & 6 & 9 \\ 0 & 4 & 3 \\ 4 & 5 & 9\end{array}\right]$
2. $B+A=\left[\begin{array}{lll}3 & 6 & 9 \\ 0 & 4 & 3 \\ 4 & 5 & 9\end{array}\right]$.
3. $A-B=\left[\begin{array}{ccc}-1 & -2 & -3 \\ -2 & 0 & -1 \\ 6 & 5 & 1\end{array}\right]$.
4. $A+C$ is not defined. If we look at our definition of matrix addition, we see that the two matrices need to be the same size. Since $A$ and $C$ have different dimensions, we don't even try to create something as an addition; we simply say that the sum is not defined.
5. $-3 A+2 B=\left[\begin{array}{ccc}1 & 2 & 3 \\ 5 & -2 & 1 \\ -17 & -15 & -7\end{array}\right]$.
6. $A-A=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]$.
7. Strictly speaking, this is $\left[\begin{array}{ccc}5 & 10 & 15 \\ -5 & 10 & 5 \\ 25 & 25 & 25\end{array}\right]+\left[\begin{array}{ccc}10 & 20 & 30 \\ 5 & 10 & 10 \\ -5 & 0 & 20\end{array}\right]=\left[\begin{array}{ccc}15 & 30 & 45 \\ 0 & 20 & 15 \\ 20 & 25 & 45\end{array}\right]$.
8. Strictly speaking, this is
$$
\begin{aligned}
5\left(\left[\begin{array}{ccc}
1 & 2 & 3 \\
-1 & 2 & 1 \\
5 & 5 & 5
\end{array}\right]+\left[\begin{array}{ccc}
2 & 4 & 6 \\
1 & 2 & 2 \\
-1 & 0 & 4
\end{array}\right]\right) & =5 \cdot\left[\begin{array}{lll}
3 & 6 & 9 \\
0 & 4 & 3 \\
4 & 5 & 9
\end{array}\right] \\
& =\left[\begin{array}{ccc}
15 & 30 & 45 \\
0 & 20 & 15 \\
20 & 25 & 45
\end{array}\right] .
\end{aligned}
$$
Our example raised a few interesting points. Notice how $A+B=B+A$. We probably aren't suprised by this, since we know that when dealing with numbers, $a+$ $b=b+a$. Also, notice that $5 A+5 B=5(A+B)$. In our example, we were careful to compute each of these expressions following the proper order of operations; knowing these are equal allows us to compute similar expressions in the most convenient way.
Another interesting thing that came from our previous example is that
$$
A-A=\left[\begin{array}{lll}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right]
$$
It seems like this should be a special matrix; after all, every entry is 0 and 0 is a special number.
In fact, this is a special matrix. We define $\mathbf{0}$, which we read as "the zero matrix," to be the matrix of all zeros. ${ }^{1}$ We should be careful; this previous "definition" is a bit ambiguous, for we have not stated what size the zero matrix should be. Is $\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$ the zero matrix? How about $\left[\begin{array}{ll}0 & 0\end{array}\right]$ ?
Let's not get bogged down in semantics. If we ever see $\mathbf{0}$ in an expression, we will usually know right away what size $\mathbf{0}$ should be; it will be the size that allows the expression to make sense. If $A$ is a $3 \times 5$ matrix, and we write $A+0$, we'll simply assume that 0 is also a $3 \times 5$ matrix. If we are ever in doubt, we can add a subscript; for instance, $\mathbf{0}_{2 \times 7}$ is the $2 \times 7$ matrix of all zeros.
Since the zero matrix is an important concept, we give it it's own definition box.
Definition 11
The following presents some of the properties of matrix addition and scalar multiplication that we discovered above, plus a few more.
## The Zero Matrix
The $m \times n$ matrix of all zeros, denoted $\mathbf{0}_{m \times n}$, is the zero matrix.
When the dimensions of the zero matrix are clear from the context, the subscript is generally omitted. Theorem 2
## Properties of Matrix Addition and Scalar Multiplication
The following equalities hold for all $m \times n$ matrices $A, B$ and $c$ and scalars $k$.
1. $A+B=B+A$ (Commutative Property)
2. $(A+B)+C=A+(B+C)$ (Associative Property)
3. $k(A+B)=k A+k B$ (Scalar Multiplication Distributive Property)
4. $k A=A k$
5. $A+\mathbf{0}=\mathbf{0}+\mathbf{A}=\mathrm{A}$ (Additive Identity)
6. $0 A=0$
Be sure that this last property makes sense; it says that if we multiply any matrix
${ }^{1}$ We use the bold face to distinguish the zero matrix, $\mathbf{0}$, from the number zero, 0 . by the number 0 , the result is the zero matrix, or $\mathbf{0}$.
We began this section with the concept of matrix equality. Let's put our matrix addition properties to use and solve a matrix equation.
Example $23 \quad$ Let
$$
A=\left[\begin{array}{cc}
2 & -1 \\
3 & 6
\end{array}\right]
$$
Find the matrix $X$ such that
$$
2 A+3 X=-4 A .
$$
Solution We can use basic algebra techniques to manipulate this equation for $X$; first, let's subtract $2 A$ from both sides. This gives us
$$
3 X=-6 A \text {. }
$$
Now divide both sides by 3 to get
$$
X=-2 A .
$$
Now we just need to compute $-2 A$; we find that
$$
X=\left[\begin{array}{cc}
-4 & 2 \\
-6 & -12
\end{array}\right] .
$$
Our matrix properties identified $\mathbf{0}$ as the Additive Identity; i.e., if you add $\mathbf{0}$ to any matrix $A$, you simply get $A$. This is similar in notion to the fact that for all numbers $a$, $a+0=a$. A Multiplicative Identity would be a matrix I where $I \times A=A$ for all matrices $A$. (What would such a matrix look like? A matrix of all 1s, perhaps?) However, in order for this to make sense, we'll need to learn to multiply matrices together, which we'll do in the next section.
## Exercises 2.1
Matrices $A$ and $B$ are given below. In Exercises $1-6$, simplify the given expression.
$$
A=\left[\begin{array}{cc}
1 & -1 \\
7 & 4
\end{array}\right] \quad B=\left[\begin{array}{cc}
-3 & 2 \\
5 & 9
\end{array}\right]
$$
1. $A+B$
2. $2 A-3 B$
3. $3 A-A$
4. $4 B-2 A$ 5. $3(A-B)+B$
5. $2(A-B)-(A-3 B)$
Matrices $A$ and $B$ are given below. In Exercises 7 - 10, simplify the given expression.
$$
A=\left[\begin{array}{l}
3 \\
5
\end{array}\right] \quad B=\left[\begin{array}{c}
-2 \\
4
\end{array}\right]
$$
7. $4 B-2 A$
8. $-2 A+3 A$
$$
\begin{aligned}
& \text { 9. }-2 A-3 A \\
& \text { 10. }-B+3 B-2 B
\end{aligned}
$$
Matrices $A$ and $B$ are given below. In Exercises $11-14$, find $X$ that satisfies the equation.
$$
A=\left[\begin{array}{cc}
3 & -1 \\
2 & 5
\end{array}\right] \quad B=\left[\begin{array}{cc}
1 & 7 \\
3 & -4
\end{array}\right]
$$
11. $2 A+X=B$
12. $A-X=3 B$
13. $3 A+2 X=-1 B$
14. $A-\frac{1}{2} X=-B$
In Exercises 15-21, find values for the scalars $a$ and $b$ that satisfy the given equation. 15. $a\left[\begin{array}{l}1 \\ 2\end{array}\right]+b\left[\begin{array}{c}-1 \\ 5\end{array}\right]=\left[\begin{array}{l}1 \\ 9\end{array}\right]$
16. $a\left[\begin{array}{c}-3 \\ 1\end{array}\right]+b\left[\begin{array}{l}8 \\ 4\end{array}\right]=\left[\begin{array}{l}7 \\ 1\end{array}\right]$
17. $a\left[\begin{array}{c}4 \\ -2\end{array}\right]+b\left[\begin{array}{c}-6 \\ 3\end{array}\right]=\left[\begin{array}{c}10 \\ -5\end{array}\right]$
18. $a\left[\begin{array}{l}1 \\ 1\end{array}\right]+b\left[\begin{array}{c}-1 \\ 3\end{array}\right]=\left[\begin{array}{l}5 \\ 5\end{array}\right]$
19. $a\left[\begin{array}{l}1 \\ 3\end{array}\right]+b\left[\begin{array}{l}-3 \\ -9\end{array}\right]=\left[\begin{array}{c}4 \\ -12\end{array}\right]$
20. $a\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]+b\left[\begin{array}{l}1 \\ 1 \\ 2\end{array}\right]=\left[\begin{array}{c}0 \\ -1 \\ -1\end{array}\right]$
21. $a\left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right]+b\left[\begin{array}{l}5 \\ 1 \\ 2\end{array}\right]=\left[\begin{array}{l}3 \\ 4 \\ 7\end{array}\right]$
### Matrix Multiplication
## AS YOU READ
1. $\mathrm{T} / \mathrm{F}$ : Column vectors are used more in this text than row vectors, although some other texts do the opposite.
2. T/F: To multiply $A \times B$, the number of rows of $A$ and $B$ need to be the same.
3. T/F: The entry in the $2^{\text {nd }}$ row and $3^{\text {rd }}$ column of the product $A B$ comes from multipling the $2^{\text {nd }}$ row of $A$ with the $3^{\text {rd }}$ column of $B$.
4. Name two properties of matrix multiplication that also hold for "regular multiplication" of numbers.
5. Name a property of "regular multiplication" of numbers that does not hold for matrix multiplication.
6. $\mathrm{T} / \mathrm{F}: A^{3}=A \cdot A \cdot A$
In the previous section we found that the definition of matrix addition was very intuitive, and we ended that section discussing the fact that eventually we'd like to know what it means to multiply matrices together.
In the spirit of the last section, take another wild stab: what do you think
$$
\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right] \times\left[\begin{array}{cc}
1 & -1 \\
2 & 2
\end{array}\right]
$$
means?
You are likely to have guessed
$$
\left[\begin{array}{cc}
1 & -2 \\
6 & 8
\end{array}\right]
$$
but this is, in fact, not right. $^{2}$ The actual answer is
$$
\left[\begin{array}{cc}
5 & 3 \\
11 & 5
\end{array}\right]
$$
If you can look at this one example and suddenly understand exactly how matrix multiplication works, then you are probably smarter than the author. While matrix multiplication isn't hard, it isn't nearly as intuitive as matrix addition is.
To further muddy the waters (before we clear them), consider
$$
\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right] \times\left[\begin{array}{ccc}
1 & -1 & 0 \\
2 & 2 & -1
\end{array}\right]
$$
Our experience from the last section would lend us to believe that this is not defined, but our confidence is probably a bit shaken by now. In fact, this multiplication is defined, and it is
$$
\left[\begin{array}{ccc}
5 & 3 & -2 \\
11 & 5 & -4
\end{array}\right]
$$
You may see some similarity in this answer to what we got before, but again, probably not enough to really figure things out.
So let's take a step back and progress slowly. The first thing we'd like to do is define a special type of matrix called a vector.
Definition 12
## Column and Row Vectors
A $m \times 1$ matrix is called a column vector.
A $1 \times n$ matrix is called a row vector.
While it isn't obvious right now, column vectors are going to become far more useful to us than row vectors. Therefore, we often omit the word "column" when refering to column vectors, and we just call them "vectors." ${ }^{3}$
${ }^{2}$ guess you could define multiplication this way. If you'd prefer this type of multiplication, write your own book.
${ }^{3}$ In this text, row vectors are only used in this section when we discuss matrix multiplication, whereas we'll make extensive use of column vectors. Other texts make great use of row vectors, but little use of column vectors. It is a matter of preference and tradition: "most" texts use column vectors more. We have been using upper case letters to denote matrices; we use lower case letters with an arrow overtop to denote row and column vectors. An example of a row vector is
$$
\vec{u}=\left[\begin{array}{llll}
1 & 2 & -1 & 0
\end{array}\right]
$$
and an example of a column vector is
$$
\vec{v}=\left[\begin{array}{l}
1 \\
7 \\
8
\end{array}\right] .
$$
Before we learn how to multiply matrices in general, we will learn what it means to multiply a row vector by a column vector.
Definition 13
ncept; an example will make things more clear.
Don't worry if this definition doesn't make immediate sense. It is really an easy
Let $\vec{u}$ be an $1 \times n$ row vector with entries $u_{1}, u_{2}, \cdots, u_{n}$ and let $\vec{v}$ be an $n \times 1$ column vector with entries $v_{1}, v_{2}, \cdots, v_{n}$. The product of $\vec{u}$ and $\vec{v}$, denoted $\vec{u} \cdot \vec{v}$ or $\vec{u} \vec{v}$, is
$$
\sum_{i=1}^{n} u_{i} v_{i}=u_{1} v_{1}+u_{2} v_{2}+\cdots+u_{n} v_{n}
$$
Example 24 Let
$$
\vec{u}=\left[\begin{array}{lll}
1 & 2 & 3
\end{array}\right], \vec{v}=\left[\begin{array}{llll}
2 & 0 & 1 & -1
\end{array}\right], \vec{x}=\left[\begin{array}{c}
-2 \\
4 \\
3
\end{array}\right], \vec{y}=\left[\begin{array}{l}
1 \\
2 \\
5 \\
0
\end{array}\right] .
$$
Find the following products.
1. $\vec{u} \vec{x}$
2. $\vec{u} \vec{y}$
3. $\vec{v} \vec{y}$
4. $\vec{U} \vec{v}$
SOLUTION
1. $\vec{u} \vec{x}=\left[\begin{array}{lll}1 & 2 & 3\end{array}\right]\left[\begin{array}{c}-2 \\ 4 \\ 3\end{array}\right]=1(-2)+2(4)+3(3)=15$ 2. $\overrightarrow{v y}=\left[\begin{array}{llll}2 & 0 & 1 & -1\end{array}\right]\left[\begin{array}{l}1 \\ 2 \\ 5 \\ 0\end{array}\right]=2(1)+0(2)+1(5)-1(0)=7$
2. $\vec{u} \vec{y}$ is not defined; Definition 13 specifies that in order to multiply a row vector and column vector, they must have the same number of entries.
3. $\vec{u} \vec{v}$ is not defined; we only know how to multipy row vectors by column vectors. We haven't defined how to multiply two row vectors (in general, it can't be done).
4. The product $\vec{x} \vec{u}$ is defined, but we don't know how to do it yet. Right now, we only know how to multiply a row vector times a column vector; we don't know how to multiply a column vector times a row vector. (That's right: $\vec{u} \vec{x} \neq \vec{x} \vec{u}$ !)
Now that we understand how to multiply a row vector by a column vector, we are ready to define matrix multiplication.
Definition 14
Matrix Multiplication
Let $A$ be an $m \times r$ matrix, and let $B$ be an $r \times n$ matrix. The matrix product of $A$ and $B$, denoted $A \cdot B$, or simply $A B$, is the $m \times n$ matrix $M$ whose entry in the $i^{\text {th }}$ row and $j^{\text {th }}$ column is the product of the $i^{\text {th }}$ row of $A$ and the $j^{\text {th }}$ column of $B$.
It may help to illustrate it in this way. Let matrix $A$ have rows $\overrightarrow{a_{1}}, \overrightarrow{a_{2}}, \cdots, \overrightarrow{a_{m}}$ and let $B$ have columns $\overrightarrow{b_{1}}, \overrightarrow{b_{2}}, \cdots, \overrightarrow{b_{n}}$. Thus $A$ looks like
$$
\left[\begin{array}{ccc}
- & \overrightarrow{a_{1}} & - \\
- & \overrightarrow{a_{2}} & - \\
& \vdots & \\
- & \overrightarrow{a_{m}} & -
\end{array}\right]
$$
where the "-" symbols just serve as reminders that the $\overrightarrow{a_{i}}$ represent rows, and $B$ looks like
$$
\left[\begin{array}{cccc}
\mid & \mid & & \mid \\
\overrightarrow{b_{1}} & \overrightarrow{b_{2}} & \cdots & \overrightarrow{b_{n}} \\
\mid & \mid & & \mid
\end{array}\right]
$$
where again, the "|" symbols just remind us that the $\overrightarrow{b_{i}}$ represent column vectors. Then
$$
A B=\left[\begin{array}{cccc}
\overrightarrow{a_{1}} \overrightarrow{b_{1}} & \overrightarrow{a_{1}} \overrightarrow{b_{2}} & \cdots & \overrightarrow{a_{1}} \overrightarrow{b_{n}} \\
\overrightarrow{a_{2}} \overrightarrow{b_{1}} & \overrightarrow{a_{2}} \overrightarrow{b_{2}} & \cdots & \overrightarrow{a_{2}} \overrightarrow{b_{n}} \\
\vdots & \vdots & \ddots & \vdots \\
\overrightarrow{a_{m}} \overrightarrow{b_{1}} & \overrightarrow{a_{m}} \overrightarrow{b_{2}} & \cdots & \overrightarrow{a_{m}} \overrightarrow{b_{n}}
\end{array}\right]
$$
Two quick notes about this definition. First, notice that in order to multiply $A$ and $B$, the number of columns of $A$ must be the same as the number of rows of $B$ (we refer to these as the "inner dimensions"). Secondly, the resulting matrix has the same number of rows as $A$ and the same number of columns as $B$ (we refer to these as the "outer dimensions").
$$
\begin{aligned}
& \text { final dimensions are the outer } \\
& \overbrace{(m \times \underbrace{r) \times(r \times n)}_{\text {these inner dimensions }}}^{\text {dimensions }} \\
& \text { must match }
\end{aligned}
$$
Of course, this will make much more sense when we see an example.
Example 25 Revisit the matrix product we saw at the beginning of this section; multiply
$$
\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\left[\begin{array}{ccc}
1 & -1 & 0 \\
2 & 2 & -1
\end{array}\right]
$$
Solution Let's call our first matrix $A$ and the second $B$. We should first check to see that we can actually perform this multiplication. Matrix $A$ is $2 \times 2$ and $B$ is $2 \times 3$. The "inner" dimensions match up, so we can compute the product; the "outer" dimensions tell us that the product will be $2 \times 3$. Let
$$
A B=\left[\begin{array}{lll}
m_{11} & m_{12} & m_{13} \\
m_{21} & m_{22} & m_{23}
\end{array}\right] .
$$
Let's find the value of each of the entries.
The entry $m_{11}$ is in the first row and first column; therefore to find its value, we need to multiply the first row of $A$ by the first column of $B$. Thus
$$
m_{11}=\left[\begin{array}{ll}
1 & 2
\end{array}\right]\left[\begin{array}{l}
1 \\
2
\end{array}\right]=1(1)+2(2)=5 .
$$
So now we know that
$$
A B=\left[\begin{array}{ccc}
5 & m_{12} & m_{13} \\
m_{21} & m_{22} & m_{23}
\end{array}\right] .
$$
Finishing out the first row, we have
$$
m_{12}=\left[\begin{array}{ll}
1 & 2
\end{array}\right]\left[\begin{array}{c}
-1 \\
2
\end{array}\right]=1(-1)+2(2)=3
$$
using the first row of $A$ and the second column of $B$, and
$$
m_{13}=\left[\begin{array}{ll}
1 & 2
\end{array}\right]\left[\begin{array}{c}
0 \\
-1
\end{array}\right]=1(0)+2(-1)=-2
$$
using the first row of $A$ and the third column of $B$. Thus we have
$$
A B=\left[\begin{array}{ccc}
5 & 3 & -2 \\
m_{21} & m_{22} & m_{23}
\end{array}\right] .
$$
To compute the second row of $A B$, we multiply with the second row of $A$. We find
$$
\begin{aligned}
& m_{21}=\left[\begin{array}{ll}
3 & 4
\end{array}\right]\left[\begin{array}{l}
1 \\
2
\end{array}\right]=11, \\
& m_{22}=\left[\begin{array}{ll}
3 & 4
\end{array}\right]\left[\begin{array}{c}
-1 \\
2
\end{array}\right]=5,
\end{aligned}
$$
and
$$
m_{23}=\left[\begin{array}{ll}
3 & 4
\end{array}\right]\left[\begin{array}{c}
0 \\
-1
\end{array}\right]=-4
$$
Thus
$$
A B=\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\left[\begin{array}{ccc}
1 & -1 & 0 \\
2 & 2 & -1
\end{array}\right]=\left[\begin{array}{ccc}
5 & 3 & -2 \\
11 & 5 & -4
\end{array}\right]
$$
Example $26 \quad$ Multiply
$$
\left[\begin{array}{cc}
1 & -1 \\
5 & 2 \\
-2 & 3
\end{array}\right]\left[\begin{array}{llll}
1 & 1 & 1 & 1 \\
2 & 6 & 7 & 9
\end{array}\right] .
$$
Solution Let's first check to make sure this product is defined. Again calling the first matrix $A$ and the second $B$, we see that $A$ is a $3 \times 2$ matrix and $B$ is a $2 \times 4$ matrix; the inner dimensions match so the product is defined, and the product will be a $3 \times 4$ matrix,
$$
A B=\left[\begin{array}{llll}
m_{11} & m_{12} & m_{13} & m_{14} \\
m_{21} & m_{22} & m_{23} & m_{24} \\
m_{31} & m_{32} & m_{33} & m_{34}
\end{array}\right] .
$$
We will demonstrate how to compute some of the entries, then give the final answer. The reader can fill in the details of how each entry was computed.
$$
\begin{gathered}
m_{11}=\left[\begin{array}{ll}
1 & -1
\end{array}\right]\left[\begin{array}{l}
1 \\
2
\end{array}\right]=-1 . \\
m_{13}=\left[\begin{array}{ll}
1 & -1
\end{array}\right]\left[\begin{array}{l}
1 \\
7
\end{array}\right]=-6 . \\
m_{23}=\left[\begin{array}{ll}
5 & 2
\end{array}\right]\left[\begin{array}{l}
1 \\
7
\end{array}\right]=19 . \\
m_{24}=\left[\begin{array}{ll}
5 & 2
\end{array}\right]\left[\begin{array}{l}
1 \\
9
\end{array}\right]=23 .
\end{gathered}
$$
$$
\begin{aligned}
& m_{32}=\left[\begin{array}{ll}
-2 & 3
\end{array}\right]\left[\begin{array}{l}
1 \\
6
\end{array}\right]=16 . \\
& m_{34}=\left[\begin{array}{ll}
-2 & 3
\end{array}\right]\left[\begin{array}{l}
1 \\
9
\end{array}\right]=25 .
\end{aligned}
$$
So far, we've computed this much of $A B$ :
$$
A B=\left[\begin{array}{cccc}
-1 & m_{12} & -6 & m_{14} \\
m_{21} & m_{22} & 19 & 23 \\
m_{31} & 16 & m_{33} & 25
\end{array}\right]
$$
The final product is
$$
A B=\left[\begin{array}{cccc}
-1 & -5 & -6 & -8 \\
9 & 17 & 19 & 23 \\
4 & 16 & 19 & 25
\end{array}\right]
$$
Example 27 Multiply, if possible,
$$
\left[\begin{array}{lll}
2 & 3 & 4 \\
9 & 8 & 7
\end{array}\right]\left[\begin{array}{cc}
3 & 6 \\
5 & -1
\end{array}\right] .
$$
Solution Again, we'll call the first matrix $A$ and the second $B$. Checking the dimensions of each matrix, we see that $A$ is a $2 \times 3$ matrix, whereas $B$ is a $2 \times 2$ matrix. The inner dimensions do not match, therefore this multiplication is not defined.
Example 28 In Example 24, we were told that the product $\vec{x} \vec{u}$ was defined, where
$$
\vec{x}=\left[\begin{array}{c}
-2 \\
4 \\
3
\end{array}\right] \quad \text { and } \vec{u}=\left[\begin{array}{lll}
1 & 2 & 3
\end{array}\right] \text {, }
$$
although we were not shown what that product was. Find $\vec{x} \vec{u}$.
Solution Again, we need to check to make sure the dimensions work correctly (remember that even though we are referring to $\vec{u}$ and $\vec{x}$ as vectors, they are, in fact, just matrices).
The column vector $\vec{x}$ has dimensions $3 \times 1$, whereas the row vector $\vec{u}$ has dimensions $1 \times 3$. Since the inner dimensions do match, the matrix product is defined; the outer dimensions tell us that the product will be a $3 \times 3$ matrix, as shown below:
$$
\vec{x} \vec{u}=\left[\begin{array}{lll}
m_{11} & m_{12} & m_{13} \\
m_{21} & m_{22} & m_{23} \\
m_{31} & m_{32} & m_{33}
\end{array}\right] .
$$
To compute the entry $m_{11}$, we multiply the first row of $\vec{x}$ by the first column of $\vec{u}$. What is the first row of $\vec{x}$ ? Simply the number -2 . What is the first column of $\vec{u}$ ? Just the number 1 . Thus $m_{11}=-2$. (This does seem odd, but through checking, you can see that we are indeed following the rules.)
What about the entry $m_{12}$ ? Again, we multiply the first row of $\vec{x}$ by the first column of $\vec{u}$; that is, we multiply $-2(2)$. So $m_{12}=-4$.
What about $m_{23}$ ? Multiply the second row of $\vec{x}$ by the third column of $\vec{u}$; multiply 4(3), so $m_{23}=12$.
One final example: $m_{31}$ comes from multiplying the third row of $\vec{x}$, which is 3 , by the first column of $\vec{u}$, which is 1 . Therefore $m_{31}=3$.
So far we have computed
$$
\vec{x} \vec{u}=\left[\begin{array}{ccc}
-2 & -4 & m_{13} \\
m_{21} & m_{22} & 12 \\
3 & m_{32} & m_{33}
\end{array}\right] .
$$
After performing all 9 multiplications, we find
$$
\vec{x} \vec{u}=\left[\begin{array}{ccc}
-2 & -4 & -6 \\
4 & 8 & 12 \\
3 & 6 & 9
\end{array}\right] .
$$
In this last example, we saw a "nonstandard" multiplication (at least, it felt nonstandard). Studying the entries of this matrix, it seems that there are several different patterns that can be seen amongst the entries. (Remember that mathematicians like to look for patterns. Also remember that we often guess wrong at first; don't be scared and try to identify some patterns.)
In Section 2.1, we identified the zero matrix 0 that had a nice property in relation to matrix addition (i.e., $A+\mathbf{0}=\boldsymbol{A}$ for any matrix $A$ ). In the following example we'll identify a matrix that works well with multiplication as well as some multiplicative properties. For instance, we've learned how $1 \cdot A=A$; is there a matrix that acts like the number 1 ? That is, can we find a matrix $X$ where $X \cdot A=A ?^{4}$
Example $29 \quad$ Let
$$
\begin{gathered}
A=\left[\begin{array}{ccc}
1 & 2 & 3 \\
2 & -7 & 5 \\
-2 & -8 & 3
\end{array}\right], \quad B=\left[\begin{array}{lll}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{array}\right] \\
C=\left[\begin{array}{lll}
1 & 0 & 2 \\
2 & 1 & 0 \\
0 & 2 & 1
\end{array}\right], \quad I=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right] .
\end{gathered}
$$
Find the following products.
1. $A B$
2. $\mathrm{AO}_{3 \times 4}$
3. IA
4. $B C$
5. $B A$
6. $A I$
7. $l^{2}$
8. $B^{2}$
${ }^{4}$ We made a guess in Section 2.1 that maybe a matrix of all 1 s would work. Solution We will find each product, but we leave the details of each computation to the reader.
1. $A B=\left[\begin{array}{ccc}1 & 2 & 3 \\ 2 & -7 & 5 \\ -2 & -8 & 3\end{array}\right]\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1\end{array}\right]=\left[\begin{array}{ccc}6 & 6 & 6 \\ 0 & 0 & 0 \\ -7 & -7 & -7\end{array}\right]$
2. $B A=\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1\end{array}\right]\left[\begin{array}{ccc}1 & 2 & 3 \\ 2 & -7 & 5 \\ -2 & -8 & 3\end{array}\right]=\left[\begin{array}{lll}1 & -13 & 11 \\ 1 & -13 & 11 \\ 1 & -13 & 11\end{array}\right]$
3. $A \mathbf{0}_{3 \times 4}=\mathbf{0}_{3 \times 4}$.
4. $A I=\left[\begin{array}{ccc}1 & 2 & 3 \\ 2 & -7 & 5 \\ -2 & -8 & 3\end{array}\right]\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]=\left[\begin{array}{ccc}1 & 2 & 3 \\ 2 & -7 & 5 \\ -2 & -8 & 3\end{array}\right]$
5. $I A=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\left[\begin{array}{ccc}1 & 2 & 3 \\ 2 & -7 & 5 \\ -2 & -8 & 3\end{array}\right]=\left[\begin{array}{ccc}1 & 2 & 3 \\ 2 & -7 & 5 \\ -2 & -8 & 3\end{array}\right]$
6. We haven't formally defined what $I^{2}$ means, but we could probably make the reasonable guess that $I^{2}=I \cdot I$. Thus
$$
I^{2}=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]
$$
7. $B C=\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1\end{array}\right]\left[\begin{array}{lll}1 & 0 & 2 \\ 2 & 1 & 0 \\ 0 & 2 & 1\end{array}\right]=\left[\begin{array}{lll}3 & 3 & 3 \\ 3 & 3 & 3 \\ 3 & 3 & 3\end{array}\right]$
8. $B^{2}=B B=\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1\end{array}\right]\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1\end{array}\right]=\left[\begin{array}{lll}3 & 3 & 3 \\ 3 & 3 & 3 \\ 3 & 3 & 3\end{array}\right]$
This example is simply chock full of interesting ideas; it is almost hard to think about where to start.
Interesting Idea \\#1: Notice that in our example, $A B \neq B A$ ! When dealing with numbers, we were used to the idea that $a b=b a$. With matrices, multiplication is not commutative. (Of course, we can find special situations where it does work. In general, though, it doesn't.)
Interesting Idea \\#2: Right before this example we wondered if there was a matrix that "acted like the number 1, " and guessed it may be a matrix of all 1 s. However, we found out that such a matrix does not work in that way; in our example, $A B \neq A$. We did find that $A I=I A=A$. There is a Multiplicative Identity; it just isn't what we thought it would be. And just as $1^{2}=1, I^{2}=I$.
Interesting Idea \\#3: When dealing with numbers, we are very familiar with the notion that "If $a x=b x$, then $a=b$." (As long as $x \neq 0$.) Notice that, in our example, $B B=B C$, yet $B \neq C$. In general, just because $A X=B X$, we cannot conclude that $A=B$.
Matrix multiplication is turning out to be a very strange operation. We are very used to multiplying numbers, and we know a bunch of properties that hold when using this type of multiplication. When multiplying matrices, though, we probably find ourselves asking two questions, "What does work?" and "What doesn't work?" We'll answer these questions; first we'll do an example that demonstrates some of the things that do work.
## Example $30 \quad$ Let
$$
A=\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right], \quad B=\left[\begin{array}{cc}
1 & 1 \\
1 & -1
\end{array}\right] \quad \text { and } \quad C=\left[\begin{array}{ll}
2 & 1 \\
1 & 2
\end{array}\right]
$$
Find the following:
1. $A(B+C)$
2. $A(B C)$
3. $A B+A C$
4. $(A B) C$
Solution We'll compute each of these without showing all the intermediate steps. Keep in mind order of operations: things that appear inside of parentheses are computed first.
1.
$$
\begin{aligned}
A(B+C) & =\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\left(\left[\begin{array}{cc}
1 & 1 \\
1 & -1
\end{array}\right]+\left[\begin{array}{ll}
2 & 1 \\
1 & 2
\end{array}\right]\right) \\
& =\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\left[\begin{array}{ll}
3 & 2 \\
2 & 1
\end{array}\right] \\
& =\left[\begin{array}{cc}
7 & 4 \\
17 & 10
\end{array}\right]
\end{aligned}
$$
2.
$$
\begin{aligned}
A B+A C & =\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\left[\begin{array}{cc}
1 & 1 \\
1 & -1
\end{array}\right]+\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\left[\begin{array}{ll}
2 & 1 \\
1 & 2
\end{array}\right] \\
& =\left[\begin{array}{cc}
3 & -1 \\
7 & -1
\end{array}\right]+\left[\begin{array}{cc}
4 & 5 \\
10 & 11
\end{array}\right] \\
& =\left[\begin{array}{cc}
7 & 4 \\
17 & 10
\end{array}\right]
\end{aligned}
$$
3.
$$
\begin{aligned}
A(B C) & =\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\left(\left[\begin{array}{cc}
1 & 1 \\
1 & -1
\end{array}\right]\left[\begin{array}{ll}
2 & 1 \\
1 & 2
\end{array}\right]\right) \\
& =\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\left[\begin{array}{cc}
3 & 3 \\
1 & -1
\end{array}\right] \\
& =\left[\begin{array}{cc}
5 & 1 \\
13 & 5
\end{array}\right]
\end{aligned}
$$
4.
$$
\begin{aligned}
(A B) C & =\left(\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\left[\begin{array}{cc}
1 & 1 \\
1 & -1
\end{array}\right]\right)\left[\begin{array}{ll}
2 & 1 \\
1 & 2
\end{array}\right] \\
& =\left[\begin{array}{cc}
3 & -1 \\
7 & -1
\end{array}\right]\left[\begin{array}{ll}
2 & 1 \\
1 & 2
\end{array}\right] \\
& =\left[\begin{array}{cc}
5 & 1 \\
13 & 5
\end{array}\right]
\end{aligned}
$$
In looking at our example, we should notice two things. First, it looks like the "distributive property" holds; that is, $A(B+C)=A B+A C$. This is nice as many algebraic techniques we have learned about in the past (when doing "ordinary algebra") will still work. Secondly, it looks like the "associative property" holds; that is, $A(B C)=(A B) C$. This is nice, for it tells us that when we are multiplying several matrices together, we don't have to be particularly careful in what order we multiply certain pairs of matrices together. $^{5}$
In leading to an important theorem, let's define a matrix we saw in an earlier example. $^{6}$
Definition 15
The $n \times n$ matrix with 1 's on the diagonal and zeros elsewhere is the $n \times n$ identity matrix, denoted $I_{n}$. When the context makes the dimension of the identity clear, the subscript is generally omitted.
Note that while the zero matrix can come in all different shapes and sizes, the
${ }^{5}$ Be careful: in computing $A B C$ together, we can first multiply $A B$ or $B C$, but we cannot change the order in which these matrices appear. We cannot multiply $B A$ or $A C$, for instance.
${ }^{6}$ The following definition uses a term we won't define until Definition 20 on page 123: diagonal. In short, a "diagonal matrix" is one in which the only nonzero entries are the "diagonal entries." The examples given here and in the exercises should suffice until we meet the full definition later. identity matrix is always a square matrix. We show a few identity matrices below.
$$
I_{2}=\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right], \quad I_{3}=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right], \quad I_{4}=\left[\begin{array}{llll}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right]
$$
In our examples above, we have seen examples of things that do and do not work. We should be careful about what examples prove, though. If someone were to claim that $A B=B A$ is always true, one would only need to show them one example where they were false, and we would know the person was wrong. However, if someone claims that $A(B+C)=A B+A C$ is always true, we can't prove this with just one example. We need something more powerful; we need a true proof.
In this text, we forgo most proofs. The reader should know, though, that when we state something in a theorem, there is a proof that backs up what we state. Our justification comes from something stronger than just examples.
Now we give the good news of what does work when dealing with matrix multiplication.
## Theorem 3
## Properties of Matrix Multiplication
Let $A, B$ and $C$ be matrices with dimensions so that the following operations make sense, and let $k$ be a scalar. The following equalities hold:
1. $A(B C)=(A B) C$ (Associative Property)
2. $A(B+C)=A B+A B$ and $(B+C) A=B A+C A$ (Distributive Property)
3. $k(A B)=(k A) B=A(k B)$
4. $A I=I A=A$
The above box contains some very good news, and probably some very surprising news. Matrix multiplication probably seems to us like a very odd operation, so we probably wouldn't have been surprised if we were told that $A(B C) \neq(A B) C$. It is a very nice thing that the Associative Property does hold.
As we near the end of this section, we raise one more issue of notation. We define $A^{0}=I$. If $n$ is a positive integer, we define
$$
A^{n}=\underbrace{A \cdot A \cdot \cdots \cdot A}_{n \text { times }}
$$
With numbers, we are used to $a^{-n}=\frac{1}{a^{n}}$. Do negative exponents work with matrices, too? The answer is yes, sort of. We'll have to be careful, and we'll cover the topic in detail once we define the inverse of a matrix. For now, though, we recognize the fact that $A^{-1} \neq \frac{1}{A}$, for $\frac{1}{A}$ makes no sense; we don't know how to "divide" by a matrix.
We end this section with a reminder of some of the things that do not work with matrix multiplication. The good news is that there are really only two things on this list.
1. Matrix multiplication is not commutative; that is, $A B \neq B A$.
2. In general, just because $A X=B X$, we cannot conclude that $A=B$.
The bad news is that these ideas pop up in many places where we don't expect them. For instance, we are used to
$$
(a+b)^{2}=a^{2}+2 a b+b^{2} .
$$
What about $(A+B)^{2}$ ? All we'll say here is that
$$
(A+B)^{2} \neq A^{2}+2 A B+B^{2}
$$
we leave it to the reader to figure out why.
The next section is devoted to visualizing column vectors and "seeing" how some of these arithmetic properties work together.
## Exercises 2.2
In Exercises 1 - 12, row and column vectors $\vec{u}$ and $\vec{v}$ are defined. Find the product $\vec{v} \vec{v}$, where possible.
1. $\vec{u}=\left[\begin{array}{ll}1 & -4\end{array}\right] \quad \vec{v}=\left[\begin{array}{c}-2 \\ 5\end{array}\right]$
2. $\vec{u}=\left[\begin{array}{ll}2 & 3\end{array}\right] \quad \vec{v}=\left[\begin{array}{c}7 \\ -4\end{array}\right]$
3. $\vec{u}=\left[\begin{array}{ll}1 & -1\end{array}\right] \quad \vec{v}=\left[\begin{array}{l}3 \\ 3\end{array}\right]$
4. $\vec{u}=\left[\begin{array}{ll}0.6 & 0.8\end{array}\right] \quad \vec{v}=\left[\begin{array}{l}0.6 \\ 0.8\end{array}\right]$
5. $\vec{u}=\left[\begin{array}{lll}1 & 2 & -1\end{array}\right] \vec{v}=\left[\begin{array}{c}2 \\ 1 \\ -1\end{array}\right]$
6. $\vec{u}=\left[\begin{array}{lll}3 & 2 & -2\end{array}\right] \vec{v}=\left[\begin{array}{c}-1 \\ 0 \\ 9\end{array}\right]$
7. $\vec{u}=\left[\begin{array}{lll}8 & -4 & 3\end{array}\right] \vec{v}=\left[\begin{array}{l}2 \\ 4 \\ 5\end{array}\right]$ 8. $\vec{u}=\left[\begin{array}{lll}-3 & 6 & 1\end{array}\right] \vec{v}=\left[\begin{array}{c}1 \\ -1 \\ 1\end{array}\right]$
8. $\vec{u}=\left[\begin{array}{llll}1 & 2 & 3 & 4\end{array}\right]$
$$
\vec{v}=\left[\begin{array}{c}
1 \\
-1 \\
1 \\
-1
\end{array}\right]
$$
10. $\vec{u}=\left[\begin{array}{llll}6 & 2 & -1 & 2\end{array}\right]$
$$
\vec{v}=\left[\begin{array}{l}
3 \\
2 \\
9 \\
5
\end{array}\right]
$$
11. $\vec{u}=\left[\begin{array}{lll}1 & 2 & 3\end{array}\right] \quad \vec{v}=\left[\begin{array}{l}3 \\ 2\end{array}\right]$
12. $\vec{u}=\left[\begin{array}{ll}2 & -5\end{array}\right] \quad \vec{v}=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$
In Exercises 13 - 27, matrices $A$ and $B$ are defined. (a) Give the dimensions of $A$ and $B$. If the dimensions properly match, give the dimensions of $A B$ and $B A$.
(b) Find the products $A B$ and $B A$, if possible.
13. $A=\left[\begin{array}{cc}1 & 2 \\ -1 & 4\end{array}\right] B=\left[\begin{array}{cc}2 & 5 \\ 3 & -1\end{array}\right]$
14. $A=\left[\begin{array}{ll}3 & 7 \\ 2 & 5\end{array}\right] \quad B=\left[\begin{array}{ll}1 & -1 \\ 3 & -3\end{array}\right]$
15. $A=\left[\begin{array}{cc}3 & -1 \\ 2 & 2\end{array}\right]$
$$
B=\left[\begin{array}{lll}
1 & 0 & 7 \\
4 & 2 & 9
\end{array}\right]
$$
16. $A=\left[\begin{array}{cc}0 & 1 \\ 1 & -1 \\ -2 & -4\end{array}\right]$
$$
B=\left[\begin{array}{cc}
-2 & 0 \\
3 & 8
\end{array}\right]
$$
17. $A=\left[\begin{array}{ccc}9 & 4 & 3 \\ 9 & -5 & 9\end{array}\right]$
$$
B=\left[\begin{array}{cc}
-2 & 5 \\
-2 & -1
\end{array}\right]
$$
18. $A=\left[\begin{array}{cc}-2 & -1 \\ 9 & -5 \\ 3 & -1\end{array}\right]$
$$
B=\left[\begin{array}{ccc}
-5 & 6 & -4 \\
0 & 6 & -3
\end{array}\right]
$$
19. $A=\left[\begin{array}{cc}2 & 6 \\ 6 & 2 \\ 5 & -1\end{array}\right]$
$$
B=\left[\begin{array}{ccc}
-4 & 5 & 0 \\
-4 & 4 & -4
\end{array}\right]
$$
20. $A=\left[\begin{array}{cc}-5 & 2 \\ -5 & -2 \\ -5 & -4\end{array}\right]$
$$
B=\left[\begin{array}{ccc}
0 & -5 & 6 \\
-5 & -3 & -1
\end{array}\right]
$$
21. $A=\left[\begin{array}{cc}8 & -2 \\ 4 & 5 \\ 2 & -5\end{array}\right]$
$$
B=\left[\begin{array}{ccc}
-5 & 1 & -5 \\
8 & 3 & -2
\end{array}\right]
$$
22. $A=\left[\begin{array}{ll}1 & 4 \\ 7 & 6\end{array}\right]$
$$
B=\left[\begin{array}{cccc}
1 & -1 & -5 & 5 \\
-2 & 1 & 3 & -5
\end{array}\right]
$$
23. $A=\left[\begin{array}{cc}-1 & 5 \\ 6 & 7\end{array}\right]$
$$
B=\left[\begin{array}{cccc}
5 & -3 & -4 & -4 \\
-2 & -5 & -5 & -1
\end{array}\right]
$$
24. $A=\left[\begin{array}{ccc}-1 & 2 & 1 \\ -1 & 2 & -1 \\ 0 & 0 & -2\end{array}\right]$
$$
B=\left[\begin{array}{ccc}
0 & 0 & -2 \\
1 & 2 & -1 \\
1 & 0 & 0
\end{array}\right]
$$
25. $A=\left[\begin{array}{ccc}-1 & 1 & 1 \\ -1 & -1 & -2 \\ 1 & 1 & -2\end{array}\right]$
$$
B=\left[\begin{array}{ccc}
-2 & -2 & -2 \\
0 & -2 & 0 \\
-2 & 0 & 2
\end{array}\right]
$$
26. $A=\left[\begin{array}{ccc}-4 & 3 & 3 \\ -5 & -1 & -5 \\ -5 & 0 & -1\end{array}\right]$
$$
B=\left[\begin{array}{ccc}
0 & 5 & 0 \\
-5 & -4 & 3 \\
5 & -4 & 3
\end{array}\right]
$$
27. $A=\left[\begin{array}{ccc}-4 & -1 & 3 \\ 2 & -3 & 5 \\ 1 & 5 & 3\end{array}\right]$
$$
B=\left[\begin{array}{ccc}
-2 & 4 & 3 \\
-1 & 1 & -1 \\
4 & 0 & 2
\end{array}\right]
$$
In Exercises 28 - 33, a diagonal matrix $D$ and a matrix $A$ are given. Find the products $D A$ and $A D$, where possible.
$$
\text { 28. } \begin{aligned}
D & =\left[\begin{array}{cc}
3 & 0 \\
0 & -1
\end{array}\right] \\
A & =\left[\begin{array}{ll}
2 & 4 \\
6 & 8
\end{array}\right] \\
\text { 29. } D & =\left[\begin{array}{cc}
4 & 0 \\
0 & -3
\end{array}\right] \\
A & =\left[\begin{array}{ll}
1 & 2 \\
1 & 2
\end{array}\right]
\end{aligned}
$$
30. $D=\left[\begin{array}{ccc}-1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3\end{array}\right]$
$$
A=\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right]
$$
31. $D=\left[\begin{array}{ccc}1 & 1 & 1 \\ 2 & 2 & 2 \\ -3 & -3 & -3\end{array}\right]$
$$
A=\left[\begin{array}{ccc}
2 & 0 & 0 \\
0 & -3 & 0 \\
0 & 0 & 5
\end{array}\right]
$$
32. $D=\left[\begin{array}{cc}d_{1} & 0 \\ 0 & d_{2}\end{array}\right]$
$$
A=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]
$$
33. $D=\left[\begin{array}{ccc}d_{1} & 0 & 0 \\ 0 & d_{2} & 0 \\ 0 & 0 & d_{3}\end{array}\right]$
$$
A=\left[\begin{array}{lll}
a & b & c \\
d & e & f \\
g & h & i
\end{array}\right]
$$
In Exercises $34-39$, a matrix $A$ and a vector $\vec{x}$ are given. Find the product $A \vec{x}$.
34. $A=\left[\begin{array}{cc}2 & 3 \\ 1 & -1\end{array}\right], \quad \vec{x}=\left[\begin{array}{l}4 \\ 9\end{array}\right]$
35. $A=\left[\begin{array}{cc}-1 & 4 \\ 7 & 3\end{array}\right], \quad \vec{x}=\left[\begin{array}{c}2 \\ -1\end{array}\right]$
36. $A=\left[\begin{array}{ccc}2 & 0 & 3 \\ 1 & 1 & 1 \\ 3 & -1 & 2\end{array}\right], \quad \vec{x}=\left[\begin{array}{l}1 \\ 4 \\ 2\end{array}\right]$
37. $A=\left[\begin{array}{ccc}-2 & 0 & 3 \\ 1 & 1 & -2 \\ 4 & 2 & -1\end{array}\right], \quad \vec{x}=\left[\begin{array}{l}4 \\ 3 \\ 1\end{array}\right]$
38. $A=\left[\begin{array}{cc}2 & -1 \\ 4 & 3\end{array}\right], \quad \vec{x}=\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]$ 39. $A=\left[\begin{array}{lll}1 & 2 & 3 \\ 1 & 0 & 2 \\ 2 & 3 & 1\end{array}\right], \quad \vec{x}=\left[\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right]$
39. Let $A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$. Find $A^{2}$ and $A^{3}$.
40. Let $A=\left[\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right]$. Find $A^{2}$ and $A^{3}$.
41. Let $A=\left[\begin{array}{ccc}-1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5\end{array}\right]$. Find $A^{2}$ and $A^{3}$.
42. Let $A=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right]$. Find $A^{2}$ and $A^{3}$.
43. Let $A=\left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 1 & 0\end{array}\right]$. Find $A^{2}$ and $A^{3}$.
44. In the text we state that $(A+B)^{2} \neq$ $A^{2}+2 A B+B^{2}$. We investigate that claim here.
(a) Let $A=\left[\begin{array}{cc}5 & 3 \\ -3 & -2\end{array}\right]$ and let $B=$ $\left[\begin{array}{cc}-5 & -5 \\ -2 & 1\end{array}\right]$. Compute $A+B$
(b) Find $(A+B)^{2}$ by using your answer from (a).
(c) Compute $A^{2}+2 A B+B^{2}$.
(d) Are the results from (a) and (b) the same?
(e) Carefully expand the expression $(A+B)^{2}=(A+B)(A+B)$ and show why this is not equal to $A^{2}+2 A B+B^{2}$.
### Visualizing Matrix Arithmetic in 2D
## AS YOU READ
1. $\mathrm{T} / \mathrm{F}$ : Two vectors with the same length and direction are equal even if they start from different places.
2. One can visualize vector addition using what law?
3. T/F: Multiplying a vector by 2 doubles its length.
4. What do mathematicians do?
5. T/F: Multiplying a vector by a matrix always changes its length and direction.
When we first learned about adding numbers together, it was useful to picture a number line: $2+3=5$ could be pictured by starting at 0 , going out 2 tick marks, then another 3 , and then realizing that we moved 5 tick marks from 0 . Similar visualizations helped us understand what $2-3$ meant and what $2 \times 3$ meant.
We now investigate a way to picture matrix arithmetic - in particular, operations involving column vectors. This not only will help us better understand the arithmetic operations, it will open the door to a great wealth of interesting study. Visualizing matrix arithmetic has a wide variety of applications, the most common being computer graphics. While we often think of these graphics in terms of video games, there are numerous other important applications. For example, chemists and biologists often use computer models to "visualize" complex molecules to "see" how they interact with other molecules.
We will start with vectors in two dimensions (2D) - that is, vectors with only two entries. We assume the reader is familiar with the Cartesian plane, that is, plotting points and graphing functions on "the $x-y$ plane." We graph vectors in a manner very similar to plotting points. Given the vector
$$
\vec{x}=\left[\begin{array}{l}
1 \\
2
\end{array}\right]
$$
we draw $\vec{x}$ by drawing an arrow whose tip is 1 unit to the right and 2 units up from its origin. $^{7}$
Figure 2.1: Various drawings of $\vec{x}$ When drawing vectors, we do not specify where you start drawing; all we specify is where the tip lies based on where we started. Figure 2.1 shows vector $\vec{x}$ drawn 3 ways. In some ways, the "most common" way to draw a vector has the arrow start at the origin, but this is by no means the only way of drawing the vector.
Let's practice this concept by drawing various vectors from given starting points.
Example $31 \quad$ Let
$$
\vec{x}=\left[\begin{array}{c}
1 \\
-1
\end{array}\right] \quad \vec{y}=\left[\begin{array}{l}
2 \\
3
\end{array}\right] \quad \text { and } \quad \vec{z}=\left[\begin{array}{c}
-3 \\
2
\end{array}\right] .
$$
Draw $\vec{x}$ starting from the point $(0,-1)$; draw $\vec{y}$ starting from the point $(-1,-1)$, and draw $\vec{z}$ starting from the point $(2,-1)$.
Solution To draw $\vec{x}$, start at the point $(0,-1)$ as directed, then move to the right one unit and down one unit and draw the tip. Thus the arrow "points" from $(0,-1)$ to $(1,-2)$.
To draw $\vec{y}$, we are told to start and the point $(-1,-1)$. We draw the tip by moving to the right 2 units and up 3 units; hence $\vec{y}$ points from $(-1,-1)$ to $(1,2)$.
To draw $\vec{z}$, we start at $(2,-1)$ and draw the tip 3 units to the left and 2 units up; $\vec{z}$ points from $(2,-1)$ to $(-1,1)$.
Each vector is drawn as shown in Figure 2.2.
Figure 2.2: Drawing vectors $\vec{x}, \vec{y}$ and $\vec{z}$ in Example 31
How does one draw the zero vector, $\vec{O}=\left[\begin{array}{l}0 \\ 0\end{array}\right] ?^{8}$ Following our basic procedure, we start by going 0 units in the $x$ direction, followed by 0 units in the $y$ direction. In other words, we don't go anywhere. In general, we don't actually draw $\overrightarrow{0}$. At best, one can draw a dark circle at the origin to convey the idea that $\overrightarrow{0}$, when starting at the origin, points to the origin.
In section 2.1 we learned about matrix arithmetic operations: matrix addition and scalar multiplication. Let's investigate how we can "draw" these operations.
${ }^{8}$ Vectors are just special types of matrices. The zero vector, $\overrightarrow{0}$, is a special type of zero matrix, $\mathbf{0}$. It helps to distinguish the two by using different notation.
## Vector Addition
Given two vectors $\vec{x}$ and $\vec{y}$, how do we draw the vector $\vec{x}+\vec{y}$ ? Let's look at this in the context of an example, then study the result.
Example $32 \quad$ Let
$$
\vec{x}=\left[\begin{array}{l}
1 \\
1
\end{array}\right] \quad \text { and } \quad \vec{y}=\left[\begin{array}{l}
3 \\
1
\end{array}\right]
$$
Sketch $\vec{x}, \vec{y}$ and $\vec{x}+\vec{y}$.
Solution A starting point for drawing each vector was not given; by default, we'll start at the origin. (This is in many ways nice; this means that the vector $\left[\begin{array}{l}3 \\ 1\end{array}\right]$ "points" to the point $(3,1)$.) We first compute $\vec{x}+\vec{y}$ :
$$
\vec{x}+\vec{y}=\left[\begin{array}{l}
1 \\
1
\end{array}\right]+\left[\begin{array}{l}
3 \\
1
\end{array}\right]=\left[\begin{array}{l}
4 \\
2
\end{array}\right]
$$
Sketching each gives the picture in Figure 2.3.
Figure 2.3: Adding vectors $\vec{x}$ and $\vec{y}$ in Example 32
This example is pretty basic; we were given two vectors, told to add them together, then sketch all three vectors. Our job now is to go back and try to see a relationship between the drawings of $\vec{x}, \vec{y}$ and $\vec{x}+\vec{y}$. Do you see any?
Here is one way of interpreting the adding of $\vec{x}$ to $\vec{y}$. Regardless of where we start, we draw $\vec{x}$. Now, from the tip of $\vec{x}$, draw $\vec{y}$. The vector $\vec{x}+\vec{y}$ is the vector found by drawing an arrow from the origin of $\vec{x}$ to the tip of $\vec{y}$. Likewise, we could start by drawing $\vec{y}$. Then, starting from the tip of $\vec{y}$, we can draw $\vec{x}$. Finally, $\operatorname{draw} \vec{x}+\vec{y}$ by drawing the vector that starts at the origin of $\vec{y}$ and ends at the tip of $\vec{x}$.
The picture in Figure 2.4 illustrates this. The gray vectors demonstrate drawing the second vector from the tip of the first; we draw the vector $\vec{x}+\vec{y}$ dashed to set it apart from the rest. We also lightly filled the parallelogram whose opposing sides are the vectors $\vec{x}$ and $\vec{y}$. This highlights what is known as the Parallelogram Law.
Figure 2.4: Adding vectors graphically using the Parallelogram Law
Key Idea 5
Parallelogram Law
To draw the vector $\vec{x}+\vec{y}$, one can draw the parallelogram with $\vec{x}$ and $\vec{y}$ as its sides. The vector that points from the vertex where $\vec{x}$ and $\vec{y}$ originate to the vertex where $\vec{x}$ and $\vec{y}$ meet is the vector $\vec{x}+\vec{y}$.
Knowing all of this allows us to draw the sum of two vectors without knowing specifically what the vectors are, as we demonstrate in the following example.
Example $33 \quad$ Consider the vectors $\vec{x}$ and $\vec{y}$ as drawn in Figure 2.5. Sketch the vector $\vec{x}+\vec{y}$.
## SOLUTION
Figure 2.5: Vectors $\vec{x}$ and $\vec{y}$ in Example 33
We'll apply the Parallelogram Law, as given in Key Idea 5. As before, we $\operatorname{draw} \vec{x}+\vec{y}$ dashed to set it apart. The result is given in Figure 2.6.
Figure 2.6: Vectors $\vec{x}, \vec{y}$ and $\vec{x}+\vec{y}$ in Example 33
## Scalar Multiplication
After learning about matrix addition, we learned about scalar multiplication. We apply that concept now to vectors and see how this is represented graphically.
Example $34 \quad$ Let
$$
\vec{x}=\left[\begin{array}{l}
1 \\
1
\end{array}\right] \quad \text { and } \quad \vec{y}=\left[\begin{array}{c}
-2 \\
1
\end{array}\right]
$$
Sketch $\vec{x}, \vec{y}, 3 \vec{x}$ and $-1 \vec{y}$.
Solution We begin by computing $3 \vec{x}$ and $-\vec{y}$ :
$$
3 \vec{x}=\left[\begin{array}{l}
3 \\
3
\end{array}\right] \quad \text { and } \quad-\vec{y}=\left[\begin{array}{c}
2 \\
-1
\end{array}\right] .
$$
All four vectors are sketched in Figure 2.7.
Figure 2.7: Vectors $\vec{x}, \vec{y}, 3 \vec{x}$ and $-\vec{y}$ in Example 34
As we often do, let us look at the previous example and see what we can learn from it. We can see that $\vec{x}$ and $3 \vec{x}$ point in the same direction (they lie on the same line), but $3 \vec{x}$ is just longer than $\vec{x}$. (In fact, it looks like $3 \vec{x}$ is 3 times longer than $\vec{x}$. Is it? How do we measure length?)
We also see that $\vec{y}$ and $-\vec{y}$ seem to have the same length and lie on the same line, but point in the opposite direction.
A vector inherently conveys two pieces of information: length and direction. Multiplying a vector by a positive scalar $c$ stretches the vectors by a factor of $c$; multiplying by a negative scalar $c$ both stretches the vector and makes it point in the opposite direction.
Knowing this, we can sketch scalar multiples of vectors without knowing specifically what they are, as we do in the following example.
Example 35 Let vectors $\vec{x}$ and $\vec{y}$ be as in Figure 2.8. Draw $3 \vec{x},-2 \vec{x}$, and $\frac{1}{2} \vec{y}$.
Figure 2.8: Vectors $\vec{x}$ and $\vec{y}$ in Example 35
Solution To draw $3 \vec{x}$, we draw a vector in the same direction as $\vec{x}$, but 3 times as long. To draw $-2 \vec{x}$, we draw a vector twice as long as $\vec{x}$ in the opposite direction; to draw $\frac{1}{2} \vec{y}$, we draw a vector half the length of $\vec{y}$ in the same direction as $\vec{y}$. We again use the default of drawing all the vectors starting at the origin. All of this is shown in Figure 2.9.
Figure 2.9: Vectors $\vec{x}, \vec{y}, 3 \vec{x},-2 x$ and $\frac{1}{2} \vec{x}$ in Example 35
## Vector Subtraction
The final basic operation to consider between two vectors is that of vector subtraction: given vectors $\vec{x}$ and $\vec{y}$, how do we draw $\vec{x}-\vec{y}$ ?
If we know explicitly what $\vec{x}$ and $\vec{y}$ are, we can simply compute what $\vec{x}-\vec{y}$ is and then draw it. We can also think in terms of vector addition and scalar multiplication: we can add the vectors $\vec{x}+(-1) \vec{y}$. That is, we can draw $\vec{x}$ and draw $-\vec{y}$, then add them as we did in Example 33. This is especially useful we don't know explicitly what $\vec{x}$ and $\vec{y}$ are.
Example 36 Let vectors $\vec{x}$ and $\vec{y}$ be as in Figure 2.10. Draw $\vec{x}-\vec{y}$.
Figure 2.10: Vectors $\vec{x}$ and $\vec{y}$ in Example 36 Solution To draw $\vec{x}-\vec{y}$, we will first draw $-\vec{y}$ and then apply the Parallelogram Law to add $\vec{x}$ to $-\vec{y}$. See Figure 2.11.
Figure 2.11: Vectors $\vec{x}, \vec{y}$ and $\vec{x}-\vec{y}$ in Example 36
In Figure 2.12, we redraw Figure 2.11 from Example 36 but remove the gray vectors that tend to add clutter, and we redraw the vector $\vec{x}-\vec{y}$ dotted so that it starts from the tip of $\vec{y} .{ }^{9}$ Note that the dotted version of $\vec{x}-\vec{y}$ points from $\vec{y}$ to $\vec{x}$. This is a "shortcut" to drawing $\vec{x}-\vec{y}$; simply draw the vector that starts at the tip of $\vec{y}$ and ends at the tip of $\vec{x}$. This is important so we make it a Key Idea.
Figure 2.12: Redrawing vector $\vec{x}-\vec{y}$
Key Idea 6
Vector Subtraction
To draw the vector $\vec{x}-\vec{y}$, draw $\vec{x}$ and $\vec{y}$ so that they have the same origin. The vector $\vec{x}-\vec{y}$ is the vector that starts from the tip of $\vec{y}$ and points to the tip of $\vec{x}$.
Let's practice this once more with a quick example.
Example 37 Let $\vec{x}$ and $\vec{y}$ be as in Figure ?? (a). Draw $\vec{x}-\vec{y}$.
Solution We simply apply Key Idea 6 : we draw an arrow from $\vec{y}$ to $\vec{x}$. We do so in Figure $2.13 ; \vec{x}-\vec{y}$ is dashed.
${ }^{9}$ Remember that we can draw vectors starting from anywhere.
(a)
(b)
Figure 2.13: Vectors $\vec{x}, \vec{y}$ and $\vec{x}-\vec{y}$ in Example 37
## Vector Length
When we discussed scalar multiplication, we made reference to a fundamental question: How do we measure the length of a vector? Basic geometry gives us an answer in the two dimensional case that we are dealing with right now, and later we can extend these ideas to higher dimensions.
Consider Figure 2.14. A vector $\vec{x}$ is drawn in black, and dashed and dotted lines have been drawn to make it the hypotenuse of a right triangle.
Figure 2.14: Measuring the length of a vector
It is easy to see that the dashed line has length 4 and the dotted line has length 3. We'll let $c$ denote the length of $\vec{x}$; according to the Pythagorean Theorem, $4^{2}+3^{2}=c^{2}$. Thus $c^{2}=25$ and we quickly deduce that $c=5$.
Notice that in our figure, $\vec{x}$ goes to the right 4 units and then up 3 units. In other words, we can write
$$
\vec{x}=\left[\begin{array}{l}
4 \\
3
\end{array}\right]
$$
We learned above that the length of $\vec{x}$ is $\sqrt{4^{2}+3^{2}} \cdot{ }^{10}$ This hints at a basic calculation that works for all vectors $\vec{x}$, and we define the length of a vector according to this rule.
${ }^{10}$ Remember that $\sqrt{4^{2}+3^{2}} \neq 4+3$ ! Definition 16
Vector Length
Let
$$
\vec{x}=\left[\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right] .
$$
The length of $\vec{x}$, denoted $\|\vec{x}\|$, is
$$
\|\vec{x}\|=\sqrt{x_{1}^{2}+x_{2}^{2}}
$$
Example $38 \quad$ Find the length of each of the vectors given below.
$$
\overrightarrow{x_{1}}=\left[\begin{array}{l}
1 \\
1
\end{array}\right] \quad \overrightarrow{x_{2}}=\left[\begin{array}{c}
2 \\
-3
\end{array}\right] \quad \overrightarrow{x_{3}}=\left[\begin{array}{l}
.6 \\
.8
\end{array}\right] \quad \overrightarrow{x_{4}}=\left[\begin{array}{l}
3 \\
0
\end{array}\right]
$$
Solution We apply Definition 16 to each vector.
$$
\begin{aligned}
& \left\|\overrightarrow{x_{1}}\right\|=\sqrt{1^{2}+1^{2}}=\sqrt{2} . \\
& \left\|\overrightarrow{x_{2}}\right\|=\sqrt{2^{2}+(-3)^{2}}=\sqrt{13} . \\
& \left\|\overrightarrow{x_{3}}\right\|=\sqrt{.6^{2}+.8^{2}}=\sqrt{.36+.64}=1 . \\
& \left\|\overrightarrow{x_{4}}\right\|=\sqrt{3^{2}+0}=3 .
\end{aligned}
$$
Now that we know how to compute the length of a vector, let's revisit a statement we made as we explored Examples 34 and 35: "Multiplying a vector by a positive scalar $c$ stretches the vectors by a factor of $c \ldots$..." At that time, we did not know how to measure the length of a vector, so our statement was unfounded. In the following example, we will confirm the truth of our previous statement.
Example $39 \quad$ Let $\vec{x}=\left[\begin{array}{c}2 \\ -1\end{array}\right]$. Compute $\|\vec{x}\|,\|3 \vec{x}\|,\|-2 \vec{x}\|$, and $\|c \vec{x}\|$, where $c$ is a scalar.
Solution We apply Definition 16 to each of the vectors.
$\|\vec{x}\|=\sqrt{4+1}=\sqrt{5}$.
Before computing the length of $\|3 \vec{x}\|$, we note that $3 \vec{x}=\left[\begin{array}{c}6 \\ -3\end{array}\right]$.
$$
\|3 \vec{x}||=\sqrt{36+9}=\sqrt{45}=3 \sqrt{5}=3|| \vec{x}\| .
$$
Before computing the length of $\|-2 \vec{x}\|$, we note that $-2 \vec{x}=\left[\begin{array}{c}-4 \\ 2\end{array}\right]$.
$\|-2 \vec{x}\|=\sqrt{16+4}=\sqrt{20}=2 \sqrt{5}=2\|\vec{x}\|$.
Finally, to compute $\|c \vec{x}\|$, we note that $c \vec{x}=\left[\begin{array}{c}2 c \\ -c\end{array}\right]$. Thus:
$$
\|c \vec{x}\|=\sqrt{(2 c)^{2}+(-c)^{2}}=\sqrt{4 c^{2}+c^{2}}=\sqrt{5 c^{2}}=|c| \sqrt{5} .
$$
This last line is true because the square root of any number squared is the absolute value of that number (for example, $\sqrt{(-3)^{2}}=3$ ).
The last computation of our example is the most important one. It shows that, in general, multiplying a vector $\vec{x}$ by a scalar $c$ stretches $\vec{x}$ by a factor of $|c|$ (and the direction will change if $c$ is negative). This is important so we'll make it a Theorem.
## Matrix - Vector Multiplication
The last arithmetic operation to consider visualizing is matrix multiplication. Specifically, we want to visualize the result of multiplying a vector by a matrix. In order to multiply a $2 \mathrm{D}$ vector by a matrix and get a $2 \mathrm{D}$ vector back, our matrix must be a square, $2 \times 2$ matrix. $^{11}$
We'll start with an example. Given a matrix $A$ and several vectors, we'll graph the vectors before and after they've been multiplied by $A$ and see what we learn.
Example $40 \quad$ Let $A$ be a matrix, and $\vec{x}, \vec{y}$, and $\vec{z}$ be vectors as given below.
$$
A=\left[\begin{array}{ll}
1 & 4 \\
2 & 3
\end{array}\right], \quad \vec{x}=\left[\begin{array}{l}
1 \\
1
\end{array}\right], \quad \vec{y}=\left[\begin{array}{c}
-1 \\
1
\end{array}\right], \quad \vec{z}=\left[\begin{array}{c}
3 \\
-1
\end{array}\right]
$$
Graph $\vec{x}, \vec{y}$ and $\vec{z}$, as well as $A \vec{x}, A \vec{y}$ and $A \vec{z}$.
## SOLUTION
${ }^{11}$ We can multiply a $3 \times 2$ matrix by a $2 \mathrm{D}$ vector and get a 3D vector back, and this gives very interesting results. See section 5.2 .
Figure 2.15: Multiplying vectors by a matrix in Example 40.
It is straightforward to compute:
$$
A \vec{x}=\left[\begin{array}{l}
5 \\
5
\end{array}\right], \quad A \vec{y}=\left[\begin{array}{l}
3 \\
1
\end{array}\right], \quad \text { and } \quad A \vec{z}=\left[\begin{array}{c}
-1 \\
3
\end{array}\right] .
$$
The vectors are sketched in Figure 2.15
There are several things to notice. When each vector is multiplied by $A$, the result is a vector with a different length (in this example, always longer), and in two of the cases (for $\vec{y}$ and $\vec{z}$ ), the resulting vector points in a different direction.
This isn't surprising. In the previous section we learned about matrix multiplication, which is a strange and seemingly unpredictable operation. Would you expect to see some sort of immediately recognizable pattern appear from multiplying a matrix and a vector? ${ }^{12}$ In fact, the surprising thing from the example is that $\vec{x}$ and $A \vec{x}$ point in the same direction! Why does the direction of $\vec{x}$ not change after multiplication by $A$ ? (We'll answer this in Section 4.1 when we learn about something called "eigenvectors.")
Different matrices act on vectors in different ways. ${ }^{13}$ Some always increase the length of a vector through multiplication, others always decrease the length, others increase the length of some vectors and decrease the length of others, and others still don't change the length at all. A similar statement can be made about how matrices affect the direction of vectors through multiplication: some change every vector's direction, some change "most" vector's direction but leave some the same, and others still don't change the direction of any vector.
How do we set about studying how matrix multiplication affects vectors? We could just create lots of different matrices and lots of different vectors, multiply, then graph, but this would be a lot of work with very little useful result. It would be too hard to find a pattern of behavior in this. ${ }^{14}$
${ }^{12}$ This is a rhetorical question; the expected answer is "No."
${ }^{13}$ That's one reason we call them "different."
${ }^{14}$ Remember, that's what mathematicians do. We look for patterns. Instead, we'll begin by using a technique we've employed often in the past. We have a "new" operation; let's explore how it behaves with "old" operations. Specifically, we know how to sketch vector addition. What happens when we throw matrix multiplication into the mix? Let's try an example.
Example $41 \quad$ Let $A$ be a matrix and $\vec{x}$ and $\vec{y}$ be vectors as given below.
$$
A=\left[\begin{array}{ll}
1 & 1 \\
1 & 2
\end{array}\right], \quad \vec{x}=\left[\begin{array}{l}
2 \\
1
\end{array}\right], \quad \vec{y}=\left[\begin{array}{c}
-1 \\
1
\end{array}\right]
$$
Sketch $\vec{x}+\vec{y}, A \vec{x}, A \vec{y}$, and $A(\vec{x}+\vec{y})$.
Solution It is pretty straightforward to compute:
$$
\vec{x}+\vec{y}=\left[\begin{array}{l}
1 \\
2
\end{array}\right] ; \quad A \vec{x}=\left[\begin{array}{l}
3 \\
4
\end{array}\right] ; \quad A \vec{y}=\left[\begin{array}{l}
0 \\
1
\end{array}\right], \quad A(\vec{x}+\vec{y})=\left[\begin{array}{l}
3 \\
5
\end{array}\right] .
$$
In Figure 2.16, we have graphed the above vectors and have included dashed gray vectors to highlight the additive nature of $\vec{x}+\vec{y}$ and $A(\vec{x}+\vec{y})$. Does anything strike you as interesting?
Figure 2.16: Vector addition and matrix multiplication in Example 41.
Let's not focus on things which don't matter right now: let's not focus on how long certain vectors became, nor necessarily how their direction changed. Rather, think about how matrix multiplication interacted with the vector addition.
In some sense, we started with three vectors, $\vec{x}, \vec{y}$, and $\vec{x}+\vec{y}$. This last vector is special; it is the sum of the previous two. Now, multiply all three by $A$. What happens? We get three new vectors, but the significant thing is this: the last vector is still the sum of the previous two! (We emphasize this by drawing dotted vectors to represent part of the Parallelogram Law.) Of course, we knew this already: we already knew that $A \vec{x}+A \vec{y}=A(\vec{x}+\vec{y})$, for this is just the Distributive Property. However, now we get to see this graphically.
In Section 5.1 we'll study in greater depth how matrix multiplication affects vectors and the whole Cartesian plane. For now, we'll settle for simple practice: given a matrix and some vectors, we'll multiply and graph. Let's do one more example.
Example 42 Let $A, \vec{x}, \vec{y}$, and $\vec{z}$ be as given below.
$$
A=\left[\begin{array}{ll}
1 & -1 \\
1 & -1
\end{array}\right], \quad \vec{x}=\left[\begin{array}{l}
1 \\
1
\end{array}\right], \quad \vec{y}=\left[\begin{array}{c}
-1 \\
1
\end{array}\right], \quad \vec{z}=\left[\begin{array}{l}
4 \\
1
\end{array}\right]
$$
Graph $\vec{x}, \vec{y}$ and $\vec{z}$, as well as $A \vec{x}, A \vec{y}$ and $A \vec{z}$.
## SOLUTION
Figure 2.17: Multiplying vectors by a matrix in Example 42.
It is straightforward to compute:
$$
A \vec{x}=\left[\begin{array}{l}
0 \\
0
\end{array}\right], \quad A \vec{y}=\left[\begin{array}{l}
-2 \\
-2
\end{array}\right], \quad \text { and } \quad A \vec{z}=\left[\begin{array}{l}
3 \\
3
\end{array}\right]
$$
The vectors are sketched in Figure 2.17.
These results are interesting. While we won't explore them in great detail here, notice how $\vec{x}$ got sent to the zero vector. Notice also that $A \vec{x}, A \vec{y}$ and $A \vec{z}$ are all in a line (as well as $\vec{x}$ !). Why is that? Are $\vec{x}, \vec{y}$ and $\vec{z}$ just special vectors, or would any other vector get sent to the same line when multiplied by $A ?^{15}$
This section has focused on vectors in two dimensions. Later on in this book, we'll extend these ideas into three dimensions (3D).
In the next section we'll take a new idea (matrix multiplication) and apply it to an old idea (solving systems of linear equations). This will allow us to view an old idea in a new way - and we'll even get to "visualize" it.
${ }^{15}$ Don't just sit there, try it out!
## Exercises 2.3
In Exercises $1-4$, vectors $\vec{x}$ and $\vec{y}$ are given. the lengths of $\vec{x}$ and $a \vec{x}$, then compare these Sketch $\vec{x}, \vec{y}, \vec{x}+\vec{y}$, and $\vec{x}-\vec{y}$ on the same Carte- lengths. sian axes.
1. $\vec{x}=\left[\begin{array}{l}1 \\ 1\end{array}\right], \vec{y}=\left[\begin{array}{c}-2 \\ 3\end{array}\right]$
2. $\vec{x}=\left[\begin{array}{l}2 \\ 1\end{array}\right], a=3$.
3. $\vec{x}=\left[\begin{array}{l}3 \\ 1\end{array}\right], \vec{y}=\left[\begin{array}{c}1 \\ -2\end{array}\right]$
4. $\vec{x}=\left[\begin{array}{l}4 \\ 7\end{array}\right], a=-2$.
5. $\vec{x}=\left[\begin{array}{c}-1 \\ 1\end{array}\right], \vec{y}=\left[\begin{array}{c}-2 \\ 2\end{array}\right]$
6. $\vec{x}=\left[\begin{array}{c}-3 \\ 5\end{array}\right], a=-1$.
7. $\vec{x}=\left[\begin{array}{l}2 \\ 0\end{array}\right], \vec{y}=\left[\begin{array}{l}1 \\ 3\end{array}\right]$
8. $\vec{x}=\left[\begin{array}{c}3 \\ -9\end{array}\right], a=\frac{1}{3}$.
In Exercises $5-8$, vectors $\vec{x}$ and $\vec{y}$ are drawn. Sketch $2 \vec{x},-\vec{y}, \vec{x}+\vec{y}$, and $\vec{x}-\vec{y}$ on the same Cartesian axes.
13. Four pairs of vectors $\vec{x}$ and $\vec{y}$ are given below. For each pair, compute $\|\vec{x}\|$, $\|\vec{y}\|$, and ||$\vec{x}+\vec{y} \|$. Use this information to answer: Is it always, sometimes, or never true that ||$\vec{x}\|+\| \vec{y}\|=\| \vec{x}+\vec{y}||$ ? If it always or never true, explain why. If it is sometimes true, explain when it is true.
(a) $\vec{x}=\left[\begin{array}{l}1 \\ 1\end{array}\right], \vec{y}=\left[\begin{array}{l}2 \\ 3\end{array}\right]$
(b) $\vec{x}=\left[\begin{array}{c}1 \\ -2\end{array}\right], \vec{y}=\left[\begin{array}{c}3 \\ -6\end{array}\right]$
(c) $\vec{x}=\left[\begin{array}{c}-1 \\ 3\end{array}\right], \vec{y}=\left[\begin{array}{l}2 \\ 5\end{array}\right]$
(d) $\vec{x}=\left[\begin{array}{l}2 \\ 1\end{array}\right], \vec{y}=\left[\begin{array}{l}-4 \\ -2\end{array}\right]$
In Exercises 14 - 17, a matrix $A$ is given. Sketch $\vec{x}, \vec{y}, A \vec{x}$ and $A \vec{y}$ on the same Cartesian axes, where
$$
\vec{x}=\left[\begin{array}{l}
1 \\
1
\end{array}\right] \text { and } \vec{y}=\left[\begin{array}{c}
-1 \\
2
\end{array}\right] .
$$
8.
14. $A=\left[\begin{array}{cc}1 & -1 \\ 2 & 3\end{array}\right]$
15. $A=\left[\begin{array}{cc}2 & 0 \\ -1 & 3\end{array}\right]$
16. $A=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$
In Exercises $9-12$, a vector $\vec{x}$ and a scalar $a$ are given. Using Definition 16, compute
17. $A=\left[\begin{array}{cc}1 & 2 \\ -1 & -2\end{array}\right]$
### Vector Solutions to Linear Systems
## AS YOU READ
1. $T / F$ : The equation $A \vec{x}=\vec{b}$ is just another way of writing a system of linear equations.
2. $\mathrm{T} / \mathrm{F}$ : In solving $A \vec{x}=\overrightarrow{0}$, if there are 3 free variables, then the solution will be "pulled apart" into 3 vectors.
3. T/F: A homogeneous system of linear equations is one in which all of the coefficients are 0 .
4. Whether or not the equation $A \vec{x}=\vec{b}$ has a solution depends on an intrinsic property of
The first chapter of this text was spent finding solutions to systems of linear equations. We have spent the first two sections of this chapter learning operations that can be performed with matrices. One may have wondered "Are the ideas of the first chapter related to what we have been doing recently?" The answer is yes, these ideas are related. This section begins to show that relationship.
We have often hearkened back to previous algebra experience to help understand matrix algebra concepts. We do that again here. Consider the equation $a x=b$, where $a=3$ and $b=6$. If we asked one to "solve for $x$," what exactly would we be asking? We would want to find a number, which we call $x$, where $a$ times $x$ gives $b$; in this case, it is a number, when multiplied by 3 , returns 6 .
Now we consider matrix algebra expressions. We'll eventually consider solving equations like $A X=B$, where we know what the matrices $A$ and $B$ are and we want to find the matrix $X$. For now, we'll only consider equations of the type $A \vec{x}=\vec{b}$, where we know the matrix $A$ and the vector $\vec{b}$. We will want to find what vector $\vec{x}$ satisfies this equation; we want to "solve for $\vec{x}$."
To help understand what this is asking, we'll consider an example. Let
$$
A=\left[\begin{array}{ccc}
1 & 1 & 1 \\
1 & -1 & 2 \\
2 & 0 & 1
\end{array}\right], \quad \vec{b}=\left[\begin{array}{c}
2 \\
-3 \\
1
\end{array}\right] \quad \text { and } \quad \vec{x}=\left[\begin{array}{l}
x_{1} \\
x_{2} \\
x_{3}
\end{array}\right]
$$
(We don't know what $\vec{x}$ is, so we have to represent it's entries with the variables $x_{1}, x_{2}$ and $x_{3}$.) Let's "solve for $\vec{x}$," given the equation $A \vec{x}=\vec{b}$.
We can multiply out the left hand side of this equation. We find that
$$
A \vec{x}=\left[\begin{array}{c}
x_{1}+x_{2}+x_{3} \\
x_{1}-x_{2}+2 x_{3} \\
2 x_{1}+x_{3}
\end{array}\right] .
$$
Be sure to note that the product is just a vector; it has just one column. Since $A \vec{x}$ is equal to $\vec{b}$, we have
$$
\left[\begin{array}{c}
x_{1}+x_{2}+x_{3} \\
x_{1}-x_{2}+2 x_{3} \\
2 x_{1}+x_{3}
\end{array}\right]=\left[\begin{array}{c}
2 \\
-3 \\
1
\end{array}\right]
$$
Knowing that two vectors are equal only when their corresponding entries are equal, we know
$$
\begin{aligned}
x_{1}+x_{2}+x_{3} & =2 \\
x_{1}-x_{2}+2 x_{3} & =-3 \\
2 x_{1}+x_{3} & =1 .
\end{aligned}
$$
This should look familiar; it is a system of linear equations! Given the matrix-vector equation $A \vec{x}=\vec{b}$, we can recognize $A$ as the coefficient matrix from a linear system and $\vec{b}$ as the vector of the constants from the linear system. To solve a matrix-vector equation (and the corresponding linear system), we simply augment the matrix $A$ with the vector $\vec{b}$, put this matrix into reduced row echelon form, and interpret the results.
We convert the above linear system into an augmented matrix and find the reduced row echelon form:
$$
\left[\begin{array}{cccc}
1 & 1 & 1 & 2 \\
1 & -1 & 2 & -3 \\
2 & 0 & 1 & 1
\end{array}\right] \quad \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{cccc}
1 & 0 & 0 & 1 \\
0 & 1 & 0 & 2 \\
0 & 0 & 1 & -1
\end{array}\right]
$$
This tells us that $x_{1}=1, x_{2}=2$ and $x_{3}=-1$, so
$$
\vec{x}=\left[\begin{array}{c}
1 \\
2 \\
-1
\end{array}\right]
$$
We should check our work; multiply out $A \vec{x}$ and verify that we indeed get $\vec{b}$ :
$$
\left[\begin{array}{ccc}
1 & 1 & 1 \\
1 & -1 & 2 \\
2 & 0 & 1
\end{array}\right]\left[\begin{array}{c}
1 \\
2 \\
-1
\end{array}\right] \text { does equal }\left[\begin{array}{c}
2 \\
-3 \\
1
\end{array}\right]
$$
We should practice.
Example $43 \quad$ Solve the equation $A \vec{x}=\vec{b}$ for $\vec{x}$ where
$$
A=\left[\begin{array}{ccc}
1 & 2 & 3 \\
-1 & 2 & 1 \\
1 & 1 & 0
\end{array}\right] \text { and }\left[\begin{array}{c}
5 \\
-1 \\
2
\end{array}\right]
$$
Solution The solution is rather straightforward, even though we did a lot of work before to find the answer. Form the augmented matrix $\left[\begin{array}{ll}A & \vec{b}\end{array}\right]$ and interpret its reduced row echelon form.
$$
\left[\begin{array}{cccc}
1 & 2 & 3 & 5 \\
-1 & 2 & 1 & -1 \\
1 & 1 & 0 & 2
\end{array}\right] \quad \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{llll}
1 & 0 & 0 & 2 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 1
\end{array}\right]
$$
In previous sections we were fine stating that the result as
$$
x_{1}=2, \quad x_{2}=0, \quad x_{3}=1,
$$
but we were asked to find $\vec{x}$; therefore, we state the solution as
$$
\vec{x}=\left[\begin{array}{l}
2 \\
0 \\
1
\end{array}\right] .
$$
This probably seems all well and good. While asking one to solve the equation $A \vec{x}=\vec{b}$ for $\vec{x}$ seems like a new problem, in reality it is just asking that we solve a system of linear equations. Our variables $x_{1}$, etc., appear not individually but as the entries of our vector $\vec{x}$. We are simply writing an old problem in a new way.
In line with this new way of writing the problem, we have a new way of writing the solution. Instead of listing, individually, the values of the unknowns, we simply list them as the elements of our vector $\vec{x}$.
These are important ideas, so we state the basic principle once more: solving the equation $A \vec{x}=\vec{b}$ for $\vec{x}$ is the same thing as solving a linear system of equations. Equivalently, any system of linear equations can be written in the form $A \vec{x}=\vec{b}$ for some matrix $A$ and vector $\vec{b}$.
Since these ideas are equivalent, we'll refer to $A \vec{x}=\vec{b}$ both as a matrix-vector equation and as a system of linear equations: they are the same thing.
We've seen two examples illustrating this idea so far, and in both cases the linear system had exactly one solution. We know from Theorem 1 that any linear system has either one solution, infinite solutions, or no solution. So how does our new method of writing a solution work with infinite solutions and no solutions?
Certainly, if $A \vec{x}=\vec{b}$ has no solution, we simply say that the linear system has no solution. There isn't anything special to write. So the only other option to consider is the case where we have infinite solutions. We'll learn how to handle these situations through examples.
Example $44 \quad$ Solve the linear system $A \vec{x}=\vec{O}$ for $\vec{x}$ and write the solution in vector form, where
$$
A=\left[\begin{array}{ll}
1 & 2 \\
2 & 4
\end{array}\right] \text { and } \quad \vec{O}=\left[\begin{array}{l}
0 \\
0
\end{array}\right]
$$
Solution (Note: we didn't really need to specify that
$$
\vec{O}=\left[\begin{array}{l}
0 \\
0
\end{array}\right]
$$
but we did just to eliminate any uncertainty.)
To solve this system, put the augmented matrix into reduced row echelon form, which we do below.
$$
\left[\begin{array}{lll}
1 & 2 & 0 \\
2 & 4 & 0
\end{array}\right] \quad \overrightarrow{\operatorname{rref}}\left[\begin{array}{lll}
1 & 2 & 0 \\
0 & 0 & 0
\end{array}\right]
$$
We interpret the reduced row echelon form of this matrix to write the solution as
$$
\begin{aligned}
& x_{1}=-2 x_{2} \\
& x_{2} \text { is free. }
\end{aligned}
$$
We are not done; we need to write the solution in vector form, for our solution is the vector $\vec{x}$. Recall that
$$
\vec{x}=\left[\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right]
$$
From above we know that $x_{1}=-2 x_{2}$, so we replace the $x_{1}$ in $\vec{x}$ with $-2 x_{2}$. This gives our solution as
$$
\vec{x}=\left[\begin{array}{c}
-2 x_{2} \\
x_{2}
\end{array}\right]
$$
Now we pull the $x_{2}$ out of the vector (it is just a scalar) and write $\vec{x}$ as
$$
\vec{x}=x_{2}\left[\begin{array}{c}
-2 \\
1
\end{array}\right]
$$
For reasons that will become more clear later, set
$$
\vec{v}=\left[\begin{array}{c}
-2 \\
1
\end{array}\right]
$$
Thus our solution can be written as
$$
\vec{x}=x_{2} \vec{v}
$$
Recall that since our system was consistent and had a free variable, we have infinite solutions. This form of the solution highlights this fact; pick any value for $x_{2}$ and we get a different solution.
For instance, by setting $x_{2}=-1,0$, and 5 , we get the solutions
$$
\vec{x}=\left[\begin{array}{c}
2 \\
-1
\end{array}\right], \quad\left[\begin{array}{l}
0 \\
0
\end{array}\right], \quad \text { and } \quad\left[\begin{array}{c}
-10 \\
5
\end{array}\right]
$$
respectively.
We should check our work; multiply each of the above vectors by $A$ to see if we indeed get $\overrightarrow{0}$.
We have officially solved this problem; we have found the solution to $A \vec{x}=\overrightarrow{0}$ and written it properly. One final thing we will do here is graph the solution, using our skills learned in the previous section.
Our solution is
$$
\vec{x}=x_{2}\left[\begin{array}{c}
-2 \\
1
\end{array}\right]
$$
This means that any scalar multiply of the vector $\vec{v}=\left[\begin{array}{c}-2 \\ 1\end{array}\right]$ is a solution; we know how to sketch the scalar multiples of $\vec{v}$. This is done in Figure 2.18.
Figure 2.18: The solution, as a line, to $A \vec{x}=\vec{O}$ in Example 44 .
Here vector $\vec{v}$ is drawn as well as the line that goes through the origin in the direction of $\vec{v}$. Any vector along this line is a solution. So in some sense, we can say that the solution to $A \vec{x}=\vec{O}$ is a line.
Let's practice this again.
Example $45 \quad$ Solve the linear system $A \vec{x}=\vec{O}$ and write the solution in vector form, where
$$
A=\left[\begin{array}{cc}
2 & -3 \\
-2 & 3
\end{array}\right]
$$
Solution Again, to solve this problem, we form the proper augmented matrix and we put it into reduced row echelon form, which we do below.
$$
\left[\begin{array}{ccc}
2 & -3 & 0 \\
-2 & 3 & 0
\end{array}\right] \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{ccc}
1 & -3 / 2 & 0 \\
0 & 0 & 0
\end{array}\right]
$$
We interpret the reduced row echelon form of this matrix to find that
$$
\begin{aligned}
& x_{1}=3 / 2 x_{2} \\
& x_{2} \text { is free. }
\end{aligned}
$$
As before,
$$
\vec{x}=\left[\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right] .
$$
Since $x_{1}=3 / 2 x_{2}$, we replace $x_{1}$ in $\vec{x}$ with $3 / 2 x_{2}$ :
$$
\vec{x}=\left[\begin{array}{c}
3 / 2 x_{2} \\
x_{2}
\end{array}\right] .
$$
Now we pull out the $x_{2}$ and write the solution as
$$
\vec{x}=x_{2}\left[\begin{array}{c}
3 / 2 \\
1
\end{array}\right] \text {. }
$$
As before, let's set
$$
\vec{v}=\left[\begin{array}{c}
3 / 2 \\
1
\end{array}\right]
$$
so we can write our solution as
$$
\vec{x}=x_{2} \vec{v} .
$$
Again, we have infinite solutions; any choice of $x_{2}$ gives us one of these solutions. For instance, picking $x_{2}=2$ gives the solution
$$
\vec{x}=\left[\begin{array}{l}
3 \\
2
\end{array}\right] .
$$
(This is a particularly nice solution, since there are no fractions. ..)
As in the previous example, our solutions are multiples of a vector, and hence we can graph this, as done in Figure 2.19.
Figure 2.19: The solution, as a line, to $A \vec{x}=\vec{O}$ in Example 45 .
Let's practice some more; this time, we won't solve a system of the form $A \vec{x}=\overrightarrow{0}$, but instead $A \vec{x}=\vec{b}$, for some vector $\vec{b}$.
Example $46 \quad$ Solve the linear system $A \vec{x}=\vec{b}$, where
$$
A=\left[\begin{array}{ll}
1 & 2 \\
2 & 4
\end{array}\right] \text { and } \quad \vec{b}=\left[\begin{array}{l}
3 \\
6
\end{array}\right]
$$
Solution (Note that this is the same matrix $A$ that we used in Example 44. This will be important later.)
Our methodology is the same as before; we form the augmented matrix and put it into reduced row echelon form.
$$
\left[\begin{array}{lll}
1 & 2 & 3 \\
2 & 4 & 6
\end{array}\right] \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{lll}
1 & 2 & 3 \\
0 & 0 & 0
\end{array}\right]
$$
Interpreting this reduced row echelon form, we find that
$$
\begin{aligned}
& x_{1}=3-2 x_{2} \\
& x_{2} \text { is free. }
\end{aligned}
$$
Again,
$$
\vec{x}=\left[\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right],
$$
and we replace $x_{1}$ with $3-2 x_{2}$, giving
$$
\vec{x}=\left[\begin{array}{c}
3-2 x_{2} \\
x_{2}
\end{array}\right] .
$$
This solution is different than what we've seen in the past two examples; we can't simply pull out a $x_{2}$ since there is a 3 in the first entry. Using the properties of matrix addition, we can "pull apart" this vector and write it as the sum of two vectors: one which contains only constants, and one that contains only " $x_{2}$ stuff." We do this below.
$$
\begin{aligned}
\vec{x} & =\left[\begin{array}{c}
3-2 x_{2} \\
x_{2}
\end{array}\right] \\
& =\left[\begin{array}{l}
3 \\
0
\end{array}\right]+\left[\begin{array}{c}
-2 x_{2} \\
x_{2}
\end{array}\right] \\
& =\left[\begin{array}{l}
3 \\
0
\end{array}\right]+x_{2}\left[\begin{array}{c}
-2 \\
1
\end{array}\right] .
\end{aligned}
$$
Once again, let's give names to the different component vectors of this solution (we are getting near the explanation of why we are doing this). Let
$$
\overrightarrow{x_{p}}=\left[\begin{array}{l}
3 \\
0
\end{array}\right] \quad \text { and } \quad \vec{v}=\left[\begin{array}{c}
-2 \\
1
\end{array}\right]
$$
We can then write our solution in the form
$$
\vec{x}=\overrightarrow{x_{p}}+x_{2} \vec{v} .
$$
We still have infinite solutions; by picking a value for $x_{2}$ we get one of these solutions. For instance, by letting $x_{2}=-1,0$, or 2 , we get the solutions
$$
\left[\begin{array}{c}
5 \\
-1
\end{array}\right], \quad\left[\begin{array}{l}
3 \\
0
\end{array}\right] \text { and }\left[\begin{array}{c}
-1 \\
2
\end{array}\right]
$$
We have officially solved the problem; we have solved the equation $A \vec{x}=\vec{b}$ for $\vec{x}$ and have written the solution in vector form. As an additional visual aid, we will graph this solution.
Each vector in the solution can be written as the sum of two vectors: $\overrightarrow{x_{p}}$ and a multiple of $\vec{v}$. In Figure 2.20, $\overrightarrow{x_{p}}$ is graphed and $\vec{v}$ is graphed with its origin starting at the tip of $\overrightarrow{x_{p}}$. Finally, a line is drawn in the direction of $\vec{v}$ from the tip of $\overrightarrow{x_{p}}$; any vector pointing to any point on this line is a solution to $A \vec{x}=\vec{b}$.
Figure 2.20: The solution, as a line, to $A \vec{x}=\vec{b}$ in Example 46 .
The previous examples illustrate some important concepts. One is that we can "see" the solution to a system of linear equations in a new way. Before, when we had infinite solutions, we knew we could arbitrarily pick values for our free variables and get different solutions. We knew this to be true, and we even practiced it, but the result was not very "tangible." Now, we can view our solution as a vector; by picking different values for our free variables, we see this as multiplying certain important vectors by a scalar which gives a different solution.
Another important concept that these examples demonstrate comes from the fact that Examples 44 and 46 were only "slightly different" and hence had only "slightly different" answers. Both solutions had
$$
x_{2}\left[\begin{array}{c}
-2 \\
1
\end{array}\right]
$$
in them; in Example 46 the solution also had another vector added to this. Was this coincidence, or is there a definite pattern here?
Of course there is a pattern! Now ... what exactly is it? First, we define a term.
Definition 17
## Homogeneous Linear System of Equations
A system of linear equations is homogeneous if the constants in each equation are zero.
Note: a homogeneous system of equations can be written in vector form as $A \vec{x}=\overrightarrow{0}$.
The term homogeneous comes from two Greek words; homo meaning "same" and genus meaning "type." A homogeneous system of equations is a system in which each equation is of the same type - all constants are 0 . Notice that the system of equations in Examples 44 and 46 are homogeneous.
Note that $A \vec{O}=\overrightarrow{0}$; that is, if we set $\vec{x}=\overrightarrow{0}$, we have a solution to a homogeneous set of equations. This fact is important; the zero vector is always a solution to a homogeneous linear system. Therefore a homogeneous system is always consistent; we need only to determine whether we have exactly one solution (just $\overrightarrow{0}$ ) or infinite solutions. This idea is important so we give it it's own box.
## Key Idea 7
## Homogeneous Systems and Consistency
All homogeneous linear systems are consistent.
How do we determine if we have exactly one or infinite solutions? Recall Key Idea 2: if the solution has any free variables, then it will have infinite solutions. How can we tell if the system has free variables? Form the augmented matrix $\left[\begin{array}{ll}A & \overrightarrow{0}\end{array}\right]$, put it into reduced row echelon form, and interpret the result.
It may seem that we've brought up a new question, "When does $A \vec{x}=\vec{O}$ have exactly one or infinite solutions?" only to answer with "Look at the reduced row echelon form of $A$ and interpret the results, just as always." Why bring up a new question if the answer is an old one?
While the new question has an old solution, it does lead to a great idea. Let's refresh our memory; earlier we solved two linear systems,
$$
A \vec{x}=\overrightarrow{0} \text { and } \quad A \vec{x}=\vec{b}
$$
where
$$
A=\left[\begin{array}{ll}
1 & 2 \\
2 & 4
\end{array}\right] \text { and } \vec{b}=\left[\begin{array}{l}
3 \\
6
\end{array}\right] .
$$
The solution to the first system of equations, $A \vec{x}=\overrightarrow{0}$, is
$$
\vec{x}=x_{2}\left[\begin{array}{c}
-2 \\
1
\end{array}\right]
$$
and the solution to the second set of equations, $A \vec{x}=\vec{b}$, is
$$
\vec{x}=\left[\begin{array}{l}
3 \\
0
\end{array}\right]+x_{2}\left[\begin{array}{c}
-2 \\
1
\end{array}\right]
$$
for all values of $x_{2}$.
Recalling our notation used earlier, set
$$
\overrightarrow{x_{p}}=\left[\begin{array}{l}
3 \\
0
\end{array}\right] \text { and let } \quad \vec{v}=\left[\begin{array}{c}
-2 \\
1
\end{array}\right] .
$$
Thus our solution to the linear system $A \vec{x}=\vec{b}$ is
$$
\vec{x}=\overrightarrow{x_{p}}+x_{2} \vec{v}
$$
Let us see how exactly this solution works; let's see why $A \vec{x}$ equals $\vec{b}$. Multiply $A \vec{x}$ :
$$
\begin{aligned}
A \vec{x} & =A\left(\overrightarrow{x_{p}}+x_{2} \vec{v}\right) \\
& =A \overrightarrow{x_{p}}+A\left(x_{2} \vec{v}\right) \\
& =A \overrightarrow{x_{p}}+x_{2}(A \vec{v}) \\
& =A \overrightarrow{x_{p}}+x_{2} \vec{O} \\
& =A \overrightarrow{x_{p}}+\overrightarrow{0} \\
& =A \overrightarrow{x_{p}} \\
& =\vec{b}
\end{aligned}
$$
We know that the last line is true, that $\overrightarrow{A \overrightarrow{x_{p}}}=\vec{b}$, since we know that $\vec{x}$ was a solution to $A \vec{x}=\vec{b}$. The whole point is that $\overrightarrow{x_{p}}$ itself is a solution to $A \vec{x}=\vec{b}$, and we could find more solutions by adding vectors "that go to zero" when multiplied by $A$. (The subscript $p$ of " $\overrightarrow{x_{p}}$ " is used to denote that this vector is a "particular" solution.)
Stated in a different way, let's say that we know two things: that $A \overrightarrow{x_{p}}=\vec{b}$ and $A \vec{v}=\overrightarrow{0}$. What is $A\left(\overrightarrow{x_{p}}+\vec{v}\right)$ ? We can multiply it out:
$$
\begin{aligned}
A\left(\overrightarrow{x_{p}}+\vec{v}\right) & =A \overrightarrow{x_{p}}+A \vec{v} \\
& =\vec{b}+\overrightarrow{0} \\
& =\vec{b}
\end{aligned}
$$
and see that $A\left(\overrightarrow{x_{p}}+\vec{v}\right)$ also equals $\vec{b}$.
So we wonder: does this mean that $A \vec{x}=\vec{b}$ will have infinite solutions? After all, if $\overrightarrow{x_{p}}$ and $\overrightarrow{x_{p}}+\vec{v}$ are both solutions, don't we have infinite solutions?
No. If $A \vec{x}=\vec{O}$ has exactly one solution, then $\vec{v}=\overrightarrow{0}$, and $\overrightarrow{x_{p}}=\overrightarrow{x_{p}}+\vec{v}$; we only have one solution.
So here is the culmination of all of our fun that started a few pages back. If $\vec{v}$ is a solution to $A \vec{x}=\vec{O}$ and $\overrightarrow{x_{p}}$ is a solution to $A \vec{x}=\vec{b}$, then $\overrightarrow{x_{p}}+\vec{v}$ is also a solution to $A \vec{x}=\vec{b}$. If $A \vec{x}=\vec{O}$ has infinite solutions, so does $A \vec{x}=\vec{b}$; if $A \vec{x}=\vec{O}$ has only one solution, so does $A \vec{x}=\vec{b}$. This culminating idea is of course important enough to be stated again. Key Idea 8
A key word in the above statement is consistent. If $A \vec{x}=\vec{b}$ is inconsistent (the linear system has no solution), then it doesn't matter how many solutions $A \vec{x}=\vec{O}$ has; $A \vec{x}=\vec{b}$ has no solution.
Enough fun, enough theory. We need to practice.
Example $47 \quad$ Let
$$
A=\left[\begin{array}{cccc}
1 & -1 & 1 & 3 \\
4 & 2 & 4 & 6
\end{array}\right] \text { and } \vec{b}=\left[\begin{array}{c}
1 \\
10
\end{array}\right]
$$
Solve the linear systems $A \vec{x}=\vec{O}$ and $A \vec{x}=\vec{b}$ for $\vec{x}$, and write the solutions in vector form.
Solution We'll tackle $A \vec{x}=\vec{O}$ first. We form the associated augmented matrix, put it into reduced row echelon form, and interpret the result.
$$
\begin{aligned}
& {\left[\begin{array}{ccccc}
1 & -1 & 1 & 3 & 0 \\
4 & 2 & 4 & 6 & 0
\end{array}\right] \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{ccccc}
1 & 0 & 1 & 2 & 0 \\
0 & 1 & 0 & -1 & 0
\end{array}\right]} \\
& x_{1}=-x_{3}-2 x_{4} \\
& x_{2}=x_{4} \\
& x_{3} \text { is free } \\
& x_{4} \text { is free }
\end{aligned}
$$
To write our solution in vector form, we rewrite $x_{1}$ and $x_{2}$ in $\vec{x}$ in terms of $x_{3}$ and $x_{4}$.
$$
\vec{x}=\left[\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4}
\end{array}\right]=\left[\begin{array}{c}
-x_{3}-2 x_{4} \\
x_{4} \\
x_{3} \\
x_{4}
\end{array}\right]
$$
Finally, we "pull apart" this vector into two vectors, one with the " $x_{3}$ stuff" and one with the " $x_{4}$ stuff."
$$
\begin{aligned}
\vec{x} & =\left[\begin{array}{c}
-x_{3}-2 x_{4} \\
x_{4} \\
x_{3} \\
x_{4}
\end{array}\right] \\
& =\left[\begin{array}{c}
-x_{3} \\
0 \\
x_{3} \\
0
\end{array}\right]+\left[\begin{array}{c}
-2 x_{4} \\
x_{4} \\
0 \\
x_{4}
\end{array}\right] \\
& =x_{3}\left[\begin{array}{c}
-1 \\
0 \\
1 \\
0
\end{array}\right]+x_{4}\left[\begin{array}{c}
-2 \\
1 \\
0 \\
1
\end{array}\right] \\
& =x_{3} \vec{u}+x_{4} \vec{v}
\end{aligned}
$$
We use $\vec{u}$ and $\vec{v}$ simply to give these vectors names (and save some space).
It is easy to confirm that both $\vec{u}$ and $\vec{v}$ are solutions to the linear system $A \vec{x}=\overrightarrow{0}$. (Just multiply $A \vec{u}$ and $A \vec{v}$ and see that both are $\overrightarrow{0}$.) Since both are solutions to a homogeneous system of linear equations, any linear combination of $\vec{u}$ and $\vec{v}$ will be a solution, too.
Now let's tackle $A \vec{x}=\vec{b}$. Once again we put the associated augmented matrix into reduced row echelon form and interpret the results.
$$
\begin{aligned}
& {\left[\begin{array}{ccccc}
1 & -1 & 1 & 3 & 1 \\
4 & 2 & 4 & 6 & 10
\end{array}\right] \quad \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{ccccc}
1 & 0 & 1 & 2 & 2 \\
0 & 1 & 0 & -1 & 1
\end{array}\right]} \\
& x_{1}=2-x_{3}-2 x_{4} \\
& x_{2}=1+x_{4} \\
& x_{3} \text { is free } \\
& x_{4} \text { is free }
\end{aligned}
$$
Writing this solution in vector form gives
$$
\vec{x}=\left[\begin{array}{l}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4}
\end{array}\right]=\left[\begin{array}{c}
2-x_{3}-2 x_{4} \\
1+x_{4} \\
x_{3} \\
x_{4}
\end{array}\right] .
$$
Again, we pull apart this vector, but this time we break it into three vectors: one with " $x_{3}$ " stuff, one with " $x_{4}$ " stuff, and one with just constants.
$$
\begin{aligned}
& \vec{x}=\left[\begin{array}{c}
2-x_{3}-2 x_{4} \\
1+x_{4} \\
x_{3} \\
x_{4}
\end{array}\right] \\
& =\left[\begin{array}{l}
2 \\
1 \\
0 \\
0
\end{array}\right]+\left[\begin{array}{c}
-x_{3} \\
0 \\
x_{3} \\
0
\end{array}\right]+\left[\begin{array}{c}
-2 x_{4} \\
x_{4} \\
0 \\
x_{4}
\end{array}\right] \\
& =\left[\begin{array}{l}
2 \\
1 \\
0 \\
0
\end{array}\right]+x_{3}\left[\begin{array}{c}
-1 \\
0 \\
1 \\
0
\end{array}\right]+x_{4}\left[\begin{array}{c}
-2 \\
1 \\
0 \\
1
\end{array}\right] \\
& =\underbrace{\overrightarrow{x_{p}}}_{\begin{array}{c}
\text { particular } \\
\text { solution }
\end{array}}+\underbrace{x_{3} \vec{u}+x_{4} \vec{v}}_{\begin{array}{c}
\text { solution to } \\
\text { homogeneous } \\
\text { equations } A \vec{x}=\overrightarrow{0}
\end{array}}
\end{aligned}
$$
Note that $A \overrightarrow{x_{p}}=\vec{b}$; by itself, $\overrightarrow{x_{p}}$ is a solution. To get infinite solutions, we add a bunch of stuff that "goes to zero" when we multiply by $A$; we add the solution to the homogeneous equations.
Why don't we graph this solution as we did in the past? Before we had only two variables, meaning the solution could be graphed in $2 \mathrm{D}$. Here we have four variables, meaning that our solution "lives" in 4D. You can draw this on paper, but it is very confusing.
Example $48 \quad$ Rewrite the linear system
$$
\begin{aligned}
& x_{1}+2 x_{2}-3 x_{3}+2 x_{4}+7 x_{5}=2 \\
& 3 x_{1}+4 x_{2}+5 x_{3}+2 x_{4}+3 x_{5}=-4
\end{aligned}
$$
as a matrix-vector equation, solve the system using vector notation, and give the solution to the related homogeneous equations.
Solution Rewriting the linear system in the form of $A \vec{x}=\vec{b}$, we have that
$$
A=\left[\begin{array}{ccccc}
1 & 2 & -3 & 2 & 7 \\
3 & 4 & 5 & 2 & 3
\end{array}\right], \quad \vec{x}=\left[\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4} \\
x_{5}
\end{array}\right] \quad \text { and } \quad \vec{b}=\left[\begin{array}{c}
2 \\
-4
\end{array}\right]
$$
To solve the system, we put the associated augmented matrix into reduced row echelon form and interpret the results.
$$
\begin{aligned}
& {\left[\begin{array}{cccccc}
1 & 2 & -3 & 2 & 7 & 2 \\
3 & 4 & 5 & 2 & 3 & -4
\end{array}\right] \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{cccccc}
1 & 0 & 11 & -2 & -11 & -8 \\
0 & 1 & -7 & 2 & 9 & 5
\end{array}\right]} \\
& x_{1}=-8-11 x_{3}+2 x_{4}+11 x_{5} \\
& x_{2}=5+7 x_{3}-2 x_{4}-9 x_{5} \\
& x_{3} \text { is free } \\
& x_{4} \text { is free } \\
& x_{5} \text { is free }
\end{aligned}
$$
We use this information to write $\vec{x}$, again pulling it apart. Since we have three free variables and also constants, we'll need to pull $\vec{x}$ apart into four separate vectors.
$$
\begin{aligned}
& \vec{x}=\left[\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4} \\
x_{5}
\end{array}\right] \\
& =\left[\begin{array}{c}
-8-11 x_{3}+2 x_{4}+11 x_{5} \\
5+7 x_{3}-2 x_{4}-9 x_{5} \\
x_{3} \\
x_{4} \\
x_{5}
\end{array}\right] \\
& =\left[\begin{array}{c}
-8 \\
5 \\
0 \\
0 \\
0
\end{array}\right]+\left[\begin{array}{c}
-11 x_{3} \\
7 x_{3} \\
x_{3} \\
0 \\
0
\end{array}\right]+\left[\begin{array}{c}
2 x_{4} \\
-2 x_{4} \\
0 \\
x_{4} \\
0
\end{array}\right]+\left[\begin{array}{c}
11 x_{5} \\
-9 x_{5} \\
0 \\
0 \\
x_{5}
\end{array}\right] \\
& =\left[\begin{array}{c}
-8 \\
5 \\
0 \\
0 \\
0
\end{array}\right]+x_{3}\left[\begin{array}{c}
-11 \\
7 \\
1 \\
0 \\
0
\end{array}\right]+x_{4}\left[\begin{array}{c}
2 \\
-2 \\
0 \\
1 \\
0
\end{array}\right]+x_{5}\left[\begin{array}{c}
11 \\
-9 \\
0 \\
0 \\
1
\end{array}\right] \\
& =\underbrace{\overrightarrow{x_{p}}}_{\begin{array}{c}
\text { particular } \\
\text { solution }
\end{array}}+\underbrace{x_{3} \vec{u}+x_{4} \vec{v}+x_{5} \vec{w}}_{\begin{array}{c}
\text { solution to homogeneous } \\
\text { equations } A \vec{x}=\overrightarrow{0}
\end{array}}
\end{aligned}
$$
So $\overrightarrow{x_{p}}$ is a particular solution; $A \overrightarrow{x_{p}}=\vec{b}$. (Multiply it out to verify that this is true.) The other vectors, $\vec{u}, \vec{v}$ and $\vec{w}$, that are multiplied by our free variables $x_{3}, x_{4}$ and $x_{5}$, are each solutions to the homogeneous equations, $A \vec{x}=\overrightarrow{0}$. Any linear combination of these three vectors, i.e., any vector found by choosing values for $x_{3}, x_{4}$ and $x_{5}$ in $x_{3} \vec{u}+x_{4} \vec{v}+x_{5} \vec{w}$ is a solution to $A \vec{x}=\overrightarrow{0}$. Example $49 \quad$ Let
$$
A=\left[\begin{array}{ll}
1 & 2 \\
4 & 5
\end{array}\right] \quad \text { and } \quad \vec{b}=\left[\begin{array}{l}
3 \\
6
\end{array}\right]
$$
Find the solutions to $A \vec{x}=\vec{b}$ and $A \vec{x}=\overrightarrow{0}$.
Solution We go through the familiar work of finding the reduced row echelon form of the appropriate augmented matrix and interpreting the solution.
$$
\begin{gathered}
{\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6
\end{array}\right] \stackrel{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{ccc}
1 & 0 & -1 \\
0 & 1 & 2
\end{array}\right]} \\
x_{1}=-1 \\
x_{2}=2
\end{gathered}
$$
Thus
$$
\vec{x}=\left[\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right]=\left[\begin{array}{c}
-1 \\
2
\end{array}\right] .
$$
This may strike us as a bit odd; we are used to having lots of different vectors in the solution. However, in this case, the linear system $A \vec{x}=\vec{b}$ has exactly one solution, and we've found it. What is the solution to $A \vec{x}=\overrightarrow{0}$ ? Since we've only found one solution to $A \vec{x}=\vec{b}$, we can conclude from Key Idea 8 the related homogeneous equations $A \vec{x}=\overrightarrow{0}$ have only one solution, namely $\vec{x}=\overrightarrow{0}$. We can write our solution vector $\vec{x}$ in a form similar to our previous examples to highlight this:
$$
\begin{aligned}
\vec{x} & =\left[\begin{array}{c}
-1 \\
2
\end{array}\right] \\
& =\left[\begin{array}{c}
-1 \\
2
\end{array}\right]+\left[\begin{array}{l}
0 \\
0
\end{array}\right] \\
& =\underbrace{\overrightarrow{x_{p}}}_{\substack{\text { particular } \\
\text { solution }}}+\underbrace{\overrightarrow{0}}_{\substack{\text { solution to } \\
A \vec{x}=\overrightarrow{0}}} .
\end{aligned}
$$
Example 50 Let
$$
A=\left[\begin{array}{ll}
1 & 1 \\
2 & 2
\end{array}\right] \text { and } \quad \vec{b}=\left[\begin{array}{l}
1 \\
1
\end{array}\right]
$$
Find the solutions to $A \vec{x}=\vec{b}$ and $A \vec{x}=\overrightarrow{0}$.
Solution To solve $A \vec{x}=\vec{b}$, we put the appropriate augmented matrix into reduced row echelon form and interpret the results.
$$
\left[\begin{array}{lll}
1 & 1 & 1 \\
2 & 2 & 1
\end{array}\right] \quad \overrightarrow{\operatorname{rref}}\left[\begin{array}{lll}
1 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]
$$
We immediately have a problem; we see that the second row tells us that $0 x_{1}+$ $0 x_{2}=1$, the sign that our system does not have a solution. Thus $A \vec{x}=\vec{b}$ has no solution. Of course, this does not mean that $A \vec{x}=\vec{O}$ has no solution; it always has a solution.
To find the solution to $A \vec{x}=\overrightarrow{0}$, we interpret the reduced row echelon form of the appropriate augmented matrix.
$$
\begin{gathered}
{\left[\begin{array}{lll}
1 & 1 & 0 \\
2 & 2 & 0
\end{array}\right] \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{lll}
1 & 1 & 0 \\
0 & 0 & 0
\end{array}\right]} \\
x_{1}=-x_{2} \\
x_{2} \text { is free }
\end{gathered}
$$
Thus
$$
\begin{aligned}
\vec{x} & =\left[\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right] \\
& =\left[\begin{array}{c}
-x_{2} \\
x_{2}
\end{array}\right] \\
& =x_{2}\left[\begin{array}{c}
-1 \\
1
\end{array}\right] \\
& =x_{2} \vec{u} .
\end{aligned}
$$
We have no solution to $A \vec{x}=\vec{b}$, but infinite solutions to $A \vec{x}=\overrightarrow{0}$.
The previous example may seem to violate the principle of Key Idea 8. After all, it seems that having infinite solutions to $A \vec{x}=\vec{O}$ should imply infinite solutions to $A \vec{x}=\vec{b}$. However, we remind ourselves of the key word in the idea that we observed before: consistent. If $A \vec{x}=\vec{b}$ is consistent and $A \vec{x}=\vec{O}$ has infinite solutions, then so will $A \vec{x}=\vec{b}$. But if $A \vec{x}=\vec{b}$ is not consistent, it does not matter how many solutions $A \vec{x}=\vec{O}$ has; $A \vec{x}=\vec{b}$ is still inconsistent.
This whole section is highlighting a very important concept that we won't fully understand until after two sections, but we get a glimpse of it here. When solving any system of linear equations (which we can write as $A \vec{x}=\vec{b}$ ), whether we have exactly one solution, infinite solutions, or no solution depends on an intrinsic property of $A$. We'll find out what that property is soon; in the next section we solve a problem we introduced at the beginning of this section, how to solve matrix equations $A X=B$.
## Exercises 2.4
In Exercises 1-6, a matrix $A$ and vectors $\vec{b}, \vec{u}$ and $\vec{v}$ are given. Verify that $\vec{u}$ and $\vec{v}$ are both solutions to the equation $A \vec{x}=\vec{b}$; that is, show that $A \vec{u}=A \vec{v}=\vec{b}$.
$$
\text { 1. } \begin{aligned}
A & =\left[\begin{array}{cc}
1 & -2 \\
-3 & 6
\end{array}\right] \\
\vec{b} & =\left[\begin{array}{l}
0 \\
0
\end{array}\right], \vec{u}=\left[\begin{array}{l}
2 \\
1
\end{array}\right], \vec{v}=\left[\begin{array}{c}
-10 \\
-5
\end{array}\right]
\end{aligned}
$$
2. $A=\left[\begin{array}{cc}1 & -2 \\ -3 & 6\end{array}\right]$,
$\vec{b}=\left[\begin{array}{c}2 \\ -6\end{array}\right], \vec{u}=\left[\begin{array}{c}0 \\ -1\end{array}\right], \vec{v}=\left[\begin{array}{l}2 \\ 0\end{array}\right]$
3. $A=\left[\begin{array}{ll}1 & 0 \\ 2 & 0\end{array}\right]$,
$\vec{b}=\left[\begin{array}{l}0 \\ 0\end{array}\right], \vec{u}=\left[\begin{array}{c}0 \\ -1\end{array}\right], \vec{v}=\left[\begin{array}{c}0 \\ 59\end{array}\right]$
4. $A=\left[\begin{array}{ll}1 & 0 \\ 2 & 0\end{array}\right]$,
$\vec{b}=\left[\begin{array}{l}-3 \\ -6\end{array}\right], \vec{u}=\left[\begin{array}{l}-3 \\ -1\end{array}\right], \vec{v}=\left[\begin{array}{c}-3 \\ 59\end{array}\right]$
5. $A=\left[\begin{array}{cccc}0 & -3 & -1 & -3 \\ -4 & 2 & -3 & 5\end{array}\right]$,
$\vec{b}=\left[\begin{array}{l}0 \\ 0\end{array}\right], \vec{u}=\left[\begin{array}{c}11 \\ 4 \\ -12 \\ 0\end{array}\right]$,
$\vec{v}=\left[\begin{array}{c}9 \\ -12 \\ 0 \\ 12\end{array}\right]$
6. $A=\left[\begin{array}{cccc}0 & -3 & -1 & -3 \\ -4 & 2 & -3 & 5\end{array}\right]$,
$\vec{b}=\left[\begin{array}{l}48 \\ 36\end{array}\right], \vec{u}=\left[\begin{array}{c}-17 \\ -16 \\ 0 \\ 0\end{array}\right]$,
$\vec{v}=\left[\begin{array}{c}-8 \\ -28 \\ 0 \\ 12\end{array}\right]$
In Exercises 7-9, a matrix $A$ and vectors $\vec{b}, \vec{u}$ and $\vec{v}$ are given. Verify that $A \vec{u}=\overrightarrow{0}, A \vec{v}=\vec{b}$ and $A(\vec{u}+\vec{v})=\vec{b}$.
7. $A=\left[\begin{array}{ccc}2 & -2 & -1 \\ -1 & 1 & -1 \\ -2 & 2 & -1\end{array}\right]$,
$\vec{b}=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right], \vec{u}=\left[\begin{array}{l}1 \\ 1 \\ 0\end{array}\right], \vec{v}=\left[\begin{array}{c}1 \\ 1 \\ -1\end{array}\right]$
8. $A=\left[\begin{array}{ccc}1 & -1 & 3 \\ 3 & -3 & -3 \\ -1 & 1 & 1\end{array}\right]$, $\vec{b}=\left[\begin{array}{c}-1 \\ -3 \\ 1\end{array}\right], \vec{u}=\left[\begin{array}{l}2 \\ 2 \\ 0\end{array}\right], \vec{v}=\left[\begin{array}{l}2 \\ 3 \\ 0\end{array}\right]$
9. $A=\left[\begin{array}{ccc}2 & 0 & 0 \\ 0 & 1 & -3 \\ 3 & 1 & -3\end{array}\right]$,
$\vec{b}=\left[\begin{array}{c}2 \\ -4 \\ -1\end{array}\right], \vec{u}=\left[\begin{array}{l}0 \\ 6 \\ 2\end{array}\right], \vec{v}=\left[\begin{array}{c}1 \\ -1 \\ 1\end{array}\right]$
In Exercises $10-24$, a matrix $A$ and vector $\vec{b}$ are given.
(a) Solve the equation $A \vec{x}=\overrightarrow{0}$.
(b) Solve the equation $A \vec{x}=\vec{b}$.
In each of the above, be sure to write your answer in vector format. Also, when possible, give 2 particular solutions to each equation.
10. $A=\left[\begin{array}{cc}0 & 2 \\ -1 & 3\end{array}\right], \vec{b}=\left[\begin{array}{l}-2 \\ -1\end{array}\right]$
11. $A=\left[\begin{array}{ll}-4 & -1 \\ -3 & -2\end{array}\right], \vec{b}=\left[\begin{array}{l}1 \\ 4\end{array}\right]$
12. $A=\left[\begin{array}{cc}1 & -2 \\ 0 & 1\end{array}\right], \vec{b}=\left[\begin{array}{c}0 \\ -5\end{array}\right]$
13. $A=\left[\begin{array}{cc}1 & 0 \\ 5 & -4\end{array}\right], \vec{b}=\left[\begin{array}{l}-2 \\ -1\end{array}\right]$
14. $A=\left[\begin{array}{cc}2 & -3 \\ -4 & 6\end{array}\right], \vec{b}=\left[\begin{array}{c}1 \\ -1\end{array}\right]$
15. $A=\left[\begin{array}{lll}-4 & 3 & 2 \\ -4 & 5 & 0\end{array}\right], \vec{b}=\left[\begin{array}{l}-4 \\ -4\end{array}\right]$
16. $A=\left[\begin{array}{ccc}1 & 5 & -2 \\ 1 & 4 & 5\end{array}\right], \vec{b}=\left[\begin{array}{l}0 \\ 1\end{array}\right]$
17. $A=\left[\begin{array}{ccc}-1 & -2 & -2 \\ 3 & 4 & -2\end{array}\right], \vec{b}=\left[\begin{array}{l}-4 \\ -4\end{array}\right]$
18. $A=\left[\begin{array}{ccc}2 & 2 & 2 \\ 5 & 5 & -3\end{array}\right], \vec{b}=\left[\begin{array}{c}3 \\ -3\end{array}\right]$
19. $A=\left[\begin{array}{cccc}1 & 5 & -4 & -1 \\ 1 & 0 & -2 & 1\end{array}\right]$,
$\vec{b}=\left[\begin{array}{c}0 \\ -2\end{array}\right]$
20. $A=\left[\begin{array}{cccc}-4 & 2 & -5 & 4 \\ 0 & 1 & -1 & 5\end{array}\right]$,
$\vec{b}=\left[\begin{array}{l}-3 \\ -2\end{array}\right]$ 21. $A=\left[\begin{array}{ccccc}0 & 0 & 2 & 1 & 4 \\ -2 & -1 & -4 & -1 & 5\end{array}\right]$ $\vec{b}=\left[\begin{array}{l}3 \\ 4\end{array}\right]$
22. $A=\left[\begin{array}{ccccc}3 & 0 & -2 & -4 & 5 \\ 2 & 3 & 2 & 0 & 2 \\ -5 & 0 & 4 & 0 & 5\end{array}\right]$, $\vec{b}=\left[\begin{array}{c}-1 \\ -5 \\ 4\end{array}\right]$
23. $A=\left[\begin{array}{ccccc}-1 & 3 & 1 & -3 & 4 \\ 3 & -3 & -1 & 1 & -4 \\ -2 & 3 & -2 & -3 & 1\end{array}\right]$, $\vec{b}=\left[\begin{array}{c}1 \\ 1 \\ -5\end{array}\right]$
24. $A=\left[\begin{array}{ccccc}-4 & -2 & -1 & 4 & 0 \\ 5 & -4 & 3 & -1 & 1 \\ 4 & -5 & 3 & 1 & -4\end{array}\right]$,
$$
\vec{b}=\left[\begin{array}{l}
3 \\
2 \\
1
\end{array}\right]
$$
In Exercises $25-28$, a matrix $A$ and vector $\vec{b}$ are given. Solve the equation $A \vec{x}=\vec{b}$, write the solution in vector format, and sketch the solution as the appropriate line on the Cartesian plane.
25. $A=\left[\begin{array}{cc}2 & 4 \\ -1 & -2\end{array}\right], \vec{b}=\left[\begin{array}{l}0 \\ 0\end{array}\right]$
26. $A=\left[\begin{array}{cc}2 & 4 \\ -1 & -2\end{array}\right], \vec{b}=\left[\begin{array}{c}-6 \\ 3\end{array}\right]$
27. $A=\left[\begin{array}{cc}2 & -5 \\ -4 & -10\end{array}\right], \vec{b}=\left[\begin{array}{l}1 \\ 2\end{array}\right]$
28. $A=\left[\begin{array}{cc}2 & -5 \\ -4 & -10\end{array}\right], \vec{b}=\left[\begin{array}{l}0 \\ 0\end{array}\right]$
### Solving Matrix Equations $A X=B$
## AS YOU READ
1. $\mathrm{T} / \mathrm{F}$ : To solve the matrix equation $A X=B$, put the matrix $\left[\begin{array}{ll}A & X\end{array}\right]$ into reduced row echelon form and interpret the result properly.
2. T/F: The first column of a matrix product $A B$ is $A$ times the first column of $B$.
3. Give two reasons why one might solve for the columns of $X$ in the equation $A X=B$ separately.
We began last section talking about solving numerical equations like $a x=b$ for $x$. We mentioned that solving matrix equations of the form $A X=B$ is of interest, but we first learned how to solve the related, but simpler, equations $A \vec{x}=\vec{b}$. In this section we will learn how to solve the general matrix equation $A X=B$ for $X$.
We will start by considering the best case scenario when solving $A \vec{x}=\vec{b}$; that is, when $A$ is square and we have exactly one solution. For instance, suppose we want to solve $A \vec{x}=\vec{b}$ where
$$
A=\left[\begin{array}{ll}
1 & 1 \\
2 & 1
\end{array}\right] \quad \text { and } \quad \vec{b}=\left[\begin{array}{l}
0 \\
1
\end{array}\right]
$$
We know how to solve this; put the appropriate matrix into reduced row echelon form and interpret the result.
$$
\left[\begin{array}{lll}
1 & 1 & 0 \\
2 & 1 & 1
\end{array}\right] \quad \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{ccc}
1 & 0 & 1 \\
0 & 1 & -1
\end{array}\right]
$$
We read from this that
$$
\vec{x}=\left[\begin{array}{c}
1 \\
-1
\end{array}\right]
$$
Written in a more general form, we found our solution by forming the augmented matrix
$$
\left[\begin{array}{ll}
A & \vec{b}
\end{array}\right]
$$
and interpreting its reduced row echelon form:
$$
\left[\begin{array}{ll}
A & \vec{b}
\end{array}\right] \quad \overrightarrow{\operatorname{rref}}\left[\begin{array}{ll}
l & \vec{x}
\end{array}\right]
$$
Notice that when the reduced row echelon form of $A$ is the identity matrix I we have exactly one solution. This, again, is the best case scenario.
We apply the same general technique to solving the matrix equation $A X=B$ for $X$. We'll assume that $A$ is a square matrix ( $B$ need not be) and we'll form the augmented matrix
$$
\left[\begin{array}{ll}
A & B
\end{array}\right] .
$$
Putting this matrix into reduced row echelon form will give us $X$, much like we found $\vec{x}$ before.
$$
\left[\begin{array}{ll}
A & B
\end{array}\right] \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{ll}
I & X
\end{array}\right]
$$
As long as the reduced row echelon form of $A$ is the identity matrix, this technique works great. After a few examples, we'll discuss why this technique works, and we'll also talk just a little bit about what happens when the reduced row echelon form of $A$ is not the identity matrix.
First, some examples.
Example $51 \quad$ Solve the matrix equation $A X=B$ where
$$
A=\left[\begin{array}{cc}
1 & -1 \\
5 & 3
\end{array}\right] \quad \text { and } \quad B=\left[\begin{array}{ccc}
-8 & -13 & 1 \\
32 & -17 & 21
\end{array}\right]
$$
Solution To solve $A X=B$ for $X$, we form the proper augmented matrix, put it into reduced row echelon form, and interpret the result.
$$
\left[\begin{array}{ccccc}
1 & -1 & -8 & -13 & 1 \\
5 & 3 & 32 & -17 & 21
\end{array}\right] \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{ccccc}
1 & 0 & 1 & -7 & 3 \\
0 & 1 & 9 & 6 & 2
\end{array}\right]
$$
We read from the reduced row echelon form of the matrix that
$$
X=\left[\begin{array}{ccc}
1 & -7 & 3 \\
9 & 6 & 2
\end{array}\right]
$$
We can easily check to see if our answer is correct by multiplying $A X$.
Example 52 Solve the matrix equation $A X=B$ where
$$
A=\left[\begin{array}{ccc}
1 & 0 & 2 \\
0 & -1 & -2 \\
2 & -1 & 0
\end{array}\right] \text { and } B=\left[\begin{array}{cc}
-1 & 2 \\
2 & -6 \\
2 & -4
\end{array}\right]
$$
Solution To solve, let's again form the augmented matrix
$$
\left[\begin{array}{ll}
A & B
\end{array}\right]
$$
put it into reduced row echelon form, and interpret the result.
$$
\left[\begin{array}{ccccc}
1 & 0 & 2 & -1 & 2 \\
0 & -1 & -2 & 2 & -6 \\
2 & -1 & 0 & 2 & -4
\end{array}\right] \rightarrow \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{ccccc}
1 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 & 4 \\
0 & 0 & 1 & -1 & 1
\end{array}\right]
$$
We see from this that
$$
X=\left[\begin{array}{cc}
1 & 0 \\
0 & 4 \\
-1 & 1
\end{array}\right]
$$
Why does this work? To see the answer, let's define five matrices.
$$
A=\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right], \vec{u}=\left[\begin{array}{l}
1 \\
1
\end{array}\right], \vec{v}=\left[\begin{array}{c}
-1 \\
1
\end{array}\right], \vec{w}=\left[\begin{array}{l}
5 \\
6
\end{array}\right] \text { and } X=\left[\begin{array}{ccc}
1 & -1 & 5 \\
1 & 1 & 6
\end{array}\right]
$$
Notice that $\vec{u}, \vec{v}$ and $\vec{w}$ are the first, second and third columns of $X$, respectively. Now consider this list of matrix products: $A \vec{u}, A \vec{v}, A \vec{w}$ and $A X$.
$$
\begin{array}{rlrl}
A \vec{u} & =\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\left[\begin{array}{l}
1 \\
1
\end{array}\right] & A \vec{v} & =\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\left[\begin{array}{c}
-1 \\
1
\end{array}\right] \\
& =\left[\begin{array}{l}
3 \\
7
\end{array}\right] & =\left[\begin{array}{l}
1 \\
1
\end{array}\right] \\
A \vec{w} & =\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\left[\begin{array}{l}
5 \\
6
\end{array}\right] & A X & =\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\left[\begin{array}{ccc}
1 & -1 & 5 \\
1 & 1 & 6
\end{array}\right] \\
& =\left[\begin{array}{l}
17 \\
39
\end{array}\right] & & =\left[\begin{array}{lll}
3 & 1 & 17 \\
7 & 1 & 39
\end{array}\right]
\end{array}
$$
So again note that the columns of $X$ are $\vec{u}, \vec{v}$ and $\vec{w}$; that is, we can write
$$
X=\left[\begin{array}{lll}
\vec{u} & \vec{v} & \vec{w}
\end{array}\right] .
$$
Notice also that the columns of $A X$ are $A \vec{u}, A \vec{v}$ and $A \vec{w}$, respectively. Thus we can write
$$
\begin{aligned}
& A X=A\left[\begin{array}{lll}
\vec{u} & \vec{v} & \vec{w}
\end{array}\right] \\
& =\left[\begin{array}{lll}
A \vec{u} & A \vec{v} & A \vec{w}
\end{array}\right] \\
& \left.=\left[\begin{array}{l}
3 \\
7
\end{array}\right] \quad\left[\begin{array}{l}
1 \\
1
\end{array}\right] \quad\left[\begin{array}{l}
17 \\
39
\end{array}\right]\right] \\
& =\left[\begin{array}{lll}
3 & 1 & 17 \\
7 & 1 & 39
\end{array}\right]
\end{aligned}
$$
We summarize what we saw above in the following statement:
The columns of a matrix product $A X$ are $A$ times the columns of $X$.
How does this help us solve the matrix equation $A X=B$ for $X$ ? Assume that $A$ is a square matrix (that forces $X$ and $B$ to be the same size). We'll let $\overrightarrow{x_{1}}, \overrightarrow{x_{2}}, \cdots \overrightarrow{x_{n}}$ denote the columns of the (unknown) matrix $X$, and we'll let $\overrightarrow{b_{1}}, \overrightarrow{b_{2}}, \cdots \overrightarrow{b_{n}}$ denote the columns of $B$. We want to solve $A X=B$ for $X$. That is, we want $X$ where
$$
\begin{aligned}
A X & =B \\
A\left[\begin{array}{llll}
\overrightarrow{x_{1}} & \overrightarrow{x_{2}} & \cdots & \overrightarrow{x_{n}}
\end{array}\right] & =\left[\begin{array}{lllll}
\overrightarrow{b_{1}} & \overrightarrow{b_{2}} & \cdots & \overrightarrow{b_{n}}
\end{array}\right] \\
{\left[\begin{array}{lllll}
A \overrightarrow{x_{1}} & A \overrightarrow{x_{2}} & \cdots & A \overrightarrow{x_{n}}
\end{array}\right] } & =\left[\begin{array}{llll}
\overrightarrow{b_{1}} & \overrightarrow{b_{2}} & \cdots & \overrightarrow{b_{n}}
\end{array}\right]
\end{aligned}
$$
If the matrix on the left hand side is equal to the matrix on the right, then their respective columns must be equal. This means we need to solve $n$ equations:
$$
\begin{gathered}
A \overrightarrow{x_{1}}=\overrightarrow{b_{1}} \\
A \overrightarrow{x_{2}}=\overrightarrow{b_{2}} \\
\vdots=\vdots \\
A \overrightarrow{x_{n}}=\overrightarrow{b_{n}}
\end{gathered}
$$
We already know how to do this; this is what we learned in the previous section. Let's do this in a concrete example. In our above work we defined matrices $A$ and $X$, and looked at the product $A X$. Let's call the product $B$; that is, set $B=A X$. Now, let's pretend that we don't know what $X$ is, and let's try to find the matrix $X$ that satisfies the equation $A X=B$. As a refresher, recall that
$$
A=\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right] \text { and } B=\left[\begin{array}{lll}
3 & 1 & 17 \\
7 & 1 & 39
\end{array}\right]
$$
Since $A$ is a $2 \times 2$ matrix and $B$ is a $2 \times 3$ matrix, what dimensions must $X$ be in the equation $A X=B$ ? The number of rows of $X$ must match the number of columns of $A$; the number of columns of $X$ must match the number of columns of $B$. Therefore we know that $X$ must be a $2 \times 3$ matrix. We'll call the three columns of $X \overrightarrow{x_{1}}, \overrightarrow{x_{2}}$ and $\overrightarrow{x_{3}}$. Our previous explanation tells us that if $A X=B$, then:
$$
\begin{aligned}
A X & =B \\
A\left[\begin{array}{lll}
\overrightarrow{x_{1}} & \overrightarrow{x_{2}} & \overrightarrow{x_{3}}
\end{array}\right] & =\left[\begin{array}{lll}
3 & 1 & 17 \\
7 & 1 & 39
\end{array}\right] \\
{\left[\begin{array}{lll}
A \overrightarrow{x_{1}} & A \overrightarrow{x_{2}} & A \overrightarrow{x_{3}}
\end{array}\right] } & =\left[\begin{array}{lll}
3 & 1 & 17 \\
7 & 1 & 39
\end{array}\right] .
\end{aligned}
$$
Hence
$$
\begin{aligned}
& A \overrightarrow{x_{1}}=\left[\begin{array}{l}
3 \\
7
\end{array}\right] \\
& A \overrightarrow{x_{2}}=\left[\begin{array}{l}
1 \\
1
\end{array}\right] \\
& A \overrightarrow{x_{3}}=\left[\begin{array}{l}
17 \\
39
\end{array}\right]
\end{aligned}
$$
To find $\overrightarrow{x_{1}}$, we form the proper augmented matrix and put it into reduced row echelon form and interpret the results.
$$
\left[\begin{array}{lll}
1 & 2 & 3 \\
3 & 4 & 7
\end{array}\right] \quad \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{lll}
1 & 0 & 1 \\
0 & 1 & 1
\end{array}\right]
$$
This shows us that
$$
\overrightarrow{x_{1}}=\left[\begin{array}{l}
1 \\
1
\end{array}\right]
$$
To find $\overrightarrow{x_{2}}$, we again form an augmented matrix and interpret its reduced row echelon form.
$$
\left[\begin{array}{lll}
1 & 2 & 1 \\
3 & 4 & 1
\end{array}\right] \quad \overrightarrow{\operatorname{rref}}\left[\begin{array}{ccc}
1 & 0 & -1 \\
0 & 1 & 1
\end{array}\right]
$$
Thus
$$
\overrightarrow{x_{2}}=\left[\begin{array}{c}
-1 \\
1
\end{array}\right]
$$
which matches with what we already knew from above.
Before continuing on in this manner to find $\overrightarrow{x_{3}}$, we should stop and think. If the matrix vector equation $A \vec{x}=\vec{b}$ is consistent, then the steps involved in putting
$$
\left[\begin{array}{ll}
A & \vec{b}
\end{array}\right]
$$
into reduced row echelon form depend only on $A$; it does not matter what $\vec{b}$ is. So when we put the two matrices
$$
\left[\begin{array}{lll}
1 & 2 & 3 \\
3 & 4 & 7
\end{array}\right] \text { and }\left[\begin{array}{lll}
1 & 2 & 1 \\
3 & 4 & 1
\end{array}\right]
$$
from above into reduced row echelon form, we performed exactly the same steps! (In fact, those steps are: $-3 R_{1}+R_{2} \rightarrow R_{2} ;-\frac{1}{2} R_{2} \rightarrow R_{2} ;-2 R_{2}+R_{1} \rightarrow R_{1}$.)
Instead of solving for each column of $X$ separately, performing the same steps to put the necessary matrices into reduced row echelon form three different times, why don't we just do it all at once ${ }^{16}$ Instead of individually putting
$$
\left[\begin{array}{lll}
1 & 2 & 3 \\
3 & 4 & 7
\end{array}\right], \quad\left[\begin{array}{lll}
1 & 2 & 1 \\
3 & 4 & 1
\end{array}\right] \text { and }\left[\begin{array}{lll}
1 & 2 & 17 \\
3 & 4 & 39
\end{array}\right]
$$
into reduced row echelon form, let's just put
$$
\left[\begin{array}{lllll}
1 & 2 & 3 & 1 & 17 \\
3 & 4 & 7 & 1 & 39
\end{array}\right]
$$
into reduced row echelon form.
$$
\left[\begin{array}{lllll}
1 & 2 & 3 & 1 & 17 \\
3 & 4 & 7 & 1 & 39
\end{array}\right] \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{ccccc}
1 & 0 & 1 & -1 & 5 \\
0 & 1 & 1 & 1 & 6
\end{array}\right]
$$
By looking at the last three columns, we see $X$ :
$$
X=\left[\begin{array}{ccc}
1 & -1 & 5 \\
1 & 1 & 6
\end{array}\right]
$$
Now that we've justified the technique we've been using in this section to solve $A X=B$ for $X$, we reinfornce its importance by restating it as a Key Idea.
## Key Idea 9
Solving $A X=B$
Let $A$ be an $n \times n$ matrix, where the reduced row echelon form of $A$ is $I$. To solve the matrix equation $A X=B$ for $X$,
1. Form the augmented matrix $\left[\begin{array}{ll}A & B\end{array}\right]$.
2. Put this matrix into reduced row echelon form. It will be of the form $\left[\begin{array}{ll}I & X\end{array}\right]$, where $X$ appears in the columns where $B$ once was.
These simple steps cause us to ask certain questions. First, we specify above that $A$ should be a square matrix. What happens if $A$ isn't square? Is a solution still possible? Secondly, we only considered cases where the reduced row echelon form of $A$ was $I$ (and stated that as a requirement in our Key Idea). What if the reduced row echelon form of $A$ isn't I? Would we still be able to find a solution? (Instead of having exactly one solution, could we have no solution? Infinite solutions? How would we be able to tell?)
${ }^{16}$ One reason to do it three different times is that we enjoy doing unnecessary work. Another reason could be that we are stupid. These questions are good to ask, and we leave it to the reader to discover their answers. Instead of tackling these questions, we instead tackle the problem of "Why do we care about solving $A X=B$ ?" The simple answer is that, for now, we only care about the special case when $B=I$. By solving $A X=I$ for $X$, we find a matrix $X$ that, when multiplied by $A$, gives the identity $I$. That will be very useful.
## Exercises 2.5
In Exercises $1-12$, matrices $A$ and $B$ are given. Solve the matrix equation $A X=B$.
$$
\text { 1. } \begin{aligned}
A & =\left[\begin{array}{cc}
4 & -1 \\
-7 & 5
\end{array}\right], \\
B & =\left[\begin{array}{cc}
8 & -31 \\
-27 & 38
\end{array}\right]
\end{aligned}
$$
2. $A=\left[\begin{array}{cc}1 & -3 \\ -3 & 6\end{array}\right]$,
$$
B=\left[\begin{array}{cc}
12 & -10 \\
-27 & 27
\end{array}\right]
$$
3. $A=\left[\begin{array}{ll}3 & 3 \\ 6 & 4\end{array}\right]$,
$$
B=\left[\begin{array}{ll}
15 & -39 \\
16 & -66
\end{array}\right]
$$
4. $A=\left[\begin{array}{cc}-3 & -6 \\ 4 & 0\end{array}\right]$,
$B=\left[\begin{array}{cc}48 & -30 \\ 0 & -8\end{array}\right]$
5. $A=\left[\begin{array}{ll}-1 & -2 \\ -2 & -3\end{array}\right]$,
$$
B=\left[\begin{array}{ccc}
13 & 4 & 7 \\
22 & 5 & 12
\end{array}\right]
$$
6. $A=\left[\begin{array}{cc}-4 & 1 \\ -1 & -2\end{array}\right]$,
$$
B=\left[\begin{array}{ccc}
-2 & -10 & 19 \\
13 & 2 & -2
\end{array}\right]
$$
7. $A=\left[\begin{array}{cc}1 & 0 \\ 3 & -1\end{array}\right], \quad B=I_{2}$
8. $A=\left[\begin{array}{ll}2 & 2 \\ 3 & 1\end{array}\right], \quad B=I_{2}$
9. $A=\left[\begin{array}{ccc}-2 & 0 & 4 \\ -5 & -4 & 5 \\ -3 & 5 & -3\end{array}\right]$,
$$
B=\left[\begin{array}{ccc}
-18 & 2 & -14 \\
-38 & 18 & -13 \\
10 & 2 & -18
\end{array}\right]
$$
10. $A=\left[\begin{array}{ccc}-5 & -4 & -1 \\ 8 & -2 & -3 \\ 6 & 1 & -8\end{array}\right]$,
$$
B=\left[\begin{array}{ccc}
-21 & -8 & -19 \\
65 & -11 & -10 \\
75 & -51 & 33
\end{array}\right]
$$
11. $A=\left[\begin{array}{ccc}0 & -2 & 1 \\ 0 & 2 & 2 \\ 1 & 2 & -3\end{array}\right], \quad B=13$
12. $A=\left[\begin{array}{ccc}-3 & 3 & -2 \\ 1 & -3 & 2 \\ -1 & -1 & 2\end{array}\right], \quad B=I_{3}$
### The Matrix Inverse
## AS YOU READ
1. $\mathrm{T} / \mathrm{F}$ : If $A$ and $B$ are square matrices where $A B=I$, then $B A=I$.
2. T/F: A matrix A has exactly one inverse, infinite inverses, or no inverse.
3. $\mathrm{T} / \mathrm{F}$ : Everyone is special. 4. $\mathrm{T} / \mathrm{F}$ : If $A$ is invertible, then $A \vec{x}=\vec{O}$ has exactly 1 solution.
4. What is a corollary?
5. Fill in the blanks: is a matrix is invertible is useful; computing the inverse
Once again we visit the old algebra equation, $a x=b$. How do we solve for $x$ ? We know that, as long as $a \neq 0$,
$$
x=\frac{b}{a}, \text { or, stated in another way, } x=a^{-1} b .
$$
What is $a^{-1}$ ? It is the number that, when multiplied by $a$, returns 1 . That is,
$$
a^{-1} a=1 .
$$
Let us now think in terms of matrices. We have learned of the identity matrix / that "acts like the number 1. " That is, if $A$ is a square matrix, then
$$
I A=A I=A .
$$
If we had a matrix, which we'll call $A^{-1}$, where $A^{-1} A=I$, then by analogy to our algebra example above it seems like we might be able to solve the linear system $A \vec{x}=\vec{b}$ for $\vec{x}$ by multiplying both sides of the equation by $A^{-1}$. That is, perhaps
$$
\vec{x}=A^{-1} \vec{b} .
$$
Of course, there is a lot of speculation here. We don't know that such a matrix like $A^{-1}$ exists. However, we do know how to solve the matrix equation $A X=B$, so we can use that technique to solve the equation $A X=I$ for $X$. This seems like it will get us close to what we want. Let's practice this once and then study our results.
Example $53 \quad$ Let
$$
A=\left[\begin{array}{ll}
2 & 1 \\
1 & 1
\end{array}\right]
$$
Find a matrix $X$ such that $A X=I$.
Solution We know how to solve this from the previous section: we form the proper augmented matrix, put it into reduced row echelon form and interpret the results.
$$
\left[\begin{array}{cccc}
2 & 1 & 1 & 0 \\
1 & 1 & 0 & 1
\end{array}\right] \quad \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{cccc}
1 & 0 & 1 & -1 \\
0 & 1 & -1 & 2
\end{array}\right]
$$
We read from our matrix that
$$
X=\left[\begin{array}{cc}
1 & -1 \\
-1 & 2
\end{array}\right]
$$
Let's check our work:
$$
\begin{aligned}
A X & =\left[\begin{array}{ll}
2 & 1 \\
1 & 1
\end{array}\right]\left[\begin{array}{cc}
1 & -1 \\
-1 & 2
\end{array}\right] \\
& =\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right] \\
& =I
\end{aligned}
$$
Sure enough, it works.
Looking at our previous example, we are tempted to jump in and call the matrix $X$ that we found " $A-1$." However, there are two obstacles in the way of us doing this.
First, we know that in general $A B \neq B A$. So while we found that $A X=1$, we can't automatically assume that $X A=I$.
Secondly, we have seen examples of matrices where $A B=A C$, but $B \neq C$. So just because $A X=I$, it is possible that another matrix $Y$ exists where $A Y=I$. If this is the case, using the notation $A^{-1}$ would be misleading, since it could refer to more than one matrix.
These obstacles that we face are not insurmountable. The first obstacle was that we know that $A X=I$ but didn't know that $X A=I$. That's easy enough to check, though. Let's look at $A$ and $X$ from our previous example.
$$
\begin{aligned}
X A & =\left[\begin{array}{cc}
1 & -1 \\
-1 & 2
\end{array}\right]\left[\begin{array}{ll}
2 & 1 \\
1 & 1
\end{array}\right] \\
& =\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right] \\
& =I
\end{aligned}
$$
Perhaps this first obstacle isn't much of an obstacle after all. Of course, we only have one example where it worked, so this doesn't mean that it always works. We have good news, though: it always does work. The only "bad" news to come with this is that this is a bit harder to prove. We won't worry about proving it always works, but state formally that it does in the following theorem.
$$
\begin{aligned}
& \text { Theorem } 5 \text { Special Commuting Matrix Products } \\
& \text { Let } A \text { be an } n \times n \text { matrix. } \\
& \text { 1. If there is a matrix } X \text { such that } A X=I_{n} \text {, then } X A=I_{n} \text {. } \\
& \text { 2. If there is a matrix } X \text { such that } X A=I_{n} \text {, then } A X=I_{n} .
\end{aligned}
$$
The second obstacle is easier to address. We want to know if another matrix $Y$ exists where $A Y=I=Y A$. Let's suppose that it does. Consider the expression $X A Y$. Since matrix multiplication is associative, we can group this any way we choose. We could group this as $(X A) Y$; this results in
$$
\begin{aligned}
(X A) Y & =I Y \\
& =Y .
\end{aligned}
$$
We could also group $X A Y$ as $X(A Y)$. This tells us
$$
\begin{aligned}
X(A Y) & =X I \\
& =X
\end{aligned}
$$
Combining the two ideas above, we see that $X=X A Y=Y$; that is, $X=Y$. We conclude that there is only one matrix $X$ where $X A=I=A X$. (Even if we think we have two, we can do the above exercise and see that we really just have one.)
We have just proved the following theorem.
So given a square matrix $A$, if we can find a matrix $X$ where $A X=I$, then we know that $X A=I$ and that $X$ is the only matrix that does this. This makes $X$ special, so we give it a special name.
Definition 18
Invertible Matrices and the Inverse of $A$
Let $A$ and $X$ be $n \times n$ matrices where $A X=I=X A$. Then:
1. $A$ is invertible.
2. $X$ is the inverse of $A$, denoted by $A^{-1}$.
Let's do an example.
Example 54 Find the inverse of $A=\left[\begin{array}{ll}1 & 2 \\ 2 & 4\end{array}\right]$.
Solution By solving the equation $A X=I$ for $X$ will give us the inverse of $A$. Forming the appropriate augmented matrix and finding its reduced row echelon form gives us
$$
\left[\begin{array}{llll}
1 & 2 & 1 & 0 \\
2 & 4 & 0 & 1
\end{array}\right] \quad \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{cccc}
1 & 2 & 0 & 1 / 2 \\
0 & 0 & 1 & -1 / 2
\end{array}\right]
$$
Yikes! We were expecting to find that the reduced row echelon form of this matrix would look like
$$
\left[\begin{array}{ll}
1 & A^{-1}
\end{array}\right]
$$
However, we don't have the identity on the left hand side. Our conclusion: $A$ is not invertible.
We have just seen that not all matrices are invertible. ${ }^{17}$ With this thought in mind, let's complete the array of boxes we started before the example. We've discovered that if a matrix has an inverse, it has only one. Therefore, we gave that special matrix a name, "the inverse." Finally, we describe the most general way to find the inverse of a matrix, and a way to tell if it does not have one.
Key Idea 10
Finding $A^{-1}$
Let $A$ be an $n \times n$ matrix. To find $A^{-1}$, put the augmented matrix
$$
\left[\begin{array}{ll}
A & I_{n}
\end{array}\right]
$$
into reduced row echelon form. If the result is of the form
$$
\left[\begin{array}{ll}
I_{n} & X
\end{array}\right]
$$
then $A^{-1}=X$. If not, (that is, if the first $n$ columns of the reduced row echelon form are not $I_{n}$ ), then $A$ is not invertible.
Let's try again.
Example 55 Find the inverse, if it exists, of $A=\left[\begin{array}{ccc}1 & 1 & -1 \\ 1 & -1 & 1 \\ 1 & 2 & 3\end{array}\right]$.
Solution We'll try to solve $A X=I$ for $X$ and see what happens.
$$
\left[\begin{array}{cccccc}
1 & 1 & -1 & 1 & 0 & 0 \\
1 & -1 & 1 & 0 & 1 & 0 \\
1 & 2 & 3 & 0 & 0 & 1
\end{array}\right] \quad \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{cccccc}
1 & 0 & 0 & 0.5 & 0.5 & 0 \\
0 & 1 & 0 & 0.2 & -0.4 & 0.2 \\
0 & 0 & 1 & -0.3 & 0.1 & 0.2
\end{array}\right]
$$
${ }^{17}$ Hence our previous definition; why bother calling $A$ "invertible" if every square matrix is? If everyone is special, then no one is. Then again, everyone is special.
## Chapter 2 Matrix Arithmetic
We have a solution, so
$$
A=\left[\begin{array}{ccc}
0.5 & 0.5 & 0 \\
0.2 & -0.4 & 0.2 \\
-0.3 & 0.1 & 0.2
\end{array}\right]
$$
Multiply $A A^{-1}$ to verify that it is indeed the inverse of $A$.
In general, given a matrix $A$, to find $A^{-1}$ we need to form the augmented matrix $\left[\begin{array}{ll}A & 1\end{array}\right]$ and put it into reduced row echelon form and interpret the result. In the case of a $2 \times 2$ matrix, though, there is a shortcut. We give the shortcut in terms of a theorem. ${ }^{18}$
Theorem 7
The Inverse of a $2 \times 2$ Matrix
Let
$$
A=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right] .
$$
$A$ is invertible if and only if $a d-b c \neq 0$.
If $a d-b c \neq 0$, then
$$
A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{cc}
d & -b \\
-c & a
\end{array}\right] .
$$
We can't divide by 0 , so if $a d-b c=0$, we don't have an inverse. Recall Example 54, where
$$
A=\left[\begin{array}{ll}
1 & 2 \\
2 & 4
\end{array}\right]
$$
Here, $a d-b c=1(4)-2(2)=0$, which is why $A$ didn't have an inverse.
Although this idea is simple, we should practice it.
Example $56 \quad$ Use Theorem 7 to find the inverse of
$$
A=\left[\begin{array}{cc}
3 & 2 \\
-1 & 9
\end{array}\right]
$$
if it exists.
${ }^{18}$ We don't prove this theorem here, but it really isn't hard to do. Put the matrix
$$
\left[\begin{array}{llll}
a & b & 1 & 0 \\
c & d & 0 & 1
\end{array}\right]
$$
into reduced row echelon form and you'll discover the result of the theorem. Alternatively, multiply $A$ by what we propose is the inverse and see that we indeed get $I$. Solution Since $a d-b c=29 \neq 0, A^{-1}$ exists. By the Theorem,
$$
\begin{aligned}
A^{-1} & =\frac{1}{3(9)-2(-1)}\left[\begin{array}{cc}
9 & -2 \\
1 & 3
\end{array}\right] \\
& =\frac{1}{29}\left[\begin{array}{cc}
9 & -2 \\
1 & 3
\end{array}\right]
\end{aligned}
$$
We can leave our answer in this form, or we could "simplify" it as
$$
A^{-1}=\frac{1}{29}\left[\begin{array}{cc}
9 & -2 \\
1 & 3
\end{array}\right]=\left[\begin{array}{cc}
9 / 29 & -2 / 29 \\
1 / 29 & 3 / 29
\end{array}\right]
$$
We started this section out by speculating that just as we solved algebraic equations of the form $a x=b$ by computing $x=a^{-1} b$, we might be able to solve matrix equations of the form $A \vec{x}=\vec{b}$ by computing $\vec{x}=A^{-1} \vec{b}$. If $A^{-1}$ does exist, then we can solve the equation $A \vec{x}=\vec{b}$ this way. Consider:
$$
\begin{aligned}
A \vec{x} & =\vec{b} & & \text { (original equation) } \\
A^{-1} A \vec{x} & =A^{-1} \vec{b} & & \text { (multiply both sides on the left by } A^{-1} \text { ) } \\
\vec{x} & =A^{-1} \vec{b} & & \left(\text { since } A^{-1} A=l\right) \\
\vec{x} & =A^{-1} \vec{b} & & (\text { since } \vec{x}=\vec{x})
\end{aligned}
$$
Let's step back and think about this for a moment. The only thing we know about the equation $A \vec{x}=\vec{b}$ is that $A$ is invertible. We also know that solutions to $A \vec{x}=\vec{b}$ come in three forms: exactly one solution, infinite solutions, and no solution. We just showed that if $A$ is invertible, then $A \vec{x}=\vec{b}$ has at least one solution. We showed that by setting $\vec{x}$ equal to $A^{-1} \vec{b}$, we have a solution. Is it possible that more solutions exist?
No. Suppose we are told that a known vector $\vec{v}$ is a solution to the equation $A \vec{x}=\vec{b}$; that is, we know that $A \vec{v}=\vec{b}$. We can repeat the above steps:
$$
\begin{aligned}
A \vec{v} & =\vec{b} \\
A^{-1} A \vec{v} & =A^{-1} \vec{b} \\
\mid \vec{v} & =A^{-1} \vec{b} \\
\vec{v} & =A^{-1} \vec{b} .
\end{aligned}
$$
This shows that all solutions to $A \vec{x}=\vec{b}$ are exactly $\vec{x}=A^{-1} \vec{b}$ when $A$ is invertible. We have just proved the following theorem. Theorem 8
A corollary ${ }^{19}$ to this theorem is: If $A$ is not invertible, then $A \vec{x}=\vec{b}$ does not have exactly one solution. It may have infinite solutions and it may have no solution, and we would need to examine the reduced row echelon form of the augmented matrix $\left[\begin{array}{ll}A & \vec{b}\end{array}\right]$ to see which case applies.
We demonstrate our theorem with an example.
Example 57 Solve $A \vec{x}=\vec{b}$ by computing $\vec{x}=A^{-1} \vec{b}$, where
$$
A=\left[\begin{array}{ccc}
1 & 0 & -3 \\
-3 & -4 & 10 \\
4 & -5 & -11
\end{array}\right] \text { and } \vec{b}=\left[\begin{array}{c}
-15 \\
57 \\
-46
\end{array}\right]
$$
Solution Without showing our steps, we compute
$$
A^{-1}=\left[\begin{array}{ccc}
94 & 15 & -12 \\
7 & 1 & -1 \\
31 & 5 & -4
\end{array}\right]
$$
We then find the solution to $A \vec{x}=\vec{b}$ by computing $A^{-1} \vec{b}$ :
$$
\begin{aligned}
\vec{x} & =A^{-1} \vec{b} \\
& =\left[\begin{array}{ccc}
94 & 15 & -12 \\
7 & 1 & -1 \\
31 & 5 & -4
\end{array}\right]\left[\begin{array}{c}
-15 \\
57 \\
-46
\end{array}\right] \\
& =\left[\begin{array}{c}
-3 \\
-2 \\
4
\end{array}\right] .
\end{aligned}
$$
We can easily check our answer:
$$
\left[\begin{array}{ccc}
1 & 0 & -3 \\
-3 & -4 & 10 \\
4 & -5 & -11
\end{array}\right]\left[\begin{array}{c}
-3 \\
-2 \\
4
\end{array}\right]=\left[\begin{array}{c}
-15 \\
57 \\
-46
\end{array}\right]
$$
${ }^{19}$ a corollary is an idea that follows directly from a theorem Knowing a matrix is invertible is incredibly useful. ${ }^{20}$ Among many other reasons, if you know $A$ is invertible, then you know for sure that $A \vec{x}=\vec{b}$ has a solution (as we just stated in Theorem 8). In the next section we'll demonstrate many different properties of invertible matrices, including stating several different ways in which we know that a matrix is invertible.
## Exercises 2.6
In Exercises $1-8, A$ matrix $A$ is given. Find $A^{-1}$ using Theorem 7 , if it exists.
1. $\left[\begin{array}{cc}1 & 5 \\ -5 & -24\end{array}\right]$
2. $\left[\begin{array}{ll}1 & -4 \\ 1 & -3\end{array}\right]$
3. $\left[\begin{array}{ll}3 & 0 \\ 0 & 7\end{array}\right]$
4. $\left[\begin{array}{ll}2 & 5 \\ 3 & 4\end{array}\right]$
5. $\left[\begin{array}{cc}1 & -3 \\ -2 & 6\end{array}\right]$
6. $\left[\begin{array}{ll}3 & 7 \\ 2 & 4\end{array}\right]$
7. $\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
8. $\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$
In Exercises $9-28$, a matrix $A$ is given. Find $A^{-1}$ using Key Idea 10, if it exists.
9. $\left[\begin{array}{cc}-2 & 3 \\ 1 & 5\end{array}\right]$
10. $\left[\begin{array}{cc}-5 & -2 \\ 9 & 2\end{array}\right]$
11. $\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$
12. $\left[\begin{array}{cc}5 & 7 \\ 5 / 3 & 7 / 3\end{array}\right]$
13. $\left[\begin{array}{ccc}25 & -10 & -4 \\ -18 & 7 & 3 \\ -6 & 2 & 1\end{array}\right]$ 14. $\left[\begin{array}{ccc}2 & 3 & 4 \\ -3 & 6 & 9 \\ -1 & 9 & 13\end{array}\right]$
14. $\left[\begin{array}{ccc}1 & 0 & 0 \\ 4 & 1 & -7 \\ 20 & 7 & -48\end{array}\right]$
15. $\left[\begin{array}{ccc}-4 & 1 & 5 \\ -5 & 1 & 9 \\ -10 & 2 & 19\end{array}\right]$
16. $\left[\begin{array}{ccc}5 & -1 & 0 \\ 7 & 7 & 1 \\ -2 & -8 & -1\end{array}\right]$
17. $\left[\begin{array}{ccc}1 & -5 & 0 \\ -2 & 15 & 4 \\ 4 & -19 & 1\end{array}\right]$
18. $\left[\begin{array}{ccc}25 & -8 & 0 \\ -78 & 25 & 0 \\ 48 & -15 & 1\end{array}\right]$
19. $\left[\begin{array}{ccc}1 & 0 & 0 \\ 7 & 5 & 8 \\ -2 & -2 & -3\end{array}\right]$
20. $\left[\begin{array}{lll}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{array}\right]$
21. $\left[\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]$
22. $\left[\begin{array}{cccc}1 & 0 & 0 & 0 \\ -19 & -9 & 0 & 4 \\ 33 & 4 & 1 & -7 \\ 4 & 2 & 0 & -1\end{array}\right]$
23. $\left[\begin{array}{cccc}1 & 0 & 0 & 0 \\ 27 & 1 & 0 & 4 \\ 18 & 0 & 1 & 4 \\ 4 & 0 & 0 & 1\end{array}\right]$
${ }^{20}$ As odd as it may sound, knowing a matrix is invertible is useful; actually computing the inverse isn't. This is discussed at the end of the next section. 25. $\left[\begin{array}{cccc}-15 & 45 & -3 & 4 \\ 55 & -164 & 15 & -15 \\ -215 & 640 & -62 & 59 \\ -4 & 12 & 0 & 1\end{array}\right]$
26. $\left[\begin{array}{cccc}1 & 0 & 2 & 8 \\ 0 & 1 & 0 & 0 \\ 0 & -4 & -29 & -110 \\ 0 & -3 & -5 & -19\end{array}\right]$
27. $\left[\begin{array}{llll}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\end{array}\right]$
28. $\left[\begin{array}{cccc}1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & -4\end{array}\right]$
In Exercises 29 - 36, a matrix $A$ and a vector $\vec{b}$ are given. Solve the equation $A \vec{x}=\vec{b}$ using Theorem 8.
29. $A=\left[\begin{array}{ll}3 & 5 \\ 2 & 3\end{array}\right], \quad \vec{b}=\left[\begin{array}{l}21 \\ 13\end{array}\right]$
30. $A=\left[\begin{array}{cc}1 & -4 \\ 4 & -15\end{array}\right], \quad \vec{b}=\left[\begin{array}{l}21 \\ 77\end{array}\right]$
31. $A=\left[\begin{array}{cc}9 & 70 \\ -4 & -31\end{array}\right], \quad \vec{b}=\left[\begin{array}{c}-2 \\ 1\end{array}\right]$ 32. $A=\left[\begin{array}{cc}10 & -57 \\ 3 & -17\end{array}\right], \quad \vec{b}=\left[\begin{array}{c}-14 \\ -4\end{array}\right]$
32. $A=\left[\begin{array}{ccc}1 & 2 & 12 \\ 0 & 1 & 6 \\ -3 & 0 & 1\end{array}\right]$,
$\vec{b}=\left[\begin{array}{c}-17 \\ -5 \\ 20\end{array}\right]$
34. $A=\left[\begin{array}{ccc}1 & 0 & -3 \\ 8 & -2 & -13 \\ 12 & -3 & -20\end{array}\right]$,
$\vec{b}=\left[\begin{array}{c}-34 \\ -159 \\ -243\end{array}\right]$
35. $A=\left[\begin{array}{ccc}5 & 0 & -2 \\ -8 & 1 & 5 \\ -2 & 0 & 1\end{array}\right]$,
$\vec{b}=\left[\begin{array}{c}33 \\ -70 \\ -15\end{array}\right]$
36. $A=\left[\begin{array}{ccc}1 & -6 & 0 \\ 0 & 1 & 0 \\ 2 & -8 & 1\end{array}\right]$,
$\vec{b}=\left[\begin{array}{c}-69 \\ 10 \\ -102\end{array}\right]$
### Properties of the Matrix Inverse
## AS YOU READ
1. What does it mean to say that two statements are "equivalent?"
2. T/F: If $A$ is not invertible, then $A \vec{x}=\vec{O}$ could have no solutions.
3. T/F: If $A$ is not invertible, then $A \vec{x}=\vec{b}$ could have infinite solutions.
4. What is the inverse of the inverse of $A$ ?
5. T/F: Solving $A \vec{x}=\vec{b}$ using Gaussian elimination is faster than using the inverse of $A$.
We ended the previous section by stating that invertible matrices are important. Since they are, in this section we study invertible matrices in two ways. First, we look at ways to tell whether or not a matrix is invertible, and second, we study properties of invertible matrices (that is, how they interact with other matrix operations).
We start with collecting ways in which we know that a matrix is invertible. We actually already know the truth of this theorem from our work in the previous section, but it is good to list the following statements in one place. As we move through other sections, we'll add on to this theorem.
Theorem 9
Invertible Matrix Theorem
Let $A$ be an $n \times n$ matrix. The following statements are equivalent.
(a) $A$ is invertible.
(b) There exists a matrix $B$ such that $B A=I$.
(c) There exists a matrix $C$ such that $A C=I$.
(d) The reduced row echelon form of $A$ is $I$.
(e) The equation $A \vec{x}=\vec{b}$ has exactly one solution for every $n \times 1$ vector $\vec{b}$.
(f) The equation $A \vec{x}=\overrightarrow{0}$ has exactly one solution (namely, $\vec{x}=\overrightarrow{0}$ ).
Let's make note of a few things about the Invertible Matrix Theorem.
1. First, note that the theorem uses the phrase "the following statements are equivalent." When two or more statements are equivalent, it means that the truth of any one of them implies that the rest are also true; if any one of the statements is false, then they are all false. So, for example, if we determined that the equation $A \vec{x}=\vec{O}$ had exactly one solution (and $A$ was an $n \times n$ matrix) then we would know that $A$ was invertible, that $A \vec{x}=\vec{b}$ had only one solution, that the reduced row echelon form of $A$ was $I$, etc.
2. Let's go through each of the statements and see why we already knew they all said essentially the same thing.
(a) This simply states that $A$ is invertible - that is, that there exists a matrix $A^{-1}$ such that $A^{-1} A=A A^{-1}=I$. We'll go on to show why all the other statements basically tell us " $A$ is invertible."
(b) If we know that $A$ is invertible, then we already know that there is a matrix $B$ where $B A=I$. That is part of the definition of invertible. However, we can also "go the other way." Recall from Theorem 5 that even if all we know is that there is a matrix $B$ where $B A=l$, then we also know that $A B=I$. That is, we know that $B$ is the inverse of $A$ (and hence $A$ is invertible).
(c) We use the same logic as in the previous statement to show why this is the same as " $A$ is invertible."
(d) If $A$ is invertible, we can find the inverse by using Key Idea 10 (which in turn depends on Theorem 5). The crux of Key Idea 10 is that the reduced row echelon form of $A$ is $I$; if it is something else, we can't find $A^{-1}$ (it doesn't exist). Knowing that $A$ is invertible means that the reduced row echelon form of $A$ is $I$. We can go the other way; if we know that the reduced row echelon form of $A$ is $I$, then we can employ Key Idea 10 to find $A^{-1}$, so $A$ is invertible.
(e) We know from Theorem 8 that if $A$ is invertible, then given any vector $\vec{b}$, $A \vec{x}=\vec{b}$ has always has exactly one solution, namely $\vec{x}=A^{-1} \vec{b}$. However, we can go the other way; let's say we know that $A \vec{x}=\vec{b}$ always has exactly solution. How can we conclude that $A$ is invertible?
Think about how we, up to this point, determined the solution to $A \vec{x}=$ $\vec{b}$. We set up the augmented matrix $\left[\begin{array}{ll}A & \vec{b}\end{array}\right]$ and put it into reduced row echelon form. We know that getting the identity matrix on the left means that we had a unique solution (and not getting the identity means we either have no solution or infinite solutions). So getting I on the left means having a unique solution; having I on the left means that the reduced row echelon form of $A$ is $I$, which we know from above is the same as $A$ being invertible.
(f) This is the same as the above; simply replace the vector $\vec{b}$ with the vector $\overrightarrow{0}$.
So we came up with a list of statements that are all equivalent to the statement " $A$ is invertible." Again, if we know that if any one of them is true (or false), then they are all true (or all false).
Theorem 9 states formally that if $A$ is invertible, then $A \vec{x}=\vec{b}$ has exactly one solution, namely $A^{-1} \vec{b}$. What if $A$ is not invertible? What are the possibilities for solutions to $A \vec{x}=\vec{b}$ ?
We know that $A \vec{x}=\vec{b}$ cannot have exactly one solution; if it did, then by our theorem it would be invertible. Recalling that linear equations have either one solution, infinite solutions, or no solution, we are left with the latter options when $A$ is not invertible. This idea is important and so we'll state it again as a Key Idea. Key Idea 11
In Theorem 9 we've come up with a list of ways in which we can tell whether or not a matrix is invertible. At the same time, we have come up with a list of properties of invertible matrices - things we know that are true about them. (For instance, if we know that $A$ is invertible, then we know that $A \vec{x}=\vec{b}$ has only one solution.)
We now go on to discover other properties of invertible matrices. Specifically, we want to find out how invertibility interacts with other matrix operations. For instance, if we know that $A$ and $B$ are invertible, what is the inverse of $A+B$ ? What is the inverse of $A B$ ? What is "the inverse of the inverse?" We'll explore these questions through an example.
Example $58 \quad$ Let
$$
A=\left[\begin{array}{ll}
3 & 2 \\
0 & 1
\end{array}\right] \text { and } B=\left[\begin{array}{cc}
-2 & 0 \\
1 & 1
\end{array}\right]
$$
Find:
1. $A^{-1}$
2. $(A B)^{-1}$
3. $(A+B)^{-1}$
4. $B^{-1}$
5. $\left(A^{-1}\right)^{-1}$
6. $(5 A)^{-1}$
In addition, try to find connections between each of the above.
## SOLUTION
1. Computing $A^{-1}$ is straightforward; we'll use Theorem 7 .
$$
A^{-1}=\frac{1}{3}\left[\begin{array}{cc}
1 & -2 \\
0 & 3
\end{array}\right]=\left[\begin{array}{cc}
1 / 3 & -2 / 3 \\
0 & 1
\end{array}\right]
$$
2. We compute $B^{-1}$ in the same way as above.
$$
B^{-1}=\frac{1}{-2}\left[\begin{array}{cc}
1 & 0 \\
-1 & -2
\end{array}\right]=\left[\begin{array}{cc}
-1 / 2 & 0 \\
1 / 2 & 1
\end{array}\right]
$$
3. To compute $(A B)^{-1}$, we first compute $A B$ :
$$
A B=\left[\begin{array}{ll}
3 & 2 \\
0 & 1
\end{array}\right]\left[\begin{array}{cc}
-2 & 0 \\
1 & 1
\end{array}\right]=\left[\begin{array}{cc}
-4 & 2 \\
1 & 1
\end{array}\right]
$$
We now apply Theorem 7 to find $(A B)^{-1}$.
$$
(A B)^{-1}=\frac{1}{-6}\left[\begin{array}{cc}
1 & -2 \\
-1 & -4
\end{array}\right]=\left[\begin{array}{cc}
-1 / 6 & 1 / 3 \\
1 / 6 & 2 / 3
\end{array}\right]
$$
4. To compute $\left(A^{-1}\right)^{-1}$, we simply apply Theorem 7 to $A^{-1}$ :
$$
\left(A^{-1}\right)^{-1}=\frac{1}{1 / 3}\left[\begin{array}{ll}
1 & 2 / 3 \\
0 & 1 / 3
\end{array}\right]=\left[\begin{array}{ll}
3 & 2 \\
0 & 1
\end{array}\right] .
$$
5. To compute $(A+B)^{-1}$, we first compute $A+B$ then apply Theorem 7:
$$
A+B=\left[\begin{array}{ll}
3 & 2 \\
0 & 1
\end{array}\right]+\left[\begin{array}{cc}
-2 & 0 \\
1 & 1
\end{array}\right]=\left[\begin{array}{ll}
1 & 2 \\
1 & 2
\end{array}\right]
$$
Hence
$$
(A+B)^{-1}=\frac{1}{0}\left[\begin{array}{cc}
2 & -2 \\
-1 & 1
\end{array}\right]=!
$$
Our last expression is really nonsense; we know that if $a d-b c=0$, then the given matrix is not invertible. That is the case with $A+B$, so we conclude that $A+B$ is not invertible.
6. To compute $(5 A)^{-1}$, we compute $5 A$ and then apply Theorem 7 .
$$
(5 A)^{-1}=\left(\left[\begin{array}{cc}
15 & 10 \\
0 & 5
\end{array}\right]\right)^{-1}=\frac{1}{75}\left[\begin{array}{cc}
5 & -10 \\
0 & 15
\end{array}\right]=\left[\begin{array}{cc}
1 / 15 & -2 / 15 \\
0 & 1 / 5
\end{array}\right]
$$
We now look for connections between $A^{-1}, B^{-1},(A B)^{-1},\left(A^{-1}\right)^{-1}$ and $(A+B)^{-1}$.
3. Is there some sort of relationship between $(A B)^{-1}$ and $A^{-1}$ and $B^{-1}$ ? A first guess that seems plausible is $(A B)^{-1}=A^{-1} B^{-1}$. Is this true? Using our work from above, we have
$$
A^{-1} B^{-1}=\left[\begin{array}{cc}
1 / 3 & -2 / 3 \\
0 & 1
\end{array}\right]\left[\begin{array}{cc}
-1 / 2 & 0 \\
1 / 2 & 1
\end{array}\right]=\left[\begin{array}{cc}
-1 / 2 & -2 / 3 \\
1 / 2 & 1
\end{array}\right] .
$$
Obviously, this is not equal to $(A B)^{-1}$. Before we do some further guessing, let's think about what the inverse of $A B$ is supposed to do. The inverse - let's call it $C$ - is supposed to be a matrix such that
$$
(A B) C=C(A B)=I .
$$
In examining the expression $(A B) C$, we see that we want $B$ to somehow "cancel" with $C$. What "cancels" $B$ ? An obvious answer is $B^{-1}$. This gives us a thought: perhaps we got the order of $A^{-1}$ and $B^{-1}$ wrong before. After all, we were hoping to find that
$$
A B A^{-1} B^{-1} \stackrel{?}{=} I,
$$
but algebraically speaking, it is hard to cancel out these terms. ${ }^{21}$ However, switching the order of $A^{-1}$ and $B^{-1}$ gives us some hope. Is $(A B)^{-1}=B^{-1} A^{-1}$ ? Let's see.
$$
\begin{aligned}
(A B)\left(B^{-1} A^{-1}\right) & =A\left(B B^{-1}\right) A^{-1} & & \text { (regrouping by the associative property) } \\
& =A I A^{-1} & & \left(B B^{-1}=I\right) \\
& =A A^{-1} & & (A I=A) \\
& =I & & \left(A A^{-1}=I\right)
\end{aligned}
$$
Thus it seems that $(A B)^{-1}=B^{-1} A^{-1}$. Let's confirm this with our example matrices.
$$
B^{-1} A^{-1}=\left[\begin{array}{cc}
-1 / 2 & 0 \\
1 / 2 & 1
\end{array}\right]\left[\begin{array}{cc}
1 / 3 & -2 / 3 \\
0 & 1
\end{array}\right]=\left[\begin{array}{cc}
-1 / 6 & 1 / 3 \\
1 / 6 & 2 / 3
\end{array}\right]=(A B)^{-1} .
$$
It worked!
4. Is there some sort of connection between $\left(A^{-1}\right)^{-1}$ and $A$ ? The answer is pretty obvious: they are equal. The "inverse of the inverse" returns one to the original matrix.
5. Is there some sort of relationship between $(A+B)^{-1}, A^{-1}$ and $B^{-1}$ ? Certainly, if we were forced to make a guess without working any examples, we would guess that
$$
(A+B)^{-1} \stackrel{?}{=} A^{-1}+B^{-1} .
$$
However, we saw that in our example, the matrix $(A+B)$ isn't even invertible. This pretty much kills any hope of a connection.
6. Is there a connection between $(5 A)^{-1}$ and $A^{-1}$ ? Consider:
$$
\begin{aligned}
(5 A)^{-1} & =\left[\begin{array}{cc}
1 / 15 & -2 / 15 \\
0 & 1 / 5
\end{array}\right] \\
& =\frac{1}{5}\left[\begin{array}{cc}
1 / 3 & -2 / 3 \\
0 & 1 / 5
\end{array}\right] \\
& =\frac{1}{5} A^{-1}
\end{aligned}
$$
Yes, there is a connection!
Let's summarize the results of this example. If $A$ and $B$ are both invertible matrices, then so is their product, $A B$. We demonstrated this with our example, and there is
${ }^{21}$ Recall that matrix multiplication is not commutative. more to be said. Let's suppose that $A$ and $B$ are $n \times n$ matrices, but we don't yet know if they are invertible. If $A B$ is invertible, then each of $A$ and $B$ are; if $A B$ is not invertible, then $A$ or $B$ is also not invertible.
In short, invertibility "works well" with matrix multiplication. However, we saw that it doesn't work well with matrix addition. Knowing that $A$ and $B$ are invertible does not help us find the inverse of $(A+B)$; in fact, the latter matrix may not even be invertible. ${ }^{22}$
Let's do one more example, then we'll summarize the results of this section in a theorem.
Example $59 \quad$ Find the inverse of $A=\left[\begin{array}{ccc}2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & -7\end{array}\right]$.
Solution We'll find $A^{-1}$ using Key Idea 10.
$$
\left[\begin{array}{cccccc}
2 & 0 & 0 & 1 & 0 & 0 \\
0 & 3 & 0 & 0 & 1 & 0 \\
0 & 0 & -7 & 0 & 0 & 1
\end{array}\right] \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{cccccc}
1 & 0 & 0 & 1 / 2 & 0 & 0 \\
0 & 1 & 0 & 0 & 1 / 3 & 0 \\
0 & 0 & 1 & 0 & 0 & -1 / 7
\end{array}\right]
$$
Therefore
$$
A^{-1}=\left[\begin{array}{ccc}
1 / 2 & 0 & 0 \\
0 & 1 / 3 & 0 \\
0 & 0 & -1 / 7
\end{array}\right]
$$
The matrix $A$ in the previous example is a diagonal matrix: the only nonzero entries of $A$ lie on the diagonal. ${ }^{23}$ The relationship between $A$ and $A^{-1}$ in the above example seems pretty strong, and it holds true in general. We'll state this and summarize the results of this section with the following theorem.
${ }^{22}$ The fact that invertibility works well with matrix multiplication should not come as a surprise. After all, saying that $A$ is invertible makes a statement about the mulitiplicative properties of $A$. It says that I can multiply $A$ with a special matrix to get $I$. Invertibility, in and of itself, says nothing about matrix addition, therefore we should not be too surprised that it doesn't work well with it.
${ }^{23}$ We still haven't formally defined diagonal, but the definition is rather visual so we risk it. See Definition 20 on page 123 for more details. Let $A$ and $B$ be $n \times n$ invertible matrices. Then:
1. $A B$ is invertible; $(A B)^{-1}=B^{-1} A^{-1}$.
2. $A^{-1}$ is invertible; $\left(A^{-1}\right)^{-1}=A$.
3. $n A$ is invertible for any nonzero scalar $n ;(n A)^{-1}=$ $\frac{1}{n} A^{-1}$.
4. If $A$ is a diagonal matrix, with diagonal entries $d_{1}, d_{2}, \cdots, d_{n}$, where none of the diagonal entries are 0 , then $A^{-1}$ exists and is a diagonal matrix. Furthermore, the diagonal entries of $A^{-1}$ are $1 / d_{1}, 1 / d_{2}, \cdots, 1 / d_{n}$.
Furthermore,
1. If a product $A B$ is not invertible, then $A$ or $B$ is not invertible.
2. If $A$ or $B$ are not invertible, then $A B$ is not invertible.
We end this section with a comment about solving systems of equations "in real life."24 Solving a system $A \vec{x}=\vec{b}$ by computing $A^{-1} \vec{b}$ seems pretty slick, so it would make sense that this is the way it is normally done. However, in practice, this is rarely done. There are two main reasons why this is the case.
First, computing $A^{-1}$ and $A^{-1} \vec{b}$ is "expensive" in the sense that it takes up a lot of computing time. Certainly, our calculators have no trouble dealing with the $3 \times 3$ cases we often consider in this textbook, but in real life the matrices being considered are very large (as in, hundreds of thousand rows and columns). Computing $A^{-1}$ alone is rather impractical, and we waste a lot of time if we come to find out that $A^{-1}$ does not exist. Even if we already know what $A^{-1}$ is, computing $A^{-1} \vec{b}$ is computationally expensive - Gaussian elimination is faster.
Secondly, computing $A^{-1}$ using the method we've described often gives rise to numerical roundoff errors. Even though computers often do computations with an accuracy to more than 8 decimal places, after thousands of computations, roundoffs
${ }^{24}$ Yes, real people do solve linear equations in real life. Not just mathematicians, but economists, engineers, and scientists of all flavors regularly need to solve linear equations, and the matrices they use are often huge.
Most people see matrices at work without thinking about it. Digital pictures are simply "rectangular arrays" of numbers representing colors - they are matrices of colors. Many of the standard image processing operations involve matrix operations. The author's wife has a "7 megapixel" camera which creates pictures that are $3072 \times 2304$ in size, giving over 7 million pixels, and that isn't even considered a "large" picture these days. can cause big errors. (A "small" 1,000 $\times 1,000$ matrix has 1, 000, 000 entries! That's a lot of places to have roundoff errors accumulate!) It is not unheard of to have a computer compute $A^{-1}$ for a large matrix, and then immediately have it compute $A A^{-1}$ and not get the identity matrix. ${ }^{25}$
Therefore, in real life, solutions to $A \vec{x}=\vec{b}$ are usually found using the methods we learned in Section 2.4. It turns out that even with all of our advances in mathematics, it is hard to beat the basic method that Gauss introduced a long time ago.
## Exercises 2.7
In Exercises 1-4, matrices $A$ and $B$ are given. Compute $(A B)^{-1}$ and $B^{-1} A^{-1}$.
1. $A=\left[\begin{array}{ll}1 & 2 \\ 1 & 1\end{array}\right], \quad B=\left[\begin{array}{ll}3 & 5 \\ 2 & 5\end{array}\right]$
2. $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right], \quad B=\left[\begin{array}{ll}7 & 1 \\ 2 & 1\end{array}\right]$
3. $A=\left[\begin{array}{ll}2 & 5 \\ 3 & 8\end{array}\right], \quad B=\left[\begin{array}{cc}1 & -1 \\ 1 & 4\end{array}\right]$
4. $A=\left[\begin{array}{ll}2 & 4 \\ 2 & 5\end{array}\right], \quad B=\left[\begin{array}{ll}2 & 2 \\ 6 & 5\end{array}\right]$
In Exercises 5-8, a $2 \times 2$ matrix $A$ is given.
Compute $A^{-1}$ and $\left(A^{-1}\right)^{-1}$ using Theorem 7.
5. $A=\left[\begin{array}{cc}-3 & 5 \\ 1 & -2\end{array}\right]$
6. $A=\left[\begin{array}{ll}3 & 5 \\ 2 & 4\end{array}\right]$
7. $A=\left[\begin{array}{ll}2 & 7 \\ 1 & 3\end{array}\right]$ 8. $A=\left[\begin{array}{ll}9 & 0 \\ 7 & 9\end{array}\right]$
8. Find $2 \times 2$ matrices $A$ and $B$ that are each invertible, but $A+B$ is not.
9. Create a random $6 \times 6$ matrix $A$, then have a calculator or computer compute $A A^{-1}$. Was the identity matrix returned exactly? Comment on your results.
10. Use a calculator or computer to compute $A A^{-1}$, where
$$
A=\left[\begin{array}{cccc}
1 & 2 & 3 & 4 \\
1 & 4 & 9 & 16 \\
1 & 8 & 27 & 64 \\
1 & 16 & 81 & 256
\end{array}\right]
$$
Was the identity matrix returned exactly? Comment on your results.
${ }^{25}$ The result is usually very close, with the numbers on the diagonal close to 1 and the other entries near 0 . But it isn't exactly the identity matrix.
## OPERATIONS ON MATRICES
In the previous chapter we learned about matrix arithmetic: adding, subtracting, and multiplying matrices, finding inverses, and multiplying by scalars. In this chapter we learn about some operations that we perform on matrices. We can think of them as functions: you input a matrix, and you get something back. One of these operations, the transpose, will return another matrix. With the other operations, the trace and the determinant, we input matrices and get numbers in return, an idea that is different than what we have seen before.
### The Matrix Transpose
## AS YOU READ
1. $\mathrm{T} / \mathrm{F}$ : If $A$ is a $3 \times 5$ matrix, then $A^{T}$ will be a $5 \times 3$ matrix.
2. Where are there zeros in an upper triangular matrix?
3. T/F: A matrix is symmetric if it doesn't change when you take its transpose.
4. What is the transpose of the transpose of $A$ ?
5. Give 2 other terms to describe symmetric matrices besides "interesting."
We jump right in with a definition. Definition 19
Let $A$ be an $m \times n$ matrix. The tranpsose of $A$, denoted $A^{T}$, is the $n \times m$ matrix whose columns are the respective rows of $A$.
Examples will make this definition clear.
Example $60 \quad$ Find the transpose of $A=\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6\end{array}\right]$.
Solution Note that $A$ is a $2 \times 3$ matrix, so $A^{T}$ will be a $3 \times 2$ matrix. By the definition, the first column of $A^{T}$ is the first row of $A$; the second column of $A^{T}$ is the second row of $A$. Therefore,
$$
A^{T}=\left[\begin{array}{ll}
1 & 4 \\
2 & 5 \\
3 & 6
\end{array}\right]
$$
Example $61 \quad$ Find the transpose of the following matrices.
$$
A=\left[\begin{array}{cccc}
7 & 2 & 9 & 1 \\
2 & -1 & 3 & 0 \\
-5 & 3 & 0 & 11
\end{array}\right] \quad B=\left[\begin{array}{ccc}
1 & 10 & -2 \\
3 & -5 & 7 \\
4 & 2 & -3
\end{array}\right] \quad C=\left[\begin{array}{lllll}
1 & -1 & 7 & 8 & 3
\end{array}\right]
$$
Solution We find each transpose using the definition without explanation. Make note of the dimensions of the original matrix and the dimensions of its transpose.
$$
A^{T}=\left[\begin{array}{ccc}
7 & 2 & -5 \\
2 & -1 & 3 \\
9 & 3 & 0 \\
1 & 0 & 11
\end{array}\right] \quad B^{T}=\left[\begin{array}{ccc}
1 & 3 & 4 \\
10 & -5 & 2 \\
-2 & 7 & -3
\end{array}\right] \quad C^{T}=\left[\begin{array}{c}
1 \\
-1 \\
7 \\
8 \\
3
\end{array}\right]
$$
Notice that with matrix $B$, when we took the transpose, the diagonal did not change. We can see what the diagonal is below where we rewrite $B$ and $B^{T}$ with the diagonal in bold. We'll follow this by a definition of what we mean by "the diagonal of a matrix," along with a few other related definitions.
$$
B=\left[\begin{array}{ccc}
1 & 10 & -2 \\
3 & -5 & 7 \\
4 & 2 & -3
\end{array}\right] \quad B^{T}=\left[\begin{array}{ccc}
1 & 3 & 4 \\
10 & -5 & 2 \\
-2 & 7 & -3
\end{array}\right]
$$
It is probably pretty clear why we call those entries "the diagonal." Here is the formal definition. Definition 20
Example 62 where
$$
A=\left[\begin{array}{lll}
1 & 2 & 3 \\
0 & 4 & 5 \\
0 & 0 & 6
\end{array}\right] \quad B=\left[\begin{array}{ccc}
3 & 0 & 0 \\
0 & 7 & 0 \\
0 & 0 & -1
\end{array}\right] \quad C=\left[\begin{array}{lll}
1 & 2 & 3 \\
0 & 4 & 5 \\
0 & 0 & 6 \\
0 & 0 & 0
\end{array}\right] .
$$
Identify the diagonal of each matrix, and state whether each matrix is diagonal, upper triangular, lower triangular, or none of the above.
Solution We first compute the transpose of each matrix.
$$
A^{T}=\left[\begin{array}{lll}
1 & 0 & 0 \\
2 & 4 & 0 \\
3 & 5 & 6
\end{array}\right] \quad B^{T}=\left[\begin{array}{ccc}
3 & 0 & 0 \\
0 & 7 & 0 \\
0 & 0 & -1
\end{array}\right] \quad C^{T}=\left[\begin{array}{llll}
1 & 0 & 0 & 0 \\
2 & 4 & 0 & 0 \\
3 & 5 & 6 & 0
\end{array}\right]
$$
Note that $I_{4}^{T}=I_{4}$.
The diagonals of $A$ and $A^{T}$ are the same, consisting of the entries 1,4 and 6 . The diagonals of $B$ and $B^{T}$ are also the same, consisting of the entries 3,7 and -1 . Finally, the diagonals of $C$ and $C^{T}$ are the same, consisting of the entries 1,4 and 6 .
The matrix $A$ is upper triangular; the only nonzero entries lie on or above the diagonal. Likewise, $A^{T}$ is lower triangular.
The matrix $B$ is diagonal. By their definitions, we can also see that $B$ is both upper and lower triangular. Likewise, $I_{4}$ is diagonal, as well as upper and lower triangular.
Finally, $C$ is upper triangular, with $C^{T}$ being lower triangular.
Make note of the definitions of diagonal and triangular matrices. We specify that a diagonal matrix must be square, but triangular matrices don't have to be. ("Most" of the time, however, the ones we study are.) Also, as we mentioned before in the example, by definition a diagonal matrix is also both upper and lower triangular. Finally, notice that by definition, the transpose of an upper triangular matrix is a lower triangular matrix, and vice-versa. There are many questions to probe concerning the transpose operations. ${ }^{1}$ The first set of questions we'll investigate involve the matrix arithmetic we learned from last chapter. We do this investigation by way of examples, and then summarize what we have learned at the end.
Example $63 \quad$ Let
$$
A=\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6
\end{array}\right] \text { and } B=\left[\begin{array}{ccc}
1 & 2 & 1 \\
3 & -1 & 0
\end{array}\right]
$$
Find $A^{T}+B^{T}$ and $(A+B)^{T}$.
Solution We note that
$$
A^{T}=\left[\begin{array}{ll}
1 & 4 \\
2 & 5 \\
3 & 6
\end{array}\right] \text { and } B^{T}=\left[\begin{array}{cc}
1 & 3 \\
2 & -1 \\
1 & 0
\end{array}\right]
$$
Therefore
$$
\begin{aligned}
A^{T}+B^{T} & =\left[\begin{array}{ll}
1 & 4 \\
2 & 5 \\
3 & 6
\end{array}\right]+\left[\begin{array}{cc}
1 & 3 \\
2 & -1 \\
1 & 0
\end{array}\right] \\
& =\left[\begin{array}{ll}
2 & 7 \\
4 & 4 \\
4 & 6
\end{array}\right] .
\end{aligned}
$$
Also,
$$
\begin{aligned}
(A+B)^{T} & =\left(\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6
\end{array}\right]+\left[\begin{array}{ccc}
1 & 2 & 1 \\
3 & -1 & 0
\end{array}\right]\right)^{T} \\
& =\left(\left[\begin{array}{lll}
2 & 4 & 4 \\
7 & 4 & 6
\end{array}\right]\right)^{T} \\
& =\left[\begin{array}{ll}
2 & 7 \\
4 & 4 \\
4 & 6
\end{array}\right]
\end{aligned}
$$
It looks like "the sum of the transposes is the transpose of the sum." ${ }^{2}$ This should lead us to wonder how the transpose works with multiplication.
Example $64 \quad$ Let
$$
A=\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right] \text { and } B=\left[\begin{array}{ccc}
1 & 2 & -1 \\
1 & 0 & 1
\end{array}\right] .
$$
${ }^{1}$ Remember, this is what mathematicians do. We learn something new, and then we ask lots of questions about it. Often the first questions we ask are along the lines of "How does this new thing relate to the old things I already know about?"
${ }^{2}$ This is kind of fun to say, especially when said fast. Regardless of how fast we say it, we should think about this statement. The "is" represents "equals." The stuff before "is" equals the stuff afterwards. Find $(A B)^{T}, A^{T} B^{T}$ and $B^{T} A^{T}$.
Solution We first note that
$$
A^{T}=\left[\begin{array}{ll}
1 & 3 \\
2 & 4
\end{array}\right] \text { and } B^{T}=\left[\begin{array}{cc}
1 & 1 \\
2 & 0 \\
-1 & 1
\end{array}\right]
$$
Find $(A B)^{T}$ :
$$
\begin{aligned}
(A B)^{T} & =\left(\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\left[\begin{array}{ccc}
1 & 2 & -1 \\
1 & 0 & 1
\end{array}\right]\right)^{T} \\
& =\left(\left[\begin{array}{lll}
3 & 2 & 1 \\
7 & 6 & 1
\end{array}\right]\right)^{T} \\
& =\left[\begin{array}{ll}
3 & 7 \\
2 & 6 \\
1 & 1
\end{array}\right]
\end{aligned}
$$
Now find $A^{T} B^{T}$ :
$$
\begin{aligned}
A^{T} B^{T} & =\left[\begin{array}{ll}
1 & 3 \\
2 & 4
\end{array}\right]\left[\begin{array}{cc}
1 & 1 \\
2 & 0 \\
-1 & 1
\end{array}\right] \\
& =\text { Not defined! }
\end{aligned}
$$
So we can't compute $A^{T} B^{T}$. Let's finish by computing $B^{T} A^{T}$ :
$$
\begin{aligned}
B^{T} A^{T} & =\left[\begin{array}{cc}
1 & 1 \\
2 & 0 \\
-1 & 1
\end{array}\right]\left[\begin{array}{ll}
1 & 3 \\
2 & 4
\end{array}\right] \\
& =\left[\begin{array}{ll}
3 & 7 \\
2 & 6 \\
1 & 1
\end{array}\right]
\end{aligned}
$$
We may have suspected that $(A B)^{T}=A^{T} B^{T}$. We saw that this wasn't the case, though - and not only was it not equal, the second product wasn't even defined! Oddly enough, though, we saw that $(A B)^{T}=B^{T} A^{T}$. ${ }^{3}$ To help understand why this is true, look back at the work above and confirm the steps of each multiplication.
We have one more arithmetic operation to look at: the inverse.
Example $65 \quad$ Let
$$
A=\left[\begin{array}{ll}
2 & 7 \\
1 & 4
\end{array}\right] .
$$
${ }^{3}$ Then again, maybe this isn't all that "odd." It is reminiscent of the fact that, when invertible, $(A B)^{-1}=$ $B^{-1} A^{-1}$.
## Chapter 3 Operations on Matrices
Find $\left(A^{-1}\right)^{T}$ and $\left(A^{T}\right)^{-1}$.
Solution We first find $A^{-1}$ and $A^{T}$ :
$$
A^{-1}=\left[\begin{array}{cc}
4 & -7 \\
-1 & 2
\end{array}\right] \text { and } A^{T}=\left[\begin{array}{ll}
2 & 1 \\
7 & 4
\end{array}\right]
$$
Finding $\left(A^{-1}\right)^{T}$ :
$$
\begin{aligned}
\left(A^{-1}\right)^{T} & =\left[\begin{array}{cc}
4 & -7 \\
-1 & 2
\end{array}\right]^{T} \\
& =\left[\begin{array}{cc}
4 & -1 \\
-7 & 2
\end{array}\right]
\end{aligned}
$$
Finding $\left(A^{T}\right)^{-1}$ :
$$
\begin{aligned}
\left(A^{T}\right)^{-1} & =\left[\begin{array}{ll}
2 & 1 \\
7 & 4
\end{array}\right]^{-1} \\
& =\left[\begin{array}{cc}
4 & -1 \\
-7 & 2
\end{array}\right]
\end{aligned}
$$
It seems that "the inverse of the transpose is the transpose of the inverse."
We have just looked at some examples of how the transpose operation interacts with matrix arithmetic operations. ${ }^{5}$ We now give a theorem that tells us that what we saw wasn't a coincidence, but rather is always true.
Theorem 11
Properties of the Matrix Transpose
Let $A$ and $B$ be matrices where the following operations are defined. Then:
1. $(A+B)^{T}=A^{T}+B^{T}$ and $(A-B)^{T}=A^{T}-B^{T}$
2. $(k A)^{T}=k A^{T}$
3. $(A B)^{T}=B^{T} A^{T}$
4. $\left(A^{-1}\right)^{T}=\left(A^{T}\right)^{-1}$
5. $\left(A^{T}\right)^{T}=A$
We included in the theorem two ideas we didn't discuss already. First, that $(k A)^{T}=$
${ }^{4}$ Again, we should think about this statement. The part before "is" states that we take the transpose of a matrix, then find the inverse. The part after "is" states that we find the inverse of the matrix, then take the transpose. Since these two statements are linked by an "is," they are equal.
${ }^{5}$ These examples don't prove anything, other than it worked in specific examples. $k A^{T}$. This is probably obvious. It doesn't matter when you multiply a matrix by a scalar when dealing with transposes.
The second "new" item is that $\left(A^{T}\right)^{T}=A$. That is, if we take the transpose of a matrix, then take its transpose again, what do we have? The original matrix.
Now that we know some properties of the transpose operation, we are tempted to play around with it and see what happens. For instance, if $A$ is an $m \times n$ matrix, we know that $A^{T}$ is an $n \times m$ matrix. So no matter what matrix $A$ we start with, we can always perform the multiplication $A A^{T}$ (and also $A^{T} A$ ) and the result is a square matrix!
Another thing to ask ourselves as we "play around" with the transpose: suppose $A$ is a square matrix. Is there anything special about $A+A^{T}$ ? The following example has us try out these ideas.
Example $66 \quad$ Let
$$
A=\left[\begin{array}{ccc}
2 & 1 & 3 \\
2 & -1 & 1 \\
1 & 0 & 1
\end{array}\right]
$$
Find $A A^{T}, A+A^{T}$ and $A-A^{T}$.
Solution Finding $A A^{T}$ :
$$
\begin{aligned}
A A^{T} & =\left[\begin{array}{ccc}
2 & 1 & 3 \\
2 & -1 & 1 \\
1 & 0 & 1
\end{array}\right]\left[\begin{array}{ccc}
2 & 2 & 1 \\
1 & -1 & 0 \\
3 & 1 & 1
\end{array}\right] \\
& =\left[\begin{array}{lll}
14 & 6 & 5 \\
6 & 4 & 3 \\
5 & 3 & 2
\end{array}\right]
\end{aligned}
$$
Finding $A+A^{T}$ :
$$
\begin{aligned}
A+A^{T} & =\left[\begin{array}{ccc}
2 & 1 & 3 \\
2 & -1 & 1 \\
1 & 0 & 1
\end{array}\right]+\left[\begin{array}{ccc}
2 & 2 & 1 \\
1 & -1 & 0 \\
3 & 1 & 1
\end{array}\right] \\
& =\left[\begin{array}{ccc}
2 & 3 & 4 \\
3 & -2 & 1 \\
4 & 1 & 2
\end{array}\right]
\end{aligned}
$$
Finding $A-A^{T}$ :
$$
\begin{aligned}
A-A^{T} & =\left[\begin{array}{ccc}
2 & 1 & 3 \\
2 & -1 & 1 \\
1 & 0 & 1
\end{array}\right]-\left[\begin{array}{ccc}
2 & 2 & 1 \\
1 & -1 & 0 \\
3 & 1 & 1
\end{array}\right] \\
& =\left[\begin{array}{ccc}
0 & -1 & 2 \\
1 & 0 & 1 \\
-2 & -1 & 0
\end{array}\right]
\end{aligned}
$$
Let's look at the matrices we've formed in this example. First, consider $A A^{T}$. Something seems to be nice about this matrix - look at the location of the 6 's, the 5 's and the 3's. More precisely, let's look at the transpose of $A A^{T}$. We should notice that if we take the transpose of this matrix, we have the very same matrix. That is,
$$
\left(\left[\begin{array}{ccc}
14 & 6 & 5 \\
6 & 4 & 3 \\
5 & 3 & 2
\end{array}\right]\right)^{T}=\left[\begin{array}{ccc}
14 & 6 & 5 \\
6 & 4 & 3 \\
5 & 3 & 2
\end{array}\right] !
$$
We'll formally define this in a moment, but a matrix that is equal to its transpose is called symmetric.
Look at the next part of the example; what do we notice about $A+A^{T}$ ? We should see that it, too, is symmetric. Finally, consider the last part of the example: do we notice anything about $A-A^{T}$ ?
We should immediately notice that it is not symmetric, although it does seem "close." Instead of it being equal to its transpose, we notice that this matrix is the opposite of its transpose. We call this type of matrix skew symmetric. ${ }^{6}$ We formally define these matrices here.
## Definition 21
## Symmetric and Skew Symmetric Matrices
A matrix $A$ is symmetric if $A^{T}=A$.
A matrix $A$ is skew symmetric if $A^{T}=-A$.
Note that in order for a matrix to be either symmetric or skew symmetric, it must be square.
So why was $A A^{T}$ symmetric in our previous example? Did we just luck out? ${ }^{7}$ Let's $^{\text {s }}$ take the transpose of $A A^{T}$ and see what happens.
$$
\begin{array}{rlrl}
\left(A A^{T}\right)^{T} & =\left(A^{T}\right)^{T}(A)^{T} & & \text { transpose multiplication rule } \\
& =A A^{T} & \left(A^{T}\right)^{T}=A
\end{array}
$$
We have just proved that no matter what matrix $A$ we start with, the matrix $A A^{T}$ will be symmetric. Nothing in our string of equalities even demanded that $A$ be a square matrix; it is always true.
We can do a similar proof to show that as long as $A$ is square, $A+A^{T}$ is a symmetric matrix. ${ }^{8}$ We'll instead show here that if $A$ is a square matrix, then $A-A^{T}$ is skew
${ }^{6}$ Some mathematicians use the term antisymmetric
${ }^{7}$ Of course not.
${ }^{8}$ Why do we say that $A$ has to be square? symmetric.
$$
\begin{aligned}
\left(A-A^{T}\right)^{T} & =A^{T}-\left(A^{T}\right)^{T} \quad \text { transpose subtraction rule } \\
& =A^{T}-A \\
& =-\left(A-A^{T}\right)
\end{aligned}
$$
So we took the transpose of $A-A^{T}$ and we got $-\left(A-A^{T}\right)$; this is the definition of being skew symmetric.
We'll take what we learned from Example 66 and put it in a box. (We've already proved most of this is true; the rest we leave to solve in the Exercises.)
## Theorem 12
## Symmetric and Skew Symmetric Matrices
1. Given any matrix $A$, the matrices $A A^{T}$ and $A^{T} A$ are symmetric.
2. Let $A$ be a square matrix. The matrix $A+A^{T}$ is symmetric.
3. Let $A$ be a square matrix. The matrix $A-A^{T}$ is skew symmetric.
Why do we care about the transpose of a matrix? Why do we care about symmetric matrices?
There are two answers that each answer both of these questions. First, we are interested in the tranpose of a matrix and symmetric matrices because they are interesting. ${ }^{9}$ One particularly interesting thing about symmetric and skew symmetric matrices is this: consider the sum of $\left(A+A^{T}\right)$ and $\left(A-A^{T}\right)$ :
$$
\left(A+A^{T}\right)+\left(A-A^{T}\right)=2 A .
$$
This gives us an idea: if we were to multiply both sides of this equation by $\frac{1}{2}$, then the right hand side would just be $A$. This means that
$$
A=\underbrace{\frac{1}{2}\left(A+A^{T}\right)}_{\text {symmetric }}+\underbrace{\frac{1}{2}\left(A-A^{T}\right)}_{\text {skew symmetric }} .
$$
That is, any matrix $A$ can be written as the sum of a symmetric and skew symmetric matrix. That's interesting.
The second reason we care about them is that they are very useful and important in various areas of mathematics. The transpose of a matrix turns out to be an important
9or: "neat," "cool," "bad," "wicked," "phat," "fo-shizzle." operation; symmetric matrices have many nice properties that make solving certain types of problems possible.
Most of this text focuses on the preliminaries of matrix algebra, and the actual uses are beyond our current scope. One easy to describe example is curve fitting. Suppose we are given a large set of data points that, when plotted, look roughly quadratic. How do we find the quadratic that "best fits" this data? The solution can be found using matrix algebra, and specifically a matrix called the pseudoinverse. If $A$ is a matrix, the pseudoinverse of $A$ is the matrix $A^{\dagger}=\left(A^{T} A\right)^{-1} A^{T}$ (assuming that the inverse exists). We aren't going to worry about what all the above means; just notice that it has a cool sounding name and the transpose appears twice.
In the next section we'll learn about the trace, another operation that can be performed on a matrix that is relatively simple to compute but can lead to some deep results.
## Exercises 3.1
In Exercises $1-24$, a matrix $A$ is given. Find $A^{T}$; make note if $A$ is upper/lower triangular, diagonal, symmetric and/or skew symmetric.
1. $\left[\begin{array}{cc}-7 & 4 \\ 4 & -6\end{array}\right]$
2. $\left[\begin{array}{cc}3 & 1 \\ -7 & 8\end{array}\right]$
3. $\left[\begin{array}{ll}1 & 0 \\ 0 & 9\end{array}\right]$
4. $\left[\begin{array}{cc}13 & -3 \\ -3 & 1\end{array}\right]$
5. $\left[\begin{array}{cc}-5 & -9 \\ 3 & 1 \\ -10 & -8\end{array}\right]$
6. $\left[\begin{array}{cc}-2 & 10 \\ 1 & -7 \\ 9 & -2\end{array}\right]$
7. $\left[\begin{array}{cccc}4 & -7 & -4 & -9 \\ -9 & 6 & 3 & -9\end{array}\right]$
8. $\left[\begin{array}{cccc}3 & -10 & 0 & 6 \\ -10 & -2 & -3 & 1\end{array}\right]$
9. $\left[\begin{array}{llll}-7 & -8 & 2 & -3\end{array}\right]$
10. $\left[\begin{array}{llll}-9 & 8 & 2 & -7\end{array}\right]$
11. $\left[\begin{array}{ccc}-9 & 4 & 10 \\ 6 & -3 & -7 \\ -8 & 1 & -1\end{array}\right]$ 12. $\left[\begin{array}{ccc}4 & -5 & 2 \\ 1 & 5 & 9 \\ 9 & 2 & 3\end{array}\right]$
12. $\left[\begin{array}{ccc}4 & 0 & -2 \\ 0 & 2 & 3 \\ -2 & 3 & 6\end{array}\right]$
13. $\left[\begin{array}{ccc}0 & 3 & -2 \\ 3 & -4 & 1 \\ -2 & 1 & 0\end{array}\right]$
14. $\left[\begin{array}{ccc}2 & -5 & -3 \\ 5 & 5 & -6 \\ 7 & -4 & -10\end{array}\right]$
15. $\left[\begin{array}{ccc}0 & -6 & 1 \\ 6 & 0 & 4 \\ -1 & -4 & 0\end{array}\right]$
16. $\left[\begin{array}{ccc}4 & 2 & -9 \\ 5 & -4 & -10 \\ -6 & 6 & 9\end{array}\right]$
17. $\left[\begin{array}{ccc}4 & 0 & 0 \\ -2 & -7 & 0 \\ 4 & -2 & 5\end{array}\right]$
18. $\left[\begin{array}{ccc}-3 & -4 & -5 \\ 0 & -3 & 5 \\ 0 & 0 & -3\end{array}\right]$
19. $\left[\begin{array}{cccc}6 & -7 & 2 & 6 \\ 0 & -8 & -1 & 0 \\ 0 & 0 & 1 & -7\end{array}\right]$
20. $\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1\end{array}\right]$
21. $\left[\begin{array}{ccc}6 & -4 & -5 \\ -4 & 0 & 2 \\ -5 & 2 & -2\end{array}\right]$
22. $\left[\begin{array}{ccc}0 & 1 & -2 \\ -1 & 0 & 4 \\ 2 & -4 & 0\end{array}\right]$
23. $\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]$
### The Matrix Trace
## AS YOU READ
1. $\mathrm{T} / \mathrm{F}$ : We only compute the trace of square matrices.
2. T/F: One can tell if a matrix is invertible by computing the trace.
In the previous section, we learned about an operation we can peform on matrices, namely the transpose. Given a matrix $A$, we can "find the transpose of $A$," which is another matrix. In this section we learn about a new operation called the trace. It is a different type of operation than the transpose. Given a matrix $A$, we can "find the trace of $A$, , which is not a matrix but rather a number. We formally define it here.
Definition 22
## The Trace
Let $A$ be an $n \times n$ matrix. The trace of $A$, denoted $\operatorname{tr}(A)$, is the sum of the diagonal elements of $A$. That is,
$$
\operatorname{tr}(A)=a_{11}+a_{22}+\cdots+a_{n n} .
$$
This seems like a simple definition, and it really is. Just to make sure it is clear, let's practice.
Example $67 \quad$ Find the trace of $A, B, C$ and $I_{4}$, where
$$
A=\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right], B=\left[\begin{array}{ccc}
1 & 2 & 0 \\
3 & 8 & 1 \\
-2 & 7 & -5
\end{array}\right] \text { and } C=\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6
\end{array}\right]
$$
Solution To find the trace of $A$, note that the diagonal elements of $A$ are 1 and 4. Therefore, $\operatorname{tr}(A)=1+4=5$.
We see that the diagonal elements of $B$ are 1,8 and -5 , so $\operatorname{tr}(B)=1+8-5=4$. The matrix $C$ is not a square matrix, and our definition states that we must start with a square matrix. Therefore $\operatorname{tr}(C)$ is not defined.
Finally, the diagonal of $I_{4}$ consists of four $1 \mathrm{~s}$. Therefore $\operatorname{tr}\left(I_{4}\right)=4$.
Now that we have defined the trace of a matrix, we should think like mathematicians and ask some questions. The first questions that should pop into our minds should be along the lines of "How does the trace work with other matrix operations?"10 We should think about how the trace works with matrix addition, scalar multiplication, matrix multiplication, matrix inverses, and the transpose.
We'll give a theorem that will formally tell us what is true in a moment, but first let's play with two sample matrices and see if we can see what will happen. Let
$$
A=\left[\begin{array}{ccc}
2 & 1 & 3 \\
2 & 0 & -1 \\
3 & -1 & 3
\end{array}\right] \text { and } B=\left[\begin{array}{ccc}
2 & 0 & 1 \\
-1 & 2 & 0 \\
0 & 2 & -1
\end{array}\right]
$$
It should be clear that $\operatorname{tr}(A)=5$ and $\operatorname{tr}(B)=3$. What is $\operatorname{tr}(A+B)$ ?
$$
\begin{aligned}
\operatorname{tr}(A+B) & =\operatorname{tr}\left(\left[\begin{array}{ccc}
2 & 1 & 3 \\
2 & 0 & -1 \\
3 & -1 & 3
\end{array}\right]+\left[\begin{array}{ccc}
2 & 0 & 1 \\
-1 & 2 & 0 \\
0 & 2 & -1
\end{array}\right]\right) \\
& =\operatorname{tr}\left(\left[\begin{array}{ccc}
4 & 1 & 4 \\
1 & 2 & -1 \\
3 & 1 & 2
\end{array}\right]\right) \\
& =8
\end{aligned}
$$
So we notice that $\operatorname{tr}(A+B)=\operatorname{tr}(A)+\operatorname{tr}(B)$. This probably isn't a coincidence.
How does the trace work with scalar multiplication? If we multiply $A$ by 4 , then the diagonal elements will be 8,0 and 12 , so $\operatorname{tr}(4 A)=20$. Is it a coincidence that this is 4 times the trace of $A$ ?
Let's move on to matrix multiplication. How will the trace of $A B$ relate to the traces of $A$ and $B$ ? Let's see:
$$
\begin{aligned}
\operatorname{tr}(A B) & =\operatorname{tr}\left(\left[\begin{array}{ccc}
2 & 1 & 3 \\
2 & 0 & -1 \\
3 & -1 & 3
\end{array}\right]\left[\begin{array}{ccc}
2 & 0 & 1 \\
-1 & 2 & 0 \\
0 & 2 & -1
\end{array}\right]\right) \\
& =\operatorname{tr}\left(\left[\begin{array}{ccc}
3 & 8 & -1 \\
4 & -2 & 3 \\
7 & 4 & 0
\end{array}\right]\right) \\
& =1
\end{aligned}
$$
${ }^{10}$ Recall that we asked a similar question once we learned about the transpose. It isn't exactly clear what the relationship is among $\operatorname{tr}(A), \operatorname{tr}(B)$ and $\operatorname{tr}(A B)$. Before moving on, let's find $\operatorname{tr}(B A)$ :
$$
\begin{aligned}
\operatorname{tr}(B A) & =\operatorname{tr}\left(\left[\begin{array}{ccc}
2 & 0 & 1 \\
-1 & 2 & 0 \\
0 & 2 & -1
\end{array}\right]\left[\begin{array}{ccc}
2 & 1 & 3 \\
2 & 0 & -1 \\
3 & -1 & 3
\end{array}\right]\right) \\
& =\operatorname{tr}\left(\left[\begin{array}{ccc}
7 & 1 & 9 \\
2 & -1 & -5 \\
1 & 1 & -5
\end{array}\right]\right) \\
& =1
\end{aligned}
$$
We notice that $\operatorname{tr}(A B)=\operatorname{tr}(B A)$. Is this coincidental?
How are the traces of $A$ and $A^{-1}$ related? We compute $A^{-1}$ and find that
$$
A^{-1}=\left[\begin{array}{ccc}
1 / 17 & 6 / 17 & 1 / 17 \\
9 / 17 & 3 / 17 & -8 / 17 \\
2 / 17 & -5 / 17 & 2 / 17
\end{array}\right] .
$$
Therefore $\operatorname{tr}\left(A^{-1}\right)=6 / 17$. Again, the relationship isn't clear. ${ }^{11}$
Finally, let's see how the trace is related to the transpose. We actually don't have to formally compute anything. Recall from the previous section that the diagonals of $A$ and $A^{T}$ are identical; therefore, $\operatorname{tr}(A)=\operatorname{tr}\left(A^{T}\right)$. That, we know for sure, isn't a coincidence.
We now formally state what equalities are true when considering the interaction of the trace with other matrix operations.
Theorem 13
## Properties of the Matrix Trace
Let $A$ and $B$ be $n \times n$ matrices. Then:
1. $\operatorname{tr}(A+B)=\operatorname{tr}(A)+\operatorname{tr}(B)$
2. $\operatorname{tr}(A-B)=\operatorname{tr}(A)-\operatorname{tr}(B)$
3. $\operatorname{tr}(k A)=k \cdot \operatorname{tr}(A)$
4. $\operatorname{tr}(A B)=\operatorname{tr}(B A)$
5. $\operatorname{tr}\left(A^{T}\right)=\operatorname{tr}(A)$
One of the key things to note here is what this theorem does not say. It says nothing about how the trace relates to inverses. The reason for the silence in these areas is that there simply is not a relationship.
${ }^{11}$ Something to think about: we know that not all square matrices are invertible. Would we be able to tell just by the trace? That seems unlikely. We end this section by again wondering why anyone would care about the trace of matrix. One reason mathematicians are interested in it is that it can give a measurement of the "size"12 of a matrix.
Consider the following $2 \times 2$ matrices:
$$
A=\left[\begin{array}{cc}
1 & -2 \\
1 & 1
\end{array}\right] \text { and } B=\left[\begin{array}{cc}
6 & 7 \\
11 & -4
\end{array}\right]
$$
These matrices have the same trace, yet $B$ clearly has bigger elements in it. So how can we use the trace to determine a "size" of these matrices? We can consider $\operatorname{tr}\left(A^{T} A\right)$ and $\operatorname{tr}\left(B^{T} B\right)$.
$$
\begin{aligned}
\operatorname{tr}\left(A^{T} A\right) & =\operatorname{tr}\left(\left[\begin{array}{cc}
1 & 1 \\
-2 & 1
\end{array}\right]\left[\begin{array}{cc}
1 & -2 \\
1 & 1
\end{array}\right]\right) \\
& =\operatorname{tr}\left(\left[\begin{array}{cc}
2 & -1 \\
-1 & 5
\end{array}\right]\right) \\
& =7 \\
\operatorname{tr}\left(B^{T} B\right) & =\operatorname{tr}\left(\left[\begin{array}{cc}
6 & 11 \\
7 & -4
\end{array}\right]\left[\begin{array}{cc}
6 & 7 \\
11 & -4
\end{array}\right]\right) \\
& =\operatorname{tr}\left(\left[\begin{array}{cc}
157 & -2 \\
-2 & 65
\end{array}\right]\right) \\
& =222
\end{aligned}
$$
Our concern is not how to interpret what this "size" measurement means, but rather to demonstrate that the trace (along with the transpose) can be used to give (perhaps useful) information about a matrix. ${ }^{13}$
${ }^{12}$ There are many different measurements of a matrix size. In this text, we just refer to its dimensions. Some measurements of size refer the magnitude of the elements in the matrix. The next section describes yet another measurement of matrix size.
${ }^{13}$ This example brings to light many interesting ideas that we'll flesh out just a little bit here.
1. Notice that the elements of $A$ are $1,-2,1$ and 1 . Add the squares of these numbers: $1^{2}+(-2)^{2}+$ $1^{2}+1^{2}=7=\operatorname{tr}\left(A^{T} A\right)$.
Notice that the elements of $B$ are $6,7,11$ and -4 . Add the squares of these numbers: $6^{2}+7^{2}+$ $11^{2}+(-4)^{2}=222=\operatorname{tr}\left(B^{T} B\right)$.
Can you see why this is true? When looking at multiplying $A^{T} A$, focus only on where the elements on the diagonal come from since they are the only ones that matter when taking the trace.
2. You can confirm on your own that regardless of the dimensions of $A, \operatorname{tr}\left(A^{T} A\right)=\operatorname{tr}\left(A A^{T}\right)$. To see why this is true, consider the previous point. (Recall also that $A^{T} A$ and $A A^{T}$ are always square, regardless of the dimensions of $A$.)
3. Mathematicians are actually more interested in $\sqrt{\operatorname{tr}\left(A^{T} A\right)}$ than just $\operatorname{tr}\left(A^{T} A\right)$. The reason for this is a bit complicated; the short answer is that "it works better." The reason "it works better" is related to the Pythagorean Theorem, all of all things. If we know that the legs of a right triangle have length $a$ and $b$, we are more interested in $\sqrt{a^{2}+b^{2}}$ than just $a^{2}+b^{2}$. Of course, this explanation raises more questions than it answers; our goal here is just to whet your appetite and get you to do some more reading. A Numerical Linear Algebra book would be a good place to start.
## Exercises 3.2
In Exercises 1-15, find the trace of the given matrix.
1. $\left[\begin{array}{cc}1 & -5 \\ 9 & 5\end{array}\right]$
2. $\left[\begin{array}{cc}-3 & -10 \\ -6 & 4\end{array}\right]$
3. $\left[\begin{array}{cc}7 & 5 \\ -5 & -4\end{array}\right]$
4. $\left[\begin{array}{cc}-6 & 0 \\ -10 & 9\end{array}\right]$
5. $\left[\begin{array}{ccc}-4 & 1 & 1 \\ -2 & 0 & 0 \\ -1 & -2 & -5\end{array}\right]$
6. $\left[\begin{array}{ccc}0 & -3 & 1 \\ 5 & -5 & 5 \\ -4 & 1 & 0\end{array}\right]$
7. $\left[\begin{array}{ccc}-2 & -3 & 5 \\ 5 & 2 & 0 \\ -1 & -3 & 1\end{array}\right]$
8. $\left[\begin{array}{ccc}4 & 2 & -1 \\ -4 & 1 & 4 \\ 0 & -5 & 5\end{array}\right]$
9. $\left[\begin{array}{ccc}2 & 6 & 4 \\ -1 & 8 & -10\end{array}\right]$
10. $\left[\begin{array}{cc}6 & 5 \\ 2 & 10 \\ 3 & 3\end{array}\right]$
11. $\left[\begin{array}{cccc}-10 & 6 & -7 & -9 \\ -2 & 1 & 6 & -9 \\ 0 & 4 & -4 & 0 \\ -3 & -9 & 3 & -10\end{array}\right]$ 12. $\left[\begin{array}{cccc}5 & 2 & 2 & 2 \\ -7 & 4 & -7 & -3 \\ 9 & -9 & -7 & 2 \\ -4 & 8 & -8 & -2\end{array}\right]$
12. $I_{4}$
13. $I_{n}$
14. A matrix $A$ that is skew symmetric.
In Exercises 16 - 19, verify Theorem 13 by:
1. Showing that $\operatorname{tr}(A)+\operatorname{tr}(B)=\operatorname{tr}(A+B)$ and
2. Showing that $\operatorname{tr}(A B)=\operatorname{tr}(B A)$.
3. $A=\left[\begin{array}{ll}1 & -1 \\ 9 & -6\end{array}\right], \quad B=\left[\begin{array}{ll}-1 & 0 \\ -6 & 3\end{array}\right]$
4. $A=\left[\begin{array}{cc}0 & -8 \\ 1 & 8\end{array}\right], \quad B=\left[\begin{array}{ll}-4 & 5 \\ -4 & 2\end{array}\right]$
5. $A=\left[\begin{array}{ccc}-8 & -10 & 10 \\ 10 & 5 & -6 \\ -10 & 1 & 3\end{array}\right]$
$B=\left[\begin{array}{ccc}-10 & -4 & -3 \\ -4 & -5 & 4 \\ 3 & 7 & 3\end{array}\right]$
19. $A=\left[\begin{array}{ccc}-10 & 7 & 5 \\ 7 & 7 & -5 \\ 8 & -9 & 2\end{array}\right]$
$B=\left[\begin{array}{ccc}-3 & -4 & 9 \\ 4 & -1 & -9 \\ -7 & -8 & 10\end{array}\right]$
### The Determinant
## AS YOU READ
1. T/F: The determinant of a matrix is always positive.
2. T/F: To compute the determinant of a $3 \times 3$ matrix, one needs to compute the determinants of $32 \times 2$ matrices.
3. Give an example of a $2 \times 2$ matrix with a determinant of 3 . In this chapter so far we've learned about the transpose (an operation on a matrix that returns another matrix) and the trace (an operation on a square matrix that returns a number). In this section we'll learn another operation on square matrices that returns a number, called the determinant. We give a pseudo-definition of the determinant here.
The determinant of an $n \times n$ matrix $A$ is a number, denoted $\operatorname{det}(A)$, that is $\operatorname{determined}$ by $A$.
That definition isn't meant to explain everything; it just gets us started by making us realize that the determinant is a number. The determinant is kind of a tricky thing to define. Once you know and understand it, it isn't that hard, but getting started is a bit complicated. ${ }^{14}$ We start simply; we define the determinant for $2 \times 2$ matrices.
Definition 23
## Determinant of $2 \times 2$ Matrices
Let
$$
A=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]
$$
The determinant of $A$, denoted by
$$
\operatorname{det}(A) \text { or }\left|\begin{array}{ll}
a & b \\
c & d
\end{array}\right|,
$$
is $a d-b c$.
We've seen the expression $a d-b c$ before. In Section 2.6, we saw that a $2 \times 2$ matrix $A$ has inverse
$$
\frac{1}{a d-b c}\left[\begin{array}{cc}
d & -b \\
-c & a
\end{array}\right]
$$
as long as $a d-b c \neq 0$; otherwise, the inverse does not exist. We can rephrase the above statement now: If $\operatorname{det}(A) \neq 0$, then
$$
A^{-1}=\frac{1}{\operatorname{det}(A)}\left[\begin{array}{cc}
d & -b \\
-c & a
\end{array}\right] .
$$
A brief word about the notation: notice that we can refer to the determinant by using what looks like absolute value bars around the entries of a matrix. We discussed at the end of the last section the idea of measuring the "size" of a matrix, and mentioned that there are many different ways to measure size. The determinant is one such way. Just as the absolute value of a number measures its size (and ignores its sign), the determinant of a matrix is a measurement of the size of the matrix. (Be careful, though: $\operatorname{det}(A)$ can be negative!)
Let's practice.
${ }^{14}$ It's similar to learning to ride a bike. The riding itself isn't hard, it is getting started that's difficult. Example $68 \quad$ Find the determinant of $A, B$ and $C$ where
$$
A=\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right], B=\left[\begin{array}{cc}
3 & -1 \\
2 & 7
\end{array}\right] \text { and } C=\left[\begin{array}{cc}
1 & -3 \\
-2 & 6
\end{array}\right]
$$
Solution Finding the determinant of $A$ :
$$
\begin{aligned}
\operatorname{det}(A) & =\left|\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right| \\
& =1(4)-2(3) \\
& =-2 .
\end{aligned}
$$
Similar computations show that $\operatorname{det}(B)=3(7)-(-1)(2)=23$ and $\operatorname{det}(C)=$ $1(6)-(-3)(-2)=0$.
Finding the determinant of a $2 \times 2$ matrix is pretty straightforward. It is natural to ask next "How do we compute the determinant of matrices that are not $2 \times 2$ ?" We first need to define some terms. ${ }^{15}$
Definition 24
Notice that this definition makes reference to taking the determinant of a matrix, while we haven't yet defined what the determinant is beyond $2 \times 2$ matrices. We recognize this problem, and we'll see how far we can go before it becomes an issue.
Examples will help.
## Example $69 \quad$ Let
$$
A=\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right] \text { and } B=\left[\begin{array}{cccc}
1 & 2 & 0 & 8 \\
-3 & 5 & 7 & 2 \\
-1 & 9 & -4 & 6 \\
1 & 1 & 1 & 1
\end{array}\right]
$$
Find $A_{1,3}, A_{3,2}, B_{2,1}, B_{4,3}$ and their respective cofactors.
${ }^{15}$ This is the standard definition of these two terms, although slight variations exist. Solution To compute the minor $A_{1,3}$, we remove the first row and third column of $A$ then take the determinant.
$$
\begin{gathered}
A=\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right] \Rightarrow\left[\begin{array}{lll}
\mathbf{1} & \mathbf{2} & \mathbf{3} \\
4 & 5 & 6 \\
7 & 8 & \mathbf{9}
\end{array}\right] \Rightarrow\left[\begin{array}{ll}
4 & 5 \\
7 & 8
\end{array}\right] \\
A_{1,3}=\left|\begin{array}{ll}
4 & 5 \\
7 & 8
\end{array}\right|=32-35=-3 .
\end{gathered}
$$
The corresponding cofactor, $C_{1,3}$, is
$$
C_{1,3}=(-1)^{1+3} A_{1,3}=(-1)^{4}(-3)=-3 .
$$
The minor $A_{3,2}$ is found by removing the third row and second column of $A$ then taking the determinant.
$$
\begin{gathered}
A=\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right] \Rightarrow\left[\begin{array}{lll}
1 & \boldsymbol{z} & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right] \Rightarrow\left[\begin{array}{ll}
1 & 3 \\
4 & 6
\end{array}\right] \\
A_{3,2}=\left|\begin{array}{ll}
1 & 3 \\
4 & 6
\end{array}\right|=6-12=-6 .
\end{gathered}
$$
The corresponding cofactor, $C_{3,2}$, is
$$
C_{3,2}=(-1)^{3+2} A_{3,2}=(-1)^{5}(-6)=6 .
$$
The minor $B_{2,1}$ is found by removing the second row and first column of $B$ then taking the determinant.
$$
\begin{gathered}
B=\left[\begin{array}{cccc}
1 & 2 & 0 & 8 \\
-3 & 5 & 7 & 2 \\
-1 & 9 & -4 & 6 \\
1 & 1 & 1 & 1
\end{array}\right] \Rightarrow\left[\begin{array}{cccc}
\mathbf{1} & 2 & 0 & 8 \\
-3 & \mathbf{5} & \mathbf{7} & \mathbf{z} \\
\mathbf{- 1} & 9 & -4 & 6 \\
\mathbf{1} & 1 & 1 & 1
\end{array}\right] \Rightarrow\left[\begin{array}{ccc}
2 & 0 & 8 \\
9 & -4 & 6 \\
1 & 1 & 1
\end{array}\right] \\
B_{2,1}=\left|\begin{array}{ccc}
2 & 0 & 8 \\
9 & -4 & 6 \\
1 & 1 & 1
\end{array}\right| \stackrel{!}{=} ?
\end{gathered}
$$
We're a bit stuck. We don't know how to find the determinate of this $3 \times 3$ matrix. We'll come back to this later. The corresponding cofactor is
$$
C_{2,1}=(-1)^{2+1} B_{2,1}=-B_{2,1},
$$
whatever this number happens to be.
The minor $B_{4,3}$ is found by removing the fourth row and third column of $B$ then taking the determinant.
$$
B=\left[\begin{array}{cccc}
1 & 2 & 0 & 8 \\
-3 & 5 & 7 & 2 \\
-1 & 9 & -4 & 6 \\
1 & 1 & 1 & 1
\end{array}\right] \Rightarrow\left[\begin{array}{cccc}
1 & 2 & \mathbf{0} & 8 \\
-3 & 5 & \mathbf{7} & 2 \\
-1 & 9 & -4 & 6 \\
\mathbf{1} & \mathbf{1} & \mathbf{1} & \mathbf{1}
\end{array}\right] \Rightarrow\left[\begin{array}{ccc}
1 & 2 & 8 \\
-3 & 5 & 2 \\
-1 & 9 & 6
\end{array}\right]
$$
$$
B_{4,3}=\left|\begin{array}{ccc}
1 & 2 & 8 \\
-3 & 5 & 2 \\
-1 & 9 & 6
\end{array}\right| \stackrel{!}{=} ?
$$
Again, we're stuck. We won't be able to fully compute $C_{4,3}$; all we know so far is that
$$
C_{4,3}=(-1)^{4+3} B_{4,3}=(-1) B_{4,3} .
$$
Once we learn how to compute determinates for matrices larger than $2 \times 2$ we can come back and finish this exercise.
In our previous example we ran into a bit of trouble. By our definition, in order to compute a minor of an $n \times n$ matrix we needed to compute the determinant of a $(n-1) \times(n-1)$ matrix. This was fine when we started with a $3 \times 3$ matrix, but when we got up to a $4 \times 4$ matrix (and larger) we run into trouble.
We are almost ready to define the determinant for any square matrix; we need one last definition.
## Definition 25
## Cofactor Expansion
Let $A$ be an $n \times n$ matrix.
The cofactor expansion of $A$ along the $i^{\text {th }}$ row is the sum
$$
a_{i, 1} c_{i, 1}+a_{i, 2} c_{i, 2}+\cdots+a_{i, n} c_{i, n} .
$$
The cofactor expansion of $A$ down the $j^{\text {th }}$ column is the sum
$$
a_{1, j} C_{1, j}+a_{2, j} C_{2, j}+\cdots+a_{n, j} C_{n, j}
$$
The notation of this definition might be a little intimidating, so let's look at an example.
## Example $70 \quad$ Let
$$
A=\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right]
$$
Find the cofactor expansions along the second row and down the first column.
Solution By the definition, the cofactor expansion along the second row is the sum
$$
a_{2,1} C_{2,1}+a_{2,2} C_{2,2}+a_{2,3} C_{2,3} \text {. }
$$
(Be sure to compare the above line to the definition of cofactor expansion, and see how the " $i$ " in the definition is replaced by " 2 " here.) We'll find each cofactor and then compute the sum.
$$
\begin{aligned}
& C_{2,1}=(-1)^{2+1}\left|\begin{array}{ll}
2 & 3 \\
8 & 9
\end{array}\right|=(-1)(-6)=6 \quad\left(\begin{array}{c}
\text { we removed the second row and } \\
\text { first column of } A \text { to compute the } \\
\text { minor }
\end{array}\right) \\
& C_{2,2}=(-1)^{2+2}\left|\begin{array}{ll}
1 & 3 \\
7 & 9
\end{array}\right|=(1)(-12)=-12 \quad\left(\begin{array}{c}
\text { we removed the second row and } \\
\text { second column of } A \text { to compute } \\
\text { the minor }
\end{array}\right) \\
& C_{2,3}=(-1)^{2+3}\left|\begin{array}{ll}
1 & 2 \\
7 & 8
\end{array}\right|=(-1)(-6)=6 \quad\left(\begin{array}{c}
\text { we removed the second row and } \\
\text { third column of } A \text { to compute the } \\
\text { minor }
\end{array}\right)
\end{aligned}
$$
Thus the cofactor expansion along the second row is
$$
\begin{aligned}
a_{2,1} c_{2,1}+a_{2,2} c_{2,2}+a_{2,3} C_{2,3} & =4(6)+5(-12)+6(6) \\
& =24-60+36 \\
& =0
\end{aligned}
$$
At the moment, we don't know what to do with this cofactor expansion; we've just successfully found it.
We move on to find the cofactor expansion down the first column. By the definition, this sum is
$$
a_{1,1} C_{1,1}+a_{2,1} c_{2,1}+a_{3,1} c_{3,1} .
$$
(Again, compare this to the above definition and see how we replaced the " $j$ " with " 1. .) We find each cofactor:
$$
\begin{aligned}
& \left.C_{1,1}=(-1)^{1+1}\left|\begin{array}{ll}
5 & 6 \\
8 & 9
\end{array}\right|=(1)(-3)=-3 \quad \begin{array}{c}
\text { we removed the first row and first } \\
\text { column of } A \text { to compute the minor }
\end{array}\right) \\
& C_{2,1}=(-1)^{2+1}\left|\begin{array}{ll}
2 & 3 \\
8 & 9
\end{array}\right|=(-1)(-6)=6 \quad(\text { we computed this cofactor above }) \\
& \left.C_{3,1}=(-1)^{3+1}\left|\begin{array}{ll}
2 & 3 \\
5 & 6
\end{array}\right|=(1)(-3)=-3 \quad \begin{array}{l}
\text { (we removed the third row and first } \\
\text { column of } A \text { to compute the minor }
\end{array}\right)
\end{aligned}
$$
The cofactor expansion down the first column is
$$
\begin{aligned}
a_{1,1} c_{1,1}+a_{2,1} c_{2,1}+a_{3,1} c_{3,1} & =1(-3)+4(6)+7(-3) \\
& =-3+24-21 \\
& =0
\end{aligned}
$$
Is it a coincidence that both cofactor expansions were 0 ? We'll answer that in a while. This section is entitled "The Determinant," yet we don't know how to compute it yet except for $2 \times 2$ matrices. We finally define it now.
\section*{| Definition 26 | The Determinant |
| :--- | :--- |}
The determinant of an $n \times n$ matrix $A$, denoted $\operatorname{det}(A)$ or $|A|$, is a number given by the following:
- if $A$ is a $1 \times 1$ matrix $A=[a]$, then $\operatorname{det}(A)=a$.
- if $A$ is a $2 \times 2$ matrix
$$
A=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]
$$
then $\operatorname{det}(A)=a d-b c$.
- if $A$ is an $n \times n$ matrix, where $n \geq 2$, then $\operatorname{det}(A)$ is the number found by taking the cofactor expansion along the first row of $A$. That is,
$$
\operatorname{det}(A)=a_{1,1} C_{1,1}+a_{1,2} C_{1,2}+\cdots+a_{1, n} C_{1, n} .
$$
Notice that in order to compute the determinant of an $n \times n$ matrix, we need to compute the determinants of $n(n-1) \times(n-1)$ matrices. This can be a lot of work. We'll later learn how to shorten some of this. First, let's practice.
Example 71 Find the determinant of
$$
A=\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right]
$$
Solution Notice that this is the matrix from Example 70. The cofactor expansion along the first row is
$$
\operatorname{det}(A)=a_{1,1} C_{1,1}+a_{1,2} C_{1,2}+a_{1,3} C_{1,3}
$$
We'll compute each cofactor first then take the appropriate sum.
Therefore the determinant of $A$ is
$$
\operatorname{det}(A)=1(-3)+2(6)+3(-3)=0 .
$$
Example 72 Find the determinant of
$$
A=\left[\begin{array}{ccc}
3 & 6 & 7 \\
0 & 2 & -1 \\
3 & -1 & 1
\end{array}\right]
$$
Solution We'll compute each cofactor first then find the determinant.
Thus the determinant is
$$
\operatorname{det}(A)=3(1)+6(-3)+7(-6)=-57 .
$$
Example $73 \quad$ Find the determinant of
$$
A=\left[\begin{array}{cccc}
1 & 2 & 1 & 2 \\
-1 & 2 & 3 & 4 \\
8 & 5 & -3 & 1 \\
5 & 9 & -6 & 3
\end{array}\right]
$$
Solution This, quite frankly, will take quite a bit of work. In order to compute this determinant, we need to compute 4 minors, each of which requires finding the determinant of a $3 \times 3$ matrix! Complaining won't get us any closer to the solution, ${ }^{16}$
${ }^{16}$ But it might make us feel a little better. Glance ahead: do you see how much work we have to do?!? so let's get started. We first compute the cofactors:
$$
\begin{aligned}
C_{1,1} & =(-1)^{1+1} A_{1,1} \\
& =1 \cdot\left|\begin{array}{ccc}
2 & 3 & 4 \\
5 & -3 & 1 \\
9 & -6 & 3
\end{array}\right| \quad \begin{array}{c}
\text { (we must compute the determinant } \\
\text { of this } 3 \times 3 \text { matrix }
\end{array} \\
& =2 \cdot(-1)^{1+1}\left|\begin{array}{cc}
-3 & 1 \\
-6 & 3
\end{array}\right|+3 \cdot(-1)^{1+2}\left|\begin{array}{cc}
5 & 1 \\
9 & 3
\end{array}\right|+4 \cdot(-1)^{1+3}\left|\begin{array}{cc}
5 & -3 \\
9 & -6
\end{array}\right| \\
& =2(-3)+3(-6)+4(-3) \\
& =-36
\end{aligned}
$$
$$
\begin{aligned}
C_{1,2} & =(-1)^{1+2} A_{1,2} \\
& =(-1) \cdot\left|\begin{array}{ccc}
-1 & 3 & 4 \\
8 & -3 & 1 \\
5 & -6 & 3
\end{array}\right| \quad \begin{array}{l}
\text { (we must compute the determinant } \\
\text { of this } 3 \times 3 \text { matrix }
\end{array} \\
& =(-1) \underbrace{\left[(-1) \cdot(-1)^{1+1}\left|\begin{array}{ll}
-3 & 1 \\
-6 & 3
\end{array}\right|+3 \cdot(-1)^{1+2}\left|\begin{array}{cc}
8 & 1 \\
5 & 3
\end{array}\right|+4 \cdot(-1)^{1+3}\left|\begin{array}{cc}
8 & -3 \\
5 & -6
\end{array}\right|\right]}_{\text {the determinate of the } 3 \times 3 \text { matrix }} \\
& =(-1)] \\
& =186
\end{aligned}
$$
$$
\begin{aligned}
C_{1,3} & =(-1)^{1+3} A_{1,3} \\
& =1 \cdot\left|\begin{array}{ccc}
-1 & 2 & 4 \\
8 & 5 & 1 \\
5 & 9 & 3
\end{array}\right| \\
& \begin{array}{l}
\text { (we must compute the determinant } \\
\text { of this } 3 \times 3 \text { matrix }
\end{array} \\
& =(-1) \cdot(-1)^{1+1}\left|\begin{array}{ll}
5 & 1 \\
9 & 3
\end{array}\right|+2 \cdot(-1)^{1+2}\left|\begin{array}{cc}
8 & 1 \\
5 & 3
\end{array}\right|+4 \cdot(-1)^{1+3}\left|\begin{array}{cc}
8 & 5 \\
5 & 9
\end{array}\right| \\
& =(-1)(6)+2(-19)+4(47) \\
& =144
\end{aligned}
$$
$$
\begin{aligned}
C_{1,4} & =(-1)^{1+4} A_{1,4} \\
& =(-1) \cdot\left|\begin{array}{ccc}
-1 & 2 & 3 \\
8 & 5 & -3 \\
5 & 9 & -6
\end{array}\right| \quad \begin{array}{l}
\text { (we must compute the determinant } \\
\text { of this } 3 \times 3 \text { matrix }
\end{array} \\
& =(-1) \underbrace{\left[(-1) \cdot(-1)^{1+1}\left|\begin{array}{cc}
5 & -3 \\
9 & -6
\end{array}\right|+2 \cdot(-1)^{1+2}\left|\begin{array}{cc}
8 & -3 \\
5 & -6
\end{array}\right|+3 \cdot(-1)^{1+3}\left|\begin{array}{cc}
8 & 5 \\
5 & 9
\end{array}\right|\right]}_{\text {the determinate of the } 3 \times 3 \text { matrix }} \\
& =(-1)] \\
& =-210
\end{aligned}
$$
We've computed our four cofactors. All that is left is to compute the cofactor expansion.
$$
\operatorname{det}(A)=1(-36)+2(186)+1(144)+2(-210)=60 .
$$
As a way of "visualizing" this, let's write out the cofactor expansion again but including the matrices in their place.
$$
\begin{aligned}
\operatorname{det}(A) & =a_{1,1} C_{1,1}+a_{1,2} C_{1,2}+a_{1,3} C_{1,3}+a_{1,4} C_{1,4} \\
& =1(-1)^{2} \underbrace{\left|\begin{array}{ccc}
2 & 3 & 4 \\
5 & -3 & 1 \\
9 & -6 & 3
\end{array}\right|}_{=-36}+2(-1)^{3} \underbrace{\left|\begin{array}{ccc}
-1 & 3 & 4 \\
8 & -3 & 1 \\
5 & -6 & 3
\end{array}\right|}_{=-186} \\
& + \\
& 1(-1)^{4} \underbrace{\left|\begin{array}{ccc}
-1 & 2 & 4 \\
8 & 5 & 1 \\
5 & 9 & 3
\end{array}\right|}_{=144}+2(-1)^{5} \underbrace{\left|\begin{array}{ccc}
-1 & 2 & 3 \\
8 & 5 & -3 \\
5 & 9 & -6
\end{array}\right|}_{=210} \\
& =60
\end{aligned}
$$
That certainly took a while; it required more than 50 multiplications (we didn't count the additions). To compute the determinant of a $5 \times 5$ matrix, we'll need to compute the determinants of five $4 \times 4$ matrices, meaning that we'll need over 250 multiplications! Not only is this a lot of work, but there are just too many ways to make silly mistakes. ${ }^{17}$ There are some tricks to make this job easier, but regardless we see the need to employ technology. Even then, technology quickly bogs down. A $25 \times 25$ matrix is considered "small" by today's standards, ${ }^{18}$ but it is essentially impossible for a computer to compute its determinant by only using cofactor expansion; it too needs to employ "tricks."
${ }^{17}$ The author made three when the above example was originally typed.
${ }^{18} \mathrm{It}$ is common for mathematicians, scientists and engineers to consider linear systems with thousands of equations and variables. In the next section we will learn some of these tricks as we learn some of the properties of the determinant. Right now, let's review the essentials of what we have learned.
1. The determinant of a square matrix is a number that is determined by the matrix.
2. We find the determinant by computing the cofactor expansion along the first row.
3. To compute the determinant of an $n \times n$ matrix, we need to compute $n$ determinants of $(n-1) \times(n-1)$ matrices.
## Exercises 3.3
In Exercises 1-8, find the determinant of the $2 \times 2$ matrix.
1. $\left[\begin{array}{cc}10 & 7 \\ 8 & 9\end{array}\right]$
2. $\left[\begin{array}{cc}6 & -1 \\ -7 & 8\end{array}\right]$
3. $\left[\begin{array}{cc}-1 & -7 \\ -5 & 9\end{array}\right]$
4. $\left[\begin{array}{cc}-10 & -1 \\ -4 & 7\end{array}\right]$
5. $\left[\begin{array}{cc}8 & 10 \\ 2 & -3\end{array}\right]$
6. $\left[\begin{array}{cc}10 & -10 \\ -10 & 0\end{array}\right]$
7. $\left[\begin{array}{cc}1 & -3 \\ 7 & 7\end{array}\right]$
8. $\left[\begin{array}{ll}-4 & -5 \\ -1 & -4\end{array}\right]$
In Exercises 9-12, a matrix $A$ is given.
(a) Construct the submatrices used to compute the minors $A_{1,1}, A_{1,2}$ and $A_{1,3}$.
(b) Find the cofactors $C_{1,1}, C_{1,2}$, and $C_{1,3}$.
9. $\left[\begin{array}{ccc}-7 & -3 & 10 \\ 3 & 7 & 6 \\ 1 & 6 & 10\end{array}\right]$
10. $\left[\begin{array}{ccc}-2 & -9 & 6 \\ -10 & -6 & 8 \\ 0 & -3 & -2\end{array}\right]$ 11. $\left[\begin{array}{ccc}-5 & -3 & 3 \\ -3 & 3 & 10 \\ -9 & 3 & 9\end{array}\right]$
11. $\left[\begin{array}{ccc}-6 & -4 & 6 \\ -8 & 0 & 0 \\ -10 & 8 & -1\end{array}\right]$
In Exercises $13-24$, find the determinant of the given matrix using cofactor expansion along the first row.
13. $\left[\begin{array}{ccc}3 & 2 & 3 \\ -6 & 1 & -10 \\ -8 & -9 & -9\end{array}\right]$
14. $\left[\begin{array}{ccc}8 & -9 & -2 \\ -9 & 9 & -7 \\ 5 & -1 & 9\end{array}\right]$
15. $\left[\begin{array}{ccc}-4 & 3 & -4 \\ -4 & -5 & 3 \\ 3 & -4 & 5\end{array}\right]$
16. $\left[\begin{array}{ccc}1 & -2 & 1 \\ 5 & 5 & 4 \\ 4 & 0 & 0\end{array}\right]$
17. $\left[\begin{array}{ccc}1 & -4 & 1 \\ 0 & 3 & 0 \\ 1 & 2 & 2\end{array}\right]$
18. $\left[\begin{array}{ccc}3 & -1 & 0 \\ -3 & 0 & -4 \\ 0 & -1 & -4\end{array}\right]$
19. $\left[\begin{array}{ccc}-5 & 0 & -4 \\ 2 & 4 & -1 \\ -5 & 0 & -4\end{array}\right]$
20. $\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 1 & 1\end{array}\right]$
21. $\left[\begin{array}{cccc}0 & 0 & -1 & -1 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & -1 & 0 \\ -1 & 0 & 1 & 0\end{array}\right]$
22. $\left[\begin{array}{cccc}2 & -1 & 4 & 4 \\ 3 & -3 & 3 & 2 \\ 0 & 4 & -5 & 1 \\ -2 & -5 & -2 & -5\end{array}\right]$
23. $\left[\begin{array}{cccc}-1 & 0 & 0 & -1 \\ -1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 \\ 1 & 0 & -1 & -1\end{array}\right]$
24. Let $A$ be a $2 \times 2$ matrix;
$$
A=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right] .
$$
23. $\left[\begin{array}{cccc}-5 & 1 & 0 & 0 \\ -3 & -5 & 2 & 5 \\ -2 & 4 & -3 & 4 \\ 5 & 4 & -3 & 3\end{array}\right]$ Show why $\operatorname{det}(A)=a d-b c$ by computing the cofactor expansion of $A$ along the first row.
### Properties of the Determinant
## AS YOU READ . .
1. Having the choice to compute the determinant of a matrix using cofactor expansion along any row or column is most useful when there are lots of what in a row or column?
2. Which elementary row operation does not change the determinant of a matrix?
3. Why do mathematicians rarely smile?
4. T/F: When computers are used to compute the determinant of a matrix, cofactor expansion is rarely used.
In the previous section we learned how to compute the determinant. In this section we learn some of the properties of the determinant, and this will allow us to compute determinants more easily. In the next section we will see one application of determinants.
We start with a theorem that gives us more freedom when computing determinants.
Theorem 14
Let $A$ be an $n \times n$ matrix. The determinant of $A$ can be computed using cofactor expansion along any row or column of A. We alluded to this fact way back after Example 70. We had just learned what cofactor expansion was and we practiced along the second row and down the third column. Later, we found the determinant of this matrix by computing the cofactor expansion along the first row. In all three cases, we got the number 0 . This wasn't a coincidence. The above theorem states that all three expansions were actually computing the determinant.
How does this help us? By giving us freedom to choose any row or column to use for the expansion, we can choose a row or column that looks "most appealing." This usually means "it has lots of zeros." We demonstrate this principle below.
Example $74 \quad$ Find the determinant of
$$
A=\left[\begin{array}{cccc}
1 & 2 & 0 & 9 \\
2 & -3 & 0 & 5 \\
7 & 2 & 3 & 8 \\
-4 & 1 & 0 & 2
\end{array}\right]
$$
Solution Our first reaction may well be "Oh no! Not another $4 \times 4$ determinant!" However, we can use cofactor expansion along any row or column that we choose. The third column looks great; it has lots of zeros in it. The cofactor expansion along this column is
$$
\begin{aligned}
\operatorname{det}(A) & =a_{1,3} C_{1,3}+a_{2,3} C_{2,3}+a_{3,3} C_{3,3}+a_{4,3} C_{4,3} \\
& =0 \cdot C_{1,3}+0 \cdot C_{2,3}+3 \cdot C_{3,3}+0 \cdot C_{4,3}
\end{aligned}
$$
The wonderful thing here is that three of our cofactors are multiplied by 0 . We won't bother computing them since they will not contribute to the determinant. Thus
$$
\begin{aligned}
\operatorname{det}(A) & =3 \cdot C_{3,3} \\
& =3 \cdot(-1)^{3+3} \cdot\left|\begin{array}{ccc}
1 & 2 & 9 \\
2 & -3 & 5 \\
-4 & 1 & 2
\end{array}\right| \\
& =3 \cdot(-147) \quad\left(\begin{array}{c}
\text { we computed the determinant of the } 3 \times 3 \text { matrix } \\
\text { without showing our work; it is }-147
\end{array}\right) \\
& =-447
\end{aligned}
$$
Wow. That was a lot simpler than computing all that we did in Example 73. Of course, in that example, we didn't really have any shortcuts that we could have employed.
Example $75 \quad$ Find the determinant of
$$
A=\left[\begin{array}{ccccc}
1 & 2 & 3 & 4 & 5 \\
0 & 6 & 7 & 8 & 9 \\
0 & 0 & 10 & 11 & 12 \\
0 & 0 & 0 & 13 & 14 \\
0 & 0 & 0 & 0 & 15
\end{array}\right] .
$$
Solution At first glance, we think "I don't want to find the determinant of a $5 \times 5$ matrix!" However, using our newfound knowledge, we see that things are not that bad. In fact, this problem is very easy.
What row or column should we choose to find the determinant along? There are two obvious choices: the first column or the last row. Both have 4 zeros in them. We choose the first column. ${ }^{19}$ We omit most of the cofactor expansion, since most of it is just 0:
$$
\operatorname{det}(A)=1 \cdot(-1)^{1+1} \cdot\left|\begin{array}{cccc}
6 & 7 & 8 & 9 \\
0 & 10 & 11 & 12 \\
0 & 0 & 13 & 14 \\
0 & 0 & 0 & 15
\end{array}\right| .
$$
Similarly, this determinant is not bad to compute; we again choose to use cofactor expansion along the first column. Note: technically, this cofactor expansion is $6 \cdot(-1)^{1+1} A_{1,1}$; we are going to drop the $(-1)^{1+1}$ terms from here on out in this example (it will show up a lot...).
$$
\operatorname{det}(A)=1 \cdot 6 \cdot\left|\begin{array}{ccc}
10 & 11 & 12 \\
0 & 13 & 14 \\
0 & 0 & 15
\end{array}\right| .
$$
You can probably can see a trend. We'll finish out the steps without explaining each one.
$$
\begin{aligned}
\operatorname{det}(A) & =1 \cdot 6 \cdot 10 \cdot\left|\begin{array}{cc}
13 & 14 \\
0 & 15
\end{array}\right| \\
& =1 \cdot 6 \cdot 10 \cdot 13 \cdot 15 \\
& =11700
\end{aligned}
$$
We see that the final determinant is the product of the diagonal entries. This works for any triangular matrix (and since diagonal matrices are triangular, it works for diagonal matrices as well). This is an important enough idea that we'll put it into a box.
Key Idea 12 The Determinant of Triangular Matrices
The determinant of a triangular matrix is the product of its diagonal elements.
It is now again time to start thinking like a mathematician. Remember, mathematicians see something new and often ask "How does this relate to things I already
${ }^{19}$ We do not choose this because it is the better choice; both options are good. We simply had to make a choice. know?" So now we ask, "If we change a matrix in some way, how is it's determinant changed?"
The standard way that we change matrices is through elementary row operations. If we perform an elementary row operation on a matrix, how will the determinant of the new matrix compare to the determinant of the original matrix?
Let's experiment first and then we'll officially state what happens.
Example $76 \quad$ Let
$$
A=\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]
$$
Let $B$ be formed from $A$ by doing one of the following elementary row operations:
1. $2 R_{1}+R_{2} \rightarrow R_{2}$
2. $5 R_{1} \rightarrow R_{1}$
3. $R_{1} \leftrightarrow R_{2}$
Find $\operatorname{det}(A)$ as well as $\operatorname{det}(B)$ for each of the row operations above.
Solution It is straightforward to compute $\operatorname{det}(A)=-2$.
Let $B$ be formed by performing the row operation in 1 ) on $A$; thus
$$
B=\left[\begin{array}{ll}
1 & 2 \\
5 & 8
\end{array}\right] .
$$
It is clear that $\operatorname{det}(B)=-2$, the same as $\operatorname{det}(A)$.
Now let $B$ be formed by performing the elementary row operation in 2 ) on $A$; that is,
$$
B=\left[\begin{array}{cc}
5 & 10 \\
3 & 4
\end{array}\right]
$$
We can see that $\operatorname{det}(B)=-10$, which is $5 \cdot \operatorname{det}(A)$.
Finally, let $B$ be formed by the third row operation given; swap the two rows of $A$. We see that
$$
B=\left[\begin{array}{ll}
3 & 4 \\
1 & 2
\end{array}\right]
$$
and that $\operatorname{det}(B)=2$, which is $(-1) \cdot \operatorname{det}(A)$.
We've seen in the above example that there seems to be a relationship between the determinants of matrices "before and after" being changed by elementary row operations. Certainly, one example isn't enough to base a theory on, and we have not proved anything yet. Regardless, the following theorem is true. Theorem 15 The Determinant and Elementary Row Operations
Let $A$ be an $n \times n$ matrix and let $B$ be formed by performing one elementary row operation on $A$.
1. If $B$ is formed from $A$ by adding a scalar multiple of one row to another, then $\operatorname{det}(B)=\operatorname{det}(A)$.
2. If $B$ is formed from $A$ by multiplying one row of $A$ by a scalar $k$, then $\operatorname{det}(B)=k \cdot \operatorname{det}(A)$.
3. If $B$ is formed from $A$ by interchanging two rows of $A$, then $\operatorname{det}(B)=-\operatorname{det}(A)$.
Let's put this theorem to use in an example.
Example $77 \quad$ Let
$$
A=\left[\begin{array}{lll}
1 & 2 & 1 \\
0 & 1 & 1 \\
1 & 1 & 1
\end{array}\right]
$$
Compute $\operatorname{det}(A)$, then find the determinants of the following matrices by inspection using Theorem 15.
$$
B=\left[\begin{array}{lll}
1 & 1 & 1 \\
1 & 2 & 1 \\
0 & 1 & 1
\end{array}\right] \quad C=\left[\begin{array}{lll}
1 & 2 & 1 \\
0 & 1 & 1 \\
7 & 7 & 7
\end{array}\right] \quad D=\left[\begin{array}{ccc}
1 & -1 & -2 \\
0 & 1 & 1 \\
1 & 1 & 1
\end{array}\right]
$$
Solution Computing $\operatorname{det}(A)$ by cofactor expansion down the first column or along the second row seems like the best choice, utilizing the one zero in the matrix. We can quickly confirm that $\operatorname{det}(A)=1$.
To compute $\operatorname{det}(B)$, notice that the rows of $A$ were rearranged to form $B$. There are different ways to describe what happened; saying $R_{1} \leftrightarrow R_{2}$ was followed by $R_{1} \leftrightarrow R_{3}$ produces $B$ from $A$. Since there were two row swaps, $\operatorname{det}(B)=(-1)(-1) \operatorname{det}(A)=$ $\operatorname{det}(A)=1$.
Notice that $C$ is formed from $A$ by multiplying the third row by 7 . Thus $\operatorname{det}(C)=$ $7 \cdot \operatorname{det}(A)=7$.
It takes a little thought, but we can form $D$ from $A$ by the operation $-3 R_{2}+R_{1} \rightarrow R_{1}$. This type of elementary row operation does not change determinants, so $\operatorname{det}(D)=$ $\operatorname{det}(A)$. Let's continue to think like mathematicians; mathematicians tend to remember "problems" they've encountered in the past, ${ }^{20}$ and when they learn something new, in the backs of their minds they try to apply their new knowledge to solve their old problem.
What "problem" did we recently uncover? We stated in the last chapter that even computers could not compute the determinant of large matrices with cofactor expansion. How then can we compute the determinant of large matrices?
We just learned two interesting and useful facts about matrix determinants. First, the determinant of a triangular matrix is easy to compute: just multiply the diagonal elements. Secondly, we know how elementary row operations affect the determinant. Put these two ideas together: given any square matrix, we can use elementary row operations to put the matrix in triangular form, ${ }^{21}$ find the determinant of the new matrix (which is easy), and then adjust that number by recalling what elementary operations we performed. Let's practice this.
Example $78 \quad$ Find the determinant of $A$ by first putting $A$ into a triangular form, where
$$
A=\left[\begin{array}{ccc}
2 & 4 & -2 \\
-1 & -2 & 5 \\
3 & 2 & 1
\end{array}\right]
$$
Solution In putting $A$ into a triangular form, we need not worry about getting leading $1 \mathrm{~s}$, but it does tend to make our life easier as we work out a problem by hand. So let's scale the first row by $1 / 2$ :
$$
\frac{1}{2} R_{1} \rightarrow R_{1} \quad\left[\begin{array}{ccc}
1 & 2 & -1 \\
-1 & -2 & 5 \\
3 & 2 & 1
\end{array}\right] .
$$
Now let's get Os below this leading 1:
$$
\begin{gathered}
R_{1}+R_{2} \rightarrow R_{2} \\
-3 R_{1}+R_{3} \rightarrow R_{3}
\end{gathered} \quad\left[\begin{array}{ccc}
1 & 2 & -1 \\
0 & 0 & 4 \\
0 & -4 & 4
\end{array}\right] .
$$
We can finish in one step; by interchanging rows 2 and 3 we'll have our matrix in triangular form.
$$
R_{2} \leftrightarrow R_{3} \quad\left[\begin{array}{ccc}
1 & 2 & -1 \\
0 & -4 & 4 \\
0 & 0 & 4
\end{array}\right] .
$$
Let's name this last matrix $B$. The determinant of $B$ is easy to compute as it is triangular; $\operatorname{det}(B)=-16$. We can use this to find $\operatorname{det}(A)$.
Recall the steps we used to transform $A$ into $B$. They are:
${ }^{20}$ which is why mathematicians rarely smile: they are remembering their problems
${ }^{21}$ or echelon form
$$
\begin{gathered}
\frac{1}{2} R_{1} \rightarrow R_{1} \\
R_{1}+R_{2} \rightarrow R_{2} \\
-3 R_{1}+R_{3} \rightarrow R_{3} \\
R_{2} \leftrightarrow R_{3}
\end{gathered}
$$
The first operation multiplied a row of $A$ by $\frac{1}{2}$. This means that the resulting matrix had a determinant that was $\frac{1}{2}$ the determinant of $A$.
The next two operations did not affect the determinant at all. The last operation, the row swap, changed the sign. Combining these effects, we know that
$$
-16=\operatorname{det}(B)=(-1) \frac{1}{2} \operatorname{det}(A) .
$$
Solving for $\operatorname{det}(A)$ we have that $\operatorname{det}(A)=32$.
In practice, we don't need to keep track of operations where we add multiples of one row to another; they simply do not affect the determinant. Also, in practice, these steps are carried out by a computer, and computers don't care about leading 1s. Therefore, row scaling operations are rarely used. The only things to keep track of are row swaps, and even then all we care about are the number of row swaps. An odd number of row swaps means that the original determinant has the opposite sign of the triangular form matrix; an even number of row swaps means they have the same determinant.
Let's practice this again.
Example $79 \quad$ The matrix $B$ was formed from $A$ using the following elementary row operations, though not necessarily in this order. Find $\operatorname{det}(A)$.
$$
B=\left[\begin{array}{lll}
1 & 2 & 3 \\
0 & 4 & 5 \\
0 & 0 & 6
\end{array}\right] \quad \begin{aligned}
& 2 R_{1} \rightarrow R_{1} \\
&
\end{aligned}
$$
Solution It is easy to compute $\operatorname{det}(B)=24$. In looking at our list of elementary row operations, we see that only the first three have an effect on the determinant. Therefore
$$
24=\operatorname{det}(B)=2 \cdot \frac{1}{3} \cdot(-1) \cdot \operatorname{det}(A)
$$
and hence
$$
\operatorname{det}(A)=-36
$$
In the previous example, we may have been tempted to "rebuild" $A$ using the elementary row operations and then computing the determinant. This can be done, but in general it is a bad idea; it takes too much work and it is too easy to make a mistake.
Let's think some more like a mathematician. How does the determinant work with other matrix operations that we know? Specifically, how does the determinant interact with matrix addition, scalar multiplication, matrix multiplication, the transpose and the trace? We'll again do an example to get an idea of what is going on, then give a theorem to state what is true.
Example $80 \quad$ Let
$$
A=\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right] \text { and } B=\left[\begin{array}{ll}
2 & 1 \\
3 & 5
\end{array}\right]
$$
Find the determinants of the matrices $A, B, A+B, 3 A, A B, A^{T}, A^{-1}$, and compare the determinant of these matrices to their trace.
Solution We can quickly compute that $\operatorname{det}(A)=-2$ and that $\operatorname{det}(B)=7$.
$$
\begin{aligned}
\operatorname{det}(A-B) & =\operatorname{det}\left(\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]-\left[\begin{array}{ll}
2 & 1 \\
3 & 5
\end{array}\right]\right) \\
& =\left|\begin{array}{cc}
-1 & 1 \\
0 & -1
\end{array}\right| \\
& =1
\end{aligned}
$$
It's tough to find a connection between $\operatorname{det}(A-B), \operatorname{det}(A)$ and $\operatorname{det}(B)$.
$$
\begin{aligned}
\operatorname{det}(3 A) & =\left|\begin{array}{cc}
3 & 6 \\
9 & 12
\end{array}\right| \\
& =-18
\end{aligned}
$$
We can figure this one out; multiplying one row of $A$ by 3 increases the determinant by a factor of 3 ; doing it again (and hence multiplying both rows by 3 ) increases the determinant again by a factor of 3 . Therefore $\operatorname{det}(3 A)=3 \cdot 3 \cdot \operatorname{det}(A)$, or $3^{2} \cdot A$.
$$
\begin{aligned}
\operatorname{det}(A B) & =\operatorname{det}\left(\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\left[\begin{array}{ll}
2 & 1 \\
3 & 5
\end{array}\right]\right) \\
& =\left|\begin{array}{cc}
8 & 11 \\
18 & 23
\end{array}\right| \\
& =-14
\end{aligned}
$$
This one seems clear; $\operatorname{det}(A B)=\operatorname{det}(A) \operatorname{det}(B)$.
$$
\begin{aligned}
\operatorname{det}\left(A^{T}\right) & =\left|\begin{array}{ll}
1 & 3 \\
2 & 4
\end{array}\right| \\
& =-2
\end{aligned}
$$
Obviously $\operatorname{det}\left(A^{T}\right)=\operatorname{det}(A)$; is this always going to be the case? If we think about it, we can see that the cofactor expansion along the first row of $A$ will give us the same result as the cofactor expansion along the first column of $A^{T} .{ }^{22}$
$$
\begin{aligned}
\operatorname{det}\left(A^{-1}\right) & =\left|\begin{array}{cc}
-2 & 1 \\
3 / 2 & -1 / 2
\end{array}\right| \\
& =1-3 / 2 \\
& =-1 / 2
\end{aligned}
$$
It seems as though
$$
\operatorname{det}\left(A^{-1}\right)=\frac{1}{\operatorname{det}(A)} .
$$
We end by remarking that there seems to be no connection whatsoever between the trace of a matrix and its determinant. We leave it to the reader to compute the trace for some of the above matrices and confirm this statement.
We now state a theorem which will confirm our conjectures from the previous example.
Theorem 16
## Determinant Properties
Let $A$ and $B$ be $n \times n$ matrices and let $k$ be a scalar. The following are true:
1. $\operatorname{det}(k A)=k^{n} \cdot \operatorname{det}(A)$
2. $\operatorname{det}\left(A^{T}\right)=\operatorname{det}(A)$
3. $\operatorname{det}(A B)=\operatorname{det}(A) \operatorname{det}(B)$
4. If $A$ is invertible, then
$$
\operatorname{det}\left(A^{-1}\right)=\frac{1}{\operatorname{det}(A)} .
$$
5. A matrix $A$ is invertible if and only if $\operatorname{det}(A) \neq 0$.
This last statement of the above theorem is significant: what happens if $\operatorname{det}(A)=$ 0 ? It seems that $\operatorname{det}\left(A^{-1}\right)=" 1 / 0$ ", which is undefined. There actually isn't a problem here; it turns out that if $\operatorname{det}(A)=0$, then $A$ is not invertible (hence part 5 of Theorem 16). This allows us to add on to our Invertible Matrix Theorem.
${ }^{22}$ This can be a bit tricky to think out in your head. Try it with a $3 \times 3$ matrix $A$ and see how it works. All the $2 \times 2$ submatrices that are created in $A^{T}$ are the transpose of those found in $A$; this doesn't matter since it is easy to see that the determinant isn't affected by the transpose in a $2 \times 2$ matrix. Let $A$ be an $n \times n$ matrix. The following statements are equivalent.
(a) $A$ is invertible.
(g) $\operatorname{det}(A) \neq 0$.
This new addition to the Invertible Matrix Theorem is very useful; we'll refer back to it in Chapter 4 when we discuss eigenvalues.
We end this section with a shortcut for computing the determinants of $3 \times 3$ matrices. Consider the matrix $A$ :
$$
\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right] .
$$
We can compute its determinant using cofactor expansion as we did in Example 71. Once one becomes proficient at this method, computing the determinant of a $3 \times 3$ isn't all that hard. A method many find easier, though, starts with rewriting the matrix without the brackets, and repeating the first and second columns at the end as shown below.
$$
\begin{array}{lllll}
1 & 2 & 3 & 1 & 2 \\
4 & 5 & 6 & 4 & 5 \\
7 & 8 & 9 & 7 & 8
\end{array}
$$
In this $3 \times 5$ array of numbers, there are 3 full "upper left to lower right" diagonals, and 3 full "upper right to lower left" diagonals, as shown below with the arrows.
The numbers that appear at the ends of each of the arrows are computed by multiplying the numbers found along the arrows. For instance, the 105 comes from multiplying $3 \cdot 5 \cdot 7=105$. The determinant is found by adding the numbers on the right, and subtracting the sum of the numbers on the left. That is,
$$
\operatorname{det}(A)=(45+84+96)-(105+48+72)=0 .
$$
To help remind ourselves of this shortcut, we'll make it into a Key Idea. Key Idea 13
We'll practice once more in the context of an example.
Example 81 where
Find the determinant of $A$ using the previously described shortcut,
$$
A=\left[\begin{array}{ccc}
1 & 3 & 9 \\
-2 & 3 & 4 \\
-5 & 7 & 2
\end{array}\right]
$$
Solution Rewriting the first 2 columns, drawing the proper diagonals, and multiplying, we get:
Summing the numbers on the right and subtracting the sum of the numbers on the left, we get
$$
\operatorname{det}(A)=(6-60-126)-(-135+28-12)=-61 .
$$
In the next section we'll see how the determinant can be used to solve systems of linear equations.
## Exercises 3.4
In Exercises 1 - 14, find the determinant of the given matrix using cofactor expansion along any row or column you choose.
2. $\left[\begin{array}{ccc}-4 & 4 & -4 \\ 0 & 0 & -3 \\ -2 & -2 & -1\end{array}\right]$
3. $\left[\begin{array}{ccc}1 & 2 & 3 \\ -5 & 0 & 3 \\ 4 & 0 & 6\end{array}\right]$
4. $\left[\begin{array}{ccc}-4 & 1 & 1 \\ 0 & 0 & 0 \\ -1 & -2 & -5\end{array}\right]$ 4. $\left[\begin{array}{ccc}0 & -3 & 1 \\ 0 & 0 & 5 \\ -4 & 1 & 0\end{array}\right]$
5. $\left[\begin{array}{ccc}-2 & -3 & 5 \\ 5 & 2 & 0 \\ -1 & 0 & 0\end{array}\right]$
6. $\left[\begin{array}{ccc}-2 & -2 & 0 \\ 2 & -5 & -3 \\ -5 & 1 & 0\end{array}\right]$
7. $\left[\begin{array}{ccc}-3 & 0 & -5 \\ -2 & -3 & 3 \\ -1 & 0 & 1\end{array}\right]$
8. $\left[\begin{array}{ccc}0 & 4 & -4 \\ 3 & 1 & -3 \\ -3 & -4 & 0\end{array}\right]$
9. $\left[\begin{array}{cccc}5 & -5 & 0 & 1 \\ 2 & 4 & -1 & -1 \\ 5 & 0 & 0 & 4 \\ -1 & -2 & 0 & 5\end{array}\right]$
10. $\left[\begin{array}{cccc}-1 & 3 & 3 & 4 \\ 0 & 0 & 0 & 0 \\ 4 & -5 & -2 & 0 \\ 0 & 0 & 2 & 0\end{array}\right]$
11. $\left[\begin{array}{cccc}-5 & -5 & 0 & -2 \\ 0 & 0 & 5 & 0 \\ 1 & 3 & 3 & 1 \\ -4 & -2 & -1 & -5\end{array}\right]$
12. $\left[\begin{array}{cccc}-1 & 0 & -2 & 5 \\ 3 & -5 & 1 & -2 \\ -5 & -2 & -1 & -3 \\ -1 & 0 & 0 & 0\end{array}\right]$
13. $\left[\begin{array}{lllll}4 & 0 & 5 & 1 & 0 \\ 1 & 0 & 3 & 1 & 5 \\ 2 & 2 & 0 & 2 & 2 \\ 1 & 0 & 0 & 0 & 0 \\ 4 & 4 & 2 & 5 & 3\end{array}\right]$
14. $\left[\begin{array}{lllll}2 & 1 & 1 & 1 & 1 \\ 4 & 1 & 2 & 0 & 2 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 3 & 2 & 0 & 3 \\ 5 & 0 & 5 & 0 & 4\end{array}\right]$
In Exercises $15-18$, a matrix $M$ and $\operatorname{det}(M)$ are given. Matrices $A, B$ and $C$ are formed by performing operations on $M$. Determine the determinants of $A, B$ and $C$ using Theorems 15 and 16, and indicate the operations used to form $A, B$ and $C$.
15. $M=\left[\begin{array}{ccc}0 & 3 & 5 \\ 3 & 1 & 0 \\ -2 & -4 & -1\end{array}\right]$, $\operatorname{det}(M)=-41$.
(a) $A=\left[\begin{array}{ccc}0 & 3 & 5 \\ -2 & -4 & -1 \\ 3 & 1 & 0\end{array}\right]$
(b) $B=\left[\begin{array}{ccc}0 & 3 & 5 \\ 3 & 1 & 0 \\ 8 & 16 & 4\end{array}\right]$
(c) $C=\left[\begin{array}{ccc}3 & 4 & 5 \\ 3 & 1 & 0 \\ -2 & -4 & -1\end{array}\right]$
16. $M=\left[\begin{array}{lll}9 & 7 & 8 \\ 1 & 3 & 7 \\ 6 & 3 & 3\end{array}\right]$, $\operatorname{det}(M)=45$.
(a) $A=\left[\begin{array}{ccc}18 & 14 & 16 \\ 1 & 3 & 7 \\ 6 & 3 & 3\end{array}\right]$
(b) $B=\left[\begin{array}{ccc}9 & 7 & 8 \\ 1 & 3 & 7 \\ 96 & 73 & 83\end{array}\right]$
(c) $C=\left[\begin{array}{lll}9 & 1 & 6 \\ 7 & 3 & 3 \\ 8 & 7 & 3\end{array}\right]$
17. $M=\left[\begin{array}{lll}5 & 1 & 5 \\ 4 & 0 & 2 \\ 0 & 0 & 4\end{array}\right]$, $\operatorname{det}(M)=-16$.
(a) $A=\left[\begin{array}{lll}0 & 0 & 4 \\ 5 & 1 & 5 \\ 4 & 0 & 2\end{array}\right]$
(b) $B=\left[\begin{array}{ccc}-5 & -1 & -5 \\ -4 & 0 & -2 \\ 0 & 0 & 4\end{array}\right]$
(c) $C=\left[\begin{array}{ccc}15 & 3 & 15 \\ 12 & 0 & 6 \\ 0 & 0 & 12\end{array}\right]$
$$
\begin{aligned}
& \text { 18. } M=\left[\begin{array}{lll}
5 & 4 & 0 \\
7 & 9 & 3 \\
1 & 3 & 9
\end{array}\right], \\
& \operatorname{det}(M)=120 .
\end{aligned}
$$
$$
\begin{aligned}
& \text { (a) } A=\left[\begin{array}{lll}
1 & 3 & 9 \\
7 & 9 & 3 \\
5 & 4 & 0
\end{array}\right] \\
& \text { (b) } B=\left[\begin{array}{ccc}
5 & 4 & 0 \\
14 & 18 & 6 \\
3 & 9 & 27
\end{array}\right] \\
& \text { (c) } C=\left[\begin{array}{lll}
-5 & -4 & 0 \\
-7 & -9 & -3 \\
-1 & -3 & -9
\end{array}\right]
\end{aligned}
$$
In Exercises $19-22$, matrices $A$ and $B$ are given. Verify part 3 of Theorem 16 by computing $\operatorname{det}(A), \operatorname{det}(B)$ and $\operatorname{det}(A B)$.
$$
\text { 19. } \begin{aligned}
A & =\left[\begin{array}{ll}
2 & 0 \\
1 & 2
\end{array}\right], \\
B & =\left[\begin{array}{cc}
0 & -4 \\
1 & 3
\end{array}\right] \\
\text { 20. } A & =\left[\begin{array}{cc}
3 & -1 \\
4 & 1
\end{array}\right], \\
B & =\left[\begin{array}{cc}
-4 & -1 \\
-5 & 3
\end{array}\right] \\
\text { 21. } A & =\left[\begin{array}{cc}
-4 & 4 \\
5 & -2
\end{array}\right], \\
B & =\left[\begin{array}{cc}
-3 & -4 \\
5 & -3
\end{array}\right] \\
\text { 22. } A & =\left[\begin{array}{cc}
-3 & -1 \\
2 & -3
\end{array}\right],
\end{aligned}
$$
$$
B=\left[\begin{array}{cc}
0 & 0 \\
4 & -4
\end{array}\right]
$$
In Exercises 23 - 30, find the determinant of the given matrix using Key Idea 13.
23. $\left[\begin{array}{ccc}3 & 2 & 3 \\ -6 & 1 & -10 \\ -8 & -9 & -9\end{array}\right]$
24. $\left[\begin{array}{ccc}8 & -9 & -2 \\ -9 & 9 & -7 \\ 5 & -1 & 9\end{array}\right]$
25. $\left[\begin{array}{ccc}-4 & 3 & -4 \\ -4 & -5 & 3 \\ 3 & -4 & 5\end{array}\right]$
26. $\left[\begin{array}{ccc}1 & -2 & 1 \\ 5 & 5 & 4 \\ 4 & 0 & 0\end{array}\right]$
27. $\left[\begin{array}{ccc}1 & -4 & 1 \\ 0 & 3 & 0 \\ 1 & 2 & 2\end{array}\right]$
28. $\left[\begin{array}{ccc}3 & -1 & 0 \\ -3 & 0 & -4 \\ 0 & -1 & -4\end{array}\right]$
29. $\left[\begin{array}{ccc}-5 & 0 & -4 \\ 2 & 4 & -1 \\ -5 & 0 & -4\end{array}\right]$
30. $\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 1 & 1\end{array}\right]$
### Cramer's Rule
## AS YOU READ
1. T/F: Cramer's Rule is another method to compute the determinant of a matrix.
2. T/F: Cramer's Rule is often used because it is more efficient than Gaussian elimination.
3. Mathematicians use what word to describe the connections between seemingly unrelated ideas?
In the previous sections we have learned about the determinant, but we haven't given a really good reason why we would want to compute it. ${ }^{23}$ This section shows one application of the determinant: solving systems of linear equations. We introduce this idea in terms of a theorem, then we will practice.
Theorem 18
Cramer's Rule
Let $A$ be an $n \times n$ matrix with $\operatorname{det}(A) \neq 0$ and let $\vec{b}$ be an $n \times 1$ column vector. Then the linear system
$$
A \vec{x}=\vec{b}
$$
has solution
$$
x_{i}=\frac{\operatorname{det}\left(A_{i}(\vec{b})\right)}{\operatorname{det}(A)},
$$
where $A_{i}(\vec{b})$ is the matrix formed by replacing the $i^{\text {th }}$ column of $A$ with $\vec{b}$.
Let's do an example.
Example 82 Use Cramer's Rule to solve the linear system $A \vec{x}=\vec{b}$ where
$$
A=\left[\begin{array}{ccc}
1 & 5 & -3 \\
1 & 4 & 2 \\
2 & -1 & 0
\end{array}\right] \text { and } \vec{b}=\left[\begin{array}{c}
-36 \\
-11 \\
7
\end{array}\right]
$$
${ }^{23}$ The closest we came to motivation is that if $\operatorname{det}(A)=0$, then we know that $A$ is not invertible. But it seems that there may be easier ways to check. Solution We first compute the determinant of $A$ to see if we can apply Cramer's Rule.
$$
\operatorname{det}(A)=\left|\begin{array}{ccc}
1 & 5 & -3 \\
1 & 4 & 2 \\
2 & -1 & 0
\end{array}\right|=49
$$
Since det $(A) \neq 0$, we can apply Cramer's Rule. Following Theorem 18, we compute $\operatorname{det}\left(A_{1}(\vec{b})\right), \operatorname{det}\left(A_{2}(\vec{b})\right)$ and $\operatorname{det}\left(A_{3}(\vec{b})\right)$.
$$
\operatorname{det}\left(A_{1}(\vec{b})\right)=\left|\begin{array}{ccc}
-36 & 5 & -3 \\
-11 & 4 & 2 \\
7 & -1 & 0
\end{array}\right|=49
$$
(We used a bold font to show where $\vec{b}$ replaced the first column of $A$.)
$$
\begin{aligned}
& \operatorname{det}\left(A_{2}(\vec{b})\right)=\left|\begin{array}{ccc}
1 & -36 & -3 \\
1 & -11 & 2 \\
2 & 7 & 0
\end{array}\right|=-245 . \\
& \operatorname{det}\left(A_{3}(\vec{b})\right)=\left|\begin{array}{ccc}
1 & 5 & -36 \\
1 & 4 & -11 \\
2 & -1 & 7
\end{array}\right|=196 .
\end{aligned}
$$
Therefore we can compute $\vec{x}$ :
$$
\begin{aligned}
& x_{1}=\frac{\operatorname{det}\left(A_{1}(\vec{b})\right)}{\operatorname{det}(A)}=\frac{49}{49}=1 \\
& x_{2}=\frac{\operatorname{det}\left(A_{2}(\vec{b})\right)}{\operatorname{det}(A)}=\frac{-245}{49}=-5 \\
& x_{3}=\frac{\operatorname{det}\left(A_{3}(\vec{b})\right)}{\operatorname{det}(A)}=\frac{196}{49}=4
\end{aligned}
$$
Therefore
$$
\vec{x}=\left[\begin{array}{l}
x_{1} \\
x_{2} \\
x_{3}
\end{array}\right]=\left[\begin{array}{c}
1 \\
-5 \\
4
\end{array}\right]
$$
Let's do another example.
Example 83 Use Cramer's Rule to solve the linear system $A \vec{x}=\vec{b}$ where
$$
A=\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right] \text { and } \vec{b}=\left[\begin{array}{c}
-1 \\
1
\end{array}\right]
$$
Solution The determinant of $A$ is -2 , so we can apply Cramer's Rule.
$$
\begin{aligned}
& \operatorname{det}\left(A_{1}(\vec{b})\right)=\left|\begin{array}{cc}
-1 & 2 \\
1 & 4
\end{array}\right|=-6 . \\
& \operatorname{det}\left(A_{2}(\vec{b})\right)=\left|\begin{array}{cc}
1 & -1 \\
3 & 1
\end{array}\right|=4 .
\end{aligned}
$$
Therefore
$$
\begin{aligned}
& x_{1}=\frac{\operatorname{det}\left(A_{1}(\vec{b})\right)}{\operatorname{det}(A)}=\frac{-6}{-2}=3 \\
& x_{2}=\frac{\operatorname{det}\left(A_{2}(\vec{b})\right)}{\operatorname{det}(A)}=\frac{4}{-2}=-2
\end{aligned}
$$
and
$$
\vec{x}=\left[\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right]=\left[\begin{array}{c}
3 \\
-2
\end{array}\right]
$$
We learned in Section 3.4 that when considering a linear system $A \vec{x}=\vec{b}$ where $A$ is square, if $\operatorname{det}(A) \neq 0$ then $A$ is invertible and $A \vec{x}=\vec{b}$ has exactly one solution. We also stated in Key Idea 11 that if $\operatorname{det}(A)=0$, then $A$ is not invertible and so therefore either $A \vec{x}=\vec{b}$ has no solution or infinite solutions. Our method of figuring out which of these cases applied was to form the augmented matrix $\left[\begin{array}{ll}A & \vec{b}\end{array}\right]$, put it into reduced row echelon form, and then interpret the results.
Cramer's Rule specifies that $\operatorname{det}(A) \neq 0$ (so we are guaranteed a solution). When $\operatorname{det}(A)=0$ we are not able to discern whether infinite solutions or no solution exists for a given vector $\vec{b}$. Cramer's Rule is only applicable to the case when exactly one solution exists.
We end this section with a practical consideration. We have mentioned before that finding determinants is a computationally intensive operation. To solve a linear system with 3 equations and 3 unknowns, we need to compute 4 determinants. Just think: with 10 equations and 10 unknowns, we'd need to compute 11 really hard determinants of $10 \times 10$ matrices! That is a lot of work!
The upshot of this is that Cramer's Rule makes for a poor choice in solving numerical linear systems. It simply is not done in practice; it is hard to beat Gaussian elimination. ${ }^{24}$
So why include it? Because its truth is amazing. The determinant is a very strange operation; it produces a number in a very odd way. It should seem incredible to the
${ }^{24} \mathrm{~A}$ version of Cramer's Rule is often taught in introductory differential equations courses as it can be used to find solutions to certain linear differential equations. In this situation, the entries of the matrices are functions, not numbers, and hence computing determinants is easier than using Gaussian elimination. Again, though, as the matrices get large, other solution methods are resorted to. reader that by manipulating determinants in a particular way, we can solve linear systems.
In the next chapter we'll see another use for the determinant. Meanwhile, try to develop a deeper appreciation of math: odd, complicated things that seem completely unrelated often are intricately tied together. Mathematicians see these connections and describe them as "beautiful."
## Exercises 3.5
In Exercises 1-12, matrices $A$ and $\vec{b}$ are given.
(a) Give $\operatorname{det}(A)$ and $\operatorname{det}\left(A_{i}\right)$ for all $i$.
(b) Use Cramer's Rule to solve $A \vec{x}=\vec{b}$. If Cramer's Rule cannot be used to find the solution, then state whether or not a solution exists.
1. $A=\left[\begin{array}{cc}7 & -7 \\ -7 & 9\end{array}\right], \quad \vec{b}=\left[\begin{array}{c}28 \\ -26\end{array}\right]$
2. $A=\left[\begin{array}{cc}9 & 5 \\ -4 & -7\end{array}\right], \quad \vec{b}=\left[\begin{array}{c}-45 \\ 20\end{array}\right]$
3. $A=\left[\begin{array}{cc}-8 & 16 \\ 10 & -20\end{array}\right], \quad \vec{b}=\left[\begin{array}{c}-48 \\ 60\end{array}\right]$
4. $A=\left[\begin{array}{cc}0 & -6 \\ 9 & -10\end{array}\right], \quad \vec{b}=\left[\begin{array}{c}6 \\ -17\end{array}\right]$
5. $A=\left[\begin{array}{cc}2 & 10 \\ -1 & 3\end{array}\right], \quad \vec{b}=\left[\begin{array}{l}42 \\ 19\end{array}\right]$
6. $A=\left[\begin{array}{cc}7 & 14 \\ -2 & -4\end{array}\right], \quad \vec{b}=\left[\begin{array}{c}-1 \\ 4\end{array}\right]$
7. $A=\left[\begin{array}{ccc}3 & 0 & -3 \\ 5 & 4 & 4 \\ 5 & 5 & -4\end{array}\right], \quad \vec{b}=\left[\begin{array}{c}24 \\ 0 \\ 31\end{array}\right]$
8. $A=\left[\begin{array}{ccc}4 & 9 & 3 \\ -5 & -2 & -13 \\ -1 & 10 & -13\end{array}\right]$,
$$
\vec{b}=\left[\begin{array}{c}
-28 \\
35 \\
7
\end{array}\right]
$$
9. $A=\left[\begin{array}{ccc}4 & -4 & 0 \\ 5 & 1 & -1 \\ 3 & -1 & 2\end{array}\right], \quad \vec{b}=\left[\begin{array}{c}16 \\ 22 \\ 8\end{array}\right]$
10. $A=\left[\begin{array}{ccc}1 & 0 & -10 \\ 4 & -3 & -10 \\ -9 & 6 & -2\end{array}\right]$,
$\vec{b}=\left[\begin{array}{c}-40 \\ -94 \\ 132\end{array}\right]$
11. $A=\left[\begin{array}{ccc}7 & -4 & 25 \\ -2 & 1 & -7 \\ 9 & -7 & 34\end{array}\right]$,
$\vec{b}=\left[\begin{array}{c}-1 \\ -3 \\ 5\end{array}\right]$
12. $A=\left[\begin{array}{ccc}-6 & -7 & -7 \\ 5 & 4 & 1 \\ 5 & 4 & 8\end{array}\right]$,
$\vec{b}=\left[\begin{array}{c}58 \\ -35 \\ -49\end{array}\right]$
## EIGENVALUES AND EIGENVECTORS
We have often explored new ideas in matrix algebra by making connections to our previous algebraic experience. Adding two numbers, $x+y$, led us to adding vectors $\vec{x}+\vec{y}$ and adding matrices $A+B$. We explored multiplication, which then led us to solving the matrix equation $A \vec{x}=\vec{b}$, which was reminiscent of solving the algebra equation $a x=b$.
This chapter is motivated by another analogy. Consider: when we multiply an unknown number $x$ by another number such as 5 , what do we know about the result? Unless, $x=0$, we know that in some sense $5 x$ will be " 5 times bigger than $x$." Applying this to vectors, we would readily agree that $5 \vec{x}$ gives a vector that is " 5 times bigger than $\vec{x} . "$ Each entry in $\vec{x}$ is multiplied by 5 .
Within the matrix algebra context, though, we have two types of multiplication: scalar and matrix multiplication. What happens to $\vec{x}$ when we multiply it by a matrix $A$ ? Our first response is likely along the lines of "You just get another vector. There is no definable relationship." We might wonder if there is ever the case where a matrix - vector multiplication is very similar to a scalar - vector multiplication. That is, do we ever have the case where $A \vec{x}=a \vec{x}$, where $a$ is some scalar? That is the motivating question of this chapter.
### Eigenvalues and Eigenvectors
## AS YOU READ . .
1. T/F: Given any matrix $A$, we can always find a vector $\vec{x}$ where $A \vec{x}=\vec{x}$.
2. When is the zero vector an eigenvector for a matrix?
3. If $\vec{v}$ is an eigenvector of a matrix $A$ with eigenvalue of 2 , then what is $A \vec{v}$ ?
4. T/F: If $A$ is a $5 \times 5$ matrix, to find the eigenvalues of $A$, we would need to find the roots of a $5^{\text {th }}$ degree polynomial. We start by considering the matrix $A$ and vector $\vec{x}$ as given below. ${ }^{1}$
$$
A=\left[\begin{array}{ll}
1 & 4 \\
2 & 3
\end{array}\right] \quad \vec{x}=\left[\begin{array}{l}
1 \\
1
\end{array}\right]
$$
Multiplying $A \vec{x}$ gives:
$$
\begin{aligned}
A \vec{x} & =\left[\begin{array}{ll}
1 & 4 \\
2 & 3
\end{array}\right]\left[\begin{array}{l}
1 \\
1
\end{array}\right] \\
& =\left[\begin{array}{l}
5 \\
5
\end{array}\right] \\
& =5\left[\begin{array}{l}
1 \\
1
\end{array}\right] !
\end{aligned}
$$
Wow! It looks like multiplying $A \vec{x}$ is the same as $5 \vec{x}$ ! This makes us wonder lots of things: Is this the only case in the world where something like this happens? ${ }^{2}$ Is $A$ somehow a special matrix, and $A \vec{x}=5 \vec{x}$ for any vector $\vec{x}$ we pick? ${ }^{3}$ Or maybe $\vec{x}$ was a special vector, and no matter what $2 \times 2$ matrix $A$ we picked, we would have $A \vec{x}=5 \vec{x} .{ }^{4}$
A more likely explanation is this: given the matrix $A$, the number 5 and the vector $\vec{x}$ formed a special pair that happened to work together in a nice way. It is then natural to wonder if other "special" pairs exist. For instance, could we find a vector $\vec{x}$ where $A \vec{x}=3 \vec{x}$ ?
This equation is hard to solve at first; we are not used to matrix equations where $\vec{x}$ appears on both sides of "=." Therefore we put off solving this for just a moment to state a definition and make a few comments.
Definition 27
## Eigenvalues and Eigenvectors
Let $A$ be an $n \times n$ matrix, $\vec{x}$ a nonzero $n \times 1$ column vector and $\lambda$ a scalar. If
$$
A \vec{x}=\lambda \vec{x},
$$
then $\vec{x}$ is an eigenvector of $A$ and $\lambda$ is an eigenvalue of $A$.
The word "eigen" is German for "proper" or "characteristic." Therefore, an eigenvector of $A$ is a "characteristic vector of $A$." This vector tells us something about $A$.
Why do we use the Greek letter $\lambda$ (lambda)? It is pure tradition. Above, we used $a$ to represent the unknown scalar, since we are used to that notation. We now switch to $\lambda$ because that is how everyone else does it. ${ }^{5}$ Don't get hung up on this; $\lambda$ is just a number.
${ }^{1}$ Recall this matrix and vector were used in Example 40 on page 75.
${ }^{2}$ Probably not.
${ }^{3}$ Probably not.
${ }^{4}$ See footnote 2 .
${ }^{5} \mathrm{An}$ example of mathematical peer pressure. Note that our definition requires that $A$ be a square matrix. If $A$ isn't square then $A \vec{x}$ and $\lambda \vec{x}$ will have different sizes, and so they cannot be equal. Also note that $\vec{x}$ must be nonzero. Why? What if $\vec{x}=\overrightarrow{0}$ ? Then no matter what $\lambda$ is, $A \vec{x}=\lambda \vec{x}$. This would then imply that every number is an eigenvalue; if every number is an eigenvalue, then we wouldn't need a definition for it. ${ }^{6}$ Therefore we specify that $\vec{x} \neq \overrightarrow{0}$.
Our last comment before trying to find eigenvalues and eigenvectors for given matrices deals with "why we care." Did we stumble upon a mathematical curiosity, or does this somehow help us build better bridges, heal the sick, send astronauts into orbit, design optical equipment, and understand quantum mechanics? The answer, of course, is "Yes." 7 This is a wonderful topic in and of itself: we need no external application to appreciate its worth. At the same time, it has many, many applications to "the real world." A simple Internet seach on "applications of eigenvalues" with confirm this.
Back to our math. Given a square matrix $A$, we want to find a nonzero vector $\vec{x}$ and a scalar $\lambda$ such that $A \vec{x}=\lambda \vec{x}$. We will solve this using the skills we developed in Chapter 2.
$$
\begin{aligned}
A \vec{x} & =\lambda \vec{x} \\
A \vec{x}-\lambda \vec{x} & =\overrightarrow{0} \\
(A-\lambda I) \vec{x} & =\overrightarrow{0}
\end{aligned}
$$
Think about this last factorization. We are likely tempted to say
$$
A \vec{x}-\lambda \vec{x}=(A-\lambda) \vec{x}
$$
but this really doesn't make sense. After all, what does "a matrix minus a number" mean? We need the identity matrix in order for this to be logical.
Let us now think about the equation $(A-\lambda I) \vec{x}=\overrightarrow{0}$. While it looks complicated, it really is just matrix equation of the type we solved in Section 2.4. We are just trying to solve $B \vec{x}=\overrightarrow{0}$, where $B=(A-\lambda I)$.
We know from our previous work that this type of equation ${ }^{8}$ always has a solution, namely, $\vec{x}=\overrightarrow{0}$. However, we want $\vec{x}$ to be an eigenvector and, by the definition, eigenvectors cannot be $\overrightarrow{0}$.
This means that we want solutions to $(A-\lambda I) \vec{x}=\vec{O}$ other than $\vec{x}=\overrightarrow{0}$. Recall that Theorem 8 says that if the matrix $(A-\lambda I)$ is invertible, then the only solution to $(A-\lambda I) \vec{x}=\overrightarrow{0}$ is $\vec{x}=\overrightarrow{0}$. Therefore, in order to have other solutions, we need $(A-\lambda I)$ to not be invertible.
Finally, recall from Theorem 16 that noninvertible matrices all have a determinant of 0 . Therefore, if we want to find eigenvalues $\lambda$ and eigenvectors $\vec{x}$, we need $\operatorname{det}(A-\lambda I)=0$.
Let's start our practice of this theory by finding $\lambda$ such that $\operatorname{det}(A-\lambda I)=0$; that is, let's find the eigenvalues of a matrix.
${ }^{6}$ Recall footnote 17 on page 107.
${ }^{7}$ Except for the "understand quantum mechanics" part. Nobody truly understands that stuff; they just probably understand it.
${ }^{8}$ Recall this is a homogeneous system of equations. Example 84 Find the eigenvalues of $A$, that is, find $\lambda$ such that $\operatorname{det}(A-\lambda I)=0$, where
$$
A=\left[\begin{array}{ll}
1 & 4 \\
2 & 3
\end{array}\right] .
$$
Solution (Note that this is the matrix we used at the beginning of this section.) First, we write out what $A-\lambda /$ is:
$$
\begin{aligned}
A-\lambda I & =\left[\begin{array}{ll}
1 & 4 \\
2 & 3
\end{array}\right]-\lambda\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right] \\
& =\left[\begin{array}{ll}
1 & 4 \\
2 & 3
\end{array}\right]-\left[\begin{array}{cc}
\lambda & 0 \\
0 & \lambda
\end{array}\right] \\
& =\left[\begin{array}{cc}
1-\lambda & 4 \\
2 & 3-\lambda
\end{array}\right]
\end{aligned}
$$
Therefore,
$$
\begin{aligned}
\operatorname{det}(A-\lambda I) & =\left|\begin{array}{cc}
1-\lambda & 4 \\
2 & 3-\lambda
\end{array}\right| \\
& =(1-\lambda)(3-\lambda)-8 \\
& =\lambda^{2}-4 \lambda-5
\end{aligned}
$$
Since we want $\operatorname{det}(A-\lambda I)=0$, we want $\lambda^{2}-4 \lambda-5=0$. This is a simple quadratic equation that is easy to factor:
$$
\begin{aligned}
\lambda^{2}-4 \lambda-5 & =0 \\
(\lambda-5)(\lambda+1) & =0 \\
\lambda & =-1,5
\end{aligned}
$$
According to our above work, $\operatorname{det}(A-\lambda I)=0$ when $\lambda=-1,5$. Thus, the eigenvalues of $A$ are -1 and 5 .
Earlier, when looking at the same matrix as used in our example, we wondered if we could find a vector $\vec{x}$ such that $A \vec{x}=3 \vec{x}$. According to this example, the answer is "No." With this matrix $A$, the only values of $\lambda$ that work are -1 and 5 .
Let's restate the above in a different way: It is pointless to try to find $\vec{x}$ where $A \vec{x}=3 \vec{x}$, for there is no such $\vec{x}$. There are only 2 equations of this form that have a solution, namely
$$
A \vec{x}=-\vec{x} \quad \text { and } \quad A \vec{x}=5 \vec{x} .
$$
As we introduced this section, we gave a vector $\vec{x}$ such that $A \vec{x}=5 \vec{x}$. Is this the only one? Let's find out while calling our work an example; this will amount to finding the eigenvectors of $A$ that correspond to the eigenvector of 5 . Example $85 \quad$ Find $\vec{x}$ such that $A \vec{x}=5 \vec{x}$, where
$$
A=\left[\begin{array}{ll}
1 & 4 \\
2 & 3
\end{array}\right]
$$
Solution Recall that our algebra from before showed that if
$$
A \vec{x}=\lambda \vec{x} \text { then }(A-\lambda I) \vec{x}=\vec{O} .
$$
Therefore, we need to solve the equation $(A-\lambda I) \vec{x}=\vec{O}$ for $\vec{x}$ when $\lambda=5$.
$$
\begin{aligned}
A-5 I & =\left[\begin{array}{ll}
1 & 4 \\
2 & 3
\end{array}\right]-5\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right] \\
& =\left[\begin{array}{cc}
-4 & 4 \\
2 & -2
\end{array}\right]
\end{aligned}
$$
To solve $(A-5 I) \vec{X}=\vec{O}$, we form the augmented matrix and put it into reduced row echelon form:
$$
\left[\begin{array}{ccc}
-4 & 4 & 0 \\
2 & -2 & 0
\end{array}\right] \quad \overrightarrow{\operatorname{rref}} \quad\left[\begin{array}{ccc}
1 & -1 & 0 \\
0 & 0 & 0
\end{array}\right]
$$
Thus
$$
\begin{aligned}
& x_{1}=x_{2} \\
& x_{2} \text { is free }
\end{aligned}
$$
and
$$
\vec{x}=\left[\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right]=x_{2}\left[\begin{array}{l}
1 \\
1
\end{array}\right] .
$$
We have infinite solutions to the equation $A \vec{x}=5 \vec{x}$; any nonzero scalar multiple of the vector $\left[\begin{array}{l}1 \\ 1\end{array}\right]$ is a solution. We can do a few examples to confirm this:
$$
\begin{gathered}
{\left[\begin{array}{ll}
1 & 4 \\
2 & 3
\end{array}\right]\left[\begin{array}{l}
2 \\
2
\end{array}\right]=\left[\begin{array}{l}
10 \\
10
\end{array}\right]=5\left[\begin{array}{l}
2 \\
2
\end{array}\right] ;} \\
{\left[\begin{array}{ll}
1 & 4 \\
2 & 3
\end{array}\right]\left[\begin{array}{l}
7 \\
7
\end{array}\right]=\left[\begin{array}{l}
35 \\
35
\end{array}\right]=5\left[\begin{array}{l}
7 \\
7
\end{array}\right] ;} \\
{\left[\begin{array}{ll}
1 & 4 \\
2 & 3
\end{array}\right]\left[\begin{array}{l}
-3 \\
-3
\end{array}\right]=\left[\begin{array}{l}
-15 \\
-15
\end{array}\right]=5\left[\begin{array}{l}
-3 \\
-3
\end{array}\right] .}
\end{gathered}
$$
Our method of finding the eigenvalues of a matrix $A$ boils down to determining which values of $\lambda$ give the matrix $(A-\lambda I)$ a determinant of 0 . In computing $\operatorname{det}(A-\lambda I)$, we get a polynomial in $\lambda$ whose roots are the eigenvalues of $A$. This polynomial is important and so it gets its own name. Definition 28
## Characteristic Polynomial
Let $A$ be an $n \times n$ matrix. The characteristic polynomial of $A$ is the $n^{\text {th }}$ degree polynomial $p(\lambda)=\operatorname{det}(A-\lambda I)$.
Our definition just states what the characteristic polynomial is. We know from our work so far why we care: the roots of the characteristic polynomial of an $n \times n$ matrix $A$ are the eigenvalues of $A$.
In Examples 84 and 85, we found eigenvalues and eigenvectors, respectively, of a given matrix. That is, given a matrix $A$, we found values $\lambda$ and vectors $\vec{x}$ such that $A \vec{x}=\lambda \vec{x}$. The steps that follow outline the general procedure for finding eigenvalues and eigenvectors; we'll follow this up with some examples.
Key Idea 14
$$
\text { Finding Eigenvalues and Eigenvectors }
$$
Let $A$ be an $n \times n$ matrix.
1. To find the eigenvalues of $A$, compute $p(\lambda)$, the characteristic polynomial of $A$, set it equal to 0 , then solve for $\lambda$.
2. To find the eigenvectors of $A$, for each eigenvalue solve the homogeneous system $(A-\lambda I) \vec{x}=\overrightarrow{0}$.
Example 86 vector where
Solution equal to 0 . Find the eigenvalues of $A$, and for each eigenvalue, find an eigen-
$$
A=\left[\begin{array}{cc}
-3 & 15 \\
3 & 9
\end{array}\right]
$$
To find the eigenvalues, we must compute $\operatorname{det}(A-\lambda I)$ and set it
$$
\begin{aligned}
\operatorname{det}(A-\lambda I) & =\left|\begin{array}{cc}
-3-\lambda & 15 \\
3 & 9-\lambda
\end{array}\right| \\
& =(-3-\lambda)(9-\lambda)-45 \\
& =\lambda^{2}-6 \lambda-27-45 \\
& =\lambda^{2}-6 \lambda-72 \\
& =(\lambda-12)(\lambda+6)
\end{aligned}
$$
Therefore, $\operatorname{det}(A-\lambda I)=0$ when $\lambda=-6$ and 12 ; these are our eigenvalues. (We
### Eigenvalues and Eigenvectors
should note that $p(\lambda)=\lambda^{2}-6 \lambda-72$ is our characteristic polynomial.) It sometimes helps to give them "names," so we'll say $\lambda_{1}=-6$ and $\lambda_{2}=12$. Now we find eigenvectors.
For $\lambda_{1}=-6$ :
We need to solve the equation $(A-(-6) I) \vec{x}=\overrightarrow{0}$. To do this, we form the appropriate augmented matrix and put it into reduced row echelon form.
$$
\left[\begin{array}{lll}
3 & 15 & 0 \\
3 & 15 & 0
\end{array}\right] \quad \overrightarrow{\operatorname{rref}} \quad\left[\begin{array}{lll}
1 & 5 & 0 \\
0 & 0 & 0
\end{array}\right]
$$
Our solution is
$$
\begin{aligned}
& x_{1}=-5 x_{2} \\
& x_{2} \text { is free; }
\end{aligned}
$$
in vector form, we have
$$
\vec{x}=x_{2}\left[\begin{array}{c}
-5 \\
1
\end{array}\right]
$$
We may pick any nonzero value for $x_{2}$ to get an eigenvector; a simple option is $x_{2}=1$. Thus we have the eigenvector
$$
\overrightarrow{x_{1}}=\left[\begin{array}{c}
-5 \\
1
\end{array}\right] .
$$
(We used the notation $\overrightarrow{x_{1}}$ to associate this eigenvector with the eigenvalue $\lambda_{1}$.)
We now repeat this process to find an eigenvector for $\lambda_{2}=12$ :
In solving $(A-12 I) \vec{x}=\overrightarrow{0}$, we find
$$
\left[\begin{array}{ccc}
-15 & 15 & 0 \\
3 & -3 & 0
\end{array}\right] \quad \overrightarrow{\operatorname{rref}} \quad\left[\begin{array}{ccc}
1 & -1 & 0 \\
0 & 0 & 0
\end{array}\right]
$$
In vector form, we have
$$
\vec{x}=x_{2}\left[\begin{array}{l}
1 \\
1
\end{array}\right] .
$$
Again, we may pick any nonzero value for $x_{2}$, and so we choose $x_{2}=1$. Thus an eigenvector for $\lambda_{2}$ is
$$
\overrightarrow{x_{2}}=\left[\begin{array}{l}
1 \\
1
\end{array}\right]
$$
To summarize, we have:
$$
\text { eigenvalue } \lambda_{1}=-6 \text { with eigenvector } \overrightarrow{x_{1}}=\left[\begin{array}{c}
-5 \\
1
\end{array}\right]
$$
and
$$
\text { eigenvalue } \lambda_{2}=12 \text { with eigenvector } \overrightarrow{x_{2}}=\left[\begin{array}{l}
1 \\
1
\end{array}\right] \text {. }
$$
We should take a moment and check our work: is it true that $\overrightarrow{A \overrightarrow{x_{1}}}=\lambda_{1} \overrightarrow{x_{1}}$ ?
$$
\begin{aligned}
A \overrightarrow{x_{1}} & =\left[\begin{array}{cc}
-3 & 15 \\
3 & 9
\end{array}\right]\left[\begin{array}{c}
-5 \\
1
\end{array}\right] \\
& =\left[\begin{array}{c}
30 \\
-6
\end{array}\right] \\
& =(-6)\left[\begin{array}{c}
-5 \\
1
\end{array}\right] \\
& =\lambda_{1} \overrightarrow{x_{1}} .
\end{aligned}
$$
Yes; it appears we have truly found an eigenvalue/eigenvector pair for the matrix $A$.
Let's do another example.
Example $87 \quad$ Let $A=\left[\begin{array}{cc}-3 & 0 \\ 5 & 1\end{array}\right]$. Find the eigenvalues of $A$ and an eigenvector for each eigenvalue.
Solution We first compute the characteristic polynomial, set it equal to 0 , then solve for $\lambda$.
$$
\begin{aligned}
\operatorname{det}(A-\lambda I) & =\left|\begin{array}{cc}
-3-\lambda & 0 \\
5 & 1-\lambda
\end{array}\right| \\
& =(-3-\lambda)(1-\lambda)
\end{aligned}
$$
From this, we see that $\operatorname{det}(A-\lambda I)=0$ when $\lambda=-3,1$. We'll set $\lambda_{1}=-3$ and $\lambda_{2}=1$.
Finding an eigenvector for $\lambda_{1}$ :
We solve $(A-(-3) I) \vec{x}=\vec{O}$ for $\vec{x}$ by row reducing the appropriate matrix:
$$
\left[\begin{array}{lll}
0 & 0 & 0 \\
5 & 4 & 0
\end{array}\right] \quad \overrightarrow{\operatorname{rref}} \quad\left[\begin{array}{ccc}
1 & 5 / 4 & 0 \\
0 & 0 & 0
\end{array}\right]
$$
Our solution, in vector form, is
$$
\vec{x}=x_{2}\left[\begin{array}{c}
-5 / 4 \\
1
\end{array}\right]
$$
Again, we can pick any nonzero value for $x_{2}$; a nice choice would eliminate the fraction. Therefore we pick $x_{2}=4$, and find
$$
\overrightarrow{x_{1}}=\left[\begin{array}{c}
-5 \\
4
\end{array}\right]
$$
Finding an eigenvector for $\lambda_{2}$ : We solve $(A-(1) I) \vec{x}=\vec{O}$ for $\vec{x}$ by row reducing the appropriate matrix:
$$
\left[\begin{array}{ccc}
-4 & 0 & 0 \\
5 & 0 & 0
\end{array}\right] \quad \overrightarrow{\operatorname{rref}} \quad\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 0 & 0
\end{array}\right]
$$
We've seen a matrix like this before, ${ }^{9}$ but we may need a bit of a refreshing. Our first row tells us that $x_{1}=0$, and we see that no rows/equations involve $x_{2}$. We conclude that $x_{2}$ is free. Therefore, our solution, in vector form, is
$$
\vec{x}=x_{2}\left[\begin{array}{l}
0 \\
1
\end{array}\right] .
$$
We pick $x_{2}=1$, and find
$$
\overrightarrow{x_{2}}=\left[\begin{array}{l}
0 \\
1
\end{array}\right]
$$
To summarize, we have:
$$
\text { eigenvalue } \lambda_{1}=-3 \text { with eigenvector } \overrightarrow{x_{1}}=\left[\begin{array}{c}
-5 \\
4
\end{array}\right]
$$
and
$$
\text { eigenvalue } \lambda_{2}=1 \text { with eigenvector } \overrightarrow{x_{2}}=\left[\begin{array}{l}
0 \\
1
\end{array}\right] \text {. }
$$
So far, our examples have involved $2 \times 2$ matrices. Let's do an example with a $3 \times 3$ matrix.
Example $88 \quad$ Find the eigenvalues of $A$, and for each eigenvalue, give one eigenvector, where
$$
A=\left[\begin{array}{ccc}
-7 & -2 & 10 \\
-3 & 2 & 3 \\
-6 & -2 & 9
\end{array}\right]
$$
Solution We first compute the characteristic polynomial, set it equal to 0 , then solve for $\lambda$. A warning: this process is rather long. We'll use cofactor expansion along the first row; don't get bogged down with the arithmetic that comes from each step; just try to get the basic idea of what was done from step to step.
${ }^{9}$ See page 31. Our future need of knowing how to handle this situation is foretold in footnote 5 .
$$
\begin{aligned}
\operatorname{det}(\boldsymbol{A}-\lambda I) & =\left|\begin{array}{ccc}
-7-\lambda & -2 & 10 \\
-3 & 2-\lambda & 3 \\
-6 & -2 & 9-\lambda
\end{array}\right| \\
& =(-7-\lambda)\left|\begin{array}{cc}
2-\lambda & 3 \\
-2 & 9-\lambda
\end{array}\right|-(-2)\left|\begin{array}{cc}
-3 & 3 \\
-6 & 9-\lambda
\end{array}\right|+10\left|\begin{array}{cc}
-3 & 2-\lambda \\
-6 & -2
\end{array}\right| \\
& =(-7-\lambda)\left(\lambda^{2}-11 \lambda+24\right)+2(3 \lambda-9)+10(-6 \lambda+18) \\
& =-\lambda^{3}+4 \lambda^{2}-\lambda-6 \\
& =-(\lambda+1)(\lambda-2)(\lambda-3)
\end{aligned}
$$
In the last step we factored the characteristic polynomial $-\lambda^{3}+4 \lambda^{2}-\lambda-6$. Factoring polynomials of degree $>2$ is not trivial; we'll assume the reader has access to methods for doing this accurately. ${ }^{10}$
Our eigenvalues are $\lambda_{1}=-1, \lambda_{2}=2$ and $\lambda_{3}=3$. We now find corresponding eigenvectors.
$$
\text { For } \lambda_{1}=-1 \text { : }
$$
We need to solve the equation $(A-(-1) I) \vec{X}=\vec{O}$. To do this, we form the appropriate augmented matrix and put it into reduced row echelon form.
$$
\left[\begin{array}{cccc}
-6 & -2 & 10 & 0 \\
-3 & 3 & 3 & 0 \\
-6 & -2 & 10 & 0
\end{array}\right] \quad \overrightarrow{\operatorname{rref}}\left[\begin{array}{cccc}
1 & 0 & -1.5 & 0 \\
0 & 1 & -.5 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]
$$
Our solution, in vector form, is
$$
\vec{x}=x_{3}\left[\begin{array}{c}
3 / 2 \\
1 / 2 \\
1
\end{array}\right]
$$
We can pick any nonzero value for $x_{3}$; a nice choice would get rid of the fractions. So we'll set $x_{3}=2$ and choose $\overrightarrow{x_{1}}=\left[\begin{array}{l}3 \\ 1 \\ 2\end{array}\right]$ as our eigenvector.
For $\lambda_{2}=2$ :
We need to solve the equation $(A-2 I) \vec{x}=\overrightarrow{0}$. To do this, we form the appropriate augmented matrix and put it into reduced row echelon form.
${ }^{10}$ You probably learned how to do this in an algebra course. As a reminder, possible roots can be found by factoring the constant term (in this case, -6 ) of the polynomial. That is, the roots of this equation could be $\pm 1, \pm 2, \pm 3$ and \pm 6 . That's 12 things to check.
One could also graph this polynomial to find the roots. Graphing will show us that $\lambda=3$ looks like a root, and a simple calculation will confirm that it is.
$$
\left[\begin{array}{cccc}
-9 & -2 & 10 & 0 \\
-3 & 0 & 3 & 0 \\
-6 & -2 & 7 & 0
\end{array}\right] \quad \overrightarrow{\operatorname{rref}}\left[\begin{array}{cccc}
1 & 0 & -1 & 0 \\
0 & 1 & -.5 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]
$$
Our solution, in vector form, is
$$
\vec{x}=x_{3}\left[\begin{array}{c}
1 \\
1 / 2 \\
1
\end{array}\right]
$$
We can pick any nonzero value for $x_{3}$; again, a nice choice would get rid of the fractions. So we'll set $x_{3}=2$ and choose $\overrightarrow{x_{2}}=\left[\begin{array}{l}2 \\ 1 \\ 2\end{array}\right]$ as our eigenvector.
For $\lambda_{3}=3$ :
We need to solve the equation $(A-3 I) \vec{X}=\overrightarrow{0}$. To do this, we form the appropriate augmented matrix and put it into reduced row echelon form.
$$
\left[\begin{array}{cccc}
-10 & -2 & 10 & 0 \\
-3 & -1 & 3 & 0 \\
-6 & -2 & 6 & 0
\end{array}\right] \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{cccc}
1 & 0 & -1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]
$$
Our solution, in vector form, is (note that $x_{2}=0$ ):
$$
\vec{x}=x_{3}\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right] \text {. }
$$
We can pick any nonzero value for $x_{3}$; an easy choice is $x_{3}=1$, so $\overrightarrow{x_{3}}=\left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right]$ as our eigenvector.
To summarize, we have the following eigenvalue/eigenvector pairs:
$$
\begin{gathered}
\text { eigenvalue } \lambda_{1}=-1 \text { with eigenvector } \overrightarrow{x_{1}}=\left[\begin{array}{l}
3 \\
1 \\
2
\end{array}\right] \\
\text { eigenvalue } \lambda_{2}=2 \text { with eigenvector } \overrightarrow{x_{2}}=\left[\begin{array}{l}
2 \\
1 \\
2
\end{array}\right] \\
\text { eigenvalue } \lambda_{3}=3 \text { with eigenvector } \overrightarrow{x_{3}}=\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right]
\end{gathered}
$$
Let's practice once more.
## Example 89
vector, where Find the eigenvalues of $A$, and for each eigenvalue, give one eigen-$$
A=\left[\begin{array}{ccc}
2 & -1 & 1 \\
0 & 1 & 6 \\
0 & 3 & 4
\end{array}\right]
$$
Solution We first compute the characteristic polynomial, set it equal to 0 , then solve for $\lambda$. We'll use cofactor expansion down the first column (since it has lots of zeros).
$$
\begin{aligned}
\operatorname{det}(A-\lambda I) & =\left|\begin{array}{ccc}
2-\lambda & -1 & 1 \\
0 & 1-\lambda & 6 \\
0 & 3 & 4-\lambda
\end{array}\right| \\
& =(2-\lambda)\left|\begin{array}{cc}
1-\lambda & 6 \\
3 & 4-\lambda
\end{array}\right| \\
& =(2-\lambda)\left(\lambda^{2}-5 \lambda-14\right) \\
& =(2-\lambda)(\lambda-7)(\lambda+2)
\end{aligned}
$$
Notice that while the characteristic polynomial is cubic, we never actually saw a cubic; we never distributed the $(2-\lambda)$ across the quadratic. Instead, we realized that this was a factor of the cubic, and just factored the remaining quadratic. (This makes this example quite a bit simpler than the previous example.)
Our eigenvalues are $\lambda_{1}=-2, \lambda_{2}=2$ and $\lambda_{3}=7$. We now find corresponding eigenvectors.
For $\lambda_{1}=-2$ :
We need to solve the equation $(A-(-2) I) \vec{x}=\overrightarrow{0}$. To do this, we form the appropriate augmented matrix and put it into reduced row echelon form.
$$
\left[\begin{array}{cccc}
4 & -1 & 1 & 0 \\
0 & 3 & 6 & 0 \\
0 & 3 & 6 & 0
\end{array}\right] \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{cccc}
1 & 0 & 3 / 4 & 0 \\
0 & 1 & 2 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]
$$
Our solution, in vector form, is
$$
\vec{x}=x_{3}\left[\begin{array}{c}
-3 / 4 \\
-2 \\
1
\end{array}\right]
$$
We can pick any nonzero value for $x_{3}$; a nice choice would get rid of the fractions. So we'll set $x_{3}=4$ and choose $\overrightarrow{x_{1}}=\left[\begin{array}{c}-3 \\ -8 \\ 4\end{array}\right]$ as our eigenvector. For $\lambda_{2}=2$ :
We need to solve the equation $(A-2 I) \vec{x}=\overrightarrow{0}$. To do this, we form the appropriate augmented matrix and put it into reduced row echelon form.
$$
\left[\begin{array}{cccc}
0 & -1 & 1 & 0 \\
0 & -1 & 6 & 0 \\
0 & 3 & 2 & 0
\end{array}\right] \quad \overrightarrow{\operatorname{rref}}\left[\begin{array}{llll}
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]
$$
This looks funny, so we'll look remind ourselves how to solve this. The first two rows tell us that $x_{2}=0$ and $x_{3}=0$, respectively. Notice that no row/equation uses $x_{1}$; we conclude that it is free. Therefore, our solution in vector form is
$$
\vec{x}=x_{1}\left[\begin{array}{l}
1 \\
0 \\
0
\end{array}\right] \text {. }
$$
We can pick any nonzero value for $x_{1}$; an easy choice is $x_{1}=1$ and choose $\overrightarrow{x_{2}}=$ $\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ as our eigenvector.
$$
\text { For } \lambda_{3}=7 \text { : }
$$
We need to solve the equation $(A-7 I) \vec{x}=\overrightarrow{0}$. To do this, we form the appropriate augmented matrix and put it into reduced row echelon form.
$$
\left[\begin{array}{cccc}
-5 & -1 & 1 & 0 \\
0 & -6 & 6 & 0 \\
0 & 3 & -3 & 0
\end{array}\right] \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & -1 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]
$$
Our solution, in vector form, is (note that $x_{1}=0$ ):
$$
\vec{x}=x_{3}\left[\begin{array}{l}
0 \\
1 \\
1
\end{array}\right]
$$
We can pick any nonzero value for $x_{3}$; an easy choice is $x_{3}=1$, so $\overrightarrow{x_{3}}=\left[\begin{array}{l}0 \\ 1 \\ 1\end{array}\right]$ as our eigenvector.
To summarize, we have the following eigenvalue/eigenvector pairs:
$$
\begin{gathered}
\text { eigenvalue } \lambda_{1}=-2 \text { with eigenvector } \overrightarrow{x_{1}}=\left[\begin{array}{c}
-3 \\
-8 \\
4
\end{array}\right] \\
\text { eigenvalue } \lambda_{2}=2 \text { with eigenvector } \overrightarrow{x_{2}}=\left[\begin{array}{l}
1 \\
0 \\
0
\end{array}\right]
\end{gathered}
$$
$$
\text { eigenvalue } \lambda_{3}=7 \text { with eigenvector } \overrightarrow{x_{3}}=\left[\begin{array}{l}
0 \\
1 \\
1
\end{array}\right]
$$
In this section we have learned about a new concept: given a matrix $A$ we can find certain values $\lambda$ and vectors $\vec{x}$ where $A \vec{x}=\lambda \vec{x}$. In the next section we will continue to the pattern we have established in this text: after learning a new concept, we see how it interacts with other concepts we know about. That is, we'll look for connections between eigenvalues and eigenvectors and things like the inverse, determinants, the trace, the transpose, etc.
## Exercises 4.1
In Exercises 1-6, a matrix $A$ and one of its eigenvectors are given. Find the eigenvalue of $A$ for the given eigenvector.
$$
\text { 1. } \begin{aligned}
A & =\left[\begin{array}{cc}
9 & 8 \\
-6 & -5
\end{array}\right] \\
\vec{x} & =\left[\begin{array}{c}
-4 \\
3
\end{array}\right]
\end{aligned}
$$
2. $A=\left[\begin{array}{cc}19 & -6 \\ 48 & -15\end{array}\right]$
$$
\vec{x}=\left[\begin{array}{l}
1 \\
3
\end{array}\right]
$$
3. $A=\left[\begin{array}{cc}1 & -2 \\ -2 & 4\end{array}\right]$
$$
\vec{x}=\left[\begin{array}{l}
2 \\
1
\end{array}\right]
$$
4. $A=\left[\begin{array}{ccc}-11 & -19 & 14 \\ -6 & -8 & 6 \\ -12 & -22 & 15\end{array}\right]$
$$
\vec{x}=\left[\begin{array}{l}
3 \\
2 \\
4
\end{array}\right]
$$
5. $A=\left[\begin{array}{ccc}-7 & 1 & 3 \\ 10 & 2 & -3 \\ -20 & -14 & 1\end{array}\right]$
$$
\vec{x}=\left[\begin{array}{c}
1 \\
-2 \\
4
\end{array}\right]
$$
6. $A=\left[\begin{array}{ccc}-12 & -10 & 0 \\ 15 & 13 & 0 \\ 15 & 18 & -5\end{array}\right]$
$$
\vec{x}=\left[\begin{array}{c}
-1 \\
1 \\
1
\end{array}\right]
$$
In Exercises 7-11, a matrix $A$ and one of its eigenvalues are given. Find an eigenvector of $A$ for the given eigenvalue.
$$
\begin{aligned}
& \text { 7. } A=\left[\begin{array}{cc}
16 & 6 \\
-18 & -5
\end{array}\right] \\
& \lambda=4 \\
& \text { 8. } A=\left[\begin{array}{cc}
-2 & 6 \\
-9 & 13
\end{array}\right] \\
& \lambda=7 \\
& \text { 9. } A=\left[\begin{array}{ccc}
-16 & -28 & -19 \\
42 & 69 & 46 \\
-42 & -72 & -49
\end{array}\right] \\
& \lambda=5 \\
& \text { 10. } A=\left[\begin{array}{ccc}
7 & -5 & -10 \\
6 & 2 & -6 \\
2 & -5 & -5
\end{array}\right] \\
& \lambda=-3 \\
& \text { 11. } \begin{aligned}
A & =\left[\begin{array}{ccc}
4 & 5 & -3 \\
-7 & -8 & 3 \\
1 & -5 & 8
\end{array}\right] \\
\lambda & =2
\end{aligned}
\end{aligned}
$$
In Exercises 12 - 28, find the eigenvalues of the given matrix. For each eigenvalue, give an eigenvector.
12. $\left[\begin{array}{ll}-1 & -4 \\ -3 & -2\end{array}\right]$
13. $\left[\begin{array}{ll}-4 & 72 \\ -1 & 13\end{array}\right]$
14. $\left[\begin{array}{cc}2 & -12 \\ 2 & -8\end{array}\right]$
15. $\left[\begin{array}{cc}3 & 12 \\ 1 & -1\end{array}\right]$
16. $\left[\begin{array}{cc}5 & 9 \\ -1 & -5\end{array}\right]$
17. $\left[\begin{array}{cc}3 & -1 \\ -1 & 3\end{array}\right]$
18. $\left[\begin{array}{cc}0 & 1 \\ 25 & 0\end{array}\right]$
19. $\left[\begin{array}{cc}-3 & 1 \\ 0 & -1\end{array}\right]$
20. $\left[\begin{array}{ccc}1 & -2 & -3 \\ 0 & 3 & 0 \\ 0 & -1 & -1\end{array}\right]$
21. $\left[\begin{array}{ccc}5 & -2 & 3 \\ 0 & 4 & 0 \\ 0 & -1 & 3\end{array}\right]$
22. $\left[\begin{array}{ccc}1 & 0 & 12 \\ 2 & -5 & 0 \\ 1 & 0 & 2\end{array}\right]$
23. $\left[\begin{array}{ccc}1 & 0 & -18 \\ -4 & 3 & -1 \\ 1 & 0 & -8\end{array}\right]$
24. $\left[\begin{array}{ccc}-1 & 18 & 0 \\ 1 & 2 & 0 \\ 5 & -3 & -1\end{array}\right]$
25. $\left[\begin{array}{ccc}5 & 0 & 0 \\ 1 & 1 & 0 \\ -1 & 5 & -2\end{array}\right]$
26. $\left[\begin{array}{ccc}2 & -1 & 1 \\ 0 & 3 & 6 \\ 0 & 0 & 7\end{array}\right]$
27. $\left[\begin{array}{ccc}3 & 5 & -5 \\ -2 & 3 & 2 \\ -2 & 5 & 0\end{array}\right]$
28. $\left[\begin{array}{lll}1 & 2 & 1 \\ 1 & 2 & 3 \\ 1 & 1 & 1\end{array}\right]$
### Properties of Eigenvalues and Eigenvectors
## AS YOU READ
1. T/F: $A$ and $A^{T}$ have the same eigenvectors.
2. $\mathrm{T} / \mathrm{F}$ : $A$ and $A^{-1}$ have the same eigenvalues.
3. T/F: Marie Ennemond Camille Jordan was a guy.
4. T/F: Matrices with a trace of 0 are important, although we haven't seen why.
5. $\mathrm{T} / \mathrm{F}$ : A matrix $A$ is invertible only if 1 is an eigenvalue of $A$.
In this section we'll explore how the eigenvalues and eigenvectors of a matrix relate to other properties of that matrix. This section is essentially a hodgepodge of interesting facts about eigenvalues; the goal here is not to memorize various facts about matrix algebra, but to again be amazed at the many connections between mathematical concepts.
We'll begin our investigations with an example that will give a foundation for other discoveries. Example $90 \quad$ Let $A=\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6\end{array}\right]$. Find the eigenvalues of $A$.
Solution To find the eigenvalues, we compute $\operatorname{det}(A-\lambda I)$ :
$$
\begin{aligned}
\operatorname{det}(A-\lambda I) & =\left|\begin{array}{ccc}
1-\lambda & 2 & 3 \\
0 & 4-\lambda & 5 \\
0 & 0 & 6-\lambda
\end{array}\right| \\
& =(1-\lambda)(4-\lambda)(6-\lambda)
\end{aligned}
$$
Since our matrix is triangular, the determinant is easy to compute; it is just the product of the diagonal elements. Therefore, we found (and factored) our characteristic polynomial very easily, and we see that we have eigenvalues of $\lambda=1,4$, and 6 .
This examples demonstrates a wonderful fact for us: the eigenvalues of a triangular matrix are simply the entries on the diagonal. Finding the corresponding eigenvectors still takes some work, but finding the eigenvalues is easy.
With that fact in the backs of our minds, let us proceed to the next example where we will come across some more interesting facts about eigenvalues and eigenvectors.
Example 91 Let $A=\left[\begin{array}{cc}-3 & 15 \\ 3 & 9\end{array}\right]$ and let $B=\left[\begin{array}{ccc}-7 & -2 & 10 \\ -3 & 2 & 3 \\ -6 & -2 & 9\end{array}\right]$ (as used in Examples 86 and 88, respectively). Find the following:
1. eigenvalues and eigenvectors of $A$ and $B$
2. eigenvalues and eigenvectors of $A^{-1}$ and $B^{-1}$
3. eigenvalues and eigenvectors of $A^{T}$ and $B^{T}$
4. The trace of $A$ and $B$
5. The determinant of $A$ and $B$
Solution We'll answer each in turn.
1. We already know the answer to these for we did this work in previous examples. Therefore we just list the answers.
For $A$, we have eigenvalues $\lambda=-6$ and 12 , with eigenvectors
$$
\vec{x}=x_{2}\left[\begin{array}{c}
-5 \\
1
\end{array}\right] \text { and } x_{2}\left[\begin{array}{l}
1 \\
1
\end{array}\right] \text {, respectively. }
$$
For $B$, we have eigenvalues $\lambda=-1,2$, and 3 with eigenvectors
$$
\vec{x}=x_{3}\left[\begin{array}{l}
3 \\
1 \\
2
\end{array}\right], x_{3}\left[\begin{array}{l}
2 \\
1 \\
2
\end{array}\right] \text { and } x_{3}\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right] \text {, respectively. }
$$
2. We first compute the inverses of $A$ and $B$. They are:
$$
A^{-1}=\left[\begin{array}{cc}
-1 / 8 & 5 / 24 \\
1 / 24 & 1 / 24
\end{array}\right] \quad \text { and } \quad B^{-1}=\left[\begin{array}{ccc}
-4 & 1 / 3 & 13 / 3 \\
-3 / 2 & 1 / 2 & 3 / 2 \\
-3 & 1 / 3 & 10 / 3
\end{array}\right]
$$
Finding the eigenvalues and eigenvectors of these matrices is not terribly hard, but it is not "easy," either. Therefore, we omit showing the intermediate steps and go right to the conclusions.
For $A^{-1}$, we have eigenvalues $\lambda=-1 / 6$ and $1 / 12$, with eigenvectors
$$
\vec{x}=x_{2}\left[\begin{array}{c}
-5 \\
1
\end{array}\right] \text { and } x_{2}\left[\begin{array}{l}
1 \\
1
\end{array}\right] \text {, respectively. }
$$
For $B^{-1}$, we have eigenvalues $\lambda=-1,1 / 2$ and $1 / 3$ with eigenvectors
$$
\vec{x}=x_{3}\left[\begin{array}{l}
3 \\
1 \\
2
\end{array}\right], x_{3}\left[\begin{array}{l}
2 \\
1 \\
2
\end{array}\right] \text { and } x_{3}\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right] \text {, respectively. }
$$
3. Of course, computing the transpose of $A$ and $B$ is easy; computing their eigenvalues and eigenvectors takes more work. Again, we omit the intermediate steps.
For $A^{T}$, we have eigenvalues $\lambda=-6$ and 12 with eigenvectors
$$
\vec{x}=x_{2}\left[\begin{array}{c}
-1 \\
1
\end{array}\right] \text { and } x_{2}\left[\begin{array}{l}
5 \\
1
\end{array}\right] \text {, respectively. }
$$
For $B^{T}$, we have eigenvalues $\lambda=-1,2$ and 3 with eigenvectors
$$
\vec{x}=x_{3}\left[\begin{array}{c}
-1 \\
0 \\
1
\end{array}\right], x_{3}\left[\begin{array}{c}
-1 \\
1 \\
1
\end{array}\right] \text { and } x_{3}\left[\begin{array}{c}
0 \\
-2 \\
1
\end{array}\right] \text {, respectively. }
$$
4. The trace of $A$ is 6 ; the trace of $B$ is 4 .
5. The determinant of $A$ is -72 ; the determinant of $B$ is -6 .
Now that we have completed the "grunt work," let's analyze the results of the previous example. We are looking for any patterns or relationships that we can find.
The eigenvalues and eigenvectors of $A$ and $A^{-1}$.
In our example, we found that the eigenvalues of $A$ are -6 and 12 ; the eigenvalues of $A^{-1}$ are $-1 / 6$ and $1 / 12$. Also, the eigenvalues of $B$ are $-1,2$ and 3 , whereas the eigenvalues of $B^{-1}$ are $-1,1 / 2$ and $1 / 3$. There is an obvious relationship here; it seems that if $\lambda$ is an eigenvalue of $A$, then $1 / \lambda$ will be an eigenvalue of $A^{-1}$. We can also note that the corresponding eigenvectors matched, too.
Why is this the case? Consider an invertible matrix $A$ with eigenvalue $\lambda$ and eigenvector $\vec{x}$. Then, by definition, we know that $A \vec{x}=\lambda \vec{x}$. Now multiply both sides by $A^{-1}$ :
$$
\begin{aligned}
A \vec{x} & =\lambda \vec{x} \\
A^{-1} A \vec{x} & =A^{-1} \lambda \vec{x} \\
\vec{x} & =\lambda A^{-1} \vec{x} \\
\frac{1}{\lambda} \vec{x} & =A^{-1} \vec{x}
\end{aligned}
$$
We have just shown that $A^{-1} \vec{x}=1 / \lambda \vec{x}$; this, by definition, shows that $\vec{x}$ is an eigenvector of $A^{-1}$ with eigenvalue $1 / \lambda$. This explains the result we saw above.
## The eigenvalues and eigenvectors of $A$ and $A^{T}$.
Our example showed that $A$ and $A^{T}$ had the same eigenvalues but different (but somehow similar) eigenvectors; it also showed that $B$ and $B^{T}$ had the same eigenvalues but unrelated eigenvectors. Why is this?
We can answer the eigenvalue question relatively easily; it follows from the properties of the determinant and the transpose. Recall the following two facts:
1. $(A+B)^{T}=A^{T}+B^{T}$ (Theorem 11) and
2. $\operatorname{det}(A)=\operatorname{det}\left(A^{T}\right)$ (Theorem 16).
We find the eigenvalues of a matrix by computing the characteristic polynomial; that is, we find $\operatorname{det}(A-\lambda I)$. What is the characteristic polynomial of $A^{T}$ ? Consider:
$$
\begin{aligned}
\operatorname{det}\left(A^{T}-\lambda I\right) & =\operatorname{det}\left(A^{T}-\lambda I^{T}\right) & & \text { since } I=I^{T} \\
& =\operatorname{det}\left((A-\lambda I)^{T}\right) & & \text { Theorem 11 } \\
& =\operatorname{det}(A-\lambda I) & & \text { Theorem 16 }
\end{aligned}
$$
So we see that the characteristic polynomial of $A^{T}$ is the same as that for $A$. Therefore they have the same eigenvalues.
What about their respective eigenvectors? Is there any relationship? The simple answer is "No."11
${ }^{11}$ We have defined an eigenvector to be a column vector. Some mathematicians prefer to use row vectors instead; in that case, the typical eigenvalue/eigenvector equation looks like $\vec{x} A=\lambda \vec{x}$. It turns out that doing things this way will give you the same eigenvalues as our method. What is more, take the transpose of the above equation: you get $(\vec{x} A)^{T}=(\lambda \vec{x})^{T}$ which is also $A^{T} \vec{x}^{T}=\lambda \vec{x}^{T}$. The transpose of a row vector is a column vector, so this equation is actually the kind we are used to, and we can say that $\vec{x}^{T}$ is an eigenvector of $A^{T}$.
In short, what we find is that the eigenvectors of $A^{T}$ are the "row" eigenvectors of $A$, and vice-versa.
## The eigenvalues and eigenvectors of $A$ and The Trace.
Note that the eigenvalues of $A$ are -6 and 12 , and the trace is 6 ; the eigenvalues of $B$ are $-1,2$ and 3 , and the trace of $B$ is 4 . Do we notice any relationship?
It seems that the sum of the eigenvalues is the trace! Why is this the case?
The answer to this is a bit out of the scope of this text; we can justify part of this fact, and another part we'll just state as being true without justification.
First, recall from Theorem 13 that $\operatorname{tr}(A B)=\operatorname{tr}(B A)$. Secondly, we state without justification that given a square matrix $A$, we can find a square matrix $P$ such that $P^{-1} A P$ is an upper triangular matrix with the eigenvalues of $A$ on the diagonal. ${ }^{12} \operatorname{Thus} \operatorname{tr}\left(P^{-1} A P\right)$ is the sum of the eigenvalues; also, using our Theorem 13, we know that $\operatorname{tr}\left(P^{-1} A P\right)=$ $\operatorname{tr}\left(P^{-1} P A\right)=\operatorname{tr}(A)$. Thus the trace of $A$ is the sum of the eigenvalues.
The eigenvalues and eigenvectors of $A$ and The Determinant.
Again, the eigenvalues of $A$ are -6 and 12 , and the determinant of $A$ is -72 . The eigenvalues of $B$ are $-1,2$ and 3 ; the determinant of $B$ is -6 . It seems as though the product of the eigenvalues is the determinant.
This is indeed true; we defend this with our argument from above. We know that the determinant of a triangular matrix is the product of the diagonal elements. Therefore, given a matrix $A$, we can find $P$ such that $P^{-1} A P$ is upper triangular with the eigenvalues of $A$ on the diagonal. Thus $\operatorname{det}\left(P^{-1} A P\right)$ is the product of the eigenvalues. Using Theorem 16, we know that $\operatorname{det}\left(P^{-1} A P\right)=\operatorname{det}\left(P^{-1} P A\right)=\operatorname{det}(A)$. Thus the determinant of $A$ is the product of the eigenvalues.
We summarize the results of our example with the following theorem.
${ }^{12}$ Who in the world thinks up this stuff? It seems that the answer is Marie Ennemond Camille Jordan, who, despite having at least two girl names, was a guy. Theorem 19
Let $A$ be an $n \times n$ invertible matrix. The following are true:
1. If $A$ is triangular, then the diagonal elements of $A$ are the eigenvalues of $A$.
2. If $\lambda$ is an eigenvalue of $A$ with eigenvector $\vec{x}$, then $\frac{1}{\lambda}$ is an eigenvalue of $A^{-1}$ with eigenvector $\vec{x}$.
3. If $\lambda$ is an eigenvalue of $A$ then $\lambda$ is an eigenvalue of $A^{T}$.
4. The sum of the eigenvalues of $A$ is equal to $\operatorname{tr}(A)$, the trace of $A$.
5. The product of the eigenvalues of $A$ is the equal to $\operatorname{det}(A)$, the determinant of $A$.
There is one more concept concerning eigenvalues and eigenvectors that we will explore. We do so in the context of an example.
Example $92 \quad$ Find the eigenvalues and eigenvectors of the matrix $A=\left[\begin{array}{ll}1 & 2 \\ 1 & 2\end{array}\right]$.
Solution To find the eigenvalues, we compute $\operatorname{det}(A-\lambda I)$ :
$$
\begin{aligned}
\operatorname{det}(A-\lambda I) & =\left|\begin{array}{cc}
1-\lambda & 2 \\
1 & 2-\lambda
\end{array}\right| \\
& =(1-\lambda)(2-\lambda)-2 \\
& =\lambda^{2}-3 \lambda \\
& =\lambda(\lambda-3)
\end{aligned}
$$
Our eigenvalues are therefore $\lambda=0,3$.
For $\lambda=0$, we find the eigenvectors:
$$
\left[\begin{array}{lll}
1 & 2 & 0 \\
1 & 2 & 0
\end{array}\right] \quad \overrightarrow{\operatorname{rref}}\left[\begin{array}{lll}
1 & 2 & 0 \\
0 & 0 & 0
\end{array}\right]
$$
This shows that $x_{1}=-2 x_{2}$, and so our eigenvectors $\vec{x}$ are
$$
\vec{x}=x_{2}\left[\begin{array}{c}
-2 \\
1
\end{array}\right]
$$
For $\lambda=3$, we find the eigenvectors:
$$
\left[\begin{array}{ccc}
-2 & 2 & 0 \\
1 & -1 & 0
\end{array}\right] \underset{\operatorname{rref}}{\longrightarrow}\left[\begin{array}{ccc}
1 & -1 & 0 \\
0 & 0 & 0
\end{array}\right]
$$
This shows that $x_{1}=x_{2}$, and so our eigenvectors $\vec{x}$ are
$$
\vec{x}=x_{2}\left[\begin{array}{l}
1 \\
1
\end{array}\right] \text {. }
$$
One interesting thing about the above example is that we see that 0 is an eigenvalue of $A$; we have not officially encountered this before. Does this mean anything significant? ${ }^{13}$
Think about what an eigenvalue of 0 means: there exists an nonzero vector $\vec{x}$ where $A \vec{x}=0 \vec{x}=\overrightarrow{0}$. That is, we have a nontrivial solution to $A \vec{x}=\overrightarrow{0}$. We know this only happens when $A$ is not invertible.
So if $A$ is invertible, there is no nontrivial solution to $A \vec{x}=\overrightarrow{0}$, and hence 0 is not an eigenvalue of $A$. If $A$ is not invertible, then there is a nontrivial solution to $A \vec{x}=\overrightarrow{0}$, and hence 0 is an eigenvalue of $A$. This leads us to our final addition to the Invertible Matrix Theorem.
## Theorem 20
## Invertible Matrix Theorem
Let $A$ be an $n \times n$ matrix. The following statements are equivalent.
(a) $A$ is invertible.
(h) $A$ does not have an eigenvalue of 0 .
This section is about the properties of eigenvalues and eigenvectors. Of course, we have not investigated all of the numerous properties of eigenvalues and eigenvectors; we have just surveyed some of the most common (and most important) concepts. Here are four quick examples of the many things that still exist to be explored.
First, recall the matrix
$$
A=\left[\begin{array}{ll}
1 & 4 \\
2 & 3
\end{array}\right]
$$
that we used in Example 84. It's characteristic polynomial is $p(\lambda)=\lambda^{2}-4 \lambda-5$. Compute $p(A)$; that is, compute $A^{2}-4 A-5 I$. You should get something "interesting," and you should wonder "does this always work?"14
${ }^{13}$ Since 0 is a "special" number, we might think so - afterall, we found that having a determinant of 0 is important. Then again, a matrix with a trace of 0 isn't all that important. (Well, as far as we have seen; it actually is). So, having an eigenvalue of 0 may or may not be significant, but we would be doing well if we recognized the possibility of significance and decided to investigate further.
${ }^{14}$ Yes. Second, in all of our examples, we have considered matrices where eigenvalues "appeared only once." Since we know that the eigenvalues of a triangular matrix appear on the diagonal, we know that the eigenvalues of
$$
A=\left[\begin{array}{ll}
1 & 1 \\
0 & 1
\end{array}\right]
$$
are " 1 and $1 ;$ " that is, the eigenvalue $\lambda=1$ appears twice. What does that mean when we consider the eigenvectors of $\lambda=1$ ? Compare the result of this to the matrix
$$
A=\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]
$$
which also has the eigenvalue $\lambda=1$ appearing twice. ${ }^{15}$
Third, consider the matrix
$$
A=\left[\begin{array}{cc}
0 & -1 \\
1 & 0
\end{array}\right]
$$
What are the eigenvalues ${ }^{16}$ We quickly compute the characteristic polynomial to be $p(\lambda)=\lambda^{2}+1$. Therefore the eigenvalues are $\pm \sqrt{-1}= \pm i$. What does this mean?
Finally, we have found the eigenvalues of matrices by finding the roots of the characteristic polynomial. We have limited our examples to quadratic and cubic polynomials; one would expect for larger sized matrices that a computer would be used to factor the characteristic polynomials. However, in general, this is not how the eigenvalues are found. Factoring high order polynomials is too unreliable, even with a computer - round off errors can cause unpredictable results. Also, to even compute the characteristic polynomial, one needs to compute the determinant, which is also expensive (as discussed in the previous chapter).
So how are eigenvalues found? There are iterative processes that can progressively transform a matrix $A$ into another matrix that is almost an upper triangular matrix (the entries below the diagonal are almost zero) where the entries on the diagonal are the eigenvalues. The more iterations one performs, the better the approximation is.
These methods are so fast and reliable that some computer programs convert polynomial root finding problems into eigenvalue problems!
Most textbooks on Linear Algebra will provide direction on exploring the above topics and give further insight to what is going on. We have mentioned all the eigenvalue and eigenvector properties in this section for the same reasons we gave in the previous section. First, knowing these properties helps us solve numerous real world problems, and second, it is fascinating to see how rich and deep the theory of matrices is.
${ }^{15}$ To direct further study, it helps to know that mathematicians refer to this as the duplicity of an eigenvalue. In each of these two examples, $A$ has the eigenvalue $\lambda=1$ with duplicity of 2 .
${ }^{16} \mathrm{Be}$ careful; this matrix is not triangular.
## Exercises 4.2
In Exercises $1-6$, a matrix $A$ is given. For each,
(a) Find the eigenvalues of $A$, and for each eigenvalue, find an eigenvector.
(b) Do the same for $A^{T}$.
(c) Do the same for $A^{-1}$.
(d) Find $\operatorname{tr}(A)$.
(e) Find $\operatorname{det}(A)$.
Use Theorem 19 to verify your results.
1. $\left[\begin{array}{cc}0 & 4 \\ -1 & 5\end{array}\right]$ 2. $\left[\begin{array}{cc}-2 & -14 \\ -1 & 3\end{array}\right]$
2. $\left[\begin{array}{cc}5 & 30 \\ -1 & -6\end{array}\right]$
3. $\left[\begin{array}{ll}-4 & 72 \\ -1 & 13\end{array}\right]$
4. $\left[\begin{array}{ccc}5 & -9 & 0 \\ 1 & -5 & 0 \\ 2 & 4 & 3\end{array}\right]$
5. $\left[\begin{array}{ccc}0 & 25 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & -3\end{array}\right]$
## GRAPHICAL EXPLORATIONS OF VECTORS
We already looked at the basics of graphing vectors. In this chapter, we'll explore these ideas more fully. One often gains a better understanding of a concept by "seeing" it. For instance, one can study the function $f(x)=x^{2}$ and describe many properties of how the output relates to the input without producing a graph, but the graph can quickly bring meaning and insight to equations and formulae. Not only that, but the study of graphs of functions is in itself a wonderful mathematical world, worthy of exploration.
We've studied the graphing of vectors; in this chapter we'll take this a step further and study some fantastic graphical properties of vectors and matrix arithmetic. We mentioned earlier that these concepts form the basis of computer graphics; in this chapter, we'll see even better how that is true.
### Transformations of the Cartesian Plane
## AS YOU READ
1. To understand how the Cartesian plane is affected by multiplication by a matrix, it helps to study how what is affected?
2. Transforming the Cartesian plane through matrix multiplication transforms straight lines into what kind of lines?
3. $\mathrm{T} / \mathrm{F}$ : If one draws a picture of a sheep on the Cartesian plane, then transformed the plane using the matrix
$$
\left[\begin{array}{cc}
-1 & 0 \\
0 & 1
\end{array}\right]
$$
one could say that the sheep was "sheared." We studied in Section 2.3 how to visualize vectors and how certain matrix arithmetic operations can be graphically represented. We limited our visual understanding of matrix multiplication to graphing a vector, multiplying it by a matrix, then graphing the resulting vector. In this section we'll explore these multiplication ideas in greater depth. Instead of multiplying individual vectors by a matrix $A$, we'll study what happens when we multiply every vector in the Cartesian plans by $A .{ }^{1}$
Because of the Distributive Property as we saw demonstrated way back in Example 41 , we can say that the Cartesian plane will be transformed in a very nice, predictable way. Straight lines will be transformed into other straight lines (and they won't become curvy, or jagged, or broken). Curved lines will be transformed into other curved lines (perhaps the curve will become "straight," but it won't become jagged or broken).
One way of studying how the whole Cartesian plane is affected by multiplication by a matrix $A$ is to study how the unit square is affected. The unit square is the square with corners at the points $(0,0),(1,0),(1,1)$, and $(0,1)$. Each corner can be represented by the vector that points to it; multiply each of these vectors by $A$ and we can get an idea of how $A$ affects the whole Cartesian plane.
Let's try an example.
Example $93 \quad$ Plot the vectors of the unit square before and after they have been multiplied by $A$, where
$$
A=\left[\begin{array}{ll}
1 & 4 \\
2 & 3
\end{array}\right]
$$
Solution The four corners of the unit square can be represented by the vectors
$$
\left[\begin{array}{l}
0 \\
0
\end{array}\right], \quad\left[\begin{array}{l}
1 \\
0
\end{array}\right], \quad\left[\begin{array}{l}
1 \\
1
\end{array}\right], \quad\left[\begin{array}{l}
0 \\
1
\end{array}\right] .
$$
Multiplying each by $A$ gives the vectors
$$
\left[\begin{array}{l}
0 \\
0
\end{array}\right], \quad\left[\begin{array}{l}
1 \\
2
\end{array}\right], \quad\left[\begin{array}{l}
5 \\
5
\end{array}\right], \quad\left[\begin{array}{l}
4 \\
3
\end{array}\right],
$$
respectively.
(Hint: one way of using your calculator to do this for you quickly is to make a $2 \times 4$ matrix whose columns are each of these vectors. In this case, create a matrix
$$
B=\left[\begin{array}{llll}
0 & 1 & 1 & 0 \\
0 & 0 & 1 & 1
\end{array}\right]
$$
Then multiply $B$ by $A$ and read off the transformed vectors from the respective columns:
$$
A B=\left[\begin{array}{llll}
0 & 1 & 5 & 4 \\
0 & 2 & 5 & 3
\end{array}\right] .
$$
${ }^{1}$ No, we won't do them one by one. This saves time, especially if you do a similar procedure for multiple matrices $A$. Of course, we can save more time by skipping the first column; since it is the column of zeros, it will stay the column of zeros after multiplication by $A$.)
The unit square and its transformation are graphed in Figure 5.1, where the shaped vertices correspond to each other across the two graphs. Note how the square got turned into some sort of quadrilateral (it's actually a parallelogram). A really interesting thing is how the triangular and square vertices seem to have changed places - it is as though the square, in addition to being stretched out of shape, was flipped.
Figure 5.1: Transforming the unit square by matrix multiplication in Example 93.
Figure 5.2: Emphasizing straight lines going to straight lines in Example 93.
To stress how "straight lines get transformed to straight lines," consider Figure 5.2. Here, the unit square has some additional points drawn on it which correspond to the shaded dots on the transformed parallelogram. Note how relative distances are also preserved; the dot halfway between the black and square dots is transformed to a position along the line, halfway between the black and square dots.
Much more can be said about this example. Before we delve into this, though, let's try one more example. Example 94 $A$, where
Plot the transformed unit square after it has been transformed by
$$
A=\left[\begin{array}{cc}
0 & -1 \\
1 & 0
\end{array}\right]
$$
Solution We'll put the vectors that correspond to each corner in a matrix $B$ as before and then multiply it on the left by $A$. Doing so gives:
$$
\begin{aligned}
A B & =\left[\begin{array}{cc}
0 & -1 \\
1 & 0
\end{array}\right]\left[\begin{array}{cccc}
0 & 1 & 1 & 0 \\
0 & 0 & 1 & 1
\end{array}\right] \\
& =\left[\begin{array}{cccc}
0 & 0 & -1 & -1 \\
0 & 1 & 1 & 0
\end{array}\right]
\end{aligned}
$$
In Figure 5.3 the unit square is again drawn along with its transformation by $A$.
Figure 5.3: Transforming the unit square by matrix multiplication in Example 94.
Make note of how the square moved. It did not simply "slide" to the left; ${ }^{2}$ nor did it "flip" across the $y$ axis. Rather, it was rotated counterclockwise about the origin $90^{\circ}$. In a rotation, the shape of an object does not change; in our example, the square remained a square of the same size.
We have broached the topic of how the Cartesian plane can be transformed via multiplication by a $2 \times 2$ matrix $A$. We have seen two examples so far, and our intuition as to how the plane is changed has been informed only by seeing how the unit square changes. Let's explore this further by investigating two questions:
1. Suppose we want to transform the Cartesian plane in a known way (for instance, we may want to rotate the plane counterclockwise $180^{\circ}$ ). How do we find the matrix (if one even exists) which performs this transformation?
2. How does knowing how the unit square is transformed really help in understanding how the entire plane is transformed?
These questions are closely related, and as we answer one, we will help answer the other.
${ }^{2}$ mathematically, that is called a translation To get started with the first question, look back at Examples 93 and 94 and consider again how the unit square was transformed. In particular, is there any correlation between where the vertices ended up and the matrix $A$ ?
If you are just reading on, and haven't actually gone back and looked at the examples, go back now and try to make some sort of connection. Otherwise - you may have noted some of the following things:
1. The zero vector $(\vec{O}$, the "black" corner) never moved. That makes sense, though; $A \vec{O}=\overrightarrow{0}$.
2. The "square" corner, i.e., the corner corresponding to the vector $\left[\begin{array}{l}1 \\ 0\end{array}\right]$, is always transformed to the vector in the first column of $A$ !
3. Likewise, the "triangular" corner, i.e., the corner corresponding to the vector $\left[\begin{array}{l}0 \\ 1\end{array}\right]$, is always transformed to the vector in the second column of $A !^{3}$
4. The "white dot" corner is always transformed to the sum of the two column vectors of $A .{ }^{4}$
Let's now take the time to understand these four points. The first point should be clear; $\vec{O}$ will always be transformed to $\vec{O}$ via matrix multiplication. (Hence the hint in the middle of Example 93, where we are told that we can ignore entering in the column of zeros in the matrix $B$.)
We can understand the second and third points simultaneously. Let
$$
A=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right], \quad \overrightarrow{e_{1}}=\left[\begin{array}{l}
1 \\
0
\end{array}\right] \quad \text { and } \quad \overrightarrow{e_{2}}=\left[\begin{array}{l}
0 \\
1
\end{array}\right] .
$$
What are $\overrightarrow{A e_{1}}$ and $\overrightarrow{A e_{2}}$ ?
$$
\begin{aligned}
A \overrightarrow{e_{1}} & =\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]\left[\begin{array}{l}
1 \\
0
\end{array}\right] \\
& =\left[\begin{array}{l}
a \\
c
\end{array}\right] \\
A \overrightarrow{e_{2}} & =\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]\left[\begin{array}{l}
0 \\
1
\end{array}\right] \\
& =\left[\begin{array}{l}
b \\
d
\end{array}\right]
\end{aligned}
$$
${ }^{3}$ Although this is less of a surprise, given the result of the previous point.
${ }^{4}$ This observation is a bit more obscure than the first three. It follows from the fact that this corner of the unit square is the "sum" of the other two nonzero corners. So by mere mechanics of matrix multiplication, the square corner $\overrightarrow{e_{1}}$ is transformed to the first column of $A$, and the triangular corner $\overrightarrow{e_{2}}$ is transformed to the second column of $A$. A similar argument demonstrates why the white dot corner is transformed to the sum of the columns of $A .^{5}$
Revisit now the question "How do we find the matrix that performs a given transformation on the Cartesian plane?" The answer follows from what we just did. Think about the given transformation and how it would transform the corners of the unit square. Make the first column of $A$ the vector where $\overrightarrow{e_{1}}$ goes, and make the second column of $A$ the vector where $\overrightarrow{e_{2}}$ goes.
Let's practice this in the context of an example.
Example 95 Find the matrix $A$ that flips the Cartesian plane about the $x$ axis and then stretches the plane horizontally by a factor of two.
Solution We first consider $\overrightarrow{e_{1}}=\left[\begin{array}{l}1 \\ 0\end{array}\right]$. Where does this corner go to under the given transformation? Flipping the plane across the $x$ axis does not change $\overrightarrow{e_{1}}$ at all; stretching the plane sends $\overrightarrow{e_{1}}$ to $\left[\begin{array}{l}2 \\ 0\end{array}\right]$. Therefore, the first column of $A$ is $\left[\begin{array}{l}2 \\ 0\end{array}\right]$.
Now consider $\overrightarrow{e_{2}}=\left[\begin{array}{l}0 \\ 1\end{array}\right]$. Flipping the plane about the $x$ axis sends $\overrightarrow{e_{2}}$ to the vector $\left[\begin{array}{c}0 \\ -1\end{array}\right]$; subsequently stretching the plane horizontally does not affect this vector. Therefore the second column of $A$ is $\left[\begin{array}{c}0 \\ -1\end{array}\right]$.
Putting this together gives
$$
A=\left[\begin{array}{cc}
2 & 0 \\
0 & -1
\end{array}\right]
$$
To help visualize this, consider Figure 5.4 where a shape is transformed under this matrix. Notice how it is turned upside down and is stretched horizontally by a factor of two. (The gridlines are given as a visual aid.)
${ }^{5}$ Another way of looking at all of this is to consider what $A \cdot I$ is: of course, it is just $A$. What are the columns of $/$ ? Just $\overrightarrow{e_{1}}$ and $\overrightarrow{e_{2}}$.
Figure 5.4: Transforming the Cartesian plane in Example 95
A while ago we asked two questions. The first was "How do we find the matrix that performs a given transformation?" We have just answered that question (although we will do more to explore it in the future). The second question was "How does knowing how the unit square is transformed really help us understand how the entire plane is transformed?"
Consider Figure 5.5 where the unit square (with vertices marked with shapes as before) is shown transformed under an unknown matrix. How does this help us understand how the whole Cartesian plane is transformed? For instance, how can we use this picture to figure out how the point $(2,3)$ will be transformed?
Figure 5.5: The unit square under an unknown transformation.
There are two ways to consider the solution to this question. First, we know now how to compute the transformation matrix; the new position of $\overrightarrow{e_{1}}$ is the first column of $A$, and the new position of $\overrightarrow{e_{2}}$ is the second column of $A$. Therefore, by looking at the figure, we can deduce that
$$
A=\left[\begin{array}{cc}
1 & -1 \\
1 & 2
\end{array}\right]
$$
${ }^{6}$ At least, $A$ is close to that. The square corner could actually be at the point $(1.01, .99)$. To find where the point $(2,3)$ is sent, simply multiply
$$
\left[\begin{array}{cc}
1 & -1 \\
1 & 2
\end{array}\right]\left[\begin{array}{l}
2 \\
3
\end{array}\right]=\left[\begin{array}{c}
-1 \\
8
\end{array}\right] .
$$
There is another way of doing this which isn't as computational - it doesn't involve computing the transformation matrix. Consider the following equalities:
$$
\begin{aligned}
{\left[\begin{array}{l}
2 \\
3
\end{array}\right] } & =\left[\begin{array}{l}
2 \\
0
\end{array}\right]+\left[\begin{array}{l}
0 \\
3
\end{array}\right] \\
& =2\left[\begin{array}{l}
1 \\
0
\end{array}\right]+3\left[\begin{array}{l}
0 \\
1
\end{array}\right] \\
& =2 \overrightarrow{e_{1}}+3 \overrightarrow{e_{2}}
\end{aligned}
$$
This last equality states something that is somewhat obvious: to arrive at the vector $\left[\begin{array}{l}2 \\ 3\end{array}\right]$,
, one needs to go 2 units in the $\overrightarrow{e_{1}}$ direction and 3 units in the $\overrightarrow{e_{2}}$ direction. To find where the point $(2,3)$ is transformed, one needs to go 2 units in the new $\overrightarrow{e_{1}}$ direction and 3 units in the new $\overrightarrow{e_{2}}$ direction. This is demonstrated in Figure 5.6.
Figure 5.6: Finding the new location of the point $(2,3)$.
We are coming to grips with how matrix transformations work. We asked two basic questions: "How do we find the matrix for a given transformation?" and "How do we understand the transformation without the matrix?", and we've answered each accompanied by one example. Let's do another example that demonstrates both techniques at once.
Example $96 \quad$ First, find the matrix $A$ that transforms the Cartesian plane by stretching it vertically by a factor of 1.5 , then stretches it horizontally by a factor of 0.5 , then rotates it clockwise about the origin $90^{\circ}$. Secondly, using the new locations of $\overrightarrow{e_{1}}$ and $\overrightarrow{e_{2}}$, find the transformed location of the point $(-1,2)$. Solution To find $A$, first consider the new location of $\overrightarrow{e_{1}}$. Stretching the plane vertically does not affect $\overrightarrow{e_{1}}$; stretching the plane horizontally by a factor of 0.5 changes $\overrightarrow{e_{1}}$ to $\left[\begin{array}{c}1 / 2 \\ 0\end{array}\right]$, and then rotating it $90^{\circ}$ about the origin moves it to $\left[\begin{array}{c}0 \\ -1 / 2\end{array}\right]$. This is the first column of $A$.
Now consider the new location of $\overrightarrow{e_{2}}$. Stretching the plane vertically changes it to $\left[\begin{array}{c}0 \\ 3 / 2\end{array}\right]$; stretching horizontally does not affect it, and rotating $90^{\circ}$ moves it to $\left[\begin{array}{c}3 / 2 \\ 0\end{array}\right]$. This is then the second column of $A$. This gives
$$
A=\left[\begin{array}{cc}
0 & 3 / 2 \\
-1 / 2 & 0
\end{array}\right] .
$$
Where does the point $(-1,2)$ get sent to? The corresponding vector $\left[\begin{array}{c}-1 \\ 2\end{array}\right]$ is found by going -1 units in the $\overrightarrow{e_{1}}$ direction and 2 units in the $\overrightarrow{e_{2}}$ direction. Therefore, the transformation will send the vector to -1 units in the new $\overrightarrow{e_{1}}$ direction and 2 units in the new $\overrightarrow{e_{2}}$ direction. This is sketched in Figure 5.7, along with the transformed unit square. We can also check this multiplicatively:
$$
\left[\begin{array}{cc}
0 & 3 / 2 \\
-1 / 2 & 0
\end{array}\right]\left[\begin{array}{c}
-1 \\
2
\end{array}\right]=\left[\begin{array}{c}
3 \\
1 / 2
\end{array}\right]
$$
Figure 5.8 shows the effects of the transformation on another shape.
Figure 5.7: Understanding the transformation in Example 96.
Figure 5.8: Transforming the Cartesian plane in Example 96 Right now we are focusing on transforming the Cartesian plane - we are making 2D transformations. Knowing how to do this provides a foundation for transforming 3D space, ${ }^{7}$ which, among other things, is very important when producing 3D computer graphics. Basic shapes can be drawn and then rotated, stretched, and/or moved to other regions of space. This also allows for things like "moving the camera view."
What kinds of transformations are possible? We have already seen some of the things that are possible: rotations, stretches, and flips. We have also mentioned some things that are not possible. For instance, we stated that straight lines always get transformed to straight lines. Therefore, we cannot transform the unit square into a circle using a matrix.
Let's look at some common transformations of the Cartesian plane and the matrices that perform these operations. In the following figures, a transformation matrix will be given alongside a picture of the transformed unit square. (The original unit square is drawn lightly as well to serve as a reference.)
## D Matrix Transformations
Horizontal stretch by a factor of $k$.
$$
\left[\begin{array}{ll}
k & 0 \\
0 & 1
\end{array}\right]
$$
Vertical stretch by a factor of $k$.
$$
\left[\begin{array}{ll}
1 & 0 \\
0 & k
\end{array}\right]
$$
${ }^{7}$ Actually, it provides a foundation for doing it in 4D, 5D, . , 17D, etc. Those are just harder to visualize. 5.1 Transformations of the Cartesian Plane
Horizontal shear by a factor of $k$.
$$
\left[\begin{array}{ll}
1 & k \\
0 & 1
\end{array}\right]
$$
Vertical shear by a factor of $k$.
$$
\left[\begin{array}{ll}
1 & 0 \\
k & 1
\end{array}\right]
$$
Horizontal reflection across the $y$ axis.
$$
\left[\begin{array}{cc}
-1 & 0 \\
0 & 1
\end{array}\right]
$$
Vertical reflection across the $x$ axis.
$$
\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right]
$$
## Diagonal reflection
across the line $y=x$.
$$
\left[\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right]
$$
Rotation around the origin by an angle of $\theta$.
$$
\left[\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right]
$$
Projection onto the $x$ axis.
(Note how the square is "squashed" down onto the $x$-axis.)
$$
\left[\begin{array}{ll}
1 & 0 \\
0 & 0
\end{array}\right]
$$
Projection onto the $y$ axis.
(Note how the square is "squashed" over onto the $y$-axis.)
$$
\left[\begin{array}{ll}
0 & 0 \\
0 & 1
\end{array}\right]
$$
Now that we have seen a healthy list of transformations that we can perform on the Cartesian plane, let's practice a few more times creating the matrix that gives the desired transformation. In the following example, we develop our understanding one more critical step.
Example 97 Find the matrix $A$ that transforms the Cartesian plane by performing the following operations in order:
1. Vertical shear by a factor of 0.5
2. Counterclockwise rotation about the origin by an angle of $\theta=30^{\circ}$ 3. Horizontal stretch by a factor of 2
3. Diagonal reflection across the line $y=x$
Solution Wow! We already know how to do this - sort of. We know we can find the columns of $A$ by tracing where $\overrightarrow{e_{1}}$ and $\overrightarrow{e_{2}}$ end up, but this also seems difficult. There is so much that is going on. Fortunately, we can accomplish what we need without much difficulty by being systematic.
First, let's perform the vertical shear. The matrix that performs this is
$$
A_{1}=\left[\begin{array}{cc}
1 & 0 \\
0.5 & 1
\end{array}\right]
$$
After that, we want to rotate everything clockwise by $30^{\circ}$. To do this, we use
$$
A_{2}=\left[\begin{array}{cc}
\cos 30^{\circ} & -\sin 30^{\circ} \\
\sin 30^{\circ} & \cos 30^{\circ}
\end{array}\right]=\left[\begin{array}{cc}
\sqrt{3} / 2 & -1 / 2 \\
1 / 2 & \sqrt{3} / 2
\end{array}\right] .
$$
In order to do both of these operations, in order, we multiply $A_{2} A_{1} \cdot{ }^{8}$
To perform the final two operations, we note that
$$
A_{3}=\left[\begin{array}{ll}
2 & 0 \\
0 & 1
\end{array}\right] \quad \text { and } \quad A_{4}=\left[\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right]
$$
perform the horizontal stretch and diagonal reflection, respectively. Thus to perform all of the operations "at once," we need to multiply by
$$
\begin{aligned}
A & =A_{4} A_{3} A_{2} A_{1} \\
& =\left[\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right]\left[\begin{array}{ll}
2 & 0 \\
0 & 1
\end{array}\right]\left[\begin{array}{cc}
\sqrt{3} / 2 & -1 / 2 \\
1 / 2 & \sqrt{3} / 2
\end{array}\right]\left[\begin{array}{cc}
1 & 0 \\
0.5 & 1
\end{array}\right] \\
& =\left[\begin{array}{cc}
(\sqrt{3}+2) / 4 & \sqrt{3} / 2 \\
(2 \sqrt{3}-1) / 2 & -1
\end{array}\right] \\
& \approx\left[\begin{array}{cc}
0.933 & 0.866 \\
1.232 & -1
\end{array}\right]
\end{aligned}
$$
${ }^{8}$ The reader might ask, "Is it important to do multiply these in that order? Could we have multiplied $A_{1} A_{2}$ instead?" Our answer starts with "Is matrix multiplication commutative?" The answer to our question is "No," so the answers to the reader's questions are "Yes" and "No," respectively. Let's consider this closely. Suppose I want to know where a vector $\vec{x}$ ends up. We claim we can find the answer by multiplying $A \vec{x}$. Why does this work? Consider:
$$
\begin{aligned}
A \vec{x} & =A_{4} A_{3} A_{2} A_{1} \vec{x} \\
& =A_{4} A_{3} A_{2}\left(A_{1} \vec{x}\right) \\
& =A_{4} A_{3}\left(A_{2} \vec{x}_{1}\right) \\
& =A_{4}\left(A_{3} \vec{x}_{2}\right) \\
& =A_{4} \vec{x}_{3} \\
& =\vec{x}_{4}
\end{aligned}
$$
(performs the vertical shear) (performs the rotation) (performs the horizontal stretch) (performs the diagonal reflection) (the result of transforming $\vec{x}$ )
Most readers are not able to visualize exactly what the given list of operations does to the Cartesian plane. In Figure 5.9 we sketch the transformed unit square; in Figure 5.10 we sketch a shape and its transformation.
Figure 5.9: The transformed unit square in Example 97.
Figure 5.10: A transformed shape in Example 97.
Once we know what matrices perform the basic transformations, ${ }^{9}$ performing complex transformations on the Cartesian plane really isn't that . . complex. It boils down to multiplying by a series of matrices.
We've shown many examples of transformations that we can do, and we've mentioned just a few that we can't - for instance, we can't turn a square into a circle. Why not? Why is it that straight lines get sent to straight lines? We spent a lot of time within this text looking at invertible matrices; what connections, if any, ${ }^{10}$ are there between invertible matrices and their transformations on the Cartesian plane?
All these questions require us to think like mathematicians - we are being asked to study the properties of an object we just learned about and their connections to things we've already learned. We'll do all this (and more!) in the following section.
## Exercises 5.1
In Exercises 1-4, a sketch of transformed unit square is given. Find the matrix $A$ that performs this transformation.
In Exercises 5-10, a list of transformations is given. Find the matrix $A$ that performs those transformations, in order, on the Cartesian plane.
5. (a) vertical shear by a factor of 2
(b) horizontal shear by a factor of 2
6. (a) horizontal shear by a factor of 2
(b) vertical shear by a factor of 2
7. (a) horizontal stretch by a factor of 3
(b) reflection across the line $y=x$
8. (a) counterclockwise rotation by an angle of $45^{\circ}$
(b) vertical stretch by a factor of $1 / 2$
9. (a) clockwise rotation by an angle of $90^{\circ}$
(b) horizontal reflection across the $y$ axis
(c) vertical shear by a factor of 1
10. (a) vertical reflection across the $x$ axis
(b) horizontal reflection across the $y$ axis
(c) diagonal reflection across the line $y=x$
In Exercises 11 - 14, two sets of transformations are given. Sketch the transformed unit square under each set of transformations. Are the transformations the same? Explain why/why not.
${ }^{10}$ By now, the reader should expect connections to exist. 11. (a) a horizontal reflection across the $y$ axis, followed by a vertical reflection across the $x$ axis, compared to
(b) a counterclockise rotation of $180^{\circ}$
12. (a) a horizontal stretch by a factor of 2 followed by a reflection across the line $y=x$, compared to
(b) a vertical stretch by a factor of 2 13. (a) a horizontal stretch by a factor of $1 / 2$ followed by a vertical stretch by a factor of 3 , compared to
(b) the same operations but in opposite order
14. (a) a reflection across the line $y=$ $x$ followed by a reflection across the $x$ axis, compared to
(b) a reflection across the the $y$ axis, followed by a reflection across the line $y=x$.
### Properties of Linear Transformations
## AS YOU READ
1. T/F: Translating the Cartesian plane 2 units up is a linear transformation.
2. $\mathrm{T} / \mathrm{F}$ : If $T$ is a linear transformation, then $T(\overrightarrow{0})=\overrightarrow{0}$.
In the previous section we discussed standard transformations of the Cartesian plane - rotations, reflections, etc. As a motivational example for this section's study, let's consider another transformation - let's find the matrix that moves the unit square one unit to the right (see Figure 5.11). This is called a translation.
Figure 5.11: Translating the unit square one unit to the right.
Our work from the previous section allows us to find the matrix quickly. By looking at the picture, it is easy to see that $\overrightarrow{e_{1}}$ is moved to $\left[\begin{array}{l}2 \\ 0\end{array}\right]$ and $\overrightarrow{e_{2}}$ is moved to $\left[\begin{array}{l}1 \\ 1\end{array}\right]$. Therefore, the transformation matrix should be
$$
A=\left[\begin{array}{ll}
2 & 1 \\
0 & 1
\end{array}\right] .
$$
However, look at Figure 5.12 where the unit square is drawn after being transformed by $A$. It is clear that we did not get the desired result; the unit square was not translated, but rather stretched/sheared in some way.
Figure 5.12: Actual transformation of the unit square by matrix $A$.
What did we do wrong? We will answer this question, but first we need to develop a few thoughts and vocabulary terms.
We've been using the term "transformation" to describe how we've changed vectors. In fact, "transformation" is synonymous to "function." We are used to functions like $f(x)=x^{2}$, where the input is a number and the output is another number. In the previous section, we learned about transformations (functions) where the input was a vector and the output was another vector. If $A$ is a "transformation matrix," then we could create a function of the form $T(\vec{x})=A \vec{x}$. That is, a vector $\vec{x}$ is the input, and the output is $\vec{x}$ multiplied by $A .{ }^{11}$
When we defined $f(x)=x^{2}$ above, we let the reader assume that the input was indeed a number. If we wanted to be complete, we should have stated
$$
f: \mathbb{R} \rightarrow \mathbb{R} \quad \text { where } \quad f(x)=x^{2} .
$$
The first part of that line told us that the input was a real number (that was the first $\mathbb{R}$ ) and the output was also a real number (the second $\mathbb{R}$ ).
To define a transformation where a $2 \mathrm{D}$ vector is transformed into another $2 \mathrm{D}$ vector via multiplication by a $2 \times 2$ matrix $A$, we should write
$$
T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} \quad \text { where } \quad T(\vec{x})=A \vec{x} .
$$
Here, the first $\mathbb{R}^{2}$ means that we are using $2 \mathrm{D}$ vectors as our input, and the second $\mathbb{R}^{2}$ means that a $2 \mathrm{D}$ vector is the output.
Consider a quick example:
$$
T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3} \quad \text { where } \quad T\left(\left[\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right]\right)=\left[\begin{array}{c}
x_{1}^{2} \\
2 x_{1} \\
x_{1} x_{2}
\end{array}\right] .
$$
Notice that this takes 2D vectors as input and returns 3D vectors as output. For instance,
$$
T\left(\left[\begin{array}{c}
3 \\
-2
\end{array}\right]\right)=\left[\begin{array}{c}
9 \\
6 \\
-6
\end{array}\right]
$$
We now define a special type of transformation (function).
${ }^{11}$ We used $T$ instead of $f$ to define this function to help differentiate it from "regular" functions. "Normally" functions are defined using lower case letters when the input is a number; when the input is a vector, we use upper case letters. Definition 29
A transformation $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is a linear transformation if it satisfies the following two properties:
1. $T(\vec{x}+\vec{y})=T(\vec{x})+T(\vec{y})$ for all vectors $\vec{x}$ and $\vec{y}$, and
2. $T(k \vec{x})=k T(\vec{x})$ for all vectors $\vec{x}$ and all scalars $k$.
If $T$ is a linear transformation, it is often said that " $T$ is linear."
Let's learn about this definition through some examples.
Example $98 \quad$ Determine whether or not the transformation $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ is a linear transformation, where
$$
T\left(\left[\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right]\right)=\left[\begin{array}{c}
x_{1}^{2} \\
2 x_{1} \\
x_{1} x_{2}
\end{array}\right]
$$
SOLUTION We'll arbitrarily pick two vectors $\vec{x}$ and $\vec{y}$ :
$$
\vec{x}=\left[\begin{array}{c}
3 \\
-2
\end{array}\right] \quad \text { and } \quad \vec{y}=\left[\begin{array}{l}
1 \\
5
\end{array}\right]
$$
Let's check to see if $T$ is linear by using the definition.
1. Is $T(\vec{x}+\vec{y})=T(\vec{x})+T(\vec{y})$ ? First, compute $\vec{x}+\vec{y}$ :
$$
\vec{x}+\vec{y}=\left[\begin{array}{c}
3 \\
-2
\end{array}\right]+\left[\begin{array}{l}
1 \\
5
\end{array}\right]=\left[\begin{array}{l}
4 \\
3
\end{array}\right] .
$$
Now compute $T(\vec{x}), T(\vec{y})$, and $T(\vec{x}+\vec{y})$ :
$$
\begin{aligned}
T(\vec{x}) & =T\left(\left[\begin{array}{c}
3 \\
-2
\end{array}\right]\right) & T(\vec{y})=T\left(\left[\begin{array}{l}
1 \\
5
\end{array}\right]\right) & T(\vec{x}+\vec{y})=T\left(\left[\begin{array}{l}
4 \\
3
\end{array}\right]\right) \\
& =\left[\begin{array}{c}
9 \\
6 \\
-6
\end{array}\right] & =\left[\begin{array}{l}
1 \\
2 \\
5
\end{array}\right] & =\left[\begin{array}{c}
16 \\
8 \\
12
\end{array}\right]
\end{aligned}
$$
Is $T(\vec{x}+\vec{y})=T(\vec{x})+T(\vec{y})$ ?
$$
\left[\begin{array}{c}
9 \\
6 \\
-6
\end{array}\right]+\left[\begin{array}{l}
1 \\
2 \\
5
\end{array}\right] \not\left[\begin{array}{c}
16 \\
8 \\
12
\end{array}\right] .
$$
Therefore, $T$ is not a linear transformation. So we have an example of something that doesn't work. Let's try an example where things do work. ${ }^{12}$
Example $99 \quad$ Determine whether or not the transformation $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ is a linear transformation, where $T(\vec{x})=A \vec{x}$ and
$$
A=\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]
$$
Solution Let's start by again considering arbitrary $\vec{x}$ and $\vec{y}$. Let's choose the same $\vec{x}$ and $\vec{y}$ from Example 98.
$$
\vec{x}=\left[\begin{array}{c}
3 \\
-2
\end{array}\right] \quad \text { and } \quad \vec{y}=\left[\begin{array}{l}
1 \\
5
\end{array}\right]
$$
If the lineararity properties hold for these vectors, then maybe it is actually linear (and we'll do more work).
1. Is $T(\vec{x}+\vec{y})=T(\vec{x})+T(\vec{y})$ ? Recall:
$$
\vec{x}+\vec{y}=\left[\begin{array}{l}
4 \\
3
\end{array}\right] .
$$
Now compute $T(\vec{x}), T(\vec{y})$, and $T(\vec{x})+T(\vec{y})$ :
$$
\begin{aligned}
& T(\vec{x})=T\left(\left[\begin{array}{c}
3 \\
-2
\end{array}\right]\right) \\
& T(\vec{y})=T\left(\left[\begin{array}{l}
1 \\
5
\end{array}\right]\right) \\
& T(\vec{x}+\vec{y})=T\left(\left[\begin{array}{l}
4 \\
3
\end{array}\right]\right) \\
& =\left[\begin{array}{c}
-1 \\
1
\end{array}\right] \\
& =\left[\begin{array}{l}
11 \\
23
\end{array}\right] \\
& =\left[\begin{array}{l}
10 \\
24
\end{array}\right]
\end{aligned}
$$
Is $T(\vec{x}+\vec{y})=T(\vec{x})+T(\vec{y})$ ?
$$
\left[\begin{array}{c}
-1 \\
1
\end{array}\right]+\left[\begin{array}{l}
11 \\
23
\end{array}\right] \stackrel{!}{=}\left[\begin{array}{l}
10 \\
24
\end{array}\right] .
$$
So far, so good: $T(\vec{x}+\vec{y})$ is equal to $T(\vec{x})+T(\vec{y})$.
${ }^{12}$ Recall a principle of logic: to show that something doesn't work, we just need to show one case where it fails, which we did in Example 98. To show that something always works, we need to show it works for all cases - simply showing it works for a few cases isn't enough. However, doing so can be helpful in understanding the situation better. 2. Is $T(k \vec{x})=k T(\vec{x})$ ? Let's arbitrarily pick $k=7$, and use $\vec{x}$ as before.
$$
\begin{aligned}
T(7 \vec{x}) & =T\left(\left[\begin{array}{c}
21 \\
-14
\end{array}\right]\right) \\
& =\left[\begin{array}{c}
-7 \\
7
\end{array}\right] \\
& =7\left[\begin{array}{c}
-1 \\
1
\end{array}\right] \\
& =7 \cdot T(\vec{x}) \quad !
\end{aligned}
$$
So far it seems that $T$ is indeed linear, for it worked in one example with arbitrarily chosen vectors and scalar. Now we need to try to show it is always true.
Consider $T(\vec{x}+\vec{y})$. By the definition of $T$, we have
$$
T(\vec{x}+\vec{y})=A(\vec{x}+\vec{y})
$$
By Theorem 3, part 2 (on page 62) we state that the Distributive Property holds for matrix multiplication. ${ }^{13}$ So $A(\vec{x}+\vec{y})=A \vec{x}+A \vec{y}$. Recognize now that this last part is just $T(\vec{x})+T(\vec{y})$ ! We repeat the above steps, all together:
$$
\begin{aligned}
T(\vec{x}+\vec{y}) & =A(\vec{x}+\vec{y}) & & \text { (by the definition of } T \text { in this example) } \\
& =A \vec{x}+A \vec{y} & & \text { (by the Distributive Property) } \\
& =T(\vec{x})+T(\vec{y}) & & \text { (again, by the definition of } T \text { ) }
\end{aligned}
$$
Therefore, no matter what $\vec{x}$ and $\vec{y}$ are chosen, $T(\vec{x}+\vec{y})=T(\vec{x})+T(\vec{y})$. Thus the first part of the lineararity definition is satisfied.
The second part is satisfied in a similar fashion. Let $k$ be a scalar, and consider:
$$
\begin{aligned}
T(k \vec{x}) & =A(k \vec{x}) & & \text { (by the definition of } T \text { is this example) } \\
& =k A \vec{x} & & \text { (by Theorem 3 part 3) } \\
& =k T(\vec{x}) & & \text { (again, by the definition of } T \text { ) }
\end{aligned}
$$
Since $T$ satisfies both parts of the definition, we conclude that $T$ is a linear transformation.
We have seen two examples of transformations so far, one which was not linear and one that was. One might wonder "Why is linearity important?", which we'll address shortly.
First, consider how we proved the transformation in Example 99 was linear. We defined $T$ by matrix multiplication, that is, $T(\vec{x})=A \vec{x}$. We proved $T$ was linear using properties of matrix multiplication - we never considered the specific values of $A$ ! That is, we didn't just choose a good matrix for $T$; any matrix $A$ would have worked. This
${ }^{13}$ Recall that a vector is just a special type of matrix, so this theorem applies to matrix-vector multiplication as well. leads us to an important theorem. The first part we have essentially just proved; the second part we won't prove, although its truth is very powerful.
Theorem 21
Definition 30
## Matrices and Linear Transformations
1. Define $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ by $T(\vec{x})=A \vec{x}$, where $A$ is an $m \times n$ matrix. Then $T$ is a linear transformation.
2. Let $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be any linear transformation. Then there exists an unique $m \times n$ matrix $A$ such that $T(\vec{x})=$ $A \vec{x}$.
The second part of the theorem says that all linear transformations can be described using matrix multiplication. Given any linear transformation, there is a matrix that completely defines that transformation. This important matrix gets its own name.
## Standard Matrix of a Linear Transformation
Let $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be a linear transformation. By Theorem 21 , there is a matrix $A$ such that $T(\vec{x})=A \vec{x}$. This matrix $A$ is called the standard matrix of the linear transformation $T$, and is denoted $[T] .^{a}$
${ }^{a}$ The matrix-like brackets around $T$ suggest that the standard matrix $A$ is a matrix "with $T$ inside."
While exploring all of the ramifications of Theorem 21 is outside the scope of this text, let it suffice to say that since 1) linear transformations are very, very important in economics, science, engineering and mathematics, and 2) the theory of matrices is well developed and easy to implement by hand and on computers, then 3 ) it is great news that these two concepts go hand in hand.
We have already used the second part of this theorem in a small way. In the previous section we looked at transformations graphically and found the matrices that produced them. At the time, we didn't realize that these transformations were linear, but indeed they were.
This brings us back to the motivating example with which we started this section. We tried to find the matrix that translated the unit square one unit to the right. Our attempt failed, and we have yet to determine why. Given our link between matrices and linear transformations, the answer is likely "the translation transformation is not a linear transformation." While that is a true statement, it doesn't really explain things all that well. Is there some way we could have recognized that this transformation wasn't linear? ${ }^{14}$
Yes, there is. Consider the second part of the linear transformation definition. It states that $T(k \vec{x})=k T(\vec{x})$ for all scalars $k$. If we let $k=0$, we have $T(0 \vec{x})=0 \cdot T(\vec{x})$, or more simply, $T(\overrightarrow{0})=\overrightarrow{0}$. That is, if $T$ is to be a linear transformation, it must send the zero vector to the zero vector.
This is a quick way to see that the translation transformation fails to be linear. By shifting the unit square to the right one unit, the corner at the point $(0,0)$ was sent to the point $(1,0)$, i.e.,
$$
\text { the vector }\left[\begin{array}{l}
0 \\
0
\end{array}\right] \text { was sent to the vector }\left[\begin{array}{l}
1 \\
0
\end{array}\right] \text {. }
$$
This property relating to $\vec{O}$ is important, so we highlight it here.
Key Idea 15
Linear Transformations and $\overrightarrow{0}$
Let $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be a linear transformation. Then:
$$
T\left(\vec{O}_{n}\right)=\vec{O}_{m}
$$
That is, the zero vector in $\mathbb{R}^{n}$ gets sent to the zero vector in $\mathbb{R}^{m}$.
The interested reader may wish to read the footnote below. ${ }^{15}$
## The Standard Matrix of a Linear Transformation
It is often the case that while one can describe a linear transformation, one doesn't know what matrix performs that transformation (i.e., one doesn't know the standard matrix of that linear transformation). How do we systematically find it? We'll need a new definition.
Definition 31
## Standard Unit Vectors
In $\mathbb{R}^{n}$, the standard unit vectors $\overrightarrow{e_{i}}$ are the vectors with a 1 in the $i^{\text {th }}$ entry and 0 s everywhere else.
${ }^{14}$ That is, apart from applying the definition directly?
${ }^{15}$ The idea that linear transformations "send zero to zero" has an interesting relation to terminology. The reader is likely familiar with functions like $f(x)=2 x+3$ and would likely refer to this as a "linear function." However, $f(0) \neq 0$, so $f$ is not "linear" by our new definition of linear. We erroneously call $f$ "linear" since its graph produces a line, though we should be careful to instead state that "the graph of $f$ is a line." We've already seen these vectors in the previous section. In $\mathbb{R}^{2}$, we identified
$$
\overrightarrow{e_{1}}=\left[\begin{array}{l}
1 \\
0
\end{array}\right] \quad \text { and } \quad \overrightarrow{e_{2}}=\left[\begin{array}{l}
0 \\
1
\end{array}\right]
$$
$\ln \mathbb{R}^{4}$, there are 4 standard unit vectors:
$$
\overrightarrow{e_{1}}=\left[\begin{array}{l}
1 \\
0 \\
0 \\
0
\end{array}\right], \quad \overrightarrow{e_{2}}=\left[\begin{array}{l}
0 \\
1 \\
0 \\
0
\end{array}\right], \quad \overrightarrow{e_{3}}=\left[\begin{array}{l}
0 \\
0 \\
1 \\
0
\end{array}\right], \quad \text { and } \quad \overrightarrow{e_{4}}=\left[\begin{array}{l}
0 \\
0 \\
0 \\
1
\end{array}\right] .
$$
How do these vectors help us find the standard matrix of a linear transformation? Recall again our work in the previous section. There, we practiced looking at the transformed unit square and deducing the standard transformation matrix $A$. We did this by making the first column of $A$ the vector where $\overrightarrow{e_{1}}$ ended up and making the second column of $A$ the vector where $\overrightarrow{e_{2}}$ ended up. One could represent this with:
$$
A=\left[\begin{array}{ll}
T\left(\overrightarrow{e_{1}}\right) & T\left(\overrightarrow{e_{2}}\right)
\end{array}\right]=[T]
$$
That is, $T\left(\overrightarrow{e_{1}}\right)$ is the vector where $\overrightarrow{e_{1}}$ ends up, and $T\left(\overrightarrow{e_{2}}\right)$ is the vector where $\overrightarrow{e_{2}}$ ends up.
The same holds true in general. Given a linear transformation $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$, the standard matrix of $T$ is the matrix whose $i^{\text {th }}$ column is the vector where $\overrightarrow{e_{i}}$ ends up. While we won't prove this is true, it is, and it is very useful. Therefore we'll state it again as a theorem.
Theorem 22
The Standard Matrix of a Linear Transformation
Let $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be a linear transformation. Then $[T]$ is the $m \times n$ matrix:
$$
[T]=\left[\begin{array}{llll}
T\left(\overrightarrow{e_{1}}\right) & T\left(\overrightarrow{e_{2}}\right) & \cdots & T\left(\overrightarrow{e_{n}}\right)
\end{array}\right]
$$
Let's practice this theorem in an example.
Example 100 Define $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{4}$ to be the linear transformation where
$$
T\left(\left[\begin{array}{l}
x_{1} \\
x_{2} \\
x_{3}
\end{array}\right]\right)=\left[\begin{array}{c}
x_{1}+x_{2} \\
3 x_{1}-x_{3} \\
2 x_{2}+5 x_{3} \\
4 x_{1}+3 x_{2}+2 x_{3}
\end{array}\right]
$$
Find $[T]$. Solution $T$ takes vectors from $\mathbb{R}^{3}$ into $\mathbb{R}^{4}$, so $[T]$ is going to be a $4 \times 3$ matrix. Note that
$$
\overrightarrow{e_{1}}=\left[\begin{array}{l}
1 \\
0 \\
0
\end{array}\right], \quad \overrightarrow{e_{2}}=\left[\begin{array}{l}
0 \\
1 \\
0
\end{array}\right] \quad \text { and } \quad \overrightarrow{e_{3}}=\left[\begin{array}{l}
0 \\
0 \\
1
\end{array}\right]
$$
We find the columns of $[T]$ by finding where $\overrightarrow{e_{1}}, \overrightarrow{e_{2}}$ and $\overrightarrow{e_{3}}$ are sent, that is, we find $T\left(\overrightarrow{e_{1}}\right), T\left(\overrightarrow{e_{2}}\right)$ and $T\left(\overrightarrow{e_{3}}\right)$.
$$
\begin{aligned}
T\left(\overrightarrow{e_{1}}\right) & =T\left(\left[\begin{array}{l}
1 \\
0 \\
0
\end{array}\right]\right) & T\left(\overrightarrow{e_{2}}\right)=T\left(\left[\begin{array}{l}
0 \\
1 \\
0
\end{array}\right]\right) & T\left(\overrightarrow{e_{3}}\right)=T\left(\left[\begin{array}{l}
0 \\
0 \\
1
\end{array}\right]\right) \\
& =\left[\begin{array}{l}
1 \\
3 \\
0 \\
4
\end{array}\right] & =\left[\begin{array}{l}
1 \\
0 \\
2 \\
3
\end{array}\right] & =\left[\begin{array}{c}
0 \\
-1 \\
5 \\
2
\end{array}\right]
\end{aligned}
$$
Thus
$$
[T]=A=\left[\begin{array}{ccc}
1 & 1 & 0 \\
3 & 0 & -1 \\
0 & 2 & 5 \\
4 & 3 & 2
\end{array}\right]
$$
Let's check this. Consider the vector
$$
\vec{x}=\left[\begin{array}{l}
1 \\
2 \\
3
\end{array}\right] .
$$
Strictly from the original definition, we can compute that
$$
T(\vec{x})=T\left(\left[\begin{array}{l}
1 \\
2 \\
3
\end{array}\right]\right)=\left[\begin{array}{c}
1+2 \\
3-3 \\
4+15 \\
4+6+6
\end{array}\right]=\left[\begin{array}{c}
3 \\
0 \\
19 \\
16
\end{array}\right] .
$$
Now compute $T(\vec{x})$ by computing $[T] \vec{x}=A \vec{x}$.
$$
A \vec{x}=\left[\begin{array}{ccc}
1 & 1 & 0 \\
3 & 0 & -1 \\
0 & 2 & 5 \\
4 & 3 & 2
\end{array}\right]\left[\begin{array}{l}
1 \\
2 \\
3
\end{array}\right]=\left[\begin{array}{c}
3 \\
0 \\
19 \\
16
\end{array}\right]
$$
They match $!^{16}$
Let's do another example, one that is more application oriented.
${ }^{16}$ Of course they do. That was the whole point. Example 101 A baseball team manager has collected basic data concerning his hitters. He has the number of singles, doubles, triples, and home runs they have hit over the past year. For each player, he wants two more pieces of information: the total number of hits and the total number of bases.
Using the techniques developed in this section, devise a method for the manager to accomplish his goal.
Solution If the manager only wants to compute this for a few players, then he could do it by hand fairly easily. After all:
total $\#$ hits $=\#$ of singles $+\#$ of doubles $+\#$ of triples $+\#$ of home runs,
and
total $\#$ bases $=\#$ of singles $+2 \times \#$ of doubles $+3 \times \#$ of triples $+4 \times \#$ of home runs.
However, if he has a lot of players to do this for, he would likely want a way to automate the work. One way of approaching the problem starts with recognizing that he wants to input four numbers into a function (i.e., the number of singles, doubles, etc.) and he wants two numbers as output (i.e., number of hits and bases). Thus he wants a transformation $T: \mathbb{R}^{4} \rightarrow \mathbb{R}^{2}$ where each vector in $\mathbb{R}^{4}$ can be interpreted as
$$
\left[\begin{array}{c}
\text { \# of singles } \\
\text { \# of doubles } \\
\text { \# of triples } \\
\text { \# of home runs }
\end{array}\right]
$$
and each vector in $\mathbb{R}^{2}$ can be interpreted as
$$
\left[\begin{array}{c}
\# \text { of hits } \\
\# \text { of bases }
\end{array}\right]
$$
To find $[T]$, he computes $T\left(\overrightarrow{e_{1}}\right), T\left(\overrightarrow{e_{2}}\right), T\left(\overrightarrow{e_{3}}\right)$ and $T\left(\overrightarrow{e_{4}}\right)$.
$$
\begin{array}{rlr}
T\left(\overrightarrow{e_{1}}\right)=T\left(\left[\begin{array}{l}
1 \\
0 \\
0 \\
0
\end{array}\right]\right) & T\left(\overrightarrow{e_{2}}\right)=T\left(\left[\begin{array}{l}
0 \\
1 \\
0 \\
0
\end{array}\right]\right) \\
=\left[\begin{array}{l}
1 \\
1
\end{array}\right] & =\left[\begin{array}{l}
1 \\
2
\end{array}\right] \\
T\left(\overrightarrow{e_{3}}\right)=T\left(\left[\begin{array}{l}
0 \\
1 \\
0
\end{array}\right]\right) & T\left(\overrightarrow{e_{4}}\right)=T\left(\left[\begin{array}{l}
0 \\
0 \\
0 \\
1
\end{array}\right]\right) \\
=\left[\begin{array}{l}
1 \\
3
\end{array}\right] & =\left[\begin{array}{l}
1 \\
4
\end{array}\right]
\end{array}
$$
(What do these calculations mean? For example, finding $T\left(\overrightarrow{e_{3}}\right)=\left[\begin{array}{l}1 \\ 3\end{array}\right]$ means that one triple counts as 1 hit and 3 bases.)
Thus our transformation matrix $[T]$ is
$$
[T]=A=\left[\begin{array}{llll}
1 & 1 & 1 & 1 \\
1 & 2 & 3 & 4
\end{array}\right]
$$
As an example, consider a player who had 102 singles, 30 doubles, 8 triples and 14 home runs. By using $A$, we find that
$$
\left[\begin{array}{llll}
1 & 1 & 1 & 1 \\
1 & 2 & 3 & 4
\end{array}\right]\left[\begin{array}{c}
102 \\
30 \\
8 \\
14
\end{array}\right]=\left[\begin{array}{l}
154 \\
242
\end{array}\right]
$$
meaning the player had 154 hits and 242 total bases.
A question that we should ask concerning the previous example is "How do we know that the function the manager used was actually a linear transformation? After all, we were wrong before - the translation example at the beginning of this section had us fooled at first."
This is a good point; the answer is fairly easy. Recall from Example 98 the transformation
$$
T_{98}\left(\left[\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right]\right)=\left[\begin{array}{c}
x_{1}^{2} \\
2 x_{1} \\
x_{1} x_{2}
\end{array}\right]
$$
and from Example 100
$$
T_{100}\left(\left[\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3}
\end{array}\right]\right)=\left[\begin{array}{c}
x_{1}+x_{2} \\
3 x_{1}-x_{3} \\
2 x_{2}+5 x_{3} \\
4 x_{1}+3 x_{2}+2 x_{3}
\end{array}\right],
$$
where we use the subscripts for $T$ to remind us which example they came from.
We found that $T_{98}$ was not a linear transformation, but stated that $T_{100}$ was (although we didn't prove this). What made the difference?
Look at the entries of $T_{98}(\vec{x})$ and $T_{100}(\vec{x}) . T_{98}$ contains entries where a variable is squared and where 2 variables are multiplied together - these prevent $T_{98}$ from being linear. On the other hand, the entries of $T_{100}$ are all of the form $a_{1} x_{1}+\cdots+a_{n} x_{n}$; that is, they are just sums of the variables multiplied by coefficients. $T$ is linear if and only if the entries of $T(\vec{x})$ are of this form. (Hence linear transformations are related to linear equations, as defined in Section 1.1.) This idea is important.
Key Idea 16
mation as
Going back to our baseball example, the manager could have defined his transfor-
$T$ is linear if and only if each entry of $T(\vec{x})$ is of the form $a_{1} x_{1}+$ $a_{2} x_{2}+\cdots a_{n} x_{n}$.
## Conditions on Linear Transformations
Let $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be a transformation and consider the entries of
$$
T(\vec{x})=T\left(\left[\begin{array}{c}
x_{1} \\
x_{2} \\
\vdots \\
x_{n}
\end{array}\right]\right)
$$
$$
T\left(\left[\begin{array}{l}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4}
\end{array}\right]\right)=\left[\begin{array}{c}
x_{1}+x_{2}+x_{3}+x_{4} \\
x_{1}+2 x_{2}+3 x_{3}+4 x_{4}
\end{array}\right] .
$$
Since that fits the model shown in Key Idea 16, the transformation $T$ is indeed linear and hence we can find a matrix $[T]$ that represents it.
Let's practice this concept further in an example.
Example 102 Using Key Idea 16, determine whether or not each of the following transformations is linear.
$$
T_{1}\left(\left[\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right]\right)=\left[\begin{array}{c}
x_{1}+1 \\
x_{2}
\end{array}\right] \quad T_{2}\left(\left[\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right]\right)=\left[\begin{array}{c}
x_{1} / x_{2} \\
\sqrt{x_{2}}
\end{array}\right]
$$
$$
T_{3}\left(\left[\begin{array}{l}
x_{1} \\
x_{2}
\end{array}\right]\right)=\left[\begin{array}{c}
\sqrt{7} x_{1}-x_{2} \\
\pi x_{2}
\end{array}\right]
$$
Solution $\quad T_{1}$ is not linear! This may come as a surprise, but we are not allowed to add constants to the variables. By thinking about this, we can see that this transformation is trying to accomplish the translation that got us started in this section - it adds 1 to all the $x$ values and leaves the $y$ values alone, shifting everything to the right one unit. However, this is not linear; again, notice how $\vec{O}$ does not get mapped to $\overrightarrow{0}$.
$T_{2}$ is also not linear. We cannot divide variables, nor can we put variabless inside the square root function (among other other things; again, see Section 1.1). This means that the baseball manager would not be able to use matrices to compute a batting average, which is (number of hits)/(number of at bats).
$T_{3}$ is linear. Recall that $\sqrt{7}$ and $\pi$ are just numbers, just coefficients.
We've mentioned before that we can draw vectors other than $2 \mathrm{D}$ vectors, although the more dimensions one adds, the harder it gets to understand. In the next section we'll learn about graphing vectors in 3D - that is, how to draw on paper or a computer screen a $3 \mathrm{D}$ vector.
## Exercises 5.2
In Exercises 1-5, a transformation $T$ is given. Determine whether or not $T$ is linear; if not, state why not.
1. $T\left(\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]\right)=\left[\begin{array}{c}x_{1}+x_{2} \\ 3 x_{1}-x_{2}\end{array}\right]$
2. $T\left(\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]\right)=\left[\begin{array}{l}x_{1}+x_{2}^{2} \\ x_{1}-x_{2}\end{array}\right]$
3. $T\left(\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]\right)=\left[\begin{array}{l}x_{1}+1 \\ x_{2}+1\end{array}\right]$
4. $T\left(\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]\right)=\left[\begin{array}{l}1 \\ 1\end{array}\right]$
5. $T\left(\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]\right)=\left[\begin{array}{l}0 \\ 0\end{array}\right]$
In Exercises 6 - 11, a linear transformation $T$ is given. Find $[T]$.
