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Effectiveness and cytotoxicity of two desensitizing agents: a dentin permeability measurement and dentin barrier testing in vitro study
Ruodan Jiang1,3,
Yongxiang Xu1,3,
Feilong Wang2,3 &
Hong Lin1,3
When evaluating the efficacy and safety of various desensitizing products in vitro, their mechanism of action and clinical utility should be considered during test model selection. This study aimed to evaluate the effects of two desensitizers, an in-office use material and an at-home use material, on dentin specimen permeability, and their dentin barrier cytotoxicity with appropriate test models.
Two materials, GLUMA desensitizer (GLU) containing glutaraldehyde and remineralizing and desensitizing gel (RD) containing sodium fluoride and fumed silica, were selected. Human dentin specimens were divided into three groups (n = 6): in groups 1 and 2, GLU was applied, and in group 3, RD was applied and immersed in artificial saliva (AS) for 24 h. Dentin specimen permeability before and after each treatment/post-treatment was measured using a hydraulic device under a pressure of 20 cm H2O. The perfusion fluid was deionized water, except in group 2 where 2% bovine serum albumin (BSA) was used. The representative specimens before and after treatment from each group were investigated using scanning electron microscopy. To measure cytotoxicity, test materials were applied to the occlusal surfaces of human dentin disks under which three-dimensional cell scaffolds were placed. After 24-h contact within the test device, cell viability was measured via 3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide (MTT) assays.
GLU significantly reduced the dentin permeability and occluded the dentinal tubules when 2% BSA was used as perfusion fluid. RD significantly reduced dentin permeability and occluded the tubules, but permeability rebounded after AS immersion. GLU significantly decreased cell viability, but RD was non-cytotoxic.
In vitro GLU application induced effective dentinal tubule occlusion only following the introduction of simulated dentinal fluid. RD provided effective tubule occlusion, but its full remineralization potential was not realized after a short period of immersion in AS. GLU may harm the pulp, whereas RD is sufficiently biocompatible.
Dentin hypersensitivity (DH) is a transient and sharp pain experienced when exposed dentin encounters thermal, mechanical, or chemical stimuli [1]. The prevalence of DH can reach over 90% depending on the population and methodology used [2,3,4]. According to the widely accepted "hydrodynamic theory" [5], the main factor causing DH is dentin permeability. Therefore, treatment strategies include sealing of the dentinal tubules to reduce dentin permeability and the activity of pulp nerves [6]. Dental materials for the occlusion of open dentinal tubules include inorganic fillers, polymers, protein denaturing materials, and tooth remineralization materials [7,8,9], which are marketed as in-office or at-home use products. With the continuous introduction of novel desensitizing agents, it is particularly important to evaluate their effectiveness and biocompatibility using appropriate in vitro models.
Scanning electron microscopy (SEM) can be used to directly observe dentinal tubule occlusion; however, the sealing shown on the dentin surface morphology does not necessarily lead to reduced permeability. The deposits produced by desensitizers may not be solid [10]. Moreover, materials that denature protein, such as glutaraldehyde, act on the dentinal fluid, which may not be observed in vitro [8]. Some studies have shown in SEM images that glutaraldehyde-containing desensitizers occlude the orifices of dentinal tubules [11, 12]. However, Pereira et al. indicated that glutaraldehyde-containing desensitizers did not produce particle precipitation at the opening or within the tubules as glutaraldehyde reacts with the plasma proteins present in the dentinal fluid [13]. Furthermore, the dentinal fluid may no longer remain after cutting and rinsing the dentin specimens in an in vitro test [14].
The evaluation of dentin permeability can directly reflect the occlusive effect of certain materials over dentinal tubules; hydraulic conductance is the main in vitro method of evaluation [6, 15]. Some studies have applied this method and designed different treatment procedures to simulate the oral environment, including its inherent chemical and mechanical challenges [8, 16,17,18,19], and others have attempted to introduce simulated dentinal fluid into the dentin specimens [8, 14]. The International Organization for Standardization has not provided standards for the evaluation of desensitization efficacy. Thus, evaluation methods close to the product mechanism and clinical conditions are more reasonable and acceptable.
Dentin desensitizing agents are materials used in direct contact with the dentin; therefore, cytotoxic substances may reach the pulp through the dentinal tubules. To achieve therapeutic effects, the long-term and repetitive use of desensitizing agents is usually recommended; therefore, these products must have good biocompatibility. Dentin desensitizing products usually must be tested for cytotoxicity before market release. These products contain various chemical components, some highly cytotoxic, such as fluoride, glutaraldehyde, and 2-hydroxyethyl methacrylate (HEMA) [20,21,22,23]. Glutaraldehyde exerts its cytotoxic effect over a wide concentration range [23]. HEMA can inhibit the proliferation of epithelial cells and pulpal fibroblasts [20]. However, some materials containing these components have no toxic or acceptable effects on dental pulp in vivo [24, 25]. Traditional in vitro cytotoxicity testing combined with a monolayer cell culture cannot simulate the three-dimensional (3D) cell growth observed in vivo, which may explain the discrepancy between in vivo and in vitro study results.
The dentin barrier test evaluates the chemical toxicity to the pulp tissue of dental materials contacting the dentin by mimicking the contact process between the materials and teeth and can predict clinical behavior with a reasonable probability, which may therefore help replace animal experiments [20,21,22, 26, 27]. At present, this test is not widely used to evaluate desensitizers. Previous studies on dentin barrier models testing desensitizers' cytotoxicity to cells in a monolayer culture revealed different degrees of cytotoxicity [20, 22].
There are some debatable results in the literature regarding glutaraldehyde-containing densensitizers, which have been shown to obtain effective dentinal tubule occlusion without the introduction of simulated dentinal fluid [11, 12]. Other studies immersed the dentin specimens in diluted bovine serum before applying the desensitizing treatment [28, 29], though this is not a realistic substitute for dentinal fluid due to bovine serum's high viscosity [30]; it is necessary to evaluate the permeability of glutaraldehyde-containing desensitizers with a suitable perfusion fluid. Products that claim the ability of remineralization should demonstrate recrystallization during the desensitizing process, which can be partially detected by permeability and SEM. There are certain advantages to investigating the remineralization of desensitizers through the evaluation of permeability. The hydraulic pressure simulates the actual physiologic conditions of the body, so only the stable remineralized crystals can be maintained, thereby reducing dentin permeability. For both types of desensitizers, a dentin barrier cytotoxicity test was more applicable than the direct contact methods. Moreover, to our knowledge, no study has combined the dentin barrier test with 3D cell cultures to assess desensitizers.
Therefore, this in vitro study aimed to evaluate the effectiveness of two desensitizing agents (GLUMA desensitizer (GLU) and Remineralizing and Desensitizing gel (RD)), an in-office use material and an at-home use material, respectively, using hydraulic conductance and SEM observation, and evaluate the cytotoxicity of these two desensitizing agents through a dentin barrier test. Two different perfusion fluids, deionized water and 2% BSA, were chosen for GLU, and a post-treatment procedure of remineralization was chosen for RD. 3D cultures of transfected rat dental papilla-derived cells were used in the cytotoxicity test. The null hypotheses were: (1) application of GLU would be effective in reducing dentin permeability, whether deionized water or 2% BSA was used as the perfusion fluid, and the dentin specimens applied with RD would maintain the same permeability reduction after and before 24-h AS immersion; (2) neither GLU nor RD would demonstrate significant cytotoxicity in a dentin barrier cytotoxicity test.
This study was approved by the Biomedical Ethics Committee of the Peking University School and Hospital of Stomatology (Process #PKUSSIRB-202060195). The authors stated that all methods were carried out in accordance with relevant guidelines and regulations. For the collection of isolated teeth, informed consent from patients was obtained.
Test materials
Two desensitizers, GLUMA Desensitizer (GLU: Heraeus Kulzer, Hanau, Germany) and Remineralizing and Desensitizing gel (RD: American Hi Teeth Science and Technology Inc., USA), were used in this study and applied to the occlusal side of the dentin disks as per the manufacturers' instructions. The details of all test and control materials are shown in Table 1.
Table 1 Tested desensitizing agents
Tooth collection and dentin disk preparation
Thirty-eight extracted sound human third molars were obtained from patients aged 20–40 years. After removing debris and soft tissues, the teeth were soaked in 70% ethanol for 15 min [31] and stored in deionized water at 4 °C until use.
Dentin disks of 500 ± 50-μm were obtained by cutting the teeth perpendicular to their long axes, close to the pulp cavities [31], using a low-speed saw, (Isomet-Buehler, Lake Bluff, IL, USA) and all dentin disks were acid-etched with 35% phosphoric acid on both sides for 30 s each, using a previously described method [21, 32]. Each crown provided one disk, with an intact test area in each disk.
Dentin permeability measurement
The dentin disks were randomly distributed into three groups (n = 6 in each group). Hydraulic conductance was measured for each acid-etched dentin disk and determined as the baseline permeability. In all groups except group 2, deionized water was used for perfusion.
Group 1: A small amount of GLU was applied to the acid-etched specimens and retained for 60 s. Subsequently, the specimens were air-dried, rinsed in deionized water, and measured immediately for permeability.
Group 2: The process was same as that for group 1; however, the perfusion fluid was replaced by 2% bovine serum albumin (BSA, Cohn Fraction V, pH 6.7; Kangyuan Biotechnology, Tianjin, China).
Group 3: RD gel was applied to acid-etched specimens and retained for 15 min. Subsequently, the specimens were rinsed in deionized water and measured immediately for permeability. These specimens were stored in artificial saliva (AS, Solarbio, Beijing, China) at 37 °C for 24 h and rinsed with deionized water; then, permeability was measured again.
AS was composed of deionized water, NaCl, KCl, Na2SO4, NH4Cl, CaCl2·2H2O, NaH2PO4·2H2O, CN2H4O, and NaF (pH 6.5). After the baseline permeability values were recorded, all treatments were conducted within 30 min. Representative SEM micrographs of the dentin disks were obtained.
Hydraulic conductance test
The hydraulic conductance equipment was made in-house, as previously described [21, 32], according to the model designed by Outhwaite et al. and Pashley et al. [15, 33] (Fig. 1). The whole equipment was filled with perfusion fluid. The water bath provided a constant pressure of 20 cm H2O (1.96 kPa) to the pulp side of the dentin disk. A pair of rubber "O" rings limited the measurement area to 0.28 cm2 at the center of the dentin disk. One tiny air bubble was injected into a 100 µL-micropipette. The experiment was conducted after the air bubble kept moving steadily for 10 min. Each measurement was completed within 20 min. The minimum division value of the micropipette was 5 µL.
Dentin permeability testing device (hydraulic conductance device) used in the present study
The volume of the perfusion fluid filtering through the dentin disk was measured by the linear displacement of the air bubble within a defined time period. Dentin permeability/hydraulic conductance, Lp (µl•min−1•cm−2•cm H2O−1), was calculated as:
$${\text{Lp}} = {\text{Jv}}/\left( {{\text{A }} \times {\text{ t }} \times {\text{ P}}} \right),$$
where Jv is the volume of fluid through the dentin disks (µL), A is the measurement area (cm2), t is the observation time (min), and P is the fluid pressure (cmH2O).
SEM (EVO 18, Zeiss, Oberkochen, Germany) was used to observe the dentin specimens transversely and longitudinally before and after the desensitization treatments. Three dentin disks were prepared for each observation group. The specimens were dried in a desiccator for 48 h, fractured into two equal parts, sputter-coated with gold, and then observed under SEM at 10 kV or 20 kV at the selected magnifications of × 750, × 10,000 and × 5000.
Dentin barrier cytotoxicity
Before the test, the hydraulic conductance of the dentin disks was assessed. The dentin disks with similar permeability were selected and divided into four groups (n = 5 in each group), ensuring that the mean value of permeability in each group was as close as possible. The four groups were randomly divided into two test and two control groups. The grouped dentin disks were used within three days.
During the cytotoxicity test, the smear layer on the occlusal sides of the dentin disks was reconstructed by grinding the occlusal sides of the dentin disks with 400-grit sandpaper at a consistent frequency and pressure for 15 s, and the dentin specimens were disinfected using 70% ethanol, as previously described [21, 32]. Concerning desensitizing agent application, GLU was applied on the occlusal surfaces of the dentin disks and kept for 60 s. Thereafter, the residual agent was air-dried. RD was applied on the occlusal surfaces of the dentin disks and retained for 15 min. The excess agent was wiped off using cotton swabs.
SV40 large T-antigen-transfected rat odontoblast-like cells, obtained from the rat dental papilla, were maintained in a minimum essential medium α medium (Gibco, USA) supplemented with 10% fetal bovine serum (Gibco, USA), 100 IU/mL penicillin, and 150 mg/mL streptomycin at 37 °C in 5% CO2 and 95% relative humidity. The 15th to 20th passage cells were used in this study.
The 8-mm diameter polystyrene 3D scaffolds (Nanjing Recongene, Nanjing, China), with four fiber layers, as previously described [21], were used for 3D cell culture. The scaffolds were placed in six-well culture plates and incubated for 48 h after adding 2 mL of cell suspension (1.5 × 105 cells/mL). The cell-seeded scaffolds were moved to 24-well culture plates and cultured for 14 ± 2 d, changing the growth medium every other day.
Dentin barrier cytotoxicity testing
The cell-seeded scaffolds were transferred into a 3D cell culture system (3D Biotek, New Jersey, USA) as previously described [21]. The main component, a polycarbonate split chamber, was a cylindrical cavity (Fig. 2). The dentin disk (occlusal side facing upward) was placed on top of the scaffold. The lower compartments of all split chambers were perfused with growth medium along with a 6 g/L HEPES buffer at 0.3 mL/h for 24 h at 37 °C. The liquid level of the growth medium covering the cell scaffolds was below the dentin disks. The test materials were applied and placed in contact with the dentin disks at 37 °C for 24 h after the perfusion was switched off [31].
Diagram (a) and photograph (b) of the split chamber
The cell-seeded scaffolds were moved to a 24-well plate containing 0.5 mL MTT solution (Amresco, USA; 1 mg MTT/mL in PBS) and incubated for 2 h. Then, 250 μL of dimethyl sulfoxide was used to dissolve the formazan precipitate, and 200 μL of this solution was transferred into a 96-well plate, to determine the spectrophotometric absorbance at 540 nm.
All statistical analyses were performed using SPSS software version 20.0 (SPSS, Chicago, IL, USA). For the dentin permeability measurement, Lp values were expressed as means and standard deviations, and the relative Lp values after treatments were expressed as percentages of baseline permeability values. The Shapiro–wilk test and Levene test were used to determine the normality and homoscedasticity of the data, respectively. The non-parametric Kruskal–Wallis test was applied to analyze the difference in baseline Lp values between the groups. The Friedman test was used to compare changes in Lp values before and after treatment/post-treatment within each group. A P-value < 0.05 was considered statistically significant. For dentin barrier cytotoxicity, results were expressed as percentages of the negative control. Statistical comparisons between groups were performed using the Mann–Whitney U test (α = 0.05).
Dentin permeability measurements
The mean Lp values (± standard deviation [SD]) and their percent changes are shown in Table 2. The Kruskal–Wallis test revealed no significant differences in the baseline permeability between the three test groups (P = 0.172). The Friedman test showed a significant difference in the Lp values before and after treatment in group 2 (P = 0.014). Specifically, using 2% BSA as the perfusion fluid, dentin permeability decreased by 82% after GLU application. However, when deionized water was used as the perfusion fluid, dentin permeability in group 1 increased by 7% after GLU application, although the difference before and after treatment was insignificant (P = 1.000). In group 3, RD significantly reduced dentin permeability by 75% after 15 min of use (P = 0.014). After subsequent immersion in AS for 24 h, mean permeability rebounded by 28%, although there was no significant difference compared with the value before AS immersion (P = 1.000).
Table 2 Dentinal permeability measurements (Lp; mean ± s.d.) of different desensitizing agents, before and after treatments
Figure 3 shows the SEM micrographs of the dentin specimen surfaces and longitudinal sections of the three groups. Almost all dentinal tubules were open after etching, indicating that the smear layer and plugs had been removed (Fig. 3a1, a2). The dentinal tubules were almost empty (Fig. 3a3).
SEM images of dentin specimen surfaces and longitudinal sections before and after treatments/post-treatment. Magnifications: × 750 (left), × 10,000 (middle) and × 5000 (right). a1–a3 SEM image obtained after acid etching in 35% phosphoric acid for 30 s, showing that the dentinal tubules are completely open and that the inside of the tubules is empty. b1–b3 SEM image obtained after GLU treatment in group 1, showing that the dentinal tubules are completely open and that the collagen mesh of the demineralized dentin presumably has a certain degree of crosslinking by glutaraldehyde, but has not collapsed. c1–c3 SEM image obtained after GLU treatment in group 2, showing that, under the effect of glutaraldehyde, nearly half of the tubule orifices are occluded due to the precipitation of the serum albumin remaining in tubules. Multiple reticular septa are observed in the lumen of the dentinal tubules. d1–d3 SEM image obtained after RD treatment in group 3, showing that most tubule orifices are occluded by deposits and that the diameter of the tubules is reduced. A small amount of granular deposit is observed on the wall of tubules. e1–e3 SEM image obtained after post-treatment by 24-h AS immersion in group 3, showing that most tubule orifices are exposed and that the amount of deposit inside the tubules has reduced. Some crystalline substances are observed inside the tubules. GLU, GLUMA Desensitizer; RD, Remineralizing and Desensitizing gel; SEM, scanning electron microscopy
Figure 3b1–b3 shows the SEM images of the dentin specimens after GLU treatment in group 1. The surface morphology was similar to that after etching; most dentinal tubules were open, clear, and free of debris. Under the action of glutaraldehyde, the collagen mesh of the demineralized dentin seemed to cross-link to a certain depth but did not collapse (Fig. 3b3).
Figure 3c1–c3 shows SEM images of the dentin specimens after GLU treatment in group 2. Nearly half of the tubule orifices were blocked (Fig. 3c1). Since 2% BSA was the perfusion fluid in group 2, the serum albumin remaining in the tubules was probably precipitated by glutaraldehyde, which occluded the orifices and the interior of the dentinal tubules (Fig. 3c2, c3). Multiple septa were observed at a certain depth in the lumen of the dentinal tubules, and the reticular-like septa were in contact with the tubular walls (Fig. 3c3).
Figure 3d1–d3 presents SEM images of the dentin specimens after RD treatment, and Fig. 3e1–e3 presents SEM images after post-treatment by immersing in AS for 24 h in group 3. After RD treatment, the dentin surface was covered with a dense layer of deposit coating obtained from reaction products, and most tubule orifices were occluded (Fig. 3d1, d2). The tubules were evidently narrowed, and a small amount of granular deposit was found inside the tubules (Fig. 3d3). After post-treatment, many occluded tubule orifices reopened (Fig. 3e1, e2), and the amount of deposit inside the tubules decreased (Fig. 3e3) but more crystalline substances were observed inside the tubules (Fig. 3e3) than that in the specimens in Fig. 3d3.
The test and statistical results are summarized in Fig. 4. GLU reduced cell viability to 11%, the result was not significantly different from that of the positive control (P = 0.310). Thus, GLU was severely cytotoxic. RD exhibited a cell viability of 90%, which was not significantly different from the result of the negative control (P = 0.421), showing the non-cytotoxicity.
Cell viability of the 3D cultures of transfected rat odontoblast-like cells. Results are expressed as the percent cell viability relative to the negative control. The indicated values are the median and 25th and 75th percentiles. Different lower-case letters indicate statistically significant differences between groups (P < 0.01). GLU, GLUMA Desensitizer; RD, Remineralizing and Desensitizing gel
Based on the results, GLU significantly reduced the dentin permeability only when 2% BSA was used as perfusion fluid. RD significantly reduced dentin permeability, but the permeability rebounded after AS immersion. Thus, the first hypothesis was rejected. GLU significantly decreased cell viability and the second hypothesis was rejected.
The present in vitro study evaluated the effectiveness and dentin barrier cytotoxicity of two desensitizing agents, an in-office use material and an at-home use material, using experimental models closer to the principle of material action and actual use in vivo.
Dentin permeability measurements revealed that when the perfusion fluid was deionized water, GLU did not occlude the dentinal tubules or affect dentin permeability. However, GLU significantly decreased the dentin permeability and occluded the dentinal tubules when the perfusion fluid was replaced by 2% BSA. The two active components of GLU are glutaraldehyde and HEMA. According to the literature, glutaraldehyde precipitates serum albumin in the dentinal fluid, and the coagulated proteins can form protein plugs that close the dentinal tubules [34, 35]. This protein coagulation leads to HEMA polymerization, and HEMA can facilitate glutaraldehyde penetration up to a depth of 200 μm in the dentinal tubules [34, 35]. Thus, dentin permeability can be significantly reduced [36, 37].
In fact, the dentinal fluid in isolated teeth is probably lost during a number of procedures, including cutting, etching, immersion, and ultrasonic cleaning. This study confirmed this point. In group 1, GLU did not close the dentinal tubules. Although glutaraldehyde appeared to cross-link the collagen mesh (Fig. 3b3), it had no effect on reducing dentin permeability. A similar collagen mesh morphology of the demineralized dentin disk was observed in a previous study using 2.5% glutaraldehyde as a fixation fluid [32]. The dentinal fluid is an important component of the pulp-dentin complex [34]. When the dentin is exposed, the outflowing dentinal fluid contains a fraction of plasma proteins [30]. Albumin is the main protein component in the plasma and dentinal fluid [38]. According to the protein fraction in the plasma and dentinal fluid, 1:3 diluted bovine serum was used to simulate the dentinal fluid in some studies [28, 38], although Özok et al. suggested that it was not a realistic substitute for dentinal fluid because of the relatively large molecular weight of the substances in the serum fraction [30]. Substances in the serum components with a molecular weight > 100,000 Da, such as globulins and lipoproteins, can reduce dentin permeability [30]. According to the literature, 2% BSA can be used as a simulated dentinal fluid [8, 14]. The bovine albumin used herein has a molecular weight of 66,000 Da; therefore, it did not affect the baseline permeability of the dentin. Our results revealed no significant differences in the baseline permeability between the groups.
Furthermore, the introduction of simulated dentinal fluid into tubules is key. Due to the capillary structure of the dentinal tubules, it is difficult for fluid to completely penetrate the tubules of the whole dentin disk during a short-term immersion. Therefore, this study used simulated dentinal fluid as perfusion fluid. After a short balance period, the albumin solution permeated the dentinal tubules, and baseline permeability was measured.
Using SEM, this study was able to detect reticular septa in the tubular lumen (Fig. 3c3), similar to the finding of Schüpbach et al. [35], where tubular occlusions were observed at a depth of 200 μm following the in vivo application of GLU and teeth extraction, and the septa led to the complete closure of the tubular lumen. This is inconsistent with Ishihata et al.'s study, which showed that no septa were found inside the dentin after it was soaked in albumin and GLU was applied [8]. The results from group 1 contradict previous studies which found that glutaraldehyde-containing desensitizers occluded the dentinal tubules or reduced dentin permeability without the introduction of simulated dentinal fluid [11, 12, 39, 40].
The main active ingredients of RD are sodium fluoride and fumed silica. Fluoride can penetrate dental hard tissue and bind with calcium salt, forming calcium fluorapatite deposits, thus occluding the dentinal tubules or reducing their diameter while promoting dentin remineralization [41]. Fumed silica, an amorphous nanoparticle, has a small particle size and large specific surface area, leading to strong surface adsorbability, large surface energy, high chemical purity, and good dispersion performance [42]. Therefore, it can be effectively deposited on the dentin surface to block the tubules. In the present study, the dentin permeability after application of RD was significantly reduced, as verified by SEM. RD may have formed a dense layer of material on the surface of the dentin (Fig. 3d1, d2). Subsequently, permeability increased after 24-h AS immersion. Notably, SD from the relative Lp values (Table 2) after 24-h AS immersion was relatively high, illustrating that the range of permeability changes was wide. According to the data, after a 24-h AS immersion, the permeability of half of the dentin disks increased compared with that after RD use; however, that of the other half decreased.
AS is usually used for in vitro studies to simulate an in vivo situation to assess erosion and remineralization in studies on DH [16, 17]. Immersion in AS may solubilize or wash away most debris from the dentin surface [16, 43], as found in this study (Fig. 3e1, e2). However, deposits inside the tubules were still observed (Fig. 3e3). Furthermore, more crystalline precipitates were found inside the tubules than before AS immersion (Fig. 3d3, e3). Phosphate groups on the surface of the collagen matrix can induce mineralization [44]. According to the report by Besinis et al., with the help of AS containing ionic calcium and phosphate, nano-silica can play the role of nucleating minerals in the collagen of demineralized dentin, enhancing the binding of calcium phosphate compounds to the collagen mesh and thus promoting remineralization [45]. Nanoscale silica can provide greater adhesion to calcium phosphate due to its large surface area, thus enhancing the potential of remineralization [46]. However, a short AS immersion time such as 24 h, may be insufficient to ensure adequate remineralization [17]. In this study, for some dentin specimens, the remineralization process after AS immersion was insufficient to maintain or enhance the occlusion of the dentinal tubules. In addition to the time factor and AS erosion, this may be related to individual differences in dentin specimens, such as tubular density and diameter. Generally, AS erosion plays a leading role, which may explain the rebound in the average relative Lp values after AS immersion. The current results were in line with a related study that also included a 24 h AS immersion [16].
The human dentin has a barrier function that can stop substances from penetrating into the pulp [20,21,22, 26]. Regarding dentin barrier cytotoxicity, GLU significantly decreased cell viability; however, the opposite was true for RD. Based on the existing research [22, 34], the high cytotoxicity of GLU exhibited in a dentin barrier test could be attributed to HEMA. Scheffel et al. found that 2.5–10% glutaraldehyde caused no obvious damage to odontoblast-like cells in a dentin barrier test, while glutaraldehyde integrated with HEMA showed high cytotoxicity [22]. In this cytotoxicity test, glutaraldehyde could react with the collagen to reduce its own concentration, and HEMA could be partially absorbed by dentin and collagen [34]. However, HEMA polymerization could not be completely induced due to inadequate serum albumin levels. It could be speculated that the cell culture medium containing fetal bovine serum cannot be fully introduced into the dentinal tubules due to the insufficient pressure on the pulp side of the dentin disks in this test device. The residual HEMA was the cause of high cytotoxicity.
HEMA has high solubility and low molecular weight, which makes it penetrate the dentin more easily. Even a small concentration of HEMA can irreversibly inhibit the cultured cells [47]. HEMA may cause DNA damage and mutation and even changes in gene expression, resulting in apoptosis [48]. Moreover, Yu et al. reported that at a concentration > 2 mmol/L, HEMA can induce the accumulation of intracellular reactive oxygen species in vitro, inhibit the proliferation and differentiation of dental mesenchymal cells, and activate the intracellular NF-κB pathway to induce autophagy formation [49]. The high cytotoxicity of GLU in this study is consistent with that observed using a similar method [20, 22] and with the precautions in the product instructions that deep cavities or areas close to the dental pulp should be properly capped.
Sodium fluoride has been proved to be cytotoxic in acidic environments [50]. A high concentration of sodium fluoride inhibited the proliferation of human epithelial cells, with different types of cells responding differently to sodium fluoride [51, 52]. Dogan et al. proved that mouse fibroblast cells were more sensitive to sodium fluoride than human keratinocytes and osteogenic sarcoma cells; they did not survive in sodium fluoride at 10 mM [52]. In the current study, RD exhibited low cytotoxicity, possibly because of its occluding effect on the dentin. Colloid components occluded dentinal tubules and precipitated, making it difficult to penetrate dentin disks. This type of desensitizer usually exhibits strong cytotoxicity in conventional in vitro tests, such as the filter diffusion and extract tests, where the contact mode between the materials and cells, cell types, and cell growth state vary from those observed in vivo. A recent study showed that desensitizers containing sodium fluoride were highly cytotoxic to monolayer gingival fibroblast cells; however, as the authors noted,the results of direct contact between materials and cells can be obviously inconsistent with clinical findings [9]. The SV40 large T-antigen-transfected rat odontoblast-like cells used in this study were obtained from rat dental papilla. This transfected cell line can be stably subcultured, have similar physiological properties to those of the dental pulp tissue, and have odontoblastic properties [53]. Combined with dynamic culture status used to simulate the blood flow of pulp tissue in vivo, the 3D culture of transfected odontoblast-like cells can reproduce the physiological and morphological characteristics of pulp cells in vivo to a certain extent [21].
The current study is limited by insufficient time for remineralization and lack of pulpal pressure in the cytotoxicity test, which can be improved in future studies. Considerably more work will need to be done to measure various types of densenitizers using test protocols that more closely mimic in vivo conditions.
Our results revealed that GLU significantly reduced the permeability of dentin disks and occluded the dentinal tubules only when simulated dentinal fluid was used instead of deionized water as the perfusion fluid, indicating that the permeability evaluation of glutaraldehyde-containing desensitizers should use simulated dentinal fluid as perfusion fluid or other equivalent methods. RD significantly reduced dentin permeability and occluded the dentinal tubules, while this effect was decreased after 24-h AS immersion. The simulation of remineralization, such as post-treatment with AS, may be necessary in the evaluation of dentinal permeability reduction by desensitizers claiming to have remineralization potential. In this dentin barrier cytotoxicity test, GLU exhibited a significant cytotoxic effect, but RD was non-cytotoxic. GLU is recommended for use after pulp capping in deep cavities. RD, as an at-home use product, is safer than GLU.