6. $T\left(\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]\right)=\left[\begin{array}{l}x_{1}+x_{2} \\ x_{1}-x_{2}\end{array}\right]$ 7. $T\left(\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]\right)=\left[\begin{array}{c}x_{1}+2 x_{2} \\ 3 x_{1}-5 x_{2} \\ 2 x_{2}\end{array}\right]$
7. $T\left(\left[\begin{array}{c}x_{1} \\ x_{2} \\ x_{3}\end{array}\right]\right)=\left[\begin{array}{c}x_{1}+2 x_{2}-3 x_{3} \\ 0 \\ x_{1}+4 x_{3} \\ 5 x_{2}+x_{3}\end{array}\right]$
8. $T\left(\left[\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right]\right)=\left[\begin{array}{l}x_{1}+3 x_{3} \\ x_{1}-x_{3} \\ x_{1}+x_{3}\end{array}\right]$
9. $T\left(\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]\right)=\left[\begin{array}{l}0 \\ 0\end{array}\right]$
10. $T\left(\left[\begin{array}{l}x_{1} \\ x_{2} \\ x_{3} \\ x_{4}\end{array}\right]\right)=\left[x_{1}+2 x_{2}+3 x_{3}+4 x_{4}\right]$
### Visualizing Vectors: Vectors in Three Dimensions
## AS YOU READ
1. $\mathrm{T} / \mathrm{F}$ : The viewpoint of the reader makes a difference in how vectors in 3D look.
2. T/F: If two vectors are not near each other, then they will not appear to be near each other when graphed.
3. T/F: The parallelogram law only applies to adding vectors in 2D.
We ended the last section by stating we could extend the ideas of drawing $2 \mathrm{D}$ vectors to drawing $3 \mathrm{D}$ vectors. Once we understand how to properly draw these vectors, addition and subtraction is relatively easy. We'll also discuss how to find the length of a vector in 3D.
We start with the basics of drawing a vector in 3D. Instead of having just the traditional $x$ and $y$ axes, we now add a third axis, the $z$ axis. Without any additional vectors, a generic 3D coordinate system can be seen in Figure 5.13.
Figure 5.13: The 3D coordinate system
In 2D, the point $(2,1)$ refers to going 2 units in the $x$ direction followed by 1 unit in the $y$ direction. In $3 D$, each point is referenced by 3 coordinates. The point $(4,2,3)$ is found by going 4 units in the $x$ direction, 2 units in the $y$ direction, and 3 units in the $z$ direction.
How does one sketch a vector on this coordinate system? As one might expect, we can sketch the vector $\vec{v}=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]$ by drawing an arrow from the origin (the point $(0,0,0)$ ) to the point $(1,2,3) .{ }^{17}$ The only "tricky" part comes from the fact that we are trying to represent three dimensional space on a two dimensional sheet of paper. However,
${ }^{17}$ Of course, we don't have to start at the origin; all that really matters is that the tip of the arrow is 1 unit in the $x$ direction, 2 units in the $y$ direction, and 3 units in the $z$ direction from the origin of the arrow. it isn't really hard. We'll discover a good way of approaching this in the context of an example.
Example $103 \quad$ Sketch the following vectors with their origin at the origin.
$$
\vec{v}=\left[\begin{array}{l}
2 \\
1 \\
3
\end{array}\right] \quad \text { and } \quad \vec{u}=\left[\begin{array}{c}
1 \\
3 \\
-1
\end{array}\right]
$$
Solution We'll start with $\vec{v}$ first. Starting at the origin, move 2 units in the $x$ direction. This puts us at the point $(2,0,0)$ on the $x$ axis. Then, move 1 unit in the $y$ direction. (In our method of drawing, this means moving 1 unit directly to the right. Of course, we don't have a grid to follow, so we have to make a good approximation of this distance.) Finally, we move 3 units in the $z$ direction. (Again, in our drawing, this means going straight "up" 3 units, and we must use our best judgment in a sketch to measure this.)
This allows us to locate the point $(2,1,3)$; now we draw an arrow from the origin to this point. In Figure 5.14 we have all 4 stages of this sketch. The dashed lines show us moving down the $x$ axis in (a); in (b) we move over in the $y$ direction; in (c) we move up in the $z$ direction, and finally in (d) the arrow is drawn.
(a)
(c)
(b)
(d)
Figure 5.14: Stages of sketching the vector $\vec{v}$ for Example 103. Drawing the dashed lines help us find our way in our representation of three dimensional space. Without them, it is hard to see how far in each direction the vector is supposed to have gone.
To draw $\vec{u}$, we follow the same procedure we used to draw $\vec{v}$. We first locate the point $(1,3,-1)$, then draw the appropriate arrow. In Figure 5.15 we have $\vec{u}$ drawn along with $\vec{v}$. We have used different dashed and dotted lines for each vector to help distinguish them.
Notice that this time we had to go in the negative $z$ direction; this just means we moved down one unit instead of up a unit.
Figure 5.15: Vectors $\vec{v}$ and $\vec{u}$ in Example 103.
As in 2D, we don't usually draw the zero vector,
$$
\vec{O}=\left[\begin{array}{l}
0 \\
0 \\
0
\end{array}\right]
$$
It doesn't point anywhere. It is a conceptually important vector that does not have a terribly interesting visualization.
Our method of drawing 3D objects on a flat surface - a 2D surface - is pretty clever. It isn't perfect, though; visually, drawing vectors with negative components (especially negative $x$ coordinates) can look a bit odd. Also, two very different vectors can point to the same place. We'll highlight this with our next two examples.
Example $104 \quad$ Sketch the vector $\vec{v}=\left[\begin{array}{c}-3 \\ -1 \\ 2\end{array}\right]$.
Solution We use the same procedure we used in Example 103. Starting at the origin, we move in the negative $x$ direction 3 units, then 1 unit in the negative $y$ direction, and then finally up 2 units in the $z$ direction to find the point $(-3,-1,2)$. We follow by drawing an arrow. Our sketch is found in Figure 5.16; $\vec{v}$ is drawn in two coordinate systems, once with the helpful dashed lines, and once without. The second drawing makes it pretty clear that the dashed lines truly are helpful.
Figure 5.16: Vector $\vec{v}$ in Example 104.
Example 105 Draw the vectors $\vec{v}=\left[\begin{array}{l}2 \\ 4 \\ 2\end{array}\right]$ and $\vec{u}=\left[\begin{array}{c}-2 \\ 1 \\ -1\end{array}\right]$ on the same coordinate system.
Solution We follow the steps we've taken before to sketch these vectors, shown in Figure 5.17. The dashed lines are aides for $\vec{v}$ and the dotted lines are aids for $\vec{u}$. We again include the vectors without the dashed and dotted lines; but without these, it is very difficult to tell which vector is which!
Figure 5.17: Vectors $\vec{v}$ and $\vec{u}$ in Example 105.
Our three examples have demonstrated that we have a pretty clever, albeit imperfect, method for drawing 3D vectors. The vectors in Example 105 look similar because of our viewpoint. In Figure 5.18 (a), we have rotated the coordinate axes, giving the vectors a different appearance. (Vector $\vec{v}$ now looks like it lies on the $y$ axis.)
Another important factor in how things look is the scale we use for the $x, y$, and $z$ axes. In 2D, it is easy to make the scale uniform for both axes; in 3D, it can be a bit tricky to make the scale the same on the axes that are "slanted." Figure 5.18 (b) again shows the same 2 vectors found in Example 105, but this time the scale of the $x$ axis is a bit different. The end result is that again the vectors appear a bit different than they did before. These facts do not necessarily pose a big problem; we must merely be aware of these facts and not make judgments about 3D objects based on one 2D image. $^{18}$
(a)
(b)
Figure 5.18: Vectors $\vec{v}$ and $\vec{u}$ in Example 105 with a different viewpoint (a) and $x$ axis scale (b).
We now investigate properties of vector arithmetic: what happens (i.e., how do we draw) when we add $3 D$ vectors and multiply by a scalar? How do we compute the length of a $3 \mathrm{D}$ vector?
## Vector Addition and Subtraction
In 2D, we saw that we could add vectors together graphically using the Parallelogram Law. Does the same apply for adding vectors in 3D? We investigate in an example.
Example $106 \quad$ Let $\vec{v}=\left[\begin{array}{l}2 \\ 1 \\ 3\end{array}\right]$ and $\vec{u}=\left[\begin{array}{c}1 \\ 3 \\ -1\end{array}\right]$. Sketch $\vec{v}+\vec{u}$.
Solution We sketched each of these vectors previously in Example 103. We sketch them, along with $\vec{v}+\vec{u}=\left[\begin{array}{l}3 \\ 4 \\ 2\end{array}\right]$, in Figure 5.19 (a). (We use loosely dashed lines for $\vec{v}+\vec{u}$.)
${ }^{18}$ The human brain uses both eyes to convey 3D, or depth, information. With one eye closed (or missing), we can have a very hard time with "depth perception." Two objects that are far apart can seem very close together. A simple example of this problem is this: close one eye, and place your index finger about a foot above this text, directly above this WORD. See if you were correct by dropping your finger straight down. Did you actually hit the proper spot? Try it again with both eyes, and you should see a noticable difference in your accuracy.
Looking at 3D objects on paper is a bit like viewing the world with one eye closed. Does the Parallelogram Law still hold? In Figure 5.19 (b), we draw additional representations of $\vec{v}$ and $\vec{u}$ to form a parallelogram (without all the dotted lines), which seems to affirm the fact that the Parallelogram Law does indeed hold.
(b)
(b)
Figure 5.19: Vectors $\vec{v}, \vec{u}$, and $\vec{v}+\vec{u}$ Example 106.
We also learned that in 2D, we could subtract vectors by drawing a vector from the tip of one vector to the other. ${ }^{19}$ Does this also work in 3D? We'll investigate again with an example, using the familiar vectors $\vec{v}$ and $\vec{u}$ from before.
Example 107 Let $\vec{v}=\left[\begin{array}{l}2 \\ 1 \\ 3\end{array}\right]$ and $\vec{u}=\left[\begin{array}{c}1 \\ 3 \\ -1\end{array}\right]$. Sketch $\vec{v}-\vec{u}$.
Solution It is simple to compute that $\vec{v}-\vec{u}=\left[\begin{array}{c}1 \\ -2 \\ 4\end{array}\right]$. All three of these vectors are sketched in Figure 5.20 (a), where again $\vec{v}$ is guided by the dashed, $\vec{u}$ by the dotted, and $\vec{v}-\vec{u}$ by the loosely dashed lines.
Does the $2 \mathrm{D}$ subtraction rule still hold? That is, can we draw $\vec{v}-\vec{u}$ by drawing an arrow from the tip of $\vec{u}$ to the tip of $\vec{v}$ ? In Figure 5.20 (b), we translate the drawing of $\vec{v}-\vec{u}$ to the tip of $\vec{u}$, and sure enough, it looks like it works. (And in fact, it really does.)
${ }^{19}$ Recall that it is important which vector we used for the origin and which was used for the tip. 5.3 Visualizing Vectors: Vectors in Three Dimensions
(a)
(b)
Figure 5.20: Vectors $\vec{v}, \vec{u}$, and $\vec{v}-\vec{u}$ from Example 107.
The previous two examples highlight the fact that even in 3D, we can sketch vectors without explicitly knowing what they are. We practice this one more time in the following example.
Example $108 \quad$ Vectors $\vec{v}$ and $\vec{u}$ are drawn in Figure 5.21. Using this drawing, sketch the vectors $\vec{v}+\vec{u}$ and $\vec{v}-\vec{u}$.
Figure 5.21: Vectors $\vec{v}$ and $\vec{u}$ for Example 108.
Solution Using the Parallelogram Law, we draw $\vec{v}+\vec{u}$ by first drawing a gray version of $\vec{u}$ coming from the tip of $\vec{v} ; \vec{v}+\vec{u}$ is drawn dashed in Figure 5.22.
To draw $\vec{v}-\vec{u}$, we draw a dotted arrow from the tip of $\vec{u}$ to the tip of $\vec{v}$.
Figure 5.22: Vectors $\vec{v}, \vec{u}, \vec{v}+\vec{u}$ and $\vec{v}-\vec{u}$ for Example 108.
## Scalar Multiplication
Given a vector $\vec{v}$ in 3D, what does the vector $2 \vec{v}$ look like? How about $-\vec{v}$ ? After learning about vector addition and subtraction in $3 \mathrm{D}$, we are probably gaining confidence in working in 3D and are tempted to say that $2 \vec{v}$ is a vector twice as long as $\vec{v}$, pointing in the same direction, and $-\vec{v}$ is a vector of the same length as $\vec{v}$, pointing in the opposite direction. We would be right. We demonstrate this in the following example.
Example $109 \quad$ Sketch $\vec{v}, 2 \vec{v}$, and $-\vec{v}$, where
$$
\vec{v}=\left[\begin{array}{l}
1 \\
2 \\
3
\end{array}\right] .
$$
## SOLUTION
Figure 5.23: Sketching scalar multiples of $\vec{v}$ in Example 109. It is easy to compute
$$
2 \vec{v}=\left[\begin{array}{l}
2 \\
4 \\
6
\end{array}\right] \quad \text { and } \quad-\vec{v}=\left[\begin{array}{l}
-1 \\
-2 \\
-3
\end{array}\right] .
$$
These are drawn in Figure 5.23. This figure is, in many ways, a mess, with all the dashed and dotted lines. They are useful though. Use them to see how each vector was formed, and note that $2 \vec{v}$ at least looks twice as long as $\vec{v}$, and it looks like $-\vec{v}$ points in the opposite direction. ${ }^{20}$
## Vector Length
How do we measure the length of a vector in 3D? In 2D, we were able to answer this question by using the Pythagorean Theorem. Does the Pythagorean Theorem apply in 3D? In a sense, it does.
Consider the vector $\vec{v}=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]$, as drawn in Figure 5.24 (a), with guiding dashed lines. Now look at part (b) of the same figure. Note how two lengths of the dashed lines have now been drawn gray, and another dotted line has been added.
(a)
(b)
Figure 5.24: Computing the length of $\vec{v}$
These gray dashed and dotted lines form a right triangle with the dotted line forming the hypotenuse. We can find the length of the dotted line using the Pythagorean Theorem.
length of the dotted line $=\sqrt{\text { sum of the squares of the dashed line lengths }}$
That is, the length of the dotted line $=\sqrt{1^{2}+2^{2}}=\sqrt{5}$.
${ }^{20}$ Our previous work showed that looks can be deceiving, but it is indeed true in this case. Now consider this: the vector $\vec{v}$ is the hypotenuse of another right triangle: the one formed by the dotted line and the vertical dashed line. Again, we employ the Pythagorean Theorem to find its length.
Thus, the length of $\vec{v}$ is (recall, we denote the length of $\vec{v}$ with $\|\vec{v}\|$ ):
$$
\begin{aligned}
\|\vec{v}\| & =\sqrt{(\text { length of gray line })^{2}+(\text { length of black line })^{2}} \\
& =\sqrt{\sqrt{5}^{2}+3^{2}} \\
& =\sqrt{5+3^{2}}
\end{aligned}
$$
Let's stop for a moment and think: where did this 5 come from in the previous equation? It came from finding the length of the gray dashed line -it came from $1^{2}+2^{2}$. Let's substitute that into the previous equation:
$$
\begin{aligned}
\|\vec{v}\| & =\sqrt{5+3^{2}} \\
& =\sqrt{1^{2}+2^{2}+3^{2}} \\
& =\sqrt{14}
\end{aligned}
$$
The key comes from the middle equation: $\|\vec{v}\|=\sqrt{1^{2}+2^{2}+3^{2}}$. Do those numbers 1, 2, and 3 look familiar? They are the component values of $\vec{v} !$ This is very similar to the definition of the length of a $2 \mathrm{D}$ vector. After formally defining this, we'll practice with an example.
## Definition 32
## D Vector Length
Let
$$
\vec{v}=\left[\begin{array}{l}
x_{1} \\
x_{2} \\
x_{3}
\end{array}\right] .
$$
The length of $\vec{v}$, denoted $\|\vec{v}\|$, is
$$
\|\vec{v}\|=\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}
$$
Example $110 \quad$ Find the lengths of vectors $\vec{v}$ and $\vec{u}$, where
$$
\vec{v}=\left[\begin{array}{c}
2 \\
-3 \\
5
\end{array}\right] \quad \text { and } \quad \vec{u}=\left[\begin{array}{c}
-4 \\
7 \\
0
\end{array}\right]
$$
Solution We apply Definition 32 to each vector:
$$
\begin{aligned}
\|\vec{v}\| & =\sqrt{2^{2}+(-3)^{2}+5^{2}} \\
& =\sqrt{4+9+25} \\
& =\sqrt{38} \\
\|\vec{u}\| & =\sqrt{(-4)^{2}+7^{2}+0^{2}} \\
& =\sqrt{16+49} \\
& =\sqrt{65}
\end{aligned}
$$
Here we end our investigation into the world of graphing vectors. Extensions into graphing 4D vectors and beyond can be done, but they truly are confusing and not really done except for abstract purposes.
There are further things to explore, though. Just as in 2D, we can transform 3D space by matrix multiplication. Doing this properly - rotating, stretching, shearing, etc. - allows one to manipulate 3D space and create incredible computer graphics.
## Exercises 5.3
In Exercises $1-4$, vectors $\vec{x}$ and $\vec{y}$ are given. In Exercises $5-8$, vectors $\vec{x}$ and $\vec{y}$ are drawn. Sketch $\vec{x}, \vec{y}, \vec{x}+\vec{y}$, and $\vec{x}-\vec{y}$ on the same Carte- Sketch $2 \vec{x},-\vec{y}, \vec{x}+\vec{y}$, and $\vec{x}-\vec{y}$ on the same sian axes.
1. $\vec{x}=\left[\begin{array}{c}1 \\ -1 \\ 2\end{array}\right], \vec{y}=\left[\begin{array}{l}2 \\ 3 \\ 2\end{array}\right]$ Cartesian axes.
2. $\vec{x}=\left[\begin{array}{c}2 \\ 4 \\ -1\end{array}\right], \vec{y}=\left[\begin{array}{l}-1 \\ -3 \\ -1\end{array}\right]$
3. $\vec{x}=\left[\begin{array}{l}1 \\ 1 \\ 2\end{array}\right], \vec{y}=\left[\begin{array}{l}3 \\ 3 \\ 6\end{array}\right]$
4. $\vec{x}=\left[\begin{array}{l}0 \\ 1 \\ 1\end{array}\right], \vec{y}=\left[\begin{array}{c}0 \\ -1 \\ 1\end{array}\right]$
Chapter 5 Graphical Explorations of Vectors
6.
8.
In Exercises $9-12$, a vector $\vec{x}$ and a scalar $a$ are given. Using Definition 32, compute the lengths of $\vec{x}$ and $a \vec{x}$, then compare these lengths.
9. $\vec{x}=\left[\begin{array}{c}1 \\ -2 \\ 5\end{array}\right], a=2$
10. $\vec{x}=\left[\begin{array}{c}-3 \\ 4 \\ 3\end{array}\right], a=-1$
11. $\vec{x}=\left[\begin{array}{l}7 \\ 2 \\ 1\end{array}\right], a=5$
12. $\vec{x}=\left[\begin{array}{c}1 \\ 2 \\ -2\end{array}\right], a=3$
## Solutions to Selected Problems
## Chapter 1
Section 1.1
1. $\mathrm{y}$
2. $y$
3. $\mathrm{n}$
4. $\mathrm{y}$
5. $y$
6. $x=1, y=-2$
7. $x=-1, y=0, z=2$
8. 29 chickens and 33 pigs
Section 1.2
1. $\left[\begin{array}{cccc}3 & 4 & 5 & 7 \\ -1 & 1 & -3 & 1 \\ 2 & -2 & 3 & 5\end{array}\right]$
2. $\left[\begin{array}{ccccc}1 & 3 & -4 & 5 & 17 \\ -1 & 0 & 4 & 8 & 1 \\ 2 & 3 & 4 & 5 & 6\end{array}\right]$
3. $\begin{aligned} x_{1}+2 x_{2} & =3 \\ -x_{1}+3 x_{2} & =9\end{aligned}$
4. $\begin{aligned} x_{1}+x_{2}-x_{3}-x_{4} & =2 \\ 2 x_{1}+x_{2}+3 x_{3}+5 x_{4} & =7\end{aligned}$
5. $\begin{aligned} x_{1}+x_{3}+7 x_{5} & =2 \\ x_{2}+3 x_{3}+2 x_{4} & =5\end{aligned}$
6. $\left[\begin{array}{ccc}2 & -1 & 7 \\ 5 & 0 & 3 \\ 0 & 4 & -2\end{array}\right]$
7. $\left[\begin{array}{ccc}2 & -1 & 7 \\ 0 & 4 & -2 \\ 5 & 8 & -1\end{array}\right]$
8. $\left[\begin{array}{ccc}2 & -1 & 7 \\ 0 & 4 & -2 \\ 0 & 5 / 2 & -29 / 2\end{array}\right]$
9. $R_{1}+R_{2} \rightarrow R_{2}$
10. $R_{1} \leftrightarrow R_{2}$
11. $x=2, y=1$
12. $x=-1, y=0$
13. $x_{1}=-2, x_{2}=1, x_{3}=2$
Section 1.3
1. (a) yes
(c) no
(b) no
(d) yes
2. (a) no
(c) yes
(b) yes
(d) yes
3. $\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
4. $\left[\begin{array}{ll}1 & 3 \\ 0 & 0\end{array}\right]$
5. $\left[\begin{array}{lll}1 & 0 & 3 \\ 0 & 1 & 7\end{array}\right]$
6. $\left[\begin{array}{ccc}1 & -1 & 2 \\ 0 & 0 & 0\end{array}\right]$
7. $\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
8. $\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
9. $\left[\begin{array}{llll}1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0\end{array}\right]$
10. $\left[\begin{array}{cccc}1 & 0 & 1 & 3 \\ 0 & 1 & -2 & 4 \\ 0 & 0 & 0 & 0\end{array}\right]$
11. $\left[\begin{array}{llllll}1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 3 & 1 & 4\end{array}\right]$
## Section 1.4
1. $x_{1}=1-2 x_{2} ; x_{2}$ is free. Possible solutions: $x_{1}=1, x_{2}=0$ and $x_{1}=-1, x_{2}=1$.
2. $x_{1}=1 ; x_{2}=2$
3. No solution; the system is inconsistent.
4. $x_{1}=-11+10 x_{3} ; x_{2}=-4+4 x_{3} ; x_{3}$ is free. Possible solutions: $x_{1}=-11$ $x_{2}=-4, x_{3}=0$ and $x_{1}=-1, x_{2}=0$ and $x_{3}=1$.
5. $x_{1}=1-x_{2}-x_{4} ; x_{2}$ is free; $x_{3}=1-2 x_{4}$; $x_{4}$ is free. Possible solutions: $x_{1}=1$, $x_{2}=0, x_{3}=1, x_{4}=0$ and $x_{1}=-2$, $x_{2}=1, x_{3}=-3, x_{4}=2$
6. No solution; the system is inconsistent.
7. $x_{1}=\frac{1}{3}-\frac{4}{3} x_{3} ; x_{2}=\frac{1}{3}-\frac{1}{3} x_{3} ; x_{3}$ is free. Possible solutions: $x_{1}=\frac{1}{3}, x_{2}=\frac{1}{3}, x_{3}=0$ and $x_{1}=-1, x_{2}=0, x_{3}=1$
8. Never exactly 1 solution; infinite solutions if $k=2$; no solution if $k \neq 2$.
9. Exactly 1 solution if $k \neq 2$; no solution if $k=2$; never infinite solutions.
## Section 1.5
## 29 chickens and 33 pigs
3. 42 grande tables, 22 venti tables
4. 30 men, 15 women, 20 kids
5. $f(x)=-2 x+10$
6. $f(x)=\frac{1}{2} x^{2}+3 x+1$
7. $f(x)=3$
8. $f(x)=x^{3}+1$
9. $f(x)=\frac{3}{2} x+1$ 17. The augmented matrix from this system is
$$
\begin{aligned}
t & =\frac{8}{3}-\frac{1}{3} f \\
x & =\frac{8}{3}-\frac{1}{3} f \\
w & =\frac{8}{3}-\frac{1}{3} f .
\end{aligned}
$$
The only time each of these variables are nonnegative integers is when $f=2$ or $f=8$. If $f=2$, then we have 2
touchdowns, 2 extra points and 2 two point conversions (and 2 field goals); this doesn't make sense since the extra points and two point conversions follow touchdowns. If $f=8$, then we have no touchdowns, extra points or two point conversions (just 8 field goals). This is the only solution; all points were scored from field goals.
19. Let $x_{1}, x_{2}$ and $x_{3}$ represent the number of free throws, 2 point and 3 point shots taken. The augmented matrix from this system is $\left[\begin{array}{llll}1 & 1 & 1 & 30 \\ 1 & 2 & 3 & 80\end{array}\right]$. From this we find the solution
$$
\begin{aligned}
& x_{1}=-20+x_{3} \\
& x_{2}=50-2 x_{3} .
\end{aligned}
$$
In order for $x_{1}$ and $x_{2}$ to be nonnegative, we need $20 \leq x_{3} \leq 25$. Thus there are 6 different scenerios: the "first" is where 20 three point shots are taken, no free throws, and 10 two point shots; the "last" is where 25 three point shots are taken, 5 free throws, and no two point shots.
21. Let $y=a x+b$; all linear functions through $(1,3)$ come in the form $y=(3-b) x+b$. Examples: $b=0$ yields $y=3 x ; b=2$ yields $y=x+2$.
22. Let $y=a x^{2}+b x+c$; we find that $a=-\frac{1}{2}+\frac{1}{2} c$ and $b=\frac{1}{2}-\frac{3}{2} c$. Examples: $c=1$ yields $y=-x+1 ; c=3$ yields $y=x^{2}-4 x+3$.
## Chapter 2
## Section 2.1
1. $\left[\begin{array}{cc}-2 & -1 \\ 12 & 13\end{array}\right]$
2. $\left[\begin{array}{cc}2 & -2 \\ 14 & 8\end{array}\right]$ 5. $\left[\begin{array}{cc}9 & -7 \\ 11 & -6\end{array}\right]$
3. $\left[\begin{array}{c}-14 \\ 6\end{array}\right]$
4. $\left[\begin{array}{l}-15 \\ -25\end{array}\right]$
5. $X=\left[\begin{array}{cc}-5 & 9 \\ -1 & -14\end{array}\right]$
6. $x=\left[\begin{array}{cc}-5 & -2 \\ -9 / 2 & -19 / 2\end{array}\right]$
7. $a=2, b=1$
8. $a=5 / 2+3 / 2 b$
9. No solution.
10. No solution.
Section 2.2
1. -22
2. 0
3. 5
4. 15
5. -2
6. Not possible.
7. $A B=\left[\begin{array}{cc}8 & 3 \\ 10 & -9\end{array}\right]$ $B A=\left[\begin{array}{cc}-3 & 24 \\ 4 & 2\end{array}\right]$
8. $A B=\left[\begin{array}{ccc}-1 & -2 & 12 \\ 10 & 4 & 32\end{array}\right]$ $B A$ is not possible.
9. $A B$ is not possible.
$$
B A=\left[\begin{array}{ccc}
27 & -33 & 39 \\
-27 & -3 & -15
\end{array}\right]
$$
19. $A B=\left[\begin{array}{ccc}-32 & 34 & -24 \\ -32 & 38 & -8 \\ -16 & 21 & 4\end{array}\right]$
$$
B A=\left[\begin{array}{cc}
22 & -14 \\
-4 & -12
\end{array}\right]
$$
21. $A B=\left[\begin{array}{ccc}-56 & 2 & -36 \\ 20 & 19 & -30 \\ -50 & -13 & 0\end{array}\right]$
$$
B A=\left[\begin{array}{cc}
-46 & 40 \\
72 & 9
\end{array}\right]
$$
23. $A B=\left[\begin{array}{cccc}-15 & -22 & -21 & -1 \\ 16 & -53 & -59 & -31\end{array}\right]$ $B A$ is not possible. 25. $A B=\left[\begin{array}{ccc}0 & 0 & 4 \\ 6 & 4 & -2 \\ 2 & -4 & -6\end{array}\right]$
$$
B A=\left[\begin{array}{ccc}
2 & -2 & 6 \\
2 & 2 & 4 \\
4 & 0 & -6
\end{array}\right]
$$
27. $A B=\left[\begin{array}{ccc}21 & -17 & -5 \\ 19 & 5 & 19 \\ 5 & 9 & 4\end{array}\right]$ $B A=\left[\begin{array}{ccc}19 & 5 & 23 \\ 5 & -7 & -1 \\ -14 & 6 & 18\end{array}\right]$
28. $D A=\left[\begin{array}{ll}4 & -6 \\ 4 & -6\end{array}\right]$
$$
A D=\left[\begin{array}{cc}
4 & 8 \\
-3 & -6
\end{array}\right]
$$
31. $D A=\left[\begin{array}{ccc}2 & 2 & 2 \\ -6 & -6 & -6 \\ -15 & -15 & -15\end{array}\right]$
$$
A D=\left[\begin{array}{ccc}
2 & -3 & 5 \\
4 & -6 & 10 \\
-6 & 9 & -15
\end{array}\right]
$$
33. $\begin{aligned} D A & =\left[\begin{array}{lll}d_{1} a & d_{1} b & d_{1} c \\ d_{2} d & d_{2} e & d_{2} f \\ d_{3} g & d_{3} h & d_{3} i\end{array}\right] \\ A D & =\left[\begin{array}{lll}d_{1} a & d_{2} b & d_{3} c \\ d_{1} d & d_{2} e & d_{3} f \\ d_{1} g & d_{2} h & d_{3} i\end{array}\right]\end{aligned}$
34. $A \vec{x}=\left[\begin{array}{c}-6 \\ 11\end{array}\right]$
35. $A \vec{x}=\left[\begin{array}{c}-5 \\ 5 \\ 21\end{array}\right]$
36. $A \vec{x}=\left[\begin{array}{c}x_{1}+2 x_{2}+3 x_{3} \\ x_{1}+2 x_{3} \\ 2 x_{1}+3 x_{2}+x_{3}\end{array}\right]$
37. $A^{2}=\left[\begin{array}{ll}4 & 0 \\ 0 & 9\end{array}\right] ; A^{3}=\left[\begin{array}{cc}8 & 0 \\ 0 & 27\end{array}\right]$
38. $A^{2}=\left[\begin{array}{lll}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{array}\right] ; A^{3}=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
39. (a) $\left[\begin{array}{cc}0 & -2 \\ -5 & -1\end{array}\right]$
(b) $\left[\begin{array}{cc}10 & 2 \\ 5 & 11\end{array}\right]$
(c) $\left[\begin{array}{cc}-11 & -15 \\ 37 & 32\end{array}\right]$
(d) No.
(e) $(A+B)(A+B)=A A+A B+B A+B B=$ $A^{2}+A B+B A+B^{2}$.
## Section 2.3
1. $\vec{x}+\vec{y}=\left[\begin{array}{c}-1 \\ 4\end{array}\right], \vec{x}-\vec{y}=\left[\begin{array}{c}3 \\ -2\end{array}\right]$
Sketches will vary depending on choice of origin of each vector.
3. $\vec{x}+\vec{y}=\left[\begin{array}{c}-3 \\ 3\end{array}\right], \vec{x}-\vec{y}=\left[\begin{array}{c}1 \\ -1\end{array}\right]$
Sketches will vary depending on choice of origin of each vector.
5. Sketches will vary depending on choice of origin of each vector.
7. Sketches will vary depending on choice of origin of each vector.
9. $\|\vec{x}\|=\sqrt{5} ;\|a \vec{x}\|=\sqrt{45}=3 \sqrt{5}$. The vector $a \vec{x}$ is 3 times as long as $\vec{x}$.
10. $\|\vec{x}\|=\sqrt{34} ;\|a \vec{x}\|=\sqrt{34}$. The vectors $a \vec{x}$ and $\vec{x}$ are the same length (they just point in opposite directions).
11. (a) $\|\vec{x}\|=\sqrt{2} ;\|\vec{y}\|=\sqrt{13}$; ||$\vec{x}+\vec{y} \|=5$.
(b) $\|\vec{x}\|=\sqrt{5} ;\|\vec{y}\|=3 \sqrt{5}$; $\|\vec{x}+\vec{y}\|=4 \sqrt{5}$.
(c) $\|\vec{x}\|=\sqrt{10} ;\|\vec{y}\|=\sqrt{29}$; ||$\vec{x}+\vec{y} \|=\sqrt{65}$.
(d) $\|\vec{x}\|=\sqrt{5} ;\|\vec{y}\|=2 \sqrt{5}$; ||$\vec{x}+\vec{y} \|=\sqrt{5}$.
The equality holds sometimes; only when $\vec{x}$ and $\vec{y}$ point along the same line, in the same direction.
15.
17.
Section 2.4 1. Multiply $A \vec{u}$ and $A \vec{v}$ to verify.
3. Multiply $A \vec{u}$ and $A \vec{v}$ to verify.
4. Multiply $A \vec{u}$ and $A \vec{v}$ to verify.
5. Multiply $A \vec{u}, A \vec{v}$ and $A(\vec{u}+\vec{v})$ to verify.
6. Multiply $A \vec{u}, A \vec{v}$ and $A(\vec{u}+\vec{v})$ to verify.
7. (a) $\vec{x}=\left[\begin{array}{l}0 \\ 0\end{array}\right]$
(b) $\vec{x}=\left[\begin{array}{c}2 / 5 \\ -13 / 5\end{array}\right]$
13. (a) $\vec{x}=\left[\begin{array}{l}0 \\ 0\end{array}\right]$
(b) $\vec{x}=\left[\begin{array}{c}-2 \\ -9 / 4\end{array}\right]$
15. (a) $\vec{x}=x_{3}\left[\begin{array}{c}5 / 4 \\ 1 \\ 1\end{array}\right]$
(b) $\vec{x}=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]+x_{3}\left[\begin{array}{c}5 / 4 \\ 1 \\ 1\end{array}\right]$
17. (a) $\vec{x}=x_{3}\left[\begin{array}{c}14 \\ -10 \\ 0\end{array}\right]$
(b) $\vec{x}=\left[\begin{array}{c}-4 \\ 2\end{array}\right]+x_{3}\left[\begin{array}{c}14 \\ -10 \\ 0\end{array}\right]$
19. (a) $\vec{x}=x_{3}\left[\begin{array}{c}2 \\ 2 / 5 \\ 1 \\ 0\end{array}\right]+x_{4}\left[\begin{array}{c}-1 \\ 2 / 5 \\ 0 \\ 1\end{array}\right]$
(b) $\vec{x}=$
21. (a) $\vec{x}=x_{2}\left[\begin{array}{c}-1 / 2 \\ 1 \\ 0 \\ 0 \\ 0\end{array}\right]+x_{4}\left[\begin{array}{c}1 / 2 \\ 0 \\ -1 / 2 \\ 1 \\ 0\end{array}\right]+$
$$
x_{5}\left[\begin{array}{c}
13 / 2 \\
0 \\
-2 \\
0 \\
1
\end{array}\right]
$$
(b) $\vec{x}=\left[\begin{array}{c}-5 \\ 0 \\ 3 / 2 \\ 0 \\ 0\end{array}\right]+x_{2}\left[\begin{array}{c}-1 / 2 \\ 1 \\ 0 \\ 0 \\ 0\end{array}\right]+$
$$
x_{4}\left[\begin{array}{c}
1 / 2 \\
0 \\
-1 / 2 \\
1 \\
0
\end{array}\right]+x_{5}\left[\begin{array}{c}
13 / 2 \\
0 \\
-2 \\
0 \\
1
\end{array}\right]
$$
23. (a) $\vec{x}=x_{4}\left[\begin{array}{c}1 \\ 13 / 9 \\ -1 / 3 \\ 1 \\ 0\end{array}\right]+x_{5}\left[\begin{array}{c}0 \\ -1 \\ -1 \\ 0 \\ 1\end{array}\right]$
(b) $\vec{x}=$
$$
\left[\begin{array}{c}
1 \\
1 / 9 \\
5 / 3 \\
0 \\
0
\end{array}\right]+x_{4}\left[\begin{array}{c}
1 \\
13 / 9 \\
-1 / 3 \\
1 \\
0
\end{array}\right]+x_{5}\left[\begin{array}{c}
0 \\
-1 \\
-1 \\
0 \\
1
\end{array}\right]
$$
25. $\vec{x}=x_{2}\left[\begin{array}{c}-2 \\ 1\end{array}\right]=x_{2} \vec{v}$
27. $\vec{x}=\left[\begin{array}{c}0.5 \\ 0\end{array}\right]+x_{2}\left[\begin{array}{c}2.5 \\ 1\end{array}\right]=\overrightarrow{x_{p}}+x_{2} \vec{v}$
## Section 2.5
1. $x=\left[\begin{array}{cc}1 & -9 \\ -4 & -5\end{array}\right]$
2. $x=\left[\begin{array}{cc}-2 & -7 \\ 7 & -6\end{array}\right]$
3. $X=\left[\begin{array}{ccc}-5 & 2 & -3 \\ -4 & -3 & -2\end{array}\right]$
4. $X=\left[\begin{array}{cc}1 & 0 \\ 3 & -1\end{array}\right]$
5. $X=\left[\begin{array}{ccc}3 & -3 & 3 \\ 2 & -2 & -3 \\ -3 & -1 & -2\end{array}\right]$
6. $X=\left[\begin{array}{ccc}5 / 3 & 2 / 3 & 1 \\ -1 / 3 & 1 / 6 & 0 \\ 1 / 3 & 1 / 3 & 0\end{array}\right]$
Section 2.6 1. $\left[\begin{array}{cc}-24 & -5 \\ 5 & 1\end{array}\right]$
3. $\left[\begin{array}{cc}1 / 3 & 0 \\ 0 & 1 / 7\end{array}\right]$
4. $A^{-1}$ does not exist.
5. $\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
6. $\left[\begin{array}{cc}-5 / 13 & 3 / 13 \\ 1 / 13 & 2 / 13\end{array}\right]$
7. $\left[\begin{array}{cc}-2 & 1 \\ 3 / 2 & -1 / 2\end{array}\right]$
8. $\left[\begin{array}{ccc}1 & 2 & -2 \\ 0 & 1 & -3 \\ 6 & 10 & -5\end{array}\right]$
9. $\left[\begin{array}{ccc}1 & 0 & 0 \\ 52 & -48 & 7 \\ 8 & -7 & 1\end{array}\right]$
10. $A^{-1}$ does not exist.
11. $\left[\begin{array}{ccc}25 & 8 & 0 \\ 78 & 25 & 0 \\ -30 & -9 & 1\end{array}\right]$
12. $\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right]$
13. $\left[\begin{array}{cccc}1 & 0 & 0 & 0 \\ -3 & -1 & 0 & -4 \\ -35 & -10 & 1 & -47 \\ -2 & -2 & 0 & -9\end{array}\right]$
14. $\left[\begin{array}{cccc}28 & 18 & 3 & -19 \\ 5 & 1 & 0 & -5 \\ 4 & 5 & 1 & 0 \\ 52 & 60 & 12 & -15\end{array}\right]$
15. $\left[\begin{array}{llll}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\end{array}\right]$
16. $\vec{x}=\left[\begin{array}{l}2 \\ 3\end{array}\right]$
17. $\vec{x}=\left[\begin{array}{c}-8 \\ 1\end{array}\right]$
18. $\vec{x}=\left[\begin{array}{c}-7 \\ 1 \\ -1\end{array}\right]$
19. $\vec{x}=\left[\begin{array}{c}3 \\ -1 \\ -9\end{array}\right]$
Section 2.7 1. $(A B)^{-1}=\left[\begin{array}{cc}-2 & 3 \\ 1 & -1.4\end{array}\right]$
3. $(A B)^{-1}=\left[\begin{array}{cc}29 / 5 & -18 / 5 \\ -11 / 5 & 7 / 5\end{array}\right]$
4. $A^{-1}=\left[\begin{array}{ll}-2 & -5 \\ -1 & -3\end{array}\right]$, $\left(A^{-1}\right)^{-1}=\left[\begin{array}{cc}-3 & 5 \\ 1 & -2\end{array}\right]$
5. $A^{-1}=\left[\begin{array}{cc}-3 & 7 \\ 1 & -2\end{array}\right]$, $\left(A^{-1}\right)^{-1}=\left[\begin{array}{ll}2 & 7 \\ 1 & 3\end{array}\right]$
6. Solutions will vary.
7. Likely some entries that should be 0 will not be exactly 0 , but rather very small values.
## Chapter 3
Section 3.1
1. $A$ is symmetric. $\left[\begin{array}{cc}-7 & 4 \\ 4 & -6\end{array}\right]$
2. $A$ is diagonal, as is $A^{T}$. $\left[\begin{array}{ll}1 & 0 \\ 0 & 9\end{array}\right]$
3. $\left[\begin{array}{ccc}-5 & 3 & -10 \\ -9 & 1 & -8\end{array}\right]$
4. $\left[\begin{array}{cc}4 & -9 \\ -7 & 6 \\ -4 & 3 \\ -9 & -9\end{array}\right]$
5. $\left[\begin{array}{c}-7 \\ -8 \\ 2 \\ -3\end{array}\right]$
6. $\left[\begin{array}{ccc}-9 & 6 & -8 \\ 4 & -3 & 1 \\ 10 & -7 & -1\end{array}\right]$
7. $A$ is symmetric. $\left[\begin{array}{ccc}4 & 0 & -2 \\ 0 & 2 & 3 \\ -2 & 3 & 6\end{array}\right]$
8. $\left[\begin{array}{ccc}2 & 5 & 7 \\ -5 & 5 & -4 \\ -3 & -6 & -10\end{array}\right]$
9. $\left[\begin{array}{ccc}4 & 5 & -6 \\ 2 & -4 & 6 \\ -9 & -10 & 9\end{array}\right]$
10. $A$ is upper triangular; $A^{T}$ is lower triangular. $\left[\begin{array}{ccc}-3 & 0 & 0 \\ -4 & -3 & 0 \\ -5 & 5 & -3\end{array}\right]$ 21. $A$ is diagonal, as is $A^{T}$. $\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1\end{array}\right]$
11. $A$ is skew symmetric. $\left[\begin{array}{ccc}0 & -1 & 2 \\ 1 & 0 & -4 \\ -2 & 4 & 0\end{array}\right]$
## Section 3.2
1. 6
2. 3
3. -9
4. 1
5. Not defined; the matrix must be square.
6. -23
7. 4
8. 0
9. (a) $\operatorname{tr}(A)=8 ; \operatorname{tr}(B)=-2 ; \operatorname{tr}(A+B)=6$
(b) $\operatorname{tr}(A B)=53=\operatorname{tr}(B A)$
19. (a) $\operatorname{tr}(A)=-1 ; \operatorname{tr}(B)=6 ; \operatorname{tr}(A+B)=5$
(b) $\operatorname{tr}(A B)=201=\operatorname{tr}(B A)$
## Section 3.3
1. 34
2. -44
3. -44
4. 28
5. (a) The submatrices are $\left[\begin{array}{cc}7 & 6 \\ 6 & 10\end{array}\right]$, $\left[\begin{array}{cc}3 & 6 \\ 1 & 10\end{array}\right]$, and $\left[\begin{array}{ll}3 & 7 \\ 1 & 6\end{array}\right]$,
respectively.
(b) $c_{1,2}=34, c_{1,2}=-24, c_{1,3}=11$
11. (a) The submatrices are $\left[\begin{array}{cc}3 & 10 \\ 3 & 9\end{array}\right]$, $\left[\begin{array}{cc}-3 & 10 \\ -9 & 9\end{array}\right]$, and $\left[\begin{array}{cc}-3 & 3 \\ -9 & 3\end{array}\right]$,
respectively.
(b) $C_{1,2}=-3, C_{1,2}=-63, C_{1,3}=18$
13. -59
14. 15
15. 3
16. 0
17. 0
18. -113 25. Hint: $C_{1,1}=d$.
## Section 3.4
1. 84
2. 0
3. 10
4. 24
5. 175
6. -200
7. 34
8. (a) $\operatorname{det}(A)=41 ; R_{2} \leftrightarrow R_{3}$
(b) $\operatorname{det}(B)=164 ;-4 R_{3} \rightarrow R_{3}$
(c) $\operatorname{det}(C)=-41 ; R_{2}+R_{1} \rightarrow R_{1}$
17. (a) $\operatorname{det}(A)=-16 ; R_{1} \leftrightarrow R_{2}$ then $R_{1} \leftrightarrow R_{3}$
(b) $\operatorname{det}(B)=-16$; $-R_{1} \rightarrow R_{1}$ and $-R_{2} \rightarrow R_{2}$
(c) $\operatorname{det}(C)=-432 ; C=3 * M$
19. $\operatorname{det}(A)=4, \operatorname{det}(B)=4, \operatorname{det}(A B)=16$
20. $\operatorname{det}(A)=-12, \operatorname{det}(B)=29$, $\operatorname{det}(A B)=-348$
21. -59
22. 15
23. 3
24. 0
## Section 3.5
1. (a) $\operatorname{det}(A)=14$, $\operatorname{det}\left(A_{1}\right)=70$, $\operatorname{det}\left(A_{2}\right)=14$
(b) $\vec{x}=\left[\begin{array}{l}5 \\ 1\end{array}\right]$
3. (a) $\operatorname{det}(A)=0, \operatorname{det}\left(A_{1}\right)=0$, $\operatorname{det}\left(A_{2}\right)=0$
(b) Infinite solutions exist.
5. (a) $\operatorname{det}(A)=16, \operatorname{det}\left(A_{1}\right)=-64$, $\operatorname{det}\left(A_{2}\right)=80$
(b) $\vec{x}=\left[\begin{array}{c}-4 \\ 5\end{array}\right]$
7. (a) $\operatorname{det}(A)=-123, \operatorname{det}\left(A_{1}\right)=-492$, $\operatorname{det}\left(A_{2}\right)=123, \operatorname{det}\left(A_{3}\right)=492$
(b) $\vec{x}=\left[\begin{array}{c}4 \\ -1 \\ -4\end{array}\right]$
9. (a) $\operatorname{det}(A)=56, \operatorname{det}\left(A_{1}\right)=224$, $\operatorname{det}\left(A_{2}\right)=0, \operatorname{det}\left(A_{3}\right)=-112$ (b) $\vec{x}=\left[\begin{array}{c}4 \\ 0 \\ -2\end{array}\right]$
10. (a) $\operatorname{det}(A)=0, \operatorname{det}\left(A_{1}\right)=147$, $\operatorname{det}\left(A_{2}\right)=-49, \operatorname{det}\left(A_{3}\right)=-49$
(b) No solution exists.
## Chapter 4
Section 4.1
1. $\lambda=3$
2. $\lambda=0$
3. $\lambda=3$
4. $\vec{x}=\left[\begin{array}{c}-1 \\ 2\end{array}\right]$
5. $\vec{x}=\left[\begin{array}{c}3 \\ -7 \\ 7\end{array}\right]$
6. $\vec{x}=\left[\begin{array}{c}-1 \\ 1 \\ 1\end{array}\right]$
7. $\lambda_{1}=4$ with $\overrightarrow{x_{1}}=\left[\begin{array}{l}9 \\ 1\end{array}\right]$; $\lambda_{2}=5$ with $\overrightarrow{x_{2}}=\left[\begin{array}{l}8 \\ 1\end{array}\right]$
8. $\lambda_{1}=-3$ with $\overrightarrow{x_{1}}=\left[\begin{array}{c}-2 \\ 1\end{array}\right]$; $\lambda_{2}=5$ with $\overrightarrow{x_{2}}=\left[\begin{array}{l}6 \\ 1\end{array}\right]$
9. $\lambda_{1}=2$ with $\overrightarrow{x_{1}}=\left[\begin{array}{l}1 \\ 1\end{array}\right]$; $\lambda_{2}=4$ with $\overrightarrow{x_{2}}=\left[\begin{array}{c}-1 \\ 1\end{array}\right]$
10. $\lambda_{1}=-1$ with $\overrightarrow{x_{1}}=\left[\begin{array}{l}1 \\ 2\end{array}\right]$; $\lambda_{2}=-3$ with $\overrightarrow{x_{2}}=\left[\begin{array}{l}1 \\ 0\end{array}\right]$
11. $\lambda_{1}=3$ with $\overrightarrow{x_{1}}=\left[\begin{array}{c}-3 \\ 0 \\ 2\end{array}\right]$; $\lambda_{2}=4$ with $\overrightarrow{x_{2}}=\left[\begin{array}{c}-5 \\ -1 \\ 1\end{array}\right]$ $\lambda_{3}=5$ with $\overrightarrow{x_{3}}=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ 23. $\lambda_{1}=-5$ with $\overrightarrow{x_{1}}=\left[\begin{array}{c}24 \\ 13 \\ 8\end{array}\right]$; $\lambda_{2}=-2$ with $\overrightarrow{x_{2}}=\left[\begin{array}{l}6 \\ 5 \\ 1\end{array}\right]$ $\lambda_{3}=3$ with $\overrightarrow{x_{3}}=\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]$
12. $\lambda_{1}=-2$ with $\overrightarrow{x_{1}}=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right]$; $\lambda_{2}=1$ with $\overrightarrow{x_{2}}=\left[\begin{array}{l}0 \\ 3 \\ 5\end{array}\right]$ $\lambda_{3}=5$ with $\overrightarrow{x_{3}}=\left[\begin{array}{c}28 \\ 7 \\ 1\end{array}\right]$
13. $\lambda_{1}=-2$ with $\vec{x}=\left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right]$; $\lambda_{2}=3$ with $\vec{x}=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$; $\lambda_{3}=5$ with $\vec{x}=\left[\begin{array}{l}0 \\ 1 \\ 1\end{array}\right]$
## Section 4.2
1. (a) $\lambda_{1}=1$ with $\overrightarrow{x_{1}}=\left[\begin{array}{l}4 \\ 1\end{array}\right]$; $\lambda_{2}=4$ with $\overrightarrow{x_{2}}=\left[\begin{array}{l}1 \\ 1\end{array}\right]$
(b) $\lambda_{1}=1$ with $\overrightarrow{x_{1}}=\left[\begin{array}{c}-1 \\ 1\end{array}\right]$; $\lambda_{2}=4$ with $\overrightarrow{x_{2}}=\left[\begin{array}{c}-1 \\ 4\end{array}\right]$
(c) $\lambda_{1}=1 / 4$ with $\overrightarrow{x_{1}}=\left[\begin{array}{l}1 \\ 1\end{array}\right]$; $\lambda_{2}=4$ with $\overrightarrow{x_{2}}=\left[\begin{array}{l}4 \\ 1\end{array}\right]$
(d) 5
(e) 4
3. (a) $\lambda_{1}=-1$ with $\overrightarrow{x_{1}}=\left[\begin{array}{c}-5 \\ 1\end{array}\right]$;
$$
\lambda_{2}=0 \text { with } \overrightarrow{x_{2}}=\left[\begin{array}{c}
-6 \\
1
\end{array}\right]
$$
(b) $\lambda_{1}=-1$ with $\overrightarrow{x_{1}}=\left[\begin{array}{l}1 \\ 6\end{array}\right]$; $\lambda_{2}=0$ with $\overrightarrow{x_{2}}=\left[\begin{array}{l}1 \\ 5\end{array}\right]$
(c) Ais not invertible. (d) -1
(e) 0
5. (a) $\lambda_{1}=-4$ with $\overrightarrow{x_{1}}=\left[\begin{array}{c}-7 \\ -7 \\ 6\end{array}\right]$;
$$
\begin{aligned}
& \lambda_{2}=3 \text { with } \overrightarrow{x_{2}}=\left[\begin{array}{l}
0 \\
0 \\
1
\end{array}\right] \\
& \lambda_{3}=4 \text { with } \overrightarrow{x_{3}}=\left[\begin{array}{c}
9 \\
1 \\
22
\end{array}\right]
\end{aligned}
$$
(b) $\lambda_{1}=-4$ with $\overrightarrow{x_{1}}=\left[\begin{array}{c}-1 \\ 9 \\ 0\end{array}\right]$;
$$
\begin{aligned}
& \lambda_{2}=3 \text { with } \overrightarrow{x_{2}}=\left[\begin{array}{c}
-20 \\
26 \\
7
\end{array}\right] \\
& \lambda_{3}=4 \text { with } \overrightarrow{x_{3}}=\left[\begin{array}{c}
-1 \\
1 \\
0
\end{array}\right]
\end{aligned}
$$
(c) $\lambda_{1}=-1 / 4$ with $\overrightarrow{x_{1}}=\left[\begin{array}{c}-7 \\ -7 \\ 6\end{array}\right]$;
$$
\begin{aligned}
& \lambda_{2}=1 / 3 \text { with } \overrightarrow{x_{2}}=\left[\begin{array}{l}
0 \\
0 \\
1
\end{array}\right] \\
& \lambda_{3}=1 / 4 \text { with } \overrightarrow{x_{3}}=\left[\begin{array}{c}
9 \\
1 \\
22
\end{array}\right]
\end{aligned}
$$
(d) 3
(e) -48
## Chapter 5
## Section 5.1
1. $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$
2. $A=\left[\begin{array}{ll}1 & 2 \\ 1 & 2\end{array}\right]$
3. $A=\left[\begin{array}{ll}5 & 2 \\ 2 & 1\end{array}\right]$
4. $A=\left[\begin{array}{ll}0 & 1 \\ 3 & 0\end{array}\right]$
5. $A=\left[\begin{array}{cc}0 & -1 \\ -1 & -1\end{array}\right]$
6. Yes, these are the same; the transformation matrix in each is $\left[\begin{array}{cc}-1 & 0 \\ 0 & -1\end{array}\right]$. 13. Yes, these are the same. Each produces the transformation matrix $\left[\begin{array}{cc}1 / 2 & 0 \\ 0 & 3\end{array}\right]$.
## Section 5.2
1. Yes
2. No; cannot add a constant.
3. Yes.
4. $[T]=\left[\begin{array}{cc}1 & 2 \\ 3 & -5 \\ 0 & 2\end{array}\right]$
5. $[T]=\left[\begin{array}{ccc}1 & 0 & 3 \\ 1 & 0 & -1 \\ 1 & 0 & 1\end{array}\right]$
6. $[T]=\left[\begin{array}{llll}1 & 2 & 3 & 4\end{array}\right]$
## Section 5.3
1. $\vec{x}+\vec{y}=\left[\begin{array}{l}3 \\ 2 \\ 4\end{array}\right], \vec{x}-\vec{y}=\left[\begin{array}{c}-1 \\ -4 \\ 0\end{array}\right]$
Sketches will vary slightly depending on orientation.
3. $\vec{x}+\vec{y}=\left[\begin{array}{l}4 \\ 4 \\ 8\end{array}\right], \vec{x}-\vec{y}=\left[\begin{array}{l}-2 \\ -2 \\ -4\end{array}\right]$
Sketches will vary slightly depending on orientation.
5. Sketches may vary slightly.
7. Sketches may vary slightly.
9. $\|\vec{x}\|=\sqrt{30},\|a \vec{x}\|=\sqrt{120}=2 \sqrt{30}$
10. $\|\vec{x}\|=\sqrt{54}=3 \sqrt{6}$, $\| a \vec{x}||=\sqrt{270}=15 \sqrt{6}$
| Textbooks |
\begin{document}
\title
[\ifdraft DRAFT: \fi Lower bounds on the mix norm of passive scalars]
{\ifdraft DRAFT: \fi Lower bounds on the mix norm of passive scalars advected by incompressible enstrophy-constrained flows.}
\begin{abstract}
Consider a diffusion-free passive scalar $\theta$ being mixed by an incompressible flow $u$ on the torus $\Tm^d$.
Our aim is to study how well this scalar can be mixed under an enstrophy constraint on the advecting velocity field.
Our main result shows that the mix-norm ($\norm{\theta(t)}_{H^{-1}}$) is bounded below by an exponential function of time.
The exponential decay rate we obtain is not universal and depends on the size of the support of the initial data.
We also perform numerical simulations and confirm that the numerically observed decay rate scales similarly to the rigorous lower bound, at least for a significant initial period of time.
The main idea behind our proof is to use recent work of Crippa and DeLellis ('08) making progress towards the resolution of Bressan's rearrangement cost conjecture.
\end{abstract}
\author{Gautam Iyer}
\address{\hskip-\parindent
Gautam Iyer\\
Department of Mathematics\\
Carnegie Mellon University \\
Pittsburgh, PA 15213}
\email{[email protected]}
\author{Alexander Kiselev}
\address{\hskip-\parindent
Alexander Kiselev\\
Department of Mathematics\\
University of Wisconsin-Madison\\
Ma\-dison, WI 53706, USA}
\email{[email protected]}
\author{Xiaoqian Xu}
\address{\hskip-\parindent
Xiaoqian Xu\\
Department of Mathematics\\
University of Wisconsin-Madison\\
Madison, WI 53706, USA}
\email{[email protected]}
\thanks{This material is based upon work partially supported by the National Science Foundation under grants
DMS-1007914,
DMS-1104415,
DMS-1159133,
DMS-1252912.
GI acknowledges partial support from an Alfred P. Sloan research fellowship. AK acknowledges partial support from a Guggenheim fellowship.
The authors also thank the Center for Nonlinear Analysis (NSF Grants No. DMS-0405343 and DMS-0635983), where part of this research was carried out. }
\maketitle
\section{Introduction}
The mixing of tracer particles by fluid flows is ubiquitous in nature, and have applications ranging from weather forecasting to food processing.
An important question that has attracted attention recently is to study ``how well'' tracers can be mixed under a constraint on the advecting velocity field, and what is the optimal choice of the ``best mixing'' velocity field (see~\cite{bblThiffeault12} for a recent review).
Our aim in this paper is to study how well passive tracers can be mixed under an \emph{enstrophy constraint} on the advecting fluid.
By passive, we mean that the tracers provide no feedback to the advecting velocity field.
Further, we assume that diffusion of the tracer particles is weak and can be neglected on the relevant time scales.
Mathematically, the density of such tracers (known as passive scalars) is modeled by the transport equation
\begin{equation}\label{eq:1}
\partial_t\theta(x,t)+u\cdot \nabla\theta=0, \,\,\,\theta(x,0) = \theta_0(x).
\end{equation}
To model stirring, the advecting velocity field~$u$ is assumed to be incompressible.
For simplicity we study~\eqref{eq:1} with periodic boundary conditions (with period $1$), mean zero initial data, and assume that all functions in question are smooth.
The first step is to quantify ``how well'' a passive scalar is mixed in our context. For \emph{diffusive} passive scalars, the decay of the variance is a commonly used measure of mixing (see for instance \cites{bblConstantinKiselevRyzhikZlatos08,bblDoeringThiffeault06,bblThiffeaultDoeringGibbon04,bblShawThiffeaultDoering07} and references there in).
But for diffusion free scalars the variance is a conserved and does not change with time.
Thus, following~\cite{bblLinThiffeaultDoering11} we quantify mixing using the $H^{-1}$-Sobolev norm: the \emph{smaller} $\norm{\theta}_{H^{-1}}$, the better mixed the scalar $\theta$ is.
The reason for using a negative Sobolev norm in this context has its roots in~\cites{bblDoeringThiffeault06,bblLinThiffeaultDoering11,bblMathewMezicPetzold05,bblShawThiffeaultDoering07}.
The motivation is that if the flow generated by the velocity field is mixing in the ergodic theory sense, then any advected quantity (in particular $\theta$) converges to $0$ weakly in $L^2$ as $t \to \infty$.
This can be shown to imply that $\norm{\theta(\cdot, t)}_{H^{s}} \to 0$ for all $s < 0$, and conversely, if $\norm{\theta(\cdot, t)}_{H^{s}} \to 0$ for some $s < 0$ then $\theta(x,t)$ converges
weakly to zero. Thus any negative Sobolev norm of $\theta$ can in principle be used to quantify its mixing properties.
In two dimensions the choice of using the $H^{-1}$ norm in particular was suggested by Lin et.\ al.~\cite{bblLinThiffeaultDoering11} as it scales like the area dominant unmixed regions; a natural length scale associated with the system.
We will work with the same Sobolev norm in any dimension $d$; the ratio of $H^{-1}$ norm to $L^2$ norm has a dimension of length, and since the $L^2$ norm of $\theta(x,t)$ is conserved, the $H^{-1}$ norm provides a natural length scale associated with the mixing process.
The questions we study in this paper are motivated by recent work of Lin et.\ al.~\cite{bblLinThiffeaultDoering11}.
In~\cite{bblLinThiffeaultDoering11}, the authors address two questions on the two dimensional torus:
\begin{itemize}
\item
The time decay of $\norm{\theta(t)}_{H^{-1}}$, given the \emph{fixed energy} constraint $\norm{u(t)}_{L^2} = U$.
\item
The time decay of $\norm{\theta(t)}_{H^{-1}}$ given a \emph{fixed enstrophy} constraint of the form $\norm{\nabla u(t)}_{L^2} = F$.
\end{itemize}
In the first case the authors prove a lower bound for~$\norm{\theta(\cdot, t)}_{H^{-1}(\mathbb T^2)}$ that is linear in $t$, with negative slope.
This suggests that it may be possible to ``mix perfectly in finite time''; namely choose $u$ in a manner that drives $\norm{\theta(\cdot, t)}_{H^{-1}}$ to zero in finite time.
This was followed by an explicit example in~\cite{bblLunasinLinNovikovMazzucatoDoering12} exhibiting finite time perfect mixing, under a finite energy constraint.
This example uses an elegant ``slice and dice'' construction, which requires the advecting velocity field to develop finer and finer scales.
Thus, while their example maintains a fixed energy constraint, the enstrophy ($\norm{\nabla u}_{L^2}$) explodes.
Together with the numerical analysis in~\cites{bblLinThiffeaultDoering11,bblLunasinLinNovikovMazzucatoDoering12} this suggests that finite time perfect mixing by an enstrophy constrained incompressible flow might be impossible.
Our main theorem settles this affirmatively.
\begin{theorem}\label{thmMain}
Let $u$ be a smooth (time dependent) incompressible periodic vector field on the $d$-dimensional torus, and let $\theta$ solve~\eqref{eq:1} with periodic boundary conditions and $L^\infty$ initial data $\theta_0$.
For any $p > 1$ and $\lambda \in (0,1)$ there exists a length scale $r_0 = r_0( \theta_0, \lambda),$ an explicit constant $\varepsilon_0 = \varepsilon_0(\lambda, d),$ and a constant $c = c(d, p)$ such that
\begin{equation}\label{eq:solve}
\norm{\theta(t)}_{H^{-1}}
\geqslant \varepsilon_0 r_0^{d/2+1} \norm{\theta_0}_{L^\infty} \exp \paren[\Big]{ \frac{-c}{m \paren{ A_\lambda }^{1/p} } \int_0^t \norm{\nabla u(s)}_{L^p} \, ds }.
\end{equation}
Here $A_\lambda$ is the super level set $\{ \theta_0 > \lambda \norm{\theta_0}_{L^\infty} \}$.
In particular, if the instantaneous enstrophy constraint $\norm{\nabla u}_{L^2} \leqslant F$ is enforced, then $\norm{\theta(t)}_{H^{-1}}$ decays at most exponentially with time.
\end{theorem}
Before commenting on the $r_0$ and $m(A_\lambda)$ dependence, we briefly mention some applications.
There are many physical situations where $\int_0^t \norm{\nabla u(s)}_{L^2} \,ds$ is well controlled.
Some examples are when $u$ satisfies the incompressible Navier-Stokes equations with $\dot H^{-1}$ forcing~\cite{bblDoeringGibbon95,bblConstantinFoias88}, the 2D incompressible Euler equations~\cite{bblBardosTiti07} or a variety of active scalar equations including the critical surface quasi-geostrophic equation~\cite{bblKiselevNazarovVolberg07,bblConstantinVicol12,bblCaffarelliVasseur10,bblChaeConstantinCordobaGancedoWu12}.
In each of these situations the passive scalars can not be mixed perfectly in finite time.
More precisely, a lower bound for the $H^{-1}$-norm of the scalar density can be read off using~\eqref{eq:solve} and the appropriate control on $\norm{\nabla u}_{L^2}$.
We also mention that the proof of this theorem is not based on energy methods.
Instead, the main idea is to relate the notion of ``mixed to scale~$\delta$'' to the $H^{-1}$ norm, and use recent progress by Crippa and DeLellis~\cite{bblCrippaDeLellis08} towards Bressan's rearrangement cost conjecture~\cite{bblBressan03}.
Some of these ideas were already suggested in~\cite{bblLunasinLinNovikovMazzucatoDoering12}.
We defer the proof of Theorem~\ref{thmMain} to Section~\ref{sxnMainTheorem}, and
pause to analyze the dependence of the bound in \eqref{eq:solve} on $r_0$ and $m(A_\lambda)$.
The length scale $r_0$ is morally the scale at which the super level set $A_\lambda$ is ``unmixed''; a notion that is made precise later.
Our proof, however, imposes a slightly stronger condition: namely, our proof will show that $r_0$ can be any length scale such that ``most'' of the super level set $A_\lambda$ occupies ``most'' of the union of disjoint balls of radius \emph{at least} $r_0$.
While we are presently unable to estimate $r_0$ in terms of a tangible norm of $\theta_0$, we remark that we at least expect a connection between $r_0$ and the ratio of the measure of $A_\lambda$ to the perimeter of $A_\lambda$ (see~\cite{bblSlepcev08} for a related notion).
On the other hand, we point out that the pre-factor in~\eqref{eq:solve} can be improved at the expense of the decay rate.
To see this, suppose for some $\kappa \in [0, 1/2)$ there exists $N$ disjoint balls of radius at least $r_1$ such that the fraction of each of these balls occupied by $A_\lambda$ is \emph{at least} $1 - \kappa$. Then our proof will show that~\eqref{eq:solve} in Theorem~\ref{thmMain} can be replaced by
\begin{gather}\label{eqnSolvePrime}
\tag{\ref{eq:solve}$'$}
\norm{\theta(t)}_{H^{-1}}
\geqslant \varepsilon_0 r_1^{d/2+1} \norm{\theta_0}_{L^\infty} \exp \paren[\Big]{ \frac{-c}{\paren{ N r_1^d }^{1/p} } \int_0^t \norm{\nabla u(s)}_{L^p} \, ds }.
\end{gather}
In this case, if $\theta_0 \in C^1$, the mean value theorem will guarantee that we can choose $N = 1$ and $r_1 > \frac{\norm{\theta_0}_{L^\infty}}{C \norm{\nabla \theta_0}_{L^\infty}}$ for a purely dimensional constant $C$.
Next we turn to the exponential decay rate.
The dependence of this on $m(A_\lambda)$ is natural.
To see this, suppose momentarily that $\theta_0$ only takes on the values $\pm1$ or $0$ representing two insoluble, immiscible fluids which are injected into a large fluid container.
Physical intuition suggests that the less the amount of fluid that is injected, the faster one can mix it.
Indeed, this is reflected in~\eqref{eq:solve} as in this case $m(A_\lambda) = \frac{1}{2} m( \supp( \theta_0 ) )$;
so the smaller the support of the initial data, the worse the lower bound~\eqref{eq:solve} is.
We mention that a bound similar to~\eqref{eq:solve} was proved in~\cite{bblSeis} using optimal transport and ideas from~\cite{bblBrenierOttoSeis}.
In~\cite{bblSeis}, however, the author only considers bounded variation ``binary phase'' initial data, where the two phases occupy the entire region;
consequently the result does not capture the dependence of the decay rate on the initial data.
We do not know if the estimate of the exponential decay provided by bound~\eqref{eq:solve} is optimal, however, numerical simulations suggest that it may be not far off.
A good candidate for the velocity field that might achieve the optimal lower bound was presented in~\cite{bblLinThiffeaultDoering11} using a steepest descent method (equation~\eqref{eqnVelocity}, below).
Due to the nonlinear nature of this formula, it is hard to rigorously prove upper bounds; but all our numerical simulations in Section~\ref{sxnNumerics} seem to indicate an exponential lower bound with a decay rate that is in a good qualitative agreement with~\eqref{eq:solve}.
However, our numerical simulations show that even if we start with initial data that is localized to a small region, it gradually spreads during mixing.
The incompressibility constraint will of course guarantee that the measure of the support of the solution is conserved in time.
But since the enstrophy constraint forbids abrupt changes in the velocity field,
the ``region occupied'' by the initial data tends to spread and is likely to eventually ``fill'' the entire torus (see Figure~\ref{fgrSolPlots}).
A very interesting question is whether there will eventually be a transition in dynamics where
the factor in the exponential decay of the mix norm depends only on the volume of the entire domain, and not the size of the region occupied by the initial data.
Our attempts to get insight into this question numerically were inconclusive, as we ran out of resolution before observing such a regime change.
What we address here, however, is an interesting link between universality of the exponential lower bound and mixing in domains with boundaries.
It has been observed formally \cite{bblGouillartKuncioDauchotDubrulleRouxThiffeault,bblGouillartDauchotDubrulleRouxThiffeault} that presence of walls with no slip conditions inhibits mixing; hyperbolic flows which usually lead to exponential decay of various mixing measures lead to only algebraic mixing rates in presence of walls.
Here, we provide an elementary and rigorous argument showing that universality of the exponential lower bound in mixing with an enstrophy constraint would lead to an algebraic in time lower bound if the initial data is compactly supported away from the boundary and the advecting velocity field vanishes at the boundary.
Agreement with earlier heuristic arguments is intriguing; but it is not clear to us if one can expect such result to be true in full generality.
It would be very interesting to know whether the efficient mixing by an incompressible enstrophy constrained flow spreads the initial data over the entire ambient volume and results in the slowdown of the exponential decay.
We plan to further investigate this issue in the future.
\subsection*{Notational convention, and plan of this paper.}
\sidenote{Sat 07/20 GI: Revisit this when the paper is actually done.}
We will assume throughout the paper that $d \geqslant 2$ is the dimension, and $\mathbb T^d$ is the $d$-dimensional torus, with side length $1$.
All periodic functions are assumed to be $1$-periodic, and we use $m$ to denote the Lebesgue measure on $\mathbb T^d$.
We will use $\norm{f}_{H^s}$ to denote the \emph{homogeneous} Sobolev norms.
This paper is organized as follows: In Section~\ref{sxnMainTheorem} we describe the notion of $\delta$-mixed data, and prove Theorem~\ref{thmMain}, modulo a few Lemmas.
In Section~\ref{sxnLemmaProofs} we prove the required lemmas.
In Section~\ref{sxnNumerics} we present numerics suggesting that the bound stated in Theorem~\ref{thmMain} is not far from optimal, at least for an initial period of time.
Finally, we conclude this paper with a scaling argument showing that an exponential lower bound on $\norm{\theta}_{H^{-1}}$ with rate independent of $\theta_0$ will imply a stronger \emph{algebraic} lower bound
for mixing with flows satisfying no-slip boundary condition.
\section{Rearrangement Costs and the Proof of the Main Theorem.}\label{sxnMainTheorem}
We devote this section to the proof of Theorem~\ref{thmMain}.
The idea behind the proof is as follows.
First, if $\norm{\theta}_{H^{-1}}$ is small enough, then its super-level sets are mixed to certain scales (Lemma~\ref{lmaMixNorm} below).
Second, any flow that starts with an ``unmixed'' set and mixes it to scale $\delta$ has to do a minimum amount of work~\cites{bblBressan03,bblCrippaDeLellis08}.
Putting these together yields Theorem~\ref{thmMain}.
We begin by describing the notion of ``mixed to scale $\delta$'', and relate this to the $H^{-1}$ Sobolev norm.
\begin{definition}\label{dfnSemiMixed}
Let $\kappa \in (0, \frac{1}{2})$ be fixed.
For $\delta > 0$, we say a set $A\subseteq \mathbb{T}^d$ is $\delta$-semi-mixed if
$$
\frac{m\paren[\big]{A \cap B(x, \delta) }}{m( B(x,\delta))}\leqslant 1-\kappa
\quad\text{for every } x \in \mathbb T^d.
$$
If additionally $A^c$ is also $\delta$-semi-mixed, then we say $A$ is $\delta$-mixed (or mixed to scale $\delta$).
\end{definition}
\begin{remark}
The parameters $\delta$ and $\kappa$ measures the scale and ``accuracy'' respectively.
The key parameter here is the scale $\delta$, and the accuracy parameter $\kappa \in (0, 1/2)$ only plays an auxiliary role.
Given a specific initial distribution to mix, $\kappa$ can be chosen to optimize the bound.
Note that the notion of a set being mixed here is the same as that of Bressan~\cite{bblBressan03}.
A set being semi-mixed is of course a weaker notion.
\end{remark}
One relation between $\delta$-semi-mixed and negative Sobolev norms is as follows.
\begin{lemma}\label{lmaMixNorm}
Let $\lambda \in (0, 1]$ and $\theta \in L^\infty(\mathbb T^d)$.
Then for any integer $n > 0$, $\kappa \in (0, \frac{\lambda}{1 + \lambda} )$ there exists an explicit constant $c_0 = c_0(d, \kappa, \lambda, n)$ such that
$$
\norm{\theta}_{H^{-n}}\leqslant \frac{\norm{\theta}_{L^\infty} \delta^{n + d/2}}{c_0}
\implies
A_\lambda \text{ is $\delta$-semi-mixed.}
$$
Here $A_\lambda$ is the super level set defined by
$A_\lambda \defeq \{\theta > \lambda \norm{\theta}_{L^\infty} \}$.
\end{lemma}
Our interest in this Lemma is mainly when $n = 1$.
Note that while Lemma~\ref{lmaMixNorm} guarantees the super level sets $A_\lambda$ are $\delta$-semi-mixed, they need not be $\delta$-mixed.
Indeed if $A_\lambda$ is very small, its complement won't be $\delta$-semi-mixed.
Also, we remark that the converse of Lemma~\ref{lmaMixNorm} need not be true.
For example the function
$$
f(x) = \sin(2 \pi x) + 10 \sin( 2\pi n x )
$$
has $\norm{f}_{H^{-1}(\mathbb T^1)} = O(1)$, and the super level set $\{ f > 5 \}$ is certainly semi-mixed to scale $1/n$ (see also~\cite{bblLinThiffeaultDoering11}).
The proof of Lemma~\ref{lmaMixNorm} follows from a duality and scaling argument.
For clarity of presentation we postpone the proof to Section~\ref{sxnLemmaProofs}.
Returning to Theorem~\ref{thmMain}, the main ingredient in its proof is a lower bound on the ``amount of work'' required to mix a set to fine scales.
This notion goes back to a conjecture of Bressan for which a \$500 prize was announced~\cite{bblBressanPrize}.
\begin{conjecture}[Bressan '03~\cite{bblBressan03}]\label{cjrBressan}
Let $H$ to be the left half of the torus, and $\Psi$ be the flow generated by
an incompressible
vector field $u$.
If after time $T$ the image of $H$ under the flow $\Psi$ is $\delta$-mixed, then there exists a constant $C$ such that
\begin{equation}\label{eqnBressan}
\int_0^T \norm{\nabla u(\cdot, t)}_{L^1} \, dt \geqslant \frac{ \abs{\ln \delta } }{C} .
\end{equation}
\end{conjecture}
We refer the reader to~\cite{bblBressan03} for the motivation of the lower bound~\eqref{eqnBressan} and further discussion.
To the best of our knowledge, this conjecture is still open.
However, Crippa and De Lellis~\cite{bblCrippaDeLellis08} made significant progress towards the resolution of this conjecture.
\begin{theorem}[Crippa, De Lellis '08~\cite{bblCrippaDeLellis08}]\label{thmCrippaDeLellis}
Using the same notation as in Conjecture~\ref{cjrBressan}, for any $p > 1$ there exists a finite positive constant $C_p$ such that
\begin{equation}\label{eqnCrippaDeLellis}
\int_0^T \norm{\nabla u(\cdot, t)}_{L^p} \, dt \geqslant \frac{\abs{\ln \delta}}{C_p}.
\end{equation}
\end{theorem}
For our purposes we will need two extensions of Theorem~\ref{thmCrippaDeLellis}.
First, we will need to start with sets other than the half torus.
Second, we will need lower bounds for the work done to \emph{semi}-mix sets to small scales.
Note that in order for a flow to $\delta$-mix a set $A$, it has to both $\delta$-semi-mix $A$ and $\delta$-semi-mix $A^c$.
Generically each of these steps should cost comparable amounts, and hence a semi-mixed version of Theorem~\ref{thmCrippaDeLellis} should follow using techniques in~\cite{bblCrippaDeLellis08}.
We state this as our next lemma.
\begin{lemma}\label{lmaSemiMixedWorkDone}
Let $\Psi$ be the flow map of an incompressible vector field $u$.
Let $A \subset \mathbb T^d$ be any measurable set and let $p > 1$.
There exist constants $r_0 = r_0(A)$ and $a = a(d, \kappa, p) > 0$, such that if for some $\delta < r_0 / 2$ and $T > 0$ the set $\Psi_T(A)$ is $\delta$-\emph{semi}-mixed, then
\begin{equation}\label{eqnSemiMixed}
\int_0^T \norm{\nabla u(\cdot, t)}_{L^p} \, dt \geqslant \frac{ m(A)^{1/p} }{a} \abs[\Big]{\ln \frac{2 \delta}{r_0} }.
\end{equation}
\end{lemma}
Morally the constant $r_0$ above should be a length scale at which set $A$ is not semi-mixed.
Our proof, however, uses a condition on $r_0$ which is slightly stronger than only requiring that $A$ is not semi-mixed to scale $r_0$.
Namely, we will require ``most'' of $A$ to occupy ``most'' of the union of disjoint balls of radius at least $r_0$.
Deferring the proof of Lemma~\ref{lmaSemiMixedWorkDone} to Section~\ref{sxnLemmaProofs}, we prove Theorem~\ref{thmMain}.
\begin{proof}[Proof of Theorem~\ref{thmMain}]
Replacing $\theta$ with $\theta / \norm{\theta}_{L^\infty}$, we may without loss of generality assume $\norm{\theta_0}_{L^\infty} = 1$.
Fix $0< \lambda \leqslant 1,$ and
$\kappa \in (0, \frac{\lambda}{1 + \lambda})$. Let $a$ be the constant from Lemma~\ref{lmaSemiMixedWorkDone}, and $c_0$ the constant from Lemma~\ref{lmaMixNorm} with $n = 1$.
Choose
$$
\delta = \paren[\Big]{ c_0 \norm{\theta(t)}_{H^{-1} } }^{\frac{2}{d+2}}.
$$
Then certainly $\norm{\theta(t)}_{H^{-1}} \leqslant \delta^{d/2+1}/ c_0$ and by Lemma~\ref{lmaMixNorm} the super level set $\{ \theta(t) > \lambda \}$ is $\delta$-semi-mixed.
Now, since $\theta$ satisfies the transport equation~\eqref{eq:1}, we know
$
\set{ \theta(t) > \lambda } = \Psi_t(A_\lambda)
$,
where $\Psi$ is the flow of the vector field $u$.
Thus, Lemma~\ref{lmaSemiMixedWorkDone} now implies
$$
\delta \geqslant \frac{r_0}{2} \exp \paren[\Big]{ \frac{-a}{m(A_\lambda)^{1/p} } \int_0^t \norm{\nabla u}_{L^p} }.
$$
Consequently
$$
\norm{\theta(t)}_{H^{-1}}
= \frac{\delta^{d/2+1} }{c_0}
\geqslant
{ \frac{r_0^{d/2+1}}{c_0 2^{d/2+1}} } \exp \paren[\Big]{ \frac{-d a}{m(A_\lambda)^{1/p} } \int_0^t \norm{\nabla u}_{L^p} },
$$
finishing the proof.
\end{proof}
\section{Proofs of Lemmas.}\label{sxnLemmaProofs}
We devote this section to the proofs of Lemmas~\ref{lmaMixNorm} and~\ref{lmaSemiMixedWorkDone}.
\begin{proof}[Proof of Lemma~\ref{lmaMixNorm}]
Suppose for the sake of contradiction that $A_\lambda$ is not $\delta$-semi-mixed.
Then by definition, there exists $x \in \mathbb T^d$ such that
\begin{equation}\label{eqnDual1}
m(A_\lambda \cap B(x,\delta))
\geqslant (1-\kappa) m(B(x,\delta))
=(1-\kappa)\pi(d) \delta^d.
\end{equation}
Here $\pi(d)$ is the volume of $d$-dimensional unit ball.
By duality
\begin{equation}\label{eqnDual2}
\norm{\theta}_{H^{-n}}
= \sup_{f\in H^{n}} \frac{1}{\norm{f}_{H^{n}} } \abs[\Big]{\int_{\mathbb{T}^d} \theta(x)f(x) \, dx }.
\end{equation}
We choose $f \in H^{n}$ to be a function which is identically equal to $1$ in $B(x,\delta)$, and which vanishes outside $B(x,(1+\varepsilon)\delta)$ for some small $\varepsilon > 0$.
A direct calculation shows that we can arrange
$$
\norm{f}_{H^{n}}\leqslant c_1(d)\cdot \varepsilon^{-n + \frac{1}{2}}\cdot \delta^{-n + \frac{d}{2}},
$$
for some (explicit) constant $c_1$ depending only on the dimension.
On the other hand using~\eqref{eqnDual1} gives
\begin{equation}\label{eq:kap}
\int_{\mathbb{T}^d}\theta(x) f(x) dx
\geqslant \pi(d) \norm{\theta}_{L^\infty} \delta^d \paren[\big]{ (1-\kappa) \lambda - \kappa - c_2(d)\varepsilon},
\end{equation}
for some (explicit) dimensional constant $c_2(d)$.
Choosing
$
\varepsilon = \frac{\lambda - (1+\lambda)\kappa}{2c_2(d)}
$
and using~\eqref{eqnDual2} we obtain
$$
\norm{\theta}_{H^{-n}}\geqslant \frac{\norm{\theta}_{L^\infty} \delta^{-n + \frac{d}{2}} }{c_0(d, \kappa, \lambda, n)}
$$
as desired.
\end{proof}
\begin{remark*}
Observe $c_0 = c_0'(d, n) (\lambda - (1 + \lambda) \kappa )^{n - \frac{1}{2}}$.
\end{remark*}
Now we turn to Lemma~\ref{lmaSemiMixedWorkDone}.
For this, we need a result from~\cite{bblCrippaDeLellis08} which controls the Lipshitz constant of the Lagrangian map except on a set of small measure.
\begin{proposition}[Crippa DeLellis '08~\cite{bblCrippaDeLellis08}]\label{ppnLipConstant}
Let $\Psi(t,x)$ be the flow map of the (incompressible) vector field $u$.
For every $p > 1$, $\eta>0$, there exists a set $E \subset \mathbb T^d$ and a constant $c = c(d, p)$ such that $m(E^c)\leqslant \eta$ and for any $t \geqslant 0$
we have
\begin{equation}\label{eqnLipConstant}
\Lip(\Psi^{-1}(t,\cdot)|_{E^c})
\leqslant \exp \paren[\Big]{ \frac{c}{\eta^{\frac{1}{p}} } \int_0^t \norm{\nabla u(s)}_{L^p} \, ds }.
\end{equation}
Here
$$
\Lip(\Psi^{-1}(t,\cdot)|_{E^c})
\defeq \sup_{\substack{x, y \in E^c\\x \neq y}} \frac{\abs{\Psi^{-1}(t, x) - \Psi^{-1}(t, y)}}{\abs{x - y}}
$$
is the Lipshitz constant of $\Psi^{-1}$ on $E^c$.
\end{proposition}
The proof of Proposition~\ref{ppnLipConstant} is built
upon the simple observation~\cite{bblAmbrosioLecumberryStefania05} that for a passive scalar $\theta(x,t)$ and smooth advecting velocity $u$ one has the inequality
\begin{equation}\label{logineq11}
\int \log_+ \abs[\big]{ \nabla \theta\paren[\big]{ t, \Psi(t,x)} } \,dx
\leqslant \int_0^t \int \abs[\big]{ \nabla u\paren[\big]{ t, \Psi(t,x) } }\,dx.
\end{equation}
This can be proved by an elementary calculation.
In fact, even the point wise bound
\[ D \log |\nabla \theta| \leqslant |\nabla u| \]
is true, where $D = \partial_t + u \cdot \nabla$ is the material derivative.
In the form \eqref{logineq11}, this inequality is not very useful.
But it turns out that the more sophisticated maximal form of this inequality \cite{bblAmbrosioLecumberryStefania05,bblCrippaDeLellis08} can be much more useful and is essentially what leads to Proposition~\ref{ppnLipConstant}.
We refer the reader to~\cite{bblCrippaDeLellis08} for the details of the proof.
We use Proposition~\ref{ppnLipConstant} to prove Lemma~\ref{lmaSemiMixedWorkDone} below.
\begin{proof}[Proof of Lemma~\ref{lmaSemiMixedWorkDone}.]
The main idea behind the proof is as follows:
Suppose first $r_0$ is some large scale at which the set $A$ is ``not semi-mixed''.
Let $T > 0$ be fixed and suppose $\Psi_T(A)$ is $\delta$-semi-mixed for some $\delta < r_0 / 2$.
Since $\Psi_T(A)$ is $\delta$-semi-mixed, there should be many points $\tilde x \in \Psi_T(A)$ and $\tilde y \in \Psi_T(A)^c$ such that $\abs{\tilde x - \tilde y} < \delta$.
Since $A$ is ``not semi-mixed'' to scale $r_0$, there should be many points $\tilde x$ and $\tilde y$ so that we additionally have $\abs{\Psi_T^{-1}(\tilde x) - \Psi_T^{-1}(\tilde y)} \geqslant r_0 / 2$.
This will force the Lipshitz constant of $\Psi_T^{-1}$ to be at least $r_0 / (2 \delta)$ on a set of large measure.
Combined with Proposition~\ref{ppnLipConstant} this will give the desired lower bound on $\int_0^t \norm{\nabla u}_{L^p}$.
We now carry out the details of the above outline.
The first step in the proof is to choose the length scale $r_0$.
Let $\varepsilon = \varepsilon( \kappa, d )$ be a small constant to be chosen later.
We claim that there exists a natural number $l$ and finitely many disjoint balls $B(x_1, r_1)$, \dots, $B(x_l, r_l)$ such that
\begin{equation}\label{eqnDisjBalls}
m\paren[\Big]{ \bigcup_{i=1}^l B(x_i, r_i) } \geqslant \frac{m(A)}{2 \cdot 3^d}
\quad\text{and}\quad
\frac{ m( A \cap B(x_j, r_j) ) }{m(B(x_j, r_j))} > 1 - \varepsilon
\end{equation}
for every $j \in \{1, \dots l\}$.
To see this, note that the metric density of $A$ is $1$ almost surely in $A$.
Thus, removing a set of measure $0$ from $A$ if necessary, we know that for every $x \in A$ there exists an $r \in (0, 1]$ such that
$$
\frac{ m(A \cap B(x, r) ) }{m(B(x, r))} > 1 - \varepsilon.
$$
Now choose $K \subset A$ compact with $m(K) > m(A) / 2$.
Since the above collection of balls is certainly a cover of $K$, we pass to a finite sub-cover.
Applying Vitali's lemma to this sub-cover we obtain a disjoint sub-family $\{B(x_i, r_i) \mid i = 1, \dots, l \}$ with $m(\cup B(x_i, r_i) ) \geqslant m(K) / 3^d$.
This immediately implies~\eqref{eqnDisjBalls}.
For convenience let $B_i = B(x_i, r_i)$, and choose $r_0 = \min\{r_1, \dots, r_l\}$.
Now let $\eta > 0$ be another small parameter that will be chosen later.
By Proposition~\ref{ppnLipConstant} we know that there exists a set $E$ with $m(E)\leqslant\eta$ such that the inequality~\eqref{eqnLipConstant} holds.
Define the set
\begin{equation}
F = \set[\big]{ x \in \mathbb T^d \;\big|\; \frac{ m( B( x ,\delta) \cap E)}{m( B(x,\delta))} > \frac{\kappa}{2} }
\end{equation}
Clearly $F \subset \{ M \Chi*E > \kappa / 2 \}$, where $M \Chi*E$ is the maximal function of $\Chi*E$.
Consequently,
$$
m( F )
\leqslant m\paren[\big]{ \{ M \Chi*E > \frac{\kappa }{2} \} }
\leqslant \frac{2 c_1}{\kappa} m(E)
$$
for some explicit constant $c_1 = c_1(d)$.
(It is well known that $c_1 = 3^d$ will suffice.)
Since $\Psi_T$ is measure preserving we know $m(\Psi_T^{-1} (E \cup F)) \leqslant (1 + 2 c_1 / \kappa) \eta$.
Thus choosing
$$
\eta = \frac{\kappa}{\kappa + 2 c_1} \paren[\Big]{ \frac{1}{4^d} - \varepsilon } \sum_{i = 1}^l m\paren[\big]{ B_i }
$$
will guarantee
$$
m( \Psi_T^{-1}( E \cup F) )
\leqslant \paren[\Big]{ \frac{1}{4^d} - \varepsilon } \sum_{i = 1}^l m\paren[\big]{ B_i }.
$$
This implies that for some $i_0 \leqslant l$ we must have
\begin{equation}\label{eqnCovering1}
m( (B_{i_0} \cap A) - \Psi_T^{-1}(E \cup F) )
\geqslant \paren[\Big]{ 1 - \frac{1}{4^d} } m\paren[\big]{ B_{i_0} }.
\end{equation}
By reordering, we may without loss of generality assume that $i_0 = 1$.
Consequently, for
$$
C_1 = \set[\Big]{ x \in (B_1 \cap A) - \Psi_T^{-1}( E \cup F) \;\Big|\; d( x, B_1^c ) > \frac{r_1}{2}}.
$$
equation~\eqref{eqnCovering1} implies
$$
m( C_1 ) \geqslant \paren[\Big]{ \frac{1}{2^d} - \frac{1}{4^d} } m( B_1 ).
$$
Now, from the collection of open balls $\{ B( \tilde x, \delta ) \mid \tilde x \in \Psi_T(C_1) \}$ the Vitali covering lemma allows us to extract a finite disjoint collection $B( \tilde x_1, \delta )$, \dots, $B( \tilde x_n, \delta )$ such that
$$
m\paren[\Big]{ \bigcup_1^n B( \tilde x_i, \delta ) } \geqslant \frac{m( C_1 )}{5^d}.
$$
Our goal is to find $\tilde y$ such that $\tilde y \in B( \tilde x_i, \delta ) - E$ for some $i$, and $\abs{\Psi_T^{-1} \tilde y - \Psi_T^{-1} x} > r_1 / 2$.
For convenience set $\tilde B_i = B( \tilde x_i, \delta )$.
Since $\Psi_T(A)$ is $\delta$-semi-mixed and $\tilde x_i \not\in F$ we have
\begin{equation}\label{eqnCovering2}
m( \Psi_T(A) \cap \tilde B_i ) \leqslant (1-\kappa) m( \tilde B_i )
\quad\text{and}\quad
m( E \cap \tilde B_i ) \leqslant \frac{\kappa}{2} m( \tilde B_i ).
\end{equation}
Also, since $\Psi_T$ is measure preserving and by the definition of $B_1$ we see
\begin{equation}\label{eqnCovering3}
m\paren[\Big]{ \bigcup_{i=1}^n \tilde B_i \cap \Psi_T\paren[\big]{ B_1 - A } }
\leqslant m( B_1 - A)
< \varepsilon m(B_1)
\end{equation}
Using the fact that $\{ \tilde B_i \}$ are all disjoint, summing~\eqref{eqnCovering2} and using~\eqref{eqnCovering3} gives
\begin{multline*}
m( \bigcup_{i=1}^n \tilde B_i \cap E \cap \Psi_T (B_1) )
< \paren[\big]{ 1 - \frac{\kappa}{2} } \sum_{i = 1}^n m( \tilde B_i ) + \varepsilon m(B_1)\\
\leqslant
\paren[\Big]{ 1 - \frac{\kappa}{2} + \varepsilon 5^d \paren[\Big]{ \frac{1}{2^d} - \frac{1}{4^d} }^{-1} } \sum_{i = 1}^n m( \tilde B_i ).
\end{multline*}
Thus choosing
$$
\varepsilon < \frac{\kappa}{2 \cdot 5^d} \paren[\Big]{ \frac{1}{2^d} - \frac{1}{4^d} }
$$
will guarantee
$$
m\paren[\big]{ \bigcup_{i=1}^n \tilde B_i \cap E \cap \Psi_T (B_1) }
< m\paren[\big]{ \bigcup_{i=1}^n \tilde B_i }
$$
This in turn will guarantee that for some $i$ we can find $\tilde y \in \tilde B_{i} - E - \Psi_T(B_1)$.
Now observe that
$$
\tilde y, \tilde x_i \not\in E,
\qquad
\abs{\tilde y - \tilde x_i} < \delta,
\qquad\text{and}\qquad
\abs{\Psi_T^{-1}( \tilde y ) - \Psi_T^{-1} (\tilde x_i ) } > \frac{r_1}{2}.
$$
The last inequality above follows because $\Psi_T^{-1}(\tilde x_i) \in C_1$ and $\Psi_T^{-1}( \tilde y ) \not\in B_1$.
This forces
$$
\Lip( \Psi_T^{-1} |_{E^c} )
\geqslant \frac{ \abs{\Psi_T^{-1}(\tilde y) - \Psi_T^{-1}(\tilde x_i)} }{\abs{\tilde y - \tilde x_i}}
> \frac{r_1}{2 \delta} \geqslant \frac{r_0}{2 \delta}.
$$
Now using~\eqref{eqnLipConstant}, and letting $a = a(d, \kappa, p)$ denote a constant that changes from line to line we obtain
\begin{equation}\label{eqnLipBd1}
\int_0^T \norm{\nabla u(t)}_{L^p} \, dt
\geqslant \frac{\eta^{\frac{1}{p}} }{a} \abs[\big]{ \log \paren[\big]{\frac{r_0}{2\delta}} }.
\end{equation}
Observe finally that
$$
\eta =
c_2 m\paren[\big]{ \bigcup_{i =1}^l B_i }
\geqslant \frac{c_2 m(A)}{2 \cdot 3^d}
$$
for some explicit constant $c_2 = c_2( d, \kappa )$.
Consequently~\eqref{eqnLipBd1} reduces to
$$
\int_0^T \norm{\nabla u(t)}_{L^p} \, dt
\geqslant \frac{m(A)^{\frac{1}{p}} }{a} \abs[\big]{ \log \paren[\big]{\frac{r_0}{2\delta}} },
$$
as desired.
\end{proof}
\section{Numerical results}\label{sxnNumerics}
In this section we present numerical results illustrating how the exponential decay rate varies with the initial data.
For numerical purposes we work on the $1$-periodic torus.
Given a parameter $a$, we define the initial data $\theta_0 = \theta_0' / \norm{\theta_0'}_{L^2}$ where
$$
\theta_0'(x, y) = \begin{dcases}
\sin\paren[\big]{ \frac{2 \pi x}{a} } \sin\paren[\big]{ \frac{2 \pi (y + \frac{a}{8}) }{a} }
& \text{for } 0 < x < \frac{a}{2} ~\text{and}~ \frac{-a}{8} < y < \frac{a}{2} - \frac{a}{8}
\\
\sin\paren[\big]{ \frac{2 \pi x}{a} } \sin\paren[\big]{ \frac{2 \pi (y - \frac{a}{8}) }{a} }
& \text{for } \frac{a}{2} < x < a ~\text{and}~ \frac{a}{8} < y < \frac{a}{2} + \frac{a}{8}\\
0 & \text{otherwise}.
\end{dcases}
$$
A figure of this is shown in~\ref{fgrSolPlots}(a).
We do not know which velocity field achieves the lower bound~\eqref{eq:solve}. However the steepest descent method introduced in~\cite{bblLinThiffeaultDoering11} provides us with a reasonable candidate.
Explicitly, their formula gives
\begin{equation}\label{eqnVelocity}
u=\frac{-\Delta^{-1}P(\theta\nabla^{-1}\theta)}{\norm{\nabla^{-1}P(\theta\nabla^{-1}\theta)}_{L^2}},
\end{equation}
where $P$ is the Leray-Hodge projection onto divergence free vector fields.
This can be derived by multiplying both sides of~\eqref{eq:1} by ${\Delta}^{-1}\theta$ and integrating by parts.
Using a pseudo-spectral method
\footnote{The code and more figures can be downloaded from~\cite{bblWebsite}.}
retaining $768$ Fourier modes in each variable we perform a numerical simulation of~\eqref{eq:1} with the initial data obtained by varying the parameter $a$ over the set $\{ 6/12, 7/12, \dots, 11/12\}$, and the velocity obtained dynamically using~\eqref{eqnVelocity}.
Plots of our solutions at various times (for $a = 11/12$) are shown in Figure~\ref{fgrSolPlots}.
\begin{figure}
\caption{Solution plots at various times for $a=11/12$.}
\label{fgrSolPlots}
\end{figure}
Figure~\ref{fgrNumerics}(a) shows graphs of $\norm{\theta(t)}_{H^{-1}}$ vs $t$ as the parameter $a$ varies over the set $\{6/12, \dots, 11/12\}$.
Figure~\ref{fgrNumerics}(b) shows graphs of $\ln \norm{\theta(t)}_{H^{-1}}$ vs $t$ for the same values of $a$.
Following a short initial ``settling down'' period, the log plots in Figure~\ref{fgrNumerics}(b) are essentially linear indicating a exponential in time decay of $\norm{\theta_0}_{H^{-1}}$.
\begin{figure}
\caption{The mix norm of the scalar density (Figures (a) \& (b)), and the negative reciprocal of the exponential decay rate vs $a$ as $a$ varies over $\{6/12, \dots, 11/12\}$ (Figure (c)).}
\label{fgrNumerics}
\end{figure}
We fit each of the log plots in Figure~\ref{fgrNumerics}(b) to a straight line, and plot the negative reciprocal of the slope vs $a$ in Figure~\ref{fgrNumerics}(c).
Since $m( \supp(\theta_0) ) = O( a^2 )$, Theorem~\ref{thmMain} predicts this graph to be linear as a function of $a$.
This is in good agreement with the observed numerics.
\section{A Scaling Argument and Universal Decay Rates.}\label{sxnScaling}
Physical intuition suggests that the exponential decay rate in~\eqref{eq:solve} should have some dependence on the size of support of $\theta_0$.
As we discussed in the introduction, the mixing process can spread around the compactly supported initial data. Whether this has to happen in the mixing process, and whether this
leads to slowdown in the mixing rate are very interesting open questions. In this section we show that exponential in time lower bound on the decay of the mix norm with the rate in the exponential independent of the initial data would have interesting consequences for mixing in domains with no slip boundaries.
\begin{proposition}\label{boundaries}
Let $I = (0, \ell)^d$ be a cube in $\mathbb R^d$.
Suppose that there exist $k\in \mathbb R$, $q \in [1, \infty]$ and $c_0 > 0$
such that
\begin{equation}\label{eqnULowerBd}
\norm{\theta(t)}_{H^{-1}}
\geqslant
B(\theta_0) \exp\paren[\Big]{ \frac{-c_0}{\ell^{d/p}} \int_0^t \norm{\nabla u}_{L^p} }
\end{equation}
for some $p \in [1, d]$, all incompressible $u$ which vanish on $\partial I$, and all initial data $\theta_0 \in C_c^\infty(I).$
Assume that there exists $\gamma \in \mathbb R$ such that the pre-factor $B(\theta_0)$ satisfies
$$
B(A\theta_0)= AB(\theta_0)
\quad\text{and}\quad
B(\theta_{0,a})= a^{-\gamma} B(\theta_0),
$$
where $\theta_{0,a}(x) = \theta_0(x/a)$ for $x \in (0,a)^d$ and $\theta_{0,a}(x) = 0$ otherwise.
Then, for any mean zero $\theta_0 \in C_c^\infty(I)$ and any smooth velocity field $u$ such that
$$
\limsup_{t \to \infty} \frac{1}{t} \int_0^t \norm{\nabla u}_{L^p} < \infty,
\quad \nabla \cdot u = 0,
\quad\text{and}\quad u = 0 \text{ on } \partial I,
$$
the decay of $\norm{\theta(t)}_{H^{-1}}$ as $t \to \infty$ is bounded below by an algebraic function of time.
\end{proposition}
\begin{proof}
We prove this using an elementary scaling argument.
Without loss of generality, assume $\ell = 1$.
Let $\theta_0 \in C_c^\infty$ have zero mean and let $f(t) = \norm{\theta(t)}_{H^{-1}}$.
Our aim is to show that for $t$ large, $f$ decays algebraically with respect to $t$.
Let $a > 0$, and define
$$
I_a = (0, a)^d,\quad
\eta( x, t ) = \Chi*{I_a} \theta\paren[\big]{ x/a, t/a },
\quad
v(x, t) = \Chi*{I_a} u\paren[\big]{ x/a, t/a }.
$$
Then $(\eta, v)$ is a solution of~\eqref{eq:1} on the \emph{unscaled} cube $I$.
Since $\theta_0$ is compactly supported in $I$ and $u = 0$ on $\partial I$, we see $\theta(t)$ remains compactly supported in $I_a$ for all $t >0$.
By our assumption,
$$
B(\eta_0) = a^{-\gamma}B(\theta_0)
\quad\text{and}\quad
\norm{\nabla v(t)}_{L^p} = a^{\frac{d}{p} - 1} \norm{\nabla u\paren[\big]{t/a } }_{L^p}.
$$
Thus the assumed lower bound~\eqref{eqnULowerBd} gives
\begin{multline*}
a^{d/2+1} f\paren[\big]{t/a}
= \norm{\eta(t)}_{H^{-1}}
\geqslant
B(\eta_0)
\exp\paren[\Big]{ -c_0 \int_0^t \norm{\nabla v}_{L^p} }\\
=
a^{-\gamma}B(\theta_0)
\exp\paren[\Big]{ -c_0 a^{\frac{d}{p} - 1} \int_0^t \norm{\nabla u\paren[\big]{ s/a }}_{L^p} \, ds }.
\end{multline*}
The first equality above follows by duality and scaling.
Hence, taking $t'=t/a$ gives
\[ f(t') \geqslant a^{-N}B(\theta_0) \exp \left(-c_0 a^{d/p} \int_0^{t'} \norm{\nabla u(s)}_{L^p} \, ds \right), \]
where $N= d/2+1+\gamma.$
This bound has to be true for every $a>0.$
Maximizing the right hand side in $a$ (and changing $t'$ to $t$), we arrive at an algebraic lower bound \begin{gather*}
f(t) \geqslant C \left( \frac{N}{\int_0^t \|\nabla u\|_{L^p}\,ds} \right)^{-pN/d}.
\qedhere \end{gather*}
\end{proof}
\section{Acknowledgements.}
The authors would like to thank Charlie Doering for introducing us to this problem and many helpful discussions.
\end{document} | arXiv |
When physicists say they have discovered a "particle" that is it's own anti-matter what does this mean?
Not being a physicist but liking to keep up with the news I read that new particles were discovered that are their own anti-particles. I can see that perhaps we can discover a particle that has an anti particle as in the case of the electrons and when they come in contact we get all energy from the "reaction". I believe they are called positrons , being the anti particle to the electron. But how can ANY particle be its own antiparticle, if that was the case would they not annihilate each other and you would never even know they ever existed since as soon as one was in the process of being created, and it is its own antiparticle it would go into non existence as fast as it came into existence and you would never know which particle it was that was its own antiparticle.
particle-physics antimatter
SedumjoySedumjoy
Just because two particles are antiparticles of each other does not mean they have to annihilate. For example, an electron and a positron can form an atom called positronium without immediately annihilating. That's because in order to collide, they need to have the right combination of momenta and position.
So what makes a particle an antiparticle? An antiparticle is simply a particle with the same properties but some of its quantum numbers are opposite its normal particle equivalent. For example, an electron has a mass of $511~\mathrm{keV}/\mathrm{c}^2$, a spin of $1/2~\hbar$, and a charge of $-\mathrm{e}$ while the positron has the same mass and spin, but a charge of $+\mathrm{e}$.
How can a particle be its own antiparticle? Simple, all the quantum numbers that change from particle to antiparticle are 0. The example you would be most familiar with is the photon. Photons are massless, have a spin of $1~\hbar$, but a charge of $0$. If you tried to create an anti-photon, it would just be a photon.
So why don't particles that are their own antiparticles decay automatically? Because in order for a particle-antiparticle pair to annihilate, you need two particles. If only one particle is created, it can't annihilate with itself.
Now, you mentioned "new particles" that are their own antiparticle. Any new particles being discovered right now are not fundamental particles like electrons or photons (unless you're talking about the Higgs, in which case you're a few years behind), they're composite particles, made up of combinations of fundamental particles. The best examples of such particles are the neutron and the proton. They are part of a class of particle called baryons, made up of three quarks. There is another class of particles, called mesons, which are composed of a quark and an antiquark. The quark and antiquark that make up a meson don't have to be of the same type (e.g. the $\pi^+$ meson is composed of an up quark and a down antiquark), so they won't annihilate each other. But some mesons are (e.g. the $\pi^0$ meson is a mix of up-antiup quarks and down-antidown quarks). Like the positronium example above, they don't immediately annihilate each other because they are in a bound state which require specific types of interactions to annihilate.
After looking it up, I've identified the new particle you were talking about (press release, Science paper, arXiv).
What's happening here is a completely different phenomenon. In condensed matter physics, they study a variety of materials. When they study those materials, they will see local excitations that behave like other objects in physics. I've seen condensed matter systems recreate black holes, create magnetic monopoles, and mimic subatomic particles. When these excitations mimic particles, they are called quasiparticles.
Among those materials are superconductors and topological insulators. A superconductor is a material that has absolutely no electrical resistance. A topological insulator is a material that either is an insulator in its volume and a conductor on its surface (for 3D materials) or an insulator on its surface and conductor on its edges (for 2D materials). What they did was take a sheet of superconductor and a sheet of topological insulator, modified the topological insulator a bit so it showed some magnetic effects, and then sandwiched the two together. This created a system where quasiparticles moved along the edges in a way that simulated fermions. These fermions were their own antiparticles.
Johnathan GrossJohnathan Gross
$\begingroup$ I was talking about "Stanford News" apparently they have evidence of the "majorana fermion", but as you pointed out it's not really a new particle , it's a fermion ? The Standard Model predicts 24 type of fermions so syas wiki but there are three classes , massive , massless and having its own antiparticle , but is this one of those types? and which types are quarks. Gosh I need a flow chart....but you answered my question...the "a particle cannot annihilate with itself. " Must be a conservation law thing? $\endgroup$ – Sedumjoy Jul 24 '17 at 3:24
$\begingroup$ Fermions are any particle with half-integer spin. Quarks, leptons (like electrons and neutrinos), baryons are all fermions. $\endgroup$ – Johnathan Gross Jul 24 '17 at 3:56
$\begingroup$ I just looked up the article and it's not really a particle. It's a quasiparticle. What they did was they took two sheets of material with different properties and stuck them together. Then they ran a magnet over it and that created excitations in the material that behaved like Majorana fermions. The excitations have no charge and have a spin of 1/2. $\endgroup$ – Johnathan Gross Jul 24 '17 at 4:09
$\begingroup$ To any condensed matter physicists out there, feel free to correct any mistakes I made in my explanation in my edit. $\endgroup$ – Johnathan Gross Jul 24 '17 at 5:36
$\begingroup$ Particles are real objects that exist. Quasiparticles are objects that mimic particles in materials. The best example would be electron holes in a conductor. If you ionize an atom in a conductor, it creates a free electron and a hole where the electron used to be. The hole behaves almost exactly like a positron. $\endgroup$ – Johnathan Gross Jul 24 '17 at 14:36
The 'charge conjugation operator' $C$ is defined so that it converts elementary particles into their antiparticles, so the charge conjugation operator acting on a state representing an electron give a state representing a positron: $$ C |e^-\rangle = |e^+\rangle \;, $$ and the operator acting on a anti-strange quark give a strange quark $$ C |\bar{s}\rangle = |s\rangle \;, $$ and so on.
But not all particles are elementary: the proton, for instance, has a valance make-up of two up quarks and a down quark $| p \rangle = | uud \rangle$. 1
What does charge conjugation do to a particle like that? It changes all the elementary particles making up the compound particle to their anti-particles.
Now consider the group of particles known as 'mesons'. They have a valance structure consisting of one quark and one anti-quark. For instance a positive pion is $$ |\pi^+\rangle = | u\bar{d} \rangle \;, $$ and the charge conjugation operator acting on one give you \begin{align} C|\pi^+\rangle &= C| u\bar{d} \rangle\\ &= | \bar{u}d \rangle\\ &= |\pi^-\rangle \;, \end{align} a negative pion.
Finally, there are some meson's who valence content is made up of a quark and an anti-quark of the same basic kind. Something like $|\pi_u^0 \rangle = | u \bar{u} \rangle$.2 When we let charge conjugation act on that we get \begin{align} C|\pi_u^0\rangle &= C| u\bar{u} \rangle\\ &= | \bar{u}u \rangle\\ &= |\pi_u^0\rangle \;, \end{align} because the ordering of the quark labels doesn't matter here. We see that the anti-particle of the $\pi_u^0$ is the same as the original particle.
1 I am going to steadfastly ignore the quark-gluon sea in this discussion.
2 Because of the nearly correct symmetry known as iso-spin a real neutral pion is actually $$ | \pi^0 \rangle = \frac{1}{\sqrt{2}}\left( | u \bar{u} \rangle - | d \bar{d} \rangle\right) \;,$$ but this complication doesn't affect the discussion or the conclusion other than making that math longer to write out.
dmckee♦dmckee
$\begingroup$ Does the math show that a particle cannot be an antiparticle of itself??? $\endgroup$ – Sedumjoy Jul 25 '17 at 1:20
$\begingroup$ No. It gives an explicit example of a particle that is it's own antiparticle. Well, the example in the text is hypothetical, but the proper $\pi^0$ from the footnote also has the same property. Though it seems there has been a copy-n-paste-o in there for nine hours which I have fixed now. $\endgroup$ – dmckee♦ Jul 25 '17 at 1:28
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Can anti-Higgs particles have formed before any baryonic matter existed or is the Higgs boson always its own anti-particle? | CommonCrawl |
Aerodynamic drag of a transiting sphere by large-scale tomographic-PIV
W. Terra1,
A. Sciacchitano1 &
F. Scarano1
Experiments in Fluids volume 58, Article number: 83 (2017) Cite this article
A method is introduced to measure the aerodynamic drag of moving objects such as ground vehicles or athletes in speed sports. Experiments are conducted as proof-of-concept that yield the aerodynamic drag of a sphere towed through a square duct in stagnant air. The drag force is evaluated using large-scale tomographic PIV and invoking the time-average momentum equation within a control volume in a frame of reference moving with the object. The sphere with 0.1 m diameter moves at a velocity of 1.45 m/s, corresponding to a Reynolds number of 10,000. The measurements in the wake of the sphere are conducted at a rate of 500 Hz within a thin volume of approximately 3 × 40 × 40 cubic centimeters. Neutrally buoyant helium-filled soap bubbles are used as flow tracers. The terms composing the drag are related to the flow momentum, the pressure and the velocity fluctuations and they are separately evaluated. The momentum and pressure terms dominate the momentum budget in the near wake up to 1.3 diameters downstream of the model. The pressure term decays rapidly and vanishes within 5 diameters. The term due to velocity fluctuations contributes up to 10% to the drag. The measurements yield a relatively constant value of the drag coefficient starting from 2 diameters downstream of the sphere. At 7 diameters the measurement interval terminates due to the finite length of the duct. Error sources that need to be accounted for are the sphere support wake and blockage effects. The above findings can provide practical criteria for the drag evaluation of generic bluff objects with this measurement technique.
Aerodynamic forces result from the relative motion of an object in air. These forces are relevant for a multitude of applications, such as aircraft systems, wind turbines and ground transportation. Aerodynamic forces are typically measured by means of wind tunnel experiments (Bacon and Reid 1924; Zdravkovich 1990 among others), where the model is immersed in a uniform air stream within accurately controlled conditions. In other cases, aerodynamic studies are conducted with the object moving in a quiescent fluid. This approach is needed for instance to study the flow behind accelerating objects (Coutanceau and Bouard 1977a) or the development of the wake of an aircraft over a large distance (Scarano et al. 2002; Von Carmer et al. 2008). Furthermore, instructive flow visualizations have been obtained of high-speed flight of a bullet in stagnant air (van Dyke 1982). Aerodynamic investigations for ground vehicles and speed sports such as cycling, skating and skiing require specific arrangements of the wind tunnel (moving floor, model supports) to achieve realistic conditions. Yet, experiments that simulate specific conditions such as curved trajectories, acceleration, or with the athlete in motion remain challenging. In contrast, experiments conducted with the object moving in the laboratory frame of reference are relatively easy to realize and can mimic more closely real-life conditions.
An aspect that has not been sufficiently covered from towed experiments, is the quantitative analysis of the flow field, such to be able to evaluate the aerodynamic forces, the drag in particular. The present work focuses on the latter question to explore the viability of drag estimation from aerodynamic data obtained during towed experiments. The results are intended as a proof-of-concept for applications in automotive industry and speed sports in particular.
In professional cycling the aerodynamic drag is routinely estimated using the mechanical power generated by the rider. These estimates, however, include other forces due to the friction between wheels and ground and other mechanical resistance. Therefore, assumptions need to be made to extract the aerodynamic drag from the total resistance (Grappe et al. 1997). Similar assumptions are made for ground vehicle testing that relies upon constant speed torque measurement (Fontaras et al. 2014) and coast down tests (Howell et al. 2002) of cars and trucks. Instrumentation of a vehicle with pressure taps to measure the overall pressure distribution requires an unrealistic multitude of sensing points. For an athlete, such approach is practically impossible.
Aerodynamic drag measurements based on wake survey are potentially of more general applicability as they do not dependent upon assumptions as discussed above and do not require extensive instrumentation of the model. The approach deals with the determination of the momentum deficit of the flow past the object. The airflow velocity is measured in the wake and the aerodynamic force is obtained from the velocity deficit compared to the incoming stream. This approach has been long practiced in wind tunnels by means of wake rakes (multi-hole Pitot probes; Goett 1939; Guglielmo and Selig 1996). In contrast to using force balances and pressure taps, a wake rake offers the advantage of yielding additional whole-field velocity information in the model's wake. The identification of the streamwise vortical structures behind cars, for instance, has been among the most significant insights in vehicle aerodynamics in the past decades (Hucho and Sovran 1993). Furthermore, relating changes in the leg orientation of a cyclist to the wake flow topology recently provided new insights for cycling aerodynamics (e.g. Crouch et al. 2014).
Particle image velocimetry (PIV) has been demonstrated as a valid instrument to replace Pitot probes for a stationary object (Kurtulus et al. 2007; Van Oudheusden 2013) as well as for moving objects like rotor blades (Ragni et al. 2011, 2012). More recently, Neeteson et al. (2016) have extended the approach to estimate the drag of a sphere freely falling in water trying to reconstruct the pressure distribution all over its surface.
The present study aim at measuring the aerodynamic drag of transiting objects using tomographic PIV measurements in the wake of the model and invoking the conservation of momentum in a control volume. Because the use of micro-sized droplets as tracers is limited to a relatively small measurement domain (Scarano 2013), using helium-filled soap bubbles (HFSB) is considered essential for the current experiment. The high light scattering efficiency and tracing fidelity of the HFSB (Bosbach et al. 2009; Scarano et al. 2015, among others) allow measurements on the meter scale (Kühn et al. 2011), preluding upscaling applications towards field tests in the automotive and speed sports.
The detailed goal of the present work is to examine the accuracy of a method that measures the aerodynamic drag of a transiting object and to assess its potential applicability for full-scale conditions. For this demonstration, a sphere is towed within a rectangular channel at a velocity of 1.45 m/s (Re = 10,000). Despite the simple geometry, the flow exhibits an unsteady, turbulent wake with complex vortex interactions (e.g. Achenbach 1972; Brücker 2001) mimicking conditions also encountered behind ground vehicles (Ahmed et al. 1984) or cyclists (Crouch et al. 2014). The experiments determine the time-average drag and a detailed comparison is made with data available from literature. The terms contributing to the overall drag are studied separately to identify a criterion for simplified measurement configurations. Finally, the experimental uncertainty related to the tomographic PIV measurements and disturbing factors such as non-homogeneous initial flow conditions, supporting strut and blockage effects are discussed.
Drag from a control volume approach
The drag force acting on a body moving in a fluid can be derived by application of the conservation of momentum in a control volume containing this body (Anderson 1991), as visualized in Fig. 1. In the incompressible flow regime, the time dependent drag force acting on the body can be written as:
Schematic description of the control volume approach within a wind-tunnel setting (stationary object)
$$D\left( t \right) = \; - \oiiint\nolimits_{v} {\frac{{\partial u}}{{\partial t}}{\rm d}{\text{V}} - \rho {\mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc} }_{{\text{S}}} \left( {\user2{v} \cdot \user2{n}} \right)u{\rm d}S - {\mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc} }_{{\text{S}}} ((p{\bf n} - \tau \cdot \user2{n}){\rm d}S)_{x} },$$
where v is the velocity vector with components [u,v,w] along the coordinate directions [x,y,z] respectively, ρ is the fluid density, p the static pressure and τ the viscous stress tensor. V is the control volume, with S its boundary and n is the outward pointing normal vector.
It can be shown that in most cases of interest the viscous stress is negligible with respect to the other contributions when the control surface is sufficiently far away from the body surface and the boundary is not aligned with the flow shear (Kurtulus et al. 2007). Furthermore, the contour S is defined as the contour abcd, with segments ad and bc approximating streamlines. When the segments ab, ad and bc are taken sufficiently far away from the model, expression (1) can be rewritten such that the only surface integral to be evaluated is that in the wake of the model S wake (segment cd):
$$D\left( t \right) = \; -\oiiint\nolimits_{v} {\frac{{\partial u}}{{\partial t}}{\rm d}{\text{V}} + \rho \mathop {\iint }\nolimits_{{{\text{S}}_{{{\text{wake}}}} }} \left( {U_{\infty } - u} \right)u{\rm d}{\text{S}} + \; \mathop {\iint }\nolimits_{{{\text{S}}_{{{\text{wake}}}} }} \left( {p_{\infty } - p} \right)\;{\rm d}{\text{S}}},$$
where \(U_{\infty }\) and \(p_{\infty }\) are, the freestream velocity and pressure, respectively.
Time-average force on a stationary model
The evaluation of the volume integral on the right hand side of Eq. (2) poses the typical problems due to limited optical access all around the object. Evaluating this integral can be avoided by considering the time-average drag instead of its instantaneous value. When decomposing the equation into the Reynolds average components and averaging both sides of the equation, the time-average drag force is obtained with the sole contribution of surface integrals:
$$\bar{D} = \rho \mathop{\iint }\nolimits_{{{\text{S}}_{{wake}} }} \left( {U_{\infty } - \bar{u}} \right)\bar{u}{\rm d}{\text{S}} - \rho \mathop {\iint }\nolimits_{{{\text{S}}_{{wake}} }} \bar{u}^{{{\prime }2}} {\rm d}{\text{S}} + \mathop{\iint }\nolimits_{{{\text{S}}_{{wake}} }} \left( {p_{\infty } - \bar{p}} \right){\rm d}{\text{S}},$$
where ū is the time-average streamwise velocity and u' the fluctuating streamwise velocity.
Time-average forces on a moving model at constant velocity
According to the principle of Galilean invariance, Eq. (3) holds in any reference frame moving at constant velocity. Following Ragni et al. (2011), the time-average drag acting on a model at constant speed U M is expressed as:
$$\bar{D} = \rho \mathop{\iint }\nolimits_{{{\text{S}}_{{{\text{wake}}}} (t)}} \left( {U_{\infty } - (\bar{u} - U_{M} )} \right)(\bar{u} - U_{M} ){\text{dS}} - \rho \mathop {\iint }\nolimits_{{{\text{S}}_{{{\text{wake}}}} (t)}} \bar{u}^{{\prime 2}} {\text{dS}} + \mathop{\iint }\nolimits_{{{\text{S}}_{{{\text{wake}}}} (t)}} \left( {p_{\infty } - \bar{p}} \right){\text{dS}},$$
where ū is the streamwise velocity measured in the laboratory frame of reference. In the moving frame of reference the freestream velocity stems from the velocity of the model relative to that of its environment \(U_{{{\text{env}}}}\):
$$U_{\infty } = U_{{\text{env}}} - U_{M}.$$
For perfectly quiescent air U env = 0. During real experiments it may occur that U env is different from zero and cannot be neglected. The value of U env needs therefore to be measured prior to the passage of the model.
Expression (4) allows deriving the time-average drag from velocity and pressure statistics in a cross section of the wake. The time-average pressure is evaluated from the velocity measurements solving the Poisson equation for pressure, according to Van Oudheusden (2013). An accurate evaluation of the pressure in a highly three-dimensional flow requires the estimation of the full velocity gradient tensor components (Ghaemi et al. 2012; Van Oudheusden 2013), which justifies the adoption of the tomographic PIV technique instead of planar stereo-PIV. The measurement volume must be large enough to encompass the full wake and reach the region of steady potential flow at its edges where Dirichlet boundary conditions can be applied based on Bernoulli law. Neumann conditions are prescribed at the inflow and outflow boundaries of the domain. The resulting approach yields the time averaged drag using only the velocity measurements and no other information.
Experimental apparatus and procedure
Measurement system and conditions
A schematic view of the system devised for the experiments is shown in Fig. 2, with a photograph of the setup in Fig. 3. The apparatus consists of a 170 cm long duct with a squared cross section of 50 × 50 cm2, where the sphere model is towed. Part of the duct has transparent walls for optical access (Fig. 3).
Schematic views of the experimental setup
Overview of the experimental system and measurement configuration
The model is a smooth sphere of diameter D = 10 cm towed at a constant speed U M = 1.45 m/s. The model is supported by an aerodynamically shaped strut with 20 mm chord and 3 mm thickness. The strut is 20 cm long and is installed onto a carriage moving on a rail beneath the bottom wall of the duct. The carriage is pulled by a linen wire connected to the shaft of a digitally controlled electric motor (Maxon Motor RE35). Markers on the surface of the sphere track its position during the transit.
The wake behind the sphere starting from rest needs time to reach a fully developed regime. For an impulsively started cylinder at low Reynolds number (Re < 100) it is reported that about 15 cylinder diameters (Coutanceau and Bouard 1977b) are needed before the wake is developed. In the present work a conservative value is taken with the model beginning 25 sphere diameters before the measurement region.
Tomographic system
The time-resolved tomo-PIV measurements are conducted using neutrally buoyant helium-filled soap bubbles (HFSB, 300 μm diameter) as tracer particles produced with an array of ten generators that yield a total of about 300,000 particles per second. The air, helium and soap fluid flow rates are controlled by a fluid supply unit provided by LaVision GmbH. The average time response of such tracer particles is expected to be below 20 s (Scarano et al. 2015). Considering the relevant flow time scale (D/U M = 70 ms) the small value of the Stokes number of the tracers indicates their adequacy for the current experiments.
The illumination is provided by a Quantronix Darwin Duo Nd:YLF laser (2 × 25 mJ/pulse at 1 kHz). The laser beam is first shaped into an elliptical cross section and is then cut into a rectangular one with light stops (Fig. 3). The size of the measurement volume is 3 cm × 40 cm × 40 cm in x, y and z direction, respectively, and is located 80 cm from the exit of the duct (Fig. 2). The tomographic imaging system consists of three Photron Fast CAM SA1 cameras (CMOS, 1024 × 1024 pixels, pixel pitch of 20 μm, 12 bits). Each camera is equipped with a 60 mm Nikkor objective set to f/8. The optical magnification is approximately 0.07. In the present conditions, the seeding concentration is approximately 3 particles/cm3 and the imaging density is 0.04 particles/pixel. PIV acquisition is performed within LaVision Davis 8.3 delivering single-exposure frames at a rate of 500 Hz.
The tunnel entrance and exit are closed to confine the HFSB seeding before the transit of the sphere. The exit is closed by a porous curtain, which maintains the seeding tracers inside, but prevents the buildup of an over-pressure. The HFSB generators are operated for approximately two minutes until the steady-state concentration is reached. Approximately quiescent conditions are achieved 30 s after the generators are switched off. The tunnel entrance wall is then opened and the model is put in motion through the duct. Image recording begins 1 s before the sphere passes through the measurement region and stops when it touches the exit curtain. Figure 4 illustrates the motion of the air by particle streaks at three positions of the model; before entering (top), inside (middle) and after leaving the measurement domain (bottom). The measurements are separated by 70 ms (35 frames) and the streaklines are obtained by averaging ten consecutive frames. A relatively high velocity is observed in the center of the measurement region behind the model (bottom image) with quiescent air conditions at the edges. An animation of raw images of the passing sphere is available at the multimedia store of the publisher (Supplementary material 1). The experiment comprises 35 repeated measurements to form a statistical estimate of the flow properties and the associated aerodynamic drag.
Particle streaks for three positions of the sphere passing through the measurement domain. The particle streaks are obtained by averaging over ten consecutive frames. The sphere positions are separated by time increments of 70 ms
The tomographic-PIV data analysis is performed with the LaVision Davis 8.3 software. Image pre-processing comprises background subtraction and Gaussian smoothing. The volume reconstruction and velocity evaluation follows the sequential MTE-MART algorithm (SMTE, Lynch and Scarano 2015), yielding a discretized object with 1074 × 1050 × 72 voxels. The interrogation is based on spatial cross-correlation volumes of 323 voxels with an overlap of 75%. The resulting velocity vector field has a density of 3 vectors/cm. Figure 5 illustrates the reconstructed intensity distribution along the depth. The SMTE algorithm returns a high reconstruction signal-to-noise ratio, indicating that the cross correlation result is not affected by ghost particles effects (Elsinga et al. 2006).
Illumination distribution along the measurement depth (evaluated over an area of 25 × 25 cm2)
A Galilean transformation of the instantaneous velocity is performed to represent the measurement in a frame of reference consistent with the object moving at a constant velocity U M . In this frame of reference, the drag is evaluated using Eq. (4). The phase-average velocity field in the laboratory frame of reference, \({\bar{\varvec{v}}}^{*}\) is obtained from the 35 repeated measurements yielding the mean velocity and its fluctuations. A three-dimensional spatial representation of the velocity field in the sphere frame of reference, \({\bar{\varvec{v}}}\) is obtained by considering the linear relation between the streamwise coordinate and the time elapsed after the passage of the sphere:
$${\bar{\varvec{v}}}\left( {x,y,z} \right) = {\bar{\varvec{v}}}^{*} (tU_{M} ,y,z),$$
where its origin matches the center of the sphere and both coordinate systems coinciding at t = 0. The procedure encompasses the streamwise direction downstream of the sphere from x/D = 0.5 till x/D = 9.5. This approach is illustrated in Fig. 6.
Illustration of the time-average streamlines after streamwise reconstruction of the velocity field (v) (in the sphere frame of reference) from phase-average time-history velocity fields (v *) (in the laboratory reference frame)
The Poisson equation for pressure is solved prescribing Neumann conditions on all boundaries. The obtained pressure field is scaled afterwards by a constant value, prescribing the freestream pressure at its top, bottom and side boundaries in the spatial range in which the velocity at these boundaries best matches stagnant conditions (x/D > 4), and excluding a small segment (5 cm wide) in the center of the bottom boundary, which is affected by the wake of the strut.
Instantaneous flow field
At a Reynolds number of 10,000 the wake of a sphere is in the unsteady regime, exhibiting vortex shedding and complex vortex interactions (Bakić et al. 2006). A snapshot of the flow structure will typically yield an asymmetric pattern, while the time-average structure is known to be axi-symmetric. Figure 7 shows the instantaneous velocity field in the laboratory frame of reference in the center YZ-plane at four consecutive time instants. A supplementary animation of the time-resolved velocity field is available as multimedia file (Supplementary material 2). Non-dimensional time is defined as \(t^{{\text{*}}} = {\text{}}t{\text{}}U_{M} /D\). Each increment \(\Delta t^{{\text{*}}} = 1\) corresponds to a translation in space of one sphere diameter in negative x-direction. At t * = 0.5, a region of accelerated flow is visible at the periphery of the wake. Furthermore, a negative peak of streamwise velocity is present in the near wake of the sphere. The maximum velocity deficit decays with time, consistently with the observations from past investigations (Jang and Lee 2008; Constantinescu and Squires 2003).
Instantaneous streamwise velocity u* in the YZ-plane at four time instants, t * = 0.5, t * = 1.5, t * = 2.5 and t * = 3.5. The measurement region is cropped to half its size along y and z for readability
Time-average flow structure
The ensemble-statistics yield the time-average velocity field, the fluctuating velocity and time-average pressure distribution. These terms are inspected to understand how the individual terms from Eq. 4 contribute to the aerodynamic drag. Figure 8 illustrates the streamwise velocity distribution in the separated wake (x/D = 0.85, top) and after the flow reattachment (x/D = 3, bottom). The velocity field in the wake is close to the axi-symmetric condition, with some slight deviations due to the supporting strut (5–10% velocity deficit). The latter will be accounted for in the section on drag derivation. Furthermore, the spatial velocity distribution shows a radial velocity directed towards the flow symmetry axis, decreasing in magnitude at increasing distance from the sphere, which is consistent with literature (e.g. Jang and Lee 2008). The expected flow reversal in the center of the wake is also captured in the present measurement (Fig. 8-top-right).
Time-average velocity vectors in the wake of the sphere at x/D = 0.85 (top-left) and x/D = 3 (bottom-left). Streamwise velocity contours (right). A rectangle (bottom-right) indicates the region where the strut drag is estimated
The streamwise velocity contour at x/D = 3 (Fig. 8 bottom-right) shows a slight asymmetry in the spatial velocity distribution outside the wake. At the top of the domain the non-dimensional streamwise velocity is about 0.98, while at the bottom it is 1.01. This asymmetry stems from the flow conditions prior to the transit of the sphere and is ascribed to the motion induced during injection of the HFSB tracers. In the derivation of the aerodynamic drag, the momentum term expresses a deficit in the wake, relative to the fluid momentum prior to the passage of the sphere (Eq. 5). Therefore any residual motion before the passage of the sphere is accounted for the drag evaluation.
The streamwise velocity distribution in the central XY-plane is depicted in Fig. 9. in the spatial range 0.5 < x/D < 3.5. The streamlines pattern yields a reattachment point at about x/D = 1.3, which is consistent with values from literature. Table 1 summarizes the relevant flow properties from other works: Jang and Lee (2008) report a recirculation length, L/D = 1.05 (Re = 11,000) obtained by PIV; Ozgoren et al. (2011) measure a value of about 1.4 (Re = 10,000) by PIV; Bakić et al. (2006) list a value of 1.5 (Re = 51,500) measured by LDV; the numerical work of Yun et al. (2006) and Constantinescu and Squires (2003), both at Re = 10,000, report a significantly longer separated wake with L/D = 1.86 and 2.2, respectively. The variability of the reattachment position can be ascribed to experimental settings, such as the model support (Bakić et al. 2006, use a single rigid support from the back of the sphere; Ozgoren et al. 2011 apply a strut from the top and Jang and Lee suspend the sphere with two thin wires forming an X-shape through the center of the sphere) as well as to the settings of the numerical simulations (i.e. the grid resolution and subgrid-scale modeling for the LES).
Reconstructed spatial distribution of the time-average velocity in the wake of the sphere in the center XY-plane (left) and the center XZ-plane (right). Flow streamlines (top) and contours of streamwise velocity (bottom). The measurement region is cropped along x for readability
Table 1 Comparison between present experimental results and values reported in literature
The maximum reverse flow velocity measured here is −0.52 occurring at x/D = 0.85 approximately on the symmetry axis (Fig. 9, top), which compares fairly well to the value of −0.4 reported by Constantinescu and Squires (2003) and − 0.427 of Bakić et al. (2006). The location of maximum reverse flow, differs from that reported by Constantinescu and Squires (2003) (x/D = 1.41), presumably due to the larger recirculation length.
Asymmetries in the mean flow are observed in both the vertical plane (Fig. 9, left) and the horizontal plane (Fig. 9-right). These may stem from a number of causes: primarily non-homogeneous flow prior to the passage of the model, but also incomplete statistical convergence and the presence of the strut.
The toroidal structure of the recirculating flow when examined in the XY-cross section yields two foci at about x/D = 0.75 and radial distance r/D = 0.45, (Fig. 9, top-left), which closely correspond with the flow topology reported by Ozgoren et al. (2011). The presence of this vortex structure is less evident in the horizontal center-plane (Fig. 9, top-right), which is ascribed to the reduced precision of the PIV measurements in low velocity regions and the limited size of the statistical ensemble. The recirculation region past a sphere features a circular focus, following Jang and Lee (2008), among others. Figure 10 illustrates that the vorticity magnitude isosurface (value selected at 6.7 rad/s) features an axi-symmetric flow structure with the annular shape shortly interrupted at the position of the supporting strut.
Streamlines of time-average velocity in the recirculation region in the wake of the sphere. Isosurface of vorticity magnitude at 6.7 rad/s (green). Velocity vectors are depicted at x/D = 0.5 and x/D = 3.5 in the at y/D = 0
The uncertainty of the time-average velocity, ε v primarily stems from the uncertainty of the measurement of the instantaneous velocity and the size of the ensemble used to estimate the time-average value. Its value decreases with the square root of the number of uncorrelated samples (N = 35 here): \(\varepsilon _{{\mathbf{v}}} = \sigma /\sqrt N\), where σ is the standard deviation of the velocity from the ensemble at the same phase. In the region outside the wake, velocity fluctuations are the smallest, and the standard uncertainty is about 0.3% of the sphere velocity. Inside the wake, the uncertainty attains a maximum level of 6.5% of the sphere velocity (as a result of the velocity fluctuations) at the shear layer locations at x/D = 1. The uncertainty of the mean velocity decreases with the distance from the model as a result of turbulence decay. At x/D = 3 the maximum uncertainty is 3.5% and at x/D = 7 it stays within 1.5%. An overview of the uncertainties of the three velocity components is given in Table 2. These uncertainties provide the baseline information to evaluate the accuracy of the aerodynamic drag estimates addressed in a later section.
Table 2 Uncertainty of the time-average velocity as a percentage of the sphere velocity along the wake
Velocity fluctuations
The distribution of turbulent fluctuations plays a role in the momentum exchange within the wake and needs to be accounted for when evaluating the aerodynamic drag as clear from the formulation in Eq. (4). Figure 11 shows the contour plots of the root-mean-square of the streamwise velocity fluctuations in the center XY-plane (top) and the center XZ-plane (bottom). The velocity fluctuations are rather symmetric in both planes. Their distribution in the XZ-plane compares well to literature data (Jang and Lee 2008; Constantinescu and Squires 2003; Yun et al. 2006), with maxima around x/D = 1 and z/D = ± 0.45, featuring two branches with peak values that diverge from the streamwise axis and decreasing in strength for x/D > 1. The distribution in the XY-plane shows less similarity to literature, likely due to the disturbance of the supporting strut. The local maxima of \(\sqrt {\bar{u}^{{\prime 2}} } /U_{M}\) are between 0.35 and 0.4, within the range reported in literature (Table 1). At x/D = 7, the fluctuations have not decayed yet and exhibit a maximum of about 0.08, indicating that the Reynolds stress term in Eq. (4) still contributes to the drag of the sphere at that distance.
Spatial distribution of streamwise velocity fluctuations in the center XY-plane (top) and the center XZ-plane (bottom)
Pressure reconstruction
The flow past a bluff body generally produces a large base drag resulting from a low-pressure region at the base of the object (Neeteson et al. 2016). After reattachment the pressure recovers towards the free-stream conditions. This variation of the pressure field is investigated to understand its contribution to the aerodynamic drag. Figure 12 depicts the distribution of the mean pressure coefficient in the center XY-plane (top) and the center XZ-plane (bottom). The spatial distribution of the time-average pressure coefficient features a minimum approximately corresponding with the focus of the toroidal recirculation (Fig. 9, bottom-left). At the reattachment point, a region of positive CP is observed. The distribution of pressure shows a slight asymmetry in both planes, but to a lesser extent compared to the velocity fields presented in Figs. 9 and 11. To the best knowledge of the authors, the pressure field in the wake of a sphere has not been evaluated in previous literature, which makes comparisons not possible. The base pressure coefficient estimated from the flow field pressure close to the solid surface is about −0.52 in the present experiment, which is comparatively higher than what is reported by Yun et al. (2006) and Constantinescu and Squires (2003) who report a value of −0.27 and Bakić et al. (2006) with a value of −0.3 at Re = 51,500. Nevertheless, in the current work, the drag is evaluated up to the far wake, where the pressure practically equals that of the quiescent flow and the pressure term is deemed negligible (|C p | < 0.004 at x/D = 7).
Spatial distribution of time-average pressure coefficient in the center XZ-plane (top) and the YZ-plane (bottom)
Aerodynamic drag
Both the model and its strut contribute to the drag. The two contributions need to be separated to obtain solely the sphere drag. The effect of the strut on the flow is visible in Fig. 8 (right). Its drag introduces a bias error for the estimation of the sphere drag. This error can be estimated by a local application of the control volume approach, which considers a region only affected by the strut. The control volume, containing a 2 cm section of the strut (−0.75 < z/D < 0.75 and − 1.4 < y/D < −1.6), is indicated by the dashed line Fig. 8, right-bottom. The contribution is evaluated at a distance of 100 strut diameters behind the model, where the pressure and Reynolds stress terms on the drag can be neglected. Afterwards, the drag of the entire strut is obtained, scaling the drag of the 2 cm section to its full length. The resulting strut drag is 0.0006 N, which is subtracted from the drag of the entire model (0.0056 N).
A second source of bias error for the drag is the finite size of the rectangular channel. Considering a blockage factor of 3.4%, the value of the drag in the experiment overestimates that of a sphere travelling in an unconfined medium (Moradian et al. 2009). The drag is therefore corrected, using the continuity equation (assuming continuous solid blockage), multiplying the measured value by a factor 0.94. The results presented in the remainder of the work refer solely to the drag of the sphere and include the correction for blockage (0.0003 N).
The time-average aerodynamic drag, derived from the velocity statistics, is expected to be independent of the distance between the measurement plane and the sphere. Given the principle stated in Eq. (4) the sum of the three terms on the right hand side is an invariant when considering steady state conditions (assumed after phase-averaging). At sufficient distance from the object, the drag is expected to be dominated solely by the momentum deficit term, as the pressure disturbance and the velocity fluctuations terms decay (Figs. 11, 12). Figure 13 shows the drag coefficient computed in the wake of the model in a streamwise range between x/D = 0.5 and x/D = 9.5. The physical location at which the drag is computed hardly affects the resulting drag coefficient value until x/D = 7, which confirms the solidity of the measurement principle. At x/D = 7, a sudden increase of the momentum contribution is observed, balanced by a negative increase of the pressure term. The latter situation is caused by the sphere hitting the porous curtain placed at the end of the channel. Given the amplitude of these effects, the measurement of the drag is not extended beyond 7 diameters, even though the drag coefficient itself appears to be affected to smaller extent.
Mean drag coefficient evaluated at varying distance behind the sphere; C D and the individual momentum, pressure and Re stress term
The momentum term is strongly negative close to the sphere, with a peak at x/D = 0.6, where it acts as a thrust term. Considering Eq. (4), this thrust originates from the reverse flow in the recirculation region and partly from the accelerated flow around the sphere periphery. The contribution of the reverse flow to the thrust ends after reattachment (x/D = 1.3), where the momentum term changes sign and increases to reach a relatively constant value after x/D = 2. The negative contribution of the momentum deficit at small x/D is mostly compensated by the pressure term, which is highest within the first diameter close to the sphere and vanishes after about x/D = 5. The Reynolds normal stresses contribute negatively to the drag by definition. A minimum is observed around x/D = 1, which corresponds to the location of the peaks of streamwise velocity fluctuations in Fig. 11. This term decays more slowly reaching a value of about − 0.05 at x/D = 7. Along the wake the contribution of the Reynolds normal stress is significant and cannot be neglected when computing the aerodynamic drag. This agrees with the study of Balachandar et al. (1997) who evaluated the contribution of the Reynolds normal stress to the wake of two-dimensional bluff bodies and suggested the presence of shear layer interactions in the vortex shedding dynamics.
The discussed spatial development of the different terms, and in particular the pressure, has a practical consequence for the measurement of the aerodynamic drag. As long as the pressure term remains significant (x/D < 5), the full velocity gradient tensor evaluation is needed for accurate pressure reconstruction, requiring tomographic-PIV or other 3D-PIV techniques. Instead, when the pressure term is not significant, stereo-PIV measurements on a single plane may be sufficient for accurate drag evaluation.
The comparison against literature data is made taking into account the main sources of uncertainty.
First, the position along the wake is considered. Although theoretically the drag may be measured at any arbitrary station in the wake, the large velocity fluctuations in the near wake increase the uncertainty of the measured time-average velocity (Table 2) and, therefore, the uncertainty of the pressure and the drag. The large variations observed for x/D < 2 suggest that reliable drag estimates should be obtained at a larger distance from the model. The computed drag coefficient is relatively constant with an average value of 0.47 for x/D > 2.
The statistical uncertainty of the mean drag coefficient, \(\varepsilon _{{\bar{C}_{D} }} N\) mostly stems from variations in the instantaneous drag, caused by large scale fluctuations in the wake. It scales inversely proportional to \(\sqrt{N}\), where N is the statistical ensemble size. The latter can be enlarged by increasing the number of passages of the sphere N P , and the amount of uncorrelated stations selected along the wake N S :
$$N = N_{P} \cdot\,N_{S}.$$
In the present experiments the statistical ensemble is built from 35 individual passages (N P = 35) of the sphere and three stations x/D = {5, 6, 7} in the wake (N S = 3), in the range where the pressure term can be neglected. Considering that the uncertainty of the drag coefficient from a single sample is \(\varepsilon _{{\bar{C}_{D} }} N\) to 0.26, the above condition reduce the statistical uncertainty to approximately \(\varepsilon _{{\bar{C}_{D} }} N\) to 0.026 at 95% coverage factor.
Any further decrease of the uncertainty of the mean drag coefficient relies on the increase of N P and N S . For the latter, the assumption of negligible environmental disturbances during the observation time must hold true. This means that after the passage of the model through the measurement region, momentum is not added by external sources or removed for instance due to wall interactions.
Considering the latter uncertainty, the presented value for the drag coefficient of 0.47 falls within the range of reported values in literature: Moradian et al. (2009) measured a CD of about 0.51 (Re = 22,000) by load cells, Achenbach (1972), a value of 0.5 (Re = 60,000) by strain gauges, while Constantinescu and Squires (2003) and Yun et al. (2006) report a value of 0.39 (Re = 10,000).
Finally, regarding the applicability of the control volume approach to other bluff body flows, it is worth mentioning that the flow over spheres becomes turbulent at a Reynolds number around 800 and remains so afterwards. Hence, the statistical approach to determine the drag can be considered also for experiments at a higher Reynolds number. Therefore, the conclusions drawn from this study can be extrapolated to model geometries other than the sphere under the assumption of similarity in the behavior of the wake.
Time resolved tomographic-PIV measurements are conducted to determine the aerodynamic drag of a transiting sphere using the control volume approach in the wake of the model. The concept is demonstrated using a newly developed system to measure the flow over a sphere with a diameter of 10 cm moving at 1.45 m/s. Velocity statistics in the wake of the sphere have been obtained from a set of 35 model transits. The obtained time-average velocity and its fluctuations are used to estimate the flow field pressure. The aerodynamic drag is evaluated via a control volume approach as the sum of these three contributions along the wake behind the sphere. The estimation of the drag coefficient is practically unaffected by the position where the momentum integral is evaluated. In particular, the time-average drag coefficient obtained 2 sphere diameters behind the model falls within the range of reported values in literature.
For practical applications of this approach, it is observed that the pressure term vanishes after 5 diameters, which can greatly simplify the measurement procedure. Three disturbing factors are worth mentioning that may affect the measurement accuracy: the flow conditions prior to the passage of the sphere need to be taken into account in the evaluation; the blockage effect due to the finite channel cross section; the contribution of the supporting strut needs to be subtracted to isolate that of the main object only.
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This work is partly funded by the TU Delft Sports Engineering Institute and the European Research Council Proof of Concept Grant "Flow Visualization Based Pressure" (no. 665477). Andrea Rubino is acknowledged for the support during experiments.
Aerospace Engineering Department, TU Delft, Delft, The Netherlands
W. Terra
, A. Sciacchitano
& F. Scarano
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Correspondence to W. Terra.
Supplementary material 1 (AVI 6187 KB)
Supplementary material 2 (AVI 22596 KB)
Terra, W., Sciacchitano, A. & Scarano, F. Aerodynamic drag of a transiting sphere by large-scale tomographic-PIV. Exp Fluids 58, 83 (2017) doi:10.1007/s00348-017-2331-0
Revised: 08 March 2017
Tomographic PIV
Momentum equation
HFSB | CommonCrawl |
\begin{document}
\title{An Identity Involving Integration with Respect\to Variable Order of Fractional Derivative}
\section{Introduction}
In 1993, Samko and Ross \cite{SamRoss} introduced the study of fractional integration and differentiation when the order is not a constant but a function. This suggestion gave rise to a number of further ideas and results \cite{malinowska2015,sun2009,valerio2011}. In particular, this implies a possibility of integration with respect to derivative's order. Here an identity is presented, in which an expression involving Riemann--Liouville fractional derivative is integrated with respect to the derivative's order.
\section{Preliminaries}
\begin{defi} Suppose $ f:[a,\infty)\to\mathbb{R}$ is an absolutely continuous function. The Riemann--Liouville (left-sided) fractional integral and derivative of order $\alpha$, $ m-1 < \alpha <m$, $ m\in\mathbb{N} $, are defined as follows: \begin{align*} J_{a+}^\alpha f(t) &= \frac{1}{\Gamma(\alpha)}\int_{a}^{t} (t-\tau)^{\alpha-1}f(\tau)d\tau,\ t> a, \\ D_{a+}^\alpha f(t) &= \frac{d^m}{dt^m} J_{a+}^{m-\alpha} f(t),\quad t>a. \end{align*} In what follows we will omit the lower limit of integration in the notation if it is equal to zero, i.e. $ J^\alpha f(t) \triangleq J_{0+}^\alpha f(t) $, $ D^\alpha f(t) \triangleq D_{0+}^\alpha f(t)$. \end{defi}
Here $ \Gamma(\alpha)$ denotes Euler's Gamma function.
\begin{prp}[Property 2.1 \cite{SamKilMar}] \label{prp1} Let $ \alpha, \beta > 0 $. Then the following identity holds: \begin{equation*} D_a^\alpha (t-a)^{\beta-1}=\frac{\Gamma(\beta)}{\Gamma(\beta-\alpha)} (t-a)^{\beta - \alpha -1}. \end{equation*} \end{prp}
In particular, it follows from Proposition \ref{prp1}, that \begin{align} D^\alpha 1 &= \frac{t^{-\alpha}}{\Gamma(1-\alpha)}, \label{eq:1}\\ D^\alpha t^{n-1} &=\frac{\Gamma(n)}{\Gamma(n - \alpha)}t^{n - \alpha - 1}, \quad n\in \mathbb{N}. \label{eq:5} \end{align}
\begin{defi} The sinc function (``Cardinal Sine'') is defined as follows: \begin{equation*} \sinc (x) = \begin{cases} \frac{\sin \pi x}{\pi x}, & x\ne 0,\\ 1, & x=0. \end{cases} \end{equation*} \end{defi}
\begin{prp} \label{prp1.5} \begin{equation} \int_{-\infty}^{\infty} \sinc(x) dx =1. \end{equation} \end{prp}
The following properties of Euler's Gamma function will be used in the sequel.
\begin{prp} \label{prp2} \begin{equation} \label{eq:2} \Gamma(1+\alpha)\Gamma(1-\alpha)=\alpha\Gamma(\alpha)\Gamma(1-\alpha)=\frac{\pi\alpha}{\sin(\pi\alpha)} = \frac{1}{\sinc(\alpha)}, \end{equation} \end{prp}
\begin{prp} \label{prp3} \begin{equation*} \Gamma(z+n) = (z)_n \Gamma(z), \end{equation*} where $(z)_n = z(z+1)\ldots (z+n-1)$ is the \textsl{Pochhammer symbol}. \end{prp}
The following theorem allowing to calculate some improper integrals with the help of contour integrals in the complex plane will also be used in the sequel. \begin{thm}[Indented Trigonometric Integrals \cite{complex}] \label{contour} Assume that $ P(z) $, $ Q(z) $, $ z\in \mathbb{C} $, are polynomials with real coefficients of degree $ m $ and $ n $, respectively, where $ n\ge m+1 $ and that $ Q(z)$ has simple zeros at the points $ t_1, \ldots, t_L $ on the $ x$-axis. If $p$ is a positive real number, and if $ f(z)=\frac{e^{ipz}P(z)}{Q(z)}$, then we can compute the Cauchy Principal Value (P.V.) of the following integral \begin{equation*} \pv \int_{-\infty}^{\infty} \frac{P(x)}{Q(x)}\sin(px)dx=\im \left( 2\pi i \sum_{j=1}^{k} \res_{z=z_j} f(z)+ \pi i \sum_{j=1}^{L} \res_{z=t_j} f(z)\right). \end{equation*} \end{thm}
\section{The Main Result}
\begin{lemma} \label{lemma} For every $ n=1,2,\ldots $ the following identity holds true: \begin{equation} \int_{-\infty}^{\infty}\frac{t^\alpha}{\Gamma(\alpha + 1)} (D^\alpha t^{n-1}) d\alpha = (2t)^{n-1}. \end{equation} \end{lemma} \begin{proof} For $ n=1$, in view of Properties \ref{prp2} and \ref{prp1.5}, we have \begin{equation*} \int_{-\infty}^{\infty} \frac{t^\alpha}{\Gamma(1+\alpha)} (D^\alpha 1) d\alpha=\int_{-\infty}^{\infty} \frac{\sin(\pi\alpha)}{\pi \alpha}d\alpha = \int_{-\infty}^{\infty} \sinc(\alpha) d\alpha = 1. \end{equation*}
Now suppose $ n=2,3,\ldots $ Then, by virtue of Properties \ref{prp2} and \ref{prp3}, we have \begin{equation*} \Gamma(n-\alpha)\Gamma(1+\alpha) = (1-\alpha)_{n-1} \Gamma(1-\alpha) \Gamma(1+\alpha)=(1-\alpha)_{n-1} \frac{\pi\alpha}{\sin(\pi\alpha)}, \end{equation*} hence \begin{equation} \label{eq:6} \begin{aligned} \int_{-\infty}^{\infty} \frac{t^\alpha}{\Gamma(1+\alpha)} (D^\alpha t^{n-1}) d\alpha &= t^{n-1} \int_{-\infty}^{\infty} \frac{\Gamma(n)}{\Gamma(n - \alpha)\Gamma(1+\alpha)}d\alpha\\
&= t^{n-1} \int_{-\infty}^{\infty} \binom{n-1}{\alpha} d\alpha\\ &=\frac{t^{n-1}}{\pi} (n-1)!\int_{-\infty}^{\infty} \frac{\sin(\pi\alpha)}{\alpha(1-\alpha)_{n-1}}d\alpha. \end{aligned} \end{equation} It follows from Theorem \ref{contour} that \begin{equation} \begin{aligned} \int_{-\infty}^{\infty} \frac{\sin(\pi\alpha)}{\alpha(1-\alpha)_{n}}d\alpha &= \int_{-\infty}^{\infty} \frac{\sin(\pi\alpha)}{\alpha(1-\alpha)(2-\alpha)\ldots (n-\alpha)}d\alpha \\ &= \im \left( \pi i \sum_{j=0}^{n} \res_{z=j} \frac{e^{i\pi z}}{z(1-z)(2-z)\ldots (n-z)} \right). \end{aligned} \end{equation} Since \begin{align*} \res_{z=0} \frac{e^{i\pi z}}{z(1-z)(2-z)\ldots (n-z)} &= \frac{1}{n!},\\ \res_{z=j} \frac{e^{i\pi z}}{z(1-z)(2-z)\ldots (n-z)} &= \frac{1}{n!}\binom{n}{j},\ j=1,\ldots, n. \end{align*} we have \begin{equation} \int_{-\infty}^{\infty} \frac{\sin(\pi\alpha)}{\alpha(1-\alpha)_{n}}d\alpha = \frac{\pi}{n!}\sum_{j=0}^{n} \binom{n}{j}=\frac{\pi 2^n}{n!}. \end{equation} Thus \begin{equation*} \int_{-\infty}^{\infty} \frac{t^\alpha}{\Gamma(1+\alpha)} (D^\alpha t^{n-1}) d\alpha=\frac{t^{n-1}}{\pi}(n-1)!\frac{\pi 2^{n-1}}{{n-1}!}=(2t)^{n-1}. \end{equation*} \end{proof}
\begin{cor} \begin{equation*} \int_{-\infty}^{\infty} \binom{n}{\alpha} d\alpha = 2^n,\quad n\in\mathbb{N}. \end{equation*} \end{cor} This Corollary follows from Lemma \ref{lemma} and \eqref{eq:6}.
\begin{cor}[Main Identity] For any function $ f(t)$ that is analytic in some neighborhood of zero $(-\varepsilon,\varepsilon)$, $ \varepsilon>0 $, the following identity holds true \begin{equation} \int_{-\infty}^{\infty}\frac{t^\alpha}{\Gamma(\alpha + 1)} [D^\alpha f(t/2)]d\alpha = f(t),\quad t\in (-\varepsilon,\varepsilon). \end{equation} \end{cor}
\end{document} | arXiv |
Use cell notation to describe galvanic cells
Describe the basic components of galvanic cells
Galvanic cells, also known as voltaic cells, are electrochemical cells in which spontaneous oxidation-reduction reactions produce electrical energy. In writing the equations, it is often convenient to separate the oxidation-reduction reactions into half-reactions to facilitate balancing the overall equation and to emphasize the actual chemical transformations.
Consider what happens when a clean piece of copper metal is placed in a solution of silver nitrate ([link]). As soon as the copper metal is added, silver metal begins to form and copper ions pass into the solution. The blue color of the solution on the far right indicates the presence of copper ions. The reaction may be split into its two half-reactions. Half-reactions separate the oxidation from the reduction, so each can be considered individually.
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\begin{array}{}\\ \\ \underset{¯}{\begin{array}{l}\text{oxidation:}\phantom{\rule{5.7em}{0ex}}\text{Cu}\left(s\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{Cu}}^{2+}\left(aq\right)+{\text{2e}}^{\text{−}}\\ \text{reduction:}\phantom{\rule{0.2em}{0ex}}\text{2}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left({\text{Ag}}^{\text{+}}\left(aq\right)+{\text{e}}^{\text{−}}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\text{Ag}\left(s\right)\right)\phantom{\rule{5em}{0ex}}\text{or}\phantom{\rule{5em}{0ex}}2{\text{Ag}}^{\text{+}}\left(aq\right)+{\text{2e}}^{\text{−}}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\text{2Ag}\left(s\right)\end{array}}\\ \text{overall:}\phantom{\rule{1.5em}{0ex}}2{\text{Ag}}^{\text{+}}\left(aq\right)+\text{Cu}\left(s\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\text{2Ag}\left(s\right)+{\text{Cu}}^{2+}\left(aq\right)\end{array}
*** Error message:
Missing # inserted in alignment preamble.
leading text: $\begin{array}{}
Missing $ inserted.
leading text: ...}{0ex}}\text{2Ag}\left(s\right)\end{array}}
Extra }, or forgotten $.
Missing } inserted.
leading text: ...text{2Ag}\left(s\right)\end{array}}\\ \text
The equation for the reduction half-reaction had to be doubled so the number electrons "gained" in the reduction half-reaction equaled the number of electrons "lost" in the oxidation half-reaction.
When a clean piece of copper metal is placed into a clear solution of silver nitrate (a), an oxidation-reduction reaction occurs that results in the exchange of Cu2+ for Ag+ ions in solution. As the reaction proceeds (b), the solution turns blue (c) because of the copper ions present, and silver metal is deposited on the copper strip as the silver ions are removed from solution. (credit: modification of work by Mark Ott)
Galvanic or voltaic cells involve spontaneous electrochemical reactions in which the half-reactions are separated ([link]) so that current can flow through an external wire. The beaker on the left side of the figure is called a half-cell, and contains a 1 M solution of copper(II) nitrate [Cu(NO3)2] with a piece of copper metal partially submerged in the solution. The copper metal is an electrode. The copper is undergoing oxidation; therefore, the copper electrode is the anode. The anode is connected to a voltmeter with a wire and the other terminal of the voltmeter is connected to a silver electrode by a wire. The silver is undergoing reduction; therefore, the silver electrode is the cathode. The half-cell on the right side of the figure consists of the silver electrode in a 1 M solution of silver nitrate (AgNO3). At this point, no current flows—that is, no significant movement of electrons through the wire occurs because the circuit is open. The circuit is closed using a salt bridge, which transmits the current with moving ions. The salt bridge consists of a concentrated, nonreactive, electrolyte solution such as the sodium nitrate (NaNO3) solution used in this example. As electrons flow from left to right through the electrode and wire, nitrate ions (anions) pass through the porous plug on the left into the copper(II) nitrate solution. This keeps the beaker on the left electrically neutral by neutralizing the charge on the copper(II) ions that are produced in the solution as the copper metal is oxidized. At the same time, the nitrate ions are moving to the left, sodium ions (cations) move to the right, through the porous plug, and into the silver nitrate solution on the right. These added cations "replace" the silver ions that are removed from the solution as they were reduced to silver metal, keeping the beaker on the right electrically neutral. Without the salt bridge, the compartments would not remain electrically neutral and no significant current would flow. However, if the two compartments are in direct contact, a salt bridge is not necessary. The instant the circuit is completed, the voltmeter reads +0.46 V, this is called the cell potential. The cell potential is created when the two dissimilar metals are connected, and is a measure of the energy per unit charge available from the oxidation-reduction reaction. The volt is the derived SI unit for electrical potential
In this equation, A is the current in amperes and C the charge in coulombs. Note that volts must be multiplied by the charge in coulombs (C) to obtain the energy in joules (J).
In this standard galvanic cell, the half-cells are separated; electrons can flow through an external wire and become available to do electrical work.
When the electrochemical cell is constructed in this fashion, a positive cell potential indicates a spontaneous reaction and that the electrons are flowing from the left to the right. There is a lot going on in [link], so it is useful to summarize things for this system:
Electrons flow from the anode to the cathode: left to right in the standard galvanic cell in the figure.
The electrode in the left half-cell is the anode because oxidation occurs here. The name refers to the flow of anions in the salt bridge toward it.
The electrode in the right half-cell is the cathode because reduction occurs here. The name refers to the flow of cations in the salt bridge toward it.
Oxidation occurs at the anode (the left half-cell in the figure).
Reduction occurs at the cathode (the right half-cell in the figure).
The cell potential, +0.46 V, in this case, results from the inherent differences in the nature of the materials used to make the two half-cells.
The salt bridge must be present to close (complete) the circuit and both an oxidation and reduction must occur for current to flow.
There are many possible galvanic cells, so a shorthand notation is usually used to describe them. The cell notation (sometimes called a cell diagram) provides information about the various species involved in the reaction. This notation also works for other types of cells. A vertical line, │, denotes a phase boundary and a double line, ‖, the salt bridge. Information about the anode is written to the left, followed by the anode solution, then the salt bridge (when present), then the cathode solution, and, finally, information about the cathode to the right. The cell notation for the galvanic cell in [link] is then
Note that spectator ions are not included and that the simplest form of each half-reaction was used. When known, the initial concentrations of the various ions are usually included.
One of the simplest cells is the Daniell cell. It is possible to construct this battery by placing a copper electrode at the bottom of a jar and covering the metal with a copper sulfate solution. A zinc sulfate solution is floated on top of the copper sulfate solution; then a zinc electrode is placed in the zinc sulfate solution. Connecting the copper electrode to the zinc electrode allows an electric current to flow. This is an example of a cell without a salt bridge, and ions may flow across the interface between the two solutions.
Some oxidation-reduction reactions involve species that are poor conductors of electricity, and so an electrode is used that does not participate in the reactions. Frequently, the electrode is platinum, gold, or graphite, all of which are inert to many chemical reactions. One such system is shown in [link]. Magnesium undergoes oxidation at the anode on the left in the figure and hydrogen ions undergo reduction at the cathode on the right. The reaction may be summarized as
\begin{array}{}\\ \underset{¯}{\begin{array}{l}\text{oxidation:}\phantom{\rule{4.1em}{0ex}}\text{Mg}\left(s\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{Mg}}^{2+}\left(aq\right)+{\text{2e}}^{\text{−}}\\ \text{reduction:}\phantom{\rule{0.2em}{0ex}}{\text{2H}}^{\text{+}}\left(aq\right)+{\text{2e}}^{\text{−}}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{H}}_{2}\left(g\right)\end{array}}\\ \text{overall:}\phantom{\rule{0.45em}{0ex}}\text{Mg}\left(s\right)+{\text{2H}}^{\text{+}}\left(aq\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{Mg}}^{2+}\left(aq\right)+{\text{H}}_{2}\left(g\right)\end{array}
leading text: ...x}}{\text{H}}_{2}\left(g\right)\end{array}}
leading text: ...t{H}}_{2}\left(g\right)\end{array}}\\ \text
The cell used an inert platinum wire for the cathode, so the cell notation is
The magnesium electrode is an active electrode because it participates in the oxidation-reduction reaction. Inert electrodes, like the platinum electrode in [link], do not participate in the oxidation-reduction reaction and are present so that current can flow through the cell. Platinum or gold generally make good inert electrodes because they are chemically unreactive.
Using Cell Notation
Consider a galvanic cell consisting of
Write the oxidation and reduction half-reactions and write the reaction using cell notation. Which reaction occurs at the anode? The cathode?
By inspection, Cr is oxidized when three electrons are lost to form Cr3+, and Cu2+ is reduced as it gains two electrons to form Cu. Balancing the charge gives
\begin{array}{}\\ \underset{¯}{\begin{array}{l}\text{oxidation:}\phantom{\rule{4.75em}{0ex}}\text{2Cr}\left(s\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{2Cr}}^{3+}\left(aq\right)+{\text{6e}}^{\text{−}}\\ \text{reduction:}\phantom{\rule{0.2em}{0ex}}{\text{3Cu}}^{2+}\left(aq\right)+{\text{6e}}^{\text{−}}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\text{3Cu}\left(s\right)\end{array}}\\ \text{overall:}\phantom{\rule{0.3em}{0ex}}\text{2Cr}\left(s\right)+{\text{3Cu}}^{2+}\left(aq\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{2Cr}}^{3+}\left(aq\right)+\text{3Cu}\left(s\right)\end{array}
leading text: ...}{0ex}}\text{3Cu}\left(s\right)\end{array}}
leading text: ...text{3Cu}\left(s\right)\end{array}}\\ \text
Cell notation uses the simplest form of each of the equations, and starts with the reaction at the anode. No concentrations were specified so: Oxidation occurs at the anode and reduction at the cathode.
By inspection, Fe2+ undergoes oxidation when one electron is lost to form Fe3+, and MnO4− is reduced as it gains five electrons to form Mn2+. Balancing the charge gives
\begin{array}{}\\ \underset{¯}{\begin{array}{l}\text{oxidation:}\phantom{\rule{9.65em}{0ex}}{\text{5(Fe}}^{2+}\left(aq\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{Fe}}^{3+}\left(aq\right)+{\text{e}}^{\text{−}}\right)\\ \text{reduction:}\phantom{\rule{1.65em}{0ex}}{\text{MnO}}_{4}{}^{\text{−}}\left(aq\right)+{\text{8H}}^{\text{+}}\left(aq\right)+{\text{5e}}^{\text{−}}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{Mn}}^{2+}\left(aq\right)+{\text{4H}}_{2}\text{O}\left(l\right)\end{array}}\\ {\text{overall: 5Fe}}^{2+}\left(aq\right)+{\text{MnO}}_{4}{}^{\text{−}}\left(aq\right)+{\text{8H}}^{\text{+}}\left(aq\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{5Fe}}^{3+}\left(aq\right)+{\text{Mn}}^{2+}\left(aq\right)+{\text{4H}}_{2}\text{O}\left(l\right)\end{array}
leading text: ...{4H}}_{2}\text{O}\left(l\right)\end{array}}
leading text: ...}_{2}\text{O}\left(l\right)\end{array}}\\ {
Cell notation uses the simplest form of each of the equations, and starts with the reaction at the anode. It is necessary to use an inert electrode, such as platinum, because there is no metal present to conduct the electrons from the anode to the cathode. No concentrations were specified so: Oxidation occurs at the anode and reduction at the cathode.
Check Your Learning
Use cell notation to describe the galvanic cell where copper(II) ions are reduced to copper metal and zinc metal is oxidized to zinc ions.
From the information given in the problem:
\begin{array}{}\\ \underset{¯}{\begin{array}{l}\text{anode (oxidation):}\phantom{\rule{5.3em}{0ex}}\text{Zn}\left(s\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{Zn}}^{2+}\left(aq\right)+{\text{2e}}^{\text{−}}\\ \text{cathode (reduction):}\phantom{\rule{0.2em}{0ex}}{\text{Cu}}^{2+}\left(aq\right)+{\text{2e}}^{\text{−}}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\text{Cu}\left(s\right)\end{array}}\\ \text{overall:}\phantom{\rule{4.7em}{0ex}}\text{Zn}\left(s\right)+{\text{Cu}}^{2+}\left(aq\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{Zn}}^{2+}\left(aq\right)+\text{Cu}\left(s\right)\end{array}
leading text: ...m}{0ex}}\text{Cu}\left(s\right)\end{array}}
leading text: ...\text{Cu}\left(s\right)\end{array}}\\ \text
Using cell notation:
The oxidation of magnesium to magnesium ion occurs in the beaker on the left side in this apparatus; the reduction of hydrogen ions to hydrogen occurs in the beaker on the right. A nonreactive, or inert, platinum wire allows electrons from the left beaker to move into the right beaker. The overall reaction is: which is represented in cell notation as:
Electrochemical cells typically consist of two half-cells. The half-cells separate the oxidation half-reaction from the reduction half-reaction and make it possible for current to flow through an external wire. One half-cell, normally depicted on the left side in a figure, contains the anode. Oxidation occurs at the anode. The anode is connected to the cathode in the other half-cell, often shown on the right side in a figure. Reduction occurs at the cathode. Adding a salt bridge completes the circuit allowing current to flow. Anions in the salt bridge flow toward the anode and cations in the salt bridge flow toward the cathode. The movement of these ions completes the circuit and keeps each half-cell electrically neutral. Electrochemical cells can be described using cell notation. In this notation, information about the reaction at the anode appears on the left and information about the reaction at the cathode on the right. The salt bridge is represented by a double line, ‖. The solid, liquid, or aqueous phases within a half-cell are separated by a single line, │. The phase and concentration of the various species is included after the species name. Electrodes that participate in the oxidation-reduction reaction are called active electrodes. Electrodes that do not participate in the oxidation-reduction reaction but are there to allow current to flow are inert electrodes. Inert electrodes are often made from platinum or gold, which are unchanged by many chemical reactions.
Write the following balanced reactions using cell notation. Use platinum as an inert electrode, if needed.
(a) (b) (c) (d)
Given the following cell notations, determine the species oxidized, species reduced, and the oxidizing agent and reducing agent, without writing the balanced reactions.
For the cell notations in the previous problem, write the corresponding balanced reactions.
Balance the following reactions and write the reactions using cell notation. Ignore any inert electrodes, as they are never part of the half-reactions.
Identify the species oxidized, species reduced, and the oxidizing agent and reducing agent for all the reactions in the previous problem.
Species oxidized = reducing agent: (a) Al(s); (b) NO(g); (c) Mg(s); and (d) MnO2(s); Species reduced = oxidizing agent: (a) Zr4+(aq); (b) Ag+(aq); (c) ; and (d)
From the information provided, use cell notation to describe the following systems:
(a) In one half-cell, a solution of Pt(NO3)2 forms Pt metal, while in the other half-cell, Cu metal goes into a Cu(NO3)2 solution with all solute concentrations 1 M.
(b) The cathode consists of a gold electrode in a 0.55 M Au(NO3)3 solution and the anode is a magnesium electrode in 0.75 M Mg(NO3)2 solution.
(c) One half-cell consists of a silver electrode in a 1 M AgNO3 solution, and in the other half-cell, a copper electrode in 1 M Cu(NO3)2 is oxidized.
Why is a salt bridge necessary in galvanic cells like the one in [link]?
Without the salt bridge, the circuit would be open (or broken) and no current could flow. With a salt bridge, each half-cell remains electrically neutral and current can flow through the circuit.
An active (metal) electrode was found to gain mass as the oxidation-reduction reaction was allowed to proceed. Was the electrode part of the anode or cathode? Explain.
An active (metal) electrode was found to lose mass as the oxidation-reduction reaction was allowed to proceed. Was the electrode part of the anode or cathode? Explain.
Active electrodes participate in the oxidation-reduction reaction. Since metals form cations, the electrode would lose mass if metal atoms in the electrode were to oxidize and go into solution. Oxidation occurs at the anode.
The mass of three different metal electrodes, each from a different galvanic cell, were determined before and after the current generated by the oxidation-reduction reaction in each cell was allowed to flow for a few minutes. The first metal electrode, given the label A, was found to have increased in mass; the second metal electrode, given the label B, did not change in mass; and the third metal electrode, given the label C, was found to have lost mass. Make an educated guess as to which electrodes were active and which were inert electrodes, and which were anode(s) and which were the cathode(s).
active electrode
electrode that participates in the oxidation-reduction reaction of an electrochemical cell; the mass of an active electrode changes during the oxidation-reduction reaction
electrode in an electrochemical cell at which oxidation occurs; information about the anode is recorded on the left side of the salt bridge in cell notation
electrode in an electrochemical cell at which reduction occurs; information about the cathode is recorded on the right side of the salt bridge in cell notation
cell notation
shorthand way to represent the reactions in an electrochemical cell
cell potential
difference in electrical potential that arises when dissimilar metals are connected; the driving force for the flow of charge (current) in oxidation-reduction reactions
galvanic cell
electrochemical cell that involves a spontaneous oxidation-reduction reaction; electrochemical cells with positive cell potentials; also called a voltaic cell
inert electrode
electrode that allows current to flow, but that does not otherwise participate in the oxidation-reduction reaction in an electrochemical cell; the mass of an inert electrode does not change during the oxidation-reduction reaction; inert electrodes are often made of platinum or gold because these metals are chemically unreactive.
voltaic cell
another name for a galvanic cell
Previous: Balancing Oxidation-Reduction Reactions
Next: Standard Reduction Potentials | CommonCrawl |
Ludwig Stickelberger
Ludwig Stickelberger (18 May 1850 – 11 April 1936) was a Swiss mathematician who made important contributions to linear algebra (theory of elementary divisors) and algebraic number theory (Stickelberger relation in the theory of cyclotomic fields).
Ludwig Stickelberger
Ludwig Stickelberger
Born18 May 1850
Buch, Schaffhausen;
Died11 April 1936 (1936-04-12) (aged 85)
Basel
NationalitySwiss
Alma materUniversity of Heidelberg
University of Berlin
Known forStickelberger relation
Frobenius–Stickelberger theorem
Scientific career
FieldsMathematics
InstitutionsUniversity of Freiburg
ThesisDe problemate quodam ad duarum bilinearium vel quadraticarum transformationem pertinente (1874)
Doctoral advisorErnst Kummer, Karl Weierstrass
Short biography
Stickelberger was born in Buch in the canton of Schaffhausen into a family of a pastor. He graduated from a gymnasium in 1867 and studied next in the University of Heidelberg. In 1874 he received a doctorate in Berlin under the direction of Karl Weierstrass for his work on the transformation of quadratic forms to a diagonal form. In the same year, he obtained his Habilitation from Polytechnicum in Zurich (now ETH Zurich). In 1879 he became an extraordinary professor in the Albert Ludwigs University of Freiburg. From 1896 to 1919 he worked there as a full professor, and from 1919 until his return to Basel in 1924 he held the title of a distinguished professor ("ordentlicher Honorarprofessor"). He was married in 1895, but his wife and son both died in 1918. Stickelberger died on 11 April 1936 and was buried next to his wife and son in Freiburg.
Mathematical contributions
Stickelberger's obituary lists the total of 14 publications: his thesis (in Latin), 8 further papers that he authored which appeared during his lifetime, 4 joint papers with Georg Frobenius and a posthumously published paper written circa 1915. Despite this modest output, he is characterized there as "one of the sharpest among the pupils of Weierstrass" and a "mathematician of high rank". Stickelberger's thesis and several later papers streamline and complete earlier investigations of various authors, in a direct and elegant way.
Linear algebra
Stickelberger's work on the classification of pairs of bilinear and quadratic forms filled in important gaps in the theory earlier developed by Weierstrass and Darboux. Augmented with the contemporaneous work of Frobenius, it set the theory of elementary divisors upon a rigorous foundation. An important 1878 paper of Stickelberger and Frobenius gave the first complete treatment of the classification of finitely generated abelian groups and sketched the relation with the theory of modules that had just been developed by Dedekind.
Number theory
Three joint papers with Frobenius deal with the theory of elliptic functions. Today Stickelberger's name is most closely associated with his 1890 paper that established the Stickelberger relation for cyclotomic Gaussian sums. This generalized earlier work of Jacobi and Kummer and was later used by Hilbert in his formulation of the reciprocity laws in algebraic number fields. The Stickelberger relation also yields information about the structure of the class group of a cyclotomic field as a module over its abelian Galois group (cf Iwasawa theory).
References
• Lothar Heffter, Ludwig Stickelberger, Jahresbericht der Deutschen Matematische Vereinigung, XLVII (1937), pp. 79–86
• Ludwig Stickelberger, Ueber eine Verallgemeinerung der Kreistheilung, Mathematische Annalen 37 (1890), pp. 321–367
External links
• Works by or about Ludwig Stickelberger at Internet Archive
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| Wikipedia |
HP-65
The HP-65 is the first magnetic card-programmable handheld calculator. Introduced by Hewlett-Packard in 1974 at an MSRP of $795[1] (equivalent to $4,717 in 2022)[2], it featured nine storage registers and room for 100 keystroke instructions. It also included a magnetic card reader/writer to save and load programs. Like all Hewlett-Packard calculators of the era and most since, the HP-65 used Reverse Polish Notation (RPN) and a four-level automatic operand stack.
HP-65
HP-65 with program card
TypeProgrammable
Introduced1974
Calculator
Entry modeRPN key stroke
Display typeRed LED seven-segment display
Display size15 digits (decimal point uses one digit), (±10±99)
CPU
Processorproprietary
Programming
Programming language(s)key codes
Memory register8 (9) plus 4-level working stack
Program steps100
Other
Power supplyInternal rechargeable battery or 115/230 V AC, 5 W
WeightCalculator: 11 oz (310 g),
recharger: 5 oz (140 g)
DimensionsLength: 6.0 inches (150 mm),
width: 3.2 inches (81 mm),
height: 0.7–1.4 inches (18–36 mm)
Bill Hewlett's design requirement was that the calculator should fit in his shirt pocket. That is one reason for the tapered depth of the calculator. The magnetic program cards are fed in at the thick end of the calculator under the LED display. The documentation for the programs in the calculator is very complete, including algorithms for hundreds of applications, including the solutions of differential equations, stock price estimation, statistics, and so forth.
Features
The HP-65 introduced the "tall", trapezoid-shaped keys that would become iconic for many generations of HP calculators. Each of the keys had up to four functions. In addition to the "normal function" printed on the key's face, a "gold" function printed on the case above the key and a "blue" function printed on the slanted front surface of the key were accessed by pushing the gold f or blue g prefix key, respectively. For example, f followed by 4 would access the sine function, or g followed by 4 would calculate $1/x$. For some mathematical functions, a gold f−1 prefix key would access the inverse of the gold-printed functions, e.g. f−1 followed by 4 would calculate the inverse sine ($\sin ^{-1}$).
Functions included square root, inverse, trigonometric (sine, cosine, tangent and their inverses), exponentiation, logarithms and factorial. The HP-65 was one of the first calculators to include a base conversion function, although it only supported octal (base 8) conversion. It could also perform conversions between degrees/minutes/seconds (sexagesimal) and decimal degree (sexadecimal) values, as well as polar/cartesian coordinate conversion.
Programming
The HP-65 had a program memory for up to 100 instructions of 6 bits which included subroutine calls and conditional branching based on comparison of x and y registers. Some (but not all) commands entered as multiple keystrokes were stored in a single program memory cell. When displaying a program, the key codes were shown without line numbers.
A program could be saved to mylar-based magnetically coated cards measuring 71 mm × 9.5 mm (2.8 in × 0.4 in), which were fed through the reader by a small electric motor through a worm gear and rubber roller at a speed of 6 cm/s (2.4 in/s).[3] The recording area used only half of the width of the card. While reversing the card to store a second program was possible, it was officially discouraged (unlike in later models such as the HP-67) because the other half of the card was touched by the rubber wheel during transport, causing extra abrasion. When inserted into an extra slot between the display and the keyboard, the printing on top of the card would correspond to the top row of keys (A - E), which served as shortcuts to the corresponding program entry points.
Cards could be write-protected by diagonally clipping the top-left corner of the card. HP also sold a number of program collections for scientific and engineering applications on sets of prerecorded (and write-protected) cards.
The HP-65 had an issue/design flaw whereby storage register R9 was corrupted whenever the user (or program) executed trigonometric functions or performed comparison tests; this kind of issue was common in many early calculators, caused by a lack of memory due to cost, power, or size considerations. Since the limitation was intended from the beginning and documented in the manual, it is not, strictly speaking, a bug.
Significant applications
During the 1975 Apollo-Soyuz Test Project, the HP-65 became the first programmable handheld calculator in outer space. Two HP-65s were carried on board the Apollo spacecraft. Calculation of parameters for the several thrusting maneuvers needed to rendezvous with the Soyuz spacecraft was done on the HP-65 and compared with the results calculated by the onboard Apollo Guidance Computer. Another program for the HP-65 allowed the crew to compute pointing angles for the spacecraft antenna for aiming at the ATS-6 communications relay satellite.
In the same year, Mitchell Feigenbaum, using the small HP-65 calculator he had been issued at the Los Alamos National Laboratory, discovered that the ratio of the difference between the values at which successive period-doubling bifurcations occur tends to a constant of around 4.6692... This "ratio of convergence" is now known as the first Feigenbaum constant.
See also
• HP-25
• HP-35
• 65 Notes
References
1. "HP Virtual Museum: Hewlett-Packard-65 programmable pocket calculator, 1974". Retrieved 2011-01-29.
2. 1634–1699: McCusker, J. J. (1997). How Much Is That in Real Money? A Historical Price Index for Use as a Deflator of Money Values in the Economy of the United States: Addenda et Corrigenda (PDF). American Antiquarian Society. 1700–1799: McCusker, J. J. (1992). How Much Is That in Real Money? A Historical Price Index for Use as a Deflator of Money Values in the Economy of the United States (PDF). American Antiquarian Society. 1800–present: Federal Reserve Bank of Minneapolis. "Consumer Price Index (estimate) 1800–". Retrieved 2023-05-28.
3. Taggart, Robert B. (May 1974). "Designing a Tiny Magnetic Card Reader" (PDF). Hewlett-Packard Journal. Retrieved 2019-05-01.
External links
• The HP-65 at an unofficial Hewlett-Packard museum (MyCalcDB); includes a photograph of the magnetic card.
• 1975 HP Calculator Christmas Guide
• HP's Virtual Museum: HP-65
• HP-65 page at the unofficial Museum of HP Calculators
• One of the HP-65s carried on the ASTP space flight is in the collection of the National Air and Space Museum.
Hewlett-Packard (HP) calculators
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| Wikipedia |
\begin{document}
\title{$L^1$ averaging lemma for transport equations with Lipschitz force fields}
\begin{abstract} The purpose of this note is to extend the $L^1$ averaging lemma of Golse and Saint-Raymond \cite{GolSR} to the case of a kinetic transport equation with a force field $F(x)\in W^{1,\infty}$. To this end, we will prove a local in time mixing property for the transport equation $\partial_t f + v.\nabla_x f + F.\nabla_v f =0$. \end{abstract}
\section*{Introduction} Let $d \in \mathbb{N}^*$ and $1<p<+\infty$. We consider $\mathbb{R}^d$ equipped with the Lebesgue measure. Let $f(x,v)$ and $g(x,v)$ be two measurable functions in $L^p(\mathbb{R}^d \times \mathbb{R}^d)$ satisfying the transport equation: \begin{equation} v.\nabla_x f = g. \end{equation}
Although transport equations are of hyperbolic nature (and thus there is a priori no regularizing effect), it was first observed for by Golse, Perthame and Sentis in \cite{GPS} and then by Golse, Lions, Perthame and Sentis \cite{GLPS} (see also Agoshkov \cite{Ago} for related results obtained independently) that the velocity average (or moment) $\rho(x)= \int f \Psi(v) dv$ with $\Psi \in \mathcal{C}^\infty_c(\mathbb{R}^d)$ is smoother than $f$ and $g$ : more specifically it belongs to some Sobolev space $W^{s,p}(\mathbb{R}^d)$ with $s>0$. These kinds of results are referred to as "velocity averaging lemma". The analogous results in the time-dependent setting also hold, that is for the equation: \begin{equation} \label{time} \partial_t f + v.\nabla_x f = g. \end{equation}
Refined results with various generalizations (like derivatives in the right-hand side, functions with different integrability in $x$ and $v$...) were obtained in \cite{DPLM}, \cite{BZ}, \cite{PS}, \cite{JV}. There exist many other interesting contributions. We refer to Jabin \cite{J} which is a rather complete review on the topic.
Velocity averaging lemmas are tools of tremendous importance in kinetic theory since they provide some strong compactness which is very often necessary to study non-linear terms (for instance when one considers an approximation scheme to build weak solutions, or for the study of asymptotic regimes). There are numerous applications of these lemmas; two emblematic results are the existence of renormalized solutions to the Boltzmann equation \cite{DPLbol} and the existence of global weak solutions to the Vlasov-Maxwell system \cite{DPLvm}. Both are due to DiPerna and Lions.
The limit case $p=1$ is actually of great interest. In general, for a sequence $(f_n)$ uniformly bounded in $L^1(dx\otimes dv)$ with $v.\nabla_x f_n$ also uniformly bounded in $L^1 (dx\otimes dv) $, the sequence of velocity averages $\rho_n=\int f_n \Psi(v)dv$ is not relatively compact in $L^1(dx)$ (we refer to \cite{GLPS} for an explicit counter-example). This lack of compactness is due to the weak compactness pathologies of $L^1$. Indeed, as soon as we add some weak compactness to the sequence (or equivalently some equiintegrability in $x$ and $v$ in view of the classical Dunford-Pettis theorem), then we recover some strong compactness in $L^1$ for the moments (see Proposition 3 of \cite{GLPS} or Proposition \ref{equiXV} below).
We recall precisely the notion of equiintegrability which is central in this paper.
\begin{definition}\label{equi}
\begin{enumerate}
\item (Local equiintegrability in $x$ and $v$)
Let $(f_\epsilon)$ be a bounded family of
$L^1_{loc}(dx\otimes dv)$. It is said locally equiintegrable in $x$ and $v$ if and only if for any $\eta>0$ and for any compact subset $K\subset \mathbb{R}^{d}\times \mathbb{R}^{d}$, there exists $\alpha>0$ such that for any measurable set $A\subset \mathbb{R}^{d}\times \mathbb{R}^{d}$ with $\vert A\vert < \alpha$, we have for any $\epsilon$ : \begin{equation} \int_A \mathbbm{1}_K(x,v)\vert f_\epsilon(x,v)\vert dv dx \leq \eta. \end{equation}
\item (Local equiintegrability in $v$)
Let $(f_\epsilon)$ be a bounded family of
$L^1_{loc}(dx\otimes dv)$. It is said locally equiintegrable in $v$ if and only if for any $\eta>0$ and for any compact subset $K\subset \mathbb{R}^{d}\times \mathbb{R}^{d}$, there exists $\alpha>0$ such that for each family $(A_x)_{x \in \mathbb{R}^d}$ of measurable sets of $\mathbb{R}^{d}$ satisfying $\sup_{x \in \mathbb{R}^d} \vert A_x\vert < \alpha$, we have for any $\epsilon$ : \begin{equation} \int \left(\int_{A_x} \mathbbm{1}_K(x,v)\vert f_\epsilon(x,v)\vert dv \right)dx \leq \eta. \end{equation}
\end{enumerate}
\end{definition}
We observe that local equiintegrability in $(x,v)$ always implies local equiintegrability in $v$, whereas the converse is false in general.
The major improvement of the paper of Golse and Saint-Raymond \cite{GolSR} is to show that actually, only equiintegrability in $v$ is needed to obtain the $L^1$ compactness for the moments. This observation was one of the key arguments of their outstanding paper \cite{GSR} which establishes the convergence of renormalized solutions to the Boltzmann equation in the sense of DiPerna-Lions to weak solutions to the Navier-Stokes equation in the sense of Leray.
More precisely, the result they prove is Theorem \ref{L1} stated afterwards, with $F=0$ (free transport case). The aim of this paper is to show that the result also holds if one adds some force field $F(x)=(F_i(x))_{1\leq i \leq d} $ with $F \in W^{1,\infty}(\mathbb{R}^d)$:
\begin{theorem}\label{L1} Let $(f_\epsilon)$ be a family bounded in $L^1_{{loc}}(dx\otimes dv)$ locally equiintegrable in $v$ and such that $v.\nabla_x f_\epsilon+ F. \nabla_v f_\epsilon$ is bounded in $L^1_{{loc}}(dx\otimes dv)$. Then : \begin{enumerate}
\item $(f_\epsilon)$ is locally equiintegrable in both variables $x$ \textbf{and} $v$.
\item For all $\Psi \in \mathcal{C}^{1}_c(\mathbb{R}^d)$, the family $\rho_\epsilon(x)=\int f_\epsilon(x,v)\Psi(v)dv$ is relatively compact in $L^1_{{loc}}(dv)$. \end{enumerate} \end{theorem}
One key ingredient of the proof for $F=0$ is the nice dispersion properties of the free transport operator. We will show in Section \ref{mix} that an analogue also holds for small times when $F\neq 0$:
\begin{proposition}
Let $ F(x) $ be a Lipschitz vector field. There exists a maximal time $\tau>0$ (depending only on $\Vert \nabla_x F \Vert _{L^{\infty}}$) such that, if $f$ is the solution to the transport equation:
\[
\left\{
\begin{array}{ll}
\partial_t f + v.\nabla_x f +F.\nabla_v f=0, \\
f(0,.,.)=f^0\in L^p(dx \otimes dv),
\end{array}
\right. \] Then:
\begin{equation} \label{intromix} \forall \vert t \vert \leq \tau, \Vert{f(t)}\Vert_{L^\infty_x(L^1_v)} \leq
\frac{2}{|t|^{d}} \Vert{f^0}\Vert_{L^1_x(L^\infty_v)}. \end{equation} \end{proposition}
Let us also mention that the main theorem generalizes to the time-dependent setting, for transport equations of the form (\ref{time}). The usual trick to deduce such a result from the stationary case is to enlarge the phase space. Indeed we can consider $x'= (t,x)$ in $\mathbb{R}^{d+1}$ endowed with the Lebesgue measure, and $v'=(t,v)$ in $\mathbb{R}^{d+1}$ endowed with the measure $\mu=\delta_{t=1} \otimes \text{Leb}$ (where $\delta$ is the dirac measure). Then such a measure $\mu$ satisfies property (2.1) of \cite{GLPS}. As a consequence, all the results of Section \ref{first} will still hold.
Nevertheless, we observe that our key local in time mixing estimate (\ref{intromix}) seems to not hold when $\mathbb{R}^{d+1}$ is equipped with the new measure $\mu$ (the main problem being that the only speed associated to the first component of $v'$ is $1$). For this reason, we can not prove that equiintegrability in $v$ implies equiintegrability in $t$. One result (among other possible variants) is the following:
\begin{theorem}
\label{L1time}
Let $F (t,x)\in \mathcal{C}^0(\mathbb{R}^+, W^{1,\infty}(\mathbb{R}^d))$. Let $(f_\epsilon)$ be a family bounded in $L^1_{{loc}}(dt\otimes dx\otimes dv)$ locally equiintegrable in $v$ and such that $\partial_t f_\epsilon + v.\nabla_x f_\epsilon+ F. \nabla_v f_\epsilon$ is bounded in $L^1_{{loc}}(dt\otimes dx\otimes dv)$. Then : \begin{enumerate}
\item $(f_\epsilon)$ is locally equiintegrable in the variables $x$ and $v$ (but not necessarily with respect to $t$).
\item For all $\Psi \in \mathcal{C}^{1}_c(\mathbb{R}^d)$, the family $\rho_\epsilon(t,x)=\int f_\epsilon(t,x,v)\Psi(v)dv$ is relatively compact with respect to the $x$ variable in $L^1_{{loc}}(dt\otimes dx)$, that is, for any compact $K \subset \mathbb{R}^+_t \times \mathbb{R}^d_x$:
\begin{equation} \lim_{\delta \rightarrow 0} \sup_{\epsilon} \sup_{\vert x' \vert \leq \delta} \Vert (\mathbbm{1}_{ K}\rho_\epsilon)(t,x+x')- (\mathbbm{1}_{ K}\rho_\epsilon)(t,x)\Vert_{L^1(dt\otimes dx)} =0. \end{equation}
\end{enumerate}
\end{theorem}
Another possibility is to assume that $f_\epsilon$ is bounded in $L^\infty_{{t,loc}}(L^1_{x,v,loc})$, in which case we will get equiintegrability in $t,x$ and $v$ and thus compactness for $\rho_\epsilon$ in $t$ and $x$. We refer to \cite{GSR}, Lemma 3.6, for such a statement in the free transport case.
\begin{remarks} \label{rk} \begin{enumerate} \item Since the result of Theorem \ref{L1} is essentially of local nature, we could slightly weaken the assumption on $F$:
For any $R>0$, \begin{equation}
\exists M(R),\forall \vert x_1\vert ,\vert x_2\vert \leq R, \quad \vert F(x_1) - F(x_2) \vert \leq M(R) \vert x_1-x_2 \vert. \end{equation}
In other words we can deal with $ F \in W^{1,\infty}_{loc}$.
\item With the same proof, we can treat the case of force fields $F(x,v) \in W^{1,\infty}_{x,v,loc}$ with zero divergence in $v$ :
\[ \operatorname{div}_v F=0. \]
Typically we may think of the Lorentz force $v\wedge B$ where $B$ is a smooth magnetic field.
\item We can handle a family of force fields $(F_\epsilon)$ depending on $\epsilon$ as soon as $(F_\epsilon)$ is uniformly bounded in $ W^{1,\infty}(\mathbb{R}^d)$.
\end{enumerate} \end{remarks}
The following of the paper is devoted to the proof of Theorem \ref{L1}. In Section \ref{first}, we prove that a family satisfying the assumptions of Theorem \ref{L1} and in addition locally equiintegrable in $x$ and $v$, has moments which are relatively strongly compact in $L^1$. In Section \ref{mix}, we investigate the local in time mixing properties of the transport equation $\partial_t f + v.\nabla_x f + F.\nabla_v f =0$. Finally in the last section, thanks to the mixing properties we establish, we show by an interpolation argument that equiintegrability in $v$ provides some equiintegrability in $x$.
\section{A first step towards $L^1$ compactness} \label{first}
The first step is to show that under the assumptions of Theorem \ref{L1}, point 1 implies point 2. Using classical averaging lemma in $L^2$ (\cite{DPLvm}, \cite{DPLM}), we first prove the following $L^2$ averaging lemma.
\begin{lemma} \label{lemL2} Let $f,g \in L^2(dx \otimes dv)$ satisfy the transport equation: \begin{equation}
v.\nabla_x f + F. \nabla_v f = g. \end{equation}
Then for all $\Psi \in \mathcal{C}^{1}_c(\mathbb{R}^d)$, $\rho(x)=\int f(x,v)\Psi(v)dv \in H^{1/4}_x$. Moreover, \begin{equation} \Vert \rho \Vert_{H^{1/4}_x} \leq C\left( \left\Vert F \right\Vert_{L^{\infty}_x} \Vert f \Vert_{L^2_{x,v}} + \Vert g \Vert_{L^2_{x,v}}\right).\end{equation} ($C$ is a constant depending only on $\Psi$.) \end{lemma}
\begin{proof} The standard idea is to consider $-F. \nabla_v f + g$ as a source. Then, since $\operatorname{div}_v F(x)=0$, we have : \begin{eqnarray*} -F. \nabla_v f + g&=& -\sum_{i=1}^d \frac{\partial}{\partial_{v_i}}\left(F_i f \right)+g. \end{eqnarray*} We conclude by applying the $L^2$ averaging lemma of \cite{DPLvm}, Theorem 3. \end{proof}
We recall now in Proposition \ref{lemV} an elementary and classical representation result, obtained by the method of characteristics.
Let $b=(v,F)$, $Z=(X,V)$. Since $F \in W^{1,\infty}$, $b$ satisfies the hypotheses of the global Cauchy-Lipschitz theorem. We therefore consider the trajectories defined by:
\begin{equation} \label{carac} \left\{
\begin{array}{ll}
Z'(t;x_0,v_0)=b(Z(t;x_0,v_0)) \\
Z(0;x_0,v_0)=(x_0,v_0).
\end{array}
\right. \end{equation}
For all time, the application $(x_0,v_0) \mapsto Z(t;x_0,v_0)=(X(t;x_0,v_0),V(t;x_0,v_0))$ is well-defined and is a $C^1$ diffeomorphism. Moreover, since $b$ does not depend explicitly on time, it is also classical that $Z(t)$ is a group. The inverse is thus given by $(x,v) \mapsto Z(-t;x,v)$.
\begin{remark}Since $\mbox{div}(b)=0$, Liouville's theorem shows that the volumes in the phase space are preserved (the jacobian determinant of $Z$ is equal to $1$). \end{remark}
\begin{proposition} \label{lemV} \begin{enumerate}
\item The time-dependent Cauchy problem :
\begin{equation} \label{transport1} \left\{
\begin{array}{ll}
\partial_t f + v.\nabla_x f +F.\nabla_v f=0, \\
f(0,.,.)=f^0\in L^p(dx \otimes dv)
\end{array}
\right. \end{equation} has a unique solution (in the distributional sense) represented by
$$f(t,x,v)=f^0(X(-t;x,v)),V(-t;x,v))\in L^p(dx\otimes dv).$$
\item For any $ \lambda > 0$, the transport equation
\begin{equation} \label{transport2} \lambda f(x,v) + v.\nabla_x f +F.\nabla_v f=g \in L^p(dx\otimes dv) \end{equation} has a unique solution (in the distributional sense) represented by: $$R_\lambda :g(x,v) \mapsto f(x,v)=\int_0^{+\infty}e^{-\lambda s}g(X(-s;x,v),V(-s;x,v))ds \in L^p(dx\otimes dv).$$ In addition, $R_\lambda$ is a linear continuous map on $L^p$ with a norm equal to $\frac{1}{\lambda}$. \end{enumerate} \end{proposition}
Using Rellich's compactness theorem, we straightforwardly have the following corollary: \begin{corollary} \label{corL2} The linear continuous map $ T_{\lambda,\Psi}$ : $$L^2(dx \otimes dv) \rightarrow L^2_{{loc}}(dx)$$ $$ g \mapsto \rho=\int{R_\lambda(g)(.,v) \Psi(v)dv}$$ is compact for all $\Psi \in \mathcal{C}^1_{c}(\mathbb{R}^d)$ and all $\lambda >0$. \end{corollary}
\begin{proof} Using Lemma \ref{lemL2} and Proposition \ref{lemV}, we have: $$ \Vert T_{\lambda,\Psi}(g) \Vert_{H^{1/4}_x} \leq C \left( 1 + \Vert F \Vert_{L^\infty} \right) \Vert g \Vert_{L^2_{x,v}}. $$ The conclusion follows. \end{proof}
Using this compactness property, as in Proposition 3 of \cite{GLPS}, we can show the next result:
\begin{proposition} \label{equiXV}
Let $\mathcal{K}$ be a bounded subset of $L^1(dx \otimes dv)$ equiintegrable in $x$ and $v$ (in view of the Dunford-Pettis theorem, it means in other words that $\mathcal{K}$ is weakly compact in $L^1$), then $T_{\lambda,\Psi}(\mathcal{K})$ is relatively strongly compact in $L^1_{loc}(dx)$. \end{proposition}
\begin{proof} We recall the proof of this result for the sake of completeness.
The proof is based on a real interpolation argument. We fix a parameter $\eta>0$. For any $g \in \mathcal{K}$ and any $\alpha
>0$, we may write : \begin{equation*} g=g_1^\alpha +g_2^\alpha, \end{equation*} with \begin{eqnarray*}
g_1^\alpha&=&\mathbbm{1}_{\{|g(x,v)|>\alpha\}}g, \\
g_2^\alpha&=&\mathbbm{1}_{\{|g(x,v)|\leq\alpha\}}g. \end{eqnarray*}
Then, by linearity of $T_{\lambda,\Psi}$, we write $u=T_{\lambda,\Psi} (g)=u_1+ u_2$, with $u_1=T_{\lambda,\Psi}(g_1^\alpha)$ and $u_2=T_{\lambda,\Psi}(g_2^\alpha)$.
Let $K$ be a fixed compact set of $\mathbb{R}^{d}_x$.
We clearly have, since $T_{\lambda,\Psi}$ is linear continuous on $L^1(K)$: \begin{equation*} \Vert u_1 \Vert_{L^1_x(K)} \leq C\Vert g_1^\alpha \Vert_1. \end{equation*} We notice that:
$$\left\vert \{(x,v), |g(x,v)|>\alpha\}\right\vert \leq \frac{1}{\alpha} \Vert g \Vert_{L^1} \leq \frac{1}{\alpha} C.$$
Since $\mathcal{K}$ is equiintegrable, there exists $ \alpha
>0$ such that for any $ g \in \mathcal{K}$ : \begin{equation*}
\int |g\mathbbm{1}_{\{|g(x,v)|>\alpha\}}|dxdv \leq \frac{\eta}{C}. \end{equation*}
Consequently for $\alpha$ large enough, we have: \begin{equation*} \Vert u_1 \Vert_{L^1_x(K)} \leq \eta. \end{equation*}
The parameter $\alpha$ being fixed, we clearly see that $\{g_2^\alpha, g\in \mathcal{K}\}$ is a bounded subset of $L^1_{x,v}\cap L^\infty_{x,v}$, and consequently of $L^2_{x,v}$. Because of Corollary \ref{corL2}, $\{u_2, u_2=T_{\lambda,\Psi}( g_2^\alpha), g \in \mathcal{K}\}$ is relatively compact in $L^2_{{loc}}(dx)$. In particular it is relatively compact in $L^1_{loc}(dx)$.
As a result, we have shown that for any $\eta >0$, there exists $\mathcal{K}_\eta \subset L^1_x(K)$ compact, such that $T_{\lambda,\Psi}(\mathcal{K}) \subset \mathcal{K}_\eta + B(0,\eta)$. So this family is precompact and consequently it is compact since $L^1_x(K)$ is a Banach space.
\end{proof} We deduce the preliminary result (which means that the first point implies the second in Theorem \ref{L1}):
\begin{theorem}
\label{L1faible} Let $(f_\epsilon)$ a family of $L^1_{loc}(dx\otimes dv)$ locally equiintegrable in $x$ and $v$ such that $(v.\nabla_x f_\epsilon + F.\nabla_v f_\epsilon)$ is a bounded family of $L^1_{loc}(dx\otimes dv)$. Then for all $\Psi \in \mathcal{C}^{1}_c(\mathbb{R}^d)$, the family $\rho_\epsilon(x)=\int f_\epsilon(x,v)\Psi(v)dv$ is relatively compact in $L^1_{{loc}}(dx)$. \end{theorem}
\begin{proof} Let $\Psi \in\mathcal{C}^{1}_c(\mathbb{R}^d)$. Let $R>0$ be a large number such that $\operatorname{Supp} \Psi \subset B(0,R)$; we intend to show that $(\mathbbm{1}_{B(0,R)}(x)\rho_\epsilon(x))$ is compact in $L^1(B(0,R))$. First of all, we can assume that the $f_\epsilon$ are compactly supported in the same compact set $K \subset \mathbb{R}^d\times \mathbb{R}^d$, with $B(0,R)\times B(0,R) \subset \mathring{K}$. Indeed we can multiply the family by a smooth function $\chi$ such that: \begin{eqnarray*}
\operatorname{supp} \chi \subset K, \\
\chi \equiv 1 \text{ on } B(0,R)\times B(0,R). \end{eqnarray*}
We observe that : \begin{eqnarray*} v.\nabla_x(\chi f_\epsilon)&=&\chi(v.\nabla_x f_\epsilon)+f_\epsilon(v.\nabla_x \chi), \\ F.\nabla_v(\chi f_\epsilon)&=& \chi F.\nabla_v( f_\epsilon)+f_\epsilon (F. \nabla_v \chi). \end{eqnarray*} Thus the family $(\chi f_\epsilon)$ satisfies the same $L^1$ boundedness properties as $(f_\epsilon)$. The equiintegrability property is also clearly preserved. Furthermore, for any $x$ in $B(0,R)$, we have : $$\int f_\epsilon(x,v) \Psi(v) dv = \int f_\epsilon(x,v)\chi(v) \Psi(v) dv$$ Consequently we are now in the case of functions supported in the same compact set.
\par We have for all $ \epsilon>0, \lambda>0$, by linearity of the resolvent $R_\lambda$ defined in Proposition \ref{lemV} : \begin{equation*} \begin{split} \int f_\epsilon(x,v) \Psi(v) dv =& \int R_\lambda (\lambda f_\epsilon + v.\nabla_x f_\epsilon + F.\nabla_v f_\epsilon)\Psi(v)dv \\ =& \lambda \int (R_\lambda f_\epsilon)(x,v)\Psi(v)dv + \int (R_\lambda (v.\nabla_x f_\epsilon + F.\nabla_v f_\epsilon))(x,v)\Psi(v)dv. \end{split} \end{equation*}
Let $\eta>0$. We take $\displaystyle{\lambda=\sup_\epsilon \frac{\Vert (v.\nabla_x f_\epsilon + F.\nabla_v f_\epsilon)\Vert_{L^1_{x,v}}\Vert \Psi\Vert_{L^\infty}}{\eta}}$.
Then we have by Proposition \ref{lemV} : \begin{equation*} \begin{split} \left\Vert \int (R_\lambda (v.\nabla_x f_\epsilon+ F.\nabla_v f_\epsilon))(x,v)\Psi(v)dv \right\Vert_{L^1_x}\leq& \Vert R_\lambda (v.\nabla_x f_\epsilon + F.\nabla_v f_\epsilon) \Vert_{L^1_{x,v}}\Vert \Psi \Vert_{L^\infty_v} \\ \leq& \frac{1}{\lambda}\Vert (v.\nabla_x f_\epsilon + F.\nabla_v f_\epsilon)\Vert_{L^1_{x,v}}\Vert \Psi\Vert_{L^\infty_v} \\ \leq& \eta. \end{split} \end{equation*}
Moreover, since $(f_\epsilon)$ is bounded in $L^1(dx \otimes dv)$ and equiintegrable in $x$ and $v$, Proposition \ref{equiXV} implies that the family $(\int R_\lambda(f_\epsilon)\Psi(v)dv)$ is relatively compact in $L^1_{x}(B(0,R))$. Finally we can argue as for the end of the proof of Proposition \ref{equiXV}: for all $\eta >0$, there exists $K_\eta \subset L^1_{x}(B(0,R))$ compact, such that $(\rho_\epsilon)\subset K_\eta + B(0,\eta)$. So this family is precompact and consequently it is compact since $L^1(B(0,R))$ is a Banach space.
\end{proof}
\section{Mixing properties of the operator $v.\nabla_x + F. \nabla_v$} \label{mix}
\subsection{Free transport case}
In the case when $F=0$, Bardos and Degond in \cite{BD} proved a mixing result (also referred to as a dispersion result for large time asymptotics) which is a key argument in the proof of Theorem \ref{L1} (with $F=0$) by Golse and Saint-Raymond \cite{GolSR}. This kind of estimate was introduced for the study of classical solutions of the Vlasov-Poisson equation in three dimensions and for small initial data.
\begin{lemma}\label{freemixing} Let $f$ be the solution to:
\begin{equation} \label{transport} \left\{
\begin{array}{ll}
\partial_t f + v.\nabla_x f =0, \\
f(0,.,.)=f^0.
\end{array}
\right. \end{equation}
Then for all $t>0$: \begin{equation} \label{esti}
\Vert{f(t)}\Vert_{L^\infty_x(L^1_v)} \leq
\frac{1}{|t|^{d}} \Vert{f^0}\Vert_{L^1_x(L^\infty_v)}. \end{equation} \end{lemma}
For further results and related questions (Strichartz estimates...), we refer to Castella and Perthame \cite{CasPer} and Salort \cite{S1}, \cite{S2}, \cite{S3}.
\begin{figure}
\caption{Mixing property for free transport}
\label{melange}
\end{figure}
When $f^0$ is the indicator function of a set with "small" measure with respect to $x$, then the previous estimate (\ref{esti}) asserts that for $t>0$, $f(t)$ is for any fixed $x$ the indicator function of a set with a "small" measure in $v$ (at least that we may estimate): this property is crucial for the following. In (\ref{esti}), there is blow-up when $t \rightarrow 0$, which is intuitive, but this does not matter since we have nevertheless a control of the left-hand side for any positive time.
Actually, for our purpose, parameter $t$ is an artificial time (it does not have the usual physical meaning). It appears as an interpolation parameter in Lemma \ref{parts}, and can be taken rather small. This is the reason why local in time mixing is sufficient. We will consequently look for local in time mixing properties. Anyway, the explicit study in Example 2 below shows that the dispersion inequality is in general false for large times when $F\neq 0$.
\begin{proof}[Proof of Lemma \ref{freemixing}] The proof of this result is based on the explicit solution to (\ref{transport}), which is: \begin{equation*} f(t,x,v)=f^0(x-tv,v)\end{equation*} We now evaluate: \begin{eqnarray*} \Vert f(t) \Vert_{L^\infty_x(L^1_v)} &=& \sup_x \int f^0(x-tv,v)dv
\\&=& \sup_x \int f^0(z,\frac{x-z}{t}) |t|^{-d}dz
\\&\leq& |t|^{-d}\int \Vert f^0(z,.) \Vert_{\infty} dz
\\&\leq& |t|^{-d}\Vert f^0 \Vert_{L^1_x(L^\infty_v)}. \end{eqnarray*} The key argument is the change of variables $x-tv \mapsto z$, the jacobian of which is equal to $t^{-d}$. \end{proof}
We intend to do the same in the more complicated case when $f$ is the solution of a transport equation with $F \neq 0$. Let us mention that in \cite{BD}, Bardos and Degond actually prove the dispersion result for non zero force fields but with a polynomial decay in time. Here, this is not the case (the field $F$ does not even depend on time $t$), but we will prove that the result holds anyway for small times.
\subsection{Study of two examples}
In the following examples, $f$ is the explicit solution to the transport equation (\ref{transport1}) with an initial condition $f^0$ and a force deriving from a potential. \\
\textbf{Example 1}
Force $F= -\nabla_x V$, with $V = -|x|^2/2$.
Let $f$ be the solution to:
\begin{equation}
\left\{
\begin{array}{ll}
\partial_t f + v.\nabla_x f + x.\nabla_v f =0, \\
f(0,.,.)=f^0.
\end{array}
\right. \end{equation} The effect of such a potential will be to make the particles escape faster to infinity. So we expect to have results very similar to those of lemma \ref{freemixing}.
After straightforward computations we get : $$f(t,x,v)=f^0\left(x \left(\frac{e^t+e^{-t}}{2}\right)+v \left(\frac{e^{-t}-e^{t}}{2} \right),x \left(\frac{e^{-t}-e^{t}}{2} \right)+v \left(\frac{e^t+e^{-t}}{2} \right) \right),$$
which allows to show the same dispersion estimate with a factor $\frac{e^t-e^{-t}}{2}$ instead of $t$.
For all $t>0$, we have: \begin{equation} \Vert{f(t)}\Vert_{L^\infty_x(L^1_v)} \leq \frac{2^d}{(e^t-e^{-t})^{d}} \Vert{f^0}\Vert_{L^1_x(L^\infty_v)}. \end{equation}
\textbf{Example 2}
(Harmonic potential) Force $F= -\nabla_x V$, with $V =|x|^2/2$.
Let $f$ be the solution to:
\begin{equation}
\left\{
\begin{array}{ll}
\partial_t f + v.\nabla_x f - x.\nabla_v f =0, \\
f(0,.,.)=f^0.
\end{array}
\right. \end{equation}
With such a potential, particles are expected to be confined and consequently do not drift to infinity. For this reason, it is hopeless to prove the analogue of Lemma \ref{freemixing} for large times (here there is no dispersion). As mentioned before, it does not matter since we only look for a result valid for small times. We expect that there is enough mixing in the phase space to prove the result.
After straightforward computations we explicitly have : $$f(t,x,v)=f^0(x\cos t - v\sin t,v\cos t + x \sin t).$$ We observe here that the solution $f$ is periodic with respect to time. Thus, as expected, it is not possible to prove any decay when $t\rightarrow +\infty$; nevertheless we can prove a mixing estimate with a factor $\vert \sin t \vert$ instead of $t$.
For all $t>0$: \begin{equation} \Vert{f(t)}\Vert_{L^\infty_x(L^1_v)} \leq \frac{1}{\vert \sin t\vert^{d}} \Vert{f^0}\Vert_{L^1_x(L^\infty_v)}. \end{equation} Of course, this estimate is useless when $t= k \pi, k \in \mathbb{N}^*$.
\begin{remark} We notice that $\frac{e^t-e^{-t}}{2} \sim_0 t$ and $\sin (t)\sim_0 t$, which seems encouraging. \end{remark}
\subsection{General case : $F$ with Lipschitz regularity}
The study of these two examples suggests that at least for small times, the mixing estimate is still satisfied, maybe with a corrector term which does not really matter.
One nice heuristic way to understand this is to see that since $F$ is quite smooth, the dynamics associated to the operator $v.\nabla_x+ F.\nabla_v$ is expected to be close to those of free transport, at least for small times.
Let $X(t;x,v)$ and $V(t;x,v)$ be the diffeomorphisms introduced in the method of characteristics in Section \ref{first} and defined in (\ref{carac}).
Using Taylor's formula, we get by definition of $X$ and $V$ : \begin{eqnarray*} X(-t;x,v)=x-tv+ \int_0^t (t-s) F( X(-s;x,v))ds. \end{eqnarray*}
We recall Rademacher's theorem which asserts that $W^{1,\infty}$ functions are almost everywhere derivable. Hence, using Lebesgue domination theorem, we get:
\begin{eqnarray} \label{id} \partial_v [X(-t;x,v)]=-tId +\int_0^t (t-s) \nabla_x F(X(-s;x,v)) \partial_v [X(-s;x,v)]ds . \end{eqnarray}
We deduce the estimate : \begin{eqnarray*} \Vert \partial_v [X(-t;x,v)]\Vert_\infty \leq t + \int_0^t (t-s) \Vert \nabla_x F \Vert_\infty \Vert \partial_v [X(-s;x,v))]\Vert_\infty ds. \end{eqnarray*}
Gronwall's lemma implies then that:
\begin{equation} \label{Gronwall} \Vert \partial_v [X(-t;x,v)]\Vert_\infty \leq t e^{\frac{t^2}{2}\Vert \nabla_x F \Vert_\infty}. \end{equation}
We can also take the determinant of identity (\ref{id}) :
\begin{equation} \label{determ} \begin{split} \det (\partial_v [X(-t;x,v)] )=& \\(-t)^d& \det \left(Id - \frac{1}{t}\int_0^t (t-s) \nabla_x F(X(-s;x,v)) \partial_v [X(-s;x,v)]ds \right). \end{split} \end{equation}
The right-hand side is the determinant of a matrix of the form $Id+A(t)$ where $A$ is a matrix whose $L^\infty$ norm is small for small times $t$ (one can use estimate (\ref{Gronwall}) to ensure that $\Vert A(t) \Vert_\infty=o(t)$). Consequently, in a neighborhood of $0$, for any fixed $x$, $\partial_v [X(-t;x,v)]$ is invertible. Furthermore the map $v\mapsto X(-t;x,v))$ is injective for small positive times. Indeed, let $v \neq v'$. We compare:
\[ X(-t;x,v')-X(-t;x,v)=t(v-v')+ \int_0^t (t-s) [F( X(-s;x,v'))-F( X(-s;x,v)) ]ds. \] Consequently we have: \[ \vert X(-t;x,v')-X(-t;x,v)\vert \leq t\vert v-v' \vert + \int_0^t (t-s) \Vert \nabla_x F \Vert_{L^\infty} \vert X(-s;x,v')- X(-s;x,v) \vert ds. \] Thus, by Gronwall inequality we obtain: \[ \vert X(-t;x,v')-X(-t;x,v)\vert \leq t\vert v-v' \vert e^{\frac{t^2}{2}\Vert \nabla_x F\Vert_{L^\infty}}. \] Finally we observe that: \begin{equation*} \begin{split} \vert X(-t;x,v')-X(-t;x,v)\vert \geq & t\vert v-v' \vert - \left\vert\int_0^t (t-s) [F( X(-s;x,v'))-F( X(-s;x,v)) ]ds \right\vert\\ \geq & t\vert v-v' \vert - \int_0^t (t-s) \Vert \nabla_x F \Vert_{L^\infty} \vert X(-s;x,v')- X(-s;x,v) \vert ds \\ \geq & \vert v-v' \vert \left(t - \int_0^t (t-s) s e^{\frac{s^2}{2}\Vert \nabla_x F\Vert_{L^\infty}} \Vert \nabla_x F \Vert_{L^\infty}ds\right). \end{split} \end{equation*} Consequently, there is a maximal time $\tau_0>0$, depending only on $\Vert \nabla_x F \Vert_{L^\infty}$ such that for any $\vert t \vert\leq \tau_0$, we have : \[ \vert X(-t;x,v')-X(-t;x,v)\vert \geq \frac t 2 \vert v-v' \vert. \] This proves our claim.
Thus, by the local inversion theorem, this map is a $\mathcal{C}^1$ diffeomorphism on its image.
We have now the following elementary quantitative estimate :
\begin{lemma} Let $t\mapsto A(t)$ be a continuous map defined on a neighborhood of $0$, such that $\Vert A(t) \Vert_\infty=o(t)$. Then for small times: \begin{equation*} \det(Id+A(t))\geq 1 - d! \Vert A(t) \Vert_\infty. \end{equation*} We recall that $d$ is the space dimension and $d!=1\times 2 \times ... \times d$. \end{lemma}
We apply this lemma to (\ref{determ}), which allows us to say that there exists a maximal time $\tau>0$ such that for any $\vert t \vert\leq \tau$, we have :
\begin{equation} \label{det}
|\det (\partial_v [X(-t;x,v)]|^{-1} \leq 2\vert t\vert^{-d}. \end{equation}
We have proved that $v \mapsto X(-t;x,v)$ is a $\mathcal{C}^1$ diffeomorphism such that the jacobian of its inverse satisfies (\ref{det}) in a neighborhood of $t=0$. We can consequently conclude as in the proof of Lemma \ref{freemixing} (by performing the change of variables $X(-t;x,v)\mapsto v$).
As a result we have proved the proposition :
\begin{proposition} \label{mixing}
Let $ F(x)$ be a Lipschitz vector field. There exists a maximal time $\tau>0$ (depending only on $\Vert \nabla_x F \Vert _{L^{\infty}}$) such that, if $f$ is the solution to the transport equation:
\begin{equation}
\left\{
\begin{array}{ll}
\partial_t f + v.\nabla_x f +F.\nabla_v f=0, \\
f(0,.,.)=f^0\in L^p(dx \otimes dv).
\end{array}
\right. \end{equation} Then:
\begin{equation} \forall \vert t \vert \leq \tau, \Vert{f(t)}\Vert_{L^\infty_x(L^1_v)} \leq
\frac{2}{|t|^{d}} \Vert{f^0}\Vert_{L^1_x(L^\infty_v)}. \end{equation} \end{proposition}
\begin{remark} If one writes down more explicit estimates, it can be easily shown that $\tau$ is bounded from below by $T$ defined as the only positive solution to the equation:
\begin{equation}
\frac{d!}{3 }
\Vert \nabla_x F\Vert_{\infty}T^2 e^{\Vert \nabla_x F\Vert_{\infty} \frac{T^2}{2}} =1.
\end{equation} \end{remark}
\begin{remark} Of course, one can replace the factor $2$ in the mixing estimate by any $q>1$ (and the maximal time $\tau$ will depend also on $q$). \end{remark}
\section{From local equiintegrability in velocity to local equiintegrability in position and velocity}
In this section, we finally proceed as in \cite{GolSR}, with some slight modifications adapted to our case. We start from the following Green's formula : \begin{lemma} \label{parts} Let $f \in L^1(dx\otimes dv)$ with compact support such that $ v.\nabla_x f + F. \nabla_v f \in L^1(dx\otimes dv)$. Then for all $\Phi^0 \in L^\infty(dx\otimes dv)$, we have for all $t \in \mathbb{R}^*_+$ : \begin{equation} \begin{split} \int\ f(x,v) \Phi^0(x,v)dxdv=&\int\ f(x,v)\Phi(t,x,v)dxdv\\-&\int_0^t\int \Phi(s,x,v)(v.\nabla_x f + F. \nabla_v f)dsdxdv, \end{split} \end{equation} where $\Phi$ is the solution to: \begin{equation} \label{trans} \left\{
\begin{array}{ll}
\partial_t \Phi+ v.\nabla_x \Phi + F.\nabla_v \Phi= 0 \\
\Phi_{\vert t=0}=\Phi_0.
\end{array}
\right. \end{equation} \end{lemma}
\begin{proof} We have for all $t>0$, $\int_\Omega f(x,v)(\partial_t+v.\nabla_x+F.\nabla_v)\Phi(s,x,v)dsdxdv=0$, where $\Omega=]0,t[\times \mathbb{R}^d\times \mathbb{R}^d$. We first have: \begin{equation*} \int_\Omega f(x,v)\partial_t \Phi(s,x,v)dsdxdv=\int\int f(x,v) \Phi(t,x,v)dxdv - \int\int f(x,v) \Phi^0(x,v)dxdv. \end{equation*} Finally, by Green's formula we obtain: \begin{equation*} \begin{split} \int_0^t \int f(x,v)(v.\nabla_x +& F.\nabla_v)\Phi(s,x,v)dsdxdv \\= -&\int_0^t \int \Phi(s,x,v)(v.\nabla_x+F.\nabla_v)f(x,v)dsdxdv. \end{split} \end{equation*} There is no contribution from the boundaries since $f$ is compactly supported. \end{proof}
\begin{lemma} \label{lem1} Let $(f_\epsilon)$ a bounded family of $L^1_{loc}(dx\otimes dv)$ locally integrable in $v$ such that $(v.\nabla_x f_\epsilon + F.\nabla_v f_\epsilon)$ is a bounded family of $L^1_{loc}(dx\otimes dv)$. Then for all $\Psi \in \mathcal{C}^1_c(\mathbb{R}^d)$, such that $\Psi \geq 0$, the family $\rho_\epsilon(x)=\int \vert f_\epsilon(x,v) \vert \Psi(v)dv $ is locally equiintegrable. \end{lemma}
\begin{proof} Let $K_1$ be a compact subset of $\mathbb{R}^d$. We want to prove that $(\mathbbm{1}_{K_1}\rho_\epsilon(x))$ is equiintegrable. Without loss of generality, we can assume as previously that the $f_\epsilon$ are supported in the same compact support $K=K_1 \times K_2$ and that $\Psi$ is compactly supported in $K_2$. Furthermore, the formula $\nabla \vert f_\epsilon \vert = \operatorname{sign}(f_\epsilon)\nabla f_\epsilon$
shows that the $\vert f_\epsilon \vert$ satisfy the same assumptions of equiintegrability and $L^1$ boundedness as the family $(f_\epsilon)$. For the sake of readability, we will thus assume that $f_\epsilon$ are almost everywhere non-negative instead of considering $\vert f_\epsilon \vert $. Finally we may assume that $\Vert \Psi \Vert_\infty=1$ (multiplying by a constant does not change the equiintegrability property).
The idea of the proof is to show that thanks to the mixing properties established previously, the equiintegrability in $v$ provides some equiintegrability in $x$. \par
Let $\eta>0$. By definition of the local equiintegrability in $v$, we obtain a parameter
$\alpha>0$ associated to $K$ and $\eta$. We also consider parameters $\alpha'>0$ and $t\in]0,\tau[$ (where $\tau$ is the maximal time in Proposition \ref{mixing}) to be fixed ultimately. We mention that $t$ will be chosen only after $\alpha'$ is fixed.
Let $A$ a bounded mesurable subset included in $K_1$ with $|A|\leq \alpha'$. We consider $\Phi^0(x,v)=\mathbbm{1}_A(x)$ and $\Phi$ the solution of the transport equation (\ref{trans}) with $ \Phi^0$ as initial data.
Observe now that we have $\Vert \Phi^0
\Vert_{L^1_x(L^\infty_v)}=|A|$. Moreover, since $\Phi^0$ takes its values in $\{0,1\}$, it is also the case for $\Phi$ (this is a plain consequence of the transport of the data). \par We define for all $s>0$ and for all $x\in \mathbb{R}^d$, the set $A(s)_x=\{v\in\mathbb{R}^d, \Phi(s,x,v)=1\}$. At this point of the proof, we make a crucial use of the mixing property stated in Proposition \ref{mixing} : \begin{eqnarray*}
\sup_x |A(t)_x|&=&\sup_x \int \Phi(t,x,v)dv \\ &=& \Vert \Phi(t,.,.) \Vert_{L^\infty_x(L^1_v)} \\
&\leq& 2|t|^{-d} \underbrace{\Vert \Phi^0
\Vert_{L^1_x(L^\infty_v)}}_{|A|\leq \alpha'} \\ &\leq& \alpha, \end{eqnarray*} if we choose $\alpha'$ satisfying $\alpha'< \frac{1}{2} t^D \alpha$.
Thanks to Lemma \ref{parts}: \begin{equation*} \begin{split} \int f(x,v) \Psi(v)\Phi^0(x,v)dxdv=&\int f(x,v)\Psi(v)\Phi(t,x,v)dxdv\\-&\int_0^t\int \Phi(s,x,v)(v.\nabla_x + F. \nabla_v )(f_\epsilon(x,v)\Psi(v))dxdvds. \end{split} \end{equation*}
In other words, the operator $\partial_t + v.\nabla_x + F.\nabla_v$ has transported the indicator function and has transformed a subset small in $x$ into a subset small in $v$.
By definition of $\rho_\epsilon$, we have: $$ \int f(x,v) \Psi(v)\Phi^0(x,v)dxdv = \int \mathbbm{1}_A(x) \rho_\epsilon(x)dx. $$ By definition of $A(t)_x$ we also have: $$ \int f(x,v)\Psi(v)\Phi(t,x,v)dxdv=\int\left(\int_{A(t)_x}f_\epsilon(x,v)\Psi(v)dv \right)dx. $$
Thus, since $(f_\epsilon)$ are locally equiintegrable in $v$ we may evaluate: \begin{equation*} \begin{split}
\int\left(\int_{A(t)_x}f_\epsilon(x,v)\Psi(v)dv \right)dx \leq& \int \int_{A(t)_x} |f_\epsilon| \mathbbm{1}_{K} \underbrace{\Vert \Psi\Vert_{\infty}}_{=1}dxdv \\ \leq &\eta . \end{split} \end{equation*}
Finally we have: \begin{equation*} \begin{split} \int \mathbbm{1}_A(x) \rho_\epsilon(x)dx=& \int\left(\int_{A(t)_x}f_\epsilon(x,v)\Psi(v)dv \right)dx \\ -& \int_0^t\int \Phi(s,x,v)(v.\nabla_x + F. \nabla_v )(f_\epsilon(x,v)\Psi(v) )dsdxdv \end{split} \end{equation*} \begin{equation*} \begin{split} \leq&\eta + \int_0^t\int
|\Phi(s,x,v)||(v.\nabla_x + F. \nabla_v
)(f_\epsilon(x,v)\Psi(v))|dsdxdv \\ \leq& \eta + t\left[\underbrace{\Vert \Psi \Phi\Vert_\infty}_{\leq1} \Vert v.\nabla_x f_\epsilon + F.\nabla_v f_\epsilon \Vert_1 + \underbrace{\Vert \Phi\Vert_\infty}_{=1} \Vert F.\nabla_v \Psi(v)\Vert_\infty \Vert f_\epsilon \Vert_1\right]\\ \leq& 2\eta, \end{split} \end{equation*}
by taking $t$ sufficiently small: $$t<\frac{\eta}{\sup_\epsilon \Vert v.\nabla_x f_\epsilon + F.\nabla_v f_\epsilon \Vert_1+\Vert F.\nabla_v \Psi(v)\Vert_\infty \Vert f_\epsilon \Vert_1}.$$
This finally proves that $(\rho_\epsilon)$ is locally equiintegrable in $x$. \end{proof}
\begin{lemma} \label{lem2} Let $(g_\epsilon)$a bounded family of $L^1_{loc}(dx\otimes dv)$ locally equiintegrable in $v$. If for all $\Psi \in
\mathcal{C}^1_c(\mathbb{R}^d)$ such that $\Psi \geq 0$, $x \mapsto \int |g_
\epsilon(x,v)|\Psi(v)dv$ is locally equiintegrable (in $x$), then $(g_\epsilon)$ is locally equiintegrable in $x$ and $v$. \end{lemma}
\begin{proof} Let $K$ be a compact subset of $\mathbb{R}^d \times \mathbb{R}^d$. We want to prove that $(\mathbbm{1}_K g_\epsilon)$ is equiintegrable in $x$ and $v$. As before, we can clearly assume that the $g_\epsilon$ are compactly supported in $K$. \par Let $\eta>0$. By definition of the local equiintegrability in $v$ for $(g_\epsilon)$, we obtain $\alpha_1>0$ associated to $\eta$ and $K$.
\par Let $\Psi \in \mathcal{C}^1_c(\mathbb{R}^d)$ a smooth non-negative and compactly supported function such that $\Psi\equiv 1$ on $p_v(K)$ (where $p_v(K)$ is the projection of $K$ on $\mathbb{R}^d_v$). By assumption, there exists $\alpha_2
>0$ such that for any $A \subset \mathbb{R}^d$ measurable set satisfying
$|A|\leq \alpha_2$, \begin{equation*}
\int_A\left(\int |g_\epsilon|\Psi dv\right)dx<\eta. \end{equation*} Let $B$ a measurable subset of $\mathbb{R}^d \times \mathbb{R}^d$
such that $|B|<\inf (\alpha_1^2,\alpha_2^2)$. We define for all $x \in \mathbb{R}^d$, $B_x=\{v \in \mathbb{R}^d, (x,v) \in B \}$.
\par We consider now $E=\{x \in \mathbb{R}^d, |B_x|\leq |B|^{1/2}\}$ : this is the subset of $x$ for which there exist few $v$ such that $(x,v)\in B$. Consequently for this subset, we can use the local equiintegrability in $v$.
Concerning $B \backslash E$, on the contrary, we can not use this property, but thanks to Chebychev's inequality we show that this subset is of small measure, which allows us to use this time the
local equiintegrability in $x$ of $\int |g_ \epsilon(x,v)|\Psi(v)dv$ : \begin{eqnarray*}
|E^c|&=&\vert \{x \in \mathbb{R}^d, |B_x|> |B|^{1/2}\} \vert \\
&\leq& \frac{|B|}{|B|^{1/2}} \\ &\leq& \alpha_2. \end{eqnarray*}
Hence we have : \begin{eqnarray*}
\int \mathbbm{1}_B |g_\epsilon|dxdv &\leq& \int_E \left(\int_{B_x}
|g_\epsilon|dv\right)dx+\int_{E^c} \left(\int |g_\epsilon|dv \right)dx \\ &\leq& \eta + \int_{E^c} \left(\int
|g_\epsilon|\Psi(v)dv \right)dx \\ &\leq& 2\eta. \end{eqnarray*}
This shows the expected result. \end{proof}
We are now able to conclude the proof of Theorem \ref{L1}.
\begin{proof}[End of the proof of Theorem \ref{L1}]
If we successively apply Lemmas \ref{lem1} and \ref{lem2}, we deduce that the family $(f_\epsilon)$ is locally
equiintegrable in $x$ and $v$.
Finally we have shown in Section \ref{first} that the first point implies the second.
\end{proof}
\end{document} | arXiv |
\begin{document}
\title{\bf Convergence in probability of an ergodic and conformal multi-symplectic numerical scheme for a damped stochastic NLS equation}
\author{
{ Jialin Hong\footnotemark[1], Lihai Ji\footnotemark[2], and Xu Wang\footnotemark[3]}\\
}
\maketitle
\footnotetext{\footnotemark[1]\footnotemark[3]Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P.R.China. J. Hong and X. Wang are supported by National Natural Science Foundation of China (NO. 91530118, NO. 91130003, NO. 11021101 and NO. 11290142).}
\footnotetext{\footnotemark[2]Institute of Applied Physics and Computational Mathematics, Beijing 100094, P.R.China. L. Ji is supported by National Natural Science Foundation of China (NO. 11471310, NO. 11601032).}
\footnotetext{\footnotemark[3]Corresponding author: [email protected].}
\begin{abstract} In this paper, we investigate the convergence order in probability of a novel ergodic numerical scheme for damped stochastic nonlinear Schr\"{o}dinger equation with an additive noise. Theoretical analysis shows that our scheme is of order one in probability under appropriate assumptions for the initial value and noise. Meanwhile, we show that our scheme possesses the unique ergodicity and preserves the discrete conformal multi-symplectic conservation law. Numerical experiments are given to show the longtime behavior of the discrete charge and the time average of the numerical solution, and to test the convergence order, which verify our theoretical results. \\
\textbf{AMS subject classification: }{\rm\small37M25, 60H35, 65C30, 65P10.}\\
\textbf{Key Words: }{\rm\small}Stochastic nonlinear Schr\"{o}dinger equation, fully discrete scheme, ergodicity, conformal multi-symplecticity, charge exponential evolution, convergence order
\end{abstract}
\section{Introduction} We consider the following weakly damped stochastic nonlinear Schr\"odinger (NLS) equation with an additive noise (see also \cite{CHW16,DO05}) \begin{equation}\label{model} \left\{ \begin{aligned}
&d\psi-\mathbf{i} (\Delta\psi+\mathbf{i}\alpha \psi+\lambda|\psi|^2\psi)dt=\epsilon QdW,\quad t\ge0,\;x\in[0,1]\subset\mathbb{R},\\ &\psi(t,0)=\psi(t,1)=0,\\ &\psi(0,x)=\psi_0(x) \end{aligned} \right. \end{equation} with a complex-valued Wiener process $W$ defined on a filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},\mathbb{P})$, a linear positive operator $Q$ on $L^2:=L^2(0,1)$ and $\lambda=\pm1$. $\alpha>0$ is the absorption coefficient, $\epsilon\ge0$ describes the size of the noise. In addition, $Q$ is assumed to commute with $\Delta$ and satisfies $Qe_k=\sqrt{\eta_k}e_k$, which $\{e_k\}_{k\ge1}$ is an eigenbasis of $\Delta$ with Dirichlet boundary condition in $L_0^2:=L^2_0(0,1)$. Throughout the paper, the subscript $0$ represents the homogenous boundary condition. Some additional assumptions on $Q$ will be given bellow. The Karhunen--Lo\`eve expansion yields \begin{align*} QW(t,x)=\sum_{k=0}^{\infty}\sqrt{\eta_k} e_k(x)\beta_k(t),\quad t\geq0,\quad x\in[0,1], \end{align*} where $\beta_k=\beta_k^1+\mathbf{i}\beta_k^2$ with $\beta_k^1$ and $\beta_k^2$ being its real and imaginary parts, respectively. In addition, we assume that $\{\beta_k^i\}_{k\ge1,i=1,2}$ is a family of $\mathbb{R}$-valued independent identified Brownian motions.
As is well known, for the deterministic cubic NLS equation, the charge of the solution is a constant, i.e., $\|\psi(t)\|_{L^{2}}=\|\psi_{0}\|_{L^{2}}$. However, for \eqref{model} with a damped term and an additive noise in addition, the charge is no longer preserved and satisfies (see Proposition 2.1 in Section 2) \begin{align}\label{ergodic}
\mathbb{E}\|\psi(t)\|_{L^2}^2=e^{-2\alpha t}\mathbb{E}\|\psi_0\|_{L^2}^2+\frac{\epsilon^{2}\eta}{\alpha}(1-e^{-2\alpha t}), \end{align} where $\eta:=\sum_{k=1}^{\infty}\eta_k<\infty$. From this equation, it can be seen that the damped term is necessary to ensure the uniform boundedness of the solution in stochastic case, which also ensures the existence of invariant measures. Indeed, if $\alpha=0$ and $\epsilon\neq0$, the $L^2$ norm grows linearly in time. Moreover, if there exists a unique invariant measure $\mu$ which satisfies \begin{align} \lim_{T\to\infty}\frac1T\int_0^T\mathbb{E} f(\psi(s))ds=\int_{H_0^1}fd\mu,\quad\forall~ f\in C_b(H_0^1) \end{align} with $H^1_0:=H^1_0(0,1)$, we say that the random process $\psi(t)$ is uniquely ergodic \cite{DO05}. The interested readers are referred to \cite{daprato,DO05} and references therein for the study of ergodicity with respect to the exact solution of stochastic PDEs. We also refer to \cite{abdulle,mattingly,talay,T02} for the study of ergodicity as well as approximate error with respect to numerical solutions of stochastic ODEs and to \cite{brehier,brehier2,brehier3} for those of parabolic stochastic PDEs. For damped stochastic NLS equation \eqref{model}, the authors in \cite{DO05} prove both the existence and the uniqueness of the invariant measure, and \cite{CHW16} presents an ergodic fully discrete scheme which is of order $2$ in space for $\lambda=0$ or $\pm1$ and order $\frac12$ in time for $\lambda=0$ or $-1$ in the weak sense.
The main goal of this work is to construct a fully discrete scheme of \eqref{model} which could inherit both the unique ergodicity and some other internal properties of the original equation, e.g., conformal multi-symplectic property (see \cite{MNS13} for a detailed description in deterministic case), and to give the optimal convergence order of the proposed scheme in probability. To this end, we first apply the central finite difference scheme to \eqref{model} in spatial direction to get a semi-discretized equation, whose solution is shown to be symplectic and uniformly bounded. Then, a splitting technique is used to discretize the semi-discretized equation and obtain an explicit fully discrete scheme. We show that the proposed scheme possesses a conformal multi-symplectic conservation law with its solution uniformly bounded. Thanks to the non-degeneracy of the additive noise, the numerical solution is also shown to be irreducible and strong Feller, which yields the uniqueness of the invariant measure. Due to the fact that the nonlinear term of \eqref{model} is not global Lipschitz, it is particularly challenging and difficulty to analyze the convergence order of the proposed scheme. Motivated by \cite{BD06,L13}, we construct a truncated equation with a global Lipschitz nonlinear term such that the proposed scheme applied to the truncated equation shows order one in mean-square sense under appropriate hypothesis on initial value and noise. We then construct a submartingale based on which we finally derive convergence order one in probability for the original equation in temporal direction. To the best of our knowledge, there has been no work in the literature which constructs schemes with both ergodicity and conformal multi-symplecticity to \eqref{model}.
The rest of the paper is organized as follows. We show the conformal multi-symplecticity and the charge evolution for \eqref{model} in section 2. In section 3, we construct a fully discrete scheme, which could inherit both the ergodicity and the conformal multi-symplecitcity of the original system. In section 4, we introduce a truncated equation, based on which we derive convergence order one in probability for the proposed scheme. Numerical experiments are carried out in section 5 to verify our theoretical results.
\section{Damped stochastic NLS equation} This section is devoted to investigate the internal properties of \eqref{model}. We define the space-time white noise $\dot{\chi}=\frac{dW}{dt}$, set $\psi=p+\mathbf{i} q$, $\dot{\chi}=\dot{\chi}_1+\mathbf{i}\dot{\chi}_2$ with $p$, $q$, $\dot{\chi}_1=\frac{dW_1}{dt}$ and $\dot{\chi}_2=\frac{dW_2}{dt}$ being real-valued functions, and rewrite \eqref{model} as \begin{equation}\label{pq} \left\{ \begin{aligned} \begin{split} p_t+q_{xx}+\alpha p+\lambda(p^2+q^2)q&=\epsilon Q\dot{\chi}_1,\\[2mm] -q_t+p_{xx}-\alpha q+\lambda(p^2+q^2)p&=-\epsilon Q\dot{\chi}_2. \end{split} \end{aligned} \right. \end{equation} Denoting $v=p_x$, $w=q_x$, $z=(p,q,v,w)^T$, above equations can be transformed into a compact form \begin{align}\label{multisym} Md_tz+K\partial_xzdt=-\alpha Mzdt+\nabla S_0(z)dt+\nabla S_1(z)\circ dW_1+\nabla S_2(z)\circ dW_2, \end{align} where \begin{equation*}M= \left( \begin{array}{cccc} 0&-1&0&0\\ 1&0&0&0\\ 0& 0&0 &0\\ 0&0&0&0\\ \end{array} \right),\quad K= \left( \begin{array}{cccc} 0&0&1&0\\ 0&0&0&1\\ -1& 0&0 &0\\ 0&-1&0&0\\ \end{array} \right) \end{equation*} and \begin{equation*} S_0(z)=-\frac{\lambda}4(p^2+q^2)^2-\frac12(v^2+w^2),\quad S_1(z)=\epsilon Qq,\quad S_2(z)=-\epsilon Qp. \end{equation*}
In the sequel, we use the notations $L^2:=L^2(0,1)$, $H^p:=H^p(0,1)$ and denote the domain of operators $\Delta^{\frac{p}2}$ with Dirichlet boundary condition by \begin{align*}
\dot{H}^p:=D(\Delta^{\frac{p}2})=\left\{u\in L^2_0\bigg{|}\|u\|_{\dot{H}^p}:=\|\Delta^{\frac{p}2}u\|_{L^2}=\sum_{k=1}^{\infty}(k\pi)^p|(u,e_k)|^2\le\infty\right\},\quad p\ge1 \end{align*}
with $(u,v):=\int_0^1u(x)\overline{v}(x)dx$ for all $u,v\in L^2$ and $e_k(x)=\sqrt{2}\sin(k\pi x)$. Furthermore, we denote the set of Hilbert-Schimdt operators from $L^2$ to $\dot{H}^p$ by $\mathcal{L}_2^p$ with norm $$\|Q\|_{\mathcal{L}_2^p}:=\sum_{k=1}^{\infty}\|Qe_k\|^2_{\dot{H}^p},\quad p\ge1.$$ Without pointing out, the equations below hold in the sense $\mathbb{P}$-a.s. We now prove that \eqref{model} possesses the stochastic conformal multi-symplectic structure, whose definition is also given in the following theorem. \begin{tm} Eq. \eqref{model} is a stochastic conformal multi-symplectic Hamiltionian system, and preserves the stochastic conformal multi-symplectic conservation law \begin{align*} d_t\omega(t,x)+\partial_x\kappa(t,x)dt=-\alpha\omega(t,x)dt, \end{align*} which means \begin{equation}\label{2forms} \begin{split} \int_{x_0}^{x_1}\omega(t_1,x)dx -\int_{x_0}^{x_1}\omega(t_0,x)dx &+\int_{t_0}^{t_1}\kappa(t,x_1)dt -\int_{t_0}^{t_1}\kappa(t,x_0)dt\\ &=-\int_{x_0}^{x_1}\int_{t_0}^{t_1}\alpha\omega(t,x)dtdx, \end{split} \end{equation} where $\omega=\frac12dz\wedge Mdz$ and $\kappa=\frac12dz\wedge Kdz$ are two differential 2-forms associated with two skew-symmetric matrices $M$ and $K$. \end{tm} \begin{proof} To simplify the proof, we denote $(z_1,z_2,z_3,z_4):=(p,q,v,w)=z^T$ and $(z_l)_t^x:=z_l(t,x)$ for $l=1,2,3,4.$ Noticing that $\omega=dz_2\wedge dz_1$ and $\kappa=dz_1\wedge dz_3+dz_2\wedge dz_4$, thus we have \begin{equation}\label{H} \begin{split} &\int_{x_0}^{x_1}\omega(t_1,x)dx -\int_{x_0}^{x_1}\omega(t_0,x)dx\\ =&\int_{x_0}^{x_1}\Big{[}d(z_2)_{t_1}^x\wedge d(z_1)_{t_1}^x-d(z_2)_{t_0}^x\wedge d(z_1)_{t_0}^x\Big{]}dx\\ =&\int_{x_0}^{x_1}\Bigg{[}\left(\sum_{l=1}^4\frac{\partial(z_2)_{t_1}^x}{\partial(z_l)_{t_0}^{x_0}}d(z_l)_{t_0}^{x_0}\right)\wedge\left(\sum_{i=1}^4\frac{\partial(z_1)_{t_1}^x}{\partial(z_i)_{t_0}^{x_0}}d(z_i)_{t_0}^{x_0}\right)\\ &-\left(\sum_{l=1}^4\frac{\partial(z_2)_{t_0}^x}{\partial(z_l)_{t_0}^{x_0}}d(z_l)_{t_0}^{x_0}\right)\wedge\left(\sum_{i=1}^4\frac{\partial(z_1)_{t_0}^x}{\partial(z_i)_{t_0}^{x_0}}d(z_i)_{t_0}^{x_0}\right)\Bigg{]}dx\\ =&\sum_{l=1}^4\sum_{i=1}^4\left[\int_{x_0}^{x_1}\left(\frac{\partial(z_2)_{t_1}^x}{\partial(z_l)_{t_0}^{x_0}}\frac{\partial(z_1)_{t_1}^x}{\partial(z_i)_{t_0}^{x_0}} -\frac{\partial(z_2)_{t_0}^x}{\partial(z_l)_{t_0}^{x_0}}\frac{\partial(z_1)_{t_0}^x}{\partial(z_i)_{t_0}^{x_0}}\right)dx\right] d(z_l)_{t_0}^{x_0}\wedge d(z_i)_{t_0}^{x_0}\\ =&:\sum_{l=1}^4\sum_{i=1}^4\mathcal{H}_{l,i}(t_1,x_1) d(z_l)_{t_0}^{x_0}\wedge d(z_i)_{t_0}^{x_0}, \end{split} \end{equation} where $\mathcal{H}_{l,i}(t_1,x_1)=\int_{x_0}^{x_1}\left(\frac{\partial(z_2)_{t_1}^x}{\partial(z_l)_{t_0}^{x_0}}\frac{\partial(z_1)_{t_1}^x}{\partial(z_i)_{t_0}^{x_0}} -\frac{\partial(z_2)_{t_0}^x}{\partial(z_l)_{t_0}^{x_0}}\frac{\partial(z_1)_{t_0}^x}{\partial(z_i)_{t_0}^{x_0}}\right)dx$. Similarly, we obtain \begin{equation}\label{M} \begin{split} &\int_{t_0}^{t_1}\kappa(t,x_1)dt -\int_{t_0}^{t_1}\kappa(t,x_0)dt\\ =&\sum_{l=1}^4\sum_{i=1}^4\Bigg{[}\int_{t_0}^{t_1}\Bigg{(} -\frac{\partial(z_1)_{t}^{x_1}}{\partial(z_i)_{t_0}^{x_0}}\frac{\partial(z_3)_{t}^{x_1}}{\partial(z_l)_{t_0}^{x_0}} +\frac{\partial(z_1)_{t}^{x_0}}{\partial(z_i)_{t_0}^{x_0}}\frac{\partial(z_3)_{t}^{x_0}}{\partial(z_l)_{t_0}^{x_0}}\\ &-\frac{\partial(z_2)_{t}^{x_1}}{\partial(z_l)_{t_0}^{x_0}}\frac{\partial(z_4)_{t}^{x_1}}{\partial(z_i)_{t_0}^{x_0}} +\frac{\partial(z_2)_{t}^{x_0}}{\partial(z_l)_{t_0}^{x_0}}\frac{\partial(z_4)_{t}^{x_0}}{\partial(z_i)_{t_0}^{x_0}}\Bigg{)}dt\Bigg{]} d(z_l)_{t_0}^{x_0}\wedge d(z_i)_{t_0}^{x_0}\\ =&:\sum_{l=1}^4\sum_{i=1}^4\mathcal{M}_{l,i}(t_1,x_1)d(z_l)_{t_0}^{x_0}\wedge d(z_i)_{t_0}^{x_0} \end{split} \end{equation} and \begin{align} \int_{x_0}^{x_1}\int_{t_0}^{t_1}\alpha\omega(t,x)dtdx =&2\alpha\sum_{l=1}^4\sum_{i=1}^4\left[\int_{x_0}^{x_1}\int_{t_0}^{t_1}\left(\frac{\partial(z_2)_{t}^x}{\partial(z_l)_{t_0}^{x_0}}\frac{\partial(z_1)_{t}^x}{\partial(z_i)_{t_0}^{x_0}}\right)dtdx\right] d(z_l)_{t_0}^{x_0}\wedge d(z_i)_{t_0}^{x_0}\nonumber\\\label{N} =&:2\alpha\sum_{l=1}^4\sum_{i=1}^4\mathcal{N}_{l,i}(t_1,x_1)d(z_l)_{t_0}^{x_0}\wedge d(z_i)_{t_0}^{x_0}. \end{align} Adding \eqref{H}, \eqref{M} and \eqref{N} together, we can find out that equation \eqref{2forms} holds if \begin{align}\label{HMN} \mathcal{H}_{l,i}(t_1,x_1)+\mathcal{M}_{l,i}(t_1,x_1)+2\alpha\mathcal{N}_{l,i}(t_1,x_1)=0 \end{align} for any $l,i=1,2,3,4$, $t_1\in\mathbb{R}_+$ and $x_1\in\mathbb{R}$. In fact, rewritting \eqref{pq} as \begin{equation*} \left\{ \begin{aligned} d_{t}z_1=&-\partial_x(z_4)dt-\alpha z_1dt+\frac{\partial S_0(z)}{\partial z_2}dt+\epsilon QdW_1,\\ d_{t}z_2=&\partial_x(z_3)dt-\alpha z_2dt-\frac{\partial S_0(z)}{\partial z_1}dt+\epsilon QdW_2 \end{aligned} \right. \end{equation*} and taking partial derivatives with respect to $(z_i)_{t_0}^{x_0}$ and $(z_l)_{t_0}^{x_0}$ respectively, we have \begin{equation*} \left\{ \begin{aligned} d_{t}\frac{\partial(z_1)_{t}^x}{\partial(z_i)_{t_0}^{x_0}}=&-\frac{\partial}{\partial x}\frac{\partial(z_4)_{t}^x}{\partial(z_i)_{t_0}^{x_0}}dt-\alpha \frac{\partial(z_1)_{t}^x}{\partial(z_i)_{t_0}^{x_0}}dt+\sum_{l=1}^4\frac{\partial S_1(z)}{\partial (z_2)_{t}^x\partial(z_l)_{t}^{x}}\frac{\partial(z_l)_t^x}{\partial(z_i)_{t_0}^{x_0}}dt,\\ d_{t}\frac{\partial(z_2)_{t}^x}{\partial(z_l)_{t_0}^{x_0}}=&\frac{\partial}{\partial x}\frac{\partial(z_3)_{t}^x}{\partial(z_l)_{t_0}^{x_0}}dt-\alpha \frac{\partial(z_2)_{t}^x}{\partial(z_l)_{t_0}^{x_0}}dt-\sum_{i=1}^4\frac{\partial S_1(z)}{\partial (z_1)_t^x\partial(z_i)_{t}^{x}}\frac{\partial(z_i)_{t}^{x}}{\partial(z_l)_{t_0}^{x_0}}dt. \end{aligned} \right. \end{equation*} Furthermore, \begin{align*} d_{t_1}\mathcal{H}_{l,i}(t_1,x_1)=&\int_{x_0}^{x_1}\frac{\partial(z_1)_{t_1}^x}{\partial(z_i)_{t_0}^{x_0}}d_{t_1}\left(\frac{\partial(z_2)_{t_1}^x}{\partial(z_l)_{t_0}^{x_0}}\right)+\frac{\partial(z_2)_{t_1}^x}{\partial(z_l)_{t_0}^{x_0}}d_{t_1}\left(\frac{\partial(z_1)_{t_1}^x}{\partial(z_i)_{t_0}^{x_0}}\right)dx\\ =&\int_{x_0}^{x_1}\frac{\partial(z_1)_{t_1}^x}{\partial(z_i)_{t_0}^{x_0}}\left(\frac{\partial}{\partial x}\frac{\partial(z_3)_{t_1}^x}{\partial(z_l)_{t_0}^{x_0}}-\alpha \frac{\partial(z_2)_{t_1}^x}{\partial(z_l)_{t_0}^{x_0}}\right)dt_1dx\\ &-\int_{x_0}^{x_1}\frac{\partial(z_2)_{t_1}^x}{\partial(z_l)_{t_0}^{x_0}}\left(\frac{\partial}{\partial x}\frac{\partial(z_4)_{t_1}^x}{\partial(z_i)_{t_0}^{x_0}}+\alpha \frac{\partial(z_1)_{t_1}^x}{\partial(z_i)_{t_0}^{x_0}}\right)dt_1dx\\ =&\frac{\partial(z_1)_{t_1}^{x_1}}{\partial(z_i)_{t_0}^{x_0}}\frac{\partial(z_3)_{t_1}^{x_1}}{\partial(z_l)_{t_0}^{x_0}} -\frac{\partial(z_1)_{t_1}^{x_0}}{\partial(z_i)_{t_0}^{x_0}}\frac{\partial(z_3)_{t_1}^{x_0}}{\partial(z_l)_{t_0}^{x_0}} -\frac{\partial(z_2)_{t_1}^{x_1}}{\partial(z_l)_{t_0}^{x_0}}\frac{\partial(z_4)_{t_1}^{x_1}}{\partial(z_i)_{t_0}^{x_0}}\\ &+\frac{\partial(z_2)_{t_1}^{x_0}}{\partial(z_l)_{t_0}^{x_0}}\frac{\partial(z_4)_{t_1}^{x_0}}{\partial(z_i)_{t_0}^{x_0}} -2\alpha\int_{x_0}^{x_1}\frac{\partial(z_1)_{t_1}^x}{\partial(z_i)_{t_0}^{x_0}}\frac{\partial(z_2)_{t_1}^x}{\partial(z_l)_{t_0}^{x_0}}dxdt_1\\ =&-d_{t_1}\mathcal{M}_{l,i}(t_1,x_1)-2\alpha d_{t_1}\mathcal{N}_{l,i}(t_1,x_1), \end{align*} which together with the fact that $\mathcal{H}_{l,i}(t_0,x_1)+\mathcal{M}_{l,i}(t_0,x_1)+2\alpha\mathcal{N}_{l,i}(t_0,x_1)=0$ yields \eqref{HMN}. We hence complete the proof. \end{proof} Next, we show that the charge of the solution $\psi(t)$, although it is not conserved anymore, satisfies an exponential type evolution law. \begin{prop}\label{exactmoment}
Assume that $\mathbb{E}\|\psi_0\|_{L^2}^2<\infty$, then the solution of \eqref{model} is uniformly bounded with \begin{align}\label{bounded}
\mathbb{E}\|\psi(t)\|_{L^2}^2=e^{-2\alpha t}\mathbb{E}\|\psi_0\|_{L^2}^2+\frac{\epsilon^{2}\eta}{\alpha}(1-e^{-2\alpha t}). \end{align} \end{prop} \begin{proof}
The It\^o's formula applied to $\|\psi(t)\|_{L^2}$ yields \begin{align*}
d\|\psi(t)\|^2_{L^2}=-2\alpha\|\psi(t)\|^2_{L^2}dt+2\epsilon\Re\left[\int_0^1\overline{\psi}QdxdW\right]+2\epsilon^{2}\eta dt, \end{align*} where $\Re[\cdot]$ denotes the real part of a complex value. Taking expectation on both sides of above equation and solving the ordinary differential equation, we derive \begin{equation*} \begin{split}
\mathbb{E}\|\psi(t)\|^2_{L^2}
=&e^{-2\alpha t}\left(\int_0^t2\epsilon^{2}\eta e^{2\alpha s}ds+\mathbb{E}\|\psi_0\|^2_{L^2}\right)\\
&=e^{-2\alpha t}\mathbb{E}\|\psi_0\|_{L^2}^2+\frac{\epsilon^{2}\eta}{\alpha}(1-e^{-2\alpha t}). \end{split} \end{equation*} This concludes the proof. \end{proof}
We hence get the conclusion that, the damped stochastic NLS equation \eqref{model} possesses the stochastic conformal multi-symplectic conservation law \eqref{2forms} with its solution being uniformly bounded \eqref{bounded}, as well as the unique ergodicity \cite{DO05}. A natural question is how to construct numerical schemes which could inherit the properties of \eqref{model} as many as possible, such as, stochastic conformal multi-symplectic conservation law, uniform boundedness of the solution and the unique ergodicity.
\section{Ergodic fully discrete scheme} We now focus on the construction of schemes which could inherit the stochastic conformal multi-symplecticity and the unique ergodicity. \subsection{Spatial semi-discretization} We apply central finite difference scheme to \eqref{model} and obtain \begin{equation}\label{fenliang} \left\{ \begin{aligned}
&d\psi_j-\mathbf{i} \left(\frac{\psi_{j+1}-2\psi_j+\psi_{j-1}}{h^2}+\mathbf{i}\alpha\psi_j+\lambda|\psi_j|^2\psi_j\right)dt=\epsilon\sum_{k=1}^P\sqrt{\eta_k}e_k(x_j)d\beta_k(t),\\ &\psi_0(t)=\psi_{J+1}(t)=0,\\ &\psi_j(0)=\psi_0(x_j), \end{aligned} \right. \end{equation} where $h$ is the uniform spatial step and $\psi_j:=\psi_j(t)$ is an approximation of $\psi(x_j,t)$ with $x_j=jh$, $j=1,2,\cdots,J$ and $(J+1)h=1$. With notations $\Psi=(\psi_1,\cdots,\psi_J)^T\in\mathbb{C}^J$, $\beta=(\beta_1,\cdots,\beta_P)^T\in\mathbb{C}^P$, $F(\Psi)=
\text{diag}\{|\psi_1|^2,\cdots,|\psi_J|^2\}$, $\Lambda=\text{diag}\{\sqrt{\eta_1^{}},\cdots,\sqrt{\eta_P^{}}\}$, \begin{equation*} A=\left( \begin{array}{cccc} -2&1 & & \\
1&-2&1 & \\ &\ddots &\ddots &\ddots \\
& &1 &-2\\ \end{array} \right)\quad\text{and}\quad \sigma= \left( \begin{array}{ccc} e_1^{}(x_1^{})&\cdots &e_P^{}(x_1^{})\\ \vdots& &\vdots\\ e_1^{}(x_J^{})&\cdots&e_P^{}(x_J^{}) \end{array} \right), \end{equation*} we rewrite \eqref{fenliang} into a finite dimensional stochastic differential equation \begin{equation}\label{space} \left\{ \begin{aligned} &d\Psi-\mathbf{i} \left(\frac1{h^2}A\Psi+\mathbf{i}\alpha\Psi+\lambda F(\Psi)\Psi\right)dt=\epsilon\sigma\Lambda d\beta,\\ &\Psi(0)=(\psi_0(x_1^{}),\cdots,\psi_0(x_J^{}))^T. \end{aligned} \right. \end{equation}
In the sequel, we denote the 2-norm for vectors or matrices by $\|\cdot\|$, i.e., $\|v\|=\left(\sum_{j=1}^J|v_j|^2\right)^{1/2}$ for a vector $v=(v_1,\cdots,v_J)^T\in\mathbb{C}^J$ and $\|A\|$=`the square root of the maximum eigenvalues of $A^TA$' for a matrix A. The solution of \eqref{space} is uniformly bounded, which is stated in the following proposition. \begin{prop}\label{semimoment}
Assume that $\mathbb{E}\|\psi_0\|_{L^2}^2<\infty$, then the solution $\Psi$ of \eqref{space} is uniformly bounded with \begin{align}
h\mathbb{E}\|\Psi(t)\|^2\le e^{-2\alpha t}h\mathbb{E}\|\Psi(0)\|^2+\frac{2\epsilon^2\eta^{(P)}}{\alpha}(1-e^{-2\alpha t}), \end{align} where $\eta^{(P)}:=\sum_{k=1}^P\eta_k^{}$. \end{prop} \begin{proof}
Similar to the proof of Proposition \ref{exactmoment}, we apply It\^o's formula to $\|\Psi(t)\|^2$ and obtain \begin{equation}\label{semiito} \begin{split}
d\|\Psi(t)\|^2&=2\Re[\overline{\Psi}^Td\Psi]+(\epsilon\sigma\Lambda d\overline{\beta})^T(\epsilon\sigma\Lambda d\beta)\\[1mm]
&=-2\alpha\|\Psi(t)\|^2dt+2\Re[\overline{\Psi}^T\epsilon\sigma\Lambda d\beta]\\ &+\epsilon^2\sum_{j=1}^J\left[\left(\sum_{k=1}^P\sqrt{\eta_k}e_k(x_j)d\overline{\beta_k}\right)^T\left(\sum_{k=1}^P\sqrt{\eta_k}e_k(x_j)d{\beta}_k\right)\right]. \end{split} \end{equation} Taking expectation on both sides of above equation leads to \begin{align*}
d\mathbb{E}\|\Psi(t)\|^2=&-2\alpha \mathbb{E}\|\Psi(t)\|^2dt+2\epsilon^2\sum_{j=1}^J\sum_{k=1}^P\eta_ke_k^2(x_j)dt. \end{align*} Thus, multiplying above equation by $he^{2\alpha t}$ and taking integral from $0$ to $t$ leads to \begin{align*}
\int_0^the^{2\alpha t}d\mathbb{E}\|\Psi(t)\|^2+\int_0^t2\alpha he^{2\alpha t}\mathbb{E}\|\Psi(t)\|^2dt =2\epsilon^2h\sum_{j=1}^J\sum_{k=1}^P\eta_ke_k^2(x_j)\int_0^te^{2\alpha t}dt. \end{align*} Based on the fact that $\sum_{j=1}^Je^2_k(x_j)\le2J\le2h^{-1}$, we have \begin{equation}\label{charge} \begin{split}
e^{2\alpha t}h\mathbb{E}\|\Psi(t)\|^2-h\mathbb{E}\|\Psi(0)\|^2 =&\frac{\epsilon^2h}{\alpha}(e^{2\alpha t}-1)\sum_{j=1}^J\sum_{k=1}^P\eta_ke_k^2(x_j)\\ \le& \frac{2\epsilon^2\eta^{(P)}}{\alpha}(e^{2\alpha t}-1) \end{split} \end{equation} which completes the proof.
In addition, noticing that for $h=1/J$ and $x_j=jh,$ $j=1,\cdots,J$, \begin{equation*} \begin{split}
\mathbb{E}\|\psi_0\|_{L^2}^2=&\mathbb{E}\sum_{j=1}^J\int_{x_{j-1}}^{x_j}|\psi_0(x)|^2dx\\
&=\mathbb{E}\sum_{j=1}^J|\psi_0(x_j)|^2h+O(h)=h\mathbb{E}\|\Psi(0)\|^2+O(h), \end{split} \end{equation*}
we get the uniform boundedness under the assumption $\mathbb{E}\|\psi_0\|_{L^2}^2<\infty$. \end{proof}
\begin{rk}\label{symeuler} Scheme \eqref{fenliang} is equivalent to the symplectic Euler scheme applied to \eqref{multisym}, i.e., \begin{equation*}\label{space2} \left\{ \begin{aligned} &p_{j+1}-p_j=hv_{j+1},\\[4mm] &q_{j+1}-q_j=hw_{j+1},\\ &v_{j+1}-v_j=h(q_j)_t+\alpha hq_j-h((p_j)^2+(q_j)^2)p_j-\epsilon\sum_{k=1}^P\sqrt{\eta_k}e_k(x_j)\frac{d\beta^2_k(t)}{dt},\\ &w_{j+1}-w_j=-h(p_j)_t-\alpha hp_j-h((p_j)^2+(q_j)^2)q_j+\epsilon\sum_{k=1}^P\sqrt{\eta_k}e_k(x_j)\frac{d\beta^1_k(t)}{dt}. \end{aligned} \right. \end{equation*} \end{rk}
\subsection{Full discretization} To construct a fully discrete scheme, which could inherit the properties of \eqref{model}, we are motivated by splitting techniques. We drop the linear terms and stochastic term for the moment and consider the following equation \begin{align}\label{nonlinear} d\Psi(t)-\mathbf{i} \lambda F(\Psi(t))\Psi(t)dt=0 \end{align}
first. Multiplying $\overline{F(\Psi(t))}$ to both sides of \eqref{nonlinear} and taking the imaginary part, we obtain $\|\Psi(t)\|^2=\|\Psi(0)\|^2$, which implies that $F(\Psi(t))=F(\Psi(0))$. Thus, \eqref{nonlinear} is shown to possess a unique solution $\Psi(t)=e^{\mathbf{i}\lambda F(\Psi(0))t}\Psi(0)$.
For linear equation \begin{align*} d\Psi(t)-\mathbf{i} \left(\frac1{h^2}A\Psi(t)+\mathbf{i}\alpha\Psi(t)\right)dt=\epsilon\sigma\Lambda d\beta, \end{align*} a modified mid-point scheme is applied to obtain its full discretization. Now we can define the following splitting schemes initialized with $\Psi^0=\Psi(0)$, \begin{align}\label{full1} &\Psi^{n+1}=e^{-\frac12\alpha\tau}\tilde{\Psi}^n+\mathbf{i}\frac{\tau}{h^2}A\frac{\Psi^{n+1}+e^{-\frac12\alpha\tau}\tilde{\Psi}^n}2-\frac12\alpha\tau\frac{\Psi^{n+1}+e^{-\frac12\alpha\tau}\tilde{\Psi}^n}2+\epsilon\sigma\Lambda\delta_{n+1}\beta,\\\label{full2} &\tilde{\Psi}^n=e^{\mathbf{i}\lambda F(\Psi^n)\tau}\Psi^n, \end{align} where $\Psi^n=(\psi_1^n,\cdots,\psi_J^n)^T\in\mathbb{C}^{J}$, $\tau$ denotes the uniform time step, $\delta_{n+1}\beta=\beta(t_{n+1})-\beta(t_n)$ and $t_n=n \tau$, $n\in\mathbb{N}$. Noticing that schemes \eqref{full1}--\eqref{full2} can be rewritten as \begin{equation}\label{full} \begin{split} \Psi^{n+1}-e^{f(\Psi^n)}\Psi^n=&\mathbf{i}\frac{\tau}{2h^2}A\left(\Psi^{n+1}+e^{f(\Psi^n)}\Psi^n\right)\\ &-\frac14\alpha\tau\left(\Psi^{n+1}+e^{f(\Psi^n)}\Psi^n\right)+\epsilon\sigma\Lambda\delta_{n+1}\beta, \end{split} \end{equation} which can also be expressed in the following explicit form \begin{equation}\label{explicitscheme} \begin{split} \Psi^{n+1}=&\left(I-\frac{\mathbf{i}\tau}{2h^2}A+\frac14\alpha\tau I\right)^{-1}\left(I+\frac{\mathbf{i}\tau}{2h^2}A-\frac14\alpha\tau I\right)e^{f(\Psi^n)}\Psi^n\\ &+\left(I-\frac{\mathbf{i}\tau}{2h^2}A+\frac14\alpha\tau I\right)^{-1}\epsilon\sigma\Lambda\delta_{n+1}\beta \end{split} \end{equation} with $I$ denoting the identity matrix and $f(\Psi^n)=\left(-\frac12\alpha I+\mathbf{i}\lambda F(\Psi^n)\right)\tau$. Thus, there uniquely exists a family of $\{\mathcal{F}_{t_n}\}_{n\ge1}$ adapted solutions $\{\Psi^n\}_{n\ge1}$ of \eqref{full} for sufficiently small $\tau$.
As for the proposed splitting schemes, \eqref{full2} coincides with the exact solution of the Hamiltonian system $d\Psi(t)-\mathbf{i} \lambda F(\Psi(t))\Psi(t)dt=0$. It then suffices to show that \eqref{full1} possesses the conformal multi-symplectic conservation law, which is stated in the following theorem. \begin{tm} Scheme \eqref{full1} possesses the discrete conformal multi-symplectic conservation law \begin{align*} &e^{-\alpha\tau}\frac{dz_j^{n+1}\wedge Mdz_j^{n+1}-dz_j^{n}\wedge Mdz_j^{n}}{\tau}+\frac{dz_j^{n+\frac12}\wedge(K_1dz_{j+1}^{n+\frac12}-K_2dz_{j-1}^{n+\frac12})}{h}\\ =&-\frac12\alpha dz_j^{n+\frac12}\wedge Mdz_j^{n+\frac12} \end{align*} with $z_j^n=(p_j^n,q_j^n,v_j^n,w_j^n)^T$, $z_j^{n+\frac12}=\frac12(z_j^{n+1}+e^{-\frac12\alpha\tau}z_j^n)$, \begin{equation*}K_1= \left( \begin{array}{cccc} 0&0&1&0\\ 0&0&0&1\\ 0& 0&0 &0\\ 0&0&0&0\\ \end{array} \right),\quad K_2= \left( \begin{array}{cccc} 0&0&0&0\\ 0&0&0&0\\ -1& 0&0 &0\\ 0&-1&0&0\\ \end{array} \right) \end{equation*} and $K_1+K_2=K$. \end{tm} \begin{proof} We denote $\tilde{\Psi}^n$ by $\Psi^n$ for convenience in this proof, and we have \begin{align*} \Psi^{n+1}=e^{-\frac12\alpha\tau}{\Psi}^n+\mathbf{i}\frac{\tau}{h^2}A\frac{\Psi^{n+1}+e^{-\frac12\alpha\tau}{\Psi}^n}2-\frac12\alpha\tau\frac{\Psi^{n+1}+e^{-\frac12\alpha\tau}{\Psi}^n}2+\epsilon\sigma\Lambda\delta_{n+1}\beta \end{align*} with $\Psi^n=\left(\psi_1^n,\cdots,\psi_J^n\right)^T\in\mathbb{C}^J$. Denote $\psi_j^n:=p_j^n+\mathbf{i} q_j^n$ with its real part $p_j^n$ and imaginary part $q_j^n$, $\delta_{n+1}\beta=\delta_{n+1}\beta^1+\mathbf{i}\delta_{n+1}\beta^2$, $v_{j+1}^{n}:=(p_{j+1}^{n}-p_j^n)h^{-1}$ and $w_{j+1}^{n}:=(q_{j+1}^{n}-q_j^n)h^{-1}$. Noticing that the $j$-th component of $h^{-2}A\Psi^n$ can be expressed as $h^{-1}(v_{j+1}^n-v_j^n)+\mathbf{i} h^{-1}(w_{j+1}^n-w_j^n)$, we decompose \eqref{full1} with its real and imaginary parts respectively and derive \begin{equation*}\label{theoformula} \left\{ \begin{aligned} \frac{p_j^{n+1}-e^{-\frac12\alpha\tau}p_j^n}{\tau}+&\frac{w_{j+1}^{n+1}-w_j^{n+1}}{2h}+e^{-\frac12\alpha\tau}\frac{w_{j+1}^{n}-w_j^{n}}{2h}\\ =&-\frac14\alpha(p_j^{n+1}+e^{-\frac12\alpha\tau}p_j^n)+\epsilon\sigma\Lambda\delta_{n+1}\beta^1,\\ \frac{q_j^{n+1}-e^{-\frac12\alpha\tau}q_j^{n}}{\tau}-&\frac{v_{j+1}^{n+1}-v_j^{n+1}}{2h}-e^{-\frac12\alpha\tau}\frac{v_{j+1}^{n}-v_j^{n}}{2h}\\ =&-\frac14\alpha(q_j^{n+1}+e^{-\frac12\alpha\tau}q_j^n)+\epsilon\sigma\Lambda\delta_{n+1}\beta^2. \end{aligned} \right. \end{equation*} Combining formulae $v_{j+1}^{n}=(p_{j+1}^{n}-p_j^n)h^{-1}$, $w_{j+1}^{n}=(q_{j+1}^{n}-q_j^n)h^{-1}$ with above equations, we get \begin{align*} M\frac{z_j^{n+1}-e^{-\frac12\alpha\tau}z_j^n}{\tau}+K_1\frac{z_{j+1}^{n+\frac12}-z_j^{n+\frac12}}{h} +K_2\frac{z_{j}^{n+\frac12}-z_{j-1}^{n+\frac12}}{h} =-\frac12\alpha Mz_j^{n+\frac12}+\xi_j^{n+\frac12}, \end{align*} where $\xi_j^{n+\frac12}:=(-\epsilon\sigma\Lambda\delta_{n+1}\beta^2,\epsilon\sigma\Lambda\delta_{n+1}\beta^1,-v_j^{n+\frac12},-w_j^{n+\frac12})^T$. Taking differential in phase space on both sides of above equation, and performing wedge product with $dz_j^{n+\frac12}$ respectively, we show the discrete conformal multi-symplectic conservation law based on the symmetry of matrix $-K_1+K_2$ and the fact $dz_j^{n+\frac12}\wedge(-K_1+K_2)dz_j^{n+\frac12}=0$, $dz_j^{n+\frac12}\wedge d\xi_j^{n+\frac12}=0$. \end{proof} \begin{rk} It is also feasible to show that schemes \eqref{full1}--\eqref{full2} are conformal symplectic in temporal direction, which together with Remark \ref{symeuler}, yields the conformal multi-symplecticity of the fully discrete scheme \eqref{full}.
\end{rk}
\begin{prop}\label{fullmoment}
Assume that $\mathbb{E}\|\psi_0\|_{L^2}^2<\infty$, $Q\in\mathcal{HS}(L^2,\dot{H}^2)$ and $P\le C_*(J+1)$ for some constant $C_*\ge1$, then the solution $\{\Psi^n\}_{n\ge1}$ of \eqref{full} is uniformly bounded, i.e., \begin{align}
h\mathbb{E}\|\Psi^n\|^2\le e^{-\alpha t_n}h\mathbb{E}\|\Psi^0\|^2+C \end{align} with $t_n=n\tau$ and constant $C$ depending on $\alpha,\epsilon,Q$ and $C_*$. \end{prop} \begin{proof} We multiply $\overline{\left(\Psi^{n+1}+e^{f(\Psi^n)}\Psi^n\right)}^T$ to \eqref{full}, take the real part and expectation, and obtain \begin{equation}\label{moment} \begin{split}
&\mathbb{E}\|\Psi^{n+1}\|^2-e^{-\alpha\tau}\mathbb{E}\|\Psi^n\|^2\\
=&-\frac14\alpha\tau\mathbb{E}\|\Psi^{n+1}+e^{f(\Psi^n)}\Psi^n\|^2+\mathbb{E}\left[\Re\left[\overline{\left(\Psi^{n+1}-e^{f(\Psi^n)}\Psi^n\right)}^T\epsilon\sigma\Lambda\Delta_{n+1}\beta\right]\right]\\
=&-\frac14\alpha\tau\mathbb{E}\|\Psi^{n+1}+e^{f(\Psi^n)}\Psi^n\|^2 +\mathbb{E}\Bigg{[}\Re\Bigg{[}\bigg{(}-\mathbf{i}\frac{\tau}{2h^2}A\overline{\left(\Psi^{n+1}+e^{f(\Psi^n)}\Psi^n\right)}\\ &-\frac14\alpha\tau\overline{\left(\Psi^{n+1}+e^{f(\Psi^n)}\Psi^n\right)}+\overline{\epsilon\sigma\Lambda\Delta_{n+1}\beta}\bigg{)}^T\epsilon\sigma\Lambda\Delta_{n+1}\beta\Bigg{]}\Bigg{]}\\
\le&-\frac14\alpha\tau\mathbb{E}\|\Psi^{n+1}+e^{f(\Psi^n)}\Psi^n\|^2+\frac18\alpha\tau\mathbb{E}\|\Psi^{n+1}+e^{f(\Psi^n)}\Psi^n\|^2\\
&+C\tau\mathbb{E}\|h^{-2}A\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2
+\frac18\alpha\tau\mathbb{E}\|\Psi^{n+1}+e^{f(\Psi^n)}\Psi^n\|^2\\
&+C\tau\mathbb{E}\|\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2+\mathbb{E}\|\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2. \end{split} \end{equation} For the smooth functions $e_k(x)$, we have \begin{align*}
\left|\Delta e_k(x_j)-\frac{e_k(x_{j+1})-2e_k(x_j)+e_k(x_{j-1})}{h^2}\right|\le Ck^4h^2\le Ck^2,\quad k\ge1 \end{align*} based on the fact $kh\le P(J+1)^{-1}\le C_*$. Thus, \begin{equation}\label{sto1} \begin{split}
&\mathbb{E}\|h^{-2}A\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2
=\epsilon^2\sum_{j=1}^J\mathbb{E}\left|\sum_{k=1}^P\sqrt{\eta_k^{}}\frac{e_k(x_{j+1})-2e_k(x_j)+e_k(x_{j-1})}{h^2}\Delta_{n+1}\beta_k\right|^2\\
&\le2\epsilon^2\sum_{j=1}^J\sum_{k=1}^P\eta_k^{}\left(|\Delta e_k(x_j)|+Ck^2\right)^2\tau \le CJ\tau\sum_{k=1}^Pk^4\eta_k \le Ch^{-1}\tau. \end{split} \end{equation}
In the last step, we have used the fact $\sum\limits_{k=1}^Pk^4\eta_k\le C\|Q\|_{\mathcal{L}_2^2}\le C$. Similarly, \begin{align}\label{sto2}
\mathbb{E}\|\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2
=\epsilon^2\sum_{j=1}^J\mathbb{E}\left|\sum_{k=1}^P\sqrt{\eta_k^{}}e_k(x_j)\Delta_{n+1}\beta_k\right|^2 \le CJ\eta\tau \le Ch^{-1}\tau. \end{align} Substituting \eqref{sto1} and \eqref{sto2} into \eqref{moment} and multiplying the result by $h$, we get \begin{align*}
h\mathbb{E}\|\Psi^{n+1}\|^2\le e^{-\alpha\tau}h\mathbb{E}\|\Psi^n\|^2+C\tau
\le e^{-\alpha t_{n+1}}h\mathbb{E}\|\Psi^0\|^2+C\tau\frac{1-e^{-\alpha t_n}}{1-e^{-\alpha\tau}}, \end{align*} which, together with the fact $\frac{1-e^{-\alpha t_n}}{1-e^{-\alpha\tau}}\le\frac1{1-(1-\alpha\tau)}=\frac1{\alpha\tau}$, completes the proof. \end{proof}
\iffalse \begin{prop} Under the assumptions in Proposition \ref{fullmoment} and $\psi_0\in\dot{H}^2$, we have in addition that \begin{align}\label{2norm}
h\mathbb{E}\left\|\frac1{h^2}A\Psi^{n}\right\|^2\le e^{-\alpha t_n}h\|\psi_0\|^2_{\dot{H}^2}+C\tau^{-1}. \end{align} \end{prop} \begin{proof} In this proof, we denote $a:=\Psi^{n+1}$ and $b:=e^{f(\Psi^n)}\Psi^n$ for convenience. Multiplying $\frac1{h^2}A\overline{\left(a-b\right)}$ to the transpose of \eqref{full} and taking both the imaginary part and the expectation, we obtain \begin{align}
&\mathbb{E}\left\|\frac1{h^2}Aa\right\|^2-\mathbb{E}\left\|\frac1{h^2}Ab\right\|^2 =\mathbb{E}\left[\Im\left[ \frac{\alpha}2\left(a+b\right)^T\frac1{h^2}A\left(\overline{a-b}\right)-\frac2{\tau}\left(\epsilon\sigma\Lambda\Delta_{n+1}\beta\right)^T\frac1{h^2}A\left(\overline{a-b}\right)\right]\right]\nonumber\\ =&\mathbb{E}\left[\frac{\alpha}2\Im\left[(a+b)^T\frac1{h^2}A\left(-\mathbf{i}\frac{\tau}{2h^2}A\left(\overline{a+b}\right)-\frac{\alpha\tau}4\left(\overline{a+b}\right)+\overline{\epsilon\sigma\Lambda\Delta_{n+1}\beta}\right)\right]\right]\nonumber\\ &-\mathbb{E}\left[\frac{2}{\tau}\Im\left[\left(\epsilon\sigma\Lambda\Delta_{n+1}\beta\right)^T\frac1{h^2}A\left(-\mathbf{i}\frac{\tau}{2h^2}A\left(\overline{a+b}\right)-\frac{\alpha\tau}4\left(\overline{a+b}\right)+\overline{\epsilon\sigma\Lambda\Delta_{n+1}\beta}\right)\right]\right]\nonumber\\
=&-\frac{\alpha\tau}4\mathbb{E}\left\|\frac1{h^2}A(a+b)\right\|^2 +\mathbb{E}\left[\frac{\alpha}{2h^2}\Im\left[(a+b)^TA(\overline{\epsilon\sigma\Lambda\Delta_{n+1}\beta})\right]\right]\nonumber\\ &+\mathbb{E}\left[\Re\left[\frac1{h^4}\left(\epsilon\sigma\Lambda\Delta_{n+1}\beta\right)^TA^2\left(\overline{a+b}\right)\right]\right] +\mathbb{E}\left[\frac{\alpha}{2h^2}\Im\left[\left(\epsilon\sigma\Lambda\Delta_{n+1}\beta\right)^TA\left(\overline{a+b}\right)\right]\right]\nonumber\\\label{eq1}
\le&\frac1{\alpha\tau}\mathbb{E}\left\|\frac1{h^2}A\epsilon\sigma\Lambda\Delta_{n+1}\beta\right\|^2\le Ch^{-1}. \end{align} based on the symmetry of matrix $A$ and \eqref{sto1}, where $\Im[\cdot]$ denotes the imaginary part of a complex number and we have used the fact that \begin{align*} \Im\left[(a+b)^TA(\overline{\epsilon\sigma\Lambda\Delta_{n+1}\beta})+\left(\epsilon\sigma\Lambda\Delta_{n+1}\beta\right)^TA\left(\overline{a+b}\right)\right]=0 \end{align*} and \begin{align*} \Re\left[\frac1{h^4}\left(\epsilon\sigma\Lambda\Delta_{n+1}\beta\right)^TA^2\left(\overline{a+b}\right)\right]
\le \frac{\alpha\tau}4\left\|\frac1{h^2}A(a+b)\right\|^2
+\frac1{\alpha\tau}\left\|\frac1{h^2}A\epsilon\sigma\Lambda\Delta_{n+1}\beta\right\|^2. \end{align*} Noticing that \begin{align*}
\left\|\frac1{h^2}Ab\right\|^2
=e^{-\alpha\tau}\left\|\frac1{h^2}Ae^{\mathbf{i}\lambda\tau F(\Psi^n)}\Psi^n\right\|^2
\le e^{-\alpha\tau}\left\|Ae^{\mathbf{i}\lambda\tau F(\Psi^n)}A^{-1}\right\|^2\left\|\frac1{h^2}A\Psi^n\right\|^2, \end{align*}
where {\color{red}$\left\|Ae^{\mathbf{i}\lambda\tau F(\Psi^n)}A^{-1}\right\|^2\le\|A\|^2\left\|e^{\mathbf{i}\lambda\tau F(\Psi^n)}\right\|_2^2\|A^{-1}\|^2=$}, and that \begin{align*}
\left\|\frac1{h^2}A\Psi^0\right\|^2, \end{align*} so we conclude from \eqref{eq1} that \begin{align*}
\left\|\frac1{h^2}A\Psi^{n+1}\right\|^2
\le e^{-\alpha\tau}\left\|\frac1{h^2}A\Psi^n\right\|^2+Ch^{-1}
\le e^{-\alpha t_{n+1}}\|\psi_0\|^2_{\dot{H}^2}+Ch^{-1}\tau^{-1}, \end{align*} which complete the proof. \end{proof} \fi
\begin{tm} Under the assumptions in Proposition \ref{fullmoment} and $\eta_k>0$ for $k=1,\cdots,P$. The solution $\{\Psi^n\}_{n\ge1}$ of \eqref{full} is uniquely ergodic with a unique invariant measure, denoted by $\mu^{\tau}_h$, satisfying \begin{align} \lim_{N\to\infty}\frac1N\sum_{n=0}^{N-1}\mathbb{E} f(\Psi^n)=\int_{\mathbb{C}^J}fd\mu_h^{\tau},\quad\forall\;f\in C_b(\mathbb{C}^J). \end{align} \end{tm} \begin{proof}
To show the existence of the invariant measures, we'll use a useful tool called Lyapunov function. For any fixed $h>0,$ we choose $V(\cdot):=h\|\cdot\|^2$ as the Lyapunov function, which satisfies that the level sets $K_c:=\{u\in\mathbb{C}^J:V(u)\le c\}$ are compact for any $c>0$ and $\mathbb{E}[V(\Psi^n)]\le V(\Psi^0)+C$ for any $n\in\mathbb{N}$. Thus, the Markov chain $\{\Psi^n\}_{n\in\mathbb{N}}$ possesses an invariant measure (see Proposition 7.10 in \cite{daprato}).
Now we show that $\{\Psi^n\}_{n\in\mathbb{N}}$ is irreducible and strong Feller (also known as the minorization condition in Assumption 2.1 of \cite{mattingly}), which yields the uniqueness of the invariant measure. In fact, for any $u,v\in\mathbb{C}^J$, we can derive from \eqref{full} that $\delta_1\beta$ can be chosen as \begin{align*} \epsilon\sigma\Lambda\delta_{1}\beta=v-e^{f(u)}u-\mathbf{i}\frac{\tau}{2h^2}A\left(v+e^{f(u)}u\right)+\frac14\alpha\tau\left(v+e^{f(u)}u\right) \end{align*} such that $\Psi^0=u,\Psi^1=v$, where we have used the fact that $\sigma$ is full rank and $\Lambda$ is invertible. Thus, we can conclude based on the homogenous property of the Markov chain $\{\Psi^n\}_{n\in\mathbb{N}}$ that the transition kernel
$P_n(u,A):=\mathbb{P}(\Psi^n\in A|\Psi^0=u)>0$, which implies the irreducibility of the chain. On the other hand, as $\delta_1\beta$ has $C^{\infty}$ density, it follows from \eqref{explicitscheme} that $\Psi^1$ also has $C^{\infty}$ density for any deterministic initial value $\Psi^0=\Psi(0)$. Then explicit construction shows that $\{\Psi^n\}_{n\in\mathbb{N}}$ possesses a family of $C^{\infty}$ density and is strong Feller. \end{proof}
The theorems above are evidently consistent with the continuous results \eqref{ergodic}, \eqref{2forms} and \eqref{bounded}, respectively. The next result concerns the error estimation of the proposed scheme, where the truncation technique will be used to deal with the non-global Lipschitz nonlinearity.
\section{Convergence order in probability} In this section, we focus on the approximate error for the proposed scheme in temporal direction. As the nonlinear term is not global Lipschitz, we consider the following truncated function first \begin{align}\label{truncate} d\Psi_R-\mathbf{i}\left(\frac1{h^2}A\Psi_R+\mathbf{i}\alpha\Psi_R+\lambda F_R(\Psi_R)\Psi_R\right)dt=\epsilon\sigma\Lambda d\beta, \end{align}
with $\Psi_R:=\Psi_R(t)=(\psi_{R,1}^{}(t),\cdots,\psi_{R,J}^{}(t))^T$ and initial value $\Psi_R(0)=\Psi(0)$. Here $F_R(v)=\theta\left(\frac{\|v\|}{R}\right)F(v)$ for any vector $v\in\mathbb{C}^J$ and a cut-off function $\theta\in C^{\infty}(\mathbb{R})$ satisfying $\theta(x)=1$ for $x\in[0,1]$ and $\theta(x)=0$ for $x\ge2$ (see also \cite{BD06,L13}). In addition, we have \begin{align*}
\|F_R(\Psi_R)\|=\theta\left(\frac{\|\Psi_R\|}{R}\right)\max_{1\le j\le J}|\psi_{R,j}^{}|^2
\le\theta\left(\frac{\|\Psi_R\|}{R}\right)\|\Psi_R\|^2 \le4R^2. \end{align*}
As a result, the nonlinear term $F_R(\Psi_R)\Psi_R$ is global Lipschitz with respect to the norm $\|\cdot\|$. The proposed scheme \eqref{explicitscheme} applied to the truncated equation \eqref{truncate} yields the following scheme \begin{equation}\label{truncatescheme} \begin{split} \Psi_R^{n+1}=&\left(I-\frac{\mathbf{i}\tau}{2h^2}A+\frac14\alpha\tau I\right)^{-1}\left(I+\frac{\mathbf{i}\tau}{2h^2}A-\frac14\alpha\tau I\right)e^{f_R(\Psi_R^n)}\Psi_R^n\\ &+\left(I-\frac{\mathbf{i}\tau}{2h^2}A+\frac14\alpha\tau I\right)^{-1}\epsilon\sigma\Lambda\Delta_{n+1}\beta, \end{split} \end{equation} where $f_R(\Psi_R^n)=\left(-\frac12\alpha I+\mathbf{i}\lambda F_R(\Psi_R^n)\right)\tau$ and $\Psi_R^n=(\psi_{R,1}^n,\cdots,\psi_{R,J}^n)^T$.
\begin{tm}\label{trunerror}
Consider Eq. \eqref{truncate} and the scheme \eqref{truncatescheme}. Assume that $\mathbb{E}\|\psi_0\|_{L^2}^2<\infty$, $Q\in\mathcal{HS}(L^2,\dot{H}^2)$, $\alpha\ge\frac12$ and $\tau=O(h^4)$. For $T=N\tau$, there exists a constant $C_R$ which depends on $\alpha,\epsilon,R,Q,\psi_0$ and is independent of $T$ and $N$ such that \begin{align*}
h\mathbb{E}\|\Psi_R(T)-\Psi_R^N\|^2\le C_R\tau^2. \end{align*} \end{tm}
\begin{proof} Denote semigroup operator $S(t):=e^{Bt}$ which is generated by the linear operator $B:=\mathbf{i}\frac1{h^2}A-\frac{\alpha}2I$, then the mild solution of \eqref{space} is \begin{equation}\label{psit} \begin{split} \Psi_R(t_{n+1})=&S(\tau)\Psi(t_n)+\int_{t_n}^{t_{n+1}}S(t_{n+1}-s)\mathbf{i}\lambda F_R(\Psi_R(s))\Psi_R(s)ds\\ &-\int_{t_n}^{t_{n+1}}S(t_{n+1}-s)\frac{\alpha}2\Psi_R(s)ds+\int_{t_n}^{t_{n+1}}S(t_{n+1}-s)\epsilon\sigma\Lambda d\beta(s). \end{split} \end{equation} Subtracting \eqref{truncatescheme} from \eqref{psit}, we obtain \begin{equation*}\label{strongerror} \begin{split} &\Psi_R(t_{n+1})-\Psi_R^{n+1}\\ =&S(\tau)\Psi_R(t_n)-\left(I-\frac12B\tau\right)^{-1}\left(I+\frac12B\tau\right)e^{f_R(\Psi_R^n)}\Psi_R^n\\ &+\int_{t_n}^{t_{n+1}}S(t_{n+1}-s)\mathbf{i}\lambda F_R(\Psi_R(s))\Psi_R(s)ds-\int_{t_n}^{t_{n+1}}S(t_{n+1}-s)\frac{\alpha}2\Psi_R(s)ds\\ &+\int_{t_n}^{t_{n+1}}\left(S(t_{n+1}-s)-\left(I-\frac12B\tau\right)^{-1}\right)\epsilon\sigma\Lambda d\beta(s)\\ =&S(\tau)\left(\Psi_R(t_n)-\Psi_R^n\right)+\left[S(\tau)-\left(I-\frac12B\tau\right)^{-1}\left(I+\frac12B\tau\right)\right]\Psi_R^n\\ &+\left(I-\frac12B\tau\right)^{-1}\left(I+\frac12B\tau\right)\left(\Psi_R^n-e^{f_R(\Psi_R^n)}\Psi_R^n\right)\\ &+\int_{t_n}^{t_{n+1}}S(t_{n+1}-s)\mathbf{i}\lambda F_R(\Psi_R(s))\Psi_R(s)ds-\int_{t_n}^{t_{n+1}}S(t_{n+1}-s)\frac12\alpha\Psi_R(s)ds\\ &+\int_{t_n}^{t_{n+1}}\left(S(t_{n+1}-s)-\left(I-\frac12B\tau\right)^{-1}\right)\epsilon\sigma\Lambda d\beta(s)\\ =&:\uppercase\expandafter{\romannumeral1}+\uppercase\expandafter{\romannumeral2}+\uppercase\expandafter{\romannumeral3}+\uppercase\expandafter{\romannumeral4}+\uppercase\expandafter{\romannumeral5}+\uppercase\expandafter{\romannumeral6}. \end{split} \end{equation*} To show the strong convergence order of \eqref{truncatescheme}, we give the estimates of above terms, respectively. For terms $\uppercase\expandafter{\romannumeral1}$ and $\uppercase\expandafter{\romannumeral2}$, we have \begin{align}\label{term1}
\mathbb{E}\|\uppercase\expandafter{\romannumeral1}\|^2
=\mathbb{E}\left\|e^{(\mathbf{i}\frac1{h^2}A-\frac{\alpha}2I)\tau}(\Psi_R(t_n)-\Psi_R^n)\right\|^2
=e^{-\alpha\tau}\mathbb{E}\|\Psi_R(t_n)-\Psi_R^n\|^2 \end{align} and \begin{equation}\label{term2} \begin{split}
\mathbb{E}\|\uppercase\expandafter{\romannumeral2}\|^2
\le C\mathbb{E}\|(B\tau)^3\Psi_R^n\|^2
&\le C\tau^6\|B^3\|^2\mathbb{E}\|\Psi_R^n\|^2\\[2mm]
&\le Ch^{-13}\tau^6\|A\|^6 \le Ch^{-13}\tau^6 \end{split} \end{equation}
based on $\left|e^x-(1-\frac{x}2)^{-1}(1+\frac{x}2)\right|=O(x^3)$ as $x\to0$ and Proposition \ref{fullmoment}. In the last step of \eqref{term2}, we also used the fact that $\|A\|$ is uniformly bounded for any dimension $J$, whose proof is not difficult and is given in the Appendix for readers' convenience. For term $\uppercase\expandafter{\romannumeral6}$, Taylor expansion yields that \begin{equation} \begin{split}
\mathbb{E}\|\uppercase\expandafter{\romannumeral6}\|^2
&\le2\mathbb{E}\left\|\int_{t_n}^{t_{n+1}}\left(S(t_{n+1}-s)-S(\tau)\right)\epsilon\sigma\Lambda d\beta(s)\right\|^2\\
&+2\mathbb{E}\left\|\left(S(\tau)-\left(I-\frac12B\tau\right)^{-1}\right)\epsilon\sigma\Lambda\Delta_{n+1}\beta\right\|^2\\[3mm]
&\le C\tau^2\mathbb{E}\|B\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2 \le Ch^{-1}\tau^3. \end{split} \end{equation} It then remains to estimate terms $\uppercase\expandafter{\romannumeral3}$, $\uppercase\expandafter{\romannumeral4}$ and $\uppercase\expandafter{\romannumeral5}$. We can obtain the following equation in the same way as that of \eqref{moment} \begin{align*}
\|\Psi_R^{n+1}\|^2-e^{-\alpha\tau}\|\Psi_R^n\|^2\le C\tau\|h^{-2}A\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2+C\|\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2. \end{align*}
Multiplying above equation by $\|\Psi_R^{n+1}\|^2$, we get \begin{equation*} \begin{split}
&\|\Psi_R^{n+1}\|^4+\left(\|\Psi_R^{n+1}\|^2-e^{-\alpha\tau}\|\Psi_R^n\|^2\right)^2-e^{-2\alpha\tau}\|\Psi_R^n\|^4\\
\le& C\tau\left(\|\Psi_R^{n+1}\|^2-e^{-\alpha\tau}\|\Psi_R^n\|^2\right)\|h^{-2}A\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2+C\tau e^{-\alpha\tau}\|\Psi_R^n\|^2\|h^{-2}A\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2\\
&+C\left(\|\Psi_R^{n+1}\|^2-e^{-\alpha\tau}\|\Psi_R^n\|^2\right)\|\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2+C e^{-\alpha\tau}\|\Psi_R^n\|^2\|\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2\\
\le& \left(\|\Psi_R^{n+1}\|^2-e^{-\alpha\tau}\|\Psi_R^n\|^2\right)^2
+\tau e^{-2\alpha\tau}\|\Psi_R^n\|^4+C\tau\|h^{-2}A\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^4+\frac{C}{\tau}\|\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^4. \end{split} \end{equation*} Based on \eqref{sto1} and \eqref{sto2}, we take expectation of above equation and derive \begin{align*}
\mathbb{E}\|\Psi_R^{n+1}\|^4
&\le(1+\tau)e^{-2\alpha\tau}\mathbb{E}\|\Psi_R^n\|^4+Ch^{-2}\tau\\[2mm]
&\le(1+\tau)^{n+1}e^{-2\alpha\tau (n+1)}\mathbb{E}\|\Psi_R^0\|^4+Ch^{-2} \le Ch^{-2} \end{align*}
for $\alpha\ge\frac12$, where in the last step we have used the fact that $\mathbb{E}\|\Psi_R^0\|^4\le(\mathbb{E}\|\Psi_R^0\|^2)^2\le Ch^{-2}$ and $(1+\tau)e^{-2\alpha\tau}<1$ for $\alpha\ge\frac12$. Similarly, we derive
$\mathbb{E}\|\Psi_R^n\|^8\le Ch^{-4},$ which implies that \begin{align*}
\mathbb{E}\|F_R(\Psi_R^n)\|^4=\mathbb{E}\left(\sum_{j=1}^J\left|\psi^n_{R,j}\right|^4\right)^2\le\mathbb{E}\|\Psi_R^n\|^8\le Ch^{-4},\quad\forall~n\in\mathbb{N}. \end{align*} Thus, by Taylor expansion, we have \begin{equation}\label{345} \begin{split} &\uppercase\expandafter{\romannumeral3}+\uppercase\expandafter{\romannumeral4}+\uppercase\expandafter{\romannumeral5}=\left(I-\frac12B\tau\right)^{-1}\left(I+\frac12B\tau\right)\Big{(}-f_R(\Psi_R^n)+O(f(\Psi_R^n)^2)\Big{)}\Psi_R^n\\ &+\int_{t_n}^{t_{n+1}}S(t_{n+1}-s)\mathbf{i}\lambda F_R(\Psi_R(s))\Psi_R(s)ds -\int_{t_n}^{t_{n+1}}S(t_{n+1}-s)\frac12\alpha\Psi_R(s)ds\\ =&\mathbf{i}\lambda \int_{t_n}^{t_{n+1}}\left[S(t_{n+1}-s)F_R(\Psi_R(s))\Psi_R(s)-\left(I-\frac12B\tau\right)^{-1}\left(I+\frac12B\tau\right)F_R(\Psi_R^n)\Psi_R^n\right]ds\\ &-\frac12\alpha\int_{t_n}^{t_{n+1}}\left[S(t_{n+1}-s)\Psi_R(s)-\left(I-\frac12B\tau\right)^{-1}\left(I+\frac12B\tau\right)\Psi_R^n\right]ds\\ &+\left(I-\frac12B\tau\right)^{-1}\left(I+\frac12B\tau\right)O(f_R(\Psi_R^n)^2)\Psi_R^n\\ :=&\tilde{\uppercase\expandafter{\romannumeral3}}+\tilde{\uppercase\expandafter{\romannumeral4}}+\tilde{\uppercase\expandafter{\romannumeral5}}. \end{split} \end{equation} Now we estimate above terms respectively. For $\tilde{\uppercase\expandafter{\romannumeral3}}$, we have \begin{align*} \tilde{\uppercase\expandafter{\romannumeral3}} =& \mathbf{i}\lambda\int_{t_n}^{t_{n+1}}\left[S(t_{n+1}-s)-\left(I-\frac12B\tau\right)^{-1}\left(I+\frac12B\tau\right)\right]F_R(\Psi_R^n)\Psi_R^nds\nonumber\\ &+ \mathbf{i}\lambda\int_{t_n}^{t_{n+1}}S(t_{n+1}-s)\Big{[}F_R(\Psi_R(s))F_R(\Psi_R(s))-F_R(\Psi_R^n)\Psi_R^n\Big{]}ds\nonumber\\ =:&\tilde{\uppercase\expandafter{\romannumeral3}}_1+\tilde{\uppercase\expandafter{\romannumeral3}}_2, \end{align*} which satisfies \begin{align*}
\mathbb{E}\|\tilde{\uppercase\expandafter{\romannumeral3}}_1\|^2
\le& \tau\int_{t_n}^{t_{n+1}}\mathbb{E}\left\|\left[S(t_{n+1}-s)-\left(I-\frac12B\tau\right)^{-1}\left(I+\frac12B\tau\right)\right]F_R(\Psi_R^n)\Psi_R^n\right\|^2ds\\
\le& Ch^{-12}\tau^8\mathbb{E}\|F_R(\Psi_R^n)\Psi_R^n\|^2
\le Ch^{-12}\tau^8\mathbb{E}\left[\sum_{j=1}^J\left|\psi_{R,j}^n\right|^6\right] \le Ch^{-15}\tau^8 \end{align*} and \begin{align*}
\mathbb{E}\|\tilde{\uppercase\expandafter{\romannumeral3}}_2\|^2
\le& \tau\int_{t_n}^{t_{n+1}}\mathbb{E}\Big{\|}S(t_{n+1}-s)\Big{[}F_R(\Psi_R(s))F_R(\Psi_R(s))-F_R(\Psi_R^n)\Psi_R^n\Big{]}\Big{\|}^2ds\\
\le&C\tau\int_{t_n}^{t_{n+1}}\mathbb{E}\|\Psi_R(s)-\Psi_R(t_n)\|^2ds+C\tau^2e^{-\alpha\tau}\mathbb{E}\|\Psi_R(t_n)-\Psi_R^n\|^2. \end{align*} Noticing that \begin{align*}
\mathbb{E}\|\Psi_R(s)-\Psi_R(t_n)\|^2
=&\mathbb{E}\bigg{\|}\int_{t_n}^sS(s-r)\mathbf{i}\lambda F_R(\Psi_R(r))\Psi_R(r)dr-\int_{t_n}^sS(s-r)\frac{\alpha}2\Psi_R(r)dr\\
&+\int_{t_n}^sS(s-r)\epsilon\sigma\Lambda d\beta(r)\bigg{\|}^2 \le h^{-3}\tau^2, \end{align*} thus \begin{align}
\mathbb{E}\|\tilde{\uppercase\expandafter{\romannumeral3}}\|^2
\le Ch^{-15}\tau^8+Ch^{-3}\tau^4+C\tau^2e^{-\alpha\tau}\mathbb{E}\|\Psi_R(t_n)-\Psi_R^n\|^2. \end{align} Term $\tilde{\uppercase\expandafter{\romannumeral4}}$ can be estimated in the same way as the estimation of $\tilde{\uppercase\expandafter{\romannumeral3}}$. Term $\tilde{\uppercase\expandafter{\romannumeral5}}$ turns to be \begin{equation}\label{term5} \begin{split}
\mathbb{E}\|\tilde{\uppercase\expandafter{\romannumeral5}}\|^2
\le& C\mathbb{E}\left\|f_R(\Psi_R^n)^2\Psi_R^n\right\|^2
\le C\tau^4\mathbb{E}\left[\sup_{1\le j\le J}\left|-\frac12\alpha+\mathbf{i}\lambda|\psi_{R,j}^n|^2\right|^4\|\Psi_R^n\|^2\right]\\
\le& C\tau^4\left(\mathbb{E}\left(\sum_{j=1}^J\left|\psi^n_{R,j}\right|^2\right)^8\right)^{\frac12}\left(\mathbb{E}\|\Psi_R^n\|^4\right)^{\frac12} \le Ch^{-5}\tau^4. \end{split} \end{equation} From \eqref{term1}--\eqref{term5}, we conclude \begin{align*}
&h\mathbb{E}\|\Psi_R(t_{n+1})-\Psi_R^{n+1}\|^2\\[2mm]
\le& h(1+C\tau^2)e^{-\alpha\tau}\mathbb{E}\|\Psi_R(t_{n})-\Psi_R^{n}\|^2+C\tau^3+Ch^{-4}\tau^4+Ch^{-12}\tau^6+Ch^{-14}\tau^8\\[2mm] \le& C\tau^{2}+Ch^{-4}\tau^{3}+Ch^{-12}\tau^5+Ch^{-14}\tau^7 \le C\tau^2, \end{align*} where in the last two steps we have used the fact that $(1+C\tau^2)e^{-\alpha\tau}<1$ for $\tau$ is sufficiently small and $\tau=O(h^4)$. \end{proof} Based on the estimates on truncated equation and its numerical scheme, we are now in the position to give the approximate error between $\Psi(t)$ and $\Psi^n$. The proof of following theorem is motivated by \cite{BD06,L13} and holds for any fixed $T>0$ without other restrictions. \begin{tm}\label{probability}
Consider Eq. \eqref{space} and scheme \eqref{explicitscheme}. Assume that $\mathbb{E}\|\psi_0\|_{L^2}^2<\infty$, $Q\in\mathcal{HS}(L^2,\dot{H}^2)$, $\alpha\ge\frac12$ and $\tau=O(h^4)$. For any $T>0$, we derive convergence order one in probability, i.e., \begin{align}
\lim_{K\to\infty}\mathbb{P}\left(\sup_{1\le n\le[T/\tau]}\sqrt{h}\|\Psi(t_{n})-\Psi^{n}\|\ge K\tau\right)=0. \end{align} \end{tm}
\begin{proof} For any $\gamma\in(0,1)$, we define
$n_{\gamma}:=\inf\{1\le n\le[T/\tau]:\|\Psi(t_n)-\Psi^n\|\ge\gamma\}$ and then deduce that \begin{align*}
&\left\{\sup_{1\le n\le[T/\tau]}\|\Psi(t_{n})-\Psi^{n}\|\ge\gamma\right\}\\
\subset&\Bigg{[}\left(\left\{\sup_{0\le n\le n_\gamma}\|\Psi(t_n)\|\ge R-1\right\}\cap\left\{\sup_{1\le n\le[T/\tau]}\|\Psi(t_{n})-\Psi^{n}\|\ge\gamma\right\}\right)\\
&\cup\left(\left\{\sup_{0\le n\le n_\gamma}\|\Psi(t_n)\|<R-1\right\}\cap\left\{\sup_{1\le n\le[T/\tau]}\|\Psi(t_{n})-\Psi^{n}\|\ge\gamma\right\}\right)\Bigg{]}\\
\subset&\Bigg{[}\left\{\sup_{0\le n\le n_\gamma}\|\Psi(t_n)\|\ge R-1\right\}\\ &\cup
\left(\left\{\sup_{0\le n\le n_\gamma}\|\Psi(t_n)\|<R-1\right\}\cap\left\{\sup_{1\le n\le[T/\tau]}\|\Psi(t_{n})-\Psi^{n}\|\ge\gamma\right\}\right)\Bigg{]}. \end{align*}
If $\left\{\sup_{0\le n\le n_\gamma}\|\Psi(t_n)\|<R-1\right\}$ happens, it is easy to show that
$\|\Psi^k\|\le\|\Psi(t_k)-\Psi^k\|+\|\Psi(t_k)\|<R-1+\gamma<R$, $F_R(\Psi_R^k)=F(\Psi_R^k)$, $\Psi_R^k=\Psi^k$ for $k=0,1,\cdots,n_\gamma-1$ and $\Psi_R(t_n)=\Psi(t_n)$ for $0\le n\le n_\gamma$. Furthermore, comparing scheme \eqref{truncatescheme} with \eqref{explicitscheme} and noticing that \begin{equation} \begin{split} f_R(\Psi_R^{n_\gamma-1})&=\left(-\frac12\alpha I+\mathbf{i}\lambda F_R(\Psi_R^{n_\gamma-1})\right)\tau\\[2mm] &=\left(-\frac12\alpha I+\mathbf{i}\lambda F(\Psi^{n_\gamma-1})\right)\tau=f(\Psi^{n_\gamma-1}), \end{split} \end{equation} we have $\Psi_R^{n_\gamma}=\Psi^{n_\gamma}$, which implies
$$\|\Psi_R(t_{n_\gamma})-\Psi_R^{n_\gamma}\|=\|\Psi(t_{n_\gamma})-\Psi^{n_\gamma}\|\ge\gamma.$$ We conclude that for any $\gamma\in(0,1)$, there exists $n_\gamma\in\mathbb{N}$ such that \begin{align*}
&\left\{\sup_{0\le n\le n_\gamma}\|\Psi(t_n)\|<R-1\right\}\cap\left\{\sup_{1\le n\le[T/\tau]}\|\Psi(t_{n})-\Psi^{n}\|\ge\gamma\right\}\\
\subset&\{\|\Psi_R(t_{n_\gamma})-\Psi_R^{n_\gamma}\|\ge\gamma\}. \end{align*} \iffalse Thus, for some constants $K,K_1>0$, choosing $\gamma=\sqrt{h^{-1}}K\tau$ and $R=\sqrt{(n_\gamma+1)K_1}$, we deduce \begin{align*}
&\mathbb{P}\left(\sup_{n\ge1}\|\Psi(t_{n})-\Psi^{n}\|\ge K\tau\right)\\
\le&\mathbb{P}\left(\sup_{0\le n\le n_\gamma}\sqrt{h}\|\Psi(t_n)\|\ge \sqrt{(n_\gamma+1) K_1}\right)+\mathbb{P}\left(\sqrt{h}\|\Psi_R(t_{n_\gamma})-\Psi_R^{n_\gamma}\|\ge K\tau\right)\\
\le&\sum_{n=0}^{n_\gamma}\mathbb{P}\left(\sqrt{h}\|\Psi(t_n)\|\ge\sqrt{(n_\gamma+1)K_1}\right)+\mathbb{P}\left(\sqrt{h}\|\Psi_R(t_{n_\gamma})-\Psi_R^{n_\gamma}\|\ge K\tau\right)\\
\le&(n_\gamma+1)\frac{h\mathbb{E}\|\Psi(t_n)\|^2}{(n_\gamma+1)K_1}+\frac{h\mathbb{E}\|\Psi_R(t_{n_\gamma})-\Psi_R^{n_\gamma}\|^2}{K^2\tau^2} \le\frac{C}{K_1}+\frac{C_R}{K^2}, \end{align*} where we have used the Chebyshev's inequality, Proposition \ref{semimoment} and Theorem \ref{trunerror}. As $C_R$ depend on $K_1$ \fi Thus, for some constants $K,K_1>0$, choosing $\gamma=\sqrt{h^{-1}}K\tau$ and $R=\sqrt{h^{-1}}K_1$, we deduce \begin{equation}\label{error} \begin{split}
&\mathbb{P}\left(\sup_{1\le n\le[T/\tau]}\sqrt{h}\|\Psi(t_{n})-\Psi^{n}\|\ge K\tau\right)\\
\le&\mathbb{P}\left(\sup_{0\le n\le n_\gamma}\sqrt{h}\|\Psi(t_n)\|\ge K_1\right)+\mathbb{P}\left(\sqrt{h}\|\Psi_R(t_{n_\gamma})-\Psi_R^{n_\gamma}\|\ge K\tau\right)\\
\le&\frac{h\mathbb{E}\Big[\sup\limits_{0\le n\le n_{\gamma}}\|\Psi(t_n)\|^2\Big]}{K_1^2}+\frac{h\mathbb{E}\|\Psi_R(t_{n_\gamma})-\Psi_R^{n_\gamma}\|^2}{K^2\tau^2}. \end{split} \end{equation}
We claim that $e^{2\alpha t}\|\Psi(t)\|^2$ is a submartingale, which ensures that \begin{equation*} \begin{split}
h\mathbb{E}\left[\sup_{0\le n\le n_{\gamma}}\|\Psi(t_n)\|^2\right]
&\le h\mathbb{E}\left[\sup_{0\le n\le n_{\gamma}}e^{2\alpha t_n}\|\Psi(t_n)\|^2\right]\\[2mm]
&\le e^{2\alpha T}h\mathbb{E}\left[\|\Psi(t_{n_\gamma})\|^2\right] \le Ce^{2\alpha T} \end{split} \end{equation*}
based on a martingale inequality and Proposition \ref{semimoment}. In fact, denoting $C_{J,P}:=\sum_{j=1}^J\sum_{k=1}^P\eta_ke_k^2(x_j)$ and applying It\^o's formula to $e^{2\alpha t}\|\Psi(t)\|^2$ similar to \eqref{semiito}, we derive \begin{align*}
e^{2\alpha t}\|\Psi(t)\|^2=\|\Psi(0)\|^2+2\int_0^te^{2\alpha s}\Re\left[\overline{\Psi}(s)\epsilon\sigma\Lambda d\beta(s)\right]+\frac{C_{J,P}}{\alpha}\left(e^{2\alpha t}-1\right) \end{align*} with $2\int_0^te^{2\alpha s}\Re\left[\overline{\Psi}(s)\epsilon\sigma\Lambda d\beta(s)\right]$ a martingale. Apparently, we have \begin{align*}
\mathbb{E}\left[e^{2\alpha t}\|\Psi(t)\|^2|\mathcal{F}_r\right]
&=\|\Psi(0)\|^2+2\int_0^re^{2\alpha s}\Re\left[\overline{\Psi}(s)\epsilon\sigma\Lambda d\beta(s)\right]+\frac{C_{J,P}}{\alpha}\left(e^{2\alpha t}-1\right)\\[2mm]
&\ge\|\Psi(0)\|^2+2\int_0^re^{2\alpha s}\Re\left[\overline{\Psi}(s)\epsilon\sigma\Lambda d\beta(s)\right]+\frac{C_{J,P}}{\alpha}\left(e^{2\alpha r}-1\right)\\[2mm]
&=e^{2\alpha r}\|\Psi(r)\|^2 \end{align*} for $r\le t$, which completes the claim. Hence, based on above claim and Theorem \ref{trunerror}, inequality \eqref{error} turns to be \begin{align*}
\mathbb{P}\left(\sup_{1\le n\le[T/\tau]}\sqrt{h}\|\Psi(t_{n})-\Psi^{n}\|\ge K\tau\right) \le\frac{Ce^{2\alpha T}}{K_1^2}+\frac{C_R}{K^2}, \end{align*} which approaches to $0$ as $K_1,K\to+\infty$ properly for any $T>0$. \end{proof}
\section{Numerical experiments} In this section, we provide several numerical experiments to illustrate the accuracy and capability of the fully discrete scheme \eqref{full}, which can be calculated explicitly. We investigate the good performance in longtime simulation of the proposed scheme and check the temporal accuracy by fixing the space step.
In the sequel, we take $\lambda=1,\,\alpha=0.5$, truncate the infinite series of Wiener process till $P=100$ and choose 500 realizations to approximate the expectation.
\begin{figure}
\caption{Evolution of the discrete charge $h\mathbb{E}\|\Psi^n\|^2$ with $t=n\tau$ for (a) $\epsilon=0$ and (b) $\epsilon=1$ ($h=0.1,\, \tau=2^{-6},\, T=35$).}
\label{charge00}
\end{figure}
{\it Charge evolution.} For the semidiscretization, the charge of the solution satisfies the evolution formula \eqref{charge}. To investigate the recurrence relation for the discrete charge of the fully discrete scheme, Fig. \ref{charge00} plots the discrete charge for different values of $\epsilon$ with initial value $\psi_0(x)=\sin(\pi x)$, $\eta_k=k^{-6}$, $h=1/J=0.1$, $\tau=2^{-5}$ and $T=35$. We can observe that the discrete charge inherits the charge dissipation law without the noise term, i.e., $\epsilon=0$, and preserves the charge dissipation law approximately with a limit $\frac{\epsilon^2h}{\alpha}\sum_{j=1}^J\sum_{k=1}^P\eta_ke_k^2(x_j)$ calculated through \eqref{charge} for $\epsilon=1$.
\begin{figure}
\caption{The temporal averages $\frac{1}{N}\sum_{n=1}^{N-1}\mathbb{E}[f(\Psi^n)]$ started from different initial values for bounded functions (a) $f=\exp(-\|\Psi\|^2)$ and (b) $f=\sin(\|\Psi\|^2)$ ($h=0.1,\,\epsilon=1,\,\tau=2^{-6},\,T=350$).}
\label{temporal_average}
\end{figure}
{\it Ergodicity.} Based on the definition of ergodicity, if numerical solution $\Psi^n$ is ergodic, its temporal averages $\frac{1}{N}\sum_{n=1}^{N-1}\mathbb{E}[f(\Psi^n)]$ started from different initial values will converge to the spatial average $\int_{\mathbb{C}^J}fd\mu_h^\tau$. To verify this property, Fig. \ref{temporal_average} shows the
temporal averages of the fully discrete scheme started from five different initial values $initial(1)=(1,~0,~\cdots,~0)^{T}$, $initial(2)=(0.0003 \mathbf{i},~0,~\cdots,~0)^{T}$, $initial(3)=(\sin\Big(\frac{1}{101}\pi\Big),~\sin\Big(\frac{2}{101}\pi\Big),~\cdots,~\sin\Big(\frac{100}{101}\pi\Big))^{T}$, $initial(4)=\frac{2+\mathbf{i}}{20}(1,~2,~\cdots,~100)^{T}$ and $initial(5)=(\exp(-\frac{\mathbf{i}}{50}),~\exp(-\frac{2\mathbf{i}}{50}),~\cdots,~\exp(-\frac{100\mathbf{i}}{50}))^{T}$. From Fig. \ref{temporal_average}, it can be seen that the time averages of \eqref{full} started from different initial values converge to the same value for two continuous and bounded functions $f$, when time $T=250$ is sufficiently large.
\begin{figure}\label{strong_error}
\end{figure}
{\it Time-independent error.}
As stated in Theorem \ref{trunerror} and \ref{probability}, the mean-square convergence error $\left(h\mathbb{E}\|\Psi_R(T)-\Psi_R^N\|^2\right)^{\frac12}$ with respect to the truncated equation \eqref{truncate} is independent of time $T$, and convergence in probability sense with respect to the original equation is also independent of time $T$. To clarify this property, by defining the mean-square convergence error as \begin{equation*}
\mathcal{E}_{h,\tau}:=\Big(h\mathbb{E}\|\Psi(T)-\Psi^N\|^{2}\Big)^{\frac{1}{2}},~T=N\tau, \end{equation*} Fig. \ref{strong_error} displays the error $\mathcal{E}_{h,\tau}$ over long time $T=10^3$ for different time step sizes: (a) $\tau=2^{-8}$ and (b) $\tau=2^{-10}$ with $h=0.25$, and shows that the mean-square convergence error is independent of time interval, which coincides with our theoretical results.
\begin{figure}
\caption{Rates of convergence of \eqref{full} for (a) $\epsilon=0$ and (b) $\epsilon=1$, respectivly ($h=0.1,\,T=1,\,\tau=2^{-l},\,11\le l\le14$). }
\label{order}
\end{figure}
{\it Convergence order.} We investigate the mean-square convergence order in temporal direction of the proposed method \eqref{full} in this experiment. Let $h=0.1$, $T=1$ and initial value $\psi_0(x)=\sin(\pi x)$. We plot
$\mathcal{E}_{h,\tau}$ against $\tau$ on a log-log scale with various combinations of $(\alpha,\epsilon)$ and take the method \eqref{full} with small time stepsize $\tau=2^{-16}$ as the reference solution. We then compare it to the method \eqref{full} evaluated with time steps $(2^{2}\tau, 2^{3}\tau, 2^{4}\tau, 2^{5}\tau)$ in order to show the rate of convergence. Fig. \ref{order} presents the mean-square convergence order for the error $\mathcal{E}_{h,\tau}$ with various sizes of $\epsilon$. Fig. \ref{order} shows that the proposed scheme \eqref{full} is of order 2 for the deterministic case, i.e., $\epsilon=0$, and of order 1 for the stochastic case with $\epsilon=1$, which coincides with the theoretical analysis.
\section{Appendix}
\begin{proof}[Proof of uniform boundedness of $\|A\|$]
Based on the definition of $\|A\|$, we only need to show that the maximum eigenvalue $\lambda_*:=\max\{\lambda:\det(\lambda I-\hat{A})=0\}$ of positive definite matrix $\hat{A}:=-A\in\mathbb{R}^{J\times J}$ is uniformly bounded with respect to dimension $J$. Let $x:=\lambda-2>-2$. Then $\lambda I-\hat{A}$ turns to be \begin{equation*}
\left( \begin{array}{cccc} x&1 & & \\
1&x&1 & \\ &\ddots &\ddots &\ddots \\
& &1 &x\\ \end{array} \right)\\ \cong\left( \begin{array}{cccc} x&1 & & \\
&x-\frac1x&1 & \\ & & x-\frac1{x-\frac1x}& \\
& & &\ddots\\ \end{array} \right)=:X. \end{equation*} We define $a_1(x):=x$, $a_{n+1}(x):=x-\frac1{a_{n}(x)}$ and $X_n(x)=\prod_{i=1}^na_i(x)$ for $n\ge1$, and deduce that \begin{align}\label{Xn} X_{n+2}(x)=xX_{n+1}(x)-X_{n}(x). \end{align} Noticing that $X_2(x)=x^2-1>0$ and $X_2(x)-X_1(x)=x^2-x-1>0$ for any $x\ge2$. We assume that $X_{j+1}(x)>0$ and $X_{j+1}(x)-X_{j}(x)>0$ for any $x\ge2$ and $1\le j\le n$, which contributes to \begin{align*} X_{n+2}(x)-X_{n+1}(x)=(x-2)X_{n+1}(x)+(X_{n+1}(x)-X_{n}(x))>0 \end{align*} and $X_{n+2}(x)>X_{n+1}(x)>0$ based on \eqref{Xn}. Then the induction yields that $X_n(x)>0$ for any $x\ge2$ and $n\in\mathbb{N}$, which implies that $\lambda_*=\max\{x:X_J(x)=0\}+2\le 4.$ \end{proof}
\nocite{*}
\end{document} | arXiv |
\begin{document}
\title{On the Optimality of Pseudo-polynomial Algorithms for Integer Programming}
\begin{abstract} In the classic {\em Integer Programming} (IP) problem, the objective is to decide whether, for a given $m \times n$ matrix $A$ and an $m$-vector $b=(b_1,\dots, b_m)$, there is a non-negative integer $n$-vector $x$ such that $Ax=b$. Solving (IP) is an important step in numerous algorithms and it is important to obtain an understanding of the precise complexity of this problem as a function of natural parameters of the input.
The classic pseudo-polynomial time algorithm of Papadimitriou~[J. ACM 1981] for instances of (IP) with a constant number of constraints was only recently improved upon by Eisenbrand and Weismantel~[SODA 2018] and Jansen and Rohwedder~[ArXiv 2018]. We continue this line of work and show that under the Exponential Time Hypothesis (ETH), the algorithm of Jansen and Rohwedder is nearly optimal. We also show that when the matrix $A$ is assumed to be non-negative, a component of Papadimitriou's original algorithm is already nearly optimal under ETH.
This motivates us to pick up the line of research initiated by Cunningham and Geelen~[IPCO 2007] who studied the complexity of solving (IP) with non-negative matrices in which the number of constraints may be unbounded, but the branch-width of the column-matroid corresponding to the constraint matrix is a constant. We prove a lower bound on the complexity of solving (IP) for such instances and obtain optimal results with respect to a closely related parameter, path-width. Specifically, we prove {\em matching} upper and lower bounds for (IP) when the {\em path-width} of the corresponding column-matroid is a constant.
\end{abstract}
\begin{abstract} In the classic {\em Integer Programming} (IP) problem, the objective is to decide whether, for a given $m \times n$ matrix $A$ and an $m$-vector $b=(b_1,\dots, b_m)$, there is a non-negative integer $n$-vector $x$ such that $Ax=b$. Solving (IP) is an important step in numerous algorithms and it is important to obtain an understanding of the precise complexity of this problem as a function of natural parameters of the input.
The classic pseudo-polynomial time algorithm of Papadimitriou~[J. ACM 1981] for instances of (IP) with a constant number of constraints was only recently improved upon by Eisenbrand and Weismantel~[SODA 2018] and Jansen and Rohwedder~[ArXiv 2018]. We continue this line of work and show that under the Exponential Time Hypothesis (ETH), the algorithm of Jansen and Rohwedder is nearly optimal. We also show that when the matrix $A$ is assumed to be non-negative, a component of Papadimitriou's original algorithm is already nearly optimal under ETH.
This motivates us to pick up the line of research initiated by Cunningham and Geelen~[IPCO 2007] who studied the complexity of solving (IP) with non-negative matrices in which the number of constraints may be unbounded, but the branch-width of the column-matroid corresponding to the constraint matrix is a constant. We prove a lower bound on the complexity of solving (IP) for such instances and obtain optimal results with respect to a closely related parameter, path-width. Specifically, we prove {\em matching} upper and lower bounds for (IP) when the {\em path-width} of the corresponding column-matroid is a constant.
\end{abstract}
\section{Introduction}\label{sec:intro}
\newcommand{A_{(\psi,c)}}{A_{(\psi,c)}} \newcommand{b_{(\psi,c)}}{b_{(\psi,c)}} \newcommand{\Delta}{\Delta} \newcommand{d}{d}
\newcommand{\|b\|_{\infty}}{\|b\|_{\infty}}
In the classic {\em Integer Programming} problem, the input is an $m\times n$ integer matrix $A$, and an $m$-vector $b=(b_1, \dots, b_m)$. We consider the feasibility version of the problem, where the objective is to find a non-negative integer $n$-vector $x $ (if one exists) such that $Ax=b$. Solving this problem, denoted by (IP)\xspace, is a fundamental step in numerous algorithms and it is important to obtain an understanding of the precise complexity of this problem as a function of natural parameters of the input.
(IP)\xspace is known to be NP-hard. However, there are two classic algorithms due to Lenstra \cite{ILP:Lenstra} and Papadimitriou ~\cite{Papadimitriou81} solving (IP)\xspace in polynomial or pseudo-polynomial time for two important cases when the number of variables and the number of constraints are bounded. These algorithms in some sense complement each other.
The algorithm of Lenstra shows that (IP)\xspace is solvable in polynomial time when the number of variables is bounded. Actually, the result of Lenstra is even stronger: (IP)\xspace is {\em fixed-parameter tractable} parameterized by the number of variables.
However, the running time of Lenstra's algorithm is doubly exponential in $n$. Later, Kannan \cite{ILP:Kannan} provided an algorithm for (IP)\xspace running in time $n^{{\mathcal{O}}(n)}$. Deciding whether the running time $n^{{\mathcal{O}}(n)}$ can be improved to $2^{{\mathcal{O}}(n)}$ is a long-standing open question.
Our work is motivated by the complexity analysis of the complementary case when the number of constraints is bounded. (IP)\xspace is NP-hard already for $m=1$ (the \textsc{Knapsack} problem) but solvable in pseudo-polynomial time. In 1981, Papadimitriou~\cite{Papadimitriou81} extended this result by showing that (IP)\xspace is solvable in pseudo-polynomial time on instances for which the number of constraints $m$ is a constant.
The algorithm of Papadimitriou consists of two steps. The first step is combinatorial, showing that if the entries of $A$ and $b$ are from $\{0,\pm 1, \dots, \pm d\},$ and (IP)\xspace has a solution, then there is also a solution which is in $\{0,1, \dots, n (md )^{2m+1}\}^n$. The second, algorithmic step shows that if (IP)\xspace has a solution with the maximum entry at most $B$, then the problem is solvable in time ${\mathcal{O}}((nB)^{m+1})$. Thus the total running time of Papadimitriou's algorithm is ${\mathcal{O}}(n^{2m+2} \cdot (md)^{(m+1)(2m+1)})$, where $d$ is an upper bound on the absolute values of the entries of $A$ and $b$. There was no algorithmic progress on this problem until the very recent breakthrough of Eisenbrand and Weismantel~\cite{EisenbrandW18}. They proved the following result.
\begin{proposition}[Theorem 2.2, \textbf{Eisenbrand and Weismantel~\cite{EisenbrandW18}}]\label{prop:Eisen} (IP)\xspace with $m\times n$ matrix $A$ is solvable in time $(m\cdot \Delta)^{{\mathcal{O}}(m)}\cdot \|b\|_{\infty}^2$, where $\Delta$ is an upper bound on the absolute values of the entries of $A$.
\end{proposition}
Then, Jansen and Rohwedder improved Proposition~\ref{prop:Eisen} and gave a matching lower bound very recently~\cite{JansenR18}.
\begin{proposition}[\textbf{Jansen and Rohwedder~\cite{JansenR18}}]\label{prop:Jansen} (IP)\xspace with $m\times n$ matrix $A$ is solvable in time ${\mathcal{O}}(m \Delta)^m \log(\Delta) \log(\Delta+\|b\|_{\infty})$.
where $\Delta$ is an upper bound on the absolute values of the entries of $A$. Assuming the Strong Exponential Time Hypothesis (SETH), there is no algorithm for (IP)\xspace\ running in time $n^{{\mathcal{O}}(1)}\cdot {\mathcal{O}}(m(\Delta+\|b\|_{\infty}))^{m-\delta}$ for any $\delta>0$. \end{proposition}
SETH is the hypothesis that CNF-SAT cannot be solved in time $(2-\epsilon)^nm^{{\mathcal{O}}(1)}$ on $n$-variable $m$-clause formulas for any constant $\epsilon$. ETH is the hypothesis that 3-SAT cannot be solved in time $2^{o(n)}$ on $n$-variable formulas. Both ETH and SETH were first introduced in the work of Impagliazzo and Paturi~\cite{ImpagliazzoP01}, which built upon earlier work of Impagliazzo, Paturi and Zane~\cite{ImpagliazzoPZ01}.
One of the natural question is whether the exponential dependence of $\|b\|_{\infty}$ can be improved
significantly
at the cost of super polynomial dependence on $n$.
Our first theorem provides a conditional lower bound indicating that any significant improvements are unlikely.
\begin{restatable}{theorem}{ETHIP}\label{thm:ETHIP}
Unless the Exponential Time Hypothesis (ETH) fails, (IP)\xspace with $m\times n$ matrix $A$ cannot be solved in time $n^{o(\frac{m}{\log m})} \cdot \|b\|_{\infty}^{o(m)}$
even when the constraint matrix $A$ is non-negative and each entry in any feasible solution is at most $2$. \end{restatable}
Let us note that since the bound in Theorem~\ref{thm:ETHIP} holds for a non-negative matrix $A$, we can always reduce (in polynomial time) the original instance of the problem to an equivalent instance where the maximum value $\Delta$ in the constraint matrix $A$ does not exceed $\|b\|_{\infty}$. Thus Theorem~\ref{thm:ETHIP} also implies the conditional lower bound $n^{o(\frac{m}{\log m})} \cdot (\Delta\cdot \|b\|_{\infty})^{o(m)}$. When $m={\mathcal{O}}( n)$, our bound also implies the lower bound $(n\cdot m)^{o(\frac{m}{\log m})} \cdot (\Delta\cdot \|b\|_{\infty})^{o(m)}$.
We complement Theorem~\ref{thm:ETHIP} by turning our focus to the dependence of algorithms solving (IP)\xspace on $m$ alone, and obtaining the following theorem.
\begin{restatable}{theorem}{ETHIPtwo}\label{thm:ETHIP2}
Unless the Exponential Time Hypothesis (ETH) fails, (IP)\xspace with $m\times n$ matrix $A$ cannot be solved in time $f(m)\cdot (n \cdot \|b\|_{\infty})^{o(\frac{m}{\log m})}$
for any computable function $f$. The result holds even when the constraint matrix $A$ is non-negative and each entry in any feasible solution is at most $1$. \end{restatable}
The difference between our first two theorems is the following. Although Theorem~\ref{thm:ETHIP} provides a better dependence on $\|b\|_{\infty}$, Theorem~\ref{thm:ETHIP2} provides much more information on how the complexity of the problem depends on $m$.
Since several parameters are involved in this running time estimation, a natural objective is to study the possible tradeoffs between them. For instance, consider the
${\mathcal{O}}(m \Delta)^m \log(\Delta) \log(\Delta+\|b\|_{\infty})$ time
algorithm (Proposition~\ref{prop:Jansen}) for (IP)\xspace. A natural follow up question is the following.
Could it be that by allowing a significantly worse dependence (a superpolynomial dependence) on $n$ and $\|b\|_{\infty}$ and an {\em arbitrary} dependence on $m$, one might be able to improve the dependence on $\Delta$ alone? Theorem~\ref{thm:ETHIP2} provides a strong argument against such an eventuality.
Indeed, since the lower bound of
Theorem~\ref{thm:ETHIP2} holds even for non-negative matrices, it rules out algorithms with running time $f(m)\cdot \Delta^{o(\frac{m}{\log m})} \cdot (n \cdot \|b\|_{\infty})^{o(\frac{m}{\log m})}$. Therefore, obtaining a subexponential dependence of
$\Delta$ on ${m}$ even at the cost of a superpolynomial dependence of $n$ and $\|b\|_{\infty}$ on $m$, and an arbitrarily bad dependence on $m$ is as hard as obtaining a subexponential algorithm for 3-SAT.
We now motivate our remaining results. We refer the reader to Figure~\ref{fig:summary} for a summary of our main results. It is straightforward to see that when the matrix $A$ happens to be non-negative, the algorithm of Papadimitriou~\cite{Papadimitriou81} runs in time ${\mathcal{O}}((n\cdot \|b\|_{\infty})^{m+1})$. Due to Theorems~\ref{thm:ETHIP} and ~\ref{thm:ETHIP2}, the dynamic programming step of the algorithm of Papadimitriou for (IP)\xspace{} when the maximum entry in a solution as well as in the constraint matrix is bounded, is already close to optimal.
Consequently, any quest for ``faster'' algorithms for (IP)\xspace must be built around the use of additional structural properties of the matrix $A$. Cunningham and Geelen~\cite{CunninghamG07} introduced such an approach by considering the \emph{branch decomposition} of the matrix $A$. They were motivated by the fact that the result of Papadimitriou can be interpreted as a result for matrices of constant \emph{rank} and {{branch-width}\xspace} is a parameter which is upper bounded by rank plus one.
For a matrix $A$, the \emph{column-matroid} of $A$ denotes the matroid whose elements are the columns of $A$ and whose independent sets are precisely the linearly independent sets of columns of $A$. We postpone the formal definitions of branch decomposition and {branch-width}\xspace till the next section. For (IP)\xspace with a \emph{non-negative} matrix $A$, Cunningham and Geelen \cite{CunninghamG07} showed that when the {branch-width}\xspace of the column-matroid of $A$ is constant, (IP)\xspace is solvable in pseudo-polynomial time. \begin{proposition}[\textbf{Cunningham and Geelen \cite{CunninghamG07}}]\label{thmCG} (IP)\xspace with { a non-negative} $m\times n$ matrix $A$ given together with a branch decomposition of its column matroid of width $k$, is solvable in time ${\mathcal{O}}((\|b\|_{\infty}+1)^{2k}mn + m^2n)$.
\end{proposition}
We analyze the complexity of (IP)\xspace parameterized by the {branch-width}\xspace of $A$, by making use of SETH
and
obtain the following lower bound(s).
\begin{theorem} \label{thm:lowbranchwidth0}
Unless SETH fails, (IP)\xspace with a non-negative $m\times n$ constraint matrix $A$ cannot be solved in time
$f({\sf bw})(\|b\|_{\infty}+1)^{(1-\epsilon){\sf bw}}(mn)^{{\mathcal{O}}(1)}$ or $f(\|b\|_{\infty})(\|b\|_{\infty}+1)^{(1-\epsilon){\sf bw}}(mn)^{{\mathcal{O}}(1)}$, for any computable function $f$. Here $\sf bw$ is the branchwidth of the column matroid of $A$. \end{theorem}
In recent years,
SETH has been used to obtain several tight conditional bounds on the running time of algorithms for various optimization problems on graphs of bounded treewidth ~\cite{LokshtanovMS11a-tw}. However, in order to be able to use SETH to prove lower bounds for (IP)\xspace in combination with the {branch-width}\xspace of matroids, we have to develop new ideas.
In fact, Theorem~\ref{thm:lowbranchwidth0} follows from stronger lower bounds we prove using the {path-width}\xspace of $A$ as our parameter of interest instead of the {branch-width}\xspace.
The parameter \emph{{path-width}\xspace} is closely related to the notion of \emph{trellis-width} of a linear code, which is a parameter commonly used in coding theory
\cite{DBLP:journals/tit/HornK96}. For a matrix $A\in {\mathbb R}^{m\times n}$, computing the {path-width}\xspace of the column matroid of $A$ is equivalent to computing the trellis-width of the linear code generated by $A$. Roughly speaking,
the {path-width}\xspace of the column matroid of $A$ is at most $k$, if there is a permutation of the columns of $A$ such that in the matrix $A'$ obtained from $A$ by applying this column-permutation, for every $1\leq i \leq n-1$, the \emph{dimension} of the subspace of ${\mathbb R}^{m}$ obtained by taking the intersection of the subspace of ${\mathbb R}^{m}$ spanned by the first $i$ columns with the subspace of ${\mathbb R}^{m}$ spanned by the remaining columns, is at most $k-1$.
\begin{figure}\label{fig:summary}
\end{figure}
The value of the parameter {{path-width}\xspace} is always at least the value of {{branch-width}\xspace} and thus Theorem~\ref{thm:lowbranchwidth0} follows from the following theorems.
\begin{theorem} \label{thm:lowbranchwidth}
Unless SETH fails, (IP)\xspace with even a non-negative $m\times n$ constraint matrix $A$ cannot be solved in time $f(k)(\|b\|_{\infty}+1)^{(1-\epsilon)k}(mn)^{{\mathcal{O}}(1)}$ for any computable function $f$ and $\epsilon>0$, where
$k$ is the {path-width}\xspace of the column matroid of $A$. \end{theorem}
\begin{theorem} \label{thm:lowentries}
Unless SETH fails, (IP)\xspace with even a non-negative $m\times n$ constraint matrix $A$ cannot be solved in time $f(\|b\|_{\infty})(\|b\|_{\infty}+1)^{(1-\epsilon)k}(mn)^{{\mathcal{O}}(1)}$ for any computable function $f$ and $\epsilon>0$, where
$k$ is the {path-width}\xspace of the column matroid of $A$. \end{theorem}
Although the proofs of both lower bounds have a similar {structure}, we believe that there are sufficiently many differences in the proofs to warrant stating and proving them separately.
Note that although there is still a gap between the upper bound of Cunningham and Geelen from Proposition~\ref{thmCG} and the lower bound provided by Theorem~\ref{thm:lowbranchwidth0}, the lower bounds given in Theorems~\ref{thm:lowentries} and \ref{thm:lowbranchwidth} are asymptotically tight in the following sense. The proof of
Cunningham and Geelen in \cite{CunninghamG07}
actually implies the upper bound stated in Theorem~\ref{thmCGlin}. We provide a self-contained proof in this paper for the reader's convenience.
\begin{theorem}\label{thmCGlin} (IP)\xspace with non-negative $m\times n$ matrix $A$ given together with a path decomposition of its column matroid of width $k$ is solvable in time ${\mathcal{O}}((\|b\|_{\infty}+1)^{k+1}mn + m^2n)$.
\end{theorem} Then by Theorem~\ref{thm:lowbranchwidth}, we cannot relax the $(\|b\|_{\infty}+1)^k$ factor in Theorem~\ref{thmCGlin} {even} if we allow in the running time an arbitrary function depending on $k$, while Theorem~\ref{thm:lowentries} shows a similar lower bound in terms of $\|b\|_{\infty}$ instead of $k$. Put together the results imply that no matter how much one is allowed to compromise on either the path-width or the bound on $\|b\|_{\infty}$,
it is unlikely that the algorithm of Theorem \ref{thmCGlin} can be improved.
The {path-width}\xspace of matrix $A$ does not exceed its rank and thus the number of constraints in (IP)\xspace. Hence, similar to Proposition~\ref{thmCG}, Theorem~\ref{thmCGlin} generalizes the result of Papadimitriou when restricted to non-negative matrices. Also we note that the assumption of non-negativity is unavoidable (without any further assumptions such as a bounded domain for the variables) in this setting because (IP)\xspace is NP-hard when the constraint matrix $A$ is allowed to have negative values (in fact even when restricted to $\{-1,0,1\}$) and the branchwidth of the column matroid of $A$ is at most 3. A close inspection of the instances they construct in their NP-hardness reduction shows that the column matroids of the resulting constraint matrices are in fact direct sums of circuits, implying that even their \emph{{path-width}\xspace} is bounded by 3.
\noindent
{\bf Organization of the paper.} The rest of the paper is organized as follows. There are two main technical parts to this paper. The first part (Section~\ref{Sec:ETHlowerbounds}) is devoted to proving Theorem~\ref{thm:ETHIP} and Theorem~\ref{thm:ETHIP2} (our ETH based lower bounds) while the second part (Section~\ref{sec:SETHLB}) is devoted to proving Theorem~\ref{thm:lowbranchwidth} and Theorem~\ref{thm:lowentries} (our SETH based lower bounds), and consequently, Theorem~\ref{thm:lowbranchwidth0}. For all our reductions, we begin by giving an overview in order to help the reader (especially in the SETH based reductions) navigate the technical details in the reductions.
We then prove Theorem~\ref{thm:lowentries} in Section~\ref{sec:lowentries} and Theorem~\ref{thmCGlin} in Section~\ref{sec:thmCGlin} (completing the results for constant {{path-width}\xspace}).
\section{Preliminaries}\label{sec:prel}
We assume that the reader is familiar with basic definitions from linear algebra, matroid theory and graph theory.
\noindent\textbf{Notations.} We use ${\mathbb Z}_{\scriptscriptstyle{\geq 0}}$ and ${\mathbb R}$ to denote the set of non negative integers and real numbers, respectively. For any positive integer $n$, we use $[n]$ and $\ZZ{n}$ to denotes the sets $\{1,\ldots,n\}$ and $\{0,1,\ldots,n-1\}$, respectively. For convenience, we say that $[0]=\emptyset$. For any two vectors $b, b'\in {\mathbb R}^m$ and $i\in [m]$, we use $b[i]$ to denote the $i^{th}$ coordinate of $b$ and we write $b'\leq b$, if $b'[i]\leq b[i]$ for all $i\in [m]$. We often use $0$ to denote the zero-vector whose length will be clear from the context. For a matrix $A\in {\mathbb R}^{m\times n}$, $I\subseteq [m]$ and $J\subseteq [n]$, $A[I,J]$ denote the submatrix of $A$ obtained by the restriction of $A$ to the rows indexed by $I$ and columns indexed by $J$. For an $m\times n$ matrix $A$ and $n$-vector $v$, we can write $Av=\sum_{i=1}^{n} A_i v[i]$, where $A_i$ is the $i^{th}$ column of $A$. Here we say that $v[i]$ is a multiplier of column $A_i$. For convenience, in this paper, we consider $0$ as an even number.
\noindent\textbf{Branch-width of matroids.} The notion of the branch-width of graphs, and implicitly of matroids, was introduced by Robertson and Seymour in \cite{RobertsonS91}. Let ${M} = (U, {\cal F})$ be a matroid with universe set $U$ and family ${\cal F}$ of independent sets over $U$. We use $r_M$ to denote the rank function of $M$. That is, for any $S\subseteq U$, $r_M(S)=\max_{S'\subseteq S, S'\in {\cal F}} \vert S'\vert$.
For $X\subseteq U$, the \emph{connectivity function}
of $M$ is defined as \[\lambda_M(X)=r_M(X) +r_M(U\setminus X) - r_M(U) +1\]
For matrix $A\in {\mathbb R}^{m\times n}$, we use $M(A)$ to denote the column-matroid of $A$. In this case the connectivity function $\lambda_{M(A)}$ has the following interpretation. For $E=\{1,\dots, n\}$ and $X\subseteq E$, we define
\[S(A,X)=\operatorname{span}(A|X)\cap\operatorname{span}(A|E\setminus X), \]
where $A|X$ is the set of columns of $A$ restricted to $X$ and $\operatorname{span}(A|X)$ is the subspace of ${\mathbb R}^{m}$
spanned by the columns $A|X$. It is easy to see that the dimension of $S(A,X)$ is equal to $\lambda_{M(A)}(X)-1$.
A tree is \emph{cubic} if its internal vertices all have degree $3$. A \emph{branch decomposition} of matroid ${M}$ with universe set $U$ is a cubic tree $T$ and mapping $\mu$ which maps elements of $U$ to leaves of $T$. Let $e$ be an edge of $T$. Then the forest $T-e$ consists of two connected components $T_1$ and $T_2$. Thus every edge $e$ of $T$ corresponds to the partitioning of $U$ into two sets
$X_e$ and $U\setminus X_e$ such that $\mu(X_e)$ are the leaves of $T_1$ and $\mu(U\setminus X_e)$ are the leaves of $T_2$. The \emph{width} of edge $e$ is $\lambda_{M}(X_e)$ and the width of branch decomposition $(T, \mu)$ is the maximum edge width, where maximum is taken over all edges of $T$. Finally, the \emph{branch-width} of $M$ is the minimum width taken over all possible branch decompositions of $M$.
The \emph{{path-width}\xspace} of a matroid is defined as follows. Recall that a \emph{caterpillar} is a tree which is obtained from a path by attaching leaves to some vertices of the path. Then the {path-width}\xspace of a matroid is the minimum width of a branch decomposition $(T,\mu)$, where $T$ is a cubic caterpillar. Let us note that every mapping of elements of a matroid to the leaves of a cubic caterpillar corresponds to an ordering of these elements.
Jeong, Kim, and Oum \cite{Jeong0O16} gave a constructive fixed-parameter tractable algorithm to construct a path decomposition of width at most $k$ for a column matroid of a given matrix.
\noindent\textbf{ETH and SETH.}
For $q\geq 3$, let $\delta_q$ be the infimum of the set of constants $c$ for which there exists an algorithm solving $q$-SAT with $n$ variables and $m$ clauses in time $2^{cn}\cdot m^{{\mathcal{O}}(1)}$. The {\em{Exponential-Time Hypothesis} (ETH)} and {\em{Strong Exponential-Time Hypothesis} (SETH)} are then formally defined as follows. ETH conjectures that $\delta_3>0$ and SETH that $\lim_{q\to \infty}\delta_q=1$.
\section{ ETH lower bounds on pseudopolynomial solvability of (IP)\xspace} \label{Sec:ETHlowerbounds}
In this section we prove Theorems~\ref{thm:ETHIP} and~\ref{thm:ETHIP2}.
\subsection{Proof of Theorem~\ref{thm:ETHIP}}\label{sec:lowrank} This subsection is devoted to the proof of Theorem~\ref{thm:ETHIP}
\ETHIP*
Our proof is by a reduction from {\sc $3$-CNF SAT} to (IP)\xspace.
There are exactly 2 variables in the (IP)\xspace\ instance for each variable (one for each literal) and clause. For each clause we define two constraints. For each variable in the 3-CNF formula, we have a constraint, which is a selection gadget.
\begin{figure}
\caption{An illustration of the matrix $A_\psi$ corresponding to the 3-CNF formula $\psi=(x_1\vee x_2\vee x_3) \wedge (\bar x_1 \vee \bar x_2 \vee x_3) \wedge (\bar x_4 \vee \bar x_2 \vee \bar x_3)$. The unfilled cells have 0 as the entry.
}
\label{fig:Apsi-illustration}
\end{figure}
We now proceed to the formal description of the reduction. From a {\sc $3$-CNF} formula $\psi$ on $n$ variables and $m$ clauses we create an equivalent (IP)\xspace{} instance $A_{\psi} x=b_{\psi},x\geq 0$, where $A_{\psi}$ is a non-negative integer $(2m+n)\times 2 (m+n)$ matrix and the largest entry in $b_{\psi}$ is $3$. Our reduction can be easily seen to be a polynomial time reduction and we do not give an explicit analysis. Let $\psi$ be the input of {\sc $3$-CNF SAT}. Let $X=\{x_1,\ldots,x_n\}$ be the set of variables in $\psi$ and ${\mathcal C}=\{C_1,\ldots,C_m\}$ be the set of clauses in $\psi$.
First we define the set of variables in the in the (IP)\xspace\ instance. For each $x_i\in X$, we have two variables $x_i$ and $\overline{x}_i$ in the (IP)\xspace{} instance $A_{\psi} x=b_{\psi},x\geq 0$. For each $C_i\in {\mathcal C}$, we have two variables $Y_i$ and $Z_i$.
Now we define the set of constraints of $A_{\psi} x=b_{\psi},x\geq 0$. For each $C_i=x\vee y \vee z$, we define two constraints \begin{eqnarray} x+y+z+Y_i&=&3 \qquad \mbox{and} \label{eqn:ci1} \\ Y_i+Z_i&=&2.\label{eqn:ci2} \end{eqnarray} \begin{eqnarray} \mbox{For each $i\in [n]$,} \qquad\qquad x_i+\overline{x}_i=1 \label{eqn:xi} \end{eqnarray} This completes the construction of (IP)\xspace{} instance $A_{\psi} x=b_{\psi},x\geq 0$. See Figure~\ref{fig:Apsi-illustration} for an illustration.
We now argue that this reduction correctly maps satisfiable 3-CNF formulas to feasible instances of (IP)\xspace and vice versa.
\begin{lemma} \label{lemma:rank_correctness} The formula $\psi$ is satisfiable if and only if $A_{\psi} x=b_{\psi},x\geq 0$ is feasible. \end{lemma} \begin{proof} Suppose that the formula $\psi$ is satisfiable and let $\phi$ be a satisfying assignment of $\psi$. We set values for the variables $\{x_i,\overline{x}_i \colon i\in [n]\}\cup \{Y_i,Z_i \colon i\in [m]\}$ and prove that $A_{\psi} x=b_{\psi}$. For any $i\in [n]$, if $\phi(x_i)=1$ we set $x_i=1$ and $\overline{x}_i=0$. Otherwise, we set $x_i=0$ and $\overline{x}_i=1$.
For every $i\in [m]$, we define \begin{equation} Y_i = \left\{ \begin{array}{ll} 0 & \mbox{if the number of literals set to $1$ in $C_{i}$ by $\phi$ is $3$,} \\ 1 & \mbox{if the number of literals set to $1$ in $C_{i}$ by $\phi$ is $2$,} \\ 2 & \mbox{otherwise,} \end{array}\right. \label{eqn:x:bottom:odd} \end{equation} and \begin{equation} Z_i = \left\{ \begin{array}{ll} 2 & \mbox{if the number of literals set to $1$ in $C_{i}$ by $\phi$ is $3$,} \\ 1 & \mbox{if the number of literals set to $1$ in $C_{i}$ by $\phi$ is $2$,} \\ 0 & \mbox{otherwise.} \end{array}\right. \label{eqn:x:bottom:even} \end{equation} We now proceed to prove that the above substitution of values to the variables is indeed a feasible solution.
Towards this, we need to show that \eqref{eqn:ci1}, \eqref{eqn:ci2}, and \eqref{eqn:xi} are satisfied. First consider \eqref{eqn:ci1}. Let $C_i=x\vee y \vee z$. Since $\phi$ is a satisfying assignment, we have that $1\leq x+y+z \leq 3$. Thus, by \eqref{eqn:x:bottom:odd}, we conclude that $x+y+z+Y_i=3$. Because of \eqref{eqn:x:bottom:odd} and \eqref{eqn:x:bottom:even}, \eqref{eqn:ci2} is satisfied. Since the values for $\{x_i,\overline{x}_i\colon i\in [n]\}$ is derived from an assignment $\phi$, \eqref{eqn:xi} is satisfied.
For the converse direction of the statement of the lemma, suppose that there exists non-negative values for the set of variables $\{x_i,\overline{x}_i \colon i\in [n]\}\cup \{Y_i,Z_i \colon i\in [m]\}$, such that \eqref{eqn:ci1}, \eqref{eqn:ci2}, and \eqref{eqn:xi} are satisfied.
Now we need to show that $\psi$ is satisfiable. Because of \eqref{eqn:xi}, we know that exactly one of $x_i$ and $\overline{x}_i$ is set to one and other is set to zero.
Next, we define an assignment $\phi$ and prove that $\phi$ is a satisfying assignment for $\psi$. For $i\in [n]$ we define \[ \phi(x_i) = \left\{ \begin{array}{ll} 1 & \mbox{if } x_i=1, \\ 0 & \mbox{if } \overline{x}_i=1. \end{array}\right. \]
We claim that $\phi$ satisfies all the clauses. Consider a clause $C_j=x\vee y \vee z$ where $j\in [m]$. Since $Y_j+Z_j=2$ (by \eqref{eqn:ci2}), we have that $Y_i\in \{0,1,2\}$. Since $Y_i\in \{0,1,2\}$, by \eqref{eqn:ci1}, at least one of $x,y$ or $z$ is set to one. This implies that $\phi$ satisfies $C_j$.
This completes the proof of the lemma. \end{proof}
By \eqref{eqn:ci2} and \eqref{eqn:xi}, we have that the value set for any variable in a feasible solution is at most $2$.
The following lemma completes the proof of the theorem.
\begin{lemma} If there is an algorithm for
(IP)\xspace{} running in time $n^{o(\frac{m}{\log m})} \|b\|_{\infty}^{o(m)}$, then
ETH fails. \end{lemma} \begin{proof} By the Sparsification Lemma~\cite{ImpagliazzoPZ01}, we know that {\sc $3$-CNF SAT} on $n'$ variables and $cn'$ clauses, where $c$ is a constant, cannot be solved in time $2^{o(n')}$ time.
Suppose there is an algorithm {\sf ALG} for (IP)\xspace{} running in time $n^{o(\frac{m}{\log m})} \|b\|_{\infty}^{o(m)}$. Then for a $3$-CNF formula $\psi$ with $n'$ variables and $m'=cn$ clauses we create an instance $A_{\psi} x=b_{\psi},$ $x\geq 0$ of (IP)\xspace{} as discussed in this section, in polynomial time, where $A_{\psi}$ is a matrix of dimension $(2cn'+n')\times (2(n'+cn'))$ and the largest entry in $b_{\psi}$ is $3$.
Then by Lemma~\ref{lemma:rank_correctness}, we can run {\sf ALG} to test whether $\psi$ is satisfiable or not. This takes time $$(2(cn'+n'))^{o(\frac{2cn'+n')}{\log (2cn'+n')})} \cdot 3^{o(2cn'+n')}=2^{o(n')},$$ hence refuting ETH. \end{proof}
\subsection{Proof of Theorem~\ref{thm:ETHIP2}}\label{sec:lowrank2} In this section we prove the following theorem.
\ETHIPtwo*
Towards proving Theorem~\ref{thm:ETHIP2} we use the ETH based lower bound result of Marx~\cite{marx-toc-treewidth} for {{\sc Partitioned Subgraph Isomorphism}}. For two graphs $G$ and $H$, a map $\phi\colon V(G)\mapsto V(H)$ is called a {\em subgraph isomorphism} from $G$ to $H$, if $\phi$ is injective and for any $\{u,v\}\in E(G)$, $\{\phi(u),\phi(v)\}\in E(H)$ (see Figure~\ref{fig:PSI_example} for an illustration).
\begin{figure}\label{fig:PSI_example}
\end{figure}
\defproblem{{\sc Partitioned Subgraph Isomorphism}}{Two graphs $G,H$, a bijection $c_G\colon V(G)\mapsto [\ell]$ and a function $c_H\colon V(H)\mapsto [\ell]$, where $\ell=\vert V(G) \vert $.} {Is there a subgraph isomorphism $\phi$ from $G$ to $H$ such that for any $v\in V(G)$, $c_G(v)=c_H(\phi(v))$?}
\begin{lemma}[Corollary 6.3~\cite{marx-toc-treewidth}] \label{lem:psimarx} If {\sc Partitioned Subgraph Isomorphism}\ can be solved in time $f(G)n^{o(\frac{k}{\log k})}$, where $f$ is an arbitrary function, $n=\vert V(H)\vert$ and $k$ is the number of edges of the smaller graph $G$, then ETH fails. \end{lemma}
To prove Theorem~\ref{thm:ETHIP2} we give a polynomial time reduction from {\sc Partitioned Subgraph Isomorphism}\ to (IP)\xspace such that for every instance $(G,H,c_G,c_H)$ of {\sc Partitioned Subgraph Isomorphism}\, the reduction outputs an instance of (IP)\xspace where the constraint matrix has dimension ${\mathcal{O}}(\vert E(G)\vert)\times{\mathcal{O}}(\vert E(H)\vert)$ and the largest value in the target vector is $\max \{ \vert E(H)\vert, \vert V(H)\vert\}$.
Let $(G,H,c_G,c_H)$ be an instance of {\sc Partitioned Subgraph Isomorphism}. Let $k=\vert E(G)\vert$ and $n=\vert V(H)\vert$. We construct an instance $Ax=b$ of (IP)\xspace from $(G,H,c_G,c_H)$ in polynomial time. Without loss of generality we assume that $[n]=V(H)$ and that there are no isolated vertices in $G$. Hence, the number of vertices in $G$ is at most $2k$. Let $m=\vert E(H)\vert$. For each $e\in E(H)$ we assign a unique integer from $[m]$. Let $\alpha\colon E(H)\mapsto [m]$ be the bijection which represents the assignment mentioned above. For any $i,j\in [\ell]$, we use $E_H(i,j)$ as a shorthand for the set of edges of $H$ between $c_H^{-1}(i)$ and $c_H^{-1}(j)$. Finally, for ease of presentation we let $\{v_1,\ldots,v_{\ell}\}=V(G)$ and $c_G(v_i)=i$ for all $i\in [\ell]$, where $\ell=\vert V(G)\vert$.
For illustrative purposes, before proceeding to the formal construction, we give an informal description of the (IP)\xspace\ instance we obtain from a specific instance of {{\sc Partitioned Subgraph Isomorphism}}. Let $H$ and $G$ be the graphs in Figure~\ref{fig:PSI_example} and consider the graph $\widehat{H}$ obtained from $H$ as depicted in Figure~\ref{fig:PSI_example_aux_graph}.
For every color $i\in [\ell]$ we have a column in $\widehat{H}$ and for every pair of distinct colors $i,j\in [\ell]$ such that $\{v_i,v_j\}\in E(G)$, we have a copy of $c_H^{-1}(i)$ in Column $i$ and Row $i$ and a copy of $c_H^{-1}(i)$ in Column $i$ and Row $j$. Thus, Column $i$ comprises at most $\ell$ copies of the vertices of $H$ whose image under $c_H$ is $i$ and Row $i$ comprises a copy of $c_H^{-1}(i)$ and additionally, a copy of every vertex $u$ of $H$ such that $v_{c_H{u}}$ is adjacent to $v_i$ in $G$. That is, the color of $u$ is ``adjacent'' to the color $i$ in $G$.
For a vertex $u\in V(H)$, we refer to the unique copy of $u$ in the $i^{th}$ row as the $i^{th}$ copy of $u$ in $\widehat H$. For every edge $e=\{a,b\}\in E(H)$ where $c_H(a)=i$, $c_H(b)=j$, and $\{v_i,v_j\}\in E(G)$, we have two copies of $e$ in $\widehat H$. The first copy of $e$ has as its endpoints, the $i^{th}$ copy of $a$ and the $i^{th}$ copy of $b$ and the second copy of $e$ has as its endpoints, the $j^{th}$ copy of $a$ and the $j^{th}$ copy of $b$. We now rephrase the {{\sc Partitioned Subgraph Isomorphism}} problem (informally) as a problem of finding a certain type of subgraph in $\widehat{H}$, which in turn will point us in the direction of our {(IP)\xspace} instance in a natural way. The rephrased problem statement is the following. Given $G$, $H$,$c_H$,$c_G$ and the resulting auxiliary graph $\widehat H$, find a set of $2|E(H)|$ edges in $\widehat H$ such that the following properties hold.
\begin{itemize}
\item (Selection) For every $\{v_i,v_j\}\in E(G)$, we pick a unique edge in $\widehat H$ with one endpoint in (Row $i$, Column $i$) and the other endpoint in (Row $i$, Column $j$) and we pick a unique edge with one endpoint in (Row $j$, Column $j$) and the other endpoint in (Row $j$, Column $i$).
\item (Consistency 1) All the edges we pick from Row $i$ of $\widehat H$ share a common endpoint at the position (Row $i$, Column $i$).
\item (Consistency 2) For any edge $e=\{a,b\}\in E(H)$ such that $c_H(a)=i$, $c_H(b)=j$, if the copy of $e$ in Row $i$ is selected in our solution then our solution contains the copy of $e$ in Row $j$ as well.
\end{itemize}
It is straightforward to see that a set of edges of $\widehat H$ which satisfy the stated properties imply a solution to our {\sc Partitioned Subgraph Isomorphism}\ instance in an obvious way. In order to obtain our (IP)\xspace\ instance, we create a variable for every edge in $\widehat H$ (or 2 for every edge in $E(H)$) and encode the properties stated above in the form of constraints.
We now formally define the (IP)\xspace\ instance output by our reduction.
\begin{figure}
\caption{An illustration of the auxiliary graph $\widehat{H}$ capturing the representation of the vertices and {\em some} edges of $H$. }
\label{fig:PSI_example_aux_graph}
\end{figure}
The set of indeterminants $x$ of the (IP)\xspace instance is $$\left\{x(\{a,b\},c_H(a),c_H(b))\colon \{a,b\}\in E(H) \right\}.$$
Notice that for any $\{a,b\}\in E(H)$, there exist an associated pair of indeterminants, namely $x(\{a,b\},c_H(a),c_H(b))$ and $x(\{a,b\},c_H(b),c_H(a))$. Thus the cardinality of $x$ is upper bounded by $2 \vert E(H)\vert =2m$. Recall that $\{v_1,\ldots,v_{\ell}\}=V(G)$ and $c_G(v_i)=i$ for all $i\in [\ell]$, where $\ell=\vert V(G)\vert$.
For each $v_i\in V(G)$ we define $2d_G(v_i)-1$ many constraints as explained below. Let $r=d_G(v_i)$ and $N_G(v_i)=\{v_{j_1},\ldots,v_{j_r}\}$. The constraints for $v_i\in V(G)$ are the following.
For all $q\in [r]$, \begin{equation}
\sum_{\substack{e\in E_H(i,j_q)}} x(e,i,j_q) = 1 \label{eqn:psi1}
\end{equation}
The constraints of the form above enforce the (Selection) property described in our informal summary.
For all $q\in [r-1]$, \begin{equation}
\sum_{\substack{\{a,b\}\in E_H(i,j_q)\\ a\in c_H^{-1}(i)}} a\cdot x(\{a,b\},i,j_q) +
\sum_{\substack{\{a,b'\}\in E_H(i,j_{q+1})\\ a \in c_H^{-1}(i)}} (n-a)\cdot x(\{a,b'\},i,j_{q+1})
= n \label{eqn:psi2} \end{equation} The constraints of the form above together enforce the (Consistency 1) property described in our informal summary.
For each $\{v_i,v_j\}\in E(G)$ with $i<j$, we define the following constraint in the (IP)\xspace instance.
\begin{eqnarray}
\sum_{\substack{ \{a,b\}\in E_H(i,j)\\ a\in c_H^{-1}(i)}} \alpha(\{a,b\}) \cdot x(\{a,b\},i,j) +\sum_{ \substack{ \{a,b\}\in E_H(i,j)\\b\in c_H^{-1}(j)}} (m-\alpha(\{a,b\})) \cdot x(\{a,b\},j,i)=m \label{eqn:psi3} \end{eqnarray} The constraints of the form above together enforce the (Consistency 2) property described in our informal summary.
This completes the construction of the (IP)\xspace instance $Ax=b,x\geq 0$. Notice that the construction of instance $Ax=b,x\geq 0$ can be done in polynomial time. Clearly, the number of rows in $A$ is $\vert E(G)\vert + \sum_{v\in V(G)} 2d_{G}(v)-1\leq 5k$ and number of columns in $A$ is $2m$. Now we prove the correctness of the reduction. \begin{lemma} \label{lem:psicor} $(G,H,c_G,c_H)$ is a {\sc Yes} instance of {\sc Partitioned Subgraph Isomorphism}\ if and only if $Ax=b,x\geq 0$ is feasible. Moreover, if $Ax=b,x\geq 0$ is feasible, then for any solution $x^*$, each entry of $x^*$ belongs to $\{0,1\}$. \end{lemma} \begin{proof} Suppose $(G,H,c_G,c_H)$ is a {\sc Yes} instance of {\sc Partitioned Subgraph Isomorphism}. Let $\phi\colon V(G)\mapsto V(H)$ be a solution to $(G,H,c_G,c_H)$. Now we define a solution $x^*\in \{0,1\}^{2m}$ to the instance $Ax=b,x\geq 0$ of (IP)\xspace. We know that for each edge $\{v_i,v_j\}\in E(G)$, $\{\phi(v_i),\phi(v_j)\}\in E(H)$. For each edge $\{v_i,v_j\}\in E(G)$, we set $x^*(\{\phi(v_i),\phi(v_j)\},i,j)=x^*(\{\phi(v_i),\phi(v_j)\},j,i)=1$. For every other indeterminant, we set its value to $0$. Now we prove that $Ax^*=b$.
Towards that first consider $(\ref{eqn:psi1})$. Fix a vertex $v_i\in V(G)$ and $v_{j_q}\in N_G(v_i)$. Since $\{v_i,v_{j_q}\}\in E(G)$, $x^*(\{\phi(v_i),\phi(v_{j_q})\},i,j_q)=1$. Moreover, since $G$ is a simple graph, for any edge $e\in E_H(i,j_q)\setminus \{\{\phi(v_i),\phi(v_{j_q})\}\}$, $x^*(e,i,j_q)=0$. This implies that $(\ref{eqn:psi1})$ is satisfied by $x^*$. Next we consider (\ref{eqn:psi2}). Fix a vertex $v_i\in V(G)$. Let $N_G(v_i)=\{v_{j_1},\ldots,v_{j_r}\}$. Also, fix $q\in [r-1]$. We know that $\{v_i,v_{j_{q}}\},\{v_i,v_{j_{q+1}}\}\in E(G)$. By the definition of $x^*$, we have that $x^*(e,i,j_q)=1$ if and only if $e=\{\phi(v_i),\phi(v_{j_q})\}$ and $x^*(e',i,{j_{q+1}})=1$ if and only if $e'=\{\phi(v_i),\phi(v_{j_{q+1}})\}$.
Thus we have that \begin{eqnarray*}
\sum_{\substack{\{a,b\}\in E_H(i,j_q)\\ a\in c_H^{-1}(i)}} a\cdot x(\{a,b\},i,j_q) +
\sum_{\substack{\{a,b'\}\in E_H(i,j_{q+1})\\ a \in c_H^{-1}(i)}} (n-a)\cdot x(\{a,b'\},i,j_{q+1})\\ =\phi(v_i)+(n-\phi(v_i))=n \end{eqnarray*} That is, $x^*$ satisfies (\ref{eqn:psi2}). Now we consider (\ref{eqn:psi3}). Fix an edge $\{v_i,v_j\}\in E(G)$ where $i<j$. Again by the definition of $x^*$, we have that $x^*(e,i,j)=1$ if and only if $e=\{\phi(v_i),\phi(v_{j})\}$ and $x^*(e,j,i)=1$ if and only if $e=\{\phi(v_i),\phi(v_{j})\}$. This implies that (\ref{eqn:psi3}) is satisfied by $x^*$. Therefore $Ax=b,x\geq 0$ is feasible.
Now we prove the converse direction of the lemma. Suppose that $Ax=b,x \geq 0$ is feasible and let $x'\in {\mathbb N}_0^{2m}$ be a solution. \begin{claim} \label{claimpsi1} Let $i,j\in [\ell]$ such that $i\neq j$ and $\{v_i,v_j\}\in E(G)$. Then there exists exactly one edge $e\in E_H(i,j)$ such that $x'(e,i,j)=x'(e,j,i)=1$. Moreover, for any $e'\in E_H(i,j)\setminus \{e\}$, $x'(e',i,j)=x'(e',j,i)=0$. \end{claim} \begin{proof}
By (\ref{eqn:psi1}), we have that there exists exactly one edge $e_1\in E_H(i,j)$ such that $x'(e_1,i,j)=1$ and for all other edges $h\in E_H(i,j)\setminus \{e_1\}$, $x'(h,i,j)=0$. Again by (\ref{eqn:psi1}), we have that there exists exactly one edge $e_2\in E_H(i,j)$ such that $x'(e_2,j,i)=1$ and for all other edges $h\in E_H(i,j)\setminus \{e_2\}$, $x'(h,j,i)=0$. By (\ref{eqn:psi3}), we have that $e_1=e_2$. This completes the proof of the claim. \end{proof}
Now we define an injection $\phi\colon V(G)\mapsto V(H)$ and prove that indeed $\phi$ is a subgraph isomorphism from $G$ to $H$. For any $i,j\in [\ell]$ with $i\neq j$ and $\{v_i,v_j\}\in E(G)$ consider the edge $e=\{a,b\}\in E_H(i,j)$ such that $x'(\{a,b\},i,j)=x'(\{a,b\},j,i)=1$ (by Claim~\ref{claimpsi1}, there exits exactly one such edge in $E_H(i,j)$). Let $a\in c_H^{-1}(i)$ and $b\in c_H^{-1}(j)$. Now we set $\phi(v_i)=a$ and $\phi(v_j)=b$. We claim that $\phi$ is well defined. Fix a vertex $v_i\in V(G)$. Let $r=d_G(v_i)$ and $N_G(v_i)=\{v_{j_1},\ldots,v_{j_r}\}$. By Claim~\ref{claimpsi1}, we know that for any $q\in [r]$, there exists exactly one edge $\{a_q,b_q\}\in E_H(i,j)$ such that $x'(\{a_q,b_q\},i,j_q)=x'(\{a_q,b_q\},j_q,i)=1$. Here, $a_q\in c_H^{-1}(i)$ and $b_q\in c_H^{-1}(j_q)$. To prove that $\phi$ is well defined, it is enough to prove that $a_1=a_2=\ldots=a_r=\phi(v_i)$. By (\ref{eqn:psi2}), we have that for any $q\in [r-1]$, $a_q=a_{q+1}$. Also since $x'(\{a_q,b_q\},i,j_q)=x'(\{a_q,b_q\},j_q,i)=1$ for all $q\in [r]$, we have that $a_1=a_2=\ldots=a_r=\phi(v_i)$. From the construction of $\phi$, we have that for any $i,j\in [\ell]$, $i\neq j$, $\phi(v_i)\in c_H^{-1}(i)$ and $\phi(v_j)\in c_H^{-1}(j)$. Moreover, $c_H^{-1}(i)\cap c_H^{-1}(j)=\emptyset$. This implies that $\phi$ is an injective map.
Now we prove that $\phi$ is an isomorphism from $G$ to $H$. Since $\phi(v_i)\in c_H^{-1}(i)$ for all $i\in[\ell]$, to prove that $\phi$ is an isomorphism, it is enough to prove that for any edge $\{v_i,v_j\}\in V(G)$, $\{\phi(v_i),\phi(v_j)\}\in E(H)$. Fix an edge $\{v_i,v_j\}\in V(G)$ with $i<j$.
By Claim~\ref{claimpsi1}, there exists exactly one edge $\{a,b\}\in E_H(i,j)$ such that $x'(\{a,b\},i,j)=x'(\{a,b\},j,i)=1$, where $a\in c_H^{-1}(i)$ and $b\in c_H^{-1}(j)$. From the definition of $\phi$, we have that $\phi(v_i)=a$ and $\phi(v_j)=b$. That is, $\{\phi(v_i),\phi(v_j)\}=\{a,b\}\in E(H)$.
By Claim~\ref{claimpsi1}, we conclude that if $Ax=b,x\geq 0$ is feasible, then for any solution $x^*$, each entry of $x^*$ belongs to $\{0,1\}$. This completes the proof of the lemma. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:ETHIP2}] Let $(G,H,c_G,c_H)$ be an instance of {\sc Partitioned Subgraph Isomorphism}. Let $Ax=b,x\geq 0$ be the instance of (IP)\xspace\ constructed from $(G,H,c_G,c_H)$ as mentioned above. We know that the construction of $Ax=b,x\geq 0$ takes time polynomial in $n$, where $n=\vert V(H)\vert$. Also, we know that the number of rows and columns in $A$ is $\leq 5\vert E(G)\vert$ and $2\vert E(H)\vert$, respectively. Moreover, the maximum entry in $b$ is $\max \{\vert V(H)\vert, \vert E(H)\vert\}$.
Suppose there is an algorithm ${\cal A}$ for (IP)\xspace, running in time $f(m')(n'\cdot d')^{o\left(\frac{m'}{\log m'}\right)}$ on instances where the constraint matrix is non-negative and is of dimension $m'\times n'$, and the maximum entry in the target vector is $d'$. Then, by running ${\cal A}$ on $Ax=b,x\geq 0$ and applying Lemma~\ref{lem:psicor}, we solve {\sc Partitioned Subgraph Isomorphism}\ in time $f(G)n^{o\left(\frac{k}{\log k}\right)}$. Thus by Lemma~\ref{lem:psimarx}, ETH fails.
This completes the proof of the theorem. \end{proof}
\label{sec:lowbranchwidth}
\section{Path-width parameterization: SETH bounds }\label{sec:SETHLB}
In this section we prove Theorems~\ref{thm:lowbranchwidth} and~\ref{thm:lowentries}. \subsection{Overview of our reductions} \label{subsec:overview} \input{overview}
\subsection{Proof of Theorem~\ref{thm:lowbranchwidth}}\label{subsec:lowbr}
In this section we provide a Proof of Theorem~\ref{thm:lowbranchwidth}, which states that unless SETH fails, (IP)\xspace{} with non-negative matrix $A$ cannot be solved in time $f(k)(\|b\|_{\infty}+1)^{(1-\epsilon)k}(mn)^{{\mathcal{O}}(1)}$ for any function $f$ and $\epsilon>0$, where $d=\max\{b[1],\ldots,b[m]\}$ and $k$ is the {path-width}\xspace of the column matroid of $A$.
Towards the proof of Theorem~\ref{thm:lowbranchwidth}, we first present the proof of our main technical lemma (Lemma~\ref{lemtechnicalintro}), which we restate here for the sake of completeness.
\lemtechnicalintro*
Let $\psi=C_1\wedge C_2\wedge\ldots \wedge C_m$ be an instance of {\sc CNF-SAT} with variable set $X=\{x_1,x_2,\ldots,x_n\}$ and let $c\geq 2$ be a fixed constant given in the statement of Lemma~\ref{lemtechnicalintro}. We construct the instance $A_{(\psi,c)} x=b_{(\psi,c)},x\geq 0$ of (IP)\xspace as follows.
\noindent {\bf Construction.} Let ${\mathcal C}=\{C_1,\ldots,C_m\}$. Without loss of generality, we assume that $n$ is divisible by $c$, otherwise we add at most $c$ dummy variables to $X$ such that $\vert X\vert$ is divisible by $c$. We divide $X$ into $c$ blocks $X_0,X_1,\ldots,X_{c-1}$. That is $X_{i}=\{x_{\frac{i \cdot n}{c}+1},x_{\frac{i\cdot n}{c}+2},\ldots, x_{\frac{(i+1)\cdot n}{c}}\}$ for each $i\in \ZZ{c}$. Let $\ell=\frac{n}{c}$ and $L=2^{\ell}$. For each block $X_{i}$, there are exactly $2^{\ell}$ assignments. We denote these assignments by $\phi_{0}(X_{i}),\phi_1(X_{i}),\ldots, \phi_{L-1}(X_{i})$.
Now, we create $m\cdot c\cdot 2^{\ell+1}$ variables; they are named $y_{C,i,a}$, where $C\in {\mathcal C}$, $i\in \ZZ{c}$ and $a\in \ZZ{2L}=\ZZ{2^{\ell+1}}$. In other words, for a clause $C$, a block $X_{i}$ and an assignment $\phi_a(X_{i})$, we create two variables; they are $y_{C,i,2a}$ and $y_{C,i,2a+1}$. Then, we create the (IP)\xspace constraints given by Equations~(\ref{eqn:sat}), (\ref{eqn:onlyone}), and (\ref{eqn:consistancy}).
This completes the construction of (IP)\xspace instance. Let $A_{(\psi,c)} y=b_{(\psi,c)}$ be the (IP)\xspace instance defined using Equations~(\ref{eqn:sat}), (\ref{eqn:onlyone}), and (\ref{eqn:consistancy}). The purpose of Equation~(\ref{eqn:sat}) is to ensure satisfiability of all the clauses. Because of Equation~(\ref{eqn:onlyone}), for each clause $C$ and for each block $X_i$, we select only one assignment. Notice, that, so far it is allowed to choose many assignments from a block $X_i$, for different clauses. To ensure the consistency of assignments in each block across clauses, we added a system of constraints (Equation~(\ref{eqn:consistancy})). Equation~(\ref{eqn:consistancy}) ensures the consistency of assignments in the adjacent clauses (in the order $C_1,\ldots,C_m$). Thus, the consistency of assignments propagates in a sequential manner. Notice that number constraints defined by Equations~(\ref{eqn:sat}), (\ref{eqn:onlyone}), and (\ref{eqn:consistancy}) are $m$, $m\cdot c$ and $(m-1)\cdot c$, respectively. The number of variables is $m\cdot c \cdot 2^{\ell+1}$. Also notice that all the coefficients in Equations~(\ref{eqn:sat}), (\ref{eqn:onlyone}) and (\ref{eqn:consistancy}) are non-negative. This implies that $A_{(\psi,c)}$ is non-negative and has dimension ${\mathcal{O}}(m)\times {\mathcal{O}}(m 2^{\frac{n}{c}})$. Thus, the property $(b.)$ of Lemma~\ref{lemtechnicalintro} is satisfied. The largest entry in $b_{(\psi,c)}$ is $L-1= 2^{\lceil \frac{n}{c}\rceil}-1$ (see Equation~(\ref{eqn:consistancy})) and hence the property $(d.)$ of Lemma~\ref{lemtechnicalintro} is satisfied. Now we prove property $(a.)$ of Lemma~\ref{lemtechnicalintro}.
\begin{center}
\textbf{
we simplify the notation by using $A$ instead of $A_{(\psi,c)} $ and $b$ instead of $ b_{(\psi,c)}$.
} \end{center}
\begin{lemma} \label{lemma:brachwidthcorrect}
Formula $\psi$ is satisfiable if and only if there exists $y^*\in {\mathbb Z}_{\scriptscriptstyle{\geq 0}}^{n'}$ such that $A y^*=b$.
where $n'= m\cdot c \cdot 2^{\ell+1}$, the number of columns in $A$.
\end{lemma}
\begin{proof}
Let $Y=\{y_{C,i,a}~|~C\in {\mathcal C},i\in \ZZ{c},a\in \ZZ{2L}\}$. Suppose $\psi$ is satisfiable.
We need to show that there is an assignment of non-negative integer values to the variables in $Y$ such that Equations~(\ref{eqn:sat}), (\ref{eqn:onlyone}) and (\ref{eqn:consistancy}) are satisfied. Let $\phi$ be a satisfying assignment of $\psi$. Then, there exist $a_0,a_1,\ldots,a_{c-1}\in \ZZ{L}$ such that
$\phi$ is the union of $\phi_{a_0}(X_0),\phi_{a_1}(X_1),\ldots,\phi_{a_{c-1}}(X_{c-1})$. Any clause $C\in {\mathcal C}$ is satisfied by at least one of the assignments
$\phi_{a_0}(X_0),\phi_{a_1}(X_1),\ldots,\phi_{a_{c-1}}(X_{c-1})$. For each
$C$, we fix an arbitrary $i \in \ZZ{c}$ such that
the assignment $\phi_{a_{i}}(X_{i})$ satisfies clause $C$.
Let $\alpha$ be a function which fixes these assignments for each clause.
That is, $\alpha: {\mathcal C} \rightarrow \ZZ{c}$ such that the assignment $\phi_{a_{\alpha(C)}}(X_{\alpha(C)})$
satisfies the clause $C$ for every $C\in{\mathcal C}$.
Now we assign values to $Y$ and
prove that these assignment satisfy Equations~(\ref{eqn:sat}), (\ref{eqn:onlyone}) and (\ref{eqn:consistancy}).
\begin{equation} y_{C,i,a} = \left\{ \begin{array}{ll} 1, & \mbox{if } \alpha(C)=i \mbox{ and } a \mbox{ is even and } \lfloor \frac{a}{2}\rfloor=a_i\\
1, & \mbox{if } \alpha(C)\neq i \mbox{ and } a \mbox{ is odd and } \lfloor \frac{a}{2}\rfloor=a_i\\ 0, & \mbox{otherwise.} \end{array}\right.\label{eqn:assignment} \end{equation} Notice that, by Equation~(\ref{eqn:assignment}), for any fixed $C\in {\mathcal C}$, exactly $c$ variables from
$\{y_{C,i,a}~|~i\in \ZZ{c},a\in [2^{\ell+1}]\}$ is set to $1$.
They are $y_{C,\alpha(C),2a_{\alpha(C)}}$ and the variables in the set $Y_C=\{y_{C,i,2a_i+1}~|~i\neq \alpha(C)\}$. This implies that in Equation~(\ref{eqn:sat}), only variable is set to $1$, and hence Equation~(\ref{eqn:sat}) is satisfied. Now consider Equation~(\ref{eqn:onlyone}) for any fixed $C\in {\mathcal C}$ and $i\in \ZZ{c}$. By equation~(\ref{eqn:assignment}),
exactly one variable from $\{y_{C,i,a}~|~a\in \ZZ{2L}\}$ is set to $1$, and hence Equation~(\ref{eqn:onlyone}) is satisfied. Now consider Equation~(\ref{eqn:consistancy}) for fixed $r\in [m-1]$ and $i\in \ZZ{c}$. By Equation~(\ref{eqn:assignment}),
exactly one variable from each set $\{y_{C_r,i,a}~|~a\in \ZZ{2L}\}$ and $\{y_{C_{r+1},i,a}~|~a\in \ZZ{2L}\}$ are set to $1$; they are one variable each from $\{y_{C_r,i, 2a_i},y_{C_r,i, 2a_i+1}\}$ and $\{y_{C_{r+1},i, 2a_i},y_{C_{r+1},i, 2a_i+1}\}$. So we get the following when we substitute values for $Y$ in Equation~(\ref{eqn:consistancy}). \begin{eqnarray*} \sum_{\substack{a \in \ZZ{2L} }} \left(\lfloor \frac{a}{2}\rfloor\cdot y_{C_r,i,a}\right) + \left((L-1-\lfloor \frac{a}{2}\rfloor)\cdot y_{C_{r+1},i,a}\right) = a_i + L-1 - a_i = L-1 \end{eqnarray*} Hence, Equation~(\ref{eqn:consistancy}) is satisfied by the assignments given in Equation~(\ref{eqn:assignment}).
Now we need to prove the converse direction. Suppose there are non-negative integer assignments to $Y$ such that Equations~(\ref{eqn:sat}), (\ref{eqn:onlyone}) and (\ref{eqn:consistancy}) are satisfied. Now we need to show that $\psi$ is satisfiable. Because of Equation~(\ref{eqn:onlyone}) all the variables in $Y$ are set to $0$ or $1$. We will extract a satisfying assignment from the values assigned to variables in $Y$. Towards that, first we prove the following claim. \begin{claim}\label{claim:extractsat} Let $y_{C_1,i,a}=1$ for some $i\in \ZZ{c}$
and $a\in \ZZ{2L}$. Then, for any $C'\in {\mathcal C}$, exactly one among $\{y_{C',i,2\lfloor \frac{a}{2}\rfloor}, y_{C',i,2\lfloor \frac{a}{2}\rfloor+1}\}$ is set to $1$. \end{claim} \begin{proof} Towards the proof, we first show that if $y_{C_r,i,a}=1$ for some $r\in [m-1]$, then exactly one among $\{y_{C_{r+1},i,2\lfloor \frac{a}{2}\rfloor}, y_{C_{r+1},i,2\lfloor \frac{a}{2}\rfloor+1}\}$ is set to $1$.
By Equation~(\ref{eqn:onlyone}) and the fact that $y_{C_r,i,a}=1$, we get that \begin{equation} \sum_{\substack{a' \in \ZZ{2L} }} \left(\lfloor \frac{a'}{2}\rfloor\cdot y_{C_r,i,a'}\right)=\lfloor \frac{a}{2}\rfloor.\label{eqn:aby2} \end{equation} Equations~(\ref{eqn:consistancy}) and (\ref{eqn:aby2}) implies that
\begin{equation} \sum_{\substack{a' \in \ZZ{2L} }} \left((L-1-\lfloor \frac{a'}{2}\rfloor)\cdot y_{C_{r+1},i,a'}\right)= L-1- \lfloor \frac{a}{2}\rfloor. \label{eqn:aby2second} \end{equation}
By Equations~(\ref{eqn:onlyone}) and (\ref{eqn:aby2second}), we get that exactly one among $\{y_{C_{r+1},i,2\lfloor \frac{a}{2}\rfloor}, y_{C_{r+1},i,2\lfloor \frac{a}{2}\rfloor+1}\}$ is set to $1$.
Thus, by applying the above arguments for $i=1,2,\ldots,m-1$, we get that for any
$C'\in {\mathcal C}\setminus \{C_1\}$, exactly one among $\{y_{C',i,2\lfloor \frac{a}{2}\rfloor}, y_{C',i,2\lfloor \frac{a}{2}\rfloor+1}\}$ is set to $1$.
Suppose $C'=C_1$. Then, by Equation~(\ref{eqn:onlyone}) and the assumption that $y_{C_1,i,a}=1$, exactly one among $\{y_{C_1,i,2\lfloor \frac{a}{2}\rfloor}, y_{C_1,i,2\lfloor \frac{a}{2}\rfloor+1}\}$ is set to $1$. \end{proof} Now we define a satisfying assignment for $\psi$. Towards that we give assignments for each blocks $X_0,\ldots,X_{c-1}$, such that the union of these assignments satisfies $\psi$. Fix any block $X_i$. By Equation~(\ref{eqn:onlyone}), exactly one among
$\{y_{C_1,i,a}~|~a\in \ZZ{2L}\}$ is set to $1$. Let $a_i\in \ZZ{2L}$ such that $y_{C_1,i,a_i}=1$. Then we choose the assignment $\phi_{\lfloor \frac{a_i}{2}\rfloor}(X_i)$ for $X_i$. Let $\phi$ be the assignment of $X$ which is the union of $\psi_{\lfloor \frac{a_1}{2}\rfloor}(X_1)$, $\psi_{\lfloor \frac{a_2}{2}\rfloor}(X_2)$,\ldots,$\psi_{\lfloor \frac{a_{c-1}}{2}\rfloor}(X_{c-1})$. By Equation~(\ref{eqn:sat}) and Claim~\ref{claim:extractsat}, $\phi$ satisfies all the clauses in ${\mathcal C}$ and hence $\psi$ is satisfiable. \end{proof}
Now we need to prove property $(c.)$ of Lemma~\ref{lemtechnicalintro}. That is the {path-width}\xspace of $A$ is at most $c+4$. Towards that we need to understand the structure of matrix $A$. We decompose the matrix $A$ into $m$ disjoint submatrices $B_1,\ldots B_m$ which are disjoint and cover all the non-zero entries in the matrix $A$. First we define some notations and fix the column indices of $A$ corresponding the the variables in the constraints.
Let $Y$ denote the set $\{y_{C,i,a}~|~C\in {\mathcal C},i\in \ZZ{c},a\in \ZZ{2L}\}$ of variables in the constraints defined by Equations~(\ref{eqn:sat}), (\ref{eqn:onlyone}) and (\ref{eqn:consistancy}). These variables can be partitioned into $\biguplus_{C\in {\mathcal C}}Y_C$, where
$Y_C=\{y_{C,i,a}~|~i\in \ZZ{c},a\in \ZZ{2L}\}$. Further for each $C\in {\mathcal C}$, $Y_C$ can be partitioned into $\bigcup_{i\in{\ZZ{c}}}Y_{C,i}$,
where $Y_{C,i}=\{y_{C,i,a}~|~a\in \ZZ{2L}\}$. The set of columns indexed by $[r\cdot c\dot 2^{\ell+1}]\setminus [(r-1)\cdot c \cdot 2^{\ell+1}]$, for any $r\in[m]$, corresponds to the set of variables in $Y_{C_r}$. Among the set of columns corresponding to $Y_C$, the first $2^{\ell+1}$ columns corresponds to the variables in $Y_{C,1}$, second $2^{\ell+1}$ columns corresponds to the variables in $Y_{C,2}$, and so on. Among the set of columns corresponds to $Y_{C,i}$ for any $C\in {\mathcal C}$ and $i\in \ZZ{c}$, the first two columns corresponds to the variable $y_{C,i,0}$ and $y_{C,i,1}$, and second two columns corresponds to the variables $y_{C,i,2}$ and $y_{C,i,3}$, and so on.
Now we move to the description of $B_j$, $j\in [m]$. The matrix $B_j$ will cover the coefficients of $Y_{C_j}$ in Equations~(\ref{eqn:sat}), (\ref{eqn:onlyone}) and (\ref{eqn:consistancy}). In other words $B_j$ covers the non-zero entries in the columns corresponding to $Y_{C_j}$, i.e, in the columns of $A$ indexed by $[j \cdot c \cdot 2^{\ell+1}]\setminus [(j-1) \cdot c \cdot 2^{\ell+1}]$. Now we explain these submatrices. Each matrix $B_j$ has $c\cdot 2^{\ell+1}$ columns; each of them corresponds to a variable in $Y_{C_j}$. Each row in $A$ corresponds to a constraint in the system of equations defined by Equations~(\ref{eqn:sat}), (\ref{eqn:onlyone}) and (\ref{eqn:consistancy}). So we use notations $f(C_1),\ldots f(C_m)$ to represents the constraints defined by
Equations~(\ref{eqn:sat}). Similarly we use notations $\{s(C,i)~|~~C\in {\mathcal C},i\in \ZZ{c}\}$ and $\{t(C,i)~|~~C\in {\mathcal C},i\in \ZZ{c}\}$ to represents the constraints defined by Equations~(\ref{eqn:onlyone}) and (\ref{eqn:consistancy}), respectively.
\noindent\emph{Matrix $B_1$.} Matrix $B_1$ is of dimension $(2c+1)\times (c\cdot 2^{\ell+1})$. In the first row of $B_1$, we have coefficients of $Y_{C_1}$ from $f(C_1)$.
For $j\in [c]$, the rows indexed by $j+1$ and $c+j+1$ are defined as follows. In the $(j+1)^{st}$ row of $B_1$, we have coefficients of $Y_{C_1}$ from $s(C_1,j)$ while in the $(c+j+1)^{st}$ row of $B_1$, we have coefficients of $Y_{C_1}$ from $t(C_1,j)$. That is the entries of $B_1$ are as follows, where $i\in \ZZ{c}$ and $a\in \ZZ{L}$. \begin{eqnarray} &&B_1[1, {i}\cdot 2^{\ell+1}+2a+1]= \left\{ \begin{array}{ll} 1 & \mbox{if } \phi_{a}(X_{i}) \mbox{ satisfies } C_1,\\ 0 & \mbox{otherwise.}
\end{array}\right.
\label{eqn:B1:evaluation}\\ &&B_1[1, {i}\cdot 2^{\ell+1}+2a+2]= 0, \text{ and} \label{eqn:B1:evaluation0}\\ &&B_1[2+{i}, {i}\cdot 2^{\ell+1}+2a+1]= B_1[2+{i}, {i}\cdot 2^{\ell+1}+2a+2] = 1,\label{eqn:B1:selection}\\ && B_1[c+2+{i}, {i}\cdot 2^{\ell+1}+2a+1]= B_1[c+2+{i}, {i}\cdot 2^{\ell+1}+2a+2] =a, \label{eqn:B1:postassignement} \end{eqnarray}
Here, Equations~(\ref{eqn:B1:evaluation}) and (\ref{eqn:B1:evaluation0}), follows from Equation~(\ref{eqn:sat}). Equations~(\ref{eqn:B1:selection}) and (\ref{eqn:B1:postassignement}) follows from Equation~(\ref{eqn:onlyone}) and (\ref{eqn:consistancy}), respectively.
All other entries in $B_1$ are zeros. That is, for all $i,i' \in \ZZ{c}$ and $g\in [2^{\ell+1}]$ such that $i \neq i'$, \begin{eqnarray} && B_1[2+i, {i'}\cdot 2^{\ell+1}+g]=B_1[c+2+i, {i'}\cdot 2^{\ell+1}+g] =0, \label{eqn:B1:zero} \end{eqnarray}
This completes the definition of $B_1$. By their role in the reduction, the matrix $B_1$ is partitioned in to three parts. The first row is called the {\em evaluation part} of $B_1$. The part composed of rows indexed by $2,3,\ldots,c+1$ is called {\em selection part} and the part composed of last $c$ rows is called {\em successor matching part} (See Figure~\ref{fig:Br:portions}).
\noindent\emph{Matrices $B_r$ for $1<r<m$.} Matrix $B_r$ is of dimension $(3c+1)\times (c\cdot 2^{\ell+1})$. The first $c$ rows are defined by Equation~(\ref{eqn:consistancy}). For $j\in [c]$, in $i^{th}$ row, we have coefficients of $Y_{C_r}$ from $t(C_{r-1},i)$. In the $(c+1)^{st}$ row of $B_r$, we have coefficients of $Y_{C_r}$ from $f(C_r)$. For $i\in [c]$, the rows indexed by $c+1+i$ and $2c+1+i$ are defined as follows. In the $(c+1+i)^{th}$ row of $B_r$, we have coefficients of $Y_{C_r}$ from $s(C_r,i)$ while in the $(2c+1+i)^{th}$ row of $B_r$, we have coefficients of $Y_{C_r}$ from $t(C_r,i)$. This completes the definition of $B_r$. By their role in the reduction, the matrix $B_r$ is partitioned in to four parts. The part composed of the first $c$ rows is called the {\em predecessor matching part}. The part composed of the row indexed by $c+1$ is called the {\em evaluation part} of $B_1$. The part composed of rows indexed by $c+2,c+3,\ldots,2c+1$ is called {\em selection part} and the part composed of last $c$ rows is called {\em successor matching part} (For illustration see Fig.~\ref{fig:Br:portionsA}). That is the entries of $B_1$ are as follows, where $i\in \ZZ{c}$ and $a\in \ZZ{L}$.
The predecessor matching part is defined by \begin{equation} B_r[{i}+1, {i}\cdot 2^{\ell+1}+2a+1]=B_r[{i}+1, {i}\cdot 2^{\ell+1}+2a+2]=L-1-a. \label{eqn:Br:preassignement} \end{equation}
The evaluation part is defined by \begin{equation} B_r[c+1, {i}\cdot 2^{\ell+1}+2a+2]=0, \label{eqn:Br:evaluation0} \end{equation} and \begin{equation} B_r[c+1, {i}\cdot 2^{\ell+1}+2a+1]= \left\{ \begin{array}{ll} 1, & \mbox{if } \phi_{a}(X_{i}) \mbox{ satisfies } C_r,\\ 0, & \mbox{otherwise.} \end{array}\right. \label{eqn:Br:evaluation} \end{equation} The selection part for $B_r$ is defined as \begin{eqnarray} &&B_r[c+2+{i}, {i}\cdot 2^{\ell+1}+2a+1]=B_r[c+2+{i}, {i}\cdot 2^{\ell+1}+2a+2]=1, \label{eqn:Br:selection} \end{eqnarray} The successor matching part for $B_r$ is defined as
\begin{eqnarray} &&B_r[2c+2+{i}, {i}\cdot 2^{\ell+1}+2a+1]=B_r[2c+2+{i}, {i}\cdot 2^{\ell+1}+2a+2]=j, \label{eqn:Br:postssignement}
\end{eqnarray}
All other entries in $B_r$, which are not listed above, are zero. That is, for all $i,i' \in \ZZ{c}$ and $g\in [2^{\ell+1}]$ such that $i \neq i'$, \begin{eqnarray} &&B_r[{i}+1, {i'}\cdot 2^{\ell+1}+g]=0 \label{eqn:Br:zero1}, \\
&&B_r[c+2+i, {i'}\cdot 2^{\ell+1}+g]=0, \text{ and } \label{eqn:Br:zeroX}\\ &&B_r[2c+2+i, {i'}\cdot 2^{\ell+1}+g]=0. \label{eqn:Br:zero2} \end{eqnarray} For an example, see Figure~\ref{fig:Br}.
\begin{figure}
\caption{Parts of $B_1$.}
\label{fig:Br:portions}
\caption{Parts of $B_r$ for $1<r\leq m$.}
\label{fig:Br:portionsA}
\caption{Parts of $B_m$.}
\label{fig:Br:portions}
\caption{Parts of $B_r$.}
\label{fig:Bm}
\end{figure}
\noindent\emph{Matrices $B_m$.} Matrix $B_m$ is of dimension $(2c+1)\times (c\cdot 2^{\ell+1})$.
For $j\in [c]$, in $i^{the}$ row, we have coefficients of $Y_{C_m}$ from $t(C_{m-1},i)$. In the $(c+1)^{st}$ row of $B_r$, we have coefficients of $Y_{C_m}$ from $f(C_m)$.
In the $(c+1+i)^{th}$ row of $B_m$, we have coefficients of $Y_r$ from $s(C_m,i)$. That is $B_m$ is obtained by deleting the successor matching part from the construction of $B_r$ above. The entries of $B_m$ are as follows, where $i\in \ZZ{c}$ and $a\in \ZZ{L}$. \begin{eqnarray} &&B_m[{i}+1, {i}\cdot 2^{\ell+1}+2a+1]=B_m[{i}+1, {i}\cdot 2^{\ell+1}+2a+2]=L-1-a, \nonumber\\ && B_m[c+1, {i}\cdot 2^{\ell+1}+2a+2]= 0, \text{ and } \nonumber\\ && B_m[c+1, {i}\cdot 2^{\ell+1}+2a+2]= \left\{ \begin{array}{ll} 1, & \mbox{if } \phi_{a}(X_{i}) \mbox{ satisfies } C_m,\\ 0, & \mbox{otherwise.} \end{array}\right. \nonumber\\ && B_m[c+2+{i}, {i}\cdot 2^{\ell+1}+2a+1]= B_m[c+2+{i}, {i}\cdot 2^{\ell+1}+2a+2]= 1, \label{eqn:Bm:selection}
\end{eqnarray}
All other entries in $B_m$ are zeros. That is, for all $i,i' \in \ZZ{c}$ and $g\in [2^{\ell+1}]$ such that $i \neq i'$, \begin{eqnarray} && B_m[1+{i}, {i'}\cdot 2^{\ell+1}+g] =0 \label{eqn:Bm:zero1}\\ && B_m[c+2+{i}, {i'}\cdot 2^{\ell+1}+g] = 0, \label {eqn:Bm:zero3}\\ && B_m[2c+2+i, {i'}\cdot 2^{\ell+1}+g] = 0. \label{eqn:Bm:zero2} \end{eqnarray}
\begin{figure}
\caption{Let $n=4,c=2,\ell=2$ and $C_r=x_1\vee \overline{x_2}\vee x_4$. The assignments are $\phi_0(X_0)=\{x_1=x_2=0\}, \phi_1(X_0)=\{x_1=0, x_2=1\}, \phi_2(X_0)=\{x_1=1, x_2=0\}, \phi_3(X_0)=\{x_1=x_2=1\}$, $\phi_0(X_1)=\{x_3=x_4=0\}, \phi_1(X_1)=\{x_3=0, x_4=1\}, \phi_2(X_1)=\{x_3=1, x_4=0\}, \phi_3(X_1)=\{x_3=x_4=1\}$. The entries defined according to $\phi_1(X_0)$ and $\phi_3(X_1)$ are colored red and blue respectively. If $1<r<m$, then the matrix on the left represents $B_r$ and if $r=1$, then $B_r$ can be obtained by deleting the yellow colored portion from the top matrix.
The matrix on the right represents $B_m$.
}
\label{fig:Br}
\end{figure}
\noindent\emph{Matrix $A$.}
Now we explain how the matrix $A$ is formed from $B_1,\ldots,B_m$. The matrices $B_1,\ldots,B_m$ are disjoint submatrices of $A$ and they cover all non zero entries of $A$. Informally, the submatrices $B_1,\ldots, B_m$ form a chain such that
the rows corresponding to the successor matching part of $B_r$ will be the same as the rows in the predecessor matching part of $B_{r+1}$ (because of Equation~(\ref{eqn:consistancy}). A pictorial representation of $A$ can be found in Fig.~\ref{fig:A}.
Formally,
let $I_1=[2c+1]$ and $I_m=[(m-1)(2c+1)+(c+1)]\setminus [(m-1)(2c+1)-c]$.
For every $1<r<m$, let $I_r=[r(2c+1)]\setminus [(r-1)(2c+1)-c]$, and for $r\in [m]$, let $J_r= [r\cdot c\cdot 2^{\ell+1}]\setminus [(r-1)\cdot c\cdot 2^{\ell+1}]$. Now for each $r\in [m]$, the matrix $A[I_r,J_r]:=B_r$. All other entries of $A$ not belonging to any of the submatrices $A[I_r,J_r]$ are zero.
Towards upper bounding the {path-width}\xspace\ of $A$, we start with some notations. We partition the set of columns of $A$ into $m$ parts $J_1,\ldots,J_m$ (we have already defined these sets) with one part per clause. For each $r\in [m]$, $J_r$ is the set of columns associated with $Y_{C_r}$. We further divide $J_r$ into $c$ equal parts, one per variable set $Y_{C_r,i}$. These parts are $$P_{r,i}=\{(r-1)c\cdot 2^{\ell+1}+ i\cdot 2^{\ell+1}+1,\ldots, (r-1)c\cdot 2^{\ell+1}+ (i+1)\cdot 2^{\ell+1}\}, \,i\in \ZZ{c}.$$ In other words, $P_{r,i}$ is the set of columns corresponding to $Y_{C_r,i}$ and $\vert P_{r,i}\vert=2^{\ell+1}$.
We also put $n'= m\cdot c \cdot 2^{\ell+1}$ to be the number of columns in $A$.
\begin{lemma} \label{lemma:pathwidthbounded} The {path-width}\xspace of the column matroid of $A$ is at most $c+4$ \end{lemma} \begin{proof} Recall that $n'=m\cdot c\cdot 2^{\ell+1}$, be the number of columns in $A$ and $m'$ be the number of rows in $A$. To prove that the {path-width}\xspace of $A$ is at most $c+4$, it is sufficient to show that
for all $j\in [n'-1]$,
\begin{equation}
\operatorname{dim}\langle \operatorname{span}(A|\{1,\dots, j\})\cap\operatorname{span}(A|\{j+1, \dots, n'\}) \rangle \leq c+3. \label{eqn:dim:intersection}
\end{equation}
The idea for proving Equation~\eqref{eqn:dim:intersection} is based on the following observation.
For $V'=A|\{1,\dots, j\}$ and $V''=A|\{j+1, \dots, n'\}$, let
$$I=\{q\in [m']~|~\mbox{there exists } v'\in V' \text{ and } v''\in V'' \text{ such that } v'[q]\neq v''[q] \neq 0\}.$$ Then the dimension of $\operatorname{span}(V')\cap \operatorname{span} (V'')$ is at most $\vert I \vert$. Thus to prove \eqref{eqn:dim:intersection}, for each $j\in [n'-1]$, we construct the corresponding set $I$ and show that its cardinality is at most $c+3$.