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
Artifical saliva
BSA:
Bovine serum albumin
DH:
GLU:
GLUMA sensitizer
HEMA:
2-Hydroxyethyl methacrylate
MTT:
3-(4,5-Dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide
RD:
Remineralizing and Desensitizing gel
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The authors would like to thank all donors of the extracted teeth.
This work was supported by the Research Foundation of Peking University School and Hospital of Stomatology (PKUSS20200112).
Department of Dental Materials, Dental Medical Devices Testing Center, Peking University School and Hospital of Stomatology, No. 22, Zhongguancun South Avenue, Haidian District, Beijing, 100081, People's Republic of China
Ruodan Jiang, Yongxiang Xu & Hong Lin
Department of Prosthodontics, Peking University School and Hospital of Stomatology & National Center of Stomatology, Beijing, People's Republic of China
Feilong Wang
National Center of Stomatology & National Clinical Research Center for Oral Diseases & National Engineering Research Center of Oral Biomaterials and Digital Medical Devices & Beijing Key Laboratory of Digital Stomatology & Research Center of Engineering and Technology for Computerized Dentistry Ministry of Health & NMPA Key Laboratory for Dental Materials, Beijing, People's Republic of China
Ruodan Jiang, Yongxiang Xu, Feilong Wang & Hong Lin
Ruodan Jiang
Yongxiang Xu
Hong Lin
RJ and HL contributed to the study conception and design. Material preparation, data collection and analysis were performed by RJ. YX and FW contributed to the SEM operation and image processing of this study. The first draft of the manuscript was written by Ruodan Jiang and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
Correspondence to Hong Lin.
The authors declare that they have no conflict of interest.
Jiang, R., Xu, Y., Wang, F. et al. Effectiveness and cytotoxicity of two desensitizing agents: a dentin permeability measurement and dentin barrier testing in vitro study. BMC Oral Health 22, 391 (2022). https://doi.org/10.1186/s12903-022-02424-7
Desensitizing agents
Dentin permeability
Dentin barrier cytotoxicity test
Dentinal fluid
Remineralization | CommonCrawl |
What is the temperature of an accretion disc surrounding a supermassive black hole?
What is the temperature of an accretion disc surrounding a supermassive black hole? Is there plasma in the disc?
supermassive-black-hole temperature accretion-discs
Declan KonroydDeclan Konroyd
$\begingroup$ This is a fantastic resource on accretion disks, and this section in particular may help with accretion rates. $\endgroup$ – HDE 226868♦ Jan 6 '15 at 23:36
It depends on the distance from the central body. This gives the temperature $T$ at a given point as a function of the distance from that point to the center ($R$): $$T(R)=\left[\frac{3GM \dot{M}}{8 \pi \sigma R^3} \left(1-\sqrt{\frac{R_{\text{inner}}}{R}} \right) \right]^{\frac{1}{4}}$$ where $G$, $\pi$, and $\sigma$ are the familiar constants, $M$ is the mass of the central body (and $\dot{M}$ is the rate of accretion onto the body), and $R_{\text{inner}}$ is the inner radius of the disk - possibly (if the object is a black hole) the Schwarzschild radius $R_s$, in which case we can simplify this a little more. So the temperature in the accretion disk is far from constant.
Whether or not there is plasma depends on the exact nature of the disk, the central object and the region around it. For example, a supermassive black hole may have different matter in its disk than that of a stellar-mass black hole. I should think, though, that black holes in binary systems accreting mass from a companion should have plasma in their accretion disks, and supermassive black holes might also have plasma from nearby stars.
$\begingroup$ I imagine, it would be more interesting to see this equation in number form. $\endgroup$ – Alexey Bobrick Jan 6 '15 at 17:22
$\begingroup$ @AlexeyBobrick With specific numbers from an actual accretion disk? $\endgroup$ – HDE 226868♦ Jan 6 '15 at 23:04
$\begingroup$ Yes, like something you can on page 9 of your reference. $\endgroup$ – Alexey Bobrick Jan 6 '15 at 23:16
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Home » Operations Research » Graphical Method to solve 2 by n game
Graphical Method to solve 2 by n game
1 Graphical method to solve $2\times n$ game
2 Step by Step Procedure to solve 2 by n game
3 2 by n game by graphical method Example 1
5 Endnote
Graphical method to solve $2\times n$ game
A game where one player has only two course of action and the other player has more than two (say $n$) course of action is called $2\times n$ game.
Graphical method can be used to solve a $2\times n$ or $m\times 2$ game or a game reducible to either $2\times n$ or $m\times 2$ after applying a dominance property.
In this tutorial we will discuss about the graphical method to solve a $2\times n$ game or a game reducible to $2\times n$ after applying dominance property.
Step by Step Procedure to solve 2 by n game
Consider the $2\times n$ game. In this game player $A$ has $2$ strategies and player $B$ has $n$ strategies. Assume that the game does not have a saddle point.
Player A \ Player B
$B_1$
$\cdots$
$B_n$
$A_1$ $a_{11}$ $a_{12}$ $\cdots$ $a_{1n}$
Following are the steps to solve $2\times n$ game using graphical method.
Apply maximin-minimax principle to check whether saddle point exists or not. If saddle point exists, then stop the method and calculate the value of the game, otherwise go to next step.
Construct two vertical axis, axis 1 and axis 2, using appropriate scales.
Axis 1 represent the payoff values of $A_2$ strategy for player $A$ and axis 2 represent payoff values of $A_1$ strategy for player $A$.
(Graph is just a sample.)
2 by n game 1
Joint the point representing $a_{1j}$ on axis 2 to the point representing $a_{2j}$ on axis 1 for all $j = 1, 2,\cdots,n$. (Graph is just a sample.)
2 by n game
Mark the lowest boundary (called Lower envelope) of the lines by thick line segment. The highest point on the lower envelope gives the maximin point $P$ and it identifies the two critical moves (strategies) of player $B$.
Suppose the lower envelope is created by $B_1$ and $B_n$ strategy. The highest point on the lower envelope gives the maximin point. This shows that the two critical moves for player $B$ are $B_1$ and $B_n$.
Using the above selection the reduced game becomes
$A_1$ $a_{11}$ $a_{1n}$
Solve the reduced game (i.e. $2\times 2$) using maximin criterion. (if saddle point exists then find the value of the game otherwise use mixed strategy technique to find the value of the game.)
2 by n game by graphical method Example 1
Solve the game with the following payoff for player A.
$A_1$ 1 3 -3 5
$A_2$ 2 5 4 -4
Given game is $2\times 4$. That is for Player $A$ has two strategies and player $B$ has four strategies. Let us check using maximin-minimax principle whether the game has a saddle point or not.
A1 1 3 -3 5
A2 2 5 4 -4
Apply maximin-minimax principle to check whether saddle point exists or not.
RowMin
A1 1 3 -3 5 -3
A2 2 5 4 -4 -4
ColMax 2 5 4 5
Thus $Max(min) = Max(-3, -4)=-3$ and $Min(max)=Min(2, 5, 4, 5)=2$. Since the $Max(min)\neq Min(max)$ for the game, the game has no saddle point.
Construct two vertical axis, axis 1 and axis 2.
Ex01-2bynA
Let the player $A$ play the strategy $A_1$ with probability $p$ and strategy $A_2$ with probability $1-p$.
Expected payoff for player $A$ for any pure strategy of player $B$ would be given by
$B$'s Strategy
Expected Payoff for $A$
$B_1$ $E_1= p+2(1-p) = -p+2$
$B_2$ $E_2= 3p+5(1-p)=-2p+5$
$B_3$ $E_3= -3p+4(1-p)=-7p+4$
$B_4$ $E_4= 5p-4(1-p)= 9p-4$
Joint the point representing $a_{1j}$ on axis 2 to the point representing $a_{2j}$ on axis 1 for all $j = 1, 2,3$.
Ex01-2bynB
Ex01-2bynC
The two critical moves for player $B$ are $B_3$ and $B_4$.
The highest point $P$ on the lower envelope gives the maximin point.
The reduced game (i.e. $2\times 2$) is
$A_1$ -3 5
$A_2$ 4 -4
Apply maximin criterion.
A1 -3 5
A2 4 -4
Select the minimum element of each row of the payoff matrix,
$$ \begin{equation*} \text{i.e., } \min_{j} a_{ij}, i=1,2,\cdots, m. \end{equation*} $$
A1 -3 5 -3
A2 4 -4 -4
For each column of the payoff matrix, select the maximum element and call it ColMax.
$$ \begin{equation*} \text{i.e., } \max_{i} a_{ij}, j=1,2,\cdots, n. \end{equation*} $$
ColMax 4 5
From each RowMin, obtain the maximum value, i.e., $Max(RowMin)$.
$$ \begin{equation*} \text{i.e., } \max_{i}\min_{j} a_{ij}=\underline{v}. \end{equation*} $$
Thus $Max(min) = Max(-3, -4)=-3$
For each ColMax, obtain the minimum value, i.e. $Min(ColMax)$.
$$ \begin{equation*} \text{i.e., } \min_{j}\max_{i} a_{ij}=\overline{v}. \end{equation*} $$
Thus $Min(max)=Min(4, 5)=4$.
Since the $Max(min)\neq Min(max)$ for the game, the game has no saddle point.
Hence the optimal strategy for the reduced game can be obtained by algebraic method.
$$ \begin{aligned} p_1 &= \frac{d-c}{(a+d)-(b+c)}\\ &=\frac{(-4)-4}{(-3-4)-(5+4))}\\ &=\frac{-8}{-16}\\ &=\frac{1}{2}\\ p_2 &= 1-p_1\\ &=\frac{1}{2} \end{aligned} $$
and the optimal strategies for player B can be determined by
$$ \begin{aligned} q_1 &= \frac{d-b}{(a+d)-(b+c)}\\ &=\frac{(-4)-5}{(-3-4)-(5+4)}\\ &=\frac{-9}{-16}\\ &=\frac{9}{16}\\ q_2 &= 1-q_1\\ &=\frac{7}{16}. \end{aligned} $$
The optimal strategies for player A can be written as
$$ \begin{aligned} S_{A} &= \begin{bmatrix} A_1 & A_2 \\ p_1 & p_2 \end{bmatrix}\\ &=\begin{bmatrix} A_1 & A_2\\ \frac{1}{2} &\frac{1}{2} \end{bmatrix} \end{aligned} $$
and the optimal strategies for player B can be written as
$$ \begin{aligned} S_{B} &= \begin{bmatrix} B_1 & B_2 & B_3 & B_4\\ 0 & 0 & q_1 & q_2 \end{bmatrix}\\ &= \begin{bmatrix} B_1 & B_2 & B_3 & B_4\\ 0 & 0 & \frac{9}{16} & \frac{7}{16} \end{bmatrix} \end{aligned} $$
And the value of the game for player A is given by
$$ \begin{aligned} V &=\frac{ad-bc}{(a+d)-(b+c)} \\ &=\frac{(-3)(-4)-(5)(4)}{(-3-4)-(5+4)} \\ &=\frac{-8}{-16}\\ &=\frac{1}{2}. \end{aligned} $$
Thus, we conclude that player $A$ may choose strategy $A_1$ with probability $\dfrac{1}{2}$ and strategy $A_2$ with probability $\dfrac{1}{2}$. Player $B$ may choose strategy $B_3$ with probability $\dfrac{9}{16}$ and strategy $B_4$ with probability $\dfrac{7}{16}$. And value of the game for player A is $\dfrac{1}{2}$ and for player B is $-\dfrac{1}{2}$.
$A_1$ 1 3 10
$A_2$ 8 6 2
Given game is $2\times 3$. That is for Player $A$ has two strategies and player $B$ has three strategies. Let us check using maximin-minimax principle whether the game has a saddle point or not.
A1 1 3 10
A2 8 6 2
A1 1 3 10 1
A2 8 6 2 2
ColMax 8 6 10
Thus $Max(min) = Max(1, 2, )=2$ and $Min(max)=Min(8, 6, 10)=6$. Since the $Max(min)\neq Min(max)$ for the game, the game has no saddle point.
Axis 1 represent the payoff values of $A_2$ strategies for player $A$ and axis 2 represent payoff values of $A_1$ strategies for player $A$.
Joint the point representing $a_{1j}$ on axis 2 to the point representing $a_{2j}$ on axis 1 for all $j = 1, 2,\cdots,n$.
$B_1$ $E_1= p+8(1-p) = -7p+8$
$B_3$ $E_3= 10p+2(1-p)=8p+2$
$A_1$ 3 10
$A_2$ 6 2
A1 3 10
A2 6 2
A1 3 10 3
ColMax 6 10
Thus $Max(min) = Max(3, 2)=3$
Thus $Min(max)=Min(6, 10)=6$.
$$ \begin{aligned} p_1 &= \frac{d-c}{(a+d)-(b+c)}\\ &=\frac{2-6}{(3+2)-(10+6)}\\ &=\frac{-4}{-11}\\ &=\frac{4}{11}\\ p_2 &= 1-p_1\\ &=\frac{7}{11} \end{aligned} $$
$$ \begin{aligned} q_1 &= \frac{d-b}{(a+d)-(b+c)}\\\\ &=\frac{2-10}{(3+2)-(10+6)}\\ &=\frac{-8}{-11}\\ &=\frac{8}{11}\\ q_2 &= 1-q_1\\ &=\frac{3}{11}. \end{aligned} $$
$$ \begin{aligned} S_{A} &= \begin{bmatrix} A_1 & A_2 \\ p_1 & p_2 \end{bmatrix}\\ &=\begin{bmatrix} A_1 & A_2\\ \frac{4}{11} &\frac{7}{11} \end{bmatrix} \end{aligned} $$
$$ \begin{aligned} S_{B} &= \begin{bmatrix} B_1 & B_2 & B_3\\ 0 & q_1 & q_2 \end{bmatrix}\\ &= \begin{bmatrix} B_1 & B_2 & B_3\\ 0 & \frac{8}{11} & \frac{3}{11} \end{bmatrix} \end{aligned} $$
$$ \begin{aligned} V &= \frac{ad-bc}{(a+d)-(b+c)}\\ &=\frac{(3)(2)-(10)(6)}{(3+2)-(10+6)} \\ &=\frac{-54}{-11}\\ &=4.9091. \end{aligned} $$
Thus, we conclude that player $A$ may choose strategy $A_1$ with probability $\dfrac{5}{12}$ and strategy $A_5$ with probability $\dfrac{7}{12}$. Player $B$ may choose strategy $B_1$ with probability $\dfrac{8}{11}$ and strategy $B_2$ with probability $\dfrac{3}{11}$. And value of the game for player A is $4.9091$ and for player B is $-4.9091$.