We proceed with the details. Let $v_1,v_2,\ldots,v_{n'}$ be the column vectors of $A$. Let $j\in [n'-1]$. Let $V_1=\{v_1,\ldots,v_j\}$ and $V_2=\{v_{j+1},\ldots,v_{n'}\}$. We need to show that
$\operatorname{dim}\langle \operatorname{span}(V_1)\cap\operatorname{span}(V_2) \rangle \leq c+3$. Let $$I'=\{q \in [m']~|~\mbox{there exists $v\in V_1$ and $v'\in V_2$ such that $v[q]\neq 0\neq v'[q]$}\}.$$ We know that $[n']$ is partitioned into parts $P_{r',i'}, r'\in [m], i'\in \ZZ{c}$.
\begin{center} \textbf{Fix $r\in [m]$ and $i\in \ZZ{c}$ such that $j \in P_{r,i}$}. \end{center}
Let $j=(r-1)c\cdot 2^{\ell+1}+i\cdot 2^{\ell+1}+g$, where $g \in [2^{\ell+1}]$. Let $q_1=\max\{0, (r-1)(2c+1)-c\}$, $q_2=r(2c+1)$, $j_1=(r-1)\cdot c\cdot 2^{\ell+1}$, and $j_2=r\cdot c\cdot 2^{\ell+1}$ Then $[q_2]\setminus [q_1]=I_r$ and $[j_2]\setminus [j_1]=J_r$ (recall the definition of sets $I_r$ and $J_r$).
By the decomposition of matrix $A$, for every $q>q_2$ and for every vector $v\in V_1$, we have $v[q]=0$. Also, for every $q\leq q_1$ and for any $v\in V_2$, we have that $v[q]=0$. This implies that $I' \subseteq [q_2]\setminus [q_1]=I_r$. Now we partition $I_r$ into $4$ parts: $R_1,R, S$, and $R_2$, These parts are defined as follows. \begin{eqnarray} R_1&=& \left\{ \begin{array}{ll} \emptyset, & \mbox{if } r=1, \\
\{(r-2)(2c+1)+i'~|~i'\in \ZZ{c}\}, & \mbox{otherwise,} \end{array}\right. \nonumber \\
R&=&\{(r-1)(2c+1)+1\}, \label{Eqn:R}\\
S&=&
\{(r-1)(2c+1)+2+ i' ~|~i'\in \ZZ{c}]\}
\nonumber \\
R_2&=& \left\{ \begin{array}{ll} \emptyset, & \mbox{if } r=m, \\
\{(r-1)(2c+1)+c+2+ i' ~|~i' \in \ZZ{c}\} ,& \mbox{otherwise }
\end{array}\right. \nonumber \end{eqnarray}
\begin{claim} \label{claim:outofwindow0} For each $r'\in [m], q\notin I_{r'}$ and $j''\in J_{r'}, v_{j''}[q]=0$. \end{claim} \begin{proof} The non-zero entries in $A$ are covered by the disjoint sub-matrices $A[I_{r'},J_{r'}]=B_{r'}, {r'}\in [m]$. Hence the claim follows. \end{proof}
\begin{claim} $\vert I'\cap R_1 \vert \leq c-(i-1)$. \label{claim:R1} \end{claim} \begin{proof} When $r=1$, $R_1=\emptyset$ and the claim trivially follows. Let $r>1$, and let $q \in R_1$ be such that $q <(r-2)(2c+1)+i$. Then $q=(r-2)(2c+1)+1+i'$ for some
$0 \leq i'< i$. Notice that $q \notin I_{r'}$ for every $r'>r$. By Claim~\ref{claim:outofwindow0}, for every $v\in \bigcup_{r'>r} J_{r'}$, $v[q]=0$. Now consider the vector $v_{j''}\in V_2 \setminus (\bigcup_{r'>r} J_r')$. Notice that $j''>j$ and $j''\in J_r$. Let $j''=j+a=(r-1)c\cdot 2^{\ell+1}+i \cdot 2^{\ell+1}+g+a$ for some $a\in [rc2^{\ell+1}-j]$.
From the decomposition of $A$, $v_{j''}[q]=B_r[i'+1, i \cdot 2^{\ell+1}+g+a]= 0$, by (\ref{eqn:Br:zero1}). Thus for every $q \in R$, $q<(r-2)(2c+1)+i$ and $v\in V_2$, $v[q]=0$.
This implies that $$\vert I'\cap R_1\vert \leq \vert \{q \geq (r-2)(2c+1)+i\}\cap R_1 \vert \leq c-(i-1).\qedhere$$ \end{proof}
\begin{claim} $\vert I'\cap R_2\vert \leq i$. \label{claim:R2} \end{claim}
\begin{proof} When $r=m$, $R_2=\emptyset$ and the claim trivially holds. So, now let $r<m$ and consider any $q \in R_2 \cap \{ q' >(r-1)(2c+1)+c+2+i\}$. Let $i'>i$ such that $q=(r-1)(2c+1)+c+2+i'$. Notice that $q\notin I_{r'}$ for any $r'<r$. Hence, by Claim~\ref{claim:outofwindow0}, for any $v\in \bigcup_{r'<r} J_{r'}$, $v[q]=0$. Now consider any vector $v_{j''}\in V_1 \setminus (\bigcup_{r'<r} J_r')$. Notice that $j''\leq j$ and $j''\in J_r$. Let $j''=(r-1)c\cdot 2^{\ell+1}+i'' \cdot 2^{\ell+1}+a$ for some $a\in [2^{\ell+1}]$ and $i''\leq i < i'$.
From the decomposition of $A$, $v_{j''}[q]=B_r[2c+2+i', i'' \cdot 2^{\ell+1}+a]= 0$, by (\ref{eqn:Br:zero2}). Hence we have shown that for any $q\in R$, $q>(r-2)(2c+1)+c+2+i$ and $v\in V_1$, $v[q]=0$. This implies that $$\vert I'\cap R_2\vert \leq \vert \{q \leq (r-1)(2c+1)+c+2+i\}\cap R_1 \vert \leq i.\qedhere$$ \end{proof} \begin{claim} $\vert I'\cap S \vert \leq 1$. \label{claim:R2'} \end{claim} \begin{proof}
Consider any $q \in S$. Let $i' \in \ZZ{c}$ such that $q=(r-1)(2c+1)+2+i'$. Notice that $q \notin I_{r'}$ for any $r'<r$, and hence, by Claim~\ref{claim:outofwindow0}, for any $v\in \bigcup_{r'<r} J_{r'}$, $v[q]=0$. Also notice that $q \notin I_{r'}$ for any $r'>r$, and hence, by Claim~\ref{claim:outofwindow0}, for any $v\in \bigcup_{r'>r+1} J_{r'}$, $v[q]=0$.
So the only potential $j''$ for which $v_{j''}[q]\neq 0$, are from $J_r$.
We claim that if $q \in I' \cap S$, then $q =(r-1)(2c+1)+2+i$. Suppose $q \in I' \cap S$ and $q <(r-1)(2c+1)+2+i$. Let $q=(r-1)(2c+1)+2+i'$, where $0\leq i' <i$. Then by the decomposition of $A$, for any $j''>j$, $v_{j''}[q]=B_r[c+2+i',j''-(r-1)c2^{\ell+1}]=B_r[c+2+i', i_1 2^{\ell+1}+a]$, where $c-1\geq i_1\geq i$ and $a\in [2^{\ell+1}]$. Thus by (\ref{eqn:Br:zeroX}), $v_{j''}[q]=B_r[c+2+i',i_1 2^{\ell+1}+a]=0$. This contradicts the assumption that $q \in I' \cap S$.
Suppose $q \in I' \cap S$ and $q >(r-1)(2c+1)+c+2+i$. Let $q=(r-1)(2c+1)+c+2+i'$, where $i<i'<c$. Then by the decomposition of $A$, for any $j''\leq j$, $v_{j''}[q]=B_r[c+2+i',j''-(r-1)c 2^{\ell+1}]=B_r[c+2+i', i_1 2^{\ell+1}+a]$, where $0\leq i_1\leq i$, $a\in [2^{\ell+1}]$. Thus by (\ref{eqn:Br:zeroX}), $v_{j''}[q]=B_r[c+2+i', i_1 2^{\ell+1}+a]=0$. This contradicts the assumption that $i\in I' \cap S$. This implies that $\vert I'\cap S \vert \leq 1$. This completes the proof of the claim. \end{proof} Therefore, we have \begin{eqnarray*} \vert I' \vert &=& \vert I' \cap I_r \vert \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad (\mbox{Because } I'\subseteq I_r)\\ &=&\vert I' \cap R_1 \vert + \vert I' \cap R \vert + \vert I' \cap S \vert + \vert I' \cap R_2 \vert \qquad \qquad\qquad (\mbox{By ~\eqref{Eqn:R}})\\
&\leq& c-(i-1)+1+1+i \qquad \qquad\qquad \qquad(\mbox{By Claims~\ref{claim:R1},\ref{claim:R2} and \ref{claim:R2'}})\\ &=&c+3 \end{eqnarray*} This completes the proof of the lemma. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:lowbranchwidth}.] We prove the theorem by assuming a fast algorithm for (IP)\xspace and use it to give a fast algorithm for {\sc CNF-SAT}, refuting SETH. Let $\psi$ be an instance of {\sc CNF-SAT} with $n_1$ variables and $m_1$ clauses. We choose a sufficiently large constant $c$ such that $(1-\epsilon)+\frac{4(1-\epsilon)}{c}+\frac{a}{c}<1$ holds.
We use the reduction mentioned in Lemma~\ref{lemtechnicalintro} and construct an instance $A_{(\psi,c)} x= b_{(\psi,c)}, x\geq 0$, of (IP)\xspace which has a solution if and only if $\psi$ is satisfiable. The reduction takes time ${\mathcal{O}}(m_1^2 2^{\frac{n_1}{c}})$. Let $\ell=\lceil \frac{n_1}{c}\rceil$. The constraint matrix $A_{(\psi,c)}$ has dimension $((m_1-1)(2c+1)+1+c)\times (m_1\cdot c\cdot 2^{\ell+1})$ and the largest entry in vector $b_{(\psi,c)}$ does not exceed $2^{\ell}-1$. The {path-width}\xspace of $M(A_{(\psi,c)})$ is at most $c+4$.
Assuming that any instance of (IP)\xspace with non-negative constraint matrix of {path-width}\xspace $k$ is solvable in time $f(k)(\|b\|_{\infty}+1)^{(1-\epsilon)k}(mn)^a$, where $d$ is the maximum value in an entry of $b$ and $\epsilon,a>0$ are constants, we have that $A_{(\psi,c)} x= b_{(\psi,c)}, x\geq 0$, is solvable in time \[ f(c+4) \cdot 2^{\ell \cdot (1-\epsilon)(c+4)} \cdot2^{\ell\cdot a} \cdot m_1^{{\mathcal{O}}(1)}= 2^{\frac{n_1}{c}(1-\epsilon)(c+4)} \cdot 2^{\frac{n_1\cdot a}{c}} \cdot m_1^{{\mathcal{O}}(1)} = 2^{n_1\left( (1-\epsilon)+\frac{4(1-\epsilon)}{c} +\frac{a}{c} \right)} \cdot m_1^{{\mathcal{O}}(1)}. \]
Here the constant $f(c+4)$ is subsumed by the term $m_1^{{\mathcal{O}}(1)}$. Hence the total running time for testing whether $\psi$ is satisfiable or not, is,
\[ {\mathcal{O}}(m_1^2 2^{\frac{n_1}{c}})+2^{n_1\left( (1-\epsilon)+\frac{4(1-\epsilon)}{c} +\frac{a}{c} \right)} m_1^{{\mathcal{O}}(1)} = 2^{n_1\left( (1-\epsilon)+\frac{4(1-\epsilon)}{c} +\frac{a}{c} \right)} m_1^{{\mathcal{O}}(1)}= 2^{\epsilon'\cdot n_1} m_1^{{\mathcal{O}}(1)}, \]
where $\epsilon'=(1-\epsilon)+\frac{4(1-\epsilon)}{c}+\frac{a}{c}<1$. This completes the proof of Theorem~\ref{thm:lowbranchwidth}.
\end{proof}
\subsection{Proof Sketch of Theorem~\ref{thm:lowentries}}\label{sec:lowentries}
In this section we prove Theorem~\ref{thm:lowentries}: (IP)\xspace{} with non-negative matrix $A$ cannot be solved in time $f(\|b\|_{\infty})(\|b\|_{\infty}+1)^{(1-\epsilon)k}(mn)^{{\mathcal{O}}(1)}$ for any function $f$ and $\epsilon>0$, unless SETH fails, where
$k$ is the {path-width}\xspace of the column matroid of $A$.
In Section~\ref{sec:lowbranchwidth}, we gave a reduction from {\sc CNF-SAT} to (IP)\xspace. However in this reduction the values in the constraint matrix $A_{(\psi,c)}$ and target vector $b_{(\psi,c)}$ can be as large as $2^{\lceil \frac{n}{c}\rceil}-1$, where $n$ is the number of variables in the {\sc CNF}-formula $\psi$ and $c$ is a constant. Let $m$ be the number of clauses in $\psi$. In this section we briefly explain how to get rid of these large values, at the cost of making large, but still bounded {path-width}\xspace. From a {\sc CNF}-formula $\psi$, we construct a matrix $A=A_{(\psi,c)}$ as described in Section~\ref{sec:lowbranchwidth}.
The only rows in $A$ which contain values strictly greater than $1$ (values other than $0$ or $1$) are the ones corresponding to the constraints defined by Equation~(\ref{eqn:consistancy}).
In other words, the values greater than $1$ are in the rows in yellow/green colored portion in Figure~\ref{fig:A}. Recall that $\ell=\lceil \frac{n}{c}\rceil$ and the largest value in $A$ is $2^{\ell}-1$. Any number less than or equal to $2^{\ell}-1$ can be represented by a binary string of length $\ell=\frac{n}{c}$. Now we rewrite the Equation~(\ref{eqn:consistancy}), by $\ell$ new equations. For each $j\in [\ell]$ and $N\in{\mathbb N}$, let $b_j(N)$, represents the $j^{th}$ bit in the $\ell$-bit binary representation of $N$. Then for each for all $r\in [m-1]$, $i\in\ZZ{c}$ and $j\in [\ell]$, we have a system of constraints \begin{eqnarray} \sum_{\substack{a \in \ZZ{2L} }} \left(b_j\left(\lfloor \frac{a}{2}\rfloor\right)\cdot y_{C_r,i,a}\right) + \left(b_j(L-1-\lfloor \frac{a}{2}\rfloor)\cdot y_{C_{r+1},i,a}\right) &=&1
\end{eqnarray}
In other words, let $P=\{(r-1)(2c+1)+c+1+i~|~r\in[m-2], i\in \ZZ{c}\}$. The rows of $A$ containing values larger than one are indexed by $P$. Now we construct a new matrix $A'$ from $A$ by replacing each row of $A$ whose index is in the set $P$ with $\ell$ rows and for any value $A[i,j], i\in P$ we write its $\ell$-bit binary representation in the column corresponding to $j$ and the newly added $\ell$ rows of $A'$. That is, for any $\gamma\in P$, we replace the row $\gamma$ with $\ell$ rows, $\gamma_{1},\ldots,\gamma_{\ell}$. Where, for any $j$, if the value $A[\gamma,j]=W$ then $A'[\gamma_{k},j]=\eta_k$, where $\eta_k$ is the $k^{th}$ bit in the $\ell$-sized binary representation of $W$.
Let $m'$ be the number of rows in $A'$. Now the target vector $b'$ is defined as $b'[i]=1$ for all $i\in [m']$. This completes the construction of the reduced (IP)\xspace{} instance $A'x=b'$. The correctness proof of this reduction is using arguments similar to those used for the correctness of Lemma~\ref{lemma:brachwidthcorrect}.
\begin{lemma} \label{lemma:pathwidthvaluesmall} The {path-width}\xspace of the column matroid of $A'$ is at most $(c+1)\frac{n}{c}+3$. \end{lemma} \begin{proof} We sketch the proof, which is similar to the proof of Lemma~\ref{lemma:pathwidthbounded}.
We define $I'_r$ and $J'_r$ for any $r\in [m]$ like $I_r$ and $J_r$ in Section~\ref{sec:lowbranchwidth}. In fact, the rows in $I'_r$ are the rows obtained from $I_r$ in the process explained above to construct $A'$ from $A$. We need to show that
$\operatorname{dim}\langle \operatorname{span}(A'|\{1,\dots, j\})\cap\operatorname{span}(A'|\{j+1, \dots, n'\}) \rangle \leq (c+1)\frac{n}{c}+2$ for all $j\in [n'-1]$, where $n'$ is the number of columns in $A'$. The proof proceeds by bounding the number of indices $I$
such that for any $q \in I$ there exist vectors $v\in A'|\{1,\dots, j\}$ and $u\in A'|\{j+1, \dots, n'\}$ with $v[q]\neq 0 \neq u[q]$. By arguments similar to the ones used in the proof of Lemma~\ref{lemma:pathwidthbounded}, we can show that for any $j\in [n'-1]$, the corresponding set $I'$ of indices is a subset of $I'_r$ for some $r\in [m]$.
Recall the partition of $I_r$ into $R_1,R,S$ and $R_2$ in Lemma~\ref{lemma:pathwidthbounded}. We partition $I'_r$ into parts $Q_1,W,U$ and $Q_2$.
Notice that $R_1,R_2\subseteq P$, where $P$ is the set of rows which covers all
values strictly greater than $1$. The set $Q_1$ and $Q_2$ are obtained from $R_1$ and $R_2$, respectively, by the process mentioned above to construct $A'$ from $A$. That is, each row in $R_i, i\in \{1,2\}$ is replaced by $\ell$ rows in $Q_i$. Rows in $W$ corresponds to rows in $R$ and $U$ corresponds to the rows in $W$. This allows us to bound the following terms for some $i \in \ZZ{c}$: \begin{eqnarray*} \vert I'\cap Q_1\vert &\leq& (c-(i -1))\ell=(c-(i-1))\ell,\\ \vert I'\cap Q_2\vert &\leq& i \cdot \ell, \\ \vert I'\cap U\vert &\leq& 1, \text{ and}\\ \vert I'\cap W\vert &\leq& 1. \end{eqnarray*} By using the fact that $I'\subseteq I'_r$ and the above system of inequalities, we can show that
$$\operatorname{dim}\langle \operatorname{span}(A'|\{1,\dots, j\})\cap\operatorname{span}(A'|\{j+1, \dots, n'\}) \rangle \leq (c+1)\lceil\frac{n}{c}\rceil+2.$$ This completes the proof sketch of the lemma. \end{proof} Now the proof of the theorem follows from Lemma~\ref{lemma:pathwidthvaluesmall} and the correctness of the reduction (it is similar to the arguments in the proof of Theorem~\ref{thm:lowbranchwidth}).
\section{Proof of Theorem~\ref{thmCGlin}}\label{sec:thmCGlin}
In this section, we sketch how the proof of Cunningham and Geelen \cite{CunninghamG07} of Theorem~\ref{thmCG}, can be adapted to prove Theorem~\ref{thmCGlin}.
Recall that a path decomposition of width $k$ can be obtained in $f(k)\cdot n^{{\mathcal{O}}(1)}$ time for some function $f$ by making use of the algorithm by Jeong et al.~\cite{Jeong0O16}. However, we do not know if such a path decomposition can be constructed in time ${\mathcal{O}} ((\|b\|_{\infty}+1)^{k+1}) n^{{\mathcal{O}}(1)}$, so the assumption that a path decomposition is given is essential.
Roughly speaking, the only difference in the proof is that when parameterized by the branch-width, the most time-consuming operation is the ``merge" operation, when we have to construct a new set of partial solutions with at most $(\|b\|_{\infty}+1)^k$ vectors from two already computed sets of sizes $(\|b\|_{\infty}+1)^k$ each. Thus to construct a new set of vectors, one has to go through all possible pairs of vectors from both sets, which takes time roughly $(\|b\|_{\infty}+1)^{2k}$.
For {path-width}\xspace parameterization, the new partial solution set is constructed from two sets, but this time one set contains at most $(\|b\|_{\infty}+1)^k$ vectors while the second contains at most $\|b\|_{\infty}+1$ vectors. This allows us to construct the new set in time roughly $(\|b\|_{\infty}+1)^{k+1}$.
Recall that for $X\subseteq [n]$, we define
$S(A,X)=\operatorname{span}(A|X)\cap\operatorname{span}(A|E \setminus X)$, where $E=[n]$. The key lemma in the proof of Theorem~\ref{thmCG} is the following. \begin{lemma}[\cite{CunninghamG07}] \label{lemma:CGboundedpartialsoln} Let $A\in \{0,1,\ldots,\|b\|_{\infty}\}^{m\times n}$ and $X\subseteq [n]$ such that $\lambda_{M(A)}(X)=k$. Then the number of vectors in $S(A,X)\cap \{0,\ldots,\|b\|_{\infty}\}^m$ is at most $(\|b\|_{\infty}+1)^{k-1}$. \end{lemma}
To prove Theorem~\ref{thmCGlin}, without loss of generality, we assume that the columns of $A$ are ordered in such a way that for every $j\in [n-1]$,
$$\operatorname{dim}\langle \operatorname{span}(A|\{1,\dots, i\})\cap\operatorname{span}(A|\{i+1, \dots, n\}) \rangle \leq k-1.$$
Let $A'=[A,b]$. That is $A'$ is obtained by appending the column-vector $b$ to the end of $A$. Then for each $i\in [n]$, \begin{equation}\label{eq_dim}
\operatorname{dim}\langle \operatorname{span}(A'|\{1,\dots, i\})\cap\operatorname{span}(A'|\{i+1, \dots, n+1\}) \rangle \leq k. \end{equation} Now we use dynamic programming to check whether the following conditions are satisfied. For $X \subseteq [n+1]$, let ${\mathcal B}(X)$ be the set of all vectors $b'\in {\mathbb Z}_{\scriptscriptstyle{\geq 0}}^m$ such that \begin{itemize} \item[(1)] $0 \leq b' \leq b$,
\item[(2)] there exists $z\in {\mathbb Z}_{\scriptscriptstyle{\geq 0}}^{\vert X \vert}$ such that $(A'|X)z=b'$, and \item[(3)] $b'\in S(A', X)$. \end{itemize} Then (IP)\xspace has a solution if and only if $b\in {\mathcal B}([n])$. Initially the algorithm computes for all $i\in [n]$, ${\mathcal B}(\{i\})$ and by Lemma~\ref{lemma:CGboundedpartialsoln},
we have that $\vert {\mathcal B}(\{i\})\vert \leq \|b\|_{\infty}+1$. In fact ${\mathcal B}(\{i\})\subseteq \{a\cdot v~|~v \mbox{ is the $i^{th}$ column vector of $A'$ and }a\in [\|b\|_{\infty}+1] \}$. Then for each $j\in [2,\ldots n]$ the algorithm computes ${\mathcal B}([j])$ in increasing order of $j$ and outputs {\sc Yes} if and only if $b\in {\mathcal B}([n])$. That is, ${\mathcal B}([j])$ is computed from the already computed sets ${\mathcal B}([j-1])$ and ${\mathcal B}(\{j\})$. Notice that $b'\in {\mathcal B}([j])$ if and only if \begin{itemize} \item[(a)] there exist $b_1\in {\mathcal B}(\{1,\ldots,j-1\})$ and $b_2\in {\mathcal B}(\{j\})$ such that $b'=b_1+b_2$, \item[(b)] $b'\leq b$ and \item[(c)] $b'\in S(A',[j])$. \end{itemize}
So the algorithm enumerates vectors $b'$ satisfying condition $(a)$, and each such vector $b'$ is included in ${\mathcal B}([j])$, if $b'$ satisfy conditions $(b)$ and $(c)$. Since by \eqref{eq_dim} and Lemma~\ref{lemma:CGboundedpartialsoln}, $\vert {\mathcal B}([j-1])\vert \leq (\|b\|_{\infty}+1)^k$ and $\vert {\mathcal B}(\{j\})\vert\leq \|b\|_{\infty}+1$, the number of vectors satisfying condition $(a)$ is $(\|b\|_{\infty}+1)^{k}$, and hence the exponential factor of the required running time follows. This provides the bound on the claimed exponential dependence in the running time of the algorithm. The bound on the polynomial component of the running time follows from exactly the same arguments as in~\cite{CunninghamG07}.
\section{Conclusion}\label{sec:concl}
In a previous version of this paper on ArXiv~\cite{FominPRS16} we pointed out that it was unknown whether the algorithm of Papadimitriou~\cite{Papadimitriou81} is asymptotically optimal. This question has now been answered by Eisenbrand and Weismantel in~\cite{EisenbrandW18}, who gave an algorithm solving (IP)\xspace with an $m\times n$ matrix $A$ in time $(m\cdot \Delta)^{{\mathcal{O}}(m)}\cdot \|b\|_{\infty}^2$, where $\Delta$ is the upper bound on the absolute values of the entries of $A$. While Theorems~\ref{thm:ETHIP} and ~\ref{thm:ETHIP2} come close to this bound,
the precise multivariate complexity of (IP)\xspace with respect to the parameters $n$, $m$, $\Delta$, and $\|b\|_{\infty}$ is not fully clear and our work leaves some unanswered questions regarding the landscape of tradeoffs between the parameters. For instance, is it possible to solve (IP)\xspace in time
\begin{itemize}
\item $ (m\cdot n\cdot \Delta)^{o(m)}\cdot (\|b\|_{\infty})^{{\mathcal{O}}(1)}$, or \item $ (m\cdot n\cdot \Delta\cdot \|b\|_{\infty})^{o(m)} $? \end{itemize}
Or could one improve our lower bound results to rule out such algorithms? While our SETH-based lower bounds for (IP)\xspace with non-negative constraint matrix are tight for {path-width}\xspace parameterization, there is a ``$(\|b\|_{\infty}+1)^k$ to $(\|b\|_{\infty}+1)^{2k}$ gap'' between lower and upper bounds for branch-width parameterization. Closing this gap is the first natural question.
The proof of Theorem~\ref{thmCG} given by Cunningham and Geelen consists of two parts.
The first part bounds the number of potential partial solutions corresponding to any edge of the branch decomposition tree by $(\|b\|_{\infty}+1)^k$. The second part is the dynamic programming over the branch decomposition using the fact that the number of potential partial solutions is bounded. The bottleneck in the algorithm of Cunningham and Geelen is the following subproblem. We are given two vector sets $A$ and $B$ of partial solutions, each set of size at most $(\|b\|_{\infty}+1)^k$. We need to construct a new vector set $C$ of partial solutions, where the set $C$ will have size at most $(\|b\|_{\infty}+1)^k$ and each vector from $C$ is the \emph{sum} of a vector from $A$ and a vector from $B$.
Thus to construct the new set of vectors, one has to go through all possible pairs of vectors from both sets $A$ and $B$, which takes time roughly $(\|b\|_{\infty}+1)^{2k}$.
A tempting approach towards improving the running time of this particular step could be the use of {\em fast subset convolution} or {\em matrix multiplication} tricks, which work very well for ``join'' operations in dynamic programming algorithms over tree and branch decompositions of graphs~\cite{Dorn06,RBR09,cut-and-count}, see also \cite[Chapter~11]{cygan2015parameterized}. Unfortunately, we have reason to suspect that these tricks may \emph{not} help for matrices: solving the above subproblem in time $(\|b\|_{\infty}+1)^{(1-\epsilon)2k}n^{{\mathcal{O}}(1)}$ for any $\epsilon>0$ would imply that $3$-SUM is solvable in time $n^{2-\epsilon}$, which is believed to be unlikely. (The $3$-SUM problem asks whether a given set of $n$ integers contains three elements that sum to zero.) Indeed, consider an equivalent version of $3$-SUM, named $3$-SUM$^\prime$, which is defined as follows. Given $3$ sets of integers $A,B$ and $C$ each of cardinality $n$, and the objective is to check whether there exist $a\in A$, $b\in B$ and $c\in C$ such that $a+b=c$. Then, $3$-SUM is solvable in time $n^{2-\epsilon}$ if and only if $3$-SUM$^\prime$ is as well (see Theorem~$3.1$ in~\cite{GajentaanO95}). However, the problem $3$-SUM$^\prime$ is equivalent to the most time consuming step in the algorithm of Theorem~\ref{thmCG}, where the integers in the input of $3$-SUM$^\prime$ can be thought of as length-one vectors. While this observation does not \emph{rule out} the existence of an algorithm solving (IP) with constraint matrices of branch-width $k$ in time $(\|b\|_{\infty}+1)^{(1-\epsilon)2k}n^{{\mathcal{O}}(1)}$, it indicates that any interesting improvement in the running time would require a completely different approach.
\end{document} | arXiv |
Abstract: Let $L$ be a linear elliptic, a pseudomonotone or a generalized monotone operator (in the sense of F. E. Browder and I. V. Skrypnik), and let $F$ be the nonlinear Nemytskij superposition operator generated by a vector-valued function $f$. We give two general existence theorems for solutions of boundary value problems for the equation $Lx=Fx$. These theorems are based on a new functional-theoretic approach to the pair $(L,F)$, on the one hand, and on recent results on the operator $F$, on the other hand. We treat the above mentioned problems in the case of strong non-linearity $F$, i.e. in the case of lack of compactness of the operator $L-F$. In particular, we do not impose the usual growth conditions on the nonlinear function $f$; this allows us to treat elliptic systems with rapidly growing coefficients or exponential non-linearities. Concerning solutions, we consider existence in the classical weak sense, in the so-called $L_\infty$-weakened sense in both Sobolev and Sobolev-Orlicz spaces, and in a generalized weak sense in Sobolev-type spaces which are modelled by means of Banach $L_\infty$-modules. Finally, we illustrate the abstract results by some applied problems occuring in nonlinear mechanics. | CommonCrawl |
# Understanding the basics of PostgreSQL
PostgreSQL is a powerful open-source relational database management system (RDBMS) that is widely used for storing and managing data. It offers a wide range of features and capabilities, making it suitable for various applications and industries.
Some key features of PostgreSQL include:
- Support for complex queries and advanced data types
- ACID (Atomicity, Consistency, Isolation, Durability) compliance for data integrity
- Extensibility through user-defined functions, data types, and operators
- Concurrency control for multi-user environments
- Replication and high availability options
- Built-in support for JSON and other NoSQL capabilities
- Security features such as role-based access control and SSL encryption
PostgreSQL follows the SQL standard and provides a rich set of SQL commands for creating, modifying, and querying databases. It also supports procedural languages like PL/pgSQL, which allows you to write stored procedures and triggers.
## Exercise
1. What are some key features of PostgreSQL?
2. What is ACID compliance?
3. Name one procedural language supported by PostgreSQL.
### Solution
1. Some key features of PostgreSQL include support for complex queries, ACID compliance, extensibility, concurrency control, replication, and security features.
2. ACID compliance refers to a set of properties that ensure reliable processing of database transactions. It stands for Atomicity, Consistency, Isolation, and Durability.
3. One procedural language supported by PostgreSQL is PL/pgSQL.
# Creating and managing database roles in PostgreSQL
In PostgreSQL, a database role is an entity that can own database objects and have specific privileges and permissions. Roles can be used to manage access control and security within a PostgreSQL database.
To create a new role in PostgreSQL, you can use the `CREATE ROLE` command. Here's an example:
```sql
CREATE ROLE myrole;
```
This will create a new role named `myrole`. By default, the new role will not have any privileges or permissions.
You can also specify additional options when creating a role, such as the ability to create databases or login to the database. Here's an example:
```sql
CREATE ROLE myrole WITH LOGIN CREATEDB;
```
This will create a new role named `myrole` with the ability to login to the database and create new databases.
- Creating a role with the ability to login and create databases:
```sql
CREATE ROLE myrole WITH LOGIN CREATEDB;
```
- Creating a role without the ability to login:
```sql
CREATE ROLE myrole;
```
## Exercise
Create a new role named `admin` with the ability to login and create databases.
### Solution
```sql
CREATE ROLE admin WITH LOGIN CREATEDB;
```
# Granting and revoking privileges for database roles
Once you have created a role in PostgreSQL, you can grant or revoke privileges and permissions to that role. Privileges control what actions a role can perform on database objects, such as tables, views, and functions.
To grant privileges to a role, you can use the `GRANT` command. Here's an example:
```sql
GRANT SELECT, INSERT, UPDATE ON mytable TO myrole;
```
This will grant the `SELECT`, `INSERT`, and `UPDATE` privileges on the table `mytable` to the role `myrole`.
To revoke privileges from a role, you can use the `REVOKE` command. Here's an example:
```sql
REVOKE SELECT, INSERT, UPDATE ON mytable FROM myrole;
```
This will revoke the `SELECT`, `INSERT`, and `UPDATE` privileges on the table `mytable` from the role `myrole`.
- Granting the `SELECT` privilege on a table to a role:
```sql
GRANT SELECT ON mytable TO myrole;
```
- Revoking the `INSERT` privilege on a table from a role:
```sql
REVOKE INSERT ON mytable FROM myrole;
```
## Exercise
Grant the `SELECT`, `INSERT`, and `UPDATE` privileges on the table `employees` to the role `manager`.
### Solution
```sql
GRANT SELECT, INSERT, UPDATE ON employees TO manager;
```
# Using access control lists in PostgreSQL
Access control lists (ACLs) in PostgreSQL allow you to define fine-grained access control for database objects. ACLs are used to specify which roles have specific privileges on a database object, such as tables, views, and functions.
To view the ACL for a database object, you can use the `\dp` command in the PostgreSQL command-line interface. Here's an example:
```sql
\dp mytable
```
This will display the ACL for the table `mytable`, showing which roles have which privileges on the table.
To modify the ACL for a database object, you can use the `GRANT` and `REVOKE` commands, similar to granting and revoking privileges. Here's an example:
```sql
GRANT SELECT, INSERT, UPDATE ON mytable TO myrole;
```
This will grant the `SELECT`, `INSERT`, and `UPDATE` privileges on the table `mytable` to the role `myrole`.
- Viewing the ACL for a table:
```sql
\dp mytable
```
- Granting the `SELECT` privilege on a table to a role:
```sql
GRANT SELECT ON mytable TO myrole;
```
## Exercise
View the ACL for the table `employees`.
### Solution
```sql
\dp employees
```
# Database security best practices
Database security is an important aspect of managing a PostgreSQL database. By following best practices, you can help protect your data from unauthorized access and ensure the integrity and confidentiality of your database.
Here are some best practices for database security in PostgreSQL:
1. Use strong passwords: Ensure that all roles and users have strong passwords to prevent unauthorized access to the database.
2. Limit access: Grant privileges and permissions only to the roles and users that require them. Avoid granting unnecessary privileges.
3. Regularly update and patch: Keep your PostgreSQL installation up to date with the latest security patches and updates to protect against known vulnerabilities.
4. Encrypt sensitive data: Use encryption to protect sensitive data, both at rest and in transit. PostgreSQL provides built-in support for SSL encryption.
5. Use secure connections: Use secure protocols, such as SSL/TLS, when connecting to the database to ensure the confidentiality and integrity of data transmitted over the network.
6. Implement access controls: Use role-based access control (RBAC) to control access to database objects. Grant privileges based on the principle of least privilege.
7. Monitor and audit: Implement monitoring and auditing mechanisms to track and log database activity. Regularly review logs for suspicious activity.
8. Backup and recovery: Regularly backup your database and test the restore process to ensure that you can recover data in the event of a security incident or data loss.
- Granting the `SELECT` privilege on a table to a role:
```sql
GRANT SELECT ON mytable TO myrole;
```
- Enabling SSL encryption for the PostgreSQL server:
```sql
ssl = on
```
## Exercise
Which best practice involves regularly backing up the database and testing the restore process?
### Solution
Backup and recovery
# Monitoring and optimizing database performance in PostgreSQL
Monitoring and optimizing database performance is crucial for ensuring the efficient operation of a PostgreSQL database. By monitoring key performance metrics and identifying areas for improvement, you can optimize the performance of your database and improve overall system performance.
Here are some key steps for monitoring and optimizing database performance in PostgreSQL:
1. Identify performance metrics: Determine the key performance metrics that are relevant to your database, such as CPU usage, memory usage, disk I/O, and query execution time.
2. Monitor performance: Use tools like pg_stat_activity, pg_stat_bgwriter, and pg_stat_user_tables to monitor the performance of your database. These tools provide valuable insights into the current state of your database and can help identify performance bottlenecks.
3. Analyze query performance: Use the EXPLAIN command to analyze the execution plan of your queries. This can help identify inefficient queries and suggest ways to optimize them, such as adding indexes or rewriting the query.
4. Optimize database configuration: Review and optimize your PostgreSQL configuration parameters to ensure they are set appropriately for your workload. This includes parameters related to memory allocation, disk I/O, and query execution.
5. Optimize schema design: Review your database schema design and make sure it is optimized for your workload. This includes considerations such as table normalization, indexing strategies, and partitioning.
6. Use performance tuning techniques: Implement performance tuning techniques such as query rewriting, query caching, and connection pooling to improve the performance of your database.
- Monitoring CPU usage:
```sql
SELECT pid, usename, datname, application_name, state, query, cpu_usage
FROM pg_stat_activity
ORDER BY cpu_usage DESC;
```
- Analyzing query execution plan:
```sql
EXPLAIN SELECT * FROM mytable WHERE column = 'value';
```
## Exercise
Which step involves reviewing and optimizing the PostgreSQL configuration parameters?
### Solution
Optimize database configuration
# Identifying and addressing performance bottlenecks
Performance bottlenecks can significantly impact the performance of a PostgreSQL database. Identifying and addressing these bottlenecks is essential for maintaining optimal database performance. By understanding the common causes of performance bottlenecks and implementing appropriate solutions, you can improve the overall performance of your database.
Here are some common performance bottlenecks in PostgreSQL and strategies for addressing them:
1. Slow queries: Identify and optimize slow queries by analyzing their execution plans, adding appropriate indexes, rewriting the queries, or tuning the database configuration.
2. Inefficient indexing: Review the indexing strategy for your database and ensure that it is optimized for your workload. Consider adding or removing indexes based on query patterns and performance requirements.
3. Insufficient hardware resources: Evaluate the hardware resources (CPU, memory, disk I/O) allocated to your database server. Consider upgrading or scaling up the hardware to meet the performance demands of your workload.
4. Lock contention: Identify and address lock contention issues by analyzing the lock activity in your database. Consider implementing strategies such as using row-level locking, reducing transaction durations, or using optimistic concurrency control.
5. Poor query design: Review the design of your queries and ensure they are efficient and optimized for your database schema. Consider rewriting queries, using appropriate join techniques, and minimizing unnecessary data retrieval.
6. Inadequate caching: Implement caching mechanisms such as query caching or result caching to reduce the load on the database server and improve query performance.
- Adding an index to improve query performance:
```sql
CREATE INDEX idx_column ON mytable (column);
```
- Implementing query caching using the pg_stat_statements extension:
```sql
shared_preload_libraries = 'pg_stat_statements'
pg_stat_statements.max = 10000
pg_stat_statements.track = all
```
## Exercise
Which performance bottleneck can be addressed by implementing caching mechanisms?
### Solution
Inadequate caching
# Indexing and query optimization in PostgreSQL
Indexing and query optimization are essential for improving the performance of a PostgreSQL database. By creating appropriate indexes and optimizing queries, you can significantly enhance the speed and efficiency of data retrieval.
Here are some key concepts and techniques for indexing and query optimization in PostgreSQL:
1. Index types: Understand the different types of indexes available in PostgreSQL, such as B-tree, hash, and GiST indexes. Choose the appropriate index type based on the data and query patterns of your database.
2. Index creation: Create indexes on columns that are frequently used in queries, especially those involved in join operations, filtering, and sorting. Consider creating composite indexes for multiple columns used together in queries.
3. Index maintenance: Regularly monitor and maintain your indexes to ensure optimal performance. This includes reindexing, vacuuming, and analyzing your database to update index statistics.
4. Query optimization techniques: Use techniques such as query rewriting, query planning, and query tuning to optimize the execution of your queries. This may involve rewriting queries to use more efficient join techniques, adding or removing indexes, or adjusting configuration parameters.
5. Explain analyze: Use the EXPLAIN ANALYZE command to analyze the execution plan and performance of your queries. This can help identify inefficient query plans and suggest optimizations.
6. Query planner statistics: Maintain accurate and up-to-date statistics for the query planner by enabling the autovacuum feature and running ANALYZE on your database. This helps the query planner make informed decisions about query execution plans.
- Creating a B-tree index on a column:
```sql
CREATE INDEX idx_column ON mytable (column);
```
- Analyzing the execution plan and performance of a query:
```sql
EXPLAIN ANALYZE SELECT * FROM mytable WHERE column = 'value';
```
## Exercise
Which technique can be used to analyze the execution plan and performance of a query?
### Solution
Explain analyze
# Backup and recovery strategies for PostgreSQL databases
Implementing backup and recovery strategies is crucial for ensuring the availability and integrity of your PostgreSQL databases. By regularly backing up your data and having a solid recovery plan in place, you can minimize the impact of data loss or system failures.
Here are some key strategies for backup and recovery in PostgreSQL:
1. Regular backups: Schedule regular backups of your PostgreSQL databases to ensure that you have up-to-date copies of your data. This can be done using tools like pg_dump or pg_basebackup.
2. Full and incremental backups: Consider using a combination of full and incremental backups to optimize backup times and storage requirements. Full backups capture the entire database, while incremental backups capture only the changes since the last full backup.
3. Backup storage: Store your backups in a secure and reliable location, such as a separate server or cloud storage. Ensure that backups are encrypted to protect sensitive data.
4. Point-in-time recovery: Enable point-in-time recovery (PITR) to restore your database to a specific point in time. This can be useful for recovering from data corruption or user errors.
5. Test the restore process: Regularly test the restore process to ensure that you can successfully recover your data in the event of a failure. This includes testing both full and incremental backups.
6. Disaster recovery planning: Develop a comprehensive disaster recovery plan that outlines the steps to be taken in the event of a major system failure or data loss. This should include procedures for restoring backups, rebuilding the database, and minimizing downtime.
- Creating a full backup using pg_dump:
```bash
pg_dump -U username -h hostname -F c -f /path/to/backup.dump dbname
```
- Enabling point-in-time recovery in the PostgreSQL configuration file:
```bash
recovery_target_time = '2022-01-01 00:00:00'
```
## Exercise
Which strategy involves capturing only the changes since the last full backup?
### Solution
Incremental backups
# Securing data at rest and in transit in PostgreSQL
Securing data at rest and in transit is essential for protecting the confidentiality and integrity of your PostgreSQL databases. By implementing appropriate encryption and security measures, you can safeguard your data from unauthorized access or interception.
Here are some key strategies for securing data at rest and in transit in PostgreSQL:
1. Encryption at rest: Implement encryption mechanisms to protect data stored on disk. This can be done using technologies such as full disk encryption or transparent data encryption (TDE). PostgreSQL also provides the pgcrypto extension for encrypting specific data within the database.
2. SSL/TLS encryption: Enable SSL/TLS encryption for client-server communication to ensure the confidentiality and integrity of data transmitted over the network. This involves configuring the PostgreSQL server to use SSL/TLS certificates and enabling secure connections.
3. Secure authentication: Use strong authentication mechanisms, such as password policies, two-factor authentication, or certificate-based authentication, to prevent unauthorized access to the database.
4. Secure key management: Implement secure key management practices to protect encryption keys used for data encryption. This includes storing keys in secure hardware modules or using key management systems.
5. Auditing and monitoring: Implement auditing and monitoring mechanisms to track and log database activity. This can help detect and respond to unauthorized access attempts or suspicious activity.
6. Regular security updates: Keep your PostgreSQL installation up to date with the latest security patches and updates. This helps protect against known vulnerabilities and security risks.
- Enabling SSL/TLS encryption in the PostgreSQL configuration file:
```bash
ssl = on
ssl_cert_file = '/path/to/server.crt'
ssl_key_file = '/path/to/server.key'
```
- Encrypting a column using the pgcrypto extension:
```sql
CREATE EXTENSION IF NOT EXISTS pgcrypto;
UPDATE mytable SET sensitive_data = pgp_sym_encrypt(sensitive_data, 'encryption_key');
```
## Exercise
Which strategy involves implementing encryption mechanisms to protect data stored on disk?
### Solution
Encryption at rest
# Advanced security features in PostgreSQL
1. Row-level security: Row-level security (RLS) allows you to control access to individual rows in a table based on specified conditions. This feature is particularly useful when you want to restrict access to certain rows based on user roles or attributes. By defining policies, you can ensure that only authorized users can view or modify specific rows of data.
2. Column-level security: Similar to row-level security, column-level security allows you to control access to individual columns in a table. This feature is useful when you want to restrict access to sensitive data within a table. By defining policies, you can specify which users or roles can access specific columns.
3. Secure function execution: PostgreSQL provides the ability to define secure functions that can be executed with elevated privileges. This feature allows you to restrict the execution of certain functions to specific roles or users. By using secure functions, you can prevent unauthorized access to sensitive operations or data.
4. Fine-grained access control: PostgreSQL allows you to define fine-grained access control policies using the policy-based access control (PBAC) feature. With PBAC, you can define complex access control rules based on various conditions, such as user attributes, time of day, or network location. This feature provides granular control over access to your database objects.
5. Auditing and logging: PostgreSQL offers robust auditing and logging capabilities that allow you to track and monitor database activity. You can configure the database to log various events, such as login attempts, queries, or modifications to data. By reviewing these logs, you can detect and investigate any suspicious activity or potential security breaches.
6. Secure data masking: PostgreSQL provides the ability to mask sensitive data when it is accessed by unauthorized users. This feature is particularly useful when you need to comply with data privacy regulations or protect sensitive information. By defining data masking policies, you can ensure that unauthorized users only see masked or obfuscated data.
- Implementing row-level security:
```sql
-- Create a security policy
CREATE POLICY my_policy
ON my_table
USING (user_id = current_user);
-- Grant access to the policy
ALTER TABLE my_table ENABLE ROW LEVEL SECURITY;
GRANT my_policy TO my_role;
```
- Defining a secure function:
```sql
-- Create a secure function
CREATE FUNCTION my_secure_function()
RETURNS void
SECURITY DEFINER
LANGUAGE plpgsql
AS $$
BEGIN
-- Perform secure operations here
END;
$$;
-- Grant execute privilege to a specific role
GRANT EXECUTE ON FUNCTION my_secure_function() TO my_role;
```
## Exercise
Which PostgreSQL feature allows you to control access to individual rows in a table based on specified conditions?
### Solution
Row-level security | Textbooks |
Home » Faculty Publications: May, 2014
Faculty Publications: May, 2014
Transport in two-dimensional disordered semimetals
Knap, Michael; Sau, Jay D.; Halperin, Bertrand I.; Demler, Eugene
Condensed Matter - Mesoscale and Nanoscale Physics, Condensed Matter - Disordered Systems and Neural Networks
5 pages, 4 figures, plus supplemental material. V2: minor clarifications and an additional reference
We theoretically study transport in two-dimensional semimetals. Typically, electron and hole puddles emerge in the transport layer of these systems due to smooth fluctuations in the potential. We calculate the electric response of the electron-hole liquid subject to zero and finite perpendicular magnetic fields using an effective medium approximation and a complimentary mapping on resistor networks. In the presence of smooth disorder and in the limit of weak electron-hole recombination rate, we find for small but finite overlap of the electron and hole bands an abrupt upturn in resistivity when lowering the temperature but no divergence at zero temperature. We discuss how this behavior is relevant for several experimental realizations and introduce a simple physical explanation for this effect.
Heisenberg-Limited Atom Clocks Based on Entangled Qubits
Kessler, E. M.; Kómár, P.; Bishof, M.; Jiang, L.; Sørensen, A. S.; Ye, J.; Lukin, M. D.
Entanglement and quantum nonlocality, Metrology, Time and frequency, Nonclassical states of the electromagnetic field including entangled photon states, quantum state engineering and measurements
2014PhRvL.112s0403K
We present a quantum-enhanced atomic clock protocol based on groups of sequentially larger Greenberger-Horne-Zeilinger (GHZ) states that achieves the best clock stability allowed by quantum theory up to a logarithmic correction. Importantly the protocol is designed to work under realistic conditions where the drift of the phase of the laser interrogating the atoms is the main source of decoherence. The simultaneous interrogation of the laser phase with a cascade of GHZ states realizes an incoherent version of the phase estimation algorithm that enables Heisenberg-limited operation while extending the coherent interrogation time beyond the laser noise limit. We compare and merge the new protocol with existing state of the art interrogation schemes, and identify the precise conditions under which entanglement provides an advantage for clock stabilization: it allows a significant gain in the stability for short averaging time.
Measurement of the B→Xsℓ+ℓ- Branching Fraction and Search for Direct CP Violation from a Sum of Exclusive Final State
Lees, J. P.; Poireau, V.; Tisserand, V.;....; Morii, M.; and 334 coauthors
Decays of bottom mesons, Charge conjugation parity time reversal and other discrete symmetries
2014PhRvL.112u1802L
We measure the total branching fraction of the flavor-changing neutral-current process B→Xsℓ+ℓ-, along with partial branching fractions in bins of dilepton and hadronic system (Xs) mass, using a sample of 471×106 ϒ(4S)→BB ¯ events recorded with the BABAR detector. The admixture of charged and neutral B mesons produced at PEP-II2 are reconstructed by combining a dilepton pair with 10 different Xs final states. Extrapolating from a sum over these exclusive modes, we measure a lepton-flavor-averaged inclusive branching fraction B(B→Xsℓ+ℓ-)=[6.73-0.64+0.70(stat)-0.25+0.34(exp syst)±0.50(model syst)]×10-6 for mℓ+ℓ-2>0.1 GeV2/c4. Restricting our analysis exclusively to final states from which a decaying B meson's flavor can be inferred, we additionally report measurements of the direct CP asymmetry ACP in bins of dilepton mass; over the full dilepton mass range, we find ACP=0.04±0.11±0.01 for a lepton-flavor-averaged sample.
tt * geometry in 3 and 4 dimensions
Cecotti, Sergio; Gaiotto, Davide; Vafa, Cumrun
Journal of High Energy Physics, Volume 2014, article id. #55
Supersymmetry and Duality, Supersymmetric gauge theory, Extended Supersymmetry, Nonperturbative Effects
2014JHEP...05..055C
We consider the vacuum geometry of supersymmetric theories with 4 supercharges, on a flat toroidal geometry. The 2 dimensional vacuum geometry is known to be captured by the tt * geometry. In the case of 3 dimensions, the parameter space is ( T 2 × ) N and the vacuum geometry turns out to be a solution to a generalization of monopole equations in 3 N dimensions where the relevant topological ring is that of line operators. We compute the generalization of the 2d cigar amplitudes, which lead to S 2 × S 1 or S 3 partition functions which are distinct from the supersymmetric partition functions on these spaces, but reduce to them in a certain limit. We show the sense in which these amplitudes generalize the structure of 3d Chern-Simons theories and 2d RCFT's. In the case of 4 dimensions the parameter space is of the form X M,N = ( T 3 × ) M × T 3 N , and the vacuum geometry is a solution to a mixture of generalized monopole equations and generalized instanton equations (known as hyper-holomorphic connections). In this case the topological rings are associated to surface operators. We discuss the physical meaning of the generalized Nahm transforms which act on all of these geometries.
Measurement of dijet cross-sections in pp collisions at 7 TeV centre-of-mass energy using the ATLAS detector
Aad, G.; Abajyan, T.; Abbott, B.; ... ; Franklin, M.;... Guimarães da Costa, J.;... Huth, J.;...; Morii, M.; and 2944 coauthors
Jets, Jet physics, Hadron-Hadron Scattering
2014JHEP...05..059A
Double-differential dijet cross-sections measured in pp collisions at the LHC with a 7 TeV centre-of-mass energy are presented as functions of dijet mass and half the rapidity separation of the two highest- p T jets. These measurements are obtained using data corresponding to an integrated luminosity of 4.5 fb-1, recorded by the ATLAS detector in 2011. The data are corrected for detector effects so that cross-sections are presented at the particle level. Cross-sections are measured up to 5 TeV dijet mass using jets reconstructed with the anti- k t algorithm for values of the jet radius parameter of 0.4 and 0.6. The cross-sections are compared with next-to-leading-order perturbative QCD calculations by NLOJet++ corrected to account for non-perturbative effects. Comparisons with POWHEG predictions, using a next-to-leading-order matrix element calculation interfaced to a parton-shower Monte Carlo simulation, are also shown. Electroweak effects are accounted for in both cases. The quantitative comparison of data and theoretical predictions obtained using various parameterizations of the parton distribution functions is performed using a frequentist method. In general, good agreement with data is observed for the NLOJet++ theoretical predictions when using the CT10, NNPDF2.1 and MSTW 2008 PDF sets. Disagreement is observed when using the ABM11 and HERAPDF1.5 PDF sets for some ranges of dijet mass and half the rapidity separation. An example setting a lower limit on the compositeness scale for a model of contact interactions is presented, showing that the unfolded results can be used to constrain contributions to dijet production beyond that predicted by the Standard Model. [Figure not available: see fulltext.]
Measurement of the production of a W boson in association with a charm quark in pp collisions at = 7 TeV with the ATLAS detector
Electroweak interaction, Hadron-Hadron Scattering, Charm physics
The production of a W boson in association with a single charm quark is studied using 4.6 fb-1 of pp collision data at = 7 TeV collected with the ATLAS detector at the Large Hadron Collider. In events in which a W boson decays to an electron or muon, the charm quark is tagged either by its semileptonic decay to a muon or by the presence of a charmed meson. The integrated and differential cross sections as a function of the pseudorapidity of the lepton from the W-boson decay are measured. Results are compared to the predictions of next-to-leading-order QCD calculations obtained from various parton distribution function parameterisations. The ratio of the strange-to-down sea-quark distributions is determined to be at Q 2 = 1.9 GeV2, which supports the hypothesis of an SU(3)-symmetric composition of the light-quark sea. Additionally, the cross-section ratio σ( W + +)/ σ( W - + c) is compared to the predictions obtained using parton distribution function parameterisations with different assumptions about the quark asymmetry. [Figure not available: see fulltext.]
Search for direct production of charginos, neutralinos and sleptons in final states with two leptons and missing transverse momentum in pp collisions at = 8TeV with the ATLAS detector
Aad, G.; Abbott, B.; Abdallah, J.;... ; Franklin, M.;... Guimarães da Costa, J.;... Huth, J.;...; Morii, M.; and 2880 coauthors
Supersymmetry, Hadron-Hadron Scattering
Searches for the electroweak production of charginos, neutralinos and sleptons in final states characterized by the presence of two leptons (electrons and muons) and missing transverse momentum are performed using 20.3 fb-1 of proton-proton collision data at = 8 TeV recorded with the ATLAS experiment at the Large Hadron Collider. No significant excess beyond Standard Model expectations is observed. Limits are set on the masses of the lightest chargino, next-to-lightest neutralino and sleptons for different lightest-neutralino mass hypotheses in simplified models. Results are also interpreted in various scenarios of the phenomenological Minimal Supersymmetric Standard Model. [Figure not available: see fulltext.]
Density-Gradient-Free Microfluidic Centrifugation for Analytical and Preparative Separation of Nanoparticles
Arosio, Paolo; Müller, Thomas; Mahadevan, L.; Knowles, Tuomas P. J.
Nano Letters, vol. 14, issue 5, pp. 2365-2371
10.1021/nl404771g
2014NanoL..14.2365A
Radio-frequency spectroscopy of polarons in ultracold Bose gases
Shashi, Aditya; Grusdt, Fabian; Abanin, Dmitry A.; Demler, Eugene
Ultracold gases trapped gases, Absorption and reflection spectra: visible and ultraviolet, Nonequilibrium gas dynamics, Scattering by phonons magnons and other nonlocalized excitations
2014PhRvA..89e3617S
Recent experimental advances enabled the realization of mobile impurities immersed in a Bose-Einstein condensate (BEC) of ultracold atoms. Here, we consider impurities with two or more internal hyperfine states, and study their radio-frequency (rf) absorption spectra, which correspond to transitions between two different hyperfine states. We calculate rf spectra for the case when one of the hyperfine states involved interacts with the BEC, while the other state is noninteracting, by performing a nonperturbative resummation of the probabilities of exciting different numbers of phonon modes. In the presence of interactions, the impurity gets dressed by Bogoliubov excitations of the BEC, and forms a polaron. The rf signal contains a δ-function peak centered at the energy of the polaron measured relative to the bare impurity transition frequency with a weight equal to the amount of bare impurity character in the polaron state. The rf spectrum also has a broad incoherent part arising from the background excitations of the BEC, with a characteristic power-law tail that appears as a consequence of the universal physics of contact interactions. We discuss both the direct rf measurement, in which the impurity is initially in an interacting state, and the inverse rf measurement, in which the impurity is initially in a noninteracting state. In the latter case, in order to calculate the rf spectrum, we solve the problem of polaron formation: a mobile impurity is suddenly introduced in a BEC, and dynamically gets dressed by Bogoliubov phonons. Our solution is based on a time-dependent variational ansatz of coherent states of Bogoliubov phonons, which becomes exact when the impurity is localized. Moreover, we show that such an ansatz compares well with a semiclassical estimate of the propagation amplitude of a mobile impurity in the BEC. Our technique can be extended to cases when both initial and final impurity states are interacting with the BEC.
Study of top quark production and decays involving a tau lepton at CDF and limits on a charged Higgs boson contribution
Aaltonen, T.; Amerio, S.; Amidei, D.;... Franklin, M.;... Guimaraes da Costa, J.;... ; and 404 coauthors
Top quarks
2014PhRvD..89i1101A
We present an analysis of top-antitop quark production and decay into a tau lepton, tau neutrino, and bottom quark using data from 9 fb-1 of integrated luminosity at the Collider Detector at Fermilab. Dilepton events, where one lepton is an energetic electron or muon and the other a hadronically decaying tau lepton, originating from proton-antiproton collisions at √s =1.96 TeV, are used. A top-antitop quark production cross section of 8.1±2.1 pb is measured, assuming standard-model top quark decays. By separately identifying for the first time the single-tau and the ditau components, we measure the branching fraction of the top quark into the tau lepton, tau neutrino, and bottom quark to be (9.6±2.8)%. The branching fraction of top quark decays into a charged Higgs boson and a bottom quark, which would imply violation of lepton universality, is limited to be less than 5.9% at a 95% confidence level [for B(H-→τν¯)=1].
Measurement of the parity-violating asymmetry parameter αb and the helicity amplitudes for the decay Λb0→J/ψΛ0 with the ATLAS detector
Bottom baryons
A measurement of the parity-violating decay asymmetry parameter, αb, and the helicity amplitudes for the decay Λb0→J/ψ(μ+μ-)Λ0(pπ-) is reported. The analysis is based on 1400 Λb0 and Λ ¯b0 baryons selected in 4.6 fb-1 of proton-proton collision data with a center-of-mass energy of 7 TeV recorded by the ATLAS experiment at the LHC. By combining the Λb0 and Λ ¯b0 samples under the assumption of CP conservation, the value of αb is measured to be 0.30±0.16(stat)±0.06(syst). This measurement provides a test of theoretical models based on perturbative QCD or heavy-quark effective theory.
Suppressing qubit dephasing using real-time Hamiltonian estimation
Shulman, Michael D.; Harvey, Shannon P.; Nichol, John M.; Bartlett, Stephen D.; Doherty, Andrew C.; Umansky, Vladimir; Yacoby, Amir
Condensed Matter - Mesoscale and Nanoscale Physics, Quantum Physics
Unwanted interaction between a quantum system and its fluctuating environment leads to decoherence and is the primary obstacle to establishing a scalable quantum information processing architecture. Strategies such as environmental and materials engineering, quantum error correction and dynamical decoupling can mitigate decoherence, but generally increase experimental complexity. Here we improve coherence in a qubit using real-time Hamiltonian parameter estimation. Using a rapidly converging Bayesian approach, we precisely measure the splitting in a singlet-triplet spin qubit faster than the surrounding nuclear bath fluctuates. We continuously adjust qubit control parameters based on this information, thereby improving the inhomogenously broadened coherence time ($T_{2}^{*}$) from tens of nanoseconds to above 2 $\mu$s and demonstrating the effectiveness of Hamiltonian estimation in reducing the effects of correlated noise in quantum systems. Because the technique demonstrated here is compatible with arbitrary qubit operations, it is a natural complement to quantum error correction and can be used to improve the performance of a wide variety of qubits in both metrological and quantum-information-processing applications.
Field-effect-induced two-dimensional electron gas utilizing modulation doping for improved ohmic contacts
Mondal, Sumit; Gardner, Geoffrey C.; Watson, John D.; Fallahi, Saeed; Yacoby, Amir; Manfra, Michael J.
Modulation-doped AlGaAs/GaAs heterostructures are utilized extensively in the study of quantum transport in nanostructures, but charge fluctuations associated with remote ionized dopants often produce deleterious effects. Electric field-induced carrier systems offer an attractive alternative if certain challenges can be overcome. We demonstrate a field-effect transistor in which the active channel is locally devoid of modulation-doping, but silicon dopant atoms are retained in the ohmic contact region to facilitate reliable low-resistance contacts. A high quality two-dimensional electron gas is induced by a field-effect and is tunable over a wide range of density. Device design, fabrication, and low temperature (T= 0.3K) transport data are reported.
BICEP2 and the Central Charge of Holographic Inflation
Larsen, Finn; Strominger, Andrew
Holographic inflation posits that the inflationary deSitter era of our universe is approximately described by a dual three-dimensional Euclidean CFT living on the spatial slice at the end of inflation. We point out that the BICEP2 results determine the central charge of this putative CFT to be given by $C_T=3 \times 10^8$.
Shaping Electromagnetic Fields
Physics - Optics
The ability to control electromagnetic fields on the subwavelength scale could open exciting new venues in many fields of science. Transformation optics provides one way to attain such control through the local variation of the permittivity and permeability of a material. Here, we demonstrate another way to shape electromagnetic fields, taking advantage of the enormous size of the configuration space in combinatorial problems and the resonant scattering properties of metallic nanoparticles. Our design does not require the engineering of a material's electromagnetic properties and has relevance to the design of more flexible platforms for probing light-matter interaction and many body physics.
A Map of Dust Reddening to 4.5 kpc from Pan-STARRS1
Schlafly, E. F.; Green, G.; Finkbeiner, D. P.; Juric, M.; Rix, H.-W.; Martin, N. F.; Burgett, W. S.; Chambers, K. C.; Draper, P. W.; Hodapp, K. W.; Kaiser, N.; Kudritzki, R.-P.; Magnier, E. A.; Metcalfe, N.; Morgan, J. S.; Price, P. A.; Stubbs, C. W.; Tonry, J. L.; Wainscoat, R. J.; Waters, C.
10 pages, 7 figures, accepted for publication in ApJ
We present a map of the dust reddening to 4.5 kpc derived from Pan-STARRS1 stellar photometry. The map covers almost the entire sky north of declination -30 degrees at a resolution of 7' to 14', and is based on the estimated distances and reddenings to more than 500 million stars. The technique is designed to map dust in the Galactic plane, where many other techniques are stymied by the presence of multiple dust clouds at different distances along each line of sight. This reddening-based dust map agrees closely with the Schlegel, Finkbeiner, and Davis (SFD; 1998) far-infrared emission-based dust map away from the Galactic plane, and the most prominent differences between the two maps stem from known limitations of SFD in the plane. We also compare the map with Planck, finding likewise good agreement in general at high latitudes. The use of optical data from Pan-STARRS1 yields reddening uncertainty as low as 25 mmag E(B-V).
Ultrahigh Transmission Optical Nanofibers
Hoffman, J. E.; Ravets, S.; Grover, J. A.; Solano, P.; Kordell, P. R.; Wong-Campos, J. D.; Orozco, L. A.; Rolston, S. L.
Physics - Optics, Physics - Atomic Physics
32 pages, 10 figures, accepted to AIP Advances
We present a procedure for reproducibly fabricating ultrahigh transmission optical nanofibers (530 nm diameter and 84 mm stretch) with single-mode transmissions of 99.95 $ \pm$ 0.02%, which represents a loss from tapering of 2.6 $\,\times \,$ 10$^{-5}$ dB/mm when normalized to the entire stretch. When controllably launching the next family of higher-order modes on a fiber with 195 mm stretch, we achieve a transmission of 97.8 $\pm$ 2.8%, which has a loss from tapering of 5.0 $\,\times \,$ 10$^{-4}$ dB/mm when normalized to the entire stretch. Our pulling and transfer procedures allow us to fabricate optical nanofibers that transmit more than 400 mW in high vacuum conditions. These results, published as parameters in our previous work, present an improvement of two orders of magnitude less loss for the fundamental mode and an increase in transmission of more than 300% for higher-order modes, when following the protocols detailed in this paper. We extract from the transmission during the pull, the only reported spectrogram of a fundamental mode launch that does not include excitation to asymmetric modes; in stark contrast to a pull in which our cleaning protocol is not followed. These results depend critically on the pre-pull cleanliness and when properly following our pulling protocols are in excellent agreement with simulations.
Deleterious passengers in adapting populations
Good, Benjamin H; Desai, Michael M
Quantitative Biology - Populations and Evolution
Most new mutations are deleterious and are eventually eliminated by natural selection. But in an adapting population, the rapid amplification of beneficial mutations can hinder the removal of deleterious variants in nearby regions of the genome, altering the patterns of sequence evolution. Here, we analyze the interactions between beneficial "driver" mutations and linked deleterious "passengers" during the course of adaptation. We derive analytical expressions for the substitution rate of a deleterious mutation as a function of its fitness cost, as well as the reduction in the beneficial substitution rate due to the genetic load of the passengers. We find that the fate of each deleterious mutation varies dramatically with the rate and spectrum of beneficial mutations, with a non-monotonic dependence on both the population size and the rate of adaptation. By quantifying this dependence, our results allow us to estimate which deleterious mutations will be likely to fix, and how many of these mutations must arise before the progress of adaptation is significantly reduced.
Compton Scattering from \nuc{6}{Li} at 86 MeV
Myers, L. S.; Ahmed, M. W.; Feldman, G.; Kafkarkou, A.; Kendellen, D. P.; Mazumdar, I.; Mueller, J. M.; Sikora, M. H.; Weller, H. R.; Zimmerman, W. R.
Nuclear Experiment
5 pages, 5 figures, submission to Phys. Rev. C
Cross sections for \nuc{6}{Li}($\gamma$,$\gamma$)\nuc{6}{Li} have been measured at the High Intensity Gamma-Ray Source (\HIGS) and the sensitivity of these cross sections to the nucleon isoscalar polarizabilities was studied. Data were collected using a quasi-monoenergetic 86 MeV photon beam at photon scattering angles of 40$^{\circ}$--160$^{\circ}$. These results are an extension of a previous measurement at a lower energy. The earlier work indicated that the \nuc{6}{Li}($\gamma$,$\gamma$)\nuc{6}{Li} reaction at 60 MeV provides a means of extracting the nucleon polarizabilities; this work demonstrates that the sensitivity of the cross section to the polarizabilities is increased at 86 MeV. A full theoretical treatment is needed to verify this conclusion and produce values of the polarizabilities.
Spin Squeezing by means of Driven Superradiance
Wolfe, Elie; Yelin, S. F.
Quantum Physics, Mathematical Physics
5+2 pages, 2 color figures. Includes analytic constructions of both the coupled first-order rate equations as well as the spin squeezing parameter explicitly in terms of the matrix elements. We appreciate and actively welcome all comments
2014arXiv1405.5288W
We discuss the possibility of generating entanglement by means of driven superradiance. In an earlier paper [Phys. Rev. Lett. 112, 140402 (2014)] the authors determined that spontaneous purely-dissipative Dicke model superradiance failed to generate any entanglement over the course of the system's time evolution. In this article we show that by adding a driving field, however, the Dicke model system can be tuned to evolve into an entangled steady state. We discuss how to optimize the driving frequency to maximize the entanglement. We show that the resulting entanglement is fairly strong, in that it leads to spin squeezing.
A Construction of stable bundles and reflexive sheaves on Calabi-Yau threefolds
Wu, Baosen; Yau, Shing Tung
Mathematics - Algebraic Geometry, High Energy Physics - Theory
We use Serre construction and deformation to construct stable bundles and reflexive sheaves on Calabi-Yau threefolds.
Mobile magnetic impurities in a Fermi superfluid: a route to designer molecules
Gopalakrishnan, Sarang; Parker, Colin V.; Demler, Eugene
Condensed Matter - Quantum Gases, Condensed Matter - Mesoscale and Nanoscale Physics, Condensed Matter - Superconductivity, Quantum Physics
5pp., 4 figures, plus appendix
A magnetic impurity in a fermionic superfluid hosts bound quasiparticle states known as Yu-Shiba-Rusinov (YSR) states. We argue here that, if the impurity is mobile (i.e., has a finite mass), the impurity and its bound YSR quasiparticle move together as a midgap molecule, which has an unusual "Mexican-hat" dispersion that is tunable via the fermion density. We map out the impurity dispersion, which consists of an "atomic" branch (in which the impurity is dressed by quasiparticle pairs) and a "molecular" branch (in which the impurity binds a quasiparticle). We discuss the experimental realization and detection of midgap Shiba molecules, focusing on lithium-cesium mixtures, and comment on the prospects they offer for realizing exotic many-body states.
A Synopsis of the Minimal Modal Interpretation of Quantum Theory
Barandes, Jacob A.; Kagan, David
Quantum Physics, General Relativity and Quantum Cosmology
7 pages + references, 2 figures; cosmetic changes, added figure, updated references, generalized conditional probabilities with attendant changes to the section on Lorentz invariance; this letter summarizes the more comprehensive companion paper at arXiv:1405.6755
We summarize a new realist interpretation of quantum theory that builds on the existing physical structure of the theory and allows experiments to have definite outcomes, but leaves the theory's basic dynamical content essentially intact. Much as classical systems have specific states that evolve along definite trajectories through configuration spaces, the traditional formulation of quantum theory asserts that closed quantum systems have specific states that evolve unitarily along definite trajectories through Hilbert spaces, and our interpretation extends this intuitive picture of states and Hilbert-space trajectories to the case of open quantum systems as well. Our interpretation---which we claim is ultimately compatible with Lorentz invariance---reformulates wave-function collapse in terms of an underlying interpolating dynamics, makes it possible to derive the Born rule from deeper principles, and resolves several open questions regarding ontological stability and dynamics.
The Minimal Modal Interpretation of Quantum Theory
73 pages + references, 9 figures; cosmetic changes, added figure, updated references, generalized conditional probabilities with attendant changes to the sections on the EPR-Bohm thought experiment and Lorentz invariance; for a concise summary, see the companion letter at arXiv:1405.6754
We introduce a realist, unextravagant interpretation of quantum theory that builds on the existing physical structure of the theory and allows experiments to have definite outcomes, but leaves the theory's basic dynamical content essentially intact. Much as classical systems have specific states that evolve along definite trajectories through configuration spaces, the traditional formulation of quantum theory asserts that closed quantum systems have specific states that evolve unitarily along definite trajectories through Hilbert spaces, and our interpretation extends this intuitive picture of states and Hilbert-space trajectories to the case of open quantum systems as well. We provide independent justification for the partial-trace operation for density matrices, reformulate wave-function collapse in terms of an underlying interpolating dynamics, derive the Born rule from deeper principles, resolve several open questions regarding ontological stability and dynamics, address a number of familiar no-go theorems, and argue that our interpretation is ultimately compatible with Lorentz invariance. Along the way, we also investigate a number of unexplored features of quantum theory, including an interesting geometrical structure---which we call subsystem space---that we believe merits further study. We include an appendix that briefly reviews the traditional Copenhagen interpretation and the measurement problem of quantum theory, as well as the instrumentalist approach and a collection of foundational theorems not otherwise discussed in the main text.
Structural stability and electronic properties of low-index surfaces of SnS
Tritsaris, Georgios A.; Malone, Brad D.; Kaxiras, Efthimios
2014JAP...115q3702T
Thin film photovoltaic cells are increasingly important for cost-effective solar energy harvesting. Layered SnS is a promising absorber material due to its high optical absorption in the visible and good doping characteristics. We use first-principles calculations based on density functional theory to study structures of low-index surfaces of SnS using stoichiometric and oxygen-containing structural models, in order to elucidate their possible effect on the efficiency of the photovoltaic device. We find that the surface energy is minimized for the surface with orientation parallel to the layer stacking direction. Compared to stoichiometric surfaces, the oxygen-containing surfaces exhibit fewer electronic states near the band gap. This reduction of near-gap surface states by oxygen should reduce recombination losses at grain boundaries and interfaces of the SnS absorber, and should be beneficial to the efficiency of the solar cell.
Tangles, generalized Reidemeister moves, and three-dimensional mirror symmetry
Córdova, Clay; Espahbodi, Sam; Haghighat, Babak; Rastogi, Ashwin; Vafa, Cumrun
Supersymmetry and Duality, Field Theories in Lower Dimensions, Extended Supersymmetry, M-Theory
Three-dimensional = 2 superconformal field theories are constructed by compactifying M5-branes on three-manifolds. In the infrared the branes recombine, and the physics is captured by a single M5-brane on a branched cover of the original ultraviolet geometry. The branch locus is a tangle, a one-dimensional knotted submanifold of the ultraviolet geometry. A choice of branch sheet for this cover yields a Lagrangian for the theory, and varying the branch sheet provides dual descriptions. Massless matter arises from vanishing size M2-branes and appears as singularities of the tangle where branch lines collide. Massive deformations of the field theory correspond to resolutions of singularities resulting in distinct smooth manifolds connected by geometric transitions. A generalization of Reidemeister moves for singular tangles captures mirror symmetries of the underlying theory yielding a geometric framework where dualities are manifest.
On the classification of 6D SCFTs and generalized ADE orbifolds
Heckman, Jonathan J.; Morrison, David R.; Vafa, Cumrun
F-Theory, Differential and Algebraic Geometry, Field Theories in Higher Dimensions
2014JHEP...05..028H
We study (1, 0) and (2, 0) 6D superconformal field theories (SCFTs) that can be constructed in F-theory. Quite surprisingly, all of them involve an orbifold singularity ℂ2/Γ with Γ a discrete subgroup of U(2). When Γ is a subgroup of SU (2), all discrete subgroups are allowed, and this leads to the familiar ADE classification of (2, 0) SCFTs. For more general U(2) subgroups, the allowed possibilities for Γ are not arbitrary and are given by certain generalizations of the A- and D-series. These theories should be viewed as the minimal 6D SCFTs. We obtain all other SCFTs by bringing in a number of E-string theories and/or decorating curves in the base by non-minimal gauge algebras. In this way we obtain a vast number of new 6D SCFTs, and we conjecture that our construction provides a full list.
Exact results for five-dimensional superconformal field theories with gravity duals
Jafferis, Daniel L.; Pufu, Silviu S.
Matrix Models, Supersymmetric gauge theory, AdS-CFT Correspondence, Extended Supersymmetry
2014JHEP...05..032J
We apply the technique of supersymmetric localization to exactly compute the S 5 partition function of several large N superconformal field theories in five dimensions that have AdS6 duals in massive type IIA supergravity. The localization computations are performed in the non-renormalizable effective field theories obtained through relevant deformations of the UV superconformal field theories. We compare the S 5 free energy to a holographic computation of entanglement entropy in the AdS6 duals and find perfect agreement. In particular, we reproduce the N 5/2 scaling of the S 5 free energy that was expected from supergravity.
The dynamics of quantum criticality revealed by quantum Monte Carlo and holography
Witczak-Krempa, William; Sørensen, Erik S.; Sachdev, Subir
Nature Physics, Volume 10, Issue 5, pp. 361-366 (2014).
(c) 2014: Nature Publishing Group
10.1038/nphys2913
2014NatPh..10..361W
Understanding the dynamics of quantum systems without long-lived excitations (quasiparticles) constitutes an important yet challenging problem. Although numerical techniques can yield results for the dynamics in imaginary time, their reliable continuation to real time has proved difficult. We tackle this issue using the superfluid-insulator quantum critical point of bosons on a two-dimensional lattice, where quantum fluctuations destroy quasiparticles. We present quantum Monte Carlo simulations for two separate lattice realizations. Their low-frequency conductivities turn out to have the same universal dependence on imaginary frequency and temperature. Using the structure of the real-time dynamics of conformal field theories described by the holographic gauge/gravity duality, we then make progress on the problem of analytically continuing the numerical data to real time. Our method yields quantitative and experimentally testable results on the frequency-dependent conductivity near the quantum critical point. Extensions to other observables and universality classes are discussed.
Search for Higgs boson decays to a photon and a Z boson in pp collisions at s=7 and 8 TeV with the ATLAS detector
Physics Letters B, Volume 732, p. 8-27.
2014PhLB..732....8A
A search is reported for a neutral Higgs boson in the decay channel H→Zγ, Z→ℓ+ℓ‑ (ℓ=e,μ), using 4.5 fb‑1 of pp collisions at s=7 TeV and 20.3 fb‑1 of pp collisions at s=8 TeV, recorded by the ATLAS detector at the CERN Large Hadron Collider. The observed distribution of the invariant mass of the three final-state particles, m, is consistent with the Standard Model hypothesis in the investigated mass range of 120–150 GeV. For a Higgs boson with a mass of 125.5 GeV, the observed upper limit at the 95% confidence level is 11 times the Standard Model expectation. Upper limits are set on the cross section times branching ratio of a neutral Higgs boson with mass in the range 120–150 GeV between 0.13 and 0.5 pb for s=8 TeV at 95% confidence level.
Invariant-mass distribution of jet pairs produced in association with a W boson in pp ¯ collisions at √s =1.96 TeV using the full CDF Run II data set
Aaltonen, T.; Amerio, S.; Amidei, D.;... ... Franklin, M.;... ; Guimaraes da Costa, J.;... and 403 coauthors
Applications of electroweak models to specific processes, Experimental tests, Other particles
We report on a study of the dijet invariant-mass distribution in events with one identified lepton, a significant imbalance in the total event transverse momentum, and two jets. This distribution is sensitive to the possible production of a new particle in association with a W boson, where the boson decays leptonically. We use the full data set of proton-antiproton collisions at 1.96 TeV center-of-mass energy collected by the Collider Detector at the Fermilab Tevatron, corresponding to an integrated luminosity of 8.9 fb-1. The data are found to be consistent with standard model expectations, and a 95% confidence level upper limit is set on the production cross section of a W boson in association with a new particle decaying into two jets.
Cross sections for the reactions e+e-→KS0KL0, KS0KL0π+π-, KS0KS0π+π-, and KS0KS0K+K- from events with initial-state radiation
Lees, J. P.; Poireau, V.; Tisserand, V.;... ... Morii, M.;... and 310 coauthors (Babar Collaboration)
Hadron production in e-e+ interactions, Decays of other mesons, Mesons
2014PhRvD..89i2002L
We study the processes e+e-→KS0KL0γ, KS0KL0π+π-γ, KS0KS0π+π-γ, and KS0KS0K+K-γ, where the photon is radiated from the initial state, providing cross section measurements for the hadronic states over a continuum of center-of-mass energies. The results are based on 469 fb-1 of data collected with the BABAR detector at SLAC. We observe the ϕ(1020) resonance in the KS0KL0 final state and measure the product of its electronic width and branching fraction with about 3% uncertainty. We present a measurement of the e +e-→KS0KL0 cross section in the energy range from 1.06 to 2.2 GeV and observe the production of a resonance at 1.67 GeV. We present the first measurements of the e+e-→KS0KL0π+π-, KS0KS0π+π-, and KS0KS0K+K- cross sections and study the intermediate resonance structures. We obtain the first observations of J/ψ decay to the KS0KL0π +π-, KS0KS0π+π-, and KS0KS0K+K- final states.
Combined Analysis of νμ Disappearance and νμ→νe Appearance in MINOS Using Accelerator and Atmospheric Neutrinos
Adamson, P.; Anghel, I.; Aurisano, A.;... Feldman, G. J.;... and 85 coauthors (Minos Collaboration)
Neutrino mass and mixing
2014PhRvL.112s1801A
We report on a new analysis of neutrino oscillations in MINOS using the complete set of accelerator and atmospheric data. The analysis combines the νμ disappearance and νe appearance data using the three-flavor formalism. We measure |Δm322|=[2.28-2.46]×10-3 eV2 (68% C.L.) and sin2θ23=0.35-0.65 (90% C.L.) in the normal hierarchy, and |Δm322|=[2.32-2.53]×10-3 eV2 (68% C.L.) and sin2θ23=0.34-0.67 (90% C.L.) in the inverted hierarchy. The data also constrain δCP, the θ23 octant degeneracy and the mass hierarchy; we disfavor 36% (11%) of this three-parameter space at 68% (90%) C.L.
Search for Invisible Decays of a Higgs Boson Produced in Association with a Z Boson in ATLAS
Standard-model Higgs bosons, Extensions of electroweak Higgs sector, Dark matter
2014PhRvL.112t1802A
A search for evidence of invisible-particle decay modes of a Higgs boson produced in association with a Z boson at the Large Hadron Collider is presented. No deviation from the standard model expectation is observed in 4.5 fb-1 (20.3 fb-1) of 7 (8) TeV pp collision data collected by the ATLAS experiment. Assuming the standard model rate for ZH production, an upper limit of 75%, at the 95% confidence level is set on the branching ratio to invisible-particle decay modes of the Higgs boson at a mass of 125.5 GeV. The limit on the branching ratio is also interpreted in terms of an upper limit on the allowed dark matter-nucleon scattering cross section within a Higgs-portal dark matter scenario. Within the constraints of such a scenario, the results presented in this Letter provide the strongest available limits for low-mass dark matter candidates. Limits are also set on an additional neutral Higgs boson, in the mass range 110<mH<400 GeV, produced in association with a Z boson and decaying to invisible particles. | CommonCrawl |
\begin{document}
\def{\mathbb R}{{\mathbb R}} \def{\mathbb Z}{{\mathbb Z}} \def{\mathbb C}{{\mathbb C}} \newcommand{\rm trace}{\rm trace} \newcommand{{\mathbb{E}}}{{\mathbb{E}}} \newcommand{{\mathbb{P}}}{{\mathbb{P}}} \newcommand{\rm {co\,}}{\rm {co\,}} \newcommand{{\mathbb E}}{{\mathbb E}} \newcommand{{\cal F}}{{\cal F}} \newtheorem{df}{Definition} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{pr}{Proposition} \newtheorem{prob}{Problem} \newtheorem{cor}{Corollary} \newtheorem{remark}{Remark} \newtheorem{problem}{Problem} \def\nu{\nu} \def\mbox{ sign }{\mbox{ sign }} \def\alpha{\alpha} \def{\mathbb N}{{\mathbb N}} \def{\cal A}{{\cal A}} \def{\cal L}{{\cal L}} \def{\cal X}{{\cal X}} \def{\cal F}{{\cal F}} \def\bar{c}{\bar{c}} \def\mbox{\rm Vol}{\mbox{\rm Vol}} \def\nu{\nu} \def\delta{\delta} \def\mbox{\rm diam}{\mbox{\rm diam}} \def\beta{\beta} \def\theta{\theta} \def\lambda{\lambda} \def\varepsilon{\varepsilon} \def{:}\;{{:}\;} \def\noindent {\bf Proof: \ }{\noindent {\bf Proof: \ }} \def\begin{flushright} $\Box $\\ \end{flushright}{\begin{flushright} $\Box $\\ \end{flushright}}
\title[Measures of sections of convex bodies]{Measures of sections of convex bodies} \author{Alexander Koldobsky}
\address{Department of Mathematics\\ University of Missouri\\ Columbia, MO 65211}
\email{koldobskiya@@missouri.edu}
\begin{abstract} This article is a survey of recent results on slicing inequalities for convex bodies. The focus is on the setting of arbitrary measures in place of volume.
\end{abstract} \maketitle
\section{Introduction}
The study of volume of sections of convex bodies is a classical direction in convex geometry. It is well developed and has numerous applications; see \cite{G3,K4}. The question of what happens if volume is replaced by an arbitrary measure on a convex body has not been considered until very recently, mostly because it is hard to believe that difficult geometric results can hold in such generality. However, in 2005 Zvavitch \cite{Zv} proved that the solution to the Busemann-Petty problem, one of the signature problems in convex geometry, remains exactly the same if volume is replaced by an arbitrary measure with continuous density. It has recently been shown \cite{K6, KM, K8, K9, K10, K11} that several partial results on the slicing problem, a major open question in the area, can also be extended to arbitrary measures. For example, it was proved in \cite{K11} that the slicing problem for sections of proportional dimensions has an affirmative answer which can be extended to the setting of arbitrary measures. It is not clear yet whether these results are representative of something bigger, or it is just an isolated event. We let the reader make the judgement.
\section{The slicing problem for measures}
The slicing problem \cite{Bo1, Bo2, Ba5, MP}, asks whether there exists an absolute constant $C$ so that for every origin-symmetric convex body $K$ in ${\mathbb R}^n$ of volume 1 there is a hyperplane section of $K$ whose $(n-1)$-dimensional volume is greater than $1/C.$ In other words, does there exist a constant $C$ so that for any $n\in {\mathbb N}$ and any origin-symmetric convex body $K$ in ${\mathbb R}^n$ \begin{equation} \label{hyper}
|K|^{\frac {n-1}n} \le C \max_{\xi \in S^{n-1}} |K\cap \xi^\bot|, \end{equation} where $\xi^\bot$ is the central hyperplane in ${\mathbb R}^n$ perpendicular to $\xi,$ and
$|K|$ stands for volume of proper dimension? The best current result $C\le O(n^{1/4})$ is due to Klartag \cite{Kla2}, who slightly improved an earlier estimate of Bourgain \cite{Bo3}. The answer is known to be affirmative for some special classes of convex bodies, including unconditional convex bodies (as initially observed by Bourgain; see also \cite{MP, J2, BN}), unit balls of subspaces of $L_p$ \cite{Ba4, J1, M1}, intersection bodies \cite[Th.9.4.11]{G3}, zonoids, duals of bodies with bounded volume ratio \cite{MP}, the Schatten classes \cite{KMP}, $k$-intersection bodies \cite{KPY, K10}. Other partial results on the problem include \cite{Ba3, BKM, DP, Da, GPV, Kla1, KlaK, Pa, EK, BaN}; see the book \cite{BGVV} for details.
Iterating (\ref{hyper}) one gets the lower dimensional slicing problem asking whether the inequality \begin{equation} \label{lowdimhyper}
|K|^{\frac {n-k}n} \le C^k \max_{H\in Gr_{n-k}} |K\cap H| \end{equation} holds with an absolute constant $C,$ where $1\le k \le n-1$ and $Gr_{n-k}$ is the Grassmanian of $(n-k)$-dimensional subspaces of ${\mathbb R}^n.$
Inequality (\ref{lowdimhyper}) was proved in \cite{K11} in the case where $k\ge \lambda n,\ 0<\lambda<1,$ with the constant $C=C(\lambda)$ dependent only on $\lambda.$ Moreover, this was proved in \cite{K11} for arbitrary measures in place of volume. We consider the following generalization of the slicing problem to arbitrary measures and to sections of arbitrary codimension.
\begin{problem} \label{prob} Does there exist an absolute constant $C$ so that for every $n\in {\mathbb N},$ every integer $1\le k < n,$ every origin-symmetric convex body $K$ in ${\mathbb R}^n,$ and every measure $\mu$ with non-negative even continuous density $f$ in ${\mathbb R}^n,$ \begin{equation}\label{main-problem}
\mu(K)\ \le\ C^k \max_{H \in Gr_{n-k}} \mu(K\cap H)\ |K|^{k/n}. \end{equation} \end{problem} Here $\mu(B)=\int_B f$ for every compact set $B$ in ${\mathbb R}^n,$ and $\mu(B\cap H)=\int_{B\cap H} f$ is the result of integration of the restriction of $f$ to $H$ with respect to Lebesgue measure in $H.$ The case of volume corresponds to $f\equiv 1.$
In some cases we will write (\ref{main-problem}) in an equivalent form \begin{equation}\label{measslicing}
\mu(K)\ \le\ C^k \frac n{n-k}\ c_{n,k} \max_{H \in Gr_{n-k}} \mu(K\cap H)\ |K|^{k/n}, \end{equation}
where $c_{n,k}= |B_2^n|^{\frac {n-k}n}/|B_2^{n-k}|,$ and $B_2^n$ is the unit Euclidean ball in ${\mathbb R}^n.$ It is easy to see that $c_{n,k}\in (e^{-k/2},1)$, and $ \frac{n}{n-k} \in (1,e^k),$ so these constants can be incorporated in the constant $C.$
Surprisingly, many partial results on the original slicing problem can be extended to the setting of arbitrary measures. Inequality (\ref{main-problem}) holds true in the following cases: \begin{itemize} \item for arbitrary $n,K,\mu$ and $k\ge \lambda n,$ where $\lambda\in (0,1),$ with the constant $C$ dependent only on $\lambda$, \cite{K11};
\item for all $n, K, \mu, k,$ with $C\le O(\sqrt{n})$, \cite{K8, K9};
\item for intersection bodies $K$ (see definition below), with an absolute constant $C$,
\cite{K6} for $k=1$, \cite{KM} for all $k;$
\item for the unit balls of $n$-dimensional subspaces of $L_p,\ p>2,$ with $C\le O(n^{1/2-1/p}),$ \cite{K10};
\item for the unit balls of $n$-dimensional normed spaces that embed in $L_p,\ p\in (-n, 2]$, with $C$ depending only on $p,$ \cite{K10};
\item for unconditional convex bodies, with an absolute constant $C,$ \cite{K11};
\item for duals of convex bodies with bounded volume ratio, with an absolute constant $C,$ \cite{K11};
\item for $k=1$ and log-concave measures $\mu$, with $C\le O(n^{1/4}),$ \cite{KZ}. \end{itemize}
The proofs of these results are based on stability in volume comparison problems introduced in \cite{K5} and developed in \cite{K6, KM, K7, K8, K9, K10,K13}. Stability reduces Problem \ref{prob} to estimating the outer volume ratio distance from a convex body to the classes of generalized intersection bodies. The concept of an intersection body was introduced by Lutwak \cite{Lu} in connection with the Busemann-Petty problem.
A closed bounded set $K$ in ${\mathbb R}^n$ is called a {star body} if every straight line passing through the origin crosses the boundary of $K$ at exactly two points different from the origin, the origin is an interior point of $K,$ and the boundary of $K$ is continuous.
For $1\le k \le n-1,$ the classes ${\cal{BP}}_k^n$ of generalized $k$-intersection bodies in ${\mathbb R}^n$ were introduced by Zhang \cite{Z3}. The case $k=1$ represents the original class of intersection bodies ${\cal{I}}_n={\cal{BP}}_1^n$ of Lutwak \cite{Lu}. We define ${\cal{BP}}_k^n$ as the closure in the radial metric of radial $k$-sums of finite collections of origin-symmetric ellipsoids (the equivalence of this definition to the original definitions of Lutwak and Zhang was established by Goodey and Weil \cite{GW} for $k=1$ and by Grinberg and Zhang \cite{GrZ} for arbitrary $k.)$ Recall that the radial $k$-sum of star bodies $K$ and $L$ in ${\mathbb R}^n$ is a new star body $K+_kL$ whose radius in every direction $\xi\in S^{n-1}$ is given by $$r^k_{K+_kL}(\xi)= r^k_{K}(\xi) + r^k_{L}(\xi).$$ The radial metric in the class of origin-symmetric star bodies is defined by
$$\rho(K,L)=\sup_{\xi\in S^{n-1}} |r_K(\xi)-r_L(\xi)|.$$ The following stability theorem was proved in \cite{K11} (see \cite{K6, KM} for slightly different versions).
\begin{theorem}\label{stab2}(\cite{K11}) Suppose that $1\le k \le n-1,$ $K$ is a generalized $k$-intersection body in ${\mathbb R}^n,$ $f$ is an even continuous non-negative function on $K,$ and $\varepsilon>0.$ If $$ \int_{K\cap H} f \ \le \varepsilon,\qquad \forall H\in Gr_{n-k}, $$ then $$
\int_K f\ \le \frac n{n-k}\ c_{n,k}\ |K|^{k/n}\varepsilon. $$ The constant is the best possible. Recall that $c_{n,k}\in (e^{-k/2},1).$ \end{theorem} Define the outer volume ratio distance from an origin-symmetric star body $K$ in ${\mathbb R}^n$ to the class ${\cal{BP}}_k^n$ of generalized $k$-intersection bodies by
$${\rm {o.v.r.}}(K,{\cal{BP}}_k^n) = \inf \left\{ \left( \frac {|D|}{|K|}\right)^{1/n}:\ K\subset D,\ D\in {\cal{BP}}_k^n \right\}.$$
Theorem \ref{stab2} immediately implies a slicing inequality for arbitrary measures and origin-symmetric star bodies.
\begin{cor} \label{lowdim} Let $K$ be an origin-symmetric star body in ${\mathbb R}^n.$ Then for any measure $\mu$ with even continuous density on $K$ we have
$$\mu(K)\le \left({\rm{ o.v.r.}}(K,{{\cal{BP}}_k^n})\right)^k \frac n{n-k}\ c_{n,k} \max_{H\in Gr_{n-k}} \mu(K\cap H)\ |K|^{k/n}.$$ \end{cor}
Thus, stability reduces Problem 1 to estimating the outer volume ratio distance from $K$ to the class of generalized $k$-intersection bodies. The results on Problem \ref{prob} mentioned above were all obtained by estimating this distance by means of various techniques from the local theory of Banach spaces. For example, the solution to the slicing problem for sections of proportional dimensions follows from an estimate obtained in \cite{KPZ}: for any origin-symmetric convex body $K$ in ${\mathbb R}^n$ and any $1\le k \le n-1,$ \begin{equation}\label{kpz} {\rm o.v.r.}(K,{\cal{BP}}_k^n) \le C_0 \sqrt{\frac{n}{k}}\left(\log\left(\frac{en}{k}\right)\right)^{3/2}, \end{equation} where $C_0$ is an absolute constant. The proof of this estimate in \cite{KPZ} is quite involved. It uses covering numbers, Pisier's generalization of Milman's reverse Brunn-Minkowski inequality, properties of intersection bodies. Combining this with Corollary \ref{lowdim}, one gets \begin{theorem}\label{proport} (\cite{K11}) If the codimension of sections $k$ satisfies $\lambda n\le k$ for some $\lambda\in (0,1),$ then for every origin-symmetric convex body $K$ in ${\mathbb R}^n$ and every measure $\mu$ with continuous non-negative density in ${\mathbb R}^n,$ $$ \mu(K)\ \le\ C^k \left(\sqrt{\frac{(1-\log \lambda)^3}{\lambda}}\right)^k
\max_{H \in Gr_{n-k}} \mu(K\cap H)\ |K|^{k/n},$$ where $C$ is an absolute constant. \end{theorem} For arbitrary $K,\mu$ and $k$ the best result so far is the following $\sqrt{n}$ estimate; see \cite{K8, K9}. By John's theorem, for any origin-symmetric convex body $K$ there exists an ellipsoid ${\cal{E}}$ so that $ \frac 1{\sqrt{n}} {\cal{E}}\subset K \subset {\cal{E}}.$ Since every ellipsoid is a generalized $k$-intersection body for every $k,$ we get that $${\rm o.v.r.}(K,{\cal{BP}}_k^n) \le \sqrt{n}.$$ By Corollary \ref{lowdim}, $$
\mu(K)\ \le\ n^{k/2} \frac n{n-k}\ c_{n,k}\max_{H\in Gr_{n-k}} \mu(K\cap H)\ |K|^{k/n}. $$
Note that the condition that the measure $\mu$ has continuous density is necessary in Problem \ref{prob}. A discrete version of inequality (\ref{main-problem}) was very recently established (with a constant depending only on the dimension) in \cite{AHZ}.
\section{The isomorphic Busemann-Petty problem} In 1956, Busemann and Petty \cite{BP} asked the following question. Let $K,L$ be origin-symmetric convex bodies in ${\mathbb R}^n$ such that \begin{equation}\label{bp-condition}
\left|K\cap\xi^\perp\right| \le \left|L\cap\xi^\perp\right|,\qquad \forall \xi\in S^{n-1}. \end{equation}
Does it necessarily follow that $\left|K\right| \le \left|L\right| ?$ The problem was solved at the end of the 1990's in a sequence of papers \cite{LR, Ba1, Gi, Bo4, Lu, P, G1, G2, Z1, K1, K2, Z2, GKS}; see \cite[p.3]{K4} or \cite[p.343]{G3} for the solution and its history. The answer is affirmative if $n\le 4$, and it is negative if $n\ge 5.$
The {lower dimensional Busemann-Petty problem} asks the same question for sections of lower dimensions. Suppose that $1\le k \le n-1,$ and $K,L$ are origin-symmetric convex bodies in ${\mathbb R}^n$ such that \begin{equation}\label{ldbp-condition}
|K\cap H|\le |L\cap H|,\qquad \forall H\in Gr_{n-k}. \end{equation}
Does it follow that $|K|\le |L|?$ It was proved in \cite{BZ} (see also \cite{K3, K4, RZ, M2} for different proofs) that the answer is negative if the dimension of sections $n-k>3.$ The problem is still open for two- and three-dimensional sections ($n-k=2,3,\ n\ge 5).$
Since the answer to the Busemann-Petty problem is negative in most dimensions, it makes sense to ask whether the inequality for volumes holds up to an absolute constant, namely, does there exist an absolute
constant $C$ such that inequalities (\ref{bp-condition}) imply $|K|\le C\ |L|\ ?$ This question is known as the isomorphic Busemann-Petty problem, and in the hyperplane case it is equivalent to the slicing problem; see \cite{MP}. A version of this problem for sections of proportional dimensions was proved in \cite{K12}. \begin{theorem}\label{bp-proport}(\cite{K12}) Suppose that $0<\lambda<1,$ $k>\lambda n,$ and $K,L$ are origin-symmetric convex bodies in ${\mathbb R}^n$ satisfying the inequalities
$$|K\cap H|\le |L\cap H|,\qquad \forall H\in Gr_{n-k}.$$ Then
$$|K|^{\frac{n-k}n}\le \left(C(\lambda)\right)^k|L|^{\frac{n-k}n},$$ where $C(\lambda)$ depends only on $\lambda.$ \end{theorem} This result implies Theorem \ref{proport} in the case of volume. It is not clear, however, whether Theorem \ref{proport} can be directly used to prove Theorem \ref{bp-proport}. \smallbreak Zvavitch \cite{Zv} has found a remarkable generalization of the Busemann-Petty problem to arbitrary measures in place of volume. Suppose that $1\le k <n,$ $\mu$ is a measure with even continuous density $f$ in ${\mathbb R}^n,$ and $K$ and $L$ are origin-symmetric convex bodies in ${\mathbb R}^n$ so that \begin{equation}\label{bp-measure} \mu(K\cap \xi^\bot) \le \mu(L\cap \xi^\bot), \qquad \forall \xi\in S^{n-1}. \end{equation} Does it necessarily follow that $\mu(K)\le \mu(L)?$ The answer is the same as for volume - affirmative if $n\le 4$ and negative if $n\ge 5.$ An isomorphic version was recently proved in \cite{KZ}, namely, for every dimension $n$ inequalities (\ref{bp-measure}) imply $\mu(K)\le \sqrt{n}\ \mu(L).$ It is not known whether the constant $\sqrt{n}$ is optimal for arbitrary measures. Also there is no known direct connection between the isomorphic Busemann-Petty problem for arbitrary measures and Problem \ref{prob}.
\section{Projections of convex bodies.} The projection analog of the Busemann-Petty problem is known as Shephard's problem, posed in 1964 in \cite{Sh}. Denote by $K\vert \xi^\bot$ the orthogonal projection of $K$ to $\xi^\bot.$ Suppose that $K$ and $L$ are origin-symmetric convex bodies in
${\mathbb R}^n$ so that $|K\vert \xi^\bot|\le |L\vert \xi^\bot|$ for every $\xi\in S^{n-1}.$ Does it follow that
$|K|\le |L|?$ The problem was solved by Petty \cite{Pe} and Schneider \cite{Sch}, independently, and the answer if affirmative only in dimension 2.
Both solutions use the fact that the answer to Shephard's problem is affirmative in every dimension under the additional assumption that $L$ is a projection body. An origin symmetric convex body $L$ in ${\mathbb R}^n$ is called a {projection body} if there exists another convex body $K$ so that the support function of $L$ in every direction is equal to the volume of the hyperplane projection of $K$ to this direction: for every $\xi\in S^{n-1},$ $$
h_{L}(\xi) = |K\vert\xi^{\bot}|. $$
The support function $h_L(\xi)=\max_{x\in L} |(\xi, x)|$ is equal to the dual norm $\|\xi\|_{L^*},$ where $L^*$ denotes the polar body of $L.$
Separation in Shephard's problem was proved in \cite{K5}. \begin{theorem} \label{main-proj1} (\cite{K5}) Suppose that $\varepsilon>0$, $K$ and $L$ are origin-symmetric convex bodies in ${\mathbb R}^n,$ and $L$ is a projection body. If
$|K\vert \xi^\bot|\le |L\vert \xi^\bot| - \varepsilon$ for every $\xi\in S^{n-1},$ then
$|K|^{\frac{n-1}n} \le |L|^{\frac{n-1}n} - c_{n,1} \varepsilon,$ where $c_{n,1}$ is the same constant as in Theorem \ref{stab2}; recall that $c_{n,1}>1/\sqrt{e}.$ \end{theorem}
Stability in Shephard's problem turned out to be more difficult, and it was proved in \cite{K13} only up to a logarithmic term and under an additional assumtpion that the body $L$ is isotropic. Recall that a convex body $D$ in
${\mathbb R}^n$ is isotropic if $|D|=1$ and $\int_{D} (x,\xi)^2 dx$ is a constant function of $\xi\in S^{n-1}.$ Every convex body has a linear image that is isotropic; see [BGVV]. \begin{theorem} \label{stabproj} (\cite{K13}) Suppose that $\varepsilon>0$, $K$ and $L$ are origin-symmetric convex bodies in ${\mathbb R}^n,$ and $L$ is a projection body which is a dilate of an isotropic body.
If $|K\vert \xi^\bot|\le |L\vert \xi^\bot| + \varepsilon$ for every $\xi\in S^{n-1},$ then
$|K|^{\frac{n-1}n} \le |L|^{\frac{n-1}n} + C \varepsilon \log^2n,$ where $C$ is an absolute constant. \end{theorem} The proof is based on an estimate for the mean width of a convex body obtained by E.Milman [M3].
The projection analog of the slicing problem reads as $$
|K|^{\frac {n-1}n} \ge c \min_{\xi\in S^{n-1}} |K\vert \xi^\bot|, $$ and it was solved by Ball [Ba2], who proved that $c$ may and has to be of the order $1/\sqrt{n}.$
The possibility of extension of Shephard's problem and related stability and separation results to arbitrary measures is an open question. Also, the lower dimensional Shephard problem was solved by Goodey and Zhang [GZ], but stability and separation for the lower dimensional case have not been established.
Stability and separation for projections have an interesting application to surface area. If $L$ is a projection body, so is $L+\varepsilon B_2^n$ for every $\varepsilon>0.$ Applying stability in Shephard's problem to this pair of bodies, dividing by $\varepsilon$ and sending $\varepsilon$ to zero, one gets a hyperplane inequality for surface area (see \cite{K7}): if $L$ is a projection body, then \begin{equation}\label{surf-proj-min}
S(L)\ge c \min_{\xi\in S^{n-1}} S(L\vert \xi^\bot)\ |L|^{\frac 1n}. \end{equation} On the other hand, applying separation to any projection body $L$ which is a dilate of a body in isotropic position (see \cite{K13}) \begin{equation}\label{surf-proj-max}
S(L)\le C\log^2n \max_{\xi\in S^{n-1}} S(L\vert \xi^\bot)\ |L|^{\frac 1n}. \end{equation} Here $c$ and $C$ are absolute constants, and $S(L)$ is surface area.
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\end{document} | arXiv |
Symplectic frame bundle
In symplectic geometry, the symplectic frame bundle[1] of a given symplectic manifold $(M,\omega )\,$ is the canonical principal ${\mathrm {Sp} }(n,{\mathbb {R} })$-subbundle $\pi _{\mathbf {R} }\colon {\mathbf {R} }\to M\,$ of the tangent frame bundle $\mathrm {F} M\,$ consisting of linear frames which are symplectic with respect to $\omega \,$. In other words, an element of the symplectic frame bundle is a linear frame $u\in \mathrm {F} _{p}(M)\,$ at point $p\in M\,,$ i.e. an ordered basis $({\mathbf {e} }_{1},\dots ,{\mathbf {e} }_{n},{\mathbf {f} }_{1},\dots ,{\mathbf {f} }_{n})\,$ of tangent vectors at $p\,$ of the tangent vector space $T_{p}(M)\,$, satisfying
$\omega _{p}({\mathbf {e} }_{j},{\mathbf {e} }_{k})=\omega _{p}({\mathbf {f} }_{j},{\mathbf {f} }_{k})=0\,$ and $\omega _{p}({\mathbf {e} }_{j},{\mathbf {f} }_{k})=\delta _{jk}\,$
for $j,k=1,\dots ,n\,$. For $p\in M\,$, each fiber ${\mathbf {R} }_{p}\,$ of the principal ${\mathrm {Sp} }(n,{\mathbb {R} })$-bundle $\pi _{\mathbf {R} }\colon {\mathbf {R} }\to M\,$ is the set of all symplectic bases of $T_{p}(M)\,$.
The symplectic frame bundle $\pi _{\mathbf {R} }\colon {\mathbf {R} }\to M\,$, a subbundle of the tangent frame bundle $\mathrm {F} M\,$, is an example of reductive G-structure on the manifold $M\,$.
See also
• Metaplectic group
• Metaplectic structure
• Symplectic basis
• Symplectic structure
• Symplectic geometry
• Symplectic group
• Symplectic spinor bundle
Notes
1. Habermann, Katharina; Habermann, Lutz (2006), Introduction to Symplectic Dirac Operators, Springer-Verlag, p. 23, ISBN 978-3-540-33420-0
Books
• Habermann, Katharina; Habermann, Lutz (2006), Introduction to Symplectic Dirac Operators, Springer-Verlag, ISBN 978-3-540-33420-0
• da Silva, A.C., Lectures on Symplectic Geometry, Springer (2001). ISBN 3-540-42195-5.
• Maurice de Gosson: Symplectic Geometry and Quantum Mechanics (2006) Birkhäuser Verlag, Basel ISBN 3-7643-7574-4.
| Wikipedia |
\begin{document}
\title{Generating entanglement with low Q-factor microcavities}
\author{A.B.~Young}\email{[email protected]}
\affiliation{Merchant Venturers School of Engineering, Woodland Road Bristol, BS8 1UB}
\author{C.Y. Hu}\affiliation{Merchant Venturers School of Engineering, Woodland Road Bristol, BS8 1UB}
\author{J.G.~Rarity}\affiliation{Merchant Venturers School of Engineering, Woodland Road Bristol, BS8 1UB}\
\date{\today}
\begin{abstract} We propose a method of generating entanglement using single photons and electron spins in the regime of resonance scattering. The technique involves matching the spontaneous emission rate of the spin dipole transition in bulk dielectric to the modified rate of spontaneous emission of the dipole coupled to the fundamental mode of an optical microcavity. We call this regime resonance scattering where interference between the input photons and those scattered by the resonantly coupled dipole transition result in a reflectivity of zero. The contrast between this and the unit reflectivity when the cavity is empty allow us to perform a non demolition measurement of the spin and to non deterministically generate entanglement between photons and spins. The chief advantage of working in the regime of resonance scattering is that the required cavity quality factors are orders of magnitude lower than is required for strong coupling, or Purcell enhancement. This makes engineering a suitable cavity much easier particularly in materials such as diamond where etching high quality factor cavities remains a significant challenge. \end{abstract} \maketitle
Entanglement is a fundamental resource for quantum information tasks, and generating entanglement between different qubit systems such as photons and single electron spins has been shown to be a key to building quantum repeaters, universal gates\cite{PhysRevLett.92.127902, Yao:2004uq,waks:153601,Yao:2005fk, Barrett:2005kx, PhysRevLett.104.160503, PhysRevB.78.085307, PhysRevB.80.205326, PhysRevB.78.125318}, and eventually large scale quantum computers\cite{PhysRevA.78.032318}. These previous proposals for generating entanglement using a deterministic spin photon interface have focussed on having the optical transitions of a spin system strongly coupled to an optical microcavity, or at least deep into the Purcell regime\cite{PhysRevB.78.085307,PhysRevB.78.125318,PhysRevB.80.205326}. Recent measurements in high quality-factor micropillars have suggested that it is hard to fulfil the requirement of strong coupling whilst maintaining the necessary input output coupling efficiency\cite{Young:2011uq}. In order to work around this we propose a non-deterministic spin photon interface that works in the low Q-factor regime where efficient in/out coupling of photons should be possible. The scheme works by operating in a regime of resonance scattering where the decay constants for the optical dipole transitions in bulk dielectric are matched to the decay parameters when resonantly coupled to an optical microcavity.
\begin{figure}
\caption{Schematic diagram of a single sided cavity coupled to a dipole. $e$ and $g$ represent the excited and ground states of the dipole transition, $\kappa$ represents the coupling rate via the input output mirror and $\kappa_{s}$ represents the loss rate from the cavity system either from the side, transmission through the back mirror, or absorption.}
\label{schematic}
\end{figure}
If we consider the single sided dipole-cavity system in Fig.\ref{schematic} then the system can be parameterised by four constants, these are: $\kappa$, the decay rate for intracavity photons via the input/output mirror (outcoupling), $\kappa_{s}$, the decay rate for intra-cavity photons into loss modes, which can include losses out the side of the cavity, transmission and absorption, $g$, the dipole-cavity field coupling rate, and $\gamma$, the linewidth of the dipole transition. We may now express the photon reflectivity when incident on the input/output mirror as\cite{PhysRevB.78.085307}:
\begin{eqnarray}\label{eqn:ref}
&&r(\omega)=|r(\omega)|e^{i\phi}\\ &=&1-\frac{\kappa(i(\omega_{d}-\omega)+\frac{\gamma}{2})}{(i(\omega_{d}-\omega)+\frac{\gamma}{2})(i(\omega_{c}-\omega)+\frac{\kappa}{2}+\frac{\kappa_{s}}{2})+g^{2}}\nonumber \end{eqnarray}
\noindent where $\omega_{d}$ and $\omega_{c}$ are the frequencies of the QD and cavity, and $\omega$ is the frequency of incident photons. If we match the linewidth of the dipole transition in bulk dielectric ($\gamma$), to the modified spontaneous emission lifetime in the cavity ($4g^{2}/\kappa$)\cite{PhysRevB.60.13276}, then any photons that are input resonant to the dipole-cavity system are scattered into lossy modes. This is due to a destructive interference between the input light and light that is scattered from the dipole.
The reflectivity for an empty cavity, and reflectivity for a cavity resonantly coupled to a dipole ($\omega_{d}=\omega_{c}$) can be seen in Fig.\ref{fig:qdrscat}. Here we consider a lossless single sided cavity ($\kappa_{s}=0$), and have set $g^{2}=\gamma\kappa/4$ (the condition for resonance scattering).
\begin{figure}\label{fig:qdrscat}
\end{figure}
We can see that for the case when the dipole transition is resonantly coupled to the cavity then there is a dip in the reflectivity spectrum ($r_{d}$), that goes to zero at zero detuning ($\omega_{c}=\omega_{d}=\omega$). This dip is a result of resonance scattering and has the linewidth of the dipole transition($\gamma$), which we have set to be $\gamma=0.1\kappa$ as an upper limit where $\gamma$ is typically $<<0.1\kappa$ for most atom-cavity \cite{tu-prl-75-4710}, and quantum dot-cavity \cite{nat-432-7014, reitzenstein:251109} experiments. For the case when the cavity is empty ($r_{c}$), all of the input light is reflected. The result is a large intensity contrast between the case of a cavity resonantly coupled to a dipole and an empty cavity.
If instead of a single dipole transition we coupled a spin system to a cavity in the resonance scattering regime, then if the two dipole transitions corresponding to the $\uparrow$, $\downarrow$ states are distinguishable in some way (energy or polarisation) we can perform a quantum-non-demolition measurement of the spin\cite{Young:2009fk}. From this QND measurement it is possible to generate entanglement non-deterministically between spins and photons. We will now move on to consider some specific spin dipole systems to outline the benefits of generating this non deterministic entanglement in the resonance scattering regime.
\section{Charged quantum dot in a pillar microcavity.}
We consider the example of a charged quantum dot where the optical transitions for orthogonal spin states couple to orthogonal circular polarisation states of light. By coupling to a pillar microcavity an incident photon would obey the following set of transformations on reflection:
\begin{eqnarray} \ket{R}\otimes\ket{\uparrow}&\rightarrow& r_{d}\ket{R}\ket{\uparrow}\\ \ket{R}\otimes\ket{\downarrow}&\rightarrow& r_{c}\ket{R}\ket{\downarrow}\\ \ket{L}\otimes\ket{\uparrow}&\rightarrow& r_{c}\ket{L}\ket{\uparrow}\\ \ket{L}\otimes\ket{\downarrow}&\rightarrow& r_{d}\ket{L}\ket{\downarrow} \end{eqnarray}
\noindent Here if the input photon has right circular polarisation $\ket{R}$, and the spin is in the state $\ket{\uparrow}$ the photon sees a dipole-coupled cavity system and has a reflectivity given by $r_{d}$. Conversely if the spin is in the state $\ket{\downarrow}$ then the input photon sees an empty cavity and has a reflectivity given by $r_{c}$.
If the input photon has left-circular polarisation $\ket{L}$ then it has the opposite interaction with the spin. In the case when the electron spin of the charged QD is in a equal superposition of spin up and spin down, and two linearly polarised (horizontal) photons are sequentially reflected from the QD-cavity then the output state will be:
\begin{eqnarray} \nonumber &&\frac{1}{\sqrt{8}}[(\ket{R}_{1}+\ket{L}_{1})\otimes(\ket{R}_{2}+\ket{L}_{2})\otimes(\ket{\uparrow}+\ket{\downarrow})]\\\nonumber &&=\frac{1}{\sqrt{8}}(r_{c}^{2}\ket{R}_{1}\ket{R}_{2}+r_{d}^{2}\ket{L}_{1}\ket{L}_{2}+\\\nonumber &&r_{c}r_{d}\ket{R}_{1}\ket{L}_{2}+r_{d}r_{c}\ket{L}_{1}\ket{R}_{2})\ket{\uparrow}\\\label{eq:pent} &+&\frac{1}{\sqrt{8}}(r_{c}^{2}\ket{L}_{1}\ket{L}_{2}+r_{d}^{2}\ket{R}_{1}\ket{R}_{2}\\\nonumber &&+r_{c}r_{d}\ket{R}_{1}\ket{L}_{2}+r_{d}r_{c}\ket{L}_{1}\ket{R}_{2})\ket{\downarrow} \end{eqnarray}
\noindent Now after a Hadamard pulse ($\pi/2$) on the electron spin we have the state:
\begin{eqnarray}\nonumber &&\ket{\psi_{out}}=\frac{1}{\sqrt{8}}[(r_{c}^{2}+r_{d}^{2})(\ket{R}_{1}\ket{R}_{2}+\ket{L}_{1}\ket{L}_{2})\\ &&+2r_{c}r_{d}(\ket{R}_{1}\ket{L}_{2}+\ket{L}_{1}\ket{R}_{2})]\ket{\uparrow}\\\nonumber +&&\frac{1}{\sqrt{8}}[(r_{c}^{2}-r_{d}^{2})\ket{R}_{1}\ket{R}_{2}+(r_{d}^{2}-r_{c}^{2})\ket{L}_{1}\ket{L}_{2}]\ket{\downarrow} \end{eqnarray}
\noindent From Fig.\ref{fig:qdrscat}, we can see the terms that are proportional to $r_{d}$ will disappear, and $r_{c}=1$. If the electron spin is then measured to be "up" ($\uparrow$) with either a third photon or using the single shot readout technique outlined in previous work\cite{Young:2009fk} then the two photon state will become:
\begin{equation} \ket{\psi_{out}}=\frac{1}{\sqrt{8}}(\ket{R}_{1}\ket{R}_{2}+\ket{L}_{1}\ket{L}_{2}) \end{equation}
\noindent which is the $\ket{\psi^{+}}$ Bell state. Alternatively if the spin is measured to be down ($\downarrow$) we will project the two photons into the state:
\begin{equation} \ket{\psi_{out}}=\frac{1}{\sqrt{8}}(\ket{R}_{1}\ket{R}_{2}-\ket{L}_{1}\ket{L}_{2}) \end{equation}
\noindent which is the $\ket{\psi^{-}}$ Bell state. Thus we have generated entangled states with unit fidelity except there is a reduced efficiency of $1/4$. In order to generate larger entangled states then we simply need to reflect more photons from the system however the efficiency scales as $1/2^{n}$, which would make the scheme intractable for entangling large numbers of photons (n).
\begin{figure}\label{spinentangler}
\end{figure}
There is an analogous procedure for entangling many spins where photons can be reflected from more than one charged-QD cavity system. Consider the case as in Fig.\ref{spinentangler} where the photon is sequentially reflected from two charged QD-cavity coupled devices operating in the resonance scattering regime. The joint two spin photon state at the output will be
\begin{eqnarray} \nonumber \ket{\psi_{out}}=&&\frac{1}{\sqrt{8}}[(\ket{\uparrow}_{1}+\ket{\downarrow}_{1})\otimes(\ket{\uparrow}_{2}+\ket{\downarrow}_{2})\otimes(\ket{R}+\ket{L})]\\\nonumber &&=\frac{1}{\sqrt{8}}(r_{c_{1}}r_{c_{2}}\ket{\uparrow}_{1}\ket{\uparrow}_{2}+r_{d_{1}}r_{d_{2}}\ket{\downarrow}_{1}\ket{\downarrow}_{2}\\\nonumber &&+r_{c_{1}}r_{d_{2}}\ket{\uparrow}_{1}\ket{\downarrow}_{2}+r_{d_{1}}r_{c_{2}}\ket{\downarrow}_{1}\ket{\uparrow}_{2})\ket{R}\\\label{eq:entresscat} +&&\frac{1}{\sqrt{8}}(r_{c_{1}}r_{c_{2}}\ket{\downarrow}_{1}\ket{\downarrow}_{2}+r_{d_{1}}r_{d_{2}}\ket{\uparrow}_{1}\ket{\uparrow}_{2}\\\nonumber &&+r_{c_{1}}r_{d_{2}}\ket{\uparrow}_{1}\ket{\downarrow}_{2}+r_{d_{1}}r_{c_{2}}\ket{\downarrow}_{1}\ket{\uparrow}_{2})\ket{L} \end{eqnarray}
\noindent Where $r_{c_{1}}$, and $r_{c_{2}}$ represent the reflectivity from empty cavity for the first and second cavities respectively, and $r_{d_{1}}$ and $r_{d_{2}}$ represent the reflectivity's from dipole-coupled-cavity systems in the resonant scattering regime for the first and send cavities respectively.
Assuming $r_{c_{1}}=r_{c_{2}}=1$, and $r_{d_{1}}=r_{d_{2}}=0$, if a Hadamard is performed on the photon (i.e. using a polarising beam splitter), upon detection of a horizontally polarised photon, the spins are projected into the state:
\begin{equation} \ket{\psi_{out}}=\frac{1}{\sqrt{8}}(\ket{\uparrow}_{1}\ket{\uparrow}_{2}+\ket{\downarrow}_{1}\ket{\downarrow}_{2}) \end{equation}
\noindent Which is the $\ket{\psi^{+}}$ Bell state. Alternatively upon detection of a vertically polarised photon, the spins are projected into the state:
\begin{equation} \ket{\psi_{out}}=\frac{1}{\sqrt{8}}(\ket{\uparrow}_{1}\ket{\uparrow}_{2}-\ket{\downarrow}_{1}\ket{\downarrow}_{2}) \end{equation}
\noindent Which is the $\ket{\psi^{-}}$ Bell state. This is identical to the photonic entanglement generated above and again has an efficiency of $1/4$ associated with photon loss. The benefit of using this technique to entangle spins is that the spin entanglement is heralded upon detection of a photon, thus it is possible to use many photons and keep reflecting them until one is detected.
\subsection{Entanglement in lossy cavities}
So far to outline this procedure we have assumed that we have a perfect cavity where all the photons escape through the input-output mode ($r_{c}=1$), or are lost through the resonant scattering process, however to make the ideas presented more realistic we must consider cavity imperfections that introduce losses. We must thus include $\kappa_{s}$ in our calculation of the reflectivity. In Fig.\ref{resscatF}.a. we can see a plot of the ratio of the rate of input-output coupling to the rate of losses ($\kappa/\kappa_{s}$) plotted against the reflectivity where the charged QD-cavity is resonantly coupled, and the probe photons are resonant with both ($\omega_{d}=\omega_{c}=\omega$). In this plot the Q-factor of the cavity remains constant, i.e. the total decay rate is not changed ($(\kappa+\kappa_{s})/\kappa_{T}=1$). We have set $g=\sqrt{\kappa_{T}\gamma/4}$, and set $\gamma=0.1\kappa$
\begin{figure}\label{resscatF}
\end{figure}
\noindent Let us first consider the case of an empty cavity given by the line $r_{c}$ in Fig.\ref{resscatF}.a. (black line) Here we can see that in the regime where $\kappa>>\kappa_{s}$, the reflectivity at zero detuning ($\omega_{c}=\omega$) is $\approx1$. As $\kappa_{s}$ is increased then the reflectivity on resonance drops corresponding to more light being lost from the cavity, until the point when $\kappa_{s}=\kappa$ at which point the reflectivity on resonance drops to $r_{c}=0$, this corresponds to the cavity resonantly transmitting light into lossy modes. As $\kappa_{s}$ is increased further then the coupling into the cavity becomes poorer, until in the regime when $\kappa_{s}>>\kappa$, when the coupling via the input output mode is negligible and the cavity behaves as a conventional mirror. For a charged quantum dot where the dipole transitions are resonantly coupled to a cavity (red dashed line), in the regime that $\kappa>>\kappa_{s}$ then $r_{d}=0$. This is what we expect to observe for the case of a resonantly coupled charged QD-cavity in the resonance scattering regime where input photons destructively interfere with scattered photons, and all of the light is lost to non-cavity modes. As $\kappa_{s}$ is increased an extra damping term is added the result is the destructive interference is no longer perfect and some light is reflected. As $\kappa_{s}$ is increased further it begins to dominate and the interference becomes constructive and the reflectivity from a dipole coupled cavity system ($r_{d}$), becomes greater than that of an empty cavity ($r_{c}$). In the limit when $\kappa_{s}>>\kappa$ then no light enters the cavity thus no light is scattered by the dipole transition and we have that $r_{d}=r_{c}$.
The effect of losses on the fidelity is that the terms proportional to $r_{d}$ in Eqn.\ref{eq:entresscat} no longer disappear and the $\ket{\psi^{+}}$ entangled state is no longer prepared with unit fidelity, but instead with a reduced fidelity given by:
\begin{equation} F_{\psi^{+}}=\frac{1}{\sqrt{1+\frac{4(r_{d}r_{c})^{2}}{(r_{d}^{2}+r_{c}^{2})^{2}}}} \end{equation}
\noindent for the case when we wish to entangle two photons with one spin with an efficiency $\eta_{\psi^{+}}$ given by:
\begin{equation}\label{eq:resscateff} \eta_{\psi^{+}}=\frac{(r_{d}^{2}+r_{c}^{2})^{2}}{4} \end{equation}
\noindent For the case when we wish to entangle two spins with one photon then we have to slightly modify these equations so that fidelity is now:
\begin{equation} F_{\psi^{+}}=\frac{1}{\sqrt{1+\frac{2(r_{d_{1}}r_{c_{2}})^{2}+2(r_{c_{1}}r_{d_{2}})^{2}}{(r_{d_{1}}r_{d_{2}}+r_{c_{1}}r_{c_{2}})^{2}}}} \end{equation}
\noindent where the efficiency is now:
\begin{equation}\label{eq:resscateff2} \eta_{\psi^{+}}=\frac{(r_{d_{1}}r_{d_{2}}+r_{c_{1}}r_{c_{2}})^{2}}{4} \end{equation}
\noindent Note that the fidelity is not influenced by the two charged-QD cavity systems having non equal values of $r_{c}$ and $r_{d}$, but only by the intensity contrast at both individual dipole cavity system. This means that both systems need not be identical a great advantage when it comes to fabrication of such structures.
The preparation of the $\ket{\psi^{-}}$ state is not affected by changes in $r_{d}$, and $r_{c}$ and always has $F=1$, but has an efficiency given by:
\begin{equation} \eta_{\psi^{-}}=\frac{(r_{d}^{2}-r_{c}^{2})^{2}}{4} \end{equation}
In Fig.\ref{resscatF}.b we can see a corresponding plot for how the fidelity and efficiency is affected by changing the ratio of $\kappa/\kappa_{s}$. Note we have maintained an overall $\kappa_{T}=const$, thus the Q-factor is constant. At the point where $r_{d}=r_{c}$ ($\kappa\approx 2\kappa_{s}$) there is a minimum in fidelity for the preparation of the $\ket{\psi^{+}}$ state, as at this point the cross terms proportional to $\ket{R}_{1}\ket{L}_{2}+\ket{L}_{1}\ket{R}_{2}$ in Eqn.\ref{eq:pent} are a maximum. When $\kappa=\kappa_{s}$ the reflectivity for an empty cavity is zero ($r_{c}=0$), therefore there is a peak in the fidelity and $F=1$, however the since $r_{d}\approx0.5$ the efficiency is low ($\eta\approx0.016$). As we move into the region where $\kappa<\kappa_{s}$ then both $r_{d}$ and $r_{c}$ increase and the efficiency increases, $r_{c}$ increases faster than $r_{d}$, until the limit when $\kappa<<\kappa_{s}$ where $r_{d}=r_{c}=1$ and $\eta=1$. However in this regime there is a minimum in fidelity ($F=1/\sqrt{2}$) for the preparation of the $\ket{\psi^{+}}$ state, again due to the two reflectivities being equal. Note the fidelity for the preparation of the $\ket{\psi^{-}}$ state remains $F=1$, however in a cavity with a large ratio of leaks to input-output coupling the efficiency $\eta_{\psi^{-}}$ drops to zero.
In order to achieve entanglement with the highest possible efficiency and fidelity for both $\ket{\psi^{+}}$ and $\ket{\psi^{-}}$ states it is necessary to have $\kappa>>\kappa_{s}$. This requirement at first sight is no different to the requirement for the deterministic spin photon-interface outlined in previous work\cite{PhysRevB.78.085307,PhysRevB.78.125318,PhysRevB.80.205326,waks:153601}. So seemingly the non-deterministic scheme outlined offers no advantage, however the required cavity Q-factors are significantly less. In order to see some of the benefits of entanglement generation using resonance scattering it is necessary to consider in more detail some experimental parameters.
We consider some of the state of the art QD pillar microcavity experiments performed by Reithmaier et. al (2004)\cite{nat-432-7014}. Here they showed strong coupling of a QD to a pillar microcavity where the QD-cavity coupling rate $g=80\mu$eV, the cavity linewidth was $\kappa_{T}=180\mu$ev (Q=7350), and the QD linewidth was $\gamma<10\mu$eV at low temperature. If we now assume the maximum value for the QD linewidth ($\gamma=10\mu$eV), then the required cavity linewidth in order to fulfil the requirement for resonance scattering is $\kappa_{T}=2.56$meV (Q=517). This a significantly smaller value than would be required for a deterministic spin-photon interface in previous work\cite{PhysRevLett.92.127902, PhysRevB.78.085307} where we would require $g>\kappa_{T}+\gamma$, meaning $\kappa_{T}=70\mu$ev (Q=18900). With the reduced Q-factor that is required to generate entanglement with resonance scattering, comes a secondary crucial benefit. The state of the art micropillars used in the experiment above and most high-Q micropillars, are limited by losses. Small diameter high Q micropillars have significant sidewall scattering and operate in the regime where $\kappa<\kappa_{s}$. Assuming the linewidth of the pillar is entirely defined by losses out the side $\kappa_{T}=\kappa_{s}=180\mu$eV. The Q-factor can then be reduced by removing, or growing fewer DBR mirror pairs. This will increase $\kappa$ whilst $\kappa_{s}$ should remain constant. Reducing the Q-factor in such a way so that $Q=517$, would result in $\kappa=2.38$meV, and $\kappa_{s}=180\mu$eV, thus have $\kappa/\kappa_{s}=13$. So by reducing the Q-factor we simultaneously increase the input-output coupling rate and move into a regime where the losses out of the side of the pillar become negligible. This means that we can entangle two spins or two photons using charged QD's coupled to such cavities, with fidelity $F>99\%$ in both the $\ket{\psi^{+}}$ and $\ket{\psi^{-}}$ states, with an efficiency $\eta=0.14$.
We have already discussed that this scheme is best employed when used to herald entanglement between many spins. Assuming perfect detection it would be necessary to send in $\approx10$ photons to ensure one was detected heralding the entanglement of two spins. The photons would have to be separated by a time greater than the spontaneous emission lifetime of the QD $\approx1ns$, so it would take approximately $10ns$ to entangle two spins. Pairs of spins could be entangled in parallel and then entanglement could be generated between pairs by repeating the process between single spins from each pair. Hence a linear cluster of N spins could be entangled in $\approx 20ns$, well within the $\mu$s coherence time of a charged QD spin. By parallelising the entanglement procedure we compensate for the non-deterministic nature of generating entanglement using resonance scattering at the expense of the complexity of the photon source required to perform the experiment.
The advantage of the non-deterministic scheme for generating entanglement is that clearly the required Q-factor is low. A knock on effect is that low Q-micropillars naturally have good input-output coupling efficiency and it is easy to achieve $\kappa>>\kappa_{s}$. To realise the spin-photon interface in the strong coupling regime requires high Q-factor low loss pillars which are much more challenging to fabricate. Further the low Q-factor means the spectral width of the cavity is large compared to the linewidth of the dipole transitions $\gamma$. This means that charged QD's in different micropillar samples have a larger range over which they can be tuned and still be resonantly coupled to the microcavity meaning it will be easier to realise the situation where both dipole transitions are at the same wavelength. Finally the low Q-factor will lessen the effects of any ellipticity or mode splitting in the cavity. Since the linewidth of the $E_{x}$ and $E_{y}$ modes will be large then any mode-spilitting as a result of fabrication error would be small in comparison
The downside to operating in the regime of resonance scattering is that the charged QD-pillar system has to be engineered so that $g^2=\kappa_{T}\gamma/4$. Since the position of self-assembled QD's is random, fulfilling this requirement will be difficult, and may require the growth of site controlled QD's with pillars etched out of the wafer around them. This is not a problem for the spin-photon interface in the strong coupling regime where the coupling rate $g$ just has to be above the threshold where $g>\kappa_{T}+\gamma$, but not have a specific value. Hence operating in the resonance scattering regime changes the nature of the engineering problem. It is easy to achieve a low loss micropillar, but it will be difficult to precisely control the structure to meet the condition for resonance scattering. One possible system that would lend itself to this sort of technique could be toroidal,or microsphere cavities where the Q-factor can effectively be tuned by changing the distance between the cavity and an evanescently coupled tapered fiber. It remains to be seen if the realisation of the structures required for this non-deterministic entanglement scheme will be any easier than the structures required for deterministic spin photon interface in the strong coupling regime.
\section{application to NV center in diamond}
\begin{figure}
\caption{{\bf a.} Energy Level diagram for $NV^{-}$ colour center in diamond showing the ground state is splitting\cite{fedor,loubser}. {\bf b.} Schematic diagram of a Barrett and Kok\cite{Barrett:2005kx} style scheme to entangle two NV centres coupled to optical microcavities in the resonance scattering regime. Photons 1 and 2 have energy corresponding to transition 1 ($\hbar\omega_{m=0}$) and are reflected from cavities 1 and 2 respectively and then interfered on a 50:50 beamsplitter.}
\label{nvscheme}
\end{figure}
The entanglement protocol outlined here for charged QD-spins, could be applied to other spin systems for example the NV-center in a photonic crystal\cite{Young:2009fk}. Here distinguishing between the two spin states can be achieved with frequency instead of polarisation. If photons were passed through an electro-optical modulator then they can be placed in a superposition of two distinct frequencies $A$ and $B$. Frequency $A$ can then be tuned to be resonant with the $^{3}A_{(m=0)} \rightarrow ^{3}E$ transition (transition (1) Fig.\ref{nvscheme}.a.), and frequency $B$ resonant with the $^{3}A_{(m=\pm1)} \rightarrow^{3}E$ transition (transition (2) Fig.\ref{nvscheme}.a). Since the linewidth of the zero phonon line at low temperature is of order MHz\cite{0953-8984-18-21-S08} then there will be two dips in the reflectivity as a result of resonance scattering corresponding to the $m=0$ and $m=\pm1$ spin states of order MHz spilt by $\approx 2.88$GHz. The distinguishability of these two dips allows us to perform a quantum non-demolition measurement of the spin\cite{Young:2009fk}, and generate entanglement using precisely the same protocol as outlined for the case of a charged quantum dot using photons in a superposition of frequency instead of polarisation.
Recent results\cite{Togan:2010fk} have also shown that the $m=\pm1$ spin states can be used as a qubit and orthogonal circular polarisations of light then couple the ground states to an excited state $A_{2}$. In this instance the resonant scattering protocol outlined for the charged QD could be directly applied to a NV-center coupled to an appropriate optical microcavity.
An alternative method to generating entanglement in this regime that is perhaps simpler for the case of the NV-center is to only use photons with frequency $\omega_{m=0}$ that are resonant with transition 1 in Fig.\ref{nvscheme}.a. In Fig.\ref{nvscheme}.b. we can see a schematic diagram of how this could work. We can take two photons 1 and 2 that are both resonant with the spin preserving transition 1, and reflect them from two cavity systems 1 and 2 that are both coupled to an NV-center in the resonance scattering regime. After the two photons are reflected they are then interfered on a 50:50 beamsplitter. The entanglement would then be generated using the exact same protocol as outlined by Barrett and Kok\cite{Barrett:2005kx}, which could lead to the formation of large cluster states. One benefit of realising this type of scheme using a resonance scattering technique is that we do not need to use photons that are generated via spontaneous emission from spin in the cavity, and can use some external source, in fact photon 1 and photon 2 can be produced from the same source. This means it should be easier to ensure that the two photons are indistinguishable, which remains a challenge\cite{Bernien:2012fk}, thus effectively removing a decoherence channel from the existing Barrett and Kok protocol. Further to produce indistinguishable phonons via spontaneous emission would require the photons produced to be transform limited. This would require some Purcell enhancement thus $g^{2}>\kappa_{T}\gamma/4$ hence the Q-factor required would need to be higher. Note that this technique is also possible for other spin cavity systems for example the charged QD system examined earlier where we would just set photons 1 and 2 to have the same circular polarisation.
Finally for illustrative purposes we can consider coupling a nitrogen vacancy centre to a photonic crystal cavity with current state of the art fabrication techniques. Recent results have shown\cite{Riedrich-Moller:2012kx} the fabrication of photonic crystals in diamond with Q-factor of $\approx700$ and a mode volume of $\approx 0.13\mu$m$^{3}$. Using this mode volume and given a typical oscillator strength for the ground to excited state triplet transitions of $f\approx0.12$\cite{nat_phs_2_408}, then we can calculate the dipole-cavity coupling rate to be $g\approx13.5\mu$eV. Since the zero phonon linewidth at low temperature is $\gamma\approx0.1\mu$eV\cite{0953-8984-16-30-R03} then the Q-factor required to meet the resonance scattering condition in such a structure would be $Q\approx256$ nearly three times smaller than has already been experimentally realised. So provided the input/output coupling rate $\kappa$ can be made much larger than the loss rate $\kappa_{s}$ then current experimentally realised structures in diamond would be suitable for generating entanglement using resonant scattering techniques.
\section{Summary}
In Summary we have shown a way to non-deterministically generate entanglement between electron spins and photons. We have shown how this can be applied to charged QD-spins, and nitrogen vacancy centers coupled to optical microcavities. The idea uses resonance scattering where orthogonal photon states are scattered and lost depending on the internal spin state of the electron spin. The advantage to this scheme is that it requires low Q micropillars where the input-output coupling rate is intrinsically high. The disadvantage is the non-deterministic nature makes scaling difficult compared to the spin-photon interface in the strong coupling regime.
\end{document} | arXiv |
\begin{document}
\title{Table of Contents}
\newtheorem{thm}{Theorem}[section]
\newtheorem{hyp}[thm]{Hypothesis}
\newtheorem{hyps}[thm]{Hypotheses}
\newtheorem{rems}[thm]{Remarks}
\newtheorem{conjecture}[thm]{Conjecture} \newtheorem{theorem}[thm]{Theorem} \newtheorem{theorem a}[thm]{Theorem A} \newtheorem{example}[thm]{Example} \newtheorem{examples}[thm]{Examples} \newtheorem{corollary}[thm]{Corollary}
\newtheorem{rem}[thm]{Remark} \newtheorem{lemma}[thm]{Lemma} \newtheorem{sublemma}[thm]{Sublemma} \newtheorem{cor}[thm]{Corollary} \newtheorem{proposition}[thm]{Proposition}
\newtheorem{exs}[thm]{Examples} \newtheorem{ex}[thm]{Example}
\newtheorem{exercise}[thm]{Exercise}
\numberwithin{equation}{section} \setcounter{part}{0}
\newcommand{\rightarrow}{\rightarrow} \newcommand{\longrightarrow}{\longrightarrow} \newcommand{\longleftarrow}{\longleftarrow} \newcommand{\Rightarrow}{\Rightarrow} \newcommand{\Leftarrow}{\Leftarrow}
\newtheorem{Thm}{Main Theorem}
\newtheorem*{thm*}{Theorem} \newtheorem{lem}[thm]{Lemma} \newtheorem{fig}[thm]{Figure} \newtheorem*{lem*}{Lemma}
\newtheorem*{prop*}{Proposition}
\newtheorem*{cor*}{Corollary} \newtheorem{dfn}[thm]{Definition} \newtheorem*{defn*}{Definition} \newtheorem{notadefn}[thm]{Notation and Definition} \newtheorem*{notadefn*}{Notation and Definition} \newtheorem{nota}[thm]{Notation} \newtheorem*{nota*}{Notation} \newtheorem{note}[thm]{Remark}
\newtheorem*{note*}{Remark} \newtheorem*{notes*}{Remarks} \newtheorem{hypo}[thm]{Hypothesis}
\newtheorem*{ex*}{Example} \newtheorem{prob}[thm]{Problems} \newtheorem{conj}[thm]{Conjecture}
\title{\v Sapovalov elements and the Jantzen sum formula for contragredient Lie superalgebras.} \author{Ian M. Musson\footnote{Research partly supported by NSA Grant H98230-12-1-0249, and Simons Foundation grant 318264.} \\Department of Mathematical Sciences\\ University of Wisconsin-Milwaukee\\ email: {\tt [email protected]}} \maketitle \begin{abstract} If $\fg$ is a contragredient Lie superalgebra and $\gamma$ is a root of $\fg,$ we prove the existence and uniqueness of \v Sapovalov elements for $\gamma$ and give upper bounds on the degrees of their coefficients. Then we use \v Sapovalov elements
to define some new highest weight modules.
If $X$ is a set of orthogonal isotropic roots and $\lambda \in \mathfrak{h}^*$ is such that $\lambda +\rho$ is orthogonal to all roots in $X$, we construct a highest weight module $M^X(\lambda)$ with character $\mathtt{e}^\lambda{p}_X$. Here $p_X$ is a partition function that counts partitions not involving roots in $X$. Examples of such modules can be constructed via parabolic induction provided $X$ is contained in the set of simple roots of some Borel subalgebra. However our construction works without this condition and provides a highest weight module for the distinguished Borel subalgebra. The main results are analogs of the \v Sapovalov determinant and the Jantzen sum formula for $M^X(\lambda)$ when $\fg$ has type A.
We also explore the behavior of \v Sapovalov elements when the Borel subalgebra is changed, relations between \v Sapovalov elements for different roots, and the survival of \v{S}apovalov elements in factor modules of Verma modules. In type A we give a closed formula for \v Sapovalov elements and give a new approach to results of Carter and Lusztig \cite{CL}.
For the proof of the main results it is enough to study the behavior for certain ``relatively general" highest weights. Using an equivalence of categories due to Cheng, Mazorchuk and Wang \cite{CMW}, the information we require is deduced from the behavior of the modules $M^X(\lambda)$ when $\fg=\fgl(2,1)$ or $\fgl(2,2)$. These low dimensional cases are studied in detail in an appendix. \end{abstract}
\noindent \ref{uv.1}. {Introduction.} \\ \ref{1s.1}. {Uniqueness of \v Sapovalov elements.} \\ \ref{1s.5}. {Proof of Theorems \ref{1Shap} and \ref{1aShap}.}\\ \ref{1sscbs}. {Changing the Borel subalgebra.} \\
\ref{RS}. {Relations between \v Sapovalov elements.} \\%7\ \ref{jaf}. {Highest weight modules with prescribed characters.}\\%9 \ref{SV}. {The submodule structure of Verma modules.} \\
\ref{1cosp}. {An (ortho) symplectic example.} \\ \ref{1s.8}. {The Type A Case.} \\ \ref{1surv}. {{Survival of \v{S}}apovalov elements in factor modules.}\\ \noindent \ref{sf}. {The Jantzen sum formula. }\\ \noindent {Appendix \ref{pip}: Anti-distinguished Borel subalgebras.}\\ \noindent {Appendix \ref{ldc}: Low Dimensional Cases}.
\section{Introduction.} \label{uv.1} Throughout this paper we work over an algebraically closed field $\mathtt{k} $
of characteristic zero. If $\fg$ is a simple Lie algebra necessary and sufficient conditions for the existence of a non-zero homomorphism from $M(\mu)$ to $M(\lambda)$ can be obtained by combining work of Verma \cite{Ve} with work of Bernstein, Gelfand and Gelfand \cite{BGG1}, \cite{BGG2}. Such maps can be described explicitly in terms of certain elements introduced by N.N.\v Sapovalov in \cite{Sh}. Verma modules are fundamental objects in the study of category $\mathcal{O}$, a study that has blossomed into an extremely rich theory in the years since these early papers appeared. We refer to the
book by Humphreys \cite{H2} for a survey. \\ \\ Significant advances have been made in the study of finite dimensional modules, and more generally modules in the category $\mathcal{O}$ for classical simple Lie superalgebras using a variety of techniques. After the early work of Kac \cite{K}, \cite{Kac3} the first major advance was made by Serganova who used geometric techniques to obtain a character formula for finite dimensional simple modules over $\fgl(m,n)$, \cite{S2}. The next development was Brundan's approach to the same problem using a combination of algebraic and combinatorial techniques \cite{Br}. For more recent developments concerning the category $\mathcal{O}$ for Lie superalgebras, see the survey article \cite{B} by Brundan and the book \cite{CW} by Cheng and Wang.
\\ \\The \v Sapovalov determinant, also introduced in \cite{Sh} has been developed in a variety of contexts, such as Kac-Moody algebras \cite{KK}, quantum groups \cite{Jo1}, generalized Verma modules \cite{KM}, and Lie superalgebras \cite{G4}, \cite{G}, \cite{G2}. The factorization of the \v Sapovalov determinant is the key ingredient in the proof of the original Jantzen sum formula, \cite{J1} Satz 5.3, see also \cite{KK}, \cite{MP} Corollary to Theorem 6.6.1 for the Kac-Moody case. \\ \\ \v Sapovalov elements have also appeared in a number of situations in representation theory. Though not given this name, they appear in the work of Carter and Lusztig \cite{CL}. Indeed determinants similar to those in our Theorem \ref{1shgl2} were introduced in \cite{CL} Equation (5), and our Corollary \ref{11.5} may be viewed as a version of \cite{CL} Theorem 2.7. Carter and Lusztig use their result to study tensor powers of the defining representation of $GL(V)$, and homomorphisms between Weyl modules in positive characteristic, see also \cite{CP} and \cite{F}. Later Carter \cite{Car} used \v Sapovalov elements to construct raising and lowering operators for $\fsl(n,{\mathbb C})$, see also \cite{Br3}, \cite{Carlin}. In \cite{Car}, these operators are used to construct orthogonal bases for non-integral Verma modules, and all finite dimensional modules for $\fsl(n,{\mathbb C})$. \\ \\ More recently Kumar and Letzter gave degrees on the coefficients of \v Sapovalov elements. In the Lie algebra case, our Theorem \ref{1Shap} is roughly equivalent to \cite{KL} Propositions 5.2 and 5.6. Kumar and Letzter use their result to obtain a new proof of the irreducibility of the Steinberg module for restricted enveloping algebras and their quantized cousins. They also apply their results in these cases to derive versions of the Jantzen sum formula originally obtained by Andersen, Jantzen and Soergel, \cite{AJS} Proposition 6.6. The Strong Linkage Principle may be deduced from this sum formula. \\ \\ The purpose of this paper is to initiate the study of these closely related topics in the super case. New phenomena arise due to the presence of isotropic roots. We note that in order to pass to positive characteristic, the results in \cite{CL} and \cite{KL} are formulated using the Kostant ${\mathbb Z}$-form of $U(\fg).$ It is easily seen that our main results can be so formulated. However we do not go in that direction. \\ \\ I would like to thank Jon Brundan for suggesting the use of noncommutative determinants to write \v Sapovalov elements for $\fgl(m,n)$ in Section \ref{1s.8}, and raising the possibility of using Theorem \ref{1shgl} to prove Theorem \ref{1shapel}. I also thank Kevin Coulembier, Volodymyr Mazorchuk and Vera Serganova for some helpful correspondence. \subsection{\bf Preliminaries.} \label{sss7.1} \noindent Before we state the main results, we introduce some notation. Throughout the paper $[n]$ denotes the set of integers $\{1,2,\ldots,n\}$. Let $\fg= \fg(A,\tau)$ be a finite dimensional contragredient Lie superalgebra with Cartan subalgebra $\mathfrak{h}$, and set of simple roots $\Pi$.
The superalgebras $\fg(A,\tau)$ coincide with the basic classical simple Lie superalgebras, except that instead of $\fpsl(n,n)$ we obtain $\fgl(n,n).$
Let $\Delta^{+}$ and \begin{equation} \label{gtri}\mathfrak{g} = \mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}^+\end{equation} be the set of positive roots containing $\Pi$, and the corresponding triangular decomposition of $\fg$ respectively. We use the Borel subalgebras $\mathfrak{b} = \mathfrak{h} \oplus \mathfrak{n}^+$ and $\mathfrak{b}^- = \mathfrak{n}^- \oplus \mathfrak{h}$.
The Verma module $M(\lambda)$ with highest weight $\lambda \in \mathfrak{h}^*$, and highest weight vector $v_\lambda$ is induced from $\mathfrak{b}$. Denote the unique simple factor of $M(\lambda)$ by $L(\lambda)$. \\ \\ We use the definition of partitions from \cite{M} Remark 8.4.3. Set $Q^+=\sum_{\alpha\in \Pi} {\mathbb N}\alpha$. If $\eta \in Q^+$, a {\it partition} of $\eta$ is a map $\pi: \Delta^+ \longrightarrow \mathbb{N} $ such that $\pi(\alpha) = 0$ or $1$ for all isotropic roots $\alpha$,
$\pi(\alpha) = 0$ for all even roots $\alpha$ such that $\alpha/2$ is a root, and
\begin{equation} \label{suma}\sum_{\alpha \in {{\Delta^+}}} \pi(\alpha)\alpha = \eta.\end{equation} For $\eta \in Q^+$, we denote by $\bf{\overline{P}(\eta)}$ the set of partitions of $\eta$. (Unlike the Lie algebra case, there can be $\eta \in Q^+$ for which $\bf{\overline{P}(\eta)}$ is empty). If $\pi \in \bf{\overline{P}(\eta)}$ the {\it degree} of $\pi$ is defined to be $|\pi| = \sum_{\alpha \in \Delta^+} \pi(\alpha).$ If $\sigma, \pi$ are partitions we say that $\sigma + \pi$ is a partition if $\sigma(\alpha)+\pi(\alpha) \le 1$ for all isotropic roots $\alpha$. In this case $\sigma + \pi$ is defined by $(\sigma + \pi)(\gamma) = \sigma(\gamma)+\pi(\gamma) $ for all positive roots $\gamma$. \\ \\ Next we introduce generating functions for certain kinds of partitions. If $X$ is any set of positive roots and $\eta\in \Delta^+$, set
\[{\bf \overline{P}}_{X}(\eta) = \{\pi \in {\bf \overline{P}}(\eta) | \pi(\alpha) = 0 \mbox{ for all } \alpha \in X\}.\]
and ${\bf p}_{X}(\eta) =|{\bf \overline{P}}_{X}(\eta)|$. Usually $X$ will be a set of pairwise orthogonal isotropic roots, but in the Jantzen sum formula $X$ will sometimes include even roots, see \eqref{yew5}. Let $X_0$ (resp. $X_1$) be the set of even (resp. odd) roots contained in $X$, and set $p_X = \sum {\bf p}_X(\eta)\mathtt{e}^{-\eta}$. Then \begin{equation} \label{pfun} p_X = \prod_{\alpha \in \Delta^+_{1}\backslash X_1} (1 + \mathtt{e}^{- \alpha})/ \prod_{\alpha \in \Delta^+_{0}\backslash X_0} (1 - \mathtt{e}^{- \alpha}).\end{equation} If $X$ is empty, set $p = p_X$, and if $X =\{\gamma\}$ is a singleton write
\begin{equation} \label{sing}{\bf \overline{P}}_{\gamma}(\eta),\quad{\bf p}_{\gamma}(\eta),\; \mbox{ and } \;p_\gamma \end{equation} instead of ${\bf \overline{P}}_{X}(\eta), {\bf p}_{X}(\eta),\mbox{ and } p_X.$ In Section \ref{jaf} we also use the usual definition of a partition. Thus for $\eta \in Q^+$, let ${\bf P}_X(\eta)$ denote the set of all functions $\pi:\Delta^+\longrightarrow {\mathbb N}$ such that \eqref{suma} holds, $\pi(\alpha) = 0$ or 1 for $\alpha$ an odd root and $\pi({\alpha}) =0$ for $\alpha\in X$. The main difference between the two notations arises in the definition of the elements $e_{-\pi}$ in \eqref{negpar} below, see \cite{M} Example 8.4.5 for the case where $\fg=\osp(1,2)$. \noindent We denote the usual BGG category $\mathcal{O}$ of $\fg_0$ modules here by $\mathcal{O}_0$ and reserve $\mathcal{O}$ for the category of ${\mathbb Z}_2$-graded $\fg$-modules which are objects of $\mathcal{O}_0$ when regarded as $\fg_0$-modules. Morphisms in $\mathcal{O}$ preserve the grading. For a module $M$ in the category $\mathcal{O}_0$, the character of $M$ is defined by ${\operatorname{ch}\:} M= \sum_{\eta \in \mathfrak{h}^*}\dim_\mathtt{k} M^{\eta}\mathtt{e}^\eta.$ Recall that the Verma module $M(\lambda)$ has character $\mathtt{e}^\lambda p.$ \\ \\ \noindent Fix a non-degenerate invariant symmetric bilinear form $(\;,\;)$ on $\mathfrak{h}^*$, and for all $\alpha \in \mathfrak{h}^*$, let $h_\alpha \in \mathfrak{h}$ be the unique element such that $(\alpha,\beta) = \beta(h_\alpha)$ for all $\beta \in \mathfrak{h}^*$. Then for all $\alpha \in \Delta^+$, choose elements $e_{\pm \alpha} \in \mathfrak{g}^{\pm \alpha}$
such that
\[ [e_{\alpha}, e_{-\alpha}] = h_{\alpha}.\] It follows that if $v_\mu$ is a highest weight vector of weight $\mu$ then \begin{equation} \label{ebl} e_\alpha e_{-\alpha} v_\mu = h_\alpha v_\mu =(\mu, \alpha)v_\mu . \end{equation} \noindent \noindent Fix an order on the set $\Delta^+$, and for $\pi$ a partition, set
\begin{equation} \label{negpar} e_{-\pi} = \prod_{\alpha \in \Delta^+} e^{\pi (\alpha)}_{-\alpha},\end{equation} the product being taken with respect to this order. In addition set \begin{equation} \label{pospar}e_\pi = \prod_{\alpha \in \Delta^+} e^{\pi(\alpha)}_\alpha,\end{equation} where the product is taken in the opposite order. The elements $e_{- \pi},$ with $\pi \in \bf{\overline{P}}(\eta)$ form a basis of $U(\mathfrak{n}^-)^{- \eta},$ \cite{M} Lemma 8.4.1. \\ \\
\noindent For a non-isotropic root $\alpha,$ we set $\alpha^\vee = 2\alpha / (\alpha, \alpha)$, and denote the reflection corresponding to $\alpha$ by $ s_\alpha.$ As usual the Weyl group $W$ is the subgroup of $GL({\mathfrak{h}}^*)$ generated by all reflections. For $u \in W$ set
\begin{equation} \label{nu} N(u) = \{ \alpha \in \Delta_0^+ | u \alpha < 0 \},\qquad \ell(u) = |N(u)|.\end{equation} We use the following well-known fact several times. \begin{lemma} If $w = s_\alpha u$ with $\ell(w)>\ell(u)$ and $\alpha$ is a isotropic root which is simple for $\fg_0$, then we have a disjoint union \begin{eqnarray} \label{Nw} N(w^{-1}) = s_\alpha N(u^{-1}) \stackrel{\cdot}{\cup} \{\alpha\}. \end{eqnarray} \end{lemma} \begin{proof} See for example \cite{H3} Chapter 1.\end{proof} \noindent Set \begin{equation}\label{rde}\rho_0=\frac{1}{2}\sum_{\alpha \in \Delta_0^+}\alpha , \quad \rho_1=\frac{1}{2}\sum_{\alpha \in \Delta_1^+}\alpha ,\quad \rho= \rho_0 -\rho_1.\end{equation} Usually we work with a fixed Borel subalgebra $\mathfrak{b}$ which we take to be either the distinguished Borel from \cite{K} Table VI, or an anti-distinguished Borel, defined in Appendix \ref{pip}. If necessary to emphasize the role played by the Borel subalgebra, we write $\rho_1 = \rho_1({\mathfrak{b}})$ and $\rho=\rho({\mathfrak{b}}).$ Note that $\mathfrak{b}_0$ is fixed throughout. A finite dimensional contragredient Lie superalgebra $\fg$ has, in general several conjugacy classes of Borel subalgebras, and this both complicates and enriches the representation theory of $\fg$. The complications are partially resolved by at first fixing a Borel subalgebra (or equivalently a basis of simple roots for $\fg$) with special properties. Then in Section \ref{1sscbs} we study the effect of changing the Borel subalgebra.
\subsection{Main Themes.} \label{s.2} {\bf A. Coefficients of \v Sapovalov elements.}\\ \\ Fix a positive root $\gamma$ and a positive integer $m$. If $\gamma$ is isotropic, assume $m=1$, and if $\gamma$ is odd non-isotropic, assume that $m$ is odd. There are two special partitions of $m\gamma$. Let $\pi^0 \in {\overline{\bf P}}(m\gamma)$ be the unique partition of $m\gamma$ such that $\pi^0(\alpha) = 0$ if $\alpha \in \Delta^+ \backslash \Pi.$ The partition $m\pi^{\gamma}$ of $m\gamma$ is given by $m\pi^{\gamma}(\gamma)=m,$ and $m\pi^{\gamma}(\alpha)= 0$ for all positive roots $\alpha$ different from $\gamma.$ We say that $\theta = \theta_{\gamma,m}\in U({\mathfrak b}^{-} )^{- m\gamma}$ is a {\it \v Sapovalov element for the pair} $(\gamma,m)$ if it has the form has the form \begin{equation} \label{rat} \theta = \sum_{\pi \in {\overline{\bf P}}(m\gamma)} e_{-\pi} H_{\pi},\end{equation} where $H_{\pi} \in U({\mathfrak h})$, $H_{\pi^0} = 1,$ and \begin{equation} \label{boo} e_{\alpha} \theta \in U({\mathfrak g})(h_{\gamma} + \rho(h_{\gamma})-m(\gamma,\gamma)/2)+U({\mathfrak g}){\mathfrak n}^+ , \; \rm{ for \; all }\;\alpha \in \Delta^+. \end{equation} We call the $H_\pi$ in \eqref{rat} the {\it coefficients} of $\theta$. For a semisimple Lie algebra, the existence of such elements was shown by \v Sapovalov, \cite{Sh} Lemma 1. The \v Sapovalov element $\theta_{\gamma,m}$ has the important property that if $\lambda$ lies on a certain hyperplane then
$\theta_{\gamma, m}v_{\lambda}$ is a highest weight vector in $M(\lambda).$ Indeed set \begin{equation} \label{vat}{\mathcal H}_{\gamma, m} = \{ \lambda \in {\mathfrak h}^*|(\lambda + \rho, \gamma) = m(\gamma, \gamma)/2 \}.\end{equation} Then $\theta_{\gamma, m} v_\lambda$ is a highest weight vector of weight $\lambda -m\gamma$ in $M(\lambda)$ for all $\lambda \in \mathcal H_{\gamma, m}$. The normalization condition $H_{\pi^0}=1$ guarantees that $\theta_{\gamma, m}v_\lambda$ is never zero. If $\gamma$ is isotropic, to simplify notation we set $\mathcal{H}_{\gamma} = \mathcal{H}_{\gamma,1},$ and denote a \v Sapovalov element for the pair $(\gamma, 1)$ by $\theta_{\gamma}$. When $X$ is an orthogonal set of isotropic roots, we set $\mathcal{H}_X= \bigcap_{\gamma\in X} \mathcal{H}_\gamma$. \\ \\
We give bounds on the degrees of the coefficients $H_{\pi}$ in (\ref{rat}). There is always a unique coefficient of highest degree, and we determine the leading term of this coefficient up to a scalar multiple. These results appear to be new even for simple Lie algebras. The exact form of the coefficients depends on the way the positive roots are ordered. Suppose $\Pi_{\rm nonisotropic},$ (resp. $\Pi_{\rm even}$) is the set of nonisotropic (resp. even) simple roots, and let $W_{\rm nonisotropic}$ (resp. $W_{\rm even}$) be the subgroup of $W$ generated by the reflections $s_\alpha,$ where $\alpha \in \Pi_{\rm nonisotropic}$ (resp. $\alpha \in \Pi_{\rm even}$). \footnote{\rm In general we have \begin{equation} \label{gag1} W_{\rm even} \subseteq W_{\rm nonisotropic}\subseteq W. \end{equation} Suppose we use the distinguished set of simple roots. Then $ W_{\rm even} =W_{\rm nonisotropic}$ if and only if $\fg\neq\osp(1,2n)$, and $W_{\rm nonisotropic} =W$ if and only if $\fg$ has type $A, C$ or $B(0,n).$
} Consider the following hypotheses. \begin{equation} \label{i} \mbox{ The set of simple roots of } \Pi \mbox{ is either distinguished or anti-distinguished.} \end{equation} \begin{equation} \label{iii} \gamma = w\beta \mbox{ for a simple root } \beta \mbox{ and } w \in W_{\rm even}.\end{equation} \begin{equation} \label{ii}
\gamma = w\beta \mbox{ for a simple root } \beta \mbox{ and } w \in W_{\rm nonisotropic}.\end{equation} When (\ref{i}) and either of (\ref{iii}) or (\ref{ii}) holds we always assume that $\ell(w)$ is minimal, and for $\alpha \in N(w^{-1}),$ we define $q(w,\alpha) = (w\beta, \alpha^\vee).$ If $\Pi = \{\alpha_i|i = 1, \ldots, t \}$ is the set of simple roots, and $\gamma = \sum_{i=1}^t a_i\alpha_i,$ then the {\it height} ${\operatorname{ht}} \gamma$ of $\gamma$ is defined to be ${\operatorname{ht}} \gamma = \sum_{i=1}^t a_i$. Let ${\mathcal I}(\mathcal H_{\gamma, m})$ be the ideal of $S(\mathfrak{h})$ consisting of functions vanishing on $\mathcal{H}_{\gamma,m}$, and $\mathcal{O} (\mathcal H_{\gamma, m})= S(\mathfrak{h})/{\mathcal I}(\mathcal H_{\gamma, m}).$
\begin{theorem} \nonumber\label{1Shap} Suppose $\fg$ is semisimple or a contragredient Lie superalgebra and \begin{itemize} \item[{{\rm(a)}}] $\gamma$ is a positive root such that $(\ref{i})$ and $(\ref{iii})$ hold. \item[{{\rm(b)}}] If $\gamma$ is isotropic assume that $m=1.$ \item[{{\rm(c)}}] If $\fg= G_2$ or $G(3)$ assume that $\gamma$ is not the highest short root of $\fg_0$.\end{itemize} Then there exists a \v Sapovalov element $\theta_{\gamma,m} \in U({\mathfrak b}^{-} )^{- m\gamma}$, which is unique modulo the left ideal $U({\mathfrak b}^{-} ){\mathcal I}(\mathcal H_{\gamma,m})$. The coefficients of $\theta_{\gamma, m}$ satisfy
\begin{equation} \label{x1} |\pi|+\deg H_{\pi} \le m{\operatorname{ht}} \gamma, \end{equation} and \begin{equation}\label{hig} H_{m\pi^{\gamma}} \mbox{ has leading term } \prod_{\alpha \in N(w^{-1})}h_\alpha^{mq(w,\alpha)}.\end{equation} \end{theorem}
\noindent If $\fg$ is a semisimple Lie algebra this result is due to Kumar and Letzter, see \cite{KL} where the case $\fg=G_2$ is also covered. \\ \\ We show that the exponents in \eqref{hig} are always positive. \begin{lemma} If $\gamma =w \beta$ with $\ell(w)$ minimal, then $q(w,\alpha)$ is a positive integer. \end{lemma} \begin{proof} By considering sub-expressions of $w$, it is enough to show this when $w = s_\alpha u$ with $\ell(w)>\ell(u)$ and $\alpha$ is a simple non-isotropic root. Here $\alpha_1= -w^{-1}\alpha$ is a positive root, so $(w\beta,\alpha^\vee) = -(\beta,\alpha_1)$ is a non-negative integer since $\beta$ is a simple root. If $(w\beta,\alpha^\vee)=0,$ then $w\beta = u\beta$ so $w$ does not have minimal length. \end{proof} \noindent If we assume hypothesis $(\ref{ii})$ instead of $(\ref{iii})$, it seems difficult to obtain the same estimates on \v Sapovalov elements as in Theorem \ref{1Shap}. However it is still possible to obtain a reasonable estimate using a different definition of the degree of a partition, at least if $m=1$. Information on $\theta_{\gamma, m}$ in general can then be deduced using Theorem \ref{1calu}. In the Theorem below we assume that $\Pi$ contains an odd non-isotropic root, since otherwise (\ref{iii}) holds and the situation is covered by Theorem \ref{1Shap}. This assumption is essential for Lemma \ref{1gag}. Likewise if $\gamma$ is odd and non-isotropic, then again (\ref{iii}) holds, so we assume that $\gamma=w\beta$ with $w \in W_{\rm nonisotropic}$ and $\beta \in \overline{\Delta}^+_{0} \cup {\overline{\Delta}}^+_{1}$, where
\begin{equation} \label{Kacroot} {\overline{\Delta}}^+_{0} = \{ \alpha \in \Delta^+_{0} | \alpha / 2 \not\in \Delta^+_{1}\}, \quad {\overline{\Delta}}^+_{1}= \{ \alpha
\in \Delta^+_{1}|2\alpha \not\in \Delta^+_{0}\}.\end{equation}
\noindent For $\alpha$ a positive root, and then for $\pi$ a partition, we define the {\it Clifford degree} of $\alpha, \pi$ by $${\rm Cdeg} (\alpha) = 2-i, \mbox{ for } \alpha \in \Delta^+_{i},\quad {\rm Cdeg} (\pi) = \sum_{\alpha \in \Delta^+} \pi(\alpha){\rm Cdeg} (\alpha).$$ The reason for this terminology is that if we set $U_n = \mbox{ span } \{e_{-\pi}|{\rm Cdeg}(\pi)\le n\},$ then $\{U_n\}_{n \ge 0}$ is the {\it Clifford filtration} on $U(\mathfrak{n}^-)$ as in \cite{M} Section 6.5. The associated graded algebra is a Clifford algebra over $S(\mathfrak{n}^-_0)$.
\begin{theorem} \label{1aShap} Suppose that $\fg$ is finite dimensional contragredient, and that $\Pi$ contains an odd non-isotropic root. Assume $\gamma$ is a positive root such that $\eqref{i}$ and $\eqref{ii}$ hold. If $\gamma$ is isotropic assume that $m=1,$ and if $\fg= G_2$ or $G(3)$ assume that $\gamma$ is not the highest short root of $\fg_0$. Then \begin{itemize} \item[{{\rm(a)}}] there exists a \v Sapovalov element $\theta_{\gamma,m} \in U({\mathfrak b}^{-} )^{- m\gamma}$, which is unique modulo the left ideal $U({\mathfrak b}^{-} ){\mathcal I}(\mathcal H_{\gamma, m})$. \item[{{\rm(b)}}] If $m=1$, the coefficients of $\theta_{\gamma}$ satisfy \begin{equation} \label{x2}
{\rm Cdeg} (\pi)+2\deg H_{\pi} \le 2\ell(w)+{\rm Cdeg}(\gamma) ,\end{equation}
and \[H_{\pi^{\gamma}} \mbox{ has leading term } \prod_{\alpha \in N(w^{-1})}h_\alpha.\] \end{itemize} \end{theorem}
\begin{rems} \label{rmd}{\rm (a) \noindent By induction using (\ref{Nw}), we have
\begin{equation} \label{xxx}{\operatorname{ht}} \gamma = 1 +\sum_{\alpha \in N(w^{-1})} q(w,\alpha).\end{equation} It is interesting to compare the inequalities (\ref{x1}) and (\ref{x2}). If $\Pi$ does not contain an odd non-isotropic root, then ${\rm Cdeg}(\pi) = 2|\pi|$ for all ${\pi \in {\overline{\bf P}}(\gamma)}$, compare Lemma \ref{1pig1} (a). Moreover from (\ref{x1}) with $m=1$ and \eqref{xxx}, we have $${\rm Cdeg}(\pi) + 2\deg H_{\pi} \le 2+ 2\sum_{\alpha \in N(w^{-1})}{q(w,\alpha)}.$$ Thus if $q(w,\alpha) = 1$ for all $\alpha \in N(w^{-1})$ we have $${\rm Cdeg}(\pi) + 2\deg H_{\pi} \le 2+2\ell(w) ,$$ and if $\gamma$ is even this equals the right side of \eqref{x2}. For example all the above assumptions hold if $\fg$ is a simply laced semisimple Lie algebra. However if $\fg =\osp(3,2)$ with (odd) simple roots $\beta=\epsilon-\delta, \alpha=\delta$, where $\alpha$ is non-isotropic and $\gamma = w(\beta)$ with $w=s_\alpha$, then $q(w,\alpha)=2$ and the upper bound for $\deg H_{\pi^{\gamma}}$ given by (\ref{x2}) is sharper than \eqref{x1}. If instead we take $\beta=\delta-\epsilon, \alpha=\epsilon$ as simple roots, and $\gamma = w(\beta)$ with $w=s_\alpha$, then again $q(w,\alpha)=2$, but (\ref{x2}) does not hold. Note in this case $\Pi$ does not contain an odd non-isotropic root. \\ \\
(b) It is often the case that equality holds in Equation \eqref{x1} for all ${\pi \in {\overline{\bf P}}(m\gamma)}$, for example this is the case in Type A, and in the case $\fg=\osp(2,4)$ when $\gamma$ is isotropic, see Sections \ref{1s.8} and \ref{1cosp} respectively. Furthermore in these cases, it is possible to order the positive roots so that all coefficients are products of linear factors. Equality also holds for all $\pi$ in \eqref{x2} when $\fg=\osp(3,2)$. However if $\fg=\osp(1,4)$ with simple roots $\alpha, \beta$ where $\beta$ is odd, then using \eqref{1f11}, it follows that the coefficient of $e_{- \alpha - \beta} e_{- \beta }$ in $\theta_{\alpha +2\beta}$ is constant. This means that the inequality in \eqref{x2} can be strict. These matters deserve further investigation. }\end{rems}
\noindent Next we mention some important consequences of Theorems \ref{1Shap} and \ref{1aShap}.
\begin{corollary} \label{lter} In Theorems \ref{1Shap} and \ref{1aShap} $(b),$ $H_{m\pi^{\gamma}}$ is the unique coefficient of highest degree in $\theta_{\gamma, m}.$ \end{corollary} \begin{proof} This follows easily from the given degree estimates, and the statements about the leading terms.\end{proof}
\noindent Suppose that $\{U(\mathfrak{n}^-)_j\}$ is either the standard or the Clifford filtration on $U(\mathfrak{n}^-)$. Then if $\mathcal{U} = U(\mathfrak{n}^-)\otimes \mathtt{k}[T]$ we define a filtration $\{\mathcal{U}_m\}$ on $\mathcal{U}$ by setting \begin{equation}\label{nuf} \mathcal{U}_m=\sum_{i+j\le m} T^iU(\mathfrak{n}^-)_j.\end{equation} For $u\in \mathcal{U}$, set $\deg_{\mathcal{U}} u= N$ if $N$ is minimal such that $u\in \mathcal{U}_N$. Choose $\xi \in \mathfrak{h}^*$ such that $(\xi, \gamma) = 0$ and $(\xi, \alpha) \neq 0$ for all roots $\alpha\in N(w^{-1})$. Then for $\lambda \in \mathcal{H}_{m\gamma}$, $\widetilde{\lambda} = \lambda + T \xi$ satisfies $(\widetilde{\lambda} + \rho, \gamma) = m(\gamma, \gamma)/2 $. Let $\theta=\theta_{\gamma, m}$ be as in Theorem \ref{1Shap} or \ref{1aShap} (b). We obtain the following result for the evaluation of $\theta$ at ${{\widetilde{\lambda} }}$.
\begin{corollary} \label{1.5} If $N = m{\operatorname{ht}} \gamma,$ we have $\theta_{\gamma, m}({{\widetilde{\lambda} }}) = \sum_{\pi \in {\overline{\bf P}}(m\gamma)} c_\pi e_{-\pi}\in \mathcal{U}_N$ with $c_\pi \in\mathtt{k}[T]$, and if $\pi =m\pi^\gamma$ or $\pi^0$ we have $\deg_{\mathcal{U}} c_\pi e_{-\pi}=N.$ \end{corollary} \begin{proof} This follows easily from \eqref{xxx}. \end{proof} \subsubsection{The case where $\gamma$ is isotropic.} We show that positive isotropic roots satisfy Hypothesis \eqref{ii}. \begin{lemma} \label{vi} Suppose that
$\Pi$ is either distinguished or anti-distinguished and that $\gamma$ is a positive isotropic root which is not simple. If $\fg=G(3)$ suppose that $\Pi$ is distinguished. Then for some non-isotropic root $\alpha$ we have $(\gamma,\alpha^\vee)> 0$, and $s_\alpha \gamma$ is a positive odd root. \end{lemma} \begin{proof} This can be shown by carrying out a case-by-case check. For types A-D, see the proof of Lemma \eqref{sim}. The Lie superalgebra $D(2,1;\alpha)$ is treated in the same way as $D(2,1)= \osp(4,2).$ For the other exceptional Lie superalgebras it is convenient to use the quiver $\mathcal{Q}$ defined as follows. Set $\mathcal{Q}_0=\Delta^1_+$ the set of odd positive roots of $\fg$. There is an arrow $\gamma' \longrightarrow \gamma$ whenever some non-isotropic root $\alpha$ we have $(\gamma,\alpha^\vee)> 0$, and $\gamma'=s_\alpha \gamma$.\
Then $\mathcal{Q}$ is a sub-quiver of the Borel quiver defined in \cite{M} Section 3.6.
When $\fg=G(3)$ this quiver appears in Exercise 4.7.10 of \cite{M}; the first and last diagrams at the bottom of page 92 correspond to the distinguished and anti-distinguished Borel subalgebra respectively. For the case of $\fg = F(4)$ see Exercise 4.7.12. Examination of these quivers easily gives the result. \end{proof} \begin{corollary} If $\gamma$ is as in the statement of Lemma \ref{vi}, then $\gamma$ satisfies Hypothesis \eqref{ii}. \end{corollary}
\noindent \v Sapovalov elements corresponding to non-isotropic roots for a basic classical simple Lie superalgebra were constructed in \cite{M} Chapter 9. This closely parallels the semisimple case. Properties of the coefficients of these elements were announced in \cite{M} Theorem 9.2.10. However the bounds on the degrees of the coefficients claimed in \cite{M} are incorrect if $\Pi$ contains a non-isotropic odd root. They are corrected by Theorem \ref{1aShap}.\\
\noindent {\bf B. Modules with prescribed characters.}\\ \\
The existence of a unique coefficient of highest degree in the \v Sapovalov element $\theta_{\gamma},$ when $\gamma$ is isotropic, is useful in the construction of some new highest weight modules $M^X(\lambda)$, as in the next result. We assume that $\Pi$ is a basis of simple roots satisfying hypothesis (\ref{i}). Let $X$ be an isotropic set of roots and set ${\mathcal{H}}_{X}= \bigcap_{\gamma\in X} {\mathcal{H}}_{\gamma}$.
Since ${\mathcal{H}}_{\gamma}={\mathcal{H}}_{-\gamma}$ for $\gamma\in X$, we may assume that all roots in $X$ are positive with respect to the basis $\Pi.$ If $\Pi$ contains (resp. does not contain) an odd non-isotropic root, we may assume that each $\gamma\in X$ satisfies \eqref{ii} (resp. \eqref{iii}) and then apply Theorem \ref{1aShap} (resp. Theorem \ref{1Shap}) to each \v Sapovalov element $\theta_{\gamma}$. These assumptions will remain in place for the remainder of this introduction, and then from Section \ref{jaf} onwards.
\begin{theorem} \label{newmodgen} Suppose that $X$ is an isotropic set of positive roots and $\lambda \in {\mathcal{H}}_{X}$. Then there exists a factor module $M^{X}(\lambda)$ of $M(\lambda)$ such that \[ {\operatorname{ch}\:} M^X(\lambda) = \mathtt{e}^{\lambda} p_{X}.\] \end{theorem}\noindent If $X$ is contained in the set of simple roots of some Borel subalgebra $\mathfrak{b}$, it is possible to a construct module with character as in the Theorem by parabolic induction. If $\mathfrak{p} = \mathfrak{b} \oplus \bigoplus_{\gamma\in X} \fg^{-\gamma}$, then provided $(\lambda+ \rho_\mathfrak{b}, \gamma)=0$ for all $\gamma\in X,$ the module ${\operatorname{Ind}}_{\;\mathfrak{p}}^{\;\fg} \; \mathtt{k}_\lambda$ induced from the one dimensional $\mathfrak{p}$-module $\mathtt{k}_\lambda$ with weight $\lambda$ will have character $\mathtt{e}^{\lambda} p_{X}.$ Our construction has however two features lacking in the approach via parabolic induction. First we do not need the condition that $X$ is contained in the set of simple roots for a Borel subalgebra. Secondly we provide a uniform construction of a highest weight module for the distinguished or anti-distinguished Borel subalgebra. This allows us to compare $M^{X}(\lambda)$ to $M^{Y}(\lambda)$ when, for example $Y\subset X$, or $Y=s_\alpha(X)$ for a reflection $s_\alpha.$ \\ \\ The construction of the modules in Theorem \ref{newmodgen} involves a process of deformation and specialization. First we extend scalars to $A = \mathtt{k}[T]$ and $B = \mathtt{k}(T)$. If $R$ is either of these algebras we set $U(\fg)_R = U(\fg) \otimes R$. Let $\mathfrak{h}_{\mathbb Q}^*$ be the rational subspace of $\mathfrak{h}^*$ spanned by the roots of $\fg$. Since ${\mathbb Q}X$ is an isotropic subspace of $\mathfrak{h}_{\mathbb Q}^*,$ it cannot contain any non-isotropic root. Thus we can choose $\xi \in \mathfrak{h}^*$ such that $(\xi, \gamma) = 0$ for all $\gamma \in X$, and $(\xi, \alpha) \neq 0$ for all even roots $\alpha$. Then for $\lambda \in \mathcal{H}_X$, \begin{equation} \label{wld} \widetilde{\lambda} = \lambda + T \xi\end{equation} satisfies $(\widetilde{\lambda} + \rho, \gamma) = 0$ for all $\gamma \in X$.\footnote{Replacing $T$ by $c\in \mathtt{k}$ in ${{\widetilde{\lambda} }}$ gives the point ${{\widetilde{\lambda} }} =\lambda+c\xi$ on the line $\lambda+ \mathtt{k} \xi$ and we think of ${{\widetilde{\lambda} }} =\lambda+T \xi$ as a generic point on this line.}
Consider the $U(\fg)_B$-module $M({{\widetilde{\lambda} }})_{B}$ with highest weight ${{\widetilde{\lambda} }} $. \footnote{ Some general remarks on base change for Lie superalgebras and their modules can be found in \cite{M} subsection 8.2.6.} The next step is to consider the factor module $M^X({{\widetilde{\lambda} }})_{B}$ of $M({{\widetilde{\lambda} }})_{B}$ obtained by setting $\theta_\gamma v_{{{\widetilde{\lambda} }}}$ equal to zero for $\gamma \in X$. Then we take the $U(\fg)_{A}$-submodule of this factor module generated by the highest weight vector $v_{{{\widetilde{\lambda} }}}$ and reduce mod $T$ to obtain the module $M^X(\lambda)$. \\ \\
We explain the idea behind the proof that $M^{X}(\lambda)$ has the character asserted in the Theorem.
If $X =\{\gamma\}$ we write $M^{\gamma}(\lambda)$ in place of $M^{X}(\lambda)$ throughout the paper. Set $\theta=\theta_\gamma.$ Let $L =U(\fg)_Bv$ a module with highest weight $\widetilde{\lambda}$ and highest weight vector $v$, such that $\theta v = 0$. Evaluation of \v Sapovalov elements at elements of $\mathfrak{h}^*\otimes B$ is defined at the start of Section \ref{1s.5}. Based on the fact that $\theta(\widetilde{\lambda})$ has a unique coefficient in $A$ of highest degree we obtain an upper bound on the dimension over $B$ of the weight space $L^{\widetilde{\lambda} -\eta}$, see Corollary \ref{ratss}. Now $\theta^2 v_{\widetilde{\lambda}} = 0$ by Theorem \ref{1zprod},
so the upper bound holds for the first and third terms in the short exact sequence \begin{equation} \label{bim} 0\longrightarrow N_B=U(\fg)_B \theta v_{\widetilde{\lambda}}\longrightarrow M({{\widetilde{\lambda} }})_{B} \longrightarrow M_B=M({{\widetilde{\lambda} }})_{B}/U(\fg)_B \theta v_{\widetilde{\lambda}} \longrightarrow 0.\end{equation}
This easily implies that the $U(\fg)_B$-modules $N_B$ and $M_B$ have characters $\mathtt{e}^{\lambda-\gamma}{p}_\gamma$ and $\mathtt{e}^\lambda{p}_\gamma$ respectively. Now let $M_A=U(\fg)_Av_{\widetilde{\lambda}}$ be the $U(\fg)_A$-submodule of $M_B$ generated by $v_{\widetilde{\lambda}}$. It follows easily that $M_A/TM_A$ is a highest weight module with weight $\lambda$ that has character $\mathtt{e}^{\lambda}{p}_\gamma$. \\ \\ The construction of $M^X(\lambda)$ in the general case is similar, but we need an extra ingredient which we illustrate in the case $X = \{\gamma,\gamma'\}$. Here we write $\theta, \theta', U_B$ in place of $\theta_\gamma$, $\theta_{\gamma'}$ and $U(\fg)_B$ respectively. We want to apply Corollary \ref{ratss} to the four factor modules arising from the series
\[ U_B\theta \theta' v_{{\widetilde{\lambda} }} \subset U_B\theta v_{{\widetilde{\lambda} }} \subset U_B\theta' v_{{\widetilde{\lambda} }} +U_B\theta v_{{\widetilde{\lambda} }} \subset M({{\widetilde{\lambda} }})_B,\] but to do this we need to know that $\theta\theta' v_{{\widetilde{\lambda} }} \in U_B\theta v_{{\widetilde{\lambda} }}$, $\theta' \theta v_{{\widetilde{\lambda} }} \in U_B\theta \theta' v_{{\widetilde{\lambda} }}$ and $\theta' \theta \theta' v_{{\widetilde{\lambda} }}= 0.$ All this will follow if we know that if $\mu \in \mathcal{H}_\gamma\cap \mathcal{H}_{\gamma'} $, and $v_{{\widetilde{\mu} }}$ is the highest weight vector in the $U(\fg)_B$ Verma module $M({{\widetilde{\mu} }})_B,$
then $\theta' \theta v_{{\widetilde{\mu} }}$ and $\theta \theta' v_{{\widetilde{\mu} }}$ are equal up to a scalar multiple. \\ \\ We give two proofs of this fact. The first, Theorem \ref{fiix2} shows that \begin{equation} \label{pee} \theta_{\gamma'}(\mu - \gamma) \theta_\gamma(\mu)=a(\mu)\theta_ \gamma(\mu - {\gamma'}) \theta_ {\gamma'}(\mu),\end{equation} for a rational function $a(\mu)$, which is a ratio of linear polynomials that are equal except for their constant terms. The computation relies on the fact that any orthogonal set of isotropic roots is $W$-conjugate to a set of simple roots in some Borel subalgebra, see \cite{DS}. \\ \\ Before turning to the second proof, we need some notation which will also be important for other results. For $\lambda \in \mathfrak{h}^*$ define
\begin{eqnarray} \label{rue} A(\lambda)_{0} &=& \{ \alpha \in \overline{\Delta}^+_{0} | (\lambda + \rho, \alpha^\vee) \in \mathbb{N} \backslash \{0\} \}\nonumber \\
A(\lambda)_{1} &=& \{ \alpha \in \Delta^+_{1} \backslash \overline{\Delta}^+_{1} | (\lambda + \rho, \alpha^\vee ) \in 2\mathbb{N} + 1 \} \\
A(\lambda) &=& A(\lambda)_{0} \cup A(\lambda)_{1}\nonumber \\
B(\lambda) &=& \{ \alpha \in \overline{\Delta}^+_{1} | (\lambda + \rho, \alpha) = 0 \} .\nonumber \end{eqnarray} Throughout the paper we say that a property holds {\it for general } $\mu$ in a closed subset $X$ of $\mathfrak{h}^*$, if it holds for all $\mu$ in a Zariski dense subset of $X$. (Such $\mu$ are the relatively general highest weights referred to in the abstract.) The second proof, which is representation theoretic, is based on the fact that for general $\mu$ such that $B(\mu) = \{ {\gamma}, {\gamma'}\}$ (that is for general $\mu\in\mathcal{H}_\gamma\cap \mathcal{H}_{\gamma'} $), the space of highest weight vectors with weight $\mu-\gamma-\gamma'$ in the Verma module ${M}(\mu)$ has dimension one, see Corollary \ref{kt1}. This implies \eqref{pee} without however the explicit computation of $a(\mu).$ \\ \\ \noindent {\bf C. The \v Sapovalov determinant and the Jantzen Sum Formula.}\\ \\ First we recall the Jantzen sum formula for Verma modules given in \cite{M} Theorem 10.3.1. The Jantzen filtration $\{{M}_{i}(\lambda)\}_{i\ge1}$ on ${M}(\lambda)$ satisfies \begin{equation} \label{1jfn}
\sum_{i > 0} {\operatorname{ch}\:} {M}_{i}(\lambda) = \sum_{\alpha \in A(\lambda)} {\operatorname{ch}\:} {M}(s_\alpha \cdot \lambda) + \sum_{\alpha \in B(\lambda)} \mathtt{e}^{\lambda -\alpha}p_\alpha.\end{equation}
We use the modules $M^{\gamma}(\lambda)$ to obtain an improvement to \eqref{1jfn}. \footnote{This improvement is mentioned by Brundan at the end of section 2 of \cite{B}.} At the same time, rather than using characters, it will be useful to rewrite the result using the Grothendieck group $K(\mathcal{O})$ of the category $\mathcal{O}$. We define $K(\mathcal{O})$ to be the free abelian group generated by the symbols $[L(\lambda)]$ for $\lambda \in \mathfrak{h}^*$. If $M\in \mathcal{O}$, the class of $M$ in $K(\mathcal{O})$ is defined as
$[M]=\sum_{\lambda \in \mathfrak{h}^*} |{M}:L(\lambda)| [L(\lambda)] ,$
where $|{M}:L(\lambda)| $ is the multiplicity of the composition factor $L(\lambda)$ in $M$.
\begin{theorem} \label{Jansum} For all $\lambda \in \mathfrak{h}^*$
\begin{equation} \label{lb} \sum_{i > 0} [{M}_{i}(\lambda)] = \sum_{\alpha \in A(\lambda)}[{M}(s_{\alpha}\cdot \lambda)] + \sum_{\gamma \in B(\lambda)} [M^{\gamma}(\lambda -\gamma)].\end{equation} \end{theorem}
\begin{proof} Combine Theorem \ref{newmodgen} with \eqref{1jfn}.\end{proof} \noindent The advantage of using this version of the formula is that $K(\mathcal{O})$ has a natural partial order. For $A, B \in \mathcal{O}$ we write $A \ge B$ if $[A]-[B]$ is a linear combination of classes of simple modules with non-negative integer coefficients. Clearly if $B$ is a subquotient of $A$ we have $A \ge B$. \footnote{In order not to lengthen this introduction, we postpone a discussion of the merits of working with characters until the start of Section \ref{SV}. }
\\ \\
This raises the question of whether there is a similar sum formula for the modules $M^\gamma(\lambda)$ and more generally for the modules $ M^X(\lambda).$ In Theorem \ref{Jansum101} we obtain such a formula for Lie superalgebras of Type A. As in the classical case the sum formula follows from the calculation of the \v Sapovalov determinant $ \det F_\eta^X $ for the modules $M^X(\lambda)$, see Theorem \ref{shapdet}. These results depend on most of the other results in the paper. As usual $ \det F_\eta^X $ factors into a product linear polynomials, each of which has an interpretation in terms of representation theory. However there is a rather subtle point about the computation of the leading term of $ \det F_\eta^X $, see Subsection \ref{com}. The issue is that there need not be a natural basis for $M^X(\lambda)$, indexed by partitions and independent of $\lambda$ as there is in the Verma module case. To remedy this we introduce another determinant $\det G^X_{\eta}$ whose leading term is more readily computed and then using \v Sapovalov elements we compare the two determinants. \\ \\ There are also three conditions on the highest weight which result in behavior that has no analog for semisimple Lie algebras. \\ \\
$\bullet$ \indent The first should be expected based on Theorem \ref{Jansum}. The set $X$ could be contained in a larger isotropic set of simple roots $Y = X\cup \{\gamma\}$, where $\gamma \notin X.$ Here we show that for general $\lambda \in \mathcal{H}_Y$, there is a non-split exact sequence \[0 \longrightarrow M^{Y}(\lambda-\gamma) \longrightarrow M^{X}(\lambda) \longrightarrow M^{Y}(\lambda)\longrightarrow 0,\] where the two end terms are simple, compare \eqref{bim}.\\ $\bullet$ \indent Secondly there could be distinct non-isotropic roots $\alpha_1, \alpha_2$ such that $\alpha_1^\vee\equiv \alpha_2^\vee \mod {\mathbb Q} X.$ Then for general $\lambda \in \mathcal{H}_X$ such that $\lambda$ is dominant integral, the module $M^X(\lambda)$ has length four, except in one exceptional case, where it has length five, see Theorems \ref{wane} and \ref{name}.\\ $\bullet$ \indent Thirdly suppose there is no pair of non-isotropic roots $\alpha_1, \alpha_2$ as in second case. There could be a non-isotropic root $\alpha$, such that there is a unique isotropic root $\gamma\in X$ with $(\gamma,\alpha^\vee)\neq0.$ Then set $\gamma' =s_\alpha \gamma$. We show that if $(\gamma,\alpha^\vee)>0$, $(\lambda+\rho,\gamma')=0$ and $Y =s_\alpha X$, then for general $\lambda\in \mathcal{H}_X\cap\mathcal{H}_Y$, there is an onto map $M^X(\lambda)\longrightarrow M^{Y}(\lambda)$ having a simple kernel, Theorem \ref{eon}. \\ \\ The Jantzen sum formula (Theorem \ref{Jansum101}) follows easily once the \v Sapovalov determinant is known. We note that, like Theorem \ref{Jansum}, the sum of characters $\sum_{i>0} {\operatorname{ch}\:} {M}^X_i({\lambda})$ is expressed as a sum of other modules with {\it positive} coefficients.
In the case of semisimple Lie algebras there are other modules which have Jantzen sum formulas, see \cite{AL}, \cite{AJS} Proposition 6.6 and \cite{J1} Satz 5.14. However these sum formulas involve terms with negative coefficients. \\ \\ {\bf D. Low Dimensional Examples and Twisting Functors.} \\ \\ In an appendix to this paper we study the modules $M^{X}(\lambda) $ in detail for the cases of $\fgl(2,1)$ and $\fgl(2,2)$. This serves two purposes. First it illustrates many of the phenomena that arise elsewhere in the paper. For example the behavior described after the second (resp. third) bullet above occurs already in the case of $\fgl(2,2)$ (resp. $\fgl(2,1)$). Secondly the results of the appendix play an important role in the evaluation of the \v Sapovalov determinant and the proof of the Jantzen sum formula for the modules $ M^X(\lambda)$ in Type A in general. Namely the representation theory needed for these results can be reduced to the case of $\fgl(2,1)$ and $\fgl(2,2)$ using an equivalence of categories due to Cheng, Mazorchuk and Wang \cite{CMW}. A similar study of low dimensional cases will presumably be necessary to extend these results to the orthosymplectic case.\\ \\ The equivalence in \cite{CMW} is obtained using twisting functors, parabolic induction and odd reflections. In Subsection \ref{agm}, we adapt this result to our needs, and present some other results on twisting functors which may be of independent interest.
\\ \\Finally our study of $\fgl(2,1)$ and $\fgl(2,2)$ leads us to make the following conjecture. \begin{conjecture} \label{conje} For any orthogonal set of isotropic roots $X$ for $\fgl(m,n)$, and any dominant weight $\lambda\in\mathcal{H}_X$, the maximal finite dimensional quotient of $M^X(\lambda)$ is simple. \end{conjecture} \noindent {\bf Organization of the paper.}\\ \\ \noindent In the next Section we discuss the uniqueness of \v Sapovalov elements. Theorems \ref{1Shap} and \ref{1aShap} are proved in subections \ref{zzw} and \ref{1s.51} respectively. The proofs depend on a rather subtle cancellation property which is illustrated in Section \ref{1cosp} in the cases of $\mathfrak{sp}(6)$ and $\osp(2,4)$. Changing the Borel subalgebra and relations between \v Sapovalov elements form the subject of Sections \ref{1sscbs} and \ref{RS} respectively. The modules $M^X(\lambda)$ are constructed and Theorem \ref{newmodgen} is proved in Section \ref{jaf}. Applications to the submodule structure of Verma modules follow in Section \ref{SV}. In Section \ref{1s.8} we give a closed formula for \v Sapovalov elements in Type A. By definition \v Sapovalov elements give rise to highest weight vectors in Verma modules. The question of when the images of these highest weight vectors in various factor modules is non-zero is studied in Section \ref{1surv}. The \v Sapovalov determinant and the Jantzen sum formula are treated in Section \ref{sf}. \\ \\ The material on the coefficients of \v Sapovalov elements has appeared in preliminary form in the unpublished preprint \cite{M1}. There is a survey of some of the results from Sections 3-8 in \cite{M2}. \section{Uniqueness of \v Sapovalov elements.} \label{1s.1} If $\mathfrak{k}$ is a simple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and $\mu, \lambda \in \mathfrak{h}^*$ we have, for $\mathfrak{k}$ Verma modules \begin{equation} \label{1kit}\dim \Hom_\mathfrak{k}(M(\mu),M(\lambda))\le 1,\end{equation} by \cite{D} Theorem 7.6.6, and it follows that the \v Sapovalov element $\theta_{\gamma, m}$ for the pair $(\gamma,m)$ is unique modulo the left ideal $U({\mathfrak b}^{-} ){\mathcal I}(\mathcal H_{\gamma, m})$. \\ \\ \noindent We do not know whether (\ref{1kit}) holds in general for Lie superalgebras, but we note that the analog of (\ref{1kit}) fails for parabolic Verma modules over simple Lie algebras, \cite{IS}, \cite{IS1}. Here we use an alternative argument to show the uniqueness of \v{S}apovalov elements.
\begin{theorem} \label{17612} Suppose $\theta_1, \theta_2$ are \v Sapovalov elements for the pair $(\gamma,m)$. Set $\mathcal{H} = \mathcal H_{\gamma, m}.$ Then \begin{itemize} \item[{{\rm(a)}}] for all $\lambda \in \mathcal H$ we have $\theta_1 v_\lambda =\theta_2 v_\lambda$ \item[{{\rm(b)}}] $\theta_1- \theta_2 \in U({\mathfrak g}){\mathfrak n}^+ + U({\mathfrak g}){\mathcal I}(\mathcal H). $ \end{itemize}\end{theorem} \begin{proof} Set
\[\Lambda = \{\lambda \in \mathcal H| A(\lambda) = \{\gamma\}, \;B(\lambda) = \emptyset \},\] if $\gamma$ is non-isotropic, and
\[\Lambda = \{\lambda \in \mathcal H| B(\lambda) = \{\gamma\}, \;A(\lambda) = \emptyset \},\] if $\gamma$ is isotropic. If $\lambda \in \Lambda$ it follows from the sum formula \eqref{1jfn} that $ M_1(\lambda)^{\lambda-m\gamma}$ is one-dimensional. Because ${M}_{1}(\lambda)$ is the unique maximal submodule of ${M}(\lambda)$, $\theta_1 v_\lambda$ and $\theta_2 v_\lambda$ are proportional. Then from the requirement that $e_{-\pi^0}$ occurs with coefficient 1 in a \v{S}apovalov element we have $\theta_1 v_\lambda =\theta_2 v_\lambda$. Since $\Lambda$ is Zariski dense in $\mathcal H$, (a) holds and (b) follows from (a) because by \cite{M} Lemma 9.4.1 we have \begin{equation} \label{1rut} \bigcap_{\lambda \in \Lambda} \ann_{U(\small{\mathfrak{g}})}v_\lambda =
U({\mathfrak g}){\mathfrak n}^+ + U({\mathfrak g}){\mathcal I}(\mathcal H).
\end{equation} \end{proof}
\section{Proof of Theorems \ref{1Shap} and \ref{1aShap}.}\label{1s.5}
\subsection{Outline of the Proof and Preliminary Lemmas.}\label{1s.4}
Theorems \ref{1Shap} and \ref{1aShap}
are proved by looking at the proofs given in \cite{H2} or \cite{M} and keeping track of the coefficients. Given $\lambda \in {\mathfrak h}^* $ we define {\it evaluation at} $\lambda$ to be the map $$\varepsilon^\lambda:U({\mathfrak b}^{-}) = U({\mathfrak n}^-)\otimes S({\mathfrak h}) \longrightarrow M (\lambda),\; \quad \sum_{i} a_i \otimes b_i \longrightarrow \sum_{i} a_i b_i(\lambda)v _{\lambda}.$$ If $B$ is a $\mathtt{k}$-algebra, then evaluation at elements of $\mathfrak{h}^*\otimes B$ is defined similarly. Let $(\gamma,m)$ be as in the statement of the Theorems and set ${\mathcal H} = {\mathcal H}_{\gamma, m}$. If $\theta =\theta_{\gamma, m}$ is as in the conclusion of the Theorem, then for any $\lambda \in {\mathcal H},$ $\theta(\lambda)v_\lambda$
is a highest weight vector in $M (\lambda)^{\lambda-m\gamma}$. Conversely suppose that $\Lambda$ is a dense subset of ${\mathcal H} $ and that for all $\lambda \in \Lambda$ we have constructed $\theta^\lambda \in U({\mathfrak n})^{-m\gamma}$ such that $\theta^\lambda v_\lambda$ is a highest weight vector in $M(\lambda)^{\lambda-m\gamma}$ and that $$\theta^\lambda = \sum_{\pi \in {\overline{\bf P}}(m\gamma)}a_{\pi, \lambda}e_{-\pi}.$$ where $a_{\pi, \lambda} $ is a polynomial function of $\lambda \in \Lambda$ satisfying suitable conditions. For $\pi \in {\overline{\bf P}}(m\gamma)$, the assignment $\lambda
\rightarrow a_{\pi, \lambda}$ for $\lambda \in \Lambda$ determines a polynomial map from ${\mathcal H}$ to
$U({\mathfrak n}^-)^{-\gamma},$ so there exists an element $H_\pi \in U({\mathfrak h})$ uniquely determined modulo ${\mathcal I}(\mathcal H)$ such that $H_{\pi}(\lambda) = a_{\pi, \lambda}$ for all $\lambda \in \Lambda$. We define the element $\theta \in U(\mathfrak{b}^-)$ by setting $$\theta= \sum_{\pi \in {\overline{\bf P}}(m\gamma)}e_{- \pi}H_{\pi}.$$ \noindent Note that $\theta$ is uniquely determined modulo the left ideal $U({\mathfrak b^-}){\mathcal I}(\mathcal H),$ and that $\theta(\lambda) = \theta^\lambda$. Also, for $\alpha \in \Delta^+$ and
$\lambda \in \Lambda$ we have $e_\alpha \theta v_\lambda = e_\alpha
\theta^\lambda v_\lambda = 0$, because $\theta^\lambda v_\lambda = 0$ is a highest weight vector, so $e_\alpha \theta \in \bigcap_{\lambda \in \Lambda} \ann_{U(\small{\mathfrak{g}})}v_\lambda.$ Thus (\ref{boo}) follows from (\ref{1rut}).\\ \\ \noindent We need to examine the polynomial nature of the coefficients of $\theta_{\gamma,m}$. The following easy observation (see \cite{D} Lemma 7.6.9), is the key to doing this. For any associative algebra and $e, a \in A$
and all $r \in \mathbb{N} $ we have, \begin{equation} \label{1cow} e^{r}a = \sum^r_{i=0} \left( \begin{array}{c}
r \\
i \end{array}\right) (({\operatorname{ad \;}} e)^i a)e^{r - i} .\end{equation} \noindent Here we should interpret $({\operatorname{ad \;}} e)a $ as $ea-ae$. The following consequence is well-known, \cite{BR} Satz 1.4, \cite{Ma} Lemma 4.2. We give the short proof for completeness. \begin{corollary} \label{1oreset} Suppose that $S$ is a multiplicatively closed subset of $A$, consisting of non-zero divisors, which is generated by
locally ${\operatorname{ad \;}} $-nilpotent elements. Then
$S$ is an Ore set in $A$. \end{corollary}
\noindent We write $A_S$ for the resulting Ore localization, or $A_e$ in the case $S=\{e^n|n \in \mathbb{N}\} $. \begin{proof} It suffices to show that for a locally ad-nilpotent generator $e\in S$, the set $\{e^n|n \in \mathbb{N}\} $ satisfies the Ore condition. Given $a \in A$ and $n \in \mathbb{N}$, suppose that $({\operatorname{ad \;}} e)^{k+1} a = 0$. Then $e^{k+n}a = a'e^n$, where \[a' = \sum^k_{i=0} \left( \begin{array}{c}
n+k \\
i \end{array}\right)
(({\operatorname{ad \;}} e)^i a)e^{k-i}.\]
\end{proof} \noindent Now suppose $\alpha \in \Pi_{\rm nonisotropic}$, and set $e = e_{-\alpha}.$ Then $e$ is a nonzero divisor in
$U = U(\mathfrak{n}^-)$, and the set $\{e^n|n \in \mathbb{N}\} $ is an Ore set in $U $ (and in $U(\fg)$) by Corollary \ref{1oreset}. Given $a \in U^{-\eta}$, and $\pi$ a partition, the coefficient of $e_{-\pi}$ in \eqref{cows} depends polynomially on $r\in\mathtt{k}$, and hence this coefficient is determined by its values on $r\in{\mathbb N}$. Also, the adjoint action of $\mathfrak{h}$ on $U$ extends to $U_e$, and in the next result we give a basis for the weight spaces of $U_e$. Let ${\widehat{\bf P}}(\eta)$ be the set of pairs $(k,\pi)$ such that $k \in \mathbb{Z}, \pi \in {\overline{\bf P}}(\eta -k\alpha)$ and $\pi(\alpha) = 0.$ Then we have \begin{lemma} \label{1uebasis} \begin{itemize} \item[{}]
\item[{{\rm(a)}}] The set $\{e_{-\pi} e^k|(k,\pi ) \in {\widehat{\bf P}}(\eta)\}$ forms a $\mathtt{k} $-basis for the weight space $U_e^{-\eta}.$
\item[{{\rm(b)}}] If $u = \sum_{(k,\pi ) \in {\widehat{\bf P}}(\eta)} c_{(k,\pi )} e_{-\pi} e^k \in U_e^{-\eta}$ with $c_{(k,\pi )} \in \mathtt{k} ,$ then $u \in U$ if and only if $c_{(k,\pi )} \neq 0$ implies $k \ge 0$. \end{itemize}
\end{lemma} \begin{proof} Throughout the proof we order the set $\Delta^+$ so that for any partition $\sigma$ we have $e_{-\sigma}= e_{-\pi}e^{\ell}$ where $\pi(\alpha)=0.$\\ (a) Suppose $u \in U_e^{-\eta}$. We need to show that $u$ is uniquely expressible in the form
\begin{equation} \label{1china} u = \sum_{(k,\pi ) \in {\widehat{\bf P}}(\eta)} c_{(k,\pi )} e_{-\pi} e^k \end{equation}
We have $ue^N \in U^{-(N\alpha+\eta)}$ for some $N$. Hence by the PBW Theorem for $U$ we have a unique expression $$ue^N = \sum_{\sigma \in {\bf\overline{P}}(\eta+N\alpha)}a_\sigma e_{-\sigma} .$$ Now if $a_\sigma \neq 0$, then $e_{-\sigma}= e_{-\pi}e^{\ell}$ where $\sigma(\alpha) = {\ell}$ and
where $\pi \in {\bf\overline{P}} (\eta+N\alpha-{\ell}\alpha)$ satisfies $\pi(\alpha) = 0$. Then $\pi$ and ${k = \ell-N}$ are uniquely determined by $\sigma$, so we set $c_{(k,\pi )} =a_\sigma$. Then clearly (\ref{1china}) holds. Given (a), (b) follows from the PBW Theorem.
\end{proof}
\noindent We remark that if $\alpha$ is a non-isotropic odd root, then we can use $e^2$ in place of $e = e_{-\alpha}$ in the above Corollary and Lemma. \noindent However we will need a version of Equation (\ref{1cow}) when $e$ is replaced by an odd element $x$ of a ${\mathbb Z}_2$-graded algebra $A$. Suppose that $z$ is homogeneous, and define $z^{[j]} = ({\operatorname{ad \;}} x)^j z$. Set $e = x^2$ and apply (\ref{1cow}) to $a = xz =[x, z] + (-1)^{\overline z}zx,$ to obtain \begin{equation} \label{1f11} x^{2\ell+1}z = \sum^\ell _{i=0} \left( \begin{array}{c}
\ell \\
i \end{array}\right) z^{[2i+1]}x^{2\ell - 2i}
+ (-1)^{\overline z}\sum^\ell _{i=0} \left( \begin{array}{c}
\ell \\ i \end{array}\right) z^{[2i]} x^{2\ell - 2i +1} .\end{equation}
\noindent The \v Sapovalov elements in Theorems \ref{1Shap} and \ref{1aShap} are constructed inductively using the next Lemma.
\begin{lemma}\label{11768} Suppose that $\mu \in \mathcal{H}_{\gamma',m}, \;\alpha \in\Pi\cap A(\mu)$ and set \begin{equation} \label{121c}\lambda = s_\alpha\cdot \mu,\;\;\gamma = s_\alpha\gamma',\;\;p = (\mu + \rho, \alpha^\vee),\;\;q = (\gamma, \alpha^\vee).\end{equation} Assume that $ q, m \in \mathbb{N}\backslash \{0\}$, and \begin{itemize} \item[{{\rm(a)}}] $\theta' \in U({\mathfrak n}^-)^{-m\gamma'}$ is such that $v = \theta'v_\mu \in M(\mu)$ is a highest weight vector. \item[{{\rm(b)}}] If ${\alpha\in \Delta^+_{1} \backslash \overline{\Delta}^+_{1} }$, then $q = 2$.
\end{itemize} \noindent Then there is a unique $\theta \in U({\mathfrak n}^-)^{-m\gamma}$ such that \begin{equation} \label{121nd} e^{p + mq}_{- \alpha}\theta' = \theta e^p_{- \alpha}. \end{equation} \end{lemma} \begin{proof} This is well-known, see for example \cite{H2} Section 4.13 or \cite{M} Theorem 9.4.3.\end{proof} \noindent Equation \eqref{121nd} is the basis for the proof of many properties of \v Sapovalov elements. We note the following variations. First under the hypothesis of the Lemma the \v Sapovalov elements $\theta_{\gamma',m}$ and $\theta_{\gamma,m}$ are related by \begin{equation} \label{121a} e^{p + mq}_{- \alpha}\theta_{\gamma',m}(\mu) = \theta_{\gamma,m}(\lambda) e^p_{- \alpha}. \end{equation} Now suppose that instead of the hypothesis $\alpha \in \Pi\cap A(\mu)$, we have $\alpha \in \Pi\cap A(\lambda -m\gamma)$, and set $r = (\lambda + \rho-m\gamma, \alpha^\vee)$. Then \begin{equation} \label{121b} \theta_{\gamma',m}(\mu) e^{r + mq}_{- \alpha} = e^r_{- \alpha} \theta_{\gamma,m}(\lambda). \end{equation} Note that \eqref{121b} becomes formally equivalent to \eqref{121a} when we set $r=-{(p + mq)}$.\\ \\ \noindent In the proofs of Theorems \ref{1Shap} and \ref{1aShap} we write $\gamma = w\beta$ for $\beta \in \Pi$ and $w\in W$. We use the Zariski dense subset $\Lambda$ of ${\mathcal H}_{\beta,m}$ defined by (see \eqref{rue} for the notation) \begin{equation} \label{1tar}\Lambda =
\{\nu \in {\mathcal H}_{\beta,m}| \Pi_{\rm nonisotropic}\subseteq A(\nu) \} .\end{equation} \begin{rem} {\rm Before giving the proofs we remark that it is also possible to construct \v Sapovalov elements directly in $U(\mathfrak{b}^-)$ instead of their evaluations in $U(\mathfrak{n}^-)$. To do this note that if $e=e_{-\alpha},$ $h\in \mathfrak{h}$ and $f(h)$ is a polynomial in $\mathfrak{h}$, then
$ef(h) = f(h+\alpha(h))e$. It follows that $\{e^n|n\in {\mathbb N}\}$ is an Ore set in $U(\mathfrak{b}^-)$, and we have $U(\mathfrak{b}^-)_e= U(\mathfrak{n}^-)_e\otimes S(\mathfrak{h}).$ }\end{rem}
\subsection{Proof of Theorem \ref{1Shap}.}\label{zzw} In this section we assume $\fg$ is contragredient and hypotheses $(\ref{i})$ and $(\ref{iii})$ hold. In particular $\gamma=w\beta$ where $\beta$ is a simple root, and $w\in W.$ Since the statement of Theorem \ref{1Shap} involves precise but somewhat lengthy conditions on the coefficients, we introduce the following definition as a shorthand. \begin{dfn} \label{1sag} {\rm We say that a family of elements $\theta^\lambda_{\gamma,m} \in U({\mathfrak n}^-)^{- m\gamma}$ is {\it in good position for $w$} if for all $\lambda \in w\cdot \Lambda$ we have} \end{dfn} \begin{equation} \label{12} \theta^\lambda_{\gamma,m} = \sum_{\pi \in {\overline{\bf P}}(m\gamma)} a_{\pi,\lambda} e_{-\pi}, \end{equation} where the coefficients $a_{\pi,\lambda} \in \mathtt{k} $ depend polynomially on $\lambda \in w\cdot \Lambda$, and \begin{itemize}
\item[{{\rm(a)}}] $\deg \; a_{\pi,\lambda} \leq m{\operatorname{ht}} \gamma - |\pi|$ \item[{{\rm(b)}}] $a_{m\pi^{\gamma},\lambda}$ is a polynomial function of $\lambda$ of degree $m({\operatorname{ht}} \gamma - 1)$ with highest term equal to $c\prod_{\alpha \in N(w^{-1})}(\lambda,\alpha)^{mq(w,\alpha)}$ for a nonzero constant $c$. \end{itemize} We show that the conditions on the coefficients in this definition are independent of the order on the positive roots $\Delta^+$ used to define the $e_{-\pi}.$
Consider two orders on $\Delta^+$, and for $\pi \in \overline{\bf P}(m\gamma)$, set $e_{-\pi} = \prod_{\alpha \in \Delta^+} e^{\pi (\alpha)}_{- \alpha}$, and $\overline{e}_{-\pi} = \prod_{\alpha \in \Delta^+} e^{\pi (\alpha)}_{- \alpha}$, the product being taken with respect to the given orders. \begin{lemma} \label{1delorder1}
Fix a total order on the set ${\overline{\bf P}}(m\gamma)$ such that if $\pi, \sigma \in {\overline{\bf P}}(m\gamma)$ and $|\pi| > |\sigma|$ then $\pi$ precedes $\sigma,$ and use this order on partitions to induce orders on the bases ${\bf B_1} = \{
e_{-\pi}|\pi \in {\overline{\bf P}}(m\gamma)\}$ and ${\bf B_2} = \{\overline{e}_{-\pi}|\pi \in {\overline{\bf P}}(m\gamma)\}$ for $U(\mathfrak{n}^-)^{-m\gamma}$. Then the change of basis matrix from the basis ${\bf B_1}$ to ${\bf B_2}$ is upper triangular with all diagonal entries equal to $\pm1.$ \end{lemma} \noindent \begin{proof} Let $\{ U_n =U_n(\mathfrak{n}^-) \}$ be the standard filtration on
$U=U({\mathfrak n}^-).$ Note that if $\pi \in {\overline{\bf P}}(m\gamma),$ then $e_{-\pi}, \overline{e}_{-\pi} \in U_{|\pi|}({\mathfrak n}^-)^{-m\gamma}.$ Also the factors of $e_{-\pi}$ supercommute modulo lower degree terms, so for all $\pi \in {\overline{\bf P}}(m\gamma),$ $e_{-\pi} \pm\overline{e}_{-\pi} \in U_{|\pi| - 1}({\mathfrak n}^-)^{-m\gamma}$. The result follows easily.
\end{proof} \begin{lemma} \label{1delorder} For $x \in U({\mathfrak n}^-)^{-m\gamma} \otimes S({\mathfrak h}),$ write \begin{equation} \label{1owl} x \; = \; \sum_{\pi \in {\overline{\bf P}}(m\gamma)} e_{-\pi}f_\pi \; = \; \sum_{\pi \in {\overline{\bf P}}(m\gamma)}
\overline{e}_{-\pi}g_\pi.\end{equation} Suppose that $f_{m\pi^{\gamma}}$ has degree $m({\operatorname{ht}} \gamma - 1),$ and that for all $\pi \in {\overline{\bf P}}(m\gamma),$ we have $\deg f_\pi \le m{\operatorname{ht}} \gamma - |\pi|.$ Then $g_{m\pi^{\gamma}}$ has the same degree and leading term as $f_{m\pi^{\gamma}}$
and
for all $\pi \in {\overline{\bf P}}(m\gamma),$ we have $\deg \;g_\pi \le m{\operatorname{ht}} \gamma - |\pi|.$ \end{lemma} \noindent \begin{proof} By Lemma \ref{1delorder1} we can write \[e_{-\pi} = \sum_{\zeta
\in {\overline{\bf P}}(m\gamma)} c_{\pi,\zeta}\overline{e}_{-\zeta},\] where $c_{\pi,\zeta} \in \mathtt{k} , c_{\pi,\pi} = \pm1$ and if $c_{\pi,\zeta} \neq 0$ with $\zeta \neq
\pi,$ then $|\zeta| < |\pi|$. Thus (\ref{1owl}) holds with $$g_\zeta = \sum_{\pi \in {\overline{\bf P}}(m\gamma)}
c_{\pi,\zeta}f_{\pi}.$$ It follows that $g_\zeta$ is a linear combination of polynomials of degree less than or equal to $m{\operatorname{ht}} \gamma - |\zeta|.$ Also $|m\pi^{\gamma}| = m,$ and for $\zeta \in {\overline{\bf P}}(m\gamma), \zeta \neq m\pi^{\gamma},$ we have $|\zeta|>m.$ Therefore $$g_{m\pi^{\gamma}} = f_{m\pi^{\gamma}} + \mbox{ a linear combination of polynomials of smaller degree.}$$ The result follows easily from this.\end{proof} \noindent If $\gamma$ is a simple root, then $\theta_{\gamma,m} = e_{-\gamma}^m$ satisfies the conditions of Theorem \ref{1Shap}. Otherwise we have $\gamma = w\beta$ for some $w \in W_{\rm even}, w \neq 1$. Write \begin{equation} \label{1wuga} w = s_{\alpha}u, \quad \gamma' = u\beta, \quad \gamma = w\beta = s_{\alpha} \gamma', \end{equation}
with $\alpha \in \Pi_{\rm even}$ and $\ell(w) =\ell(u) + 1.$ Suppose $\nu \in \Lambda$ and set \begin{equation} \label{1wuga1} \mu = u\cdot \nu, \quad \lambda = w\cdot \nu = s_{\alpha}\cdot \mu.
\end{equation} \noindent For the remainder of this subsection we assume that the positive roots are ordered so that for any partition $\pi$ we have $e_{-\pi} = e_{-\sigma}e_{-\alpha}^k$ for a non-negative integer $k$ and
a partition $\sigma$ with $\sigma(\alpha)=0$. The next Lemma is the key step in establishing the degree estimates in the proof of Theorem \ref{1Shap}. The idea is to use Equation (\ref{121nd}) and the fact that $\theta \in U({\mathfrak n}^-)$, rather than a localization of $U({\mathfrak n}^-)$, to show that certain coefficients cancel. Then using induction and (\ref{121nd}) we obtain the required degree estimates. Since the proof of the Lemma is rather long we break it into a number of steps. \begin{lemma} \label{1wpfg} Suppose that $p, m, q$ are as in Lemma \ref{11768}, $\alpha \in \Pi_{\rm even}$ and \begin{equation} \label{12nd} e^{p+mq}_{- \alpha}\theta^{\mu}_{\gamma',m} = \theta^\lambda_{\gamma,m}e^p_{- \alpha}. \end{equation} Then the family $\theta^\lambda_{\gamma,m} $ is in good position for $w$ if the family $\theta^\mu_{\gamma',m}$ is in good position for $u$. \end{lemma} \begin{proof} {\it Step 1. Setting the stage.}\\ \\
Suppose that \begin{equation} \label{14cha7} \theta^{\mu}_{\gamma',m} = \sum_{\pi' \in {\overline{\bf P}}(m\gamma')} a'_{\pi',\mu} e_{-\pi'}, \end{equation} and let
\begin{equation} \label{1qwer} e_{- \pi'}^{(j)} = ({\operatorname{ad \;}} e_{- \alpha})^j e_{- \pi'} \in U_{|\pi'|}(\mathfrak{n}^-)^{-(m\gamma' + j \alpha )},\end{equation} for all $j \geq 0,$ and $\pi' \in {\overline{\bf P}}(m\gamma')$. Then by Equation (\ref{1cow}) \begin{eqnarray} \label{149} e^{p + mq}_{- \alpha} e_{- \pi'} = \sum_{i \geq 0} \left( \begin{array}{c} p + mq \\ j \end{array} \right) e^{(j)}_{- \pi'} \; e^{p + mq-j}_{- \alpha}. \end{eqnarray} Choose $N$ so that $e^{(N+1)}_{- \pi'} = 0,$ for all $\pi' \in {\overline{\bf P}}(m\gamma').$ Then for all such $\pi' $ and $j = 0, \ldots, N$ we can write \begin{eqnarray} \label{199} e_{- \pi'}^{(j)}e^{N-j}_{- \alpha} = \sum_{\zeta \in {\overline{\bf P}}(m\gamma' + N \alpha )} b_{j, \zeta}^{\pi'}e_{- \zeta} , \end{eqnarray} with $ b_{j, \zeta}^{\pi'} \in \mathtt{k} .$ Furthermore if $ b_{j, \zeta}^{\pi'} \neq 0,$ then since
$e_{- \pi'}^{(j)}e^{N-j}_{- \alpha} \in U_{|\pi'| + N - j}$,
(\ref{1qwer}) gives \begin{eqnarray} \label{198}
|\zeta| \leq |\pi'| + N - j. \end{eqnarray} {\it Step 2. The cancellation step.}\\ \\ By Equations (\ref{14cha7}) and (\ref{149}) \begin{eqnarray} \label{1199} e^{p + mq}_{- \alpha} \theta^{\mu}_{\gamma',m} & = & \sum _{\pi' \in {\overline{\bf P}}(m\gamma')} a'_{\pi',\mu} e^{p + mq}_{- \alpha}e_{-\pi'}\\ & = & \sum_{ \pi' \in {\overline{\bf P}}(m\gamma'),\;j \geq 0} \left( \begin{array}{c} p + mq \nonumber\\ j \end{array} \right) a'_{\pi',\mu} e_{- \pi'}^{(j)}e^{p + mq-j}_{- \alpha}. \end{eqnarray} Now collecting coefficients, set \begin{equation}\label{159} c_{ \zeta,\lambda}
= \sum_{\pi' \in {\overline{\bf P}}(m\gamma'),\;j \geq 0} \left( \begin{array}{c} p + mq \\ j \end{array} \right) a'_{\pi',\mu} b_{j, \zeta}^{\pi'}. \end{equation} Then using Equations (\ref{199}) and (\ref{1199}), we have in $U_e,$ where $e = e_{-\alpha}$, \begin{eqnarray} \label{1lab} e^{p + mq}_{- \alpha} \theta^{\mu}_{\gamma',m} & = & \sum_{\zeta \in {\overline{\bf P}}(m\gamma' + N\alpha )}
c_{ \zeta, \lambda}e_{-\zeta}e^{p + mq-N}_{- \alpha}. \end{eqnarray} By (\ref{12nd}) and Lemma \ref{1uebasis}, $c_{ \zeta, \lambda} = 0$ unless $\zeta(\alpha) \geq N-mq.$\\ \\
{\it Step 3. The coefficients $a_{\pi,\lambda}$.}\\ \\ It remains to deal with the nonzero terms $c_{ \zeta, \lambda}$. There is a bijection
\[f:{\overline{\bf P}}(m\gamma) \longrightarrow \{\zeta \in {\overline{\bf P}}(m\gamma' +N\alpha)\;|\;\zeta(\alpha) \geq N-mq\},\] defined by \begin{equation} \label{1yam}
(f\pi)(\sigma)=\left\{ \begin{array}
{ccl}\pi(\sigma)&\mbox{if} \;\; \sigma \neq \alpha, \\\pi(\alpha) + N - mq &\mbox{if} \;\; \sigma = \alpha. \end{array} \right. \end{equation} Moreover if $f\pi = \zeta,$ then
\begin{eqnarray} \label{1119}|\zeta| = |\pi| + N -mq \end{eqnarray} and $e_{-\pi} = e_{-\zeta}e^{mq-N}_{- \alpha}$. Thus in Equation (\ref{12nd}) the coefficients $a_{\pi,\lambda}$ of $\theta^\lambda_{\gamma,m} $ (see (\ref{12})) are given by \begin{equation} \label{1cwp2} a_{\pi,\lambda} = c_{f(\pi),\lambda}.\end{equation} {\it Step 4. Completion of the proof.}\\ \\ We now show that the family $\theta^\lambda_{\gamma,m} $ is in good position for $w.$ For this we use Equations (\ref{159}) and (\ref{1cwp2}), noting that $p = -(\lambda + \rho,\alpha)$ depends linearly on $\lambda.$ It is clear that the coefficients $a_{\pi,\lambda}$ are polynomials in $\lambda$.
By induction $\deg a'_{\pi',\mu} \le m{\operatorname{ht}} \gamma' - |\pi'|.$ Thus using
(\ref{159}), \begin{equation} \label{1fox} \; \mbox{deg} \;\; a_{\pi,\lambda} = \;
\mbox{deg} \; c_{\zeta,\lambda} \leq \max\{j + \; \mbox{deg} \; a'_{\pi',\mu}\;|\; b_{j, \zeta}^{\pi'} \neq 0\}. \end{equation} Now if $b_{j, \zeta}^{\pi'} \neq 0$ then Equation (\ref{198}) holds. Therefore by Equation (\ref{1119}) \begin{eqnarray}
\mbox{deg} \;\; a_{\pi,\lambda} \nonumber &\leq&
\mbox{deg} \;\; a'_{\pi',\mu}+ |\pi'| + N - |\zeta|
\\ &=& \mbox{deg} \;\; a'_{\pi',\mu}+ |\pi'| - |\pi| +mq\nonumber
\end{eqnarray} Finally since $\gamma = \gamma' + q\alpha$, induction gives
(a) in Definition \ref{1sag}.\\ \\ \noindent Also, modulo terms of lower degree \begin{equation} \label{1Nw1} a_{m\pi^{\gamma},\lambda} \equiv \left( \begin{array}{c} p + mq\\ mq \end{array} \right) a'_{m\pi^{\gamma'},\mu} .\end{equation} Note that the above binomial coefficient is a polynomial of degree $mq$ in $p$.
By induction
$a'_{m\pi^{\gamma'},\mu}$ has highest term $c'\prod_{\tau \in N(u^{-1})}(\mu,\tau)^{mq(u,\tau)}$ as a polynomial in $\mu$, for a nonzero constant $c'$. Now $(\mu + \rho,\tau) = (\lambda +\rho, s_\alpha \tau)$, and $(\mu + \rho,\tau)- (\mu,\tau)$, $(\lambda +\rho, s_\alpha \tau) - (\lambda, s_\alpha \tau)$ are independent of $\mu, \lambda$. Therefore as a polynomial in $\lambda$, $a'_{m\pi^{\gamma'},\mu}$ has highest term $$c'\prod_{\tau \in N(u^{-1})}(\lambda,s_\alpha\tau)^{mq(u,\tau)} = c'\prod_{\tau \in N(u^{-1})}(\lambda,s_\alpha\tau)^{mq(w,s_\alpha\tau)}.$$
Since we assume that $\gamma$ is not the highest short root of $\fg_0$ if $\fg= G_2$ or $G(3)$, it follows from \cite{M} Table 3.4.1, that the $\alpha$-string through $\gamma'$ has length $q+1$, and $\gamma'-\alpha$ is not a root. Thus from the representation theory of $\fsl(2)$, it follows that $({\operatorname{ad \;}} e_{- \alpha})^q(\fg^{-\gamma'}) = \fg^{-\gamma}$. Hence, as $e_{-\gamma}$ is not used in the construction of $\theta_{\gamma',m},$ we can choose the notation so that $e^{(mq)}_{- m\gamma'} = e_{- m\gamma}$. Then $e^{(mq+1)}_{- m\gamma'} = 0.$ Now $q(w,\alpha) = (\gamma, \alpha^\vee) = q,$ and the degree of the binomial coefficient in (\ref{1Nw1}) as a polynomial in $p$ is $mq$, so the claim about the leading term of $a_{m\pi^{\gamma},\lambda} $ in Definition \ref{1sag} (b) follows from Equations (\ref{1Nw1}) and (\ref{Nw}).\end{proof} \begin{theorem} \label{1localshap} Suppose $\gamma = w\beta$ with $w \in W_{\rm even}$ and $\beta$ simple. Define $\Lambda$ as in \eqref{1tar}. Then there exists a family of elements $\theta^\lambda_{\gamma,m} \in U({\mathfrak n}^-)^{- m\gamma}$ for all $\lambda \in w\cdot \Lambda$ which is in good position for $w$, such that
\[\theta^\lambda_{\gamma,m}v_{\lambda} \; \mbox{is a highest weight vector in}\; M(\lambda)^{\lambda - m\gamma}.\]
\end{theorem} \noindent \begin{proof} We use induction on the length of $w.$ If $w = 1,$ we take $\theta^\lambda_{\gamma,m} = e_{-\beta}^m$ for all $\lambda.$ Now assume that $w \neq 1,$ and use the notation of Equations (\ref{1wuga}) and (\ref{1wuga1}). Suppose $\lambda =s_{\alpha}\cdot \mu =w\cdot \nu\in w\cdot\Lambda,$ and set \[p = (\mu + \rho, \alpha^{\vee}) = (\nu + \rho, u^{-1}\alpha^{\vee}), \quad(\gamma, \alpha^{\vee}) = q.\] Then $p$ and $q$ are positive integers, and $\lambda = \mu - p\alpha$ and $\gamma = \gamma' +q\alpha.$ Now $U({\mathfrak n}^-)e^p_{- \alpha} v_{\mu}$ is a submodule of $M(\mu)$ which is isomorphic to $M(\lambda)$. Also $M(\lambda)$ is uniquely embedded in $M(\mu)$, by \cite{M} Theorem 9.3.2, so we set $M(\lambda)=U({\mathfrak n}^-)e^p_{- \alpha} v_{\mu}$. Induction gives elements $\theta^{\mu}_{\gamma',m} \in U({\mathfrak n}^-)^{- \gamma'}$ which are in good position for $u,$ such that
\[v = \theta^{\mu}_{\gamma',m}v_{\mu} \in M(\mu)^{\mu - m\gamma'} \mbox{is a highest weight vector}. \]
By Lemma \ref{11768} there exists a unique element $\theta^{\lambda}_{\gamma,m} \in U({\mathfrak n}^-)^{- m\gamma}$ such that (\ref{12nd}) holds and therefore
\[e^{p+mq}_{- \alpha} v = \theta^{\lambda}_{\gamma,m} e^{p}_{- \alpha} v_{\mu} \in U({\mathfrak n}^-)e^p_{- \alpha} v_{\mu} = M(\lambda).\]
It follows from Lemma \ref{1wpfg} that the family $\theta^\lambda_{\gamma,m}$ is in good position for $w$. \end{proof} \noindent {\it Proof of Theorem \ref{1Shap}.} Let $\theta^\lambda_{\gamma,m}$ be the family of elements from Theorem \ref{1localshap}. The existence of the elements $$\theta_{\gamma, m} = \sum_{\pi \in {\overline{\bf P}}(m\gamma)} e_{-\pi} H_{\pi} \in U(\mathfrak{b}^-),$$ with $\theta_{\gamma, m}(\lambda) = \theta_{\gamma, m}^\lambda$ for all $ \lambda \in w\cdot\Lambda$ follows since $ w \cdot\Lambda$ is Zariski dense in ${\mathcal H}_{\gamma,m}.$ The claims about the coefficients $H_{\pi}$ hold since the family $\theta^\lambda_{\gamma,m}$ is in good position for $w$.
$\Box$ \subsection{Proof of Theorem \ref{1aShap}.} \label{1s.51} Now suppose that $\fg$ is contragredient, with $\Pi$ as in $(\ref{i})$. We assume that $\Pi$ contains an odd non-isotropic root and $\gamma=w\beta$ with $w \in W_{\rm nonisotropic}$, $\;\beta \in \overline{\Delta}^+_{0}\cup {\overline{\Delta}}^+_{1}$. Let $\{ U_n =U_n(\mathfrak{n}^-) \}$ be the Clifford filtration on $U=U({\mathfrak n}^-).$ Our assumptions have the following consequence. \begin{lemma} \label{1gag} Let $\gamma$ be as above and $\alpha \in \Pi_{\rm nonisotropic}$ is such that $(\gamma,\alpha)\neq 0.$ Then \begin{itemize} \item[{{\rm(a)}}] If $\alpha$ is even then $(\gamma,\alpha^\vee) = \pm 1$ \item[{{\rm(b)}}] If $\alpha$ is odd then $(\gamma,\alpha^\vee) = \pm 2$. \end{itemize}\end{lemma} \begin{proof} Left to the reader. The assumption that $\Pi$ contains an odd non-isotropic root is crucial to (a). Without this $\osp(3,2)$ would be a counterexample.\end{proof}
\begin{lemma} \label{1it} The Clifford filtration on $U(\mathfrak{n}^-)$ is stable under the adjoint action of $\mathfrak{n}_0^-,$ and satisfies ${\operatorname{ad \;}} \mathfrak{n}_1^- (U_n) \subseteq U_{n+1}$. \end{lemma} \begin{proof} Left to the reader.\end{proof}\noindent
\noindent Fix a total order on the set ${\overline{\bf P}}(\gamma)$ such that if $\pi, \sigma \in {\overline{\bf P}}(\gamma)$ and $|\pi| > |\sigma|,$ or if
$|\pi| = |\sigma|$ and ${\rm Cdeg}(\pi) > {\rm Cdeg}(\sigma)$ then $\pi$ precedes $\sigma$. \\ \\
\noindent Next we prove a Lemma relating ${\rm Cdeg}(\pi)$ to $|\pi|$ and the order defined above. \begin{lemma} Set $\Xi = {\Delta}^+_{0} \backslash \overline{\Delta}^+_{0},$ and suppose $\pi, \sigma \in {\overline{\bf P}}(\gamma)$. Let $a(\pi) = \sum_{2\delta \in \Xi}\pi(\delta).$
\begin{itemize} \label{1pig1}\item[{{\rm(a)}}] We have $2|\pi| -{\rm Cdeg}(\pi) = a(\pi)$.\end{itemize} \begin{itemize} \item[{{\rm(b)}}] $a(\pi) \le 2.$ \item[{{\rm(c)}}] If $\sigma$ precedes $\pi$, then ${\rm Cdeg}(\pi) \le {\rm Cdeg}(\sigma).$ \end{itemize} \end{lemma} \begin{proof} (a) follows since
\[ |\pi| =a(\pi)+ \sum_{\alpha \in \overline{\Delta}_{0}^+}\pi(\alpha) + \sum_{\alpha \; \rm non-isotropic}\pi(\alpha) ,\] and \[ {\rm Cdeg}(\pi) =a(\pi)+ 2\sum_{\alpha \in \overline{\Delta}_{0}^+}\pi(\alpha)+ \sum_{\alpha \; \rm non-isotropic}\pi(\alpha) .\] For (b) we note that the Lie superalgebras that have
an odd non-isotropic root $\delta$ are $G(3)$ and the family $\osp(2m+1,2n)$.
Define a group homomorphism $f:\bigoplus_{\alpha \in \Pi}{\mathbb Z}\alpha \longrightarrow \mathbb{Z}$ by setting $f( \delta) = 1$ and $f(\alpha) = 0$ for any $\alpha \in \Pi$, $\alpha \neq \delta.$ It can be checked on a case-by-case basis that if $\delta\in \Pi$, then $\delta$ occurs with coefficient at most two when a positive root $\gamma$ is written as a linear combination of simple roots. Since $a(\pi) = f(\gamma)$ for $\pi\in {\overline{\bf P}}(\gamma)$,
(b) follows. If (c) is false, then by definition of the order, we must have $|\pi| < |\sigma|$ and
${\rm Cdeg}(\pi) > {\rm Cdeg}(\sigma).$ But then by (a) this implies that $$a(\sigma) =2|\sigma| -{\rm Cdeg}(\sigma) \ge 2|\pi| -{\rm Cdeg}(\pi)+3 = a(\pi) + 3\ge 3$$ which contradicts (b). \end{proof} \begin{rem} {\rm We remark that if $\fg=G(3)$ and we work with the anti-distinguished system of simple roots, then
$W'=W_{\rm nonisotropic}$ has order 4 and is generated by the commuting reflections $s_{-\delta}$ and $s_{-3\alpha_1-2\alpha_2}$ (notation as in \cite{M} section 4.4). Thus in particular if $\beta$ is the simple isotropic root, then $|W'\beta|=4$. We leave it to the reader to check the assertions in Theorem \ref{1aShap} in this case, using Equation \eqref{1f11} and Lemma \ref{11768}. \footnote{If $\beta$ is simple non-isotropic, then $W'\beta\cap \Delta^+ =\{\beta\}$. If $\fg=G(3)$ it is more interesting to use the distinguished system of simple roots.} For the rest of this section we assume that $\fg\neq G(3)$. Thus since we assume that $\Pi$ contains an odd non-isotropic root, $\fg=\osp(2m+1,2n)$ for some $m, n$. In the proof of Lemma \ref{1ant} we use the explicit description of the roots of $\fg$ given in, for example \cite{M} Section 2.3.} \end{rem}
\begin{dfn} \label{1sog} We say that a family of elements $\theta^\lambda_{\gamma} \in U({\mathfrak n}^-)^{- \gamma}$ is {\it in good position for $w$} if for all $\lambda \in w\cdot \Lambda$ we have \begin{equation} \label{12os} \theta^\lambda_{\gamma} = \sum_{\pi \in {\overline{\bf P}}(\gamma)} a_{\pi,\lambda} e_{-\pi}, \end{equation} where the coefficients $a_{\pi,\lambda} \in \mathtt{k} $ depend polynomially on $\lambda$, and \begin{itemize} \item[{{\rm(a)}}] $2\deg \; a_{\pi,\lambda} \leq 2\ell(w) +{\rm Cdeg}(\gamma) - {\rm Cdeg}(\pi)$ \item[{{\rm(b)}}] $a_{\pi^{\gamma},\lambda}$ is a polynomial function of $\lambda$ of degree $\ell(w)$ with highest term equal to $c\prod_{\alpha \in N(w^{-1})}(\lambda,\alpha)$ for a nonzero constant $c$. \end{itemize}\end{dfn} \noindent Consider two orders on $\Delta^+$, and for $\pi \in \overline{\bf P}(\gamma)$, set $e_{-\pi} = \prod_{\alpha \in \Delta^+} e^{\pi (\alpha)}_{- \alpha}$, and $\overline{e}_{-\pi} = \prod_{\alpha \in \Delta^+} e^{\pi (\alpha)}_{- \alpha}$, the product being taken with respect to the given orders.
\begin{lemma} \label{1ant}Use the order on ${\overline{\bf P}}(\gamma)$ defined just before Lemma \ref{1pig1} to induce orders on two bases ${\bf B_1} = \{e_{-\pi}|\pi \in {\overline{\bf P}}(\gamma)\}$ and
${\bf B_2} = \{\overline{e}_{-\pi}|\pi \in {\overline{\bf P}}(\gamma)\}$ for $U(\mathfrak{n}^-)^{-\gamma}$. Write $x \in U({\mathfrak n}^-)^{-\gamma}\otimes S({\mathfrak h})$ as \[x \; = \; \sum_{\pi \in {\overline{\bf P}}(\gamma)}e_{-\pi}f_\pi \; = \; \sum_{\pi \in {\overline{\bf P}}(\gamma)}\overline{e}_{-\pi}g_\pi.\] as in Equation {\rm (\ref{1owl})}. If the coefficients $f_\pi$ satisfy \begin{itemize} \item[{{\rm(a)}}] $2\deg \; f_{\pi} \leq 2\ell(w) +{\rm Cdeg}(\gamma) - {\rm Cdeg}(\pi)$ \item[{{\rm(b)}}] $f_{\pi^{\gamma}}$ is a polynomial of degree $\ell(w)$ with highest term equal to $c\prod_{\alpha \in N(w^{-1})}h_\alpha$ for a nonzero constant $c$. \end{itemize} then the coefficients $g_\pi$ satisfy the same conditions.\end{lemma} \begin{proof} First we claim that the analog of Lemma \ref{1delorder1} holds, that is changing the order on $\Delta^+$ in the definition of the $e_{-\pi}$ requires only the introduction of terms $e_{-\sigma}$ where $\pi$ precedes $\sigma$. It is enough to check this when the two orders differ only in that two neighboring roots are switched. By definition of the Clifford filtration any even root vector is central modulo lower degree terms. Also for the distinguished or anti-distinguished Borel subalgebra the commutator of two root vectors corresponding to roots in $\overline{\Delta}^+_{1}$ is zero.
So it is enough to check the case where the two roots are $\delta_i$ and $\delta_j$ with $i\neq j$. Because an even power of $e_{-\delta_i}$ or $ e_{-\delta_j}$ is central modulo lower degree terms it is enough to check the case where $\pi({\delta_i}) = \pi({\delta_j})=1.$ Here we have $[e_{-\delta_i}, e_{-\delta_j}] = e_{-\delta_i-\delta_j}$ up to a non-zero scalar multiple. Then \[{e}_{-\pi} =-\overline{e}_{-\pi} \pm\overline{e}_{-\sigma} \mbox{ modulo terms of lower degree in the Clifford filtration,} \] where $\sigma$ is defined by \begin{equation} \label{2yam}
\sigma(\alpha)=\left\{ \begin{array}
{ccl}\pi(\alpha) -1 &\mbox{if} \;\; \alpha = \delta_i \mbox{ or } \delta_j, \\ \pi(\alpha) + 1 &\mbox{if} \;\; \alpha = \delta_i + \delta_j,\\ \pi(\alpha)& \mbox{ otherwise.} \end{array} \right.\nonumber \end{equation}
\noindent Now the claim follows since ${\rm Cdeg}(\pi) = {\rm Cdeg}(\sigma)$, but $|\pi| > |\sigma|$. The rest of the proof is the same as the proof of Lemma {\rm \ref{1delorder}}.\end{proof}
\noindent If $\gamma$ is a simple root, then $\theta_{\gamma} = e_{-\gamma}$ satisfies the conditions of Theorem \ref{1aShap}. We recall for convenience Equations \eqref{1wuga} and \eqref{1wuga1} with minor modifications. Write $\gamma = w\beta$ for some $w \in W_{\rm nonisotropic}, w\neq 1$. Assume that \begin{equation} w =s_{\alpha}u, \quad \gamma' = u\beta, \quad \gamma = w\beta = s_{\alpha} \gamma',\nonumber \end{equation}
with $\alpha \in \Pi_{\rm nonisotropic}$ and $\ell(w) =\ell(u) + 1.$ Suppose $\nu \in \Lambda$ and set \begin{equation} \mu = u\cdot \nu, \quad \lambda = w\cdot \nu = s_{\alpha}\cdot \mu.\nonumber
\end{equation} As before we assume that the positive roots are ordered so that for any partition $\pi$ we have $e_{-\pi} = e_{-\sigma}e_{-\alpha}^k$ for a non-negative integer $k$ and
a partition $\sigma$ with $\sigma(\alpha)=0$.
\begin{lemma} \label{1wpfg1} Suppose that $p, q$ and $\alpha$ are as in Lemma \ref{11768} and \begin{equation} \label{13nd} e^{p+q}_{- \alpha}\theta^{\mu}_{\gamma'} = \theta^\lambda_{\gamma}e^p_{- \alpha}. \end{equation} Then the family $\theta^\lambda_{\gamma} $ is in good position for $w$ if the family $\theta^\mu_{\gamma'}$ is in good position for $u$. \end{lemma} \begin{proof}
If $\alpha$ is odd, then $p = 2\ell -1$ is odd, and by Lemma \ref{1gag}, $q =2$. Write $\theta^{\mu}_{\gamma'}$ as in (\ref{14cha7}) and then define the $e_{- \pi'}^{(j)}$ as in (\ref{1qwer}). Set $\varepsilon(\gamma') = 1$ if $\gamma'$ is an even root and $\varepsilon(\gamma') = -1$ if $\gamma'$ is odd. Then instead of (\ref{149}) we have, by (\ref{1f11}) \begin{equation} \label{1f12} e_{-\alpha}^{2\ell+1}e_{- \pi'} = \sum^\ell _{i=0} \left( \begin{array}{c}
\ell \\
i \end{array}\right) e_{- \pi'}^{[2i+1]}e_{-\alpha}^{2(\ell - i)}
+ \varepsilon(\gamma')\sum^\ell _{i=0} \left( \begin{array}{c}
\ell \\ i \end{array}\right) e_{- \pi'}^{[2i]} e_{-\alpha}^{2\ell - 2i +1} .\end{equation} \noindent Parallel to the definition of the $b_{j, \zeta}^{\pi'}$ in (\ref{199}), we set for sufficiently large $N$
\[e_{- \pi'}^{[j]}e^{N-j}_{- \alpha}= \sum_{\zeta \in {\overline{\bf P}}(\gamma'+N\alpha)} b_{j, \zeta}^{\pi'}e_{- \zeta}.\] For $x \in \mathbb{R}$ we denote the largest integer not greater than $x$ by $\left\lfloor x \right\rfloor$. Then if $ b_{j, \zeta}^{\pi'} \neq 0,$ we have \begin{eqnarray} \label{198o} {\rm Cdeg}(\zeta) \leq {\rm Cdeg}(\pi') + N - 2\left\lfloor \frac{j}{2} \right\rfloor. \end{eqnarray}
Indeed this holds because by Lemma \ref{1it}, we have for such $j$ \begin{equation} \label{1la1} e_{- \pi'}^{[j]}e^{N-j}_{- \alpha} \in U_{{\rm Cdeg}(\pi') + N - 2\left\lfloor \frac{j}{2} \right\rfloor}.\end{equation}
Define coefficients $a'_{\pi',\mu}$ by \begin{equation} \theta^{\mu}_{\gamma',m} = \sum_{\pi' \in {\overline{\bf P}}(m\gamma')} a'_{\pi',\mu} e_{-\pi'}, \end{equation} as in \eqref{14cha7}. Then replacing (\ref{159}) we set, \begin{equation}\label{159o} c_{ \zeta,\lambda} = \sum_{\pi' \in {\overline{\bf P}}(\gamma'),\;i \geq 0} \left( \begin{array}{c} \ell \\i\end{array} \right) a'_{\pi',\mu} b_{2i+1, \zeta}^{\pi'} + \varepsilon(\gamma')\sum_{\pi' \in {\overline{\bf P}}(\gamma'),\;i \geq 0} \left( \begin{array}{c} \ell \\i\end{array} \right) a'_{\pi',\mu} b_{2i, \zeta}^{\pi'}. \end{equation} Then we obtain the following variant of Equation (\ref{1lab}) \begin{eqnarray} \label{1labos} e^{2\ell+1}_{- \alpha} \theta^{\mu}_{\gamma'} & = & \sum_{\zeta \in {\overline{\bf P}}(m\gamma + N\alpha )} c_{ \zeta, \lambda}e_{-\zeta}e^{2\ell+1-N}_{- \alpha}. \end{eqnarray} In the cancellation step we find that $c_{ \zeta,\lambda}=0$ unless $\zeta(\alpha)\ge N-2$, and the bijection
\[f:{\overline{\bf P}}(\gamma) \longrightarrow \{\zeta \in {\overline{\bf P}}(\gamma' +N\alpha)\;|\;\zeta(\alpha) \geq N-2\},\] is defined as in Equation (\ref{1yam}) with $m=1$ and $q=2.$ Then the coefficients $a_{\pi,\lambda}$ are defined as in (\ref{1cwp2}). Instead of Equations (\ref{1119}) and (\ref{1fox}) we have, when $f\pi = \zeta,$ \begin{eqnarray} \label{1119o}{\rm Cdeg}(\zeta) = {\rm Cdeg}(\pi) + N -2, \end{eqnarray} and \begin{equation} \label{1foxa} \; \mbox{deg} \;\; a_{\pi,\lambda} \;
\leq \max\{\lfloor j/2 \rfloor + \; \mbox{deg} \; a'_{\pi',\mu}\;|\; b_{j, \zeta}^{\pi'} \neq 0\}. \end{equation} Hence using (\ref{198o}) in place of (\ref{198}), and then (\ref{1119o}) we obtain, \begin{eqnarray} \label{1scat} 2\mbox{deg} \;\; a_{\pi,\lambda} &\leq& 2\mbox{deg} \;\; a'_{\pi',\mu}+ {\rm Cdeg}(\pi') + N - {\rm Cdeg}(\zeta)\\ &=& 2\mbox{deg} \;\; a'_{\pi',\mu}+ {\rm Cdeg}(\pi') - {\rm Cdeg}(\pi) +2.\nonumber \end{eqnarray} Therefore by induction, and since ${\rm Cdeg}(\gamma') ={\rm Cdeg}(\gamma) $, \begin{eqnarray} \label{1scat1} 2\mbox{deg} \;\; a_{\pi,\lambda} &\leq& 2\ell(u)+{\rm Cdeg}(\gamma') - {\rm Cdeg}(\pi) + 2\nonumber\\ &=& 2\ell(w)+{\rm Cdeg}(\gamma) - {\rm Cdeg}(\pi).\nonumber \end{eqnarray} giving condition (a) in Definition \ref{1sog}. \\ \\ The proof in the case where $\alpha$ is an even root is the same as in Section \ref{1s.5} apart from the inequalities. If $ b_{j, \zeta}^{\pi'} \neq 0,$ then instead of (\ref{198}), we have \begin{eqnarray} \label{198e} {\rm Cdeg}(\zeta) \leq {\rm Cdeg}(\pi') + 2(N - j). \end{eqnarray} Define a bijection
\[f:{\overline{\bf P}}(\gamma) \longrightarrow \{\zeta \in {\overline{\bf P}}(\gamma' +N\alpha)\;|\;\zeta(\alpha) \geq N-1\},\] as in \eqref{1yam} with $m=q=1$.
Then condition (a) follows since in place of (\ref{1119}) we have, if $f\pi =\zeta$ \begin{eqnarray} \label{11199}{\rm Cdeg}(\zeta) = {\rm Cdeg}(\pi) + 2(N -1). \end{eqnarray} We leave the proof that (b) holds in Definition \ref{1sog} to the reader. \end{proof} \noindent Theorem \ref{1aShap} follows from Lemma \ref{1ant} in the same way that Theorem \ref{1Shap} follows from Lemma \ref{1wpfg}.
\begin{rem} \label{cap}{\rm \v Sapovalov elements and their evaluations can also be constructed inside a Kostant ${\mathbb Z}-$form of the algebras $U(\mathfrak{b}^-)$ and $U(\mathfrak{n}^-)$. This is done for Lie algebras in \cite{KL}. We briefly indicate how to do it for Lie superalgebras using the ${\mathbb Z}-$form given by Fioresi and Gavarini \cite{FG}. \footnote{The point of doing this is to allow for a change in the base ring, and in particular to enable passage to positive characteristic.} For $e \in \mathfrak{n}_0^-$, $h\in \mathfrak{h}\oplus \mathtt{k}$ and $b, c\in {\mathbb N}$, define \[e^{(b)}=\frac{e^b}{b !}\in U(\mathfrak{n}^-) \mbox{ and } \left(\begin{array}{c} h \\ c \end{array}\right) = \frac{h(h-1)\ldots (h-c+1)}{c!}\in U(\mathfrak{h}).\] Let $\{x_1, \ldots x_p\}$, $\{e_1, \ldots e_q\}$ and $\{h_1, \ldots h_n\}$ be bases for $\mathfrak{n}_1^-, \mathfrak{n}_0^-$ and $\mathfrak{h}$ respectively. Then for $A\in {\mathbb Z}_2^p, B \in {\mathbb N}^q$ and $C \in {\mathbb N}^n$, consider the elements $x_A, e_B, h_C$ of $U(\mathfrak{b}^-)$ given by \[x_A = x_1^{a_1}\ldots x_p^{a_p}, \quad e_B = \frac{e_1}{b_1 !}\ldots \frac{e_q}{b_q !}, \quad h_C = \left(\begin{array}{c} h_1 \\ c_1 \end{array}\right)\ldots \left(\begin{array}{c} h_n \\ c_n \end{array}\right).\] By \cite{FG} Theorem 4.7, the products $x_A e_B h_C$ (resp. $x_A e_B $, $h_C$) form a $\mathtt{k}$-basis of $U(\mathfrak{b}^-)$ (resp. $U(\mathfrak{n}^-)$, $S(\mathfrak{h})$)
and the ${\mathbb Z}$-span of the set of all such products form a ${\mathbb Z}$-algebra $U_{\mathbb Z}(\mathfrak{b}^-)$ (resp. $U_{\mathbb Z}(\mathfrak{n}^-)$, $S_{\mathbb Z} (\mathfrak{h})$). If $({\operatorname{ad \;}} e)^{(b)}=\frac{({\operatorname{ad \;}} e)^b}{b !},$ then by the proof of \cite{H} Proposition 25.5, the operator $({\operatorname{ad \;}} e)^{(b)}$ leaves $U_{\mathbb Z}(\mathfrak{n}^-)$ and $U_{\mathbb Z}(\mathfrak{b}^-)$ invariant. Now \eqref{1cow} may be written in the form \begin{equation}
e^{(r)}a = \sum^r_{i=0} (({\operatorname{ad \;}} e)^{(i)} a)e^{(r - i)}.\nonumber\end{equation} and \eqref{121nd} is equivalent to
\begin{equation} e^{(p)}_{- \alpha}e^{mq}_{- \alpha}\theta_{\gamma',m} = \theta_{\gamma,m} e^{(p)}_{- \alpha}.\nonumber \end{equation} Turning to \eqref{1f11} and using the notation introduced immediately before that equation, we define \begin{eqnarray} x^{(2i)} = e^{(i)}= e^i/i!= x^{2i+1}/i! && x^{(2i+1)} = x^{(2i+1)}/i!\nonumber \\ z^{^{(2i)}} =\frac{({\operatorname{ad \;}} x)^{2i}z}{i!} &&z^{(2i+1)} = \frac{({\operatorname{ad \;}} x)^{2i+1}z}{i!}\nonumber \end{eqnarray} Then \eqref{1f11} can be written in the form \begin{equation} x^{(2\ell+1)}z = \sum^\ell _{i=0} z^{(2i+1)}x^{(2\ell - 2i)}
+ (-1)^{\overline z}\sum^\ell _{i=0} z^{(2i)} x^{(2\ell - 2i +1)}.\nonumber \end{equation} Finally we note that if $e= e_{-\alpha}$ is a root vector and $h \in \mathfrak{h},$ we have \[e\left(\begin{array}{c} h \\ c \end{array}\right) = \left(\begin{array}{c} h +\alpha(h)\\ c \end{array}\right)e,\] and by \cite{H} Lemma 26.1, $\left(\begin{array}{c} h +\alpha(h)\\ c \end{array}\right) \in S_{\mathbb Z} (\mathfrak{h})$. Using these equations and the inductive construction, compare \eqref{121nd} and \eqref{13nd}, it is easy to construct a \v{S}apovalov element
$\theta_{\gamma,m}$ inside $U_{\mathbb Z}(\mathfrak{b}^-).$ }\end{rem}
\section{Changing the Borel subalgebra.}\label{1sscbs}
\subsection{Adjacent Borel subalgebras.}
We consider the behavior of \v Sapovalov elements when the Borel subalgebra is changed. Let ${\mathfrak{b}'},{\mathfrak{b}}''$ be arbitrary adjacent Borel subalgebras,
and suppose \begin{equation} \label{1fggb} \fg^\alpha \subset {\mathfrak{b}}', \quad \fg^{-\alpha} \subset {\mathfrak{b}''}\end{equation} for some isotropic root $\alpha$. Let $S$ be the intersection of the sets of roots of $\mathfrak{b}'$ and $\mathfrak{b}''$, $\mathfrak{p} =\mathfrak{b}' + \mathfrak{b}''$ and $\mathfrak{r} = \bigoplus_{\beta \in S} \mathtt{k} e_{-\beta}.$ Then $\mathfrak{r}, \mathfrak{p}$ are subalgebras of $\fg$ with $\fg = \mathfrak{p} \oplus \mathfrak{r}$. Furthermore $\mathfrak{r}$ is stable under ${\operatorname{ad \;}} e_{\pm \alpha},$ and consequently, so is $U(\mathfrak{r})$.
Note that \begin{equation} \label{1rgb}\rho({\mathfrak{b}}'') = \rho({\mathfrak{b}'}) + \alpha.\end{equation} Suppose $v_\mu$ is a highest weight vector for $\mathfrak{b}'$ with weight $\mu=\mu(\mathfrak{b}')$, and let $N$ be the module generated by $v_\mu$. Then \begin{equation} \label{1ebl} e_\alpha e_{-\alpha} v_\mu = h_\alpha v_\mu =(\mu, \alpha)v_\mu . \end{equation} As is well known (see for example Corollary 8.6.3 in \cite{M}), if $(\mu +\rho({\mathfrak{b}'}), \alpha) \neq 0$, then \begin{equation} \label{1mon}\mu({\mathfrak{b}''}) + \rho({\mathfrak{b}''}) = \mu({\mathfrak{b}}') + \rho({\mathfrak{b}}').\end{equation} In this situation we call the change of Borel subalgebras from ${\mathfrak{b}'}$ to ${\mathfrak{b}}''$ (or vice-versa) a {\it typical change of Borels for $N$}. Suppose that $\gamma$ is a positive root of both ${\mathfrak{b}'}$ and ${\mathfrak{b}}''$, and that $\theta_{\gamma,m}$ is a \v Sapovalov element
corresponding to the pair $(\gamma, m)$ using the negatives of the roots of ${\mathfrak{b}}''$. Consider the Zariski dense subset $\Lambda_{\gamma, m}$ of ${\mathcal H}_{\gamma, m}$ given by \[\Lambda_{\gamma, m} = \{ \mu \in
{\mathcal H}_{\gamma, m}|(\mu + \rho, \alpha) \notin \mathbb{Z} \mbox{ for all positive roots } \alpha \neq \gamma\}.\]
Since the coefficients of $\theta_{\gamma,m}$ are polynomials, $\theta_{\gamma,m}$ is determined (as usual modulo a left ideal) by the values of $\theta_{\gamma,m}v_\mu$ for $\mu \in \Lambda_{\gamma, m}$. \\ \\ Assume $\mu \in \Lambda_{\gamma, m}$, and for brevity set $\theta =\theta_{\gamma,m}(\mu).$ Let $v_{\mu'}$ be a highest weight vector in a Verma module $M_{\small{\mathfrak{b}}'}(\mu')$ for ${\mathfrak{b}'}$ with highest weight ${\mu'}\in \Lambda_{\gamma, m}.$ Then $v_\mu =e_{-\alpha} v_{\mu'}$ is a highest weight vector for ${\mathfrak{b}}''$ which also generates $M_{{ \small{\mathfrak{b}}'}}(\mu')$. Thus we can write \[M_{{ \small{\mathfrak{b}}'}}(\mu')=M_{{ \small{\mathfrak{b}}''}}(\mu).\] Next note that $u = \theta e_{-\alpha} v_{\mu'}$ is a highest weight vector for ${\mathfrak{b}}''$, and $e_{\alpha} \theta e_{-\alpha} v_{\mu'} $ is a highest weight vector for ${\mathfrak{b}'}$ of weight $\mu' -m\gamma$ that generates the same submodule of $M_{{ \small{\mathfrak{b}}'}}(\mu')$ as $u$. We can write $\theta$ in a unique way as $\theta=e_\alpha \theta_{1} + \theta_{2}$ with $\theta_i \in U(\mathfrak{r})$. Then \begin{eqnarray} \label{1slab}
e_{\alpha}\theta e_{-\alpha} v_{\mu'} &=& e_{\alpha}\theta_2 e_{-\alpha} v_{\mu'} \nonumber\\
&=& \theta'_1 e_{-\alpha} v_{\mu'} \pm \theta'_2 v_{\mu'} \nonumber
\end{eqnarray} where $\theta'_1 = [e_{\alpha},\theta_2], \; \; \theta'_2 = (\mu',\alpha)\theta_2 \in U(\mathfrak{r}).$
Note that the term $e_{-m\pi^\gamma}$ cannot occur in $e_\alpha \theta_1$ or $\theta'_1 e_{-\alpha}. $
Allowing for possible re-order of positive roots used to define the $e_{-\pi}$
(compare Lemma \ref{1delorder}) we conclude that modulo terms of lower degree,
the coefficient of $e_{-m\pi^\gamma}$ in $e_{\alpha}\theta e_{-\alpha} v_{\mu'}$ is equal to $\pm(\mu',\alpha)$ times the coefficient of $e_{-m\pi^\gamma}$ in $\theta_2 e_\alpha v_{\mu'}$.
\subsection{Chains of Borel subalgebras.} Using adjacent Borel subalgebras it is possible to give an alternative construction of \v Sapovalov elements corresponding to an isotropic root $\gamma$ which is a simple root for some Borel subalgebra. This condition always holds in type A, but for other types, it is quite restrictive: if $\fg = \osp(2m,2n+1)$ the assumption only holds for roots of the form $\pm(\epsilon_i-\delta_j)$, while if $\fg = \osp(2m,2n)$ it holds only for these roots and the root $\epsilon_m+\delta_n$. (Theorems \ref{1Shap} and \ref{1aShap} on the other hand apply to any positive isotropic root, provided we choose the appropriate Borel subalgebra satisfying Hypothesis (\ref{i}).) \\ \\ Suppose that $\mathfrak{b}$ is a distinguished or anti-distinguished Borel subalgebra, and let $\mathfrak{b}'\in \mathcal{B}.$ Consider a sequence
\begin{equation} \label{distm} \mathfrak{b} = \mathfrak{b}^{(0)}, \mathfrak{b}^{(1)}, \ldots, \mathfrak{b}^{(r)}=\mathfrak{b}' \end{equation}
of Borel subalgebras such that $\mathfrak{b}^{(i-1)}$ and $\mathfrak{b}^{(i)}$ are adjacent for $1 \leq i \leq r$.
It follows from \cite{M} Theorem 3.1.3, that such a chain always exists. If there is no chain of adjacent Borel subalgebras connecting $\mathfrak{b}$ and $\mathfrak{b}'$ of shorter length than (\ref{distm}), we set $d(\mathfrak{b},\mathfrak{b}')=r.$
In this case there are isotropic roots $\beta_i$ such that
\begin{equation} \label{bar}\fg^{\beta_i} \subset \mathfrak{b}^{(i-1)}, \quad \fg^{-\beta_i} \subset \mathfrak{b}^{(i)}\end{equation}
for $i\in [r]$, and $\beta_1,\ldots,\beta_r$ are distinct positive roots of $\mathfrak{b}.$
Now suppose that $\gamma$ is a positive isotropic root for $\mathfrak{b}$ and a simple root in some Borel subalgebra. Then define \begin{equation} d(\gamma)= \min\left\{\;r \;\vline \mbox{ for some chain as in \eqref{distm}, } \gamma \mbox{ is a simple root in } \mathfrak{b}^{(r)}\right\}. \end{equation} \noindent Suppose that $\gamma$ is a simple isotropic root of $\mathfrak{b}^{(r)}$,
where $d(\gamma)=r,$ and suppose that $\lambda$ is a {\it general} element of $\mathcal{H}_\gamma$. Specifically we take this to mean that each change of Borels in the chain \eqref{distm} is typical for the Verma modules $M(\lambda), M(\lambda-\gamma)$, induced from $\mathfrak{b}.$ The set of all such $\lambda$ is Zariski dense in $\mathcal{H}_\gamma$.
Then set
\begin{equation} \label{tom}v_0 = v_\lambda, \;\;\lambda_{0} = \lambda, \;\; v_i= e_{-\beta_i}v_{i-1},\;\; \lambda_{i} = \lambda_{i-1}-\beta_i \;\mbox{ for }\; i \in [r],\end{equation} and \begin{equation} \label{rom} u_r =e_{-\gamma}v_r,\;\; u_i = e_{\beta_{i+1}}\ldots e_{\beta_r}u_r \;\;\mbox{ for } \;\;0\le i \le r-1.\end{equation}
Then $u_i$ and $v_i$ are highest weight vectors for the Borel subalgebra $\mathfrak{b}^{(i)}$ with weights $\lambda_{i}$ and $\lambda_{i}-\gamma$ respectively. Since each change of Borels is typical, it follows that $(\lambda_i+\rho_i,\gamma)=0$ for all $i$, where $\rho_i$ is the analog of $\rho$ for $\mathfrak{b}^{(i)}.$ The observations of the previous Subsection give the following. \begin{lemma}\label{1sea} The leading term of the coefficient of $e_{-\gamma}v_\lambda$ in $$e_{\beta_{1}}\ldots e_{\beta_r}e_{-\gamma}e_{-\beta_{r}}\ldots e_{-\beta_1}v_\lambda$$ is, up to a constant multiple, equal to $\prod_{i=1}^r (\lambda,\beta_i).$ \end{lemma} \noindent \subsection{Relation to the Weyl group.} When $\gamma$ is a simple root of some Borel subalgebra, we relate the approach to \v Sapovalov elements by change of Borel to the approach using the Weyl group. The analysis reveals a set of roots $N(\gamma)$ which has properties analogous to those of the set $N(w^{-1})$ from \eqref{nu}. \\ \\ Suppose that $\mathfrak{b}$ is the distinguished or anti-distinguished Borel subalgebra, and let $\mathfrak{b}'\in \mathcal{B}.$ Consider the chain of Borels from \eqref{distm} and the roots $\beta_1,\ldots,\beta_r$ from \eqref{bar}. \noindent Recall the definition of $q(w,\alpha)$ before Theorem \ref{1Shap}. The main result of this Subsection is the following. \begin{proposition} \label{1alp}Suppose $\gamma = w\beta$ with $w \in W_{\rm nonisotropic}$ and $\beta$ a simple isotropic root, and that $\gamma$ is a simple root of the Borel subalgebra $\mathfrak{b}^{(r)}$, where $d(\gamma)=r$. Then
\[\{\gamma-\beta_i| i=1, \ldots, r, (\gamma,\beta_i) \neq 0\} = N(w^{-1}). \] Moreover $q(w,\alpha) =1$ for all $\alpha \in N(w^{-1}).$ \end{proposition} \noindent A corollary will be used in the next Subsection. Before proving the Proposition we need several preparatory results. The first is familiar from the Lie algebra case. \begin{lemma} \label{H84} If $\gamma, \alpha$ are roots with $\alpha$ non-isotropic, and the $\alpha$-string through $\gamma$ is \[ \gamma -p\alpha, \ldots, \gamma, \ldots, \gamma + q\alpha\] then $(\gamma, \alpha^\vee) = p-q$. \end{lemma} \begin{proof} Let $\mathfrak{s}$ be the subalgebra of $\fg$ generated by the root vectors $e_\alpha$ and $e_{-\alpha}$. Then $\mathfrak{s}$ is isomorphic to $\fsl(2)$ or $\osp(1,2)$. In the first case this is proved as in the Lie algebra case, \cite{H} Prop. 8.4 (e) using the representation theory of $\fsl(2)$. Essentially the same proof also works when $\mathfrak{s} \cong \osp(1,2)$.\end{proof}
\begin{lemma} \label{new}Suppose $d(\gamma) =r$ and $\gamma$ is a simple root of the Borel subalgebra $\mathfrak{b}^{(r)}$. Then $\alpha:= \gamma-\beta_r$ is a simple even root of $\mathfrak{b}^{(r-1)}$ and $\gamma':= s_\alpha \gamma = \gamma-\alpha = \beta_r$. \end{lemma} \begin{proof} We show that the diagram below on the left (resp. right) is part of the Dynkin-Kac diagram for $\mathfrak{b}^{(r-1)}$ (resp. $\mathfrak{b}^{(r)}$).
\begin{picture}(60,40)(-53,-20) \thinlines \put(44,3){\line(1,0){39.8}} \put(39.7,2.3){\circle{8}} \put(83,-15){$\beta_r$} \put(38,-15){$\alpha$} \put(205,-15){$-\beta_r$} \put(170,-15){$\gamma$} \put(169,-0.3){$\otimes$} \put(177,3){\line(1,0){36}} \put(212.7,-0.3){$\otimes$} \put(83.7,-0.3){$\otimes$} \end{picture}
\noindent Since $d(\gamma)=r, \gamma$ is not a simple root of $\mathfrak{b}^{(r-1)}$ and thus $\gamma$ is connected to $-\beta_r$ as in the second diagram. By comparing the diagrams to those in Table 3.4.1 and the first row of Table 3.5.1 in \cite{M}, we see that the first diagram is the only possibility,
and the $\alpha$-string through $\beta_r$ consists of $\beta_r$ and $\gamma$. Therefore by Lemma \ref{H84} $(\gamma,\alpha^\vee)=1,$ and the assertions about $\gamma'$ follow easily.\end{proof}
\noindent Now suppose $\mathfrak{b}$ is any Borel in $\mathcal{B}$ and set
\begin{equation} Q(\mathfrak{b})=\{\alpha|\alpha \in \Delta_1^+(\mathfrak{b}^{\operatorname{dist}})\cap -\Delta_1^+(\mathfrak{b})\}.\end{equation} and $Q(\gamma) = Q(\mathfrak{b})$ where $d(\gamma)= d(\mathfrak{b}^{\operatorname{dist}},\mathfrak{b}).$
Next define
\begin{equation} N(\gamma) = \{\alpha \in Q(\gamma) |(\gamma, \alpha)\neq 0\}.\end{equation} This set is analogous to the set $N(w^{-1})$ from \eqref{nu}.
\begin{lemma} \label{nef} In the situation of Lemma \ref{new}, we have a disjoint union \[N(\gamma) = s_\alpha N(\gamma')\stackrel{\cdot}{\cup} \{\gamma'\}.\] \end{lemma} \begin{proof}
\noindent Without loss of generality, we can assume that $\fg$ has type A, $\alpha= \epsilon_{i}-\epsilon_{i+1}$, $\gamma'= \epsilon_{i+1}-\delta_{j'}$ and $\gamma = \epsilon_{i}-\delta_{j'}$. Set
\[R(\gamma) = \{(k, \ell')| \epsilon_{k}-\delta_{\ell'} \in N(\gamma)\}.\] Then we need to show \begin{equation} \label{def}R(\gamma) = \tau_{i,i+1}R(\gamma')\stackrel{\cdot}{\cup} \{(i+1,j')\},\end{equation} where $\tau_{i,i+1}$ is the transposition switching $i$ and $i+1$. We can write the shuffles corresponding to $\gamma$ and $\gamma'$ as the concatenations of \begin{equation} \label{hon}A,A',i,j',i+1, B, B'\;\; \mbox{ and }\;\; A,i,A',i+1, j',B,B'\end{equation} respectively, where \[A = (1,\ldots,i-1), \;\;B= (i+2,\ldots, m), \] \[ A' = (1',\ldots,j'-1),\quad B'=(j'+1,\ldots,n').\] Thus \begin{eqnarray} R(\gamma) &=& \{i\}\ti A' \;\;\bigcup \;\;(\{i+1\}\stackrel{\cdot}{\cup} B) \;\;\ti\{j'\}, \nonumber\\
R(\gamma') &=& \{i+1\}\ti A' \;\;\bigcup B \;\;\ti\{j'\}\nonumber,\end{eqnarray} \noindent and this clearly gives \eqref{def}. \end{proof}
\noindent {\it Proof of Proposition \ref{1alp}}. Suppose $\gamma = w\beta$ with $w \in W_{\rm nonisotropic}$ and $\beta$ simple isotropic. The first statement is clearly true for $\gamma$ simple. As in Lemma \ref{new} we assume that $\gamma'= s_\alpha \gamma = \gamma-\alpha = \beta_r$.
Set $A_w = \{\gamma-\beta_i| i=1, \ldots, r, (\gamma,\beta_i) \neq 0\} $ and $u= s_\alpha w$.
We show by induction on $|A_w|$ that $A_w = s_\alpha A_u \stackrel{\cdot}{\cup} \{\alpha\},$ a disjoint union.
The first equality below comes from the definition of $A_w$, the second from Lemma \ref{nef}, the third is a simple rearrangement, and the fourth holds by induction.
\begin{eqnarray} \label{uni} A_w &=& \{\gamma-\sigma|\sigma\in N(\gamma)\} \nonumber\\
&=& \{\gamma-\sigma|\sigma\in s_\alpha N(\gamma')\} \stackrel{\cdot}{\cup} \{\gamma-\gamma'\} \nonumber\\
&=& s_\alpha\{\gamma'-\tau|\tau\in N(\gamma')\} \stackrel{\cdot}{\cup} \{\alpha\} \nonumber\\ &=& s_\alpha A_u \stackrel{\cdot}{\cup} \{\alpha\}.\end{eqnarray}
This shows that $A_w$ satisfies the same recurrence as $N(w^{-1})$ from Equation (\ref{Nw}), and this gives the first statement. (Note also that since the union in \eqref{uni} is disjoint, we have $\ell(w) = \ell(u)+1$.) By induction $q(u,\sigma)=1$ for all $\alpha \in N(u^{-1})$. Thus $q(w, s_\alpha\sigma)=1$ for all such $\sigma$. Since $q(w,\alpha) =(w\beta, \alpha^\vee)=(\gamma,\alpha^\vee)=1$ by Lemma \ref{new}, the second statement holds.
$\Box$ \begin{corollary} \label{ax}The leading term of the coefficient of $e_{-\gamma}v_\lambda$ in $\theta_\gamma v_\lambda $ is given by $\prod^r_{i=1, (\gamma,\beta_i) \neq 0} (\lambda,\beta_i)$
up to a scalar multiple, independently of $\lambda\in \mathcal{H}_\gamma$.\end{corollary} \begin{proof} We work always up to a scalar multiple. Under the hypothesis of Theorem \ref{1aShap} the leading term equals $\prod_{\beta \in N(w^{-1})}(\lambda,\beta)$. If instead we have the hypothesis of Theorem \ref{1Shap}, then since $q(w,\alpha) =1$ for all $\alpha \in N(w^{-1})$ by Proposition \ref{1alp}, we obtain the same leading term. Furthermore \[\prod_{\beta \in N(w^{-1})}(\lambda,\beta)=\prod_{i: (\gamma,\beta_i) \neq 0} (\lambda,\gamma-\beta_i).\] \noindent Since $(\lambda+\rho,\gamma) =0$ this gives the result.\end{proof}
\subsection{\v Sapovalov elements and change of Borel. } There exists a unique (modulo a suitable left ideal in $U(\fg)$) \v Sapovalov element $\theta_\gamma^{(i)}$ for the Borel subalgebra $\mathfrak{b}^{(i)}$ and polynomials $g_i(\lambda), h_i(\lambda)$ such that \begin{equation} \label{1tlc} e_{-\beta_i}\theta_\gamma^{(i-1)} e_{\beta_i}v_{i} = g_i(\lambda)\theta_\gamma^{(i)}v_{i}, \end{equation} and \begin{equation} \label{1tlc1} e_{\beta_i}\theta_\gamma^{(i)}e_{-\beta_i}v_{i-1} = h_i(\lambda)\theta_\gamma^{(i-1)}v_{i-1}. \end{equation} \begin{lemma} \label{1112} For general $\lambda\in \mathcal{H}_\gamma,$ \begin{equation} \label{zxy}g_i(\lambda)h_i(\lambda)= (\lambda +\rho,\beta_{i})(\lambda+\rho-\gamma,\beta_{i}).\end{equation} \end{lemma} \begin{proof} We have using first \eqref{1ebl} and the definition of $v_i$, then \eqref{1tlc} and \eqref{1tlc1}, \begin{eqnarray} \label{1fff} (\lambda_{i-1}, \beta_{i} )(\lambda_{i-1}-\gamma,\beta_{i})\theta^{(i-1)}_\gamma v_{i-1} &=& e_{\beta_{i}}e_{-\beta_{i}} \theta_\gamma^{(i-1)} e_{\beta_{i}}e_{-\beta_{i}}v_{i-1} \nonumber \\ &=& e_{\beta_{i}}e_{-\beta_{i}} \theta_\gamma^{(i-1)} e_{\beta_{i}}v_{i} \nonumber \\ &=& g_i(\lambda)e_{\beta_{i}} \theta_\gamma^{(i)} e_{-\beta_{i}} v_{i-1} \nonumber \\ &=& g_i(\lambda)h_i(\lambda)\theta_\gamma^{(i-1)}v_{i-1} .\nonumber \end{eqnarray} \noindent Now each change of Borels is typical and $\beta_{i}$ is simple isotropic for $\mathfrak{b}^{(i-1)}$, so we have $(\lambda_{i-1}, \beta_{i} )$=$(\lambda +\rho,\beta_{i}).$ The result follows. \end{proof}
\begin{theorem} \label{1rot} Set $F(\gamma) = \{i\in [r]|(\gamma,\beta_i)=0 \}$. There is a nonzero $c\in \mathtt{k}$ such that for all $\lambda \in \mathcal{H}_\gamma$,
\[e_{\beta_{1}}\ldots e_{\beta_r}e_{-\gamma}e_{-\beta_{r}}\ldots e_{-\beta_1}v_\lambda= c\prod_{i \in F(\gamma)}(\lambda+\rho,\beta_i) \theta_\gamma v_\lambda.\]
\end{theorem} \begin{proof} It is enough to show this for all $\lambda$ in a Zariski dense subset of $\mathcal{H}_\gamma$. Thus we may assume that each change of Borels in \eqref{distm} is typical for $M(\lambda)$. Since $v_i= e_{-\beta_i}v_{i-1},$ and $u_{i-1}= e_{\beta_{i}}u_{i},$ Equation (\ref{1tlc1}) and reverse induction on $i$ yield
\begin{equation} \label{1pt} u_{i-1}=\prod_{j=i}^r h_j(\lambda)\theta_\gamma^{(i-1)}v_{i-1}. \end{equation} Hence
\begin{equation} \label{1nat} u_0 = \prod_{j=1}^r h_j(\lambda) \theta_\gamma v_{0}. \end{equation} Therefore by comparing the coefficient of $e_{-\gamma} v_\lambda$ on both sides of (\ref{1nat}), and using Lemma \ref{1sea} and Corollary \ref{ax}, we have modulo terms of lower degree, that \begin{equation} \label{1tat} \prod_{j=1}^r h_j(\lambda) = \prod_{i\in F(\gamma)} (\lambda,\beta_i). \end{equation} Now none of the functions $\lambda \longrightarrow (\lambda,\beta_i)$, for $i\in [r] $, on $\mathcal{H}_\gamma$ is a multiple of another. It follows that, up to a constant multiple, $h_j(\lambda) = (\lambda,\beta_{j}) + \mbox{ a constant}$
if $j \in F(\gamma)$, and $h_j(\lambda)$ is constant if
$j \notin F(\gamma).$ However we know from Lemma \ref{1112} that if $(\lambda,\beta_{j}) = 0$ then $h_j(\lambda)$ divides $(\lambda+\rho,\beta_{j}).$ Thus the result follows. \end{proof} \noindent Next suppose that $\mathfrak{b}'', \mathfrak{b}'$ are adjacent Borel subalgebras as in Equation {\rm (\ref{1fggb})}, and that $d(\mathfrak{b},\mathfrak{b}'')= d(\mathfrak{b},\mathfrak{b}')+1$. We can find a sequence of Borel subalgebras as in (\ref{distm}) such that $\mathfrak{b}'=\mathfrak{b}^{(i-1)}$ and
$\mathfrak{b}''=\mathfrak{b}^{(i)}$ . Adopting the notation of Equations (\ref{1tlc}) and (\ref{1tlc1}),
we can now clarify the relationship between the \v Sapovalov elements $\theta_\gamma^{(i-1)}$ and $\theta_\gamma^{(i)}$. \begin{corollary} With the above notation, we have up to constant multiples,
\begin{itemize} \item[{{\rm(a)}}] If $(\gamma,\beta_i) = 0$, then $h_i(\lambda) = g_i(\lambda) = (\lambda +\rho,\beta_{i})$. \item[{{\rm(b)}}] If $(\gamma,\beta_{i}) \neq 0$, then $h_i(\lambda) = 1$ and $g_i(\lambda) = (\lambda +\rho,\beta_{i})(\lambda +\rho-\gamma,\beta_{i}).$ \end{itemize}\end{corollary} \begin{proof} From the last paragraph of the proof it follows that, up to a constant multiple, $h_j(\lambda) = (\lambda+\rho,\beta_{j})$
if $j \in F(\gamma)$. We obtain the result by looking at the degrees of both sides in
\eqref{1tat}.\end{proof}
\section{Relations between \v Sapovalov elements.} \label{RS} \subsection{Powers of \v Sapovalov elements.} \label{1zzprod}
\subsubsection{Isotropic Roots.}\noindent First we record an elementary but important property of the \v Sapovalov element $\theta_{\gamma}$ corresponding to an isotropic root ${\gamma}$. \begin{theorem} \label{1zprod} If $\lambda \in \mathcal{H}_\gamma,$ then $ \theta^2_{\gamma} v_\lambda = 0.$ Equivalently,
$\theta_{\gamma}(\lambda-\gamma)\theta_{\gamma}(\lambda) =0.$\end{theorem} \noindent \begin{proof} It is enough to show this for all $\lambda$ in a Zariski dense subset of $\mathcal{H}_\gamma$. We assume $\gamma = w\beta$ for $\beta \in \Pi$ where $\beta$ is isotropic, and $w\in W_{\rm nonisotropic}$.
Let $\Lambda$ be the subset of ${\mathcal H}_{\beta}$ defined by Equation (\ref{1tar}), and suppose $\lambda \in w\cdot \Lambda$. The proof is by induction on the length of $w.$ We can assume that $w \neq 1.$ Replace $\mu$ with $\mu - \gamma'$ and $\lambda = s_\alpha\cdot \mu$ with $s_\alpha\cdot (\mu - \gamma') = \lambda - \gamma$ in Equation (\ref{12nd}) or (\ref{13nd}). Then $p$ is replaced by $p + q$ and we obtain \[ e^{p + 2q}_{- \alpha}\theta^{\mu - \gamma'}_{\gamma'} = \theta^{\lambda - \gamma}_{\gamma}e^{p + q}_{- \alpha}.\]
Combining this with Equation (\ref{12nd}) and using induction we have \[0 = e^{p+ 2q}_{-\alpha}\theta^{\mu - \gamma'}_{\gamma'}\theta^{\mu}_{\gamma'} = \theta^{\lambda - \gamma}_{\gamma} \theta^\lambda_{\gamma}e^{p}_{-\alpha} .\] The result follows since $e_{-\alpha}$ is not a zero divisor in $U({\mathfrak n}^-).$\end{proof} \noindent \begin{rem} {\rm From the Theorem, if $\lambda \in \mathcal{H}_\gamma,$ there is a sequence of maps \begin{equation} \label{1let} \ldots M(\lambda-\gamma)
\stackrel{\psi_{\lambda,\gamma}}{\longrightarrow} M(\lambda)
\stackrel{\psi_{\lambda+\gamma,\gamma}}{\longrightarrow} M(\lambda+\gamma) \ldots \end{equation} such that the composite of two successive maps is zero. The map $\psi_{\lambda,\gamma}$ in (\ref{1let}) is defined by $\psi_{\lambda,\gamma}(x v_{\lambda-\gamma}) = x\theta_{\gamma}v_\lambda$. The complex \eqref{1let} can have non-zero homology see Remark \ref{dig} (d).}\end{rem} \subsubsection{Evaluation of \v{S}apovalov elements.} \label{nir} Up to this point we have only evaluated \v{S}apovalov element $\theta_{\gamma,m}$ at points $\lambda \in \mathcal{H}_{m,\gamma}$.
However to study the behavior of powers of \v Sapovalov elements for non-isotropic roots, we need to evaluate at arbitrary points, $\lambda \in \mathfrak{h}^*$. Another situation where it is useful to do this is in the work of Carter \cite{Car} on the construction of orthogonal bases for non-integral Verma modules and simple modules in type A. However some care must be taken since the \v{S}apovalov element $\theta_{\gamma,m}$ is only defined modulo the ideal $U({\mathfrak b}^{-} ){\mathcal I}(\mathcal H_{\gamma, m})$. \\ \\ \noindent Suppose $e$ is locally ${\operatorname{ad \;}} $-nilpotent. Equation \eqref{1cow} leads to a formula for conjugation by $e^r$ which extends to the $1$-parameter family of automorphism $\Theta_r, r\in \mathtt{k}$ of $A_S,$ given by \begin{equation} \label{cows} \Theta_{r}(a) = \sum_{i\ge0} \left( \begin{array}{c}
r \\
i \end{array}\right) (({\operatorname{ad \;}} e)^i a)e^{- i} .\end{equation} For a generalization, see \cite{Ma} Lemma 4.3. \\ \\ Fix a non-isotropic root $\alpha$ and let $S$ be the multiplicative subset $S=\{e_{-\alpha}^n\}$ in $U=U(\mathfrak{n}^-)$, and let $U_S$ be the corresponding Ore localization. We use the generalized conjugation automorphisms $\Theta_x$ (for $x\in \mathtt{k}$) of $U_S$ defined in \eqref{cows} with $e = e_{-\alpha}$. For $x\in{\mathbb N}$ we have $\Theta_x(a) = e_{-\alpha}^x a e_{-\alpha}^{-x}.$ Thus $\Theta_x(e_{-\alpha}) = {e_{-\alpha}}$, and \begin{equation}\label{yam} \Theta_x(e_{-\alpha}ue_{-\alpha}^{-1}) = \Theta_{x+1}(u). \end{equation} As noted earlier, the value of $\Theta_x(u)$ for $x\in \mathtt{k}$ is determined by its values for $x\in{\mathbb N}$. To stress the dependence of $\Theta_x(u)$ on the root $\alpha$ we sometimes denote it by $\Theta^\alpha_x(u).$ \begin{lemma} Suppose that $\mu \in \mathcal{H}_{\gamma'}, $ $\lambda = s_\alpha\cdot \mu,\;\;\gamma = s_\alpha\gamma',\;\;q = (\gamma, \alpha^\vee).$ Assume that $ q \in \mathbb{N}\backslash \{0\}$, and if ${\alpha\in \Delta^+_{1} \backslash \overline{\Delta}^+_{1} }$, that $q = 2$. If $x = (\mu + \rho, \alpha^\vee)$ we have \begin{equation} \label{bam} \theta_{\gamma}(\lambda)= \Theta_{x}(e^{q}_{- \alpha}\theta_{\gamma'}(\mu)).\end{equation}\end{lemma} \begin{proof} If $\alpha \in\Pi\cap A(\mu)$ this follows from Lemma \ref{11768}.\end{proof} The issue is that we need to show that using this definition, we need to show that for arbitrary points, $\lambda \in \mathfrak{h}^*$, we have $\theta_{\gamma,m}(\lambda)\in U$, and not just that $ \theta_{\gamma,m}(\lambda)\in U_S,$ see Corollary \ref{c4}. Fortunately there is an easy way to ensure this in all non-exceptional cases.\\ \\
For $\beta \in \Pi$, let $W_\beta$ be the subgroup of $W$ generated by all simple reflections $s_\alpha$ where $\alpha$ is non-isotropic and $\alpha\neq\beta$. We consider a stronger version of hypotheses \eqref{iii} and \eqref{ii}. \begin{eqnarray} \label{iv} && \mbox{The root }\gamma \in \Delta^+ \mbox{ is such that for some }\\&\;\;& \beta \in \Pi, \mbox{ and } w\in W_\beta \mbox{ we have } \gamma=w\beta. \nonumber\end{eqnarray}
\begin{lemma} \label{sim} Hypothesis \eqref{iv} holds provided \begin{itemize} \item[{{\rm(a)}}] $\fg$ is a simple Lie algebra of type A-D, and $\gamma$ is a positive root. \item[{{\rm(b)}}] $\fg$ is a contragredient Lie superalgebra, the basis $\Pi$ of positive roots satisfies \eqref{i} and $\gamma$ is a positive isotropic root such that \eqref{ii} holds. \item[{{\rm(c)}}] $\fg$ is a contragredient Lie superalgebra of type A-D, the basis $\Pi$ of positive roots satisfies \eqref{i} and $\gamma$ is a positive root such that \eqref{iii} or \eqref{ii} holds. \end{itemize} \end{lemma} \begin{proof} To prove (a), it suffices to exhibit a chain of roots of the form \begin{equation} \label{jim} \gamma=\gamma_0, \gamma_1, \ldots , \gamma_m =\beta\end{equation} such that $\beta$ is simple, and that for $i \in [m]$, we have $\gamma_i = s_{\beta_i} \gamma_{i-1}$ for some $\beta_i \in \Pi \backslash \{\beta\}$ with $(\gamma_i,\beta_i^\vee) > 0$. We use the same notation as Bourbaki \cite{Bo} Chapter 6. We treat the case where $\fg$ has type $B_n$ or $C_n$ first. Here there are two root lengths and all but one of the simple roots has the same length, the exception being the rightmost root $\alpha_n$ of the Dynkin diagram. If $\gamma$ and $\alpha_n$ have the same length, then by \cite{H} Lemma 10.2, the result holds with $\beta= \alpha_ n$. If this is not the case we have $\gamma= \epsilon_i \pm \epsilon_j$ for some $i, j \in [n]$ with $i<j$. By induction we can assume that $i=1$. If $\gamma= \epsilon_1 + \epsilon_j$, the sequence of roots is \begin{eqnarray} \gamma&=& \epsilon_1 + \epsilon_j,\; \epsilon_1 + \epsilon_{j+1},\; \ldots,\; \epsilon_1 + \epsilon_n, \nonumber\\ &&\epsilon_1 - \epsilon_n,\; \epsilon_1 - \epsilon_{n-1},\; \ldots,\; \epsilon_1 - \epsilon_2 =\beta. \nonumber\end{eqnarray} It is easy to see that each term in the above sequence is obtained from its predecessor by a simple reflection, and that none of these reflections fixes the hyperplane orthogonal to $\beta$. If $\gamma = \epsilon_1 - \epsilon_j$ we need only a terminal subsequence of the roots in the second row. The same applies if $\fg$ has type $A_n$. If $\fg$ has type $D_n$, one such sequence is \begin{eqnarray} \label{gym} \gamma&=& \epsilon_1 + \epsilon_j,\; \epsilon_1 + \epsilon_{j+1},\; \ldots,\; \epsilon_1 + \epsilon_{n-1},\; \epsilon_1 + \epsilon_n, \\ &&\epsilon_1 - \epsilon_{n-1},\; \epsilon_1 - \epsilon_{n-2},\; \ldots,\; \epsilon_1 - \epsilon_2 =\beta. \nonumber\end{eqnarray} Next note that (b) follows immediately since for $\beta$ isotropic we have $W_\beta = W_{\rm nonisotropic}.$ Finally to prove (c) we can assume that $\gamma$ is non-isotropic, and then under the additional assumptions in (c) the we reduce to the statement in (a).\end{proof} \noindent Now suppose that $\gamma=w\beta$ is non-isotropic and satisfies \eqref{iii} or \eqref{ii}. Suppose also that \eqref{iv} holds. We observe that in the inductive construction of the \v{S}apovalov element $\theta_{\gamma,m}$, the assumption that $\lambda \in \mathcal{H}_{m,\gamma}$ is only used for the base case of the induction. It follows that \begin{lemma} If $\Pi_{\rm nonisotropic} \backslash \{\alpha\} =\{\alpha_1,\ldots, \alpha_r\}$, and $\lambda,\bar\lambda \in \mathfrak{h}^*$ satisfy \begin{equation}\label{yob}(\lambda+\rho,\alpha_i) =(\bar\lambda+\rho,\alpha_i) \mbox{ for } i\in[r],\end{equation}
then $\theta_{\gamma,m}(\lambda) = \theta_{\gamma,m}(\bar\lambda).$ \end{lemma} \begin{corollary} \label{c4} For all $\lambda \in \mathfrak{h}^*,$ and $j>0,$ we have $ \theta_{\gamma,j}(\lambda)\in U$.\end{corollary} \begin{proof} Suppose $\bar\lambda_j \in \mathcal{H}_{\gamma,j}$ satisfies \eqref{yob} and note that $\theta_{\gamma,j}(\bar\lambda) \in U$ by the cancellation step in the proof of Theorem \ref{1Shap}.\end{proof} \begin{rem} {\rm We end this Subsection with a discussion of the uniqueness of the evaluations of \v Sapovalov elements constructed above. We first observe that even if $\fg=\fsl(3)$, and $\gamma$ is the non-simple positive root, then unless $\lambda\in \mathcal{H}_{\gamma,m}$ the construction depends on which of the simple roots we call $\beta$. We adopt the notation of \cite{M} Exercise 9.5.2. Thus the simple the positive roots are $\alpha, \beta$ and $\gamma = \alpha +\beta.$ Choose negative root vectors $e_{-\alpha} = e_{32}, e_{-\beta} = e_{21}$ and $e_{-\gamma} = e_{31}.$ Suppose that $\mu =s_\alpha \cdot\lambda$, $\nu = s_\beta \cdot\lambda$ and that $p=(\mu+\rho,\alpha^\vee) =-(\lambda+\rho,\alpha^\vee) $, $q=(\nu+\rho,\beta^\vee) =-(\mu+\rho,\beta^\vee)$. Then using $\beta,$ (resp. $\alpha$) as the simple root in question, we obtain evaluations $\theta_{\gamma, 1}^{(\beta)}(\lambda)$ (resp. $\theta_{\gamma, 1}^{(\alpha)}(\lambda)$) $$\theta_{\gamma, 1}^{(\beta)}(\lambda) = e_{-\beta}e_{-\alpha}+(p+1)e_{-\gamma} $$ $$\theta_{\gamma, 1}^{(\alpha)}(\lambda) = e_{-\alpha} e_{-\beta}-(q+1)e_{-\gamma}. $$ These are equal iff $p+q+1=0$, that is iff $\lambda \in \mathcal{H}_{\gamma,1}.$ Coming back to the general case, we now fix the simple root $\beta$. If $\fg$ does not has type $A, B$ or $C$, then the sequence of roots in \eqref{jim} is uniquely determined by $\gamma$. For $\fg = D_n$ the sequence is unique except that the subsequence \[ \gamma_1=\epsilon_1 + \epsilon_{n-1},\; \gamma_2=\epsilon_1 + \epsilon_n,\;\gamma_3=\epsilon_1 - \epsilon_{n-1}\] of \eqref{gym} can be replaced by \begin{eqnarray} \gamma_1=\epsilon_1 + \epsilon_{n-1},\; \gamma_2'=\epsilon_1 - \epsilon_n,\; \gamma_3=\epsilon_1 - \epsilon_{n-1}.\nonumber\end{eqnarray} This is related to the fact that for $\fg= D_n$, the natural module is not uniserial as a module for a Borel subalgebra of $\fg$, \cite{M} Exercise 3.7.6. Let $\alpha' = \epsilon_{n-1}- \epsilon_n$ and $\alpha = \epsilon_{n-1} + \epsilon_n,$ be the simple roots corresponding to the two rightmost nodes of the Dynkin diagram. We have
\[ \gamma_1= \gamma_2 + \alpha',\quad \gamma_2=\gamma_3 + \alpha, \] \[ \gamma_1= \gamma_2' + \alpha,\quad \gamma_2'=\gamma_3 + \alpha'.\] Suppose that $\nu = s_\alpha s_{\alpha'} \cdot \mu.$ The construction of $\theta_{\gamma_1,m}(\nu) $ from $
\theta_{\gamma_3,m}(\mu) $ in 2 steps can be carried out in 2 different ways. We claim each way gives the same result. Suppose that $\lambda = s_{\alpha} \mu,$\;$ \lambda' = s_{\alpha'} \mu,$\; $ x = (\mu+\rho, \alpha^\vee)$ and $y = (\mu+\rho, (\alpha')^\vee).$ Then we need to show that \begin{equation} \label{num} \Theta^\alpha_x (e_{-\alpha}^m\Theta^{\alpha'}_y(e_{-\alpha'}^m\theta_{\gamma_3,m}(\mu))) = \Theta^{\alpha'}_y (e_{-\alpha'}^m\Theta^{\alpha}_x(e_{-\alpha}^m\theta_{\gamma_3,m}(\mu))).\end{equation}
To show this set $\Theta^{\alpha,\alpha'}_{x,y} = \Theta^\alpha_x \circ \Theta^{\alpha'}_y.$ Since the roots $\alpha$ and $\alpha'$ are orthogonal, we have $\Theta^{\alpha,\alpha'}_{x,y} = \Theta^{\alpha',\alpha}_{y,x}$. Also $\Theta_x^\alpha(e_{-\alpha'}) = {e_{-\alpha'}}$ and $\Theta_y^{\alpha'}(e_{-\alpha}) = {e_{-\alpha}}$. Therefore both sides of \eqref{num} are equal to
\[\Theta^{\alpha,\alpha'}_{x,y} (e_{-\alpha}^m e_{-\alpha'}^m\theta_{\gamma_3,m}(\mu)).\] }\end{rem}
\subsubsection{Non-Isotropic Roots.} \begin{theorem} \label{1calu} If $\lambda \in \mathcal{H}_{\gamma,m}$, then \begin{equation}\label{1CL}
\theta_{\gamma,m}(\lambda) =\theta_{\gamma,1}(\lambda-(m-1)\gamma)\ldots \theta_{\gamma,1}(\lambda-\gamma)\theta_{\gamma,1}(\lambda). \end{equation} Equivalently if $v_\lambda$ is a highest weight vector of weight $\lambda$ in the Verma module $M(\lambda)$ we have $\theta_{\gamma,m}v_\lambda = \theta_{\gamma,1}^m v_\lambda$. If $\lambda \in \mathcal{H}_{\gamma,m}$ this is a highest weight vector which is independent of the choice of $\beta$ in the inductive construction. \end{theorem} \begin{proof} Clearly \eqref{1CL} holds if $\gamma$ is a simple root. Suppose that \eqref{121c} holds, and assume that $\lambda\in w\cdot \Lambda$ where $\Lambda$ is defined in \eqref{1tar}. For $i=0,\ldots, m-1$ we have $$(\mu + \rho-i\gamma', \alpha^\vee)= p+iq,$$ so by the inductive definition \eqref{yam} and \eqref{bam}, \begin{eqnarray} \label{1inp} \theta_{\gamma,1}(\lambda-i\gamma)&=& \Theta_{ p + iq}(e^q _{- \alpha}\theta_{\gamma',1}(\mu-i\gamma'))\nonumber\\ &=& \Theta_{ p }(e^{(i+1)q}_{- \alpha}\theta_{\gamma',1}(\mu-i\gamma')e^{-iq}_{- \alpha}).\end{eqnarray} Now using the corresponding result for $\theta_{\gamma',m}(\mu)$ we have \begin{eqnarray} \label{tea}e^{mq}_{- \alpha}\theta_{\gamma',m}(\mu) &=& e^{mq}_{- \alpha}\theta_{\gamma',1}(\mu-(m-1)\gamma')\ldots \theta_{\gamma',1}(\mu-\gamma')\theta_{\gamma',1}(\mu)\nonumber\\
&=& e^{(m-1)q}_{- \alpha}(e^{q}_{- \alpha}\theta_{\gamma',1}(\mu-(m-1)\gamma'))e^{-(m-1)q}_{- \alpha}\cdot\nonumber\\ &&\cdot e^{(m-2)q}_{- \alpha}(e^{q}_{- \alpha}\theta_{\gamma',1}(\mu-(m-2)\gamma'))e^{-(m-2)q}_{- \alpha}\cdot\nonumber\\ &\cdots &\nonumber\\ & & \cdot \;e^{kq}_{- \alpha}(e^{q}_{- \alpha}\theta_{\gamma',1}(\mu-k\gamma') )e^{-kq}_{- \alpha}\cdot\nonumber\\ &\cdots & \nonumber\\ & &\cdot \;e^{q}_{- \alpha}(e^{q}_{- \alpha}\theta_{\gamma',1}(\mu-\gamma'))e^{-q}_{- \alpha}\cdot e^{q}_{- \alpha}\theta_{\gamma',1}(\mu).\nonumber\end{eqnarray} The result follows by applying the automorphism $\Theta_p$ to both sides and using \eqref{bam} and \eqref{1inp}.\end{proof}
\subsection{Pairs of Roots.} Next we consider relations between \v Sapovalov elements coming from different isotropic roots $\gamma$ and $\gamma'$. There are two cases depending on whether or not $(\gamma,\gamma') =0.$ \subsubsection{The non-orthogonal case.}
Let $\Pi_{\rm nonisotropic}$ be the set of nonisotropic simple roots, and ${Q}^+_{0} = \sum_{\alpha \in \Pi_{\rm nonisotropic}} \mathbb{N}
\alpha.$ If $\gamma,\gamma'$ are positive non-orthogonal isotropic roots, then $\gamma'=s_\alpha \gamma$ for some non-isotropic $\alpha\in {Q}^+_{0}$. In this situation the next result relates \v Sapovalov elements for $\gamma, \gamma'$ and $\alpha$. We set $\theta_{\alpha,0} =1.$
\begin{theorem} \label{man}Let $\gamma$ be a positive isotropic root and $\alpha$ a non-isotropic root contained in ${Q}^+_{0}.$ Let $v_{\lambda}$ be a highest weight vector in a Verma module with highest weight $\lambda,$ and set $\gamma' = s_\alpha\gamma.$ Let $p = (\lambda+\rho, \alpha^\vee) $, and assume $q = (\gamma, \alpha^\vee) \in \mathbb{N} \backslash \{0\}$. Suppose $\alpha \in A(\lambda)$ and $p = (\lambda + \rho, \alpha^\vee)$.
Then \begin{itemize} \item[{{\rm(a)}}] If $(\lambda+\rho,\gamma') = 0$ we have \begin{equation} \label{pin}\theta_{\gamma}\theta_{\alpha,p} v_{\lambda}= \theta_{\alpha,p+q} \theta_{\gamma'} v_{\lambda}.\end{equation} \item[{{\rm(b)}}] If $(\lambda+\rho,\gamma) = 0$, and $p-q\ge 0,$ we have
\begin{equation} \label{pun}\theta_{\gamma'}\theta_{\alpha,p} v_{\lambda}= \theta_{\alpha,p-q} \theta_{\gamma} v_{\lambda}.\end{equation}\end{itemize} \end{theorem}
\begin{proof} It suffices to prove (a) for all $\lambda$ in the Zariski dense subset $\Lambda$ of $\mathcal{H}_{\gamma'} \cap \mathcal{H}_{{\alpha,p}}$ given by
\[\Lambda=\{ \lambda \in \mathcal{H}_{\gamma'} \cap \mathcal{H}_{{\alpha,p}}|A(\lambda) = \{\alpha\},\; B(\lambda) = \{\gamma'\}\}.\] However for $\lambda \in \Lambda$, $M(\lambda)$ contains a unique highest weight vector of weight $s_\alpha \cdot \lambda -\gamma.$ Therefore, $\theta_{\gamma}\theta_{\alpha,p} v_{\lambda}$ and $ \theta_{\alpha,p+q} \theta_{\gamma'} v_{\lambda}$ are equal up to a scalar multiple. Now if $\pi^0$ is the partition of $\gamma+p\alpha$ with $\pi^0(\sigma)=0$ for all non-simple roots $\sigma,$ then it follows easily from the definition of \v Sapovalov elements, that $e_{-\pi^0}v_{\lambda}$ occurs with coefficient equal to one in both $\theta_{\gamma}\theta_{\alpha,p} v_{\lambda}$ and $ \theta_{\alpha,p+q} \theta_{\gamma'} v_{\lambda}$, and from this we obtain the desired conclusion. The proof of (b) is similar. \end{proof}
\begin{rems} \rm{ (i) Perhaps the most interesting case of Equation (\ref{pin}) arises when $p=0$, since then we have an inclusion between submodules of a Verma module obtained by multiplying the highest weight vector $v_{\lambda}$ by $\theta_{\gamma}$ and $\theta_{\gamma'}$. Similarly the most interesting case of Equation (\ref{pun}) is when $p=q.$ \\ (ii) In the case that $\alpha$ is a simple root, \eqref{pin} reduces to Equation (\ref{12nd}).}\end{rems} \subsubsection{The orthogonal case for special pairs.}\label{oc1}
\noindent \noindent Now we consider two isotropic roots $\gamma_{1}, {\gamma_{2}}$ such that $(\gamma_{1},{\gamma_{2}}) =0.$ In Subsection \ref{oc} we show that for $\lambda \in \mathcal{H}_{\gamma_{1},\gamma_{2}}:=\mathcal{H}_{\gamma_{1}} \cap \mathcal{H}_{{\gamma_{2}}}$ the highest weight vectors $\theta_{\gamma_{2}}\theta_{\gamma_{1}} v_\lambda$ and $\theta_{\gamma_{1}}\theta_{\gamma_{2}}v_\lambda$ are equal up to a constant multiple, the constant being a ratio of linear polynomials in $\lambda$ differing only in their constant terms, see Equation (\ref{kat}) for the exact statement. Here we consider some properties of such pairs of roots.\\ \\ If $\fg$ has two orthogonal isotropic roots then $\fg$ has defect at least two, see \cite{KaWa}. In particular $\fg$ cannot be exceptional. So we assume that $\fg = \fgl(m,n), \osp(2m,2n)$ or $\osp(2m+1,2n)$ with $m, n \ge 2$. As has become quite standard we express the roots of $\fg$ in terms of linear forms $\epsilon_i, \delta_i \in \mathfrak{h}^*$, see \cite{K} 2.5.4 or \cite{M} Chapter 2. The odd isotropic roots have the form
\begin{equation} \label{or} \overline{\Delta}_1 = \{ \pm (\epsilon_i -\delta_j)|i\in [m], \; j \in [n]\},\end{equation} if $\fg = \fgl(m,n)$ or
\begin{equation} \label{nor} \overline{\Delta}_1 = \{ \pm \epsilon_i \pm \delta_j|i \in [m], \; j \in [n]\},\end{equation} if $\fg$ is orthosymplectic. We assume the bilinear form $(\;,\;)$ on $\mathfrak{h}^*$ satisfies \begin{equation} \label{edform}(\epsilon_i,\epsilon_j) = \delta_{i,j} = - (\delta_i, \delta_j).\end{equation} We say that $\Pi$ has $\osp$ type ($\spo$ type) if it contains the simple root $\epsilon_m-\delta_1$ (resp. $\delta_n-\epsilon_1)$. We also say that an isotropic root is of type $\spo$ if it does not have the form $\epsilon_i-\delta_j$, and type $\osp$ if it does not have the form $\delta_i- \epsilon_j$. Roots of the form $\epsilon_i+\delta_j$ are of both $\osp$ and $\spo$ type. The following remarks and Lemma can easily be checked using \cite{K} Table VI, see also \cite{M} Table 3.4.3. The distinguished set of simple roots for types $B(m,n)$ with $m\ge 1$ and $D(m,n)$ have type $\spo$, and those of type $A(m,n)$ and $C(n)$ have type $\osp.$ The anti-distinguished set of simple roots, when it exists, has the opposite type. Note that the Lie superalgebra $C(n)$ has no anti-distinguished set of simple roots. We have
\begin{lemma} If $\Pi$ is either distinguished or anti-distinguished and $X$ is an orthogonal set of positive isotropic roots,
then all roots in $X$ have the same type.
\end{lemma} \noindent We say that an orthogonal pair of isotropic roots $\gamma_1, \gamma_2$ is {\it special} if both roots are simple for the same Borel subalgebra. If this is the case, we can if necessary perform an odd reflection (see \cite{PS}, \cite{S1}, Section 3) and replace one of the roots by its negative, to assume that both $\gamma_1$ and $ \gamma_2$ have the same type. From \eqref{or} or \eqref{nor}, there are positive even roots $\alpha_1, \alpha_2$ such that \begin{equation}\label{et2}\iota_1:= \gamma_1+ \alpha_1 = \gamma_2+ \alpha_2\end{equation} is a root. It follows that \begin{equation}\label{et1}\iota_2 := \gamma_1- \alpha_2= \gamma_2- \alpha_1\end{equation} is also a root, and that $\iota_1, \iota_2$ are positive orthogonal isotropic roots. Note that $\gamma_1+ \gamma_2=\iota_1+ \iota_2.$ Thus $0=(\iota_2,\iota_1+ \iota_2)= (\iota_2,\gamma_1+ \gamma_2),$ and we assume that \begin{equation} \label{rrr}(\gamma_2, \iota_2)= 1 = -(\gamma_1,\iota_2).\end{equation}
\noindent Now we can state the main result on \v Sapovalov elements for special pairs of isotropic roots. If the functions $a(\lambda)$ and $b(\lambda)$ are proportional we write $a(\lambda)\doteq b(\lambda)$. \begin{theorem} \label{fiix} If $\gamma_{1}, {\gamma_{2}}$ is a special pair, and $\iota=\iota_2$ is as in \eqref{et1}, then for all $\lambda \in \mathcal{H}_{\gamma_{1},\gamma_{2}}$ we have \begin{equation} \label{kat1}[(\lambda+\rho,\iota)-1]\theta_{\gamma_{2}}(\lambda - \gamma_{1})\theta_{\gamma_{1}}(\lambda)\ \doteq [(\lambda+\rho,\iota)+1]\theta_{\gamma_{1}}(\lambda - {\gamma_{2}}) \theta_ {\gamma_{2}}(\lambda).\end{equation}
\end{theorem} \noindent We introduce some notation that is needed for the proof. For the remainder of Section \ref{RS}, we assume that $\mathfrak{b}$ is either the distinguished or anti-distinguished Borel subalgebra. Consider a graph with vertices the set of Borel subalgebras having the same even part as $\mathfrak{b}$, with an edge connecting two Borels if they are connected by an odd reflection. If $\gamma$ is an isotropic root, set $d(\gamma)=r$ if $r$ is the shortest length of a path connecting $\mathfrak{b}$ to a Borel containing $\gamma$. Such a path will be called a {\it path leading to $\gamma.$} Thus if $d(\gamma_{1}) = r, d(\gamma_{2})= s$, there are chains of adjacent Borel subalgebras, compare \eqref{distm}
\begin{equation} \label{dir} \mathfrak{b} = \mathfrak{b}^{(0)}, \mathfrak{b}^{(1)}, \ldots,
\mathfrak{b}^{(r)}, \end{equation} and
\begin{equation} \label{dis} \mathfrak{b} = \mathfrak{b}^{[0]}, \mathfrak{b}^{[1]}, \ldots, \mathfrak{b}^{[s]} \end{equation} such that $\gamma_{1}$ and ${\gamma_{2}}$ are simple roots of $\mathfrak{b}^{(r)}$ and $ \mathfrak{b}^{[s]}$ respectively.
There are odd roots $\beta_i$, for $i \in [r]$ and $\beta_{[i]}$, for $ i \in [s]$, such that for all $i$, $$\fg^{\beta_i} \subset \mathfrak{b}^{(i-1)}, \quad \fg^{-\beta_i} \subset \mathfrak{b}^{(i)}, \quad \quad \fg^{\beta_{[i]}} \subset \mathfrak{b}^{[i-1]}, \quad \fg^{-\beta_{[i]}} \subset \mathfrak{b}^{[i]}.$$ Set
\[ F(1)= \{i \in [r] |(\gamma_{1},\beta_i)=0\}, \quad F(2)= \{i \in [s]| (\gamma_{2},\beta_{[i]})=0\}.\] We can arrange that the paths leading to $\gamma_1$ and $\gamma_2$ share an initial segment which is as long as possible. This means that for some $t$, we will have $\beta_i= \beta_{[i]},$ for $i \in [t+1]$. \\ \\ \noindent {\it Proof of Theorem \ref{fiix}}.
Consider the Zariski dense subset $\Lambda_{\gamma_{1},\gamma_{2}}$ of $\mathcal{H}_{\gamma_{1},\gamma_{2}}$ given by
\[\Lambda_{\gamma_{1},\gamma_{2}}= \{\mu \in \mathcal{H}_{\gamma_{1},\gamma_{2}}|(\mu + \rho, \alpha) \notin \mathbb{Z} \mbox{ for all positive isotropic roots } \alpha \neq \gamma_{1}, \gamma_{2}\}.\] It is enough to prove the result for $\lambda \in \Lambda_{\gamma_{1},\gamma_{2}}.$ This ensures that each change of Borels in what follows is typical for $M(\lambda)$, except where one of the roots $\gamma_{1}, \gamma_{2}$ is replaced by its negative. Next we define the following products of root vectors \[e_T = e_{\beta_{1}} \dots e_{\beta_{t+1}}, \quad e_{-T} = e_{-\beta_{t+1}} \dots e_{-\beta_{1}},\]
\[e_R = e_{\beta_{t+2}} \dots e_{\beta_r}, \quad e_{-R} = e_{-\beta_{r}} \dots e_{-\beta_{t+2}}, \] \[e_S = e_{\beta_{[t+2]}} \dots e_{\beta_{[s]}},\quad e_{-S} = e_{-\beta_{[s]}} \dots e_{-\beta_{[t+2]}}. \] The root $\iota := \beta_{t+1}$ is a simple root for the Borel subalgebra $\mathfrak{b}^{(t)}$, and so $\iota$ corresponds to a node of the Dynkin-Kac diagram for $ \mathfrak{b}^{(t)}$. Let $\mathfrak{k}$ (resp. $\mathfrak{l}$) be the subalgebra of $\fg$ generated by root vectors (positive and negative) corresponding to the nodes to the left (resp. right) of this node. We have $[\mathfrak{k},\mathfrak{l}]=0$, and hence \begin{equation} \label{bats}[e_{\pm S},e_{\pm R}] = [e_{\pm S},e_{-\gamma_{1}}]=[e_{\pm R},e_{-\gamma_{2}}] = 0.\end{equation} Set $$\Phi_S =e_Se_{-{\gamma_{2}}}e_{-S} ,\quad \;\Phi_R=e_Re_{-\gamma_{1}}e_{-R}.$$ We claim that \begin{equation}\label{god}\Phi_S \Phi_R=\pm \Phi_R \Phi_S.\end{equation} Indeed by (\ref{bats}), \begin{eqnarray} \label{ma1} \Phi_R \Phi_S &=& \pm e_Re_{-{\gamma_{1}}}e_{-R}e_Se_{-{\gamma_{2}}}e_{-S} \nonumber \\ &=& \pm e_Re_{-{\gamma_{1}}}e_Se_{-R}e_{-{\gamma_{2}}}e_{-S} \nonumber \\ &=& \pm e_Re_Se_{-{\gamma_{1}}}e_{-\gamma_{2}}e_{-R}e_{-S} \nonumber \\ &=& \pm e_Re_Se_{-{\gamma_{2}}}e_{-\gamma_{1}}e_{-R}e_{-S} \nonumber\\ &=& \pm e_Se_{-{\gamma_{2}}}e_Re_{-S}e_{-\gamma_{1}}e_{-R} \nonumber\\ &=&\pm \Phi_S \Phi_R .\nonumber \end{eqnarray} \noindent From Theorem \ref{1rot} we obtain \begin{equation} \label{nod} e_{T}\Phi_Re_{-T}v_\lambda \doteq \prod_{i \in F(1)}(\lambda+\rho,\beta_i)\theta_{\gamma_{1}} v_\lambda. \end{equation} Similarly using the fact that the expression in (\ref{nod}) is a highest weight vector of weight $\lambda-\gamma_{1}$, we have \begin{equation} \label{odd1}e_T\Phi_S e_{-T}e_{T}\Phi_Re_{-T}v_\lambda \doteq a_{\gamma_{1}}\theta_{\gamma_{2}}\theta_{\gamma_1}v_\lambda, \end{equation} where \[a_{\gamma_{1}}=\prod_{i \in F(1)}(\lambda+\rho,\beta_i) \prod_{i \in F(2)}(\lambda+\rho-\gamma_{1},\beta_{[i]}),\] Next set \begin{equation} \label{cod} b_{\gamma_{1}}=[(\lambda+\rho,\iota)+1]\prod_{i=1}^t (\lambda+\rho-\gamma_{1},\beta_i),\end{equation} and $c_{\gamma_{1}} = a_{\gamma_{1}}/b_{\gamma_{1}}.$ Switching the roles of $\gamma_1$ and $\gamma_2$, define similarly, \[a_{\gamma_{2}}=\prod_{i \in F(2)}(\lambda+\rho,\beta_{[i]}) \prod_{i \in F(1)}(\lambda+\rho-\gamma_{2},\beta_i), \quad b_{\gamma_{2}}=[(\lambda+\rho,\iota)-1]\prod_{i=1}^t (\lambda+\rho-\gamma_{2},\beta_i),\] and $c_{\gamma_{2}} = a_{\gamma_{2}}/b_{\gamma_{2}}.$ Since the leftmost factor in $e_T$ is $e_{\beta_{1}}$, we can write
$e_{T}\Phi_Re_{-T}v_\lambda =e_{\beta_{1}}w$ for some $w$ which is a highest weight vector for $\mathfrak{b}^{(1)}$ of weight $\lambda-\gamma_1-\beta_1$. On the other hand, the rightmost factor of $e_{-T}$ is $e_{-\beta_{1}}$, and from (\ref{ebl})
we have $e_{-\beta_{1}}e_{\beta_{1}}w =(\lambda+\rho-\gamma_{1},\beta_1)w$. Continuing like this with the other
factors of $e_T$ and using \eqref{rrr}, gives $e_{-T}e_{T}\Phi_Re_{-T}v_\lambda \doteq b_{\gamma_{1}}\Phi_Re_{-T}v_\lambda. $ \footnote{ The factor $(\lambda+\rho,\iota)+1$ in $b_{\gamma_{1}}$, which plays a crucial role in \eqref{kat1}, arises at this point using \eqref{rrr}, since $e_{\beta_{t+1}}=e_\iota$ is a factor of $e_T$. The differences between the first factors in the definitions of $b_{\gamma_{1}}$ and $b_{\gamma_{2}}$ are due to \eqref{rrr}.} Therefore
\begin{equation} \label{odd}e_T\Phi_S e_{-T}e_{T}\Phi_Re_{-T}v_\lambda \doteq b_{\gamma_{1}} e_T\Phi_S \Phi_Re_{-T}v_\lambda. \end{equation}
Combining this with Equation (\ref{odd1}) we have
\begin{equation} \label{rod1} e_T\Phi_S\Phi_Re_{-T}v_\lambda \doteq c_{\gamma_{1}} \theta_{\gamma_{2}}\theta_{\gamma_1}v_\lambda.\end{equation} Similarly
\begin{equation} \label{sod}e_T\Phi_R\Phi_Se_{-T}v_\lambda \doteq c_{\gamma_{2}} \theta_{\gamma_{1}}\theta_{\gamma_2}v_\lambda.\end{equation} Together with Equation (\ref{god}) this gives
\begin{equation} \label{pod}c_{\gamma_{1}} \theta_{\gamma_{2}}\theta_{\gamma_1}v_\lambda \doteq c_{\gamma_{2}} \theta_{\gamma_{1}}\theta_{\gamma_2}v_\lambda.\end{equation} The proof of Theorem \ref{fiix} is completed by the Lemma below. Since the proof is rather technical, it is followed by an example illustrating some of the notation.
$\Box$
\begin{lemma} \label{cop} We have $[(\lambda+\rho,\iota)+1]c_{\gamma_{1}} = [(\lambda+\rho,\iota)-1]c_{\gamma_{2}}$. \end{lemma} \noindent Set $G(j)= F(j)\cap [t+1]$ and $H(j) =F(j)\backslash G(j)$ for $j=1,2$. First we show \begin{sublemma} If $k\in H(1),$ then $(\gamma_2,\beta_k)=0$ and similarly if $k\in H(2),$ then $(\gamma_1,\beta_{[k]})=0.$ \end{sublemma} \begin{proof} To show this, we assume that $\gamma_1 = \epsilon_i -\delta_{i'}$ and $ \gamma_2 = \epsilon_j -\delta_{j'}$ with $i<j$ and $i'<j'$, compare also the first bullet in Theorem \ref{hog}. (Note that \eqref{et2}-\eqref{rrr} hold with $\beta_1 = \delta_{i'} -\delta_{j'}, \beta_2 = \epsilon_i-\epsilon_j,$ $\iota_1 = \epsilon_i -\delta_{j'}$ and $\iota_2 = \epsilon_j -\delta_{i'}.$) Set $\iota = \iota_2$. \\ \\ In \cite{M} 3.3, the Borel subalgebras with the same even part as $\mathfrak{b}^{\operatorname{dist}}$ are described in terms of shuffles. Here the notation is slightly different. We write a permutation $\sigma$ of the set $\{1,2, \ldots,m, 1',2',\ldots,n'\}$ in one-line notation as \[ \underline{\sigma} = (\sigma(1), \sigma(2), \ldots ,\sigma(m), \sigma(1'),\sigma(2'),\ldots,\sigma(n')).\] Then we say that $\sigma$ is a {\it shuffle} if $1,2, \ldots,m$ and $ 1',2',\ldots,n'$ are subsequences of $\underline{\sigma}$. We can write the shuffles corresponding to $\mathfrak{b}^{(t+1)}$ and $\mathfrak{b}^{(r)}$ as the concatenations of \[A,i,B,A',i',j,C,B'\;\; \mbox{ and }\;\; A,A',i,i',B,j,C,B'\] respectively, where \[A = (1,\ldots,i-1), \;\;B= (i+1,\ldots, j-1), \;\;C = (j+1,\ldots,m),\]\[ A' = (1',\ldots,i'-1),\quad B'=(i'+1,\ldots,n').\]
Then $\{\beta_k|k\in H(1)\}$ consists of all odd roots $\alpha$ which are roots of $\mathfrak{b}^{(r)}$ but not roots of $\mathfrak{b}^{(t)}$, such that $(\gamma_1,\alpha)=0$. Looking at pairs of entries which occur in opposite orders in the two shuffles, it follows that any such root $\alpha$ is contained in the set of roots
\[\{\epsilon_i-\delta_{a'}, \epsilon_b-\delta_{a'}, \epsilon_b-\delta_{i'} | a' \in A', b\in B\},\] and clearly all roots in this set are orthogonal to $\gamma_2.$ \end{proof}
\noindent{\it Proof of Lemma \ref{cop}.}
By the Sublemma, if $$z = \prod_{i \in H(1)}(\lambda+\rho,\beta_i) \prod_{i \in H(2)}(\lambda+\rho,\beta_{[i]}),$$ then \[a_{\gamma_{1}}=z\prod_{i \in G(1)}(\lambda+\rho,\beta_i) \prod_{i \in G(2)}(\lambda+\rho-\gamma_{1},\beta_{[i]}),\] and \[a_{\gamma_{2}}=z\prod_{i \in G(2)}(\lambda+\rho,\beta_{[i]}) \prod_{i \in G(1)}(\lambda+\rho-\gamma_{2},\beta_i).\]
Now denote the complements of $G(1), {G(2)}$ in $[1\ldots t+1]$ by $\overline{G(1)}, \overline{G(2)}$ respectively, and set
\[I_1 = {G(1)} \cap {G(2)}, \; I_2 = {G(1)} \cap \overline{G(2)}, \; I_3 = \overline{G(1)} \cap {G(2)}.\] Note also that $\overline{G(1)} \cap \overline{G(2)} =\{t+1\},$ but since $\alpha_{t+1} = \iota$ is not orthogonal to $\gamma_1$ it does not contribute to the product defining $a_{\gamma_{1}} $. Hence
\begin{eqnarray} a_{\gamma_{1}} &=&z\prod_{i\in I_1}(\lambda+\rho,\beta_i)^2 \prod_{i\in I_2} (\lambda+\rho,\beta_i) \prod_{i\in I_3}(\lambda+\rho-\gamma_{1},\beta_i). \nonumber \end{eqnarray} and $b_{\gamma_{1}}=[(\lambda+\rho,\iota)+1]\prod_{j = 1}^3 b_{\gamma_{1}}^{(j)}$, where for $1\le j \le 3,$
\begin{equation} \label{cads} b^{(j)}_{\gamma_{1}}=\prod_{i \in I_j} (\lambda+\rho-\gamma_1,\beta_i).\end{equation} Canceling common factors of $a_{\gamma_{1}}$ and $b_{\gamma_{1}}$, it follows that $$[(\lambda+\rho,\iota)+1]c_{\gamma_{1}} = z\prod_{i\in I_1}(\lambda+\rho,\beta_i),$$ and similarly this is equal to $[(\lambda+\rho,\iota)-1]c_{\gamma_{2}}.$
$\Box$\\
\noindent This concludes the proof of Theorem \ref{fiix}. \begin{example} {\rm Let $\fg = \fgl(4,4), \gamma_1 = \epsilon_1 -\delta_{1}, \gamma_2 = \epsilon_3 -\delta_{3}.$ Then $\iota = \epsilon_3 -\delta_{1}.$ Below we give four Dynkin-Kac diagrams for $\fg$. The first is the anti-distinguished diagram, and the single grey node corresponds to the simple root $\epsilon_4 -\delta_{1}.$ In the notation of Theorem \ref{fiix} the second, third and fourth diagrams correspond to the Borel subalgebras $\mathfrak{b}^{(t)}, \mathfrak{b}^{(r)},\mathfrak{b}^{[s]}$, and so $\iota, \gamma, \gamma'$ are simple roots of these subalgebras respectively. The node corresponding to the root $\iota$ is indicated by a square. If $\lambda \in \Lambda$, and $v_\lambda$ is a highest weight vector in a Verma module with highest weight $\lambda$, then $e_{-T}v_\lambda, e_{-R}e_{-T}v_\lambda$ and $e_{-S}e_{-T}v_\lambda$ are highest weight vectors for the Borel subalgebras $\mathfrak{b}^{(t)}, \mathfrak{b}^{(r)},\mathfrak{b}^{[s]}$ respectively.
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\noindent
We have $t=1, r=3, s= 5$, \[\beta_1 = \epsilon_4-\delta_1, \quad \beta_2 = \epsilon_3-\delta_1, \quad \beta_3 = \epsilon_2-\delta_1,\]\[ \beta_{[3]} = \epsilon_4-\delta_2, \quad \beta_{[4]} = \epsilon_4-\delta_3, \quad \beta_{[5]} = \epsilon_3-\delta_2,\] \[F(1)=\emptyset, \quad G(2) = \{1\}, \quad H(2) = \{3\}\] and \[e_T = e_{45}e_{35},\quad e_{-T}=e_{53}e_{54}, \] \[e_R = e_{25},\quad e_{-R}=e_{52} ,\] \[e_S = e_{46} e_{47} e_{36} ,\quad e_{-S}=e_{63} e_{74} e_{64}.\] } \end{example}
\subsubsection{On a result of Duflo and Serganova.}\label{cry} To extend Theorem \ref{fiix}, we need a variant of a result of Duflo and Serganova, \cite{DS} Lemma 4.4 (2). \noindent First some definitions. Suppose that $\Pi$ is a basis of simple roots for $\fg$ satisfying (\ref{i}) and recall the group $W_{\rm nonisotropic}$ from Equation (\ref{ii}), defined using $\Pi$. Let $W_1$ (resp. $W_2$) be the Weyl group of $\mathfrak{o}(\ell)$ with $\ell=2m$ or $2m+1$ (resp. the Weyl group of $\mathfrak{sp}(2n)$). Then we have $W_{\rm nonisotropic}=W_1\times S_n$ or $S_m \times W_2$ if $\Pi$ has type
$\spo$ or $\osp$ respectively.
\begin{theorem} \label{hog}Suppose $X$ is a set of orthogonal isotropic roots all having the same type as $\Pi$. Then there exists $w\in W_{\rm nonisotropic},$ such that $wX$ is contained in the set of simple roots for some Borel subalgebra. In fact if $X=|k|$, there exists $w\in W$ such that $wX$ has one if the following forms. \begin{itemize} \item $\{\epsilon_i -\delta_i\}_{i=1}^k $ if $\fg$ is of type A, or $\Pi$ is of $\osp$ type \item $\{\delta_{i}- \epsilon_i\}_{i=1}^k$ if $\Pi$ is of $\spo$ type, and either $\fg= \osp(2m,2n)$\\ with $ k<m$ or $\fg = \osp(2m+1,2n)$
\item $\{\delta_{n-k+i} -\epsilon_{i}, \delta_{n} \pm \epsilon_{m}\}_{i=1}^{k-1}$ if $\Pi$ is of $\spo$ type and $\fg=\osp(2k,2n)$. \end{itemize} \end{theorem}
\begin{proof}
First suppose \begin{equation} \label{clr} X = \{\epsilon_{f(i)} - a_{i}\delta_{h(i)}\}_{i=1}^k \end{equation}
where $a_{i} =\pm 1$ and $f:[k]\longrightarrow [m],$ and $h:[k]\longrightarrow [n]$. Since $X$ is orthogonal, $f$ and $h$ are injective, so reordering $X$, we may assume that $f$ is increasing. Then using the $W$-action we can assume $f(i)= h(i)=i$ for all $i\in [k]$. If $ \fg$ has type A, then $a_{i}=1$ for all $i$, and we have shown $X$ is conjugate to $\{\epsilon_i -\delta_i\}_{i=1}^k$. By changing the signs of the $\delta_i$ we also have the result for $\fg$ is orthosymplectic, and $\Pi$ of $\osp$ type. If $\Pi$ is of $\spo$ type, we start instead with $X = \{\delta_{h(i)}- a_{i}\epsilon_{f(i)}\}_{i=1}^k$ and argue similarly unless $\fg=\osp(2m,2n),$ where we are only allowed to change an even number of signs of the $\epsilon_i$. If $k<m$ there is still enough room for the argument to go through. Otherwise $n \ge m = k$ and we see that $X$ is conjugate to \[ \{\delta_{n-k+1} -\epsilon_{1}, \ldots \delta_{n-1} -\epsilon_{m-1}, \delta_{n} + (-1)^b \epsilon_{m}\},\] where $b$ is the number of $a_{i}$ that are negative. \end{proof}
\subsubsection{The orthogonal case in general.}\label{oc} \noindent \noindent Let $X=\{\kappa_{1},\kappa_{2}\}$ be a set of two orthogonal isotropic roots both having the same type as $\Pi$, and let $w$ be as in Theorem \ref{hog}. Then set
$\gamma_{1}=w\kappa_1,\gamma_{2}=w\gamma_2$ and let $\iota =\iota_2$ be as in \eqref{et1}. We obtain \eqref{pee} in a more precise form. \begin{theorem} \label{fiix2} For $\lambda \in \mathcal{H}_{\kappa_{1},\kappa_{2}}$ we have \begin{equation} \label{kat}[(\lambda+\rho,w^{-1}\iota)-1]\theta_{\kappa_{2}}(\lambda - \kappa_{1})\theta_{\kappa_{1}}(\lambda)\ \doteq [(\lambda+\rho,w^{-1}\iota)+1]\theta_{\kappa_{1}}(\lambda - {\kappa_{2}}) \theta_ {\kappa_{2}}(\lambda).\end{equation}
\end{theorem} \begin{proof} We use induction on the length $\ell(w)$ of $w\in W_{\rm nonisotropic}$, Theorem \ref{fiix} giving the result when $w=1$. Write $w = us_\alpha$ where $\ell(u)=\ell(w)-1$ and $\alpha$ is a simple root. Assume that $(\kappa_{1},\alpha^\vee)= q,$ $(\kappa_{2},\alpha^\vee)= q',$ and define $s_\alpha \kappa_{1} = \tau_1 =\kappa_{1}-q\alpha$, $s_\alpha \kappa_{2} = \tau_2 =\kappa_{2}-q'\alpha$. If $(\mu+\rho,\alpha^\vee)= p,$ we assume that $p,q$ and $q'$ are non-negative integers. Then set $\lambda = s_\alpha\cdot\mu = \mu -p\alpha$, $v_\lambda=e_{-\alpha}^p v_\mu$. By induction \begin{equation} \label{bad}[(\mu+\rho,u^{-1}\iota)-1]\theta_{\tau_2}(\mu - \tau_1)\theta_{\tau_1}(\mu)=[(\mu+\rho,u^{-1}\iota)+1]\theta_{\tau_1}(\mu - {\tau_2}) \theta_ {\tau_2}(\mu).\end{equation}
Then by Equation (\ref{12nd})
\begin{equation} \label{nut} \theta_{\kappa_{1}}e_{-\alpha}^{p}v_\mu=e_{-\alpha}^{p+q}\theta_{\tau_1}v_\mu,\end{equation} and
\begin{equation} \label{not} \theta_{\kappa_{2}}e_{-\alpha}^{p+q}\theta_{\tau_1}v_\mu=e_{-\alpha}^{p+q+q'}\theta_{\tau_2}\theta_{\tau_1}v_\mu. \end{equation} Hence \begin{eqnarray} \label{seat} \theta_{\kappa_{2}}\theta_{\kappa_{1}}v_\lambda &=& \theta_{\kappa_{2}}\theta_{\kappa_{1}}e_{-\alpha}^{p}v_\mu \nonumber\\ &=& \theta_{\kappa_{2}}e_{-\alpha}^{p+q}\theta_{\tau_1}v_\mu \quad \mbox{by (\ref{nut})} \nonumber\\ &=& e_{-\alpha}^{p+q+q'} \theta_{\tau_2}\theta_{\tau_1}v_\mu \quad \;\mbox{by (\ref{not})}. \end{eqnarray} Similarly by first interchanging the pairs $(\tau_1,\kappa_{1})$ and $(\tau_2,\kappa_{2})$ in Equations (\ref{not}) and (\ref{nut}) we obtain \begin{equation}\label{yea}\theta_{\kappa_{1}}\theta_{\kappa_{2}}v_\lambda=e_{-\alpha}^{p+q+q'}\theta_{\tau_1}\theta_{\tau_2}v_\mu.\end{equation} Since $(\lambda+\rho,w^{-1}\iota)= (\mu+\rho,u^{-1}\iota)$, we obtain the result from Equations (\ref{bad}), (\ref{seat}) and (\ref{yea}).\end{proof}
\section{Highest weight modules with prescribed characters.}\label{jaf} In this Subsection we assume that $\fg$ is a basic classical simple Lie superalgebra and $\Pi$ is a basis of simple roots satisfying hypothesis (\ref{i}) and either of (\ref{iii}) or (\ref{ii}). Our goal is to prove Theorem \ref{newmodgen}. Let $X$ be an orthogonal set of positive isotropic roots. When $(\lambda + \rho, \gamma) = 0$ for all $\gamma \in X,$ we construct some highest weight modules $M^X(\lambda)$ with highest weight $\lambda$ and character $ \mathtt{e}^{\lambda} p_X$. \\ \\ We begin with a sketch of the construction. Define $\xi$ and $\widetilde{\lambda}$ as in \eqref{wld}. The main step in the proof of Theorem \ref{newmodgen} is the proof that the module $M^X({\widetilde{\lambda}})_B$ defined below in \eqref{tuv} has character $\mathtt{e}^{\widetilde{\lambda}}p_X$. First we show in Corollary \ref{ratss} that the weight spaces of $M^X(\widetilde{\lambda})$ satisfy $\dim_B M^X(\widetilde{\lambda})^{\widetilde{\lambda}- \eta} \leq {\bf p}_{X}(\eta)$ for all $\eta \in Q^+$. Then if $X = Y \cup \{\gamma\}$ where $\gamma\notin Y,$ we show that there is an exact sequence \begin{equation} \label{ses}0 \longrightarrow L\longrightarrow {M^Y}(\widetilde{\lambda})_B \longrightarrow N\longrightarrow 0,\end{equation} where $L$, $N$ are homomorphic images of ${M^X}(\widetilde{\lambda}-\gamma)_B$ and ${M^X}(\widetilde{\lambda})_B$ respectively, see Equations \eqref{rod} and \eqref{yim}. The claim about the character of
$M^{X}({\widetilde{\lambda} })_{B}$ then follows by induction on $|X|$. Now let $M^{X}({\widetilde{\lambda} })_{A}$ be the submodule of $M^{X}({\widetilde{\lambda} })_{B}$ generated by the highest weight vector. We show that Theorem \ref{newmodgen} holds with \begin{equation} \label{hut} M^{X}(\lambda) = M^{X}({\widetilde{\lambda} })_{A}/TM^{X}({\widetilde{\lambda} })_{A}.\end{equation}
\subsection{An upper bound for the dimension of certain weight spaces.}\label{jf1} Let $X$ be an orthogonal set of isotropic roots. In this Subsection, $M=U(\fg)_Bv$ is a module with highest weight $\widetilde{\lambda}$ and highest weight vector $v$. We assume $\theta_\gamma v = 0$ for all $\gamma \in X.$
In this Subsection we use the usual notation for partitions, see Subsection \ref{sss7.1}. Let $L$ be the subspace of $M^{\widetilde{\lambda} -\eta}$ generated by all products $e_{-\pi} v$ with $\pi\in {\bf {P}}_{X}(\eta).$ We fix an order on the positive roots such that for any partition $\pi$ the factors of the form $e_{-\gamma}$, with
$\gamma \in \Delta_1^+$ occur to the right of the other factors in Equation (\ref{negpar}) and among these factors, those with $\gamma\in X$ occur farthest to the right.
\begin{proposition} \label{rats} For all $\pi \in {\bf {P}}(\eta)$ we have $e_{-\pi}v \in L$. \end{proposition}
\noindent For $S\subseteq \Delta^+_1$ we set $e_{-S} =\prod_{\gamma \in S}e_{-\gamma}$, and $\|S\| = \sum_{\gamma \in S}\gamma$. For any partition $\pi \in {\bf {P}}(\eta) $ we have a unique decomposition
\begin{equation} \label{unique decomposition} e_{-\pi} =e_{-\sigma}e_{-S},\end{equation} where $S\subseteq X$ and $\sigma \in {\bf {P}}_X(\eta-\|S\|)$. Because of the way we have ordered root vectors, $e_{-S}v \in L$ implies
$e_{-\pi}v \in L$. So it is enough to prove the result when $\eta = \|S\|$, equivalently $e_{-\sigma}=1$. If this is the case and \eqref{unique decomposition} holds, we set $S_\pi = S.$ \\ \\ Suppose $a_1,\ldots, a_r\in \mathfrak{n}_1$ and $x_1,\ldots,x_s\in \mathfrak{n}_0$. If $J=\{j_1<\ldots<j_t\}$ is a subset of $[s]$, we set \[x_J = x_{j_1}\ldots x_{j_t}, \quad [a_i,x]_J = [[\ldots[a_i, x_{j_1}]\ldots ]x_{j_t}].\] \begin{lemma} \label{0.1} We have \begin{eqnarray}\label{fla} a_1\ldots a_r x_1\ldots x_s &=& \sum x_{J(0)} [a_1,x]_{J(1)}\ldots [a_r,x]_{J(r)} \end{eqnarray} where the sum is over all partitions $[s]= J(0)\cup J(1)\cup \ldots \cup J(r)$ of $[s]$. \footnote{We admit the possibility that some of the $J(i)$ are empty.}
\end{lemma} \begin{proof} An easy induction.\end{proof}
\begin{lemma} \label{dfg} If $R$ and $S$ are subsets of $X$, and $\|R\|=\|S\|$ then $R=S.$
\end{lemma} \begin{proof} This is clear if $|X|=1$. Otherwise we can assume that $\fg$ is not exceptional, and then the result follows using an explicit description of the roots, compare \eqref{clr}. It is crucial that $X$ is an orthogonal set of roots.\end{proof}
\noindent {\it Proof of Proposition \ref{rats}.}
By \eqref{unique decomposition} and induction on $\eta$, it suffices to prove the result when $\eta=\|S_\pi\|$. We use induction on $s_\pi=|S_\pi|$. If $s_\pi = 0,$ then $\pi \in {\bf {P}}_X(\eta)$ and the result holds by the definition of $L$. Assume the result holds whenever $s_\sigma<s_\pi=s$. Suppose $S_\pi=\{\gamma_1,\ldots, \gamma_s\}$. Then
\begin{equation} \label{fp} 0=\theta_{\gamma_{1}}\ldots \theta_{\gamma_{s}}v =\sum_{\langle \sigma\rangle} p_{\langle \sigma\rangle}e_{-\sigma(1)} \ldots e_{-\sigma(s)}v,\end{equation} where the sum is over all $s$-tuples $\langle \sigma\rangle=(\sigma(1), \ldots , {\sigma(s)})$ with $\sigma(i) \in {\bf {P}}(\gamma_i)$ for $i\in[s]$, and $p_{\langle \sigma\rangle}=p_{\langle \sigma\rangle}(T) \in A.$ We write the term corresponding to ${\langle \sigma\rangle} $ as \begin{equation} \label{lst} e_{\langle \sigma\rangle}v=e_{-\sigma(1)} \ldots e_{-\sigma(s)}v.\end{equation}
Now one term in \eqref{fp} is $e_{-\pi}v=e_{-\gamma_1}\ldots e_{-\gamma_s}v.$ We write this term $e_{-\pi}v$ also as $e_{\langle \pi\rangle}v$. If ${\langle \sigma\rangle} \neq {\langle \pi\rangle}$ we show that $ e_{\langle \sigma\rangle}v$ is, modulo terms that can be treated by induction, a $\mathtt{k}$-multiple of $e_{\langle \pi\rangle}v$, see \eqref{ban}. Since $\deg p_{\langle \sigma\rangle}<\deg p_{\langle \pi\rangle}$ by Theorem \ref{1Shap} or Theorem \ref{1aShap}, this will give the result. \\ \\ We have $\sigma(i)(\beta_i)\neq0$ for a unique odd root $\beta_i$, and $\beta_i\le\gamma_i$. Also $e_{-\sigma(i)}=e_{-\kappa_i}e_{-\beta_i}$ for some partition $\kappa_i$ of ${\gamma_i}-\beta_i$. Thus \begin{equation} \label{m29} e_{\langle \sigma\rangle}v= e_{-\kappa_1}e_{-\beta_1}\ldots e_{-\kappa_s}e_{-\beta_s}v,\end{equation} and by repeated use of Lemma \ref{0.1}, we move all odd root vectors $e_{-\beta_i}$ to the right in \eqref{m29}. We do not however need a formula as explicit as \eqref{fla}. Instead consider the multiset
\[\mathcal{A} = \bigcup_{i=1}^s \{\alpha^{\kappa_i(\alpha)}|\alpha\in \Delta_0^+\}.\] ({The notation means that the root $\alpha$ appears $\kappa_i(\alpha)$ times.}) Then we see that modulo terms already known to satisfy the conclusion of the Proposition, $e_{\langle \sigma\rangle}v$ is a $\mathtt{k}$-linear combination of products of the form \begin{equation} \label{expr1} e_{-\omega_1} \ldots e_{-\omega_s}v,\end{equation}
where the $e_{-\omega_i} \in ({\operatorname{ad \;}} U(\mathfrak{n}_0))e_{-\beta_i}$ are odd root vectors. Thus there is a multiset partition $\bigcup_{i=1}^s \mathcal{A}_i$ of $\mathcal{A}$ such that $\omega_i=\beta_i + \sum_{\alpha\in \mathcal{A}_i}\alpha$. Note that since the bracket of two odd root vectors is even, and $[\mathfrak{n}^-_0,\mathfrak{n}_1^-]\subseteq \mathfrak{n}_1^-$, we can assume by induction on $\eta$, that any terms $x_{J(0)}$ arising from \eqref{fla} are in fact constant. We can further require that there is a partition $\omega$ of $\eta$ such that the expression in \eqref{expr1} is equal to $e_{-\omega}$ up to a permutation of the factors. Then by induction on $s_\pi$, we can assume that the $\Omega =\{\omega_1, \ldots, \omega_s\}$ is a subset of $X$, and so by Lemma \ref{dfg}, $\Omega=\{\gamma_1, \ldots, \gamma_s\}$. Now if $i\neq j$, then $[e_{-\gamma_i},e_{-\gamma_j}]=0$, so it follows that after all the rewriting we have
\begin{equation} \label{ban} e_{\langle \sigma\rangle}v = a_\sigma e_{-\pi}v+b_\sigma\end{equation}
with $a_\sigma \in \mathtt{k}$ and $b_\sigma \in L$. By \eqref{fp}, $p(T)e_{\langle \pi\rangle}\in L$ for some polynomial $p$ with
$\deg p = p_{\langle \pi\rangle}$. The result follows.
$\Box$
\begin{example} {\rm We examine the proof in the case $\fg=\fgl(2,2)$ using the notation for roots and root vectors from Section \ref{Ch8}. Let $X=\{\beta,\alpha+\beta+\gamma\}$, $\eta = \alpha+2\beta+\gamma$ and $M={M^X}(\widetilde{\lambda})_B^{\widetilde{\lambda}-\eta}$. Then Proposition \ref{rats} claims that $M$ is spanned over $B$ by all $e_{-\pi}v$ with ${\pi \in {{\bf P}_X(\eta)}}$. The most interesting stage of the proof arises when in the notation of Equation \eqref{unique decomposition}
we have $e_{-\pi} =e_{-S}$ with $S\subseteq X$ and we have already shown by induction that the result holds when $|S| =1$. Then we consider the product $0= e_{-\beta}\theta_{\alpha+\beta+\gamma} v.$ By Theorem \ref{1Shap} the coefficient of $e_{-\beta}e_{-\alpha-\beta-\gamma}v$ in $\theta_{\alpha+\beta+\gamma}v$ is quadratic in $T$. However the above product also contains the term $e_{-\beta} e_{-\alpha} e_{-\gamma} e_{-\beta}v$ with constant coefficient. Reordering creates the term $e_{-\beta}e_{-\alpha-\beta-\gamma}v$, and adding this term does not change the quadratic nature of the coefficient of $e_{-\beta}\theta_{\alpha+\beta+\gamma}v$. The product also contains the terms $e_{-\beta} e_{-\alpha} e_{-\beta-\gamma} v$ and $e_{-\beta} e_{-\gamma} e_{-\alpha-\beta}v$ both with linear coefficient in $T$. Reordering these terms creates the new terms $e_{-\beta-\gamma} e_{-\alpha-\beta}v$, $e_{-\alpha} e_{-\beta} e_{-\beta-\gamma} v$ and $e_{-\gamma} e_{-\beta} e_{-\alpha-\beta}v$. The first of these has the form $e_{-\pi}v$ with $\pi \in {{\bf{{P}}_{X}(\eta)}}$ while the other two are contained in $M$ by induction. It follows that $e_{-\beta}e_{-\alpha-\beta-\gamma}v\in M.$ } \end{example}
\begin{corollary} \label{ratss} With the same notation as the Proposition.
\begin{itemize} \item[{{\rm(a)}}] The weight space $M^{\widetilde{\lambda} -\eta}_B$ is spanned over $B$ by all $e_{-\pi}v$ with $\pi \in {{\bf{{P}}}}_{X}(\eta)$. \item[{{\rm(b)}}] $\dim_B M^{\widetilde{\lambda}- \eta} \leq {\bf p}_{X}(\eta)$.\end{itemize}\end{corollary} \begin{proof} Immediate. \end{proof}
We can in fact deduce more from the proof of the Proposition. Let $L_A$ be the subspace of ${M_A}$ spanned over $A$ by all products $e_{-\pi} v$ where $\pi\in {\bf{P}}_X =\cup_{\eta \in Q^+} {\bf{P}}_{X}(\eta)$. We are interested in situations with $L_A=M_A$. This equality does not always hold, see Theorem \ref{AA2} (f). However we show that the condition holds if $\lambda$ is replaced by $\lambda+c\xi$ for all but finitely many values of $c$. \begin{corollary} \label{nob} Given $\lambda, \xi$ as in Proposition \ref{rats}, set $\lambda_c = \lambda +c\xi$. Then for all but finitely many $c \in \mathtt{k}$, the weight space $M^{\widetilde{\lambda} -\eta}_A$ is spanned over $A$ by all $e_{-\pi}v$ with $\pi \in {\bf {P}}_{X}(\eta)$, for all $\eta \in Q^+$. \end{corollary}
\begin{proof} For simplicity we assume the hypotheses of Theorem \ref{1aShap} hold. By the choice of $\xi$ we have $(\alpha^\vee, \xi) = a_\alpha \neq 0$ for all $\alpha\in \Delta^+_0$. The leading term of $\theta_\gamma$ evaluated at $\lambda_c$ is up to a non-zero scalar multiple equal to \begin{equation} \label{grt} H_{\pi^{\gamma}}(\lambda_c) = \prod_{\alpha \in N(w^{-1})}((\alpha^\vee,\lambda) + ca_\alpha),\end{equation} and each term in the product is zero for exactly one value of $c$. For any other coefficient of $H_\pi$ of $\theta_\gamma$, $H_\pi(\lambda_c)$ is a polynomial in $c$ which has degree strictly lower than the polynomial in \eqref{grt}. With these remarks the proof is essentially the same as the proof of Theorem \ref{rats}. The role of the indeterminate $T$ is played by $c$. \end{proof}
\subsection{The Modules $M^{X}(\lambda).$} Suppose $\lambda \in \mathcal{H}_X$, and with $\widetilde{\lambda}$ as in \eqref{wld} define \begin{equation} \label{tuv} M^X({\widetilde{\lambda}})_B = M({\widetilde{\lambda}})_B/ \sum_{\gamma \in X} U(\fg)_B \theta_\gamma v_{{\widetilde{\lambda}}}.\end{equation} Then $M^X({\widetilde{\lambda}})_B$ is a $U(\fg)_B$-module generated by a highest weight vector $v^X_{{\widetilde{\lambda}}}$ (the image of $v_{{\widetilde{\lambda}}}$) with weight $\widetilde{\lambda}$. Set $M^X({\widetilde{\lambda}})_A = U(\fg)_Av^X_{{\widetilde{\lambda}}} \subset M^X({\widetilde{\lambda}})_B.$ Then
\[M^X({\widetilde{\lambda}})_A\otimes_A B = M^X({\widetilde{\lambda}})_B.\] \begin{theorem}\label{zoo}
The set \begin{equation} \label{adb} \{e_{-\pi} v^X_{\widetilde{\lambda}}|\pi \in {{\bf{\overline{P}}}}_{X}(\eta)\},\end{equation} is a $B$-basis for
the weight space $M^X({\widetilde{\lambda}})_B^{\widetilde{\lambda} -\eta}$.
\end{theorem} \begin{proof} By Proposition \ref{rats} the listed elements span $M^X({\widetilde{\lambda}})_B^{\widetilde{\lambda} -\eta}$, so it suffices to show $\dim_B M^X(\widetilde{\lambda})_B^{\widetilde{\lambda} - \eta} ={\bf p}_{X}(\eta ).$ Suppose that $X = Y \cup \{\gamma\}$ where $\gamma\notin Y.$ We show there is an exact sequence of $U(\fg)_B$-modules
\begin{equation}\label{rod} 0 \longrightarrow L \longrightarrow {M^Y}(\widetilde{\lambda})_B \longrightarrow N\longrightarrow 0\end{equation} and surjective maps
\begin{equation}\label{yim} {M^X}(\widetilde{\lambda}-\gamma)_B \longrightarrow L, \quad {M^X}(\widetilde{\lambda})_B \longrightarrow N\end{equation} Indeed if $L=U(\fg)_Bw$ where $w= \theta_\gamma v^Y_{{\widetilde{\lambda}}}$, then $L$ is a highest weight module of weight ${\widetilde{\lambda}}-\gamma.$ From Theorem \ref{fiix2}, and Theorem \ref{1zprod} we have $\theta_{\gamma'}w = 0$ for all $\gamma'\in X.$ Thus $L$ is an image of ${M^X}(\widetilde{\lambda}-\gamma)_B$. On the other hand, if $N$ is the cokernel of the inclusion of $L$ into ${M^Y}(\widetilde{\lambda})_B$, it is clear that $N$ is an image of
${M^X}(\widetilde{\lambda})_B.$ Hence using induction on $|X|$ for the first equality below, and then Equations \eqref{rod}, \eqref{yim} and Proposition \ref{rats} we obtain \begin{eqnarray} {\bf p}_{Y}(\eta) &=& \dim_B {M^Y}(\widetilde{\lambda})_B^{\widetilde{\lambda}-\eta} = \dim_B L^{\widetilde{\lambda} - \eta} +\dim_B N^{\widetilde{\lambda} - \eta}\nonumber\\ &\le& \dim_B {M^X}(\widetilde{\lambda}-\gamma)_B^{\widetilde{\lambda} - \eta} + \dim_B M^X(\widetilde{\lambda})_B^{\widetilde{\lambda} - \eta} \nonumber\\ &\le& {\bf p}_{X}(\eta - \gamma)+{\bf p}_{X}(\eta ) ={\bf p}_{Y}(\eta). \nonumber \end{eqnarray} It follows that equality holds throughout, and that the maps in (\ref{yim}) are isomorphisms. Thus the dimension of $M^X(\widetilde{\lambda})_B^{\widetilde{\lambda} - \eta}$ is as claimed.
\end{proof} \noindent {\it Proof of Theorem \ref{newmodgen}.} \noindent The module $M^{X}(\lambda)$ defined in \eqref{hut} is generated by the image of $v^X_{{{\widetilde{\lambda}}}}$ which is a highest weight vector of weight $\lambda$.
Finally the claim about the character of $M^{X}(\lambda)$ follows from Theorem \ref{zoo} and the following considerations applied to the weight spaces $K, L$ of the modules $M^{X}({\widetilde{\lambda} })_{R}$ for $R = A, B$ respectively. If $K$ is an ${A}$-submodule of a finite dimensional $B$-module $L$ such that $K\otimes_{A} B = L$, then $\dim_\mathtt{k} K/TK = \dim_B L.$
$\Box$ \begin{corollary} \label{hco} Suppose that $\gamma$ is an odd isotropic root and $\lambda \in {\mathcal{H}}_{\gamma}$. Then the kernel of the natural map $M(\lambda)\longrightarrow M^{\gamma}(\lambda)$ contains $U(\mathfrak{g})\theta_{\gamma}v_{{{\widetilde{\lambda}}}}$. \end{corollary} \begin{proof} This follows from \eqref{tuv} since $\theta_{\gamma}v^X_{{{\widetilde{\lambda}}}} \in U(\mathfrak{g})_{B}\theta_\gamma v_{\widetilde{\lambda} }^X \cap U(\fg)_{A}v_{\widetilde{\lambda} }^X.$ \end{proof}
\begin{rem}\label{bod} {\rm We note some variations on \eqref{1let}. Suppose $\lambda\in \mathcal{H}_\gamma$ for $\gamma$ an isotropic root. By \eqref{rod} and \eqref{yim} with $Y=\emptyset$, we have an exact sequence \begin{equation} \label{duh} 0\longrightarrow M^{\gamma}({\widetilde{\lambda} -\gamma})_{B}\longrightarrow {M}({\widetilde{\lambda} })_{B} \longrightarrow M^{\gamma}(\widetilde{\lambda})_{B}\longrightarrow 0\end{equation} where the first map sends $x v_{{\widetilde{\lambda} }-\gamma}$ to $ x\theta_{\gamma}v_{\widetilde{\lambda} }$. \\ \\ The first map in \eqref{duh} also induces a map $M^{\gamma}({\widetilde{\lambda} -\gamma})_{A}\longrightarrow {M}({\widetilde{\lambda} })_{A}$. Clearly the kernel $N\cap T{M}({\widetilde{\lambda} })_{A}$ of the combined map
\begin{equation} \label{bd} N=M^{\gamma}({\widetilde{\lambda} -\gamma})_{A}\longrightarrow {M}({\widetilde{\lambda} })_{A} \longrightarrow {M}({\widetilde{\lambda} })_{A}/T{M}({\widetilde{\lambda} })_{A} \cong
M(\lambda) \end{equation} contains $TN$, but the containment can be strict, see Theorem \ref{A2} (c). If this is the case the highest weight module $M^{\gamma}({{\lambda} -\gamma})=N/TN$ will not embed in the Verma module
$M(\lambda)$. }\end{rem}
\subsection{Behavior in the most general cases.}\label{gb}
In the most general case, for $\lambda \in \mathcal{H}_X$ the modules $M^{X}(\lambda) $ are simple. Beyond this case we are interested in the behavior of $M^{X}(\lambda) $ when $\lambda$ lies on certain hyperplanes in $\mathcal{H}_X$. For example in Proposition \ref{need}, we describe the general behavior when $\lambda\in\mathcal{H}_Y$ for an orthogonal set of roots $Y$ containing $X$ such that $|Y|=|X|+1$.
\begin{lemma} \label{wry}\begin{itemize} \item[{{\rm(a)}}] Any orthogonal set of isotropic roots is linearly independent.
\item[{{\rm(b)}}] If $Y$ is an orthogonal set of isotropic roots and $\beta\in \Delta^+\backslash Y,$ then $\beta \notin \mathtt{k} Y.$ \end{itemize} \end{lemma}
\begin{proof} Part (a) is left to the reader. It is similar to the proof of (b). Suppose $\beta\in \mathtt{k} Y.$ Then $\beta$ is isotropic since $\mathtt{k} Y$ is an isotropic subspace of $\mathfrak{h}^*$. Also $(\beta,Y)\neq 0$ by (a). The result is clear if $|Y|=1$, so we assume that $|Y|>1$. This implies that $\fg$ is not exceptional. Thus we have $Y = \{\pm(\epsilon_{p_j}+a_j\delta_{q_j})\}_{j=1}^k,$ $\beta=(\epsilon_{p}\pm\delta_{q})$ where $a_j=\pm 1,$ and $ p, p_j \in [m], q, q_j \in [n]$ for some $m, n$. Suppose we have a relation \[(\epsilon_{p}\pm\delta_{q}) +\sum_{j=1}^k b_j(\epsilon_{p_j}+a_j\delta_{q_j}) =0.\] Since $\epsilon_{1},\ldots ,\epsilon_{m}$ are linearly independent we have $p=p_j$ for some $j$, and $b_j=-1$, $b_\ell = 0$ for $\ell \neq j$. Thus the relation is equivalent to $\epsilon_{p}\pm\delta_{q}= \epsilon_{p_j}+a_j\delta_{q_j}.$ But this implies $q=q_j$ and $\beta \in Y$, a contradiction. \end{proof}
\noindent Suppose that $Y$ is an orthogonal set of (positive) isotropic roots. We introduce a suitable Zariski dense subset $\Lambda_Y$ of $\mathcal{H}_Y$. For $S \subseteq {\overline{Y}}= \Delta^+_1\backslash Y,$ we have, for any highest weight vector $v_\lambda$, that $e_Se_{-S}v_\lambda=p_S(\lambda)v_\lambda$ for some $p_S\in S(\mathfrak{h})$ with leading term $\prod_{\beta\in S} h_\beta$. It follows from Lemma \ref{wry} that if $V(p_S)$ is the zero locus of $p_S$, then $V(p_S) \cap \mathcal{H}_Y$ is a proper closed subset of $\mathcal{H}_Y$. (We remark that if $R$ is a subset of $S$ it need not be the case that $p_R$ divides $p_S$.) Thus \[ \Lambda_Y' = \mathcal{H}_Y \backslash \bigcup_{S \subseteq {\overline{Y}}} V(p_S)\] is a non-empty open subset of $\mathcal{H}_Y.$ Now set
\[\Lambda_Y=\{\lambda \in \Lambda_Y'|(\lambda+\rho_0,\alpha^\vee) \notin \mathbb{Z} \mbox{ for all non-isotropic roots } \alpha \}.\]
Since $\Lambda_Y$ is obtained from $\Lambda_Y'$ by deleting a discrete countable union of hyperplanes, $\Lambda_Y$ is Zariski dense in $\mathcal{H}_Y$. \\ \\ If $M$ is a $U(\fg)$-module, a {\it $\fg_0$-Verma flag} on $M$ is a filtration by $\fg_0$-submodules \begin{equation} \label{end} 0 = M_0 \subset M_1 \subset \ldots \subset M_k = M\end{equation} such that for $i = 1,\ldots, k$ $M_i/M_{i-1}$ is a Verma module $M^0(\mu_i)$ for $U(\fg_0)$ with highest weight $\mu_i$. \begin{proposition} \label{need} Suppose $Y$ is an isotropic and $Y = X\cup \{\gamma\}$, where $\gamma \notin X.$ \begin{itemize} \item[{{\rm(a)}}] If $\lambda\in \Lambda_Y$, then $M^{Y}(\lambda)$ is simple. \item[{{\rm(b)}}] If $\lambda,\lambda-\gamma \in \Lambda_Y$, there is a non-split exact sequence \[0 \longrightarrow M^{Y}(\lambda-\gamma) \longrightarrow M^{X}(\lambda) \longrightarrow M^{Y}(\lambda)\longrightarrow 0\] \end{itemize} \end{proposition}
\begin{proof}
By \cite{M} Theorem 10.4.5, $M(\lambda)$ has a $\fg_0$-Verma flag. The assumption on $\lambda$ implies that if $\mu$ is the highest weight of any factor in this series, then $(\mu+\rho_0,\alpha^\vee)$ is not an integer for any non-isotropic root $\alpha$. Thus the $U(\fg_0)$-Verma module $M^0(\mu)$ is simple. It follows that all $U(\fg_0)$-module composition factors of $M^Y(\lambda)$ are Verma modules. In addition we can order the subsets $S$ of ${\overline{Y}}$ so that the $U(\fg_0)$-submodules $N_R(\lambda) = \sum_{R\le S} U(\fg_0)e_{-S}v_\lambda$ form a Verma flag in $M^Y(\lambda)$. If $L$ is a non-zero submodule of $M^Y(\lambda)$ we can choose $R$ so that $L\cap N_R(\lambda) \neq 0,$ but $L\cap N_S(\lambda) =0$ for any $S$ which properly contains $R.$ This implies that $e_{-R}v_\lambda^Y$ is a $\fg_0$-highest weight vector in $L$. Hence (a) holds because $e_Re_{-R}v_\lambda^Y$ is a non-zero multiple of $v_\lambda^Y$. \\ \\ To prove (b), note first that $\theta_\gamma v^X_{{{\lambda}}}$ is a highest weight vector in $M^{X}(\lambda)$ so $U(\fg)\theta_\gamma v^X_{{{\lambda}}}$ has a factor module which is isomorphic to $L(\lambda-\gamma)$. However $M^{Y}(\lambda-\gamma)$ is a highest weight module with highest weight $\lambda-\gamma$. Hence $M^{Y}(\lambda-\gamma) \cong U(\fg)\theta_\gamma v^X_{{{\lambda}}}$. A similar argument shows that $M^{X}(\lambda)/U(\fg)\theta_\gamma v^X_{{{\lambda}}} \cong L(\lambda).$ This gives the sequence in (b). It does not split since $M^{X}(\lambda)$ has a unique maximal submodule. \end{proof}
\section{The submodule structure of Verma modules.} \label{SV}\ In this section we apply Theorem \ref{Jansum} to determine the structure of Verma modules in the simplest cases. First however we make some remarks concerning the Grothendieck group $K(\mathcal{O})$ and characters. For $\lambda \in \mathfrak{h}^*$, set $D(\lambda) = \lambda - Q^+$ and let $\mathcal{E}$ be the set of functions on $\mathfrak{h}^*$ which are zero outside of a finite union of sets of the form $D(\lambda)$. Elements of $\mathcal{E}$ can be written as formal linear combinations $\sum_{\lambda \in \mathfrak{h}^*} c_{\lambda} \mathtt{e}^{\lambda}$ where $\mathtt{e}^{\lambda}(\mu) = \delta_{\lambda\mu}$. We can make $\mathcal{E}$ into an algebra using the convolution product, see \cite{M} Section 8.4 for details.
Let $C(\mathcal{O})$ be the additive subgroup of $\mathcal{E}$ generated by the characters ${\operatorname{ch}\:} L(\lambda)$ for $\lambda \in \mathfrak{h}^*$. It well known that there is an isomorphism from the group $K(\mathcal{O})$ to $C(\mathcal{O})$ sending $[M]$ to ${\operatorname{ch}\:} M$ for all modules $M \in \mathcal{O},$ see \cite{J1} Satz 1.11, \cite{M} Theorem 8.4.6. Hence we can work either in $K(\mathcal{O})$ or in $\mathcal{C}(\mathcal{O})$ as it suits us. A reason for doing the former was mentioned in Section C of the introduction. However if we wish to carry out computations involving partition functions for example, then it is natural and easier to work with characters.
\begin{lemma} \label{sat} Suppose $\lambda \in \mathfrak{h}^*$. \begin{itemize} \item[{{\rm(a)}}] If $A(\lambda) = \{\alpha\}$ and $B(\lambda) = \emptyset$, then ${M}_{1}(\lambda) = {M}(s_{\alpha}\cdot \lambda)$ and ${M}_{2}(\lambda) = 0,$ \item[{{\rm(b)}}] If $B(\lambda) = \{\gamma\}$ and $A(\lambda) = \emptyset$, then ${M}_{1}(\lambda) \cong M^{\gamma}(\lambda - \gamma)$ and ${M}_{2}(\lambda) = 0.$ \end{itemize} In both cases the submodule ${M}_{1}(\lambda)$ is simple. \end{lemma}
\begin{proof} Parts (a) and (b) follow easily from (\ref{lb}). The last statement holds because ${M}_{1}(\lambda)$ is a self dual highest weight module. \end{proof} \noindent If $M$ is a finitely generated $U(\fg)$-module, then $M$ is finitely generated as a $U(\fg_0)$-module. We can give $M$ a good filtration and then define the Gelfand-Kirillov
dimension $d(M)$ and Bernstein number $e(M)$ of $M$ using the Hilbert polynomial of the associated graded $S(\fg_0)$-module.
For a Verma module $M= M(\lambda)$, we set $\mathtt{d} =d(M),$ and $\mathtt{e} = e(M)$.
Then $\mathtt{d} =|\Delta^+_{0}|,$ and $\mathtt{e} = 2^{|{\Delta^+_{1}}|}$. For a more general result on induced modules, see
\cite{M} Lemma 7.3.12. Also
if the module $N$ has character $\mathtt{e}^{\mu} {p}_{\gamma}$, we have $d(N)=\mathtt{d}$ and $e(N)=\mathtt{e}/2.$
\begin{lemma} \label{nl} If $ 0 = N_0 \subset N_1 \subset \ldots \subset N_k = {M}(\lambda)$ is a series of submodules of $M$, then $\sum_i e(N_i/N_{i-1}) = \mathtt{e}$ where the sum is over all factors $N_i/N_{i-1}$ with $d(N_i/N_{i-1})= \mathtt{d}$.\end{lemma} \begin{proof} See \cite{KrLe} Theorem 7.7.\end{proof} \noindent We say that a $U(\fg)$-module $M$ is {\it homogeneous} if $d(N) =d(M)$ for any non-zero submodule $N$ of $M.$ Any $\fg_0$ Verma module contains a unique minimal submodule, which is itself a Verma module, it follows that a $\fg_0$ Verma module is homogeneous. \begin{lemma} \label{hom} If the $U(\fg)$-module $M$ has a $\fg_0$-Verma flag, then $M$ is homogeneous. \end{lemma} \begin{proof} The argument is well-known, but we outline the proof for convenience. Consider a filtration as in \eqref{end}.
Let $N$ be a nonzero submodule of $M$ and choose $i$ minimal such that $N \cap M_i \neq 0$. Then $N \cap M_i$ is isomorphic to a nonzero submodule of $M^0(\mu_i)$ which is a homogeneous $U(\fg_0)$-module as observed above. Hence \begin{eqnarray*} d(M^0(\mu_i)) &=& d(N \cap M_i) \\
& \leq & d(N) \leq d(M) . \end{eqnarray*} The result follows since $d(M^0(\mu_i)) = d(M) = \mathtt{d}$. \end{proof}
\begin{corollary} \label{veho} Any Verma module $M= M(\lambda)$ for $U(\fg)$ is homogeneous. \end{corollary} \begin{proof} By \cite{M} Theorem 10.4.5, $M$ has a $\fg_0$-Verma flag. \end{proof}
\begin{lemma} \label{you}Suppose $A(\lambda) = \{\alpha\}$ and $B(\lambda) = \{\kappa\}$, and set $\mu =s_\alpha \cdot \lambda.$ Then
\begin{itemize} \item[{{\rm(a)}}] $A(\mu) = \emptyset$ and $B(\mu) = \{s_\alpha \kappa\}$. \item[{{\rm(b)}}] The unique maximal ${}$ submodule of $\;M(\mu)\;$ is simple and isomorphic to $\;{M}^{s_\alpha \kappa}(s_\alpha\cdot(\lambda -\kappa)).$ \item[{{\rm(c)}}] If $N$ is a submodule of ${M}(\lambda)$ whose character is equal to $ {\operatorname{ch}\:} M^\kappa({\lambda -\kappa})$ then
${M}(\mu)\cap N$ is the unique proper submodule of ${M}(\mu).$ \end{itemize}
\end{lemma}
\begin{proof} First $B(\mu) = \{s_\alpha \kappa\}$ since the bilinear form $(\;,\;)$, is $W$-invariant. We prove $A(\mu) = \emptyset,$ in the case that $\Delta$ does not contain a non-isotropic odd root. Let $\Delta_\lambda = \{ \gamma \in \Delta_0| (\lambda+\rho,\gamma^\vee) \in {\mathbb Z}\}$ be the integral subroot system determined by $\lambda$. Then $\Delta_\lambda = \Delta_\mu$, and to show $A(\mu) = \emptyset$ we may assume that $\Delta_\lambda = \Delta_0$. Let $\mathtt P$ be the set of indecomposable roots in $\Delta_0^+$. Then $\mathtt P = \{\omega_1, \omega_2, \ldots, \omega_r\}$ is a basis for the root system $\Delta_0$. Since $A(\lambda) = \{\alpha\}$ it follows that $(\lambda+\rho, \omega_i^\vee)>0$ for some $i$ and then that $\alpha=\omega_i$. But then $s_\alpha$ permutes the positive roots other than $\alpha$ and $A(\mu) = \emptyset$ follows.
Part (b) follows from Lemma \ref{sat} (b) applied to $M(\mu)$. To prove (c) use the sum formula (\ref{lb}) in the form \begin{equation} \label{lib} \sum_{i > 0} [{M}_{i}(\lambda)] =[{M}(\mu)] + [M^{\kappa}(\lambda -\kappa)],\end{equation} Set $N' = {M}(\mu)\cap N.$ If $N'= 0$, then $M_1$ contains ${M}(\mu) \oplus N$. Combined with (\ref{lib}) and the hypothesis on ${\operatorname{ch}\:} N$ this implies that $M_2=0$ and ${M}(\mu) \oplus N=M_1$. However $M_1/M_2$ is self-dual, which is impossible since it has ${M}(\mu)$ as a direct summand, and by (b) ${M}(\mu)$ is a non-simple highest weight module. Now $\mathtt{e}(N) = \mathtt{e}/2,$ so $N$ cannot contain a Verma submodule. Since
$M(\mu)$ has length two by (b) the result follows.
\end{proof}
\noindent Next for $q = (\gamma, \alpha^\vee) \in \mathbb{N} \backslash \{0\}$ where $\gamma$ is isotropic, set $\gamma'=s_\alpha \gamma = \gamma-q\alpha.$ We consider Verma modules $M(\lambda)$ such that \begin{equation} \label{but} A(\lambda) =\emptyset, \quad B(\lambda) =\{\gamma, \gamma'\} .\end{equation} Note that $B(\lambda) =\{\gamma, \gamma'\}$ implies that $(\lambda + \rho, \alpha^\vee)=0$. Now consider the set
\begin{equation} \label{Gld}\Lambda=\{\lambda| (\ref{but}) \mbox{ holds and } A(\lambda-\gamma) =\emptyset, B(\lambda-\gamma) =\{\gamma\}, A(\lambda+\gamma') =\emptyset, B(\lambda+\gamma') =\{\gamma'\} \}.\end{equation}
Since $(\lambda+\rho-\gamma, \alpha^\vee) = (\lambda+\rho+\gamma', \alpha^\vee)=-q$, $\Lambda$ is a Zariski dense subset in $\mathcal H_{\gamma} \cap \mathcal H_{\gamma'}.$
\begin{lemma} \label{nid} Suppose $\lambda \in \Lambda$, and that $K$ is the kernel of the map
${M}(\lambda)\longrightarrow {M}^{\gamma'}(\lambda).$ Let $L$ be the socle of $M(\lambda)$. Then the Jantzen filtration on ${M}(\lambda)$ satisfies
$${M}_{3}(\lambda)=0\subset L={M}_{2}(\lambda)\subset K={M}_{1}(\lambda)\subset{M}(\lambda).$$ This is the unique composition series for $M(\lambda)$. Furthermore $K=U(\fg)\theta_{\gamma'} v_\lambda,\;$ $L=U(\fg)\theta_\gamma v_\lambda\cong M^{\gamma}(\lambda- \gamma) \cong L(\lambda- \gamma)$, $K/L \cong L(\lambda-\gamma')$ and $d(K/L)<\mathtt{d}$. \end{lemma} \noindent \begin{proof} The hypotheses and Lemma \ref{sat} imply that $M(\lambda-\gamma)$ has length two with simple top $M^\gamma(\lambda-\gamma)$ and $M(\lambda+\gamma')$ has length two with simple socle $M^{\gamma'}(\lambda)$. Thus $M^{\gamma'}(\lambda)$ is the unique simple image of $M(\lambda)$, so $M_1(\lambda)=K$. Note $K$ has the same character as $M^{\gamma'}(\lambda -\gamma')$. Thus $d(K) =\mathtt{d}$ and $e(K) = \mathtt{e}/2.$ Now the socle of $M(\lambda)$ contains a copy of $M^\gamma(\lambda-\gamma)$, and we have
$d(M^\gamma(\lambda-\gamma)) =\mathtt{d}$ and $e(M^\gamma(\lambda-\gamma)) = \mathtt{e}/2.$ From Lemma \ref{hom} we see that $L$ is isomorphic to $M^\gamma(\lambda-\gamma)\cong L(\lambda-\gamma)$. Since $\theta_\gamma v_\lambda$ is a highest weight vector with weight $\lambda-\gamma$ we have $L = U(\fg)\theta_\gamma v_\lambda$. It follows from this and (\ref{lb}) that in the Grothendieck group $K(\mathcal{O})$
\begin{eqnarray} \label{lob} \sum_{i > 0} [ {M}_{i}(\lambda)] &=& [ M^{\gamma}(\lambda -\gamma)]+ [ M^{\gamma'}(\lambda -\gamma')]\nonumber\\ &=& [K]+[L].\nonumber\end{eqnarray} Therefore $\sum_{i >1} [ {M}_{i}(\lambda)] = [L]$, and this gives the statements about ${M}_2(\lambda)$ and ${M}_3(\lambda)$. Finally $\mathtt{d}(K/L) <\mathtt{d}$ by Lemma \ref{nl}. \end{proof} \begin{proposition} \label{fad} Suppose that $A(\nu) = \{\alpha\}$, $B(\nu) = \{\gamma\}$ with $p=(\nu + \rho, \alpha^\vee) \in {\mathbb N} \backslash\{0\}$, and that $\lambda = \nu-\gamma\in \Lambda$, as defined in \eqref{Gld}. Set $\mu = s_\alpha \cdot \nu$. If $p =(\gamma, \alpha^\vee),$ then the lattice of submodules of $M={M}(\nu)$ is as in Figure \ref{fig1}, where \begin{equation} \label{cid} V_1 = U(\fg)\theta_{\alpha,p}v_{\nu} \cong M(\mu), \quad V_2 =\Ker M(\nu)\longrightarrow M^\gamma(\nu), \quad V_3 = U(\fg) \theta_{s_\alpha\gamma}\theta_{\alpha,p}v_{\nu}.\end{equation} The unique maximal submodule of $V_i$ is $L_i$ where \begin{equation} \label{cfcrs} L_1=L(\mu)\cong {M}^{s_\alpha \gamma}(\mu),\quad L_2 = L(\lambda-\gamma'),\quad L_3=L(\lambda)\cong {M}^{s_\alpha \gamma}(\lambda).\end{equation} The Jantzen filtration is given by \begin{equation} \label{lid}{M}_{3}(\nu) = 0, \quad {M}_{2}(\nu) = V_3 =V_1 \cap V_2, \quad {M}_{1}(\nu) = V_1+ V_2.\end{equation} \end{proposition}
\begin{proof} Define the $V_i$ and $L_i$ by \eqref{cid} and \eqref{cfcrs}. Note that $(\lambda+ \rho, \alpha^\vee) =0$, so $s_\alpha \cdot \lambda=\lambda.$ Since $(\nu + \rho, \alpha^\vee) \in {\mathbb N} \backslash\{0\},$ $M(s_\alpha \cdot \nu)$ embeds in $M(\nu)$, and by Lemma \ref{sat}, $V_1= {M}(\mu)$ has length two with socle $V_3$.
Now $\mathtt{e}(V_3)= \mathtt{e}(M(\nu)/V_3) = \mathtt{e}/2$, so the same argument as in Lemma \ref{nid} yields that $V_3$ is the socle of $M(\nu).$ Hence since $V_2$ has length two, $V_3 \subseteq V_1 \cap V_2 \subseteq V_1$
and $V_2$ cannot contain $V_1$, by looking at characters. Thus $V_3 =V_1 \cap V_2$. Now by (\ref{lb}) we have in the Grothendieck group $K(\mathcal{O})$
\begin{equation} \label{sob} \sum_{i > 0} [ {M}_{i}(\nu)] = [ M^{\gamma}(\nu -\gamma)]+ [ M(\mu)].\end{equation} Since $d(L_3)=d(L_1)= \mathtt{d}$, and $e(L_3)=e(L_1)= \mathtt{e}/2,$ we have
\begin{equation} \label{zob} \mathtt{e}\ge e (M_1(\nu)) \ge |{M}_{1}(\nu):L_3| + |{M}_{1}(\nu):L_1| \ge \mathtt{e}.\end{equation} Thus $|{M}_{1}(\nu):L_3|\le 1$, but by (\ref{sob}),
$\sum_{i > 0} |M_{i}(\nu):L_3| =2,$ so $V_3\subseteq {M}_{2}(\nu).$ Hence
\begin{eqnarray} \label{rob} [ {M}_{1}(\nu)]+[V_3] +\sum_{i \ge 3} [ {M}_{i}(\nu)] &\le& \sum_{i > 0} [ {M}_{i}(\nu)] =[ M(\mu)]+ [ V_2]\nonumber\\ &=& [ M(\mu)+ V_2] +[ M(\mu)\cap V_2] \\ &\le& [ {M}_{1}(\nu)]+[V_3]. \nonumber\end{eqnarray} Therefore ${M}_{3}(\nu)=0, {M}_{2}(\nu) = V_3$ and ${M}_{1}(\nu) = V_2+V_1.$ The only thing left to show is that $V_2/V_3\cong L(\lambda-\gamma').$ However we have ${\operatorname{ch}\:} V_2 = \mathtt{e}^{\nu-\gamma} {p}_\gamma$, and by Lemma \ref{nid} with $\lambda=\nu-\gamma$, this is also the character of the length two module $M(\lambda)/M_2(\lambda)$. Thus $V_2$ has length two and $V_2/V_3$ is simple. We have ${\operatorname{ch}\:} V_2/V_3= {\operatorname{ch}\:} V_2-{\operatorname{ch}\:} V_3$ and this can easily be calculated, compare Lemma \ref{ink}, and this gives the correct highest weight. \end{proof} \begin{rems} {\rm \label{2.6} \noindent \begin{itemize} \item[{{\rm(a)}}] In the situation of Proposition \ref{fad}, the modules $V_2$ and $M(\lambda-\gamma)/M_2(\lambda-\gamma)$ need not be isomorphic, see Theorem \ref{A2} with $n=1$. The same example shows that $V_2$ need not be a highest weight module. \item[{{\rm(b)}}] The hypothesis in Lemma \ref{nid} holds in Theorem \ref{AA2} when $n=0,$ with $\gamma'=\beta.$
\end{itemize}}\end{rems} \noindent In the Theorem below, both cases arise when $\fg =\fsl(2,1)$, (or $\fgl(2,1)$) using the anti-distinguished Borel subalgebra. In the notation of Subsection \ref{C8}, take $\kappa =\gamma$ if $q = 1,$ and $\kappa = \beta$ if $q=-1.$
\begin{theorem} Suppose $q= (\kappa, \alpha^\vee) = \pm1$ and $p = (\lambda + \rho, \alpha^\vee)\in \mathbb{N} \backslash \{0\}$ where $\kappa$, $\alpha$ are isotropic and non-isotropic respectively. Then for general $\lambda$ such that $A(\lambda) = \{\alpha\}$, $B(\lambda) = \{\kappa\}$,
the lattice of submodules of $M={M}(\lambda)$ is as in Figure \ref{fig1} with the submodules $V_i$ as in $(\ref{cid}).$ Moreover the Jantzen filtration is given by $(\ref{lid})$ and if $p>q=1$ or $q=-1$, then $V_2 = U(\fg) \theta_{\kappa}v_\lambda$. \end{theorem}
\begin{proof} Define $V_1, V_2, V_3$ by Equation (\ref{cid}) with $\nu$ replaced by $\lambda$, and $\gamma$ by $\kappa$. Set $\mu = s_\alpha \cdot \lambda.$ Then by Lemma \ref{you} $V_1 = M(\mu)$ has length 2 with socle $V_3$. We use induction on $p$. If $\nu = \lambda-q\kappa,$ then $(\nu + \rho, \alpha^\vee)=p-1$. Different proof strategies are necessary depending on the sign of $q$. If $q=1$ we use the map from $M(\nu)$ to $M(\lambda)$ sending $v_\nu$ to $\theta_\kappa v_{\lambda} $, and if $q=-1$, we use the map from $M(\lambda)$ to $M(\nu)$ sending $v_\lambda$ to $\theta_\kappa v_{\nu} $. \\ \\ Suppose that $q=1.$ If $p=1$, then all the assertions hold by Proposition \ref{fad}, so assume $p\ge 2$. By Proposition \ref{fad} if $p=2$, or induction if $p>2$, $M(\nu)$ has a length two factor module with character $\mathtt{e}^\nu {p}_\kappa = {\operatorname{ch}\:} V_2$. Hence $V_2$ has length two with socle isomorphic to $V_3$. Also $U(\fg) \theta_\kappa v_{\lambda} \subseteq V_2$ by Corollary \ref{hco}. Since $\theta_\kappa v_{\lambda}$ is a highest weight vector with weight $\lambda-\kappa$, and the only highest weight of $V_3$ is $s_\alpha\cdot(\lambda-\kappa) \neq \lambda-\kappa$, it follows that $U(\fg) \theta_\kappa v_{\lambda} =V_2$. Because $e(M(s_\alpha\cdot\lambda)) =\mathtt{e}$, $L(s_\alpha\cdot(\lambda-\kappa))=\mathtt{e}/2$
and $M(s_\alpha\cdot\lambda)$ contains
$L(s_\alpha\cdot(\lambda-\kappa))$ with multiplicity one, we have $|M(\lambda):L(s_\alpha\cdot(\lambda-\kappa))|=|{M}_{2}(\lambda):L(s_\alpha\cdot(\lambda-\kappa))|=1.$
Now the sum formula \eqref{lb} takes the form \[ \sum_{i\ge1}[M_i(\lambda)] = [V_1]+ [V_2] = [L(s_\alpha\cdot\lambda)]+ [L(\lambda-\kappa)]+ 2[L(s_\alpha\cdot(\lambda-\kappa))]. \] This easily gives $M_1(\lambda) = V_1+V_2$, $M_2(\lambda) = V_1\cap V_2=V_3$ and $M_3(\lambda) = 0.$ \\ \\ Now suppose $q=-1.$ If $p=1,$ the conditions of Lemma \ref{nid} hold in the general case with $\nu, \kappa'=s_\alpha\cdot\kappa$ and $\kappa$ in place of $\lambda$ and $\gamma$ and $\gamma'$ respectively. Thus $\theta_{\kappa'} v_{\nu}$ generates $M_2(\nu)$, and the map $xv_\lambda \longrightarrow x\theta_{\kappa} v_\nu$ induces an isomorphism $M(\lambda)/V_2\longrightarrow M_1(\nu)=U(\fg) \theta_\kappa v_{\nu}\cong M^\kappa(\nu-\kappa)$.
Therefore
\begin{equation} \label{rid} {\operatorname{ch}\:} (M(\lambda)/V_2) ={\operatorname{ch}\:} M^\kappa(\nu-\kappa) = \mathtt{e}^\lambda {p}_\kappa. \end{equation} If $p>1$ then (\ref{rid}) holds by induction and a similar argument. Now as in the proof of Proposition \ref{fad}, (see Equation \eqref{zob}) we see that $|{M}_{1}(\lambda):V_3|\le 1$ and (\ref{rob}) holds. \\ \\
Also from (\ref{rid}) $M(\lambda)/V_2$ has length two. We know $M_1(\lambda)/M_2(\lambda) = V_1/V_3 \oplus V_2/V_3$ where $V_1/V_3\cong L(s_\alpha\cdot \lambda)$ and $V_2/V_3$ is a highest weight module with highest weight $\lambda-\kappa$ such that $|V_2/V_3: L(s_\alpha\cdot \lambda)| =0.$ Hence $V_2/V_3$ is a self-dual highest weight module and so is simple. \end{proof}
\noindent Next we consider the structure of $M(\lambda)$ when $B(\lambda)= \{\gamma, {\gamma'}\}$ for orthogonal roots $\gamma$ and $\gamma'$.
\begin{lemma} \label{fix} If $\lambda\in \mathcal{H}_{\gamma} \cap \mathcal{H}_{\gamma'},$ then for all but only finitely many $c \in \mathtt{k}$ we have \begin{equation} \label{we1}\theta_\gamma(\lambda +c\xi - \gamma)\theta_{{\gamma'}}(\lambda+c\xi)\neq 0\end{equation} and \begin{equation} \label{we2}\theta_{{\gamma'}}(\lambda +c\xi - \gamma)\theta_\gamma(\lambda+c\xi)\neq 0.\end{equation}\end{lemma} \begin{proof} Set $\widetilde{\lambda}= \lambda +T\xi$. It follows from Corollary \ref{rmd} that when $\theta_\gamma(\widetilde{\lambda}- \gamma) \theta_{{\gamma'}}(\widetilde{\lambda})v_{\widetilde{\lambda}}$
is written as a $A$-linear combination of terms $e_{-\pi}v_{\widetilde{\lambda}}$, the coefficient of $e_{-\gamma}e_{\gamma'} v_{\widetilde{\lambda}}$ is a polynomial in $T$ of degree $d_\gamma +d_{{\gamma'}}$. Hence (\ref{we1}) holds for all but finitely many $c$, and a similar argument applies to (\ref{we2}).\end{proof} \noindent \begin{lemma} \label{tri} For general $\lambda \in \mathcal{H}_{\gamma} \cap \mathcal{H}_{{\gamma'}}$ we have $$[M(\lambda):L(\lambda-\gamma-\gamma')]=1.$$ \end{lemma}
\begin{proof} \footnote{This proof is due to Vera Serganova.}
We require $\theta_\gamma(\lambda - \gamma)\theta_{{\gamma'}}(\lambda)\neq 0$ or $\theta_{{\gamma'}}(\lambda - \gamma)\theta_\gamma(\lambda)\neq 0$, as well as some further conditions that arise in the proof. This implies $[M(\lambda):L(\lambda-\gamma-\gamma')]\ge 1.$ By \cite{M}, Theorem 10.4.5 $M(\lambda)$ has a series with factors which are Verma modules $M^0(\mu)$ for $\fg_0$, and $M^0(\lambda-2\rho_1)$ occurs exactly once as a factor in this series. Furthermore $M^0(\lambda-2\rho_1) =L^0(\lambda-2\rho_1)$ for general $\lambda$, so as a $\fg_0$-module $$[M(\lambda):L^0(\lambda-2\rho_1)]=1.$$ So it is enough to show that \begin{equation} \label{tag}[L(\lambda-\gamma-\gamma'):L^0(\lambda-2\rho_1)]\ge1.\end{equation} or equivalently $[L({{{\overline{\lambda}}}}):L^0({\overline{\lambda}}-\eta)]\ge1$
where $\overline{\lambda} = \lambda-\gamma-\gamma',$ and $-\eta =\gamma+\gamma' -2\rho_1$. Let $X$ be the set of all positive isotropic roots $\sigma$ different from $\gamma$ and $ \gamma'$, and $e_{-X}$ be the ordered product of all root vectors $e_{-\sigma}$ where $\sigma\in X$. The weight space $M(\overline{\lambda})^{\overline{\lambda}-\eta}$ has a basis consisting of vectors of the form $e_{-\pi}v_{\overline{\lambda}}$, with $\pi$ a partition of $\eta$, and
$e_{-X}v_{\overline{\lambda}}$ is one such basis element. We claim that for general ${\overline{\lambda}}$,
\begin{equation} \label{tug}e_{-X} v_{\overline{\lambda}} \notin \mathfrak{n}_0^-M({\overline{\lambda}}) + I_{\overline{\lambda}}\end{equation} where $I_{\overline{\lambda}}$ is the maximal submodule of $M({\overline{\lambda}}).$
Let $(\;,\;)$ be a contravariant form on $M({\overline{\lambda}})^{{\overline{\lambda}}-\eta}$ with radical equal to $I({\overline{\lambda}})^{{\overline{\lambda}}-\eta}$. By the proof of \cite{M} Lemma 10.1.2, we see that as a polynomial in ${\overline{\lambda}}$, the degree of $g({\overline{\lambda}}) =(e_{-X}v_{\overline{\lambda}}, e_{-X}v_{\overline{\lambda}})$ is greater than the degree of $(wv_{\overline{\lambda}},e_{-X}v_{\overline{\lambda}})$ for any $w \in (\mathfrak{n}_0^-U(\mathfrak{n}^-)v_{\overline{\lambda}})^{{\overline{\lambda}}-\eta}$. Moreover the leading term of $g({\overline{\lambda}})$ is $\prod_{\sigma \in X}({\overline{\lambda}},\sigma)$ which is non-zero at general elements of $\mathcal{H}_{\gamma} \cap \mathcal{H}_{\gamma'}$. This implies (\ref{tug}). We deduce from this that the image of $e_{-X}v_{\overline{\lambda}}$ in $H_0(\mathfrak{n}_0^-,L({\overline{\lambda}}))^{\overline{\lambda}-\eta}$ is non-zero. This says that $L({\overline{\lambda}})$ contains a $\fg_0$ highest weight vector of weight ${\overline{\lambda}-\eta}$,
and (\ref{tag}) follows. \end{proof}
\noindent This gives a representation theoretic proof of \eqref{pee}. \begin{corollary} \label{kt1} There is a rational function $a$ of $\lambda \in \mathcal{H}_{\gamma} \cap \mathcal{H}_{{\gamma'}}$ such that \begin{equation} \label{kt} \theta_{\gamma'}(\lambda - \gamma) \theta_\gamma(\lambda)=a(\lambda)\theta_ \gamma(\lambda - {\gamma'}) \theta_ {\gamma'}(\lambda).\end{equation} \end{corollary}
\begin{proof} Up to a scalar multiple, $M(\lambda)$ can contain at most one highest weight vector with weight $\lambda-\gamma-{\gamma'}$. However both $\theta_{\gamma'}(\lambda - \gamma) \theta_\gamma(\lambda)v_\lambda$ and $\theta_ \gamma(\lambda - {\gamma'}) \theta_ {\gamma'}(\lambda)v_\lambda$ are both highest weight vectors with this weight, so (\ref{kt}) holds. \end{proof}
\section{An (ortho) symplectic example.} \label{1cosp} \begin{example} \label{1ex2.4} {\rm A crucial step in the construction of \v Sapovalov elements was the observation in the proofs of Lemmas \ref{1wpfg} and \ref{1wpfg1} that the term $c_{ \zeta,\lambda}$ defined in Equation (\ref{159}) are zero unless $\zeta(\alpha) \geq N-mq,$ (using the notation of the Lemmas). We give an example where the individual terms on the right of Equation (\ref{159}) are not identically zero, and verify directly that the sum itself is zero. This cannot happen in Type A. The key difference in the examples below seems to be that it is necessary to apply Equation (\ref{121nd}) more than once with the same simple root $\alpha$. Consider the Dynkin-Kac diagram below for the Lie superalgebra $\fg = \osp(2,4)$.}\end{example}
\begingroup \setlength{\unitlength}{0.10in} \begin{picture}(-30,-10) \thicklines \put(14.414,0.0){\line(1,0){9.23}} \put(25.95,1.0){\line(1,0){10.1}} \put(25.95,-1.0){\line(1,0){10.1}} \put(30,0){\line(1,-1){1.5}} \put(30,0){\line(1,1){1.5}} \put(12,1){\line(1,-1){2}} \put(12,-1){\line(1,1){2}} \put(13,0){\circle{2.828}} \put(25,0){\circle{2.828}} \put(37,0){\circle{2.828}} \put(11.6,-3.0){$\epsilon-\delta_1$} \put(23.1,-3.0){$\delta_1-\delta_2$} \put(36.2,-3.0){$2\delta_2$} \end{picture}
\endgroup
\noindent Let $\beta = \epsilon - \delta_1, \alpha_1 = \delta_1 - \delta_2, \alpha_2 = 2\delta_2$, be the corresponding simple roots. If we change the grey node to a white node we obtain the Dynkin diagram for $\mathfrak{sp}(6)$. In this case the simple roots are $\beta=\delta_{0} - \delta_{1}, \alpha_1 = \delta_{1} - \delta_{2} $ and $\alpha_2=2 \delta_{2}$. Let $e_{-\beta}, e_{-\alpha_1}, e_{-\alpha_2} $ be the negative simple root vectors. The computation of the \v Sapovalov elements $\theta_1, \theta_2, \theta_3$ for the roots $\beta + \alpha_1, \beta + \alpha_1+ \alpha_2$ and $\beta + 2\alpha_1+ \alpha_2$ respectively, is the same for $\osp(2,4)$ and for $\mathfrak{sp}(6)$. Then define the other negative root vectors by $$e_{- \alpha_1 - \alpha_2} = [e_{-\alpha_1},e_{- \alpha_2 } ], \quad \quad e_{- 2\alpha_1 - \alpha_2} = [e_{-\alpha_1}, e_{- \alpha_1 - \alpha_2}], $$ $$e_{-\beta- \alpha_1} =[e_{-\alpha_1} ,e_{-\beta} ], \quad e_{-\beta- \alpha_1-\alpha_2} = [e_{- \alpha_2 } ,e_{-\beta- \alpha_1}], \quad e_{-\beta- 2\alpha_1 - \alpha_2} = [e_{-\alpha_1},e_{-\beta- \alpha_1-\alpha_2}]. $$ It follows from the Jacobi identity that $$[e_{-\beta}, e_{- \alpha_1 - \alpha_2}] = e_{-\beta- \alpha_1-\alpha_2}, \quad \quad [e_{- \alpha_1 - \alpha_2},e_{-\beta- \alpha_1}] = e_{-\beta- 2\alpha_1 - \alpha_2},$$ and $$[e_{-\beta},e_{- 2\alpha_1 - \alpha_2}] = 2e_{-\beta- 2\alpha_1-\alpha_2}.$$ We order the set of positive roots so that for any partition $\pi$, $e_{-\alpha_1}$ occurs first if at all in $e_{-\pi}$, and any root vector $e_{-\sigma}$ with $\sigma$ an odd root occurs last.\\ \\ Let $s_1, s_2$ be the reflections corresponding to the simple roots $\alpha_1, \alpha_2.$ Then for $\lambda \in \mathfrak{h}^*$ define $\lambda_1 =s_1\cdot\lambda, \;\lambda_2 = s_2\cdot\lambda, \;\mu = s_1\cdot\lambda_2$. Let $$(\lambda + \rho,\alpha^\vee_1) = p = -(\mu + \rho,(\alpha_1+\alpha_2)^\vee) $$ and \[(\lambda_1 + \rho,\alpha^\vee_2) = (\lambda + \rho,(2\alpha_1 + \alpha_2)^\vee) = q= -(\mu + \rho,(2\alpha_1+\alpha_2)^\vee) ,\] $$(\lambda_2 + \rho,\alpha^\vee_1) = (\lambda + \rho,(\alpha_1+\alpha_2)^\vee) = r = -(\mu + \rho,\alpha_1^\vee) .$$ Then $r = 2q - p.$ Let $\gamma$ be any positive root that involves $\beta$ with non-zero coefficient when expressed as a linear combination of simple roots. We compute the \v Sapovalov elements $\theta_{\gamma,1}$ for $\mathfrak{sp}(6)$ and $\theta_{\gamma}$ for $\osp(2,4).$ To do this we use Equation (\ref{121nd}). We can assume $\gamma \neq \beta.$ Suppose that $p, q, r$ are nonnegative integers. Then $$e_{-\alpha_1}^{p+1}e_{-\beta} = \theta_1e_{-\alpha_1}^{p}$$ $$e_{- \alpha_2 }^{q+1}\theta_1 = \theta_2e_{- \alpha_2 }^{q}$$ $$e_{- \alpha_1 }^{r+1}\theta_2 = \theta_3e_{- \alpha_1 }^{r}.$$ In the computations below we write $e_{-\pi}$, for $\pi$ a partition (resp. $\theta_i$ for $i=1,2,3$) in place of $e_{-\pi}v_\lambda$ (resp. $\theta_i v_\lambda$). First note that $$[e_{-\alpha_1}^{p+1} ,e_{-\beta} ] = (p+1)e_{-\beta- \alpha_1} e^p_{-\alpha_1}$$ $$[e_{- \alpha_2 }^{q+1} ,e_{-\beta- \alpha_1}] = (q+1)e_{-\beta- \alpha_1-\alpha_2} e_{- \alpha_2 }^q$$ $$[e_{- \alpha_2 }^{q+1} ,e_{-\alpha_1}] =-(q+1)e_{- \alpha_1 - \alpha_2} e_{- \alpha_2 }^q.$$ This easily gives
$$\theta_1 = (p+1)e_{-\beta- \alpha_1} + e_{-\beta}e_{-\alpha_1} = pe_{-\beta- \alpha_1} + e_{-\alpha_1}e_{-\beta}.$$ We order the set of positive roots so that for any partition $\pi$, $e_{-\alpha_2}$ occurs last if at all in $e_{-\pi}$, and any root vector $e_{-\sigma}$ with $\sigma$ an odd root occurs first. \begin{eqnarray}\label{1la2} \theta_2 &=& (p+1)[(q+1)e_{- \beta - \alpha_1 - \alpha_2} +e_{-\beta- \alpha_1}e_{-\alpha_2}] +e_{-\beta}[ e_{-\alpha_1}e_{- \alpha_2} - (q+1)e_{- \alpha_1 - \alpha_2} ]. \nonumber \end{eqnarray} Next order the set of positive roots so that for any partition $\pi$, $e_{-\alpha_1}$ occurs last if at all in $e_{-\pi}$, and any root vector $e_{-\sigma}$ with $\sigma$ an odd root occurs first. To find $\theta_3$ we use $$[e_{-\alpha_1}^{r+1} ,e_{-\beta-\alpha_1-\alpha_2} ] = (r+1)e_{-\beta- 2\alpha_1 -\alpha_2} e^r_{-\alpha_1},$$ $$[e_{-\alpha_1}^{r+1} ,e_{-\beta-\alpha_1}e_{-\alpha_2} ] = (r+1)e_{-\beta- \alpha_1}e_{-\alpha_1-\alpha_2}e^ {r}_{-\alpha_1} +\left( \begin{array}{c}
r+1 \\
2 \end{array}\right)e_{-\beta- \alpha_1}e_{-2\alpha_1 -\alpha_2} e^{r-1}_{-\alpha_1},$$ $$[e_{-\alpha_1}^{r+1} ,e_{-\beta}e_{-\alpha_1-\alpha_2} ] = (r+1)[e_{-\beta}e_{- 2\alpha_1 -\alpha_2} e^r_{-\alpha_1}
+e_{-\beta- \alpha_1}e_{-\alpha_1 -\alpha_2} e^r_{-\alpha_1} +re_{-\beta-\alpha_1}e_{-2\alpha_1-\alpha_2} e_{\alpha_1}^{r-1}],$$ $$e_{-\alpha_1}^{r+1} e_{-\beta}e_{-\alpha_2}e_{-\alpha_1}
= e_{-\beta}[e_{-\alpha_2} e_{- \alpha_1}^{2} + (r+1)e_{- \alpha_1-\alpha_2}e_{-\alpha_1} +\left( \begin{array}{c}
r+1 \\
2 \end{array}\right)e_{- 2\alpha_1 -\alpha_2}] e^{r}_{-\alpha_1}$$ $$+ (r+1)e_{-\beta- \alpha_1}[e_{-\alpha_2} e_{- \alpha_1} + re_{- \alpha_1-\alpha_2}]e^{r}_{-\alpha_1} +(r-1)\left( \begin{array}{c}
r+1 \\
2 \end{array}\right)e_{-\beta}e_{- 2\alpha_1 -\alpha_2} e^{r-1}_{-\alpha_1}.$$ The above equations allow us to write $e_{- \alpha_1 }^{r+1}\theta_2$ in terms of elements $e_{-\pi}$ with $\pi$ a partition of $\beta + (r+2)\alpha_1 + \alpha_2$. We see that the term $e_{-\beta- \alpha_1}e_{- 2\alpha_1 -\alpha_2} e^{r-1}_{-\alpha_1}$ occurs in $e_{- \alpha_1 }^{r+1}\theta_2$ with coefficient \[\left( \begin{array}{c}
r+1 \\
2 \end{array}\right)[(p+1) -2q + (r-1)] = 0.\] This is predicted by the cancellation step in the proof of Lemma \ref{1wpfg}. In the remaining terms, $e_{- \alpha_1 }^{r}$ can be factored on the right, and this yields \begin{eqnarray}\label{1th3} \theta_3 &=& (p+1)(q+1)(r+1)e_{-\beta- 2\alpha_1 -\alpha_2} +(p+1)(q+1)e_{-\beta- \alpha_1-\alpha_2}e_{-\alpha_1} \\ &+& (q+1)(r+1)e_{-\beta- \alpha_1}e_{-\alpha_1 -\alpha_2}-(p/2)(r+1)e_{-\beta}e_{- 2\alpha_1 -\alpha_2}\nonumber \\ &+& 2(q+1)e_{-\beta- \alpha_1}e_{-\alpha_2}e_{-\alpha_1}+ (r-q+1)e_{-\beta}e_{-\alpha_1-\alpha_2}e_{-\alpha_1}+e_{-\beta}e_{-\alpha_2} e_{- \alpha_1}^{2}. \nonumber \end{eqnarray} Using the opposite orders on positive roots to those used above to define the $e_{-\pi}$ we obtain \begin{eqnarray}\label{1la3} \theta_2 &=& p[qe_{- \beta - \alpha_1 - \alpha_2} + e_{- \alpha_2 } e_{-\beta- \alpha_1}] +[ e_{-\alpha_2} e_{- \alpha_1 } - qe_{- \alpha_1 - \alpha_2} ]e_{-\beta}, \nonumber \end{eqnarray} and \begin{eqnarray}\label{1tho} \theta_3 &=& pqr e_{-\beta- 2\alpha_1 -\alpha_2} +pq e_{-\alpha_1} e_{-\beta- \alpha_1-\alpha_2}\\ &+& qr e_{-\alpha_1 -\alpha_2}e_{-\beta- \alpha_1} -(r/2)(p+1)e_{- 2\alpha_1 -\alpha_2}e_{-\beta}\nonumber \\ &+& 2q e_{-\alpha_1}e_{-\alpha_2}e_{-\beta- \alpha_1} +
(r-q-1)e_{-\alpha_1}e_{-\alpha_1-\alpha_2}e_{-\beta} +e_{- \alpha_1}^{2} e_{-\alpha_2}e_{-\beta}.\nonumber \end{eqnarray} \begin{rem} {\rm It seems remarkable that all the coefficients of $\theta_3$ in (\ref{1th3}) and (\ref{1tho}) are products of linear factors. This is also true in the Type A case, see Equations (\ref{1shtpa}) and (\ref{1shtpb}). A partial explanation of this phenomenon is given by specializing these coefficients to zero. Vanishing of these coefficients gives rise to factorizations of $\theta_3$ as in the examples below. } \end{rem} \begin{itemize} \item[{{\rm(a)}}] If $p=(\lambda + \rho,\alpha_1^\vee) =0$, then $r = 2q$ and we have \begin{eqnarray}\label{1la4} \theta_3 &=&
2q^2 e_{-\alpha_1 -\alpha_2}e_{-\beta- \alpha_1} -qe_{- 2\alpha_1 -\alpha_2}e_{-\beta}\nonumber \\ &+& 2q e_{-\alpha_1}e_{-\alpha_2}e_{-\beta- \alpha_1} +
(q-1)e_{-\alpha_1}e_{-\alpha_1-\alpha_2}e_{-\beta} +e_{- \alpha_1}^{2} e_{-\alpha_2}e_{-\beta} \nonumber \\&=& \theta_{\alpha_1+\alpha_2}\theta_{\beta+\alpha_1}.\nonumber \end{eqnarray} \item[{{\rm(b)}}] If $q=(\lambda + \rho,(2\alpha_1 + \alpha_2)^\vee) =0$ then $p = -r$, $\theta_2= \theta_{\beta +\alpha_1+\alpha_2} = \theta_{\alpha_2}\theta_{\beta+\alpha_1}$, and we have \begin{eqnarray}\label{1la5} \theta_3 &=& [(p/2)(p+1)e_{- 2\alpha_1 -\alpha_2}
-(p+1)e_{-\alpha_1}e_{-\alpha_1-\alpha_2} +e_{- \alpha_1}^{2} e_{-\alpha_2}]e_{-\beta}\nonumber \\&=& \theta_{2\alpha_1+\alpha_2}\theta_{\beta}.\nonumber \end{eqnarray} \item[{{\rm(c)}}] If $r=(\lambda + \rho,(\alpha_1 + \alpha_2)^\vee) = 0$, then $p = 2q$,
and we have \begin{eqnarray}\label{1la6} \theta_3 &=& 2q^2 e_{-\alpha_1} e_{-\beta- \alpha_1-\alpha_2}\nonumber \\ &+& 2q e_{-\alpha_1}e_{-\alpha_2}e_{-\beta- \alpha_1} -
(q+1)e_{-\alpha_1}e_{-\alpha_1-\alpha_2}e_{-\beta} +e_{- \alpha_1}^{2} e_{-\alpha_2}e_{-\beta}\nonumber \\
&=& e_{-\alpha_1}[2q^2 e_{-\beta- \alpha_1-\alpha_2}
2q e_{-\alpha_2}e_{-\beta- \alpha_1} -
(q+1)e_{-\alpha_1-\alpha_2}e_{-\beta} +e_{- \alpha_1} e_{-\alpha_2}e_{-\beta}]\nonumber \\&=& \theta_{\alpha_1}\theta_{\beta+\alpha_1+\alpha_2}.\nonumber \end{eqnarray} \end{itemize} Similarly if $p=-1,$ (resp. $q=-1,$ $r=-1$) then (\ref{1tho}) yields the factorizations $\theta_3=\theta_{\beta+\alpha_1}\theta_{\alpha_1+\alpha_2}$, (resp. $\theta_2=
\theta_{\beta+\alpha_1} \theta_{\alpha_2},$ $\theta_3=\theta_{\beta}\theta_{2\alpha_1+\alpha_2}$, and $\theta_3=\theta_{\beta+\alpha_1+\alpha_2}\theta_{\alpha_1}$). On the other hand we see that $p$ divides the coefficients of $e_{-\beta- 2\alpha_1 -\alpha_2}$ and $e_{-\alpha_1} e_{-\beta- \alpha_1-\alpha_2}$ in (\ref{1tho}) since when $p = 0$, $\theta_3=\theta_{\alpha_1+\alpha_2}\theta_{\beta+\alpha_1}$ can be written as a linear combination of different $e_{-\pi}$. In this way we obtain explanations for all the linear factors in (\ref{1th3}) and (\ref{1tho}) with the exception of the coefficients $r-q\pm 1$ of $e_{-\beta}e_{-\alpha_1-\alpha_2}e_{-\alpha_1}$ and $e_{-\alpha_1}e_{-\alpha_1-\alpha_2}e_{-\beta}.$ At this point it may be worthwhile mentioning that $r-q =(\lambda + \rho,\alpha_2^\vee)$. In addition equality holds in the upper bounds given in Theorem \ref{1Shap} for the degrees of all the coefficients in (\ref{1th3}) and (\ref{1tho}).
\section{The Type A Case.}\label{1s.8}
\subsection{Lie Superalgebras.}\label{1far} We construct the elements $\theta_{\gamma}$ in Theorem \ref{1Shap} explicitly when ${\mathfrak g} = \fgl(m,n).$ Suppose that $\gamma = \epsilon_r - \delta_{s}.$ For $1 \leq i<j \leq m$ and $1 \leq k<\ell \leq n$ define roots $\sigma_{i,j}, \tau_{k,\ell}$ by \[ \sigma_{i,j} = \epsilon_{i} - \epsilon_{j}, \quad \tau_{k,\ell} = \delta_k - \delta_\ell. \] Suppose $B = (b_{i,j})$ is a $k \times \ell$ matrix with entries in $U({\mathfrak n}^{-})$, $I \subseteq \{1, \ldots , k \}=[k]$ and $J \subseteq \{1, \ldots , \ell \}.$ We denote the submatrix of $B$ obtained by deleting the $ith$ row for $i \in I,$ and the $jth$
column for $j \in J$ by $_IB_J.$ If either set is empty, we omit the corresponding subscript. When $I = \{ i\},$ we write $_iB$ in place of $_IB$ and likewise when $|J| = 1.$ If $I$ or $J$ equals $[p]$ we write $_{[p]}B$ or $B_{[p]}$. If $k = \ell$ we define two noncommutative determinants of $B,$ the first working from left to right, and the second working from right to left. \begin{equation} \label{lr}{\stackrel{\longrightarrow }{{\rm det}}}(B) = \sum_{w \in \mathcal{S}_k} sign(w) b _{w(1),1} \ldots b _{w(k),k}, \end{equation} \begin{equation} \label{rl}{\stackrel{\longleftarrow}{{\rm det}}}(B) = \sum_{w \in \mathcal{S}_k} sign(w) b _{w(k),k} \ldots b _{w(1),1}.\end{equation} If $k = 0$ we make the convention that ${\stackrel{\longleftarrow}{{\rm det}}}(B) = {\stackrel{\longrightarrow}{{\rm det}}}(B) = 1.$ We call ${\stackrel{\longrightarrow}{{\rm det}}}(B) $ and ${\stackrel{\longleftarrow}{{\rm det}}}(B)$ the {\it LR and RL determinants} of $B$ respectively. We note the following cofactor expansions down the first and last columns
\begin{eqnarray} {\stackrel{\longrightarrow }{{\rm det}}}\;B &=& \sum_{j=1}^k (-1)^{j+1} b_{j1} {\stackrel{\longrightarrow }{{\rm det}}}({}_{j}B_{1})\nonumber\\ &=& \sum_{j=1}^k (-1)^{k+j+1} {\stackrel{\longrightarrow }{{\rm det}}}({}_{j}B_{k})b_{jk}.\nonumber \end{eqnarray} These are easily derived from \eqref{lr} by grouping terms. There is a similar formula for cofactor expansion down the last column.
\\ \\ Consider the following matrices with entries in $U(\mathfrak{n}^-)$
\begin{equation}\label{wok} A^+(\lambda, r) = \left[ {\begin{array}{ccccc} e_{r+1,r} & e_{r+2,r} & \hdots & e_{m,r}\\ -a_1 & e_{r+2,r+1} & \hdots & e_{m,r+1} \\ 0&-a_2 & \hdots & e_{m,r+2} \\
\vdots & \vdots & \ddots & \vdots \\ 0& \hdots & \ldots& e_{m,m-1} \\ 0 & \hdots & 0& -a_{m-r}\\ \end{array}}
\right],\end{equation}
\begin{equation} \label{gov} A^-(\lambda, s) = \left[ {\begin{array}{ccccc} e_{m+s,m+s-1}&e_{m+s,m+s-2} & \hdots & e_{m+s,m+2} & e_{m+s,m+1} \\
b_{s-1} &e_{m+s-1,m+s-2} & \hdots & e_{m+s-1,m+2} & e_{m+s-1,m+1} \\ 0& b_{s-2} & \hdots & e_{m+s-2,m+2} & e_{m+s-2,m+1} \\
\vdots & \vdots & \ddots & \vdots& \vdots \\ 0&0& \hdots & b_2& e_{m+2,m+1} \\ 0& 0& \hdots & \hdots 0& b_1 \end{array}}
\right], \end{equation} where for $\lambda\in \mathcal{H}_\gamma$ we set \begin{equation} \label{eat} a_i=(\lambda + \rho,\sigma_{r,r+i}^\vee ) \mbox{ and } b_i = (\lambda + \rho,\tau_{i,s}^\vee ).\end{equation} Also let $B^+(\lambda, r)$ (resp. $B^-(\lambda, s)$) be the matrices obtained from $A^+(\lambda, r)$ (resp. $A^-(\lambda, s)$) by replacing each $a_i$ by $a_i +1$ (resp. replacing each $b_i$ by $b_i +1$). \\ \\
Observe that in $A^+(\lambda, r)$ and $A^-(\lambda, s)$ the number of rows exceeds the number of columns by one. We also consider two degenerate cases. In general $A^+(\lambda, r)$ and $B^+(\lambda, r)$ are $m-r+1\ti m-r$ matrices, so if $r=m$ then $A^+(\lambda, r)$ and $B^+(\lambda, r)$ are ``matrices with zero columns"
In this case we ignore the summation over $j$ in the following formulas, replacing ${\stackrel{\longrightarrow}{{\rm det}}}(_jA^+(\lambda, r))$ by 1, $e_{m+i,j+r-1}$ by $e_{m+i,m}$ and $i+j+r+m$ by $i+1+r+m$. Similar remarks apply to the case where $s = 1.$ \\ \\ Below we present two determinantal formulas for the \v Sapovalov element $\theta_\gamma$ evaluated at $\lambda\in \mathcal{H}_\gamma$. The key differences are the placement of the odd root vectors $e_{m+i,j+r-1}$ and the types of the determinants used.
\begin{theorem} \label{1shgl} For $\lambda \in \mathcal{H}_\gamma$, we have \begin{equation} \label{1shtpa} \theta_{\gamma}(\lambda) = {\sum_{j=1}^{m-r+1} \sum_{i= 1}^s (-1)^{i+1}{\stackrel{\longrightarrow}{{\rm det}}}(_jA^+(\lambda, r)) \;{\stackrel{\longrightarrow}{{\rm det}}}( _{s+1-i}A^-(\lambda, s))}e_{m+i,j+r-1}.\quad \quad \end{equation} \begin{equation} \label{1shtpb}= {\sum_{j=1}^{m-r+1} \sum_{i= 1}^s e_{m+i,j+r-1}\;(-1)^{i+1}{\stackrel{\longleftarrow}{{\rm det}}}(_jB^+(\lambda, r))\;{\stackrel{\longleftarrow}{{\rm det}}}( _{s+1-i}B^-(\lambda, s))}.\quad \quad \end{equation} \end{theorem} \begin{proof} We prove (\ref{1shtpa}). The proof of (\ref{1shtpb}) is similar.
For the isotropic simple root $\beta = \epsilon_m-\delta_1$, (\ref{1shtpa}) reduces to $\theta_\beta(\lambda) = e_{m+1, m}.$ Note that the overall sign in \eqref{1shtpa} is determined by the condition that the coefficient of $e_{-\pi^0}$ in $\theta_{\gamma}(\lambda)$ is equal to 1, and that this term arises when the last rows of $A^-(\lambda, s)$ and $A^+(\lambda, r))$ are deleted before taking determinants. Suppose that $\alpha = \delta_{s} -\delta_{s+1}, \gamma = \epsilon_r - \delta_{s}$ and $\gamma' = \epsilon_r - \delta_{s+1} = s_\alpha \gamma.$ Assuming the result for $\gamma$ we prove it for $\gamma'$. The result for $\epsilon_{r -1} - \delta_{s}$ can be deduced in a similar way. Set $e_{-\alpha} = e_{m+s+1,m+s}.$ For $1 \leq i \leq s-1$ we have \[ \tau_{i,s+1} = \delta_i - \delta_{s+1} = s_\alpha \tau_{i,s}.\] Consider the matrix
$A^-(\lambda',s+1)$ below, which replaces the matrix $A^-(\lambda, s)$ in Equation (\ref{1shtpa}) in the analogous expression for $\theta_{\gamma'}(\lambda')$. $$ \left[ {\begin{array}{ccccc} e_{m+s+1,m+s}&e_{m+s+1,m+s-1} & \hdots & e_{m+s+1,m+2} & e_{m+s+1,m+1} \\
(\lambda' + \rho,\tau_{s,s+1}^\vee ) &e_{m+s,m+s-1} & \hdots & e_{m+s,m+2} & e_{m+s,m+1} \\ 0& (\lambda' + \rho,\tau_{s-1,s+1}^\vee ) & \hdots & e_{m+s-1,m+2} & e_{m+s-1,m+1} \\
\vdots & \vdots & \ddots & \vdots& \vdots \\ 0&0& \hdots & (\lambda' + \rho,\tau_{2,s+1}^\vee ) & e_{m+2,m+2} \\ 0& 0& \hdots & 0 & (\lambda' + \rho,\tau_{1,s+1}^\vee ) \end{array}}
\right]. $$ Suppose that $(\lambda + \rho,\alpha^\vee )= p$ and let $\lambda' = s_\alpha\cdot\lambda.$ Then \[ (\lambda' + \rho,\tau_{i,s+1}^\vee ) = (\lambda + \rho,\tau_{i,s}^\vee ),\] for $1 \leq i \leq s-1$ , and this means that the last $s-1$ subdiagonal entries of $A^-(\lambda',s+1)$ and $A^-(\lambda, s)$ are equal. Also the entry in second row first column of $A^-(\lambda',s+1)$ is equal to $-p.$ If we remove the first row from $A^-(\lambda',s+1)$ the first column of the resulting matrix will have only one non-zero entry $-p$. If in addition we remove this column, we obtain the matrix $A^-(\lambda, s)$. Therefore
\begin{equation} \label{1stra2} -p\;{\stackrel{\longrightarrow}{{\rm det}}}(_1A^{-}(\lambda,s)) = {{\stackrel{\longrightarrow}{{\rm det}}}}( {}_1A^-(\lambda',s+1)).\end{equation} Similarly by removing the second row from $A^-(\lambda',s+1)$ and noting that $e_{m+s+1,m+s} $ commutes with all entries in $ {}_1A^{-}(\lambda,s)$, we see that \begin{equation} \label{1stra1} e_{m+s+1,m+s} {\stackrel{\longrightarrow}{{\rm det}}}( {}_1A^{-}(\lambda,s))= {\stackrel{\longrightarrow}{{\rm det}}}( {}_1A^{-}(\lambda,s)) e_{m+s+1,m+s} \quad = {\stackrel{\longrightarrow}{{\rm det}}}( _2A^-(\lambda',s+1)). \end{equation} Equation (\ref{121a}) in this situation takes the form
\begin{equation} \label{122nd} e^{p + 1}_{- \alpha}\theta_\gamma(\lambda) = \theta_{\gamma'}(\lambda') e^p_{- \alpha}. \end{equation} If $r \leq k \leq m +s -1$ we have
\begin{equation} \label{123nd} e^{p + 1}_{- \alpha} e_{m+s,k} = (pe_{m+s+1,k} +
e_{m+s+1,m+s} e_{m+s,k})e^p_{- \alpha}. \end{equation} We now consider two cases: in the first entries in ${\stackrel{\longrightarrow}{{\rm det}}}( _{s+1-i}A^-(\lambda, s))$ are replaced by entries in ${\stackrel{\longrightarrow}{{\rm det}}}( _{\{1,s+2-i\}}A^-(\lambda',s+1)_1)$. Suppose $1 \leq i \leq s-1$, and $r \leq k \leq m$.
Then $ e_{- \alpha}$ commutes with $e_{m+i,k}$ and all entries in the matrix $_{s+1-i}A^-(\lambda, s)$ except for those in the first row. Replacing $e_{m+s,k}$ in the matrix $_{s+1-i}A^-(\lambda, s)$ by $e_{m+s+1,k}$ yields the matrix $_{\{1,s+2-i\}}A^-(\lambda',s+1)_1.$ Hence Equation (\ref{123nd}) gives the first equality below. For the second we use cofactor expansion down the first column
\begin{eqnarray} \label{124nd} e^{p + 1}_{- \alpha}\;{\stackrel{\longrightarrow}{{\rm det}}}( _{s+1-i}A^-(\lambda, s))e_{m+i,k} &=& [p\;{\stackrel{\longrightarrow}{{\rm det}}}(_{\{1,s+2-i\}}A^-(\lambda',s+1)_1)\nonumber \\ & +& e_{m+s+1,m+s}{\stackrel{\longrightarrow}{{\rm det}}}(_{s+1-i}A^-(\lambda, s)) ]e_{m+i,k}e^p_{- \alpha}\nonumber\\ &=& \;{\stackrel{\longrightarrow}{{\rm det}}}( _{s+2-i}A^-(\lambda',s+1))e_{m+i,k} e^p_{- \alpha}. \end{eqnarray} In the second case entries in ${\stackrel{\longrightarrow}{{\rm det}}}( _1A^-(\lambda, s))$
are unchanged but the factor $e_{m+s,k}$ is replaced. If $r \leq k \leq m$, then all entries in the matrix $_1A^-(\lambda, s),$ commute with $e_{m+s,k}$ and $e_{- \alpha}, $ so by Equation (\ref{123nd}) we get the first equality below, and the second equality comes from Equations (\ref{1stra2}) and (\ref{1stra1}) \begin{equation}\label{12576} e^{p + 1}_{- \alpha} \;{\stackrel{\longrightarrow}{{\rm det}}}( _1A^-(\lambda, s))e_{m+s,k}e^{-p}_{- \alpha}= \;{\stackrel{\longrightarrow}{{\rm det}}}( {}_1A^{-}(\lambda,s))
[pe_{m+s+1,k} +
e_{m+s+1,m+s}e_{m+s,k}] \nonumber \end{equation} \begin{equation}\quad\quad\quad\quad\quad\quad\quad\quad\quad = -{\stackrel{\longrightarrow}{{\rm det}}}( _{1}A^-(\lambda',s+1))e_{m+s+1,k} + {\stackrel{\longrightarrow}{{\rm det}}}(_{2} A^-(\lambda',s+1))e_{m+s,k}.\end{equation}
Since ${\stackrel{\longrightarrow}{{\rm det}}}(_jA^+(\lambda, r))$ commutes with $e_{- \alpha}$ and $e_{m+j,i}$ for all $i,j$, it follows from the induction assumption and Equations (\ref{122nd}), (\ref{124nd}) and (\ref{12576}) that \begin{equation} \label{1shtpab} \theta_{\gamma'}(\lambda')= \sum_{j=1}^{m-r+1} \sum_{i= 1}^{s+1} (-1)^{i+1}{\stackrel{\longrightarrow}{{\rm det}}}(_jA^+(\lambda, r)) \; {\stackrel{\longrightarrow}{{\rm det}}}( _{s+2-i}A^-(\lambda',s+1))e_{m+i,j+r-1},\quad \quad \nonumber \end{equation} as desired. \end{proof}
\noindent By inserting odd root vectors as an extra column in either $A^+(\lambda, r)$ or $A^-(\lambda, s)$, we obtain a variation of Equation \eqref{1shtpa}
where the odd root vectors are inserted into one of the determinants. Note that the two determinants in \eqref{1shtpa} commute. We give the details only for $A^+(\lambda, r)$. For $i\in [s]$, let $C_{(i)}(\lambda, r)$ be the matrix obtained from $A^+(\lambda, r)$ by adjoining the vector \begin{equation}\label{hug} (e_{m+i,r}, e_{m+i,r+1}, \ldots, e_{m+i,m} )^{\operatorname{transpose}}\end{equation} as the last column.
\begin{theorem} \label{1thgl}With the above notation, suppose that $\gamma= \epsilon_r - \delta_{s}$ and $\lambda\in\mathcal{H}_\gamma$. Then we have
\begin{equation} \label{1thtpa} \theta_{\gamma}(\lambda) = \sum_{i= 1}^s (-1)^{i+1}{\stackrel{\longrightarrow}{{\rm det}}}(_{s+1-i} A^-(\lambda, s))\;{\stackrel{\longrightarrow}{{\rm det}}}(C_{(i)}(\lambda, r)). \quad \quad \end{equation} \end{theorem} \begin{proof} This follows by cofactor expansion of ${\stackrel{\longrightarrow}{{\rm det}}}(C_{(i)}(\lambda, r))$ down the last column. If $r=m$, ${\stackrel{\longrightarrow}{{\rm det}}}(C_{(i)}(\lambda, r))$ should be interpreted as $e_{m+i,m}$.
\end{proof}
\begin{rem} \label{did} {\rm For convenience we record the facts. \begin{itemize} \item[{{\rm(a)}}] If $A''$ is the matrix obtained from $A^-(\lambda, s)$ by deleting the row and column containing $b_k$, then there are no entries in $A''$ of the form $e_{m+k,*}$ or $e_{*,m+k}$. \item[{{\rm(b)}}] If
$C_{(i)}''$ is the matrix obtained from $C_{(i)}(\lambda, r)$ by deleting the row and column containing $a_\ell$, then there are no entries in $C_{(i)}''$ of the form $e_{\ell,*}$ or $e_{*,\ell}$. \end{itemize}}\end{rem}
\subsection{Lie Algebras.} Let $\fg= \fgl(m)$, and $\alpha = \epsilon_r -\epsilon_\ell $. We give a determinantal formula for the \v{S}apovalov element $\theta_{\alpha,1}$. Set $\sigma_{i,j} = \epsilon_{i} - \epsilon_{j}$. Consider the following matrices with entries in $U(\mathfrak{n}^-)$.
$$ C_\lambda = \left[ {\begin{array}{ccccc} e_{r+1,r} & e_{r+2,r} & \hdots & e_{\ell,r}\\ -a_1 & e_{r+2,r+1} & \hdots & e_{\ell,r+1} \\ 0&-a_2 & \hdots & e_{\ell,r+2} \\
\vdots & \vdots & \ddots & \vdots \\ 0& \hdots & -a_{j-1}& e_{\ell,\ell-1} \end{array}}
\right],\; D_\lambda= \left[ {\begin{array}{cccc} e_{\ell,\ell-1}&e_{\ell,\ell-2} & \hdots & e_{\ell,r} \\
1-a_{1} &e_{\ell -1,\ell-2} & \hdots & e_{\ell -1,r} \\ 0& 1-a_{2} & \hdots & e_{\ell -2,r} \\
\vdots & \vdots & \ddots & \vdots \\ 0&0& 1-a_{j-1} & e_{r+1,r} \end{array}}
\right]. $$ where $j=\ell-r$ and $a_k=(\lambda + \rho,\sigma_{r,r+k}^\vee ) $ for $k\in [j-1]$.
\begin{theorem} \label{1shgl2} The \v{S}apovalov element $\theta_{\alpha,1}$ is given by \[ \theta_{\alpha,1}(\lambda) = {\stackrel{\longrightarrow}{{\rm det}}}\;C_\lambda = {\stackrel{\longrightarrow}{{\rm det}}}\;D_\lambda, \] for $\lambda$ such that $ (\lambda + \rho,\sigma_{r,\ell}^\vee )=1$.\end{theorem} \begin{proof} Similar to the proof of Theorem \ref{1shgl}. \end{proof} \noindent Note that if $ (\lambda + \rho,\sigma_{r,\ell}^\vee )=1$ and $ (\lambda + \rho,\sigma_{r,s}^\vee )=0$, then $ (\lambda + \rho,\sigma_{s,\ell}^\vee )=1$ and $ (\lambda + \rho- \sigma_{s,\ell},\sigma_{r,s}^\vee )=1$. \begin{corollary} \label{1sapc} In the above situation, for any highest weight vector $v_\lambda$ of weight $\lambda$, we have
$\theta_{\epsilon_{r,1} -\epsilon_{\ell,1}}v_\lambda = \theta_{\epsilon_{r,1} -\epsilon_{s,1}} \theta_{\epsilon_{s,1} -\epsilon_{\ell,1}}v_\lambda$.
\end{corollary} \begin{proof} Under the given hypothesis the matrix $C_\lambda$ in Theorem \ref{1shgl2} is block upper triangular.\end{proof} \noindent \begin{corollary} \label{11.5} For $p\ge 1$ and $\lambda\in \mathcal{H}_{\alpha,p}$, we have \[\theta_{\alpha,p}(\lambda) = {\stackrel{\longrightarrow}{{\rm det}}}\;C_{\lambda-(p-1)\alpha}\ldots {\stackrel{\longrightarrow}{{\rm det}}}\;C_{\lambda-\alpha}\;{\stackrel{\longrightarrow}{{\rm det}}}\;C_{\lambda}.\] \end{corollary} \begin{proof} Combine Theorems \ref{1calu} and \ref{1shgl2}. \end{proof} \noindent The above result may be viewed as a version of \cite{CL} Theorem 2.7.
\subsection{Expansions of the Determinantal Formulas.}\label{expa} Returning to $\fg= \fgl(m,n),$ we obtain two expansions of the determinantal formula from \eqref{1thtpa}. In the first case we assume that $\gamma, \gamma' \in B(\lambda)$, where $\gamma' = s_\alpha\gamma = \gamma -\alpha$. This means that $(\lambda + \rho,\alpha^\vee ) =0$ and by Equation \eqref{pin} we have, in this situation $\theta_{\gamma}v_{\lambda}= \theta_{\alpha,1} \theta_{\gamma'} v_{\lambda}$. Theorem \ref{stc} below could be considered as a refinement of this equation when $\lambda$ is replaced by ${{\widetilde{\lambda}}}=\lambda+T \xi$. It will be used to study the modules ${M^X}({\lambda})$ in Subsection \ref{pea}. The idea is to express the extra term that arises as a \v Sapovalov element for a proper subalgebra of $\fg$. There is a second case, which will be used in a similar way in Subsection \ref{pie}. \subsubsection{Hessenberg Matrices.}
A matrix $B=(b_{ij})$ is {\it $($upper$)$ Hessenberg} if $b_{ij}=0$ unless $i\le j+1$. If $B$ is $n\ti n$ Hessenberg, we say that $B=(b_{ij})$ is {\it Hessenberg of order $n$}.
The determinant of such a matrix is a signed sum of $2^{n-1}$ (suitably ordered) terms $ \prod_{i=1}^n b_{\nu(i),i}$ for certain permutations $\nu$. \begin{lemma} \label{hes} Suppose that $B$ is Hessenberg of order $n$. \begin{itemize} \item[{{\rm(a)}}] For a fixed $q\in [n-1],$ let $\mathtt{T} = b_{q+1q}.$ Then
\begin{equation} \label{sam}{\stackrel{\longrightarrow}{{\rm det}}} B = -\mathtt{T} {\stackrel{\longrightarrow}{{\rm det}}}B'' + {\stackrel{\longrightarrow}{{\rm det}}}B',\end{equation} where $B'$ and $B''$ are obtained from $B$ by setting $\mathtt{T} =0,$ and by deleting the row and column containing $\mathtt{T}$ respectively. \item[{{\rm(b)}}] The matrix $B'$ is block upper triangular, with two diagonal blocks which are upper Hessenberg of order $q$ and $n-q$. \item[{{\rm(c)}}] The matrix $B''$ is upper Hessenberg of order $n-1$. Also any term in the expression \eqref{lr} for ${\stackrel{\longrightarrow }{{\rm det}}}(B)$ which contains a factor of the form $b_{iq}$ or $b_{q+1j}$
cannot occur in ${\stackrel{\longrightarrow}{{\rm det}}} B''$. \item[{{\rm(d)}}] If $B$ is
$(n+1)\ti n$ Hessenberg with $b_{q+1,q}=0$, and $i\in [q]$, then the submatrix obtained from $B$ by deleting row $i$ is singular $($both determinants are zero$)$. \end{itemize} \end{lemma} \begin{proof} Part (a) follows by separating the products in \eqref{lr} that contain $\mathtt{T}$ from those that do not. Note that $\mathtt{T}$ commutes with all entries in $B$, and that the order of all other factors of the products is unchanged. The rest is easy. \end{proof} \noindent
\subsubsection{Two Factorizations.}
At first we do not make any assumptions on $\lambda\in \mathfrak{h}^*$. However as in Subsection \ref{nir}, we need to evaluate \v Sapovalov elements at arbitrary points, $\lambda \in \mathfrak{h}^*$. To do this we take Equation \eqref{1thtpa} as the definition.
We write $C_{(i)}'$ and $C_{(i)}''$ for the matrices obtained from $C_{(i)}(\lambda, r)$ by setting $\mathtt{T} =0,$ and by deleting the row and column containing $\mathtt{T}$ respectively. Similarly we write $A'$ and $A''$ for the matrices obtained from $A=A^-(\lambda, s)$ by setting $\mathtt{S}=0$, and by deleting the row and column containing $\mathtt{S}$ respectively. The same notation is used for submatrices of $C_{(i)}$ and $A$. Because $-\mathtt{T}$ is on the subdiagonal, we have by Lemma \ref{hes}
\begin{eqnarray} \label{2thtpa} {\stackrel{\longrightarrow}{{\rm det}}}C_{(i)} &=&\mathtt{T} \;{\stackrel{\longrightarrow}{{\rm det}}}C_{(i)}'' + {\stackrel{\longrightarrow}{{\rm det}}}C_{(i)}'. \end{eqnarray}
\noindent For $1\le i\le s$, let $E_{(i)}= {}_{[j]}{C_{(i)}}{}_{[j]}$ be the matrix obtained from $C_{(i)}$ by deleting the first $j=\ell-r$ rows and columns.
Also let $F=\; _{[s-k]}A^-(\lambda, s)_{[s-k]} $.
For an even root $\alpha$, we abbreviate $\theta_{\alpha,1}$ to $\theta_{\alpha}$. \begin{lemma} \label{haha} \begin{itemize} \item[{{\rm(a)}}] If $k+1\le i\le s$, then $_{s+1-i}A'$ is singular. \item[{{\rm(b)}}] If $i\in [k]$, the matrix $_{s+1-i}A'$ is block upper triangular with upper triangular block having LR determinant $\theta_{\alpha_2}$
and lower triangular block $_{k+1-i}F$. Hence \begin{equation} \label{hoho}{\stackrel{\longrightarrow}{{\rm det}}} _{s+1-i}A' = \theta_{\alpha_2}\ti{\stackrel{\longrightarrow}{{\rm det}}}_{k+1-i}F.\end{equation}
\item[{{\rm(c)}}]
For all $i\in [s]$,
the matrix $C_{(i)}'$ is block upper triangular with with upper triangular block having LR determinant $\theta_{\alpha_1}$ and lower triangular block $E_{(i)}$. Hence \begin{equation}\label{hihi}{\stackrel{\longrightarrow}{{\rm det}}}C_{(i)}'=\theta_{\alpha_1}\ti{\stackrel{\longrightarrow}{{\rm det}}}E_{(i)}.\end{equation} \end{itemize} \end{lemma} \begin{proof} Part (a) follows from Lemma \ref{hes} (d). By Lemma \ref{hes} (b) $_{s+1-i}A'$ is block upper triangular, and using Theorem \ref{1shgl2}, the upper triangular block has determinant equal to $\theta_{\alpha_2}$. This gives (b) and the proof of (c) is similar.\end{proof} \noindent We apply Lemma \ref{hes} in the case that $B= {}_{s+1-i}A$.
\begin{lemma} \label{sky} \begin{itemize} \item[{{\rm(a)}}] If $i\leq k-1$, then \begin{equation} \label{cue}{\stackrel{\longrightarrow}{{\rm det}}} _{s+1-i}A = -\mathtt{S} {\stackrel{\longrightarrow}{{\rm det}}} _{s-i}A'' + {\stackrel{\longrightarrow}{{\rm det}}} _{s+1-i}A'.\end{equation} \item[{{\rm(b)}}] If $i\geq k+1$, then \begin{equation} \label{men}{\stackrel{\longrightarrow}{{\rm det}}} _{s+1-i}A = \mathtt{S} {\stackrel{\longrightarrow}{{\rm det}}} _{s+1-i}A''.\end{equation} \item[{{\rm(c)}}] \begin{equation} \label{kip}{\stackrel{\longrightarrow}{{\rm det}}} _{s+1-k}A = {\stackrel{\longrightarrow}{{\rm det}}} _{s+1-k}A'.\end{equation} \end{itemize}\end{lemma} \begin{proof} Note that if $p<q$, then row $q$ of $A$ is row $q-1$ of $_p A$. If $i\leq k-1$ it follows that $_{s-i}(A'')$ is obtained from $_{s+1-i}A$ by deleting the row and column containing $\mathtt{S}$, that is $_{s-i}(A'') = (_{s+1-i}A)''$. Similarly if $i\geq k+1$ then $_{s+1-i}(A'') = (_{s+1-i}A)''$. In both cases we obtain $_{s+1-i}(A') = (_{s+1-i}A)' $, but by Lemma \ref{haha} (a) $_{s+1-i}A'$ is singular if $i\geq k+1$. The different signs in \eqref{cue} and \eqref{men} arise since $\mathtt{S}$ is on the subdiagonal of $_{s+1-i}A$ in case (a), and on the diagonal of $_{s+1-i}A$ in case (b). \end{proof}
\subsubsection{Some General Linear Subalgebras.} To state the results, we require a bit more notation. For the rest of this Subsection, the element $\rho$ defined for $\fgl(m,n)$ in Equation \eqref{rde} will be denoted by $\rho_{m,n}.$ In addition let $(\;,\;)_{m,n}$ be the bilinear form on $\mathfrak{h}^*$ defined by \[(\epsilon_i,\epsilon_j)_{m,n} = - (\delta_i,\delta_j)_{m,n} = \delta_{i,j}\] for all relevant indices $i, j$.
Set $({\lambda} + \rho,\alpha^\vee ) =T$. The extra term in \eqref{nice} below comes from a element for a subalgebra of $\fg$ isomorphic to $\fgl(m-1,n)$. Suppose that $V=\mathtt{k}^{m|n}$ is a super vector space of dimension $(m|n)$ and identify $\fgl(m,n)$ and $\fgl(V)$ by means of the standard basis $e_1,\ldots,e_{m+n}.$ Do the same for
$\fgl(m-1,n)=\fgl(\mathtt{k}^{m-1|n})$ using the standard basis $\overline{e}_1,\ldots,\overline{e}_{m+n-1}$ for $\mathtt{k}^{m-1|n}$. Fix $k, \ell $ with $1\le \ell \le m$, and $1\le k\le n$. Then let ${\overline\psi}^\ell :\overline{\fg} = \fgl(m-1,n)\longrightarrow \fg=\fgl(m, n)$
be the embedding induced by the map $V=\mathtt{k}^{m-1|n}\longrightarrow \mathtt{k}^{m|n}$ sending $\overline{e}_i$ to $e_i$ for $i<\ell$ and $\overline{e}_i$ to $e_{i+1}$ for $i\ge \ell$ (so that $e_\ell$ is not in the image of this map). \\ \\ The embeddings of general linear subalgebras defined in the above paragraph are all we need for Theorem \ref{stc}, but for Theorem \ref{nex} we need two variations.
Define maps $\mathtt{k}^{m|n-1}\longrightarrow \mathtt{k}^{m|n}$, (resp. $\mathtt{k}^{m-1|n-1}\longrightarrow \mathtt{k}^{m|n}$) of superspaces using basis elements in a similar way, with $e_{m+k}$ not in the image ( resp. $e_\ell$ and $e_{m+k}$ not in the image). Then let $\underline{\psi}_k:\underline{\fg} =\fgl(m,n-1)\longrightarrow \fg$ and ${\underline{\overline{\psi}}_k^\ell }: \underline{\overline{\fg}} = \fgl(m-1,n-1)\longrightarrow \fg$
be the embeddings of Lie superalgebras induced by these maps. Finally, let ${\mathfrak{h}}, {\overline{\mathfrak{h}}}$ and $\underline{\overline{\mathfrak{h}}}$ be the diagonal Cartan subalgebras of ${\fg}, {\overline{\fg}}$ and $\underline{\overline{\fg}}$, and let \[{\overline\phi^\ell }:\mathfrak{h}^*\longrightarrow {\overline{\mathfrak{h}}^*}, \quad {\underline\phi_k}:\mathfrak{h}^*\longrightarrow {\underline{\mathfrak{h}}^*} \; \mbox{ and }\; \underline{\overline{\phi}}_k^\ell :\mathfrak{h}^*\longrightarrow \underline{\overline{\mathfrak{h}}}^*\]
be the maps dual to the restriction of ${\overline\psi}^\ell, \underline{\psi}_k$ and $\underline{\overline{\psi}}_k^\ell $ to ${\overline{\mathfrak{h}}}, \underline{\mathfrak{h}}$ and $\underline{\overline{\mathfrak{h}}}$ respectively. \\ \\ We fix $\ell \in [m]$ and $k \in [n]$, and then use the shorthand \[ \overline{\psi} =\overline{\psi}^\ell,\quad \underline{\psi} = \underline{{\psi}}_k,\quad \underline{\overline{\psi}} =\underline{\overline{\psi}}_k^\ell, \quad \overline{\phi} =\overline{\phi}^\ell,\quad \underline{\phi} = \underline{{\phi}}_k,\quad \underline{\overline{\phi}} =\underline{\overline{\phi}}_k^\ell.\] For $\alpha\in \mathfrak{h}^*$ we define \[\overline{\alpha} =\overline{\phi}(\alpha),\quad \underline{\alpha} = \underline{{\phi}}(\alpha),\quad \underline{\overline{\alpha}} =\underline{\overline{\phi}}(\alpha).\]
Observe that $\;{\overline\phi}({\epsilon_{\ell}}) =0$ and the restriction of ${\overline\phi}$ to
$\mathfrak{h}'={\operatorname{span}}\{ \epsilon_p,\delta_q|p\neq \ell\}$ is an isomorphism onto ${\overline\mathfrak{h}}^*.$ Given $\lambda\in \mathfrak{h}^*$, we define ${\lambda}_1\in {\overline{\mathfrak{h}}}^*$, by \begin{equation} \label{flx} ({\lambda}_1+\rho_{m-1,n}, {\overline\beta})_{m-1,n} = ({\lambda}+\rho_{m,n},\beta)_{m,n}\end{equation}
for $\beta \in \mathfrak{h}'$.
Similarly we define ${\lambda}_2\in \underline{{\mathfrak{h}}}^*, \; {\lambda}_3\in \underline{\overline{\mathfrak{h}}}^*$ by \begin{equation} \label{fax} ({\lambda}_2+\rho_{m,n-1},{\underline\beta})_{m,n-1} = ({\lambda}+\rho_{m,n},\beta)_{m,n},\end{equation}
\begin{equation} \label{fox} ({\lambda}_3+\rho_{m-1,n-1},\underline{\overline\beta})_{m-1,n-1} = ({\lambda}+\rho_{m,n},\beta)_{m,n}.\end{equation}
In \eqref{fax} and \eqref{fox} we assume that
$$\beta \in {\operatorname{span}}\{ \epsilon_p,\delta_q|q\neq k\} \mbox{ and } \beta \in {\operatorname{span}}\{ \epsilon_p,\delta_q|p\neq \ell, q\neq k\}$$ respectively. \subsubsection{Pullbacks.} From now on $A= A^-(\lambda, s) $. Let $A''= {}_{s+1-k}A_{s-k} $. This is the matrix obtained from $A$ by deleting the row and column containing $\mathtt{S}$.
By Remark \ref{did}, all entries from $\fg$ in $A''$ are in the image of the map $\;{\underline\psi}_{k}:\underline{\fg} = \fgl(m,n-1)\longrightarrow \fg=\fgl(m, n)$. This allows us to define a matrix $A[k]$ with entries in $U({\underline{\fg}})$ called the {\it pullback of $A''$ under ${\underline{{\psi}}}_k$} in the following way. \begin{itemize} \item[{{\rm(i)}}] The matrix $A[k]$ is Hessenberg of the same size as $A''$. \item[{{\rm(ii)}}] All subdiagonal entries of $A[k]$ and $A''$ are equal. \item[{{\rm(iii)}}] If $j \ge i$, then the entry in row $i$ and column $j$ of $A[k]$ is the element of ${\underline{\fg}}$ which is mapped by ${\underline\psi}$ to the entry in row $i$ and column $j$ of $A''$. \end{itemize} We sometimes write $\underline{\psi}(A[k])= A''$ in this situation. We have $\underline{\psi}(_{s-i}A[k]) =_{s-i}A''.$
Note that this is consistent with \eqref{fax}. In fact the latter just says that (ii) holds. The point of Equations \eqref{flx}-\eqref{fox} is that they make it clear that $\gamma\in B(\lambda)$ implies $\overline{\gamma} \in B(\lambda_1)$ etc, which we need for the applications. \\ \\ Next we list the other pullbacks we require.
Then we explain why these pullbacks are defined. Let \begin{equation} \label{lax} f(i) = \left\{ \begin{array}
{ccc} i & \mbox{ if } & i\le k-1\\ i+1 & \mbox{ if } & i\ge k. \end{array} \right.\end{equation} We define \begin{equation} \label{sax} C_{(i)}[k] \mbox{ (resp. } E_{(i)}[k]) \mbox{ to be the pullback } \mbox{ of } C_{(f(i))} \mbox{ (resp. } E_{(f(i))}) \mbox{ under } {\underline{{\psi}}} ,\end{equation}
\begin{equation} \label{six} F[{\ell}] \mbox{ (resp. } C_{(i)}[{\ell}]) \mbox{ to be the pullback of } F \mbox{ (resp. } C_{(i)}'') \mbox{ under } \overline{\psi},\end{equation} \begin{equation} \label{sox} A''[k,{\ell}] \mbox{ (resp. } C_{(i)}[k,{\ell}]) \mbox{ to be the pullback of } A'' \mbox{ (resp. } C''_{(f(i))}) \mbox{ under } \underline{\overline{\psi}}. \end{equation} The pullbacks of $C_{(i)}''$ and both matrices in \eqref{sox} exist by Remark \ref{did}. For \eqref{sax} note that all entries in $A^+(\lambda, r)$ are elements of $ \fgl(m)\oplus 0 \subseteq \fg_0$. Also if $i\neq k$, the vector in \eqref{hug} contains no element of the form $e_{m+k,*}$ or $e_{*,m+k}$, so the same is true for $C_{(i)}$, and $E_{(i)}, i\neq k$. Note that $C'_{(k)}$ is not considered in \eqref{sax}. The argument for $F$ is similar but easier, all entries in $F={}_{[s-k]}A^-(\lambda, s)_{[s-k]} $ are elements of $0\oplus \fgl(n) \subseteq \fg_0$. Thus $F$ contains no entries of the form $e_{\ell,*}$ or $e_{*,\ell}$.
\subsubsection{\v Sapovalov elements.} \label{dry} We can use Theorem \ref{1thgl} to express the {\v Sapovalov elements} for
\begin{equation} \gamma_1, \quad {\beta_1} = {\underline{\gamma_1+\alpha_2}}, \quad {\beta_2} = {\overline{\alpha_1+\gamma_1}}, \quad
{\underline{\overline{\gamma}}} = {{\underline{\overline\phi} (\gamma)}}.
\end{equation} in terms of the matrices introduced above. Note that these are roots for different Lie superalgebras. \begin{lemma} Given $\lambda\in \mathfrak{h}^*$, define $\lambda_1-\lambda_3$ by \eqref{flx}-\eqref{fox}. Then the \v Sapovalov elements for the roots $ \gamma_1, \beta_1, \beta_2$ and ${\underline{\overline{\gamma}}}$ are given by
\begin{equation} \label{moe} \theta_{\gamma_1}(\lambda) =\sum_{i= 1}^k (-1)^{i+1}{\stackrel{\longrightarrow}{{\rm det}}}_{k+1-i}F\ti {\stackrel{\longrightarrow}{{\rm det}}}E_{(i)},
\end{equation} \begin{equation}\label{mad} \theta_{\beta_1}({{\lambda}_1})= \sum_{i= 1}^{s-1} (-1)^{i}{\stackrel{\longrightarrow}{{\rm det}}}_{s-i}A[k] \ti {\stackrel{\longrightarrow}{{\rm det}}}\;E_{(i)}[k],\end{equation}
\begin{equation}\label{mud}\theta_{{\beta_2}}({{\lambda}_2})=\sum_{i= 1}^k (-1)^{i+1}{\stackrel{\longrightarrow}{{\rm det}}}_{k+1-i}F[\ell] \ti{\stackrel{\longrightarrow}{{\rm det}}}{C}''_{(i)}[\ell],\end{equation}
\begin{equation} \label{mix}\theta_{\underline{{\overline{\gamma}}}}({{\lambda}_3})= \sum_{i= 1}^{s-1}(-1)^{i+1}\;{\stackrel{\longrightarrow}{{\rm det}}}_{s-i}A''[k,{\ell}]\ti
{\stackrel{\longrightarrow}{{\rm det}}}C_{(i)}''[k,{\ell}]. \end{equation} \end{lemma} \begin{proof} This follows from Theorem \ref{1thgl} with $F$ (resp. $A[k]$, $F[\ell], A''[k,{\ell}]$) playing the role of $A^-(\lambda, s)$, and $E_{(i)},$ (resp. $E_{(i)}[k],$ ${C}''_{(i)}[\ell], C_{(i)}''[k,{\ell}])$ playing the role of $C_{(i)}(\lambda, r)$. \end{proof} \subsubsection{Determinantal Expansions: Main Results.} \label{det} Assume that $\gamma = \epsilon_r- \delta_{s},$ $\alpha = \epsilon_r- \epsilon_{\ell }$ and $\gamma'=s_\alpha\gamma=\epsilon_\ell- \delta_{s},$ where $r<\ell=r+j\le m.$ \footnote{ The case were $\alpha$ is a root of $\fgl(n)$ can be handled similarly.}
Note that $\;{\overline\phi}({\epsilon_{\ell}}) =0$ and the restriction of ${\overline\phi}$ to $\mathfrak{h}' ={\operatorname{span}}\{ \epsilon_i,\delta_j|i\neq \ell\}$ is an isomorphism onto ${\overline\mathfrak{h}}.$ Define $a_i, b_i$ as in \eqref{eat}.
\begin{theorem} \label{stc} With the above notation, set $\mathtt{T}=({\lambda}+\rho,\alpha^\vee)=a_j.$ Then if $\overline{\gamma} = {\overline\phi} (\gamma)$, we have \begin{equation}\label{nice}\theta_\gamma v_{{\lambda}} = [\overline{\psi}(\theta_{\overline{\gamma}}( {{\lambda_1}}))\mathtt{T} +\theta_{\alpha,1}\theta_{\gamma'}]v_{{\lambda}}. \end{equation}\end{theorem}
\noindent We remark that in the definition of $A^+({{\lambda} }, r)$, and hence also in $C_{(i)}({{\lambda} }, r)$ we have We give the proof after proving Theorem \ref{nex}. \\ \\
\noindent Next we suppose $\alpha_1, \alpha_2$ (resp. $\gamma_1, \gamma_2$) are distinct positive even (resp. odd) roots such that $\alpha_1 +\gamma_1 +\alpha_2 \;=\gamma_2$. Then set $\;\gamma=\gamma_2$. Without loss of generality we may assume that $\alpha_1 = \epsilon_r- \epsilon_{\ell }$, $\alpha_2 = \delta_k- \delta_{s},$ $\gamma_1 =\epsilon_{\ell }- \delta_k$, where $r<\ell=r+j\le m.$ In the result below, we treat $\mathtt{T}= a_j,$ and $ \mathtt{S}= b_k$ as indeterminates, and obtain expansions of Equation \eqref{1thtpa}. \begin{theorem} \label{nex} We have \begin{equation}\label{mice} \theta_{\gamma_2}v_{{\lambda}} = [\theta_{\alpha_2}\theta_{\alpha_1}\theta_{\gamma_1} -\theta_{{{\alpha_1}}}\overline{\psi}(\theta_{{\beta_1}}({{\lambda_1}}))\mathtt{S} + \theta_{{\alpha_2}}\underline{\psi}(\theta_{{\beta_2}}({{\lambda_2}}))\mathtt{T}
-\underline{\overline\psi}(\theta_{\underline{{\overline{\gamma}}}}({{\lambda_3}}))\mathtt{S}\mathtt{T}] v_{{\lambda}}.\end{equation} \end{theorem}
\begin{rem} \label{dan} {\rm We need to know that certain products in $U(\mathfrak{n}^-)$ commute, and this can be shown based on a consideration of the weights of their factors. Write $\epsilon_{m+i} =\delta_i$ for $i\in [n]$. Then we say that a product $u=u_1 \ldots u_t$ has {\it top weight} $\epsilon_p -\epsilon_q,$ where $1\le p< q\le m+n,$ if for $i\in [t]$, $u_i$ has weight $\epsilon_{p_i} - \epsilon_{q_i}$ (with $p_i<q_i$) and $\sum_{i=1}^{t}\epsilon_{p_i} - \epsilon_{q_i} =\epsilon_{p} - \epsilon_{q}$. Clearly $v, w$ commute if they are respectively linear combinations of elements having top weights $\epsilon_p -\epsilon_q$ and $\epsilon_a -\epsilon_b$ with $q<a$. }\end{rem} \noindent {\it Proof of Theorem \ref{nex}.} We substitute \eqref{2thtpa} and the expressions for $_{s+1-i}A^-({\lambda}, s)$ from Lemma \ref{sky} into \eqref{1thtpa} to obtain
\begin{eqnarray} \label{skn} \theta_{\gamma}({\lambda})
&=&\sum_{i= 1}^{k-1}(-1)^{i+1} (-\mathtt{S} {\stackrel{\longrightarrow}{{\rm det}}} _{s-i}A'' + {\stackrel{\longrightarrow}{{\rm det}}} _{s+1-i}A') \ti(\mathtt{T} \;{\stackrel{\longrightarrow}{{\rm det}}}C_{(i)}'' + {\stackrel{\longrightarrow}{{\rm det}}}C_{(i)}')\nonumber\\ &+&\sum_{i= k+1}^s(-1)^{i+1} \mathtt{S} {\stackrel{\longrightarrow}{{\rm det}}} _{s+1-i}A''\ti(\mathtt{T} \;{\stackrel{\longrightarrow}{{\rm det}}}C_{(i)}'' + {\stackrel{\longrightarrow}{{\rm det}}}C_{(i)}')\nonumber\\ &+&(-1)^{k+1} {\stackrel{\longrightarrow}{{\rm det}}} _{s+1-k}A'\ti(\mathtt{T} \;{\stackrel{\longrightarrow}{{\rm det}}}C_{(k)}'' + {\stackrel{\longrightarrow}{{\rm det}}}C_{(k)}').\end{eqnarray} To complete the proof we compute the coefficients of $\mathtt{S}\mathtt{T}, \mathtt{S}, \mathtt{T}$ and the constant term in \eqref{skn}. \\ \\
\noindent (i) The constant coefficient in \eqref{skn} is the first expression below. We use Lemma \ref{haha} (a), then \eqref{hoho}, \eqref{hihi} and then the fact that $\theta_{\alpha_1}$ commutes with all entries in ${\stackrel{\longrightarrow}{{\rm det}}}_{k+1-i}F$ (Remark \ref{dan}). The final equality holds by \eqref{moe} \begin{eqnarray} \label{2thtpc} \sum_{i= 1}^s (-1)^{i+1}{\stackrel{\longrightarrow}{{\rm det}}}_{s+1-i}A'\ti {\stackrel{\longrightarrow}{{\rm det}}}C_{(i)} &=& \sum_{i= 1}^k(-1)^{i+1}{\stackrel{\longrightarrow}{{\rm det}}}_{s+1-i}A'\ti {\stackrel{\longrightarrow}{{\rm det}}}C_{(i)}'\nonumber\\ &=& \sum_{i= 1}^k (-1)^{i+1} \theta_{\alpha_2}\ti{\stackrel{\longrightarrow}{{\rm det}}}_{k+1-i}F\ti \theta_{\alpha_1}\ti{\stackrel{\longrightarrow}{{\rm det}}}E_{(i)}\nonumber\\ &=& \sum_{i= 1}^k (-1)^{i+1}\theta_{\alpha_2}\theta_{\alpha_1}\ti{\stackrel{\longrightarrow}{{\rm det}}}_{k+1-i}F\ti {\stackrel{\longrightarrow}{{\rm det}}}E_{(i)}\nonumber\\ &=& \theta_{\alpha_2}\theta_{\alpha_1}\theta_{\gamma_1}v_{{\lambda}}. \end{eqnarray}
\\ \\ \noindent (ii) Using first (a pullback of) \eqref{hihi}, then the fact that $\theta_{{\underline\alpha_1}}$ commutes with all entries in $A[k]$, and then \eqref{mad} we have \begin{eqnarray} \label{4thtpa} \sum_{i= 1}^{s-1}(-1)^{i}{\stackrel{\longrightarrow}{{\rm det}}}_{s-i}A[k]\ti {\stackrel{\longrightarrow}{{\rm det}}}C_{(i)}'[k] &=& \sum_{i= 1}^{s-1} (-1)^{i}{\stackrel{\longrightarrow}{{\rm det}}}_{s-i}A[k]\ti \theta_{{\underline\alpha_1}}\ti{\stackrel{\longrightarrow}{{\rm det}}}\;E_{(i)}[k]\nonumber\\ &=& \sum_{i= 1}^{s-1} (-1)^{i}\theta_{{\underline\alpha_1}}\ti{\stackrel{\longrightarrow}{{\rm det}}}_{s-i}A[k] \ti {\stackrel{\longrightarrow}{{\rm det}}}\;E_{(i)}[k]\nonumber\\ &=& -\theta_{{\underline\alpha_1}}\theta_{{\beta_1}}({{\lambda_1}}). \end{eqnarray} By \eqref{sax} and the fact that $\underline{{\psi}}$ maps ${\;_{s-i}}A[k]\;$ to
$\;_{s-i}A''$, $\underline{{\psi}}$ maps \eqref{4thtpa} to \begin{equation} \sum_{i= 1}^{k-1}(-1)^{i}\;{\stackrel{\longrightarrow}{{\rm det}}} _{s-i}A'' \ti {\stackrel{\longrightarrow}{{\rm det}}}C_{(i)}' +\sum_{i= k}^{s-1}(-1)^{i}\;{\stackrel{\longrightarrow}{{\rm det}}} _{s-i}A'' \ti {\stackrel{\longrightarrow}{{\rm det}}}C_{(i+1)}'\nonumber \end{equation} and this shows that the coefficient of $\mathtt{S}$ in as stated in \eqref{skn}. \\ \\ \noindent (iii) Next, by \eqref{hoho} and \eqref{mud} we have
\begin{eqnarray} \sum_{i= 1}^{k}(-1)^{i+1}{\stackrel{\longrightarrow}{{\rm det}}}_{s+1-i}A'[\ell]\ti {\stackrel{\longrightarrow}{{\rm det}}}C''_{(i)}[\ell] &=&\sum_{i= 1}^k (-1)^{i+1}\theta_{\overline{\alpha_2}} {\stackrel{\longrightarrow}{{\rm det}}}_{k+1-i}F[\ell] \ti{\stackrel{\longrightarrow}{{\rm det}}}{C}''_{(i)}[\ell]\nonumber\\ & =& \theta_{{\overline\alpha_2}}\theta_{{\beta_2}}({{\lambda_2}}),\nonumber \end{eqnarray} and $\overline{{\psi}}$ maps this to \begin{equation}\sum_{i= 1}^k (-1)^{i+1}
{\stackrel{\longrightarrow}{{\rm det}}}_{s+1-i} A' \ti {\stackrel{\longrightarrow}{{\rm det}}}C''_{(i)},\nonumber\\ \end{equation} which is the coefficient of $\mathtt{T}$ in \eqref{skn}.
\\ \\ (iv) Finally, let \[A''= {}_{s+1-k}A_{s-k} = \underline{\overline{\psi}}(A[k,\ell]). \] By \eqref{mix}, \begin{equation} -\theta_{\underline{{\overline{\gamma}}}}({{\lambda}_3})= \sum_{i= 1}^{s-1}(-1)^{i}\;{\stackrel{\longrightarrow}{{\rm det}}}_{s-i}A''[k,{\ell}]\ti
{\stackrel{\longrightarrow}{{\rm det}}}C_{(i)}''[k,{\ell}],\nonumber \end{equation}
and $\underline{\overline{\psi}}$ maps this to
\begin{equation}\sum_{i= 1}^{k-1}(-1)^{i}\;{\stackrel{\longrightarrow}{{\rm det}}}_{s-i}A'' \ti {\stackrel{\longrightarrow}{{\rm det}}}C_{(i)}'' +\sum_{i= k}^{s-1}(-1)^{i}\;{\stackrel{\longrightarrow}{{\rm det}}}_{s-i}A'' \ti {\stackrel{\longrightarrow}{{\rm det}}}C_{(i+1)}'' \nonumber \end{equation} which is the coefficient of $\mathtt{S}\mathtt{T}$ in \eqref{skn}.
$\Box$ \\ \\ {\it Proof of Theorem \ref{stc}} The proof is similar to the proof of Theorem \ref{nex}, but easier. Instead of \eqref{skn} we have
\begin{equation} \label{sin} \theta_{\gamma}(\widetilde{\lambda}) = \sum_{i= 1}^s(-1)^{i+1}{\stackrel{\longrightarrow}{{\rm det}}}(_{s+1-i}A^-(\widetilde{\lambda}, s)) \ti(-\mathtt{T} \;{\stackrel{\longrightarrow}{{\rm det}}}C_{(i)}'' + {\stackrel{\longrightarrow}{{\rm det}}}C_{(i)}')\nonumber.\end{equation} The proof is completed by computing coefficients as before.
$\Box$
\begin{rem}{\rm Suppose that $(\lambda+\rho, \alpha^\vee_1) = (\lambda+\rho, \alpha^\vee_2)=1$ and assume that $(\xi,\gamma_1)= (\xi,\gamma_2)=0$, and $(\xi,\alpha^\vee_1) = (\xi,\alpha^\vee_2) = 1.$ Then set ${{\widetilde{\lambda}}}=\lambda+T \xi$. Suppose that $\alpha_1 = \epsilon_r- \epsilon_{\ell }$, $\alpha_2 = \delta_k- \delta_{s},$ $\gamma_1 =\epsilon_{\ell }- \delta_k$, where $r<\ell=r+j\le m.$ Then $(\widetilde{\lambda}+\rho, \alpha_1^\vee) = (\widetilde{\lambda}+\rho, \alpha_2^\vee) = T+1$. Hence in the notation of Theorem \ref{nex} we have $\mathtt{S}=\mathtt{T} = T+1$. Thus if $\theta_{\gamma_1}v_{\widetilde{\lambda}}=\theta_{\gamma_2}v_{\widetilde{\lambda}} =0$, then \eqref{mice} yields, over $B$ \begin{equation}\label{mice2} 0=[\underline{\overline\psi}(\theta_{\underline{{\overline{\gamma}}}}({{{\widetilde{\lambda}}_3}}))(T+1) +\theta_{{\alpha_1}}\underline{\psi}(\theta_{{\beta_1}}({{{\widetilde{\lambda}}_2}})) -\theta_{{{\alpha_2}}}\overline{\psi}(\theta_{{\beta_2}}({{{\widetilde{\lambda}}_1}}))] v_{\widetilde{\lambda}}.\end{equation} This gives a generalization of \eqref{rat2}. }\end{rem}
\section{{Survival of \v{S}}apovalov elements in factor modules.}\label{1surv}
Let $v_{\lambda}$ be a highest weight vector in a Verma module $M(\lambda)$ with highest weight $\lambda,$ and suppose $\gamma$ is an odd root with $(\lambda+\rho, \gamma) =0$. We are interested in the condition that the image of $\theta_{\gamma}v_{\lambda}$ is non-zero in various factor modules of $M(\lambda)$. \subsection{Independence of \v{S}apovalov elements.} Given $\lambda \in \mathfrak{h}^*$ recall the set $B(\lambda)$ defined in Section \ref{1s.1}, and define a ``Bruhat order" $\le$ on $B(\lambda)$ by $\gamma' \le \gamma$ if $\gamma-\gamma'$ is a sum of positive even roots. Then introduce a relation $\downarrow$ on $B(\lambda)$ by $\gamma' \downarrow \gamma$ if $\gamma' \le \gamma$ and $(\gamma,\gamma') \neq 0.$ If $\gamma \in B(\lambda)$, we say that $\gamma$ is $\lambda$-{\it minimal} if $\gamma' \downarrow \gamma$ with $\gamma' \in B(\lambda)$ implies that $\gamma' = \gamma$. For $\gamma \in B(\lambda)$ set $B(\lambda)^{-\gamma} = B(\lambda)\backslash \{\gamma\}$. We say $\gamma$ is {\it independent at } $\lambda$ if $$\theta_{\gamma}v_{\lambda}\notin \sum_{\gamma' \in B(\lambda)^{-\gamma} }U(\fg)\theta_{\gamma'} v_{\lambda}.$$ \begin{proposition} \label{1pot}If $\gamma' \downarrow \gamma$ with $\gamma' \in B(\lambda)$ and $\gamma' < \gamma$, then $\theta_{\gamma}v_{\lambda}\in U(\fg)\theta_{\gamma'} v_{\lambda}.$ \end{proposition} \begin{proof} The hypothesis implies that $(\gamma,\alpha^\vee)>0$ and $\gamma =s_\alpha \gamma'$. Thus the result follows from Theorem \ref{man}.\end{proof} \noindent By the Proposition, if we are interested in the independence of the \v{S}apovalov elements $\theta_\gamma$ for distinct isotropic roots, it suffices to study only $\lambda$-minimal roots $\gamma$. \\ \\ For the rest of this section we assume that $\fg = \fgl(m,n)$. We use Equation (\ref{1shtpa}), and order the positive roots of $\fg$ so that each summand in this equation is a constant multiple of $e_{-\pi}$ for some $\pi \in {\overline{\bf P}}(\gamma)$. For such $\pi$ the odd root vector is the rightmost factor of $e_{-\pi}$, that is we have $e_{-\pi}\in U(\mathfrak{n}^-_0)\mathfrak{n}^-_1$. \begin{lemma} \label{1car}If If $\gamma$ is $\lambda$-minimal, then $e_{-\gamma} v_\lambda$ occurs with non-zero coefficient in $\theta_\gamma v_\lambda$. \end{lemma} \begin{proof} Assume $\gamma = \epsilon_r - \delta_{s}.$ Then if $\alpha = \epsilon_r - \epsilon_i$ with $r<i$, or $\alpha = \delta_j - \delta_s$ with $j<s$ we have $(\lambda + \rho, \alpha^\vee) \neq 0$, since $\gamma$ is $\lambda$-minimal. Thus the entries on the superdiagonals of $A^+(\lambda, r)$ and $A^-(\lambda, s)$ are non-zero. Thus the result follows from Theorem \ref{1shgl}.\end{proof} \begin{theorem} \label{1boy} The isotropic root $\gamma$ is independent at $\lambda$ if and only if $\gamma$ is $\lambda$-{minimal}. \end{theorem} \begin{proof} Set $B=B(\lambda)^{-\gamma} $. If $\gamma$ is not $\lambda$-{minimal} then $\gamma$ is not independent at $\lambda$ by Proposition \ref{1pot}. Suppose that $\gamma$ is $\lambda$-{minimal} and $$\theta_{\gamma}v_{\lambda}\in \sum_{\gamma' \in B}U(\fg)\theta_{\gamma'} v_{\lambda} = \sum_{\gamma' \in B}U(\mathfrak{n}^-)\theta_{\gamma'} v_{\lambda},$$ then by comparing weights \begin{equation} \label{1la}\theta_{\gamma}v_{\lambda}\in \sum_{\gamma' \in B}U(\mathfrak{n}^-_0)e_{-\gamma'} v_{\lambda},\end{equation} But Lemma \ref{1car} implies that $$\theta_{\gamma}v_{\lambda}\equiv ce_{-\gamma}v_{\lambda}\mod \sum_{\gamma' \in B}U(\mathfrak{n}^-_0)e_{-\gamma'} v_{\lambda}$$ for some non-zero constant $c.$ By (\ref{1la}) this contradicts the PBW Theorem. \end{proof} \subsection{Survival of \v{S}apovalov elements in Kac modules.} For $\fg = \fgl(m,n)$ we have $\mathfrak{g}_1 = \mathfrak{g}_1^+ \oplus \mathfrak{g}_1^-,$ where $\mathfrak{g}_1^+ $ (resp. $\mathfrak{g}_1^-$) is the set of block upper (resp. lower) triangular matrices. Let $\mathfrak{h}$ be the Cartan subalgebra of $\fg$ consisting of diagonal matrices, and set $\mathfrak{p} = \mathfrak{g}_0 \oplus \mathfrak{g}_1^{+}.$ Next let
\begin{eqnarray*}
P^+ &=&
\{ \lambda \in \mathfrak{h}^*|(\lambda, \alpha^\vee) \in \mathbb{Z}, (\lambda, \alpha^\vee) \geq 0 \quad
\mbox{for all}\quad \alpha \in \Delta^+_0 \}
\end{eqnarray*} For $\lambda \in P^+,$ let $L^0(\lambda)$ be the (finite dimensional) simple $\mathfrak{g}_0$-module with highest weight $\lambda$. Then $L^0(\lambda)$ is naturally a $\mathfrak{p}$-module and we define the Kac module $K(\lambda)$ by \[K(\lambda) = U(\mathfrak{g}) \otimes_{U(\mathfrak{p})} L^0(\lambda).\]
Note that as a $\mathfrak{g}_0$-module \[K(\lambda) = \Lambda(\mathfrak{g}^-_1) \otimes L^0(\lambda).\] The next result is well-known. Indeed two methods of proof are given in Theorem 4.37 of \cite{Br}. The second of these is based on Theorem 5.5 in \cite{S2}. We give a short proof using Theorem \ref{1shgl}. We now assume that the roots are ordered as in Equation (\ref{1shtpb}), that is with the odd root vector first. \begin{theorem} \label{1shapel} If $\lambda$ and $\lambda - \epsilon_r+\delta_s$ belong to $P^+$ and $(\lambda+\rho,\epsilon_r - \delta_s) = 0$, then $$[K(\lambda):L(\lambda-\epsilon_r+\delta_s)] \neq 0.$$ \end{theorem}
\begin{proof} Set $\gamma = \epsilon_r - \delta_s$.
Let $\theta_\gamma(\lambda)$ be as in Theorem \ref{1shgl}. Then $w = \theta_\gamma(\lambda)v_\lambda$ is a highest weight vector in the Verma module $M(\lambda)$ with weight $\lambda - \gamma$. It suffices to show that the image of $w$ in the Kac module $K(\lambda)$ is nonzero. We have an embedding of $\fg_0$-modules \[ \fg^-_1 \otimes L^0(\lambda) \subseteq \Lambda \fg^-_1 \otimes L^0(\lambda). \] The elements $e_{m+j,i+r-1}$ in Equation (\ref{1shtpb}) form part of a basis for $\fg^-_1$. Furthermore the coefficient of $e_{m+j,i+r-1}$ belong to $U(\mathfrak{n}^-_0)$. Therefore it suffices to show that the coefficient of $e_{m+s,r}$ in this equation is nonzero. This coefficient is found by deleting the first column of the matrix $B^+(\lambda, r)$ and the last row of $B^-(\lambda, s)$ and taking determinants of the resulting matrices, which have only zero entries above the main diagonal. We find that the coefficient of $e_{m+s,r}$ is \[ \pm \prod_{k=1}^{m-r} (1 - (\lambda + \rho, \sigma_{r,r+k}^\vee)) \prod^{s-1}_{k=1} (1 - (\lambda + \rho, \tau_{k,s}^\vee )). \] Since $\lambda \in P^+, (\lambda + \rho, \sigma_{r,r+k}^\vee) \geq 1$ with equality if and only if $k = 1$ and $(\lambda, \epsilon_r - \epsilon_{r+1}) = 0$. This cannot happen if $\lambda - \gamma \in P^+$, so the first product above is nonzero, and similarly so is the second. \end{proof}
\section{The Sum Formula.}\label{sf}
\subsection{An analog of the \v Sapovalov determinant.} \label{7.4} Throughout this Section we assume that $\fg= \fgl(m,n).$ Although ${\bf{\overline{P}}}_X(\eta) = {\bf P}_X(\eta)$ for all $\eta,$ we continue to use the former notation because some arguments hold outside of Type A. The goal is to give a Jantzen sum formula for the modules ${M^X}({\lambda})$ introduced in Section \ref{jaf}. This is done by first computing a \v Sapovalov determinant for these modules. We define an $A$-valued bilinear form on ${M^X}(\widetilde{\lambda})_A$ as in \cite{M} Corollary 8.2.11 and Equation (8.2.14). Let $F_\eta^X(\widetilde{\lambda})$ be the restriction of this form to the weight space $M^X(\widetilde{\lambda})_A^{\widetilde{\lambda}-\eta}.$ (Equation \eqref{dff} below can be taken as the definition of $F_\eta^X(\widetilde{\lambda}))$. The determinant of this form has the important property, see \cite{J1} Lemma 5.1 or \cite{M} Lemma 10.2.1, \begin{equation} \label{yet}v_{T}(\det F^X_\eta(\widetilde{\lambda})) = \sum_{i>0} \dim M_i^{X}(\lambda)^{\lambda - \eta},\end{equation} where $\{\dim M_i^{X}(\lambda)\}$ is the Jantzen filtration. However there is a related determinant $\det G^X_\eta(\widetilde{\lambda})$ whose leading term is easier to compute. The elements $e_{-\pi}v_{\widetilde{\lambda}}^X$ with $\pi \in {\bf{\overline{P}}}_{X}(\eta)$ belong to ${M^X}(\widetilde{\lambda})_A^{\widetilde{\lambda} - \eta}$, and form a $B$-basis for ${M^X}(\widetilde{\lambda})_A^{\widetilde{\lambda} - \eta}\otimes_A B$, but they do not in general form a basis for ${M^X}(\widetilde{\lambda})_A^{\widetilde{\lambda} - \eta}$ as an $A$-module. We define $G^X_\eta(\widetilde{\lambda})$ to be the $A$-bilinear form on $M^X(\widetilde{\lambda})_A^{\widetilde{\lambda} - \eta}$ such that for ${\pi,\sigma \in {\bf{\overline{P}}}_{X}(\eta)}$,
\begin{equation} \label{shy} G^X_\eta(\widetilde{\lambda})(e_{-\sigma}v_{\widetilde{\lambda}}^X,e_{-\pi}v_{\widetilde{\lambda}}^X)=[\zeta_A(e_{\sigma}e_{-\pi})(\widetilde{\lambda})],\end{equation} where $\zeta_A:U(\fg)_A \longrightarrow S(\mathfrak{h})_A$ is the Harish-Chandra projection, \cite{M} (8.2.13). We note that $\det G_\eta^X$ depends on the ordering of the basis, as can be seen already in the case of $\fgl(2,1)$. However its leading term, which we denote by $\operatorname{LT} \det G_\eta^X$ is well-defined up to a scalar multiple.\\ \\ Our goal in this section is to compute the determinants $\det F_\eta^X$ and $ \operatorname{LT} \det G_\eta^X$. We point out at the outset some complications that arise which are not present in the classical case \cite{H2} Theorem 5.8, \cite{M} Theorem 10.2.5. The first is a rather minor point: these determinants should really be considered as elements of $\mathcal{O}(\mathcal{H}_X)=S(\mathfrak{h})/\mathcal{I}(\mathcal{H}_X),$ but we shall express them as elements of $S(\mathfrak{h})$ which map to the corresponding elements of $\mathcal{O}(\mathcal{H}_X)$. Remarkably the determinant $F_\eta^X$ factors a product of linear terms with leading terms of the form $h_\alpha$ with $\alpha$ a root, see Theorem \ref{shapdet}. Apart from having to deal with two determinants, the first real complication arises since it is possible to have distinct non-isotropic positive roots $\alpha_1,$ ${\alpha_2}$ such that $h_{\alpha_1}$ and $h_{\alpha_2}$ are proportional mod $\mathcal{I}(\mathcal{H}_X).$
Indeed suppose that there are distinct orthogonal isotropic roots
$\gamma_1,$ ${\gamma_2}\in X$ and consider the hypotheses
\begin{equation} \label{had} \alpha_1^\vee \equiv -\alpha_2^\vee \mod {\mathbb Q} \gamma_1 +{\mathbb Q} \gamma_2. \end{equation} or \begin{equation} \label{cad} \alpha_1^\vee \equiv \alpha_2^\vee \mod {\mathbb Q} \gamma_1 +{\mathbb Q} \gamma_2 \end{equation} If either \eqref{cad} or \eqref{had} holds, no third non-isotropic positive root $\alpha_3$ satisfies $\alpha_1^\vee \equiv \pm \alpha_3^\vee \mod {\mathbb Q} X$. \\ \\ \noindent For example in type A, suppose that $i<j, k <\ell,$
\begin{equation}\label{wan} \alpha_1 = \epsilon_i -\epsilon_j, \;\alpha_2 =\delta_k-\delta_\ell\end{equation} and either \begin{equation} \label{won} \gamma_1 = \epsilon_j -\delta_k, \; \gamma_2 = \epsilon_i -\delta_\ell
\end{equation} or \begin{equation}\label{wun} \gamma_1 = \epsilon_j-\delta_\ell, \;\gamma_2 = \epsilon_i -\delta_k.\end{equation} If we have \eqref{won}, then $\alpha_1 +\gamma_1 +\alpha_2 =\gamma_2$ so \eqref{had} holds. If we have \eqref{wun} instead, then $\alpha_1+ \gamma_1 = \alpha_2 +\gamma_2$, so \eqref{cad} holds. Note that for $\lambda\in \mathcal{H}_X$ we have, $(\lambda+\rho,\alpha_1^\vee)= -(\lambda+\rho,\alpha_2^\vee)$ if \eqref{had} holds, and $(\lambda+\rho,\alpha_1^\vee)= (\lambda+\rho,\alpha_2^\vee)$ if \eqref{cad} holds. The former case is easily dealt with. To deal with the latter, let $E_{X}$ be the set of pairs $(\alpha_1,\alpha_2)$ such that \eqref{cad} holds. In this case we assume $\alpha_1 +\gamma_1 +\alpha_2 =\gamma_2$, and we frequently let $\alpha$ denote $\alpha_1$, especially to avoid double subscripts. Similarly for compactness we write $[\alpha] =(\alpha_1,\alpha_2)$. A subalgebra of $\fg$ which is a direct sum of root spaces and is isomorphic to $\fgl(2,2)$ will be called a $\fgl(2,2)$-{\it subalgebra}. When \eqref{cad} holds we let $\mathfrak{k}[\alpha]$ denote the $\fgl(2,2)$-subalgebra whose positive part is generated by the root vectors for the roots $\alpha_1, \alpha_2$ and $ \gamma_1$.
Let $W'(\alpha)$ (resp. $V'(\alpha)$) be the set of all (resp. all odd) positive roots of this subalgebra, and set $W(\alpha)=W'(\alpha)\cup X$, $V(\alpha)=V'(\alpha)\cup X$. For $\lambda \in \mathcal{H}_X$ define
\begin{equation}\label{efg} E_X(\lambda) = \{(\alpha_1,\alpha_2) \in E_{X}|(\lambda+\rho,\alpha_1^\vee) =1\}.\end{equation} There are two more situations to deal with. First set
$$B_X= \{\gamma \in (\Delta^+_1\; \backslash \; X)| \gamma \mbox{ is isotropic and } (\gamma ,X) =0\}.$$ Finally let $C_X$ be the set of positive non-isotropic roots $\alpha$, such that there is a unique isotropic root $\gamma\in X$ with $(\gamma,\alpha^\vee)\neq0,$ and set $\gamma = \Gamma(\alpha)$. In this circumstance, since $X$ is an orthogonal set of roots, it follows that $s_\alpha \gamma \notin X.$ Note that if $\alpha \in C_X$ and $\gamma=\Gamma(\alpha)$ we have $h_{{s_\alpha \gamma} } \equiv \pm h_\alpha$ mod $\mathcal{I}(\mathcal{H}_X)$. Let $C^+_X$ be the subset of $C_X$ consisting of those $\alpha$ for which $(\Gamma(\alpha),\alpha^\vee) >0.$ Thus for $\alpha \in C^+_X,$ we have $s_\alpha \gamma =\gamma-\alpha.$ For $\alpha \in C_X^+$, set $Z(\alpha) = X \cup s_\alpha X= X\cup \{s_\alpha\gamma\}$. Then for $\lambda\in \mathfrak{h}^*$, define
\[C_X(\lambda) = \{\alpha \in C_X^+|Z(\alpha) \subseteq B(\lambda)\}.\] Note that if $\alpha\in C_X(\lambda)$ we have $(\lambda+\rho,\alpha)=0$.
\\ \\
\noindent Next fix $\eta$ and consider the following products \begin{eqnarray} D'_0&=& \prod_{\alpha \in C^+_{X},
\gamma = \Gamma(\alpha)}h_{\alpha}^{2 {\bf p}_{Z(\alpha) }(\eta-\gamma)} \label{yaws}\\ D''_0&=& \prod_{[\alpha] \in E_{X}}h_{\alpha}^{2({\bf p}_{V(\alpha)}(\eta -\gamma_2)+{\bf p}_{V(\alpha)}(\eta -\gamma_1-\gamma_2)-{\bf p}_{W(\alpha)}(\eta -\gamma_1-\gamma_2))},\label{yews}\\ D_{1} & = & \prod_{\alpha \in {\Delta}^+_{0}} \;\; \prod^\infty_{r=1}
(h_{\alpha} + (\rho, \alpha) - r(\alpha,\alpha)/2)^
{{\bf p}_{s_\alpha X}(\eta - r \alpha)}, \label{yew1}\\
D_2 & = & \prod_{\gamma \in B_X } (h_{\gamma } +(\rho, \gamma ))^{{\bf p}_{X\cup\{\gamma \}}(\eta - \gamma )},\label{yew3}\\ D_3 &=&\prod_{\alpha \in C^+_X, \gamma = \Gamma(\alpha)} (h_{\alpha}+ (\rho, \alpha) )^{{\bf p}_X(\eta)-{\bf p}_{s_{\alpha} X}(\eta)},\label{yew4}\\ D_4&=& \prod_{[\alpha] \in E_{X}}(h_{\alpha}+ (\rho, \alpha_1) - (\alpha_1,\alpha_1)/2) ^{2{\bf p}_{W(\alpha)}(\eta -\gamma_1-\gamma_2)}.\label{yew5} \end{eqnarray} Then set $D_0=D'_0D''_0$ and $D^X_{\eta}=D_{1}D_{2}D_{3}D_4.$ Our main result on the \v Sapovalov determinant is as follows. \begin{theorem} \label{shapdet}
\begin{itemize} \item[{{\rm(a)}}] Modulo the ideal defining $\mathcal{H}_X$, and up to a nonzero constant factor \begin{equation} \label{la8} \det F^X_{\eta} =D^X_{\eta}.\end{equation} \item[{{\rm(b)}}] Up to a nonzero constant factor, $\det G^X_{\eta}$ has the same leading term as $D_0\det F^X_{\eta}.$\end{itemize} \end{theorem}
\begin{rems} \begin{itemize} \item[{{\rm(a)}}]{\rm The factors in $D_0$ come from comparing the bilinear forms, see Theorems \ref{son} and \ref{epin}, also \eqref{hid} (resp. \eqref{rat3}-\eqref{cat1}) for the case of $\fgl(2,1)$ (resp. $\fgl(2,2)$).
The other factors come from representation theory. The factors in $D_1$ are analogs of the classical ones, those in $D_2$ come from isotropic roots orthogonal to $X$ and those in $D_3$ come from Lemma \ref{eon}. For $D_4$ see Theorem \ref{name}. \item[{{\rm(b)}}] We note that the contributions from $D''_0$ and $D_4$ to the leading term simplify when combined, since $W(\alpha)$ is eliminated. We obtain \begin{eqnarray}\label{yew} \operatorname{LT}\;D''_0D_4&=& \prod_{[\alpha] \in E_{X}}h_{\alpha}^{2({\bf p}_{V(\alpha)}(\eta -\gamma_2)+{\bf p}_{V(\alpha)}(\eta -\gamma_1-\gamma_2))}.\;\;\;\;\;\;\;\end{eqnarray} \item[{{\rm(c)}}] \noindent To compare the factors in the leading term of
$\det G^X_{\eta}$ and $D_0D^X_{\eta}$, we use some shorthand. Thus if $H\in S(\mathfrak{h})$ and $h\in \mathfrak{h},$ let $|H:h|$ denote the multiplicity of $h$ in the leading term of $H$. Similarly $|H/H':h|$ means $|H:h|-|H':h|$. Note that if $\alpha \in \Delta^+$, then $\alpha$ can belong to at most one of the sets $B_X, C_X, E_X$. Hence to find
$|D_\eta^X:h_\alpha|$, it is enough to find $|D_1D_i:h_\alpha|$ for $i=2,3,4$, see also Remark \ref{dab}. }\end{itemize}
\end{rems}
\begin{example} {\rm Suppose $\fg =\fgl(3,2)$ and set $\gamma_1=\epsilon_2-\delta_1, \gamma_2 =\epsilon_1-\delta_2, \alpha_1 = \epsilon_1-\epsilon_2,$ and $\alpha_2=\delta_1-\delta_2.$ Then $\alpha_1 +\gamma_1 +\alpha_2 =\gamma_2$. There are three $W$-orbits on the set of sets of orthogonal isotropic roots of size two and we consider a representative from each orbit. If $X =\{\gamma_1, \gamma_2\}$, then $E_X=\{(\alpha_1,\alpha_2)\}$ and $s_{\alpha_1}X = s_{\alpha_2}X = \{\epsilon_1-\delta_1, \epsilon_2-\delta_2 \}$. We have \[ C_X =\{\epsilon_1-\epsilon_3, \epsilon_2-\epsilon_3 \} = C_X^+, \;\Gamma(\epsilon_1-\epsilon_3) = \epsilon_1-\delta_1, \mbox{and } \Gamma(\epsilon_2-\epsilon_3) = \epsilon_2-\delta_2.\]
If $X' =\{\epsilon_1-\delta_1, \epsilon_3-\delta_2 \}$ and
$X'' =\{\epsilon_2-\delta_1, \epsilon_3-\delta_2 \}$, then $C_{X'}^+ =\{\epsilon_1-\epsilon_2 \}$ and $C_{X''}^+$ is empty. We have $|C_{X'}| =|C_{X''}|=2$ and $E_{X'}, E_{X''}$ are singletons. }\end{example}
\subsection{Comparison of the Bilinear Forms.}\label{com} \noindent Our aim is to compare the $T$-adic valuation of the determinants of the bilinear forms $F^X_\eta(\widetilde{\lambda})$ and $G^X_\eta(\widetilde{\lambda})$ from Subsection \ref{7.4}. The main results, Theorems \ref{son} and \ref{epin} explain the presence of the terms $D_0'$ and $D_0''$ respectively in Theorem \ref{shapdet}. Theorem \ref{son} generalizes the behavior in the case of $\fgl(2,1)$, see \eqref{hid} with $n=0$, while Theorem \ref{epin} generalizes the behavior in the case of $\fgl(2,2)$, see \eqref{rat3}. The proof of both results follows the same pattern, which we outline first. In this subsection we denote the highest weight vector $v_{\widetilde{\lambda}}^X$ in ${M^X}(\widetilde{\lambda})_B$ simply by $v_{\widetilde{\lambda}}$. \subsubsection{Strategy of the Proofs.}
\noindent As before, $A=\mathtt{k}[T]$ and $B=\mathtt{k}(T)$. \noindent Let $C$ be the local ring $C=A_{(T)}$ with maximal ideal $(T)=TC$.
As far as the $T$-adic valuation is concerned, we may work over $C$ rather than over $A$.
If $\eta\in Q^+$, then by Theorem \ref{zoo}, a non-zero $p \in {M^X}(\widetilde{\lambda})_B^{\widetilde{\lambda}-\eta}$ can be written uniquely in the form
\begin{equation} \label{sea} p=\sum_{\tau \in {{\bf{\overline{P}}}_X(\eta)}}a_\tau e_{-\tau}v_{\widetilde{\lambda}}^X \in
{M^X}(\widetilde{\lambda})_B^{\widetilde{\lambda}-\eta},\end{equation} with $a_\tau \in B$, and we set ${\operatorname{Supp}}\; p =\{\tau \in {\bf{\overline{P}}}_X(\eta)|a_\tau \neq 0\}$. \\ \\ Also if $\alpha\in \Delta_0^+$, an {\it $\alpha$-principal part} of $p$ is a partition $\sigma\in {\operatorname{Supp}}\; p$ such that $\sigma(\alpha)>\tau(\alpha)$ for all $\tau\in {\operatorname{Supp}}\; p, \tau \neq \sigma$. Clearly the $\alpha$-principal part of $p$ is unique if it exists. \\ \\
Fix $\lambda\in \mathcal{H}_X$, $\eta\in Q^+,$ and set $M_C = {M^X}(\widetilde{\lambda})_C^{\widetilde{\lambda}-\eta}$. Let $N_C$ the $C$-submodule of $M_C$ spanned by the set $\{e_{-\pi}v_{\widetilde{\lambda}}| \pi \in {\bf{\overline{P}}}_{X}(\eta)\}$. By the elementary divisor theorem, there is a $C$-basis $v_1,\ldots, v_k$ for $M_C$ and integers $0\le z_1 \le \ldots \le z_k$ such that
\begin{equation} \label{rex} w_1=T^{z_1}v_1,\ldots, w_k=T^{z_k}v_k\end{equation}
is a $C$-basis for $N_C$. Let $z_{\lambda,\eta} = 2\sum_{i=1}^k z_i$. Suppose we use these bases to find $\det G^X_{\eta}(\widetilde\lambda)$ and $\det F^X_{\eta}(\widetilde\lambda)$. Then if we factor $T^{\sum_{i=1}^k z_i}$ from both the rows and the columns of the Gram matrix of $G^X_{\eta}(\widetilde\lambda)$ we obtain the Gram matrix of $F^X_{\eta}(\widetilde\lambda)$. Therefore
\begin{equation} \label{ron} z_{\lambda,\eta} = v_T(\det G^X_{\eta}(\widetilde\lambda))-v_T(\det F^X_{\eta}(\widetilde\lambda)). \end{equation} We need to consider more general bilinear forms. Suppose that ${\bf V}$ is a basis for $M_B= M_C\otimes B$ contained in $M_C$, and define
$$G^X_{\eta,{\bf V}}(\widetilde\lambda) =\det (\zeta(b^t b'))_{b,b'\in {\bf V}}. $$ Thus if {\bf W} is a $C$-basis for $N_C$, then $v_T(\det G^X_{\eta,{\bf W}}(\widetilde\lambda))=v_T(\det G^X_{\eta}(\widetilde\lambda))$. At the other extreme, suppose that ${\bf U}$ is a basis of $M_C$ as a $C$-module, then \begin{equation}\label{dff} v_T(\det G^X_{\eta,{\bf U}}(\widetilde\lambda))=v_T(\det F^X_{\eta}(\widetilde\lambda)).\end{equation} Consider bases ${\bf W}= \{w_1,\ldots, w_k\}$ and ${\bf V}= \{v_1,\ldots, v_k\}$ such that \eqref{rex} holds and
\begin{equation} \label{nub} N_C={\operatorname{C-span}}\;{\bf W}\subseteq {\operatorname{C-span}}\;
{\bf V} \subseteq {\operatorname{C-span}}\; {\bf U}.\end{equation} Then we have
\begin{equation} \label{uuu} z_{\lambda,\eta} \ge v_T(\det G^X_{\eta}(\widetilde\lambda))-v_T(\det G^X_{\eta, {\bf V}}(\widetilde\lambda)),\end{equation} and we show that when the hypothesis of Theorem \ref{son} (resp. Theorem \ref{epin}) hold for an even root $\alpha$, the right side of
\eqref{epin} is bounded below by $|\operatorname{LT} D'_0:h_\alpha|$ (resp. $|\operatorname{LT} D''_0:h_\alpha| $). \\ \\ \noindent At this point the proofs of the two results begin to diverge. For Theorem \ref{son}, to construct {\bf V, W} we use the explicit expressions for $\theta_\gamma v_{\widetilde{\lambda}}$ given in Theorem \ref{stc}. This yields non-zero elements $p, q\in {M^X}(\widetilde{\lambda})_C^{\widetilde{\lambda}-\gamma}$ such that $p+qT=\theta_\gamma v_{\widetilde{\lambda}}=0$. Then for certain partitions $\sigma$ we set $p_\sigma= e_{-\sigma}p$ and $q_\sigma= e_{-\sigma}q$. The basis {\bf W} contains all of the $p_\sigma$, and ${\bf V}$ is obtained from ${\bf W} $ by replacing each $p_\sigma$ by $q_\sigma$.
Then to prove Theorem \ref{son}, we simply have to count the number of partitions affected. The situation for Theorem \ref{epin} is a bit more complicated. We begin with Theorem \ref{nex}, but then there are two different cases involved and certain partitions need to be excluded.
\subsubsection{The Factor $D_0'$.}\label{pea} We start with the factor $D_0'$ from Equation \eqref{yaws}. Suppose that $\alpha\in C^+_X$, $\gamma = \Gamma(\alpha)$ and set $Z=Z(\alpha)=X \cup s_\alpha X$. We adopt the notation of Theorem \ref{stc}, with $\mathtt{T}=T$. Thus $\gamma = \epsilon_r- \delta_{s},$ $\alpha = \epsilon_r- \epsilon_{\ell }$ and $\gamma'=s_\alpha\gamma=\epsilon_\ell- \delta_{s}.$ Set $j=\ell-r$.
\begin{theorem} \label{son} For general $\lambda \in \mathcal{H}_X \cap \mathcal{H}_{\alpha, 0}$, we have
\begin{equation} \label{laid}z_{\lambda,\eta} \ge 2 \;{\bf p}_{Z(\alpha)}(\eta-\gamma) = |\operatorname{LT} D'_0:h_\alpha| .\end{equation} \end{theorem}
\noindent The equality in \eqref{laid} holds by the definition of $D_0'$, see \eqref{yaws}. To prove the inequality, we need some preparation. The hypothesis in the Theorem implies that $\gamma, \gamma'=s_\alpha \gamma =\gamma-\alpha \in B(\lambda)$ for some odd root $\gamma$. If $p= \theta_{\alpha,1}\theta_{\gamma'}v_{\widetilde{\lambda}}$ and $q=\overline{\psi}(\theta_{\overline{\gamma}}(\overline{\phi}({\widetilde{\lambda}})))v_{\widetilde{\lambda}}$, then by \eqref{nice} $\theta_\gamma v_{\widetilde{\lambda}}= p+qT$. Suppose $\sigma \in {\bf{\overline{P}}}_{Z}(\eta-\gamma)$, define $p_\sigma, q_\sigma$ as in the previous subsection and set $\sigma'=\sigma+\pi^\alpha+\pi^{\gamma'}$. Thus \[\sigma'(\xi) =\left\{ \begin{array}
{cl}
\sigma(\alpha)+1 &\mbox{if} \;\; \xi=\alpha \\
1 &\mbox{if} \;\; \xi=\gamma' \\ \sigma(\xi) &\mbox{otherwise}. \end{array} \right. \] Note that $\sigma'\in {\bf{\overline{P}}}_{X}(\eta),$ since $\gamma =\gamma'+\alpha$, $Z=X \cup \{\gamma'\}$ and $\sigma\in {\bf{\overline{P}}}_{Z}(\eta-\gamma)$. \begin{lemma} \label{app} For general $\lambda \in \mathcal{H}_X \cap \mathcal{H}_{\alpha, 0}$ \begin{itemize} \item[{{\rm(a)}}] The coefficient of $e_{-\sigma'}v_{\widetilde{\lambda}}$ in $\theta_{\alpha,1}\theta_{\gamma'}v_{\widetilde{\lambda}}$ is invertible in $C$. \item[{{\rm(b)}}] The $\alpha$-principal part of $p_\sigma$ is $\sigma'$. \end{itemize}\end{lemma}
\begin{proof} Define $a_i$ and $b_j$ as in \eqref{eat}. Then \begin{eqnarray} \widetilde{a_i}&=&(\widetilde{\lambda} + \rho,\sigma_{r,r+i}^\vee ) ={a_i} + T(\xi,\sigma_{r,r+i}^\vee )\\ \widetilde{b_i} &=& (\widetilde{\lambda} + \rho,\tau_{i,s}^\vee )\;\;\;\; = b_i +T(\xi,\tau_{i,s}^\vee ).\nonumber \end{eqnarray} To find the coefficients of $p$ and $q$ we evaluate the determinants appearing in the determinants \eqref{wok} and \eqref{gov} with $\lambda$ replaced by $\widetilde{\lambda}$, that is with $a_i,b_i$ replaced by $\widetilde{a_i}, \widetilde{b_i}$ respectively. Thus if the $b_i,$ and the $a_i$ are non-zero, then the $\widetilde{b_i},$ and the $\widetilde{a_i}$ (with $i\neq j$) are units in $C$. Thus for general ${\lambda}$ the coefficients of $p$ and $q$ are units in $C$. \\ \\ Now to prove (b), observe first that $\pi^\alpha +\pi^{\gamma'} \in {\operatorname{Supp}}\; \theta_{\alpha,1}\theta_{\gamma'}v_{\widetilde{\lambda}}$, and for general $\lambda$, $e_{-\sigma}e_{-\alpha}e_{-\gamma'}v_{\widetilde{\lambda}}$ occurs in $e_{-\sigma}\theta_{\alpha,1}\theta_{\gamma'}v_{\widetilde{\lambda}}$ with a coefficient that is a unit in $C$. Recall that odd root vectors occur last in the product $e_{-\sigma}$, and that the root vectors for odd positive roots anti-commute since we are working in type $A.$ Thus the only factor in $e_{-\sigma}e_{-\alpha}e_{-\gamma'}v_{\widetilde{\lambda}}$ that could be out of order is the middle root vector $e_{-\alpha}$. Also for any term $e_{-\omega}$ that arises by taking commutators of $e_{-\alpha}$ with root vectors for roots that are involved in $\sigma$, the partition $\omega$ satisfies $\omega(\alpha)<\sigma'(\alpha)$.\\ \\ We consider in more detail the odd root vectors $e_{-\omega}$ that arise in the way described in the above paragraph, using commutators with $e_{-\alpha}$. We need to check that for such $\omega$ we have $\omega(\kappa)=0$ for all $\kappa\in X$. (Clearly this also holds if $e_{-\omega}$ arises from commutators of $e_{-\alpha}$ with other even roots.) Since $\alpha\in C^+_X$, we have $(\gamma,\alpha^\vee)=1.$ Thus without loss of generality, we can assume that $\alpha=\epsilon_j-\epsilon_k, \gamma' = \epsilon_k-\delta_r $ and $\gamma =\gamma'+\alpha=\epsilon_j-\delta_r$. Then set
\[I(\sigma) =\{s\in [n]\;|\; \sigma(\epsilon_k-\delta_s) = 1, \sigma(\epsilon_j-\delta_s) = 0, s\neq r\}\] and for $\sigma \in {\bf{\overline{P}}}_{Z}(\eta-\gamma)$ and $s\in I(\sigma)$, define a partition $ \sigma^{(s)}$ of $\eta$ by \[\sigma^{(s)}(\xi) =\left\{ \begin{array}
{cl}
0 &\mbox{if} \;\; \xi=\epsilon_k-\delta_s \\
1 &\mbox{if} \;\; \xi=\epsilon_j-\delta_s \\ 1 &\mbox{if} \;\; \xi=\gamma' \\ \sigma(\xi) &\mbox{otherwise}. \end{array} \right. \] Note that $\epsilon_j-\delta_s\notin X$ because $(\epsilon_j-\delta_s,\gamma)\neq 0$ and $X$ is an orthogonal set of roots containing $\gamma$. Since $\sigma \in {\bf{\overline{P}}}_{Z}(\eta-\gamma)$ it follows that $\sigma^{(s)} \in {\bf{\overline{P}}}_{X}(\eta)$. Now we have, modulo the span of terms obtained from commutators of $e_{-\alpha}$ with even root vectors,
\begin{equation} \label{007} e_{-\sigma}e_{-\alpha}e_{-\gamma'} \equiv e_{-\sigma'} + \sum_{s\in I(\sigma)} c_\sigma e_{-\sigma^{(s)}}, \end{equation}
\noindent with non-zero $c_\sigma\in \mathtt{k}$, and $\sigma^{(s)}(\alpha) = \sigma(\alpha)=\sigma'(\alpha)-1$ for all $s\in I(\sigma)$. Thus $\sigma'$ is the $\alpha$-principal part of $p_\sigma$. \end{proof}
\begin{example} {\rm Suppose that $\fg=\fgl(3,3)$, $\alpha=\epsilon_1-\epsilon_3, \gamma'=\epsilon_3 -\delta_2, \gamma=\epsilon_3 -\delta_2,$ $e_{-\sigma} = e_{63}e_{43}$ and $X=\{\gamma\}$.
Suppose we want to collect all odd root vectors on the right. (Note that all root vectors for positive odd roots commute). Then $e_{-\alpha}e_{-\gamma'}v_{\widetilde{\lambda}}$ occurs with non-zero coefficient in $\theta_{\alpha,1}\theta_{\gamma'}v_{\widetilde{\lambda}}$. However, since the root vectors $e_{63}$ and $e_{43}$ are odd, $e_{-\alpha}$ is out of order in $e_{-\sigma}e_{-\alpha}e_{-\gamma'} v_{\widetilde{\lambda}}=e_{63}e_{43}e_{31}e_{53}v_{\widetilde{\lambda}}$. In the notation of the Lemma, $j=1, k = 3, r =2$ and $I(\sigma)= \{1, 3\}$, and we have \begin{eqnarray} e_{-\sigma}e_{-\alpha}e_{-\gamma'}
&=& e_{31}e_{63}e_{43}e_{53}\nonumber\\ &+& e_{61}e_{43}e_{53}\nonumber\\
&+& e_{63}e_{41}e_{53}.\nonumber\end{eqnarray} Now all terms are in the correct order, and $e_{-\alpha}$ occurs only in the first term.} \end{example}
\noindent Next recall that $N_C= {\operatorname{C-span}}\; \{e_{-\pi}v_{\widetilde{\lambda}}| \pi \in {\bf{\overline{P}}}_{X}(\eta)\}$, and set
\[{\bf W}_{1} = \{e_{-\pi}v_{\widetilde{\lambda}}|\pi\in{\bf \overline{P}}_{X}(\eta), \pi\neq \sigma' \mbox{ for any } \sigma\in{\bf \overline{P}}_{Z}(\eta-\gamma)\}.\]
\begin{lemma} \label{0,111} For general $\lambda \in \mathcal{H}_X \cap \mathcal{H}_{\alpha, 0}$ the set
\[{\bf W}= \{p_\sigma v_{\widetilde{\lambda}}|\sigma\in{\bf \overline{P}}_{Z}(\eta-\gamma)\}\cup {\bf W}_1\] is a $B$-basis for ${M^X}(\widetilde{\lambda})_B^{\widetilde{\lambda}-\eta}$ and ${\operatorname{C-span}}\; {\bf W} = N_C$. \end{lemma}
\begin{proof} Since $|{\bf W}| = \dim_B {M^X}(\widetilde{\lambda})_B^{\widetilde{\lambda}-\eta}= {\operatorname{rank}}_C N_C$, it suffices to prove the last statement. Obviously $L={\operatorname{C-span}}\; {\bf W}\subseteq N_C$. If $\pi \in {\bf \overline{P}}_{X}(\eta)$ we show by induction on $\pi(\alpha)$ that $e_{-\pi}v_{\widetilde{\lambda}} \in L$. By definition of ${\bf W}$ we only have to show this when $\pi=\sigma'$ for some $\sigma \in {\bf \overline{P}}_{Z}(\eta-\gamma)$. Write \begin{eqnarray} \label{sos} p_\sigma &=& u_\sigma e_{-\sigma}e_{-\alpha}e_{-\gamma'} v_{\widetilde{\lambda}} + \sum_{\omega \in {\bf{\overline{P}}}_{X}(\eta):\omega\neq \sigma'} a_{\sigma,\;\omega} e_{-\omega}v_{\widetilde{\lambda}}\nonumber\\ &=& u_\sigma e_{-\sigma'} v_{\widetilde{\lambda}} + \sum_{\omega \in {\bf{\overline{P}}}_{X}(\eta):\omega\neq \sigma'} b_{\sigma,\;\omega} e_{-\omega}v_{\widetilde{\lambda}}, \end{eqnarray}
\noindent where $a_{\sigma,\;\omega}, b_{\sigma,\;\omega} \in C,$ and $u_\sigma $ is a unit in $C$. Each term $e_{-\omega}v_{\widetilde{\lambda}}$ in the sum \eqref{sos} is obtained by expanding $e_{-\sigma}\theta_{\alpha,1}\theta_{\gamma'}v_{\widetilde{\lambda}}$ and re-ordering (as in the proof of the previous lemma). Observe that when this is done, any $\omega$ arising from a commutator of $e_{-\alpha}$ with another root vector satisfies $\omega(\alpha)<\pi(\alpha)$.
Note that for general $\lambda,$ $e_{-\gamma'}v_{\widetilde{\lambda}}$ occurs in $\theta_{\gamma'}v_{\widetilde{\lambda}}$ with a coefficient which is a unit in $C$. Since $\sigma(\gamma')=0$, any term $e_{-\omega}v_{\widetilde{\lambda}}$ in the expansion with $\omega =\tau'$ (where $\tau \in {\bf{\overline{P}}_{Z}}(\eta-\gamma)$) must be a multiple of $e_{-\gamma'} v_{\widetilde{\lambda}} \in {\operatorname{Supp}} \;\theta_{\gamma'}v_{\widetilde{\lambda}}$. Using Lemma \ref{app}, it follows that for all $\omega$ such that $b_{\sigma,\;\omega}\neq 0$ in \eqref{sos} we have either $\omega \neq \tau'$ or $\omega(\alpha)<\pi(\alpha).$ In either case $e_{-\omega}v_{\widetilde{\lambda}} \in L$, so since $u_\sigma$ is a unit in $C$, we obtain the result from \eqref{sos}. \end{proof}
\noindent {\it Proof of Theorem \ref{son}.} We claim first that the set
$\{p_\sigma v_{\widetilde{\lambda}}|\sigma\in{\bf \overline{P}}_{Z}(\eta-\gamma)\}$ is $B$-linearly independent. Suppose that \[ \sum_{\sigma\in{\bf \overline{P}}_{Z}(\eta-\gamma)} a_\sigma p_\sigma =0, \]
and set $k=\max\{\sigma(\alpha)|a_\sigma \neq 0\}.$ Then from \eqref{sos} we have \[ \sum_{\sigma\in{\bf \overline{P}}_{Z}(\eta-\gamma):\sigma(\alpha)=k} c_\sigma e_{-\sigma'} =0,\] where $c_\sigma =a_\sigma u_\sigma,$ for $u_\sigma$ as in \eqref{sos}. But by Theorem \ref{zoo} the elements $e_{-\pi}v_{\widetilde{\lambda}}$ with $\pi\in{\bf \overline{P}}_{X}(\eta)$ are $B$-linearly independent, and $\sigma'\in {\bf{\overline{P}}}_{X}(\eta)$ by Lemma \ref{app}. Thus all $c_\sigma$ are zero, a contradiction which proves the claim. \\ \\ Next set
\[{\bf V}= \{q_\sigma |\sigma\in{\bf \overline{P}}_{Z}(\eta-\gamma)\}\cup {\bf W}_1.\] Since $p_\sigma +q_\sigma T =0,$ it follows from Lemma \ref{0,111} that
there are bases $ w_1,\ldots, w_k$ and $ v_1,\ldots, v_k$ for $N_C$ and ${\operatorname{C-span}}\;{\bf V} $ respectively, and integers $0\le z_1 \le \ldots \le z_k$ such that \eqref{rex} holds and at least ${\bf p}_{Z}(\eta-\gamma)$ of the $z_i$ are positive. Hence $$v_T(\det G^X_{\eta}(\widetilde\lambda))-v_T(\det G^X_{\eta, {\bf V}}(\widetilde\lambda))\ge 2 \;{\bf p}_{Z}(\eta-\gamma)$$ and combined with \eqref{uuu}, this yields \eqref{laid}.
$\Box$
\subsubsection{The Factor $D_0''$.}\label{pie}
We now turn our attention to the factor $D_0''$ from Equation \eqref{yews}. If $(\alpha_1,\alpha_2)\in E_X$ there are roots $\gamma_1,\gamma_2\in X$ such that $\alpha_1 +\gamma_1 +\alpha_2 =\gamma_2$. Set $\alpha =\alpha_1$. The main result is as follows. \begin{theorem} \label{epin} If $\eta \ge \gamma$ then for general $\lambda \in \mathcal{H}_X\cap \mathcal{H}_{\alpha,1}$ we have
\begin{equation} \label{laid2}z_{\lambda,\eta} \ge 2({\bf p}_{V(\alpha)}(\eta -\gamma_2)+{\bf p}_{V(\alpha)}(\eta -\gamma_1-\gamma_2)-{\bf p}_{W(\alpha)}(\eta -\gamma_1-\gamma_2)) = |\operatorname{LT} D''_0:h_\alpha| .\nonumber\end{equation} \end{theorem} \noindent Again the equality follows from the definition of $D_0''$, see \eqref{yews}. For the rest we first introduce some notation. The proof is similar to the proof of Theorem \ref{son}, so we omit some of the details. First define
\[\Theta_\eta^{(1)} =\{\pi\in{\bf{\overline{P}}}_{V(\alpha)}(\eta -\gamma_1-\gamma_2)|\pi \notin {\bf{\overline{P}}}_{W(\alpha)}(\eta -\gamma_1-\gamma_2)\}, \quad \Theta_\eta^{(2)}= {\bf{\overline{P}}}_{V(\alpha)}(\eta -\gamma_2),\]
and set $\Theta_\eta= \Theta_\eta^{(1)} \cup \Theta_\eta^{(2)}$. \\ \\ Since $(\lambda+\rho, \alpha^\vee) =1,$ we have $(\widetilde{\lambda}+\rho, \alpha^\vee) =T+1.$ Adopting the notation of Subsection \ref{det} we define $\lambda_1,\lambda_2,$ and $ \lambda_3$ as in \eqref{flx}, \eqref{fax} and \eqref{fox}. Then from Equation \eqref{mice} we see that if $\theta_{\gamma_2}v_{\widetilde{\lambda}} =\theta_{\gamma_1} v_{\widetilde{\lambda}}= 0$ we have, (compare \eqref{rat3} for the case $\fgl(2,2)$) \\ \begin{equation} \label{ump} [\underline{\overline\psi}(\theta_{\underline{{\overline{\gamma}}}}(\underline{\overline{\phi}}({\widetilde{\lambda}})))(T +1)+ \theta_{{\alpha_1}}\underline{\psi}(\theta_{{\beta_1}}({{\lambda_2}})) -\theta_{{{\alpha_2}}}\overline{\psi}(\theta_{{\beta_2}}({{\lambda_1}}))]v_{\widetilde{\lambda}}=0.\end{equation} Let \[p_1 =\theta_{{\alpha_1}}\underline{\psi}(\theta_{{\beta_1}}({{\lambda_2}}))v_{\widetilde{\lambda}},\quad p_2 =\theta_{{{\alpha_2}}}\overline{\psi}(\theta_{{\beta_2}}({{\lambda_1}}))v_{\widetilde{\lambda}},\quad q= \underline{\overline\psi}(\theta_{\underline{{\overline{\gamma}}}}({{\lambda_3}}))v_{\widetilde{\lambda}} \] and $p=p_1-p_2+q.$ Then in $U(\mathfrak{n}^-)_A^{-\gamma_2}v_{\widetilde{\lambda}}$ we can write \eqref{ump} as $p+qT=0,$
\begin{lemma} \label{log} \begin{itemize} \item[{{\rm(a)}}] $|\Theta_\eta^{(1)}|= {\bf p}_{V(\alpha)}(\eta -\gamma_1-\gamma_2)-{\bf p}_{W(\alpha)}(\eta -\gamma_1-\gamma_2)$. \item[{{\rm(b)}}] If $\sigma \in \Theta_\eta^{(1)}$, then either $\sigma(\alpha_1)>0$ or $\sigma(\alpha_2)>0.$ \item[{{\rm(c)}}] If $\sigma \in {\bf{\overline{P}}}_{V(\alpha)}(\eta -\gamma_1-\gamma_2)$, then $e_{-\sigma}p\neq 0$. \end{itemize} \end{lemma} \begin{proof} The first two parts follow easily from the definitions, so we turn to (c). If $\pi_1= \pi^{\alpha_1}+\pi^{(\gamma_1 + \alpha_2)}$ and $\pi_2=\pi^{\alpha_2}+\pi^{(\alpha_1+\gamma_1)}$, \footnote{ We can assume $\alpha_1 = \epsilon_r -\epsilon_\ell, \;\alpha_2 =\delta_k-\delta_s, \gamma_1 = \epsilon_\ell- \delta_k$ and $\gamma_2 = \epsilon_r -\delta_s.$ Then putting odd root vectors last, we have $e_{-\pi_1}= e_{\ell,r} e_{m+s,\ell}$ and $e_{-\pi_2}= e_{m+s,m+k}e_{m+k,r}$.} we have by Lemma \ref{hes} (c)
\begin{equation} \label{p1p} \pi_1\in {\operatorname{Supp}} \; p_1,\; \pi_1\notin {\operatorname{Supp}} \; p_2,\; \pi_1\notin {\operatorname{Supp}}\; q,\end{equation} and \begin{equation} \label{p2p} \pi_2\in {\operatorname{Supp}} \; p_2,\; \pi_2\notin {\operatorname{Supp}} \; p_1,\; \pi_2\notin {\operatorname{Supp}}\; q.\end{equation}
\noindent Next suppose $\sigma \in {\bf{\overline{P}}}_{V(\alpha)}(\eta -\gamma_1-\gamma_2)$. Then
$\sigma(\alpha_1+\gamma_1)= \sigma(\gamma_1 + \alpha_2)=0.$
Combined with \eqref{p1p} and \eqref{p2p}, this shows that $e_{-\sigma}p\neq 0$. \end{proof} \noindent \begin{lemma} \label{axe} Suppose $\sigma\in {\bf{\overline{P}}}_{V(\alpha)}(\eta -\gamma_1-\gamma_2)-{\bf{\overline{P}}}_{W(\alpha)}(\eta -\gamma_1-\gamma_2)$, and define \[\sigma' = \sigma +\pi^{\gamma_1 + \alpha_2}+\pi^{\alpha_1+\gamma_1} \;\in \;{\bf{\overline{P}}}_{X}(\eta).\] Then for general $\lambda$ and for $i=1,$ or 2, $\sigma'$ is the $\alpha_1$ {\rm(}resp. $\alpha_2${\rm)} principal part of $e_{-\sigma}e_{-(\gamma_1 + \alpha_2)}e_{-(\alpha_1+\gamma_1)}v_{\widetilde{\lambda}}$. \end{lemma} \begin{proof} By Lemma \ref{log} we have either $\sigma(\alpha_1)> 0$, or $\sigma(\alpha_2)> 0$, and we assume the former. Define the partition $\hat \sigma$ by \[\hat \sigma(\xi) =\left\{ \begin{array}
{ccl}\sigma(\alpha_1)- 1 &\mbox{if} \;\; \xi=\alpha_1 \\\sigma(\xi) &\mbox{otherwise} \end{array} \right. \] Then set $p_\sigma = e_{-\hat \sigma} e_{-(\alpha_1+\gamma_1)}p$ and $q_\sigma = e_{-\hat \sigma} e_{-(\alpha_1+\gamma_1)}q$. \\ \\ We have, up to re-ordering terms \begin{equation} \label{ess1} e_{-\sigma}e_{-(\gamma_1 + \alpha_2)}e_{-(\alpha_1+\gamma_1)}v_{\widetilde{\lambda}}= e_{-\hat \sigma} e_{-\alpha_1}e_{-(\gamma_1 + \alpha_2)}e_{-(\alpha_1+\gamma_1)}v_{\widetilde{\lambda}}\end{equation} Up to re-ordering these elements are in ${\operatorname{Supp}}\; p_\sigma$ and for a general $\lambda\in \mathcal{H}_X\cap \mathcal{H}_{\alpha,1}$, they occur in $p_\sigma$ with coefficient that is a unit in $C$. The result follows. (Compare \eqref{dog1} and \eqref{cat1} for the case of $\fgl(2,2).$) \end{proof} \noindent We need to consider additional relations arising in the following way. For $\sigma \in \Theta_\eta^{(2)}={\bf{\overline{P}}}_{V(\alpha)}(\eta -\gamma_2)$ we have the relation $e_{-\sigma}(p+qT)=0.$ Set $p_\sigma =e_{-\sigma}p$ and $q_\sigma = e_{-\sigma}q.$ Set $\sigma' = \sigma+ \pi^{\alpha_1}+\pi^{(\gamma_1 + \alpha_2)} \in{\bf \overline{P}}_{X}(\eta).$ As in Lemma \ref{axe} it can be shown that for general $\lambda$, $\sigma'$ is the $\alpha_1$ principal part of $p_\sigma$. \begin{lemma} \label{eee} Set
\[{\bf W}_{1} = \{e_{-\pi}v_{\widetilde{\lambda}}|\pi\in{\bf \overline{P}}_{X}(\eta), \pi\neq \sigma' \mbox{ for any } \sigma\in\Theta_\eta\}.\] Then Equation \eqref{nub} holds with
\[{\bf W}= \{p_\sigma |\sigma\in\Theta_\eta\}\cup {\bf W}_1, \quad
{\bf V} =\{q_\sigma |\sigma\in\Theta_\eta\}\cup {\bf W}_1.\] Furthermore {\bf W} and {\bf V} are $B$-bases for ${M^X}(\widetilde{\lambda})_B^{\widetilde{\lambda}-\eta}$.\end{lemma}
\begin{proof} Similar to the proof of Lemma \ref{0,111}.\end{proof}
\noindent {\it Proof of Theorem \ref{epin}.} First we use Theorem \ref{zoo} to show the elements of {\bf W} are linearly independent and $N_C={\operatorname{C-span}}\;{\bf W}$. Suppose that \[\sum_{\sigma \in \Theta_\eta^{(1)}} a_\sigma p_\sigma + \sum_{\sigma \in \Theta_\eta^{(2)}} b_\sigma p_\sigma =0,\] with coefficients $a_\sigma, b_\sigma \in B.$ Then both sums must be zero because for $\sigma$ in the first sum we have \[\sigma'(\alpha_1+\gamma_1) =\sigma'( \gamma_1 + \alpha_2)=1,\] but for $\sigma$ in the second sum we have \[\mbox{either } \sigma'(\alpha_1+\gamma_1) =0 \mbox{ or } \sigma'( \gamma_1 + \alpha_2)=0.\] Arguing as in the proof of Theorem \ref{son}, it follows that all coefficients $a_\sigma, b_\sigma$ are zero. Hence \eqref{rex} holds for the bases {\bf W} and {\bf V}, and at least \[{\bf p}_{V(\alpha)}(\eta -\gamma_2)+{\bf p}_{V(\alpha)}(\eta -\gamma_1-\gamma_2)-{\bf p}_{W(\alpha)}(\eta -\gamma_1-\gamma_2) \]
of the $z_i$ are positive. The proof is concluded as before.
$\Box$ \begin{corollary} For all $\alpha\in \Delta^+_0$,
\begin{equation} \label{(c)}|\operatorname{LT} \det G^X_\eta:h_\alpha| \ge |\operatorname{LT} D_0 :h_\alpha| + |\operatorname{LT} \det F^X_\eta :h_\alpha|.\end{equation} \end{corollary} \begin{proof} Combine Theorems \ref{son} and \ref{epin}.\end{proof} \begin{rem}{\rm In this Subsection we have been concerned with situations where $z_{\lambda,\eta} $ is as large as possible. However for general $\lambda\in \mathcal{H}_X$, we have $z_{\lambda,\eta} = 0$. Indeed if set $\lambda_c = \lambda +c\xi$ for $c\in \mathtt{k}$, it follows from Corollary \ref{nob} that for all but finitely many $c$, $z_{\lambda_c,\eta} = 0$ for all $\eta,$ that is \begin{equation} \label{poe} u\det G^X_\eta(\widetilde\lambda_c) = \det F^X_\eta(\widetilde\lambda_c),\end{equation} for some unit $u$ in $C$.} \end{rem}
\subsection{Partition Identities.} \label{Partitions.}
We refer to $p, p_X$ and $p_\alpha$ as defined by \eqref{pfun} as {\it partition functions}. They are functions in the following sense. Suppose $\nu, \mu\in Q^+$. If the ${\mathbb Z}$-linear function $\mathtt{e}^{\nu}$ from ${\mathbb Z} Q$ to ${\mathbb Z}$ is defined by $\mathtt{e}^{\nu}(\mu) = \delta_{\nu,\mu}$, then ${p_X}(-\mu) = \sum_\eta {\bf p}_X(\eta)\mathtt{e}^{-\eta}(-\mu) = {\bf p}_X(\mu)$. Note that $\mathtt{e}^{-\alpha}p_X(-\mu) = {\bf p}_X(\mu-\alpha)$.
We have the following relations between partition functions.
\begin{lemma} \label{ink} Suppose $\alpha \in C_{X}, \gamma = \Gamma(\alpha)\in X$ with $\gamma' = s_\alpha\gamma
\notin X.$ Set $Y = s_\alpha X,$ $ Z = X\cup Y$, and $\mu_0 = \mu-\gamma$. Then \begin{itemize} \item[{{\rm(a)}}] If $\gamma'= \gamma -\alpha,$ then \[({p}_X- {p}_{s_\alpha X})= {\mathtt{e}^{-\gamma'}(1 - \mathtt{e}^{- \alpha})}{p}_Z,\] and \[{{\bf p}_{X}(\mu)}-{{\bf p}_{s_\alpha X}(\mu)} ={\bf p}_Z{(\mu_0 +\alpha)}-{{\bf p}_Z(\mu_0 )}.\] \item[{{\rm(b)}}] If $\gamma'= \gamma +\alpha,$ then \[{{\bf p}_{X}(\mu)}-{{\bf p}_{s_\alpha X}(\mu)} ={\bf p}_Z{(\mu_0 -\alpha)}-{{\bf p}_Z(\mu_0)}.\] \end{itemize} \noindent Now suppose $(\alpha_1,\alpha_2) \in E_{X}$ and $Y = s_{\alpha_1} X= s_{\alpha_2} X$. Write $\alpha_1 +\gamma_1 +\alpha_2 =\gamma_2$. Then if $V=X\cup Y$, \begin{itemize} \item[{{\rm(c)}}] \[{p}_Y-{p}_{X}= [(1 + \mathtt{e}^{-\gamma_1})(1 + \mathtt{e}^{-\gamma_2})-(1 + \mathtt{e}^{-\gamma_1- \alpha_1})(1 + \mathtt{e}^{-\gamma_1- \alpha_2})]{p}_{V}.\] \item[{{\rm(d)}}] \[{\bf p}_Y{(\mu)}-{{\bf p}_{X}(\mu)}= {{\bf p}_{V}(\mu-\gamma_1)} +{{\bf p}_{V}(\mu-\gamma_2)} - {{\bf p}_{V}(\mu-\alpha_1-\gamma_1)}-{{\bf p}_{V}(\mu-\alpha_2-\gamma_1)}.\] \end{itemize}\end{lemma} \begin{proof} The first statement in (a) follows since \begin{eqnarray} p_X - p_{s_\alpha X} &=&[(1 + \mathtt{e}^{-\gamma'})-(1 + \mathtt{e}^{-\gamma})]p_Z\nonumber\\ &=&{\mathtt{e}^{-\gamma'}(1 - \mathtt{e}^{- \alpha})}{p}_Z.\nonumber\end{eqnarray} The second follows by evaluation at $-\mu$. The proofs of the other parts are similar.\end{proof}
\subsection{The Leading Term.}\label{lete}
We make a direct computation of the leading term of $ \det G^X_{\eta}$ by adapting the proof of \cite{M} Lemma 10.1.3, and show the result is consistent with Theorem \ref{shapdet}. We prove Theorem \ref{shapdet} by showing that for each factor of $D_\eta$ we have \begin{equation} \label{por} |D_\eta:h_\alpha|\ge |G_\eta:h_\alpha|.\end{equation} Since $D_\eta$ and $G_\eta$ have the same leading term, it follows that there can be no factors of $D_\eta$ other than those listed. We need to consider the cases where $\lambda$ lies on a hyperplane inside $\mathcal{H}_X$, and among such $\lambda$ it suffices to look at the most general behavior. In such cases we will have equality in \eqref{por}. \\ \\ In the next result we use the notation $B_X, C_X, E_X, V(\alpha)$
as in the statement of Theorem \ref{shapdet}.
\begin{lemma} \label{gro} The leading term of $ \det G^X_{\eta}$ has the form $\operatorname{LT} \det G^X_\eta =G_{1}G_{2}G_{3}G_4$ where \begin{eqnarray} \label{la1} G_1&=&\prod_{\alpha \in {\Delta}^+_{0}}\prod^{\infty}_{r=1}h_\alpha^{{\bf p}_{X}(\eta-r\alpha)},\quad G_3 =\prod_{\alpha \in C_{X}, \gamma = \Gamma(\alpha)} \prod_{\pi \in {\bf \overline{P}}_{X}(\eta)}
h_{{\alpha} }^{\pi({s_\alpha \gamma} )},\nonumber\\ G_2&=&\prod_{\gamma \in B_X } h_{\gamma }^{{\bf p}_{X\cup\{\gamma \}}(\eta - \gamma )}, \; G_4 =\prod_{[\alpha] \in E_{X}} \prod_{\pi \in {\bf \overline{P}}_{X}(\eta)} h_{{\alpha} }^{\pi({\alpha_1+\gamma_1} )+\pi({\alpha_2+\gamma_1} )}.\nonumber \end{eqnarray}
\begin{rem} \label{dab} {\rm Note that \begin{equation} \label{nat}G_3 =\prod_{\alpha \in C_{X}} h_{{\alpha} }^{{\bf{\overline{p}}}_{X}(\eta)-{\bf{\overline{p}}}_{s_\alpha X}(\eta)}\end{equation} and \begin{equation} \label{nit}G_4 =\prod_{[\alpha] \in E_{X}} h_{{\alpha} }^{{\bf{\overline{p}}}_{V(\alpha)}(\eta-\alpha_1-\gamma_1 )+ {{\bf{\overline{p}}}}_{V(\alpha)}(\eta-\alpha_2-\gamma_1) +2{\bf{\overline{p}}}_{V(\alpha)}(\eta-\gamma_1-\gamma_2)}.\end{equation} The exponent on $G_3$ here is the same as the exponent on $D_3$ in \eqref{yew4}, however the product is over $C_X$ in the former and $C_X^+$ in the latter. It is worth noting that $D_2$ and $G_2$ are the only terms in Theorem \ref{shapdet} and Lemma \ref{gro} that involve odd roots. Also the sets $C_X$ and $E_X$ are disjoint. This means that in checking the multiplicities of the factors of the product in Theorem \ref{shapdet} (as well as Lemmas \ref{gro} and \ref{xyz}), we can study 3 cases separately. Note that we need to consider $D_1$ and $G_1$ in both Cases 2 and 3. \begin{itemize} \item Case 1. Factors of $D_2$ and $G_2$. \item Case 2. Factors of $D_0'', D_1, D_4$, $G_1$ and $ G_4$.
\item Case 3. Factors of $D_0', D_1, D_3$, $G_1$ and $G_3$. \end{itemize} }\end{rem} \end{lemma} \noindent {\it Proof of Lemma \ref{gro}.} We adapt the proof of \cite{M} Lemma 10.1.3. The leading term of $\det G^X_{\eta}$ is the same as the leading term of the product of the diagonal entries, and this equals
\begin{equation} \label{why} \prod_{\alpha \in(\Delta^+\; \backslash \; X)}\; \prod_{\pi \in {\bf \overline{P}}_{X}(\eta)} h_{\alpha }^{\pi(\alpha )}.\end{equation} We have to consider various possibilities for roots $\alpha \in \Delta^+\; \backslash \; X$. Suppose first that $\alpha \in {\Delta}^+_{0}$. If $\alpha\notin C_X$ and neither \eqref{cad} or \eqref{had} hold. then the multiplicity of $h_\alpha$ is $\sum^\infty_{r=1}{\bf p}_{X}(\eta - r\alpha)$ exactly as in \cite{M} Lemma 10.1.3, while if \eqref{had} holds the multiplicity is
\begin{equation} \label{add}\sum^\infty_{r=1}{\bf p}_{X}(\eta - r\alpha_1)+ \sum^\infty_{r=1} {\bf p}_{X}(\eta - r \alpha_2).\end{equation} This accounts for the exponent in the inner product in \eqref{why}, and gives the term $G_1$.
However if $\alpha\in C_X$ and $ \gamma = \Gamma(\alpha)$, then since $h_\alpha \equiv \pm h_{{s_\alpha \gamma} }$
we get the additional contribution coming from $G_3.$ This deals with Case 3, and we turn to Case 2. If $(\alpha_1,\alpha_2) \in E_{X}$ we need to add \[{\bf{\overline{p}}}_{V(\alpha)}(\eta-\alpha_1-\gamma_1 )+ {{\bf{\overline{p}}}}_{V(\alpha)}(\eta-\alpha_2-\gamma_1) +2{{\bf{\overline{p}}}}_{V(\alpha)}(\eta-\gamma_1-\gamma_2)\] to \eqref{add} to get the multiplicity of $h_\alpha$, whence the term $G_4$. Finally Case 1 is straightforward: if $\alpha \in B_X $ as in $G_2$, the multiplicity of $h_{\alpha }$ in the leading term of $\det G^X_{\eta}$ is \[\sum_{\pi \in {\bf \overline{P}}_{X}(\eta)} \pi(\alpha) = {{\bf p}_{X\cup\{\alpha \}}(\eta - \alpha )}. \]
$\Box$ \begin{lemma} \label{xyz} Modulo the ideal defining $\mathcal{H}_X$, we have
$|\det G^X_{\eta}:h| = |D_0D^X_{\eta}:h|$ for all $h \in\mathfrak{h}.$
\end{lemma} \begin{proof} If $\alpha\in {\Delta}^+_{0}
$ and $s_\alpha X = X$, it is easily checked that $ h_{\alpha}$ has the same multiplicity in $G_\eta$ and $ D_\eta$. Next if ${\alpha \in B_X }$, that is $\alpha$ is an isotropic root which is orthogonal to $X$, then $ h_{\alpha}$ has the same multiplicity in the leading terms of $G_2$ and $ D_2$. Indeed $G_2$ {\it is} the leading term of $D_2$. The multiplicity of $h_\alpha$ in other terms is zero. \\ \\ Next consider the case where $\alpha\in C_X $, that is Case 3 in Remark \ref{dab}. Suppose that $\gamma = \Gamma(\alpha)\in X,$ but $\gamma'=s_\alpha \gamma \notin X$. Set $Z=X\cup \{\gamma'\}$ and $\eta_0 =\eta-\gamma$. Then using Lemma \ref{ink} we obtain telescoping sums
\begin{eqnarray} \label{stu}|G_1 :h_\alpha| -| D_1 :h_\alpha| &=&\sum_{r\ge 1}{{\bf p}_{X}(\eta - r \alpha)}-{{\bf p}_{s_\alpha X}(\eta - r \alpha)}\nonumber\\ &=& \left\{ \begin{array}{cr} {\bf p}_Z(\eta_0)& \mbox{if } {\alpha \in C^+_X}, \\ -{\bf p}_Z(\eta_0 - \alpha) & \mbox{otherwise}. \end{array} \right. \end{eqnarray} Next note that if $\gamma\in X$, and $s_\alpha \gamma\notin X$ then
\begin{eqnarray} \label{G4}|G_3 :h_\alpha| &=& \sum_{\pi \in {\bf \overline{P}}_{X}(\eta)} {\pi({s_\alpha \gamma} )} \nonumber\\
&=&|{\pi \in {\bf \overline{P}}_{X}(\eta)}|{\pi({s_\alpha \gamma} )}=1|\;=\;{\bf p}_Z(\eta- {s_\alpha \gamma})\nonumber\\ &=& \left\{ \begin{array}{ll} {\bf p}_Z(\eta_0 + \alpha)\;\;\; \mbox{if} & {\alpha \in C^+_X, \gamma = \Gamma(\alpha)}\\ {\bf p}_Z(\eta_0 - \alpha) & \mbox{otherwise}. \end{array} \right. \end{eqnarray} Also if $\alpha \in C^+_X$ and $\gamma = \Gamma(\alpha)$ then by Lemma \ref{ink} and \eqref{yaws},
\begin{equation} \label{G5} | D_3 :h_\alpha|= {\bf p}_Z(\eta_0 + \alpha)-{\bf p}_Z(\eta_0) \mbox{ and } | D_0' :h_\alpha|= 2{\bf p}_Z(\eta_0). \end{equation} Therefore using \eqref{stu}, \eqref{G4} and \eqref{G5} we have
\begin{eqnarray} && |G_1 :h_\alpha| -|D_1 :h_\alpha| +|G_3 :h_\alpha|-|D_0' :h_\alpha| -|D_3 :h_\alpha|\nonumber\\ &=& {\bf p}_Z(\eta_0) +{\bf p}_Z(\eta_0 + \alpha) -2{\bf p}_Z(\eta_0)+ {\bf p}_Z(\eta_0)-{\bf p}_Z(\eta_0 + \alpha)=0. \nonumber \end{eqnarray} showing that $| D_\eta:h_\alpha|= |G_\eta:h_\alpha|.$ The case where $\alpha \in C_X \backslash C^+_X$ is similar, but easier since the multiplicity of $h_\alpha$ in $D_0'$ and $D_3$ is zero. \\ \\
\noindent Now consider the case of a pair $(\alpha_1,\alpha_2) \in E_{X}$, that is Case 2 in Remark \ref{dab}. Set $\alpha=\alpha_1, Y = s_{\alpha}X$ and $V=X\cup Y$. We claim that
\begin{equation} \label{cri} | D_1/G_1 :h_\alpha| = -2{{\bf p}_{V}(\eta - \gamma_2)}+\sum_{i=1}^2{\bf p}_{V}(\eta - \alpha_i-\gamma_1).\end{equation} Indeed by \eqref{yew1} and the expression for $G_1$ in Lemma \ref{gro},
\[ | D_1/G_1 :h_\alpha| = \sum^\infty_{r=1} \sum_{i=1}^2{{\bf p}_{Y}(\eta - r \alpha_i)}-
{{\bf p}_{X}(\eta - r \alpha_i)}\nonumber \]
Next using Lemma \ref{ink},
\begin{eqnarray} | D_1/G_1 :h_\alpha| =\sum^\infty_{r=1} \sum_{i=1}^2 {\bf p}_{V}(\eta - r\alpha_i-\gamma_1)
&+&
\sum^\infty_{r=1} \sum_{i=1}^2 {\bf p}_{V}(\eta - r\alpha_i-\gamma_2)\nonumber\\ -\sum^\infty_{r=1} \sum_{i=1}^2 {\bf p}_{V}(\eta - r\alpha_i- \alpha_1 -\gamma_1)&-& \sum^\infty_{r=1} \sum_{i=1}^2 {\bf p}_{V}(\eta - r\alpha_i-\alpha_2-\gamma_1).\nonumber\end{eqnarray} Introducing $\gamma_2 =\alpha_1 +\gamma_1 +\alpha_2$ into the second line above, we obtain the following, which evidently collapses to the right side of \eqref{cri}
\begin{eqnarray} | D_1/G_1 :h_\alpha| =\sum^\infty_{r=1} \sum_{i=1}^2 {\bf p}_{V}(\eta - r\alpha_i-\gamma_1)
&+&
\sum^\infty_{r=1} \sum_{i=1}^2 {\bf p}_{V}(\eta - r\alpha_i-\gamma_2)\nonumber\\ -\sum^\infty_{r=1} \sum_{i=1}^2 {\bf p}_{V}(\eta - (r+1)\alpha_i-\gamma_1)&-& \sum^\infty_{r=1} \sum_{i=1}^2 {\bf p}_{V}(\eta - (r-1)\alpha_i-\gamma_2).\nonumber\end{eqnarray}
Taking into account the contributions given by $D_0''D_4$ from \eqref{yew}, and $G_4$ from Lemma \ref{gro}
we obtain \begin{equation}| D_1/G_1 :h_\alpha| +|D_0''D_4:h_\alpha| -|G_4 :h_\alpha|=0 \label{hope}\end{equation}
\end{proof}
\begin{corollary} \label{tcp} To complete the proof of Theorem \ref{shapdet}, it suffices to show that $\det F^X_{\eta}$ is divisible by each of the factors $D_1,\ldots,D_4$ defined in Equations \eqref{yew1}-\eqref{yew5}.\end{corollary}
\begin{proof} Immediate. Since the leading terms match there can be no more factors.\end{proof}
\begin{corollary} \label{fara} For all
roots $\alpha$, $|D^X_{\eta}:h_\alpha|\ge |\det F^X_{\eta}:h_\alpha|$. \end{corollary} \begin{proof} Combine Lemma \ref{xyz} with \eqref{(c)}. \end{proof}
\subsection{Parabolic Induction and Twisting Functors.}\label{agm}
\subsubsection{Introduction.} Let $X$ be an orthogonal set of isotropic roots. The aim of this Subsection is to construct certain modules that appear in Jantzen sum formula Theorem \ref{Jansum101}.
In the next Subsection we consider a special case where the modules can be constructed by parabolic induction. In Subsection \ref{nmvtf} the general result will be deduced using twisting functors. \\ \\ Our next goal is to show that if $[\alpha] =(\alpha_1,\alpha_2)\in E_X(\lambda)$ there is a module in the category $\mathcal{O}$ with character $\mathtt{e}^\lambda p_{W(\alpha)}$. \footnote{See \eqref{efg} for notation.} \footnote{This is the first and only time in this paper that we consider modules with a prescribed character of the form $\mathtt{e}^\lambda p_Y$ where $Y$ contains even roots.} These modules play an important role in the Jantzen sum formula for $M^X(\lambda)$. We briefly recall what this means. The hypothesis $[\alpha] \in E_X$ means that there exist odd roots $\gamma_1, \gamma_2\in X$ and even roots $\alpha_1, \alpha_2$, such that $\alpha_1^\vee \equiv \alpha_2^\vee \mod {\mathbb Q} \gamma_1 +{\mathbb Q} \gamma_2$. This happens when $\gamma_2 = \alpha_1 +\gamma_1+\alpha_2$. Using the fact that $X$ is orthogonal, it is easy to see that if $\alpha_3$ is any other positive even root such that $\alpha_1^\vee \equiv \alpha_3^\vee \mod {\mathbb Q} X$, then $\alpha_3=\alpha_2$, that is either $\alpha_1$ or $\alpha_2$ in $[\alpha] =(\alpha_1,\alpha_2)\in E_X(\lambda)$ determines the other. The additional hypothesis on $\lambda$ is that $(\lambda+\rho,\alpha_1^\vee) =1$. Let $\mathfrak{k}[\alpha]$ be the $\fgl(2,2)$ subalgebra associated to $[\alpha]$, and $W'(\alpha)$ (resp. $V'(\alpha)$) be the set of all (resp. all odd) positive roots of $\mathfrak{k}[\alpha]$. We assume that $X'$ is an orthogonal set of isotropic roots which is orthogonal to $W'(\alpha)$ and there are disjoint unions $X=V'(\alpha)\stackrel{\cdot}{\cup} X' $ and $W(\alpha)=W'(\alpha) \stackrel{\cdot}{\cup} X'.$ \\ \\
In the special case where $\mathfrak{k}[\alpha]=\mathfrak{k}$
is the subalgebra $\mathfrak{k}$ from \eqref{str} with $(k,\ell)=(2,2)$, and $X'=\{\epsilon_{k+i}- \delta_{\ell+i}\}_{i=1}^{r}$ the result follows from Proposition \ref{propmain}. In general we need to prove the result when $[\alpha]$ and $\lambda$ are replaced by $v[\alpha] =(v\alpha_1,v\alpha_2)\in E_{vX}(v\cdot\lambda)$ for $v\in W$. \footnote{ Note that in Type A, $v\rho_1=\rho_1$ and hence $v\cdot\lambda= v\circ\lambda$.}
For this we use twisting functors. The result is as follows.
\begin{theorem} \label{con1} Suppose $\mathfrak{k}[\alpha]$ is as above, ${[\alpha] \in E_{X}(\lambda)}$, and $v\in W$ is such that $N(v)$ is disjoint from $\Delta^+(\mathfrak{k})$. Then the module $T_v {\operatorname{Ind}}_{\mathfrak{p}}^{\fg} \;\mathtt{k}_\lambda$ has character $\mathtt{e}^{v\cdot\lambda} p_{vW(\alpha)}$. \end{theorem} \begin{proof} This follows from the more general Theorem \ref{con2} below. \end{proof} \subsubsection{Parabolic Induction.}\label{pari} In the next Subsection, we use twisting functors to construct some new modules that we need for the Jantzen sum formula. The underlying reason for the existence of these modules is the fact that $\fgl(1,1)$ has many one dimensional modules. Here we study a special case where the modules can be constructed by parabolic induction. Suppose that $\mathfrak{k},\mathfrak{l}$ are subalgebras of $\fg$, such that $\mathfrak{k}$ is isomorphic to $\fgl(k,\ell)$ and for some $r,t_1, t_2$
\begin{equation}\label{str}\mathfrak{l} = \mathfrak{k} \oplus \fgl(1,1)^r\oplus\fgl(1|0)^{t_1}\oplus \fgl(0|1)^{t_2}.\end{equation} \noindent
First we explain how $\mathfrak{l}$ is embedded in $\fg=\fgl(m,n)$.
To do this it is convenient to introduce a certain partial flag in $\mathtt{k}^{m|n}$. This will also allow us to define some
important subalgebras of $\fg$. Let $e_1, \ldots, e_m,$ $e_{1'}, \ldots ,e_{n'}$ be the standard basis for $\mathtt{k}^{m|n}$
and consider the diagram below. The $i^{th}$ set of numbers (for $i\ge 0$) separated by a pair of vertical line is used as the index set for a ${\mathbb Z}_2$-graded subspace ${\mathbb W}^{(i)}$ of $\mathtt{k}^{m|n}$. \begingroup
\setlength{\unitlength}{0.14in} \begin{picture}(-15,-10)(18.2,0) \label{fig2}
\put(17.6,-1){\line(0,1){3}} \put(23.6,-1){\line(0,1){3}} \put(23.6,-1){\line(0,1){3}} \put(18,-1){$1',2',\ldots ,\ell'$} \put(18,1){$1,\;2,\;\ldots ,k$} \put(24,-1){$(\ell +1)'$} \put(24,1){$\;\;k+1\;$} \put(27.6,-1){\line(0,1){3}} \put(28.5,0){$\cdots$} \put(30.6,-1){\line(0,1){3}} \put(31,-1){$(\ell +r)'$} \put(31,1){$\;k+r\;$} \put(34.6,-1){\line(0,1){3}} \put(35.2,0){$k+r+1$} \put(39.7,-0.4){\line(0,1){1.5}} \put(40.5,0){$\cdots$} \put(42.5,-0.4){\line(0,1){1.5}} \put(43.1,0){$m$} \put(44.3,-0.4){\line(0,1){1.5}} \put(44.9,0){$(\ell +r+1)'$} \put(51.2,0){$\cdots$} \put(50.5,-0.4){\line(0,1){1.5}} \put(53.8,0){$n'$} \put(53.2,-0.4){\line(0,1){1.5}} \put(55,-0.4){\line(0,1){1.5}} \end{picture}
\endgroup
\noindent We assume that $\mathfrak{l}$ contains $\mathfrak{h}$ (equivalently $m= k+r+t_1$ and $n=\ell+r+ t_2$). This gives $t+1$ subspaces (where $t:=m -k + t_2$), which we number as
\[{\mathbb W}^{(0)} ={\operatorname{span}}\{e_1,\ldots, e_k,e_{1'},\ldots, e_{\ell'}\},\; {\mathbb W}^{(1)} ={\operatorname{span}}\{e_{k+1}, e_{(\ell+1)'}\}, \ldots, {\mathbb W}^{(t)}=\mathtt{k} e_{n'}.\] \noindent Now set ${\mathbb V}^{(-1)}=0,$ and ${\mathbb V}^{(i)} = {\mathbb W}^{(0)}\oplus\ldots \oplus {\mathbb W}^{(i)}$ for $0\le i\le t$. In this notation, let
\[\mathfrak{k} = \{x\in \fg|x{\mathbb W}^{(0)}\subseteq {\mathbb W}^{(0)}, \; x{\mathbb W}^{(i)} =0 \mbox{ for } i>0\},\]
\[\mathfrak{p} = \{x\in \fg|x{\mathbb V}^{(i)}\subseteq {\mathbb V}^{(i)} \mbox{ for all } i\}, \quad
\mathfrak{l} = \{x\in \fg|x{\mathbb W}^{(i)}\subseteq {\mathbb W}^{(i)} \mbox{ for all } i\}.\] Next define
\[\mathfrak{v}_0 = \{x\in \fg_0|x{\mathbb W}_j^{(i)}\subseteq {\mathbb V}_j^{(i-1)},\;\;\mbox{ for all } i \mbox{ and } j=1,2\},\]
\[\mathfrak{v}_1^+ = \{x\in \fg|x{\mathbb W}_1^{(i)}\subseteq {\mathbb V}_0^{(i-1)},\;\;x{\mathbb W}_0^{(i)} = 0 \mbox{ for all } i\},\]
\[\mathfrak{v}_1^- = \{x\in \fg|x{\mathbb W}_0^{(i)}\subseteq {\mathbb V}_1^{(i-1)},\;\;x{\mathbb W}_1^{(i)} = 0 \mbox{ for all } i\},\] and then set $\mathfrak{v}_1= \mathfrak{v}_1^+\oplus\mathfrak{v}_1^-$, $\mathfrak{v}=\mathfrak{v}_0\oplus\mathfrak{v}_1$. Note that $\mathfrak{p}$ is the subalgebra of $\fg$ stabilizing the partial flag \[ 0 \subseteq {\mathbb V}^{(0)} \subseteq {\mathbb V}^{(1)} \subseteq \ldots \subseteq {\mathbb V}^{(t) },\] and $\mathfrak{v}$ is the ideal in $\mathfrak{p}$ consisting of all elements that act nilpotently on the flag. We have $\mathfrak{p}=\mathfrak{l}\oplus \mathfrak{v}$. Also $\mathfrak{k}\cong \fgl(k,l)$. If $\mathfrak{c}$ is an ad-$\mathfrak{h}$ stable subspace of $\fg$, let $\Delta(\mathfrak{c})$ be the set of roots of $\mathfrak{c}$. The set of even and odd roots in $\Delta(\mathfrak{c})$ is denoted by $\Delta_0(\mathfrak{c})$ and $\Delta_1(\mathfrak{c})$ respectively. Let $\mathfrak{w}$ be the ad-$\mathfrak{h}$ stable subspace of $\fg$ with $\Delta(\mathfrak{w}) = - \Delta(\mathfrak{v})$. Then $\fg= \mathfrak{w} \oplus \mathfrak{p}$.
\begin{example} \label{54}{\rm The first diagram yields a flag in $\mathtt{k}^{5|4}$ The second diagram then shows the resulting subspaces of $\fg=\fgl(5,4)$, constructed using the above recipes.}\end{example} \begingroup
\setlength{\unitlength}{0.14in} \begin{picture}(-15,-10)(18.2,1) \label{fig101}
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\put(34.6,1){$4$} \put(37.1,0){$5$} \put(39.6,0){$4'$} \end{picture}
\endgroup
\[ \begin{tabular}{|c|c|c|c|c||c|c|c|c|}\hline $ \mathfrak{k}$ &$\mathfrak{k}$ & $\mathfrak{v}_0$ & $\mathfrak{v}_0$ & $\mathfrak{v}_0$ &$ \mathfrak{k}$ &$\mathfrak{v}_1^+$ & $\mathfrak{v}_1^+$ & $\mathfrak{v}_1^+$ \\ \hline $ \mathfrak{k}$ &$\mathfrak{k}$ & $\mathfrak{v}_0$ & $\mathfrak{v}_0$ & $\mathfrak{v}_0$ & $ \mathfrak{k}$ &$\mathfrak{v}_1^+$ & $\mathfrak{v}_1^+$ & $\mathfrak{v}_1^+$ \\ \hline $ 0$ &0 & $\fgl(1,1)^{(1)}$ & $\mathfrak{v}_0$ & $\mathfrak{v}_0$ & 0&$\fgl(1,1)^{(1)}$ & $\mathfrak{v}_1^+$ & $\mathfrak{v}_1^+$ \\ \hline $ 0$ &0& 0& $\fgl(1,1)^{(2)}$ & $\mathfrak{v}_0$& 0&0 & $\fgl(1,1)^{(2)}$ & $\mathfrak{v}_1^+$ \\ \hline $ 0$ &0& 0& 0 & $\fgl(1,0)$ & 0&0 & 0 & $\mathfrak{v}_1^+$ \\ \hline \hline $\mathfrak{k}$ &$\mathfrak{k}$ & $\mathfrak{v}_1^{-}$ & $\mathfrak{v}_1^{-}$ & $\mathfrak{v}_1^{-}$ & $\mathfrak{k}$&$\mathfrak{v}_0$ & $\mathfrak{v}_0$ & $\mathfrak{v}_0$ \\ \hline 0&0& $\fgl(1,1)^{(1)}$ & $\mathfrak{v}_1^{-}$ & $\mathfrak{v}_1^{-}$ & 0 &$\fgl(1,1)^{(1)}$ & $\mathfrak{v}_0$ & $\mathfrak{v}_0$ \\ \hline 0&0& 0 & $\fgl(1,1)^{(2)}$ & $\mathfrak{v}_1^{-}$ & 0&0 & $\fgl(1,1)^{(2)}$ & $\mathfrak{v}_0$ \\ \hline 0&0& 0& 0& 0& 0&0& 0& $\fgl(0,1)$ \\ \hline
\end{tabular} \] Non-zero entries in the second diagram correspond to the non-zero matrix entries in $\mathfrak{p}$. Superscripts are used to distinguish the two copies of $\fgl(1,1).$ \\ \\ \noindent Now set ${\mathbb W} = {\mathbb W}^{(0)}\oplus {\mathbb W}^{(r+1)}\oplus\ldots \oplus {\mathbb W}^{(t)}$, and
\[\mathfrak{u}^+ = \{x\in \fg|x{\mathbb W}_1^{(i)}\subseteq {\mathbb W}_0^{(i)},\;\;x{\mathbb W}_0^{(i)} = 0 \mbox{ for } i \in [r], \;\;x{\mathbb W}=0 \},\]
\[\mathfrak{u}^- = \{x\in \fg|x{\mathbb W}_0^{(i)}\subseteq {\mathbb W}_1^{(i)},\;\;x{\mathbb W}_1^{(i)} = 0 \mbox{ for } i \in [r], \;\;x{\mathbb W}=0 \}.\] If $\mathfrak{u} =\mathfrak{u}^+\oplus\mathfrak{u}^-$, we have $\mathfrak{l}=\mathfrak{k}\oplus\mathfrak{u}+\mathfrak{h}$. Note that the entries in $\mathfrak{u}^+$ (resp. $\mathfrak{u}^-$) occur in the copies of $\fgl(1,1)$ above (resp. below) the main diagonal. The set of weights of $\mathfrak{u}^+$ is
\begin{equation} \label{lip} \{\epsilon_{k+j} - \delta_{(l+j)'}|j \in [r]\}. \end{equation} Next define
\[Z_0 = \Delta^+(\mathfrak{v}_0), \quad Z_1^\pm=\Delta(\mathfrak{v}_1^\pm), \quad Z_1 = Z_1^+\cup Z_1^-, \quad \widehat{Z}_1 = Z_1^+\cup -Z_1^- .\]
\begin{lemma} \begin{itemize} \item[{}] \item[{{\rm(a)}}] If $L_0={\prod_{\alpha \in \Delta_0^+}} (1 -\mathtt{e}^{ -\alpha})$, we have
\begin{equation} \label{abc} {\prod_{\alpha \in \Delta({\mathfrak{l}_0^+})} (1 - \mathtt{e}^{ -\alpha})}{\prod_{\alpha \in Z_{0}} (1 - \mathtt{e}^{ -\alpha})} = L_0.\end{equation} \noindent \item[{{\rm(b)}}] Set $\sigma = \sum_{\alpha\in Z^-_1}\alpha$. Then \begin{equation} \label{lem} \prod_{\alpha \in Z_{1}} (1 +\mathtt{e}^{- \alpha})=\mathtt{e}^{-\sigma}\prod_{\alpha \in \widehat{Z}_1}(1 +\mathtt{e}^{- \alpha}).\end{equation} \item[{{\rm(c)}}] If $\rho(\mathfrak{b}')$ is the analog of $\rho$ for $\mathfrak{b}'$ we have $\rho(\mathfrak{b}')=\rho + \sigma. $ \item[{{\rm(d)}}] There is a unique Borel subalgebra $\mathfrak{b}'$ contained in $\mathfrak{p}$ such that
\[ d(\mathfrak{b},\mathfrak{b}') =\min\{d(\mathfrak{b},\mathfrak{b}'')| \mathfrak{b}'' \mbox{ is a Borel subalgebras contained in } \mathfrak{p}\}\]
\item[{{\rm(e)}}] We have $\Delta^+(\mathfrak{b}') = Z_1\cup \{\epsilon_{k+j} - \delta_{(l+j)'}|j \in [r]\}.$ \end{itemize} \end{lemma}
\begin{proof} For (a) note we have a disjoint union $\Delta_0^+ = \Delta(\mathfrak{l}^+_0) \stackrel{\cdot}{\cup} Z_0,$ from which
\eqref{abc} follows. Similarly (b) holds since $Z_{1}^+$ is common to $Z_{1}$ and $\widehat{Z}_1$. For (c) note that $Z_1^-=\Delta(\mathfrak{v}_1^-)$ is precisely the set of odd roots of $\mathfrak{b}'$ which are not roots of $\mathfrak{b}^{\operatorname{dist}}.$ We leave (d), (e) to the reader. \end{proof} \noindent Note that \begin{equation} \label{rad} (\lambda+\rho(\mathfrak{b}^{\operatorname{dist}}),X) = (\lambda'+\rho(\mathfrak{b}'),X) = 0.\end{equation}
\noindent The following lemma is straightforward. \begin{lemma} \label{oil} If $\lambda\in \mathfrak{h}^*$, then $\lambda$ defines a one dimensional representation of $\mathfrak{l}$ if and only if $(\lambda+\rho,\gamma)=0$ for all odd $\gamma \in X,$
and $(\lambda,\alpha)=0$ for each even simple root $\alpha$ of $\mathfrak{l}$. \end{lemma} \begin{proposition} \label{propmain} Assume that $\lambda\in \mathfrak{h}^*$ defines a one dimensional representation $\mathtt{k}_\lambda$ of $\mathfrak{l},$ and regard $\mathtt{k}_\lambda$ as a $\mathfrak{p}$-module by allowing $\mathfrak{v}$ to act trivially. Let $X=\Delta^+(\mathfrak{l})$. Then \begin{itemize} \item[{{\rm(a)}}] The module ${\operatorname{Ind}}_{\mathfrak{p}}^{\fg} \mathtt{k}_\lambda$ has character \begin{equation} \label{sot} {\operatorname{ch}\:} {\operatorname{Ind}}_{\mathfrak{p}}^{\fg} \;\mathtt{k}_\lambda =\frac{{}\prod_{\alpha \in Z_{1}} (1 +\mathtt{e}^{- \alpha})} {\prod_{\alpha \in Z_{0}} (1 -\mathtt{e}^{ -\alpha})}\;\mathtt{e}^\lambda = \frac{{}\prod_{\alpha \in \widehat{Z}_1} (1 +\mathtt{e}^{- \alpha})}{ \prod_{\alpha \in Z_{0}} (1 -\mathtt{e}^{ -\alpha})}\;\mathtt{e}^{\lambda-\sigma}=\mathtt{e}^\lambda p_X.\end{equation} \item[{{\rm(b)}}] The module ${\operatorname{Ind}}_{\mathfrak{p}_0}^{\fg_0}\;\mathtt{k}_\lambda $ has character \begin{eqnarray} \label{wed} {\operatorname{ch}\:} {\operatorname{Ind}}_{\mathfrak{p}_0}^{\fg_0}\;\mathtt{k}_\lambda &=& {\prod_{\alpha \in Z_{0}} (1 -\mathtt{e}^{ -\alpha})}^{-1}\;\mathtt{e}^\lambda = {\prod_{\alpha \in \Delta({\mathfrak{l}_0^+})} (1 -\mathtt{e}^{ -\alpha})}\mathtt{e}^\lambda/L_0. \end{eqnarray} \end{itemize}
\end{proposition}
\begin{proof} (a) Let $\mathfrak{w}$ be the ${\operatorname{ad \;}} \mathfrak{h}$-stable subalgebra of $\fg$ such that $\Delta(\mathfrak{w})=-\Delta(\mathfrak{v})$ and $\fg=\mathfrak{w}\oplus \mathfrak{p}$. By \cite{M} Corollary 6.1.5, if $S(\mathfrak{w})$ is the supersymmetric algebra on $\mathfrak{w}$, we can write $U(\fg) = S(\mathfrak{w})\otimes U(\mathfrak{p})$ as right $U(\mathfrak{p})$-modules. Now $\Delta(\mathfrak{w})= Z_0 \cup Z_1$, Hence the first equality in (a) since $S(\mathfrak{w})$ is the tensor product of the exterior algebra on $\mathfrak{w}_1$ with the symmetric algebra on $\mathfrak{w}_0$. The second equality comes from \eqref{lem}, and the third is an easy consequence. The proof of the first equality in (b) is similar, and the second follows from \eqref{abc}.
\end{proof} \noindent Let $X=\Delta^+_1(\mathfrak{l})$. We note the following general behavior of ${\operatorname{Ind}}_{\mathfrak{p}}^{\fg} \;\mathtt{k}_\lambda $. \begin{lemma} \label{clo} If $B(\lambda)=X$, then $M={\operatorname{Ind}}_{\mathfrak{p}}^{\fg} \;\mathtt{k}_\lambda$ is a highest weight module for the distinguished Borel subalgebra of $\fg$ with highest weight $\lambda'=\lambda-\sigma$. \end{lemma}
\begin{proof} Consider the Borel subalgebra $\mathfrak{b}$ with the same even part as the distinguished Borel subalgebra $\mathfrak{b}^{\operatorname{dist}}$ of $\fg$, and with $$\mathfrak{b}^{\operatorname{dist}}_1 \cap \mathfrak{b}_1=(\mathfrak{b}^{\operatorname{dist}}_\mathfrak{k})_1\oplus\mathfrak{u}^+\oplus\mathfrak{v}_1,$$ where $\mathfrak{b}^{\operatorname{dist}}_\mathfrak{k}$ is the distinguished Borel subalgebra of $\mathfrak{k}$.
Then $\mathfrak{b} \subseteq \mathfrak{p}$, so $M$ is a highest weight module for $\mathfrak{b}$ with highest weight $\lambda$. Also there is a sequence
\begin{equation} \label{1distm1} \mathfrak{b} = \mathfrak{b}^{(0)}, \mathfrak{b}^{(1)}, \ldots,
\mathfrak{b}^{(u)}. \end{equation}
of Borel subalgebras of $\fg$, such that $\mathfrak{b}^{(i-1)}$ and
$\mathfrak{b}^{(i)}$ are adjacent for $1 \leq i \leq u$, and $\mathfrak{b}^{(u)} = \mathfrak{b}^{\operatorname{dist}}$. The assumption $B(\lambda)=X$ ensures that each change of Borels from $\mathfrak{b}$ to $ \mathfrak{b}^{\operatorname{dist}}$ is typical for $M$, see \eqref{1mon}. Now $M$ is generated by a highest weight vector $v_\lambda$ with respect to $\mathfrak{b}$, and there are odd root vectors $e_{-\alpha_1},\ldots, e_{-\alpha_u}$ such that $m=e_{-\alpha_u} \ldots e_{-\alpha_1}v_\lambda$ is a highest weight vector for $\mathfrak{b}^{\operatorname{dist}}$. The assumption $B(\lambda)=X$ also implies that $m$ generates $M$ and the result follows. \end{proof} \noindent \subsubsection{New Modules via Twisting Functors.}\label{nmvtf} Now consider any $\mathfrak{h}$-stable subalgebra $\mathfrak{l}'$ of $\fg=\fgl(m,n)$ such that $\mathfrak{l}' \cong \mathfrak{k} \oplus \fgl(1,1)^r$, with $\mathfrak{k}$ a $\fgl(k, \ell)$-subalgebra of $\mathfrak{l}$. By abuse of notation we write $\fgl(1,1)^r$ for the isomorphic copy of this subalgebra inside $\mathfrak{l}'$. We denote the set of even (resp. odd) simple roots of $\mathfrak{k}$ by $\Pi_0(\mathfrak{k})$, (resp. $\Pi_1(\mathfrak{k})$) and do likewise for $\mathfrak{l}'$. \begin{lemma} We have \begin{itemize} \item[{{\rm(a)}}] $$\Pi_0(\mathfrak{k})=\{\epsilon_{s_i}- \epsilon_{s_{i+1}}\}_{i=1}^{k-1} \cup \{\delta_{t_i}- \delta_{t_{i+1}}\}_{i=1}^{\ell-1},$$
$\Pi_1(\mathfrak{k})=\{\epsilon_{s_k}- \delta_{t_{1}}\}$, $\Pi_0(\mathfrak{l}')=\Pi_0(\mathfrak{k})$, and
$\Pi_1(\mathfrak{l}')=\Pi_1(\mathfrak{k})\cup\{\epsilon_{c_i}- \delta_{d_{i}}\}_{i=1}^r,$ where $1\le s_1<s_2<\ldots s_k\le m$, $1\le t_1<t_2<\ldots t_\ell\le n$ and $c_i \notin \{{s_i}\}_{i=1}^k$, $d_i \notin \{{t_i}\}_{i=1}^\ell$ \item[{{\rm(b)}}] Let $i' = n+i$ for $i\in [\ell]$. Suppose $w\in W=S_m \ti S_n$ satisfies $w(k+i) = c_i$ for $i\in[r]$, $w(\ell+i)' = d_i'$ for $i\in[r]$ $wi = s_i,$ for $i\in [k]$, $wi' = t_i$ $i\in [\ell]$. Then $w\Delta^+(\mathfrak{l})= \Delta^+(\mathfrak{l}')$
and $N(w)\cap \Delta^+_0(\mathfrak{l})=\emptyset.$ \end{itemize}\end{lemma} \begin{proof} Left to the reader. \end{proof}
\begin{theorem} \label{con2} If $\lambda$ defines a one-dimensional representation of $\mathfrak{p}$, then
the module $T_w {\operatorname{Ind}}_{\mathfrak{p}}^{\fg} \;\mathtt{k}_\lambda$ has character $\mathtt{e}^{w\cdot\lambda} p_{\Delta^+(\mathfrak{l}')}$. \end{theorem} \begin{proof} See \cite{M101}. \end{proof}
\subsection{An Equivalence of Categories and Generic Cases.}\label{lab} \subsubsection{The Equivalence of Categories.}\label{gcvtf} In \cite{CMW}, twisting functors are used to show that every block of the category $\mathcal{O}$ for a Lie superalgebra in Type A is equivalent to an integral block of some (possibly smaller) Lie superalgebra of Type A. We show how this result can be used to reduce the study of the modules $M^X(\lambda)$ in the most general cases to the study of $\fgl(2,1)$ and $\fgl(2,2).$ The main result of \cite{CMW} states that \begin{theorem}\label{thmmain} Every block $\mathcal{O}_\lambda$, $\lambda\in \mathfrak{h}^*$, is equivalent to an integral block of the category $\mathcal{O}$ for some direct sum of general linear Lie superalgebras. \end{theorem} \noindent The weight $\lambda$ determines a subroot system $\Delta_\lambda$ of $\Delta$ with Weyl group $W_\lambda$ which is the subgroup of the Weyl group $W$ generated by ${s_\alpha}$ with $\alpha$ an even root of $\Delta_\lambda$.
The proof in \cite{CMW} shows that there is a Lie superalgebra $\fg_{[\widehat{\lambda}]}$ and an integral weight $\widehat{\lambda}$ for $\fg_{[\widehat{\lambda}]}$ such that $\mathcal O_\lambda$ is equivalent to a block $\mathcal{O}_{\widehat{\lambda}}^\mathfrak{l}$ of the category of $\mathcal{O}^{\mathfrak{l}}$ for $\mathfrak{l} = \fg_{[\widehat{\lambda}]}$. The block $\mathcal{O}^{\mathfrak{l}}_{{\widehat{\lambda}}}$ is the Serre subcategory of $\mathcal{O}^{\mathfrak{l}}$ generated by simple objects of the form $L_{\mathfrak{l}}(\mu)$ with $\mu=w\cdot(\lambda-\sum_{j}k_j\alpha_j)$, for $w\in W_{\lambda}$ and $\{\alpha_j\}$ a set of mutually orthogonal odd isotropic roots satisfying $(\lambda+\rho_{\mathfrak{l}},\alpha_j)=0$ and $k_j\in{\mathbb Z}$.
Under this equivalence, the Verma module $M(w\cdot \lambda)$ corresponds to the Verma module $M_\mathfrak{l}(w\cdot \widehat{\lambda})$ for $\mathfrak{l}$ with highest weight $w\cdot \widehat{\lambda}$, and the simple module $L(w\cdot\lambda)$ to $L_\mathfrak{l}(w\cdot\widehat{\lambda})$ for all $w\in W_\lambda$. In this set-up $\mathfrak{l}$ is a Levi subalgebra of $\fg$ containing a Cartan subalgebra $\mathfrak{h}$ of $\fg$.
\begin{proposition} \label{ec} Suppose that $B(\lambda) = X$ and $\mathfrak{l} = \fg_{[\widehat{\lambda}]}$ is as in \eqref{str}. \begin{itemize} \item[{{\rm(a)}}] There is an equivalence of categories between the block $\mathcal{O}_{\widehat{\lambda}}^\mathfrak{l}$ and the block $\mathcal{O}_{\widehat{\lambda}}^{\mathfrak{k}+\mathfrak{h}}$. \item[{{\rm(b)}}] There is an equivalence of categories $\mathcal{O}_{\widehat{\lambda}}^{\mathfrak{k}+\mathfrak{h}}\longrightarrow \mathcal{O}_{{\lambda}}$obtained by combining $(a)$ with Theorem {\rm \ref{thmmain}}. This equivalence sends $N$ to ${\operatorname{Ind}}^\fg_\mathfrak{p} N$. \end{itemize} \end{proposition}
\begin{proof} The key points are as follows: any simple $\mathfrak{l}$-module is the tensor product of a simple $\mathfrak{k}$-module and simple modules for the copies of $\fgl(1,1), \fgl(1,0)$ and $\fgl(0,1)$ in \eqref{str}. From the definition of $\mathcal{O}_{\widehat{\lambda}}^\mathfrak{l}$ as a Serre subcategory and the fact that $X'' \subseteq B(\lambda)$ we see that the only simple $\fgl(1,1)$-modules that can appear are one dimensional, and are annihilated by $\mathfrak{u}$. Such modules arise canonically from simple modules in $\mathcal{O}_{\widehat{\lambda}}^{\mathfrak{k}+\mathfrak{h}}$. This proves (a) and (b) holds by construction.\end{proof}
\subsubsection{Lessons learned from the $\fgl(2,2)$ case: Roots in $E_X$.} \noindent Now suppose that $X$ is an orthogonal set of isotropic roots and $(\alpha_1,\alpha_2)\in E_{X}$. Suppose that $\alpha_1 +\gamma_1 +\alpha_2 =\gamma_2$ where $\gamma_1, \gamma_2\in X$. Combining the equivalence of categories from Theorem \ref{thmmain}, with Proposition \ref{propmain} we deduce most of the following results from the $\fgl(2,2)$ case, see Theorems \ref{genst}, \ref{stz1} and \ref{mnt}. However, the fact that \v Sapovalov elements give highest weight vectors is of course true in general. Also the statements about the Jantzen filtration hold because the modules in question are multiplicity free, and each quotient $M^X_i(\lambda)/M^X_{i+1}(\lambda)$ is contragredient with composition factors of multiplicity one, and therefore semisimple. The phrase ``for general $\lambda$" means that, up to $W$-conjugacy, $\mathfrak{l} = \fg_{[\widehat{\lambda}]}$ is as in \eqref{str} with $\mathfrak{k} =\fgl(2,2)$, and we assume (without any loss of generality) that \[\Delta_0^+(\mathfrak{k}) =\{\alpha_1,\alpha_2\}, \; B(\lambda) = X \mbox{ and } \{\gamma_1,\gamma_2\} \subseteq \Delta_1^+(\mathfrak{k})\cap X.\]
Note that the number of $\fgl(1,1)$ terms in \eqref{str} is $r=|X|-2$. \begin{theorem}\label{wane} Let $n$ be an integer greater than 1. For general $\lambda \in \mathcal{H}_X\cap \mathcal{H}_{\alpha_1,n}$ where $(\alpha_1,\alpha_2) \in E_{X}$, \begin{itemize} \item[{{\rm(a)}}] The socle $\mathcal{S}$ of $M^X(\lambda)$ is $M^X(s_1s_2\cdot \lambda) \cong L(s_1s_2\cdot \lambda).$ \item[{{\rm(b)}}] We have \[U(\fg)\theta_{\alpha_1,n}v_\lambda \cong M^Y(s_1\cdot \lambda), \quad U(\fg)\theta_{\alpha_2,n}v_\lambda \cong M^Y(s_2\cdot \lambda)\]
\[M^X_1(\lambda)=U(\fg)\theta_{\alpha_1,n}v_\lambda+U(\fg)\theta_{\alpha_2,n}v_\lambda\] and \begin{equation} \label{tan} \mathcal{S}=U(\fg)\theta_{\alpha_1,n}\theta_{\alpha_2,n}v_\lambda =U(\fg)\theta_{\alpha_1,n}v_\lambda\cap U(\fg)\theta_{\alpha_2,n}v_\lambda \subseteq M_2^X(\lambda).\end{equation} \end{itemize} The lattice of submodules of $M^X(\lambda)$ is as in Figure \ref{fig1} with $V_1=U(\fg)\theta_{\alpha_1,n}v_\lambda$ and $V_2 =U(\fg)\theta_{\alpha_2,n}v_\lambda.$ \end{theorem} \begin{theorem} \label{name} For general $\lambda \in \mathcal{H}_X\cap \mathcal{H}_{\alpha_1,1}$ where $(\alpha_1,\alpha_2) \in E_{X}$, \begin{itemize} \item[{{\rm(a)}}] the socle $\mathcal{S}$ of $M^X(\lambda)$ satisfies \begin{equation} \label{stz} \mathcal{S} = L(\lambda-\gamma_1-\gamma_2)\oplus L(s_1s_2\cdot \lambda) = U(\fg)\theta_{\alpha_1,1}v_\lambda\cap U(\fg)\theta_{\alpha_2,1}v_\lambda, \end{equation} and \begin{equation} \label{ton} \mathcal{S} \subseteq M^X_2(\lambda).\end{equation} \item[{{\rm(b)}}] We have \[U(\fg)\theta_{\alpha_1,1}v_\lambda/L(\lambda-\gamma_1-\gamma_2) \cong M^Y(s_1\cdot \lambda), \quad U(\fg)\theta_{\alpha_2,1}v_\lambda/L(\lambda-\gamma_1-\gamma_2) \cong M^Y(s_2\cdot \lambda),\] \[L(s_1s_2\cdot \lambda) =M^X(s_1 s_2\cdot \lambda),\]
and \[U(\fg)\theta_{\alpha_1,1}v_\lambda+U(\fg)\theta_{\alpha_2,1}v_\lambda=M^X_1(\lambda).\]
\end{itemize} \end{theorem} \begin{rem} {\rm It follows from the sum formula, Theorem \ref{Jansum101}, that equality holds in \eqref{tan} and \eqref{ton}, that is $\mathcal{S} = M^X_2(\lambda)$ in both cases. In addition we have $M^X_3(\lambda)=0$.}\end{rem}
\begin{theorem} \label{bits} Suppose that \eqref{had} holds, and that $m=(\lambda+\rho,\alpha_1^\vee)\in{\mathbb Z}.$ Set $Y=s_{\alpha_1} X$.
Then for general $\lambda$ in $\mathcal{H}_X\cap \mathcal{H}_{{\alpha_1,m}}$ \begin{itemize} \item[{{\rm(a)}}] if $m>0$ then ${M}^X(\lambda)$ has a unique proper submodule $M^{Y}(s_{\alpha_1}\cdot\lambda)={M}^X_{1}(\lambda).$ \item[{{\rm(b)}}] If $m< 0$, then $M^X(\lambda)$ has a unique proper submodule $M^{Y}(s_{\alpha_2}\cdot\lambda)={M}^X_{1}(\lambda).$ \end{itemize} \end{theorem}
\subsubsection{Lessons learned from the $\fgl(2,1)$ case: Roots in $C_X$.} \noindent Next we consider the case where $\alpha\in C_X$. Most of the following result can be deduced using twisting functors from the $\fgl(2,1)$ case, but since parts of it seem to work outside type A, we give a direct proof. The highest weight vectors of $M^X(\lambda)$ and $ M^{Y}(\lambda)$ are denoted by $v^X_\lambda$ and $v^{Y}_\lambda$ respectively.
\begin{theorem} \label{eon} Suppose $\alpha \in C_{X}, \gamma = \Gamma(\alpha)$ and set $\gamma' = s_\alpha\gamma, Y =s_\alpha X$, ${Z(\alpha)}=X \cup Y$. \begin{itemize} \item[{{\rm(a)}}] If $\alpha\in C_X\backslash C_X^+$, then for general $\lambda$ with $ B(\lambda) = X\cup Y$, $M^X(\lambda)$ is simple. \item[{{\rm(b)}}] If $\alpha \in C_{X}(\lambda)$, we have an onto map $M^X(\lambda)\longrightarrow M^{Y}(\lambda)$ sending $v^X_\lambda$ to $v^{Y}_\lambda$. \item[{{\rm(c)}}] Let ${K}^{X,\alpha}({\lambda})$ be the kernel of the map $M^X(\lambda)\longrightarrow M^{s_\alpha X}(\lambda)$. Then \begin{equation} \label{cut} {\operatorname{ch}\:} {K}^{X,\alpha}({\lambda}) = \mathtt{e}^{\lambda}({p}_X- {p}_{s_\alpha X}) = {\mathtt{e}^{\lambda-\gamma'}(1 - \mathtt{e}^{- \alpha})}{p}_{Z(\alpha)}.\end{equation}
Equivalently \[\dim {K}^{X,\alpha}({\lambda})^{\lambda-\eta} ={\bf p}_{Z(\alpha)}(\eta-\gamma') - {\bf p}_{Z(\alpha)}(\eta-\gamma).\] \item[{{\rm(d)}}] For general $\lambda$ such that $\alpha\in C_{X}(\lambda)$, ${K}^{X,\alpha}({\lambda})=M_1^X(\lambda)$, and this submodule is simple. \end{itemize} \end{theorem}
\begin{proof} We deduce (a) from the $\fgl(2,1)$ case. Let $\mathfrak{k}=\fgl(2,1)$ in \eqref{str}, and denote highest weight modules for $\mathfrak{k}$ by $M_\mathfrak{k}$. Then switching to the notation of the Appendix, what has to be checked is that the $\mathfrak{k}$-module $M_\mathfrak{k}^\beta(-\rho)$ is simple. However from Theorem \ref{AA2} (e), $M_\mathfrak{k}^\beta(-\rho)\cong M_\mathfrak{k}(-\rho)/V_1$ where $V_1$ is the maximal submodule of $M_\mathfrak{k}(-\rho)$. It is clear that the map in (b) is surjective if it is well-defined. The domain and target of the proposed map are both obtained as factors of $M(\lambda)$ by imposing certain relations. To show that it is well-defined we need to show that the relations satisfied by $M^X(\lambda)$ are also satisfied by $M^{s_\alpha X}(\lambda)$. The relations needed to define $M^X(\widetilde{\lambda})_B$ and $M^{s_\alpha X}(\widetilde{\lambda})_B$ are the same except that we require $\theta_\gamma v_{\widetilde{\lambda}} = 0$ in $M^X(\lambda)_B$ and $\theta_{\gamma'} v_{\widetilde{\lambda}} = 0$ in $M^{s_\alpha X}(\lambda)_B$. Now in any module in which $\theta_{\gamma'} v_{\widetilde{\lambda}} = 0$ we have $\theta_{\alpha,1}\theta_{\gamma'}v_{\widetilde{\lambda}}=0$, and hence by Theorem \ref{stc}, $\theta_\gamma v_{\widetilde{\lambda}} \in TU(\fg)_A v_{\widetilde{\lambda}}$. Reducing mod $T$ we obtain the statement in (b).
\\ \\
The first equality in \eqref{cut} follows from (b), the second from Lemma \ref{ink}. For (d), note that by (a) $M^Y(\lambda)$ is simple for general $\lambda$, hence ${K}^{X,\alpha}({\lambda})=M_1^X(\lambda)$. Again we compare to the $\fgl(2,1)$ case. Then in the notation of Theorem \ref{AA2}, the map $M^X(\lambda)\longrightarrow M^{Y}(\lambda)$ is the map $M_\mathfrak{k}(\lambda)/V_2 \longrightarrow M_\mathfrak{k}(\lambda)/V_1$ and the kernel of this map is simple.
\end{proof}
\subsection{Proof of Theorem \ref{shapdet}.}\label{pfs} We use Corollary \ref{tcp} and consider the factors $D_1,\ldots,D_4$ defined in Equations \eqref{yew1}-\eqref{yew5}. We show that the multiplicity $m(\alpha,k,\eta)^X$ of $h_{\alpha}-k$ in $\det F^X_\eta$ is exactly that predicted in the Theorem for each relevant root $\alpha$. To do this it is enough to show that the multiplicity of $h_{\alpha}$ in the leading term of $\det F^X_\eta$ is as predicted. First an easy Lemma, see \cite{KK}. \begin{lemma} \label{eek} Suppose $M^X_1(\lambda)$ is simple and isomorphic to $M^Y(\lambda-\mu)$ for some $\mu \in Q^+$. Then \begin{itemize} \item[{{\rm(a)}}]
the Jantzen filtration on $M^X(\lambda)$ takes the form \begin{equation}\label{wad} M^X_{1}(\lambda) = M^X_{2}(\lambda) \ldots= M^X_{c_k}(\lambda) \neq 0 =M^X_{c_k+1}(\lambda). \end{equation} \item[{{\rm(b)}}] the multiplicity of $h_\alpha-k$ in $\det F^X_\eta$ is $c_{k}{{\bf p}_{Y}(\eta - \mu)}.$ \end{itemize} \end{lemma} \begin{proof} Obviously (a) holds, and then (b) follows from \eqref{yet}.\end{proof} \noindent {\it Proof of Theorem \ref{shapdet}.} We consider 3 cases.\\ \\ \noindent {\bf Case 1.} First consider the factors of $D_2$. Suppose that $\gamma$ is isotropic $(\gamma,X)=0$.
Suppose that $\lambda$ satisfies the conditions of Lemma \ref{need} (b), and use the notation of the Lemma. Then $M^X_{1}(\lambda) = M^Y(\lambda-\gamma)$ is simple. Hence by \eqref{yet} and \eqref{wad}, the multiplicity of $h_{\gamma} + (\rho, \gamma)$ in $\det F^X_\eta$ is $v_{T}(\det F^X_\eta(\widetilde{\lambda})) = a{\bf p}_{X}(\eta - \gamma)$. The computation of the leading term in Lemma \ref{gro} implies $a=1$ as claimed. \\ \\ {\bf Case 2.} Now consider the case where \eqref{cad} holds. Without loss we can assume that $\alpha_1 +\gamma_1 +\alpha_2 =\gamma_2$. Suppose that $\lambda_0$ is a general point in $\mathcal{H}_X\cap \mathcal{H}_{\alpha_1,0}.$ Then if $\lambda_k =\lambda_0-k\gamma_1,$ for $k\ge1$ we have $(\lambda_k+\rho,\alpha^\vee)=k$. We use Theorems \ref{wane} and \ref{name}. The socle $\mathcal{S}_k$ of $M^X(\lambda_k)$ satisfies $\mathcal{S}_k\subseteq M^X_2(\lambda_k)$. If $k>1$, then the lattice of submodules of $M^X(\lambda_k)$
is as in Figure \ref{fig1} with \begin{equation} \label{pup} V_1(\lambda_k)=M^Y(s_1\cdot\lambda_k), \;V_2(\lambda_k)=M^Y(s_2\cdot\lambda_k), \mbox{ and } V_3(\lambda_k)=M^X(s_1s_2\cdot\lambda_k),\end{equation}
where $Y = s_1 X=s_2X$. Thus the composition factors of $M^X_1(\lambda_k)$ are \[L_1(\lambda_k)=L(s_1\cdot\lambda_k), \;L_2(\lambda_k)=L(s_2\cdot\lambda_k) \mbox{ and } L_3(\lambda_k)=L(s_1s_2\cdot\lambda_k),\] Also $[L_1(\lambda_k)], [L_2(\lambda_k)],$ and $[L_3(\lambda_k)]$ generate the same subgroup of the Grothendieck group $K(\mathcal{O})$ as $[V_1(\lambda_k)], [V_2(\lambda_k)],$ and $[V_3(\lambda_k)]$. If $k=1$, then the subgroup of the Grothendieck group $K(\mathcal{O})$ generated by the composition factors of $M_1^X(\lambda)$ is the same as that generated by $[L(\lambda_1-\gamma_1-\gamma_2)]$ and $[V_1(\lambda_1)], [V_2(\lambda_1)],$ $[V_3(\lambda_1)]$ (as in \eqref{pup}).
Thus the composition factors of $M^X_1(\lambda_k)$ are \[L_1(\lambda_k)=L(s_1\cdot\lambda_k), \;L_2(\lambda_k)=L(s_2\cdot\lambda_k) \mbox{ and } L_3(\lambda_k)=L(s_1s_2\cdot\lambda_k),\] Also $[L_1(\lambda_k)], [L_2(\lambda_k)],$ and $[L_3(\lambda_k)]$. Note that $[M^X(s_1s_2\cdot\lambda_k)]=[L_3(\lambda_k)]$.
\\ \\ In the classical case, the next step is to use the linear independence of certain partition functions \cite{KK}, \cite{M} Theorem 10.2.5. Here we use a generalization of this argument. We have \begin{equation} \label{mik} \sum_{i>0} [M_i(\lambda_k)]= a_k[V_1(\lambda_k)]+b_k[V_2(\lambda_k)]+c_k[V_3(\lambda_k)] +\delta_{k,1}d[L(\lambda-\gamma_1-\gamma_2)], \end{equation} for integers $a_k,b_k, c_k, d$. Since $\mathcal{S}_1 \subseteq M^X(\lambda_1)$ we have $d\ge2$. Thus the multiplicity $m(\alpha_1,k,\eta)^X$ of $h_{\alpha_1}-k$ in $\det F^X_\eta$ is \[ a_k{{\bf p}_{Y}(\eta - k\alpha_1)} +b_k{{\bf p}_{Y}(\eta - k\alpha_2)} +c_k{{\bf p}_{X}(\eta - k\alpha_1-k\alpha_2)} +\delta_{k,1}d{\bf p}_{W(\alpha)}(\eta-\gamma_1-\gamma_2).\] Now the multiplicity of $h_\alpha$ in $\operatorname{LT} \det F^X_{\eta}$ is at least $\sum_{k\ge 1}m(\alpha_1,k,\eta)^X.$ We claim that \begin{equation} \label{expr}\sum_{k\ge 1}{{\bf p}_{Y}(\eta - k\alpha_1)} +{{\bf p}_{Y}(\eta - k\alpha_2)}+2{\bf p}_{W(\alpha)}(\eta-\gamma_1-\gamma_2)\ge \sum_{k\ge 1}m(\alpha_1,k,\eta)^X.\end{equation} By \eqref{yew1} and \eqref{yew5}, the multiplicity of $h_{\alpha_1}$ in the leading term of $D^X_\eta$ equals the left side of \eqref{expr}. Thus the claim follows from Corollary \ref{fara}. \\ \\ Now if $\eta = i\alpha_1$ for $i\ge 1$, then ${{\bf p}_{Y}(\eta - k\alpha_2)}={\bf p}_{W(\alpha)}(\eta-\gamma_1-\gamma_2) =0$ for all $k\ge 1$. Thus we obtain a system of inequalities \[\sum_{k\ge 1}{{\bf p}_{Y}((i- k)\alpha_1)}\ge \sum_{k\ge 1} a_k{{\bf p}_{Y}((i - k)\alpha_1)}.\] Taking $i=1, 2,\ldots$ we have $${\bf p}_{Y}(0)\ge a_1{\bf p}_{Y}(0), \;\; {\bf p}_{Y}(0) +{\bf p}_{Y}(\alpha_1) \ge a_2{\bf p}_{Y}(0) +a_1{\bf p}_{Y}(\alpha_1) \mbox{ etc. }$$
Since $|M^X_1(\lambda_k):L_j(\lambda_k)|>0$ for $j=1,2$, it follows that $a_k$ and $b_k$ are positive.
We conclude that $a_k=1$ and similarly $b_k=1$ for all $k\ge1$. Since $|M^X_1(\lambda_k):L_j(\lambda_k)|=1$ and $\mathcal{S}_k\subseteq M^X_2(\lambda_k)$ for all $k,$ we have from \eqref{mik} that $c_k \ge 0$ and \[2{\bf p}_{W(\alpha)}(\eta-\gamma_1-\gamma_2)\ge d{\bf p}_{W(\alpha)}(\eta-\gamma_1-\gamma_2) +\sum_{k\ge 1}c_k{{\bf p}_{X}(\eta - k\alpha_1-k\alpha_2)}. \] Hence arguing as before, using $\eta =i(\alpha_1+\alpha_2)$ for $i\ge 1$, we find $c_k=0$ for all $k$, and
$d=2$. \\ \\ {\bf Case 3.} \noindent
Suppose next that $\alpha=\alpha_1 \in {\Delta}^+_{0}$ and \eqref{cad} {\it does not} hold. We can handle several cases simultaneously by introducing some more notation. First we define the index set $I$ by \[ I= \left\{ \begin{array}{cl} {\mathbb Z} & \mbox{if there is a root } \alpha_2 \mbox{ such that } \alpha_1^\vee \equiv -\alpha_2^\vee \mod {\mathbb Q} X, \\ {\mathbb N}& \mbox{ if } \alpha\in C_X^+\\ {\mathbb N} \backslash \{0\} & \mbox{otherwise}. \end{array} \right.\] Next for $r\in I$, define ${\bf q}_{Y}(\eta - r \alpha)$ by \[ {\bf q}_{Y}(\eta - r \alpha) =\left\{ \begin{array}{crc} {\bf p}_{Y}(\eta - r \alpha_1)& \mbox{if } r>0, \\ {\bf p}_{Y}(\eta + r \alpha_2)& \mbox{if } I ={\mathbb Z} &\mbox{ and } r<0\\ 0 & \mbox{otherwise}.& \end{array} \right.\]
Suppose $r\in I, r\neq 0$ and choose a general $\lambda\in \mathcal{H}_X$ such that $r=(\lambda+\rho,\alpha_1^\vee).$ Then ${M}^X_{1}(\lambda)$ is simple by Theorem \ref{bits}. Hence by Lemma \ref{eek}, the Jantzen filtration has the form \eqref{wad} and the multiplicity of $h_{\alpha} + \rho(h_{\alpha}) - r(\alpha, \alpha)/2$ in $\det F^X_\eta$ is equal to $c_r {\bf q}_{Y}(\eta - r \alpha)$, for some positive integer $c_r$.
If $r=0$, and $\alpha \in C^+_{X}$, set $\gamma = \Gamma(\alpha)$ and $\gamma' = s_\alpha\gamma$. Then by Theorem \ref{eon},
${M}^X_{1}(\lambda) = K^{X,\alpha}(\lambda),$ is simple, and for some positive integer $c_0$, the multiplicity of $h_{\alpha} + \rho(h_{\alpha})$ in $\det F^X_\eta$ is equal to \[ c_0( {\bf p}_{Z(\alpha)}(\eta-\gamma') - {\bf p}_{Z(\alpha)}(\eta-\gamma)) .\]
Thus using Corollary \ref{fara}, the multiplicity of $h_{\alpha}$ in the leading term of $\det F^X_\eta$ is at least \[c_0( {\bf p}_{Z(\alpha)}(\eta-\gamma') - {\bf p}_{Z(\alpha)}(\eta-\gamma)) +
\sum_{r\in I, r\neq 0}c_r{\bf q}_{Y}(\eta - r \alpha)=|\operatorname{LT} \det F^X_\eta:h_\alpha| \le |D^X_\eta :h_\alpha|\] On the other hand the multiplicity of $h_\alpha$ in the leading term of $D^X_\eta$ is
\[|D_1D_3:h_\alpha|= {\bf p}_{Z(\alpha)}(\eta-\gamma') - {\bf p}_{Z(\alpha)}(\eta-\gamma) + \sum_{r\in {\mathbb Z}, r\neq 0}{\bf q}_{Y}(\eta - r \alpha).\]
Reasoning as in Case 2, we deduce that $c_r=1$ for all $r\in I,$ $c_r =0$ otherwise. So the multiplicity of
$(h_{\alpha} + (\rho, \alpha) - r(\alpha,\alpha)/2)$ is as claimed. Taken together, Cases 1-3 show that $D^X_{\eta}$ divides $\det F^X_{\eta}$. Therefore the result follows from Corollary \ref{fara}.
$\Box$ \subsection{The Jantzen sum formula. }\label{jasu} \noindent {\rm We have the following analog of the Jantzen sum formula.
\begin{theorem} \label{Jansum101} For all $\lambda \in \mathcal{H}_X$, we have
\begin{eqnarray} \label{tin} \sum_{i>0} {\operatorname{ch}\:} {M}^X_i({\lambda}) &=& \sum_{\alpha \in A(\lambda)} {\operatorname{ch}\:} {M}^{s_\alpha X} (s_\alpha \cdot \lambda)
+\sum_{\gamma \in B_X(\lambda)} {\operatorname{ch}\:} {M}^{X\cup\{\gamma \}}({\lambda}-\gamma ) \\ &+& \sum_{\alpha \in C_X(\lambda)} {\operatorname{ch}\:} {K}^{X,\alpha}({\lambda})+\sum_{[\alpha] \in E_{X}(\lambda)} {\operatorname{ch}\:} M^{W(\alpha)}(\lambda-\gamma_1-\gamma_2).\nonumber\quad \quad\end{eqnarray} \end{theorem}
\noindent \begin{proof} We have \begin{equation}\label{24}\sum_{i>0} {\operatorname{ch}\:} {M}^X_{i} (\lambda) = \sum_{i>0} \sum_{\eta} \dim {M}^{X}_{i}(\lambda)^{\lambda - \eta} \mathtt{e}^{\lambda - \eta}, \end{equation} and by \eqref{yet},
\begin{equation}\label{25} \sum_{i>0} \dim {M}^{X}_{i}(\lambda)^{\lambda - \eta} = v_{T}( \det F_\eta^X (\widetilde{\lambda})). \end{equation}
\noindent Now use the factorization of $ \det F_\eta^X $ given by Theorem \ref{shapdet}. For $\alpha \in {\Delta}^+_0$
\[ v_{T}((h_{\alpha} + \rho(h_{\alpha}) - r(\alpha,\alpha)/2) (\widetilde{\lambda})) = \left\{ \begin{array}{ll} 1 \;\;\; \mbox{if} & (\lambda + \rho, \alpha^\vee) = r \\ 0 & \;\;\;\;\; \mbox{otherwise}. \end{array} \right . \]
\noindent and for $\gamma \in B_X$
\[ v_{T}((h_{\gamma } + \rho(h_{\gamma })) (\widetilde{\lambda})) = \left\{ \begin{array}{ll} 1 \;\;\; \mbox{if} & (\lambda + \rho, \gamma ) = 0
\\ 0 & \;\;\;\;\; \mbox{otherwise.} \end{array} \right . \] We have similar expressions if $\alpha\in C_X^+$ and $(\alpha_1,\alpha_2) \in E_{X}$.
\noindent From this we obtain
\begin{eqnarray}\label{26} v_{T}( \det F_\eta^X (\widetilde{\lambda})) &=& \sum_{\alpha \in A(\lambda)} {\bf p}_{s_\alpha X}(\eta - (\lambda + \rho, \alpha^\vee) \alpha ) + \sum_{\gamma \in B_{X}(\lambda)} {\bf p}_{X\cup \{\gamma \}}(\eta - \gamma ).\nonumber\\ &+& \sum_{\alpha \in C_X(\lambda)}\sum_{\eta} ({\bf p}_{Z(\alpha)}(\eta-\gamma') - {\bf p}_{Z(\alpha)}(\eta-\gamma) )\\ &+&\sum_{[\alpha] \in E_{X}(\lambda)} {\bf p}_{W(\alpha)}(\eta-\gamma_1-\gamma_2).\nonumber \end{eqnarray}
\noindent We combine Equations (\ref{24}) - (\ref{26}) to conclude
\begin{eqnarray}
\sum_{i>0} {\operatorname{ch}\:} {M}^X_i({\lambda}) & = & \sum_{\alpha \in A(\lambda)} \sum_{\nu} {\bf p}_{s_\alpha X}(\nu)\mathtt{e}^{s_\alpha \cdot \lambda - \nu} + \sum_{\alpha \in C_X(\lambda)}\sum_{\eta} ({\bf p}_X(\eta)-{\bf p}_{s_{\alpha} X}(\eta)) \mathtt{e}^{\lambda - \eta} \nonumber\\ & + & \sum_{\gamma \in B_{X}(\lambda)} \sum_{\nu} {\bf p}_{X\cup \{\gamma \}}(\nu)\mathtt{e}^{\lambda - \gamma - \nu} +\sum_{[\alpha] \in E_{X}(\lambda)} {\bf p}_{W(\alpha)}(\eta-\gamma_1-\gamma_2)\mathtt{e}^{\lambda - \eta} \nonumber \end{eqnarray} and this easily yields \eqref{tin}. \end{proof}
\pagebreak
\appendix \begin{center} {\Large\bf Appendices} \end{center}
\section{Anti-distinguished Borel subalgebras.} \label{pip} \noindent In \cite{K} Table VI, Kac gave a particular diagram for each contragredient Lie superalgebra that we will call {\it distinguished.} \footnote{The term distinguished Borel subalgebra was later introduced by Kac to refer to the Borel subalgebras corresponding to these diagrams, \cite{Kac2} Proposition 1.5.} \footnote{We remark that there are some omissions in this table. Corrections appear in \cite{FSS} (and elsewhere).} The distinguished diagram contains at most one grey node, attached to a simple isotropic root vector. Unless $\fg = \osp(1,2n)$, $\osp(2,2n), D(2,1,\alpha)$ or $F(4)$ there is exactly one other diagram with this property. This diagram will be called {\it anti-distinguished.} The anti-distinguished diagram for $\fgl(m,n)$ is the same as the distinguished diagram for $\fgl(n,m)$. If $\fg = \osp(1,2n)$ there is only diagram, and it contains no grey node, while if $\fg = \osp(2,2n)$ the anti-distinguished diagram contains exactly two grey nodes. If $\fg = F(4)$, then apart from the distinguished diagram, there are two other diagrams with a unique grey node. We call all these diagrams {\it anti-distinguished}. \\ \\ Except in type A, the anti-distinguished diagrams are given in the table below. For $\fg=F(4)$ or $G(3)$ we follow the notation of \cite{FSS}. In the first two rows, the symbol $\begin{picture}(10,10)(50,0) \thinlines \put(55,2.8){\circle{6}} \put(52.7,.10){$\bullet$} \end{picture}$ represents a node which is either grey or white. Each diagram in these rows has a unique grey vertex. Each anti-distinguished diagram corresponds to one or more Borel subalgebras, which we also call {\it anti-distinguished}. The number of such Borels is indicated in the table. For $\fg$ of type A, the grey node in the distinguished (resp. anti-distinguished diagram) corresponds to the simple root $\epsilon_m-\delta_1$ (resp. $\delta_n-\epsilon_1$). The corresponding Borel subalgebras consist of the upper (resp. lower) triangular matrices. If $\fg=D(2,1,\alpha)$ there are three Borel subalgebras that share the same diagram as the distinguished diagram for $\osp(4,2)$. We arbitrarily declare one of these to be distinguished and another anti-distinguished. Without doing this, some results would not apply to the Lie superalgebra $D(2,1,\alpha)$.
\begin{tabular}{|c|c|c|} \hline {}
algebra &
\# Borels & anti-distinguished diagram \\ \hline & & \\ & & \\ $\begin{array}{c}
B(m,n) \\
m >0 \end{array}$ & 1& \begin{picture}(310,10)(26,0) \thinlines \put(55,3){\circle{6}}
\put(58,3){\line(1,0){46}}
\put(162,3){\line(1,0){46}}
\put(214,3){\line(1,0){46}}
\put(265.2,5){\line(1,0){46}} \put(265.2,1){\line(1,0){46}}
\put(308,-2.3){\huge $\bullet$} \put(130,3){$\ldots$} \put(211,3){\circle{6}} \put(263,3){\circle{6}} \put(208.3,0.30){$\bullet$} \put(260.6,0.30){$\bullet$} \put(52.6,.30){$\bullet$} \put(295,3){\line(-1,-1){10}}
\put(295,3){\line(-1,1){10}}
\put(335,20){} \end{picture}\\& & \\
\hline
& & \\ & & \\ $C(n)$ & 1& \begin{picture}(310,10)(26,0) \thinlines \put(55,3){\circle{6}}
\put(58,3){\line(1,0){46}}
\put(162,3){\line(1,0){46}}
\put(214,3){\line(1,0){46}} \put(130,3){$\ldots$} \put(211,3){\circle{6}} \put(263,3){\circle{6}}
\put(296.5,-19){$\otimes$} \put(296,17){$\otimes$}
\put(266.0,4){\line(2,1){31}}
\put(265,0.5){\line(2,-1){31.5}}
\put(300,-12.5){\line(0,1){29.0}}
\put(340,0){}
\end{picture}\\& & \\ & & \\
\hline
& & \\ & & \\ $D(m,1)$ & 2& \begin{picture}(310,10)(26,0) \thinlines \put(55,3){\circle{6}}
\put(58,3){\line(1,0){46}}
\put(162,3){\line(1,0){46}}
\put(214,3){\line(1,0){46}}
\put(266.7,3){\line(1,0){45}}
\put(315,3){\circle{6}} \put(130,3){$\ldots$} \put(211,3){\circle{6}} \put(259,.3){$\otimes$} \put(295,3){\line(-1,-1){10}}
\put(295,3){\line(-1,1){10}}
\put(231,3){\line(1,-1){10}}
\put(231,3){\line(1,1){10}}
\put(335,20){} \end{picture}\\& & \\
\hline & & \\ & & \\ $\begin{array}{c}
D(m,n) \\
m, n >1 \end{array}$ & 2& \begin{picture}(310,10)(26,0) \thinlines \put(55,3){\circle{6}}
\put(58,3){\line(1,0){46}}
\put(162,3){\line(1,0){46}}
\put(214,3){\line(1,0){46}}
\put(265.2,5){\line(1,0){46}} \put(265.2,1){\line(1,0){46}}
\put(314,3){\circle{6}} \put(130,3){$\ldots$} \put(211,3){\circle{6}} \put(263,3){\circle{6}} \put(208.6,0.30){$\bullet$}
\put(52.6,.30){$\bullet$}
\put(285,3){\line(1,1){10}}
\put(285,3){\line(1,-1){10}}
\put(335,20){} \end{picture}\\& & \\
\hline & & \\ & & \\ $F(4)$
& 1& \begin{picture}(310,10)(26,0) \thinlines \put(55,3){\circle{8}}
\put(59,3){\line(1,0){82}}
\put(58,0.5){\line(1,0){84}}
\put(58,5.5){\line(1,0){84}}
\put(148.4,3){\line(1,0){76}}
\put(232,5.2){\line(1,0){80}}
\put(232,.8){\line(1,0){80}} \put(315,3){\circle{8}}
\put(55,3){\circle{8}}
\put(140.7,.3){$\otimes$} \put(229,3){\circle{8}}
\put(266,3){\line(1,-1){10}} \put(266,3){\line(1,1){10}} \put(105,3){\line(-1,-1){10}}
\put(105,3){\line(-1,1){10}}
\put(335,20){} \end{picture}\\& & \\
\hline
& & \\ & & \\ $F(4)$
& 1& \begin{picture}(310,10)(26,0) \thinlines \put(55,3){\circle{8}}
\put(59,3){\line(1,0){82}}
\put(58,0.5){\line(1,0){84}}
\put(58,5.5){\line(1,0){84}}
\put(233,3){\line(1,0){78}} \put(147,5.3){\line(1,0){79}} \put(147.5,.8){\line(1,0){78}}\put(315,3){\circle{8}}
\put(55,3){\circle{8}} \put(140.7,.3){$\otimes$} \put(229,3){\circle{8}} \put(105,3){\line(-1,-1){10}}
\put(105,3){\line(-1,1){10}}
\put(186,3){\line(1,-1){10}} \put(186,3){\line(1,1){10}} \end{picture}\\& & \\
\hline
& & \\ & & \\ $G(3)$
& 1& \begin{picture}(310,10)(26,0) \thinlines
\put(99,5.2){\line(1,0){85.5}}
\put(99,0.5){\line(1,0){85.5}}
\put(190,3){\line(1,0){79}} \put(189.5,5.3){\line(1,0){80}} \put(189.5,.8){\line(1,0){80}} \put(273,3){\circle{8}} \put(89.7,-3.3){\Huge $\bullet$} \put(182.5,.3){$\otimes$}
\put(225,3){\line(1,-1){10}} \put(225,3){\line(1,1){10}} \end{picture}\\& & \\
\hline
\end{tabular}
\section{Low Dimensional Cases}\label{ldc}
This appendix serves two purposes. First it illustrates the general theory developed in the rest of the paper. Secondly and more fundamentally, for the factorization of the \v{S}apovalov determinant for $\fgl(m,n)$ we need to know the structure of the modules $M^X(\lambda)$ in certain generic cases. This is done by a reduction to the cases of $\fgl(2,1)$ and $\fgl(2,2)$.
\subsection{The Cases $\fsl(2,1)$ and $\fgl(2,1)$.}\label{C8} We suppose that $\mathfrak{g} = \fsl (2,1)$, though everything we say works with minor modifications for $\fgl(2,1)$. Let\[h = e_{1,1} - e_{2,2},\; \;z = e_{1,1} + e_{2,2} + 2e_{3,3} \; \; \mbox{and} \; \;\mathfrak{h} = \mbox{ span } \{h,z\}.\] The notation used here is the same as \cite{M} Exercises 10.5.3 and 10.5.4. We use the distinguished set of positive roots ${\Delta^+} = \{ \alpha, \beta, \gamma = \alpha + \beta \},$ with $\alpha$ even and $\beta$ odd simple roots. We define negative root vectors \[e_{-\alpha} = e_{2,1}, \; \; e_{-\beta} = e_{3,2},\;\; e_{-\gamma} = e_{3,1}.\] \noindent The \v{S}apovalov element $\theta_\gamma$ is given by
\[\theta_\gamma = e_{-\beta}e_{-\alpha} +e_{-\gamma}h.\] Note that $\fg$ has three Borel subalgebras whose even part is the standard upper triangular Borel subalgebra of $\fg_0$. We label these as $\mathfrak{b}^{(1)}$-$\mathfrak{b}^{(3)}$ where $\mathfrak{b}^{(1)}, \mathfrak{b}^{(2)}, \mathfrak{b}^{(3)}$ respectively have set of simple roots \[ \{\alpha,\beta\}, \quad \{\gamma,-\beta\}, \quad \{-\gamma,\alpha\}.\] We also consider the parabolic subalgebras \[\mathfrak{p} = \mathfrak{b}^{(1)}+\mathfrak{b}^{(2)},\quad \mathfrak{q} = \mathfrak{b}^{(2)}+ \mathfrak{b}^{(3)}.\] Note that $\fg =\mathfrak{p} \oplus \mathfrak{m} =\mathfrak{q} \oplus \mathfrak{n}$ where $\mathfrak{m}$ and $\mathfrak{n}$ are abelian subalgebras with roots $-\alpha, -\gamma$ and $-\alpha, \beta$ respectively. Thus $U(\mathfrak{m}) = \mathtt{k}[e_{-\alpha}, e_{-\gamma}]$ and $U(\mathfrak{n}) = \mathtt{k}[e_{-\alpha}, e_{\beta}]$. If $\mu\in \mathfrak{h}^*$ defines a one-dimensional $\mathfrak{p}$-module, we write ${\operatorname{Ind}}_\mathfrak{p}^\fg \;\mathtt{k}_\mu $ for the induced $\fg$-module. Similar notation is used for one-dimensional modules induced from $\mathfrak{q}.$ \\ \\ \noindent The figure below
will be used repeatedly when we study the submodule structure of Verma modules and the modules $M^X(\lambda)$ in certain cases. \[ \xymatrix{ &M\ar@{-}[d]&\\
&V_1+V_2&\\%\ar@{-}[dr]\\ V_{1} \ar@{-}[dr] \ar@{-}[ur] && V_{2} \ar@{-}[dl] \ar@{-}[ul]&\\
& V_3= V_1\cap V_2\ar@{-}[d] & \\ &0&} \] \begin{fig} \label{fig1}\end{fig}
\noindent Note that using the notation of the figure, we have in $K(\mathcal{O})$ that \begin{equation} \label{tod} [V_1+V_2] +[V_3] = [V_1] +[V_2].\end{equation}
\noindent Suppose that $\lambda = \beta+n\gamma$ where $n$ is a non-negative integer. This is the most interesting case, because $\lambda +\rho=n\gamma$ a multiple of the non-simple odd positive root $\gamma.$
\begin{theorem} \label{A2} If $\lambda = \beta+n\gamma$ with $n\ge 1$, we have \begin{itemize} \item[{{\rm(a)}}] The lattice submodules of $M={M}(\lambda)$ is as in Figure \ref{fig1}, where $V_i = U(\mathfrak{g})v_i$ for \begin{equation} \label{kid} v_1 = e_{-\alpha}^{n}v_\lambda,\quad v_2 = e_{-\gamma}e_{-\beta}v_\lambda,\quad \mbox{ and } v_3 = e_{-\beta}v_1 = e_{-\alpha}^{n-1}e_{\beta}v_2.\end{equation}
\item[{{\rm(b)}}] The module $M/V_2$ has character $\mathtt{e}^{\lambda} {p}_{\gamma}.$ Thus $M^\gamma(\lambda)=M/V_2$ is the factor module of $M$ obtained from the specialization process described in Theorem \ref{newmodgen}. \item[{{\rm(c)}}] The following are equivalent \begin{itemize} \item[{{\rm(i)}}] $n>1$ \item[{{\rm(ii)}}] $V_2 = U(\fg)\theta_\gamma v_\lambda$.
\item[{{\rm(iii)}}] $V_2$ is a highest weight module. \item[{{\rm(iv)}}] The kernel $N\cap T{M}({\widetilde{\lambda} })_{A}$ of the map from $N= M^{\gamma}({\widetilde{\lambda} -\gamma})_{A}$ to ${M}({{\lambda} })$ in \eqref{bd} equals $TN.$ \end{itemize} \end{itemize} \end{theorem} \begin{proof} For the statement about the lattice of submodules see \cite{M3} or \cite{M} Exercise 10.5.4.
We consider
highest weight modules obtained by specialization. Let ${\widetilde{\lambda}} = \lambda +T\gamma$ and consider the $U(\fg)_B$-module $M({\widetilde{\lambda}})_{B}$. We have
\begin{eqnarray} \label{hid} u &:=& \theta_\gamma v_{{\widetilde{\lambda}}} = (e_{-\beta} e_{-\alpha}+(T+n-1)e_{-\gamma} )v_{{\widetilde{\lambda}}}\\ &=& (e_{-\alpha} e_{-\beta}+(T+n)e_{-\gamma} )v_{\widetilde{\lambda}}.\nonumber \end{eqnarray} Let $\bar{v}_{\widetilde{\lambda}}$ be the image of $v_{\widetilde{\lambda}}$ modulo $U(\fg)_B u$. Since $T+k$ is invertible in $B$ for any $k$, it follows that $$U(\fg)_B u = B[e_{-\alpha},e_{-\beta}]u \;\mbox{ and }\; U(\fg)_B \bar{v}_{\widetilde{\lambda}} = B[e_{-\alpha},e_{-\beta}]\bar{v}_{\widetilde{\lambda}}.$$
Thus if $N=U(\fg)_Au$ and $N'=U(\fg)_A\bar{v}_{\widetilde{\lambda}},$
the proof of Theorem \ref{newmodgen} shows that for any $n$, $N/TN$ and $N'/TN'$ are highest weight modules with characters $\mathtt{e}^{\lambda -\gamma} {p}_{\gamma}$ and $\mathtt{e}^{\lambda} {p}_{\gamma}$ respectively. \\ \\ To prove (b) first note that $e_{-\beta}v_{{\lambda} } $ is a highest weight vector for $\mathfrak{b}^{(2)}$ which generates $M(\lambda).$ However $h_\gamma e_{-\beta}v_{\lambda} = 0$, so $e_{-\gamma} e_{-\beta}v_{{\lambda} } $ generates a proper submodule. It follows easily that \begin{equation} \label{inm}V_2\cong{\operatorname{Ind}}_\mathfrak{q}^\fg \;\mathtt{k}_{\lambda-\beta-\gamma} = U(\mathfrak{n})\mathtt{k}_{\lambda-\beta-\gamma},\mbox{ and } M/V_2\cong{\operatorname{Ind}}_\mathfrak{q}^\fg \;\mathtt{k}_{\lambda-\beta} = U(\mathfrak{n})\mathtt{k}_{\lambda-\beta},\end{equation} which yields the first statement in (b). The second statement holds since $V_2$ is the unique submodule with this character. \\ \\ Now we prove (c). Suppose first that $n>1$.
Since $h_\beta v_2 = (1-n) v_2$ it follows from \eqref{hid} that (ii) and (iii) hold, so $V_2 \cong M^\gamma(\lambda-\gamma)$. Thus $N/TN\cong M^\gamma(\lambda-\gamma)$ embeds in $M(\lambda)$, so $N\cap T{M}({\widetilde{\lambda} })_{A}=TN.$ \\ \\ Now suppose $n = 1.$ Then the submodule $V_3$ generated by $\theta_\gamma v_{{\lambda}} =v_3$ has character $\mathtt{e}^{\beta} {p}_{\beta} \neq \mathtt{e}^{\lambda -\gamma} {p}_{\gamma}.$ Since $e_{\beta}v_2\neq 0$, $V_2$ is not a highest weight module. We have shown that (ii), (iii) do not hold. Note in addition that by Equation \eqref{hid}, $e_{-\beta}uv_{{\widetilde{\lambda} }} = Te_{-\beta}e_{-\gamma}v_{\widetilde{\lambda} } \in N\cap T{M}({\widetilde{\lambda} })_{A}$. Since $e_{-\beta}e_{-\gamma}v_{\widetilde{\lambda} } \notin N$, (iv) does not hold. \end{proof} \begin{rem} {\rm The element $v_2\in M(\lambda)$ is not a highest weight vector for the distinguished Borel subalgebra. However, compare \eqref{hid},
\begin{equation} \label{suv}e_{\beta}v_2 = \theta_\gamma v_{{{\lambda}}}=(e_{-\beta} e_{-\alpha}+(n-1)e_{-\gamma} )v_{{{\lambda}}}= (e_{-\alpha} e_{-\beta}+ne_{-\gamma} )v_{{\lambda}}.\end{equation}} \end{rem} \begin{theorem} \label{AA2} Suppose that $n= 0$, that is $\lambda=\beta =-\rho$. Then \begin{itemize}
\item[{{\rm(d)}}] ${M}(\lambda)$ has a unique composition series of length 3
$${M}(\lambda) \supset V_1 = U(\mathfrak{g}) e_{-\beta} v_{{\lambda}}
\supset V_2 = U(\mathfrak{g})e_{-\alpha} e_{-\beta} v_{{\lambda}}
\supset 0,$$ and $V_1/V_2$ is isomorphic to the trivial module.
\item[{{\rm(e)}}] $M(\lambda)/V_1$ and $M(\lambda)/V_2$ have characters $\mathtt{e}^{\lambda} {p}_{\beta}$ and $\mathtt{e}^{\lambda} {p}_{\gamma}$ respectively.
\item[{{\rm(f)}}] The module $M^\gamma(\lambda)=M(\lambda)/V_2$ has a basis consisting of the images of the elements $e_{-\alpha}^mv_{ \lambda}$, $e_{-\alpha}^me_{-\gamma} v_{\lambda}$ with $m \ge 0$
and $e_{-\beta}v_{\lambda}$. \end{itemize} \end{theorem} \begin{proof} First (d) is shown in \cite{M3} or \cite{M} Exercise 10.5.4. It is easy to see that $V_1\cong{\operatorname{Ind}}_\mathfrak{p}^\fg \;\mathtt{k}_{\lambda-\beta} = U(\mathfrak{m})\mathtt{k}_{\lambda-\beta},$ and $M/V_1\cong{\operatorname{Ind}}_\mathfrak{p}^\fg \;\mathtt{k}_{\lambda} = U(\mathfrak{m})\mathtt{k}_{\lambda}.$ This gives ${\operatorname{ch}\:} M(\lambda)/V_1 = \mathtt{e}^{\lambda} {p}_{\beta}$ so by Lemma \ref{ink}, ${\operatorname{ch}\:} M(\lambda)/V_2 = \mathtt{e}^{\lambda} {p}_{\gamma}$. Finally (f) holds because in the factor module $M^{\gamma}(\lambda)_A$ of $M({\widetilde{\lambda}})_A$ we have \[e_{-\alpha}^{m+1} e_{-\beta}v_{\widetilde{\lambda}}= -Te_{-\alpha}^m e_{-\gamma} v_{\widetilde{\lambda}},\] so that $M^{\gamma}(\lambda)_A^{\widetilde{\lambda}-\gamma-m\alpha}$ has $A$-basis $e_{-\alpha}^m e_{-\gamma}v_{\widetilde{\lambda}}$. \end{proof}
\begin{rems} \label{dig}{\rm \begin{itemize} \item[{{\rm(a)}}] Many of the submodules and factor modules of $M(\lambda)$ can be constructed using induced modules and their duals. When $n=1$ in addition to \eqref{inm}, we have $V_3\cong{\operatorname{Ind}}_\mathfrak{p}^\fg \;\mathtt{k}_{s_\alpha \cdot\lambda-\beta} = U(\mathfrak{m})\mathtt{k}_{s_\alpha\cdot\lambda-\beta}$ and $V_1\cong M(s_\alpha \cdot \lambda)$. \item[{{\rm(b)}}] When $n=0$ we have $V_1\cong{\operatorname{Ind}}_\mathfrak{p}^\fg \;\mathtt{k}_{\lambda-\beta} = U(\mathfrak{m})\mathtt{k}_{\lambda-\beta}$ and $M/V_1\cong{\operatorname{Ind}}_\mathfrak{p}^\fg \;\mathtt{k}_{s_\alpha \cdot\lambda} = U(\mathfrak{m})\mathtt{k}_{s_\alpha\cdot\lambda}$ and $V_2\cong{\operatorname{Ind}}_\mathfrak{q}^\fg \;\mathtt{k}_{\lambda-\beta-\gamma} = U(\mathfrak{n})\mathtt{k}_{\lambda-\beta-\gamma}.$ \item[{{\rm(c)}}]
When $n=0$, the module $M^\gamma(\lambda)=M(\lambda)/V_2$ has a one dimensional trivial submodule This shows that unlike the case of Verma modules, the modules $M^\gamma(\lambda)$ need not have a filtration with factors which are Verma modules for $\fg_0$, compare \cite{M} Theorem 10.4.5. This example raises issues for the Jantzen filtration, see Lemma \ref{eon} and the factor $D_3$ in Theorem \ref{shapdet}. Finally we note that $$(M(\lambda)/V_2)^\vee \cong{\operatorname{Ind}}_\mathfrak{q}^\fg \;\mathtt{k}_{\lambda-\beta} = U(\mathfrak{n})\mathtt{k}_{\lambda-\beta}.$$ This is not a highest weight module for the standard Borel. \item[{{\rm(d)}}] We claim that the complex \eqref{1let} is exact at $M(\lambda)$ iff $n\neq 0,1$. First if $n=0$, it follows easily from \eqref{suv} that ${\operatorname{Im}\;} {\psi_{\lambda,\gamma}}= V_3$ and ${\operatorname{Ker}\;}{\psi_{\lambda+\gamma,\gamma}}=V_2.$ If $n=1$, then as noted in the proof of Theorem \ref{A2}, ${\operatorname{Im}\;} {\psi_{\lambda,\gamma}}= V_3$. We have $${\psi_{\lambda+\gamma,\gamma}}(v_2) =e_{-\gamma }e_{-\beta}(e_{-\beta}e_{-\alpha}+ e_{-\gamma})v_{\lambda+\gamma}=0.$$ Thus $V_2\subseteq{\operatorname{Ker}\;}{\psi_{\lambda+\gamma,\gamma}},$ and it is not hard to see that equality holds. If $n>1$, then by Theorems \ref{A2} and that ${\operatorname{Im}\;} {\psi_{\lambda,\gamma}}= V_2\subseteq {\operatorname{Ker}\;}{\psi_{\lambda+\gamma,\gamma}}.$ Equality must hold since any submodule strictly containing $V_2$ has finite codimension. Similar arguments can be used for the case $\lambda = \beta+n\gamma$ with $n$ negative. \end{itemize}} \end{rems}
\subsection{The Case of $\fgl(2,2)$.}\label{Ch8} The Jantzen sum formula has some terms which are not Verma modules. This suggests that we look at some more modules. Suppose that $X$ is a set of odd orthogonal roots, and $(\lambda + \rho, \alpha) = 0$ for all $\alpha \in X.$ We consider the modules $M^X(\lambda)$ with
character $ \mathtt{e}^{\lambda} p_X$ defined in Section \ref{jaf}.
For the remainder of this appendix, let $\mathfrak{g}$ be the Lie superalgebra $\fgl(2,2)$. Then $\mathfrak{g} = \mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}$ where $\mathfrak{n}^-, \mathfrak{h}$ and $\mathfrak{n}$ are the subalgebras of lower triangular, diagonal and upper triangular matrices. Let $$e_\alpha = e_{1,2},\;\; e_\beta = e_{2,3},\;\;e_\gamma = e_{3,4},$$ so that the simple roots of $\mathfrak{n}$ are $\alpha,\;\; \beta,\;\; \gamma.$ Also define $$e_{\alpha + \beta} = e_{1,3},\;\; e_{\beta +\gamma} = e_{2,4},\;\; e_{\alpha + \beta + \gamma} = e_{1,4}.$$ For each positive root $\eta,$ let $e_{-\eta}$ be the transpose of $e_\eta.$ Let $\mathfrak{h}^*$ have basis $\epsilon_{1}, \epsilon_{2}, \delta_{1}, \delta_{2} $ as usual with $(\epsilon_{i}, \epsilon_{j}) = - (\delta_{i}, \delta_{j}) = \delta_{i,j}.$ The inner product of the roots is given by the table \[
\begin{tabular} {|c|c|c|c|} \hline
&$\alpha$ & $\beta$ & $\gamma$\\ \hline $\alpha$ & 2& -1& 0\\ \hline
$\beta$ &-1 & 0 & 1\\ \hline $\gamma$ & 0 & 1 & -2\\ \hline
\end{tabular}
\] Let $\mathfrak{h}$ be the Cartan subalgebra with basis \[h_{\alpha} = e_{1,1} - e_{2,2},\;\;\;h_{\gamma} = e_{4,4} - e_{3,3}, \;\;\; h_{\beta} = e_{3,3} + e_{2,2}. \] Note that with these definitions $<h_\lambda,h_\mu> = (\lambda, \mu)$ where $<\;,\;>$ is the supertrace, so $h_\lambda v_\mu = (\lambda, \mu)v_\mu.$
Let \[ \mathfrak{b} = \mathfrak{b}_1 = \left[ \begin{array}{cccc} * & * & * & * \\ 0 & * & * & * \\ 0 & 0 & * & * \\ 0 & 0 & 0 & * \end{array} \right] \] be the distinguished Borel, and \[ \mathfrak{b}_2 = \left[ \begin{array}{cccc} * & * & * & * \\ 0 & * & $0$ & * \\ 0 & * & * & *\\
0 & 0 & 0 & * \end{array} \right] .\] Note that $\mathfrak{b}, \mathfrak{b}_2$ are adjacent Borels. We also consider the subalgebras \[\mathfrak{p} = \left[ \begin{array}{cccc}
* & * & * & * \\
0 & * & 0 & * \\
* & * & * & * \\
$0$ & * & 0 & * \end{array} \right] , \mathfrak{m} = \left[ \begin{array}{cccc}
$0$ & $0$ & 0 & 0 \\
* & $0$ & * & 0 \\
$0$ & $0$ & $0$ & $0$ \\
* & $0$ & * & $0$ \end{array} \right] \] and $\mathfrak{q} = \mathfrak{b}_1 + \mathfrak{b}_2$. Note that the supertrace, ${\operatorname{Str}\;}$ is given by \[{\operatorname{Str}\;} = \epsilon_1+\epsilon_2-\delta_1-\delta_2.\] We think of ${\operatorname{Str}\;}$ as a one dimensional representation of $\fg$ with kernel $\fsl(2,2)$. Let \begin{equation} \label{per} X = \{\beta, \alpha +\beta+\gamma\}, \quad Y = \{\alpha +\beta, \beta+\gamma\}.\end{equation} Each of $X,Y$ is an isotropic set of positive roots, but only $Y$ is the set of simple roots for some Borel subalgebra. Note that $s_\alpha(X) =s_\gamma(X)= Y$. When $(\lambda+\rho,\beta)=0$, $\lambda$ defines a one-dimensional $\mathfrak{q}$-module, $\mathtt{k} v_\lambda$, and we define $M^\beta(\lambda)= {\mbox{ Ind}}^{\small{\mathfrak{g}}}_{\small{\mathfrak{q}}} \;\mathtt{k} v_\lambda.$ \subsection{\v Sapovalov Elements.} \label{1s} Suppose $\lambda \in \mathfrak{h}^*$ satisfies \begin{equation} \label{ustare} h_{\alpha} v_{\lambda} = av_{\lambda}, \quad \quad h_{\beta} v_{\lambda} = 0, \quad \quad h_{\gamma} v_{\lambda} = -cv_{\lambda} \end{equation} and $$(\lambda, \alpha^\vee) = a,\;\;\; (\lambda, \gamma^\vee) = c.$$ Set $\sigma = s_\alpha \cdot \lambda, \;\tau = s_\gamma \cdot \lambda, \;\nu = s_\alpha s_\gamma \cdot \lambda.$ Then \[(\sigma+\rho, \alpha+\beta)= (\tau+\rho, \beta+\gamma)= (\nu+\rho, \alpha+\beta+\gamma)=0.\] Also \[(\sigma+\rho, \alpha^\vee) = -(a+1),\;\;\; (\sigma+\rho, \gamma^\vee) = c+1. \] \[(\tau+\rho, \alpha^\vee) = a+1,\;\;\; (\tau+\rho, \gamma^\vee) = -(c+1). \] \[(\nu+\rho, \alpha^\vee) = -(a+1),\;\;\; (\nu+\rho, \gamma^\vee) = -(c+1). \] First to find the \v Sapovalov element for the root $\alpha + \beta,$ note that $$ e_{-\alpha}^{p+1} e_{-\beta} = [e_{-\alpha}e_{-\beta}- pe_{-\alpha -\beta }] e_{-\alpha}^{p}.$$ Therefore using \eqref{121nd} with $p=(s_\alpha\cdot\lambda +\rho, \alpha^\vee) = -(a+1),$ we obtain \[\theta_{\alpha +\beta}(\sigma)= e_{-\beta}e_{-\alpha}- (a+2)e_{-\alpha -\beta} . \] So \begin{equation} \label{a+b}\theta_{\alpha +\beta}= e_{-\beta}e_{-\alpha}+ e_{-\alpha -\beta }h_\alpha. \end{equation} Similarly we have
\begin{equation} \label{c+b} \theta_{\beta+\gamma}= e_{-\beta}e_{-\gamma}+ e_{-\beta-\gamma}h_\gamma. \end{equation}
\begin{equation} \label{a+b+c} \theta_{\alpha+\beta+\gamma}= e_{-\beta}e_{-\gamma}e_{-\alpha}+ e_{-\alpha -\beta}e_{-\gamma}h_\alpha +e_{-\beta-\gamma}e_{-\alpha}h_\gamma +e_{-\alpha-\beta-\gamma}h_\alpha h_\gamma. \end{equation} This gives
\begin{lemma} With $a, c$ as in \eqref{ustare} the \v Sapovalov element for the root $\alpha +\beta +\gamma$ satisfies \[\theta_{\alpha +\beta +\gamma}(\lambda) = e_{-\alpha}e_{-\gamma}e_{-\beta} -(a+1)(c+1)e_{-\alpha -\beta-\gamma} + (a+1)e_{-\gamma}e_{-\alpha -\beta} -(c+1)e_{-\alpha}e_{-\beta -\gamma}.\]\end{lemma}
We will be especially interested in the deformed case with $a=c$. Note that $(\rho, \alpha^\vee) = (\rho, \gamma^\vee) = 1$ and $(\rho, \alpha+\beta+\gamma) = (\rho, \beta) = 0.$ Thus for ${{\lambda}\in\mathcal{H}_X }$ we can take ${\widetilde{\lambda} }=\lambda+T\rho.$ Then when $a=c$, we have \[\theta_{\alpha +\beta +\gamma}(\widetilde{\lambda}) = e_{-\alpha}e_{-\gamma}e_{-\beta} -(a+T+1)^2e_{-\alpha -\beta-\gamma} + (a+T+1)e_{-\gamma}e_{-\alpha -\beta} -(a+T+1)e_{-\alpha}e_{-\beta -\gamma}.\]
\noindent In the deformed setting when $e_{- \beta}v_{{\widetilde{\lambda} } }=0$, and we can always cancel the factor $a+T+1$ in $M^\beta({{\widetilde{\lambda}}})_B$. Hence \begin{eqnarray} \label{ing}\widetilde{w}:&=&[(a+T+1)e_{-\alpha -\beta-\gamma} - e_{-\gamma}e_{-\alpha -\beta} +e_{-\alpha}e_{-\beta -\gamma}] v_{\widetilde{\lambda}}.\\ &=& [(a+T)e_{-\alpha -\beta-\gamma} - e_{-\gamma}e_{-\alpha -\beta }+e_{-\beta -\gamma}e_{-\alpha}]v_{\widetilde{\lambda}} \nonumber \\ &=& [(a+T-1)e_{-\alpha -\beta-\gamma} - e_{-\alpha -\beta }e_{-\gamma}+e_{-\beta -\gamma}e_{-\alpha}]v_{\widetilde{\lambda}} \in U(\fg)_B \theta_{\alpha +\beta +\gamma} v_{{\widetilde{\lambda}}}\nonumber
\nonumber \end{eqnarray}
\subsection{Change of Borel.}\label{cob}
In the table below we define 6 Borel subalgebras of $\fg$ by listing the corresponding sets of simple roots. Adjacent entries in the table correspond to adjacent Borel subalgebras.
\[
\begin{tabular} {|c|c|c|} \hline
$\mathfrak{b}^{(1)} \;\; \{\alpha, \beta, \gamma\}$& &\\ \hline
$\mathfrak{b}^{(2)} \;\; \{ \alpha + \beta, -\beta, \beta + \gamma \}$ &
$\mathfrak{b}^{(3)} \;\; \{ -\alpha -\beta, \alpha, \beta + \gamma\}$& \\ \hline
$\mathfrak{b}^{(4)} \;\; \{\alpha + \beta , \gamma, -\beta - \gamma\}$ &
$\mathfrak{b}^{(5)} \;\; \{
-\alpha - \beta, \alpha + \beta+ \gamma,- \beta -\gamma\}$&
$\mathfrak{b}^{(6)} \;\; \{\gamma, -\alpha - \beta -\gamma, \alpha\}$\\
\hline
\end{tabular}
\] \noindent Next let \begin{equation} \label{fff} f_1 = e_{-\alpha - \beta},\quad f_2 = e_{- \beta - \gamma},\quad f_3 = e_{-\alpha - \beta - \gamma}.\end{equation} The negatives of the weights of these elements are \[\sigma_1 = {\alpha + \beta},\; \sigma_2 = { \beta + \gamma}, \; \sigma_3 = {\alpha + \beta+ \gamma}.\]
Suppose that $(\lambda+\rho,\beta)= (\lambda+\rho,\alpha+\beta+\gamma)=0$, and let $v_\lambda$ be a highest weight vector in $M= M^\beta(\lambda)$ or $M^X(\lambda)$ with weight $\lambda$. We list some highest weight vectors in $M$ for various Borel subalgebras. These elements will be used later to analyze the structure of the modules $M^\beta(\lambda)$ and $M^X(\lambda)$. Starting from $\mathfrak{b}^{(1)}$ and $ \mathfrak{b}^{(2)}$ we perform odd reflections, until we arrive at a highest weight vector $v_\kappa$ for $\mathfrak{b}^{(5)}.$ It turns out that $v_\kappa$ is
a highest weight vector for $ \mathfrak{b}^{(6)}$ in $M^\beta(\lambda)$, and also a $\fg$ highest weight vector which maps to zero in $M^X(\lambda)$, see Theorem \ref{gnn}. Working now only in $M^\beta(\lambda)$
we reverse the process until we return to a highest weight vector $w$ for $\mathfrak{b}^{(1)}$ and $ \mathfrak{b}^{(2)}$. The element $w$ arises from a \v Sapovalov element, see \eqref{ing} and Lemma \ref{w}. The elements in the table below are highest weight vectors for $\fg_0$ since $\mathfrak{b}^{(1)}, \ldots,\mathfrak{b}^{(6)}$ all have the same even part. These elements will be used to study the decomposition $M^\beta(\lambda)$ and $M^X(\lambda)$ as $\fg_0$-modules.
\[ \begin{tabular}{|c|c|c|c|c|c|c|} \hline Highest weight vector& Borel subalgebra &Weight \\ \hline $v_\lambda$& $\mathfrak{b}^{(1)}, \mathfrak{b}^{(2)}$&
$\lambda$ \\ \hline $e_{-\alpha - \beta}v_\lambda$& $\mathfrak{b}^{(3)}$& $\lambda - \sigma_1$ \\ \hline $e_{- \beta - \gamma}v_\lambda$& $\mathfrak{b}^{(4)}$& $\lambda - \sigma_2$ \\ \hline $e_{-\alpha - \beta}e_{-\beta-\gamma}v_\lambda$& $\mathfrak{b}^{(5)}$& $\lambda - \sigma_1 - \sigma_2$ \\ \hline
$v_\kappa = e_{-\alpha - \beta - \gamma}e_{-\beta-\gamma}e_{-\alpha - \beta}v_\lambda$& $\mathfrak{b}^{(5)}$ and $\mathfrak{b}^{(6)}$& $\lambda - \sigma_1 - \sigma_2 -\sigma_3$
\\ \hline
$x = e_{\alpha + \beta}v_\kappa$& $\mathfrak{b}^{(4)}$& $\lambda - \sigma_2 -\sigma_3$
\\ \hline $y = -e_{\beta+\gamma}v_\kappa$& $\mathfrak{b}^{(3)}$& $\lambda - \sigma_1 -\sigma_3$ \\ \hline
$e_{\beta+\gamma}e_{\alpha + \beta }v_\kappa$& $\mathfrak{b}^{(2)}$ and $\mathfrak{b}^{(1)}$& $\lambda - \sigma_3,$ see Lemma \ref{quack} \\ \hline $w$& $\mathfrak{b}^{(1)}, \mathfrak{b}^{(2)}$& $\lambda-\sigma_3$ \\ \hline
\end{tabular}
\]
\subsection{General Computations in $M^\beta(\lambda)$.}\label{cms}
Suppose $\lambda = a(\alpha+\beta+\gamma).$\footnote{ We could consider more generally $\mu = a(\alpha+\beta+\gamma)+b\beta,$ but the analysis is very similar. This is because \[M^Y(\lambda) \otimes {\operatorname{Str}\;}^{\otimes c} \cong M^Y(\lambda + c(\alpha+2\beta+\gamma) ).\] }At first there is no condition on $a$. We will construct $M^X(\lambda)$ as a factor module of $M^\beta(\lambda)$. Before doing this it will help to introduce some special elements of $M^\beta(\lambda)$. Since $(\lambda, \alpha + \beta + \gamma) = (\lambda, \beta) = 0,$ that is $(\lambda,X) = 0,$ $\lambda$ defines a one dimensional $\mathfrak{q}$-module $\mathtt{k} v_{\lambda}.$ Now \eqref{ustare} becomes \begin{equation} \label{sstar} h_{\alpha} v_{\lambda} = av_{\lambda}, \quad \quad h_{\beta} v_{\lambda} = 0, \quad \quad h_{\gamma} v_{\lambda} = -av_{\lambda}.\end{equation} Also we have $$(\lambda, \alpha^\vee) = (\lambda, \gamma^\vee)=a.$$
Let $M = M^\beta(\lambda)$ be the induced $\fg$-module: $M={\operatorname{Ind}}^\fg_\mathfrak{q} \;\mathtt{k}_\lambda$ and \begin{equation}\label{vk}v_\kappa = e_{-\alpha-\beta-\gamma}e_{-\beta-\gamma }
e_{-\alpha-\beta }v_\lambda.\end{equation} The module $M$ has character $\mathtt{e}^{\lambda}p_\beta.$ \begin{lemma} \label{w} The element \begin{eqnarray} \label{ww} w &=& [(a+1)e_{-\alpha -\beta-\gamma} - e_{-\gamma}e_{-\alpha -\beta } + e_{-\alpha}e_{-\beta -\gamma}] v_\lambda \nonumber\\ &=& [ae_{-\alpha -\beta-\gamma} - e_{-\gamma}e_{-\alpha -\beta } +e_{-\beta -\gamma}e_{-\alpha}]v_\lambda\\ &=& [(a-1)e_{-\alpha -\beta-\gamma} - e_{-\alpha -\beta }e_{-\gamma}+e_{-\beta -\gamma}e_{-\alpha}]v_\lambda \nonumber \end{eqnarray}
is a highest weight vector for $\fg,$ and \[w = [e_{-\alpha -\beta-\gamma}(h_\alpha+1) - e_{-\gamma}e_{-\alpha -\beta } + e_{-\alpha}e_{-\beta -\gamma}] v_\lambda \] \end{lemma} \begin{proof} This follows from \eqref{ing} or a direct computation.\end{proof} \noindent Next using (\ref{sstar}) \begin{eqnarray} \label{dog} x =e_{\alpha + \beta }v_\kappa &=& e_{-\gamma} e_{-\alpha-\beta } e_{-\beta-\gamma }v_\lambda - ae_{-\alpha-\beta-\gamma} e_{-\beta-\gamma }v_\lambda, \nonumber\\ &=& e_{-\alpha-\beta } e_{-\beta-\gamma }e_{-\gamma} v_\lambda - (a-1)e_{-\alpha-\beta-\gamma} e_{-\beta-\gamma }v_\lambda \end{eqnarray} and \begin{eqnarray} \label{cat} y = -e_{\beta+\gamma }v_\kappa &=& e_{-\alpha} e_{-\alpha-\beta } e_{-\beta-\gamma }v_\lambda + ae_{-\alpha-\beta-\gamma} e_{-\alpha-\beta}v_\lambda\nonumber \\ &=& e_{-\alpha-\beta } e_{-\beta-\gamma }e_{-\alpha} + (a-1)e_{-\alpha-\beta-\gamma} e_{-\alpha-\beta}v_\lambda. \end{eqnarray}
\begin{rem} {\rm In the deformed case, when $a=0$ and $(T\xi,\alpha^\vee) = T$, we obtain from \eqref{ing} that modulo $U(\fg)_B \theta_{\alpha +\beta +\gamma} v_{{\widetilde{\lambda}}}$ \begin{eqnarray}\label{rat2} Te_{-\alpha -\beta-\gamma}v_{\widetilde{\lambda}} &\equiv& [e_{-\gamma}e_{-\alpha -\beta }-e_{-\beta -\gamma}e_{-\alpha}]v_{\widetilde{\lambda}}\\ &\equiv & [e_{-\alpha -\beta }e_{-\gamma}-e_{-\alpha}e_{-\beta -\gamma}]v_{\widetilde{\lambda}}\nonumber\end{eqnarray} Equivalently \begin{equation}\label{rat3} (T+1)e_{-\alpha -\beta-\gamma} v_{\widetilde{\lambda}}
\equiv [e_{-\alpha -\beta }e_{-\gamma} -e_{-\beta -\gamma}e_{-\alpha}]v_{\widetilde{\lambda}}\end{equation}
Similarly when $a=0$ we obtain from \eqref{dog} and \eqref{cat} that
\begin{equation} \label{dog1} e_{-\gamma} e_{-\alpha-\beta } e_{-\beta-\gamma }v_{\widetilde{\lambda}} \equiv Te_{-\alpha-\beta-\gamma} e_{-\beta-\gamma }v_{\widetilde{\lambda}} \end{equation} and
\begin{equation} \label{cat1} e_{-\alpha} e_{-\beta-\gamma } e_{-\alpha-\beta}v_{\widetilde{\lambda}} \equiv T e_{-\alpha-\beta} e_{-\alpha-\beta-\gamma}v_{\widetilde{\lambda}}.\end{equation} For a generalization of \eqref{rat3} see \eqref{mice2}. These congruences are important when we compare the determinants of the two bilinear forms $F^X_\eta$ and $G^X_\eta$ in Subsection \ref{com}. (Note that we can obtain \eqref{dog1} and \eqref{cat1} by multiplying \eqref{rat2} by $e_{-\beta -\gamma}$ and $e_{-\alpha -\beta }$ respectively.)}\end{rem} \begin{lemma} \label{wot} We have \begin{equation} \label{kwik} e_{-\alpha-\beta }w = y \end{equation} \begin{equation} \label{quick} e_{-\beta-\gamma }w = x\end{equation} \begin{equation} \label{kwak} e_{-\alpha-\beta }x = \pm (a-1) v_\kappa \end{equation} \begin{equation} \label{quock} e_{-\beta-\gamma }y = \pm (a-1) v_\kappa.\end{equation} Therefore if $a\neq 1$, $v_\kappa\in U(\fg)w$.\end{lemma} \begin{proof} To prove \eqref{kwik} note that by \eqref{ww} and \eqref{cat}, \[e_{-\alpha-\beta }w = f_1[(1-a)f_3+f_2e_{-\alpha}]v_\lambda=y.\] The proof of \eqref{quick} is similar. The proofs of \eqref{kwak} and \eqref{quock} are easier. When \eqref{dog} is multiplied by $e_{-\alpha - \beta}$ one of the terms becomes zero and the other a multiple of $v_\kappa$, giving \eqref{kwak}. The last statement follows immediately. \end{proof} \begin{corollary} \label{II} For any $a$ we have $x, y\in U(\fg)w \cap U(\fg)v_\kappa$. \end{corollary} \begin{proof} This follows from \eqref{dog}, \eqref{cat}, \eqref{kwik} and \eqref{quick}. \end{proof} \noindent Next set \begin{equation} \label{k11} z = e_{-\gamma}f_3f_1v_\lambda -e_{-\alpha} f_3f_2v_\lambda.\end{equation} \begin{lemma} \label{yamo} We have \begin{itemize} \item[{{\rm(a)}}] $e_\beta v_\kappa = \pm z.$ \item[{{\rm(b)}}] $ e_{\alpha}z = \pm x, \;\; e_{\gamma}z = \pm y.$ \item[{{\rm(c)}}] $ e_{\beta}z = 0.$ \item[{{\rm(d)}}] $e_{-\beta} e_{-\alpha}v_\lambda = f_1 v_\lambda$, and $e_{-\beta} e_{-\gamma}v_\lambda = -f_2 v_\lambda$ \end{itemize} \end{lemma} \begin{proof} For (a) we note that \[e_\beta v_\kappa = \pm f_3[e_\beta, f_1f_2]v_\lambda = \pm f_3(e_{-\alpha}f_2 - e_{-\gamma}f_1)v_\lambda = \pm z.\] Then (c) follows from (a), (b) is an easy verification, and (d) follows since $e_{-\beta}v_\lambda= 0$. \end{proof}
\begin{lemma} \label{quack} \begin{equation}\label{zin} e_{\beta+\gamma }x = (1-a)w.\end{equation} \end{lemma} \begin{proof} First note that
$$e_{\beta+\gamma }e_{-\alpha-\beta } e_{-\beta-\gamma}v_\lambda = -e_{-\alpha - \beta}h_{\beta+\gamma}v_\lambda = ae_{-\alpha-\beta}v_\lambda.$$ Therefore $$e_{\beta+\gamma }e_{-\gamma} e_{-\alpha-\beta } e_{-\beta-\gamma }v_\lambda = (e_{-\gamma}e_{\beta+\gamma } +e_{\beta})e_{-\alpha - \beta}e_{-\beta-\gamma}v_\lambda.$$ $$=[(a-1)e_{-\gamma} e_{-\alpha-\beta } + e_{-\alpha} e_{-\beta-\gamma} + e_{-\alpha-\beta-\gamma}]v_\lambda.$$ and $$e_{\beta+\gamma }e_{-\alpha-\beta-\gamma} e_{-\beta-\gamma} v_\lambda = [ae_{-\alpha-\beta-\gamma} +e_{-\alpha} e_{-\beta-\gamma}]v_\lambda \;\;\quad \mbox{using (\ref{sstar})}.$$ Combining these equations gives $$e_{\beta+\gamma }x = e_{\beta+\gamma}e_{\alpha + \beta }e_{-\alpha-\beta-\gamma}e_{-\alpha- \beta}e_{-\beta-\gamma }v_\lambda$$ $$= [(a-1)e_{-\gamma} e_{-\alpha-\beta } + e_{-\alpha} e_{-\beta-\gamma} + e_{-\alpha-\beta-\gamma}]v_\lambda $$ $$-a[ae_{-\alpha-\beta-\gamma} +e_{-\alpha} e_{-\beta-\gamma}]v_\lambda $$ $$=[(1-a^2)e_{-\alpha-\beta-\gamma} +(a-1)e_{-\gamma} e_{-\alpha-\beta } +(1 - a)e_{-\alpha} e_{-\beta-\gamma} ]v_\lambda.$$ $$ = (1-a)w.$$\end{proof} \begin{corollary} \label{san} $e_{\alpha + \beta }y = (a-1)w.$ \end{corollary} \begin{proof} Immediate from \eqref{dog}, \eqref{cat} and Lemma \ref{quack}. \end{proof} \noindent \begin{corollary} \label{cor33} If $a \neq -1,$ then $$U(\mathfrak{g}_0)w \; \oplus \; U(\mathfrak{g}_0) f_1v_\lambda \; \oplus \; U(\mathfrak{g}_0)f_2v_\lambda = U(\mathfrak{g}_0)f_3v_\lambda
\; \oplus \; U(\mathfrak{g}_0) f_1v_\lambda \; \oplus \; U(\mathfrak{g}_0)f_2v_\lambda. $$ The LHS is a direct sum of $U(\fg_0)$ highest weight modules. \end{corollary} \begin{proof} By Lemma \ref{w}, $w=[(a+1)f_3- e_{-\gamma}f_1 + e_{-\alpha}f_2] v_\lambda $ and this gives the first statement. The second follows from Subsection \ref{cob}.\end{proof} \begin{corollary} \label{Ian} If $a\neq 1$ then $w, x, y$ and $v_\kappa$ all generate the same submodule of $M(a).$ \end{corollary} \begin{proof} This follows from Lemmas \ref{wot}, \ref{quack}, \eqref{dog} and \eqref{cat}. \end{proof}
\begin{lemma} \label{l35} Suppose that $a \neq 0.$ Then we have \begin{itemize} \item[{{\rm(a)}}] \[ U(\mathfrak{g}_0) f_2f_1v_\lambda \; \oplus \; U(\mathfrak{g}_0) x
= U(\mathfrak{g}_0) f_2f_1v_\lambda \; \oplus \; U(\mathfrak{g}_0) f_3f_2v_\lambda, \] \item[{{\rm(b)}}] \[ U(\mathfrak{g}_0) f_2f_1v_\lambda \; \oplus \; U(\mathfrak{g}_0) y
= U(\mathfrak{g}_0) f_2f_1v_\lambda \; \oplus \; U(\mathfrak{g}_0) f_3f_1v_\lambda. \] \item[{{\rm(c)}}] The left sides are direct sums of $\fg_0$ highest weight modules. \end{itemize} \end{lemma} \begin{proof} (a) follows immediately from \eqref{quick}, and (b) is proved in a similar
way. Note that $x, y$ are $\fg_0$ highest weight vectors since they appear in the second table in Subsection \ref{cob}. \end{proof} \begin{lemma} \label{r} If $a=1$, then $e_\beta v_\kappa$ is a non-zero multiple of $e_{-\alpha}e_{-\gamma}e_{-\beta}w.$ \end{lemma} \begin{proof} Note that $$v_\kappa = e_{-(\alpha + \beta + \gamma)}e_{-(\gamma + \beta) }e_{-(\alpha + \beta) }v_\lambda.$$ An easy computation shows that $e_\beta v_\kappa $ is a non-zero multiple of $e_{-\alpha -\beta-\gamma}w.$
If $a=1$ then $w$ has weight zero as a weight vector for $\fsl(2)\ti\fsl(2)$. Hence $e_{-\alpha} w = e_{-\gamma}w =0$. Another computation shows that $e_{-\alpha}e_{-\gamma}e_{-\beta}w$ is also a non-zero multiple of $e_{-\alpha -\beta-\gamma}w.$ \end{proof} \noindent For the convenience of the reader, we record that the elements $v_\kappa, w, x, y, z$ are defined in this Section in Equations \eqref{vk}, \eqref{ww}, \eqref{dog}, \eqref{cat}, \eqref{k11} respectively. All these elements except for $z$ are $\fg_0$ highest weight vectors that also appear in the table at the end of Section \ref{cob}. For the elements $f_1, f_2, f_3$ see \eqref{fff}. \subsection{$M^X(\lambda)$ as a factor module of $M^\beta(\lambda)$.} \label{3}
We assume throughout this section that $\lambda \in\mathcal{H}_X$, and $(\lambda,\alpha^\vee)=a.$ \noindent Let $N=N^X(\lambda)$ be the kernel of the natural map from $M^\beta(\lambda)$ onto $M^X(\lambda),$ and let $\mathcal{N}^X(\lambda)$ be the submodule of $M^\beta(\lambda)$ generated by $w$ and $v_\kappa$. The goal of this section is to show that $N^X(\lambda)=\mathcal{N}^X(\lambda)$.
\begin{theorem} \label{gnn} The submodule $N^X(\lambda)$ of $M^\beta(\lambda)$ is equal to
$\mathcal{N}^X(\lambda)$, and we have $x, y \in \mathcal{N}^X(\lambda)$. Furthermore $N^X(\lambda)$ is a highest weight module iff $a\neq 1$. \end{theorem} \begin{proof} Recall the element ${\widetilde{w}} \in U(\fg)_B \theta_{\alpha +\beta +\gamma} v_{{\widetilde{\lambda}}}$ from \eqref{ing}.
After setting $T=0$, ${\widetilde{w}} $ specializes to $w$ as in \eqref{ww}. Hence $w \in N^X(\lambda)$. By \eqref{kwik} and \eqref{quick} $x, y \in N^X(\lambda),$ and if $a\neq 1$ we have $v_\kappa \in N^X(\lambda)$ by \eqref{kwak} or \eqref{quock}, so by Lemma \ref{w} $N^X(\lambda)$ is a highest weight module generated by $w$. Now suppose that $a= 1$. Then
\begin{equation} \label{ing1} e_{-\alpha -\beta}e_{-\beta -\gamma}{\widetilde{w}} =\pm Tv_\kappa \in U(\fg)_B \theta_{\alpha +\beta +\gamma} v_{{\widetilde{\lambda}}}.\end{equation} Hence $v_\kappa \in U(\fg)_B \theta_{\alpha +\beta +\gamma} v_{{\widetilde{\lambda}}}$, so $\mathcal{N}^X(\lambda) \subseteq N^X(\lambda).$
To complete the proof it suffices to show that the factor module $ M^\beta(\lambda)/\mathcal{N}^X(\lambda)$ has character $\mathtt{e}^{\lambda}p_X.$ This is done in Theorems \ref{3.7}, \ref{3.10} and \ref{3.11}. \end{proof} \noindent For the remainder of this section we fix $\lambda \in \mathcal{H}_X$ and suppose $a$ is as in \eqref{sstar}.
\begin{theorem} \label{3.7} Suppose $a \neq 0, -1$ then \begin{itemize} \item[{{\rm(a)}}] $M^\beta(\lambda)$ is a direct sum of 8 Vermas for $\fg_0.$ The highest weight vectors are $$v_\lambda, \; f_1v_\lambda, \; f_2v_\lambda, \; f_2f_1v_\lambda, \; v_\kappa =f_3f_2f_1v_\lambda, \; x, \; y, \; w.$$ \item[{{\rm(b)}}] The factor module $ M^\beta(\lambda)/\mathcal{N}^X(\lambda) $ is the direct sum of the 4 Vermas for $\fg_0$ with highest weight vectors \begin{equation} \label{ext}v_\lambda, \; f_1v_\lambda, \; f_2v_\lambda, \; f_2f_1v_\lambda.\end{equation} \item[{{\rm(c)}}] We have ${\operatorname{ch}\:} \mathcal{N}^X(\lambda)=\mathtt{e}^{\lambda - \sigma_3}p_X$ and ${\operatorname{ch}\:} M^\beta(\lambda)/\mathcal{N}^X(\lambda) =\mathtt{e}^{\lambda}p_X.$ Thus $N^X(\lambda)=\mathcal{N}^X(\lambda)$. \end{itemize} \end{theorem} \begin{proof} (a) is immediate from Corollary \ref{cor33} and Lemma \ref{l35}. For (b) note that $\mathcal{N}^X(\lambda)= U(\fg) w$ contains the $U(\fg_0)$ highest weight vectors $x,y$ and $v_\kappa$ by Lemma \ref{wot}.
The statement follows since $w$ is a highest weight vector for $\fg$. Finally (c) is a computation based on (b). \end{proof}
\subsubsection{The case $a=0$.} \label{A=0} \begin{theorem} \label{3.10} For the case $a = 0,$ we have as $\fg_0$-modules \begin{itemize} \item[{{\rm(a)}}] $M^\beta(\lambda) = U(\mathfrak{g}_0) v_\lambda \; \oplus \; U(\mathfrak{g}_0) f_1v_\lambda \; \oplus \; U(\mathfrak{g}_0)f_2v_\lambda \; \oplus \;U(\mathfrak{g}_0)w \; \oplus \; U(\mathfrak{g}_0) v_\kappa \; \oplus \; I $ where $I$ is an indecomposable with Verma socle $U(\mathfrak{g}_0)f_1 f_2v_\lambda$ and highest weight vectors $f_1f_3v_\lambda$ and $f_2f_3v_\lambda$ in the top.
\item[{{\rm(b)}}] We have $$\mathcal{N}^X(\lambda) = U(\mathfrak{g}_0)w
\; \oplus \; U(\mathfrak{g}_0) v_\kappa \; \oplus \;J$$ where $J$
is an indecomposable $U(\mathfrak{g}_0)$-module containing
$J'=(U(\mathfrak{g}_0)x + U(\mathfrak{g}_0)y)$ and with $J/J'=U(\mathfrak{g}_0)z$, a Verma module with highest weight $\lambda-2(\alpha+\beta+\gamma)$. \item[{{\rm(c)}}] The character of $\mathcal{N}^X(\lambda)$ is \[ \mathtt{e}^{\lambda - \sigma_3}(1 + \mathtt{e}^{-\alpha - \beta})(1 + \mathtt{e}^{-\beta-\gamma})/ \prod_{\sigma \in \Delta^+_{0}} (1 - \mathtt{e}^{- \sigma}) = \mathtt{e}^{\lambda - \sigma_3}p_X.\] Thus $N^X(\lambda)=\mathcal{N}^X(\lambda)$. \end{itemize} \end{theorem} \begin{proof} (a) Note that \begin{equation} \label{k10} e_{\gamma}f_3f_2v_\lambda =f_1f_2v_\lambda \mbox{ and } e_{\alpha}f_3f_1v_\lambda = \pm f_1f_2v_\lambda.\end{equation} For example the first equation holds since $e_{- \beta}v_\lambda = 0,$ and $$[e_{\gamma} , e_{-\alpha -\beta - \gamma}e_{-\beta-\gamma}] = e_{-\alpha - \beta}e_{-\beta-\gamma} - e_{-\alpha- \beta - \gamma}e_{- \beta}.$$ In addition it is easy to see that \[e_{\alpha}f_3f_2v_\lambda = e_{\gamma}f_3f_1v_\lambda = 0.\] Now the first five summands in the expression for $M^\beta(\lambda)$ are generated by $\fg_0$-highest weight vectors, and their sum $K$ is direct. Thus $M^\beta(\lambda)/K$ has the same character as the direct sum of $\fg_0$ Vermas with highest weights $\lambda- \sigma_1-\sigma_2$, $\lambda- \sigma_1-\sigma_3$ and $\lambda- \sigma_2 -\sigma_3$. The claim follows. \\ \\ (b) By Lemmas \ref{wot} and Lemma \ref{yamo} (a) all summands on the right side are contained in the left. Also the sum is direct by comparing weights. (Note that the weights of $v_\kappa$ involve $\beta$ three times, $x, y, z$ twice and $w$ only once). Lemma \ref{yamo} also implies that $z$ is a highest weight vector mod $J'$. The weight of $z$ is antidominant, so $z$ generates a $\fg_0$ simple Verma module. \\ \\ (c) If $a = 0,$ then $$x = e_{-\gamma} e_{-\alpha-\beta } e_{-\beta-\gamma }v_\lambda,$$ and $$y = \pm e_{-\alpha} e_{-\alpha-\beta } e_{-\beta-\gamma }v_\lambda.$$
Hence $U(\mathfrak{g}_0)x \cap U(\mathfrak{g}_0)y = U(\mathfrak{g}_0)e_{-\alpha} e_{-\gamma}f_1f_2v_\lambda.$
Therefore $U(\mathfrak{g}_0)x + U(\mathfrak{g}_0)y$ has character \begin{equation} \label{badc} \mathtt{e}^{\lambda - \sigma_3}( \mathtt{e}^{-\sigma_1} + \mathtt{e}^{-\sigma_2} - \mathtt{e}^{-\sigma_3})/ \prod_{\sigma \in \Delta^+_{0}}(1 - \mathtt{e}^{- \sigma}). \end{equation} The result follows since $z$ has weight $\lambda - 2\sigma_3,$ $v_\kappa $ has weight $\lambda - \sigma_1 - \sigma_2 - \sigma_3,$ and $w$ has weight $\lambda - \sigma_3.$\end{proof} \noindent When $a=0$ the elements $z, x, y$ from Subsection \ref{cms} can be written as follows. \begin{equation} \label{fem} z= e_{-\gamma}f_3 f_1v_\lambda - e_{-\alpha} f_3 f_2v_\lambda, \quad x =e_{-\gamma}f_1 f_2v_\lambda, \quad y= \pm e_{-\alpha} f_1 f_2v_\lambda.\end{equation} It follows that $J\subset I$ in Theorem \ref{3.10}. Also \[e_{-\gamma}y = \pm e_{-\alpha} x\] The table below gives a complete list of $\fg_0$-singular vectors in $M/N$. In order to show the Weyl group symmetry, for each $\fg_0$-singular vector we list the shifted weight (i.e. the $\rho$-shifted weight) of $s$ as the ordered pair \begin{equation} \label{rot}(({\operatorname{wt}} \; s +\rho, \alpha^\vee), ({\operatorname{wt}} \;s +\rho, \gamma^\vee))\end{equation} This makes evident the singular vectors with the same central character. \[
\begin{tabular} {|c|c|} \hline $\fg_0$-singular vector $s$&
shifted weight of $s$\\ \hline $v_\lambda$ & $(1,1)$\\ \hline $f_1f_2v_\lambda$ & $(-1,-1)$\\ \hline $f_3v_\lambda$ &$(0,0)$\\ \hline
$f_1v_\lambda$ & $(0, 2)$\\ \hline $f_2v_\lambda$ &$(2,0)$\\ \hline $f_1f_2f_3v_\lambda$ &$(0,0)$\\ \hline
$f_1f_3v_\lambda$ & $(-1, 1)$\\ \hline $f_2f_3v_\lambda$ &$(1,-1)$\\ \hline $x$ & $(1,-1)$\\ \hline $y$ & $(-1,1)$\\ \hline $z$ &$(-1,-1)$\\ \hline $v_\kappa$& $(0,0)$\\ \hline
\end{tabular}
\] The highest weight vectors of the Verma composition factors of $I$ and their weights for $[\fg_0,\fg_0] \cong \fsl(2)\ti\fsl(2)$ are displayed in the diagram below, see \eqref{k10}. We write $f_1f_2$ in place of $ f_1f_2 v_\lambda $ etc.
\begin{displaymath}
\xymatrix{ & f_1f_3 \;\; (-1,1)&& f_2f_3 \;\;(1,-1)&\\ && f_1 f_2 \;\;(1, 1)\ar@{-}[ul] \ar@{-}[ur]&&
} \end{displaymath} The weights of the elements $x, y, z \in J$ are displayed in the diagram below. We note that as a $\fg_0$-module, $J$ does not have a Verma flag see \eqref{badc}, and also Corollary \ref{bdc}. \begin{displaymath}
\xymatrix{ &&z\;\; (-1,-1)&&\\ & x \;\; (1, -1)\ar@{-}[ur]&& y \;\; (-1,1)\ar@{-}[ul]
} \end{displaymath} \subsubsection{ The case $a=-1$.}\label{3.111} When $a=-1$, $(\lambda+\rho,\alpha^\vee)=0$, and as we shall see $M^X(\lambda)$ is simple.
\begin{theorem} \label{3.11} For the case $a = -1,$ we have as $\fg_0$-modules \begin{itemize} \item[{{\rm(a)}}] $M^\beta(\lambda) = U(\mathfrak{g}_0) v_\lambda \; \oplus \; U(\mathfrak{g}_0)x \; \oplus \; U(\mathfrak{g}_0)y\; \oplus \; U(\mathfrak{g}_0)f_2f_1v_\lambda \;\oplus \; U(\mathfrak{g}_0) v_\kappa
\; \oplus \; I$ where $I$ is an indecomposable having Verma socle $\mathcal{V}= U(\mathfrak{g}_0)f_1 v_\lambda \oplus U(\mathfrak{g}_0)f_2v_\lambda$ and such that the image of $f_3v_\lambda$ is a highest weight vector which generates the $\fg_0$-Verma module $I/\mathcal{V}$. \item[{{\rm(b)}}] We have \begin{equation} \label{wvk}\mathcal{N}^X(\lambda) = U(\mathfrak{g}_0)x \;\oplus \; U(\mathfrak{g}_0)y \; \oplus \;
U(\mathfrak{g}_0)w
\; \oplus \; U(\mathfrak{g}_0) v_\kappa, \end{equation} where $U(\mathfrak{g}_0)w \subset \mathcal{V}$. \item[{{\rm(c)}}]
The character of $\mathcal{N}^X(\lambda) $ is \[ \mathtt{e}^{\lambda - \sigma_3}(1 + \mathtt{e}^{-\alpha - \beta})(1 + \mathtt{e}^{-\beta-\gamma})/ \prod_{\sigma \in \Delta^+_{0}} (1 - \mathtt{e}^{- \sigma}) =\mathtt{e}^{\lambda - \sigma_3}p_X.\] Hence $\mathcal{N}^X(\lambda) = N^X(\lambda)$. \item[{{\rm(d)}}] Set $M=M^\beta(\lambda)$, $N= N^X(\lambda)$, $u_1=e_{-\gamma}f_1v_\lambda$ and $u_2 =e_{-\alpha}f_2v_\lambda$. Then as a $\fg_0$-module \[M/N = [U(\mathfrak{g}_0) v_\lambda \; \oplus U(\mathfrak{g}_0)f_2f_1v_\lambda \oplus \; J],\] where $J$ is an indecomposable with socle $J_1= U(\mathfrak{g}_0)u_1 =
U(\mathfrak{g}_0)u_2$, and unique maximal submodule $J_2 =U(\mathfrak{g}_0)f_1v_\lambda + U(\mathfrak{g}_0)f_2v_\lambda$ . The quotient $J_2/J_1$ is the direct sum of two highest weight modules with weights $(-2,0)$ and $(0,-2)$. The second $($resp. first$)$ copy of $\fsl(2)$ acts trivially on the first $($resp. second$)$ module. \end{itemize} \end{theorem} \begin{proof} (a) For the structure of $I$, note that $e_\alpha f_3v_\lambda = - f_2v_\lambda$
and $e_\gamma f_3v_\lambda = f_1v_\lambda$ by an easy calculation. For the other terms use Lemma \ref{l35}.\\ (b) Equation \eqref{wvk} follows easily from Lemma \ref{wot}. By Lemma \ref{w}, $$w = (e_{-\alpha}e_{-\beta-\gamma }-e_{-\gamma}e_{-\alpha-\beta } )v_\lambda= (e_{-\alpha}f_2-e_{-\gamma}f_1)v_\lambda= u_2-u_1\in \mathcal{V}$$ is a highest weight vector for $\fg$, and by Lemma \ref{wot}, $\mathcal{N}^X(\lambda)= U(\fg)w$. Finally (c) and (d) follow from (b). \end{proof} \noindent The table below gives a complete list of $\fg_0$-singular vectors in $M/N$. The weights are shifted as in \eqref{rot}. \[
\begin{tabular} {|c|c|} \hline $\fg_0$-singular vector $s$&
shifted weight of $s$\\ \hline $v_\lambda$ & $(0,0)$\\ \hline $f_1f_2v_\lambda$ & $(0,0)$\\ \hline $f_3v_\lambda$ &$(-1,-1)$\\ \hline $u=e_{-\alpha}f_2v_\lambda = e_{-\gamma}f_1v_\lambda$ & $(-1,-1)$\\ \hline $f_1v_\lambda$ & $(-1, 1)$\\ \hline $f_2v_\lambda$ &$(1,-1)$\\ \hline
\end{tabular}
\] \begin{theorem} \label{riz} The module $M^X(\lambda)=M/N$ is simple. \end{theorem} \begin{proof} It suffices to show that no linear combination of the singular vectors (other than $v_\lambda$) in the table is a highest weight vector. We have $e_\beta u= e_{-\alpha}e_{-\gamma}v_\lambda$, as noted in the proof of Theorem \ref{3.11}. Since $e_\beta$ commutes with $f_3$, it follows that $e_\beta f_3v_\lambda = 0$. However $f_3$ is not a $\fg_0$ highest weight vector.\\ \\ For the rest of the proof we show that no linear combination, other than $v_\lambda$ is killed by $e_\beta$. \\ \\
We have $[e_\beta,f_1f_2] = e_{-\alpha}f_2 - f_1e_{-\gamma},$ and
it follows that $e_\beta f_1f_2v_\lambda = (2u-f_3)v_\lambda\neq 0.$ So there is no highest weight vector with weight $(-1, -1)$.\\ \\ Finally
we have $[e_\beta,f_1]v_\lambda = e_{-\alpha}v_\lambda\neq 0$, and $[e_\beta,f_2]v_\lambda = e_{-\gamma}v_\lambda\neq 0.$ \end{proof}
\subsection{The Maximal Finite Quotient.} \label{mfq} Let $n=a+1>0$, and suppose $(\lambda+\rho, \alpha^\vee) = n$. Then $M^X(\lambda)/(U(\fg)e_{-\alpha}^n v_\lambda)$ is $\alpha$-finite. If also $(\lambda+\rho, \gamma^\vee)=n$, then $M^X(\lambda)/(U(\fg)e_{-\alpha}^n v_\lambda +U(\fg)e_{-\gamma}^n v_\lambda)$ is finite (dimensional). Now the maximal finite quotient of $M(\lambda)$ is the Kac module $K(\lambda)$. We use known results on the structure of $K(\lambda)$ to determine the maximal finite quotient of $M^X(\lambda).$ From \cite{SZ}, the Jantzen filtration on $K(\lambda)$ is the unique Loewy filtration, and hence coincides with both the socle and radical series of $K(\lambda).$ For $\fg =\fgl(2,2)$, this filtration takes the form \[K(\lambda) = K^0(\lambda) \supset K^1(\lambda) \supset K^2(\lambda) \supset K^3(\lambda) =0.\] Setting $K_i(\lambda) =K^i(\lambda)/K^{i+1}(\lambda)$ we have $K_0(\lambda) = L(\lambda)$. Concerning $K_1(\lambda)$ and $K_2(\lambda)$ there are three cases. \begin{itemize} \item[{{\rm(a)}}] If $n>2$, then \[K_1(\lambda) = L(\lambda-\beta) \oplus L(\lambda-\alpha-\beta-\gamma),\quad K_2(\lambda) = L(\lambda-\alpha-2\beta-\gamma). \] \item[{{\rm(b)}}] If $n=2$, then \[K_1(\lambda) = L(\lambda-\beta) \oplus L(\lambda-\alpha-\beta-\gamma) \oplus L(\lambda-2\alpha-3\beta-2\gamma),\] and $K_2(\lambda)$ is as in case (a). However $K(\lambda)$ does not contain a highest weight vector of weight $\lambda-2\alpha-3\beta-2\gamma$. \item[{{\rm(c)}}] If $n=1$, then \[K_1(\lambda) = L(\lambda-\beta),\quad K_2(\lambda) = L(\lambda-2(\alpha+\beta+\gamma)).\] \end{itemize} These facts can also be deduced from \cite{MS}. Until Theorem \ref{simp}, $v_\lambda$ will denote the highest weight vector in $K(\lambda)$. To describe the highest weight and singular vectors in $K(\lambda)$, recall the element $v_\kappa$ from \eqref{vk}, now regarded as an element of $K(\lambda)$ and consider the following products, (which are zero in some cases) \[v_1 =e_{-\beta} v_\lambda,\quad v_2 = \theta_{\alpha +\beta+\gamma}v_\lambda,\quad v_3= \theta_{\alpha +\beta+\gamma}e_{-\beta} v_\lambda,\quad v_4=f_1 f_2 f_3 e_{-\beta} v_\lambda.\] Then we have \begin{itemize} \item[{{\rm(a)}}] In case (a), $v_1, v_2$ are highest weight vectors
which together generate $K^1(\lambda)$, and $v_3$ is a highest weight vector which generates $K^2(\lambda)$. \item[{{\rm(b)}}] In case (b), $v_1, v_2$ are highest weight vectors and $v_\kappa$ is a highest weight vector mod $K^2(\lambda)$, see Lemma \ref{r}. Together these elements generate $K^1(\lambda)$. Also $v_3$ is a highest weight vector which generates $K^2(\lambda)$. \item[{{\rm(c)}}] In case (c), $v_1$ is a highest weight vector
which generates $K^1(\lambda)$, and $v_4$ is a highest weight vector which generates $K^2(\lambda)$. Note that $\lambda-\alpha-\beta-\gamma$ is not dominant in this case. \end{itemize} For convenience we record the dimensions of the finite dimensional irreducible $\fg$-modules. The same recursive method can be used to give an explicit expression for their characters, see also \eqref{chl}. First note that if $\lambda, \mu\in \mathcal{H}_X$ and $(\lambda+\rho, \alpha^\vee) = (\mu+\rho, \alpha^\vee) = n$, then $L(\lambda)$ is isomorphic to a tensor product of $L(\mu)$ with a one dimensional module. Hence $b_n= \dim L(\lambda)$ depends only on $n$. Also $\dim K(\lambda) = 16n^2.$ Now if $n= 1,$ then $L(\lambda)$ is trivial as a $[\fg_0,\fg_0]$-module, and so $b_1=1.$ Then from (a)-(c) above we have \begin{eqnarray} 16 &=& 2b_1 +b_2\nonumber\\ 64 &=& 2b_1 + 2b_2 + b_3\nonumber\\ 16n^2 &=& b_{n-1}+ 2b_{n}+b_{n+1} \mbox{ for } n\ge 3. \nonumber \end{eqnarray} Solving these equations we see that \begin{equation} \label{beq} b_{n} = 4n^2 -2\mbox{ for } n \ge2. \end{equation} The next result supports Conjecture \ref{conje}. \begin{theorem} \label{simp} If $\lambda \in \mathcal{H}_X$ is dominant, the maximal finite quotient of $M^X(\lambda)$ is simple. Thus the maximal submodule of $M^X(\lambda)$ is $U(\fg)e_{-\alpha}^n v_\lambda +U(\fg)e_{-\gamma}^n v_\lambda$. \end{theorem}
\begin{proof} This is shown by an analysis of cases (a)-(c) above. \end{proof} \noindent Theorem \ref{simp} is perhaps surprising since for any $\lambda \in \mathcal{H}_X$ we have in $K(\mathcal{O})$ \begin{equation} \label{leg} [M(\lambda)] = [M^X(\lambda)] +[M^X(\lambda- \alpha-\beta-\gamma)] +[M^X(\lambda- \beta)] +[M^X(\lambda- \alpha-2\beta-\gamma)].\end{equation} In fact $M^X(\lambda)$ has a filtration with factors having the same characters as the modules on the right of this equation. Thus since a Kac module $K(\lambda)$ can have 3, 4 or 5 composition factors, it is worth observing what happens to these composition factors relative to this filtration. First we note that $M(\lambda)$ has a submodule $M^\beta(\lambda-\beta)$ with factor module $M^\beta(\lambda).$ The first (resp. last) pair of summands on the right side of \eqref{leg} appear in a filtration of $M^\beta(\lambda)$ (resp. $M^\beta(\lambda-\beta)$). The most interesting case is (b) since if $n=2,$ the Kac module $K(\lambda)$ has greatest length. Here $v_1,$ and $v_3$ map to zero in $M^\beta(\lambda)$ and $v_2, v_\kappa \in N^X(\lambda)$. Thus the only composition factor of the Kac module that survives in $M^X(\lambda)$ is $L(\lambda),$ and $N^X(\lambda)$ is not a highest weight module. The analysis of cases (a) and (c) is straightforward once it is noted that in case (a), the weight $\lambda- \alpha-\beta-\gamma$ is not dominant. \subsection{On the structure of $M^X(\lambda)$.}\label{smx} \begin{theorem} \label{ns} The module $M^X(a)$ is simple unless $a \in{\mathbb N}$. \end{theorem} \begin{proof} We assume that $a\neq -1,$ since that case has just been covered. Then if $a$ is not a non-negative integer, then none of the vectors in \eqref{ext} are dominant. Thus $M^X(\lambda)$ is a direct sum of simple $\fg_0$-modules. The result now follows from easy calculations. For example, since $h_{\alpha+\beta}v_\lambda = av_\lambda$, $f_1v_\lambda$ does not generate a proper submodule.\end{proof}
\begin{theorem} \label{genst} If $(\lambda+\rho,\alpha^\vee)=n>1$, then \begin{itemize} \item[{{\rm(a)}}] The socle $\mathcal{S}$ of $M^X(\lambda)$ is $M^X(s_\alpha s_\gamma \cdot \lambda) \cong L(s_\alpha s_\gamma \cdot \lambda).$ \item[{{\rm(b)}}] We have \[U(\fg)e_{-\alpha}^n v_\lambda \cong M^Y(s_\alpha \cdot \lambda), \quad U(\fg)e_{-\gamma}^n v_\lambda \cong M^Y(s_\gamma \cdot \lambda)\] and \[L(s_\alpha s_\gamma \cdot \lambda) =M^X(s_\alpha s_\gamma\cdot \lambda).\] \item[{{\rm(c)}}] \[M^X_1(\lambda)=U(\fg)e_{-\alpha}^n v_\lambda+U(\fg)e_{-\gamma}^n v_\lambda
\quad {\rm and } \;\; \mathcal{S} \subseteq M^X_2(\lambda).\] \end{itemize} The lattice of submodules of $M^X(\lambda)$ is as in Figure \ref{fig1} with $V_1= U(\fg)e_{-\alpha}^n v_\lambda$ and $V_2= U(\fg)e_{-\gamma}^n v_\lambda.$ \end{theorem} \begin{proof} Omitted. It is similar to, but easier than the proof of Theorem \ref{stz1}.\end{proof}
\noindent Now we discuss the structure of $M^\beta(\lambda)$ and $M^X(\lambda)$ in the exceptional cases. \subsubsection{The case $a=0$.}\label{a=0} We continue with the notation from Subsection \ref{A=0}.
If $n=(\lambda+\rho,\alpha^\vee) =1$, equivalently $a=0,$ then from Theorem \ref{3.10}, \begin{equation} \label{1fad} M^\beta(\lambda)/N^X(\lambda) = U(\mathfrak{g}_0) v_\lambda \; \oplus \; U(\mathfrak{g}_0) f_1v_\lambda \; \oplus \; U(\mathfrak{g}_0)f_2v_\lambda \; \oplus I/J.\end{equation} \begin{corollary} \label{bdc} Let $\fg'=[\fg_0,\fg_0]$. Then as a $\fg'$-module, $I/J$ has composition factors with highest weights $(0,0), (-2,0)$ and $(0,-2)$ each with multiplicity one. \end{corollary} \begin{lemma} \label{shi} The element $p=f_1 f_2v_\lambda$ is a $\fg$ highest weight vector in $M/N$ with weight $\lambda -\alpha-2\beta -\gamma$ and spans a 1-dimensional trivial $\fg$-module. We have $p \in U(\fg)e_{-\alpha}v_\lambda \cap U(\fg) e_{-\gamma} v_\lambda$.\end{lemma} \begin{proof} As noted in Subsection \ref{cob}, $f_1 f_2v_\lambda=-f_2 f_1v_\lambda$ is a highest weight vector for $\mathfrak{b}_5$. In particular it is a $\fg_0$ highest weight vector. Also we have in $M$ that \[ e_{\beta}f_2f_1 v_\lambda =\pm(e_{-\beta-\gamma}e_{-\alpha} - e_{-\gamma} e_{-\alpha-\beta }) v_\lambda = \pm w \in N^X(\lambda),\] see Lemma \ref{w}. In addition $e_{-\alpha}p = e_{-\gamma}p= 0$ by \eqref{fem} and Theorem \ref{3.10}. Also $f_3p =v_\kappa \in N^X(\lambda)$. Since $\fg_0$, $f_3$ and $e_\beta$ generate $[\fg,\fg]$ we obtain the first statement. The second follows since $p = \pm e_{-\beta-\gamma}e_{-\beta} e_{-\alpha}v_\lambda =\pm e_{-\alpha-\beta}e_{-\beta} e_{-\gamma}v_\lambda $. \end{proof} \begin{rem}{\rm In contrast to the case $n>1$, when $n =1$ the submodule $U(\fg)e_{-\alpha}^n v_\lambda$ of $M^X(\lambda)$ is not isomorphic to $M^Y(s_\alpha \cdot \lambda)$. Indeed, if $u =e_{-\alpha} v_\lambda$ then $e_{-\beta}u$ is a highest weight vector for
$\mathfrak{b}^{(2)}$ which generates the same submodule of $M^X(\lambda)$ as $u$. So from $U(\fg)e_{-\alpha} v_\lambda \cong M^Y(s_\alpha \cdot \lambda)$ we would have $U(\fg)e_{-\alpha} v_\lambda \cong {\operatorname{Ind}}_\mathfrak{p}^\fg \;\mathtt{k} e_{-\beta}e_{-\alpha} v_\lambda $, see Theorem \ref{2} (a) and hence $e_{-\beta-\gamma}u=0$. However $e_{-\beta-\gamma}u =\pm f_2f_1 v_\lambda$. What this shows is the following. \begin{lemma} \label{k9} There are non-split exact sequences \[0 \longrightarrow \mathtt{k} f_2f_1 v_\lambda \longrightarrow U(\fg)e_{-\alpha} v_\lambda \longrightarrow M^Y(s_\alpha \cdot \lambda) \longrightarrow 0\] and \[0 \longrightarrow \mathtt{k} f_2f_1 v_\lambda \longrightarrow U(\fg)e_{-\gamma} v_\lambda \longrightarrow M^Y(s_\gamma \cdot \lambda) \longrightarrow 0.\] \end{lemma} }\end{rem} \begin{lemma} \label{yak} \begin{itemize} \item[{{\rm(a)}}] $p_Y - p_X = \mathtt{e}^{-\beta}.$ \item[{{\rm(b)}}] When $(\lambda, \alpha) = (\lambda, \gamma)=(\lambda, \beta) = 0$ we have $$[L(\lambda)] + [L(\lambda-\alpha -2\beta-\gamma)]+ [M^Y(s_\alpha \cdot \lambda)] +[M^Y(s_\gamma \cdot \lambda)] -[M^X(w_0\cdot\lambda)]=M^X(\lambda).$$ \item[{{\rm(c)}}] When $(\lambda, \alpha^\vee) = (\lambda, \gamma^\vee) > 0=(\lambda, \beta)$ we have $$[L(\lambda)] + [M^Y(s_\alpha \cdot \lambda)] +[M^Y(s_\gamma \cdot \lambda)] -[M^X(w_0\cdot\lambda)]=M^X(\lambda).$$ \end{itemize} \end{lemma} \begin{proof} (a) follows from Lemma \ref{ink} (a). To prove (b) and (c), we identify the class of a module in the Grothendieck group $K(\mathcal{O})$ with its character. Then (b) follows since by (a), \begin{eqnarray} && [M^Y(s_\alpha \cdot \lambda)] +[M^Y(s_\gamma \cdot \lambda)] -[M^X(w_0\cdot\lambda)]-M^X(\lambda)\nonumber\\ &=& \mathtt{e}^\lambda[(\mathtt{e}^{-\alpha} +\mathtt{e}^{-\gamma})(p_X+ \mathtt{e}^{-\beta}) -p_X -\mathtt{e}^{-\alpha-\gamma}p_X]\nonumber\\ &=& \mathtt{e}^\lambda[(\mathtt{e}^{-\alpha}-\mathtt{e}^{-\gamma})\mathtt{e}^{-\beta}-p_X (1 - \mathtt{e}^{-\alpha}) (1 - \mathtt{e}^{-\gamma})]\nonumber\\ &=& -\mathtt{e}^\lambda - \mathtt{e}^{\lambda-\alpha-2\beta-\gamma}\nonumber \\ &=& -[L(\lambda)] - [L(\lambda-\alpha -2\beta-\gamma)].\end{eqnarray} Similarly if $(\lambda+\rho,\alpha^\vee)=n>1$, then setting $x=\mathtt{e}^{-\alpha}, y=\mathtt{e}^{-\beta}, z =\mathtt{e}^{-\gamma}$ and $\Xi=(1 - x^{-\alpha}) (1 - z^{-\gamma}),$ we have \begin{eqnarray} \label{chl}
&& [M^X(\lambda)]+[M^X(w_0\cdot\lambda)] -[M^Y(s_\alpha \cdot \lambda)] -[M^Y(s_\gamma \cdot \lambda)]\\ &=& \mathtt{e}^\lambda[(1+ x^nz^ n)(1+xy)(1+yz) - (x^n+z^n)(1+y)(1+xyz)]/\Xi\nonumber.\end{eqnarray} Taking limits as $x,y,z\longrightarrow 1$ we obtain $4n^2-2$, which equals $\dim L(\lambda)$ by \eqref{beq}. \end{proof} \begin{theorem} \label{stz1} \begin{itemize} \item[{{\rm(a)}}] The socle $\mathcal{S}$ of $M^X(\lambda)$ satisfies \[\mathcal{S} = \mathtt{k} f_2f_1 v_\lambda \oplus U(\fg)e_{-\alpha}e_{-\gamma} v_\lambda \; = \; U(\fg)e_{-\alpha}v_\lambda \cap U(\fg) e_{-\gamma} v_\lambda. \] \item[{{\rm(b)}}] We have \[U(\fg)e_{-\alpha}v_\lambda/\mathtt{k} f_2f_1 v_\lambda \cong M^Y(s_\alpha\cdot \lambda), \quad U(\fg)e_{-\gamma}v_\lambda/\mathtt{k} f_2f_1 v_\lambda \cong M^Y(s_\gamma\cdot \lambda) \] and \[U(\fg)e_{-\alpha}e_{-\gamma}v_\lambda\cong M^X(s_\alpha s_\gamma\cdot \lambda).\] \item[{{\rm(c)}}] $\mathcal{S} \subseteq M_2^X(\lambda).$\end{itemize} \end{theorem} \begin{proof} First observe that $U(\fg)e_{-\alpha}e_{-\gamma} v_\lambda \cong M^X(s_\alpha s_\gamma \cdot \lambda)$ is simple by Theorem \ref{ns}. The other statements in (b) hold by Lemma \ref{k9}. Also by Lemma \ref{shi}, $\mathtt{k} f_2f_1 v_\lambda $ spans a one-dimensional submodule of $M^X(\lambda)$. This can also be shown using Lemma \ref{yamo} (d). Thus $\mathcal{S}' =\mathtt{k} f_2f_1 v_\lambda \oplus U(\fg)e_{-\alpha}e_{-\gamma} v_\lambda \subseteq \mathcal{S}$. Furthermore
$M^X(\lambda)/(U(\fg)e_{-\alpha}v_\lambda +U(\fg) e_{-\gamma} v_\lambda)$ is the one dimensional module $L(\lambda)$ by Theorem \ref{simp}. Next we show that $L= (U(\fg)e_{-\alpha}v_\lambda \cap U(\fg) e_{-\gamma} v_\lambda)/\mathcal{S}'$ is zero. Set $M_\alpha = U(\fg)e_{-\alpha}v_\lambda, M_\gamma = U(\fg)e_{-\gamma}v_\lambda.$ Then in the Grothendieck group $K(\mathcal{O})$ we have \begin{equation} \label{k7}
[M_\alpha /M_\alpha\cap M_\gamma] = [M^Y(s_\alpha)/M^X(w_0\cdot\lambda)] -[L], \end{equation} \begin{equation} \label{k8} [M_\gamma /M_\alpha\cap M_\gamma] = [M^Y(s_\gamma)/M^X(w_0\cdot\lambda)] -[L] .\end{equation} For example using Lemma \ref{k9} \begin{eqnarray} \label{k6}
[M_\alpha /M_\alpha\cap M_\gamma] &=& [M_\alpha]- [L(\lambda-\alpha -2\beta-\gamma)]-[L]-[M^X(w_0\cdot\lambda)]\nonumber\\ &=& [M^Y(s_\alpha)/M^X(w_0\cdot\lambda)] -[L].\nonumber \end{eqnarray} In addition using \eqref{k7} and \eqref{k8}, \begin{equation} \label{k5}
[M_\alpha\cap M_\gamma] = [L]+[L(\lambda-\alpha -2\beta-\gamma)]+ [M^X(w_0\cdot\lambda)]. \end{equation} Finally \begin{eqnarray} \label{k4}
[M_\alpha+M_\gamma] &-&[M_\alpha\cap M_\gamma]\\ &=& [M^Y(s_\alpha \cdot \lambda)/M^X(w_0\cdot\lambda)]+ [M^Y(s_\gamma \cdot \lambda)/M^X(w_0\cdot\lambda)].\nonumber\end{eqnarray} Combining \eqref{k4} and \eqref{k5} with Theorem \ref{simp} and comparing the result with Lemma \ref{yak} (b), we obtain $L=0.$ Finally (c) holds since the module $M_1^X(\lambda)/M_2^X(\lambda)$ is self-dual. \end{proof} \subsection{The module $M^Y(\lambda)$.}\label{mmt} The main result of this subsection is the following. \begin{theorem} \label{mnt}If $\lambda\in \mathcal{H}_Y$ and $(\lambda+\rho,\alpha^\vee) = a \in {\mathbb Z}$ , then $M_1^Y(\lambda)$ is simple and \[M_1^Y(\lambda) \cong \left\{ \begin{array}{cl} M^X(s_\alpha\cdot \lambda) & \mbox{ if } a \ge 0 \\ M^X(s_\gamma\cdot \lambda)& \mbox{ if } a<0. \end{array} \right.\] \end{theorem} \noindent \begin{proof} This follows from Theorems \ref{gnat} and Corollary \ref{lcor} below. \end{proof} \noindent We show that if $a\neq 0,$ we can construct $M^Y(\lambda)$ using induction from the parabolic subalgebra $\mathfrak{p}= \mathfrak{b}+\mathfrak{b}^{(2)}.$ As in the footnote in Subsection \ref{cms}, there is no loss of generality in assuming that $\mu = a(\alpha+\beta),$ with $a \in \mathbb{Z}.$ Let $\lambda= \mu +\beta$.
Then $(\mu, \alpha +\beta) = (\mu, \beta+\gamma) = 0,$ so $\mu$ defines a one dimensional $\mathfrak{p}$-module $\mathtt{k} v_{\mu}.$ Also \begin{equation} \label{k3}(\lambda +\rho, \alpha^\vee) = -(\lambda +\rho, \gamma^\vee)= a.\end{equation}
\noindent
Note also that \begin{equation} \label{star}h_{\alpha} v_{\mu} = h_{\gamma} v_{\mu} = av_{\mu}, \quad \quad h_{\beta} v_{\mu} = -av_{\mu}\end{equation} and that $$(\mu, \alpha^\vee) = -(\mu, \gamma^\vee).$$ Set $R^Y(a)={\operatorname{Ind}}_\mathfrak{p}^\fg \;\mathtt{k} v_\mu$ and $v_\lambda=e_{\beta}v_\mu.$ Then $v_\lambda, v_\mu$ are highest weight vectors for the Borels $\mathfrak{b}, \mathfrak{b}^{(2)} $ respectively. If $a=0$, $v_\lambda$ generates a proper submodule.
\begin{lemma} \label{Ma1} For all $a$ we have a direct sum of $U(\mathfrak{m}_0)$-modules $$R^Y(a) = U(\mathfrak{m}_0)v_{\mu} \; \oplus \; U(\mathfrak{m}_0)e_{-\alpha-\beta-\gamma} v_{\mu} \; \oplus \; U(\mathfrak{m}_0)e_{\beta} v_{\mu} \; \oplus \; U(\mathfrak{m}_0)e_{-(\alpha + \beta + \gamma)} e_{\beta} v_{\mu} .$$ \end{lemma} \noindent \begin{proof} Since $R^Y(a)$ is induced from $\mathfrak{p}$ and $\fg = \mathfrak{p} \oplus \mathfrak{m},$ this follows at once from the PBW Theorem. \end{proof}
\begin{lemma} \label{sub} Suppose $(\mu,Y)=0,$ so that $\mu$ defines a character of $\mathfrak{p}.$ Then \[{\operatorname{ch}\:} {\operatorname{Ind}}_\mathfrak{p}^\fg \;\mathtt{k} v_\mu = \mathtt{e}^{\mu +\beta}p_Y.\] Thus $R^Y(a)$ and $M^Y(a)$ have the same character. \end{lemma} \begin{proof} This follows from Lemma \ref{Ma1} because \[\mathtt{e}^\mu(1 + \mathtt{e}^{\beta})(1 + \mathtt{e}^{-\alpha - \beta -\gamma}) = \mathtt{e}^{\mu + \beta} (1 + \mathtt{e}^{ -\beta})(1 + \mathtt{e}^{-\alpha - \beta -\gamma}). \] \end{proof}
\begin{lemma} \label{udef} Set $$u = (e_{-\alpha}e_{-\gamma} + a e_{-\alpha-\beta-\gamma}e_{\beta})v_\mu$$ \[u'=(e_{-\alpha}e_{-\gamma} e_{-\beta}- a^2 e_{-\alpha-\beta-\gamma})v_{\lambda} \] Then $u, u'$ are $\fg_0$ highest weight vectors. \end{lemma} \begin{proof} A computation. Note that the substitution $e_{-\beta}v_\lambda = a v_\mu$ gives $au=u'.$ \end{proof}
\begin{lemma} \label{107} The elements $e_{-\alpha-\beta}e_{-\beta}v_{{{\lambda}}}$ and $e_{-\beta-\gamma}e_{-\beta}v_\lambda=0$ are in the kernel of the natural map $M(\lambda)\longrightarrow M^Y(a)$. \end{lemma} \begin{proof} By \eqref{a+b} we have in $M^Y(a)_A$ that \[0 = \theta_{\alpha +\beta}v_{{\widetilde{\lambda}}} = (e_{-\beta}e_{-\alpha}+ e_{-\alpha -\beta}h_\alpha)v_{{\widetilde{\lambda}}} = (e_{-\beta}e_{-\alpha}+ (T+a)e_{-\alpha -\beta})v_{{\widetilde{\lambda}}}.\] Multiplying both sides by $e_{-\beta}$ and using the fact $M^Y(a)_A$ is $\mathtt{k}[T]$ torsion-free, we obtain $e_{-\beta} e_{-\alpha-\beta}v_{{\widetilde{\lambda}}}=0$ in $M^Y(a)_A$. Hence $e_{-\beta} e_{-\alpha-\beta}v_{{{\lambda}}}=-e_{-\alpha-\beta}e_{-\beta}v_{{{\lambda}}}=0$ in $M^Y(a)$. Similarly using the expression for $\theta_{\beta+\gamma}$ in \eqref{c+b}, we see that $e_{-\beta-\gamma}e_{-\beta}v_\lambda=0$ in $M^Y(a)$. \end{proof} \noindent Now fix $a$ and set $M=R^Y(a)$. We will improve Lemma \ref{Ma1} to obtain decompositions into $\fg_0$-modules, see Theorems \ref{2} and \ref{2.5}. \subsubsection{The case $a>0$.} Because of \eqref{star} and \eqref{k3}, we can restrict our attention to the cases where $a\ge 0.$ If $a < 0$ we can carry out a similar analysis using $\gamma$ in place of $\alpha.$ Note that $v_\lambda$ generates a proper submodule of $M$ iff $a = 0.$
\begin{theorem} \label{2} \begin{itemize} \item[{{\rm(a)}}] If $a \neq 0,$ then $M$ is a highest weight module for the distinguished Borel with highest weight $\lambda=\mu+\beta $. \item[{{\rm(b)}}] Furthermore \begin{eqnarray} M &=& U(\mathfrak{m}_0)v_{\mu} \; \oplus \; U(\mathfrak{m}_0)u \; \oplus \; U(\mathfrak{m}_0)e_{-\alpha-\beta-\gamma} v_{\mu} \; \oplus \; U(\mathfrak{m}_0)e_{\beta}v_{\mu},\nonumber\\ &=& U(\mathfrak{m}_0)e_{-\beta}v_{\lambda}\; \oplus \; U(\mathfrak{m}_0)(e_{-\alpha}e_{-\gamma} e_{-\beta}- a^2 e_{-\alpha-\beta-\gamma})v_{\lambda} \nonumber\\ &\oplus& U(\mathfrak{m}_0)e_{-\alpha-\beta-\gamma}e_{-\beta}v_{\lambda}\; \oplus \; U(\mathfrak{m}_0)v_{\lambda},\end{eqnarray} and these are direct sums of $\fg_0$-modules. In addition $M$ is generated as a $\mathfrak{g}$-module by $v_{\lambda}$ which is a highest weight vector for $\mathfrak{b}.$\end{itemize} \end{theorem} \begin{proof} First (a) follows from Lemma \ref{Ma1}. By Lemma \ref{udef} we have for $a \neq 0,$ $$U(\mathfrak{m}_0)v_{\mu} \; \oplus \; U(\mathfrak{m}_0)e_{-\alpha-\beta-\gamma} e_{\beta} v_{\mu} = U(\mathfrak{m}_0)v_{\mu} \; \oplus \; U(\mathfrak{m}_0)u.$$ The first decomposition in (b) follows from Lemma \ref{Ma1}, and the second from $e_{-\beta}v_\lambda = a v_\mu$.\end{proof} \noindent Since $(\lambda + \rho, \alpha^\vee) = a $, $v_{\eta} = e_{-\alpha}^{a}v_{\lambda}$ is a highest weight vector for $\mathfrak{b}$ with weight $\eta = s_{\alpha}\cdot\lambda.$
Let $N$ be the submodule generated by $v_{\eta}$. Theorem \ref{2} gives a decomposition of $M$ into $\fg_0$-submodules, and we give a compatible decomposition of $N$. \begin{theorem} \label{gnat} Set $v_\kappa= e_{-\alpha}^{a+1}v_{\mu}$. Then \begin{itemize} \item[{{\rm(a)}}] we have a direct sum of $\fg_0$-modules $$ N = U(\mathfrak{m}_0)v_\kappa
\; \oplus \; U(\mathfrak{m}_0) e_{-\beta - \gamma}v_\eta \; \oplus \; U(\mathfrak{m}_0)v_\eta \; \oplus \; U(\mathfrak{m}_0)e_{-\beta - \gamma}v_\kappa,$$ with \begin{eqnarray} U(\mathfrak{m}_0)v_\kappa
&\subseteq& U(\mathfrak{m}_0)v_{\mu},\quad \; \;U(\mathfrak{m}_0) e_{-\beta - \gamma}v_\eta \subseteq\; U(\mathfrak{m}_0)u\label{one}\\ U(\mathfrak{m}_0)v_\eta &\subseteq& U(\mathfrak{m}_0)e_{\beta}v_{\mu},\;\;
U(\mathfrak{m}_0)e_{-\beta - \gamma}v_\kappa\subseteq U(\mathfrak{m}_0)e_{-\alpha - \beta - \gamma} v_{\mu} .\label{two}\end{eqnarray}
\item[{{\rm(b)}}] The character of $N$ is \[ \mathtt{e}^\kappa(1 + \mathtt{e}^{\alpha + \beta})(1 + \mathtt{e}^{-\beta-\gamma})/ \prod_{\sigma \in \Delta^+_{0}} (1 - \mathtt{e}^{- \sigma}) \] \[ = \mathtt{e}^{\eta}(1 + \mathtt{e}^{-\alpha - \beta})(1 + \mathtt{e}^{-\beta-\gamma})/ \prod_{\sigma \in \Delta^+_{0}} (1 - \mathtt{e}^{- \sigma}) = \mathtt{e}^{s_\alpha\cdot\lambda} p_X.\] \item[{{\rm(c)}}] $N$ and $M/N$ are simple. \end{itemize} \end{theorem} \begin{proof} Note that $v_\kappa$
is a highest weight vector for $\mathfrak{b}^{(2)}$ with weight $$\kappa = s_\alpha \cdot \mu = \mu - (a+1)\alpha.$$ Also $e_{-\alpha -\beta }v_\eta=-v_\kappa$ and $e_{\alpha + \beta }v_\kappa = -(a+1)v_\eta,$ so $v_\eta$ and $v_\kappa$ both generate the same submodule. Hence $N \cap U(\mathfrak{m}_0)v_{\lambda}$ contains $v_\eta,$ which has weight \begin{equation} \label{ad}\eta = \kappa + \alpha + \beta = s_\alpha \cdot (\mu + \beta) = s_\alpha\cdot\lambda.\end{equation} Next the inclusions in \eqref{one} and \eqref{two} follow since \begin{eqnarray} e_{-\beta - \gamma}v_\eta &=& e_{-\beta-\gamma}e_{-\alpha}^{a}e_\beta v_\mu,\nonumber\\ &=&(e_{-\alpha}^{a}e_{-\beta-\gamma} + ae_{-\alpha}^{a-1}e_{-\alpha-\beta-\gamma})e_\beta v_\mu\nonumber\\ &=&e_{-\alpha}^{a-1}(e_{-\alpha}e_{-\gamma} + ae_{-\alpha-\beta-\gamma} e_{\beta})v_\mu\nonumber\\ &=& e_{-\alpha}^{a-1}u.\nonumber\end{eqnarray} and \begin{eqnarray} e_{-\beta-\gamma}v_\kappa &=& e_{-\beta-\gamma}e_{-\alpha}^{a+1}v_\mu\nonumber\\ &=& (a+1)e_{-\alpha}^{a}e_{-\alpha-\beta-\gamma}v_\mu.\nonumber\end{eqnarray} From these computations we also see that $e_{-\beta-\gamma}v_\eta$ is a $\fg_0$ highest weight vector in $N \cap U(\mathfrak{m}_0)e_{-(\alpha + \beta + \gamma)} v_{\lambda}$
with weight $s_\alpha \cdot (\mu - \alpha - \gamma ) = \kappa + \alpha - \gamma,$ and that $e_{-\beta-\gamma}v_\kappa$ is a $\fg_0$ highest weight vector in $N \cap U(\mathfrak{m}_0)e_{-\alpha-\beta-\gamma}v_\mu$ with weight $s_\alpha \cdot (\mu - \alpha - \beta -\gamma ) = \kappa -\beta -\gamma.$ Also $e_{\alpha + \beta }e_{-\alpha}^{a}e_{-\alpha-\beta-\gamma}v_\mu = e_{-\alpha}^{a-1}u.$ Equality in \eqref{one} and \eqref{two} as well as the statements about simplicity follow easily by looking at the highest weights and using $\fsl(2)$ theory. Finally (b) follows from (a) and (\ref{ad}). \end{proof} \begin{corollary} If $a\neq 0$, then $R^Y(a)\cong M^Y(a)$.\end{corollary} \begin{proof} \noindent As $\mathfrak{p} = \mathfrak{b}\oplus {\operatorname{span}}\{e_{-\alpha-\beta}, e_{-\beta-\gamma}\}$, $M(\lambda)={\operatorname{Ind}}_\mathfrak{b}^\fg \;\mathtt{k} v_\lambda$ and $R^Y(a)={\operatorname{Ind}}_\mathfrak{p}^\fg \;\mathtt{k} e_{-\beta}v_\lambda$,
it follows that $R^Y(a)$ is the (universal) module obtained from $M(\lambda)$ by setting $e_{-\alpha-\beta} e_{-\beta}v_\lambda$ and $e_{-\beta-\gamma}e_{-\beta}v_\lambda$ equal to zero. Since the same relations hold in $M^Y(a)$ by Lemma \ref{107}, there is an onto map from $R^Y(a)\longrightarrow M^Y(a)$. Because both modules have the same character, it must be an isomorphism. \end{proof}
\subsubsection{The case $a=0$.} \label{a=0v2} From \eqref{a+b} we have \[(e_{-\alpha }e_{-\beta }+(T+a)e_{-\alpha -\beta} v_{{\widetilde{\lambda}}}) =\theta_{\alpha +\beta } v^Y_{{\widetilde{\lambda}}}=0.\] \noindent Now set $M=R^Y(\lambda)$, where $\lambda=0$ and $v_\mu= e_{-\beta}v_\lambda$.\ \begin{theorem} \label{2.5} We have \begin{itemize} \item[{{\rm(a)}}] If $I= U(\mathfrak{m}_0)v_{\mu} \; \oplus \;U(\mathfrak{m}_0)e_{-(\alpha +\beta + \gamma)} e_{\beta} v_{\mu},$ we have \begin{equation} \label{fyj} M = U(\mathfrak{m}_0) e_{\beta }v_\mu\; \oplus \; U(\mathfrak{m}_0) e_{-\alpha-\beta-\gamma}v_\mu \; \oplus \; I,\end{equation} a direct sum of $\fg_0$-modules. Also $I$ is an indecomposable containing the Verma submodule $S=U(\mathfrak{g}_0)v_\mu$, such that $I/S$ is a Verma module with highest weight vector $e_{-\alpha-\beta-\gamma}e_{\beta}v_\mu$. \item[{{\rm(b)}}] The unique maximal $\fg$-submodule of $M$ is \[N = U(\mathfrak{m}_0)e_\beta v_{\mu} \; \oplus \; U(\mathfrak{m}_0)e_{-\alpha-\beta-\gamma} v_{\mu} \oplus J,\] where $J$ is an indecomposable, codimension one submodule of $I$ which fits into an exact sequence \[0\longrightarrow U(\mathfrak{m}_0)\mathfrak{m}_0v_{\mu} \longrightarrow J\longrightarrow U(\mathfrak{m}_0)e_{-(\alpha + \beta + \gamma)} e_{\beta} v_{\mu} \longrightarrow 0.\]
\item[{{\rm(c)}}] The character of $N$ is $\mathtt{e}^{\beta}p_X,$ and $M/N$ is the trivial module. \item[{{\rm(d)}}] The module $N$ is a simple highest weight module with highest weight $\beta$. Thus $N\cong L(\beta)$.\end{itemize} \end{theorem} \begin{proof} First \eqref{fyj} follows from Lemma \ref{Ma1}. Note that $s_\alpha\cdot \mu = \mu - \alpha$, $s_\gamma\cdot \mu = \mu - \gamma$. Also $e_{-\alpha}v_{\lambda} = e_{\beta}e_{-\alpha}v_{\mu},\; e_{-\gamma} v_{\lambda} = e_{\beta}e_{-\gamma}v_{\mu} .$ It is easy to check that $e_{\beta} v_{\mu} $ and $e_{-\alpha-\beta-\gamma} v_{\mu}$ are highest weight vectors for $\fg_0$ and that \[e_\alpha e_{-\alpha-\beta-\gamma} e_{\beta} v_{\mu} = -e_{-\gamma} v_{\mu},\quad \quad
e_\gamma e_{-\alpha-\beta-\gamma} e_{\beta} v_{\mu} = -e_{-\alpha} v_{\mu}\] \noindent The character of $U(\mathfrak{m}_0)\mathfrak{m}_0v_{\mu}$ is \[ (\mathtt{e}^{-\alpha} + \mathtt{e}^{-\gamma} - \mathtt{e}^{-\alpha - \gamma})/ \prod_{\sigma \in \Delta^+_{0}}(1 - \mathtt{e}^{- \sigma}). \] In addition $N$ contains the $\fg_0$ highest weight vectors $e_{-\alpha-\beta-\gamma} v_{\mu},\; e_{\beta} v_{\mu}, e_{-\alpha-\beta-\gamma} e_{\beta} v_{\mu}$ with weights $ -\alpha - \beta - \gamma, \beta$ and $-\alpha -\gamma$ respectively. Adding the characters of the $\fg_0$-modules generated by these elements we obtain \[{\operatorname{ch}\:} N = (\mathtt{e}^{-\alpha} + \mathtt{e}^{-\gamma} + \mathtt{e}^{\beta} + \mathtt{e}^{-\alpha -\beta -\gamma})/ \prod_{\sigma \in \Delta^+_{0}}(1 - \mathtt{e}^{- \sigma}) = \mathtt{e}^{\beta}p_X. \] Now (c) follows since by (b), Lemma \ref{2} (a) and Lemma \ref{yak} we have \[{\operatorname{ch}\:} M/N = {\operatorname{ch}\:} M -{\operatorname{ch}\:} N = \mathtt{e}^{\beta} ( p_Y - p_X) =1.\]
\noindent Finally, using $$e_{-\beta-\gamma}e_\beta v_\mu =\pm e_{-\gamma}v_\mu, \; e_{-\alpha -\beta}e_{-\gamma}v_\mu =\pm e_{-\alpha}v_\mu, \; e_{-\alpha-\beta}e_{-\beta-\gamma}e_{\beta}v_\mu = \pm e_{-(\alpha +\beta + \gamma)} v_{\mu},$$
it is easy to check that $e_\beta v_\mu$ is a highest weight vector which generates $N$. Since $(\lambda+\rho, \alpha^\vee) = \; (\lambda +\rho, \gamma^\vee) = 0.$ the results of Subsection \ref{3.111} apply, and in particular (d) follows from Theorem \ref{riz}. \end{proof} \begin{corollary} In the Grothendieck group $K(\mathcal{O})$, we have $[R^Y(0)] = [M^Y(0)]$. Thus $M^Y(0)$ has length two with unique maximal $\fg$-submodule isomorphic to the module $N$ in Theorem \ref{2.5} $(b)$. However $R^Y(0)$ and $M^Y(0)$ are not isomorphic. \end{corollary} \begin{proof} The first statement holds since $R^Y(0)$ and $M^Y(0)$ have the same character. For the last statement note that $M^Y(0)$ is a highest weight module for the distinguished Borel but $R^Y(0)$ is not. By Theorem \ref{2.5}, the submodule $N$ of $R^Y(0)$ has codimension one. On the other hand the elements $e_{-\alpha} v_\lambda, e_{-\beta} v_\lambda$ and $e_{-\gamma} v_\lambda$ generate a proper submodule of $M^Y(0)$ with codimension one. The result follows since $R^Y(0)$ and $M^Y(0)$ have the same character. \end{proof} \begin{corollary} \label{lcor} If $a=0,$ the unique maximal submodule of $M^Y(\lambda)$ is simple and isomorphic to $M^X(\lambda).$\end{corollary} \begin{proof} Combine the previous corollary and Theorem \ref{2.5}.\end{proof}
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Toward a mechanistic understanding of adsorption behavior of phenol onto a novel activated carbon composite
Phenol adsorption on high microporous activated carbons prepared from oily sludge: equilibrium, kinetic and thermodynamic studies
N. Mojoudi, N. Mirghaffari, … J. Bedia
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Lead ferrite-activated carbon magnetic composite for efficient removal of phenol from aqueous solutions: synthesis, characterization, and adsorption studies
Esmaeil Allahkarami, Abolfazl Dehghan Monfared, … Guilherme Luiz Dotto
Study of the hydrogen physisorption on adsorbents based on activated carbon by means of statistical physics formalism: modeling analysis and thermodynamics investigation
Manel Ben Yahia & Sarra Wjihi
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Daniel Dittmann, Paul Eisentraut, … Ulrike Braun
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Esmaeil Allahkarami1,2,
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Luis Felipe O. Silva3 &
Guilherme Luiz Dotto4
Scientific Reports volume 13, Article number: 167 (2023) Cite this article
In this research, the solid–liquid adsorption systems for MSAC (PbFe2O4 spinel-activated carbon)-phenol and pristine activated carbon-phenol were scrutinized from the thermodynamics and statistical physics (sta-phy) viewpoints. Experimental results indicated that MSAC composite outperformed pristine AC for the uptake of phenol from waste streams. By increasing the process temperature, the amount of phenol adsorbed onto both adsorbents, MSAC composite and pristine AC, decreased. Thermodynamic evaluations for MSAC demonstrated the spontaneous and exothermic characteristics of the adsorption process, while positive values of ΔG for pristine AC indicated a non-spontaneous process of phenol adsorption in all temperatures. In a mechanistic investigation, statistical physics modeling was applied to explore the responsible mechanism for phenol adsorption onto the MSAC composite and pristine AC. The single-layer model with one energy was the best model to describe the experimental data for both adsorbents. The adsorption energies of phenol onto both adsorbents were relatively smaller than 20 kJ/mol, indicating physical interactions. By increasing temperature from 298 to 358 K, the value of the absorbed amount of phenol onto the MSAC composite and pristine AC at saturation (Qsat) decreased from 158.94 and 138.91 to 115.23 and 112.34 mg/g, respectively. Mechanistic studies confirm the significant role of metallic hydroxides in MSAC to facilitate the removal of phenol through a strong interaction with phenol molecules, as compared with pristine activated carbon.
Phenols are important contaminants in natural waters due to their toxicity and carcinogenicity properties1,2. The concentration of phenol in industrial wastes varies from 50 to 2000 mg/L3. Phenolic compounds, even at low concentrations, are considered health-threatening organic pollutants due to their water-soluble and acidic nature4,5,6. Therefore, the elimination of these compounds from the waste streams before their discharge to the environment is vital. In this context, some biological and physicochemical methods, such as ion exchange6,7, adsorption8,9, chemical reduction10, solvent extraction11,12, and reverse osmosis13, were utilized to remove the phenolic-based compounds from different solutions. However, adsorption process has gained much attention due to its ease and convenience in operation, simple design, and economic considerations10,14,15. In this regard, various adsorbents such as activated carbon16,17, minerals18,19, and polymers20,21 have been applied to investigate phenol uptake. For example, Lammini et al.22 synthesized cobalt oxide Co3O4 and applied it for adsorption of phenol from solution. According to their results, the highest capacity and removal percentage of phenol at pH 4 were obtained 8.10 mg/g and 98%, respectively. Dehmani et al.23 activated Moroccan clay with sulfuric acid and used it to uptake phenol from the aqueous solution. They indicated that clay chemical activation enhanced its adsorption capability. Mohammed et al.24 applied Faujasite-Type Y Zeolite for the phenol uptake from aqueous solution. They found that the equilibrium data for the uptake of phenol by the developed adsorbent were in accordance with the Langmuir adsorption isotherm.
Empirical, semi-empirical, and theoretical adsorption isotherm models can predict the experimental data obtained from these studies. Therefore, the analysis of experimental data with these models provides important information for the design of the adsorption process as an operation unit and attributes further comprehension into the mechanism of the adsorption process and economic considerations25,26,27. In general, experimental data obtained in the adsorption of phenol by various adsorbents have been discussed using Freundlich, Langmuir, Dubinin–Radushkevich, Temkin, and Liu models28,29. However, the interpretation of phenol adsorption properties using these models is limited due to the disability of these models to understand the complete adsorption process and the parameters affecting the process. Statistical physics modeling can be applied to explain the phenol adsorption isotherms30 to overcome these limitations. Indeed, the analysis of experimental data via statistical physics fundaments provides useful information about the parameters related to the steric and energetic state of the adsorption process that controls the adsorption mechanism31. These models assume that the phenol adsorption onto the prepared adsorbent occurs by forming a variable number of layers or a constant number (1 or 2) of layers32. In addition, they assume that a variable number of adsorbate molecules is linked to receptor sites of the adsorbent (unit mass)31. In general, statistical physics models are useful tools for obtaining new interpretations about the adsorption mechanism of adsorbate onto the developed adsorbent at the molecular level.
Recently, the activated carbon (AC) has attracted great attention for wastewater treatment purposes33,34,35 due to its large surface area, high adsorption performance, micro-porous nature, and ease of application and availability. Most of the studies in the literature have been focused on the preparation of activated carbon from different types of precursors to improve the physical properties of AC. Pal et al.36 synthesized four AC samples from two biomass precursors namely waste palm trunk and mangrove applying KOH as an activating agent. Their results indicated that the biomass-derived activated carbons had high surface area and large pore volume leading to improve the uptake of CO2. However, there are the limited studies about improving the surface properties of AC. On the other hand, activated carbon is extremely difficult to separate from the solution37. To overcome this limitation, the introduction of ferromagnetism in mesoporous activated carbon particles has been performed for obtaining magnetic activated carbon. Magnetic activated carbon is of great interest as offers an easy way to separate the sorbent from aqueous solution for reuse and attractive catalytic performance38,39. Metal ferrites (MFe2O4, M = Zn, Co, Pb, Sr, etc.) have face-centered cubic structures with M2+ and Fe3+ cations filling in the tetrahedral and octahedral coordination sites. Incorporation of the mentioned metallic cations could improve the magnetic properties of the developed composite40,41. Therefore, the main aim of this work is to present a mechanistic study on the lead ferrite-AC composite as a new material for wastewater decontamination purposes.
This mechanistic study applied a novel PbFe2O4 spinel—activated carbon composite (MSAC) prepared by co-precipitation to remove phenol from solutions. The adsorption system MSAC-phenol was compared with pristine activated carbon from the thermodynamic and statistical physics viewpoints. Adsorption studies investigated the influence of temperature on phenol adsorption capacity, in which five isotherm curves were constructed. From the thermodynamic perspective, the variations of Gibbs free energy, enthalpy, and entropy were evaluated. Concerning the sta-phy, some models were tested, and the steric and energetic parameters were interpreted. The present research contributes to new theoretical insights on phenol removal by MSAC composite and pristine activated carbon through statistical physics modeling.
Solid–liquid adsorption system
The adsorbents used were PbFe2O4 spinel -activated carbon composite (MSAC) prepared by co-precipitation and pristine activated carbon. Pristine AC (received from Korea; Hanil, P60,) was applied in its original form with no any purification. The adsorbate used was phenol. First, the stock solution of phenol (Merck, Darmstadt, Germany) was prepared by dissolving specific amounts of adsorbate into double-distilled water. Then, the desired concentrations of phenol for doing adsorption experiments were obtained by diluting the stock solution. HCl, NaOH, FeCl3, PbCl2, and oleic acid (supplied by Sigma–Aldrich) were analytical grade.
The batch adsorption experiments were performed in closed glass flasks of 100 mL filled with the solutions of pH-adjusted adsorbate at ambient temperature. The phenol losses by vaporizing was hindered by immediate sealing of flasks34. The pH of aqueous phase was changed with drops of HCl and NaOH (both are 0.1 M). All samples were filtered after equilibration to prevent interfering the carbon fines with the analysis. Finally, the concentrations of phenol in the supernatant solutions were determined by UV-spectrophotometry (271 nm). The results presented in the manuscript were based on the average of three tests, which were carried out three times for each condition.
To calculate the amount of the phenol adsorbed by sorbent (\({\mathrm{q}}_{\mathrm{e}}\)) the following expression was applied:
$${\mathrm{q}}_{\mathrm{e}}=\frac{({\mathrm{C}}_{0}-{\mathrm{C}}_{\mathrm{e}})\mathrm{V}}{\mathrm{m}}$$
in which, the adsorbate concentration at initial and equilibrium states are denoted by C0 and Ce in mg/L, V is the solution volume in L, and the grams of the sorbent materials is shown by m.
MSAC preparation
A mixture of 25 mL of 0.4 M Fe3+ solution plus 25 mL of 0.2 M Pb2+ solution was prepared. 2.5 g AC was added into the Pb2+/Fe3+ solution during a stirring stage37. NaOH was then added dropwise until pH became greater than 12. Following this stage, the Oleic acid was added42. The resulted precipitate was centrifuged and was then washed repeatedly with ethanol and water to obtain the MSAC adsorbent. The process was followed by a drying stage in an oven at 60 °C. Then, the obtained composite was pyrolysized in a high temperature of 700 °C under argon atmosphere. The time and the process rate was 1 h and 10 °C/min.
The high resolution pictures of the developed sorbent was prepared by Scanning electron microscopy (SEM) (Seron Technology, AIS2100). The elemental composition of the developed adsorbent was detected by energy dispersive X-ray (EDX, Shimadzu). For SEM and EDX analysis, around 50 mg mass of adsorbent was applied. The surface chemical properties of developed adsorbent were detected by FTIR spectroscopy (Shimadzu IR instrument). The samples (MSAC before and after adsorption) were prepared in KBr discs in the usual way from alright dried mixtures of about 4% (w/w). FTIR spectra were recorded by the buildup of a minimum of 64 scans with a resolution of 4 cm−1 per sample. BET analysis was applied to sepcify the pore size and specific surface area of the developed MSAC using Belsorp mini II model. Around 50 mg mass of each adsorbent is applied for BET analysis. In this experiment, the sample tube was submerged into the liquid N2 reservoir at 77 K during the N2 gas adsorption. Thermo-gravimetric analysis (using PT1000, LINSEIS, Germany) was conducted under sample heating from 20 to 900 °C at the heating rate of 10 °C/min. In this experiment, around 5 mg mass of adsorbent was used. The experiment was conducted in an N2 atmosphere from 20 to 800 °C and using a synthetic air atmosphere from 800 to 900 °C.
Adsorption modeling
Adsorption equilibrium
To further explore the adsorption mechanism, the adsorption isotherms of phenol onto MSAC and pristine activated carbon adsorbents were investigated using Tempkin43, Freundlich44, and Langmuir39 isotherm models which their equations are as follows:
$${\text{Tempkin}}\;q_{e} = B\ln (K_{T} C_{e} );\;\;\;B = RT/b_{T}$$
$${\text{Freundlich}}\;q_{e} = K_{F} C_{e}^{1/n}$$
$${\text{Langmuir}}\;q_{e} = \frac{{q_{m} K_{L} C_{e} }}{{1 + K_{L} C_{e} }}$$
where KT is constant with the unit of reciprocal of concentration (L/mg), and interconnected to maximum binding energy; B is the Tempkin constant corresponding to the adsorption heat (mg/g); 1/n is known as the exponent of Freundlich; KF is Freundlich model constant related to the adsorption capacity of adsorbent (mg/g)(mg/L)−1/n; and KL and qm are Langmuir constant (L/mg) and the theoretical saturation capacity (mg/g), respectively.
Adsorption thermodynamics
Temperature significantly affects the adsorption process. Therefore, the adsorbent properties could be modified, and the phenol properties in solutions could change. The thermodynamic parameters include \(\Delta\)G (Gibbs free energy change of adsorption) (kJ/mol), \(\Delta\)S (entropy change of adsorption) (kJ/mol K), and \(\Delta\)H (enthalpy change of adsorption) (kJ/mol), which can be calculated using equilibrium constants at diverse temperatures. These parameters are obtained as follow45:
$$\Delta \mathrm{G}= \Delta \mathrm{H}-\mathrm{T}\Delta \mathrm{S}$$
$$\mathrm{K}=\frac{{\mathrm{C}}_{\mathrm{a}}}{{\mathrm{C}}_{\mathrm{b}}}$$
$$\mathrm{lnK }=-\frac{\Delta \mathrm{H}}{\mathrm{RT}}+\frac{\Delta \mathrm{S}}{\mathrm{R}}$$
where K is the thermodynamic equilibrium constant (–), T is the temperature (K), R is the universal gas constant (kJ/mol K), Ca and Cb (mg/L) are the equilibrium concentration of phenol on the adsorbent and adsorbate phases, respectively.
Statistical physics evaluation
To theoretically understand the adsorption mechanism of phenol, statistical physics modeling was applied. This method assumes that a changeable amount of molecules is linked to NM receptor sites of the adsorbent (unit mass)31. The reaction of adsorption for adsorbate molecules (P) onto receptor sites of adsorbent (M) is as follows:
$$\mathrm{nP}+\mathrm{M}\to {\mathrm{P}}_{\mathrm{n}}\mathrm{M}$$
where n is the stoichiometric coefficient of reaction, the n parameter can be an integer value or not. The values of n smaller than the unity indicate that the multi-anchorage adsorption mechanism may occur, while those greater than the unity represent the assumption of a multi-molecular adsorption mechanism46. In this context, a single layer adsorption process and or two or more adsorption layers may be considered adsorption isotherm. This work used two sta-phy models: single-layer (Eq. 6) and double-layer (Eq. 7). The variation of the adsorbed quantity of adsorbate (Q (mg/g)) against the equilibrium concentration (C (mg/L)) for these models is given by:
$$\mathrm{Q}=\frac{{\mathrm{nN}}_{\mathrm{M}}}{1+{\left(\frac{{\mathrm{C}}_{1/2}}{\mathrm{C}}\right)}^{\mathrm{n}}}$$
$$\mathrm{Q}={\mathrm{nN}}_{\mathrm{M}}\frac{{\left(\frac{\mathrm{C}}{{\mathrm{C}}_{1}}\right)}^{\mathrm{n}}+{2\left(\frac{\mathrm{C}}{{\mathrm{C}}_{2}}\right)}^{2\mathrm{n}}}{1+{\left(\frac{\mathrm{C}}{{\mathrm{C}}_{1}}\right)}^{\mathrm{n}}+{\left(\frac{\mathrm{C}}{{\mathrm{C}}_{2}}\right)}^{2\mathrm{n}}}$$
where n denotes the number of phenol molecules connected per receptor sites of the developed adsorbent; C1/2 is the concentration at half-saturation; NM indicates the receptor areas density of adsorbent; Qsat is the absorbed amount of molecule at saturation which is equal to:
$${\mathrm{Q}}_{\mathrm{sat}}={\mathrm{nN}}_{\mathrm{M}}\left(1+{\mathrm{N}}_{2}\right)$$
In the monolayer adsorption process, N2 = 0.
C1 and C2 are respectively the first and second layer concentrations at half-saturation.
MSAC characterization
Figure 1a shows the zeta potential on the surface of MSAC. The pHpzc value of MSAC being 6.7 causes net charge of the developed adsorbent below this pH to be positive. At pH > pHpzc, the net charge of the developed adsorbent is negative. Therefore, protonated species will attract each other, and thus, the adsorption yield will increase.
(a) Zeta potential and (b) FTIR spectra (transmitance mode) of MSAC composite before and after phenol adsorption (T = 25 °C).
FTIR spectra of adsorbent were conducted to detect the surface chemical properties of MSAC adsorbent before and after the adsorption of phenol (Fig. 1b). The observed peak at 1601 cm−1 was attributed to the bending vibration of H–O. In addition, the adsorbed water has usually shown a peak in the region of 3600–3200 cm−1 owing to stretching vibration of H–O, \({\text{v}}_{\text{s}}\) OH, as shown by 3402 cm−1 in the FTIR spectra. The peaks emerged at 2920 was related to methylene group asymmetric (\({\text{v}}_{\text{as}}{{\text{CH}}}_{2}\)), and methylene group symmetric (\({\text{v}}_{\text{s}}{{\text{CH}}}_{2}\)) stetching vibration. The small sharp bands revealed at 2940–2840 cm−1 region was resulted from C–H stertching vibration (\({\text{v}}_{\text{s}}\) CH). For AC, the vibration of O–C–O and C–O usually produces bands in the 1180–1110 cm−1 region47. The presence of the lead and ferrite was confirmed by the observed bands at 700–600 cm−1 attributed to \({\text{v}}_{\text{s}}\) Fe-oxide and \({\text{v}}_{\text{s}}\) Pb–O-Pb. However, following the adsorption process these vibration peaks of metal–oxygen bonds were subjected to a small shift from 643 to 584 cm−1, and 687 to 588 cm−1, respectively. The phenol adsorption onto AC was demonstrated through the band apperaing at about 1384 cm−1. In addition, the emergence of new bonds at the region of 3110–2938 cm−1 can be assigned to the C–H peak of phenol. Also, the adsorbed phenol produced some additional bands in the fingerprint region (centered at 807 cm−1 and 774 cm−1) that proved a mono-substitution aromatic.
The morphology of MSAC adsorbent (Fig. 2a) was investigated by scanning electron microscopy. The SEM photograph coupled with X-ray mapping for the developed adsorbent are presented in Fig. 2b–e. Figure 2b shows the aggregation of the precipitates of Pb and Fe on the AC surface. From Fig. 2d,e illustrate the individual detection of Pb and Fe elements on the AC surface. In fact, these elements were homogeneously distributed on the prepared absorbent surface. To further understand the individual component percentages in the MSAC composite, Scanning electron microscopy-energy-dispersive X-ray spectroscopy (SEM–EDX) analysis was performed (Fig. 2c). EDX spectrum of the MSAC composite shows that the percentages of Pb and Fe were 14.96% and 7.21%, respectively, that of carbon was 69.94% and that of oxygen was 7.79%. The determined atomic ratio of Pb to Fe in the developed adsorbent was 1:2.02, which was in close proximity to the theoretical ratio in PbFe2O4. The results of BET analysis showed the pristine AC and MSAC have specific surface area of 1023.9 m2/g and 774.53 m2/g, respectively. Also, the pore size values of pristine AC and MSAC composite were 4.013 nm and 11.89 nm, respectively. The change in specific surface area and pore size of MSAC composite could be attributed to the blockage of some pores following the precipitation of Pb and Fe on the surface of activated carbon. The blockage was more likely to be occurred for small pores. Therefore, as the number of small pores in the course of MSAC preparation decreased, the specific surface area was reduced. However, the remaining relatively larger pores resulted in higher value for pore size of MSAC, as compared to pristine AC.
(a) Real picture, (b): SEM image, (c): EDX analysis of MSACadsorbent, and X-ray mapping of (d): lead, and (e): iron in MSAC.
Literature review indicated the phenol adsorption is governed mainly by micropore filling through the π–π dispersion interaction in micropores smaller than double molecular diameter of phenol48. Taking into account phenol molecular size (0.43–0.57 nm) and the pore size values of pristine AC and MSAC composite that are 4.013 nm and 11.89 nm, respectively, it can be found that the molecular size of phenol is less than the pore size obtaining for both sorbents. Therefore, the phenol molecules transference inside the adsorbent is ensured.
TGA analysis was applied to study the thermo-stability of organic/inorganic components in the developed MSAC adsorbent against the heating in a range of 20 °C to 900 °C. Stepwise heating process was performed in a nitrogen atmosphere at a rate of 10 °C/min. In addition, the TGA curve achieved continuously during the mass loss was also differentiated, which provides the derivative thermo-gravimetric (DTG) plot (Fig. 3). A slight weight loss at the initial stage of heating (in temperature range less that about 100–150 °C) is usually observed in some cases and can be due to the burn-off of volatile constituents. A significant weight loss is occurred after the initial stage in the temperature range of 200–450 °C, indicated by a sharp peak in DTG curve. At the final stage, by completion of the heating the metal compounds are retained.
TGA and DTG curves for MSAC adsorbent.
Analysis of thermo-gravimetric curves demonstrated three main portions of constituents for the developed MSAC: the higher portion composed of carbon-based material (70% by weight), the second one includes 9% weight volatile materials, and the third portion comprised of material with higher thermos-stability such as metals and other impurities.
Environmental Protection Agency (EPA) guidelines state that Pb and Fe in drinking water should not exceed 50 and 100 μg/L, respectively. In this regard, the adsorbent and water were mixed together for twenty-four hours using a magnetic stirrer at various initial pH values, ranging from 2 to 14. As seen in Fig. 4, the concentrations of lead and iron at each of the four pH values were far lower than what is considered acceptable.Therefore, it should be possible to successfully use this highly stable adsorbent for the purposes of treating wastewater.
Concentration of lead and iron in the leach liquor solution at different pH values (pH = 2–14, T = 25 °C).
Thermodynamic interpretation
Figure 5 shows the effect of temperature on the phenol adsorption onto MSAC composite and unmodified activated carbon at different initial phenol concentrations, i.e., the isotherm curves. The curves presented a favorable profile, with high phenol adsorption capacities even at low equilibrium concentrations. At Ce values lower than 200 mg/L, the curves are more inclined, revealing a good affinity between phenol and the magnetic activated carbon. It can be seen that, as Ce increases, this inclination decreases, and qe tends to a constant value (the plateau). This profile is common for the temperature range from 298 to 358 K.
Influence of temperature on phenol adsorption isotherms onto (a) MSAC and (b) pristine activated carbon (Initial phenol concentration = 100 mg/L, pH = 7 and adsorbent dosage = 1.5 g/L).
As the temperature of the process increased, the amount of phenol adsorbed onto both adsorbents, MSAC and pristine AC, decreased. Figure 6 shows the plotting ln K versus [1/T] for 100 mg/L. Table 1 displays the thermodynamic parameters of the phenol adsorption onto MSAC and pristine activated carbon. An increasing trend of temperature from 298 to 358 K during the adsorption process revealed negative amounts of ΔH and ΔG, indicating the spontaneous and exothermic nature of the process. Also, the negative ∆S exhibited the stability of the solution–solid interface during the adsorption of phenol onto MSAC composite. The negative values of ΔG indicated the spontaneous adsorption of phenol onto the developed adsorbent. Therefore, it was deduced that phenol adsorption onto MSAC at low temperatures is favored (Fig. 6).
Vant' Hoff plot for the phenol adsorption onto (a) MSAC and (b) pristine activated carbon.
Table 1 Thermodynamic parameters of phenol adsorption onto MSAC and pristine activated carbon at different temperature.
As the temperature of the process increased, the adsorption of phenol onto pristine activated carbon decreased. As given in Table 1, positive ΔG values describe that adsorption is non-spontaneous under examined conditions. In the case of pristien AC application for phenol adsorption, an increase in temperature from 298 to 358 led to the negative amounts of ΔH and ΔS, confirming the exothermic and associative nature of the process. Results of the thermodynamic studies for two adsorption processes of phenol onto MSAC and pristine activated carbon shows that lead ferrite coating on activated carbon causes the spontaneous adsorption process. In fact, lead ferrite coating on activated carbon changes the nature of the process from non-spontaneous toward spontaneous state.
Another thing is that the magnitude of ΔH was lower than 80 kJ/mol (Table 1). This observation is indicative that the phenol adsorption onto both adsorbents is controlled mainly by physical forces. Finally, it can be seen that the values of ΔG for MSAC composite increased with T, so the more negative value was found at 298 K, confirming that the phenol adsorption is favored at this temperature.
The possible explanation for this temperature-dependent behavior can be based on the adsorbent or adsorbate. It was encountered that the adsorbent is little affected by the temperature in this range (by analytical techniques and experiments). Its main characteristics, including surface chemistry and textural properties, keep the same from 298 to 358 K under aqueous media. In this sense, the temperature effect could be explained by the phenol characteristics. The temperature increase leads to an increase in phenol solubility. Consequently, phenol prefers to be water-diluted than be adsorbed into the adsorbent surface, leading to exothermic adsorption, as observed in this research.
Equilibrium studies
To better understand the detailed processes about the influence of temperature on the adsorption equilibrium, the experimental data were analyzed by Langmuir, Freundlich and Tempkin models. Figure 7 reveals the equilibrium curves for the adsorption of phenol onto MSAC composite and pristine activated carbon at different temperatures. It can be found that the adsorption capacity of both adsorbents decreased with an increase in temperature, being the highest capacities found at 298 K. Table 2 presents the isothermal parameters for the adsorption process in the case of both MSAC as well as pristine AC. The values of R2, R2adj and MSE indicated that the Langmuir and Tempkin models can represent the experimental data for MSAC composite in all temperatures 298–358 K. Analyzing isothermal data indicated that the experimental data for pristine activated carbon were best fitted in the Langmuir, Freundlich and Tempkin models.
Adsorption isotherm curves for phenol removal using (a) MSAC and (b) pristine activated carbon (pH = 7, concentration = 1.5 g/L).
Table 2 Equilibrium adsorption parameters for phenol removal by MSAC and pristine AC.
Statistical physics inferences
Sta-phy models assessed new insights about the phenol adsorption onto both adsorbents. Single-layer (Eq. 6) and double-layer (Eq. 7) models were selected. The model selection was based on the coefficient of determination (R2) and the number of models parameters. Table 3 depicts the R2 values for both models in all tested temperatures. It can be found that both models agree with the experimental isotherms since good R2 values were found. However, a detailed analysis showed that the single-layer model presented higher R2 values for all temperatures. Besides, the double-layer model has four parameters, while the single-layer has only three.
Table 3 Coefficient of determination (R2) for different statistical physics models for MSAC and pristine activated carbon.
In conclusion, the single-layer model was considered as the best sta-phy model to predict the phenol adsorption isotherms on both adsorbents. Therefore, the sta-phy parameters estimated by this model were applied for adsorption systems interpretation based on this result. Thus, the parameters are n (number of phenol molecules per site), NM (density of receptor sites), Qsat (adsorbed amount at saturation), and E (adsorption energy). Table 4 lists the mentioned parameters.
Table 4 Statistical physics parameters for the phenol adsorption on MSAC and pristine activated carbon based on the single-layer model.
Statistical physics model parameters (n and NM)
The examination of the number of phenol molecules linked per adsorbent receptor site provides valuable data to understand the adsorption mechanism of phenol on the adsorbent. This parameter can outline the geometrical position of phenol at different temperatures. It is enough to compare the values of n by the unity to identify the adsorption position. If the values of n are smaller than one, it describes the position of adsorption as a parallel because a part of phenol molecules is linked to the adsorbent. On the other hand, if the values of n are equal or higher than one, one or more phenol molecules are linked to each binding site of the adsorbent31. The effect of temperature on the number of phenol molecules linked per adsorbent site is shown in Fig. 8.
Change in n parameter with the variation of temperature for MSAC and pristine AC.
The values of n changed from 0.88 to 1.04 for the adsorption system of phenol- MSAC (Table 4 and Fig. 8). The value of n greater than one implied that the active site of MSAC adsorbent interacted with more than one phenol, and therefore inclined orientation was deduced31. The main adsorption site of the MSAC could capture a fraction of phenol molecule when n < 1. The later suggested that the orientation of molecules attached to the adsorbent surface was horizontal31. The behavior of this parameter showed that the phenol orientation on MSAC composite was changed as n raised up to 1.04 (Fig. 8). This phenomenon could be assigned to thermal agitation. Moreover, the physicochemical properties of the developed adsorbent is supposed to be responsible for the transition of phenol adsorption. In fact, the molecules of phenol establish a multi-molecular interaction with the surface of adsorbent. Also, the transition of phenol molecule on the adsorbent surface from parallel to vertical position can be possibly related to the adsorption of –OH group on the functional groups of the developed adsorbent.
As given in Table 4, the values of n for the adsorption system of phenol-pristine AC varied from 0.85 to 0.99. Considering the values of n for the mentioned system (n < 1), it can be found that pristine AC could capture a fraction of phenol molecule. Also, it was concluded that phenol molecule was docked with parallel orientation on pristine AC. The behavior of this parameter indicated that phenol molecule could not move freely on the surface of tested adsorbent (Fig. 9).
Schematic diagram for the change of phenol orientation onto MSAC and pristine activated carbon due to thermal agitation.
The density of receptor sites (NM) of tested adsorbent is an important parameter for the uptake of phenol onto the adsorbent. Figure 10 shows the evolution of the density of receptor sites of MSAC composite and pristine activated carbon at various temperatures. It can be found that the number of anchorages increased upon raising the temperature. This behavior expresses an antagonistic effect. As the amount of phenol molecules per site enhances, the accessible active site of the adsorbent for phenol removal decreases.
Dependence of the density of receptor sites on temperature for MSAC and pristine activated carbon.
Adsorbed amount at saturation (Qsat)
The parameter of adsorbed quantity at saturation provides valuable information about the adsorbent, which can be affected by external variables (pH, adsorbate concentration, and temperature). Figure 11 shows the change in the amount of adoption at the saturation state as temperature varies. From an analytical point of view, its values are equal to the product of the two parameters, n, and NM. As given in Table 4, as the temperature rose from 298 to 358 K, the value of Qsat for MSAC and pristine AC decreased from 158.94 and 138.91 to 115.23 and 112.34 mg/g indicating that the phenol adsorption onto both mentioned adsorbents is exothermic.
Influence of temperature on adsorbed quantity at saturation for MSAC and pristine activated carbon.
Results indicated that at all examined temperatures, the adsorption quantities at saturation for pristine activated carbon were smaller than those of MSAC composite, confirmed the experimental data. This discrepancy is related to a variation in the structure of MSAC as it has additional metallic hydroxides in comparison with pristine AC. These hydroxides can facilitate the adsorption of phenol through establishment of a stronger interaction.
Adsorption energy (E)
It is essential to calculate the adsorption energy during the adsorption process to identify the phenol removal mechanism. The C1/2 (i.e., half-saturation concentration) is associated with the binding energy of adsorbed layer, considering the expression of the monolayer adsorption process. From this parameter (C1/2), the adsorption energy value can be estimated and used to interpret the temperature effect. The expression of adsorption energy is written as:
$$\mathrm{E}=\mathrm{RTln}\left(\frac{{\mathrm{C}}_{\mathrm{s}}}{{\mathrm{C}}_{1/2}}\right)$$
where Cs denotes the solubility of phenol in water.
Table 4 presents the adsorption energies of phenol onto both adsorbents, which were relatively smaller than 20 kJ/mol. Figure 12 shows the evolution of adsorption energy at various temperatures. The obtained findings revealed that the adsorption of phenol from the aqueous phase using MSAC composite and pristine AC could occur via physical adsorption. Furthermore, as the temperature of the process increased, the adsorption energy increased, indicating that the nature of phenol adsorption was exothermic, which was agreement with the experimental data.
Influence of temperature on adsorption energy (E) for MSAC and pristine activated carbon.
According to the interpretation of single-layer model, it was concluded that the principal reason that enhanced the adsorption of phenol onto MSAC composite compared with the pristine AC is the metal hydroxides coated on activated carbon. These metal hydroxides from MSAC composite can contribute to a strong interaction with phenol molecule compared to the interactions between pristine AC and phenol molecules. In fact, the dominant mechanism for the adsorption of phenol on MSAC is attributed to the attractive electrostatic interaction that emerges between the metal hydroxides and phenol's –OH acidic group. It is noted that all the adsorption capacities of phenol onto MSAC are higher than those of pristine AC due to the stronger interactions between phenol molecule and MSAC composite, especially the additional interaction provided by metal hydroxides.
Proposed mechanism for the interaction of phenol with MSAC
There are various proposed mechanisms such as electrostatic attraction, pi-pi attraction between the phenolic ring and activated carbon basal planes, donor–acceptor complex formation, and hydrogen bonding between phenol molecules, and the presence of suitable functional groups on the adsorbent surface21,49. The attractive electrostatic interaction of metal hydroxides and phenol's –OH may affect the adsorption of phenol onto MSAC composite. Mechanistic studies showed that the values of the number of phenol molecules linked per adsorbent receptor site varied from 0.88 to 1.04 for the adsorption system of phenol- MSAC composite. In fact, the transition of phenol adsorption position was interconnected to the physicochemical characteristics of the developed adsorbent. The molecules of phenol establish a multi-molecular interaction with the surface of adsorbent. Also, the transition of phenol molecule on the adsorbent surface from parallel to vertical position can possibly be related to the adsorption of –OH interacting with the functional groups of the developed adsorbent. The latter is responsible for the higher adsorption quantities at saturation for MSAC composite than those of pristine activated carbon. This discrepancy is related to the structure of the adsorbent, as MSAC composite has additional –OH groups comparing to pristine AC. These groups enhance the phenol removal by creating a strong interaction with phenol molecules. In addition, functional groups on the AC surface may participate in the adsorption process. H-bond formation between these functional groups (i.e., OH and CO) (H-bond acceptor) phenol's –OH (H-bond donor), and also dipole–dipole attractions are the dominant interactions of phenol with the adsorbent. In addition to the electrostatic attraction, the interaction of phenol atomic rings with the adsorbent surface through pi-pi electron donor–acceptor interactions can result in the phenol adsorption by the developed adsorbent. The proposed mechanism for phenol adsorption onto MSAC composite is shown in Fig. 13.
Proposed mechanism for phenol adsorption onto MSAC composite.
Reusability of PbFe2O4 spinel-activated carbon composite
Reusability of an adsorbent is an important parameter in the adsorption process. There are different methods, such as heating or chemical regeneration and solvent washing for reusing the adsorbent. In this research, solvent washing, as a well-known technique, was used to regenerate the PbFe2O4 spinel-activated carbon composite. The regeneration of developed adsorbent for six successive cycles (adsorption–desorption) was carried out by using 1 mol/L NaOH. Figure 14 shows the phenol removal capacity of MSAC composite during different adsorption/desorption cycles. Results indicted that the sorbent preserved 85 percent of its maximum capacity after these six cycles. The interesting adsorption potential of MSAC composite after six cycles of regeneration makes it practically/commercially attractive and green.
Adsorption capacity of phenol onto MSAC composite in successive cycles (C0 = 300 mg/L, pH = 7, concentration = 1.5 g/L).
This research reports the theoretical analysis of phenol adsorption onto PbFe2O4 spinel—magnetic activated carbon (MSAC) and pristine activated carbon. The main characteristics of MSAC are a surface area of 774.53 m2/g, a pore size of 11.89 nm, particle diameter lower than 0.25 mm, and a zeta potential of 6.7. The main functional groups on the surface are OH, aromatic rings, COC, CH2, Pb–O–Pb, and Fe–O–Fe. The adsorption efficiency of MSAC composite decreased with increasing temperature, indicating the exothermic and associative nature of the process. Comparison of the thermodynamic parameters of two adsorption process of phenol onto MSAC and pristine activated carbon shows that lead ferrite coating on activated carbon change the nature of the process from a non-spontaneous toward a spontaneous state. Furthermore, to theoretically understand the adsorption mechanism of phenol, statistical physics modeling was applied. The results indicated that a good relationship resulted from the single-layer model. After that, n, NM, Qsat, and E parameters were obtained from this model. The values of the number of phenol molecules linked per adsorbent receptor site varied from 0.88 to 1.04 for the adsorption system of phenol- MSAC. The behavior of this parameter showed a change of phenol orientation on the MSAC composite due to thermal agitation. Also, the values of n for the adsorption system of phenol-pristine AC varied from 0.85 to 0.99, indicating that pristine AC could capture a fraction of phenol molecules. It was concluded that the phenol molecule was docked with parallel orientation on pristine AC. By increasing the number of phenol molecules per site, the accessible active site of the developed adsorbent for phenol removal decreases. Therefore, variation of temperature from 298 to 358 K, the value of Qsat decreased from 158.94 and 138.91 to 115.23 and 112.34 mg/g for MSAC and pristine AC, respectively. Also, the values of phenol adsorption energy were reduced, indicating that the phenol adsorption onto both mentioned adsorbents is exothermic. According to the interpretation of single-layer model, it was concluded that the principal reason that enhanced the adsorption of phenol onto MSAC composite compared with the pristine AC could be attributed to the functional role –OH groups coated on AC. These metal hydroxides from MSAC composite can contribute to establish a stronger interaction with phenols compared to the interactions between pristine AC and phenol molecules. The significant role of metallic hydroxides in MSAC to facilitate the removal of phenol is also confirmed by mechanistic studies. The main mechanism is the attractive electrostatic interaction of metal hydroxides with phenol's –OH acidic group.
The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.
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Department of Petroleum Engineering, Faculty of Petroleum, Gas and Petrochemical Engineering, Persian Gulf University, Bushehr, 75169-13817, Iran
Esmaeil Allahkarami & Abolfazl Dehghan Monfared
Persian Gulf Star Oil Company, Bandar Abbas, Iran
Esmaeil Allahkarami
Department of Civil and Environmental, Universidad de la Costa, CUC, Calle 58 # 55–66, Barranquilla, Atlántico, Colombia
Luis Felipe O. Silva
Chemical Engineering Department, Federal University of Santa Maria–UFSM, Santa Maria, RS, Brazil
Guilherme Luiz Dotto
Abolfazl Dehghan Monfared
E.A.: Experimental design and conduction, Modeling and Software, Preparing the original draft. A.D.M.: Managing and supervising the project, Conceptualization, Data curation, Investigation, Writing—Reviewing and Editing. L.F.O.S.: Software, Validation, Visualization, Editing. G.L.D.: Validation, Visualization, Editing, Modeling and Software.
Correspondence to Abolfazl Dehghan Monfared.
Allahkarami, E., Dehghan Monfared, A., Silva, L.F.O. et al. Toward a mechanistic understanding of adsorption behavior of phenol onto a novel activated carbon composite. Sci Rep 13, 167 (2023). https://doi.org/10.1038/s41598-023-27507-5 | CommonCrawl |
Newman's lemma
In mathematics, in the theory of rewriting systems, Newman's lemma, also commonly called the diamond lemma, states that a terminating (or strongly normalizing) abstract rewriting system (ARS), that is, one in which there are no infinite reduction sequences, is confluent if it is locally confluent. In fact a terminating ARS is confluent precisely when it is locally confluent.[1]
Equivalently, for every binary relation with no decreasing infinite chains and satisfying a weak version of the diamond property, there is a unique minimal element in every connected component of the relation considered as a graph.
Today, this is seen as a purely combinatorial result based on well-foundedness due to a proof of Gérard Huet in 1980.[2] Newman's original proof was considerably more complicated.[3]
Diamond lemma
In general, Newman's lemma can be seen as a combinatorial result about binary relations → on a set A (written backwards, so that a → b means that b is below a) with the following two properties:
• → is a well-founded relation: every non-empty subset X of A has a minimal element (an element a of X such that a → b for no b in X). Equivalently, there is no infinite chain a0 → a1 → a2 → a3 → .... In the terminology of rewriting systems, → is terminating.
• Every covering is bounded below. That is, if an element a in A covers elements b and c in A in the sense that a → b and a → c, then there is an element d in A such that b ∗→ d and c ∗→ d, where ∗→ denotes the reflexive transitive closure of →. In the terminology of rewriting systems, → is locally confluent.
The lemma states that if the above two conditions hold, then → is confluent: whenever a ∗→ b and a ∗→ c, there is an element d such that b ∗→ d and c ∗→ d. In view of the termination of →, this implies that every connected component of → as a graph contains a unique minimal element a, moreover b ∗→ a for every element b of the component.[4]
Notes
1. Franz Baader, Tobias Nipkow, (1998) Term Rewriting and All That, Cambridge University Press ISBN 0-521-77920-0
2. Gérard Huet, "Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems", Journal of the ACM (JACM), October 1980, Volume 27, Issue 4, pp. 797–821.
3. Harrison, p. 260, Paterson (1990), p. 354.
4. Paul M. Cohn, (1980) Universal Algebra, D. Reidel Publishing, ISBN 90-277-1254-9 (See pp. 25–26)
References
• M. H. A. Newman. On theories with a combinatorial definition of "equivalence". Annals of Mathematics, 43, Number 2, pages 223–243, 1942.
• Paterson, Michael S. (1990). Automata, languages, and programming: 17th international colloquium. Lecture Notes in Computer Science. Vol. 443. Warwick University, England: Springer. ISBN 978-3-540-52826-5.
Textbooks
• Term Rewriting Systems, Terese, Cambridge Tracts in Theoretical Computer Science, 2003. (book weblink)
• Term Rewriting and All That, Franz Baader and Tobias Nipkow, Cambridge University Press, 1998 (book weblink)
• John Harrison, Handbook of Practical Logic and Automated Reasoning, Cambridge University Press, 2009, ISBN 978-0-521-89957-4, chapter 4 "Equality".
External links
• Diamond lemma at PlanetMath.
• "Newman's Proof of Newman's Lemma", a PDF on the original proof, Archived July 6, 2017, at the Wayback Machine
| Wikipedia |
\begin{document}
\date{} \title{Learning Mean-Field Games}
\begin{abstract} This paper presents a general mean-field game (GMFG) framework for simultaneous learning and decision-making in stochastic games with a large population. It first establishes the existence of a unique Nash Equilibrium to this GMFG, and explains that naively combining Q-learning with the fixed-point approach in classical MFGs yields unstable algorithms. It then proposes a Q-learning algorithm with Boltzmann policy (GMF-Q), with analysis of convergence property and computational complexity. The experiments on repeated Ad auction problems demonstrate that this GMF-Q algorithm is
efficient and robust in terms of convergence and learning accuracy. Moreover, its performance is superior in convergence, stability, and learning ability, when compared with existing algorithms for multi-agent reinforcement learning.
\end{abstract} \section{Introduction} \label{introduction}
\paragraph{Motivating example.} This paper is motivated by the following Ad auction problem for an advertiser. An Ad auction is a stochastic game on an Ad exchange platform among a large number of players, the advertisers. In between the time a web user requests a page and the time the page is displayed, usually within a millisecond, a Vickrey-type of second-best-price auction is run to incentivize interested advertisers to bid for an Ad slot to display advertisement. Each advertiser has limited information before each bid: first, her own {\it valuation} for a slot depends on an unknown conversion of clicks for the item; secondly, she, should she win the bid, only knows the reward {\textit{after}} the user's activities on the website are finished. In addition, she has a budget constraint in this repeated auction.
The question is, how should she bid in this online sequential repeated game when there is a {\textit{large}} population of bidders competing on the Ad platform, with {\textit{unknown}} distributions of the conversion of clicks and rewards?
Besides the Ad auction, there are many real-world problems involving a large number of players and unknown systems. Examples include massive multi-player online role-playing games \cite{MMORPG}, high frequency tradings \cite{algo_trade_mfg}, and the sharing economy \cite{sharing_eco}.
\paragraph{Our work.}
Motivated by these problems, we consider a general framework of simultaneous learning and decision-making in stochastic games with a large population. We formulate a general mean-field-game (GMFG) with incorporation of action distributions, (randomized) relaxed policies, and with unknown rewards and dynamics. This {\color{black}general} framework can also be viewed as a generalized version of MFGs of McKean-Vlasov type \cite{MKV}, which is a different paradigm from the classical MFG. It is also beyond the scope of the existing Q-learning framework for Markov decision problem (MDP) with unknown distributions, as MDP is technically equivalent to a single player stochastic game.
On the theory front, this {\color{black}general} framework differs from all existing MFGs. We establish under appropriate technical conditions, the existence and uniqueness of the Nash equilibrium (NE) to this GMFG. On the computational front, we show that naively combining Q-learning with the three-step fixed-point approach in classical MFGs yields unstable algorithms. We then propose a Q-learning algorithm with Boltzmann policy (GMF-Q), establish its convergence property and analyze its computational complexity. Finally, we apply this GMF-Q algorithm to the Ad auction problem, where this GMF-Q algorithm demonstrates its
efficiency and robustness in terms of convergence and learning. Moreover,
its performance is superior, when compared with existing algorithms for multi-agent reinforcement learning for convergence, stability, and learning accuracy.
\paragraph{Related works.} On learning large population games with mean-field approximations, \cite{YYTXZ2017} focuses on inverse reinforcement learning for MFGs without decision making, \cite{YLLZZW2018} studies an MARL problem with a {\color{black}first-order} mean-field approximation term modeling the interaction between one player and all the other finite players, and \cite{KC2013} and \cite{YMMS2014} consider model-based adaptive learning for MFGs {\color{black}in specific models (\textit{e.g.}, linear-quadratic and oscillator games)}. {\color{black}More recently, \cite{Manymany} studies the local convergence of actor-critic algorithms on finite time horizon MFGs, and \cite{rl_mfg_local} proposes a policy-gradient based algorithm and analyzes the so-called local NE for reinforcement learning in infinite time horizon MFGs.} For learning large population games without mean-field approximation, see \cite{MARL_literature2, MARL_literature1} and the references therein. In the specific topic of learning auctions with a large number of advertisers, \cite{CRZMWYG2017} and \cite{JSLGWZ2018} explore reinforcement learning techniques to search for social optimal solutions with real-word data, and \cite{IJS2011} uses MFGs to model the auction system with unknown conversion of clicks within a Bayesian framework.
{\color{black}However, none of these works consider the problem of simultaneous learning and decision-making in a general MFG framework. Neither do they establish the existence and uniqueness of the {\color{black}(global)} NE, nor do they present model-free learning algorithms with complexity analysis and convergence to the NE.} Note that in principle, global results are harder to obtain compared to local results.
\section{Framework of General MFG (GMFG)}\label{n2mfg}
\subsection{Background: classical $N$-player Markovian game and MFG}\label{classical} Let us first recall the classical $N$-player game. There are $N$ players in a game.
At each step $t$, the state of player $i \ \ (=1, 2, \cdots, N)$ is $s^i_t\in\mathcal{S}\subseteq\mathbb{R}^d$ and she takes an action $a^i_t\in\mathcal{A}\subseteq\mathbb{R}^p$. Here $d, p$ are positive integers, and $\mathcal{S}$ and $\mathcal{A}$ are compact (for example, finite) state space and action space, respectively.
Given the current state profile of $N$-players ${\bf s}_t=(s^1_t,\dots,s^N_t)\in\mathcal{S}^N$ and the action $a^i_t$, player $i$ will receive a reward $r^i({\bf s}_t, a^i_t)$ and her state will change to $s^i_{t+1}$ according to a transition probability function $P^i({\bf s}_t, a^i_t)$.
A Markovian game further restricts the admissible policy/control for player $i$ to be of the form
$a^i_t=\pi^i_t({\bf s}_t)$. That is, $\pi^i_t:\mathcal{S}^N\rightarrow \mathcal{P}(\mathcal{A})$ maps each state profile ${\bf s}\in\mathcal{S}^N$ to a randomized action, with {\color{black}$\mathcal{P}(\mathcal{X})$ the space of probability measures on space $\mathcal{X}$}.
The accumulated reward (a.k.a. the value function) for player $i$, given the initial state profile ${\bf s}$ and the policy profile sequence $\pmb{\pi} :=\{\pmb{\pi}_t\}_{t=0}^{\infty} $ with $\pmb{\pi}_t=(\pi^1_t,\dots,\pi^N_t)$, is then defined as \begin{eqnarray}\label{game}
V^i({\bf s},\pmb{\pi}):=\mathbb{E}\left[\sum_{t=0}^{\infty}\gamma^t r^i({\bf s}_t,a^i_t)\Big| {\bf s}_0={\bf s}\right], \end{eqnarray} where $\gamma\in(0,1)$ is the discount factor, $a^i_t\sim \pi^i_t({\bf s}^t)$, and $s^i_{t+1}\sim P^i({\bf s}_t, a_t^i)$. The goal of each player is to maximize her value function over all admissible policy sequences.
In general, this type of stochastic $N$-player game is notoriously hard to analyze, especially when $N$ is large {\color{black}\cite{PR05}}. Mean field game (MFG), pioneered by \cite{HMC2006} and \cite{LL2007} {\color{black}in the continuous settings and later developed in \cite{MFG_n_conv, MFG_gomes, MFG_binact, MFG_discrete_time, MFG_discrete_time2} for discrete settings}, provides an ingenious and tractable aggregation approach to approximate the otherwise challenging $N$-player stochastic games.
The basic idea for an MFG goes as follows. Assume all players are identical, indistinguishable and interchangeable, when $N\to \infty$, one can view the limit of other players' states ${\bf s}_t^{-i}=(s_t^1,\dots,s_t^{i-1},s_t^{i+1},\dots,s_t^N)$ as a population state distribution {\color{black}$\mu_t$ with $\mu_t(s):=\lim_{N \rightarrow \infty}\frac{\sum_{j=1, j\neq i}^N \textbf{I}_{s_t^j=s}}{N}$}.\footnote{{\color{black}Here the indicator function $\textbf{I}_{s_t^j=s}=1$ if $s_t^j=s$ and $0$ otherwise.}} Due to the homogeneity of the players, one can then focus on a single (representative) player. That is, in an MFG, one may consider instead the following optimization problem, \[ \begin{array}{ll}
\text{maximize}_{\pmb{\pi}} & V(s,\pmb{\pi},\pmb{\mu}):=\mathbb{E}\left[\sum\limits_{t=0}^\infty \gamma^t r(s_t,a_t,\mu_t)|s_0=s\right]\\ \text{subject to} & s_{t+1}\sim P(s_t,a_t,\mu_t), \quad a_t\sim \pi_t(s_t,\mu_t), \end{array} \]
where $\pmb{\pi}:={\{\pi_t\}_{t=0}^{\infty}}$ denotes the policy sequence and $\pmb{\mu} := \{\mu_t\}_{t=0}^\infty$ the distribution flow. In this MFG setting, at time $t$, after the representative player chooses her action $a_t$ according to some policy $\pi_t$, she will receive reward $r(s_t,a_t,\mu_t)$ and her state will evolve under a \textit{controlled stochastic dynamics} of a mean-field type $P(\cdot|s_t,a_t,\mu_t)$. Here the policy $\pi_t$ depends on both the current state $s_t$ and the current population state distribution $\mu_t$ such that $\pi:\mathcal{S}\times \mathcal{P}(\mathcal{S})\rightarrow \mathcal{P}(\mathcal{A})$.
\subsection{General MFG (GMFG)}\label{mfg-set-up} In the classical MFG setting, the reward and the dynamic for each player are known. They depend only on $s_t$ the state of the player, $a_t$ the action of this particular player, and $\mu_t$ the population state distribution. In contrast, in the motivating auction example, the reward and the dynamic are unknown; they rely on the actions of {\it all} players, as well as on $s_t$ and $\mu_t$.
We therefore define the following general MFG (GMFG) framework. At time $t$, after the representative player chooses her action $a_t$ according to some policy $\pi:\mathcal{S}\times \mathcal{P}(\mathcal{S})\rightarrow \mathcal{P}(\mathcal{A})$, she will receive a reward $r(s_t,a_t,\mathcal{L}_t)$ and her state will evolve according to $P(\cdot|s_t,a_t,\mathcal{L}_t)$, where $r$ and $P$ are possibly unknown. The objective of the player is to solve the following control problem: \begin{equation}\label{mfg} \begin{array}{ll}
\text{maximize}_{\pmb{\pi}} & V(s,\pmb{\pi},\pmb{\mathcal{L}}):=\mathbb{E}\left[\sum\limits_{t=0}^\infty \gamma^t r(s_t,a_t,{\color{black}\mathcal{L}_t})|s_0=s\right]\\ \text{subject to} & s_{t+1}\sim P(s_t,a_t,{\color{black}\mathcal{L}_t}),\quad a_t\sim \pi_t(s_t,\mu_t). \end{array}\tag{GMFG} \end{equation} Here, $\pmb{\mathcal{L}}:=\{\mathcal{L}_t\}_{t=0}^{\infty}$, with $\mathcal{L}_t=\mathbb{P}_{s_t,a_t}\in \mathcal{P}(\mathcal{S}\times \mathcal{A})$ the joint distribution of the state and the action (\textit{i.e.}, the \text{population state-action pair}). $\mathcal {L}_t$ has marginal distributions $\alpha_t$ for the population action and $\mu_t$ for the population state. Notice that $\{\mathcal{L}_t\}_{t=0}^{\infty}$ could depend on time. Namely, an infinite time horizon MFG could still have time-dependent NE solution due to the mean information process (game interaction) in the MFG. This is fundamentally different from the theory of single-agent MDP where the optimal control, if exists uniquely, would be time independent in an infinite time horizon setting.
In this framework, we adopt the well-known Nash Equilibrium (NE) for analyzing stochastic games.
\begin{definition}[NE for GMFGs]\label{nash2} In \eqref{mfg}, a player-population profile $(\pmb{\pi}^{\star},\pmb{\mathcal{L}}^{\star}):=(\{\pi_t^\star\}_{t=0}^{\infty},\{\mathcal{L}_t^\star\}_{t=0}^{\infty})$
is called an NE if \begin{enumerate}
\item (Single player side) Fix $\pmb{\mathcal{L}}^{\star}$, for any policy sequence $\pmb{\pi}:=\{\pi_t\}_{t=0}^{\infty}$ and initial state $s\in \mathcal{S}$, \begin{equation} V\left(s,\pmb{\pi}^{\star},\pmb{\mathcal{L}}^{\star}\right)\geq V\left(s,\pmb{\pi},\pmb{\mathcal{L}}^{\star}\right). \end{equation}
\item (Population side) $\mathbb{P}_{s_t,a_t}= {\mathcal{L}_t^{\star}}$ for all $t\geq 0$, where $\{s_t,a_t\}_{t=0}^{\infty}$ is the dynamics under the policy sequence $\pmb{\pi}^{\star}$ starting from $s_0 \sim \mu_0^{\star}$, with $a_t\sim\pi_t^\star(s_t,{\color{black}\mu_t^{\star}})$, $s_{t+1}\sim P(\cdot|s_t,a_t,{\color{black}\mathcal{L}_t^\star})$, and $\mu_t^{\star}$ being the population state marginal of $\mathcal{L}_t^\star$. \end{enumerate} \end{definition} The single player side condition captures the optimality of $\pmb{\pi}^\star$, when the population side is fixed. The population side condition ensures the ``consistency'' of the solution: it guarantees that the state and action distribution flow of the single player does match the population state and action sequence $\pmb{\mathcal{L}}^{\star}$.
\subsection{Example: GMFG for the repeated auction} \label{section:example} Now, consider the repeated Vickrey auction with a budget constraint in Section \ref{introduction}. Take a representative advertiser in the auction. Denote $s_t \in \{0,1,2,\cdots,s_{\max}\}$ as the budget of this player at time $t$, where $s_{\max}\in \mathbb{N}^+$ is the maximum budget allowed on the Ad exchange with a unit bidding price. Denote $a_t$ as the bid price submitted by this player and $\alpha_t$ as the bidding/(action) distribution of the population. The reward for this advertiser with bid $a_t$ and budget $s_t$ is
\begin{eqnarray} \label{reward} r_t = {\color{black}\textbf{I}_{w_t^M=1}}\left[(v_t-a^M_t)-(1+\rho){\color{black}\textbf{I}_{s_{t}<a^M_t}}(a^M_t-s_{t})\right]. \end{eqnarray} Here $w^M_t$ takes values $1$ and $0$, with $w_t^M=1$ meaning this player winning the bid and $0$ otherwise. The probability of winning the bid would depend on $M$, the index for the game intensity, and $\alpha_t$. {\color{black}(See discussion on $M$ in Appendix \ref{intensity}.)} The conversion of clicks at time $t$ is $v_t$ and follows an unknown distribution. $a^M_t$ is the value of the second largest bid at time $t$, taking values from $0$ to $s_{\max}$, and depends on both $M$ and $\mathcal{L}_t$. Should the player win the bid, the reward $r_t$ consists of two parts, corresponding to the two terms in (\ref{reward}). The first term is the profit of wining the auction, as the winner only needs to pay for the second best bid {\color{black}$a_t^M$} in a Vickrey auction. The second term is the penalty of overshooting if the payment exceeds her budget, with a penalty rate $\rho$. At each time $t$, the budget dynamics $s_t$ follows, \begin{eqnarray*}
{s}_{t+1}=\left\{
\begin{array}{ll}
s_t, \qquad & w^M_t \neq 1,\\
s_t - a^M_t, \qquad & w^M_t=1 \text{ and } a^M_t \leq s_t,\\
0, \qquad & w^M_t =1 \text{ and } a^M_t > s_t.
\end{array}
\right.
\end{eqnarray*} That is, if this player does not win the bid, the budget will remain the same. If she wins and has enough money to pay, her budget will decrease from $s_t$ to $s_t - a^M_t$. However, if she wins but does not have enough money, her budget will be $0$ after the payment and there will be a penalty in the reward function. Note that in this game, both the rewards $r_t$ and the dynamics $s_t$ are unknown {\it a priori}.
In practice, one often modifies the dynamics of $s_{t+1}$ with a non-negative random budget fulfillment $\Delta({s}_{t+1})$ after the auction clearing \cite{GKP2012}, such that
$\hat{s}_{t+1} = {s}_{t+1} + \Delta({s}_{t+1})$.
One may see some particular choices of $\Delta({s}_{t+1})$ in the experiment section (Section \ref{experiments}).
\section{Solution for GMFGs}\label{MFG_basic} We now establish the existence and uniqueness of the NE to (\ref{mfg}), by generalizing the classical fixed-point approach for MFGs to this GMFG setting. (See \cite{HMC2006} and \cite{LL2007} for the classical case). It consists of three steps. \paragraph{Step A.} Fix $\pmb{\mathcal{L}}:=\{\mathcal{L}_t\}_{t=0}^{\infty}$, (\ref{mfg}) becomes the classical optimization problem. Indeed, with $\pmb{\mathcal{L}}$ fixed, the population state distribution sequence $\pmb{\mu}:=\{\mu_t\}_{t=0}^{\infty}$ is also fixed, hence the space of admissible policies is reduced to the single-player case. Solving (\ref{mfg}) is now reduced to finding a policy sequence
$\pi_{t,\pmb{\mathcal{L}}}^\star\in \Pi:=\{\pi\,|\,\pi:\mathcal{S}\rightarrow\mathcal{P}(\mathcal{A})\}$
over all admissible $\pmb{\pi}_{\pmb{\mathcal{L}}}=\{\pi_{t,\pmb{\mathcal{L}}}\}_{t=0}^{\infty}$,
to maximize
\[ \begin{array}{ll}
V(s,\pmb{\pi}_{\pmb{\mathcal{L}}}, \pmb{\mathcal{L}}):= &\mathbb{E}\left[\sum\limits_{t=0}^{\infty}\gamma^tr(s_t,a_t,\mathcal{L}_t)|s_0=s\right],\\ \text{subject to}& s_{t+1}\sim P(s_t,a_t,\mathcal{L}_t),\quad a_t\sim\pi_{t,\pmb{\mathcal{L}}}(s_t). \end{array} \] Notice that with $\pmb{\mathcal{L}}$ fixed, one can safely suppress the dependency on $\mu_t$ in the admissible policies. Moreover, given this fixed $\pmb{\mathcal{L}}$ sequence and the solution $\pmb{\pi}_{\pmb{\mathcal{L}}}^\star:=\{\pi_{t,\pmb{\mathcal{L}}}^\star\}_{t=0}^{\infty}$, one can define a mapping from the fixed population distribution sequence $\pmb{\mathcal{L}}$ to an arbitrarily chosen optimal randomized policy sequence. That is, $$\Gamma_1:\{\mathcal{P}(\mathcal{S}\times \mathcal{A})\}_{t=0}^{\infty}\rightarrow\{\Pi\}_{t=0}^{\infty}, $$ such that $\pmb{\pi}_{\pmb{\mathcal{L}}}^\star=\Gamma_1(\pmb{\mathcal{L}})$. Note that this $\pmb{\pi}_{\pmb{\mathcal{L}}}^\star$ sequence satisfies the single player side condition in Definition \ref{nash2} for the population state-action pair sequence $\pmb{\mathcal{L}}$. That is, $ V\left(s,\pmb{\pi}_{\pmb{\mathcal{L}}}^\star,{\color{black}\pmb{\mathcal{L}}}\right)\geq V\left(s,\pmb{\pi},{\color{black}\pmb{\mathcal{L}}}\right),$ for any policy sequence $\pmb{\pi} = \{\pi_t\}_{t=0}^{\infty}$ and any initial state $s\in\mathcal{S}$.
As in the MFG literature \cite{HMC2006}, a feedback regularity condition is needed for analyzing Step A. \begin{assumption}\label{policy_assumption} There exists a constant $d_1\geq 0$,
such that for any $\pmb{\mathcal{L}}, \pmb{\mathcal{L}}^{\prime} \in \{\mathcal{P}(\mathcal{S}\times\mathcal{A})\}_{t=0}^{\infty}$, \begin{equation}\label{Gamma1_lip} D(\Gamma_1(\pmb{\mathcal{L}}),\Gamma_1( \pmb{\mathcal{L}}^{\prime})) \leq d_1\mathcal{W}_1(\pmb{\mathcal{L}}, \pmb{\mathcal{L}}^{\prime}), \end{equation} where \begin{equation} \begin{split} D(\pmb{\pi},\pmb{\pi}^{\prime})&:=\sup_{s\in\mathcal{S}}\mathcal{W}_1(\pmb{\pi}(s),\pmb{\pi}^{\prime}(s))=\sup_{s\in\mathcal{S}}\sup_{t\in\mathbb{N}}W_1(\pi_t(s),\pi_t'(s)),\\ \mathcal{W}_1(\pmb{\mathcal{L}}, \pmb{\mathcal{L}}^{\prime})&:=\sup_{t\in\mathbb{N}}W_1(\mathcal{L}_t,\mathcal{L}_t'), \end{split} \end{equation} and $W_1$ is the $\ell_1$-Wasserstein distance between probability measures \cite{metrics_prob, COT_cuturi, OT_ON}. \end{assumption}
\paragraph{Step B.} Based on the analysis in Step A and $\pmb{\pi}_{\pmb{\mathcal{L}}}^\star=\{\pi_{t,\pmb{\mathcal{L}}}^\star\}_{t=0}^{\infty}$, update the initial sequence $\pmb{\mathcal{L}}$ to $ \pmb{\mathcal{L}}^{\prime}$ following the controlled dynamics $P(\cdot|s_t, a_t,\mathcal{L}_t)$.
Accordingly, for any admissible policy sequence $\pmb{\pi} \in \{\Pi\}_{t=0}^{\infty}$ and a joint population state-action pair sequence $\pmb{\mathcal{L}}\in \{\mathcal{P}(\mathcal{S}\times \mathcal{A})\}_{t=0}^{\infty}$, define a mapping $\Gamma_2:\{\Pi\}_{t=0}^{\infty}\times \{\mathcal{P}(\mathcal{S}\times\mathcal{A})\}_{t=0}^{\infty}\rightarrow \{\mathcal{P}(\mathcal{S}\times\mathcal{A})\}_{t=0}^{\infty}$ as follows:
\begin{eqnarray} \Gamma_2(\pmb{\pi},\pmb{\mathcal{L}}):= \pmb{\hat{\mathcal{L}}} = \{\mathbb{P}_{s_t,a_t}\}_{t=0}^{\infty}, \end{eqnarray}
where $s_{t+1}\sim \mu_t P(\cdot|\cdot,a_t,\mathcal{L}_t)$, $a_t\sim \pi_t(s_t)$, $s_0 \sim \mu_0$, and $\mu_t$ is the population state marginal of $\mathcal{L}_t$.
One needs a standard assumption in this step. \begin{assumption}\label{population_assumption} There exist constants $d_2,~d_3\geq 0$, such that for any admissible policy sequences $\pmb{\pi},\pmb{\pi}^1,\pmb{\pi}^2$ and joint distribution sequences $\pmb{\mathcal{L}}, \pmb{\mathcal{L}}^{1}, \pmb{\mathcal{L}}^{2}$, \begin{equation}\label{Gamma2_lip1} \mathcal{W}_1(\Gamma_2(\pmb{\pi}^1,\pmb{\mathcal{L}}),\Gamma_2(\pmb{\pi}^2,\pmb{\mathcal{L}})) \leq d_2 D(\pmb{\pi}^1,\pmb{\pi}^2), \end{equation} \begin{equation}\label{Gamma2_lip2} \mathcal{W}_1(\Gamma_2(\pmb{\pi},\pmb{\mathcal{L}}^1{\color{black})},\Gamma_2(\pmb{\pi},\pmb{\mathcal{L}}^2)) \leq d_3 \mathcal{W}_1(\pmb{\mathcal{L}}^1,\pmb{\mathcal{L}}^2). \end{equation} \end{assumption} Assumption \ref{population_assumption} can be reduced to Lipschitz continuity and boundedness of the transition dynamics $P$. (See the Appendix for more details.)
\paragraph{Step C.} Repeat Step A and Step B until $ \pmb{\mathcal{L}}^{\prime}$ matches $\pmb{\mathcal{L}}$.
This step is to take care of the population side condition. To ensure the convergence of
the combined step A and step B, it suffices if $\Gamma:\{\mathcal{P}(\mathcal{S}\times\mathcal{A})\}_{t=0}^{\infty}\rightarrow \{\mathcal{P}(\mathcal{S}\times\mathcal{A})\}_{t=0}^{\infty}$ is a contractive mapping under the $\mathcal{W}_1$ distance, with $\Gamma(\pmb{\mathcal{L}}):=\Gamma_2(\Gamma_1(\pmb{\mathcal{L}}), \pmb{\mathcal{L}})$. Then by the Banach fixed point theorem {\color{black}and the completeness of the related metric spaces}, there exists a unique NE to the GMFG.
In summary, we have
\begin{theorem}[Existence and Uniqueness of GMFG solution] \label{thm1} Given Assumptions \ref{policy_assumption} and \ref{population_assumption}, and assuming that $d_1d_2+d_3< 1$, there exists a unique NE to \eqref{mfg}.
\end{theorem}
\section{RL Algorithms for {\color{black}(stationary)} GMFGs}\label{AIQL}
In this section, we design the computational algorithm for the GMFG. Since the reward and transition distributions are unknown, this is simultaneously learning the system and finding the NE of the game.
We will focus on the case with finite state and action spaces, \textit{i.e.}, $|\mathcal{S}|,|\mathcal{A}|<\infty$. We will look for stationary (time independent) NEs. Accordingly, we abbreviate $\pmb{\pi}:=\{\pi\}_{t=0}^{\infty}$ and $\pmb{\mathcal{L}}:=\{\mathcal{L}\}_{t=0}^{\infty}$ as $\pi$ and $\mathcal{L}$, respectively. This stationarity property enables developing appropriate time-independent Q-learning algorithm, suitable for an infinite time horizon game. Modification from the GMFG framework to this special stationary setting is straightforward, and is left to Appendix \ref{stat_mfg_app}. Note that the assumptions to guarantee the existence and uniqueness of GMFG solutions are slightly different between the stationary and non-stationary cases. For instance, one can compare \eqref{Gamma2_lip1}-\eqref{Gamma2_lip2} with \eqref{Gamma2_lip1_stat}-\eqref{Gamma2_lip2_stat}.
The algorithm consists of two steps, parallel to Step $A$ and Step $B$ in Section \ref{MFG_basic}. \paragraph{Step 1: Q-learning with stability for fixed $\mathcal{L}$.} With $\mathcal{L}$ fixed, it becomes a standard learning problem for an infinite horizon MDP. We will focus on the Q-learning algorithm \cite{Sutton, Ben_tutorial}.
The Q-learning algorithm approximates the value iteration by stochastic approximation. At each step with the state $s$ and an action $a$, the system reaches state $s'$ according to the controlled dynamics and the Q-function is updated according to \begin{equation}\label{Q-learning} Q_{\mathcal{L}}(s,a) \leftarrow ~(1-\beta_t(s,a))Q_{\mathcal{L}}(s,a)+ \beta_t(s,a) \left[r(s,a,\mathcal{L})\right.+\left.\gamma \max\nolimits_{\tilde{a}} Q_{\mathcal{L}}(s^{\prime},\tilde{a})\right], \end{equation}
where the step size $\beta_t(s,a)$ can be chosen as (\textit{cf}. \cite{Q-rate}) \[ \beta_t(s,a)= \begin{cases}
|\#(s,a,t)+1|^{-h}, & (s,a)=(s_t,a_t),\\ 0, & \text{otherwise}. \end{cases} \] with $h\in(1/2,1)$. Here $\#(s,a,t)$ is the number of times up to time $t$ that one visits the pair $(s,a)$. The algorithm then proceeds to choose action $a'$ based on $Q_{\mathcal{L}}$ with appropriate exploration strategies, including the $\epsilon$-greedy strategy.
After obtaining the approximate $\hat{Q}_{\mathcal{L}}^\star$, in order to retrieve an approximately optimal policy,
it would be natural to define an \textbf{argmax-e} operator so that actions with equal maximum Q-values would have equal probabilities to be selected. Unfortunately, the discontinuity and sensitivity of \textbf{argmax-e} could lead to an unstable algorithm (see Figure \ref{fig:naive} for the corresponding naive Algorithm \ref{AQL_MFG} in Appendix).
\footnote{\textbf{argmax-e} is not continuous: Let $x=(1,1)$, then $\textbf{argmax-e}(x)=(1/2,1/2)$. For any $\epsilon>0$, let $y=(1,1-\epsilon)$, then $\textbf{argmax-e}(y)=(1,0)$.}
Instead, we consider a Boltzmann policy based on the operator $\textbf{softmax}_c:\mathbb{R}^n\rightarrow\mathbb{R}^n$, defined as \begin{eqnarray}\label{eqn:soft_max_operator} \textbf{softmax}_c(x)_i=\frac{\exp(cx_i)}{\sum_{j=1}^n\exp(cx_j)}. \end{eqnarray} This operator is smooth and close to the \textbf{argmax-e} (see Lemma \ref{soft-arg-diff} in the Appendix). Moreover, even though Boltzmann policies are not optimal, the difference between the Boltzmann and the optimal one can always be controlled by choosing the hyper-parameter $c$ appropriately in the \textbf{softmax} operator. {\color{black}Note that other smoothing operators (\textit{e.g.}, Mellowmax \cite{Mellowmax}) may also be considered in the future.}
\paragraph{Step 2: error control in updating $\mathcal {L}$.} Given the sub-optimality of the Boltzmann policy, one needs to characterize the difference between the optimal policy and the non-optimal ones. In particular, one can define the action gap between the best action and the second best action in terms of the Q-value as $\delta^s( \mathcal{L}):=\max_{a'\in\mathcal{A}}Q^\star_{\mathcal{L}}(s,a')-\max_{a\notin \text{argmax}_{a\in\mathcal{A}}Q^\star_{\mathcal{L}}(s,a)}Q^\star_{\mathcal{L}}(s,a)>0$. Action gap is important for approximation algorithms \cite{gap-increase}, and are closely related to the problem-dependent bounds for regret analysis in reinforcement learning and multi-armed bandits, and advantage learning algorithms including A2C \cite{A2C}.
The problem is: in order for the learning algorithm to converge in terms of $\mathcal{L}$ (Theorem \ref{conv_AIQL}), one needs to ensure a definite differentiation between the optimal policy and the sub-optimal ones. This is problematic as the infimum of $\delta^s(\mathcal{L})$ over an infinite number of $\mathcal{L}$ can be $0$. To address this, the population distribution at step $k$, say $\mathcal{L}_k$,
needs to be projected to a finite grid, called $\epsilon$-net. The relation between the $\epsilon$-net and \text{action gaps} is as follows:
\noindent\textit{For any $\epsilon>0$, there exist a positive function $\phi(\epsilon)$ and an $\epsilon$-net $S_{\epsilon}:=\{\mathcal{L}^{(1)},\dots,\mathcal{L}^{(N_{\epsilon})}\}$ $\subseteq$ $\mathcal{P}(\mathcal{S}\times\mathcal{A})$, with the properties that $\min_{i=1,\dots,N_{\epsilon}} d_{TV}(\mathcal{L},\mathcal{L}^{(i)})\leq \epsilon$ for any $\mathcal{L}\in\mathcal{P}(\mathcal{S}\times\mathcal{A})$, and that $\max_{a'\in\mathcal{A}}Q^\star_{\mathcal{L}^{(i)}}(s,a')-Q^\star_{\mathcal{L}^{(i)}}(s,a)\geq \phi(\epsilon)$
for any $i=1,\dots,N_{\epsilon}$, $s\in\mathcal{S}$, and any $a\notin \text{argmax}_{a\in\mathcal{A}}Q^\star_{\mathcal{L}^{(i)}}(s,a)$. }
Here the existence of $\epsilon$-nets is trivial due to the compactness of the probability simplex $\mathcal{P}(\mathcal{S}\times\mathcal{A})$, and the existence of $\phi(\epsilon)$ comes from the finiteness of the action set $\mathcal{A}$. In practice, $\phi(\epsilon)$ often takes the form of $D\epsilon^{\alpha}$ with $D>0$ and the exponent $\alpha>0$ characterizing the decay rate of the action gaps.
Finally, to enable Q-learning, it is assumed that one has access to a population simulator (See \cite{Batch_MARL, MARL_PDO}). That is, for any policy $\pi\in\Pi$, given the current state $s\in\mathcal{S}$, for any population distribution $\mathcal{L}$, one can obtain the next state $s'\sim P(\cdot|s,\pi(s,\mu),\mathcal{L})$, a reward $r= r(s,\pi(s,\mu),\mathcal{L})$, and the next population distribution $\mathcal{L}'=\mathbb{P}_{s',\pi(s',\mu)}$. For brevity, we denote the simulator as $(s',r, \mathcal{L}')=\mathcal{G}(s,\pi,\mathcal{L})$. Here $\mu$ is the state marginal distribution of $\mathcal{L}$.
In summary, we propose the following Algorithm \ref{AIQL_MFG}.
\begin{algorithm}[h]
\caption{\textbf{Q-learning for GMFGs (GMF-Q)}}
\label{AIQL_MFG} \begin{algorithmic}[1]
\STATE \textbf{Input}: Initial $\mathcal{L}_0$, tolerance $\epsilon>0$.
\FOR {$k=0, 1, \cdots$}
\STATE Perform Q-learning for $T_k$ iterations to find the approximate Q-function $\hat{Q}_k^\star(s,a)=\hat{Q}^\star_{\mathcal{L}_k}(s,a)$ of an MDP with dynamics $P_{\mathcal{L}_k}(s'|s,a)$ and rewards $r_{\mathcal{L}_k}(s,a)$.
\STATE Compute $\pi_k\in\Pi$ with $\pi_k(s)=\textbf{softmax}_c(\hat{Q}^\star_k(s,\cdot))$.
\STATE Sample $s\sim \mu_k$ ($\mu_k$ is the population state marginal of $\mathcal{L}_k$), obtain $\tilde{\mathcal{L}}_{k+1}$ from $\mathcal{G}(s,\pi_k,\mathcal{L}_k)$.
\STATE Find $\mathcal{L}_{k+1}=\textbf{Proj}_{S_{\epsilon}}(\tilde{\mathcal{L}}_{k+1})$ \ENDFOR \end{algorithmic} \end{algorithm}
Note that \textbf{softmax} is applied only at the end of each outer iteration when a good approximation of $Q$ function is obtained. Within the outer iteration for the MDP problem with fixed mean-field information, standard Q-learning method is applied.
Here $\textbf{Proj}_{S_{\epsilon}}(\mathcal{L})=\text{argmin}_{\mathcal{L}^{(1)},\dots,\mathcal{L}^{(N_{\epsilon})}}d_{TV}(\mathcal{L}^{(i)},\mathcal{L})$. {\color{black}For computational tractability, it would be sufficient to choose $S_{\epsilon}$ as a truncation grid so that projection of $\tilde{\mathcal{L}}_k$ onto the epsilon-net reduces to truncating $\tilde{\mathcal{L}}_k$ to a certain number of digits. For instance, in our experiment, the number of digits is chosen to be 4. The choices of the hyper-parameters $c$ and $T_k$ can be found in Lemma \ref{Q-finite-bd} and Theorem \ref{conv_AIQL}. In practice, the algorithm is rather robust with respect to these hyper-parameters.}
In the special case when the rewards $r_{\mathcal{L}}$ and transition dynamics $P(\cdot|s,a,\mathcal{L})$ are known, one can replace the Q-learning step in the above Algorithm \ref{AIQL_MFG} by a value iteration, resulting in the GMF-V Algorithm \ref{VI} in the Appendix.
We next show the convergence of this GMF-Q algorithm (Algorithm \ref{AIQL_MFG}) to an $\epsilon$-Nash of \eqref{mfg}, with complexity analysis.
\begin{theorem}[Convergence and complexity of GMF-Q]\label{conv_AIQL} Assume the same conditions in {\color{black}Theorem \ref{thm1_stat}} and Lemma \ref{Q-finite-bd} in the Appendix. For any tolerances $\epsilon,~\delta>0$, set $\delta_k=\delta/ K_{\epsilon,\eta}$, $\epsilon_k=(k+1)^{-(1+\eta)}$ for some $\eta\in(0,1]$ $(k=0,\dots,K_{\epsilon,\eta}-1)$,
$T_k=T^{\mathcal{M}_{\mathcal{L}_k}}(\delta_k,\epsilon_k)$ (defined in Lemma \ref{Q-finite-bd} in the Appendix) and $c=\frac{\log(1/\epsilon)}{\phi(\epsilon)}$. Then with probability at least $1-2\delta$, $W_1(\mathcal{L}_{K_{\epsilon,\eta}},\mathcal{L}^\star)\leq C\epsilon$. \\
Moreover, the total number of iterations $T=\sum_{k=0}^{K_{\epsilon,\eta} -1}T^{\mathcal{M}_{\mathcal{L}_k}}(\delta_k,\epsilon_k)$ is bounded by \footnote{{\color{black}Let $h=\frac{3}{4}$, $\eta=1$, the bound reduces to $T=O(K_{\epsilon}^{\frac{19}{3}}(\log(\frac{K_{\epsilon}}{\delta}))^{\frac{41}{3}})$. Note that this bound may not be tight.}} \begin{equation}\label{Tbound} T=O\left(K_{\epsilon,\eta}^{1+\frac{4}{h}}\left(\log(K_{\epsilon,\eta}/\delta)\right)^{\frac{2}{1-h}+\frac{2}{h}+3}\right). \end{equation} Here $K_{\epsilon,\eta}:=\left\lceil 2\max\left\{(\eta\epsilon{\color{black}/c})^{-1/\eta},\log_d(\epsilon/{\color{black}\max\{\text{diam}(\mathcal{S})\text{diam}(\mathcal{A}),{\color{black}c}\}})+1\right\}\right\rceil$ is the number of outer iterations, $h$ is the step-size exponent in Q-learning (defined in Lemma \ref{Q-finite-bd} in the Appendix), and the constant $C$ is independent of $\delta$, $\epsilon$ and $\eta$. \end{theorem}
The proof of Theorem \ref{conv_AIQL} in the Appendix depends on the Lipschitz continuity of the \textbf{softmax} operator \cite{softmax}, the closeness between \textbf{softmax} and the \textbf{argmax-e} (Lemma~\ref{soft-arg-diff} in the Appendix), and the complexity of Q-learning for the MDP (Lemma~\ref{Q-finite-bd} in the Appendix).
\section{Experiment: repeated auction game}\label{experiments}
In this section, we report the performance of the proposed GMF-Q Algorithm. The objectives of the experiments include 1) testing the convergence, stability, and learning ability of GMF-Q in the GMFG setting, and 2) comparing GMF-Q with existing multi-agent reinforcement learning algorithms, including IL algorithm and MF-Q algorithm.
We take the GMFG framework for the repeated auction game from Section \ref{section:example}. Here each advertiser learns to bid in the auction with a budget constraint.
\paragraph{Parameters.}
The model parameters are set as: $|\mathcal{S}|=|\mathcal{A}|=10$, the overbidding penalty $\rho=0.2$,
the distributions of the conversion rate $v \sim$ uniform(${\color{black}\{1,2,3,4\}})$,
and the competition intensity index $M=5$.
The random fulfillment is chosen as: if $s < s_{\max}$, $\Delta(s)=1$ with probability $\frac{1}{2}$ and $\Delta(s)=0$ with probability $\frac{1}{2}$; if $s=s_{\max}$, $\Delta(s)=0$.
The algorithm parameters are (unless otherwise specified): the temperature parameter $c=4.0$, the discount factor $\gamma=0.8$, the parameter $h$ from Lemma \ref{Q-finite-bd} in the Appendix being $h=0.87$, and the baseline inner iteration being $2000$. {Recall that for GMF-Q, both $v$ and the dynamics of $P$ for $s$ are unknown {\it a priori}.} {\color{black}The $90\%$-confidence intervals are calculated with $20$ sample paths.} \paragraph{Performance evaluation in the GMFG setting.} Our experiment shows that the GMF-Q Algorithm is efficient and robust, and learns well. \paragraph{{\it Convergence and stability of GMF-Q.}} GMF-Q is efficient and robust. First, GMF-Q converges after about $10$ outer iterations; secondly, as the number of inner iterations increases, the error decreases (Figure~\ref{fig:inexact}); and finally, the convergence is robust with respect to both the change of number of states and the initial population distribution (Figure~\ref{fig:different_state}).
In contrast, the Naive algorithm does not converge even with $10000$ inner iterations, and the {\color{black}joint distribution $\mathcal{L}_t$} keeps fluctuating (Figure~\ref{fig:naive}).
\paragraph{{\it Learning accuracy of GMF-Q.}} GMF-Q learns well. Its learning accuracy is tested against its special form GMF-V {\color{black}(Appendix \ref{GMF-V})}, with the latter assuming a known distribution of conversion rate $v$ and the dynamics $P$ for the budget $s$. The relative $L_2$ distance between the Q-tables of these two algorithms is
$\Delta Q := \frac{\|Q_{\text{GMF-V}}-Q_{\text{GMF-Q}}\|_2}{\|Q_{\text{GMF-V}}\|_2}=0.098879$. This implies that GMF-Q learns the true GMFG solution with $90$-percent accuracy with $10000$ inner iterations.
The heatmap in Figure~\ref{fig:q_learn_table} is the Q-table for GMF-Q Algorithm after $20$ outer iterations. Within each outer iteration, there are $T_k^{\text{GMF-Q}} = 10000$ inner iterations. The heatmap in Figure~\ref{fig:q_iteration_table} is the Q-table for GMF-Q Algorithm after $20$ outer iterations. Within each outer iteration, there are $T_k^{\text{GMF-V}} = 5000$ inner iterations.
\begin{table}
\begin{center}
\caption{Q-table with $T_k^{\text{GMF-V}} = 5000$.}
\label{tab:q_comparison}
\begin{tabular}{c|c|c|c|c}
$T_k^{\text{GMF-Q}} $ &1000& 3000 & 5000&10000 \\
\hline
$\Delta Q$ &0.21263 & 0.1294& 0.10258&0.0989
\end{tabular}
\end{center}
\end{table}
\begin{figure}
\caption{Q-tables: GMF-Q vs. GMF-V.}
\label{fig:q_learn_table}
\label{fig:q_iteration_table}
\label{fig:q_tables}
\end{figure}
\begin{figure}
\caption{Convergence with different \\\hspace{\textwidth} number of inner iterations.}
\label{fig:inexact}
\caption{Convergence with different \\\hspace{\textwidth}number of states.}
\label{fig:different_state}
\end{figure}
\begin{figure}
\caption{Fluctuations of Naive Algorithm (30 sample paths).}
\label{fig:naive_1}
\label{fig:naive_1}
\label{fig:naive}
\end{figure}
\begin{figure}
\caption{Learning accuracy based on $C(\pmb{\pi})$.}
\label{fig:comparison1}
\label{fig:comparison1}
\label{fig:comparison2}
\label{fig:comparison}
\end{figure}
\paragraph{Comparison with existing algorithms for $N$-player games.} To test the effectiveness of GMF-Q for approximating $N$-player games, we next compare GMF-Q with IL algorithm and MF-Q algorithm. IL algorithm \cite{T1993} considers $N$ independent players and each player solves a decentralized reinforcement learning problem ignoring other players in the system. The MF-Q algorithm \cite{YLLZZW2018} extends the NASH-Q Learning algorithm for the $N$-player game introduced in \cite{HW2003}, adds the aggregate actions $(\bar{\pmb{a}}_{-i}=\frac{\sum_{j \ne i}a_j}{{\color{black}N}-1})$ from the opponents, and works for the class of games where the interactions are only through the average actions of $N$ players.
\paragraph{{\it Performance metric.}} We adopt the following metric to measure the difference between a given policy $\pi$ and an NE (here $\epsilon_0>0$ is a safeguard, and is taken as $0.1$ in the experiments):
$$C(\pmb{\pi}) = \frac{1}{N|\mathcal{S}|^N}\sum\nolimits_{i=1}^N \sum\nolimits_{\pmb{s} \in \mathcal{S}^N }\dfrac{\max_{{\pi}^i} V_i(\pmb{s},({\pmb{\pi}^{-i}},\pi^i))-V_i(\pmb{s},{\pmb{\pi}})}{|\max_{{\pi}^i} V_i(\pmb{s},({\pmb{\pi}^{-i}},\pi^i))|+\epsilon_0}.$$ Clearly $C(\pmb{\pi}) \geq 0$, and $C(\pmb{\pi}^*)=0$ if and only if $\pmb{\pi}^*$ is an NE.
Policy $\arg \max _{{\pi}_i} V_i(\pmb{s},({\pmb{\pi}^{-i}},\pi_i))$ is called the best response to $\pmb{\pi}^{-i}$. A similar metric without normalization has been adopted in \cite{PPP2018}.
Our experiment (Figure \ref{fig:comparison}) shows that GMF-Q is superior in terms of convergence rate, accuracy, and stability for approximating an $N$-player game: GMF-Q converges faster than IL and MF-Q, with the smallest error, and with the lowest variance, as {\color{black}$\epsilon$-net improves the stability.}
For instance, when $N=20$, IL Algorithm converges with the largest error $0.220$.
The error from MF-Q is $0.101$, smaller than IL but still bigger than the error from GMF-Q. The GMF-Q converges with the lowest error $0.065$. {\color{black} Moreover, as $N$ increases, the error of GMF-Q deceases while the errors of both MF-Q and IL increase significantly. As $|\mathcal{S}|$ and $|\mathcal{A}|$ increase, GMF-Q is robust with respect to this increase of dimensionality, while
both MF-Q and IL clearly suffer from the increase of the dimensionality with decreased convergence rate and accuracy.} Therefore, GMF-Q is more scalable than IL and MF-Q, when the system is complex and the number of players $N$ is large.
\section{Conclusion} This paper builds a GMFG framework for simultaneous learning and decision-making, establishes the existence and uniqueness of NE, and proposes a Q-learning algorithm GMF-Q with convergence and complexity analysis. Experiments demonstrate superior performance of GMF-Q.
\section*{Acknowledgment} We thank Haoran Tang for the insightful early discussion on stabilizing the Q-learning algorithm and sharing the ideas of his work on soft-Q-learning \cite{softQ}, which motivates our adoption of the soft-max operators. {\color{black}We also thank the anonymous NeurIPS 2019 reviewers for the valuable suggestions.}
\appendix
\section{Distance metrics and completeness} This section reviews some basic properties of the Wasserstein distance. It then proves that the metrics defined in the main text are indeed distance functions and define complete metric spaces.
\paragraph{$\ell_1$-Wasserstein distance and dual representation.} The $\ell_1$ Wasserstein distance over $\mathcal{P}(\mathcal{X})$ for $\mathcal{X}\subseteq\mathbb{R}^k$ is defined as \begin{equation}\label{W1}
W_1(\nu,\nu'):=\inf_{M\in\mathcal{M}(\nu,\nu')}\int_{\mathcal{X}\times\mathcal{X}}\|x-y\|_2\text{d}M(x,y). \end{equation} where $\mathcal{M}(\nu,\nu')$ is the set of all measures (couplings) on $\mathcal{X}\times\mathcal{X}$, with marginals $\nu$ and $\nu'$ on the two components, respectively.
The Kantorovich duality theorem enables the following equivalent dual representation of $W_1$: \begin{equation}\label{W1_dual}
W_1(\nu,\nu')=\sup_{\|f\|_L\leq 1}\left|\int_{\mathcal{X}}fd\nu-\int_{\mathcal{X}}fd\nu'\right|, \end{equation}
where the supremum is taken over all $1$-Lipschitz functions $f$, \textit{i.e.}, $f$ satisfying $|f(x)-f(y)|\leq \|x-y\|_2$ for all $x,y\in\mathcal{X}$.
The Wasserstein distance $W_1$ can also be related to the total variation distance via the following inequalities \cite{metrics_prob}: \begin{equation}\label{W1_tv} d_{\min}(\mathcal{X})d_{TV}(\nu,\nu')\leq W_1(\nu,\nu')\leq \text{diam}(\mathcal{X})d_{TV}(\nu,\nu'), \end{equation}
where $d_{\min}(\mathcal{X})=\min_{x\neq y\in\mathcal{X}}\|x-y\|_2$, which is guaranteed to be positive when $\mathcal{X}$ is finite.
When $\mathcal{S}$ and $\mathcal{A}$ are compact, for any compact subset $\mathcal{X}\subseteq\mathbb{R}^k$, and for any $\nu,\nu'\in\mathcal{P}(\mathcal{X})$, $W_1(\nu,\nu')\leq \text{diam}(\mathcal{X})d_{TV}(\nu,\nu')\leq \text{diam}(\mathcal{X})<\infty$, where $\text{diam}(\mathcal{X})=\sup_{x,y\in\mathcal{X}}\|x-y\|_2$ and $d_{TV}$ is the total variation distance. Moreover, one can verify \begin{lemma}\label{metrics} Both $D$ and $\mathcal{W}_1$ are distance functions, and they are finite for any input distribution pairs. In addition, both $(\{\Pi\}_{t=0}^{\infty},D)$ and $(\{\mathcal{P}(\mathcal{S}\times\mathcal{A})\}_{t=0}^{\infty}, \mathcal{W}_1)$ are \textit{complete metric spaces}. \end{lemma} These facts enable the usage of Banach fixed-point mapping theorem for the proof of existence and uniqueness (Theorems \ref{thm1} and \ref{thm1_stat}). \begin{proof}[Proof of Lemma \ref{metrics}] It is known that for any compact set $\mathcal{X}\subseteq\mathbb{R}^k$, $(\mathcal{P}(\mathcal{X}), W_1)$ defines a complete metric space \cite{wass_comp}. Since $W_1(\nu,\nu')\leq \text{diam}(\mathcal{X})$ is uniformly bounded for any $\nu,~\nu'\in\mathcal{P}(\mathcal{X})$, we know that $\mathcal{W}_1(\pmb{\mathcal{L}},\pmb{\mathcal{L}}')\leq \text{diam}(\mathcal{X})$ and $D(\pmb{\pi},\pmb{\pi'})\leq \text{diam}(\mathcal{X})$ as well, so they are both finite for any input distribution pairs. It is clear that they are distance functions based on the fact that $W_1$ is a distance function.
Finally, we show the completeness of the two metric spaces $(\{\Pi\}_{t=0}^{\infty},D)$ and $(\{\mathcal{P}(\mathcal{S}\times\mathcal{A})\}_{t=0}^{\infty}, \mathcal{W}_1)$. Take $(\{\Pi\}_{t=0}^{\infty},D)$ for example. Suppose that $\pmb{\pi}^k$ is a Cauchy sequence in $(\{\Pi\}_{t=0}^{\infty},D)$. Then for any $\epsilon>0$, there exists a positive integer $N$, such that for any $m,~n\geq N$, \begin{equation} D(\pmb{\pi}^n,\pmb{\pi}^{m})\leq \epsilon\Longrightarrow W_1(\pi_t^n(s),\pi_t^{m}(s))\leq \epsilon\text{ for any $s\in\mathcal{S}$, $t\in\mathbb{N}$}, \end{equation}
which implies that $\pi_t^k(s)$ forms a Cauchy sequence in $(\mathcal{P}(\mathcal{A}),W_1)$, and hence by the completeness of $(\mathcal{P}(\mathcal{A}),W_1)$, $\pi_t^k(s)$ converges to some $\pi_t(s)\in\mathcal{P}(\mathcal{A})$. As a result, $\pmb{\pi}^n\rightarrow\pmb{\pi}\in\{\Pi\}_{t=0}^{\infty}$ under metric $D$, which shows that $(\{\Pi\}_{t=0}^{\infty},D)$ is complete.
The completeness of $(\{\mathcal{P}(\mathcal{S}\times\mathcal{A})\}_{t=0}^{\infty}, \mathcal{W}_1)$ can be proved similarly.
\end{proof}
The same argument for Lemma \ref{metrics} shows that both $D$ and $W_1$ are distance functions and are finite for any input distribution pairs, with both $(\Pi, D)$ and $(\mathcal{P}(\mathcal{S}\times\mathcal{A}),W_1)$ again complete metric spaces.
\section{Existence and uniqueness for stationary NE of GMFGs} \label{stat_mfg_app} \begin{definition}[Stationary NE for GMFGs]\label{nash2_stat} In \eqref{mfg}, a player-population profile ($\pi^\star$, $\mathcal{L}^\star$)
is called a stationary NE if \begin{enumerate}
\item (Single player side) For any policy $\pi$ and any initial state $s\in \mathcal{S}$, \begin{equation} V\left(s,\pi^\star,{\color{black}\mathcal{L}^\star}\right)\geq V\left(s,\pi,{\color{black}\mathcal{L}^\star}\right). \end{equation}
\item (Population side) $\mathbb{P}_{s_t,a_t}= {\mathcal{L}^{\star}}$ for all $t\geq 0$, where $\{s_t,a_t\}_{t=0}^{\infty}$ is the dynamics under the policy $\pi^\star$ starting from $s_0 \sim \mu^{\star}$, with $a_t\sim\pi^\star(s_t,{\color{black}\mu^{\star}})$, $s_{t+1}\sim P(\cdot|s_t,a_t,{\color{black}\mathcal{L}^\star})$, and $\mu^{\star}$ being the population state marginal of $\mathcal{L}^\star$. \end{enumerate} \end{definition}
The existence and uniqueness of the NE to (\ref{mfg}) in the stationary setting can be established by modifying appropriately the same fixed-point approach for the GMFG in the main text.
\paragraph{Step 1.} Fix $\mathcal{L}$, the GMFG becomes the classical optimization problem. That is, solving (\ref{mfg}) is now reduced to finding a policy
$\pi_{\mathcal{L}}^\star\in \Pi:=\{\pi\,|\,\pi:\mathcal{S}\rightarrow\mathcal{P}(\mathcal{A})\}$ to maximize
\[ \begin{array}{ll}
V(s,\pi_{\mathcal{L}}, \mathcal{L}):= &\mathbb{E}\left[\sum\limits_{t=0}^{\infty}\gamma^tr(s_t,a_t,\mathcal{L})|s_0=s\right],\\ \text{subject to}& s_{t+1}\sim P(s_t,a_t,\mathcal{L}),\quad a_t\sim\pi_{\mathcal{L}}(s_t). \end{array} \] Now given this fixed $\mathcal{L}$ and the solution $\pi_{\mathcal{L}}^\star$ to the above optimization problem, one can again define $$\Gamma_1:\mathcal{P}(\mathcal{S}\times \mathcal{A})\rightarrow\Pi, $$ such that $\pi_{\mathcal{L}}^\star=\Gamma_1(\mathcal{L})$. Note that this $\pi_{\mathcal{L}}^\star$ satisfies the single player side condition for the population state-action pair $L$, \begin{equation} V\left(s,\pi_{\mathcal{L}}^\star,\mathcal{L}\right)\geq V\left(s,\pi,\mathcal{L}\right), \end{equation} for any policy $\pi$ and any initial state $s\in\mathcal{S}$.
Accordingly, a similar feedback regularity condition is needed in this step. \begin{assumption}\label{policy_assumption_stat} There exists a constant $d_1\geq 0$,
such that for any $\mathcal{L}, \mathcal{L}' \in \mathcal{P}(\mathcal{S}\times\mathcal{A})$, \begin{equation}\label{Gamma1_lip} D(\Gamma_1(\mathcal{L}),\Gamma_1(\mathcal{L}'){\color{black})} \leq d_1W_1(\mathcal{L}, \mathcal{L}'), \end{equation} where \begin{equation} \begin{split} D(\pi,\pi')&:=\sup_{s\in\mathcal{S}}W_1(\pi(s),\pi'(s)), \end{split} \end{equation} and $W_1$ is the $\ell_1$-Wasserstein distance (a.k.a. earth mover distance) between probability measures. \end{assumption}
\paragraph{Step 2.} Based on the analysis of Step 1 and $\pi_{\mathcal{L}}^\star$, update the initial $\mathcal{L}$ to $\mathcal{L}'$ following the controlled dynamics $P(\cdot|s_t, a_t,\mathcal{L})$.
Accordingly, define a mapping $\Gamma_2:\Pi\times \mathcal{P}(\mathcal{S}\times\mathcal{A})\rightarrow \mathcal{P}(\mathcal{S}\times\mathcal{A})$ as follows:
\begin{eqnarray} \Gamma_2(\pi, \mathcal{L}):=\hat{\mathcal{L}}= \mathbb{P}_{s_1,a_1}, \end{eqnarray}
where {\color{black}$a_1\sim \pi(s_1)$}, $s_{1}\sim \mu P(\cdot|\cdot,a_0,\mathcal{L})$, $a_0\sim\pi(s_0)$, $s_0 \sim \mu$, and $\mu$ is the population state marginal of $\mathcal{L}$.
One also needs a similar assumption in this step. \begin{assumption}\label{population_assumption_stat} There exist constants $d_2,~d_3\geq 0$, such that for any admissible policies $\pi,\pi_1,\pi_2$ and joint distributions $\mathcal{L}, \mathcal{L}_1, \mathcal{L}_2$, \begin{equation}\label{Gamma2_lip1_stat} W_1(\Gamma_2(\pi_1,\mathcal{L}),\Gamma_2(\pi_2,\mathcal{L})) \leq d_2 D(\pi_1,\pi_2),\\ \end{equation} \begin{equation}\label{Gamma2_lip2_stat} W_1(\Gamma_2(\pi,\mathcal{L}_1),\Gamma_2(\pi,\mathcal{L}_2)) \leq d_3 W_1(\mathcal{L}_1,\mathcal{L}_2). \end{equation} \end{assumption}
\paragraph{Step 3.} Repeat until $\mathcal{L}'$ matches $\mathcal{L}$.
This step is to ensure the population side condition. To ensure the convergence of
the combined step one and step two, it suffices if $\Gamma:\mathcal{P}(\mathcal{S}\times\mathcal{A})\rightarrow \mathcal{P}(\mathcal{S}\times\mathcal{A})$ with $\Gamma(\mathcal{L}):=\Gamma_2(\Gamma_1(\mathcal{L}), \mathcal{L})$ is a contractive mapping (under the $W_1$ distance).