In this tutorial, you learned about graphical method to solve $2\times n$ game and how to use graphical method to solve $2\times n$ game with illustrated examples.
To learn more about different methods to solve a game please refer to the following tutorials:
Let me know in the comments if you have any questions on Graphical method to solve $2\times n$ game and your thought on this article.
Categories Game Theory, Operations Research Tags game theory, graphical method to solve game
Graphical Method to solve m by 2 game
Graphical Method for Linear Programming Problem | CommonCrawl |
\begin{document}
\begin{abstract} We present a number of lower bounds for the $h$--vectors of $k$--CM, broken circuit and independence complexes. These lead to bounds on the coefficients of the characteristic and reliability polynomials of matroids. The main techniques are the use of series and parallel constructions on matroids and the short simplicial $h$--vector for pure complexes.
\end{abstract}
\maketitle
\section{Introduction}
Based on the ideas of Whitney \cite{Wh} and Rota \cite{Ro}, the broken circuit complex of a graph was introduced by Wilf in ``What polynomials are chromatic?'' \cite{Wi} Extended to matroids by Brylawski \cite{Bry4}, its $f$--vector corresponds to the coefficients of the characteristic polynomial of the matroid. The $h$--vector encodes the same information in a different way. From the point of view of matroids, Wilf's original question becomes, ``What are the possible $f$--vectors, or equivalently $h$--vectors, of broken circuit complexes of matroids?''
Cohen--Macaulay complexes cover a wide variety of examples. In addition to the broken circuit and independence complexes of matroids covered here, Cohen--Macaulay complexes also include all triangulations of homology balls and spheres. In contrast to broken circuit complexes, the possible $h$--vectors (and hence $f$--vectors) of Cohen--Macaulay complexes have been completely characterized (see, for instance, \cite[Theorem II.3.3, pg. 59]{St}). Introduced by Baclawski, doubly Cohen--Macualay complexes are Cohen--Macaulay complexes which neither lose a dimension nor lose the Cohen--Macaulay property when any vertex is removed. Spheres are doubly Cohen--Macaulay but balls are not. More generally, a Cohen--Macaulay complex is $k$--CM if it retains its dimension and is still Cohen--Macaulay whenever $k-1$ or fewer vertices are removed. In addition to the independence complexes considered below, the order complex of a geometric lattice with the top and bottom points removed is $k$--CM if every line has at least $k$ points \cite{Ba2}.
The $h$--vectors of independence complexes of matroids are contained in the intersection of $h$--vectors of broken circuit complexes and $k$--CM complexes. Precisely, the cone on any independence complex is a broken circuit complex. In addition, if the smallest cocircuit of the matroid has cardinality $k,$ then its independence complex is a $k$--CM complex. The close connection between $h$--vectors of independence complexes of matroids and reliability problems has been studied by a number of authors. See \cite{CC} for a recent survey.
Upper bounds on all of the above complexes have been studied. As they are all Cohen--Macaulay they share a common absolute upper bound of $h_i \le \binom{n-r-1+i}{i},$ where $n$ is the number of vertices and $(r-1)$ is the dimension of the complex. In addition, they all satisfy the relative upper bound $h_{i+1} \le h^{<i>}_i$ (see Section \ref{k-CM} for a definition of $h^{<i>}_i$).
Our main purpose is to analyze absolute and relative lower bounds for the $h$--vectors of $k$--CM, broken circuit and independence complexes. Section \ref{face enumeration} contains the basic facts of the short--simplicial $h$--vector. The main tool for providing relative lower bounds is equation (\ref{del by h}). The broken circuit and independence complex of a matroid are described in section \ref{matroids}. Sections \ref{k-CM}, \ref{BC complexes} and \ref{ind complexes} contain absolute and relative lower bounds for $k$--CM, broken circuit and independence complexes respectively.
Throughout the paper $\Delta$ is an $(r-1)$--dimensional simplicial
complex with vertex set $V, |V|=n.$ The link of a vertex $v \in V$ is $lk_{\Delta} v,$ or just $lk \ v$ if no confusion is possible. We use $\Delta - v$ for the complex obtained by removing $v$ and all of the faces which contain $v$ from $\Delta.$ Similarly, if $A \subseteq V,$ then, $\Delta -A$ is the complex obtained by removing all of the vertices in $A$ and any faces which contain one or more of those vertices.
\section{Face enumeration} \label{face enumeration}
The combinatorics of a simplicial complex $\Delta$ can be encoded in
several ways. The most direct is to let $f_i(\Delta)$ be the number of
faces of cardinality $i.$ For an $(r-1)$--dimensional complex the
$h$--vector of $\Delta$ is the sequence $(h_0(\Delta), \dots,
h_r(\Delta)),$ where \begin{equation} \label{f by h}
h_i(\Delta) = \sum^i_{j=0} (-1)^{i-j} \binom{r-j}{r-i} f_j(\Delta). \end{equation} Equivalently, \begin{equation} \label{h by f}
f_j(\Delta) = \sum^j_{i=0} \binom{r-i}{r-j} h_i(\Delta). \end{equation}
\noindent By convention, $h_i(\Delta) = f_i(\Delta) =0 $ if $i <0$ or $i > r.$ The {\it short simplicial} $h$--vector was introduced in \cite{HN} as a simplicial analogue of the short cubical $h$--vector in \cite{Ad}. It is the sum of the $h$--vectors of the links of the vertices. As far as we know, (\ref{short by h}) was first stated in \cite{Mc}. However, only a proof for shellable $\Delta$ was given there. So, we include a proof for arbitrary pure complexes for the sake of completeness.
\begin{defn}
Let $\Delta$ be a pure simplicial complex . Define \begin{equation} \label{short h}
\tilde{h}_i(\Delta) = \sum_{v \in V} h_i ( lk \ v). \end{equation} \end{defn}
\begin{lem} \cite{HN}
Let $\Delta$ be a pure simplicial complex. For all
$i, 0 \le i \le r-1,$
\begin{equation} \label{short by f}
\tilde{h}_i(\Delta) = \sum^i_{j=0} (-1)^{i-j} (j+1) \binom{r-j-1}{r-i-1}
f_{j+1}.
\end{equation} \end{lem}
\begin{prop}
Let $\Delta$ be a pure simplicial complex. Then,
\begin{equation} \label{short by h}
\tilde{h}_{i-1}(\Delta) = i \ h_i(\Delta) + (r-i+1) h_{i-1}(\Delta).
\end{equation} If $\dim (\Delta-v) = r-1$ for every vertex $v,$ then
\begin{equation} \label{del by h}
\sum_{v \in V} h_i(\Delta - v) = (n-i) h_i(\Delta) - (r-i+1)
h_{i-1}(\Delta).
\end{equation} \end{prop}
\begin{proof}
Combining (\ref{h by f}) and (\ref{short by f}), $$ \begin{array}{lcl} \tilde{h}_{i-1}(\Delta) & = & \displaystyle\sum^{i-1}_{j=0} (-1)^{i-j-1} (j+1) \binom{r-j-1}{r-i} \displaystyle\sum^{j+1}_{k=0} \binom{r-k}{r-j-1} h_k(\Delta) \\ \ & = & \displaystyle\sum^i_{k=0} h_k(\Delta) \left\{ \displaystyle\sum^{i-1}_{j=k-1} (-1)^{i-j-1} (j+1) \binom{r-j-1}{r-i} \binom{r-k}{r-j-1} \right\} \\ \ & = & \displaystyle\sum^i_{k=0} h_k(\Delta) \left\{ \displaystyle\sum^{i-1}_{j=k-1} (-1)^{i-j-1} (j+1) \binom{r-j-1}{i-j-1} \binom{r-k}{j+1-k} \right\}. \end{array}$$ Substituting $s=j-k+1$ and $t=i-j-1,$ $$\begin{array}{lcl} \tilde{h}_{i-1}(\Delta) & = & \displaystyle\sum^i_{k=0} h_k(\Delta) \left\{ \sum_{s+t=i-k} (-1)^t (i-t) \binom{r+t-i}{t} \binom {r+s+t-i}{s} \right\} \\ \ & = & \displaystyle\sum^i_{k=0} h_k(\Delta) \left\{ \sum_{s+t=i-k} (-1)^t (i-t) \frac{A}{s! t!} \right\}, \end{array}$$ where $A$ is the falling factorial $(r-k) \cdot (r-k-1) \cdots (r-i+1).$
For a fixed $i$, define $c_k$ by $$c_k= \sum_{s+t=i-k} (-1)^t (i-t) \frac{1}{s! t!}.$$ Equation (\ref{short by h}) is equivalent to showing that $c_i = i, c_{i-1} = 1$ and $c_k = 0$ in all other cases. This can be seen by recognizing $c_{i-k}$ as the $k^{th}$ term in the generating series for $$(i+x)e^{-x} \cdot e^x = \left( \sum^\infty_{t=0} (-1)^t \frac{(i-t)}{t!} x^t \right) \left( \sum^\infty_{s=0} \frac{1}{s!} x^s \right).$$
In order to prove that (\ref{del by h}) holds, we first notice that the hypothesis implies that $h_i(\Delta) = h_i(\Delta-v) + h_{i-1}(lk \ v)$ for every vertex $v.$ Now sum this equation over all the vertices and apply equation (\ref{short by h}). \end{proof}
The above proposition makes precise the idea that, taken together, $h_{i-1}(\Delta)$ and $h_i(\Delta)$ measure the ``average contribution of $h_{i-1}( lk \ v)$ to $h_i(\Delta).$'' Another consequence of (\ref{short by h}) is that if the automorphism group of a pure $(r-1)$--dimensional complex $\Delta$ is transitive, or more generally if $h_{i-1}(lk \ v)$ is independent of $v,$ then $n$ divides $\{i\ h_i(\Delta) + (r-i+1) h_{i-1}(\Delta)\}.$
\section{Broken circuit and independence complexes of matroids} \label{matroids}
We follow \cite{O} for matroid terminology. Unless otherwise specified, $M$ is always a rank $r$ matroid with ground set $E$ (or $E(M)$ if necessary) and $|E|=n.$ There are many equivalent ways of defining matroids. The most convenient for us is the following.
A {\it matroid}, $M,$ is a pair $(E, \mathcal{I}), E$ a non-empty
finite ground set and $\mathcal{I}$ a distinguished set of subsets
of $E.$ The members of $\mathcal{I}$ are called the {\it
independent} subsets of $M$ and are required to satisfy:
\begin{enumerate}
\item The empty set is in $\mathcal{I}.$
\item If $B$ is an independent set and $A \subseteq B,$ then $A$ is
an independent set.
\item If $A$ and $B$ are independent sets such that $|A| < |B|,$
then there exists an element $x \in B-A$ such that $A \cup x$ is
independent. \end{enumerate} Matroid theory was introduced by Whitney \cite{W}. The prototypical example of a matroid is a finite subset of a vector space with the canonical independent sets. Another example is the cycle matroid of a graph. Here the ground set is the edge set of the graph and a collection of edges is independent if and only if it is acyclic.
An element $e$ of a matroid is a {\it loop} if it is not contained in any independent set. The {\it circuits} of a matroid are its minimal dependent sets. Every loop of $M$ is a circuit. A maximal independent set is called a {\it basis}, and any element which is contained in every basis is a {\it coloop} of the matroid. Every basis of $M$ has the same cardinality. The {\it rank} of $M,$ or $r(M),$ is that common cardinality. Similarly, the rank of a subset $A$ of $E$ is the cardinality of any maximal independent subset of $A$ and is denoted $r(A).$ The {\it deletion} of $M$ at $e$ is denoted $M - e.$ It is the matroid whose finite set is $E - e$ and whose independent sets are simply those members of $\mathcal{I}$ which do not contain $e.$ The {\it contraction} of $M$ at $e$ is denoted $M/e.$ It is a matroid whose ground set is also $E - e.$ If $e$ is a loop or a coloop of $M$ then $M/e = M - e.$ Otherwise, a subset $I$ of $E-e$ is independent in $M/e$ if and only if $I \cup e$ is independent in $M.$ Deletion and contraction for a subset $A$ of $E$ is defined by repeatedly deleting or contracting each element of $A.$
The {\it dual} of $M$ is $M^\star.$ It is the matroid whose ground set is the same as $M$ and whose bases are the complements of the bases of $M.$ For example, $U_{i,j}$ is the matroid defined by $E=\{1,2,\dots,j\}$ and
$\mathcal{I}=\{A \subseteq E: |A| \le i.\}$ So, $U_{i,j}^\star = U_{j-i,j}.$
Two non-loop elements $e,f \in E$ are {\it parallel} if they form a circuit. The relation ``is parallel to'' is an equivalence relation on $E$ and the corresponding equivalence classes are the parallel classes of $M.$ If $P$ is a parallel class of $M$, then for any $e \in P$ all of the members of $P-e$ are loops in $M/e.$ A parallel class in $M^\star$ is a {\it series class} of $M.$ If $S$ is a series class of $M,$ then for any $e \in S,$ all of the members of $S - e$ are coloops in $M-e.$
Let $M=(E,\mathcal{I})$ and $M^\prime=(E^\prime,\mathcal{I}^\prime)$ be two matroids with $E \cap E^\prime = \emptyset.$ Then $M \oplus M^\prime$ is the direct sum of $M$ and $M^\prime.$ It is the matroid whose ground set is $E \cup E^\prime$ and whose independent sets are those subsets of the form $I \cup I^\prime, I \in \mathcal{I}, I^\prime \in \mathcal{I}^\prime.$ A matroid is {\it connected} if it is not the direct sum of two smaller matroids. Every matroid can be written uniquely (up to order) as a direct sum $M = M_1 \oplus \dots \oplus M_k$ of connected matroids. The {\it components} of $M$ are the summands of this decomposition.
The {\it independence complex} of $M$ is $$\Delta(M) = \{A \subseteq E: A \mbox{ is independent}\}.$$ Evidently, $\Delta(M)$ is a pure $(r-1)$--dimensional complex, where $r$ is the rank of $M.$ In addition, $\Delta(M-e) = \Delta(M) -e$ and if $e$ is not a loop of $M,$ then $\Delta(M/e) = lk_{\Delta(M)} \ e.$
In order to define the broken circuit complex for $M$, we first choose a linear order $\omega$ on the elements of the matroid. Given such an order, a {\it broken circuit} is a circuit with its least element removed. The {\it broken circuit complex} is the simplicial complex whose simplices are the subsets of $E$ which do not contain a broken circuit. We denote the broken circuit complex of $M$ and $\omega$ by $\Delta^{BC}(M),$ or $\Delta^{BC}(M,\omega).$ Different orderings may lead to different complexes, see \cite[Example 7.4.4]{Bj}. However, $f_i(\Delta^{BC}(M,\omega))$ does not depend on $\omega$ (see Theorem \ref{h,b and tp} below). Conversely, distinct matroids can have the same broken circuit complex. For instance, let $E=\{e_1,e_2,e_3,e_4,e_5,e_6\},$ and let $\omega$ be the obvious order. Let $M_1$ be the matroid on $E$ whose bases are all triples except $\{e_1,e_2,e_3\}$ and $\{e_4,e_5,e_6\}$ and let $M_2$ be the matroid on $E$ whose bases are all triples except $\{e_1,e_2,e_3\}$ and $\{e_1,e_5,e_6\}.$ Then $M_1$ and $M_2$ are non--isomorphic matroids but their broken circuit complexes are identical.