Similar to the proof of Theorem \ref{thm1}, again by the Banach fixed point theorem and the completeness of the related metric spaces, there exists a unique stationary NE of the GMFG. That is,
\begin{theorem}[Existence and Uniqueness of stationary MFG solution] \label{thm1_stat} Given Assumptions \ref{policy_assumption_stat} and \ref{population_assumption_stat}, and assume $d_1d_2+d_3< 1$. Then there exists a unique stationary NE to \eqref{mfg}. \end{theorem}
\section{Additional comments on assumptions} As mentioned in the main text, the single player side Assumption \ref{policy_assumption} and its counterpart Assumption \ref{policy_assumption_stat} for the stationary version correspond to the feedback regularity condition in the classical MFG literature. Here we add some comments on the population side Assumption \ref{population_assumption} and its stationary version Assumption \ref{population_assumption_stat}. For simplicity and clarity, let us consider the stationary case with finite state and action spaces. Then we have the following result. \begin{lemma}\label{assumption2_exp}
Suppose that $\max_{s,a,\mathcal{L},s'}P(s'|s,a,\mathcal{L})\leq c_1$, and that $P(s'|s,a,\cdot)$ is $c_2$-Lipschitz in $W_1$, \textit{i.e.}, \begin{equation}
|P(s'|s,a,\mathcal{L}_1)-P(s'|s,a,\mathcal{L}_2)|\leq c_2W_1(\mathcal{L}_1,\mathcal{L}_2). \end{equation} Then in Assumption \ref{population_assumption_stat}, $d_2$ and $d_3$ can be chosen as \begin{equation}
d_2=\frac{2\text{diam}(\mathcal{S})\text{diam}(\mathcal{A})|\mathcal{S}|c_1}{d_{\min}(\mathcal{A})} \end{equation}
and $d_3=\frac{\text{diam}(\mathcal{S})\text{diam}(\mathcal{A})c_2}{2}$, respectively. \end{lemma}
Lemma \ref{assumption2_exp} provides an explicit characterization of the population side assumptions based only on the boundedness and Lipschitz properties of the transition dynamics $P$. In particular, $c_1$ becomes smaller when the transition dynamics becomes more diverse and the state space becomes larger.
\begin{proof} (Lemma \ref{assumption2_exp}) We begin by noticing that $\mathcal{L}'=\Gamma_2(\pi,\mathcal{L})$ can be expanded and computed as follows: \begin{equation}
\mu'(s')=\sum\nolimits_{s\in\mathcal{S},a\in\mathcal{A}}\mu(s)P(s'|s,a,\mathcal{L})\pi(s,a),\quad \mathcal{L}'(s',a')=\mu'(s')\pi(s',a'), \end{equation} where $\mu$ is the state marginal distribution of $\mathcal{L}$.
Now by the inequalities (\ref{W1_tv}), we have \begin{equation} \begin{split} W_1&(\Gamma_2(\pi_1,\mathcal{L}),\Gamma_2(\pi_2,\mathcal{L}))\leq \text{diam}(\mathcal{S}\times\mathcal{A})d_{TV}(\Gamma_2(\pi_1,\mathcal{L}),\Gamma_2(\pi_2,\mathcal{L}))\\
=&\dfrac{\text{diam}(\mathcal{S}\times\mathcal{A})}{2}\sum_{s'\in\mathcal{S},a'\in\mathcal{A}}\left|\sum_{s\in\mathcal{S},a\in\mathcal{A}}\mu(s)P(s'|s,a,\mathcal{L})\left(\pi_1(s,a)\pi_1(s',a')-\pi_2(s,a)\pi_2(s',a')\right)\right|\\
\leq& \dfrac{\text{diam}(\mathcal{S}\times\mathcal{A})}{2}\max_{s,a,\mathcal{L},s'}P(s'|s,a,\mathcal{L})\sum_{s,a,s',a'}\mu(s)(\pi_1(s,a)+\pi_2(s,a))|\pi_1(s',a')-\pi_2(s',a')|\\
\leq& \dfrac{\text{diam}(\mathcal{S}\times\mathcal{A})}{2}\max_{s,a,\mathcal{L},s'}P(s'|s,a,\mathcal{L})\sum_{s',a'}|\pi_1(s',a')-\pi_2(s',a')|\cdot (1+1)\\
=&2\text{diam}(\mathcal{S}\times\mathcal{A})\max_{s,a,\mathcal{L},s'}P(s'|s,a,\mathcal{L})\sum_{s'}d_{TV}(\pi_1(s'),\pi_2(s'))\\
\leq & \frac{2\text{diam}(\mathcal{S}\times\mathcal{A})\max_{s,a,\mathcal{L},s'}P(s'|s,a,\mathcal{L})|\mathcal{S}|}{d_{\min}(\mathcal{A})}D(\pi_1,\pi_2)= \frac{2\text{diam}(\mathcal{S})\text{diam}(\mathcal{A})|\mathcal{S}|c_1}{d_{\min}(\mathcal{A})}D(\pi_1,\pi_2). \end{split} \end{equation}
Similarly, we have \begin{equation} \begin{split} W_1&(\Gamma_2(\pi,\mathcal{L}_1),\Gamma_2(\pi,\mathcal{L}_2))\leq \text{diam}(\mathcal{S}\times\mathcal{A})d_{TV}(\Gamma_2(\pi,\mathcal{L}_1),\Gamma_2(\pi,\mathcal{L}_2))\\
=&\dfrac{\text{diam}(\mathcal{S}\times\mathcal{A})}{2}\sum_{s'\in\mathcal{S},a'\in\mathcal{A}}\left|\sum_{s\in\mathcal{S},a\in\mathcal{A}}\mu(s)\pi(s,a)\pi(s',a')\left(P(s'|s,a,\mathcal{L}_1)-P(s'|s,a,\mathcal{L}_2)\right)\right|\\
\leq & \dfrac{\text{diam}(\mathcal{S}\times\mathcal{A})}{2}\sum_{s,a,s',a'}\mu(s)\pi(s,a)\pi(s',a')\left|P(s'|s,a,\mathcal{L}_1)-P(s'|s,a,\mathcal{L}_2)\right|\\ \leq & \frac{\text{diam}(\mathcal{S})\text{diam}(\mathcal{A})c_2}{2}.
\end{split} \end{equation} This completes {\color{black}the} proof. \end{proof}
\section{Proof of Theorems \ref{thm1} and \ref{thm1_stat}}
For notational simplicity, we only present the proof for the stationary case (Theorem \ref{thm1_stat}). The proof of Theorems \ref{thm1} is the same with appropriate notational changes.
First by Definition \ref{nash2_stat} and the definitions of $\Gamma_i$ $(i=1,2)$, $(\pi,\mathcal{L})$ is a stationary NE iff $\mathcal{L}=\Gamma(\mathcal{L})=\Gamma_2(\Gamma_1(\mathcal{L}),\mathcal{L})$ and $\pi=\Gamma_1(\mathcal{L})$, where $\Gamma(\mathcal{L})=\Gamma_2(\Gamma_1(\mathcal{L}),\mathcal{L})$. This indicates that for any $\mathcal{L}_1,\mathcal{L}_2\in\mathcal{P}(\mathcal{S}\times\mathcal{A})$, \begin{equation} \begin{split}
&W_1(\Gamma(\mathcal{L}_1),\Gamma(\mathcal{L}_2))=W_1(\Gamma_2(\Gamma_1(\mathcal{L}_1),\mathcal{L}_1),\Gamma_2(\Gamma_1(\mathcal{L}_2),\mathcal{L}_2))\\
&\leq W_1(\Gamma_2(\Gamma_1(\mathcal{L}_1),\mathcal{L}_1),\Gamma_2(\Gamma_1(\mathcal{L}_2),\mathcal{L}_1))+W_1(\Gamma_2(\Gamma_1(\mathcal{L}_2),\mathcal{L}_1),\Gamma_2(\Gamma_1(\mathcal{L}_2),\mathcal{L}_2))\\
&\leq (d_1d_2+d_3)W_1(\mathcal{L}_1,\mathcal{L}_2).
\end{split} \end{equation} And since $d_1d_2+d_3\in[0,1)$, by the Banach fixed-point theorem, we conclude that there exists a unique fixed-point of $\Gamma$, or equivalently, a unique stationary MFG solution to \eqref{mfg}.
\section{Proof of Theorem \ref{conv_AIQL}}
The proof of Theorem \ref{conv_AIQL} relies on the following lemmas.
\begin{lemma}[\cite{softmax}]\label{softmax-lip}
The softmax function is $c$-Lipschitz, \textit{i.e.}, $\|\textbf{softmax}_c(x)-\textbf{softmax}_c(y)\|_2\leq c\|x-y\|_2$ for any $x,~y\in\mathbb{R}^n$. \end{lemma}
{\color{black} Notice that for a finite set $\mathcal{X}\subseteq\mathbb{R}^k$ and any two (discrete) distributions $\nu,~\nu'$ over $\mathcal{X}$, we have \begin{equation} \begin{split}
W_1(\nu,\nu')&\leq \text{diam}(\mathcal{X})d_{TV}(\nu,\nu')=\frac{\text{diam}(\mathcal{X})}{2}\|\nu-\nu'\|_1\leq \frac{\text{diam}(\mathcal{X})}{2}\|\nu-\nu'\|_2, \end{split} \end{equation}
where in computing the $\ell_1$-norm, $\nu,~\nu'$ are viewed as vectors of length $|\mathcal{X}|$.
Hence Lemma \ref{softmax-lip} implies that for any $x,~y\in\mathbb{R}^{|\mathcal{X}|}$, when $\textbf{softmax}_c(x)$ and $\textbf{softmax}_c(y)$ are viewed as probability distributions over $\mathcal{X}$, we have \[
W_1(\textbf{softmax}_c(x),\textbf{softmax}_c(y))\leq \frac{\text{diam}(\mathcal{X})c}{2}\|x-y\|_2\leq \frac{\text{diam}(\mathcal{X})\sqrt{|\mathcal{X}|}c}{2}\|x-y\|_{\infty}.
\]}
\begin{lemma}\label{soft-arg-diff} The distance between the softmax and the argmax mapping is bounded by \[
\|\textbf{softmax}_c(x)-\textbf{argmax-e}(x)\|_2\leq 2n\exp(-c\delta), \] where {\color{black}$\delta=x_{\max}-\max_{x_j<x_{\max}}x_j$, $x_{\max}=\max_{i=1,\dots,n}x_i$, and} $\delta:=\infty$ when all $x_j$ are equal. \end{lemma}
{\color{black} Similar to Lemma \ref{softmax-lip}, Lemma \ref{soft-arg-diff} implies that
for any $x\in\mathbb{R}^{|\mathcal{X}|}$, viewing $\textbf{softmax}_c(x)$ as probability distributions over $\mathcal{X}$ leads to \[
W_1(\textbf{softmax}_c(x),\textbf{argmax-e}(x))\leq \text{diam}(\mathcal{X})|\mathcal{X}|\exp(-c\delta). \]
\begin{proof}[Proof of Lemma~\ref{soft-arg-diff}] Without loss of generality, assume that $x_1=x_2=\dots= x_m=\max_{i=1,\dots,n}x_i=x^\star>x_j$ for all $m<j\leq n$. Then \[ \textbf{argmax-e}(x)_i= \begin{cases} \frac{1}{m}, & i \leq m,\\ 0, & otherwise. \end{cases} \] \[ \textbf{softmax}_c(x)_i= \begin{cases} \frac{e^{cx^\star}}{me^{cx^\star}+\sum_{j=m+1}^ne^{cx_j}}, & i\leq m, \\ \frac{e^{cx_i}}{me^{cx^\star}+\sum_{j=m+1}^ne^{cx_j}}, & otherwise. \end{cases} \] Therefore \begin{equation*} \begin{split}
\|\textbf{soft}&\textbf{max}_c(x)-\textbf{argmax-e}(x)\|_2\leq \|\textbf{softmax}_c(x)-\textbf{argmax-e}(x)\|_1\\ =&m\left(\frac{1}{m}-\frac{e^{cx^\star}}{me^{cx^\star}+\sum_{j=m+1}^ne^{cx_j}}\right)+\frac{\sum_{i=m+1}^ne^{cx_i}}{me^{cx^\star}+\sum_{j=m+1}^ne^{cx_j}}\\ =& \frac{2\sum_{i=m+1}^ne^{cx_i}}{me^{cx^\star}+\sum_{i=m+1}^ne^{cx_i}}= \frac{2\sum_{i=m+1}^ne^{-c\delta_i}}{m+\sum_{i=m+1}^ne^{-c\delta_i}}\\ \leq &\frac{2}{m}\sum_{i=m+1}^ne^{-c\delta_i}\leq \frac{2(n-m)}{m}e^{-c\delta}\leq 2ne^{-c\delta}, \end{split} \end{equation*} with $\delta_i=x_i-x^\star$. \end{proof}
\begin{lemma}[\cite{Q-rate}]\label{Q-finite-bd} For an MDP, say $\mathcal{M}$, suppose that the Q-learning algorithm takes step-sizes \[ \beta_t(s,a)= \begin{cases}
|\#(s,a,t)+1|^{-h}, & (s,a)=(s_t,a_t),\\ 0, & \text{otherwise}. \end{cases} \]
with $h\in(1/2,1)$. Here $\#(s,a,t)$ is the number of times up to time $t$ that one visits the state-action pair $(s,a)$. Also suppose that the covering time of the state-action pairs is bounded by $L$ with probability at least {\color{black}$1-p$ for some $p\in(0,1)$}. Then $\|Q_{T^{\mathcal{M}}(\delta,\epsilon)}-Q^\star\|_{\infty}\leq \epsilon$ with probability at least $1-2\delta$. Here $Q_T$ is the $T$-th update in Q-learning, and $Q^\star$ is the (optimal) Q-function, given that \[ \begin{split}
T^{\mathcal{M}}(\delta,&\epsilon)=\Omega\left(\left(\dfrac{L\log_p(\delta)}{\beta}\log\dfrac{V_{\max}}{\epsilon}\right)^{\frac{1}{1-h}}+\left(\dfrac{\left(L\log_p(\delta)\right)^{1+3h}V_{\max}^2\log\left(\frac{|\mathcal{S}||\mathcal{A}|V_{\max}}{\delta\beta\epsilon}\right)}{\beta^2\epsilon^2}\right)^{\frac{1}{h}}\right), \end{split} \] where $\beta=(1-\gamma)/2$, $V_{\max}=R_{\max}/(1-\gamma)$, and $R_{\max}$ is an upper bound on the extreme difference between the expected rewards, \textit{i.e.}, $\max_{s,a,\mu}{r}(s,a,\mu)-\min_{s,a,\mu}{r}(s,a,\mu)\leq R_{\max}$.
\end{lemma}
Here the covering time {\color{black}$L$} of a state-action pair sequence is defined to be the number of steps needed to visit all state-action pairs starting from any arbitrary state-action pair, and {\color{black} $T^{\mathcal{M}}(\delta,\epsilon)$ is the number of inner iterations $T_k$ set in Algorithm~\ref{AIQL_MFG}. This will guarantee the convergence in Theorem \ref{conv_AIQL}.}
{\color{black}Also notice that the $l_{\infty}$ norm above is defined in an element-wise sense, \textit{i.e.}, for $M\in\mathbb{R}^{|\mathcal{S}|\times|\mathcal{A}|}$, we have $\|M\|_\infty=\max_{s\in\mathcal{S},a\in\mathcal{A}}|M(s,a)|$.}
\begin{proof}[Proof of Theorem \ref{conv_AIQL}] Define $\hat{\Gamma}_1^k(\mathcal{L}_k):=\textbf{softmax}_c\left(\hat{Q}^\star_{\mathcal{L}_k}\right)$. In the following, $\pi=\textbf{softmax}_c(Q_{\mathcal{L}})$ is understood as the policy $\pi$ with $\pi(s)=\textbf{softmax}_c(Q_{\mathcal{L}}(s,\cdot))$. Let $\mathcal{L}^\star$ be the population state-action pair in a stationary NE of \eqref{mfg}. Then $\pi_k=\hat{\Gamma}_1^k(\mathcal{L}_k)$. Denoting $d:=d_1d_2+d_3$, we see \begin{equation*}
\begin{split}
W_1(\tilde{\mathcal{L}}_{k+1}&,\mathcal{L}^\star)=W_1(\Gamma_2(\pi_k,\mathcal{L}_k),\Gamma_2(\Gamma_1(\mathcal{L}^\star),\mathcal{L}^\star))\\
\leq& W_1(\Gamma_2(\Gamma_1(\mathcal{L}_k),\mathcal{L}_k),\Gamma_2(\Gamma_1(\mathcal{L}^\star),\mathcal{L}^\star))+W_1(\Gamma_2(\Gamma_1(\mathcal{L}_k),\mathcal{L}_k),\Gamma_2(\hat{\Gamma}_1^k(\mathcal{L}_k),\mathcal{L}_k))\\
\leq& W_1(\Gamma(\mathcal{L}_k),\Gamma(\mathcal{L}^\star))+d_2D(\Gamma_1(\mathcal{L}_k),\hat{\Gamma}_1^k(\mathcal{L}_k))\\
\leq & (d_1d_2+d_3)W_1(\mathcal{L}_k,\mathcal{L}^\star)+d_2D(\textbf{argmax-e}(Q_{\mathcal{L}_k}^\star),\textbf{softmax}_c(\hat{Q}_{\mathcal{L}_k}^\star))\\
\leq& dW_1(\mathcal{L}_k,\mathcal{L}^\star)+d_2D(\textbf{softmax}_c(\hat{Q}_{\mathcal{L}_k}^\star),\textbf{softmax}_c(Q_{\mathcal{L}_k}^\star))\\
&+d_2D(\textbf{argmax-e}(Q_{\mathcal{L}_k}^\star),\textbf{softmax}_c(Q_{\mathcal{L}_k}^\star))\\
\leq & dW_1(\mathcal{L}_k,\mathcal{L}^\star)+\frac{cd_2\text{diam}(\mathcal{A})\sqrt{|\mathcal{A}|}}{2}\|\hat{Q}_{\mu_k}^\star-Q_{\mu_k}^\star\|_{\infty}\\
&+d_2D(\textbf{argmax-e}(Q_{\mathcal{L}_k}^\star),\textbf{softmax}_c(Q_{\mathcal{L}_k}^\star)).
\end{split} \end{equation*} Then since $\mathcal{L}_k\in S_{\epsilon}$ by the projection step, Lemma \ref{soft-arg-diff}, and Lemma \ref{Q-finite-bd} with the choice of $T_k=T^{\mathcal{M}_\mu}(\delta_k,\epsilon_k)$), we have, with probability at least $1-2\delta_k$, \begin{equation}
W_1(\tilde{\mathcal{L}}_{k+1},\mathcal{L}^\star)\leq dW_1(\mathcal{L}_k,\mathcal{L}^\star)+\frac{cd_2\text{diam}(\mathcal{A})\sqrt{|\mathcal{A}|}}{2}\epsilon_k+d_2\text{diam}(\mathcal{A})|\mathcal{A}|e^{-c\phi(\epsilon)}. \end{equation}
Finally, it is clear that with probability at least $1-2\delta_k$, \begin{equation*}
\begin{split}
W_1(\mathcal{L}_{k+1},\mathcal{L}^\star)&\leq W_1(\tilde{\mathcal{L}}_{k+1},\mathcal{L}^\star)+ W_1(\tilde{\mathcal{L}}_{k+1},\textbf{Proj}_{S_{\epsilon}}(\tilde{\mathcal{L}}_{k+1})) \\
&\leq dW_1(\mathcal{L}_k,\mathcal{L}^\star)+\frac{cd_2\text{diam}(\mathcal{A})\sqrt{|\mathcal{A}|}}{2}\epsilon_k+d_2\text{diam}(\mathcal{A})|\mathcal{A}|e^{-c\phi(\epsilon)}+\epsilon.
\end{split} \end{equation*} By telescoping, this implies that with probability at least $1-2\sum_{k=0}^{{\color{black}K-1}}\delta_k$, \begin{equation} \begin{split}
W_1(\mathcal{L}_K, \mathcal{L}^\star)\leq &d^KW_1(\mathcal{L}_0,\mathcal{L}^\star)+\frac{cd_2\text{diam}(\mathcal{A})\sqrt{|\mathcal{A}|}}{2}\sum_{k=0}^{K-1}d^{K-k}\epsilon_k \\
&+\dfrac{(d_2\text{diam}(\mathcal{A})|\mathcal{A}|e^{-c\phi(\epsilon)}+\epsilon)(1-d^{\color{black}K})}{1-d}. \end{split} \end{equation}
Since $\epsilon_k$ is summable, hence $\sup_{k\geq 0}\epsilon_k<\infty$, $\sum_{k=0}^{K-1}d^{K-k}\epsilon_k\leq \dfrac{\sup_{k\geq 0}\epsilon_k}{1-d}d^{\lfloor(K-1)/2\rfloor}+\sum_{k=\lceil(K-1)/2\rceil}^{\infty}\epsilon_k$.
Now plugging in $K=K_{\epsilon,\eta}$, with the choice of $\delta_k$ and $c=\frac{\log(1/\epsilon)}{\phi(\epsilon)}$, and noticing that $d\in[0,1)$, it is clear that with probability at least $1-2\delta$, \begin{equation} \begin{split} W_1(\mathcal{L}_{K_{\epsilon,\eta}},\mathcal{L}^\star)\leq &d^{K_{\epsilon,\eta}}W_1(\mathcal{L}_0,\mathcal{L}^\star)\\
&+\frac{cd_2\text{diam}(\mathcal{A})\sqrt{|\mathcal{A}|}}{2}\left( \dfrac{\sup_{k\geq 0}\epsilon_k}{1-d}d^{\lfloor(K_{\epsilon,\eta}-1)/2\rfloor}+\sum_{k=\lceil(K_{\epsilon,\eta}-1)/2\rceil}^{\infty}\epsilon_k\right)\\
&+\dfrac{(d_2\text{diam}(\mathcal{A})|\mathcal{A}|+1)\epsilon}{1-d}. \end{split} \end{equation}
Setting $\epsilon_k=(k+1)^{-(1+\eta)}$, then when $K_{\epsilon,\eta}\geq 2(\log_d(\epsilon{\color{black}/c})+1)$, \[
\dfrac{\sup_{k\geq 0}\epsilon_k}{1-d}d^{\lfloor(K_{\epsilon,\eta}-1)/2\rfloor}\leq \frac{\epsilon{\color{black}/c}}{1-d}. \] Similarly, when $K_{\epsilon,\eta}\geq 2(\eta\epsilon{\color{black}/c})^{-1/\eta}$, $\sum_{k=\left\lceil\frac{K_{\epsilon,\eta}-1}{2}\right\rceil}^{\infty}\epsilon_k\leq \epsilon{\color{black}/c}$.
Finally, when $K_{\epsilon,\eta}\geq \log_d(\epsilon/(\text{diam}(\mathcal{S})\text{diam}(\mathcal{A})))$, $d^{K_{\epsilon,\eta}}W_1(\mathcal{L}_0,\mathcal{L}^\star)\leq \epsilon$, since $W_1(\mathcal{L}_0,\mathcal{L}^\star)\leq \text{diam}(\mathcal{S}\times\mathcal{A}){\color{black}=\text{diam}(\mathcal{S})\text{diam}(\mathcal{A})}$.
In summary, if $K_{\epsilon,\eta}=\lceil 2\max\{(\eta\epsilon{\color{black}/c})^{-1/\eta}$, $\log_d(\epsilon/\max\{\text{diam}(\mathcal{S})\text{diam}(\mathcal{A}),{\color{black}c}\})+1\}\rceil$, then with probability at least $1-2\delta$, \[ \begin{split}
&W_1(\mathcal{L}_{K_{\epsilon,\eta}},\mathcal{L}^\star)\leq \left(1+\frac{d_2\text{diam}(\mathcal{A})\sqrt{|\mathcal{A}|}(2-d)}{2(1-d)}+\dfrac{(d_2\text{diam}(\mathcal{A})|\mathcal{A}|+1)}{1-d}\right)\epsilon=O(\epsilon). \end{split}
\]
Finally, plugging in $\epsilon_k$ and $\delta_k$ into $T^{\mathcal{M}_L}(\delta_k,\epsilon_k)$, and noticing that $k {\color{black} \leq } K_{\epsilon,\eta}$ and $\sum_{k=0}^{K_{\epsilon,\eta}-1}(k+1)^{\alpha}\leq \frac{K_{\epsilon,\eta}^{\alpha+1}}{\alpha+1}$, we immediately arrive at \begin{equation*}\label{Tbound_complex} \begin{split} T= O&\left(\left(\log(K_{\epsilon,\eta}/\delta)\right)^{\frac{1}{1-h}}K_{\epsilon,\eta}\left(\log K_{\epsilon,\eta}\right)^{\frac{1}{1-h}} +\left(\log(K_{\epsilon,\eta}/\delta)\right)^{\frac{1}{h}+3}\frac{K_{\epsilon,\eta}^{1+\frac{2(1+\eta)}{h}}}{1+\frac{2(1+\eta)}{h}}\left(\log (K_{\epsilon,\eta}/\delta)\right)^{\frac{1}{h}}\right). \end{split} \end{equation*} By further relaxing $\eta$ to $1$ and merging the terms, (\ref{Tbound}) follows. \end{proof}
\section{Naive algorithm}
The Naive iterative algorithm (Algorithm \ref{AQL_MFG}) is to replace Step A in the three-step fixed-point approach of GMFGs with Q-learning iterations. The limitation of this Naive algorithm has been discussed in the main text {\color{black}(Step 1, Section \ref{AIQL})} and {\color{black}empirically verified in Section \ref{experiments}} (Figure \ref{fig:naive}). \begin{algorithm}
\caption{\textbf{Alternating Q-learning for GMFGs (Naive)}}
\label{AQL_MFG} \begin{algorithmic}[1]
\STATE \textbf{Input}: Initial population state-action pair $L_0$
\FOR {$k=0, 1, \cdots$}
\STATE Perform Q-learning to find the Q-function $Q_k^\star(s,a)=Q_{L_k}^\star(s,a)$ of an MDP with dynamics $P_{L_k}(s'|s,a)$ and rewards $r_{L_k}(s,a)$.
\STATE Solve $\pi_k\in \Pi$ with $\pi_k(s)=\textbf{argmax-e}\left(Q_k^\star(s,\cdot)\right)$.
\STATE Sample $s\sim \mu_k$, where $\mu_k$ is the population state marginal of $L_k$, and obtain $L_{k+1}$ from $\mathcal{G}(s,\pi_k,L_k)$. \ENDFOR \end{algorithmic} \end{algorithm}
\section{GMF-V}\label{GMF-V} GMF-V, briefly mentioned in Section \ref{AIQL}, is the value-iteration version of our main algorithm GMF-Q. GMF-V applies to the GMFG setting with fully known transition dynamics $P$ and rewards $r$.
\begin{algorithm}[H]
\caption{\textbf{Value Iteration for GMFGs (GMF-V)}}
\label{VI} \begin{algorithmic}[1]
\STATE \textbf{Input}: Initial $L_0$, tolerance $\epsilon>0$.
\FOR {$k=0, 1, \cdots$}
\STATE Perform value iteration for $T_k$ iterations to find the approximate Q-function $Q_{L_k}$ and value function $V_{L_k}$:
\FOR {$t=1,2,\cdots,T_k$}
\FOR {all $s \in \mathcal{S}$ and $s \in \mathcal{A}$}
\STATE $ Q_{L_k}(s,a) \leftarrow \mathbb{E}[r(s,a,L_k)]+\gamma \sum_{s^{\prime}}P(s^{\prime}|s,a,L_k)V_{L_k}(s^{\prime})$
\STATE $V_{L_k}(s)\leftarrow\max_a Q_{L_k}(s,a)$
\ENDFOR
\ENDFOR
\STATE Compute a policy $\pi_k\in\Pi$: \\ ${ \hspace{1cm}\pi_k(s)=\textbf{softmax}_c({Q}_{L_k}(s,\cdot))}$.
\STATE Sample $s\sim \mu_k$, where $\mu_k$ is the population state marginal of $L_k$, and obtain $\tilde{L}_{k+1}$ from $\mathcal{G}(s,\pi_k,L_k)$.
\STATE Find $L_{k+1}=\textbf{Proj}_{S_{\epsilon}}(\tilde{L}_{k+1})$ \ENDFOR \end{algorithmic} \end{algorithm}
\section{More details for the experiments} \subsection{Competition intensity index $M$.}\label{intensity} In the experiment, the competition index $M$ is interpreted and implemented as the number of selected players in each auction competition. That is, in each round, $M-1$ players will be randomly
selected from the population to compete with the {\it representative} advertiser for the auction. Therefore, the population distribution $\mathcal{L}_t$, the winner indicator {\color{black}$w_t^M$}, and second-best price $a_t^M$ all depend on $M$. This parameter $M$ is also referred to as the {\it auction thickness} in the auction literature \cite{IJS2011}. \subsection{Adjustment for Algorithm MF-Q.}
For MF-Q, \cite{YLLZZW2018} assumes all $N$ players have a joint state $s$. In the auction experiment, we make the following adjustment for MF-Q for computational efficiency and model comparability: each player $i$ makes decision based on her own private state and table $Q^i$ is a functional of $s^i$, $a^i$ and $\frac{\sum_{j \neq i}a^j}{N-1}$.
}
\end{document} | arXiv |
\begin{document}
\title{An Estimator for Matching Size in Low Arboricity Graphs with Two Applications \footnote{This work is supported by the Iranian Institute for Research in Fundamental Sciences (IPM), Project Number 98050014.}}
\author{Hossein Jowhari \footnote{ Department of Computer Science and Statistics, Faculty of Mathematics, K. N. Toosi University of Technology. Email: [email protected]} } \maketitle
\begin{abstract} In this paper, we present a new simple degree-based estimator for the size of maximum matching in bounded arboricity graphs. When the arboricity of the graph is bounded by $\alpha$, the estimator gives a $\alpha+2$ factor approximation of the matching size. For planar graphs, we show the estimator does better and returns a $3.5$ approximation of the matching size.
Using this estimator, we get
new results for approximating the matching size of planar graphs in the streaming and distributed models of computation. In particular, in the vertex-arrival streams, we get a randomized $O(\frac{\sqrt{n}}{\varepsilon^2}\log n)$ space algorithm for approximating the matching size within $(3.5+\varepsilon)$ factor in a planar graph on $n$ vertices. Similarly, we get a simultaneous protocol in the vertex-partition model for approximating the matching size within $(3.5+\varepsilon)$ factor using $O(\frac{n^{2/3}}{\varepsilon^2}\log n)$ communication from each player.
In comparison with the previous estimators, the estimator in this paper does not need to know the arboricity of the input graph and improves the approximation factor for the case of planar graphs. \end{abstract}
\section{Introduction}
A matching in a graph $G=(V,E)$ is a subset of edges $M \subseteq E$ where no two edges in $M$ share an endpoint. A maximum matching of $G$ has the maximum number of edges among all possible matchings. Here we let $m(G)$ denote the matching size of $G$, {\em i.e. } the size of a maximum matching in $G$. In this paper, we present algorithms for approximating $m(G)$ in the sublinear models of computation. In particular, our results fit the vertex-arrival stream model (also known as the adjacency list streams). In the vertex-arrival model, in contrast with the edge-arrival version
where the input stream is an arbitrary ordering of the edges, here each item in the stream is a vertex of the graph followed by a list of its neighbors.
We also focus on graphs with bounded arboricity. A graph $G=(V,E)$ has arboricity bounded by $\alpha$ if the edge set $E$ can be partitioned into at most $\alpha$ forests. A well-known fact (known as the Nash-William theorem \cite{NW64}) states that a graph has arboricity $\alpha$, if and only if every induced subgraph on $t$ vertices has at most $\alpha (t-1)$ number of edges. Graphs with low arboricity cover a wide range of graphs such as constant degree graphs, planar graphs, and graphs with small tree-widths. In particular planar graphs have arboricity bounded by $3$.
A simple reduction from counting distinct elements implies that computing $m(G)$ exactly requires $\Omega(n)$ space complexity even for trees and randomized algorithms (see \cite{alon1999space} for the lower bound on distinct elements problem.) This has initiated the study of finding {\em computationally-light} estimators for $m(G)$ that take small space to compute. With this focus, following the work by Esfandiari {\em et al. } \cite{EHLMO15}, there has been a series of papers \cite{MV16,CormodeJMM17,MV18,BuryGMMSVZ19}
that have designed estimators for the matching size based on the degrees of vertices, edges and the arboricity of the input graph.
In this paper, we design another degree-based estimator for $m(G)$ in low arboricity graphs that has certain advantages in comparison with the previous works and leads to new algorithmic results. Before describing our estimator we briefly review some of the previous ideas. In the discussions below, we assume $G$ has arboricity bounded by $\alpha$.
\paragraph*{Shallow edges, high degree vertices} Esfandiari {\em et al. } \cite{EHLMO15}
were first to observe that one can approximately
characterize the matching size of low arboricity graphs
based on the degree information of the vertices and the local neighborhood
od the edges.
Let $H$ denote the set of vertices with degree more than
$h=2\alpha+3$ and let $F$ denote the set of edges with both
endpoints having degree at most $h$. Esfandiari {\em et al. } have shown that
$m(G) \le |H|+|F| \le (5\alpha+9)m(G).$
Based on this estimator, the authors in \cite{EHLMO15} have designed a $\tilde{O}(\varepsilon^{-2}\alpha n^{2/3})$
space algorithm for approximating $m(G)$ within
$5\alpha+9+\varepsilon$ factor in the edge-arrival model.
\paragraph*{Fractional matchings}
By establishing an interesting connection with fractional matchings
and the Edmonds Polytope theorem,
Mcgregor and Vorotnikova \cite{MV16} have shown the following quantity
approximates
$m(G)$ within $(\alpha+2)$ factor.
$$(\alpha+1)\sum_{(u,v) \in E }\min \{\frac1{\deg(u)}, \frac1{deg(v)},\frac1{\alpha+1}\}.$$
Based on this estimator, the authors in \cite{MV16} have given a $\tilde{O}(\varepsilon^{-2}n^{2/3})$
space streaming
algorithm (in the edge-arrival model) that approximate $m(G)$ within
$\alpha+2+\varepsilon$ factor. Also in the same work,
another degree-based estimator is given that
returns a $\frac{(\alpha+2)^2}2$ factor approximation of $m(G)$.
A notable property of this estimator is that it can be implemented
in the vertex-arrival stream model in $O(\log n)$ bits of space.
\paragraph*{$\alpha$-Last edges} Cormode {\em et al. } \cite{CormodeJMM17}
(later revised by Mcgregor and Vorotnikova \cite{MV18}) have designed an
estimator that depends on a given
ordering of the edges. Given a stream of edges $S=e_1,\ldots,e_m$,
let $E_\alpha(S)$ denote a subset of edges where $(u,v)\in E_\alpha(S)$ iff the vertices $u$ and $v$ both appear at most $\alpha$ times in $S$ after the edge $(u,v)$.
It is shown that $m(G) \le |E_\alpha(S)| \le (\alpha+2)m(G).$ Moreover a $O(\frac1{\varepsilon^2} \log^2n)$ space
streaming algorithm is given that approximates $|E_\alpha(S)|$ within $1+\varepsilon$ factor in the edge-arrival model.
\subsection{The estimator in this paper} The new estimator
is purely based on the degree of the vertices in the graph without any dependence on $\alpha$. To estimate the matching size, we count the number of what we call {\em locally superior} vertices in the graph. Namely,
\begin{definition} In graph $G=(V,E)$, we call $u \in V$ a locally superior vertex
if $u$ has a neighbor $v$ such that $deg(u) \ge deg(v)$. We let $\ell(G)$ denote the number of locally superior vertices in $G$. \end{definition}
We show if the arboricity of $G$ is bounded by $\alpha$, then $\ell(G)$ approximates $m(G)$ within $(\alpha+2)$ factor (Lemma \ref{lem:locally superior}.) This repeats the same bound obtained by the estimators in \cite{MV16} and \cite{CormodeJMM17},
however for planar graphs, we show that the approximation factor is
at most $3.5$ which beats the previous bounds (Lemma \ref{lem:planar})
\footnote{We do not have tight examples for our analysis.
In fact, we conjecture that $\ell(G)$ approximates $m(G)$
within $3$ factor when $G$ is planar.}.
As an evidence, consider the 4-regular planar graph on $9$ vertices. Both of the estimators in \cite{MV16} and \cite{CormodeJMM17}, report $18$ as the estimation for $m(G)$ while the exact answer is $4$. It follows their approximation factor is at least $4.5$.
Unfortunately, the new estimator, in spite of its simplicity, does not seem to be applicable in the edge-arrival model without an extra pass over the stream. To decide if a vertex is locally superior, we need to know its neighbors and learn their degrees which becomes burdonsome in one pass. However in the vertex-partition model, we can obtain this information in one pass and consequently can achieve sublinear space bounds. More formally, we get a randomized $O(\frac{\sqrt{n}}{\varepsilon^2}\log n)$ space algorithm for approximating $m(G)$ within $(3.5+\varepsilon)$ factor in this model. In terms of approximation factor, this improves over existing sublinear algorithms \cite{MV16,MV18}.
As another application of our estimator, we get a sublinear simultaneous
protocol in the {\em vertex-partition} model for approximating $m(G)$ when $G$ is planar. In this model, vertex set $V$ is partitioned into $t$ subsets $V_1,\ldots,V_t$ where each subset is given to a player.
The $i$-th player knows the edges on $V_i$.
The players do not communicate with each other. They only send one message to a {\em referee}
whom at the end computes an approximation of the matching size. (The referee does not get any part of the input.)
We assume the referee and the players have a shared source of randomness. Within this setting, we design a protocol that approximates
$m(G)$ within $3.5+\varepsilon$ factor using $O(\frac{n^{2/3}}{\varepsilon^2}\log n)$ communication
from each player.
Note that for $t > 3$ and $t = o(n^{1/3})$, this result is non-trivial. The best previous result implicit in the works of \cite{CCEHMMV16, MV16} computes a $5+\varepsilon$ factor approximation using $\tilde{O}(n^{4/5})$ communication from each player. We should also mention that, based on the estimator in \cite{MV16}, there is a simultaneous protocol that reports a $12.5$ factor approximation of $m(G)$ using $O(\log n)$ communication.
\section{Graph properties} \label{sec:est} In the following proofs, we let $M \subseteq E$ denote a maximum matching in graph $G$. When the underlying graph is clear from the context, for the vertex set $S$, we use $N(S)$ to denote the neighbors of the vertices in $S$ excluding $S$ itself. For vertex $u$, we simply use $N(x)$ to denote the neighbors of $u$. The vertex $v$ is a neighbor of the edge $(x,y)$ if $v$ is adjacent with $x$ or $y$. When $x$ is paired with $y$ in the matching $M$, abusing the notation, we define $M(x)=y$.
\begin{lemma} \label{lem:locally superior} Let $G=(V,E)$ be a graph with arboricity $\alpha$. We have $$m(G) \le \ell(G) \le (\alpha + 2) m(G).$$ \end{lemma}
\begin{proof} The left hand side of the inequality is easy to show. For every edge in $E$, at least one of the endpoints is locally superior. Since edges in $M$ are disjoint,
at least $|M|$ number of endpoints must be locally superior. This proves $m(G)\le \ell(G)$.
To show the right hand side, we use a charging argument. Let $L$ denote the locally superior vertices in $G$. Our goal is to show an upper bound on $|L|$ in terms
of $|M|$ and $\alpha$. Let $X \subseteq L$ be the set of locally superior vertices that are NOT endpoints of a matching edge.
The challenge is to prove an upper on $|X|$.
The vertices in $X$ do not contribute to the maximum matching. However all the vertices in $N(X)$ must be endpoints of matching edges (otherwise $M$ would not be a maximal matching.) For the same reason, there cannot be an edge between the vertices in $X$.
To prove an upper bound on $|X|$, in the first step, using an assignment procedure, we assign a subset of
vertices in $X$ to edges in $M$ in a way any target edge gets at most $\alpha-1$ locally superior vertices. We do the assignments in the following way.
\paragraph*{The Assignment Procedure} If we find a $y \in N(X)$ with at most $\alpha-1$ neighbors in $X$, we assign all the neighbors of $y$ in $X$ to the matching edge $(y,M(y))$.
We repeat this process, every time picking a vertex in $N(X)$ with less than $\alpha$ neighbors in $X$ and do the assignment that we just described,
until we cannot find such a vertex in $N(X)$.
Note that when we assign a locally superior vertex
$x$, we remove the edges on $x$ before continuing the procedure.
Here we emphasize the fact that if $y$ has a neighbor $x \in X$, then $M(y)$ cannot have neighbors in $X \setminus \{x\}$ (otherwise it would create an augmenting path and contradict the optimality of $M$.)
Let $X_1 \subseteq X$ be the assigned locally superior vertices and
$M_1 \subseteq M$ be the used matching edges in the assignment procedure.
We have \begin{equation} \label{eqn:X1}
|X_1| \le (\alpha-1)|M_1|. \end{equation}
Let $X_2 = X \setminus X_1$ be the unassigned vertices in $X$. Now we try to prove an upper bound on $|X_2|$. For this, we need to make a few observations.
\begin{observation}
\label{obs:Y}
Let $Y_2=N(X_2)$.
The pair $y$ and $M(y)$ cannot be both in $Y_2$.
\end{observation} \begin{proof} Suppose $y$ and $M(y)$ are both in $N(X_2)$. Let $B$ and $C$ be the neighbors of $y$ and $M(y)$ in $X_2$
respectively. If $|B \cup C| > 1$, then one can find an augmenting path of length $3$ (with respect to $M$.)
A contradiction.
On the other hand, if $|B \cup C| =1$, then $y$ and $M(y)$ have only a shared neighbor $x \in X_2$ which means the edge $e=(y,M(y))$ should have been used by the assignment procedure and as result $x \in X_1$. Another contradiction. \end{proof}
\begin{observation}
\label{obs:X2degree} Every vertex $x \in X_2$ has degree at least $\alpha+1$.
\end{observation}
\begin{proof}
Consider $x \in X_2$. Suppose, for the sake
of contradiction, $\deg(x)$
is $k$ where $k \le \alpha$. Since
$x$ is a locally superior vertex, there must be a $y \in N(x)$ with degree
at most $k$ in $G$. We know that $y$ is an endpoint of a matching edge.
In the assignments procedure, whenever we used an edge $e \in M$
all the neighbors of its endpoints (in $X$) were assigned. Since $x$ is not
assigned yet,
it means the edge $(y,M(y))$ has not been used. Consequently
$y$ must have at least $\alpha$ neighbors in $X_2$. Counting the edge
$(y,M(y))$, we should have $\deg(y) \ge \alpha+1$. A contradiction.
\end{proof}
Let $G'=(X_2 \cup Y_2,E')$ be a bipartite graph where $E'$ is the set
of edges between $X_2$ and $Y_2$.
From Observation \ref{obs:X2degree}, we have
\begin{equation} \label{eq:lowerboundE'}
(\alpha+1)|X_2|\le |E'|. \end{equation}
Since $G'$ is a subgraph of $G$, its arboricity is bounded by $\alpha$. As result,
\begin{equation} \label{eq:upperboundE'}
|E'| \le \alpha(|X_2|+|Y_2|). \end{equation}
Recall that $Y_2$ are endpoints of matching edges. Let $M_2$ be those matching edges.
Observation \ref{obs:Y} implies that $|Y_2|=|M_2|$. As result, combining (\ref{eq:lowerboundE'}) and (\ref{eq:upperboundE'}), we get the following.
\begin{equation}
|X_2| \le \alpha|Y_2|=\alpha|M_2|. \end{equation}
To prove an upper bound on $|L|$, we also need to count the locally superior vertices that are endpoints of matching edges. Let
$Z = L\setminus X$. We have
$|Z| \le 2|M|.$ Summing up, we get \begin{align*}
|L|&=|X_1| + |X_2| + |Z| \\
& \le (\alpha -1)|M_1| + \alpha|M_2| + 2|M| \\
& = \alpha(|M_1| +|M_2|) + 2|M|-|M_1| \\
& \le (\alpha +2)|M| - |M_1| \\
& \le (\alpha +2)|M| \end{align*} This proves the lemma. \end{proof}
\begin{lemma} \label{lem:planar} Let $G=(V,E)$ be a planar graph. We have $ \ell(G) \le 3.5 m(G) $. \end{lemma} \begin{proof} For planar graphs, similar to what we did in the proof of Lemma \ref{lem:locally superior}, we first try to assign some of the vertices in $X$ to the matching edges using a simple assignment procedure. (Recall that $X$ is the set of vertices in $L$ that are not endpoints of edges in $M$.)
\paragraph{The Assignment Procedure} Let $Y_1 =\emptyset$. If we find a $y \in N(X)$ with only $1$ neighbor $x \in X$, we assign $x$ to the matching edge $(y,M(y))$. Also we add $y$ to $Y_1$.
We continue the procedure until we cannot find such a vertex in $N(X)$.
Note that when we assign a locally superior vertex $x$, we remove the edges on $x$.
Let $X_1 \subseteq X$ be the assigned locally superior vertices and
$M_1 \subseteq M$ be the used matching edges in the assignment procedure. Note that $|Y_1|=|M_1|$.
We have \begin{equation} \label{eqn:planarX1}
|X_1| \le |M_1|. \end{equation}
Let $X_2 = X \setminus X_1$. Using a similar argument that we used for proving Observation \ref{obs:X2degree}, we can show every vertex in $X_2$ has degree at least $3$. Also letting $Y_2=N(X_2)$, we observe that $y \in Y_2$ and $M(y)$ cannot be both in $Y_2$ as we noticed in the Observation \ref{obs:Y}. Let $M_2 \subseteq M$ be the matching
edges with one endpoint in $Y_2$. We have $|Y_2|=|M_2|$.
Now consider the bipartite graph $G'=(X_2 \cup Y_2,E')$ where $E'$ is the set
of edges between $X_2$ and $Y_2$. Every planar bipartite graph with $n$ vertices has at most $2n-4$ edges {\footnote {For a short proof of this, combine the
Euler's formula $|V|-|E|+|F|=2$
with the inequality $2|E|\ge 4|F|$ caused by each face having at least 4
sides (since there are no odd cycles) and we get $|E|\le 2|V|-4.$}}. Since $G'$ is a bipartite planar graph, it follows,
\begin{equation} \label{eqn:planarE'}
3|X_2| \le |E'| < 2(|X_2|+|Y_2|)= 2(|X_2|+|M_2|). \end{equation}
This shows $|X_2| < 2|M_2|$. Letting $Z=L \setminus X$ and $M_3=M\setminus(M_1 \cup M_2)$, we get
\begin{equation} \label{eqn:R_upperbound1}
|L|=|X_1| + |X_2| + |Z| \le |M_1| + 2|M_2| + 2|M|
\le 3|M| +|M_2| -|M_3|. \end{equation}
This already proves $|L|$ is bounded by $4|M|$. To prove the bound claimed in the lemma,
we also show that $|L| \le 3|M|+|M_1|+|M_3|$. Combined with the
inequality (\ref{eqn:R_upperbound1}), this proves the lemma.
Let $Y=Y_1 \cup Y_2$. Note that $Y$ are one side of
the matching edges in $M_1 \cup M_2$. Let
$Y'=\{ M(y) \; | \; y \in Y\}.$ We use a special subset of $Y'$, named $Y''$ which is defined as follows. We let $Y''$ denote the locally superior vertices in $Y'$ that have degree $2$ or they are adjacent with both endpoints of an edge in $M_3$. We make the following observation regarding the vertices in $Y''$.
\begin{observation} \label{obs:Y''} We can assign each vertex $y' \in Y''$ to a distinct $e \in Y_1 \cup M_3$ where $e$ has no neighbor in $Y'\setminus \{y'\}$. \end{observation}
\begin{proof} Consider $y' \in Y''$. If $y'$ is adjacent with both endpoints of an edge $e=(z,z') \in M_3$, we assign $y'$ to $e$ (when there are multiple edges with this condition we pick one of them arbitrarily.) Note that $z$ and $z'$ cannot have neighbors in $Y'$ other than $y'$ because otherwise it would create an augmenting path.
Now suppose $y'$ has degree $2$. Since $y'$ is a locally superior vertex, it must have a neighbor $z$ of degree at most $2$. The neighbor $z$ cannot be in $Y_2 \cup X_2$ because the vertices in $Y_2 \cup X_2$ have degree at least $3$.
We distinguish between two cases. \begin{itemize}
\item $M(y') \in Y_2$. In this case, $z$ cannot be in $Y_1$ either because the vertices in $Y_1$ are already of degree $2$ without $y'$. Also $z \notin X_1$ because otherwise
it would create an augmenting path. The only possibility is that $z$ is an endpoint of a matching
edge in $M_3$. We assign $y'$ to the matching edge $(z,z') \in M_3$. Note that
$z'$ cannot have a neighbor in $Y'\setminus \{y'\}$ because it would create an augmenting path.
\item $M(y') \in Y_1$. Here $z$ could be in $X_1$. If this is the case, then $M(y')$ cannot have a neighbor in $Y' \setminus \{y'\}$ because it would create an augmenting path. In this case, we assign $y'$ to $M(y')$. If $z = M(y')$, then again we assign $y'$ to $M(y')$. The only remaining possibility is that $z$ an endpoint of a matching edge in $M_3$ which we handle it similar to the previous case. \end{itemize} \end{proof}
\begin{figure}
\caption{ A demonstration of the construction in the proof of lemmas \ref{lem:locally superior} and \ref{lem:planar}. Thick edges represent matching edges. The unfilled vertices belong to the set $Y''$. }
\label{fig_graph}
\end{figure}
Now, assume we assign the vertices in $Y''$ to the elements in $Y_1 \cup M_3$ according to the above observation. Let $Y_{1}' \subseteq Y_1$ and $M_{3}' \subseteq M_3$ be the vertices and edges that were used in the assignment. Let $Y'''$ be the remaining locally superior vertices in $Y'$. Namely, $Y'''= (L\cap Y')\setminus Y''$. Before making the final point, we observe that only one endpoint of the edges in $M_3$ are adjacent with vertices in $Y'''$. Let $Y_3$ be the endpoint of edges in $M_3 \setminus M_3'$ that have neighbors in $Y'''$.
Consider the bipartite graph $G''(V'',E'')$ where $$V''=(X_2 \cup Y''') \cup \big(Y_2 \cup (Y_1 \setminus Y_{1}') \cup Y_3\big)$$ and $E''$ is the set of edges between $X_2$ and $Y_2$, and the edges between $Y'''$ and $ Y_2 \cup (Y_1 \setminus Y_1')\cup Y_3$.
Relying on the facts that $G''$ is a planar bipartite graph, $Y'''$ is composed of vertices with degree at least $3$, and the edges on $Y'''$ are all in $E''$, we have
$$ 3|X_2| + 3|Y'''| \le |E''| \le 2 (|X_2|+|Y_2|+|Y_1\setminus Y_1'|+|Y'''|+|Y_3|).$$ It follows, \begin{align*}
|X_2| + |Y'''| & \le 2(|Y_2|+|Y_1\setminus Y_1'|+|Y_3|) \\
& \le 2(|M_2|+|M_1|-|Y_1'|+|M_3|-|M_3'|) \\
& = 2(|M|-|Y_1'|-|M_3'|) \end{align*}
Since $|Y''|=|Y'_1|+|M'_3|$, we get \begin{equation} \label{eq:Y'''}
|X_2| + |Y'''| \le 2|M|-2|Y''| \end{equation}
Let $Z_1$, $Z_2$ and $Z_3$ denote the locally superior vertices that are endpoints of matching edges in $M_1$, $M_2$ and $M_3$ respectively. From the definition of $Y''$ and $Y'''$, we have \begin{equation} \label{eq:Z1Z2}
|Z_1| + |Z_2| \le |M_1| + |M_2| + |Y''| +|Y'''| \end{equation} From (\ref{eq:Y'''}) and (\ref{eq:Z1Z2}), we get \begin{align*}
|L| & = |X_1| + |X_2| + |Z_1| + |Z_2| + |Z_3| \\
& \le |M_1| + |X_2| + (|M_1| + |M_2| + |Y''| +|Y'''|) + 2|M_3|\\
& = 2|M_1| + (|X_2|+|Y'''|) + |M_2| + |Y''| + 2|M_3|\\
& \le 2|M_1| + |M_2| + 2|M| - |Y''| + 2|M_3|\\
& = 3|M|+ |M_1| + |M_3| -|Y''|\\
& \le 3|M|+ |M_1| + |M_3| \end{align*} This finishes the proof of the lemma. \end{proof}
\section{Algorithms} \label{sec:alg} We first present a high-level sampling-based estimator for $\ell(G)$. Then we show how this estimator can be implemented in the streaming and distributed settings
using small space and communication. For our
streaming result, we use a combination of the estimator
for $\ell(G)$ and the greedy maximal matching algorithm. For the simultaneous protocol, we use the estimator for $\ell(G)$ in combination with the edge-sampling primitive in \cite{CCEHMMV16} and an estimator in \cite{MV16}.
The high-level estimator (described in Algorithm \ref{alg:locally superior}) samples a
subset of vertices $S \subseteq V$ and computes the locally superior vertices in $S$.
The quantity $\ell(G)$ is estimated from the scaled
ratio of the locally superior vertices in the sample set.
\begin{algorithm}[th]
Run the following estimator $r=\lceil \frac 8{\epsilon^2}\rceil $ number of times in parallel. In the end, report the average of the outcomes.
\hspace{1cm} 1. Sample $s$ vertices (uniformly at random) from $V$ without replacement.
\hspace{1cm} 2. Let $S$ be the set of sampled vertices.
\hspace{1cm} 3. Compute $S'$ where $S'$ is the set of locally superior vertices in $S$.
\hspace{1cm} 4. Return $\frac{n}{s}|S'|$ as an estimation for $\ell(G)$.
\caption{The high-level description of the estimator for $\ell(G)$}
\label{alg:locally superior} \end{algorithm}
\begin{lemma} \label{lem:locally superioralg} Assuming $s \ge \frac{n}{\ell(G)}$, the high-level estimator in Algoirthm \ref{alg:locally superior}
returns a $1+\varepsilon$ factor approximation of $\ell(G)$ with probability
at least $7/8$. \end{lemma}
\begin{proof}
Fix a parallel repetition of the algorithm and let $X$ denote the outcome of the associated estimator. Assuming an arbitrary ordering on the locally superior vertices, let $X_i$ denote the random variable associated with $i$-th locally superior vertex. We define $X_i=1$ if the $i$-th locally superior vertex has been sampled, otherwise $X_i=0$. We have $X = \frac{n}{s} \sum_{i=1}^{\ell(G)} X_i$. Since $Pr(X_i=1)=\frac{s}{n}$, we get $E[X]=\ell(G)$.
Further we have \begin{align*} E[X^2]= \frac{n^2}{s^2}E\Big[\sum_{i,j}^{\ell(G)}X_iX_j\Big] & = \frac{n^2}{s^2}\Big[\sum_i^{\ell(G)}E[X_i^2]+\sum_{i\neq j}^{\ell(G)}E[X_iX_j]\Big] \\
& = \frac{n^2}{s^2}\Big[\frac{s}{n}\ell(G)+{\ell(G) \choose 2}\frac{s(s-1)}{n(n-1)}\Big] \\
& = \frac{n}{s}\ell(G)+{\ell(G) \choose 2}\frac{n(s-1)}{s(n-1)} \\
& < \frac{n}{s}\ell(G) + \ell^2(G) \end{align*}
Consequently, $Var[X]=E[X^2]-E^2[X]< \frac{n}{s}\ell(G).$
Let $Y$ be the average of the outcomes of $r$ parallel and independent repetitions of the basic estimator. We have $E[Y]=\ell(G)$ and $Var[Y]<\frac{n}{sr}\ell(G)$. Using the Chebyshev's inequality,
$$Pr(|Y-E[Y]|\ge \varepsilon E[Y])\le \frac{Var[Y]}{\varepsilon^2E^2[X]}<\frac{n/s}{r\varepsilon^2 \ell(G)}.$$ Setting $r=\frac8{\varepsilon^2}$ and $s \ge \frac{n}{\ell(G)}$, the above probability will be less than $1/8$. \end{proof}
\subsection{The streaming algorithm} We first note that we can implement the high-level estimator of Algorithm \ref{alg:locally superior} in the vertex-arrival stream model using $O(\frac{s}{\varepsilon^2}\log n)$ space. Consider a single repetition of the estimator.
The sampled set $S$ is selected in the beginning of the algorithm (before the stream.) This can be done using a reservoir sampling strategy \cite{Vitter85} in $O(|S|\log n)$ space. To decide if $u \in S$ is locally superior or not, we just need to store $\deg(u)$ and
the minimum degree of the neighbors that are visited so far. This takes $O(\log n)$ bits of space. As result, the whole space needed to implement a single repetition is $O(s\log n)$ bits.
The streaming algorithm runs two threads in parallel. In one thread it runs the streaming implementation of Algorithm \ref{alg:locally superior} after setting $s=\lceil \sqrt{n} \:\rceil$. In the other thread, it runs a greedy algorithm to find a maximal matching in the input graph. We stop the greedy algorithm whenever the size of the discovered matching $F$
exceeds $\sqrt{n}$. In the end, if $|F| < \sqrt{n}$, we output $|F|$ as an approximation for $m(G)$, otherwise we report the outcome of the first thread.
Note that if $|F| < \sqrt{n}$, $F$ is a maximal matching in $G$. Hence
$|F| \ge \frac12 m(G)$. Assume $|F| \ge \sqrt{n}$. In this case the algorithm outputs the result of first thread. In this case, by Lemma \ref{lem:locally superior}, we know $\ell(G) \ge \sqrt{n}$. Consequently, it follows from Lemma \ref{lem:locally superioralg}, the first thread returns a $1+O\varepsilon$ approximation of $\ell(G)$ and consequently it returns a $3.5+O(\varepsilon)$ approximation of $m(G)$. Since the greedy algorithm takes at most $O(\sqrt{n})$ space, the space complexity of the algorithm is dominated by the space usage of the first thread. We get the following result.
\begin{theorem} \label{thm:planar} Let $G$ be a planar graph. There is a streaming algorithm (in the vertex-arrival model) that returns a $3.5+\epsilon$ factor approximation of $m(G)$ using $O(\frac{\sqrt{n}}{\epsilon^2})$ space. \end{theorem}
\subsection{A simultaneous communication protocol} To describe the simultaneous protocol, we consider two cases separately: (a) when the matching size is low; to be precise, when it is smaller than some fixed value $k=n^{1/3}$, and (b) when the matching size is high, {\em i.e. } at least $\Omega(k)$. For each case, we describe a separate solution. The overall protocol will be these solutions (run in parallel) combined with a sub-protocol (in parallel) to distinguish between the cases.
\paragraph*{Graphs with large matching size}
In the case when matching size is large, similar to what was done in the streaming model, we run an implementation of Algorithm 1 in the simultaneous model. To see the implementation, in the simultaneous model all the players (including the referee) know the sampled set $S$. This results from access to the shared randomness. For each $u \in S$, the players send the minimum degree of the neighbors of $u$ in his input to the referee. The player that owns $u$, also sends $\deg(u)$ to the referee.
Having received this information, the referee can decide
if $u$ is a locally superior vertex or not.
As result, we can implement Algorithm \ref{alg:locally superior} in
the simultaneous model using a protocol with
$O(\frac{s}{\varepsilon^2}\log n)$ message size.
\paragraph*{Graphs with small matching size} In the case where the matching size is small, we use the edge-sampling method of \cite{CCEHMMV16}. We review their basic sampling primitive in its general form. Given a graph $G(V,E)$, let $c:V \rightarrow [b]$ be a totally random function that assigns each vertex in $V$ a random number (color) in $[b]=\{1,\ldots,b\}$. The set $\textrm{Sample}_{b,d,1}$ is a random subset of $E$ picked in the following way. Given a subset $K \subseteq [b]$ of size $d \in \{1,2\}$, let $E_K$ be the edges of $G$ where the color of their endpoints matches $K$. For example when $K=\{3,4\}$, the set $E_{\{3,4\}}$ contains all edges $(u,v)$ such that $\{c(u),c(v)\}=\{3,4\}$. For all $K \subseteq [b]$ of size $d$, the set $\textrm{Sample}_{b,d,1}$ picks a random edge from $E_{K}$. Finally, the random set $\textrm{Sample}_{b,d,r}$ is the union of $r$ independent instances of $\textrm{Sample}_{b,d,1}$. We have the following lemma from \cite{CCEHMMV16} (see Theorems 4 in the reference.)
\begin{lemma} \label{lem:Sample} Let $G=(V,E)$ be a graph. When $m(G) \le k$, with probability $1-1/{\mathrm{poly}}(k)$, the random set $\textrm{Sample}_{100k,2,O(\log k)}$ contains a matching of size $m(G)$.
\end{lemma}
Note that, in the simultaneous vertex-partition model, the referee can obtain an instance of $\textrm{Sample}_{b,d,1}$
via a protocol with $O(b^d\log n)$ message size. To see this, using the shared randomness, the players pick the random function $c: V \rightarrow [b]$. Let $E^{(i)}$ be the subset of edges owned by the $i$-th player. We have $E=\bigcup_{i=1}^t E^{(i)}$. To pick a random edge from $E_K$ for a given $K \subseteq [b]$, the $i$-th player randomly picks an edge $e \in E_K \cap E^{(i)}$
and sends it along with $|E_K \cap E^{(i)}|$ to the referee. After receiving this information from all the players, the referee can generate a random element of $E_K$. Since there are $O(b^d)$ different $d$-subsets of $[b]$, the size of the message from a player to the referee is bounded by $O(b^d\log n)$ bits. Consequently, the referee can produce a rightful instance of $\textrm{Sample}_{b,d,r}$ using $O(rb^d\log n)$ communication from each player.
\paragraph*{How to distinguish between the cases?} To accomplish this, here we use a degree-based estimator by Mcgregor and Vorotnikova \cite{MV16} described
in the following lemma. \begin{lemma} \label{lem:vorotnikova} Let $G$ be a planar graph. We have $$m(G)\le A'(G)=\sum_{u\in V}\min\{\deg(u)/2,4-\deg(u)/2\} \le 12.5 \:m(G).$$ \end{lemma}
It is easy to see that, in the simultaneous vertex-partition model, we can implement this estimator with $O(\log n)$ bits communication from each player.
\paragraph*{The final protocol} Let $k=\lceil n^{1/3} \rceil$. We run the following threads in parallel.
\begin{enumerate} \item A protocol that implements the high-level estimator (Algorithm \ref{alg:locally superior}) with $s= \lceil 12.5 n/k\rceil$ as its input parameter according to the discussions above. Let $z_1$ be the output of this protocol. \item A protocol to compute an instance of $\textrm{Sample}_{b,d,r}$ for $b=100k$ and $d=2$ and $r=O(\log k)$. Let $z_2$ be the size of maximum matching in the sampled set.
\item A protocol to compute $A'(G)$. Let $z_3$ be the output of this thread. \end{enumerate}
In the end, if $z_3 \ge \frac{k}{12.5} $, the referee outputs $z_1$ as an approximation for $m(G)$, otherwise the referee reports $z_2$ as the final answer.
\begin{theorem} Let $G$ be a planar graph on $n$ vertices. The above simultaneous protocol with probability $3/4$ returns a $3.5+O(\varepsilon)$ approximation
of $m(G)$ where each player sends $O(\frac{n^{2/3}}{\varepsilon^2})$ bits
to the referee. \end{theorem}
\begin{proof} First we note that by choosing the constants large enough, we can assume the thread (2) errs with probability at most $1/8$. If $z_3 \ge \frac{k}{12.5} $, then we know $m(G) \ge \frac{k}{12.5} $. This follows from Lemma \ref{lem:vorotnikova}. Consequently by Lemma \ref{lem:locally superior}, we have $\ell(G)\ge \frac{k}{12.5}$. Therefore from Lemma \ref{lem:locally superioralg}, we
have $|z_1-\ell(G)|\le \varepsilon \ell(G)$ with probability at least $7/8$. It follows from Lemma \ref{lem:planar} that $(1-\varepsilon)m(G) \le z_1 \le (3.5 +3.5\varepsilon) m(G).$
On the other hand, if $z_3 < \frac{k}{12.5}$, by Lemma \ref{lem:vorotnikova} we know that $m(G)$ must be less than $k$. Having this, from Lemma \ref{lem:Sample}, with probability at least $7/8$, we get $z_2 = m(G)$. In this case the protocol computes the exact matching size of the graph.
The communication complexity each player is dominated by the cost of the first thread which is $O(n^{2/3}\varepsilon^{-2}\log n)$. The total error probability is bounded by $1/4$. This finishes the proof. \end{proof}
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\end{document} | arXiv |
Links between DNA methylation and nucleosome occupancy in the human genome
Clayton K. Collings1 &
John N. Anderson2
DNA methylation is an epigenetic modification that is enriched in heterochromatin but depleted at active promoters and enhancers. However, the debate on whether or not DNA methylation is a reliable indicator of high nucleosome occupancy has not been settled. For example, the methylation levels of DNA flanking CTCF sites are higher in linker DNA than in nucleosomal DNA, while other studies have shown that the nucleosome core is the preferred site of methylation. In this study, we make progress toward understanding these conflicting phenomena by implementing a bioinformatics approach that combines MNase-seq and NOMe-seq data and by comprehensively profiling DNA methylation and nucleosome occupancy throughout the human genome.
The results demonstrated that increasing methylated CpG density is correlated with nucleosome occupancy in the total genome and within nearly all subgenomic regions. Features with elevated methylated CpG density such as exons, SINE-Alu sequences, H3K36-trimethylated peaks, and methylated CpG islands are among the highest nucleosome occupied elements in the genome, while some of the lowest occupancies are displayed by unmethylated CpG islands and unmethylated transcription factor binding sites. Additionally, outside of CpG islands, the density of CpGs within nucleosomes was shown to be important for the nucleosomal location of DNA methylation with low CpG frequencies favoring linker methylation and high CpG frequencies favoring core particle methylation. Prominent exceptions to the correlations between methylated CpG density and nucleosome occupancy include CpG islands marked by H3K27me3 and CpG-poor heterochromatin marked by H3K9me3, and these modifications, along with DNA methylation, distinguish the major silencing mechanisms of the human epigenome.
Thus, the relationship between DNA methylation and nucleosome occupancy is influenced by the density of methylated CpG dinucleotides and by other epigenomic components in chromatin.
The genomes of eukaryotic organisms are packaged into tightly condensed arrangements of nucleoprotein complexes referred to as chromatin. At the primary level of chromatin compaction, a 147-base-pair segment of DNA spirals nearly twice around an octamer of histone proteins to form a structure known as the nucleosome [1, 2]. The degree of nucleosome occupancy that occurs along DNA in chromatin is important because it can dictate the accessibility of DNA to the transcriptional machinery and to other proteins involved in genome regulation [3]. With advances in high-throughput sequencing technologies, nucleosome maps have revealed differential nucleosome occupancy patterns over entire genomes for a variety of species and cell types [4]. For example, nucleosome-depleted regions are observed overlapping transcription start sites of active genes, while high nucleosome occupancy is found to encompass the promoters of silent genes [5]. Furthermore, several genome‐wide explorations in conjunction with biochemical modifications have elucidated mechanisms that have been evoked to explain the differences in nucleosome occupancy detected across intragenic and intergenic chromatin.
In vertebrate cells, the most common mode of DNA methylation entails the addition of a methyl group to a cytosine residue in the context of a CpG dinucleotide. CpG methylation is perhaps the best understood epigenetic mark and is maintained through cell division during DNA replication primarily by the DNA methyltransferase Dnmt1 with some assistance from the de novo methyltransferases, Dnmt3a and Dnmt3b [6]. This modification has been linked to gene silencing and is considered to be an important factor in the formation of constitutive and facultative heterochromatin [7]. Additionally, DNA methylation has also been shown to be essential for normal tissue-specific development. During embryonic stem cell differentiation, select CpGs throughout the genome become methylated by the de novo DNA methyltransferases, and through DNA methylation's epigenetic influence on chromatin structure and gene regulation, the inheritability of diverse cellular phenotypes within higher eukaryotic species is sustained [8, 9].
Although CpG dinucleotides are underrepresented in mammals, they are not randomly distributed, and regions with high CpG density, referred to as CpG islands, are typically unmethylated and cover the promoters of many housekeeping genes [10]. The chromatin architecture of active CpG island promoters is characterized by nucleosome depletion, histone acetylation, H3K4 methylation, but not H3K36 methylation [10]. Additionally, numerous transcription factors bind to CpG islands, and proteins with CXXC domains, which target unmethylated CpGs, are especially enriched in CpG Islands [10]. Some examples include the transcription factor Sp1 [11], the H3K36me2 demethylase Kdm2a [12], and the H3K4me3 methyltransferase subunit Cfp1 [13]. These features of active transcription are thought to protect CpG islands from de novo methylation [10].
Downstream of active promoters, Setd2 catalyzes the methylation of H3K36 with elongating Pol II in the bodies of transcribed genes [5]. H3K36me3 is enriched in exons over introns and has been proposed to be associated with co-transcriptional splicing mechanisms [14]. With the transfer of Pol II, histone H3K4 demethylation is performed by Kdm5 and Kdm1 [15]. In parallel, the de novo DNA methyltransferases preferentially bind to unmethylated H3K4 and to H3K36me3 [16–19], and recently, the presence of H3K36me3 was shown to be linked to the enrichment of binding and de novo methylation by DNMT3b over DNMT3a in gene bodies in vivo [20].
We previously examined the effects of DNA methylation on the stability of a large heterogeneous population of nucleosomes [21]. Specifically, with bacterial artificial chromosomes (BACs), the CpG methyltransferase M. SssI, isolated histones, and micrococcal nuclease, we conducted nucleosome reconstitution experiments in conjunction with high‐throughput sequencing on ~1 MB of mammalian DNA that was unmethylated or methylated. The features by which DNA methylation was found to increase the stability of nucleosomes (already positioned by nucleotide sequence) were elevated CpG frequency and a tendency for the minor grooves of CpGs to be rotationally oriented toward the histone surface, and these methylation‐sensitive nucleosomes were found to be enriched in exons and in CpG islands [21]. Our in vitro nucleosome data reflected nucleosomal DNA methylation patterns observed in vivo in terms of the co‐enrichment of DNA methylation and nucleosome occupancy in exons [22–25], the increased nucleosome occupancy associated with methylated CpG islands [25, 26], and the rotational orientation of methylated CpGs in Arabidopsis nucleosomes [22].
In order to extend our research beyond our in vitro experiments, we sought to understand the relationship between DNA methylation and nucleosome occupancy in the cell. In this study, we first perform an integrated analysis of MNase-seq and NOMe-seq data. Through this approach, we survey chromatin landscapes from the perspective of the nucleosome and find an underlying positive correlation between methylated CpG density and nucleosome occupancy. We also acknowledge exceptions to this pattern that can be linked to the presence or absence of other epigenetic factors. Finally, we extensively characterize the chromatin in CpG islands and at conserved transcription factor binding sites to reveal regulation of DNA methylation and nucleosome occupancy in the vicinities of these genomic landmarks.
DNA methylation and nucleosome occupancy at the genome level
A common strategy used to study chromatin from genome-wide high-throughput sequencing data involves designating boundary elements and then characterizing the markers surrounding these sites [27]. An example of this approach is given in Additional file 1: Figure S1, which displays average DNA methylation and nucleosome occupancy levels amid computationally predicted sites for the transcription factor, CTCF. These CTCF recognition sequences often mark the boundaries of topologically associated domains [28]. The results show that these presumptive CTCF sites are flanked by a series of regularly spaced nucleosomes and that methylation at CpG dinucleotides occurs preferentially in the linker regions of these phased nucleosomes [29–31]. The elucidation of these patterns raises the question of what can constitute a chromatin boundary.
Approximately 15 million nucleosomes are positioned along the haploid human genome. The primary bioinformatics approach utilized in this work treated each nucleosome identified by MNase-seq as a chromatin boundary element. Using the flanking DNA centered on nucleosome midpoints, the sequence content, DNA methylation levels, and nucleosome occupancies were measured for the entire genome and for several subgenomic regions. In order to compare DNA methylation patterns derived from nucleosomes reconstituted in vitro to those formed in the cell, BS-seq [32] and MNase-seq [33], data derived from leukocytes were used (Fig. 1a–f). For the analysis of nucleosome occupancy and DNA methylation, in vivo MNase-seq and NOMe-seq data from fetal lung fibroblasts (IMR90 cells) were used (Fig. 1g–j) [29].
Genome-wide nucleosomal DNA sequence and methylation patterns in leukocytes and IMR90 cells. a For the in vitro MNase-seq data from leukocytes, average occurrences of select dinucleotides were computed from forward and reverse complement sequences aligned to nucleosome midpoints. Using BS-seq and MNase-seq data from leukocytes, average occurrences of CpGs (b), methylated and unmethylated CpGs (c, d) and mCpG/CpG fractions (e, f) were computed for both the in vitro and in vivo libraries from forward and reverse complement sequences aligned to nucleosome midpoints. Using MNase-seq and NOMe-seq data from IMR90 cells, average mHCG/HCG fractions (g, h) and uGCH/GCH fractions (i, j) were computed from forward and reverse complement sequences aligned to MNase-seq derived nucleosome midpoints. Nucleosomes within CpG Islands were excluded from this analysis
A hallmark of rotationally positioned nucleosomal DNA is the ∼10-base-pair periodicity of A/T-rich sequences with minor grooves facing toward the histone octamer alternating with G/C-rich sequences whose minor grooves face away [34–36]. This arrangement is illustrated using the in vitro data presented in Fig. 1a. It was suggested long ago that this pattern facilitates the winding of DNA around the histone octamer. According to this widely accepted model, the minor grooves of CpG dinucleotides should face outwards at positions ±10, ±20, ±30, ±40, ±50, ±60, and ±70 base pairs from the nucleosome midpoint, and this pattern has been observed in S. cerevisiae and C. elegans which lack CpG methylation [37–39]. To our knowledge, such a clear pattern for total CpG occurrence has never been observed in mammalian nucleosomal DNA, and the lack of this periodic pattern is illustrated by the in vitro and in vivo datasets in Fig. 1b. However, a weak ~10-bp periodicity of the methylated CpG fraction is detected in the in vitro data (Fig. 1c), which becomes magnified when the fraction of CpGs that are methylated are quantified in Fig. 1e. This periodicity is indicative of an unusual rotational orientation as the minor grooves of the methylated CpG dinucleotides face the histone surface, in agreement with previous reports in Arabidopsis and human DNA sequences [21, 22, 40]. Note that the frequency of CpGs is ~1.6-fold higher in the in vitro data as compared to the in vivo data (Fig. 1b). Also note that nucleosomes from CpG islands were excluded from the analysis for Fig. 1 and for all other analyses unless otherwise specified.
Comparisons between the in vitro and in vivo datasets yield striking differences in nucleosomal DNA methylation patterns. Preferential methylation is observed in nucleosome cores from the in vitro dataset (Fig. 1c, e), while a linker preference is found in the in vivo dataset (Fig. 1d, e). This linker preference is extended to nucleosomes that flank the fixed nucleosomes as DNA methylation levels peak between phased nucleosomes (Fig. 1f). On the other hand, nucleosome phasing is not observed in the in vitro data, presumably because the in vitro reconstitutions were carried out in moderate DNA excess. This condition may also explain the enrichment of mCpGs in the nucleosome core from the in vitro dataset since, as shown in Figs. 2 and 3, CpG occurrence is positively correlated with DNA methylation levels and nucleosome occupancy, and as noted earlier, the in vitro data are enriched in CpGs relative to the in vivo data (Fig. 1b). The NOMe-seq data aligned to MNase-seq derived nucleosome midpoints (Fig. 1g–j) mirror the results for the combined BS-seq and MNase-seq data (Fig. 1e, f). Additionally, the nucleosome phasing (Fig. 1j), the enrichment of DNA methylation in linker DNA (Fig. 1h), and the 10-bp periodic patterns of DNA methylation levels with the unique rotational orientation (Fig. 1g) give us confidence that the integration of MNase-seq and NOMe-seq data was precisely executed.
Average DNA methylation levels and nucleosome occupancy as a function of nucleosome core CpG frequency. Nucleosome sublibraries were generated for the in vitro and in vivo leukocyte and IMR90 MNase-seq libraries based on the number of CpG occurrences between positions −61 to +61 relative to nucleosome midpoints. Using BS-seq and MNase-seq data from leukocytes, average mCpG/CpG fractions were computed from forward and reverse complement sequences aligned to nucleosome midpoints for each in vitro (a) and in vivo (b) nucleosome sublibrary. Using MNase-seq and NOMe-seq data from IMR90 cells, average mHCG/HCG fractions (c) and uGCH/GCH fractions (d) were computed from forward and reverse complement sequences aligned to MNase-seq derived nucleosome midpoints for each nucleosome sublibrary. Nucleosomes within CpG Islands were excluded from this analysis
Effects of increasing methylated CpG density in the nucleosome core on nucleosome occupancy and the ratio of methylated CpG density in core versus linker DNA. Boxplots display the distribution of nucleosome occupancy as a function of the number of CpGs (a), the number of methylated CpGs (c), the number of unmethylated CpGs (d), G + C content (g), and G + C-CpG content (i) for all nucleosomes in the genome outside of CpG islands. Boxplots also show the distribution of DNA methylation levels (b), G + C content (f), and G + C-CpG content (h) as a function of the number of CpGs. e Boxplots show the distribution of the ratios of methylated CpG density per base pair in the core versus linker DNA as a function of the number of CpGs. See "Methods" section for more details
In an attempt to understand the relationship between DNA methylation and nucleosome occupancy, we first divided the MNase-seq derived nucleosome sequences into sublibraries based on the numbers of CpGs. The average mCpG/CpG fractions in the in vitro and in vivo sublibraries from leukocytes were determined, and the results are plotted in panels a and b of Fig. 2, respectively. In both cases, low CpG densities (1–3 CpGs per fragment) correspond to preferred linker methylation, while higher frequencies correspond to preferential methylation in the nucleosome core. These results can explain the preferential linker methylation in the total in vivo data and the preferential core methylation in the total in vitro data since the overall CpG content in the unfractionated in vitro library is 1.6-fold higher than what is observed in the in vivo data (Fig. 1b, e, f). Figure 2c shows essentially the same results using the MNase-seq derived sublibraries and NOMe-seq data from IMR90 cells, and Fig. 2d shows a corresponding graded increase in nucleosome occupancy as a function of increasing CpG content in the central selected nucleosome, and this increase does not appreciably extend into the two flanking nucleosomes.
To further characterize the relationship between DNA methylation and nucleosome occupancy, boxplots were used to show the distributions of nucleosome occupancy in different sequence contexts using all nucleosomes in the genome excluding those within CpG islands (Fig. 3). See "Methods" section for more details. Figure 3a shows an apparent linear relationship between CpG frequency and nucleosome occupancy, while a cooperative relationship between CpG frequency and DNA methylation is indicated in Fig. 3b [20, 39]. Combining mCpG frequency and nucleosome occupancy data in Fig. 3c yields a stronger correlation compared to total CpG frequency (Fig. 3a), suggesting that the presence of mCpGs has a role in dictating nucleosome occupancy levels. This interpretation is further supported by the negative relationship between uCpGs and occupancy (Fig. 3d). An additional feature that is dependent on CpG content is the ratio of mCpG levels in the core versus linker, such that linker preference gradually gives rise to core preference as a function of increasing CpG content (Fig. 3e).
In order to provide balance for the analyses conducted in Figs. 2 and 3, we also investigated the effects of CpG density in linker DNA on nucleosome occupancy and on DNA methylation. Interestingly, with increasing linker CpG density, average nucleosome occupancies remained constant while DNA methylation percentages, for the most part, appeared to drop in the core (Additional file 1: Figure S2). Moreover, when comparing the effects of increasing CpG density in linker versus nucleosome core DNA on mCpG density and on nucleosome occupancy side by side, it becomes apparent that mCpG density in nucleosomal DNA but not linker DNA is correlated with nucleosome occupancy (Additional file 1: Figure S3).
Many studies have implicated G + C content in the control of nucleosome occupancy [41, 42]. However, these models have shown limited applicability in mammalian cells [32]. It is conceivable that the increase in nucleosome occupancy that we observe upon increasing CpG is actually due to an increase in G + C content since CpG-rich fragments tend to be rich in G + C content as computed in Fig. 3f. However, this increase in G + C content is accompanied by only a marginal increase in nucleosome occupancy (Fig. 3g), and when CpGs are removed from the G + C content, there is essentially no considerable effect on nucleosome occupancy as shown in Fig. 3h, i.
DNA methylation and nucleosome occupancy within various genomic features
The MNase-seq derived nucleosome midpoints from IMR90 cells were annotated by HOMER [43] in order to determine whether the results for the entire genome displayed in Figs. 2 and 3 are observed within 20 different genomic features (Fig. 4; Additional file 1: Figures S4–S11). We first carried out each of the analyses portrayed by Fig. 3a–i using exons and featureless intergenic sequences in place of the sequences used from the total genome, and the results are similar to those presented in Fig. 3 (Additional file 1: Figures S4, S5). Average profiles of CpG occurrence, DNA methylation, and nucleosome occupancy aligned to nucleosomes within these features reveal a diverse set of chromatin landscapes (Additional file 1: Figures S6–S11), but within each of these genomic features, positive correlations are observed when plotting the distributions of nucleosome occupancy as function mCpG frequency as exhibited in Additional file 1: Figure S12. These results provide additional evidence for the dependence of nucleosome occupancy on mCpG density and imply that this phenomenon is a universal or a nearly universal property in the genome.
Frequency profiles of CpGs, DNA methylation levels, and nucleosome occupancy surrounding nucleosomes positioned within exons, introns, and SINE-Alu elements. Using MNase-seq and NOMe-seq data from IMR90 cells, average occurrences of CpGs (a), mHCG/HCG fractions (b) and uGCH/GCH fractions (c) were computed from forward and reverse complement sequences aligned to MNase-seq derived nucleosome midpoints
Among the genomic features with the highest nucleosome occupancies and mCpG density include exons [22–24] and SINE-Alu transposable elements [44]. Characteristics of these elements along with intron sequences are shown in Fig. 4. As compared to flanking and bulk sequences, both exons and SINE-Alu elements are enriched in CpGs, possess high levels of DNA methylation within the nucleosome core, and display an enrichment in nucleosome occupancy. In contrast, intron sequences are similar to bulk DNA, possessing low CpG content, average nucleosome occupancy, and higher methylation levels in linker DNA. The results in Additional file 1: Figure S13 also show that there are sharp increases in CpG occurrence, DNA methylation, and nucleosome occupancy in exons over introns at both the 5′ and 3′ exon–intron junctions. The higher CpG occurrences in exons as compared to introns are at least partially due to coding constraints. The potential consequences of this effect in terms of the control of co-transcriptional splicing by DNA methylation have been discussed previously [24].
DNA methylation and nucleosome occupancy in domains of selected histone modifications
Nucleosomes containing posttranslationally modified and variant histones are considered to represent key players in the control of the genome [5, 6, 9, 23, 45]. It was therefore of interest to characterize the DNA methylation status of these nucleosomes in order to determine whether or not their occupancies are proportional to mCpG density as it is in the bulk of the genome. Before performing this analysis, we first annotated every base pair in the peaks of 12 well-studied histone modifications and the variant H2A.Z using IMR90 ChIP-seq data from the Roadmap Epigenomics Project [46]. The annotation data shown in Additional file 1: Figure S14 show where these modifications are enriched or depleted relative to the entire genome. Figure 5a–c displays the CpG content, DNA methylation levels, and nucleosome occupancy data surrounding nucleosome midpoints that are located within peaks of the histone modifications listed in the key of panel d. For each histone modification, the average nucleosome occupancy values were plotted as a function of the number of mCpGs per nucleosome (Fig. 5d).
Analysis of nucleosome occupancy and mCpG density in differentially marked chromatin across the genome. Using MNase-seq, NOMe-seq, and Roadmap ChIP-seq data from IMR90 cells, average occurrences of CpGs (a), mHCG/HCG fractions (b), and uGCH/GCH fractions (c) were computed from forward and reverse complement sequences aligned to MNase-seq derived nucleosome midpoints within several histone modification peaks indicated in the key. The 12 histone modifications (and variant H2A.Z) in the key are ordered by decreasing average DNA methylation at the nucleosome midpoint. d Using data between positions −61 to +61 relative to the nucleosome midpoint, the average nucleosome occupancy for each histone modification was plotted against the corresponding average number of methylated CpGs in the central 123 base pairs of the nucleosome. e–g Nucleosome occupancy (e) and mCpGs density (f) were weighted by ChIP-seq Z-scores for both single and paired histone modifications (see "Methods" section). g Plots of nucleosome occupancy and mCpG density from the heatmaps in e and f show that H3K9me3-modified chromatin (blue dots) deviates from the correlation observed in the genome-wide data
Nucleosomes marked by H3K4me3 are highly enriched in promoters, 5'UTRs, and CpG Islands (Additional file 1: Figure S12) and as expected contain high levels of unmethylated CpGs and possess low nucleosome occupancy values (Fig. 5). On the other hand, nucleosomes marked by H3K36me3 are located in gene bodies and are enriched by a factor of 2 in exons over introns (Additional file 1: Figure S12). Accordingly, H3K36me3-modified nucleosomes contain moderately high levels of methylated CpGs are associated with high nucleosome occupancy (Fig. 5). Regardless of histone modification, occupancy levels for most nucleosomes are correlated with their mCpG frequencies and DNA methylation levels in this analysis (Fig. 5d; Additional file 1: Figure S15). The only major exception is observed with nucleosomes marked by H3K9me3, which display very low levels of CpGs and low DNA methylation levels yet possess the highest nucleosome occupancy out of all the examined histone modifications (Fig. 5d). H3K9me3-modified nucleosomes are preferentially associated with constitutive heterochromatin where they play critical roles in DNA silencing [47].
In order to further investigate the relationship between DNA methylation and nucleosome occupancy within modified nucleosomes, we expanded the analysis in Fig. 5a–d by assessing the pairwise data among the 12 histone modifications and the variant H2A.Z (Fig. 5e–g). Comparison of the heatmaps displaying nucleosome occupancy (Fig. 5e) and mCpG density (Fig. 5f) as well as DNA methylation levels (Additional file 1: Figure S15) reveals a strong correspondence for most histone modification pairs and is supported by the high correlation given in Fig. 5g. The major exceptions to this correspondence include pairs marked by H3K9me3 indicated by the blue dots in Fig. 5g, which reflect the findings in Fig. 5d.
DNA methylation and nucleosome occupancy in CpG islands
Over half of the promoters in the human genome reside within CpG islands. These CpG- and G + C-rich segments serve as platforms for the assembly of unstable nucleosomes and sites for attracting regulatory proteins leading to regions of chromatin that are permissive for transcriptional activation [10]. We characterized the chromatin in CpG islands in IMR90 cells by analyzing the relationships among nucleosome occupancy, DNA methylation, and the various histone modifications analyzed in Fig. 5. Figure 6a shows that, like the bulk of the genome, the frequencies of mCpG are positively correlated with nucleosome occupancies in CpG islands. The CpG island nucleosomes were then divided into two groups, non-TSS and TSS, based on whether or not their midpoints occurred within 500 bp of a TSS. For each MNase-seq derived nucleosome, nucleosome occupancy and DNA methylation values were plotted in 2D color-coded scatterplots (Fig. 6b, c). These results reveal an exception to the theme observed with bulk DNA in that there are significant numbers of the sequences that display low methylation but high nucleosome occupancy.
Characterization of chromatin at CpG islands. a Boxplots display the distribution of nucleosome occupancy as a function of the number of methylated CpGs. b, c Nucleosome with midpoints within CpG islands was divided into two groups, non-TSS and TSS. For each nucleosome, nucleosome occupancy and DNA methylation values were plotted in the 2D color-coded scatterplots. Using NOMe-seq and Roadmap ChIP-seq data from IMR90 cells, nucleosome occupancy, DNA methylation levels, and Z-scores from 12 histone modifications and the histone variant H2A.Z were aligned to TSSs that overlapped CpG Islands (d) and to the centers of CpG islands located in gene bodies and intergenic regions (e). Heatmap values were computed in 101 21-bp bins surrounding the TSSs and non-TSS CpG island centers, and sites (heatmap rows) were sorted by nucleosome occupancy
The heatmaps representing CpG islands displayed in Fig. 6d, e, which consist of NOMe-seq, DNase-seq, and ChIP-seq data sorted by nucleosome occupancy, were constructed in order to elucidate the link between the low DNA methylation levels and high nucleosome occupancies observed in a subset of CpG island nucleosomes (Fig. 6b, c). Figure 6d displays data aligned to TSSs that are overlapped by CpG islands, and Fig. 6e shows data aligned to CpG island centers in non-TSS regions, which were subdivided into intragenic and intergenic groups. An open chromatin configuration at promoters is signified by the void of nucleosomes and the enrichment in DNaseI hypersensitivity at the TSSs in Fig. 6d. The heatmaps appear to show a strong overall correspondence between nucleosome occupancy and DNA methylation in gene bodies and intergenic regions but not at TSSs with high nucleosome occupancy. Most histone modifications appear to follow the overall positive correlation between DNA methylation and nucleosome occupancy. For example, the nucleosomes marked by acetylated histones as well as by H3K4me2/3 are located in unmethylated CpG islands with low nucleosome occupancy, while nucleosomes marked by H3K36me3 are located in methylated CpG islands with moderate to high levels of both occupancy and methylation in gene bodies. One noticeable exception in Fig. 6d is enrichment of H3K27me3, which is an epigenetic mark of polycomb-repressed genes [48]. However, when the heatmaps are sorted by H3K27me3 or H3K9me3 signals, it becomes apparent that both of these epigenetic modifications are characterized by low levels of methylation and moderate to high nucleosome occupancy in CpG islands in all three annotations (Additional file 1: Figures S16, S17) with a stronger anti-correlation associated with the presence of H3K27me3 [49].
DNA methylation and nucleosome occupancy at conserved transcription factor binding sites
In order to further evaluate the control of DNA methylation and nucleosome occupancy at regulatory elements in an unbiased manner, we studied all transcription factor binding sites (TFBSs) that are conserved in mammals and that contain CpG in their recognition sequences [50–52]. In this analysis, we characterized the chromatin at TFBSs in HCT116 cells using NOMe-seq [53], BS-seq [54], and also ChIP-seq data from ENCODE (Fig. 7) [55]. All CpG-containing TFBSs provided by the UCSC genome browser were divided into unmethylated and methylated sites (see "Methods" section), and the data above were aligned to these coordinates and sorted by decreasing nucleosome occupancy (Fig. 7a). Nearly all methylated sites display high nucleosome occupancy, while a large fraction of unmethylated sites are nucleosome depleted, DNaseI hypersensitive, and marked by H3K27ac (Fig. 7a, b), in agreement with several studies that have characterized the chromatin at active enhancers [56–58]. We also examined the occupancy levels of 10 transcription factors from ENCODE at their respective unmethylated and methylated CpG-containing TFBSs (Fig. 7c, d). All ten transcription factors exhibit binding at their unmethylated sites and appear nearly or completely absent at their methylated sites (Fig. 7d). Further analysis of the SP1 transcription factor shows that its occupancy is associated with low nucleosome occupancy, unmethylated CpGs, DNaseI hypersensitivity and H3K27 acetylation (Fig. 7c).
Characterization of chromatin at unmethylated and methylated conserved transcription factor binding sites. All conserved transcription factor binding sites (TFBSs) from the UCSC genome browser were divided into unmethylated and methylated sites (see "Methods" section). a Using NOMe-seq, BS-seq, and ENCODE ChIP-seq data from HCT116 cells, nucleosome occupancy, DNA methylation levels, and DNase-seq and H3K27ac signals were aligned to unmethylated and methylated conserved TFBSs. b Average uGCH/GCH fractions were also computed from forward and reverse complement sequences aligned to these unmethylated and methylated conserved TFBSs. c Similar to a, nucleosome occupancy, DNA methylation levels, and DNase-seq, H3K27ac, and SP1 signals were aligned to unmethylated and methylated SP1 conserved TFBSs. d Using ENCODE ChIP-seq data for 10 transcription factors, including SP1, average signals were plotted with respect to their corresponding unmethylated and methylated conserved TFBSs. In the heatmaps, average signals were computed in 51 21-bp bins surrounding each conserved site. Scales for DNase-seq, H3K27ac, and the transcription factors represent occupancy levels in reads per million
We extended the characterization of the chromatin shown in Fig. 7a by examining 5 additional histone modifications and the variant H2A.Z using ChIP-seq data from the Jones laboratory (Additional file 1: Figure S18) [53]. For the methylated sites, the enrichment of H3K36 trimethylation and depletion of H3K27 acetylation and H3K4 trimethylation imply that several methylated sites are located in the gene bodies and deficient in active promoters or enhancers (Additional file 1: Figure S18). On the other hand, the unmethylated sites can be divided into two main groups. One group is enriched in CpG islands as well as active promoters and enhancers, which are indicated by low nucleosome occupancy and DNA methylation with high H3K27 acetylation and H3K4 methylation, and the second group is less CpG rich and possesses low to moderate levels of DNA methylation and high nucleosome occupancy (Additional file 1: Figure S18). Thus, the second group represents another exception to the positive correlation between methylated CpGs and nucleosome occupancy. Interestingly, the second group exhibits an enrichment of H3K9 and H3K27 methylation (Additional file 1: Figure S18), which are the same modifications linked to the exceptions observed in CpG-poor genomic regions and in CpG islands (Figs. 5, 6). These modifications along with the presence of H3K4me1 suggest that some of the sites in the second group may signify poised enhancers (Additional file 1: Figure S18) [56–58].
Relationships between DNA methylation and nucleosome occupancy in vitro and in vivo
In order to explore links between DNA methylation and nucleosome occupancy, we relied heavily on data derived from NOMe-seq because this methodology, developed by Jones and coworkers, enables the simultaneous measurement of nucleosome occupancy and endogenous methylation for cell populations and single cells [29, 53, 59]. In comparison with NOMe-seq, MNase-seq requires higher coverage and relies on enrichment-based measurements of nucleosome occupancy, and these estimations can be skewed by sequence biases generated by enzyme cutting preferences, extents of digestion, and library amplification steps [60]. However, MNase-seq's primary advantage comes from its capability to determine the positions of nucleosome midpoints at near base-pair resolution, and this precision can be enhanced if paired-end sequencing is applied [21, 60]. By exploiting the strengths of both MNase-seq and NOMe-seq, we were able to confidently examine the effects of DNA methylation on nucleosome occupancy (Figs. 1, 2, 3; Additional file 1: Figures S2, S3).
Epigenetic factors must be maintained during development and cell differentiation, for without this characteristic they would be diluted during cell division. Two major epigenetic factors that satisfy this criterion are DNA methylation and stable posttranslationally modified histones. The metabolically stable histone methylation of H3K9 and H3K27, in contrast to histones modified by acetylation or phosphorylation, has half-lives measured in hours or longer and is commonly viewed as memory markers [45, 61–64]. These stable histone modifications along with DNA methylation are also involved in chromatin silencing, which raises the question as to whether or not their silencing mechanisms display similarities. The results of this investigation provide implications for this question, for the modes of action of these factors, and for epigenetics in general.
Our results suggest that the nucleosome serves as an effector arm of epigenetic mechanisms and that enhanced nucleosome occupancy is causally related to silencing induced by DNA methylation and stable histone modifications. The results in our previous studies demonstrated that DNA methylation enhances the stability of nucleosomes in a fraction of the human genome in vitro that contains multiple CpGs arranged in an unusual rotation with their minor grooves facing toward the histone octamer [21]. The studies described in this report elaborated on these features and extended the findings to the entire genome under in vivo conditions. The results show that nucleosome occupancy is correlated with mCpG frequency in the total genome and in subgenomic regions defined by CpG frequency (Figs. 2, 3; Additional file 1: Figures S2, S3) and by the coordinates of annotated features (Figs. 4, 6; Additional file 1: Figures S4–S13, S16, S17, S19), histone modification domains (Fig. 5; Additional file 1: Figure S15), and CpG-containing transcription factor binding sites (Fig. 7; Additional file 1: Figure S18). However, our analyses uncovered two exceptions to the generalization between mCpG levels and nucleosome occupancy, which were nucleosomes marked by H3K9me3 and H3K27me3 (Figs. 5, 6; Additional file 1: Figures S16–S18). These nucleosomes displayed low levels of mCpG but high nucleosome occupancy values, which is indicative of some feature, other than DNA methylation, being responsible for their high nucleosome residency.
Modes of action for mCpG
The results in this report raise the question of why increasing methylated CpG density is associated with increasing nucleosome occupancy. One possible answer could be derived from the influential silencing mechanisms of methyl-binding proteins [65, 66]. It is also conceivable that highly methylated CpG-rich DNA could directly enhance nucleosome stability before nucleosomes are assembled as demonstrated in vitro [21, 67, 68] or become more stable after de novo methylation. We proposed in our MNase-seq study with in vitro reconstituted nucleosomes [21] that the hydrophobic, bulky methyl groups in the accessible major groove could cause narrowing of the corresponding minor groove. Indeed, in a recent DNase-seq experiment conducted on naked DNA, cutting frequencies were shown to be influenced by the methylation-induced narrowing of the minor groove [69]. The consequence of this action could strengthen the interactions between positive-charged histone arginines and the negative-charged DNA phosphate backbone, thereby enhancing nucleosome stability and in turn, increasing nucleosome occupancy in the cell [21, 69].
Although previous studies have indicated that the nucleosome core is the preferred site of DNA methylation [21, 22, 70, 71], others have suggested that methylation occurs preferentially in the linker regions [30, 72]. The results in Figs. 2 and 3 provide a possible explanation for this apparent discrepancy since a transition of preferential methylation from linker to core as a function of increasing CpG content was revealed along with a corresponding increase in nucleosome occupancy. For example, nucleosomes in CpG-rich exonic DNA and SINE-Alu sequences display higher mCpG density and nucleosome occupancy levels in the core, while nucleosomes in CpG-poor intronic DNA show selective methylation in linker DNA (Fig. 4). In light of this trend, the in vivo data displayed in Fig. 1d imply that methylated CpG dinucleotides have virtually no effect on nucleosome occupancy or positioning at the unfractionated genome level. However, in increasingly CpG-rich DNA, the effect of methylated CpG density on nucleosome occupancy becomes more apparent, and therefore, the majority of this effect is likely limited to a small fraction of the genome where CpG density is high. In fact, only ~7% of the genome is represented by 4 or more CpGs in 123-base-pair sliding windows.
The transition of preferential methylation from linker to nucleosome core may reflect the cooperative binding and enzymatic activities of the de novo DNA methyltransferases DNMT3a and DNMT3b, which have been shown to increase with CpG density outside of CpG islands [20]. Similar to bulk genomic DNA, we find a similar cooperative relationship between DNA methylation levels and CpG frequency in nucleosomal DNA (Fig. 3b). This cooperative mode of binding and methylation may be due to the heteromeric nature of the DNMT3 complexes in which two DNMT active sites display a spacing equivalent to about 10 nucleotides of DNA [73]. Thus, in the domains of high CpG density, clusters of CpGs would tend to become preferentially methylated, conceivably promoting nucleosome assembly or repositioning to the site. The unusual rotational orientation of mCpGs in a 10-nucleotide period reported previously [20, 21] and shown in Fig. 1 may also facilitate the action of the DNMTs since multiple mCpGs with major grooves facing away from the histone surface should be the most assessable to these enzymes.
It is important to emphasize that in methylated CpG dinucleotides, the methyl groups reside within the major groove, and it is likely that the major grooves of some CpGs along nucleosomal DNA are less accessible to the de novo methyltransferases. Indeed, it was proposed that the rotational orientation observed in methylated CpGs in Arabidopsis nucleosomes is a product of major groove accessibility [22]. The results of the de novo methyltransferase studies conducted by the Schubeler group imply that in domains of low CpG density, de novo methyltransferase binding and activity is reduced, and consequently, the DNA methylation levels in these regions are expected to be less efficiently maintained [20]. Moreover, with decreasing CpG density, the methylation of nucleosome core DNA may be more effectively inhibited due to the relatively CpG-rich substrate preference of the de novo methyltransferases and due to a decrease in probability of major groove accessibility to CpG dinucleotides. Accordingly, the DNA flanking CTCF sites (Additional file 1: Figure S1), encompassing partially methylated domains (Additional file 1: Figure S19), and in the bulk of the genome (Fig. 1), where DNA methylation levels are higher in linker DNA, are located in CpG-deficient regions. Furthermore, the positive correlation between mCpG density and nucleosome occupancy, as well as the lack of an effect of increasing linker mCpG density on nucleosome occupancy levels, suggests that methylated CpG dinucleotides in adjacent linker DNA are not significantly influencing nucleosome formation (Figs. 2, 3; Additional file 1: Figures S2–S3).
Genome silencing by DNA methylation, H3K9Me3 and H3K27Me3
A central question concerning epigenetic mechanisms is the source and nature of the primary signals for epigenetic silencing. The signals must ultimately reside in the DNA sequence, but their nature is poorly understood. The signals for relating nucleosome occupancy to DNA methylation are the patterns of CpG dinucleotides, which are encoded in the sequence, and the factors that dictate the methylation status of CpGs such as nucleosome core versus linker localization and the rotational orientation of the CpGs in the nucleosome core. Furthermore, it is most often assumed that the initial signal for posttranslational modification of histones originates with specific regulatory proteins like transcription factors that recognize specific sequences in order to elicit a chain of events that lead to the final modification.
An alternative view is that simple DNA sequence patterns are directly responsible for the initial recognition process [74, 75]. For example, the clustering of unmethylated CpGs in G + C-rich regions is thought to serve as a signal for recognition of polycomb group complexes, which in turn results in the methylation of lysine 27 on histone H3 and ultimately chromatin silencing [48, 75]. Likewise, A + T-rich oligonucleotide sequences have been proposed to play a role in the recognition of H3K9 methylases leading to heterochromatization and repression [74, 75]. It has also been proposed that abundant nuclear proteins such as the HMG box proteins recognize these sites, and it is interesting to note that HMG proteins preferentially bind to AT duplex sequences of the form (ATATAT)N as compared to (AAAAAA)N in physiological ionic strength and temperature and that this AT-heteropolymeric specificity is shared by nucleosomes that contain H3K9Me3 [74, 76].
DNA methylation, transcription factors, and enhancer chromatin
Active enhancer chromatin typically contains multiple bound transcription factors, certain histone marks such as H3K27ac, undermethylated DNA and an open structure as evidence by DNaseI hypersensitivity [56–58]. We attempted to simplify our analysis by characterizing evolutionally conserved transcription factor chromatin that contain at least one CpG in the recognition sequence. There exists apparent heterogeneity in the dataset as seen by the presence of H3K9me3, H3K27me3, partially methylated DNA, and deficiencies in H3K27ac (Additional file 1: Figure S18), which are characteristics of poised enhancers, sequences which are inactive but have potential for activation. The results in Fig. 7d show that 10 out of 10 transcription factors were preferentially associated with unmethylated DNA sequences, which raises the question whether this specificity is reflected in the binding of transcription factors to naked DNA. There are numerous examples where methylation blocks the in vitro binding of transcription factors that have CpG in their binding sites [77–79], but there are also cases including Sp1 where methylation is without effect on transcription factor binding [79, 80]. The molecular complexity of enhancer chromatin also makes it difficult to unravel cause and effect relationships. It is conceivable that the undermethylation of enhancer DNA is responsible for the reduced nucleosome occupancy and more open chromatin structure of enhancer sequences which would be consistent with the results presented in this study. However, there are alternative explanations to this proposal including the ability of transcription factors to induce loss of methylation at CpG sites, bound transcription factors excluding DNA methyltransferases, and passive DNA demethylation by DNA replication in the absence of maintenance methylation [80, 81].
Cancer, DNA methylation, and the nucleosome
A conserved property of cancer cells and tumors is global genomic hypomethylation and the local hypermethylation of some CpG Islands [82–84]. The global hypomethylation of the genome is viewed as a driving force for genomic instability in cancer, which characterizes the disease [83]. The activation of many mobile elements such as the SINE-Alu sequences observed in almost all cancers provides examples of this phenomenon [82, 83]. In fact, it has been suggested that the main carcinogenic effect of global DNA hypomethylation in cancer is mediated by its ability to create genomic instability [84]. The results of this study may be relevant to these observations since a reduction in mCpGs levels by as little as a one CpG per nucleosome results in detectable decreases in nucleosome occupancy. A decrease in the occupancy or stability of such nucleosomes might be expected to have effects on the positions of these nucleosomes as well as the positions of nucleosome arrays adjacent to these nucleosomes, which are prevalent in the genome. Changes in the stabilities and positioning of nucleosomes in a significant fraction of the chromatin, as predicted from the present study, are expected to have profound, inheritable effects on the expression of the genome.
Previous studies have suggested that DNA methylation directly enhances the stability of nucleosomes in vitro. However, the relationship between DNA methylation and nucleosome occupancy is poorly understood. In this study, we implemented a bioinformatics approach that combines MNase-seq and NOMe-seq data to study links between DNA methylation and nucleosome occupancy throughout the human genome. Using this approach, we demonstrated that increasing mCpG density is correlated with nucleosome occupancy and that in mCpG-rich nucleosomes, methylation levels are greater in the core than in the adjacent linker DNA. These nucleosomal DNA methylation patterns were detected not only in total genomic DNA but also within most subgenomic regions. Prominent exceptions to the positive correlation between mCpG density and nucleosome occupancy included CpG islands marked by H3K27me3 and CpG-poor heterochromatin marked by H3K9me3, and these modifications, along with DNA methylation, characterize the major silencing mechanisms of mammalian chromatin. Thus, the density of methylated CpG dinucleotides may be an important factor in regulating nucleosome occupancy levels the human genome.
Previously aligned in vivo and in vitro MNase-seq nucleosome and control data from leukocyte cells (neutrophil granulocytes) were acquired from GEO under accession number GSE25133 (GSM678045-63) [33]. Processed ENCODE BS-seq data from leukocyte cells (GM12878) were obtained from EMBL-EBI and are associated with GEO accession number GSE40832 (GSM1002650) [32]. Previously aligned MNase-seq data and processed NOMe-seq data from IMR90 cells were obtained from GEO under accession numbers GSE21823 (GSM543311) and GSE40770 (GSM1001125), respectively [29]. Previously aligned DNase-seq and ChIP-seq data for 12 histone modifications and the histone variant H2A.Z from IMR90 cells were obtained from the UCSD Human Reference Epigenome Mapping Project (Roadmap, GSE16256) [46]. Processed NOMe-seq data (for Fig. 7; Additional file 1: Figure S16) and ChIP-seq data (for Additional file 1: Figure S17) from HCT116 cells were obtained from GEO (GSE58638) [53]. Processed BS-seq data (for Fig. 7; Additional file 1: Figure S16) and previously aligned ChIP-seq and DNase-seq data (for Fig. 7) from HCT116 cells were obtained from GEO (GSE60106, GSM1465024) [54] and the ENCODE project [55], respectively. Exon, CpG island, and conserved TFBS coordinates were obtained from the UCSC genome browser [85] and/or the HOMER software [43], and computationally predicated CTCF sites were obtained from the CTCF Database 2.0 [86].
Identification and annotation of nucleosome midpoints
Using MNase-seq data from leukocyte and IMR90 cells, nucleosome midpoints were determined by adding or subtracting 73 base pairs from the 5′ end of each read that aligned to plus or minus strands, respectively. For the MNase-seq in vitro and in vivo nucleosome and control data from leukocyte cells, coverage was calculated in 147-base-pair windows across the genome, and fractions of coverage between the control and the nucleosome libraries were computed. Subsequently, the numbers of reads at the nucleosome midpoints were normalized by these fractions of coverage in order to subtract background that could be generated by MNase cutting biases. Regardless of whether or not the MNase control data were subtracted from the nucleosome data, the dinucleotide and DNA methylation frequency profiles in Fig. 1a–f appeared nearly identical to ones where control data were not subtracted (data not shown). Nucleosome midpoints from IMR90 cells were annotated using the HOMER software package, and all frequency profiles surrounding nucleosome midpoints were generated using in-house scripts.
Data analysis for Fig. 3
Using MNase-seq and NOMe-seq data from IMR90 cells, the number of CpGs, G + C content, average mHCG/HCG fraction, and average uGCH/GCH fraction were computed for each nucleosome outside of CpG islands between positions −61 and +61 relative to the MNase-seq derived nucleosome midpoint. Methylation data for at least 2 cytosines within HCGs and GCHs regardless of strand were required for inclusion of a nucleosome in the analysis. With these data, boxplots were used to display the distributions presented in the figure. The numbers of methylated and unmethylated CpGs were determined by multiplying the average mHCG/HCG and uHCG/HCG fractions, respectfully, by the number of CpGs and rounding to the nearest integer. For the analysis in panel e, the number of CpGs and the average mHCG/HCG ratio were computed for the linker DNA of each included nucleosome between positions (±85 to ±115) relative to the nucleosome midpoint. Methylation data for at least 2 cytosines within HCGs in the linker DNA regardless of strand were required for inclusion of a nucleosome in this analysis. With these data, boxplots show the distribution of the ratios of methylated CpG density per base pair in the core versus linker DNA as a function of the number of CpGs. Methylation CpG density per base pair in a nucleosome and its linker DNA were computed by multiplying the average mHCG/HCG ratio by the number of CpGs in the core and linker separately and dividing by 123 and 62 base pairs, respectfully.
Identification and annotation of ChIP-seq peaks
Peaks for ChIP-seq data from IMR90 cells were identified using SICER with default parameters [87]. Each base pair in every peak called by SICER for the 12 histone modifications and the histone variant H2A.Z was annotated by the HOMER software.
For the analysis represented by panels e–g, the number of CpGs, average mHCG/HCG fraction, and average uGCH/GCH fraction were determined for each nucleosome between positions −61 and +61 relative to each MNase-seq derived nucleosome midpoint. ChIP-seq Z-scores for the 12 histone modifications and the variant H2A.Z were also determined at each midpoint. For each histone modification, averages values were weighted by Z-scores using the following formula.
$$\bar{x}\left( h \right) = \frac{{\mathop \sum \nolimits_{i}^{n} x_{i} \times z\left( h \right)_{i} }}{{\mathop \sum \nolimits_{i}^{n} z\left( h \right)_{i} }}\; \quad {\text{if}}\; z\left( h \right)_{i} > 0$$
For histone modification pairs, average values were weighted by the modification with the smaller Z-score.
$${\text{if}}\; z\left( {h_{A} } \right)_{i} \;{\text{and}}\; z\left( {h_{B} } \right)_{i} > 0$$
$$\bar{x}\left( {h_{A,B} } \right) = \frac{{\mathop \sum \nolimits_{i}^{n} x_{i} \times z\left( {h_{A} } \right)_{i} + \mathop \sum \nolimits_{i}^{n} x_{i} \times z\left( {h_{B} } \right)_{i} }}{{\mathop \sum \nolimits_{i}^{n} z\left( {h_{A} } \right)_{i} + \mathop \sum \nolimits_{i}^{n} z\left( {h_{B} } \right)_{i} }}$$
$$z\left( {h_{A} } \right)_{i} = 0\; \quad {\text{if}}\;z\left( {h_{A} } \right)_{i} > z\left( {h_{B} } \right)_{i}$$
$$z\left( {h_{B} } \right)_{i} = 0\; \quad {\text{if}}\;z\left( {h_{B} } \right)_{i} > z\left( {h_{A} } \right)_{i}$$
Using MNase-seq and NOMe-seq data from IMR90 cells, the same procedure described for Fig. 3 was carried out for nucleosomes positioned in CpG islands. CpG island nucleosomes were divided into two groups, non-TSS and TSS, based on whether or not their midpoints occurred within 500 bp of a TSS. These information were used to generate the plots in panels a, b, and c. For panel d, if a CpG island overlapped a RefSeq TSS, the CpG island was included. For panel e, if a CpG island center was annotated as an exon or intron (gene body) or as intergenic by HOMER and if the CpG island was not used in panel d, the CpG island was included. Subsequently, NOMe-seq data in bigwig format were aligned to CpG island TSSs and CpG island gene body and intergenic centers using an unpublished Perl script written by Yaping Liu, and in-house scripts were used to express the occupancy and methylation levels in 101 21-base pair bins. Subsequently, the CpG islands were sorted by decreasing average occupancy across the 101 bins. Using Roadmap ChIP-seq data from IMR90 cells in bed format and another Perl script written by Yaping Liu, RPM values minus input were computed across the genome for each dataset, and these values were transformed into Z-scores. These Z-scores were then aligned to the TSSs and CpG islands centers sorted by decreasing occupancy and binned in the same way as above. Heatmaps were generated using Java Tree View [88].
Data analysis for Fig. 7 and Additional file 1: Figure S18
BS-seq data from HCT116 were used instead of NOMe-seq data to evaluate the DNA methylation levels at conserved TFBSs so that more sites could be analyzed. All hg19 conserved TFBSs from the UCSC genome browser with methylation data for a cytosine in at least one CpG were divided into unmethylated and methylated sites depending on whether or not the average mCpG/CpG fraction was greater than or equal to 0.5. These conserved TFBSs do not include CTCF conserved sites. Alignment of the NOMe-seq occupancy data to conserved TFBSs was executed in the same way as described for Fig. 6. For Fig. 7, ENCODE ChIP-seq and DNase-seq data in bam format were aligned to conserved TFBSs using ngs.plot [89], but RPM levels were not transformed into Z-scores. For Additional file 1: Figure S18, previously generated ChIP-seq Z-scores in bigwig format were aligned to the conserved TFBSs, and these values were expressed in 21-bp bins. All heatmaps were generated using Java Tree View [88].
Supplementary methods
More detailed bioinformatics procedures and in-house scripts used in this study are provided in Additional file 2.
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CKC and JNA conceived the ideas and analyses conducted for this study. CKC and JNA wrote the manuscript. CKC wrote in-house scripts and generated figures. Both authors read and approved the final manuscript.
We would like to thank the Purdue Bioinformatics Core for the use of their computational resources. We would also like to thank Yaping Lui for sharing his bioinformatics tools, which were used in some analyses.
All data analyzed in this study were derived from previously published work (see "Methods" section). Detailed bioinformatics protocols and in-house scripts are provided in Additional file 2.
This work was funded by the Department of Biological Sciences at Purdue University.
Department of Biochemistry and Molecular Genetics, Northwestern University Feinberg School of Medicine, 320 E. Superior Street, Chicago, IL, 60611, USA
Clayton K. Collings
Department of Biological Sciences, Purdue University, 915 W. State Street, West Lafayette, IN, 47907, USA
John N. Anderson
Search for Clayton K. Collings in:
Search for John N. Anderson in:
Correspondence to John N. Anderson.
13072_2017_125_MOESM1_ESM.docx
Additional file 1. Additional figures.
13072_2017_125_MOESM2_ESM.zip
Additional file 2. Detailed bioinformatics procedures and in-house scripts used in this study are enclosed in the zip file.
Collings, C.K., Anderson, J.N. Links between DNA methylation and nucleosome occupancy in the human genome. Epigenetics & Chromatin 10, 18 (2017). https://doi.org/10.1186/s13072-017-0125-5
Nucleosome
NOMe-seq
MNase-seq | CommonCrawl |
Corporate Finance & Accounting Financial Ratios
Incremental Capital Output Ratio – ICOR Definition
By Adam Hayes
What Is the Incremental Capital Output Ratio (ICOR)?
The incremental capital output ratio (ICOR) is a frequently used tool that explains the relationship between the level of investment made in the economy and the consequent increase in GDP. ICOR indicates the additional unit of capital or investment needed to produce an additional unit of output.
The utility of ICOR is that with more and more investment, the capital output ratio itself may change and hence the usual capital output ratio will not be useful. It is a metric that assesses the marginal amount of investment capital necessary for a country or other entity to generate the next unit of production.
Overall, a higher ICOR value is not preferred because it indicates that the entity's production is inefficient. The measure is used predominantly in determining a country's level of production efficiency.
The Formula the Incremental Capital Output Ratio (ICOR)
ICOR=Annual InvestmentAnnual Increase in GDPICOR=\frac{\text{Annual Investment}}{\text{Annual Increase in GDP}}ICOR=Annual Increase in GDPAnnual Investment
What Does the Incremental Capital Output Ratio Tell You?
Some critics of ICOR have suggested that its uses are restricted as there is a limit to how efficient countries can become as their processes become increasingly advanced. For example, a developing country can theoretically increase its GDP by a greater margin with a set amount of resources than its developed counterpart can. This is because the developed country is already operating with the highest level of technology and infrastructure.
Any further improvements would have to come from more costly research and development, whereas the developing country can implement existing technology to improve its situation.
For example, suppose that Country X has an incremental capital output ratio (ICOR) of 10. This implies that $10 worth of capital investment is necessary to generate $1 of extra production. Furthermore, if country X's ICOR was 12 last year, this implies that Country X has become more efficient in its use of capital.
The incremental capital output ratio (ICOR) explains the relationship between the level of investment made in the economy and the consequent increase in GDP.
The utility of ICOR is that with more and more investment, the capital output ratio itself may change and hence the usual capital output ratio will not be useful.
Some critics of ICOR have suggested that its uses are restricted as there is a limit to how efficient countries can become as their processes become increasingly advanced.
Example of How to Use the Incremental Capital Output Ratio
As a real-world example of using ICOR, take the example of India. The planning commission working group in India put out the required rate of investment that would be needed to achieve different growth outcomes in the 12th Five-Year Plan. For a growth rate of 8%, the investment rate at market price would need to be at 30.5%, while for a growth rate of 9.5%, an investment rate of 35.8% would be required.
Savings rates in India dropped from the level of 36.8% of the gross domestic product in the year 2007-08 to 30.8% in 2012-13. The rate of growth during the same period fell from 9.6% to 6.2%. The growth is further expected to fall to the level of 5% in the current financial year with a savings rate of 30%.
Clearly, the drop in India's growth rate is more dramatic and steeper than the fall in the savings rates. Therefore, there are reasons beyond savings and investment rate that would explain the drop in the rate of growth in the Indian economy. Otherwise, the economy is getting increasingly inefficient.
Limitations of the Incremental Capital Output Ratio
For advanced economies, accurately estimating ICOR is subject to a myriad of issues. A primary complaint of critics is its inability to adjust to the new economy – an economy ever-more-driven by intangible assets, which are difficult to measure or record.
For instance, in the 21st century, businesses are impacted ever more by design, branding, R&D and software, all of which are more challenging to factor into investment levels and GDP than their predecessor tangible assets, like machinery, buildings and computers – a hallmark of industrial periods.
On-demand options such as software-as-a-service have greatly driven down the need for investment in fixed assets. This can be extended even further with the rise of "as-a-service" models for nearly everything. It all adds up to businesses increasing their production levels with items that are now expensed, and not capitalized – and thus, considered an investment.
Even the ICOR's denominator, GDP, isn't immune to necessary adjustments in for changes in modern economic output measurement.
Productivity measures the efficiency of production in macroeconomics, and is typically expressed as a ratio of GDP to hours worked.
Gross Domestic Product – GDP
Gross Domestic Product (GDP) is the monetary value of all finished goods and services made within a country during a specific period.
The Capital Asset Pricing Model is a model that describes the relationship between risk and expected return.
Production Efficiency Definition
Production efficiency describes a maximum capacity level in which an entity can no longer produce more of a good without lowering the production of another.
Internet of Energy (IoE)
The Internet of Energy refers to the automation of electricity infrastructures for energy producers, often allowing energy to flow more efficiently.
Net Foreign Factor Income (NFFI) Definition
Net foreign factor income (NFFI) is the difference between a nation's gross national product (GNP) and gross domestic product (GDP).
Reverse Engineering Return On Equity
Are Social Security Payments Included in the U.S. GDP Calculation?
Top 20 Economies in the World
Why the USSR Collapsed Economically
Catch on to the CCAPM
Economics Report: Compare and Contrast India vs. Brazil | CommonCrawl |
\begin{document}
\title{Linear Non-Transitive Temporal Logic, Knowledge Operations, Algorithms for Admissibility }
\author{Vladimir Rybakov} \authorrunning{V.Rybakov} \titlerunning{Agent's Knowledge Operations in Non-Transitive Temporal Logic
}
\institute{School of Computing, Mathematics and DT,
Manchester Metropolitan University,
John Dalton Building, Chester Street, Manchester, M1 5GD, U.K,
\email{[email protected]}
}
\toctitle{Lecture Notes in Computer Science}
\mainmatter
\maketitle
\newcommand{\alpha}{\alpha} \newcommand{\beta}{\beta} \newcommand{\gamma}{\gamma} \newcommand{\delta}{\delta} \newcommand{\lambda}{\lambda} \newcommand{{\cal F}or}{{\cal F}or} \newcommand{{\cal A}}{{\cal A}} \newcommand{{\cal B}}{{\cal B}} \newcommand{{\cal C}}{{\cal C}} \newcommand{{\cal M}}{{\cal M}} \newcommand{{\cal M}}{{\cal M}} \newcommand{{\cal B}}{{\cal B}} \newcommand{{\cal R}}{{\cal R}} \newcommand{{\cal F}}{{\cal F}} \newcommand{{\cal Y}}{{\cal Y}} \newcommand{{\cal X}}{{\cal X}} \newcommand{{\cal D}}{{\cal D}} \newcommand{{\cal W}}{{\cal W}} \newcommand{{\cal Z}}{{\cal Z}} \newcommand{{\cal L}}{{\cal L}} \newcommand{{\cal W}}{{\cal W}} \newcommand{{\cal K}}{{\cal K}} \newcommand{{\varphi}}{{\varphi}}
\newcommand{\ii}[0] {\rightarrow}
\newcommand{\ri}[0] {\mbox{$\Rightarrow$}}
\newcommand{\lri}[0] {\mbox{$\Leftrightarrow$}}
\newcommand{\lr}[0] {\mbox{$\Longleftrightarrow$}}
\newcommand{\ci}[1]{\cite{#1}}
\newcommand{{\sl Proof}}{{\sl Proof}}
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\newcommand{\dd}[0]{ \rule{1.5mm}{1.5mm}}
\newcommand{{\cal LT \hspace*{-0.05cm}L}}{{\cal LT \hspace*{-0.05cm}L}}
\newcommand{{\bf N}}{{\bf N}}
\newcommand{{{\bf N^{-1}}}}{{{\bf N^{-1}}}}
\newcommand{{{\bf U}}}{{{\bf U}}}
\newcommand{{{\bf S}}}{{{\bf S}}}
\newcommand{{{\bf B}}}{{{\bf B}}}
\newcommand{{\cal N}}{{\cal N}} \newcommand{{{\bf S}}}{{{\bf S}}} \newcommand{{\cal LT \hspace*{-0.05cm}L}_K(Z)}{{\cal LT \hspace*{-0.05cm}L}_K(Z)} \newcommand{{{\cal Z}_C}}{{{\cal Z}_C}}
\date{}
\begin{abstract} The paper studies problems of satisfiability, decidability and admissibility of inference rules, conceptions of knowledge and agent's knowledge in non-transitive temporal linear logic $LTL_{Past,m}$. We find algorithms solving mentioned problems, justify our approach to consider linear non-transitive time with several examples.
Main, most complicated, technical new result is {it \bf decidability of $LTL_{Past,m}$
w.r.t. admissible rules}. We discuss several ways to formalize conceptions of knowledge and agent's knowledge within given approach in non-transitive linear logic with models directed to {\em past}.
\end{abstract}
{\bf Keywords:} temporal logic, non-transitive accessibility relations,
knowledge,
\ parameterized knowledge operations, satisfiability, admissible
rules, deciding
\ algorithms
\section{Introduction}
No question that there is no more interesting and mysterious object as conception of time. But nowadays it works fine in prose of the life -- for example, it has many interpretations in CS (eg. for interpretation of spread the set of check points in computational runs, etc.).
Historically, investigations of temporal logic in mathematical/philosophical logic
based at modal systems was originated by Arthur Prior in the late 1950s.
Since then temporal logic has been (and is) very active area in mathematical logic and information sciences, AI
and CS (cf. eg. -- Gabbay and Hodkinson\cite{ghr,gbho,gbho2}). It was observed that temporal logic has important applications in formal verification, where it is used to state requirements of hardware or software systems.
In particular, linear temporal logic $ {\cal LT \hspace*{-0.05cm}L} $ (with Until and Next) is very useful instrument(cf. Manna, Pnueli \ci{ma1,ma2}, Vardi \ci{va1,va2}) (${\cal LT \hspace*{-0.05cm}L}$ was used for analyzing protocols of computations, check of consistency, etc.). The decidability and satisfiability problems for ${\cal LT \hspace*{-0.05cm}L}$, so to say main problems, were in focus of investigations and were successfully resolved (cf. references above).
The conception of knowledge, and especially the one implemented via multi-agent approach is a popular area in Logic in Computer Science. Various aspects including interaction and autonomy, effects of cooperation etc were investigated (cf. eg. Woldridge et al \cite{wl1,wl2,wl3}, Lomuscio et al \cite{lo1,be11}). In particular, a multi-agent logic with distances were studied and satisfiability problem for it was solved (Rybakov et al \cite{ry10t}); conception of Chance Discovery in multi-agent's environment was considered (Rybakov \cite{ry11a,r12t});
a logic modeling uncertainty via agent's views was investigated (cf. McLean et al \cite{mcln1}); representation of agents interaction (as a dual of common knowledge) was suggested in Rybakov \cite{ry09t,vr179}.
Conception of refined common knowledge was suggested in Rybakov \cite{ry2003}. Historically the conception of common knowledge was formalized and profoundly analyzed in 1990x, cf. eg. Fagin et al \cite{fag1}, using as a base agent's knowledge (S5-like) modalities. The approach to model knowledge in terms of symbolic logic, probably, may be dated to the end of 1950. At 1962 Hintikka \cite{jh4} wrote the book: {\em Knowledge and Belief}, the first book-length work to suggest using modalities to capture the semantics of knowledge.
In contemporary study, the field of knowledge representation and reasoning in logical terms is very wide and active area. Frequently modal and multi-modal logics were used for formalizing agent's reasoning. Such logics were, in particular, suggested in Balbiani et al \cite{pb1}, Vakarelov \cite{dv}, Fagin et al \cite{fag1}, Rybakov \cite{ry2003,vr179}. The book Fagin et al \cite{fag1} contains summarized to that time systematic approach to study the notion of common knowledge. Some contemporary study of knowledge and believes in terms of single-modal logic may be found in
Halpern et al \cite{hal1}. Modern approach to knowledge frequently uses conception of justification in terms of epistemic logic (cf. eg. Atremov et al \cite{art1,art2}, Halpern \cite{hal1}). We will suggest some views on knowledge and knowledge in terms of multi-agent logic being based at refined version of linear temporal logic $LTL$. We will need some technique borrowed from tools for verification of admissibility for inference rules.
The problem of admissibility for inference rules and unifiability problem were already addressed to linear temporal logic. The admissibility problem (to determine for any given rule if this rule is admissible for a given logic) was in focus of interest for many logicians. Active research in the area may be dated to Harvey Friedman problem \cite{fr}: if there is an algorithm of verification for admissibility in the intuitionistic propositional logic $\mathbf{IPC}$ (this problem was first solved by Rybakov in 1984, \cite{ry84}). Since then many logicians were interested to study admissibility from various viewpoints and in many logical systems related to non-classical propositional logics
(cf. V.~Rybakov \cite{r4,Rybakov92,Rybakov2001,r6,r7,ry08a}, Rybakov, et al \cite{babry,babry1}, R.~Iemhoff \cite{r1,r3},
R.~Iemhoff and G.~Metcalfe \cite{r322},
E.~Jerabek \cite{r61,r62,r63,r61a}); prime questions were recognizing admissibility, study of bases for inference rules.
Only a necessary condition for admissibility of inference rules in the branching-time temporal logic $T_{S4}$ was found in \cite{ry08aa}, though for linear temporal logic LTL the problem was solved in full \cite{ry08a}.
Complexity problem for admissibility in intuitionistic logic and some modal logics was first studied and
re-solved in Jerabek \cite{r61bb}.
Effective approach to study admissible rules was offered by S.~Ghilardi via unification technique; at (\cite{ghi1}, 1999) it was first found and algorithm writing out a complete set of unifiers for any unifiable in $\mathbf{IPC}$ formula, and this gives another solution for admissibility problem. Since then, unification in propositional modal logics over K4
was extensively studied by S.~Ghilardi~\cite{ghi1,ghi2,ghi10}. He developed a novel method, based on L\"owenheim approach, which has proved to be also useful in dealing with admissibility and bases of admissible rules.
In (Babenyshev and Rybakov \cite{babry}) we found solution of the unification problem in the linear temporal logic LTL. (the case of ${\cal LT \hspace*{-0.05cm}L}$ with no Until was solved; the case of linear temporal logic with future and past easy follows because we may model in this logic the universal modality (cf. Rybakov \cite{r7})).
Our paper studies a non-transitive version of the linear temporal logic ${\cal LT \hspace*{-0.05cm}L}$ - the linear temporal logic $LTL_{Past,m}$ based at non-transitive linear frames. We consider the frames diverted to {\em past} and since that we as the basic operation {\em Since} instead of {\em Until}. Based at the property of operation {\em Since} (to keep truth values of a statements since a specified property happened to be true) we define a conception to be {\em knowledge} in several possible ways including {\em parameterized knowledge} and {\em knowledge via agent's viewpoint, including voting, preceding events etc.}. We find this to be very plausible and useful interpretation.
We begin form construction the logic $LTL_{Past,m}$: we define its language, syntax, semantic models. Then we shortly comment why these models look plausible and give several examples. Next, we develop all necessary technical instruments, including notion of valid inference rules, and reduced normal forms for inference rues. Based at this, we solve satisfiability and decidability problems for $LTL_{Past,m}$ (these results
similar to the ones submitted to a conference \cite{vr14g}, for the case non-uniform bounds of intransitivity).
Then we approach to the main problem solved in this paper: we show that $LTL_{Past,m}$ is decidable w.r.t. admissible rules and we find an algorithm solving admissibility problem. This is the main (and most complicated) new technical result of this paper. We, in the concluding part, comment how this approach might be used in interpretation of logical knowledge and agent's knowledge operations. The paper is about self contained and use only technique which is explained and defined here (proofs however due to length are usually omitted).
\section{Necessary preliminary information}
We will base our approach at a new non-transitive version of the linear temporal logic ${\cal LT \hspace*{-0.05cm}L}$ (motivation for non-transitivity will be given at short separate section below). Therefore we start from a sort recall of notation and definitions concerning ${\cal LT \hspace*{-0.05cm}L}$ . The language of the Linear Temporal Logic ($ {\cal LT \hspace*{-0.05cm}L} $ in the sequel) extends the language of Boolean logic by operations ${\bf N}$ (next) and ${{\bf U}}$ (until). The formulas of ${\cal LT \hspace*{-0.05cm}L}$ are built up from a set $Prop$ of atomic
propositions (synonymously - propositional letters)
and are closed under applications of Boolean
operations, the unary operation ${\bf N}$ (next) and the binary
operation ${{\bf U}}$ (until). The formula ${\bf N}{\varphi}$ has meaning: the statement ${\varphi}$ holds in the next time point (state); the formula ${\varphi} {{\bf U}} \psi$ means: ${\varphi}$ holds until $\psi$ will be true. Semantics for ${\cal LT \hspace*{-0.05cm}L}$ consists of {\em infinite
transition systems (runs, computations)}; formally they are
represented as
linear Kripke structures based on natural numbers. The infinite linear Kripke structure is a
quadruple \( {\cal M}:=\langle {\cal N}, \leq, \mathrm{Next}, V
\rangle,\)
where ${\cal N}$ is the set of all natural
numbers; $\leq$ is the
standard order on ${\cal N}$, $\mathrm{Next}$ is the binary relation,
where $a \ \mathrm{Next} \ b$ means $b$ is the number
next to $a$. $V$ is a valuation of a subset $S$ of
$Prop$. Hence the valuation $V$
assigns truth values to elements of $S$.
So, for any
$p\in S$, $V(p)\subseteq {\cal N}$, $V(p)$ is the
set of all $n$ from ${\cal N}$
where $p$ is true (w.r.t. $V$). All elements of ${\cal N}$ are called to be {\em states} (worlds), $\leq$ is the {\em transition} relation (which is linear in our case), and $V$ can be interpreted as {\em labeling} of the states with atomic propositions. The triple $\langle {\cal N}, \leq, \mathrm{Next}
\rangle$ is a Kripke frame which we will denote for short by
${\cal N}$.
The truth values in any Kripke structure
${\cal M}$, can be extended from propositions of $S$ to arbitrary formulas constructed from these propositions as follows:
\begin{definition} Computational rules for logical operations:
\begin{itemize}
\item $ \forall p\in Prop \ ({\cal M},a)\vv_V p \ \lri \ a\in {\cal N} \wedge \ a\in V(p);$
\item $ ({\cal M},a)\vv_V ({\varphi}\wedge \psi) \ \lri \ $ $({\cal M},a)\vv_V {\varphi} \wedge ({\cal M},a)\vv_V \psi;$
\item $ ({\cal M},a)\vv_V \neg {\varphi} \ \lri \ not [({\cal M},a)\vv_V {\varphi}] ;$
\item $ ({\cal M},a)\vv_V {\bf N} {\varphi} \ \lri [\ \forall b [(a \ \mathrm{Next} \ b) \ri ({\cal M},b)\vv_V {\varphi}]] ;$
\item $ ({\cal M},a) \vv_V ({\varphi} {{\bf U}} \psi) \ \lri \ \exists b [ (a\leq b)\wedge (({\cal M},b)\vv_V \psi) \wedge $ \item $
\ \ \ \ \ \ \ \ \ \ \forall c [(a\leq c < b) \ \ \ri \ \ ({\cal M},c)\vv_V {\varphi} ]]. $
\end{itemize}
\end{definition}
{ For a Kripke structure ${\cal M}:=\langle \mathcal{N}, \leq, \mathrm{Next},V
\rangle$ and a formula ${\varphi}$ with letters from the domain of $V$,
we say
${\varphi}$ is valid in ${\cal M}$ (denotation -- ${\cal M}\vv {\varphi}$) if, \ for any $b$ of ${\cal M}$ ($b\in \mathcal{N}$),
the formula ${\varphi}$ is true at $b$ (denotation: $({\cal M},b)\vv_V {\varphi})$.}
The linear temporal logic ${\cal LT \hspace*{-0.05cm}L}$ is the set of all formulas
which are valid in all infinite temporal linear Kripke structures
${\cal M}$ based on ${\cal N}$ with standard $\leq$ and $\mathrm{Next}$.
Now we will modify the models for the ones with non-transitive time.
\section{Possible words models with non-transitive time}
Our approach will need a dual of ${\cal LT \hspace*{-0.05cm}L}$ -- the logic with {\em since} operation.
Actually we may interpret this dual is as a standard LTL, but with time diverted to past,
and yet based at non-transitive models. We may introduce this dual as follows. The formulas are constructed as earlier, but with the binary logical operation ${{\bf S}}$ (since) instead of ${{\bf U}}$ (until).
The frame ${\cal N}^{-}$ is $\langle N, \geq, \mathrm{Next} \rangle$,
and $V$ as before is a valuation of a subset $S$ of
$Prop$ on the set $N$. So, we take the language of ${\cal LT \hspace*{-0.05cm}L}$, delete ${{\bf U}}$ and replace it with
the binary operation ${{\bf S}}$. The definition of the truth relation for
${{\bf S}}$ is as follows:
\[ ({\cal N}^{-},a) \vv_V ({\varphi} {{\bf S}} \psi) \ \lri \exists b [ (b\geq a)\wedge (({\cal N}^{-},b)\vv_V \psi) \wedge \]
\[
\hspace*{0.1cm}\forall c [(a \leq c < b) \ri ({\cal N}^{-},c)\vv_V {\varphi} ]].
\] So, ${{\bf S}}$ is just the dual of ${{\bf U}}$ (and note that it acts exactly as ${{\bf U}}$, we simply interpret it to {\em past}). Operation {\em Next} (notation ${\bf N}$) will act as earlier, but again directed to the past, {\em next one means next in past}. That is
\[ ({\cal N}^{-},a) \vv_V {\bf N} {\varphi} \ \lri \ ({\cal N}^{-}, a+1)\vv_V {\varphi}. \] (We will not use standard notation for LTL with {\em until} because it will break our approach to model knowledge in later part in this paper.)
\begin{definition} {\it \bf A non-transitive possible-worlds linear frames with uniform non-transitivity} (which upon we will base our approach) is a freame:
\[{\cal F} := \langle N, \geq, \mathrm{Next}, \bigcup_{ i \in N} R_i \rangle,\] where each $R_i$ is the standard linear order ($\geq$) on the interval $[i,i+m]$, where $m$ is a fixed natural number (measure of intransitivity). \end{definition}
For any set of letters $P$ we may define an arbitrary valuation $V$ on ${\cal F}$ in standard way, and ${\cal F}$ with a valuation is called a model ${\cal M}$. Thus, for any $p \in P$, we have $V(p) \subseteq N$ and we may extend $V$
to all boolean formulas built up from $P$ as usual. The same way as earlier we define truth values for formulas of kind $ {\bf N} {\varphi}$. But for formulas $ {\varphi} {{\bf S}} \psi$ the definition is new one:
\begin{definition} Computation rule for {\bf weak bounded since}:
$$({\cal M},a) \vv_V ({\varphi} \ {{\bf S}} \ \psi) \ \ \ \lri \ \ \ $$
$$ \exists b [ (b R_{a} a)\wedge (({\cal M},b)\vv_V \psi) \wedge \forall c [(a \leq c < b) \ri ({\cal M},c)\vv_V {\varphi} ]].
$$
\end{definition}
\begin{definition} The logic $LTL_{Past,m}$ is the set of all formulas which are valid at any model ${\cal M}$ with the measre of intransitivity $m$. \end{definition}
The relation $\bigcup_{ i \in N} R_i $ is evidently non-transitive and right now we will explain why this approach to time is accepted. Just to immediately comment, briefly note that ${{\bf S}}$ works similar to usual ${{\bf U}}$ (but only within the stet of states accessible from the current one, -- since our accessibility relations are non-transitive); and yet it acts as {\em Since} regarding our definition of the time accessibility relation directed to past.
\section{Why we consider that time might be non-transitive}
The approach described above considers the time directed to past (which, so to say, relays to our memory about past). Computationally (form CS viewpoint), we analyze the behavior of a computation, computational runs, being given by protocols of events which happened (already happened) while computation.
{\bf View (i)}. { \em Computations view}. Inspections of protocols for computations are limited by time resources and have non-uniform length. Therefore, if we interpret our models as the ones reflecting inspection of protocols for computation, the amount of check points is finite. In any point of inspection we may refer to stored protocols, and any one has limited length. Thus the inspections look as non-transitive accessibility relations.
{\bf View (ii)}. {\em Agent's-admin's view.} We may consider states (worlds of our model) as checkpoints of admins (agents) for any inspection of state of network in past. Any admin has allowed amount of inspections for previous states, but only within the areas of its(his/her) responsibility (by security or another reasons). So, the accessibility is not transitive again, the admin (a1) can reach a state, and there in, the admin (a2) responsible for this state (it may be a new one or the same yet), has again some allowed amount of inspections to past. But, in total, (a1) cannot inspect all states accessible for (a2).
{\bf View (iii)}. {\em Agent's-users's view.} If we consider the sates of the models as the content of web pages admissible for users, any surf step is accessibility relation, and starting from any web page user may achieve, using links in hypertext(s) some foremost available web site. The latter one may have web links which are available only for individuals possessing passwords for accessibility. And these one (having password) may continue web surf, etc. Clearly that in this approach, web surfing looks as non-transitive relation. Here we interpret web surf as time, but diverted to past, instead of opposite. The models suggested above again serve well these approach.
{\bf View (iv)}. {\em View on time in past for collecting knowledge.} In human perception, the some finite time in past (not in future) is only available to individuals to inspect evens and knowledge collected to current time state. The time is past in our feelings looks as linear, and, as an individual, the same as mankind in total, have finite amount of memory to remember information and events. There, in past, at foremost available (memorable) time point, individuals again had a memorable interval of time with collected information. And so forth... So, the time in past for mankind overall is not transitive.
{\bf View (v)}. {\em View in past for individuals as agents with opposition.} Here the picture is similar to the case (iv) above, but we may consider the knowledge as the collection of facts which about majority experts (agents) have affirmative positive opinion. And, in past time, the voted opinion of experts about facts could be different at distinct time points, Besides the time intervals memorable by experts might be very variative in past. Therefore in this approach time nohow may look transitive.
\section{Satisfiability, decidability, admissible rules, computation algorithms, }
Since we introduced a sensibly innovated linear temporal logic (modification of LTL, a very popular in CS logic), which is non-transitive, we would like to address first to this logic basic computational problems for any logic: satisfiability and decidability problems. It is immediately seen that the old standard techniques to solve these problems do not work because the accessibility relation is not transitive. Therefore the standard technique of rarefication as eg. at \cite{ry09t,vr179} or direct usage of automatons technique do not work here. If we would try to use some variants of filtration technique than a technical obstacle is possible nested operations ${{\bf S}}$ and non-transitivity again: how we would define accessibility relations $R_i$ then.
Therefore we will need some preliminary work to avoid nested operations. In fact we will be based on a modernization of the technique used already, eg. in \cite{ry09t},
Most gain from this approach is that we will then consider only very simple and uniform formulas which are formulas without nested temporal operations. For this, we will need a transformation of formulas into rules in reduced form. Recall that a (sequential) (inference) rule is
an expression (statement)
\[ {\bf r}:= \frac{\varphi_1( x_1, \dots , x_n), \dots , \varphi_l( x_1, \dots , x_n)}{\psi(x_1, \dots , x_n)}, \] where $\varphi_1( x_1, \dots , x_n), \dots , \varphi_l( x_1, \dots , x_n)$ and $\psi(x_1, \dots , x_n)$ are
formulas constructed out of letters $x_1, \dots , x_n$. The letters $x_1, \dots , x_n$ are the variables of ${\bf r}$, we use the notation $x_i \in Var(\bf r)$. A meaning of a rule {\bf r} is that the statement (formula) $ \psi(x_1, \dots , x_n)$ (which is called conclusion) follows (logically follows) from statements (formulas) $\varphi_1( x_1, \dots , x_n),$ $\dots ,$ $\varphi_l( x_1, \dots , x_n)$ which are called premisses.
\begin{definition} {\it A rule
${\bf r}$ is said to be {\em valid} in a model $\langle { {\cal M} }, V \rangle$ (we will use the notation ${{\cal M}}\vv_V \ {\sl r}$) if
\[ [\forall a \ (({{\cal M},a}) \vv_V \bigwedge_{1\leq i \leq l}\varphi_i)] \Rightarrow [ \forall a \ (({{\cal M}},a) \vv_V \psi)].\] Otherwise we say ${\bf r}$ is {\sl refuted} in ${\cal M}$, or {\sl refuted in ${\cal M}$ by $V$}, and write ${{\cal M} }\nv_V {\bf r}$. A rule ${\bf r}$ is {\sl\ valid } in a frame ${{\cal M}}$ (notation ${{\cal M}}\vv \ {\bf r}$) if, for any valuation $V$, the following holds ${{\cal M}}\vv_V {\bf r}$}. \end{definition}
For any formula ${\varphi}$, we can transform ${\varphi}$ into the
rule $x \ii x / {\varphi}$ and employ a technique of reduced normal forms for inference rules as follows.
\begin{lemma} \label{p1} {For any formula ${\varphi}$, ${\varphi}$ is a theorem of $LTL_{Past,m}$ iff the rule $({x \ii x / {\varphi}})$ is valid in any frame ${{\cal M}}$ .} \end{lemma}
\begin{definition}
A rule ${\bf r}$ is said to be in {\em reduced normal form} if \( {\bf r}= \varepsilon / x_1\) where
\[\varepsilon :=
\bigvee_{1\leq j \leq l}
[ \bigwedge_{1\leq i \leq n}
x_i^{t(j,i,0)} \wedge \bigwedge_{1\leq i \leq n}
({\bf N} x_i)^{t(j,i,1)} \wedge \bigwedge_{1\leq i,k \leq n, i \neq k} (x_i {{\bf S}} x_k)^{t(j,i,k,1)}] \]
and,
for any formula $\alpha$ above,
$\alpha^0 := \alpha$, $\alpha^1:= \neg \alpha$. \end{definition}
\begin{definition} {\it Given a rule ${\bf r_{nf}}$ in reduced normal form, ${\bf r_{nf}}$ is said to be a {\it normal reduced form for a rule ${\bf r }$} iff, for any frame ${\cal F}$ for $LTL_{Past,m}$,
\[ {\cal M} \vv {\bf r} \ \lri {\cal M} \vv {\bf r_{nf}} .\]} \end{definition}
\begin{theorem} \label{mt3} {\it There exists an algorithm running in (single) exponential time, which, for any given rule ${\bf r}$, constructs its normal reduced form ${\bf r_{nf}}$}. \end{theorem}
Now we need we need special finite frames ${\cal F}(N^{-,m})$ having the structure resembling the structure of frames ${\cal F}$ but not linear discrete ones (which are not the frames which we used to define
$LTL_{Past,m}$). We have no space to define their structure explicitly due to paper space limitation. Note only that the accessibility relations at frames ${\cal F}(N^{-,m})$ are non-transitive and with measure of transitivity $m$ also.
\begin{lemma} \label{oo1} {\sl For any given rule $\bf r_{nf}$ in reduced normal form, $\bf r_{nf}$ is refuted in a frame of ${\cal F}$ iff $\bf r_{nf}$ can be refuted in some finite frame ${\cal F}(N^{-,m})$ by a valuation $V$, where the size of the frame ${\cal F}(N^{-,m})$ has size effectively computable from the size of $\bf r_{nf}$. } \end{lemma}
It is clear that a formula ${\varphi}$ is satisfiable (${\varphi}$ is true at a state of a model for $LTL_{Past}$) if and only if $\neg {\varphi} \not\in LTL_{Past,m}$. Therefore based at Theorem \ref{mt3}, Lemma \ref{p1} and Lemma \ref{oo1} we may prove:
\begin{theorem} \label{bn1} {\it The logic
$LTL_{Past,m}$ is decidable; the satisfiability problem for the non-transitive linear temporal logic
$LTL_{Past,m}$ is decidable: for any formula we can compute if it is satisfiable and to compute the valuation of the model satisfying this formula.} \end{theorem}
The computational algorithm, in its final stage, consists of definition a computable valuation in some initial part of $N$ which finally is resulted in a total valuation satisfying the formula. Now on we have enough technique to proceed to admissibility of inference rules.
\begin{definition}{\em An inference rule
\( {\bf r}:= {\varphi_1( x_1, \dots , x_n), \dots , \varphi_l( x_1, \dots , x_n)} / {\psi(x_1, \dots , x_n)}, \)
is said to be
{\em admissible }
in a logic $L$ if, for any tuple of formulas
$\alpha_1 , \dots , \alpha_n $, the following
holds
\( [ \bigwedge_{1\leq i \leq l} \varphi_i(\alpha_1, \dots , \alpha_n)\in L]
\ \ \Rightarrow \ \ [ \psi(\alpha_1, \dots , \alpha_n) \in L ]. \) } \end{definition}
Thus, for any admissible rule,
any instance into the premises making all of them theorems of a logic $L$ makes also the conclusion to be a theorem. Using the same algorithm of construction reduced normal form $\bf r_{nf}$ for any given rule $\bf r$ as at Theorem $\ref{mt3}$ we may obtain
\begin{lemma} \label{ooxx} {\sl For any given rule $\bf r$, $\bf r$ is admissible in $LTL_{Past,m}$ iff $\bf r_{nf}$ is admissible in $LTL_{Past,m}$ }. \end{lemma}
Next necessary for our approach result is the following statement:
\begin{lemma} \label{oox1} {\sl For any given rule $\bf r_{nf}$
in reduced normal form, $\bf r_{nf}$ is not admissible in the logic $LTL_{Past}$ if and only if $\bf r_{nf}$ is refuted in a {\bf special} finite frame
by a valuation $V$ possessing {\bf some special properties} (and the size of the this frame, as earlier, has an effective bound computable from the size of $\bf r_{nf}$). } \end{lemma}
Based at Lemmas \ref{ooxx} and \ref{oox1} we obtain
\begin{theorem} \label{ooxx} {\sl The logic $LTL_{Past,m}$ is decidable w.r.t. admissible rules. There is an algorithm verifying for any given inference rule if it is admissible in $LTL_{Past,m}$}. \end{theorem}
It is good time to give examples admissible for $LTL_{Past,m}$ but invalid rules. For example the rules
\[ {\bf N} x / x, \ \ {\bf N} x_1 \ii {\bf N} x_2/ x_1 \ii x_2, \ \ {\bf N} x_1 \ {{\bf U}} \ {\bf N} x_2 / x_1 {{\bf U}} x_2 \] are admissible but invalid in $LTL_{Past,m}$. This is because
\begin{theorem}{\em If $\varphi(p_1, \dots , p_n, q_1, \dots , q_m)$ is an arbitrary boolean formula constructed from propositional letters $p_1, \dots ,
p_n, q_1, \dots , q_m$, then the rule
\[ \frac{ \varphi({\bf N} x_1, \dots , {\bf N} x_n,
{\bf N} y_1 {{\bf U}} {\bf N} z_1 \dots , {\bf N} y_m {{\bf U}} {\bf N} z_m ) }
{ \varphi(x_1, \dots , x_n,
y_1 {{\bf U}} z_1 \dots , y_m {{\bf U}} z_m ) ) } \] is admissible in $LTL_{Past,m}$} \end{theorem}
We may show it exactly the same way as it was proved for the logic $LTL$ itself in \cite{ry08a}.
This theorem gives an infinite set of admissible in $LTL_{Past,m}$ rules, were some infinite part of this set consists of rules invalid in $LTL_{Past,m}$. Actually these rules alow to withdraw operation ${\bf N}$ from formulas of the premisses.
It is clear that $\Box x \ii \Box\Box x \ \in {\cal LT \hspace*{-0.05cm}L}$ but $ \Box x \ii \Box\Box x \ \notin LTL_{Past,m}$. Thus, we immediately see, there are admissible in ${\cal LT \hspace*{-0.05cm}L}$ rules which are not admissible in $LTL_{Past,m}$. However even vise versa,
\begin{theorem} \label{art2} There are not passive inference rules (which means their premisses are unifiable)
which are admissible
in $LTL_{Past,m}$ but not admissible in
${\cal LT \hspace*{-0.05cm}L}$.
\end{theorem} This statement is already not immediate or trivial. Thus non-transitive temporal linear logics ${\cal LT \hspace*{-0.05cm}L}$ and $LTL_{Past,m}$ differ w.r.t. admissible inference rules: no one set is enclosed into the another one.
Now we would like to comment the case of temporal linear non-transitive logic with non-uniform bound of intransitivity. We reported about this logic in \cite{vr14g}. Just to recall the definition and results,
\begin{definition} A {\it \bf non-transitive possible-worlds frame} is \[{\cal M} := \langle N, \geq, \mathrm{Next}, \bigcup_{ i \in N} R_i \rangle,\] where each $R_i$ is the standard linear order ($\geq$) on the interval $[i,m_i]$, where $m_i \in N, m_i > i$ and $m_{i+1} > m_i$. \end{definition}
Again as earlier we may define a model ${\cal M}$ on ${\cal F}$ by introducing a valuation $V$ on ${\cal F}$ and extend it on all formulas as earlier. In particular,
for formulas $ {\varphi} {{\bf S}} \psi$:
\begin{definition} Computation rule for {\bf weak, non-unform since}:
$$({\cal M},a) \vv_V ({\varphi} \ {{\bf S}} \ \psi) \ \ \ \lri \ \ \ $$
$$ \exists b [ (b R_{a} a)\wedge (({\cal M},b)\vv_V \psi) \wedge \forall c [(a \leq c < b) \ri ({\cal M},c)\vv_V {\varphi} ]].
$$
\end{definition}
\begin{definition} The logic $LTL_{Past}$ is the set of all formulas which are valid at any model ${\cal M}$ with any
valuation. \end{definition}
The relation $\bigcup_{ i \in N} R_i $ is again non-transitive and the bounds non-transitivity in frames is arbitrary, - non-uniform. Using an approach similar to the one used in this paper above we stated the following:
\begin{lemma} \label{oo1} {\sl For any given rule $\bf r_{nf}$ in reduced normal form, $\bf r_{nf}$ is refuted in a frame of ${\cal F}$ iff $\bf r_{nf}$ can be refuted in some finite frame ${\cal F}(N^{-})$ by a valuation $V$, where the size of the frame ${\cal F}(N^{-})$ has size effectively computable from the size of $\bf r_{nf}$. } \end{lemma}
And based at this we deducted:
\begin{theorem} \cite{vr14g}\label{bn1} {\it The logic
$LTL_{Past}$ is decidable; the satisfiability problem for the non-transitive linear temporal logic
$LTL_{Past}$ is decidable: for any formula we can compute if it is satisfiable and to compute the valuation of the model satisfying this formula.} \end{theorem}
But at \cite{vr14g} we were not able to solve in $LTL_{Past}$ the problem of admissibility for inference rules. Here we solved admissibility problem for $LTL_{Past,m}$, but the presence of uniform bound $m$ for non-transitivity was impotent and necessary to develop indispensable technique.
\section{Applications: knowledge of agent's}
In this section we would like to describe applications of our technique and obtained results for formalization the conception of {\em knowledge} in logical terms (we commented at the introduction, that there is a extensive research devoted to study {\em knowledge} in logical framework, and -- especially epistemic modal logic; here we would suggest somewhat natural and simple, but anyway it seems new).
We start from a trivial statement
that
{\em knowledge} is not absolute and depends on opinions of individuals (agents) who accept a statement as safely true or not. Yet, it is not unequally defined what we actually, in fact, consider as knowledge. First, would like to look at it via temporal perspective. Some evident trivial observations are that
{\em (i) Human beings remember (at least some) past, but
(ii) they do not know future at all (rather could surmise what will happen in
immediate proximity time steps);
(iii) individual memory tells to us that the time in past was linear
(though it might be only our perception).
}
Therefore it looks meaningful to look for the interpretation of {\em knowledge} in
past linear temporal logic - $LTL_{Past}$. Here below we will use the unary logical operations $K_i$ with meaning - it is a knowledge operation. So, $K_i {\varphi}$ says that the statement ${\varphi}$ is {\em knowledge}.
{\bf (i) approach: when knowledge holds stable:}
\[ ({\cal N}^{-},a) \vv_V K_1 {\varphi} \ \ \ \lri \ \ \ \exists b [ (b\geq a)\wedge (({\cal N}^{-},b)\vv_V {\varphi}) \wedge \]
\[ \forall c [(a \leq c < b) \ri ({\cal N}^{-},c)\vv_V {\varphi} ]. \]
That is
\[ ({\cal N}^{-},a) \vv_V K_1 {\varphi} \ \ \ \lri \ \ \ ({\cal N}^{-},a) \vv_V {\varphi} S {\varphi} .\]
This is an unusual but rather plausible interpretation.
In time being, we say that ${\varphi}$ is a knowledge if one day in past it happened and since then is true until now, today. The only disturbing point here is that $b$ could be equal $a$ - so, $b$ is today, so then knowledge has very sort time support... Therefore such interpretation yet needs a refining.
{\bf (ii) approach: knowledge if always was true: }
\[ ({\cal N}^{-},a) \vv_V K_2 {\varphi} \ \ \ \lri \ \ \ ({\cal N}^{-},a) \vv_V \neg ( \top S \neg {\varphi}) . \]
This is close to standard interpretation in epistemic logic offered quit a while ago: we consider a fact to be knowledge if it has been true always (in our approach -- in past).
{\bf (iii) approach: via parameterized knowledge: }
\[ ({\cal N}^{-},a) \vv_V K_{\psi} {\varphi} \ \ \ \lri \ \ \ ({\cal N}^{-},a) \vv_V {\varphi} S \psi. \]
This means ${\varphi}$ has a stable value true, -- since some event happened in past, which is modeled now by $\psi$ to be true at a state. Thus, as soon as $\psi$ happened to be true, ${\varphi}$ always has been true until now. Clearly this approach generalizes first two suggested above and yet it is more flexible. From technical standpoint we just use standard {\em Since} operation of the linear temporal logic LTL but diverted to past.
This approach is looking very natural; it gives new view angle on the problem and yet uses some respectful and well established technique.
{\bf (iv) approach: via agents knowledge as voted truth for
\ \ \ \ \ the valuation:}
This is very well established area, cf. the book Fagin et al \cite{fag1} and more contemporary publications e.g. - Rybakov \cite{ry2003,vr179}. Though here we would like to look at it from an another standpoint. Earlier logical knowledge operations (agents knowledge) were just unary logical operations $K_i$ interpreted as $S5$-modalities,
and knowledge operations were introduced via the vote of agents about truth the statement, etc. We would like to suggest here somewhat very simple, but it seems natural and new.
Here, in order to implement multi-agent's framework we assume that all agents have theirs own valuations at the frame $N^{-}$. So, we have $n$ much agents, and $n$-much valuations $V_i$, and as earlier the truth values w.r.t. $V_i$ of any propositional letter $p_j$ at any world $a\in N$. For applications viewpoint, $V_i$ correspond to agents information about truth of $p_j$ (they may be different). So, $V_i$ is just individual {\em information} .
How the information can be turned to {\em local} knowledge? One way is the voted value of truth: we consider a new valuation $V$, w.r.t. which letters $p_j$ are true at $a$ if majority, biggest part (which one is negotiable) of agents, believes that $p_j$ is true at $a$. Then we achieve a model with a single (standard) valuation $V$. Then we can apply {\em any of proposed upper approaches} to introduce, so to say, logical operations of global knowledge $K$. If we accept it, our technique works and we may compute true laws of for statements about knowledge logical operations.
{\bf (v) approach: via agents knowledge as conflict resolution:
\ \ \ \ \ at evaluating point}
Here we suggest a way starting similar as in the case (iv) above until introduction of different valuations $V_i$ of agent's opinion. But then we suggest
\[ ({\cal N}^{-},a) \vv_V K_{\psi} {\varphi} \ \ \ \lri \ \ \ \forall i [({\cal N}^{-}, a) \vv_{V_i} K_{\psi} {\varphi} ]. \]
In this case, if we will allow nested knowledge operations together with several valuations $V_i$ for agent's information and yet derivative valuation $V$ for all cases when we
evaluate $K_{\psi}{\varphi}$ (regardless for which agent (i.e. $V_i$)),
no decision procedure is known, and our technique does not work. We think that to solve this open problem it is an interesting open question.
\section{Open problems}
There are many remaining open interesting problems in our suggested framework. E.g.the problem described at the final part of Section 6 (approach (v)) for multi-agent case. Yet one more open interesting question is to extend the suggested approach to linear logic based at all integer numbers $Z$ (which means we have potentially infinite past and infinite future). Then the knowledge will be interpret by a stable truth on reasonably long intervals of time in past and future, so we will need to use both operations ${{\bf S}}$ and ${{\bf U}}$. Next, it would be good to study such approach to the case of continuous time (in past, or both - past and future). Yet the case of admissibility problem for rules in $LTL_{past}$, as well as the ones for rules with coefficients remain open.
\end{document} | arXiv |
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NMR absorption at a chemical shift greater than d 3, and the following 13C NMR spectrum: d 24.4, d 26.5, d 44.2, and d 212.6. The resonances at d 44.2 and d 212.6 have very low in-tensity. 19.6 The 13C NMR spectrum of 2-ethylbutanal consists of the following
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13C NMR peak loion in butanone. The above is the correct X 13 X 2 2 13 C - N M R spectrum of butanone. In the X 13 X 2 2 13 C - N M R spectrum of butanone, I figure that the peak …
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1/8/2022· 13C NMR peak loion in butanone organic-chemistry molecular-structure spectroscopy nmr-spectroscopy 1,692 No, substitution by an alkyl group produces a small downfield shift in both proton and carbon nmr. Therefore, in a typical hydrocarbon a The
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the wide range of chemical shifts in 13 c nuclei and the extreme narrowing of the resonance lines (which are most often intentionally broadened for spectral noise reduction purposes) would make 13
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2-Butanone is a manufactured chemical but it is also present in the environment from natural sources. It is a colorless liquid with a sharp, sweet odor. It is also known as methyl ethyl ketone (MEK). 2-Butanone is produced in large quantities. Nearly half of its use is
23/11/2021· Generally, NMR impurities are found in trace concentration and therefore they are relatively easy to identify, as their NMR peaks show very low intensities compared to those of the analyte. To make it easier to characterize impurities, laboratories often make use of tables that summarize the chemical shifts of the most common impurities. 19 , 20 , 21
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Interpreting the C-13 NMR spectrum of butanone As you can see from the diagram above there are 4 chemical shift linesin the C-13 NMR spectrum of butanone indiing 4 different chemical environments of the carbon atoms. CH3COCH2CH3 (Note the 4 colours indiing the 4 different chemical environment of the carbon atoms in butanone).
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Chemical Environments The nuer of chemical environment a molecule has is the nuer of peaks in the 13C NMR spectrum.For example: IR has told you that you have an alcohol functional group, but you don't know if the alcohol is propan-1-ol or propan-2-ol. The 13C NMR will allow you to tell the difference.
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25/7/2018· 1 H and 13 C NMR spectra were recorded on a Bruker Avance III-400 MHz superconducting NMR spectrometer (400.1 MHz for 1 H and 100.6 MHz for 13 C) using DMSO- d6 as solvent at a concentration of approximately 5% (w/v) for 1 H NMR and approximately 20% (w/v) for 13 C NMR. All spectra were recorded at room temperature (298 K).
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21/4/2022· The Chemical Shift of Connected to sp3 Hybridized Carbons. We can see in the table that sp3 hybridized C – H bonds in alkanes and cycloalkanes give signal in the upfield region (shielded, low resonance frequency) at the range of 1–2 ppm. The only peak that comes before saturated C-H protons is the signal of the protons of tetramethylsilane
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List of mountains in Pakistan. K2, the 2nd highest in the world. Nanga Parbat, the 9th highest in the world. Pakistan is home to 108 peaks above 7,000 metres and 4555 [1] above 6,000 m. There is no count of the peaks above 5,000 and 4,000 m. Five of the 14 highest independent peaks in the world (the eight-thousanders) are in Pakistan (four of
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In Carbon-13 NMR, you get six peaks image from bmse000351 Hexanoic Acid at BMRB using correlation spectroscopy between the proton and carbon NMR, you can see how these peaks overlap: image of 2-d HSQC from bmse000351 Hexanoic Acid at … | CommonCrawl |
\begin{document}
\title{A Combinatorial Discussion on Finite Dimensional Leavitt Path Algebras } \author{A. Ko\c{c}$^{(1)}$, S. Esin$^{(2)}$, \.{I}. G\"{u}lo\u{g}lu$^{(2)}$, M. Kanuni$^{(3)}$ \\
$^{(1)}$ \.{I}stanbul K\"{u}lt\"{u}r University\\ Department of Mathematics and Computer Sciences\\ $^{(2)}$ Do\u{g}u\c{s} University\\ Department of Mathematics\\ $^{(3)}$ Bo\u{g}azi\c{c}i University\\ Department of Mathematics} \maketitle
\begin{abstract} Any finite dimensional semisimple algebra A over a field K is isomorphic to a direct sum of finite dimensional full matrix rings over suitable division rings. In this paper we will consider the special case where all division rings are exactly the field K. All such finite dimensional semisimple algebras arise as a finite dimensional Leavitt path algebra. For this specific finite dimensional semisimple algebra $A$ over a field $K,$ we define a uniquely detemined specific graph - which we name as a truncated tree associated with $A$ - whose Leavitt path algebra is isomorphic to $A$. We define an algebraic invariant $\kappa (A)$ for $A\ $and count the number of isomorphism classes of Leavitt path algebras with $\kappa (A)=n.$
Moreover, we find the maximum and the minimum $K$-dimensions of the Leavitt path algebras of possible trees with a given number of vertices and determine the number of distinct Leavitt path algebras of a line graph with a given number of vertices. \end{abstract}
\textit{Keywords:} \textit{Finite dimensional semisimple algebra, Leavitt path algebra, Truncated trees, Line graphs.}
\section{Introduction}
By the well-known Wedderburn-Artin Theorem \ \cite{2}, any finite dimensional semisimple algebra $A$ over a field $K$ is isomorphic to a direct sum of finite dimensional full matrix rings over suitable division rings. In this paper we will consider the special case where \ all division rings are exactly the field $K.$ All such finite dimensional semisimple algebras arise as a finite dimensional Leavitt path algebra as studied in \cite{1}. The Leavitt path algebras are introduced by Abrams and Aranda Pino in 2005, \cite{3}. Many papers on Leavitt path algebras appeared in literature since then. In the following discussion, we are particularly interested in answering some combinatorial questions on the finite dimensional Leavitt path algebras.
We start by recalling the definitions of a path algebra and a Leavitt path algebra, see \cite{1}. A \textit{directed graph} $E=(E^{0},E^{1},r,s)$ consists of two countable sets $E^{0},E^{1}$ and functions $ r,s:E^{1}\rightarrow E^{0}$. The elements $E^{0}$ and $E^{1}$ are called \textit{vertices} and \textit{edges}, respectively. For each $e\in E^{0},$ $ s(e)$ is the source of $e$ and $r(e)$ is the range of $e.$ If $s(e)=v$ and $ r(e)=w,$ then we say that $v$ emits $e$ and that $w$ receives $e.$ A vertex which does not receive any edges is called a \textit{source,} and a vertex which emits no edges is called a \textit{sink.} A graph is called \textit{ row- finite} if $s^{-1}(v)$ is a finite set for each vertex $v$. For a row-finite graph the edge set $E^{1}$ of $E~$is finite if its set of vertices $E^{0}$ is finite. Thus, a row-finite graph is finite if $E^{0}$ is a finite set.
A path in a graph $E$ is a sequence of edges $\mu =e_{1}\ldots e_{n}$ such that $r(e_{i})=s(e_{i+1})$ for $i=1,\ldots ,n-1.$ In such a case, $s(\mu ):=s(e_{1})$ is the \textit{source }of $\mu $ and $r(\mu ):=r(e_{n})$ is the \textit{range} of $\mu $, and $n$ is the \textit{length }of $\mu ,$ i.e., $ l(\mu )=n.$
If $s(\mu )=r(\mu )$ and $s(e_{i})\neq s(e_{j})$ for every $i\neq j$, then $ \mu $ is called a \textit{cycle}. If $E$ does not contain any cycles, $E$ is called \textit{acyclic}.
For $n\geq 2,$ define $E^{n}$ to be the set of paths of length $n,$ and $ E^{\ast }=\bigcup\limits_{n\geq 0}E^{n}$ the set of all paths.
The path $K$-algebra over $E$ is defined as the free $K$-algebra $ K[E^{0}\cup E^{1}]$ with the relations:
\begin{enumerate} \item[(1)] $v_{i}v_{j}=\delta _{ij}v_{i}$ \ for every $v_{i},v_{j}\in E^{0}.$
\item[(2)] $e_{i}=e_{i}r(e_{i})=s(e_{i})e_{i}$ $\ $for every $e_{i}\in E^{1}. $ \end{enumerate}
This algebra is denoted by $KE$. Given a graph $E,$ define the extended graph of $E$ as the new graph $\widehat{E}=(E^{0},E^{1}\cup (E^{1})^{\ast },r^{\prime },s^{\prime })$ where $(E^{1})^{\ast }=\{e_{i}^{\ast
}~|~e_{i}\in E^{1}\}$ and the functions $r^{\prime }$ and $s^{\prime }$ are defined as \begin{equation*}
r^{\prime }|_{E^{1}}=r,~~~~s^{\prime }|_{E^{1}}=s,~~~~r^{\prime }(e_{i}^{\ast })=s(e_{i})~~~~~~\text{and~~~~~}s^{\prime }(e_{i}^{\ast })=r(e_{i}). \end{equation*} The Leavitt path algebra of $E$ with coefficients in $K$ is defined as the path algebra over the extended graph $\widehat{E},$ with relations:
\begin{enumerate} \item[(CK1)] $e_{i}^{\ast }e_{j}=\delta _{ij}r(e_{j})$ \ for every $e_{j}\in E^{1}$ and $e_{i}^{\ast }\in (E^{1})^{\ast }.$
\item[(CK2)] $v_{i}=\sum_{\{e_{j}\in E^{1}~|~s(e_{j})=v_{i}\}}e_{j}e_{j}^{\ast }$ \ for every $v_{i}\in E^{0}$ which is not a sink. \end{enumerate}
This algebra is denoted by $L_{K}(E)$. The conditions (CK1) and (CK2) are called the Cuntz-Krieger relations. In particular condition (CK2) is the Cuntz-Krieger relation at $v_{i}$. If $v_{i}$ is a sink, we do not have a (CK2) relation at $v_{i}$. Note that the condition of row-finiteness is needed in order to define the equation (CK2).
The main structure theorem in\ \cite{1} can be summarized as follows:
For any $v\in E^{0},$ we define $n(v)=\left\vert \left\{ \alpha \in E^{\ast
}~|~r(\alpha )=v\right\} \right\vert .$
\begin{proposition} \label{prop1}:
\begin{enumerate} \item The Leavitt path algebra $L_{K}(E)$ is a finite-dimensional $K$ -algebra if and only if $E$ is a finite and acyclic graph.
\item If $A=\bigoplus\limits_{i=1}^{s}M_{n_{i}}(K)$ , then $A\cong L_{K}(E)$ for a graph $E$ having $s$ connected components each of which is an oriented line graph with $n_{i}$ vertices, \linebreak $i=1,2,\cdots ,s.$
\item A finite dimensional $K$-algebra $A$ arises as a $L_{K}(E)$ for a graph $E$ if and only if $A=\bigoplus\limits_{i=1}^{s}M_{n_{i}}(K).$
\item If $A=\bigoplus\limits_{i=1}^{s}M_{n_{i}}(K)$ and $A\cong L_{K}(E)$ for a finite, acyclic graph $E$, then the number of sinks of $E$ is equal to $s$, and each sink $v_{i}$ $(i=1,2,\cdots ,s)$ has $n(v_{i})=n_{i}$ with a suitable indexing of the sinks. \end{enumerate} \end{proposition}
\section{Truncated Trees}
For a finite dimensional Leavitt path algebra $L_{K}(E)$ of a graph $E$, we would like to construct a distinguished graph $F$ having the Leavitt path algebra isomorphic to $L_{K}(E)$ as follows:
\begin{theorem} \label{thm 2}Let $E$ be a finite, acyclic graph with no isolated points. Let
\linebreak $s=|S(E)|$ where $S(E)$ is the set of sinks of $E$ and $N=\max
\{n(v)~|~v\in S(E)\}$. Then there exists a unique (up to isomorphism) tree $ F $ with exactly one source and $s+N-1$ vertices such that $L_{K}(E)\cong L_{K}(F)$. \end{theorem}
\begin{proof} Let the sinks $v_{1},v_{2},\ldots ,v_{s}$ of $E$ be indexed \ such that \begin{equation*} 2\leq n(v_{1})\leq n(v_{2})\leq \ldots \leq n(v_{s})=N. \end{equation*} Define a graph $F=(F^{0},F^{1},r,s)$ as follows: \begin{eqnarray*} F^{0} &=&\{u_{1},u_{2},\ldots ,u_{N},w_{1},w_{2},\ldots w_{s-1}\} \\ F^{1} &=&\{e_{1},e_{2},\ldots ,e_{N-1},f_{1},f_{2},\ldots ,f_{s-1}\} \\ s(e_{i}) &=&u_{i}\text{ \ \ \ \ and \ \ \ }r(e_{i})=u_{i+1}\text{\ \ \ \ \ \ \ \ \ \ \ \ }i=1,\ldots ,N-1 \\ s(f_{i}) &=&u_{n(v_{i})-1}\text{ \ \ \ \ \ \ and \ \ \ \ }r(f_{i})=w_{i} \text{ \ \ \ \ \ \ }i=1,\ldots ,s-1. \end{eqnarray*} \FRAME{dhF}{3.7317in}{1.1467in}{0pt}{}{}{Figure}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "T";width 3.7317in;height 1.1467in;depth 0pt;original-width 12.8961in;original-height 3.9271in;cropleft "0";croptop "1";cropright "1";cropbottom "0";tempfilename 'LXQCLQ00.wmf';tempfile-properties "XPR";}}Clearly, $F$ is a directed tree with unique source $u_{1}$ and $s+N-1$ vertices. $F$ has exactly $s$ sinks, namely $u_{N},w_{1},w_{2},\ldots w_{s-1}$ with $n(u_{N})=N,$ $ n(w_{i})=n(v_{i}),$ \ \ $i=1,\ldots ,s-1.$ Therefore, $L_{K}(E)\cong L_{K}(F).$
For the uniqueness part, take a tree $T$ with exactly one source and \linebreak $s+N-1$ vertices such that $L_{K}(E)\cong L_{K}(T)$. Since $
N=\max \{n(v)~|~v\in S(E)\}$ which is equal to the square root of the maximum of the $K$-dimensions of the minimal ideals of $L_{K}(E)$ and hence $
L_{K}(T),$ there exists a sink $v$ in $T$ with $\left\vert \{\mu _{i}\in T^{\ast }~|~r(\mu _{i})=v\}\right\vert =N.$ On the other hand, since $T$ is a tree with a unique source and hence any vertex is connected to the unique source by a uniquely determined path, we see that the unique path joining $v$ to the source must contain exactly $N$ \ vertices, say $a_{1},...,a_{N-1},v$ \ where $a_{1}$ is the unique source and the length of the path joining $ a_{k}$ to $a_{1}$ being equal to $k-1$ for any \linebreak $k=1,2,...,N-1$. As $L_{K}(E)=\bigoplus\limits_{i=1}^{s}M_{n_{i}}(K)$ with $s$ summands, the remaining $s-1$ vertices must then all be sinks by Proposition \ref{prop1} (4), say $b_{1},...,b_{s-1}.$ Since for any vertex $a$ different from the unique source we have $n(a)>1$ we see that for each $i=1,\ldots ,s-1$ there exists an edge $g_{i}$ with $r(g_{i})=b_{i}.$ Since $s(g_{i})$ is not a sink we see that $s(g_{i})\in \{a_{1},a_{2},...,a_{N-1}\},$ more precisely $ s(g_{i})=a_{n(b_{i})-1},$ $i=1,2,...,s-1.$ Thus \ $T$ is isomorphic to $F.$ \end{proof}
Observe that the $F$ constructed in Theorem \ref{thm 2} is the tree with one source and smallest possible number of vertices $(s+N-1)$ having $L_{K}(F)$ isomorphic to $L_{K}(E).$ We call $F$ constructed in Theorem \ref{thm 2} as the\textit{\ truncated tree associated with} $E.$
\begin{proposition} With the above definition of $F$, there is no tree $T$ with \linebreak $
|T^{0}|<|F^{0}|$ such that $L_{K}(T)\cong L_{K}(F).$ \end{proposition}
\begin{proof} Notice that since $T$ is a tree, any vertex contributing to a sink represents a unique path ending at that sink. Assume on the contrary there exists a tree $T$ with $n$ vertices and $L_{K}(T)\cong A=\bigoplus\limits_{i=1}^{s}M_{n_{i}}(K)$ such that $n<s+N-1.$ Since $N$ is the maximum of $n_{i}$'$s$ there exists a sink with $N$ vertices contributing. But in $T$ the number $n-s$ of vertices which are not sinks is less than $N-1.$ Hence the maximum contribution to any sink can be at most $ n-s+1$ which is strictly less than $N.$ This is the desired contradiction. \end{proof}
However if we omit the tree assumption then it is possible to find a graph $ G $ with smaller number of vertices having $L_{K}(G)$ isomorphic to $ L_{K}(E) $ as the next example illustrates.
\begin{example} \ \ \ \FRAME{dhF}{3.1678in}{0.7697in}{0pt}{}{}{Figure}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "T";width 3.1678in;height 0.7697in;depth 0pt;original-width 8.4475in;original-height 2.0314in;cropleft "0";croptop "1";cropright "1";cropbottom "0";tempfilename
'LXL3CQ04.wmf';tempfile-properties "XPR";}}Both $L_{K}(G)\cong M_{3}(K)\cong L_{K}(F)$ and $|G^{0}|=2$ where as $|F^{0}|=3$. \end{example}
Given $F_{1}$, $F_{2}$ truncated trees associated with graphs $G_{1}$ and $ G_{2}$ respectively, then $F_{1}\cong F_{2}$ iff $L_{K}(F_{1})\cong L_{K}(F_{2})$ so there is a one-to-one correspondence between the Leavitt path algebra and truncated trees.
For a given finite dimensional Leavitt path algebra $A=\bigoplus \limits_{i=1}^{s}M_{n_{i}}(K)$ with $2\leq n_{1}\leq n_{2}\leq \ldots \leq n_{s}=N,$ the number $s$ is the number of minimal ideals of $A$ and $N^{2}$ is the maximum of the dimensions of these ideals. Therefore $\kappa (A)=s+N-1 $ is a uniquely determined algebraic invariant of $A$. Given $ m\geq 2$, the number of isomorphism classes of finite dimensional Leavitt path algebras $A$ which do not have any ideals isomorphic to $K$ and $\kappa (A)=m$ is equal to the number of distinct truncated trees with $m$ vertices by the previous paragraph. The next proposition computes this number.
\begin{definition} Define a function $d:E^{0}\rightarrow
\mathbb{N}
$ such that for any $u\in E^{0}$, \begin{equation*}
d(u)=\left\vert \{v~|~~n(v)\leq n(u)\}\right\vert . \end{equation*} \end{definition}
Observe that in a truncated tree, the restriction of the function $d$ on the set of vertices which are not sinks is one to one.
\begin{proposition} The number of distinct truncated trees with $n$ vertices is $2^{n-2}.$ \end{proposition}
\begin{proof} For every truncated tree $E$ with $n$ vertices we assign an $n$-vector \linebreak $\alpha (E)=(\alpha _{1},\alpha _{2},\cdots ,\alpha _{n})$ where $ \alpha _{i}\in \{0,1\}$ as follows:
\begin{itemize} \item $\alpha (E)$ contains exactly $N-1$ many $1$'s where $N-1$ is the number of non-sinks of $E$.
\item To define that vector it is sufficient to know which component\ is $1.$
\item To each vertex $v$ which is not a sink, we assign a $1$ appearing in the $d(v)$-$th$ component.
\item Remaining components are all zero. \end{itemize}
Hence $\alpha (E)$ starts with $1$ and ends with $0$.
Given any $\{0,1\}$ sequence $\beta $ of length $n$ starting with $1$ and ending with $0$, there exists clearly a unique truncated tree $E$ with $n$ vertices such that $\alpha (E)=\beta .$ Hence the number of distinct truncated trees with $n$ vertices is equal to the number of all $\{0,1\}$ -sequences of length $n$ in which the first and last components are constant which is equal to $2^{n-2}.$ \end{proof}
For a tree $F$ with $n$ vertices the $K$-dimension of $L_{K}(F)$ is not uniquely determined by the number of vertices only. However, we can compute the maximum and the minimum $K$-dimensions of $L_{K}(F)$ where $F$ ranges over all possible trees with $n$ vertices.
\begin{lemma} \label{lemma7} The maximum $K$-dimension of $L_{K}(E)$ where $E$ ranges over all possible trees with $n$ vertices and $s$ sinks is equal to $s(n-s+1)^{2} $. \end{lemma}
\begin{proof} Assume $E$ is a tree with $n$ vertices. Then $L_{K}(E)\cong \bigoplus\limits_{i=1}^{s}M_{n_{i}}(K),$ by Proposition \ref{prop1} (3) where $s$ is the number of sinks in $E$ and $n_{i}\leq n-s+1$ for all $ i=1,\ldots s.$ Hence \begin{equation*} \dim L_{K}(E)=\sum\limits_{i=1}^{s}n_{i}^{2}\leq s(n-s+1)^{2}. \end{equation*} Notice that there exists a tree $E$ as sketched below\FRAME{dhF}{3.1254in}{ 0.921in}{0pt}{}{}{Figure}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "T";width 3.1254in;height 0.921in;depth 0pt;original-width 8.3022in;original-height 2.4267in;cropleft "0";croptop "1";cropright "1";cropbottom "0";tempfilename 'LXL3CQ00.wmf';tempfile-properties "XPR";}}with $n$ vertices and\ $s$ sinks such that $\dim L_{K}(E)=s(n-s+1)^{2}.$ \end{proof}
\begin{theorem} The maximum $K$-dimension of $L_{K}(E)$ where $E$ ranges over all possible trees with $n$ vertices is given by $f(n)$ where \begin{equation*} f(n)=\left\{ \begin{array}{ccc} \dfrac{n(2n+3)^{2}}{27} & if & n\equiv 0\text{ \ }(\func{mod}3) \\ \text{ \ \ \ } & & \\ \dfrac{1}{27}\left( n+2\right) \left( 2n+1\right) ^{2} & if & n\equiv 1\text{ \ }(\func{mod}3) \\ \text{ \ \ \ } & & \\ \dfrac{4}{27}(n+1)^{3} & if & n\equiv 2\text{ \ }(\func{mod}3) \end{array} \right. \end{equation*} \end{theorem}
\begin{proof} Assume $E$ is a tree with $n$ vertices. Then $L_{K}(E)\cong \bigoplus\limits_{i=1}^{s}M_{n_{i}}$ where $s$ is the number of sinks in $E$ . Now, to find $\max \dim L_{K}(E)$ we need only to determine maximum value of the function $f(s)=s(n-s+1)^{2}$ for \linebreak $s=1,2,\ldots ,n-1.$ Extending the domain of $f(s)$ to real numbers $1\leq s\leq n-1$ we get a continuous function, hence we can find its maximum value. \begin{equation*} f(s)=s(n-s+1)^{2}\Rightarrow \frac{d}{ds}\left( s(n-s+1)^{2}\right) =\left( n-3s+1\right) \left( n-s+1\right) \end{equation*} Then $s=\dfrac{n+1}{3}$ is the only critical point in the interval $\left[ 1,n-1\right] $ and since $\dfrac{d^{2}f}{ds^{2}}(\dfrac{n+1}{3})<0,$ it is a local maximum. In particular $f$ is increasing on $\left[ 1,\dfrac{n+1}{3} \right] $ and decreasing on $\left[ \dfrac{n+1}{3},n-1\right] $ . We have three cases:
\textbf{Case 1: } $n\equiv 2$\ \ $(\func{mod}3).$ In this case $s=\dfrac{n+1 }{3}$ is an integer and maximum $K$-dimension of $L_{K}(E)$ is $f\left( \dfrac{n+1}{3}\right) =\dfrac{4}{27}\left( n+1\right) ^{3}$ and we have $ n_{i}=\dfrac{2(n+1)}{3},$ for each $i=1,2,\ldots ,s.
$
\textbf{Case 2: }$n\equiv 0$ \ $(\func{mod}3).$ Then we have: $\frac{n}{3} =t<t+\dfrac{1}{3}=s<t+1$ and\newline \begin{equation*} f\left( \frac{n}{3}\right) =\frac{(2n+3)^{2}n}{27}=\alpha _{1}\text{ and } f\left( \frac{n}{3}+1\right) =\frac{4n^{2}(n+3)}{27}=\alpha _{2}. \end{equation*} Note that, $\alpha _{1}>\alpha _{2}$. So $\alpha _{1}$ is maximum $K$ -dimension of $L_{K}(E)$ and we have $n_{i}=\dfrac{2}{3}n+1,$ for each $ i=1,2,\ldots ,s$.$
$
\textbf{Case 3:} $n\equiv 1$ \ $(\func{mod}3).$ Then $\dfrac{n-1}{3}=$ $t<t+ \dfrac{2}{3}=s<t+1$ and \newline \begin{equation*} f\left( \frac{n-1}{3}\right) =\frac{4}{27}\left( n+2\right) ^{2}\left( n-1\right) =\beta _{1}\text{ } \end{equation*} and \begin{equation*} f\left( \frac{n+2}{3}\right) =\frac{1}{27}\left( 2n+1\right) ^{2}\left( n+2\right) =\beta _{2}. \end{equation*} In this case $\beta _{2}>\beta _{1}$ and so $\beta _{2}$ gives the maximum $ K $-dimension of $L_{K}(E)$ and we have $n_{i}=\dfrac{2n+1}{3},$ for each $ i=1,2,\ldots ,s$. \end{proof}
\begin{theorem} The minimum $K$-dimension of $L_{K}(E)$ where $E$ ranges over all possible trees with $n$ vertices and $s$ sinks is equal to $r(q+2)^{2}+(s-r)(q+1)^{2}$ , where $n-1=qs+r,~~0\leq r<s.$ \end{theorem}
\begin{proof} We call a graph a \textit{bunch tree} if it is obtained by identifiying the unique sources of the finitely many oriented finite line graphs.\FRAME{dhF}{ 2.3756in}{1.6864in}{0pt}{}{}{Figure}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "T";width 2.3756in;height 1.6864in;depth 0pt;original-width 6.4584in;original-height 4.5731in;cropleft "0";croptop "1";cropright "1";cropbottom "0";tempfilename 'LXL3CQ01.wmf';tempfile-properties "XPR";}} Let $\mathcal{E}(n,s)$ be the set of all bunch trees with $n$ vertices and $ s $ sinks.
Every element of $\mathcal{E}(n,s)$ can be uniquely represented by an $s$ -tuple $(t_{1},t_{2},...,t_{s})$ where each $t_{i}$ is the number of vertices contributing only to the $i$-th$^{\text{ }}$sink with $1\leq t_{1}\leq t_{2}\leq ...\leq t_{s}$ and $t_{1}+t_{2}+...+t_{s}=n-1.$
Let $E\in \mathcal{E}(n,s)$ with $t_{s}-t_{1}\leq 1$. This $E$ is represented by the $s$-tuple $(q,\ldots ,q,\underset{}{q+1,\ldots ,q}+1)$ where $n-1=sq+r$, $0\leq r<s.$
Now we claim that the dimension of $E$ is the minimum of the set \begin{equation*} \left\{ \dim L_{K}(F):F\text{ tree with }s\text{ sinks and }n\text{ vertices} \right\} . \end{equation*} If we represent $U\in \mathcal{E}(n,s)$ by the $s$-tuple $ (u_{1},u_{2},...,u_{s})$ then $E\neq U$ implies that $u_{s}-u_{1}\geq 2.$
Consider the $s$-tuple $(t_{1},t_{2},...,t_{s})$ where $ (t_{1},t_{2},...,t_{s})$ is obtained from \linebreak $ (u_{1}+1,u_{2},...,u_{s-1},u_{s}-1)$ by reordering the components in increasing order.
In this case the dimension $d_{U}$ of $U$ is \begin{equation*} d_{U}=(u_{1}+1)^{2}+\ldots +(u_{s}+1)^{2}. \end{equation*} Similarly, the dimension $d_{T}$ of the bunch graph $T$ represented by the $ s $-tuple $(t_{1},t_{2},...,t_{s}),$ is \begin{equation*} d_{T}=(t_{1}+1)^{2}+\ldots +(t_{s}+1)^{2}=(u_{1}+2)^{2}+\ldots +u_{s-1}^{2}+u_{s}{}^{2}. \end{equation*} Hence \begin{equation*} d_{U}-d_{T}=2(u_{s}-u_{1})-2>0. \end{equation*} Repeating this process sufficiently many times we see that the process has to end at the exceptional bunch tree $E$ showing that its dimension is the smallest among the dimensions of all elements of $\mathcal{E}(n,s)$.
Now let $F$ be an arbitrary tree with $n$ vertices and $s$ sinks. As above we assign to $F$ the $s$-tuple $(n_{1},n_{2},...,n_{s})$ with $ n_{i}=n(v_{i})-1$ where the sinks $v_{i},~i=1,2,\ldots ,s$ are indexed in such a way that $n_{i}\leq n_{i+1},~i=1,\ldots ,s-1.$ Observe that $ n_{1}+n_{2}+\cdots +n_{s}\geq n-1$. Let $\beta =\dsum\limits_{i=1}^{s}n_{i}-(n-1).$ Since $s\leq n-1$, $\beta \leq \dsum\limits_{i=1}^{s}(n_{i}-1).$ Either $n_{1}-1\geq \beta $ or there exists a unique $k\in \left\{ 2,\ldots ,s\right\} $ such that $ \dsum\limits_{i=1}^{k-1}(n_{i}-1)<\beta \leq \dsum\limits_{i=1}^{k}(n_{i}-1)$ . If $n_{1}-1\geq \beta ,$ then let \begin{equation*} m_{i}=\left\{ \begin{array}{ccc} n_{1}-\beta & , & i=1 \\ n_{i} & , & i>1 \end{array} \right. . \end{equation*} Otherwise, let \begin{equation*} m_{i}=\left\{ \begin{array}{ccc} 1 & , & i\leq k-1 \\ n_{k}-\left( \beta -\sum\limits_{i=1}^{k-1}(n_{i}-1)\right) & , & i=k \\ n_{i} & , & i\geq k+1 \end{array} \right. . \end{equation*} In both cases, the $s$-tuple $(m_{1},m_{2},\ldots ,m_{s})$ that satisfies $ 1\leq m_{i}\leq n_{i}$, \linebreak $m_{1}\leq m_{2}\leq \cdots \leq m_{s}$ and $m_{1}+m_{2}+\cdots +m_{s}=n-1$ is obtained. So, there exists a bunch tree $M$ namely the one corresponding uniquely to $(m_{1},m_{2},\ldots ,m_{s})$ which has dimension $d_{M}\leq d_{F}.$ This implies that $d_{F}\geq d_{E}.$
Hence the result follows. \end{proof}
\begin{lemma} The minimum $K$-dimension of $L_{K}(E)$ where $E$ ranges over all possible trees with $n$ vertices occurs when the number of sinks is $n-1$ and is equal to $4(n-1)$. \end{lemma}
\begin{proof} By the previous theorem we see that \begin{equation*} \dim L_{K}(E)\geq r(q+2)^{2}+(s-r)(q+1)^{2} \end{equation*} where $n-1=qs+r,~~0\leq r<s.$ We have \begin{equation*} r(q+2)^{2}+(s-r)(q+1)^{2}=(n-1)(q+2)+qr+r+s. \end{equation*} Thus \begin{equation*} (n-1)(q+2)+qr+r+s-4(n-1)=(n-1)(q-2)+qr+r+s\geq 0\text{ \ }if\text{ \ }q\geq 2. \end{equation*} If $q=1,$then $-(n-1)+2r+s=-(n-1)+r+(n-1)=r\geq 0.$ Hence $\dim L_{K}(E)\geq 4(n-1).$
Notice that there exists a truncated tree $E$ with $n$ vertices and \linebreak $\dim L_{K}(E)=4(n-1)$ as sketched below : \FRAME{dhF}{1.3828in}{ 1.0136in}{0pt}{}{}{Figure}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "T";width 1.3828in;height 1.0136in;depth 0pt;original-width 3.4065in;original-height 2.4898in;cropleft "0";croptop "1";cropright "1";cropbottom "0";tempfilename 'LXL3CQ02.wmf';tempfile-properties "XPR";}} \end{proof}
\section{Line Graphs}
The \textit{total-degree} of the vertex $v$ is the number of edges that either have $v$ as its source or as its range, that is, $tot\deg (v)=\left\vert s^{-1}(v)\cup r^{-1}(v)\right\vert .$ A finite graph $E$ is a \textit{line graph} if it is connected, acyclic and $totdeg(v)\leq 2$ for every $v\in E^{0}.$
\begin{remark} In \cite{1}, the proposition 5.7 shows that a semisimple finite dimensional algebra $A=\bigoplus\limits_{i=1}^{s}M_{n_{i}}(K)$ over the field $K$ can be described as a Leavitt path algebra $L(E)$ defined \ by a line graph $E,$ if and only if $A$ has no ideals of $K-$dimension $1$ and the number of minimal ideals of $A$ of $K$ dimension $2^{2}$ is at most $2.$ On the other hand, if $A\cong L(E)$ for some $n$ line graph $E$ then $n-1=\sum \limits_{i=1}^{s}(n_{i}-1),$ that is, $n$ is an algebraic invariant of $A.$ \end{remark}
Therefore the following proposition answers a reasonable question.
\begin{proposition} The number $A_{n}$ of isomorphism classes of Leavitt path algebras defined by line graphs having exactly $n$ vertices is \begin{equation*} A_{n}=P(n-1)-P(n-4) \end{equation*} where $P(m)$ is the number of partitions of the natural number $m.$ \end{proposition}
\begin{proof} Any $n$-line graph has $n-1$ edges. In a line graph, for any edge $e$ there exists a unique sink $v$ so that there exists a path from $s(e)$ to $v.$ In this case we say that $e$ is directed towards $v$. The number of edges directed towards $v$ is clearly equal to $n(v)-1.$ Let $E$ and $F$ be two $n$ -line graphs. $L_{K}(E)\cong L_{K}(F)$ if and only if there exists a bijection $\phi :S(E)\rightarrow S(F)$ such that for each $v$ in $S(E),$ we have $n(v)=n(\phi (v)).$ Therefore the number of isomorphism classes of Leavitt path algebras determined by $n$-line graphs is the number of partitions of $n-1$ edges in which the number of parts having exactly one edge is at most two. Since the number of partitions of $k$ objects having at least three parts each of which containing exactly one element is $P(k-3)$, we get the result $A_{n}=P(n-1)-P(n-4).$ \end{proof}
\end{document} | arXiv |
In triangle $ABC$, let angle bisectors $BD$ and $CE$ intersect at $I$. The line through $I$ parallel to $BC$ intersects $AB$ and $AC$ at $M$ and $N$, respectively. If $AB = 17$, $AC = 24$, and $BC = 33$, then find the perimeter of triangle $AMN$.
Since $MN$ is parallel to $BC$, $\angle MIB = \angle IBC$. But $BI$ is an angle bisector, so $\angle IBC = \angle IBM$. Hence, triangle $MIB$ is isosceles with $MI = MB$. By the same argument, triangle $NIC$ is isosceles, with $NI = NC$.
[asy]
import geometry;
unitsize(1 cm);
pair A, B, C, I, M, N;
A = (1,3);
B = (0,0);
C = (4,0);
I = incenter(A,B,C);
M = extension(I, I + B - C, A, B);
N = extension(I, I + B - C, A, C);
draw(A--B--C--cycle);
draw(B--I--C);
draw(M--N);
label("$A$", A, dir(90));
label("$B$", B, SW);
label("$C$", C, SE);
label("$I$", I, dir(90));
label("$M$", M, NW);
label("$N$", N, NE);
[/asy]
Therefore, the perimeter of triangle $AMN$ is simply \begin{align*}
AM + AN + MN &= AM + AN + MI + NI \\
&= AM + AN + MB + NC \\
&= (AM + MB) + (AN + NC) \\
&= AB + AC \\
&= 17 + 24 \\
&= \boxed{41}.
\end{align*} | Math Dataset |
An end-to-end graph convolutional kernel support vector machine
Padraig Corcoran ORCID: orcid.org/0000-0001-9731-33851
A novel kernel-based support vector machine (SVM) for graph classification is proposed. The SVM feature space mapping consists of a sequence of graph convolutional layers, which generates a vector space representation for each vertex, followed by a pooling layer which generates a reproducing kernel Hilbert space (RKHS) representation for the graph. The use of a RKHS offers the ability to implicitly operate in this space using a kernel function without the computational complexity of explicitly mapping into it. The proposed model is trained in a supervised end-to-end manner whereby the convolutional layers, the kernel function and SVM parameters are jointly optimized with respect to a regularized classification loss. This approach is distinct from existing kernel-based graph classification models which instead either use feature engineering or unsupervised learning to define the kernel function. Experimental results demonstrate that the proposed model outperforms existing deep learning baseline models on a number of datasets.
The world contains much implicit structure which can be modelled using a graph. For example, an image can be modelled as a graph where objects (e.g. person, chair) are modelled as vertices and their pairwise relationships (e.g. sitting) are modelled as edges (Krishna et al. 2017). This representation has led to useful solutions for many vision problems including image captioning and visual question answering (Chen et al. 2019). Similarly, a street network can be modelled as a graph where locations are modelled as vertices and street segments are modelled as edges. This representation has led to useful solutions for many transportation problems including the placement of electrical vehicle charging stations (Gagarin and Corcoran 2018).
Given the ubiquity of problems which can be modelled in terms of graphs, performing machine learning on graphs represents an area of great research interest. Advances in the application of deep learning or neural networks to sequence spaces in the context of natural language processing and fixed dimensional vector spaces in the context of computer vision has led to much interest in applying deep learning to graphs. There exist many types of machine learning tasks one may wish to perform on graphs. These include vertex classification, graph classification, graph generation (You et al. 2018) and learning implicit/hidden structures (Franceschi et al. 2019). In this work we focus on the task of graph classification. Examples of graph classification tasks include human activity recognition where human pose is modelled using a skeleton graph (Yan et al. 2018), visual scene understanding where the scene is modelled using a scene graph (Xu et al. 2017) and semantic segmentation of three dimensional point clouds where the point cloud is modelled as a graph of geometrically homogeneous elements (Landrieu and Simonovsky 2018).
Graph convolutional is the most commonly used deep learning architecture applied to graphs. This architecture consists of a sequence of convolutional layers where each layer iteratively updates a vector space representation of each vertex. In their seminal work, Gilmer et al. (2017) demonstrated that many different convolutional layers can be formulated in terms of a framework containing two steps. In the first step, message passing is performed where each vertex receives messages from adjacent vertices regarding their current representation. In the second step, each vertex performs an update of its representation which is a function of its current representation and the messages it received in the previous step. In order to perform graph classification given a sequence of convolutional layers, the set of vertex representations output from this sequence must be integrated to form a graph representation. This graph representation can subsequently be used to predict a corresponding class label. We refer to this task of integrating vertex representations as vertex pooling and it represents the focus of this article. Note that, Gilmer et al. (2017) refers to this task as readout.
Performing vertex pooling is made challenging by the fact that different sets of vertex representations corresponding to different graphs may contain different numbers of elements. Furthermore, the elements in a given set are unordered. Therefore one cannot directly apply a feed-forward or recurrent architecture because these require an input lying in a vector space or sequence space respectively. To overcome this challenge most solutions involve mapping the sets of vertex representations to either a vector or sequence space which can then form the input to a feed-forward or recurrent architecture respectively. There exists a wide array of such solutions ranging from computing simple summary statistics such as mean vertex representation to more complex clustering based methods (Ying et al. 2018).
In this article we propose a novel binary graph classification model which performs vertex pooling by mapping a set of vertex representations to an element in a reproducing kernel Hilbert space (RKHS). A RKHS is a function space for which there exists a corresponding kernel function equalling the dot product in this space. Being a function space where the domain of functions in this space is a Euclidean Space, the RKHS in question is of infinite dimension and in turn has high model capacity. However, the infinite nature of this space makes it challenging to work directly in this space. To overcome this challenge, we use the corresponding kernel function which allows us to implicitly compute the dot product in this space without explicitly mapping to the space in question. This is a commonly used strategy known as the kernel trick. More specifically, the kernel corresponding to the RKHS is used within a support vector machine (SVM) to perform binary graph classification. A useful feature of the proposed pooling method is that the mapping to a RKHS is parameterized by a scale parameter which controls the degree to which different sets of vertex representations can be discriminated.
The proposed graph classification model is trained in a supervised end-to-end manner where the convolutional layers, the kernel function and SVM parameters are jointly optimized with respect to a regularized classification loss. This approach is distinct from existing kernel-based models which instead use feature engineering or unsupervised learning to define the kernel function and only optimize the parameters of the classification method in a supervised manner (Yanardag and Vishwanathan 2015). Using feature engineering can result in diagonal dominance whereby a graph is determined to only be similar to itself, but not to any other graph (Yanardag and Vishwanathan 2015). Although unsupervised learning can overcome this problem and improve performance, the kernel may not be optimal for the task at hand given it was learned in an unsupervised as opposed to supervised manner (Ivanov and Burnaev 2018). The proposed solution of optimizing in an end-to-end manner overcomes these limitations.
The remainder of this paper is structured as follows. "Related work" section reviews related work on graph kernels and vertex pooling methods. "Methodology" section describes the proposed graph classification model. "Evaluation" section presents an evaluation of this model through comparison to 12 baseline models on 4 datasets. Finally, "Conclusions and future work" section draws some conclusions from this work and discusses possible future research directions.
In this work we propose a novel vertex pooling method which performs vertex pooling by mapping to a RKHS. In the following two sections we review related work on vertex pooling methods and graph kernels.
Vertex pooling
As discussed in the introduction to this article, existing vertex pooling methods generally map the set of vertex representations to a fixed dimensional vector space or sequence space. The simplest methods for performing vertex pooling compute a summary statistic of the set of vertex representations. Commonly used summary statistics include mean, max and sum (Duvenaud et al. 2015). Despite the simple nature of these methods, a recent study by Luzhnica et al. (2019) demonstrated that in some cases they can outperform more complex methods. Zhang et al. (2018a) proposed a vertex pooling method which first performs a sorting of vertex representations based on the Weisfeiler-Lehman graph isomorphism algorithm. A subset of these vertex representations are then selected based on this ranking, where the size of this subset is a user specified parameter. Li et al. (2016) proposed a vertex pooling method which outputs an element in sequence space. Gilmer et al. (2017) proposed to perform vertex pooling by applying the set2set model from Vinyals et al. (2016). The set2set model maps the set of vertex representations to fixed dimensional vector space representation which is invariant to the order of elements in the set. Ying et al. (2018) proposed a vertex pooling method which uses clustering to iteratively integrate vertex representations and outputs an element in a fixed dimensional vector space. Kearnes et al. (2016) proposed a vertex pooling method which creates a fuzzy histogram of the vertex representations and outputs an element in a fixed dimensional vector space.
Graph kernels
As described in the introduction to this article, existing kernel-based graph classification methods use either feature engineering or unsupervised learning to define the kernel. We now review each of these approaches in turn.
The most common approach for feature engineering kernels is the \(\mathcal {R}\)-convolution framework where the kernel function of two graphs is defined in terms of the similarity of their respective substructures (Haussler 1999). This framework is similar to the bag-of-words framework used in natural language processing. Substructures used in the \(\mathcal {R}\)-convolution framework to define kernels include graphlets (Shervashidze et al. 2009), shortest path properties (Borgwardt and Kriegel 2005) and random walk properties (Sugiyama and Borgwardt 2015).
The Weisfeiler-Lehman framework is a framework for feature engineering kernels which is inspired by the Weisfeiler-Lehman test of graph isomorphism. In this framework the vertex representations of a given graph are iteratively updated in a similar manner to graph convolution to give a sequence of graphs. A kernel is then defined with respect to this sequence by summing the application of a given kernel, known as the base kernel, to each graph in the sequence. Shervashidze et al. (2011) proposed a family of kernels using this framework by considering a set of base kernels including one which measures the similarity of shortest path properties. Rieck et al. (2019) proposed a kernel using this framework by considering a base kernel which measures the similarity of topological properties.
Kriege et al. (Kriege et al. 2016) proposed another framework for feature engineering kernels known as assignment kernels which computes an optimal assignment between graph substructures and sums over a kernel applied to each correspondence in the assignment. The authors proposed a number of kernels using this framework including one based on the Weisfeiler-Lehman graph isomorphism algorithm. Kondor and Pan (2016) proposed a multiscale kernel which considers vertex features plus topological information through the graph Laplacian. Zhang et al. (2018c) proposed a kernel-based on the return probabilities of random walks. The authors used an approximation of the kernel function so that the method can be applied to large datasets (Rahimi and Recht 2008).
To overcome the limitations of feature engineering and improve performance, recent works in the field of graph kernels have considered unsupervised learning techniques. These methods generally learn a graph representation in an unsupervised manner and subsequently use this representation to define a kernel. Yanardag and Vishwanathan (2015) proposed a kernel which uses the \(\mathcal {R}\)-convolution framework to define a set of substructures and subsequently learns an embedding of these substructures in an unsupervised manner using a word2vec type model. Ivanov and Burnaev (2018) proposed a kernel which determines two graphs to be similar if their vertices have similar neighbourhoods measured in terms of anonymous walks which are a generalization of random walks. Learning is performed in an unsupervised manner using a word2vec type model. Nikolentzos et al. (2017) proposed a graph kernel which first computes sets of vertex representations corresponding to the graphs in question in an unsupervised manner. The similarity of these sets are then computed using the earth mover's distance. The authors noted that these similarities do not yield a positive semidefinite kernel matrix preventing it from being used in some kernel-based classification methods. To overcome this issue the authors use a version of the support vector machine for indefinite kernel matrices. Similar to Nikolentzos et al. (2017) and Wu et al. (2019) proposed a graph kernel which first computes sets of vertex representations corresponding to the graphs in questions in an unsupervised manner. The resulting set of embeddings are in turn used to embed the graph in question by measuring the disturbance distance to sets of embeddings corresponding to random graphs. Finally, this graph representation is used to define a kernel. Nikolentzos et al. (2018) proposed a method that performs an unsupervised clustering of the input graph into components and subsequently learns a kernel function which takes as input these components.
The proposed graph classification model consists of the following three steps. In the first step, a sequence of graph convolutional layers are applied to the graph in question to generate a corresponding set of vertex representations. In the second step, this set of vertex representations is mapped to a RKHS. In the final step, graph classification is performed using a SVM. Each of these three steps are described in turn in the first three subsections of this section. In the final subsection we describe how the parameters of each step are optimized jointly in an end-to-end manner. Before that, we first introduce some notation and formally define the problem of graph classification.
A graph is a tuple (V,E) where V is a set of vertices and E⊆(V×V) is a set of edges. Let \(\mathcal {G}\) denote the space of graphs. Let l:V→Σ denote a vertex labelling function. In this work we assume that Σ is a finite set. Let \(\mathbb {G} = \lbrace \mathcal {G}_{1}, \mathcal {G}_{2}, \dots, \mathcal {G}_{n} \rbrace \) denote a set of n graphs and \(\mathbb {Y} = \lbrace \mathcal {Y}_{1}, \mathcal {Y}_{2}, \dots, \mathcal {Y}_{n} \rbrace \) denote a corresponding set of graph labels. In this work we assume that graph labels take elements in the set {0,1}. In this work we consider the problem of binary graph classification where given \(\mathbb {G}\) and \(\mathbb {Y}\) we wish to learn a map \(\mathcal {G} \rightarrow \lbrace 0, 1\rbrace \).
Graph convolution layers
A large number of different graph convolutional layers have been proposed. Broadly speaking a graph convolutional layer will update the representation of each vertex in a given graph where this update is a function of the current representation of that vertex plus the representations of its adjacent neighbours. In this section we only briefly review existing graph convolutional layers but the interested reader can find a more indepth analysis in the following review papers (Zhang et al. 2018b; Wu et al. 2019).
Gilmer et al. (2017) showed that many different convolutional layers may be reformulated in terms of a framework called Message Passing Neural Networks defined in terms of a message function M and an update function U. In this framework vertex representations are updated according to Eq. 1 where \(h_{v}^{t}\) denotes the representation of vertex v output from the t-th convolutional layer and N(v) denotes the set of vertices adjacent to v. Each vertex representation \(h_{v}^{t}\) is an element of \(\mathbb {R}^{m}\) where the dimension m may vary from layer to layer. For the input layer, that is t=1, vertex representations equal a one-hot encoding of the vertex labelling function l and therefore the corresponding dimension is |Σ|. For all subsequent layers the corresponding dimension is a model hyper-parameter.
$$ \begin{aligned} m^{t+1}_{v} & = \sum_{w \in N(v)} M\left(h_{v}^{t}, h_{w}^{t}\right) \\ h^{t+1}_{v} & = U\left(h^{t}_{v}, m^{t+1}_{v}\right) \end{aligned} $$
In the proposed graph classification model we use the functions M and U originally proposed by Hamilton et al. (2017) and defined in Eq. 2. Here CONCAT is the horizontal vector concatenation operation, Wt and bt are the weights and biases respectively for the t-th convolutional layer, and ReLU is the real valued rectified linear unit non-linearity.
$$ \begin{aligned} M\left(h_{v}^{t}, h_{w}^{t}\right) & = h_{w}^{t} \\ U\left(h^{t}_{v}, m^{t+1}_{v}\right) & = \text{ReLU} \left(\mathbf{W}^{t} \cdot \text{CONCAT} \left(h_{v}^{t-1}, m^{t+1}_{v} \right) + \mathbf{b}^{t} \right) \end{aligned} $$
A sequence of two convolutional layers were used in the proposed model. A number of studies have found that the use of two layers empirically gives the best performance (Kipf and Welling 2017). This sequence of layers will map a graph \(\mathcal {G}_{i}=(V,E)\) to a set of |V| points in \(\mathbb {R}^{m}\) where m is the dimension of the final convolutional layer. Since the number of vertices in a graph may vary the number of points in \(\mathbb {R}^{m}\) may in turn vary. Let us denote by Set the space of sets of points in \(\mathbb {R}^{m}\). Given this, the sequence of convolutions layers defines a map \(\mathcal {G} \rightarrow \text {Set}\).
Mapping to RKHS
The output from the sequence of convolutional layers defined in the previous subsection is an element in the space Set. In this section we propose a method for mapping elements in this space to a reproducing kernel Hilbert space (RKHS). We in turn define a kernel between elements in this space.
A Hilbert space is a vector space with an inner product such that the induced norm turns the space into a complete metric space. A positive-semidefinite kernel on a set \(\mathcal {X}\) is a function \(k: \mathcal {X} \times \mathcal {X} \rightarrow \mathbb {R}\) such that there exists a feature space \(\mathcal {H}\) and a map \(\phi : \mathcal {X} \rightarrow \mathcal {H}\) such that k(x,y)=〈ϕ(x),ϕ(y)〉 where \(x,y \in \mathcal {X}\) and 〈·,·〉 denotes the dot product in \(\mathcal {H}\). Equivalently, a function \(k: \mathcal {X} \times \mathcal {X} \rightarrow \mathbb {R}\) is a kernel if and only if for every subset \(\left \lbrace x_{1}, \dots, x_{q} \right \rbrace \subseteq \mathcal {X}\), the q×q matrix K with entries Kij=k(xi,xj) is positive semi-definite (Schölkopf et al. 2002). Given a kernel k, one can define a map \(\mathcal {X} \to \mathbb {R}^{\mathcal {X}}\) as Eq. 3 where codomain of this map is the space of real valued functions on \(\mathcal {X}\). Such a space is called a function space. Given this, it can be proven that k(x,y)=〈k(·,x),k(·,y)〉. By virtue of this property, \(\mathbb {R}^{\mathcal {X}}\) is called a reproducing kernel Hilbert space (RKHS) corresponding to the kernel k (Schölkopf et al. 2002).
$$ x \mapsto k(\cdot, x) $$
Let \(k^{R}_{\sigma }: \mathbb {R}^{m} \times \mathbb {R}^{m} \rightarrow \mathbb {R}\) be the Gaussian kernel function defined in Eq. 4 which is parameterized by \(\sigma \in \mathbb {R}_{\geq 0}\).
$$ k^{R}_{\sigma}(u,v) = \exp\left(-\Vert u-v \Vert^{2}_{2} /2\sigma^{2}\right) $$
Given \(k^{R}_{\sigma }\), we define a map \(F: \text {Set} \times \mathbb {R} \rightarrow \mathbb {R}^{\mathbb {R}^{m}}\) in Eq. 5 where \(\mathbb {R}^{\mathbb {R}^{m}}\) is the space of real valued functions on \(\mathbb {R}^{m}\). To illustrate this map consider the element of Set displayed in Fig. 1a where the dimension m equals 2. Recall that elements in the space Set correspond to sets of points in \(\mathbb {R}^{m}\). Figures 1b and c display the elements of \(\mathbb {R}^{\mathbb {R}^{m}}\) resulting from applying the map F to this element of Set with σ parameter values of 0.001 and 0.0005 respectively.
$$ F(x, \sigma) = \sum_{v \in x} k^{R}_{\sigma}(v,\cdot) $$
An element of the space Set is displayed in (a) where the dimension m equals 2 and each point in \(\mathbb {R}^{m}\) is represented by a red dot. The elements of \(\mathbb {R}^{\mathbb {R}^{m}}\), which are themselves functions \(\mathbb {R}^{m} \rightarrow \mathbb {R}\), resulting from applying the map F to this element with σ parameter values of 0.001 and 0.0005 are displayed in (b) and (c) respectively
The parameter σ of the map F is a scale parameter and may be interpreted as follows. As the value of σ approaches 0, F(x,σ) becomes a sum of a set indicator functions applied to x. In this case distinct elements of the space Set map to distinct elements of \(\mathbb {R}^{\mathbb {R}^{m}}\) where the distance between these functions measured by the Lp norm is greater than zero. On the other hand, as σ approaches ∞, differences between the functions are gradually smoothed out and in turn the distance between the functions gradually reduces. Therefore, one can view the parameter σ as controlling the discrimination power of the method.
Given the map F defined in Eq. 5, we define the kernel \(k^{L}_{\sigma }: \mathbb {R}^{\mathbb {R}^{m}} \times \mathbb {R}^{\mathbb {R}^{m}} \rightarrow \mathbb {R}\) in Eq. 6. Note that, the final equality in this equation follows from the reproducing property of the RKHS related to \(k^{R}_{\sigma }\) and the bilinearity of the inner product (Paulsen and Raghupathi 2016). By examination of Eq. 6, we see that the kernel \(k^{L}_{\sigma }\) equals the dot product between elements in the codomain of the map F which is an infinite dimensional function space. That is, the kernel allows us to operate in this codomain without the computational complexity of explicitly mapping into it. In Theorem 1 we prove that \(k^{L}_{\sigma }\) is a valid positive-semidefinite kernel.
$$ \begin{aligned} & k^{L}_{\sigma}(F(x_{i}, \sigma), F(x_{j}, \sigma)) = \langle F(x_{i}, \sigma), F(x_{j}, \sigma) \rangle \\ & = \left\langle \sum_{v \in x_{i}} k^{R}_{\sigma}(v,\cdot), \sum_{u \in x_{j}} k^{R}_{\sigma}(u,\cdot) \right\rangle = \sum_{v \in x_{i}} \sum_{u \in x_{j}} k^{R}_{\sigma}(v, u) \end{aligned} $$
Theorem 1
The kernel \(k^{L}_{\sigma }\) is a positive-semidefinite kernel.
The kernel \(k^{L}_{\sigma }\) is a positive-semidefinite kernel because it is defined in Eq. 6 to equal the dot product in the space \(\mathbb {R}^{\mathbb {R}^{m}}\). □
The kernel \(k^{L}_{\sigma }\) has a specific scale which is specified by σ. In order to adopt a multi-scale approach we consider a set of s scales \(\Sigma = \lbrace \sigma _{1}, \dots, \sigma _{s} \rbrace \) to define a corresponding set of kernels \(\left \lbrace k^{L}_{\sigma _{1}}, \dots, k^{L}_{\sigma _{s}} \right \rbrace \). We combine these kernels using a linear combination defined in Eq. 7 where \(\lbrace \beta _{1}, \dots, \beta _{s} \rbrace \in \mathbb {R}^{s}_{\geq 0}\). In Theorem 2 we prove that \(k^{L}_{\sigma }\) is a valid positive-semidefinite kernel.
$$ k^{L}_{\Sigma}(F(x_{i}), F(x_{j})) = \sum_{l=1}^{s} \beta_{l} k^{L}_{\sigma_{l}}(F(x_{i}), F(x_{j})) $$
The kernel \(k^{L}_{\sigma }\) is a positive-semidefinite kernel because it is the sum of positive-semidefinite kernels and the coefficients \(\lbrace \beta _{1}, \dots, \beta _{s} \rbrace \) are all positive (see proposition 13.1 in Schölkopf et al. (2002)). □
Recall that we consider the problem of graph classification whereby given \(\mathbb {G} = \lbrace \mathcal {G}_{1}, \mathcal {G}_{2}, \dots, \mathcal {G}_{n} \rbrace \) and \(\mathbb {Y} = \lbrace \mathcal {Y}_{1}, \mathcal {Y}_{2}, \dots, \mathcal {Y}_{n} \rbrace \) we wish to learn a map \(\mathcal {G} \rightarrow \lbrace 0, 1\rbrace \).
Let \(f: \text {Set} \rightarrow \mathbb {R}\) be a map from which we obtain a decision function by sgn (f). That is, if f returns a positive value we classify the graph in question as 1 and otherwise we classify it as 0. We determine a suitable map f lying in the RKHS \(\mathcal {H}\) corresponding to the kernel \(k^{L}_{\sigma }\) by Eq. 8. Note that, the first term in this sum corresponds to the soft margin loss (Schölkopf et al. 2002) and the second term is a regularization term.
$$ \hat{f} = {\underset{f \in \mathcal{H}}{\text{arg min}}} \sum_{i=1}^{n} \max(0, 1-y_{i}f(x_{i})) + \lambda \Vert f \Vert_{\mathcal{H}}^{2} $$
By the representer theorem any solution to Eq. 8 can be written in the form of Eq. 9 where \(\lbrace \alpha _{1}, \dots, \alpha _{n} \rbrace \in \mathbb {R}^{n}\) (Paulsen and Raghupathi 2016).
$$ f(\cdot) = \sum_{j=1}^{n} \alpha_{j} k^{L}_{\Sigma}(\cdot, x_{j}) $$
Substituting this into Eq. 8 we obtain Eq. 10 where optimization of the function f is performed with respect to \(\lbrace \alpha _{1}, \dots, \alpha _{n} \rbrace \in \mathbb {R}^{n}\). Here \(K^{L}_{i,j} = k^{L}_{\Sigma }(x_{i}, x_{j})\), ⊙ is the elementwise multiplication operator (Hadamard product), \(\vec {0}\) is a vector of zeros of size n and \(\vec {1}\) is a vector of ones of size n.
$$ \begin{aligned} \hat{f} & = \underset{\alpha \in \mathbb{R}^{n}}{\text{arg min}} \sum_{i=1}^{n} \max\left(0, 1-y_{i} \sum_{j=1}^{n} \alpha_{j} k^{L}_{\Sigma}(x_{i}, x_{j})\right) \\ & \qquad \qquad + \lambda \Vert \sum_{j=1}^{n} \alpha_{j} k^{L}_{\Sigma}(\cdot, x_{j}) \Vert_{\mathcal{H}}^{2} \\ & = \underset{\alpha \in \mathbb{R}^{n}}{\text{arg min}} \sum_{i=1}^{n} \max\left(0, 1-y_{i} \sum_{j=1}^{n} \alpha_{j} k^{L}_{\Sigma}(x_{i}, x_{j})\right) \\ & \qquad \qquad + \lambda \sum_{i,j=1}^{n} \alpha_{i} \alpha_{j} k^{L}_{\Sigma}(x_{i}, x_{j}) \\ & = \underset{\alpha \in \mathbb{R}^{n}}{\text{arg min}} \Vert \max\left(\vec{0}, \vec{1} - y \odot K^{L} \alpha\right) \Vert_{1}\\ & \qquad \qquad + \lambda \alpha^{T} K^{L} \alpha \end{aligned} $$
End-to-end optimization
As described in the previous subsections, the proposed classification model contains three steps with each having corresponding parameters which require optimization with respect to the objective function defined in Eq. 10. The parameters in question are the sets of convolutional layer parameters Wt and bt defined in Eq. 2, the sets of kernel parameters σl and βl defined in Eq. 7, and the set of SVM parameters αj defined in Eq. 9. All of these parameters are unconstrained real values apart from the sets of kernel parameters σl and βl which are constrained to be positive real values. As such, the optimization problem in question is a constrained optimization problem. In this work we wish to optimize all the above model parameters jointly in an end-to-end manner. We refer to this as the end-to-end optimization problem. Note that, if only the SVM parameters were optimized and all other parameters were fixed, the optimization problem could be formulated as a quadratic program by taking the dual and solved in closed-form (Schölkopf et al. 2002). This is the most commonly used method for optimizing the parameters of an SVM.
In order to solve the end-to-end optimization problem we use a gradient based optimization method. Such methods are the most commonly used methods for optimizing neural network parameters (Goodfellow et al. 2016). There are two main approaches that can be used to apply a gradient based optimization method to a constrained optimization problem. The first approach is to project the result of each gradient step back into the feasible region. The second approach is to transform the constrained optimization problem into an unconstrained optimization problem and solve this problem. Such a transformation can be achieved using the Karush-Kuhn-Tucker (KKT) method (Nocedal and Wright 2006). In this work we use the former approach. In practice this reduces to passing the parameters σl and βl through the function max(·,0) after each gradient step. The above optimization can be used in conjunction with any gradient based optimization method such as stochastic gradient descent. In this work the Adam method was used (Kingma and Ba 2014).
In this section we present an evaluation of the proposed end-to-end graph classification model with respect to current state-of-the-art models. This section is structured as follows. "Implementation details" section provides implementation details for the proposed model. "Baseline methods" section describes the baseline models used to compare the proposed model against. Finally, "Datasets and results" section describes the datasets used in this evaluation and compares the performance of all models on these datasets.
The parameters of the proposed model were initialized as follows. The convolutional layer weights Wt and biases bt in Eq. 2 were initialized using Kaiming initialization (He et al. 2015) and to a value of 0 respectively. The kernel parameters \(\lbrace \sigma _{1}, \dots, \sigma _{s} \rbrace \) and \(\lbrace \beta _{1}, \dots, \beta _{s} \rbrace \) and s in Eq. 7 were all initialized to a value of 1.
The model hyper-parameters were set as follows. The dimension of the convolutional hidden layers was set equal to 25. The Adam optimizer learning rate was set to its default value of 0.001 and training was performed for 300 epochs. The hyper-parameters λ in Eq. 10 and s in Eq. 7 were selected from the sets {0.0,0.5,1.0,1.5,2.0,2.5,3.0} and {1,2} respectively by considering classification accuracy on a validation set. Larger values for the hyper-parameter s were not considered to ensure scalability of the model to medium sized datasets.
The time and space complexity of classifying a given graph is O(n) where n is the number of graphs in the training dataset. This is a consequence of the summation in Eq. 9 over all training examples. The time and space complexity of performing an update of the method parameters using backprop is O(n2) because this step computes the complete kernel matrix K in Eq. 10. Note that, the above time complexity analysis assumes that each element in the kernel matrix K can be computed in constant time. In reality, if we assume that each graph contains m vertices the time complexity of computing each element in K is m2 (see Eq. 6). In this case the time complexity of classifying a given graph is O(nm2) while the time complexity of performing an update of the method parameters using backprop is O(n2m2).
Baseline methods
As described in the related work section of this paper, existing models for graph classification belong to two main categories of feature engineered kernel and end-to-end deep learning models. For the purposes of this evaluation, we compared the model proposed in this work to baseline models in each of these categories.
A set of 17 baseline models were considered where this set contains 5 feature engineered kernel models and 12 end-to-end deep learning models. We considered so many baseline models to ensure we were comparing to state of the art; many existing models claim to outperform each other so it is difficult to determine which models are in fact state of the art. The end-to-end deep learning baseline models considered in the evaluation are end-to-end models but not are kernel-based models. The proposed model is the first end-to-end kernel-based model for graphs.
In order to cover the breadth of different feature engineered kernel models, we considered three kernel functions in the \(\mathcal {R}\)-convolution framework, one kernel function in the Weisfeiler-Lehman framework and one kernel function which uses unsupervised learning. The kernel functions in question are entitled Graphlet by Shervashidze et al. (2009), Shortest Path by Borgwardt and Kriegel (2005), Vertex Histogram by Sugiyama and Borgwardt (2015), Weisfeiler Lehman by Shervashidze et al. (2011) and Pyramid Match by Nikolentzos et al. (2017). These kernel functions are described in the Appendix section of this article. For each kernel function, classification was performed using a Support Vector Machine (SVM) with a kernel matrix precomputed using the kernel function in question. Implementations for the kernel functions were obtained from the GraKeL Python library (Siglidis et al. 2020).
The end-to-end deep learning baseline models considered are entitled GCN, GCNWithJK, GIN, GIN0, GINWithJK, GIN0WithJK, GraphSAGE, GraphSAGEWithJK, DiffPool, GlobalAttentionNet, Set2SetNet and SortPool. The architectures of these models are described in the Appendix section of this article. Implementations for these models were obtained from the PyTorch Geometric Python library (Fey and Lenssen 2019); these can be downloaded directly from the benchmark section of the PyTorch Geometric websiteFootnote 1. For each end-to-end deep learning baseline model the corresponding model parameters were optimized using the Adam optimizer with the default learning rate of 0.001 and run for 300 epochs. In all cases a negative log likelihood loss function was used. Model hyper-parameters corresponding to the number and dimension of hidden layers were selected from the sets {1,2,3,4,5} and {16,32,64,128} respectively by considering the loss on a validation set.
Datasets and results
To evaluate the proposed graph classification model we considered five commonly used graph classification datasets obtained from the TU Dortmund University graph dataset repository (Kersting et al. 2016)Footnote 2. Summary statistics for each of these datasets are displayed in Table 1. The MUTAG dataset contains graphs corresponding to chemical compounds and the binary classification problem concerns predicting a particular characteristic of the chemical (Debnath et al. 1991). The PTC_MR dataset contains graphs corresponding to chemical compounds and the binary classification problem concerns predicting a carcinogenicity property. The BZR_MD dataset contains graphs corresponding to chemical compounds and the binary classification problem concerns predicting a particular characteristic of the chemical (Sutherland et al. 2003). The PTC_FM dataset contains graphs corresponding to chemical compounds and the binary classification problem and concerns predicting a carcinogenicity property. The COX2 dataset contains graphs corresponding to molecules and the binary classification problem concerns predicting if a given molecule is active or inactive (Sutherland et al. 2003).
Table 1 Summary statistics for each dataset used in the evaluation
Stratified k-folds cross-validation with a k value of 10 was used to split the data into training, validation and testing sets. During each of the k training steps, one of the k−1 folds in the training set was randomly selected to be a validation set and classification accuracy on this set was used to select model hyper-parameters. It has been shown that different training, validation and testing set splits of the data can lead to quite different rankings of graph classification models (Shchur et al. 2018). However averaging the performance of k different splits, as done in this work, helps to reduce this instability. In our analysis the same training, testing and validation splits were used for all graph classification models considered. This is an important point because the performance of a given model may vary as a function of the split used. For each dataset we computed the mean accuracy on the test sets for each method. The results of this analysis are displayed in Table 2. For two of the five datasets, the proposed graph classification model achieved the best mean performance and outperformed most other models by a significant margin. For the remaining three datasets, the proposed method achieved a better mean performance than many but not all baseline methods. These positive results demonstrate the utility of the proposed model. In most cases the proposed model achieved best performance on the validation set with the hyper-parameter s having a value of 2. Recall, from Eq. 7, that this hyper-parameter equals the number of individual kernels integrated by the model. This demonstrates the utility of integrating multiple kernels.
Table 2 For each dataset, the mean classification accuracy plus standard deviation of 10-fold cross validation for each graph classification model are displayed
It is important to note that the proposed method was compared against a large number of benchmark methods (17). This makes it challenging for any single method to perform best on all datasets. It is difficult to interpret exactly why one deep learning architecture performs better or worse than another on a particular dataset. However, one limitation of the proposed method that may limit its ability to accurately discriminate is that it only models the distribution of node embeddings and not the position of these nodes in the graph. The recent work by You et al. (2019) suggests position information is important. The DiffPool method which performed best on the BZR_MD dataset actually uses node position information when performing clustering in the pooling step (this is illustrated in Figure 1 of the original paper by Ying et al. (2018)). We hypothesize that position information may not be important for some graph classification tasks while being important for others. This may explain why the proposed method does not uniformly outperform all others. It is also worth noting that the proposed method achieved similar performance to the GIN method on the BZR_MD dataset. In a recent paper by Errica et al. (2019), the authors found the GIN method to achieve best results on a number of datasets. Finally, it is interesting to note that the end-to-end deep learning models did not uniformly outperform the feature engineered kernel models. In fact, the best mean performance on the BZR_MD dataset was achieved by a feature engineered kernel model.
Conclusions and future work
This article proposes a novel kernel-based support vector machine (SVM) for graph classification. Unlike existing kernel-based models, the proposed model is trained in a supervised end-to-end manner whereby the convolutional layers, the kernel function and SVM parameters are jointly optimized. The proposed model outperforms existing deep learning models on a number of datasets which demonstrates the utility of the model.
Despite these positive results, the proposed model is not a suitable candidate solution for all graph classification problems. Like all kernel-based models, the proposed model does not natively scale to large datasets. This is a consequence of the fact that training the model requires computation and storing of the kernel matrix whose size is quadratic in the number of training examples. This limitation may potentially be overcome by performing an approximation of the kennel function (Rahimi and Recht 2008). The authors plan to investigate this research direction in future work.
We briefly describe the kernel functions corresponding to the feature engineered kernel baseline models considered in the work.
Graphlet - This is a kernel in the \(\mathcal {R}\)-convolution framework which uses substructures based on Graphlets and was proposed by Shervashidze et al. (2009).
Shortest Path - This is a kernel in the \(\mathcal {R}\)-convolution framework which uses substructures based on shortest paths and was proposed by Borgwardt and Kriegel (2005).
Vertex Histogram - This is a kernel in the \(\mathcal {R}\)-convolution which uses substructures based on random walks and was proposed by Sugiyama and Borgwardt (2015).
Weisfeiler Lehman - This is a kernel in the Weisfeiler-Lehman framework which uses the Vertex Histogram Kernel as the base kernel (Sugiyama and Borgwardt 2015) and was proposed by Shervashidze et al. (2011).
Pyramid Match - This kernel uses unsupervised learning and was proposed by Nikolentzos et al. (2017).
We briefly describe the architectures corresponding to the end-to-end deep learning baseline models considered in the work. More specific implementation details can be found at the benchmark section of the PyTorch Geometric website.
GCN - This model consists of graph convolutional layers proposed by Kipf and Welling (2017), followed by mean pooling, followed by a non-linear layer, followed by a dropout layer, followed by a linear layer, followed by a softmax layer.
GCNWithJK - This model is equal to GCN but with the addition of jump or skip connections before mean pooling as proposed by Xu et al. (2018).
GIN - This model consists of the graph convolutional layers proposed by Xu et al. (2019), followed by mean pooling, followed by a non-linear layer, followed by a dropout layer, followed by a linear layer, followed by a softmax layer. The convolution layer in question has a parameter ε which is learned.
GIN0 - This model is equal to GIN with the exception that the parameter ε is not learned and instead is set to a value of 0.
GINWithJK - This model is equal to GIN but with the addition of jump or skip connections before mean pooling as proposed by Xu et al. (2018).
GIN0WithJK - This model is equal to GIN0 but with the addition of jump or skip connections before mean pooling as proposed by Xu et al. (2018).
GraphSAGE - This model consists of the graph convolutional layers proposed by Hamilton et al. (Hamilton et al. 2017), followed by a mean pooling layer, followed by a non-linear layer, followed by a dropout layer, followed by a linear layer, followed by a softmax layer.
GraphSAGEWithJK - This model is equal to GraphSAGE but with the addition of jump or skip connections before mean pooling as proposed by Xu et al. (2018).
DiffPool - This model consists of the graph convolutional layers proposed by Hamilton et al. (2017), followed by the pooling method proposed by Ying et al. (2018), followed by a non-linear layer, followed by a dropout layer, followed by a linear layer, followed by a softmax layer.
GlobalAttentionNet - This model consists of the graph convolutional layers proposed by Hamilton et al. (2017), followed by the pooling layer proposed by Li et al. (2016), followed by a dropout layer, followed by a non-linear layer, followed by a linear layer, followed by a softmax layer.
Set2SetNet - This model consists of the graph convolutional layers proposed by Hamilton et al. (2017), followed by the pooling layer proposed by Vinyals et al. (2016), followed by a non-linear layer, followed by a dropout layer, followed by a linear layer, followed by a softmax layer.
SortPool - This model consists of the graph convolutional layers proposed by Hamilton et al. (2017), followed by the pooling layer proposed by Zhang et al. (2018a), followed by a non-linear layer, followed by a dropout layer, followed by linear layer, followed by a softmax layer.
All data used in this work is publicly available from the TU Dortmund University graph dataset repository.
https://github.com/rusty1s/pytorch_geometric/tree/master/benchmark
https://chrsmrrs.github.io/datasets/
SVM:
RKHS:
Reproducing kernel Hilbert space
KKT:
Karush-Kuhn-Tucker
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The author would like to acknowledge the many useful discussions he had with Bertrand Gauthier concerning kernel methods.
Nothing to declare.
School of Computer Science & Informatics, Cardiff University, Queen's Buildings, 5 The Parade, Roath, Cardiff, CF24 3AA, UK
Padraig Corcoran
Padraig Corcoran was solely responsible for all work presented in this paper. The author(s) read and approved the final manuscript.
Correspondence to Padraig Corcoran.
The author declares that they have no competing interests.
Corcoran, P. An end-to-end graph convolutional kernel support vector machine. Appl Netw Sci 5, 39 (2020). https://doi.org/10.1007/s41109-020-00282-2
Graph neural network
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\begin{document}
\title{Deformed super brackets on forms of a manifold}
\thispagestyle{plain}
\section{Introduction} Let \(M\) be a differentiable manifold and \(\tbdl{M} \) and \(\cbdl{M}\) be the tangent and cotangent bundle of \(M\) (or the spaces of the sections). In addition to the typical example of Lie superalgebra \( \mathfrak{g} = \sum _{p=1} ^{\textrm{dim} M} \gaiseki{p} \) with the Schouten bracket,
the space of differential forms \( \mathfrak{h} = \sum_{q=0}^{\textrm{dim} M} \cgaiseki{q} \) with the bracket \begin{equation} \BktU{}{\alpha}{\beta} = \parity{a} d( \alpha \wedge \beta )\quad \text{where} \alpha \in \cgaiseki{a}\ \text{ and }\ \beta \in \cgaiseki{b} \end{equation} becomes a Lie superalgebra, where the grading of \( \cgaiseki{a} \) is \( - a -1 \), and is often referred to as \(a'\) (cf.\ \cite{Mik:Miz:superForms}). The grading of \( \gaiseki{a} \) is \( a -1 \), and is also represented by \(a'\).
There is a notion of deformation of the exterior differentiation \(d\) by a 1-form \(\phi\) defined by \( d_{t} \alpha = d \alpha + t \phi \wedge \alpha \) where \(t\) is a scalar parameter runs at least \([0,1]\) interval, and it is well-known that \( d_{t} \circ d_{t} = 0 \) if \(\phi\) is a 1-cocycle. It is natural to expect \(d_{\phi}\) defines a Lie superalgebra structure, namely \begin{equation*}
\BktT{\alpha}{\beta}= \parity{a} d_{t} ( \alpha \wedge \beta ) = \BktU{}{\alpha}{\beta} + {\alpha} \wedge(t\phi)\wedge {\beta} \end{equation*} will be a super bracket for each \(t\). Super symmetry holds good. About super Jacobi identity,
\begin{align*} & \quad \BktT{ \BktT{\alpha}{\beta}}{\gamma} \\ & = \Pkt{\Pkt{\alpha}{\beta}}{\gamma} + \parity{a+1} d ( \alpha\wedge\beta \wedge t \phi \wedge \gamma) + \parity{a} d({\alpha}\wedge{\beta}) \wedge t \phi \wedge \gamma \\ &= \Pkt{\Pkt{\alpha}{\beta}}{\gamma} + \parity{b+c+1} \alpha\wedge\beta \wedge d \gamma\wedge t \phi + \parity{b+1} \alpha\wedge\beta \wedge \gamma\wedge d( t \phi) \\ &= \Pkt{\Pkt{\alpha}{\beta}}{\gamma} + \parity{b+c+1} \alpha\wedge\beta \wedge d \gamma\wedge t \phi \quad \text{if \(\phi\) is closed.}
\\ &\quad \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a'c'} \BktT{ \BktT{\alpha}{\beta}}{\gamma} \\ &=\parity{a'c'} \Pkt{\Pkt{\alpha}{\beta}}{\gamma} + \parity{a'c'+b+c+1} \alpha\wedge\beta \wedge d \gamma\wedge t \phi
+ \parity{a'c'+ b+1} \alpha\wedge\beta \wedge \gamma\wedge d( t \phi) \\&= \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a c } d( \alpha) \wedge\beta \wedge d \gamma \wedge t \phi + 3 \parity{ac+a+b+c } \alpha \wedge \beta \wedge \gamma \wedge d( t \phi ) \\&= \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a c } d( \alpha) \wedge\beta \wedge d \gamma \wedge t \phi \quad \text{if \(\phi\) is closed.} \end{align*}
So far, there is no affirmative statement in general setting.
On the other hand, there is a result of deformation of the Schouten bracket by
D.~Iglesias and J.~C.~Marrero
in \cite{Igles:Marrero}. They say for a 1-cocycle \(\phi\),
\begin{equation} \Sbt{P}{Q}^{\phi} = \Sbt{P}{Q} +\parity{p} P(\phi) \wedge (q-1) Q + (p-1)P \wedge Q(\phi) \quad \text{ where }\quad P(\phi) = \iota_{\phi} P \label{defn:deformedSbt} \end{equation}
satisfies the axioms of bracket of Lie superalgebra. \begin{enumerate}
\item \( \SbtPhi{P}{Q} = - \parity{(p-1)(q-1)} \SbtPhi{Q}{P} \) \item \( \SbtPhi{P}{ \SbtPhi{Q}{R} } = \SbtPhi{\SbtPhi{P}{Q}}{R} + \parity{(p-1)(q-1)} \SbtPhi{Q}{\SbtPhi{P}{R}} \) \; . \kmcomment{ \item \( \SbtPhi{P}{ Q \wedge R} = \SbtPhi{P}{Q} \wedge {R} + \parity{(p-1)q} Q \wedge \SbtPhi{P}{R} + \parity{p} P(\phi) \wedge Q \wedge R \)
\item \(\SbtPhi{f}{P} = - P(d_{\phi} f)\) for \(f \in C^{\infty}(M)\)\; . } \end{enumerate}
Inspired by the work above, in this note for a given 1-form \(\phi\), we fix properties of a function \(F\) so that \begin{equation} \parity{a} d ( \alpha \wedge \beta ) + F(a,b)
{\alpha} \wedge (t \phi)\wedge {\beta} \quad \quad ( \alpha \in \cgaiseki{a}, \; \beta \in \cgaiseki{b}) \; \label{deform:bkt:forms}
\end{equation} becomes super bracket for \(t\). The function \(F\) should be defined on \( \{ (a,b) \in \ensuremath{\mathbb{Z}}^{2}\mid a+b \leqq \textrm{dim} M - 1 \}\).
Main results in this note is that there are deformations of two super brackets on the space \( \mathfrak{h} = \sum_{q=0}^{\textrm{dim} M} \cgaiseki{q} \), and there is a natural extension to a subalgebra of \(\tbdl{M}\).
Claim 1: A deformation of the trivial bracket: For a given 1-form \(\phi\) \begin{equation*} \BktU{t,\phi }{ \alpha }{\beta } = F(a,b)
{\alpha} \wedge (t \phi)\wedge {\beta} \quad \quad ( \alpha \in \cgaiseki{a}, \; \beta \in \cgaiseki{b}) \;
\end{equation*} is a super bracket on \(\mathfrak{h}\) when \(F\) is a symmetric function.
Claim 2: A deformation of the standard bracket: For a given closed 1-form \(\phi\), \begin{equation*} \BktU{t,\phi}{\alpha}{\beta} = \parity{a}d({\alpha}\wedge{\beta}) + \tfrac{a+b+2}{2} \alpha \wedge t \phi \wedge \beta \quad \quad ( \alpha \in \cgaiseki{a}, \; \beta \in \cgaiseki{b}) \; \end{equation*} is a super bracket on \(\mathfrak{h}\).
Claim 3: An extension of the
deformation of the standard bracket: For a given closed 1-form \(\phi\), Claim 2 says \(\mathfrak{h},\BktU{t,\phi}{\cdot}{\cdot } \) are Lie superalgebra. Let \(\frakgN{0} ' = \{ X\in\tbdl{M} \mid \Lb{X}\phi = 0 \} \), which is a subalgebra of \(\tbdl{M}\). Then \((\mathfrak{h},\BktU{t,\phi}{\cdot}{\cdot } ) \oplus\frakgN{0}'\) becomes a Lie superalgebra naturally by the Lie derivative for each \(t\).
Based on these results, there are many issues to be studied. We would like to develop homology theory of deformed superalgebra.
When a Lie group \(G\) acts on \(M\), we get \( \sum_{p=1}^{\textrm{dim} M} \Lambda^{p} \tbdli{M}{G} \) of \(G\)-invariant multivector fields, and \( \sum_{q=0}^{\textrm{dim} M} \Lambda^{q}\cbdli{M}{G} \) of \(G\)-invariant differential forms. Since the action preserves the Jacobi-Lie bracket, the Schouten bracket is preserved by the action. Also the action commutes with the differentiation \(d\), the Lie superalgebra bracket is preserved by the action. In short, \( \sum_{p=1}^{\textrm{dim} M} \Lambda^{p} \tbdli{M}{G} \) and \( \sum_{q=0}^{\textrm{dim} M} \Lambda^{q}\cbdli{M}{G} \) have Lie superalgebra structures. The simplest case is a Lie group acts on itself. \(\overline{\mathfrak{g}} = \sum_{p=1}^{n} \Lambda^{p} \mathfrak{g}\) where \(\mathfrak{g} =\)Lie algebra of \(G\), has a Lie superalgebra structure by the Schouten bracket (cf.\cite{Mik:Miz:superLowDim}). The differential \(d\) gives
a Lie superalgebra structure on \(\overline{\mathfrak{h}} = \sum_{p=0}^{n} \Lambda^{p} \mathfrak{h}\), where \( \mathfrak{h} = \mathfrak{g}^{*} \) (cf.\ \cite{Mik:Miz:superForms}). Concrete and fancy examples are presented from those superalgebras.
\section{Deformation from the trivial bracket} In this section, we study deformed super bracket of the trivial bracket on \(\mathfrak{h}\), namely \begin{equation} \label{def:def:from:triv} \BktU{t,\phi }{ \alpha }{\beta } = F(a,b)
{\alpha} \wedge (t \phi)\wedge {\beta} \quad \quad ( \alpha \in \cgaiseki{a}, \; \beta \in \cgaiseki{b}) \; .
\end{equation} Then the symmetric property of \(F\) implies the super symmetric property of \( \BktU{t,\phi }{ \cdot }{\cdot }\) because of \begin{equation*} \BktU{t,\phi }{ \alpha }{\beta } + \parity{ (1+a)(1+b) } \BktU{t,\phi }{\beta } { \alpha } = (F(a,b) - F(b,a)) \alpha \wedge (t \phi) \wedge \beta \;. \end{equation*}
The super Jacobi identity holds automatically because \begin{align*} \BktU{t,\phi }{ \BktU{t,\phi }{ \alpha }{\beta } } {\gamma} &= \BktU{t,\phi }{ F(a,b) { \alpha } \wedge (t \phi) \wedge {\beta } } {\gamma} = F(a,b) F(a+b+1,c) { \alpha } \wedge (t \phi) \wedge {\beta } \wedge (t\phi) \wedge \gamma = 0 \; . \end{align*} \begin{prop} The bracket \eqref{def:def:from:triv} is a super bracket on \(\mathfrak{h} \) when \(F\) is a symmetric function on \( \{ (a,b) \in \ensuremath{\mathbb{Z}}^{2}\mid a+b \leqq \textrm{dim} M - 1 \}\). \end{prop} \begin{remark} In the proposition above, \(\phi\) is not necessarily closed.
Like nilpotent subalgebras of a Lie algebra, \( \BktU{t,\phi} { \BktU{t,\phi} {\mathfrak{h}} {\mathfrak{h}}} {\mathfrak{h}} = 0\) holds. In contrast, \( \parity{a' c'} \BktU{ }{ \BktU{ }{ \alpha }{\beta } }{\gamma } = \parity{a' c'} \alpha \wedge d\beta \wedge \gamma - \parity{b' a'} \beta \wedge d\gamma \wedge d \alpha \) holds for \( \BktU{}{\alpha }{\beta } = \parity{a} d( \alpha \wedge \beta )\). \end{remark}
We already know that the superalgebra \(\mathfrak{h} = \sum_{i=0}^{\textrm{dim} M} \Lambda^{i} \cbdl{M} \) with the bracket \( \Pkt{\alpha}{\beta} = \parity{a} d ( \alpha\wedge\beta)\) has an extension by \(\frakgN{0} = \tbdl{M} \) through Lie derivative in \cite{Mik:Miz:superForms}. Here we study the deformed superalgebra given by the bracket \eqref{def:def:from:triv} has an extension by a subalgebra of \(\tbdl{M}\) through the Lie derivative \( \Lb{X} = \iota_{X}\circ d + d \circ \iota_{X}\) with respect to \(X\).
Let \(F(a,b)\) be a symmetric function on \(\ensuremath{\mathbb{Z}}^{2}\) and \(\phi\) be a 1-form on \(M\) and \begin{equation} \BktU{t,\phi}{\alpha}{\beta} = F(a,b) \alpha \wedge t \phi \wedge \beta \quad \quad ( \alpha \in \cgaiseki{a}, \; \beta \in \cgaiseki{b}) \;. \end{equation} For 1-vector field \(X\), we define \( \BktU{t,\phi} {X}{\alpha} = \Lb{X}\alpha\) and
\( \BktU{t,\phi} {\alpha}{X} = - \Lb{X}\alpha\). So the super symmetry holds
good. About super Jacobi identity, we check two cases: one is \(X,Y, \alpha\) and the other is \(X, \alpha, \beta\). The first case the super Jacobi identity is just the formula \( \Lb{\Sbt{X}{Y}} = \Lb{X}\Lb{Y} - \Lb{Y} \Lb{X} \).
So the super symmetry holds good. We treat the other case: Since \begin{align*} & \quad
\BktU{t,\phi}{X}{ \BktU{t,\phi}{\alpha}{\beta}}
- \BktU{t,\phi}{ \BktU{t,\phi}{X}{\alpha}}{\beta}
- \BktU{t,\phi}{\alpha} { \BktU{t,\phi}{X}{\beta}}
\\ &= t F(a,b) \Lb{X} ( \alpha \wedge \phi \wedge \beta )
- t F(a,b) \Lb{X}\alpha \wedge \phi \wedge \beta
- t F(a,b) \alpha \wedge \phi \wedge \Lb{X}{\beta}
\\&= t F(a,b)
\alpha \wedge ( \Lb{X} \phi) \wedge \beta \end{align*} we see that \( \Lb{X} \phi = 0 \) is an efficient condition for super Jacobi identity. \begin{prop} Let \(\frakgN{0}' = \{ X\in \frakgN{0} \mid
\Lb{X} \phi = 0 \} \), which is a subalgebra of \(\frakgN{0}\). Then
( \( \mathfrak{h} \oplus \frakgN{0}', \BktU{t,\phi}{\cdot}{\cdot} \)) is
a deformed superalgebra.
If \(\phi\) is a 1-cocycle, then
( \( \mathfrak{h} \oplus \frakgN{0}'', \BktU{t,\phi}{\cdot}{\cdot} \)) is
a deformed superalgebra, where
\(\frakgN{0}'' = \{ X\in \frakgN{0} \mid
\inn{X} \phi = 0 \} \), which is a subalgebra of \(\frakgN{0}'\). \end{prop}
Some concrete example will appear in the tail of the next section.
\section{Deformation from the standard bracket} It is known in \cite{Mik:Miz:superForms} that \( \Pkt{\alpha}{\beta} = \parity{a} d ( \alpha\wedge \beta ) \) defines a super bracket on \(\sum_{i=0}^{\textrm{dim} M} \Lambda^{i} \cbdl{M} \). Looking at the deformed Schouten bracket, the bracket we expect is of form \( \BktU{t,\phi}{\alpha}{\beta} = \Pkt{\alpha} {\beta} + F(a,b) \alpha \wedge t \phi \wedge \beta\) for some function \(F\) on \( \{ (a,b) \in \ensuremath{\mathbb{Z}}^{2}\mid a+b \leqq \textrm{dim} M - 1 \}\).
About super symmetric property, we have \begin{equation} \BktU{t,\phi}{\alpha}{\beta}+ \parity{a' b'} \BktU{t,\phi}{\beta}{\alpha} = ( F(a,b) - F(b,a)) \alpha \wedge t \phi \wedge \beta \end{equation} About Jacobi identity, we see \begin{align*} &\quad \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a'c'}\left( \BktU{t,\phi}{ \BktU{t,\phi}{\alpha}{\beta}}{\gamma} -
\Pkt{ \Pkt{\alpha}{\beta}}{\gamma} \right) \\ &= \left( F (1+ b+c,a ) -F ( a,b ) -F ( b,c) -F ( c,a ) +F (1+ c+a,b ) \right)
\left( -1 \right) ^{ac+a+c} \left( \alpha \wedge t \phi\wedge\beta\wedge d
\left( \gamma \right) \right)
\\ &
- \left( -1 \right) ^{ac+a+c} \left( F (1+ a+b,c ) +F \left(1+ b+c,a \right) -F ( a,b
) -F \left( b,c \right) -F \left( c,a \right) \right)
\left( \alpha \wedge d \left( \beta \right) \wedge t \phi \wedge \gamma \right) \\ &
+ \left( F ( a,b ) +F ( b,c ) +F ( c,a )
\right) \left( -1 \right) ^{ac+a+b+c}
\left( \alpha \wedge d \left(t \phi \right) \wedge \beta \wedge \gamma \right) \\ & - \left( -1 \right) ^{ac+c}
\left( F (1+ a+b,c ) -F ( a,b ) -F ( b,c
) -F ( c,a ) +F (1+ c+a,b ) \right)
\left( d \left( \alpha \right) \wedge \beta \wedge t \phi \wedge\gamma \right)
\;.
\end{align*} Thus, if \(\phi\) is a 1-cocycle, then we get the following sufficient conditions for super Jacobi identity
\begin{align} 0&= F (1+ b+c,a ) +F (1+ c+a,b ) - F ( a,b ) -F ( b,c ) -F ( c,a ) \;, \label{eq:x:1} \\ 0&= F (1+ a+b,c ) +F (1+ b+c,a ) -F ( a,b ) -F ( b,c ) -F ( c,a ) \;, \label{eq:x:2}
\\ 0&= F(1+ c+a,b ) + F (1+ a+b,c )- F ( a,b ) -F ( b,c ) -F ( c,a ) \; . \end{align} The difference \( \eqref{eq:x:1} - \eqref{eq:x:2} =0 \) implies \( F (1+ c+a,b ) = F (1+ a+b,c )\). Putting \(c=0\), we have
\(F (1+ a,b ) = F (1+ a+b,0 ) \) and \(\ensuremath{\displaystyle } F(a,0) = F(0,0) ( \frac{a} {2}
+ 1)\), we see that the symmetric function \(F\) satisfying the above 3 conditions is \begin{equation} F(a,b) = \kappa (a+b+2)\ \ \text{where}\ \kappa \text{ is a constant.} \end{equation}
Assume 1-form \(\phi\) is not closed. Then we get the following sufficient conditions for super Jacobi identity
\begin{align}
0&= F(1+ b+c,a ) + F(1+ c+a,b ) \label{nococycle:a} \\ 0&= F(1+ a+b,c ) + F(1+ b+c,a ) \label{nococycle:c} \\ 0&= F( a,b ) + F( b,c ) +F( c,a ) \label{nococycle:z} \\ 0&= F(1+ c+a,b ) + F(1+ a+b,c ) \label{nococycle:b} \\\shortintertext{We conclude a symmetric \(F\) satisfying the 4 above conditions is trivial as follows. Putting \(c=0\) in \eqref{nococycle:z}} F(a,b) &= - G(a) - G(b) \quad \text{where}\ G(a) = F(0,a) = F(a,0)\;. \\\shortintertext{Applying this expression to \eqref{nococycle:z}, we have} 0 &= - 2 ( G(a)+ G(b) + G(c) )\;, \ \text{and so }\; G(a) = 0\;,\ F(a,b) = 0 \;, \ \text{i.e., trivial.} \end{align} We summarize above discussion. \begin{thm} The super symmetry of the bracket \eqref{deform:bkt:forms} \[\parity{a} d ( \alpha \wedge \beta ) + F(a,b)
{\alpha} \wedge (t \phi)\wedge {\beta} \]
yields \(F(a,b)\) is a symmetric function, i.e., \( F(a,b) = F(b,a) \).
The super Jacobi identity implies if \(\phi\) is not a cocycle, i.e., not exact then \( F(a,b) = 0\). If \(\phi\) is a cocycle, i.e., if \(\phi\) is exact then the super Jacobi identify yields \( F(a,b) = \kappa (a+b+2)\). \end{thm}
\kmcomment{ Super Jacobi identity yield \begin{align} 0& = \mathop{ \mathfrak{S}}_{a,b,c} \parity{a}F(a,b)- \parity{b}F(b,c+a+1)- \parity{c}F(c,a+b+1)\;. \end{align}
\parity{a}F(a,b)+ \parity{b}F(b,c)+ \parity{c}F(c,a)- \parity{b}F(b,c+a+1)- \parity{c}F(c,a+b+1)\;. kmcomment}
\begin{kmCor} Let \(\phi\) be an exact 1-form. \begin{equation} \BktU{t,\phi}{\alpha}{\beta} = \Pkt{\alpha}{\beta} + \tfrac{a+b+2}{2} \alpha \wedge t \phi \wedge \beta \label{defn:def:bkt:forms} \end{equation} where \(\Pkt{\alpha}{\beta} = \parity{a}d({\alpha}\wedge{\beta}) \), \( \alpha \in \Lambda^{a}\cbdl{M}\) and \( \beta \in \Lambda^{b}\cbdl{M}\). This bracket satisfies super symmetry and super Jacobi identity, and the space \(\mathfrak{h} \) with this bracket is a Lie superalgebra. \end{kmCor} \begin{remark} This bracket is a super bracket from Theorem above. If one proceed to reconfirm that this bracket is a super bracket, it is one page exercise. Symbol calculus, Maple has a package \texttt{difforms} and it is helpful to study of properties of this bracket. \end{remark}
\kmcomment{ \textbf{Proof:} For simplicity, we denote \( t \phi\) by \(\phi\) here. \begin{align*} \BktU{t,\phi}{\alpha}{\beta}+ \parity{a' b'} \BktU{t,\phi}{\beta}{\alpha} &= \Pkt{\alpha}{\beta} + \tfrac{a'+b'}{2}\alpha \wedge \phi \wedge \beta + \parity{a' b'} \Pkt{\beta}{\alpha} + \parity{a' b'} \tfrac{a'+b'}{2} \beta \wedge \phi \wedge \alpha \\ &=(\Pkt{\alpha}{\beta}+ \parity{a' b'} \Pkt{\beta}{\alpha}) + \tfrac{a'+b'} {2}( \parity{ b+ab+a }+ \parity{a' b'})\beta \wedge \phi \wedge \alpha \\ &= 0 \;. \end{align*} \begin{align*} \BktU{t,\phi}{ \BktU{t,\phi}{\alpha}{\beta}}{\gamma} & = \Pkt{ \BktU{t,\phi}{\alpha}{\beta}} {\gamma} + \tfrac{a'+b'+c'}{2} \BktU{t,\phi}{\alpha}{\beta} \wedge \phi \wedge \gamma \\ &= \Pkt{ \Pkt{\alpha}{\beta} + \tfrac{a'+b'}{2} \alpha \wedge \phi \wedge \beta} {\gamma} + \tfrac{a'+b'+c'}{2} ( \Pkt{\alpha}{\beta} + \tfrac{a'+b'}{2} \alpha \wedge \phi \wedge \beta) \wedge \phi\wedge \gamma \\ &= \Pkt{\Pkt{\alpha}{\beta}}{\gamma} + \tfrac{a'+b'}{2} \Pkt{\alpha\wedge\phi\wedge\beta}{\gamma} + \tfrac{a'+b'+c'}{2} \parity{a} d({\alpha}\wedge{\beta}) \wedge \phi\wedge \gamma \\ &= \Pkt{\Pkt{\alpha}{\beta}}{\gamma} +\tfrac{a'+b'}{2} \parity{a+1+b} d ( \alpha\wedge\phi \wedge\beta \wedge \gamma) +\tfrac{a'+b'+c'}{2} \parity{a} d({\alpha}\wedge{\beta}) \wedge \phi\wedge \gamma \\ &= \Pkt{\Pkt{\alpha}{\beta}}{\gamma} +\tfrac{a'+b'}{2} \parity{1+b'} \phi \wedge d(\alpha\wedge\beta \wedge \gamma) +\tfrac{a'+b'+c'}{2} \parity{b'} \phi \wedge d(\alpha\wedge\beta)\wedge \gamma \end{align*} Since \(\ensuremath{\displaystyle } \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a'c'} \Pkt{ \Pkt{\alpha}{\beta}}{\gamma} = 0 \), in order to show \(\ensuremath{\displaystyle } \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a'c'} \BktU{t,\phi}{ \BktU{t,\phi}{\alpha}{\beta}}{\gamma}\) vanishes, we treat the cycle sum of the rest of the last equation after ignoring \( \phi \wedge\). Namely \begin{align*} \text{Final target} &= \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a' c'} \left(\tfrac{a'+b'}{2} \parity{1+b'} d(\alpha\wedge\beta \wedge \gamma) +\tfrac{a'+b'+c'}{2} \parity{b'} d(\alpha\wedge\beta)\wedge \gamma \right) \\ &= \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a' c'} \tfrac{a'+b'}{2} \parity{1+b'} \left(d(\alpha\wedge\beta) \wedge \gamma + \parity{a'+b'} \alpha\wedge\beta \wedge d \gamma\right) \\& + \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a' c'} \tfrac{a'+b'+c'}{2} \parity{b'} d(\alpha\wedge\beta)\wedge \gamma \\ &= \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a' c'} \tfrac{a'+b'}{2} \parity{1+a'} \alpha\wedge\beta \wedge d \gamma + \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a' c'} \tfrac{c'}{2} \parity{b'} d(\alpha\wedge\beta)\wedge \gamma \\ &= \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a' c'} \tfrac{a'+b'}{2} \parity{1+a'} \alpha\wedge\beta \wedge d \gamma \\&\quad + \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a' c'} \tfrac{c'}{2} \parity{b'}\left((d \alpha) \wedge\beta + \parity{a}\alpha\wedge
d \beta\right) \wedge \gamma \\ &= \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a' c'} \tfrac{a'+b'}{2} \parity{1+a'} \alpha\wedge\beta \wedge d \gamma \\&\quad + \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a' c'} \tfrac{c'}{2} \parity{b'}\left((d \alpha) \wedge\beta \right) \wedge \gamma
+ \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a' c'} \tfrac{c'}{2} \parity{b'}\left(\parity{a}\alpha\wedge
d \beta\right) \wedge \gamma \\ &= \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a' c'} \tfrac{a'+b'}{2} \parity{1+a'} \alpha\wedge\beta \wedge d \gamma \\&\quad + \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a' c'} \tfrac{c'}{2} \parity{b'+ a'(b'+c')}\beta \wedge \gamma \wedge d \alpha
+ \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a' c'} \tfrac{c'}{2} \parity{b'+ a+b' c+ ac }\gamma\wedge \alpha\wedge d \beta \\ &= \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a' c'} \tfrac{a'+b'}{2} \parity{1+a'} \alpha\wedge\beta \wedge d \gamma \\&\quad + \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{c' b'} \tfrac{b'}{2} \parity{a'+ c'(a'+b')}\alpha \wedge \beta \wedge d \gamma
+ \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{b' a'} \tfrac{a'}{2} \parity{c'+ b+c' a+ ba }\alpha\wedge \beta\wedge d \gamma \\ &= \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a' c'} \tfrac{a'+b'}{2} \parity{1+a'} \alpha\wedge\beta \wedge d \gamma \\& \quad + \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a' c'} \tfrac{b'}{2} \parity{a'}\alpha \wedge \beta \wedge d \gamma
+ \mathop{\mathfrak{S}}_{\alpha,\beta,\gamma} \parity{a' c'} \tfrac{a'}{2} \parity{a'}\alpha\wedge \beta\wedge d \gamma \\&= 0 \;. \end{align*} \qed }
\begin{exam} Take a 2-dimensional Lie algebra with the Lie bracket relations \( \Sbt{\yb{1}}{\yb{2}} = \yb{1}\) and the \( \zb{1}, \zb{2}\) is the dual basis so that \( d \zb{1} = - \zb{1} \wedge \zb{2}\) and \( d \zb{2} = 0\). The chain complex of weight \(-3\) is given by \( \wtedC{1}{-3} = \ensuremath{\mathbb{R}}(\zb{1}\wedge \zb{2})\), \( \wtedC{2}{-3} = \ensuremath{\mathbb{R}}(\zb{1}\bigtriangleup 1)+ \ensuremath{\mathbb{R}} (\zb{2}\bigtriangleup 1)\), \( \wtedC{3}{-3} = \ensuremath{\mathbb{R}}(1\bigtriangleup 1 \bigtriangleup 1) = \ensuremath{\mathbb{R}} ( \bigtriangleup^{3} 1 )\). We refer to the appendix or \cite{Mik:Miz:superForms} about odd notations. Fix \(\phi = \zb{2}\). \(\partial (\zb{1}\wedge \zb{2}) = 0\) for 1-chain. \begin{align*}
\partial (\zb{j}\bigtriangleup 1 )& = \BktU{t, \zb{2}}{\zb{j}}{1} = \begin{cases*} c_{1} \delta_{j}^{1} t \zb{1} \wedge \zb{2} & \(c_{1}\) is constant, deform of trivial \\ \delta_{j}^{1} ( 1+ \frac{3}{2}t ) \zb{1} \wedge \zb{2} & deform of standard \end{cases*} \\ \partial ( \bigtriangleup ^{3} 1 ) &= \tbinom{3}{2} \BktU{t, \zb{2}}{1}{1} \bigtriangleup 1 = \begin{cases*} c_{0} t \zb{2}\bigtriangleup 1 & \(c_{0}\) is constant, deform of trivial \\
t \zb{2} \bigtriangleup 1 & deform of standard \end{cases*}
\end{align*} We summarize the kernel dimensions and Betti numbers as follows, we assume \( c_{0} c_{1} \ne 0\)).
\renewcommand{0.7}{0.8} \[
\begin{array} {c | *{3}{c}} \text{trivial} & 1 & 2 & 3 \\\hline \textrm{dim} & 1 & 2 & 1 \\\hline \ker \textrm{dim} & 1 & 1 + \delta_{t}^0 & \delta_{t}^0 \\ \text{Betti} & \delta_{t}^0 & 2 \delta_{t}^0 & \delta_{t}^0 \end{array}
\hspace{20mm}
\begin{array} {c | *{3}{c}} \text{standard} & 1 & 2 & 3 \\\hline \textrm{dim} & 1 & 2 & 1 \\\hline \ker \textrm{dim} & 1 & 1 + \delta_{2+3t}^0 & \delta_{t}^0 \\ \text{Betti} & \delta_{2+3t}^0 & \delta_{2+3t}^0 + \delta_{t}^0 & \delta_{t}^0 \end{array} \] \renewcommand{0.7}{1.0}
\label{exam:w3:dim2} \end{exam}
\subsection{An extension}
We mentioned before that the superalgebra \(\mathfrak{h} = \sum_{i=0}^{\textrm{dim} M} \Lambda^{i} \cbdl{M} \) with the bracket \( \Pkt{\alpha}{\beta} = \parity{a} d ( \alpha\wedge\beta)\) has an extension by \(\tbdl{M} \) through Lie derivation in \cite{Mik:Miz:superForms}. Here we study two deformed superalgebras have an extension by a subalgebra of \( \frakgN{0} = \tbdl{M} \) through the Lie derivative \( \Lb{X} = \iota_{X}\circ d + d \circ \iota_{X}\) with respect to \(X\).
The superalgebra \(\mathfrak{h} \) has the deformed bracket \( \BktU{t, \phi} {\alpha}{\beta} = \parity{a} d ( \alpha\wedge\beta)+ \frac{a+b+2}{2} \alpha \wedge t \phi \wedge \beta \), where \(\phi\) is a 1-cocycle.
Let \( \Ekt{\cdot}{\cdot} \) be a candidate of superbracket on \(\mathfrak{h} \oplus \tbdl{M}\), i.e., \( \Ekt{\alpha}{\beta} = \BktU{t,\phi}{\alpha}{\beta} \) , \( \Ekt{X}{Y} = \Sbt{X}{Y} \), \( \Ekt{X}{\beta} = - \Ekt{\beta}{X} = \Lb{X}{\beta} \) for forms \(\alpha, \beta\) and 1-vectors \(X,Y\). We have to check super Jacobi identity for two cases. Again we abbreviate \(t\phi\) by \(\phi\). One case is all right as following. \begin{align*} &\quad \Ekt{ \Ekt{X}{Y} }{\alpha} +\Ekt{ \Ekt{Y}{\alpha} }{X} +\Ekt{ \Ekt{\alpha}{X} }{Y} \\=& \Ekt{ \Sbt{X}{Y} }{ \alpha } - \Lb{X} \Lb{Y} \alpha + \Lb{Y} \Lb{X} \alpha =(\Lb{ \Sbt{X}{Y} } - \Lb{X} \Lb{Y} + \Lb{Y} \Lb{X}) \alpha = 0 \; . \end{align*} We try the other. \begin{align*} &\quad \Ekt{X}{\Ekt{\alpha}{\beta} } +\Ekt{\alpha}{ \Ekt{\beta}{X } } +\parity{b'a'}\Ekt{\beta}{\Ekt{X}{\alpha} } \\ =& \Lb{X} \Ekt{\alpha}{\beta}
+ \parity{a} d( \alpha \wedge \Ekt{\beta}{X} )
+ \frac{a'+b'}{2} \alpha \wedge \phi \wedge \Ekt{ \beta}{X} \\ & + \parity{1+a' b'} \left( \parity{b'} d ( \beta \wedge \Lb{X}{\alpha} ) - \frac{a'+b'}{2} \beta\wedge \phi \wedge \Lb{X}{\alpha} \right) \\ =& \Lb{X}( \parity{a} d( \alpha\wedge\beta ) + \frac{a'+b'}{2} \alpha\wedge \phi\wedge \beta)
+ \parity{a+1} d( \alpha \wedge \Lb{X}\beta )
\\& - \frac{a'+b'}{2} \alpha \wedge \phi \wedge \Lb{X}\beta
+ \parity{1+a' b'} \left( \parity{b'} d ( \beta \wedge \Lb{X}{\alpha} ) - \frac{a'+b'}{2} \beta\wedge \phi \wedge \Lb{X}{\alpha} \right) \\ =& \parity{a} d \Lb{X} ( \alpha\wedge\beta ) + \frac{a'+b'}{2}\Lb{X}(\alpha\wedge \phi\wedge \beta) \\ & + \parity{a+1} d( \alpha \wedge\Lb{X}\beta ) - \frac{a'+b'}{2} \alpha \wedge \phi \wedge (\Lb{X}\beta ) \\ & + \parity{1+a b'}
d ( \beta \wedge \Lb{X}{\alpha} ) + \parity{a' b'}\frac{a'+b'}{2} \beta\wedge \phi \wedge ( \Lb{X}{\alpha} + \inner{X}{00} \alpha ) \\ =& + \frac{a'+b'}{2}\Lb{X}(\alpha\wedge \phi\wedge \beta)
+ \parity{a' b'}\frac{a'+b'}{2} \beta\wedge \phi \wedge
\Lb{X}{\alpha} - \frac{a'+b'}{2} \alpha \wedge \phi \wedge (\Lb{X}\beta ) \\ & + \parity{a} d \Lb{X}( \alpha\wedge\beta ) + \parity{a+1} d( \alpha \wedge\Lb{X}\beta )
+ \parity{a+1 }
d ( \Lb{X}{\alpha} \wedge \beta ) \\ =& + \frac{a'+b'}{2}\Lb{X}(\alpha\wedge \phi\wedge \beta) + \parity{a' b'}\frac{a'+b'}{2}(\beta\wedge \phi \wedge
\Lb{X}{\alpha} ) - \frac{a'+b'}{2}(\alpha \wedge \phi \wedge \Lb{X}\beta ) \\ =& + \frac{a'+b'}{2}\left(\Lb{X}(\alpha\wedge \phi\wedge \beta) - \Lb{X}{\alpha}\wedge \phi \wedge \beta - \alpha \wedge \phi \wedge \Lb{X}\beta \right) \\ =& + \frac{a'+b'}{2}\left(\alpha\wedge \Lb{X}(\phi) \wedge \beta \right) \end{align*} This vanishes if \(\Lb{X}\phi=0\). We see that \(\{ X\in \tbdl{M} \mid \Lb{X}\phi=0\}\) forms a \(\ensuremath{\mathbb{R}}\) subalgebra of \(\tbdl{M}\). Thus we have the following result.
\begin{thm}
The superalgebra \((\mathfrak{h},\BktU{t,\phi}{\cdot}{\cdot}) \) allows an extension by \(\frakgN{0}' = \{ X\in \tbdl{M} \mid \Lb{X}\phi=0\}\), namely, \( \mathfrak{h} \oplus \frakgN{0}' \) becomes a Lie superalgebra extension of \(\mathfrak{h}\). \end{thm}
\begin{exam} We extend \(\mathfrak{h}\) in Example \ref{exam:w3:dim2} by \( \frakgN{0}\) and show the \((- 3)\)-weighted chain complex. We denote \( \yb{1}\bigtriangleup \yb{2}\) by \(U\), and
\( \zb{1}\wedge \zb{2}\) by \(V \).
\[
\begin{array}[t] {c|*{5}c | c} \wtedCR{m}{3} & 1 & 2 & 3 & 4 & 5 & \text{ if }\\\hline \textrm{dim} & 1 & 4 & 6 & 4 & 1 \\\hline \text{basis} & V & { \renewcommand{0.7}{0.7} \begin{array}{c} \zb{j}\bigtriangleup 1 \\ \yb{i}\bigtriangleup V \end{array} } & { \renewcommand{0.7}{0.7} \begin{array} {c} 1^{3} \\ \yb{i}\bigtriangleup \zb{j} \bigtriangleup 1 \\ U \bigtriangleup V \end{array} } & { \renewcommand{0.7}{0.7} \begin{array}{c} \yb{i}\bigtriangleup 1^{3} \\ U \bigtriangleup \zb{j} \bigtriangleup 1 \end{array} } & U \bigtriangleup 1^{3} \\\hline \ker\textrm{dim} & 1 & 3 & \begin{array}{c} 4 \\ 3 \end{array}
& \begin{array}{c} 2 \\ 1 \end{array}
& 0 & \begin{array}{c} t(1+3t/2) = 0 \\ t(1+3t/2) \ne 0 \end{array} \\
\hline \text{Betti} & 0 & \begin{array}{c} 1 \\ 0 \end{array}
& \begin{array}{c} 2 \\ 0 \end{array}
& \begin{array}{c} 1 \\ 0 \end{array}
& 0 & \begin{array}{c} t(1+3t/2) = 0 \\ t(1+3t/2) \ne 0 \end{array} \\ \end{array} \]
By the direct computation, we see that the boundary image is spanned as follows. \begin{align*} \partial \wtedC{1}{-3} & = \{ 0 \}\;, \qquad \partial \wtedC{2}{-3} = \{ (1+3t/2) V, V \}\\ \partial \wtedC{3}{-3} &= \{ 3t \zb{2}\bigtriangleup 1,
\zb{2}\bigtriangleup 1 + (1+3t/2) \yb{1} \bigtriangleup V,
\zb{1}\bigtriangleup 1 - (1+3t/2) \yb{2} \bigtriangleup V \}\;, \\ \partial \wtedC{4}{-3} & = \{ t \yb{1} \bigtriangleup \zb{2}\bigtriangleup 1, t \yb{2} \bigtriangleup \zb{2}\bigtriangleup 1, - \yb{2} \bigtriangleup \zb{2}\bigtriangleup 1 + (1+3t/2) \yb{1} \bigtriangleup \yb{2} \bigtriangleup V,
\yb{1} \bigtriangleup \zb{2}\bigtriangleup 1\}\;, \\ \partial \wtedC{5}{-3} & = \{ \yb{1} \bigtriangleup^{3} 1 + 3t \yb{1} \bigtriangleup \yb{2} \bigtriangleup \zb{2} \bigtriangleup 1 \}\; . \end{align*} Thus, the kernel dimensions of \(\partial\) for m=1,3,5 are 1,3,0 and those of m = 3,4 are 4,2 if \(t(1+3t/2) = 0\) else 3,1. Finally the Betti numbers are 0,1,2,1, 0 if \(t(1+3t/2) = 0\) else 0,0,0,0,0 .
To get the weighted homology groups of Lie algebras, even of 2-dimensional, we need hard work. We prepare reporting the general weighted homology groups of 2-dimensional Lie algebra. \end{exam}
In \cite{Mik:Miz:super2} and \cite{Mik:Miz:super3}, we introduced double weight for the algebra of homogeneous polynomial coefficient multi-vector fields on \(\ensuremath{\mathbb{R}}^{n}\). By the similar way, we get examples of double weighted super algebras of homogeneous polynomial coefficient forms and 1-vector fields on \(\ensuremath{\mathbb{R}}^{n}\) in \cite{Mik:Miz:superForms}. It may be interesting to study those deformed double weighted superalgebras and their homology groups.
\appendix \defAppendix: \Alph{section}{Appendix: \Alph{section}}
\section{Quick review of the homology groups of Lie superalgebra}
Let \( \mathfrak{g} = \sum_{i\in\ensuremath{\mathbb{Z}}} \frakgN{i} \) be a Lie superalgebra. From super symmetry \( \Sbt{X}{Y} + \parity{ x y } \Sbt{Y}{X} = 0 \) for \(
X \in \frakgN{x}, Y \in \frakgN{y} \), \(m\)-th chain space is given by \(\myCS{m} = \otimes ^{m} \mathfrak{g} /\text{Ideal of} ( {X} \otimes {Y} + \parity{ x y } {Y} \otimes {X} )\). We denote the class of \( A_{1} \otimes \cdots \otimes A_{p}\) by \( A_{1} \bigtriangleup \cdots \bigtriangleup A_{p}\).
Let \( \widetilde{A} = A_{1} \bigtriangleup \cdots \bigtriangleup A_{p}\) and
\( \widetilde{B} = B_{1} \bigtriangleup \cdots \bigtriangleup B_{q}\).
The boundary operator \(\ensuremath{\displaystyle } \partial_{} :\myCS{m}\to \myCS{m-1}\) called (\textit{boundary homomorphism}) is defined by \begin{align}
\partial ( Y_{1}\bigtriangleup \cdots \bigtriangleup Y_{m} ) &= \sum_{i<j} (-1)^{ i-1 + y_{i}(\mathop{\sum}_{i< s<j} y_{s}) } Y_{1} \bigtriangleup \cdots \widehat{ Y_{i} } \cdots \bigtriangleup \underbrace{\Sbt{Y_{i}}{Y_{j}}}_{j} \bigtriangleup \cdots \bigtriangleup Y_{m} \label{triv:1} \end{align} for a decomposable element, where \(\ensuremath{\displaystyle } y_{i}\) is the degree of homogeneous element
\(Y_{i}\), i.e., \(\ensuremath{\displaystyle } Y_{i} \in \mathfrak{g}_{y_{i}} \). It is clear that \(\ensuremath{\displaystyle } \partial\circ \partial = 0\) and we have the homology groups \(\ensuremath{\displaystyle } \myHom{m}(\mathfrak{g}, \ensuremath{\mathbb{R}}) = \ker(\partial : \myCS{m} \rightarrow \myCS{m-1})/ \partial ( \myCS{m+1} ) \).
We say
a non-zero \(m\)-th decomposable element
\( Y_{1}\bigtriangleup \cdots \bigtriangleup Y_{m} \) has the weight
\( \sum_{i=1}^{m} \yb{i} \) where \(\ensuremath{\displaystyle } Y_{i} \in \frakgN{\yb{i}} \). The weight is preserved by \(\partial\), i.e., \( \partial ( \wtedC{m}{w} ) \subset \wtedC{m-1}{w}\) where
\(\wtedC{m}{w} = \) the subspace of \(w\)-weighted \(m\)-th chains and
we have the weighted homology groups.
\begin{align} \shortintertext{ If all $y_{i}$ are even in \eqref{triv:1}, then }
\partial ( Y_{1}\bigtriangleup \cdots \bigtriangleup Y_{m} ) & = - \sum_{i<j} (-1)^{ i+j }
\Sbt{Y_{i}}{Y_{j}} \bigtriangleup
Y_{1} \bigtriangleup \cdots\widehat{Y_{i}}\cdots \widehat{Y_{j}}\cdots \bigtriangleup Y_{m} \;. \label{all:even} \\ \shortintertext{ If all $y_{i}$ are odd in \eqref{triv:1}, then }
\partial ( Y_{1}\bigtriangleup \cdots \bigtriangleup Y_{m} ) & = \sum_{i<j}
\Sbt{Y_{i}}{Y_{j}} \bigtriangleup
Y_{1} \bigtriangleup \cdots\widehat{Y_{i}}\cdots \widehat{Y_{j}}\cdots \bigtriangleup Y_{m} \;. \label{all:odd} \end{align} \begin{defn} Let \(A =A_{1} \bigtriangleup \cdots \bigtriangleup A_{\bar{a}} \) (\( A_{i}\in \mathfrak{g}_{a_{i}}\)) and \(B =B_{1} \bigtriangleup \cdots \bigtriangleup B_{\bar{b}} \) ( \( B_{j}\in \mathfrak{g}_{b_{j}}\)).
Define \begin{equation} \SbtES{A}{B} = \partial ( A\bigtriangleup B ) - (\partial A)\bigtriangleup B - (-1)^{\bar{a}} A \bigtriangleup \partial B \;. \label{bdary:bunkai} \end{equation} \end{defn} It satisfies \begin{align} \SbtES{A}{B} &= \sum_{i,j} (-1)^{i+ a_{i} \sum\limits_{s>i} a_{s}+ j+ b_{j}(1+ \sum\limits_{s=1}^{j} b_{s})} A_{1} \bigtriangleup \cdots \widehat{ A_{i} }\cdots A_{\bar{a}} \bigtriangleup \Sbt{A_{i}}{B_{j}} \bigtriangleup B_{1} \bigtriangleup \widehat{ B_{j} } \cdots \bigtriangleup B_{\bar{b}}\;. \label{defn:super:Schouten} \\ \noalign{ If all \(a_{i}\) are even and all \(b_{j}\) are odd in \eqref{defn:super:Schouten}, then} \SbtES{A}{B} & = \sum_{i,j} (-1)^{i+1} A_{1}\bigtriangleup\cdots\widehat{ A_{i} }\cdots A_{\bar{a}} \bigtriangleup \Sbt{A_{i}}{B_{j}} \bigtriangleup B_{1} \bigtriangleup \cdots \widehat{ B_{j} }\cdots \bigtriangleup B_{\bar{b}}\;. \\\shortintertext{ Let \(C =C_{1} \bigtriangleup \cdots \bigtriangleup C_{\bar{c}} \) (\( C_{k}\in \mathfrak{g}_{c_{k}}\)) with \( c_{k}\) are all even. Then } \SbtES{A}{C\bigtriangleup B} & = \sum_{i,k} (-1)^{i+k} A_{1}\bigtriangleup\cdots\widehat{ A_{i} }\cdots A_{\bar{a}} \bigtriangleup \Sbt{A_{i}}{C_{k}} \bigtriangleup C_{1} \bigtriangleup \cdots \widehat{ C_{k} }\cdots \bigtriangleup C_{\bar{c}} \bigtriangleup B \\&\quad + \sum_{i,j} (-1)^{i+1} A_{1}\bigtriangleup\cdots\widehat{ A_{i} }\cdots A_{\bar{a}} \bigtriangleup C \bigtriangleup \Sbt{A_{i}}{B_{j}} \bigtriangleup B_{1} \bigtriangleup \cdots \widehat{ B_{j} }\cdots \bigtriangleup B_{\bar{b}} \;. \notag \\ & = \sum_{i,k} (-1)^{i+1} A_{1}\bigtriangleup\cdots\widehat{ A_{i} }\cdots A_{\bar{a}} \bigtriangleup C_{1} \bigtriangleup \cdots \bigtriangleup \Sbt{A_{i}}{C_{k}} \bigtriangleup \cdots \bigtriangleup C_{\bar{c}} \bigtriangleup B \\&\quad + \sum_{i,j} (-1)^{i+1} A_{1}\bigtriangleup\cdots\widehat{ A_{i} }\cdots A_{\bar{a}} \bigtriangleup C \bigtriangleup \Sbt{A_{i}}{B_{j}} \bigtriangleup B_{1} \bigtriangleup \cdots \widehat{ B_{j} }\cdots \bigtriangleup B_{\bar{b}} \;. \notag \end{align}
\nocite{Mik:Miz:super2} \nocite{Mik:Miz:super3}
\end{document} | arXiv |
1. Analysing Data
1.01 Types of data
1.02 Describing data
1.03 Associations between categorical variables
1.04 Associations between numerical variables
INVESTIGATION: Statistical investigation process
AustraliaAU
Bivariate data: explanatory and response variables
The goal of bivariate data analysis is see if two variables are associated in some way. The two variables that we study in bivariate statistics are called the explanatory variable and the response variable.
An experiment is to be conducted in which a researcher surveys men and women as to their preference for coffee or tea or neither. The researcher would then analyse the results to find whether gender is a predictor for hot beverage preference. Identify the explanatory and response variables in this experiment.
Think: To identify which is the explanatory variable and which is the response variable we consider the questions "Does gender explain hot beverage preference?" or "Does hot beverage preference explain the gender of a person?".
Do: So we have that
Explanatory variable: Gender
Response variable: Hot beverage preference
Explanatory Variable - the variable which we expect to explain or predict the value of the response variable.
Response Variable - the variable which we expect to respond to the value of the explanatory variable.
When displaying bivariate data graphically the explanatory variable is plotted on the horizontal axis (the $x$x-axis), and the response variable on the vertical axis (the $y$y-axis).
A single coordinate point in a bivariate data set might be written in the form $\left(x,y\right)$(x,y), and it would be understood that $x$x is the explanatory variable and $y$y is the response variable.
Plotting explanatory and response variables
Explanatory variable - plotted on the horizontal $x$x-axis
Response variable - plotted on the vertical $y$y-axis
Consider the following variables:
Temperature ($^\circ$°C)
Number of ice cream cones sold
Which of the following statements makes sense?
A change in temperature affects the number of ice cream cones sold.
A change in the number of ice cream cones sold affects the temperature.
Which is the explanatory variable and which is the response variable?
EV: number of ice cream cones sold
RV: temperature
EV: temperature
RV: number of ice cream cones sold
Which of the following images has the variables placed in the correct positions?
The scatter plot shows the relationship between sea temperature and the amount of healthy coral.
Which variable is the response variable?
Level of healthy coral
Which variable is the explanatory variable?
Analysing two-way frequency tables: row and column percentages
Two-way tables allow us to display and examine the relationship between two sets of categorical data. The categories are labelled at the top and the left side of the table, and the frequency of the different characteristics appear in the interior of the table. Often the totals of each row and column are also shown.
The following are the statistics of the passengers and crew who sailed on the Titanic on its fateful maiden voyage in 1912.
$202$202 $118$118 $178$178 $212$212 $710$710
$123$123 $167$167 $528$528 $696$696 $1514$1514
Row percentages table
Although it is interesting to know that $202$202 First Class passengers survived, it is far more useful to know the percentage break up of survivors from each class. To do this we need to calculate row percentages. To find the percentage divide the value in the table by the row total and multiply by $100%$100%. The calculations for the first row in the table are shown below.
Think: We need the fraction of first class survivors out of total number of survivors (row total) rounded to the nearest percentage.
Percentage of survivors that were first class $=$= $\frac{\text{first class survivors}}{\text{total number survivors}}$first class survivorstotal number survivors
$=$= $\frac{202}{710}$202710
$\approx$≈ $28%$28%
$\frac{202}{710}\times100%\approx28%$202710×100%≈28% $\frac{118}{710}\times100%\approx17%$118710×100%≈17% $\frac{178}{704}\times100%\approx25%$178704×100%≈25% $\frac{212}{710}\times100%\approx30%$212710×100%≈30% $100%$100%
$\frac{123}{1514}\times100%\approx8%$1231514×100%≈8% $11%$11% $35%$35% $46%$46% $100%$100%
Reflect: This percentage frequency table now gives us more useful information than the raw data. The first row gives us the percentage breakdown of survivors by class. We can now easily read that $46%$46% of the people who died were crew members whereas only $8%$8% of the people who died were in first class.
Column percentages table
We can also calculate the percentage in each class type that survived or died. To do this we calculate column percentages. To find the percentage divide the value in the table by the column total and multiply by $100%$100%. The calculations are shown in the table below:
$\frac{202}{325}\times100%\approx62%$202325×100%≈62% $\frac{118}{285}\times100%\approx41%$118285×100%≈41% $\frac{178}{706}\times100%\approx25%$178706×100%≈25% $\frac{212}{908}\times100%\approx23%$212908×100%≈23%
$\frac{123}{325}\times100%\approx38%$123325×100%≈38% $59%$59% $75%$75% $77%$77%
$100%$100% $100%$100% $100%$100% $100%$100%
Reflect: This percentage frequency table now gives us more useful information. The first column gives us the percentage breakdown of survivals and deaths in first class. From the raw data we can see that a similar number of first class ($202$202) and third class ($178$178) passengers survived. However this can be misleading. The percentage frequency table shows us that $62%$62% of first class passengers survived whereas only $25%$25% of third class passengers survived.
Maria surveyed a group of people about the type of job they had. She recorded the data in the following graph.
Complete the following two-way table displaying the row percentage.
Give your answers correct to $1$1 decimal place.
$17.6%$17.6% $\editable{}$
$%$% $29.4%$29.4% $41.2%$41.2% $100%$100%
$22.2%$22.2% $27.8%$27.8% $\editable{}$
$%$% $\editable{}$
$%$% $100%$100%
Complete the following two-way table displaying the column percentage.
$\editable{}$
$%$% $28.6%$28.6% $62.5%$62.5% $\editable{}$
$%$%
$%$% $37.5%$37.5% $\editable{}$
Finding the association between variables
To find if there is an association between the variables in the Titanic table we can ask the question "Is survival rate dependent on the class of the passenger?"
In order to find this we must first identify the explanatory variable. In this problem it is the class of passenger. The explanatory variable forms the heading of each column, therefore the column percentage frequency table will best indicate any patterns.
$62%$62% $41%$41% $25%$25% $23%$23%
When we read across the first row in the column percentage frequency table and look at the numbers we can see a clear difference in the percentages of passengers that lived or died in each class. This suggests there is an association between the class of passenger and the rate of survival. It appears that the higher the class of passenger, the higher the rate of survival.
Which percentage frequency table to use?
If the explanatory variable forms the column headings then we use the column percentage frequency table to look for association.
If the explanatory variable forms the row headings then we use the row percentage frequency table to look for association.
How do we tell if there is an association?
If it's a column percentage table then look across the rows for differences in the values. If the values are similar then we say there is NO clear association.
If it's a row percentage table then look down the columns for differences in the values. If the values are similar then we say there is NO clear association.
How to describe the association?
First state if there is or is not an association apparent. For example: There appears to be an association between the variables.
Next describe the association. For example: The class of passenger affects the survival rate of the passengers.
Finally give an example. For example: The higher the class of passenger, the more likely the passenger was to survive.
An overview of the percentages in two-way tables can bring to light clear associations. The presence of more subtle associations and an objective measure of the significance of such associations requires additional analysis and methods from further studies in statistics.
Note: The term 'association' is used to describe a relationships between variables. An association does not mean one variable causes the other variable to change but that a change in one variable appears to affect the other.
Glen surveyed all the students in Year $12$12 at his school and summarised the results in the following table:
Play netball
Do not play netball
Height$\ge$≥$170$170 cm
$33$33 $72$72 $105$105
Height$<$<$170$170 cm
$13$13 $30$30 $43$43
$46$46 $102$102 $148$148
To examine if there is an association between height and playing netball, should Glen use a column or row percentage frequency table?
Complete the row percentage frequency table for this data.
Round your answers to the nearest percentage.
Looking at the columns of the completed table, does there appear to be an association between height and playing netball?
No, there does not appear to be any association as the numbers are similar.
Yes, there appears to be an association as the numbers are quite different. It seems that taller people like to play netball.
Members of a gym were asked what kind of training they do. Each responder only did one kind of training. The table shows the results.
$11$11 $26$26
Type of training
To examine if there is an association between the type of training and gender, should we use a column or row percentage frequency table?
Looking at the columns of the completed table, does there appear to be an association between the type of training and the gender of gym members?
Yes, there appears to be an association as the numbers are quite different. Men seem to prefer weights, while women seem to prefer cardio.
No, there does not appear to be any association as the numbers are different.
Does a person's gender cause them to choose a certain type of training?
Yes. As we saw, women prefer to do cardio and men prefer to do weights.
No, association is not causation. There appears to be an association but we cannot say whether one variable causes the other.
100% stacked column graphs
Association between variables can often be seen more clearly in a stacked column graph. Below is a stacked column graph (also called segmented column graph) for the data from the Titanic table earlier. When we look at each column we can see the proportion of blue in each column is different. This indicates there is an association between the variables.
If there is no association then the proportion of the sections in each column are the same. When we look at the graph below we can see that each column is divided into similar size sections. This indicates there is NO clear association between household composition and distribution of money.
How to draw a stacked column graph
Label the horizontal axis with the explanatory variables.
Label the vertical axis with percentages from $0%$0% to $100%$100%.
Draw a column for each explanatory variable that reaches the height of $100%$100% on the vertical axis.
To divide each column into the percentages as shown in the frequency table start from the bottom of the column, count to the first percentage and draw a horizontal line to mark it off, then count up to the second percentage from the horizontal line and then mark off again, until all sections are complete
Write the label and percentage in each section of the columns which indicates the response variables displayed or provide a key.
$170$170 people were surveyed about their music preference. The results have been recorded in the table below.
Musical Preferences by Gender
Music Preference
$8$8 $15$15 $23$23
$6$6 $2$2 $8$8
$6$6 $9$9 $15$15
What is the explanatory variable in this data set?
Which of the following $100%$100% stacked column charts should be used to look for an association between the variables?
Does this stacked column chart suggest that there is an association between music preference and gender?
No. The data does not suggest any association, as the corresponding segments are of similar sizes.
Yes. The data suggests there is an association, as the corresponding segments are of similar sizes.
No. The data does not suggest any association, as the corresponding segments are of different sizes.
Yes. The data suggests there is an association, as the corresponding segments are of different sizes.
A group of year 12 students surveyed their class and recorded the hair colour and eye colour for each student. The results are displayed in the $100%$100% stacked column chart shown below.
What is the explanatory variable for this chart?
Does the chart suggest an association between eye colour and hair colour?
Yes, as the corresponding segments are similar in size.
Yes, as the corresponding segments are of different sizes.
No, as the corresponding segments are similar in size.
No, as the corresponding segments are of different sizes.
Can we say that having blue eyes causes a high chance of having blonde hair?
Yes. The data shows that students with blue eyes are more likely to have blonde hair.
No. There appears to be an association, but we cannot say that one causes the other.
ACMGM049
construct two-way frequency tables and determine the associated row and column sums and percentages
use an appropriately percentaged two-way frequency table to identify patterns that suggest the presence of an association
describe an association in terms of differences observed in percentages across categories in a systematic and concise manner, and interpret this in the context of the data
identify the response variable and the explanatory variable | CommonCrawl |
Auxiliary function
In mathematics, auxiliary functions are an important construction in transcendental number theory. They are functions that appear in most proofs in this area of mathematics and that have specific, desirable properties, such as taking the value zero for many arguments, or having a zero of high order at some point.[1]
Definition
Auxiliary functions are not a rigorously defined kind of function, rather they are functions which are either explicitly constructed or at least shown to exist and which provide a contradiction to some assumed hypothesis, or otherwise prove the result in question. Creating a function during the course of a proof in order to prove the result is not a technique exclusive to transcendence theory, but the term "auxiliary function" usually refers to the functions created in this area.
Explicit functions
Liouville's transcendence criterion
Because of the naming convention mentioned above, auxiliary functions can be dated back to their source simply by looking at the earliest results in transcendence theory. One of these first results was Liouville's proof that transcendental numbers exist when he showed that the so called Liouville numbers were transcendental.[2] He did this by discovering a transcendence criterion which these numbers satisfied. To derive this criterion he started with a general algebraic number α and found some property that this number would necessarily satisfy. The auxiliary function he used in the course of proving this criterion was simply the minimal polynomial of α, which is the irreducible polynomial f with integer coefficients such that f(α) = 0. This function can be used to estimate how well the algebraic number α can be estimated by rational numbers p/q. Specifically if α has degree d at least two then he showed that
$\left|f\left({\frac {p}{q}}\right)\right|\geq {\frac {1}{q^{d}}},$
and also, using the mean value theorem, that there is some constant depending on α, say c(α), such that
$\left|f\left({\frac {p}{q}}\right)\right|\leq c(\alpha )\left|\alpha -{\frac {p}{q}}\right|.$
Combining these results gives a property that the algebraic number must satisfy; therefore any number not satisfying this criterion must be transcendental.
The auxiliary function in Liouville's work is very simple, merely a polynomial that vanishes at a given algebraic number. This kind of property is usually the one that auxiliary functions satisfy. They either vanish or become very small at particular points, which is usually combined with the assumption that they do not vanish or can't be too small to derive a result.
Fourier's proof of the irrationality of e
Another simple, early occurrence is in Fourier's proof of the irrationality of e,[3] though the notation used usually disguises this fact. Fourier's proof used the power series of the exponential function:
$e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}.$
By truncating this power series after, say, N + 1 terms we get a polynomial with rational coefficients of degree N which is in some sense "close" to the function ex. Specifically if we look at the auxiliary function defined by the remainder:
$R(x)=e^{x}-\sum _{n=0}^{N}{\frac {x^{n}}{n!}}$
then this function—an exponential polynomial—should take small values for x close to zero. If e is a rational number then by letting x = 1 in the above formula we see that R(1) is also a rational number. However, Fourier proved that R(1) could not be rational by eliminating every possible denominator. Thus e cannot be rational.
Hermite's proof of the irrationality of er
Hermite extended the work of Fourier by approximating the function ex not with a polynomial but with a rational function, that is a quotient of two polynomials. In particular he chose polynomials A(x) and B(x) such that the auxiliary function R defined by
$R(x)=B(x)e^{x}-A(x)$
could be made as small as he wanted around x = 0. But if er were rational then R(r) would have to be rational with a particular denominator, yet Hermite could make R(r) too small to have such a denominator, hence a contradiction.
Hermite's proof of the transcendence of e
To prove that e was in fact transcendental, Hermite took his work one step further by approximating not just the function ex, but also the functions ekx for integers k = 1,...,m, where he assumed e was algebraic with degree m. By approximating ekx by rational functions with integer coefficients and with the same denominator, say Ak(x) / B(x), he could define auxiliary functions Rk(x) by
$R_{k}(x)=B(x)e^{kx}-A_{k}(x).$
For his contradiction Hermite supposed that e satisfied the polynomial equation with integer coefficients a0 + a1e + ... + amem = 0. Multiplying this expression through by B(1) he noticed that it implied
$R=a_{0}+a_{1}R_{1}(1)+\cdots +a_{m}R_{m}(1)=a_{1}A_{1}(1)+\cdots +a_{m}A_{m}(1).$
The right hand side is an integer and so, by estimating the auxiliary functions and proving that 0 < |R| < 1 he derived the necessary contradiction.
Auxiliary functions from the pigeonhole principle
Main article: Siegel's lemma
The auxiliary functions sketched above can all be explicitly calculated and worked with. A breakthrough by Axel Thue and Carl Ludwig Siegel in the twentieth century was the realisation that these functions don't necessarily need to be explicitly known – it can be enough to know they exist and have certain properties. Using the Pigeonhole Principle Thue, and later Siegel, managed to prove the existence of auxiliary functions which, for example, took the value zero at many different points, or took high order zeros at a smaller collection of points. Moreover they proved it was possible to construct such functions without making the functions too large.[4] Their auxiliary functions were not explicit functions, then, but by knowing that a certain function with certain properties existed, they used its properties to simplify the transcendence proofs of the nineteenth century and give several new results.[5]
This method was picked up on and used by several other mathematicians, including Alexander Gelfond and Theodor Schneider who used it independently to prove the Gelfond–Schneider theorem.[6] Alan Baker also used the method in the 1960s for his work on linear forms in logarithms and ultimately Baker's theorem.[7] Another example of the use of this method from the 1960s is outlined below.
Auxiliary polynomial theorem
Let β equal the cube root of b/a in the equation ax3 + bx3 = c and assume m is an integer that satisfies m + 1 > 2n/3 ≥ m ≥ 3 where n is a positive integer.
Then there exists
$F(X,Y)=P(X)+Y*Q(X)$
such that
$\sum _{i=0}^{m+n}u_{i}X^{i}=P(X),$
$\sum _{i=0}^{m+n}v_{i}X^{i}=Q(X).$
The auxiliary polynomial theorem states
$\max _{0\leq i\leq m+n}{(|u_{i}|,|v_{i}|)}\leq 2b^{9(m+n)}.$
A theorem of Lang
In the 1960s Serge Lang proved a result using this non-explicit form of auxiliary functions. The theorem implies both the Hermite–Lindemann and Gelfond–Schneider theorems.[8] The theorem deals with a number field K and meromorphic functions f1,...,fN of order at most ρ, at least two of which are algebraically independent, and such that if we differentiate any of these functions then the result is a polynomial in all of the functions. Under these hypotheses the theorem states that if there are m distinct complex numbers ω1,...,ωm such that fi (ωj ) is in K for all combinations of i and j, then m is bounded by
$m\leq 20\rho [K:\mathbb {Q} ].$
To prove the result Lang took two algebraically independent functions from f1,...,fN, say f and g, and then created an auxiliary function which was simply a polynomial F in f and g. This auxiliary function could not be explicitly stated since f and g are not explicitly known. But using Siegel's lemma Lang showed how to make F in such a way that it vanished to a high order at the m complex numbers ω1,...,ωm. Because of this high order vanishing it can be shown that a high-order derivative of F takes a value of small size one of the ωis, "size" here referring to an algebraic property of a number. Using the maximum modulus principle Lang also found a separate way to estimate the absolute values of derivatives of F, and using standard results comparing the size of a number and its absolute value he showed that these estimates were contradicted unless the claimed bound on m holds.
Interpolation determinants
After the myriad of successes gleaned from using existent but not explicit auxiliary functions, in the 1990s Michel Laurent introduced the idea of interpolation determinants.[9] These are alternants – determinants of matrices of the form
${\mathcal {M}}=\left(\varphi _{i}(\zeta _{j})\right)_{1\leq i,j\leq N}$
where φi are a set of functions interpolated at a set of points ζj. Since a determinant is just a polynomial in the entries of a matrix, these auxiliary functions succumb to study by analytic means. A problem with the method was the need to choose a basis before the matrix could be worked with. A development by Jean-Benoît Bost removed this problem with the use of Arakelov theory,[10] and research in this area is ongoing. The example below gives an idea of the flavour of this approach.
A proof of the Hermite–Lindemann theorem
One of the simpler applications of this method is a proof of the real version of the Hermite–Lindemann theorem. That is, if α is a non-zero, real algebraic number, then eα is transcendental. First we let k be some natural number and n be a large multiple of k. The interpolation determinant considered is the determinant Δ of the n4×n4 matrix
$\left(\{\exp(j_{2}x)x^{j_{1}-1}\}^{(i_{1}-1)}{\Big |}_{x=(i_{2}-1)\alpha }\right).$
The rows of this matrix are indexed by 1 ≤ i1 ≤ n4/k and 1 ≤ i2 ≤ k, while the columns are indexed by 1 ≤ j1 ≤ n3 and 1 ≤ j2 ≤ n. So the functions in our matrix are monomials in x and ex and their derivatives, and we are interpolating at the k points 0,α,2α,...,(k − 1)α. Assuming that eα is algebraic we can form the number field Q(α,eα) of degree m over Q, and then multiply Δ by a suitable denominator as well as all its images under the embeddings of the field Q(α,eα) into C. For algebraic reasons this product is necessarily an integer, and using arguments relating to Wronskians it can be shown that it is non-zero, so its absolute value is an integer Ω ≥ 1.
Using a version of the mean value theorem for matrices it is possible to get an analytic bound on Ω as well, and in fact using big-O notation we have
$\Omega =O\left(\exp \left(\left({\frac {m+1}{k}}-{\frac {3}{2}}\right)n^{8}\log n\right)\right).$
The number m is fixed by the degree of the field Q(α,eα), but k is the number of points we are interpolating at, and so we can increase it at will. And once k > 2(m + 1)/3 we will have Ω → 0, eventually contradicting the established condition Ω ≥ 1. Thus eα cannot be algebraic after all.[11]
Notes
1. Waldschmidt (2008).
2. Liouville (1844).
3. Hermite (1873).
4. Thue (1977) and Siegel (1929).
5. Siegel (1932).
6. Gel'fond (1934) and Schneider (1934).
7. Baker and Wüstholz (2007).
8. Lang (1966).
9. Laurent (1991).
10. Bost (1996).
11. Adapted from Pila (1993).
References
• Waldschmidt, Michel. "An Introduction to Irrationality and Transcendence Methods" (PDF).
• Liouville, Joseph (1844). "Sur des classes très étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationnelles algébriques". J. Math. Pures Appl. 18: 883–885, and 910–911.
• Hermite, Charles (1873). "Sur la fonction exponentielle". C. R. Acad. Sci. Paris. 77.
• Thue, Axel (1977). Selected Mathematical Papers. Oslo: Universitetsforlaget.
• Siegel, Carl Ludwig (1929). "Über einige Anwendungen diophantischer Approximationen". Abhandlungen Akad. Berlin. 1: 70.
• Siegel, Carl Ludwig (1932). "Über die Perioden elliptischer Funktionen". Journal für die reine und angewandte Mathematik. 1932 (167): 62–69. doi:10.1515/crll.1932.167.62. S2CID 199545608.
• Gel'fond, A. O. (1934). "Sur le septième Problème de D. Hilbert". Izv. Akad. Nauk SSSR. 7: 623–630.
• Schneider, Theodor (1934). "Transzendenzuntersuchungen periodischer Funktionen. I. Transzendend von Potenzen". J. Reine Angew. Math. 172: 65–69.
• Baker, Alan; Wüstholz, G. (2007), "Logarithmic forms and Diophantine geometry", New Mathematical Monographs, Cambridge University Press, vol. 9, p. 198
• Lang, Serge (1966). Introduction to Transcendental Numbers. Addison–Wesley Publishing Company.
• Laurent, Michel (1991). "Sur quelques résultats récents de transcendance". Astérisque. 198–200: 209–230.
• Bost, Jean-Benoît (1996). "Périodes et isogénies des variétés abéliennes sur les corps de nombres (d'après D. Masser et G. Wüstholz)". Astérisque. 237: 795.
• Pila, Jonathan (1993). "Geometric and arithmetic postulation of the exponential function". J. Austral. Math. Soc. A. 54: 111–127. doi:10.1017/s1446788700037022.
| Wikipedia |
\begin{definition}[Definition:Convergent Sequence/Note on Domain of N]
Let $\sequence {x_k}$ be a sequence.
:$\ds \lim_{n \mathop \to \infty} x_n \to l$
be the limit of $\sequence {x_k}$.
That is:
:$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \map d {x_n, l} < \epsilon$
Some sources insist that $N \in \N$ but this is not strictly necessary and can make proofs more cumbersome.
\end{definition} | ProofWiki |
\begin{document}
\title{Weighted sum formulas of multiple zeta values with even arguments}
\date{\today\thanks{The first author is supported by the National Natural Science Foundation of China (Grant No. 11471245) and Shanghai Natural Science Foundation (grant no. 14ZR1443500).} }
\author{Zhonghua Li \quad and \quad Chen Qin}
\address{School of Mathematical Sciences, Tongji University, No. 1239 Siping Road, Shanghai 200092, China}
\email{zhonghua\[email protected]}
\address{School of Mathematical Sciences, Tongji University, No. 1239 Siping Road, Shanghai 200092, China}
\email{2014chen\[email protected]}
\keywords{Multiple zeta values, Multiple zeta-star values, Bernoulli numbers, Weighted sum formulas}
\subjclass[2010]{11M32,11B68}
\begin{abstract} We prove a weighted sum formula of the zeta values at even arguments, and a weighted sum formula of the multiple zeta values with even arguments and its zeta-star analogue. The weight coefficients are given by (symmetric) polynomials of the arguments. These weighted sum formulas for the zeta values and for the multiple zeta values were conjectured by L. Guo, P. Lei and J. Zhao. \end{abstract}
\maketitle
\section{Introduction}\label{Sec:Intro}
For a positive integer $n$ and a sequence $\mathbf{k}=(k_1,\ldots,k_n)$ of positive integers with $k_1>1$, the multiple zeta value $\zeta(\mathbf{k})$ and the multiple zeta-star value $\zeta^{\star}(\mathbf{k})$ are defined by the following infinite series $$\zeta(\mathbf{k})=\zeta(k_1,\ldots,k_n)=\sum\limits_{m_1>\cdots>m_n\geqslant 1}\frac{1}{m_1^{k_1}\cdots m_n^{k_n}}$$ and $$\zeta^{\star}(\mathbf{k})=\zeta^{\star}(k_1,\ldots,k_n)=\sum\limits_{m_1\geqslant \cdots\geqslant m_n\geqslant 1}\frac{1}{m_1^{k_1}\cdots m_n^{k_n}},$$ respectively. The number $n$ is called the depth. In depth one case, both $\zeta(\mathbf{k})$ and $\zeta^{\star}(\mathbf{k})$ are special values of the Riemann zeta function at positive integer arguments.
The study of these values may be traced back to L. Euler. Among other things, L. Euler found the following sum formula $$\sum\limits_{i=2}^{k-1}\zeta^{\star}(i,k-i)=(k-1)\zeta(k),\quad k\geqslant 3,$$ or equivalently, $$\sum\limits_{i=2}^{k-1}\zeta(i,k-i)=\zeta(k),\quad k\geqslant 3.$$ There are many generalizations and variations of the sum formula, among which we mention some weighted sum formulas at even arguments. In \cite{Gangl-Kaneko-Zagier}, the following formula $$\sum\limits_{i=1}^{k-1}\zeta(2i,2k-2i)=\frac{3}{4}\zeta(2k)$$ was proved by using the regularized double shuffle relations of the double zeta values. M. E. Hoffman considered the sum $$\sum\limits_{k_1+\cdots+k_n=k\atop k_j\geqslant 1}\zeta(2k_1,\ldots,2k_n)$$ in \cite{Hoffman2017}, and we showed in \cite{Li-Qin} that the formulas given by M. E. Hoffman in \cite{Hoffman2017} are consequences of the regularized double shuffle relations of the multiple zeta values. Later in \cite{Guo-Lei-Zhao}, new families of weighted sum formulas of the forms $$ \sum\limits_{k_1+k_2=k\atop k_j\geqslant 1}F(k_1,k_2)\zeta(2k_1)\zeta(2k_2),\quad \sum\limits_{k_1+k_2+k_3=k\atop k_j\geqslant 1}G(k_1,k_2,k_3)\zeta(2k_1)\zeta(2k_2)\zeta(2k_3) $$ and $$\sum\limits_{k_1+k_2=k\atop k_j\geqslant 1}F(k_1,k_2)\zeta(2k_1,2k_2),\quad \sum\limits_{k_1+k_2+k_3=k\atop k_j\geqslant 1}G(k_1,k_2,k_3)\zeta(2k_1,2k_2,2k_3)$$ were given, where $F(x,y)$ and $G(x,y,z)$ are (symmetric) polynomials with rational coefficients. And in the end of \cite{Guo-Lei-Zhao}, L. Guo, P. Lei and J. Zhao proposed the following general conjecture.
\begin{conj}[{\cite[Conjecture 4.7]{Guo-Lei-Zhao}}]\label{Conj:WeightedSum} Let $F(x_1,\ldots,x_n)\in\mathbb{Q}[x_1,\ldots,x_n]$ be a symmetric polynomial of degree $r$. Set $d=\deg_{x_1}F(x_1,\ldots,x_n)$. Then for every positive integer $k\geqslant n$ we have \begin{align} \sum\limits_{k_1+\cdots+k_n=k\atop k_j\geqslant 1}F(k_1,\ldots,k_n)\zeta(2k_1)\cdots\zeta(2k_n)=\sum\limits_{l=0}^Te_{F,l}(k)\zeta(2l)\zeta(2k-2l), \label{Eq:WeightedSum-Zeta-Conj}\\ \sum\limits_{k_1+\cdots+k_n=k\atop k_j\geqslant 1}F(k_1,\ldots,k_n)\zeta(2k_1,\ldots,2k_n)=\sum\limits_{l=0}^Tc_{F,l}(k)\zeta(2l)\zeta(2k-2l), \label{Eq:WeightedSum-MZV-Conj} \end{align} where $T=\max\{[(r+n-2)/2],[(n-1)/2]\}$, $e_{F,l}(x),c_{F,l}(x)\in\mathbb{Q}[x]$ depend only on $l$ and $F$, $\deg e_{F,l}(x)\leqslant r-1$ and $\deg c_{F,l}(x)\leqslant d$. \end{conj}
Here as usual, for a real number $x$, we denote by $[x]$ the greatest integer that not exceeding $x$.
The purpose of this paper is to give a proof of Conjecture \ref{Conj:WeightedSum}. In fact, we also prove the zeta-star analogue of \eqref{Eq:WeightedSum-MZV-Conj}. To prove \eqref{Eq:WeightedSum-Zeta-Conj}, as in \cite{Guo-Lei-Zhao}, we first establish a weighted sum formula of the Bernoulli numbers. In \cite{Guo-Lei-Zhao}, L. Guo, P. Lei and J. Zhao used certain zeta functions to study the Bernoulli numbers. Here we just use the generating function of the Bernoulli numbers. Hence our method seems more elementary. After getting the weighted sum formula of the Bernoulli numbers, we obtain the weighted sum formula \eqref{Eq:WeightedSum-Zeta-Conj} by using Euler's evaluation formula of the zeta values at even arguments. Finally, applying the symmetric sum formulas of M. E. Hoffman \cite{Hoffman1992}, we obtain the weighted sum formula \eqref{Eq:WeightedSum-MZV-Conj} and its zeta-star analogue from the formula \eqref{Eq:WeightedSum-Zeta-Conj}.
The paper is organized as follows. In Section \ref{Sec:WeightSum-Bernoulli}, we deal with the weighted sum of the Bernoulli numbers. In Section \ref{Sec:WeightSum-Zeta}, we prove the weighted sum formula \eqref{Eq:WeightedSum-Zeta-Conj}. And in Section \ref{Sec:WeightSum-MZV}, we prove the weighted sum formula \eqref{Eq:WeightedSum-MZV-Conj} and its zeta-star analogue. Finally, in Section \ref{Sec:RegDouble-WeightSum}, we show that the weighted sum formulas obtained in this paper can be deduced from the regularized double shuffle relations of the multiple zeta values.
\section{A weighted sum formula of Bernoulli numbers}\label{Sec:WeightSum-Bernoulli}
The Bernoulli numbers $\{B_i\}$ are defined by $$\sum\limits_{i=0}^{\infty} \frac{B_i}{i!}t^i=\frac{t}{e^t-1}.$$ It is known that $B_0=1$, $B_1=-\frac{1}{2}$ and $B_i=0$ for odd $i\geqslant 3$. We set \begin{align*} &f(t)=\frac{t}{e^t-1}-1+\frac{1}{2}t=\sum\limits_{i=1}^\infty\frac{B_{2i}}{(2i)!}t^{2i},\\ &g(t)=\frac{t}{e^t-1}+\frac{1}{2}t=\sum\limits_{i=0}^\infty\frac{B_{2i}}{(2i)!}t^{2i}. \end{align*} We compute the derivatives of the even function $f(t)$. Let $D=t\frac{d}{dt}$ and $h(t)=\frac{t}{e^t-1}$. Then using the formula $$h'(t)=\frac{1-t}{e^t-1}-\frac{t}{(e^t-1)^2},$$ we find that for any nonnegative integer $m$, \begin{align} D^mf(t)=\sum\limits_{i=0}^{m+1}f_{mi}(t)h(t)^i. \label{Eq:Diff-f} \end{align} Here $f_{mi}(t)$ are polynomials determined by $f_{00}(t)=\frac{1}{2}t-1$, $f_{01}(t)=1$ and the recursive formulas \begin{align} \begin{cases} f_{m0}(t)=tf_{m-1,0}'(t) & \text{for\;} m\geqslant 1,\\ f_{m,m+1}(t)=-mf_{m-1,m}(t) & \text{for\;} m\geqslant 1,\\ f_{mi}(t)=tf'_{m-1,i}(t)+i(1-t)f_{m-1,i}(t)-(i-1)f_{m-1,i-1}(t) & \text{for\;} 1\leqslant i\leqslant m. \end{cases} \label{Eq:Recursive-fmi} \end{align} In particular, for any integers $m,i$ with $1\leqslant i\leqslant m+1$, we have $f_{mi}(t)\in\mathbb{Z}[t]$. From \eqref{Eq:Recursive-fmi}, it is easy to see that for any nonnegative integer $m$, we have $$f_{m0}(t)=\frac{1}{2}t-\delta_{m,0},\quad f_{m,m+1}(t)=(-1)^mm!.$$
\begin{lem} For any integers $m,i$ with $1\leqslant i\leqslant m+1$, we have $\deg f_{mi}(t)=m+1-i$, and the leading coefficient $c_{mi}$ of $f_{mi}(t)$ satisfies the condition $(-1)^mc_{mi}>0$. \end{lem}
\noindent {\bf Proof.\;} We use induction on $m$. Assume that $m\geqslant 1$. The result for $i=m+1$ follows from $f_{m,m+1}(t)=(-1)^mm!$. Now assume the integer $i$ satisfies the condition $1\leqslant i\leqslant m$, and $$f_{m-1,i}(t)=c_{m-1,i}t^{m-i}+\text{lower degree terms}$$ with $(-1)^{m-1}c_{m-1,i}>0$. Let $c_{m0}=\frac{1}{2}$. Then we have $$f_{mi}(t)=(-ic_{m-1,i}-(i-1)c_{m-1,i-1})t^{m+1-i}+\text{lower degree terms}.$$ As \begin{align*} &(-1)^m(-ic_{m-1,i}-(i-1)c_{m-1,i-1})\\ =&i(-1)^{m-1}c_{m-1,i}+(i-1)(-1)^{m-1}c_{m-1,i-1}>0, \end{align*} we get the result. \qed
Therefore we have $$f_{m0}(t)=c_{m0}t-\delta_{m,0}$$ with $c_{m0}=\frac{1}{2}$, and for any integers $m,i$ with the condition $1\leqslant i\leqslant m+1$, we have $$f_{mi}(t)=c_{mi}t^{m+1-i}+\text{lower degree terms},$$ with the recursive formula $$c_{mi}=-ic_{m-1,i}-(i-1)c_{m-1,i-1},\quad (1\leqslant i\leqslant m)$$ and $c_{m,m+1}=(-1)^mm!$.
\begin{cor} For any nonnegative integer $m$, we have $c_{m1}=(-1)^m$. \end{cor}
For later use, we need the following lemma.
\begin{lem} For any nonnegative integer $m$, we have \begin{align} \sum\limits_{i=1}^{m+1}(-1)^{i-1}f_{mi}(t)t^{i-1}=1. \label{Eq:Sum-fmi} \end{align} In particular, we have \begin{align} \sum\limits_{i=1}^{m+1}(-1)^{i-1}c_{mi}=\delta_{m,0}. \label{Eq:Sum-cmi} \end{align} \end{lem}
\noindent {\bf Proof.\;} We proceed by induction on $m$ to prove \eqref{Eq:Sum-fmi}. The case of $m=0$ follows from the fact $f_{01}(t)=1$. Now assume that $m\geqslant 1$, using the recursive formula \eqref{Eq:Recursive-fmi}, we have \begin{align*} &\sum\limits_{i=1}^{m+1}(-1)^{i-1}f_{mi}(t)t^{i-1}=\sum\limits_{i=1}^m(-1)^{i-1}f_{m-1,i}'(t)t^{i}+\sum\limits_{i=1}^{m}(-1)^{i-1}if_{m-1,i}(t)t^{i-1}\\ &\qquad+\sum\limits_{i=1}^m(-1)^iif_{m-1,i}(t)t^i+\sum\limits_{i=1}^m(-1)^{i}(i-1)f_{m-1,i-1}(t)t^{i-1}+m!t^m\\ =&\sum\limits_{i=1}^m(-1)^{i-1}(f_{m-1,i}(t)t^{i})'+(-1)^mmf_{m-1,m}(t)t^m+m!t^m\\ =&\frac{d}{dt}\sum\limits_{i=1}^m(-1)^{i-1}f_{m-1,i}(t)t^{i}. \end{align*} Then we get \eqref{Eq:Sum-fmi} from the induction assumption. Finally, comparing the coefficients of $t^m$ of both sides of \eqref{Eq:Sum-fmi}, we get \eqref{Eq:Sum-cmi}. \qed
Now we want to express $h(t)^i$ by $D^mg(t)$. For this purpose, we use matrix computations. For any nonnegative integer $m$, let $A_m(t)$ be a $(m+1)\times (m+1)$ matrix defined by $$A_m(t)=\begin{pmatrix} f_{01}(t) & &&\\ f_{11}(t) & f_{12}(t) &&\\ \vdots & \vdots & \ddots &\\ f_{m1}(t) & f_{m2}(t) & \cdots & f_{m,m+1}(t) \end{pmatrix}.$$ Note that for $m\geqslant 1$, we have $$A_m(t)=\begin{pmatrix} A_{m-1}(t) & 0\\ \alpha_m(t) & (-1)^mm! \end{pmatrix}$$ with $\alpha_m(t)=(f_{m1}(t),\ldots,f_{mm}(t))$. From linear algebra, we know that the matrix $\begin{pmatrix} A & 0\\ C & B \end{pmatrix}$ is invertible with $$\begin{pmatrix} A & 0\\ C & B \end{pmatrix}^{-1}=\begin{pmatrix} A^{-1} & 0\\ -B^{-1}CA^{-1} & B^{-1} \end{pmatrix},$$ provided that $A$ and $B$ are invertible square matrices. Therefore by induction on $m$, we find that for all nonnegative integer $m$, the matrices $A_m(t)$ are invertible, and the inverses satisfy the recursive formula \begin{align} A_m(t)^{-1}=\begin{pmatrix} A_{m-1}(t)^{-1} & 0\\ (-1)^{m+1}\frac{1}{m!}\alpha_m(t)A_{m-1}(t)^{-1} & (-1)^m\frac{1}{m!} \end{pmatrix},\quad (m\geqslant 1). \label{Eq:Recursive-AmInverse} \end{align}
For any nonnegative integer $m$, set $$A_m(t)^{-1}=\begin{pmatrix} g_{01}(t) & &&\\ g_{11}(t) & g_{12}(t) &&\\ \vdots & \vdots & \ddots &\\ g_{m1}(t) & g_{m2}(t) & \cdots & g_{m,m+1}(t) \end{pmatrix}.$$
\begin{lem} Let $m$ and $i$ be integers. \begin{itemize}
\item [(1)] For any $m\geqslant 0$, we have $g_{m,m+1}(t)=(-1)^m\frac{1}{m!}$;
\item [(2)] If $1\leqslant i\leqslant m$, we have the recursive formula
\begin{align}
g_{mi}(t)=(-1)^{m+1}\frac{1}{m!}\sum\limits_{j=i}^{m}f_{mj}(t)g_{j-1,i}(t);
\label{Eq:Recursive-gmi}
\end{align}
\item [(3)] If $1\leqslant i\leqslant m+1$, we have $g_{mi}(t)\in\mathbb{Q}[t]$ with $\deg g_{mi}(t)\leqslant m+1-i$;
\item [(4)] For $1\leqslant i\leqslant m+1$, set
$$g_{mi}(t)=d_{mi}t^{m+1-i}+\text{lower degree terms}.$$
Then we have $d_{m,m+1}=(-1)^m\frac{1}{m!}$ and
\begin{align}
d_{mi}=(-1)^{m+1}\frac{1}{m!}\sum\limits_{j=i}^{m}c_{mj}d_{j-1,i}
\label{Eq:Recursive-dmi}
\end{align}
for $1\leqslant i\leqslant m$. \end{itemize} \end{lem}
\noindent {\bf Proof.\;} The assertions in items (1) and (2) follow from \eqref{Eq:Recursive-AmInverse}. To prove the item (3), we proceed by induction on $m$. For the case of $m=0$, we get the result from $g_{01}(t)=1$. Assume that $m\geqslant 1$, then $g_{m,m+1}(t)=(-1)^m\frac{1}{m!}\in\mathbb{Q}[t]$ with degree zero. For $1\leqslant i\leqslant j\leqslant m$, using the induction assumption, we may set $$g_{j-1,i}(t)=d_{j-1,i}t^{j-i}+\text{lower degree terms}\in\mathbb{Q}[t].$$ Since $$f_{mj}(t)=c_{mj}t^{m+1-j}+\text{lower degree terms}\in\mathbb{Z}[t],$$ we get $$f_{mj}(t)g_{j-1,i}(t)=c_{mj}d_{j-1,i}t^{m+1-i}+\text{lower degree terms}\in\mathbb{Q}[t].$$ Using \eqref{Eq:Recursive-gmi}, we finally get $$g_{mi}(t)=\left((-1)^{m+1}\frac{1}{m!}\sum\limits_{j=i}^{m}c_{mj}d_{j-1,i}\right)t^{m+1-i}+\text{lower degree terms}\in\mathbb{Q}[t].$$ The item (4) follows from the above proof. \qed
\begin{cor} For any nonnegative integer $m$, we have $d_{m1}=(-1)^m$. \end{cor}
\noindent {\bf Proof.\;} We use induction on $m$. If $m\geqslant 1$, using \eqref{Eq:Recursive-dmi} and the induction assumption, we get $$d_{m1}=(-1)^{m+1}\frac{1}{m!}\sum\limits_{j=1}^m(-1)^{j-1}c_{mj}.$$ By \eqref{Eq:Sum-cmi}, we have $$d_{m1}=(-1)^{m+1}\frac{1}{m!}(\delta_{m,0}-(-1)^{m}c_{m,m+1}),$$ which implies the result. \qed
To get $h(t)^i$, we rewrite \eqref{Eq:Diff-f} as \begin{align} \begin{pmatrix} g(t)\\ Dg(t)\\ \vdots\\ D^mg(t) \end{pmatrix}-\frac{1}{2}t\begin{pmatrix} 1\\ 1\\ \vdots\\ 1 \end{pmatrix}=A_m(t)\begin{pmatrix} h(t)\\ h(t)^2\\ \vdots\\ h(t)^{m+1} \end{pmatrix}, \label{Eq:G-h-Matrix} \end{align} and rewrite \eqref{Eq:Sum-fmi} as $$A_{m}(t)\begin{pmatrix} 1\\ -t\\ t^2\\ \vdots\\ (-1)^mt^m \end{pmatrix}=\begin{pmatrix} 1\\ 1\\ \vdots\\ 1 \end{pmatrix}.$$ Therefore we find $$ \begin{pmatrix} h(t)\\ h(t)^2\\ \vdots\\ h(t)^{m+1} \end{pmatrix}=A_m(t)^{-1}\begin{pmatrix} g(t)\\ Dg(t)\\ \vdots\\ D^mg(t) \end{pmatrix}-\frac{1}{2}t\begin{pmatrix} 1\\ -t\\ t^2\\ \vdots\\ (-1)^mt^m \end{pmatrix}. $$ Then for any positive integer $i$, we get \begin{align} h(t)^i=\sum\limits_{j=1}^ig_{i-1,j}(t)D^{j-1}g(t)+\frac{1}{2}(-1)^it^i. \label{Eq:h-i} \end{align}
For the later use, we prepare a lemma.
\begin{lem}\label{Lem:G-linearInd} For a nonnegative integer $m$, the functions $1,g(t),D g(t),\ldots,D^m g(t)$ are linearly independent over the rational function field $\mathbb{Q}(t)$. \end{lem}
\noindent {\bf Proof.\;} Let $p(t),p_0(t),p_1(t),\ldots,p_m(t)\in\mathbb{Q}(t)$ satisfy $$p(t)+p_0(t)g(t)+p_1(t)D g(t)+\cdots+p_m(t)D^mg(t)=0.$$ Using \eqref{Eq:G-h-Matrix}, we get $$p(t)+\frac{1}{2}t\sum\limits_{j=0}^mp_j(t)+(p_0(t),\ldots,p_m(t))A_m(t)\begin{pmatrix} h(t)\\ \vdots\\ h(t)^{m+1} \end{pmatrix}=0.$$ Since $e^t$ is transcendental over $\mathbb{Q}(t)$, we know $e^t-1$, and then $h(t)$ is transcendental over $\mathbb{Q}(t)$. Hence we have $$p(t)+\frac{1}{2}t\sum\limits_{j=0}^mp_j(t)=0,\qquad (p_0(t),\ldots,p_m(t))A_m(t)=0,$$ which implies that all $p_j(t)$ and $p(t)$ are zero functions as the matrix $A_m(t)$ is invertible. \qed
From now on let $n$ be a fixed positive integer, and $m_1,\ldots,m_n$ be fixed nonnegative integers. We want to compute $D^{m_1}f(t)\cdots D^{m_n}f(t)$. On the one hand, using \eqref{Eq:Diff-f}, we have $$D^{m_1}f(t)\cdots D^{m_n}f(t)=\sum\limits_{i=0}^{m_1+\cdots+m_n+n}f_i(t)h(t)^i,$$ with $$f_i(t)=\sum\limits_{i_1+\cdots+i_n=i\atop 0\leqslant i_j\leqslant m_j+1}f_{m_1i_1}(t)\cdots f_{m_ni_n}(t).$$
\begin{lem} We have $$f_0(t)=\prod\limits_{j=1}^n\left(\frac{1}{2}t-\delta_{m_j,0}\right),$$ and $\deg f_i(t)\leqslant m_1+\cdots+m_n+n-i$ for any nonnegative integer $i$. \end{lem}
\noindent {\bf Proof.\;} For integers $i_1,\ldots,i_n$ with the conditions $i_1+\cdots+i_n=i$ and $0\leqslant i_j\leqslant m_j+1$, we have $$\deg(f_{m_1i_1}(t)\cdots f_{m_ni_n}(t))\leqslant \sum\limits_{j=1}^n(m_j+1-i_j)=m_1+\cdots+m_n+n-i,$$ which deduces that $\deg f_i(t)\leqslant m_1+\cdots+m_n+n-i$. \qed
Then using \eqref{Eq:h-i}, we get \begin{align*} &D^{m_1}f(t)\cdots D^{m_n}f(t)\\ =&\sum\limits_{i=1}^{m_1+\cdots+m_n+n}f_i(t)\left(\sum\limits_{j=1}^ig_{i-1,j}(t)D^{j-1}g(t)+\frac{1}{2}(-1)^it^i\right)+f_0(t)\\ =&\sum\limits_{j=1}^{m_1+\cdots+m_n+n}F_j(t)D^{j-1}g(t)+F_0(t) \end{align*} with $$F_0(t)=f_0(t)+\frac{1}{2}\sum\limits_{i=1}^{m_1+\cdots+m_n+n}(-1)^if_i(t)t^i$$ and $$F_j(t)=\sum\limits_{i=j}^{m_1+\cdots+m_n+n}f_i(t)g_{i-1,j}(t),\quad (1\leqslant j\leqslant m_1+\cdots+m_n+n).$$
\begin{lem} Let $j$ be a nonnegative integer with $j\leqslant m_1+\cdots+m_n+n$. Then \begin{itemize}
\item [(1)] the function $F_j(t)$ is even;
\item [(2)] we have
$$F_0(t)=\frac{1}{2}\prod\limits_{j=1}^n\left(\frac{1}{2}t-\delta_{m_j,0}\right)+\frac{1}{2}(-1)^n \prod\limits_{j=1}^n\left(\frac{1}{2}t+\delta_{m_j,0}\right).$$ In particular, $\deg F_0(t)\leqslant n$;
\item [(3)] if $j>0$, we have $\deg F_j(t)\leqslant m_1+\cdots+m_n+n-j$. Moreover, we have $\deg F_1(t)\leqslant m_1+\cdots+m_n+n-2$ provided that $n$ is even or $m_1,\ldots,m_n$ are not all zero. \end{itemize} \end{lem}
\noindent {\bf Proof.\;} Since $D^{m}f(t)$ and $D^m g(t)$ are even, we have $$\sum\limits_{j=1}^{m_1+\cdots+m_n+n}F_j(t)D^{j-1}g(t)+F_0(t)=\sum\limits_{j=1}^{m_1+\cdots+m_n+n}F_j(-t)D^{j-1}g(t)+F_0(-t).$$ Then by Lemma \ref{Lem:G-linearInd}, we know all $F_j(t)$ are even functions.
By the definition of $f_i(t)$, we have $$\sum\limits_{i=0}^{m_1+\cdots+m_n+n}(-1)^if_i(t)t^i=\prod\limits_{j=1}^n\sum\limits_{i_j=0}^{m_j+1}(-1)^{i_j}f_{m_ji_j}(t)t^{i_j}.$$ Using \eqref{Eq:Sum-fmi}, we find $$\sum\limits_{i=0}^{m_1+\cdots+m_n+n}(-1)^if_i(t)t^i=\prod\limits_{j=1}^n(f_{m_j0}(t)-t).$$ Then we get (2) from the fact that $f_{m0}(t)=\frac{1}{2}t-\delta_{m,0}$ and the expression of $f_0(t)$.
Since $$\deg f_i(t)g_{i-1,j}(t)\leqslant (m_1+\cdots+m_n+n-i)+(i-j)=m_1+\cdots+m_n+n-j,$$ we get $\deg F_j(t)\leqslant m_1+\cdots+m_n+n-j$.
If we set $$\widetilde{c}_{mi}=\begin{cases} \frac{1}{2}\delta_{m,0} & \text{if\;} i=0,\\ c_{mi} & \text{if\;} i\neq 0, \end{cases}$$ then the coefficient of $t^{m+1-i}$ in $f_{mi}(t)$ is $\widetilde{c}_{mi}$ for any integers $m,i$ with the condition $0\leqslant i\leqslant m+1$. Since $$F_1(t)=\sum\limits_{i=1}^{m_1+\cdots+m_n+n}\sum\limits_{i_1+\cdots+i_n=i\atop 0\leqslant i_j\leqslant m_j+1}f_{m_1i_1}(t)\cdots f_{m_ni_n}(t)g_{i-1,1}(t),$$ and $d_{i-1,1}=(-1)^{i-1}$, we find the coefficient of $t^{m_1+\cdots+m_n+n-1}$ in $F_1(t)$ is \begin{align*} &\sum\limits_{i=1}^{m_1+\cdots+m_n+n}\sum\limits_{i_1+\cdots+i_n=i\atop 0\leqslant i_j\leqslant m_j+1}(-1)^{i-1}\widetilde{c}_{m_1i_1}\cdots \widetilde{c}_{m_ni_n}\\ =&\widetilde{c}_{m_10}\cdots \widetilde{c}_{m_n0}-\prod\limits_{j=1}^n\sum\limits_{i_j=0}^{m_j+1}(-1)^{i_j}\widetilde{c}_{m_ji_j}, \end{align*} which is $$\widetilde{c}_{m_10}\cdots \widetilde{c}_{m_n0}-\prod\limits_{j=1}^n(\widetilde{c}_{m_j0}-\delta_{m_j,0})$$ by \eqref{Eq:Sum-cmi}. Then the coefficient of $t^{m_1+\cdots+m_n+n-1}$ in $F_1(t)$ is $$\left(\frac{1}{2}\right)^n(1-(-1)^n)\delta_{m_1,0}\cdots\delta_{m_n,0},$$ which is zero if $n$ is even or at least one $m_i$ is not zero. \qed
Now for a positive integer $j$ with $j\leqslant m_1+\cdots+m_n+n$, let $a_{jl}\in\mathbb{Q}$ be the coefficient of $t^{2l}$ in the polynomial $F_j(t)$. Then we have \begin{align} F_j(t)=\sum\limits_{l=0}^{\left[\frac{m_1+\cdots+m_n+n-j}{2}\right]}a_{jl}t^{2l}. \label{Eq:Fj} \end{align} If $n$ is even or $m_1,\ldots,m_n$ are not all zero, we have $$F_1(t)=\sum\limits_{l=0}^{\left[\frac{m_1+\cdots+m_n+n-2}{2}\right]}a_{1l}t^{2l}.$$ Hence we have $$D^{m_1}f(t)\cdots D^{m_n}f(t)=\sum\limits_{j=1}^{m_1+\cdots+m_n+n}\sum\limits_{l=0}^{\left[\frac{m_1+\cdots+m_n+n-j}{2}\right]}a_{jl}t^{2l}D^{j-1}g(t)+F_0(t).$$ Changing the order of the summation, we have $$D^{m_1}f(t)\cdots D^{m_n}f(t)=\sum\limits_{l=0}^{T}\sum\limits_{j=1}^{m_1+\cdots+m_n+n-2l}a_{jl}t^{2l}D^{j-1}g(t)+F_0(t),$$ where $$T=\begin{cases} \left[\frac{n-1}{2}\right] & \text{if\;} m_1=\cdots=m_n=0,\\ \left[\frac{m_1+\cdots+m_n+n-2}{2}\right] & \text{otherwise}. \end{cases}$$ Since $$D^{j-1}g(t)=\sum\limits_{i=0}^\infty(2i)^{j-1}\frac{B_{2i}}{(2i)!}t^{2i},$$ we get \begin{align*} &D^{m_1}f(t)\cdots D^{m_n}f(t)\\ =&\sum\limits_{k=0}^\infty\sum\limits_{l=0}^{\min\{T,k\}}\left(\sum\limits_{j=1}^{m_1+\cdots+m_n+n-2l}a_{jl}(2k-2l)^{j-1}\right)\frac{B_{2k-2l}}{(2k-2l)!}t^{2k}+F_0(t). \end{align*} Then the coefficient of $t^{2k}$ in $D^{m_1}f(t)\cdots D^{m_n}f(t)$ is \begin{align} \sum\limits_{l=0}^{\min\{T,k\}}\left(\sum\limits_{j=1}^{m_1+\cdots+m_n+n-2l}2^{j-1}a_{jl}(k-l)^{j-1}\right)\frac{B_{2k-2l}}{(2k-2l)!}, \label{Eq:Coeff-Right} \end{align} provided that $k\geqslant n$.
On the other hand, since $$D^mf(t)=\sum\limits_{i=1}^{\infty}(2i)^m\frac{B_{2i}}{(2i)!}t^{2i},$$ we find the coefficient of $t^{2k}$ in $D^{m_1}f(t)\cdots D^{m_n}f(t)$ is \begin{align} \sum\limits_{k_1+\cdots+k_n=k\atop k_j\geqslant 1}(2k_1)^{m_1}\cdots(2k_n)^{m_n}\frac{B_{2k_1}\cdots B_{2k_n}}{(2k_1)!\cdots(2k_n)!}. \label{Eq:Coeff-Left} \end{align}
Finally, comparing \eqref{Eq:Coeff-Right} with \eqref{Eq:Coeff-Left}, we get a weighted sum formula of the Bernoulli numbers.
\begin{thm}\label{Thm:WeightedSum-Bernoulli} Let $n,k$ be positive integers with $k\geqslant n$. Then for any nonnegative integers $m_1,\ldots,m_n$, we have \begin{align} &\sum\limits_{k_1+\cdots+k_n=k\atop k_j\geqslant 1}k_1^{m_1}\cdots k_n^{m_n}\frac{B_{2k_1}\cdots B_{2k_n}}{(2k_1)!\cdots(2k_n)!}\nonumber\\ =&\sum\limits_{l=0}^{\min\{T,k\}}\left(\sum\limits_{j=1}^{m_1+\cdots+m_n+n-2l}\frac{a_{jl}}{2^{m_1+\cdots+m_n-j+1}}(k-l)^{j-1}\right)\frac{B_{2k-2l}}{(2k-2l)!}, \label{Eq:WeightedSum-Bernoulli} \end{align} where $T=\max\{[(m_1+\cdots+m_n+n-2)/2],[(n-1)/2]\}$ and $a_{jl}$ are determined by \eqref{Eq:Fj}. \end{thm}
Note that in \cite[Theorem 1]{Petojevic-Srivastava}, A. Petojevi\'{c} and H. M. Srivastava had considered the case of $m_1=\cdots=m_n=0$. See also \cite[Theorems 1 and 2]{Dilcher}.
In the end of this section, we list some explicit examples of $n=4$. Note that some examples of $n=2$ and $n=3$ were given in \cite{Guo-Lei-Zhao}.
\begin{exam}\label{Exe:Bernoulli} Let $k$ be a positive integer with $k\geqslant 4$. Set $\sum=\sum\limits_{k_1+k_2+k_3+k_4=k\atop k_j\geqslant 1}$. We have \begin{align*} &\sum\frac{B_{2k_1}B_{2k_2}B_{2k_3}B_{2k_4}}{(2k_1)!(2k_2)!(2k_3)!(2k_4)!}=-\frac{(k+1)(2k+1)(2k+3)}{3}\frac{B_{2k}}{(2k)!} -\frac{2k}{3}\frac{B_{2k-2}}{(2k-2)!},\\ &\sum k_1^2\frac{B_{2k_1}B_{2k_2}B_{2k_3}B_{2k_4}}{(2k_1)!(2k_2)!(2k_3)!(2k_4)!}=-\frac{k(k+1)(2k+1)(2k+3)(4k+3)}{120}\frac{B_{2k}}{(2k)!}\\ &\qquad\qquad-\frac{k(4k^2-6k+3)}{24}\frac{B_{2k-2}}{(2k-2)!}-\frac{2k-5}{160}\frac{B_{2k-4}}{(2k-4)!},\\ &\sum k_1^3\frac{B_{2k_1}B_{2k_2}B_{2k_3}B_{2k_4}}{(2k_1)!(2k_2)!(2k_3)!(2k_4)!}=-\frac{k(k+1)(2k+1)(2k+3)(4k^2+6k+1)}{240} \frac{B_{2k}}{(2k)!}\\ &\qquad-\frac{k(12k^3-12k^2-11k+9)}{96}\frac{B_{2k-2}}{(2k-2)!}-\frac{(2k-5)(13k-9)}{960}\frac{B_{2k-4}}{(2k-4)!}. \end{align*} Set $B_{k_1,k_2,k_3,k_4}=\frac{B_{2k_1}B_{2k_2}B_{2k_3}B_{2k_4}}{(2k_1)!(2k_2)!(2k_3)!(2k_4)!}$. Using the formulas \begin{align*} &\sum k_1 B_{k_1,k_2,k_3,k_4} =\frac{k}{4}\sum B_{k_1,k_2,k_3,k_4},\\ &\sum k_1k_2 B_{k_1,k_2,k_3,k_4} =\frac{k^2}{12}\sum B_{k_1,k_2,k_3,k_4}-\frac{1}{3}\sum k_1^2 B_{k_1,k_2,k_3,k_4},\\ &\sum k_1^2k_2 B_{k_1,k_2,k_3,k_4} =\frac{k}{3}\sum k_1^2B_{k_1,k_2,k_3,k_4}-\frac{1}{3}\sum k_1^3 B_{k_1,k_2,k_3,k_4},\\ &\sum k_1k_2k_3 B_{k_1,k_2,k_3,k_4} =\frac{k^3}{24}\sum B_{k_1,k_2,k_3,k_4}-\frac{k}{2}\sum k_1^2 B_{k_1,k_2,k_3,k_4}\\ &\qquad\qquad\qquad\qquad+\frac{1}{3}\sum k_1^3 B_{k_1,k_2,k_3,k_4}, \end{align*} one can work out other weighted sum formulas of the Bernoulli numbers with the condition $m_1+m_2+m_3+m_4\leqslant 3$. \end{exam}
\section{Weighted sum formulas of zeta values at even arguments}\label{Sec:WeightSum-Zeta}
Euler's formula claims that for any positive integer $k$, \begin{align} \zeta(2k)=(-1)^{k+1}\frac{B_{2k}}{2(2k)!}(2\pi)^{2k}. \label{Eq:Euler-Formula} \end{align} Then from Theorem \ref{Thm:WeightedSum-Bernoulli}, we get the following weighted sum formula for zeta values at even arguments.
\begin{thm}\label{Thm:WeightedSum-Zeta} Let $n,k$ be positive integers with $k\geqslant n$. Then for any nonnegative integers $m_1,\ldots,m_n$, we have \begin{align} &\sum\limits_{k_1+\cdots+k_n=k\atop k_j\geqslant 1}k_1^{m_1}\cdots k_n^{m_n}\zeta(2k_1)\cdots\zeta(2k_n)=(-1)^n\sum\limits_{l=0}^{\min\{T,k\}}\frac{(2l)!}{B_{2l}}\nonumber\\ &\times\left(\sum\limits_{j=1}^{m_1+\cdots+m_n+n-2l}\frac{a_{jl}}{2^{m_1+\cdots+m_n+n-j-1}}(k-l)^{j-1}\right)\zeta(2l)\zeta(2k-2l), \label{Eq:WeightedSum-Zeta} \end{align} where $T=\max\{[(m_1+\cdots+m_n+n-2)/2],[(n-1)/2]\}$ and $a_{jl}$ are determined by \eqref{Eq:Fj}. \end{thm}
Finally, we obtain the weighted sum formula \eqref{Eq:WeightedSum-Zeta-Conj}.
\begin{thm}\label{Thm:WeightedSum-Zeta-General} Let $n,k$ be positive integers with $k\geqslant n$. Let $F(x_1,\ldots,x_n)\in\mathbb{Q}[x_1,\ldots,x_n]$ be a polynomial of degree $r$. Then we have $$\sum\limits_{k_1+\cdots+k_n=k\atop k_j\geqslant 1}F(k_1,\ldots,k_n)\zeta(2k_1)\cdots \zeta(2k_n)=\sum\limits_{l=0}^{\min\{T,k\}}e_{F,l}(k)\zeta(2l)\zeta(2k-2l),$$ where $T=\max\{[(r+n-2)/2],[(n-1)/2]\}$, $e_{F,l}(x)\in\mathbb{Q}[x]$ depends only on $l$ and $F$, and $\deg e_{F,l}(x)\leqslant r+n-2l-1$. \end{thm}
Note that the polynomial $F(x_1,\ldots,x_n)$ in Theorem \ref{Thm:WeightedSum-Zeta-General} need not be symmetric, and the upper bound for the degree of the polynomial $e_{F,l}(x)$ is different from that in Conjecture \ref{Conj:WeightedSum}. In Conjecture \ref{Conj:WeightedSum}, the upper bound for $\deg e_{F,l}(x)$ is $r-1$, which should be a typo. See the examples below.
\begin{exam} Set $\sum=\sum\limits_{k_1+k_2+k_3+k_4=k\atop k_j\geqslant 1}$. For a positive integer $k$ with $k\geqslant 4$, we have \begin{align*} &\sum \zeta(2k_1)\zeta(2k_2)\zeta(2k_3)\zeta(2k_4)=\frac{(k+1)(2k+1)(2k+3)}{24}\zeta(2k) -2k\zeta(2)\zeta(2k-2),\\ &\sum k_1^2\zeta(2k_1)\zeta(2k_2)\zeta(2k_3)\zeta(2k_4)=\frac{k(k+1)(2k+1)(2k+3)(4k+3)}{960}\zeta(2k)\\ &\qquad\qquad-\frac{k(4k^2-6k+3)}{8}\zeta(2)\zeta(2k-2)+\frac{9(2k-5)}{8}\zeta(4)\zeta(2k-4),\\ &\sum k_1^3\zeta(2k_1)\zeta(2k_2)\zeta(2k_3)\zeta(2k_4)=\frac{k(k+1)(2k+1)(2k+3)(4k^2+6k+1)}{1920} \zeta(2k)\\ &\quad-\frac{k(12k^3-12k^2-11k+9)}{32}\zeta(2)\zeta(2k-2)+\frac{3(2k-5)(13k-9)}{16}\zeta(4)\zeta(2k-4), \end{align*} which can deduce all other weighted sums \eqref{Eq:WeightedSum-Zeta} under the conditions $n=4$ and $m_1+m_2+m_3+m_4\leqslant 3$ as explained in Example \ref{Exe:Bernoulli}. \end{exam}
\section{Weighted sum formulas of multiple zeta values with even arguments}\label{Sec:WeightSum-MZV}
To treat the weighted sum of the multiple zeta values with even arguments and its zeta-star analogue, we recall the symmetric sum formulas of M. E. Hoffman \cite[Theorems 2.1 and 2.2]{Hoffman1992}. For a partition $\Pi=\{P_1,P_2,\ldots,P_i\}$ of the set $\{1,2,\ldots,n\}$, let $l_j=\sharp P_j$ and $$c(\Pi)=\prod\limits_{j=1}^i (l_j-1)!,\quad \tilde{c}(\Pi)=(-1)^{n-i}c(\Pi).$$ We also denote by $\mathcal{P}_n$ the set of all partitions of the set $\{1,2,\ldots,n\}$. Then the symmetric sum formulas are \begin{align} \sum\limits_{\sigma\in S_n}\zeta(k_{\sigma(1)},\ldots,k_{\sigma(n)})=\sum\limits_{\Pi\in\mathcal{P}_n}\tilde{c}(\Pi)\zeta(\mathbf{k},\Pi) \label{Eq:SymSum-MZV} \end{align} and \begin{align} \sum\limits_{\sigma\in S_n}\zeta^{\star}(k_{\sigma(1)},\ldots,k_{\sigma(n)})=\sum\limits_{\Pi\in\mathcal{P}_n}c(\Pi)\zeta(\mathbf{k},\Pi), \label{Eq:SymSum-MZSV} \end{align} where $\mathbf{k}=(k_1,\ldots,k_n)$ is a sequence of positive integers with all $k_i>1$, $S_n$ is the symmetric group of degree $n$ and for a partition $\Pi=\{P_1,\ldots,P_i\}\in\mathcal{P}_n$, $$\zeta(\mathbf{k},\Pi)=\prod\limits_{j=1}^i\zeta\left(\sum\limits_{l\in P_j}k_l\right).$$
Now let $\mathbf{k}=(2k_1,\ldots,2k_n)$ with all $k_i$ positive integers. Using \eqref{Eq:SymSum-MZV} and \eqref{Eq:SymSum-MZSV}, we have \begin{align} &\sum\limits_{\sigma\in S_n}\zeta(2k_{\sigma(1)},\ldots,2k_{\sigma(n)})\nonumber\\ =&\sum\limits_{i=1}^n(-1)^{n-i}\sum\limits_{l_1+\cdots+l_i=n\atop l_1\geqslant \cdots\geqslant l_i\geqslant 1}\prod\limits_{j=1}^i(l_j-1)!\sum\limits_{\Pi=\{P_1,\ldots,P_i\}\in\mathcal{P}_n\atop \sharp{P_j}=l_j}\zeta(\mathbf{k},\Pi) \label{Eq:SymSum-2-MZV} \end{align} and \begin{align} &\sum\limits_{\sigma\in S_n}\zeta^{\star}(2k_{\sigma(1)},\ldots,2k_{\sigma(n)})\nonumber\\ =&\sum\limits_{i=1}^n\sum\limits_{l_1+\cdots+l_i=n\atop l_1\geqslant \cdots\geqslant l_i\geqslant 1 }\prod\limits_{j=1}^i(l_j-1)!\sum\limits_{\Pi=\{P_1,\ldots,P_i\}\in\mathcal{P}_n\atop \sharp{P_j}=l_j}\zeta(\mathbf{k},\Pi). \label{Eq:SymSum-2-MZSV} \end{align}
From now on, let $k,n$ be fixed positive integers with $k\geqslant n$, and let $F(x_1,\ldots,x_n)$ be a fixed symmetric polynomial with rational coefficients. It is easy to see that \begin{align*} &\sum\limits_{k_1+\cdots+k_n=k\atop k_j\geqslant 1}F(k_1,\ldots,k_n)\sum\limits_{\sigma\in S_n}\zeta(2k_{\sigma(1)},\ldots,2k_{\sigma(n)})\\ =&n!\sum\limits_{k_1+\cdots+k_n=k\atop k_j\geqslant 1}F(k_1,\ldots,k_n)\zeta(2k_1,\ldots,2k_n) \end{align*} and \begin{align*} &\sum\limits_{k_1+\cdots+k_n=k\atop k_j\geqslant 1}F(k_1,\ldots,k_n)\sum\limits_{\sigma\in S_n}\zeta^{\star}(2k_{\sigma(1)},\ldots,2k_{\sigma(n)})\\ =&n!\sum\limits_{k_1+\cdots+k_n=k\atop k_j\geqslant 1}F(k_1,\ldots,k_n)\zeta^{\star}(2k_1,\ldots,2k_n). \end{align*} On the other hand, for a partition $\Pi=\{P_1,\ldots,P_i\}\in\mathcal{P}_n$ with $\sharp P_j=l_j$, we have \begin{align} &\sum\limits_{k_1+\cdots+k_n=k\atop k_j\geqslant 1}F(k_1,\ldots,k_n)\zeta(\mathbf{k},\Pi)\nonumber\\ =&\sum\limits_{t_1+\cdots+t_i=k\atop t_j\geqslant 1}\sum\limits_{{{k_1+\cdots+k_{l_1}=t_1\atop\vdots}\atop k_{l_1+\cdots+l_{i-1}+1}+\cdots+k_n=t_i}\atop k_j\geqslant 1}F(k_1,\ldots,k_n)\zeta(2t_1)\cdots\zeta(2t_i). \label{Eq:F-times-zeta} \end{align} To treat the inner sum about $F(k_1,\ldots,k_n)$ in the right-hand side of \eqref{Eq:F-times-zeta}, we need the following lemmas.
\begin{lem}\label{Lem:PowerSum-2} For any positive integer $k$ and any nonnegative integers $p_1,p_2$, we have \begin{align} &\sum\limits_{i=1}^{k-1} i^{p_1}(k-i)^{p_2}\nonumber\\ =&\sum\limits_{i,j\geqslant 0\atop p_1\leqslant i+j\leqslant p_1+p_2}(-1)^{j+p_1}\binom{i+j}{i}\binom{p_2}{i+j-p_1}\frac{B_i}{j+1}(k-1)^{j+1}k^{p_1+p_2-i-j}. \label{Eq:PowerSum-2} \end{align} In particular, the right-hand side of \eqref{Eq:PowerSum-2} is a polynomial of $k$ with rational coefficients of degree $p_1+p_2+1$. \end{lem}
\noindent {\bf Proof.\;} Let $S_{p_1,p_2}(k)=\sum\limits_{i=1}^{k-1} i^{p_1}(k-i)^{p_2}$ and let $$G_k(t_1,t_2)=\sum\limits_{p_1,p_2\geqslant 0}S_{p_1,p_2}(k)\frac{t_1^{p_1}t_2^{p_2}}{p_1!p_2!}$$ be the generating function. We have $$G_k(t_1,t_2)=\sum\limits_{i=1}^{k-1}e^{it_1+(k-i)t_2}=\frac{(1-e^{(k-1)(t_1-t_2)})e^{kt_2}}{e^{t_2-t_1}-1}.$$ Using the definition of the Bernoulli numbers, we get \begin{align*} G_k(t_1,t_2)=&\sum\limits_{i\geqslant 0,j\geqslant 1,l\geqslant 0}(-1)^{i}\frac{B_i}{i!j!l!}(k-1)^jk^l(t_1-t_2)^{i+j-1}t_2^l\\ =&\sum\limits_{i,j,l\geqslant 0}(-1)^{i}\frac{B_i}{i!(j+1)!l!}(k-1)^{j+1}k^l(t_1-t_2)^{i+j}t_2^l. \end{align*} Finally, we obtain the expansion $$G_k(t_1,t_2)=\sum\limits_{i,j,l\geqslant 0\atop 0\leqslant m\leqslant i+j}(-1)^{j+m}\binom{i+j}{m}\frac{B_i}{i!(j+1)!l!}(k-1)^{j+1}k^lt_1^mt_2^{i+j+l-m}.$$ Comparing the coefficient of $\frac{t_1^{p_1}t_2^{p_2}}{p_1!p_2!}$, we get \eqref{Eq:PowerSum-2}.
Then as a polynomial of $k$, the degree of the right-hand side of \eqref{Eq:PowerSum-2} is less than or equal to $p_1+p_2+1$, and the coefficient of $k^{p_1+p_2+1}$ is $$\sum\limits_{j=p_1}^{p_1+p_2}(-1)^{j+p_1}\binom{p_2}{j-p_1}\frac{1}{j+1},$$ which is $$\sum\limits_{j=0}^{p_2}(-1)^{j}\binom{p_2}{j}\frac{1}{j+p_1+1}.$$ Since $$x^{p_1}(1-x)^{p_2}=\sum\limits_{j=0}^{p_2}(-1)^{j}\binom{p_2}{j}x^{j+p_1},$$ we find the coefficient of $k^{p_1+p_2+1}$ is $$\int_0^1x^{p_1}(1-x)^{p_2}dx=B(p_1+1,p_2+1)=\frac{p_1!p_2!}{(p_1+p_2+1)!},$$ which is nonzero. \qed
More generally, we have
\begin{lem}\label{Lem:PowerSum} Let $k$ and $n$ be integers with $k\geqslant n\geqslant 1$, and let $p_1,\ldots,p_n$ be nonnegative integers. Then there exists a polynomial $f(x)\in\mathbb{Q}[x]$ of degree $p_1+\cdots+p_n+n-1$, such that $$\sum\limits_{k_1+\cdots+k_n=k\atop k_j\geqslant 1}k_1^{p_1}\cdots k_n^{p_n}=f(k).$$ \end{lem}
\noindent {\bf Proof.\;} We proceed by induction on $n$. If $n=1$, we may take $f(x)=x^{p_1}$. For $n>1$, since $$\sum\limits_{k_1+\cdots+k_n=k\atop k_j\geqslant 1}k_1^{p_1}\cdots k_n^{p_n}=\sum\limits_{k_1+k_2=k\atop k_j\geqslant 1}\left(\sum\limits_{l_1+\cdots+l_{n-1}=k_1\atop l_j\geqslant 1}l_1^{p_1}\cdots l_{n-1}^{p_{n-1}}\right)k_2^{p_n},$$ using the induction assumption we have $$\sum\limits_{k_1+\cdots+k_n=k\atop k_j\geqslant 1}k_1^{p_1}\cdots k_n^{p_n}=\sum\limits_{k_1+k_2=k\atop k_j\geqslant 1}g(k_1)k_2^{p_n},$$ where $g(x)\in\mathbb{Q}[x]$ is of degree $p_1+\cdots+p_{n-1}+n-2$. Then the result follows from Lemma \ref{Lem:PowerSum-2}. \qed
Now we return to the computation of the right-hand side of \eqref{Eq:F-times-zeta}. Using Lemma \ref{Lem:PowerSum}, there exists a polynomial $f_{t_1,\ldots,t_i}(x_1,\ldots,x_i)\in\mathbb{Q}[x_1,\ldots,x_i]$ of degree $\deg F+n-i$, such that \begin{align*} &\sum\limits_{k_1+\cdots+k_n=k\atop k_j\geqslant 1}F(k_1,\ldots,k_n)\zeta(\mathbf{k},\Pi)\\ =&\sum\limits_{t_1+\cdots+t_i=k\atop t_j\geqslant 1}f_{t_1,\ldots,t_i}(t_1,\ldots,t_i)\zeta(2t_1)\cdots\zeta(2t_i). \end{align*} Therefore we get \begin{align*} &\sum\limits_{k_1+\cdots+k_n=k\atop k_j\geqslant 1}F(k_1,\ldots,k_n)\zeta(2k_1,\ldots,2k_n) =\frac{1}{n!}\sum\limits_{i=1}^n(-1)^{n-i}\sum\limits_{l_1+\cdots+l_i=n\atop l_1\geqslant\cdots \geqslant l_i\geqslant 1}\\ &\times\prod\limits_{j=1}^i(l_j-1)!n(l_1,\ldots,l_i)\sum\limits_{t_1+\cdots+t_i=k\atop t_j\geqslant 1}f_{t_1,\ldots,t_i}(t_1,\ldots,t_i)\zeta(2t_1)\cdots\zeta(2t_i) \end{align*} and \begin{align*} &\sum\limits_{k_1+\cdots+k_n=k\atop k_j\geqslant 1}F(k_1,\ldots,k_n)\zeta^{\star}(2k_1,\ldots,2k_n) =\frac{1}{n!}\sum\limits_{i=1}^n\sum\limits_{l_1+\cdots+l_i=n\atop l_1\geqslant\cdots\geqslant l_i\geqslant 1}\\ &\times\prod\limits_{j=1}^i(l_j-1)!n(l_1,\ldots,l_i)\sum\limits_{t_1+\cdots+t_i=k\atop t_j\geqslant 1}f_{t_1,\ldots,t_i}(t_1,\ldots,t_i)\zeta(2t_1)\cdots\zeta(2t_i), \end{align*} where $$n(l_1,\ldots,l_i)=\frac{n!}{\prod\limits_{j=1}^il_j!\prod\limits_{j=1}^n\sharp\{m\mid 1\leqslant m\leqslant i,k_m=j\}!}$$ is the number of partitions $\Pi=\{P_1,\ldots,P_i\}\in\mathcal{P}_n$ with the conditions $\sharp P_j=l_j$ for $j=1,2,\ldots,i$.
Applying Theorem \ref{Thm:WeightedSum-Zeta-General}, we then prove the weighted sum formula \eqref{Eq:WeightedSum-MZV-Conj} and its zeta-star analogue.
\begin{thm}\label{Thm:WeightedSum-MZV} Let $n,k$ be positive integers with $k\geqslant n$. Let $F(x_1,\ldots,x_n)\in\mathbb{Q}[x_1,\ldots,x_n]$ be a symmetric polynomial of degree $r$. Then we have $$\sum\limits_{k_1+\cdots+k_n=k\atop k_j\geqslant 1}F(k_1,\ldots,k_n)\zeta(2k_1,\ldots,2k_n)=\sum\limits_{l=0}^{\min\{T,k\}}c_{F,l}(k)\zeta(2l)\zeta(2k-2l)$$ and $$\sum\limits_{k_1+\cdots+k_n=k\atop k_j\geqslant 1}F(k_1,\ldots,k_n)\zeta^{\star}(2k_1,\ldots,2k_n)=\sum\limits_{l=0}^{\min\{T,k\}}c_{F,l}^{\star}(k)\zeta(2l)\zeta(2k-2l),$$ where $T=\max\{[(r+n-2)/2],[(n-1)/2]\}$, $c_{F,l}(x),c_{F,l}^{\star}(x)\in\mathbb{Q}[x]$ depend only on $l$ and $F$, and $\deg c_{F,l}(x),\deg c_{F,l}^{\star}(x)\leqslant r+n-2l-1$. \end{thm}
Note that in Theorem \ref{Thm:WeightedSum-MZV}, the upper bound for the polynomial $c_{F,l}(x)$ is different from that in Conjecture \ref{Conj:WeightedSum}. In Conjecture \ref{Conj:WeightedSum}, the upper bound for $\deg c_{F,l}(x)$ is $\deg_{x_1}F(x_1,\ldots,x_n)$. It seems that one may obtain this upper bound but need more efforts.
\begin{exam} After getting the weighted sum formulas \eqref{Eq:WeightedSum-Zeta} with $n=2$ and $n=3$, we can obtain the weighted sum formulas of the multiple zeta values (resp. the multiple zeta-star values) of depth four. Here are some examples. For multiple zeta values, we have \begin{align*} &\sum\zeta(2k_1,2k_2,2k_3,2k_4)=\frac{35}{64}\zeta(2k)-\frac{5}{16}\zeta(2)\zeta(2k-2),\\ &\sum\left(k_1^2+k_2^2+k_3^2+k_4^2\right)\zeta(2k_1,2k_2,2k_3,2k_4)=\frac{7k(10k-3)}{128}\zeta(2k)\\ &\qquad-\frac{10k^2+9k-30}{32}\zeta(2)\zeta(2k-2)+\frac{3(2k-5)}{16}\zeta(4)\zeta(2k-4),\\ &\sum\left(k_1^3+k_2^3+k_3^3+k_4^3\right)\zeta(2k_1,2k_2,2k_3,2k_4)=\frac{7k(40k^2-18k+3)}{512}\zeta(2k)\\ &\qquad-\frac{40k^3+54k^2-174k+15}{128}\zeta(2)\zeta(2k-2)+\frac{3(2k-5)(3k+2)}{32}\zeta(4)\zeta(2k-4), \end{align*} and for multiple zeta-star values, we have \begin{align*} &\sum\zeta^{\star}(2k_1,2k_2,2k_3,2k_4)=\frac{(4k-5)(8k^2-20k+3)}{192}\zeta(2k)\\ &\qquad\qquad-\frac{4k-7}{16}\zeta(2)\zeta(2k-2),\\ &\sum\left(k_1^2+k_2^2+k_3^2+k_4^2\right)\zeta^{\star}(2k_1,2k_2,2k_3,2k_4)\\ =&\frac{k(128k^4-600k^3+920k^2-600k+227)}{1920}\zeta(2k)\\ &\qquad-\frac{(2k-3)(16k^2-63k+68)}{96}\zeta(2)\zeta(2k-2)-\frac{2k-5}{16}\zeta(4)\zeta(2k-4),\\ &\sum\left(k_1^3+k_2^3+k_3^3+k_4^3\right)\zeta^{\star}(2k_1,2k_2,2k_3,2k_4)\\ =&\frac{k(256k^5-1440k^4+2760k^3-2400k^2+1664k-435)}{7680}\zeta(2k)\\ &\qquad-\frac{32k^4-184k^3+318k^2-136k-51}{128}\zeta(2)\zeta(2k-2)\\ &\qquad+\frac{15(k-4)(2k-5)}{32}\zeta(4)\zeta(2k-4). \end{align*} Here $k$ is a positive integer with $k\geqslant 4$ and $\sum=\sum\limits_{k_1+k_2+k_3+k_4=k\atop k_j\geqslant 1}$. \end{exam}
\section{Regularized double shuffle relations and weighted sum formulas}\label{Sec:RegDouble-WeightSum}
In this section, we briefly explain that the weighted sum formulas in Theorems \ref{Thm:WeightedSum-Zeta}, \ref{Thm:WeightedSum-Zeta-General} and \ref{Thm:WeightedSum-MZV} can be deduced from the regularized double shuffle relations of the multiple zeta values (For the details of the regularized double shuffle relations, one can refer to \cite{Ihara-Kaneko-Zagier,Racinet} or \cite{Li-Qin}).
We get Theorem \ref{Thm:WeightedSum-Zeta} and hence Theorem \ref{Thm:WeightedSum-Zeta-General} just from \eqref{Eq:WeightedSum-Bernoulli} and Euler's formula \eqref{Eq:Euler-Formula}. While \eqref{Eq:WeightedSum-Bernoulli} is an equation about the Bernoulli numbers and Euler's formula can be deduced from the regularized double shuffle relations (\cite{Li-Qin}). Hence we get Theorems \ref{Thm:WeightedSum-Zeta} and \ref{Thm:WeightedSum-Zeta-General} from the regularized double shuffle relations.
We get Theorem \ref{Thm:WeightedSum-MZV} from Theorem \ref{Thm:WeightedSum-Zeta-General} and the symmetric sum formulas. While the symmetric sum formulas are consequences of the harmonic shuffle products (\cite[Theorem 2.3]{Hoffman2015}). In fact, let $Y=\{z_k\mid k=1,2,\ldots\}$ be an alphabet with noncommutative letters and let $Y^{\ast}$ be the set of all words generated by letters in $Y$, which contains the empty word $1_Y$. Let $\mathfrak{h}^1=\mathbb{Q}\langle Y\rangle$ be the noncommutative polynomial algebra over $\mathbb{Q}$ generated by $Y$. As in \cite{Hoffman1997,Muneta}, we define two bilinear commutative products $\ast$ and $\bar{\ast}$ on $\mathfrak{h}^1$ by the rules \begin{align*} &1_Y\ast w=w\ast 1_Y=w,\\ &z_kw_1\ast z_lw_2=z_k(w_1\ast z_lw_2)+z_l(z_kw_1\ast w_2)+z_{k+l}(w_1\ast w_2);\\ &1_Y \,\bar{\ast}\, w=w\,\bar{\ast}\, 1_Y=w,\\ &z_kw_1\,\bar{\ast}\, z_lw_2=z_k(w_1\,\bar{\ast}\, z_lw_2)+z_l(z_kw_1\,\bar{\ast}\, w_2)-z_{k+l}(w_1\,\bar{\ast}\, w_2), \end{align*} where $w,w_1,w_2\in Y^{\ast}$ and $k,l$ are positive integers. Let $$\mathfrak{h}^0=\mathbb{Q}1_Y+\sum\limits_{n,k_1,\ldots,k_n\geqslant 1\atop k_1\geqslant 2}\mathbb{Q}z_{k_1}\cdots z_{k_n}$$ be a subalgebra of $\mathfrak{h}^1$, which is also a subalgebra with respect to either the product $\ast$ or the product $\bar{\ast}$. Let $Z:\mathfrak{h}^0\rightarrow\mathbb{R}$ and $Z^{\star}:\mathfrak{h}^0\rightarrow\mathbb{R}$ be the $\mathbb{Q}$-linear maps determined by $Z(1_Y)=Z^{\star}(1_Y)=1$ and $$Z(z_{k_1}\cdots z_{k_n})=\zeta(k_1,\ldots,k_n),\quad Z^{\star}(z_{k_1}\cdots z_{k_n})=\zeta^{\star}(k_1,\ldots,k_n),$$ where $n,k_1,\ldots,k_n\geqslant 1$ with $k_1\geqslant 2$. It is known that both the maps $Z:(\mathfrak{h}^0,\ast)\rightarrow \mathbb{R}$ and $Z^{\star}:(\mathfrak{h}^0,\bar{\ast})\rightarrow \mathbb{R}$ are algebra homomorphisms. Hence from the following lemma, we know that the symmetric sum formulas are consequences of the harmonic shuffle products. And therefore Theorem \ref{Thm:WeightedSum-MZV} is also deduced from the regularized double shuffle relations.
\begin{lem} Let $n$ be a positive integer and $\mathbf{k}=(k_1,\ldots,k_n)$ be a sequence of positive integers. We have \begin{align} \sum\limits_{\sigma\in S_n}z_{k_{\sigma(1)}}\cdots z_{k_{\sigma(n)}}=\sum\limits_{\Pi=\{P_1,\ldots,P_i\}\in\mathcal{P}_n}\tilde{c}(\Pi)z_{\mathbf{k},P_1}\ast\cdots\ast z_{\mathbf{k},P_i} \label{Eq:SymSum-ast} \end{align} and \begin{align} \sum\limits_{\sigma\in S_n}z_{k_{\sigma(1)}}\cdots z_{k_{\sigma(n)}}=\sum\limits_{\Pi=\{P_1,\ldots,P_i\}\in\mathcal{P}_n}c(\Pi)z_{\mathbf{k},P_1}\bar{\ast}\cdots\bar{\ast} z_{\mathbf{k},P_i}, \label{Eq:SymSum-sast} \end{align} where $z_{\mathbf{k},P_j}=z_{\sum\limits_{l\in P_j}k_l}$. \end{lem}
\noindent {\bf Proof.\;} To be self contained, we give a proof here. We prove \eqref{Eq:SymSum-ast} and one can prove \eqref{Eq:SymSum-sast} similarly. We proceed by induction on $n$. The case of $n=1$ is obvious. Now assume that \eqref{Eq:SymSum-ast} is proved for $n$. Let $\mathbf{k}=(k_1,\ldots,k_n)$ and $\mathbf{k}'=(k_1,\ldots,k_{n+1})$. Since \begin{align*} &z_{k_{n+1}}\ast \sum\limits_{\sigma\in S_n}z_{k_{\sigma(1)}}\cdots z_{k_{\sigma(n)}}=\sum\limits_{\sigma\in S_n}\sum\limits_{j=1}^{n+1}z_{k_{\sigma(1)}}\cdots z_{k_{\sigma(j-1)}}z_{k_{n+1}}z_{k_{\sigma(j)}}\cdots z_{k_{\sigma(n)}}\\ &\qquad+\sum\limits_{\sigma\in S_n}\sum\limits_{j=1}^{n}z_{k_{\sigma(1)}}\cdots z_{k_{\sigma(j-1)}}z_{k_{\sigma(j)}+k_{n+1}}z_{k_{\sigma(j+1)}}\cdots z_{k_{\sigma(n)}}\\ =&\sum\limits_{\sigma\in S_{n+1}}z_{k_{\sigma(1)}}\cdots z_{k_{\sigma(n+1)}}+\sum\limits_{j=1}^n\sum\limits_{\sigma\in S_n}z_{k^{(j)}_{\sigma(1)}}\cdots z_{k^{(j)}_{\sigma(n)}} \end{align*} with $$\mathbf{k}^{(j)}=(k_1,\ldots,k_{j-1},k_j+k_{n+1},k_{j+1},\ldots,k_n)=(k_1^{(j)},\ldots,k_n^{(j)}),$$ using the induction assumption on $\mathbf{k}$ and $\mathbf{k}^{(j)}$ with $j=1,\ldots,n$, we have \begin{align*} &\sum\limits_{\sigma\in S_{n+1}}z_{k_{\sigma(1)}}\cdots z_{k_{\sigma(n+1)}}=\sum\limits_{\Pi=\{P_1,\ldots,P_i\}\in\mathcal{P}_n}\tilde{c}(\Pi)z_{\mathbf{k},P_1}\ast\cdots\ast z_{\mathbf{k},P_i}\ast z_{k_{n+1}}\\ &\qquad\quad-\sum\limits_{j=1}^n\sum\limits_{\Pi=\{P_1,\ldots,P_i\}\in\mathcal{P}_n}\tilde{c}(\Pi)z_{\mathbf{k}^{(j)},P_1}\ast\cdots\ast z_{\mathbf{k}^{(j)},P_i}. \end{align*} Because any $\Pi\in\mathcal{P}_{n+1}$ must satisfy and can only satisfy one of the following two conditions: \begin{itemize}
\item [(i)] there exists one $P\in\Pi$, such that $P=\{k_{n+1}\}$;
\item [(ii)] for any $P\in \Pi$, $P\neq \{k_{n+1}\}$, \end{itemize}
we see that the right-hand side of the above equation is just $$\sum\limits_{\Pi=\{P_1,\ldots,P_i\}\in\mathcal{P}_{n+1}}\tilde{c}(\Pi)z_{\mathbf{k}',P_1}\ast\cdots\ast z_{\mathbf{k}',P_i}.$$ Hence we get \eqref{Eq:SymSum-ast}. \qed
\end{document} | arXiv |
\begin{document}
\title{Grouped Domination Parameterized by Vertex Cover, Twin Cover, and Beyond\texorpdfstring{\thanks{ Partially supported by JSPS KAKENHI Grant Numbers JP17H01698, JP17K19960, JP18H04091, JP20H05793, JP20H05967, JP21K11752, JP21H05852, JP21K17707, JP21K19765, and JP22H00513.
}}{}}
\titlerunning{Grouped Domination Parameterized by Structural Parameters}
\author{Tesshu Hanaka\inst{1}\orcidID{0000-0001-6943-856X} \and Hirotaka Ono\inst{2}\orcidID{0000-0003-0845-3947} \and Yota Otachi\inst{2}\orcidID{0000-0002-0087-853X} \and Saeki Uda\inst{2}}
\authorrunning{T. Hanaka et al.}
\institute{Kyushu University, Fukuoka, Japan \email{[email protected]} \and Nagoya University, Nagoya, Japan \email{[email protected], [email protected], [email protected]}}
\maketitle
\begin{abstract} A dominating set $S$ of graph $G$ is called an \emph{$r$-grouped dominating set} if $S$ can be partitioned into $S_1,S_2,\ldots,S_k$ such that the size of each unit $S_i$ is $r$ and the subgraph of $G$ induced by $S_i$ is connected. The concept of $r$-grouped dominating sets generalizes several well-studied variants of dominating sets with requirements for connected component sizes, such as the ordinary dominating sets ($r=1$), paired dominating sets ($r=2$), and connected dominating sets ($r$ is arbitrary and $k=1$). In this paper, we investigate the computational complexity of \textsc{$r$-Grouped Dominating Set}, which is the problem of deciding whether a given graph has an $r$-grouped dominating set with at most $k$ units. For general $r$, \textsc{$r$-Grouped Dominating Set} is hard to solve in various senses because the hardness of the connected dominating set is inherited. We thus focus on the case in which $r$ is a constant or a parameter, but we see that \textsc{$r$-Grouped Dominating Set} for every fixed $r>0$ is still hard to solve. From the observations about the hardness, we consider the parameterized complexity concerning well-studied graph structural parameters. We first see that \textsc{$r$-Grouped Dominating Set} is fixed-parameter tractable for $r$ and treewidth, which is derived from the fact that the condition of $r$-grouped domination for a constant $r$ can be represented as monadic second-order logic (\mso{2}). This fixed-parameter tractability is good news, but the running time is not practical. We then design an $O^*(\min\{(2\tau(r+1))^{\tau},(2\tau)^{2\tau}\})$-time algorithm for general $r\ge 2$, where $\tau$ is the twin cover number, which is a parameter between vertex cover number and clique-width. For paired dominating set and trio dominating set, i.e., $r \in \{2,3\}$, we can speed up the algorithm, whose running time becomes $O^*((r+1)^\tau)$. We further argue the relationship between FPT results and graph parameters, which draws the parameterized complexity landscape of \textsc{$r$-Grouped Dominating Set}.
\keywords{Dominating Set \and Paired Dominating Set \and Parameterized Complexity \and Graph Structural Parameters.} \end{abstract}
\section{Introduction} \subsection{Definition and motivation}\label{sec:intro1} Given an undirected graph $G=(V,E)$, a vertex set $S\subseteq V$ is called a \emph{dominating set} if every vertex in $V$ is either in $S$ or adjacent to a vertex in $S$. The dominating set problem is the problem of finding a dominating set with the minimum cardinality. Since the definition of dominating set, i.e., covering all the vertices via edges, is natural, many practical and theoretical problems are modeled as dominating set problems with additional requirements; many variants of dominating set are considered and investigated. Such variants somewhat generalize or extend the ordinary dominating set based on theoretical or applicational motivations.
In this paper, we focus on variants that require the dominating set to satisfy specific connectivity and size constraints. One example considering connectivity is the connected dominating set. A dominating set is called a \emph{connected dominating set} if the subgraph induced by a dominating set is connected. Another example is the paired dominating set. A paired dominating set is a dominating set of a graph such that the subgraph induced by it admits a perfect matching.
This paper introduces the $r$-grouped dominating set, which generalizes the connected dominating set, the paired dominating set, and some other variants. A dominating set $S$ is called an \emph{$r$-grouped dominating set} if $S$ can be partitioned into $\{S_1,S_2,\ldots,S_k\}$ such that each $S_i$ is a set of $r$ vertices and $G[S_i]$ is connected. We call each $S_i$ a \emph{unit}. The $r$-grouped dominating set generalizes both the connecting dominating set and the paired dominating set in the following sense: a connecting dominating set with $r$ vertices is equivalent to an $r$-grouped dominating set of one unit, and a paired dominating set with $k$ pairs is equivalent to a $2$-grouped dominating set with $k$ units.
This paper investigates the parameterized complexity of deciding whether a given graph has an $r$-grouped dominating set with $k$ units. The parameters that we focus on are so-called graph structural parameters, such as vertex cover number and twin-cover number. The results obtained in this paper are summarized in Our Contribution (Section \ref{sec:contribution}).
\subsection{Related work} An enormous number of papers study the dominating set problem, including the ones strongly related to the $r$-grouped dominating set.
The dominating set problem is one of the most important graph optimization problems. Due to its NP-hardness, its tractability is finely studied from several aspects, such as approximation, solvable graph classes, fast exact exponential-time solvability, and parameterized complexity. Concerning the parameterized complexity, the dominating set problem is W[2]-complete for solution size $k$; it is unlikely to be fixed-parameter tractable~\cite{CyganFKLMPPS15}.
On the other hand, since the dominating set can be expressed in \mso{1}, it is FPT when parametrized by clique-width or treewidth (see, e.g., \cite{Kreutzer11}).
The connected dominating set is a well-studied variant of dominating set. This problem arises in communication and computer networks such as mobile ad hoc networks. It is also W[2]-hard when parameterized by the solution size \cite{CyganFKLMPPS15}. Furthermore, the connected dominating set also can be expressed in \mso{1}; it is FPT when parametrized by clique-width and treewidth as in the ordinary dominating set problem. Furthermore, single exponential-time algorithms for connected dominating set parameterized by treewidth can be obtained by the Cut \& Count technique \cite{CNPMMVW2022} or the rank-based approach \cite{BCKN2015}.
The notion of the paired dominating set is introduced in \cite{HS1995:paired,HS1998:paired} by Haynes and Slater as a model of dominating sets with pairwise backup. It is NP-hard on split graphs, bipartite graphs~\cite{CLZ2010:paierd}, graphs of maximum degree 3~\cite{CLZ2009:paired:approx}, and planar graphs of maximum degree 5 \cite{TripathiKPPW22}, whereas it can be solved in polynomial time on strongly-chordal graphs~\cite{CLZ2009:paierd}, distance-hereditary graphs~\cite{LKH2020:paired}, and AT-free graphs~\cite{TripathiKPPW22}. There are several graph classes (e.g., strongly orderable graphs \cite{PP2019:paired}) where the paired dominating set problem is tractable, whereas the ordinary dominating set problem remains NP-hard. For other results about the paired dominating set, see a survey~\cite{Desormeaux2020}.
\subsection{Our contributions}\label{sec:contribution} This paper provides a unified view of the parameterized complexity of dominating set problem variants with connectivity and size constraints.
As mentioned above, an $r$-grouped dominating set of $G$ with $1$ unit is equivalent to a connected dominating set with size $r$, which implies that some hardness results of \textsc{$r$-Grouped Dominating Set} for general $r$ are inherited directly from \textsc{Connected Dominating Set}. From these, we mainly consider the case where $r$ is a constant or a parameter.
Unfortunately, \textsc{$r$-Grouped Dominating Set} for $r=1,2$ is also hard to solve again because \textsc{$1$-Grouped Dominating Set} and \textsc{$2$-Grouped Dominating Set} are respectively the ordinary dominating set problem and the paired dominating set problem. Thus, it is worth considering whether a larger but constant $r$ enlarges, restricts, or leaves unchanged the graph classes for which similar hardness results hold. A way to classify or characterize graphs of certain classes is to focus on graph-structural parameters. By observing that the condition of $r$-grouped dominating set can be represented as monadic second-order logic (\mso{2}), we can see that \textsc{$r$-Grouped Dominating Set} is fixed-parameter tractable for $r$ and treewidth. Recall that the condition of the connected dominating set can be represented as monadic second-order logic (\mso{1}), which implies that there might exist a gap between $r=1$ and $2$, or between $k=1$ and $k>1$. Although this FPT result is good news, its time complexity is not practical. From these observations, we focus on less generalized graph structural parameters, vertex cover number $\nu$ or twin cover number $\tau$ as a parameter, and design single exponential fixed-parameter algorithms for \textsc{$r$-Grouped Dominating Set}.
Our algorithm is based on dynamic programming on nested partitions of a vertex cover, and its running time is $O^*(\min\{(2\nu(r+1))^{\nu},(2\nu)^{2\nu}\})$ for general $r\ge 2$. For paired dominating set and trio dominating set, i.e., $r \in \{2,3\}$, we can tailor the algorithm to run in $O^*((r+1)^\nu)$ time by observing that the nested partitions of a vertex cover degenerate in some sense.
We then turn our attention to a more general parameter, the twin cover number. We show that, given a twin cover, \textsc{$r$-Grouped Dominating Set} admits an optimal solution in which twin-edges do not contribute to the connectivity of $r$-units. This observation implies that these edges can be removed from the graph, and thus we can focus on the resultant graph of bounded vertex cover number. Hence, we can conclude that our algorithms still work when the parameter $\nu$ in the running time is replaced with twin cover number $\tau$.
We further argue the relationship between FPT results and graph parameters. The perspective is summarized in Figure \ref{fig:parameters0}, which draws the parameterized complexity landscape of \textsc{$r$-Grouped Dominating Set}.
\begin{figure}
\caption{The complexity of $r$-\textsc{Grouped Dominating Set} with respect to structural graph parameters.
An edge between two parameters indicates that there is a function in the one above
that lower-bounds the one below (e.g., $\text{treewidth} \le \text{pathwidth}$).}
\label{fig:parameters0}
\end{figure}
\section{Preliminaries}
Let $G=(V,E)$ be an undirected graph. For a vertex subset $V'\subseteq V$, the subgraph induced by $V'$ is denoted by $G[V']$. Also, let us denote by $N(v)$ and $N[v]$ the open neighborhood and the closed neighborhood of $v$, respectively. The degree of a vertex $v$ is defined by $d(v) = |N(v)|$. The maximum degree of $G$ is denoted by $\Delta$.
A vertex set $S$ is a \emph{vertex cover} of $G$ if for every edge $\{u,v\}\in E$, at least one of $u,v$ is in $S$. The \emph{vertex cover number} $\nu$ of $G$ is defined by the size of a minimum vertex cover of $G$. A minimum vertex cover of $G$ can be found in $O^*(1.2738^{\nu})$ time~\cite{CKX2010}.\footnote{The $O^*$ notation suppresses the polynomial factors of the input size.}
Two vertices $u$ and $v$ are (\emph{true}) \emph{twins} if $N[u]=N[v]$. An edge $\{u,v\}\in E$ is a \emph{twin edge} if $u$ and $v$ are true twins. A vertex set $S$ is a \emph{twin cover} if for every edge $\{u,v\}\in E$, either $\{u,v\}$ is a twin edge, or at least one of $u,v$ is in $S$. The size $\tau$ of a minimum twin cover of $G$ is called the \emph{twin cover number} of $G$. A minimum twin cover of $G$ can be found in $O^*(1.2738^{\tau})$ time~\cite{Ganian2015}.
We briefly introduce basic terminology of parameterized complexity. Given an input size $n$ and a parameter $k$, a problem is \emph{fixed-parameter tractable (FPT)} if it can be solved in $f(k)n^{O(1)}$ time where $f$ is some computable function. Also, a problem is \emph{slice-wise polynomial (XP)} if it can be solved in $n^{f(k)}$ time. See standard textbooks (e.g., \cite{CyganFKLMPPS15}) for more details.
\subsection{$r$-Grouped Dominating Set}
An \emph{$r$-grouped dominating set with $k$ units} in $G$ is a family $\mathcal{D} = \{D_1, \ldots, D_k\}$ of subsets of $V$ such that $D_i$'s are mutually disjoint, $|D_i|=r$, $G[D_i]$ is connected for $1\le i\le k$, and $\bigcup_{D\in \mathcal{D}} D$ is a dominating set of $G$. For simplicity, let $\bigcup \mathcal{D}$ denote $\bigcup_{D\in \mathcal{D}} D$.
We say that $\mathcal{D}$ is a minimum $r$-grouped dominating set if it is an $r$-grouped dominating set with the minimum number of units.
\begin{myproblem}
\problemtitle{$r$-\textsc{Grouped Dominating Set}}
\probleminput{A graph $G$ and positive integers $r$ and $k$.}
\problemquestion{Is there an $r$-grouped dominating set with at most $k$ units in $G$?} \end{myproblem}
\section{Basic Results} In this section, we prove $r$-\textsc{Grouped Dominating Set} is W[2]-hard but XP when parameterized by $k+r$ and it is NP-hard even on planar bipartite graphs of maximum degree 3.
We first observe that finding an $r$-grouped dominating set with at most $1$ unit is equivalent to finding a connected dominating set of size $r$. Thus, the W[2]-hardness of \textsc{$r$-Grouped Dominating Set} parameterized by $r$ follows the one of \textsc{Connected Dominating Set} parameterized by the solution size. Also, the case $r = 1$ follows immediately from the hardness of the ordinary \textsc{Dominating Set}, which is W[2]-complete on split graphs and bipartite graphs \cite{Raman2008}. In the remaining part of this section, we discuss the hardness results only for the cases $r\ge 2$ and $k\ge 2$.
\begin{theorem}\label{thm:W[2]:r} For every fixed $k\ge 1$, \textsc{$r$-Grouped Dominating Set} is W[2]-hard when parameterized by $r$ even on split graphs. \end{theorem}
\begin{proof}
We give a reduction from \textsc{Dominating Set} on split graphs. Let $\langle G=(C\cup I, E),r\rangle$ be an instance of \textsc{Dominating Set} where $C$ forms a clique and $I$ forms an independent set. Without loss of generality, we suppose that $|C|\ge 2$ and $|I|\ge 2$. We create $k$ copies $G_1=(V_1, E_1), \ldots, G_k=(V_k, E_k)$ of $G$ where $V_i = \{v^{(i)} \mid v\in V\}$ and $E_i = \{e^{(i)} \mid e\in E\}$. Note that $V_i = C_i \cup I_i$. Finally, we make $\bigcup_i C_i$ a clique.
The resulting graph $G'$ is clearly a split graph.
We show that there is a dominating set of size at most $r$ in $G$ if and only if there is an $r$-grouped dominating set with at most $k$ units in $G'$.
Suppose that there is a dominating set $D$ of size at most $r$ in $G$. Without loss of generality, we can assume that $D\subseteq C$ \cite{Bertossi1984}. Then we define $D'_i=\{v^{(i)}\mid v\in D\}$ for $1\le i\le k$ and $D' = \bigcup_i D'_i$. Since $D$ is a dominating set in $G$, so is $D'_i$ on $G_i$ for each $i$. Thus, $D'$ is a dominating set in $G'$. Because $G_i$ is a split graph and a clique and $D'_i\subseteq C_i$, $D'_i$ is a connected dominating set of $G_i$ of size at most $r$.
If $|D'_i|<r$, we arbitrarily add $r-|D'_i|$ vertices in $G_i$ to $D'_i$. Then, we have a connected dominating set $D'_i$ of $G_i$ of size exactly $r$ for each $i$, which can be regarded as a unit of size $r$ of an $r$-grouped dominating set. Clearly, $\{D'_i\mid 1\le i\le k\}$ is an $r$-grouped dominating set with $k$ units in $G'$
Conversely, suppose that there is an $r$-grouped dominating set $\mathcal{D}$ with at most $k$ units in $G'$.
Then there is a vertex set $D_i=\bigcup \mathcal{D}\cap V_i$ of size at most $r$ in some $G_i$ by $|\bigcup \mathcal{D}|\le rk$. Since $|I_i|\ge 2$, $D_i$ contains at least one vertex in $C_i$. Moreover, any vertex not in $V_i$ cannot dominate vertices in $I_i$. This means that $D_i$ is a dominating set in $G_i$. Since $G_i$ is a copy of $G$, there is a dominating set of size at most $r$ in $G$. \end{proof}
By a similar reduction, we also show that \textsc{$r$-Grouped Dominating Set} is W[2]-hard when parameterized by $k$.
\begin{theorem}\label{thm:W[2]:k:split} For every fixed $r\ge 1$, \textsc{$r$-Grouped Dominating Set} is W[2]-hard when parameterized by $k$ even on split graphs. \end{theorem}
\begin{proof}
We give a reduction from \textsc{Dominating Set} on split graphs. Let $\langle G=(C\cup I, E),k\rangle$ be an instance of \textsc{ Dominating Set} where $C$ forms a clique and $I$ forms an independent set. Without loss of generality, we suppose that $|C|\ge 2$ and $|I|\ge 2$. We create $r$ copies $G_1=(V_1, E_1), \ldots, G_r=(V_r, E_r)$ of $G$ where $V_i = \{v^{(i)} \mid v\in V\}$ and $E_i = \{e^{(i)} \mid e\in E\}$. Note that $V_i = C_i \cup I_i$. Then we make $\bigcup_i C_i$ a clique. The resulting graph $G'$ is clearly a split graph.
We show that there is a dominating set of size at most $k$ in $G$ if and only if there is an $r$-grouped dominating set with at most $k$ units in $G'$.
Suppose that there is a dominating set $D$ of size at most $k$ in $G$. Without loss of generality, we can assume that $D\subseteq C$.
For each $v\in D$, we define $D_v=\{v^{(i)}\mid 1\le i\le r\}$. Furthermore, let $\mathcal{D}=\{D_v \mid v\in D\}$. We see that $\mathcal{D}$ is an $r$-grouped dominating set with at most $k$ units in $G'$. Since $D$ is a dominating set in $G$ and $G'$ consists of $r$ copies of $G$, $\bigcup \mathcal{D}$ is clearly a dominating set in $G'$. Furthermore, because $\bigcup_i C_i$ is a clique, each $D_v$ forms a clique of size $r$, which can be regarded as a unit.
By the assumption that $|D|\le k$, we conclude that $\mathcal{D}$ is an $r$-grouped dominating set with at most $k$ units in $G'$.
Conversely, suppose that there is an $r$-grouped dominating set $\mathcal{D}$ with at most $k$ units in $G'$. We see that there is a dominating set $D$ of size at most $k$ in some $G_i$. Indeed, $\bigcup \mathcal{D}\cap V_i$ is a dominating set $D$ of size at most $k$ in $G_i$ because
$|\bigcup \mathcal{D}|\le rk$ and there is a vertex in $\bigcup \mathcal{D}\cap C_i$ by $|I|\ge 2$. This completes the proof. \end{proof}
Furthermore, we show the W[2]-hardness of \textsc{$r$-Grouped Dominating Set} on bipartite graphs.
\begin{theorem}\label{thm:W[2]:r:bipartite} For every fixed $k\ge 1$, \textsc{$r$-Grouped Dominating Set} is W[2]-hard when parameterized by $r$ even on bipartite graphs. \end{theorem}
\begin{proof} We reduce \textsc{Dominating Set} on split graphs to \textsc{$r$-Grouped Dominating Set}.
We are given an instance $\langle G=C\cup I, E),r\rangle$ of \textsc{Dominating Set}. Without loss of generality, we assume $|C|\ge 2$, $|I|\ge 2$. Moreover, we assume that if $\langle G=C\cup I, E),r\rangle$ is a yes-instance, there is a dominating set $D$ of size at most $r$ such that $D\subseteq C$ \cite{Bertossi1984}.
We first delete all the edges in the clique $C$, and add two edges $\{s_1,t\},\{s_2,t\}$ and connect $t$ to all the vertices in $C$. The obtained graph is bipartite. We then create $k$ copies $G_1=(V_1\cup \{s_1^{(1)},s_2^{(1)},t^{(1)}\},E_1), \ldots, G_k=(V_k\cup \{s_1^{(k)},s_2^{(k)},t^{(k)}\},E_k)$ of the graph where $V_i=\{u^{(i)}\mid u\in V\}$ for $1\le i\le k$. To connect $G_1, \ldots, G_k$, we add edges $\{s_1^{(i)},s_1^{(i+1)}\}$ for $1\le i\le k-1$. The resulting graph denoted by $G'$ remains bipartite.
In the following, we show that there is a dominating set of size at most $r$ in $G$ if and only if there is an $(r+1)$-grouped dominating set with $k$ units in $G'$. Suppose that there is a dominating set $D\subseteq C$ of size at most $r$ in $G$. We assume that $|D|=r$ because otherwise we only have to add $r-|D|$ vertices in $G$ to $D$ arbitrarily. For each graph $G_i$, define $D_i = \{v^{(i)}\mid v\in D\}$. Since $t^{(i)}$ is connected to $s_1^{(i)}$, $s_2^{(i)}$, and all the vertices in the clique part $C_i$ of $G_i$ and $D_i$ is a dominating set in $G_i[V_i]$, $D_i\cup \{t^{(i)}\}$ is a connected dominating set of size $r+1$ in $G_i$. Therefore, $\{D_i\cup \{t^{(i)}\}\mid 1\le i\le k\}$ is an $(r+1)$-grouped dominating set with $k$ units in $G'$.
Conversely, let $\mathcal{D}$ be an $(r+1)$-grouped dominating set with $k$ units in $G'$. Since $|\bigcup \mathcal{D}|\le (r+1)k$, some $G_i$ satisfies $|\bigcup \mathcal{D}\cap V_i\cup \{s_1^{(i)},s_2^{(i)},t^{(i)}\}|\le (r+1)$. To dominate $s_2^{(i)}$, $\bigcup \mathcal{D}$ must contains $t^{(i)}$. Note that $r\ge 2$. Thus, $|\bigcup \mathcal{D}\cap V_i|\le r$.
Furthermore, $G_i$ is bipartite and $|I|\ge 2$, hence there is a vertex $v^{(i)}$ in $\bigcup \mathcal{D}\cap C_i$. Let $D\subseteq V$ be a set in $G$ corresponding to $\bigcup \mathcal{D}\cap V_i$. Then $D$ is a dominating set of size $r$ in $G$. Indeed, since $\bigcup \mathcal{D}\cap V_i$ dominates the independent set part of $G_i$, $D$ also dominates the independent set part of $G$. Moreover, since $G$ is a split graph, vertex $v\in V$ corresponding to $v^{(i)}$ dominates all the vertices in $D$. This completes the proof. \end{proof}
\begin{theorem}\label{thm:W[2]:k:bipartite} For every fixed $r\ge 1$, \textsc{$r$-Grouped Dominating Set} is W[2]-hard when parameterized by $k$ even on bipartite graphs. \end{theorem}
\begin{proof} We reduce \textsc{Dominating Set} on split graphs to \textsc{$r$-Grouped Dominating Set}.
We are given an instance $\langle G=C\cup I, E),k \rangle$ of \textsc{Dominating Set}. Without loss of generality, if $\langle G=C\cup I, E),k\rangle$ is a yes-instance, there is a dominating set $D$ of size at most $k$ such that $D\subseteq C$ \cite{Bertossi1984}. Then we construct a bipartite graph $G'$ as follows. First, delete all the edges in the clique $C$. Then $G$ becomes a bipartite graph. We next add $k$ paths of length $r$ and connect an endpoint of each path to all the vertices in $C$. Let $P_i = (u^{(i)}_1, \ldots, u^{(i)}_r)$ denote such paths for $1\le i\le k$, and $u^{(i)}_1$'s are the endpoints connected to $C$. The resulting graph $G'$ is bipartite.
Suppose that $G$ has a dominating set $D=\{v_1, \ldots, v_{|D|}\}\subseteq C$ of size at most $k$. From $D$, we construct an $r$-grouped dominating set with $k$ units. For each $v_i\in D$, we choose a path $v_i, u^{(i)}_1, \ldots, u^{(i)}_{r-1}$ of length $r$ as one unit of the $r$-grouped dominating set. If $|D| < k$, we choose the remaining $k-|D|$ paths $u^{(i)}_1, \ldots, u^{(i)}_{r}$ for $|D|+1\le i\le r$. Let $\mathcal{D}$ be the set of such $k$ paths. Then $\mathcal{D}$ is an $r$-grouped dominating set with $k$ units because the length of each path in $\mathcal{D}$ is $r$ and $\bigcup \mathcal{D}$ contains $D$, which dominates all the vertices in the original $G$ and the vertices in $P_i$'s.
Conversely, let $\mathcal{D}$ be an $r$-grouped dominating set with $k$ units in $G'$. To dominate an endpoint $u^{(i)}_r$ in $P_i$, $\bigcup \mathcal{D}$ must contain $u^{(i)}_{r-1}$, which implies $|\bigcup \mathcal{D} \cap \bigcup_i P_i|\ge k(r-1)$. Thus, we have $|(C\cup I) \cap \bigcup \mathcal{D}|\le k$. Since $\bigcup \mathcal{D}$ is a dominating set of $G'$ and any vertex in $P_i$'s cannot dominate $I$, $(C\cup I) \cap \bigcup \mathcal{D}$ is a dominating set of size $k$ in $G$. \end{proof}
On the other hand, we can show that the problem is XP when parameterized by $k+r$. \begin{theorem} \textsc{$r$-Grouped Dominating Set} can be solved in $O^*(\Delta^{O(kr^2)})$ time. \end{theorem} \begin{proof} We guess the candidates of $r$-grouped dominating sets with at most $k$ units. We first pick an arbitrary vertex $v$ and branch $d(v)+1$ cases. One case is that $v$ is contained in $\mathcal{D}$. The vertices in the unit containing $v$ is reachable from $v$ via at most $r-1$ edges. Since the number of such vertices is at most $\Delta^{r-1}$, the choice of the other $r-1$ vertices is at most $\binom{\Delta^{r-1}}{r-1}=\Delta^{O(r^2)}$. Thus the number of candidates of units that contains $v$ is $\Delta^{O(r^2)}$. Another case is that $v$ is not contained in $\mathcal{D}$. Then at least one neighbor of $v$ is contained in $\mathcal{D}$. The number of candidates of units that contain it is also $\Delta^{O(r^2)}$. Therefore, the total number of candidates of units that dominate $v$ is $\Delta^{O(r^2)}$. After guessing one unit, we repeatedly pick a non-dominated vertex and branch as above. The repetition occurs at most $k$ times. Thus, the total running time is $\Delta^{O(kr^2)}$.
\end{proof}
\begin{corollary} \textsc{$r$-Grouped Dominating Set} belongs to XP when parameterized by $k+r$. \end{corollary}
Tripathi et al.~\cite{TripathiKPPW22} showed that \textsc{Paired Dominating Set} (equivalently, \textsc{$2$-Grouped Dominating Set}) is NP-complete for planar graphs with maximum degree~5. We show that \textsc{$r$-Grouped Dominating Set} is NP-hard even on planar bipartite graphs of maximum degree 3 for every fixed $r\ge 1$. This strengthens the result by Tripathi et al.~\cite{TripathiKPPW22}.
\begin{theorem}\label{thm:NP-c:planar} For every fixed $r\ge 1$, \textsc{$r$-Grouped Dominating Set} is NP-complete on planar bipartite graphs of maximum degree 3. \end{theorem} \begin{proof}
We reduce \textsc{Restricted Planar 3-SAT} to \textsc{$r$-Grouped Dominating Set}. \textsc{Restricted Planar 3-SAT} is a variant of \textsc{Planar 3-SAT} such that each variable occurs in exactly three clauses, in at most two clauses positively and in at most two clauses negatively. It is known that \textsc{Restricted Planar 3-SAT} is NP-complete~\cite{MiddendorfP93}.
Let $\phi$ be an instance of \textsc{Restricted Planar 3-SAT}, $n$ and $m$ be the number of variables and clauses of $\phi$, respectively. The incidence graph of $\phi$ is a bipartite graph such that it consists of variable vertices $v_{x_i}$'s corresponding to variables and clause vertices $c_j$'s corresponding to clauses. A variable vertex $v_{x_i}$ is connected to a clause variable $c_j$ if $C_j$ has a literal of $x_i$. The incidence graph of $\phi$ is planar.
For the incidence graph of $\phi$, we construct the graph $G=(V,E)$ by replacing variable vertices by variable gadgets. For each variable $x_i$, its variable gadget is constructed as follows. We create three vertices $v_{x_i}, v_{\bar{x}_i}, y_i$, and then add edges $\{v_{x_i}, y_i\}$, $\{v_{\bar{x}_i}, y_i\}$. Furthermore, we attach a path $P^r_i = y_i z^{(1)}_i z^{(2)}_i\cdots z^{(r-1)}_i$ of length $r-1$ for each $y_i$. Here, we define $z^{(0)}_i = y_i$.
Let $V_X = \{v_{x_i}, v_{\bar{x}_i} \mid i\in \{1,\ldots,n\}\}$ and $V_C = \{c_j \mid j\in \{1,\ldots,m\}\}$. For each variable $x_i$, $v_{x_i}$ is connected to $c_j$ if $C_j$ has a positive literal of $x_i$, and $v_{\bar{x}_i}$ is connected to $c_j$ if $C_j$ has a negative literal of $x_i$. We complete the construction of the graph $G=(V,E)$. Figure \ref{fig:NP-hard_planar} shows a concrete example of $G=(V,E)$ for $\phi$. Notice that $G$ is bipartite because $V_X$ and $V_C$ form independent sets, respectively, and $P^r_i$ is a path. Furthermore, $G$ is planar because the incidence graph of $\phi$ and the variable gadgets are planar. Finally, since each variable occurs in exactly three clauses, in at most two clauses positively and in at most two clauses negatively, the maximum degree of $G$ is at most 3.
\begin{figure}
\caption{The graph $G$ obtained by the reduction from an instance $\phi = (x_1\lor x_2 \lor \bar{x}_4)(\bar{x}_1\lor x_2\lor \bar{x}_3)(\bar{x}_2\lor x_3\lor x_4)(\bar{x}_1\lor \bar{x}_3\lor \bar{x}_4)$ of \textsc{Restricted Planar 3-SAT} to \textsc{$3$-Grouped Dominating Set}. }
\label{fig:NP-hard_planar}
\end{figure}
We are ready to show that $\phi$ is a yes-instance if and only if there is an $r$-grouped dominating set with at most $n$ in $G$.
Suppose that we are given a truth assignment of $\phi$. For each variable $x_i$, we select path $v_{x_i}y_i z^{(1)}_i z^{(2)}_i\cdots z^{(r-2)}_i$ as a unit of an $r$-grouped dominating set if $x_i$ is assigned to true. Otherwise, we select path $v_{\bar{x}_i}y_i z^{(1)}_i z^{(2)}_i\cdots z^{(r-2)}_i$. The number of vertices in each unit is $r$. The unit of $x_i$ dominates vertex $z^{(r-1)}_i$. Since each clause has at least one truth literal for the truth assignment, each clause vertex $c_j$ is dominated by some unit. Therefore, the set of selected paths is an $r$-grouped dominating set with at most $n$ units.
Conversely, we are given an $r$-grouped dominating set $\mathcal{D}$ with at most $n$ in $G$. For each $i$, $z^{(r-2)}_i$ must be contained in $\bigcup \mathcal{D}$. If not, $z^{(r-1)}_i$ is not dominated because of $r\ge 2$. Since $P^r_i$ is a path of length $r-1$ and the number of units is $n$, the vertices of a unit are selected from $\{v_{x_i},v_{\bar{x}_i}\} \cup \{y_i, z^{(1)}_i, z^{(2)}_i, \cdots, z^{(r-1)}_i\}$ for each $i$ and the unit forms a path of length $r$. If vertex $z^{(r-1)}_i$ is contained in $\bigcup \mathcal{D}$, $\bigcup \mathcal{D}$ does not contain $v_{x_i}$ and $v_{\bar{x}_i}$. Thus, we can remove $z^{(r-1)}_i$ from $\bigcup \mathcal{D}$ and add either $v_{x_i}$ or $v_{\bar{x}_i}$ arbitrarily to $\bigcup \mathcal{D}$. Since $v_{x_i}y_i z^{(1)}_i z^{(2)}_i\cdots z^{(r-2)}_i$ is a path of length $r$, this replacement does not collapse the property of $r$-grouped dominating set. Thus, we can suppose that the path of $i$th unit of $\mathcal{D}$ has either $v_{x_i}$ or $v_{\bar{x}_i}$ as an endpoint. Since $\bigcup \mathcal{D}$ is a dominating set, each vertex in $V_C$ has at least one vertex in $\mathcal{D}\cap V_X$ as a neighbor. This implies that the assignment corresponding to the selection of endpoints of units is a truth assignment. This completes the proof. \end{proof}
In the proof of Theorem \ref{thm:NP-c:planar}, the size of the constructed graph for $\phi$ is $O(rn+m)$. Thus, we have the following corollary. \begin{corollary}\label{cor:ETH:hard} For every fixed $r\ge 1$, \textsc{$r$-Grouped Dominating Set} cannot be solved in time $2^{o(n+m)}$ on bipartite graphs unless ETH fails. \end{corollary}
\section{Fast Algorithms Parameterized by Vertex Cover Number and by Twin Cover Number} \label{sec:fast-algorithms}
In this section, we present FPT algorithms for \textsc{$r$-Grouped Dominating Set} parameterized by vertex cover number $\nu$. Our algorithm is based on dynamic programming on nested partitions of a vertex cover, and its running time is $O^*((2\nu(r+1))^{\nu})$ for general $r\ge 2$. For the cases of $r \in \{2,3\}$, we can tailor the algorithm to run in $O^*((r+1)^\nu)$ time by focusing on the fact that the nested partitions of a vertex cover degenerate in some sense.
We then turn our attention to a more general parameter twin cover number. We show that, given a twin cover, \textsc{$r$-Grouped Dominating Set} admits an optimal solution in which twin-edges do not contribute to the connectivity of $r$-units. This implies that these edges can be removed from the graph, and thus we can focus on the resultant graph of bounded vertex cover number. Hence, we can conclude that our algorithms still work when the parameter $\nu$ in the running time is replaced with twin cover number $\tau$.
\begin{theorem} \label{thm:r+tau_fpt} For graphs of twin cover number $\tau$, \textsc{$r$-Grouped Dominating Set} can be solved in $O^*((2\tau(r+1))^{\tau})$ time. For the cases of $r \in \{2,3\}$, it can be solved in $O^*((r+1)^\tau)$ time. \end{theorem}
With a simple observation, Theorem~\ref{thm:r+tau_fpt} implies that \textsc{$r$-Grouped Dominating Set} parameterized solely by $\tau$ is fixed-parameter tractable. \begin{corollary} \label{cor:tau^tau} For graphs of twin cover number $\tau$, \textsc{$r$-Grouped Dominating Set} can be solved in $O^*((2\tau)^{2\tau})$ time. \end{corollary} \begin{proof}
If $r < 2\tau -1$, then the problem can be solved in $O^*((2\tau)^{2\tau})$ time by Theorem~\ref{thm:r+tau_fpt}. Assume that $r \ge 2\tau-1$. Let $C$ be a connected component of the input graph. If $|V(C)| < r$, then we have a trivial no-instance. Otherwise, we construct a connected dominating set $D$ of $C$ with size exactly $r$, which works as a unit dominating $C$. We initialize $D$ with a non-empty twin cover of size at most $\tau$. Note that such a set can be found in $O^*(1.2738^{\tau})$ time:
if $C$ is a complete graph, then we pick an arbitrary vertex $v \in V(C)$ and set $D = \{v\}$; otherwise, just find a minimum twin cover. Since $C$ is connected, $D$ is a dominating set of $C$. If $C[D]$ is not connected, we update $D$ with a new element $v$ adjacent to at least two connected components of $C[D]$. Since $|D| \le \tau$ at the beginning, we can repeat this update at most $\tau -1$ times, and after that $C[D]$ becomes connected and $|D| \le 2\tau - 1 \le r$. We finally add $r - |D|$ vertices arbitrarily and obtain a desired set. \end{proof}
In the next subsection, we first present an algorithm for \textsc{2-Grouped Dominating Set} parameterized by vertex cover number, which gives a basic scheme of our dynamic programming based algorithms. We then see how we extend the idea to \textsc{3-Grouped Dominating Set}. As explained above, these algorithms are based on dynamic programming (DP), and they compute certain function values on partitions of a vertex cover. Unfortunately, it is not obvious how to extend the strategy to general $r$. Instead, we consider nested partitions of a vertex cover for DP tables, which makes the running time a little slower though. In the last subsection, we see how a vertex cover can be replaced with a twin cover in the same running time in terms of order.
\subsection{Algorithms parameterized by vertex cover number} \subsubsection{Algorithm for \textsc{2-Grouped Dominating Set}} We first present an algorithm for the simplest case $r=2$, i.e., the paired dominating set. Let $G=(V,E)$ be a graph and $J$ be a vertex cover of $G$. Then, $I=V\setminus J$ is an independent set. The basic scheme of our algorithm follows the algorithm for the dominating set problem by Liedloff~\cite{liedloff}, which focuses on a partition of a given vertex cover $J$. For a minimum dominating set $D$, the vertex cover $J$ is partitioned into three parts: $J\cap D$; $(J\setminus D)\cap N(J\cap D)$, that is, the vertices in $J\setminus D$ that are dominated by $J\cap D$; and $J\setminus N[J\cap D]$, that is, the remaining vertices. Note that the remaining vertices in $J\setminus N[J\cap D]$ are dominated by $I\cap D$. Once $J\cap D$ is fixed, a minimum $I\cap D$ is found by solving the set cover problem that reflects the condition that $J\setminus N[J\cap D]$ must be dominated by $I\cap D$. The algorithm computes a minimum dominating set by solving set cover problems defined by all candidates of $J\cap D$.
To adjust the algorithm to \textsc{2-Grouped Dominating Set}, we need to handle the condition that a dominating set contains a perfect matching.
For each subset $J_D\subseteq J$, we find a subset $I_D\subseteq I$ (if any exists) of the minimum size such that $J_D \cup I_D$ can form a 2-grouped dominating set. Let $X$ and $Y$ be disjoint subsets of $J$, and let $I=\{v_1,v_2,\ldots,v_{|I|}\}$ (see Fig.~\ref{fig:pd}). \begin{figure}
\caption{Partitioning a vertex cover into three parts.}
\label{fig:pd}
\end{figure}
For $j=0, \ldots,|I|$, we define an auxiliary table $A[X,Y,j]$ as the minimum size of $I'\subseteq \{v_1,v_2,\ldots,v_j\}$ that satisfies the following conditions. \begin{enumerate}
\item $Y\subseteq N(I')$,
\item $I'\cup X$ has a partition $\mathcal D^{(2)}=\{D^{(2)}_{1},D^{(2)}_{2},\ldots,D^{(2)}_{p}\}$ with $p\le k$ such that for all $i=1,\ldots,p$, $|D^{(2)}_{i}|=2$ and $G[D^{(2)}_{i}]$ is connected. \end{enumerate} We set $A[X,Y,j]=\infty$ if no $I'\subseteq \{v_1,v_2,\ldots,v_j\}$ satisfies the conditions. We can easily compute $A[X,Y,0] \in \{0, \infty\}$ as $A[X,Y,0] = 0$ if and only if $G[X]$ has a perfect matching and $Y = \emptyset$.
Now the following recurrence formula computes $A$: \begin{align*} A[X,Y,j+1] & = \min \left\{
A[X,Y,j], \
\underset{u\in N(v_{j+1}) \cap X}{\min} A[X \setminus \{ u \},Y \setminus N(v_{j+1}),j]+1 \right\}. \end{align*}
The recurrence finds the best way under the condition that we can use vertices from $v_1,v_2,\ldots,v_j, v_{j+1}$ in a dominating set: not using $v_{j+1}$, or pairing $v_{j+1}$ with $u \in N(v_{j+1}) \cap X$. We can compute all entries of $A$ in $O^*(3^{|J|})$ time in a DP manner as there are only $3^{|J|}$ ways for choosing disjoint subsets $X$ and $Y$ of $J$.
Now we compute the minimum number of units in a $2$-grouped dominating set of $G$ (if any exists) by looking up some appropriate table entries of $A$. Let $\mathcal{D}$ be a 2-grouped dominating set of $G$ with $J_{D} = J \cap \bigcup \mathcal{D}$ and $I_{D} = I \cap \bigcup \mathcal{D}$. Since $\bigcup \mathcal{D}$ is a dominating set with no isolated vertex in $G[\bigcup \mathcal{D}]$, $J_{D}$ dominates all vertices in $I$. Let $J_Y = J \setminus N[J_D]$.
Then the definition of $A$ implies that $A[J_D,J_Y,|I|] = |I_{D}|$. Conversely, if $X \subseteq J$ dominates $I$, $Y = J \setminus N[X]$, and $A[X, Y, |I|] \ne \infty$, then there is a $2$-grouped dominating set with $(|X| + A[X, Y, |I|])/2$ units. Therefore, the minimum number of units in a $2$-grouped dominating set of $G$ is $\min\{(|X| + A[X, J \setminus N[X],|I|])/2 \mid X \subseteq J \text{ and } I \subseteq N(X)\}$,
which can be computed in $O^{*}(2^{|J|})$ time given the table $A$.
Thus the total running time of the algorithm is $O^{*}(3^{|J|})$.
\subsubsection{Algorithm for \textsc{3-Grouped Dominating Set}} Next, we consider the case $r=3$, i.e., the trio dominating set. Let $G=(V,E)$ be a graph, $J$ be a vertex cover of $G$, and $I=V\setminus J$. The basic idea is the same as the case $r=2$ except that we partition the vertex cover into four parts in the DP, and thus the recurrence formula for $A$ is different. In the DP, the vertex cover $J$ is partitioned into four parts depending on the partial solution corresponding to each table entry.
For each subset $J_D\subseteq J$, we find a subset $I_D\subseteq I$ (if any exists) of the minimum size such that $J_D \cup I_D$ can form a 3-grouped dominating set. Intuitively, the set $F$ represents partial units that will later be completed to full units. Let $X$, $F$, and $Y$ be disjoint subsets of $J$, and let $I=\{v_1,v_2,\ldots,v_{|I|}\}$.
For $j=0,\ldots,|I|$, we define $A[X,F,Y,j]$ as the minimum size of $I'\subseteq \{v_1,v_2,\ldots,v_j\}$ that satisfies the following conditions: \begin{enumerate}
\item $Y\subseteq N(I')$,
\item $I'$ can be partitioned into two parts $I'_2,I'_3$ satisfying the following conditions:
\begin{itemize}
\item $I'_2\cup F$ has a partition $\mathcal D^{(2)}=\{D^{(2)}_1,D^{(2)}_2,\ldots,D^{(2)}_p\}$ with $p\le k$ such that for all $i=1,\ldots,p$, $|D^{(2)}_{i}|=2$ and $G[D^{(2)}_{i}]$ is connected.
\item $I'_3\cup X$ has a partition $\mathcal D^{(3)}=\{D^{(3)}_1,D^{(3)}_2,\ldots,D^{(3)}_q\}$ with $q\le k$ such that for all $i=1,\ldots,q$, $|D^{(3)}_{i}|=3$ and $G[D^{(3)}_{i}]$ is connected.
\end{itemize} \end{enumerate} We set $A[X,F,Y,j]=\infty$ if no $I'\subseteq \{v_1,v_2,\ldots,v_j\}$ satisfies the conditions. We can easily compute $A[X,F,Y,0] \in \{0, \infty\}$ as $A[X,F,Y,0] = 0$ if and only if $F = Y = \emptyset$ and
$G[X]$ admits a partition into connected graphs of $3$-vertices. The last condition can be checked in $O(2^{|J|} \cdot |J|^{3})$ time for all $X \subseteq J$
by recursively considering all possible ways for removing three vertices from $X$; that is, $A[X,\emptyset,\emptyset,0] = \min_{\{x,y,z\} \in \binom{X}{3}} A[X \setminus \{x,y,z\}, \emptyset, \emptyset, 0]$ if $|X| \ge 3$.
The following recurrence formula holds: $A[X,F,Y,j+1] = \min \{f_1, f_2, f_3, f_4\}$, where
\begin{align*}
f_1 &= A[X,F,Y,j],
\\
f_2 &= \min_{\alpha,\beta\in X,|E(G[\{\alpha,\beta,v_{j+1}\}])|\geq 2} A[X\setminus\{\alpha,\beta\},F, Y\setminus \{N(\{\alpha,\beta,v_{j+1}\})\},j]+1,
\\
f_3 &= \min_{\alpha\in X\cap N(v_{j+1}) } A[X\setminus\{\alpha\},F\cup \{\alpha\},Y\setminus N(v_{j+1}),j]+1,
\\
f_4 &= \min_{\beta\in F\cap N(v_{j+1})} A[X,F\setminus \{\beta\},Y\setminus N(\{\beta,v_{j+1}\}),j]+1. \end{align*} The four options $f_{1},f_{2},f_{3}$, and $f_{4}$ assume different ways of the role of $v_{j+1}$ and compute the optimal value under the assumptions (see Fig.~\ref{fig:tdre}).
Concretely, $f_{1}$ reflects the case when $v_{j+1}$ does not belong to the solution, and $f_{2}$ reflects the case when $v_{j+1}$ belongs to the solution together with two vertices in $J$ in a connected way. In $f_{3}$, it reflects that $v_{j+1}$ forms a triple in the solution with a vertex in $F$ and a vertex in $I_j$. In $f_{4}$, it reflects that $v_{j+1}$ currently forms a pair in $J$ and will form a triple with a vertex in $I\setminus I_j$.
We can compute all entries of $A$ in $O^*(4^{|J|})$ time as the number of combinations of three disjoint sets $X,F,Y$ of $J$ is $4^{|J|}$.
\begin{figure}
\caption{How $v_{j+1}$ is used. (The white vertices belong to a dominating set.)}
\label{fig:tdre}
\end{figure}
Similarly to the previous case of $r = 2$, we can compute the minimum number of units in a $3$-grouped dominating set as $\min\{(|X| + A[X, \emptyset, J \setminus N[X],|I|])/3 \mid X \subseteq J \text{ and } I \subseteq N(X)\}$.
Given the table $A$, this can be done in $O^{*}(2^{|J|})$ time. Thus the total running time of the algorithm is $O^{*}(4^{|J|})$.
\subsubsection{Algorithm for \textsc{$r$-Grouped Dominating Set}} We now present our algorithm for general $r\ge 4$. Let $G=(V,E)$ be a graph, $J$ be a vertex cover of $G$, and $I=V\setminus J$.
This case still allows an algorithm based on a similar framework to the previous cases, though connected components of general $r$ can be built up from smaller fragments of connected components; this yields an essential difference that worsens the running time. In the DP, the vertex cover $J$ is partitioned into $r+1$ parts depending on the partial solution corresponding to each table entry, and then some of the parts in the partition are further partitioned into smaller subsets. In other words, each table entry corresponds to a nested partition of the vertex cover.
As in the previous algorithms, for each subset $J_D\subseteq J$, we find a subset $I_D \subseteq I$ (if any exists) of the minimum size such that $J_D \cup I_D$ can form an $r$-grouped dominating set. Let $X$, $F^{(r-1)},\dots,F^{(3)},F^{(2)},Y$ be disjoint subsets of $J$, and let $I=\{v_1,v_2,\ldots,v_{|I|}\}$.
For $i=2,\ldots,r-1$, let $\mathcal{F}^{(i)}$ be a partition of $F^{(i)}$, where $\mathcal{F}^{(i)}=\{F^{(i)}_1,F^{(i)}_2,\ldots,F^{(i)}_{|\mathcal{F}^{(i)}|}\}$. The number of such nested partitions $(X,\mathcal{F}^{(r-1)},\dots,\mathcal{F}^{(2)},Y)$ is at most $(r+1)^{|J|}|J|^{|J|}$. \begin{figure}
\caption{A nested partition of a vertex cover.}
\label{fig:rd}
\end{figure}
For $j=0,\ldots,|I|$, we define $A[X,\mathcal{F}^{(r-1)},\dots,\mathcal{F}^{(2)},Y,j]$ as the minimum size of $I'\subseteq \{v_1,v_2,\ldots,v_j\}$ that satisfies the following conditions: \begin{enumerate}
\item $Y\subseteq N(I')$,
\item $I'$ can be partitioned into $r-1$ parts $I'_2,I'_3,\dots,I'_r$ satisfying the following conditions:
\begin{itemize}
\item for $i=2,\ldots,r-1$, $I'_i \cup F^{(i)}$ has a partition $\mathcal D^{(i)}=\{D^{(i)}_1,D^{(i)}_2,\ldots,D^{(i)}_{|\mathcal{F}^{(i)}|}\}$ such that
for all $p=1,\ldots, |\mathcal{F}^{(i)}|$, $D^{(i)}_p$ includes at least one vertex of $I'$ and is a superset of ${F}^{(i)}_p$, and $|D^{(i)}_{p}|=i$ and $G[D^{(i)}_{p}]$ is connected.
\item $I'_r\cup X$ has a partition $\mathcal D^{(r)}=\{D^{(r)}_1,D^{(r)}_2,\ldots,D^{(r)}_{q}\}$ such that for all $i=1,\ldots,q$, $|D^{(r)}_{i}|=r$ and $G[D^{(r)}_{i}]$ is connected.
\end{itemize} \end{enumerate} We set $A[X,\mathcal{F}^{(r-1)},\dots,\mathcal{F}^{(2)},Y,j] = \infty$ if no $I'\subseteq \{v_1,v_2,\ldots,v_j\}$ satisfies the conditions. We can compute $A[X,\mathcal{F}^{(r-1)},\dots,\mathcal{F}^{(2)},Y,0]$, which is $0$ or $\infty$, as it is $0$ if and only if ${F}^{(r-1)} = \dots = {F}^{(2)} = Y = \emptyset$ and
$G[X]$ admits a partition into connected graphs of $r$ vertices. The last condition can be checked in $O(|J|^{|J|})$ time for all $X \subseteq J$ by checking all possible partitions of $J$.
Assume that all entries of $A$ with $j \le c$ for some $c$ are computed. Since the degree of $v_{c+1}$ is at most $|J|$, the number of possible ways of how $v_{c+1}$ extends a partial solution is at most $2^{|J|}$. Thus from each table entry of $A$ with $j = c$, we obtain at most $2^{|J|}$ candidates of the table entries with $j = c+1$. Thus, we can compute all entries of $A$ in $O^{*}(2^{|J|}(r+1)^{|J|}|J|^{|J|})$ time.
Given $A$, we can compute the minimum number of units in an $r$-grouped dominating set as $\min\{(|X| + A[X, \emptyset, \dots, \emptyset, J \setminus N[X],|I|])/r \mid X \subseteq J \text{ and } I \subseteq N(X)\}$. Again this takes only $O^{*}(2^{|J|})$ time. Thus the total running time of the algorithm is $O^{*}(2^{|J|}(r+1)^{|J|}|J|^{|J|})=O^{*}((2\nu(r+1))^{\nu})$.
\subsection{Algorithms parameterized by twin cover number} In this subsection, we show that the algorithms presented above still work when the parameter $\nu$ in the running time is replaced with twin cover number $\tau$. To show this, we prove the following lemma. It says that twin-edges do not contribute to the connectivity of units for some minimum $r$-grouped dominating sets and can be removed from the graph. As a result, the vertex cover number can be replaced with the twin cover number. \begin{lemma}\label{lem:twin-K} Let $G$ be a graph and $K$ be a twin cover of $G$. If $G$ has an $r$-grouped dominating set, then there exists a minimum $r$-grouped dominating set such that every unit has at least one vertex in $K$. \end{lemma} \begin{proof} Let $G=(V,E)$ be a graph, and $K$ be a twin cover of $G$. Suppose that a minimum $r$-grouped dominating set $\mathcal D$ exists and one of its units $D=\{v_1,v_2,\ldots,v_r\}$ has no vertex in $K$. Since $K$ is a twin cover, $N[v_1]=N[v_2]=\dots=N[v_r]$ holds. Let $K_{D} = K\cap N(v_1)$. Then, there is at least one vertex $x$ in $K_{D}$ such that $x \notin \bigcup \mathcal D$. Suppose to the contrary that there is no such $x$, and thus all vertices in $K_{D}$ belong to $\bigcup \mathcal D$. This implies that no vertex is dominated only by $D$ and that $D$ itself is dominated by some vertices in $K_{D}$. Thus, $\mathcal{D} \setminus \{D\}$ is an $r$-grouped dominating set. This contradicts the minimality of $\mathcal{D}$. Let $D' = D\setminus\{v_1\}\cup \{x\}$, then $\mathcal D'=\mathcal D\setminus \{D\}\cup \{D'\}$ is also a minimum $r$-grouped dominating set of $G$ (see Fig. \ref{fig:tclem}). By repeating this process, we can obtain a minimum $r$-grouped dominating set such that every unit has at least one vertex in $K$. \end{proof} \begin{figure}
\caption{An example for exchange. White vertices belong to a dominating set. }
\label{fig:tclem}
\end{figure}
\section{Beyond Vertex Cover and Twin Cover}
In this section, we further explore the parameterized complexity of $r$-\textsc{Grouped Dominating Set} with respect to structural graph parameters that generalize vertex cover number and twin cover number. We show that if we do not try to optimize the running time of algorithms, then we can use known algorithmic meta-theorems that
automatically give fixed-parameter algorithms parameterized by certain graph parameters.
For the sake of brevity, we define only the parameters for which we need their definitions. For example, we do not need the definition of treewidth for applying the meta-theorem described below. On the other hand, to contrast the results here with the ones in the previous sections, it is important to see the picture of the relationships between the parameters. See \figurename~\ref{fig:parameters0} for the hierarchy of the graph parameters we deal with.
Roughly speaking, the algorithmic meta-theorems we use here say that if a problem can be expressed in a certain logic (e.g., $\mathsf{FO}${}, \mso{1}, or \mso{2}), then the problem is fixed-parameter tractable parameterized by a certain graph parameter (e.g., twin-width, treewidth, or clique-width). Such theorems are extremely powerful and used widely for designing fixed-parameter algorithms~\cite{Kreutzer11}. On the other hand, the generality of the meta-theorems unfortunately comes with very high dependency on the parameters~\cite{FrickG04}. When our target parameter is vertex cover number, the situation is slightly better, but still a double-exponential $2^{2^{\mathrm{\Omega}(\nu)}}$ lower bound of the parameter dependency is known under ETH~\cite{Lampis12}. This implies that our ``slightly superexponential'' $2^{O(\tau \log \tau)}$ algorithm in Section~\ref{sec:fast-algorithms} cannot be obtained by applications of known meta-theorems.
In the rest of this section, we first introduce $\mathsf{FO}${}, \mso{1}, and \mso{2} on graphs. We then observe that the problem can be expressed in $\mathsf{FO}${} when $r$ and $k$ are part of the parameter and in \mso{2} when $r$ is part of the parameter. These observations combined with known meta-theorems immediately imply that $r$-\textsc{Grouped Dominating Set} is fixed-parameter tractable when \begin{itemize}
\item parameterized by $r+ k$ on nowhere dense graph classes;
\item parameterized by $r + k + \textrm{twin-width}$
if a contraction sequence of the minimum width is given as part of the input; and
\item parameterized by $r + \text{treewidth}$. \end{itemize}
We then consider the parameter $k + \text{treewidth}$ and show that this case is intractable. More strongly, we show that $r$-\textsc{Grouped Dominating Set} is W[1]-hard when the parameter is $k + \text{treedepth} + \text{feedback vertex set number}$.
We finally consider the parameter modular-width, a generalization of twin cover number, and show that $r$-\textsc{Grouped Dominating Set} parameterized by modular-width is fixed-parameter tractable.
\subsection{Results based on algorithmic meta-theorems} \label{sec:logic} The \emph{first-order logic} on graphs ($\mathsf{FO}$) allows variables representing vertices of the graph under consideration. The atomic formulas are the equality $x=y$ of variables and the adjacency $E(x,y)$ meaning that $\{x,y\} \in E$. The $\mathsf{FO}${} formulas are defined recursively from atomic formulas with the usual Boolean connectives ($\lnot$, $\land$, $\lor$, $\Rightarrow$, $\Leftrightarrow$), and quantification of variables ($\forall$, $\exists$). We also use the existential quantifier with a dot ($\dot{\exists}$) to quantify distinct objects. For example, $\dot{\exists} a,b \colon \phi$ means $\exists a,b \colon (a \ne b) \land \phi$. We write $G \models \phi$ if $G$ satisfies (or \emph{models}) $\phi$. Given a graph $G$ and an $\mathsf{FO}${} formula $\phi$, \textsc{$\mathsf{FO}${} Model Checking} asks whether $G \models \phi$.
It is straightforward to express the property of having an $r$-grouped dominating set of $k$ units with an $\mathsf{FO}${} formula whose length depends only on $r+k$: \begin{align*}
\phi_{r,k}
&=
\dot{\exists} v_{1}, v_{2}, \dots, v_{rk} \colon \\
&\qquad
\mathsf{dominating}(v_{1}, \dots, v_{rk}) \land {}
\bigwedge_{0 \le i \le k-1}
\mathsf{connected}(v_{ir+1}, \dots, v_{ir+r}), \end{align*} where $\mathsf{dominating}(\cdot\cdot\cdot)$ is a subformula expressing that the $rk$ vertices form a dominating set and $\mathsf{connected}(\cdot\cdot\cdot)$ is the one expressing that the $r$ vertices induce a connected subgraph (see Section~\ref{sec:subformulas} for the expressions of the subformulas). This implies that \textsc{$r$-Grouped Dominating Set} parameterized by $r + k$ is fixed-parameter tractable on graph classes on which
\textsc{$\mathsf{FO}${} Model Checking} parameterized by the formula length $|\phi|$ is fixed-parameter tractable. Such graph classes include nowhere dense graph classes~\cite{GroheKS17} and graphs of bounded twin-width (given with so called contraction sequences)~\cite{BonnetKTW22}. \begin{corollary} \label{cor:r+k_nwd} $r$-\textsc{Grouped Dominating Set} parameterized by $r+k$ is fixed-parameter tractable on nowhere dense graph classes. \end{corollary} \begin{corollary} \label{cor:r+k+twin-width} $r$-\textsc{Grouped Dominating Set} parameterized by $r + k + \textrm{twin-width}$ is fixed-parameter tractable if a contraction sequence of the minimum width is given as part of the input. \end{corollary}
The \emph{monadic second-order logic} on graphs (\mso{1}) is an extension of $\mathsf{FO}${} that additionally allows variables representing vertex sets and the inclusion predicate $X(x)$ meaning that $x \in X$. \mso{2} is a further extension of \mso{1} that also allows edge variables, edge-set variables, and an atomic formula $I(e,x)$ representing the edge-vertex incidence relation. Given a graph $G$ and an \mso{1} (\mso{2}, resp.) formula $\phi(X)$ with a free set variable $X$, \mso{1} (\mso{2}, resp.) \textsc{Optimization} asks to find a minimum set $S$ such that $G \models \phi(S)$.
It is not difficult to express the property of a vertex set being the union of $r$-units of a $r$-grouped dominating set with an \mso{2} formula whose length depending only on $r$:\footnote{ Note that there is no equivalent \mso{1} formula of length depending only on $r$. This is because $G \models \psi_{2}(V)$ expresses the property of having a perfect matching, for which an \mso{1} formula does not exist (see e.g., \cite{CourcelleE12}).} \begin{align*}
\psi_{r}(X)
={}&
\mathsf{dominating}(X) \land {}
\\
&\left(\exists F \subseteq E \colon \mathsf{span}(F,X) \land
(\forall C \subseteq X\colon
\mathsf{cc}(F,C) \Rightarrow \mathsf{size}_{r}(C))\right), \end{align*} where $\mathsf{dominating}(X)$ is a subformula expressing that $X$ is a dominating set, $\mathsf{span}(F,X)$ is the one expressing that $X$ is the set of all endpoints of the edges in $F$, $\mathsf{cc}(F,C)$ expresses that $C$ is the vertex set of a connected component of the subgraph induced by $F$, and $\mathsf{size}_{r}(C)$ means that $C$ contains exactly $r$ elements (again, see Section~\ref{sec:subformulas} for the expressions of the subformulas). Since \textsc{\mso{2} Optimization} parameterized by treewidth is fixed-parameter tractable~\cite{ArnborgLS91,BoriePT92,Courcelle90mso1}, we have the following result. \begin{corollary} \label{cor:r+tw} $r$-\textsc{Grouped Dominating Set} parameterized by $r + \text{treewidth}$ is fixed-parameter tractable. \end{corollary}
\subsection{Hardness parameterized by $k + \text{treewidth}$} Now the natural question regarding treewidth and $r$-\textsc{Grouped Dominating Set} would be the complexity parameterized by $k + \text{treewidth}$. Unfortunately, this case is W[1]-hard even if treewidth is replaced with a possibly much larger parameter $\text{pathwidth} + \text{feedback vertex set number}$ and the graphs are restricted to be planar. Furthermore, if the planarity is not required, we can replace pathwidth in the parameter with treedepth. \begin{theorem} $r$-\textsc{Grouped Dominating Set} parameterized by $k + \text{pathwidth} + \text{feedback vertex set number}$ is W[1]-hard on planar graphs. \end{theorem} \begin{proof} Given a graph $G = (V,E)$ and an integer $r \ge 2$,
\textsc{Equitable Connected Partition} asks whether there exists a partition of $V$ into $k = |V|/r$ sets $V_{1}, \dots, V_{k}$
such that $G[V_{i}]$ is connected and $|V_{i}| = r$ for $1 \le i \le k$. It is known that \textsc{Equitable Connected Partition} parameterized by $k + \text{pathwidth} + \text{feedback vertex set number}$ is W[1]-hard even on planar graphs~\cite{EncisoFGKRS09}. We reduce this problem to ours.
Let $\langle G = (V,E), r \rangle$ be an instance of \textsc{Equitable Connected Partition}. To each vertex $v$ of $G$, we attach a new vertex of degree~$1$, which we call a \emph{pendant} at $v$. This modification does not change the feedback vertex number and may increase the pathwidth by at most $1$ (see e.g., \cite[Lemma~A.2]{BelmonteHKKKKLO22}). Let $H$ be the resultant graph, which is planar. To prove the lemma, it suffices to show that $\langle H, k \rangle$ is a yes-instance of $r$-\textsc{Grouped Dominating Set} if and only if $\langle G, r \rangle$ is a yes-instance of \textsc{Equitable Connected Partition}.
To prove the if direction, assume that $\langle G, r \rangle$ is a yes-instance of \textsc{Equitable Connected Partition} and that $V_{1}, \dots, V_{k}$ certificate it. Clearly, $\{V_{1}, \dots, V_{k}\}$ is an $r$-grouped dominating set of $H$.
To prove the only-if direction, assume that $H$ has an $r$-grouped dominating set $\mathcal{D}$ with at most $k$ units. Let $v \in V$ and $p$ be the pendant at $v$. Observe that $\bigcup \mathcal{D}$ contains exactly one of $v$ and $p$ since it needs at least one of them for dominating $p$
and $|\bigcup \mathcal{D}| \le rk = |V|$. Furthermore, the assumption $r \ge 2$ implies that
$\bigcup \mathcal{D}$ cannot contain $p$ as it has no neighbor other than $v$. This implies that $\bigcup \mathcal{D} = V$ and that $\mathcal{D}$ contains exactly $|V|/r = k$ units. Therefore, the family $\mathcal{D}$ is a certificate that $\langle G, r \rangle$ is a yes-instance of \textsc{Equitable Connected Partition}. \end{proof}
It is known that on general (not necessarily planar) graphs, \textsc{Equitable Connected Partition} parameterized by $k + \text{treedepth} + \text{feedback vertex set number}$ is W[1]-hard~\cite{GimaO22_arxiv}. Since adding pendants to all vertices increases treedepth by at most $1$ (see e.g., \cite{NesetrilO2012}), the same reduction shows the following hardness. \begin{theorem} \label{thm:k+td+fvs} $r$-\textsc{Grouped Dominating Set} parameterized by $k + \text{treedepth} + \text{feedback vertex set number}$ is W[1]-hard. \end{theorem}
\subsection{Fixed-parameter tractability parameterized by modular-width}
Let $G = (V,E)$ be a graph. A set $M \subseteq V$ is a \emph{module} if for each $v \in V \setminus M$, either $M \subseteq N(v)$ or $M \cap N(v) = \emptyset$ holds. The \emph{modular-width} of $G$, denoted $\mathtt{mw}(G)$, is the minimum integer $k$
such that either $|V| \le k$ or there exists a partition of $V$ into at most $k$ modules $M_{1}, \dots, M_{k'}$ of $G$ such that each $G[M_{i}]$ has modular-width at most $k$. It is known that the modular-width of a graph and a recursive partition certificating it can be computed in linear time~\cite{CournierH94,HabibP10,TedderCHP08}.
Observe that if $V$ is partitioned into modules $M_{1}, \dots, M_{k}$ of $G$, then for two distinct modules $M_{i}$ and $M_{j}$, we have either no or all possible edges between them. If there are all possible edges between $M_{i}$ and $M_{j}$, then we say that $M_{i}$ and $M_{j}$ are adjacent.
\begin{theorem} \label{thm:mw} $r$-\textsc{Grouped Dominating Set} parameterized by modular-width is fixed-parameter tractable. \end{theorem} \begin{proof} Let $\langle G = (V,E), k \rangle$ be an instance of $r$-\textsc{Grouped Dominating Set}. We only consider the case of $r \ge 2$ since the other case of $r=1$ is known (see \cite{CourcelleMR00,GajarskyLO13}). We may assume that $G$ is connected since otherwise we can solve the problem on each connected component separately. We also assume that $G$ has at least $r$ vertices as otherwise the problem is trivial. Let $M_{1}, \dots, M_{\mu}$ be a partition of $V$ into modules with $2 \le \mu \le \mathtt{mw}(G)$. For each module $M_{i}$, there is at least one adjacent module $M_{j}$ as $G$ is connected.
We first assume that $r \ge \mu$. Let $D \subseteq V$ be an arbitrary set of size $r$ that takes at least one vertex from each module $M_{i}$. Recall that we have either no or all possible edges between two distinct modules. Thus, the connectivity of $G$ implies that $G[D]$ is connected and $D$ is a dominating set of $G$. This implies that $\{D\}$ is an $r$-grouped dominating set with one unit.
Next assume that $r < \mu$. In this case, we show below that if $G$ has an $r$-grouped dominating set, then $G$ has an $r$-grouped dominating set with at most $\mu$ units. This implies that $r+k < 2\mu$, and thus the problem can be solved as \textsc{$\mathsf{FO}${} Model Checking} with a formula of length depending only on $\mu$, which is fixed-parameter tractable parameterized by $\mu$ (see~\cite{CourcelleMR00,GajarskyLO13}).
Before proving the upper bound of $k$, we show that if $G$ has an $r$-grouped dominating set, then there is a minimum one such that no unit is entirely contained in a module $M_{i}$. Assume that $\mathcal{D}$ is a minimum $r$-grouped dominating set of $G$. If $D \subseteq M_{i}$ holds for some $i$ and $D \in \mathcal{D}$, then there is a vertex $v$ in a module $M_{j}$ adjacent to $M_{i}$ that does not belong to $\bigcup \mathcal{D}$. This is because, otherwise, $\mathcal{D} \setminus \{D\}$ is still an $r$-grouped dominating set. Let $u$ be an arbitrary vertex in $D$
and set $D' = D \setminus \{u\} \cup \{v\}$. As $r \ge 2$, $D'$ intersects both $M_{i}$ and $M_{j}$. Also we can see that $|D'| = r$, $D'$ is connected (as $u$ is adjacent to all vertices in $M_{i}$), and all vertices dominated by $D$ are dominated by $D'$ as well. Thus, $\mathcal{D} \setminus \{D\} \cup \{D'\}$ is an $r$-grouped dominating set. We can repeat this process until we have the claimed property.
As discussed above, it suffices to show the upper bound $k$ for the number of units. Let $\mathcal{D}$ be a minimum $r$-grouped dominating set of $G$ such that no unit is entirely contained in a module $M_{i}$. We say that a module $M_{i}$ is \emph{private} for a unit $D \in \mathcal{D}$
if $D$ is the only one in $\mathcal{D}$ that intersects $M_{i}$. Suppose to the contrary that $|\mathcal{D}| > \mu$. Then, there is $D \in \mathcal{D}$ such that no module $M_{i}$ is private for $D$.
If a module $M_{i}$ is adjacent to a module $M_{j}$ that intersects $D$, then since $M_{j}$ is not private for $D$, $\mathcal{D} \setminus \{D\}$ contains a unit intersecting $M_{j}$, which dominates $M_{i}$.
If a module $M_{i}$ intersects $D$, then since $D$ intersects at least two modules and $G[D]$ is connected, there is a module $M_{j}$ adjacent to $M_{i}$ and intersecting $D$. Hence, as the previous case, $\mathcal{D} \setminus \{D\}$ contains a unit dominating $M_{i}$.
Therefore, we can conclude that $\mathcal{D} \setminus \{D\}$ is an $r$-grouped dominating set. This contradicts the minimality of $\mathcal{D}$. \end{proof}
\subsection{Auxiliary subformulas} \label{sec:subformulas} Here we present $\mathsf{FO}${} or \mso{2} expressions of some of the subformulas in Section~\ref{sec:logic}. All of them are standard and presented only to show basic ideas.
The following formulas expressing dominating sets are almost direct translation of the definition and should be easy to read. \begin{align*}
\mathsf{dominating}(X)
&=
\forall u \; \exists v \colon X(v) \land ((u = v) \lor E(u,v)).
\\
\mathsf{dominating}(v_{1}, \dots, v_{p})
&=
\forall u \colon (u = v_{1}) \lor (u = v_{2}) \lor \dots \lor (u = v_{p}) \\
& \qquad\quad \lor E(u, v_{1}) \lor E(u, v_{2}) \lor \dots \lor E(u, v_{p}). \end{align*}
The connectivity of $G[X]$ is a little bit tricky to state. We state that for each nonempty proper subset $Y$ of $X$, there is an edge between $Y$ and $X \setminus Y$. See e.g., \cite{CyganFKLMPPS15} for the full expression of $\mathsf{connected}$. The $\mathsf{FO}${} version of $\mathsf{connected}$ can be expressed based on the same idea but the length of the formula depends on the number $r$ of vertices it can take (which is fine for us as $r$ is part of the input when we use this formula).
In \cite{CyganFKLMPPS15}, an \mso{2} formula expressing the connectivity of the graph induced by an edge set is also presented. We call it $\mathsf{connectedE}$ and use it below.
Recall that $\mathsf{span}(F,X)$ expresses that $X$ is the set of all endpoints of the edges in $F$ and that $\mathsf{cc}(F,C)$ expresses that $C$ is the vertex set of a connected component of the subgraph induced by $F$. They can be stated as follows: \begin{align*}
\mathsf{span}(F,X)
&=
\forall v \colon X(v) \Leftrightarrow (\exists e \colon F(e) \land I(e, v)),
\\[.5ex]
\mathsf{cc}(F,C) &=
\exists F' \colon (F' \subseteq F) \land \mathsf{span}(F', C) \land \mathsf{connectedE}(F') \\
& \qquad\quad \land (\forall F'' \colon (F' \subseteq F'' \subseteq F) \land \lnot \mathsf{connectedE}(F'')), \end{align*} where the inclusion relation $F \subseteq F'$ can be stated as $\forall e \colon F(e) \Rightarrow F'(e)$.
Finally, when $r$ is part of the parameter,
$\mathsf{size}_{r}(C)$ meaning that $|C| = r$ can be stated as follows: \begin{align*}
\mathsf{size}_{r}(C) &= \dot{\exists} v_{1}, \dots, v_{r} \colon
\bigwedge_{1 \le i \le r} C(v_{i}) \land \left(\lnot \exists v \colon C(v) \land \bigwedge_{1 \le i \le r} v \ne v_{i} \right). \end{align*}
\end{document} | arXiv |
Conservative vector field
In vector calculus, a conservative vector field is a vector field that is the gradient of some function.[1] A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the vector field under the line integral being conservative. A conservative vector field is also irrotational; in three dimensions, this means that it has vanishing curl. An irrotational vector field is necessarily conservative provided that the domain is simply connected.
Conservative vector fields appear naturally in mechanics: They are vector fields representing forces of physical systems in which energy is conserved.[2] For a conservative system, the work done in moving along a path in a configuration space depends on only the endpoints of the path, so it is possible to define potential energy that is independent of the actual path taken.
Informal treatment
In a two- and three-dimensional space, there is an ambiguity in taking an integral between two points as there are infinitely many paths between the two points—apart from the straight line formed between the two points, one could choose a curved path of greater length as shown in the figure. Therefore, in general, the value of the integral depends on the path taken. However, in the special case of a conservative vector field, the value of the integral is independent of the path taken, which can be thought of as a large-scale cancellation of all elements $d{R}$ that don't have a component along the straight line between the two points. To visualize this, imagine two people climbing a cliff; one decides to scale the cliff by going vertically up it, and the second decides to walk along a winding path that is longer in length than the height of the cliff, but at only a small angle to the horizontal. Although the two hikers have taken different routes to get up to the top of the cliff, at the top, they will have both gained the same amount of gravitational potential energy. This is because a gravitational field is conservative.
Intuitive explanation
M. C. Escher's lithograph print Ascending and Descending illustrates a non-conservative vector field, impossibly made to appear to be the gradient of the varying height above ground (gravitational potential) as one moves along the staircase. The force field experienced by the one moving on the staircase is non-conservative in that one can return to the starting point while ascending more than one descends or vice versa, resulting in nonzero work done by gravity. On a real staircase, the height above the ground is a scalar potential field: one has to go upward exactly as much as one goes downward in order to return to the same place, in which case the work by gravity totals to zero. This suggests path-independence of work done on the staircase; equivalently, the force field experienced is conservative (see the later section: Path independence and conservative vector field). The situation depicted in the print is impossible.
Definition
A vector field $\mathbf {v} :U\to \mathbb {R} ^{n}$, where $U$ is an open subset of $\mathbb {R} ^{n}$, is said to be conservative if and only if there exists a $C^{1}$ (continuously differentiable) scalar field $\varphi $[3] on $U$ such that
$\mathbf {v} =\nabla \varphi .$
Here, $\nabla \varphi $ denotes the gradient of $\varphi $. Since $\varphi $ is continuously differentiable, $\mathbf {v} $ is continuous. When the equation above holds, $\varphi $ is called a scalar potential for $\mathbf {v} $.
The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of a conservative vector field and a solenoidal field.
Path independence and conservative vector field
Main article: Gradient theorem
Path independence
A line integral of a vector field $\mathbf {v} $ is said to be path-independent if it depends on only two integral path endpoints regardless of which path between them is chosen:[4]
$\int _{P_{1}}\mathbf {v} \cdot d\mathbf {r} =\int _{P_{2}}\mathbf {v} \cdot d\mathbf {r} $
for any pair of integral paths $P_{1}$ and $P_{2}$ between a given pair of path endpoints in $U$.
The path independence is also equivalently expressed as
$\int _{P_{c}}\mathbf {v} \cdot d\mathbf {r} =0$
for any piecewise smooth closed path $P_{c}$ in $U$ where the two endpoints are coincident. Two expressions are equivalent since any closed path $P_{c}$ can be made by two path; $P_{1}$ from an endpoint $A$ to another endpoint $B$, and $P_{2}$ from $B$ to $A$, so
$\int _{P_{c}}\mathbf {v} \cdot d\mathbf {r} =\int _{P_{1}}\mathbf {v} \cdot d\mathbf {r} +\int _{P_{2}}\mathbf {v} \cdot d\mathbf {r} =\int _{P_{1}}\mathbf {v} \cdot d\mathbf {r} -\int _{-P_{2}}\mathbf {v} \cdot d\mathbf {r} =0$
where $-P_{2}$ is the reverse of $P_{2}$ and the last equality holds due to the path independence $ \int _{P_{1}}\mathbf {v} \cdot d\mathbf {r} =\int _{-P_{2}}\mathbf {v} \cdot d\mathbf {r} .$
Conservative vector field
A key property of a conservative vector field $\mathbf {v} $ is that its integral along a path depends on only the endpoints of that path, not the particular route taken. In other words, if it is a conservative vector field, then its line integral is path-independent. Suppose that $\mathbf {v} =\nabla \varphi $ for some $C^{1}$ (continuously differentiable) scalar field $\varphi $[3] over $U$ as an open subset of $\mathbb {R} ^{n}$ (so $\mathbf {v} $ is a conservative vector field that is continuous) and $P$ is a differentiable path (i.e., it can be parameterized by a differentiable function) in $U$ with an initial point $A$ and a terminal point $B$. Then the gradient theorem (also called fundamental theorem of calculus for line integrals) states that
$\int _{P}\mathbf {v} \cdot d{\mathbf {r} }=\varphi (B)-\varphi (A).$
This holds as a consequence of the definition of a line integral, the chain rule, and the second fundamental theorem of calculus. $\mathbf {v} \cdot d\mathbf {r} =\nabla {\varphi }\cdot d\mathbf {r} $ in the line integral is an exact differential for an orthogonal coordinate system (e.g., Cartesian, cylindrical, or spherical coordinates). Since the gradient theorem is applicable for a differentiable path, the path independence of a conservative vector field over piecewise-differential curves is also proved by the proof per differentiable curve component.[5]
So far it has been proven that a conservative vector field $\mathbf {v} $ is line integral path-independent. Conversely, if a continuous vector field $\mathbf {v} $ is (line integral) path-independent, then it is a conservative vector field, so the following biconditional statement holds:[4]
For a continuous vector field $\mathbf {v} :U\to \mathbb {R} ^{n}$, where $U$ is an open subset of $\mathbb {R} ^{n}$, it is conservative if and only if its line integral along a path in $U$ is path-independent, meaning that the line integral depends on only both path endpoints regardless of which path between them is chosen.
The proof of this converse statement is the following.
$\mathbf {v} $ is a continuous vector field which line integral is path-independent. Then, let's make a function $\varphi $ defined as
$\varphi (x,y)=\int _{a,b}^{x,y}\mathbf {v} \cdot d{\mathbf {r} }$
over an arbitrary path between a chosen starting point $(a,b)$ and an arbitrary point $(x,y)$. Since it is path-independent, it depends on only $(a,b)$ and $(x,y)$ regardless of which path between these points is chosen.
Let's choose the path shown in the left of the right figure where a 2-dimensional Cartesian coordinate system is used. The second segment of this path is parallel to the $x$ axis so there is no change along the $y$ axis. The line integral along this path is
$\int _{a,b}^{x,y}\mathbf {v} \cdot d{\mathbf {r} }=\int _{a,b}^{x_{1},y}\mathbf {v} \cdot d{\mathbf {r} }+\int _{x_{1},y}^{x,y}\mathbf {v} \cdot d{\mathbf {r} }.$
By the path independence, its partial derivative with respect to $x$ (for $\varphi $ to have partial derivatives, $\mathbf {v} $ needs to be continuous.) is
${\frac {\partial \varphi }{\partial x}}={\frac {\partial }{\partial x}}\int _{a,b}^{x,y}\mathbf {v} \cdot d{\mathbf {r} }={\frac {\partial }{\partial x}}\int _{a,b}^{x_{1},y}\mathbf {v} \cdot d{\mathbf {r} }+{\frac {\partial }{\partial x}}\int _{x_{1},y}^{x,y}\mathbf {v} \cdot d{\mathbf {r} }=0+{\frac {\partial }{\partial x}}\int _{x_{1},y}^{x,y}\mathbf {v} \cdot d{\mathbf {r} }$
since $x_{1}$ and $x$ are independent to each other. Let's express $\mathbf {v} $ as $\mathbf {v} }=P(x,y)\mathbf {i} +Q(x,y)\mathbf {j} $ where $\mathbf {i} $ and $\mathbf {j} $ are unit vectors along the $x$ and $y$ axes respectively, then, since $d\mathbf {r} =dx\mathbf {i} +dy\mathbf {j} $,
${\frac {\partial }{\partial x}}\varphi (x,y)={\frac {\partial }{\partial x}}\int _{x_{1},y}^{x,y}\mathbf {v} \cdot d\mathbf {r} ={\frac {\partial }{\partial x}}\int _{x_{1},y}^{x,y}P(t,y)dt=P(x,y)$
where the last equality is from the second fundamental theorem of calculus.
A similar approach for the line integral path shown in the right of the right figure results in $ {\frac {\partial }{\partial y}}\varphi (x,y)=Q(x,y)$ so
$\mathbf {v} =P(x,y)\mathbf {i} +Q(x,y)\mathbf {j} ={\frac {\partial \varphi }{\partial x}}\mathbf {i} +{\frac {\partial \varphi }{\partial y}}\mathbf {j} =\nabla \varphi $
is proved for the 2-dimensional Cartesian coordinate system. This proof method can be straightforwardly expanded to a higher dimensional orthogonal coordinate system (e.g., a 3-dimensional spherical coordinate system) so the converse statement is proved. Another proof is found here as the converse of the gradient theorem.
Irrotational vector fields
Let $n=3$ (3-dimensional space), and let $\mathbf {v} :U\to \mathbb {R} ^{3}$ be a $C^{1}$ (continuously differentiable) vector field, with an open subset $U$ of $\mathbb {R} ^{n}$. Then $\mathbf {v} $ is called irrotational if and only if its curl is $\mathbf {0} $ everywhere in $U$, i.e., if
$\nabla \times \mathbf {v} \equiv \mathbf {0} .$
For this reason, such vector fields are sometimes referred to as curl-free vector fields or curl-less vector fields. They are also referred to as longitudinal vector fields.
It is an identity of vector calculus that for any $C^{2}$ (continuously differentiable up to the 2nd derivative) scalar field $\varphi $ on $U$, we have
$\nabla \times (\nabla \varphi )\equiv \mathbf {0} .$
Therefore, every $C^{1}$ conservative vector field in $U$ is also an irrotational vector field in $U$. This result can be easily proved by expressing $\nabla \times (\nabla \varphi )$ in a Cartesian coordinate system with Schwarz's theorem (also called Clairaut's theorem on equality of mixed partials).
Provided that $U$ is a simply connected open space (roughly speaking, a single piece open space without a hole within it), the converse of this is also true: Every irrotational vector field in a simply connected open space $U$ is a $C^{1}$ conservative vector field in $U$.
The above statement is not true in general if $U$ is not simply connected. Let $U$ be $\mathbb {R} ^{3}$ with removing all coordinates on the $z$-axis (so not a simply connected space), i.e., $U=\mathbb {R} ^{3}\setminus \{(0,0,z)\mid z\in \mathbb {R} \}$. Now, define a vector field $\mathbf {v} $ on $U$ by
$\mathbf {v} (x,y,z)~{\stackrel {\text{def}}{=}}~\left(-{\frac {y}{x^{2}+y^{2}}},{\frac {x}{x^{2}+y^{2}}},0\right).$
Then $\mathbf {v} $ has zero curl everywhere in $U$ ($\nabla \times \mathbf {v} \equiv \mathbf {0} $ at everywhere in $U$), i.e., $\mathbf {v} $ is irrotational. However, the circulation of $\mathbf {v} $ around the unit circle in the $xy$-plane is $2\pi $; in polar coordinates, $\mathbf {v} =\mathbf {e} _{\phi }/r$, so the integral over the unit circle is
$\oint _{C}\mathbf {v} \cdot \mathbf {e} _{\phi }~d{\phi }=2\pi .$
Therefore, $\mathbf {v} $ does not have the path-independence property discussed above so is not conservative even if $\nabla \times \mathbf {v} \equiv \mathbf {0} $ since $U$ where $\mathbf {v} $ is defined is not a simply connected open space.
Say again, in a simply connected open region, an irrotational vector field $\mathbf {v} $ has the path-independence property (so $\mathbf {v} $ as conservative). This can be proved directly by using Stokes' theorem,
$\oint _{P_{c}}\mathbf {v} \cdot d\mathbf {r} =\iint _{A}(\nabla \times \mathbf {v} )\cdot d\mathbf {a} =0$
for any smooth oriented surface $A$ which boundary is a simple closed path $P_{c}$. So, it is concluded that In a simply connected open region, any $C^{1}$ vector field that has the path-independence property (so it is a conservative vector field.) must also be irrotational and vise versa.
Abstraction
More abstractly, in the presence of a Riemannian metric, vector fields correspond to differential $1$-forms. The conservative vector fields correspond to the exact $1$-forms, that is, to the forms which are the exterior derivative $d\phi $ of a function (scalar field) $\phi $ on $U$. The irrotational vector fields correspond to the closed $1$-forms, that is, to the $1$-forms $\omega $ such that $d\omega =0$. As $d^{2}=0$, any exact form is closed, so any conservative vector field is irrotational. Conversely, all closed $1$-forms are exact if $U$ is simply connected.
Vorticity
The vorticity ${\boldsymbol {\omega }}$ of a vector field can be defined by:
${\boldsymbol {\omega }}~{\stackrel {\text{def}}{=}}~\nabla \times \mathbf {v} .$
The vorticity of an irrotational field is zero everywhere.[6] Kelvin's circulation theorem states that a fluid that is irrotational in an inviscid flow will remain irrotational. This result can be derived from the vorticity transport equation, obtained by taking the curl of the Navier-Stokes Equations.
For a two-dimensional field, the vorticity acts as a measure of the local rotation of fluid elements. Note that the vorticity does not imply anything about the global behavior of a fluid. It is possible for a fluid that travels in a straight line to have vorticity, and it is possible for a fluid that moves in a circle to be irrotational.
Conservative forces
If the vector field associated to a force $\mathbf {F} $ is conservative, then the force is said to be a conservative force.
The most prominent examples of conservative forces are a gravitational force and an electric force associated to an electrostatic field. According to Newton's law of gravitation, a gravitational force $\mathbf {F} _{G}$ acting on a mass $m$ due to a mass $M$ located at a distance $r$ from $m$, obeys the equation
$\mathbf {F} _{G}=-{\frac {GmM}{r^{2}}}{\hat {\mathbf {r} }},$
where $G$ is the gravitational constant and ${\hat {\mathbf {r} }}$ is a unit vector pointing from $M$ toward $m$. The force of gravity is conservative because $\mathbf {F} _{G}=-\nabla \Phi _{G}$, where
$\Phi _{G}~{\stackrel {\text{def}}{=}}-{\frac {GmM}{r}}$
is the gravitational potential energy. In other words, the gravitation field ${\frac {\mathbf {F} _{G}}{m}}$ associated with the gravitational force $\mathbf {F} _{G}$ is the gradient of the gravitation potential ${\frac {\Phi _{G}}{m}}$ associated with the gravitational potential energy $\Phi _{G}$. It can be shown that any vector field of the form $\mathbf {F} =F(r){\hat {\mathbf {r} }}$ is conservative, provided that $F(r)$ is integrable.
For conservative forces, path independence can be interpreted to mean that the work done in going from a point $A$ to a point $B$ is independent of the moving path chosen (dependent on only the points $A$ and $B$), and that the work $W$ done in going around a simple closed loop $C$ is $0$:
$W=\oint _{C}\mathbf {F} \cdot d{\mathbf {r} }=0.$
The total energy of a particle moving under the influence of conservative forces is conserved, in the sense that a loss of potential energy is converted to the equal quantity of kinetic energy, or vice versa.
See also
• Beltrami vector field
• Conservative force
• Conservative system
• Complex lamellar vector field
• Helmholtz decomposition
• Laplacian vector field
• Longitudinal and transverse vector fields
• Solenoidal vector field
References
1. Marsden, Jerrold; Tromba, Anthony (2003). Vector calculus (Fifth ed.). W.H.Freedman and Company. pp. 550–561.
2. George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists, 6th edition, Elsevier Academic Press (2005)
3. For $\mathbf {v} =\nabla \varphi $ to be path-independent, $\varphi $ is not necessarily continuously differentiable, the condition of being differentiable is enough, since the Gradient theorem, that proves the path independence of $\nabla \varphi $, does not require $\varphi $ to be continuously differentiable. There must be a reason for the definition of conservative vector fields to require $\varphi $ to be continuously differentiable.
4. Stewart, James (2015). "16.3 The Fundamental Theorem of Line Integrals"". Calculus (8th ed.). Cengage Learning. pp. 1127–1134. ISBN 978-1-285-74062-1.
5. Need to verify if exact differentials also exist for non-orthogonal coordinate systems.
6. Liepmann, H.W.; Roshko, A. (1993) [1957], Elements of Gas Dynamics, Courier Dover Publications, ISBN 0-486-41963-0, pp. 194–196.
Further reading
• Acheson, D. J. (1990). Elementary Fluid Dynamics. Oxford University Press. ISBN 0198596790.
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\title{\Large\bf Quantization of the shift of argument subalgebras\\ in type $A$}
\author{{Vyacheslav Futorny\quad and\quad Alexander Molev}}
\date{} \maketitle
\begin{abstract} Given a simple Lie algebra $\mathfrak{g}$ and an element $\mu\in\mathfrak{g}^*$, the corresponding shift of argument subalgebra of $ {\rm S}(\mathfrak{g})$ is Poisson commutative. In the case where $\mu$ is regular, this subalgebra is known to admit a quantization, that is, it can be lifted to a commutative subalgebra of $ {\rm U}(\mathfrak{g})$. We show that if $\mathfrak{g}$ is of type $A$, then this property extends to arbitrary $\mu$, thus proving a conjecture of Feigin, Frenkel and Toledano Laredo. The proof relies on an explicit construction of generators of the center of the affine vertex algebra at the critical level.
\end{abstract}
\noindent Department of Mathematics, University of S\~{a}o Paulo,\newline Caixa Postal 66281, S\~{a}o Paulo, SP 05315-970, Brazil\newline [email protected]
\noindent School of Mathematics and Statistics\newline University of Sydney, NSW 2006, Australia\newline [email protected]
\section{Introduction} \label{sec:int} \setcounter{equation}{0}
\paragraph{Shift of argument subalgebras.}
Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}\tss$ with basis elements $Y_1,\dots,Y_l$ and the corresponding structure constants $c_{ij}^{\hspace{1pt} k}$. The symmetric algebra $ {\rm S}(\mathfrak{g})$ can be equipped with the {\it Lie--Poisson bracket\/} defined on the elements of the Lie algebra by \beql{liepoisson} \{Y_i,Y_j\}=\sum_{k=1}^l\hspace{1pt} c_{ij}^{\hspace{1pt} k}\, Y_k. \end{equation}
Let $P=P(Y_1,\dots,Y_l)$ be an element of $ {\rm S}(\mathfrak{g})$ of a certain degree $d$. Fix any element $\mu\in\mathfrak{g}^*$ and let $z$ be a variable. Make the substitution $Y_i\mapsto Y_i+z\,\mu(Y_i)$ and expand as a polynomial in $z$, \begin{equation*} P\big(Y_1+z\,\mu(Y_1),\dots,Y_l+z\,\mu(Y_l)\big) =P^{(0)}+P^{(1)} z+\dots+P^{(d)} z^d \end{equation*} to define elements $P^{(i)}\in {\rm S}(\mathfrak{g})$ associated with $P$ and $\mu$. Denote by $\overline\mathcal{A}_{\mu}$ the subalgebra of $ {\rm S}(\mathfrak{g})$ generated by all elements $P^{(i)}$ associated with all $\mathfrak{g}$-invariants $P\in {\rm S}(\mathfrak{g})^{\mathfrak{g}}$. The subalgebra $\overline\mathcal{A}_{\mu}$ of $ {\rm S}(\mathfrak{g})$ is known as the {\it Mishchenko--Fomenko subalgebra\/} or {\it shift of argument subalgebra\/}. Its key property observed in \cite{mf:ee} states that $\overline\mathcal{A}_{\mu}$ is Poisson commutative; that is, $\{R,S\}=0$ for any elements $R,S\in\overline\mathcal{A}_{\mu}$.
We will identify $\mathfrak{g}^*$ with $\mathfrak{g}$ via a symmetric invariant bilinear form (see \eqref{killi} below) and let $n$ denote the rank of $\mathfrak{g}$. An element $\mu\in\mathfrak{g}^*\cong\mathfrak{g}$ is called {\em regular}, if the centralizer $\mathfrak{g}^{\mu}$ of $\mu$ in $\mathfrak{g}$ has minimal possible dimension; this minimal dimension coincides with $n$. The subalgebra $ {\rm S}(\mathfrak{g})^{\mathfrak{g}}$ admits a family $P_1,\dots,P_n$ of algebraically independent generators of respective degrees $d_1,\dots,d_n$. If the element $\mu\in\mathfrak{g}^*$ is regular, then $\overline\mathcal{A}_{\mu}$ has the properties: \begin{enumerate} \item[$i)$] the subalgebra $\overline\mathcal{A}_{\mu}$ of $ {\rm S}(\mathfrak{g})$ is maximal Poisson commutative; \item[$ii)$]\label{pki} the elements $P^{(i)}_{k}$ with $k=1,\dots,n$ and $i=0,1,\dots,d_k-1$, are algebraically independent generators of $\overline\mathcal{A}_{\mu}$. \end{enumerate}
Property $i)$ is a theorem of Panyushev and Yakimova~\cite{py:as}; the case of regular semisimple $\mu$ is due to Tarasov~\cite{t:ms}. Property $ii)$ is due to Bolsinov~\cite{b:cf}; the regular semisimple case goes back to the original paper \cite{mf:ee}. Another proof of $ii)$ was given in~\cite{fft:gm}.
\paragraph{Vinberg's problem.}
The universal enveloping algebra $ {\rm U}(\mathfrak{g})$ is equipped with a canonical filtration and the associated graded algebra $ {\rm gr}\, {\rm U}(\mathfrak{g})$ is isomorphic to $ {\rm S}(\mathfrak{g})$. Given that the subalgebra $\overline\mathcal{A}_{\mu}$ of $ {\rm S}(\mathfrak{g})$ is Poisson commutative, one could look for a commutative subalgebra $\mathcal{A}_{\mu}$ of $ {\rm U}(\mathfrak{g})$ which ``quantizes" $\overline\mathcal{A}_{\mu}$ in the sense that $ {\rm gr}\,\mathcal{A}_{\mu}=\overline\mathcal{A}_{\mu}$. This quantization problem was raised by Vinberg in \cite{v:sc}, where, in particular, some commuting families of elements of $ {\rm U}(\mathfrak{g})$ were produced. A positive solution of Vinberg's problem was given by Rybnikov~\cite{r:si} (for regular semisimple $\mu$) and Feigin, Frenkel and Toledano Laredo~\cite{fft:gm} (for any regular $\mu$) with the use of the center of the associated affine vertex algebra at the critical level (also known as the {\em Feigin--Frenkel center}). To briefly outline the solution, equip $\mathfrak{g}$ with a standard symmetric invariant bilinear form $\langle\,\,,\,\rangle$ defined as the normalized Killing form \beql{killi} \langle X,Y\rangle=\frac{1}{2\hspace{1pt} h^{\vee}}\, {\rm tr}\,\big({\rm{ad}\,}\hspace{1pt} X\,{\rm{ad}\,}\hspace{1pt} Y\big), \end{equation} where $h^{\vee}$ is the {\it dual Coxeter number\/} for $\mathfrak{g}$. The corresponding {\it affine Kac--Moody algebra\/} $\widehat\mathfrak{g}$ is the central extension \beql{km} \widehat\mathfrak{g}=\mathfrak{g}\hspace{1pt}[t,t^{-1}]\oplus\mathbb{C}\tss K, \end{equation} where $\mathfrak{g}[t,t^{-1}]$ is the Lie algebra of Laurent polynomials in $t$ with coefficients in $\mathfrak{g}$. For any $r\in\mathbb{Z}\tss$ and $X\in\mathfrak{g}$ we set $X[r]=X\, t^r$. The commutation relations of the Lie algebra $\widehat\mathfrak{g}$ have the form \begin{equation*} \big[X[r],Y[s]\big]=[X,Y][r+s]+r\,\delta_{r,-s}\langle X,Y\rangle\, K, \qquad X, Y\in\mathfrak{g}, \end{equation*} and the element $K$ is central in $\widehat\mathfrak{g}$. For any $\kappa\in\mathbb{C}\tss$ denote by $ {\rm U}_{\kappa}(\widehat\mathfrak{g})$ the quotient of $ {\rm U}(\widehat\mathfrak{g})$ by the ideal generated by $K-\kappa$. The value $\kappa=-h^{\vee}$ corresponds to the {\it critical level\/}. Let ${\rm I}$ denote the left ideal of $ {\rm U}_{-h^{\vee}}(\widehat\mathfrak{g})$ generated by $\mathfrak{g}[t]$ and let ${\rm Norm\tss}\hspace{1pt}{\rm I}$ be its normalizer, \begin{equation*}
{\rm Norm\tss}\hspace{1pt}{\rm I}=\{v\in {\rm U}_{-h^{\vee}}(\widehat\mathfrak{g})\ |\ {\rm I}\hspace{1pt} v\subseteq {\rm I}\}. \end{equation*} The normalizer is a subalgebra of $ {\rm U}_{-h^{\vee}}(\widehat\mathfrak{g})$, and ${\rm I}$ is a two-sided ideal of ${\rm Norm\tss}\hspace{1pt}{\rm I}$. The {\it Feigin--Frenkel center\/} $\mathfrak{z}(\widehat\mathfrak{g})$ is the associative algebra defined as the quotient \beql{ffnorm} \mathfrak{z}(\widehat\mathfrak{g})={\rm Norm\tss}\hspace{1pt}{\rm I}/{\rm I}. \end{equation} By the Poincar\'e--Birkhoff--Witt theorem, the quotient of the algebra $ {\rm U}_{-h^{\vee}}(\widehat\mathfrak{g})$ by the left ideal ${\rm I}$ is isomorphic to the universal enveloping algebra $ {\rm U}\big(t^{-1}\mathfrak{g}[t^{-1}]\big)$, as a vector space. Hence, we have a vector space embedding \begin{equation*} \mathfrak{z}(\widehat\mathfrak{g})\hookrightarrow {\rm U}\big(t^{-1}\mathfrak{g}[t^{-1}]\big). \end{equation*} Since $ {\rm U}\big(t^{-1}\mathfrak{g}[t^{-1}]\big)$ is a subalgebra of $ {\rm U}_{-h^{\vee}}(\widehat\mathfrak{g})$, the embedding is an algebra homomorphism so that the Feigin--Frenkel center $\mathfrak{z}(\widehat\mathfrak{g})$ can be regarded as a subalgebra of $ {\rm U}\big(t^{-1}\mathfrak{g}[t^{-1}]\big)$. In fact, this subalgebra is {\it commutative\/} which is not immediate from the definition, but can be seen by identifying $\mathfrak{z}(\widehat\mathfrak{g})$ with the center of the affine vertex algebra at the critical level. Furthermore, by a theorem of Feigin and Frenkel~\cite{ff:ak} (see \cite{f:lc} for a detailed exposition), there exist elements $S_1,\dots,S_n\in \mathfrak{z}(\widehat\mathfrak{g})$ such that \beql{genz}
\mathfrak{z}(\widehat\mathfrak{g})=\mathbb{C}\tss[T^{\hspace{1pt} r}S_l\ |\ l=1,\dots,n,\ \ r\geqslant 0], \end{equation} where $T$ is the derivation of the algebra $ {\rm U}\big(t^{-1}\mathfrak{g}[t^{-1}]\big)$ which is determined by the property that its commutator with the operator of left multiplication by $X[r]$ is found by \begin{equation*} \big[T,X[r]\big]=-r\hspace{1pt} X[r - 1],\qquad X\in\mathfrak{g},\quad r<0. \end{equation*} We will call such family $S_1,\dots,S_n$ a {\em complete set of Segal--Sugawara vectors for $\mathfrak{g}$}. Another derivation $D$ of the algebra $ {\rm U}\big(t^{-1}\mathfrak{g}[t^{-1}]\big)$ is determined by the property \begin{equation*} \big[D,X[r]\big]=-r\hspace{1pt} X[r],\qquad X\in\mathfrak{g},\quad r<0; \end{equation*} and $D$ defines a grading on $ {\rm U}\big(t^{-1}\mathfrak{g}[t^{-1}]\big)$.
Given any element $\mu\in\mathfrak{g}^*$ and a nonzero $z\in\mathbb{C}\tss$, the mapping \beql{evalr} \varrho^{}_{\,\mu,z}: {\rm U}\big(t^{-1}\mathfrak{g}[t^{-1}]\big)\to {\rm U}(\mathfrak{g}), \qquad X[r]\mapsto X z^r+\delta_{r,-1}\,\mu(X),\quad X\in\mathfrak{g}, \end{equation} defines an algebra homomorphism. The image of $\mathfrak{z}(\widehat\mathfrak{g})$ under $\varrho^{}_{\,\mu,z}$ is a commutative subalgebra of $ {\rm U}(\mathfrak{g})$. It does not depend on $z$ and is denoted by $\mathcal{A}_{\mu}$. If $S\in {\rm U}\big(t^{-1}\mathfrak{g}[t^{-1}]\big)$ is an element of degree $d$ with respect to the grading defined by $D$, then regarding $\varrho^{}_{\,\mu,z}(S)$ as a polynomial in $z^{-1}$, define the elements $S^{(i)}\in {\rm U}(\mathfrak{g})$ by the expansion \beql{rhoexp} \varrho^{}_{\,\mu,z}(S)=S^{(0)} z^{-d}+\dots+S^{(d-1)} z^{-1}+S^{(d)}. \end{equation} If $\mu\in\mathfrak{g}^*$ is regular then the following holds: \begin{enumerate} \item[$i)$] the subalgebra $\mathcal{A}_{\mu}$ of $ {\rm U}(\mathfrak{g})$ is maximal commutative; \item[$ii)$] if $S_1,\dots,S_n\in \mathfrak{z}(\widehat\mathfrak{g})$ are elements of the respective degrees $d_1,\dots,d_n$ satisfying \eqref{genz}, then the elements $S^{(i)}_{k}$ with $k=1,\dots,n$ and $i=0,1,\dots,d_k-1$ are algebraically independent generators of $\mathcal{A}_{\mu}$; \item[$iii)$] $ {\rm gr}\,\mathcal{A}_{\mu}=\overline\mathcal{A}_{\mu}$. \end{enumerate} This is derived with the use of the respective properties of the algebra $\overline\mathcal{A}_{\mu}$; see \cite{fft:gm} for proofs. The subalgebra $\mathcal{A}_{\mu}$ was further studied in \cite{ffr:oi} where its spectra in finite-dimensional irreducible representations of $\mathfrak{g}$ were described.
Note that both algebras $\mathcal{A}_{\mu}$ and $\overline\mathcal{A}_{\mu}$ are defined for arbitrary elements $\mu\in\mathfrak{g}^*$. Given that the property $iii)$ holds for all regular $\mu$, it was conjectured in \cite[Conjecture~1]{fft:gm}, that this property is valid for all $\mu$. As a consequence of our main result, we obtain a proof of this conjecture for type $A$; see the Main Theorem below. In particular, this gives another proof of $iii)$ for regular $\mu$. More precisely, we will work with the reductive Lie algebra $\mathfrak{g}=\mathfrak{gl}_n$ and consider the respective subalgebras $\overline\mathcal{A}_{\mu}\subset {\rm S}(\mathfrak{gl}_n)$ and $\mathcal{A}_{\mu}\subset {\rm U}(\mathfrak{gl}_n)$. The proof will be based on the use of explicit formulas for generators of $\mathcal{A}_{\mu}$.
\paragraph{Generators of $\mathcal{A}_{\mu}$.} For the Lie algebras $\mathfrak{g}$ of type $A$, a few families of explicit generators $S_1,\dots,S_n$ of $\mathfrak{z}(\widehat\mathfrak{g})$, and hence generators of the subalgebra $\mathcal{A}_{\mu}$, were produced by Chervov and Talalaev~\cite{ct:qs} by extending Talalaev's work \cite{t:qg}; see also \cite{cm:ho} and \cite{mr:mm} where more direct proofs were given. In types $B$, $C$ and $D$ such explicit generators were constructed in \cite{m:ff}. Note also earlier work of Nazarov and Olshanski~\cite{no:bs}, where maximal commutative subalgebras of $ {\rm U}(\mathfrak{g})$ were produced with the use of Yangians; they quantize the Poisson algebras $\overline\mathcal{A}_{\mu}$ in all classical types for the case of regular semisimple $\mu$. In a different form, a quantization of $\overline\mathcal{A}_{\mu}$ in type $A$ was provided by Tarasov~\cite{t:cs} via a symmetrization map.
We will work with a particular family of generators of $\mathfrak{z}(\widehat\mathfrak{gl}_n)$ which we recall below in Sec.~\ref{sec:gff}. They allow us to define the associated family of generators $\phi^{(k)}_m$ with $m=1,\dots,n$ and $k=0,\dots,m-1$ of the subalgebra $\mathcal{A}_{\mu}\subset {\rm U}(\mathfrak{gl}_n)$; see \eqref{defvp} below. There generators are algebraically independent if $\mu$ is regular.
Our main result provides a way to choose an algebraically independent family of generators $\phi^{(k)}_m$ of $\mathcal{A}_{\mu}$ for an arbitrary element $\mu$. To describe this subset, we will identify $\mathfrak{gl}^*_n$ with $\mathfrak{gl}_n$ via a symmetric bilinear form and regard $\mu$ as an $n\times n$ matrix. Suppose that the distinct eigenvalues of $\mu$ are $\lambda_1,\dots,\lambda_r$ and the Jordan canonical form of $\mu$ is the direct sum of the respective Jordan blocks $J_{\alpha^{(i)}_j}(\lambda_i)$ of sizes
$\alpha^{(i)}_1\geqslant \alpha^{(i)}_2\geqslant\dots \geqslant \alpha^{(i)}_{s_i}\geqslant 1$. We let $\alpha^{(i)}$ denote the corresponding Young diagram whose $j$-th row is $\alpha^{(i)}_j$ and let $|\alpha^{(i)}|$ be the number of boxes of $\alpha^{(i)}$. Given these data, introduce another Young diagram $\gamma=(\gamma_1,\gamma_2,\dots)$ by setting \beql{gal} \gamma_l=\sum_{i=1}^r\sum_{j\geqslant l+1} \alpha^{(i)}_j, \end{equation} so that $\gamma_l$ is the total number of boxes which are strictly below the $l$-th rows in all diagrams $\alpha^{(i)}$. Furthermore, associate the elements of the family $\phi^{(k)}_m$ with boxes of the diagram $\Gamma=(n,n-1,\dots,1)$ so that the $(i,j)$ box of $\Gamma$ corresponds to $\phi^{(n-i-j+1)}_{n-j+1}$, as illustrated: \beql{Ga} \Gamma\quad=\qquad \begin{matrix} \phi^{(n-1)}_n & \phi^{(n-2)}_{n-1} & \dots & \phi^{(1)}_2 & \phi^{(0)}_1\\[0.5em] \phi^{(n-2)}_n & \phi^{(n-3)}_{n-1} & \dots & \phi^{(0)}_2 & \\ \dots & \dots & \dots & & \\[0.5em] \phi^{(1)}_n\phantom{-} & \phi^{(0)}_{n-1}\phantom{-} & & &\\[0.5em] \phi^{(0)}_n\phantom{-} & & & & \end{matrix} \end{equation} Note that the diagram $\gamma$ is contained in $\Gamma$. We can now state our main theorem, where $\mu$ is an arbitrary element of $\mathfrak{gl}_n$ and $\Gamma/\gamma$ is the associated skew diagram.
\begin{mthm} The elements $\phi^{(k)}_m$ corresponding to the boxes of the skew diagram $\Gamma/\gamma$ are algebraically independent generators of the subalgebra $\mathcal{A}_{\mu}$. Moreover, the subalgebra $\mathcal{A}_{\mu}$ is a quantization of $\overline\mathcal{A}_{\mu}$ so that $ {\rm gr}\,\mathcal{A}_{\mu}=\overline\mathcal{A}_{\mu}$. \end{mthm}
By considering some other complete sets of Segal--Sugawara vectors, we also show that the first part of the Main Theorem remains valid if the elements $\phi^{(k)}_m$ are replaced with those of other families; see Corollaries~\ref{cor:othgen} and \ref{cor:othgentwo} below.
\begin{example}\label{ex:gensk} Take $n=6$ and let $\mu$ be a nilpotent matrix with the Jordan blocks of sizes $(2,2,1,1)$. Then $\gamma=(4,2,1)$ and the skew diagram $\Gamma/\gamma$ is \begin{equation*} \young(::::\mbox{}\mbox{},::\mbox{}\mbox{}\mbox{},:\mbox{}\mbox{}\mbox{},\mbox{}\mbox{}\mbox{},\mbox{}\mbox{},\mbox{}) \end{equation*} so that the algebraically independent generators of $\mathcal{A}_{\mu}$ are those corresponding to the boxes of $\Gamma$, excluding $\phi^{(2)}_3$, $\phi^{(3)}_4$, $\phi^{(3)}_5$, $\phi^{(4)}_5$, $\phi^{(3)}_6$, $\phi^{(4)}_6$ and $\phi^{(5)}_6$. \qed \end{example}
Note also two extreme cases. If $\mu$ is regular, then all Jordan blocks correspond to distinct eigenvalues so that each $\alpha^{(i)}$ is a singe row diagram. Therefore, $\gamma=\varnothing$, so that all generators $\phi^{(k)}_m$ associated with the boxes of $\Gamma$ are algebraically independent. On the other hand, for scalar matrices $\mu$ we have $\gamma=(n-1,n-2,\dots,1)$. In this case, $\mathcal{A}_{\mu}$ is generated by $\phi^{(0)}_1,\dots,\phi^{(0)}_n$ and it coincides with the center of $ {\rm U}(\mathfrak{gl}_n)$.
Our proofs rely on {\em Bolsinov's completeness criterion} \cite[Theorem~3.2]{b:cf} which applies to the shift of argument subalgebras associated with an arbitrary Lie algebra $\mathfrak{g}$. The required condition for reductive Lie algebras is the equality \beql{beconj} {\rm{ind}\,}\mathfrak{g}={\rm{ind}\,}\mathfrak{g}^{\mu} \end{equation} of the indices of $\mathfrak{g}$ and the centralizer $\mathfrak{g}^{\mu}$ of $\mu$ in $\mathfrak{g}$, where the {\em index} of an arbitrary Lie algebra $\mathfrak{g}$ is the minimal dimension of the stabilizers $\mathfrak{g}^x$, $x\in\mathfrak{g}^*$, for the coadjoint representation. In the case $\mathfrak{g}=\mathfrak{gl}_n$ and arbitrary $\mu\in\mathfrak{g}$ this equality was claimed to be verified by Bolsinov \cite[Sec.~3]{b:cp} (and was suggested to be extendable to arbitrary semisimple Lie algebras) and by Elashvili (private communication), but details were not published. The first published proof is due to Yakimova~\cite{y:ic}, which extends to all classical Lie algebras. The equality \eqref{beconj} is widely referred to as the {\em Elashvili conjecture}, but should rather be called the {\em Bolsinov--Elashvili conjecture}\footnote{ A.~Elashvili kindly informed us that the conjectural equality had emerged from A.~Bolsinov's questions to him and so it should also be attributed to the author of \cite{b:cf}.}; see e.g. \cite{cm:ic} for its proof covering all simple Lie algebras and more references.
We are grateful to Alexey Bolsinov, Alexander Elashvili, Leonid Rybnikov and Alexander Veselov for useful discussions. The first author was supported in part by the CNPq grant (301320/2013-6) and by the Fapesp grant (2010/50347-9). This work was completed during the second author's visit to the University of S\~{a}o Paulo. He would like to thank the Department of Mathematics for the warm hospitality.
\section{Generators of $\mathfrak{z}(\widehat\mathfrak{gl}_n)$} \label{sec:gff} \setcounter{equation}{0}
For $i,j\in\{1,\dots,n\}$ we will denote by $E_{ij}$ the standard basis elements of $\mathfrak{gl}_n$. We extend the form \eqref{killi} to the invariant symmetric bilinear form on $\mathfrak{gl}_n$ which is given by \begin{equation*} \langle X,\, Y\rangle= {\rm tr}\hspace{1pt}(X\hspace{1pt} Y)-\frac{1}{n}\, {\rm tr}\hspace{1pt} X \, {\rm tr}\hspace{1pt} Y,\qquad X, Y\in\mathfrak{gl}_n, \end{equation*} where $X$ and $Y$ are regarded as $n\times n$ matrices. Note that the kernel of the form is spanned by the element $E_{11}+\dots+E_{nn}$, and its restriction to the subalgebra $\mathfrak{sl}_n$ is given by \begin{equation*} \langle X,\, Y\rangle= {\rm tr}\hspace{1pt}(X\hspace{1pt} Y),\qquad X, Y\in\mathfrak{sl}_n. \end{equation*} The affine Kac--Moody algebra $\widehat\mathfrak{gl}_n=\mathfrak{gl}_n[t,t^{-1}]\oplus\mathbb{C}\tss K$ has the commutation relations \beql{commrel} \big[E_{ij}[r],E_{kl}[s\hspace{1pt}]\hspace{1pt}\big] =\delta_{kj}\, E_{i\hspace{1pt} l}[r+s\hspace{1pt}] -\delta_{i\hspace{1pt} l}\, E_{kj}[r+s\hspace{1pt}] +r\hspace{1pt}\delta_{r,-s}\, K\Big(\delta_{kj}\hspace{1pt}\delta_{i\hspace{1pt} l} -\frac{\delta_{ij}\hspace{1pt}\delta_{kl}}{n}\Big), \end{equation} and the element $K$ is central. The critical level $-n$ coincides with the negative of the dual Coxeter number for $\mathfrak{sl}_n$. We will work with the extended Lie algebra $\widehat\mathfrak{gl}_n\oplus\mathbb{C}\tss\tau$ where the additional element $\tau$ satisfies the commutation relations \beql{taur} \big[\tau,X[r]\hspace{1pt}\big]=-r\, X[r-1],\qquad \big[\tau,K\big]=0. \end{equation} For any $r\in\mathbb{Z}\tss$ combine the elements $E_{ij}[r]$ into the matrix $E[r]$ so that \beql{matrer} E[r]=\sum_{i,j=1}^n e_{ij}\otimes E_{ij}[r]\in {\rm{End}\,}\mathbb{C}\tss^n\otimes {\rm U}, \end{equation} where the $e_{ij}$ are the standard matrix units and $ {\rm U}$ stands for the universal enveloping algebra of $\widehat\mathfrak{gl}_n\oplus\mathbb{C}\tss\tau$. For each $a\in\{1,\dots,m\}$ introduce the element $E[r]_a$ of the algebra \beql{tenprka} \underbrace{{\rm{End}\,}\mathbb{C}\tss^{n}\otimes\dots\otimes{\rm{End}\,}\mathbb{C}\tss^{n}}_m{}\otimes {\rm U} \end{equation} by \beql{matnota} E[r]_a=\sum_{i,j=1}^{n} 1^{\otimes(a-1)}\otimes e_{ij}\otimes 1^{\otimes(m-a)}\otimes E_{ij}[r]. \end{equation} We let $H^{(m)}$ and $A^{(m)}$ denote the respective images of the symmetrizer $h^{(m)}$ and anti-symmetrizer $a^{(m)}$ in the group algebra for the symmetric group $\mathfrak S_m$ under its natural action on $(\mathbb{C}\tss^{n})^{\otimes m}$. The elements $h^{(m)}$ and $a^{(m)}$ are the idempotents in the group algebra $\mathbb{C}\tss[\mathfrak S_m]$ defined by \begin{equation*} h^{(m)}=\frac{1}{m!}\,\sum_{s\in\mathfrak S_m} s \qquad\text{and}\qquad a^{(m)}=\frac{1}{m!}\,\sum_{s\in\mathfrak S_m} {\rm sgn}\ts s\cdot s. \end{equation*} We will identify $H^{(m)}$ and $A^{(m)}$ with the respective elements $H^{(m)}\otimes 1$ and $A^{(m)}\otimes 1$ of the algebra \eqref{tenprka}. Define the elements $\phi^{}_{m\hspace{1pt} a},\psi^{}_{m\hspace{1pt} a}, \theta^{}_{m\hspace{1pt} a}\in {\rm U}\big(t^{-1}\mathfrak{gl}_n[t^{-1}]\big)$ by the expansions \begin{align}\label{deftra} {\rm tr}_{1,\dots,m}\, A^{(m)} \big(\tau+E[-1]_1\big)\dots \big(\tau+E[-1]_m\big) &=\phi^{}_{m\tss0}\,\tau^m+\phi^{}_{m\tss1}\,\tau^{m-1} +\dots+\phi^{}_{m\hspace{1pt} m},\\[0.7em] \label{deftrh} {\rm tr}_{1,\dots,m}\, H^{(m)} \big(\tau+E[-1]_1\big)\dots \big(\tau+E[-1]_m\big) &=\psi^{}_{m\tss0}\,\tau^m+\psi^{}_{m\tss1}\,\tau^{m-1} +\dots+\psi^{}_{m\hspace{1pt} m}, \end{align} where the traces are taken with respect to all $m$ copies of ${\rm{End}\,}\mathbb{C}\tss^n$ in \eqref{tenprka}, and \beql{deftracepa} {\rm tr}\, \big(\tau+E[-1]\big)^m=\theta^{}_{m\tss0}\,\tau^m+\theta^{}_{m\tss1}\,\tau^{m-1} +\dots+\theta^{}_{m\hspace{1pt} m}. \end{equation} Expressions like $\tau+E[-1]$ are understood as matrices, where $\tau$ is regarded as the scalar matrix $\tau\hspace{1pt} 1$. Furthermore, introduce the {\em column-determinant} of the matrix $\tau+E[-1]$ by \beql{cdet} {\rm cdet}\, \big(\tau+E[-1]\big)=\sum_{\sigma\in\mathfrak S_n} {\rm sgn}\ts\sigma\cdot \big(\tau+E[-1]\big)_{\sigma(1)\hspace{1pt} 1}\dots \big(\tau+E[-1]\big)_{\sigma(n)\hspace{1pt} n} \end{equation} and expand it as a polynomial in $\tau$, \beql{coldetal} {\rm cdet}\, \big(\tau+E[-1]\big)=\tau^n+\phi^{}_{1}\,\tau^{n-1} +\dots+\phi^{}_{n},\qquad \phi^{}_{m}\in {\rm U}\big(t^{-1}\mathfrak{gl}_n[t^{-1}]\big). \end{equation} We have the expansion of the noncommutative characteristic polynomial, \beql{idtrd} {\rm cdet}\, \big(u+\tau+E[-1]\big)=\sum_{m=0}^n u^{n-m}\, {\rm tr}_{1,\dots,m}\, A^{(m)} \big(\tau+E[-1]_1\big)\dots \big(\tau+E[-1]_m\big), \end{equation} where $u$ is a variable. This implies the relations \beql{chaam} \phi^{}_{m\hspace{1pt} a}=\binom{n-a}{m-a}\,\phi^{}_{a},\qquad 0\leqslant a\leqslant m\leqslant n. \end{equation} In particular, $\phi^{}_{m\hspace{1pt} m}=\phi^{}_{m}$ for $m=1,\dots,n$.
\begin{thm}\label{thm:allff} All elements $\phi^{}_{m}$, $\psi^{}_{m\hspace{1pt} a}$ and $\theta^{}_{m\hspace{1pt} a}$ belong to the Feigin--Frenkel center $\mathfrak{z}(\widehat\mathfrak{gl}_n)$. Moreover, each of the families \begin{equation*} \phi^{}_{1},\dots,\phi^{}_{n},\qquad \psi^{}_{1\hspace{1pt} 1},\dots,\psi^{}_{n\hspace{1pt} n} \qquad\text{and}\qquad \theta^{}_{1\hspace{1pt} 1},\dots,\theta^{}_{n\hspace{1pt} n} \end{equation*} is a complete set of Segal--Sugawara vectors for $\mathfrak{gl}_n$. \qed \end{thm}
This theorem goes back to \cite{ct:qs}, where the elements $\phi^{}_{m}$ were first discovered (in a slightly different form). A direct proof of the theorem was given in \cite{cm:ho}. The elements $\psi^{}_{m\hspace{1pt} a}$ are related to $\phi^{}_{m\hspace{1pt} a}$ through the quantum MacMahon Master Theorem of \cite{glz:qm}, while a relationship between the $\phi^{}_{m\hspace{1pt} a}$ and $\theta^{}_{m\hspace{1pt} a}$
is provided by a Newton-type identity given in \cite[Theorem~15]{cfr:ap}. Note that super-versions of these relations between the families of Segal--Sugawara vectors for the Lie superalgebra $\mathfrak{gl}_{m|n}$ were given in the paper \cite{mr:mm}, which also provides simpler arguments in the purely even case.
\section{Generators of $\mathcal{A}_{\mu}$} \label{sec:gamu} \setcounter{equation}{0}
In accordance with the results which we recalled in the Introduction, the application of the homomorphism \eqref{evalr} to elements of $\mathfrak{z}(\widehat\mathfrak{gl}_n)$ provided by Theorem~\ref{thm:allff} yields the corresponding families of elements of the subalgebra $\mathcal{A}_{\mu}\subset {\rm U}(\mathfrak{gl}_n)$ through the expansion \eqref{rhoexp}. To give explicit formulas, we will use the tensor product algebra \eqref{tenprka}, where $ {\rm U}$ will now denote the algebra of differential operators whose elements are finite sums of the form \begin{equation*} \sum_{k,l\geqslant 0} u^{}_{kl}\, z^{-k}\hspace{1pt} \partial^{\, l}_z, \qquad u^{}_{kl}\in {\rm U}(\mathfrak{gl}_n). \end{equation*} Note that $\partial_z$ emerges here as the image of the element $-\tau$ under the extension of the homomorphism \eqref{evalr}. As in \eqref{matrer}, we set \begin{equation*} E=\sum_{i,j=1}^n e_{ij}\otimes E_{ij}\in {\rm{End}\,}\mathbb{C}\tss^n\otimes {\rm U}(\mathfrak{gl}_n), \end{equation*} and extend the notation \eqref{matnota} to the matrices $E$, $\mu$ and $M=-\partial_z+\mu+Ez^{-1}$. Assuming that $\mu\in\mathfrak{gl}_n$ is arbitrary, introduce the polynomials $\phi^{}_{m\hspace{1pt} a}(z)$, $\psi^{}_{m\hspace{1pt} a}(z)$ and $\theta^{}_{m\hspace{1pt} a}(z)$ in $z^{-1}$ (depending on $\mu$) with coefficients in $ {\rm U}(\mathfrak{gl}_n)$ by the expansions \begin{equation*} \begin{aligned} {\rm tr}_{1,\dots,m}\, A^{(m)} M_1\dots M_m&=\phi^{}_{m\tss0}(z)\,\partial_z^{\, m}+\phi^{}_{m\tss1}(z)\,\partial_z^{\, m-1} +\dots+\phi^{}_{m\hspace{1pt} m}(z),\\[0.5em] {\rm tr}_{1,\dots,m}\, H^{(m)} M_1\dots M_m&=\psi^{}_{m\tss0}(z)\,\partial_z^{\, m}+\psi^{}_{m\tss1}(z)\,\partial_z^{\, m-1} +\dots+\psi^{}_{m\hspace{1pt} m}(z), \end{aligned} \end{equation*} and \begin{equation*} {\rm tr}\, M^m =\theta^{}_{m\tss0}(z)\,\partial_z^{\, m}+\theta^{}_{m\tss1}(z)\,\partial_z^{\, m-1} +\dots+\theta^{}_{m\hspace{1pt} m}(z). \end{equation*} Furthermore, following \eqref{coldetal} define the polynomials $\phi^{}_{a}(z)$ by expanding the column-determinant \beql{cdetcoma} {\rm cdet}\, M=\phi^{}_{0}(z)\,\partial_z^{\, n}+\phi^{}_{1}(z) \,\partial_z^{\, n-1} +\dots+\phi^{}_{n}(z). \end{equation} By \eqref{chaam} we have \begin{equation*} \phi^{}_{m\hspace{1pt} a}(z)=\binom{n-a}{m-a}\,\phi^{}_{a}(z),\qquad 0\leqslant a\leqslant m\leqslant n, \end{equation*} and so $\phi^{}_{m\hspace{1pt} m}(z)=\phi^{}_{m}(z)$ for all $m$. Introduce the coefficients of polynomials by \begin{equation*} \begin{aligned} \phi^{}_{m}(z)&=\phi^{\hspace{1pt}(0)}_{m}z^{-m} +\dots+\phi^{\hspace{1pt}(m-1)}_{m}z^{-1} +\phi^{\hspace{1pt}(m)}_{m},\\[0.5em] \psi^{}_{m\hspace{1pt} m}(z)&=\psi^{\hspace{1pt}(0)}_{m\hspace{1pt} m}z^{-m} +\dots+\psi^{\hspace{1pt}(m-1)}_{m\hspace{1pt} m}z^{-1} +\psi^{\hspace{1pt}(m)}_{m\hspace{1pt} m}, \end{aligned} \end{equation*} and \begin{equation*} \theta^{}_{m\hspace{1pt} m}(z)=\theta^{\hspace{1pt}(0)}_{m\hspace{1pt} m}z^{-m} +\dots+\theta^{\hspace{1pt}(m-1)}_{m\hspace{1pt} m}z^{-1} +\theta^{\hspace{1pt}(m)}_{m\hspace{1pt} m}. \end{equation*}
By Theorem~\ref{thm:allff} and the general results of \cite{fft:gm} and \cite{r:si} we get the following.
\begin{thm}\label{thm:comsera} Given any $\mu\in\mathfrak{gl}_n$, all coefficients of the polynomials $\phi^{}_{m}(z)$, $\psi^{}_{m\hspace{1pt} a}(z)$ and $\theta^{}_{m\hspace{1pt} a}(z)$ belong to the commutative subalgebra $\mathcal{A}_{\mu}$ of $ {\rm U}(\mathfrak{gl}_n)$. Moreover, the elements of each of the families \begin{equation*} \phi^{\hspace{1pt}(k)}_{m},\qquad \psi^{\hspace{1pt}(k)}_{m\hspace{1pt} m}\qquad\text{and}\qquad \theta^{\hspace{1pt}(k)}_{m\hspace{1pt} m} \end{equation*} with $m=1,\dots,n$ and $k=0,1,\dots,m-1$, are generators of the algebra $\mathcal{A}_{\mu}$. If $\mu$ is regular, then each of these families is algebraically independent. \qed \end{thm}
\begin{example}\label{ex:tracom} Using the family $\theta^{\hspace{1pt}(k)}_{m\hspace{1pt} m}$ we get the following algebraically independent generators of the algebra $\mathcal{A}_{\mu}$ for regular $\mu$: \begin{equation*} \begin{aligned} \text{for}\quad\mathfrak{gl}_2: &\qquad {\rm tr}\, E,\quad {\rm tr}\, \mu\hspace{1pt} E,\quad {\rm tr}\, E^2\\[0.3em] \text{for}\quad\mathfrak{gl}_3: &\qquad {\rm tr}\, E,\quad {\rm tr}\, \mu\hspace{1pt} E,\quad {\rm tr}\, \mu^2 E, \quad {\rm tr}\, E^2,\quad {\rm tr}\, \mu\hspace{1pt} E^2,\quad {\rm tr}\, E^3\\[0.3em] \text{for}\quad\mathfrak{gl}_4: &\qquad {\rm tr}\, E,\quad {\rm tr}\, \mu\hspace{1pt} E,\quad {\rm tr}\, \mu^2 E, \quad {\rm tr}\, \mu^3 E, \quad {\rm tr}\, E^2,\quad {\rm tr}\, \mu\hspace{1pt} E^2,\\[0.3em] &\qquad\qquad 2\, {\rm tr}\, \mu^2 E^2+ {\rm tr}\,(\mu\hspace{1pt} E)^2,\quad {\rm tr}\, E^3,\quad {\rm tr}\, \mu\hspace{1pt} E^3,\quad {\rm tr}\, E^4. \end{aligned} \end{equation*} \end{example}
\section{Proof of the Main Theorem} \label{sec:ptab} \setcounter{equation}{0}
Note that $M=-\partial_z+\mu+Ez^{-1}$ is a {\em Manin matrix} and therefore the polynomials $\phi^{}_{m\hspace{1pt} a}(z)$ and $\psi^{}_{m\hspace{1pt} a}(z)$ admit expressions in terms of noncommutative minors and permanents. In more detail, given two subsets $B=\{b_1,\dots,b_k\}$ and $C=\{c_1,\dots,c_k\}$ of $\{1,\dots,n\}$ we will consider the corresponding column-minor \begin{equation*} M^B_C=\sum_{\sigma\in\mathfrak S_k} {\rm sgn}\ts\sigma\cdot M_{b_{\sigma(1)}c_1}\dots M_{b_{\sigma(k)}c_k}. \end{equation*} By \cite[Proposition~18]{cfr:ap} (see also \cite[Proposition~2.1]{mr:mm}) we have \begin{equation*} A^{(m)} M_1\dots M_m = A^{(m)} M_1\dots M_m\hspace{1pt} A^{(m)}, \end{equation*} which implies \beql{vpkm}
{\rm tr}_{1,\dots,m}\, A^{(m)} M_1\dots M_m=\sum_{I,\, |I|=m} M^I_I, \end{equation} summed over the subsets $I=\{i_1,\dots,i_m\}$ with $i_1<\dots<i_m$. By Theorem~\ref{thm:comsera}, the algebra $\mathcal{A}_{\mu}$ is generated by the coefficients $\phi^{(k)}_{m}$ of the constant term of the differential operator, \begin{equation*} \phi^{\hspace{1pt}(0)}_{m}z^{-m} +\dots+\phi^{\hspace{1pt}(m-1)}_{m}z^{-1}
+\phi^{\hspace{1pt}(m)}_{m}=\sum_{I,\, |I|=m} M^I_I\, 1, \end{equation*} assuming that $\partial_z\, 1=0$. This implies the formula \beql{defvp}
\phi^{(k)}_{m}=z^{m-k}\,\sum_{I,\, |I|=m}
\sum_{\atopn{B,C\subset I}{|B|=|C|=k}}\, {\rm sgn}\ts\sigma\cdot{\mu\hspace{1pt}}^B_C\, \big[{-}\partial_z+Ez^{-1}\big]^{I\setminus B}_{I\setminus C}\, 1, \end{equation} where $\sigma$ denotes the permutation of the set $I$ given by \begin{equation*} \sigma=\binom{B,\ I\setminus B}{C,\ I\setminus C} =\binom{b_1,\dots,b_k,i_1,\dots,\widehat b_1,\dots,\widehat b_k,\dots,i_m} {c_1,\dots,c_k,i_1,\dots,\widehat c_1,\dots,\widehat c_k,\dots,i_m}, \end{equation*} and we assume that $b_1<\dots<b_k$ and $c_1<\dots<c_k$ for the respective elements of the subsets $B$ and $C$ in $I$.
For each $l=1,\dots,n$ introduce the polynomial in a variable $t$ with coefficients in $\mathcal{A}_{\mu}$ by \beql{polphi} \Phi_l(t,\mu)=\phi^{(0)}_{l}(\mu)\hspace{1pt} t^{\, n-l}+\phi^{(1)}_{l+1}(\mu)\hspace{1pt} t^{\, n-l-1} +\dots+\phi^{(n-l)}_{n}(\mu), \end{equation} where the elements $\phi^{(k)}_{m}=\phi^{(k)}_{m}(\mu)$ are defined in \eqref{defvp} and we indicated dependence of $\mu$. The coefficients of $\Phi_l(t,\mu)$ are the elements of the $l$-th row of the diagram $\Gamma$; see \eqref{Ga}.
\begin{lem}\label{lem:relshi} For any $a\in\mathbb{C}\tss$ we have the relation \begin{equation*} \Phi_l(t,\mu+a\hspace{1pt} 1)=\Phi_l(t+a,\mu). \end{equation*} \end{lem}
\begin{proof} We have \begin{equation*} {\rm tr}_{1,\dots,m}\, A^{(m)} (a+M_1)\dots (a+M_m)=\sum_{p=0}^m a^p \sum_{i_1<\dots<i_{m-p}} {\rm tr}_{1,\dots,m}\, A^{(m)} M_{i_1}\dots M_{i_{m-p}}. \end{equation*} Furthermore, $A^{(m)}= {\rm sgn}\ts p\cdot A^{(m)}\hspace{1pt} P$ for any $p\in\mathfrak S_m$, where $P$ denotes the image of $p$ in the algebra \eqref{tenprka} under the action of $\mathfrak S_m$. Hence, applying conjugations by appropriate elements $P$ and using the cyclic property of trace, we can write the expression as \begin{equation*} \sum_{p=0}^m \binom{m}{p}\hspace{1pt} a^p\, {\rm tr}_{1,\dots,m}\, A^{(m)} M_1\dots M_{m-p}. \end{equation*} The partial trace of the anti-symmetrizer over the $m$-th copy of ${\rm{End}\,}\mathbb{C}\tss^n$ is found by \beql{partr} {\rm tr}_{m}\, A^{(m)}=\frac{n-m+1}{m} \, A^{(m-1)} \end{equation} which implies \begin{equation*} {\rm tr}_{m-p+1,\dots,m}\, A^{(m)}=\frac{(n-m+p)!\,(m-p)!}{(n-m)!\, m!}\, A^{(m-p)}. \end{equation*} Hence, \begin{equation*} {\rm tr}_{1,\dots,m}\, A^{(m)} (a+M_1)\dots (a+M_m)=\sum_{p=0}^m \binom{n-m+p}{p}\hspace{1pt} a^p\, {\rm tr}_{1,\dots,m-p}\, A^{(m-p)} M_1\dots M_{m-p}. \end{equation*} Now equate the constant terms of the differential operators on both sides and take the coefficients of $z^{-m+k}$ to get the relation \begin{equation*} \phi^{(k)}_{m}(\mu+a\hspace{1pt} 1)=\sum_{p=0}^k \binom{n-m+p}{p}\hspace{1pt} a^p\, \phi^{(k-p)}_{m-p}(\mu). \end{equation*} Therefore, for the polynomial $\Phi_l(t,\mu+a\hspace{1pt} 1)$ we find \begin{equation*} \begin{aligned} \Phi_l(t,\mu+a\hspace{1pt} 1)&=\sum_{k=0}^{n-l} \phi^{(k)}_{l+k}(\mu+a\hspace{1pt} 1)\hspace{1pt} t^{\, n-l-k} =\sum_{k=0}^{n-l} \,\sum_{p=0}^k \binom{n-l-k+p}{p}\hspace{1pt} a^p\, \phi^{(k-p)}_{l+k-p}(\mu)\hspace{1pt} t^{\, n-l-k}\\ {}&=\sum_{p=0}^{n-l}a^p\, \sum_{r=0}^{n-l-p} \binom{n-l-r}{p}\, \phi^{(r)}_{l+r}(\mu)\hspace{1pt} t^{\, n-l-p-r}, \end{aligned} \end{equation*} which coincides with \begin{equation*} \sum_{p=0}^{n-l}\frac{a^p}{p!}\, \Big(\frac{d}{d\hspace{1pt} t}\Big)^p\, \Phi_l(t,\mu)=\Phi_l(t+a,\mu), \end{equation*} as claimed. \end{proof}
\begin{lem}\label{lem:zecor} Suppose that $\mu$ has the form of a block-diagonal matrix \beql{mucan} \mu=\begin{bmatrix}J_{\alpha}(0)&\text{\rm O}\,\\ \text{\rm O}&\widetilde\mu\, \end{bmatrix},
\end{equation} where $J_{\alpha}(0)$ is the nilpotent Jordan matrix associated with a diagram $\alpha=(\alpha_1,\alpha_2,\dots)$ and $\widetilde\mu$ is an arbitrary square matrix of size $q$ such that $|\alpha|+q=n$. Then for any $l\geqslant 1$ we have \begin{equation*} \phi^{(k)}_{l+k}=0\qquad \text{for all}\quad n-l-\delta_l+1\leqslant k\leqslant n-l, \end{equation*} where $\delta_l=\alpha_{l+1}+\alpha_{l+2}+\dots$ is the number of boxes of $\alpha$ below its row $l$. \end{lem}
\begin{proof} The generator $\phi^{(k)}_{l+k}$ is found by \eqref{defvp} for $m=l+k$. The internal sum is a linear combination of $k\times k$ minors of the matrix $\mu$ satisfying the condition that the union $B\cup C$ of the row and column indices of each minor is a set of size not exceeding $k+l$. On the other hand, with the given condition on $k$, the minor ${\mu\hspace{1pt}}^B_C$ can be nonzero only if the union of row and column indices is of the size at least $k+l+1$. Indeed, this follows from the observation that if $p$ is a positive integer, then any nonzero $p\times p$ minor of a nilpotent Jordan block has the property that the minimal possible size of the union of its row and column indices is $p+1$. However, the condition $k\geqslant n-l-\delta_l+1$ means that $k\geqslant \alpha_1+\dots+\alpha_l-l+1+q$. Therefore, a nonzero $k\times k$ minor must involve at least $l+1$ Jordan blocks. \end{proof}
In the following we use the notation of the Main Theorem. In addition, for each diagram $\alpha^{(i)}$ we denote by $\delta^{(i)}_l$ the corresponding parameter $\delta_l$, as defined in Lemma~\ref{lem:zecor}, so that for the number $\gamma_l$ defined in \eqref{gal} we have \begin{equation*} \gamma_l=\sum_{i=1}^r \delta^{(i)}_l. \end{equation*}
\begin{cor}\label{cor:vanish} The polynomial $\Phi_l(t,\mu)$ admits the factorization \begin{equation*} \Phi_l(t,\mu)=(t+\lambda_1)^{\delta^{(1)}_l}\dots (t+\lambda_r)^{\delta^{(r)}_l}\, \widetilde\Phi_l(t,\mu) \end{equation*} for a certain polynomial $\widetilde\Phi_l(t,\mu)$ in $t$. \end{cor}
\begin{proof} The algebra $\mathcal{A}_{\mu}$ is known to depend only on the adjoint orbit of $\mu$; see \cite{fft:gm}. More precisely, as we can see from formulas \eqref{vpkm}, the elements $\phi^{(k)}_m$ are unchanged under the simultaneous replacements $\mu\mapsto g\hspace{1pt}\mu\hspace{1pt} g^{-1}$ and $E\mapsto g\hspace{1pt} E\hspace{1pt} g^{-1}$ for $g\in {\rm GL}_n$. This implies that $\mathcal{A}_{g\hspace{1pt}\mu\hspace{1pt} g^{-1}}$ can be identified with the algebra $\mathcal{A}_{\mu}$ associated with the image of $ {\rm U}(\mathfrak{gl}_n)$ under the automorphism sending $E$ to $g\hspace{1pt} E\hspace{1pt} g^{-1}$.
For any $i\in\{1,\dots,r\}$ the Jordan canonical form of $\mu-\lambda_i 1$ is a matrix of the form \eqref{mucan}, where $\alpha=\alpha^{(i)}$. By Lemma~\ref{lem:zecor}, the polynomial $\Phi_l(t,\mu-\lambda_i 1)$ is divisible by $t^{\delta^{(i)}_l}$. Hence, by Lemma~\ref{lem:relshi}, the polynomial $ \Phi_l(t,\mu)=\Phi_l(t+\lambda_i,\mu-\lambda_i 1) $ is divisible by $(t+\lambda_i)^{\delta^{(i)}_l}$. \end{proof}
We can now complete the proof of the Main Theorem. First, Corollary~\ref{cor:vanish} implies that for any $l=1,\dots,n$ the generators $\phi^{(k)}_{l+k}$ with $n-l-\gamma_l+1\leqslant k\leqslant n-l$ are linear combinations of those generators with $k=0,1,\dots,n-l-\gamma_l$. Therefore, the elements $\phi^{(k)}_{l+k}$ corresponding to the boxes of the skew diagram $\Gamma/\gamma$ generate the algebra $\mathcal{A}_{\mu}$. It remains to verify that these generators are algebraically independent.
Consider the elements $\overline\phi^{\,(k)}_{m}\in {\rm S}(\mathfrak{gl}_n)$ which are defined by \beql{bardefvp}
\overline\phi^{\,(k)}_{m}=\sum_{I,\, |I|=m}
\sum_{\atopn{B,C\subset I}{|B|=|C|=k}}\, {\rm sgn}\ts\sigma\cdot{\mu\hspace{1pt}}^B_C\, {E\hspace{1pt}}^{I\setminus B}_{I\setminus C}, \end{equation} with the notation as in \eqref{defvp}, where the entries of the matrix $E$ are now regarded as elements of the symmetric algebra $ {\rm S}(\mathfrak{gl}_n)$. Equivalently, the elements $\overline\phi^{\,(k)}_{m}$ are found by \beql{phiz} {\rm tr}_{1,\dots,m}\, A^{(m)} \big(\mu^{}_1+E^{}_1z^{-1}\big)\dots \big(\mu^{}_m+E^{}_mz^{-1}\big) =\overline\phi^{\,(0)}_{m}z^{-m} +\dots+\overline\phi^{\,(m-1)}_{m}z^{-1} +\overline\phi^{\,(m)}_{m}. \end{equation} They are generators of the subalgebra $\overline\mathcal{A}_{\mu}$. The arguments of this section (including Lemmas~\ref{lem:relshi}, \ref{lem:zecor} and Corollary~\ref{cor:vanish}) applied to these generators instead of the $\phi^{(k)}_{m}$ show that the elements $\overline\phi^{\,(k)}_{m}$ corresponding to the boxes of the skew diagram $\Gamma/\gamma$ generate the algebra $\overline\mathcal{A}_{\mu}$. Furthermore, we have the following.
\begin{lem}\label{lem:algin} The generators $\overline\phi^{\,(k)}_{m}$ of the subalgebra $\overline\mathcal{A}_{\mu}$ corresponding to the boxes of the skew diagram $\Gamma/\gamma$ are algebraically independent. \end{lem}
\begin{proof} Regarding the elements $\overline\phi^{\,(k)}_{m}$ as polynomials in the variables $E_{ij}$, we will see that their differentials $d\,\overline\phi^{\,(k)}_{m}$ are linearly independent at a certain point. Since these elements generate $\overline\mathcal{A}_{\mu}$, the linear span of the differentials $d\,\overline\phi^{\,(k)}_{m}$ at any point coincides with the linear span of all differentials
\begin{equation*} d\,\overline\mathcal{A}_{\mu}=\text{span of}\ \{d\phi\ |\ \phi\in \overline\mathcal{A}_{\mu}\}. \end{equation*} On the other hand, Bolsinov's criterion \cite[Theorem~3.2]{b:cf} implies that the relation \begin{equation*} \dim d\,\overline\mathcal{A}_{\mu}=\text{\rm rank\,}\mathfrak{gl}_n+\frac12 \big(\hspace{-1pt}\dim\mathfrak{gl}_n-\dim\mathfrak{gl}_n^{\,\mu}\big) \end{equation*} holds at a certain regular point if and only if the equality \eqref{beconj} holds for $\mathfrak{g}=\mathfrak{gl}_n$; see also \cite[Theorem~2.7]{cm:ic} for a concise exposition of this result. This equality does hold \cite{y:ic}, and so, to show that the differentials $d\,\overline\phi^{\,(k)}_{m}$ of the generators are linearly independent at a certain point, we only need to verify that the number of boxes of the skew diagram $\Gamma/\gamma$ coincides with \begin{equation*} \text{\rm rank\,}\mathfrak{gl}_n+\frac12\big(\hspace{-1pt}\dim\mathfrak{gl}_n-\dim\mathfrak{gl}_n^{\,\mu}\big)= n+\frac12\big(n^2-\dim\mathfrak{gl}_n^{\,\mu}\big).
\end{equation*} Since $|\Gamma|=n(n+1)/2$, the desired formula is equivalent to the relation \beql{relcen}
\dim\mathfrak{gl}_n^{\,\mu}=2\hspace{1pt} |\gamma|+n. \end{equation} For the dimension of the centralizer we have \begin{equation*} \dim\mathfrak{gl}_n^{\,\mu}=\sum_{i=1}^r\dim\mathfrak{gl}_{n_i}^{\,\mu^{(i)}}, \end{equation*} where $\mu^{(i)}$ denotes the direct sum of all Jordan blocks of $\mu$ with the eigenvalue $\lambda_i$, and $n_i$ is the size of $\mu^{(i)}$. Hence, by the definition of $\gamma$, the verification of \eqref{relcen} reduces to the case where $\mu$ has only one eigenvalue. Let $\alpha_1\geqslant\dots\geqslant\alpha_s$ be the respective sizes of the Jordan blocks of such matrix $\mu$. Then $ \dim\mathfrak{gl}_n^{\,\mu}=\alpha_1+3\hspace{1pt} \alpha_2+\dots+(2\hspace{1pt} s-1)\hspace{1pt}\alpha_s, $ while \begin{equation*}
|\gamma|=\alpha_2+2\hspace{1pt}\alpha_3+\dots+(s-1)\hspace{1pt} \alpha_s\qquad\text{and}\qquad n=\alpha_1+\dots+\alpha_s, \end{equation*} thus implying \eqref{relcen}. \end{proof}
Now consider the generators $\phi^{(k)}_{m}$ of the algebra $\mathcal{A}_{\mu}$ associated with the boxes of the diagram $\Gamma/\gamma$. By Lemma~\ref{lem:algin}, the corresponding elements $\overline\phi^{\,(k)}_{m}$ are nonzero, so that the image of $\phi^{(k)}_{m}$ in the $(m-k)$-th component of $ {\rm gr}\, {\rm U}(\mathfrak{gl}_n)\cong {\rm S}(\mathfrak{gl}_n)$ coincides with $\overline\phi^{\,(k)}_{m}$. Moreover, the generators $\phi^{(k)}_{m}$ corresponding to the boxes of the diagram $\Gamma/\gamma$ are algebraically independent. This completes the proof of the first part of the Main Theorem, and the second part also follows.
Finally, we will extend the first part of the Main Theorem by providing some other families of algebraically independent generators of the algebra $\mathcal{A}_{\mu}$. To this end, introduce the families $\overline\psi^{\,(k)}_{m}$ and $\overline\theta^{\,(k)}_{m}$ of generators of the algebra $\overline\mathcal{A}_{\mu}$ by the respective expansions \beql{psiz} {\rm tr}_{1,\dots,m}\, H^{(m)} \big(\mu^{}_1+E^{}_1z^{-1}\big)\dots \big(\mu^{}_m+E^{}_mz^{-1}\big) =\overline\psi^{\,(0)}_{m}z^{-m} +\dots+\overline\psi^{\,(m-1)}_{m}z^{-1} +\overline\psi^{\,(m)}_{m} \end{equation} and \beql{thez} {\rm tr}\,\big(\mu+E\hspace{1pt} z^{-1}\big)^m=\overline\theta^{\,(0)}_{m}z^{-m} +\dots+\overline\theta^{\,(m-1)}_{m}z^{-1} +\overline\theta^{\,(m)}_{m}, \end{equation} where \begin{equation*} E=\sum_{i,j=1}^n e_{ij}\otimes E_{ij}\in {\rm{End}\,}\mathbb{C}\tss^n\otimes {\rm S}(\mathfrak{gl}_n), \end{equation*} and extend the notation \eqref{matnota} to matrices $E$ and $\mu$. The polynomials $\overline\phi_{m}(z)$, $\overline\psi_{m}(z)$ and $\overline\theta_{m}(z)$ in $z^{-1}$ given by the respective expressions in \eqref{phiz}, \eqref{psiz} and \eqref{thez} are related by the classical MacMahon Master Theorem and Newton's identities: \beql{mm} \sum_{l=0}^m(-1)^l\, \overline\phi_{l}(z)\, \overline\psi_{m-l}(z)=0 \end{equation} and \beql{newton} m\, \overline\phi_{m}(z)=\sum_{l=1}^m (-1)^{l-1}\, \overline\theta_{l}(z) \, \overline\phi_{m-l}(z) \end{equation} for $m\geqslant 1$. Writing the relations \eqref{mm} and \eqref{newton} in terms of the coefficients of the polynomials, we find that each of the generators $\overline\psi^{\,(k)}_{m}$ and $\overline\theta^{\,(k)}_{m}$ with $m=1,\dots,n$ and $k=0,1,\dots,m-1$ will be presented in the form \begin{equation*} c\cdot \overline\phi^{\,(k)}_{m}+\text{linear combination of}\quad \overline\phi^{\,(k_1)}_{m_1}\dots \overline\phi^{\,(k_s)}_{m_s},\quad s\geqslant 2, \end{equation*} for a nonzero constant $c$, where $m_1+\dots+m_s=m$ and $k_1+\dots+k_s=k$. As we pointed out above, the elements $\overline\phi^{\,(k)}_{m}$ corresponding to a certain row of the diagram $\Gamma$ are linear combinations of the elements of this row in the skew diagram $\Gamma/\gamma$. This implies that each of the families of generators $\overline\psi^{\,(k)}_{m}$ and $\overline\theta^{\,(k)}_{m}$ associated with the boxes of $\Gamma/\gamma$ as in \eqref{Ga}, is algebraically independent. This leads to the following corollary, where, as before, $\mu\in\mathfrak{gl}_n$ is an arbitrary matrix.
\begin{cor}\label{cor:othgen} The elements of each of the two families $\psi^{(k)}_{m\hspace{1pt} m}$ and $\theta^{\hspace{1pt}(k)}_{m\hspace{1pt} m}$ associated with the boxes of the skew diagram $\Gamma/\gamma$ as in \eqref{Ga} are algebraically independent generators of the algebra $\mathcal{A}_{\mu}$. \qed \end{cor}
To construct two more families of generators of the algebra $\mathcal{A}_{\mu}$, define the elements $\varphi^{(k)}_m, \psi^{(k)}_m\in {\rm U}(\mathfrak{gl}_n)$ by the expansions \begin{equation*} \begin{aligned} {\rm tr}_{1,\dots,m}\, A^{(m)} \big(\mu^{}_1+E^{}_1z^{-1}\big)\dots \big(\mu^{}_m+E^{}_mz^{-1}\big) {}&=\varphi^{\hspace{1pt}(0)}_{m}z^{-m} +\dots+\varphi^{\hspace{1pt}(m-1)}_{m}z^{-1} +\varphi^{\hspace{1pt}(m)}_{m},\\[0.5em] {\rm tr}_{1,\dots,m}\, H^{(m)} \big(\mu^{}_1+E^{}_1z^{-1}\big)\dots \big(\mu^{}_m+E^{}_mz^{-1}\big){}&=\psi^{\hspace{1pt}(0)}_{m}z^{-m} +\dots+\psi^{\hspace{1pt}(m-1)}_{m}z^{-1} +\psi^{\hspace{1pt}(m)}_{m}. \end{aligned} \end{equation*} It is easy to verify that each of the families $\varphi^{(k)}_m$ and $\psi^{(k)}_m$ with $m=1,\dots,n$ and $k=0,\dots,m-1$ generates the algebra $\mathcal{A}_{\mu}$. Indeed, by Theorem~\ref{thm:comsera}, the algebra $\mathcal{A}_{\mu}$ is generated by the coefficients $\phi^{(k)}_{m}$ of the constant term of the differential operator, \begin{multline} {\rm tr}_{1,\dots,m}\, A^{(m)} \big({-}\partial_z+\mu^{}_1+E^{}_1z^{-1}\big)\dots \big({-}\partial_z+\mu^{}_m+E^{}_mz^{-1}\big)\, 1\\[0.7em] {}=\phi^{\hspace{1pt}(0)}_{m}z^{-m} +\dots+\phi^{\hspace{1pt}(m-1)}_{m}z^{-1} +\phi^{\hspace{1pt}(m)}_{m}. \nonumber \end{multline} Hence, $\phi^{(k)}_{m}$ is found as the coefficient of $z^{-m+k}$ in the expression \begin{equation*} \sum_{i_1<\dots<i_k} \sum_{j_1<\dots<j_{m-k}} \, {\rm tr}_{1,\dots,m}\, A^{(m)} \mu^{}_{i_1}\dots \mu^{}_{i_k} \big({-}\partial_z+E^{}_{j_1}z^{-1}\big)\dots \big({-}\partial_z+E^{}_{j_{m-k}}z^{-1}\big)\, 1, \end{equation*} summed over disjoint subsets of indices $\{i_1,\dots,i_k\}$ and $\{j_1,\dots,j_{m-k}\}$ of $\{1,\dots,m\}$. Therefore, \begin{equation*} \phi^{(k)}_{m}=z^{m-k}\,\binom{m}{k}\, {\rm tr}_{1,\dots,m} \, A^{(m)} \mu^{}_1\dots \mu^{}_k \big({-}\partial_z+E^{}_{k+1}z^{-1}\big)\dots \big({-}\partial_z+E^{}_mz^{-1}\big)\, 1. \end{equation*} By calculating the partial trace of the anti-symmetrizer with the use of \eqref{partr}, we get \begin{equation*} \phi^{(k)}_{m}=\binom{m}{k}\, {\rm tr}_{1,\dots,m}\, A^{(m)} \mu^{}_1\dots \mu^{}_k \, E^{}_{k+1}\dots E^{}_m+\sum_{r=k+1}^{m-1} c_r\, {\rm tr}_{1,\dots,r}\, A^{(r)} \mu^{}_1\dots \mu^{}_k \, E^{}_{k+1}\dots E^{}_r \end{equation*} for certain constants $c_r$. The same argument applied to the expansion defining the elements $\varphi^{(k)}_{m}$ gives \begin{equation*} \varphi^{(k)}_{m}=\binom{m}{k}\, {\rm tr}_{1,\dots,m}\, A^{(m)} \mu^{}_1\dots \mu^{}_k \, E^{}_{k+1}\dots E^{}_m. \end{equation*} This yields a triangular system of linear relations \begin{equation*} \phi^{(k)}_{m}=\varphi^{(k)}_{m}+\sum_{r=k+1}^{m-1} c_r\, \varphi^{(k)}_{r}. \end{equation*} Since $\phi^{(k)}_{k+1}=\varphi^{(k)}_{k+1}$, we can conclude that the elements $\varphi^{(k)}_{m}$ are generators of $\mathcal{A}_{\mu}$. The argument for the elements $\psi^{(k)}_{m}$ is quite similar. Taking into account the properties of the elements $\overline\phi^{\,(k)}_{m}$ and $\overline\psi^{\,(k)}_{m}$, we come to another corollary.
\begin{cor}\label{cor:othgentwo} The elements of each of the two families $\varphi^{(k)}_{m}$ and $\psi^{(k)}_{m}$ associated with the boxes of the skew diagram $\Gamma/\gamma$ as in \eqref{Ga} are algebraically independent generators of the algebra $\mathcal{A}_{\mu}$. \qed \end{cor}
\end{document} | arXiv |
NuSTAR Reveals the Comptonizing Corona of the Broad-Line Radio Galaxy 3C 382
D. R. Ballantyne, J. M. Bollenbacher, L. W. Brenneman, K. K. Madsen, M. Balokovic, S. E. Boggs, Finn Erland Christensen, W. W. Craig, P. Gandhi, C. J. Hailey, F. A. Harrison, A. M. Lohfink, A. Marinucci, C. B. Markwardt, D. Stern, D. J. Walton, W. W. Zhang
National Space Institute
Broad-line radio galaxies (BLRGs) are active galactic nuclei that produce powerful, large-scale radio jets, but appear as Seyfert 1 galaxies in their optical spectra. In the X-ray band, BLRGs also appear like Seyfert galaxies, but with flatter spectra and weaker reflection features. One explanation for these properties is that the X-ray continuum is diluted by emission from the jet. Here, we present two NuSTAR observations of the BLRG 3C 382 that show clear evidence that the continuum of this source is dominated by thermal Comptonization, as in Seyfert 1 galaxies. The two observations were separated by over a year and found 3C 382 in different states separated by a factor of 1.7 in flux. The lower flux spectrum has a photon-index of $\Gamma=1.68^{+0.03}_{-0.02}$, while the photon-index of the higher flux spectrum is $\Gamma=1.78^{+0.02}_{-0.03}$. Thermal and anisotropic Comptonization models provide an excellent fit to both spectra and show that the coronal plasma cooled from $kT_e=330\pm 30$ keV in the low flux data to $231^{+50}_{-88}$ keV in the high flux observation. This cooling behavior is typical of Comptonizing corona in Seyfert galaxies and is distinct from the variations observed in jet-dominated sources. In the high flux observation, simultaneous Swift data are leveraged to obtain a broadband spectral energy distribution and indicates that the corona intercepts $\sim 10$% of the optical and ultraviolet emitting accretion disk. 3C 382 exhibits very weak reflection features, with no detectable relativistic Fe K$\alpha$ line, that may be best explained by an outflowing corona combined with an ionized inner accretion disk.
https://doi.org/10.1088/0004-637X/794/1/62
Ballantyne, D. R., Bollenbacher, J. M., Brenneman, L. W., Madsen, K. K., Balokovic, M., Boggs, S. E., ... Zhang, W. W. (2014). NuSTAR Reveals the Comptonizing Corona of the Broad-Line Radio Galaxy 3C 382. Astrophysical Journal, 794(1). https://doi.org/10.1088/0004-637X/794/1/62
Ballantyne, D. R. ; Bollenbacher, J. M. ; Brenneman, L. W. ; Madsen, K. K. ; Balokovic, M. ; Boggs, S. E. ; Christensen, Finn Erland ; Craig, W. W. ; Gandhi, P. ; Hailey, C. J. ; Harrison, F. A. ; Lohfink, A. M. ; Marinucci, A. ; Markwardt, C. B. ; Stern, D. ; Walton, D. J. ; Zhang, W. W. / NuSTAR Reveals the Comptonizing Corona of the Broad-Line Radio Galaxy 3C 382. In: Astrophysical Journal. 2014 ; Vol. 794, No. 1.
@article{965bf730c02f492390313dbea2298204,
title = "NuSTAR Reveals the Comptonizing Corona of the Broad-Line Radio Galaxy 3C 382",
abstract = "Broad-line radio galaxies (BLRGs) are active galactic nuclei that produce powerful, large-scale radio jets, but appear as Seyfert 1 galaxies in their optical spectra. In the X-ray band, BLRGs also appear like Seyfert galaxies, but with flatter spectra and weaker reflection features. One explanation for these properties is that the X-ray continuum is diluted by emission from the jet. Here, we present two NuSTAR observations of the BLRG 3C 382 that show clear evidence that the continuum of this source is dominated by thermal Comptonization, as in Seyfert 1 galaxies. The two observations were separated by over a year and found 3C 382 in different states separated by a factor of 1.7 in flux. The lower flux spectrum has a photon-index of $\Gamma=1.68^{+0.03}_{-0.02}$, while the photon-index of the higher flux spectrum is $\Gamma=1.78^{+0.02}_{-0.03}$. Thermal and anisotropic Comptonization models provide an excellent fit to both spectra and show that the coronal plasma cooled from $kT_e=330\pm 30$ keV in the low flux data to $231^{+50}_{-88}$ keV in the high flux observation. This cooling behavior is typical of Comptonizing corona in Seyfert galaxies and is distinct from the variations observed in jet-dominated sources. In the high flux observation, simultaneous Swift data are leveraged to obtain a broadband spectral energy distribution and indicates that the corona intercepts $\sim 10${\%} of the optical and ultraviolet emitting accretion disk. 3C 382 exhibits very weak reflection features, with no detectable relativistic Fe K$\alpha$ line, that may be best explained by an outflowing corona combined with an ionized inner accretion disk.",
author = "Ballantyne, {D. R.} and Bollenbacher, {J. M.} and Brenneman, {L. W.} and Madsen, {K. K.} and M. Balokovic and Boggs, {S. E.} and Christensen, {Finn Erland} and Craig, {W. W.} and P. Gandhi and Hailey, {C. J.} and Harrison, {F. A.} and Lohfink, {A. M.} and A. Marinucci and Markwardt, {C. B.} and D. Stern and Walton, {D. J.} and Zhang, {W. W.}",
doi = "10.1088/0004-637X/794/1/62",
publisher = "IOP Publishing",
Ballantyne, DR, Bollenbacher, JM, Brenneman, LW, Madsen, KK, Balokovic, M, Boggs, SE, Christensen, FE, Craig, WW, Gandhi, P, Hailey, CJ, Harrison, FA, Lohfink, AM, Marinucci, A, Markwardt, CB, Stern, D, Walton, DJ & Zhang, WW 2014, 'NuSTAR Reveals the Comptonizing Corona of the Broad-Line Radio Galaxy 3C 382', Astrophysical Journal, vol. 794, no. 1. https://doi.org/10.1088/0004-637X/794/1/62
NuSTAR Reveals the Comptonizing Corona of the Broad-Line Radio Galaxy 3C 382. / Ballantyne, D. R.; Bollenbacher, J. M.; Brenneman, L. W.; Madsen, K. K.; Balokovic, M.; Boggs, S. E.; Christensen, Finn Erland; Craig, W. W.; Gandhi, P.; Hailey, C. J.; Harrison, F. A.; Lohfink, A. M.; Marinucci, A.; Markwardt, C. B.; Stern, D.; Walton, D. J.; Zhang, W. W.
In: Astrophysical Journal, Vol. 794, No. 1, 2014.
T1 - NuSTAR Reveals the Comptonizing Corona of the Broad-Line Radio Galaxy 3C 382
AU - Ballantyne, D. R.
AU - Bollenbacher, J. M.
AU - Brenneman, L. W.
AU - Madsen, K. K.
AU - Balokovic, M.
AU - Boggs, S. E.
AU - Christensen, Finn Erland
AU - Craig, W. W.
AU - Gandhi, P.
AU - Hailey, C. J.
AU - Harrison, F. A.
AU - Lohfink, A. M.
AU - Marinucci, A.
AU - Markwardt, C. B.
AU - Stern, D.
AU - Walton, D. J.
AU - Zhang, W. W.
N2 - Broad-line radio galaxies (BLRGs) are active galactic nuclei that produce powerful, large-scale radio jets, but appear as Seyfert 1 galaxies in their optical spectra. In the X-ray band, BLRGs also appear like Seyfert galaxies, but with flatter spectra and weaker reflection features. One explanation for these properties is that the X-ray continuum is diluted by emission from the jet. Here, we present two NuSTAR observations of the BLRG 3C 382 that show clear evidence that the continuum of this source is dominated by thermal Comptonization, as in Seyfert 1 galaxies. The two observations were separated by over a year and found 3C 382 in different states separated by a factor of 1.7 in flux. The lower flux spectrum has a photon-index of $\Gamma=1.68^{+0.03}_{-0.02}$, while the photon-index of the higher flux spectrum is $\Gamma=1.78^{+0.02}_{-0.03}$. Thermal and anisotropic Comptonization models provide an excellent fit to both spectra and show that the coronal plasma cooled from $kT_e=330\pm 30$ keV in the low flux data to $231^{+50}_{-88}$ keV in the high flux observation. This cooling behavior is typical of Comptonizing corona in Seyfert galaxies and is distinct from the variations observed in jet-dominated sources. In the high flux observation, simultaneous Swift data are leveraged to obtain a broadband spectral energy distribution and indicates that the corona intercepts $\sim 10$% of the optical and ultraviolet emitting accretion disk. 3C 382 exhibits very weak reflection features, with no detectable relativistic Fe K$\alpha$ line, that may be best explained by an outflowing corona combined with an ionized inner accretion disk.
AB - Broad-line radio galaxies (BLRGs) are active galactic nuclei that produce powerful, large-scale radio jets, but appear as Seyfert 1 galaxies in their optical spectra. In the X-ray band, BLRGs also appear like Seyfert galaxies, but with flatter spectra and weaker reflection features. One explanation for these properties is that the X-ray continuum is diluted by emission from the jet. Here, we present two NuSTAR observations of the BLRG 3C 382 that show clear evidence that the continuum of this source is dominated by thermal Comptonization, as in Seyfert 1 galaxies. The two observations were separated by over a year and found 3C 382 in different states separated by a factor of 1.7 in flux. The lower flux spectrum has a photon-index of $\Gamma=1.68^{+0.03}_{-0.02}$, while the photon-index of the higher flux spectrum is $\Gamma=1.78^{+0.02}_{-0.03}$. Thermal and anisotropic Comptonization models provide an excellent fit to both spectra and show that the coronal plasma cooled from $kT_e=330\pm 30$ keV in the low flux data to $231^{+50}_{-88}$ keV in the high flux observation. This cooling behavior is typical of Comptonizing corona in Seyfert galaxies and is distinct from the variations observed in jet-dominated sources. In the high flux observation, simultaneous Swift data are leveraged to obtain a broadband spectral energy distribution and indicates that the corona intercepts $\sim 10$% of the optical and ultraviolet emitting accretion disk. 3C 382 exhibits very weak reflection features, with no detectable relativistic Fe K$\alpha$ line, that may be best explained by an outflowing corona combined with an ionized inner accretion disk.
U2 - 10.1088/0004-637X/794/1/62
DO - 10.1088/0004-637X/794/1/62
Ballantyne DR, Bollenbacher JM, Brenneman LW, Madsen KK, Balokovic M, Boggs SE et al. NuSTAR Reveals the Comptonizing Corona of the Broad-Line Radio Galaxy 3C 382. Astrophysical Journal. 2014;794(1). https://doi.org/10.1088/0004-637X/794/1/62
10.1088/0004-637X/794/1/62
0004 637X 794 1 62Final published version, 584 KB | CommonCrawl |
Norwegian Mathematical Society
The Norwegian Mathematical Society (Norwegian: Norsk matematisk forening, NMF) is a professional society for mathematicians. It was formed in 1918, with Carl Størmer elected as its first president.[1] It organizes mathematical contests and the annual Abel symposium and also awards the Viggo Brun Prize to young Norwegian mathematicians for outstanding research in mathematics, including mathematical aspects of information technology, mathematical physics, numerical analysis, and computational science.[2] The 2018 Prize winner was Rune Gjøringbø Haugseng.[3] The NMF is a member of the International Council for Industrial and Applied Mathematics and provides the Norwegian National Committee in the International Mathematical Union.[4]
Norwegian Mathematical Society
Norsk matematisk forening
FormationNovember 2, 1918 (1918-11-02)
FounderPoul Heegaard, Arnfinn Palmstrom, Richard Birkeland, and Carl Størmer
PurposeTo promote the study of the mathematical sciences
Membership
Individuals and Institutions or Companies
President (Formann)
Hans Munthe-Kaas
Parent organization
International Mathematical Union
Websiteweb.matematikkforeningen.no
Past Presidents and Honorary Members
Term of office Name
1st President 1918-1925 Carl Størmer
2nd President 1925-1928 Alf Guldberg
3rd President 1928-1935 Poul Heegaard
4th President 1935-1946 Ingebrigt Johansson
5th President 1946-1951 Jonas E. Fjeldstad
6th President 1951-1953 Viggo Brun
7th President 1953-1959 Ralph Tambs Lyche
8th President 1960-1966 Karl Egil Aubert
9th President 1967-1971 Jens Erik Fenstad
10th President 1972-1974 Per Holm
11th President 1975-1982 Erling Størmer
12th President 1983-1985 Dag Normann
13th President 1985-1988 Bernt Øksendal
14th President 1989-1991 Ragni Piene
15th President 1991-1995 Geir Ellingsrud
16th President 1995-2000 Bent Birkeland
17th President 2000-2003 Dag Normann
18th President 2003-2007 Kristian Seip
19th President 2007-2011 Brynjulf Owren
20th President 2011-2015 Sigmund Selberg
21st President 2015-2019 Petter Andreas Bergh
22nd President since 2019 Hans Munthe-Kaas
The Society elected two Honorary Members: Carl Størmer (elected 22 February 1949) and Viggo Brun (elected 30 May 1974).[5]
References
1. "Some History". Norwegian Mathematical Society. Archived from the original on 21 November 2007. Retrieved 14 September 2018.
2. "Norsk Matematisk Forening: Viggo Brun-Prisen". Retrieved 8 July 2020.
3. "Norsk Matematisk Forening: Viggo Brun Prize Winner 2018". 7 September 2018. Retrieved 8 July 2020.
4. "Norsk Matematisk Forening: Internasjonalt samarbeid". Retrieved 8 July 2020.
5. "Norsk Matematisk Forening: Foreningens Historie". Retrieved 8 July 2020.
External links
• Official website
Authority control
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| Wikipedia |
\begin{document}
\preprint{APS/123-QED}
\title{Construction of genuinely entangled multipartite subspaces from bipartite ones by reducing the total number of separated parties }
\author{K. V. Antipin}
\email{[email protected]}
\affiliation{Faculty of Physics, M. V. Lomonosov Moscow State University,\\ Leninskie gory, Moscow 119991, Russia}
\date{\today}
\begin{abstract} Construction of genuinely entangled multipartite subspaces with certain characteristics has become a relevant task in various branches of quantum information. Here we show that such subspaces can be obtained from an arbitrary collection of bipartite entangled subspaces under joining of their adjacent subsystems. In addition, it is shown that direct sums of such constructions under certain conditions are genuinely entangled. These facts are then used in detecting entanglement of tensor products of mixed states and constructing subspaces that are distillable across every bipartite cut, where for the former application we include an example with the analysis of genuine entanglement of a tripartite state obtained from two Werner states. \end{abstract}
\keywords{Entangled subspace; genuine multipartite entanglement; quantum channel; tensor network}
\maketitle
\section{Introduction}
Entangled subspaces have become an object of intensive research in recent years due to their potential utility in the tasks of quantum information processing. Ref.~\cite{Parth04}, the work by K. R. Parthasarathy, where completely entangled subspaces~(CESs) were described, can be thought of as a starting point for developing this direction. CESs are subspaces that are free of fully product vectors. This concept was later generalized to genuinely entangled subspaces~(GESs)~\cite{DemAugWit18,CMW08} -- those entirely composed of states in which entanglement is present in every bipartite cut of a compound system.
Genuine multipartite entanglement~(GME), being the strongest form of entanglement, has found many applications in quantum protocols~\cite{YeCh06,MP08,MEO18}. In this connection genuinely entangled subspaces are useful since they can serve as a source of GME states. As an example, it is known that any state entirely supported on GME is genuinely entangled. Another example is connected with detection of genuine entanglement: a state having significant overlap with a GES is genuinely entangled~\cite{DemAugQut19,KVAnt21}, and certain entanglement measures can be estimated for such a state~\cite{KVAnt21}. There are also some indications that GESs can be used in quantum cryptography~\cite{SheSrik19} and quantum error correction~\cite{HuGra20}.
There are several approaches to construction of GESs~\cite{DemAugWit18, AgHalBa19, DemAug20, KVAnt21,Dem21}, including those of maximal possible dimensions. While the problem of constructing maximal GESs for any number of parties and any local dimensions seems to be solved recently in Ref.~\cite{Dem21}, it is of significant interest to build entangled subspaces with certain useful for quantum protocols characteristics such as given values of entanglement measures, distillability property, robustness of entanglement under external noise, etc. It is the task we concentrate on in the present paper, following the path of compositional construction started in Ref.~\cite{KVAnt21}. We investigate a special operation when bipartite completely entangled subspaces are combined together with the use of tensor products with subsequent joining the adjacent subsystems~(parties). We show that such an operation can generate GESs and that its compositional character together with the freedom of choice of the input subspaces opens the possibility to control the parameters of the output GESs. Such construction can be relevant for quantum networks~\cite{Sim17,BFD19,KSYG21}. In particular, when two states are combined, this operation corresponds to the star configuration~\cite{TenNet2021}. Combination of two subspaces in turn can be associated with a superposition of several quantum networks.
The paper is structured as follows. In Section~\ref{sec::prel} we give necessary definitions and provide some mathematical background. In Section~\ref{sec::main} the main lemmas concerning the properties of tensor products of entangled subspaces are stated and proved. In Section~\ref{sec::app} it is shown how the established properties can be applied in several tasks such as constructing GESs with certain useful properties, detecting entanglement of tensor products of mixed states. In Section~\ref{sec::disc} we conclude and propose possible directions of further research.
\section{Preliminaries}\label{sec::prel}
Throughout this paper we consider finite dimensional Hilbert spaces and their tensor products. We begin with more precise definitions of entangled states and subspaces.
A pure $n$-partite state is \emph{entangled} if it cannot be written as a tensor product of states for every subsystem, i.~e., \begin{equation}
\ket{\psi}\ne\ket{\phi}_1\otimes\ldots\otimes\ket{\phi}_n. \end{equation}
A \emph{bipartite cut}~(\emph{bipartition}) $A|\bar A$ of an $n$-partite state is defined by specifying a subset $A$ of the set of $n$ parties as well as its complement $\bar A$ in this set.
A pure $n$-partite state $\ket{\psi}$ is called \emph{biseparable} if it can be written as a tensor product \begin{equation}
\ket{\psi} = \ket{\phi}_A\otimes\ket{\chi}_{\bar{A}} \end{equation}
with respect to some bipartite cut $A|\bar A$. On the contrary, a multipartite pure state is called \emph{genuinely entangled} if it is not biseparable with respect to any bipartite cut.
Similarly, a mixed multipartite state is called \emph{biseparable} if it can be decomposed into a convex sum of biseparable pure states, not necessarily with respect to the same bipartite cut. In the opposite case it is called \emph{genuinely entangled}.
A subspace of a multipartite Hilbert space is called \emph{completely entangled}~(CES) if it consists only of entangled states. A \emph{genuinely entangled subspace}~(GES) is a subspace composed entirely of genuinely entangled states.
Next we recall some measures of entanglement.
The geometric measure of entanglement of a bipartite pure state $\ket{\psi}$ is defined by \begin{equation}\label{geom}
G(\psi) \coloneqq 1 - \max_i\{\lambda_i\}, \end{equation} where $\lambda_i$ is the $i$-th Schmidt coefficient squared as in the Schmidt decomposition $\ket{\psi} = \sum_i\,\sqrt{\lambda_i}\ket{i}\otimes\ket{i}$. This measure is generalized~\cite{Guhnetall20} to detect genuine multipartite entanglement as \begin{equation}\label{geomGME}
G_{GME}(\psi) \coloneqq \min_{A|\bar A} G_{A|\bar A}(\psi), \end{equation}
where the minimization runs over all possile bipartite cuts $A|\bar A$ and $G_{A|\bar A}(\psi)$ -- the geometric measure~(\ref{geom}) with respect to bipartite cut $A|\bar A$.
For mixed multipartite states the geometric measure of genuine entanglement is defined via the convex roof construction \begin{equation}\label{cfGME}
G_{GME}(\rho) \coloneqq \min_{\{(p_j,\,\psi_j)\}}\,\sum_j\,p_j\,G_{GME}(\psi_j), \end{equation} where the minimum is taken over all ensemble decompositions $\rho = \sum_j p_j\dyad{\psi_j}$.
To quantify entanglement of a subspace $\mathcal S$, we will use the entanglement measure $EM$ of its least entangled vector: \begin{equation}
EM(\mathcal S) \coloneqq \min_{\ket{\psi}\in\mathcal S}EM(\psi). \end{equation}
In place of $EM$ here can be used the geometric measure $G_{A|\bar A}$ across a specific bipartite cut, as well as the genuine entanglement measure $G_{GME}$ of Eq.~(\ref{geomGME}).
We proceed to quantum channels and their connections with entangled subspaces.
Let $\mathcal L(\mathcal H)$ denote the set of all linear operators on $\mathcal H$. A \emph{quantum channel} $\Phi_{A\rightarrow B}$ is a linear, completely positive and trace-preserving map between $\mathcal L(\mathcal H_A)$ and $\mathcal L(\mathcal H_B)$ \cite{Wilde13}, for two finite dimensional Hilbert spaces $\mathcal H_A$ and $\mathcal H_B$.
A crucial property used in the present work is the correspondence between quantum channels and linear subspaces of composite Hilbert spaces~\cite{AubSz17}. Consider an isometry $V\,\colon\, \mathcal H_A\rightarrow \mathcal H_B\otimes \mathcal H_C$ whose range is $W$, some subspace of $\mathcal H_B\otimes\mathcal H_C$. The corresponding quantum channel $\mathrm\Phi_{A\rightarrow B}\colon\,\mathcal L(\mathcal H_A)\rightarrow\mathcal L(\mathcal H_B)$ can be introduced by \begin{equation}\label{IsoRep}
\mathrm\Phi_{A\rightarrow B}(\rho) = \mathrm{Tr}_{\mathcal H_C} (V\rho V^{\dagger}). \end{equation} If we trace out subsystem $B$ instead, a \emph{complementary}~\cite{ShorDev} to $\Phi$ quantum channel $\Phi^C_{A\rightarrow C}$ is obtained: \begin{equation}\label{IsoRepCom}
\mathrm\Phi^C_{A\rightarrow C}(\rho) = \mathrm{Tr}_{\mathcal H_B} (V\rho V^{\dagger}). \end{equation} The correspondence works in the opposite direction as well: by Stinespring's dilation theorem~\cite{Stine55}, for any channel $\mathrm\Phi_{A\rightarrow B}$ there exists some subspace $W\subset \mathcal H_{B}\otimes \mathcal H_{C}$ such that $\mathrm\Phi_{A\rightarrow B}$ is determined by Eq.~(\ref{IsoRep}).
Eqs.~(\ref{IsoRep}) and (\ref{IsoRepCom}) are represented diagrammatically on Fig.~\ref{fig:chancom}. In this paper we use tensor diagram notation and the corresponding tools for diagrammatic reasoning from Ref.~\cite{CoeKis17}, which include the discarding symbol depicting tracing out a particular subsystem and various line deformations denoting linear algebra operations. Refs.~\cite{BiamonteEtAll15, Biamonte19} are also good sources on application of tensor diagrams in quantum information theory.
\begin{figure}
\caption{ A representation of channel $\mathrm\Phi_{A\rightarrow B}$ and its complementary channel $\Phi^C_{A\rightarrow C}$ both acting on a pure state $\ket{\psi}\in\mathcal H_A$: the isometry $V$ takes the state to $\mathcal H_B\otimes \mathcal H_C$, then one of the two subsystems is traced out~(which is denoted by the discarding symbol).}
\label{fig:chancom}
\end{figure}
An important characteristic of a quantum channel $\Phi$ is the \emph{maximal output norm}~\cite{AmHolWer2000} defined by \begin{equation}\label{OutNorm}
\nu_p(\mathrm\Phi) = \sup_{\rho\in\mathcal D(\mathcal H)}\norm{\mathrm\Phi(\rho)}_p,\,\,p>1, \end{equation} where $\norm{\rho}_p = (\mathrm{Tr}(\abs{\rho}^p))^{1/p}$ is the $p$-norm and $\mathcal D(\mathcal H)$ is the set of density operators on $\mathcal H$. The supremum in Eq.~(\ref{OutNorm}) can be taken over pure input states due to convexity of the $p$-norm. The quantity $\nu_p(\mathrm\Phi)$ also characterizes the entanglement of the subspace $W$ corresponding to the channel $\Phi$: $W$ is completely entangled iff $\nu_p(\mathrm\Phi) < 1$.
\begin{figure}
\caption{ An isometry $V$ acting on subsystem $B$ of a pure bipartite state $\ket{\psi}$ from a completely entangled subspace of a tensor product Hilbert space $\mathcal H_A\otimes\mathcal H_B$. Acting of a properly chosen isometry on each state in the subspace generates a genuinely entangled subspace of a tripartite Hilbert space $\mathcal H_A\otimes\mathcal H_C\otimes\mathcal H_D$. }
\label{fig:iso23}
\end{figure}
Let us mention another crucial property concerning the maximal output norm. Consider a product channel $I\otimes\Phi$, where $I$ is the identity map~(the ideal channel). Then \begin{equation}\label{idealcomb}
\nu_p(I\otimes\Phi) = \nu_p(\Phi),\quad 1\leqslant p\leqslant\infty. \end{equation} It was proved in Ref.~\cite{AmHolWer2000}.
Ref.~\cite{KVAnt21} provides a simple approach to constructing tripartite genuinely entangled subspaces with the use of composition of bipartite completely entangled subspaces and quantum channels of certain types. The approach is presented on Fig.~\ref{fig:iso23}, where an isometry $V$ is acting on one of the two subsystems of each state from a completely entangled subspace of $\mathcal H_A\otimes\mathcal H_B$. It was shown that, when the isometry corresponds to a quantum channel $\Phi$ with $\nu_p(\mathrm\Phi) < 1$ for $p>1$~(i.~e., the isometry has a CES as its range), a genuinely entangled subspace of $\mathcal H_A\otimes\mathcal H_C\otimes\mathcal H_D$ is generated.
Interestingly enough, there are other types of isometries that can generate GESs via the scheme on Fig.~\ref{fig:iso23}, and they don't necessarily have completely entangled ranges. In the present paper, though, we will use those of the described above type.
There will be a lot of joining of subsystems in the present paper. Let $A$ and $B$ be two systems with Hilbert spaces $\mathcal H_A$ and $\mathcal H_B$, respectively, and $\dim(\mathcal H_A)=d_A$, $\dim(\mathcal H_B)=d_B$. Let $C$ be a larger system such that $\dim(\mathcal H_C)=d_A d_B$. We say that $A$ and $B$ are joined into $C=AB$ if, given fixed computational bases $\{\ket{i}_A\}$ and $\{\ket{j}_B\}$ of $\mathcal H_A$ and $\mathcal H_B$ respectively, there is a mapping between the product basis of $\mathcal H_A\otimes\mathcal H_B$ and a fixed computational basis $\{\ket{k}_C\}$ of $\mathcal H_C$: \begin{equation}
\ket{i}_A\otimes\ket{j}_B\rightarrow\ket{k'}_C,\quad
k'=i\,d_B + j, \end{equation} i.~e., the bases are joined in the lexicographic order. The mapping is extended on all other vectors of $\mathcal H_A\otimes\mathcal H_B$ by linearity.
\section{\label{sec::main}Entangled states and subspaces from tensor product} We begin the section with a simple observation. \begin{lemma} Let $\ket{\phi}_{AB_1}$ and $\ket{\chi}_{B_2C}$ be two pure bipartite entangled states on $\mathcal H_A \otimes\mathcal H_{B_1}$ and $\mathcal H_{B_2}\otimes\mathcal H_C$, respectively. Let $\ket{\psi}_{ABC}$ be a tripartite pure state on $\mathcal H_A\otimes\mathcal H_B\otimes\mathcal H_C$ that is obtained from taking the tensor product $\ket{\phi}_{AB_1}\otimes\ket{\chi}_{B_2C}$ with subsequent joining subsystems $B_1$ and $B_2$ into a larger one, $B=B_1B_2$~(see Fig.~\ref{fig:twostates}). Then $\ket{\psi}_{ABC}$ is genuinely entangled. \end{lemma} \begin{proof}
One needs to check that the tripartite state is entangled across all three bipartitions $A|BC$, $B|AC$, $C|AB$, which can be conveniently seen from the diagrammatic representation. For bipartition $B|AC$, as shown on Fig.~\ref{fig:twostatesbip}, tracing out subsystem $B$~(i. e., subsystems $B_1$ and $B_2$) results in a state equal to $\rho^{\phi}_A \otimes \rho^{\chi}_C$, where \begin{equation}\label{parden}
\rho^{\phi}_A = \mathrm{Tr}_{B_1}\{\dyad{\phi}_{AB_1}\},\quad \rho^{\chi}_C = \mathrm{Tr}_{B_2}\{\dyad{\chi}_{B_2C}\}. \end{equation} The bipartite states $\ket{\phi}_{AB_1}$ and $\ket{\chi}_{B_2C}$ are entangled, and hence the corresponding one party states $\rho^{\phi}_A$ and $\rho^{\chi}_C$ are mixed. As a tensor product of mixed states, the resulting state is also mixed. The other two bipartitions are analyzed similarly. \end{proof}
\begin{figure}
\caption{ Tensor product of two bipartite entangled pure states generates a tripartite genuinely entangled state after joining subsystems $B_1$ and $B_2$.}
\label{fig:twostates}
\end{figure}
\begin{figure}
\caption{ Entanglement in bipartition $B|AC$ of a tripartite pure state $\ket{\psi}_{ABC}$: after tracing out subsystem $B$ the resulting state is a tensor product of two mixed states $\rho^{\phi}_A$ and $\rho^{\chi}_C$.}
\label{fig:twostatesbip}
\end{figure}
What is more interesting is that two bipartite entangled subspaces can be combined in a similar way to generate a genuinely entangled subspace.
\begin{lemma}\label{gesprod} Let $\mathcal S_{AB_1}$ be a completely entangled subspace of $\mathcal H_A\otimes\mathcal H_{B_1}$, and $\mathcal G_{B_2C}$ -- a completely entangled subspace of $\mathcal H_{B_2}\otimes\mathcal H_C$. Then their tensor product $\mathcal S_{AB_1}\otimes\mathcal G_{B_2C}$, after joining subsystems $B_1$ and $B_2$ into $B=B_1B_2$, is a genuinely entangled subspace of $\mathcal H_A\otimes\mathcal H_B\otimes\mathcal H_C$, with the geometric measure of genuine entanglement \begin{equation}\label{gmeges}
G_{GME}(\mathcal S_{AB_1}\otimes\mathcal G_{B_2C}) = \min\left(G(\mathcal S_{AB_1}),\,G(\mathcal G_{B_2C})\right). \end{equation} \end{lemma}
\begin{proof} The argument follows from diagrammatic reasoning involving the correspondence between bipartite subspaces and quantum channels.
Let $\ket{\psi_1}_{AB_1},\,\ldots,\,\ket{\psi_n}_{AB_1}$ be basis vectors in $\mathcal S_{AB_1}$, and $\ket{\chi_1}_{B_2C},\,\ldots,\,\ket{\chi_k}_{B_2C}$ -- basis vectors in $\mathcal G_{B_2C}$. The elements $\{\ket{\psi_i}_{AB_1}\otimes\ket{\chi_j}_{B_2C}\}$ then span $\mathcal S_{AB_1}\otimes\mathcal G_{B_2C}$.
Consider also Hilbert spaces $\mathcal H_D$ and $\mathcal H_E$ with $\dim(\mathcal H_D)=\dim(\mathcal S_{AB_1})$, $\dim(\mathcal H_E)=\dim(\mathcal G_{B_2C})$ and basis states $\ket{\mu_1}_{D},\,\ldots,\,\ket{\mu_n}_{D}$ and $\ket{\nu_1}_{E},\,\ldots,\,\ket{\nu_k}_{E}$, respectively.
Let $V_1\colon\,\mathcal H_{D}\rightarrow\mathcal H_{A}\otimes\mathcal H_{B_1}$ be an isometry that maps the states $\{\ket{\mu_i}_{D}\}$ to the states $\{\ket{\psi_i}_{AB_1}\}$, and $V_2\colon\,\mathcal H_{E}\rightarrow\mathcal H_{B_2}\otimes\mathcal H_{C}$ -- an isometry mapping $\{\ket{\nu_j}_{E}\}$ to $\{\ket{\chi_j}_{B_2C}\}$. The ranges of $V_1$ and $V_2$ are then the completely entangled subspaces $\mathcal S_{AB_1}$ and $\mathcal G_{B_2C}$, respectively.
A particular element $\ket{\psi_i}_{AB_1}\otimes\ket{\chi_j}_{B_2C}$ of $\mathcal S_{AB_1}\otimes\mathcal G_{B_2C}$ can be written as \begin{multline}
\ket{\psi_i}_{AB_1}\otimes\ket{\chi_j}_{B_2C} \\
= \left(V_1\otimes V_2\right)\left(\ket{\mu_i}_{ D}\otimes\ket{\nu_j}_{E}\right), \end{multline} and hence the whole subspace $\mathcal S_{AB_1}\otimes\mathcal G_{B_2C}$ can be presented as the result of action of the isometry $V_1\otimes V_2$ on each state from the tensor product Hilbert space $\mathcal H_D\otimes\mathcal H_E$ spanned by $\{\ket{\mu_i}_{D}\otimes\ket{\nu_j}_{E}\}$~(see Fig.~\ref{fig:gesgenod}).
\begin{figure}\label{fig:gesgenod}
\end{figure}
We can use this diagrammatic representation of a general state from $\mathcal S_{AB_1}\otimes\mathcal G_{B_2C}$ for the analysis of entanglement. Consider now bipartition $A|BC$. Tracing out subsystems $B=B_1B_2$ and $C$ of the state has the same effect as tracing out subsystem $E$ of the corresponding state $\ket{\phi}_{DE}$ from $\mathcal H_D\otimes\mathcal H_E$ with subsequent action of the quantum channel $\Phi\colon\mathcal H_D\rightarrow\mathcal H_A$ associated with the isometry $V_1$~(see Fig.~\ref{fig:ges_bip1}). The isometry $V_2$ gets completely traced out and has no effect here. The channel $\Phi$ hence acts on a state $\rho^{\phi}_D=\mathrm{Tr}_E\{\dyad{\phi}_{DE}\}$. Being a convex function, the output norm $\norm{\Phi(\rho^{\phi}_D)}_{\infty}$ attains its maximal value, $\nu_{\infty}(\Phi)$, on pure $\rho^{\phi}_D$~(and, correspondingly, on separable $\ket{\phi}_{DE}$). Consequently, for the geometric measure of entanglement of the subspace $\mathcal S_{AB_1}\otimes\mathcal G_{B_2C}$ across bipartition $A|BC$ we have \begin{equation}
G_{A|BC}(\mathcal S_{AB_1}\otimes\mathcal G_{B_2C}) = 1 - \nu_{\infty}(\Phi). \end{equation} On the other hand, the channel $\Phi$ corresponds to the isometry $V_1$ whose range is $\mathcal S_{AB_1}$, and so $\nu_{\infty}(\Phi)$ is equal to the maximum of the first Schmidt coefficient squared taken over all states in $\mathcal S_{AB_1}$. In other words, $G(\mathcal S_{AB_1})=1 - \nu_{\infty}(\Phi)$, and hence \begin{equation}
G_{A|BC}(\mathcal S_{AB_1}\otimes\mathcal G_{B_2C}) = G(\mathcal S_{AB_1}). \end{equation}
The analysis of bipartition $C|AB$, conducted similarly, yields \begin{equation}
G_{C|AB}(\mathcal S_{AB_1}\otimes\mathcal G_{B_2C}) = G(\mathcal G_{B_2C}). \end{equation}
\begin{figure}\label{fig:ges_bip1}
\end{figure}
Consider bipartition $B|AC$. Tracing out subsystem $B=B_1B_2$ is equivalent to action of two quantum channels: $\Phi_1\colon\mathcal H_D\rightarrow\mathcal H_A$ and $\Phi_2\colon\mathcal H_E\rightarrow\mathcal H_C$, associated with the isometries $V_1$ and $V_2$ and applied to subsystems $D$ and $E$ of $\ket{\phi}_{DE}$, respectively~(see Fig.~\ref{fig:ges_bip2}). \begin{figure}
\caption{Tracing out subsystem $B$ of a state from $\mathcal S_{AB_1}\otimes\mathcal G_{B_2C}$ is equivalent to action of quantum channels $\Phi_1$ and $\Phi_2$ on parties $D$ and $E$ of the corresponding state $\ket{\phi}_{DE}$ from $\mathcal H_D\otimes\mathcal H_E$.}
\label{fig:ges_bip2}
\end{figure} Analytically this state can be presented as \begin{equation}\label{phistate}
(\Phi_1\otimes\Phi_2)\dyad{\phi}_{DE} = (I\otimes\Phi_2)\,\tau_{DE}, \end{equation} where $\tau_{DE} = (\Phi_1\otimes I)\dyad{\phi}_{DE}$. For the output norm of this state we have \begin{equation}\label{geombound}
\norm{(I\otimes\Phi_2)\,\tau_{DE}}_{\infty}\leqslant\nu(I\otimes\Phi_2)_{\infty} = \nu(\Phi_2)_{\infty}, \end{equation}
where the last equality is due to the property~(\ref{idealcomb}). From Eq.~(\ref{geombound}) it follows that $G_{B|AC}\geqslant G(\mathcal G_{B_2C})$. Actually, $\Phi_1$ and $\Phi_2$ enter Eq.~(\ref{phistate}) symmetrically, and hence another bound for the geometric measure can be written: $G_{B|AC}\geqslant G(\mathcal S_{AB_1})$. Combining the two results, we have: \begin{equation}\label{bipbound}
G_{B|AC}(\mathcal S_{AB_1}\otimes\mathcal G_{B_2C})\geqslant\max\left(G(\mathcal S_{AB_1}),\,G(\mathcal G_{B_2C})\right). \end{equation} Gathering the results across three bipartitions, we obtain Eq.~(\ref{gmeges}). \end{proof}
\begin{remark}\label{remchan}
The bound in Eq.~(\ref{bipbound}) is not optimal. The geometric measure across bipartition $B|AC$ is directly connected with the maximal output norm of a tensor product of two channels~(as in Eq.~(\ref{phistate})) and the problem of multiplicativity of the maximal output norm, which was investigated in Refs.~\cite{AmHolWer2000,HolWern02,King03,HaydWin08}. In general, the norm is not multiplicative, and $\nu_p(\Phi_1\otimes\Phi_2)\geqslant\nu_p(\Phi_1)\nu_p(\Phi_2)$. In some particular cases, for example, when one of two channels is entanglement breaking, multiplicativity holds~\cite{King03}. In relation to Lemma~\ref{gesprod} this means that, when one of the completely entangled subspaces in tensor product corresponds to an entanglement breaking channel~(with output purity strictly less than $1$), the geometric measure across bipartition $B|AC$ attains its maximal possible value \begin{multline}
G_{B|AC}(\mathcal S_{AB_1}\otimes\mathcal G_{B_2C})=G(\mathcal S_{AB_1}) + G(\mathcal G_{B_2C})\\ - G(\mathcal S_{AB_1})G(\mathcal G_{B_2C}). \end{multline} \end{remark}
Lemma~\ref{gesprod} can be extended to the case where $(n+1)$-partite GESs are constructed from tensor product of $n$ bipartite CESs with subsequent joining the adjacent subsystems. \begin{corollary} Let $\mathcal{S}^{(1)}_{A_1A_2},\,\mathcal{S}^{(2)}_{A_3A_4},\,\ldots,\,\mathcal{S}^{(n)}_{A_{2n-1}A_{2n}}$ be a system of $n$ bipartite completely entangled subspaces of tensor product Hilbert spaces $\mathcal H_{A_1}\otimes\mathcal H_{A_2},\,\mathcal H_{A_3}\otimes\mathcal H_{A_4},\,\ldots,\,\mathcal H_{A_{2n-1}}\otimes\mathcal H_{A_{2n}}$, respectively~($n\geqslant 2$). Let
\begin{equation*}
\mathcal W_{A_1 A'_2A'_3\ldots A'_n A_{2n}}\coloneqq\mathcal{S}^{(1)}_{A_1A_2}\otimes\ldots\otimes\mathcal{S}^{(n)}_{A_{2n-1}A_{2n}}
\end{equation*}
be a subspace of an $(n+1)$-partite tensor product Hilbert space $\mathcal H_{A_1}\otimes\mathcal H_{A'_2}\otimes\mathcal H_{A'_3}\otimes\ldots\otimes\mathcal H_{A'_n}\otimes\mathcal H_{A_{2n}}$,
after taking tensor products and joining subsystems $A_2$ and $A_3$, $A_4$ and $A_5$, \ldots, $A_{2n-2}$ and $A_{2n-1}$ into $A'_2=A_2A_3$, $A'_3=A_4A_5$, \ldots, $A'_n=A_{2n-2}A_{2n-1}$, respectively. Then $\mathcal W_{A_1 A'_2A'_3\ldots A'_n A_{2n}}$ is genuinely entangled, with the geometric measure of genuine entanglement
\begin{equation}\label{ngeom}
G_{GME}(\mathcal W_{A_1 A'_2A'_3\ldots A'_n A_{2n}})
=\min\left(G_1,\,\ldots,\, G_n\right),
\end{equation}
where $G_i$ -- the geometric measure of entanglement of the subspace $\mathcal{S}^{(i)}_{A_{2i-1}A_{2i}}$, $1\leqslant i\leqslant n$. \end{corollary} \begin{proof} In analogy with the proof of Lemma~\ref{gesprod}~(see Fig.~\ref{fig:gesgenod}), the $(n+1)$-partite subspace under consideration is the result of action of $n$ isometries $\{V_i\}$ on each state $\ket{\phi}$ from an $n$-partite tensor product Hilbert space $\mathcal H_{C_1}\otimes\ldots\otimes\mathcal H_{C_n}$, with subsequent joining the adjacent subsystems $A_2$ and $A_3$, \ldots, $A_{2n-2}$ and $A_{2n-1}$ into $A'_2$, \ldots, $A'_n$, respectively~(see Fig.~\ref{fig:nges}). Here the isometry $V_i$ is associated with the subspace $\mathcal{S}^{(i)}_{A_{2i-1}A_{2i}}$ for $1\leqslant i\leqslant n$.
Now, analyzing entanglement in each of the $2^n-1$ possible bipartite cuts in a way similar to that in the proof of Lemma~\ref{gesprod}, we obtain $2^n-1$ values and lower bounds~(written with the $\geqslant$ signs) for the geometric measure in these cuts: \begin{multline}\label{allbips}
G_1,\, G_2,\,\ldots,\,G_n; \geqslant\max(G_1,\,G_2),\,\geqslant\max(G_1,\,G_3),\\\ldots;\,\geqslant\max(G_1,\,G_2,\,G_3),\,\ldots;\,\geqslant\max\left(G_1,\,\ldots,\, G_n\right). \end{multline} Here, for example, the value $G_i$ appears in a bipartite cut where, after tracing out appropriate subsystems, only the isometry $V_i$ is left partially traced out, with all other isometries $\{V_k\}$, $k\ne i$ being completely traced out. This situation is analogous to that in the proof of Lemma~\ref{gesprod} shown on Fig.~\ref{fig:ges_bip1}. For another example, the lower bound $\max(G_1,\,G_2,\,G_3)$ appears for a cut where, after tracing out appropriate subsystems, only partially traced out isometries $V_1$, $V_2$, $V_3$ are left, and the rest isometries are completely traced out. This case is analogous to that shown on Fig.~\ref{fig:ges_bip2}~(the difference is that here are three isometries instead of those two presented on the figure).
Noting that the value $\min\left(G_1,\,\ldots,\, G_n\right)$ is the minimum among those in Eq.~(\ref{allbips}), we obtain the equality in Eq.~(\ref{ngeom}). \end{proof}
\begin{figure}\label{fig:nges}
\end{figure}
Next we consider some situations where GESs are constructed from direct sums of tensor products of CESs. The following property will be useful here. \begin{lemma}\label{lemCES} Let $\mathcal S_{AB_1}$ be a completely entangled subspace of a tensor product Hilbert space $\mathcal H_A\otimes\mathcal H_{B_1}$. Then the tensor product $\mathcal S_{AB_1}\otimes\mathcal H_{B_2}$, after joining subsystems $B_1$ and $B_2$ into $B=B_1B_2$, is a completely entangled subspace of $\mathcal H_A\otimes\mathcal H_B$. \end{lemma} \begin{proof} Assume that $\mathcal S_{AB_1}$ is spanned by vectors $\ket{\psi_1}_{AB_1},\,\ldots,\,\ket{\psi_n}_{AB_1}$. Let $\ket{\nu_1}_{B_2},\,\ldots,\,\ket{\nu_k}_{B_2}$ be an orthonormal basis in $\mathcal H_{B_2}$. The elements $\{\ket{\psi_i}_{AB_1}\otimes\ket{\nu_j}_{B_2}\}$ are linearly independent due to orthonormality of the system $\{\ket{\nu_j}_{B_2}\}$ and linear independence of $\{\ket{\psi_i}_{AB_1}\}$. Let us check that any linear combination of these elements yields an entangled state in $\mathcal H_A\otimes\mathcal H_{B_1B_2}$. If, in some linear combinations, there are elements with the same vector from $\mathcal H_{B_2}$, they can be combined into one term, as in the following example: \begin{multline}
c_i\ket{\psi_i}_{AB_1}\otimes\ket{\nu_l}_{B_2} + c_j\ket{\psi_j}_{AB_1}\otimes\ket{\nu_l}_{B_2} \\
= c'\ket{\phi}_{AB_1}\otimes\ket{\nu_l}_{B_2}, \end{multline} where \begin{equation*}
c'\ket{\phi}_{AB_1} = c_i\ket{\psi_i}_{AB_1} + c_j\ket{\psi_j}_{AB_1}, \end{equation*} with $\ket{\phi}_{AB_1}$ being a normalized state and $c'$ -- some normalization factor. As a linear combination of vectors from a CES, the vector $\ket{\phi}_{AB_1}$ is entangled. Therefore, without loss of generality, one can consider linear combinations \begin{equation}\label{lincom}
\sum_{i=1}^k\,c_i\ket{\phi_i}_{AB_1}\otimes\ket{\nu_i}_{B_2},\quad\sum_{i=1}^k\,\abs{c_i}^2 = 1, \end{equation} where all the terms have distinct vectors $\{\ket{\nu_i}\}$ from $\mathcal H_{B_2}$, and $\{\ket{\phi_i}_{AB_1}\}$ -- some normalized vectors from the given bipartite CES $\mathcal S_{AB_1}$. Next, tracing out subsystem $B=B_1B_2$ in Eq.~(\ref{lincom}), with the use of the orthonormality property $\mathrm{Tr}_{B_2}\{\ket{\nu_i}\bra{\nu_j}_{B_2}\}=\delta_{ij}$, one obtains the reduced density operator on $\mathcal H_A$: \begin{multline}\label{mixedreas}
\sum_{i,\,j=1}^k\,c_i c_j^{*}\,\mathrm{Tr}_{B_1}\{\ket{\phi_i}\bra{\phi_j}_{AB_1}\}\, \mathrm{Tr}_{B_2}\{\ket{\nu_i}\bra{\nu_j}_{B_2}\}\\
= \sum_{i=1}^k\,\abs{c_i}^2\,\mathrm{Tr}_{B_1}\{\ket{\phi_i}\bra{\phi_i}_{AB_1}\}\\
\equiv\,\sum_{i=1}^k\,\abs{c_i}^2\,\rho^{\phi_i}_A. \end{multline} As a convex sum of mixed states $\rho^{\phi_i}_A$, this state is mixed, and hence the linear combination in Eq.~(\ref{lincom}) yields an entangled state in $\mathcal H_A\otimes\mathcal H_B$. \end{proof}
The statement of Lemma~\ref{lemCES} can now be slightly changed with the aim to consider direct sums of tensor products.
\begin{corollary}\label{sumprodces} Let $\mathcal S^{(1)}_{AB_1},\,\ldots,\,\mathcal S^{(n)}_{AB_1}$ be a system of completely entangled subspaces of $\mathcal H_A\otimes\mathcal H_{B_1}$. Let $\mathcal P^{(1)}_{B_2},\,\ldots,\,\mathcal P^{(n)}_{B_2}$ be a system of mutually orthogonal subspaces of $\mathcal H_{B_2}$. Then the direct sum of tensor products \begin{equation}\label{dsum}
\left(\mathcal S^{(1)}_{AB_1}\otimes\mathcal P^{(1)}_{B_2}\right)\oplus\ldots\oplus\left(\mathcal S^{(n)}_{AB_1}\otimes\mathcal P^{(n)}_{B_2}\right), \end{equation} after joining subsystems $B_1$ and $B_2$ into $B=B_1B_2$, is a completely entangled subspace of $\mathcal H_A\otimes\mathcal H_B$. \end{corollary} \begin{proof} Let $\mathcal S^{(r)}_{AB_1}$ be spanned by a system of vectors $\ket{\psi_1^{(r)}}_{AB_1},\,\ldots,\,\ket{\psi_{l_r}^{(r)}}_{AB_1}$ and let $\mathcal P^{(r)}_{AB_1}$ be spanned by an orthonormal system of vectors $\ket{\nu_1^{(r)}}_{B_2},\,\ldots,\,\ket{\nu_{k_r}^{(r)}}_{B_2}$, for each $r\colon 1\leqslant r\leqslant n$. An arbitrary vector $\ket{\chi}_{AB}$ that belongs to the direct sum~(\ref{dsum}) can be decomposed as \begin{equation}\label{lindifces}
\ket{\chi}_{AB} = \sum_{r=1}^n\sum_{i=1}^{k_r}\, c_i^{(r)}\ket{\phi_i^{(r)}}_{AB_1}\otimes\ket{\nu_i^{(r)}}_{B_2},
\end{equation}
where the terms with distinct vectors $\nu$ were gathered and each $\ket{\phi_i^{(r)}}_{AB_1}$, being a linear combination of $\ket{\psi_1^{(r)}}_{AB_1},\,\ldots,\,\ket{\psi_{l_r}^{(r)}}_{AB_1}$, is entangled. All vectors $\nu$ are mutually orthogonal: $\bra{\nu_i^{(r)}}\ket{\nu_j^{(s)}}=\delta_{ij}\delta_{rs}$, and hence the linear combination in Eq.~(\ref{lindifces}) has the same structure as that in Eq.~(\ref{lincom}). Repeating the same reasoning as in Eq.~(\ref{mixedreas}), we obtain that $\ket{\chi}_{AB}$ is entangled. \end{proof}
\begin{lemma}\label{sumprod} Let $\mathcal S^{(1)}_{AB_1},\,\ldots,\,\mathcal S^{(n)}_{AB_1}$ be a system of completely entangled subspaces of $\mathcal H_A\otimes\mathcal H_{B_1}$, and $\mathcal G^{(1)}_{B_2C},\,\ldots,\,\mathcal G^{(n)}_{B_2C}$ -- a system of mutually orthogonal completely entangled subspaces of $\mathcal H_{B_2}\otimes\mathcal H_C$ whose direct sum $\Sigma_{B_2C}\coloneqq\mathcal G^{(1)}_{B_2C}\oplus\,\ldots\oplus\,\mathcal G^{(n)}_{B_2C}$ is also completely entangled. Then the direct sum of tensor products \begin{equation}\label{dsumges}
\left(\mathcal S^{(1)}_{AB_1}\otimes\mathcal G^{(1)}_{B_2C}\right)\oplus\ldots\oplus\left(\mathcal S^{(n)}_{AB_1}\otimes\mathcal G^{(n)}_{B_2C}\right), \end{equation} after joining subsystems $B_1$ and $B_2$ into $B=B_1B_2$, is a genuinely entangled subspace of $\mathcal H_A\otimes\mathcal H_B\otimes\mathcal H_C$. \end{lemma} \begin{proof} Let $\mathcal H_{B_2}$ be a Hilbert space of dimension equal to the dimension of $\Sigma_{B_2C}$. Consider an isometry $V\colon\,\mathcal H_{B_2}\rightarrow\mathcal H_{B_2}\otimes\mathcal H_C$ that maps $\mathcal H_{B_2}$ to $\Sigma_{B_2C}$. The isometry has a CES as its range, and so it corresponds to a quantum channel with output purity strictly less than $1$. By Eq.~(\ref{idealcomb}), so does the isometry $I_{\scriptscriptstyle B_1}\otimes V_{\scriptscriptstyle B_2\rightarrow B_2C}$.
Let $\mathcal P^{(1)}_{B_2},\,\ldots,\,\mathcal P^{(n)}_{B_2}$ be a system of mutually orthogonal subspaces of $\mathcal H_{B_2}$. By Corollary~\ref{sumprodces}, \begin{equation*}
\Omega_{AB}\coloneqq\left(\mathcal S^{(1)}_{AB_1}\otimes\mathcal P^{(1)}_{B_2}\right)\oplus\ldots\oplus\left(\mathcal S^{(n)}_{AB_1}\otimes\mathcal P^{(n)}_{B_2}\right) \end{equation*} is a CES of $\mathcal H_A\otimes\mathcal H_B$. The subspace in Eq.~(\ref{dsumges}) is obtained from $\Omega_{AB}$ by the action of the isometry $I_{\scriptscriptstyle B_1}\otimes V_{\scriptscriptstyle B_2\rightarrow B_2C}$ on subsystem $B=B_1B_2$~(see Fig.~\ref{fig:gesgen}). This situation corresponds to the scheme on Fig.~\ref{fig:iso23}. Therefore, the generated subspace is genuinely entangled. \end{proof}
\begin{figure}
\caption{ Action of the isometry $I_{\scriptscriptstyle B_1}\otimes V_{\scriptscriptstyle B_2\rightarrow B_2C}$ on each state from $\Omega_{AB}$ generates the GES presented in Eq.~(\ref{dsumges}).}
\label{fig:gesgen}
\end{figure}
Note that the CESs $S^{(1)}_{AB_1},\,\ldots,\,\mathcal S^{(n)}_{AB_1}$ in the above statement can be arbitrary, and they can have arbitrary relations to each other~(e.~g, intersect or not intersect). In particular, each of them can be spanned by just one entangled vector.
\begin{corollary}\label{corent} Let $\ket{\psi_1}_{AB_1},\,\ldots,\,\ket{\psi_n}_{AB_1}$ be some entangled vectors in $\mathcal H_A\otimes\mathcal H_{B_1}$, and $\ket{\chi_1}_{B_2C},\,\ldots,\,\ket{\chi_n}_{B_2C}$ -- mutually orthogonal vectors spanning a completely entangled subspace of $\mathcal H_{B_2}\otimes\mathcal H_C$. Then a system of vectors $\ket{\psi_1}_{AB_1}\otimes\ket{\chi_1}_{B_2C},\,\ldots,\,\ket{\psi_n}_{AB_1}\otimes\ket{\chi_n}_{B_2C}$ spans a genuinely entangled subspace of $\mathcal H_A\otimes\mathcal H_B\otimes\mathcal H_C$. \end{corollary}
\section{Applications}\label{sec::app}
The established properties can have several applications.
\subsection{Tensor products of mixed bipartite entangled states}
In Refs.~\cite{ShenChen20,SunChen21} it was stated as a conjecture that a tensor product of two mixed bipartite entangled states, $\alpha_{AB_1}\otimes\beta_{B_2C}$, after joining $B_1$ and $B_2$, is a genuinely entangled tripartite state. Later the conjecture was disproved in Ref.~\cite{TenNet2021} by finding an example with two entangled isotropic states whose tensor product is not GE. In this connection, it is interesting to search for sufficient conditions of genuine entanglement of such tensor products.
One condition of this type can be obtained from combining the properties of tensor products of CESs with a particular witness of genuine entanglement connected with projection on some GES, namely, in Ref.~\cite{KVAnt21} it was shown that if, for a multipartite state $\rho$ and a genuinely entangled subspace $W$, the inequality \begin{equation}\label{EntWit}
\Tr{\rho\,\Pi_W} + G_{GME}(W) - 1 >0 \end{equation} holds, then $\rho$ is genuinely entangled. Here $\Pi_W$ -- an orthogonal projector onto $W$. \begin{lemma}\label{StProd} Let $\alpha_{AB_1}$ and $\beta_{B_2C}$ be two bipartite mixed states on $\mathcal H_{A}\otimes\mathcal H_{B_1}$ and $\mathcal H_{B_2}\otimes\mathcal H_{C}$, respectively. Let $W_1$ and $W_2$ be two completely entangled subspaces of $\mathcal H_{A}\otimes\mathcal H_{B_1}$ and $\mathcal H_{B_2}\otimes\mathcal H_{C}$, respectively. Then the tensor product $\alpha_{AB_1}\otimes\beta_{B_2C}$, after joining $B_1$ and $B_2$ into $B=B_1B_2$, is a genuinely entangled tripartite state on $\mathcal H_A\otimes\mathcal H_B\otimes\mathcal H_C$ if \begin{multline}\label{EnCon}
\Tr{\alpha_{AB_1}\,\Pi_{W_1}}\Tr{\beta_{B_2C}\,\Pi_{W_2}} \\ > 1 - \min\left(G(W_1),\,G(W_2)\right). \end{multline} \end{lemma} \begin{proof} We can use condition~(\ref{EntWit}) with respect to the state $\alpha_{AB_1}\otimes\beta_{B_2C}$ and the subspace $W_1\otimes W_2$ of the tensor product Hilbert space $\mathcal H_A\otimes\mathcal H_B\otimes\mathcal H_C$. By Lemma~\ref{gesprod}, $W_1\otimes W_2$ is a GES, with the GME geometric measure \begin{equation}\label{minG}
G_{GME}(W_1\otimes W_2) = \min\left(G(W_1),\,G(W_2)\right). \end{equation} In addition, \begin{multline}\label{TrF} \Tr{\alpha_{AB_1}\otimes\beta_{B_2C}\,\Pi_{W_1\otimes W_2}} \\ = \Tr{\alpha_{AB_1}\,\Pi_{W_1}}\Tr{\beta_{B_2C}\,\Pi_{W_2}}. \end{multline} Combining Eqs.~(\ref{EntWit}), (\ref{minG}), and (\ref{TrF}), we obtain sufficient condition~(\ref{EnCon}) for genuine entanglement of $\alpha_{AB_1}\otimes\beta_{B_2C}$. \end{proof} \begin{remark} Lower bounds on two GME entanglement measures, the concurrence and the convex-roof extended negativity~(CREN), can be also obtained in connection with this entanglement witness. For example, if condition~(\ref{EnCon}) holds, Eq.~(64) from Ref.~\cite{KVAnt21} yields the bound for the CREN of the state $\alpha_{AB_1}\otimes\beta_{B_2C}$: \begin{multline}\label{CREN}
N_{GME}(\alpha_{AB_1}\otimes\beta_{B_2C})\\ \geqslant\frac{\Tr{\alpha_{AB_1}\,\Pi_{W_1}}\Tr{\beta_{B_2C}\,\Pi_{W_2}} + G_{12} - 1}{2\,(1 - G_{12})}, \end{multline} where $G_{12}=\min\left(G(W_1),\,G(W_2)\right)$. \end{remark}
\subsubsection*{Example: tensor product of two Werner states}
Consider the Werner states family on $\mathbb{C}^d\otimes\mathbb{C}^d$: \begin{equation}
\rho_{\scriptscriptstyle\mathcal W}(p,d) = \frac1{d^2 + pd}\left(I_d\otimes I_d + p\sum_{i,\,j=0}^{d-1}\,\ket{i,\,j}\bra{j,\,i}\right). \end{equation}
In Ref.~\cite{SunChen21} it was proved that $\rho_{\scriptscriptstyle\mathcal W}(p_1,2)\otimes\rho_{\scriptscriptstyle\mathcal W}(p_2,2)$, when viewed as a tripartite state on $\mathbb{C}^2\otimes\mathbb{C}^4\otimes\mathbb{C}^2$, is genuinely entangled in the region \begin{equation}\label{oldDom}
-1\,\leqslant\,p_1\,\leqslant -0.940198;\quad -1\,\leqslant\,p_2\,\leqslant\,-0.94066. \end{equation} With the use of Lemma~\ref{StProd} this domain can be extended.
Let us consider the tensor product $\rho_{\scriptscriptstyle\mathcal W}(p_1,d)\otimes\rho_{\scriptscriptstyle\mathcal W}(p_2,d)$ of two Werner states on $\mathbb{C}^d\otimes\mathbb{C}^d$. With the use of relations \begin{equation}
\Pi_{\mathcal A} = \frac{I-\mathrm{SWAP}}2;\quad\Pi_S=\frac{I+\mathrm{SWAP}}2, \end{equation} where $\Pi_{\mathcal A},\,\Pi_S$ -- the projectors onto the antisymmetric and the symmetric subspaces of $\mathbb{C}^d\otimes\mathbb{C}^d$ respectively, and \begin{equation}
\mathrm{SWAP} = \sum_{i,\,j=0}^{d-1}\,\ket{i,\,j}\bra{j,\,i}, \end{equation} the operator that exchanges qudits, the Werner state itself can be rewritten as \begin{equation}
\rho_{\scriptscriptstyle\mathcal W}(p,d) = \frac1{d^2+pd}\left[(1+p)\Pi_S + (1-p)\Pi_{\mathcal A}\right]. \end{equation} For our analysis it is more convenient to reparameterize it with a new variable $s$ related to $p$ as \begin{equation}\label{sp}
\frac{2s}{d(d-1)} = \frac{1-p}{d(p+d)}, \end{equation} so that \begin{equation}\label{NpW}
\rho_{\scriptscriptstyle\mathcal W}(s,d) = \frac{2(1-s)}{d(d+1)}\Pi_S + \frac{2s}{d(d-1)}\Pi_{\mathcal A}. \end{equation} Let us apply Lemma~\ref{StProd} and condition~(\ref{EnCon}) to the state $\rho_{\scriptscriptstyle\mathcal W}(s_1,d)\otimes\rho_{\scriptscriptstyle\mathcal W}(s_2,d)$, with both $W_1$ and $W_2$ chosen to be the antisymmetric subspace $\mathcal A$ of $\mathbb{C}^d\otimes\mathbb{C}^d$, which has dimension equal to $d(d-1)/2$. It is known~\cite{Vid02} that the geometric measure $G(\mathcal{A}) = 1/2$~(see also \cite{KVAnt20}). From Eq.~(\ref{NpW}) it follows that $$\Tr{\rho_{\scriptscriptstyle\mathcal W}(s,d)\,\Pi_{\mathcal A}} = s,$$ and thus condition~(\ref{EnCon}) takes a simple form: \begin{equation}\label{domEn}
s_1 s_2 > \frac12. \end{equation} \begin{figure}
\caption{ Shaded regions represent the domains in $s_1,\,s_2$ where the state $\rho_{\scriptscriptstyle\mathcal W}(s_1,d)\otimes\rho_{\scriptscriptstyle\mathcal W}(s_2,d)$ is genuinely entangled: the area above the graph $s_2=1/(2s_1)$ and its maximal square subdomain.}
\label{fig:plot}
\end{figure} Eq.~(\ref{domEn}) defines the domain of genuine entanglement of the state $\rho_{\scriptscriptstyle\mathcal W}(s_1,d)\otimes\rho_{\scriptscriptstyle\mathcal W}(s_2,d)$~(the whole shaded area above the graph $s_2 = 1/(2s_1)$ depicted on Fig.~\ref{fig:plot}). In particular, we can specify the maximal square subdomain~(also shown on Fig.~\ref{fig:plot}) \begin{equation}
\frac1{\sqrt 2} < s_1 \leqslant 1;\quad \frac1{\sqrt 2} < s_2 \leqslant 1, \end{equation} where the parameters $s_1$ and $s_2$ vary independently and where this state is GE. With the use of Eq.~(\ref{sp}), this region can be rewritten in terms of $p_1,\,p_2$. For $d=2$ we obtain \begin{equation}
-1\leqslant p_1,\,p_2 < 3\sqrt2 -5\approx -0.757359, \end{equation} which extends the domain in Eq.~(\ref{oldDom}). For larger $d$ the region becomes even wider: \begin{equation}
-1\leqslant p_1,\,p_2 < \frac{d(1-\sqrt2)-1}{\sqrt2 +d -1}, \end{equation} with the upper bound tending to $1-\sqrt2\approx -0.414213$, when $d\rightarrow\infty$.
In addition, Eq.~(\ref{CREN}) yields a lower bound on the negativity of the state, which reads as \begin{equation}
N_{GME}(\rho_{\scriptscriptstyle\mathcal W}(s_1,d)\otimes\rho_{\scriptscriptstyle\mathcal W}(s_2,d)) \geqslant s_1s_2 - \frac12, \end{equation} or, by Eq.~(\ref{sp}), \begin{multline}
N_{GME}(\rho_{\scriptscriptstyle\mathcal W}(p_1,d)\otimes\rho_{\scriptscriptstyle\mathcal W}(p_2,d)) \\\geqslant
\frac{(d-1)^2(1-p_1)(1-p_2)}{4(p_1+d)(p_2+d)} - \frac12. \end{multline}
\subsection{Construction of multipartite NPT and distillable subspaces}
An important aspect in the tasks of quantum information processing is the possibility to extract pure entangled states from mixed ones. The states from which pure entanglement can be obtained are called distillable~\cite{BenVinSmWoot96}.
More formally, a state $\rho$ on $\mathcal H_A\otimes\mathcal H_B$ is $1$-distillable~(or one-copy distillable)~\cite{VSSTT00} if there exists a pure Schmidt rank $2$ bipartite state $\ket{\psi}$ such that \begin{equation}
\bra{\psi}\rho^{T_A}\ket{\psi} < 0, \end{equation} where $T_A$ -- the transpose operation applied on subsystem $A$~(the partial transpose). Next, a state $\rho$ is $n$-distillable if $\rho^{\otimes n}$ is $1$-distillable.
All distillable states are necessarily NPT - those with partial transpose having at least one negative eigenvalue~(non-positive partial transpose). It is an open question whether the converse is true.
A multipartite subspace is called NPT with respect to some bipartite cut if any density operator with support in the subspace is NPT across this bipartite cut.
Such subspaces can serve as a source of various mixed NPT states that could potentially be distillable. There are several known constructions of multipartite subspaces that are NPT with respect to certain bipartite cuts~\cite{SAS14,JLP19}. In particular, Ref.~\cite{JLP19} provides the method of construction of maximal multipartite subspaces that are NPT across at least one bipartite cut.
In this subsection we show that $(n+1)$-partite subspaces that are NPT with respect to \emph{any} bipartite cut can be constructed from $n$ bipartite NPT subspaces. We call a multipartite subspace $1$-distillable across some bipartite cut if any density operator supported on the subspace is $1$-distillable across this cut.
\begin{lemma}\label{sprodnpt}
Let $\mathcal{S}^{(1)}_{A_1A_2},\,\mathcal{S}^{(2)}_{A_3A_4},\,\ldots,\,\mathcal{S}^{(n)}_{A_{2n-1}A_{2n}}$ be a system of $n$ bipartite NPT subspaces of tensor product Hilbert spaces $\mathcal H_{A_1}\otimes\mathcal H_{A_2},\,\mathcal H_{A_3}\otimes\mathcal H_{A_4},\,\ldots,\,\mathcal H_{A_{2n-1}}\otimes\mathcal H_{A_{2n}}$, respectively~($n\geqslant 2$). Let
\begin{equation*}
\mathcal W_{A_1 A'_2A'_3\ldots A'_n A_{2n}}\coloneqq\mathcal{S}^{(1)}_{A_1A_2}\otimes\ldots\otimes\mathcal{S}^{(n)}_{A_{2n-1}A_{2n}}
\end{equation*}
be a subspace of an $(n+1)$-partite tensor product Hilbert space $\mathcal H_{A_1}\otimes\mathcal H_{A'_2}\otimes\mathcal H_{A'_3}\otimes\ldots\otimes\mathcal H_{A'_n}\otimes\mathcal H_{A_{2n}}$,
after taking tensor products and joining subsystems $A_2$ and $A_3$, $A_4$ and $A_5$, \ldots, $A_{2n-2}$ and $A_{2n-1}$ into $A'_2=A_2A_3$, $A'_3=A_4A_5$, \ldots, $A'_n=A_{2n-2}A_{2n-1}$, respectively. Then $\mathcal W_{A_1 A'_2A'_3\ldots A'_n A_{2n}}$ is NPT across any bipartite cut. If, in addition, each of bipartite subspaces $\mathcal S$ is $1$-distillable, then $\mathcal W_{A_1 A'_2A'_3\ldots A'_n A_{2n}}$ is $1$-distillable across any bipartite cut.
\end{lemma}
See Appendix~\ref{app:lemp} for the proof.
\subsubsection*{Example: construction of a tripartite subspace $1$-distillable across any bipartite cut}
We construct this example from tensor product of two $1$-distillable bipartite subspaces. To find such bipartite subspaces, we use the argument from Ref.~\cite{AgHalBa19} which combines the results of Refs.~\cite{NJohn13,ChenDjok16}. In Ref.~\cite{NJohn13} it was shown that for a bipartite $\mathbb C^{d_1}\otimes\mathbb C^{d_2}$ system NPT subspaces of dimension up to $(d_1-1)(d_2-1)$ can be constructed. The NPT subspace $\mathcal S$ of maximal dimension reads as
\begin{multline}\label{nptex}
\mathcal S\coloneqq\mathrm{span}\{\ket{j}\ket{k+1} - \ket{j+1}\ket{k}\},\\
0\leqslant j\leqslant d_1-2,\quad 0\leqslant k\leqslant d_2-2.
\end{multline}
(Theorem~1 of Ref.~\cite{NJohn13}).
Next, in Ref.~\cite{ChenDjok16} it was shown that any rank 4 NPT state is $1$-distillable, which, combined with the results of Ref.~\cite{ChenDjok11}, means that all NPT states of rank \emph{at most} 4 are $1$-distillable. Therefore, bipartite subspaces~(\ref{nptex}) of dimensions up to $4$ are $1$-distillable.
Using the above facts, we can take a subspace $\mathcal S$ of Eq.~(\ref{nptex}) with $d_1=d_2=3$, such that $\dim(\mathcal S)=4$. Let $\mathcal W$ denote a subspace obtained from the tensor product $\mathcal S\otimes\mathcal S$ of $\mathcal S$ with itself, with subsequent joining the two adjacent subsystems. $\mathcal W$ is hence a $16$-dimensional subspace of a tripartite $3\otimes 9\otimes 3$ Hilbert space. According to Lemma~\ref{sprodnpt}, $\mathcal W$ is $1$-distillable across any of the three bipartite cuts.
The subspace $\mathcal W$ is spanned by the system of $16$ vectors obtained from all possible tensor products of vectors from $\mathcal S$ with each other. After taking the tensor products the two adjacent subsystems are to be joined according to the lexicographic order:
\begin{multline}
\ket{0}\ket{0}\rightarrow \ket{0},\quad\ket{0}\ket{1}\rightarrow\ket{1},\\
\ldots,\quad\ket{2}\ket{2}\rightarrow\ket{8},
\end{multline}
or, more generally,
\begin{equation}\label{order}
\ket{i}\ket{j}\rightarrow \ket{3i+j}.
\end{equation} The tensor product of two vectors from~(\ref{nptex}) \begin{equation}
\left(\ket{j}\ket{k+1} - \ket{j+1}\ket{k}\right)\otimes\left(\ket{l}\ket{m+1} - \ket{l+1}\ket{m}\right), \end{equation} indexed by $(j,\,k)$ and $(l,\,m)$ respectively, yields, by Eq.~(\ref{order}), a generic vector from the system of vectors spanning $\mathcal W$: \begin{multline}
\ket{j}\ket{3(k+1)+l}\ket{m+1}-\ket{j}\ket{3(k+1)+l+1}\ket{m}\\
-\ket{j+1}\ket{3k+l}\ket{m+1}+\ket{j+1}\ket{3k+l+1}\ket{m},\\
0\leqslant j,\,k,\,l,\,m\leqslant1. \end{multline}
\subsection{Entanglement criterion}
Corollary~\ref{corent} can be combined with some known results to give entanglement conditions for mixed states supported on tensor products. We give one such example using the result of Ref.~\cite{DRA21}, a simple sufficient condition for a subspace to be completely entangled: \begin{theorem}[Ref.~\cite{DRA21}]\label{exttheor} Let $V$ be a subspace spanned by $k$ pairwise orthogonal pure bipartite states $\{\ket{\phi_i}\}$ such that \begin{equation}
\sum_{i=1}^k\,G(\ket{\psi_i}) - (k-1) > 0, \end{equation} where $G$ -- the geometric measure of entanglement. Then $V$ is a completely entangled subspace. \end{theorem}
Combining it with Corollary~\ref{corent}, we obtain some sort of an entanglement criterion.
\begin{lemma} Let $\rho =\sum_{i=1}^n\,\dyad{\psi_i}$ be a density operator on a tripartite tensor product Hilbert space $\mathcal H_A\otimes\mathcal H_B\otimes\mathcal H_C$, where each state $\ket{\psi_i}$ is obtained from tensor product $\ket{\phi_i}_{AB_1}\otimes\ket{\chi_i}_{B_2C}$ of pure states $\ket{\phi_i}_{AB_1}\in\mathcal H_A\otimes\mathcal H_{B_1}$ and $\ket{\chi_i}_{B_2C}\in\mathcal H_{B_2}\otimes\mathcal H_C$, with subsequent joining subsystems $B_1$ and $B_2$ into $B$. Suppose that each $\ket{\phi_i}_{AB_1}$ is entangled. Suppose that $\{\ket{\chi_i}_{B_2C}\}$ are mutually orthogonal and such that \begin{equation*}
\sum_{i=1}^n\,G(\ket{\chi_i}) - (n-1) > 0. \end{equation*} Then $\rho$ is a genuinely entangled state. \end{lemma} \begin{proof} By Corollary~\ref{corent} and Theorem~\ref{exttheor} the states $\{\ket{\psi_i}\}$ span a GES. As a state supported on a GES, $\rho$ is genuinely entangled. \end{proof}
\section{Discussion}\label{sec::disc}
We have presented several properties of genuinely entangled subspaces obtained from the tensor product structure.
The advantage of such a construction is the possibility to control such useful characteristics of states supported on the output GESs as various measures of entanglement, distillability across some or all bipartite cuts, robustness of entanglement under mixing with external noise~(not covered here, but it easily follows from Eqs.~(68)-(71) of Ref.~\cite{KVAnt21}). In particular, highly entangled subspaces can be generated in this way. In addition, if a tripartite GES is constructed from two CESs with given geometric measures of entanglement, and one of them corresponds to an entanglement breaking channel, then, according to Remark on page~\pageref{remchan}, the exact values of the geometric measure across all three bipartite cuts are known for the resulting GES.
It has also been shown that, under certain conditions, GESs can be obtained from the direct sum of tensor products of bipartite CESs~(Lemma~\ref{sumprod}). Such a structure reminds of the inner product of vectors in the Euclidean space, although here in Lemma~\ref{sumprod} the conditions are not symmetric with respect to the left and the right subspaces in tensor products. In addition, as it was shown in Ref.~\cite{KVAnt21}, the scheme of Fig.~\ref{fig:iso23}, used in the proof of the lemma, cannot generate GESs of maximal possible dimensions, although the dimensions of output GESs asymptotically approach the maximal ones when local dimensions of subsystems are high. Therefore, the construction of Lemma~\ref{sumprod} doesn't generate maximal GESs either. A possible direction of further research can be the generalization of Lemma~\ref{sumprod} with the aim to obtain more symmetric conditions on bipartite subspaces as well as conditions sufficient for construction of maximal GESs.
\begin{acknowledgments} The author thanks M. V. Lomonosov Moscow State University for supporting this work. \end{acknowledgments}
\appendix
\section{\label{app:lemp}Proof of Lemma~\ref{sprodnpt}}
\begin{proof}
We prove the lemma for $n=2$, the case of arbitrary $n$ can be considered in a similar way.
Let $\rho$ be a density operator supported on
$\mathcal W_{A_1 A'_2A_4}=\mathcal{S}^{(1)}_{A_1A_2}\otimes\mathcal{S}^{(2)}_{A_3A_4}$, where $A'_2=A_2A_3$~(we use $A'_2$ and $A_2A_3$ interchangeably), so that $\rho$ has an ensemble decomposition
\begin{equation}\label{dec}
\rho = \sum_i\,p_i\,\dyad{\psi_i}_{A_1A'_2A_4},
\end{equation}
with $\ket{\psi_i}_{A_1A'_2A_4}$ being decomposed as
\begin{equation}\label{vecdec}
\ket{\psi_i}_{A_1A'_2A_4} = \sum_{jk}\,c^{(i)}_{jk}\,\ket{\phi_j}_{A_1A_2}\otimes\ket{\chi_k}_{A_3A_4},
\end{equation}
where $\ket{\phi_j}_{A_1A_2}\in\mathcal{S}^{(1)}_{A_1A_2}$, $\ket{\chi_k}_{A_3A_4}\in\mathcal{S}^{(2)}_{A_3A_4}$, and $c^{(i)}_{jk}\in\mathbb C$.
For the bipartite cut $A_1|A'_2A_4$ we choose the partial transpose to act on subsystem $A_1$. We want to show that there is a pure state $\ket{\Gamma}\in\mathcal H_{A_1}\otimes\mathcal H_{A'_2}\otimes\mathcal H_{A_4}$ such that
\begin{equation}\label{npt}
\bra{\Gamma}\rho^{T_{A_1}}\ket{\Gamma} < 0.
\end{equation}
We can take $\ket{\Gamma}$ to have structure
\begin{equation}\label{gammarepr}
\ket{\Gamma}_{A_1A_2A_3A_4} = \ket{\Phi}_{A_1A_2}\otimes\ket{\tau}_{A_3A_4},
\end{equation}
(before joining $A_2$ and $A_3$), with some pure states $\ket{\Phi}_{A_1A_2}\in\mathcal H_{A_1}\otimes\mathcal H_{A_2}$, $\ket{\tau}_{A_3A_4}\in\mathcal H_{A_3}\otimes\mathcal H_{A_4}$. Now, for each term in decomposition~(\ref{dec}), it can be noted that in expression
\begin{multline}\label{transfppt}
\bra{\Gamma}\left(\dyad{\psi_i}_{A_1A'_2A_4}\right)^{T_{A_1}}\ket{\Gamma}\\
=\bra{\Phi}\otimes\bra{\tau}\left(\dyad{\psi_i}_{A_1A'_2A_4}\right)^{T_{A_1}}\ket{\Phi}_{A_1A_2}\otimes\ket{\tau}_{A_3A_4}
\end{multline}
the operations $T_{A_1}$ and scalar product with $\ket{\tau}_{A_3A_4}$ can be taken independently~(as acting on different subsystems). So, first taking a partial scalar product of $\ket{\psi_j}$ with $\ket{\tau}$, with the use of Eq.~(\ref{vecdec}) we obtain
\begin{multline}
\bra{\tau}_{A_3A_4}\ket{\psi_i}_{A_1A'_2A_4}=\sum_{jk}\,c^{(i)}_{jk}\,\ket{\phi_j}_{A_1A_2} \bra{\tau}\ket{\chi_k}_{A_3A_4}\\
=\sum_j\,\Tilde{c}^{(i)}_j\ket{\phi_j}_{A_1A_2}=n_i\ket{\eta_i}_{A_1A_2},
\end{multline}
where $\ket{\eta_i}_{A_1A_2}$ -- some normalized state from the subspace $\mathcal{S}^{(1)}_{A_1A_2}$ and $n_i>0$ -- the corresponding normalization constant. Now the left part of Eq.~(\ref{npt}) can be written as
\begin{equation}
\bra{\Gamma}\rho^{T_{A_1}}\ket{\Gamma} = c\bra{\Phi}\sigma^{T_{A_1}}\ket{\Phi}_{A_1A_2}, \end{equation} where \begin{equation}\label{sig}
\sigma = \sum_i\,\Tilde{p_i}\dyad{\eta_i}_{A_1A_2}, \end{equation} a state entirely supported on $\mathcal{S}^{(1)}_{A_1A_2}$, with \begin{equation}
\Tilde{p_i}=\frac{n_i^2 p_i}c,\quad c = \sum_i\,n_i^2 p_i. \end{equation} Since the state $\sigma$ is NPT, choosing in Eq.~(\ref{gammarepr}) the state $\Phi$ such that \begin{equation}\label{partnpt} \bra{\Phi}\sigma^{T_{A_1}}\ket{\Phi}_{A_1A_2}<0, \end{equation}
we obtain the state $\ket{\Gamma}$ for which condition~(\ref{npt}) is satisfied, and this shows that $\mathcal W_{A_1 A'_2A_4}$ is NPT across bipartite cut $A_1|A'_2A_4$. \begin{figure}\label{fig:ppt1}
\end{figure}
The reasoning in Eqs.~(\ref{transfppt})-(\ref{sig}) can be conveniently represented diagrammatically, as shown on Fig.~\ref{fig:ppt1}.
If, in addition, subspace $\mathcal{S}^{(1)}_{A_1A_2}$ is $1$-distillable, then there exists a Schmidt rank $2$ state $\ket{\Phi}_{A_1A_2}$ such that condition~(\ref{partnpt}) is satisfied. Using this state in Eq.~(\ref{gammarepr}), we construct a Schmidt rank $2$ state $\ket{\Gamma}$~(again, after joining $A_2$ and $A_3$) such that condition~(\ref{npt}) is satisfied, thus proving $1$-distillability of $\mathcal W_{A_1 A'_2A_4}$ across bipartite cut $A_1|A'_2A_4$.
The same holds for bipartite cut $A_1A'_2|A_4$~(subspaces $\mathcal{S}^{(1)}_{A_1A_2}$ and ${S}^{(2)}_{A_3A_4}$ enter the lemma symmetrically).
Consider now bipartite cut $A'_2|A_1A_4$. This time we choose the partial transpose to act on joint subsystem $A_1A_4$. This operation reduces to taking transposes on subsystems $A_1$ and $A_4$ independently: $T_{A_1A_4}=T_{A_1}\otimes T_{A_4}$.
For the state $\ket{\Gamma}$ we can take the structure~(\ref{gammarepr}) requiring the state $\ket{\tau}_{A_3A_4}$ to be a product state: \begin{equation}\label{prodstr}
\ket{\tau}_{A_3A_4} = \ket{\mu}_{A_3}\otimes\ket{\nu}_{A_4}, \end{equation} with some pure states $\ket{\mu}_{A_3}\in\mathcal H_{A_3}$ and $\ket{\nu}_{A_4}\in\mathcal H_{A_4}$.
Now, for each term in Eq.~(\ref{dec}), the partial scalar product of $\ket{\tau}$ with the transposed projector $\dyad{\psi_i}$ can be written as \begin{multline} \bra{\tau}\left(\dyad{\psi_i}_{A_1A'_2A_4}\right)^{T_{A_1}\otimes T_{A_4}}\ket{\tau}_{A_3A_4}\\ =\bra{\mu}\otimes\bra{\nu}\left(\dyad{\psi_i}_{A_1A'_2A_4}\right)^{T_{A_1}\otimes T_{A_4}}\ket{\mu}_{A_3}\otimes\ket{\nu}_{A_4}\\ =\bra{\mu}\otimes\bra{\nu^*}\left(\dyad{\psi_i}_{A_1A'_2A_4}\right)^{T_{A_1}}\ket{\mu}_{A_3}\otimes\ket{\nu^*}_{A_4}, \end{multline} where we took advantage of the product structure~(\ref{prodstr}) to eliminate the second transpose operation $T_{A_4}$~(see also Fig.~\ref{fig:ppt2}). Here $\ket{\nu^*}$ denotes the vector with components equal to complex conjugated components of the vector $\ket{\nu}$ with respect to the computational basis. \begin{figure}
\caption{ The second transpose operation, $T_{A_4}$, can be eliminated on a product state $\ket{\tau}_{A_3A_4} = \ket{\mu}_{A_3}\otimes\ket{\nu}_{A_4}$. The rest of the calculations are analogous to those on the diagram of Fig.~\ref{fig:ppt1}.}
\label{fig:ppt2}
\end{figure}
Now it can be easily seen that this case is reduced to the previous one of bipartite cut $A_1|A'_2A_4$ with the state $\ket{\tau}$ replaced with $\ket{\mu}_{A_3}\otimes\ket{\nu^*}_{A_4}$: we can repeat the reasoning starting from Eq.~(\ref{transfppt}) on and obtain that $\mathcal W_{A_1 A'_2A_4}$ is NPT across bipartite cut $A'_2|A_1A_4$. If, in addition, subspace $\mathcal{S}^{(1)}_{A_1A_2}$ is $1$-distillable, then $\mathcal W_{A_1 A'_2A_4}$ is $1$-distillable across $A'_2|A_1A_4$.
When $n>2$, each possible bipartite cut can be analyzed similarly: choosing appropriate product structure of the state $\ket{\Gamma}$, we reduce the case with many transposes acting on different subsystems to the situation where there is only one partial transpose acting on some state that is entirely supported on one of the subspaces $\mathcal S$, then repeat the above reasoning.
\end{proof}
\nocite{*}
\end{document} | arXiv |
Topological Methods in Nonlinear Analysis
On semiclassical ground states for Hamiltonian elliptic system with critical growth
Vol 49, No 1 (March 2017) /
Xianhua Tang
Wen Zhang
Hamiltonian elliptic systems, semiclassical ground states, concentration, critical growth
In this paper, we study the following Hamiltonian elliptic system with gradient term and critical growth: \begin{equation*} \begin{cases} -\epsilon^{2}\Delta \psi +\epsilon b\cdot \nabla \psi +\psi=K(x)f(|\eta|)\varphi+W(x)|\eta|^{2^*-2}\varphi &\hbox{in} \mathbb{R}^{N},\\ -\epsilon^{2}\Delta \varphi -\epsilon b\cdot \nabla \varphi +\varphi=K(x)f(|\eta|)\psi+W(x)|\eta|^{2^*-2}\psi &\hbox{in} \mathbb{R}^{N}, \end{cases} \end{equation*} where $\eta=(\psi,\varphi)\colon \mathbb{R}^{N}\rightarrow\mathbb{R}^{2}$, $K, W\in C(\mathbb{R}^{N}, \mathbb{R})$, $\epsilon$ is a small positive parameter and $b$ is a constant vector. We require that the nonlinear potentials $K$ and $W$ have at least one global maximum. Combining this with other suitable assumptions on $f$, we prove the existence, exponential decay and concentration phenomena of semiclassical ground state solutions for all sufficiently small $\epsilon> 0$
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Vol 49, No 1 (March 2017)
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\begin{document}
\title{The basin of infinity of tame polynomials} \author{Jan Kiwi and Hongming Nie}
\address{Facultad de Matem\'aticas, Pontificia Universidad Cat\'olica de Chile} \email{[email protected]} \address{Institute for Mathematical Sciences, Stony Brook University}
\email{[email protected]} \date{\today} \maketitle \begin{abstract}
Let $\mathbb{C}_v$ be a
characteristic zero algebraically closed field which is complete with respect to a non-Archimedean absolute value. We provide a necessary and sufficient condition for two tame polynomials in $\mathbb{C}_v[z]$ of degree $d \ge 2$ to be analytically conjugate on their basin of infinity. In the space of monic centered polynomials,
tame polynomials with all their critical points in the basin of infinity
form the tame shift locus. We show that a tame map $f\in\mathbb{C}_v[z]$ is in the closure of
the tame shift locus if and only if the Fatou set of $f$ coincides with the basin of infinity. \end{abstract}
\tableofcontents
\section{Introduction}\label{sec:intro}
Let $\mathbb{C}_v$ be a characteristic $0$ algebraically closed field with residue characteristic $p\ge 0$
which is complete with respect to a non-Archimedean absolute value $|\cdot|$, that is assumed to be non-trivial.
We regard nonconstant polynomials $f \in \mathbb{C}_v[z]$ as dynamical systems acting on the Berkovich analytic line $\mathbb{A}^1_{an}$ over $\mathbb{C}_v$. The dynamics of a degree $d \ge 2$ polynomial $f$ partitions $\mathbb{A}^1_{an}$ into two sets: the \emph{basin of infinity} ${\pazocal{B}}(f)$ consisting of all $x \in \mathbb{A}^1_{an}$ with unbounded orbit and, its complement, the \emph{filled Julia set} ${\pazocal{K}}(f)$, which is compact. The theme of this paper is to investigate when the actions of two polynomials on their basins of infinity are analytically conjugate. However, in this introduction we start discussing some consequences of our investigation and finish describing directly related results.
It is well known that the dynamics of a polynomial $f$ is strongly influenced by the dynamical behavior of points $x$ where $f$ fails to be locally injective. The set of such points $x \in \mathbb{A}^1_{an}$ is called the \emph{ramification locus} ${\pazocal{R}}(f) \subset \mathbb{A}^1_{an}$ (c.f.~\cite{Faber13II}). We will focus on the more tractable class of \emph{tame polynomials} consisting on maps $f$ for which ${\pazocal{R}}(f)$ is a locally finite tree (c.f.~\cite{Trucco14}). We say that a tame polynomial lies in the \emph{tame shift locus} if ${\pazocal{R}}(f)$ is contained in the basin of infinity, equivalently, if the critical points of $f$ are contained in the basin of infinity. In this case, the unique Fatou component of $f$ is the basin of infinity and the Julia set ${\pazocal{J}}(f) := \partial {\pazocal{K}}(f)$ is formed by type I points. That is, ${\pazocal{J}}(f)$ is contained in the (classical) affine line $\mathbb{A}^1$. Moreover, $f: {\pazocal{J}}(f) \to {\pazocal{J}}(f)$ is topologically conjugate to the one-sided shift on $d$ symbols, (c.f.~\cite[Theorem 3.1]{Kiwi06}).
In parameter space, for simplicity, we work in the space ${\operatorname{Poly}}_d$ of \emph{monic and centered polynomials of degree $d$} which, via coefficients, is naturally identified with $\mathbb{C}_v^{d-1}$.
Unless otherwise stated, we work with the metric topology in ${\operatorname{Poly}}_d$ induced by the (sup) norm on $\mathbb{C}_v^{d-1}$.
Our investigation regarding the basin of infinity will allow us to characterize the location in parameter space of tame polynomials whose basin of infinity agree with the Fatou set or, equivalently, the Julia set is contained in the classical affine line $\mathbb{A}^1$.
\begin{introtheorem}
\label{ithr:A}
Consider a tame polynomial $f \in {\operatorname{Poly}}_d$.
Then ${\pazocal{J}} (f) \subset \mathbb{A}^1$ if and only if $f$ is in the closure of the tame shift locus. \end{introtheorem}
Note that each polynomial in the tame shift locus is expanding on its Julia set (c.f~\cite{Benedetto01,Lee19}). Hence we show that tame polynomials with ${\pazocal{J}} (f) \subset \mathbb{A}^1$ can be perturbed to maps which, in a certain sense, have hyperbolic (expanding) dynamics over their Julia sets.
When a tame polynomial $f$ is in the closure of the tame shift locus, it is not difficult to deduce from Trucco's work~\cite{Trucco14} that ${\pazocal{J}} (f) \subset \mathbb{A}^1$. Thus, our contribution is to show that Julia critical points (if any) of a tame polynomial $f$ with ${\pazocal{J}} (f) \subset \mathbb{A}^1$ simultaneaously become escaping under an appropriate arbitrarily small perturbation of $f$. We say that a Julia critical point $c \in {\pazocal{J}}(f)$ is \emph{active at $f$} if there exists a polynomial $g$ arbitrarily close to $f$ with a critical point $c'$ close to $c$ such that $c'$ lies in the basin of infinity under iterations of $g$.
(We must warn the reader that our definition of active/passive critical points differs from the one by Irokawa~\cite{Irokawa19}.)
\begin{introcorollary}
\label{icor:B}
Assume that $f \in {\operatorname{Poly}}_d$ is a
tame polynomial such that ${\pazocal{J}}(f) \subset \mathbb{A}^1$.
If a critical point $c$ lies in the Julia set ${\pazocal{J}}(f)$,
then $c$ is active at $f$. \end{introcorollary}
Without any assumption on $f \in {\operatorname{Poly}}_d$, we conjecture that if a critical point $c$ lies in ${\pazocal{J}}(f)$, then $c$ is active at $f$.
That is, we conjecture that bifurcations must occur at maps with Julia critical points. On the other hand, $J$-stability and ``hyperbolicity'' results for maps whose Julia set is critical point free have been obtained by Benedetto and Lee~\cite{Benedetto01,Lee19, Benedetto22}, and T. Silverman~\cite{Silverman17,Silverman19}. Although useful, analogies with open problems and results in complex dynamics should be taken with extreme caution. In fact, the hypothesis that $c$ is a Julia critical point is \emph{stronger} in the non-Archimedean context than in the complex setting since it immediately implies a control on the geometry around $c$ (i.e. there exists a sequence of nested Fatou annuli around $c$ with divergent sum of moduli). So although the ``analogue'' conjecture in complex dynamics has proven to be hard and elusive, even for quadratic polynomials, it might be the case that establishing activity for Julia critical points is more accessible in the non-Archimedean setting.
To study perturbations of a given polynomial it is convenient to consider analytic families of critically marked monic and centered polynomials $\{(f_\lambda, c_1(\lambda), \dots, c_{d-1}(\lambda)) \}$ parametrized by a disk $\Lambda \subset \mathbb{C}_v$ (see \S~\ref{sec:analytic-family}). In contrast with T. Silverman's work, for our purpose, we only consider parameters $\lambda \in \mathbb{C}_v$ and polynomials with coefficients in $\mathbb{C}_v$, that is we do not work with (non-classical) parameters in the Berkovich disk associated to $\Lambda$ (c.f.~\cite{Silverman17,Silverman19}). A critical point $c_i(\lambda)$ is called \emph{passive in $\Lambda$} if either $c_i(\lambda) \in {{\pazocal{B}}(\lambda)}:={\pazocal{B}}(f_\lambda)$ for all $\lambda \in \Lambda$ or $c_i(\lambda) \notin {\pazocal{B}}(\lambda)$ for all $\lambda \in \Lambda$. Analytic conjugacies between actions on basins of infinity must respect critical orbits and agree, up to normalization, with a conjugacy furnished near infinity by the B\"ottcher coordinates. Thus, it is convenient to prescribe the locations of the critical points in ${\pazocal{B}}(\lambda)$ with the aid of the B\"ottcher coordinate $\phi_\lambda$ which conjugates $f_\lambda$ with $z \mapsto z^d$ in a neighborhood of $\infty$ (see \S~\ref{sec:poly}). We say that the \emph{B\"ottcher coordinate of a critical point $c_i(\lambda) \in {\pazocal{B}}(\lambda)$
is constant in $\Lambda$} if $\phi_\lambda(\iter{f_\lambda}{n}(c_i(\lambda)))$ is a constant function of $\lambda \in \Lambda$ for some $n$ sufficiently large. Along analytic families with passive critical points having constant B\"ottcher coordinates, the analytic dynamics in the basin of infinity is constant:
\begin{introtheorem}
\label{ithr:C}
Consider an analytic family $\{(f_\lambda, c_1(\lambda), \dots, c_{d-1}(\lambda)) \}$ of critically marked tame monic and centered polynomials
parametrized by an open disk $ \Lambda \subset \mathbb{C}_v$.
Assume that all critical points are passive and that the B\"ottcher coordinates of escaping critical points are constant in $\Lambda$.
Then for all $\lambda_1, \lambda_2 \in \Lambda$, the maps $f_{\lambda_1}:{\pazocal{B}}(\lambda_1) \to {\pazocal{B}}(\lambda_1)$ and $f_{\lambda_2}:{\pazocal{B}}(\lambda_2) \to {\pazocal{B}}(\lambda_2)$ are analytically conjugate.
Moreover, if in addition, ${\pazocal{J}}(f_{\lambda_0}) \subset \mathbb{A}^1$ for some $\lambda_0\in\Lambda$, then $f_\lambda = f_{\lambda_0}$ for all $\lambda \in \Lambda$. \end{introtheorem}
The moreover part of the statement above is a consequence of the fact from analytic geometry that compact subsets of $\mathbb{C}_v$ are analytically removable (see Theorem~\ref{thr:removability} or \cite[Proposition~2.7.13]{Fresnel04}).\footnote{ This fact is analogous to the removability of absolute measure $0$ subsets of the complex plane, e.g. see~\cite{McMullen94}.} In complex polynomial dynamics DeMarco and Pilgrim~\cite{DeMarco11} studied the map which assigns to each element of moduli space its dynamics on the basin of infinity (modulo analytic conjugacy). In this language, removability implies that, for maps in the closure of the tame shift locus, the analytic dynamics on their basins of infinity determines the map uniquely (modulo affine conjugacy). For a general $f_0 \in {\operatorname{Poly}}_d$ it would be interesting to describe the locus of maps $f \in {\operatorname{Poly}}_d$ whose action on ${\pazocal{B}}(f)$ is analytically conjugate to ${\pazocal{B}}(f_0)$ (c.f.~\cite{DeMarco11}).
Given a tame polynomial $f$, we define the \emph{dynamical core} of $f$ as the smallest forward invariant set ${\pazocal{A}}_f$ containing the non-classical ramification points in ${\pazocal{B}}(f)$. It is not difficult to show that ${\pazocal{A}}_f$ is a locally finite tree. Then, given any pair of tame polynomials $f$ and $g$ in ${\operatorname{Poly}}_d$ we introduce the notion of an \emph{extendable conjugacy} $h: {\pazocal{A}}_f \to {\pazocal{A}}_g$. Loosely speaking, an extendable conjugacy is a diameter preserving isometric conjugacy that is locally a translation and agrees with a B\"ottcher coordinate change near infinity.
For a precise definition see
\S~\ref{sec:conjugacy}.
Analytic geometry will imply that, modulo a root of unity, an
analytic conjugacy
$\varphi: {\pazocal{B}}(f) \to {\pazocal{B}}(g)$ restricts to an
extendable conjugacy $h: {\pazocal{A}}_f \to {\pazocal{A}}_g$ and, conversely, via an analytic continuation argument we prove that extendable conjugacies upgrade to analytic conjugacies:
\begin{introtheorem}
\label{ithr:D}
Let $f, g \in {\operatorname{Poly}}_d$ be tame polynomials.
There exists an extendable conjugacy between $f: {\pazocal{A}}_f \to {\pazocal{A}}_f$ and $g: {\pazocal{A}}_g \to {\pazocal{A}}_g$ if and only if there exists an analytic conjugacy
between $f: {\pazocal{B}} (f) \to {\pazocal{B}}(f)$ and $g: {\pazocal{B}} (g) \to {\pazocal{B}}(g)$
which
extends the B\"ottcher coordinate change. \end{introtheorem}
Let us now give an overview of the paper.
Section~\ref{sec:preliminaries} is devoted to preliminaries. We start summarizing basic facts and notation related to the Berkovich affine line $\mathbb{A}^1_{an}$ in \S~\ref{sec:berkovich-space}. The space of monic and centered polynomials ${\operatorname{Poly}}_d$ is introduced in \S~\ref{sec:poly}. Here we also discuss basic (dynamical) objects associated to a given polynomial such as the \emph{base point} and \emph{B\"ottcher coordinates}, and introduce the \emph{dynamical core}. Analytic families of polynomials is the topic discussed in \S~\ref{sec:analytic-family} with emphasis on definitions such as passive/active critical points and constant B\"ottcher coordinates.
Section~\ref{sec:extendable} is devoted to the proof of Theorem~\ref{ithr:D}. Extendable conjugacies are defined in~\S~\ref{sec:conjugacy}. In \S~\ref{sec:analytic2extendable}, one direction of Theorem~\ref{ithr:D} is established, namely that analytic conjugacies restrict to extendable conjugacies between dynamical cores (Corollary~\ref{c:ithr-D-easy}). The other direction is obtained via an ``analytic continuation'' argument in \S~\ref{s:maps-annuli} which employs a lemma proven in Appendix~\ref{appendix}.
In Section~\ref{sec:existence} we prove a slightly stronger version of Theorem~\ref{ithr:C} which relies on the notion of $\rho$-close B\"ottcher coordinates. The stronger version of Theorem~\ref{ithr:C} will be needed to establish Theorem~\ref{ithr:A}. Intuitively, given two polynomials $f, g$ we introduce $\rho \in ]0,\infty]$ to quantify how close are the B\"ottcher coordinates of escaping critical points. With this notion, $\rho=\infty$ corresponds to having the same B\"ottcher coordinates. Most of the proof is in \S~\ref{sec:passive2extendable} after we introduce the $\rho$-trimmed dynamical core in~\S~\ref{sec:trimmed} and use it to define the notion of $\rho$-close B\"ottcher coordinates in~\S~\ref{sec:close}.
Section~\ref{sec:rigidity} is devoted to prove Theorem~\ref{ithr:A} and Corollary~\ref{icor:B}. In \S~\ref{sec:bounded} we deduce from Trucco's work \cite{Trucco14} that tame polynomials in the closure of the tame shift locus do not have bounded Fatou components. In \S~\ref{sec:perturbation}, we finish the proof of the aforementioned results constructing suitable analytic families to perturb into the shift locus a tame polynomial $f$ such that ${\pazocal{J}}(f) \subset \mathbb{A}^1$.
{
\section{Preliminaries} \label{sec:preliminaries} For future reference we introduce basic definitions, notation and results regarding Berkovich space {and analytic maps} in \S~\ref{sec:berkovich-space}, regarding the dynamics of monic and centered polynomials in \S~\ref{sec:poly} and, regarding analytic families of such polynomials in \S~\ref{sec:analytic-family}.
\subsection{Berkovich space {and analytic maps}} \label{sec:berkovich-space} The elements of $\mathbb{A}^1_{an}$ are the multiplicative seminorms in $\mathbb{C}_v [z]$
that extend the absolute value $| \cdot |$ on $\mathbb{C}_v$. As customary we write elements of $\mathbb{A}^1_{an}$ simply by $x$ and, when necessary, the corresponding seminorm of
$f(z) \in \mathbb{C}_v [z]$ is denoted by $\| f(z) \|_x$. Also, to ease notations, let
$|x| := \|z\|_x$. Via the identification of $ x\in \mathbb{A}^1$ with the seminorm
$\|f(z)\|_x = |f(x)|$, we may regard $\mathbb{A}^1$ as a subset of $\mathbb{A}^1_{an}$. Most of our work occurs in $\mathbb{A}^1_{an}$, however sometimes it is convenient to regard $\mathbb{A}^1_{an}$ as naturally embedded in the Berkovich projective line $\mathbb{P}^1_{an}$ which (set theoretically) is obtained by adding one point, denoted by $\infty$, to $\mathbb{A}^1_{an}$. The default topology for $\mathbb{A}^1_{an}$ and $\mathbb{P}^1_{an}$ will be the Gel'fand topology. With this topology $\mathbb{A}^1_{an}$ is locally compact and $\mathbb{P}^1_{an}$ is the one-point compactification of $\mathbb{A}^1_{an}$. Moreover, $\mathbb{A}^1$ and $\mathbb{P}^1$ are dense in $\mathbb{A}^1_{an}$ and $\mathbb{P}^1_{an}$, respectively. We refer the reader to~\cite{Baker10, Benedetto19} for general background on the Berkovich affine and projective line specially adapted to one dimensional non-Archimedean dynamics.
Given $x \in \mathbb{A}^1_{an}$, let
$$\overline{D}_x := \left\{ y \in \mathbb{A}^1_{an} : \|f(z)\|_y \le \|f(z)\|_x, \,\, \forall f(z) \in \mathbb{C}_v[z] \right\}.$$ According to Berkovich's classification of seminorms, $\overline{D}_x \cap \mathbb{A}^1$ is
a (possibly degenerate) $\mathbb{C}_v$-closed disk of radius $r \ge 0$ or the empty set. If $r=0$ (resp. $r \in |\mathbb{C}_v^\times|$, $r \notin |\mathbb{C}_v^\times|$), then $x$ is called a type I (resp. II, III) point. When $\overline{D}_x \cap \mathbb{A}^1$ is empty, $x$ is called a type IV point. The \textit{Gauss point}, denoted by $x_G$, is the unique point whose associated disk
$\overline{D}_{x_G}$ is $\{ z \in \mathbb{C}_v : |z| \le 1 \}$. It follows that $x_G$ is of type II.
Any point in $\mathbb{A}^1_{an}$ can be represented by the cofinal equivalence class of a decreasing sequence of $\mathbb{C}_v$-closed disks, so $\mathbb{A}^1_{an}$ is equipped with a partial order $\prec$ that extends the inclusion relation on the $\mathbb{C}_v$-closed disks, in particular, for $x_1,x_2\in\mathbb{A}^1_{an}$ of type I, II or III, $x_1\prec x_2$ if and only if $\overline{D}_{x_1} \cap \mathbb{A}^1\subset\overline{D}_{x_2} \cap \mathbb{A}^1$. The partial order $\prec$ extends to $\mathbb{P}^1_{an}$ by declaring $\infty$ to be the unique maximal element in $\mathbb{P}^1_{an}$. This partial order $\prec$ endows $\mathbb{P}^1_{an}$ and $\mathbb{A}^1_{an}$ with a tree structure. Given two distinct points $y_1,y_2\in\mathbb{P}^1_{an}$, denote by $[y_1,y_2]$ (resp. $]y_1,y_2[$) the closed (resp. open) segment in $\mathbb{P}^1_{an}$ connecting $y_1$ and $y_2$.
For $a \in \mathbb{C}_v$ and $r >0$, the \emph{Berkovich open and closed disks} with radii $r$ containing $a$, respectively are \begin{eqnarray*}
D(a,r) & : = & \{ x \in \mathbb{A}^1_{an} : \| z -a \|_x < r \}, \\
\overline{D}(a,r) & : = & \{ x \in \mathbb{A}^1_{an} : \| z -a \|_x \le r \}. \end{eqnarray*}
For any $x\in\mathbb{A}^1_{an}$, after defining the \emph{diameter} of $x$ by
$$\mathrm{diam}(x) := \inf \{ \| z - a \|_x : a \in \mathbb{C}_v \},$$
we have $\overline{D}_x= \overline{D}(a,\mathrm{diam}(x))$ for all $a\in\overline{D}_x \cap \mathbb{A}^1$ if $x$ is of type I, II, or III, and therefore $\mathrm{diam}(x)$ coincides with the diameter $r$ of $\overline{D}_x \cap \mathbb{A}^1$; in this case we write $x=x_{a,r}$. If $x$ is of type IV, then $\mathrm{diam}(x)>0$, while $\overline{D}_x \cap \mathbb{A}^1$ is empty.
The space ${\mathbb{H}}:=\mathbb{A}^1_{an} \setminus \mathbb{A}^1$ is equipped with a metric $\mathrm{dist}_\H$ called \emph{the hyperbolic metric} which is invariant under affine
transformations. With respect to this metric ${\mathbb{H}}$ is an $\mathbb{R}$-tree. The subspace topology in ${\mathbb{H}}$ is strictly coarser than the metric topology. We normalize $\mathrm{dist}_\H$ so that $$\mathrm{dist}_\H(x_{0,r}, x_{0,s}) = \log (s/r)$$ for all $0<r<s$.
We say that a closed and connected subset $\mathbb{A}^1_{an}$ or $\mathbb{P}^1_{an}$ is a \emph{subtree}. Unless explicitly stated, a subtree is equipped with neither vertices nor edges. However, for a subtree $\Gamma\subset\mathbb{A}^1_{an}$, the \emph{valence} of $\Gamma$ at $x\in\Gamma$ is the number (maybe $\infty$) of connected components of $\Gamma\setminus\{x\}$. We denote by $\partial\Gamma$ the set of points in $\Gamma$ having valence $1$. A nonempty connected subset $S\subset\mathbb{A}^1_{an}$ is an \emph{open subtree} if $S=\overline{S}\setminus\partial\overline{S}$; so in this case $S$ is contained in $\mathbb{H}$ and any point in $S$ has valence at least $2$, moreover, for simplicity, set $\partial S:=\partial\overline{S}$. An (open) subtree in $\mathbb{A}^1_{an}$ is \emph{locally finite} if each point has finite valence and a neighborhood containing finitely many branch points of this (open) subtree. Given an open and connected set $U \subsetneq \mathbb{A}^1_{an}$, the \emph{skeleton} of $U$, denoted by $\operatorname{sk} U$, is the set formed by all $x \in U$ that separate the complement of $U$ in $\mathbb{P}^1_{an}$. It follows that $\operatorname{sk} U$ is a locally finite open subtree in $\mathbb{H}$.
Given a point $x \in \mathbb{A}^1_{an}$, a connected component of $\mathbb{A}^1_{an} \setminus \{x\}$ corresponds to a \emph{direction $\vec{v}$ at $x$}. Denote by $D_x(\vec{v})$ the component of $\mathbb{A}^1_{an} \setminus \{x\}$ corresponding to the direction $\vec{v}$ at $x$. When the point $x$ is clear from context or irrelevant, we will sometimes write $D(\vec{v})$ for $D_x(\vec{v})$. The set of all directions at $x$ forms the \emph{space of directions or (projectivized) tangent space at $x$}, denoted by $T_x \mathbb{A}^1_{an}$. It will also be convenient to denote the unique direction at $x$ `containing' $z\in\mathbb{A}^1_{an}$ by $\vec{v}_x(z)$ and denote by $\vec{v}_x(\infty)$ the unique direction at $x$ such that $D(\vec{v}_x(\infty))$ is unbounded. At type I and IV points, the tangent space is a singleton and at type III, consists of two directions. However, at type II points the tangent space is naturally endowed with the structure of a projective line over the residue field of $\mathbb{C}_v$ with a distinguished point $\infty$.
Given an open set $U \subset \mathbb{A}^1_{an}$, an \emph{analytic map} $\psi:U \to \mathbb{A}^1_{an}$ is a morphism between the corresponding analytic structures. Analytic maps can be described in a more concrete manner as follows. An \emph{affinoid} $X$ in $\mathbb{A}^1$ is a subset of the form $\overline{B}(a,r) \setminus \cup B(a_i,r_i)$ where $r, r_i \in |\mathbb{C}_v^\times|$ and $a, a_i \in \mathbb{A}^1$. The corresponding Banach $\mathbb{C}_v$-algebra ${\pazocal{A}}(X)$ is formed by the uniform limits of rational maps with poles outside $X$ endowed with the sup-norm $\|\cdot\|_X$ on $X$. Then $X_{an}$ is the space of all multiplicative seminorms on
${\pazocal{A}}(X)$ bounded by $\|\cdot\|_X$ that agree with $|\cdot|$ on the constants, endowed with the Gel'fand topology. Restriction to $\mathbb{C}_v[z]$ identifies $X_{an}$
with $\overline{D}(a,r) \setminus \cup D(a_i,r_i) \subset \mathbb{A}^1_{an}$. Analytic maps $\psi: X_{an} \to \mathbb{A}^1_{an}$ are naturally identified with ${\pazocal{A}}(X)$. That is, given $\varphi \in {\pazocal{A}}(X)$, $x \in X_{an}$ and $q(z) \in \mathbb{C}_v[z]$, then $\|q(z)\|_{\varphi(x)} = \|q(\varphi(z))\|_x$ defines a morphism $\varphi: X_{an} \to \mathbb{A}^1_{an}$. It follows that for any given arbitrarily large hyperbolic ball in ${\mathbb{H}}$, the action of $\varphi: X_{an} \to \mathbb{A}^1_{an}$ agrees with that of a rational map with poles outside $X$. Given an open set $U \subset \mathbb{A}^1_{an}$, we have that $\psi: U \to \mathbb{A}^1_{an}$ is analytic if its restriction to every affinoid $X$ contained in $U$ is analytic. Assuming that $U$ is connected, nonconstant analytic maps $\psi: U \to \mathbb{A}^1_{an}$ are open maps with isolated zeros. Therefore, analytic maps that coincide on an open subset of $U$ are equal. Furthermore, we have the following analytic removability result (c.f \cite[Proposition~2.7.13]{Fresnel04}):
\begin{theorem}\label{thr:removability}
Assume that $K \subset \mathbb{A}^1$ is a compact set contained in a disk $D \subset \mathbb{A}^1_{an}$.
If $f: D \setminus K \to \mathbb{A}^1_{an}$ is a bounded analytic map, then
$f$ extends to a analytic map $\bar{f}: D \to \mathbb{A}^1_{an}$. \end{theorem}
Assume that $U$ is an open set containing $x \in \mathbb{A}^1_{an}$ and $\psi: U \to \mathbb{A}^1_{an}$ is a non-constant analytic map. Then $\psi$ has a well defined local degree at every $x \in U$ which we denote by $\deg_x \psi$. Write $y:= \psi(x)$, and define
the \emph{tangent map}
$T_x\psi: T_x \mathbb{A}^1_{an} \to T_y \mathbb{A}^1_{an}$ of $\psi$ at $x$
by the property that there exists a neighborhood
$V$ of $x$ such that $\psi(D_x(\vec{v}) \cap V) \subset D_{y} (T_x\psi(\vec{v}))$ for any $\vec{v}\in T_x \mathbb{A}^1_{an}$. The direction $T_x\psi(\vec{v})\in T_y\mathbb{A}^1_{an}$ exists and it is independent of the choice of $V$. When $x$ is a type II point, the map $T_x\psi$ is a non-constant rational map over the residue field of $\mathbb{C}_v$. Moreover, the local degree $\deg_x\psi$ of $\psi$ at $x$ coincides with the degree of $T_x\psi$.
{An analytic map is piecewise linear with respect to the hyperbolic metric. The scaling factors or ``slopes'' are determined by the local degrees (c.f. \cite[Theorem 9.35]{Baker10}):}
\begin{lemma}\label{lem:factor}
Let $\psi$ be a non-constant analytic map on an open set $U\subset \mathbb{A}^1_{an}$. If $\psi$ has constant local degree on a segment $]x_1,x_2[\subset U$, then for any $x\in]x_1,x_2[$
$$\mathrm{dist}_\H(\psi(x_1),\psi(x_2))=\deg_x\psi\cdot\mathrm{dist}_\H(x_1,x_2).$$
\end{lemma}
\subsection{Polynomial dynamics}\label{sec:poly}
We denote by ${\operatorname{Poly}}_d$ the space of \emph{monic and centered polynomials} of degree $d \ge 2$ which is naturally identified via coefficients with $\mathbb{C}_v^{d-1}$, since the elements of ${\operatorname{Poly}}_d$ are of the form $$f(z) = z^d + a_{d-2} z^{d-2} + \cdots + a_{0} \in \mathbb{C}_v[z].$$ Observe that every degree $d$ polynomial is affinely conjugate to a unique element of ${\operatorname{Poly}}_d$, modulo conjugacy given by multiplication by a $(d-1)$-th root of unity.
For $f$ as above, $$f'(z) = d \cdot \prod_{i=1}^{d-1} (z -c_i),$$ where $\sum_{i=1}^{d-1} c_i =0$ and we say that $(f,c_1, \dots, c_{d-1})$ is \emph{a critically marked polynomial}. In fact, $\operatorname{Crit}(f) := \{ c_1, \dots, c_{d-1} \}$ is the set of critical points of $f$. Two critically marked polynomials $(f,c_1,\dots,c_{d-1})$ and $(g, c_1', \dots, c_{d-1}')$ are \emph{affinely conjugate} if there exists an affine map $A$ such that $ A \circ f = g \circ A$ and $A(c_i) = c_i'$ for all $i$. When the marking is clear from context, sometimes we simply say that $f$ is a critically marked polynomial.
Recall that the \emph{ramification locus} of $f\in {\operatorname{Poly}_d}$ is $${\pazocal{R}}(f):=\{x\in\mathbb{A}^1_{an}: \deg_xf\ge 2\}.$$ According to Faber~\cite{Faber13} this set is unbounded, connected and $\operatorname{Crit}(f) = {\pazocal{R}}(f)\cap\mathbb{A}^1$. If ${\pazocal{R}}(f)\cap{\mathbb{H}}$ is a locally finite open subtree, then we say that $f$ is a \emph{tame polynomial}. In this case, ${\pazocal{R}}(f)\cup\{\infty\}$ is the convex hull of $\operatorname{Crit}(f)\cup\{\infty\}$ in $\mathbb{P}^1_{an}$. Let $p$ be the residue characteristic of $\mathbb{C}_v$. If $p=0$, then all polynomials are tame. If $p\neq 0$, then $f$ is tame if and only if for all $x \in \mathbb{A}^1_{an}$ we have $p \nmid \deg_x f$ or, equivalently, the degree of $f: D \mapsto f(D)$ is not divisible by $p$ for all disks $D \subset \mathbb{A}^1$. A thorough study of ${\pazocal{R}}(f)$ is contained in~\cite{Faber13, Faber13II}.
For tame polynomials, we have the following Riemann-Hurwitz formula: \begin{lemma}[{\cite[Propositions 2.6 and 2.8]{Trucco14}}]\label{lem:RH} Let $f\in {\operatorname{Poly}_d}$ be a tame polynomial. For any $x\in\mathbb{A}^1_{an}$, $$\deg_xf-1=\sum_{z\in\overline{D}_x\cap\operatorname{Crit}(f)}(\deg_z f-1)$$ \end{lemma}
From \cite{Faber13II} we deduce below that perturbations preserving multiplicities of critical points preserve tameness. Thus, sometimes it will be convenient to work, in the corresponding parameter space. Namely, consider $k \ge 1$ and $\mathbf{d}:=(d_1, \dots, d_k)$ where $d_i \ge 2$ for all $i$ and $$d_1+\cdots+d_k -k = d-1.$$ We denote by ${\operatorname{Poly}}(\mathbf{d})$ the space formed by all the pairs $(f,c_1,\dots,c_k)$ where $f \in {\operatorname{Poly}_d}$, $c_i \in \mathbb{A}^1$ and $\deg_{c_i} f = d_i$ for all $i=1, \dots,k$. The space ${\operatorname{Poly}}(\mathbf{d})$ is naturally identified via $(f,c_1,\dots,c_k) \mapsto (c_1,\dots, c_k, f(0))$ with the space formed by the elements $(c_1,\dots,c_k,b)$ of $ \mathbb{C}_v^k \times \mathbb{C}_v$ such that $\sum d_i c_i =0$ and $c_i \neq c_j$ if $i \neq j$. Sometimes we abuse of notation and simply say that $f \in {\operatorname{Poly}}(\mathbf{d})$ and denote by $c_i(f)$ the corresponding marked critical points.
\begin{lemma}\label{lem:tame} Tame polynomials are an open {(possibly empty)} subset of ${\operatorname{Poly}}(\mathbf{d})$ for any $\mathbf{d}$. \end{lemma} \begin{proof} {Let us assume that there exists a tame polynomial in ${\operatorname{Poly}}(\mathbf{d})$.} First observe that if $k=1$, then the elements of ${\operatorname{Poly}}(\mathbf{d})$
are unicritical polynomials which are tame if and only if $p \nmid d$.
So we may assume that $k\ge 2$ and let $f \in {\operatorname{Poly}}(\mathbf{d})$
be a tame polynomial. Set $d_0 =d$ and $r_i =\min\{|\binom{d_i}{\ell}|:0\le \ell\le d_i\}$ for $i=0, \dots, k$. Note that $d_i\ge 2$ and $0<r_i\le 1$.
Let $z_0$ be a type II point of sufficiently large diameter so that
all the critical points of $f$ are contained in ${D}_{z_0}$. From the Riemman-Hurwitz formula, it follows that $\deg_{z_0} f =d$.
Consider an open $\mathrm{dist}_\H$-disk $B\subset{\mathbb{H}}$ of finite radius $R$ centered at $z_0$ such that $B$ contains all the branch points of ${\pazocal{R}}(f)$ and for all $1\le i\le k$, there exist
$x_i, z_i \in ]c_i(f),z_0] \cap B$ such that $c_i(f)$ is the unique critical point of $f$ in $\overline{D}_{z_i}$ and $x_i \prec z_i$ with $\mathrm{diam}(x_i) < r_i \mathrm{diam}(z_i)$. We may also assume that $R > 1/r_0$ and pick $x _0 \in B$ such that
$x_0 \succ z_0$ and $\mathrm{diam}(x_0) > \mathrm{diam}(z_0)/r_0$.
Consider a neighborhood ${\pazocal{U}}\subset{\operatorname{Poly}}(\mathbf{d})$ of $f$ such that for all $g \in {\pazocal{U}}$ we have $g(x)=f(x)$ for all $x \in B$ and $|c_i(g) - c_i(f)| < \mathrm{diam}(x_i)$ for all $i\ge 1$. It follows that ${\pazocal{R}}(g) \cap B = {\pazocal{R}}(f) \cap B$. Moreover, $c_i(g)\prec x_i$ is the unique critical point of $g$ in $\overline{D}_{z_i}$ for all $i\ge 1$. We now compute the local degrees of $g\in{\pazocal{U}}$ at points in $\overline{D}_{x_i}$ applying \cite[Section 4]{Faber13II}. Observing that $d_i=\deg_{z_i}f=\deg_{z_i}g$ and $\deg_{c_i(g)}g=d_i$, we conclude that $\deg_yg=d_i$ for all $y\in[c_i(g),z_i]$. Changing coordinates on the source and target by affine maps, we may assume that $c_i(g)=0=g(c_i(g))$ and $z_i=x_G$, which implies that $x_i=x_{0,s_i}$ with $s_i< r_i$. Then $0$ is the unique critical point of $g$ in $\overline{D}_{x_G}$ and $\deg_yg=d_i$ for all $y\in[0,x_G]$. Since $p\nmid d_i$, the conclusion in \cite[Section 4.1]{Faber13II} implies that ${\pazocal{R}}(g)\cap\overline{D}_{x_{0,s_i}}=[0,x_{0,s_i}]$. Let $\overline{D} = \mathbb{A}^1_{an} \setminus D_{x_0}$. Via the change of coordinates of the form $z \mapsto 1/(az)$, we may also conclude that ${\pazocal{R}}(g) \cap \overline{D} = [x_0,\infty[$. Now let $ V= B \cup \overline{D} \cup\left(\bigcup_{1\le i\le d-1} \overline{D}_{x_i}\right)$. Then ${\pazocal{R}}(g) \subset V$, since ${\pazocal{R}}(g)$ is connected.
Hence, ${\pazocal{R}}(g)$ is the union of the arcs $[c_i(g),\infty[$ for $1 \le i \le k$. Therefore, $g$ is tame for all $g \in {\pazocal{U}}$. \end{proof}
{\begin{remark} The space ${\operatorname{Poly}}(\mathbf{d})$ may contain no tame polynomials if $\mathbb{C}_v$ has positive residue characteristic $p>0$. For example, if an entry $d_i$ of $\mathbf{d}$ is divisible by $p$, then all polynomials in ${\operatorname{Poly}}(\mathbf{d})$ are not tame. \end{remark}}
Given $f \in {\operatorname{Poly}_d}$, the \emph{basin of infinity} of $f$ $${\pazocal{B}}(f) := \{ x \in \mathbb{A}^1_{an} : f^{\circ n} (x) \to \infty \}$$ is an open, connected and unbounded subset of $\mathbb{A}^1_{an}$. Its complement $${\pazocal{K}}(f) := \mathbb{A}^1_{an} \setminus {\pazocal{B}}(f)$$ is the \emph{filled Julia set} of $f$ whose boundary $${\pazocal{J}}(f) := \partial {\pazocal{K}}(f),$$ is the \emph{Julia set} of $f$. The Fatou set is ${\pazocal{F}}(f) := \mathbb{A}^1_{an} \setminus {\pazocal{J}}(f)$. Every connected component of ${\pazocal{K}}(f)$ is either a point or a closed disk. In the latter case, every maximal open disk contained in the component of ${\pazocal{K}}(f)$ is a bounded Fatou component.
Let $${\pazocal{R}}_\infty (f) := ({\pazocal{R}}(f)\cap{\pazocal{B}}(f)) \setminus \operatorname{Crit}(f)$$ be the set formed by the non-classical ramification points in the basin of inifinity. We say that the forward invariant set $${\pazocal{A}}_f:=\bigcup_{j\ge 0}f^{\circ j} ({\pazocal{R}}_\infty(f))$$ is the \emph{dynamical core of $f$ in ${\pazocal{B}}(f)$}. If in addition we assume that $f$ is tame, then after denoting the postcritical set by $$\operatorname{Post}(f):=\bigcup_{j\ge 0}f^{\circ j}(\operatorname{Crit}(f))$$ and considering the open and connected subset of $\mathbb{A}^1_{an}$ $${\pazocal{B}}_0(f):={\pazocal{B}}(f)\setminus\operatorname{Post}(f),$$ we have that $${\pazocal{A}}_f \subset \operatorname{sk}{\pazocal{B}}_0(f).$$ Thus, the dynamical core ${\pazocal{A}}_f$ of a tame polynomial $f$ is a locally finite open subtree of ${\mathbb{H}}$ and any point in $\partial{\pazocal{A}}_f$ is either a postcritical point in ${\pazocal{B}}(f)$ or a point in ${\pazocal{J}}(f)\cap{\mathbb{H}}$.
Given a monic and centered polynomial $f(z) = a_0 + a_1 z + \cdots + z^d$
let $$R_f := \max \{ 1, |a_i|^{1/(d-i)}: i=0,\dots, d-2 \}.$$ Then $$x_f := x_{0, R_f}$$ is called the \emph{base point of $f$}. The base point is characterized by the fact that $\overline{D}_{x_f}$ is the minimal closed Berkovich disk containing ${\pazocal{K}}(f)$, see \cite[Section 6.1]{Rivera00}. It follows that $\deg_{x_f}f=d$ and
$x_f\in{\pazocal{R}}(f)$. If moreover $f$ is tame, then every $c\in\operatorname{Crit}(f)$ is contained in $\overline{D}_{x_f}$. Keep in mind that $R_f = |x_f|$.
A polynomial $f\in{\operatorname{Poly}}_d$ is \emph{simple} if its Julia set is a singleton; {otherwise, we say $f$ is \emph{nonsimple}. For simple $f\in{\operatorname{Poly}}_d$,} the unique point in ${\pazocal{J}}(f)$ is the Gauss point which is also the base point of $f$ and its dynamical core is the open segment connecting the base point to $\infty$. Moreover, a tame polynomial is simple if and only if its basin of infinity is critical point free. {If $f\in{\operatorname{Poly}}_d$ is nonsimple, } the base point $x_f$ lies in ${\pazocal{B}}(f)$ and hence in ${\pazocal{A}}_f$, see \cite[Proposition 4.3]{Trucco14}.
In an appropriate coordinate near $\infty$, called the \emph{B\"ottcher coordinate}, a degree $d$ monic and tame polynomial $f$ acts as the monomial $z \mapsto z^d$. More precisely, given a tame $f \in {\operatorname{Poly}_d}$ there exists an analytic isomorphism, $$\phi_f : \mathbb{A}^1_{an} \setminus \overline{D} (0,R_f) \to \mathbb{A}^1_{an} \setminus \overline{D} (0,R_f)$$ such that $\phi_f\circ f = \phi_f^d$ and $\phi_f(z)/z \to 1$ as $\mathbb{A}^1 \ni z \to \infty$
(e.g.~see~\cite{Ingram13,DeMarco19,Salerno20}). The germ of $\phi_f$ at infinity is uniquely determined by the above properties. It follows that $|\phi_f (z)| = |z|$ for all $z \in \mathbb{A}^1_{an} \setminus \overline{D} (0,R_f)$. Although $\phi_f$ does not extend to ${\pazocal{B}}(f)$, its absolute value
$|\phi_f|$ extends via the functional equation $|\phi_f|(z)= |\phi (f(z))|^{1/d}$ to a well defined function on the whole basin of infinity ${\pazocal{B}}(f)$. Provided that $f \in {\operatorname{Poly}}_d$ is tame and nonsimple, it is not difficult to show that \begin{equation}
\label{eq:diameter}
R_f = \max \{ |\phi_f| (c) : c \in \operatorname{Crit}(f) \cap {\pazocal{B}}(f) \} {>1,} \end{equation} and
{$$R_f^d = \max \{ |\phi_f| (f(c)) : c \in \operatorname{Crit}(f) \cap {\pazocal{B}}(f) \},$$ see \cite[Proposition 3.4]{Trucco14}.}
To end this subsection, we show that B\"ottcher coordinates $\phi_f$ depend analytically on $f$. We mention here that such analyticity
appears in \cite[Proposition 2.13]{Favre22} described in terms of Green function. To be more precise, let us introduce
the Tate algebra associated to a polydisk (e.g. see~\cite{Fresnel04}).
For $n \ge 1$, considering
$\mathbf{r}=(r_1,\dots,r_n)$ where $r_i \in |\mathbb{C}_v^\times|$ for all $i$, we denote by $P(\mathbf{r})$ the rational polydisk in $\mathbb{C}_v^n$ formed by all
$(\lambda_1, \dots, \lambda_n) \in \mathbb{C}_v^n$ such that $|\lambda_i| \le r_i$ for all $i$; then the \emph{Tate algebra} $T_n[\mathbf{r}]$ is
the one formed by all (formal) power series in $\mathbb{C}_v \llbracket \lambda_1, \dots, \lambda_n \rrbracket$ convergent in $P(\mathbf{r})$. Endowed with the sup norm, the Tate algebra $T_n[\mathbf{r}]$ is a Banach algebra over $\mathbb{C}_v$ \cite[Theorem 3.2.1]{Fresnel04}. Recall that $p\ge 0$ denotes the residue characteristic of $\mathbb{C}_v$. \begin{lemma}\label{lem:B-analytic}
Assume that $p\nmid d$ and pick $\mathbf{r}=(r_1,\dots,r_{d-1})$ with $r_i \in |\mathbb{C}_v^\times|$ for all $1\le i\le d-1$. Suppose that there exists $R\in |\mathbb{C}_v^\times|$ with $R>1$ such that $|x_f|<R$ for any $f\in P(\mathbf{r})\subset{\operatorname{Poly}}_d\cong\mathbb{C}_v^{d-1}$. Then the map
$$\Phi:P(\mathbf{r})\times\{|z|\ge R\}\to\{|z|\ge R\},$$
sending $(f,z)$ to $\phi_f(z)$ is analytic.
More precisely, setting $\Phi_\infty(f,z)=\Phi(f,z)/z$ we have $\Phi_\infty(f,1/z)\in T_d[\tilde{\mathbf{r}}]$ where $\tilde{\mathbf{r}}=(r_1,\dots,r_{d-1},R^{-1})$.
\end{lemma}
\begin{proof}
First it follows from \cite[Lemma 8]{Ingram13} the map $\Phi$ is well-defined. Now to prove the analyticity, we apply a standard convergence argument as in \cite[Lemma 7]{Ingram13}. For given $f\in P(\mathbf{r})$ and $|z| \ge R$, define $\Phi_n(f,z)$ to be the $d^n$-th rood of $f^{\circ n}(z)/z^{d^n}$ such that $\Phi_n(f,z)-1\in\mathbb{C}_v\llbracket z^{-1}\rrbracket $. According to~\cite{Ingram13}$\Phi_n(f,z) \to \phi_f(z)/z$ as $n \to \infty$. The lemma will follow after showing that the convergence is uniform.
Since $p\nmid d$,
$$\left|\Phi_{n+1}(f,z)-\Phi_n(f,z)\right|=\left|\left(\Phi_{n+1}(f,z)\right)^{d^{n+1}}-\left(\Phi_n(f,z)\right)^{d^{n+1}}\right|.$$
It follows that
\begin{align*}
\left|\Phi_{n+1}(f,z)-\Phi_n(f,z)\right|&=\left|\frac{f^{\circ{n+1}}(z)}{z^{d^{n+1}}}-\left(\frac{f^{\circ n}(z)}{z^{d^n}}\right)^d\right|=\left|z^{-d^n}\right|\left|f(f^{\circ n}(z))-f^{\circ n}(z)\right|\\
&=\left|z^{-d^{n+1}}\right|\left|\sum_{j=0}^{d-2}a_j(f)\left(f^{\circ n}(z)\right)^j\right|,
\end{align*}
where $a_j(f)$ is the coefficients of $z^j$ in $f$.
Let $\| f \|:=\max_{0\le j\le d-2}\{|a_j(f)|\}$ and $M:=\sup\{||f||: f\in P(\mathbf{r})\}<\infty$, we conclude that for sufficiently large $n$,
\begin{equation*}
\left|\Phi_{n+1}(f,z)-\Phi_n(f,z)\right| \le\left|z\right|^{-d^{n+1}} \cdot \| f\| \cdot |z|^{d^n(d-2)} = |z|^{-2d^n} \cdot \| f\| \le MR^{-2d^n}.
\end{equation*}
Note that $z \Phi_n (f,z) \to \phi_f(z)$ for all $f \in P(\mathbf{r})$ and $|z| \ge R$.
Moreover, $\{\Phi_{n}(f,1/z) \}$ is a Cauchy sequence in the Banach algebra $T_d[\tilde{\mathbf{r}}]$
and therefore its limit $\Phi_\infty(f,1/z)$ is analytic.
\end{proof}
\subsection{Analytic family} \label{sec:analytic-family}
We consider analytic families parametrized by an open disk $\Lambda$ in $\mathbb{C}_v$ of radius in $|\mathbb{C}_v^\times|$.
A family $\{f_\lambda\}_{\lambda\in\Lambda}\subset{\operatorname{Poly}}_d$ is \emph{analytic} if $$f_\lambda(z) = z^d + \sum_{i=0}^{d-2} a_i (\lambda) z^i$$ where $a_i(\lambda)$ is analytic for all $i$ (i.e. a power series in $\lambda$ which converges for all $\lambda \in \Lambda$). In this case, we say that $\{(f_\lambda,c_1(\lambda), \dots, c_{d-1}(\lambda))\}$ is a \emph{critically marked
analytic family} if $(f_\lambda,c_1(\lambda), \dots, c_{d-1}(\lambda))$ is a critically marked polynomial for all $\lambda \in \Lambda$ and $c_i(\lambda)$ is analytic in $\Lambda$ for all $i$. To simplify notation, we sometimes just label by $\lambda$ the objects associated to $f_\lambda$. Also we will often omit the markings $c_i(\lambda)$ from the notation and simply say that $\{f_\lambda\}$ is a critically marked analytic family implicitly assuming that the markings are $c_1(\lambda), \dots, c_{d-1}(\lambda)$.
When $\{f_\lambda\}_{\lambda\in\Lambda}\subset{\operatorname{Poly}}_d$ is an analytic family and
$c: \Lambda \to \mathbb{C}_v$ is an analytic function such that $c(\lambda)$ is a critical point of $f_\lambda$ for all $\lambda$ we say that $c(\lambda)$ is a \emph{marked critical point} of the family $\{f_\lambda\}$.
\begin{definition}
Let $\{f_\lambda\}_{\lambda\in\Lambda}$ be an analytic family with a
marked critical point $c(\lambda)$.
We say that $c(\lambda)$ is a \emph{passive critical point in $\Lambda$} if $\{\lambda\in\Lambda : c(\lambda)\in{\pazocal{B}}(\lambda)\}=\Lambda$ or $\emptyset$. We say that a critically marked analytic family parametrized by $\Lambda$ is \emph{passive in $\Lambda$}
if all its critical points are passive in $\Lambda$. \end{definition}
We emphasize that our definition of passive is not a local property, it depends on the domain $\Lambda$ of the analytic family $\{f_\lambda\}$.
For an analytic family $\{f_\lambda\}_{\lambda\in\Lambda}$, denote by $\phi_{\lambda}: \mathbb{A}^1_{an} \setminus \overline{D} (0,R_\lambda) \to \mathbb{A}^1_{an} \setminus\overline{D} (0,R_\lambda)$ the B\"ottcher coordinate of $f_\lambda$. Given $\lambda_0, \lambda_1 \in \Lambda$ and $R=\max\{R_{\lambda_0},R_{\lambda_1}\}$ we say that $\phi_{\lambda_1}^{-1}\circ\phi_{\lambda_0}: \mathbb{A}^1_{an} \setminus \overline{D} (0,R) \to \mathbb{A}^1_{an} \setminus\overline{D} (0,R)$ is the \emph{B\"ottcher coordinate change} between the dynamical space of $f_{\lambda_0}$ and $f_{\lambda_1}$.
\begin{definition}\label{def:constant-B}
Consider a passive marked critical point
$c(\lambda)$ in ${\pazocal{B}}(\lambda)$
of an analytic family $\{f_\lambda\}$ of tame polynomials
parametrized by $\Lambda$.
We say that the \emph{B\"ottcher coordinate of $c(\lambda)$ is constant} if
for all $\lambda_0, \lambda_1 \in \Lambda$, there exists $n$ such that
the corresponding B\"ottcher coordinate change
sends $f^{\circ n}_{\lambda_0}(c(\lambda_0))$ to $f^{\circ n}_{\lambda_1}(c(\lambda_1))$. \end{definition}
As an immediate consequence of~\eqref{eq:diameter} we have that base points of passive families are constant:
\begin{lemma}
\label{l:passive-constant}
Let $\{f_\lambda\}_{\lambda\in\Lambda}$ be a passive critically marked analytic family of tame polynomials parametrized by a disk $\Lambda$.
Then the base point of $f_\lambda$ is independent of
$\lambda\in\Lambda$. \end{lemma}
\begin{proof}
We may assume that the maps involved are nonsimple.
Pick an element $\lambda_0 \in \Lambda$ and apply \eqref{eq:diameter}
to find a critical point $c_i(\lambda_0) \in {\pazocal{B}}(\lambda_0)$ such
that $R_{\lambda_0} = |\phi_{\lambda_0}|(c_i(\lambda_0))$.
Keep in mind that $|c_i(\lambda)| \le R_\lambda$ for all $\lambda$.
Let $v(\lambda) =f_\lambda (c_i(\lambda))$. Then
$|v(\lambda_0)| = R^d_{\lambda_0}$.
Observe that
$v(\lambda)-c_i(\lambda) \neq 0$ for all $\lambda$, since the critical points are all passive. Therefore, $|v(\lambda)-c_i(\lambda)|$ has constant
value $R_{\lambda_0}^d$.
Thus, for all $\lambda$,
$$R_{\lambda_0}^d = |v(\lambda)-c_i(\lambda)| \le R_\lambda^d$$
where the last inequality follows from
$|c_i(\lambda)| \le R_\lambda$ and $|v(\lambda)| \le R_\lambda^d$.
Since this occurs for all $\lambda_0 \in \Lambda$, we have that $R_\lambda$ is constant. \end{proof}
{Observe that if the base point $x_\Lambda$ of an analytic family $\{f_\lambda\}$ of tame polynomials is independent of $\lambda$, then the B\"ottcher coordinate change $\phi^{-1}_\lambda \circ \phi_{\lambda_0}(z)$ is also analytic in $\lambda$ for all
$z$ such that $|z| > |x_\Lambda|$ {(see Lemma \ref{lem:B-analytic})}. Hence, if a passive marked critical point $c(\lambda)$ in ${\pazocal{B}}(\lambda)$ has constant B\"ottcher coordinate,
then the B\"ottcher coordinate change sends $f^{\circ m}_{\lambda_0}(c(\lambda_0))$ to $f^{\circ m}_{\lambda}(c(\lambda))$ for all $m \ge 1$ such that $|f^{\circ m}_{\lambda_0}(c(\lambda_0))| > |x_\Lambda|$.}
\section{Extendable and Analytic Conjugacies} \label{sec:extendable} The aim of this section is to discuss the notion of extendable conjugacies and prove Theorem~\ref{ithr:D}.
\subsection{Extendable conjugacies}\label{sec:conjugacy} We shall investigate conditions under which the conjugacy $\phi^{-1}_g \circ \phi_f$ furnished near $\infty$ by the B\"ottcher coordinates of two polynomials $f,g \in {\operatorname{Poly}_d}$ extends to an analytic conjugacy $\varphi: {\pazocal{B}} (f) \to {\pazocal{B}}(g)$.
\begin{definition}\label{def:conjugacy}
Consider tame polynomials $f, g \in {\operatorname{Poly}_d}$ and locally finite unbounded
open subtrees ${\pazocal{T}}_f, {\pazocal{T}}_g \subset {\mathbb{H}}$ such that $f({\pazocal{T}}_f) \subset {\pazocal{T}}_f$ and $g({\pazocal{T}}_g) \subset {\pazocal{T}}_g$. A $\mathrm{dist}_\H$-isometry $h: {\pazocal{T}}_f \to {\pazocal{T}}_g$
is an \emph{extendable conjugacy} if
\begin{enumerate}
\item (conjugacy) $h \circ g = f \circ h$,
\item (B\"ottcher coordinate change)
$\phi_g^{-1} \circ \phi_f (x) = h(x)$ for all $x \in {\pazocal{T}}_f$ in a neighborhood of $\infty$, and
\item (locally a translation)
for all $x \in {\pazocal{T}}_f$, there exists a translation
$\tau_x (z) =z +b_x$ such that $\tau_x (y) = h(y)$ for all $y\in{\pazocal{T}}_f$ in a neighborhood of $x$.
\end{enumerate} \end{definition}
The following result implies that an extendable conjugacy has a well-defined tangent map. \begin{lemma}
\label{l:unique-admissible}
Let ${\pazocal{T}}_0,{\pazocal{T}}_1$ be two locally finite open subtrees in ${\mathbb{H}}$ and let $\iota: {\pazocal{T}}_0 \to {\pazocal{T}}_1$ be locally a translation.
Consider $x \in {\pazocal{T}}_0$ and two translations $\tau$ and $\tau'$ which agree with $\iota$ in a neighborhood of $x$. Then $T_x \tau = T_x \tau'$. \end{lemma}
\begin{proof} Noting that the valence of ${\pazocal{T}}_0$ at $x$ is at least $2$, since $\tau=\iota=\tau'$ in a neighborhood of $x$, we obtain that $T_x \tau$ and $T_x \tau'$ agree in at least two directions in $T_x\mathbb{A}^1_{an}$ and therefore $T_x \tau = T_x \tau'$.
\end{proof}
By Lemma \ref{l:unique-admissible}, if $h$ is an extendable conjugacy as in Definition \ref{def:conjugacy}, given $x\in{\pazocal{T}}_f$ and setting $y=h(x)$, we denote by $T_x h : T_x \mathbb{A}^1_{an} \to T_y \mathbb{A}^1_{an}$ the unique map that agrees with the tangent map of a translation {$\tau_x$} locally coincident with $h$, that is $T_x h=T_x\tau_x$.
\subsection{From analytic to extendable conjugacies} \label{sec:analytic2extendable} In this subsection we discuss general results about analytic maps between open subsets of $\mathbb{A}^1_{an}$ and prove one direction of Theorem~\ref{ithr:D} (see Corollary \ref{c:ithr-D-easy}).
An open annulus in $\mathbb{A}^1_{an}$ is a set $A$ of the form $D (a,r) \setminus \overline{D}(a,s)$ for some $0 < s < r$. The image of a non-constant analytic map $\psi: A \to \mathbb{A}^1_{an}$ is either a disk or an annulus (c.f.\cite[Proposition~9.44 and Lemma~9.45]{Baker10}). In the latter case, $\psi$ has a well defined degree, and in a certain sense, $T_x \psi$ is ``constant'' along type II points $x \in \operatorname{sk} A$:
\begin{lemma}
\label{l:constant-skeleton}
Let $A_0$ and $A_1$ be two open annuli contained in $\mathbb{A}^1_{an}$, and
suppose that $\psi : A_0 \to A_1$ is a surjective analytic map of degree $\delta$.
Then there exist $\sigma \in
\{+\delta,-\delta\}$ and $0\not=a\in \mathbb{C}_v$ such that for arbitrary $b\in\mathbb{A}^1$ in the bounded component of $\mathbb{A}^1_{an}\setminus A_1$ and arbitrary $c\in\mathbb{A}^1$ in the bounded component of $\mathbb{A}^1_{an}\setminus A_0$, setting $\gamma(z) = a (z-c)^\sigma + b$, we have that $\psi(x)=\gamma(x)$ and $T_x \psi = T_x \gamma$
for all $ x \in \operatorname{sk} (A_0)$. \end{lemma}
\begin{proof}
Pick any $c \in \mathbb{A}^1$ in the bounded component of $\mathbb{A}^1_{an}\setminus A_0$ and any $b \in \mathbb{A}^1$ in the bounded component of $\mathbb{A}^1_{an}\setminus A_1$. Modulo the translation $z\mapsto z-c$ in the domain and the translation $z\mapsto z-b$ in the range, we may assume that $\operatorname{sk}(A_0)$ and $\operatorname{sk}(A_1)$ are contained in $]0,\infty[$.
Then, from the Mittag-Leffler decomposition of $\psi$ (see~\cite[Proposition~2.2.6]{Fresnel04}), we have $\psi (z) = \sum_{-\infty}^{+\infty} a_n z^n$.
Moreover, for some $\sigma \in \{+\delta,-\delta\}$ and for all $r$
such that $|z| = r$ is contained in $A_0$, we have
that $|\psi(z)| = |a_{\sigma}|r^\sigma$ and $|a_j| r^j < |a_\sigma|r^\sigma$ for all $j \neq \sigma$ since $\psi$ maps $\operatorname{sk}(A_0)$ onto $\operatorname{sk}(A_1)$ and has degree $\delta$.
Hence, $$|\psi (z) - a_\sigma z^\sigma| < |a_\sigma|r^\sigma.$$
Let $\gamma (z) = a_\sigma z^\sigma$. It follows that $\psi(x)=\gamma(x)$ and $T_x \psi = T_x \gamma$ for all $ x \in \operatorname{sk}(A_0)$.
\end{proof}
\begin{corollary}
\label{l:t-criteriumB}
Let $A_0$ and $A_1$ be two open annuli in $\mathbb{A}^1_{an}$, and suppose
that $\psi_1, \psi_2: A_0 \to A_1$ are surjective analytic maps.
If there exists a type II point $x_0 \in \operatorname{sk} A_0$ such that $\psi_1(x_0) = \psi_2(x_0)$ and
$T_{x_0} \psi_1 = T_{x_0} \psi_2$, then $\psi_1(x) = \psi_2(x)$ and
$T_{x} \psi_1 = T_{x} \psi_2$ for all $x \in \operatorname{sk} A_0$. \end{corollary}
\begin{proof}
Modulo affine maps in the domain and the target, we can assume that $\operatorname{sk}(A_0)$ and $\operatorname{sk}(A_1)$ are contained in $]0,\infty[$ and $x_0=x_G=\psi_1(x_0) = \psi_2(x_0)$. Since the point $0$ is contained in both the bounded component of $\mathbb{A}^1_{an}\setminus A_0$ and the bounded component of $\mathbb{A}^1_{an}\setminus A_1$, by Lemma \ref{l:constant-skeleton}, there exists $\gamma_1(z)=az^\delta$ and $\gamma_2(z)=a'z^{\delta'}$ such that $\gamma_1(x)=\psi_1(x)$, $\gamma_2(x)=\psi_2(x)$ and $T_{x}\gamma_1=T_{x}\psi_1$, $T_{x}\gamma_2=T_{x}\psi_2$ for all $x\in\operatorname{sk}(A_0)$. Since $x_0=x_G\in\operatorname{sk}(A_0)$ and $\gamma_1(x_0)=x_0=\gamma_2(x_0)$, we obtain $|a-a'|<|a|=|a'|=1$ and $\delta=\delta'$. Then for any $r$ with $|z|=r$ contained in $A_0$, we have
$$|\gamma_1(z)-\gamma_2(z)|=|(a-a')z^\delta|<|\gamma_1(z)|.$$
It follows that $|(\gamma_1 -\gamma_2)(x)|<|\gamma_1(x)|$ and $T_x\gamma_1 = T_x\gamma_2$ for all $ x \in \operatorname{sk}(A_0)$. Thus the conclusion holds. \end{proof}
We will also need the following observation:
\begin{lemma}
\label{l:constant-ball}
Suppose that $\psi_1, \psi_2: U \to \mathbb{A}^1_{an}$
are analytic maps defined in a neighborhood $U$ of a type $II$ point $x_0$.
If $\psi_1 (x_0) = \psi_2(x_0)$ and $T_{x_0} \psi_1 = T_{x_0} \psi_2$, then there exists $\varepsilon >0$ such that for all $x\in U$ with $\mathrm{dist}_\H (x, x_0) < \varepsilon$ we have that $\psi_1(x)=\psi_2(x)$ and
$T_x \psi_1 = T_x\psi_2 $. \end{lemma}
\begin{proof}
Let $y_0 = \psi_1(x_0) = \psi_2(x_0)$. Taking into account that $x_0$ is a type II point,
given such maps $\psi_1, \psi_2$ we have that
$|\psi_1(z) - \psi_2(z)|_{x_0} < \operatorname{diam} (y_0)$. Since $\operatorname{diam} (y)$ is continuous with respect to
the metric topology in ${\mathbb{H}}$ and $|\psi_1(z) - \psi_2(z)|_{x}$ is continuous in the Berkovich topology which is weaker than the metric topology, it follows
that $|\psi_1(z) - \psi_2(z)|_{x} < \operatorname{diam} (\psi_1(x))$ for all $x$ in a ${\mathbb{H}}$-metric disk around $x_0$. Hence, $\psi_2(x) = \psi_1(x)$ and
$T_x \psi_2 = T_x \psi_1$ for all $x$ in this $\mathrm{dist}_\H$-disk. \end{proof}
Analytic isomorphisms between open and connected sets which agree with a translation in a neighborhood of an skeleton point of the domain, are locally a translation at all skeleton points:
\begin{proposition}
\label{p:iso-adm}
For $b \in \mathbb{C}_v$, let $\tau_b(z) = z+b$.
Consider two open and connected sets $U_0, U_1\subsetneq \mathbb{A}^1_{an}$.
Assume that $\psi: U_0 \to U_1$ is an analytic isomorphism such that $\psi(x_0)=\tau_b(x_0)$ and
$T_{x_0} \psi = T_{x_0} \tau_b$ for some $x_0 \in \operatorname{sk} U_0$ and
some $b \in \mathbb{C}_v$.
Then for any $x\in \operatorname{sk} U_0$, there exist $b_x\in \mathbb{C}_v$ and a neighborhood $V\subset \operatorname{sk} U_0$ of $x$ such that for all $y\in V$, $\psi(y)=\tau_{b_x}(y)$ and
$T_{y} \psi = T_{y} \tau_{b_x}$. \end{proposition}
\begin{proof}
Let $\Gamma \subset \operatorname{sk} U_0$ be the set formed by all $x \in \operatorname{sk} U_0$
with the following property:
there exist $b_x\in\mathbb{C}_v$ and a neighborhood $V\subset \operatorname{sk} U_0$ of $x$ such that for
all $y \in V$ we have $\psi(y)=\tau_{b_x}(y)$ and $T_y \psi = T_y \tau_{b_x}$. By Lemma~\ref{l:constant-ball}, if a branch point $x'\in\operatorname{sk} U_0$ is contained in $\Gamma$ (resp. $\operatorname{sk} U_0\setminus \Gamma$), then there exists an open hyperbolic ball $V'\subset U_0$ around $x'$ such that $V'\cap \operatorname{sk} U_0$ is contained in $\Gamma$ (resp.
in $(\operatorname{sk} U_0) \setminus \Gamma$). Given any two branch points $x_1$ and $x_2$ of $\operatorname{sk} U_0$ such that $]x_1,x_2[$ contains no branch point of $\operatorname{sk} U_0$, by Lemma~\ref{l:constant-skeleton} and \ref{l:constant-ball}, the segment $]x_1,x_2[$ is either contained in $\Gamma$ or in $(\operatorname{sk} U_0) \setminus \Gamma$. Therefore, $\Gamma$ is a non-empty clopen subset of $\operatorname{sk} U_0$ and the proposition follows. \end{proof}
Given a tame polynomial $f \in {\operatorname{Poly}}_d$, recall from Section \ref{sec:poly} that the dynamical core ${\pazocal{A}}_f$ is contained in the skeleton of the open and connected set ${\pazocal{B}}_0(f)$.
\begin{corollary}
\label{c:ithr-D-easy}
Let $f, g\in{\operatorname{Poly}_d}$ be two tame polynomials. Assume that $\psi: {\pazocal{B}} (f) \to {\pazocal{B}}(g)$ is an analytic conjugacy between $f: {\pazocal{B}} (f) \to {\pazocal{B}}(f)$ and $g: {\pazocal{B}} (g) \to {\pazocal{B}}(g)$ that extends the corresponding B\"ottcher coordinate change.
Then $\psi: {\pazocal{A}}_f \to {\pazocal{A}}_g$ is an extendable conjugacy between $f: {\pazocal{A}}_f \to {\pazocal{A}}_f$ and $g: {\pazocal{A}}_g \to {\pazocal{A}}_g$. \end{corollary}
\begin{proof}
By Definition \ref{def:conjugacy}, it suffices to show $\psi$ is locally a translation on ${\pazocal{A}}_f$. Observe that $c$ is a critical point
of $f$ in ${\pazocal{B}}(f)$ if and only if $\psi(c)$ is a critical point
of $g$ in ${\pazocal{B}}(g)$. Hence, $\psi:{\pazocal{B}}_0(f)\to{\pazocal{B}}_0(g)$ is an analytic isomorphism. Since ${\pazocal{A}}_f \subset \operatorname{sk}{\pazocal{B}}_0(f)$, by Proposition \ref {p:iso-adm}, $\psi$ is locally a translation on ${\pazocal{A}}_f$. \end{proof}
Corollary~\ref{c:ithr-D-easy} establishes one direction of Theorem~\ref{ithr:D}.
\subsection{From extendable to analytic conjugacies} \label{s:maps-annuli}
In this subsection, we discuss results about maps between open sets and prove the remaining direction of Theorem~\ref{ithr:D}.
Recall that $p \ge 0$ denotes the characteristic of the residue field of $\mathbb{C}_v$. We begin with the following well-known facts about prime-to-$p$ \'etale maps between annuli. Proofs are supplied for the sake of completeness.
\begin{lemma}
\label{l:isomorphic-d}
Let $\delta \ge 1$ be an integer not divisible by $p$, and let $A_0 = \{ r^{1/\delta} < |z| < s^{1/\delta} \}$ and $A_1 = \{ r < |z| < s \}$ be two open annuli in $\mathbb{A}^1_{an}$ for some $0 < r < s$. Consider an analytic map $\psi: A \to A_1$ of degree $\delta$,
where $A \subset \mathbb{A}^1_{an}$ is an open annulus.
Then there exists an analytic isomorphism $\varphi: A \to A_0$
such that $\varphi (z)^\delta = \psi(z)$. Moreover, $\varphi$ is unique up to multiplication by $\mu \in \mathbb{C}_v$ where $\mu^\delta =1$. \end{lemma}
\begin{proof}
Write $A = \{ r' < |z-a| < s' \}$ for some $a\in\mathbb{C}_v$. Then there exists $b_n\in\mathbb{C}_v$ such that
$$\psi(z) = \sum_{-\infty}^{+\infty} b_n (z-a)^n$$
where, for all $r' < t < s'$, either $|b_\delta| t^\delta > |a_n| t^n$ for all $n \neq \delta$ or $|b_{-\delta}| t^{-\delta} > |b_n| t^n$ for all $n \neq -\delta$.
Without loss of generality, assume the former. Then
$$\psi(z) = b_\delta (z-a)^\delta (1 + \varepsilon(z))$$ with $|\varepsilon(z)| < 1$ for all $z \in A$. Consider
$\beta \in \mathbb{C}_v$ such that $\beta^\delta =b_\delta$. Since
$p$ does not divide $d$, there exists a
unique function $\gamma(z)$ such that $\gamma(z)^\delta = 1 + \varepsilon(z)$
and $|\gamma(z) -1| <1$. Then $\varphi(z) = \beta (z-a) \gamma(z)$ is such that $\varphi(z)^\delta = g(z)$. The lemma follows. \end{proof}
\begin{corollary}
\label{c:identity}
Suppose that $\psi: A \to A_1$ is an analytic map of degree $\delta \ge 2$ between two open annuli $A, A_1\subset\mathbb{A}^1_{an}$ such that
$\delta$ is not divisible by $p$.
Assume that $\psi_1: A \to A $ is an analytic isomorphism such that the following hold:
\begin{enumerate}
\item $$\psi \circ \psi_1 = \psi.$$
\item $\psi_1(x) = x$ and $T_x \psi_1 = \operatorname{id}$ for some type II point $x$ in $\operatorname{sk}(A)$.
\end{enumerate}
Then $\psi_1 = \operatorname{id}$. \end{corollary}
\begin{proof}
After translation we may assume $A_1 =
\{ r < |z| < s \}$. Let
$A_0$ and $\varphi: A \to A_0 $ be as in Lemma \ref{l:isomorphic-d}. Then $\varphi(z)^\delta=\psi(z)$. It follows that $(\varphi (\psi_1(z)))^\delta =\psi(\psi_1(z))$. By assumption (1), we have $(\varphi (\psi_1(z)))^\delta=\psi(\psi_1(z))= \psi(z)=\varphi(z)^\delta$. Thus $\varphi \circ \psi_1(z) = (\mu \varphi) (z)$ for some
$\mu\in\mathbb{C}_v$ with $\mu^\delta =1$.
By assumption (2), we conclude $\varphi(x)=(\mu\varphi)(x)$ and
$T_x \varphi = T_x (\mu\varphi)$. Thus $\mu=1$. It follows that $\varphi \circ \psi_1 = \varphi$ which implies $\psi_1 = \operatorname{id}$. \end{proof}
\begin{corollary} \label{c:extension-annulus} Let $A,A'$ and $A_1$ be open annuli in $\mathbb{A}^1_{an}$. Suppose that $\psi_1: A \to A_1$ and $\psi_2: A' \to A_1$ are degree $\delta \ge 1$ analytic maps such that
$\delta$ is not divisible by $p$. Then there exist exactly $\delta$ isomorphisms $\varphi: A \to A'$ such that $\psi_2\circ \varphi =\psi_1$. Moreover, if $B_1$ is a subannulus of $A_1$ with $\operatorname{sk}(B_1) \subset \operatorname{sk}(A_1)$, then every isomorphism $\psi: \psi_1^{-1} (B_1) \to \psi_2^{-1}(B_1)$ such that $\psi_2 \circ \psi =\psi_1$ extends to a unique isomorphism $\varphi: A \to A'$ such that $\psi_2\circ \varphi =\psi_1$. \end{corollary}
\begin{proof}
After change of coordinates, unique modulo multiplication by a $\delta$-th root of unity, $\psi_1$ and $\psi_2$ become
$z \mapsto z^\delta$. It follows that there are exactly $\delta$ isomorphisms
$\varphi$ as in the statement. Similarly, there are exactly $\delta$
isomorphisms $\psi$ as in the statement. Thus, every such $\psi$ is the restriction of an isomorphism $\varphi$.
\end{proof}
Now we provide a criterium for the tangent maps of two analytic functions to agree at a type II point.
\begin{lemma}
\label{l:t-criteriumA}
Let $D\subset\mathbb{A}^1_{an}$ be an open Berkovich disk and $x_0\in D$ be a type II point.
Suppose that $\psi_1, \psi_2: D \to \mathbb{A}^1_{an}$ are analytic maps satisfying the following:
\begin{enumerate}
\item $p\nmid\deg_{x_0}\psi_1$ and $p\nmid\deg_{x_0}\psi_2$,
\item $\deg_{\vec{v}} \psi_1 = \deg_{\vec{v}} \psi_2$ for all $\vec{v} \in T_{x_0} \mathbb{A}^1_{an}$,
\item there exists $x\in\mathbb{A}^1_{an}$ with $x_0\precneqq x$ such that
\begin{enumerate}
\item $\psi_1 (y_0) = \psi_2(y_0)$ for all $x_0\prec y_0 \prec x$, and
\item $T_{y_0} \psi_1 = T_{y_0} \psi_2$ for all $x_0\precneqq y_0 \prec x$,
\end{enumerate}
\item $T_{x_0} \psi_1 (\vec{v}_0) = T_{x_0} \psi_2 (\vec{v}_0)$ for at least one direction $\vec{v}_0 \in T_{x_0} \mathbb{A}^1_{an}\setminus\{\vec{v}_{x_0}(\infty)\}$.
\end{enumerate}
Then $T_{x_0} \psi_1 = T_{x_0} \psi_2$. \end{lemma} \begin{proof}
From (3a), continuity yields that $\psi_1(x_0) = \psi_2(x_0)$. Using the same
coordinate changes for $\psi_1$ and $\psi_2$,
we may assume that $x_0$ and $\psi_1(x_0)=\psi_2(x_0)$ are the Gauss point. Thus, it is sufficient to show that the reductions
$\tilde{\psi}_1$ and $\tilde{\psi}_2$ of $\psi_1$ and $\psi_2$ coincide.
While statement (1) implies that the polynomials
$\tilde{\psi}_1$ and $\tilde{\psi}_2$
have finitely many critical points, statement (2) implies that
$\tilde{\psi}_1$ and $\tilde{\psi}_2$ have the same critical points counting multiplicities. In particular, $\deg_{x_0}\psi_1=\deg_{x_0}\psi_2$.
From (3b), it is not difficult to conclude that the leading coefficients of $\tilde{\psi}_1$ and $\tilde{\psi}_2$ coincide.
From (4), the constant terms of $\tilde{\psi}_1$ and $\tilde{\psi}_2$ agrees. Therefore, $\tilde{\psi}_1 =\tilde{\psi}_2$.
\end{proof}
Through an analytic continuation argument along ${\pazocal{A}}_f$, we will prove the following:
\begin{proposition}\label{p:ithr-D-hard} Let $f, g \in {\operatorname{Poly}}_d$ be tame polynomials. If $h: {\pazocal{A}}_f \to {\pazocal{A}}_g$ is an extendable conjugacy between $f: {\pazocal{A}}_f \to {\pazocal{A}}_f$ and $g: {\pazocal{A}}_g \to {\pazocal{A}}_g$, then there exists an analytic conjugacy $\phi: {\pazocal{B}}(f)\to{\pazocal{B}}(g)$ between $f: {\pazocal{B}} (f) \to {\pazocal{B}}(f)$ and $g: {\pazocal{B}} (g) \to {\pazocal{B}}(g)$
which
agrees with the B\"ottcher coordinate change near infinity
and with $h$ on ${\pazocal{A}}_f$. \end{proposition}
Given a tame polynomial $f \in {\operatorname{Poly}}_d$, recall from Section~\ref{sec:poly} that $\phi_f$ denotes the B\"ottcher coordinate and that
$|\phi_f|$ extends to ${\pazocal{B}}(f)$.
For $s>1$, let $$V_f (s) := \{ z \in \mathbb{A}^1_{an} : |\phi_f| (z) > s \}$$ and $${\pazocal{A}}_f (s) := {\pazocal{A}}_f \cap V_f (s).$$ Observe that ${\pazocal{A}}_f (s)$ is forward invariant. Moreover, ${\pazocal{A}}_f(s)$ is contained in the skeleton of $$V'_f(s):=V_f(s) \setminus \operatorname{Post}(f).$$
\begin{proof}[Proof of Proposition \ref{p:ithr-D-hard}]
For all $z \in \mathbb{A}^1_{an}$ such that $|z|$ is sufficiently large, we have $|\phi_f (z)| = |z| = |\phi_g (z)|$. Hence, for $s_0\gg1$ sufficiently large, $\phi (z) = \phi_g^{-1} \circ \phi_f (z): V_f(s_0) \to V_g (s_0) $ is an analytic isomorphism and $\phi=h$ on ${\pazocal{A}}_f(s_0)$. Moreover, $\phi$ is an analytic conjugacy between $f: V_f(s_0) \to V_f(s_0)$ and $g: V_g(s_0) \to V_g(s_0)$.
We assume that there exists $s_\star > 1$ such that $\phi$ extends to $V_f({s_\star})$, that is, we have an analytic isomorphism $\phi: V_f ({s_\star}) \to V_g({s_\star})$ extending the B\"ottcher coordinate change and conjugating $f: {\pazocal{A}}_f(s_\star) \to {\pazocal{A}}_f(s_\star)$ with $g: {\pazocal{A}}_g(s_\star) \to {\pazocal{A}}_g(s_\star)$ such that $\phi(x)=h(x)$ for all $x \in {\pazocal{A}}_f({s_\star})$. To prove the proposition it is sufficient to show that $\phi$ in fact extends to an analytic isomorphism $\overline{\phi}: V_f (s) \to V_g (s)$ for some $s < s_\star$ such that $\overline{\phi}(x)=h(x)$ for all $x \in {\pazocal{A}}_f (s)$. It automatically follows that $\overline{\phi}$ is a conjugacy since $\overline{\phi}^{-1}\circ g \circ \overline{\phi} (z) = f(z)$ in an open set.
Consider $x_0 \in \partial V_f (s_\star)$. Keep in mind that there are only finitely many such $x_0$. Moreover, $]x_0, \infty[ \subset V_f (s_\star)$ and $\phi (]x_0,\infty[) = ]x_0',\infty[$ for some $x_0' \in \partial V_g (s_\star)$. The main step is to extend $\phi$ to a neighborhood of $x_0$ in $\mathbb{A}^1_{an}$. We consider three cases.
\emph{Case 1. $\deg_{x_0} f =1$.} Consider $ x \succ x_0$ close to $x_0$ such that $x$ is not in the ramification tree of $f$. Set $x' := \phi (x)$. Then $\deg_x f =1$ and, therefore, $\deg_{x'} g =1$, since $\phi$ is a conjugacy in $V_f(s_\star)$. Let $U_0$ (resp. $V_0$) be the Berkovich open disk formed by points in the direction of $x_0$ (resp. $x'_0$) at $x$ (resp. $x'$). We may denote by $G$ the inverse of $g: V_0 \to g(V_0)$. It follows that $\overline{\phi}:=G \circ \phi \circ f : U_0 \cap V_f (s_\star^{1/d}) \to V_0 \cap V_g (s_\star^{1/d})$ is a well defined analytic isomorphism that coincides with $\phi$ in $U_0 \cap V_f (s_\star)$ and agrees with $h$ on the points of ${\pazocal{A}}_f$ contained in $U_0 \cap V_f (s_\star^{1/d})$.
\emph{Case 2. $\deg_{x_0} f \ge 2$ and $x_0$ is a type II point.} In particular, $x_0$ lies in the ramification tree and therefore in ${\pazocal{A}}_f$. Our assumption that $\phi = h$ on ${\pazocal{A}}_f(s_\star)$ guarantees that $\phi$ maps $V'_f(s_\star)$ onto $V'_g(s_\star)$. The B\"ottcher coordinate change $\phi$ is asymptotic to the identity at infinity, therefore, $\phi(x)=x$ and $T_{x}\phi=\operatorname{id}$ for all $x\in{\pazocal{A}}_f(s_\star)$ sufficiently close to $\infty$. By Proposition~\ref{p:iso-adm} and Lemma~\ref{l:unique-admissible} we conclude that $T_x \phi = T_x h$ for all $x \in {\pazocal{A}}_f(s_\star) \subset \operatorname{sk} V'_f(s_\star)$.
Let $x_1=f(x_0)$ and consider translations $\tau_0, \tau_1$ that locally agree with $h$ around $x_0, x_1$, respectively. After checking that (1)--(4) of Lemma~\ref{l:t-criteriumA} hold for $\psi_1=g \circ \tau_0$ and $\psi_2=\tau_1 \circ f$, we will conclude that $T_{x_0} (g \circ \tau_0) = T_{x_0} (\tau_1 \circ f)$. Tameness of $f$ and $g$ guarantees that $p\nmid\deg_{x_0} (g \circ \tau_0)$ and $p\nmid\deg_{x_0} (\tau_1 \circ f)$, and hence (1) of Lemma~\ref{l:t-criteriumA} follows. Now for (2), pick $\vec{v}\in T_{x_0}\mathbb{A}^1_{an}$ and let $\vec{u}:=T_{x_0} \tau_0(\vec{v})$. We must show that $\deg_{\vec{u}} g = \deg_{\vec{v}} f$. Since $h$ agrees with $\tau_0$ on a neigborhood of $x_0$, the direcion $D(\vec{v})$ is disjoint from ${\pazocal{A}}_f$ if and only if the direction $D(\vec{u})$ is disjoint from ${\pazocal{A}}_g$. When $D(\vec{v})$ and $D(\vec{u})$ are disjoint from ${\pazocal{A}}_f$ and ${\pazocal{A}}_g$, respectively, $\deg_{\vec{v}}f=1=\deg_{\vec{u}}g$. When $D(\vec{v})$ intersects ${\pazocal{A}}_f$, consider a small arc $]x,x_0] \subset {\pazocal{A}}_f$ in the direction $D(\vec{v})$. Then the hyperbolic length of $]x,x_0]$ under $h \circ f$ (resp. $g \circ h$) is multiplied by a factor of $\deg_{\vec{v}}f$ (resp. $\deg_{\vec{u}}g$), since $h$ is an isometry. But $h$ is also a conjugacy, so these factors must agree. That is, $\deg_{\vec{v}}f = \deg_{\vec{u}}g$. Thus (2) of Lemma~\ref{l:t-criteriumA} holds. For $x \in]x_0,\infty[ \subset V_f(s_\star)$ sufficiently close to $x_0$, we have $T_x (\phi \circ f) = T_x (g \circ \phi)$. Therefore, $T_x (\tau_1 \circ f) = T_x (g \circ \tau_0)$ since $T_x\phi = T_x h$ and $T_x h = T_x \tau_i$ for $x \in ]x_i,\infty[$.
That is, (3) of Lemma~\ref{l:t-criteriumA} holds. Finally (4) also holds since ${\pazocal{A}}_f$ has points in at least one bounded direction at $x_0$. From Lemma~\ref{l:t-criteriumA} now we have $T_{x_0} (g \circ \tau_0) = T_{x_0} (\tau_1 \circ f)$.
By Lemma \ref{lem:lift} in the appendix, there exists an analytic map $\psi$ defined in a neighborhood $V$ of $x_0$ such that $g \circ \tau_0 \circ \psi = \tau_1 \circ f$ in $V$ and $\psi(y_0) = y_0$, $T_{y_0} \psi = \operatorname{id}$ for $y_0\in V$ with $\mathrm{dist}_\H (y_0,x_0)$ small enough. For a small annulus $A\subset V$ with $\operatorname{sk} A=]x_0,x[$ for some $x_0\precneqq x\in V$ sufficiently close to $x_0$, considering the map $f:A\to f(A)$ and setting $\psi_1:=\phi^{-1}\circ \tau_0 \circ \psi$, we obtain that $\psi_1: A\to A$ satisfies $f\circ\psi_1=f\circ\phi^{-1}\circ \tau_0 \circ \psi=\tau_1^{-1}\circ g\circ \tau_0 \circ \psi=\tau_1^{-1}\circ \tau_1 \circ f=f$ in $A$, moreover, for any $y_0\in\operatorname{sk} A$, we have $\psi_1(y_0)=\phi^{-1}\circ \tau_0 \circ \psi(y_0)=\phi^{-1}\circ \tau_0(y_0)=h^{-1}\circ h(y_0)=y_0$ and $T_{y_0}\psi_1=T_{y_0}(\phi^{-1}\circ \tau_0 \circ \psi)=T_{y_0}(\phi^{-1}\circ \tau_0)=T_{h(y_0)}(\tau_{y_0}^{-1})T_{y_0}\tau_0=\operatorname{id}$. Then by Corollary~\ref{c:identity}, we conclude that $\psi_1=\operatorname{id}$ and hence $\phi = \tau_0 \circ \psi$ in $A$. Thus, $\phi$ extends to a neighborhood of $x_0$ and the extension $\overline{\phi}:=\tau_0 \circ \psi$ agrees with $\tau_0=h$ on the points of ${\pazocal{A}}_f$ sufficiently close to $x_0$.
\emph{Case 3. $\deg_{x_0} f \ge 2$ and $x_0$ is a type III point.} Let $A$ be a small annulus such that $x_0\in A$ and $\operatorname{sk} A\subset{\pazocal{A}}_f$. Denote by $A_1:=\phi\circ f(A)$ and let $A'$ be the connected component of $g^{-1}(A_1)$ containing $h(x_0)$. Applying Corollary~\ref{c:extension-annulus} to $\phi\circ f: A\to A_1$ and $g: A'\to A_1$, we obtain a unique analytic isomorphism $\overline{\phi}: A\to A'$ that is the extension of $\phi:A\cap V_f (s_\star)\to A'\cap V_g (s_\star)$ satisfying $g\circ\overline{\phi}=\phi\circ f$ in $A$. Observe that $\overline{\phi}=h$ in $\operatorname{sk} A$ since $h(x_0) = \overline{\phi}(x_0)$ and $\overline{\phi}$ acts as an isometry on $\operatorname{sk} A$.
Thus for $x_0 \in \partial V_f (s_\star)$, we obtain an extension $\overline{\phi}$ of $\phi$ in a neighborhood of $x_0$ such that $\overline{\phi}=h$ on the points of ${\pazocal{A}}_f$ in this neighborhood. \end{proof}
\begin{proof}[Proof of Theorem~\ref{ithr:D}] Theorem~\ref{ithr:D} is an immediate consequence of Corollary \ref{c:ithr-D-easy} and Proposition \ref{p:ithr-D-hard}. \end{proof}
\section{Existence of extendable conjugacies} \label{sec:existence}
The goal of this section is to prove Propositions~\ref{prop:close} and \ref{prop:p-to-a} which together become a slightly stronger version of Theorem \ref{ithr:C}.
\subsection{Trimmed dynamical core} \label{sec:trimmed} Given a tame polynomial $f {\in{\operatorname{Poly}_d}}$, recall that the dynamical core is $${\pazocal{A}}_f = {\pazocal{B}}(f) \cap \bigcup_{c \in \operatorname{Crit}(f), n \ge 0} ]f^{\circ n} (c), \infty[.$$
Sometimes
it will be convenient to consider the subtree of ${\pazocal{A}}_f$
formed by the elements that escape to infinity through a hyperbolic $\rho$-neighborhood of the axis $]x_f,\infty[$ for some $\rho >0$ (possibly $\infty$). Note that since $f$ is tame, {in $|z| > |x_f|$} the distance to the axis is preserved by dynamics. More precisely,
for all $x \in {\mathbb{H}}$ such that $|x| > |x_f|$ we have that $$\mathrm{dist}_\H(x,]x_f,\infty[) = \mathrm{dist}_\H(f(x),]x_f,\infty[).$$
\begin{definition}
\label{d:trimmed}
Consider a tame polynomial $f {\in{\operatorname{Poly}_d}}$.
Given $\rho \in ]0,\infty]$, we say that the \emph{$\rho$-trimmed dynamical core of $f$}, denoted by ${\pazocal{T}}_f$, is the set formed by all $x \in {\pazocal{A}}_f$ such that $\mathrm{dist}_\H (f^{\circ n} (x), ]x_f,\infty[) < \rho$ for all $n$ sufficiently large. \end{definition}
Keep in mind that ${\pazocal{A}}_f$ is the $\rho$-trimmed dynamical core of $f$ with $\rho=\infty$. {Also, from the previous discussion, $x \in {\pazocal{A}}_f$ with $|x|>|x_f|$ lies in the $\rho$-trimmed dynamical core ${\pazocal{T}}_f$ if and only if
$\mathrm{dist}_\H(x,]x_f,\infty[) < \rho$, equivalently, $\log (|x|/\mathrm{diam}(x)) < \rho$.}
\begin{lemma}
Let $f \in {\operatorname{Poly}_d}$ be a tame polynomial. For any $\rho \in ]0,\infty]$, the
corresponding $\rho$-trimmed dynamical core ${\pazocal{T}}_f$ is a locally finite open
subtree of ${\mathbb{H}}$ which is forward invariant (i.e. $f ({\pazocal{T}}_f) \subset {\pazocal{T}}_f$).
Moreover, if $x \in {\pazocal{T}}_f$, then $]x,\infty[ \subset {\pazocal{T}}_f$. Furthermore,
if a critical point $c$ lies in
a direction $D_x(\vec{v})$ at some $x \in {\pazocal{T}}_f$, then $D_x(\vec{v}) \cap {\pazocal{T}}_f \neq \emptyset$. \end{lemma}
\begin{proof}
Directly from its definition we conclude that ${\pazocal{T}}_f$ is forward invariant.
Moreover, ${\pazocal{T}}_f$ is obtained from ${\pazocal{A}}_f$ by cutting off some of its ``branches''.
Indeed,
if $x \in {\pazocal{A}}_f \setminus {\pazocal{T}}_f$ and $y \prec x$, then $f^{\circ n}(y) \prec f^{\circ n}(x)$ for all $n \ge 0$. Hence, for $n$ sufficiently large $f^{\circ n}(y)$
is at distance at least $\rho$ from the axis and therefore $y \notin {\pazocal{T}}_f$. It follows that ${\pazocal{T}}_f$ is connected and relatively open in ${\pazocal{A}}_f$. Therefore
${\pazocal{T}}_f$ is also a locally finite open subtree of ${\mathbb{H}}$.
If a direction $D_x(\vec{v})$ contains a critical point $c$ for some
$x \in {\pazocal{T}}_f$, then $D_x(\vec{v}) \cap {\pazocal{T}}_f \neq \emptyset$, since
${\pazocal{T}}_f$ is open in ${\pazocal{A}}_f$ and $]c,\infty[ \cap {\pazocal{B}}(f) \subset {\pazocal{A}}_f$. \end{proof}
Given a $\rho$-trimmed dynamical core ${\pazocal{T}}_f$, we say that $x \in {\pazocal{T}}_f$ is a \emph{vertex} of ${\pazocal{T}}_f$ if $f^{\circ n }(x)$ is a branch point of ${\pazocal{T}}_f$ for some $n \ge 0$. The set formed by the vertices of ${\pazocal{T}}_f$ is denoted by ${\pazocal{V}}_f$. An \emph{edge} of ${\pazocal{T}}_f$ is a connected component of ${\pazocal{T}}_f \setminus {\pazocal{V}}_f$.
\begin{proposition}[Vertices of ${\pazocal{T}}_f$]
\label{p:vertices-a}
Consider a non-simple tame polynomial $f\in{\operatorname{Poly}_d}$ with base point $x_f$ {and $\rho \in ]0,\infty]$}.
Let ${\pazocal{V}}_f$ be the set of vertices of {the} $\rho$-trimmed dynamical core ${\pazocal{T}}_f$.
Then $x_f \in {\pazocal{V}}_f$ and $f({\pazocal{V}}_f) \subset {\pazocal{V}}_f$. Moreover,
the set of accumulation points of ${\pazocal{V}}_f$ in $\mathbb{A}^1_{an}$ is contained
in the Julia set ${\pazocal{J}}(f)$. \end{proposition}
\begin{proof}
We claim that if $x \in {\pazocal{T}}_f$ is a branch point of ${\pazocal{T}}_f$ then $f(x)$ is also a branch point. Indeed,
let us proceed by contradiction assuming that $x$ is branch point and $f(x)$ is not. Then all bounded directions at $x$ containing elements of ${\pazocal{T}}_f$
are contained in the preimage of a single direction $\vec{w}$ at $f(x)$.
By the previous lemma, $\vec{w}$ is the unique critical value direction of
$T_xf$ and therefore $T_xf$ is a monomial. Hence
the preimage of $\vec{w}$ is a singleton and $x$ is not a branch point.
In view of the previous paragraph, if $f^{\circ n} (x)$ is a branch point of ${\pazocal{T}}_f$, then $f^{\circ n}(f(x))$ is a branch point. Thus, $f ({\pazocal{V}}_f) \subset {\pazocal{V}}_f$.
Forward invariance of the axis $[x_f,\infty[$ implies that $x_f \in {\pazocal{T}}_f$.
Since $f$ is nonsimple and tame we may apply \eqref{eq:diameter} to conclude that there exists a critical value $v$
of $f$ such that $|v| = |f(x_f)|$. It follows that $f(x_f)$ is a branch point of ${\pazocal{A}}_f$. Therefore, $x_f \in {\pazocal{V}}_f$.
Let us say that two escaping critical points $c, c'$
are eventually in the same direction if there exists
$m, n \ge 0$ such that $|x_f|< |f^{\circ n}(c)| = |f^{\circ m}(c')| = R$
and $|f^{\circ n}(c) - f^{\circ m}(c')|< R$ (i.e. they lie in the same direction
at $x_{0,R}$).
There exists $N \ge 0$ such that for all pairs of critical points
$c, c'$ which are eventually in the same direction, then $n,m$ above can be chosen to be at most $N$.
Now consider $X : = \{ x : |f^{\circ N}(x_f)| < |x| \le |f^{\circ N+1}(x_f)| \}$. It follows that ${\pazocal{A}}_f \cap X$ maps homoemorphically onto its image under $f$. Moreover, ${\pazocal{A}}_f \cap X$ is the union of finitely many arcs of the form $]z,f^{\circ N+1}(x_f)]$ with the arc $]f^{\circ N}(x_f),f^{\circ N+1}(x_f)]$. In particular, ${\pazocal{A}}_f \cap X$ contains finitely many branch points.
Also, $X$ is a ``fundamental domain'' for the action of $f$ on ${\pazocal{B}}(f)$.
Namely, ${\pazocal{B}}(f)$ is the disjoint union of the sets $f^{\circ n}(X)$ for $n \in \mathbb{Z}$.
Since every branch point of ${\pazocal{T}}_f$ is the iterated image or preimage of one in ${\pazocal{A}}_f \cap X$, the {conclusion} follows. \end{proof}
Now we introduce an increasing collection $\{{\pazocal{T}}_f^{(j)}\}_{j \ge 0}$
of subtrees of ${\pazocal{T}}_f$ that saturate ${\pazocal{T}}_f$. Set ${\pazocal{T}}_f^{(0)}:= {\pazocal{T}}_f \cap \{ x : |x| > |x_f| \}$. Recursively, the level $j+1$ subtree ${\pazocal{T}}_f^{(j+1)}$ is obtained by adjoining to ${\pazocal{T}}_f^{(j)}$ all the elements $x \in {\pazocal{T}}_f$ such that there exists $y \in {\pazocal{T}}_f^{(j)}$ with the property that $]x,y[$ contains at most one vertex of ${\pazocal{T}}_f$.
\begin{corollary}
Consider a nonsimple tame polynomial $f\in{\operatorname{Poly}_d}$ and $\rho \in ]0,\infty]$.
Let ${\pazocal{T}}_f$ be the $\rho$-trimmed
dynamical core of $f$.
Then $${\pazocal{T}}_f = \bigcup_{j \ge 0} {\pazocal{T}}_f^{(j)}.$$
Moreover,
$$f ({\pazocal{T}}_f^{(j+1)}) \subset {\pazocal{T}}_f^{(j)},$$
for all $j \ge 0$. \end{corollary}
\begin{proof}
For all $x \in {\pazocal{T}}_f$, observe that $]x,x_f[ \cap {\pazocal{V}}_f$ is finite, since $[x,x_f]$ is free of accumulation points of ${\pazocal{V}}_f$,
by Proposition~\ref{p:vertices-a}.
Given $x \notin {\pazocal{T}}_f^{(0)}$ and $j \ge 1$, note that $x \in {\pazocal{T}}_f^{(j)}$
if and only if $]x,x_f]$ contains at most $j$ elements of ${\pazocal{V}}_f$.
By the previous paragraph, we have that every element of ${\pazocal{T}}_f$ lies in
some ${\pazocal{T}}_f^{(j)}$.
By definition, $f( {\pazocal{T}}_f^{(0)}) \subset {\pazocal{T}}_f^{(0)}$.
For all $j \ge 1$ and $x \in {\pazocal{T}}_f^{(j)} \setminus {\pazocal{T}}_f^{(0)}$, our map
$f$ acts bijectively on $]x,x_f]$ and therefore
$]f(x),f(x_f)]$ contains
at most $j$ elements of ${\pazocal{V}}_f$.
If $f (x) \notin {\pazocal{T}}_f^{(0)}$,
then $]f(x), f(x_f)]$ is the disjoint union of $]f(x), x_f]$
and $]x_f, f(x_f)]$. Hence, $]f(x), x_f]$ contains at most $j-1$ elements of ${\pazocal{V}}_f$ and $f(x) \in {\pazocal{T}}_f^{(j-1)}$.
It follows that $f( {\pazocal{T}}_f^{(j)}) \subset {\pazocal{T}}_f^{(j-1)}$. \end{proof}
\begin{proposition}[Edges of ${\pazocal{T}}_f$]
\label{l:edges-a}
Consider a nonsimple tame polynomial $f$ and $\rho \in ]0,\infty]$. Denote by ${\pazocal{T}}_f$ the $\rho$-trimmed
dynamical core of $f$.
Let $e$ be an edge of ${\pazocal{T}}_f$. Then there exist $a \in {\pazocal{V}}_f \cup \partial {\pazocal{T}}_f$ and $b \in {\pazocal{V}}_f$
such that $a \prec b$ and $e = ]a,b[$.
Moreover,
$f(e) = ]f(a),f(b)[$ is also an edge of ${\pazocal{T}}_f$
and $\deg_x f = \deg_a f$, for all
$x \in e$. Furthermore, for all $y, y' \in \overline{e}$,
$$\mathrm{dist}_\H (f(y),f(y')) = \deg_af \cdot \mathrm{dist}_\H (y,y').$$ \end{proposition}
\begin{proof}
An edge $e$ is, by definition, a connected component of ${\pazocal{T}}_f \setminus {\pazocal{V}}_f$. If the extreme points are $a$ and $b$, then $]a,\infty[$ and $]b,\infty[$ are contained in ${\pazocal{T}}_f$ and, $]a,b[$ is branch point free. It follows that $a \prec b$ or $b \prec a$. Without loss of generality we assume that $a \prec b$.
By definition if a preimage $w$ of a vertex lies in ${\pazocal{T}}_f$, then it is a vertex. Therefore, $f(e)$ is an edge. For all $x \in {\mathbb{H}}$, since $f$ is tame, $\deg_x f -1$ agrees with the number of critical points in the disk $\overline{D}_x$,
counted with multiplicities {(see Lemma \ref{lem:RH})}. This number is constant along edges and therefore $\deg_x f = \deg_af$ for all $x \in e$.
Thus, the hyperbolic metric is expanded by a factor $\deg_af$ along $e$
{(see Lemma \ref{lem:factor})}. \end{proof}
\subsection{Close B\"ottcher coordinates of escaping critical points} \label{sec:close}
We start by introducing a notion that quantifies how close are the B\"ottcher coordinates of escaping critical points with the agreement that $\exp(-\infty):=0$.
\begin{definition}
\label{def:close}
Consider two tame critically marked polynomials $f$ and
$g$ in ${\operatorname{Poly}_d}$ with the same base point $x_0$. Let $\phi_f, \phi_g : \mathbb{A}^1_{an} \setminus \overline{D}(a,|x_0|) \to \mathbb{A}^1_{an} \setminus \overline{D}(a,|x_0|)$ be the corresponding B\"ottcher coordinates.
Given $\rho \in ]0,\infty]$ we say that the \emph{escaping critical points of $f$ and $g$ have $\rho$-close
B\"ottcher coordinates} if $|f^{\circ m}(c_i(f))| > |x_0|$ if and only if $|g^{\circ m}(c_i(g))| > |x_0|$ and in this case:
\begin{equation}
\label{eq:rho}
|\phi_f(f^{\circ m}(c_i(f)))- \phi_g(g^{\circ m}(c_i(g)))| \le
\exp(-\rho) |f^{\circ m}(c_i(f))|. \end{equation}
\end{definition}
{For any} passive critically marked analytic family of tame polynomials, observe that escaping critical points have $\infty$-close B\"ottcher coordinates if and only if they have constant B\"ottcher coordinates, by Definition~\ref{def:constant-B}.
Note that if $f$ and $g$ have $\rho$-close B\"ottcher coordinates and \eqref{eq:rho} holds for $i$ and $m$, then
$$|f^{\circ m}(c_i(f))|= |\phi_f(f^{\circ m}(c_i(f)))| = |\phi_g(g^{\circ m}(c_i(g)))|=|g^{\circ m}(c_i(g))|.$$
For every escaping critical point $c_i(f)$ let $m_i \ge 1$ be such that $ |x_0|<|f^{\circ m_i}(c_i(f))| \le |x_0|^d$, equivalently $m_i$ is the smallest integer such that $ |x_0|<|f^{\circ m_i}(c_i(f))|$. If for all such escaping $c_i(f)$
we have that $|g^{\circ m_i}(c_i(g))| > |x_0|$ and \eqref{eq:rho} holds for $m=m_i$, then $f$ and $g$ have $\rho$-close B\"ottcher coordinates, since $z \mapsto z^d$ is a $\mathrm{dist}_\H$-isometry on $D_{x_G}(\infty) \setminus \, ]x_G,\infty[$.
\begin{proposition}\label{prop:close}
Let $\{f_\lambda \}$ be an analytic family of critically marked tame polynomials in ${\operatorname{Poly}}_d$ parametrized by an open disk $\Lambda \subset \mathbb{C}_v$
with radius in $|\mathbb{C}_v^\times|$.
Assume that all the critical points are passive in $\Lambda$. Then
for every closed disk $\overline{\Omega}$ contained in $\Lambda$,
there exists $\rho>0$ such that for all
$\lambda_1, \lambda_2 \in \overline{\Omega} $, the escaping critical points
of $f_{\lambda_1}$ and $f_{\lambda_2}$ have $\rho$-close B\"ottcher coordinates. \end{proposition}
\begin{proof}
From Lemma~\ref{l:passive-constant} we know that the base point $x_\Lambda$ is independent of $\lambda \in \Lambda$.
Denote by $r_\Lambda$ the diameter of $x_\Lambda$.
Consider a closed disk $\overline{\Omega} \subset \Lambda$ and pick a reference parameter $\lambda_1 \in \overline{\Omega}$.
Assume that $|f_{\lambda_1}^{\circ n} (c_i(\lambda_1))| > r_\Lambda$ where $n$ is the smallest number with this property. To simplify notation let $\alpha_i(\lambda) = f_{\lambda}^{\circ n} (c_i(\lambda))$. Then $|\alpha_i(\lambda)|$ takes a constant value, say $r_i$, because otherwise $\alpha_i(\lambda)-c_i(\lambda)$ would vanish at some parameter.
Moreover, for the same reason, the direction of $\alpha_i(\lambda)$ at $x_{0,r_i}$ is also constant in $\Lambda$. Since the tangent map of $\phi_\lambda$
at $x_{0,r_i}$ is the identity, in standard coordinates,
$$|\phi_{\lambda} (\alpha(\lambda))-\phi_{\lambda_1} (\alpha(\lambda_1))|
< r_i$$
for all $\lambda \in \Lambda$.
Since $\phi_\lambda(\alpha_i(\lambda))$ is analytic, there exists
$\rho >0$ such that for all $\lambda_2 \in \overline{\Omega}$ and all $i$:
\begin{equation*}
|\phi_{\lambda_2} (\alpha_i(\lambda_2))-\phi_{\lambda_1} (\alpha_i(\lambda_1))| \leq \exp(-\rho) \cdot r_i.
\end{equation*}
\end{proof}
\subsection{From passive critical points to extendable conjugacies} \label{sec:passive2extendable}
To prove Theorem \ref{ithr:C}, we first show the following result.
\begin{proposition}
\label{prop:p-to-a}
Let $\{f_\lambda \}$ be a passive analytic family of critically marked tame polynomials in ${\operatorname{Poly}}_d$ parametrized by a disk $\Lambda \subset \mathbb{C}_v$. Consider $\rho >0$ and denote by ${\pazocal{T}}_\lambda$ the $\rho$-trimmed dynamical core of $f_\lambda$. Also let
$${\pazocal{C}} := \{ \alpha: \Lambda \to \mathbb{A}^1 : \alpha(\lambda) = f^\ell_\lambda (c_i(\lambda)) \text{ for some } \ell \ge 0, 1 \le i < d \}. $$
Suppose that the B\"ottcher coordinates of escaping critical points of $f_{\lambda_0}$ and $f_\lambda$ are $\rho$-close for all $\lambda \in \Lambda$. Then there exists a unique map $h_\lambda : {\pazocal{T}}_{\lambda_0} \to {\pazocal{T}}_\lambda$ such that
if $x = x_{\alpha(\lambda_0), r} \in {\pazocal{T}}_{\lambda_0}$ for some $\alpha \in {\pazocal{C}}$ and $r >0$, then
$$h_\lambda (x) = x_{\alpha(\lambda), r}.$$
Moreover, $h_\lambda$ is an extendable conjugacy between $f_{\lambda_0}:{\pazocal{T}}_{\lambda_0}\to{\pazocal{T}}_{\lambda_0}$ and $f_{\lambda}:{\pazocal{T}}_{\lambda}\to{\pazocal{T}}_{\lambda}$. \end{proposition}
\begin{proof}
In view of Lemma~\ref{l:passive-constant}, denote by $x_\Lambda$ the base point of $f_\lambda$, which is independent of $\lambda \in \Lambda$.
Observe that for any $\lambda\in\Lambda$, every point in ${\pazocal{T}}_\lambda$ has form $x_{\alpha(\lambda),r}$ for some $\alpha\in{\pazocal{C}}$ and $r>0$. So if $h_\lambda$ exists, it is unique.
The existence of $h_\lambda$ will follow once we prove
by induction on $j \ge 0$ the following assertions:
(1) if $\alpha, \beta \in {\pazocal{C}}$ and $r >0$ are such that $x_{\alpha(\lambda_0),r} = x_{\beta(\lambda_0),r} \in {\pazocal{T}}_{\lambda_0}^{(j)}$, then $x_{\alpha(\lambda),r} = x_{\beta(\lambda),r} \in {\pazocal{T}}_{\lambda}^{(j)}$ for all $\lambda \in \Lambda$.
In this case, we let $h_\lambda (x_{\alpha(\lambda_0),r}) := x_{\alpha(\lambda),r}$.
(2) $h_\lambda: {\pazocal{T}}_{\lambda_0}^{(j)} \to {\pazocal{T}}_{\lambda}^{(j)}$ is an extendable
conjugacy.
To simplify notation we set $\lambda_0:=0$ and employ subscripts accordingly.
We start with the case in which $j=0$.
Consider $\alpha \in {\pazocal{C}}$ such that, $|\alpha(0)| > |x_\Lambda|$.
Set $r_\alpha := |\phi_0|(\alpha(0))$ and $R_\alpha := \exp(-\rho) r_\alpha$.
Since the
B\"ottcher coordinates of
escaping critical points are $\rho$-close, it follows that
$r_\alpha=|\phi_\lambda|(\alpha(\lambda)) = |\alpha(\lambda)|$ for all $\lambda$.
The inequality
$$|\phi_\lambda(\alpha(\lambda)) - \phi_0(\alpha(0))| \le R_\alpha$$
which by definition holds for all $\alpha$ such that $|x_\Lambda| < |\alpha(0)| \le |x_\Lambda|^d$, extends to all $\alpha \in {\pazocal{C}}$
such that $|\alpha(0)|>|x_\Lambda|$. This follows since $\phi_\lambda$ and $z \mapsto z^d$
are isometries in the hyperbolic metric on connected components of the complement of $]x_\Lambda,\infty[$ not containing $z=0$.
Note that $x_{\alpha(\lambda),r} \in {\pazocal{T}}^{(0)}_\lambda $
if and only if $r > R_\alpha$. In this case,
$\phi_0(x_{\alpha(0),r})= \phi_\lambda(x_{\alpha(\lambda),r})$ since B\"ottcher coordinates are diameter preserving.
Therefore, $h_\lambda := \phi_\lambda^{-1} \circ \phi_0 : {\pazocal{T}}^{(0)}_0 \to {\pazocal{T}}^{(0)}_\lambda$ is well defined and (1) holds for all $\alpha, \beta$ such that
$|\alpha(0)| > |x_\Lambda|$ and $|\beta(0)|> |x_\Lambda|$. Statement (1) extends to arbitrary $\alpha, \beta \in {\pazocal{C}}$ since in the rest of the cases
$x_{\alpha(\lambda),r} = x_{0,r}$ for some $r > |x_\Lambda|$ whenever $x_{\alpha(\lambda),r} \in {\pazocal{T}}^{(0)}_\lambda $.
Moreover,
the open set with skeleton ${\pazocal{T}}^{(0)}_0$ is mapped by $\phi_\lambda^{-1} \circ \phi_0$ onto the open set
with skeleton ${\pazocal{T}}^{(0)}_\lambda$. Therefore, by Proposition~\ref{p:iso-adm},
$h_\lambda$
is an extendable conjugacy and (2) also holds for $j=0$.
Now we assume that assertions (1) and (2) hold for some $j \ge 0$ and proceed
to establish their validity for $j+1$.
Consider an endpoint $x_0 \in \partial{\pazocal{T}}_{0}^{(j)}$ such that $x_0 \in {\pazocal{T}}_{0}^{(j+1)}$ and
denote by $r_0$
its diameter. We will establish assertions (1) and (2) for $x_0$ and the points on the edges of ${\pazocal{T}}_{0}^{(j+1)}$ growing from $x_0$.
A key observation will be the following.
Assume that $\alpha, \beta \in {\pazocal{C}}$ are such that
$\alpha(0)$ and $\beta(0)$ lie in the same bounded direction at $x_0$ (i.e., $\alpha(0) \prec x_0$, $\beta(0) \prec x_0$ and
$|\alpha (0) - \beta(0)| <r_0$). Then
\begin{equation}
\label{eq:direction}
|\alpha(\lambda) - \beta(\lambda)| < r_0 \text{ for all }\lambda \in \Lambda.
\end{equation}
Indeed, $x_{\alpha(\lambda),r} = x_{\beta(\lambda),r} \in {\pazocal{T}}_{\lambda}^{(j)}$ for all $r >r_0$. Thus, $|\alpha(\lambda) - \beta(\lambda)| \le r_0$ for all $\lambda \in \Lambda$.
By the Maximum Principle, the last inequality is strict, as claimed.
Now, let us suppose that there exists
a point $x_0' \prec x_0$ such that $e_0 = ]x_0',x_0[$ is an edge of ${\pazocal{T}}_{0}^{(j+1)}$ which is not in ${\pazocal{T}}_{0}^{(j)}$. Denote the diameter of $x_0'$ by $r_0'$. By construction, there exists
$\alpha \in {\pazocal{C}}$ such that $\alpha(0) \prec x_0'$. For all
$\lambda \in \Lambda$, let $x_\lambda$ (resp. $x_\lambda'$) be the point of diameter
$r_0$ (resp. $r_0'$) such that $\alpha(\lambda) \prec x_\lambda' \prec x_\lambda$.
Let us first prove that (1) holds if
$]x_\lambda',x_\lambda[$ is an edge of ${\pazocal{T}}_{\lambda}^{(j+1)}$.
Indeed, assuming that $e_\lambda=]x_\lambda',x_\lambda[$ is an edge of ${\pazocal{T}}_{\lambda}^{(j+1)}$,
if $x_{\alpha(0),r} = x_{\beta(0), r} \in e_0$ for some $\beta \in {\pazocal{C}}$, then $|\beta(\lambda) - \alpha(\lambda)| < r_0$ for all $\lambda \in \Lambda$, in view of \eqref{eq:direction}. Therefore, $|\beta(\lambda) - \alpha(\lambda)| \le r_0'$ since the edge $e_\lambda$ is contained
in $]\beta(\lambda), x_\lambda[$. Thus, $x_{\alpha(\lambda),r} = x_{\beta(\lambda), r}$ because $r > r_0'$.
To finish the proof of (1) for $j+1$ we must show that $e_\lambda$ is an edge of ${\pazocal{T}}_{\lambda}^{(j+1)}$ for all $\lambda \in \Lambda$. Let $e_\lambda'= ]a_\lambda,x_\lambda[ \subset ]\alpha(\lambda), x_\lambda[$ be the edge of ${\pazocal{T}}_{\lambda}^{(j+1)}$ in the direction of $\alpha(\lambda)$ at $x_\lambda$.
Since $e_\lambda$ is an arc with constant hyperbolic length contained in $]\alpha(\lambda), x_\lambda[$ and $e'_0=e_0$,
to show that $e'_\lambda = e_\lambda$ for all $\lambda \in \Lambda$ it suffices
to stablish that the hyperbolic length of $e_\lambda'$ is also independent of $\lambda$.
Denote by $D$ the disk corresponding the direction $\alpha(\lambda)$ at $x_\lambda$.
Note that {by Lemma \ref{lem:RH}}, $f_\lambda : D \to f_\lambda(D)$ has degree $m+1$, where $m$ is the number of critical points in $D$, counted with multiplicity which is independent of $\lambda$ in view of \eqref{eq:direction}.
Observe that
$f_\lambda (e_\lambda')$ is the edge of ${\pazocal{T}}^{(j)}_\lambda$ contained
in $]f_\lambda(\alpha(\lambda)), f_\lambda(x_\lambda)[$ with one endpoint at $f_\lambda(x_\lambda)$. By the inductive hypothesis (1), $h_\lambda$ is an isometry between $f_0 (e_0')$ and $f_\lambda (e_\lambda')$ and therefore the hyperbolic
length of $f_\lambda (e_\lambda')$ is a constant independent of $\lambda$, say $L$.
Thus, if $L_\lambda$ denotes the length of
$e_\lambda'$ then $L_\lambda \cdot (m+1) = L$ {by Lemma \ref{lem:factor}}. Therefore, the length of
$e_\lambda'$ is constant equal to the length of $e_0' =e_0$. It follows that $e_\lambda' = e_\lambda$ for all $\lambda$.
To show that $h_\lambda : {\pazocal{T}}^{(j+1)}_0 \to {\pazocal{T}}_\lambda^{(j+1)}$ is an extendable conjugacy it only remains to show that it is locally a translation. For this, consider any $x \in {\pazocal{T}}^{(j+1)}_0 \setminus {\pazocal{T}}^{(j)}_0 $. Denote by $r$ the diameter of $x$. Let $\alpha \in {\pazocal{C}}$ be such that $\alpha(0) \prec x$. For $\lambda \in \Lambda$, consider the traslation $$\tau_\lambda (z) := z + \alpha(\lambda) - \alpha(0).$$
We claim that $\tau_\lambda$ agrees with $h_\lambda$ in a neighborhood of $x$.
Let $\beta \in {\pazocal{C}}$ be such that $\beta(0) \prec x$. Then, for all $\lambda \in \Lambda$, $$|\beta (\lambda) - \alpha(\lambda)| \le r.$$
Therefore,
$|\beta(\lambda) - \tau_\lambda (\beta(0))| \leq r$. But since $\beta(0) = \tau_\lambda (\beta(0))$ we may apply the Maximum Principle to conclude that
$$|\beta(\lambda) - \tau_\lambda (\beta(0))| < r$$
for all $\lambda \in \Lambda$. Thus, $\tau_\lambda$ maps the direction of $\beta(0)$ at $x$,
to the direction of $\beta(\lambda)$ at $\tau_\lambda(x)$. Hence, it coincides with
$h_\lambda$ in a small segment with one endpoint at $x$ contained in $]\beta(0),x[$. Since
this occurs for all $\beta \in {\pazocal{C}}$, the claim follows concluding the proof of the proposition.
\end{proof}
Theorem \ref{ithr:C} is an immediate consequence of the combination of Proposition \ref{prop:p-to-a}, Theorem \ref{ithr:D} and a classical removability result \cite[Proposition 2.7.13]{Fresnel04}.
\begin{proof}[Proof of Theorem \ref{ithr:C}]
The first assertion follows immediately from Proposition \ref{prop:p-to-a} and Theorem \ref{ithr:D}. If ${\pazocal{J}}(\lambda_0) \subset \mathbb{A}^1$,
then ${\pazocal{J}}(\lambda_0)$ is analytically removable, by Theorem~\ref{thr:removability} (see \cite[Proposition 2.7.13]{Fresnel04}). Thus any analytic conjugacy between $f_{\lambda_1}:{\pazocal{B}}(\lambda_1) \to {\pazocal{B}}(\lambda_1)$ and $f_{\lambda_2}:{\pazocal{B}}(\lambda_2) \to {\pazocal{B}}(\lambda_2)$ extends to an analytic automorphism of $\mathbb{A}^1_{an}$. Therefore $f_{\lambda_0}$ and $f_\lambda$ are affinely conjugate for all $\lambda\in\Lambda$. Since there are finitely many elements in ${\operatorname{Poly}}_d$ that are affinely conjugate to $f_{\lambda_0}$, we conclude that $f_\lambda=f_{\lambda_0}$ for all $\lambda\in\Lambda$. Thus the second assertion holds.
\end{proof}
\section{Polynomials with Julia critical points} \label{sec:rigidity}
The aim of this section is to prove Theorem~\ref{ithr:A} and Corollary~\ref{icor:B}.
\subsection{Bounded Fatou components of tame polynomials} \label{sec:bounded}
For a point $x\in\mathbb{A}^1_{an}$ and a polynomial $f$, denote by $\omega(x)$ the $\omega$-limit set of $x$ under $f$.
The following result, due to Trucco, shows that
the orbit of $x \in {\pazocal{J}}(f) \cap {\mathbb{H}}$
accumulates inside a hyperbolic ball around the base point $x_f$ of radius proportional
to $\mathrm{dist}_\H(x,x_f)$.
\begin{proposition}[{\cite[Proposition 7.1]{Trucco14}}]\label{p:distance}
Suppose that $f$ is a tame polynomial of degree at least $2$.
If $x\in{\pazocal{J}}(f) \cap {\mathbb{H}}$, then for all $y\in\omega(x)$
we have that $y \in {\mathbb{H}}$ and $$\mathrm{dist}_\H (y,x_f)\le d^{d-1}\mathrm{dist}_\H (x,x_f).$$ \end{proposition}
One direction of Theorem \ref{ithr:A} is a consequence of the following:
\begin{corollary}\label{coro:oneA}
If $f \in {\operatorname{Poly}}_d$ is a tame polynomial in the closure of the tame shift locus, then ${\pazocal{J}}(f) \subset \mathbb{A}^1$. \end{corollary}
\begin{proof}
Suppose on the contrary that ${\pazocal{J}}(f)\cap{\mathbb{H}}\not=\emptyset$. {Pick $x_0\in{\pazocal{J}}(f)\cap{\mathbb{H}}$.}
By Proposition \ref{p:distance}, for all $y \in \omega(x_0)$
we have that $\mathrm{dist}_\H (f^{\circ n}(y), x_f) \le R$ for some $R >0$.
Let ${\pazocal{D}}$ be a neighborhood of $f$ in ${\operatorname{Poly}}_d$ such that for all $g \in {\pazocal{D}}$
we have that $f(x)=g(x)$ for all $x$ such that $\mathrm{dist}_\H (x, x_f)\le R$.
It follows that every $y \in \omega(x_0) \subset {\mathbb{H}}$ lies in the filled Julia set ${\pazocal{K}}(g)$
for all $g \in {\pazocal{D}}$. Therefore, ${\pazocal{D}}$ is free of polynomials in the tame shift locus. \end{proof}
A necessary and sufficient condition for the abscence of bounded Fatou
components is provided by our following result:
\begin{proposition}\label{p:Julia type I} Let $f\in{\operatorname{Poly}_d}$ be a tame polynomial. Then
$\operatorname{Crit}(f)\subset{\pazocal{B}}(f)\cup{\pazocal{J}}(f)$ if and only if ${\pazocal{J}}(f)\subset\mathbb{A}^1$. \end{proposition} \begin{proof}
If ${\pazocal{J}}(f) \subset \mathbb{A}^1$, then $\mathbb{A}^1_{an} = {\pazocal{B}}(f)\cup{\pazocal{J}}(f)$ clearly contains all the critical points of $f$.
We assume that $\operatorname{Crit}(f)\subset{\pazocal{B}}(f)\cup{\pazocal{J}}(f)$ and proceed by contradiction to show that ${\pazocal{J}}(f)\subset\mathbb{A}^1$.
Hence, suppose that there exists $x\in{\pazocal{J}}(f)\setminus\mathbb{A}^1$. For each $j\ge 0$, set $x_j:=f^{\circ j}(x)$ and regard
$]x_j,x_f[$ as an open subtree equipped with the vertices $\{x_j^{(n)}\}_{n\ge 1}$ formed by all iterated preimages of $x_f$ in
$]x_j,x_f[$ in decreasing order (i.e. $ x_j^{(n)} \succ x_j^{(n+1)}$ for all $n$).
Then $f^{\circ n} (x_j^{(n)}) = x_f$ and $x_j^{(n)}\to x_j$ as $n\to\infty$, see \cite[Proposition 3.6]{Trucco14}.
We claim that there exists a uniform $N\ge 1$ such that the degree of $f$ on $]x_j, x_j^{(n)}[$ is $1$ for all $j\ge 0$ and $n\ge N$. For otherwise, there would exist a critical point $c$, $n_k \to \infty$ and $j_k \ge 0$
such that $\{x_{j_k}^{(n_k)}\}_k \in ]c, x_f]$ is a decreasing sequence. Then both $x_{j_k}$ and $x_{j_k}^{(n_k)}$ would converge, as $k \to \infty$,
to some $y \succeq c$. By
Proposition \ref{p:distance}, we would have that $y \in {\pazocal{J}}(f)\cap {\mathbb{H}} $ and therefore $\overline{D}_y \subset {\pazocal{K}}(f)$, which
contradicts that $\operatorname{Crit}(f)\subset{\pazocal{B}}(f)\cup{\pazocal{J}}(f)$ since $\overline{D}_y\setminus\{y\}$ is contained in ${\pazocal{F}}(f)\setminus{\pazocal{B}}(f)$.
Now from Lemma \ref{lem:factor}, it follows that $f^{\circ(\ell+1)}$ maps $]x_0^{(N+\ell+1)},x_0^{(N+\ell)}[$ isometrically onto $]x_{\ell+1}^{(N)},x_{\ell+1}^{(N-1)}[$ for any $\ell\ge 0$. Morover, there are only finitely many arcs of the form $]x_{\ell+1}^{(N)},x_{\ell+1}^{(N-1)}[$. Therefore, the
length of $]x_0^{(N+\ell+1)},x_0^{(N+\ell)}[$ is uniformly bounded below for all $\ell$. We conclude that
$$\mathrm{dist}_\H(x,x_f)\ge\sum_{\ell\ge 0}\mathrm{dist}_\H(x_0^{(N+\ell+1)},x_0^{(N+\ell)})=\infty,$$
and hence $x\in\mathbb{A}^1$, which contradicts the assumption that $x\in{\pazocal{J}}(f)\setminus\mathbb{A}^1$. \end{proof}
\subsection{Moving critical points to the basin of infinity} \label{sec:perturbation}
The following result establishes a strong form of density of polynomials with all their critical points passive. To state the result let us agree that if $\Lambda \subset \mathbb{C}_v$ is a disk and $\lambda_0 \in \Lambda$, a \emph{maximal disk $\Lambda'$ in $\Lambda \setminus \{\lambda_0\}$} is an open disk of radius $s$ such that $s = |\lambda' - \lambda_0|$ for all $\lambda' \in \Lambda'$.
\begin{lemma}\label{lem:small-disk}
Let $\Lambda$ be a disk of radius $r>0$ parametrizing a critically
marked analytic family $\{f_\lambda\}$ of tame polynomials in
${\operatorname{Poly}}_d$ with the same base point $x_\Lambda$. Given $\lambda_0 \in \Lambda$, {for any $s <r$ sufficiently close to $r$,}
there exist a maximal disk $\Lambda'\subset\Lambda \setminus \{\lambda_0\}$ of radius $s$ such that the following hold:
\begin{enumerate}
\item all the critical points of $\{f_\lambda\}$ are passive in $\Lambda'$; and
\item if $c_i(\lambda)$ is active in $\Lambda$, then $c_i(\lambda') \in {\pazocal{B}}(\lambda')$ for all $\lambda' \in \Lambda'$.
\end{enumerate} \end{lemma}
\begin{proof}
Let $c_1(\lambda), \dots, c_k(\lambda)$ be the active critical points
in $\Lambda$. For all $j$ we have let $g_{j,n} (\lambda):=f^{\circ n}_\lambda (c_j(\lambda))$. Note that $\sup \{ |g_{j,n}(\lambda)| : \lambda \in \Lambda \}$
converges to $\infty$ as $n \to \infty$. Thus, there
exists $N_j$ and $s_j<r$ such that $|x_\Lambda| < r_{j,n} (t):=\sup \{ |g_{j,n}(\lambda)| : |\lambda-\lambda_0| \le t \}$
for all $s_j < t < r$ and $n \ge N_j$. Pick $s<r$ arbitrarily close to $r$
in the value group so that $s>s_j$ for all $j=1, \dots, k$.
Then $g_{j,N_j}$ maps all but finitely maximal open disks contained in $|\lambda-\lambda_0| =s$ onto a maximal open disk contained in the ``sphere''
$S_j:=\{z \in \mathbb{A}^1: |z|=r_{j,N_j} (s) \}$.
Since $S_j \subset {\pazocal{B}}(\lambda)$ for all $\lambda \in \Lambda$,
we may choose a maximal open disks $\Lambda'$
contained in $|\lambda-\lambda_0| =s$ such that for all $j=1, \dots, k$, we have $g_{j,N_j}(\Lambda') \subset S_j$
and the conclusions of the lemma hold. \end{proof}
\begin{corollary}\label{coro:passive}
Suppose $\{f_\lambda\}$ is a critically marked analytic family of tame polynomials in ${\operatorname{Poly}}_d$
parametrized by a disk $\Lambda$ with the same base point $x_\Lambda$.
Assume that $c_1(\lambda), \dots, c_k(\lambda)$ are passive critical points in $\Lambda$
and that $c_{k+1}(\lambda), \dots, c_{d-1}(\lambda)$ are active critical points in $\Lambda$.
Given $\lambda_0 \in \Lambda$ there exists {a disk} $\Lambda' \subset \Lambda$. such that the following hold:
\begin{enumerate}
\item all the critical points are passive in $\Lambda'$;
\item for all $k+1\le j\le d-1$ and all $\lambda' \in \Lambda'$,
we have $c_j(\lambda') \in {\pazocal{B}}(\lambda')$; and
\item for all $1\le j \le k$, if $c_j(\lambda_0) \in {\pazocal{J}}(\lambda_0)$, then
$c_j(\lambda') \in {\pazocal{J}}(\lambda')$ for all $\lambda' \in \Lambda'$.
\end{enumerate} \end{corollary}
\begin{proof}
Given $\lambda_0$, consider a disk
$\Lambda'$ furnished by Lemma \ref{lem:small-disk}. Hence (1) and (2) hold for $\Lambda'$. Assume that $c_j(\lambda_0) \in {\pazocal{J}}(\lambda_0)$ and
$c_j(\lambda)$ is passive in $\Lambda$. Then $c_j(\lambda) \in {\pazocal{K}}(\lambda)$
for all $\lambda \in \Lambda$. Denote by $\delta(\lambda) \ge 0$ the diameter
of the component of ${\pazocal{K}}(\lambda)$ containing $c_j(\lambda)$.
To establish (3) for $c_j$ we must show that $\delta(\lambda)=0$ for all $\lambda \in \Lambda'$. We proceed by contradiction.
Suppose that there exists $\lambda' \in \Lambda'$ so
that $\delta(\lambda') >0$. Then {by Propositions \ref{prop:close} and \ref{prop:p-to-a}} for every closed disk $\overline{\Lambda''} \subset \Lambda'$ {with $\lambda'\in\Lambda''$}, there exists $\rho > 0$ such that the dynamics of $f_\lambda$ over the corresponding $\rho$-trimmed dynamical core ${\pazocal{T}}_\lambda$ is isometrically conjugate to the action of $f_{\lambda'}$ on ${\pazocal{T}}_{\lambda'}$ for all $\lambda\in\Lambda''$. In particular, the iterated preimages of $x_\Lambda$ in $]c_j(\lambda), x_\Lambda]$ isometrically correspond
to those of $x_\Lambda$ in $]c_j(\lambda'), x_\Lambda]$. It follows that
$\delta(\lambda) = \delta(\lambda')$ for all $\lambda \in \overline{\Lambda''}$ for every closed disk $\overline{\Lambda''}$ contained in $\Lambda'$.
Therefore, $\delta(\lambda) = \delta(\lambda')$ for all $\lambda \in \Lambda'$.
Modulo change of coordinates, we may assume that $c_j(\lambda) =0$ for all $\lambda \in \Lambda$. Then, given $y$ with $|y| \le \delta(\lambda')$ and $n \ge 1$,
we have $|f^{\circ n}_\lambda (y)| \le |x_\Lambda|$ for all $\lambda \in \Lambda'$. Since $\Lambda'$ is a maximal open disk in $B=\{\lambda : |\lambda-\lambda_0| \le s \}$ and $\lambda \mapsto f^{\circ n}_\lambda (y)$ is analytic on $B$, we have $|f^{\circ n}_\lambda (y)| \le |x_\Lambda|$ for all $\lambda \in B$. In particular, since $\lambda_0 \in B$, we have that $\delta(\lambda_0) \ge \delta(\lambda')$ which contradicts our assumption that $c_j(\lambda_0) \in {\pazocal{J}}(\lambda_0)$. \end{proof}
{Recall that the space of polynomials ${\operatorname{Poly}}(\mathbf{d})$ with constant critical multiplicity was introduced in Section \ref{sec:poly}. Given $f \in {\operatorname{Poly}}(\mathbf{d})$ with at least one non-escaping critical point, the following result guarantees the existence of a one parameter analytic family passing through $f$ with certain properties that will be need in the proofs of Theorem \ref{ithr:A} and Corollary \ref{icor:B}.}
\begin{lemma}
\label{l:analytic}
Assume that $(f, c_1, \dots, c_k)$ is a nonsimple tame polynomial in
${\operatorname{Poly}}(\mathbf{d})$ such that $c_i \in {\pazocal{B}}(f)$ for $1 \le i \le j$.
If $j <k$, then there exists a nonconstant
analytic family of tame polynomials
$\{(f_\lambda,c_1(\lambda), \dots,c_k(\lambda))\}$ parametrized by a disk
$\Lambda \subset \mathbb{C}_v$ with $0 \in \Lambda$ such that
$(f,c_1,\dots,c_k)=(f_0,c_1(0), \dots,c_k(0))$, the base point of $f_\lambda$ is independent of $\lambda \in \Lambda$ and the
B\"ottcher coordinates of $c_1(\lambda), \dots, c_j(\lambda)$ are constant. \end{lemma}
\begin{proof}
For each $i \le j$,
let $n_i$ be such that $|f^{\circ n_i}(c_i)| > |x_f|$.
Recall that ${\operatorname{Poly}}(\mathbf{d})$ is
naturally identified with elements of a hyperplane in $\mathbb{C}_v^k \times \mathbb{C}_v$ which are
in the complement
of a finite collection of linear subspaces of $\mathbb{C}_v^k \times \mathbb{C}_v$ and
that tame polynomials are open in ${\operatorname{Poly}}(\mathbf{d})$ (endowed with the sup-norm). Thus we may assume that there exists a polydisk $P \subset
{\operatorname{Poly}}(\mathbf{d})$ of tame polynomials
containing $f$. Shrinking $P$ if necessary, we may also assume that for all $g \in P$
the basepoint of
$g$ is $x_f$ and that for every $i \le j$ we have $|g^{\circ n_i}(c_i(g))| > |x_f|$. Without loss of generality we
identify $P$ with $\overline{\mathbb{D}}^k$ where $\overline{\mathbb{D}} = \{ z \in \mathbb{C}_v : |z| \le 1 \}$.
For $n \ge 1$, consider
$\mathbf{r}=(r_1,\dots,r_n)$ where $r_i \in |\mathbb{C}_v^\times|$ for all $i$. Recall
that $P(\mathbf{r})$ denotes the corresponding polydisk and $T_n[\mathbf{r}]$ is the associated Tate algebra, as in the end of Section \ref{sec:poly}. If $r_i =1$ for all $i$, we simply write $T_n$ for the Tate algebra and $\overline{\mathbb{D}}^n$ for the corresponding polydisk.
Note that the maximal spectrum $\operatorname{Sp}(T_n)$ is in natural bijection with $\overline{\mathbb{D}}^n$.
By Lemma \ref{lem:B-analytic},
we have that $F_i(g):=1/\phi_g (g^{\circ n_i}(c_i(g)))$ lies in the Tate algebra $T_k$ for all $i\le j$.
Let $I$ be the ideal generated by $F_1(g)-F_1(f), \dots,
F_j(g)-F_j(f)$
in $T_k$, which we may assume to be a radical ideal.
Our aim is to find the analytic image of one-dimensional disk contained in the vanishing locus $V \subset \overline{\mathbb{D}}^k$ of $I$.
Since $j <k$, it follows that $T_k/I$ is a reduced affinoid algebra of dimension at least $1$. Enlarge $I$, if necessary, to obtain an ideal $J$ such that $A=T_k/J$ is reduced and has dimension~$1$.
Let $B$ be a normalisation of $A$. That is, $B$ is an integrally closed affinoid algebra of dimension~$1$ and $A \hookrightarrow B$ is a finite morphism. Take a maximal ideal $\mathfrak{m}_f$ in $B$ which maps onto $f$ via
$\operatorname{Sp}(B) \to \operatorname{Sp}(A) \hookrightarrow \operatorname{Sp}(T_k) \to \overline{\mathbb{D}}^k$.
Since the dimension of $B$ is $1$, the localization $B_{\mathfrak{m}_f}$ is regular (e.g. see~\cite[Proposition 9.2]{Atiyah69}). By~\cite[Theorem~3.6.3]{Fresnel04}, $B$ can be represented by
$T_n/(G_1,\dots,G_s)$ such that a $(n-1)\times(n-1)$-minor of the
Jacobian matrix
$(\partial G_t/\partial \lambda_m)$, modulo $(G_1,\dots,G_s)$, is not in $\mathfrak{m}_f$. To fix ideas, let us assume that $\mathfrak{m}_f =(\lambda_1, \dots, \lambda_n)$ (i.e. it corresponds to the origin in $\overline{\mathbb{D}}^n$) and let $J':=(G_1,\dots,G_s)$.
By the Implicit Function Theorem (e.g. see~\cite[II.III.10]{Serre64}),
after relabeling if necessary, there exists $\epsilon_i >0$ in $|\mathbb{C}_v^\times|$
such that $B_\epsilon = T_n[(\epsilon_i)]/J'$ is isomorphic to $T_1[\epsilon_1]$. Thus,
$T_k \to A \to B \to B_\epsilon \to T_1[\epsilon_1]$ gives a nonconstant analytic map $\lambda \mapsto f_\lambda$
from $\{ \lambda : |\lambda | \le \epsilon_1 \}$ into $\overline{\mathbb{D}}^k$
such that $f_0 = f$ and $f_\lambda \in V$ for all $\lambda$
(i.e. the escaping critical points $c_1(\lambda), \dots, c_j(\lambda)$ of $f_\lambda$ have constant B\"ottcher coordinates). \end{proof}
Now we can prove Theorem \ref{ithr:A} and Corollary \ref{icor:B}. \begin{proof}[Proof of Theorem \ref{ithr:A}]
{Since we have already established Corollary \ref{coro:oneA},} now assume that $f \in {\operatorname{Poly}}_d$ is a tame polynomial such that ${\pazocal{J}}(f) \subset \mathbb{A}^1$ and $\operatorname{Crit}(f) \cap J(f) \neq \emptyset$. To prove that $f$ is in the closure of the tame shift locus, it will be sufficient to show that there exists an arbitrarily close tame polynomial $g \in {\operatorname{Poly}}_d$ such that ${\pazocal{J}}(g) \subset \mathbb{A}^1$ and $\# \operatorname{Crit}(g) \cap J(g) < \# \operatorname{Crit}(f) \cap J(f)$.
By Lemma \ref{l:analytic} we can consider a nonconstant one-dimensional critically marked analytic family $\{f_\lambda\}_{\lambda\in\Lambda}$ in ${\operatorname{Poly}}_d$ parametrized by an open disk with $0\in\Lambda \subset \mathbb{C}_v$ such that $f_0=f$, the base point $x_\Lambda$ is of $f_\lambda$ is independent of $\lambda\in\Lambda$ and the B\"ottcher coordinates of escaping critical points of $f_0$ are constant. Since ${\pazocal{J}}(f_0)\subset\mathbb{A}^1$, we conclude that at least one critical point of $f_\lambda$ is active in $\Lambda$; for otherwise, by Theorem \ref{ithr:C}, the map $f_\lambda$ would be affine conjugate to $f_0$ for all $\lambda\in\Lambda$, which is a contradiction since affine conjugacy classes in ${\operatorname{Poly}}_d$ are finite. By Corollary \ref{coro:passive}, there exists $\lambda' \in \Lambda$ such that all the critical points of $f_{\lambda'}$ are in ${\pazocal{B}}(f_{\lambda'})$ or in ${\pazocal{J}}(f_{\lambda'})$ and the number of Julia critical points of $f_{\lambda'}$ is strictly smaller than the number of Julia critical points of $f_{0}$. It follows that $g=f_{\lambda'}$ is such that ${\pazocal{J}}(g) \subset \mathbb{A}^1$ and $\#\left(\operatorname{Crit}(g) \cap{\pazocal{J}}(g)\right) < \#\left(\operatorname{Crit}(f) \cap{\pazocal{J}}(f)\right)$. \end{proof}
\begin{proof}[Proof of Corollary \ref{icor:B}] Suppose on the contrary that $c_i$ is passive. Corollary \ref{coro:passive} implies that the critical point $c_i(g)$ is contained in ${\pazocal{J}}(g)$ for all $g$ in a sufficiently small neighborhood of $f$, which contradicts Theorem \ref{ithr:A}. \end{proof}
\appendix \section{A local lemma} \label{appendix} For the sake of completeness we present here the proof of a result which easily follows along the lines of Hensel's Lemma.
\begin{lemma} \label{lem:lift} Consider two polynomials $f,g\in\mathbb{C}_v[z]$ and
suppose that \begin{enumerate} \item $f(x_G)=x_G=g(x_G)$,
\item $T_{x_G}f=T_{x_G}g$, and \item $f'(x_G)=x_G$. \end{enumerate} Then there exist an affinoid $V$ containing $x_G$ and an analytic function $h: V\to h(V)$ such that \begin{enumerate}[label=(\alph*)]
\item $f\circ h=g$ in $V$, and
\item $h(x_G)=x_G$ and $T_{x_G}h= \operatorname{id}$. \end{enumerate} \end{lemma}
\begin{proof}
Denote by $d \ge 2$ the degree of $f$.
For $\rho >1$ sufficiently close to $1$, there exists $0 < s <1$ such that
$|f(z) - g(z)| \le s$ for all $z \in \mathbb{A}^1$ with $|z| \le \rho$.
Choose $s < \mu < 1$ and consider $1 < r < \rho$ such that:
$$\mu r^d < 1 \text{ and } sr^d < \mu.$$
Let $V$ be a closed (rational) affinoid contained in $D(0,r)$ and containing $x_G$ in its interior such that $\mathrm{diam}(x) > \mu$ for all $x \in \partial V$ and the following hold for all
$z \in V \cap \mathbb{A}^1$:
$$\mu r^d < |f'(z)| \text{ and } sr^d < \mu \cdot |f'(z)|^2.$$
Observe that if $z \in V \cap \mathbb{A}^1$ and $w \in \mathbb{A}^1$ is such that $|w| \le \mu$ then $z+w \in V$ and, moreover,
$$|f'(z +w)| = |f'(z)|;$$
indeed, for some polynomials $a_j(z)$ of degree at most $d$ with coefficients in the closed unit ball $\mathfrak{O} \subset \mathbb{C}_v$,
$$|f'(z+w) -f'(z)| = |\sum_{j \ge 1} a_j(z) w^j| \le r^d |w| < |f'(z)|.$$
Given $x \in V \cap \mathbb{A}^1$, for $n\ge 0$, let
\begin{eqnarray*}
z_0 (x) &= & x,\\
w_n(x) &=& - \dfrac{f(z_n(x))-g(x)}{f'(z_n(x))},\\
z_{n+1}(x) & = & z_n(x) + w_n(x).
\end{eqnarray*}
Note that $|w_0(x)| \le s/|f'(x)| < s \sqrt{\mu/sr^d} < \mu$.
We claim that for all $n \ge 0$,
\begin{eqnarray*}
|f(z_{n+1}(x))-g(x)| & < & \mu |f(z_{n}(x))-g(x)|,\\
|w_{n+1}(x)| &<& \mu |w_n(x)|.
\end{eqnarray*} Let us proceed by induction. We will omit the case $n=0$ for the first inequality since the inductive step is very similar, so consider $n \ge 1$ and
assume that the inequalities hold for all $k < n$. Observe that
$$f(z_n(x) + w_n(x))- g(x) = \sum_{j\ge 2} f_j(z_n(x)) w_n(x)^j$$
for some $f_j(z) \in \mathfrak{O}[z]$ of degrees at most $d$.
Hence
\begin{eqnarray*}
|f(z_{n+1}(x)) - g(x)| &\le& r^d |w_n(x)|^2 = r^d \dfrac{|f(z_n(x))-g(x)|^2}{
|f'(z_n(x))|^2}\\
& \le & r^d |f(z_n(x))-g(x)| \dfrac{s}{|f'(z_n(x))|^2}\\
& < & \mu |f(z_n(x))-g(x)|,
\end{eqnarray*}
where the last inequality follows from $|f'(x)| = |f'(z_n(x))|$ since
$|w_k(x)| < \mu$ for all $k < n$.
Similarly, {noting that $|w_n(x)| < \mu$ and hence $|f'(z_{n+1}(x))|= |f'(z_n(x))|$, we have} \begin{eqnarray*}
|w_{n+1}(x)| &\le& \dfrac{r^d |w_n(x)|^2}{|f'(z_n(x))|} \\
& = & |w_n(x)| \dfrac{r^d |f(z_n(x))-g(x)| }{|f'(z_n(x))|^2}\\
&\le& |w_n(x)| \dfrac{r^d s}{|f'(z_n(x))|^2} < \mu |w_n(x)| . \end{eqnarray*}
To finish the proof observe that $w_n(x)$ converges to $0$ uniformly in $V$ as $n\to \infty$, and hence $z_n(x)$ converges to an analytic function $z(x)$ in $V$. Moreover, $$f(z(x))-g(x)=\lim_{n\to\infty}f(z_n(x))-g(x)=0.$$
Also $|z(x)-x|<\mu$ since $|z_n(x)-x|<\mu$ for all $n$. The conclusion of the lemma follows immediately by letting $h(x)=z(x)$.
\end{proof}
\end{document} | arXiv |
Ilona Palásti
Ilona Palásti (1924–1991) was a Hungarian mathematician who worked at the Alfréd Rényi Institute of Mathematics. She is known for her research in discrete geometry, geometric probability, and the theory of random graphs.[1] With Alfréd Rényi and others, she was considered to be one of the members of the Hungarian School of Probability.[2]
Contributions
In connection to the Erdős distinct distances problem, Palásti studied the existence of point sets for which the $i$th least frequent distance occurs $i$ times. That is, in such points there is one distance that occurs only once, another distance that occurs exactly two times, a third distance that occurs exactly three times, etc. For instance, three points with this structure must form an isosceles triangle. Any $n$ evenly-spaced points on a line or circular arc also have the same property, but Paul Erdős asked whether this is possible for points in general position (no three on a line, and no four on a circle). Palásti found an eight-point set with this property, and showed that for any number of points between three and eight (inclusive) there is a subset of the hexagonal lattice with this property. Palásti's eight-point example remains the largest known.[3][4][E]
Another of Palásti's results in discrete geometry concerns the number of triangular faces in an arrangement of lines. When no three lines may cross at a single point, she and Zoltán Füredi found sets of $n$ lines, subsets of the diagonals of a regular $2n$-gon, having $n(n-3)/3$ triangles. This remains the best lower bound known for this problem, and differs from the upper bound by only $O(n)$ triangles.[3][D]
In geometric probability, Palásti is known for her conjecture on random sequential adsorption, also known in the one-dimensional case as "the parking problem". In this problem, one places non-overlapping balls within a given region, one at a time with random locations, until no more can be placed. Palásti conjectured that the average packing density in $d$-dimensional space could be computed as the $d$th power of the one-dimensional density.[5] Although her conjecture led to subsequent research in the same area, it has been shown to be inconsistent with the actual average packing density in dimensions two through four.[6][A]
Palásti's results in the theory of random graphs include bounds on the probability that a random graph has a Hamiltonian circuit, and on the probability that a random directed graph is strongly connected.[7][B][C]
Selected publications
A.
Palásti, Ilona (1960), "On some random space filling problems", Magyar Tud. Akad. Mat. Kutató Int. Közl., 5: 353–360, MR 0146947
B.
Palásti, I. (1966), "On the strong connectedness of directed random graphs", Studia Scientiarum Mathematicarum Hungarica, 1: 205–214, MR 0207588
C.
Palásti, I. (1971), "On Hamilton-cycles of random graphs", Periodica Mathematica Hungarica, 1 (2): 107–112, doi:10.1007/BF02029168, MR 0285437, S2CID 122925690
D.
Füredi, Z.; Palásti, I. (1984), "Arrangements of lines with a large number of triangles", Proceedings of the American Mathematical Society, 92 (4): 561–566, doi:10.2307/2045427, JSTOR 2045427, MR 0760946
E.
Palásti, I. (1989), "Lattice-point examples for a question of Erdős", Periodica Mathematica Hungarica, 20 (3): 231–235, doi:10.1007/BF01848126, MR 1028960, S2CID 123415960
References
1. Former Members of the Institute, Alfréd Rényi Institute of Mathematics, retrieved 2018-09-13.
2. Johnson, Norman L.; Kotz, Samuel (1997), "Rényi, Alfréd", Leading personalities in statistical sciences: From the seventeenth century to the present, Wiley Series in Probability and Statistics: Probability and Statistics, New York: John Wiley & Sons, pp. 205–207, doi:10.1002/9781118150719.ch62, ISBN 0-471-16381-3, MR 1469759. See in particular p. 205.
3. Bárány, Imre (2006), "Discrete and convex geometry", in Horváth, János (ed.), A panorama of Hungarian mathematics in the twentieth century. I, Bolyai Soc. Math. Stud., vol. 14, Springer, Berlin, pp. 427–454, doi:10.1007/978-3-540-30721-1_14, MR 2547518 See in particular p. 444 and p. 449.
4. Konhauser, Joseph D. E.; Velleman, Dan; Wagon, Stan (1996), Which Way Did the Bicycle Go?: And Other Intriguing Mathematical Mysteries, Dolciani Mathematical Expositions, vol. 18, Cambridge University Press, Plate 3, ISBN 9780883853252.
5. Solomon, Herbert (1986), "Looking at life quantitatively", in Gani, J. M. (ed.), The craft of probabilistic modelling: A collection of personal accounts, Applied Probability, New York: Springer-Verlag, pp. 10–30, doi:10.1007/978-1-4613-8631-5_2, ISBN 0-387-96277-8, MR 0861127. See in particular p. 23.
6. Blaisdell, B. Edwin; Solomon, Herbert (1982), "Random sequential packing in Euclidean spaces of dimensions three and four and a conjecture of Palásti", Journal of Applied Probability, 19 (2): 382–390, doi:10.2307/3213489, JSTOR 3213489, MR 0649975
7. Bollobás, Béla (2001), Random graphs, Cambridge Studies in Advanced Mathematics, vol. 73 (2nd ed.), Cambridge, UK: Cambridge University Press, doi:10.1017/CBO9780511814068, ISBN 0-521-80920-7, MR 1864966. See in particular p. 198 and p. 201.
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| Wikipedia |
\begin{document}
\title{Generation and complete nondestructive analysis of hyperentanglement assisted by nitrogen-vacancy centers in resonators }
\author{Qian Liu and Mei Zhang\footnote{Corresponding author: [email protected]} }
\address{Department of Physics, Applied Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing 100875, China}
\date{\today }
\begin{abstract} We present two efficient schemes for the deterministic generation and the complete nondestructive analysis of hyperentangled Bell states in both the polarization and spatial-mode degrees of freedom (DOFs) of two-photon systems, assisted by the nitrogen-vacancy (NV) centers in diamonds coupled to microtoroidal resonators as a result of cavity quantum electrodynamics (QED). With the input-output process of photons, two-photon polarization-spatial hyperentangled Bell states can be generated in a deterministic way and their complete nondestructive analysis can be achieved. These schemes can be generalized to generate and analyze hyperentangled Greenberger-Horne-Zeilinger states of multi-photon systems as well. Compared with previous works, these two schemes relax the difficulty of their implementation in experiment as it is not difficult to obtain the $\pi$ phase shift in single-sided NV-cavity systems. Moreover, our schemes do not require that the transmission for the uncoupled cavity is balanceable with the reflectance for the coupled cavity. Our calculations show that these schemes can reach a high fidelity and efficiency with current technology, which may be a benefit to long-distance high-capacity quantum communication with two DOFs of photon systems. \end{abstract} \pacs{03.67.Bg, 03.67.Hk, 42.50.Pq}
\maketitle
\section{Introduction} \label{sec1}
Recently, hyperentanglement, which is defined as the entanglement in multiple degrees of freedom (DOFs) of a quantum system \cite{heper1,heper2,heper3}, has attracted much attention as it has some important applications in quantum information processing. It can speedup quantum computation (e.g., hyper-parallel photonic quantum computation \cite{hypercomputation,hypercomputation2}) and it can be used to assist the complete Bell-state analysis and entanglement purification. In 2003, Walborn \emph{et al.} \cite{Walborn} presented a simple linear-optical scheme for the complete Bell-state analysis of photons with hyperentanglement. In 2006, Schuck \emph{et al.} \cite{Schuck} demonstrated the complete deterministic analysis of polarization Bell states with only linear optics assisted by polarization-time-bin hyperentanglement. In 2007, Barbieri \emph{et al.} \cite{Barbieri} demonstrated the complete and deterministic discrimination of polarization Bell states assisted by momentum entanglement. In 2009, Wilde and Uskov \cite{quantumec} proposed a quantum-error-correcting-code scheme assisted by linear-optical hyperentanglement. In 2010, some deterministic entanglement purification protocols (EPPs) \cite{EPPsheng2,EPPsheng3,EPPdeng} were proposed with the polarization-spatial hyperentanglement of photon systems, which can solve the troublesome problem that a large amount of quantum resources would be sacrificed in the conventional EPPs, and they are very useful in practical quantum repeaters. In 2012, Wang, Song, and Long \cite{repeater4} proposed an important quantum repeater protocol with polarization-spatial hyperentanglement.
The hyperentanglement of photon systems can also be used to largely increase the channel capacity of quantum communication. For example, Barreiro \emph{et al.} \cite{HESC} beat the channel capacity limit of photonic superdense coding with polarization-orbital-angular-momentum hyperentanglement in linear optics in 2008. In 2010, Sheng, Deng, and Long \cite{kerr} gave the first scheme for the complete hyperentangled-Bell-state analysis (HBSA) for quantum communication, and they designed the pioneering model for the quantum teleportation with two DOFs of photon pairs, resorting to cross-Kerr nonlinearity. In 2011, Pisenti \emph{et al}. \cite{HBSA1} pointed out the limitations for manipulation and measurement of entangled systems with inherently linear unentangling devices. In 2012, Ren \emph{et al.} \cite{HBSA2} proposed another interesting scheme for the complete HBSA for photon systems by using the giant nonlinear optics in quantum dot-cavity systems and they presented the entanglement swapping with photonic polarization-spatial hyperentanglement. In 2012, Wang, Lu, and Long \cite{HBSA3} introduced an interesting scheme for the complete HBSA for photon systems by the giant circular birefringence induced by double-sided quantum-dot-cavity systems. In 2013, Ren, Du, and Deng \cite{HECP} proposed the parameter-splitting method to extract the polarization-spatial maximally hyerentangled photons when the coefficients of the initial partially hyperentangled states are known, and this fascinating method can be achieved with the maximum success probability by performing the protocol only once, resorting to linear-optical elements only. They \cite{HECP} also gave the first hyperentanglement concentration protocol (hyper-ECP) for unknown polarization-spatial less-hyperentangled states with linear-optical elements only \cite{HECP}. Ren and Deng \cite{HyperEPP} presented the first hyperentanglement purification protocol (hyper-EPP) for two-photon systems in polarization-spatial hyperentangled states, and it is very useful in the high-capacity quantum repeaters with hyperentanglement. In 2014, Ren, Du, and Deng \cite{RenHEPP2} gave a two-step hyper-EPP for polarization-spatial hyperentangled states with the quantum-state-joining method, and it has a far higher efficiency. In the same time, Ren and Long \cite{hyperecpgeneral} proposed a general hyper-ECP for photon systems assisted by quantum dot spins inside optical microcavities. Recently, Li and Ghose presented an interesting hyper-ECP resorting to linear optics \cite{lixhyperconcentration} and another efficient hyper-ECP for the multipartite hyperentangled state via the cross-Kerr nonlinearity \cite{lixhyperconcentration2}.
Another attractive candidate for solid-state quantum information processing is the nitrogen-vacancy (NV) center in a diamond, owing to its long decoherence time even at room temperature \cite{Jelezko}, and its spin can be initialized and readout via a highly stable optical transition \cite{Davies,Gruber}. By using the coherent manipulation of an electron spin and nearby individual nuclear spins, Dutt \emph{et al.} \cite{register} demonstrated a controllable quantum register in NV centers in 2007. Decoherence-protected quantum gates for a hybrid solid-state register \cite{register1} was also experimentally demonstrated on a single NV center. As this system allows for high-fidelity polarization and detection of single electron and nuclear spin states even under ambient conditions \cite{PD1,PD2,Gruber,PD3}, the multipartite entanglement among single spins in diamond was demonstrated by Neumann \emph{et al.} \cite{Neumann} in 2008. In 2010, Togan \emph{et al.} \cite{QEG} realized the quantum entanglement generation of an optical photon and an NV center. Photon Fock states on-demand can be implemented in a low-temperature solid-state quantum system with an NV center in a nano-diamond coupled to a nearby high-Q optical cavity \cite{Fock}.
Recently, a combination of NV centers and microcavities, a promising solid-state cavity quantum electrodynamics (QED) system, has gained widespread attention \cite{transfer,s33,s34,s35,s36,s37,s38,WeigateNV,wang}. One of the microcavities is called microtoroidal resonator (MTR) with a quantized whispering-gallery mode (WGM) and required to be of a high Q factor and a small mode volume \cite{MTR1,MTR2}. However, when MTR couples to a fiber, its Q factor is surely degraded \cite{s34}. The single-photon input-output process from a MTR in experiment also has been demonstrated \cite{s34}. In 2009, the quantum nondemolition measurement on a single spin of an NV center has been proposed with a low error rate \cite{s37} and it was experimentally demonstrated through Faraday rotation \cite{s38} in 2010. In 2011, Chen \emph{et al.} \cite{s36} proposed an efficient scheme to entangle separate NV centers by coupling to MTRs. In 2013, Wei and Deng \cite{WeigateNV} proposed some interesting schemes for compact quantum gates on electron-spin qubits assisted by diamond NV centers inside cavities.
In this paper, we present two efficient schemes to generate deterministically hyperentangled states, i.e., hyperentangled Bell states and hyperentangled Greenberger-Horne-Zeilinger (GHZ) states, in which photons are entangled in both the polarization and spatial-mode DOFs, assisted by the NV centers in diamonds coupled to MTRs and the input-output process of photons as a result of cavity QED. We also propose a scheme to distinguish completely the 16 polarization-spatial hyperentangled Bell states, and it works in a nondestructive way. After analyzing the hyperentangled Bell states, the photon systems can be used for other tasks in quantum information processing. Compared with previous works \cite{kerr,HBSA2,HBSA3}, these two schemes relax the difficulty of their implementation in experiment as it is not difficult to obtain the $\pi$ phase shift in single-sided NV-cavity systems. Moreover, they do not require that the transmission for the uncoupled cavity is balanceable with the reflectance for the coupled cavity. Our calculations show that these schemes can work with a high fidelity and efficiency with current experimental techniques, which may be beneficial to long-distance high-capacity quantum communication, such as quantum teleportation, quantum dense coding, and quantum superdense coding with two DOFs of photon systems.
This article is organized as follows. In Sec. \ref{sec2}, we will introduce the diamond-NV-center system and its single-photon input-output process. The generation of hyperentangled Bell states and hyperentangled GHZ states, and the complete nondestructive HBSA are presented in Secs. \ref{sec3} and \ref{sec4}. A discussion and a summary are given in Sec. \ref{sec5}.
\begin{figure}
\caption{(Color online) (a) Schematic diagram for an NV center inside an MTR. (b) The electron energy-level configuration of an NV center in an MTR with the relevant transitions driven by different polarized photons. $L$ ($R$) represents the left(right) circularly polarized photon. }
\label{fig1}
\end{figure}
\section{A nitrogen-vacancy center in microtoroidal resonator} \label{sec2}
As shown in Fig. \ref{fig1}(a), an NV center, composed of a substitutional nitrogen atom and an adjacent vacancy in diamond lattice, is coupled to an MTR with a WGM. The NV center is negatively charged with two unpaired electrons located at the vacancy, and the energy-level structure of the NV center coupling to the cavity mode is shown in Fig. \ref{fig1}(b). The ground state is a spin triplet with the splitting at $2.87$ GHz between the levels
$|m=0\rangle$ and $|m=\pm1\rangle$ owing to spin-spin interactions. The specifically excited state, which is one of the six eigenstates of the full Hamitonian including spin-orbit and spin-spin interactions in the absence of any perturbation, such as by an external magnetic field or crystal strain, is labeled as \cite{QEG}
$|A_{2}\rangle=|E_{-}\rangle|+\rangle+|E_{+}\rangle|-\rangle$, where
$|E_{+}\rangle$ and $|E_{-}\rangle$ are the orbital states with the angular momentum projections $+1$ and $-1$ along the NV axis, respectively. The optical transition is allowed between the ground state $|m=\pm1\rangle$ and the excited state $|A_{2}\rangle$ owing to the total angular momentum conservation \cite{QEG,Lenef}.
An NV center can be modeled as a $\Lambda$-type three-level structure with the ground state $|-\rangle=|m=-1\rangle$ and
$|+\rangle=|m=1\rangle$, and the excited state is
$|1\rangle=|A_{2}\rangle$. The transitions
$|-\rangle\leftrightarrow|1\rangle$ and
$|+\rangle\leftrightarrow|1\rangle$ in the NV center are resonantly coupled to the right (R) and the left (L) circularly polarized photons with the identical transition frequency, respectively. Considering the structure in Fig.\ref{fig1}, which can be modeled as a single-sided cavity, one can write down the Heisenberg equations of motion for this system as follows: \begin{eqnarray} \begin{split} \frac{da}{dt}&=-[i(\omega_{c}-\omega)+\frac{\kappa}{2}+\frac{\kappa_{s}}{2}]\,a-\text{g}\sigma_{\!-}-\sqrt{\kappa}\,a_{in},\\ \frac{d\sigma_{\!-}}{dt}&=-[i(\omega_{0}-\omega)+\frac{\gamma}{2}]\sigma_{\!-}-\text{g}\sigma_{\!z}a,\\ a_{out}&=a_{in}+\sqrt{\kappa}\,a, \end{split} \end{eqnarray} where $\omega$, $\omega_{c}$, and $\omega_{0}$ are the frequencies of the photon, cavity mode, and the atomic-level transition, respectively. $\text{g}$ is the coupling strength between the NV center and the cavity mode. $\frac{\gamma}{2}$ and $\frac{\kappa}{2}$ are the decay rates of the NV center and the cavity field, respectively. $\frac{\kappa_{s}}{2}$ is the side leakage rate of the cavity. $a_{in}$ and $a_{out}$ are the input and the output field operators.
In the weak excitation approximation, the reflection coefficient in the steady state can be obtained, \begin{eqnarray}\label{eq1} r(\omega)=1-\frac{\kappa[i(\omega_{0}-\omega)+\frac{\gamma}{2}]}{[i(\omega_{0}-\omega)+ \frac{\gamma}{2}][i(\omega_{c}-\omega)+\frac{\kappa}{2}+\frac{\kappa_{s}}{2}]+\text{g}^{2}}.\nonumber\\ \end{eqnarray} For $\text{g}=0$, the reflection coefficient $r_{0}(\omega)$ is \begin{eqnarray}\label{eq2} r_{0}(\omega)=\frac{i(\omega_{c}-\omega)-\frac{\kappa}{2}+\frac{\kappa_{s}}{2}}{i(\omega_{c}-\omega)+\frac{\kappa}{2}+\frac{\kappa_{s}}{2}}. \end{eqnarray} From Eqs. (\ref{eq1}) and (\ref{eq2}), one can see that if $\omega_{0}=\omega_{c}=\omega$, \begin{eqnarray} r(\omega)=\frac{(\kappa_{s}-\kappa)\gamma+4\text{g}^{2}}{(\kappa_{s}+\kappa)\gamma+4\text{g}^{2}},\;\;\;\;\;\;\;\; r_{0}(\omega)=\frac{\kappa_{s}-\kappa}{\kappa_{s}+\kappa}. \end{eqnarray}
If the NV center is in the initial state $|-\rangle$ ($|+\rangle$)
and a single polarized photon $|L\rangle$ ($|R\rangle$) is input, the photon will experience a phase shift $e^{i\phi}$ owing to the Faraday rotation. However, if the initial state of the NV center is
$|-\rangle$ ($|+\rangle$), the input photon with $|R\rangle$
($|L\rangle$) polarization will get a phase shift $e^{i\phi_{0}}$. In the resonant condition $\omega_{0}=\omega_{c}=\omega$, when $\kappa_{s}\ll\kappa$ and $4\text{g}^{2}\gg\kappa\gamma$, we approximately have $\phi=0$ and $\phi_{0}=\pi$ from Eq. (\ref{eq1}). The change of the input photon can be summarized as follows \cite{WeigateNV}: \begin{eqnarray}
|R\rangle|+\rangle&\rightarrow&|R\rangle|+\rangle,\;\;\;\;\;\;\,
|R\rangle|-\rangle\;\rightarrow\;-|R\rangle|-\rangle,\nonumber\\
|L\rangle|+\rangle&\rightarrow&-|L\rangle|+\rangle,\;\;\;\;
|L\rangle|-\rangle\;\rightarrow\;|L\rangle|-\rangle.\label{eq5} \end{eqnarray}
\section{Photonic hyperentanglement generation} \label{sec3}
We first describe how to generate two-photon polarization-spatial hyperentangled Bell states assisted by NV centers coupled to MTRs as a result of cavity QED, and then extend this approach for the generation of three-photon polarization-spatial hyperentangled GHZ states.\\
\subsection{Generation of two-photon hyperentangled Bell states} \label{sec31}
A two-photon hyperentangled Bell state in both the polarization and the spatial-mode DOFs can be expressed as \begin{eqnarray}
|\eta_{1}^{+}\rangle_{PS} = \frac{1}{2}(|RR\rangle+|LL\rangle)_{ab} (|a_{1}b_{1}\rangle+|a_{2}b_{2}\rangle)_{ab}. \end{eqnarray}
Here, $|R\rangle$ and $|L\rangle$ denote the right-circular polarization and the left-circular polarization of photons, respectively. $a_{1}$ ($b_{1}$) and $a_{2}$ ($b_{2}$) are the different spatial modes for photon $a$ (b). The subscripts $P$ and $S$ denote the polarization and the spatial-mode DOFs, respectively. $a$ and $b$ represent the two photons in the hyperentangled state. The four Bell states in the polarization DOF can be expressed as \begin{eqnarray} \begin{split}
|\Phi_{1}^{\pm}\rangle_{P}&=\frac{1}{\sqrt{2}}(|RR\rangle\pm|LL\rangle),\\
|\Phi_{2}^{\pm}\rangle_{P}&=\frac{1}{\sqrt{2}}(|LR\rangle\pm|RL\rangle), \end{split} \end{eqnarray} and those in the spatial-mode DOF can be written as \begin{eqnarray} \begin{split}
|\Phi_{1}^{\pm}\rangle_{S}&=\frac{1}{\sqrt{2}}(|a_{1}b_{1}\rangle\pm|a_{2}b_{2}\rangle),\\
|\Phi_{2}^{\pm}\rangle_{S}&=\frac{1}{\sqrt{2}}(|a_{2}b_{1}\rangle\pm|a_{1}b_{2}\rangle). \end{split} \end{eqnarray}
The principle of our scheme for the two-photon polarization-spatial hyperentangled Bell states generation (HBSG) assisted by NV centers coupled to MTRs as a result of cavity QED is shown in Fig.\ref{fig2}. Here SW is an optical switch and BS represents a $50:50$ beam splitter which can accomplish the following transformation in the spatial-mode DOF of the photons, \begin{eqnarray} \begin{split} K_{a_{1}(b_{2})}^{\dag}&\rightarrow \frac{1}{\sqrt{2}}(K_{c_{1}(d_{1})}^{\dag}+K_{c_{2}(d_{2})}^{\dag}),\\ K_{a_{2}(b_{1})}^{\dag}&\rightarrow \frac{1}{\sqrt{2}}(K_{c_{1}(d_{1})}^{\dag}-K_{c_{2}(d_{2})}^{\dag}). \end{split} \end{eqnarray}
\begin{figure*}
\caption{(Color online) Schematic diagram for two-photon polarization-spatial HBSG. PBS represents a circular polarization beam splitter which is used to transmit the $L$ polarized photon and reflect the $R$ polarized photon, respectively. SW represents an optical switch and BS is a 50:50 beam splitter. HWP represents a half-wave plate which is used to perform a bit-flip operation
$X=|R\rangle\langle L|+|L\rangle\langle R|$ on the photon in the polarization DOF. NV$_1$ and NV$_2$ represent two NV centers. $k_1$ and $k_2$ ($k=a,b,c,d$) represent different spatial modes.}
\label{fig2}
\end{figure*}
Suppose that two NV centers NV$_1$ and NV$_2$ are initialed to the superposition states
$|\varphi^{+}\rangle_{1}=|\varphi^{+}\rangle_{2}=\frac{1}{\sqrt{2}}(|-\rangle+|+\rangle)$, and the two photons $a$ and $b$ with the same frequency are prepared in the same initial state
$|\phi\rangle_{a}=|\phi\rangle_{b}=\frac{1}{\sqrt{2}}(|R\rangle+|L\rangle)$. Photons $a$ and $b$ are successively sent into the device shown in Fig.\ref{fig2}. A time interval $\Delta t$ exists between two photons, and $\Delta t$ should be less than the spin coherence time $T$. When photon $a$ passes through the cavity, the optical switch (SW) is switched to await photon $b$. After passing through the two NV centers, the two photons $a$ and $b$ can be entangled with the electron spins in the NV centers in the two cavities. The corresponding transformations on the states can be described as follows:
\begin{eqnarray}\label{eq10}
\!\!\vert
\Phi\rangle_0\!\!\!\!&=&\!\!|\phi\rangle_{a}|\phi\rangle_{b}|\varphi^{+}\rangle_{1}|\varphi^{+}\rangle_{2}\nonumber\\
\!\!\!\!&\xrightarrow{P\!B\!S\!s}&\!\!\!\frac{1}{2}(|L\rangle|a_{1}\!\rangle
\!+\! |R\rangle|a_{2}\!\rangle)(|L\rangle|b_{1}\!\rangle
\!+\! |R\rangle|b_{2}\!\rangle)|\varphi^{+}\!\rangle_{1}|\varphi^{+}\!\rangle_{2},\nonumber\\ &\!\!\!\! \xrightarrow{NV_1}&\!\!\! \frac{1}{2}
[(|LL\rangle|a_{1}b_{1}\rangle+|RR\rangle|a_{2}b_{2}\rangle)|\varphi^{+}\rangle_{1}\nonumber\\ &&
-(|LR\rangle|a_{1}b_{2}\rangle+|RL\rangle|a_{2}b_{1}\rangle)|\varphi^{-}\rangle_{1} ]|\varphi^{+}\rangle_{2},\nonumber\\ &\!\!\!\! \xrightarrow{B\!S}&\!\!\!\frac{1}{4}
[|\Phi_{1}^{+}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}|\varphi^{+}\rangle_{1}
+|\Phi_{1}^{-}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}|\varphi^{+}\rangle_{1}\nonumber\\ &&
-|\Phi_{2}^{+}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}|\varphi^{-}\!\rangle_{1}
\!+\!|\Phi_{2}^{-}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}|\varphi^{-}\!\rangle_{1}]|\varphi^{+}\!\rangle_{2},\nonumber\\ &\!\!\!\!
\xrightarrow{NV_2\;}&\!\!\!\frac{1}{4}(-|\Phi_{1}^{-}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}|\varphi^{+}\rangle_{1}|\varphi^{-}\rangle_{2} \nonumber\\
&&+\,|\Phi_{1}^{-}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}|\varphi^{+}\rangle_{1}|\varphi^{+}\rangle_{2}\nonumber\\ &&
+\,|\Phi_{2}^{-}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}|\varphi^{-}\rangle_{1}|\varphi^{-}\rangle_{2} \nonumber\\
&&+\,|\Phi_{2}^{-}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}|\varphi^{-}\rangle_{1}|\varphi^{+}\rangle_{2}).
\end{eqnarray}
Here
$|\varphi^{\pm}\rangle=\frac{1}{\sqrt{2}}(|-\rangle\pm|+\rangle)$.
From Eq. (\ref{eq10}), one can see that the injecting photons $a$
and $b$ pass through PBS in sequence. Each of them is split into two wave-packets. The part of photon $a$ ($b$) in the state $|L\rangle$
is transmitted to path $a_{1}$ ($b_{1}$), while the part in the state $|R\rangle$ is reflected by PBS to path $a_{2}$ ($b_{2}$) and subsequently it interacts with the first NV center (named NV$_1$). The wave-packets from the spatial modes $a_{1}$ and $a_{2}$ ($b_{1}$ and $b_{2}$) are mixed at the beam splitter (BS). HWP is used to keep the polarization state of the photon unchanged. After BSs, the states of the two-photon systems are divided into two groups, according to the state of NV$_2$ when
it is measured with the basis $\{\vert \varphi^+\rangle, \vert \varphi^-\rangle\}$. So do the outcomes of the measurement on NV$_1$. The relationship between the measurement outcomes of these two NV centers and the polarization-spatial hyperentangled Bell states of the two photons is shown in Table \ref{table1}.
\begin{table}[htb] \centering \caption{The relation between the outcomes of the two NV centers and the final polarization-spatial hyperentangled Bell states.}
\begin{tabular}{ccc}
\hline\hline
NV$_1$ & $\;\;\;\;\;\;\;\;$ NV$_2$ $\;\;\;\;\;\;\;\;$ & hyperentangled Bell states \\
\hline
$|\varphi^{+}\rangle_1$ & $|\varphi^{-}\rangle_2$ & $|\Phi_{1}^{-}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$ \\
$|\varphi^{+}\rangle_1$ & $|\varphi^{+}\rangle_2$ & $|\Phi_{1}^{-}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}$ \\
$|\varphi^{-}\rangle_1$ & $|\varphi^{-}\rangle_2$ & $|\Phi_{2}^{-}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$ \\
$|\varphi^{-}\rangle_1$ & $|\varphi^{+}\rangle_2$ & $|\Phi_{2}^{-}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}$ \\ \hline\hline \end{tabular}\label{table1}
\end{table}
From Table \ref{table1}, one can see that if NV$_1$ is in the state $|\varphi^{+}\rangle_{1}$ and NV$_2$ is in the state
$|\varphi^{-}\rangle_{2}$, the two-photon system $ab$ is in the hyperentangled Bell state
$|\Phi_{1}^{-}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$. When the two NV centers are in the states $|\varphi^{+}\rangle_{1}$ and
$|\varphi^{+}\rangle_{2}$, respectively, the two-photon system is in the hyperentangled Bell state
$|\Phi_{1}^{-}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}$. The hyperentangled Bell states
$|\Phi_{2}^{-}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$ and
$|\Phi_{2}^{-}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}$ also correspond to different combinations of the states of the two NV centers. Therefore, one can generate the polarization-spatial hyperentangled Bell states of the two-photon system by measuring the states of the two NV centers. By applying a Hadamard operation on the electron-spin state, its states
$\frac{1}{\sqrt{2}}(|-\rangle+|+\rangle)$ and
$\frac{1}{\sqrt{2}}(|-\rangle-|+\rangle)$ can be rotated to
$|+\rangle$ and $|-\rangle$, respectively. The measurement on spin readout can be achieved with the resonant optical excitation \cite{Gaebel}. Other polarization-spatial hyperentangled Bell states can be obtained in a similar way or by resorting to the single-qubit operations and the acquired hyperentangled Bell state.
\begin{figure*}
\caption{(Color online) Schematic diagram for the generation of three-photon polarization-spatial hyperentangled GHZ states.}
\label{fig3}
\end{figure*}
\subsection{Generation of three-photon hyperentangled GHZ states} \label{sec32}
Similar to the case for two-photon polarization-spatial hyperentangled Bell states, we denote a three-photon polarization-spatial hyperentangled GHZ state as \begin{eqnarray}\label{eq11}
|\zeta_{1}^{+}\rangle_{PS} = \frac{1}{2}(|RRR\rangle\!+\!|LLL\rangle)_{abc}(|a_{1}b_{1}c_{1}\rangle\!+\!|a_{2}b_{2}c_{2}\rangle)_{abc}.\nonumber\\ \end{eqnarray} Here $a$, $b$, and $c$ represent the three photons in the hyperentangled state. The eight three-photon GHZ states in the polarization DOF can be expressed as \begin{eqnarray}\label{eq12} \begin{split}
|\Psi_{1}^{\pm}\rangle_{P}&=\frac{1}{\sqrt{2}}(|RRR\rangle\pm|LLL\rangle),
\\
|\Psi_{2}^{\pm}\rangle_{P}&=\frac{1}{\sqrt{2}}(|LRR\rangle\pm|RLL\rangle), \\
|\Psi_{3}^{\pm}\rangle_{P}&=\frac{1}{\sqrt{2}}(|RLR\rangle\pm|LRL\rangle), \\
|\Psi_{4}^{\pm}\rangle_{P}&=\frac{1}{\sqrt{2}}(|RRL\rangle\pm|LLR\rangle). \end{split} \end{eqnarray} and the eight GHZ states in the spatial-mode DOF are \begin{eqnarray}\label{eq13} \begin{split}
|\Psi_{1}^{\pm}\rangle_{S}&=\frac{1}{\sqrt{2}}(|a_{1}b_{1}c_{1}\rangle\pm|a_{2}b_{2}c_{2}\rangle),\\
|\Psi_{2}^{\pm}\rangle_{S}&=\frac{1}{\sqrt{2}}(|a_{2}b_{1}c_{1}\rangle\pm|a_{1}b_{2}c_{2}\rangle),\\
|\Psi_{3}^{\pm}\rangle_{S}&=\frac{1}{\sqrt{2}}(|a_{1}b_{2}c_{1}\rangle\pm|a_{2}b_{1}c_{2}\rangle),\\
|\Psi_{4}^{\pm}\rangle_{S}&=\frac{1}{\sqrt{2}}(|a_{1}b_{1}c_{2}\rangle\pm|a_{2}b_{2}c_{1}\rangle). \end{split} \end{eqnarray}
The principle of our scheme for the generation of a three-photon polarization-spatial hyperentangled GHZ state is shown in Fig.\ref{fig3}. Considering all the three NV centers are in the initial state
$|\varphi^{+}\rangle_{1}=|\varphi^{+}\rangle_{2}=|\varphi^{+}\rangle_{3}=\frac{1}{\sqrt{2}}(|-\rangle+|+\rangle)$, and the flying photons $a$, $b$, and $c$ are in the superposition state
$|\phi\rangle_{a}=|\phi\rangle_{b}=|\phi\rangle_{c}=\frac{1}{\sqrt{2}}(|R\rangle+|L\rangle)$. A brief description of our scheme for hyperentangled-GHZ-state generation can be written as follows.
Each of the three photons $a$, $b$, and $c$ is split into two wave-packets by PBS. The photon in the state $|L\rangle$ is injected into the pathes $a_{1}$, $b_{1}$, and $c_{1}$, while the photon in the state $|R\rangle$ is sent into the pathes $a_{2}$, $b_{2}$, and $c_{2}$. The photons in the pathes $a_{2}$ and $b_{2}$ interact with
NV$_1$, and those in the pathes $b_{1}$ and $c_{2}$ interact with NV$_2$. BSs mix the spatial modes $a_{1}$ and $a_{2}$, $b_{1}$ and $b_{2}$, and $c_{1}$ and $c_{2}$. The states of the three photons are divided into four groups, according to the states of NV$_1$ and NV$_2$. Under the condition that NV$_3$ is imported, the hyperentangled states of the photons can be determined by measuring the states of the NV centers. The evolution of the whole system can be described as
\begin{eqnarray}\label{eq14}
\vert \Psi\rangle_0\!\!\!\!&=&\!\!|\phi\rangle_{a}|\phi\rangle_{b}|\phi\rangle_{c}|\varphi^{+}\rangle_{1}|\varphi^{+}\rangle_{2}|\varphi^{+}\rangle_{3}\nonumber\\ \!\!\!\!\!\!&\xrightarrow{U_T}&\!\!
\frac{1}{2\sqrt{2}}(|\Psi_{1}^{+}\rangle_{P}|\Psi_{2}^{+}\rangle_{S})|\varphi^{+}\rangle_{1}|\varphi^{-}\rangle_{2}|\varphi^{-}\rangle_{3} \nonumber\\
&& -|\Psi_{1}^{+}\rangle_{P}|\Psi_{2}^{-}\rangle_{S}|\varphi^{+}\rangle_{1}|\varphi^{-}\rangle_{2}|\varphi^{+}\rangle_{3}\nonumber\\
&& +|\Psi_{4}^{+}\rangle_{P}|\Psi_{2}^{+}\rangle_{S}|\varphi^{+}\rangle_{1}|\varphi^{+}\rangle_{2}|\varphi^{-}\rangle_{3} \nonumber\\
&& +|\Psi_{4}^{+}\rangle_{P}|\Psi_{2}^{-}\rangle_{S}|\varphi^{+}\rangle_{1}|\varphi^{+}\rangle_{2}|\varphi^{+}\rangle_{3}\nonumber\\
&& +|\Psi_{3}^{+}\rangle_{P}|\Psi_{2}^{+}\rangle_{S}|\varphi^{-}\rangle_{1}|\varphi^{+}\rangle_{2}|\varphi^{-}\rangle_{3} \nonumber\\
&& -|\Psi_{3}^{+}\rangle_{P}|\Psi_{2}^{-}\rangle_{S}|\varphi^{-}\rangle_{1}|\varphi^{+}\rangle_{2}|\varphi^{+}\rangle_{3}\nonumber\\
&& +|\Psi_{2}^{+}\rangle_{P}|\Psi_{1}^{+}\rangle_{S}|\varphi^{-}\rangle_{1}|\varphi^{-}\rangle_{2}|\varphi^{-}\rangle_{3} \nonumber\\
&& -|\Psi_{2}^{+}\rangle_{P}|\Psi_{2}^{-}\rangle_{S}|\varphi^{-}\rangle_{1}|\varphi^{-}\rangle_{2}|\varphi^{+}\rangle_{3}).
\end{eqnarray}
Here $U_T$ represents the total operation by PBS, NV$_1$, NV$_2$, BS, and NV$_3$ in sequence.
$|\varphi^{\pm}\rangle=\frac{1}{\sqrt{2}}(|-\rangle\pm|+\rangle)$. The relation between the outcomes of the measurements on the three NV centers and the obtained final polarization-spatial hyperentangled GHZ states is shown in Table \ref{table2}.
\begin{table}[htb] \centering \caption{The relation between the outcomes of the three NV centers and the final hyperentangled GHZ states}
\begin{tabular}{cccc} \hline\hline
NV$_1$ & $\;\;\;\; $ NV$_2$ $\;\;\; $ & $\;\;\; $ NV$_3$ $\;\;\;\; $ & hyperentangled GHZ states \\
\hline
$|\varphi^{+}\rangle_{1}$&$|\varphi^{-}\rangle_{2}$&$|\varphi^{-}\rangle_{3}$&$|\Psi_{1}^{+}\rangle_{P}|\Psi_{2}^{+}\rangle_{S}$ \\
$|\varphi^{+}\rangle_{1}$&$|\varphi^{-}\rangle_{2}$&$|\varphi^{+}\rangle_{3}$&$|\Psi_{1}^{+}\rangle_{P}|\Psi_{2}^{-}\rangle_{S}$ \\
$|\varphi^{+}\rangle_{1}$&$|\varphi^{+}\rangle_{2}$&$|\varphi^{-}\rangle_{3}$&$|\Psi_{4}^{+}\rangle_{P}|\Psi_{2}^{+}\rangle_{S}$ \\
$|\varphi^{+}\rangle_{1}$&$|\varphi^{+}\rangle_{2}$&$|\varphi^{+}\rangle_{3}$&$|\Psi_{4}^{+}\rangle_{P}|\Psi_{2}^{-}\rangle_{S}$ \\
$|\varphi^{-}\rangle_{1}$&$|\varphi^{+}\rangle_{2}$&$|\varphi^{-}\rangle_{3}$&$|\Psi_{3}^{+}\rangle_{P}|\Psi_{2}^{+}\rangle_{S}$ \\
$|\varphi^{-}\rangle_{1}$&$|\varphi^{+}\rangle_{2}$&$|\varphi^{+}\rangle_{3}$&$|\Psi_{3}^{+}\rangle_{P}|\Psi_{2}^{-}\rangle_{S}$ \\
$|\varphi^{-}\rangle_{1}$&$|\varphi^{-}\rangle_{2}$&$|\varphi^{-}\rangle_{3}$&$|\Psi_{2}^{+}\rangle_{P}|\Psi_{1}^{+}\rangle_{S}$ \\
$|\varphi^{-}\rangle_{1}$&$|\varphi^{-}\rangle_{2}$&$|\varphi^{+}\rangle_{3}$&$|\Psi_{2}^{+}\rangle_{P}|\Psi_{2}^{-}\rangle_{S}$ \\ \hline\hline \end{tabular}\label{table2}
\end{table}
Table \ref{table2} shows that if the three NV centers are in the states $|\varphi^{+}\rangle_{1}$, $|\varphi^{-}\rangle_{2}$, and
$|\varphi^{-}\rangle_{3}$, respectively, the three-photon system $abc$ is in the polarization-spatial hyperentangled GHZ state
$|\Phi_{1}^{+}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}$. When the NV centers are in the state $|\varphi^{+}\rangle_{1}$,
$|\varphi^{-}\rangle_{2}$, and $|\varphi^{+}\rangle_{3}$, the final hyperentangled GHZ state of the three photons is
$|\Phi_{1}^{+}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}$. Our scheme can in principle generate the eight deterministic hyperentangled three-photon GHZ states by measuring the states of the NV centers.