In order to easily distinguish the $h$--vectors of $\Delta(M)$ and $\Delta^{BC}(M)$ we use the following notation.
\begin{defn} Let $M$ be a rank $r$ matroid. \label{h,b,w defs}
\begin{itemize}
\item
$h_i(M) = h_i(\Delta(M)).$
\item
$b_i(M) = h_{r-i}(\Delta^{BC}(M)).$
\item
$w_i(M) = f_{r-i}(\Delta^{BC}(M)).$
\item
$b^\star_i(M) = b_i(M^\star) = h_{n-r-i}(\Delta^{BC}(M^\star))$
\end{itemize} \end{defn}
We will suppress the $M$ when there is no danger of confusion. The invariants $h_i, b_i, w_i, b^\star_i$ are closely related to the Tutte polynomial of $M.$ The {\it Tutte polynomial} is a two--variable polynomial invariant of $M$ defined by
$$T(M;x,y) = \displaystyle\sum_{A \subseteq E} (x-1)^{r(M)-r(A)}
(y-1)^{|A|-r(A)}.$$
\begin{thm} \cite{Bj} \label{h,b and tp}
Suppose $M$ has $k$ components and $j$ coloops. Then, \begin{itemize}
\item[a.]
$T(M;x,1) = h_0 x^r + h_1 x^{r-1} + \dots + h_{r-j} x^j.$
\item[b.]
$T(M;x,0) = b_r x^r + b_{r-1} x^{r-1} + \dots + b_k x^k.$
\item[c.]
$T(M;0,y) = b^\star_{n-r} x^{n-r} + \dots + b^\star_k x^k.$
\item[d.]
$(-1)^r T(M;1-x,0) = w_0 x^r - w_1 x^{r-1} + \dots + (-1)^r w_r.$ \end{itemize} \end{thm}
The $w_i$ are the unsigned Whitney numbers of the first kind. The {\it characteristic polynomial} of $M$ is $(-1)^r T(M;1-x,0).$ The characteristic polynomial of a matroid has a number of applications including graph coloring and flows, linear coding theory and hyperplane arrangements. See \cite{BO} for a survey.
Properties [a]-[d] of $b_i$ and $h_i$ listed below follow immediately from corresponding properties of the Tutte polynomial which can be found in \cite{Bry2}. The parallel and series connection of two (pointed) matroids is described in \cite[Section 7.1]{O}.
\begin{thm}[Tutte recursion] \ \label{tuttepoly}
\begin{itemize}
\item[a.]
If $M$ has $j$ coloops, then $h_i (M) = h_i (\tilde{M}),$ where $\tilde{M}$ is $M$ with the coloops deleted. In particular, $h_i(M) > 0$ if and only if
$0 \le i \le r-j.$
\item[b.]
If $M$ has $k$ components and no loops, then $b_i > 0$
if and only if $ k \le i \le r.$
\item[c.]
If $e$ is neither a loop nor a coloop of $M,$ then $h_i(M) =
h_i(M-e) + h_{i-1}(M/e)$ and $b_i(M) = b_i(M-e) + b_i(M/e).$
\item[d.]
If $M = M_1 \oplus M_2,$ then $h_i(M) = \displaystyle\sum_{j+k=i}
h_j(M_1) h_k(M_2)$ and \\ $b_i(M) = \displaystyle\sum_{j+k = i} b_j(M_1)
b_k(M_2).$
\item[e.]
Suppose that $P$ is a parallel class of $M.$ Let $\tilde{M}$
be $M$ with all but one element, say $e,$ of $P$ deleted. Then, $h_i(M) = h_i(\tilde{M}) + (|P|-1) h_{i-1}(\tilde{M}/e).$
\item[f.]
Let $S$ be a series class of $M.$ Let $\tilde{M}$ be $M$ with all but one element, say $e,$ of $S$
contracted. Then $b_i(M) = b_i(\tilde{M}) + \sum^{|S|-1}_{j=1} b_{i-j}(\tilde{M}-e).$
\item[g.]
Let $M$ be a parallel connection of $A$ and $B,$ where the rank of
$A$ is $r(A)$ and the rank of $B$ is $r(B).$ The rank of $M$ is
$r(A) + r(B) -1.$ In addition, $b_i(M) = \displaystyle\sum_{j+k=i+1} b_j(A) b_k(B).$ If $A$ and $B$ are connected, then $M$ is also
connected. \end{itemize} \end{thm}
\begin{proof}
Property [g] follows from the fact that if $M$ is a parallel connection of $A $ and $B,$ then $T(M;x,0) = T(A;x,0) * T(B;x,0)/x$ \cite[pg. 179--182]{Bry2}. Both [e] and [f] are proved by deleting and contracting all the elements of the given parallel or series class except $e.$
\end{proof}
One of the consequences of [a] and [f] above is that if we increase the size of a series class of cardinality $k$ in $M$ by one, then $b_1,\dots,b_k$ are unchanged, while $b_i$ for $i > k$ may increase.
\section{Cohen--Macaulay and $k$--CM complexes} \label{k-CM}
There are several equivalent definitions of Cohen--Macaulay complexes. The following will suffice for our purposes.
\begin{defn}
A pure $(r-1)$--dimensional complex $\Delta$ is {\it Cohen--Macaulay} if
for every face $F \in \Delta$ and $i < \dim (lk \ F),
\tilde{H}_i(lk \ F;\mathbb{Q})=0.$ \end{defn}
A numerical description of all possible $h$--vectors of Cohen--Macaulay complexes can be given using the following operator. Given any positive integers $h$ and $i$ there is a unique way of writing $$h=\binom{a_i}{i} + \binom{a_{i-1}}{i-1} + \dots + \binom{a_j}{j}$$ so that $a_i > a_{i-1} > \dots > a_j \ge j \ge 1.$ Define $$h^{<i>} = \binom{a_i+1}{i+1} + \binom{a_{i-1}+1}{i} + \dots + \binom{a_j+1}{j+1}$$
\begin{thm} \cite{St} \label{CM h-vectors}
A sequence of non--negative integers $(h_0, \dots, h_r)$ is the
$h$--vector of some Cohen--Macaulay complex if and only if $h_0=1$ and
$h_{i+1} \le h^{<i>}_i$ for all $1 \le i \le r-1.$ \end{thm}
The notion of $k$--CM complexes was introduced by Baclawski \cite{Ba2}.
\begin{defn}
Let $\Delta$ be a pure $(r-1)$--dimensional simplicial complex with
vertex set $V$ and $k \ge 1.$ We say
that $\Delta$ is $k$--CM if for all $A \subseteq V$ with $|A|<k, \Delta -A$ is Cohen--Macaulay of dimension $(r-1).$ \end{defn}
Examples of $2$--CM complexes include order complexes of geometric lattices, finite buildings and triangulations of spheres. Several examples and constructions involving $k$--CM complexes, especially for order complexes of posets, are contained in \cite{Ba2}. Since $lk_{\Delta} v - A = lk_{\Delta - A} v,$ the link of any vertex of a $k$--CM complex is $k$--CM, and removing a vertex from a $k$--CM complex leaves a $(k-1)$--CM complex (as long as $k > 1$).
The independence and broken circuit complexes of a matroid are Cohen--Macaulay \cite{St1}. So, $\Delta(M)$ is $k$--CM if and only if every hyperplane of $M$ has cardinality at most $n-k$. Equivalently, the smallest cocircuit of $M$ has at least $k$ elements. However, $\Delta^{BC}(M)$ is a cone on the least element, hence it is only $1$--CM. If the cone point is removed, then the remaining complex is also Cohen--Macaulay, but may still be only $1$--CM. For example, let $M$ be the cycle matroid of the theta--graph with three paths each of length 2. Direct computation shows that the $h$--vector of $\Delta^{BC}(M)$ is $(1,2,3,1).$ Removing the cone point leaves a $2$--dimensional complex with $5$ points and the same $h$--vector. By Corollary \ref{relative k-cm} below, $(1,2,3,1)$ is not the $h$--vector of any $2$--dimensional $2$--CM complex with $5$ points.
Theorem \ref{CM h-vectors} gives an upper bound for possible $h$--vectors of Cohen--Macaulay complexes. It also makes it clear that there are no lower bounds. For $k$--CM complexes we have the following absolute lower bound. Recall that $U_{r,n}$ is the rank $r$ matroid with $n$ elements such that every $r$--element subset is a basis.
\begin{prop} \label{k-CM abs}
Let $\Delta$ be an $(r-1)$--dimensional $k$--CM complex.
Then,
$$h_i(\Delta) \ge h_i(U_{r,r+k-1}).$$ \end{prop}
\begin{proof}
Induction on $n$ and $k$. When $k=1,$ the theorem is simply the
statement that $h_i(\Delta) \ge 0$ for $i \ge 1,$ and $h_0(\Delta) \ge
1.$ For fixed $k,$ the definition of $k$--CM forces $n \ge r+k-1.$ Suppose
$n=r+k-1.$ Since the removal of any subset of vertices of cardinality
$k-1$ does not lower the dimension of $\Delta,$ every subset of vertices
of cardinality $r$ must be a face of $\Delta.$ So, $\Delta =
\Delta(U_{r,r+k-1}).$ For the induction step, let $v$
be any vertex of $\Delta.$ Then $$h_i(\Delta) = h_i(\Delta -v) + h_{i-1}(lk_{\Delta} v) \ge h_i(U_{r,r+k-2}) + h_{i-1}(U_{r-1,r+k-2}) = h_i(U_{r,r+k-1}).$$ \end{proof}
Minimizing $h$--vectors is closely related to the problem of finding the least reliable graph. Let $G$ be a connected graph with $r+1$ vertices and $n$ edges. Thus $M(G),$ the cycle matroid of $G,$ has rank $r$ and cardinality $n.$ Suppose that each edge of $G$ has equiprobability $p, 0 < p < 1$ of being deleted. Then the probability that $G$ remains connected is $R_G(p) = (1-p)^r [h_0(M(G)^\star) + h_1 (M(G)^\star) p + \dots + h_{n-r}(M(G)^\star) p^{n-r}].$ Boesch, Satyanarayana and Suffel posed the problem of finding the minimum of $R_G(p)$ among all connected simple graphs with $r+1$ vertices and $n$ edges. They also conjectured that a particular graph, which they called $L(r+1,n),$ would attain that lower bound \cite{BSS}. Brown, Colbourn and Devitt further conjectured that the $h$--vector of $L(r+1,n)$ would be an absolute lower bound for the $h$--vector of $M(G)^\star$ among all connected simple graphs with $r+1$ vertices and $n$ edges \cite{BCD}. The original conjecture of Boesch et. al. was confirmed for $n$ greater than $\binom{r-1}{2}$ in \cite{PSS}. The corresponding problem in the category of matroids is to find among all rank $r$ cosimple matroids of cardinality $n$ one which minimizes the $h$--vector. Since $M$ is cosimple if and only if $\Delta(M)$ is $3$--CM, the above proposition shows that $U_{0,n-r-2} \oplus U_{r,r+2}$ is the solution to this problem.
Combining the above proposition with (\ref{del by h}) immediately gives a relative lower bound.
\begin{cor} \label{relative k-cm}
Let $\Delta$ be an $(r-1)$--dimensional $k$--CM complex with $n$
vertices. Then, $$(n-i) h_i \ge (r-i+1) h_{i-1} + n \binom{i+k-3}{i}.$$ \end{cor}
\begin{proof} For every vertex $v, \Delta - v$ is $(k-1)$--CM. Now combine (\ref{del by h}), Proposition \ref{k-CM abs} and the fact that $h_i(U_{r,r+k-2}) = \binom{i+k-3}{i}.$ \end{proof}
\begin{prob}
Given $r,n,k$ and $i,$ what is the minimum of $h_i(\Delta)$ over all
$(r-1)$--dimensional $k$--CM complexes with $n$ vertices? Does there
exist a $\Delta$ which attains these values? \end{prob}
Conjecture II.6.2 in \cite{St} would imply that for $2$--CM complexes with $n$ equal to $r+2, h_i(\Delta) \ge h_i(\Delta(U_{1,2} \oplus U_{r-1,r})).$ In section \ref{ind complexes} we will give an answer to this problem for independence complexes of matroids when $n$ is sufficiently large.
\section{Broken circuit complexes} \label{BC complexes}
In this section we assume that $M$ has no loops. An absolute upper bound for $b_i$ when $1 \le i \le r$ is $\binom{n-i-1}{r-i}$ and this is achieved by $U_{n,r}.$ Theorem \ref{CM h-vectors} gives a relative upper bound of $b_{r-i} \le b_{r-i+1}^{<i-1>} .$ Absolute lower bounds for $b_i$ were determined by Brylawski.
\begin{thm} \cite{Bry3}
If $M$ is as above, then $b_i \ge n-r$ for all $i, 2 \le i \le r-1.$ \end{thm}
In order to find relative lower bounds for $b_1$ we introduce the following definition.
\begin{defn}
Let $S$ be a series class of a connected matroid $M.$
Then $S$ is a {\bf regular series} class of $M$ if $M-S$ is connected. \end{defn}
\begin{prop}
If $M$ is connected and contains more than one series class, then $M$ contains at least three regular series classes. \end{prop} \begin{proof}
Induction on $m$, the number of series classes in $M.$ A matroid
with exactly two series class is not connected. If $m=3,$ then $M$ is
the cycle matroid of a theta graph with exactly three paths. In this case all three of the series classes are regular.