\section{Complete nondestructive polarization-spatial hyperentangled-Bell-state analysis} \label{sec4}
The principle of our scheme for distinguishing the 16 hyperentangled Bell states in both the polarization and the spatial-mode DOFs of entangled photon pairs is shown in Fig. \ref{fig4}. Here, we use four NV centers coupled to four MTRs and a few linear-optical elements to achieve the complete nondestructive analysis of hyperentangled Bell states. Considering that two NV centers are both initialized in the superposition state
$|\varphi^{+}\rangle_{1}=|\varphi^{+}\rangle_{2}=\frac{1}{\sqrt{2}}(|-\rangle+|+\rangle)$, and a hyperentangled photon pair is in one of the 16 hyperentangled Bell states. As shown in Fig. \ref{fig4}, one can let photon $a$ pass through the cavities first and then photon $b$. SW is used for the arrival of photon $b$ after photon $a$ passes through the cavity. $\Delta t$ is the time interval between photon $a$ and photon $b$ which is smaller than the spin coherence time of the NV center.
\begin{figure*}
\caption{(Color online) Schematic diagram for the complete nondestructive analysis of two-photon polarization-spatial hyperentangled
Bell states. }
\label{fig4}
\end{figure*}
According to Eq.(\ref{eq5}), after the photons pass through NV$_1$ and NV$_2$, the evolution of the system composed of the two photons and the two NV centers is shown in Table \ref{table3}. One can see that the 16 polarization-spatial hyperentangled states can be divided into four groups. If
both the spins of NV$_1$ and NV$_2$ are changed to be
$|\varphi^{-}\rangle_{1}|\varphi^{-}\rangle_{2}$, the photon pair $ab$ is in one of the four hyperentangled Bell states
$|\Phi_{1}^{\pm}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}$ and
$|\Phi_{2}^{\pm}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$. If both NV$_1$ and NV$_2$ stay in the initial state
$|\varphi^{+}\rangle_{1}|\varphi^{+}\rangle_{2}$, the photon pair $ab$ is in one of the four hyperentangled Bell states
$|\Phi_{1}^{\pm}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}$ and
$|\Phi_{2}^{\pm}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}$. Otherwise, when NV$_1$ and NV$_2$ are in the state of
$|\varphi^{-}\rangle_{1}|\varphi^{+}\rangle_{2}$ (or
$|\varphi^{+}\rangle_{1}|\varphi^{-}\rangle_{2}$), the photon pair $ab$ is in one of the four hyperentangled Bell states
$|\Phi_{2}^{\pm}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}$ and
$|\Phi_{1}^{\pm}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$ (or
$|\Phi_{1}^{\pm}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}$ and
$|\Phi_{2}^{\pm}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}$).
\begin{table}[htb] \centering \caption{The relation between the initial states and the final states of the system after the photons pass through NV$_{1}$ and NV$_{2}$.}
\begin{tabular}{cc} \hline\hline
$\;\;\; $ Initial states $\;\;\; $ & $\;\;\;\;\;\;\;\;\;$ Final states $\;\;\; $ \\
\hline
$|\Phi_{1}^{\pm}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}|\varphi^{+}\rangle_{1}|\varphi^{+}\rangle_{2}$
&$\;\;\;\;$ $|\Phi_{1}^{\pm}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}|\varphi^{-}\rangle_{1}|\varphi^{-}\rangle_{2}$ \\
$|\Phi_{2}^{\pm}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}|\varphi^{+}\rangle_{1}|\varphi^{+}\rangle_{2}$
&$\;\;\;\;$ $|\Phi_{2}^{\mp}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}|\varphi^{-}\rangle_{1}|\varphi^{+}\rangle_{2}$ \\
$|\Phi_{1}^{\pm}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}|\varphi^{+}\rangle_{1}|\varphi^{+}\rangle_{2}$
&$\;\;\;\;$ $|\Phi_{1}^{\mp}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}|\varphi^{-}\rangle_{1}|\varphi^{+}\rangle_{2}$ \\
$|\Phi_{2}^{\pm}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}|\varphi^{+}\rangle_{1}|\varphi^{+}\rangle_{2}$
&$\;\;\;\;$ $|\Phi_{2}^{\pm}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}|\varphi^{-}\rangle_{1}|\varphi^{-}\rangle_{2}$ \\
$|\Phi_{1}^{\pm}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}|\varphi^{+}\rangle_{1}|\varphi^{+}\rangle_{2}$
&$\;\;\;\;$ $|\Phi_{1}^{\mp}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}|\varphi^{+}\rangle_{1}|\varphi^{-}\rangle_{2}$ \\
$|\Phi_{2}^{\pm}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}|\varphi^{+}\rangle_{1}|\varphi^{+}\rangle_{2}$
&$\;\;\;\;$ $|\Phi_{2}^{\pm}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}|\varphi^{+}\rangle_{1}|\varphi^{+}\rangle_{2}$ \\
$|\Phi_{1}^{\pm}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}|\varphi^{+}\rangle_{1}|\varphi^{+}\rangle_{2}$
&$\;\;\;\;$ $|\Phi_{1}^{\pm}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}|\varphi^{+}\rangle_{1}|\varphi^{+}\rangle_{2}$ \\
$|\Phi_{2}^{\pm}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}|\varphi^{+}\rangle_{1}|\varphi^{+}\rangle_{2}$
&$\;\;\;\;$ $|\Phi_{2}^{\mp}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}|\varphi^{+}\rangle_{1}|\varphi^{-}\rangle_{2}$ \\ \hline\hline \end{tabular}\label{table3}
\end{table}
Subsequently, one can let the photons pass through the BS and QWP which are used to transform the phase information discrimination to parity information discrimination both for the polarization DOF and the spatial-mode DOF. According to Table \ref{table3}, after the photons pass through BS and QWP, the initial hyperentangled Bell state $|\Phi_{1}^{\pm}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}$ and
$|\Phi_{2}^{\pm}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$ for the same group will become \begin{eqnarray}\label{eq16}
|\Phi_{1}^{+}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}&\xrightarrow[]{NV_1, NV_2, BS, QWP}&
|\Phi_{1}^{+}\rangle_{P}|\Phi_{1}^{+}\rangle_{S},\nonumber\\
|\Phi_{1}^{-}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}&\xrightarrow[]{NV_1, NV_2, BS, QWP}&
|\Phi_{2}^{+}\rangle_{P}|\Phi_{1}^{+}\rangle_{S},\nonumber\\
|\Phi_{2}^{+}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}&\xrightarrow[]{NV_1, NV_2, BS, QWP}&
|\Phi_{1}^{-}\rangle_{P}|\Phi_{2}^{+}\rangle_{S},\nonumber\\
|\Phi_{2}^{-}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}&\xrightarrow[]{NV_1, NV_2, BS, QWP}&
|\Phi_{2}^{-}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}.\;\;\;\; \end{eqnarray} From Table \ref{table3}, one can see that the final hyperentangled Bell states of two-photon systems in the same group will be changed into four different groups. By letting the photons pass through NV$_3$ and NV$_4$ which has the same syndetic connection with NV$_1$ and NV$_2$, one can distinguish those hyperentangled Bell states. The hyperentangled Bell states in the other three groups will have the same conditions. After passing through all the elements shown in Fig. \ref{fig4}, the final states of the two photons become those ones shown in Table \ref{table4}.
\begin{table}[htb] \centering \caption{The relation between the initial hyperentangled Bell states and the final states of the two photons after passing through all the elements.}
\begin{tabular}{cccc} \hline\hline
Initial states $\; $ & Final states & $\; $ Initial states $\; $ & $\; $ Final states \\
\hline
$|\Phi_{1}^{+}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}$ &$\; $
$|\Phi_{1}^{+}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}$ &$\; $
$|\Phi_{1}^{-}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}$ &$\; $ $|\Phi_{2}^{-}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}$ \\
$|\Phi_{2}^{+}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}$ & $\; $
$|\Phi_{2}^{-}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}$ &$\; $
$|\Phi_{2}^{-}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}$ &$\; $ $|\Phi_{1}^{-}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}$ \\
$|\Phi_{1}^{+}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$ &$\; $
$|\Phi_{2}^{+}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$ &$\; $
$|\Phi_{1}^{-}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$ &$\; $ $|\Phi_{1}^{-}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}$ \\
$|\Phi_{2}^{+}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$ &$\; $
$|\Phi_{1}^{+}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$ &$\; $
$|\Phi_{2}^{-}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$ &$\; $ $|\Phi_{2}^{-}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}$ \\
$|\Phi_{1}^{+}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}$ &$\; $
$|\Phi_{2}^{+}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}$ &$\; $
$|\Phi_{1}^{-}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}$ &$\; $ $|\Phi_{1}^{-}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$ \\
$|\Phi_{2}^{+}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}$ &$\; $
$|\Phi_{1}^{+}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}$ &$\; $
$|\Phi_{2}^{-}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}$ &$\; $ $|\Phi_{2}^{-}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$ \\
$|\Phi_{1}^{+}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}$ &$\; $
$|\Phi_{1}^{+}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}$ &$\; $
$|\Phi_{1}^{-}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}$ &$\; $ $|\Phi_{2}^{-}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}$ \\
$|\Phi_{2}^{+}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}$ &$\; $
$|\Phi_{2}^{+}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}$ &$\; $
$|\Phi_{2}^{-}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}$ &$\; $ $|\Phi_{1}^{-}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}$ \\ \hline\hline \end{tabular}\label{table4}
\end{table}
In this time, by using linear-optical elements, one can transform the state of the two-photon system into its initial hyperentangled Bell state. That is, one can completely distinguish the 16 hyperentangled Bell states with the measurement outcomes of the states of the four NV centers rather than using single-photon detectors to proceed destructive measurement. The relation between the initial hyperentangled Bell states and the measurement outcomes of the states of the NV centers is shown in Table \ref{table5}.
\begin{table}[htb] \centering \caption{The relation between the initial hyperentangled Bell states and the measurement outcomes of the states of the NV centers.}
\begin{tabular}{ccccc} \hline\hline
Bell states & $\;\;$ NV$_1$ $\;\;$ & $\;\;$ NV$_2$ $\;\;$ & $\;\;$ NV$_3$ $\;\;$ & $\;\;$ NV$_4$ $\;\;$ \\
\hline
$|\Phi_{1}^{+}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}$ & $|\varphi^{-}\rangle_{1}$ & $|\varphi^{-}\rangle_{2}$ & $|\varphi^{-}\rangle_{3}$ & $|\varphi^{-}\rangle_{4}$ \\
$|\Phi_{1}^{-}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}$ & $|\varphi^{-}\rangle_{1}$ & $|\varphi^{-}\rangle_{2}$ & $|\varphi^{-}\rangle_{3}$ & $|\varphi^{+}\rangle_{4}$ \\
$|\Phi_{2}^{+}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}$ & $|\varphi^{-}\rangle_{1}$ & $|\varphi^{+}\rangle_{2}$ & $|\varphi^{+}\rangle_{3}$ & $|\varphi^{-}\rangle_{4}$ \\
$|\Phi_{2}^{-}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}$ & $|\varphi^{-}\rangle_{1}$ & $|\varphi^{+}\rangle_{2}$ & $|\varphi^{+}\rangle_{3}$ & $|\varphi^{+}\rangle_{4}$ \\
$|\Phi_{1}^{+}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$ & $|\varphi^{-}\rangle_{1}$ & $|\varphi^{+}\rangle_{2}$ & $|\varphi^{-}\rangle_{3}$ & $|\varphi^{-}\rangle_{4}$ \\
$|\Phi_{1}^{-}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$ & $|\varphi^{-}\rangle_{1}$ & $|\varphi^{+}\rangle_{2}$ & $|\varphi^{-}\rangle_{3}$ & $|\varphi^{+}\rangle_{4}$ \\
$|\Phi_{2}^{+}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$ & $|\varphi^{-}\rangle_{1}$ & $|\varphi^{-}\rangle_{2}$ & $|\varphi^{+}\rangle_{3}$ & $|\varphi^{-}\rangle_{4}$ \\
$|\Phi_{2}^{-}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$ & $|\varphi^{-}\rangle_{1}$ & $|\varphi^{-}\rangle_{2}$ & $|\varphi^{+}\rangle_{3}$ & $|\varphi^{+}\rangle_{4}$ \\
$|\Phi_{1}^{+}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}$ & $|\varphi^{+}\rangle_{1}$ & $|\varphi^{-}\rangle_{2}$ & $|\varphi^{+}\rangle_{3}$ & $|\varphi^{+}\rangle_{4}$ \\
$|\Phi_{1}^{-}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}$ & $|\varphi^{+}\rangle_{1}$ & $|\varphi^{-}\rangle_{2}$ & $|\varphi^{+}\rangle_{3}$ & $|\varphi^{-}\rangle_{4}$ \\
$|\Phi_{2}^{+}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}$ & $|\varphi^{+}\rangle_{1}$ & $|\varphi^{+}\rangle_{2}$ & $|\varphi^{-}\rangle_{3}$ & $|\varphi^{+}\rangle_{4}$ \\
$|\Phi_{2}^{-}\rangle_{P}|\Phi_{2}^{+}\rangle_{S}$ & $|\varphi^{+}\rangle_{1}$ & $|\varphi^{+}\rangle_{2}$ & $|\varphi^{-}\rangle_{3}$ & $|\varphi^{-}\rangle_{4}$ \\
$|\Phi_{1}^{+}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}$ & $|\varphi^{+}\rangle_{1}$ & $|\varphi^{+}\rangle_{2}$ & $|\varphi^{+}\rangle_{3}$ & $|\varphi^{+}\rangle_{4}$ \\
$|\Phi_{1}^{-}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}$ & $|\varphi^{+}\rangle_{1}$ & $|\varphi^{+}\rangle_{2}$ & $|\varphi^{+}\rangle_{3}$ & $|\varphi^{-}\rangle_{4}$ \\
$|\Phi_{2}^{+}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}$ & $|\varphi^{+}\rangle_{1}$ & $|\varphi^{-}\rangle_{2}$ & $|\varphi^{-}\rangle_{3}$ & $|\varphi^{+}\rangle_{4}$ \\
$|\Phi_{2}^{-}\rangle_{P}|\Phi_{2}^{-}\rangle_{S}$ & $|\varphi^{+}\rangle_{1}$ & $|\varphi^{-}\rangle_{2}$ & $|\varphi^{-}\rangle_{3}$ & $|\varphi^{-}\rangle_{4}$ \\ \hline\hline \end{tabular}\label{table5}
\end{table}
From the analysis above, one can see that the hyperentangled Bell states in both the polarization and the spatial-mode DOFs can be completely distinguished assisted by NV centers in diamonds confined in MTRs, and our analysis is nondestructive. Our scheme can be generalized to the complete analysis of multi-photon polarization-spatial hyperentangled GHZ states by importing more NV-center-cavity systems.
\section{Discussion and summary}\label{sec5}
In our schemes, the reflection coefficient of input photon pulse and the phase shift induced on the output photon play a crucial role. Under the resonant condition $\omega_{0}=\omega_{c}=\omega$, if the cavity side leakage is neglected, the fidelities of our schemes can reach 100\% in the strong-coupling regime with $r(\omega)\cong1$ and $r_{0}(\omega)\cong-1$. If the cavity leakage is taken into account, the spin-selective optical transition rules employed in our work become \begin{eqnarray}
|R\rangle|+\rangle&\rightarrow&r|R\rangle|+\rangle,\;\;\;\;\;\;\,
|R\rangle|-\rangle\;\rightarrow\;r_{0}|R\rangle|-\rangle,\nonumber\\
|L\rangle|+\rangle&\rightarrow&r_{0}|L\rangle|+\rangle,\;\;\;\;
|L\rangle|-\rangle\;\rightarrow\;r|L\rangle|-\rangle.\label{eq16} \end{eqnarray} Considering the practical implementation of the system, we numerically simulate the relation between the fidelities (the efficiencies) and the coupling strength $\text{g}$, the cavity decay rate $\kappa$, and the NV center decay rate $\gamma$. Defining the fidelity of the process for generating or completely analyzing hyperentangled states in our schemes as
$F=|\langle\psi_{f}|\psi\rangle|^{2}$. Here $|\psi_{f}\rangle$ is the final state by considering the cavity side leakage and
$|\psi\rangle$ denotes the final state with an ideal condition. We calculate the fidelity of our scheme for generating the hyperentangled state
$|\Phi_{1}^{-}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$ and its fidelity is \begin{eqnarray}\label{eq17} F_{1}=\frac{(r-r_{0})^{2}(r^{2}+r_{0}^{2}+2)^{2}}{8(r^{2}+r_{0}^{2})(r^{4}+r_{0}^{4}+2)}. \end{eqnarray} By computations it is found, since the hyperentangled Bell states
$|\Phi_{1}^{+}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}$,
$|\Phi_{1}^{+}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$, and
$|\Phi_{1}^{-}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}$ have the same parity conditions with the hyperentangled Bell state
$|\Phi_{1}^{-}\rangle_{P}|\Phi_{1}^{-}\rangle_{S}$, the fidelities of our scheme for their generations are also $F_{1}$. $F_{2}$, $F_{3}$, and $F_{4}$ correspond to the fidelities for generating the hyperentangled Bell states
$|\Phi_{1}^{\pm}\rangle_{P}|\Phi_{2}^{\pm}\rangle_{S}$,
$|\Phi_{2}^{\pm}\rangle_{P}|\Phi_{1}^{\pm}\rangle_{S}$, and
$|\Phi_{2}^{\pm}\rangle_{P}|\Phi_{2}^{\pm}\rangle_{S}$, respectively. Here \begin{eqnarray}\label{eq18} \begin{split} F_{2}&=\frac{(r^{2}+r_{0}^{2}+2)^{4}}{16(r^{4}+r_{0}^{4}+2)^{2}}, \\ F_{3}&=\frac{(r-r_{0})^{4}}{4(r^{2}+r_{0}^{2})^{2}}, \\ F_{4}&=\frac{(1-rr_{0})^{2}(r-r_{0})^{2}}{4(1+r^{2}r_{0}^{2})(r^{2}+r_{0}^{2})}. \end{split} \end{eqnarray} The fidelities of our HBSG scheme varies with the parameter $\text{g}/\sqrt{\kappa\gamma}$, shown in
Fig. \ref{fig5} (a). For our HBSG scheme, the efficiency, which is
defined as the ratio of the number of the output photons to the
input photons, can be written as \begin{eqnarray}\label{eq19} \eta_{1}=\frac{1}{2^{8}}(r^{2}+r_{0}^{2}+2)^{4}. \end{eqnarray} The efficiency of our HBSG scheme varies with the parameter $\text{g}/\sqrt{\kappa\gamma}$, shown in
Fig. \ref{fig5} (b).
\begin{figure}
\caption{(Color online) (a) The fidelities of the HBSG scheme for the generation of the hyperentangled states
$|\Phi_{1}^{\pm}\rangle_{P}|\Phi_{1}^{\pm}\rangle_{S}$ ($F_{1}$),
$|\Phi_{1}^{\pm}\rangle_{P}|\Phi_{2}^{\pm}\rangle_{S}$ ($F_{2}$),
$|\Phi_{2}^{\pm}\rangle_{P}|\Phi_{1}^{\pm}\rangle_{S}$ ($F_{3}$), and $|\Phi_{2}^{\pm}\rangle_{P}|\Phi_{2}^{\pm}\rangle_{S}$ ($F_{4}$) vs the parameter $\text{g}/\sqrt{\kappa\gamma}$ for the leakage rates $\kappa_{s}/\kappa=0.03$, respectively. (b) The efficiency of the HBSG scheme vs the parameter $\text{g}/\sqrt{\kappa\gamma}$ for different leakage rates $\kappa_{s}/\kappa$.}
\label{fig5}
\end{figure}
\begin{figure}
\caption{(Color online) (a) Fidelity of the present HBSA scheme for the hyperentangled Bell state
$|\Phi_{1}^{+}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}$ versus the parameter $\text{g}/\sqrt{\kappa\gamma}$ for different leakage rates $\kappa_{s}/\kappa$. (b) The efficiency of the HBSA scheme vs the parameter $\text{g}/\sqrt{\kappa\gamma}$ for different leakage rates $\kappa_{s}/\kappa$.}
\label{fig6}
\end{figure}
From Fig. \ref{fig5}(a), one can see that when $\text{g}/\sqrt{\kappa\gamma}=1.5$, the fidelities are $F_{1}=96.68\%$, $F_{2}=94.69\%$, $F_{3}=98.70\%$ , and $F_{1}=97.43\%$ for the leakage rates $\kappa_{s}/\kappa=0.03$, and the efficiency of our scheme is $\eta_{1}=53.96\%$ for the leakage rates $\kappa_{s}/\kappa=0.06$. When the parameter $\text{g}/\sqrt{\kappa\gamma}$ is larger than 3, the fidelities of our HBSG scheme for the hyperentangled Bell states
$|\Phi_{2}^{\pm}\rangle_{P}|\Phi_{1}^{\pm}\rangle_{S}$ will be $F_{3}\approx1$, while other fidelities are higher than $99\%$, and the efficiency of our HBSG scheme will be higher than $\eta_{1}=71.75\%$. In our protocol for generating hyperentangled GHZ states, we use three NV-center-cavity systems rather than two NV-center-cavity systems, which makes the fidelity and the efficiency of the scheme lower than those in the HBSG protocol.
The fidelity of our HBSA protocol for the hyperentangled Bell state
$|\Phi_{1}^{+}\rangle_{P}|\Phi_{1}^{+}\rangle_{S}$ is given by \begin{eqnarray} F=\frac{\varepsilon^{8}}{4(\alpha+2\beta)}. \end{eqnarray} Here \begin{eqnarray} \begin{split} \alpha &=(r^{2}-r_{0}^{2})^{2}[(r-r_{0})^{2}(r^{2}+1)(r_{0}^{2}+1)\\ &\;\;\;\; +4rr_{0}(r^{2}+r_{0}^{2})],\\ \beta &=(r^{2}+r_{0}^{2})^{2}[(r^{2}+r_{0}^{2})^{2}+4r^{2}r_{0}^{2}],\\ \varepsilon &=r-r_{0}. \end{split} \end{eqnarray} The efficiency of our HBSA protocol is \begin{eqnarray} \eta=\frac{1}{2^{16}}(r^{2}+r_{0}^{2}+2)^{8}. \end{eqnarray} Both the fidelity and efficiency of our HBSA scheme vary with the parameter $\text{g}/\sqrt{\kappa\gamma}$, shown in Fig. \ref{fig6}. The plots indicates that for $\text{g}/\sqrt{\kappa\gamma}\geq0.5$ the higher fidelities and efficiencies of our HBSG scheme depend on the higher NV-cavity coupling strength. When the parameter $\text{g}/\sqrt{\kappa\gamma}$ is larger than 3, the fidelity and the efficiency will be higher than $F=99.58\%$ and $\eta=51.48\%$ for the leakage rates $\kappa_{s}/\kappa=0.06$.
The fidelity of our HBSG and HBSA schemes can be reduced a few percent because of the imperfection of electron spin population, the imperfection of the NV electron spins readout, and the imperfection of frequency-selective microwave manipulation. For $m_{s}=0$, the preparation fidelity is $99.7\pm0.1\%$, and $99.2\pm0.1\%$ for $m_{s}=\pm1$ \cite{tgaebel}. The average readout fidelity of the NV electron spins is $93.2\pm0.5\%$ \cite{tgaebel}. By using isotopically purified diamonds and polarizing the nitrogen nuclear spin, we can reduce the microwave manipulation imperfection. The spin decoherence may also reduce the fidelity of our schemes. In our schemes, the single photons are successively sent into the devices, the time interval $\Delta t$ between two photons should be less than the electron-spin coherence time $T$. In experimental cases, the spin relaxation time $T_{1}$ of NV centers in diamond scales from microseconds to seconds at low temperature and the dephasing time $T_{2}$ is about 2 ms in an isotopically pure diamond \cite{Neumann,r43}. The electron-spin coherence time $T$ is $>10$ ms \cite{HBernien}, which is much longer than the photon coherence time $\sim10$ ns and the subnanosecond electron-spin manipulation control \cite{GDFuchs}.
For the practical operations, the photon loss due to absorption and scatting, the mismatch between the input pulse mode and MTR, and the inefficiency of the detectors will bring ineffectiveness to our schemes. Interestingly, the photon loss will just affect the success efficiency, rather than the fidelity. The present experimental technology can generate 300000 high-quality single photons within 30 s \cite{MH}. Another key ingredient of our protocols is the coupling between NV centers and MTRs. In realistic experiments, the strong coupling between the NV centers and the WGM has been demonstrated in different kinds of microcavities \cite{s33,nvcavity2,nvcavity3,nvcavity4,nvcavity5,nvcavity6}. The coupling strength between NV centers and the WGM can reach $\text{g}/2\pi\sim0.3-1$GHz \cite{s33,nvcavity3,nvcavity4,nvcavity5}. The Q factor of the chip-based microcavity is higher than 25000. Considering the parameters $[\text{g}_{ZPL}, \kappa, \gamma_{total}, \gamma_{ZPL}]/2\pi=[0.30, 26, 0.013, 0.0004]$ GHz of an NV center coupled to a microdisk \cite{nvcavity3}, we have $\text{g}\approx3\sqrt{\kappa\gamma}$ and the fidelities of our HBSG and HBSA schemes can exceed 99\%. Therefore our protocols are feasible in experiment.
Compared with previous works \cite{kerr,HBSA2,HBSA3}, these two schemes relax the difficulty of their implementation in experiment as it is not difficult to generate the $\pi$ phase shift in single-sided NV-cavity systems. Moreover, single-sided NV-cavity systems have a long coherence time even at the room temperature (1.8 ms) \cite{r43}, different from quantum-dot-cavity systems. The first HBSA scheme by Sheng, Deng, and Long \cite{kerr} is achieved with cross-Kerr nonlinearity. It is perfect in theory. At present, a clean cross-Kerr nonlinearity in the optical single-photon regime is still a controversial assumption with current technology \cite{kerr2,kerr3}. The second HBSA scheme by Ren \emph{et al.} \cite{HBSA2} with single-sided quantum-dot-cavity systems requires the $\pi$ phase difference of the Faraday rotation between a hot cavity and an empty cavity, and it is not easy to acquire the phase difference with only one nonlinear interaction between a photon and a quantum dot. Compared with the work by Wang, Lu, and Long \cite{HBSA3}, our schemes do not require that the transmission for the uncoupled cavity is balanceable with the reflectance for the coupled cavity. Moreover, the coherent manipulation of the spin of a single NV center to accomplish quantum information and computation tasks at room temperature has been presented \cite{nvregister,algorithm}, which provides the basis for the current schemes.
In summary, we have proposed two efficient schemes for the deterministic generation and the complete nondestructive analysis of hyperentangled Bell states in both polarization and spatial-mode DOFs assisted by NV centers in MTRs. The HBSG protocol can also be extended to achieve the generation of multi-photon hyperentangled GHZ states efficiently. Compared with previous works \cite{kerr,HBSA2,HBSA3}, our schemes relax the difficulty of their implementation in experiment. Our calculations show that the proposed schemes can work with a high fidelity and efficiency under the current experimental techniques, which may be a benefit to long-distance high-capacity quantum communication, such as quantum teleportation, quantum dense coding, and quantum superdense coding with two DOFs of photon systems.
\section*{ACKNOWLEDGMENTS}
This work is supported by the National Natural Science Foundation of China under Grant No. 11475021, the National Key Basic Research Program of China under Grant No. 2013CB922000.
\end{document} | arXiv |
Dual graphs and modified Barlow-Bass resistance estimates for repeated barycentric subdivisions
DCDS-S Home
February 2019, 12(1): 1-26. doi: 10.3934/dcdss.2019001
Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets
Kevin Arfi and Anna Rozanova-Pierrat ,
Laboratoire de Mathématiques et Informatique pour la Complexité et les Systèmes, CentralSupélec, Université Paris Saclay, Bȃtiment Bouygues, 3 rue Joliot-Curie, 91190 Gif-sur-Yvette, France
* Corresponding author: Anna Rozanova-Pierrat
Received February 2017 Revised June 2017 Published July 2018
Figure(1)
In the framework of the Laplacian transport, described by a Robin boundary value problem in an exterior domain in $\mathbb{R}^n$, we generalize the definition of the Poincaré-Steklov operator to $d$-set boundaries, $n-2< d<n$, and give its spectral properties to compare to the spectra of the interior domain and also of a truncated domain, considered as an approximation of the exterior case. The well-posedness of the Robin boundary value problems for the truncated and exterior domains is given in the general framework of $n$-sets. The results are obtained thanks to a generalization of the continuity and compactness properties of the trace and extension operators in Sobolev, Lebesgue and Besov spaces, in particular, by a generalization of the classical Rellich-Kondrachov Theorem of compact embeddings for $n$ and $d$-sets.
Keywords: Poincaré-Steklov operator, d-sets, Laplacian transport, fractal.
Mathematics Subject Classification: Primary: 35J25, 46E35; Secondary: 47A10.
Citation: Kevin Arfi, Anna Rozanova-Pierrat. Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 1-26. doi: 10.3934/dcdss.2019001
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Figure 1. Example of the considered domains: $\Omega_0$ (the von Koch snowflake) is the bounded domain, bounded by a compact boundary $\Gamma$, which is a $d$-set (see Definition 2.3) with $d = \log 4/ \log 3>n-1 = 1$. The truncated domain $\Omega_S$ is between the boundary $\Gamma$ and the boundary $S$ (presented by the same von Koch fractal as $\Gamma$). The boundaries $\Gamma$ and $S$ have no an intersection and here are separated by the boundary of a ball $B_r$ of a radius $r>0$. The domain, bounded by $S$, is called $\Omega_1 = \overline{\Omega}_0\cup \Omega_S$, and the exterior domain is $\Omega = \mathbb{R}^n\setminus \overline{\Omega}_0$
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\begin{document}
\title{A Family of Quantum Codes with Exotic Transversal Gates}
\author{Eric Kubischta} \thanks{ These authors contributed equally to this work.} \author{Ian Teixeira} \thanks{ These authors contributed equally to this work.}
\affiliation{Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742 USA}
\begin{abstract}
Recently it has been shown that the binary icosahedral group $2I$ together with a certain involution forms the most efficient single-qubit universal gate set. In order for this to be viable, one must construct a quantum code with transversal gate group $2I$, however, no such code has ever been demonstrated explicitly. We fill this void by constructing a novel family of quantum codes that all have transversal gate group $2I$. \end{abstract}
\maketitle
\section{Introduction}
Let $((n,K,d))$ denote an $n$-qubit quantum error correcting code with a codespace of dimension $K$ and with distance $d$. The Eastin-Knill theorem \cite{eastinKnill} shows that for any $ ((n,K,d)) $ code with $ d \geq 2 $, the logical operations in $ SU(K) $ which can be implemented transversally are always a \textit{finite} subgroup $ G \subset SU(K) $. A logical gate $g$ is called transversal if $g$ can be implemented as $U_1 \otimes \cdots \otimes U_n$ where each $U_i \in U(2)$. Transversal gates are considered naturally fault tolerant because errors cannot propagate between physical qubits.
In what follows we will concern ourselves with $K = 2$. That is, we encode a single logical qubit into $n$ physical qubits. \textit{Maximal} finite subgroups of $SU(2)$ are especially interesting in the context of universality because, by definition, only a single gate outside of the subgroup is needed to generate all logical gates. There are only two maximal finite subgroups of $SU(2)$: $2O$ and $2I$. Here $2O$ is the binary octahedral group, which corresponds to the single qubit Clifford group, and $2I$ is the binary icosahedral group.
Many codes exist with transversal gate group $ 2O $. For example, the $ [[7,1,3]] $ Steane code, and more generally all doubly even self dual CSS codes, have transversal gate group $ 2O $. However, no code has ever been demonstrated to have transversal gate group $2I$.
This omission is particularly glaring given the role $2I$ plays in the ``optimal absolute Super-Golden-Gate set" proposed in \cite{superGoldenGates} as the best single qubit universal gate set.
A Super-Golden-Gate set $G + \mathcal{T}$ is a universal gate set that consists of a finite group of gates $G$, which are considered ``cheap" to implement, together with a single gate $\mathcal{T}$ outside the group, which is considered ``expensive." These gates can approximate any gate in $SU(2)$ and the efficiency is judged by the number of expensive $\mathcal{T}$ gates needed in the approximation procedure. The original example of a super-golden gate set is the universal gate set Clifford + $ T $, or equivalently $2O + T$. Here $T$ is the square root of the phase gate. For this gate set, Clifford operations are indeed ``cheap" since some codes can implement them transversally, e.g., the $ [[7,1,3]] $ Steane code. Implementing the $ T $ gate is ``expensive" since it requires a procedure which is not \textit{naturally} fault tolerant, such as magic state distillation \cite{magicstatedist}.
The icosahedral golden gate set $2I + T_{60}$ defined in \cite{superGoldenGates} consists of $2I$ as the ``cheap" gates together with a certain involution $T_{60}$ that is considered ``expensive." This is an even more efficient universal gate set than Clifford + $ T $. Indeed in \cite{fast2I} it is shown that the fast navigation algorithm for $2I + T_{60}$ is an improvement on the best known navigation algorithm for the Clifford + $ T $ golden gate set by a factor of $\log_2(59) \approx 5.88$.
However, a practical implementation of the work of \cite{superGoldenGates}, and in particular the algorithm of \cite{fast2I}, requires a ``cheap" way to implement gates from $ 2I $. The most natural solution is to proceed as in the case of the Clifford + $ T $ golden gate set and find a quantum error correcting code with transversal gate group $ 2I $. However, as previously mentioned, no code has ever been demonstrated to have transversal gate group $ 2I $.
In answer to this need, we showcase for the first time a family of codes having $2I$ as the transversal gate set. The next step in a practical implementation of the icosahedral Golden Gate set would be a fault tolerant implementation of the $ T_{60} $ gate, perhaps using some icosahedral analog of magic state distillation.
\section{Preliminaries}
As usually written, most single qubit quantum gates are not in $SU(2)$ but rather in $U(2)$. However, we can ``specialize" them by multiplying by a global phase factor. This will not in general be unique, so we must be careful to fix a convention. We will denote matrices from $SU(2)$ by sans serif font.
The single qubit Pauli matrices in $U(2)$ are \[ I = \smqty( 1 & 0 \\ 0 & 1), \quad X = \smqty(0 & 1 \\ 1 & 0), \quad Y = \smqty(0 & i \\ -i & 0), \quad Z = \smqty(1 & 0 \\ 0 & -1). \] The Pauli matrices in $SU(2)$ are \[
\mathsf{X} = -i X ,\quad \mathsf{Y} = -i Y, \quad \mathsf{Z} = -i Z. \] The Hadamard gate and its specialized version are \[ H = \tfrac{1}{\sqrt{2}}\smqty(1 & 1 \\ 1 & -1), \qquad \mathsf{H} = - i H. \] The phase gate and its specialized version are \[ S = \smqty(1 & 0 \\ 0 & i), \qquad \mathsf{S} = e^{- i \pi /4} S. \] A less common gate is \[
\mathsf{M} = \mathsf{H} \mathsf{S}^\dagger = \tfrac{1}{\sqrt{2}}\smqty( 1 & -i \\
1 & i). \] The conjugation action of $ \mathsf{M} $ on Paulis is by cycling: \[ \mathsf{X} \to \mathsf{Y}, \quad \mathsf{Y} \to \mathsf{Z}, \quad \mathsf{Z} \to \mathsf{X}. \]
The single qubit Clifford group $\mathsf{C}$ is generated as $\mathsf{C} = \expval{ \mathsf{X}, \mathsf{Z}, \mathsf{M}, \mathsf{H}, \mathsf{S} }$ and has $48$ elements. This group is isomorphic to $2O$. Although there are infinitely many (conjugate) realizations of $2O$ in $SU(2)$, this is the only version that contains the Pauli group $\mathsf{P} = \expval{\mathsf{X}, \mathsf{Z}}$ so it is a canonical choice. A subgroup of $2O$ is the binary tetrahedral group $2T$ with $ 24 $ elements. Again, $ 2T $ has infinitely many realizations, but the canonical realization of $ 2T $ is $\expval{ \mathsf{X}, \mathsf{Z}, \mathsf{M}}$, showcasing the role of the $\mathsf{M}$ gate.
The binary icosahedral group $2I$ of order $120$ also has infinitely many realizations. But unlike $2O$ and $ 2T $, there is not a single canonical choice, but rather two, related by Clifford conjugation. We will focus on the version $2I \cong \expval{ \mathsf{X}, \mathsf{Z}, \mathsf{M}, \mathsf{\Phi} }$ where $$ \mathsf{\Phi} = \tfrac{1}{2} \smqty( \phi + i \phi^{-1} & 1 \\ -1 & \phi - i \phi^{-1} ) $$ is a novel gate and $ \phi = (1+\sqrt{5})/2$ is the golden ratio. Although we won't be concerned with it here, the conjugate canonical choice of $ 2I $ subgroup is $\expval{\mathsf{X}, \mathsf{Z}, \mathsf{M}, \mathsf{S} \mathsf{\Phi} \mathsf{S}^\dagger }$.
The character table for $2I$ is given in \cref{tab:chartable}. The irrep $ \pi_i $ denotes the restriction to $ 2I $ of the unique $ i $ dimensional irrep of $ SU(2) $. An overbar denotes the conjugate irrep under the outer automorphism of $ 2I $. We use primes to distingiush other irreps of the same dimension. In all cases, the subscript of the irrep is the dimension.
\begin{table}[htp]
\centering
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|} \hline
& 1 & 1 & 20 & 30 & 12 & 12 & 20 & 12 & 12 \\ \hline \hline
$\pi_1$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
$\pi_2$ & 2 & -2 & -1 & 0 & $\phi^{-1}$ & $-\phi$ & 1 & $\phi$ & $-\phi^{-1}$ \\
$\overline{\pi_2 }$ & 2 & -2 & -1 & 0 & $-\phi$ & $\phi^{-1}$ & 1 & $-\phi^{-1}$ & $\phi$ \\
$\pi_3$ & 3 & 3 & 0 & -1 & $-\phi^{-1}$ & $\phi$ & 0 & $\phi$ & $-\phi^{-1}$ \\
$\overline{\pi_3}$ & 3 & 3 & 0 & -1 & $\phi$ & $-\phi^{-1}$ & 0 & $-\phi^{-1}$ & $\phi$ \\
$\pi_4$ & 4 & -4 & 1 & 0 & -1 & -1 & -1 & 1 & 1 \\
$\pi_{4'}$ & 4 & 4 & 1 & 0 & -1 & -1 & 1 & -1 & -1\\
$\pi_5$ & 5 & 5 & -1 & 1 & 0 & 0 & -1 & 0 & 0 \\
$\pi_6$ & 6 & -6 & 0 & 0 & 1 & 1 & 0 & -1 & -1 \\ \hline
\end{tabular}
\caption{$2I$ Character Table}
\label{tab:chartable} \end{table}
There are only two irreps with dimension $2$, namely, $\pi_2$ and its conjugate $\overline{\pi_2}$. A code with $K=2$ and transversal $2I$ must transform in one of these irreps.
\section{A Binary Icosahedral Code}
There is an $n=7$ qubit code that admits a $2I$ transversal gate set. The parameters are $ ((7,2,3)) $, i.e., it encodes a single logical qubit into $ 7 $ physical qubits and corrects arbitrary single qubit errors.
An (unnormalized) basis for the codespace is: \begin{subequations}\label{code:us} \begin{align*}
\ket{0_L} &= 15 \ket{\text{wt 0}} + \sqrt{5}\; \overline{ \ket{\text{wt 2}}}_{21} \numberthis \\
& \quad + 3 \overline{ \ket{ \text{wt 4}}}_{35}- 3 \sqrt{5} \, \overline{ \ket{ \text{wt 6}}}_7, \\
\ket{1_L} &= - 3 \sqrt{5} \; \overline{ \ket{ \text{wt 1}}}_7 + 3 \overline{ \ket{\text{wt 3}}}_{35} \numberthis \\
&\quad + \sqrt{5}\, \overline{ \ket{\text{wt 5}}}_{21} + 15 \ket{\text{wt 7}}. \end{align*} \end{subequations} Here the bar notation denotes an (unnormalized) uniform sum over all distinct permutations. For example $ \overline{ \ket{\text{wt 4}}}_{35} $ is a uniform sum over all $35={7 \choose 4} $ distinct permutations of weight $4$ kets (e.g., $\ket{0001111}$).
The weight enumerator coefficients \cite{quantumMacWilliams} of the $ ((7,2,3)) $ $ 2I $ code are \begin{align*}
A &= \qty(1,0,7,0,7,0,49,0), \\
B &= \qty(1,0,7,42,7,84,49,66). \end{align*} We immediately observe that $d = 3$ since $A_i = B_i$ for each $i < 3$.
\subsection*{Transversality}
Logical $ X $ is exactly transversal, i.e., $ X^{\otimes 7} $ implements logical $ X $. So $ \ket{1_L}=X^{\otimes 7}\ket{0_L} $ and the coefficients of $ \ket{0_L} $ totally determine the coefficients of $ \ket{1_L}$. Also note that, because of global phase, logical $ \mathsf{X} $ is actually implemented by the physical gate $ (-\mathsf{X})^{\otimes 7} $.
Similarly, logical $ Z $ is exactly transversal. And logical $ \mathsf{Z} $ is implemented by $ (-\mathsf{Z})^{\otimes 7} $. The support of $ \ket{0_L} $ is all $ 64 $ even weight kets and the support of $ \ket{1_L} $ is all $ 64 $ odd weight kets. It is always true when $ Z $ is exactly transversal that $\ket{0_L}$ consists exclusively of even weight bit strings and $\ket{1_L}$ consists of odd weight bit strings.
Logical $\mathsf{M}$ is $\mathsf{M}^*$-strongly transversal, where $ * $ denotes complex conjugation. In other words the physical gate $ (\mathsf{M}^*)^{\otimes 7} $ implements logical $ \mathsf{M} $. Note that the $[[7,1,3]]$ Steane code also implements logical $\mathsf{M}$ in this way.
Logical $\mathsf{\Phi}$ is $\mathsf{\Phi}^\star$-strongly transversal where \[ \mathsf{\Phi}^\star=\tfrac{1}{2} \smqty( -\phi^{-1} + i \phi & 1 \\ -1 & -\phi^{-1} - i \phi ) \] is the same as the $ \mathsf{\Phi} $ gate except we make the replacement $ \phi \mapsto -1/\phi $ and we take the complex conjugate. In other words, the physical gate $ (\mathsf{\Phi}^\star)^{\otimes 7} $ implements logical $ \mathsf{\Phi} $. No code to date has been demonstrated that admits a transversal $\mathsf{\Phi}$ (or a transversal $\mathsf{\Phi}'$).
The code in \cref{code:us} transforms in $\overline{\pi_2}$. This explains why logical $\mathsf{\Phi}$ is implemented by $(\mathsf{\Phi}^\star)^{\otimes 7}$.
\subsection*{Other Properties}
The $ ((7,2,3)) $ code is not a stabilizer code or even a codeword stabilized (CWS) code \cite{CWS}. Any stabilizer or CWS code with $ Z $ exactly transversal must have $ \ket{0_L}$ and $ \ket{1_L} $ as stabilizer states. However the logical kets $ \ket{0_L}, \ket{1_L} $ given in \cref{code:us} cannot be stabilizer states since the coefficients with respect to the computational basis are not in integer ratios.
Next, notice that the code is permutationally invariant, yet still has distance $ d=3 $. This unusual combination of properties highlights the nonadditive nature of the code, as \cite{automorph} shows that permutationally invariant stabilizer codes can have at most distance $ 2 $. Note well that permutational invariance is not preserved under the usual notion of code equivalence so it is only meaningful for the particular code representative we have chosen (e.g. applying $ ZIIIIII $ would produce an equivalent code which is no longer permutationally invariant).
\subsection*{Relation to Permutationally Invariant Codes}
Having found a $ ((7,2,3)) $ $ 2I $ code which happened to be permutationally invariant, the relevance of the work in \cite{2004permutation} subsequently came to our attention. The focus therein was not on transversality but rather on finding $ n $ qubit codes that were permutationally invariant. The authors find two such $((7,2,3))$ codes. Unbeknownst to them, the two codes are equivalent. One of their codes is not only equivalent the code in \cref{code:us}, but exactly the same. To the best of our knowledge, the code in \cite{2004permutation} has never been shown to have any transversal gates outside of the Pauli group.
\subsection*{Relation to Spin Codes}
In \cite{gross1}, the author considers the problem of encoding a qubit into a single large spin. A spin-$s$ system corresponds to the unique $ 2s+1 $ dimensional irrep of $ SU(2) $. Encoding a qubit into this space means choosing a $2$-dimensional subspace. Each $g \in SU(2)$ has a natural action on the $2s+1$ dimensional spin space through the Wigner $ D $-matrix $ D^s(g) $. And any such $ D^s(g) $ that also preserves the $2$-dimensional codespace implements a logical gate.
Spin codes are a finite dimensional analog of the continuous variable GKP codes \cite{gkp}. As such, they work with an alternative error model, aiming to correct (local) lowest order perturbations rather than the (global) low weight Pauli errors usually considered for multiqubit codes.
Nevertheless, one can associate a spin code with a multiqubit code in a natural way. The Dicke state mapping expresses a state of a spin $ s $ system as a permutationally invariant state of a system of $ n=2s $ qubits. The Dicke state mapping for a spin $ s=7/2 $ system is given in \cref{tab:dicke}. We borrow the overline notation from earlier. \begin{table}[htp]
\centering \renewcommand{1.5}{1.5}
\begin{tabular}{|c|c|} \hline
Spin state & Dicke state \\ \hline \hline
$\ket{\tfrac{7}{2}, \tfrac{7}{2}}$ & $\ket{\text{wt 0}}$ \\
$\ket{\tfrac{7}{2}, \tfrac{5}{2}}$ & $\tfrac{1}{\sqrt{7}}\overline{ \ket{\text{wt 1}}}_{7}$ \\
$\ket{\tfrac{7}{2}, \tfrac{3}{2}}$ & $\tfrac{1}{\sqrt{21}}\overline{ \ket{\text{wt 2}}}_{21}$ \\
$\ket{\tfrac{7}{2}, \tfrac{1}{2}}$ & $\tfrac{1}{\sqrt{35}}\overline{ \ket{\text{wt 3}}}_{35}$ \\
$\ket{\tfrac{7}{2}, -\tfrac{1}{2}}$ & $\tfrac{1}{\sqrt{35}}\overline{ \ket{\text{wt 4}}}_{35}$\\
$\ket{\tfrac{7}{2}, -\tfrac{3}{2}}$ & $\tfrac{1}{\sqrt{21}}\overline{ \ket{\text{wt 5}}}_{21}$\\
$\ket{\tfrac{7}{2}, -\tfrac{5}{2}}$ & $\tfrac{1}{\sqrt{7}}\overline{ \ket{\text{wt 6}}}_{7}$ \\
$\ket{\tfrac{7}{2}, -\tfrac{7}{2}}$ & $\ket{\text{wt 7}}$ \\ \hline
\end{tabular}
\caption{Correspondence between Spin states and Dicke states}
\label{tab:dicke} \end{table}
The Dicke state mapping $\mathscr{D}$ behaves well with respect to the natural action of $ SU(2) $ on a spin-$ s $ system and the natural action of $ SU(2) $ on an $ n=2s $ qubit system (via the tensor product). In particular $$ \mathscr{D}\Big[D^s(g)\ket{\psi}\Big]=g^{\otimes n} \mathscr{D}[\,\ket{\psi}\,] , $$ where $ g \in SU(2) $, $ \ket{\psi} $ is a state of a spin $ s $ system, $ D^s $ is a Wigner $ D $-matrix for a spin $ s $ system, and $ g^{\otimes n} $ is the $ n $-fold tensor product of $ g $ with itself. The main implication of this property is that $\mathscr{D}$ converts logical gates of a spin code into logical \textit{transversal} gates of the corresponding (permutationally symmetric) multiqubit code.
The most surprising feature of the Dicke state mapping $\mathscr{D}$ is that it also behaves well with respect to error correcting properties. It can be shown that when spin codes satisfy certain extra conditions (see below) then they are mapped by $ \mathscr{D} $ to an error \textit{correcting} multiqubit code, i.e., $ d\geq 3$. These observations only recently came to our attention but the recent paper \cite{gross2} makes this explicit.
Nearly all codes in \cite{gross1} are constructed to be representations of the Clifford group $ 2O $. However, the author also constructs one code which is instead a representation of $ 2I $. This code is given by \begin{subequations}\label{code:jon} \begin{align}
\ket{0_L} &= \sqrt{\tfrac{3}{10}} \ket{\tfrac{7}{2}, \tfrac{7}{2}} + \sqrt{\tfrac{7}{10}} \ket{\tfrac{7}{2}, -\tfrac{3}{2}} , \\
\ket{1_L}&= \sqrt{\tfrac{7}{10}} \ket{\tfrac{7}{2}, \tfrac{3}{2}} - \sqrt{\tfrac{3}{10}} \ket{\tfrac{7}{2}, -\tfrac{7}{2}}. \end{align} \end{subequations}
Applying the Dicke state mapping $\mathscr{D}$ to this spin code yields a multiqubit code which is equivalent by a non-entangling gate to the code we provided in \cref{code:us}.
However, unlike the permutation code from \cite{2004permutation}, the image under $\mathscr{D}$ of the code in \cref{code:jon} does not have an identical codespace to the code we gave in \cref{code:us}. As a result, the multiqubit code corresponding to \cref{code:jon} is transversal for a version of $2I$ that has small intersection with the Clifford group: neither logical $X$ nor logical $Z$ have transversal implementations for this version of the code.
\section{An Infinite Family of $ 2I $ Codes}
Although the code in \cref{code:us} was discovered by independent means (and with different motivation), we came to understand that the methods used to construct the spin code in \cref{code:jon} could also be used to construct an infinite family of $2I$ transversal codes. The main tool in this construction is the following lemma.
\begin{lemma} \cite{gross1,gross2} \label{lem:spincodecond}
A spin code with binary tetrahedral $2T$ symmetry and $ \mel{0_L}{J_Z}{0_L}=0 $ corresponds under $ \mathscr{D} $ to a multiqubit code with $ d \geq 3 $. \end{lemma}
It is easy to see that the binary tetrahedral group $2T$ is a subgroup of $ 2I $. So all spin codes with $ 2I $ symmetry also have $ 2T $ symmetry, and the lemma applies.
Our use of the method of \cite{gross1,gross2} proceeds as follows. First, construct a spin code with $ 2I $ symmetry that satisfies $ \mel{0_L}{J_Z}{0_L}=0 $. Then applying $ \mathscr{D} $ will produce a multiqubit code with transversal gate group $ 2I $ and $ d \geq 3 $.
One goal of \cite{gross2} is to make $2O$ transversal multiqubit codes. The authors construct spin codes by projecting onto irreps of $2O$. They must choose irreps with multiplicity so that they can take non-trivial linear combinations of the form $\ket{0_L} = \sum_i c_i \ket{v_i}$ where $\{ \ket{v_i} \}$ span the irrep. At this point they must solve a system of equations to see if there is any choice of $c_i$ for which $\bra{0_L} J_Z \ket{0_L} = 0$.
In contrast, we have a much stronger result for $2I$. Let \[
\Pi_\pi =\frac{\dim(\pi)}{120} \sum_{g \in 2I} \chi_\pi^*(g) D^{s}(g) \] be the spin projector onto the irrep $\pi$ of $2I$ where $\chi_\pi$ is the character corresponding to $\pi$. We have shown (numerically) that \[
\Pi_{\overline{\pi_2}} J_z \Pi_{\overline{\pi_2}} = 0 \] for all values of $n=2s$ up to $501$. We expect this to hold for all values of $n$. Note that $ \Pi_{\overline{\pi_2}} $ vanishes identically for all even $ n $ as well as the odd values $ n=1,3,5,9,11,15,21 $.
Applying the Dicke state mapping $\mathscr{D}$, we can use \cref{lem:spincodecond} to conclude the startling result that \textit{every} spin code transforming in the $ \overline{\pi_2} $ irrep of $ 2I $ corresponds to a $ d \geq 3 $ multiqubit code with transversal gate group $ 2I $.
Thus there exist $ ((n,2,3)) $ codes with transversal gate group $ 2I $ for all odd numbers of qubits $ n $ with the exception of $ n=1,3,5,9,11,15,21 $. We are not familiar with any other families of error correcting codes which exists for (essentially) every odd number of qubits. This is quite miraculous, only occurring for the $ \overline{\pi_2} $ irrep of $ 2I $ and not for the $ \pi_2 $ irrep of $ 2I $ or for any of the irreps of $ 2O $ or $ 2T $.
\subsection{Examples}
The first few values of $s$ for which a $\overline{\pi_2}$ irrep of $2I$ appears are $s= 7/2, 13/2, 17/2$. These correspond to qubit codes with $n = 7, 13, 17$ respectively. We already saw the $n = 7$ case in \cref{code:us}. Now we list the codewords for the other two cases.
An (unnormalized) basis for the $((13,2,3)))$ transversal $2I$ code is: \begin{align*}
\ket{0_L} &= 495 \ket{\text{wt 0}} + 33 \sqrt{5} \, \overline{\ket{\text{wt 2}} }_{78} + 3 \overline{ \ket{\text{wt 4}}}_{715} \\
&\quad - 3 \sqrt{5} \, \overline{ \ket{\text{wt 6}}}_{1716} -25 \overline{ \ket{\text{wt 8}}}_{1287} \\
& \quad + 9 \sqrt{5} \, \overline{ \ket{\text{wt 10}}}_{286} - 165 \sqrt{5}\, \overline{ \ket{\text{wt 12}}}_{13} , \\
\ket{1_L} &= X^{\otimes 13} \ket{0_L}. \end{align*}
An (unnormalized) basis for the $((17,2,3)))$ transversal $2I$ code is: \begin{align*}
\ket{0_L} &= 780 \ket{\text{wt 0}}
+ -13\sqrt{5} \, \overline{\ket{\text{wt 2}} }_{136}
+ 26 \overline{ \ket{\text{wt 4}}}_{2380} \\
&+\sqrt{5} \, \overline{ \ket{\text{wt 6}}}_{12376}
+8 \overline{ \ket{\text{wt 8}}}_{24310}
-\sqrt{5} \, \overline{ \ket{\text{wt 10}}}_{19448} \\
& -10 \overline{ \ket{\text{wt 12}}}_{6188}
+13\sqrt{5} \, \overline{ \ket{\text{wt 14}}}_{680}
+260 \overline{ \ket{\text{wt 16}}}_{17} \\
\ket{1_L} &= X^{\otimes 17} \ket{0_L}. \end{align*}
\subsection{An Infinite Family of Infinite Families of $ 2I $ codes}
Assuming that $\Pi_{\overline{\pi_2}} J_z \Pi_{\overline{\pi_2}} = 0$ holds for all $n$, then this method supplies not only an infinite family of $ 2I $ codes but actually a countably infinite family of uncountably infinite families of $ 2I $ codes. For values of $ n $ for which the $ \overline{\pi_2} $ irrep appears with multiplicity $ \mu $ this method yields a complex projective space $\mathbb{C}P^{\mu-1} $ worth of distinct $ 2I $ codes. That is, each point on the manifold corresponds to a unique $2I$ code (with respect to code equivalence by non-entangling gates).
When $ \mu=1 $ that means there is a unique code up to code equivalence (for example the case of $ n=7,13,17 $ given above). But for any larger value of $ \mu $, there is infinitely many distinct $ 2I $ codes. The first value of $ n $ for which this is relevant is $ n=37 $ which has $ \mu=2 $ and thus there are a $ \mathbb{C}P^{1}\simeq S^2 $ worth of inequivalent codes. In general $ \mu \approx 1+ n/30$. So for larger numbers of qubits this method produces larger and larger manifolds worth of $ ((n,2,3)) $ codes with transversal gate group $ 2I $.
Every single code constructed in this way has the same transversality properties as the $ ((7,2,3)) $ code in \cref{code:us}; logical $ \mathsf{X} $, logical $ \mathsf{Z} $, logical $ \mathsf{M} $, and logical $ \mathsf{\Phi} $ all have strongly transversal implementations.
\subsection*{Another Family}
So far we have only discussed $ 2I $ codes transforming in the $ \overline{\pi_2} $ representation. The reason for this is the astonishing fact that $ \Pi_{\overline{\pi_2}} J_z \Pi_{\overline{\pi_2}} = 0 $.
However, it is worth noting that although $ \pi_2 $ does not behave as well $ \overline{\pi_2} $ it is still possible to create $ d=3 $ codes transforming in the $ \pi_2 $ irrep of $ 2I $. These are akin to the $2O$ codes found in \cite{gross2} in the sense that the irrep $\pi_2$ must occur with multiplicity in order to solve for a choice of logical zero that satisfies $\bra{0_L} J_Z \ket{0_L} = 0$. The first value of $n$ for which $\pi_2$ occurs with multiplicity is $n = 31$, and indeed a $ ((31,2,3)) $ code with transversal $2I$ can be obtained in this manner.
\section{Conclusion}
A uniting theme of this paper is the novel gate $ \mathsf{\Phi} $, a gate which is transversal for all codes herein. The gate $ \mathsf{\Phi} $ is exotic in the sense that it lies outside the Clifford hierarchy and thus any code with transversal $ \mathsf{\Phi} $ must be nonadditive \cite{transuniv,restrictions}.
In the 26 years since the first nonadditive code was discovered \cite{nonadditive}, the demonstrated advantage over stabilizer codes has been confined to marginal improvements in the parameter $ K $ relative to fixed $ n $ and $ d $.
However, our $ 2I $ code family motivates the study of nonadditive codes from a different, and much stronger, perspective. Namely, we show that nonadditive codes can achieve transversality properties which are fundamentally inaccessible to any stabilizer code.
\section{Acknowledgments} We would like to thank J. Maxwell Sylvester, Michael Gullans, and Victor Albert for helpful conversations. This research was supported in part by NSF QLCI grant OMA-2120757.
\end{document} | arXiv |
A special issue
Dedicated to Peter W. Bates on the occasion of his 60th birthday
Jibin Li, Kening Lu, Junping Shi and Chongchun Zeng
This special issue of Discrete and Continuous Dynamical Systems-A is dedicated to Peter W. Bates on the occasion of his 60th birthday, and in recognition of his outstanding contributions to infinite dimensional dynamical systems and the mathematical theory of phase transitions.
Peter Bates was born in Manchester, England on December 27, 1947. He graduated from the University of London in mathematics in 1969 after which he moved to United States with his family. Later, he attended the University of Utah and received his Ph.D. in 1976. Following his graduation, Peter moved to Texas and taught at University of Texas at Pan American and Texas A&M University. He returned to Utah in 1984 and taught at Brigham Young University until 2004. He is currently a professor of mathematics at Michigan State University.
Jibin Li, Kening Lu, Junping Shi, Chongchun Zeng. Preface. Discrete & Continuous Dynamical Systems - A, 2009, 24(3): i-ii. doi: 10.3934/dcds.2009.24.3i.
A semilinear wave equation with smooth data and no resonance having no continuous solution
José F. Caicedo and Alfonso Castro
We prove that a boundary value problem for a semilinear wave equation with smooth nonlinearity, smooth forcing, and no resonance cannot have continuous solutions. Our proof shows that this is due to the non-monotonicity of the nonlinearity.
Jos\u00E9 F. Caicedo, Alfonso Castro. A semilinear wave equation with smooth dataand no resonance having no continuous solution. Discrete & Continuous Dynamical Systems - A, 2009, 24(3): 653-658. doi: 10.3934/dcds.2009.24.653.
Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity
Fengxin Chen
The uniqueness and stability of traveling wave solutions for system of nonlocal evolution equations with bistable nonlinearity are established. It is also proved that traveling waves are monotone and exponentially asymptotically stable, up to translation.
Fengxin Chen. Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2009, 24(3): 659-673. doi: 10.3934/dcds.2009.24.659.
Model reference control for SIRS models
Shui-Nee Chow and Yongfeng Li
The purpose of this paper is to introduce the model reference control method (MRC) in system biology. We review the main framework of MRC based on neural networks and some research issues. The model reference control for some model biological systems plant is considered.
Shui-Nee Chow, Yongfeng Li. Model reference control for SIRS models. Discrete & Continuous Dynamical Systems - A, 2009, 24(3): 675-697. doi: 10.3934/dcds.2009.24.675.
Quadratic perturbations of a class of quadratic reversible systems with two centers
B. Coll, Chengzhi Li and Rafel Prohens
Quadratic perturbations of a one-parameter family of quadratic reversible systems with two centers (without other singularities in finite plane) are studied. The exact upper bound of the number of limit cycles, the configurations of limit cycles, and the bifurcation diagrams for different range of the parameter are given.
B. Coll, Chengzhi Li, Rafel Prohens. Quadratic perturbations of a class of quadratic reversible systems with two centers. Discrete & Continuous Dynamical Systems - A, 2009, 24(3): 699-729. doi: 10.3934/dcds.2009.24.699.
Peak solutions for the Dirichlet problem of an elliptic system
E. N. Dancer, Danielle Hilhorst and Shusen Yan
We study a system of elliptic equations arising from biology with a chemotaxis term. This system is non-variational. Using a reduction argument, we show that the system has solutions with peaks near the boundary and inside the domain.
E. N. Dancer, Danielle Hilhorst, Shusen Yan. Peak solutions for the Dirichlet problem of an elliptic system. Discrete & Continuous Dynamical Systems - A, 2009, 24(3): 731-761. doi: 10.3934/dcds.2009.24.731.
Traveling wave solutions in parametric forms for a diffusion model with a nonlinear rate of growth
Zhaosheng Feng and Goong Chen
In this paper, we study a model of insect and animal dispersal where both density-dependent diffusion and nonlinear rate of growth are present. We analyze the existence of bounded traveling wave solution under certain parametric conditions by using the qualitative theory of dynamical systems. An explicit traveling wave solution is obtained by means of the first integral method. Traveling wave solutions in parametric forms for three particular cases are established by the Lie symmetry method.
Zhaosheng Feng, Goong Chen. Traveling wave solutions in parametric forms for a diffusion model with a nonlinear rate of growth. Discrete & Continuous Dynamical Systems - A, 2009, 24(3): 763-780. doi: 10.3934/dcds.2009.24.763.
Time averaging in turbulence settings may reveal an infinite hierarchy of length scales
Paul Fife, Joseph Klewicki and Tie Wei
The problem of discerning key features of steady turbulent flow adjacent to a wall has drawn the attention of some of the most noted fluid dynamicists of all time. Standard examples of such features are found in the mean velocity profiles of turbulent flow in channels, pipes or boundary layers. The aim of this article is to explain and further develop the recent concept of scaling patch for the time-averaged equations of motion of incompressible flow made highly turbulent by friction at a fixed boundary (introduced in recent papers by Wei et al, Fife et al, and Klewicki et al.) Besides outlining ways to identify the patches, which provide the scaling structure of mean profiles, a critical comparison will be made between that approach and more traditional ones.
Our emphasis will be on the question of how and how well these arguments supply insight into the structure of the mean flow profiles. Although empirical results may initiate the search for explanations, they will be viewed simply as means to that end.
Paul Fife, Joseph Klewicki, Tie Wei. Time averaging in turbulence settings may reveal aninfinite hierarchy of length scales. Discrete & Continuous Dynamical Systems - A, 2009, 24(3): 781-807. doi: 10.3934/dcds.2009.24.781.
On the stability of high Lewis number combustion fronts
Anna Ghazaryan and Christopher K. R. T. Jones
We consider wavefronts that arise in a mathematical model for high Lewis number combustion processes. An efficient method for the proof of the existence and uniqueness of combustion fronts is provided by geometric singular perturbation theory. The fronts supported by the model with very large Lewis numbers are small perturbations of the front supported by the model with infinite Lewis number. The question of stability for the fronts is more complicated. Besides discrete spectrum, the system possesses essential spectrum up to the imaginary axis. We show how a geometric approach which involves construction of the Stability Index Bundles can be used to relate the spectral stability of wavefronts with high Lewis numbers to the spectral stability of the front in the case of infinite Lewis number. We discuss the implication for nonlinear stability of fronts with high Lewis numbers. This work builds on the ideas developed by Gardner and Jones [12] and generalized in the papers by Bates, Fife, Gardner and Jones [3, 4].
Anna Ghazaryan, Christopher K. R. T. Jones. On the stability of high Lewis number combustion fronts. Discrete & Continuous Dynamical Systems - A, 2009, 24(3): 809-826. doi: 10.3934/dcds.2009.24.809.
On the number of limit cycles of a cubic Near-Hamiltonian system
Junmin Yang and Maoan Han
For the near-Hamiltonian system $\dot{x}=y+\varepsilon P(x,y),\dot{y}=x-x^2+\varepsilon Q(x,y)$, where $P$ and $Q$ are polynomials of $x,y$ having degree 3 with varying coefficients we obtain 5 limit cycles.
Junmin Yang, Maoan Han. On the number of limit cycles of a cubic Near-Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 2009, 24(3): 827-840. doi: 10.3934/dcds.2009.24.827.
Critical periods of a periodic annulus linking to equilibria at infinity in a cubic system
Zhirong He and Weinian Zhang
In this paper we investigate critical periods for a planar cubic differential system with a periodic annulus linking to equilibria at infinity. The monotonicity of the period function is decided by the sign of the second order derivative of a Abelian integral. We derive a Picard-Fuchs equation from a system of Abelian integrals and further give an induced Riccati equation for a ratio of derivatives of Abelian integrals. The number of critical points of the period function for periodic annulus is determined by discussing an planar autonomous system, the orbits of which describe solutions of the Riccati equation.
Zhirong He, Weinian Zhang. Critical periods of a periodic annuluslinking to equilibria at infinity in a cubic system. Discrete & Continuous Dynamical Systems - A, 2009, 24(3): 841-854. doi: 10.3934/dcds.2009.24.841.
Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains
Jianhua Huang and Wenxian Shen
The current paper is devoted to the study of pullback attractors for general nonautonomous and random parabolic equations on non-smooth domains $D$. Mild solutions are considered for such equations. We first extend various fundamental properties for solutions of smooth parabolic equations on smooth domains to solutions of general parabolic equations on non-smooth domains, including continuous dependence on parameters, monotonicity, and compactness, which are of great importance in their own. Under certain dissipative conditions on the nonlinear terms, we prove that mild solutions with initial conditions in $L_q(D)$ exist globally for $q$ » $1$. We then show that pullback attractors for nonautonomous and random parabolic equations on non-smooth domains exist in $L_q(D)$ for $1$ « $q$ < $\infty$.
Jianhua Huang, Wenxian Shen. Pullback attractors for nonautonomous and random parabolicequations on non-smooth domains. Discrete & Continuous Dynamical Systems - A, 2009, 24(3): 855-882. doi: 10.3934/dcds.2009.24.855.
Co-existence of traveling waves for a model of microbial growth and competition in a flow reactor
Wenzhang Huang
Consider a reaction-diffusion model for a microbial flow reactor with two competing species. Suppose that the amount of nutrient is input in a constant velocity at one end of the flow reactor and is washed out at the other end of the reactor. We study the dynamical behavior of population growth of these two species. In particular we are interested in the problem on the coexistence of traveling waves that best describes the long time dynamical behavior. By developing shooting method and continuation argument with the aid of an appropriately Liapunov function, we obtain the sufficient conditions for the coexistence of traveling waves as well as the minimum wave speed.
Wenzhang Huang. Co-existence of traveling waves for a model of microbial growth and competition in a flow reactor. Discrete & Continuous Dynamical Systems - A, 2009, 24(3): 883-896. doi: 10.3934/dcds.2009.24.883.
Family of nonlinear wave equations which yield loop solutions and solitary wave solutions
Jibin Li
2009, 24(3): 897-907 doi: 10.3934/dcds.2009.24.897 +[Abstract](1643) +[PDF](1087.2KB)
This paper gives a family of nonlinear wave equations, which can yield so called loop solution, cusp wave solution and solitary wave solution depending on the values of parameter $A$. For two third order systems, the dynamical behavior of these solutions are considered. The exact explicit parametric representations of solitary wave solutions and periodic wave solutions are given. It concerns with the properties of singular traveling wave systems.
Jibin Li. Family of nonlinearwave equations which yield loop solutions and solitary wavesolutions. Discrete & Continuous Dynamical Systems - A, 2009, 24(3): 897-907. doi: 10.3934/dcds.2009.24.897.
Regularization of simultaneous binary collisions and solutions with singularity in the collinear four-body problem
Tiancheng Ouyang and Zhifu Xie
The purpose of this paper is to analyze the asymptotic properties of collision orbits of Newtonian $N$-body problems. We construct new coordinates and time transformation that regularize the singularities of simultaneous binary collisions in the collinear four-body problem. The motion in the new coordinates and time scale across simultaneous binary collisions at least $C^2$. The explicit formulae are given in detail for the transformations and the extension of solutions. Furthermore, we study the behaviors of the motion approaching, across and after the simultaneous binary collision. Numerical simulations have been conducted for the special case in which the bodies are distributed symmetrically about the center of mass.
Tiancheng Ouyang, Zhifu Xie. Regularization of simultaneous binary collisions and solutions with singularity in the collinear four-body problem. Discrete & Continuous Dynamical Systems - A, 2009, 24(3): 909-932. doi: 10.3934/dcds.2009.24.909.
An eigenvalue variation problem of magnetic Schrödinger operator in three dimensions
Xing-Bin Pan
This paper concerns the lowest eigenvalue $\mu(b\N^Q)$ of the Schrödinger operator in three-dimensions with a magnetic potential $b\N^Q$, where the vector field $\N^Q$ depends on a matrix $Q$ varying in $SO(3)$ and $b$ is a real parameter. The eigenvalue variation problem is to minimize the lowest eigenvalue among all $Q$ in $SO(3)$. This problem arises in the phase transitions of smectic liquid crystals. We give an estimate of the minimum value inf${\mu(b\N^Q):~Q\in SO(3)\}$ for large $b$, and examine its dependence on geometry of the domain surface.
Xing-Bin Pan. An eigenvalue variation problem of magneticSchr\u00F6dinger operator in three dimensions. Discrete & Continuous Dynamical Systems - A, 2009, 24(3): 933-978. doi: 10.3934/dcds.2009.24.933.
Shell structure as solution to a free boundary problem from block copolymer morphology
Xiaofeng Ren
2009, 24(3): 979-1003 doi: 10.3934/dcds.2009.24.979 +[Abstract](2100) +[PDF](300.3KB)
A shell like structure is sought as a solution of a free boundary problem derived from the Ohta-Kawasaki theory of diblock copolymers. The boundary of the shell satisfies an equation that involves its mean curvature and the location of the entire shell. A variant of Lyapunov-Schmidt reduction process is performed that rigorously reduces the free boundary problem to a finite dimensional problem. The finite dimensional problem is solved numerically. The problem has two parameters: $a$ and $\gamma$. When $a$ is small, there are a lower bound and a sequence such that if $\gamma$ is greater than the lower bound and stays away from the sequence, there is a shell like solution.
Xiaofeng Ren. Shell structure as solution to a free boundary problem from block copolymer morphology. Discrete & Continuous Dynamical Systems - A, 2009, 24(3): 979-1003. doi: 10.3934/dcds.2009.24.979.
Existence theorems for periodic Markov process and stochastic functional differential equations
Daoyi Xu, Yumei Huang and Zhiguo Yang
In this paper, an effective existence theorem for periodic Markov process is first established. Using the theorem, we consider a class of periodic $It\hat{o}$ stochastic functional differential equations, and some sufficient conditions for the existence of periodic solution of the equations are given. To overcome the difficulties created by the special features possessed by the periodic stochastic differential equations with delays, as one will see, several lemmas are introduced. These existence theorems are rather general and therefore have great power in applications. Especially, our results are natural generalization of some classical periodic theorems on the model without stochastic perturbation. An example is worked out to demonstrate the advantages of our results.
Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochasticfunctional differential equations. Discrete & Continuous Dynamical Systems - A, 2009, 24(3): 1005-1023. doi: 10.3934/dcds.2009.24.1005.
Multidimensional periodic traveling waves in infinite cylinders
Guangyu Zhao
We study the existence and uniqueness, as well as various qualitative properties of periodic traveling waves for a reaction-diffusion equation in infinite cylinders. We also investigate the spectrum of the operator obtained by linearizing with respect to such a traveling wave. A detailed description of the spectrum is obtained.
Guangyu Zhao. Multidimensional periodic traveling waves in infinite cylinders. Discrete & Continuous Dynamical Systems - A, 2009, 24(3): 1025-1045. doi: 10.3934/dcds.2009.24.1025. | CommonCrawl |
\begin{document}
\title[Number of points]{On the number of rational points \\ on special families of curves \\ over function fields}
\author{Douglas Ulmer} \address{School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA} \email{[email protected]}
\author{Jos\'e Felipe Voloch} \address{School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand} \email{[email protected]}
\begin{abstract}
We construct families of curves which provide counterexamples for a
uniform boundedness question. These families generalize those studied
previously by several authors in \cite{Ulmer14a}, \cite{AIMgroup},
and \cite{ConceicaoUlmerVoloch12}. We show, in detail, what fails
in the argument of Caporaso, Harris, Mazur that uniform boundedness
follows from the Lang conjecture. We also give a direct proof that
these curves have finitely many rational points and give explicit
bounds for the heights and number of such points. \end{abstract}
\maketitle
\section{Unboundedness of rational points}
The question of whether there is a uniform bound for the number of rational points on curves of fixed genus greater than one over a fixed number field has been considered by several authors. In particular, in \cite{CaporasoHarrisMazur97} Caporaso et al.\ showed that this would follow from the Bombieri-Lang conjecture that the set of rational points on a variety of general type over a number field is not Zariski dense. In \cite{ConceicaoUlmerVoloch12}, Concei\c c\~ao and the present authors gave examples over function fields of families of smooth curves of fixed genus whose number of rational points is unbounded. Our first point is that these examples are part of a more general family.
Fix a prime $p$ and a power $q$ of $p$, let ${\mathbb{F}_q}$ be the field of $q$ elements, and let ${\mathbb{F}_q}(t)$ be the rational function field over ${\mathbb{F}_q}$. Choose an integer $r>1$ and prime to $p$, and let $h(x)\in{\mathbb{F}_q}[x]$ be a polynomial of positive degree which is not the $e$-th power of another element of ${\mathbb{F}_q}(t)$ for any divisor $e>1$ of $r$. We also assume that $h(0)\neq0$. For $a\in{\mathbb{F}_q}(t)\setminus{\mathbb{F}_q}$, let $X=X_{h,r,a}$ be the smooth projective curve over ${\mathbb{F}_q}(t)$ associated to the equation $$X:\quad y^r=h(x)h(a/x).$$ Our hypotheses imply that $X$ is absolutely irreducible and its genus is independent of $a$. In the case where $h$ has distinct roots and degree $s$ with $(r,s)=1$, the Riemann-Hurwitz formula shows that $X$ has genus $g=(r-1)s$.
\begin{thm}
Assume that $r$ divides $q^f+1$ for some $f\ge1$. Then as $a$
varies through ${\mathbb{F}_q}(t)\setminus{\mathbb{F}_q}$, the number of rational points
of the curve $X_{h,r,a}$ over $\mathbb{F}_q(t)$ is unbounded. \end{thm}
\begin{proof}
We first note that if $d=q^n+1$, $r$ divides $d$, and $a=t^d$, then
we have a rational point $(x,y)=(t,h(t)^{d/r})$ on $X$. Second, if
$m$ divides $n$ and $n/m$ is odd, then $d'=q^m+1$ divides $d$. If
$r$ divides $d'$, setting $e=d/d'$, we have another rational point
$(x,y)=(t^e,h(t^{e})^{d'/r})$ on $X$. Thus if we take $n$ to be a
multiple of $f$ such that $n/f$ is odd and has many factors, we have
many points. \end{proof}
\begin{exs} Up to change of coordinates, the case $r=2$, $h(x)=x+1$ is
the elliptic curve studied in \cite{Ulmer14a}, the case $r>1$,
$h(x)=x+1$ is the curve of genus $r-1$ whose Jacobian is the subject
of \cite{AIMgroup}, and the case $r=2$, $h(x)=x^s+1$ with $s$ odd is
the curve of genus $s$ studied in \cite{ConceicaoUlmerVoloch12}. \end{exs}
\begin{rem}
Fixing $h$ and $r$, here we consider a family of curves $X_a$ over a
fixed field ${\mathbb{F}_q}(t)$. It is sometimes more convenient to consider
the fixed curve $y^r=h(x)h(t/x)$ over extensions
${\mathbb{F}_q}(u)/{\mathbb{F}_q}(t)$ where $t$ is a varying rational function of $u$. \end{rem}
Consider the case $r=2$, $h(x)=x^s+1$ with $s$ odd. Let $\mathcal{X}$ be the smooth projective surface with affine model $$y^2=x(x^s+1)(x^s+t^s)$$ and consider the fibration $\mathcal{X} \to \mathbb{P}^1$, $(x,y,t) \mapsto t$. Its generic fiber is isomorphic to $X_{h,r,t}$ over ${\mathbb{F}_q}(t)$. As remarked in \cite{ConceicaoUlmerVoloch12}, the results of \cite{CaporasoHarrisMazur97} show that the fibration has a fibered power which covers a variety of general type. However, since this fibration is defined over a finite field, the variety of general type will also be defined over a finite field. Moreover it may have a Zariski dense set of $\mathbb{F}_p(t)$-rational points, so the rest of the argument of \cite{CaporasoHarrisMazur97} does not apply. (See \cite{AbramovichVoloch96} for a general discussion, including the function field case).
We can be more specific: In the next section, we will see that for many choices of $h$ and $r$, $X_{h,r,t}$ has a model over $\mathbb{P}^1_t$ which is already a variety of general type.
\section{Geometry of a regular proper model of $X$} When a curve $X$ over ${\mathbb{F}_q}(t)$ has a model $\mathcal{X}\to\mathbb{P}^1_t$ such that $\mathcal{X}$ is dominated by a product of curves, many questions about $X$ become much simpler. For example, the Tate conjecture on divisors holds for $\mathcal{X}$, the conjecture of Birch and Swinnerton-Dyer holds for the Jacobian of $X$, and it is often possible to compute or estimate the rank of the N\'eron-Severi group of $\mathcal{X}$ and the rank of group of rational points on the Jacobian. (This observation is mainly due to Shioda \cite{Shioda86} with further elaboration in \cite{Ulmer07b}.)
Fix a polynomial $h(x)\in{\mathbb{F}_q}[t]$ and an integer $r$ with hypotheses as in the first section. Fix also an integer $d$ prime to $p$, and let $X=X_{h,r,t^d}$ be the smooth projective curve over ${\mathbb{F}_q}(t)$ associated to the equation $$y^r=h(x)h(t^d/x).$$ Let $\mathcal{X}$ be a smooth projective surface equipped with a morphism to $\mathbb{P}^1$ whose generic fiber is isomorphic to $X$. (The construction is elementary; see \cite[Ch.~2]{Ulmer14b} for details.) In this section, we will show that $\mathcal{X}$ is dominated by a product of curves and give two applications: $\mathcal{X}$ is often of general type, and $X$ is non-isotrivial.
Let $\mathcal{C}=\mathcal{C}_{h,r,d}$ be the smooth projective curve over ${\mathbb{F}_q}$ associated to $$w^r=h(z^d).$$ Our hypotheses on $h$ and $r$ imply that $\mathcal{C}$ is absolutely irreducible. Note that $\mathcal{C}$ admits an action (over ${\overline{\mathbb{F}}_q}$) of the group $G:=\mu_r\times\mu_d$.
\begin{prop} The surface $\mathcal{X}$ is birational to the quotient of $\mathcal{C}\times\mathcal{C}$ by the action of $G$, where $G$ acts ``anti-diagonally,'' i.e., by the action above on the first factor and by its inverse on the second factor. \end{prop}
\begin{proof} The surface $\mathcal{X}$ is birational to the (quasi-) affine surface given by $$\mathcal{Y}:\quad y^r=h(x)h(t^d/x).$$ We define a rational map $\phi$ from $\mathcal{C}\times\mathcal{C}$ to $\mathcal{Y}$ by setting \begin{align*} \phi^*(x)&=z_1^d\\ \phi^*(y)&=w_1w_2\\ \phi^*(t)&=z_1z_2 \end{align*} It is evident that $\phi$ factors through $(\mathcal{C}\times\mathcal{C})/G$ where $G$ acts anti-diagonally, and a consideration of degrees shows that the induced rational map from $(\mathcal{C}\times\mathcal{C})/G$ to $\mathcal{Y}$ is birational. \end{proof}
We note that $(\mathcal{C}\times\mathcal{C})/G$, and therefore $\mathcal{X}$, contains infinitely many rational curves. Indeed the images in the quotient of the graphs of $q^n$-power Frobenius maps $\mathcal{C}\to\mathcal{C}$ and their transposes are rational curves. This gives a Zariski dense set of rational curves on $\mathcal{X}$.
Note that when $d=q^n+1$, the image of the graph of the $q^n$-power Frobenius $\mathcal{C}\to\mathcal{C}$ in $\mathcal{X}$ is the section of $\mathcal{X}\to\mathbb{P}^1$ corresponding to the point $(t,h(t)^{d/r})$, and the image of the transpose of Frobenius corresponds to the point $(t^{d-1},h(t)^{d/r})$. In some sense, this ``explains'' these points.
Our next result shows that $\mathcal{X}$ has general type as soon as $\mathcal{C}$ has genus $>1$. (See also \cite[\S7.1]{Granville07} for another proof of this fact.) If $h$ has degree $s$ with $(r,s)=1$ and distinct, non-zero roots, and if $r|d$, then the genus of $\mathcal{C}$ is $(r-1)(ds-2)/2$ which is $>1$ for large $d$ as soon as $r>1$ and $s\ge1$.
\begin{lemma}\label{lemma:general-type}
Let $C$ be a curve of genus $g(C)>1$ over a field $k$. Let $G$ be a
finite abelian group of automorphisms of $C$ with the order of $G$
prime to the characteristic of $k$. Let $Y = C \times C$ and let
$G$ act on $Y$ ``anti-diagonally": $g(y_1,y_2) = (gy_1,g^{-1}y_2)$.
Then the quotient $Y/G$ is of general type. \end{lemma}
Note that $Y/G$ is normal with isolated singular points, so it makes sense to speak of the canonical bundle and the plurigenera of $Y/G$.
\begin{proof}
We will show that $Y/G$ has Kodaira dimension 2, i.e., that the
plurigenera of $Y/G$ grow quadratically. Let
$V_n=H^0(C,K_C^{\otimes n})$. Since $g(C)>1, \dim V_n$ grows
linearly with $n: \dim V_n \ge cn$ for some $c>0$.
Decompose $V_n$ into eigenspaces for the action of $G$. At least
one of them has dimension $\ge \dim(V_n)/|G|$. Call it $V_{n,\rho}$
(where $\rho$ is the character by which $G$ acts on this subspace).
Since $G$ acts anti-diagonally, the image of
$V_{n,\rho}\otimes V_{n,\rho} \to H^0(Y,K_Y^{\otimes n})$ (via
pull-back and wedge product) lands in the $G$-invariant subspace,
which we denote $H^0(Y,K_Y^{\otimes n})^G$. The map is injective,
so
$$\dim H^0(Y,K_Y^{\otimes n})^G\ge(\dim(V_n)/|G|)^2.$$
This last expression is $\ge c'n^2$ for some $c'>0$.
Since $|G|$ is prime to the characteristic of $k$, we have $$H^0(Y/G,K_{Y/G}^{\otimes n})=H^0(Y,K_Y^{\otimes n})^G.$$ Thus $\dim H^0(Y/G,K_{Y/G}^{\otimes n}) \ge c'n^2$, as required. \end{proof}
Now we show that $X$ is not isotrivial, i.e., there does not exist a curve $X_0$ defined over a finite field $k$ and an isomorphism $$X\times_{{\mathbb{F}_q}(t)}\overline{{\mathbb{F}_q}(t)}\cong X_0\times_k\overline{{\mathbb{F}_q}(t)}.$$
\begin{prop}
The curve $X=X_{h,r,a}$ is not isotrivial for any
$a\in{\mathbb{F}_q}(t)\setminus{\mathbb{F}_q}$. \end{prop}
\begin{proof} From the definition of isotrivial, it clearly suffices to prove that $X_{h,r,t}$ is not isotrivial, so we assume $a=t$ for the rest of the proof. We will use the domination of a regular proper model $\mathcal{X}$ of $X$ by $\mathcal{C}\times\mathcal{C}$ where $\mathcal{C}$ is the curve associated to $w^r=h(z)$.
Let $Z\subset\mathcal{C}\times\mathcal{C}$ be the locus where $z_1z_2=0$. Since $h(0)\neq0$, this is the union of $2r$ curves each isomorphic to $\mathcal{C}$ meeting transversally at $r^2$ points.
Let $\widetilde{\mathcal{C}\times\mathcal{C}}$ be the blow up of $\mathcal{C}\times\mathcal{C}$ at the closed points where either $(z_1=0,z_2=\infty)$ or $(z_1=\infty,z_2=0)$. Let $\tilde Z$ be the strict transform of $Z$ in $\widetilde{\mathcal{C}\times\mathcal{C}}$.
The anti-diagonal action of $G:=\mu_r$ on $\mathcal{C}\times\mathcal{C}$ lifts uniquely to $\widetilde{\mathcal{C}\times\mathcal{C}}$, it preserves $\tilde Z$, and it has no fixed points on $\tilde Z$. (Again we use that $h(0)\neq0$.) It follows that $\tilde Z/G$ is the union of two copies of $\mathcal{C}$ meeting transversally at $r$ points. In particular, $\tilde Z/G$ is a semistable curve. It also follows that $\widetilde{\mathcal{C}\times\mathcal{C}}/G$ is regular in a neighborhood of $\tilde Z/G$.
Let $\mathcal{C}\times\mathcal{C}{\dashrightarrow}\mathbb{P}^1_t$ be the rational map defined by $t=z_1z_2$. This induces a morphism $\widetilde{\mathcal{C}\times\mathcal{C}}\to\mathbb{P}^1$ which factors through $\pi:\widetilde{\mathcal{C}\times\mathcal{C}}/G\to\mathbb{P}^1_t$. Moreover, the generic fiber of $\pi$ is $X$, and $\pi^{-1}(0)$ is precisely $\tilde Z/G$.
We have thus constructed a regular proper model of $X$ in a neighborhood of $t=0$ such that the special fiber is a non-smooth, semi-stable curve. This proves that the moduli map $\mathbb{P}^1_t\to\overline{\mathcal M}_g$ associated to $X$ is non-constant, and so $X$ is non-isotrivial. \end{proof}
\section{Height bounds} The finiteness of $X({\mathbb{F}_q}(t))$ when $X$ has genus $>1$ is of course a consequence of the Mordell conjecture for function fields. We will use the ABC theorem to give a direct, effective proof of this fact for a subclass of the curves studied above, namely a common generalization of the curves in \cite{ConceicaoUlmerVoloch12} and \cite{AIMgroup}.
For the rest of the paper, we fix positive integers $r$ and $s$ prime to one another and to $p$, we let $h(x)=x^s+1$, and we study the curve $$X:\quad y^r=h(x)h(t^d/x)=\frac{(x^s+1)(x^s+t^{ds})}{x^s}$$ over ${\mathbb{F}_q}(t)$ where $d$ is prime to $p$. As noted above, the genus $g$ of $X$ is $(r-1)s$.
Note that if $(x,y)$ is an ${\mathbb{F}_q}(t)$-rational point on $X$ and $x$ is a $p$-th power, then $(t^d/x,y)$ is another point and $t^d/x$ is not a $p$-th power.
In this section, we prove the following height bound.
\begin{thm}\label{thm:ht-bound}
Suppose that the genus $g$ of $X$ is $>2$. Let $(x,y)$ be an
${\mathbb{F}_q}(t)$-rational point on $X$, write $x=u/v$ with $u,v\in{\mathbb{F}_q}[t]$,
$(u,v)=1$, and let $\delta=max\{\deg u,\deg v\}$. If $x$ is not
a $p$-th power, then $$\delta\le\frac{dg-1}{g-2}$$
and if $x$ is a $p$-th power, then $$\delta\le\frac{2d(g-1)-1}{g-2}.$$ \end{thm}
\begin{proof}
The case when $x$ is a $p$-th power follows immediately from the
case when $x$ is not a $p$-th power after replacing $x$ with
$t^d/x$, so we may assume $x$ is not a $p$-th power.
We write $a$ for $t^d$. The hypotheses imply that $$\frac{(u^s+v^s)(u^s+a^sv^s)}{u^sv^s}$$ is an $r$-th power in ${\mathbb{F}_q}(t)$. Since $u$ and $v$ are relatively prime, we have
$$\left.\gcd(u^s,u^s+a^sv^s)\right|a^s$$ and
$$\left.\gcd(u^s+v^s,u^s+a^sv^s)\right|(a^s-1),$$ and all of the other terms in the displayed quantity are pairwise relatively prime, i.e., $$\gcd(u^s,v^s)=\gcd(u^s,u^s+v^s)= \gcd(v^s,u^s+v^s)=\gcd(v^s,u^s+a^sv^s)=1.$$ Therefore, $v$ is an $r$-th power, $t^iu$ is an $r$-th power for some $i\in\{0,\dots,r-1\}$, and $f(u^s+v^s)$ is an $r$-th power for some $f$ dividing $(a^s-1)^{r-1}$.
Next, we recall the ABC theorem in the following form (a special case of \cite[Chapter 6, Lemma 10]{Mason84}).
\begin{ABC}
If $A,B \in \mathbb{F}_q[t]$ are not both $p$-th powers, $(A,B)=1$, and
$C=A+B$, then we have $$\max \{\deg A,\deg B, \deg C \} \le \deg N(ABC) -1,$$ where $N(P)$ is the product of irreducible factors of $P$. \end{ABC}
Apply this with $A=u^s$, $B=v^s$. We have $\deg N(A)\le(\delta+r-1)/r$, $\deg N(B)\le \delta/r$, and $$\deg N(C)\le(\delta s+\deg f)/r\le(\delta s+ds(r-1))/r.$$ Since $A$ and $B$ are relatively prime, $N(ABC)=N(A)N(B)N(C)$ and we find that $$\delta s\le\frac{\delta(s+2)+ds(r-1)-1}{r}$$ and so $$\frac{\delta((r-1)s-2)}{r}\le\frac{ds(r-1)-1}{r}.$$ Assuming that $g-2=(r-1)s-2>0$, we find that $$\delta\le\frac{ds(r-1)-1}{(r-1)s-2}=\frac{dg-1}{g-2}$$ as desired. \end{proof}
We note that when $d=p^n+1$, we have points on $X(\mathbb{F}_p(t))$ with $x$
coordinate equal to $t^{(p^n+1)/(p^m+1)}$, which are not $p$-th powers, and for $m|n$, with $n/m$ odd, equal to $t^{(p^n+1)p^m/(p^m+1)}$, which are $p$-th powers. This shows that no major improvement of the inequality of the theorem can be expected.
\section{Cardinality bounds} We continue to study the curve $$X:\quad y^r=\frac{(x^s+1)(x^s+t^{ds})}{x^s}$$ over ${\mathbb{F}_q}(t)$ where $p$, $r$, and $s$ are pairwise relatively prime and $d$ is prime to $p$. Theorem~\ref{thm:ht-bound} yields an explicit bound on the number of points on $X({\mathbb{F}_q}(t))$ which is independent of $q$:
\begin{cor}
There is a constant $C$ depending only on $r$ and $s$, such that,
for any power $q$ of $p$ and any $d$ prime to $p$, we have
$\# X(\mathbb{F}_q(t)) \le C^d$. \end{cor}
\begin{proof}
Theorem~\ref{thm:ht-bound} shows that the $x$ coordinate of any
affine point has degree $O(d)$ and likewise the $y$ coordinate.
A curve that has infinitely many points of bounded
height (with coefficients in the algebraic closure of $\mathbb{F}_q$)
is isotrivial by \cite[Proposition 2]{Lang60}
and the remark that immediately follows,
so we get finiteness this way without appealing to the
Mordell conjecture. But we get more: The conditions on the $O(d)$
coefficients of the numerator and denominator of $x$ and $y$ for the
point to lie on $X$ is a system of $O(d)$ equations in $O(d)$
variables and each equation has degree at most $r+2s$. We can
consider this system over the algebraic closure of $\mathbb{F}_p$ and, by
the above argument, it has finitely many solutions, so by B\'ezout's
theorem it has at most $(r+2s)^{O(d)}$ solutions, proving the
corollary. \end{proof}
The main result of \cite{PachecoPazuki13} implies a bound similar to that of the theorem but with $C$ depending on $r$, $s$, and $p$. The cardinality of the set of points constructed in \cite{ConceicaoUlmerVoloch12} (and reviewed in Section~1 above) when $d=p^n+1$ is bounded by a multiple of the number of divisors of $n$, so there is a huge gap between the known upper bounds for the number of points and the number of points we can produce. It would be very interesting to narrow this gap or perhaps identify all the rational points.
Finally, we note that it is possible to improve the exponent when $d$ is large with respect to $q$. Indeed, the degree of conductor of the Jacobian of $X$ is $O(d)$ (with a constant depending only on $r$ and $s$). It follows from the arguments in \cite[\S11]{Ulmer07b} (generalizing \cite{Brumer92}) that the order of vanishing at $s=1$ of the $L$-function of $X$, and therefore the rank of the Mordell-Weil group of the Jacobian of $X$, is $O(d/\log d)$ (with a constant depending on $r$, $s$, and $q$). Applying \cite{BuiumVoloch96}, we find that the number of points on $X$ is at most $C_1^{d/\log d}$ where $C_1$ depends on $r$, $s$, and $q$. These bounds, and in particular the exact value of the rank, can in many cases be determined more precisely using the domination by a product of curves in Section~3 and arguments as in \cite{Ulmer13a}.
\emph{ Acknowledgements:} Both authors thank the Simons Foundation for financial support under grants \#359573 and \#234591. We also thank Igor Shparlinski for comments on an earlier version of the paper.
{}
\end{document} | arXiv |
More accurate demand forecasts are obviously good as far as inventory optimization is concerned. However, the quantitative assessment of the financial gains generated by an increase of the forecasting accuracy typically remains a fuzzy area for many retailers and manufacturers. This article details how to compute the benefits generated by an improved forecast.
The viewpoint adopted in this article is a best fit for high turnover inventories, with turnovers above 15. For high turnover values, the dominant effect is not so much stockouts, but rather the sheer amount of inventory, and its reduction through better forecasts. If such is not your case, you can check out our alternative formula for low turnover.
$D$ the turnover (total annual sales).
$\alpha$ the cost of stockout to gross margin ratio.
$p$ the service level achieved with the current error level (and current stock level).
$\sigma$ the forecast error of the system in place, expressed in MAPE (mean absolute percentage error).
$\sigma_n$ the forecast error of the new system being benchmarked (hopefully lower than $\sigma$).
It is possible to replace the MAPE error measurements by MAE (mean absolute error) measures within the formula. This replacement is actually strongly advised if slow movers exist in your inventory.
Let's consider a large retail network that can obtain a 10% reduction of the (relative) forecast error through a new forecasting system.
Based on the formula above, we obtain a gain at $B=1,800,000€$ per year. If we assume that the overall profitability of the retailer is 5%, then we see that a 10% improvement in forecasting accuracy already contribute to 4% of the overall profitability.
At a fundamental level, inventory optimization is a tradeoff between excess inventory costs vs. excess stockout costs.
Let's assume, for now, that, for a given stock level, the stockout frequency is proportional to the forecasting error. This point will be demonstrated in the next section.
The total volume of sales lost through stockouts is simple to estimate: it's $D(1-p)$, at least for any reasonably high value of $p$. In practice, this estimation is very good if $p$ is greater than 90%.
Hence, the total volume of margin lost through stock-outs is $D(1-p)m$.
Then, in order to model the real cost of the stock out, which is not limited to the loss of margin (think loss of customer loyalty for example), we introduce the coefficient $\alpha$. So the total economical loss caused by stock outs becomes $D(1-p)m\alpha$.
Based the assumption (demonstrated below) that stockouts are proportional to the error, we need to apply the factor $(\sigma - \sigma_n) / \sigma$ as the evolution of the stockout cost caused by the new average forecast error.
Let's demonstrate now the statement that, for a given inventory level, stockouts are proportional to the forecasting error.
In order to do that, let's start with service levels at 50% ($p=0.5$). In this context, the safety stock formula indicates that safety stocks are at zero. Several variants exist for the safety stock formula, but they are all behaving similarly in this respect.
With zero safety stocks, it becomes easier to evaluate the loss caused by forecast errors. When the demand is greater than the forecast (which happens here 50% of the time by definition of $p=0.5$), then the average percentage of sales lost is $\sigma$. Again, this is only the consequence of $\sigma$ being the mean absolute percentage error. However, with the new forecasting system, the loss is $\sigma_n$ instead.
Thus, we see that with $p=0.5$, stockouts are indeed proportional to the error. The reduction of the stockouts when replacing the old forecast with the new one will be $\sigma_n / \sigma$.
Now, what about $p \not= 0.5$? By choosing a service level distinct from 50%, we are transforming the mean forecasting problem into a quantile forecasting problem. Thus, the appropriate error metric for quantile forecasts becomes the pinball loss function, instead of the MAPE.
However, since we can assume here that the two mean forecasts (the old one, and the new one) will be extrapolated as quantile (to compute the reorder point), though the same formula, the ratio of the respective errors will remain the same. In particular, if the safety stock is small (say less than 20%) compared to the primary stock, then this approximation is excellent in practice.
The factor $\alpha$ has been introduced to reflect the real impact of a stockout on the business. A minima, we have $\alpha = 1$ because the loss caused by an extra stockout is at least equal to the volume of gross margin being lost. Indeed, when considering the marginal cost of a stockout, all infrasture and manpower costs are fixed, hence the gross margin should be considered.
a loss of client loyaulty.
a loss of supplier trust.
more erratic stock movements, stressing supply chain capacities (storage, transport, ...).
overhead efforts for downstream teams who try to mitigate stockouts one way or another.
Among several large food retail networks, we have observed that, as a rule thumb, practionners are assuming $\alpha=3$. This high cost for stockouts is also the reason why, in the first place, the same retail networks typically seek high service levels, above 95%.
In this section, we debunk one recurrent misconception about the impact of an extra accuracy, which can be expressed as extra accuracy only reduces safety stocks.
Looking at the safety stock formula, one might be tempted to think that the impact of a reduced forecasting error will be limited to lowering the safety stock; all other variables remaining unchanged (stockouts in particular). This is a major misunderstanding.
the primary stock, equal to the lead demand, that is to say the average forecast demand multiplied by the lead time.
the safety stock, equal to the demand error multiplied by a safety coefficient that depends mostly of $p$, the service level.
Let's go back to the situation where the service level equals 50%. In this situation, safety stocks are at zero (as seen before). If the forecast error was only impacting the safety stock component, then it would imply that the primary stock was immune to poor forecast. However, since there is no inventory here beyond the primary stock, we end-up with the absurd conclusion that the whole inventory has become immune to arbitrarily bad forecasts. Obviously, this does not make sense. Hence, the initial assumption, that only safety stocks were impacted is wrong.
Despite being incorrect, the safety stock only assumption is tempting because when looking at the safety stock formula, it looks like one immediate consequence. However, one should not jump to conclusions too hastily: this is not the only one consequence. The primary stock is built on top of the demand forecast as well, and it's the first one to be impacted by a more accurate forecast.
In section, we delve in further details that have been omitted in the discussion above for the sake of clarity and simplicity.
The formula above indicates that reducing the forecast error at 0% should bring stockouts at zero as well. On one hand, if customer demand could be anticipated with 100% accuracy 1 year in advance, achieving near-perfect inventory levels would seem less outstanding. One the other hand, some factors such as the varying lead time complicates the task. Even if the demand is perfectly known, an varying timing of delivery might generate further uncertainties.
In practice, we observe that the uncertainty related to the lead time is typically small compared to the uncertainty related to the demand. Hence, neglecting the impact of varying lead time is reasonable as long as forecasts remain somewhat inaccurate (say for MAPEs higher than 10%).
Delivering superior forecasts is the number one priority for Lokad. For companies with advance forecasting systems in place, benchmarks performed by our clients indicate that we typically reduce the relative forecasting error by 10% or more. For companies with little practices in place, the gain can go up to 30%. However, don't take our word for granted, and benchmark yourself for free your inventory practices with our forecasting engine using our 30-day free trial. | CommonCrawl |
\begin{definition}[Definition:Category of Presheaves on Topological Space]
Let $X$ be a topological space.
Let $\mathbf C$ be a category.
The '''category of $\mathbf C$-valued presheaves on $X$''' is the category with:
{{DefineCategory
|ob = $\mathbf C$-valued presheaves on $X$
|mor = morphisms of presheaves
|id = identity morphisms on presheaves
|comp = composition of morphisms of presheaves
}}
\end{definition} | ProofWiki |
How to calculate the generators of a list of stabilizer?
Let's say we have a list of stabilizers: {'YZY*', 'XYY*', 'YZYZ', 'Z*Z*', '*XZ*', 'XZX*', 'ZX*Z', 'XZXZ', 'YYX*', 'Z*ZZ'}. Is there any existed formula or function (eg. in qiskit) that can calculate its generators?
qiskit error-correction stabilizer-state
glS♦
peachnutspeachnuts
$\begingroup$ Note that there isn't a single set of generators. It is rather like giving a basis for a subspace ... there are many equivalent ones. $\endgroup$
– Markus Heinrich
The way that I'd do it is to write out the stabilizers in a $10\times 8$ matrix in this case (number of rows= number of stabilizers, number of columns is double the number of qubits). For each row, take a stabilizer and write out, for the first 4 columns, if there's an $X$ on a given qubit, and in the last 4 columns, if there's a $Z$ on a given qubit (remember, $Y$ contains both $X$ and $Z$).
Then, I'd simply run row reduction on my matrix (modulo 2). I don't know how to do this in qiskit, but Mathematica has the handy function RowReduce that can work modulo 2.
In essence, you're running Gram-Schmidt asking if each stabilizer in turn can be created as a product of the stabilizers already in your set.
$\begingroup$ Thanks! I used your method and did the RowReduce. I got a matrix: $\endgroup$
– peachnuts
$\begingroup$ Yes, you should have a matrix that specifies the (max) 4 stabilizers that generate the given stabilizers. (Note that the result is non-unique). $\endgroup$
– DaftWullie
$\begingroup$ Thanks! Here is the matrix that I got {{1,0,1,0,0,1,0,0},{0,1,0,0,0,0,0,0},{0,0,0,0,1,0,0,0},{0,0,0,0,0,0,1,0},{0,0,0,0,0,0,0,1},{0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0}} It has 5 terms that contain non-zero entry. In my understanding, they correspond to 5 generators: XZXI, IXII, ZIII, IIZI, IIIZ. . But you said the maximum number is 4, right? How can I eliminate them? $\endgroup$
$\begingroup$ Some of those don't commute. That suggests either your initial "stabilizers" don't commute with each other, or you've made a typo in entering the matrix. $\endgroup$
$\begingroup$ I got the answer {{1, 0, 1, 0, 0, 1, 0, 0}, {0, 1, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 1, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}}. $\endgroup$
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Qiskit Implementation of Grover's Algorithm to search a list | CommonCrawl |
\begin{definition}[Definition:Empty Topological Space]
Let $\O$ denote the empty set.
The topological space $\struct {\O, \set \O}$ is called the '''empty topological space'''.
\end{definition} | ProofWiki |
DOE PAGES Journal Article: Interacting dark energy: possible explanation for 21-cm absorption at cosmic dawn
Title: Interacting dark energy: possible explanation for 21-cm absorption at cosmic dawn
In this paper, a recent observation points to an excess in the expected 21-cm brightness temperature from cosmic dawn. In this paper, we present an alternative explanation of this phenomenon, an interaction in the dark sector. Interacting dark energy models have been extensively studied recently and there is a whole variety of such in the literature. Here we particularize to a specific model in order to make explicit the effect of an interaction.
Costa, André A. [1]; Landim, Ricardo C. G. [2];
Search DOE PAGES for author "Landim, Ricardo C. G."
Wang, Bin [3]; Abdalla, Elcio [1]
Univ. de Sao Paulo, Sao Paulo (Brazil)
Univ. de Sao Paulo, Sao Paulo (Brazil); SLAC National Accelerator Lab., Menlo Park, CA (United States)
YangZhou Univ., Yangzhou (China)
SLAC National Accelerator Lab., Menlo Park, CA (United States)
SLAC-PUB-17304
Journal ID: ISSN 1434-6044; PII: 6237
AC02-76SF00515; 150254/2017-2; 208206/2017-5; 013/26496-2
European Physical Journal. C, Particles and Fields
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
Costa, André A., Landim, Ricardo C. G., Wang, Bin, and Abdalla, Elcio. Interacting dark energy: possible explanation for 21-cm absorption at cosmic dawn. United States: N. p., 2018. Web. doi:10.1140/epjc/s10052-018-6237-7.
Costa, André A., Landim, Ricardo C. G., Wang, Bin, & Abdalla, Elcio. Interacting dark energy: possible explanation for 21-cm absorption at cosmic dawn. United States. doi:10.1140/epjc/s10052-018-6237-7.
Costa, André A., Landim, Ricardo C. G., Wang, Bin, and Abdalla, Elcio. Tue . "Interacting dark energy: possible explanation for 21-cm absorption at cosmic dawn". United States. doi:10.1140/epjc/s10052-018-6237-7. https://www.osti.gov/servlets/purl/1475481.
title = {Interacting dark energy: possible explanation for 21-cm absorption at cosmic dawn},
author = {Costa, André A. and Landim, Ricardo C. G. and Wang, Bin and Abdalla, Elcio},
abstractNote = {In this paper, a recent observation points to an excess in the expected 21-cm brightness temperature from cosmic dawn. In this paper, we present an alternative explanation of this phenomenon, an interaction in the dark sector. Interacting dark energy models have been extensively studied recently and there is a whole variety of such in the literature. Here we particularize to a specific model in order to make explicit the effect of an interaction.},
doi = {10.1140/epjc/s10052-018-6237-7},
journal = {European Physical Journal. C, Particles and Fields},
DOI: 10.1140/epjc/s10052-018-6237-7
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A small amount of mini-charged dark matter could cool the baryons in the early Universe
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Coupled dark matter-dark energy in light of near universe observations
Honorez, Laura Lopez; Reid, Beth A.; Mena, Olga
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The European Physical Journal C, Vol. 79, Issue 6
Listening to the sound of dark sector interactions with gravitational wave standard sirens
Yang, Weiqiang; Vagnozzi, Sunny; Valentino, Eleonora Di
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Mena, Olga; Palomares-Ruiz, Sergio; Villanueva-Domingo, Pablo
Physical Review D, Vol. 100, Issue 4
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Poulin, Vivian; Smith, Tristan L.; Grin, Daniel
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Landim, Ricardo G.
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Neutrino Mass Ordering from Oscillations and Beyond: 2018 Status and Future Prospects
de Salas, Pablo F.; Gariazzo, Stefano; Mena, Olga
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Constraining the Dark Matter Vacuum Energy Interaction Using the EDGES 21 cm Absorption Signal
Wang, Yuting; Zhao, Gong-Bo
DOI: 10.3847/1538-4357/aaeb9c
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Brownsberger, Sasha R.; Stubbs, Christopher W.; Scolnic, Daniel M.
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The effects of the ionosphere on ground-based detection of the global 21 cm signal from the cosmic dawn and the dark ages
Journal Article Datta, Abhirup ; Burns, Jack O. ; Bradley, Richard ; ... - Astrophysical Journal
Detection of the global H i 21 cm signal from the Cosmic Dawn and the Epoch of Reionization is the key science driver for several ongoing ground-based and future ground-/space-based experiments. The crucial spectral features in the global 21 cm signal (turning points) occur at low radio frequencies ≲100 MHz. In addition to the human-generated radio frequency interference, Earth's ionosphere drastically corrupts low-frequency radio observations from the ground. In this paper, we examine the effects of time-varying ionospheric refraction, absorption, and thermal emission at these low radio frequencies and their combined effect on any ground-based global 21 cm experiment. Itmore » should be noted that this is the first study of the effect of a dynamic ionosphere on global 21 cm experiments. The fluctuations in the ionosphere are influenced by solar activity with flicker noise characteristics. The same characteristics are reflected in the ionospheric corruption to any radio signal passing through the ionosphere. As a result, any ground-based observations of the faint global 21 cm signal are corrupted by flicker noise (or 1/f noise, where f is the dynamical frequency) which scales as ν{sup −2} (where ν is the frequency of radio observation) in the presence of a bright galactic foreground (∝ν{sup −s}, where s is the radio spectral index). Hence, the calibration of the ionosphere for any such experiment is critical. Any attempt to calibrate the ionospheric effects will be subject to the inaccuracies in the current ionospheric measurements using Global Positioning System (GPS) ionospheric measurements, riometer measurements, ionospheric soundings, etc. Even considering an optimistic improvement in the accuracy of GPS–total electron content measurements, we conclude that Earth's ionosphere poses a significant challenge in the absolute detection of the global 21 cm signal below 100 MHz.« less
DOI: 10.3847/0004-637X/831/1/6
Astro2020 Science White Paper: First Stars and Black Holes at Cosmic Dawn with Redshifted 21-cm Observations
Journal Article Mirocha, Jordan ; et al. - TBD
The "cosmic dawn" refers to the period of the Universe's history when stars and black holes first formed and began heating and ionizing hydrogen in the intergalactic medium (IGM). Though exceedingly difficult to detect directly, the first stars and black holes can be constrained indirectly through measurements of the cosmic 21-cm background, which traces the ionization state and temperature of intergalactic hydrogen gas. In this white paper, we focus on the science case for such observations, in particular those targeting redshifts zmore » $$\gtrsim$$ 10 when the IGM is expected to be mostly neutral. 21-cm observations provide a unique window into this epoch and are thus critical to advancing first star and black hole science in the next decade.« less
Evaluating the QSO contribution to the 21-cm signal from the Cosmic Dawn
Journal Article Ross, Hannah E. ; Dixon, Keri L. ; Ghara, Raghunath ; ... - Monthly Notices of the Royal Astronomical Society
The upcoming radio interferometer Square Kilometre Array (SKA) is expected to directly detect the redshifted 21-cm signal from the neutral hydrogen present during the Cosmic Dawn. Temperature fluctuations from X-ray heating of the neutral intergalactic medium can dominate the fluctuations in the 21-cm signal from this time. This heating depends on the abundance, clustering, and properties of the X-ray sources present, which remain highly uncertain. We present a suite of three new large-volume, 349\,Mpc a side, fully numerical radiative transfer simulations including QSO-like sources, extending the work previously presented in Ross et al. (2017). The results show that our QSOs have a modest contribution to the heating budget, yet significantly impact the 21-cm signal. Initially, the power spectrum is boosted on large scales by heating from the biased QSO-like sources, before decreasing on all scales. Fluctuations from images of the 21-cm signal with resolutions corresponding to SKA1-Low at the appropriate redshifts are well above the expected noise for deep integrations, indicating that imaging could be feasible for all the X-ray source models considered. The most notable contribution of the QSOs is a dramatic increase in non-Gaussianity of the signal, as measured by the skewness and kurtosis of the 21-cm probability distribution functions. Furthermore, in the case of late Lyman-more » $$\alpha$$ saturation, this non-Gaussianity could be dramatically decreased particularly when heating occurs earlier. We conclude that increased non-Gaussianity is a promising signature of rare X-ray sources at this time, provided that Lyman-$$\alpha$$ saturation occurs before heating dominates the 21-cm signal.« less
DOI: 10.1093/mnras/stz1220
THE IMPACT OF THE SUPERSONIC BARYON-DARK MATTER VELOCITY DIFFERENCE ON THE z {approx} 20 21 cm BACKGROUND
Journal Article McQuinn, Matthew ; O'Leary, Ryan M - Astrophysical Journal
Recently, Tseliakhovich and Hirata showed that during the cosmic Dark Ages the baryons were typically moving supersonically with respect to the dark matter with a spatially variable Mach number. Such supersonic motion may source shocks that inhomogeneously heat the universe. This motion may also suppress star formation in the first halos. Even a small amount of coupling of the 21 cm signal to this motion has the potential to vastly enhance the 21 cm brightness temperature fluctuations at 15 {approx}< z {approx}< 40, as well as to imprint distinctive acoustic oscillations in this signal. We present estimates for the sizemore » of this coupling, which we calibrate with a suite of cosmological simulations of the high-redshift universe using the GADGET and Enzo codes. Our simulations, discussed in detail in a companion paper, are initialized to self-consistently account for gas pressure and the dark matter-baryon relative velocity, v {sub bc} (in contrast to prior simulations). We find that the supersonic velocity difference dramatically suppresses structure formation on 10-100 comoving kpc scales, it sources shocks throughout the universe, and it impacts the accretion of gas onto the first star-forming minihalos (even for halo masses as large as 10{sup 7} M {sub Sun }). However, prior to reheating by astrophysical sources, we find that the v {sub bc}-sourced temperature fluctuations can contribute only as much as Almost-Equal-To 10% of the fluctuations in the 21 cm signal. We do find that v {sub bc} in certain scenarios could source an O(1) component in the power spectrum of the 21 cm background on observable scales via the X-ray (but not ultraviolet) backgrounds produced once the first stars formed. In a scenario in which {approx}10{sup 6} M {sub Sun} minihalos reheated the universe via their X-ray backgrounds, we find that the pre-reionization 21 cm signal would be larger than previously anticipated and exhibit more significant acoustic features. Such features would be a direct probe of the first stars and black holes. In addition, we show that structure formation shocks are unable to heat the universe sufficiently to erase a strong 21 cm absorption trough at z {approx} 20 that is found in most models of the sky-averaged 21 cm intensity.« less
Topics in Theoretical Physics
Technical Report Cohen, Andrew ; Schmaltz, Martin ; Katz, Emmanuel ; ...
This award supported a broadly based research effort in theoretical particle physics, including research aimed at uncovering the laws of nature at short (subatomic) and long (cosmological) distances. These theoretical developments apply to experiments in laboratories such as CERN, the facility that operates the Large Hadron Collider outside Geneva, as well as to cosmological investigations done using telescopes and satellites. The results reported here apply to physics beyond the so-called Standard Model of particle physics; physics of high energy collisions such as those observed at the Large Hadron Collider; theoretical and mathematical tools and frameworks for describing the laws ofmore » nature at short distances; cosmology and astrophysics; and analytic and computational methods to solve theories of short distance physics. Some specific research accomplishments include + Theories of the electroweak interactions, the forces that give rise to many forms of radioactive decay; + Physics of the recently discovered Higgs boson. + Models and phenomenology of dark matter, the mysterious component of the universe, that has so far been detected only by its gravitational effects. + High energy particles in astrophysics and cosmology. + Algorithmic research and Computational methods for physics of and beyond the Standard Model. + Theory and applications of relativity and its possible limitations. + Topological effects in field theory and cosmology. + Conformally invariant systems and AdS/CFT. This award also supported significant training of students and postdoctoral fellows to lead the research effort in particle theory for the coming decades. These students and fellows worked closely with other members of the group as well as theoretical and experimental colleagues throughout the physics community. Many of the research projects funded by this grant arose in response to recently obtained experimental results in the areas of particle physics and cosmology. We describe a few of these below. Relativity is founded on a symmetry property of nature called "Lorentz Invariance". Like all symmetry properties, it is essential to determine precisely how symmetric nature actually is; that is, do the laws of nature fully respect the symmetry or is there room for tiny symmetry violating effects? An important consequence of Lorentz invariance is the existence of a universal limiting velocity for all physical particles. Light travels at this limiting velocity so it is frequently referred to as simply "the speed of light", but relativity requires that ALL particles travel more slowly than this speed. Once the Higgs particle was discovered in 2012 a natural question was whether or not this particle's speed was consistent with relativity. Although the speed of the Higgs particle is not measurable directly, Cohen has shown that, if the maximal speed of the Higgs particle was not precisely the same as the speed of light, then the Higgs would have some unusual properties. In some cases the Higgs would be unstable to some unusual decay modes; in other cases the interactions of the Higgs with other particles would change the properties of these other particles in ways that could be observed in so-called cosmic rays, very energetic particles (such as photons, protons and other atomic nuclei) coming from space. Once these particles hit the upper atmosphere they produce a "shower" of particles that can be seen by ground-based instruments. If the Higgs has a maximal speed that differs even a tiny bit from the speed of light these showers would look quite different from what is observed. In this way Cohen was able to establish that the Higgs travels with a maximal speed that cannot differ from the speed of light by more than one part in a thousand-trillion. This is by far the most precisely determined property of the Higgs particle. Cohen and Schmaltz reviewed evidence from the Large Hadron Collider (LHC), a particle physics experiment operating at the CERN laboratory near Geneva, for a new particle sometimes called a W'. This evidence included certain unexpected by-products in collisions of protons at very high energy. While the evidence was not significant enough to claim a discovery, it was sufficiently intriguing that many particle theorists worked to construct explanations for this signal. Cohen and Schmaltz were able to determine that such explanations are highly constrained by previous experiments involving collisions of very energetic particles. Nevertheless they were able to construct a theory that adequately explains the LHC data and remain consistent with prior experiments. Their explanation predicts the existence of yet another new particle, called a Z', with a mass slightly greater than that of the W'. This additional particle, if it exists, should be seen as more data is collected from the LHC. Amusingly, there is one collision by-product that has already been seen by the CMS experiment at the LHC that supports the existence of this new particle; however, it is not unlikely that this single event is a so-called "background" event, that is a somewhat atypical by-product of a conventional Standard Model process. This theory for the anomalous LHC data will either be confirmed or excluded with further data-taking at the LHC. The ratio of the number of electrons produced in bottom quark decays over the number of muons produced has been measured at the LHC. This ratio is interesting because it can be predicted very precisely from a basic property of the Standard Model: lepton universality. If lepton universality is correct, the ratio of electrons to muons is predicted to be equal to 1. The first measurements of this ratio find a value different from 1 with a statistical significance of about 3 standard deviations. Schmaltz and collaborator proposed a new extension of the Standard Model which can explain the new data. In addition, Schmaltz and collaborators proposed several new measurements of ratios of decay rates which can confirm or rule out the surprising results from the earlier LHC data. The most recent and precise measurements of the cosmic microwave background from the Planck satellite, from a combination of measurements of the dark matter distribution in the universe, and from a measurement of the expansion rate of the universe today show some disagreement when interpreted in terms of the so-called LambdaCDM model. Schmaltz and collaborators proposed an alternative model to LambdaCDM in which the usual cold dark matter is replaced by a new ``dark sector". This sector consists of a cold dark matter particle which interacts with a newly postulated dark radiation component of the universe. The dark radiation can help explain the discrepancy in measurements of the expansion rate, and the dark matter interactions subtly modify the clumping of dark matter at large scales, thus potentially explaining both kinds of tensions in the data. In two publications Schmaltz described the new model and then performed a precision comparison of the predictions of the model with all currently available cosmological data. The results favor the new model at the level of three standard deviations with current data. Quantum Field Theory (QFT) is the language we use to describe quantum systems which are consistent with Einstein's theory of Special Relativity. In particular, the requirement of Einstein's theory that signals not travel faster than the speed of light constrains the types of interactions which particles can engage in. One consequence of relativity is that these interactions cannot preserve particle number. The stronger the interactions, the more severe the particle number violation in a given Relativistic QFT. When particle number violation is strong, it becomes very difficult to adequately parameterize the quantum wave function (which characterizes the state of a quantum system). For example, though we can formulate the QFT which describes the strong force as a set of interactions between quark and gluon particles, we have no clear idea how to express the proton state in terms of these quarks and gluons. This is because the proton, though a bound state of quarks and gluons, is not a state of a fixed number of particles due to strong interactions. Yet, understanding the proton state is very important in order to theoretically predict the reaction rates observed at the LHC in Geneva, which is a proton-proton collider. Katz has formulated a new approach to QFT, which among other things offers a way to adequately approximate the quantum wave function of a bound state at strong coupling. The approximation scheme is related to the fact that any sensible QFT (including that of the strong interactions) is at short distances approximately self-similar upon rescaling of space and time. It turns out that keeping track of the response upon this rescaling is important in efficiently parameterizing the state. Katz and collaborators have used this observation to approximate the state of the proton in toy versions of the strong force. In the late 60s Sheldon Glashow, Abdus Salam and Steven Weinberg (1979 Nobel Prize awardees) proposed a theory unifying weak and electromagnetic interaction which assumed the existence of new particles, the W and Z bosons. The W and Z bosons were eventually detected in high-energy collision in a particle accelerator at CERN, and the recent discovery of the Higgs meson at the Large Hadron Collider (LHC), always at CERN, completed the picture. However, deep theoretical considerations indicate that the theory by Glashow, Weinberg and Salam, often referred to as "the standard model" cannot be the whole story: the existence of new particles and new interactions at yet higher energies is widely anticipated. The experiments at the LHC are looking for these, while theorists, like Brower, Rebbi and collaborators, are investigating models for these new interactions. Working in a large national collaboration with access to the most powerful DOE computers Brower, Rebbi and colleagues have been using calculational techniques, similar to those successfully employed for many years to investigate the interactions among quarks in nucleons, to study theories that can describe the expected "beyond the standard model" (BSM) interactions. Their results, which include also a model for dark matter, have been published in several refereed papers in prestigious journals. Various ideas in topologically interesting field theories predict hypothetical objects such as fractional charges and Majorana excitations. However, such fascinating objects have not been seen in particle physics. Nevertheless, these objects demonstrate possible phenomena that quantum field theory can support. Pi used condensed matter physics as a laboratory to study possible realizations and observable effects of these objects predicted by quantum field theory. In recent times there has developed considerable interest among condensed matter field theorists in precisely the same geometrical and topological structures, which were first discovered in particle physics field theories. From particle physicists' point of view, this is an interesting development, since condensed matter provides an arena in which one can concretely realize particle physics ideas. Moreover, particle physicists can learn new ideas from condensed matter physics. Higgs phenomenon is precisely an important particle physics realization of condensed matter ideas. In contrast to the small distance characterizing condensed matter systems, field theory also describes large distance physics characterizing cosmology. Pi worked on various geometrical effects in the standard theory of cosmology, viz general relativity.« less
DOI: 10.2172/1327249 | CommonCrawl |
\begin{document}
\title{Geometric filtrations of \\classical link concordance}
\begin{abstract} This paper studies \emph{grope} and \emph{Whitney tower} filtrations on the set of concordance classes of classical links in terms of \emph{class} and \emph{order} respectively. Using the tree-valued intersection theory of Whitney towers, the associated graded quotients are shown to be finitely generated abelian groups under a well-defined connected sum operation.
\emph{Twisted Whitney towers} are also studied, along with a corresponding quadratic enhancement of the intersection theory for framed Whitney towers that measures Whitney-disk framing obstructions. The obstruction theory in the framed setting is strengthened, and the relationships between the twisted and framed filtrations are described in terms of exact sequences which show how higher-order Sato-Levine and higher-order Arf invariants are obstructions to framing a twisted Whitney tower.
The results from this paper combine with those in \cite{CST2,CST3,CST4} to give a classifications of the filtrations; see our survey \cite{CST0} as well as the end of the introduction below. UPDATE: The results of this paper have been completely subsumed into the paper \emph{Whitney tower concordance of classical links} \cite{WTCCL}. \end{abstract}
\author[J. Conant]{James Conant} \email{[email protected]} \address{Dept. of Mathematics, University of Tennessee, Knoxville, TN}
\author[R. Schneiderman]{Rob Schneiderman} \email{[email protected]} \address{Dept. of Mathematics and Computer Science, Lehman College, City University of New York, Bronx, NY}
\author[P. Teichner]{Peter Teichner} \email{[email protected]} \address{Dept. of Mathematics, University of California, Berkeley, CA and} \address{Max-Planck Institut f\"ur Mathematik, Bonn, Germany}
\keywords{Whitney towers, gropes, link concordance, trees, higher-order Arf invariants, higher-order Sato-Levine invariants, twisted Whitney disks}
\maketitle
\section{Introduction}\label{sec:intro} Several key theorems and conjectures in low-dimensional topology can be stated in terms of certain 2-complexes known as \emph{gropes}. These are geometric embodiments of commutators of group elements, see e.g.~\cite{Can, COT, CT1,CT2,FQ,FT2,T1, T2}.
Gropes are built by gluing together surfaces along collections of embedded essential circles, and come in several different types with varying measures of complexity.
The {\em height} of a grope corresponds to the derived series and was used (in the presence of caps) in \cite{FQ,FT2} to formulate the main open problem for topological $4$--manifolds: the 4-dimensional surgery and s-cobordism theorems for arbitrary fundamental groups are equivalent to a certain statement about capped gropes of height $\geq 2$. In the uncapped setting, gropes of increasing height were used in \cite{COT} to define a new filtration of the knot concordance group.
In this paper, we shall study a more basic measure of the complexity, namely the \emph{class} of a grope, which corresponds to commutator length and the lower central series (Figure~\ref{fig:class4gropeA}). In the 3-dimensional setting it was used in \cite{CT1,CT2} to give a geometric interpretation of Vassiliev's finite type invariants of knots and (the first nonvanishing term of) the Kontsevich integral.
\begin{figure}
\caption{A class $4$ grope.}
\label{fig:class4gropeA}
\end{figure}
In the 4-dimensional setting it is a fundamental open problem to determine under what conditions the components of a link in the $3$--sphere bound class $n$ gropes disjointly embedded in the $4$--ball, and this paper will describe a program for computing the \\
\emph{Grope concordance filtration} (by class) \begin{equation} \tag{$\mathbb{G}$} \dots \subseteq \mathbb{G}_{3} \subseteq \mathbb{G}_{2} \subseteq \mathbb{G}_{1} \subseteq \mathbb{G}_{0} \subseteq \mathbb{L} \end{equation} on the set $\mathbb{L}=\mathbb{L}(m)$ of framed links in $S^3$ with $m$ components. Here $\mathbb{G}_n=\mathbb{G}_n(m)$ is defined to be the set of framed links that bound class~$(n+1)$ framed gropes disjointly embedded in $B^4$. The index shift provides compatibility with other filtrations introduced below and for the same reason, we {\em define} $\mathbb{G}_0$ to be the set of evenly framed links.
The intersection of all $\mathbb{G}_n$ contains all slice links because a 2-disk is a grope of arbitrary large class. In fact, this filtration factors through link concordance; and we shall use this fact implicitly at various places.
Recall that the connected sum of links is not a well-defined operation because of the choices of connecting bands that are involved. As a consequence, there is no direct definition of the associated graded quotients of the filtration $\mathbb{G}_n$. However, one can define an equivalence relation on links by using the notion of {\em class~$n$ grope concordance} between two links. This is obtained by using framed gropes built on annuli connecting link components in $S^3\times I$. We then define the {\em associated graded} $\G_n=\G_n(m)$ as the quotient of $\mathbb{G}_n$ modulo grope concordance of class $n+2$. Our first result is the following corollary of Theorem~\ref{thm:R-onto-G} below.
\begin{cor} \label{cor:gropes} For all $m,n\in\mathbb{N}$, the sets $\G_n(m)$ are finitely generated abelian groups, under a well-defined connected sum $\#$. Moreover, $\G_n$ is the set of framed links $L\in\mathbb{G}_n$ modulo the relation that $[L_1]=[L_2]\in\G_n$ if and only if $L_1 \# -L_2$ lies in $\mathbb{G}_{n+1}$, for some choice of connected sum $\#$. Here $-L$ is the mirror image of $L$ with reversed framing. \end{cor}
For example, $\G_0(m) \cong\mathbb{Z}^k$ where $k=m(m+1)/2$ is the number of possible linking numbers and framings (on the diagonal) of a link with $m$ components. This follows from the fact that disjointly embedded class 2 gropes in $B^4$ are framed surfaces which induce the zero framings on their boundary and show the vanishing of all linking numbers. We also note that $\G_1(1) \cong \mathbb{Z}_2$ is given by the Arf invariant, and that $\mathbb{G}_2(1) =\mathbb{G}_n(1)$ for all $n\geq 2$, by \cite{S2}.
The proof of Corollary~\ref{cor:gropes} can be most succinctly formulated by defining a sequence of finitely generated abelian groups $\mathcal{T}_n=\mathcal{T}_n(m)$ in terms of certain trees. These groups have previously appeared in the study of graph cohomology, Feynman diagrams and finite type invariants of links and $3$--manifolds.
\begin{defn}\label{def:Tau}\cite{ST2} In this paper, a {\em tree} will always refer to an oriented unitrivalent tree, where the {\em orientation} of a tree is given by cyclic orientations at all trivalent vertices. The \emph{order} of a tree is the number of trivalent vertices. Univalent vertices will usually be labeled from the set $\{1,2,3,\ldots,m\}$ corresponding to the link components, and we consider trees up to isomorphisms preserving these labelings. We define $\mathcal{T}=\mathcal{T}(m)$ to be the free abelian group on such trees, modulo the antisymmetry (AS) and Jacobi (IHX) relations shown in Figure~\ref{fig:ASandIHXtree-relations}. \begin{figure}
\caption{Local pictures of the \emph{antisymmetry} (AS) and \emph{Jacobi} (IHX) relations
in $\mathcal{T}$. Here all trivalent orientations are induced from a fixed orientation of the plane, and univalent vertices possibly extend to subtrees which are fixed in each equation.}
\label{fig:ASandIHXtree-relations}
\end{figure} Since the AS and IHX relations are homogeneous with respect to order, $\mathcal{T}$ inherits a grading $\mathcal{T}=\oplus_n\mathcal{T}_n$, where $\mathcal{T}_n=\mathcal{T}_n(m)$ is the free abelian group on order $n$ trees, modulo AS and IHX relations. \end{defn}
Then a construction similarl to Cochran's iterated Bing-doubling construction for realizing Milnor invariants \cite{C} (and equivalent to `simple clasper surgery along trees' in the sense of Goussarov and Habiro) leads to our {\em realization map} \[ R_n : \mathcal{T}_n \to \G_n \] and Theorem~\ref{thm:R-onto-G} says that this map is well-defined and onto. For example, if $ i\neq j$ then $R_0$ sends the order zero tree $i -\!\!\!-\!\!\!-\!\!\!- \,j $ to a disjoint union of an $(m-2)$-component unlink and a Hopf link with components numbered $i$ and $j$, all components being zero-framed. If $i=j$, we get an $m$-component unlink where the $i$th component has framing 2 and all other components are zero-framed.
In fact, it will turn out that $\G_n$ is \emph{isomorphic} to $\mathcal{T}_n$ when $n$ is even (and to a quotient by certain $2$-torsion in the odd case) via a map that is essentially defined by forgetting the roots of the rooted trees which correspond to iterated commutators determined by the grope branches.
The graded groups $\G_n$ are reminiscent of the groups of knots arising in the theory of finite type invariants, first discovered by Goussarov \cite{Gu1}, defined as a quotient of the monoid of knots by $n$-equivalence. These groups were later constructed more conceptually using claspers by both Habiro and Goussarov \cite{H, Gu2}. For the case of string links with $m$-strands, in \cite{CST} we defined $G_n(m)$ to be the monoid of $m$-string links modulo ($3$-dimensional) grope cobordism of class $n+1$, and the associated graded quotients are finitely generated abelian groups. As in the current paper, we defined a surjective realization map $$\Phi\colon \mathcal B^g_n(m)\to G_n(m)/G_{n+1}(m)$$ where $\mathcal B^g_n(m)$ is the group of Feynman diagrams with grope degree $n$, and showed that rationally it is an isomorphism. Clearly there is a surjection $\rho\colon G_n(m)/G_{n+1}(m)\twoheadrightarrow {\sf G}_n(m)$, and indeed our realization maps $R_n$ are the compositions $\rho\circ\Phi$, since graphs with loops give rise to null-concordant links and are therefore in the kernel of $\rho\circ\Phi$. (See also Remark~\ref{rem:finite-type-IHX} below.)
Although we could take the composition $\rho\circ\Phi$ as our definition of $R_n$, and prove the required properties by modifying the $3$-dimensional methods developed for knots by the first and third authors in \cite{CT1,CT2}, we take a different approach here which is directly 4-dimensional and significantly simpler.
\subsection{The Whitney tower filtration}\label{subsec:W-tower-filtration} We consider a second filtration on the set of framed links with $m$ components, defined by using (framed) {\em Whitney towers} in place of (framed) gropes. Just like gropes are iterated surface stages glued to each other in a specified way, a Whitney tower is constructed by adding layers of immersed framed Whitney disks that pair up the intersection points of lower stages (Figure~\ref{fig:order3Whitneytower-and-withTrees} and Section~\ref{sec:w-towers}).
Whitney towers also come with a measure of complexity, the {\em order}, which determines how often intersection points are paired up. For example, an order zero Whitney tower is just a union of framed immersed disks in $B^4$ bounded by a link in $S^3$ with the induced framing, and an order~$1$ Whitney tower would also have all intersections among the immersed disks paired by Whitney disks. We define $\mathbb{W}_n=\mathbb{W}_n(m)$ to be the set of framed $m$-component links that bound a Whitney tower of order~$n$ in $B^4$. On the set of framed links, we thus get the\\ {\em Whitney tower filtration} (by order) \begin{equation} \tag{$\mathbb{W}$} \dots \subseteq \mathbb{W}_{3} \subseteq \mathbb{W}_{2} \subseteq \mathbb{W}_{1} \subseteq \mathbb{W}_{0} \subseteq \mathbb{L} =\mathbb{L}(m) \end{equation} \begin{figure}
\caption{Part of a Whitney tower in $4$--space (left), and shown with part of an associated tree (right).
Whitney disks pair up transverse intersections of opposite sign.}
\label{fig:order3Whitneytower-and-withTrees}
\end{figure}
As for gropes above, the equivalence relation of order $n+1$ {\em Whitney tower concordance} (see Section~\ref{sec:realization-maps}) then leads to the `associated graded' $\W_n=\W_n(m)$. \begin{thm} \label{thm:R-onto-G} For each $m,n$, there are surjective realization maps \[ R_n=R_n(m) : \mathcal{T}_n(m) \twoheadrightarrow\W_n(m) \] and the sets $\W_n(m)$ are finitely generated abelian groups under the well-defined operation of connected sum $\#$. Moreover, $\W_n$ is the set of framed links $L\in\mathbb{W}_n$ modulo the relation that $[L_1]=[L_2]\in\W_n$ if and only if $L_1 \# -L_2$ lies in $\mathbb{W}_{n+1}$, for some choice of connected sum $\#$. Here $-L$ is the mirror image of $L$ with reversed framing. \end{thm}
Note that $\mathbb{W}_0$ consists of links that are evenly framed because a component has even framing if and only if it bounds a framed immersed disk in $B^4$. It is well known that the signed number of self-intersection points of such a framed disk equals twice the framing on the boundary. As a consequence, if a knot bounds the next type of Whitney tower (of order one) then all the intersections of the first stage (order zero) disk are paired up by Whitney disks and so the framing is zero. In fact, $\W_0(1) \cong 2\mathbb{Z}$, given by the framing; and $\W_1(1) \cong\mathbb{Z}_2$, detected by the Arf invariant.
The resemblance between the two realization maps above is no coincidence: It is a result of the second author in \cite{S1} that $\mathbb{G}_n=\mathbb{W}_n$ and $\G_n=\W_n$ by completely geometric constructions that lead from gropes to Whitney towers and back (Figure~\ref{fig:wtower-grope}). \begin{figure}
\caption{Controlled conversions between order~$n$ Whitney towers and class~$(n+1)$ gropes are described in detail in \cite{S1}.}
\label{fig:wtower-grope}
\end{figure}
This means that we can continue to study either filtration and in this paper, we decide to take the Whitney tower point of view. This choice is motivated by the fact that Whitney towers come with an obstruction theory which is accompanied by higher-order intersection invariants taking values in $\mathcal{T}$. As described in Section~\ref{sec:w-towers} and hinted at in the right side of Figure~\ref{fig:order3Whitneytower-and-withTrees}, any Whitney tower $\mathcal{W}$ of order~$n$ has an order $n$ intersection invariant $\tau_n(\mathcal{W})\in\mathcal{T}_n$ which is defined by summing the trees pictured in Figure~\ref{fig:order3Whitneytower-and-withTrees}. Order $n$ intersection invariants vanish on Whitney towers of order $n+1$, and the key step in the proof of Theorem~\ref{thm:R-onto-G} (given in Section~\ref{sec:realization-maps}) is the following ``raising the order'' result:
\begin{thm}[\cite{CST,S1,ST2}]\label{thm:raise} A link bounds a Whitney tower $\mathcal{W}$ of order~$n$ with $\tau_n(\mathcal{W})=0$, if and only if it bounds a Whitney tower of order $n+1$. \end{thm}
This theorem follows from realizing all relations in $\mathcal{T}_n$ geometrically by controlled maneuvers on Whitney towers. It was a very pleasant surprise that, say, the Jacobi identity (or IHX relation) has an incarnation as a certain sequence of moves on Whitney disks and their boundary-arcs in a Whitney tower \cite{CST}. It has the following important corollary whose proof is in Section~\ref{def:wtc}.
\begin{cor}\label{cor:tau=w-concordance} Links $L_0$ and $L_1$ represent the same element of $\W_n$ if and only if there exist order $n$ Whitney towers $\mathcal{W}_i$ in $B^4$ with $\partial\mathcal{W}_i=L_i$ and $\tau_n(\mathcal{W}_0)=\tau_n(\mathcal{W}_1)\in\mathcal{T}_n$. \end{cor}
Our realization map $R_n$ (and its surjectivity) can then be explained extremely well by the following commutative diagram: \begin{equation} \xymatrix{ \mathfrak{W}_n\ar@{->>}[d]_{\tau_n}\ar@{->>}[r]^{\partial} & \mathbb{W}_n \ar@{->>}[d]\\ \mathcal{T}_n \ar@{->>}[r]^{R_n} &\W_n } \end{equation} An element in the set $\mathfrak{W}_n$ is an order $n$ Whitney tower $\mathcal{W}$ in $B^4$ whose boundary $\partial \mathcal{W}$ lies in $\mathbb{W}_n$ by definition of our filtration. The map $R_n$ then arises directly from above the diagram via Corollary~\ref{cor:tau=w-concordance} and the surjectivity of the intersection invariant $\tau_n$.
The proof of Theorem~\ref{thm:R-onto-G} will be completed via a Bing doubling (and internal band sum) construction applied to the Hopf link, showing that $\tau_n$ is surjective.
As a consequence of Theorem~\ref{thm:R-onto-G} together with results in \cite{CST2,CST3,CST4}, it turns out that $\tau_{2n}(\mathcal{W})$ only depends on the concordance class of the link $L=\partial\mathcal{W}$, and not the Whitney tower $\mathcal{W}$ it bounds; and that the intersection invariants $\tau_{2n}$ induce inverses to the realization maps $R_{2n}$. In particular, we have following classification theorem for even orders: \begin{thm}[\cite{CST4}] \label{thm:even} The maps $R_{2n}:\mathcal{T}_{2n}\twoheadrightarrow \W_{2n}$ are isomorphisms. \end{thm}
To describe the situation in odd orders, we first introduce the {\em reduced} version $\widetilde\mathcal{T}_{2n-1}$ of $\mathcal{T}_{2n-1}$ by dividing out the {\em framing relations}. These relations are the images of homomorphisms \[ \Delta_{2n-1} : \mathbb{Z}_2 \otimes \mathcal{T}_{n-1} \to \mathcal{T}_{2n-1} \] defined by sending an order $n-1$ tree $t$ to the sum of trees gotten by doubling the subtree adjacent to each univalent vertex of $t$ (see Figure~\ref{fig:Delta trees} and the precise definition in Section~\ref{sec:proof-thm-odd}).
\begin{thm} \label{thm:odd} The odd order realization maps $R_{2n-1}$ vanish on the image of $\Delta_{2n-1}$ and give surjections \[ \widetilde R_{2n-1} : \widetilde\mathcal{T}_{2n-1} \twoheadrightarrow\W_{2n-1} \] Moreover, if a link $L$ bounds a Whitney tower $\mathcal{W}$ of order~$2n-1$ with $\tau_{2n-1}(\mathcal{W}) \in \operatorname{Im}(\Delta_{2n-1})$ then $L$ bounds a Whitney tower of order $2n$. \end{thm} Here the results of \cite{CST2,CST3,CST4} also apply to the cases where $n$ is \emph{even}, implying that $\widetilde \tau_{4k-1}(\mathcal{W}) \in \widetilde \mathcal{T}_{4k-1}$ only depends on the link $L$ and not the Whitney tower $\mathcal{W}$, and that $\widetilde \tau_{4k-1}$ induces an inverse to the realization map $\widetilde R_{4k-1}$. We get the following classification in these orders: \begin{thm} [\cite{CST4}]\label{thm:4k-1}
$\widetilde R_{4k-1}:\mathcal{T}_{4k-1}\twoheadrightarrow\widetilde \W_{4k-1}$ are isomorphisms. \end{thm}
The framing relations in odd orders arise from a subtle interplay between the IHX relations and certain framing obstructions associated to Whitney disks, as will be described below. As suggested by theorems~\ref{thm:odd} and \ref{thm:4k-1}, we believe that these relations do indeed capture all the relevant geometric aspects regarding the filtration. Setting $\widetilde{\mathcal{T}}_{2n}:=\mathcal{T}_{2n}$, we conjecture: \begin{conj} \label{conj:4k-3} The realization maps give $\W_n\cong\widetilde \mathcal{T}_n$ for all $n$. \end{conj} As discussed below and described in detail in \cite{CST2,CST4}, the affirmation of this conjecture is equivalent to the non-triviality of certain \emph{higher-order Arf invariants} which complete the classification of $\W_{4k-3}$.
The big picture is best described by introducing the following generalization of framed Whitney towers.
\subsection{The Twisted Whitney tower filtration}\label{subsec:twisted-W-tower-filtration} The proof of Theorem~\ref{thm:odd} given in Section~\ref{sec:proof-thm-odd} uses in addition to IHX type maneuvers another well known geometric move on a Whitney disk, namely the \emph{boundary twist}. A {\em framed} Whitney disk changes into a {\em twisted} Whitney disk by this move, at the cost of creating a new interior intersection in the Whitney disk (Figure~\ref{boundary-twist-and-section-fig}).
This motivated us to introduce yet another filtration of the set of links by looking at those links $\mathbb{W}^\iinfty_n=\mathbb{W}^\iinfty_n(m)$ with $m$ components that bound {\em twisted} Whitney towers of order $n$ (rather than {\em framed} Whitney towers as in the definition of $\mathbb{W}_n$). These are Whitney towers, except that certain Whitney disks are allowed to be twisted. We arrive at the\\ {\em Twisted Whitney tower filtration} (by order) \begin{equation} \tag{$\mathbb{W}^\iinfty$} \dots \subseteq \mathbb{W}^\iinfty_{3} \subseteq \mathbb{W}^\iinfty_{2} \subseteq \mathbb{W}^\iinfty_{1} \subseteq \mathbb{W}^\iinfty_{0}=\mathbb{L} \end{equation} We refer to Section~\ref{sec:realization-maps} for a precise definition, also of the associated graded $\W^\iinfty_n=\W^\iinfty_n(m)$; and to Section~\ref{sec:w-towers} for details on twisted Whitney towers, including the associated \emph{twisted intersection invariant} $\tau_n^\iinfty(\mathcal{W})\in\mathcal{T}^\iinfty_n= \mathcal{T}^\iinfty_n (m)$. The groups $\mathcal{T}^\iinfty_{2n-1}$ are defined as quotients of $\mathcal{T}_{2n-1}$ by the subgroups generated by trees of the form \[
i\,-\!\!\!\!\!-\!\!\!<^{\,J}_{\,J} \] where $J$ is a subtree of order~$n-1$. These \emph{boundary-twist relations} correspond to the intersections created by performing a boundary-twist on an order $n$ Whitney disk. The groups $\mathcal{T}^\iinfty_{2n}$ include certain ``twisted'' \emph{$\iinfty$-trees} representing framing obstructions on order $n$ Whitney disks, which are not required to be framed in an order $2n$ twisted Whitney tower. In \cite{CST4} we show that $\mathcal{T}$ is the universal home for invariant symmetric bilinear forms on quasi-Lie algebras, and that $\mathcal{T}^\iinfty_{2n}$ is the universal (symmetric quadratic) refinement of this form in order $n$.
Here and in the following the symbol $\iinfty$ represents a {\em twist}, and in particular does {\em not} stand for ``infinity''.
\begin{thm} \label{thm:twisted} For each $m,n$, there are surjective realization maps \[ R^\iinfty_n=R^\iinfty_n(m) : \mathcal{T}^\iinfty_n(m) \twoheadrightarrow\W^\iinfty_n(m) \] and the sets $\W^\iinfty_n(m)$ are finitely generated abelian groups under the well-defined connected sum $\#$ operation. Moreover, $\W^\iinfty_n$ is the set of framed links $L\in\mathbb{W}^\iinfty_n$ modulo the relation that $[L_1]=[L_2]\in\W^\iinfty_n$ if and only if $L_1 \# -L_2$ lies in $\mathbb{W}^\iinfty_{n+1}$, for some choice of connected sum $\#$. Here $-L$ is the mirror image of $L$ with reversed framing. \end{thm} As before, the key step in proving this result is the following criterion for {\em raising the order of a twisted Whitney tower}: \begin{thm} \label{thm:twisted-raising-intro} A link bounds a twisted Whitney tower $\mathcal{W}$ of order~$n$ with $\tau^{\iinfty}_{n}(\mathcal{W})=0$ if and only if it bounds a twisted Whitney tower of order $n+1$. \end{thm} Theorem~\ref{thm:twisted-raising-intro} follows from the more general Theorem~\ref{thm:twisted-order-raising-on-A} in Section~\ref{sec:w-towers}, which is proved in Section~\ref{sec:proof-twisted-thm} using ``twisted Whitney moves'' (Lemma~\ref{lem:twistedIHX}) as well as boundary-twists and a construction for ``geometrically cancelling'' twisted Whitney disks.
In the twisted setting we also have classifications of the associated graded groups of links in three out of four cases: \begin{thm}[\cite{CST2,CST4}] \label{thm:twisted-isos} For $n\equiv 0,1,3\mod 4$, the realization maps $R^\iinfty_{n}:\mathcal{T}^\iinfty_n \twoheadrightarrow \W^\iinfty_n$ are isomorphisms. \end{thm} In fact, in these orders the $\W^\iinfty_n$ are isomorphic to image $\sD_n$ of the first nonvanishing order $n$ (length $n+2$) Milnor link invariants \cite{CST2,CST4}. Theorem~\ref{thm:twisted} (as well as theorems~\ref{thm:even} and \ref{thm:4k-1} above) depends in an essential way on our resolution \cite{CST3} of a combinatorial conjecture of J. Levine involving a map from unrooted trees representing Whitney tower intersections to rooted trees representing iterated commutators determined by link longitudes. As described in \cite{CST2,CST3}, the kernel of this map is generated by certain symmetric $\iinfty$-trees that are not detected by Milnor invariants. These trees are related to what we call {\em higher-order Arf invariants} of the twisted filtration in \cite{CST2}, and the conjectured non-triviality of these invariants is shown in \cite{CST4} to be equivalent to the above Conjecture~\ref{conj:4k-3} as well as the following: \begin{conj} \label{conj:twisted} The maps $R^\iinfty_{4k-2}:\mathcal{T}^\iinfty_{4k-2} \twoheadrightarrow \W^\iinfty_{4k-2}$ are isomorphisms. \end{conj}
\subsection{Framed versus twisted Whitney towers}\label{subsec:overview} There is a surprisingly simple relation between twisted and framed Whitney towers of various orders that's very well expressed in terms of the following result:
\begin{prop}\label{prop:exact sequence} For any $n\in\mathbb{N}$, there is an exact sequence \[ \xymatrix{ 0\ar[r] & \W_{2n} \ar[r] & \W^\iinfty_{2n} \ar[r] & \W_{2n-1} \ar[r] & \W^\iinfty_{2n-1} \ar[r] & 0
} \] where all maps are induced by the identity on the set of links. \end{prop}
To see why this is true, observe that it is easier to find a twisted Whitney tower than a framed one and hence there is a natural inclusion $\mathbb{W}_{n} \subseteq \mathbb{W}_{n}^\iinfty$ between our two filtrations $(\mathbb{W})$ and $(\mathbb{W}^\iinfty)$ on the set of links. Moreover, by Definition~\ref{def:twisted-W-towers} we actually have \[ \mathbb{W}_{2n-1}^\iinfty = \mathbb{W}_{2n-1} \] because we found that a twisted Whitney disk of order~$n$ is most naturally associated to an order $2n$ Whitney tower, compare Remark~\ref{rem:motivate-twisted-towers}. One then needs to show that indeed $\mathbb{W}_{2n}^\iinfty \subseteq \mathbb{W}_{2n-1}$, which is accomplished in Lemma~\ref{lem:boundary-twisted-IHX} of Section~\ref{sec:proof-twisted-thm} using boundary-twists and twisted Whitney moves. Finally, by the `Moreover' parts of Theorems \ref{thm:R-onto-G} and \ref{thm:twisted}, we may say that \[ \W_n = \mathbb{W}_n/ \mathbb{W}_{n+1} \quad \text{ and } \quad \W^\iinfty_n = \mathbb{W}^\iinfty_n/ \mathbb{W}^\iinfty_{n+1} \] which implies the exact sequence in Proposition~\ref{prop:exact sequence}.
If our above conjectures above hold, then for every $n$ the various realization maps should lead to the analogous exact sequence for our groups defined by trees. The following result shows that this is indeed the case and even gives more information on kernels (respectively cokernels). It is best expressed in terms of the \emph{free quasi-Lie algebra} on $m$ generators: \[ \sL'=\sL'(m)= \bigoplus_{n\in\mathbb{N}} \sL'_n \] \begin{defn}[compare \cite{L3}]\label{def:L'} The abelian group $\sL'_{n+1}=\sL'_{n+1}(m)$ is generated by order~$n$ (trivalent, oriented) \emph{rooted} trees, each having a specified univalent vertex, labeled as {\em root}, and all other univalent vertices labelled by elements of $\{1,\dots,m\}$, modulo the AS and IHX relations of Figure~\ref{fig:ASandIHXtree-relations}. \end{defn} Here the prefix `quasi' reflects the fact that, although the IHX relation corresponds to the Jacobi identity via the usual identification of rooted trees with non-associative brackets, the usual Lie algebra self-annihilation relation $[X,X]=0$ does {\em not} hold in $\sL'$. It is replaced by the weaker anti-symmetry (AS) relation $[Y,X]=-[X,Y]$.
\begin{thm}[\cite{CST4}]\label{thm:Tau-sequences} For any $m,n\in\mathbb{N}$, there are short exact sequences \[ \xymatrix{ 0\ar[r] & \mathcal{T}_{2n} \ar[r] & \mathcal{T}^\iinfty_{2n} \ar[r] & \mathbb{Z}_2 \otimes \sL'_{n+1} \ar[r] & 0
} \] and \[ \xymatrix{ 0\ar[r] & \mathbb{Z}_2 \otimes \sL'_{n+1} \ar[r] & \widetilde\mathcal{T}_{2n-1} \ar[r] & \mathcal{T}^\iinfty_{2n-1} \ar[r] & 0
} \] \end{thm} This result is proved in \cite{CST4} using the \emph{universality} of $ \mathcal{T}^\iinfty_{2n}$ as the target of quadratic refinements of the canonical `inner product' pairing \[ \langle \ , \ \rangle: \sL'_{n+1} \times \sL'_{n+1} \to \mathcal{T}_{2n} \] given by gluing the roots of two rooted trees, see Definition~\ref{def:trees}. We can connect these short exact sequences to a single exact sequence \[ \xymatrix{ 0\ar[r] & \mathcal{T}_{2n} \ar[r] & \mathcal{T}^\iinfty_{2n} \ar[r] & \widetilde\mathcal{T}_{2n-1} \ar[r] & \mathcal{T}^\iinfty_{2n-1} \ar[r] & 0
} \] just like in Proposition~\ref{prop:exact sequence}. However, Theorem~\ref{thm:Tau-sequences} makes the additional predictions that \[ \operatorname{Cok} ( \W_{2n} \to \W^\iinfty_{2n}) \cong \mathbb{Z}_2 \otimes \sL'_{n+1} \cong \operatorname{Ker}( \W_{2n-1} \to \W^\iinfty_{2n-1} ) \] assuming the conjectures in this paper. As a consequence of these conjectures, we would obtain new concordance invariants with values in $\mathbb{Z}_2 \otimes \sL'_{n+1}$ and defined on $\mathbb{W}^\iinfty_{2n}$, as the obstructions for a link to bound a framed Whitney tower of order~$2n$. In \cite{CST4} we show that a quotient of $\mathbb{Z}_2 \otimes \sL'_{n+1}$, namely $\mathbb{Z}_2 \otimes \sL_{n+1}$, is indeed detected by what we call {\em higher-order Sato-Levine invariants}. Here the free quasi-Lie algebra $\sL'$ is replaced by the usual \emph{free Lie algebra} $\sL$ satisfying the Jacobi identity and self-annihilation relations $[X,X]=0$.
Levine showed in \cite{L2} that for odd $n$ the squaring map $X\mapsto [X,X]$ induces an isomorphism $$ \mathbb{Z}_2 \otimes \sL_{\frac{n+1}{2}} \cong\operatorname{Ker}(\mathbb{Z}_2 \otimes \sL'_{n+1}\twoheadrightarrow\mathbb{Z}_2 \otimes \sL_{n+1}) $$ We thus conjecture that this group is also detected by concordance invariants which are the above-mentioned {\em higher-order Arf invariants}. These would generalize the usual Arf invariants of the link components, which (as shown in \cite{CST2}) is the case $n=1$.
It is interesting to note that the case $n=0$ leads to the prediction $$ \operatorname{Cok} ( \W_{0} \to \W^\iinfty_{0}) \cong \mathbb{Z}_2 \otimes \sL_{1} \cong (\mathbb{Z}_2)^m $$ This is indeed the group of framed $m$-component links modulo those with even framings! In fact, the consistency of this computation was the motivating factor to consider filtrations of the set of {\em framed} links $\mathbb{L}$, rather than just oriented links.
As described in \cite{CST4}, the higher-order Sato-Levine invariants turn out to be determined by the Milnor invariants, providing a new interpretation for these classical invariants as obstructions to ``untwisting'' a Whitney tower. On the other hand, the higher-order Arf invariants do not appear to correspond to any known invariants. They take values in a (currently unknown) quotient of $\mathbb{Z}_2 \otimes \sL_{\frac{n+1}{2}}$ and we show in \cite{CST4} that together with the Milnor invariants they give a complete characterization of our three filtrations.
\subsection{Summary of all papers in this series} Paper \cite{CST0} gives an overview of the results of this paper together with the closely related papers \cite{CST2,CST3,CST4,CST5}. The classifications of the geometric filtrations of link concordance defined in the current paper are achieved in a sequence of steps:
\begin{enumerate} \item The current paper extends our previous intersection theory of Whitney towers \cite{CST,ST2} to the reduced (and twisted) settings, and proves that our obstruction theory works: If the order $n$ intersection invariant of a (twisted) Whitney tower $\mathcal{W}$ vanishes in $\widetilde{\mathcal{T}}$
(resp. $\mathcal{T}^\iinfty$) then $\mathcal{W}$ ``can be raised'' from order $n$ to order $n+1$, without changing the link $\partial \mathcal{W}$. As a consequence, we obtain the realization epimorphisms: \[ \widetilde R_n: \widetilde\mathcal{T}_n \twoheadrightarrow\W_n \quad \mbox{and} \quad R^\iinfty_{n}:\mathcal{T}^\iinfty_n \twoheadrightarrow \W^\iinfty_n \] We also introduce the exact sequences which explain the relationship between framed and twisted Whitney towers, motivating the definitions of the higher-order Sato-Levine and higher-order Arf invariants.
\item In \cite{CST2} we first show that the order $n$ Milnor invariants for links can be thought of as an epimorphism \[ \mu_n:\W^\iinfty_n \twoheadrightarrow \sD_n:=\ker([\cdot,\cdot]:\sL_1\otimes \sL_{n+1}\to \sL_{n+2}) \] Note that $\sD_n$ is a free abelian group whose rank can be computed via the Hall basis algorithm. Secondly, we use the geometric notion of {\em grope duality} to show that the composition \[ \eta_n: \mathcal{T}^\iinfty_n \overset{R^\iinfty_n}{\twoheadrightarrow} \W^\iinfty_n \overset{\mu_n}{\twoheadrightarrow} \sD_n \] can be described combinatorially in a very simple way: it is given by summing over all ways of choosing a root on a given tree. This map $\eta_n$ is closely related to the procedure for converting a Whitney tower into a grope which gives the isomorphism $\W_n\cong\G_n$. The construction of boundary links realizing the image of the higher-order Arf invariants leads to new geometric characterizations of links with vanishing Milnor invariants through length $2n$.
\item In \cite{CST3} we use combinatorial Morse theory to prove the \emph{Levine Conjecture}, that a map $\eta_n'$ (analogous to $\eta_n$), which is also defined by summing over all root choices, gives an isomorphism $\mathcal{T}_n \cong \sD'_n$, where $\sD'_n$ is the bracket kernel on the free quasi-Lie algebra $\sL'$ (analogous to $\sD_n$). This result has implications in the study of $3$-dimensional homology cylinders, where Levine originally formulated his conjecture, as well as playing a key role in providing the algebraic framework for the classifications of $\W_n$ and $\W^\iinfty_n$ completed in \cite{CST4}.
\item In \cite{CST4}, we assemble the relevant groups into commutative diagrams of exact sequences and complete the classifications of the geometric filtrations of link concordance defined in the current paper. For instance, our resolution in \cite{CST3} of the Levine Conjecture is used to show that $\eta_n$ is an isomorphism for $n\equiv 0,1,3\mod 4$ and that its kernel is $\mathbb{Z}_2\otimes \sL_k$ for $n=4k-2$. This leads to the formulations of the higher-order Arf invariants in both the framed and twisted settings, as well as a demonstration of their equivalence, and their relation to the higher-order Sato-Levine invariants. The geometric filtrations are shown to be classified by the Milnor and higher-order Arf invariants. This allows us to give complete proofs of Theorems~\ref{thm:even}, \ref{thm:4k-1} and \ref{thm:twisted} above.
Moreover, we show in what sense the twisted intersection invariant $\tau_n^\iinfty$ is the universal quadratic refinement of its framed partner $\tau_n$, and complete the algebraic description of the relationship between the framed and twisted Whitney tower filtrations (Theorem~\ref{thm:Tau-sequences} above). \item Further applications to filtrations of string links and homology cylinders are described in \cite{CST5}. \end{enumerate}
We emphasize that although the Milnor and higher-order Arf invariants completely classify the groups $\W_n$ and $\W^\iinfty_n$, there remains the question of determining the exact (finite) 2-torsion group which is the range of the higher-order Arf invariants, as touched on above and elaborated on throughout these papers.
{\bf Acknowledgments:} This paper was partially written while the first two authors were visiting the third author at the Max-Planck-Institut f\"ur Mathematik in Bonn. They all thank MPIM for its stimulating research environment and generous support. The exposition of this paper was significantly improved by a careful and insightful anonymous referee. The first author was also supported by NSF grant DMS-0604351 and the last author was also supported by NSF grants DMS-0806052 and DMS-0757312. The second author was partially supported by PSC-CUNY research grant PSCREG-41-386.
\section{Whitney towers}\label{sec:w-towers} We sketch here the relevant theory of Whitney towers as developed in \cite{CST,S1,ST2}, giving details for the new notion of \emph{twisted} Whitney towers. We work in the \emph{smooth oriented} category (with orientations usually suppressed from notation), even though all our results hold in the locally flat topological category by the basic results on topological immersions in Freedman--Quinn \cite{FQ}. In fact, it can be shown that the filtrations $\mathbb G_n$, $\mathbb W_n$ and $\mathbb W^\iinfty_n$ are identical in the smooth and locally flat settings. This is because a topologically flat surface can be promoted to a smooth surface at the cost of only creating unpaired intersections of arbitrarily high order (see Remark~\ref{rem:locally-flat-and-smooth}).
\subsection{Operations on trees}\label{subsec:trees} To describe Whitney towers it is convenient to use the bijective correspondence between formal non-associative bracketings of elements from the index set $\{1,2,3,\ldots,m\}$ and rooted trees, trivalent and oriented as in Definition~\ref{def:Tau}, with each univalent vertex labeled by an element from the index set, except for the \emph{root} univalent vertex which is left unlabeled.
\begin{defn}\label{def:trees} Let $I$ and $J$ be two rooted trees. \begin{enumerate} \item The \emph{rooted product} $(I,J)$ is the rooted tree gotten by identifying the root vertices of $I$ and $J$ to a single vertex $v$ and sprouting a new rooted edge at $v$. This operation corresponds to the formal bracket (Figure~\ref{inner-product-trees-fig} upper right). The orientation of $(I,J)$ is inherited from those of $I$ and $J$ as well as the order in which they are glued.
\item The \emph{inner product} $\langle I,J \rangle $ is the unrooted tree gotten by identifying the roots of $I$ and $J$ to a single non-vertex point. Note that $\langle I,J \rangle $ inherits an orientation from $I$ and $J$, and that all the univalent vertices of $\langle I,J \rangle $ are labeled. (Figure~\ref{inner-product-trees-fig} lower right.)
\item The \emph{order} of a tree, rooted or unrooted, is defined to be the number of trivalent vertices. \end{enumerate} \end{defn} The notation of this paper will not distinguish between a bracketing and its corresponding rooted tree (as opposed to the notation $I$ and $t(I)$ used in \cite{S1,ST2}). In \cite{S1,ST2} the inner product is written as a dot-product, and the rooted product is denoted by $*$.
\begin{figure}
\caption{The \emph{rooted product} $(I,J)$ and \emph{inner product} $\langle I,J \rangle$ of $I=(I_1,I_2)$ and $J=(J_1,J_2)$. All trivalent orientations correspond to a clockwise orientation of the plane.}
\label{inner-product-trees-fig}
\end{figure}
\subsection{Whitney disks and higher-order intersections}\label{subsec:order-zero-w-towers-and-ints} A collection $A_1,\ldots,A_m\looparrowright (M,\partial M)$ of connected surfaces in a $4$--manifold $M$ is a \emph{Whitney tower of order zero} if the $A_i$ are \emph{properly immersed} in the sense that the boundary is embedded in $\partial M$ and the interior is generically immersed in $M \smallsetminus \partial M$.
To each order zero surface $A_i$ is associated the order zero rooted tree consisting of an edge with one vertex labeled by $i$, and to each transverse intersection $p\in A_i\cap A_j$ is associated the order zero tree $t_p:=\langle i,j \rangle$ consisting of an edge with vertices labelled by $i$ and $j$. Note that for singleton brackets (rooted edges) we drop the bracket from notation, writing $i$ for $(i)$.
The order 1 rooted Y-tree $(i,j)$, with a single trivalent vertex and two univalent labels $i$ and $j$, is associated to any Whitney disk $W_{(i,j)}$ pairing intersections between $A_i$ and $A_j$. This rooted tree can be thought of as being embedded in $M$, with its trivalent vertex and rooted edge sitting in $W_{(i,j)}$, and its two other edges descending into $A_i$ and $A_j$ as sheet-changing paths. (The cyclic orientation at the trivalent vertex of the bracket $(i,j)$ corresponds to an orientation of $W_{(i,j)}$ via a convention described below in \ref{subsec:w-tower-orientations}.)
Recursively, the rooted tree $(I,J)$ is associated to any Whitney disk $W_{(I,J)}$ pairing intersections between $W_I$ and $W_J$ (see left-hand side of Figure~\ref{WdiskIJandIJKint-fig}); with the understanding that if, say, $I$ is just a singleton $i$, then $W_I$ denotes the order zero surface $A_i$. Note that a $W_{(I,J)}$ can be created by a finger move pushing $W_J$ through $W_I$.
To any transverse intersection $p\in W_{(I,J)}\cap W_K$ between $W_{(I,J)}$ and any $W_K$ is associated the un-rooted tree $t_p:=\langle (I,J),K \rangle$ (see right-hand side of Figure~\ref{WdiskIJandIJKint-fig}).
\begin{figure}
\caption{On the left, (part of) the rooted tree $(I,J)$ associated to a Whitney disk $W_{(I,J)}$. On the right, (part of) the unrooted tree $t_p=\langle (I,J),K \rangle$ associated to an intersection $p\in W_{(I,J)}\cap W_K$. Note that $p$ corresponds to where the roots of $(I,J)$ and $K$ are identified to a (non-vertex) point in $\langle (I,J),K \rangle$.}
\label{WdiskIJandIJKint-fig}
\end{figure}
\begin{defn}\label{def:int-and-Wdisk-order} The \emph{order of a Whitney disk} $W_I$ is defined to be the order of the rooted tree $I$, and the \emph{order of a transverse intersection} $p$ is defined to be the order of the tree $t_p$. \end{defn}
\begin{defn}\label{def:framed-tower} A collection $\mathcal{W}$ of properly immersed surfaces together with higher-order Whitney disks is an \emph{order $n$ Whitney tower} if $\mathcal{W}$ contains no unpaired intersections of order less than $n$. \end{defn} The Whitney disks in $\mathcal{W}$ must have disjointly embedded boundaries, and generically immersed interiors. All Whitney disks and order zero surfaces must also be \emph{framed} (as discussed next).
\subsection{Twisted Whitney disks and framings}\label{subsec:twisted-w-disks} The normal disk-bundle of a Whitney disk $W$ in $M$ is isomorphic to $D^2\times D^2$, and comes equipped with a canonical nowhere-vanishing \emph{Whitney section} over the boundary given by pushing $\partial W$ tangentially along one sheet and normally along the other, avoiding the tangential direction of $W$ (see Figure~\ref{Framing-of-Wdisk-fig}, and e.g.~1.7 of \cite{Sc}). Pulling back the orientation of $M$ with the requirement that the normal disks have $+1$ intersection with $W$ means the Whitney section determines a well-defined (independent of the orientation of $W$) relative Euler number $\omega(W)\in\mathbb{Z}$ which represents the obstruction to extending the Whitney section across $W$. Following traditional terminology, when $\omega(W)$ vanishes $W$ is said to be \emph{framed}. (Since $D^2\times D^2$ has a unique trivialization up to homotopy, this terminology is only mildly abusive.) In general when $\omega(W)=k$, we say that $W$ is $k$-\emph{twisted}, or just \emph{twisted} if the value of $\omega(W)$ is not specified. So a $0$-twisted Whitney disks is a framed Whitney disk.
\begin{figure}
\caption{The Whitney section over the boundary of a framed Whitney disk is
indicated by the dotted loop shown on the left for a clean Whitney disk $W$ in
a 3-dimensional slice of 4-space. On the right is shown an embedding into $3$--space of the normal
disk-bundle over $\partial W$, indicating how the Whitney section determines a well-defined nowhere
vanishing section which lies in the $I$-sheet and is normal to the $J$-sheet. }
\label{Framing-of-Wdisk-fig}
\end{figure}
Note that a {\em framing} of $\partial A_i$ (respectively $A_i$) is by definition a trivialization of the normal bundle of the immersion. If the ambient $4$-manifold is oriented, this is equivalent to an orientation and a nonvanishing normal vector field on $\partial A_i$ (respectively $A_i$). The twisting $\omega(A_i)\in\mathbb{Z}$ of an order zero surface is also defined when a framing of $\partial A_i$ is given, and $A_i$ is said to be \emph{framed} when $\omega(A_i)=0$.
\subsection{Twisted Whitney towers}\label{subsec:intro-twisted-w-towers} In the definition of an order $n$ Whitney tower given just above (following \cite{CST,S1,S2,ST2}) all Whitney disks and order zero surfaces are required to be framed. It turns out that the natural generalization to twisted Whitney towers involves allowing twisted Whitney disks only in at least ``half the order'' as follows:
\begin{defn}\label{def:twisted-W-towers} A \emph{twisted Whitney tower of order $0$} is a collection of properly immersed surfaces in a $4$--manifold (without any framing requirement).
For $n>0$, a \emph{twisted Whitney tower of order $(2n-1)$} is just a (framed) Whitney tower of order $(2n-1)$ as in Definition~\ref{def:framed-tower} above.
For $n>0$, a \emph{twisted Whitney tower of order $2n$} is a Whitney tower having all intersections of order less than $2n$ paired by Whitney disks, with all Whitney disks of order less than $n$ required to be framed, but Whitney disks of order at least $n$ allowed to be twisted. \end{defn}
\begin{rem}\label{rem:framed-is-twisted} Note that, for any $n$, an order $n$ (framed) Whitney tower is also an order $n$ twisted Whitney tower. We may sometimes refer to a Whitney tower as a \emph{framed} Whitney tower to emphasize the distinction, and will always use the adjective ``twisted'' in the setting of Definition~\ref{def:twisted-W-towers}. \end{rem}
\begin{rem}\label{rem:motivate-twisted-towers} The convention of allowing only order $\geq n$ twisted Whitney disks in order $2n$ twisted Whitney towers is explained both algebraically and geometrically in \cite{CST2}. In any event, an order $2n$ twisted Whitney tower can always be modified so that all its Whitney disks of order $>n$ are framed, so the twisted Whitney disks of order equal to $n$ are the important ones. \end{rem}
\subsection{Whitney tower orientations}\label{subsec:w-tower-orientations}
Orientations on order zero surfaces in a Whitney tower $\mathcal{W}$ are fixed, and required to induce the orientations on their boundaries. After choosing and fixing orientations on all the Whitney disks in $\mathcal{W}$, the associated trees are embedded in $\mathcal{W}$ so that the vertex orientations are induced from the Whitney disk orientations, with the descending edges of each trivalent vertex enclosing the \emph{negative intersection point} of the corresponding Whitney disk, as in Figure~\ref{WdiskIJandIJKint-fig}. (In fact, if a tree $t$ has more than one trivalent vertex which corresponds to the same Whitney disk, then $t$ will only be immersed in $\mathcal{W}$, but this immersion can be taken to be a local embedding around each trivalent vertex of $t$ as in Figure~\ref{WdiskIJandIJKint-fig}.)
This ``negative corner'' convention, which differs from the positive corner convention in \cite{CST,ST2}, will turn out to be compatible with commutator conventions for use in \cite{CST2}.
With these conventions, different choices of orientations on Whitney disks in $\mathcal{W}$ correspond to anti-symmetry relations (as explained in \cite{ST2}).
\subsection{Intersection invariants for Whitney towers}\label{subsec:intro-w-tower-int-invariants} We recall from Definition~\ref{def:Tau} that the abelian group $\mathcal{T}_n$ is the free abelian group on labeled vertex-oriented order $n$ trees, modulo the AS and IHX relations, see Figure~\ref{fig:ASandIHXtree-relations}. The obstruction theory of \cite{ST2} in the current simply connected setting works as follows. \begin{defn} The \emph{order $n$ intersection invariant} $\tau_n(\mathcal{W})$ of an order $n$ Whitney tower $\mathcal{W}$ is defined to be $$ \tau_n(\mathcal{W}):=\sum \epsilon_p\cdot t_p \in\mathcal{T}_n $$ where the sum is over all order $n$ intersections $p$, with $\epsilon_p=\pm 1$ the usual sign of a transverse intersection point. \end{defn}
As stated in Theorem~\ref{thm:raise} in the introduction, if $L$ bounds $\mathcal{W}\subset B^4$ with $\tau_n(\mathcal{W})=0\in \mathcal{T}_n$, then $L$ bounds a Whitney tower of order $n+1$. This is a special case of the simply connected version of the more general Theorem~2 of \cite{ST2}. We will use the following version of Theorem~2 of \cite{ST2} where the order zero surfaces are either properly immersed disks in $B^4$ or properly immersed annuli in $S^3\times I$:
\begin{thm}[\cite{ST2}]\label{thm:framed-order-raising-on-A} If a collection $A$ of properly immersed surfaces in a simply connected $4$--manifold supports an order $n$ Whitney tower $\mathcal{W}$ with $\tau_n(\mathcal{W})=0\in\mathcal{T}_n$, then $A$ is regularly homotopic (rel $\partial$) to $A'$ which supports an order $n+1$ Whitney tower. \end{thm}
\subsection{Intersection invariants for twisted Whitney towers}\label{subsec:intro-twisted-w-tower-int-invariants}
The intersection invariants for Whitney towers are extended to twisted Whitney towers as follows:
\begin{defn}\label{def:T-infty-odd} The abelian group $\mathcal{T}^{\iinfty}_{2n-1}$ is the quotient of $\mathcal{T}_{2n-1}$ by the \emph{boundary-twist relations}: \[ \langle (i,J),J \rangle \,=\, i\,-\!\!\!\!\!-\!\!\!<^{\,J}_{\,J}\,\,=\,0 \] Here $J$ ranges over all order $n-1$ rooted trees. (This is the same as taking the quotient of $\widetilde{\mathcal{T}}_{2n-1}=\mathcal{T}_{2n-1}/\operatorname{Im}(\Delta_{2n-1})$ by boundary-twist relations since $\operatorname{Im}(\Delta_{2n-1})$ is contained in the span of boundary-twist relations -- see Section~\ref{sec:proof-thm-odd}). \end{defn}
The boundary-twist relations correspond geometrically to the fact that performing a boundary twist (Figure~\ref{boundary-twist-and-section-fig}) on an order $n$ Whitney disk $W_{(i,J)}$ creates an order $2n-1$ intersection point $p\in W_{(i,J)}\cap W_J$ with associated tree $t_p=\langle (i,J),J \rangle $ (which is 2-torsion by the AS relations) and changes $\omega (W_{(i,J))})$ by $\pm1$. Since order $n$ twisted Whitney disks are allowed in an order $2n$ Whitney tower such trees do not represent obstructions to the existence of the next order twisted tower.
For any rooted tree $J$ we define the corresponding {\em $\iinfty$-tree}, denoted by $J^\iinfty$, by labeling the root univalent vertex with the symbol ``$\iinfty$'': $$ J^\iinfty := \iinfty\!-\!\!\!- J $$
\begin{defn}\label{def:T-infty-even} The abelian group $\mathcal{T}^{\iinfty}_{2n}$ is the free abelian group on order $2n$ trees and order $n$ $\iinfty$-trees, modulo the following relations: \begin{enumerate}
\item AS and IHX relations on order $2n$ trees (Figure~\ref{fig:ASandIHXtree-relations})
\item \emph{symmetry} relations: $(-J)^\iinfty = J^\iinfty$
\item \emph{twisted IHX} relations: $I^\iinfty=H^\iinfty+X^\iinfty- \langle H,X\rangle $
\item {\em interior twist} relations: $2\cdot J^\iinfty=\langle J,J\rangle $ \end{enumerate} \end{defn}
Here the AS and IHX relations are as usual, but they only apply to non-$\iinfty$ trees. The \emph{symmetry relation} corresponds to the fact that the relative Euler number $\omega(W)$ is independent of the orientation of the Whitney disk $W$, with $-J$ denoting the ``opposite'' orientation of $J$ (meaning that the trivalent orientations differ at an odd number of vertices). The \emph{twisted IHX relation} corresponds to the effect of performing a Whitney move in the presence of a twisted Whitney disk, as described below in Lemma~\ref{lem:twistedIHX}. The \emph{interior-twist relation} corresponds to the fact that creating a $\pm1$ self-intersection in a $W_J$ changes the twisting by $\mp 2$ (Figure~\ref{InteriorTwistPositiveEqualsNegative-fig}).
\begin{rem}\label{rem:quadratic-form} The symmetry, twisted IHX, and interior twist relations in $\mathcal T^\iinfty_{2n}$ have a surprisingly natural algebraic interpretation that we explain in \cite{CST4}. The idea is to extend the map $J\mapsto J^\iinfty$ to a {\em symmetric quadratic refinement} $q$ of the bilinear form $\langle \cdot,\cdot\rangle$ on the free quasi-Lie algebra of rooted trees (the intersection form on Whitney disks) by defining $q(J)=J^\iinfty$ and extending to linear combinations by the formula \[ q(J+K):=J^\iinfty+K^\iinfty+\langle J,K\rangle \] Expanding $q(I-H+X)=0$ leads to the 6-term IHX relation \[ I^\iinfty+H^\iinfty+X^\iinfty=\langle I,H \rangle-\langle I,X \rangle+\langle H,X \rangle \] which is equivalent to the twisted IHX relation in the presence of the interior-twist relations. Those in turn follow by setting $K:=-J$ from the symmetry relation. \end{rem}
\begin{rem}\label{rem:finite-type-IHX}
We discovered in \cite{CST} that the (framed) IHX relation can be realized in three dimensions as well as four, and it is interesting to note that many of the relations that we obtain for twisted Whitney towers in four dimensions can also be realized by rooted clasper surgeries (grope cobordisms) in three dimensions. Here the twisted Whitney disk corresponds to a $\pm1$ framed leaf of a clasper. For example the relation $I^\iinfty=H^\iinfty+X^\iinfty-\langle H,X\rangle$ has the following clasper explanation. $I^\iinfty$ represents a clasper with one isolated twisted leaf. By the topological IHX relation, one can replace $I^\iinfty$ by two claspers of the form $H^\iinfty$ and $(-X)^\iinfty=X^\iinfty$ embedded in a regular neighborhood of the original clasper with leaves parallel to the leaves of the original. The twisted leaves are now linked together, so applying Habiro's zip construction (which complicates the picture considerably) one gets three tree claspers, of the form $H^\iinfty$, $X^\iinfty$ and $\langle H,-X\rangle$ respectively.
Similarly, the relation $2\cdot J^\iinfty=\langle J,J\rangle$ has an interpretation where one takes a clasper which represents $J^\iinfty$ and splits off a geometrically cancelling parallel copy, representing the tree $J^\iinfty$. Again, because the twisted leaves link, we also get the term $\langle J,-J\rangle.$
These observations will be enlarged upon in \cite{CST5} to analyze filtrations on homology cylinders. \end{rem}
Recall from Definition~\ref{def:twisted-W-towers} (and Remark~\ref{rem:motivate-twisted-towers}) that twisted Whitney disks only occur in even order twisted Whitney towers, and only those of half-order are relevant to the obstruction theory. \begin{defn}\label{def:tau-infty} The \emph{order $n$ intersection intersection invariant} $\tau_{n}^{\iinfty}(\mathcal{W})$ of an order $n$ twisted Whitney tower $\mathcal{W}$ is defined to be $$ \tau_{n}^{\iinfty}(\mathcal{W}):=\sum \epsilon_p\cdot t_p + \sum \omega(W_J)\cdot J^\iinfty\in\mathcal{T}^{\iinfty}_{n} $$ where the first sum is over all order $n$ intersections $p$ and the second sum is over all order $n/2$ Whitney disks $W_J$ with twisting $\omega(W_J)\in\mathbb{Z}$. For $n=0$, recall from \ref{subsec:order-zero-w-towers-and-ints} above our notational convention that $W_j$ denotes $A_j$, and that $\omega(A_j)\in\mathbb{Z}$ is the relative Euler number of the normal bundle of $A_j$ with respect to the given framing of $\partial A_j$ as in \ref{subsec:twisted-w-disks} .
\end{defn}
By splitting the twisted Whitney disks, as explained in subsection~\ref{subsec:split-w-towers} below, for $n>0$ we may actually assume that all non-zero $\omega(W_J)\in\{\pm 1\}$, just like the signs $\epsilon_p$.
As in the framed case, the vanishing of $\tau_{n}^{\iinfty}$ is sufficient for the existence of a twisted Whitney tower of order $(n+1)$, and the proof of Theorem~\ref{thm:twisted} in Section~\ref{sec:proof-twisted-thm} will be based on the following analogue of the framed order-raising Theorem~\ref{thm:framed-order-raising-on-A} to the twisted setting: \begin{thm}\label{thm:twisted-order-raising-on-A} If a collection $A$ of properly immersed surfaces in a simply connected $4$--manifold supports an order $n$ twisted Whitney tower $\mathcal{W}$ with $\tau_n^\iinfty(\mathcal{W})=0\in\mathcal{T}^\iinfty_n$, then $A$ is regularly homotopic (rel $\partial$) to $A'$ which supports an order $n+1$ twisted Whitney tower. \end{thm} The proof of Theorem~\ref{thm:twisted-order-raising-on-A} is given in Section~\ref{sec:proof-twisted-thm} below.
Proofs of the ``order-raising'' Theorems \ref{thm:twisted-order-raising-on-A} and \ref{thm:framed-order-raising-on-A} (and its strengthening Theorem~\ref{thm:framed-order-raising-mod-Delta} below) depend on realizing the relations in the target groups by controlled manipulations of Whitney towers. The next two subsections introduce combinatorial notions useful for describing the algebraic effect of such geometric constructions.
\emph{For the rest of this section we assume our Whitney towers are of positive order for convenience of notation.}
\subsection{Intersection forests}\label{subsec:int-forests} Recall that the trees associated to intersections and Whitney disks in a Whitney tower can be considered to be immersed in the Whitney tower, with vertex orientations induced by the Whitney tower orientation, as in Figure~\ref{WdiskIJandIJKint-fig}.
\begin{defn}\label{def:intersection forests} The \emph{intersection forest} $t(\mathcal{W})$ of a framed Whitney tower $\mathcal{W}$ is the disjoint union of signed trees associated to all unpaired intersections $p$ in $\mathcal{W}$:
\[
t(\mathcal{W})=\amalg_p\ \epsilon_p \cdot t_p
\] with $\epsilon_p$ the sign of the intersection point $p$. For $\mathcal{W}$ of order~$n$, we can think of the signed order $n$ trees in $t(\mathcal{W})$ as an ``abelian word'' in the generators $\pm t_p$ which represents $\tau_n(\mathcal{W})\in\mathcal{T}_n$. More precisely, $t(\mathcal{W})$ is an element of the free abelian monoid, with unit $\emptyset$, generated by (isomorphism classes of) signed trees, trivalent, labeled and vertex-oriented as usual. We emphasize that there are no cancellations or other relations here.
\begin{rem} In the older papers \cite{CST,S1,ST2} we referred to $t(\mathcal{W})$ as the ``geometric intersection tree'' (and to the group element $\tau_n(\mathcal{W})$ as the order $n$ intersection ``tree'', rather than ``invariant''), but the term ``forest'' better describes the disjoint union of (signed) trees $t(\mathcal{W})$. \end{rem}
Similarly to the framed case, the \emph{intersection forest} $t(\mathcal{W})$ of a {\em twisted} Whitney tower $\mathcal{W}$ is the disjoint union of signed trees associated to all unpaired intersections $p$ in $\mathcal{W}$ and integer-coefficient $\iinfty$-trees associated to all non-trivially twisted Whitney disks $W_J$ in $\mathcal{W}$: \[ t(\mathcal{W})=\amalg_p \ \epsilon_p \cdot t_p \,\, + \amalg_J \ \omega(W_J)\cdot J^\iinfty \] with $\omega(W_J)\in\mathbb{Z}$ the twisting of $W_J$. Again, there are no cancellations or relations (and the informal ``$+$'' sign in the expression is purely cosmetic). \end{defn} We will see in the next subsection that all the trees can be made to be disjoint in $\mathcal{W}$, with all non-zero $\omega(W_J)=\pm 1$, so that $t(\mathcal{W})$ is also a topological disjoint union which corresponds to an element in the free abelian monoid generated by (isomorphism classes of) signed trees, and signed $\iinfty$-trees.
\subsection{Splitting twisted Whitney towers}\label{subsec:split-w-towers} A framed Whitney tower is \emph{split} if the set of singularities in the interior of any Whitney disk consists of either a single point, or a single boundary arc of a Whitney disk, or is empty. This can always be arranged, as observed in Lemma~13 of \cite{ST2} (Lemma~3.5 of \cite{S1}), by performing finger moves along Whitney disks guided by arcs connecting the Whitney disk boundary arcs. Implicit in this construction is that the finger moves preserve the Whitney disk framings (by not twisting relative to the Whitney disk that is being split -- see Figure~\ref{twist-split-Wdisk-fig}). A Whitney disk $W$ is \emph{clean} if the interior of $W$ is embedded and disjoint from the rest of the Whitney tower. In the setting of twisted Whitney towers, it will simplify the combinatorics to use ``twisted'' finger moves to similarly split-off twisted Whitney disks into $\pm 1$-twisted clean Whitney disks.
We call a twisted Whitney tower \emph{split} if all of its non-trivially twisted Whitney disks are clean and have twisting $\pm 1$, and all of its framed Whitney disks are split in the usual sense (as for framed Whitney towers). \begin{lem}\label{lem:split-w-tower} If $A$ supports an order $n$ twisted Whitney tower $\mathcal{W}$, then $A$ is homotopic (rel $\partial$) to $A'$ which supports a split order $n$ twisted Whitney tower $\mathcal{W}'$, such that: \begin{enumerate} \item The disjoint union of non-$\iinfty$ trees $\amalg_p \ \epsilon_p \cdot t_p \subset t(\mathcal{W})$ is isomorphic to the disjoint union of non-$\iinfty$ trees $\amalg_{p'} \ \epsilon_{p'} \cdot t_{p'} \subset t(\mathcal{W}')$.
\item Each $\omega(W_J)\cdot J^\iinfty$ in $t(\mathcal{W})$ gives rise to the disjoint union of exactly $|\omega(W_J) |$-many $\pm 1\cdot J^\iinfty$ in $\mathcal{W}'$, where the sign $\pm$ corresponds to the sign of $\omega(W_J)$. \end{enumerate} \end{lem}
\begin{proof} \begin{figure}
\caption{A neighborhood of a twisted finger move which splits a Whitney disk into two Whitney disks.
The vertical black arcs are slices of
the new Whitney disks, and the grey arcs are slices of extensions of the Whitney sections.
The finger-move is supported in a neighborhood of an arc in the original Whitney disk running
from a point in the Whitney disk boundary on the ``upper'' surface sheet to a point
in the Whitney disk boundary on the ``lower'' surface sheet. (Before the finger-move this guiding arc would
have been visible in the middle picture as a vertical black arc-slice of the original
Whitney disk.)}
\label{twist-split-Wdisk-fig}
\end{figure} Illustrated in Figure~\ref{twist-split-Wdisk-fig} is a local picture of a twisted finger move, which splits one Whitney disk into two, while also changing twistings. If the original Whitney disk in Figure~\ref{twist-split-Wdisk-fig} was framed, then the two new Whitney disks will have twistings $+1$ and $-1$, respectively. In general, if the arc guiding the finger move splits the twisting of the original Whitney disk into $\omega_1$ and $\omega_2$ zeros of the extended Whitney section, then the two new Whitney disks will have twistings $\omega_1+1$ and $\omega_2-1$, respectively. Thus, by repeatedly splitting off framed corners into $\pm 1$-twisted Whitney disks, any
$\omega$-twisted Whitney disk ($\omega \in\mathbb{Z}$) can be split into $|\omega |$-many $+1$-twisted or $-1$-twisted clean Whitney disks, together with split framed Whitney disks containing any interior intersections in the original twisted Whitney disk. Combining this with the untwisted splitting of the framed Whitney disks as in Lemma~13 of \cite{ST2} gives the result. \end{proof}
\begin{rem}\label{rem:locally-flat-and-smooth} We sketch here a brief explanation of why the smooth and locally flat filtrations are equal. A locally flat surface can be made smooth by a small perturbation, which after introducing cusps as necessary can be assumed to be a regular (locally flat) homotopy. By a general position argument, this regular homotopy can be assumed to be a finite number of finger moves, which are guided by arcs and lead to canceling self-intersection pairs which admit small disjointly embedded Whitney disks (which are `inverses' to the finger moves). These Whitney disks are only locally flat, but can be perturbed to be smooth, again only at the cost of creating paired self-intersections, and iteration of this process leads to an arbitrarily high-order smooth sub-Whitney tower pairing all intersections created by the original surface perturbation. \end{rem}
\section{The realization maps}\label{sec:realization-maps} This section contains clarifications and proofs of Theorems \ref{thm:R-onto-G} and \ref{thm:twisted} from the introduction which state the existence of surjections $R_n\colon\mathcal{T}_n\to\W_n$ and $R^\iinfty_n\colon\mathcal{T}^\iinfty_n\to\W^\iinfty_n$ for all $n$, in particular exhibiting the sets $\W_n$ and $\W^\iinfty_n$ as finitely generated abelian groups under connected sum.
All proofs in this section apply in the reduced setting as well, and the constructions described here also define the surjections $\widetilde{R}_n\colon\widetilde{\mathcal{T}}_n\to\W_n$ described in the introduction.
Recall that our manifolds are assumed oriented, but orientations are suppressed from the discussion as much as possible. In the following an orientation is fixed once and for all on $S^3$; and a \emph{framed link} has oriented components, each equipped with a nowhere-vanishing normal section. \begin{defn}\label{def:links-bounding-towers} A framed link $L\subset S^3=\partial B^4$ \emph{bounds} an order $n$ Whitney tower $\mathcal{W}$ if $\mathcal{W}\subset B^4$ is an order $n$ Whitney tower whose order zero surfaces are immersed disks bounded by the components of $L$, as in Definition~\ref{def:framed-tower}.
Similarly, a framed link $L\subset S^3=\partial B^4$ \emph{bounds} an order $n$ twisted Whitney tower $\mathcal{W}$ if $\mathcal{W}\subset B^4$ is an order $n$ twisted Whitney tower whose order zero surfaces are immersed disks bounded by the components of $L$, as in Definition~\ref{def:twisted-W-towers}. \end{defn}
\begin{defn}\label{def:wtc} For $n\geq 1$, framed links $L_0$ and $L_1$ in $S^3$ are \emph{Whitney tower concordant of order $n$} if the $i$th components of $L_0\subset S^3\times\{0\}$ and $-L_1\subset S^3\times\{1\}$ cobound an immersed annulus $A_i$ for each $i$ such that the $A_i$ are transverse and support an order $n$ Whitney tower. If the $A_i$ support a \emph{twisted} order $n$ Whitney tower then $L_0$ and $L_1$ are said to be \emph{twisted Whitney tower concordant of order $n$}. \end{defn} Note that a (twisted) Whitney tower concordance preserves framings on on $L_0$ and $L_1$ (as links in $S^3$) since all self-intersections of the $A_i$ are paired by Whitney disks in any (twisted) Whitney tower of order $n\geq 1$.
Recall from the introduction that the set of $m$-component framed links in $S^3$ which bound order $n$ (twisted) Whitney towers in $B^4$ is denoted by $\mathbb{W}_n=\mathbb{W}_n(m)$ (resp. $\mathbb{W}^\iinfty_n$); and and the quotient of $\mathbb{W}_n$ by the equivalence relation of order $n+1$ (twisted) Whitney tower concordance is denoted by $\W_n$ (resp. $\W^\iinfty_n$).
Throughout this section the twisted setting mirrors the framed setting, with discussions and arguments given simultaneously.
We first need to show our essential criterion, Corollary~\ref{cor:tau=w-concordance}, for links to represent equal elements in the associated graded $\W_n$: Links $L_0$ and $L_1$ in $\mathbb{W}_n$ represent the same element of $\W_n$ if and only if there exist order $n$ Whitney towers $\mathcal{W}_i$ in $B^4$ with $\partial\mathcal{W}_i=L_i$ and $\tau_n(\mathcal{W}_0)=\tau_n(\mathcal{W}_1)\in\mathcal{T}_n$.
\begin{proof}[Proof of Corollary~\ref{cor:tau=w-concordance}] If $L_0$ and $L_1$ are equal in $\W_n$ then they cobound $A$ supporting an order $n+1$ Whitney tower $\mathcal{V}$ in $S^3\times I$, and any order $n$ Whitney tower $\mathcal{W}_1$ in $B^4$ bounded by $L_1$ can be extended by $\mathcal{V}$ to form an order $n$ Whitney tower $\mathcal{W}_0$ in $B^4$ bounded by $L_0$, with $\tau_n(\mathcal{W}_0)=\tau_n(\mathcal{W}_1)\in\mathcal{T}_n$ since $\tau_n(\mathcal{V})$ vanishes.
Conversely, suppose that $L_0$ and $L_1$ bound order $n$ Whitney towers $\mathcal{W}_0$ and $\mathcal{W}_1$ in $4$--balls $B_0^4$ and $B_1^4$, with $\tau_n(\mathcal{W}_0)=\tau_n(\mathcal{W}_1)$. Then constructing $S^3\times I$ as the connected sum $B_0^4\# B_1^4$ (along balls in the complements of $\mathcal{W}_0$ and $\mathcal{W}_1$), and tubing together the corresponding order zero disks of $\mathcal{W}_0$ and $\mathcal{W}_1$, and taking the union of the Whitney disks in $\mathcal{W}_0$ and $\mathcal{W}_1$, yields a collection $A$ of properly immersed annuli connecting $L_0$ and $L_1$ and supporting an order $n$ Whitney tower $\mathcal{V}$. Since the orientation of the ambient $4$--manifold has been reversed for one of the original Whitney towers, say $\mathcal{W}_1$, which results in a global sign change for $\tau_n(\mathcal{W}_1)$, it follows that $\mathcal{V}$ has vanishing order $n$ intersection invariant: $$ \tau_n(\mathcal{V})=\tau_n(\mathcal{W}_0)-\tau_n(\mathcal{W}_1)=\tau_n(\mathcal{W}_0)-\tau_n(\mathcal{W}_0)=0\in\mathcal{T}_n $$ So by Theorem~\ref{thm:framed-order-raising-on-A}, $A$ is homotopic (rel $\partial$) to $A'$ supporting an order $n+1$ Whitney tower, and hence $L_0$ and $L_1$ are equal in $\W_n$. \end{proof}
\begin{rem}\label{rem:tau=w-concordance} The analogous statement and proof of Corollary~\ref{cor:tau=w-concordance} holds in the twisted case (with Theorem~\ref{thm:twisted-order-raising-on-A} playing the role of Theorem~\ref{thm:framed-order-raising-on-A}). For this case, we'll spell out the statement carefully but in several instances below we will just state that the twisted case is analogous: Links $L_0$ and $L_1$ in $\mathbb{W}_n^\iinfty$ represent the same element of $\W^\iinfty_n$ if and only if there exist order $n$ twisted Whitney towers $\mathcal{W}_0$ and $\mathcal{W}_1$ in $B^4$ bounded by $L_0$ and $L_1$ respectively such that $\tau^\iinfty_n(\mathcal{W}_0)=\tau^\iinfty_n(\mathcal{W}_1)\in\mathcal{T}^\iinfty_n$. \end{rem}
\begin{rem}\label{rem:reduced-tau=w-concordance} Remark~\ref{rem:tau=w-concordance} similarly applies to the reduced setting by Theorem~\ref{thm:framed-order-raising-mod-Delta} below, although we will omit further reference to $\widetilde{\mathcal{T}}$ in this section. \end{rem}
\subsection{Band sums of links}\label{subsec:band-sum} The \emph{band sum} $L\#_\beta L'\subset S^3$ of oriented $m$-component links $L$ and $L'$ along bands $\beta$ is defined as follows: Form $S^3$ as the connected sum of $3$--spheres containing $L$ and $L'$ along balls in the link complements. Let $\beta$ be a collection of disjointly embedded oriented bands joining like-indexed link components such that the band orientations are compatible with the link orientations. Take the usual connected sum of each pair of components along the corresponding band. Although it is well-known that the concordance class of $L\#_\beta L'$ depends in general on $\beta$, it turns out that the image of $L\#_\beta L'$ in $\W_n$ (or in $\W^\iinfty_n$) does not depend on $\beta$:
\begin{lem}\label{lem:link-sum-well-defined} For links $L$ and $L'$ representing elements of $\W_n$, any band sum $L\#_\beta L'$ represents an element of $\W_n$ which only depends on the equivalence classes of $L$ and $L'$ in $\W_n$. The same statement holds in $\W^\iinfty_n$. \end{lem}
\begin{proof} We shall only give the proof in the framed case, the twisted case is analogous. If $L_0$ and $L_1$ represent the same element of $\W_n$, and if $L'_0$ and $L'_1$ represent the same element of $\W_n$, then by Corollary~\ref{cor:tau=w-concordance} above, for $i=0,1$, there are order $n$ Whitney towers $\mathcal{W}_i$ and $\mathcal{W}'_i$ bounding $L_i$ and $L'_i$ such that $\tau_n(\mathcal{W}_0)=\tau_n(\mathcal{W}_1)$ and $\tau_n(\mathcal{W}'_0)=\tau_n(\mathcal{W}'_1)$. By Lemma~\ref{lem:exists-tower-sum} just below, $L_i\#_{\beta_i} L'_i$ bounds $\mathcal{W}_i^\#$ for $i=0,1$, with $$ \tau_n(\mathcal{W}_0^\#)=\tau_n(\mathcal{W}_0)+\tau_n(\mathcal{W}'_0)=\tau_n(\mathcal{W}_1)+\tau_n(\mathcal{W}'_1)=\tau_n(\mathcal{W}_1^\#) $$ so again by Corollary~\ref{cor:tau=w-concordance}, $L_0\#_{\beta_0} L'_0$ is order $n+1$ Whitney tower concordant to $L_1\#_{\beta_1} L'_1$, hence $L_0\#_{\beta_0} L'_0$ and $L_1\#_{\beta_1} L'_1$ represent the same element of $\W_n$. \end{proof}
\begin{lem}\label{lem:exists-tower-sum} If $L$ and $L'$ bound order $n$ (twisted) Whitney towers $\mathcal{W}$ and $\mathcal{W}'$ in $B^4$, then for any $\beta$ there exists an order $n$ (twisted) Whitney tower $\mathcal{W}^\#\subset B^4$ bounded by $L\#_\beta L'$, such that $t(\mathcal{W}^\#)=t(\mathcal{W})\amalg t(\mathcal{W}')$, where $t(\mathcal{V})$ denotes the intersection forest of a Whitney tower $\mathcal{V}$ as above in subsection~\ref{subsec:int-forests}. \end{lem} \begin{proof} Let $B$ and $B'$ be the $3$--balls in the link complements used to form the $S^3$ containing $L\#_\beta L'$. Then gluing together the two $4$--balls containing $\mathcal{W}$ and $\mathcal{W}'$ along $B$ and $B'$ forms $B^4$ containing $L\#_\beta L'$ in its boundary. Take $\mathcal{W}^\#$ to be the boundary band sum of $\mathcal{W}$ and $\mathcal{W}'$ along the order zero disks guided by the bands $\beta$, with the interiors of the bands perturbed slightly into the interior of $B^4$. It is clear that $t(\mathcal{W}^\#)$ is just the disjoint union $t(\mathcal{W})\amalg t(\mathcal{W}')$ since no new singularities have been created. \end{proof}
\subsection{The realization maps}\label{subsec:realization-maps} The realization maps $R_n$ are defined as follows: Given any group element $g\in\mathcal{T}_n$, by Lemma~\ref{lem:realization-of-geometric-trees} just below there exists an $m$-component link $L\subset S^3$ bounding an order $n$ Whitney tower $\mathcal{W}\subset B^4$ such that $\tau_n(\mathcal{W})=g\in\mathcal{T}_n$. Define $R_n(g)$ to be the class determined by $L$ in $\W_n$. This is well-defined (does not depend on the choice of such $L$) by Corollary~\ref{cor:tau=w-concordance}. The twisted realization map $R^\iinfty_n$ is defined the same way using twisted Whitney towers.
\begin{lem}\label{lem:realization-of-geometric-trees} For any disjoint union $\amalg_p\ \epsilon_p \cdot t_p \,\, + \amalg_J\ \omega (W_J) \cdot J^\iinfty$ there exists an $m$-component link $L$ bounding a twisted Whitney tower $\mathcal{W}$ with intersection forest $t(\mathcal{W})= \amalg_p\ \epsilon_p \cdot t_p \,\, + \amalg_J \ \omega (W_J) \cdot J^\iinfty$. If the disjoint union contains no $\iinfty$-trees then all Whitney disks in $\mathcal{W}$ are framed. \end{lem} Note that if in the disjoint union all non-$\iinfty$ trees are order at least $n$ and all $\iinfty$-trees are order at least $n/2$ then $\mathcal{W}$ will have order $n$.
\begin{proof} It suffices to consider the cases where the disjoint union consists of just a single (signed) tree or $\iinfty$-tree since by Lemma~\ref{lem:exists-tower-sum} any sum of such trees can then be realized by band sums of links.
The following algorithm, in the untwisted case, is the algorithm called "Bing-doubling along a tree" by Cochran and used in Section 7 of \cite{C} and Theorem 3.3 of \cite{C1} to produce links in $S^3$ with prescribed (first non-vanishing) Milnor invariants.
\begin{figure}
\caption{Pushing into $B^4$ from left to right: A Hopf link in $S^3=\partial B^4$ bounds embedded disks $D_1\cup D_2\subset B^4$ which intersect in a point $p$, with $t_p=\langle 1,2 \rangle$. }
\label{fig:Hopf-disk}
\end{figure}
\textbf{Realizing order zero trees and $\iinfty$-trees.} A 0-framed Hopf link bounds an order zero Whitney tower $\mathcal{W}=D_1\cup D_2\subset B^4$, where the two embedded disks $D_1$ and $D_2$ have a single interior intersection point $p$ with $t_p=\langle 1,2 \rangle = 1 -\!\!\!-\!\!\!- \,2 $ (see Figure~\ref{fig:Hopf-disk}). Assuming appropriate fixed orientations of $B^4$ and $S^3$, the sign $\epsilon_p$ associated to $p$ is the usual sign of the Hopf link. So taking a 0-framed $(m-2)$-component trivial link together with a Hopf link (as the $i$th and $j$th components) gives an $m$-component link $L$ bounding $\mathcal{W}$ with $t(\mathcal{W})=\epsilon_p\cdot\langle i,j \rangle = \epsilon_p\cdot i -\!\!\!- \,j $, for any $\epsilon_p=\pm 1$, and $i\neq j$.
To realize the tree $\pm\ i -\!\!\!-\!\!\!- \,i $, we can use the unlink with framings 0, except that the component labeled by the index $i$ has framing $\pm 2$. Similarly, if the component has framing $\pm 1$ then the resulting tree is $\pm \ \iinfty -\!\!\!-\!\!\!-\,i $.
\begin{figure}
\caption{Pushing into $B^4$ from left to right: The disks $ D_{i_2}$ and $D_j$ extend to the
right-most picture where they are completed by capping off the unlink. The disk $D_{i_1}$ only extends to the middle picture where the intersections between $D_{i_1}$ and $ D_{i_2}$ are paired by the Whitney disk $W_{(i_1,i_2)}$, that has a single interior intersection $p\in W_{(i_1,i_2)}\cap D_j$ with $t_p=\langle (i_1,i_2),j \rangle$.}
\label{fig:Borromean-Bing-Hopf}
\end{figure}
\textbf{Realizing order $1$ trees.} Consider now a link $L$ whose $i$th and $j$th components form a Hopf link $L^i\cup L^j$ bounding disks $D_i\cup D_j\subset B^4$ with transverse intersection $p=D_i\cap D_j$. Assume that $D_i\cup D_j$ extends to an order zero Whitney tower $\mathcal{W}$ bounded by $L$ with $t(\mathcal{W})=\epsilon_p\cdot t_p=\epsilon_p\cdot\langle i,j \rangle$.
Replacing $L^i$ by an untwisted Bing-double $L^{i_1}\cup L^{i_2}$ results in a new sublink of Borromean rings $L^{i_1}\cup L^{i_2}\cup L^j$ bounding disks $D_{i_1}\cup D_{i_2}\cup D_j$ as indicated in Figure~\ref{fig:Borromean-Bing-Hopf}, with $D_{i_1}$ and $ D_{i_2}$ intersecting in a canceling pair of intersections paired by an order $1$ Whitney disk $W_{(i_1,i_2)}$, which can be formed from $D_i$ with a small collar removed, so that $W_{(i_1,i_2)}$ has a single intersection with $D_j$ corresponding to the original $p=D_i\cap D_j$. (One can think of $D_{i_1}$ and $ D_{i_2}$ as being formed by the trace of the obvious pulling-apart homotopy that shrinks $L^{i_1}$ and $L^{i_2}$ down in a tubular neighborhood of $L^i$, with the canceling pair of intersections between $D_{i_1}$ and $D_{i_2}$ being created as the clasps are pulled apart.)
The effect of this Bing-doubling operation on the intersection forest is that the original order zero $t_p=\langle i,j \rangle$ has given rise to the order~$1$ tree $\langle (i_1,i_2),j \rangle$. Switching the orientation on one of the new components changes the sign of $p$, as can be checked using our orientation conventions. By relabeling and/or banding together components of this new link any labels on this order~$1$ tree can be realized. Since the doubling was untwisted, $W_{(i_1,i_2)}$ is framed (see Figures \ref{fig:Bing-unlink-W-disk} and \ref{fig:Bing-unlink-W-disk-twisting}), so the Whitney tower bounded by the new link is order $1$.
\begin{figure}
\caption{Pushing into $B^4$ from left to right: An $i$- and $j$-labeled $n$-twisted Bing-double (case $n=2$) of the unknot in
$S^3=\partial B^4$
bounds disks $D_i$ and $D_j$ whose intersections are paired by a Whitney disk $W_{(i,j)}$.
$D_j$ extends to the right-hand picture but $D_i$ only extends to the middle picture, where the boundary of $W_{(i,j)}$ is indicated by the dark arcs. The rest of $W_{(i,j)}$ extends into the right-hand picture where disjointly embedded disks bounded by the unlink complete both $W_{(i,j)}$ and $D_j$. The interior of $W_{(i,j)}$ is embedded and disjoint from both $D_i$ and $D_j$. Figure~\ref{fig:Bing-unlink-W-disk-twisting} shows that $W_{(i,j)}$ is twisted, with $\omega(W_{(i,j)})=n$.}
\label{fig:Bing-unlink-W-disk}
\end{figure}
\textbf{Realizing order $n$ trees.} Since any order $n$ tree can be gotten from some order $n-1$ tree by attaching two new edges to a univalent vertex as in the previous paragraph, it follows inductively that any order $n$ tree is the intersection forest of a Whitney tower bounded by some link. (First create a distinctly-labeled tree of the desired `shape' by doubling, then correct the labels by interior band-summing.)
\textbf{Realizing $\iinfty$-trees of order $1$.} As illustrated (for the case $n=2$) in Figures~\ref{fig:Bing-unlink-W-disk} and \ref{fig:Bing-unlink-W-disk-twisting}, the $n$-twisted Bing-double of the unknot (with components labeled $i$ and $j$) bounds an order $2$ twisted Whitney tower $\mathcal{W}$ with $t(\mathcal{W})=n\cdot ( i,j )^\iinfty=n\cdot \iinfty \!-\!\!\!\!\!-\!\!\!\!<^{\,i}_{\,j}$. Banding together the two components would yield a knot realizing $(i,i)^\iinfty$.
\begin{figure}
\caption{The Whitney section over $\partial W_{(i,j)}$ (from Figure~\ref{fig:Bing-unlink-W-disk}) is indicated by the dashed arcs on the left. The twisting
$\omega (W_{(i,j)})=n$ (the obstruction to extending the Whitney section across the Whitney disk) corresponds to the
$n$-twisting of the Bing-doubling operation.}
\label{fig:Bing-unlink-W-disk-twisting}
\end{figure}
\textbf{Realizing $\iinfty$-trees of order $n$.} By applying iterated untwisted Bing-doubling operations to the $i$- and $j$-labeled components of the order $1$ case, one can construct for any rooted tree $( I,J )$ a link bounding a twisted Whitney tower $\mathcal{W}$ with $t(\mathcal{W})=n \cdot ( I,J )^\iinfty$. For instance, if in the construction of Figure~\ref{fig:Bing-unlink-W-disk} the $j$-labeled link component is replaced by an untwisted Bing-double, then the disk $D_j$ in that construction would be replaced by a (framed) Whitney disk $W_{(j_1,j_2)}$, and the $n$-twisted $W_{(i,j)}$ would be replaced by an $n$-twisted $W_{(i,(j_1,j_2))}$. (As for non-$\iinfty$ trees above, first create a distinctly-labeled tree of the desired `shape' by doubling, then correct the labels by interior band-summing.) \end{proof}
\subsection{Proofs of Theorem~\ref{thm:R-onto-G} and Theorem~\ref{thm:twisted}}\label{subsec:realization-maps} Recall the content of Theorem~\ref{thm:R-onto-G}: The realization maps $R_n: \mathcal{T}_n \to\W_n$ are epimorphisms, with the group operation on $\W_n$ given by band sum $\#$. Moreover, $\W_n$ is the set of framed links $L\in\mathbb{W}_n$ modulo the relation that $[L_1]=[L_2]\in\W_n$ if and only if $L_1 \# -L_2$ lies in $\mathbb{W}_{n+1}$, for some choice of connected sum $\#$, where $-L$ is the mirror image of $L$ with reversed framing. The content of Theorem~\ref{thm:twisted} in the twisted setting is analogous.
From Lemma~\ref{lem:link-sum-well-defined} the band sum of links gives well-defined operations in $\W_n$ and $\W^\iinfty_n$ which are clearly associative and commutative, with the $m$-component unlink representing an identity element. The realization maps are homomorphisms by Lemma~\ref{lem:exists-tower-sum} and surjectivity is proven as follows: Given any link $L\in \mathbb{W}_n$, choose a Whitney tower $\mathcal{W}$ of order $n$ with boundary $L$ and compute $\tau:=\tau_n(\mathcal{W})$. Then take $L':= R_n(\tau)$, a link that's obviously in the image of $R_n$ and for which we know a Whitney tower $\mathcal{W}'$ with boundary $L'$ and $\tau(\mathcal{W}') = \tau$. By Corollary~\ref{cor:tau=w-concordance} it follows that $L$ and $L'$ represent the same element in $\W_n$.
Considering the second ``Moreover...'' statements of Theorem~\ref{thm:R-onto-G} and Theorem~\ref{thm:twisted}, first assume that $L_0$ and $L_1$ represent the same element of $\W_n$ (resp. $\W^\iinfty_n$). Then by Corollary~\ref{cor:tau=w-concordance}, there exist order $n$ (twisted) Whitney towers $\mathcal{W}_0$ and $\mathcal{W}_1$ in $B^4$ bounded by $L_0$ and $L_1$ respectively such that $\tau_n(\mathcal{W}_0)=\tau_n(\mathcal{W}_1)\in\mathcal{T}_n$ (resp. $\tau^\iinfty_n(\mathcal{W}_0)=\tau^\iinfty_n(\mathcal{W}_1)\in\mathcal{T}^\iinfty_n$). We want to show that $L_0\#-L_1$ bounds an order~$n+1$ (twisted) Whitney tower, which will follow from Lemma~\ref{lem:exists-tower-sum} and the ``order-raising'' Theorem~\ref{thm:framed-order-raising-on-A} (respectively Theorem~\ref{thm:twisted-order-raising-on-A}) if $-L_1$ bounds an order $n$ (twisted) Whitney tower $\overline{\mathcal{W}_1}$ such that $\tau_n(\overline{\mathcal{W}_1})=-\tau_n(\mathcal{W}_1)\in\mathcal{T}_n$ (resp. $\tau^\iinfty_n(\overline{\mathcal{W}_1})=-\tau^\iinfty_n(\mathcal{W}_1)\in\mathcal{T}^\iinfty_n$). If $r$ denotes the reflection on $S^3$ which sends $L_1$ to $-L_1$, then the product $r\times\operatorname{id}$ of $r$ with the identity is an involution on $S^3\times I$, and the image $r\times\operatorname{id}(\mathcal{W}_1)$ of $\mathcal{W}_1$ is such a $\overline{\mathcal{W}_1}$. To see this, observe that $r\times\operatorname{id}$ switches the signs of all transverse intersection points, and is an isomorphism on the oriented trees in $\mathcal{W}_1$; and hence switches the signs of all Whitney disk framing obstructions (which can be computed as intersection numbers between Whitney disks and their push-offs) -- note that $r\times\operatorname{id}$ is only being applied to $\mathcal{W}_1$, while $S^3\times I$ is fixed.
For the other direction of the ``Moreover...'' statements, assume that $L_0\#-L_1\subset S^3$ bounds an order~$n+1$ (twisted) Whitney tower $\mathcal{W}\subset B^4$. By the definition of connected sum, $S^3$ decomposes as the union of two disjoint $3$--balls $B_0$ and $B_1$ containing $L_0$ and $-L_1$, joined together by the $S^2\times I$ through which passes the bands guiding the connected sum. Taking another $4$--ball with the same decomposition of its boundary $3$--sphere, and gluing the $4$--balls together by identifying the boundary $2$--spheres of the $3$--balls, and identifying the $S^2\times I$ subsets by the identity map, forms $S^3\times I$ containing an order $n+1$ (twisted) Whitney tower concordance between $L_0$ and $-L_1$ which consists of $\mathcal{W}$ together with the parts of the connected-sum bands that are contained in $S^2\times I$.
\section{Proof of Theorem~\ref{thm:twisted-order-raising-on-A} and the twisted IHX lemma}\label{sec:proof-twisted-thm} This section contains a proof of the ``twisted order-raising'' Theorem~\ref{thm:twisted-order-raising-on-A} of Section~\ref{sec:w-towers}, which was used (along with Corollary~\ref{cor:tau=w-concordance}) in Section~\ref{sec:realization-maps} to prove Theorem~\ref{thm:twisted} of the introduction. A key step in the proof involves a geometric realization of the twisted IHX relation as described in Lemma~\ref{lem:twistedIHX} below.
At the end of the section, the proof of Proposition~\ref{prop:exact sequence} is completed by Lemma~\ref{lem:boundary-twisted-IHX} in \ref{subsec:boundary-twisted-IHX-lemma} which shows how any order $2n$ twisted Whitney tower can be converted into an order $2n-1$ framed Whitney tower.
\subsection{Proof of Theorem~\ref{thm:twisted-order-raising-on-A}}\label{subsec:proof-of-twisted-order-raising-thm} Recall the statement of Theorem~\ref{thm:twisted-order-raising-on-A}: If a collection $A$ of properly immersed surfaces in a simply connected $4$--manifold supports an order $n$ twisted Whitney tower $\mathcal{W}$ with $\tau^\iinfty_n(\mathcal{W})=0\in\mathcal{T}^\iinfty_n$, then $A$ is regularly homotopic (rel $\partial$) to $A'$ supporting an order $n+1$ twisted Whitney tower.
Recall also from subsection~\ref{subsec:int-forests} that the intersection forest $t(\mathcal{W})$ of an order $n$ twisted Whitney tower $\mathcal{W}$ is a disjoint union of signed trees which can be considered to be immersed in $\mathcal{W}$. The order $n$ trees in $t(\mathcal{W})$ (together with the order $n/2$ $\iinfty$-trees if $n$ is even) represent $\tau_n^\iinfty(\mathcal{W})\in\mathcal{T}^\iinfty_n$, and the proof of Theorem~\ref{thm:twisted-order-raising-on-A} involves controlled manipulations of $\mathcal{W}$ which first convert $t(\mathcal{W})$ into ``algebraically canceling'' pairs of isomorphic trees with opposite signs, and then exchange these for ``geometrically canceling'' intersection points which are paired by a new layer of Whitney disks. We pause here to clarify these notions:
\subsubsection{Algebraic versus geometric cancellation}\label{subsubsec:alg-vs-geo-cancellation}
If Whitney disks $W_I$ and $W_J$ in $\mathcal{W}$ intersect transversely in a pair of points $p$ and $p'$, then $t_p$ and $t_{p'}$ are isomorphic (as labeled, oriented trees). If $p$ and $p'$ have opposite signs, and if the ambient $4$--manifold is simply connected, then there exists a Whitney disk $W_{(I,J)}$ pairing $p$ and $p'$, and we say that $\{\,p\,,\,p'\,\}$ is a \emph{geometrically canceling} pair.
In this setting we also refer to $\{\,\epsilon_p\cdot t_p\,,\,\epsilon_{p'}\cdot t_{p'}\,\}$ as a geometrically canceling pair of signed trees in $t(\mathcal{W})$ (regarding them as subsets of $\mathcal{W}$ associated to the geometrically canceling pair of points).
On the other hand, given transverse intersections $p$ and $p'$ in $\mathcal{W}$ with $ t_p = t_{p'}$ (as labeled oriented trees) and $\epsilon_p=-\epsilon_{p'}$, we say that $\{\,p\,,\,p'\,\}$ is an \emph{algebraically canceling} pair of intersections, and similarly call $\{\,\epsilon_p\cdot t_p\,,\,\epsilon_{p'}\cdot t_{p'}\,\}$ an algebraically canceling pair of signed trees in $t(\mathcal{W})$. Changing the orientations at a \emph{pair} of trivalent vertices in any tree $t_p$ does not change its value in $\mathcal{T}$ by the AS relations, and (as discussed in 3.4 of \cite{ST2}) such orientation changes can be realized by changing orientations of Whitney disks in $\mathcal{W}$ together with our orientation conventions (\ref{subsec:w-tower-orientations}).
Any geometrically canceling pair is also an algebraically canceling pair, but the converse is clearly not true as an algebraically canceling pair can have \emph{corresponding trivalent vertices} lying in \emph{different Whitney disks}. A process for converting algebraically canceling pairs into geometrically canceling pairs by manipulations of the Whitney tower is described in 4.5 and 4.8 of \cite{ST2}.
Similarly, if a pair of twisted Whitney disks $W_{J_1}$ and $W_{J_2}$ have isomorphic (unoriented) trees $J^\iinfty_1$ and $J^\iinfty_2$ with opposite twistings $\omega(W_{J_1})=-\omega (W_{J_2})$, then the Whitney disks form an \emph{algebraically canceling} pair (as do the corresponding signed $\iinfty$-trees in $t(\mathcal{W})$). Note that the orientations of the $\iinfty$-trees are not relevant here by the independence of $\omega(W)$ from the orientation of $W$ and the symmetry relations in $\mathcal{T}^\iinfty$. A geometric construction for eliminating algebraically canceling pairs of twisted Whitney disks from a twisted Whitney tower will be described below.
\textbf{Outline of the proof:} To motivate the proof of Theorem~\ref{thm:twisted-order-raising-on-A} we summarize here how the methods of \cite{CST,S1,ST2} (as described in Section~4 of \cite{ST2}) apply in the framed setting to prove the analogous order-raising theorem in framed setting (Theorem~\ref{thm:framed-order-raising-on-A} of Section~\ref{sec:w-towers}): The first part of the proof changes the intersection forest $t(\mathcal{W})$ so that all trees occur in algebraically canceling pairs by using the $4$--dimensional IHX construction of \cite{CST} to realize IHX relators, and by adjusting Whitney disk orientations as necessary to realize AS relations. The second part of the proof uses the Whitney move IHX construction of \cite{S1} to ``simplify'' the shape of the algebraically canceling pairs of trees. Then the third part of the proof uses controlled homotopies to exchange the simple algebraic canceling pairs for geometrically canceling intersection points which are paired by a new layer of Whitney disks as described in 4.5 of \cite{ST2}.
Extending these methods to the present twisted setting will require two variations: realizing the new relators in $\mathcal{T}_n^\iinfty$, and achieving an analogous geometric cancellation for twisted Whitney disks corresponding to algebraically canceling pairs of (simple) $\iinfty$-trees. We will concentrate on these new variations, referring the reader to \cite{CST,S1,ST2} for the other parts just mentioned.
\textbf{Notation and conventions:}
By Lemma~\ref{lem:split-w-tower} it may be assumed that $\mathcal{W}$ is split at each stage of the constructions throughout the proof, so that all trees in $t(\mathcal{W})$ are embedded in $\mathcal{W}$.
In spite of modifications, $\mathcal{W}$ will not be renamed during the proof.
Throughout this section we will notate elements of $t(\mathcal{W})$ as formal sums, representing disjoint union by juxtaposition.
Note that if $\mathcal{W}$ is an order $n$ twisted Whitney tower, then the intersection forest $t(\mathcal{W})$ may contain higher order trees and $\iinfty$-trees in addition to those representing $\tau_n^\iinfty(\mathcal{W})$. These higher-order elements of $t(\mathcal{W})$ can be ignored throughout the proof for the following reasons: On the one hand, in a split $\mathcal{W}$ all the constructions leading to the elimination of unpaired order $n$ intersections (and twisted Whitney disks of order $n/2$) of $\mathcal{W}$ can be carried out away from any higher-order elements of $t(\mathcal{W})$. Alternatively, one could first exchange all twisted Whitney disks of order greater than $n/2$ for unpaired intersections of order greater than $n$ by boundary-twisting (Figure~\ref{boundary-twist-and-section-fig}). Then, all intersections of order greater than $n$ can be converted into into many algebraically canceling pairs of order $n$ intersections by repeatedly ``pushing down'' unpaired intersections until they reach the order zero disks, as illustrated for instance in Figure~12 of \cite{S2} (assuming, as we may, that $\mathcal{W}$ contains no Whitney disks of order greater than $n$).
Thus, we can and will assume throughout the proof that $t(\mathcal{W})$ represents $\tau_n^\iinfty(\mathcal{W})$.
\textbf{The odd order case:} Given $\mathcal{W}$ of order $2n-1$ with $\tau^\iinfty_{2n-1}(\mathcal{W})=0\in\mathcal{T}^\iinfty_{2n-1}$, it will suffice to modify $\mathcal{W}$ --- while only creating unpaired intersections of order at least $2n$ and twisted Whitney disks of order at least $n$ --- so that all order $2n-1$ trees in $t(\mathcal{W})$ come in algebraically canceling pairs of trees (since by \cite{ST2} the corresponding algebraically canceling pairs of order $2n-1$ intersection points can be exchanged for geometrically canceling intersections which are paired by Whitney disks, as mentioned just above).
\begin{figure}
\caption{Boundary-twisting a Whitney disk $W$ changes $\omega (W)$ by $\pm 1$ and creates an intersection point with one of the sheets paired by $W$. The horizontal arcs trace out part of the sheet, the dark non-horizontal arcs
trace out the newly twisted part of a collar of $W$, and the grey arcs indicate part of the Whitney section over $W$. The bottom-most
intersection in the middle picture corresponds to the $\pm 1$-twisting created by the move.}
\label{boundary-twist-and-section-fig}
\end{figure}
Since $\tau_{2n-1}^\iinfty(\mathcal{W})=0\in\mathcal{T}^\iinfty_{2n-1}$, the intersection forest $t(\mathcal{W})$ is in the span of IHX and boundary-twist relators, after choosing Whitney disk orientations to realize AS relations as necessary. By locally creating intersection trees of the form $+I -H +X$ using the 4-dimensional geometric IHX theorem of \cite{CST} (and by choosing Whitney disk orientations to realize AS relations as needed), $\mathcal{W}$ can be modified so that all order $2n-1$ trees in $t(\mathcal{W})$ either come in algebraically canceling pairs, or are boundary-relator trees of the form $\pm\langle (i,J),J \rangle$.
For each tree of the form $t_p=\pm\langle (i,J),J \rangle$ we can create an algebraically canceling
$t_{p'}=\mp\langle (i,J),J \rangle$ at the cost of only creating order~$n$ $\iinfty$-trees as follows. First use Lemma~14 of \cite{ST2} (Lemma~{3.6} of \cite{S1}) to move the unpaired intersection point $p$ so that $p\in W_{(i,J)}\cap W_J$. Now, by boundary-twisting $W_{(i,J)}$ into its supporting Whitney disk $W'_J$ (Figure~\ref{boundary-twist-and-section-fig}), an algebraically canceling intersection $p'\in W_{(i,J)}\cap W'_J$ can be created at the cost of changing the twisting $\omega (W_{(i,J)})$ by $\pm 1$.
Since $\langle (i,J),J \rangle$ has an order $2$ symmetry, the canceling sign can always be realized by a Whitney disk orientation choice. This algebraic cancellation of $t_p$ has been achieved at the cost of only adding to $t(\mathcal{W})$ the order $n$ $\iinfty$-tree $(i,J)^\iinfty$ corresponding to the $\pm 1$-twisted order $n$ Whitney disk $W_{(i,J)}$.
Having arranged that all the order $2n-1$ trees in $t(\mathcal{W})$ occur in algebraically canceling pairs, applying the tree-simplification and geometric cancellation described in \cite{ST2} to all these algebraically canceling pairs yields an order $2n$ twisted Whitney tower $\mathcal{W}'$.
\textbf{The even order case:} For $\mathcal{W}$ of order $2n$ with $\tau^\iinfty_{2n}(\mathcal{W})=0\in\mathcal{T}^\iinfty_{2n}$, we arrange for $t(\mathcal{W})$ to consist of only algebraically canceling pairs of generators by realizing all relators in $\mathcal{T}_{2n}^\iinfty$, then construct an order $2n+1$ twisted Whitney tower by introducing a new method for geometrically cancelling the pairs of twisted Whitney disks (while the algebraically canceling pairs of non-$\iinfty$ trees lead to geometrically canceling intersections as before):
First of all, the order $0$ case corresponding to linking numbers is easily checked, so we will assume $n\geq 1$.
The IHX relators and AS relations for non-$\iinfty$ trees can be realized as usual.
Note that any signed tree $\epsilon\cdot J^\iinfty\in t(\mathcal{W})$ does not depend on the orientation of the tree $J$ because changing the orientation on the corresponding twisted Whitney disk $W_J$ does not change $\omega (W_J)$.
For any rooted tree $J$ the relator $\langle J,J \rangle-2\cdot J^\iinfty$ corresponding to the interior-twist relation can be realized as follows. Use finger moves to create a clean framed Whitney disk $W_J$. Performing a positive interior twist on $W_J$ as in Figure~\ref{InteriorTwistPositiveEqualsNegative-fig} creates a self-intersection $p\in W_J\cap W_J$ with $t_p=\langle J,J \rangle$ and changes the twisting $\omega(W_J)$ of $W_J$ to $-2$.
The negative of the relator is similarly constructed starting with a negative twist. \begin{figure}
\caption{A $+1$ interior twist on a Whitney disk changes the twisting by $-2$, as is seen in the pair of $-1$ intersections between the black vertical slice of the Whitney disk and the grey slice of a Whitney-parallel copy. Note that the pair of (positive) black-grey intersections near the $+1$ intersection is just an artifact of the immersion of the normal bundle into $4$--space and does not contribute to the relative Euler number.}
\label{InteriorTwistPositiveEqualsNegative-fig}
\end{figure}
The relator $-I^\iinfty+H^\iinfty+X^\iinfty-\langle H,X \rangle$ corresponding to the twisted IHX relation is realized as follows. For any rooted tree $I$, create a clean framed Whitney disk $W_I$ by finger moves. Then split this framed Whitney disk using the twisted finger move of Lemma~\ref{lem:split-w-tower} into two clean twisted Whitney disks with twistings $+1$ and $-1$, and associated signed $\iinfty$-trees $+I^\iinfty$ and $-I^\iinfty$, respectively. The next step is to perform a $+1$-twisted version (described in Lemma~\ref{lem:twistedIHX} below) of the ``Whitney move IHX'' construction of Lemma~{7.2} in \cite{S1}, which will replace the $+1$-twisted Whitney disk by two $+1$-twisted Whitney disks having $\iinfty$-trees $+H^\iinfty$ and $+X^\iinfty$, and containing a single negative intersection point with tree $-\langle H,X \rangle$, where $H$ and $X$ differ locally from $I$ as in the usual IHX relation.
Thus, any Whitney tower can be modified to create exactly the relator $-I^\iinfty+H^\iinfty+X^\iinfty-\langle H,X \rangle$, for any rooted tree $I$. The negative of the relator can be similarly realized by using Lemma~\ref{lem:twistedIHX} applied to the $-1$-twisted $I$-shaped Whitney disk.
So since $\tau_{2n}^\iinfty(\mathcal{W})$ vanishes, it may be arranged, by realizing relators as above, that all the trees in $t(\mathcal{W})$ occur in algebraically canceling pairs. Now, by repeated applications of Lemma~\ref{lem:twistedIHX} below, the algebraically canceling pairs of clean $\pm 1$-twisted Whitney disks can be exchanged for (many) algebraically canceling pairs of clean $\pm 1$-twisted Whitney disks, all of whose trees are \emph{simple} (right- or left-normed), with the $\iinfty$-label at one end of the tree as illustrated in Figure~\ref{simple-infty-tree-fig} -- this also creates more algebraically canceling pairs of non-$\iinfty$ trees (the ``error term'' trees in Lemma~\ref{lem:twistedIHX}).
As in the odd case, all algebraically canceling pairs of intersections with non-$\iinfty$ trees can be exchanged for geometrically canceling pairs by \cite{ST2}. To finish building the desired order~$2n+1$ twisted Whitney tower, we will describe how to eliminate the remaining algebraically canceling pairs of clean twisted order~$n$ Whitney disks (all having simple trees) using a construction that bands together Whitney disks and is additive on twistings. This construction is an iterated elaboration of a construction originally from Chapter~10.8 of \cite{FQ} (which was used to show that that $\tau_1 \otimes \mathbb{Z}_2$ did not depend on choices of pairing intersections by Whitney disks).
\begin{figure}
\caption{The simple twisted tree $J_n^\infty$.}
\label{simple-infty-tree-fig}
\end{figure}
Consider an algebraically canceling pair of clean $\pm 1$-twisted Whitney disks $W_{J_n}$ and $W'_{J_n}$, whose simple $\iinfty$-trees $+ J_n^\iinfty$ and $-J_n^\iinfty$ are as in Figure~\ref{simple-infty-tree-fig}, using the notation $J_n=(\cdots ((i,j_1),j_2),\cdots , j_n)$. Each trivalent vertex corresponds to a Whitney disk, and we will work from left to right, starting with the order one Whitney disks $W_{(i,j_1)}$ and $W'_{(i,j_1)}$, banding together Whitney disks of the same order from the two trees, while only creating new unpaired intersections of order greater than $2n$. At the last step, $W_{J_n}$ and $W'_{J_n}$ will be banded together into a single framed clean Whitney disk, providing the desired geometric cancellation. (The reason for working with \emph{simple} trees is that the construction for achieving geometric cancellation requires \emph{connected} surfaces for certain steps. For instance, the following construction only gets started because the left most trivalent vertices of an algebraically canceling pair of simple trees correspond to Whitney disks which pair the connected order zero surfaces $D_i$ and $D_{j_1}$.)
To start the construction consider the Whitney disks $W_{(i,j_1)}$ and $W'_{(i,j_1)}$, pairing intersections between the order zero immersed disks $D_i$ and $D_{j_1}$. Let $V$ be another Whitney disk for a canceling pair consisting of one point from each of the points paired by $W_{(i,j_1)}$ and $W'_{(i,j_1)}$. Figure~\ref{bandedWdisksWithTwistsNoHigherInts} illustrates how a parallel copy $V'$ of $V$ can be banded together with $W_{(i,j_1)}$ and $W'_{(i,j_1)}$ to form a Whitney disk $W''_{(i,j_1)}$ for the remaining canceling pair. The twisting of $W''_{(i,j_1)}$ is the sum of the twistings on $W_{(i,j_1)}$, $W'_{(i,j_1)}$, and $V$; so $W''_{(i,j_1)}$ is framed if $V$ is framed, since both $W_{(i,j_1)}$ and $W'_{(i,j_1)}$ are framed for $n>1$ (and in the $n=1$ case $W_{(i,j_1)}=W_{J_n}$ and $W'_{(i,j_1)}=W'_{J_n}$ contribute canceling $\pm 1$ twistings). If $V$ is both framed and clean, then the result of replacing $W_{(i,j_1)}$ and $W'_{(i,j_1)}$ by $V$ and $W''_{(i,j_1)}$ preserves the order of $\mathcal{W}$ and creates no new intersections.
So if $n=1$, then $W_{J_n}$ and $W'_{J_n}$ have been geometrically canceled, meaning that their corresponding $\iinfty$-trees have been eliminated from $t(\mathcal{W})$ without creating any new unpaired order $2n$ intersections or new twisted order $n$ Whitney disks.
\begin{figure}
\caption{The Whitney disks $W_{(i,j_1)}$, $W'_{(i,j_1)}$, and $V'$ are
banded together to form the Whitney disk
$W''_{(i,j_1)}$ pairing the outermost pair of intersections between $D_i$ and $D_{j_1}$.
In the cases $n>1$, the interior of $W''_{(i,j_1)}$ contains two pairs of canceling intersections with $D_{j_2}$ (which are not shown), and supports the sub-towers consisting of the rest of the higher-order Whitney disks (that were supported by $W_{(i,j_1)}$ and $W'_{(i,j_1)}$) corresponding to the trivalent vertices in both trees $\pm J_n^\infty$.}
\label{bandedWdisksWithTwistsNoHigherInts}
\end{figure}
The next step shows how $V$ can be arranged to be framed and clean, at the cost of only creating intersections of order greater than $2n$: Any twisting $\omega(V)$ can be killed by boundary twisting $V$ into $D_{j_1}$. Then, using the construction shown in Figure~\ref{bandedWdisksWithPushDown}, any interior intersection between $V$ and any $K$-sheet (e.g.~an intersection with $D_{j_1}$ from boundary-twisting) can be pushed down into $D_i$ and paired by a thin Whitney disk $W_{(K,i)}$, which in turn has intersections with the $D_{j_1}$-sheet that can be paired by a Whitney disk $W_{K_1}:=W_{((K,i),j_1)}$ made from a Whitney-parallel copy of $W_{(i,j_1)}$. Now, parallel copies of the Whitney disks from the sub-tower supported by $W_{(i,j_1)}$ can be used to build a sub-tower on $W_{K_1}$: Using the notation $K_{r+1}=(K_r,i)$, for $r=1,2,3,\ldots n$, the Whitney disk $W_{K_{r+1}}$ is built from a Whitney-parallel copy of $W_{J_r}$, and pairs intersections between $W_{K_r}$ and $j_r$. Note that the order of each $W_{K_r}$ is at least $r$. The top order $W_{K_{n+1}}$ inherits the $\pm 1$-twisting from $W_{J_n}$, and has a single interior intersection with tree $\langle K_{n+1}, J_n \rangle$ which is of order at least $2n+1$.
For multiple intersections between $V$ and various $K$-sheets this part of the construction can be carried out simultaneously using nested parallel copies of the thin Whitney disk in Figure~\ref{bandedWdisksWithPushDown} and more Whitney-parallel copies of the sub-towers described in the previous paragraph.
The result of the construction so far is that the left-most trivalent vertices of the trees $+ J_n^\iinfty$ and $-J_n^\iinfty$ now correspond to the \emph{same} order $1$ Whitney disk $W''_{J_1}=W''_{(i,j_1)}$, at the cost of having created (after splitting-off) a clean twisted Whitney disk of order at least $n+1$, and an unpaired intersection of order at least $2n+1$. In particular, this completes the proof for the case $n=1$.
\begin{figure}\label{bandedWdisksWithPushDown}
\end{figure}
For the cases $n>1$, observe that since $W''_{J_1}$ is \emph{connected}, this construction can be repeated, with $W''_{J_1}$ playing the role of $D_i$, and $D_{j_2}$ playing the role of $D_{j_1}$, to get a single order $2$ Whitney disk $W''_{J_2}$ which corresponds to the second trivalent vertices from the left in both trees $+ J_n^\iinfty$ and $-J_n^\iinfty$. By iterating the construction, eventually we band together $W_{J_n}$ and $W'_{J_n}$ into a single framed clean Whitney disk at the last step, having created only clean twisted Whitney disks of order at least $n+1$, and unpaired intersections of order at least $2n+1$.
\subsection{The twisted IHX lemma}\label{subsec:twistedIHX}
The proof of Theorem~\ref{thm:twisted-order-raising-on-A} is completed by the following lemma which describes a twisted geometric IHX relation based on the framed version in Lemma~{7.2} of \cite{S1}.
\begin{lem}\label{lem:twistedIHX} Any split twisted Whitney tower $\mathcal{W}$ containing a clean $+1$-twisted Whitney disk with signed $\iinfty$-tree $+I^\iinfty$ can be modified (in a neighborhood of the Whitney disks and local order zero sheets corresponding to $I^\iinfty$) to a twisted Whitney tower $\mathcal{W}'$ such that $t(\mathcal{W}')$ differs from $t(\mathcal{W})$ exactly by replacing $+I^\iinfty$ with the signed trees $+H^\iinfty$, $+X^\iinfty$, and $-\langle H,X \rangle$, where $+I-H+X$ is a Jacobi relator.
Similarly, a clean $-1$-twisted Whitney disk with $\iinfty$-tree $-I^\iinfty$ in $t(\mathcal{W})$ can be replaced by $-H^\iinfty$, $-X^\iinfty$, and $+\langle H,X \rangle$ in $t(\mathcal{W}')$. \end{lem}
\begin{proof} Before describing how to adapt the construction and notation of \cite{S1} to give a detailed proof of Lemma~\ref{lem:twistedIHX}, we explain why the framed relation $+I=+H-X$ leads to the twisted relation $+I^\iinfty=+H^\iinfty+X^\iinfty-\langle H,X \rangle$. In the framed case, a Whitney disk with tree $I$ is replaced by Whitney disks with trees $H$ and $X$, such that the new Whitney disks are parallel copies of the original using the Whitney framing, and inherit the framing of the original. In order to preserve the trivalent vertex orientations of the trees, the orientation of the H-Whitney disk is the same as the original I-Whitney disk, and the orientation of the X-Whitney disk is the opposite of the I-Whitney disk. Now, if the original I-Whitney disk was $+1$-twisted, then both the H- and X-Whitney disks will inherit this same $+1$-twisting, because the twisting -- which is a self-intersection number -- is independent of the Whitney disk orientation. The H- and X-Whitney disks will also intersect in a single point with sign $-1$, since they inherited opposite orientations from the I-Whitney disk. Thus, (after splitting) a twisted Whitney tower can be modified so that a $+I^\iinfty$ is replaced by exactly $+H^\iinfty+X^\iinfty-\langle H,X \rangle$ in the intersection forest. Similarly, a $-I^\iinfty$ can be replaced exactly by $-H^\iinfty-X^\iinfty+\langle H,X \rangle$.
The framed IHX Whitney-move construction is described in detail in \cite{S1} (over four pages, including six figures). We describe here how to adapt that construction to the present twisted case, including the relevant modification of notation. Orientation details are not given in \cite{S1}, but all that needs to be checked is that the X-Whitney disk inherits the opposite orientation as the H-Whitney disk (given that the tree orientations are preserved, and using our negative-corner orientation convention in \ref{subsec:w-tower-orientations} above). In Lemma~{7.2} of \cite{S1}, the ``split sub-tower $\mathcal{W}_p$'' refers to the Whitney disks and order zero sheets containing the tree $t_p$ of an unpaired intersection $p$ in a split Whitney tower $\mathcal{W}$. In the current setting, a clean $+1$-twisted Whitney disk $W$ plays the role of $p$, and the construction will modify $\mathcal{W}$ in a neighborhood of the Whitney disks and order zero sheets containing the $\iinfty$-tree associated to $W$. In the notation of Figure~18 of \cite{S1}, the sub-tree of the I-tree denoted by $L$ contains $p$, so to interpret the entire construction in our case only requires the understanding that this sub-tree contains the $\iinfty$-label sitting in $W$. (Note that in Figure~18 of \cite{S1} the labels $I$, $J$, $K$ and $L$ denote \emph{sub}-trees, and in particular the $I$-labeled sub-tree should not be confused with the ``I-tree'' in the IHX relation.)
In the case where the $L$-labeled sub-tree is order zero, then $L$ is just the $\iinfty$-label, and the upper trivalent vertex of the I-tree in Figure~18 of \cite{S1} corresponds to the clean $+1$-twisted $W$, with $\iinfty$-tree $((I,J),K)^\iinfty$. Then the construction, which starts by performing a Whitney move on the framed Whitney disk $W_{(I,J)}$ corresponding to the lower trivalent vertex of the I-tree, yields the $+1$-twisted H- and X-Whitney disks as discussed in the first paragraph of this proof, with $\iinfty$-trees $(I,(J,K))^\iinfty$ and $(J,(I,K))^\iinfty$, and non-$\iinfty$ tree $\langle (I,(J,K)),(J,(I,K))\rangle$ corresponding to the resulting unpaired intersection (created by taking Whitney-parallel copies of the twisted $W$ to form the H- and X-Whitney disks).
In the case where the $L$-labeled sub-tree is order $1$ or greater, then the upper trivalent vertex of the I-tree in Figure~18 of \cite{S1} corresponds to a framed Whitney disk, and Whitney-parallel copies of this framed Whitney disk and the other Whitney disks corresponding to the $L$-labeled sub-tree are also used to construct the sub-towers containing the $+1$-twisted Whitney disks with H and X $\iinfty$-trees (which will again will lead to a single unpaired intersection as before). \end{proof}
\subsection{Twisted even and framed odd order Whitney towers.}\label{subsec:boundary-twisted-IHX-lemma} To complete the proof of Proposition~\ref{prop:exact sequence} in the introduction, the following lemma implies that $\mathbb{W}^\iinfty_{2n}\subset \mathbb{W}_{2n-1}$: \begin{lem}\label{lem:boundary-twisted-IHX} If a collection $A$ of properly immersed surfaces in a simply connected $4$--manifold supports an order $2n$ \emph{twisted} Whitney tower, then $A$ is homotopic (rel $\partial$) to $A'$ which supports an order $2n-1$ \emph{framed} Whitney tower. \end{lem} \begin{proof} Let $\mathcal{W}$ be any order $2n$ twisted Whitney tower $\mathcal{W}$ supported by $A$. If $\mathcal{W}$ contains no order $n$ non-trivially twisted Whitney disks, then $\mathcal{W}$ is an order $2n$ framed Whitney tower, hence also is an order $2n-1$ framed Whitney tower. If $\mathcal{W}$ does contain order $n$ non-trivially twisted Whitney disks, they can be eliminated at the cost of only creating intersections of order at least $2n-1$ as follows:
Consider an order $n$ twisted Whitney disk $W_J\subset\mathcal{W}$ with twisting $\omega(W_J)=k\in\mathbb{Z}$. If $W_J$ pairs intersections between an order zero surface $A_i$ and an order $n-1$ Whitney disk $W_I$ then $J=(i,I)$, and by performing $|k|$ boundary-twists of $W_J$ into $W_I$, $W_J$ can be made to be framed at the cost of only creating $|k|$ order $2n-1$ intersections, whose corresponding trees are of the form $\langle\,(i,I),I\,\rangle$.
If $W_J$ pairs intersections between two Whitney disks, then by applying the twisted geometric IHX move of Lemma~\ref{lem:twistedIHX} as necessary, $W_J$ can be replaced by a union of order $n$ twisted Whitney disks each having a boundary arc on an order zero surface as in the previous case, at the cost of only creating unpaired intersections of order $2n$, each of which is an error term in Lemma~\ref{lem:twistedIHX}. \end{proof}
\section{Proof of Theorem~\ref{thm:odd}}\label{sec:proof-thm-odd} This section defines the doubling map $\Delta_{2n-1}:\mathbb{Z}_2\otimes\mathcal{T}_{n-1}\rightarrow \mathcal{T}_{2n-1}$ which determines the framing relations described in the introduction, and strengthens the obstruction theory for framed Whitney towers described in \cite{ST2} by showing that the vanishing of $\tau_{2n-1}(\mathcal{W})$ in the reduced group $\widetilde{\mathcal{T}}_{2n-1}:=\mathcal{T}_{2n-1}/\operatorname{Im}(\Delta_{2n-1})$ is sufficient for the promotion of $\mathcal{W}$ to a Whitney tower of order $2n$. This means that $\mathcal{T}_n$ can be replaced everywhere by $\widetilde{\mathcal{T}}_n$ (with $\widetilde{\mathcal{T}}_{2n}:=\mathcal{T}_{2n}$) throughout Section~\ref{sec:realization-maps}, and in particular proves Theorem~\ref{thm:odd} of the introduction.
\begin{defn}\label{def:Delta} The \emph{doubling map} $\Delta_{2n-1}:\mathbb{Z}_2\otimes\mathcal{T}_{n-1}\rightarrow \mathcal{T}_{2n-1}$ is defined for generators $t\in\mathcal{T}_{n-1}$ by $$\Delta (t):=\sum_{v\in t} \langle i(v),(T_v(t),T_v(t))\rangle$$ where $T_v(t)$ denotes the rooted tree gotten by replacing $v$ with a root, and the sum is over all univalent vertices of $t$, with $i(v)$ the original label of the univalent vertex $v$. \end{defn} That $\Delta_{2n-1}$ is well-defined as a homomorphism on $\mathcal{T}_{n-1}$ is clear since AS and IHX relations go to doubled relations. The image of $\Delta_{2n-1}$ is $2$-torsion by AS relations and hence it factors through $\mathbb{Z}_2\otimes\mathcal{T}_{n-1}$. See Figure~\ref{fig:Delta trees} for explicit illustrations of $\Delta_1$ and $\Delta_3$.
\begin{figure}
\caption{The map $\Delta_{2n-1}:\mathbb{Z}_2\otimes\mathcal{T}_{n-1}\rightarrow \mathcal{T}_{2n-1}$ in the cases $n=1$ and $n=2$.}
\label{fig:Delta trees}
\end{figure}
The following theorem strengthens Theorem~\ref{thm:framed-order-raising-on-A} in Section~\ref{sec:w-towers}. \begin{thm}\label{thm:framed-order-raising-mod-Delta} If a collection $A$ of properly immersed surfaces in a simply connected $4$--manifold supports a framed
Whitney tower $\mathcal{W}$ of order $(2n-1)$ with $\tau_{2n-1}(\mathcal{W})\in\operatorname{Im}(\Delta_{2n-1})$, then $A$ is homotopic (rel $\partial$) to $A'$ which supports a framed Whitney tower of order $2n$. \end{thm}
\begin{proof} As discussed above in the outline the proof of Theorem~\ref{thm:twisted-order-raising-on-A} (just after subsubsection~\ref{subsubsec:alg-vs-geo-cancellation}), to prove Theorem~\ref{thm:framed-order-raising-mod-Delta} it will suffice to show that the intersection forest $t(\mathcal{W})$ can be changed by trees representing any element in $\operatorname{Im}(\Delta_{2n-1})<\mathcal{T}_{2n-1}$ at the cost of only introducing trees of order greater than or equal to $2n$, so that the order $2n-1$ trees in $t(\mathcal{W})$ all occur in algebraically canceling pairs. Note that $\operatorname{Im}(\Delta_{2n-1})$ is $2$-torsion by the AS relations, so orientations and signs are not an issue here. As in Section~\ref{sec:proof-twisted-thm}, elements of $t(\mathcal{W})$ will be denoted by formal sums, and $\mathcal{W}$ will not be renamed as modifications are made.
\textbf{The case $n=1$:} Given any order zero tree $\langle i,j \rangle$, create a clean framed Whitney disk $W_{( i,j )}$ by performing a finger move between the order zero surfaces $A_i$ and $A_j$. Then use a twisted finger move (Figure~\ref{twist-split-Wdisk-fig}) to split $W_{( i,j )}$ into two twisted Whitney disks with associated trees $(i,j)^\iinfty -(i,j)^\iinfty$. Now boundary-twist each Whitney disk into a different sheet to recover the framing and add \[ \langle i,(i,j) \rangle + \langle j,(i,j) \rangle = \Delta_1( \langle i,j \rangle ) \]
to $t(\mathcal{W})$. Alternatively, after creating the framed $W_{( i,j )}$, perform an interior twist on $W_{( i,j )}$ to get $\omega(W_{( i,j )})=\pm2$, then kill $\omega(W_{( i,j )})$ by two boundary-twists, one into each sheet, again adding $\langle i,(i,j) \rangle + \langle j,(i,j) \rangle$ to $t(\mathcal{W})$. Note that $\operatorname{Im}\Delta_1$ in $\mathcal{T}_1$ corresponds to the order $1$ FR framing relation of \cite{S3,ST1}.
\begin{figure}
\caption{Multiple $\infty$-roots attached to a tree represent sums (disjoint unions) of trees. On the left: the two trees that result from twist-splitting a clean $W_{( i, (I_1, I_2))}$ in the case $\langle i,(I_1, I_2) \rangle=\langle i, ((j,k),(a,(b,c))) \rangle$. Each arrow indicates an application of a twisted IHX Whitney move, which pushes $\infty$-roots towards the univalent vertices. The right-most sum of trees becomes the image of $\langle i,(I_1, I_2) \rangle$ under $\Delta$ after applying boundary-twists to the associated twisted Whitney disks.}
\label{fig:Delta-infty-tree-example}
\end{figure} \textbf{The cases $n>1$:} For any order $n-1$ tree $\langle i,(I_1, I_2) \rangle$, create a clean $W_{( i, (I_1, I_2))}$ by finger moves. (Here we are taking any order $n-1$ tree, choosing an $i$-labeled univalent vertex, and writing it as the inner product of the order zero rooted tree $i$ and the remaining order $n-1$ tree.) Then split $W_{( i, (I_1, I_2))}$ using a twisted finger move to get two twisted Whitney disks each having associated $\iinfty$-tree $( i, (I_1, I_2))^\iinfty$. Leave one of these twisted Whitney disks alone, and to the other apply the twisted geometric IHX Whitney-move (Lemma~\ref{lem:twistedIHX} of Section~\ref{sec:proof-twisted-thm}) to replace $( i, (I_1, I_2))^\iinfty$ by $( I_1, (I_2, i))^\iinfty+( I_2, (i,I_1))^\iinfty-\langle ( I_1, (I_2, i)),( I_2, (i,I_1))\rangle$ in $t(\mathcal{W})$. Note that the tree $\langle ( I_1, (I_2, i)),( I_2, (i,I_1))\rangle$ is order $2n$. If $I_1$ and $I_2$ are not both order zero then continue to apply the twisted geometric IHX Whitney-move (pushing the $\iinfty$-labeled vertices away from the $\iinfty$-labeled vertex that is adjacent to the original $i$-labeled vertex) until the resulting union of trees has all $\iinfty$-labeled vertices adjacent to a univalent vertex (all twisted Whitney disks have a boundary arc on an order zero surface) -- see Figure~\ref{fig:Delta-infty-tree-example} for an example. Then, boundary-twisting each twisted Whitney disk into the order zero surface recovers the framing on each Whitney disk and the resulting change in $t(\mathcal{W})$ is a sum of trees as in the right hand side of the equation in Definition~\ref{def:Delta} representing the image of $\langle i,(I_1, I_2) \rangle$ under $\Delta_{2n-1}$, together with trees of order at least $2n$. \end{proof}
\end{document} | arXiv |
3-DNF proves the algorithm is in P class
To understand fully, please read link
After, reading the link we will take a look at how we recover our solutions to a constrained Sudoku Puzzle.
If we assume that a sudoku puzzle was generated with this procedure we can now create a "semi"-solver. I say "semi" because we need the $3 \times 3$ grid $M_{2,2}$ already solved for us. Let's assume we have this. As an example I will assume we are provided:
$$\begin{bmatrix} 5 & 9 & 6\\ 1 & 2 & 4\\ 3 & 7 & 8 \end{bmatrix}$$
Now we will flatten it into: $[5,9,6,1,2,4,3,7,8]$ and permute as follows:
[8, 5, 9, 6, 1, 2, 4, 3, 7]-----list 1
Now for each list, we will turn them into a $3 \times 3$ grid using the same mapping in step 2 above. For example list 1 would get mapped to
$$\begin{bmatrix} 8 & 5 & 9 \\ 6 & 1 & 2 \\ 4 & 3 & 7 \end{bmatrix}$$
Now we position these in the game board the same way we did as step 3 above. For example our layout would be as follows:
**list1** **list4** **list7**
In the prior example this would give us the correct solution:
$$M = \begin{bmatrix} 8 & 5 & 9 & 4 & 3 & 7 & 6 & 1 & 2\\ 6 & 1 & 2 & 8 & 5 & 9 & 4 & 3 & 7\\ 4 & 3 & 7 & 6 & 1 & 2 & 8 & 5 & 9\\ 7 & 8 & 5 & 2 & 4 & 3 & 9 & 6 & 1\\ 9 & 6 & 1 & 7 & 8 & 5 & 2 & 4 & 3\\ 2 & 4 & 3 & 9 & 6 & 1 & 7 & 8 & 5\\ 3 & 7 & 8 & 1 & 2 & 4 & 5 & 9 & 6\\ 5 & 9 & 6 & 3 & 7 & 8 & 1 & 2 & 4\\ 1 & 2 & 4 & 5 & 9 & 6 & 3 & 7 & 8\\ \end{bmatrix}$$
Then we have list 9 (our input) will always give you correct solution in quadratic time.
For further illustration, I intend to prove that the algorithm aforementioned is in P class in two ways.
Here, we'll take a look at 3-DNF.
(L1 ∧ L2 ∧ L3) | (L4 ∧ L5 ∧ L6) | (L7 ∧ L8 ∧ L9)
Let L1=list1, L2 = list2,...
Therefore, the algorithm generates grids and recovers correct solutions easily.
Now, lets say I want to check the satsifiability of the algorithm's circular shifts. Here, I generate 3 more grids to show that there is a 3x3 positive 3-satisfying permutes.
l = [8, 5, 9, 6, 1, 2, 4, 3, 7]
[5, 9, 6, 1, 2, 4, 3, 7, 8]-l1
x = [5, 9, 6, 1, 2, 4, 3, 7, 8]
[9, 6, 1, 2, 4, 3, 7, 8, 5]-x1
y = [9, 6, 1, 2, 4, 3, 7, 8, 5]
[6, 1, 2, 4, 3, 7, 8, 5, 9]-y1
Here, I demonstrate that the 3x3 shift meets satisfiability for 9! Sudoku grids generated by the algorithm. At the end of the question I prove that the expression is always meets satisfiability when given the correct inputs.
(l1 ∨ x9 ∨ y8) ∧ (l2 ∨ x1 ∨ y9)
l1 = [5, 9, 6, 1, 2, 4, 3, 7, 8]
x9 = [5, 9, 6, 1, 2, 4, 3, 7, 8]
y8 = [5, 9, 6, 1, 2, 4, 3, 7, 8]
All the listed elements above have their defined variables within these expressions. All the expressions hold true.
(𝑙1∨𝑥9∨𝑦8)∧(𝑙2∨𝑥1∨𝑦9)∧(𝑙3∨𝑥2∨𝑦1)∧(𝑙4∨𝑥3∨𝑦2)∧(𝑙5∨𝑥4∨𝑦3)∧(𝑙6∨𝑥5∨𝑦4)∧(𝑙7∨𝑥6∨𝑦5)∧(𝑙8∨𝑥7∨𝑦6)∧(𝑙9∨𝑥8∨𝑦7)∧(𝑙1∨𝑥9∨𝑦8)∧(𝑙2∨𝑥1∨𝑦9)
Here is a chart showing the 3-satsifiability of the algorithm. Proving that the 3x3 shift overlaps all 9! valid grids that the algorithm can generate
Overall, are these proofs correct that constrained Sudoku is in P class?
satisfiability decision-problem
Travis Wells
Travis WellsTravis Wells
As I already explained earlier, any algorithm on a 9x9 input takes constant time. Thus it is in P. This is not very interesting or useful.
When people talk about Sudoku being NP-hard, they don't actually mean Sudoku, they are referring to a generalization on a grid of arbitrary size. Your question doesn't prove that this generalized problem is in P.
General comment: It looks like you're immersed in details, but haven't got a solid grasp on the fundamentals/basics yet. I encourage you to spend some more time learning about the definition of languages, decision problems, P, NP, NP-complete, NP-hard, and reductions before trying to take your Sudoku "project" any further. As it stands some of your statements appear to reflect a misunderstanding of basic concepts, and so you're spending time on things that are a dead end or reflect some basic misconceptions. (For instance, an algorithm can't be in P, and a proof can't be in; a problem can be.) I hope you'll take this as aimed to help you learn, rather than an attempt to criticize you personally or tear you down.
D.W.♦D.W.
$\begingroup$ Okay, I was just going to build upon a new questions after reading through reductibility and etc on Wikipedia. And, I'm building upon the same subject on every new question. I'll accept your answer while I formulate a new question as I'm getting more clarification step by step(both reading and asking) $\endgroup$ – Travis Wells Apr 24 '19 at 19:29
$\begingroup$ @TravisWells, Cool! I'd suggest finding a good textbook or an online course on algorithms or complexity theory. Wikipedia isn't a great resource. Don't expect it to be something you can learn in one sitting. Have fun -- it's a beautiful subject! $\endgroup$ – D.W.♦ Apr 24 '19 at 19:34
$\begingroup$ I'm in the process of formulating a question about NM-3sat. Which is basically no mixed instances. (eg. L1 or x9 or y8) and (l2 or x1 or y1) No negations unless all variables are negated. I was wondering if it would be possible to reduce DNF to NM-3sat showing that constrained puzzles can be just as hard to solve. $\endgroup$ – Travis Wells Apr 24 '19 at 19:40
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\begin{document}
\title{An optimal bound on the number of moves for open Mancala}
\author[A. Musesti] {Alessandro Musesti} \address[Alessandro Musesti]{Dipartimento di Matematica e Fisica ``Niccol\`o Tartaglia'',
Universit\`a Cattolica del Sacro Cuore, Via dei Musei 41, 25121
Brescia, Italy} \email{[email protected]} \author[M. Paolini] {Maurizio Paolini} \address[Maurizio Paolini]{Dipartimento di Matematica e Fisica ``Niccol\`o Tartaglia'',
Universit\`a Cattolica del Sacro Cuore, Via dei Musei 41, 25121
Brescia, Italy} \email{[email protected]} \author[C. Reale] {Cesco Reale} \address[Cesco Reale]{Festival Italiano di Giochi Matematici} \email{[email protected]} \date{\today}
\begin{abstract} We determine the optimal bound for the maximum number of moves required to reach a periodic configuration of \emph{open mancala} (also called open owari), inspired by a popular African game. A mancala move can be interpreted as a map from the set of compositions of a given integer in itself, thus relating our result to the study of the corresponding finite dynamical system. \end{abstract}
\maketitle \section{Introduction} \textit{Mancala} is a family of traditional African games played in many versions and under many names. It comprises a circular list of \textit{holes} containing zero or more \textit{seeds}. A move consists of selecting a nonempty hole, taking all its seeds and \textit{sowing} them in the subsequent holes, one seed per hole.
Here we study an idealized version of this called \emph{open mancala} (see \cite{Bou1}, where the game is called \textit{open owari}). We assume there is an infinite sequence of holes. Further we assume that the nonempty holes are consecutive; such a configuration is described by a sequence of positive integers $\lambda_i$, $i = 1,\dots,\ell$, giving the number of seeds in each nonempty hole, starting from the leftmost one. We assume a move always selects the leftmost nonempty hole, sowing to the right. Thus the leftmost nonempty hole always advances of one position. It is clear that the sequence of configurations becomes periodic in a finite time. The periodic configurations have been completely classified (\cite{Bou1,Bru,Bou2}): see Section \ref{sec:notations}.
This game is related to a card game called \emph{Bulgarian Solitaire}, discussed by Gardner \cite{Gar} in a 1983 \emph{Scientific American} column. Indeed, that game is isomorphic to the case of open mancala where $\lambda_i \geq \lambda_j$ for $i \leq j$ (monotone mancala), see Section \ref{sec:bulgarian}. Gardner mentioned a conjecture on the maximum number of moves before the onset of periodicity when the number of cards is of the form $k(k+1)/2$. The conjecture was proved independently a few years later by Igusa \cite{Igu} and Etienne \cite{Eti}. In 1998, Griggs \& Ho \cite{GrHo} revisited the result and proposed a new conjecture in the case of a generic number of cards. Due to the isomorphism between monotone mancala and Bulgarian solitaire, our result is strongly related to this conjecture. However, our results apply to open mancala, not to the monotone variant, hence the conjecture about the Bulgarian Solitaire remains unproven.
The main result of the paper is the optimal bound for the maximum number of moves, before open mancala reaches periodicity, as a function of $n$, the number of seeds. In Section \ref{sec:lower} we prove the lower bound by producing, for any $n$, a configuration that requires exactly that number of moves. In Section \ref{sec:smon} we introduce the main tool in proving the upper bound, namely {\em $s$-monotonicity}. In Section \ref{sec:upper} we prove that every configuration reaches periodicity within that number of moves.
\section{Notations and setting} \label{sec:notations} We denote by ${\mathbb N}$ the set of nonnegative integers and by ${\mathbb N}^*$ the set of strictly positive integers. By $T_k$, $k \in {\mathbb N}$, we denote the $k$-th triangular number \[ T_k = \sum_{i=0}^k i = \frac{k(k+1)}{2}, \] where in particular $T_0 = 0$.
\begin{definition}
A \emph{configuration} $\lambda$ is a sequence of nonnegative
numbers $\lambda_i$, $i \geq 1$, that is $\lambda : {\mathbb N}^*
\to {\mathbb N}$. The \emph{support} $\support(\lambda)$ of a configuration $\lambda$ is the set of indices $\{i \in {\mathbb N}^* : \lambda_i > 0\}$. \end{definition}
\begin{definition} A \emph{mancala configuration} $\lambda$ is a configuration having a \emph{connected} support of the form $\{1, \dots, \ell\}$ for some $\ell = \length(\lambda) \in {\mathbb N}$. The index $\length(\lambda)$ is called the \emph{length} of the configuration. In the special case of the \emph{zero} configuration, we shall conventionally define its length to be zero. We shall denote by $\configurations$ the set of all mancala configurations. \end{definition}
\begin{definition} If $\lambda \in \configurations$, its \emph{mass} is the number $$
|\lambda| =
\sum_{i=1}^\infty \lambda_i =
\sum_{i=1}^{\length(\lambda)} \lambda_i. $$ We denote by $\configurations_n$ the set of all mancala configurations of mass $n$. \end{definition}
The \emph{mancala} game, in our setting, is a discrete dynamical system associated with a function $\map : \configurations \to \configurations$ mapping the space of \emph{mancala} configurations $\configurations$ in itself.
\begin{definition}[Sowing] \label{def:move} The \emph{sowing} of a mancala configuration $\lambda$ is an operation on $\lambda$ that results in a new configuration $\mu = \map(\lambda)$ defined as follows:
$$ \mu_i = \begin{cases} \lambda_{i+1} + 1 \qquad & \text{if $1 \leq i \leq \lambda_1$}, \\ \lambda_{i+1} \qquad & \text{if $i > \lambda_1$} . \end{cases} $$
Conventionally, $\map$ maps the empty configuration into itself. \end{definition}
It is clear that $\map$ preserves the mass of a configuration, so that it can be restricted to $\configurations_n$.
It is convenient to define the (right) \emph{shift} operator $\shift$, that acts on generic sequences simply by a change in the indices.
\begin{definition}[Shift operator] \label{def:shift} If $\lambda : {\mathbb N}^* \to {\mathbb N}$ is a sequence, the sequence $\shift(\lambda)$ is defined by
$$ \shift(\lambda)_i = \begin{cases} 0 \qquad & \text{if $i = 1$}, \\ \lambda_{i-1} \qquad& \text{if $i > 1$}. \end{cases} $$
\end{definition} Clearly, the result of the shift operator is never a mancala configuration (with the exception of the empty configuration).
Both $\map$ and $\shift$ can be iterated, the symbol $\map^k$ (resp.\ $\shift^k$) denoting the result of $k$ repeated applications of $\map$ (resp.\ of $\shift$). The shift operator has a left inverse $\shift^{-1}$, defined by $\shift^{-1} (\lambda)_i = \lambda_{i+1}$ and satisfying $\shift^{-1}(\shift(\lambda)) = \lambda$ for any configuration $\lambda$.
\begin{definition}[Partial ordering and sum] We say that $\lambda \leq \mu$ if $\lambda_i \leq \mu_i$ for all $i \geq 1$. Moreover we say that $\lambda < \mu$ if $\lambda\leq \mu$ and $\lambda\neq \mu$.
If $\lambda$, $\mu : {\mathbb N}^* \to {\mathbb Z}$ are two integer sequences (in particular if any of them is a configuration in $\configurations$), we define the sum and difference $\lambda \pm \mu$ componentwise: $(\lambda \pm \mu)_i = \lambda_i \pm \mu_i$. \end{definition}
\begin{remark}[Comparison]
It is easy to check that if $\lambda$, $\mu \in \configurations$ and
$\lambda \leq \mu$, then $\map(\lambda) \leq \map(\mu)$. \end{remark}
\begin{definition} \label{defn:monotone} A configuration is called \emph{monotone}
if it is weakly decreasing, i.e. if $\lambda_i \geq \lambda_j$ whenever $i \leq j$. \end{definition}
\begin{remark}[Monotone mancala] If $\lambda$ is a monotone configuration, then so it is $\map(\lambda)$. Moreover, if $\lambda \in \configurations$ has length $\ell$, then $\map^{\ell-1} (\lambda)$ is monotone. \end{remark}
The following special configurations play an important role.
\begin{definition}[Marching group]\label{def:mg} For a given $k \in {\mathbb N}$ we define the special mancala configuration $\marching^k$ with mass $T_k$ and length $k$, called \emph{marching group} \footnote{Please note that this is \emph{not} a group in the mathematical sense.} of order $k$, as follows:
$$ \marching^k_i = \begin{cases}
k - i + 1 \qquad & \text{if $i \leq k$} , \\ 0 . \end{cases} $$
\end{definition}
\begin{definition}[Augmented marching group]\label{def:amg} A mancala configuration $\lambda$ such that $\marching^k \leq \lambda < \marching^{k+1}$ for some $k \in {\mathbb N}$ is called an \emph{augmented marching group}
of order $k$. It satisfies $T_k \leq |\lambda| < T_{k+1}$. \end{definition} Notice that a marching group is a particular case of an augmented marching group. The following important theorem about augmented marching groups is readily proved. See \cite[Theorems 1 and 2]{Bou1}.
\begin{theorem}\label{teo:periodic}
Augmented marching groups are the \emph{only} periodic
configurations for $\map$ and, if $\marching^k \leq \lambda
< \marching^{k+1}$, then the period of $\lambda$ is a
divisor of $k+1$.
Marching groups are the \emph{only} fixed points for $\map$. \end{theorem}
A remarkable fact about open mancala is that every configuration becomes an augmented marching group, and hence periodic, after a finite number of moves. The paper is devoted to find an optimal bound on such a number of moves which depends only on the mass of the initial configuration. \begin{definition}[Depth and diameter]\label{def:depthdiameter}
The \emph{depth} of a configuration $\lambda$ is its distance
from the periodic configuration, {\em i.e.} the number of
moves needed to reach periodicity.
For $n\geq 0$ we call the \emph{depth of $\configurations_n$}, denoted by $\depth(n)$,
the maximal depth of all configurations $\lambda$ with
$|\lambda|=n$.
The \emph{diameter} of a configuration is the number of moves
before the first repetition.
It is equal to its depth plus the length of the period minus one.
The maximal diameter of all configurations of mass $n$ is called the
\emph{diameter of $\configurations_n$}. \end{definition}
Clearly if $n = T_k$ the diameter of $\configurations_n$ equals its depth $\depth(n)$, whereas if $n = T_k + r$, $1 \leq r \leq k$ the diameter of $\configurations_n$ is bounded above by $\depth(n) + k$. \footnote{It is also strictly larger than $\depth(n)$, more precisely a lower bound is given by $\depth(n) + m - 1$ where $m$ is the smallest divisor of $k+1$ larger than $1$. }
The number $\depth(n)$ can be seen also as the depth of a graph. Indeed, for a given $n \geq 1$ we can construct the directed graph $\graph_n$ having the configurations in $\configurations_n$ as nodes and an arc from $\lambda$ to $\mu$ whenever $\mu = \map(\lambda)$. The graph $\graph_n$ contains exactly $2^{n-1}$ nodes. A \emph{cycle} of $\graph_n$ correspond to sequences of moves that repeat periodically. A configuration is periodic if it belongs to a cycle.
In the case $n\geq 2$, if we remove all arcs connecting two periodic configurations (arcs that belong to a cycle), the remaining graph $\graph_n^o$ is a disjoint union of trees rooted at a periodic configuration, and in each tree all arcs point toward the root. In this setting, $\depth(n)$ represents the depth of $\graph_n$, {\em i.e.} the maximal depth of the trees of $\graph_n^o$.
\subsection{Energy levels interpretation}
We can view the mancala configurations and mancala moves in a way reminiscent of the energy levels for the electrons in an atom.
Let us consider the subset $L \subset {\mathbb N} \times {\mathbb N}$ given by
\[ L = \{ (i, j) \in {\mathbb N} \times {\mathbb N} : 1 \leq i \leq j \} . \]
It is convenient to think of the integers $i$ and $j$ as a numbering of square cells instead of coordinates of points, in a way similar to the familiar sea-battle game. Each square cell in $L$ can either be empty or contain a single seed.
The \emph{column} index $i$ corresponds to one of the holes in the mancala game, so that $\lambda_i$, where $\lambda \in \configurations$, is the total number of occupied cells in column $i$ of $L$. We shall also associate an \emph{energy} to a seed positioned in cell $(i,j)$, given by its second coordinate $j$. The $k$-th level of $L$ is the set $\{(1,k), \dots, (k,k)\}$ of cells having energy $k$. Finally we let the seeds free to immediately \emph{fall} down without changing their column, but decreasing their energy (the second coordinate of their current position) provided the new position (and all those in between) are free.
The mancala configuration $\lambda \in \configurations$ then corresponds to a positioning of seeds in $L$ such that $(i,j)$ is occupied by a seed if and only if $1 \leq j \leq \lambda_i$ (see Figure \ref{fig:energyexample}).
\begin{figure}\label{fig:energyexample}
\end{figure}
A \emph{mancala move} can now be viewed in this setting as follows. Firstly, each seed is moved one position to the left without changing its energy level, while seeds in column $1$ are \emph{rotated} to the last column of that energy level. Namely, a seed at position $(i,j)$ is moved in the new position $(i-1, j)$ if $i > 1$, $(j,j)$ if $i = 1$. This is just a left rotation of the seeds of each level, so that cells will receive at most one seed. Secondly we let gravity pack all the seeds in each column at the lowest energy possible.
We notice that if all energy levels up to $k$ are completely filled, then a mancala move does not change those levels (up to a rearrangement of the seeds), and all the action takes place from the lowest level that is not completely filled, say $k+1$, up to the highest level that is not completely empty. The \emph{lowest active level} is the lowest level that is not completely filled.
The overall energy of a configuration (the sum of the individual energy of every seed) does not change after the first part of the move (the rotation within each energy level) whereas it can decrease during the second part (vertical downward shift).
\begin{definition}[gap]\label{def:gap} We shall call \emph{gap} an empty cell at the lowest energy level, say $k$, that is not completely filled. We shall think of a gap as an \emph{absent seed}, and in this respect we can \emph{follow} a gap through mancala moves as it rotates left along its energy level. \end{definition} A gap can be filled when it receives a seed that falls from above after a move, and this can happen only when the gap is in column $k$.
\begin{remark}
The filling of a gap at the lowest energy level that is not
completely filled is permanent, since there is no possibility of
further energy decrease for seeds at that energy level. All gaps
will be eventually filled, provided there are sufficiently many
seeds at energy levels larger than $k$. \end{remark}
\section{Lower bound}\label{sec:lower}
In order to prove a good lower bound $\ldepth(n)$ for the depth $\depth(n)$ we need to find appropriate configurations and to compute the number of moves required to reach a periodic configuration (which is an augmented marching group).
\subsection{The biaugmented marching group}
Let $n = T_k + r$ with $2\leq r \leq k+1$. A \emph{$q$-biaugmented
marching group} is a mancala configuration $\lambda \in \configurations_n$ of the form $\lambda = \marching^k + \varepsilon$ where $\marching^k$ is the marching group of order $k$ and $\varepsilon = (\varepsilon_i)_i$ is an increment of the form \[ \begin{cases} \varepsilon_1 = 2\\ \varepsilon_i = 1 & \text{for $2 \leq i \leq q+1$}\\ \varepsilon_{q+2} = 0\\ \varepsilon_i \in \{0,1\} & \text{for $q+3 \leq i \leq k+1$} \end{cases} \] for some $0 \leq q < k$. The second (resp. fourth) row in the definition is void if $q = 0$ (resp. if $q = k-1$). Clearly such a configuration is not an augmented marching group. Suitable choices of $q$ and $\varepsilon$ (compatible with the definition above) allow to obtain all the values of $r$ such that $2 \leq r \leq k + 1$.
\begin{proposition} \label{prop:biaugmented1} A $q$-biaugmented marching group has depth $q(k+2) + 2$, {\em i.e.}\ it becomes an augmented marching group exactly after $q(k+2) + 2$ moves. \end{proposition}
\begin{proof}
If $q = 0$, a direct check shows that after two moves we obtain an
augmented marching group. If $q > 0$, let us \emph{color} the
seed\footnote{\label{fn:bouchet}The idea of the colored seed is due
to Bouchet~\cite[Section 3]{Bou2}} in the first hole which is at
the energy level $k+2$ (the highest energy level); we agree that the
colored seed is the last to be sown in the mancala move, so that it
moves at the energy level $k+2$ until it falls on the level $k+1$,
when the configuration becomes an augmented marching group. Notice
that the lowest active level is $k+1$.
The colored seed is again at the leftmost position after exactly $k+2$ moves, while $k+1$ is the period of the seeds moving in the lowest active level. Hence the first ``hole'' at level $k+1$, which were in position $q+2$ at the beginning, gets one step closer to the colored seed every $k+2$ moves. At the end, the configuration becomes a $0$-biaugmented marching group in $q(k+2)$ moves, and in two further moves it becomes periodic. \end{proof}
As an example, Figure \ref{fig:energyexample} shows the energy-level interpretation of a $2$-biaugmented marching group with $k=3$ and its evolution. The gray circle in the pictures is the colored seed of the proof.
\subsection{The biaugmented marching group of the second kind}
Let $n = T_k + r = T_{k-1} + k + r$ with $0 \leq r \leq k-2$. A \emph{$q$-biaugmented marching group of the second kind} is a configuration $\lambda \in \configurations_n$ of the form $\lambda = \marching^{k-1} + \varepsilon$, where $\marching^{k-1}$ is the marching group of order $k-1$ and $\varepsilon = (\varepsilon_i)_i$ is an increment of the form \[ \begin{cases} \varepsilon_i \in \{1,2\} & \text{for $1 \leq i < k-q-1$}\\ \varepsilon_{k-q-1} = 2\\ \varepsilon_i = 1 & \text{for $k-q \leq i < k$}\\ \varepsilon_{k} = 0 \end{cases} \] for some $0 \leq q < k-1$. Again, such a configuration is not an augmented marching group.
\begin{proposition} The depth of a $q$-biaugmented marching group of the second kind is given by $t=(q+1)k$, {\em i.e.}\ it becomes an augmented marching group precisely after $(q+1)k$ moves. \end{proposition}
\begin{proof}
Let us color red the seeds which are at the beginning in the energy
level $k+1$, and blue the ones in the energy level $k$, which is the
lowest active level. As in the proof of the previous proposition, we
agree that the colored seeds are the last to be sown in every
mancala move, the red seed (if present) being sown after the blue
one. In that way, the blue seeds remain forever at the level $k$,
while one of the red seeds will eventually lose one energy level,
when the configuration becomes an augmented marching group.
We track the evolution of the colored seeds, which move in the two
highest energy levels. The pattern of red seeds rotates with a
period $k+1$, while the pattern of blue seeds rotate with a period
of $k$; hence the pattern of red seeds slowly slides one position to
the right with respect to the pattern of blue seeds every $k$ moves,
and the value of $q$ decreases by one. When $q = 0$, it is easy to
see that after $k$ moves a red seed reaches the lower level. Hence
the initial configuration takes exactly $(q+1)k$ moves in order to reach
periodicity. \end{proof}
\begin{figure}\label{fig:biaug2}
\end{figure}
\subsection{Evolution of the Heaviside configuration}
\begin{definition}[Heaviside configuration] We define the \emph{Heaviside configuration} $\heaviside^n \in \configurations_n$ by \[ \heaviside^n_i = \begin{cases} 1 \qquad & \text{if $1 \leq i \leq n$}, \\ 0 \qquad & \text{if $i > n$}. \end{cases} \] \end{definition} We shall also study the special configuration $\heaviside^\infty$ with a single seed in all holes $i \geq 1$: although $\heaviside^\infty$ is not in $\configurations$ since it has unbounded support, the sowing $\map$ is still well-defined.
\begin{proposition}\label{theorem:ones} Let $k\geq 1$ and $t = T_k + r$ with $0 \leq r \leq k$. Then the configuration after $t$ moves starting from $\heaviside^\infty$ is given by \begin{equation}\label{eq:heaviside} \lambda_i^{(t)} = \begin{cases} k - i + 2 & \text{if } 1 \leq i \leq k - r + 1 \\ k - i + 3 & \text{if } k - r + 1 < i \leq k + 1 \\ 1 & \text{if } i > k + 1. \end{cases} \end{equation} \end{proposition}
\begin{proof}
It is easy to check that exactly at every move $T_{j-1}$ the
configuration becomes greater than the marching group $\marching^j$,
hence we have the result for $r=0$. In particular, for any $0\leq r\leq
k$ the leftmost element of the
sequence at time $t=T_k+r$ is given by $\lambda_1^{(t)} = k +
1$. Hence in the subsequent move we have to add one seed in the holes with
indices $i = 2, \dots, k+2$ and then shift all piles of one position
to the left (dropping the leftmost). If $r < k$ we get $$ \lambda_i^{(t+1)} = \lambda_{i+1}^{(t)} + 1 \qquad \forall i = 1, \dots, k + 1 $$ and the result follows by induction. If $r = k$, then $t+1 = T_k + k + 1 = T_{k+1}$ and we have to add one seed to the piles with indices $i = 2, \dots, k+2$, which contain the values $k+1, k, k-1, \dots$, to begin with. Again the result follows immediately. \end{proof}
\begin{remark}\label{rem:truncatedheaviside} Notice that at time $\bar t = T_{k-1} + r$, $0 \leq r \leq k$ we moved at most $k$ seeds ($\lambda_1^{(\bar t - 1)} \leq k)$, so that we do not touch the piles at position $i > k$, which came from $\bar t$ shifts to the left. This means that Proposition \ref{theorem:ones} holds for a starting configuration
$\heaviside^n$ with $n = T_k + r$ (we denote it by \verb|1^n|) until time $\bar t$, with truncation to zero for elements with index $i > k$. Moreover, the number of seeds in the leftmost pile (index $i=1$) is exactly $k$ at times between $t=T_{k-1}$ and $t=T_k - 1$ and is $k+1$ at time $t=T_k$ if $r > 0$. \end{remark}
\begin{remark}\label{rem:truncatedheaviside2} In the special case $n = T_k + 1$ ($r = 1$) we can exactly describe the resulting configuration at time $t = T_k$. This is done by observing that the number of seeds in the first pile (Remark \ref{rem:truncatedheaviside}) is the same as for the full $\heaviside^\infty$ sequence up to time $T_k$, meaning that the sowing process is exactly the same. We can thus recover the resulting sequence at time $T_k$ by subtracting from \eqref{eq:heaviside} the missing seeds in their expected position ($2, 3, \dots$) obtaining \[ \lambda_i^{(T_k)} = \begin{cases} k + 1 & \text{if } i = 1 \\ k - i + 1 & \text{if } 1 < i \leq k + 1 \\ 0 & \text{if } i > k + 1 \end{cases} \] or equivalently $\lambda^{(T_k)} = \marching^k + \delta^1$ where $\delta^1$ is the sequence with one in position 1 and zero elsewhere. \end{remark}
\subsection{The augmented Heaviside sequences}\label{sec:augheaviside}
Now we want to analyze the evolution of the augmented sequence obtained from $\heaviside^n$ by adding a seed at the position with index $m$. We can write the sequence as \[ \heaviside^n + \delta^m \] where $\delta^m$ denotes the sequence with a $1$ at the position $m$ and zero elsewhere. The analysis can be conveniently done by using again Bouchet's trick, see Footnote~\ref{fn:bouchet}. In this way the evolution of the augmented configuration coincides exactly with the evolution of the non-augmented configuration with the addition of the colored seed, to be positioned appropriately.
At each sowing the colored seed moves one position to the left; when it reaches the leftmost pile (index $i=1$), at the next move its new position depends on how large the pile is. We however know precisely how the Heaviside sequence evolves, so that we can explicitly predict the position of the colored seed at each time.
We are particularly interested in the case where $n = T_k + 1$ and $m = T_p +1$ with $1 \leq p \leq k$.
\begin{lemma}\label{teo:augmentedheaviside} The \emph{colored} seed in the evolution of the augmented heaviside sequence $\heaviside^n + \delta^m$ with $n = T_k + 1$ and $m = T_p + 1$, $1 \leq p \leq k$ is located at position $i = k - p + 1$ at time $t = T_k$.
\end{lemma}
\begin{proof} After $T_p$ moves the colored seed reaches the leftmost pile which, according to Proposition \ref{theorem:ones}, will contain also $p+1$ noncolored seeds. This is true even in the special case $p = k$, where the colored seed is added at the rightmost nonempty pile of the sequence $\heaviside^n$, see Remark \ref{rem:truncatedheaviside}. This covers the case $p = k$. If $p < k$, at time $T_p + 1$ the colored seed will go into the hole with index $p+2$ and will be again into the leftmost pile at time $t = T_p + p + 2 = T_{p+1} + 1$. If $p+1 < k$ we can again resort to Proposition \ref{theorem:ones} and conclude that the leftmost pile will contain the colored seed and $p+2$ noncolored seeds at that time. This argument can be repeated for $q$ cycles as long as $p + q < k$ with the leftmost pile containing $p + q + 1$ noncolored seeds at time $t = T_{p+q} + q$. By taking $q = k - p - 1$, the largest admissible value, we have the colored seed in the leftmost pile at time $t = T_{k-1} + k - p - 1$ together with $k$ noncolored seed. At the next time $t = T_{k-1} + k - p$ the colored seed will be sown in position $k+1$ and will move left of one position in the next $p$ moves so that at time $t = T_{k-1} + k = T_k$ it will be in the hole at position $k - p + 1$, which concludes the proof. \end{proof}
The method of coloring seeds can be also used with more than one seed, provided that we do not place two or more colored seeds in the same hole and as long as the colored seeds do not interact (staying all in different piles). Thanks to Lemma \ref{teo:augmentedheaviside} we indeed have noninteracting colored seeds for $T_k$ moves if we place them in positions with indices of the form $T_p + 1$ for a set of values $1 \leq p \leq k$: if two of them would land in the same pile during the evolution, then the expected position given by Lemma \ref{teo:augmentedheaviside} would be the same, which is not the case. We can thus exactly predict their position at time $t = T_k$, there will be a colored seed in position $s$, if and only if there was a colored seed in position $T_{k-s+1} + 1$. If this is true for $s=1,\dots,q+1$ (with the addition of $q + 1$ colored seeds), recalling Remark \ref{rem:truncatedheaviside2} we obtain a $q$-biaugmented marching group. By Proposition~\ref{prop:biaugmented1}, the biaugmented marching group will become periodic after exactly $q(k+2) + 2$ further moves.
We want to construct, for a given $n = T_k + r$, a configuration that takes as long as possible to become periodic. In view of the previous discussion to achieve this goal it is mostly convenient to place the colored $r$ seeds in the rightmost positions with index of the form $T_p + 1$.
\begin{definition} We call \emph{augmented Heaviside sequence} the configuration of mass $n = T_k + r$, $1 \leq r \leq k+1$, and length $\ell = T_k + 1$, which is obtained from $\heaviside^{\ell}$ by adding a seed in the rightmost $r-1$ positions having index of the form $T_p + 1$ (in particular there will be no added seeds if $r=1$). \end{definition} As an example, the augmented Heaviside sequence with mass $n=19=T_5+4$ is given by \begin{center} {\tt 1 1 1 1 1 1 2 1 1 1 2 1 1 1 1 2}. \end{center}
The preceding arguments give a proof of the following
\begin{proposition}\label{teo:lower1} Let $n = T_k + r$ with $2\leq r \leq k+1$. Then the corresponding augmented Heaviside sequence becomes an $(r-2)$-biaugmented marching group after $T_k$ moves. In particular, its evolution becomes periodic after exactly $\ldepth_1(n)$ moves with
\begin{equation}\label{eq:lower1} \ldepth_1(n) := T_k + (r-2)(k+2) + 2 . \end{equation} \end{proposition}
The choice $r = k + 1$ is a limit case corresponding to $n = T_{k+1}$, and we have $\ldepth_1(T_{k+1}) = T_k + (k-1)(k+2) + 2 = 3 T_k$. Moreover if we write $n = T_{k+1} - s$ with $0 \leq s < k$, then $s=k+1-r$ and $\ldepth_1(n)$ in \eqref{eq:lower1} can be equivalently written as $\ldepth_1(n) = 3 T_k - s(k+2)$.
Direct inspection allows to show that for $r < k+1$ the particular biaugmented marching group at which we arrive at time $T_k$ evolves into an augmented marching group having period of length exactly $k+1$ (and not a proper divisor). Hence the diameter of the evolution is $\ldepth_1(n) + k$.
\subsection{Truncated augmented Heaviside sequences}
We now consider the augmented Heaviside sequence corresponding to $n = T_k + k + 1 = T_{k+1}$: it has length $k+1$, contains an increment at all the positions of the form $T_p + 1$, $1 \leq p \leq k$ and the total number $n$ of seeds is itself a triangular number. In the rightmost positions, this sequence has a $2$ followed by $k-1$ ones and a final $2$. Given $0\leq r\leq k-1$, let us remove from this sequence the rightmost $k-r$ piles, obtaining a configuration with mass $T_k+r$. Equivalently, we can obtain the same configuration starting from the augmented Heaviside sequence corresponding to $n = T_{k-1} + k = T_k$ and joining to the right a sequence of $r$ piles with one seed each.
\begin{definition} Let us call \emph{truncated augmented Heaviside sequence} such a configuration. \end{definition} As an example, the truncated augmented Heaviside sequence with mass $n=17=T_5+2$ is given by \begin{center} {\tt 1 2 1 2 1 1 2 1 1 1 2 1 1}. \end{center}
\begin{proposition} \label{prop:truncated} The truncated augmented Heaviside sequence with $n = T_k + r$, $0 \leq r \leq k-2$, becomes a $(k-r-2)$-biaugmented marching group of the second kind after $T_{k-1}$ moves. \end{proposition}
\begin{proof} Simply color the rightmost sequence of ones and track their position during the first $T_{k-1}$ moves of the augmented Heaviside sequence of Proposition \ref{teo:lower1} with $k-1$ in place of $k$ and $r = k$. \end{proof}
\begin{corollary}\label{teo:lower2} A truncated augmented Heaviside sequence with $n = T_k + r$, $0 \leq r \leq k-2$, becomes periodic after exactly
\begin{equation}\label{eq:lower2} \ldepth_2(n): = T_k + k(k-r-2) = 3 T_{k-1} - rk \end{equation}
moves. \end{corollary}
If $n = T_k + k$ or $n = T_k + k -1$, the function $\ldepth_2(n)$ is not defined by the previous Corollary, and we shall conventionally set it to $-1$.
As for the augmented Heaviside sequences of Section \ref{sec:augheaviside}, the special biaugmented marching group of the second kind at which we arrive if $r > 0$ evolves to an augmented marching group with period exactly $k+1$, and again we have a diameter given by $\ldepth_2(n) + k$.
\subsection{The special ``plateau'' sequence}
In the special case $n = T_{2r} + r$, $r \geq 1$, we need a further type of configuration.
\begin{definition} For $n = T_{2r} + r$, $r \geq 1$, we define the \emph{biaugmented Heaviside sequence} in the following way: take the truncated augmented Heaviside sequence for $n-1 = T_{2r} + r - 1$, and then add a third seed in the pile with two seeds at position $i = T_{2r-1} + 1$. \end{definition} As an example, $n = 24 = T_6 + 3$ produces the sequence \begin{center} {\tt 1 2 1 2 1 1 2 1 1 1 2 1 1 1 1 3 1 1}. \end{center}
\begin{proposition}\label{teo:lower3} The biaugmented Heaviside sequence with $n = T_{2r}+r$ becomes periodic after exactly
\begin{equation}\label{eq:lower3} \ldepth_3(n) := T_{2r} + 2r(r-1) \end{equation} moves. \end{proposition}
\begin{proof} Let us color the extra added third seed in the pile at position $i = T_{2r-1} + 1$. By construction, without the colored seed the biaugmented Heaviside sequence becomes a truncated augmented Heaviside sequence with mass $T_{2r}+r-1$, hence by Proposition~\ref{prop:truncated}, after $T_{2r-1}$ moves we have an $(r-1)$-biaugmented marching group of the second kind with an extra seed (corresponding to an increment of $+3$ with respect to the marching group) in the leftmost pile. The argument we used to study the biaugmented marching groups can be adapted to the present case, proving that the periodic configuration is reached exactly after the same number of moves as for the biaugmented marching group of the second kind: $(q+1)k$ with $q = r-1$ and $k=2r$. Hence the initial configuration becomes periodic after \[ T_{2r-1}+2r^2=T_{2r-1}+2r+2r(r-1)=T_{2r}+2r(r-1).\qedhere \] \end{proof}
Also in this case a direct inspection of the resulting augmented marching group allows to show that its period is exactly $k+1$, and again the diameter of the special plateau sequence is $\ldepth_3(n) + k$.
The lower bound $\ldepth_3(n)$ is only defined for special values of $n$, of the form $n = T_{2r} + r = 4 T_r$, and we conventionally set it to $-1$ in all the other cases.
We can now gather the three lower bounds and define
\begin{equation} \ldepth(n) := \max \{ \ldepth_1(n), \ldepth_2(n), \ldepth_3(n) \} \end{equation}
so that the previous results can be summarized as the following: \begin{theorem}[Lower bound] \begin{equation} \depth(n) \geq \ldepth(n) . \end{equation} \end{theorem}
\begin{proof} Simply gather the results obtained in Propositions \ref{teo:lower1}, \ref{teo:lower3} and Corollary \ref{teo:lower2}. \end{proof}
A plot of the function $\ldepth(n)$ for $n\leq 45$ can be seen in Figure~\ref{fig:ldepth}.
\begin{figure}
\caption{The values of the function $\ldepth(n)$ for $n\leq 45$. We have $\ldepth(45)=108$.}
\label{fig:ldepth}
\end{figure}
\begin{remark} There is an explicit way to express the lower bound $\ldepth(n)$. Indeed, writing $n\geq 1$ in the form $n=T_k+r$ with $k\geq 0$ and $1\leq r\leq k+1$, one can prove that \[ \ldepth(n)= \begin{cases} \ldepth_2(n)= T_k + k(k-r-2) & \text{if $r < k/2$}\\ \ldepth_3(n)=T_{2r} + 2r(r-1) & \text{if $r = k/2$}\\ \ldepth_1(n)= T_k + (r-2)(k+2) + 2 & \text{if $r > k/2$}. \end{cases} \] Notice that the middle case occurs only when $k$ is even. \end{remark}
\section{\texorpdfstring{$s$-monotonicity}{s-monotonicity}} \label{sec:smon} In this section we introduce one of the main notions of the paper, namely \emph{$s$-monotonicity}. Firstly we define a sequence which is strictly related to the energy levels interpretation of the mancala game.
\begin{definition}[Energy sequence]\label{def:energyseq} Given a mancala configuration $\lambda \in \configurations$ we define the \emph{energy sequence} $e = \energy(\lambda) : {\mathbb Z} \to {\mathbb N}$ as follows: setting $\period = \period(\lambda) = \max\{\lambda_1, \length(\lambda)\}$, we define \begin{equation} e_i = \lambda_i + i -1 \qquad i = 1,\dots ,\period \end{equation}
and extend it on ${\mathbb Z}$ as a periodic function of period $\period$. \end{definition} Notice that if $\lambda_1 > \length(\lambda)$ we actually use values of $\lambda$ outside its support, so that we have $e_i = i - 1$ for $\length(\lambda) < i \leq \lambda_1$.
\begin{remark} We have that $\lambda \geq \marching^k$ if and only if $\energy(\lambda)_i \geq k$ for all $i \in {\mathbb Z}$. \end{remark}
\begin{definition}[$s$-monotonicity]\label{def:smon} Given $s \geq 1$ and $\lambda \in \configurations$, we say that $\lambda$ is \emph{$s$-monotone} if: \begin{itemize} \item[(i)] $\lambda_1 \leq \length(\lambda) + 1$, and \item[(ii)] its energy sequence $e = \energy(\lambda)$ satisfies the inequality
\begin{equation} e_j \leq e_i + 1 \qquad \forall~ i < j \leq i + s \end{equation} \end{itemize} (notice that the first requirement $\lambda_1 \leq \length(\lambda) + 1$ is redundant if $s\geq 2$).
It is convenient to introduce also the (void) notion of $0$-monotonicity, satisfied by all mancala configurations. \end{definition}
\begin{definition}[Increasing plateau]\label{def:increasingplateau} Let $s\geq 0$. An \emph{increasing plateau of length $s$} is a subset of $(s+2)$ consecutive values $e_i,e_{i+1},\dots,e_{i+s+1}$ of an energy sequence such that \begin{align*} & e_{i}<e_{i+1}=\dots=e_{i+s}<e_{i+s+1} && \text{if $s\geq
1$}\\ & e_{i} \leq e_{i+1}-2 && \text{if $s=0$}. \end{align*} \end{definition} It is readily proved that a configuration is $s$-monotone if and only if each increasing plateau has length at least $s$.
Let us give a few examples of $s$-monotone configurations: \begin{itemize} \item {\tt 1 2}\quad is $0$-monotone but not $1$-monotone \item {\tt 1 1 1}\quad is $1$-monotone but not $2$-monotone \item {\tt 4 1 1}\quad and\quad {\tt 2 2 1 1}\quad are $2$-monotone but not $3$-monotone \item {\tt 5 2 2 1}\quad and\quad {\tt 3 3 2 1 1}\quad are $3$-monotone but not $4$-monotone \item {\tt 6 3 3 2 1}\quad and\quad {\tt 4 4 3 2 1 1}\quad are $4$-monotone but not $5$-monotone. \end{itemize}
\begin{remark} If a mancala configuration is $s_2$-monotone, then it is also $s_1$-monotone for any $s_1 < s_2$. Moreover, a configuration is an augmented marching group if and only if it is $s$-monotone for all $s \in {\mathbb N}$. \end{remark}
\begin{remark}\label{rem:weakstrong} The notions of monotonicity, in the sense of Definition~\ref{defn:monotone}, and of $1$-monotonicity, in the sense of the previous definition, are slightly different. For instance, the sequence {\tt 4 2} is monotone but not $1$-monotone. However, if a configuration $\lambda$ is monotone, then $\map(\lambda)$ is $1$-monotone. Hence, in view also of next theorem, during the evolution of a configuration there is at most one time when it is monotone but not $1$-monotone.
In passing, we observe that any $1$-monotone sequence has a predecessor that is merely monotone, {\em i.e.}\ a $1$-monotone configuration can be obtained by applying a move to some monotone configurations. Hence the maximal depth/diameter in the family of monotone configurations is exactly one plus the maximal depth/diameter in the family of $1$-monotone configurations. \end{remark}
The following theorem shows that the class of $s$-monotone configurations is closed under mancala moves.
\begin{theorem}\label{teo:smonclosed} If $\lambda$ is $s$-monotone, then $\map(\lambda)$ is $s$-monotone. \end{theorem}
\begin{proof} Let us set $\period = \max\{\lambda_1, \length(\lambda)\}$, $\mu = \map(\lambda)$, $e = \energy(\lambda)$ and $f = \energy(\mu)$.
Clearly any mancala configuration is at least $0$-monotone, hence the set of $0$-monotone configurations is closed under mancala moves. A direct check shows that the property $\lambda_1 \leq \length(\lambda) + 1$ is preserved under a mancala move whenever $\lambda$ is a monotone configuration, indeed we have $\lambda_2 \leq \lambda_1 \leq \length(\mu)$, whence $\mu_1 = \lambda_2 + 1 \leq \length(\mu) + 1$. Consequently the set of $1$-monotone configurations is closed under mancala moves.
Now assume that $s > 1$, hence we have to check only the values of the energy sequence. We divide the proof in five cases.
\textbf{Case $\lambda_1 \geq \length(\lambda)$.} We have $\period = \length(\mu) = \lambda_1$ and the new period is $\period(\mu) = \max\{\mu_1, \length(\mu)\} = \max\{\lambda_2+1, \lambda_1\}$. Since $s > 1$, then $\lambda$ is monotone and $\lambda_2 \leq \lambda_1$, hence we have $\period \leq \period(\mu) \leq \period + 1$ and the new energy sequence $f$ is obtained by a left-shift of $e$ possibly followed by the insertion of a new value. Indeed $f_i = \mu_i + i - 1 = \lambda_{i+1} + 1 + i - 1 = e_{i+1}$ for $i = 1, \dots, \period-1$ and $f_\period = \mu_\period + \period - 1 = \period$, since $\mu_\period = 1$. If $\lambda_2 < \lambda_1$, then $\period(\mu)=m$ and the new energy sequence $f$ is exactly a left-shift of $e$, so that $\mu$ is itself $s$-monotone. If $\lambda_2 = \lambda_1$, then $\period(\mu)=m+1$ and the new value to be inserted is $f_{\period+1} = \mu_{\period+1} + \period + 1 - 1 = \period$ (being $\mu_{m+1}=0$) and it is equal to the previous value $f_\period$. Since inserting a value in the sequence cannot destroy the $s$-monotonicity property if it is equal to the previous (or the next) value, we can conclude that $\mu$ is $s$-monotone.
\textbf{Case $\lambda_1 \leq \length(\lambda) - 3$.} We have $\period = \length(\lambda)$ and $\length(\mu) = \length(\lambda) - 1$. Since $\lambda_2 \leq \lambda_1$ we have $\mu_1 \leq \lambda_1 + 1 \leq \length(\lambda) - 2 = \length(\mu) - 1$, so that $f_1 \leq \period - 2$ and $\period(\mu) = \period - 1$. Moreover, $\lambda_1 < \period - 1$ implies that $\mu_{\period-1} = \lambda_\period$, so that $f_{\period-1} = \mu_{\period-1} + \period - 2 = \lambda_\period + \period - 2 \geq \period - 1$. Using the inequalities above we conclude
\begin{equation}\label{eq:invloc1} f_1 < f_{\period-1} = f_0. \end{equation}
The energy sequence $f$ is obtained from the energy sequence $e$ with the following operations: \begin{enumerate} \item translate the values of $e$ to the left of one position $e'_i = e_{i+1}$. This has no impact on the $s$-monotonicity; \item decrease by one the values with indices in the range $\lambda_1 + 1, \dots, \period$; \item remove the value at index $\period$ and extend to ${\mathbb Z}$ with period $\period-1$. \end{enumerate} The last two operations can only impact $s$-monotonicity in the case of indices $i \leq 0 < j$ with $j - i \leq s$ (up to addition of multiples of $\period-1$). However using \eqref{eq:invloc1} we have $f_j - f_i < (f_j - f_1) + (f_0 - f_i) \leq 1 + 1$.
\textbf{Case $\lambda_1 = \length(\lambda) - 2$.} The argument is similar to the previous, but \eqref{eq:invloc1} now becomes
\begin{equation}\label{eq:invloc2} f_1 \leq f_{\period-1} = f_0 . \end{equation}
Moreover the set of indices (within the period $\{1,\dots,\period\}$) where we decrease the energy by one (step 2 above) reduces to $\{ \period-1, \period\}$ and is further reduced to the single index $\{ \period-1 \}$ after removal of the energy value with index $\period$ (step 3 above). Now suppose by contradiction that $f_j - f_i \geq 2$ with $i \leq 0 < j$, $j - i \leq s$. This is only possible if both $f_j - f_1 = 1$ and $f_0 - f_i = 1$. The latter equation implies $i < 0$ and, in view of the discussion above, it can be rewritten as $1 = f_{\period-1} - f_{\period-1+i} = e_\period - 1 - e_{\period+i}$ which contradicts the $s$-monotonicity of $\lambda$.
\textbf{Case $\lambda_1 = \length(\lambda) - 1$ and $\lambda_2 < \lambda_1$.} It turns out that $\period(\mu) = \length(\lambda) - 1 = \period - 1$. Moreover $e_0 = e_\period = \lambda_\period + \period - 1 \geq \period$ whereas $e_1 = \lambda_1 < \period$, so that $e_1 < e_0$. The resulting energy $f$ can be obtained by a left shift of $e$ followed by removal of element at index $\period$, which is the same as first removing the element at index $1$ of $e$ and then performing a left shift. The fact that $e_0 > e_1$ allows to ensure that the removal of the value $e_1$ does not impact the $s$-monotonicity, which is then unaffected by the final left shift.
\textbf{Case $\lambda_1 = \length(\lambda) - 1$ and $\lambda_2 = \lambda_1$.} It turns out that in this case
$\period(\mu) = \period(\lambda) = \period = \length(\lambda)$ and that the energy sequence of $\mu$ is just a left-shift of the energy sequence of $\lambda$: $f_i = e_{i+1}$.
These five cases cover all possible situations so that we conclude the proof. \end{proof}
For particular configurations it is possible to prove that repeated mancala moves actually increase the order of monotonicity.
The following is our first important result for which the distinction between monotonicity and $1$-monotonicity is crucial. For instance, the result is false for the sequence \texttt{4 2}, which is monotone but not $1$-monotone.
\begin{lemma}\label{teo:incrmon} Let $\lambda \in \configurations$, $s \in {\mathbb N}$ and $2 \leq q \in {\mathbb N}$ such that \begin{enumerate} \item[(i)] $\lambda$ is $s$-monotone; \item[(ii)] $\lambda \geq \marching^{q-1}$; \item[(iii)] $\length(\lambda) = q-1$, (i.e. $\lambda_q = 0$). \end{enumerate} Then $\map^q (\lambda)$ (the configuration after $q$ moves) is $(s+1)$-monotone. \end{lemma}
\begin{proof} Denote by $\mu$ the configuration obtained after $q$ moves. Since we apply a number of moves that is larger than the length of the sequence, it follows that $\mu$ is at least $1$-monotone (it is already monotone after $q-2$ moves, hence $1$-monotone after $q-1$ moves by Remark \ref{rem:weakstrong}). This covers the case $s=0$, and we can suppose $s \geq 1$. In particular we can assume that $q-1\leq \lambda_1\leq q$ where the first inequality follows from requirement (ii) and the second inequality comes from (iii) and constraint (i) of Definition \ref{def:smon}.
Using the energy level interpretation of the game, we want to track the position after $q$ moves of the seeds with energy larger than $q$. First observe that requirement (ii) implies that all energy levels up to the $(q-1)$-th are completely filled and that seeds at level $q$ (which is not completely filled) are subjected to $q$ rotations to the left and end up at the same initial position. A seed in column $i$ (notice that requirement (iii) implies that $i < q$) at level $q + j$ is also rotated $q$ times to the left, however the $(q+j)$-th energy level has length $q+j$, so that it will end up in column $i + j$ and can possibly decrease its energy (that is, its height) once or more than once. Requirements (i) and (iii) imply that there is no seed at height $q+1$ in the first $s$ columns, whereas seeds in column $s+1$ and height at least $q+1$ get moved to the right after the set of $q$ moves. This allows to conclude that $f_j - f_i \leq 1$ for all $i \leq 0 < j$, $j - i \leq s+1$, where $f = \energy(\mu)$ is the energy sequence of $\mu$ given by Definition \ref{def:energyseq}. If there were no energy decrease, then all increasing plateaus would increase their length by $1$ and the $(s+1)$-monotonicity would follow. The decrease in energy after each move involves all seeds in the final columns and it can be easily checked that it maintains the $(s+1)$-monotonicity. \end{proof}
\begin{definition}[$q$-canonical configurations]\label{def:qcanonical} Let $2 \leq q \in {\mathbb N}$, $n\geq T_q$ and $\lambda \in \configurations_n$. We say that $\lambda$ is $q$-canonical if \begin{enumerate} \item $\lambda \geq \marching^{q-1}$; \item $\length(\lambda) = q-1$, in particular $\lambda$ \textbf{is not} $\geq \marching^q$; \footnote{Using requirement (1), the energy level $q$ is the lowest that is not completely filled. } \item the gap (see Definition \ref{def:gap}) in the rightmost position (column $q$) in the energy level at height $q$ is the last to be filled after repeated moves. Notice that requirement $n \geq T_q$ implies that the $q$-th energy level will eventually fill up completely. \end{enumerate}
\end{definition}
Due to the central importance of $q$-canonical configurations a few remarks are in order.
\begin{remark} The value $q$ of a $q$-canonical configuration is necessarily the height of the lowest active energy level (the lowest level that is not completely filled). This follows from requirements (1) and (2) of the above definition. Hence a configuration cannot be $q$-canonical with two different values of $q$. \end{remark}
\begin{remark} In view of the comments following Definition \ref{def:gap}, in particular the fact that \emph{filling a gap} is permanent, we can indeed define a \emph{filling order} on the gaps at level $q$. All the gaps (at level $q$) will be eventually filled due to requirement $n \geq T_q$, and we can identify the gap that will be filled last. \end{remark}
\begin{remark} An augmented marching group cannot be $q$-canonical, since if $\lambda \in \configurations_n$ is an augmented marching group with $n \geq T_q$ then $\lambda \geq \marching^q$. This is incompatible with requirement (2) of Definition \ref{def:qcanonical}. \end{remark}
\begin{remark}\label{rem:sameq} Since the gap in the rightmost position of a $q$-canonical configuration $\lambda$ returns in the same position after every sequence of exactly $q$ moves, it follows that the $q$-th energy level fills up after exactly $rq$ moves, for some $r \in {\mathbb N}^*$. In particular $\map^{rq}(\lambda) \geq \marching^q$. Moreover $\map^{iq}(\lambda)$ is itself a $q$-canonical configuration for all $i = 0,\dots,r-1$ and there are no other $q$-canonical configurations in between. \end{remark}
As an example, let us consider the configuration $\lambda= \texttt{5 5 3 6 2}$, it satisfies requirements (1) and (2) of Definition \ref{def:qcanonical} with $q = 6$ (see Figure \ref{fig:ex55362}
for an energy-level interpretation), it has three gaps at the energy level $6$ with three seeds at higher level, enough to completely fill the sixth energy level ($|\lambda| = 21 = T_6$). However this configuration is not a $6$-canonical configuration because the rightmost gap in column $6$ is not the last filled. This can only be seen by letting the configuration evolve and annotating the filling order of the three gaps. In this example only seven mancala moves are sufficient to see that the gap in column $6$ is filled first, whereas the gap that is filled last is the one originally in column $3$. The configuration $\lambda = \texttt{9 5 3 2 2}$ (obtained from the previous one with three mancala moves) is indeed a $6$-canonical configuration. As noted in Remark \ref{rem:sameq}, after six mancala moves we have either another $6$-canonical configuration or the sixth level becomes completely filled. It turns out that we have a total of four other $6$-canonical configurations, one after each sequence of six moves starting from the $6$-canonical configuration \texttt{9 5 3 2 2}.
\begin{figure}\label{fig:ex55362}
\end{figure}
\begin{theorem}\label{teo:maxmoncanon} A $q$-canonical configuration cannot be $(q-1)$-monotone. In other words it is at most $(q-2)$-monotone. \end{theorem}
\begin{proof} Let $\lambda$ be a $q$-canonical configuration, $q \geq 2$; hence $\lambda_1\geq q-1$. Now suppose by contradiction that $\lambda$ is also $(q-1)$-monotone; this implies that $\lambda_1\leq q$ by (i) of Definition~\ref{def:smon}. Set $e = \energy(\lambda)$ (the energy sequence), with period $m(\lambda) = \lambda_1$, hence \[ q-1\leq m(\lambda)\leq q. \] Either $e_1 = \lambda_1 = q-1$ or $e_q = q-1$. Energy levels strictly larger than $q$ must also be present, otherwise $\lambda$ would be an augmented marching group, incompatible with $q$-canonicity. Hence there must exist an increasing plateau, and its length cannot be longer than $m(\lambda) - 2$ due to the periodicity of $e$, which is at most $q-2$. On the other hand, $(q-1)$-monotonicity implies that the length of every increasing plateau is at least $q - 1$, a contradiction. \end{proof}
\section{Upper bound}\label{sec:upper}
\subsection{Analyzing the evolution of a configuration}
In this section we shall fix $n \geq T_k$ for some $k \in {\mathbb N}$ and consider a mancala configuration $\lambda \in \configurations_n$. We let $\lambda$ evolve (applying mancala moves) until it reaches an augmented marching group; we denote by $\lambda(t) = \map^t(\lambda)$ the result of $t \in {\mathbb N}$ mancala moves, so that in particular $\lambda(0) = \lambda$. Let $\overline t$ be the smallest time $t$ for which $\lambda(t)$ is an augmented marching group, in particular $\lambda(\overline t) \geq \marching^k$. We also denote by $t^* \leq \overline t$ the smallest time $t$ such that $\lambda(t) \geq \marching^k$.
During the evolution $\lambda(t)$ will occasionally become a $q$-canonical configuration. Specifically we indicate with $t_1 < t_2 < \dots < t_s < t^*$ the times before $t^*$ when we have a $q$-canonical configuration for some $q \in {\mathbb N}$, $\lambda(t^*)$ can be itself $q$-canonical, however $\lambda(t^*) \geq \marching^k$ would imply $q > k$, hence $n \geq T_{k+1}$. In other words, if we restrict $T_k \leq n < T_{k+1}$, then $\lambda(t^*)$ cannot be $q$-canonical.
We call such times \textit{critical times}. Specifically we have a sequence $q_1 \leq q_2 \leq \dots \leq q_s$ such that $\lambda(t_i)$ is a $q_i$-canonical configuration.
Consider two consecutive critical times $t_i$ and $t_{i+1}$. Remark \ref{rem:sameq} applied to $\lambda(t_i)$ tells us that after $q_i$ moves, we have either another $q_i$-canonical configuration, with no other $q$-canonical configurations in between, in which case $t_{i+1} = t_i + q_i$ and $q_{i+1} = q_i$, or we have just filled the $q_i$-th energy level so that we have $t_{i+1} \geq t_i + q_i$ and $q_{i+1} > q_i$. In both cases we have $t_{i+1} - t_i \geq q_i$. We can now apply Lemma \ref{teo:incrmon} to $\lambda(t_i)$ with $q = q_i$ and conclude that the $s$-monotonicity of $\lambda(t_{i+1})$ increases strictly. Since $\lambda(t_1)$ is $0$-monotone, we obtain by induction that $\lambda(t_{i+1})$ is $i$-monotone and that $\lambda(t^*)$ is $s$-monotone.
It is possible for the set of critical times to be empty. In this case we shall see that the filling up of the energy levels is very fast.
\begin{lemma}\label{teo:stepcanon}
If $\lambda \geq \marching^{q-1}$ and $|\lambda| \geq T_q$, then there exists $0 \leq i < q$ such that either $\map^i(\lambda) \geq \marching^q$ or $\map^i(\lambda)$ is a $q$-canonical configuration. If $\lambda$ is itself $q$-canonical, the result holds true with
$i = q$ (besides the trivial value $i=0$). \end{lemma}
\begin{proof} If the $q$-th energy level is full, then there is nothing to prove, otherwise locate the gap that will be filled last in column $j$. If $j = q$ we already have a $q$-canonical sequence, otherwise we apply $j$ moves in order to rotate the gap position into column $q$, now we either have a $q$-canonical sequence or we filled the $q$-th energy level. The second part of the Lemma easily follows by first performing a single move and then reasoning as in the first part. \end{proof}
\begin{corollary}\label{teo:emptycrit} If the set of critical times is empty ($s = 0$), then one has $t^* \leq T_{k-1}$, i.e. \[ \lambda(T_{k-1})\geq \marching^k. \] \end{corollary}
\begin{proof} It can be readily seen by repeated applications of the previous Lemma. \end{proof}
Similarly, the $q_1$-canonical sequence $\lambda(t_1)$ is reached after at most $T_{q_1 - 1}$ moves, i.e. $t_1 \leq T_{q_1 - 1}$.
Now let \begin{gather*} I = \{i\in{\mathbb N}:\ 1\leq i\leq s-1,\ q_{i+1} > q_i \},\\ J = \{i\in{\mathbb N}:\ 1\leq i\leq s-1,\ q_{i+1} = q_i \} \end{gather*} (notice that $I\cup J=\{1,\dots,s-1\}$), and decompose
\[ t_s = t_1 + \sum_{i \in I} (t_{i+1} - t_i) + \sum_{j \in J} (t_{j+1} - t_j) . \] If $i \in I$ we first apply one move to $\lambda(t_i)$ and then invoke Lemma \ref{teo:stepcanon} $(q_{i+1} - q_i)$ times, concluding that
\[
t_{i+1} - t_i \leq 1 + T_{q_{i+1} - 1} - T_{q_i - 1} .
\]
On the other hand, if $j \in J$ then $t_{j+1} - t_j = q_j \leq q_s$ (Remark \ref{rem:sameq}). In the end we have the estimate
\begin{equation}\label{eq:tpbound}
t_s \leq T_{q_s - 1} + |I| + q_s |J| \leq T_{q_s - 1} + q_s (s-1) . \end{equation}
At each critical time we gain one level of monotonicity, so that $\lambda(t_s)$ is $(s-1)$-monotone. The maximal possible value of $q_s$ is $q_s = k$, which gives an upper bound $k-2$ for the $s$-monotonicity of $\lambda(t_s)$ (Theorem \ref{teo:maxmoncanon}) and hence we have $s \leq k-1$. We get \[ t_s \leq T_{q_s-1} + q_s (k-2). \]
To obtain the configuration $\lambda(t^*) \geq \marching^k$ we need to apply a move to $\lambda(t_s)$ and again invoke Lemma \ref{teo:stepcanon} (more than once if $q_s < k$) for a total of at most $1 + (k-1) + \dots + (q_s-1)$ moves. We notice that the last term in the previous sum is also present in the sum that defines $T_{q_s - 1}$, so that using \eqref{eq:tpbound} we arrive at
\[ t^* \leq T_{k-1} + 1 + (q_s-1) + q_s (s-1)
= T_{k-1} + s q_s \leq T_{k-1} + k s. \]
We summarize the previous analysis in the following lemma.
\begin{lemma}\label{teo:balance} Let $n \geq T_k$ and $\lambda \in \configurations_n$. Then there exists $0 \leq s \leq k-1$ such that after at most $t = T_{k-1} + k s$ moves the resulting configuration $\lambda(t)$ satisfies: \begin{enumerate} \item $\lambda(t) \geq \marching^k$; \item $\lambda(t)$ is $s$-monotone. \end{enumerate} \end{lemma}
If $n = T_k$ or $n = T_k + 1$ then the resulting configuration is an augmented marching group.
\begin{definition}\label{def:functionW} Given $0 < b \leq a$ two integers, we define $V(a,b)$ by \[ V(a,b) = \sum_{i\in{\mathbb N}^*: a-ib > 0} (a - i b) . \] The value of $V$ can be explicitly computed by first expressing $a = q b + p$ with $q = \left\lfloor \frac{a}{b} \right \rfloor$ and $0 \leq p < b$ (Euclidean division) and then using the formula for the sum of the first natural numbers. We end up with $V(a,b) = qb + \frac{b}{2}q(q-1)$. We have the special values \begin{itemize} \item $V(a,a) = 0$, $V(a,b) = a-b$ if $a \leq 2b$ (only one term in the above sum), \item $V(a,b) = 2a - 3b > a-b$ if $2b < a \leq 3b$ (two terms in the above sum), \item $V(a,b) > 2a - 3b$ if $a > 3b$ (more than two terms in the above sum). \end{itemize} Moreover, if $b < a$ then $V$ is strictly increasing with respect to $a$ and strictly decreasing with respect to $b$. We can then ``invert'' $V$ with respect to $b$ and define the function \[ W(a,v) = \max \{b \in {\mathbb N}^*: b \leq a,\ V(a,b) \geq v\} , \] which is (weakly) decreasing with respect to $v$ and (weakly) increasing with respect to $a$. We have the special values \begin{itemize} \item $W(a,v) = a - v$ if $1 \leq v \leq \frac{a}{2} + 1$, \item $W(a,v) = \left\lfloor
\frac{2a - v}{3}
\right\rfloor$, if $\frac{a}{2} \leq v \leq a + 3$. \end{itemize} \end{definition}
Referring to Figure \ref{fig:vmap} the value of $V(a,b)$ can be graphically interpreted as the number of unit squares completely included in the right triangle having base (long cathetus) of length $a$ and height (short cathetus) of length $\frac{a}{b}$.
\begin{figure}\label{fig:vmap}
\end{figure}
With the same graphical interpretation in mind function $W(a,v)$ can be viewed as the maximal value of $b$ such that the right triangle of sides $a$ and $\frac{a}{b}$ contains at least $v$ unit squares.
For $n\geq T_k$, let us consider a $k$-canonical configuration. By Lemma \ref{teo:stepcanon}, after a cycle of $k$ moves we have that the resulting configuration is either itself $k$-canonical or it has just become larger than $\marching^k$. Iterating this procedure, we want to relate the number of cycles needed to become larger than $\marching^k$ with the degree of monotonicity of the configuration.
\begin{lemma}\label{teo:cyclenumbound} Let $\lambda \in \Lambda_n$ with $n = T_k + r$, $r \geq 1$ and suppose that \begin{enumerate} \item $\lambda$ is a $k$-canonical configuration; \item $\lambda$ is $s$-monotone for some $s \in {\mathbb N}$. \end{enumerate} Let $\sigma \geq 1$ denote the number of cycles of $k$ moves that are required to become larger than $\marching^k$ and suppose that $s + \sigma \geq 2$. Then \[ r \leq V(k-1,s+\sigma-1) - 1. \] Equivalently, reasoning by contradiction, we have the upper bound \[ \sigma \leq \max\{W(k-1,r+1) + 1 - s, 0\} . \] \end{lemma}
\begin{proof} First we analyze the case $\sigma = 1$, so that $s \geq 1$. Since the configuration is $s$-monotone and $k$-canonical, in this case $\lambda_1 \leq k$, hence $e_1, \dots, e_s \leq k$, $e_{s+1}, \dots,$ $e_{2s} \leq k+1$ and so on, where $e$ is the energy sequence $e = \energy(\lambda)$. Then the total number of seeds with energy above $k$ cannot be larger than $V(k-1,s)$. This is obtained by observing that we have a sequence of $q$ plateaus of length $s$ and increasing height from $0$ to $q-1$ and a final top plateau of length $p$ and height $q$. This implies that $r \leq V(k-1,s) - 1$ where the ``$-1$'' is a consequence of the fact that the $k$-th energy level is not completely full. For a generic value of $\sigma$ we simply perform $\sigma - 1$ cycles of $k$ moves obtaining a configuration with level of monotonicity $s + \sigma - 1$ with still $\lambda_k = 0$ and we can resort to the special case discussed above. \end{proof}
\subsection{Optimal upper bound}
The following lemma will be fundamental in proving the optimal upper bound.
\begin{lemma}\label{teo:magico} Let $\lambda \in \configurations_n$ with $n = T_k + r$, $0 \leq r \leq k$, and denote by $\sigma_q$ the number of $q$-canonical configurations encountered during the evolution of $\lambda$, with $q = 2, \dots, k$.
Then the total number $s = \sum_q \sigma_q$ of canonical configurations encountered, and hence the least level of monotonicity of the first configuration that becomes larger than (or equal to) $\marching^k$, is bounded by \[ s \leq W(k-1,r+1) + 1 . \] In particular, for $r < \frac{k}{2}$ we have
\begin{equation}\label{eq:magico2} s \leq k - r - 1 \end{equation}
and for $\frac{k-3}{2} \leq r \leq k$ we have
\begin{equation}\label{eq:magico3} s \leq \left\lfloor
\frac{2k-r}{3}
\right\rfloor, \end{equation}
where we used the special values of function $W$ computed in Definition \ref{def:functionW}. \end{lemma}
\begin{proof} If $\sigma_q = 0$ for all $q$, then we can resort to Corollary \ref{teo:emptycrit} and conclude immediately. Otherwise, set $s_q = \sum_{i \leq q} \sigma_i$ and let $\bar q$ be the largest value of $q$ such that $\sigma_q > 0$. We apply Lemma \ref{teo:cyclenumbound} with $k=\bar q$, $r = n - T_{\bar q}$ and $s = \sum_{i < \bar q} \sigma_i = s_{\bar q-1}$ and obtain an upper bound for $\sigma_{\bar q}$ that, since $\sigma_{\bar q} > 0$ can be written as \[ \sigma_{\bar q} \leq W(\bar q-1,n-T_{\bar q} + 1) + 1 - s_{\bar q-1} \] or equivalently as \[ s = s_{\bar q} \leq W(\bar q-1,n-T_{\bar q} + 1) + 1 \leq W(k-1,r+1) + 1 \] where the last inequality follows from the monotonicity of $W$ with respect to its arguments. \end{proof}
The evolution of a configuration $\lambda \in \Lambda_n$, with $n = T_k + r$ and $1 \leq r \leq k$, will be decomposed into two stages: \begin{enumerate} \item the evolution from the beginning up to the first time $t$ when $\lambda(t) \geq \marching^k$, where as usual $\lambda(t) = \map^t(\lambda)$; \item the subsequent evolution until we reach an augmented marching group. \end{enumerate} The duration of each of these stages will be estimated from above with bounds that depend upon a few parameters and we shall find a uniform bound valid for all possible feasible choices of such parameters finally obtaining an upper bound for $\depth(n)$.
We need some further lemmas.
\begin{lemma}\label{teo:augmentedifrmonotone} Let $\lambda \in \Lambda_n$ and let $k$ be the largest integer such that $\lambda \geq \marching^k$. Write $n = T_k + r$ and suppose that $\lambda$ is $r$-monotone. Then $\lambda$ is an augmented marching group. \end{lemma}
\begin{proof} By contradiction suppose that $\lambda$ is not an augmented marching group, then the energy sequence has an element with value $k$ (otherwise $\lambda \geq \marching^{k+1}$) and an element with value $k+2$ (otherwise $\lambda$ would be an augmented marching group). This implies the existence of an increasing plateau at level $k+1$, which must have length at least $r$. An energy level of $k+1$ at position $i$ is possible without using one of the $r$ exceeding seeds only if $i = k+1$, however if that position is part of the increasing plateau, then the seed at level $k+2$ must necessarily stay on top of a seed at level $k+1$. In all cases we need at least $r+1$ seeds to build such an increasing plateau, which contradicts the assumptions. \end{proof}
\begin{lemma}\label{teo:nolongersuboptimal} Let $\lambda \in \configurations_n$ with $n = T_k + r$, $0 \leq r \leq k$, $\lambda \geq \marching^k$ and $s$-monotone. If $\map^{t^\dag}(\lambda)$ becomes an augmented marching group, then \[
t^\dag \leq \max \{r - 1 + (r - s - 1)(k+1), 0\} . \] \end{lemma}
\begin{proof} Using the energy level interpretation we identify the gap at level $k+1$ that will be filled last, and call $i_0$ its column index.
We consider two cases.
\textbf{Case $i_0 < r$ or $i_0 = k+1$.}
The first term $r-1$ in the estimate is then the maximal number of moves that are required to move that gap into the rightmost $(k+1)$-th column. We observe that starting from that configuration the augmented marching group will be obtained exactly after a number of moves multiple of $k+1$
After each cycle of $k+1$ moves the level of monotonicity increases at least of one unit (Lemma \ref{teo:incrmon}), so that after $r - s - 1$ cycles of $k+1$ moves we obtain a configuration $\mu$ that is at least $(r-1)$-monotone.
Now we want to prove that $(r-1)$-monotonicity (and not higher) is actually impossible under the circumstances. This would imply at least $r$-monotonicity that in turn implies that we have an augmented marching group thanks to Lemma \ref{teo:augmentedifrmonotone}.
Suppose then by contradiction that $\mu$ is $(r-1)$-monotone but not $r$-monotone and hence it is not an augmented marching group. This means that the gap at level $k+1$, column $k+1$ is still empty. The energy sequence of $\mu$ has then both elements at level $k$ and elements at level $k+2$, so that there exists at least an increasing plateau at level $k+1$ (i.e. a sequence of energy elements as contiguous positions at level $k+1$ following an element at level $k$ and preceding an element at level $k+2$). Such plateau has length at least $r-1$ due to the $(r-1)$-monotonicity. Now recall that we have an excess of $r$ seeds with energy larger or equal to $k+1$ (levels $k$ and below are completely filled thanks to the assumptions). Column $k+2$ is the only one having energy $k+1$ with no need to use one of the excess seeds, however we have barely enough seeds to form an increasing plateau of length $r-1$ which requires $r+1$ seeds that reduce to $r$ only if we take advantage of the energy step at column $k+2$.
Recalling that column $k+1$ has energy level $k$ (the gap is still empty), the only possible disposition of the $r$ excess seeds is the one with $r-1$ of them in columns $1$ to $r-1$ (energy $k+1$) and the last excess seed itself in column $r-1$ (then with energy $k+2$). This configuration becomes an augmented marching group after exactly $r$ moves, however we know that an augmented marching group is reached after a multiple of $k+1$ moves and hence $r = k+1$, a contradiction.
Since $(r-1)$-monotonicity is impossible for $\mu$, then we have at least $r$-monotonicity and hence we have an augmented marching group.
\textbf{Case $r \leq i_0 < k + 1$.}
Let us consider the result of the evolution as soon as it becomes an augmented marching group and call it $\mu$, the position of the ``marked'' gap is then column $k+1$ and it has just been filled with the last of the $r$ seeds the were originally at energy levels larger then $k+1$. Immediately to the left of this seed there is a contiguous sequence of, say, $\rho-1$ seeds, $1 \leq \rho \leq r$ at level $k+1$, preceded by a gap in column $k + 1 - \rho$. Clearly this latter gap (let us call it ``gap number $2$'') can be traced back to its corresponding position in the initial configuration $\lambda$, and will originally be empty in $\lambda$ and will continue to be empty during the whole evolution. It is located in column $i_0 - \rho$ of $\lambda$, to the left of the marked gap (in column $i_0$).
No seed at energy larger then $k+1$ (they are the ``active'' seeds, the only ones that ``move'' with respect to the $k+1$ level) can ever cross or reach the column containing gap number $2$, so that the active seeds originally to the left of gap number $2$ or to the right of the marked gap (call this region the ``outer region'') can ever interact with the region between gap number $2$ and the active gap.
Without loss of generality we can then suppose that the outer region has reached its final configuration already in $\lambda$. This actually means that there are no active seeds to the right of the marked gap.
The final step is then to observe that in this situation $\lambda$ can be interpreted as the result of the evolution (with $k+1-i_0$ moves) from a configuration $\bar\lambda$ that itself satisfies the hypotheses of this Lemma and the marked gap already in the $k+1$ position (in other words we can ``steal'' a few moves) and we can resort to the first case of the Lemma.
The proof is thus complete. \end{proof}
We now analyze the evolution of $\lambda \in \Lambda_n$, $n = T_k + r$, $0 \leq r \leq k$. We denote with $s \in {\mathbb N}$ the number of $q$-canonical configurations encountered during the evolution with $q \leq k$.
Using Lemma \ref{teo:magico} we obtain the constraint $s \leq W(k-1,r+1) + 1$, which in particular gives \begin{equation}\label{eq:ssconstraints}
\begin{cases}
0 \leq s \leq k - r - 1 \qquad & \text{if $r < \frac{k}{2}$},
\\
0 \leq s \leq \left\lfloor \frac{2k-r}{3}
\right\rfloor
\qquad & \text{if $\frac{k-3}{2} \leq r \leq k+1$},
\\
0 \leq s \leq r \qquad & \text{if $r = \frac{k}{2}$}.
\end{cases} \end{equation}
The third estimate is actually just a special case of the second one; the first and second estimates coincide for $r$ in the common interval $\frac{k-3}{2} \leq r < \frac{k}{2}$.
As usual we denote by $\lambda(t) = \map^t(\lambda)$ and we set \begin{itemize} \item
$t_A$ the smallest time such that $\lambda(t_A) \geq \marching^k$ \item
$t^\dag$ the smallest time such that $\lambda(t^\dag)$ is an augmented marching group. \end{itemize}
\begin{lemma}\label{teo:timesestimates} We have the following estimates: \begin{enumerate} \item $t_A \leq T_{k-1} + k s$; \item $t^\dag - t_A = 0$ if $s \geq r$; \item $t^\dag - t_A \leq r - 1 + (r - s - 1)(k+1)$ if $s < r$. \end{enumerate} \end{lemma}
\begin{proof} The first point is exactly Lemma \ref{teo:balance}.
Point (2) is simply Lemma \ref{teo:augmentedifrmonotone}.
Point (3) follows from Lemma \ref{teo:nolongersuboptimal} \end{proof}
We then have a total estimate of $t^\dag = t_A + (t^\dag - t_A)$ as a function of $s$ that reads as follows:
\begin{equation}\label{eq:endtime} t^\dag \leq \begin{cases}
T_{k-1} + (r - 1)(k+2) - s
& \qquad \text{if } s < r , \\
T_{k-1} + k s
& \qquad \text{if } s \geq r . \end{cases} \end{equation}
Clearly in the case $s < r$ it is most convenient to take $s = 0$, whereas if $s \geq r$ the maximum value is achieved by selecting the largest possible value of $s$, which is dictated by the constraints \eqref{eq:ssconstraints}.
We recall from Section \ref{sec:lower} the definition of $\ldepth(n)$ as
\[ \ldepth(n) = \begin{cases}
3 T_{k-1} - k r
& \qquad \text{if } r < \frac{k}{2} , \\
3 T_{k-1} - k (r - 1)
& \qquad \text{if } r = \frac{k}{2} , \\
T_{k-1} + (k + 2)(r - 1)
& \qquad \text{if } r > \frac{k}{2} , \end{cases} \]
\begin{theorem}[Depth of $\configurations_n$]\label{teo:mainresult} Let $n = T_k + r$ with $k \geq 2$ and $0 \leq r \leq k$. Then $\depth(n) = \ldepth (n)$. The case $k \leq 1$ corresponds to $n \leq 2$, easily studied manually. \end{theorem}
\begin{proof} In view of the results of Section \ref{sec:lower} it suffices to prove the upper bound $\depth(n) \leq \ldepth(n)$. Let $\lambda \in \Lambda_n$. We invoke Lemma \ref{teo:balance} so that there exists $s$ satisfying the constraints \eqref{eq:ssconstraints} such that after at most $T_{k-1} + k s$ moves the resulting configuration is $s$-monotone and $\geq \marching^k$. Now we enforce Lemma \ref{teo:timesestimates} and distinguish three cases.
\textbf{Case $r < \frac{k}{2}$.}
If $s \geq r$, we use the first inequality of \eqref{eq:ssconstraints} in \eqref{eq:endtime} to conclude
\[ t^\dag \leq T_{k-1} + k (k - r - 1)
= \ldepth (n) . \]
If $s < r$, using $s \geq 0$, and $2r \leq k - 1$, i.e. $r \leq k - r - 1$ we have
\[ t^\dag \leq T_{k-1} + (r - 1)(k + 2)
\leq T_{k-1} + rk + 2r - k - 2
< T_{k-1} + k(k - r - 1) - 2 \]
so that in any case $t^\dag \leq \ldepth (n)$.
\textbf{Case $r = \frac{k}{2}$.}
Again we distinguish the case $s \geq r$, together with the third inequality of \eqref{eq:ssconstraints} to obtain
\[ t^\dag \leq T_{k-1} + k r
= 3 T_{k-1} - k(k-r-1) = \ldepth(n) . \]
Whereas if $s < r$, again using $s \geq 0$ we have
\[ t^\dag \leq T_{k-1} + (r - 1)(k + 2)
= 3 T_{k-1} - k(r-1) - 2
< \ldepth(n) . \]
\textbf{Case $r > \frac{k}{2}$.}
We start with the case $s < r$ and again use $s \geq 0$ to get
\[ t^\dag \leq T_{k-1} + (r-1)(k+2) = \ldepth(n) . \]
Finally, if $s \geq r$, enforcing the second inequality in \eqref{eq:ssconstraints}, using $2r - k \geq 1$ we obtain
\[ \begin{aligned} t^\dag & \leq T_{k-1} + k \frac{2k-r}{3} \\ & =
T_{k-1} + (r-1)(k+2) - (2r - k)\left(\frac{2}{3}k + 1\right) + 2
< \ldepth(n) . \end{aligned} \]
We have covered all the possible cases and the proof is complete. \end{proof}
The previous Theorem \ref{teo:mainresult}, and in particular the upper bound $\depth(n) \leq \ldepth(n)$ also allows to compute the maximal diameter of $\configurations_n$:
\begin{corollary}[Diameter of $\configurations_n$] The diameter of $\configurations_n$ (see Definition \ref{def:depthdiameter}) is given by $\ldepth(n) + P(n) - 1$ where $P(n)$ is the maximal period of an augmented marching group, i.e. $P(n) = 1$ if $n = T_k$ is a triangular number, otherwise $P(n) = k$ if $n = T_k + r$, with $0 < r \leq k$, is not a triangular number. \end{corollary}
\begin{proof} That the diameter cannot exceed $\ldepth(n) + P(n) - 1$ immediately follows from Theorem \ref{teo:mainresult}. The reverse inequality follows from the constructions of Section \ref{sec:lower}, in particular we make use of Proposition \ref{teo:lower1}, Corollary \ref{teo:lower2}, Proposition \ref{teo:lower3} and the discussion about the period of the resulting augmented marching groups following each of these results. \end{proof}
\section{Monotone mancala and Bulgarian solitaire} \label{sec:bulgarian}
The notion of $s$-monotonicity naturally introduces a sort of graduation on the graph of moves. Indeed, Theorem \ref{teo:smonclosed} asserts that the subgraph obtained by considering only the $s$-monotone configurations for some $s \in {\mathbb N}$ is closed under mancala moves (no arc leaves this subset). It is then natural to ask what is the depth of such subgraphs.
More precisely, for $n, s \in {\mathbb N}$ let us denote by $\configurations_{n,s}$ and $\configurations_{n,\text{mon}}$ the subsets of $\configurations_n$ given by
\begin{equation*} \configurations_{n,s} = \{ \lambda \in \configurations_n : \lambda \text{ is $s$-monotone}\} , \quad \configurations_{n,\text{mon}} = \{ \lambda \in \configurations_n : \lambda \text{ is monotone}\} . \end{equation*}
Similarly, with $\graph_{n,s}$ and $\graph_{n,\text{mon}}$ we denote the subgraphs of $\graph_n$ having nodes in $\configurations_{n,s}$ and $\configurations_{n,\text{mon}}$ respectively. Then the \emph{depth} of $\graph_{n,s}$, denoted by $\depth(n,s)$, is the maximal distance from a node of $\graph_{n,s}$ to a periodic configuration. In this context, recalling that $0$-monotonicity is a void notion, our main Theorem \ref{teo:mainresult} gives a value for $\depth(n,0) = \depth(n)$. It would be desirable to have a result about the depth $\depth(n,s)$ for any value of $s$; unfortunately, at the moment we are not able to extend our result to nontrivial values of $s$.
Especially the value $s=1$ is interesting: indeed, Remark \ref{rem:weakstrong} relates $\graph_{n,1}$ to $\graph_{n,\text{mon}}$ and in particular their respective depth. Adding one to $\depth(n,1)$ we obtain the depth of the graph of monotone configurations. This is of special interest in view of the following \begin{remark} The monotone mancala graph is isomorphic to the graph of the Bulgarian solitaire \cite{Eti},\cite{GrHo},\cite{Igu}. \end{remark}
The Bulgarian solitaire, already mentioned in the Introduction, is played as follows. The starting position consists of a number of piles, each with a (possibly different) number of cards. Each move consists in removing one card from each pile and collecting the removed cards in a new pile. Piles that become empty are simply neglected and the order of the piles (and of the cards in a pile) is inessential. We can associate a mancala configuration to a Bulgarian solitaire configuration by defining $\lambda_i$ equal to the number of piles with at least $i$ cards. In this way we obtain a valid mancala configuration which is clearly monotone. It can be readily shown that this correspondence is one to one and that it is consistent with the respective game rules.
Hence, the depth of the Bulgarian solitaire subgraph is given by $1+\depth(n,1)$. The following conjecture is stated in \cite{GrHo}:
\begin{conjecture} Let $n = T_k + r$ with $k \geq 2$ and $0 \leq r \leq k$. The depth of $\graph_{n,1}$ is given by
\begin{equation*} \depth(n,1) = \begin{cases} 2T_{k-1} - rk - 1
& \qquad \text{if } 0 \leq r <
\left\lfloor \frac{k}{2} \right\rfloor \\ T_{k-1} + r - 1
& \qquad \text{if } r =
\left\lfloor \frac{k}{2} \right\rfloor
\text{ or }
\left\lfloor \frac{k}{2} \right\rfloor + 1 \\ 2T_k - (k + 1 - r)(k + 2) - 1
& \qquad \text{if }
\left\lfloor \frac{k}{2} \right\rfloor + 1
< r \leq k+1. \end{cases} \end{equation*}
\end{conjecture}
In \cite{GrHo} the authors proved this result for some special values of $n$, while the above expression is proved to be a lower bound. Our computer simulations validates the conjecture for values up to $n=169$, improving the previous limit of $n=36$.
\end{document} | arXiv |
\begin{document}
\begin{abstract} Among the Ramsey-type hierarchies, namely, Ramsey's theorem, the free set, the thin set and the rainbow Ramsey theorem, only Ramsey's theorem is known to collapse in reverse mathematics. A promising approach to show the strictness of the hierarchies would be to prove that every computable instance at level~$n$ has a low$_n$ solution. In particular, this requires effective control of iterations of the Turing jump.
In this paper, we design some variants of Mathias forcing to construct solutions to cohesiveness, the Erd\H{o}s-Moser theorem and stable Ramsey's theorem for pairs, while controlling their iterated jumps. For this, we define forcing relations which, unlike Mathias forcing, have the same definitional complexity as the formulas they force. This analysis enables us to answer two questions of Wei Wang, namely, whether cohesiveness and the Erd\H{o}s-Moser theorem admit preservation of the arithmetic hierarchy, and can be seen as a step towards the resolution of the strictness of the Ramsey-type hierarchies. \end{abstract}
\title{Controlling iterated jumps of solutions to combinatorial problems}
\section{Introduction}
Effective forcing is a very powerful tool in the computational analysis of mathematical statements. In this framework, lowness is achieved by deciding formulas during the forcing argument, while ensuring that the whole construction remains effective. Thus, the definitional strength of the forcing relation is very sensitive in effective forcing. We present a new forcing argument enabling one to control iterated jumps of solutions to Ramsey-type theorems. Our main motivation is reverse mathematics.
\subsection{Reverse mathematics}
Reverse mathematics is a vast mathematical program whose goal is to classify ordinary theorems in terms of their provability strength. It uses the framework of subsystems of second order arithmetic, which is sufficiently rich to express in a natural way many theorems. The base system, $\rca$ standing for Recursive Comprehension Axiom, contains the basic first order Peano arithmetic together with the~$\Delta^0_1$ comprehension scheme and the~$\Sigma^0_1$ induction scheme. Thanks to the equivalence between~$\Delta^0_1$-definable sets and computable sets, $\rca$ can be considered as capturing ``computable mathematics''. The proof-theoretic analysis of the theorems in reverse mathematics is therefore closely related to their computability-theoretic content. See Simpson~\cite{Simpson2009Subsystems} for a formal introduction to reverse mathematics.
Early reverse mathematics results support two main empirical observations: First, many ordinary (i.e.\ non set-theoretic) theorems require very weak set existence axioms. Second, most of those theorems are in fact \emph{equivalent} to one of four main subsystems, which together with $\rca$ are known as the ``Big Five''. However, among the theorems studied in reverse mathematics, a notable class of theorems fails to support those observations, namely, Ramsey-type theorems. This article focuses on consequences of Ramsey's theorem below the arithmetic comprehension axiom ($\aca$). See Hirschfeldt~\cite{Hirschfeldt2015Slicing} for a gentle introduction to the reverse mathematics below~$\aca$.
\subsection{Controlling iterated jumps}
Among the hierarchies of combinatorial principles, namely, Ramsey's theorem~\cite{Jockusch1972Ramseys,Seetapun1995strength,Cholak2001strength}, the rainbow Ramsey theorem~\cite{Csima2009strength,Wang2014Cohesive,Patey2015Somewhere}, and the free set and thin set theorems~\cite{Cholak2001Free,Wang2014Some} -- only Ramsey's theorem is known to collapse within the framework of reverse mathematics. The above mentioned hierarchies satisfy the lower bounds of Jockusch~\cite{Jockusch1972Ramseys}, that is, there exists a computable instance at every level~$n \geq 2$ with no $\Sigma^0_n$ solution. Thus, a possible strategy for proving that a hierarchy is strict consists of showing the existence, for every computable instance at level~$n$, of a low${}_n$ solution.
The solutions to combinatorial principles are often built by Mathias forcing, whose forcing relation is known to be of higher definitional strength than the formula it forces~\cite{Cholak2014Generics}. Therefore there is a need for new notions of forcing with a better-behaving forcing relation. In this paper, we design three notions of forcing to construct solutions to cohesiveness, the Erd\H{o}s-Moser theorem and stable Ramsey's theorem for pairs, respectively. We define a forcing relation with the expected properties, and which formalises the first and the second jump control of Cholak, Jockusch and Slaman~\cite{Cholak2001strength}. This can be seen as a step toward the resolution the strictness of the Ramsey-type hierarchies. We take advantage of this new analysis of Ramsey-type statements to prove two conjectures of Wang about the preservation of the arithmetic hierarchy.
\subsection{Preservation of the arithmetic hierarchy}
The notion of preservation of the arithmetic hierarchy has been introduced by Wang in~\cite{Wang2014Definability}, in the context of a new analysis of principles in reverse mathematics in terms of their definitional strength.
\begin{definition}[Preservation of definitions]\ \begin{itemize}
\item[1.] A set $Y$ \emph{preserves $\Xi$-definitions} (relative to $X$) for $\Xi$ among $\Delta^0_{n+1}, \Pi^0_n, \Sigma^0_n$ where $n > 0$, if every properly $\Xi$ (relative to $X$) set is properly $\Xi$ relative to $Y$ ($X \oplus Y$). $Y$ \emph{preserves the arithmetic hierarchy} (relative to $X$) if $Y$ preserves $\Xi$-definitions (relative to $X$) for all $\Xi$ among $\Delta^0_{n+1}, \Pi^0_n, \Sigma^0_n$ where $n > 0$.
\item[2.] Suppose that $\Phi = (\forall X) (\exists Y) \varphi(X,Y)$ and $\varphi$ is arithmetic. $\Phi$ \emph{admits preservation of $\Xi$-definitions} if for each $Z$ and $X \leq_T Z$ there exists $Y$ such that $Y$ preserves $\Xi$-definitions relative to $Z$ and $\varphi(X,Y)$ holds. $\Phi$ \emph{admits preservation of the arithmetic hierarchy} if for each $Z$ and $X \leq_T Z$ there exists $Y$ such that $Y$ preserves the arithmetic hierarchy relative to $Z$ and $\varphi(X,Y)$ holds. \end{itemize} \end{definition}
The preservation of the arithmetic hierarchy seems closely related to the problem of controlling iterated jumps of solutions to combinatorial problems. Indeed, a proof of such a preservation usually consists of noticing that the forcing relation has the same strength as the formula it forces, and then deriving a diagonalization from it. See Lemma~\ref{lem:diagonalization} for a case-in-point. Wang proved in~\cite{Wang2014Definability} that weak König's lemma ($\wkl$), the rainbow Ramsey theorem for pairs ($\rrt^2_2$) and the atomic model theorem ($\amt$) admit preservation of the arithmetic hierarchy. He conjectured that this is also the case for cohesiveness and the Erd\H{o}s-Moser theorem. We prove the two conjectures via the following concatenation of Theorems~\ref{thm:coh-preservation-arithmetic-hierarchy} and~\ref{thm:em-preserves-arithmetic}, where~$\coh$ stands for cohesiveness and~$\emo$ for the Erd\H{o}s-Moser theorem.
\begin{theorem} $\coh$ and~$\emo$ admit preservation of the arithmetic hierarchy. \end{theorem}
\subsection{Definitions and notation}
Fix an integer $k \in \omega$. A \emph{string} (over $k$) is an ordered tuple of integers $a_0, \dots, a_{n-1}$ (such that $a_i < k$ for every $i < n$). The empty string is written $\epsilon$. A \emph{sequence} (over $k$) is an infinite listing of integers $a_0, a_1, \dots$ (such that $a_i < k$ for every $i \in \omega$). Given $s \in \omega$, $k^s$ is the set of strings of length $s$ over~$k$ and $k^{<s}$ is the set of strings of length $<s$ over~$k$. As well, $k^{<\omega}$ is the set of finite strings over~$k$ and $k^{\omega}$ is the set of sequences (i.e. infinite strings) over~$k$.
Given a string $\sigma \in k^{<\omega}$, we use $|\sigma|$ to denote its length. Given two strings $\sigma, \tau \in k^{<\omega}$, $\sigma$ is a \emph{prefix} of $\tau$ (written $\sigma \preceq \tau$) if there exists a string $\rho \in k^{<\omega}$ such that $\sigma \rho = \tau$. Given a sequence $X$, we write $\sigma \prec X$ if $\sigma = X {\upharpoonright} n$ for some $n \in \omega$. A \emph{binary string} (resp. real) is a \emph{string} (resp. sequence) over $2$. We may identify a real with a set of integers by considering that the real is its characteristic function.
A tree $T \subseteq k^{<\omega}$ is a set downward-closed under the prefix relation. A \emph{binary} tree is a set $T \subseteq 2^{<\omega}$. A set $P \subseteq \omega$ is a \emph{path} through~$T$ if for every $\sigma \prec P$, $\sigma \in T$. A string $\sigma \in k^{<\omega}$ is a \emph{stem} of a tree $T$ if every $\tau \in T$ is comparable with~$\sigma$. Given a tree $T$ and a string $\sigma \in T$, we denote by $T^{[\sigma]}$ the subtree $\{\tau \in T : \tau \preceq \sigma \vee \tau \succeq \sigma\}$.
Given two sets $A$ and $B$, we denote by $A < B$ the formula $(\forall x \in A)(\forall y \in B)[x < y]$. We write $A \subseteq^{*} B$ to mean that $A - B$ is finite, that is, $(\exists n)(\forall a \in A)(a \not \in B \rightarrow a < n)$. A \emph{Mathias condition} is a pair $(F, X)$ where $F$ is a finite set, $X$ is an infinite set and $F < X$. A condition $(F_1, X_1)$ \emph{extends } $(F, X)$ (written $(F_1, X_1) \leq (F, X)$) if $F \subseteq F_1$, $X_1 \subseteq X$ and $F_1 \smallsetminus F \subset X$. A set $G$ \emph{satisfies} a Mathias condition $(F, X)$ if $F \subset G$ and $G \smallsetminus F \subseteq X$.
\section{Cohesiveness preserves the arithmetic hierarchy}
Cohesiveness plays a central role in reverse mathematics. It appears naturally in the standard proof of Ramsey's theorem, as a preliminary step to reduce an instance of Ramsey's theorem over $(n+1)$-tuples into a non-effective instance over $n$-tuples.
\begin{definition}[Cohesiveness] An infinite set $C$ is $\vec{R}$-cohesive for a sequence of sets $R_0, R_1, \dots$ if for each $i \in \omega$, $C \subseteq^{*} R_i$ or $C \subseteq^{*} \overline{R_i}$. A set $C$ is \emph{cohesive} (resp. \emph{p-cohesive}, \emph{r-cohesive}) if it is $\vec{R}$-cohesive where $\vec{R}$ is the sequence of all the c.e. sets (resp. primitive recursive sets, computable sets). $\coh$ is the statement ``Every uniform sequence of sets $\vec{R}$ admits an infinite $\vec{R}$-cohesive set.'' \end{definition}
Mileti~\cite{Mileti2004Partition} and Jockusch and Lempp [unpublished] proved that $\coh$ is a consequence of Ramsey's theorem for pairs over $\rca$. The computational power of $\coh$ is relatively well understood. A Turing degree $\mathbf{d}$ \emph{bounds} $\coh$ if every computable sequence of sets~$R_0, R_1, \dots$, has an $\vec{R}$-cohesive set bounded by $\mathbf{d}$. Jockusch and Stephan characterized in~\cite{Jockusch1993cohesive} the degrees bounding $\coh$ as the degrees whose jump is PA relative to $\emptyset'$. The author~\cite{Patey2015weakness} extended this characterization to an instance-wise correspondance between cohesiveness and the statement ``For every~$\Delta^0_2$ tree~$T$, there is a set whose jump computes a path through~$T$''. Wang~\cite{Wang2014Definability} conjectured that $\coh$ admits preservation of the arithmetic hierarchy. We prove his conjecture by using a new forcing argument.
\begin{theorem}\label{thm:coh-preservation-arithmetic-hierarchy} $\coh$ admits preservation of the arithmetic hierarchy. \end{theorem}
Before proving Theorem~\ref{thm:coh-preservation-arithmetic-hierarchy}, we state an immediate corollary.
\begin{corollary} There exists a cohesive set preserving the arithmetic hierarchy. \end{corollary} \begin{proof} Jockusch~\cite{Jockusch1972Degreesa} proved that every PA degree computes a sequence of sets containing, among others, all the computable sets. Wang proved in~\cite{Wang2014Definability} that $\wkl$ preserves the arithmetic hierarchy. Therefore there exists a uniform sequence of sets $\vec{R}$ containing all the computable sets and preserving the arithmetic hierarchy. By Theorem~\ref{thm:coh-preservation-arithmetic-hierarchy} relativized to $\vec{R}$, there exists an infinite $\vec{R}$-cohesive set $C$ preserving the arithmetic hierarchy relative to~$\vec{R}$. In particular $C$ is r-cohesive and preserves the arithmetic hierarchy. By~\cite{Jockusch1993cohesive}, the degrees of r-cohesive and cohesive sets coincide. Therefore $C$ computes a cohesive set which preserves the arithmetic hierarchy. \end{proof}
Given a uniformly computable sequence of sets~$R_0, R_1, \dots$, the construction of an $\vec{R}$-cohesive set is usually done with computable Mathias forcing, that is, using conditions~$(F, X)$ in which~$X$ is computable. The construction starts with~$(\emptyset, \omega)$ and interleaves two kinds of steps. Given some condition~$(F,X)$, \begin{itemize}
\item[(S1)] the \emph{extension} step consists of taking an element $x$ from $X$ and adding it to~$F$,
therefore forming the extension $(F \cup \{x\}, X \smallsetminus [0,x])$;
\item[(S2)] the \emph{cohesiveness} step consists of deciding which one of $X \cap R_i$
and $X \cap \overline{R}_i$ is infinite, and taking the chosen one as the new reservoir. \end{itemize}
Cholak, Dzhafarov, Hirst and Slaman~\cite{Cholak2014Generics} studied the definitional complexity of the forcing relation for computable Mathias forcing. They proved that it has good definitional properties for the first jump, but not for iterated jumps. Indeed, given a computable Mathias condition~$c = (F, X)$ and a $\Sigma^0_1$ formula~$(\exists x)\varphi(G, x)$, one can $\emptyset'$-effectively decide whether there is an extension~$d$ forcing~$(\exists x)\varphi(G, x)$ by asking the following question: \begin{quote} Is there an extension~$d = (E, Y) \leq c$ and some~$n \in \omega$ such that~$\varphi(E, n)$ holds? \end{quote} If there is such an extension, then we can choose it to be a \emph{finite extension}, that is, such that~$Y =^* X$. Therefore, the question is $\Sigma^{0,X}_1$. Consider now a $\Pi^0_2$ formula~$(\forall x)(\exists y)\varphi(G, x, y)$. The question becomes \begin{quote} For every extension~$d \leq c$ and every~$m \in \omega$, is there some extension~$e = (E, Y) \leq d$ and some~$n \in \omega$ such that~$\varphi(E, m, n)$ holds? \end{quote} In this case, the extension~$d$ is not usually a finite extension and therefore the question cannot be presented in a $\Pi^0_2$ way. In particular, the formula ``$Y$ is an infinite subset of~$X$'' is definitionally complex. In general, deciding iterated jumps of a generic set requires to be able to talk about the future of a given condition, and in particular to describe by simple means the formula ``$d$ is a valid condition'' and the formula ``$d$ is an extension of~$c$''.
Thankfully, in the case of cohesiveness, we do not need the full generality of the computable Mathias forcing. Indeed, the reservoirs have a very special shape. After the first application of stage~(S2), the set $X$ is, up to finite changes, of the form $\omega \cap R_0$ or $\omega \cap \overline{R_0}$. After the second application of (S2), it is in one of the following forms: $\omega \cap R_0 \cap R_1$, $\omega \cap R_0 \cap \overline{R}_1$, $\omega \cap \overline{R}_0 \cap R_1$, $\omega \cap \overline{R}_0 \cap \overline{R}_1$, and so on. More generally, after $n$ applications of (S2), a condition~$c = (F, X)$ is characterized by a pair~$(F, \sigma)$ where $\sigma$ is a string of length~$n$ representing the choices made during (S2). Given a string~$\sigma \in 2^{<\omega}$, let~$R_\sigma = \bigcap_{\sigma(i) = 0} \overline{R}_i \bigcap_{\sigma(i) = 1} R_i$. In particular, $R_\varepsilon = \omega$, where~$\varepsilon$ is the empty string.
Even within this restricted partial order, the decision of the~$\Pi^0_2$ formula remains too complicated sinces it requires deciding if $R_\sigma$ is infinite. However, notice that the~$\sigma$'s such that~$R_\sigma$ is infinite are exactly the initial segments of the $\Pi^{0, \emptyset'}_1$ class~$\mathcal{C}(\vec{R})$ defined as the collection of the reals~$X$
such that~$R_\sigma$ has more than~$|\sigma|$ elements for every~$\sigma \prec X$. We can therefore use a compactness argument at the second level to decrease the definitional strength of the forcing relation, as Wang~\cite{Wang2014Definability} did for weak K\"onig's lemma.
\subsection{The forcing notion}
We let~$\mathbb{T}$ denote the collection of all the infinite $\emptyset'$-primitive recursive trees~$T$ such that~$[T] \subseteq \mathcal{C}(\vec{R})$. By $\emptyset'$-primitive recursive, we mean the class of functions Add a comment to this line obtained by adding the characteristic function of $\emptyset'$ to the basic primitive recursive functions, and closing under the standard primitive recursive operations. Note that~$\mathbb{T}$ is a computable set. Given two finite sets~$E, F$ and some string~$\sigma \in 2^{<\omega}$, we write~$E \leq_\sigma F$ to say that $F \subseteq E \subseteq F \cup (R_\sigma \cap [\max F, \infty))$. In other words, $E \leq_\sigma F$ if and only if $(E, R_\sigma)$ is a valid Mathias extension of $(F, R_\sigma)$, where~$R_\sigma$ might be finite. We are now ready to defined our partial order.
\begin{definition} Let $\mathbb{P}$ be the partial order whose conditions are tuples $(F, \sigma, T)$ where~$F \subseteq \omega$ is a finite set, $\sigma \in 2^{<\omega}$ and~$T \in \mathbb{T}$ with stem~$\sigma$. A condition~$d = (E, \tau, S)$ \emph{extends} $c = (F, \sigma, T)$ (written~$d \leq c$) if~$E \leq_\sigma F$, $\tau \succeq \sigma$ and $S \subseteq T$. \end{definition}
Given a condition~$c = (F, \sigma, T)$, the string~$\sigma$ imposes a finite restriction on the possible extensions of the set~$F$. The condition $c$ intuitively denotes the Mathias condition $(F, R_\sigma \cap (\max F, \infty))$ with some additional constraints on the extensions of~$\sigma$ represented by the tree~$T$. Accordingly, set~$G$ \emph{satisfies} $(F, \sigma, T)$ if it satisfies the induced Mathias condition, that is, if~$F \subseteq G \subseteq F \cup (R_\sigma \cap (\max F, \infty))$. We let~$\operatorname{Ext}(c)$ be the collection of all the extensions of~$c$.
Note that although we did not explicitely require $R_\sigma$ to be infinite, this property holds for every condition~$(F, \sigma, T) \in \mathbb{P}$. Indeed, since $[T] \subseteq \mathcal{C}(\vec{R})$, then $R_\tau$ is infinite for every extensible node~$\tau \in T$. Since $\sigma$ is a stem of~$T$, it is extensible and therefore $R_\sigma$ is infinite.
\subsection{Preconditions and forcing $\Sigma^0_1$ ($\Pi^0_1$) formulas}
When forcing complex formulas, we need to be able to consider all possible extensions of some condition~$c$. Checking that some $d = (E, \tau, S)$ is a valid condition extending $c$ requires to decide whether the $\emptyset'$-p.r.\ tree $S$ is infinite, which is a $\Pi^0_2$ question. At some point, we will need to decide a $\Sigma^0_1$ formula without having enough computational power to check that the tree part is infinite (see clause~(ii) of Definition~\ref{def:forcing-condition}). As the tree part of a condition is not accurate for such formulas, we may define the corresponding forcing relation over a weaker notion of condition where the tree is not required to be infinite.
\begin{definition}[Precondition] A \emph{precondition} is a condition~$(F, \sigma, T)$ without the assumption that $T$ is infinite. \end{definition}
In particular, $R_\sigma$ may be a finite set. The notion of condition extension can be generalized to the preconditions. The set of all preconditions is computable, contrary to the set~$\mathbb{P}$. Given a precondition~$c = (F, \sigma, T)$, we denote by~$\operatorname{Ext}_1(c)$ the set of all preconditions~$(E, \tau, S)$ extending~$c$ such that~$\tau = \sigma$ and~$T = S$. Here, $T = S$ in a strong sense, that is, the Turing indices of~$T$ and~$S$ are the same. This fact is used in clause~a) of Lemma~\ref{lem:extension-complexity}. We let~$\mathbb{A}$ denote the collection of all the finite sets of integers. The set $\mathbb{A}$ can be thought of as representing the set of finite approximations of the generic set~$G$. We also fix a uniformly computable enumeration $\mathbb{A}_0 \subseteq \mathbb{A}_1 \subseteq \dots$ of finite subsets of~$\mathbb{A}$ such that~$\bigcup_s \mathbb{A}_s = \mathbb{A}$. We denote by~$\operatorname{Apx}(c)$ the set $\{ E \in \mathbb{A} : (E, \sigma, T) \in \operatorname{Ext}_1(c) \}$. In particular, $\operatorname{Apx}(c)$ is collection of all finite sets~$E$ satisfying~$c$, that is, $\operatorname{Apx}(c) = \{ E \in \mathbb{A} : E \leq_\sigma F \}$. Last, we let~$\operatorname{Apx}_s(c) = \operatorname{Apx}(c) \cap \mathbb{A}_s$. We start by proving a few trivial statements.
\begin{lemma}\label{lem:basic-statements} Fix a precondition $c = (F, \sigma, T)$. \begin{itemize}
\item[1)] If $c$ is a condition then $\operatorname{Ext}_1(c) \subseteq \operatorname{Ext}(c)$.
\item[2)] If $c$ is a condition then $\operatorname{Apx}(c) = \{ E : (E, \tau, S) \in \operatorname{Ext}(c) \}$.
\item[3)] If $d$ is a precondition extending $c$ then
$\operatorname{Apx}(d) \subseteq \operatorname{Apx}(c)$ and~$\operatorname{Apx}_s(d) \subseteq \operatorname{Apx}_s(c)$. \end{itemize} \end{lemma} \begin{proof}\ \begin{itemize}
\item[1)] By definition, if $c$ is a condition, then $T$ is infinite. If $d \in \operatorname{Ext}_1(c)$ then $d = (E, \sigma, T)$ for some $E \in \operatorname{Apx}(c)$. As $d$ is a precondition and $T$ is infinite, $d$ is a condition.
\item[2)] By definition,
$\operatorname{Apx}(c) = \{ E : (E, \sigma, T) \in \operatorname{Ext}_1(c) \} \subseteq \{ E : (E, \tau, S) \in \operatorname{Ext}(c) \}$.
In the other direction, fix an extension $(E, \tau, S) \in \operatorname{Ext}(c)$.
By definition of an extension, $E \leq_\tau F$, so $E \leq_\sigma F$.
Therefore $(E, \sigma, T) \in \operatorname{Ext}_1(c)$ and by definition of $\operatorname{Apx}(c)$, $E \in \operatorname{Apx}(c)$.
\item[3)]
Fix some $(E, \tau, S) \in \operatorname{Ext}_1(d)$. As $d$ extends $c$, $\tau \succeq \sigma$.
By definition of an extension, $E \leq_\tau F$, so $E \leq_\sigma F$,
hence $(E, \sigma, T) \in \operatorname{Ext}_1(c)$.
Therefore $\operatorname{Apx}(d) = \{ E : (E, \tau, S) \in \operatorname{Ext}_1(d) \} \subseteq \{ E : (E, \sigma, T) \in \operatorname{Ext}_1(c) \} = \operatorname{Apx}(c)$. For any~$s \in \omega$,
$\operatorname{Apx}_s(d) = \operatorname{Apx}(d) \cap \mathbb{A}_s \subseteq \operatorname{Apx}(c) \cap \mathbb{A}_s = \operatorname{Apx}_s(c)$. \end{itemize} \end{proof}
Note that although the extension relation has been generalized to preconditions, $\operatorname{Ext}(c)$ is defined to be the set of all the \emph{conditions} extending $c$. In particular, if $c$ is a precondition which is not a condition, $\operatorname{Ext}(c) = \emptyset$, whereas at least $c \in \operatorname{Ext}_1(c)$. This is why clause 1 of Lemma~\ref{lem:basic-statements} gives the useful information that whenever~$c$ is a true condition, so are the members of $\operatorname{Ext}_1(c)$.
\begin{definition}\label{def:forcing-precondition} Fix a precondition~$c = (F, \sigma, T)$ and a $\Sigma^0_0$ formula $\varphi(G, x)$. \begin{itemize}
\item[(i)] $c \Vdash (\exists x)\varphi(G, x)$ iff $\varphi(F, w)$ holds for some~$w \in \omega$
\item[(ii)] $c \Vdash (\forall x)\varphi(G, x)$ iff $\varphi(E, w)$ holds for every~$w \in \omega$
and every set~$E \in \operatorname{Apx}(c)$. \end{itemize} \end{definition}
As explained, $\sigma$ restricts the possible extensions of the set~$F$ (see clause 3 of Lemma~\ref{lem:basic-statements}), so this forcing notion is stable by condition extension. The tree $T$ itself restricts the possible extensions of $\sigma$, but has no effect in deciding a $\Sigma^0_1$ formula (Lemma~\ref{lem:tree-no-effect-first-level}).
The following trivial lemma expresses the fact that the tree part of a precondition has no effect in the forcing relation for a $\Sigma^0_1$ or $\Pi^0_1$ formula.
\begin{lemma}\label{lem:tree-no-effect-first-level} Fix two preconditions $c = (F, \sigma, T)$ and $d = (F, \sigma, S)$, and some $\Sigma^0_1$ or $\Pi^0_1$ formula~$\varphi(G)$. $$ c \Vdash \varphi(G) \hspace{10pt} \mbox{ if and only if } \hspace{10pt} d \Vdash \varphi(G) $$ \end{lemma} \begin{proof} Simply notice that the tree part of the condition does not occur in the definition of the forcing relation, and that $\operatorname{Apx}(c) = \operatorname{Apx}(d)$. \end{proof}
As one may expect, the forcing relation for a precondition is closed under extension.
\begin{lemma}\label{lem:forcing-extension-precondition} Fix a precondition $c$ and a $\Sigma^0_1$ or $\Pi^0_1$ formula $\varphi(G)$. If $c \Vdash \varphi(G)$ then for every precondition $d \leq c$, $d \Vdash \varphi(G)$. \end{lemma} \begin{proof} Fix a precondition $c = (F, \sigma, T)$ such that $c \Vdash \varphi(G)$ and an extension $d = (E, \tau, S) \leq c$. \begin{itemize}
\item If $\varphi \in \Sigma^0_1$ then $\varphi(G)$ can be expressed as $(\exists x)\psi(G, x)$ where $\psi \in \Sigma^0_0$. As $c \Vdash \varphi(G)$, then by clause (i) of Definition~\ref{def:forcing-precondition}, there exists a $w \in \omega$ such that $\psi(F, w)$ holds. By definition of $d \leq c$, $E \leq_\sigma F$, so $\psi(E, w)$ holds, hence $d \Vdash \varphi(G)$.
\item If $\varphi \in \Pi^0_1$ then $\varphi(G)$ can be expressed as $(\forall x)\psi(G, x)$ where $\psi \in \Sigma^0_0$. As $c \Vdash \varphi(G)$, then by clause (ii) of Definition~\ref{def:forcing-precondition}, for every $w \in \omega$ and every $H \in \operatorname{Apx}(c)$, $\varphi(H, w)$ holds. By clause 3 of Lemma~\ref{lem:basic-statements}, $\operatorname{Apx}(d) \subseteq \operatorname{Apx}(c)$ so $d \Vdash \varphi(G)$. \end{itemize} \end{proof}
\subsection{Forcing higher formulas}
We are now able to define the forcing relation for any arithmetic formula. The forcing relation for arbitrary arithmetic formulas is induced by the forcing relation for~$\Sigma^0_1$ formulas. However, the definitional strength of the resulting relation is too high with respect to the formula it forces. We therefore design a custom relation with better definitional properties, and which still preserve the expected properties of a forcing relation, that is, the density of the set of conditions forcing a formula or its negation, and the preservation of the forced formulas under condition extension.
\begin{definition}\label{def:forcing-condition} Let~$c = (F, \sigma, T)$ be a condition and~$\varphi(G)$ be an arithmetic formula. \begin{itemize}
\item[(i)] If $\varphi(G) = (\exists x)\psi(G, x)$ where~$\psi \in \Pi^0_{n+1}$ then~$c \Vdash \varphi(G)$
iff there is a $w < |\sigma|$ such that~$c \Vdash \psi(G, w)$
\item[(ii)] If $\varphi(G) = (\forall x)\psi(G, x)$ where~$\psi \in \Sigma^0_1$ then $c \Vdash \varphi(G)$
iff for every~$\tau \in T$, every $E \in \operatorname{Apx}_{|\tau|}(c)$
and every~$w < |\tau|$, $(E, \tau, T^{[\tau]}) \not \Vdash \neg \psi(G, w)$
\item[(iii)] If~$\varphi(G) = \neg \psi(G, x)$ where~$\psi \in \Sigma^0_{n+3}$ then $c \Vdash \varphi(G)$
iff $d \not \Vdash \psi(G)$ for every~$d \leq c$. \end{itemize} \end{definition}
Note that in clause (ii) of Definition~\ref{def:forcing-condition}, there may be some $\tau \in T$ such that $T^{[\tau]}$ is finite, hence $(E, \tau, T^{[\tau]})$ is not necessarily a condition. This is where we use the generalization of forcing of $\Sigma^0_1$ and $\Pi^0_1$ formulas to preconditions. We now prove that this relation enjoys the main properties of a forcing relation.
\begin{lemma}\label{lem:forcing-extension} Fix a condition $c$ and an arithmetic formula $\varphi(G)$. If $c \Vdash \varphi(G)$ then for every condition $d \leq c$, $d \Vdash \varphi(G)$. \end{lemma} \begin{proof} We prove by induction over the complexity of the formula $\varphi(G)$ that for every condition $c$, if $c \Vdash \varphi(G)$ then for every condition $d \leq c$, $d \Vdash \varphi(G)$. Fix a condition $c = (F, \sigma, T)$ such that $c \Vdash \varphi(G)$ and an extension $d = (E, \tau, S)$. \begin{itemize}
\item If $\varphi \in \Sigma^0_1 \cup \Pi^0_1$ then it follows from Lemma~\ref{lem:forcing-extension-precondition}.
\item If $\varphi \in \Sigma^0_{n+2}$ then $\varphi(G)$ can be expressed as $(\exists x)\psi(G, x)$
where $\psi \in \Pi^0_{n+1}$.
By clause~(i) of Definition~\ref{def:forcing-condition}, there exists a $w \in \omega$
such that $c \Vdash \psi(G, w)$. By induction hypothesis, $d \Vdash \psi(G, w)$
so by clause~(i) of Definition~\ref{def:forcing-condition}, $d \Vdash \varphi(G)$.
\item If $\varphi \in \Pi^0_2$ then $\varphi(G)$ can be expressed as $(\forall x)\psi(G, x)$ where $\psi \in \Sigma^0_1$.
By clause~(ii) of Definition~\ref{def:forcing-condition}, for every $\rho \in T$, every $w < |\rho|$,
and every $H \in \operatorname{Apx}_{|\rho|}(c)$, $(H, \rho, T^{[\rho]}) \not \Vdash \neg \psi(G, w)$.
As $S \subseteq T$ and $\operatorname{Apx}(d) \subseteq \operatorname{Apx}(c)$, for every $\rho \in S$, every $w < |\rho|$,
and every $H \in \operatorname{Apx}_{|\rho|}(d)$, $(H, \rho, T^{[\rho]}) \not \Vdash \neg \psi(G, w)$.
By Lemma~\ref{lem:tree-no-effect-first-level}, $(H, \rho, S^{[\rho]}) \not \Vdash \neg \psi(G, w)$
hence by clause~(ii) of Definition~\ref{def:forcing-condition}, $d \Vdash \varphi(G)$.
\item If $\varphi \in \Pi^0_{n+3}$ then $\varphi(G)$ can be expressed as $\neg \psi(G)$ where $\psi \in \Sigma^0_{n+3}$.
By clause~(iii) of Definition~\ref{def:forcing-condition}, for every $e \in \operatorname{Ext}(c)$, $e \not \Vdash \psi(G)$.
As $\operatorname{Ext}(d) \subseteq \operatorname{Ext}(c)$, for every $e \in \operatorname{Ext}(d)$, $e \not \Vdash \psi(G)$,
so by clause~(iii) of Definition~\ref{def:forcing-condition}, $d \Vdash \varphi(G)$. \end{itemize} \end{proof}
\begin{lemma}\label{lem:forcing-dense} For every arithmetic formula $\varphi$, the following set is dense $$ \{c \in \mathbb{P} : c \Vdash \varphi(G) \mbox{ or } c \Vdash \neg \varphi(G) \} $$ \end{lemma} \begin{proof} We prove by induction over $n > 0$ that if $\varphi$ is a $\Sigma^0_n$ ($\Pi^0_n$) formula then the following set is dense $$ \{c \in \mathbb{P} : c \Vdash \varphi(G) \mbox{ or } c \Vdash \neg \varphi(G) \} $$ It suffices to prove it for the case where $\varphi$ is a $\Sigma^0_n$ formula, as the case where $\varphi$ is a $\Pi^0_n$ formula is symmetric. Fix a condition $c = (F, \sigma, T)$. \begin{itemize}
\item In case $n = 1$, the formula $\varphi$ is of the form $(\exists x)\psi(G, x)$ where $\psi \in \Sigma^0_0$.
Suppose there exist a $w \in \omega$ and a set $E \in \operatorname{Apx}(c)$ such that $\psi(E, w)$ holds.
The precondition $d = (E, \sigma, T)$ is a condition extending $c$ by clause~1 of Lemma~\ref{lem:basic-statements}
and by definition of $\operatorname{Apx}(c)$.
Moreover $d \Vdash (\exists x)\psi(G, x)$ by clause~(i) of Definition~\ref{def:forcing-precondition} hence $d \Vdash \varphi(G)$.
Suppose now that for every $w \in \omega$ and every $E \in \operatorname{Apx}(c)$, $\psi(E, w)$ does not hold.
By clause~(ii) of Definition~\ref{def:forcing-precondition},
$c \Vdash (\forall x)\neg \psi(G, x)$, hence $c \Vdash \neg \varphi(G)$.
\item In case $n = 2$, the formula $\varphi$ is of the form $(\exists x)\psi(G, x)$ where
$\psi \in \Pi^0_1$. Let
$$
S = \{ \tau \in T : (\forall w < |\tau|)(\forall E \in \operatorname{Apx}_{|\tau|}(c))(E, \tau, T^{[\tau]}) \not \Vdash \psi(G, w) \}
$$
The set $S$ is obviously $\emptyset'$-p.r. We prove that it is a subtree of $T$.
Suppose that $\tau \in S$ and $\rho \preceq \tau$.
Fix a $w < |\rho|$ and $E \in \operatorname{Apx}_{|\rho|}(c)$. In particular $w < |\tau|$ and $E \in \operatorname{Apx}_{|\tau|}(c)$
so $(E, \tau, T^{[\tau]}) \not \Vdash \psi(G, w)$.
Note that~$(E, \tau,T^{[\tau]})$ is a precondition extending~$(E, \rho, T^{[\rho]})$,
so by the contrapositive of Lemma~\ref{lem:forcing-extension-precondition}, $(E, \rho, T^{[\rho]}) \not \Vdash \psi(G, w)$.
Therefore $\rho \in S$. Hence $S$ is a tree, and as $S \subseteq T$, it is a subtree of $T$.
If $S$ is infinite, then $d = (F, \sigma, S)$
is an extension of $c$ such that for every $\tau \in S$, every $w < |\tau|$
and every $E \in \operatorname{Apx}_{|\tau|}(c)$, $(E, \tau, T^{[\tau]}) \not \Vdash \psi(G, w)$.
By Lemma~\ref{lem:tree-no-effect-first-level}, for every $E \in \operatorname{Apx}_{|\tau|}(c)$,
$(E, \tau, S^{[\tau]}) \not \Vdash \psi(G, w)$ and by clause 3 of Lemma~\ref{lem:basic-statements},
$\operatorname{Apx}_{|\tau|}(d) \subseteq \operatorname{Apx}_{|\tau|}(c)$.
Therefore, by clause~(ii) of Definition~\ref{def:forcing-condition}, $d \Vdash (\forall x)\neg \psi(G, x)$
so $d \Vdash \neg \varphi(G)$. If $S$ is finite, then pick some $\tau \in T \smallsetminus S$
such that $T^{[\tau]}$ is infinite. By choice of $\tau \in T \smallsetminus S$, there exist a $w < |\tau|$
and an $E \in \operatorname{Apx}_{|\tau|}(c)$ such that $(E, \tau, T^{[\tau]}) \Vdash \psi(G, w)$.
$d = (E, \tau, T^{[\tau]})$ is a valid condition extending $c$
and by clause~(i) of Definition~\ref{def:forcing-condition} $d \Vdash \varphi(G)$.
\item In case $n > 2$, density follows from clause~(iii) of Definition~\ref{def:forcing-condition}. \end{itemize} \end{proof}
Any sufficiently generic filter~$\mathcal{F}$ induces a unique generic real~$G$ defined by $$ G = \bigcup \{ F \in \mathbb{A} : (F, \sigma, T) \in \mathcal{F} \} $$ The following lemma informally asserts that the forcing relation is \emph{sound} and \emph{complete}. Sound because whenever a property is forced at some point, then this property actually holds over the generic real~$G$. The forcing is also complete in that every property which holds over~$G$ is forced at some point whenever the filter is sufficiently generic.
\begin{lemma}\label{lem:coh-holds-filter} Suppose that $\mathcal{F}$ is a sufficiently generic filter and let~$G$ be the corresponding generic real. Then for each arithmetic formula $\varphi(G)$, $\varphi(G)$ holds iff $c \Vdash \varphi(G)$ for some $c \in \mathcal{F}$. \end{lemma} \begin{proof} We prove by induction over the complexity of the arithmetic formula $\varphi(G)$ that $\varphi(G)$ holds iff $c \Vdash \varphi(G)$ for some $c \in \mathcal{F}$. Note that thanks to Lemma~\ref{lem:forcing-dense}, it suffices to prove that if $c \Vdash \varphi(G)$ for some $c \in \mathcal{F}$ then $\varphi(G)$ holds. Indeed, conversely if $\varphi(G)$ holds, then by genericity of $G$ either $c \Vdash \varphi(G)$ or $c \Vdash \neg \varphi(G)$ for some~$c \in \mathcal{F}$, but if $c \Vdash \neg \varphi(G)$ then $\neg \varphi(G)$ holds, contradicting the hypothesis. So $c \Vdash \varphi(G)$.
We proceed by case analysis on the formula~$\varphi$. Note that in the above argument, the converse of the~$\Sigma$ case is proved assuming the~$\Pi$ case. However, in our proof, we use the converse of the~$\Sigma^0_{n+3}$ case to prove the $\Pi^0_{n+3}$ case. We need therefore to prove the converse of the~$\Sigma^0_{n+3}$ case without Lemma~\ref{lem:forcing-dense}. Fix a condition $c = (F, \sigma, T) \in \mathcal{F}$ such that $c \Vdash \varphi(G)$. \begin{itemize}
\item If $\varphi \in \Sigma^0_1$ then $\varphi(G)$ can be expressed as $(\exists x)\psi(G, x)$ where $\psi \in \Sigma^0_0$.
By clause~(i) of Definition~\ref{def:forcing-precondition}, there exists a $w \in \omega$ such that
$\psi(F, w)$ holds. As $F \subseteq G$ and~$G \smallsetminus F \subseteq (\max F, \infty)$, then by continuity $\psi(G, w)$ holds, hence $\varphi(G)$ holds.
\item If $\varphi \in \Pi^0_1$ then $\varphi(G)$ can be expressed as $(\forall x)\psi(G, x)$ where $\psi \in \Sigma^0_0$.
By clause~(ii) of Definition~\ref{def:forcing-precondition}, for every $w \in \omega$
and every $E \in \operatorname{Apx}(c)$, $\psi(E, w)$ holds. As $\{E \subset_{fin} G : E \supseteq F \} \subseteq \operatorname{Apx}(c)$,
then for every $w \in \omega$, $\psi(G, w)$ holds, so $\varphi(G)$ holds.
\item If $\varphi \in \Sigma^0_{n+2}$ then $\varphi(G)$ can be expressed as $(\exists x)\psi(G, x)$
where $\psi \in \Pi^0_{n+1}$.
By clause~(i) of Definition~\ref{def:forcing-condition}, there exists a $w \in \omega$
such that $c \Vdash \psi(G, w)$. By induction hypothesis, $\psi(G, w)$ holds, hence $\varphi(G)$ holds.
Conversely, suppose that $\varphi(G)$ holds. Then there exists a $w \in \omega$ such that $\psi(G, w)$ holds,
so by induction hypothesis $c \Vdash \psi(G, w)$ for some $c \in \mathcal{F}$,
so by clause~(i) of Definition~\ref{def:forcing-condition}, $c \Vdash \varphi(G)$.
\item If $\varphi \in \Pi^0_2$ then $\varphi(G)$ can be expressed as $(\forall x)\psi(G, x)$ where $\psi \in \Sigma^0_1$.
By clause~(ii) of Definition~\ref{def:forcing-condition}, for every $\tau \in T$, every $w < |\tau|$,
and every $E \in \operatorname{Apx}_{|\tau|}(c)$, $(E, \tau, T^{[\tau]}) \not \Vdash \neg \psi(G, w)$.
Suppose by way of contradiction that $\psi(G, w)$ does not hold for some $w \in \omega$.
Then by induction hypothesis, there exists a $d \in \mathcal{F}$ such that $d \Vdash \neg \psi(G, w)$.
Let $e = (E, \tau, S) \in \mathcal{F}$ be such that $e \Vdash \neg \psi(G, w)$, $|\tau| > w$
and $e$ extends both $c$ and $d$. The condition $e$ exists by Lemma~\ref{lem:forcing-extension-precondition}.
We can furthermore require that $E \in \operatorname{Apx}_{|\tau|}(c)$,
so $e \not \Vdash \neg \psi(G, w)$ and $e \Vdash \neg \psi(G, w)$. Contradiction.
Hence for every $w \in \omega$, $\psi(G, w)$ holds, so $\varphi(G)$ holds.
\item If $\varphi \in \Pi^0_{n+3}$ then $\varphi(G)$ can be expressed as $\neg \psi(G)$ where $\psi \in \Sigma^0_{n+3}$.
By clause~(iii) of Definition~\ref{def:forcing-condition}, for every $d \in \operatorname{Ext}(c)$, $d \not \Vdash \psi(G)$.
By Lemma~\ref{lem:forcing-extension}, $d \not \Vdash \psi(G)$ for every~$d \in \mathcal{F}$, and by
a previous case, $\psi(G)$ does not hold, so $\varphi(G)$ holds. \end{itemize} \end{proof}
We now prove that the forcing relation enjoys the desired definitional properties, that is, the complexity of the forcing relation is the same as the complexity of the formula forced. We start by analysing the complexity of some components of this notion of forcing.
\begin{lemma}\label{lem:extension-complexity}\ \begin{itemize}
\item[a)] For every precondition $c$, $\operatorname{Apx}(c)$ and $\operatorname{Ext}_1(c)$ are $\Delta^0_1$ uniformly in~$c$.
\item[b)] For every condition $c$, $\operatorname{Ext}(c)$ is $\Pi^0_2$ uniformly in~$c$. \end{itemize} \end{lemma} \begin{proof}\ \begin{itemize}
\item[a)] Fix a precondition $c = (F, \sigma, T)$.
A set $E \in \operatorname{Apx}(c)$ iff the following $\Delta^0_1$ predicate holds:
$$
(F \subseteq E) \wedge (\forall x \in E \smallsetminus F)[x > \max F \wedge x \in R_\sigma]
$$
Moreover, $(E, \tau, S) \in \operatorname{Ext}_1(c)$ iff the $\Delta^0_1$ predicate
$E \in \operatorname{Apx}(c) \wedge \tau = \sigma \wedge S = T$ holds.
As already mentioned, the equality~$S = T$ is translated into ``the indices
of~$S$ and~$T$ coincide'' which is a $\Sigma^0_0$ statement.
\item[b)] Fix a condition $c = (F, \sigma, T)$.
By clause 2) of Lemma~\ref{lem:basic-statements},
$(E, \tau, S) \in \operatorname{Ext}(c)$ iff the following $\Pi^0_2$ formula holds
$$
\begin{array}{ll}
E \in \operatorname{Apx}(c) \wedge \sigma \preceq \tau \\
\wedge (\forall \rho \in S)(\forall \xi)[\xi \preceq \rho \rightarrow \xi \in S] & \mbox{ ($S$ is a tree)}\\
\wedge (\forall n)(\exists \rho \in 2^n)\rho \in S) & \mbox{ ($S$ is infinite) }\\
\wedge (\forall \rho \in S)(\sigma \prec \rho \vee \rho \preceq \sigma) & \mbox{ ($S$ has stem $\sigma$)}\\
\wedge (\forall \rho \in S)(\rho \in T) & \mbox{ ($S$ is a subset of $T$) }\\
\end{array}
$$
\end{itemize} \end{proof}
\begin{lemma}\label{lem:complexity-forcing} Fix an arithmetic formula $\varphi(G)$. \begin{itemize}
\item[a)] Given a precondition $c$, if $\varphi(G)$ is a $\Sigma^0_1$ ($\Pi^0_1$) formula
then so is the predicate $c \Vdash \varphi(G)$.
\item[b)] Given a condition $c$, if $\varphi(G)$ is a $\Sigma^0_{n+2}$ ($\Pi^0_{n+2}$) formula
then so is the predicate $c \Vdash \varphi(G)$. \end{itemize} \end{lemma} \begin{proof} We prove our lemma by induction over the complexity of the formula $\varphi(G)$. Fix a (pre)condition $c = (F, \sigma, T)$. \begin{itemize}
\item If $\varphi(G) \in \Sigma^0_1$ then it can be expressed as $(\exists x)\psi(G, x)$ where $\psi \in \Sigma^0_0$.
By clause~(i) of Definition~\ref{def:forcing-precondition}, $c \Vdash \varphi(G)$ if and only if
the formula $(\exists w \in \omega)\psi(F, w)$ holds. This is a $\Sigma^0_1$ predicate.
\item If $\varphi(G) \in \Pi^0_1$ then it can be expressed as $(\forall x)\psi(G, x)$ where $\psi \in \Sigma^0_0$.
By clause~(ii) of Definition~\ref{def:forcing-precondition}, $c \Vdash \varphi(G)$ if and only if
the formula $(\forall w \in \omega)(\forall E \in \operatorname{Apx}(c))\psi(E, w)$ holds.
By clause~a) of Lemma~\ref{lem:extension-complexity}, this is a $\Pi^0_1$ predicate.
\item If $\varphi(G) \in \Sigma^0_{n+2}$ then it can be expressed as $(\exists x)\psi(G, x)$ where $\psi \in \Pi^0_{n+1}$.
By clause~(i) of Definition~\ref{def:forcing-condition}, $c \Vdash \varphi(G)$ if and only if
the formula $(\exists w < |\sigma|)c \Vdash \psi(G, w)$ holds. This is a $\Sigma^0_{n+2}$ predicate
by induction hypothesis.
\item If $\varphi(G) \in \Pi^0_2$ then it can be expressed as $(\forall x)\psi(G, x)$ where $\psi \in \Sigma^0_1$.
By clause~(ii) of Definition~\ref{def:forcing-condition}, $c \Vdash \varphi(G)$ if and only if
the formula $(\forall \tau \in T)(\forall w < |\tau|)(\forall E \in \operatorname{Apx}_{|\tau|}(c))
(E, \tau, T^{[\tau]}) \not \Vdash \neg \psi(G, w)$ holds.
By induction hypothesis, $(E, \tau, T^{[\tau]}) \not \Vdash \neg \psi(G, w)$ is a $\Sigma^0_1$ predicate,
hence by clause~a) of Lemma~\ref{lem:extension-complexity}, $c \Vdash \varphi(G)$ is a $\Pi^0_2$ predicate.
\item If $\varphi(G) \in \Pi^0_{n+3}$ then it can be expressed as $\neg \psi(G)$ where $\psi \in \Sigma^0_{n+3}$.
By clause~(iii) of Definition~\ref{def:forcing-condition}, $c \Vdash \varphi(G)$ if and only if
the formula $(\forall d)(d \not \in \operatorname{Ext}(c) \vee d \not \Vdash \psi(G))$ holds.
By induction hypothesis, $d \not \Vdash \psi(G)$ is a $\Pi^0_{n+3}$ predicate.
Hence by clause~b) of Lemma~\ref{lem:extension-complexity},
$c \Vdash \varphi(G)$ is a $\Pi^0_{n+3}$ predicate. \end{itemize} \end{proof}
\subsection{Preserving the arithmetic hierarchy}
The following lemma asserts that every sufficiently generic real for this notion of forcing preserves the arithmetic hierarchy. The argument deeply relies on the fact that this notion of forcing admits a forcing relation with good definitional properties.
\begin{lemma}\label{lem:diagonalization} If $A \not \in \Sigma^0_{n+1}$ and $\varphi(G, x)$ is $\Sigma^0_{n+1}$, then the set of $c \in \mathbb{P}$ satisfying the following property is dense: $$ [(\exists w \in A)c \Vdash \neg \varphi(G, w)] \vee [(\exists w \not \in A)c \Vdash \varphi(G, w)] $$ \end{lemma} \begin{proof} Fix a condition $c = (F, \sigma, T)$. \begin{itemize}
\item In case $n = 0$, $\varphi(G, w)$ can be expressed as $(\exists x)\psi(G, w, x)$ where $\psi \in \Sigma^0_0$.
Let $U = \{ w \in \omega : (\exists E \in \operatorname{Apx}(c))(\exists u)\psi(E, w, u) \}$.
By clause~a) of Lemma~\ref{lem:extension-complexity}, $U \in \Sigma^0_1$, thus $U \neq A$.
Fix $w \in U \Delta A$. If $w \in U \smallsetminus A$ then by definition of~$U$,
there exist an $E \in \operatorname{Apx}(c)$ and a $u \in \omega$ such that $\psi(E, w, u)$ holds.
By definition of $\operatorname{Apx}(c)$ and clause~1) of Lemma~\ref{lem:basic-statements}, $d = (E, \sigma, T)$
is a condition extending $c$. By clause~(i) of Definition~\ref{def:forcing-precondition},
$d \Vdash \varphi(G, w)$.
If $w \in A \smallsetminus U$, then for every $E \in \operatorname{Apx}(c)$ and every $u \in \omega$,
$\psi(E, w, u)$ does not hold, so by clause~(ii) of Definition~\ref{def:forcing-precondition},
$c \Vdash (\forall x)\neg \psi(G, w, x)$, hence $c \Vdash \neg \varphi(G, w)$.
\item In case $n = 1$, $\varphi(G, w)$ can be expressed as $(\exists x)\psi(G, w, x)$ where $\psi \in \Pi^0_1$.
Let $U = \{ w \in \omega : (\exists s)(\forall \tau \in 2^s \cap T)
(\exists u < s)(\exists E \in \operatorname{Apx}_s(c))(E, \tau, T^{[\tau]}) \Vdash \psi(G, w, u) \}$.
By Lemma~\ref{lem:complexity-forcing} and clause~a) of Lemma~\ref{lem:extension-complexity},
$U \in \Sigma^0_2$, thus $U \neq A$. Fix $w \in U \Delta A$.
If $w \in U \smallsetminus A$ then by definition of~$U$, there exist an $s \in \omega$,
a $\tau \in 2^s \cap T$, a $u < s$ and an $E \in \operatorname{Apx}_s(c)$ such that $T^{[\tau]}$ is infinite
and $(E, \tau, T^{[\tau]}) \Vdash \psi(G, w, u)$. Thus $d = (E, \tau, T^{[\tau]})$ is a condition
extending $c$ and by clause~(i) of Definition~\ref{def:forcing-condition}, $d \Vdash \varphi(G, w)$.
If $w \in A \smallsetminus U$, then let
$S = \{ \tau \in T : (\forall u < |\tau|)(\forall E \in \operatorname{Apx}_{|\tau|}(c) (E, \tau, T^{[\tau]}) \not \Vdash \psi(G, w, u) \}$.
As proven in Lemma~\ref{lem:forcing-dense}, $S$ is a $\emptyset'$-p.r. subtree of~$T$
and by $w \not \in U$, $S$ is infinite. Thus $d = (F, \sigma, S)$ is a condition extending $c$.
By clause~3) of Lemma~\ref{lem:basic-statements}, $\operatorname{Apx}(d) \subseteq \operatorname{Apx}(c)$,
so for every $\tau \in S$, every $u < |\tau|$, and every $E \in \operatorname{Apx}_{|\tau|}(d)$,
$(E, \tau, T^{[\tau]}) \not \Vdash \psi(G, w, u)$. By Lemma~\ref{lem:tree-no-effect-first-level},
$(E, \tau, S^{[\tau]}) \not \Vdash \psi(G, w, u)$, so by clause~(ii) of Definition~\ref{def:forcing-condition},
$d \Vdash (\forall x)\neg \psi(G, w, u)$ hence $d \Vdash \neg \varphi(G, w)$.
\item In case $n > 1$, let $U = \{ w \in \omega : (\exists d \in \operatorname{Ext}(c)) d \Vdash \varphi(G, w) \}$.
By clause~b) of Lemma~\ref{lem:extension-complexity} and Lemma~\ref{lem:complexity-forcing},
$U \in \Sigma^0_n$, thus $U \neq A$.
Fix $w \in U \Delta A$. If $w \in U \smallsetminus A$ then by definition of~$U$,
there exists a condition $d$ extending $c$ such that $d \Vdash \varphi(G, w)$.
If $w \in A \smallsetminus U$, then for every $d \in \operatorname{Ext}(c) d \not \Vdash \varphi(G, w)$
so by clause~(iii) of Definition~\ref{def:forcing-condition}, $c \Vdash \neg \varphi(G, w)$. \end{itemize} \end{proof}
We are now ready to prove Theorem~\ref{thm:coh-preservation-arithmetic-hierarchy}.
\begin{proof}[Proof of Theorem~\ref{thm:coh-preservation-arithmetic-hierarchy}] Let~$C$ be a set and~$R_0, R_1, \dots$ be a uniformly $C$-computable sequence of sets. Let~$T_0$ be a $C'$-primitive recursive tree such that~$[T_0] \subseteq \mathcal{C}(\vec{R})$. Let~$\mathcal{F}$ be a sufficiently generic filter containing~$c_0 = (\emptyset, \epsilon, T_0)$. and let $G$ be the corresponding generic real. By genericity, the set~$G$ is an infinite $\vec{R}$-cohesive set. By Lemma~\ref{lem:diagonalization} and Lemma~\ref{lem:complexity-forcing}, $G$ preserves non-$\Sigma^0_{n+1}$ definitions relative to~$C$ for every~$n \in \omega$. Therefore, by Proposition 2.2 of~\cite{Wang2014Definability}, $G$ preserves the arithmetic hierarchy relative to~$C$. \end{proof}
\section{The Erd\H{o}s Moser theorem preserves the arithmetic hierarchy}
We now extend the previous result to the Erd\H{o}s-Moser theorem. The Erd\H{o}s-Moser theorem is a statement coming from graph theory. It can be used with the ascending descending principle~($\ads$) to provide an alternative proof of Ramsey's theorem for pairs ($\rt^2_2$). Indeed, every coloring~$f : [\omega]^2 \to 2$ can be seen as a tournament~$R$ such that~$R(x,y)$ holds if~$x < y$ and~$f(x,y) = 1$, or~$x > y$ and~$f(y, x) = 0$. Every infinite transitive subtournament induces a linear order whose infinite ascending or descending sequences are homogeneous for~$f$.
\begin{definition}[Erd\H{o}s-Moser theorem] A tournament $T$ on a domain $D \subseteq \omega$ is an irreflexive binary relation on~$D$ such that for all $x,y \in D$ with $x \not= y$, exactly one of $T(x,y)$ or $T(y,x)$ holds. A tournament $T$ is \emph{transitive} if the corresponding relation~$T$ is transitive in the usual sense. A tournament $T$ is \emph{stable} if $(\forall x \in D)(\exists n)[(\forall s > n) T(x,s) \vee (\forall s > n) T(s, x)]$. $\emo$ is the statement ``Every infinite tournament $T$ has an infinite transitive subtournament.'' $\semo$ is the restriction of $\emo$ to stable tournaments. \end{definition}
Bovykin and Weiermann proved in \cite{Bovykin2005strength} that $\emo + \ads$ is equivalent to $\rt^2_2$ over $\rca$, and $\semo + \sads$ is equivalent to~$\srt^2_2$ over~$\rca$. Lerman et al.~\cite{Lerman2013Separating} proceeded to a combinatorial and effective analysis of the Erd\H{o}s-Moser theorem, and proved in particular that there is an $\omega$-model of $\emo$ which is not a model of~$\srt^2_2$. The author simplified their proof in~\cite{Patey2015Iterative} and showed in~\cite{Patey2015Somewhere} that $\rca \vdash \emo \rightarrow [\sts^2 \vee \coh]$, where $\sts^2$ stands for the stable thin set theorem for pairs. In particular, since Wang~\cite{Wang2014Definability} proved that $\sts^2$ does not admit preservation of the arithmetic hierarchy, Theorem~\ref{thm:coh-preservation-arithmetic-hierarchy} follows from Theorem~\ref{thm:em-preserves-arithmetic}. From a definitional point of view, Wang proved in~\cite{Wang2014Definability} that $\emo$ admits preservation of $\Delta^0_2$ definitions and preservation of definitions beyond the $\Delta^0_2$ level. He conjectured that $\emo$ admits preservation of the arithmetic hierarchy. The balance of this section proves his conjecture.
\begin{theorem}\label{thm:em-preserves-arithmetic} $\emo$ admits preservation of the arithmetic hierarchy. \end{theorem}
Again, the core of the proof consists of finding a good forcing notion whose generics will preserve the arithmetic hierarchy. For simplicity, we will restrict ourselves to stable tournaments even though it is clear that the forcing notion can be adapted to arbitrary tournaments. The proof of Theorem~\ref{thm:em-preserves-arithmetic} will be obtained by composing the proof that cohesiveness and the stable Erd\H{o}s-Moser theorem admit preservation of the arithmetic hierarchy.
The following notion of \emph{minimal interval} plays a fundamental role in the analysis of $\emo$. See~\cite{Lerman2013Separating} for a background analysis of $\emo$.
\begin{definition}[Minimal interval] Let $T$ be an infinite tournament and $a, b \in T$ be such that $T(a,b)$ holds. The \emph{interval} $(a,b)$ is the set of all $x \in T$ such that $T(a,x)$ and $T(x,b)$ hold. Let $F \subseteq T$ be a finite transitive subtournament of $T$. For $a, b \in F$ such that $T(a,b)$ holds, we say that $(a,b)$ is a \emph{minimal interval of $F$} if there is no $c \in F \cap (a,b)$, i.e., no $c \in F$ such that $T(a,c)$ and $T(c,b)$ both hold. \end{definition}
We must introduce an preliminary variant of Mathias forcing which is more suited to the Erd\H{o}s-Moser theorem.
\subsection{Erd\H{o}s Moser forcing}
The following notion of Erd\H{o}s-Moser forcing was implicitly first used by Lerman, Solomon and Towsner~\cite{Lerman2013Separating} to separate the Erd\H{o}s-Moser theorem from stable Ramsey's theorem for pairs. The author formalized this notion of forcing in~\cite{Patey2015Degrees} to construct a low${}_2$ degree bounding the Erd\H{o}s-Moser theorem.
\begin{definition} An \emph{Erd\H{o}s Moser condition} (EM condition) for an infinite tournament $R$ is a Mathias condition $(F, X)$ where \begin{itemize}
\item[(a)] $F \cup \{x\}$ is $R$-transitive for each $x \in X$
\item[(b)] $X$ is included in a minimal $R$-interval of $F$. \end{itemize} \end{definition}
The Erd\H{o}s-Moser extension is the usual Mathias extension. EM conditions have good properties for tournaments as shown by the following lemmas. Given a tournament $R$ and two sets $E$ and $F$, we denote by $E \to_R F$ the formula $(\forall x \in E)(\forall y \in F) R(x,y) \mbox{ holds}$.
\begin{lemma}[Patey~\cite{Patey2015Degrees}]\label{lem:emo-cond-beats} Fix an EM condition $(F, X)$ for a tournament $R$. For every $x \in F$, $\{x\} \to_R X$ or $X \to_R \{x\}$. \end{lemma}
\begin{lemma}[Patey~\cite{Patey2015Degrees}]\label{lem:emo-cond-valid} Fix an EM condition $c = (F, X)$ for a tournament $R$, an infinite subset $Y \subseteq X$ and a finite $R$-transitive set $F_1 \subset X$ such that $F_1 < Y$ and $[F_1 \to_R Y \vee Y \to_R F_1]$. Then $d = (F \cup F_1, Y)$ is a valid extension of $c$. \end{lemma}
\subsection{Partition trees}
Given a string $\sigma \in k^{<\omega}$, we denote by $\mathrm{set}_\nu(\sigma)$ the set $\{ x < |\sigma| : \sigma(x) = \nu \}$ where $\nu < k$. The notion can be extended to sequences $P \in k^{\omega}$ where $\mathrm{set}_\nu(P) = \{ x \in \omega : P(x) = \nu \}$.
\begin{definition}[Partition tree] A \emph{$k$-partition tree of $[t, \infty)$} for some $k, t \in \omega$ is a tuple $(k, t, T)$ such that $T$ is a subtree of $k^{<\omega}$. A \emph{partition tree} is a $k$-partition tree of $[t, \infty)$ for some $k, t \in \omega$. \end{definition}
To simplify our notation, we may use the same letter $T$ to denote both a partition tree $(k, t, T)$ and the actual tree $T \subseteq k^{<\omega}$. We then write $\mathrm{dom}(T)$ for $[t, \infty)$ and $\mathrm{parts}(T)$ for~$k$. Given a p.r.\ partition tree~$T$, we write~$\#T$ for its Turing index, and may refer to it as its \emph{code}.
\begin{definition}[Refinement] Given a function $f : \ell \to k$, a string $\sigma \in \ell^{<\omega}$ \emph{$f$-refines} a string $\tau \in k^{<\omega}$
if $|\sigma| = |\tau|$ and for every $\nu < \ell$, $\mathrm{set}_\nu(\sigma) \subseteq \mathrm{set}_{f(\nu)}(\tau)$. A p.r.\ $\ell$-partition tree $S$ of $[u,\infty)$ \emph{$f$-refines} a p.r.\ $k$-partition tree $T$ of $[t, \infty)$ (written $S \leq_f T$) if $\#S \geq \#T$, $\ell \geq k$, $u \geq t$ and for every $\sigma \in S$, $\sigma$ $f$-refines some $\tau \in T$. \end{definition}
The partition trees will act as the reservoirs in the forcing conditions defined in the next section. Consequently, refining a partition tree restricts the reservoir, as desired when extending a condition. The collection of partition trees is equipped with a partial order $\leq$ such that $(\ell, u, S) \leq (k, t, T)$ if there exists a function $f : \ell \to k$ such that $S \leq_f T$. Given a $k$-partition tree of $[t, \infty)$ $T$, we say that part $\nu$ of $T$ is \emph{acceptable} if there exists a path $P$ through $T$ such that $\mathrm{set}_\nu(P)$ is infinite. Moreover, we say that part $\nu$ of $T$ is \emph{empty} if $(\forall \sigma \in T)[dom(T) \cap \mathrm{set}_\nu(\sigma) = \emptyset]$. Note that each partition tree has at least one acceptable part since for every path~$P$ through~$T$, $\mathrm{set}_\nu(P)$ is infinite for some~$\nu < k$. It can also be the case that part~$\nu$ of~$T$ is non-empty, while for every path $P$ through~$T$, $\mathrm{set}_\nu(P) \cap \mathrm{dom}(T) = \emptyset$. However, in this case, we can choose the infinite computable subtree~$S = \{\sigma \in T : \mathrm{set}_\nu(\sigma) \cap \mathrm{dom}(T) = \emptyset\}$ of~$T$ which has the same collection of infinite paths and such that part~$\nu$ of~$S$ is empty.
Given a $k$-partition tree $T$, a finite set $F \subseteq \omega$ and a part $\nu < k$, define $$
T^{[\nu, F]} = \{ \sigma \in T : F \subseteq \mathrm{set}_\nu(\sigma) \vee |\sigma| < \max F \} $$ The set $T^{[\nu, F]}$ is a (possibly finite) subtree of~$T$ which id-refines $T$ and such that~$F \subseteq \mathrm{set}_\nu(P)$ for every path~$P$ through $T^{[\nu, F]}$.