For the induction step, let $S$ be a series class which is not regular. Let $\tilde{M}$ be the matroid obtained by contracting
all but one of the elements of $S.$ Let $e$ be the remaining element of $S.$
Since $\tilde{M}$ is connected, but $\tilde{M}-e$ is not connected, $\tilde{M}$
is the series connection of two connected matroids $A$ and $B$ at $e$ \cite[Theorem 7.1.16]{O}. Both $A$ and $B$ must contain more than one series class, otherwise they would be contained in $S.$ Therefore, the induction
hypothesis applies to $A$ and $B.$ Even if $\{e\}$ is contained in
a regular series class in $A$ and $B,$ both $A$ and $B$ contain two other regular series classes. All four of these series classes are regular in $M.$ \end{proof}
\begin{thm} \label{char by r}
If $M$ is connected and $1 \le i \le r$, then \begin{equation} \label{eq1} b_i \le \binom{r-2}{i-1} b_1 + \binom{r-2}{i-2}. \end{equation} \end{thm}
\begin{proof}
The proof is by
induction on $n,$ the initial case being the three--point line. Let $S$ be a series class of $M.$ If $S$ is the only series class of $M,$ then $M$ is a circuit and (\ref{eq1}) holds. Otherwise, by the previous proposition, we may choose $S$ to be a regular series class. In particular, $M-S$ is connected. We break the induction step into three cases.
\begin{enumerate}
\item
$M-S$ and $M/S$ are connected:
Let $s = |S|.$ If $s > i,$
then $b_i(M) =
b_i(\hat{M})$ and $b_1(M) = b_1(\hat{M}),$
where $\hat{M}$ is $M$ with $S$ contracted down to a series class
of cardinality $i.$ So, we will assume that $s \le i.$ Let $\tilde{M}$ be $M$
with $S$ contracted down to a single element $e.$ Since $M$ is connected, $e$ is neither a loop nor a coloop of $M.$ Applying Tutte recursion to $M$ and then again to $\tilde{M}$ we see that
$$b_i(M) = b_i(\tilde{M}/e) + \sum^{s-1}_{j=0} b_{i-j}(\tilde{M}-e).$$
Now, since $\tilde{M}/e = M/S$ is a rank $r-s$ connected matroid and $\tilde{M}-e = M - S$ is a rank $r-s+1$ connected matroid, the induction hypothesis implies that the above expression is bounded above by $$\binom{r-s-2}{i-1} b_1(\tilde{M}/e) + \binom{r-s-1}{i-2} + \sum^{s-1}_{j=0} \binom{r-s-1}{i-j-1} b_1(\tilde{M}-e) + \sum^{s-1}_{j=0} \binom{r-s-1}{i-j-2} $$
$$\le \binom{r-2}{i-1} b_1(\tilde{M}/e) + \binom{r-2}{i-1} b_1 (\tilde{M}-e) + \binom{r-2}{i-2} +$$ $$\left\{ \binom{r-s-2}{i-1} - \binom{r-2}{i-1} \right\} b_1(\tilde{M}/e) + \binom{r-s-2}{i-2}.$$
\noindent Since $\tilde{M}/e$ is connected, $b_1(\tilde{M}/e) \ge 1.$ Thus, the last row is non--positive and (\ref{eq1}) is satisfied. To see the last inequality, note that
$$\sum^{s-1}_{j=0} \binom{r-s-1}{i-j-1} \le \sum^{s-1}_{j=0} \binom{r-s-1}{i-j-1} \binom{s-1}{j}= \binom{r-2}{i-1},$$
\noindent and similarly,
$$\sum^{s-1}_{j=0} \binom{r-s-1}{i-j-2} \le \sum^{s-1}_{j=0} \binom{r-s-1}{i-j-2} \binom{s-1}{j}= \binom{r-2}{i-2}.$$
\item
$S = \{e\}, M-e$ is connected, but $M/e$ is not connected:
Then, $M$ is the parallel connection of two connected
matroids $A$ and $B$ with $r(A) + r(B) -1 = r$ \cite[Theorem 7.1.16]{O}. By Theorem \ref{tuttepoly}
and the
induction hypothesis,
$$b_i(M) = \sum_{j+k-1=i} b_j(A) b_k(B)$$ $$ \begin{array}{ll} \le & \displaystyle\sum_{j+k-1=i} \left\{ \binom{r(A) -2}{j-1} b_1(A) + \binom{r(A)-2}{j-2} \right\} \left\{ \binom{r(B) -2}{k-1} b_1(B) + \binom{r(B)-2}{k-2} \right\} \\ \ & \ \\
= & \displaystyle\sum_{j+k-1=i} \left\{ \binom{r(A)-2}{j-1} \binom{r(B)-2}{k-1} b_1(A) b_1(B) + \binom{r(A)-2}{j-1} \binom{r(B)-2}{k-2} b_1(A) \right\} + \\ \ & \\ \ & \displaystyle\sum_{j+k-1=i} \left\{ \binom{r(A)-2}{j-2} \binom{r(B)-2}{k-1} b_1(B) + \binom{r(A)-2}{j-2} \binom{r(B)-2}{k-2} \right\} \\ \ & \\ = & \displaystyle\binom{r-3}{i-1} b_1(A) b_1(B) + \binom{r-3}{i-2} b_1(A) + \binom{r-3}{i-2} b_1(B) + \binom{r-3}{i-3}. \end{array}$$ \\
\noindent Therefore,
$$\begin{array}{ll} \ & \displaystyle\binom{r-2}{i-1} b_1(M) + \binom{r-2}{i-2} - b_i(M) \\ \ge & \displaystyle\left\{ \binom{r-2}{i-1} - \binom{r-3}{i-1} \right\} b_1(A) b_1(B) + \left\{ \binom{r-2}{i-2} - \binom{r-3}{i-3} \right\} - \\ \ & \displaystyle\binom{r-3}{i-2} \left\{ b_1(A) + b_1(B) \right\} \\ = & \displaystyle\binom{r-3}{i-2} \left( b_1(A) b_1(B) + 1 - b_1(A) - b_1(B) \right) \ge 0. \end{array}$$
\item Finally, suppose that $S$ is a non--trivial series, $M-S$ is connected, but $M/S$ is not connected. Let $s, \tilde{M}$ and $e$ be as above. Since $\tilde{M}/e$ is not connected, $b_1(\tilde{M})=b_1(\tilde{M}-e).$ Therefore,
$$b_i(M) = b_i(\tilde{M}) + \sum^{s-1}_{j=1} b_{i-j}(\tilde{M}-e) $$ $$\le b_1(M) \left\{ \sum^{s-1}_{j=0} \binom{r-s-1}{i-j-1} \right\} + \sum^{s-1}_{j=0} \binom{r-s-1}{i-j-2} $$ $$ \le \binom{r-2}{i-1} b_1(M) + \binom{r-2}{i-2}.$$
\end{enumerate} \end{proof}
\begin{cor} \label{disconnected}
Let $M$ be a rank $r$ matroid with $k$ components, $r-k \ge 2.$
Let $2 \le i \le r-k.$ Then, \begin{equation} \label{eqdisc} b_{i+k-1}(M) \le \binom{r-k-1}{i-1} b_k(M) + \binom{r-k-1}{i-2}. \end{equation} \end{cor}
\begin{proof}
Since $k=1$ is the previous theorem we assume that $M$ is not connected.
Let $M=M_1 \oplus \dots \oplus M_k$ be a direct sum decomposition of $M$ into connected matroids. Define $\tilde{M}_1=M_1.$ Given $\tilde{M}_i$ let $\tilde{M}_{i+1}$ be any parallel connection of $\tilde{M}_i$ and $M_{i+1}.$ Then $\tilde{M}_k$ is a connected matroid of rank $r-k+1.$ Furthermore, by Theorem \ref{tuttepoly} $b_{i+k-1}(M) = b_i(\tilde{M}_k).$ Since (\ref{eq1}) holds for the connected $\tilde{M},$ (\ref{eqdisc}) holds for $M.$ \end{proof}
When does equality occur in the above theorem? The proof shows that if equality occurs, then it must also occur in the minors of $M$ used in the induction. Combining this with an induction argument shows that if $b_i(M) = \binom{r-2}{i-1} b_1(M) + \binom{r-2}{i-2},$ then $b_j(M) = \binom{r-2}{j-1} b_1(M) + \binom{r-2}{j-2}$ for all $1 \le j \le i.$ Brylawski proved (\ref{eq1}) for $i=r-1.$ He also showed that given $b_1$ and $r,$ then equality occurs if $M$ is the parallel connection of a $(b_1+2)$--point line and $r-1$ three--point lines. Hence, (\ref{eq1}) is optimal, although a complete description of the matroids which satisfy equality in this corollary remains unknown \cite{Bry3}.
The coefficient $b_1(M)$ is also known as $\beta(M),$ the {\it beta} invariant of $M.$ Brylawski identified matroids with beta invariant 1 as series--parallel matroids \cite{Bry5} while Oxley classified matroids with $2 \le \beta(M) \le 4$ \cite{Ox2}.
\begin{thm}
Assume $r \ge 2$ and let $\beta=b_1(M).$ Then, for all $i, 0 \le i \le r,$
$$w_i \le \sum^i_{j=0} \binom{r-j}{r-i} \left\{ \binom{r-2}{r-i-1} \beta + \binom{r-2}{r-i-2} \right\}.$$ \end{thm}
\begin{proof} This follows immediately from (\ref{h by f}) and Theorem \ref{char by r}. \end{proof}
It is also possible to estimate $b_i$ in terms of $n-r.$ For positive integers $i$ and $x$ define $$\phi_i(x)=\binom{x-2}{i-1} \binom{x-1}{0} + \binom{x-2}{i-2} \binom{x}{1} + \dots + \binom{x-2}{0} \binom{x+i-2}{i-1}.$$
\begin{thm}
Suppose $M$ is connected. Then, \begin{equation} \label{eq2}
b_i(M) \le \phi_i(n-r) b_1(M) + \phi_{i-1}(n-r). \end{equation} \end{thm}
\begin{proof}
We can assume that every series class of $M$ has exactly $i$ elements. Indeed, by [a] and [c] of Theorem \ref{tuttepoly}, any series class with more than $i$ elements can be contracted down to cardinality $i$ without changing either side of (\ref{eq2}), while expanding any class with fewer than $i$ elements may increase the left--hand side of (\ref{eq2}) but will not alter the right--hand side. Let $\tilde{M}$ be the matroid obtained from $M$ by contracting all of the series classes down to one element. The dual of the formula on the top of page 185 of \cite{Bry2} is
\begin{equation} \label{eq3} T(M;x,0) = (x^{i-1} + \dots x +1)^{n-r} T(\tilde{M}; x^i, \frac{x^{i-1} + \dots x}{x^{i-1} + \dots x + 1}) \end{equation}
Using (\ref{eq3}), we see that,
\begin{equation} \label{eq4}
b_i(M) = \displaystyle\sum^i_{j=1} \binom{n-r+i-j-1}{i-j} b^\star_j(\tilde{M}). \end{equation}
\noindent Since $b^\star_1(\tilde{M}) = b_1(M),$ (\ref{eq2}) follows from (\ref{eq4}) by applying (\ref{eq1}) to $\tilde{M}^\star.$
\end{proof}
Inequality (\ref{eq2}) is as optimal as can be expected in the sense that given $n-r,i$ and $b_1$ there are matroids which satify equality. Take any matroid which satisfies equality in (\ref{eq1}) and expand every series class to cardinality $i.$ Then, equality in (\ref{eq2}) holds. Of course, since $b_r=1$ and $\phi_i$ is increasing in $i$, no matroid can satisfy equality in (\ref{eq2}) for all $i.$
\section{Independence complexes} \label{ind complexes}
Suppose the smallest cocircuit of $M$ has cardinality $k.$ As pointed out in Section \ref{k-CM}, $\Delta(M)$ is a $k$--CM complex. So, we can apply those methods to $\Delta(M).$ In addition to the previously mentioned absolute upper bound $h_i(M) \le \binom{n-r+i-1}{i}$ and relative upper bound $h_{i+1} \le h^{<i>}_i,$ the $h$--vectors of independence complexes of matroids satisfy an analogue of the $g$--theorem for simplicial polytopes.
\begin{thm} \cite{Sw3} Assume that $M$ has no coloops. Let $g_i(M)=h_i(M)-h_{i-1}(M).$ Then for all $i, 1 \le i \le (r+1)/2,$ $$g_{i+1}(M) \le g_i^{<i>}(M).$$ \end{thm}
\noindent The above theorem was proved independently by Hausel and Sturmfels for matroids representable over the rationals using toric hyperk\"{a}ler varieties \cite{HS}.
Relative lower bounds, also reminiscent of the $g$--threorem for simplicial polytopes, were originally established by Chari using a PS--ear decomposition of $\Delta(M).$ See \cite{Ch} for the definition of PS--ear decompositions and a proof of the following theorem.
\begin{thm}
Suppose $M$ has no coloops. Then for all $i, 0 \le
i \le r/2.$ $$ h_{i-1} \le h_i,$$ $$ h_i \le h_{r-i}.$$ \end{thm}
\begin{prob}
Do $2$--CM complexes satisfy the inequalities in the previous two
theorems? \end{prob}
An affirmative answer to this question would, with the addition of the Dehn--Sommerville equations, give a complete description of all possible $h$--vectors of simplicial homology spheres \cite[Conjecture II.6.2]{St}.
In \cite{BC2} Brown and Colbourn conjectured that for co-graphic $M,$ the complex zeros of $T(M;x,1)$ were contained in the closed unit disk. While this has since proven to be false \cite{RS}, attempts to prove it led to a couple of relative lower bounds for $h$--vectors of independence complexes of any matroid.
\begin{thm} Suppose $M$ has no coloops.
\begin{enumerate}
\item \cite{BC2} For all $i \le r,$
$$ h_i \ge \sum^i_{j=1} (-1)^{j-1} h_{i-j}$$
\item \cite{Wa}
Let $ I_j$ be the number of independent subsets of $M$ of cardinality $j.$ Then for all $0 \le k \le r,$ $$ \sum^r_{j=k} \binom{j}{k} (-2)^{r-j} I_j \ge 0.$$
\end{enumerate} \end{thm}
Stanley used the notion of a level ring to establish the relative lower bound $ h_{j-i}(M) \le h_i(M) h_j(M)$ whenever $0 \le i,j \le r.$ In particular, setting $j=r,$ we find that $h_{r-i}(M)\le \binom{n-r+i-1}{i} h_r(M).$ By applying (\ref{del by h}) we can obtain similar relative lower bounds for $h_{i-j}(M)$ in terms of $h_i(M)$ and we can also determine when equality occurs.
\begin{prop}
Assume that $M$ has no coloops. Then
for all $i, 1 \le j < i \le r,$ \begin{equation} \label{max h}
h_{i-j}(M) \le \frac{\binom{n-i+j-1}{r-i+j}}{\binom{n-i-1}{r-i}} h_i (M). \end{equation}
Furthermore, equality occurs if and only if every series class of $M$ has cardinality greater than $r-i+j.$ \end{prop}
\begin{proof} Since $M$ has no coloops, $\Delta(M)$ is a $2$--CM complex. Therefore, (\ref{del by h}) implies $(r-i+1) h_{i-1}(M) \le (n-i) h_i(M).$ In order for equality to occur, $h_i(M-e)$ must be zero for every $e$ in $E.$ By Theorem \ref{tuttepoly} [a], this is equivalent to every series class of $M$ having cardinality greater than $r-i+1.$ The proposition follows by induction on $j.$ \end{proof}
In \cite{BC} Brown and Colbourn proved the relative lower bound $h_{r-1} (M) \le r h_r(M)$ which only involves the rank of $M.$ This can be improved using Theorem \ref{char by r}.
\begin{thm}
Let $M$ be a rank $r$ matroid without coloops. Then, \begin{equation} \label{ind by r} h_{r-i} \le \binom{r-1}{i} h_r + \binom{r-1}{i-1}. \end{equation} \end{thm}
\begin{proof}
By \cite{Bry4}, $h_i(M)$ equals $b_{r-i+1}$ of the free coextension of
$M.$ Since the latter matroid has rank $r+1$ and is connected,
(\ref{ind by r}) is an immediate consequence of Theorem \ref{char by r}. \end{proof}
As in the case of Theorem \ref{char by r}, if $h_{r-i}(M) = \binom{r-1}{i} h_r(M)+ \binom{r-1}{i-1},$ then $h_{r-j}(M) \le \binom{r-1}{j} h_r(M) + \binom{r-1}{j-1}$ for all $0 \le j \le i.$ A routine deletion--contraction induction shows that for a given $r$ and $h_r,$ $$M=U_{1,h_r+1} \oplus \underbrace{U_{1,2} \oplus \dots \oplus U_{1,2}}_{r-1}$$ satisfies equality in (\ref{ind by r}).
\begin{cor}
Let $M$ be a rank $r$ matroid without coloops. Let $I_j$ be the number
of independent subsets of $M$ of cardinality $j.$ Then, $$I_j \le \displaystyle\sum^j_{i=0} \binom{r-i}{r-j} \left\{ \binom{r-1}{i} h_r + \binom{r-1}{i-1} \right\}.$$ \end{cor}
\begin{proof} Apply the above theorem to (\ref{f by h}). \end{proof}
In section \ref{k-CM} we posed the problem of finding absolute lower bounds for a $k$--CM complex given $n$ and $r.$ Here we examine this problem for independence complexes. Consider the special case of a rank two matroid $M$ without loops. The simplification of $M$ is isomorphic to $U_{2,m}$ where $m$ is the number of parallel classes of $M.$ Therefore, $M$ is specified up to isomorphism by a partition $n = p_1 + \dots + p_m,$ where the $p_i$'s are the sizes of the parallel classes of $M.$ Since $h_0 = 1$ and $h_1 = n-r,$ minimizing the $h$--vector of $M$ is equivalent to minimizing the number of bases of $M.$ As noted earlier, $M$ is $k$--CM if and only if every hyperplane of $M$ has cardinality at most $n-k.$ Equivalently, each $p_i \le n-k.$ The number of bases of $M$ is
$$\binom{n}{2} - \displaystyle\sum^m_{i=1} \binom{p_i}{2}.$$
\noindent This is minimized by setting $m = \lceil n/(n-k) \rceil, p_i = n-k$ for $i \le m-1,$ and $p_m = n - (m-1)(n-k).$ Note that this implies that when $n \ge 2k, h_2(M)$ is bounded below by $h_2(U_{1,n-k} \oplus U_{1,k}).$
An independence complexes is $2$--CM if and only if it has no coloops. In \cite{Bj} Bj\"{o}rner showed that for any matroid without coloops $h_i \ge n-r$ for $0 < i < r.$ While it is not specifically stated, the proof implies that $h_r \ge n-2r+1.$ In general, given $n$ and $r$ there may be no single coloop-free matroid that achieves all of these bounds. For example, if $n=8$ and $ r=4,$ then the only matroid without coloops such that $h_4(M) = 1$ is $M =U_{1,2} \oplus U_{1,2} \oplus U_{1,2} \oplus U_{1,2}.$ However, $h_2(M) = 6 > n-r.$ If we restrict our attention to $i < r, $ then $U_{1,n-r} \oplus U_{r-1,r}$ does satisfy $h_i = n-r$ for $0 < i < r.$
\begin{defn}
$M(r,n,k) =U_{1,n-r-k+2} \oplus U_{r-1,r+k-2}$ \end{defn}
\noindent Direct computation shows that $h_i(M(r,n,k)) = \binom{k+i-2}{i} + (n-r-k+1) \binom{k+i-3}{i-1}.$ In addition, $\Delta(M(r,n,k))$ is $k$--CM as long as $n \ge r+2k-2.$
\begin{thm} \label{matroid k-cm}
Fix $r \ge 2$ and $k \ge 3$ . There exists $N(k,r)$ such that if $M$ is a matroid without loops whose smallest cocircuit has cardinality at least $k$ and $n \ge N(k,r),$ then for all $i, 0 \le i \le r,$
\begin{equation} \label{long term} h_i(M) \ge h_i(M(r,n,k)). \end{equation}
\end{thm}
\begin{proof}
First we show that if $n > k(r+1),$ then there exists $e \in M$ such that $\Delta(M-e)$ is still $k$--CM. Let $\mathbf{H}$ be the set of hyperplanes of $M$ of cardinality $n-k.$ If $\mathbf{H}$ is empty, then any $e$ will do since no hyperplane of $M-e$ will have size greater than $n-k-1.$ Otherwise, let $B$ be the intersection of all of the hyperplanes in $\mathbf{H}.$ Since $B$ is a flat of $M$ there exists $H_1, \dots, H_{r+1},$ not necessarily distinct, in $\mathbf{H}$ such that $H_1 \cap \dots \cap H_{r+1} = B.$ Therefore, $|B| \ge n-k(r+1)$ and $B$ is not empty. But, for any $e \in B, \Delta(M-e)$ is $k$--CM.
As noted above, when $r=2, N(2,k) = 2k$ works. So, assume that $r \ge 3.$ Let $M^\prime$ be a contraction of $M$ and let $n^\prime=|E(M^\prime)|.$ By Proposition \ref{k-CM abs}, $h_i(M^\prime) \ge h_i(U_{r-1,r+k-2}).$ In fact, if $n > r+k-2,$ then $h_i(M^\prime)$ is strictly greater than $h_i(U_{r-1,r+k-2})$ for $1 \le i \le r.$ Indeed, this is proved by Tutte recursion as in Proposition \ref{k-CM abs}. The base case compares the $h$--vectors of $U_{2,4}$ and any five element rank two matroid whose smallest cocircuit has at least three elements. The $h$--vector of $U_{2,4}$ is $(1,2,3).$ From the discussion of rank two matroids, the $h$--vectors of the latter group of matroids is bounded below by $(1,3,4),$ the $h$--vector of the matroid whose simplification is $U_{2,3}$ and whose parallel classes have cardinality $2,2$ and $1.$ Note that this claim is not true when $k=2.$ In particular, $U_{1,2} \oplus U_{r-1,r}$ is a rank $r$ matroid without coloops and $r+2$ elements whose $h_r$ is not strictly less than $h_r$ of $U_{r,r+1}.$
To finish the proof, we find $N(r,k,i)$ such that the theorem holds for just $h_i$ and then let $N(r,k)$ be the maximum of the all of the $N(r,k,i).$ Since $h_0(M)=1$ and $h_1(M) = n-r, r+k-1$ works for $N(r,k,0)$ and $N(r,k,1).$ So fix $i \ge 2.$ Let $N$ be the minimum of $h_i(\bar{M})$ for all loopless matroids $\bar{M}$ such that $|E(\bar{M})| = k(r+1) +1, r(\bar{M}) = r$ and the smallest cocircuit of $\bar{M}$ has at least $k$ elements. Let $N(r,k,i) = k(r+1) + 1 + h_i(M(r,k(r+1)+1,k)) - N.$ \\
{\bf Claim:} If $n \ge N(k,r,i),$ then $h_i(M) \ge h_i(M(r,n,k)).$ \\
{\it Proof of claim:} Choose $e_1 \in M$ such that the smallest cocircuit of $M-e_1$ has cardinality greater than or equal to $k.$ Given $e_j$ choose $e_{j+1}$ so that the smallest cocircuit of $M - \{e_1,\dots,e_j,e_{j+1}\}$ has size at least $k.$ This can be done up to $j = n-k(r+1)-1.$ Deleting and contracting on each deletion,
$$h_i(M) = h_i(\tilde{M}) + \displaystyle\sum_j h_{i-1}(M - \{e_1,\dots,e_{j-1}\}/e_j),$$
\noindent where $\tilde{M}$ is $M - \{e_1,\dots,e_{n-k(r+1)-1}\}.$ By construction, $|E(\tilde{M})| = k(r+1)+1, r(\tilde{M}) = r$ and the smallest cocircuit of $\tilde{M}$ has at least $k$ elements. In addition, the rank of each contraction is $r-1$ and its independence complex is $k$--CM. There are two possibilities.
\begin{itemize}
\item
Every contraction has more than $r+k-2$ non-loop elements. In this case $h_i(M) \ge h_i(\tilde{M}) + (n-k(r+1)-1)[ h_{i-1}(U_{r-1,r+k-2})+1].$ Compare this to computing $h_i(M(r,n,k))$ by deleting and contracting down to $U_{1,rk-k+3} \oplus U_{r-1,r+k-2}.$ The definition of $N(r,k,i)$ insures that $h_i(M)$ is bounded below by $h_i(M(r,n,k)).$
\item
At least one contraction, say $M - \{e_1,\dots,e_{j-1}\}/e_j)$ has exactly $r+k-2$ elements. Since this contraction is a rank $r-1$ matroid whose smallest cocircuit has at least $k$ elements it must be equal to $U_{r-1,r+k-2}.$ Therefore, $M - \{e_1,\dots,e_{j-1}\}$ has one non--trivial parallel class which contains $e_j$ and the simplification of $M - \{e_1,\dots,e_{j-1}\}$ is a one--element coextension of $U_{r-1,r+k-2}.$ The one--element coextension of $U_{r-1,r+k-2}$ which minimizes $h_i(M - \{e_1,\dots,e_{j-1}\})$ is the one obtained by adding a coloop to $U_{r-1,r+k-2}.$ Hence, $h_i(M - \{e_1,\dots,e_{j-1}\})$ is bounded below by $h_i(M(r,n-j,k)).$ However, this implies that $h_i(M) \ge h_i(M(r,n-r,k) + j \ h_{i-1}( U_{r-1,r+k-2}) = h_i(M(r,n,k)).$
\end{itemize}
\end{proof}
Some lower bound on $n$ is necessary in order for (\ref{long term}) to hold. For instance, let $M= U_{1,3} \oplus U_{1,3} \oplus U_{1,3}.$ Then $r=3, k=3$ and $n=9.$ The $h$--vector of $M$ is $(1,6,12,8),$ while the $h$--vector of $M(3,9,3) = U_{1,5} \oplus U_{2,4}$ is $(1,6,11,12).$
\noindent As usual, absolute lower bounds yield relative lower bounds via (\ref{del by h}). \begin{cor}
Fix $r \ge 2$ and $k \ge 3$ . There exists $N(k,r)$ such that if $M$ is a matroid without loops whose smallest cocircuit has cardinality $k$ and $n \ge N(k,r),$ then for all $i, 0 \le i \le r,$
$$(r-i+1) h_{i-1}(M) + n \ h_i(M(r,n-1,k-1)) \le (n-i) h_i(M).$$ \end{cor}
\noindent {\it Acknowledgment:} An anonymous referee's comments and suggestions dramatically improved the exposition in several places.
\end{document} | arXiv |
\begin{document}
\begin{frontmatter}
\title{Limit distributions for large P\'olya urns\protect\thanksref{T1}} \runtitle{Limit laws for large P\'olya urns} \thankstext{T1}{Supported by ANR-05-BLAN-0372-02 ``Structures Al\' eatoires Discr\`etes et Algorithmes.''}
\begin{aug} \author[A]{\fnms{Brigitte} \snm{Chauvin}\corref{}\ead[label=e1]{[email protected]}\ead[label=u1,url]{http://www.math.uvsq.fr/\texttildelow chauvin/}}, \author[B]{\fnms{Nicolas} \snm{Pouyanne}\ead[label=e2]{[email protected]}\ead[label=u2,url]{http://www.math.uvsq.fr/\texttildelow pouyanne/}} \and \author[C]{\fnms{Reda} \snm{Sahnoun}\ead[label=e3]{[email protected]}} \runauthor{B. Chauvin, N. Pouyanne and R. Sahnoun} \affiliation{INRIA Rocquencourt and Universit\'e de Versailles, Universit\'e de Versailles and Universit\'e de Versailles}
\address[A]{B. Chauvin\\ INRIA Rocquencourt, project Algorithms\\ Domaine de Voluceau B.P.105\\ 78153 Le Chesnay Cedex\\ France \\ and \\ Laboratoire de Math\'ematiques\\\quad de Versailles \\ CNRS, UMR 8100 \\ Universit\'e de Versailles, St. Quentin \\ 45, avenue des Etats-Unis\\ 78035 Versailles Cedex\\ France\\ \printead{e1}\\ \printead{u1}} \address[B]{N. Pouyanne\\ Laboratoire de Math\'ematiques\\\quad de Versailles\\ CNRS, UMR 8100\\ Universit\'e de Versailles, St. Quentin \\ 45, avenue des Etats-Unis\\ 78035 Versailles Cedex\\ France\\ \printead{e2}\\ \printead{u2}} \address[C]{R. Sahnoun\\ Laboratoire de Math\'ematiques de Versailles\\ CNRS, UMR 8100\\ Universit\'e de Versailles, St. Quentin\\ 45, avenue des Etats-Unis\\ 78035 Versailles Cedex\\ France\\ \printead{e3}} \end{aug}
\received{\smonth{9} \syear{2009}} \revised{\smonth{3} \syear{2010}}
\begin{abstract} We consider a two-color P\'olya urn in the case when a fixed number $S$ of balls is added at each step. Assume it is a \emph{large} urn that is, the second eigenvalue $m$ of the replacement matrix satisfies $1/2<m/S\leq1$. After $n$ drawings, the composition vector has asymptotically a first deterministic term of order $n$ and a second random term of order $n^{m/S}$. The object of interest is the limit distribution of this random term.