We denote by $\mathbb{U}$ the set of all ordered pairs $(\nu, T)$ such that $T$ is an infinite, primitive recursive $k$-partition tree of $[t, \infty)$ for some $t, k \in \omega$ and $\nu < k$. The set $\mathbb{U}$ is equipped with a partial ordering $\leq$ such that $(\mu, S) \leq (\nu, T)$ if $S$ $f$-refines $T$ and $f(\mu) = \nu$ for some~$f$. In this case we say that \emph{part $\mu$ of $S$ refines part $\nu$ of~$T$}. Note that the domain of $\mathbb{U}$ and the relation $\leq$ are co-c.e. We denote by $\mathbb{U}[T]$ the set of all $(\nu, S) \leq (\mu, T)$ for some $(\mu, T) \in \mathbb{U}$.
\begin{definition}[Promise for a partition tree]\label{def:em-promise} Fix a p.r. $k$-partition tree of $[t,\infty)$ $T$. A class $\mathcal{C} \subseteq \mathbb{U}[T]$ is a \emph{promise for $T$} if \begin{itemize}
\item[a)] $\mathcal{C}$ is upward-closed under the $\leq$ relation restricted to $\mathbb{U}[T]$
\item[b)] for every infinite p.r. partition tree $S \leq T$,
$(\mu, S) \in \mathcal{C}$ for some non-empty part $\mu$ of~$S$. \end{itemize} \end{definition}
A promise for $T$ can be seen as a two-dimensional tree with at first level the acyclic digraph of refinement of partition trees. Given an infinite path in this digraph, the parts of the members of this path form an infinite, finitely branching tree. The following lemma holds for every $\emptyset'$-computable promise. However, we shall work later with conditions containing $\emptyset'$-primitive recursive promises in order to lower the definitional complexity of being a valid condition and to be able to prove Lemma~\ref{lem:em-complexity-forcing}. We therefore focus on $\emptyset'$-p.r.\ promises.
\begin{lemma}\label{lem:em-refinement-complexity} Let $T$ and $S$ be p.r.\ partition trees such that $S \leq_f T$ for some function $f : \mathrm{parts}(S) \to \mathrm{parts}(T)$ and let $\mathcal{C}$ be a $\emptyset'$-p.r.\ promise for $T$. \begin{itemize}
\item[a)] The predicate ``$T$ is an infinite $k$-partition tree of $[t, \infty)$'' is $\Pi^0_1$ uniformly in $T$, $k$ and~$t$.
\item[b)] The relations ``$S$ $f$-refines $T$''
and ``part $\nu$ of $S$ $f$-refines part $\mu$ of $T$''
are $\Pi^0_1$ uniformly in $S$, $T$ and $f$.
\item[c)] The predicate ``$\mathcal{C}$ is a promise for $T$'' is $\Pi^0_2$ uniformly in an index for~$\mathcal{C}$ and~$T$. \end{itemize} \end{lemma} \begin{proof}\ \begin{itemize}
\item[a)] $T$ is an infinite $k$-partition tree of $[t, \infty)$ if and only if the $\Pi^0_1$ formula $[(\forall \sigma \in T)(\forall \tau \preceq \sigma) \tau \in T \cap k^{<\infty}]
\wedge [(\forall n)(\exists \tau \in k^n) \tau \in T]$ holds.
\item[b)] Suppose that $T$ is a $k$-partition tree of $[t,\infty)$ and $S$ is an $\ell$-partition tree of $[u,\infty)$. $S$ $f$-refines $T$ if and only if the $\Pi^0_1$ formula holds: $$
u \geq t \wedge \ell \geq k \wedge [(\forall \sigma \in S)(\exists \tau \in k^{|\sigma|} \cap T) (\forall \nu < u)set_\nu(\sigma) \subseteq \mathrm{set}_{f(\nu)}(\tau)] $$ Part $\nu$ of $S$ $f$-refines part $\mu$ of $T$ if and only if $\mu = f(\nu)$ and $S$ $f$-refines $T$.
\item[c)] Given $k, t \in \omega$, let $PartTree(k, t)$ denote
the $\Pi^0_1$ set of all the infinite p.r.\ $k$-partition trees of $[t, \infty)$.
Given a $k$-partition tree $S$ and a part $\nu$ of $S$,
let $Empty(S, \nu)$ denote the $\Pi^0_1$ formula ``part $\nu$ of $S$ is empty'',
that is the formula $(\forall \sigma \in S) \mathrm{set}_\nu(\sigma) \cap \mathrm{dom}(S) = \emptyset$.
$\mathcal{C}$ is a promise for $T$ if and only if the following $\Pi^0_2$ formula holds:
$$
\begin{array}{l}
(\forall \ell, u)(\forall S \in PartTree(\ell,u))[S \leq T \rightarrow (\exists \nu < \ell) \neg Empty(S, \nu) \wedge (\nu, S) \in \mathcal{C})] \\
\wedge (\forall \ell', u')(\forall V \in PartTree(\ell', u'))(\forall g : \ell \to \ell')[ S \leq_g V \leq T \rightarrow \\
(\forall \nu < \ell)((\nu, S) \in \mathcal{C} \rightarrow (g(\nu), V) \in \mathcal{C})]
\end{array}
$$ \end{itemize} \end{proof}
Given a promise $\mathcal{C}$ for $T$ and some infinite p.r. partition tree $S$ refining $T$, we denote by $\mathcal{C}[S]$ the set of all $(\nu, S') \in \mathcal{C}$ below some $(\mu, S) \in \mathcal{C}$, that is, $\mathcal{C}[S] = \mathcal{C} \cap \mathbb{U}[S]$. Note that by clause b) of Lemma~\ref{lem:em-refinement-complexity}, if $\mathcal{C}$ is a $\emptyset'$-p.r. promise for $T$ then $\mathcal{C}[S]$ is a $\emptyset'$-p.r. promise for~$S$.
Establishing a distinction between the acceptable parts and the non-acceptable ones requires a lot of definitional power. However, we prove that we can always find an extension where the distinction is $\Delta^0_2$. We say that an infinite p.r. partition tree $T$ \emph{witnesses its acceptable parts} if its parts are either acceptable or empty.
\begin{lemma}\label{lem:em-promise-extension-witnessing-acceptable} For every infinite p.r.\ $k$-partition tree $T$ of $[t, \infty)$, there exists an infinite p.r.\ $k$-partition tree $S$ of $[u, \infty)$ refining $T$ with the identity function and such that $S$ witnesses its acceptable parts. \end{lemma} \begin{proof} Given a partition tree $T$, we let $I(T)$ be the set of its empty parts. Let~$T$ be a fixed infinite p.r.\ $k$-partition tree of $[t, \infty)$. It suffices to prove that if $\nu$ is a non-empty and non-acceptable part of~$T$, then there exists an infinite p.r.\ $k$-partition tree $S$
refining $T$ with the identity function, such that $\nu \in I(S) \smallsetminus I(T)$. As $I(T) \subseteq I(S)$ and $|I(S)| \leq k$, it suffices to iterate the process at most $k$ times to obtain a refinement witnessing its acceptable parts.
So fix a non-empty and non-acceptable part $\nu$ of~$T$. By definition of being non-acceptable, there exists a path $P$ through $T$ and an integer $u > \max(t, \mathrm{set}_\nu(P))$. Let $S = \{ \sigma \in T : \mathrm{set}_\nu(\sigma) \cap [u, \infty) = \emptyset \}$. The set $S$ is a p.r. $k$-partition tree of $[u, \infty)$ refining $T$ with the identity function and such that part $\nu$ of $S$ is empty. Moreover, $S$ is infinite since~$P \in [S]$. \end{proof}
The following lemma strengthens clause b) of Definition~\ref{def:em-promise}.
\begin{lemma}\label{lem:promise-keeps-acceptable-parts} Let $T$ be a p.r. partition tree and $\mathcal{C}$ be a promise for $T$. For every infinite p.r. partition tree $S \leq T$, $(\mu, S) \in \mathcal{C}$ for some acceptable part $\mu$ of~$S$. \end{lemma} \begin{proof} Fix an infinite p.r. $\ell$-partition tree $S \leq T$. By Lemma~\ref{lem:em-promise-extension-witnessing-acceptable}, there exists an infinite p.r. $\ell$-partition tree $S' \leq_{id} S$ witnessing its acceptable parts. As $\mathcal{C}$ is a promise for $T$ and $S' \leq T$, there exists a non-empty (hence acceptable) part $\nu$ of $S'$ such that $(\nu, S') \in \mathcal{C}$. As $\mathcal{C}$ is upward-closed, $(\nu, S) \in \mathcal{C}$. \end{proof}
\subsection{Forcing conditions}
We now describe the forcing notion for the Erd\H{o}s-Moser theorem. Recall that an EM condition for an infinite tournament~$R$ is a Mathias condition $(F, X)$ where $F \cup \{x\}$ is $R$-transitive for each $x \in X$ and $X$ is included in a minimal $R$-interval of $F$.
\begin{definition} We denote by~$\mathbb{P}$ the forcing notion whose conditions are tuples $(\vec{F}, T, \mathcal{C})$ where \begin{itemize}
\item[(a)] $T$ is an infinite p.r.\ partition tree
\item[(b)] $\mathcal{C}$ is a $\emptyset'$-p.r. promise for $T$
\item[(c)] $(F_\nu, \mathrm{dom}(T))$ is an EM condition for $R$ and each $\nu < \mathrm{parts}(T)$ \end{itemize} A condition $d = (\vec{E}, S, \mathcal{D})$ \emph{extends} $c = (\vec{F}, T, \mathcal{C})$ (written $d \leq c$) if there exists a function $f : \ell \to k$ such that $\mathcal{D} \subseteq \mathcal{C}$ and the followings hold: \begin{itemize}
\item[(i)] $(E_\nu, \mathrm{dom}(S))$ EM extends $(F_{f(\nu)}, \mathrm{dom}(T))$ for each $\nu < \mathrm{parts}(S)$
\item[(ii)] $S$ $f$-refines $\bigcap_{\nu < \mathrm{parts}(S)} T^{[f(\nu), E_\nu]}$ \end{itemize} \end{definition}
We may think of a condition $c = (\vec{F}, T, \mathcal{C})$ as a collection of EM conditions $(F_\nu, H_\nu)$ for~$R$, where $H_\nu = \mathrm{dom}(T) \cap \mathrm{set}_\nu(P)$ for some path $P$ through $T$. $H_\nu$ must be infinite for at least one of the parts $\nu < \mathrm{parts}(T)$. At a higher level, $\mathcal{D}$ restricts the possible subtrees $S$ and parts $\mu$ refining some part of $T$ in the condition~$c$. Given a condition $c = (\vec{F}, T, \mathcal{C})$, we write $\mathrm{parts}(c)$ for $\mathrm{parts}(T)$.
\begin{lemma}\label{lem:em-forcing-infinite} For every condition $c = (\vec{F}, T, \mathcal{C})$ and every $n \in \omega$, there exists an extension $d = (\vec{E}, S, \mathcal{D})$
such that $|E_\nu| \geq n$ on each acceptable part $\nu$ of~$S$. \end{lemma} \begin{proof} It suffices to prove that for every condition $c = (\vec{F}, T, \mathcal{C})$ and every acceptable part $\nu$ of $T$, there exists an extension $d = (\vec{E}, S, \mathcal{D})$ such that $S \leq_{id} T$
and $|E_\nu| \geq n$. Iterating the process at most $\mathrm{parts}(T)$ times completes the proof. Fix an acceptable part $\nu$ of $T$ and a path $P$ trough $T$ such that $\mathrm{set}_\nu(P)$ is infinite. Let $F'$ be an $R$-transitive subset of $\mathrm{set}_\nu(P) \cap \mathrm{dom}(T)$ of size $n$. Such a set exists by the classical Erd\H{o}s-Moser theorem. Let $\vec{E}$ be defined by $E_\mu = F_\mu$ if $\mu \neq \nu$ and $E_\nu = F_\nu \cup F'$ otherwise. As the tournament~$R$ is stable, there exists some~$u \geq t$ such that $(E_\nu, [u, \infty))$ is an EM condition and therefore EM extends $(F_\nu, \mathrm{dom}(T))$. Let $S$ be the p.r.\ partition tree $T^{[\nu, E_\nu]}$ of $[u, \infty)$. The condition $(\vec{E}, S, \mathcal{C}[S])$ is the desired extension. \end{proof}
Given a condition~$c \in \mathbb{P}$, we denote by $\operatorname{Ext}(c)$ the set of all its extensions.
\subsection{The forcing relation}
The forcing relation at the first level, namely, for $\Sigma^0_1$ and $\Pi^0_1$ formulas, is parameterized by some part of the tree of the considered condition. Thanks to the forcing relation we will define, we can build an infinite decreasing sequence of conditions which decide $\Sigma^0_1$ and~$\Pi^0_1$ formulas effectively in~$\emptyset'$. This sequence yields a $\emptyset'$-computably bounded $\emptyset'$-computable tree of (possibly empty) parts. Therefore, any PA degree relative to~$\emptyset'$ is sufficient to control the first jump of an infinite transitive subtournament of a stable infinite computable tournament.
We cannot do better since Kreuzer proved in~\cite{Kreuzer2012Primitive} the existence of an infinite, stable, computable tournament with no low infinite transitive subtournament. If we ignore the promise part of a condition, the careful reader will recognize the construction of Cholak, Jockusch and Slaman~\cite{Cholak2001strength} of a low${}_2$ infinite subset of a $\Delta^0_2$ set or its complement by the first jump control. The difference, which at first seems only notational, is in fact one of the key features of this notion of forcing. Indeed, forcing iterated jumps requires a definitionally weak description of the set of extensions of a condition, and it requires much less computational power to describe a primitive recursive tree than an infinite reservoir of a Mathias condition.
\begin{definition}\label{def:em-forcing-precondition} Fix a condition $c = (\vec{F}, T, \mathcal{C})$, a $\Sigma^0_0$ formula $\varphi(G, x)$ and a part $\nu < \mathrm{parts}(T)$. \begin{itemize}
\item[1.] $c \Vdash_\nu (\exists x)\varphi(G, x)$ iff there exists a $w \in \omega$ such that $\varphi(F_\nu, w)$ holds.
\item[2.] $c \Vdash_\nu (\forall x)\varphi(G, x)$ iff
for every $\sigma \in T$, every $w < |\sigma|$ and every $R$-transitive set $F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma)$,
$\varphi(F_\nu \cup F', w)$ holds. \end{itemize} \end{definition}
We start by proving some basic properties of the forcing relation over~$\Sigma^0_1$ and $\Pi^0_1$ formulas. As one may expect, the forcing relation at first level is closed under the refinement relation.
\begin{lemma}\label{lem:em-forcing-extension-level1} Fix a condition $c = (\vec{F}, T, \mathcal{C})$ and a $\Sigma^0_1$ ($\Pi^0_1$) formula $\varphi(G)$. If $c \Vdash_\nu \varphi(G)$ for some $\nu < \mathrm{parts}(T)$, then for every $d = (\vec{E}, S, \mathcal{D}) \leq c$ and every part $\mu$ of $S$ refining part $\nu$ of $T$, $d \Vdash_{\mu} \varphi(G)$. \end{lemma} \begin{proof}We have two cases. \begin{itemize}
\item If $\varphi \in \Sigma^0_1$ then it can be expressed as $(\exists x)\psi(G, x)$
where $\psi \in \Sigma^0_0$. By clause~1 of Definition~\ref{def:em-forcing-precondition},
there exists a $w \in \omega$ such that
$\psi(F_\nu, w)$ holds. By property (i) of the definition of an extension, $E_\mu \supseteq F_\nu$
and $(E_\mu \smallsetminus F_\nu) \subset \mathrm{dom}(T)$, therefore $\psi(E_\mu, w)$ holds by continuity,
so by clause~1 of Definition~\ref{def:em-forcing-precondition}, $d \Vdash_\mu (\exists x)\psi(G, x)$.
\item If $\varphi \in \Pi^0_1$ then it can be expressed as $(\forall x)\psi(G, x)$
where $\psi \in \Sigma^0_0$. Fix a $\tau \in S$, a $w < |\tau|$ and an $R$-transitive set
$F' \subseteq \mathrm{dom}(S) \cap \mathrm{set}_\mu(\tau)$.
It suffices to prove that $\varphi(E_\mu \cup F')$ holds to conclude
that $d \Vdash_\mu (\forall x)\psi(G, x)$ by clause~2 of Definition~\ref{def:em-forcing-precondition}.
By property~(ii) of the definition of an extension,
there exists a $\sigma \in T^{[\nu, E_\mu]}$
such that $|\sigma| = |\tau|$ and $\mathrm{set}_\mu(\tau) \subseteq \mathrm{set}_\nu(\sigma)$.
As $\mathrm{dom}(S) \subseteq \mathrm{dom}(T)$, $F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma)$.
As $\sigma \in T^{[\nu, E_\mu]}$, $E_\mu \subseteq \mathrm{set}_\nu(\sigma)$
and by property~(i) of the definition of an extension, $E_\mu \subseteq \mathrm{dom}(T)$.
So $E_\mu \cup F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma)$.
As $w < |\tau| = |\sigma|$ and~$E_\mu \cup F'$ is an $R$-transitive subset of~$\mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma)$,
then by clause~2 of Definition~\ref{def:em-forcing-precondition} applied to $c \Vdash_\nu (\forall x)\psi(G, x)$,
$\varphi(F_\nu \cup (E_\mu \smallsetminus F_\nu) \cup F', w)$ holds, hence $\varphi(E_\mu \cup F')$ holds. \end{itemize} \end{proof}
Before defining the forcing relation at higher levels, we prove a density lemma for $\Sigma^0_1$ and $\Pi^0_1$ formulas. It enables us in particular to reprove that every degree PA relative to $\emptyset'$ computes the jump of an infinite $R$-transitive set.
\begin{lemma}\label{lem:em-forcing-dense-level1} For every $\Sigma^0_1$ ($\Pi^0_1$) formula $\varphi$, the following set is dense $$ \{ c = (\vec{F}, T, \mathcal{C}) \in \mathbb{P} : (\forall \nu < \mathrm{parts}(T))[ c \Vdash_\nu \varphi(G) \vee c \Vdash_\nu \neg \varphi(G)] \} $$ \end{lemma} \begin{proof} It suffices to prove the statement for the case where $\varphi$ is a $\Sigma^0_1$ formula, as the case where $\varphi$ is a $\Pi^0_1$ formula is symmetric. Fix a condition $c = (\vec{F}, T, \mathcal{C})$ and let $I(c)$ be the set of the parts $\nu < \mathrm{parts}(T)$ such that $c \not \Vdash_\nu \varphi(G)$ and $c \not \Vdash_\nu \neg \varphi(G)$. If $I(c) = \emptyset$ then we are done, so suppose $I(c) \neq \emptyset$ and fix some $\nu \in I(c)$. We will construct an extension $d$ of~$c$ such that $I(d) \subseteq I(c) \smallsetminus \{\nu\}$. Iterating the operation completes the proof.
The formula $\varphi$ is of the form $(\exists x)\psi(G, x)$ where $\psi \in \Sigma^0_0$. Define $f : k+1 \to k$ as $f(\mu) = \mu$ if $\mu < k$ and $f(k) = \nu$ otherwise. Let $S$ be the set of all $\sigma \in (k+1)^{<\omega}$ which $f$-refine some $\tau \in T \cap k^{|\sigma|}$
and such that for every $w < |\sigma|$, every part $\mu \in \{\nu, k\}$ and every finite $R$-transitive set $F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\mu(\sigma)$, $\varphi(F_\nu \cup F', w)$ does not hold.
Note that $S$ is a p.r. partition tree of $[t, \infty)$ refining $T$ with witness function~$f$. Suppose that $S$ is infinite. Let $\vec{E}$ be defined by $E_\mu = F_\mu$ if $\mu < k$ and $E_k = F_\nu$
and consider the extension $d = (\vec{E}, S, \mathcal{C}[S])$. We claim that~$\nu, k \not \in I(d)$. Fix a part $\mu \in \{\nu, k\}$ of $S$. By definition of $S$, for every $\sigma \in S$, every $w < |\sigma|$ and every $R$-transitive set $F' \subseteq \mathrm{dom}(S) \cap \mathrm{set}_\mu(\sigma)$, $\varphi(E_\mu \cup F', w)$ does not hold. Therefore, by clause~2 of Definition~\ref{def:em-forcing-precondition}, $d \Vdash_\mu (\forall x)\neg \psi(G, x)$, hence $d \Vdash_\mu \neg \varphi(G)$. Note that $I(d) \subseteq I(c) \smallsetminus \{\nu\}$.
Suppose now that $S$ is finite. Fix a threshold $\ell \in \omega$ such that $(\forall \sigma \in S)|\sigma| < \ell$ and a $\tau \in T \cap k^\ell$ such that $T^{[\tau]}$ is infinite. Consider the 2-partition $E_0 \sqcup E_1$ of $\mathrm{set}_\nu(\tau) \cap \mathrm{dom}(T)$ defined by $E_0 = \{ i \geq t : \tau(i) = \nu \wedge (\exists n)(\forall s > n) R(i, s) \mbox{ holds}\}$ and $E_1 = \{ i \geq t : \tau(i) = \nu \wedge (\exists n)(\forall s > n) R(s, i) \mbox{ holds}\}$. This is a 2-partition since the tournament~$R$ is stable. As there exists no $\sigma \in S$ which $f$-refines $\tau$, there exists a $w < \ell$ and an $R$-transitive set $F' \subseteq E_0$ or $F' \subseteq E_1$ such that $\varphi(F_\nu \cup F', w)$ holds. By choice of the partition, there exists a $t' > t$ such that $F' \to_R [t', \infty)$ or $[t', \infty) \to_R F'$. By Lemma~\ref{lem:emo-cond-valid}, $(F_\nu \cup F', [t', \infty))$ is a valid EM extension for $(F_\nu, [t, \infty))$. As $T^{[\tau]}$ is infinite, $T^{[\nu, F']}$ is also infinite. Let $\vec{E}$ be defined by $E_\mu = F_\mu$ if $\mu \neq \nu$ and $E_\mu = F_\nu \cup F'$ otherwise. Let $S$ be the $k$-partition tree $(k, t', T^{[\nu, F']})$. The condition $d = (\vec{E}, S, \mathcal{C}[S])$ is a valid extension of $c$. By clause~1 of Definition~\ref{def:em-forcing-precondition}, $d \Vdash_\mu \varphi(G)$. Therefore $I(d) \subseteq I(c) \smallsetminus \{\nu\}$. \end{proof}
As in the previous notion of forcing, the following trivial lemma expresses the fact that the promise part of a condition has no effect in the forcing relation for a $\Sigma^0_1$ or $\Pi^0_1$ formula.
\begin{lemma}\label{lem:em-promise-no-effect-first-level} Fix two conditions $c = (\vec{F}, T, \mathcal{C})$ and $d = (\vec{F}, T, \mathcal{D})$, and a $\Sigma^0_1$ ($\Pi^0_1$) formula. For every part $\nu$ of $T$, $c \Vdash_\nu \varphi(G)$ if and only if $d \Vdash_\nu \varphi(G)$. \end{lemma} \begin{proof} If $\varphi \in \Sigma^0_1$ then $\varphi(G)$ can be expressed as $(\exists x)\psi(G, x)$ where $\psi \in \Sigma^0_0$. By clause~1 of Definition~\ref{def:em-forcing-precondition}, $c \Vdash_\nu \varphi(G)$ iff there exists a $w \in \omega$ such that $\psi(F_\nu, w)$ holds, iff $d \Vdash_\nu \varphi(G)$. Similarily, if $\varphi \in \Pi^0_1$ then $\varphi(G)$ can be expressed as $(\forall x)\psi(G, x)$ where $\psi \in \Sigma^0_0$. By clause~2 of Definition~\ref{def:em-forcing-precondition},
$c \Vdash_\nu \varphi(G)$ iff for every $\sigma \in T$, every $w < |\sigma|$ and every $R$-transitive set $F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma)$, $\varphi(F_\nu \cup F', w)$ holds, iff $d \Vdash_\nu \varphi(G)$. \end{proof}
We are now ready to define the forcing relation for an arbitrary arithmetic formula. Again, the natural forcing relation induced by the forcing of~$\Sigma^0_0$ formulas is too complex, so we design a more effective relation which still enjoys the main properties of a forcing relation.
\begin{definition}\label{def:em-forcing-condition} Fix a condition $c = (\vec{F}, T, \mathcal{C})$ and an arithmetic formula $\varphi(G)$. \begin{itemize}
\item[1.] If $\varphi(G) = (\exists x)\psi(G, x)$ where $\psi \in \Pi^0_1$ then
$c \Vdash \varphi(G)$ iff for every part $\nu < \mathrm{parts}(T)$ such that
$(\nu, T) \in \mathcal{C}$ there exists a $w < \mathrm{dom}(T)$ such that $c \Vdash_\nu \psi(G, w)$
\item[2.] If $\varphi(G) = (\forall x)\psi(G, x)$ where $\psi \in \Sigma^0_1$ then
$c \Vdash \varphi(G)$ iff for every infinite p.r.\ $k'$-partition tree $S$,
every function~$f : \mathrm{parts}(S) \to \mathrm{parts}(T)$,
every $w$ and $\vec{E}$ smaller than $\#S$ such that the followings hold
\begin{itemize}
\item[i)] $(E_\nu, \mathrm{dom}(S))$ EM extends $(F_{f(\nu)}, \mathrm{dom}(T))$ for each $\nu < \mathrm{parts}(S)$
\item[ii)] $S$ $f$-refines $\bigcap_{\nu < \mathrm{parts}(S)} T^{[f(\nu), E_\nu]}$
\end{itemize}
for every $(\mu, S) \in \mathcal{C}$, $(\vec{E}, S, \mathcal{C}[S]) \not \Vdash_\mu \neg \psi(G, w)$
\item[3.] If $\varphi(G) = (\exists x)\psi(G, x)$ where $\psi \in \Pi^0_{n+2}$ then
$c \Vdash \varphi(G)$ iff there exists a $w \in \omega$ such that $c \Vdash \psi(G, w)$
\item[4.] If $\varphi(G) = \neg \psi(G,x)$ where $\psi \in \Sigma^0_{n+3}$
then $c \Vdash \varphi(G)$ iff $d \not \Vdash \psi(G)$ for every $d \in \operatorname{Ext}(c)$. \end{itemize} \end{definition}
Notice that, unlike the forcing relation for $\Sigma^0_1$ and $\Pi^0_1$ formulas, the relation over higher formuals does not depend on the part of the relation. The careful reader will have recognized the combinatorics of the second jump control introduced by Cholak, Jockusch and Slaman in~\cite{Cholak2001strength}. We now prove the main properties of this forcing relation.
\begin{lemma}\label{lem:em-forcing-extension} Fix a condition $c$ and a $\Sigma^0_{n+2}$ ($\Pi^0_{n+2}$) formula $\varphi(G)$. If $c \Vdash \varphi(G)$ then for every $d \leq c$, $d \Vdash \varphi(G)$. \end{lemma} \begin{proof} We prove the statement by induction over the complexity of the formula $\varphi(G)$. Fix a condition $c = (\vec{F}, T, \mathcal{C})$ such that $c \Vdash \varphi(G)$ and an extension $d = (\vec{E}, S, \mathcal{D})$ of~$c$. \begin{itemize}
\item If $\varphi \in \Sigma^0_2$ then $\varphi(G)$ can be expressed as $(\exists x)\psi(G, x)$
where $\psi \in \Pi^0_1$. By clause~1 of Definition~\ref{def:em-forcing-condition},
for every part $\nu$ of $T$ such that $(\nu, T) \in \mathcal{C}$, there exists a $w < \mathrm{dom}(T)$
such that $c \Vdash_\nu \psi(G, w)$. Fix a part $\mu$ of $S$ such that $(\mu, S) \in \mathcal{D}$.
As $\mathcal{D} \subseteq \mathcal{C}$, $(\mu, S) \in \mathcal{C}$. By upward-closure of $\mathcal{C}$,
part $\mu$ of $S$ refines some part $\nu$ of $\mathcal{C}$ such that $(\nu, T) \in \mathcal{C}$.
Therefore by Lemma~\ref{lem:em-forcing-extension-level1}, $d \Vdash_\mu \psi(G, w)$,
with $w < \mathrm{dom}(T) \leq \mathrm{dom}(S)$. Applying again clause~1 of Definition~\ref{def:em-forcing-condition},
we deduce that $d \Vdash (\forall x)\psi(G, x)$, hence $d \Vdash \varphi(G)$.
\item If $\varphi \in \Pi^0_2$ then $\varphi(G)$ can be expressed as $(\forall x)\psi(G, x)$ where $\psi \in \Sigma^0_1$.
Suppose by way of contradiction that $d \not \Vdash (\forall x)\psi(G, x)$.
Let $f : \mathrm{parts}(S) \to \mathrm{parts}(T)$ witness the refinement $S \leq T$.
By clause~2 of Definition~\ref{def:em-forcing-condition}, there exists an infinite p.r.\
$k'$-partition tree $S'$, a function~$g : \mathrm{parts}(S') \to \mathrm{parts}(S)$, a $w \in \omega$, and $\vec{H}$
smaller than the code of $S'$ such that
\begin{itemize}
\item[i)] $(H_\nu, \mathrm{dom}(S'))$ EM extends $(E_{g(\nu)}, \mathrm{dom}(S))$ for each $\nu < \mathrm{parts}(S')$
\item[ii)] $S'$ $g$-refines $\bigcap_{\nu < \mathrm{parts}(S')} S^{[g(\nu), H_\nu]}$
\item[iii)] there exists a $(\mu, S') \in \mathcal{D}$ such that $(\vec{H}, S', \mathcal{D}[S']) \Vdash_\mu \neg \psi(G, w)$.
\end{itemize}
To deduce by clause~2 of Definition~\ref{def:em-forcing-condition} that
$c \not \Vdash (\forall x)\psi(G, x)$ and derive a contradiction, it suffices to prove
that the same properties hold with respect to $T$.
\begin{itemize}
\item[i)] By property (i) of the definition of an extension,
$(E_{g(\nu)}, \mathrm{dom}(S))$ EM extends $(F_{f(g(\nu))}, \mathrm{dom}(T))$
and $(H_\nu, \mathrm{dom}(S')$ EM extends $(E_{g(\nu)}, \mathrm{dom}(S))$,
then $(H_\nu, \mathrm{dom}(S'))$ EM extends $(F_{f(g(\nu))}, \mathrm{dom}(T))$.
\item[ii)] By property (ii) of the definition of an extension,
$S$ $f$-refines $\bigcap_{\nu < \mathrm{parts}(S')} T^{[f(g(\nu)), E_{g(\nu)}]}$
and $S'$ $g$-refines $\bigcap_{\nu < \mathrm{parts}(S')} S^{[g(\nu), H_\nu]}$,
then $S'$ $(g \circ f)$-refines $\bigcap_{\nu < \mathrm{parts}(S')} T^{[(g \circ f)(\nu), H_\nu]}$.
\item[iii)] As $\mathcal{D} \subseteq \mathcal{C}$, there exists a part $(\mu, S') \in \mathcal{C}$
such that $(\vec{H}, S', \mathcal{D}[S']) \Vdash_\mu \neg \psi(G, w)$.
By Lemma~\ref{lem:em-promise-no-effect-first-level}, $(\vec{H}, S', \mathcal{C}[S']) \Vdash_\mu \neg \psi(G, w)$.
\end{itemize}
\item If $\varphi \in \Sigma^0_{n+3}$ then $\varphi(G)$ can be expressed as $(\exists x)\psi(G, x)$
where $\psi \in \Pi^0_{n+2}$.
By clause~3 of Definition~\ref{def:em-forcing-condition}, there exists a $w \in \omega$
such that $c \Vdash \psi(G, w)$. By induction hypothesis, $d \Vdash \psi(G, w)$
so by clause~3 of Definition~\ref{def:em-forcing-condition}, $d \Vdash \varphi(G)$.
\item If $\varphi \in \Pi^0_{n+3}$ then $\varphi(G)$ can be expressed as $\neg \psi(G)$ where $\psi \in \Sigma^0_{n+3}$.
By clause~4 of Definition~\ref{def:em-forcing-condition}, for every $e \in \operatorname{Ext}(c)$, $e \not \Vdash \psi(G)$.
As $\operatorname{Ext}(d) \subseteq \operatorname{Ext}(c)$, for every $e \in \operatorname{Ext}(d)$, $e \not \Vdash \psi(G)$,
so by clause~4 of Definition~\ref{def:em-forcing-condition}, $d \Vdash \varphi(G)$. \end{itemize} \end{proof}
\begin{lemma}\label{lem:em-forcing-dense} For every $\Sigma^0_{n+2}$ ($\Pi^0_{n+2}$) formula $\varphi$, the following set is dense $$ \{c \in \mathbb{P} : c \Vdash \varphi(G) \mbox{ or } c \Vdash \neg \varphi(G) \} $$ \end{lemma} \begin{proof} We prove the statement by induction over~$n$. It suffices to treat the case where $\varphi$ is a $\Sigma^0_{n+2}$ formula, as the case where $\varphi$ is a $\Pi^0_{n+2}$ formula is symmetric. Fix a condition $c = (\vec{F}, T, \mathcal{C})$. \begin{itemize}
\item In case $n = 0$, the formula $\varphi$ is of the form $(\exists x)\psi(G, x)$ where
$\psi \in \Pi^0_1$.
Suppose there exist an infinite p.r. $k'$-partition tree $S$
for some $k' \in \omega$, a function~$f : \mathrm{parts}(S) \to \mathrm{parts}(T)$ and a $k'$-tuple of finite sets $\vec{E}$ such that
\begin{itemize}
\item[i)] $(E_\nu, [\ell, \infty))$ EM extends $(F_{f(\nu)}, \mathrm{dom}(T))$ for each $\nu < \mathrm{parts}(S)$.
\item[ii)] $S$ $f$-refines $\bigcap_{\nu < \mathrm{parts}(S)} T^{[f(\nu), E_\nu]}$
\item[iii)] for each non-empty part $\nu$ of $S$ such that $(\nu, S) \in \mathcal{C}$,
$(\vec{E}, S, \mathcal{C}[S]) \Vdash_\nu \psi(G, w)$ for some $w < \#S$
\end{itemize}
We can choose $\mathrm{dom}(S)$ so that $(E_\nu, \mathrm{dom}(S))$ EM extends $(F_{f(\nu)}, \mathrm{dom}(T))$ for each $\nu < \mathrm{parts}(S)$.
Properties i-ii) remain trivially true.
By Lemma~\ref{lem:em-forcing-extension-level1} and Lemma~\ref{lem:em-promise-no-effect-first-level}, property iii) remains true too.
Let $\mathcal{D} = \mathcal{C}[S] \smallsetminus \{(\nu, S') \in \mathcal{C} : \mbox{ part } \nu \mbox{ of } S' \mbox{ is empty} \}$.
As $\mathcal{C}$ is an $\emptyset'$-p.r. promise for $T$,
$\mathcal{C}[S]$ is an $\emptyset'$-p.r. promise for $S$.
As $\mathcal{D}$ is obtained from $\mathcal{C}[S]$ by removing only empty parts, $\mathcal{D}$ is also an $\emptyset'$-p.r. promise for $S$.
By clause~1 of Definition~\ref{def:em-forcing-condition},
$d = (\vec{E}, S, \mathcal{D}) \Vdash (\exists x)\psi(G, x)$ hence $d \Vdash \varphi(G)$.
We may choose a coding of the p.r. trees such that
the code of $S$ is sufficiently large to witness $\ell$ and $\vec{E}$.
So suppose now that for every infinite p.r. $k'$-partition tree $S$,
every function~$f : \mathrm{parts}(S) \to \mathrm{parts}(T)$ and $\vec{E}$ smaller than the code of $S$ such that properties i-ii) hold,
there exists a non-empty part $\nu$ of $S$ such that $(\nu, S) \in \mathcal{C}$
and $(\vec{E}, S, \mathcal{C}) \not \Vdash_\nu \psi(G, w)$ for every $w < \ell$.
Let $\mathcal{D}$ be the collection of all such $(\nu, S)$. The set $\mathcal{D}$ is $\emptyset'$-p.r.\
since by Lemma~\ref{lem:em-complexity-forcing}, both $(\vec{E}, S, \mathcal{C}) \not \Vdash_\nu \psi(G, w)$
and ``part $\nu$ of~$S$ is non-empty'' are $\Sigma^0_1$.
By Lemma~\ref{lem:em-forcing-extension-level1} and since we require that~$\#S \geq \#T$
in the definition of~$S \leq T$, $\mathcal{D}$ is upward-closed under the refinement relation,
hence is a promise for~$T$. By clause~2
of Definition~\ref{def:em-forcing-condition}, $d = (\vec{F}, T, \mathcal{D}) \Vdash (\forall x) \neg \psi(G, x)$,
hence $d \Vdash \neg \varphi(G)$.
\item In case $n > 0$, density follows from clause~4 of Definition~\ref{def:em-forcing-condition}. \end{itemize} \end{proof}
By Lemma~\ref{lem:promise-keeps-acceptable-parts}, given any filter~$\mathcal{F} = \{c_0, c_1, \dots \}$ with $c_s = (\vec{F}_s, T_s, \mathcal{C}_s)$, the set of the acceptable parts~$\nu$ of~$T_s$ such that~$(\nu, T_s) \in \mathcal{C}_s$ forms an infinite, directed acyclic graph~$\mathcal{G}(\mathcal{F})$. Whenever~$\mathcal{F}$ is sufficiently generic, the graph~$\mathcal{G}(\mathcal{F})$ has a unique infinite path~$P$. The path~$P$ induces an infinite set~$G = \bigcup_s F_{P(s), s}$. We call~$P$ the \emph{generic path} and $G$ the \emph{generic real}.
\begin{lemma}\label{lem:em-generic-level1} Suppose that $\mathcal{F}$ is sufficiently generic and let~$P$ and~$G$ be the generic path and the generic real, respectively. For any $\Sigma^0_1$ ($\Pi^0_1$) formula $\varphi(G)$, $\varphi(G)$ holds iff $c_s \Vdash_{P(s)} \varphi(G)$ for some $c_s \in \mathcal{F}$. \end{lemma} \begin{proof} Fix a condition $c_s = (\vec{F}, T, \mathcal{C}) \in \mathcal{F}$ such that $c \Vdash_{P(s)} \varphi(G)$, and let~$\nu = P(s)$. \begin{itemize}
\item If $\varphi \in \Sigma^0_1$ then $\varphi(G)$ can be expressed as $(\exists x)\psi(G, x)$ where $\psi \in \Sigma^0_0$.
By clause~1 of Definition~\ref{def:em-forcing-precondition}, there exists a $w \in \omega$ such that
$\psi(F_\nu, w)$ holds. As $\nu = P(s)$, $F_\nu = F_{P(s)} \subseteq G$
and~$G \smallsetminus F_\nu \subseteq (\max F_\nu, \infty)$, so $\psi(G, w)$ holds by continuity, hence $\varphi(G)$ holds.
\item If $\varphi \in \Pi^0_1$ then $\varphi(G)$ can be expressed as $(\forall x)\psi(G, x)$ where $\psi \in \Sigma^0_0$.
By clause~2 of Definition~\ref{def:em-forcing-precondition}, for every $\sigma \in T$, every $w < |\sigma|$
and every $R$-transitive set $F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma)$, $\psi(F_\nu \cup F', w)$ holds.
For every $F' \subseteq G \smallsetminus F_\nu$, and $w \in \omega$ there exists a $\sigma \in T$
such that $w < |\sigma|$ and $F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma)$. Hence $\psi(F_\nu \cup F', w)$ holds.
Therefore, for every $w \in \omega$, $\psi(G, w)$ holds, so $\varphi(G)$ holds. \end{itemize} The other direction holds by Lemma~\ref{lem:em-forcing-dense-level1}. \end{proof}
\begin{lemma}\label{lem:em-generic-higher-levels} Suppose that $\mathcal{F}$ is sufficiently generic and let~$P$ and~$G$ be the generic path and the generic real, respectively. For any $\Sigma^0_{n+2}$ ($\Pi^0_{n+2}$) formula $\varphi(G)$, $\varphi(G)$ holds iff $c_s \Vdash \varphi(G)$ for some $c_s \in \mathcal{F}$. \end{lemma} \begin{proof} Assuming the reversal, we first show that if $\varphi(G)$ holds, then $c_s \Vdash \varphi(G)$ for some $c_s \in \mathcal{F}$. Indeed, by Lemma~\ref{lem:em-forcing-dense} and by genericity of $\mathcal{F}$ either $c_s \Vdash \varphi(G)$ or $c_s \Vdash \neg \varphi(G)$, but if $c \Vdash \neg \varphi(G)$ then $\neg \varphi(G)$ holds, contradicting the hypothesis. So $c_s \Vdash \varphi(G)$. We now prove the forward implication by induction over the complexity of the formula $\varphi(G)$. Fix a condition $c_s = (\vec{F}, T, \mathcal{C}) \in \mathcal{F}$ such that $c_s \Vdash \varphi(G)$. We proceed by case analysis on $\varphi$. \begin{itemize}
\item If $\varphi \in \Sigma^0_2$ then $\varphi(G)$ can be expressed as $(\exists x)\psi(G, x)$
where $\psi \in \Pi^0_1$.
By clause~1 of Definition~\ref{def:em-forcing-condition}, for every part $\nu$ of $T$
such that $(\nu, T) \in \mathcal{C}$, there exists a $w < \mathrm{dom}(T)$
such that $c_s \Vdash_\nu \psi(G, w)$. In particular $(P(s), T) \in \mathcal{C}$, so $c_s \Vdash_{P(s)} \psi(G, w)$.
By Lemma~\ref{lem:em-generic-level1}, $\psi(G, w)$ holds, hence $\varphi(G)$ holds.
\item If $\varphi \in \Pi^0_2$ then $\varphi(G)$ can be expressed as $(\forall x)\psi(G, x)$ where $\psi \in \Sigma^0_1$.
By clause~2 of Definition~\ref{def:em-forcing-condition}, for every infinite $k'$-partition
tree $S$, every function~$f : \mathrm{parts}(S) \to \mathrm{parts}(T)$,
every $w$ and $\vec{E}$ smaller than the code of $S$ such that the followings hold
\begin{itemize}
\item[i)] $(E_\nu, \mathrm{dom}(S))$ EM extends $(F_{f(\nu)}, \mathrm{dom}(T))$ for each $\nu < \mathrm{parts}(S)$
\item[ii)] $S$ $f$-refines $\bigcap_{\nu < \mathrm{parts}(S)} T^{[f(\nu), E_\nu]}$
\end{itemize}
for every $(\mu, S) \in \mathcal{C}$, $(\vec{E}, S, \mathcal{C}[S]) \not \Vdash_\mu \neg \psi(G, w)$.
Suppose by way of contradiction that $\psi(G, w)$ does not hold for some $w \in \omega$.
Then by Lemma~\ref{lem:em-generic-level1}, there exists a $d_t \in \mathcal{F}$
such that $d_t \Vdash_{P(t)} \neg \psi(G, w)$.
Since~$\mathcal{F}$ is a filter, there is a condition $e_r = (\vec{E}, S, \mathcal{D}) \in \mathcal{F}$
extending both~$c_s$ and~$d_t$.
Let~$\mu = P(r)$. By choice of~$P$, $(\mu, S) \in \mathcal{C}$, so
by clause ii), $(\vec{E}, S, \mathcal{C}[S]) \not \Vdash_\mu \psi(G, w)$,
hence by Lemma~\ref{lem:em-promise-no-effect-first-level}, $e_r \not \Vdash_\mu \neg \psi(G, w)$.
However, since part~$\mu$ of~$S$ refines part~$P(t)$ of~$d_t$,
then by Lemma~\ref{lem:em-forcing-extension-level1}, $e_r \Vdash_\mu \neg \psi(G, w)$. Contradiction.
Hence for every $w \in \omega$, $\psi(G, w)$ holds, so $\varphi(G)$ holds.
\item If $\varphi \in \Sigma^0_{n+3}$ then $\varphi(G)$ can be expressed as $(\exists x)\psi(G, x)$
where $\psi \in \Pi^0_{n+2}$.
By clause~3 of Definition~\ref{def:em-forcing-condition}, there exists a $w \in \omega$
such that $c_s \Vdash \psi(G, w)$. By induction hypothesis, $\psi(G, w)$ holds, hence $\varphi(G)$ holds.
Conversely, if $\varphi(G)$ holds, then there exists a $w \in \omega$ such that $\psi(G, w)$ holds,
so by induction hypothesis $c_s \Vdash \psi(G, w)$ for some $c_s \in \mathcal{F}$,
so by clause~3 of Definition~\ref{def:em-forcing-condition}, $c_s \Vdash \varphi(G)$.
The proof of the reversal is not redundant with the first paragraph of the proof since
it is used in the next case at the same rank.
\item If $\varphi \in \Pi^0_{n+3}$ then $\varphi(G)$ can be expressed as $\neg \psi(G)$ where $\psi \in \Sigma^0_{n+3}$.
By clause~4 of Definition~\ref{def:em-forcing-condition}, for every $d \in \operatorname{Ext}(c_s)$, $d \not \Vdash \psi(G)$.
By Lemma~\ref{lem:em-forcing-extension}, $d \not \Vdash \psi(G)$ for every~$d \in \mathcal{F}$
and by the previous case, $\psi(G)$ does not hold, so $\varphi(G)$ holds. \end{itemize} \end{proof}
We now prove that the forcing relation has good definitional properties as we did with the notion of forcing for cohesiveness.
\begin{lemma}\label{lem:em-extension-complexity} For every condition $c$, $\operatorname{Ext}(c)$ is $\Pi^0_2$ uniformly in~$c$. \end{lemma} \begin{proof} Recall from Lemma~\ref{lem:em-refinement-complexity} that given $k, t \in \omega$, $PartTree(k, t)$ denotes the $\Pi^0_1$ set of all the infinite p.r.\ $k$-partition trees of $[t, \infty)$, and given a $k$-partition tree $S$ and a part $\nu$ of $S$, the predicate $Empty(S, \nu)$ denotes the $\Pi^0_1$ formula ``part $\nu$ of $S$ is empty'', that is, the formula $(\forall \sigma \in S)[\mathrm{set}_\nu(\sigma) \cap \mathrm{dom}(S) = \emptyset]$. If $T$ is p.r. then so is $T^{[\nu, H]}$ for some finite set $H$.
Fix a condition $c = (\vec{F}, (k, t, T), \mathcal{C})$. By definition, $(\vec{H}, (k', t', S), \mathcal{D}) \in \operatorname{Ext}(c)$ iff the following formula holds: $$ \begin{array}{l@{\hskip 0.5in}r} (\exists f : k' \to k)\\ (\forall \nu < k')(H_\nu, [t', \infty)) \mbox{ EM extends } (F_{f(\nu)}, [t, \infty)) & (\Pi^0_1)\\ \wedge S \in PartTree(k', t') \wedge S \leq_f \bigwedge_{\nu < k'} T^{[f(\nu), H_{\nu}]} & (\Pi^0_1)\\ \wedge \mathcal{D} \mbox{ is a promise for } S \wedge \mathcal{D} \subseteq \mathcal{C} & (\Pi^0_2)\\ \end{array} $$ By Lemma~\ref{lem:em-refinement-complexity} and the fact that $\bigwedge_{\nu < k'} T^{[f(\nu), H_{\nu}]}$ is p.r. uniformly in $T$, $f$, $\vec{H}$ and $k'$, the above formula is $\Pi^0_2$. \end{proof}
\begin{lemma}\label{lem:em-complexity-forcing} Fix an arithmetic formula $\varphi(G)$, a condition $c = (\vec{F}, T, \mathcal{C})$ and a part $\nu$ of $T$. \begin{itemize}
\item[a)] If $\varphi(G)$ is a $\Sigma^0_1$ ($\Pi^0_1$) formula
then so is the predicate $c \Vdash_\nu \varphi(G)$.
\item[b)] If $\varphi(G)$ is a $\Sigma^0_{n+2}$ ($\Pi^0_{n+2}$) formula
then so is the predicate $c \Vdash \varphi(G)$. \end{itemize} \end{lemma} \begin{proof} We prove our lemma by induction over the complexity of the formula $\varphi(G)$. \begin{itemize}
\item If $\varphi(G) \in \Sigma^0_1$ then it can be expressed as $(\exists x)\psi(G, x)$ where $\psi \in \Sigma^0_0$.
By clause~1 of Definition~\ref{def:em-forcing-precondition}, $c \Vdash_\nu \varphi(G)$ if and only if
the formula $(\exists w \in \omega)\psi(F_\nu, w)$ holds. This is a $\Sigma^0_1$ predicate.
\item If $\varphi(G) \in \Pi^0_1$ then it can be expressed as $(\forall x)\psi(G, x)$ where $\psi \in \Sigma^0_0$.
By clause~2 of Definition~\ref{def:em-forcing-precondition}, $c \Vdash_\nu \varphi(G)$ if and only if
the formula $(\forall \sigma \in T)(\forall w < |\sigma|)(\forall F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma))
[F'\ R\mbox{-transitive} \rightarrow \psi(F_\nu \cup F', w)]$ holds. This is a $\Pi^0_1$ predicate.
\item If $\varphi(G) \in \Sigma^0_2$ then it can be expressed as $(\exists x)\psi(G, x)$ where $\psi \in \Pi^0_1$.
By clause~1 of Definition~\ref{def:em-forcing-condition}, $c \Vdash \varphi(G)$ if and only if
the formula $(\forall \nu < \mathrm{parts}(T))(\exists w < \mathrm{dom}(T))[(\nu, T) \in \mathcal{C} \rightarrow c \Vdash_\nu \psi(G, w)]$ holds.
This is a $\Sigma^0_2$ predicate by induction hypothesis and the fact that $\mathcal{C}$ is $\emptyset'$-computable.
\item If $\varphi(G) \in \Pi^0_2$ then it can be expressed as $(\forall x)\psi(G, x)$ where $\psi \in \Sigma^0_1$.
By clause~2 of Definition~\ref{def:em-forcing-condition}, $c \Vdash \varphi(G)$ if and only if
for every infinite $k'$-partition tree $S$, every function~$f : \mathrm{parts}(S) \to \mathrm{parts}(T)$,
every $w$ and $\vec{E}$ smaller than the code of $S$ such that the followings hold
\begin{itemize}
\item[i)] $(E_\nu, \mathrm{dom}(S))$ EM extends $(F_{f(\nu)}, \mathrm{dom}(T))$ for each $\nu < \mathrm{parts}(S)$
\item[ii)] $S$ $f$-refines $\bigcap_{\nu < \mathrm{parts}(S)} T^{[f(\nu), E_\nu]}$
\end{itemize}
for every $(\mu, S) \in \mathcal{C}$, $(\vec{E}, S, \mathcal{C}[S]) \not \Vdash_\mu \neg \psi(G, w)$.
By Lemma~\ref{lem:em-refinement-complexity}, Properties i-ii) are $\Delta^0_2$.
Moreover, the predicate $(\mu, S) \in \mathcal{C}$ is $\Delta^0_2$.
By induction hypothesis, $(\vec{E}, S, \mathcal{C}) \not \Vdash_\mu \neg \psi(G, w)$ is $\Sigma^0_1$.
Therefore $c \Vdash \varphi(G)$ is a $\Pi^0_2$ predicate.
\item If $\varphi(G) \in \Sigma^0_{n+3}$ then it can be expressed as $(\exists x)\psi(G, x)$ where $\psi \in \Pi^0_{n+2}$.
By clause~3 of Definition~\ref{def:em-forcing-condition}, $c \Vdash \varphi(G)$ if and only if
the formula $(\exists w \in \omega)c \Vdash \psi(G, w)$ holds. This is a $\Sigma^0_{n+3}$ predicate
by induction hypothesis.
\item If $\varphi(G) \in \Pi^0_{n+3}$ then it can be expressed as $\neg \psi(G)$ where $\psi \in \Sigma^0_{n+3}$.
By clause~4 of Definition~\ref{def:em-forcing-condition}, $c \Vdash \varphi(G)$ if and only if
the formula $(\forall d)(d \not \in \operatorname{Ext}(c) \vee d \not \Vdash \psi(G))$ holds.
By induction hypothesis, $d \not \Vdash \psi(G)$ is a $\Pi^0_{n+3}$ predicate.
By Lemma~\ref{lem:em-extension-complexity}, the set $\operatorname{Ext}(c)$ is $\Pi^0_2$-computable uniformly in $c$,
thus $c \Vdash \varphi(G)$ is a $\Pi^0_{n+3}$ predicate. \end{itemize} \end{proof}
\subsection{Preserving the arithmetic hierarchy}
We now prove the core lemmas showing that every sufficiently generic real preserves the arithmetic hierarchy. The proof is split into two lemmas since the forcing relation for $\Sigma^0_1$ and~$\Pi^0_1$ formulas depends on the part of the condition, and therefore has to be treated separately.
\begin{lemma}\label{lem:em-diagonalization-level1} If $A \not \in \Sigma^0_1$ and $\varphi(G, x)$ is $\Sigma^0_1$, then the set of $c = (\vec{F}, T, \mathcal{C}) \in \mathbb{P}$ satisfying the following property is dense: $$ (\forall \nu < \mathrm{parts}(T))[(\exists w \in A)c_s \Vdash_\nu \neg \varphi(G, w)] \vee [(\exists w \not \in A)c_s \Vdash_\nu \varphi(G, w)] $$ \end{lemma} \begin{proof} The formula $\varphi(G, w)$ can be expressed as $(\exists x)\psi(G, w, x)$ where $\psi \in \Sigma^0_0$. Given a condition $c = (\vec{F}, T, \mathcal{C})$, let $I(c)$ be the set of the parts $\nu$ of $T$ such that for every $w \in A$, $c \not \Vdash_\nu \neg \varphi(G, w)$ and for every $w \in \overline{A}$, $c \not \Vdash_\nu \varphi(G, w)$. If $I(c) = \emptyset$ then we are done, so suppose $I(c) \neq \emptyset$ and fix some $\nu \in I(c)$. We will construct an extension $d$ such that $I(d) \subseteq I(c) \smallsetminus \{\nu\}$. Iterating the operation completes the proof.
Say that $T$ is a $k$-partition tree of $[t, \infty)$ for some $k, t \in \omega$. Define $f : k+1 \to k$ as $f(\mu) = \mu$ if $\mu < k$ and $f(k) = \nu$ otherwise. Given an integer $w \in \omega$, let $S_w$ be the set of all $\sigma \in (k+1)^{<\omega}$
which $f$-refine some $\tau \in T \cap k^{|\sigma|}$ and such that for every $u < |\sigma|$, every part $\mu \in \{\nu, k\}$ and every finite $R$-transitive set $F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\mu(\sigma)$, $\varphi(F_\nu \cup F', w, u)$ does not hold.
The set $S_w$ is a p.r.\ (uniformly in $w$) partition tree of $[t, \infty)$ refining $T$ with witness function~$f$. Let $U = \{ w \in \omega : S_w \mbox{ is finite } \}$. $U \in \Sigma^0_1$, thus $U \neq A$. Fix some $w \in U \Delta A$. Suppose first that $w \in A \smallsetminus U$. By definition of $U$, $S_w$ is infinite. Let $\vec{E}$ be defined by $E_\mu = F_\mu$ if $\mu < k$ and $E_k = F_\nu$, and consider the extension $d = (\vec{E}, S_w, \mathcal{C}[S_w])$. We claim that $I(d) \subseteq I(c) \smallsetminus \{\nu\}$.
Fix a part $\mu \in \{\nu, k\}$ of $S_w$. By definition of $S_w$, for every $\sigma \in S_w$, every $u < |\sigma|$ and every $R$-transitive set $F' \subseteq \mathrm{dom}(S_w) \cap \mathrm{set}_\mu(\sigma)$, $\varphi(E_\mu \cup F', w, u)$ does not hold. Therefore, by clause~2 of Definition~\ref{def:em-forcing-precondition}, $d \Vdash_\mu (\forall x)\neg \psi(G, w, x)$, hence $d \Vdash_\mu \neg \varphi(G, w)$, and this for some $w \in A$. Thus $I(d) \subseteq I(c) \smallsetminus \{\nu\}$.
Suppose now that $w \in U \smallsetminus A$, so $S_w$ is finite. Fix an $\ell \in \omega$ such that $(\forall \sigma \in S)|\sigma| < \ell$ and a $\tau \in T \cap k^\ell$ such that $T^{[\tau]}$ is infinite. Consider the 2-partition $E_0 \cup E_1$ of $\mathrm{set}_\nu(\tau) \cap \mathrm{dom}(T)$ defined by $E_0 = \{i \geq t : \tau(i) = \nu \wedge (\exists n)(\forall s > n) R(i, s) \mbox{ holds}\}$ and $E_0 = \{i \geq t : \tau(i) = \nu \wedge (\exists n)(\forall s > n) R(s, i) \mbox{ holds}\}$. As there exists no $\sigma \in S_w$ which $f$-refines $\tau$, there exists a $u < \ell$ and an $R$-transitive set $F' \subseteq E_0$ or $F' \subseteq E_1$ such that $\varphi(F_\nu \cup F', w, u)$ holds. By choice of the partition, there exists a $t' > t$ such that $F' \to_R [t', \infty)$ or $[t', \infty) \to_R F'$. By Lemma~\ref{lem:emo-cond-valid}, $(F_\nu \cup F', [t', \infty))$ is a valid EM extension of $(F_\nu, [t, \infty))$. As $T^{[\tau]}$ is infinite, $T^{[\nu, F']}$ is also infinite. Let $\vec{E}$ be defined by $E_\mu = F_\mu$ if $\mu \neq \nu$ and $E_\mu = F_\nu \cup F'$ otherwise. Let $S$ be the $k$-partition tree $(k, t', T^{[\nu, F']})$. The condition $d = (\vec{E}, S, \mathcal{C}[S])$ is a valid extension of $c$. By clause~1 of Definition~\ref{def:em-forcing-precondition}, $d \Vdash_\mu \varphi(G, w)$ with $w \not \in A$. . Therefore $I(d) \subseteq I(c) \smallsetminus \{\nu\}$. \end{proof}
\begin{lemma}\label{lem:em-diagonalization} If $A \not \in \Sigma^0_{n+2}$ and $\varphi(G, x)$ is $\Sigma^0_{n+2}$, then the set of $c \in \mathbb{P}$ satisfying the following property is dense: $$ [(\exists w \in A)c \Vdash \neg \varphi(G, w)] \vee [(\exists w \not \in A)c \Vdash \varphi(G, w)] $$ \end{lemma} \begin{proof} Fix a condition $c = (\vec{F}, T, \mathcal{C})$. \begin{itemize}
\item In case $n = 0$, $\varphi(G, w)$ can be expressed as $(\exists x)\psi(G, w, x)$ where $\psi \in \Pi^0_1$.
Let $U$ be the set of integers $w$ such that there exists an infinite p.r. $k'$-partition tree $S$
for some $k' \in \omega$, a function~$f : \mathrm{parts}(S) \to \mathrm{parts}(T)$ and a $k'$-tuple of finite sets $\vec{E}$ such that
\begin{itemize}
\item[i)] $(E_\nu, [\ell, \infty))$ EM extends $(F_{f(\nu)}, \mathrm{dom}(T))$ for each $\nu < \mathrm{parts}(S)$.
\item[ii)] $S$ $f$-refines $\bigcap_{\nu < \mathrm{parts}(S)} T^{[f(\nu), E_\nu]}$
\item[iii)] for each non-empty part $\nu$ of $S$ such that $(\nu, S) \in \mathcal{C}$,
$(\vec{E}, S, \mathcal{C}[S]) \Vdash_\nu \psi(G, w, u)$ for some $u < \#S$
\end{itemize}
By Lemma~\ref{lem:em-complexity-forcing} and Lemma~\ref{lem:em-refinement-complexity},
$U \in \Sigma^0_2$, thus $U \neq A$. Let $w \in U \Delta A$.
Suppose that $w \in U \smallsetminus A$.
We can choose $\mathrm{dom}(S)$ so that $(E_\nu, \mathrm{dom}(S))$ EM extends $(F_{f(\nu)}, \mathrm{dom}(T))$ for each $\nu < \mathrm{parts}(S)$.
By Lemma~\ref{lem:em-forcing-extension-level1} and Lemma~\ref{lem:em-promise-no-effect-first-level}, properties i-ii) remain true.
Let $\mathcal{D} = \mathcal{C}[S] \smallsetminus \{(\nu, S') \in \mathcal{C} : \mbox{ part } \nu \mbox{ of } S' \mbox{ is empty} \}$.
As $\mathcal{C}$ is an $\emptyset'$-p.r. promise for $T$,
$\mathcal{C}[S]$ is an $\emptyset'$-p.r. promise for $S$.
As $\mathcal{D}$ is obtained from $\mathcal{C}[S]$ by removing only empty parts, $\mathcal{D}$ is also an $\emptyset'$-p.r. promise for $S$.
By clause~1 of Definition~\ref{def:em-forcing-condition},
$d = (\vec{E}, S, \mathcal{D}) \Vdash (\exists x)\psi(G, w, x)$ hence $d \Vdash \varphi(G, w)$ for some $w \not \in A$.
We may choose a coding of the p.r. trees such that
the code of $S$ is sufficiently large to witness $u$ and $\vec{E}$.
So suppose now that $w \in A \smallsetminus U$. Then for every infinite p.r. $k'$-partition tree $S$,
every $\ell$ and $\vec{E}$ smaller than the code of $S$ such that properties i-ii) hold,
there exists a non-empty part $\nu$ of $S$ such that $(\nu, S) \in \mathcal{C}$
and $(\vec{E}, S, \mathcal{C}) \not \Vdash_\nu \psi(G, w, u)$ for every $u < \ell$.
Let $\mathcal{D}$ be the collection of all such $(\nu, S)$. The set $\mathcal{D}$ is $\emptyset'$-p.r.
By Lemma~\ref{lem:em-forcing-extension-level1} and since~$\#S \geq \#T$
whenever~$S \leq_f T$, $\mathcal{D}$ is upward-closed under the refinement relation,
hence it is a promise for~$T$. By clause~2.
of Definition~\ref{def:em-forcing-condition}, $d = (\vec{F}, T, \mathcal{D}) \Vdash (\forall x) \neg \psi(G, w, x)$,
hence $d \Vdash \neg \varphi(G, w)$ for some $w \in A$.
\item In case $n > 0$, let $U = \{ w \in \omega : (\exists d \in \operatorname{Ext}(c)) d \Vdash \varphi(G, w) \}$.
By Lemma~\ref{lem:em-extension-complexity} and Lemma~\ref{lem:em-complexity-forcing},
$U \in \Sigma^0_{n+2}$, thus $U \neq A$.
Fix some $w \in U \Delta A$. If $w \in U \smallsetminus A$ then by definition of~$U$,
there exists a condition $d$ extending $c$ such that $d \Vdash \varphi(G, w)$.
If $w \in A \smallsetminus U$, then for every $d \in \operatorname{Ext}(c)$, $d \not \Vdash \varphi(G, w)$
so by clause~4 of Definition~\ref{def:em-forcing-condition}, $c \Vdash \neg \varphi(G, w)$. \end{itemize} \end{proof}
We are now ready to prove Theorem~\ref{thm:em-preserves-arithmetic}. It follows from the preservation of the arithmetic hierarchy for cohesiveness and the stable Erd\H{o}s-Moser theorem.
\begin{proof}[Proof of Theorem~\ref{thm:em-preserves-arithmetic}] Since $\rca \vdash \coh \wedge \semo \rightarrow \emo$, then by Theorem~\ref{thm:coh-preservation-arithmetic-hierarchy} it suffices to prove that $\semo$ admits preservation of the arithmetic hierarchy. Fix some set~$C$ and a $C$-computable stable infinite tournament~$R$. Let $\mathcal{C}_0$ be the $C'$-p.r. set of all $(\nu, T) \in \mathbb{U}$ such that $(\nu, T) \leq (0, 1^{<\omega})$. Let~$\mathcal{F}$ be a sufficiently generic filter containing~$c_0 = (\{\emptyset\}, 1^{<\omega}, \mathcal{C}_0)$. Let~$P$ and $G$ be the corresponding generic path and generic real, respectively. By definition of a condition, the set~$G$ is $R$-transitive. By Lemma~\ref{lem:em-forcing-infinite}, $G$ is infinite. By Lemma~\ref{lem:em-diagonalization-level1} and Lemma~\ref{lem:em-complexity-forcing}, $G$ preserves non-$\Sigma^0_1$ definitions relative to~$C$. By Lemma~\ref{lem:em-diagonalization} and Lemma~\ref{lem:em-complexity-forcing}, $G$ preserves non-$\Sigma^0_{n+2}$ definitions relative to~$C$ for every~$n \in \omega$. Therefore, by Proposition 2.2 of~\cite{Wang2014Definability}, $G$ preserves the arithmetic hierarchy relative to~$C$. \end{proof}
\section{$\mathsf{D}^2_2$ preserves higher definitions}
Among the Ramsey-type hierarchies, the $\mathsf{D}$ hierarchy is conceptually the simplest one. It is therefore natural to study it in order to understand better the control of iterated jumps and focus on the core combinatorics without the technicalities specific to another hierarchy.
\begin{definition} For every~$n, k \geq 1$, $\mathsf{D}^n_k$ is the statement ``Every~$\Delta^0_n$ $k$-partition of the integers has an infinite subset in one of its parts''. \end{definition}
In particular, $\mathsf{D}^1_k$ is nothing but $\rt^1_k$ for computable colorings. Cholak et al.~\cite{Cholak2001strength} proved that~$\mathsf{D}^2_k$ and stable Ramsey's theorem for pairs and~$k$ colors ($\srt^2_k$) are computably equivalent and that the proof is formalizable over~$\rca+\bst$. Later, Chong et al.~\cite{Chong2010role} proved that~$\mathsf{D}^2_2$ implies~$\bst$ over~$\rca$, showing therefore that~$\rca \vdash \mathsf{D}^2_k \leftrightarrow \srt^2_\ell$ for every~$k, \ell \geq 2$. Wang~\cite{Wang2014Definability} studied Ramsey's theorem within his framework of preservation of definitions and proved that $\mathsf{D}^2_2$ admits preservation of $\Xi$ definitions simultaneously for all $\Xi$ in $\{\Sigma^0_{n+2}, \allowbreak \Pi^0_{n+2}, \allowbreak \Delta^0_{n+2} : n \in \omega \}$, but not~$\Delta^0_2$ definitions. More precisely, he prove that $\sads$, which is a consequence of $\mathsf{D}^2_2$, does not admit preservation of $\Delta^0_2$ definitions. He used for this a combination of the first jump control of Cholak, Jockusch and Slaman~\cite{Cholak2001strength} and a relativization of the preservation of the arithmetic hierarchy by~$\wkl$.
In this section, we design a notion of forcing for~$\mathsf{D}^2_2$ with a forcing relation which has the same definitional complexity as the formula it forces. It enables us to reprove that $\mathsf{D}^2_2$ admits preservation of $\Xi$ definitions simultaneously for all $\Xi$ in $\{\Sigma^0_{n+2}, \allowbreak \Pi^0_{n+2}, \allowbreak \Delta^0_{n+3} : n \in \omega \}$. The proof is significantly more involved than the previous proofs of preservation of the arithmetic hierarchy.
\subsection{Sides of a sequence of sets}
A main feature in the construction of a solution to an instance $R_0, R_1$ of $\mathsf{D}^2_2$ is the parallel construction of a subset of $R_0$ and a subset of $R_1$. The intrinsic disjunction in the forcing argument prevents us from applying the same strategy as for the Erd\H{o}s-Moser theorem and obtain a preservation of the arithmetic hierarchy. Given some~$\alpha < 2$, we shall refer to $R_\alpha$ or simply $\alpha$ as a \emph{side} of $\vec{R}$. We also need to define a relative notion of acceptation and emptiness of a part.
\begin{definition}Fix a $k$-partition tree $T$ of $[t, \infty)$ and a set $X$. We say that part $\nu$ of $T$ is \emph{$X$-acceptable} if there exists a path $P$ through $T$ such that $\mathrm{set}_\nu(P) \cap X$ is infinite. We say that part $\nu$ of $T$ is \emph{$X$-empty} if $(\forall \sigma \in T)[\mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma) \cap X = \emptyset]$. \end{definition}
The intended uses of those notions will be $R_\alpha$-acceptation and $R_\alpha$-emptiness. Every partition tree has an $R_\alpha$-acceptable part for some $\alpha < 2$. The notion of $X$-emptiness is $\Pi^{0,X}_1$, and therefore $\Pi^0_2$ if $X$ is $\Delta^0_2$, which raises new problems for obtaining a forcing relation of weak definitional complexity. We would like to define a stronger notion of ``witnessing its acceptable parts'' and prove that for every infinite p.r.\ partition tree~$T$, there is a p.r.\ refined tree~$S$ such that for each side $\alpha$ and each part~$\nu$ of~$S$, either~$\nu$ is $R_\alpha$-empty in~$S$, or~$\nu$ is $R_\alpha$-acceptable. However, the resulting tree~$S$ would be $\emptyset'$-p.r.\ since~$R_\alpha$ is $\emptyset'$-computable. Thankfully, we will be able to circumvent this problem in Lemma~\ref{lem:cohzp-validity-exists}.
\subsection{Forcing conditions}
Fix a $\Delta^0_2$ 2-partition~$R_0 \cup R_1 = \omega$. We now describe the notion of forcing to build an infinite subset of~$R_0$ or of~$R_1$.
\begin{definition} We denote by~$\mathbb{P}$ the forcing notion whose conditions are tuples
$((F_\nu^\alpha : \alpha < 2, \nu < k), T, \mathcal{C})$ where \begin{itemize}
\item[(a)] $T$ is an infinite, p.r.\ $k$-partition tree
\item[(b)] $\mathcal{C}$ is a $\emptyset'$-p.r.\ promise for $T$
\item[(c)] $(F^\alpha_\nu, \mathrm{dom}(T))$ is a Mathias condition for each $\nu < k$ and $\alpha < 2$ \end{itemize} A condition $d = (\vec{E}, S, \mathcal{D})$ \emph{extends} $c = (\vec{F}, T, \mathcal{C})$ (written $d \leq c$) if there exists a function $f : \mathrm{parts}(S) \to \mathrm{parts}(T)$ such that $\mathcal{D} \subseteq \mathcal{C}$ and the followings hold \begin{itemize}
\item[(i)] $(E^\alpha_\nu, \mathrm{dom}(S) \cap R_\alpha)$ Mathias extends $(F^\alpha_{f(\nu)}, \mathrm{dom}(T) \cap R_\alpha)$
for each $\nu < \mathrm{parts}(S)$ and $\alpha < 2$
\item[(ii)] $S$ $f$-refines $\bigcap_{\nu < \mathrm{parts}(S), \alpha < 2} T^{[f(\nu), E^\alpha_\nu]}$ \end{itemize} \end{definition}
In the whole construction, the index $\alpha$ indicates that we are constructing a set which is almost included in $R_\alpha$. Given a condition $c = (\vec{F}, T, \mathcal{C})$, we write again $\mathrm{parts}(c)$ for $\mathrm{parts}(T)$. The following lemma shows that we can force our constructed set to be infinite if we choose it among the acceptable parts.
\begin{lemma}\label{lem:cohzp-forcing-infinite} For every condition $c = (\vec{F}, T, \mathcal{C})$ and every $n \in \omega$, there exists an extension $d = (\vec{E}, S, \mathcal{D})$
such that $|E^\alpha_\nu| \geq n$ on each $R_\alpha$-acceptable part $\nu$ of~$S$ for each $\alpha < 2$. \end{lemma} \begin{proof} It suffices to prove that for every condition $c = (\vec{F}, T, \mathcal{C})$, every side $\alpha < 2$ and every $R_\alpha$-acceptable part $\nu$ of $T$, there exists an extension $d = (\vec{E}, S, \mathcal{D})$ such that $S \leq_{id} T$
and $|E^\alpha_\nu| \geq n$. Iterating the process at most $\mathrm{parts}(T) \times 2$ times completes the proof. Fix an $R_\alpha$-acceptable part $\nu$ of $T$ and a path $P$ through $T$ such that $\mathrm{set}_\nu(P) \cap R_\alpha$ is infinite. Let $F'$ be a subset of $\mathrm{set}_\nu(P) \cap \mathrm{dom}(T) \cap R_\alpha$ of size $n$. Let $\vec{E}$ be defined by $E^\beta_\mu = F^\beta_\mu$ if $\mu \neq \nu \vee \beta \neq \alpha$ and $E^\alpha_\nu = F^\alpha_\nu \cup F'$ otherwise. Let $S$ be the p.r.\ partition tree obtained from $T^{[\nu, E^\alpha_\nu]}$ by restricting its domain so that $(E^\alpha_\nu, \mathrm{dom}(S) \cap R_\alpha)$ Mathias extends $(F^\alpha_\nu, \mathrm{dom}(T) \cap R_\alpha)$. The condition $(\vec{E}, S, \mathcal{C}[S])$ is the desired extension. \end{proof}
Given a condition~$c$, we denote by~$\operatorname{Ext}(c)$ the set of all its extensions.
\subsection{Forcing relation}
We need to define two forcing relations at the first level: the ``true'' forcing relation, i.e., the one having the good density properties but whose decision requires too much computational power, and a ``weak'' forcing relation having better computational properties, but which does not behave well with respect to the forcing. We start with the definition of the true forcing relation.
\begin{definition}[True forcing relation]\label{def:cohzp-true-forcing-precondition} Fix a condition $c = (\vec{F}, T, \mathcal{C})$, a $\Sigma^0_0$ formula $\varphi(G, x)$, a part $\nu < \mathrm{parts}(T)$, and a side $\alpha < 2$. \begin{itemize}
\item[1.] $c \Vvdash^\alpha_\nu (\exists x)\varphi(G, x)$ iff there exists a $w \in \omega$ such that $\varphi(F^\alpha_\nu, w)$ holds.
\item[2.] $c \Vvdash^\alpha_\nu (\forall x)\varphi(G, x)$ iff
for every $\sigma \in T$ such that $T^{[\sigma]}$ is infinite,
every $w < |\sigma|$ and every set $F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma) \cap R_\alpha$,
$\varphi(F^\alpha_\nu \cup F', w)$ holds. \end{itemize} \end{definition}
Given a condition $c$, a side $\alpha < 2$, a part $\nu$ of $c$ and a $\Pi^0_1$ formula $\varphi$, the relation $c \Vvdash^\alpha_\nu \varphi(G)$ is $\Pi^{0, \emptyset' \oplus R_\alpha}_1$, hence $\Pi^0_2$ as $R_\alpha$ is $\Delta^0_2$. This relation enjoys the good properties of a forcing relation, that is, it is downward-closed under the refinement relation (Lemma~\ref{lem:cohzp-true-forcing-extension-level1}), and the set of the conditions forcing either a $\Sigma^0_1$ formula or its negation is dense (Lemma~\ref{lem:cohzp-true-forcing-dense-level1}).
\begin{lemma}\label{lem:cohzp-true-forcing-extension-level1} Fix a condition $c = (\vec{F}, T, \mathcal{C})$ and a $\Sigma^0_1$ ($\Pi^0_1$) formula $\varphi(G)$. If $c \Vvdash^\alpha_\nu \varphi(G)$ for some $\nu < \mathrm{parts}(T)$ and $\alpha < 2$, then for every $d = (\vec{E}, S, \mathcal{D}) \leq c$ and every part $\mu$ of $S$ refining part $\nu$ of $T$, $d \Vvdash^\alpha_\mu \varphi(G)$. \end{lemma} \begin{proof}\ \begin{itemize}
\item If $\varphi \in \Sigma^0_1$ then it can be expressed as $(\exists x)\psi(G, x)$
where $\psi \in \Sigma^0_0$. By clause~1 of Definition~\ref{def:cohzp-true-forcing-precondition},
there exists a $w \in \omega$ such that
$\psi(F^\alpha_\nu, w)$ holds. By property (i) of the definition of an extension, $E^\alpha_\mu \supseteq F^\alpha_\nu$
and $(E^\alpha_\mu \smallsetminus F^\alpha_\nu) \subset \mathrm{dom}(T) \cap R_\alpha$,
therefore by continuity $\psi(E^\alpha_\mu, w)$ holds,
so by clause~1 of Definition~\ref{def:cohzp-true-forcing-precondition},
$d \Vvdash^\alpha_\mu (\exists x)\psi(G, x)$.
\item If $\varphi \in \Pi^0_1$ then it can be expressed as $(\forall x)\psi(G, x)$
where $\psi \in \Sigma^0_0$.
Fix a $\tau \in S$ such that $S^{[\tau]}$ is infinite, a $w < |\tau|$ and a set
$F' \subseteq \mathrm{dom}(S) \cap \mathrm{set}_\mu(\tau) \cap R_\alpha$. Let~$f$ be the function witnesing~$d \leq c$.
By property~(ii) of the definition of an extension, $\tau$ $f$-refines a $\sigma \in T^{[\nu, E^\alpha_\mu]}$.
We claim that we can even choose~$\sigma$ to be extendible in $T^{[\nu, E^\alpha_\mu]}$.
Indeed, since~$\tau$ is extendible in~$S$, let~$P$ be a path through~$S$ extending~$\tau$
and let~$U$ be the set of~$\sigma$'s in~$T$ such that~$P {\upharpoonright} s$ $f$-refines~$\sigma$ for some~$s$.
The set~$U$ is an infinite subtree of~$T$. Let~$\sigma$ be a string of length~$|\tau|$ and extendible in~$U$, hence in~$T$.
By definition of~$U$, $\tau$ $f$-refines $\sigma$. By definition of a refinement,
such that $|\sigma| = |\tau|$ and $\mathrm{set}_\mu(\tau) \subseteq \mathrm{set}_\nu(\sigma)$. As $w < |\tau|$
and $\mathrm{dom}(S) \subseteq \mathrm{dom}(T)$, $F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma) \cap R_\alpha$.
As $\sigma \in T^{[\nu, E^\alpha_\mu]}$, $E^\alpha_\mu \subseteq \mathrm{set}_\nu(\sigma)$
and by property~(i) of the definition of an extension, $E^\alpha_\mu \subseteq \mathrm{dom}(T) \cap R_\alpha$
so $E^\alpha_\mu \subseteq \mathrm{dom}(T) \cap R_\alpha$.
Therefore $E^\alpha_\mu \cup F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma) \cap R_\alpha$.
By clause~2 of Definition~\ref{def:cohzp-true-forcing-precondition} applied to $c \Vvdash^\alpha_\nu (\forall x)\psi(G, x)$,
$\psi(F^\alpha_\nu \cup (E^\alpha_\mu \smallsetminus F^\alpha_\nu) \cup F', w)$ holds, hence $\psi(E^\alpha_\mu \cup F', w)$ holds
and still by clause~2 of Definition~\ref{def:cohzp-true-forcing-precondition}, $d \Vvdash_\mu (\forall x)\psi(G, x)$. \end{itemize} \end{proof}
\begin{lemma}\label{lem:cohzp-true-forcing-dense-level1} For every $\Sigma^0_1$ ($\Pi^0_1$) formula $\varphi$, the following set is dense in $\mathbb{P}$: $$ \{c \in \mathbb{P} : (\forall \nu < \mathrm{parts}(c))(\forall \alpha < 2)
[c \Vvdash^\alpha_\nu \varphi(G) \mbox{ or } c \Vvdash^\alpha_\nu \neg \varphi(G)] \} $$ \end{lemma} \begin{proof} It suffices to prove the statement for the case where $\varphi$ is a $\Sigma^0_1$ formula, as the case where $\varphi$ is a $\Pi^0_1$ formula is symmetric. Fix a condition $c = (\vec{F}, T, \mathcal{C})$ and let $I(c)$ be the set of pairs $(\nu, \alpha) \in \mathrm{parts}(T) \times 2$ such that $c \not \Vvdash^\alpha_\nu \varphi(G)$ and $c \not \Vvdash^\alpha_\nu \neg \varphi(G)$. If $I(c) = \emptyset$ we are done, so suppose $I(c) \neq \emptyset$. Fix some $(\alpha, \nu) \in I(c)$. We will construct an extension $d$ such that $I(d) \subseteq I(c) \smallsetminus \{(\alpha, \nu)\}$. Iterating the operation completes the proof.
The formula $\varphi$ is of the form $(\exists x)\psi(G, x)$ where $\psi \in \Sigma^0_0$. Suppose there exists a $\sigma \in T$ such that $T^{[\sigma]}$ is infinite, a $w < |\sigma|$ and a set $F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma) \cap R_\alpha$ such that $\psi(F_\nu^\alpha \cup F', w)$ holds. In this case, letting $\vec{E}$ be defined by $E_\mu^\beta = F_\mu^\beta$ if $\mu \neq \nu \vee \beta \neq \alpha$ and $E_\nu^\alpha = F_\nu^\alpha \cup F'$, and letting~$S$ be the tree~$T^{[\sigma]}$ where the domain is restricted so that $(E_\nu^\alpha, \mathrm{dom}(S))$ Mathias extends $(F_\nu^\alpha, \mathrm{dom}(T))$, by clause 1 of Definition~\ref{def:cohzp-true-forcing-precondition}, the condition $d = (\vec{E}, S, \mathcal{C}[S])$ is a valid extension of $c$ such that $d^\alpha_\nu \Vvdash \varphi(G)$.
Suppose now that for every $\sigma \in T$ such that $T^{[\sigma]}$
is infinite, every $w < |\sigma|$ and every set $F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma) \cap R_\alpha$, $\psi(F_\nu^\alpha \cup F', w)$ does not hold. In this case, by clause 2 of Definition~\ref{def:cohzp-true-forcing-precondition}, $c \Vvdash^\alpha_\nu \neg \varphi(G)$. \end{proof}
We now define the weak forcing relation which is almost the same as the true one, except that the set~$F'$ is not required to be a subset of~$R_\alpha$ and that $T^{[\sigma]}$ might be finite. Because of this, whenever a condition forces a $\Pi^0_1$ formula by the weak forcing relation, so does it by the strong forcing relation.
\begin{definition}[Weak forcing relation]\label{def:cohzp-forcing-precondition} Fix a condition $c = (\vec{F}, T, \mathcal{C})$, a $\Sigma^0_0$ formula $\varphi(G, x)$, a part $\nu < \mathrm{parts}(T)$ and a side $\alpha < 2$. \begin{itemize}
\item[1.] $c \Vdash^\alpha_\nu (\exists x)\varphi(G, x)$ iff there exists a $w \in \omega$ such that $\varphi(F^\alpha_\nu, w)$ holds.
\item[2.] $c \Vdash^\alpha_\nu (\forall x)\varphi(G, x)$ iff
for every $\sigma \in T$, every $w < |\sigma|$ and every set $F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma)$,
$\varphi(F^\alpha_\nu \cup F', w)$ holds. \end{itemize} \end{definition}
As one may expect, the weak forcing relation at the first level is also closed under the refinement relation.
\begin{lemma}\label{lem:cohzp-forcing-extension-level1} Fix a condition $c = (\vec{F}, T, \mathcal{C})$ and a $\Sigma^0_1$ ($\Pi^0_1$) formula $\varphi(G)$. If $c \Vdash^\alpha_\nu \varphi(G)$ for some $\nu < \mathrm{parts}(T)$ and $\alpha < 2$, then for every $d = (\vec{E}, S, \mathcal{D}) \leq c$ and every part $\mu$ of $S$ refining part $\nu$ of $T$, $d \Vdash^\alpha_\mu \varphi(G)$. \end{lemma} \begin{proof}\ \begin{itemize}
\item If $\varphi \in \Sigma^0_1$ then this is exactly clause 1 of Lemma~\ref{lem:cohzp-true-forcing-extension-level1}
since the definition of the weak and the true forcing relations coincide for~$\Sigma^0_1$ formulas.
\item If $\varphi \in \Pi^0_1$ then it can be expressed as $(\forall x)\psi(G, x)$
where $\psi \in \Sigma^0_0$. Fix a $\tau \in S$, a $w < |\tau|$ and a set
$F' \subseteq \mathrm{dom}(S) \cap \mathrm{set}_\mu(\tau)$. By property~(ii) of the definition of an extension,
there exists a $\sigma \in T^{[\nu, E^\alpha_\mu]}$
such that $|\sigma| = |\tau|$ and $\mathrm{set}_\mu(\tau) \subseteq \mathrm{set}_\nu(\sigma)$. As $w < |\tau|$
and $\mathrm{dom}(S) \subseteq \mathrm{dom}(T)$, $F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma)$.
As $\sigma \in T^{[\nu, E^\alpha_\mu]}$, $E^\alpha_\mu \subseteq \mathrm{set}_\nu(\sigma)$
and by property~(i) of the definition of an extension, $E^\alpha_\mu \subseteq \mathrm{dom}(T)$.
Therefore $E^\alpha_\mu \cup F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma)$.
By clause~2 of Definition~\ref{def:cohzp-forcing-precondition} applied to $c \Vdash^\alpha_\nu (\forall x)\psi(G, x)$,
$\psi(F^\alpha_\nu \cup (E^\alpha_\mu \smallsetminus F^\alpha_\nu) \cup F', w)$ holds, hence $\psi(E^\alpha_\mu \cup F', w)$ holds
and still by clause~2 of Definition~\ref{def:cohzp-forcing-precondition}, $d \Vdash^\alpha_\mu (\forall x)\psi(G, x)$. \end{itemize} \end{proof}
The following trivial lemma simply reflects the fact that the promise $\mathcal{C}$ is not part of the definition of the weak forcing relation for $\Sigma^0_1$ or~$\Pi^0_1$ formulas, and therefore has no effect on it.
\begin{lemma}\label{lem:cohzp-promise-no-effect-first-level} Fix two conditions $c = (\vec{F}, T, \mathcal{C})$ and $d = (\vec{E}, T, \mathcal{D})$ and a $\Sigma^0_1$ ($\Pi^0_1$) formula. For every part $\nu$ of $T$ such that $F^\alpha_\nu = E^\alpha_\nu$, $c \Vdash^\alpha_\nu \varphi(G)$ if and only if $d \Vdash^\alpha_\nu \varphi(G)$. \end{lemma} \begin{proof} If $\varphi \in \Sigma^0_1$ then $\varphi(G)$ can be expressed as $(\exists x)\psi(G, x)$ where $\psi \in \Sigma^0_0$. By clause~1 of Definition~\ref{def:cohzp-forcing-precondition}, $c \Vdash^\alpha_\nu \varphi(G)$ iff there exists a $w \in \omega$ such that $\psi(F^\alpha_\nu, w)$ holds. As $F^\alpha_\nu = E^\alpha_\nu$, $c \Vdash^\alpha_\nu \varphi(G)$ iff $d \Vdash^\alpha_\nu \varphi(G)$. Similarily, if $\varphi \in \Pi^0_1$ then $\varphi(G)$ can be expressed as $(\forall x)\psi(G, x)$ where $\psi \in \Sigma^0_0$. By clause~2 of Definition~\ref{def:cohzp-forcing-precondition},
$c \Vdash^\alpha_\nu \varphi(G)$ iff for every $\sigma \in T$, every $w < |\sigma|$ and every set $F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma)$, $\psi(F^\alpha_\nu \cup F', w)$ holds. As $F^\alpha_\nu = E^\alpha_\nu$, $c \Vdash^\alpha_\nu \varphi(G)$ iff $d \Vdash^\alpha_\nu \varphi(G)$. \end{proof}
We can now define the forcing relation over higher formulas. It is defined inductively, starting with $\Sigma^0_1$ and~$\Pi^0_1$ formulas. We extend the weak forcing relation instead of the true one for effectiveness purposes. We shall see later that the weak forcing relation behaves like the true one for some parts and some sides of a condition, and therefore that it tells us something about the truth of the formula over some carefully defined generic real~$G$. Note that the forcing relation over higher formulas is still parameterized by the side~$\alpha$ of the condition.
\begin{definition}\label{def:cohzp-forcing-condition} Fix a condition $c = (\vec{F}, T, \mathcal{C})$, a side $\alpha < 2$ and an arithmetic formula $\varphi(G)$. \begin{itemize}
\item[1.] If $\varphi(G) = (\exists x)\psi(G, x)$ where $\psi \in \Pi^0_1$ then
$c \Vdash^\alpha \varphi(G)$ iff for every part $\nu$ of $T$ such that
$(\nu, T) \in \mathcal{C}$ there exists a $w < \mathrm{dom}(T)$ such that $c \Vdash^\alpha_\nu \psi(G, w)$
\item[2.] If $\varphi(G) = (\forall x)\psi(G, x)$ where $\psi \in \Sigma^0_1$ then
$c \Vdash^\alpha \varphi(G)$ iff for every infinite p.r.\ $k'$-partition tree $S$,
every function $f : \mathrm{parts}(S) \to \mathrm{parts}(T)$,
every $w$ and $\vec{E}$ smaller than $\#S$ such that the followings hold
\begin{itemize}
\item[i)] $E^\beta_\nu = F^\beta_{f(\nu)}$ for each $\nu < \mathrm{parts}(S)$ and $\beta \neq \alpha$
\item[ii)] $(E^\alpha_\nu, \mathrm{dom}(S) \cap R_\alpha)$ Mathias extends
$(F^\alpha_{f(\nu)}, \mathrm{dom}(T) \cap R_\alpha)$ for each $\nu < \mathrm{parts}(S)$
\item[iii)] $S$ $f$-refines $\bigcap_{\nu < \mathrm{parts}(S)} T^{[f(\nu), E^\alpha_\nu]}$
\end{itemize}
for every $(\mu, S) \in \mathcal{C}$, $(\vec{E}, S, \mathcal{C}[S]) \not \Vdash^\alpha_\mu \neg \psi(G, w)$
\item[3.] If $\varphi(G) = (\exists x)\psi(G, x)$ where $\psi \in \Pi^0_{n+2}$ then
$c \Vdash^\alpha \varphi(G)$ iff there exists a $w \in \omega$ such that $c \Vdash^\alpha \psi(G, w)$
\item[4.] If $\varphi(G) = \neg \psi(G)$ where $\psi \in \Sigma^0_{n+3}$
then $c \Vdash^\alpha \varphi(G)$ iff $d \not \Vdash^\alpha \psi(G)$ for every $d \in \operatorname{Ext}(c)$. \end{itemize} \end{definition}
Note that clause 2.ii) of Definition~\ref{def:cohzp-forcing-condition} seems to be~$\Pi^0_2$ since~$R_\alpha$ is $\Delta^0_2$. However, in fact, one just needs to ensure that~$\mathrm{dom}(S) \subseteq \mathrm{dom}(T)$ and~$E^\alpha_\nu \smallsetminus F^\alpha_{f(\nu)} \subseteq \mathrm{dom}(T) \cap R_\alpha$. This is a $\Delta^0_2$ predicate, and so is its negation, so one can already easily check that the forcing relation over a $\Pi^0_2$ formula will be also $\Pi^0_2$. Before proving the usual properties about the forcing relation, we need to discuss the role of the sides in the forcing relation. We are now ready to prove that the forcing relation is closed under extension.
\begin{lemma}\label{lem:cohzp-forcing-extension} Fix a condition $c$, a side $\alpha < 2$ and a $\Sigma^0_{n+2}$ ($\Pi^0_{n+2}$) formula $\varphi(G)$. If $c \Vdash^\alpha \varphi(G)$ then for every $d \leq c$, $d \Vdash^\alpha \varphi(G)$. \end{lemma} \begin{proof} We prove the statement by induction over the complexity of the formula $\varphi(G)$. Fix a condition $c = (\vec{F}, T, \mathcal{C})$ and a side $\alpha < 2$ such that $c \Vdash^\alpha \varphi(G)$. Fix an extension $d = (\vec{E}, S, \mathcal{D})$ of $c$. \begin{itemize}
\item If $\varphi \in \Sigma^0_2$ then $\varphi(G)$ can be expressed as $(\exists x)\psi(G, x)$
where $\psi \in \Pi^0_1$. By clause~1 of Definition~\ref{def:cohzp-forcing-condition},
for every part $\nu$ of $T$ such that $(\nu, T) \in \mathcal{C}$, there exists a $w < \mathrm{dom}(T)$
such that $c \Vdash^\alpha_\nu \psi(G, w)$. Fix a part $\mu$ of $S$ such that $(\mu, S) \in \mathcal{D}$.
As $\mathcal{D} \subseteq \mathcal{C}$, $(\mu, S) \in \mathcal{C}$. By upward-closure of $\mathcal{C}$,
part $\mu$ of $S$ refines some part $\nu$ of $\mathcal{C}$ such that $(\nu, T) \in \mathcal{C}$.
Therefore by Lemma~\ref{lem:cohzp-forcing-extension-level1}, $d \Vdash^\alpha_\mu \psi(G, w)$,
with $w < \mathrm{dom}(T) \leq \mathrm{dom}(S)$. Applying again clause~1 of Definition~\ref{def:cohzp-forcing-condition},
we deduce that $d \Vdash (\forall x)\psi(G, x)$, hence $d \Vdash^\alpha \varphi(G)$.
\item If $\varphi \in \Pi^0_2$ then $\varphi(G)$ can be expressed as $(\forall x)\psi(G, x)$ where $\psi \in \Sigma^0_1$.
Suppose by way of contradiction that $d \not \Vdash^\alpha (\forall x)\psi(G, x)$.
Let $f : \mathrm{parts}(S) \to \mathrm{parts}(T)$ witness the refinement $S \leq T$.
By clause~2 of Definition~\ref{def:cohzp-forcing-condition},
there exist an infinite p.r.\ $k'$-partition tree $S'$, a function~$g : \mathrm{parts}(S') \to \mathrm{parts}(S)$,
a $w \in \omega$, and a $2k'$-tuple of finite sets $\vec{H}$ smaller than the code of $S'$ such that
\begin{itemize}
\item[i)] $H^\beta_\nu = E^\beta_{g(\nu)}$ for each $\nu < \mathrm{parts}(S')$ and $\beta \neq \alpha$
\item[ii)] $(H^\alpha_\nu, \mathrm{dom}(S') \cap R_\alpha)$ Mathias extends $(E^\alpha_{g(\nu)}, \mathrm{dom}(S) \cap R_\alpha)$
for each $\nu < \mathrm{parts}(S')$
\item[iii)] $S'$ $g$-refines $\bigcap_{\nu < \mathrm{parts}(S')} S^{[g(\nu), H^\alpha_\nu]}$
\item[iv)] there exists a $(\mu, S') \in \mathcal{D}$ such that
$(\vec{H}, S', \mathcal{D}[S']) \Vdash^\alpha_\mu \neg \psi(G, w)$.
\end{itemize}
To deduce by clause~2 of Definition~\ref{def:cohzp-forcing-condition} that
$c \not \Vdash^\alpha (\forall x)\psi(G, x)$ and derive a contradiction, it suffices to prove
that the same properties hold with respect to~$T$. Let $\vec{H}'$ be defined by
$H^{'\beta}_\nu = F^\beta_{f(g(\nu))}$ for each $\nu < \mathrm{parts}(S')$ and $\beta \neq \alpha$
and $H^{'\alpha}_\nu = H^\alpha_\nu$.
\begin{itemize}
\item[i)] It trivially holds by choice of $\vec{H}'$.
\item[ii)] By property (i) of the definition of an extension,
$(E^\alpha_{g(\nu)}, \mathrm{dom}(S))$ Mathias extends $(F^\alpha_{f(g(\nu))}, \mathrm{dom}(T))$.
Moreover $(H^\alpha_\nu, \mathrm{dom}(S')$ Mathias extends $(E^\alpha_{g(\nu)}, \mathrm{dom}(S))$,
so $(H^{'\alpha}_\nu, \mathrm{dom}(S')) = (H^\alpha_\nu, \mathrm{dom}(S'))$ Mathias extends $(F_{f(g(\nu))}, \mathrm{dom}(T))$.
\item[iii)] As by property (ii) of the definition of an extension,\\
$S$ $f$-refines $\bigcap_{\nu < \mathrm{parts}(S')} T^{[f(g(\nu)), E^\alpha_{g(\nu)}]}$, and \\
$S'$ $g$-refines $\bigcap_{\nu < \mathrm{parts}(S')} S^{[g(\nu), H^\alpha_\nu]}$ then \\
$S'$ $(g \circ f)$-refines $\bigcap_{\nu < \mathrm{parts}(S')} T^{[g(\nu), H^{'\alpha}_\nu]}$.
\item[iv)] As $\mathcal{D} \subseteq \mathcal{C}$, there exists a part $(\mu, S') \in \mathcal{C}$
such that $(\vec{H}, S', \mathcal{D}[S']) \Vdash^\alpha_\mu \neg \psi(G, w)$.
By Lemma~\ref{lem:cohzp-promise-no-effect-first-level}, $(\vec{H}', S', \mathcal{C}[S']) \Vdash^\alpha_\mu \neg \psi(G, w)$.
\end{itemize}
\item If $\varphi \in \Sigma^0_{n+3}$ then $\varphi(G)$ can be expressed as $(\exists x)\psi(G, x)$
where $\psi \in \Pi^0_{n+2}$.
By clause~3 of Definition~\ref{def:cohzp-forcing-condition}, there exists a $w \in \omega$
such that $c \Vdash^\alpha \psi(G, w)$. By induction hypothesis, $d \Vdash^\alpha \psi(G, w)$
so by clause~3 of Definition~\ref{def:cohzp-forcing-condition}, $d \Vdash^\alpha \varphi(G)$.
\item If $\varphi \in \Pi^0_{n+3}$ then $\varphi(G)$ can be expressed as $\neg \psi(G)$ where $\psi \in \Sigma^0_{n+3}$.
Suppose by way of contradiction that $d \not \Vdash^\alpha \varphi(G)$.
By clause~4 of Definition~\ref{def:cohzp-forcing-condition},
there exists an $e \in \operatorname{Ext}(d)$ such that $e \Vdash^\alpha \psi(G)$.
In particular, $e \in \operatorname{Ext}(c) $, so by clause~4 of Definition~\ref{def:cohzp-forcing-condition}, $e \not \Vdash^\alpha \psi(G)$
since $c \Vdash^\alpha \varphi(G)$.
Contradiction. \end{itemize} \end{proof}
Although the weak forcing relation does not satisfy the density property, the forcing relation over higher formulas does. The reason is that the extended forcing relation does not involve the weak forcing relation over~$\Sigma^0_1$ formulas in the clause 2 of Definition~\ref{def:cohzp-forcing-condition}, but uses instead the weaker statement ``$c$ does not force the negation of the~$\Sigma^0_1$ formula''. The link between this statement and the statement ``$c$ has an extension which forces the $\Sigma^0_1$ formula'' is used when proving that $\varphi(G)$ holds iff $c \Vdash \varphi(G)$ for some condition belonging to a sufficiently generic filter. We now prove the density of the forcing relation for higher formulas.
\begin{lemma}\label{lem:cohzp-forcing-dense} For every $\Sigma^0_{n+2}$ ($\Pi^0_{n+2}$) formula $\varphi$, the following set is dense in $\mathbb{P}$: $$ \{c \in \mathbb{P} : (\forall \alpha < 2)[c \Vdash^\alpha \varphi(G) \mbox{ or } c \Vdash^\alpha \neg \varphi(G)] \} $$ \end{lemma} \begin{proof} We prove the statement by induction over~$n$. It suffices to treat the case where $\varphi$ is a $\Sigma^0_{n+2}$ formula, as the case where $\varphi$ is a $\Pi^0_{n+2}$ formula is symmetric. Moreover, it is enough to prove that for every condition $c$ and every $\alpha < 2$, there exists an extension $d \leq c$ such that $d \Vdash^\alpha \varphi(G) \mbox{ or } d \Vdash^\alpha \neg \varphi(G)$. Iterating the process at most twice completes the proof. Fix a condition $c = (\vec{F}, T, \mathcal{C})$ and a part $\alpha < 2$. \begin{itemize}
\item In case $n = 0$, the formula $\varphi$ is of the form $(\exists x)\psi(G, x)$ where
$\psi \in \Pi^0_1$.
Suppose there exists an infinite p.r.\ $k'$-partition tree $S$
for some $k' \in \omega$, a function~$f : \mathrm{parts}(S) \to \mathrm{parts}(T)$,
and a $2k'$-tuple of finite sets $\vec{E}$ such that
\begin{itemize}
\item[i)] $E^\beta_\nu = F^\beta_{f(\nu)}$ for each $\nu < \mathrm{parts}(S)$ and $\beta \neq \alpha$
\item[ii)] $(E^\alpha_\nu, \mathrm{dom}(S) \cap R_\alpha)$ Mathias extends
$(F^\alpha_{f(\nu)}, \mathrm{dom}(T) \cap R_\alpha)$ for each $\nu < \mathrm{parts}(S)$.
\item[iii)] $S$ $f$-refines $\bigcap_{\nu < \mathrm{parts}(S)} T^{[f(\nu), E^\alpha_\nu]}$
\item[iv)] for each non-empty part $\nu$ of $S$ such that $(\nu, S) \in \mathcal{C}$,
$(\vec{E}, S, \mathcal{C}[S]) \Vdash^\alpha_\nu \psi(G, w)$ for some~$w < \#S$
\end{itemize}
Let $\mathcal{D} = \mathcal{C}[S] \smallsetminus \{(\nu, S') \in \mathcal{C} : \mbox{ part } \nu \mbox{ of } S' \mbox{ is empty} \}$.
As $\mathcal{C}$ is an $\emptyset'$-p.r.\ promise for $T$,
$\mathcal{C}[S]$ is an $\emptyset'$-p.r.\ promise for $S$.
As $\mathcal{D}$ is obtained from $\mathcal{C}[S]$ by removing only empty parts, $\mathcal{D}$ is also an $\emptyset'$-p.r.\ promise for $S$.
By clause~1 of Definition~\ref{def:cohzp-forcing-condition},
$d = (\vec{E}, S, \mathcal{D}) \Vdash^\alpha (\exists x)\psi(G, x)$ hence $d \Vdash^\alpha \varphi(G)$.
We may choose a coding of the p.r.\ trees such that
the code of $S$ is sufficiently large to witness $f$, $\ell$ and $\vec{E}$.
So suppose now that for every infinite p.r.\ $k'$-partition tree $S$,
every function~$f : \mathrm{parts}(S) \to \mathrm{parts}(T)$, $\ell \in \omega$ and $\vec{E}$
smaller than the code of $S$ such that properties i-iii) hold,
there exists a non-empty part $\nu$ of $S$ such that $(\nu, S) \in \mathcal{C}$
and $(\vec{E}, S, \mathcal{C}[S]) \not \Vdash^\alpha_\nu \psi(G, w)$ for every $w < \ell$.
Let $\mathcal{D}$ be the collection of all such $(\nu, S)$. $\mathcal{D}$ is $\emptyset'$-p.r.
By Lemma~\ref{lem:cohzp-forcing-extension-level1} and since we require that~$\#S \geq \#T$
in the definition of~$S \leq T$, $\mathcal{D}$ is upward-closed, hence is a promise for~$T$. By clause~2
of Definition~\ref{def:cohzp-forcing-condition}, $d = (\vec{F}, T, \mathcal{D}) \Vdash^\alpha (\forall x) \neg \psi(G, x)$,
hence $d \Vdash^\alpha \neg \varphi(G)$.
\item In case $n > 0$, density follows from clause~4 of Definition~\ref{def:cohzp-forcing-condition}. \end{itemize} \end{proof}
We now prove that the weak forcing relation extended to any arithmetic formula enjoys the desired definability properties. For this, we start with a lemma showing that the extension relation is $\Pi^0_2$. Therefore, only the first two levels have to be treated independently, since the extension relation does not add some extra complexity to the forcing relation for higher formulas.
\begin{lemma}\label{lem:cohzp-extension-complexity} For every condition $c$, $\operatorname{Ext}(c)$ is $\Pi^0_2$ uniformly in~$c$. \end{lemma} \begin{proof} Recall from Lemma~\ref{lem:em-refinement-complexity} that given $k, t \in \omega$, the set $PartTree(k, t)$ denotes the $\Pi^0_1$ set of all the infinite p.r.\ $k$-partition trees of $[t, \infty)$, and given a $k$-partition tree $S$ and a part $\nu$ of $S$, the predicate $Empty(S, \nu)$ denotes the $\Pi^0_1$ formula ``part $\nu$ of $S$ is empty'', that is, the formula $(\forall \sigma \in S)[\mathrm{set}_\nu(\sigma) \cap \mathrm{dom}(S) = \emptyset$]. If $T$ is p.r.\ then so is $T^{[\nu, H]}$ for some finite set $H$.
Fix a condition $c = (\vec{F}, (k, t, T), \mathcal{C})$. $(\vec{H}, (k', t', S), \mathcal{D}) \in \operatorname{Ext}(c)$ iff the following formula holds: $$ \begin{array}{l@{\hskip 0.5in}r} (\exists f : k' \to k)\\ (\forall \nu < k')(\forall \alpha < 2) (H^\alpha_\nu, [t', \infty) \cap R_\alpha) \mbox{ Mathias extends } (F^\alpha_{f(\nu)}, [t, \infty) \cap R_\alpha) & (\Pi^0_2)\\ \wedge S \in PartTree(k', t') \wedge S \leq_f \bigwedge_{\nu < k', \alpha < 2} T^{[f(\nu), H^\alpha_{\nu}]} & (\Pi^0_1)\\ \wedge \mathcal{D} \mbox{ is a promise for } S \wedge \mathcal{D} \subseteq \mathcal{C} & (\Pi^0_2)\\ \end{array} $$ The formula $(H^\alpha_\nu, [t', \infty) \cap R_\alpha) \mbox{ Mathias extends } (F^\alpha_{f(\nu)}, [t, \infty) \cap R_\alpha)$ can be written $(\forall x < t)[x \in H^\alpha_\nu \leftrightarrow x \in F^\alpha_{f(\nu)}] \wedge t' \geq t \wedge (\forall x \in H^\alpha_\nu \smallsetminus F^\alpha_{f(\nu)}) x \in R_\alpha$ and therefore is $\Pi^0_2$. By Lemma~\ref{lem:em-refinement-complexity} and the fact that $\bigwedge_{\nu < k', \alpha < 2} T^{[f(\nu), H^\alpha_{\nu}]}$ is p.r.\ uniformly in $T$, $f$, $\vec{H}$ and $k'$, the above formula is~$\Pi^0_2$. \end{proof}
\begin{lemma}\label{lem:cohzp-complexity-forcing} Fix an arithmetic formula $\varphi(G)$, a condition $c = (\vec{F}, T, \mathcal{C})$, a side $\alpha < 2$ and a part $\nu$ of $T$. \begin{itemize}
\item[a)] If $\varphi(G)$ is a $\Sigma^0_1$ ($\Pi^0_1$) formula
then so is the predicate $c \Vdash^\alpha_\nu \varphi(G)$.
\item[b)] If $\varphi(G)$ is a $\Sigma^0_{n+2}$ ($\Pi^0_{n+2}$) formula
then so is the predicate $c \Vdash^\alpha \varphi(G)$. \end{itemize} \end{lemma} \begin{proof} We prove our lemma by induction over the complexity of the formula $\varphi(G)$. \begin{itemize}
\item If $\varphi(G) \in \Sigma^0_1$ then it can be expressed as $(\exists x)\psi(G, x)$ where $\psi \in \Sigma^0_0$.
By clause~1 of Definition~\ref{def:cohzp-forcing-precondition}, $c \Vdash^\alpha_\nu \varphi(G)$ if and only if
the formula $(\exists w \in \omega)\psi(F^\alpha_\nu, w)$ holds. This is a $\Sigma^0_1$ predicate.
\item If $\varphi(G) \in \Pi^0_1$ then it can be expressed as $(\forall x)\psi(G, x)$ where $\psi \in \Sigma^0_0$.
By clause~2 of Definition~\ref{def:cohzp-forcing-precondition}, $c \Vdash^\alpha_\nu \varphi(G)$ if and only if
the formula $(\forall \sigma \in T)(\forall w < |\sigma|)(\forall F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma))
\psi(F^\alpha_\nu \cup F', w)$ holds. This is a $\Pi^0_1$ predicate.
\item If $\varphi(G) \in \Sigma^0_2$ then it can be expressed as $(\exists x)\psi(G, x)$ where $\psi \in \Pi^0_1$.
By clause~1 of Definition~\ref{def:cohzp-forcing-condition}, $c \Vdash^\alpha \varphi(G)$ if and only if
the formula $(\forall \nu < \mathrm{parts}(T)(\exists w < \mathrm{dom}(T))[(\nu, T) \in \mathcal{C} \rightarrow c \Vdash^\alpha_\nu \psi(G, w)]$ holds.
This is a $\Sigma^0_2$ predicate by induction hypothesis and the fact that $\mathcal{C}$ is $\emptyset'$-computable.
\item If $\varphi(G) \in \Pi^0_2$ then it can be expressed as $(\forall x)\psi(G, x)$ where $\psi \in \Sigma^0_1$.
By clause~2 of Definition~\ref{def:cohzp-forcing-condition}, $c \Vdash \varphi(G)$ if and only if
for every infinite $k'$-partition tree $S$, every function $f : \mathrm{parts}(S) \to \mathrm{parts}(T)$,
every $w$ and $\vec{E}$ smaller than the code of $S$ such that the followings hold
\begin{itemize}
\item[i)] $(E_\nu, \mathrm{dom}(S) \cap R_\alpha)$ Mathias extends $(F_{f(\nu)}, \mathrm{dom}(T) \cap R_\alpha)$ for each $\nu < \mathrm{parts}(S)$
\item[ii)] $S$ $f$-refines $\bigcap_{\nu < \mathrm{parts}(S)} T^{[f(\nu), E_\nu]}$
\end{itemize}
for every $(\mu, S) \in \mathcal{C}$, $(\vec{E}, S, \mathcal{C}[S]) \not \Vdash^\alpha_\mu \neg \psi(G, w)$.
By Lemma~\ref{lem:em-refinement-complexity}, Properties i-ii) are $\Delta^0_2$.
Moreover the predicate $(\mu, S) \in \mathcal{C}$ is $\Delta^0_2$ since~$\mathcal{C}$ is $\emptyset'$-p.r.
By induction hypothesis, $(\vec{E}, S, \mathcal{C}[S]) \not \Vdash^\alpha_\mu \neg \psi(G, w)$ is $\Sigma^0_1$.
Therefore $c \Vdash^\alpha \varphi(G)$ is a $\Pi^0_2$ predicate.
\item If $\varphi(G) \in \Sigma^0_{n+3}$ then it can be expressed as $(\exists x)\psi(G, x)$ where $\psi \in \Pi^0_{n+2}$.
By clause~3 of Definition~\ref{def:cohzp-forcing-condition}, $c \Vdash^\alpha \varphi(G)$ if and only if
the formula $(\exists w \in \omega)c \Vdash^\alpha \psi(G, w)$ holds. This is a $\Sigma^0_{n+3}$ predicate
by induction hypothesis.
\item If $\varphi(G) \in \Pi^0_{n+3}$ then it can be expressed as $\neg \psi(G)$ where $\psi \in \Sigma^0_{n+3}$.
By clause~4 of Definition~\ref{def:cohzp-forcing-condition}, $c \Vdash^\alpha \varphi(G)$ if and only if
the formula $(\forall d)(d \not \in \operatorname{Ext}(c) \vee d \not \Vdash^\alpha \psi(G))$ holds.
By induction hypothesis, $d \not \Vdash^\alpha \psi(G)$ is a $\Pi^0_{n+3}$ predicate.
By Lemma~\ref{lem:cohzp-extension-complexity}, the set $\operatorname{Ext}(c)$ is $\Pi^0_2$-computable uniformly in $c$,
thus $c \Vdash^\alpha \varphi(G)$ is a $\Pi^0_{n+3}$ predicate. \end{itemize} \end{proof}
\subsection{Validity}
As we already saw, we have two candidate forcing relations for $\Sigma^0_1$ and $\Pi^0_1$ formulas: \begin{itemize}
\item[1.] The ``true'' forcing relation $c \Vvdash^\alpha \varphi(G)$.
This relation has been shown to have the expected density properties through Lemma~\ref{lem:cohzp-true-forcing-dense-level1}.
However deciding such a relation requires too much computational power.
\item[2.] The ``weak'' forcing relation $c \Vdash^\alpha \varphi(G)$.
Deciding such a relation requires the same definitional power as the formula it forces.
It provides a sufficient condition for forcing the formula $\varphi(G)$
as $c \Vdash^\alpha \varphi(G)$ implies $c \Vvdash^\alpha \varphi(G)$,
but the converse does not hold and we cannot prove the density property in the general case. \end{itemize}
Thankfully, there exist some sides and parts of any condition on which those two forcing relations coincide. This leads to the notion of validity.
\begin{definition}[Validity] Fix an enumeration $\varphi_0(G), \varphi_1(G), \dots$ of all $\Pi^0_1$ formulas. Fix a condition $c = (\vec{F}, T, \mathcal{C})$, a side $\alpha < 2$, and a part $\nu$ of $T$. We say that \emph{side $\alpha$ is $n$-valid in part $\nu$ of $T$} for some~$n \in \omega$ if part $\nu$ of $T$ is $R_\alpha$-acceptable and for every~$i < n$, $c \Vvdash^\alpha_\nu \varphi_i(G)$ iff $c \Vdash^\alpha_\nu \varphi_i(G)$. \end{definition}
The following lemma shows that given some~$n \in \omega$, we can restrict $\mathcal{C}$ so that it ``witnesses its $n$-valid parts''.
\begin{lemma}\label{lem:cohzp-validity-exists} For every $n \in \omega$, the following set is dense in~$\mathbb{P}$: $$ \{ (\vec{F}, T, \mathcal{C}) \in \mathbb{P} : (\forall \nu)(\exists \alpha < 2)
[(\nu, T) \in \mathcal{C} \rightarrow \mbox{side } \alpha \mbox{ is } n\mbox{-valid in part } \nu \mbox{ of } T] \} $$ \end{lemma} \begin{proof} Given a condition $c = (\vec{F}, T, \mathcal{C})$, let $I(c)$ be the set of the parts $\nu$ of $T$ such that $(\nu, T) \in \mathcal{C}$ and no $\alpha < 2$ is valid for $\varphi$ in part $\nu$ of $T$. Fix a condition $c = (\vec{F}, T, \mathcal{C}) \in \mathbb{P}$. By iterating Lemma~\ref{lem:cohzp-forcing-dense}, we can assume without loss of generality that for each~$i < n$, $$ (\forall \alpha < 2)[c \Vdash^\alpha (\exists x)\varphi_i(G) \mbox{ or } c \Vdash^\alpha (\forall x)\neg \varphi_i(G)] $$ The dummy variable~$x$ ensures that the forcing relation for~$\Sigma^0_2$ and~$\Pi^0_2$ is applied. It suffices to prove that for every $\nu \in I(c)$, there exists an extension $d = (\vec{E}, S, \mathcal{D})$
such that $I(d) \subseteq I(c) \smallsetminus \{\nu\}$. Iterating the process at most $|\mathrm{parts}(T)|$ times completes the proof.
Fix a part $\nu \in I(c)$ and let $\mathcal{D}$ be the set of $(\mu, S) \in \mathcal{C}$ such that part $\mu$ of $S$ does not refine part $\nu$ of $T$. The set $\mathcal{D}$ is a $\emptyset'$-p.r.\ upward-closed subset of $\mathcal{C}$. It suffices to prove that for every infinite p.r.\ partition tree $S \leq T$, there exists a non-empty part $\mu$ of $S$ such that $(\mu, S) \in \mathcal{D}$ to deduce that $\mathcal{D}$ is a promise for $T$ and obtain an extension $d = (\vec{E}, T, \mathcal{D})$ of $c$ such that $I(d) \subseteq I(c) \smallsetminus \{\nu\}$.
Fix an infinite p.r.\ partition tree $S \leq_g T$ for some~$g$ and let~$\mu$ be a part of~$S$ $g$-refining part $\nu$ of $T$. By choice of $\nu$, for every $\alpha < 2$, either $\mu$ is not $R_\alpha$-acceptable in $S$, or $c \Vvdash^\alpha_\nu \varphi_{i_\alpha}(G)$ but $c \not \Vdash^\alpha_\nu \varphi_{i_\alpha}(G)$ for some~$i_\alpha < n$. In the latter case, by choice of $c$, $c \Vdash^\alpha_\nu (\forall x)\neg \varphi_{i_\alpha}(G)$.
We now assume that $S$ has $k$ parts, among which $m$ parts $g$-refine~$\nu$. Let~$f : k+m \to k$ be the function such that~$f(\mu) = \mu$ for each part~$\mu$ of~$S$ not $g$-refining part~$\nu$ of~$T$, and such that $f(\mu_\alpha) = \mu$ for each part~$\mu$ of~$S$ $g$-refining part~$\nu$ of~$T$ and each $\alpha < 2$. In other words, $f$ forks each part~$\mu$ of~$S$ $g$-refining the part~$\mu$ of~$T$ into $2$ parts~$\mu_0$ and~$\mu_1$. Let~$P$ be a path through~$S$, and let~$t \in \omega$ be large enough to ``witness the non~$R_\alpha$-acceptable sides''. More formally, let~$t$ be such that for every~$\alpha < 2$, either~$\mathrm{set}_\mu(P) \cap R_\alpha \cap [t, \infty) = \emptyset$ for each part~$\mu$ of~$S$ $g$-refining part~$\nu$ of~$T$, or $c \Vvdash^\alpha_\nu \varphi_{i_\alpha}(G)$ but $c \not \Vdash^\alpha_\nu \varphi_{i_\alpha}(G)$.
Let~$S'$ be the p.r.\ tree of all the $\tau$'s $f$-refining some~$\sigma \in S$ and such that for each~$\alpha < 2$ and each part $\mu$ of $S$ $g$-refining part~$\nu$ of~$T$, either~$\mathrm{set}_{\mu_\alpha}(\tau) \cap [t, \infty) = \emptyset$, or $\varphi_{i_\alpha}(F^\alpha_\nu \cup F')$ holds for each~$F' \subseteq \mathrm{dom}(S) \cap \mathrm{set}_{\mu_\alpha}(\tau)$. The tree~$S'$ is a $(k+m)$-partition tree of~$[t, \infty)$ $f$-refining~$S$. We claim that~$S'$ is infinite. Fix some~$s \in \omega$, we will prove that~$\tau \in S'$ for some string~$\tau$ of length~$s$. Let~$\sigma = P {\upharpoonright} s$. In particular, $S^{[\sigma]}$ is infinite, so for every~$\alpha < 2$ and every part $\mu$ of~$S$ $g$-refining part~$\nu$ of~$T$, either~$\mathrm{set}_\mu(P) \cap R_\alpha \cap [t, \infty) = \emptyset$ by definition of~$t$, or, unfolding clause 2 of Definition~\ref{def:cohzp-true-forcing-precondition} for $c \Vvdash^\alpha_\nu \varphi_{i_\alpha}(G)$ and since $\sigma$ $g$-refines some extendible node in~$T$, for every set~$F' \subseteq \mathrm{dom}(S) \cap \mathrm{set}_\mu(\sigma) \cap R_\alpha$, $\varphi(F^\alpha_\nu \cup F')$ holds. Let~$\tau$ be the string refining~$\sigma$ such that $\mathrm{set}_{\mu_\alpha}(\tau) = \mathrm{set}_\mu(\sigma) \cap R_\alpha$ for each~$\alpha < 2$ and each part~$\mu$ of $S$ $g$-refining part~$\nu$ of~$T$. By definition of~$S'$, $\tau \in S'$. Therefore~$S'$ is infinite. Moreover, by definition of~$S$', for each~$\alpha < 2$, either $\mu_\alpha$ is empty in $S'$ or~$(\vec{E}, S', \mathcal{C}[S']) \Vdash^\alpha_{\mu_\alpha} \varphi_{i_\alpha}(G)$, where $\vec{E}$ is obtained by duplicating the sets in~$\vec{F}$ according the forks of~$g \circ f$.
By definition of $c \Vdash^\alpha_\nu (\forall x)\neg \varphi_{i_\alpha}(G)$, $(\vec{E}, S', \mathcal{C}[S']) \not \Vdash^\alpha_{\mu_\alpha} \varphi_{i_\alpha}(G)$ for each~$\alpha < 2$ and each part $\mu$ of $S$ $g$-refining part $\nu$ of~$T$ such that $(\mu_\alpha, S') \in \mathcal{C}$. Then, for each~$\alpha < 2$, either~$\mu_\alpha$ is empty in~$S'$, or $(\mu_\alpha, S') \not \in \mathcal{C}$, as otherwise it would contradict $(\vec{E}, S', \mathcal{C}[S']) \Vdash^\alpha_{\mu_\alpha} \varphi(G)$. So there must exists a non-empty part $\mu$ of $S'$ not refining part $\nu$ of $T$ such that $(\mu, S') \in \mathcal{C}$, and by upward closure of a promise, there exists a non-empty part $\mu$ of $S$ not refining part $\nu$ of $T$ such that $(\mu, S) \in \mathcal{C}$. By definition of $\mathcal{D}$, $(\mu, S) \in \mathcal{D}$. Therefore $\mathcal{D}$ is a promise for $T$ and we conclude. \end{proof}
Given any filter~$\mathcal{F} = \{c_0, c_1, \dots \}$ with $c_s = (\vec{F}_s, T_s, \mathcal{C}_s)$ the set of pairs $(\alpha, \nu_s)$ such that $(\nu_s, T_s) \in \mathcal{C}_s$ forms again an infinite, directed acyclic graph~$\mathcal{G}(\mathcal{F})$. By Lemma~\ref{lem:cohzp-validity-exists}, whenever~$\mathcal{F}$ is sufficiently generic, the graph~$\mathcal{G}(\mathcal{F})$ yields a sequence of parts $P$ such that for every~$s$ if~$c_s$ refines $c_t$, then part~$P(s)$ of~$c_s$ refines part~$P(t)$ of~$c_t$, and such that for every $n$, there is some~$s$ and some side~$\alpha < 2$ such that the side~$\alpha$ is $n$-valid in part $P(s)$ of~$c_s$. The path~$P$ induces an infinite set~$G = \bigcup \{ F^\alpha_{P(s), s} : s \in \omega \}$. Since whenever $\alpha$ is $n$-valid in part~$P(s)$ of~$c_s$, then it is $m$-valid in part~$P(s)$ of~$c_s$ for every~$m < n$, we can fix an $\alpha < 2$ such that for every $n$, there is some~$s$ such that the side~$\alpha$ is $n$-valid in part $P(s)$ of~$c_s$. We call $\alpha$ the \emph{generic side}, $P$ the \emph{generic path} and $G$ the \emph{generic real}.
By choosing a generic path that goes through valid sides and parts of the conditions, we recovered the density property for the weak forcing relation and can therefore prove that a property holds over the generic real if and only if it can be forced by some condition belonging to the generic filter.
\begin{lemma}\label{lem:cohzp-generic-level1} Suppose that $\mathcal{F}$ is sufficiently generic and let $\alpha$,~$P$ and~$G$ be the generic side, the generic path and the generic real, respectively. For every $\Sigma^0_1$ ($\Pi^0_1$) formula $\varphi(G)$, $\varphi(G)$ holds iff $c_s \Vdash^\alpha_{P(s)} \varphi(G)$ for some $c_s \in \mathcal{F}$. \end{lemma} \begin{proof} Thanks to validity, it suffices to prove that if $c_s \Vdash^\alpha_\nu \varphi(G)$ for some~$c_s \in \mathcal{F}$, then $\varphi(G)$ holds. Indeed, if $\varphi(G)$ holds, then by genericity of $\mathcal{F}$, $c_s \Vvdash^\alpha_{P(s)} \varphi(G)$ or $c_s \Vvdash^\alpha_{P(s)} \neg \varphi(G)$ for some $c_s \in \mathcal{F}$. By validity of side $\alpha$ in part $P(s)$ of $c_s$, $c_s \Vdash^\alpha_{P(s)} \varphi(G)$ or $c_s \Vdash^\alpha_{P(s)} \neg \varphi(G)$. If $c_s \Vdash^\alpha_{P(s)} \neg \varphi(G)$ then $\neg \varphi(G)$ holds, contradicting the hypothesis. So $c_s \Vdash^\alpha_{P(s)} \varphi(G)$. Fix a condition $c_s = (\vec{F}, T, \mathcal{C}) \in \mathcal{F}$ such that $c_s \Vdash^\alpha_\nu \varphi(G)$, where~$\nu = P(s)$. \begin{itemize}
\item If $\varphi \in \Sigma^0_1$ then $\varphi(G)$ can be expressed as $(\exists x)\psi(G, x)$ where $\psi \in \Sigma^0_0$.
By clause~1 of Definition~\ref{def:cohzp-forcing-precondition}, there exists a $w \in \omega$ such that
$\psi(F^\alpha_\nu, w)$ holds. As $\nu = P(s)$, $F^\alpha_\nu = F^\alpha_{P(s)} \subseteq G$
and~$G \smallsetminus F^\alpha_\nu \subseteq (\max F^\alpha_\nu, \infty)$, so $\psi(G, w)$ holds by continuity, hence $\varphi(G)$ holds.
\item If $\varphi \in \Pi^0_1$ then $\varphi(G)$ can be expressed as $(\forall x)\psi(G, x)$ where $\psi \in \Sigma^0_0$.
By clause~2 of Definition~\ref{def:cohzp-forcing-precondition}, for every $\sigma \in T$, every $w < |\sigma|$
and every set $F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma)$, $\psi(F^\alpha_\nu \cup F', w)$ holds.
For every $F' \subseteq G \smallsetminus F^\alpha_\nu$, and $w \in \omega$ there exists a $\sigma \in T$
such that $w < |\sigma|$ and $F' \subseteq \mathrm{dom}(T) \cap \mathrm{set}_\nu(\sigma)$. Hence $\psi(F^\alpha_\nu \cup F', w)$ holds.
Therefore, for every $w \in \omega$, $\psi(G, w)$ holds, so $\varphi(G)$ holds. \end{itemize} \end{proof}
\begin{lemma} Suppose that $\mathcal{F}$ is sufficiently generic and let $\alpha$ and~$G$ be the generic side and the generic real, respectively. For every $\Sigma^0_{n+2}$ ($\Pi^0_{n+2}$) formula $\varphi(G)$, $\varphi(G)$ holds iff $c_s \Vdash^\alpha \varphi(G)$ for some $c_s \in \mathcal{F}$. \end{lemma} \begin{proof} This lemma uses validity implicitly by calling Lemma~\ref{lem:cohzp-generic-level1}, where it was used explicitly. Emulating the proof of Lemma~\ref{lem:em-generic-higher-levels}, it suffices to prove that if $c_s \Vdash^\alpha \varphi(G)$ for some $c_s \in \mathcal{F}$ then $\varphi(G)$ holds. Let~$P$ be the generic path induced by the generic filter~$\mathcal{F}$. Fix a condition $c_s = (\vec{F}, T, \mathcal{C}) \in \mathcal{F}$ such that $c_s \Vdash^\alpha \varphi(G)$. We proceed by case analysis on $\varphi$. \begin{itemize}
\item If $\varphi \in \Sigma^0_2$ then $\varphi(G)$ can be expressed as $(\exists x)\psi(G, x)$
where $\psi \in \Pi^0_1$.
By clause~1 of Definition~\ref{def:cohzp-forcing-condition}, for every part $\nu$ of $T$
such that $(\nu, T) \in \mathcal{C}$, there exists a $w < \mathrm{dom}(T)$
such that $c_s \Vdash^\alpha_\nu \psi(G, w)$. Since $(P(s), T) \in \mathcal{C}$,
$c_s \Vdash^\alpha_{P(s)} \psi(G, w)$. By Lemma~\ref{lem:cohzp-generic-level1}, $\psi(G, w)$ holds, hence $\varphi(G)$ holds.
\item If $\varphi \in \Pi^0_2$ then $\varphi(G)$ can be expressed as $(\forall x)\psi(G, x)$ where $\psi \in \Sigma^0_1$.
By clause~2 of Definition~\ref{def:cohzp-forcing-condition}, for every infinite $k'$-partition
tree $S$, every function~$f : \mathrm{parts}(S) \to \mathrm{parts}(T)$,
every $w$ and $\vec{E}$ smaller than the code of $S$ such that the followings hold
\begin{itemize}
\item[i)] $(E_\nu, \mathrm{dom}(S) \cap R_\alpha)$ Mathias extends $(F_{f(\nu)}, \mathrm{dom}(T) \cap R_\alpha)$ for each $\nu < \mathrm{parts}(S)$
\item[ii)] $S$ $f$-refines $\bigcap_{\nu < \mathrm{parts}(S)} T^{[f(\nu), E_\nu]}$
\end{itemize}
for every $(\mu, S) \in \mathcal{C}$, $(\vec{E}, S, \mathcal{C}[S]) \not \Vdash^\alpha_\mu \neg \psi(G, w)$.
Suppose by way of contradiction that $\psi(G, w)$ does not hold for some $w \in \omega$.
Then by Lemma~\ref{lem:cohzp-generic-level1}, there exists a $c_t \in \mathcal{F}$
such that $c_t \Vdash^\alpha_{P(t)} \neg \psi(G, w)$. Since~$\mathcal{F}$ is a filter,
there is a condition~$c_e = (\vec{E}, S, \mathcal{D}) \in \mathcal{F}$ extending~$c_s$ and~$c_t$.
By choice of~$P$, $(P(e), S) \in \mathcal{C}$, so by clause ii), $(\vec{E}, S, \mathcal{C}[S]) \not \Vdash^\alpha_{P(e)} \psi(G, w)$,
hence by Lemma~\ref{lem:cohzp-promise-no-effect-first-level}, $c_e \not \Vdash^\alpha_{P(e)} \psi(G, w)$.
However, since part~$P(e)$ of~$c_e$ refines part~$P(t)$ of~$c_t$,
then by Lemma~\ref{lem:cohzp-forcing-extension-level1}, $c_e \Vdash^\alpha_{P(e)} \psi(G, w)$.
Contradiction. Hence, for every~$w \in \omega$, $\psi(G, w)$ holds, so~$\varphi(G)$ holds.
\item If $\varphi \in \Sigma^0_{n+3}$ then $\varphi(G)$ can be expressed as $(\exists x)\psi(G, x)$
where $\psi \in \Pi^0_{n+2}$.
By clause~3 of Definition~\ref{def:cohzp-forcing-condition}, there exists a $w \in \omega$
such that $c_s \Vdash^\alpha \psi(G, w)$. By induction hypothesis, $\psi(G, w)$ holds, hence $\varphi(G)$ holds.
Conversely, if $\varphi(G)$ holds, then there exists a $w \in \omega$ such that $\psi(G, w)$ holds,
so by induction hypothesis $c_s \Vdash^\alpha \psi(G, w)$ for some $c_s \in \mathcal{F}$,
so by clause~3 of Definition~\ref{def:cohzp-forcing-condition}, $c_s \Vdash^\alpha \varphi(G)$.
\item If $\varphi \in \Pi^0_{n+3}$ then $\varphi(G)$ can be expressed as $\neg \psi(G)$ where $\psi \in \Sigma^0_{n+3}$.
By clause~4 of Definition~\ref{def:cohzp-forcing-condition}, for every $d \in \operatorname{Ext}(c_s)$, $d \not \Vdash^\alpha \psi(G)$.
By Lemma~\ref{lem:cohzp-forcing-extension}, $d \not \Vdash^\alpha \psi(G)$ for every~$d \in \mathcal{F}$,
and by a previous case, $\psi(G)$ does not hold, so $\varphi(G)$ holds. \end{itemize} \end{proof}
\subsection{Preserving definitions}
The following (and last) lemma shows that every sufficiently generic real preserves higher definitions. This preservation property cannot be proved in the case of non-$\Sigma^0_1$ sets since the weak forcing relation does not have the good density property in general.
\begin{lemma}\label{lem:cohzp-diagonalization} If $A \not \in \Sigma^0_{n+2}$ and $\varphi(G, x)$ is $\Sigma^0_{n+2}$, then the set of $c \in \mathbb{P}$ satisfying the following property is dense: $$ (\forall \alpha < 2)[(\exists w \in A)c \Vdash^\alpha \neg \varphi(G, w)] \vee [(\exists w \not \in A)c \Vdash^\alpha \varphi(G, w)] $$ \end{lemma} \begin{proof} It is sufficient to find, given a condition $c$ and a side $\alpha < 2$, an extension $d$ of~$c$ such that the following holds: $$ [(\exists w \in A)c \Vdash^\alpha \neg \varphi(G, w)] \vee [(\exists w \not \in A)c \Vdash^\alpha \varphi(G, w)] $$ Fix a condition $c = (\vec{F}, T, \mathcal{C})$ and a side $\alpha < 2$. \begin{itemize}
\item In case $n = 0$, $\varphi(G, w)$ can be expressed as $(\exists x)\psi(G, w, x)$ where $\psi \in \Pi^0_1$.
Let $U$ be the set of integers $w$ such that there exists an infinite p.r.\ $k'$-partition tree $S$
for some $k' \in \omega$, a function~$f : \mathrm{parts}(S) \to \mathrm{parts}(T)$ and a $2k'$-tuple of finite sets $\vec{E}$ such that
\begin{itemize}
\item[i)] $E^\beta_\nu = F^\beta_{f(\nu)}$ for each $\nu < \mathrm{parts}(S)$ and $\beta \neq \alpha$
\item[ii)] $(E^\alpha_\nu, \mathrm{dom}(S) \cap R_\alpha)$ Mathias extends $(F^\alpha_{f(\nu)}, \mathrm{dom}(T))$ for each $\nu < \mathrm{parts}(S)$.
\item[iii)] $S$ $f$-refines $\bigcap_{\nu < \mathrm{parts}(S)} T^{[f(\nu), E^\alpha_\nu]}$
\item[iv)] for each non-empty part $\nu$ of $S$ such that $(\nu, S) \in \mathcal{C}$,
$(\vec{E}, S, \mathcal{C}[S]) \Vdash^\alpha_\nu \psi(G, w, u)$ for some~$u < \#S$
\end{itemize}
By Lemma~\ref{lem:cohzp-complexity-forcing} and Lemma~\ref{lem:em-refinement-complexity},
$U \in \Sigma^0_2$, thus $U \neq A$. Let $w \in U \Delta A$.
Suppose that $w \in U \smallsetminus A$.
Let $\mathcal{D} = \mathcal{C}[S] \smallsetminus \{(\nu, S') \in \mathcal{C} : \mbox{ part } \nu \mbox{ of } S' \mbox{ is empty} \}$.
As $\mathcal{C}$ is an $\emptyset'$-p.r.\ promise for $T$,
$\mathcal{C}[S]$ is an $\emptyset'$-p.r.\ promise for $S$.
As $\mathcal{D}$ is obtained from $\mathcal{C}[S]$ by removing only empty parts, $\mathcal{D}$ is also an $\emptyset'$-p.r.\ promise for $S$.
By Lemma~\ref{lem:cohzp-promise-no-effect-first-level}, for every part~$\nu$ of~$S$
such that~$(\nu, S) \in \mathcal{D} \subseteq \mathcal{C}$, $(\vec{E}, S, \mathcal{D}) \Vdash^\alpha_\nu \psi(G, w, u)$ for some~$u < \mathrm{dom}(S)$,
hence by clause~1 of Definition~\ref{def:cohzp-forcing-condition},
$d = (\vec{E}, S, \mathcal{D}) \Vdash^\alpha (\exists x)\psi(G, w, x)$.
In other words, $d \Vdash^\alpha \varphi(G)$ for some $w \not \in A$.
We may choose a coding of the p.r.\ trees such that
the code of $S$ is sufficiently large to witness $w$ and $\vec{E}$.
So suppose now that $w \in A \smallsetminus U$. Then for every infinite p.r.\ $k'$-partition tree $S$,
every function~$f : \mathrm{parts}(S) \to \mathrm{parts}(T)$
and every $\vec{E}$ smaller than the code of $S$ such that properties i-iii) hold,
there exists a non-empty part $\nu$ of $S$ such that $(\nu, S) \in \mathcal{C}$
and $(\vec{E}, S, \mathcal{C}[S]) \not \Vdash^\alpha_\nu \psi(G, w, u)$ for every $u < \#S$.
Let $\mathcal{D}$ be the collection of all such $(\nu, S)$. $\mathcal{D}$ is $\emptyset'$-p.r.
By Lemma~\ref{lem:cohzp-forcing-extension-level1} and since~$\#S \geq \#T$
whenever~$S \leq T$, $\mathcal{D}$ is upward-closed under the refinement relation, hence is a promise for~$T$. By clause~2
of Definition~\ref{def:cohzp-forcing-condition}, $d = (\vec{F}, T, \mathcal{D}) \Vdash^\alpha (\forall x) \neg \psi(G, w, x)$,
hence $d \Vdash^\alpha \neg \varphi(G, w)$ for some $w \in A$.
\item
In case $n > 0$, let $U = \{ w \in \omega : (\exists d \in \operatorname{Ext}(c)) d \Vdash^\alpha \varphi(G, w) \}$.
By Lemma~\ref{lem:em-extension-complexity} and Lemma~\ref{lem:em-complexity-forcing},
$U \in \Sigma^0_{n+2}$, thus $U \neq A$.
Fix $w \in U \Delta A$. If $w \in U \smallsetminus A$ then by definition of~$U$,
there exists a condition $d$ extending $c$ such that $d \Vdash^\alpha \varphi(G, w)$.
If $w \in A \smallsetminus U$, then for every $d \in \operatorname{Ext}(c)$, $d \not \Vdash^\alpha \varphi(G, w)$
so by clause~4 of Definition~\ref{def:cohzp-forcing-condition}, $c \Vdash^\alpha \neg \varphi(G, w)$. \end{itemize} \end{proof}
We are now ready to reprove Corollary~3.29 from Wang~\cite{Wang2014Definability}.
\begin{theorem}[Wang~\cite{Wang2014Definability}] $\rt^2_2$ admits preservation of $\Xi$ definitions simultaneously for all $\Xi$ in $\{\Sigma^0_{n+2}, \allowbreak \Pi^0_{n+2}, \allowbreak \Delta^0_{n+3} : n \in \omega \}$. \end{theorem} \begin{proof} Since $\rca \vdash \coh \wedge \mathsf{D}^2_2 \rightarrow \rt^2_2$, and $\coh$ admits preservation of the arithmetic hierarchy, it suffices to prove that~$\mathsf{D}^2_2$ admits preservation of $\Xi$ definitions simultaneously for all $\Xi$ in $\{\Sigma^0_{n+2}, \allowbreak \Pi^0_{n+2}, \allowbreak \Delta^0_{n+3} : n \in \omega \}$. Fix some set~$C$ and a $\Delta^{0,C}_2$ 2-partition $R_0 \cup R_1 = \omega$. Let $\mathcal{C}_0$ be the $C'$-p.r.\ set of all $(\nu, T) \in \mathbb{U}$ such that $(\nu, T) \leq (0, 1^{<\omega})$. Let~$\mathcal{F}$ be a sufficiently generic filter containing~$c_0 = (\{\emptyset, \emptyset\}, 1^{<\omega}, \mathcal{C}_0)$. Let $G$ be the corresponding generic real. By definition of a condition, the set~$G$ is $\vec{R}$-cohesive. By Lemma~\ref{lem:cohzp-diagonalization} and Lemma~\ref{lem:cohzp-complexity-forcing}, $G$ preserves non-$\Sigma^0_{n+2}$ definitions relative to~$C$ for every~$n \in \omega$. Therefore, by Proposition 2.2 of~\cite{Wang2014Definability}, $G$ preserves $\Xi$ definitions relative to~$C$ simultaneously for all $\Xi$ in $\{\Sigma^0_{n+2}, \Pi^0_{n+2}, \Delta^0_{n+3} : n \in \omega \}$. \end{proof}
\noindent \textbf{Acknowledgements}. The author is thankful to Wei Wang for interesting comments and discussions. The author is funded by the John Templeton Foundation (`Structure and Randomness in the Theory of Computation' project). The opinions expressed in this publication are those of the author(s) and do not necessarily reflect the views of the John Templeton Foundation.
\end{document} | arXiv |
\begin{document}
\title{f Stratified-algebraic vector bundles} \thispagestyle{empty}
\begin{abstract} We investigate stratified-algebraic vector bundles on a real algebraic variety $X$. A~stratification of $X$ is a finite collection of pairwise disjoint, Zariski locally closed subvarieties whose union is $X$. A topological vector bundle $\xi$ on $X$ is called a stratified-algebraic vector bundle if, roughly speaking, there exists a stratification $\mathcal{S}$ of $X$ such that the restriction of $\xi$ to each stratum $S$ in $\mathcal{S}$ is an algebraic vector bundle on $S$. In particular, every algebraic vector bundle on $X$ is stratified-algebraic. It turns out that stratified-algebraic vector bundles have many surprising properties, which distinguish them from algebraic and topological vector bundles. \end{abstract}
\keywords{Real algebraic variety, stratification, stratified-algebraic vector bundle, stratified-regular map.}
\subjclass{14P25, 14P99, 14F25, 19A49.}
\tableofcontents
\section{Introduction and main results}\label{sec-1} In real algebraic geometry, the role of algebraic, semi-algebraic and Nash vector bundles is firmly established. Vector bundles of a new type, called stratified-algebraic vector bundles, are introduced and investigated in this paper. Stratified-algebraic vector bundles form an intermediate category between algebraic and semi-algebraic vector bundles. They have many desirable features of algebraic vector bundles, but are more flexible. Some of their properties and applications are quite unexpected.
For background material on real algebraic geometry we refer to \cite{bib9}. The term \emph{real algebraic variety} designates a locally ringed space isomorphic to an algebraic subset of $\mathbb{R}^N$, for some $N$, endowed with the Zariski topology and the sheaf of real-valued regular functions (such an object is called an affine real algebraic variety in \cite{bib9}). The class of real algebraic varieties is identical with the class of quasi-projective real varieties, cf. \cite[Proposition~3.2.10, Theorem~3.4.4]{bib9}. Morphisms of real algebraic varieties are called \emph{regular maps}. Each real algebraic variety carries also the Euclidean topology, which is induced by the usual metric on $\mathbb{R}$. Unless explicitly stated otherwise, all topological notions relating to real algebraic varieties refer to the Euclidean topology.
Let $X$ be a real algebraic variety. By a \emph{stratification} of $X$ we mean a finite collection $\mathcal{X}$ of pairwise disjoint, Zariski locally closed subvarieties whose union is $X$. Each subvariety in $\mathcal{X}$ is called a \emph{stratum} of $\mathcal{X}$; a stratum can be empty. The stratification $\mathcal{X}$ is said to be \emph{nonsingular} if each stratum in it is a nonsingular subvariety. A stratification $\mathcal{X}'$ of $X$ is said to be a \emph{refinement} of $\mathcal{X}$ if each stratum of $\mathcal{X}'$ is contained in some stratum of $\mathcal{X}$. There is a nonsingular stratification of $X$ which is a refinement of $\mathcal{X}$. If $\mathcal{X}_1$ and $\mathcal{X}_2$ are stratifications of $X$, then the collection $\{ S_1 \cap S_2 \mid S_1 \in \mathcal{X}_1, S_2 \in \mathcal{X}_2 \}$ is a stratification of $X$ that is a refinement of $\mathcal{X}_i$ for $i=1,2$. These facts will be frequently tacitly used. The stratification $\{X\}$ of $X$, consisting of only one stratum, is said to be \emph{trivial}.
Let $\mathbb{F}$ stand for $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$ (the quaternions). All $\mathbb{F}$-vector spaces will be left $\mathbb{F}$-vector spaces. When convenient, $\mathbb{F}$ will be identified with $\mathbb{R}^{d(\mathbb{F})}$, where ${d(\mathbb{F}) = \dim_{\mathbb{R}}\mathbb{F}}$.
For any topological $\mathbb{F}$-vector bundle $\xi$ on $X$, denote by $E(\xi)$ its total space and by $\pi(\xi) \colon E(\xi) \to X$ the bundle projection. The fiber of $\xi$ over a point $x$ in $X$ is ${E(\xi)_x \coloneqq \pi(\xi)^{-1}(x)}$. Denote by $\varepsilon_X^n(\mathbb{F})$ the standard trivial $\mathbb{F}$-vector bundle on $X$ with total space $X \times \mathbb{F}^n$, where $X \times \mathbb{F}^n$ is regarded as a real algebraic variety. By an \emph{algebraic $\mathbb{F}$-vector bundle} on $X$ we mean an algebraic $\mathbb{F}$-vector subbundle of $\varepsilon_X^n(\mathbb{F})$ for some $n$ (cf. \cite[Chapters~12 and 13]{bib9} for various characterizations of algebraic $\mathbb{F}$-vector bundles). In particular, if $\xi$ is an algebraic $\mathbb{F}$-vector subbundle of $\varepsilon_X^n(\mathbb{F})$, then $E(\xi)$ is a Zariski closed subvariety of $X \times \mathbb{F}^n$, $\pi(\xi)$ is the restriction of the canonical projection $X \times \mathbb{F}^n \to X$, and $E(\xi)_x$ is an $\mathbb{F}$-vector subspace of $\{x\}\times\mathbb{F}^n$ for each point $x$ in $X$.
Given a stratification $\mathcal{X}$ of $X$, we now introduce the crucial notion for this paper.
\begin{definition}\label{def-1-1} An \emph{$\mathcal{X}$-algebraic $\mathbb{F}$-vector bundle} on $X$ is a topological
$\mathbb{F}$-vector subbundle $\xi$ of $\varepsilon_X^n(\mathbb{F})$, for some $n$, such that the restriction $\xi|_S$ of $\xi$ to each stratum $S$ of $\mathcal{X}$ is an algebraic $\mathbb{F}$-vector subbundle of $\varepsilon_S^n(\mathbb{F})$. If $\xi$ and $\eta$ are $\mathcal{X}$-algebraic $\mathbb{F}$-vector bundles on $X$, an \emph{$\mathcal{X}$-algebraic morphism} $\varphi \colon \xi \to \eta$ is a morphism of topological
$\mathbb{F}$-vector bundles which induces a morphism of algebraic $\mathbb{F}$-vector bundles $\varphi_S \colon \xi|_S \to \eta|_S$ for each stratum $S$ in $\mathcal{X}$. \end{definition}
The conditions imposed on $\varphi$ mean that $\varphi \colon E(\xi) \to E(\eta)$ is a continuous map, ${\pi(\xi) = \pi(\eta) \circ \varphi}$, the restriction $\varphi_x \colon E(\xi)_x \to E(\eta)_x$ of $\varphi$ is an $\mathbb{F}$-linear transformation for each point $x$ in $X$, and the restriction $\varphi_S \colon \pi(\xi)^{-1}(S) \to \pi(\eta)^{-1}(S)$ of $\varphi$ is a regular map of real algebraic varieties for each stratum $S$ in $\mathcal{X}$.
If $\xi$ is as in Definition~\ref{def-1-1}, we also say that $\xi$ is an \emph{$\mathcal{X}$-algebraic $\mathbb{F}$-vector subbundle of $\varepsilon_X^n(\mathbb{F})$}.
One readily checks that $\mathcal{X}$-algebraic $\mathbb{F}$-vector bundles on $X$ (together with $\mathcal{X}$-algebraic morphisms) form a category. An $\mathcal{X}$-algebraic morphism of $\mathcal{X}$-algebraic $\mathbb{F}$-vector bundles is an isomorphism if and only if it is a bijective map. In particular, the category of algebraic $\mathbb{F}$-vector bundles on $X$ coincides with the category of $\{X\}$-algebraic $\mathbb{F}$-vector bundles on $X$.
\begin{definition}\label{def-1-2} A \emph{stratified-algebraic $\mathbb{F}$-vector bundle} on $X$ is an $\mathcal{S}$-algebraic $\mathbb{F}$-vector bundle for some stratification $\mathcal{S}$ of $X$. If $\xi$ and $\eta$ are stratified-algebraic $\mathbb{F}$-vector bundles on $X$, a \emph{stratified-algebraic morphism} $\varphi \colon \xi \to \eta$ is an $\mathcal{S}$-algebraic morphism for some stratification $\mathcal{S}$ of $X$ such that both $\xi$ and $\eta$ are $\mathcal{S}$-algebraic $\mathbb{F}$-vector bundles. \end{definition}
A stratified-algebraic $\mathbb{F}$-vector subbundle of $\xi_X^n(\mathbb{F})$ is defined in an obvious way.
Stratified-algebraic $\mathbb{F}$-vector bundles on $X$ (together with stratified-algebraic morphisms) form a category. A stratified-algebraic morphism of stratified-algebraic $\mathbb{F}$-vector bundles is an isomorphism if and only if it is a bijective map.
Theory of $\mathcal{X}$-algebraic and stratified-algebraic $\mathbb{F}$-vector bundles is developed in the subsequent sections. In the present section, we only announce seven rather surprising results.
Denote by $\mathbb{S}^d$ the unit $d$-sphere,
\begin{equation*} \mathbb{S}^d = \{ (x_0, \ldots, x_d) \in \mathbb{R}^{d+1} \mid x_0^2 + \cdots + x_d^2 = 1 \}. \end{equation*}
\begin{theorem}\label{th-1-3} Let $X$ be a compact real algebraic variety homotopically equivalent to $\mathbb{S}^d$. Then each topological $\mathbb{F}$-vector bundle on $X$ is isomorphic to a stratified-algebraic $\mathbb{F}$-vector bundle. \end{theorem}
Theorem~\ref{th-1-3} makes it possible to demonstrate an essential difference between algebraic and stratified-algebraic $\mathbb{F}$-vector bundles.
\begin{example}\label{ex-1-4} It is well known that each topological $\mathbb{F}$-vector bundle on $\mathbb{S}^d$ is isomorphic to an algebraic $\mathbb{F}$-vector bundle, cf. \cite[Theorem~11.1]{bib44} and \cite[Proposition~12.1.12; pp.~325, 326, 352]{bib9}. This is no longer true if $\mathbb{S}^d$ is replaced by a nonsingular real algebraic variety diffeomorphic to $\mathbb{S}^d$. Indeed, for every positive integer $k$, there exists a nonsingular real algebraic variety $\Sigma^{4k}$ that is diffeomorphic to $\mathbb{S}^{4k}$ and each algebraic $\mathbb{F}$-vector bundle on $\Sigma^{4k}$ is topologically stably trivial, cf. \cite[Theorem~9.1]{bib8}. However, on $\mathbb{S}^{4k}$, and hence on $\Sigma^{4k}$, there are topological $\mathbb{F}$-vector bundles that are not stably trivial, cf. \cite{bib28}. On the other hand, according to Theorem~\ref{th-1-3}, each topological $\mathbb{F}$-vector bundle on $\Sigma^{4k}$ is isomorphic to a stratified-algebraic $\mathbb{F}$-vector bundle. \end{example}
Stratified-algebraic vector bundles on a compact real algebraic variety are in some sense stable with respect to stratifications. This is made precise in the following result.
\begin{theorem}\label{th-1-5} Any compact real algebraic variety $X$ admits a nonsingular stratification $\mathcal{S}$ such that each stratified-algebraic $\mathbb{F}$-vector bundle on $X$ is isomorphic (in the category of stratified-algebraic $\mathbb{F}$-vector bundles on $X$) to an $\mathcal{S}$-algebraic $\mathbb{F}$-vector bundle. \end{theorem}
It is remarkable that the stratification $\mathcal{S}$ in Theorem~\ref{th-1-5} is suitable for all stratified-algebraic $\mathbb{F}$-vector bundles on $X$. In general one cannot take as $\mathcal{S}$ the trivial stratification $\{ X \}$ of $X$, even if $X$ is a ``simple'' nonsingular real algebraic variety, cf. Example~\ref{ex-1-4}.
A \emph{multiblowup} of a real algebraic variety $X$ is a regular map $\pi \colon X' \to X$ which is the composition of a finite number of blowups with nonsingular centers. If no additional restrictions on the centers of blowups are imposed, $\pi$ need not be a birational map (for example, a blowup of $X$ with center of dimension $\dim X$ is not a birational map).
\begin{theorem}\label{th-1-6} For any compact real algebraic variety $X$, there exists a birational multiblowup $\pi \colon X' \to X$, with $X'$ a nonsingular variety, such that for each stratified-algebraic $\mathbb{F}$-vector bundle $\xi$ on $X$, the pullback $\mathbb{F}$-vector bundle $\pi^*\xi$ on $X'$ is isomorphic (in the category of stratified-algebraic $\mathbb{F}$-vector bundles on $X'$) to an algebraic $\mathbb{F}$-vector bundle on $X'$. \end{theorem}
The multiblowup $\pi \colon X' \to X$ in Theorem~\ref{th-1-6} is chosen in a universal way, that is, it does not depend on $\xi$.
Any topological $\mathbb{F}$-vector bundle $\xi$ can be regarded as an $\mathbb{R}$-vector bundle, which is indicated by $\xi_{\mathbb{R}}$. If $\xi$ is $\mathcal{X}$-algebraic, then so is $\xi_{\mathbb{R}}$.
\begin{theorem}\label{th-1-7} Let $X$ be a compact real algebraic variety. A topological $\mathbb{F}$-vector bundle $\xi$ on $X$ is isomorphic to a stratified-algebraic $\mathbb{F}$-vector bundle if and only if the topological $\mathbb{R}$-vector bundle $\xi_{\mathbb{R}}$ is isomorphic to a stratified-algebraic $\mathbb{R}$-vector bundle. \end{theorem}
This result is unexpected since it may happen that a topological $\mathbb{F}$-vector bundle $\xi$ (with $\mathbb{F}=\mathbb{C}$ or $\mathbb{F}=\mathbb{H}$) is not isomorphic to any algebraic $\mathbb{F}$-vector bundle, whereas the $\mathbb{R}$-vector bundle $\xi_{\mathbb{R}}$ is isomorphic to an algebraic $\mathbb{R}$-vector bundle, cf. the $\mathbb{C}$-line bundle $\lambda^{\mathbb{C}}$ in Example~\ref{ex-1-11}.
Theorems~\ref{th-1-5}, \ref{th-1-6} and \ref{th-1-7} are equivalent to certain approximation results, cf. Theorems~\ref{th-4-8}, \ref{th-4-9} and \ref{th-6-6}.
It is possible to give, in some cases, a simple geometric criterion for a topological vector bundle to be isomorphic to a stratified-algebraic vector bundle.
Let $X$ be a compact nonsingular real algebraic variety. A smooth (of class~$\mathcal{C}^{\infty}$) \mbox{$\mathbb{F}$-vector} bundle $\xi$ on $X$ is said to be \emph{adapted} if there exists a smooth section ${u \colon X \to \xi}$ transverse to the zero section and such that its zero locus $Z(u) \coloneqq \{ {x \in X} \mid {u(x)=0} \}$, which is a compact smooth submanifold of $X$, is smoothly isotopic to a nonsingular Zariski locally closed subvariety $Z$ of $X$. In that case, $Z$ is a closed subset of $X$ in the Euclidean topology, but need not be Zariski closed. If $\func{rank} \xi = \dim X$, then $\xi$ is adapted since the zero locus of any smooth section of $\xi$ that is transverse to the zero section is a finite set.
\begin{theorem}\label{th-1-8} Let $X$ be a compact nonsingular real algebraic variety. If a smooth $\mathbb{F}$-line bundle $\xi$ on $X$ is adapted, then it is topologically isomorphic to a stratified-algebraic $\mathbb{F}$-line bundle. \end{theorem}
Theorem~\ref{th-1-8} with $\mathbb{F} = \mathbb{R}$ is not interesting since then a stronger result, asserting that $\xi$ is isomorphic to an algebraic $\mathbb{R}$-line bundle, is known, cf. \cite[Theorem~12.4.6]{bib9}. However, $\xi$ need not be isomorphic to any algebraic $\mathbb{F}$-line bundle for $\mathbb{F}=\mathbb{C}$ or $\mathbb{F}=\mathbb{H}$, cf. the $\mathbb{F}$-line bundle $\lambda^{\mathbb{F}}$ in Example~\ref{ex-1-11}.
Assuming that $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C}$, the $r$th exterior power of any $\mathbb{F}$-vector bundle of rank $r$ is denoted by $\det \xi$. Thus, $\det \xi$ is an $\mathbb{F}$-line bundle.
\begin{theorem}\label{th-1-9} Let $X$ be a compact nonsingular real algebraic variety and let $\xi$ be a smooth $\mathbb{F}$-vector bundle of rank $2$ on $X$, where $\mathbb{F} = \mathbb{R}$ or $\mathbb{F}=\mathbb{C}$. If both $\mathbb{F}$-vector bundles $\xi$ and $\det \xi$ are adapted, then $\xi$ is topologically isomorphic to a stratified-algebraic $\mathbb{F}$-vector bundle. \end{theorem}
It is not a serious restriction that the vector bundle $\xi$ in the last two theorems is smooth. In fact, any topological $\mathbb{F}$-vector bundle on a smooth manifold is topologically isomorphic to a smooth $\mathbb{F}$-vector bundle, cf. \cite{bib27}. Actually, suitably refined versions of Theorems~\ref{th-1-8} and~\ref{th-1-9} hold true even if the variety $X$ is possibly singular, cf. Theorems~\ref{th-6-9} and~\ref{th-6-10}.
We select one more result for this section.
\begin{theorem}\label{th-1-10} Let $X = X_1 \times \cdots \times X_n$, where each $X_i$ is a compact real algebraic variety homotopically equivalent to the unit $d_i$-sphere for $1 \leq i \leq n$. If $\mathbb{F}=\mathbb{C}$ or $\mathbb{F}=\mathbb{H}$, then each topological $\mathbb{F}$-vector bundle on $X$ is isomorphic to a stratified-algebraic $\mathbb{F}$-vector bundle. If $\xi$ is a topological $\mathbb{R}$-vector bundle on $X$, then the direct sum $\xi \oplus \xi$ is isomorphic to a stratified-algebraic $\mathbb{R}$-vector bundle. \end{theorem}
We do not know if in Theorem~\ref{th-1-10} each topological $\mathbb{R}$-vector bundle on $X$ is isomorphic to a stratified-algebraic $\mathbb{R}$-vector bundle. This remains an open problem even for $\mathbb{R}$-vector bundles on the standard $n$-torus
\begin{equation*} \mathbb{T}^n = \mathbb{S}^1 \times \cdots \times \mathbb{S}^1 \quad \textrm{(the $n$-fold product)}. \end{equation*}
It is worthwhile to contrast the results above with the behavior of algebraic $\mathbb{F}$-vector bundles on $\mathbb{T}^n$. As usual, the $k$th Chern class of a $\mathbb{C}$-vector bundle $\xi$ will be denoted by $c_k(\xi)$. Any $\mathbb{F}$-vector bundle $\zeta$ can be regarded as a $\mathbb{K}$-vector bundle, denoted $\zeta_{\mathbb{K}}$, where $\mathbb{K} \subseteq \mathbb{F}$ and $\mathbb{K}$ stands for $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. In particular, $\zeta_{\mathbb{F}}=\zeta$. If $\zeta$ is an $\mathbb{H}$-vector bundle, then $\zeta_{\mathbb{R}} = (\zeta_{\mathbb{C}})_{\mathbb{R}}$.
\begin{example}\label{ex-1-11} Every algebraic $\mathbb{C}$-vector bundle on $\mathbb{T}^n$ is algebraically stably trivial, cf. \cite{bib11} or \cite[Corollary~12.6.6]{bib9}. In particular, every algebraic $\mathbb{C}$-line bundle on $\mathbb{T}^n$ is algebraically trivial. Consequently, for each algebraic $\mathbb{F}$-vector bundle $\zeta$ on $\mathbb{T}^n$, where $\mathbb{F}=\mathbb{C}$ or $\mathbb{F}=\mathbb{H}$, one has $c_k(\zeta_{\mathbb{C}}) = 0$ for every $k \geq 1$. Furthermore, for each algebraic $\mathbb{R}$-vector bundle $\eta$ on $\mathbb{T}^n$, the direct sum $\eta \oplus \eta$ is algebraically stably trivial (it suffices to consider the complexification $\mathbb{C} \otimes \eta$ of $\eta$ and make use of the equality $(\mathbb{C} \otimes \eta)_{\mathbb{R}} = \eta \oplus \eta)$. On the other hand, if $1 \leq n\leq 3$, then each topological $\mathbb{R}$-vector bundle on $\mathbb{T}^n$ is isomorphic to an algebraic $\mathbb{R}$-vector bundle \cite{bib13}.
Assume now that $\mathbb{F}=\mathbb{C}$ or $\mathbb{F}=\mathbb{H}$. We choose a smooth $\mathbb{F}$-line bundle $\theta^{\mathbb{F}}$ on $\mathbb{T}^{d(\mathbb{F})}$ such that $c_1(\theta^{\mathbb{C}}) \neq 0$ and $c_2((\theta^{\mathbb{H}})_{\mathbb{C}}) \neq 0$. If $n \geq d(\mathbb{F})$ and $p_{\mathbb{F}} \colon \mathbb{T}^n = \mathbb{T}^{d(\mathbb{F})} \times \mathbb{T}^{n - d(\mathbb{F})} \to \mathbb{T}^{d(\mathbb{F})}$ is the canonical projection, then the smooth $\mathbb{F}$-line bundle
\begin{equation*} \lambda^{\mathbb{F}} \coloneqq p_{\mathbb{F}}^* \theta^{\mathbb{F}} \end{equation*}
on $\mathbb{T}^n$ is not topologically isomorphic to any algebraic $\mathbb{F}$-line bundle. This assertion holds since $c_1(\lambda^{\mathbb{C}}) \neq 0$ and $c_2((\lambda^{\mathbb{H}})_{\mathbb{C}}) \neq 0$. Obviously, $\theta^{\mathbb{F}}$ is adapted, and hence $\lambda^{\mathbb{F}}$ is adapted too. Furthermore, the $\mathbb{R}$-vector bundle $(\lambda^{\mathbb{C}})_{\mathbb{R}}$ on $\mathbb{T}^n$ is isomorphic to an algebraic $\mathbb{R}$-vector bundle since the $\mathbb{R}$-vector bundle $(\theta^{\mathbb{C}})_{\mathbb{R}}$ on $\mathbb{T}^2$ is isomorphic to an algebraic $\mathbb{R}$-vector bundle, $p_{\mathbb{C}}$ is a regular map, and
\begin{equation*} (\lambda^{\mathbb{C}})_{\mathbb{R}} = p_{\mathbb{C}}^* ( (\theta^{\mathbb{C}})_{\mathbb{R}} ). \end{equation*}
Finally, if $\xi = (\lambda^{\mathbb{H}})_{\mathbb{R}}$, then the $\mathbb{R}$-vector bundle $\xi \oplus \xi$ on $\mathbb{T}^n$ is not isomorphic to any algebraic $\mathbb{R}$-vector bundle. Indeed, supposing otherwise, the complexification $\mathbb{C} \otimes (\xi \oplus \xi)$ of $\xi \oplus \xi$ would be isomorphic to an algebraic $\mathbb{C}$-vector bundle, and hence $c_2(\mathbb{C} \otimes (\xi \oplus \xi)) =0$. However, one has $c_1( (\lambda^{\mathbb{H}} \oplus \lambda^{\mathbb{H}})_{\mathbb{C}} ) = 0$, which implies
\begin{equation*} c_2( \mathbb{C} \otimes (\xi \oplus \xi) ) = c_2( \mathbb{C} \otimes (\lambda^{\mathbb{H}} \oplus \lambda^{\mathbb{H}})_{\mathbb{R}} ) = 2 c_2( (\lambda^{\mathbb{H}} \oplus \lambda^{\mathbb{H}})_{\mathbb{C}} ) = 4 c_2 ( (\lambda^{\mathbb{H}})_{\mathbb{C}} ) \neq 0, \end{equation*}
cf. \cite[Corollary~15.5]{bib38}. \end{example}
There are other significant differences between algebraic and stratified-algebraic vector bundles. The interested reader can identify them by consulting papers devoted to algebraic vector bundles on real algebraic varieties \cite{bib5, bib6, bib7, bib8, bib9, bib10, bib13, bib14}. It should be mentioned that there are topological $\mathbb{F}$-line bundles which are not isomorphic to stratified-algebraic $\mathbb{F}$-line bundles. For instance, such $\mathbb{F}$-line bundles exist on some nonsingular real algebraic varieties diffeomorphic to $\mathbb{T}^n$, cf. Example~\ref{ex-7-10}.
The paper is organized as follows. In Section~\ref{sec-2}, we define in a natural way stratified-regular maps. Some homotopical properties of such maps imply Theorem~\ref{th-1-3}. In Section~\ref{sec-3}, we study relationships between vector bundles of various types and appropriate finitely generated projective modules. As a model serve classical results due to Serre \cite{bib41} and Swan \cite{bib43}. Subsequently, we prove Theorems~\ref{th-1-5} and~\ref{th-1-6} by applying \cite{bib44} and some results from K-theory. Section~\ref{sec-4} is devoted to maps with values in a Grassmannian (actually, a multi-Grassmannian). The main topic is the approximation of continuous maps by stratified-regular maps. Approximation results of this kind are essential for the rest of the paper. As the starting point for these results the extension theorem of Koll\'ar and Nowak \cite{bib31} is indispensable. In Section~\ref{sec-5}, we show how certain properties of a vector bundle can be deduced from the behavior of its restrictions to Zariski closed subvarieties of the base space. This is useful in proofs by induction in Section~\ref{sec-6}. In particular, Theorems~\ref{th-1-7}, \ref{th-1-8} and~\ref{th-1-9} are proved in Section~\ref{sec-6}. Algebraic and $\mathbb{C}$-algebraic cohomology classes have many applications in real algebraic geometry, cf. \cite{bib1, bib2, bib5, bib6, bib8, bib9, bib10, bib12, bib13, bib14, bib16, bib17, bib18, bib19, bib32, bib34}. We introduce stratified-algebraic and stratified-$\mathbb{C}$-algebraic cohomology classes in Section~\ref{sec-7} and Section~\ref{sec-8}, respectively. They prove to be very useful in our investigation of stratified-algebraic vector bundles. In particular, stratified-$\mathbb{C}$-algebraic cohomology classes play a key role in the proof of Theorem~\ref{th-1-10} given in Section~\ref{sec-8}.
\section{Stratified-regular maps}\label{sec-2}
Throughout this section, $X$ and $Y$ denote real algebraic varieties, and $\mathcal{X}$ denotes a stratification of $X$.
To begin with, we introduce maps that will be crucial for the investigation of $\mathcal{X}$-algebraic and stratified-algebraic vector bundles.
\begin{definition}\label{def-2-1} A map $f \colon X \to Y$ is said to be \emph{$\mathcal{X}$-regular} if it is continuous and its restriction to each stratum in $\mathcal{X}$ is a regular map. Furthermore, $f$ is said to be \emph{stratified-regular} if it is $\mathcal{S}$-regular for some stratification $\mathcal{S}$ of $X$. \end{definition}
In particular, $f$ is $\{X\}$-regular if and only if it is a regular map. Following \cite{bib31,bib32,bib34}, we say that $f \colon X \to Y$ is a \emph{continuous rational} map if $f$ is continuous and its restriction to some Zariski open and dense subvariety of $X$ is a regular map.
By a \emph{filtration} of $X$ we mean a finite sequence $\mathcal{F} = (X_{-1}, X_0, \ldots, X_m)$ of Zariski closed subvarieties satisfying
\begin{equation*} \varnothing = X_{-1} \subseteq X_0 \subseteq \ldots \subseteq X_m = X. \end{equation*}
The collection $\overline{\mathcal{F}} \coloneqq \{ X_i \setminus X_{i-1} \mid 0 \leq i \leq m \}$ is a stratification of $X$.
\begin{proposition}\label{prop-2-2} For a map $f \colon X \to Y$, the following conditions are equivalent:
\begin{conditions} \item\label{prop-2-2-a} The map $f$ is stratified-regular.
\item\label{prop-2-2-b} There exists a filtration $\mathcal{F}$ of $X$ such that $f$ is $\overline{\mathcal{F}}$-regular and each stratum in $\overline{\mathcal{F}}$ is nonsingular and equidimensional. \setcounter{favoritecondition}{\value{conditionsi}} \suspend{conditions} \begin{altconditions}[start=\value{favoritecondition}] \item\label{prop-2-2-b1} There exists a filtration $\mathcal{F}'$ of $X$ such that $f$ is $\overline{\mathcal{F}}'$-regular. \end{altconditions} \resume{conditions}
\item\label{prop-2-2-c} For each Zariski closed subvariety $Z$ of $X$, the restriction $f|_Z \colon Z \to Y$ is a continuous rational map. \setcounter{favoritecondition}{\value{conditionsi}} \end{conditions}
\end{proposition}
\begin{proof} Suppose that condition (\ref{prop-2-2-c}) holds. Since the map $f$ is continuous rational, there exists a Zariski open and dense subvariety $X^0$ of $X$ such that the restriction of $f$ to $X^0$ is a regular map. We can choose such an $X^0$ disjoint from the singular locus of $X$. Let
$Z$ be the union of $X \setminus X^0$ and all the irreducible components of $X$ of dimension strictly less than $\dim X$. Then $Z$ is a Zariski closed subvariety of $X$ with $\dim Z < \dim X$, and $X \setminus Z$ is nonsingular of pure dimension. Since the map $f|_Z$ is continuous rational, the construction above can be repeated with $X$ replaced by $Z$. By continuing this process, we conclude that (\ref{prop-2-2-b}) holds.
If $V$ is an irreducible Zariski closed subvariety of $X$, then there exists a stratum $S$ in $\mathcal{X}$ such that the intersection $S \cap V$ is nonempty and Zariski open in $V$ (hence Zariski dense in $V$). It follows that (\ref{prop-2-2-a}) implies (\ref{prop-2-2-c}).
It is obvious that (\ref{prop-2-2-b}) implies (\ref{prop-2-2-b1}), and (\ref{prop-2-2-b1}) implies (\ref{prop-2-2-a}). \end{proof}
\begin{remark}\label{rem-2-3} Proposition~\ref{prop-2-2} (with $Y=\mathbb{R}$) appears in the paper of Koll\'ar and Nowak \cite{bib31} as a comment following Definition~8. In \cite{bib31}, the attention is focused on functions satisfying condition~(\ref{prop-2-2-c}), called continuous hereditarily rational functions. By \cite[Proposition~7]{bib31}, if the variety $X$ is nonsingular, then condition (\ref{prop-2-2-c}) is equivalent to
\begin{altconditions}[start=\value{favoritecondition}] \item\label{rem-2-3-c'}\textit{The map $f$ is continuous rational.} \end{altconditions}
If $X$ is singular, then (\ref{rem-2-3-c'}) need not imply (\ref{prop-2-2-c}), cf. \cite[Examples~2 and~3]{bib31}. Continuous rational maps defined on nonsingular real algebraic varieties are investigated in \cite{bib32, bib34}. According to \cite[Theorem~9]{bib31} and \cite[Corolaire~4.40]{bib23}, stratified-regular maps coincide with ``\emph{applications r\'egulues}'' between real algebraic varieties, which are introduced in a more general framework in \cite{bib23}. Notions introduced and methods developed in \cite{bib35, bib36, bib37, bib39} are important in the study of the geometry of ``\emph{fonctions r\'egulues}'', cf. \cite{bib23}. \end{remark}
The proof of Theorem~\ref{th-1-3} requires some knowledge of homotopy classes represented by stratified-regular maps with values in the unit $d$-sphere $\mathbb{S}^d$, cf. Theorem~\ref{th-2-5}. We first need the following technical result.
\begin{lemma}\label{lem-2-4} Assume that the variety $X$ is compact of dimension $d$. Let $A$ be the union of the singular locus of $X$ and all the irreducible components of $X$ of dimension at most $d -1$. Let $\varphi \colon X \to \mathbb{R}^d$ be a continuous map which is constant in a neighborhood of $A$ and smooth on $X \setminus A$. Assume that $0$ in $\mathbb{R}^d$ is a regular value of
$\varphi|_{X \setminus A}$, and the inverse image $V \coloneqq \varphi^{-1}(0)$ is disjoint from $A$. Then there exists a regular map $\psi \colon X \to \mathbb{R}^d$ such that $\psi$ is equal to $\varphi$ on $A$, $\psi^{-1}(0)=V$, and the differentials $d\psi_x$ and $d\varphi_x$ are equal for every point $x$ in $V$. \end{lemma}
\begin{proof} We may assume that $X$ is a Zariski closed subvariety of $\mathbb{R}^n$ for some $n$. Then we can find a smooth map $f \colon \mathbb{R}^n \to \mathbb{R}^d$ which is an extension of $\varphi$ and is constant on a neighborhood $U$ of $A$ in $\mathbb{R}^n$. Thus $U \subseteq f^{-1}(c)$ for some point $c=(c_1, \ldots, c_d)$ in $\mathbb{R}^d$. Since $V$ is a finite set, there exists a polynomial map $g \colon \mathbb{R}^n \to \mathbb{R}^d$ such that $g(x)=f(x)$ and $dg_x = df_x$ for every point $x$ in $V$, and $A \subseteq g^{-1}(c)$. Indeed, $g$ can be constructed as follows. First we choose a polynomial map $\eta \colon \mathbb{R}^n \to \mathbb{R}^d$ with $\eta(x)=f(x)$ and $d\eta_x=df_x$ for every $x$ in $V$. If $\alpha \colon \mathbb{R}^n \to \mathbb{R}$ is a polynomial function satisfying $V \subseteq \alpha^{-1}(0)$ and $A \subseteq \alpha^{-1}(1)$, then the map $g \coloneqq \eta - \alpha^2(\eta-c)$ satisfies all the requirements.
For any subset $Z$ of $\mathbb{R}^n$, we denote by $I(Z)$ the ideal of all polynomial functions on $\mathbb{R}^n$ vanishing on $Z$. Let $p_1, \ldots, p_r$ be generators of the ideal $I(V)^2I(A)$.
\begin{assertion} There exist smooth maps $\lambda_1 \colon \mathbb{R}^n \to \mathbb{R}^d, \ldots, \lambda_r \colon \mathbb{R}^n \to \mathbb{R}^d$ such that $f=g+p_1 \lambda_1 + \cdots + p_r\lambda_r$. \end{assertion}
The Assertion can be proved as follows. For any point $x$ in $\mathbb{R}^n$, denote by $\mathcal{C}^{\infty}_x$ the ring of germs at $x$ of smooth functions on $\mathbb{R}^n$. Let $f = (f_1, \ldots, f_d)$, $g = (g_1, \ldots, g_d)$ and $h_i = f_i - g_i$ for $1 \leq i \leq d$. For every point $x$ in $\mathbb{R}^n \setminus A$, the germ $(h_i)_x$ of $h_i$ at $x$ belongs to the ideal $I(V)^2I(A)\mathcal{C}^{\infty}_x$. Obviously, $g_i - c_i$ belongs to $I(A)$. If $x$ is a point in $A$, then $f_i - c_i$ vanishes in a neighborhood of $x$, and hence $(h_i)_x = ( (f_i)_x - c_i) - ( (g_i)_x - c_i)$ belongs to $I(V)^2I(A)\mathcal{C}^{\infty}_x$. Making use of partition of unity, we obtain smooth functions $\lambda_{i1}, \ldots, \lambda_{ir}$ on $\mathbb{R}^n$ for which $h_i = p_1\lambda_{i1} + \cdots + p_r\lambda_{ir}$. Thus, the Assertion holds with $\lambda_j = (\lambda_{1j}, \ldots \lambda_{dj})$ for $1 \leq j \leq r$.
Now, let $q_1 \colon \mathbb{R}^n \to \mathbb{R}^d, \ldots, q_r \colon \mathbb{R}^n \to \mathbb{R}^d$ be polynomial maps and let $\psi \colon X \to \mathbb{R}^d$ be the restriction of the map $g + p_1q_1 + \cdots p_rq_r$. Then $\psi = \varphi$ on $A$, $V \subseteq \psi^{-1}(0)$, and $d\psi_x = d\varphi_x$ for every point $x$ in $V$. According to the Stone--Weierstrass theorem, given $\varepsilon > 0$, we can find polynomial maps $q_j$ such that
\begin{equation*} \@ifstar{\oldnorm}{\oldnorm*}{\lambda_j(x) - q_j(x)} + \sum_{k=1}^{n} \@ifstar{\oldnorm}{\oldnorm*}{ \frac{\partial \lambda_j}{\partial x_k}(x) - \frac{\partial q_j}{\partial x_k}(x)} < \varepsilon \end{equation*}
for all $x$ in $X$ and $1 \leq j \leq r$. If $\varepsilon$ is sufficiently small, then $\psi^{-1}(0) = V$, and hence $\psi$ satisfies all the requirements. \end{proof}
\begin{theorem}\label{th-2-5} If the variety $X$ is compact of dimension $d$, then each continuous map from $X$ into $\mathbb{S}^d$ is homotopic to a stratified-regular map. \end{theorem}
\begin{proof} Let $A$ be the union of the singular locus of $X$ and all the irreducible components of $X$ of dimension at most $d-1$. Since $(X,A)$ is a polyhedral pair \cite[Corollary~9.3.7]{bib9}, the restriction of $f$ to some neighborhood of $A$ is null homotopic. Hence, according to the homotopy extension theorem \cite[p.~90, Theorem~1.4]{bib27}, the map $f$ can be deformed without affecting its homotopy class so that
$f$ is constant in a compact neighborhood $L$ of $A$. Furthermore, we may assume that $f$ is smooth on $X \setminus A$. By Sard's theorem, there exists a regular value $y$ in $\mathbb{S}^d$ of the smooth map $f|_{X \setminus A}$ such that both points $y$ and $-y$ are in $\mathbb{S}^d \setminus f(L)$. In particular, the set $f^{-1}(y)$ is finite. We choose a compact neighborhood $K$ of $f^{-1}(y)$ in $X$ which is disjoint from $L \cup f^{-1}(-y)$ and such that each point in $K$ is a regular point of $f$. Let $p \colon \mathbb{S}^d \setminus \{-y\} \to \mathbb{R}^d$ be the stereographic projection (in particular, $p(y)=0$). Let $\kappa \colon X \to \mathbb{R}$ be a continuous function, smooth on $X \setminus A$, with $\kappa = 1$ on $K \cup L$ and $\kappa = 0$ in a neighborhood of $f^{-1}(-y)$. The map $\xi = (\xi_1, \ldots, \xi_d) \colon X \to \mathbb{R}^d$, defined by
\begin{equation*} \xi(x) =
\begin{cases} \kappa(x)p(f(x)) & \textrm{for } x \textrm{ in } X \setminus f^{-1}(-y)\\ 0 & \textrm{for } x \textrm{ in } f^{-1}(-y), \end{cases}
\end{equation*}
is continuous, smooth on $X \setminus A$, $\xi^{-1}(0)\cap L = \varnothing$, $f^{-1}(y) \subseteq \xi^{-1}(0)$, and each point in $K$ is a regular point of $\xi$. Let $\lambda \colon X \to \mathbb{R}$ be a continuous function, smooth on $X \setminus A$, with $\lambda^{-1}(0) = K \cup L$ (such a function exists, cf. for instance \cite[Theorem~14.1]{bib21}). The map $\eta \colon X \times \mathbb{R}^d \to \mathbb{R}^d$, defined by
\begin{equation*} \eta( x, (s_1,\ldots s_d) ) = ( \xi_1(x) + s_1\lambda(x)^2, \ldots, \xi_d(x) + s_d\lambda(x)^2 ), \end{equation*}
is continuous, smooth on $(X \setminus A) \times \mathbb{R}^d$, and $0$ in $\mathbb{R}^d$ is a regular value of the restriction of $\eta$ to $(X \setminus A) \times \mathbb{R}^d$. According to the parametric transversality theorem, we can choose a point $(s_1, \ldots, s_d)$ in $\mathbb{R}^d$ such that if $\varphi \colon X \to \mathbb{R}^d$ is defined by $\varphi(x) = \eta( x, (s_1, \ldots, s_d) )$ for all $x$ in $X$, then $0$ in $\mathbb{R}^d$ is a regular value of the restriction of $\varphi$ to $X \setminus A$. By construction, $\varphi$ is constant in a neighborhood of $A$, and the set $V \coloneqq \varphi^{-1}(0)$ is disjoint from $A$. Furthermore, $V$ is a finite set containing $f^{-1}(y)$. Let $\psi \colon X \to \mathbb{R}^d$ be a regular map as in Lemma~\ref{lem-2-4}. We choose a regular function $\alpha \colon X \to \mathbb{R}$ with $\alpha^{-1}(0) = W$ and $\alpha^{-1}(1) = f^{-1}(y)\cup A$, where $W \coloneqq V \setminus f^{-1}(y)$. For example, we can take $\alpha = \alpha_1^2 / (\alpha_1^2 + \alpha_2^2)$, where $\alpha_1$ and $\alpha_2$ are regular functions on $X$ satisfying $\alpha_1^{-1}(0) = W$ and $\alpha_2^{-1}(0) = f^{-1}(y) \cup A$. By the \L{}ojasiewicz inequality \cite[Corollary~2.6.7]{bib9}, there exist a neighborhood $U$ of $W$ in $X$, a positive real number $c$, and a positive integer $k$ for which
\begin{equation*} \@ifstar{\oldnorm}{\oldnorm*}{\psi(x)} \geq c\alpha(x)^{2k} \quad \textrm{for all $x$ in $U$}. \end{equation*}
Let $\bar{\psi}(x) = (1 / \alpha(x)^{2k+1}) \psi(x)$ for $x$ in $X \setminus W$. The map $g \colon X \to \mathbb{S}^d$, defined by
\begin{equation*} g(x)=
\begin{cases} p^{-1}(\bar{\psi}(x)) & \textrm{for $x$ in $X \setminus W$}\\ -y & \textrm{for $x$ in $W$}, \end{cases}
\end{equation*}
is continuous. Since $p$ is a biregular isomorphism, the restriction
$g|_{X \setminus W}$ is a regular map, and hence $g$ is a stratified-regular map. It suffices to prove that $f$ is homotopic to $g$.
The map $\bar{\psi} \colon X \setminus W \to \mathbb{R}^d$ has the following properties: $\psi^{-1}(0) = f^{-1}(y) = V \setminus W$ and $d\bar{\psi}_x = d\varphi_x$ for every point $x$ in $f^{-1}(y)$. If $U_1$ is a sufficiently small open neighborhood of $f^{-1}(y)$ in $X$, then
\begin{equation*} (1-t) \varphi(x) + t\bar{\psi}(x) \neq 0 \quad \textrm{for all $(x,t)$ in $(U_1 \setminus f^{-1}(y)) \times [0,1]$}. \end{equation*}
We may assume that $U_1 \subseteq K$, and hence $\varphi(x) = p(f(x))$ for all $x$ in $U_1$. Let $U_2$ be an open neighborhood of $f^{-1}(y)$ whose closure $\overline{U_2}$ is contained in $U_1$. Choose a continuous function $\mu \colon X \to [0,1]$, smooth on $X \setminus A$, with $\mu = 1$ on $\overline{U_2}$ and $\mu = 0$ in a neighborhood of $X \setminus U_1$. Then the map $F \colon X \times [0,1] \to \mathbb{S}^p$, defined by
\begin{equation*} F(x,t)=
\begin{cases} p^{-1} ( (1 - t\mu(x)) \varphi(x) + t\mu(x) \bar{\psi}(x) ) & \textrm{for $(x,t)$ in $U_1 \times [0,1]$} \\ f(x) & \textrm{for $(x,t)$ in $(X \setminus U_1) \times [0,1]$}, \end{cases}
\end{equation*}
is continuous. Furthermore, if $F_t \colon X \to \mathbb{S}^p$ is defined by $F_t(x) = F(x,t)$, then $F_t^{-1}(y) = f^{-1}(y) = g^{-1}(y)$ for every $t$ in $[0,1]$, and $F_0 = f$. It remains to prove that the map $\bar{f} \coloneqq F_1$ is homotopic to $g$. This can be done as follows. Note that $\bar{f}=g$ on $U_2$. If $q \colon \mathbb{S}^d \setminus \{y\} \to \mathbb{R}^d$ is the stereographic projection, then
\begin{equation*} G(x,t)=
\begin{cases} q^{-1} ( (1-t)q( \bar{f}(x) ) + tq(g(x)) ) & \textrm{for $(x,t)$ in $(X \setminus f^{-1}(y)) \times [0,1]$}\\ \bar{f}(x) & \textrm{for $(x,t)$ in $U_2 \times [0,1]$} \end{cases}
\end{equation*}
is a homotopy between $\bar{f}$ and $g$. \end{proof}
A special case of Theorem~\ref{th-2-5}, with $X$ nonsingular, is contained in \cite[Theorem~1.2]{bib32}.
Recall that each regular map from $\mathbb{T}^n$ into $\mathbb{S}^n$ is null homotopic, cf. \cite{bib11} or \cite{bib12}. In particular, Theorem~\ref{th-2-5} shows that stratified-regular maps are more flexible than regular maps. Other results illustrating this point can be found in \cite{bib32, bib34}.
For any continuous map $f \colon X \to Y$ and any topological $\mathbb{F}$-vector subbundle $\theta$ of $\varepsilon_Y^n(\mathbb{F})$, the pullback $f^*\theta$ will be regarded as a topological $\mathbb{F}$-vector subbundle of $\varepsilon_X^n(\mathbb{F})$.
\begin{proposition}\label{prop-2-6} If the map $f$ is $\mathcal{X}$-regular, and the $\mathbb{F}$-vector bundle $\theta$ is algebraic, then $f^*\theta$ is $\mathcal{X}$-algebraic. Similarly, if $f$ is stratified-regular and $\theta$ is stratified-algebraic, then $f^*\theta$ is stratified-algebraic. \end{proposition}
\begin{proof} The first assertion is obvious. For the second assertion, let $\mathcal{T}$ be a stratification of $X$ such that the map $f$ is $\mathcal{T}$-regular, and let $\mathcal{Y}$ be a stratification of $Y$ such that the $\mathbb{F}$-vector bundle $\theta$ is $\mathcal{Y}$-algebraic. Then $\mathcal{S} \coloneqq \{ T \cap f^{-1}(P) \mid T \in \mathcal{T}, P \in \mathcal{Y} \}$ is a stratification of $X$, and $f^*\theta$ is $\mathcal{S}$-algebraic. \end{proof}
\begin{proof}[Proof of Theorem~\ref{th-1-3}.] Let $h \colon X \to \mathbb{S}^d$ be a homotopy equivalence. By Theorem~\ref{th-2-5}, $h$ is homotopic to a stratified-regular map $f \colon X \to \mathbb{S}^d$. According to Proposition~\ref{prop-2-6}, if $\theta$ is an algebraic $\mathbb{F}$-vector bundle on $\mathbb{S}^d$, then $f^*\theta$ is a stratified-algebraic $\mathbb{F}$-vector bundle on $X$. The proof is complete since every topological $\mathbb{F}$-vector bundle on $\mathbb{S}^d$ is isomorphic to an algebraic $\mathbb{F}$-vector bundle (cf. Example~\ref{ex-1-4}). \end{proof}
\section[Basic properties of stratified-algebraic vector bundles]{\texorpdfstring{Basic properties of stratified-algebraic\\ vector bundles}{Basic properties of stratified-algebraic vector bundles}}\label{sec-3}
Throughout this section, $X$ denotes a real algebraic variety, and $\mathcal{X}$ is a stratification of $X$. All modules that appear below are left modules.
Vector bundles are often investigated by means of maps into Grassmannians, cf. \cite{bib3, bib5, bib9, bib10, bib25, bib27, bib28, bib29}. As in \cite{bib9, bib10}, the Grassmannian $\mathbb{G}_k(\mathbb{F}^n)$ of $k$-dimensional $\mathbb{F}$-vector subspaces of $\mathbb{F}^n$ will be regarded as a real algebraic variety. The tautological $\mathbb{F}$-vector bundle on $\mathbb{G}_k(\mathbb{F}^n)$ will be denoted by $\gamma_k(\mathbb{F}^n)$. If $\xi$ is a topological (resp. algebraic) $\mathbb{F}$-vector subbundle of $\varepsilon_X^n(\mathbb{F})$ of rank $k$, then the map $f \colon X \to \mathbb{G}_k(\mathbb{F}^n)$ defined by
\begin{equation*} E(\xi)_x = \{ x \} \times f(x) \quad \textrm{for all $x$ in $X$} \end{equation*}
is continuous (resp. regular).
Let $K$ be a finite nonempty collection of nonnegative integers and $n$ an integer such that $n \geq k$ for every $k$ in $K$. We denote by $\mathbb{G}_K(\mathbb{F}^n)$ the disjoint union of all $\mathbb{G}_k(\mathbb{F}^n)$ for $k$ in~$K$. The tautological $\mathbb{F}$-vector bundle $\gamma_K(\mathbb{F}^n)$ on $\mathbb{G}_K(\mathbb{F}^n)$ is the bundle whose restriction to $\mathbb{G}_k(\mathbb{F}^n)$ is $\gamma_k(\mathbb{F}^n)$ for each $k$ in $K$. In particular, $\mathbb{G}_K(\mathbb{F}^n)$ is a real algebraic variety, and $\gamma_K(\mathbb{F}^n)$ is an algebraic $\mathbb{F}$-vector subbundle of the standard trivial $\mathbb{F}$-vector bundle on $\mathbb{G}_K(\mathbb{F}^n)$ with fiber $\mathbb{F}^n$. We call $\mathbb{G}_K(\mathbb{F}^n)$ the $(K,n)$-\emph{multi-Grassmannian}. It is explained below why we need this notion.
Any algebraic $\mathbb{F}$-vector bundle on $X$ has constant rank on each Zariski connected component of $X$. This observation can be partially generalized as follows.
\begin{proposition}\label{prop-3-1} Assume that the variety $X$ is nonsingular. Then any stratified-algebraic $\mathbb{F}$-vector bundle on $X$ has constant rank on each irreducible component of $X$. \end{proposition}
\begin{proof} We may assume that $X$ is irreducible. It suffices to note that in each stratification of $X$, one can find a stratum which is nonempty and Zariski open in $X$. \end{proof}
However, we encounter a different phenomenon for vector bundles on singular varieties.
\begin{example}\label{ex-3-2} The real algebraic curve
\begin{equation*} C \coloneqq \{ (x,y) \in \mathbb{R}^2 \mid y^2 = x^2(x-1) \} \end{equation*}
is irreducible, and hence Zariski connected. It has two connected components in the Euclidean topology, $S_1 = \{(0,0)\}$ and $S_2 = C \setminus S_1$. The collection $\mathcal{C} = \{S_1,S_2\}$ is a stratification of $C$. Let $\xi$ be the topological $\mathbb{F}$-vector subbundle of
$\varepsilon_C^2(\mathbb{F})$ such that $E(\xi|_{S_1}) = \{(0,0)\} \times (\mathbb{F}
\times \{0\})$ and $\xi|_{S_2} = \varepsilon_{S_2}^2(\mathbb{F})$. Then $\xi$ is $\mathcal{C}$-algebraic and it does not have constant rank. Furthermore, if $K = \{1,2\}$ and $f \colon C \to \mathbb{G}_K(\mathbb{F}^2)$ is a map such that $f(S_1) \subseteq \mathbb{G}_1(\mathbb{F}^2)$ and $f(S_2) \subseteq \mathbb{G}_2(\mathbb{F}^2)$, then $f$ is $\mathcal{C}$-regular with $\xi = f^*\gamma_K(\mathbb{F}^2)$. \end{example}
\begin{proposition}\label{prop-3-3} Any $\mathcal{X}$-algebraic $\mathbb{F}$-vector bundle $\xi$ on $X$ is of the form $\xi = f^* \gamma_K(\mathbb{F}^n)$ for some multi-Grassmannian $\mathbb{G}_K(\mathbb{F}^n)$ and $\mathcal{X}$-regular map $f \colon X \to \mathbb{G}_K(\mathbb{F}^n)$. \end{proposition}
\begin{proof} Let $\xi$ be an $\mathcal{X}$-algebraic $\mathbb{F}$-vector subbundle of $\varepsilon_X^n(\mathbb{F})$. For each stratum $S$ in $\mathcal{X}$, the function
\begin{equation*} S \to \mathbb{Z}, \quad x \to \dim_{\mathbb{F}} E(\xi)_x \end{equation*}
is locally constant in the Zariski topology, the $\mathbb{F}$-vector bundle
$\xi|_S$ being algebraic. Hence, the set $K \coloneqq \{ \dim_{\mathbb{F}} E(\xi)_x \mid x \in X \}$ is finite. The map $f \colon X \to \mathbb{G}_K(\mathbb{F}^n)$ defined by
\begin{equation*} E(\xi)_x = \{x\} \times f(x) \quad \textrm{for all $x$ in $X$} \end{equation*}
is continuous and $\xi = f^*\gamma_K(\mathbb{F}^n)$. Furthermore, the map $f|_S$ is regular. Thus $f$ is $\mathcal{X}$-regular, as required. \end{proof}
\begin{proposition}\label{prop-3-4} Any stratified-algebraic $\mathbb{F}$-vector bundle $\xi$ on $X$ is of the form
\begin{equation*} \xi = f^* \gamma_K(\mathbb{F}^n) \end{equation*}
for some multi-Grassmannian $\mathbb{G}_K(\mathbb{F}^n)$ and stratified-regular map ${f \colon X \to \mathbb{G}_K(\mathbb{F}^n)}$. \end{proposition}
\begin{proof} It suffices to apply Proposition~\ref{prop-3-3}. \end{proof}
\begin{proposition}\label{prop-3-5} For a topological $\mathbb{F}$-vector bundle $\xi$ on $X$, the following conditions are equivalent:
\begin{conditions} \item\label{prop-3-5-a} The bundle $\xi$ is stratified-algebraic.
\item\label{prop-3-5-b} There exists a filtration $\mathcal{F}$ of $X$ such that $\xi$ is $\overline{\mathcal{F}}$-algebraic and each stratum in $\overline{\mathcal{F}}$ is nonsingular and equidimensional. \setcounter{favoritecondition}{\value{conditionsi}} \end{conditions} \begin{altconditions}[start=\value{favoritecondition}] \item\label{prop-3-5-b'} There exists a filtration $\mathcal{F}'$ of $X$ such that $\xi$ is $\overline{\mathcal{F}}'$-algebraic. \end{altconditions}
\end{proposition}
\begin{proof} By Proposition~\ref{prop-3-4}, if the $\mathbb{F}$-vector bundle $\xi$ is stratified-algebraic, then
\begin{equation*} \xi=f^*\gamma_K(\mathbb{F}^n) \end{equation*}
for some multi-Grassmannian $\mathbb{G}_K(\mathbb{F}^n)$ and stratified-regular map $f \colon X \to \mathbb{G}_K(\mathbb{F}^n)$. According to Proposition~\ref{prop-2-2}, there exists a filtration $\mathcal{F}$ of $X$ such that $f$ is $\overline{\mathcal{F}}$-regular and each stratum in $\overline{\mathcal{F}}$ is nonsingular and equidimensional. It follows that $\xi$ is $\overline{\mathcal{F}}$-algebraic. Consequently, (\ref{prop-3-5-a}) implies (\ref{prop-3-5-b}). It is obvious that (\ref{prop-3-5-b}) implies (\ref{prop-3-5-b'}), and (\ref{prop-3-5-b'}) implies (\ref{prop-3-5-a}). \end{proof}
Recall that $d(\mathbb{F}) = \dim_{\mathbb{R}}\mathbb{F}$. As a consequence of Theorem~\ref{th-2-5}, we obtain the following result on vector bundles on low-dimensional varieties.
\begin{corollary}\label{cor-3-6} Assume that the variety $X$ is compact and $\dim X \leq d(\mathbb{F})$. Then any topological $\mathbb{F}$-vector bundle of constant rank on $X$ is topologically isomorphic to a stratified-algebraic $\mathbb{F}$-vector bundle. \end{corollary}
\begin{proof} Let $\xi$ be a topological $\mathbb{F}$-vector bundle of rank $r \geq 1$ on $X$. Since $\dim X \leq d(\mathbb{F})$, the bundle $\xi$ splits off a trivial vector bundle of rank $r-1$. Moreover, if $\dim X < d(\mathbb{F})$, then $\xi$ is topologically trivial. These are well known topological facts, cf. \cite[p.~99]{bib28}. Hence we may assume without loss of generality that $r=1$ and $\dim X = d(\mathbb{F})$. Then there exists a continuous map $f \colon X \to \mathbb{G}_1(\mathbb{F}^2)$ such that the pullback $\mathbb{F}$-line bundle $f^*\gamma_1(\mathbb{F}^2)$ is isomorphic to $\xi$. Recall that $\mathbb{G}_1(\mathbb{F}^2)$ is biregularly isomorphic to the unit $d(\mathbb{F})$-sphere. Consequently, according to Theorem~\ref{th-2-5}, the map $f$ can be assumed to be stratified-regular. Thus, it suffices to apply Proposition~\ref{prop-2-6}. \end{proof}
Subsequent results require some preparation. For any real algebraic variety $Y$, denote by $\mathcal{C}(X,Y)$ the set of all continuous maps from $X$ into $Y$. There are the following inclusions:
\begin{equation*} \mathcal{R}(X,Y) \subseteq \mathcal{R}_{\mathcal{X}}(X,Y) \subseteq \mathcal{R}^0(X,Y) \subseteq \mathcal{C}(X,Y), \end{equation*}
where $\mathcal{R}(X,Y)$ (resp. $\mathcal{R}_{\mathcal{X}}(X,Y)$, $\mathcal{R}^0(X,Y)$) is the set of all regular (resp. $\mathcal{X}$-regular, stratified-regular) maps. Each of the sets $\mathcal{R}(X,\mathbb{F})$, $\mathcal{R}_{\mathcal{X}}(X,\mathbb{F})$ and $\mathcal{R}^0(X,\mathbb{F})$ is a subring of the ring $\mathcal{C}(X,\mathbb{F})$. We next discuss various aspects of the Serre--Swan construction \cite{bib41, bib43}, relating vector bundles and finitely generated projective modules.
If $\xi$ is a topological $\mathbb{F}$-vector bundle on $X$, then the set $\Gamma(\xi)$ of all (global) continuous sections of $\xi$ is a $\mathcal{C}(X,\mathbb{F})$-module. If $\varphi \colon \xi \to \eta$ is a morphism of topological $\mathbb{F}$-vector bundles on $X$, then
\begin{equation*} \Gamma(\varphi) \colon \Gamma(\xi) \to \Gamma(\eta), \quad \Gamma(\varphi)(\sigma) = \varphi \circ \sigma \end{equation*}
is a homomorphism of $\mathcal{C}(X,\mathbb{F})$-modules. Since $X$ is homotopically equivalent to a compact subset of $X$ (cf. \cite[Corollary~9.3.7]{bib9}), it follows form \cite{bib43} that $\Gamma$ is an equivalence of the category of topological $\mathbb{F}$-vector bundles on $X$ with the category of finitely generated projective $\mathcal{C}(X,\mathbb{F})$-modules. We give below suitable counterparts of this result for $\mathcal{X}$-algebraic and stratified-algebraic $\mathbb{F}$-vector bundles on $X$.
If $\xi$ is an $\mathcal{X}$-algebraic $\mathbb{F}$-vector bundle on $X$, an
\emph{$\mathcal{X}$-algebraic section} $u \colon X \to \xi$ is a continuous section whose restriction $u|_S \colon S \to \xi|_S$ to each stratum $S$ in $\mathcal{X}$ is an algebraic section. In other words, $u \colon X \to E(\xi)$ is a continuous map such that $\pi(\xi) \circ u$ is the identity map of
$X$, and the restriction $u|_S \colon S \to \pi(\xi)^{-1}(S)$ is a regular map of real algebraic varieties for each stratum $S$ in $\mathcal{X}$. The set $\Gamma_{\mathcal{X}}(\xi)$ of all (global) $\mathcal{X}$-algebraic sections of $\xi$ is an $\mathcal{R}_{\mathcal{X}}(X,\mathbb{F})$-module. For any $\mathcal{X}$-algebraic morphism $\varphi \colon \xi \to \eta$ of $\mathcal{X}$-algebraic $\mathbb{F}$-vector bundles on X,
\begin{equation*} \Gamma_{\mathcal{X}}(\varphi) \colon \Gamma_{\mathcal{X}}(\xi) \to \Gamma_{\mathcal{X}}(\eta), \quad \Gamma_{\mathcal{X}}(\varphi)(u)=\varphi \circ u \end{equation*}
is a homomorphism of $\mathcal{R}_{\mathcal{X}}(X,\mathbb{F})$-modules.
The $\mathcal{R}(X,\mathbb{F})$-module $\Gamma_{\mathcal{X}}(\varepsilon_X^n(\mathbb{F}))$ is canonically isomorphic to the direct sum $\mathcal{R}_{\mathcal{X}}(X,\mathbb{F})^n$ of $n$ copies of $\mathcal{R}_{\mathcal{X}}(X,\mathbb{F})$. If $\xi$ is an $\mathcal{X}$-algebraic $\mathbb{F}$-vector subbundle of $\varepsilon_X^n(\mathbb{F})$, then $\Gamma_{\mathcal{X}}(\xi)$ will be regarded as a submodule of $\Gamma_{\mathcal{X}}(\varepsilon_X^n(\mathbb{F}))$.
For any topological $\mathbb{F}$-vector subbundle $\xi$ of $\varepsilon_X^n(\mathbb{F})$, let $\xi^{\perp}$ denote its orthogonal complement with respect to the standard inner product $\mathbb{F}^n \times \mathbb{F}^n \to \mathbb{F}$. Thus, $\xi^{\perp}$ is a topological $\mathbb{F}$-vector subbundle of $\varepsilon_X^n(\mathbb{F})$ and $\xi \oplus \xi^{\perp} = \varepsilon_X^n(\mathbb{F})$. The orthogonal projection $\varepsilon_X^n(\mathbb{F}) \to \xi$ is a topological morphism of $\mathbb{F}$-vector bundles.
\begin{proposition}\label{prop-3-7} If $\xi$ is an $\mathcal{X}$-algebraic $\mathbb{F}$-vector subbundle of $\varepsilon_X^n(\mathbb{F})$, then $\xi^{\perp}$ also is an $\mathcal{X}$-algebraic $\mathbb{F}$-vector subbundle of $\varepsilon_X^n(\mathbb{F})$, and the orthogonal projection $\varepsilon_X^n(\mathbb{F}) \to \xi$ is an $\mathcal{X}$-algebraic morphism. In particular, $\xi$ is generated by $n$ (global) $\mathcal{X}$-algebraic sections. \end{proposition}
\begin{proof}
Let $S$ be a stratum in $\mathcal{X}$. One readily checks that $(\xi|_S)^\perp =
(\xi^{\perp})|_S$ is an algebraic $\mathbb{F}$-vector subbundle of
$\varepsilon_S^n(\mathbb{F}) = \varepsilon_X^n(\mathbb{F})|_S$, and the orthogonal projection $\varepsilon_S^n(\mathbb{F}) \to \xi|_S$ is an algebraic morphism. The last assertion in the proposition follows immediately. \end{proof}
\begin{proposition}\label{prop-3-8} If $\xi$ is an $\mathcal{X}$-algebraic $\mathbb{F}$-vector bundle on $X$, then the $\mathcal{R}_{\mathcal{X}}(X,\mathbb{F})$-module $\Gamma_{\mathcal{X}}(\xi)$ is finitely generated and projective. Furthermore, $\Gamma_{\mathcal{X}}$ is an equivalence of the category of $\mathcal{X}$-algebraic $\mathbb{F}$-vector bundles on $X$ with the category of finitely generated projective $\mathcal{R}_{\mathcal{X}}(X,\mathbb{F})$-modules. \end{proposition}
\begin{proof} According to Proposition~\ref{prop-3-7}, if $\xi$ is an $\mathcal{X}$-algebraic $\mathbb{F}$-vector subbundle of $\varepsilon_X^n(\mathbb{F})$, then
\begin{equation*} \Gamma_{\mathcal{X}}(\xi) \oplus \Gamma_{\mathcal{X}}(\xi^{\perp}) = \Gamma_{\mathcal{X}}(\varepsilon_X^n(\mathbb{F})). \end{equation*}
Hence, $\Gamma_{\mathcal{X}}(\xi)$ is a finitely generated projective $\mathcal{R}_{\mathcal{X}}(X,\mathbb{F})$-module. Furthermore, $\Gamma_{\mathcal{X}}$ is an equivalence of categories since, in view of Proposition~\ref{prop-3-7}, the proof given in \cite[pp.~30, 31]{bib3} that $\Gamma$ is an equivalence of categories in the topological framework can easily be adapted to $\Gamma_{\mathcal{X}}$. \end{proof}
If $\xi$ is an algebraic $\mathbb{F}$-vector bundle on $X$, then the set $\Gamma_{\mathrm{alg}}(\xi)$ of all (global) algebraic sections of $\xi$ is an $\mathcal{R}(X,\mathbb{F})$-module. For any morphism $\varphi \colon \xi \to \eta$ of algebraic $\mathbb{F}$-vector bundles on $X$,
\begin{equation*} \Gamma_{\mathrm{alg}}(\varphi) \colon \Gamma_{\mathrm{alg}} (\xi) \to \Gamma_{\mathrm{alg}}(\eta), \quad \Gamma_{\mathrm{alg}} (\varphi) (\sigma) = \varphi \circ \sigma \end{equation*}
is a homomorphism of $\mathcal{R}(X, \mathbb{F})$-modules. It is well known that $\Gamma_{\mathrm{alg}}$ is an equivalence of the category of algebraic $\mathbb{F}$-vector bundles on $X$ with the category of finitely generated projective $\mathcal{R}(X,\mathbb{F})$-modules, cf. \cite[Proposition~12.1.12]{bib9}. This result is a special case of Proposition~\ref{prop-3-8} since algebraic $\mathbb{F}$-vector bundles on $X$ coincide with $\{X\}$-algebraic $\mathbb{F}$-vector bundles, and $\mathcal{R}(X,\mathbb{F}) = \mathcal{R}_{\{X\}}(X,\mathbb{F})$.
There is also a version of Proposition~\ref{prop-3-8} for stratified-algebraic vector bundles. If $\xi$ is a stratified-algebraic $\mathbb{F}$-vector bundle on $X$, a \emph{stratified-algebraic section} $u \colon X \to \xi$ is an $\mathcal{S}$-algebraic section for some stratification $\mathcal{S}$ of $X$ such that $\xi$ is an $\mathcal{S}$-algebraic $\mathbb{F}$-vector bundle. The set $\Gamma_{\mathrm{str}}(\xi)$ of all (global) stratified-algebraic sections of $\xi$ is an $\mathcal{R}^0(X, \mathbb{F})$-module. For any stratified-algebraic morphism $\varphi \colon \xi \to \eta$ of stratified-algebraic $\mathbb{F}$-vector bundles on $X$,
\begin{equation*} \Gamma_{\mathrm{str}}(\varphi) \colon \Gamma_{\mathrm{str}}(\xi) \to \Gamma_{\mathrm{str}}(\eta), \quad \Gamma_{\mathrm{str}} (\varphi) (\sigma) = \varphi \circ \sigma \end{equation*}
is a homomorphism of $\mathcal{R}^0(X,\mathbb{F})$-modules.
\begin{proposition}\label{prop-3-9} If $\xi$ is a stratified-algebraic $\mathbb{F}$-vector bundle on $X$, then the $\mathcal{R}^0(X,\mathbb{F})$-module $\Gamma_{\mathrm{str}}(\xi)$ is finitely generated and projective. Furthermore, $\Gamma_{\mathrm{str}}$ is an equivalence of the category of stratified-algebraic $\mathbb{F}$-vector bundles on $X$ with the category of finitely generated projective $\mathcal{R}^0(X,\mathbb{F})$-modules. \end{proposition}
\begin{proof} One proceeds as in the proof of Proposition~\ref{prop-3-8}. \end{proof}
We identify the direct sum $\varepsilon_X^n(\mathbb{F}) \oplus \varepsilon_X^m(\mathbb{F})$ with $\varepsilon_X^{n+m}(\mathbb{F})$. Consequently, if $\xi \subseteq \varepsilon_X^n(\mathbb{F})$ and $\eta \subseteq \varepsilon_X^m(\mathbb{F})$ are $\mathbb{F}$-vector subbundles, then $\xi \oplus \eta \subseteq \varepsilon_X^{n+m}(\mathbb{F})$ is an $\mathbb{F}$-vector subbundle.
It is convenient to bring into play the sets of isomorphism classes of vector bundles of types considered above. Denote by $\func{VB}_{\mathbb{F}\mathrm{\mhyphen alg}}(X)$, $\func{VB}_{\mathbb{F}\mhyphen\mathcal{X}}(X)$, $\func{VB}_{\mathbb{F}\mathrm{\mhyphen str}}(X)$ and $\func{VB}_{\mathbb{F}}(X)$ the sets of isomorphism classes (in the appropriate category) of algebraic, $\mathcal{X}$-algebraic, stratified-algebraic and topological $\mathbb{F}$-vector bundles on $X$. Each of these sets of isomorphism classes is a commutative monoid with operation induced by direct sum of $\mathbb{F}$-vector bundles. There are obvious canonical homomorphisms of monoids
\begin{equation*} \func{VB}_{\mathbb{F}\mathrm{\mhyphen alg}}(X) \to \func{VB}_{\mathbb{F} \mhyphen \mathcal{X}}(X) \to \func{VB}_{\mathbb{F}\mathrm{\mhyphen str}}(X) \to \func{VB}_{\mathbb{F}}(X). \end{equation*}
For example, if $\xi$ is an $\mathcal{X}$-algebraic $\mathbb{F}$-vector bundle on $X$, then $\func{VB}_{\mathbb{F} \mhyphen \mathcal{X}}(X) \to \func{VB}_{\mathbb{F}\mathrm{\mhyphen str}}(X)$ sends the isomorphism class of $\xi$ in the category of $\mathcal{X}$-algebraic $\mathbb{F}$-vector bundles on $X$ to the isomorphism class of $\xi$ in the category of stratified-algebraic $\mathbb{F}$-vector bundles on $X$. Any composition of these homomorphisms will also be called a canonical homomorphism. Note that $\func{VB}_{\mathbb{F}\mathrm{\mhyphen alg}}(x) = \func{VB}_{\mathbb{F} \mhyphen \{X\}}(X)$.
For any ring $R$ (associative with $1$), the set $P(R)$ of isomorphism classes of finitely generated projective $R$-modules is a commutative monoid, with operation induced by direct sum of $R$-modules. If $R$ is a subring of a ring $R'$, then there is a canonical homomorphism
\begin{equation*} P(R) \to P(R'), \end{equation*}
induced by the correspondence which assigns to an $R$-module $M$ the $R'$-module $R' \otimes_R M$.
There are canonical homomorphisms of monoids
\begin{align*} &\Gamma_{\mathrm{alg}} \colon \func{VB}_{\mathbb{F}\mathrm{\mhyphen alg}}(X) \to P(\mathcal{R}(X,\mathbb{F})), &&\Gamma_{\mathcal{X}} \colon \func{VB}_{\mathbb{F} \mhyphen \mathcal{X}}(X) \to P(\mathcal{R}_{\mathcal{X}}(X,\mathbb{F})), \\ &\Gamma_{\mathrm{str}} \colon \func{VB}_{\mathbb{F}\mathrm{\mhyphen str}}(X) \to P(\mathcal{R}^0(X,\mathbb{F})), &&\Gamma \colon \func{VB}_{\mathbb{F}}(X) \to P(\mathcal{C}(X,\mathbb{F})), \end{align*}
induced by the global section functor in the appropriate category of $\mathbb{F}$-vector bundles on $X$. For example, $\Gamma_{\mathcal{X}} \colon \func{VB}_{\mathbb{F} \mhyphen \mathcal{X}}(X) \to P(\mathcal{R}_{\mathcal{X}}(X,\mathbb{F}))$ sends the isomorphism class of an $\mathcal{X}$-algebraic $\mathbb{F}$-vector bundle $\xi$ on $X$ to the isomorphism class of the $\mathcal{R}_{\mathcal{X}}(X,\mathbb{F})$-module $\Gamma_{\mathcal{X}}(\xi)$ (cf. Proposition~\ref{prop-3-8}).
\begin{theorem}\label{th-3-10} The diagram
\begin{diagram} \func{VB}_{\mathbb{F}\mathrm{\mhyphen alg}}(X) & \rTo & \func{VB}_{\mathbb{F} \mhyphen \mathcal{X}}(X) & \rTo & \func{VB}_{\mathbb{F}\mathrm{\mhyphen str}}(X) & \rTo & \func{VB}_{\mathbb{F}}(X) \\ \dTo^{\Gamma_{\mathrm{alg}}} && \dTo^{\Gamma_{\mathcal{X}}} && \dTo^{\Gamma_{\mathrm{str}}} && \dTo^{\Gamma} \\ P(\mathcal{R}(X,\mathbb{F})) & \rTo & P(\mathcal{R}_{\mathcal{X}}(X,\mathbb{F})) & \rTo & P(\mathcal{R}^0(X,\mathbb{F})) & \rTo & P(\mathcal{C}(X,\mathbb{F})) \end{diagram}
is commutative, and the vertical maps are all bijective. Furthermore, if the variety $X$ is compact, then the horizontal maps are all injective. \end{theorem}
\begin{proof} Let $\xi$ be an $\mathcal{X}$-algebraic $\mathbb{F}$-vector subbundle of $\varepsilon_X^n(\mathbb{F})$. Tensoring
\begin{equation*} \Gamma_{\mathcal{X}}(\xi) \oplus \Gamma_{\mathcal{X}} (\xi^{\perp}) = \Gamma_{\mathcal{X}} (\varepsilon_X^n(\mathbb{F})) \end{equation*}
with $\mathcal{R}^0(X,\mathbb{F})$ (over the ring $\mathcal{R}_{\mathcal{X}}(X,\mathbb{F})$) and identifying $\mathcal{R}^0(X,\mathbb{F}) \otimes \Gamma_{\mathcal{X}}(\varepsilon_X^n(\mathbb{F}))$ with $\Gamma_{\mathrm{str}}(\varepsilon_X^n(\mathbb{F}))$, one readily checks that the $\mathcal{R}^0(X,\mathbb{F})$-modules $\mathcal{R}^0(X,\mathbb{F}) \otimes \Gamma_{\mathcal{X}}(\xi)$ and $\Gamma_{\mathrm{str}}(\xi)$ are isomorphic. This implies that the middle square in the diagram is commutative. Similar arguments show that the other two squares also are commutative. According to Propositions~\ref{prop-3-8} and~\ref{prop-3-9}, the maps $\Gamma_{\mathrm{alg}}$, $\Gamma_{\mathcal{X}}$ and $\Gamma_{\mathrm{str}}$ are bijective. By \cite{bib41}, the map $\Gamma$ is bijective (since $X$ is homotopically equivalent to a compact subset of $X$, cf. \cite[Corollary~9.3.7]{bib9}).
Suppose that the variety $X$ is compact. It suffices to prove that each map in the bottom row of the diagram is injective. This follows from Swan's theorem \cite[Theorem~2.2]{bib44}. Indeed, $\mathcal{C}(X,\mathbb{F})$ is a topological ring with topology induced by the $\sup$ norm. Each subring of $\mathcal{C}(X,\mathbb{F})$ is a topological ring with the subspace topology. By the Weierstrass approximation theorem, $\mathcal{R}(X,\mathbb{F})$ is dense in $\mathcal{C}(X,\mathbb{F})$. Consequently, Swan's theorem is applicable. \end{proof}
Denote by $K_{\mathbb{F}\mathrm{\mhyphen alg}}(X)$, $K_{\mathbb{F} \mhyphen \mathcal{X}}(X)$, $K_{\mathbb{F}\mathrm{\mhyphen str}}(X)$ and $K_{\mathbb{F}}(X)$ the Grothendieck group of the commutative monoids $\func{VB}_{\mathbb{F}\mathrm{\mhyphen alg}}(X)$, $\func{VB}_{\mathbb{F} \mhyphen \mathcal{X}}(X)$, $\func{VB}_{\mathbb{F}\mathrm{\mhyphen str}}(X)$ and $\func{VB}_{\mathbb{F}}(X)$. Note that $K_{\mathbb{F}\mathrm{\mhyphen alg}}(X) = K_{\mathbb{F} \mhyphen \{X\}}(X)$.
As usual, for any ring $R$, the Grothendieck group of the commutative monoid $P(R)$ will be denoted by $K_0(R)$.
\begin{corollary}\label{cor-3-11} The commutative diagram in Theorem~\ref{th-3-10} gives rise to a commutative diagram
\begin{diagram} K_{\mathbb{F}\mathrm{\mhyphen alg}}(X) & \rTo & K_{\mathbb{F} \mhyphen \mathcal{X}}(X) & \rTo & K_{\mathbb{F}\mathrm{\mhyphen str}}(X) & \rTo & K_{\mathbb{F}}(X) \\ \dTo^{\Gamma_{\mathrm{alg}}} && \dTo^{\Gamma_{\mathcal{X}}} && \dTo^{\Gamma_{\mathrm{str}}} && \dTo^{\Gamma} \\ K_0(\mathcal{R}(X,\mathbb{F})) & \rTo & K_0(\mathcal{R}_{\mathcal{X}}(X,\mathbb{F})) & \rTo & K_0(\mathcal{R}^0(X,\mathbb{F})) & \rTo & K_0(\mathcal{C}(X,\mathbb{F})) \end{diagram}
in which the vertical homomorphisms are all isomorphisms. Furthermore, if the variety $X$ is compact, then the horizontal homomorphisms are all monomorphisms. \end{corollary}
\begin{proof} It suffices to make use of Theorem~\ref{th-3-10}. \end{proof}
\begin{corollary}\label{cor-3-12} Assume that the variety $X$ is compact. For a stratified-algebraic $\mathbb{F}$-vector bundle $\xi$ on $X$, the following conditions are equivalent:
\begin{conditions} \item $\xi$ is isomorphic in the category of stratified-algebraic $\mathbb{F}$-vector bundles on $X$ to an $\mathcal{X}$-algebraic $\mathbb{F}$-vector bundle on $X$.
\item $\xi$ is isomorphic in the category of topological $\mathbb{F}$-vector bundles on $X$ to an $\mathcal{X}$-algebraic $\mathbb{F}$-vector bundle on $X$.
\item The class of $\xi$ in $K_{\mathbb{F}\mathrm{\mhyphen str}}(X)$ belongs to the image of the canonical homomorphism $K_{\mathbb{F} \mhyphen \mathcal{X}}(X) \to K_{\mathbb{F}\mathrm{\mhyphen str}}(X)$.
\item The class of $\xi$ in $K_{\mathbb{F}}(X)$ belongs to the image of the canonical homomorphism $K_{\mathbb{F} \mhyphen \mathcal{X}}(X) \to K_{\mathbb{F}}(X)$. \end{conditions}
\end{corollary}
\begin{proof} In view of Theorem~\ref{th-3-10} and Corollary~\ref{cor-3-11}, it suffices to apply Swan's theorem \cite[Theorem~2.2]{bib44}. \end{proof}
In the same way, we obtain the next two corollaries, which are recorded for the sake of completeness.
\begin{corollary}\label{cor-3-13} Assume that the variety $X$ is compact. For a topological $\mathbb{F}$-vector bundle $\xi$ on $X$, the following conditions are equivalent:
\begin{conditions} \item $\xi$ is isomorphic in the category of topological $\mathbb{F}$-vector bundles on $X$ to an $\mathcal{X}$-algebraic $\mathbb{F}$-vector bundle on $X$.
\item The class of $\xi$ in $K_{\mathbb{F}}(X)$ belongs to the image of the canonical homomorphism $K_{\mathbb{F} \mhyphen \mathcal{X}}(X) \to K_{\mathbb{F}}(X)$. \end{conditions}
\end{corollary}
\begin{corollary}\label{cor-3-14} Assume that the variety $X$ is compact. For a topological $\mathbb{F}$-vector bundle $\xi$ on $X$, the following conditions are equivalent:
\begin{conditions} \item $\xi$ is isomorphic in the category of topological $\mathbb{F}$-vector bundles on $X$ to a stratified-algebraic $\mathbb{F}$-vector bundle on $X$.
\item The class of $\xi$ in $K_{\mathbb{F}}(X)$ belongs to the image of the canonical homomorphism $K_{\mathbb{F}\mathrm{\mhyphen str}}(X) \to K_{\mathbb{F}}(X)$. \end{conditions}
\end{corollary}
Now we are in a position to prove two results announced in Section~\ref{sec-1}.
\begin{proof}[Proof of Theorem~\ref{th-1-5}] Recall that the group $K_{\mathbb{F}}(X)$ is finitely generated (cf. \cite[Exercise~III.7.5]{bib29} or the spectral sequence in \cite{bib4, bib21}). According to Corollary~\ref{cor-3-11}, the group $K_{\mathbb{F}\mathrm{\mhyphen str}}(X)$ also is finitely generated. Hence there exist stratified-algebraic $\mathbb{F}$-vector bundles $\xi_1, \ldots, \xi_r$ on $X$ whose classes in $K_{\mathbb{F}\mathrm{\mhyphen str}}(X)$ generate this group. We can find a stratification $\mathcal{S}$ of $X$ such that each $\xi_i$ is $\mathcal{S}$-algebraic for $1 \leq i \leq r$. Consequently, the canonical homomorphism $K_{\mathbb{F} \mhyphen \mathcal{S}}(X) \to K_{\mathbb{F}\mathrm{\mhyphen str}}(X)$ is surjective. The proof is complete in view of Corollary~\ref{cor-3-12}. \end{proof}
\begin{proof}[Proof of Theorem~\ref{th-1-6}] As in the proof of Theorem~\ref{th-1-5}, we obtain stratified-algebraic $\mathbb{F}$-vector bundles $\xi_1, \ldots, \xi_r$ on $X$ whose classes in $K_{\mathbb{F}\mathrm{\mhyphen str}}(X)$ generate this group. According to Proposition~\ref{prop-3-4}, each $\xi_i$ is of the form $\xi_i = f_i^* \gamma_{K_i} (\mathbb{F}^{n_i})$ for some multi-Grassmannian $\mathbb{G}_{K_i}(\mathbb{F}^{n_i})$ and stratified-regular map $f_i \colon X \to \mathbb{G}_{K_i}(\mathbb{F}^{n_i})$. Set
\begin{equation*} f \coloneqq (f_1, \ldots, f_r) \colon X \to G \coloneqq \mathbb{G}_{K_1}(\mathbb{F}^{n_1}) \times \cdots \times \mathbb{G}_{K_r}(\mathbb{F}^{n_r}). \end{equation*}
By Hironaka's theorem on resolution of singularities \cite{bib25, bib30}, there exists a birational multiblowup $\rho \colon \tilde{X} \to X$ such that the variety $\tilde{X}$ is nonsingular. The composite map $f \circ \rho \colon \tilde{X} \to G$ is stratified-regular, and hence, in view of Proposition~\ref{prop-2-2}, it is continuous rational. Now, Hironaka's theorem on resolution of points of indeterminacy \cite{bib26, bib30} implies the existence of a birational multiblowup $\sigma \colon X' \to \tilde{X}$ such that the composite map $f \circ \rho \circ \sigma \colon X' \to G$ is regular. The variety $X'$ is nonsingular and $\pi \coloneqq \rho \circ \sigma \colon X' \to X$ is a birational multiblowup. Since the composite map $f_i \circ \pi$ is regular, the pullback
\begin{equation*} \pi^*\xi_i = \pi^*(f_i^* \gamma_{K_i}(\mathbb{F}^{n_i})) = (f_i \circ \pi)^* \gamma_{K_i} (\mathbb{F}^{n_i}) \end{equation*}
is an algebraic $\mathbb{F}$-vector bundle on $X'$ for $1 \leq i \leq r$. Since the classes of $\xi_1, \ldots, \xi_r$ generate the group $K_{\mathbb{F}\mathrm{\mhyphen str}}(X)$, it follows that the image of the homomorphism \begin{equation*} \pi^* \colon K_{\mathbb{F} \mathrm{\mhyphen str}} (X) \to K_{\mathbb{F} \mathrm{\mhyphen str}}(X') \end{equation*}
induced by $\pi$ is contained in the image of the canonical homomorphism
\begin{equation*} K_{\mathbb{F} \mathrm{\mhyphen alg}} (X') = K_{\mathbb{F} \mhyphen \{X'\}} (X') \to K_{\mathbb{F} \mathrm{\mhyphen str}} (X'). \end{equation*}
Hence, in view of Corollary~\ref{cor-3-11} (with $X=X'$), for any stratified-algebraic $\mathbb{F}$-vector bundle $\xi$ on $X$, the pullback $\pi^*\xi$ is isomorphic in the category of stratified-algebraic $\mathbb{F}$-vector bundles on $X'$ to an algebraic $\mathbb{F}$-vector bundle on $X'$. This proves Theorem~\ref{th-1-6}. \end{proof}
We conclude this section by showing that the categories of vector bundles considered here are closed under standard operations. This is made precise in Proposition~\ref{prop-3-15} and its proof. In fact it was already done for the direct sum $\oplus$ when the monoids $\func{VB}_{\mathbb{F} \mhyphen \mathcal{X}}(X)$ and $\func{VB}_{\mathbb{F}\mathrm{\mhyphen str}}(X)$ were introduced.
\begin{proposition}\label{prop-3-15} Let $\xi$ and $\eta$ be $\mathcal{X}$-algebraic $\mathbb{F}$-vector bundles on $X$. Then the $\mathbb{F}$-vector bundles $\xi \oplus \eta$, $\func{Hom}(\xi, \eta)$ and $\xi^{\vee}$ (dual bundle) are $\mathcal{X}$-algebraic. If $\mathbb{F} = \mathbb{R}$ or $\mathbb{F} = \mathbb{C}$, then the $\mathbb{F}$-vector bundles $\xi \otimes \eta$ and $\bigwedge^k\xi$ ($k$th exterior power) are $\mathcal{X}$-algebraic. Furthermore, the analogous statements hold true for stratified-algebraic $\mathbb{F}$-vector bundles. \end{proposition}
\begin{proof} Suppose that $\xi$ is an $\mathcal{X}$-algebraic subbundle of $\varepsilon_X^n(\mathbb{F})$ and $\eta$ is an $\mathcal{X}$-algebraic subbundle of $\varepsilon_X^m(\mathbb{F})$. We already know that $\xi \oplus \eta$ is regarded as a subbundle of
\begin{equation*} \varepsilon_X^n(\mathbb{F}) \oplus \varepsilon_X^m(\mathbb{F}) \cong \varepsilon_X^{n+m}(\mathbb{F}). \end{equation*}
Since $\xi \oplus \xi^{\perp} = \varepsilon_X^n(\mathbb{F})$ and $\eta \oplus \eta^{\perp} = \varepsilon_X^m(\mathbb{F})$, (cf. Proposition~\ref{prop-3-7}), $\func{Hom}(\xi,\eta)$ can be regarded as a subbundle of $\func{Hom}(\varepsilon_X^n(\mathbb{F}), \varepsilon_X^m(\mathbb{F})) \cong \varepsilon_X^{nm}(\mathbb{F})$. By dualizing the orthogonal projection $\varepsilon_X^n(\mathbb{F}) \to \xi$, we regard $\xi^{\vee}$ as a subbundle of $\varepsilon_X^n(\mathbb{F})^{\vee} \cong \varepsilon_X^n(\mathbb{F})$. If $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C}$, then $\xi \otimes \eta$ can be regarded as a subbundle of $\varepsilon_X^n(\mathbb{F}) \otimes \varepsilon_X^m(\mathbb{F}) \cong \varepsilon_X^{nm}(\mathbb{F})$, and $\bigwedge^k\xi$ can be regarded as a subbundle of $\bigwedge^k\varepsilon_X^n(\mathbb{F}) \cong \varepsilon_X^q(\mathbb{F})$, where $q = \binom{n}{k}$ is the binomial coefficient. After these identifications, each of the vector bundles under consideration becomes an $\mathcal{X}$-algebraic subbundle of $\varepsilon_X^p(\mathbb{F})$ for an appropriate $p$. The same argument works for stratified-algebraic vector bundles. \end{proof}
\section{Approximation by stratified-regular maps}\label{sec-4} As in Section~\ref{sec-3}, for any real algebraic variety $X$ with stratification $\mathcal{X}$, and any real algebraic variety $Y$, we have the following inclusions:
\begin{equation*} \mathcal{R}(X,Y) \subseteq \mathcal{R}_{\mathcal{X}}(X,Y) \subseteq \mathcal{R}^0(X,Y) \subseteq \mathcal{C}(X,Y). \end{equation*}
A challenging problem is to find a useful description of the closure of each of the sets $\mathcal{R}(X,Y)$, $\mathcal{R}_{\mathcal{X}}(X,Y)$ and $\mathcal{R}^0(X,Y)$ in the space $\mathcal{C}(X,Y)$, endowed with the compact-open topology. In other words, the problem is to find a characterization of these maps in $\mathcal{C}(X,Y)$ that can be approximated by either regular or $\mathcal{X}$-regular or stratified-regular maps. Approximation by regular maps is investigated in \cite{bib9, bib10, bib12, bib13, bib16, bib18}. As demonstrated in \cite{bib32, bib34}, stratified-regular maps are much more flexible than regular maps. In the present section, we prove approximation theorems for maps with values in multi-Grassmannians. It is important for applications to obtain results in which approximating maps satisfy some extra conditions.
We first recall a key extension result due to Koll\'ar and Nowak \cite[Theorem~9, Proposition~10]{bib31}.
\begin{theorem}[\cite{bib31}]\label{th-4-1}
Let $X$ be a real algebraic variety. Let $A$ and $B$ be Zariski closed subvarieties of $X$ with $B \subseteq A$. For any stratified-regular function $f \colon A \to \mathbb{R}$ whose restriction $f|_{A \setminus B}$ is a regular function, there exists a stratified-regular function $F \colon X
\to \mathbb{R}$ such that $F|_A = f$ and $F|_{X \setminus B}$ is a regular function. \end{theorem}
It should be mentioned that in \cite{bib31}, Theorem~\ref{th-4-1} is stated in terms of hereditarily rational functions (cf. Remark~\ref{rem-2-3}).
The following terminology will be convenient. We say that a topological $\mathbb{F}$-vector bundle on a real algebraic variety $X$ \emph{admits an algebraic} (resp. \emph{$\mathcal{X}$-algebraic}, \emph{stratified-algebraic}) structure if it is topologically isomorphic to an algebraic (resp. $\mathcal{X}$-algebraic, stratified-algebraic) $\mathbb{F}$-vector bundle on $X$. If $Z$ is a Zariski closed subvariety of $X$, we say that a multiblowup $\pi \colon X' \to X$ of $X$ is \emph{over} $Z$ if the restriction $\pi_Z \colon X' \setminus \pi^{-1}(Z) \to X \setminus Z$ of $\pi$ is a biregular isomorphism.
\begin{notation}\label{not-4-2} In the remainder of this section, $X$ denotes a compact real algebraic variety, and $A$ is a Zariski closed subvariety of $X$. Moreover, $\mathbb{G}_K(\mathbb{F}^n)$ is the $(K,n)$-multi-Grassmannian and $\gamma_K(\mathbb{F}^n)$ is the tautological $\mathbb{F}$-vector bundle on $\mathbb{G}_K(\mathbb{F}^n)$, for some fixed $(K,n)$ (cf. Section~\ref{sec-2}). \end{notation}
Our basic approximation result is the following.
\begin{lemma}\label{lem-4-3} Let $B$ be a Zariski closed subvariety of $X$ with $B \subseteq A$. Let $f
\colon X \to \mathbb{R}$ be a continuous function such that $f|_A$ is a stratified-regular function and $f|_{A \setminus B}$ is a regular function. For every $\varepsilon > 0$, there exists a stratified-regular function $g \colon X \to \mathbb{R}$ such that $g|_A = f|_A$, the restriction
$g|_{X \setminus B}$ is a regular function, and
\begin{equation*} \@ifstar{\oldabs}{\oldabs*}{g(x)-f(x)} < \varepsilon \quad \textrm{for all $x$ in $X$.} \end{equation*}
\end{lemma}
\begin{proof}
According to Theorem~\ref{th-4-1}, there exists a stratified-regular function $h \colon X \to \mathbb{R}$ such that $h|_A = f|_A$ and the restriction
$h|_{X \setminus B}$ is a regular function. The function $\varphi
\coloneqq f-h$ is continuous and $\varphi|_A=0$. Hence, by \cite[Lemma~2.1]{bib15}, one can find a regular function $\psi \colon X
\to \mathbb{R}$ satisfying $\psi|_A=0$ and
\begin{equation*} \@ifstar{\oldabs}{\oldabs*}{\varphi(x) - \psi(x)} < \varepsilon \quad \textrm{for all $x$ in $X$.} \end{equation*}
The function $g \coloneqq h + \psi$ has the required properties. \end{proof}
For any stratification $\mathcal{X}$ of $X$ and any Zariski closed subvariety $B$ of $X$, we set
\begin{equation*} \mathcal{X} \cap B \coloneqq \{ S \cap B \mid S \in \mathcal{X} \} \quad \textrm{and} \quad \mathcal{X} \setminus B \coloneqq \{ S \setminus B \mid S \in \mathcal{X} \}. \end{equation*}
Then
\begin{equation*} \mathcal{X}_B \coloneqq (\mathcal{X} \cap B) \cup \{ \mathcal{X} \setminus B \} \end{equation*}
is a stratification of $X$. Obviously $\mathcal{X}_B \cap A$ is a stratification of $A$.
\begin{theorem}\label{th-4-4} Let $\mathcal{X}$ be a stratification of $X$ and let $B$ be a Zariski closed subvariety of~$X$. Let $f \colon X \to \mathbb{G}_K(\mathbb{F}^n)$
be a continuous map whose restriction $f|_A$ is $(\mathcal{X}_B \cap A)$-regular. If $B \subseteq A$ and $A \setminus B$ is a stratum in $\mathcal{X}_B$, then the following conditions are equivalent:
\begin{conditions} \item\label{th-4-4-a} Each neighborhood of $f$ in $\mathcal{C}(X, \mathbb{G}_K(\mathbb{F}^n))$ contains an $\mathcal{X}_B$-regular map
\begin{equation*} g \colon X \to \mathbb{G}_K(\mathbb{F}^n) \end{equation*}
with $g|_A=f|_A$.
\item\label{th-4-4-b} The map $f$ is homotopic to an $\mathcal{X}_B$-regular map $h \colon X \to
\mathbb{G}_K(\mathbb{F}^n)$ with $h|_A = f|_A$.
\item\label{th-4-4-c} The pullback $\mathbb{F}$-vector bundle $f^*\gamma_K(\mathbb{F}^n)$ on $X$ admits an $\mathcal{X}_B$-algebraic structure. \end{conditions}
\end{theorem}
\begin{proof} It is a standard fact that (\ref{th-4-4-a}) implies (\ref{th-4-4-b}). If (\ref{th-4-4-b}) holds, then the $\mathcal{X}_B$-algebraic $\mathbb{F}$-vector bundle $h^*\gamma_K(\mathbb{F}^n)$ is topologically isomorphic to $f^*\gamma_K(\mathbb{F}^n)$, and hence (\ref{th-4-4-c}) holds. It suffices to prove that (\ref{th-4-4-c}) implies (\ref{th-4-4-a}).
Suppose that (\ref{th-4-4-c}) holds. In order to simplify notation, we set $G = \mathbb{G}_K(\mathbb{F}^n)$, $\gamma=\gamma_K(F^n)$, $\mathcal{A}=\mathcal{X}_B\cap A$, and $\varepsilon_X^l = \varepsilon_X^l(\mathbb{F}^n)$ for any nonnegative integer $l$. Condition (\ref{th-4-4-c}) implies the existence of a topological isomorphism $\varphi \colon \xi \to f^*\gamma$, where $\xi$ is an $\mathcal{X}_B$-algebraic $\mathbb{F}$-vector subbundle of $\varepsilon_X^k$ for some $k$. Regarding $f^*\gamma$ as an $\mathbb{F}$-vector subbundle of $\varepsilon_X^n$, we get a continuous section $w \colon X \to \func{Hom}(\xi, \varepsilon_X^n)$, defined by $w(x)(e)=\varphi(e)$ for all points $x$ in $X$ and all vectors $e$ in the fiber $E(\xi)_x$. In particular, $w(x) \colon E(\xi)_x \to \{x\} \times \mathbb{F}^n$ is an injective $\mathbb{F}$-linear transformation satisfying
\begin{equation*} w(x)(E(\xi)_x) = \{x\} \times f(x). \end{equation*}
Making use of the equalities $\varepsilon_X^k = \xi \oplus \xi^{\perp}$ and $\varepsilon_X^n = f^*\gamma \oplus f^* \gamma^{\perp}$, we find a continuous section $\bar{w} \colon X \to \func{Hom}(\varepsilon_X^k, \varepsilon_X^n)$ satisfying $\bar{w}(x) (E(\varepsilon_X^k)_x)
\subseteq E(f^*\gamma)_x$ and $\bar{w}(x)|_{E(\xi)_x} = w(x)$ for all $x$ in $X$. Since $X$ is compact and the $\mathbb{F}$-vector bundle $\func{Hom}(\varepsilon_X^k, \varepsilon_X^n)$ can be identified with $\varepsilon_X^{kn}$, the Weierstrass approximation theorem implies the existence of an algebraic section $v \colon X \to \func{Hom}(\varepsilon_X^k, \varepsilon_X^n)$ arbitrarily close to $\bar{w}$ in the compact-open topology. If $\rho \colon \varepsilon_X^n \to f^*\gamma$ is the orthogonal projection and $\iota \colon f^*\gamma \hookrightarrow \varepsilon_X^n$ is the inclusion morphism, then $\bar{v} \coloneqq \iota \circ \rho \circ v \colon X \to \func{Hom}(\varepsilon_X^k, \varepsilon_X^n)$ is a continuous section close to $\bar{w}$. If $v$ is sufficiently close to $\bar{w}$, then the $\mathbb{F}$-linear transformation $\bar{v}(x) \colon E(\xi)_x \to \{x\} \times \mathbb{F}^n$ is injective and
\begin{equation*} \bar{v}(x)(E(\xi)_x) = \{x\} \times f(x) \end{equation*}
for all $x$ in $X$. Since the map $f|_A$ is $\mathcal{A}$-regular, the
$\mathbb{F}$-vector bundle $(f|_A)^*\gamma = (f^*\gamma)|_A$ on $A$ is
$\mathcal{A}$-algebraic. Hence, in view of Proposition~\ref{prop-3-7}, the restriction $\rho_A \colon \varepsilon_X^n|_A \to (f^*\gamma)|_A$ of $\rho$ is an $\mathcal{A}$-algebraic morphism. Consequently, the restriction
$\bar{v}|_A \colon A \to \func{Hom}(\varepsilon_X^k, \varepsilon_X^n)$ of $\bar{v}$ is an $\mathcal{A}$-algebraic section. By Lemma \ref{lem-4-3}, there exists a continuous section
\begin{equation*} u \colon X \to \func{Hom}(\varepsilon_X^k, \varepsilon_X^n) \cong \varepsilon_X^{kn} \end{equation*}
such that $u|_A = \bar{v}|_A$, the restriction $u|_{X \setminus B}$ is an algebraic section, and $u$ is close to $\bar{v}$. If $u$ is sufficiently close to $\bar{v}$, then the $\mathbb{F}$-linear transformation $u(x) \colon E(\xi)_x \to \{x\} \times \mathbb{F}^n$ is injective for all $x$ in $X$. Now, the map $g \colon X \to G$, defined by
\begin{equation*} u(x)(E(\xi)_x) = \{x\} \times g(x) \end{equation*}
for all $x$ in $X$, is continuous and close to $f$, and $g|_A = f|_A$. If $T$ is a stratum in $\mathcal{X}_B$, then $u|_T$ is an algebraic section and
$\xi|_T$ is an algebraic $\mathbb{F}$-vector subbundle of $\varepsilon_X^k|_T$, and hence the map $g|_T$ is regular (cf. \cite[Proposition~3.4.7]{bib9}). Consequently, $g$ is an $\mathcal{X}_B$-regular map. Thus, (\ref{th-4-4-c}) implies (\ref{th-4-4-a}). \end{proof}
It is worthwhile to state several consequences of Theorem~\ref{th-4-4}.
\begin{corollary}\label{cor-4-5} Let $f \colon X \to \mathbb{G}_K(\mathbb{F}^n)$ be a continuous map whose restriction
$f|_A$ is regular. Then the following conditions are equivalent:
\begin{conditions}
\item Each neighborhood of $f$ in $\mathcal{C}(X, \mathbb{G}_K(\mathbb{F}^n))$ contains a regular map $g \colon X \to \mathbb{G}_K (\mathbb{F}^n)$ with $g|_A=f|_A$.
\item The map $f$ is homotopic to a regular map $h \colon X \to
\mathbb{G}_K(\mathbb{F}^n)$ with $h|_A=f|_A$.
\item The pullback $\mathbb{F}$-vector bundle $f^*\gamma_K(\mathbb{F}^n)$ on $X$ admits an algebraic structure. \end{conditions}
\end{corollary}
\begin{proof} If $\mathcal{X}=\{X\}$ and $B = \varnothing$, then $\mathcal{X}_B = \{X\}$ and $\mathcal{X}_B\cap A = \{A\}$. Hence, it suffices to apply Theorem~\ref{th-4-4}. \end{proof}
Corollary~\ref{cor-4-5} with $A = \varnothing$ is well known, cf. \cite{bib10} or \cite[Theorem~13.3.1]{bib9}.
\begin{corollary}\label{cor-4-6} Let $\mathcal{X}$ be a stratification of $X$ such that $\mathcal{A} \coloneqq \{ S \in \mathcal{X} \mid S \subseteq A\}$ is a stratification of $A$. Let $f \colon X \to
\mathbb{G}_K(\mathbb{F}^n)$ be a continuous map whose restriction $f|_A$ is $\mathcal{A}$-regular. Then the following conditions are equivalent:
\begin{conditions} \item Each neighborhood of $f$ in $\mathcal{C}(X, \mathbb{G}_K(\mathbb{F}^n))$ contains an
$\mathcal{X}$-regular map $g \colon X \to \mathbb{G}_K(\mathbb{F}^n)$ with $g|_A = f|_A$.
\item The map $f$ is homotopic to an $\mathcal{X}$-regular map $h \colon X \to
\mathbb{G}_K(\mathbb{F}^n)$ with $h|_A=f|_A$.
\item The pullback $\mathbb{F}$-vector bundle $f^*\gamma_K(\mathbb{F}^n)$ on $X$ admits an $\mathcal{X}$-algebraic structure. \end{conditions}
\end{corollary}
\begin{proof} If $B=A$, then $\mathcal{X}_B=\mathcal{X}$ and $\mathcal{X}_B\cap A = \mathcal{A}$, and hence it suffices to apply Theorem~\ref{th-4-4}. \end{proof}
\begin{corollary}\label{cor-4-7} Let $\mathcal{X}$ be a stratification of $X$. For a continuous map $f \colon X \to \mathbb{G}_K(\mathbb{F}^n)$, the following conditions are equivalent:
\begin{conditions} \item The map $f$ can be approximated by $\mathcal{X}$-regular maps.
\item The map $f$ is homotopic to an $\mathcal{X}$-regular map.
\item The pullback $\mathbb{F}$-vector bundle $f^*\gamma_K(\mathbb{F}^n)$ on $X$ admits an $\mathcal{X}$-algebraic structure. \end{conditions}
\end{corollary}
\begin{proof} This is Corollary~\ref{cor-4-6} with $A = \varnothing$. \end{proof}
Corollary~\ref{cor-4-7} makes it possible to interpret Theorems~\ref{th-1-5} and~\ref{th-1-6} as approximation results.
\begin{theorem}\label{th-4-8} There exists a stratification $\mathcal{S}$ of $X$ such that any stratified-regular map from $X$ into $\mathbb{G}_K(\mathbb{F}^n)$ can be approximated by $\mathcal{S}$-regular maps. \end{theorem}
\begin{proof} Let $\mathcal{S}$ be a stratification of $X$ as in Theorem~\ref{th-1-5}. By Proposition~\ref{prop-2-6}, if
\begin{equation*} f \colon X \to \mathbb{G}_K(\mathbb{F}^n) \end{equation*}
is a stratified-regular map, then the $\mathbb{F}$-vector bundle $f^*\gamma_K(\mathbb{F}^n)$ on $X$ is stratified-algebraic. It follows that $f^*\gamma_K(\mathbb{F}^n)$ admits an $\mathcal{S}$-algebraic structure. According to Corollary~\ref{cor-4-7}, $f$ can be approximated by $\mathcal{S}$-regular maps. \end{proof}
\begin{theorem}\label{th-4-9} There exists a birational multiblowup $\pi \colon X' \to X$, with $X'$ a nonsingular variety, such that for any stratified-regular map $f \colon X \to \mathbb{G}_K(\mathbb{F}^n)$, the composite map $f \circ \pi \colon X' \to \mathbb{G}_K(\mathbb{F}^n)$ can be approximated by regular maps. \end{theorem}
\begin{proof} Let $\pi \colon X' \to X$ be a birational multiblowup as in Theorem~\ref{th-1-6}. By Proposition~\ref{prop-2-6}, if $f \colon X \to \mathbb{G}_K(\mathbb{F}^n)$ is a stratified-regular map, the $\mathbb{F}$-vector bundle $f^*\gamma_K(\mathbb{F}^n)$ on $X$ is stratified-algebraic. It follows that the $\mathbb{F}$-vector bundle
\begin{equation*} \pi^* ( f^* \gamma_K (\mathbb{F}^n) ) = (f \circ \pi)^* \gamma_K (\mathbb{F}^n) \end{equation*}
on $X'$ admits an algebraic-structure. According to Corollary~\ref{cor-4-7} (with $X=X'$ and $\mathcal{X}=\{X'\}$), the map $f \circ \pi \colon X' \to \mathbb{G}_K(\mathbb{F}^n)$ can be approximated by regular maps. \end{proof}
It follows from Corollary~\ref{cor-4-7} and Propositions~\ref{prop-2-6} and~\ref{prop-3-4} that Theorem~\ref{th-4-9} is equivalent to Theorem~\ref{th-1-6}.
\begin{theorem}\label{th-4-10} Let $f \colon X \to \mathbb{G}_K(\mathbb{F}^n)$ be a continuous map whose restriction
$f|_A$ is stratified-regular. Then the following conditions are equivalent:
\begin{conditions} \item\label{th-4-10-a} Each neighborhood of $f$ in $\mathcal{C}(X, \mathbb{G}_K(\mathbb{F}^n))$ contains a stratified-regular map
\begin{equation*} g \colon X \to \mathbb{G}_K(\mathbb{F}^n) \end{equation*}
with $g|_A=f|_A$.
\item\label{th-4-10-b} The map $f$ is homotopic to a stratified-regular map $h \colon X
\to \mathbb{G}_K(\mathbb{F}^n)$ with $h|_A=f|_A$.
\item\label{th-4-10-c} The pullback $\mathbb{F}$-vector bundle $f^*\gamma_K(\mathbb{F}^n)$ on $X$ admits a stratified-algebraic structure. \end{conditions}
\end{theorem}
\begin{proof} As in the case of Theorem~\ref{th-4-4}, it suffices to prove that (\ref{th-4-10-c}) implies (\ref{th-4-10-a}).
Suppose that (\ref{th-4-10-c}) holds. Let $\mathcal{T}$ be a stratification of $X$ such that the $\mathbb{F}$-vector bundle $f^*\gamma_K(\mathbb{F}^n)$ admits a
$\mathcal{T}$-algebraic structure, and let $\mathcal{P}$ be a stratification of $A$ such that the map $f|_A$ is $\mathcal{P}$-regular. The collection
\begin{equation*} \mathcal{X} \coloneqq \{ P \cap T \mid P \in \mathcal{P}, T \in \mathcal{T} \} \cup \{ T \cap (X \setminus A) \mid T \in \mathcal{T} \} \end{equation*}
is a stratification of $X$, and $\mathcal{A} \coloneqq \{ S \in \mathcal{X} \mid S
\subseteq A \}$ is a stratification of $A$. By construction, $\mathcal{X}$ is a refinement of $\mathcal{T}$, and $\mathcal{A}$ is a refinement of $\mathcal{P}$. Consequently, the map $f|_A$ is $\mathcal{A}$-regular, and the $\mathbb{F}$-vector bundle $f^*\gamma_K(\mathbb{F}^n)$ on $X$ admits an $\mathcal{X}$-algebraic structure. The proof is complete in view of Corollary~\ref{cor-4-6}. \end{proof}
In the results above, approximation by $\mathcal{X}$-regular or stratified-regular maps is equiva\-lent to certain conditions on pullbacks of the tautological vector bundle. It is often convenient (see Sections~\ref{sec-5} and~\ref{sec-6}) to have these conditions expressed in terms involving multiblowups.
\begin{proposition}\label{prop-4-11} Let $f \colon X \to \mathbb{G}_K(\mathbb{F}^n)$ be a continuous map whose restriction
$f|_A$ is stratified-regular. Assume the existence of a multiblowup $\pi \colon X' \to X$ over $A$ such that the pullback $\mathbb{F}$-vector bundle $(f
\circ \pi)^* \gamma_K(\mathbb{F}^n)$ on $X'$ admits an algebraic structure. Then each neighborhood of $f$ in $\mathcal{C}(X, \mathbb{G}_K(\mathbb{F}^n))$ contains a stratified-regular map $g \colon X \to \mathbb{G}_K(\mathbb{F}^n)$ such that $g|_A =
f|_A$ and the restriction $g|_{X \setminus A}$ is a regular map. In particular, $f$ is homotopic to a stratified-regular map $h \colon X \to
\mathbb{G}_K(\mathbb{F}^n)$ such that $h|_A = f|_A$ and the restriction $h|_{X \setminus A}$ is a regular map. \end{proposition}
\begin{proof}
The map $f|_A$ is $\mathcal{A}$-regular for some stratification $\mathcal{A}$ of $A$. The collection \linebreak $\mathcal{A}' \coloneqq \{ \pi^{-1}(S) \mid S \in \mathcal{A} \}$ is a stratification of $A' \coloneqq \pi^{-1}(A)$, whereas $\mathcal{X}' \coloneqq \mathcal{A}' \cup \{X' \setminus A'\}$ is a stratification of $X'$. Since the map $f \circ \pi \colon X' \to \mathbb{G}_K(\mathbb{F}^n)$ is continuous and its restriction
$(f \circ \pi)|_{A'}$ is $\mathcal{A}'$-regular, according to Corollary~\ref{cor-4-6} (with $X=X'$, $A=A'$, $\mathcal{X}=\mathcal{X}'$, $\mathcal{A}=\mathcal{A}'$, $f=f \circ \pi$), there exists an $\mathcal{X}'$-regular map $\varphi \colon X' \to \mathbb{G}_K(\mathbb{F}^n)$ that is arbitrarily close to $f \circ \pi$ and satisfies
$\varphi|_{A'} = (f \circ \pi)|_{A'}$. In particular, the restriction
$\varphi|_{X' \setminus A'}$ is a regular map. By assumption, the restriction $\pi_A \colon X' \setminus A' \to X \setminus A$ of $\pi$ is a biregular isomorphism. Since the map $\pi$ is proper, the map $g \colon X \to \mathbb{G}_K(\mathbb{F}^n)$, defined by
\begin{equation*} g(x)=
\begin{cases} \varphi(\pi_A^{-1}(x)) & \textrm{for $x$ in $X \setminus A$} \\ f(x) & \textrm{for $x$ in $A$}, \end{cases}
\end{equation*}
is continuous. Moreover, $g$ is close to $f$, $g|_A = f|_A$, and the restriction $g|_{X \setminus A}$ is a regular map. If $g$ is sufficiently close to $f$, then $g$ is homotopic to $f$, and hence the existence of $h$ follows. \end{proof}
\begin{corollary}\label{cor-4-12} Let $f \colon X \to \mathbb{G}_K(\mathbb{F}^n)$ be a continuous map whose restriction
$f|_A$ is stratified-regular. If the variety $X \setminus A$ is nonsingular, then the following conditions are equivalent:
\begin{conditions} \item\label{cor-4-12-a} Each neighborhood of $f$ in $\mathcal{C}(X, \mathbb{G}_K(\mathbb{F}^n))$ contains a stratified-regular map
\begin{equation*} g \colon X \to \mathbb{G}_K(\mathbb{F}^n) \end{equation*}
such that
$g|_A=f|_A$ and the restriction $g|_{X \setminus A}$ is a regular map.
\item\label{cor-4-12-b} The map $f$ is homotopic to a stratified-regular map $h \colon X \to \mathbb{G}_K(\mathbb{F}^n)$ such that $h|_A = f|_A$ and the restriction $h|_{X \setminus A}$ is a regular map.
\item\label{cor-4-12-c} There exists a multiblowup $\pi \colon X' \to X$ over $A$ such that $X'$ is a nonsingular variety and the pullback $\mathbb{F}$-vector bundle $(f \circ \pi)^* \gamma_K(\mathbb{F}^n)$ on $X'$ admits an algebraic structure. \end{conditions}
\end{corollary}
\begin{proof} It is a standard fact that (\ref{cor-4-12-a}) implies (\ref{cor-4-12-b}). In view of Proposition~\ref{prop-4-11}, (\ref{cor-4-12-c}) implies (\ref{cor-4-12-a}). It remains to prove that (\ref{cor-4-12-b}) implies (\ref{cor-4-12-c}).
Suppose that (\ref{cor-4-12-b}) holds. Since the variety $X \setminus A$ is nonsingular, Hironaka's theorem on resolution of singularities \cite{bib25, bib30} implies the existence of a multiblowup $\rho \colon \tilde{X} \to X$ over $A$ such that $\tilde{X}$ is a nonsingular variety and the set $\tilde{X} \setminus \rho^{-1}(A)$ is Zariski dense in $\tilde{X}$. The map $h \circ \rho \colon \tilde{X} \to \mathbb{G}_K(\mathbb{F}^n)$ is continuous and its restriction to $\tilde{X} \setminus \rho^{-1}(A)$ is a regular map. Hence, according to Hironaka's theorem on resolution of points of indeterminacy \cite{bib25, bib30}, there exists a multiblowup $\sigma \colon X' \to \tilde{X}$ over $\rho^{-1}(A)$ such that the map $h \circ \pi \colon X' \to \mathbb{G}_K(\mathbb{F}^n)$ is regular, where $\pi \coloneqq \rho \circ \sigma$. Consequently, $(h \circ \pi)^* \gamma_K(\mathbb{F}^n)$ is an algebraic $\mathbb{F}$-vector bundle on $X'$. The variety $X'$ is nonsingular, and $\pi \colon X' \to X$ is a multiblowup over $A$. Since the maps $h \circ \pi$ and $f \circ \pi$ are homotopic, the $\mathbb{F}$-vector bundles $(h \circ \pi)^* \gamma_K(\mathbb{F}^n)$ and $(f \circ \pi)^*\gamma_K(\mathbb{F}^n)$ are topologically isomorphic. Thus, (\ref{cor-4-12-b}) implies (\ref{cor-4-12-c}). \end{proof}
We also have the following modification of Proposition~\ref{prop-4-11}.
\begin{proposition}\label{prop-4-13} Let $f \colon X \to \mathbb{G}_K(\mathbb{F}^n)$ be a continuous map whose restriction
$f|_A$ is stratified-regular. Assume the existence of a multiblowup $\pi \colon X' \to X$ over $A$ such that the pullback $\mathbb{F}$-vector bundle $(f
\circ \pi)^* \gamma_K(\mathbb{F}^n)$ on $X'$ admits a stratified-algebraic structure. Then each neighborhood of $f$ in $\mathcal{C}(X, \mathbb{G}_K(\mathbb{F}^n))$ contains a stratified-regular map $g \colon X \to \mathbb{G}_K(\mathbb{F}^n)$ with $g|_A = f|_A$. In particular, $f$ is homotopic to a stratified-regular map $h \colon X
\to \mathbb{G}_K(\mathbb{F}^n)$ with $h|_A = f|_A$. \end{proposition}
\begin{proof}
The map $f|_A$ is $\mathcal{A}$-regular for some stratification $\mathcal{A}$ of $A$, and the $\mathbb{F}$-vector bundle $(f \circ \pi)^* \gamma_K (\mathbb{F}^n)$ admits an $\mathcal{S}$-algebraic structure for some stratification $\mathcal{S}$ of $X'$. The collection $\mathcal{P} \coloneqq \{ \pi^{-1}(T) \mid T \in \mathcal{A} \}$ is a stratification of $A' \coloneqq \pi^{-1}(A)$, and the map $(f \circ
\pi)|_{A'}$ is $\mathcal{P}$-regular. Moreover, the collection
\begin{equation*} \mathcal{S}' \coloneqq \{ P \cap S \mid P \in \mathcal{P}, S \in \mathcal{S} \} \cup \{ (X' \setminus A') \cap S \mid S \in \mathcal{S} \} \end{equation*}
is a stratification of $X'$, and the collection
\begin{equation*} \mathcal{A}' \coloneqq \{ S' \in \mathcal{S}' \mid S' \subseteq A' \} \end{equation*}
is a stratification of $A'$. By construction, $\mathcal{A}'$ is a refinement of $\mathcal{P}$, and $\mathcal{S}'$ is a refinement of $\mathcal{S}$. Consequently, the map $(f
\circ \pi)|_{A'}$ is $\mathcal{A}'$-regular, and the $\mathbb{F}$-vector bundle $(f \circ \pi)^* \gamma_K(\mathbb{F}^n)$ on $X'$ admits an $\mathcal{S}'$-algebraic structure. According to Corollary~\ref{cor-4-6} (with $X=X'$, $A=A'$, $\mathcal{X}=\mathcal{S}'$, $\mathcal{A}=\mathcal{A}'$, $f=f \circ \pi$), there exists an $\mathcal{S}'$-regular map $\varphi
\colon X' \to \mathbb{G}_K(\mathbb{F}^n)$ that is arbitrarily close to $f \circ \pi$ and satisfies $\varphi|_{A'} = (f \circ \pi)|_{A'}$. The restriction $\pi_A \colon X' \setminus A' \to X \setminus A$ of $\pi$ is a biregular isomorphism. Since the map $\pi$ is proper, the map $g \colon X \to \mathbb{G}_K(\mathbb{F}^n)$, defined by
\begin{equation*} g(x)=
\begin{cases} \varphi(\pi_A^{-1}(x)) & \textrm{for $x$ in $X\setminus A$}\\ f(x) & \textrm{for $x$ in $A$,} \end{cases}
\end{equation*}
is continuous. By construction, $g$ is close to $f$ and $g|_A = f|_A$. Moreover,
\begin{equation*} \mathcal{X} \coloneqq \mathcal{A} \cup \{ \pi_A(S') \mid S' \in \mathcal{S}', S' \subseteq X' \setminus A' \} \end{equation*}
is a stratification of $X$, and the map $g$ is $\mathcal{X}$-regular. If $g$ is sufficiently close to $f$, then $g$ is homotopic to $f$, and hence the existence of $h$ follows. \end{proof}
\section[Topological versus stratified-algebraic vector bundles]{\texorpdfstring{Topological versus stratified-algebraic\\ vector bundles}{Topological versus stratified-algebraic vector bundles}}\label{sec-5}
Throughout this section, $X$ denotes a compact real algebraic variety. By making use of the notion of filtration, introduced in Section~\ref{sec-2}, we demonstrate how the behavior of a vector bundle on $X$ can be deduced from the behavior of its restrictions to Zariski closed subvarieties of $X$. This is crucial for, in particular, the proofs of Theorems~\ref{th-1-7}, \ref{th-1-8} and~\ref{th-1-9}, given in the next section.
\begin{proposition}\label{prop-5-1} Let $\xi$ be a topological $\mathbb{F}$-vector bundle on $X$ and let
\begin{equation*} \mathcal{F} = (X_{-1}, X_0, \ldots, X_m) \end{equation*}
be a filtration of $X$. Assume that for each $i = 0, \ldots, m$, there exists a multiblowup $\pi_i \colon X'_i \to X_i$ over $X_{i-1}$ such that the pullback $\mathbb{F}$-vector bundle $\pi_i^*
(\xi|_{X_i})$ on $X'_i$ admits an algebraic structure. Then $\xi$ admits an $\overline{\mathcal{F}}$-algebraic structure. \end{proposition}
\begin{proof} Since $X$ is compact, we may assume that $\xi$ is of the form $\xi = f^*\gamma_K(\mathbb{F}^n)$ for some multi-Grassmannian $\mathbb{G}_K(\mathbb{F}^n)$ and continuous map $f \colon X \to \mathbb{G}_K(\mathbb{F}^n)$. It suffices to prove that $f$ is homotopic to an $\overline{\mathcal{F}}$-regular map, cf. Corollary~\ref{cor-4-7}.
\begin{assertion} For each $i=0,\ldots,m$, there exists a continuous map $g_i \colon X \to \mathbb{G}_K(\mathbb{F}^n)$ homotopic to $f$ and such that the restriction of $g_i$ to $X_j \setminus X_{j-1}$ is a regular map for $0 \leq j \leq i$. \end{assertion}
If the Assertion holds, then the map $g_m$ is $\overline{\mathcal{F}}$-regular and the proof is complete. We prove the Assertion by induction on $i$. Recall that $(X, X_i)$ is a polyhedral pair for \linebreak $0 \leq i \leq m$, cf. \cite[Corollary~9.3.7]{bib9}. Since $X'_0=X_0$ and $\pi_0 \colon X'_0
\to X_0$ is the identity map, the $\mathbb{F}$-vector bundle $\xi|_{X_0}$ on $X_0$ admits an algebraic structure. According to Proposition~\ref{prop-4-11} (with $X=X_0$, $A=\varnothing$), the map
$f|_{X_0}$ is homotopic to a regular map $\varphi \colon X_0 \to \mathbb{G}_K(\mathbb{F}^n)$. The homotopy extension theorem \cite[p.~118, Corollary~5]{bib42} implies the existence of a continuous map $g_0
\colon X \to \mathbb{G}_K(\mathbb{F}^n)$ homotopic to $f$ and satisfying $g_0|_{X_0} = \varphi$. We now suppose that $0 \leq i \leq m-1$ and the map $g_i$
satisfying the required conditions has already been constructed. In particular, the map $g_i|_{X_i}$ is stratified-regular. Since the maps
$g_i|_{X_{i+1}}$ and $f|_{X_{i+1}}$ are homotopic, the $\mathbb{F}$-vector bundles $(g_i|_{X_{i+1}})^* \gamma_K (\mathbb{F}^n)$ and $(f|_{X_{i+1}})^*
\gamma_K(\mathbb{F}^n) = \xi|_{X_{i+1}}$ are topologically isomorphic. Consequently, the $\mathbb{F}$-vector bundle
\begin{equation*}
\pi_{i+1}^* ( (g_i|_{X_{i+1}})^* \gamma_K (\mathbb{F}^n) ) = ( (g_i|_{X_{i+1}}) \circ \pi_{i+1} )^* \gamma_K(\mathbb{F}^n) \end{equation*}
on $X'_{i+1}$ admits an algebraic structure, being topologically isomorphic to $\pi_{i+1}^*(\xi|_{X_{i+1}})$. According to Proposition~\ref{prop-4-11} (with $X = X_{i+1}$, $A = X_i$), the map
$g_i|_{X_{i+1}}$ is homotopic to a continuous map $\psi \colon X_{i+1}
\to \mathbb{G}_K(\mathbb{F}^n)$ such that $\psi|_{X_i} = g_i|_{X_i}$ and the restriction of $\psi$ to $X_{i+1} \setminus X_i$ is a regular map. By the homotopy extension theorem, there exists a continuous map $g_{i+1} \colon X \to \mathbb{G}_K(\mathbb{F}^n)$ homotopic to $g_i$ (hence homotopic to $f$) and satisfying
$g_{i+1}|_{X_{i+1}} = \psi$. This completes the proof of the Assertion. \end{proof}
\begin{theorem}\label{th-5-2} Let $\xi$ be a topological $\mathbb{F}$-vector bundle on $X$ and let $\mathcal{F} = (X_{-1}, X_0, \ldots, X_m)$ be a filtration of $X$. If the stratification $\overline{\mathcal{F}}$ is nonsingular, then the following conditions are equivalent:
\begin{conditions} \item\label{th-5-2-a} The $\mathbb{F}$-vector bundle $\xi$ admits an $\overline{\mathcal{F}}$-algebraic structure.
\item\label{th-5-2-b} For each $i=0,\ldots,m$, there exists a multiblowup $\pi_i \colon X'_i \to X_i$ over $X_{i-1}$ such that $X'_i$ is a nonsingular variety and the pullback $\mathbb{F}$-vector bundle
$\pi_i^*(\xi|_{X_i})$ on $X'_i$ admits an algebraic structure. \end{conditions}
\end{theorem}
\begin{proof} Suppose that (\ref{th-5-2-a}) holds. In view of Propositions~\ref{prop-2-2} and~\ref{prop-3-3}, we may assume that $\xi$ is of the form $\xi = f^*\gamma_K(\mathbb{F}^n)$ for some multi-Grassmannian $\mathbb{G}_K(\mathbb{F}^n)$ and $\overline{\mathcal{F}}$-regular map $f \colon X \to \mathbb{G}_K(\mathbb{F}^n)$. Set $f_i
\coloneqq f|_{X_i}$ for $0 \leq i \leq m$. The map $f_i|_{X_{i-1}}$ is stratified-regular, and the restriction of $f_i$ to $X_i \setminus X_{i-1}$ is a regular map. The existence of $\pi_i$ as in (\ref{th-5-2-b}) follows from Corollary~\ref{cor-4-12} (with $X=X_i$, $A=X_{i-1}$). Thus, (\ref{th-5-2-a}) implies (\ref{th-5-2-b}). According to Proposition~\ref{prop-5-1}, (\ref{th-5-2-b}) implies (\ref{th-5-2-a}). \end{proof}
Theorem~\ref{th-5-2} implies the following characterization of topological $\mathbb{F}$-vector bundles on $X$ admitting a stratified-algebraic structure.
\begin{corollary}\label{cor-5-3} For a topological $\mathbb{F}$-vector bundle $\xi$ on $X$, the following conditions are equivalent:
\begin{conditions} \item The $\mathbb{F}$-vector bundle $\xi$ admits a stratified-algebraic structure.
\item There exists a filtration $\mathcal{F} = (X_{-1}, X_0, \ldots, X_m)$ of $X$, with $\overline{\mathcal{F}}$ a nonsingular strati\-fi\-ca\-tion, and for each $i = 0, \ldots, m$, there exists a multiblowup $\pi_i \colon X'_i \to X_i$ over $X_{i-1}$ such that $X'_i$ is a nonsingular variety and the pullback
$\mathbb{F}$-vector bundle $\pi_i^*(\xi|_{X_i})$ on $X_i'$ admits an algebraic structure. \end{conditions}
\end{corollary}
\begin{proof} It suffices to combine Proposition~\ref{prop-3-5} and Theorem~\ref{th-5-2}. \end{proof}
The following result will also prove to be useful.
\begin{theorem}\label{th-5-4} For a topological $\mathbb{F}$-vector bundle $\xi$ on $X$, the following conditions are equivalent:
\begin{conditions} \item\label{th-5-4-a} The $\mathbb{F}$-vector bundle $\xi$ admits a stratified-algebraic structure.
\item\label{th-5-4-b} There exists a filtration $\mathcal{F} = (X_{-1}, X_0,
\ldots, X_m)$ of $X$, and for each $i=0,\ldots,m$, there exists a multiblowup $\pi_i \colon X'_i \to X_i$ over $X_{i-1}$ such that the pullback $\mathbb{F}$-vector bundle $\pi_i^*(\xi|_{X_i})$ on $X'_i$ admits a stratified-algebraic structure. \end{conditions}
\end{theorem}
\begin{proof} By Corollary~\ref{cor-5-3}, (\ref{th-5-4-a}) implies (\ref{th-5-4-b}). It suffices to prove that (\ref{th-5-4-b}) implies (\ref{th-5-4-a}). The proof is similar to the proof of Proposition~\ref{prop-5-1}.
Suppose that (\ref{th-5-4-b}) holds. Since $X$ is compact, we may assume that $\xi$ is of the form
\begin{equation*} \xi = f^* \gamma_K(\mathbb{F}^n) \end{equation*}
for some multi-Grassmannian $\mathbb{G}_K(\mathbb{F}^n)$ and continuous map $f \colon X \to \mathbb{G}_K(\mathbb{F}^n)$. It remains to prove that $f$ is homotopic to a stratified-regular map, cf. Theorem~\ref{th-4-10} (with $A = \varnothing$).
\begin{assertion} For each $i=0, \ldots, m$, there exists a continuous map $g_i \colon X \to \mathbb{G}_K(\mathbb{F}^n)$ homotopic to $f$ and such that the restriction
$g_i|_{X_i}$ is a stratified-regular map. \end{assertion}
If the Assertion holds, then the map $g_m$ is stratified-regular and the proof is complete. We prove the Assertion by induction on $i$. Since $X'_0=X_0$ and $\pi_0 \colon X'_0 \to X_0$ is the identity map, the
$\mathbb{F}$-vector bundle $\xi|_{X_0}$ on $X_0$ admits a stratified-algebraic structure. According to Proposition~\ref{prop-4-13} (with $X=X_0$ and
$A=\varnothing$), the map $f|_{X_0}$ is homotopic to a stratified-regular map $\varphi \colon X_0 \to \mathbb{G}_K(\mathbb{F}^n)$. The homotopy extension theorem implies the existence of a continuous map $g_0 \colon X \to \mathbb{G}_K(\mathbb{F}^n)$ homotopic to $f$ and satisfying $g_0|_{X_0} = \varphi$. We now suppose that $0 \leq i \leq m-1$ and the map $g_i$ satisfying the required conditions has already been constructed. In particular, the map
$g_i|_{X_i}$ is stratified-regular. Since the maps $g_i|_{X_{i+1}}$ and $f|_{X_{i+1}}$
are homotopic, the $\mathbb{F}$-vector bundles $(g_i|_{X_{i+1}})^* \gamma_K
(\mathbb{F}^n)$ and $(f|_{X_{i+1}})^* \gamma_K (\mathbb{F}^n) = \xi|_{X_{i+1}}$ are topologically isomorphic. Consequently, the $\mathbb{F}$-vector bundle
\begin{equation*}
\pi_{i+1}^* ( ( g_i|_{X_{i+1}} )^* \gamma_K (\mathbb{F}^n) ) = ( (g_i|_{X_{i+1}}) \circ \pi_{i+1} )^* \gamma_K (\mathbb{F}^n) \end{equation*}
on $X'_{i+1}$ admits a stratified-algebraic structure, being topologically isomorphic to \linebreak $\pi_{i+1}^* (\xi|_{X_{i+1}})$. According to Proposition~\ref{prop-4-13} (with $X = X_{i+1}$, $A=X_i$), the map
$g_i|_{X_{i+1}}$ is homotopic to a stratified-regular map $\psi \colon X_{i+1} \to \mathbb{G}_K(\mathbb{F}^n)$ satisfying $\psi|_{X_i} = g_i|_{X_i}$. By the homotopy extension theorem, there exists a continuous map $g_{i+1}
\colon X \to \mathbb{G}_K(\mathbb{F}^n)$ homotopic to $g_i$ (hence homotopic to $f$) and satisfying $g_{i+1}|_{X_{i+1}} = \psi$. This completes the proof of the Assertion. \end{proof}
\section{Blowups and vector bundles}\label{sec-6} We need certain constructions involving smooth (of class $\mathcal{C}^{\infty}$) manifolds. All smooth manifolds are assumed to be paracompact and without boundary. Submanifolds are supposed to be closed subsets of the ambient manifold. For any smooth manifold $M$, the total space of the tangent bundle to $M$ is denoted by $TM$. If $Z$ is a smooth submanifold of $M$, then $N_ZM$ denotes the total space of the normal bundle to $Z$ in $M$. Thus for each point $x$ in $Z$, the fiber $(N_ZM)_x$ is equal to $TM_x/TZ_x$.
Let $\theta$ be a smooth $\mathbb{R}$-vector bundle of rank $r$ on $M$, and let $s \colon M \to \theta$ be a smooth section transverse to the zero section. Then the zero locus $Z(s)$ of $s$ is a smooth submanifold of $M$, which is either empty or of codimension $r$. In order to have a convenient reference, we record the following well known fact.
\begin{lemma}\label{lem-6-1}
With notation as above, the restriction $\theta|_{Z(s)}$ is smoothly isomorphic to the normal bundle to $Z(s)$ in $M$. \end{lemma}
\begin{proof} Let $E$ denote the total space of $\theta$, and let $Z \coloneqq Z(s)$. We regard $M$ as a smooth submanifold of $E$, identifying it with the image by the zero section. We identify $\theta$ with the normal bundle to $M$ in $E$. Since $Z= s^{-1}(M)$ and $s$ is transverse to $M$ in $E$, for each point $z$ in $Z$, the differential $ds_z \colon TM_z \to TE_z$ induces an $\mathbb{R}$-linear isomorphism
\begin{equation*} (N_ZM)_z \to (N_ME)_z = E_z. \end{equation*}
The proof is complete. \end{proof}
For any smooth submanifold $D$ of $M$ of codimension $1$, there exists a smooth $\mathbb{R}$-line bundle $\lambda(D)$ on $M$ with a smooth section $s_D \colon M \to \lambda(D)$ such that $Z(s_D) = D$ and $s_D$ is transverse to the zero section. Both $\lambda(D)$ and $s_D$ are constructed as in algebraic geometry, by regarding $D$ as a $\mathcal{C}^{\infty}$ divisor on $M$. Explicitly, if $\{U_i\}$ is an open cover of $M$ and $\{ f_i \colon U_i \to \mathbb{R} \}$ is a collection of smooth local equations for $D$, then $\lambda(D)$ is determined by the transition functions $g_{ij} \coloneqq f_i / f_j \colon U_i \cap U_j \to \mathbb{R} \setminus \{0\}$, whereas $s_D$ is determined by the functions $f_i$. On the other hand, if $\lambda(D)$ and $s_D$ are already given, then $s_D$ gives rise to smooth local equations for $D$, and hence $\lambda(D)$ is uniquely determined up to smooth isomorphism. It is also convenient to set $\lambda(\varnothing) \coloneqq \varepsilon^1_M(\mathbb{R})$.
For any smooth submanifold $Z$ of $M$, we denote by
\begin{equation*} \pi(M,Z) \colon B(M,Z) \to M \end{equation*}
the blowup of $M$ with center $Z$ (cf. \cite{bib2} for basic properties of this construction). Recall that as a point set, $B(M,Z)$ is the union of $M \setminus Z$ and the total space $\mathbb{P}(N_ZM)$ of the projective bundle associated with the normal bundle to $Z$ in $M$. The map $\pi(M,Z)$ is the identity on $M \setminus Z$ and the bundle projection $\mathbb{P}(N_ZM) \to Z$ on $\mathbb{P}(N_ZM)$. On $B(M,Z)$ there is a natural smooth manifold structure, and $\pi(M,Z)$ is a smooth map. If $Z \neq \varnothing$ and $\func{codim}_M Z \geq 1$, then $\pi(M,Z)^{-1}(Z) = \mathbb{P}(N_ZM)$ is a smooth submanifold of $B(M,Z)$ of codimension $1$.
\begin{proposition}\label{prop-6-2} Let $M$ be a smooth manifold. Let $\theta$ be a smooth $\mathbb{R}$-vector bundle of positive rank on $M$, and let $s \colon M \to \theta$ be a smooth section transverse to the zero section. If $Z \coloneqq Z(s)$ and $\pi(M,Z) \colon B(M,Z) \to M$ is the blowup of $M$ with center $Z$, then the pullback $\mathbb{R}$-vector bundle $\pi(M,Z)^*\theta$ on $B(M,Z)$ contains a smooth $\mathbb{R}$-line subbundle smoothly isomorphic to $\lambda(D)$, where $D \coloneqq \pi(M,Z)^{-1}(Z)$. \end{proposition}
\begin{proof} Let $E$ be the total space of $\xi$ and let $p \colon E \to M$ be the bundle projection. We regard $M$ as a smooth submanifold of $E$ and identify the normal bundle to $M$ in $E$ with $\theta$. Thus as a point set $B(E,M)$ is the union of $E \setminus M$ and $\mathbb{P}(E)$, while
\begin{equation*} \pi(E,M) \colon B(E,M) \to E \end{equation*}
is the identity on $E \setminus M$ and the bundle projection $\mathbb{P}(E) \to M$ on $\mathbb{P}(E)$. Here $\mathbb{P}(E)$ is the total space of the projective bundle on $M$ associated with $\theta$. The pullback smooth $\mathbb{R}$-vector bundle $(p \circ \pi(E,M))^*\theta$ on
$B(E,M)$ contains a smooth $\mathbb{R}$-line subbundle $\lambda$ defined as follows. The fiber of $\lambda$ over a point $e$ in $E \setminus M$ is the line $\{e\} \times (\mathbb{R} e)$, and the restriction $\lambda|_{\mathbb{P}(E)}$ is the tautological $\mathbb{R}$-line bundle on $\mathbb{P}(E)$.
Since $s$ is transverse to $M$ in $E$, for each point $z$ in $Z$, the differential
\begin{equation*} ds_z \colon TM_z \to (TE)_z \end{equation*}
induces an $\mathbb{R}$-linear isomorphism
\begin{equation*} \bar{d}s_z \colon (N_ZM)_z \to (N_ME)_z = E_z \end{equation*}
between the fibres over $z$ of the normal bundle to $Z$ in $M$ and the normal bundle to $M$ in $E$. Define $\bar{s} \colon B(M,Z) \to B(E,M)$ by $\bar{s}(x)=x$ for all $x$ in $M \setminus Z$ and $\bar{s}(l) = \bar{d}s_z(l)$ for all $l$ in $\mathbb{P}(N_ZM)_z$ (thus, $\bar{s}(l)$ is in the fiber $\mathbb{P}(E)_z$). By construction, $\bar{s}$ is a smooth map satisfying
\begin{equation*} p \circ \pi(E,M) \circ \bar{s} = \pi(M,Z). \end{equation*}
Hence $\bar{s}^*\lambda$ is a smooth $\mathbb{R}$-line subbundle of
\begin{equation*} \bar{s}^*( (p \circ \pi(E,M) )^* \theta ) = (p \circ \pi(E,M) \circ \bar{s})^* \theta = \pi(M,Z)^*\theta. \end{equation*}
It remains to prove that the $\mathbb{R}$-line bundles $\bar{s}^*\lambda$ and $\lambda(D)$ are smoothly isomorphic. To this end, it suffices to construct a smooth section $u \colon B(M,Z) \to \bar{s}^*\lambda$ that is transverse to the zero section and satisfies $Z(u) = D$. Such a section can be obtained as follows. The smooth section $v \colon B(E,M) \to \lambda$, defined by $v(e)=(e,e)$ for all $e$ in $E \setminus M$ and
$v|_{\mathbb{P}(E)}=0$, is transverse to the zero section and satisfies $Z(v) = \mathbb{P}(E)$. On the other hand, the smooth map $\bar{s} \colon B(M,Z) \to B(E,M)$ is transverse to $\mathbb{P}(E)$ in $B(E,M)$ and $\bar{s}^{-1}(\mathbb{P}(E)) = \pi(M,Z)^{-1}(Z) = D$. Consequently, the smooth section
\begin{equation*} u \coloneqq \bar{s}^*v \colon B(M,Z) \to \bar{s}^*\lambda \end{equation*}
satisfies the required conditions. \end{proof}
Let $X$ be a nonsingular real algebraic variety. For any nonsingular Zariski closed subvariety $D$ of $X$ of codimension $1$, there exists an algebraic $\mathbb{R}$-line bundle $\lambda(D)$ on $X$ with an algebraic section $s_D \colon X \to \lambda(D)$ such that $Z(s_D)=D$ and $s_D$ is transverse to the zero section. The $\mathbb{R}$-line bundle $\lambda(D)$ is uniquely determined up to algebraic isomorphism.
Recall that any algebraic $\mathbb{R}$-vector bundle $\theta$ on $X$ has an algebraic section transverse to the zero section. Indeed, $\theta$ is generated by finitely many global algebraic sections $s_1, \ldots, s_n$, cf. \cite[Theorem~12.1.7]{bib9} or Proposition~\ref{prop-3-7}. According to the transversality theorem, for a general point $(t_1, \ldots, t_n)$ in $\mathbb{R}^n$, the algebraic section $t_1 s_1 + \cdots + t_n s_n$ is transverse to the zero section.
If $Z$ is a nonsingular Zariski closed subvariety of $X$, the $\mathcal{C}^{\infty}$ blowup
\begin{equation*} \pi(X,Z) \colon B(X,Z) \to X \end{equation*}
can be identified with the algebraic blowup, cf. \cite[Lemma~2.5.5]{bib2}.
The following consequence of Proposition~\ref{prop-6-2} will play an important role.
\begin{corollary}\label{cor-6-3} Let $X$ be a nonsingular real algebraic variety. Let $\theta$ be a smooth $\mathbb{R}$-vector bundle of positive rank on $X$, and let $s \colon X \to \theta$ be a smooth section transverse to the zero section. Assume that $Z \coloneqq Z(s)$ is a nonsingular Zariski closed subvariety of $X$. If $\pi \colon X' \to X$ is the blowup with center $Z$, then the smooth $\mathbb{R}$-vector bundle $\pi^*\theta$ on $X'$ contains a smooth $\mathbb{R}$-line subbundle $\lambda$ which is smoothly isomorphic to an algebraic $\mathbb{R}$-line bundle on $X'$. In particular, $\lambda$ admits an algebraic structure. \end{corollary}
\begin{proof} Let $D \coloneqq \pi^{-1}(Z)$. Either $D = \varnothing$ or $D$ is a nonsingular Zariski closed subvariety of $X'$ of codimension $1$. Consequently, $\lambda(D)$ is an algebraic $\mathbb{R}$-line bundle on $X'$. The proof is complete in view of Proposition~\ref{prop-6-2}. \end{proof}
There is also an algebraic-geometric counterpart of Proposition~\ref{prop-6-2}.
\begin{proposition}\label{prop-6-4} Let $X$ be a nonsingular real algebraic variety. Let $\theta$ be an algebraic $\mathbb{R}$-vector bundle of positive rank on $X$, and let $s \colon X \to \theta$ be an algebraic section transverse to the zero section. If $\pi \colon X' \to X$ is the blowup with center $Z \coloneqq Z(s)$, then the pullback $\mathbb{R}$-vector bundle $\pi^*\theta$ on $X'$ contains an algebraic $\mathbb{R}$-line subbundle which is algebraically isomorphic to $\lambda(D)$, where $D \coloneqq \pi^{-1}(Z)$. \end{proposition}
\begin{proof} The proof of Proposition~\ref{prop-6-2} can be carried over to the algebraic-geometric setting. \end{proof}
For any $\mathbb{R}$-vector bundle $\theta$, we denote by $\mathbb{C} \otimes \theta$ and $\mathbb{H} \otimes \theta$ the complexification and the quaternionification of $\theta$. In order to have the uniform notation, we also set $\mathbb{R} \otimes \theta \coloneqq \theta$. Thus, $\mathbb{F} \otimes \theta$ is an $\mathbb{F}$-vector bundle.
The following lemma leads directly to the proof of Theorem~\ref{th-1-7}.
\begin{lemma}\label{lem-6-5} Let $X$ be a compact nonsingular real algebraic variety. If $\xi$ is a topological $\mathbb{F}$-vector bundle on $X$ such that the $\mathbb{R}$-vector bundle $\xi_{\mathbb{R}}$ admits an algebraic structure, then $\xi$ admits a stratified-algebraic structure as an $\mathbb{F}$-vector bundle. \end{lemma}
\begin{proof} We may assume without loss of generality that the variety $X$ is irreducible. Then $\xi$ is of constant rank since $\xi_{\mathbb{R}}$ admits an algebraic structure. We use induction on $r \coloneqq \func{rank} \xi$. Obviously, the assertion holds if $r=0$. Suppose that $r \geq 1$ and the assertion holds for all topological $\mathbb{F}$-vector bundles of rank at most $r-1$, defined on compact nonsingular real algebraic varieties. Let $\theta$ be an algebraic $\mathbb{R}$-vector bundle on $X$ that is topologically isomorphic to $\xi_{\mathbb{R}}$. Let $s \colon X \to \theta$ be an algebraic section transverse to the zero section and let $Z \coloneqq Z(s)$. Then either $Z = \varnothing$ or $Z$ is a nonsingular Zariski closed subvariety of $X$ of codimension $d(\mathbb{F})r$. Let $\pi \colon X' \to X$ be the blowup of $X$ with center $Z$. According to Proposition~\ref{prop-6-4}, the pullback $\mathbb{R}$-vector bundle $\pi^*\theta$ on $X'$ contains an algebraic $\mathbb{R}$-line subbundle $\lambda$. Hence, $\pi^*(\xi_{\mathbb{R}}) = (\pi^*\xi)_{\mathbb{R}}$ contains a topological $\mathbb{R}$-line subbundle $\mu$ that is topologically isomorphic to $\lambda$. Since $\pi^*\xi$ is an $\mathbb{F}$-vector bundle, it contains a topological $\mathbb{F}$-line subbundle $\lambda'$ isomorphic to $\mathbb{F} \otimes \mu$. By construction, $\lambda'$ is topologically isomorphic to the algebraic $\mathbb{F}$-line bundle $\mathbb{F} \otimes \lambda$. In particular, the $\mathbb{F}$-line bundle $\lambda'$ admits an algebraic structure, and the $\mathbb{F}$-vector bundle $\pi^*\xi$ can be expressed as
\begin{equation*} \pi^* \xi = \lambda' \oplus \xi', \end{equation*}
where $\xi'$ is a topological $\mathbb{F}$-vector bundle on $X'$ of rank $r-1$. Note that the $\mathbb{R}$-vector bundle
\begin{equation*} \pi^*(\xi_{\mathbb{R}}) = (\pi^*\xi)_{\mathbb{R}} = \lambda'_{\mathbb{R}} \oplus \xi'_{\mathbb{R}} \end{equation*}
admits an algebraic structure. Since $\lambda'_{\mathbb{R}}$ admits an algebraic structure, there exists an algebraic $\mathbb{R}$-vector bundle $\eta$ on $X'$ such that the direct sum $\eta \oplus \lambda'_{\mathbb{R}}$ is topologically isomorphic to a trivial algebraic $\mathbb{R}$-vector bundle $\varepsilon$ on $X'$, cf. \cite[Theorem~12.1.7]{bib9} or Proposition~\ref{prop-3-7}. Consequently, the $\mathbb{R}$-vector bundle $\varepsilon \oplus \xi'_{\mathbb{R}}$ admits an algebraic structure, being topo\-logi\-cal\-ly isomorphic to $\eta \oplus \pi^*(\xi_{\mathbb{R}})$. According to \cite[Proposition~12.3.5]{bib9} or Corollary~\ref{cor-3-13}, the $\mathbb{R}$-vector bundle $\xi'_{\mathbb{R}}$ admits an algebraic structure. Now, by the induction hypothesis, the $\mathbb{F}$-vector bundle $\xi'$ on $X'$ admits a stratified-algebraic structure. Consequently, the $\mathbb{F}$-vector bundle $\pi^*\xi$ on $X'$ admits a stratified-algebraic structure.
Consider the topological $\mathbb{F}$-vector bundle $\xi|_Z$ on $Z$. Since the $\mathbb{R}$-vector bundle
\begin{equation*}
(\xi|_Z)_{\mathbb{R}} = (\xi_{\mathbb{R}})|_Z \end{equation*}
is topologically isomorphic to $\theta|_Z$, the construction above can be repeated with
$\xi|_Z$ substituted for $\xi$. By continuing this process, we obtain a filtration $\mathcal{F} = (X_{-1}, X_0, \ldots, X_m)$ of $X$ such that for each $i=0, \ldots, m$, the following two conditions are satisfied:
\begin{starconditions} \item $X_i$ is a nonsingular Zariski closed subvariety of $X$;
\item If $\pi_i \colon X'_i \to X_i$ is the blowup of $X_i$ with center
$X_{i-1}$, then the pullback $\mathbb{F}$-vector bundle $\pi_i^* (\xi|_{X_i})$ on $X'_i$ admits a stratified-algebraic structure. \end{starconditions}
According to Theorem~\ref{th-5-4}, the topological $\mathbb{F}$-vector bundle $\xi$ admits a stratified-algebraic structure, as required. \end{proof}
\begin{proof}[Proof of Theorem~\ref{th-1-7}] Let $\xi$ be a topological $\mathbb{F}$-vector bundle on $X$. Obviously, if $\xi$ admits a stratified-algebraic structure, then so does $\xi_{\mathbb{R}}$.
Suppose now that $\xi_{\mathbb{R}}$ admits a stratified-algebraic structure. We prove by induction on $\dim X$ that $\xi$ admits a stratified-algebraic structure. In view of Theorem~\ref{th-1-6}, there exists a birational multiblowup $\pi \colon X' \to X$ such that $X'$ is a nonsingular variety and the $\mathbb{R}$-vector bundle $\pi^*(\xi_{\mathbb{R}}) = (\pi^*\xi)_{\mathbb{R}}$ on $X'$ admits an algebraic structure. By Lemma~\ref{lem-6-5}, the $\mathbb{F}$-vector bundle $\pi^*\xi$ on $X'$ admits a stratified-algebraic structure. Let $A$ be a Zariski closed subvariety such that $\dim A < \dim X$ and $\pi$ is a multiblowup over
$A$. The $\mathbb{R}$-vector bundle $(\xi|_A)_{\mathbb{R}} = (\xi_{\mathbb{R}})|_A$ on $A$ admits a stratified-algebraic structure. Hence, by the induction hypothesis, the
$\mathbb{F}$-vector bundle $\xi|_A$ on $A$ admits a stratified-algebraic structure. Note that $\mathcal{F} = (\varnothing, A, X)$ is a filtration of $X$. According to Theorem~\ref{th-5-4}, the $\mathbb{F}$-vector bundle $\xi$ admits a stratified-algebraic structure. \end{proof}
We show next that Theorem~\ref{th-1-7} can be restated as an approximation result. If $K$ is a nonempty finite collection of nonnegative integers, we set $d(\mathbb{F})K \coloneqq \{ d(\mathbb{F})k \mid k \in K \}$. Any $\mathbb{F}$-vector subspace $V$ of $\mathbb{F}^n$ can be regarded as an $\mathbb{R}$-vector subspace of $\mathbb{R}^{d(\mathbb{F})n}$, which is indicated by $V_{\mathbb{R}}$. The correspondence $V \to V_{\mathbb{R}}$ gives rise to a regular map
\begin{equation*} i \colon \mathbb{G}_K(\mathbb{F}^n) \to \mathbb{G}_{d(\mathbb{F})K}(\mathbb{R}^{d(\mathbb{F})n}). \end{equation*}
\begin{theorem}\label{th-6-6} Let $X$ be a compact real algebraic variety. For a continuous map
\begin{equation*} f \colon X \to \mathbb{G}_K(\mathbb{F}^n), \end{equation*}
the following conditions are equivalent:
\begin{conditions} \item\label{th-6-6-a} The map $f \colon X \to \mathbb{G}_K(\mathbb{F}^n)$ can be approximated by stratified-regular maps.
\item\label{th-6-6-b} The map $i \circ f \colon X \to \mathbb{G}_{d(\mathbb{F})K} (\mathbb{R}^{d(\mathbb{F})n})$ can be approximated by stratified-regular maps. \end{conditions}
\end{theorem}
\begin{proof} Condition (\ref{th-6-6-a}) implies (\ref{th-6-6-b}), the map $i$ being regular. Since
\begin{equation*} (\gamma_K(\mathbb{F}^n))_{\mathbb{R}} = i^*( \gamma_{d(\mathbb{F})K} (\mathbb{R}^{d(\mathbb{F})n}) ), \end{equation*}
we get
\begin{equation*} ( f^* \gamma_K (\mathbb{F}^n) )_{\mathbb{R}} = f^* ( (\gamma_K(\mathbb{F}^n))_{\mathbb{R}} ) = (i \circ f)^* \gamma_{d(\mathbb{F})K} (\mathbb{R}^{d(\mathbb{F})n}). \end{equation*}
Hence, according to Theorem~\ref{th-4-10} (with $\mathbb{F}=\mathbb{R}$, $f = i \circ f$, $A = \varnothing$), condition (\ref{th-6-6-b}) implies that the $\mathbb{R}$-vector bundle $(f^*\gamma_K(\mathbb{F}^n))_{\mathbb{R}}$ admits a stratified-algebraic structure. In view of Theorem~\ref{th-1-7}, the $\mathbb{F}$-vector bundle $f^*\gamma_K(\mathbb{F}^n)$ admits a stratified-algebraic structure. Making again use of Theorem~\ref{th-4-10} (with $A = \varnothing$), we conclude that (\ref{th-6-6-b}) implies (\ref{th-6-6-a}). \end{proof}
In view of Theorem~\ref{th-4-10}, one readily sees that Theorem~\ref{th-6-6} implies Theorem~\ref{th-1-7}.
Proofs of Theorems~\ref{th-1-8} and~\ref{th-1-9} require further preparation.
\begin{lemma}\label{lem-6-7} Let $X$ be a compact nonsingular real algebraic variety and let $\xi$ be an adapted smooth $\mathbb{F}$-vector bundle on $X$. Then there exists a smooth section $s \colon X \to \xi$ transverse to the zero section and such that $Z(s)$ is a nonsingular Zariski locally closed subvariety of $X$. \end{lemma}
\begin{proof} According to the definition of an adapted vector bundle, we can choose a smooth section $u \colon X \to \xi$ transverse to the zero section and such that its zero locus $Z(u)$ is smoothly isotopic to a nonsingular Zariski locally closed subvariety $Z$ of $X$. Since any isotopy can be extended to a diffeotopy \cite[p.~180, Theorem~1.3]{bib27}, there exists a smooth diffeomorphism $h \colon X \to X$ that is homotopic to the identity map $\mathbbm{1}_X$ of $X$ and satisfies $h(Z) = Z(u)$. The pullback section $h^*u \colon X \to h^*\xi$ is transverse to the zero section and $Z(h^*u)=Z$. The proof is complete since the $\mathbb{F}$-vector bundle $h^*\xi$ is smoothly isomorphic to $\xi$, the maps $h$ and $\mathbbm{1}_X$ being homotopic. \end{proof}
\begin{lemma}\label{lem-6-8} Let $X$ be a nonsingular real algebraic variety. Let $\xi$ be a smooth $\mathbb{F}$-vector bundle on $X$, and let $s \colon X \to \xi$ be a smooth section transverse to the zero section and such that $Z \coloneqq Z(s)$ is a nonsingular Zariski locally closed subvariety of $X$. Let $V$ be the Zariski closure of $Z$ in $X$ and let $W \coloneqq V \setminus Z$. Then there exists a multiblowup $\rho \colon \tilde{X} \to X$ over $W$ such that $\tilde{Z} \coloneqq \rho^{-1}(Z)$ is a nonsingular Zariski closed subvariety of $\tilde{X}$, the pullback section $\rho^*s \colon \tilde{X} \to \rho^*\xi$ is transverse to the zero section, and
$Z(\rho^*s) = \tilde{Z}$. Furthermore, if the variety $X$ is compact and the $\mathbb{F}$-vector bundles $\xi|_W$ on $W$ and $\rho^*\xi$ on $\tilde{X}$ admit stratified-algebraic structures, then $\xi$ admits a stratified-algebraic structure. \end{lemma}
\begin{proof} Since $Z$ is closed in the Euclidean topology, it readily follows that $W$ is the singular locus of $V$. According to Hironaka's theorem on resolution of singularities \cite{bib25, bib30}, there exists a multiblowup $\rho \colon \tilde{X} \to X$ over $W$ such that the Zariski closure $\tilde{Z}$ of $\rho^{-1}(Z)$ in $\tilde{X}$ is a nonsingular subvariety. In the case under consideration, $\tilde{Z} = \rho^{-1}(Z)$, the set $Z$ being closed in the Euclidean topology. Since the restriction $\rho_W \colon \tilde{X} \setminus \rho^{-1}(W) \to X \setminus W$ of $\rho$ is a biregular isomorphism, the pullback section $\rho^*s \colon \tilde{X} \to \rho^*\xi$ is transverse to the zero section. By construction, $Z(\rho^*s) = \tilde{Z}$. Note that $\mathcal{F} = (\varnothing, W, X)$ is a filtration of $X$. The last assertion in Lemma~\ref{lem-6-8} follows from Theorem~\ref{th-5-4}. \end{proof}
\begin{proof}[Proof of Theorem~\ref{th-1-8}] Let $\xi$ be an adapted smooth $\mathbb{F}$-line bundle on $X$. According to Lemma~\ref{lem-6-7}, there exists a smooth section $s \colon X \to \xi$ transverse to the zero section and such that $Z \coloneqq Z(s)$ is a nonsingular Zariski locally closed subvariety of $X$. Let $V$ be the Zariski closure of $Z$ in $X$ and let $W \coloneqq V \setminus Z$. Since
$\xi$ is of rank $1$ and $Z(s|_W) = \varnothing$, it follows that the
$\mathbb{F}$-vector bundle $\xi|_W$ on $W$ is topologically trivial. In particular, $\xi|_W$ admits a stratified-algebraic structure.
In view of Lemma~\ref{lem-6-8}, we may assume without loss of generality that $Z$ is a nonsingular and Zariski closed subvariety of $X$ (that is, $W=\varnothing$). Let $\pi \colon X' \to X$ be the blowup with center $Z$. According to Corollary~\ref{cor-6-3}, the smooth $\mathbb{R}$-vector bundle $\pi^*(\xi_{\mathbb{R}}) = (\pi^*\xi)_{\mathbb{R}}$ on $X'$ contains a smooth $\mathbb{R}$-line subbundle $\lambda$ admitting an algebraic structure. Since $\pi^*\xi$
is an $\mathbb{F}$-line bundle, it follows that it is topologically isomorphic to $\mathbb{F} \otimes \lambda$. Consequently, $\pi^*\xi$ admits an algebraic structure. The restriction $(\xi_{\mathbb{R}})|_Z$ is smoothly isomorphic to the normal bundle to $Z$ in $X$, cf. Lemma~\ref{lem-6-1}. In particular, the $\mathbb{R}$-vector bundle $(\xi|_Z)_{\mathbb{R}} = (\xi_{\mathbb{R}})|_Z$
admits an algebraic structure. By Theorem~\ref{th-1-7}, the $\mathbb{F}$-line bundle $\xi|_Z$ admits a stratified-algebraic structure. Since $\mathcal{F} = (\varnothing, Z, X)$ is a filtration of $X$, Theorem~\ref{th-5-4} implies that $\xi$ admits a stratified-algebraic structure. \end{proof}
\begin{proof}[Proof of Theorem~\ref{th-1-9}] Assume that both $\mathbb{F}$-vectors bundles $\xi$ and $\det\xi$ are adapted. According to Theorem~\ref{th-1-8}, the $\mathbb{F}$-line bundle $\det\xi$ admits a stratified algebraic structure.
By Lemma~\ref{lem-6-7}, there exists a smooth section $s \colon X \to \xi$ transverse to the zero section and such that $Z \coloneqq Z(s)$ is a nonsingular Zariski locally closed subvariety of $X$. Let $V$ be the Zariski closure of $Z$ in $X$ and let $W \coloneqq V \setminus Z$. Since
$\xi$ is of rank $2$ and $Z(s|_W) = \varnothing$, it follows that
$\xi|_W$ is topologically isomorphic to $\varepsilon^1_W(\mathbb{F})\oplus\mu$ for some topological $\mathbb{F}$-line bundle $\mu$ on $W$. We have
\begin{equation*}
(\det\xi)|_W = \det(\xi|_W) \cong \det(\varepsilon^1_W(\mathbb{F}) \oplus \mu) \cong \varepsilon^1_W(\mathbb{F}) \otimes \mu \cong \mu. \end{equation*}
Consequently, the $\mathbb{F}$-vector bundle $\xi|_W$ on $W$ admits a stratified-algebraic structure.
In view of Lemma~\ref{lem-6-8}, we may assume without loss of generality that $Z$ is a nonsingular and Zariski closed subvariety of $X$ (that is, $W = \varnothing$). Let $\pi \colon X' \to X$ be the blowup with center $Z$. According to Corollary~\ref{cor-6-3}, the smooth $\mathbb{R}$-vector bundle $\pi^*(\xi_{\mathbb{R}}) = (\pi^*\xi)_{\mathbb{R}}$ contains a smooth $\mathbb{R}$-line subbundle $\lambda$ admitting an algebraic structure. Since $\pi^*\xi$ is an $\mathbb{F}$-vector bundle, it contains a smooth $\mathbb{F}$-line subbundle $\xi_1$ isomorphic to $\mathbb{F} \otimes \lambda$. Hence
\begin{equation*} \pi^*\xi = \xi_1 \oplus \xi_2, \end{equation*}
where $\xi_1$ and $\xi_2$ are smooth $\mathbb{F}$-line bundles, and $\xi_1$ admits an algebraic structure. We have
\begin{equation*} \pi^*(\det\xi) = \det(\pi^*\xi) = \det(\xi_1\oplus\xi_2) \cong \xi_1 \otimes \xi_2, \end{equation*}
which implies
\begin{equation*} \xi_1^{\vee} \otimes \pi^* (\det\xi) \cong \xi_1^{\vee} \otimes \xi_1 \otimes \xi_2 \cong \xi_2. \end{equation*}
Thus, according to Propositions~\ref{prop-2-6} and~\ref{prop-3-15}, $\xi_2$ admits a stratified-algebraic structure. Consequently, $\pi^*\xi$ admits a stratified-algebraic structure. The restriction
$(\xi_{\mathbb{R}})|_Z$ is smooth\-ly isomorphic to the normal bundle to $Z$ in $X$, cf. Lemma~\ref{lem-6-1}. In particular, the $\mathbb{R}$-vector bundle
$(\xi|_Z)_{\mathbb{R}} = (\xi_{\mathbb{R}})|_Z$ admits an algebraic structure. By Theorem~\ref{th-1-7}, the $\mathbb{F}$-vector bundle $\xi|_Z$ admits a stratified-algebraic structure. Since $\mathcal{F} = (\varnothing, Z, X)$ is a filtration of $X$, Theorem~\ref{th-5-4} implies that $\xi$ admits a stratified-algebraic structure. \end{proof}
There are natural generalizations of Theorems~\ref{th-1-8} and~\ref{th-1-9} to vector bundles on an arbitrary compact real algebraic variety (not necessarily nonsingular).
First we recall a topological fact. If $M$ is a smooth manifold, then each topological $\mathbb{F}$-vector bundle $\eta$ on $M$ is topologically isomorphic to a smooth $\mathbb{F}$-vector bundle $\eta^{\infty}$, which is uniquely determined up to smooth isomorphism, cf. \cite[p.~101, Theorem~3.5]{bib27}.
A topological $\mathbb{F}$-vector bundle $\xi$ on a compact nonsingular real algebraic variety $Y$ is said to be \emph{adapted} if the smooth $\mathbb{F}$-vector bundle $\xi^{\infty}$ is adapted. In particular, $\xi$ is adapted if it admits an algebraic structure.
\begin{theorem}\label{th-6-9} Let $X$ be a compact real algebraic variety and let $\xi$ be a topological $\mathbb{F}$-line bundle on $X$. Then the following conditions are equivalent:
\begin{conditions} \item\label{th-6-9-a} The $\mathbb{F}$-line bundle $\xi$ admits a stratified-algebraic structure.
\item\label{th-6-9-b} There exists a filtration $\mathcal{F} = (X_{-1}, X_0, \ldots, X_m)$ of $X$ with $\overline{\mathcal{F}}$ a nonsingular strati\-fi\-ca\-tion, and for each $i = 0, \ldots, m$, there exists a multiblowup $\pi_i \colon X'_i \to X_i$
over $X_{i-1}$ such that $X'_i$ is a nonsingular variety and the pullback $\mathbb{F}$-line bundle $\pi_i^*(\xi|_{X_i})$ on $X'_i$ is adapted. \end{conditions}
\end{theorem}
\begin{proof} According to Corollary~\ref{cor-5-3}, (\ref{th-6-9-a}) implies (\ref{th-6-9-b}). In view of Theorems~\ref{th-5-4} and~\ref{th-1-8}, (\ref{th-6-9-b}) implies (\ref{th-6-9-a}). \end{proof}
\begin{theorem}\label{th-6-10} Let $X$ be a compact real algebraic variety and let $\xi$ be a topological $\mathbb{F}$-vector bundle of rank $2$ on $X$, where $\mathbb{F}=\mathbb{R}$ of $\mathbb{F}=\mathbb{C}$. Then the following conditions are equivalent:
\begin{conditions} \item\label{th-6-10-a} The $\mathbb{F}$-vector bundle $\xi$ admits a stratified-algebraic structure.
\item\label{th-6-10-b} There exists a filtration $\mathcal{F} = (X_{-1}, X_0, \ldots, X_m)$ of $X$ with $\overline{\mathcal{F}}$ a nonsingular stratification, and for each $i = 0, \ldots, m$, there exists a multiblowup $\pi_i \colon X'_i \to X_i$ over $X_{i-1}$ such that $X'_i$ is a nonsingular variety and the pullback $\mathbb{F}$-vector bundles
$\pi_i^*(\xi|_{X_i})$ and $\pi_i^*((\det\xi)|_{X_i})$ on $X'_i$ are adapted. \end{conditions}
\end{theorem}
\begin{proof} According to Corollary~\ref{cor-5-3}, (\ref{th-6-10-a}) implies (\ref{th-6-10-b}). In view of Theorems~\ref{th-5-4} and~\ref{th-1-9}, (\ref{th-6-10-b}) implies (\ref{th-6-10-a}). \end{proof}
\section{Stratified-algebraic cohomology classes}\label{sec-7} To begin with, we summarize basic properties of algebraic cohomology classes. Let $V$ be a compact nonsingular real algebraic variety. A cohomology class $v$ in $H^k(V;\mathbb{Z}/2)$ is said to be \emph{algebraic} if the homology class Poincar\'e dual to it can be represented by a Zariski closed subvariety of $V$ of codimension $k$. The set $H_{\mathrm{alg}}^k (V; \mathbb{Z}/2)$ of all algebraic cohomology classes in $H^k(V; \mathbb{Z}/2)$ forms a subgroup. The direct sum $H_{\mathrm{alg}}^*(V; \mathbb{Z}/2)$ of the groups $H_{\mathrm{alg}}^k (V; \mathbb{Z}/2)$, for $k \geq 0$, is a subring of the cohomology ring $H^* (V; \mathbb{Z}/2)$. For any algebraic $\mathbb{R}$-vector bundle $\xi$ on $V$, its $k$th Stiefel--Whitney class $w_k(\xi)$ belongs to $H_{\mathrm{alg}}^k(V; \mathbb{Z}/2)$ for every $k \geq 0$. If $h \colon V \to W$ is a regular map between compact nonsingular real algebraic varieties, then
\begin{equation*} h^*(H_{\mathrm{alg}}^*(W; \mathbb{Z}/2)) \subseteq H_{\mathrm{alg}}^* (V; \mathbb{Z}/2). \end{equation*}
For proofs of the facts listed above we refer to \cite{bib20} or \cite{bib1, bib6, bib9, bib17}.
We now introduce the main notion of this section. Let $X$ be a real algebraic variety. A cohomology class $u$ in $H^k(X; \mathbb{Z}/2)$ is said to be \emph{stratified-algebraic} if there exists a stratified-regular map $\varphi \colon X \to V$, into a compact nonsingular real algebraic variety $V$, such that $u = \varphi^*(v)$ for some cohomology class $v$ in $H_{\mathrm{alg}}^k (V; \mathbb{Z}/2)$.
\begin{proposition}\label{prop-7-1} For any real algebraic variety $X$, the set $H_{\mathrm{str}}^k (X; \mathbb{Z}/2)$ of all stratified-algebraic cohomology classes in $H^k(X; \mathbb{Z}/2)$ forms a subgroup. Furthermore, the direct sum
\begin{equation*} H_{\mathrm{str}}^* (X; \mathbb{Z}/2) = \bigoplus_{k \geq 0} H_{\mathrm{str}}^k (X; \mathbb{Z}/2) \end{equation*}
is a subring of the cohomology ring $H^*(X;\mathbb{Z}/2)$. \end{proposition}
\begin{proof} Let $\varphi_i \colon X \to V_i$ be a stratified-regular map into a compact nonsingular real algebraic variety $V_i$ for $i=1,2$. The stratified-regular map $(\varphi_1, \varphi_2) \colon X \to V_1 \times V_2$ satisfies $p_i \circ (\varphi_1, \varphi_2) = \varphi_i$, where $p_i \colon V_1 \times V_2 \to V_i$ is the canonical projection. If $v_i$ (resp. $w_i$) is a cohomology class in $H^k(V_i; \mathbb{Z}/2)$ (resp. $H^{k_i} (V_i; \mathbb{Z}/2)$) for $i=1,2$, then
\begin{align*} \varphi_1^*(v_1) + \varphi_2^*(v_2) &= (\varphi_1, \varphi_2)^* (p_1^*(v_1) + p_2^*(v_2)), \\ \varphi_1^*(w_1) \mathbin{\smile} \varphi_2^*(w_2) &= (\varphi_1, \varphi_2)^* (p_1^*(w_1) \mathbin{\smile} p_2^*(w_2)), \end{align*}
where $\mathbin{\smile}$ stands for the cup product. If $v_i$ and $w_i$ are algebraic cohomology classes, then the cohomology classes $p_1^*(v_1) + p_2^*(v_2)$ and $p_1^*(w_1) \mathbin{\smile} p_2^*(w_2)$ are algebraic too. The proof is complete. \end{proof}
\begin{proposition}\label{prop-7-2} If $f \colon X \to Y$ is a stratified-regular map between real algebraic varieties, then
\begin{equation*} f^*(H_{\mathrm{str}}^*(Y; \mathbb{Z}/2)) \subseteq H_{\mathrm{str}}^* (X; \mathbb{Z}/2). \end{equation*}
\end{proposition}
\begin{proof} Let $\psi \colon Y \to W$ be a stratified-regular map into a compact nonsingular real algebraic variety $W$. For any cohomology class $w$ in $H^k(W; \mathbb{Z}/2)$,
\begin{equation*} f^*(\psi^*(w)) = (\psi \circ f)^*(w). \end{equation*}
The proof is complete since $\psi \circ f$ is a stratified-regular map. \end{proof}
\begin{proposition}\label{prop-7-3} Let $\xi$ be a stratified-algebraic $\mathbb{R}$-vector bundle on a real algebraic variety $X$. Then the $k$th Stiefel--Whitney class $w_k(\xi)$ of $\xi$ belongs to $H_{\mathrm{str}}^k(X; \mathbb{Z}/2)$ for every $k \geq 0$. \end{proposition}
\begin{proof} According to Proposition~\ref{prop-3-4}, we may assume that $\xi$ is of the form $\xi = f^*\gamma_K(\mathbb{R}^n)$ for some multi-Grassmannian $\mathbb{G}_K(\mathbb{R}^n)$ and stratified-regular map $f \colon X \to \mathbb{G}_K(\mathbb{R}^n)$. Then $w_k(\xi) = f^*(w_k (\gamma_K (\mathbb{R}^n) ) )$. The proof is complete in view of Proposition~\ref{prop-7-2} since
\begin{equation*} H_{\mathrm{alg}}^* (\mathbb{G}_K(\mathbb{R}^n); \mathbb{Z}/2) = H^*(\mathbb{G}_K(\mathbb{R}^n); \mathbb{Z}/2), \end{equation*}
cf. \cite[Proposition~11.3.3]{bib9}. \end{proof}
For any real algebraic variety $X$, let $\func{VB}^1_{\mathbb{R}}(X)$ denote the group of isomorphism classes of topological $\mathbb{R}$-line bundles on $X$ (with operation induced by tensor product). The first Stiefel--Whitney class gives rise to an isomorphism
\begin{equation*} w_1 \colon \func{VB}^1_{\mathbb{R}}(X) \to H^1(X; \mathbb{Z}/2). \end{equation*}
Similarly, denote by $\VB_{\R\mhyphen\mathrm{str}}^1(X)$ the group of isomorphism classes of stratified-algebraic $\mathbb{R}$-line bundles on $X$. Now, the first Stiefel--Whitney class gives rise to a homomorphism
\begin{equation*} w_1 \colon \VB_{\R\mhyphen\mathrm{str}}^1(X) \to H^1(X; \mathbb{Z}/2). \end{equation*}
\begin{proposition}\label{prop-7-4} For any real algebraic variety $X$,
\begin{equation*} w_1(\VB_{\R\mhyphen\mathrm{str}}^1(X)) = H_{\mathrm{str}}^1(X; \mathbb{Z}/2). \end{equation*}
Furthermore, if the variety $X$ is compact, then the homomorphism
\begin{equation*} w_1 \colon \VB_{\R\mhyphen\mathrm{str}}^1(X) \to H^1(X; \mathbb{Z}/2) \end{equation*}
is injective. \end{proposition}
\begin{proof} The inclusion
\begin{equation*} w_1(\VB_{\R\mhyphen\mathrm{str}}^1(X)) \subseteq H_{\mathrm{str}}^1(X; \mathbb{Z}/2) \end{equation*}
follows from Proposition~\ref{prop-7-3}. If $u$ is a cohomology class in $H_{\mathrm{str}}^1(X; \mathbb{Z}/2)$, then there exists a stratified-regular map $\varphi \colon X \to V$, into a compact nonsingular real algebraic variety $V$, such that $u = \varphi^*(v)$ for some cohomology class $v$ in $H_{\mathrm{alg}}^1(V; \mathbb{Z}/2)$. By \cite[Theorem~12.4.6]{bib9}, $v$ is of the form $v = w_1(\lambda)$ for some algebraic $\mathbb{R}$-line bundle $\lambda$ on $V$. The pullback $\mathbb{R}$-line bundle $\varphi^*\lambda$ on $X$ is stratified-algebraic, cf. Proposition~\ref{prop-2-6}. Since $u = \varphi^*(w_1(\lambda)) = w_1(\varphi^*\lambda)$, we get
\begin{equation*} H_{\mathrm{str}}^1(X; \mathbb{Z}/2) \subseteq w_1(\VB_{\R\mhyphen\mathrm{str}}^1(X)). \end{equation*}
If the variety $X$ is compact, then the canonical homomorphism $\VB_{\R\mhyphen\mathrm{str}}^1(X) \to \func{VB}_{\mathbb{R}}^1(X)$ is injective (cf. Theorem~\ref{th-3-10}), and hence the last assertion in the proposition follows. \end{proof}
Proposition~\ref{prop-7-4} can be interpreted as an approximation result for maps into real projective $n$-space $\mathbb{P}^n(\mathbb{R}) = \mathbb{G}_1(\mathbb{R}^{n+1})$ with $n \geq 1$.
\begin{proposition}\label{prop-7-5} Let $X$ be a compact real algebraic variety. For a continuous map $f \colon X \to \mathbb{P}^n(\mathbb{R})$, the following conditions are equivalent:
\begin{conditions} \item The map $f$ can be approximated by stratified-regular maps.
\item The cohomology class $f^*(u_n)$ belongs to $H_{\mathrm{str}}^1 (X; \mathbb{Z}/2)$, where $u_n$ is a generator of the group $H^1(\mathbb{P}^n(\mathbb{R}); \mathbb{Z}/2) \cong \mathbb{Z}/2$. \end{conditions}
\end{proposition}
\begin{proof} Since $u_n = w_1(\gamma_1(\mathbb{R}^{n+1}))$, we have $f^*(u_n) = w_1(f^*\gamma_1(\mathbb{R}^{n+1}))$. Thus, it suffices to combine Theorem~\ref{th-4-10} (with $A=\varnothing$) and Proposition~\ref{prop-7-4}. \end{proof}
\begin{corollary}\label{cor-7-6} Let $X$ be a compact real algebraic variety. For a continuous map \linebreak $f \colon X \to \mathbb{S}^1$, the following conditions are equivalent:
\begin{conditions} \item The map $f$ can be approximated by stratified-regular maps.
\item The cohomology class $f^*(u)$ belongs to $H_{\mathrm{str}}^1(X; \mathbb{Z}/2)$, where $u$ is a generator of the group $H^1(\mathbb{S}^1; \mathbb{Z}/2) \cong \mathbb{Z}/2$. \end{conditions}
\end{corollary}
\begin{proof} It suffices to apply Proposition~\ref{prop-7-5} since $\mathbb{S}^1$ is biregularly isomorphic to $\mathbb{P}^1(\mathbb{R})$. \end{proof}
We next look for relationships between the groups $H_{\mathrm{str}}^k (-; \mathbb{Z}/2)$ and $H_{\mathrm{alg}}^k(-; \mathbb{Z}/2)$.
\begin{proposition}\label{prop-7-7} If $X$ is a compact nonsingular real algebraic variety, then
\begin{equation*} H_{\mathrm{str}}^*(X; \mathbb{Z}/2) = H_{\mathrm{alg}}^*(X; \mathbb{Z}/2). \end{equation*}
\end{proposition}
\begin{proof} The inclusion
\begin{equation*} H_{\mathrm{alg}}^*(X; \mathbb{Z}/2) \subseteq H_{\mathrm{str}}^*(X; \mathbb{Z}/2) \end{equation*}
is obvious. According to \cite[Propostion~1.3]{bib32}, if $\varphi \colon X \to V$ is a continuous rational map into a compact nonsingular real algebraic variety $V$, then
\begin{equation*} \varphi^* ( H_{\mathrm{alg}}^*(V; \mathbb{Z}/2) ) \subseteq H_{\mathrm{alg}}^*(X; \mathbb{Z}/2). \end{equation*}
Consequently, the inclusion
\begin{equation*} H_{\mathrm{str}}^* (X; \mathbb{Z}/2) \subseteq H_{\mathrm{alg}}^*(X; \mathbb{Z}/2) \end{equation*}
follows since each stratified-regular map from $X$ into $V$ is continuous rational, cf. Proposition~\ref{prop-2-2}. \end{proof}
For an arbitrary compact real algebraic variety $X$, the group $H_{\mathrm{str}}^1(X; \mathbb{Z}/2)$ can be described as follows.
\begin{proposition}\label{prop-7-8} Let $X$ be a compact real algebraic variety. For a cohomology class $u$ in $H^1(X; \mathbb{Z}/2)$, the following conditions are equivalent:
\begin{conditions} \item The cohomology class $u$ belongs to $H_{\mathrm{str}}^1(X; \mathbb{Z}/2)$.
\item There exists a filtration $\mathcal{F} = (X_{-1}, X_0, \ldots, X_m)$ of $X$ with $\overline{\mathcal{F}}$ a nonsingular strati\-fication, and for each $i =0, \ldots, m$, there exists a multiblowup $\pi_i \colon X'_i \to X_i$
over $X_{i-1}$ such that $X'_i$ is a nonsingular variety and the cohomology class $\pi_i^*(u|_{X_i})$ belongs to $H_{\mathrm{alg}}^1(X'_i; \mathbb{Z}/2)$. Here $u|_{X_i}$ is the image of $u$ under the homomorphism \linebreak $H^1(X; \mathbb{Z}/2) \to H^1(X_i; \mathbb{Z}/2)$ induced by the inclusion map $X_i \hookrightarrow X$. \end{conditions}
\end{proposition}
\begin{proof} Let $\mathcal{F} = (X_{-1}, X_0, \ldots, X_m)$ be a filtration of $X$ with $\overline{\mathcal{F}}$ a nonsingular stratification, and for each $i=0,\ldots,m$, let $\pi_i \colon X'_i \to X_i$ be a multiblowup over $X_{i-1}$ such that $X'_i$ is a nonsingular variety. If $\xi$ is a topological $\mathbb{R}$-line bundle on $X$ with $w_1(\xi)=u$, then
$w_1(\xi|_{X_i}) = u|_{X_i}$ and $w_1(\pi_i^*(\xi|_{X_i})) =
\pi_i^*(u|_{X_i})$ for $0 \leq i \leq m$. In view of
\cite[Theorem~12.4.6]{bib9}, the $\mathbb{R}$-line bundle $\pi_i^*(\xi|_{X_i})$
on $X'_i$ admits an algebraic structure if and only if the cohomology class $\pi_i^*(u|_{X_i})$ belongs to $H_{\mathrm{alg}}^1(X'_i; \mathbb{Z}/2)$. Consequently, it suffices to combine Corollary~\ref{cor-5-3} (with $\mathbb{F}=\mathbb{R}$) and Propositions~\ref{prop-7-4} and~\ref{prop-7-7}. \end{proof}
In \cite{bib19, bib-star}, algebraic cohomology classes are interpreted as obstructions to representing homotopy classes by regular maps. Some results contained in these papers can be strengthened by transferring them to the framework of stratified objects, cf. Theorem~\ref{th-7-9} below. First it is convenient to introduce some notation.
For any $n$-dimensional compact smooth manifold $M$, let $\llbracket M \rrbracket$ denote its fundamental class in $H_n(M; \mathbb{Z}/2)$. If $M$ is a smooth submanifold of a smooth manifold $N$, let $\llbracket M \rrbracket_N$ denote the homology class in $H_n(N; \mathbb{Z}/2)$ represented by $M$.
\begin{theorem}\label{th-7-9} Let $Y$ be a real algebraic variety with $H_{\mathrm{str}}^k(Y; \mathbb{Z}/2) \neq 0$ for some ${k \geq 1}$. Then there exist a compact connected nonsingular real algebraic variety $X$ and a continuous map $f \colon X \to Y$ such that $\dim X = k+1$ and $f$ is not homotopic to any stratified-regular map. \end{theorem}
\begin{proof} Let $w$ be a nonzero cohomology class in $H_{\mathrm{str}}^k(Y; \mathbb{Z}/2)$. Choose a homology class $\beta$ in $H_k(Y; \mathbb{Z}/2)$ which satisfies
\begin{equation*} \langle w, \beta \rangle \neq 0 \end{equation*}
and is represented by a singular cycle with support contained in one connected component of $Y$. By \cite[Theorem~2.1]{bib19}, $\beta$ can be expressed as
\begin{equation*} \beta = h_*(\llbracket K \rrbracket), \end{equation*}
where $K$ is a $k$-dimensional compact connected smooth manifold that is the boundary of a compact smooth manifold with boundary and $h \colon K \to Y$ is a continuous map. Let $p_0$ be a point in $\mathbb{S}^1$. Since the normal bundle of $K \times \{p_0\}$ in $K \times \mathbb{S}^1$ is trivial, according to \cite[Theorem~2.6, Proposition~2.5]{bib19}, there exist a nonsingular real algebraic variety $X$ and a smooth diffeomorphism $\varphi \colon X \to K \times \mathbb{S}^1$ such that the homology class $\alpha \coloneqq \llbracket \varphi^{-1} (K \times \{p_0\}) \rrbracket_X$ satisfies
\begin{equation*} \langle u, \alpha \rangle = 0 \end{equation*}
for every cohomology class $u$ in $H_{\mathrm{alg}}^k(X; \mathbb{Z}/2)$.
Let $\pi \colon K \times \mathbb{S}^1 \to K$ be the canonical projection. It suffices to prove that the continuous map
\begin{equation*} f \coloneqq h \circ \pi \circ \varphi \colon X \to Y \end{equation*}
is not homotopic to any regular map. This can be done as follows. We have $w = \psi^*(v)$ for some stratified-regular map $\psi \colon Y \to W$ into a compact nonsingular real algebraic variety $V$ and a cohomology class $v$ in $H_{\mathrm{alg}}^k(V; \mathbb{Z}/2)$. Since $(\pi \circ \varphi)_* (\alpha) = \pi_* (\varphi_* (\alpha) ) = \llbracket K \rrbracket$, we get
\begin{equation*} (\psi \circ f)_* (\alpha) = (\psi \circ h \circ \pi \circ \varphi)_* (\alpha) = \psi_* (h_* (\llbracket K \rrbracket) ) =\psi_*(\beta). \end{equation*}
Consequently,
\begin{equation*} \langle (\psi \circ f)^*(v), \alpha \rangle = \langle v, (\psi \circ f)_* (\alpha) \rangle = \langle v, \psi_*(\beta) \rangle = \langle \psi^*(v), \beta \rangle = \langle w, \beta \rangle \neq 0. \end{equation*}
It follows that the cohomology class $(\psi \circ f)^*(v)$ does not belong to $H_{\mathrm{alg}}^k(X; \mathbb{Z}/2)$. However, if the map $f \colon X \to Y$ were homotopic to a stratified-regular map $g \colon X \to Y$, then $\psi \circ f$ would be homotopic to a stratified-regular map $\psi \circ g$, and hence the cohomology class $(\psi \circ f)^* (v) = (\psi \circ g)^* (v)$ would be in $H_{\mathrm{alg}}^k(X; \mathbb{Z}/2)$, cf. Propositions~\ref{prop-7-2} and~\ref{prop-7-7}. This completes the proof. \end{proof}
We conclude this section by giving an example of an $\mathbb{F}$-line bundle not admitting a stratified-algebraic structure. Recall that $d(\mathbb{F}) = \dim_{\mathbb{R}} \mathbb{F}$ and $\mathbb{T}^n = \mathbb{S}^1 \times \cdots \times \mathbb{S}^1$ (the $n$-fold product).
\begin{example}\label{ex-7-10} For any integer $n > d(\mathbb{F})$, there exist a nonsingular real algebraic variety $X$ and a topological $\mathbb{F}$-line bundle $\xi$ on $X$ such that $X$ is diffeomorphic to $\mathbb{T}^n$ and $\xi$ does not admit a stratified-algebraic structure. This assertion can be proved as follows. Let $d \coloneqq d(\mathbb{F})$ and let $y_0$ be a point in $\mathbb{T}^{n-d}$. Let $\alpha$ be the homology class in $H_d(\mathbb{T}^n; \mathbb{Z}/2)$ represented by the smooth submanifold $K \coloneqq \mathbb{T}^d \times \{y_0\}$ of $\mathbb{T}^n$. Set
\begin{equation*} A \coloneqq \{ u \in H^d(\mathbb{T}^n; \mathbb{Z}/2) \mid \langle u, \alpha \rangle = 0 \}. \end{equation*}
Since the normal bundle of $K$ in $\mathbb{T}^n$ is trivial and $K$ is the boundary of a compact smooth manifold with boundary, it follows from \cite[Propositon~2.5, Theorem~2.6]{bib19} that there exist a nonsingular real algebraic variety $X$ and a smooth diffeomorphism $\varphi \colon X \to \mathbb{T}^n$ with
\begin{equation*} H_{\mathrm{alg}}^d(X; \mathbb{Z}/2) \subseteq \varphi^*(A). \end{equation*}
For the $\mathbb{F}$-line bundle $\gamma_1(\mathbb{F}^2)$ on $\mathbb{G}_1(\mathbb{F}^2)$, one has $w_d(\gamma_1(\mathbb{F}^2)_{\mathbb{R}}) \neq 0$ in $H^d(\mathbb{G}_1(\mathbb{F}^2); \mathbb{Z}/2)$. Choosing a continuous map $h \colon \mathbb{T}^d \to \mathbb{G}_1(\mathbb{F}^2)$ for which the induced homomorphism
\begin{equation*} h^* \colon H^d(\mathbb{G}_1(\mathbb{F}^2); \mathbb{Z}/2) \to H^d(\mathbb{T}^d; \mathbb{Z}/2) \end{equation*}
is an isomorphism, we obtain a topological $\mathbb{F}$-line bundle $\lambda \coloneqq h^* \gamma_1 (\mathbb{F}^2)$ with $w_d(\lambda_{\mathbb{R}}) \neq 0$ in $H^d(\mathbb{T}^d; \mathbb{Z}/2)$. If $p \colon \mathbb{T}^n = \mathbb{T}^d \times \mathbb{T}^{n-d} \to \mathbb{T}^d$ is the canonical projection and $\eta \coloneqq p^*\lambda$, then
\begin{equation*} w_d(\eta_{\mathbb{R}}) \notin A. \end{equation*}
Consequently, for the $\mathbb{F}$-line bundle $\xi \coloneqq \varphi^* \eta$ on $X$, we have
\begin{equation*} w_d(\xi_{\mathbb{R}}) \notin H_{\mathrm{alg}}^d(X; \mathbb{Z}/2). \end{equation*}
In view of Propositions~\ref{prop-7-3} and~\ref{prop-7-7}, the $\mathbb{R}$-vector bundle $\xi_{\mathbb{R}}$ cannot admit a stratified-algebraic structure. Hence $\xi$ does not admit a stratified-algebraic structure. \end{example}
\section{\texorpdfstring{Stratified-$\mathbb{C}$-algebraic cohomology classes}{Stratified-C-algebraic cohomology classes}}\label{sec-8}
We first recall the construction of $\mathbb{C}$-algebraic cohomology classes. Let $V$ be a compact nonsingular real algebraic variety. A \emph{nonsingular projective complexification} of $V$ is a pair $(\mathbb{V}, \iota)$, where $\mathbb{V}$ is a nonsingular projective scheme over $\mathbb{R}$ and $\iota \colon V \to \mathbb{V}(\mathbb{C})$ is an injective map such that $\mathbb{V}(\mathbb{R})$ is Zariski dense in $\mathbb{V}$, $\iota(V) = \mathbb{V}(\mathbb{R})$ and $\iota$ induces a biregular isomorphism between $V$ and $\mathbb{V}(\mathbb{R})$. Here the set $\mathbb{V}(\mathbb{R})$ of real points of $\mathbb{V}$ is regarded as a subset of the set $\mathbb{V}(\mathbb{C})$ of complex points of $\mathbb{V}$. The existence of $(\mathbb{V}, \iota)$ follows from Hironaka's theorem on resolution of singularities \cite{bib26, bib30}. We identify $\mathbb{V}(\mathbb{C})$ with the set of complex points of the scheme $\mathbb{V}_{\mathbb{C}} \coloneqq \mathbb{V} \times_{\func{Spec} \mathbb{R}} \func{Spec} \mathbb{C}$ over $\mathbb{C}$. The cycle map
\begin{equation*} \func{cl}_{\mathbb{C}} \colon A^*(\mathbb{V}_{\mathbb{C}}) = \bigoplus_{k \geq 0} A^k(\mathbb{V}_{\mathbb{C}}) \to H^{\mathrm{even}} (\mathbb{V}(\mathbb{C});\mathbb{Z}) = \bigoplus_{k \geq 0} H^{2k}(\mathbb{V}(\mathbb{C}); \mathbb{Z}) \end{equation*}
is a ring homomorphism defined on the Chow ring of $\mathbb{V}_{\mathbb{C}}$, cf. \cite{bib20} or \cite[Corollary~19.2]{bib24}. Hence
\begin{equation*} H_{\mathrm{alg}}^{2k}(\mathbb{V}(\mathbb{C}); \mathbb{Z}) \coloneqq \func{cl}_{\mathbb{C}}(A^k(\mathbb{V}_{\mathbb{C}})) \end{equation*}
is the subgroup of $H^{2k}(\mathbb{V}(\mathbb{C});\mathbb{Z})$ that consists of the cohomology classes corresponding to algebraic cycles (defined over $\mathbb{C}$) on $\mathbb{V}_{\mathbb{C}}$ of codimension $k$. By construction,
\begin{equation*} H_{\CB\mhyphen\mathrm{alg}}^{2k} (V; \mathbb{Z}) \coloneqq \iota^* (H_{\mathrm{alg}}^{2k} (\mathbb{V}(\mathbb{C}); \mathbb{Z}) ) \end{equation*}
is a subgroup of $H^{2k}(V;\mathbb{Z})$, and
\begin{equation*} H_{\CB\mhyphen\mathrm{alg}}^{\mathrm{even}} (V; \mathbb{Z}) \coloneqq \bigoplus_{k \geq 0} H_{\CB\mhyphen\mathrm{alg}}^{2k} (V; \mathbb{Z}) \end{equation*}
is a subring of $H^{\mathrm{even}}(V;\mathbb{Z})$. The subring $H_{\CB\mhyphen\mathrm{alg}}^{\mathrm{even}} (V; \mathbb{Z})$ does not depend on the choice of $(\mathbb{V}, \iota)$. If the variety $V$ is irreducible, then $H_{\CB\mhyphen\mathrm{alg}}^0(V; \mathbb{Z})$ is the subgroup generated by $1 \in H^0(V;\mathbb{Z})$. A cohomology class $v$ in $H^{2k}(V;\mathbb{Z})$ is said to be \emph{$\mathbb{C}$-algebraic} if it belongs to $H_{\CB\mhyphen\mathrm{alg}}^{2k}(V;\mathbb{Z})$. For any algebraic $\mathbb{C}$-vector bundle $\xi$ on $V$, its $k$th Chern class $c_k(\xi)$ is in $H_{\CB\mhyphen\mathrm{alg}}^{2k}(V;\mathbb{Z})$ for every $k \geq 0$. If $h \colon V \to W$ is a regular map between compact nonsingular real algebraic varieties, then
\begin{equation*} h^*(H_{\CB\mhyphen\mathrm{alg}}^{\mathrm{even}}(W;\mathbb{Z})) \subseteq H_{\CB\mhyphen\mathrm{alg}}^{\mathrm{even}}(V;\mathbb{Z}). \end{equation*}
Proofs of these properties of $H_{\CB\mhyphen\mathrm{alg}}^{\mathrm{even}}(-;\mathbb{Z})$ are given in \cite{bib8}. Numerous applications of $\mathbb{C}$-algebraic cohomology classes can be found in \cite{bib8, bib12, bib16, bib18, bib33}.
Let $X$ be a real algebraic variety. A cohomology class $u$ in $H^{2k}(X;\mathbb{Z})$ is said to be \emph{stratified-$\mathbb{C}$-algebraic} if there exists a stratified-regular map $\varphi \colon X \to V$, into a compact nonsingular real algebraic variety $V$, such that $u = \varphi^*(v)$ for some cohomology class $v$ in $H_{\CB\mhyphen\mathrm{alg}}^{2k}(V;\mathbb{Z})$.
\begin{proposition}\label{prop-8-1} For any real algebraic variety $X$, the set $H_{\CB\mhyphen\mathrm{str}}^{2k}(X;\mathbb{Z})$ of all stratified-$\mathbb{C}$-algebraic cohomology classes in $H^{2k}(V;\mathbb{Z})$ forms a subgroup. Furthermore, the direct sum
\begin{equation*} H_{\CB\mhyphen\mathrm{str}}^{\mathrm{even}} (X;\mathbb{Z}) \coloneqq \bigoplus_{k \geq 0} H_{\CB\mhyphen\mathrm{str}}^{2k} (X;\mathbb{Z}) \end{equation*}
is a subring of $H^{\mathrm{even}}(X;\mathbb{Z})$. \end{proposition}
\begin{proof} It suffices to adapt the proof of Proposition~\ref{prop-7-1}. \end{proof}
\begin{proposition}\label{prop-8-2} If $f \colon X \to Y$ is a stratified-regular map between real algebraic varieties, then
\begin{equation*} f^*(H_{\CB\mhyphen\mathrm{str}}^{\mathrm{even}}(Y;\mathbb{Z})) \subseteq H_{\CB\mhyphen\mathrm{str}}^{\mathrm{even}}(X;\mathbb{Z}). \end{equation*}
\end{proposition}
\begin{proof} The proof of Proposition~\ref{prop-7-2} also works in the case under consideration here. \end{proof}
\begin{proposition}\label{prop-8-3} Let $\xi$ be a stratified-algebraic $\mathbb{C}$-vector bundle on a real algebraic variety $X$. Then the $k$th Chern class $c_k(\xi)$ of $\xi$ belongs to $H_{\CB\mhyphen\mathrm{str}}^{2k}(X; \mathbb{Z})$ for every $k \geq 0$. \end{proposition}
\begin{proof} One can copy the proof of Proposition~\ref{prop-7-3}. It suffices to observe that
\begin{equation*} H_{\CB\mhyphen\mathrm{alg}}^{\mathrm{even}} (\mathbb{G}_K(\mathbb{C}^n); \mathbb{Z}) = H^{\mathrm{even}} (\mathbb{G}_K(\mathbb{C}^n); \mathbb{Z}), \end{equation*}
cf. \cite[Example~5.5]{bib33}. \end{proof}
\begin{corollary}\label{cor-8-4} Let $\xi$ be a stratified-regular $\mathbb{R}$-vector bundle on a real algebraic variety $X$. Then the $k$th Pontryagin class $p_k(\xi)$ of $\xi$ belongs to $H_{\CB\mhyphen\mathrm{str}}^{4k}(X; \mathbb{Z})$ for every $k \geq 0$. \end{corollary}
\begin{proof} Since $p_k(\xi) = (-1)^k c_{2k}(\mathbb{C} \otimes \xi)$, is suffices to make use of Proposition~\ref{prop-8-3}. \end{proof}
For any topological $\mathbb{F}$-vector bundle $\xi$ on $X$, one can interpret $\func{rank} \xi$ as an element of $H^0(X;\mathbb{Z})$. Then the following holds.
\begin{proposition}\label{prop-8-5} Let $X$ be a real algebraic variety. If $\xi$ is a stratified-algebraic $\mathbb{F}$-vector bundle on $X$, then $\func{rank} \xi$ belongs to $H_{\CB\mhyphen\mathrm{str}}^0(X;\mathbb{Z})$. Conversely, each cohomology class in $H_{\CB\mhyphen\mathrm{str}}^0(X;\mathbb{Z})$ is of the form $\func{rank} \eta$ for some stratified-algebraic $\mathbb{F}$-vector bundle $\eta$ on $X$, whose restriction to each connected component of $X$ is topologically trivial. \end{proposition}
\begin{proof} By Proposition~\ref{prop-3-4}, $\xi$ is of the form $\xi = f^*\gamma_K(\mathbb{F}^n)$ for some multi-Grassmannian $\mathbb{G}_K(\mathbb{F}^n)$ and stratified-regular map $f \colon X \to \mathbb{G}_K(\mathbb{F}^n)$. Since $\func{rank} \xi = f^*(\func{rank} \gamma_K(\mathbb{F}^n))$ and
\begin{equation*} H_{\CB\mhyphen\mathrm{alg}}^0(\mathbb{G}_K(\mathbb{F}^n);\mathbb{Z}) = H^0(\mathbb{G}_K(\mathbb{F}^n);\mathbb{Z}), \end{equation*}
it follows that $\func{rank}\xi$ belongs to $H_{\CB\mhyphen\mathrm{str}}^0(X;\mathbb{Z})$. The second part of the proposition readily follows from the description of $H_{\CB\mhyphen\mathrm{str}}^0(X;\mathbb{Z})$. \end{proof}
For any real algebraic variety $X$, let $\func{VB}_{\mathbb{C}}^1(X)$ denote the group of isomorphism classes of topological $\mathbb{C}$-line bundles on $X$ (with operation induced by tensor product). The first Chern class gives rise to an isomorphism
\begin{equation*} c_1 \colon \func{VB}_{\mathbb{C}}^1(X) \to H^2(X;\mathbb{Z}). \end{equation*}
Similarly, denote by $\VB_{\CB\mhyphen\mathrm{str}}^1(X)$ the group of isomorphism classes of stratified-algebraic $\mathbb{C}$-line bundles on $X$. Now, the first Chern class gives rise to a homomorphism
\begin{equation*} c_1 \colon \VB_{\CB\mhyphen\mathrm{str}}^1(X) \to H^2(X;\mathbb{Z}). \end{equation*}
\begin{proposition}\label{prop-8-6} For any real algebraic variety $X$,
\begin{equation*} c_1(\VB_{\CB\mhyphen\mathrm{str}}^1(X)) = H_{\CB\mhyphen\mathrm{str}}^2(X;\mathbb{Z}). \end{equation*}
Furthermore, if the variety $X$ is compact, then the homomorphism
\begin{equation*} c_1 \colon \VB_{\CB\mhyphen\mathrm{str}}^1(X) \to H^2(X;\mathbb{Z}) \end{equation*}
is injective. \end{proposition}
\begin{proof} The inclusion
\begin{equation*} c_1(\VB_{\CB\mhyphen\mathrm{str}}^1(X)) \subseteq H_{\CB\mhyphen\mathrm{str}}^2(X;\mathbb{Z}) \end{equation*}
follows from Proposition~\ref{prop-8-3}. If $u$ is a cohomology class in $H_{\CB\mhyphen\mathrm{str}}^2(X;\mathbb{Z})$, then there exists a stratified-regular map $\varphi \colon X \to V$, into a compact nonsingular real algebraic variety $V$, such that $u = \varphi^*(v)$ for some cohomology class $v$ in $H_{\CB\mhyphen\mathrm{alg}}^2(V;\mathbb{Z})$. By \cite[Remark~5.4]{bib8}, $v$ is of the form $v = c_1(\lambda)$ for some algebraic $\mathbb{C}$-line bundle $\lambda$ on $V$. The pullback $\mathbb{C}$-line bundle $\varphi^*\lambda$ on $X$ is stratified-algebraic, cf. Proposition~\ref{prop-2-6}. Since $u = \varphi^*(c_1(\lambda)) = c_1(\varphi^*\lambda)$, we get
\begin{equation*} H_{\CB\mhyphen\mathrm{str}}^2(X;\mathbb{Z}) \subseteq c_1(\VB_{\CB\mhyphen\mathrm{str}}^1(X)). \end{equation*}
If the variety $X$ is compact, then the canonical homomorphism $\VB_{\CB\mhyphen\mathrm{str}}^1(X) \to \func{VB}_{\mathbb{C}}^1(X)$ is injective (cf. Theorem~\ref{th-3-10}), and hence the last assertion in the proposition follows. \end{proof}
Proposition~\ref{prop-8-6} can be interpreted as an approximation result for maps into complex projective $n$-space $\mathbb{P}^n(\mathbb{C}) = \mathbb{G}_1(\mathbb{C}^{n+1})$ for $n \geq 1$.
\begin{proposition}\label{prop-8-7} Let $X$ be a compact real algebraic variety. For a continuous map $f \colon X \to \mathbb{P}^n(\mathbb{C})$, the following conditions are equivalent:
\begin{conditions} \item The map $f$ can be approximated by stratified-regular maps.
\item The cohomology class $f^*(v_n)$ belongs to $H_{\CB\mhyphen\mathrm{str}}^2(X;\mathbb{Z})$, where $v_n$ is a generator of the group $H^2(\mathbb{P}^n(\mathbb{C});\mathbb{Z}) \cong \mathbb{Z}$. \end{conditions}
\end{proposition}
\begin{proof} We may assume that $v_n = c_1(\gamma_1(\mathbb{C}^{n+1}))$. Since $f^*(v_n) = c_1(f^*\gamma_1(\mathbb{C}^{n+1}))$, it suffices to make use of Theorem~\ref{th-4-10} (with $A \neq \varnothing$) and Proposition~\ref{prop-8-6}. \end{proof}
\begin{corollary}\label{cor-8-8} Let $X$ be a compact real algebraic variety. For a continuous map\linebreak $f \colon X \to \mathbb{S}^2$, the following conditions are equivalent:
\begin{conditions} \item The map $f$ can be approximated by stratified-regular maps.
\item The cohomology class $f^*(s_2)$ belongs to $H_{\CB\mhyphen\mathrm{str}}^2(X;\mathbb{Z})$, where $s_2$ is a generator of the group $H^2(\mathbb{S}^2;\mathbb{Z}) \cong \mathbb{Z}$. \end{conditions}
\end{corollary}
\begin{proof} It suffices to apply Proposition~\ref{prop-8-7} since $\mathbb{S}^2$ is biregularly isomorphic to $\mathbb{P}^1(\mathbb{C})$. \end{proof}
Any real algebraic variety $X$ is homotopically equivalent to a compact polyhedron \cite[Corollary~9.6.7]{bib9}, and hence the Chern character
\begin{equation*} \func{ch} \colon K_{\mathbb{C}}(X) \to H^{\mathrm{even}}(X;\mathbb{Q}) \end{equation*}
induces an isomorphism
\begin{equation*} \func{ch} \colon K_{\mathbb{C}}(X) \otimes \mathbb{Q} \to H^{\mathrm{even}}(X;\mathbb{Q}), \end{equation*}
cf. \cite{bib4} or \cite[p.~255, Theorem~A]{bib25}. On the other hand, the Chern character
\begin{equation*} \func{ch} \colon K_{\CB\mhyphen\mathrm{str}}(X) \to H^{\mathrm{even}}(X;\mathbb{Q}) \end{equation*}
induces a homomorphism
\begin{equation*} \func{ch} \colon K_{\CB\mhyphen\mathrm{str}}(X) \otimes \mathbb{Q} \to H^{\mathrm{even}}(X;\mathbb{Q}). \end{equation*}
We next describe the image of the last homomorphism. Denote by $H_{\CB\mhyphen\mathrm{str}}^{\mathrm{even}}(X;\mathbb{Q})$ the image of $H_{\CB\mhyphen\mathrm{str}}^{\mathrm{even}}(X;\mathbb{Z})\otimes\mathbb{Q}$ by the canonical isomorphism $H^{\mathrm{even}}(X;\mathbb{Z}) \otimes\mathbb{Q} \to H^{\mathrm{even}}(X;\mathbb{Q})$.
\begin{proposition}\label{prop-8-9} For any real algebraic variety $X$,
\begin{equation*} \func{ch}(K_{\CB\mhyphen\mathrm{str}}(X) \otimes \mathbb{Q}) = H_{\CB\mhyphen\mathrm{str}}^{\mathrm{even}}(X;\mathbb{Q}). \end{equation*}
Furthermore, if the variety $X$ is compact, then the homomorphism
\begin{equation*} \func{ch} \colon K_{\CB\mhyphen\mathrm{str}}(X) \otimes \mathbb{Q} \to H^{\mathrm{even}}(X;\mathbb{Q}) \end{equation*}
is injective. \end{proposition}
\begin{proof} The inclusion
\begin{equation*} \func{ch}(K_{\CB\mhyphen\mathrm{str}}(X) \otimes \mathbb{Q}) \subseteq H_{\CB\mhyphen\mathrm{str}}^{\mathrm{even}}(X;\mathbb{Q}) \end{equation*}
follows from Propositions~\ref{prop-8-3} and~\ref{prop-8-5}. If $u$ is a cohomology class in $H_{\CB\mhyphen\mathrm{str}}^{\mathrm{even}}(X;\mathbb{Q})$, then there exists a stratified-regular map $\varphi \colon X \to V$, into a compact nonsingular real algebraic variety $V$, such that $u = \varphi^*(v)$ for some cohomology class $v$ in $H_{\CB\mhyphen\mathrm{alg}}^{\mathrm{even}}(V; \mathbb{Q})$. By \cite[Proposition~5.4]{bib8}, $v$ is of the form $v = \func{ch}(\alpha)$ for some element $\alpha$ in $K_{\CB\mhyphen\mathrm{alg}}(X)\otimes\mathbb{Q}$. In view of Proposition~\ref{prop-2-6}, $\varphi^*(\alpha)$ is in $K_{\CB\mhyphen\mathrm{str}}(X)\otimes\mathbb{Q}$. Since $u = \varphi^*(\func{ch}(\alpha)) = \func{ch}(\varphi^*(\alpha))$, we get
\begin{equation*} H_{\CB\mhyphen\mathrm{str}}^{\mathrm{even}}(X;\mathbb{Q}) \subseteq \func{ch}(K_{\CB\mhyphen\mathrm{str}}(X)\otimes\mathbb{Q}). \end{equation*}
If the variety $X$ is compact, then the canonical homomorphism $K_{\CB\mhyphen\mathrm{str}}(X) \to K_{\mathbb{C}}(X)$ is injective (cf. Corollary~\ref{cor-3-11}), and hence the last assertion in the proposition follows. \end{proof}
One can also prove a sharper result than the first part of Proposition~\ref{prop-8-9}.
\begin{proposition}\label{prop-8-10} Let $X$ be a real algebraic variety and let $k$ be a positive integer. For any cohomology class $u$ in $H_{\CB\mhyphen\mathrm{str}}^{2k}(X;\mathbb{Z})$, there exists a stratified-algebraic $\mathbb{C}$-vector bundle $\xi$ on $X$ with $c_i(\xi)=0$ for $1 \leq i \leq k-1$ and $c_k(\xi) = (-1)^{k-1}(k-1)!u$. \end{proposition}
\begin{proof} We first consider $\mathbb{C}$-algebraic cohomology classes.
\begin{assertion} Let $V$ be a compact nonsingular real algebraic variety and let $v$ be a cohomology class in $H_{\CB\mhyphen\mathrm{alg}}^{2k}(V;\mathbb{Z})$, where $k \geq 1$. Then there exists an algebraic $\mathbb{C}$-vector bundle $\eta$ on $V$ with $c_i(\eta)=0$ for $1 \leq i \leq k-1$ and $c_k(\eta) = (-1)^{k-1}(k-1)!v$. \end{assertion}
The Assertion can be proved as follows. Let $(\mathbb{V},\iota)$ be a nonsingular projective complexification of $V$. We may assume that $V = \mathbb{V}(\mathbb{R})$ and $\iota \colon V \hookrightarrow \mathbb{V}(\mathbb{C})$ is the inclusion map. By definition, $v = \iota^*(\func{cl}_{\mathbb{C}}(z))$ for some element $z$ in the Chow group $A^k(\mathbb{V}_{\mathbb{C}})$. Let $\mathbb{U}$ be an affine Zariski open subscheme of $\mathbb{V}$ containing $V$. According to Grothendieck's formula \cite[Example~15.3]{bib24} (see also \cite[Propostition~19.1.2]{bib24}), there exists an algebraic vector bundle $\mathbb{E}$ on $\mathbb{U}_{\mathbb{C}}$ with $c_i(\mathbb{E})=0$ for $1 \leq i \leq k-1$ and $c_k(\mathbb{E}) = (-1)^{k-1}(k-1)!j^*(\func{cl}_{\mathbb{C}}(z))$, where $j \colon \mathbb{U}(\mathbb{C}) \to \mathbb{V}(\mathbb{C})$
is the inclusion map. The restriction $\eta \coloneqq \mathbb{E}|_V$ is an algebraic $\mathbb{C}$-vector bundle on $V$ satisfying all the conditions stated in the Assertion.
We can now easily complete the proof. The cohomology class $u$ is of the form ${u = \varphi^*(v)}$, where $\varphi \colon X \to V$ is a stratified-regular map into a compact nonsingular real algebraic variety $V$, and $v$ is a cohomology class in $H_{\CB\mhyphen\mathrm{alg}}^{2k}(V;\mathbb{Z})$. If $\eta$ is an algebraic $\mathbb{C}$-vector bundle on $V$ as in the Assertion, then the pullback $\varphi^*\eta$ is a stratified-algebraic $\mathbb{C}$-vector bundle on $X$ (cf. Proposition~\ref{prop-2-6}) having all the required properties. \end{proof}
Under some assumptions we obtain a nice characterization of topological $\mathbb{C}$-vector bundles admitting a stratified-algebraic structure.
\begin{theorem}\label{th-8-11} Let $X$ be a compact real algebraic variety. Assume that the group $H^*(X;\mathbb{Z})$ has no torsion and the quotient group $H^{2k}(X;\mathbb{Z}) / H_{\CB\mhyphen\mathrm{str}}^{2k}(X;\mathbb{Z})$ has no $(k-1)!$-torsion elements for every $k \geq 1$. For a topological $\mathbb{C}$-vector bundle $\xi$ on $X$, the following conditions are equivalent:
\begin{conditions} \item\label{th-8-11-a} $\xi$ admits a stratified-algebraic structure.
\item\label{th-8-11-b} $\func{rank}\xi$ belongs to $H_{\CB\mhyphen\mathrm{str}}^0(X;\mathbb{Z})$, and $c_k(\xi)$ belongs to $H_{\CB\mhyphen\mathrm{str}}^{2k}(X;\mathbb{Z})$ for every $k \geq 1$. \end{conditions}
\end{theorem}
\begin{proof} According to Propositions~\ref{prop-8-3} and~\ref{prop-8-5}, (\ref{th-8-11-a}) implies (\ref{th-8-11-b}). For the proof of the reversed implication, we first establish the following:
\begin{assertion} Let $\theta$ be a topological $\mathbb{C}$-vector bundle on $X$ and let $k$ be a positive integer such that $c_i(\theta)=0$ for $1 \leq i \leq k-1$ and $c_k(\theta)$ belongs to $H_{\CB\mhyphen\mathrm{str}}^{2k}(X;\mathbb{Z})$. Then there exists a stratified-algebraic $\mathbb{C}$-vector bundle $\theta_k$ on $X$ with $c_i(\theta_k)=0$ for $1 \leq i \leq k-1$ and $c_k(\theta \oplus \theta_k) = 0$. \end{assertion}
Recall that $X$ is a compact polyhedron, cf. \cite[Corollary~9.6.7]{bib9}. Since the group $H^*(X;\mathbb{Z})$ has no torsion, the $k$th Chern class $c_k(\theta)$ is of the form
\begin{equation*} c_k(\theta) = (-1)^{k-1}(k-1)!u \end{equation*}
for some cohomology class $u$ in $H^{2k}(X;\mathbb{Z})$, cf. \cite[p.~19]{bib4}. By assumption, $(-1)^{k-1}(k-1)!u$ is in $H_{\CB\mhyphen\mathrm{str}}^{2k}(X;\mathbb{Z})$, and hence $u$ is in $H_{\CB\mhyphen\mathrm{str}}^{2k}(X;\mathbb{Z})$, the quotient group
\begin{equation*} H^{2k}(X;\mathbb{Z})/H_{\CB\mhyphen\mathrm{str}}^{2k}(X;\mathbb{Z}) \end{equation*}
having no $(k-1)!$-torsion elements. According to Proposition~\ref{prop-8-10}, there exists a stratified-algebraic $\mathbb{C}$-vector bundle $\theta_k$ on $X$ with $c_i(\theta_k)=0$ for $1 \leq i \leq k-1$ and $c_k(\theta_k) = (-1)^{k-1}(k-1)!(-u)$. Thus, $\theta_k$ satisfies all the conditions stated in the Assertion.
If (\ref{th-8-11-b}) holds, then making use of the Assertion, we obtain a stratified-algebraic $\mathbb{C}$-vector bundle $\zeta$ on $X$ with $c_i(\xi \oplus \zeta) = 0$ for every $i \geq 1$. If $\eta$ is a stratified-algebraic $\mathbb{C}$-vector bundle on $X$ such that the direct sum $\zeta \oplus \eta$ is a trivial vector bundle (cf. Proposition~\ref{prop-3-7}), then $c_i(\xi) = c_i(\eta)$ for every $i \geq 1$. Now, in view of the second part of Proposition~\ref{prop-8-5}, there exists a stratified-algebraic $\mathbb{C}$-vector bundle $\xi'$ on $X$ such that $\func{rank} \xi - \func{rank} \xi'$ is constant, and $c_i(\xi) = c_i(\xi')$ for every $i \geq 1$. Consequently, the $\mathbb{C}$-vector bundles $\xi$ and $\xi'$ are topologically stably equivalent, the group $H^{\mathrm{even}}(X;\mathbb{Z})$ having no torsion, cf. \cite{bib40}. Hence, according to Corollary~\ref{cor-3-14}, $\xi$ admits a stratified-algebraic structure. \end{proof}
Any $\mathbb{H}$-vector bundle $\xi$ can be regarded as a $\mathbb{C}$-vector bundle, which is indicated by $\xi_{\mathbb{C}}$.
\begin{corollary}\label{cor-8-12} Let $X$ be a compact real algebraic variety. Assume that the group $H^*(X; \mathbb{Z})$ has no torsion and the quotient group $H^{2k}(X;\mathbb{Z}) / H_{\CB\mhyphen\mathrm{str}}^{2k}(X;\mathbb{Z})$ has no $(k-1)!$-torsion elements for every $k \geq 1$. For a topological $\mathbb{H}$-vector bundle $\xi$ on $X$, the following conditions are equivalent:
\begin{conditions} \item\label{cor-8-12-a} $\xi$ admits a stratified-algebraic structure.
\item\label{cor-8-12-b} $\func{rank} \xi$ belongs to $H_{\CB\mhyphen\mathrm{str}}^0(X;\mathbb{Z})$, and $c_{2k}(\xi_{\mathbb{C}})$ belongs to $H_{\CB\mhyphen\mathrm{str}}^{4k}(X;\mathbb{Z})$ for every $k \geq 1$. \end{conditions}
\end{corollary}
\begin{proof} According to Propositions~\ref{prop-8-3} and~\ref{prop-8-5}, (\ref{cor-8-12-a}) implies (\ref{cor-8-12-b}). Since $\xi$ is an $\mathbb{H}$-vector bundle, it follows that $c_{2i+1}(\xi)=0$ for every $i \geq 0$. Hence, in view of Theorem~\ref{th-8-11}, if (\ref{cor-8-12-b}) holds, then the $\mathbb{C}$-vector bundle $\xi_{\mathbb{C}}$ admits a stratified-algebraic structure. Consequently, the $\mathbb{R}$-vector bundle $\xi_{\mathbb{R}} = (\xi_{\mathbb{C}})_{\mathbb{R}}$ admits a stratified-algebraic structure. Thus, by Theorem~\ref{th-1-7}, (\ref{cor-8-12-b}) implies (\ref{cor-8-12-a}). \end{proof}
For $\mathbb{R}$-vector bundles we can only obtain a weaker result.
\begin{corollary}\label{cor-8-13} Let $X$ be a compact real algebraic variety and let $\xi$ be a topological $\mathbb{R}$-vector bundle on $X$. Assume that the group $H^*(X;\mathbb{Z})$ has no torsion and the quotient group $H^{2k}(X;\mathbb{Z}) / H_{\CB\mhyphen\mathrm{str}}^{2k}(X;\mathbb{Z})$ has no $(k-1)!$-torsion elements for every $k \geq 1$. If $\func{rank} \xi$ belongs to $H_{\CB\mhyphen\mathrm{str}}^0(X;\mathbb{Z})$, and $p_k(\xi)$ belongs to $H_{\CB\mhyphen\mathrm{str}}^{4k}(X;\mathbb{Z})$ for every $k \geq 1$, then the direct sum $\xi \oplus \xi$ admits a stratified-algebraic structure. \end{corollary}
\begin{proof} We have $c_{2i+1}(\mathbb{C}\otimes\xi)=0$ for every $i \geq 0$ since $c_{2i+1}(\mathbb{C}\otimes\xi)$ is always an element of order at most $2$ (cf. \cite[p.~174]{bib38}) and the group $H^{\mathrm{even}}(X;\mathbb{Z})$ has no torsion. Moreover, $c_{2k}(\mathbb{C}\otimes\xi) = (-1)^kp_k(\xi)$ for every $k \geq 1$. According to Theorem~\ref{th-8-11}, the $\mathbb{C}$-vector bundle $\mathbb{C}\otimes\xi$ admits a stratified-algebraic structure. The proof is complete since ${(\mathbb{C}\otimes\xi)_{\mathbb{R}} = \xi \oplus \xi}$. \end{proof}
We next identify some classes of real algebraic varieties which satisfy the assumptions of the last three results.
\begin{lemma}\label{lem-8-14} If $X$ is a compact real algebraic variety of even dimension $2k$, then
\begin{equation*} H_{\CB\mhyphen\mathrm{str}}^{2k}(X;\mathbb{Z}) = H^{2k}(X;\mathbb{Z}). \end{equation*}
\end{lemma}
\begin{proof} Let $s_{2k}$ be a generator of the cohomology group $H^{2k}(\mathbb{S}^{2k};\mathbb{Z}) \cong \mathbb{Z}$. Each cohomology class $u$ in $H^{2k}(X;\mathbb{Z})$ is of the form $u=f^*(s_{2k})$ for some continuous map $f \colon X \to \mathbb{S}^{2k}$. According to Theorem~\ref{th-2-5}, we may assume that $f$ is stratified-regular. The proof is complete since $H_{\CB\mhyphen\mathrm{alg}}^{2k}(\mathbb{S}^{2k};\mathbb{Z}) = H^{2k}(\mathbb{S}^{2k}; \mathbb{Z})$, cf. \cite[Propostion~4.8]{bib8}. \end{proof}
\begin{proposition}\label{prop-8-15} Let $X = X_1 \times \cdots \times X_n$, where each $X_i$ is a compact real algebraic variety homotopically equivalent to the unit $d_i$-sphere for $1 \leq i \leq n$. Then
\begin{equation*} H_{\CB\mhyphen\mathrm{str}}^{\mathrm{even}}(X;\mathbb{Z}) = H^{\mathrm{even}}(X;\mathbb{Z}). \end{equation*}
\end{proposition}
\begin{proof} Let $p_i \colon X \to X_i$ be the canonical projection, $1 \leq i \leq n$. For each pair $(j,l)$ of integers satisfying $1 \leq j \leq l \leq n$, let $q_{jl} \colon X \to X_j \times X_l$ be the canonical projection. The $\mathbb{Z}$-algebra $H^{\mathrm{even}}(X;\mathbb{Z})$ is generated by all $p_i^*(H^{d_i}(X_i; \mathbb{Z}))$ with $d_i$ even and all $q_{jl}^*( H^{d_j + d_l} (X_j \times X_l; \mathbb{Z}) )$ with $d_j$ and $d_l$ odd. The proof is complete in view of Lemma~\ref{lem-8-14} and Propositions~\ref{prop-8-1} and~\ref{prop-8-2}. \end{proof}
The reader may compare Proposition~\ref{prop-8-15} and results of \cite{bib14, bib16, bib18} to see a sharp difference between $H_{\CB\mhyphen\mathrm{str}}^{\mathrm{even}}(-;\mathbb{Z})$ and $H_{\CB\mhyphen\mathrm{alg}}^{\mathrm{even}}(-;\mathbb{Z})$.
\begin{proof}[Proof of Theorem~\ref{th-1-10}] It suffices to combine Theorem~\ref{th-8-11}, Corollaries~\ref{cor-8-12}, \ref{cor-8-13}, and Proposition~\ref{prop-8-15}. \end{proof}
Theorem~\ref{th-1-10} can be interpreted as an approximation result for maps into Grassmannians.
\begin{theorem}\label{th-8-16} Let $X = X_1 \times \cdots X_n$, where each $X_i$ is a compact real algebraic variety homotopically equivalent to the unit $d_i$-sphere for $1 \leq i \leq n$. If $\mathbb{F} = \mathbb{C}$ or $\mathbb{F} = \mathbb{H}$, then for any pair $(k,m)$ of integers satisfying $1 \leq k \leq m$, each continuous map from $X$ into the Grassmannian $\mathbb{G}_k(\mathbb{F}^m)$ can be approximated by stratified-regular maps. \end{theorem}
\begin{proof} If suffices to make use of Theorem~\ref{th-1-10} and Theorem~\ref{th-4-10} (with $A = \varnothing$). \end{proof}
As an interesting special case, we obtain the following approximation result for maps with values in the unit spheres $\mathbb{S}^2$ or $\mathbb{S}^4$.
\begin{corollary}\label{cor-8-17} Let $X = X_1 \times \cdots \times X_n$, where each $X_i$ is a compact real algebraic variety homotopically equivalent to the unit $d_i$-sphere for $1 \leq i \leq n$. If $k=2$ or $k=4$, then each continuous map from $X$ into the unit $k$-sphere $\mathbb{S}^k$ can be approximated by stratified-regular maps. \end{corollary}
\begin{proof} Recall that $d(\mathbb{F}) = \dim_{\mathbb{R}}\mathbb{F}$, and $\mathbb{G}_1(\mathbb{F}^2)$ is biregularly isomorphic to $\mathbb{S}^{d(\mathbb{F})}$. Thus, Corollary~\ref{cor-8-17} is a special case of Theorem~\ref{th-8-16}. \end{proof}
We also have the following result analogous to Proposition~\ref{prop-8-15}.
\begin{theorem}\label{th-8-18} Let $X$ be a compact real algebraic variety with
\begin{equation*} H_{\mathrm{str}}^1(X; \mathbb{Z}/2) = H^1(X; \mathbb{Z}/2). \end{equation*}
Assume that for a positive integer $k$, the cohomology group $H^{2k}(X;\mathbb{Z})$ is generated by the cup products of cohomology classes belonging to $H^1(X;\mathbb{Z})$. Then
\begin{equation*} H_{\CB\mhyphen\mathrm{str}}^{2k}(X;\mathbb{Z}) = H^{2k}(X;\mathbb{Z}). \end{equation*}
\end{theorem}
\begin{proof} Each cohomology class in $H^1(X;\mathbb{Z})$ is of the form $f^*(s_1)$, where $f \colon X \to \mathbb{S}^1$ is a continuous map and $s_1$ is a generator of the group $H^1(X; \mathbb{Z}) \cong \mathbb{Z}$, cf. \cite[pp.~425, 428]{bib42}. Consequently, the group $H^{2k}(X; \mathbb{Z})$ is generated by the cohomology classes of the form $g^*(t_{2k})$, where $g$ is a continuous map from $X$ into the $2k$-torus $\mathbb{T}^{2k} = \mathbb{S}^1 \times \cdots \times \mathbb{S}^1$, and $t_{2k}$ is a generator of the group $H^{2k}(\mathbb{T}^{2k}; \mathbb{Z}) \cong \mathbb{Z}$. In view of Corollary~\ref{cor-7-6} and the equality $H_{\mathrm{str}}^1(X;\mathbb{Z}/2) = H^1(X; \mathbb{Z}/2)$, we may assume that the map $g$ is stratified-regular. By Proposition~\ref{prop-8-15}, $H_{\CB\mhyphen\mathrm{str}}^{2k}(\mathbb{T}^{2k}; \mathbb{Z}) = H^{2k}(\mathbb{T}^{2k};\mathbb{Z})$. In order to complete the proof it suffices to apply Propositions~\ref{prop-8-1} and~\ref{prop-8-2}. \end{proof}
\begin{corollary}\label{cor-8-19} Let $X$ be a compact real algebraic variety with
\begin{equation*} H_{\mathrm{str}}^1(X;\mathbb{Z}/2) = H^1(X;\mathbb{Z}/2). \end{equation*}
Assume that $X$ is homotopically equivalent to the $n$-torus $\mathbb{S}^1 \times \cdots \times \mathbb{S}^1$ ($n$ factors). Then
\begin{equation*} H_{\CB\mhyphen\mathrm{str}}^{\mathrm{even}}(X;\mathbb{Z}) = H^{\mathrm{even}}(X;\mathbb{Z}). \end{equation*}
If $\mathbb{F} = \mathbb{C}$ or $\mathbb{F} = \mathbb{H}$, then each topological $\mathbb{F}$-vector bundle on $X$ admits a stratified-algebraic structure. If $\xi$ is a topological $\mathbb{R}$-vector bundle on $X$, then the direct sum $\xi \oplus \xi$ admits a stratified-algebraic structure. \end{corollary}
\begin{proof} Obviously, $H_{\CB\mhyphen\mathrm{str}}^0(X; \mathbb{Z}) = H^0(X;\mathbb{Z}) \cong \mathbb{Z}$, the group $H^*(X;\mathbb{Z})$ has no torsion, and the $\mathbb{Z}$-algebra $H^*(X; \mathbb{Z})$ is generated by $H^1(X;\mathbb{Z})$. It suffices to apply Theorem~\ref{th-8-18}, Theorem~\ref{th-8-11} and Corollaries~\ref{cor-8-12} and~\ref{cor-8-13}. \end{proof}
We conclude this section by giving a different description of the cohomology group $H_{\CB\mhyphen\mathrm{str}}^2(X;\mathbb{Z})$, which in turn leads to a new interpretation of our results on $\mathbb{C}$-line bundles. First some topological facts will be recalled for the convenience of the reader.
Let $M$ be a smooth manifold and let $N$ be a smooth submanifold of $M$ of codimension~$k$. By convention, submanifolds are assumed to be closed subsets of the ambient manifold. Assume that the normal bundle of $N$ in $M$ is oriented and denote by $\tau_N^M$ the Thom class of $N$ in the cohomology group $H^k(M, M \setminus N; \mathbb{Z})$, cf. \cite[p.~118]{bib38}. The image of $\tau_N^M$ by the restriction homomorphism $H^k(M, M \setminus N; \mathbb{Z}) \to H^k(M;\mathbb{Z})$, induced by the inclusion map $M \hookrightarrow (M; M \setminus N)$, will be denoted by $[N]^M$ and called the cohomology class represented by $N$. If $M$ is compact and oriented, and $N$ is endowed with the compatible orientation, then $[N]^M$ is up to sign Poincar\'e dual to the homology class in $H_*(M; \mathbb{Z})$ represented by $N$, cf. \cite[p.~136]{bib38}.
Let $P$ be a smooth manifold and let $Q$ be a smooth submanifold of $P$. Let $f \colon M \to P$ be a smooth map transverse to $Q$. If the normal bundle of $Q$ in $P$ is oriented and the normal bundle of the smooth submanifold $N \coloneqq f^{-1}(Q)$ of $M$ is endowed with the orientation induced by $f$, then $\tau_N^M = f^*(\tau_Q^P)$, where $f$ is regarded as a map from $(M, M \setminus N)$ into $(P, P \setminus Q)$ (this follows form \cite[p.~117, Theorem~6.7]{bib27}). In particular, ${[N]^M = f^*([Q]^P)}$.
Let $\xi$ be an oriented smooth $\mathbb{R}$-vector bundle of rank $k$ on $M$. If $s \colon M \to \xi$ is a smooth section transverse to the zero section, and the normal bundle of ${Z(s) = \{ x \in M \mid s(x) = 0 \}}$ is endowed with the orientation induced by $s$ from the orientation of $\xi$ (cf. the proof of Lemma~\ref{lem-6-1}), then
\begin{equation*} e(\xi) = [Z(s)]^M, \end{equation*}
where $e(\xi)$ stands for the Euler class of $\xi$. Indeed, let $E$ be the total space of $\xi$ and $p \colon E \to M$ the bundle projection. Identify $M$ with the image of the zero section of $\xi$. The section $s$ is transverse to $M$ and $Z(s) = s^{-1}(M)$. Consequently, $[Z(s)]^M = s^*([M]^E)$. Hence
\begin{equation*} p^*([Z(s)]^M) = p^*(s^*([M]^E)) = (s \circ p)^* ([M]^E) = [M]^E, \end{equation*}
where the last equality holds since $s \circ p \colon E \to E$ is homotopic to the identity map. On the other hand, $p^*(e(\xi)) = [M]^E$, cf. \cite[p.~98]{bib38}. It follows that $e(\xi) = [Z(s)]^M$ since $p^*$ is an isomorphism.
If $\xi$ is a $\mathbb{C}$-vector bundle of rank $k$, then
\begin{equation*} e(\xi_{\mathbb{R}}) = c_k(\xi), \end{equation*}
where $\xi_{\mathbb{R}}$ is endowed with the orientation determined by the complex structure of $\xi$, cf. \cite[p.~158]{bib38}.
\begin{lemma}\label{lem-8-20} Let $M$ be a smooth manifold, and let $N$ be a smooth codimension $2$ submanifold of $M$ with oriented normal bundle. Let $\lambda$ be a smooth $\mathbb{C}$-line bundle on $M$ with $c_1(\lambda) = [N]^M$. Then there exists a smooth section $s \colon M \to \lambda$ transverse to the zero section and satisfying $Z(s)=N$. \end{lemma}
\begin{proof} Let $\tau = (T, \rho, N)$ be a tubular neighborhood of $N$ in $M$. The smooth section $v \colon T \to \rho^*\tau$, defined by $v(x) = (x,x)$ for all $x$ in $T$, is transverse to the zero section and satisfies
$Z(v)=N$. The differential of $v$ induces an isomorphism between the normal bundle to $N$ in $M$ and $(\rho^*\tau)|_N \cong \tau$ (cf. the proof of Lemma~\ref{lem-6-1}). Via this isomorphism, $\tau$ is endowed with an orientation. Actually, $\tau$ can be regarded as a smooth $\mathbb{C}$-line bundle, being oriented of rank $2$ (recall that $SO(2) \cong U(1)$). The restriction of $\rho^*\tau$ to $T \setminus N$ is a trivial smooth $\mathbb{C}$-line bundle. Consequently, the $\mathbb{C}$-line bundle $\rho^*\tau$ on $T$ and the standard trivial $\mathbb{C}$-line bundle $\varepsilon$ on $M \setminus N$ can be glued over $T \setminus N$. The resulting smooth $\mathbb{C}$-line bundle $\mu$ on $M$ has a smooth section $w \colon M \to \mu$, obtained by gluing $v$ and a nowhere zero section of $\varepsilon$, which is transverse to the zero section and satisfies $Z(w)=N$. It follows from the facts recalled before Lemma~\ref{lem-8-20} that $c_1(\mu) = [N]^M$. The proof is complete since the smooth $\mathbb{C}$-line bundles $\lambda$ and $\mu$ are isomorphic. \end{proof}
We now return to real algebraic geometry. Let $X$ be a compact nonsingular real algebraic variety. We say that a cohomology class $u$ in $H^2(X;\mathbb{Z})$ is \emph{adapted} if it is of the form $u=[Z]^X$, where $Z$ is a nonsingular Zariski locally closed subvariety of $X$ of codimension $2$, which is closed in the Euclidean topology and whose normal bundle in $X$ is oriented.
Recall that the definition of an adapted smooth vector bundle is given in Section~\ref{sec-1}, and then extended to topological vector bundles in Section~\ref{sec-6} (before Theorem~\ref{th-6-9}).
\begin{proposition}\label{prop-8-21} For a topological $\mathbb{C}$-line bundle $\xi$ on a compact nonsingular real algebraic variety, the following conditions are equivalent:
\begin{conditions} \item\label{prop-8-21-a} The $\mathbb{C}$-line bundle $\xi$ is adapted.
\item\label{prop-8-21-b} The cohomology class $c_1(\xi)$ is adapted. \end{conditions}
\end{proposition}
\begin{proof} We may assume that the $\mathbb{C}$-line bundle $\xi$ is smooth. It follows form the definition of an adapted $\mathbb{C}$-line bundle that (\ref{prop-8-21-a}) implies (\ref{prop-8-21-b}). If (\ref{prop-8-21-b}) holds, then $\xi$ is adapted in view of Lemma~\ref{lem-8-20}. \end{proof}
Denote by $G(X)$ the subgroup of $H^2(X;\mathbb{Z})$ generated by all adapted cohomology classes.
\begin{theorem}\label{th-8-22} For any compact nonsingular real algebraic variety $X$,
\begin{equation*} G(X) \subseteq H_{\CB\mhyphen\mathrm{str}}^2(X; \mathbb{Z}). \end{equation*}
In particular, if $\xi$ is a topological $\mathbb{C}$-line bundle on $X$ with $c_1(\xi)$ in $G(X)$, then $\xi$ admits a stratified-algebraic structure. \end{theorem}
\begin{proof} The inclusion $G(X) \subseteq H_{\CB\mhyphen\mathrm{str}}^2(X;\mathbb{Z})$ follows form Theorem~\ref{th-1-8} (with $\mathbb{F}=\mathbb{C}$) and Propositions~\ref{prop-8-6} and~\ref{prop-8-21}. Hence, the second assertion is a consequence of Proposition~\ref{prop-8-6}. \end{proof}
Theorem~\ref{th-8-22} is of interest since the group $G(X)$ is easier to compute directly than the group $H_{\CB\mhyphen\mathrm{str}}^2(X;\mathbb{Z})$. Adapted cohomology classes can also be used to give a complete description of $H_{\CB\mhyphen\mathrm{str}}^2(X;\mathbb{Z})$.
\begin{theorem}\label{th-8-23} Let $X$ be a compact real algebraic variety. For a cohomology class $u$ in $H^2(X; \mathbb{Z})$, the following conditions are equivalent:
\begin{conditions} \item The cohomology class $u$ belongs to $H_{\CB\mhyphen\mathrm{str}}^2(X;\mathbb{Z})$.
\item There exists a filtration $\mathcal{F} = (X_{-1}, X_0, \ldots, X_m)$ of $X$ with $\overline{\mathcal{F}}$ a nonsingular stratification, and for each $i = 0, \ldots, m$, there exists a multiblowup $\pi_i \colon X'_i \to X_i$
over $X_{i-1}$ such that $X'_i$ is a nonsingular variety and the cohomology class $\pi_i^*(u|_{X_i})$ in $H^2(X'_i;\mathbb{Z})$ is adapted. Here
$u|_{X_i}$ is the image of $u$ under the homomorphism $H^2(X;\mathbb{Z}) \to H^2(X_i;\mathbb{Z})$ induced by the inclusion map $X_i \hookrightarrow X$. \end{conditions}
\end{theorem}
\begin{proof} Let $\mathcal{F} = (X_{-1}, X_0, \ldots, X_m)$ be a filtration of $X$ with $\overline{\mathcal{F}}$ a nonsingular stratification, and for each $i = 0, \ldots, m$, let $\pi_i \colon X'_i \to X_i$ be a multiblowup over $X_{i-1}$ such that $X'_i$ is a nonsingular variety. If $\xi$ is a topological $\mathbb{C}$-line bundle on $X$ with $c_1(\xi) = u$, then
${c_1(\xi|_{X_i}) = u|_{X_i}}$ and $c_1(\pi_i^*(\xi|_{X_i})) =
\pi_i^*(u|_{X_i})$ for $0 \leq i \leq m$. In view of Proposition~\ref{prop-8-21}, the $\mathbb{C}$-line bundle $\pi_i^*(\xi|_{X_i})$ on $X'_i$ is adapted if and only if the cohomology class
$\pi_i^*(u|_{X_i})$ is adapted. Consequently, it suffices to combine Theorem~\ref{th-6-9} (with $\mathbb{F}=\mathbb{C}$) and Proposition~\ref{prop-8-6}. \end{proof}
\cleardoublepage \phantomsection \addcontentsline{toc}{section}{\refname}
\end{document} | arXiv |
Is "playing dumb" considered un-helpful?
It is fairly common for someone to post a question with an easy-to-fix notational error, for example, writing an integral and forgetting to put $dx$ at the end. Some users will just edit the post and fix the notation, but others seem to prefer commenting something like, "I don't understand what you've written. Is there meant to be a $dx$ at then end of that integral?"
This strikes me as disingenuous and rude. The user commenting this way almost certainly does know what the OP means, and what they're really trying to say is, "hey, you forgot part of the notation." I understand that this particular form of disingenuousness is common enough in the mathematical community - certain teachers like to use it - but is it fair to say that it does not create the type of atmosphere we wish to maintain at a site such as this?
Thanks in advance for your thoughts on this matter.
EDIT: It's apparent this question has produced a good deal of misunderstanding. I'm not asking how to respond to a commenter who I think is "playing dumb". I'm not asking if, or suggesting that, I can distinguish it from honest questioning. I'm asking whether doing it is a good idea. Is it helpful, or does it do more harm than good? I know that it happens sometimes, and I wonder if it is sound pedagogy.
discussion etiquette
G Tony Jacobs
G Tony JacobsG Tony Jacobs
$\begingroup$ If you're eager to guess what an OP means, where's the problem? I find it rude if an OP lets me guess. Tastes are different, obviously. $\endgroup$ – Professor Vector Aug 30 '17 at 18:22
$\begingroup$ Simply sliding in and making a quick edit to include $dx$ when an integral sign is used, and disappearing may go unnoticed by the OP. I'd make such an edit, but also post a comment: "Don't forget to include $dx$ whenever you write an integral." I make such an explicit comment because on a text, an asker might integrate successfully a difficult integral, but $dx$ is missing, or in the case of an indefinite integral, $+C$ is missing, and that could undermine a user's test score for such errors. $\endgroup$ – Namaste Aug 30 '17 at 18:59
$\begingroup$ This comment seems to be based on a strange misunderstanding of my original question. Not sure what to say. $\endgroup$ – G Tony Jacobs Aug 30 '17 at 19:20
$\begingroup$ @amWhy, I was not referring to your comment in mine. Your comment makes perfect sense to me. $\endgroup$ – G Tony Jacobs Aug 30 '17 at 19:51
$\begingroup$ I really enjoy these ambiguous things$$\int_0^1t^x$$Option choices are $(t-1)\ln(t)$ or $\frac1{x+1}$. Please choose one $\ddot\smile$ $\endgroup$ – Simply Beautiful Art Aug 30 '17 at 19:52
$\begingroup$ GTonyJacobs No worries; I was really suggesting a better way commenting & editing, than the one you speak of, on which I agree with your concern. I agree, some users can be pretty "snarky" in their comments. $\endgroup$ – Namaste Aug 30 '17 at 19:55
$\begingroup$ Questions that are actually ambiguous are another matter, one I was not attempting to address with this question. $\endgroup$ – G Tony Jacobs Aug 30 '17 at 19:57
$\begingroup$ I think this is a particular type of snark that is peculiar to math education. I imagine it has its defenders; it is certainly common enough. I wonder if we'll hear from them. $\endgroup$ – G Tony Jacobs Aug 30 '17 at 19:59
$\begingroup$ @SimplyBeautifulArt: The answer is obviously $t^x$, since I'm integrating with respect to a third and independent variable $y$. :) I suspect you could get anything at all if you choose $y$ dependent on $t$ and $x$ in a clever way! $\endgroup$ – user14972 Aug 30 '17 at 23:20
$\begingroup$ One may also ask how to evaluate ∫<sub>0</sub><sup>1</sup>t<sup>x</sup>e<sup>-t</sup>dt, but the trailing part was lost during cut-and-paste and then someone familiar with MathJax edited the truncated expression to $\int_0^1 t^x$. Things like this occur from time to time. They don't occur frequently, but they are not rare either. I don't think it's fair to assume that the commenter was playing dumb. $\endgroup$ – user1551 Aug 31 '17 at 3:48
$\begingroup$ I'm not talking about assuming the commenter is playing dumb. I'm asking whether the commenter ever should play dumb. Is it a valid pedagogical tool, or does it do more harm than good? $\endgroup$ – G Tony Jacobs Aug 31 '17 at 3:57
$\begingroup$ To fully 'play dump' is IMO usually not a good idea for this site; but IMO the example you give does not fully 'play dump' as it give a clear indication of what commenter thinks the problem is. Would it be only the first sentence I'd say it is unhelpful. Whether "I don't understand what you've written. Is there meant to be a dx at then end of that integral?" or "hey, you forgot part of the notation." goes over better is a matter of personal taste. Personally I'd find the 'hey' a bit odd for example. $\endgroup$ – quid♦ Aug 31 '17 at 18:51
$\begingroup$ I think it's clear that my example wasn't optimal to illustrate my question. :/ $\endgroup$ – G Tony Jacobs Aug 31 '17 at 18:54
$\begingroup$ I apologize: I wasn't aware at the time (before your edit) that you are speaking of "pedagogy" (rough translation from Greek: to guide children). People will be grateful to know how you see them, but I'm not interested, I'm a mathematician. If somebody gives me three ambiguous terms of a series and expects I should guess the general form and give them the sum in closed form, I find that rude, and I may tell that person, with or without your permission. Playing dumb I leave to others (and not all are just playing). $\endgroup$ – Professor Vector Aug 31 '17 at 19:16
$\begingroup$ @ProfessorVector, I have no idea why you're commenting here. Nothing you've written has been relevant or helpful. Your allusion to my "permission" is bizarre, and the example you cite is utterly irrelevant to this discussion. Your etymological "reasoning" is beneath anyone who would call himself or herself a mathematician. Why you feel the need to "contribute" to this discussion is utterly unclear. $\endgroup$ – G Tony Jacobs Aug 31 '17 at 19:28
Having been on the Internet a while now, I know that no matter how genuinely confused I am, and no matter how politely I try to phrase my question, there can still be someone who thinks I am just trying to be a jerk. Wikipedia has a policy, not observed as universally as it should be, that one should assume that others are acting in good faith. I think this is a good policy, not just on Internet message boards but in life.
Here is one of many examples. The querent had used the symbol ≤ to compare two groups. I asked for clarification:
What do you mean by ≤ here? Does that mean that (group A) is a subgroup of (group B)?
I would have been very unhappy if the reply had been some variation on "of course it means that, stop being an ass". (It wasn't; the reply was flawless: "Yes, that is what I meant".)
Even if I had intended to be an ass, nothing would be gained from a rude or defensive response. Simply replying "Yes, that is what I meant", as the querent did, is always superior. This was a successful interaction. I think we both got it right.
Summary: On the Internet, it can be hard to tell sometimes if people are trying to be jerks. Sometimes they are trying to be jerks. But it is nevertheless better to act, as much as possible, as though they are not. Haters gonna hate; we don't have to let them bring everyone else down too.
MJDMJD
$\begingroup$ From the Be nice policy: "Be welcoming, be patient, and assume good intentions." $\endgroup$ – Simply Beautiful Art Aug 30 '17 at 19:48
$\begingroup$ Yes. But also, even if you suspect the intentions might be bad, it is often best to behave as if you thought they were good. Because, if you were mistaken, and they really were good, it is obviously better, and even if your suspicions were correct and the intentions were bad, at least only one person is acting like a jerk. $\endgroup$ – MJD Aug 30 '17 at 19:52
$\begingroup$ I am aware of the "assume good faith" principle, and have argued for it voluminously on Wikipedia. Unfortunately, that's not what I was asking about here. $\endgroup$ – G Tony Jacobs Aug 30 '17 at 20:00
$\begingroup$ Having now reread your question, I still think my answer addresses it directly. $\endgroup$ – MJD Aug 30 '17 at 20:02
$\begingroup$ Then it's apparent that I need to clarify my original question. $\endgroup$ – G Tony Jacobs Aug 30 '17 at 20:05
$\begingroup$ There's a difference between, "how should we respond to behavior X?", and on the other hand, "is behavior X considered helpful?" I am in complete agreement with you about the answer to the first question; I was asking the second one. $\endgroup$ – G Tony Jacobs Aug 30 '17 at 20:09
$\begingroup$ And my point is that neither you nor anyone else can be sure of recognizing behavior X when you see it, so there is little factual basis for any opinions about the behavior. $\endgroup$ – MJD Aug 30 '17 at 20:33
$\begingroup$ I'm not claiming I can be sure of recognizing it. I'm addressing the prior question, of whether one should engage in it in the first place. Maybe this behavior is good and helpful, and I should incorporate it into my technique? There are people who will cheerfully admit to doing it, and defend it, so there's no need to engage in forensics to recognize it in this context. You may find the behavior unhelpful, but that's not a universally shared view. $\endgroup$ – G Tony Jacobs Aug 30 '17 at 21:25
$\begingroup$ It's not about recognizing it. I was trying, with this question, to generate a conversation about the pros and cons of "playing dumb" as a pedagogical technique. $\endgroup$ – G Tony Jacobs Aug 30 '17 at 21:26
I'd say it's completely down to the particular tone the user uses. Using "playing dumb" to point out user mistakes, on its own, is not bad in my opinion.
I think, so long as it isn't taken to the extreme of making fun of the OP, this is a good way to teach the OP to be exact when using mathematical language. Often, if a user learns how to properly use mathematical language, he has learnt a much more valuable lesson than just how to solve some particular integral - and after all, this site is here to help people be better at math moreso than solving their homework. Naturally, if the OP then edits and improves his question, the answer must either be deleted or edited - if not, it certainly is un-helpful.
I think it is related to providing "technically correct" answers, such as if a user asks "Prove there is no number $x$ such that $x^2=2$", and the first answer will be "We cannot prove that, because $x=\sqrt{2}$ solves that equation".
No, I don't think the practice is bad in itself, however overdoing it can prove unhelpful. Going overboard and doing it when someone forgets a $dx$ is taking it too far, but that doesn't mean the whole concept is bad.
Trevor Gunn
5xum5xum
$\begingroup$ "Naturally, if the OP then edits and improves his question, the answer must either be deleted or edited - if not, it certainly is un-helpful." There is a problem there, because in general it is frowned upon to edit a question in such a way that it invalidates existing answer posts. It is thus better not to give actual answers to questions that one believes to be misstated (a comment is fine though). Except maybe if one also anticipates the correction. $\endgroup$ – quid♦ Aug 30 '17 at 16:35
$\begingroup$ I prefer TL;DR to be up front, so I don't read the entire thing before getting to it. $\endgroup$ – Simply Beautiful Art Aug 30 '17 at 19:45
$\begingroup$ In the spirit of @quid's comment, I agree that questions or suggestions about clarifying the questions are better placed in comments than in answers. $\endgroup$ – G Tony Jacobs Aug 30 '17 at 21:32
$\begingroup$ @GTonyJacobs true but a long winded hint on specific cases can exceed the 599 character limit of a comment so without making multiple comments sometimes there may be no other easy options as well. $\endgroup$ – user451844 Aug 31 '17 at 1:59
$\begingroup$ I have never tried to leave a hint that exceeded 599 characters. One for the bucket list, eh? $\endgroup$ – G Tony Jacobs Aug 31 '17 at 4:56
$\begingroup$ @GTonyJacobs Hyperlinks are pretty long $\endgroup$ – Simply Beautiful Art Sep 12 '17 at 21:43
Sometimes these things can be borderline as written, and it's necessary to ask for clarification. Because of the differences in our backgrounds, something like "Is there meant to be a $dx$ at then end of that integral?" could be intended in all good faith.
But also much hinges on presentation. Text on the internet is, in general, a notoriously ambiguous mode of communication.
As a matter of courtesy, one should try to soften such corrections. Actually, I don't think the example you gave is so very bad. It's certainly within the limits of slack which you cut people you don't know very well.
On the other hand, I have seen things where posters, who should have known better, have said things along the lines of "Your question about conditions on a set $X$ to be a group is utterly wrong because what if $X=\emptyset$?" when there was, in all likelihood, an innocent omission on the poster's part. That sort of obtuse comment is counterproductive, but it could be presented in a better way.
rschwiebrschwieb
$\begingroup$ Just recently there was a question about the topological properties of a certain subset of $\Bbb R$, which neglected to specify the topology. Someone who should have known better posted a completely po-faced reply about the properties of the set in the discrete topology. I was happy to see this deliberately unhelpful answer downvoted into oblivion. $\endgroup$ – MJD Aug 30 '17 at 18:15
$\begingroup$ @MJD Exactly${}{}{}$ $\endgroup$ – rschwieb Aug 30 '17 at 18:29
$\begingroup$ I remember that question, about whether $(0,1)$ and $[0,1]$ were homeomorphic. Doing that stuff about the discrete topology is fine, if you also talk about why specifying the topology is important, and answer the question from the perspective of the usual topology on $\mathbb{R}$ as well. $\endgroup$ – G Tony Jacobs Aug 31 '17 at 14:45
"Playing dumb", as you put it, is a principal and important method of teaching, not restricted to mathematics. I've found it works well when used correctly. I use it in my office hours, and I use it on math.stackexchange.
The situation where it applies is where a little error signifies to me a principle point of misunderstanding. In such a situation, I ask a simple question about the format or meaning of the original question, either to the student in my office hours or in a comment/answer here on math.stackexchange. My intent is to force the asker to ponder what they actually meant to ask, and sometimes this is all that it takes for them to reach understanding.
Now, this can be hard to judge, and sometimes it doesn't work. But often it does, as is proved on math.stackexchange when various answerers miss the point and post answers which go over the OP's head, whereas I "play dumb" and my comment/answer hits the point.
Lee MosherLee Mosher
$\begingroup$ This is the kind of answer I was hoping for, even though I don't, a priori, agree with you. I asked this question to challenge my own preconceived notions. Can you please cite an example, preferably from Stack Exchange, where this strategy is used correctly and effectively? My personal experience with it has been negative, so that's why I ask. $\endgroup$ – G Tony Jacobs Sep 1 '17 at 15:03
$\begingroup$ I'll try to find one, though I admit that it doesn't come up often and it may be hard to find. $\endgroup$ – Lee Mosher Sep 1 '17 at 15:04
I have very frequently added the $dx$ and done similar things. If I were to see $\displaystyle \int f(x,y)$ then I might ask whether that was intended to be $\displaystyle \int f(x,y) \, dx$ or $\displaystyle \int f(x,y)\, dy.$
There are actual cases where I don't know what is meant and I ask. If someone asks what is intended I would not start from the assumption that it's sarcasm. Once on Wikipedia someone said they objected to the existence of a certain Wikipedia article on the grounds that a mathematical equation should not be the subject of a Wikipedia article unless it's an earth-shaking new discovery. I expressed some objection to that position and he allowed that I had a point, but then added: "but you know what I meant." I didn't know what he meant. Further discussion revealed that he thought it was obvious that a mathematical equation should not be the topic of a Wikipedia article, and he assumed that would be obvious to me, and that he didn't know what an "equation" is. It would have been easy for me to think he was being sarcastic, and maybe he thought I was being sarcastic. Sometimes someone who claims not to understand what you meant does not understand what you meant.
Michael HardyMichael Hardy
$\begingroup$ Michael, I remember you from Wikipedia. Are you still active there? I was GTBacchus. I hear what you're saying here, but I think I can generally tell the difference between genuine requests for clarification and this particular rhetorical technique. In any event, my question is about the rhetorical technique, not about honest questions. $\endgroup$ – G Tony Jacobs Aug 30 '17 at 14:59
$\begingroup$ "Sometimes someone who claims not to understand what you meant does not understand what you meant." You mean sometimes, people actually mean what they say?! This is mind blowing, we have to tell the world! $\endgroup$ – 5xum Aug 30 '17 at 14:59
$\begingroup$ I'm not sure how I came across as assuming all requests for clarification are sarcasm. I don't assume that at all. $\endgroup$ – G Tony Jacobs Aug 30 '17 at 15:00
$\begingroup$ @GTonyJacobs : I didn't think you thought that ALL such requests are sarcasm. $\endgroup$ – Michael Hardy Aug 30 '17 at 17:00
$\begingroup$ @GTonyJacobs : I still frequently edit Wikipedia articles but no longer often create new ones. $\endgroup$ – Michael Hardy Aug 30 '17 at 17:01
$\begingroup$ I don't think I phrased my initial question as well as I could have. I knew at the outset that many requests for clarification are simply that. I was trying to ask about a very specific rhetorical technique that many mathematicians use. I call it, "playing dumb". The answers I'm getting pointing out that sometimes requests for clarification are simply that indicates that I didn't phrase my question well. $\endgroup$ – G Tony Jacobs Aug 30 '17 at 17:03
$\begingroup$ @GTonyJacobs, even after reading your comments, I still don't know what you meant by "playing dumb" if you don't mean "requesting clarification when a question is ambiguously phrased and unclear." I suppose you will think that I'm playing dumb, though. $\endgroup$ – Wildcard Sep 6 '17 at 1:13
$\begingroup$ No, I don't think that. I mean situations where a commenter points out an error in notation, not by saying, "I think you mean X", nor by asking, "do you mean X", but by pretending to be confused or misled when they're really not. The OP means Y, but types X. The commenter knows what they meant, but just responds to X anyway, without letting on that they suspect an error. That's what I'm talking about. Some people do that, on purpose. $\endgroup$ – G Tony Jacobs Sep 6 '17 at 1:28
I think some principles apply:
It's incumbent on people seeking help to make their questions clear
Careful use of notation is very important in mathematical questions. Sometimes even tiny changes can radically alter what is being asked.
We have a role not just in answering questions but in helping people learn and understand, and in being able to ask better, more precise (well-formulated) questions.
A better formulated question is often more readily answered by the asker. Numerous times I have been preparing to ask a question and have found that proper preparation in asking the question (care over notation and definitions, clearer expression of the issue etc) has quite a few times led to me answering it myself. Helping people improve their ability to do this -- to formulate questions properly, with all the benefits that brings -- is crucial to helping them get better, faster answers and with their own development as users of mathematics. That is, helping people to arrive at better questions is definitely part of helping them.
There are numerous aspects of doing that; part of it will be editing (at least if it's obvious what the intent is), part will be commenting to explain the issue. Part of it can include "playing dumb" in that comment, as long as it's not done in a rude way - it's a form of instruction with a long history and is at least sometimes quite effective. It does, for example, make it clearer that the responsibility for the question is their own - they should not post any old nonsense and expect others to do all the lifting to make it work as a question.
So I think playing dumb - at least in some situations - is fine, as long as we keep in mind the point: firstly to help the user end up with a good question and secondly to help them write better questions in the future. If it's doing that without putting posters off, I think it's completely fine.
In terms of the original question, it's disingenuous by the ordinary dictionary definition of the word ("not candid or sincere, typically by pretending that one knows less about something than one really does") but I don't think it's automatically rude.
Sometimes it's actually useful to approach it that way because sometimes at least what it seems the user is asking is not what they actually wanted to ask, and presuming you know less than you think you do may - in some cases - be more accurate, and perhaps less rude.
Glen_bGlen_b
$\begingroup$ This is a great answer. Too bad it's also a late answer, and so isn't likely to get the attention it deserves. $\endgroup$ – Wildcard Sep 6 '17 at 1:16
$\begingroup$ @Wildcard :-/ I want a bounty on meta. $\endgroup$ – Simply Beautiful Art Sep 12 '17 at 21:47
In math, I know that there is a lot that I don't know. If I make an assumption about something, I could very well be wrong.
For example, I think I understand the basics of calculus, but I'm not secure enough in that knowledge to teach it to someone else nor even to correctly guess what is meant when something is omitted.
In your integral example, I wouldn't be sure whether $dx$ or $dt$ is meant. Without Michael Hardy's answer, I wouldn't have even thought of $dy$ as a possibility.
For basic arithmetic, I might be more confident if I think the asker is unaware of operator precedence, but I still would not assume that someone who writes a - b/c meant (a - b)/c.
On the other hand, I think it would really be playing dumb if I were to say I honestly think $$\prod_{i = 1}^n 1 - \frac{1}{p_i}$$ could possibly mean $$\left(\prod_{i = 1}^n 1\right) - \frac{1}{p_i}.$$
Although of course coming from someone else that could be a genuinely sincere query that is misunderstood as sarcastic.
Maybe instead of saying "I don't understand X" it would be better to say "I think you meant to write Y rather than X." Then hopefully the response is either "I really did mean X" or "You're right, I'll change it accordingly," not "What the hell is your problem?"
Robert SoupeRobert Soupe
$\begingroup$ I'm really beginning to regret how I phrased this question. In my example, there was not meant to be a $t$ anywhere in sight. The point was to ask whether intentionally playing dumb, when you really do know what they meant, is more helpful or more harmful. A couple of people have answered saying that they think it's a good strategy. I wanted to know why. Personally, I agree with you that, "I think you meant Y rather than X, is that right?", is almost universally better. $\endgroup$ – G Tony Jacobs Sep 2 '17 at 22:24
$\begingroup$ I would say harmful. In the black and white of text, there are few cues to distinguish the helpful teacher from the jerk. $\endgroup$ – Robert Soupe Sep 2 '17 at 23:06
$\begingroup$ "...I might be more confident if I think the asker is unaware of operator precedence, but I still would not assume that someone who writes a - b/c meant (a - b)/c." - good grief, and just when I was starting to see that "90% of people will get this arithmetic question wrong" stupidity again in a number of other places. :o $\endgroup$ – J. M. is a poor mathematician Sep 3 '17 at 5:04
Some people are not comfortable telling directly to others they are wrong. It can be because they are afraid to sound rude. Or simply because there is still a small possibility of they are the one that are wrong and are don't want to loose face. Then, "playing dumb" is not a snarky way to dismiss the OP and point out an obvious mistake, but a polite way to speak and a form of etiquette.
A real-life example: A highly reputed Japanese professor told me about his PhD student in that way: "He is really brilliant and very passionate. In life, he will always respect me and stay polite, but when we do maths together, he has no hesitation telling me directly when I am wrong".
A SE example: on another SE site about language with many non-native English users, someone answered one of my question starting by "I was trolling on the web to find an answer..." and I commented something like "There is probably a typo. You meant strolling I think". It appeared that I was wrong and the use of to troll here was correct. I would have felt embarassed if I had made a more direct comment like "Hey, you meant strolling on the web"
My opinion is that we should not think too much about people playing dumb. Most of the time, the OP will get the content of the comment, understand (s)he is wrong and correct the question. Or someone else will do it.
TaladrisTaladris
$\begingroup$ Yeah, this question was not intended to be about others playing dumb. This question was intended to be about whether or not I should do it. Is it a good way to teach? Totally different thrust, you know? $\endgroup$ – G Tony Jacobs Sep 13 '17 at 3:19
$\begingroup$ Ah sorry, I understood the question as "Shall we do it or not?" as a collective. I would suggest to always give the other the benefit of the doubt and "playing dumb" a little will not hurt you and may make the other person more comfortable. $\endgroup$ – Taladris Sep 13 '17 at 3:31
$\begingroup$ In my experience as a student, it has made me more uncomfortable, but I wonder if that's just me, or just the way that teacher did it. Interesting. $\endgroup$ – G Tony Jacobs Sep 13 '17 at 3:32
$\begingroup$ I guess it is a problem of degree, personal taste and culture (starting a comment by "hey" sounds rude to my French ears). I tend to play dumb in my first comment, and become more direct afterwards if I feel it necessary. $\endgroup$ – Taladris Sep 13 '17 at 3:37
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