The method consists in embedding the discrete-time urn in continuous time, getting a two-type branching process. The dislocation equations associated with this process lead to a system of two differential equations satisfied by the Fourier transforms of the limit distributions. The resolution is carried out and it turns out that the Fourier transforms are explicitly related to Abelian integrals over the Fermat curve of degree $m$. The limit laws appear to constitute a new family of probability densities supported by the whole real line. \end{abstract}
\begin{keyword}[class=AMS] \kwd[Primary ]{60C05} \kwd[; secondary ]{60J80} \kwd{05D40}. \end{keyword}
\begin{keyword} \kwd{P\'olya urn} \kwd{urn model} \kwd{martingale} \kwd{characteristic function} \kwd{embedding in continuous time} \kwd{multitype branching process} \kwd{Abelian integrals over Fermat curves}. \end{keyword}
\end{frontmatter} \setcounter{footnote}{1}
\section{Introduction}\label{intro}
Consider a two-color P\'olya--Eggenberger urn random pro\-cess, with replacement matrix $R=\left({ a\atop c}\enskip {b\atop d }\right)$: the urn starts with a finite number of red and black balls as initial composition (possibly monochromatic). At each discrete time $n$, one draws a ball uniformly at random, notices its color, puts it back into the urn and adds balls according to the following rule: if the ball drawn is red, one adds $a$ red balls and $b$ black balls; if the ball drawn is black, one adds $c$ red balls and $d$ black balls. The integers $a,b,c,d$ are assumed to be nonnegative\footnote{One admits classically negative values for $a$ and~$d$, together with arithmetical conditions on $c$ and~$b$. Nevertheless, the paper deals with so-called \emph{large} urns, for which this never happens.} and the urn is assumed to be \textit{balanced}, which means that the total number of balls added at each step is a constant $S=a+b=c+d$. The composition vector of the urn at time $n$ is denoted by
\[ U^{\mathit{DT}}(n)=\pmatrix{ \# \mbox{ red balls at time $n$}\cr \# \mbox{ black balls at time $n$} }. \]
It is a random vector and the article deals with its asymptotics when $n$ tends to infinity. Throughout the paper, the qualifier DT is used to refer to \emph {discrete-time} objects while CT will refer to \emph{continuous-time} ones.
Since P\'olya's original paper \cite{Pol}, this question has been extensively studied so that citing all contributions is hopeless. The following references give however a good idea of the variety of methods: combinatorics with many papers by Mahmoud (see his recent book \cite {Mah08}), probabilistic methods by means of embedding the process in continuous time (see Janson \cite{Jan}), analytic combinatorics by Flajolet {et al.} \cite{FlajDP} and a more algebraic approach in \cite{Pou08}. The union of these papers is sufficiently well documented, guiding the reader to a quasi exhaustive bibliography.
The asymptotic behavior of $U^{\mathit{DT}}(n)$ is closely related to the spectral decomposition of the replacement matrix. In case of two colors, $R$ is equivalent to $\left({ S\atop 0}\enskip {0\atop m} \right)$, where the largest eigenvalue is the balance $S$ and the smallest eigenvalue is the integer $m = a-c = d-b$. We denote by $\sigma$ the ratio between the two eigenvalues:
\[ \sigma= \frac mS\leq1. \]
It is well known that the asymptotics of the process has two different behaviors depending on the position of $\sigma$ with respect to the value $1/2$. Briefly:\vspace*{1pt}
(i) when $\sigma<\frac{1}{2} $, the urn is called \emph{small} and, except when $R$ is triangular, the composition vector is asymptotically Gaussian\footnote{The case $\sigma=1/2$ is similar to this one, the normalization being $\sqrt{n\log n}$ instead of $\sqrt n$.}:
\[ \frac{U^{\mathit{DT}}(n) -nv_1}{\sqrt n} \mathop{\hbox to 23pt{\rightarrowfill}}\limits _{n\to\infty}\limits^{\mathcal{D}}{\mathcal N}(0,\Sigma^2), \]
where $v_1$ is a suitable eigenvector of $^t\! R$ relative to $S$ and $\Sigma^2$ has a simple closed form;
(ii) when $\frac{1}{2} < \sigma< 1 $, the urn is called \emph {large} and the composition vector has a quite different strong asymptotic form:
\begin{equation} U^{\mathit{DT}}(n) = nv_1 + n^{\sigma} W^{\mathit{DT}} v_2 + o(n^{\sigma}), \end{equation}
where $v_1, v_2$ are suitable eigenvectors of $^t\! R$ relative to the eigenvalues $S$ and $m$, $W^{\mathit{DT}}$ is a real-valued random variable arising as the limit of a martingale, the little $o$ being almost sure and in any $L^p, p\geq1$. The moments of $W^{\mathit{DT}}$ can be recursively calculated but they have no known global closed form \cite{Pou08}.
The particular case $\sigma= 1$ is the original P\'olya urn; it corresponds to taking $R=S\operatorname{Id}$ as replacement matrix. It has been well known, since Gouet \cite{Gouet}, that the composition vector admits an almost sure asymptotics of order one: $U^{\mathit{DT}}(n) = n D +o(n)$ where the random vector $D$ has a Dirichlet density (explicitly given in~\cite{Gouet}).
In the present article, the object of interest is the distribution of $W^{\mathit{DT}}$ for large urns.
The first step consists in classically embedding the discrete-time process $(U^{\mathit{DT}}(n))_{n\geq0}$ into a continuous-time Markov process $(U^{\mathit{CT}}(t))_{t\geq0}$, by equipping each ball with an exponential clock. At any $n$th jump time $\tau_n$, the continuous-time process $U^{\mathit{CT}}(\tau_n)$ has the same distribution as $U^{\mathit{DT}}(n)$. This connection between both processes is the key point, allowing us to work on the continuous-time process, where independence properties have been gained.
In Theorem \ref{continuousurn}, we show that, in the case of large urns, the continuous-time process satisfies, when $t$ tends to infinity, the following asymptotics,
\begin{equation} U^{\mathit{CT}}(t) = e^{S t}\xi v_1 \bigl(1+o(1)\bigr)+ e^{m t} W^{\mathit{CT}} v_2\bigl(1+o(1)\bigr), \end{equation}
where $v_1$ and $v_2$ are the same vectors as above, $\xi$ and $W^{\mathit{CT}}$ are real-valued random variables that arise as limits of martingales, with $o(\cdot)$ meaning ``almost sure and in any $L^p, p\geq1$.'' Moreover, we prove that $\xi$ is Gamma-distributed. These results are based on the spectral decomposition of the infinitesimal generator of the continuous-time process on spaces of two-variable polynomials.
Thanks to the embedding connection, the two random variables $W^{\mathit{DT}}$ and $W^{\mathit{CT}}$ are connected (Theorem~\ref{thmartingaleconnection}):
\[ W^{\mathit{CT}} = \xi^{\sigma} W^{\mathit{DT}} \qquad\mbox{a.s.}, \]
$\xi$ and $W^{\mathit{DT}}$ being independent. Since $\xi^{\sigma}$ is invertible,\footnote{A probability distribution $A$ is called \emph{invertible} when, for any probability distributions $A$ and $B$, the equation $AX=B$ admits a unique solution $X$ independent of $A$, see, for instance, Chaumont and Yor
\cite{ChaumontYor}. The invertibility of any power of a Gamma distribution can be shown by elementary considerations on Fourier transforms.} the attention is focused on the determination of the distribution of $W^{\mathit{CT}}$.
Because of the nonnegativity of $R$ entries, $(U^{\mathit{CT}}(t))_{t\geq0}$ is a two-type branching process, visualized as a random tree: the branching property gives rise to dislocation equations on $U^{\mathit{CT}}(t)$. If one denotes by $\mathcal{F}$ ({resp.}, $\mathcal{G}$) the characteristic function of $W^{\mathit{CT}}$ starting from one red ball and no black ball ({resp.}, no red ball, one black ball), the independence of the subtrees in the branching process implies that the characteristic function of \emph{any} $W^{\mathit{CT}}$ starting from $\alpha$ red balls and $\beta$ black balls is the product $\mathcal{F}^{\alpha}\mathcal{G}^{\beta}$. Furthermore, the dislocation equations on $U^{\mathit{CT}}(t)$ lead to the following differential system
\begin{equation} \label{premiersystemeFourier} \cases{ \mathcal{F}(x)+mx\mathcal{F}'(x)=\mathcal{F}(x)^{a+1}\mathcal {G}(x)^b, \cr \mathcal{G}(x)+mx\mathcal{G}'(x)=\mathcal{F}(x)^{c}\mathcal{G}(x)^{d+1}, } \end{equation}
together with suitable boundary conditions. Notice that the corresponding exponential moment generating series (Laplace series) are also solutions of (\ref{premiersystemeFourier}), but their radius of convergence is equal to 0. This is detailed in Section \ref{serieLaplace}.
The solution of system (\ref{premiersystemeFourier}) is obtained in Section \ref{resolution}, where it is shown that $\mathcal{F}$ and $\mathcal{G}$ can be made explicit in terms of inverse functions of Abelian integrals over the Fermat curve of degree $m$. Indeed, for any complex $z$ in a suitable open subset of $\mathbb{C}$, let
\[ I_{m,S,b}(z)=\int_{[z,z\infty)}(1+u^m)^{b/m}\frac {du}{u^{S+1}}, \]
where $[z,z\infty)$ denotes the ray $\{ tz, t\geq1\}$. The function $I_{m,S,b}$ defines a conformal mapping on the open sector $ \mathcal{V}_m=\{ z\neq0, 0<\arg(z)<\pi/m\} $. If $J_{m,S,b}$ denotes the holomorphic function, defined on the lower half-plane as left inverse function of $I_{m,S,b}$ and extended to the slit plane by conjugacy, the closed form of $\mathcal{F}$ and $\mathcal{G}$ are given in the following result.
\begin{Th*} For any $x>0$,\vspace*{-2pt}
\[ \cases{ \mathcal{F}(x)=Kx^{-1/m}J_{m,S,b}\biggl( C_0+\dfrac {K^S}Sx^{- S/m}\biggr), \cr \mathcal{G} (x)=Kx^{-1/m}J_{m,S,c}\biggl( C_0+\dfrac {K^S}Sx^{- S/m}\biggr) , }\vspace*{-2pt}
\]
where $K\in\mathbb{C}$ and $C_0<0$ are explicit constants. \end{Th*}
For precise statements and proofs, see Section \ref{calculdeF} and Theorem \ref{characteristicFunctions}.
The solution of system (\ref{premiersystemeFourier}) is effected by a ramified change of variable and functions, leading to the following monomial system:
\begin{equation} \label{monom} \cases{ f'=f^{a+1}g^b, \cr g'=f^cg^{d+1}. } \end{equation}
This remarkable fact is evocative of the case of \emph{small} urns and discrete time, as considered in a beautiful study of Flajolet {et al.} \cite{Flaj}. The method of \cite{Flaj} leads directly to the same system (\ref{monom}) on generating functions. The assumption $\sigma<1/2$, when expressed in term of the four parameters $a$, $b$, $c$ and $d$, does not fundamentally affect the system but requires completely different analytic considerations.
The limit laws of the $W^{\mathit{CT}}$ appear to constitute a new family of probability distributions, indexed by three parameters $S,m,b$ subject to assumptions (\ref{hypSmb}) and by initial conditions $\alpha, \beta$. We prove in Section \ref{densiteWCT} that they admit densities that can be expressed by means of the inverse Fourier transforms of their characteristic function derivatives. Furthermore, the laws are infinitely divisible and their support is the whole real line, the radius of convergence of their exponential moment generating series being equal to $0$.
Many questions remain open. For instance, are these distributions characterized by their moments? What is the precise asymptotics of their densities at infinity (tails)? It is shown in \cite{Sahnoun2} that, for triangular and nondiagonal replacement matrices, the discrete-time limit law $W^{\mathit{DT}}$ is never infinitely divisible; does this situation persist in the present nontriangular case?
\section{The model}\label{model}
\subsection{Definition of the process}\label{sec2.1}
Let $a$, $b$, $c$ and $d$ be nonnegative integers such that $a+b=c+d =:S$ and $R$ be the matrix
\begin{equation} \label{matriceUrne} R:= \pmatrix{ a&b\cr c&d } = \pmatrix{ a&S-a\cr S-d&d }.
\end{equation}
The discrete-time P\'olya--Eggenberger urn process associated with the replacement matrix $R$, which has been informally described in the \hyperref[intro]{Introduction}, is the Markov chain $(U^{\mathit{DT}}(n), n\in\mathbb{N})$, having $\mathbb{N}^2\setminus \{ (0,0)\}$ as state space and
\begin{equation} \label{probasTransition}
\frac x{x+y}\delta_{(x+a,y+b)}+\frac y{x+y}\delta_{(x+c,y+d)} \end{equation}
as transition probability at any nonzero point $(x,y)\in\mathbb {N}^2$. In this formula, $\delta_v$ denotes Dirac point mass at $v\in\mathbb{N}^2$. This means that $(U^{\mathit{DT}}(n), n\in\mathbb{N})$ is a random walk in $\mathbb{N}^2\setminus\{ (0,0)\}$ (or in the two-dimensional one-column nonzero matrices with nonnegative integer entries, we will use both notations) recursively defined by the conditional probabilities
\[ \cases{ \mathrm{P}
\biggl( U^{\mathit{DT}}(n+1) = U^{\mathit{DT}}(n) + \pmatrix{ a\cr b }
| U^{\mathit{DT}}(n)=\pmatrix{ x\cr y }\biggr) =\dfrac{x}{x+y},
\cr \mathrm{P} \biggl( U^{\mathit{DT}}(n+1) = U^{\mathit{DT}}(n) + \pmatrix{ c\cr d }
| U^{\mathit{DT}}(n)=\pmatrix{ x\cr y }\biggr) =\dfrac{y}{x+y}. } \] In the sequel,
\[ \bigl( U_{(\alpha,\beta)}^{\mathit{DT}}(n),n\geq0\bigr) \]
will denote the process starting from the nonzero vector $(\alpha ,\beta)$ and
\[ u:=\alpha+\beta \]
will denote the total number of balls at time $0$. Notice that the balance property $S=a+b=c+d$ implies that the total number of balls at time $n$, when $U^{\mathit{DT}}(n)=(x,y)$, is the (nonrandom) number $x+y=u+nS$.
Denote by $w_1={ a\choose b }$ and $w_2={ c\choose d }$ the increment vectors of the walk. The transition operator $\Phi$ is defined, for any function $f$ on $\mathbb{N}^2$ and for any $v={ x\choose y }$, by
\[ \Phi(f)(v) = x[ f(v+w_1) - f(v)] + y[ f(v+w_2) - f(v)] . \]
Conditioning on $({\mathcal F}_n, n\geq0)$, which is the filtration associated with the process $(U^{\mathit{DT}}(n) , n\geq0)$, one gets
\[ \mathbb{E}^{{\mathcal F}_n}\bigl( f\bigl( U^{\mathit{DT}}(n+1)\bigr)\bigr) = \biggl( I + \frac {\Phi }{u+nS}\biggr) (f) ( U^{\mathit{DT}}(n) ). \]
In particular,
\begin{equation} \label{mart} \mathbb{E}^{{\mathcal F}_n}\bigl( U^{\mathit{DT}}(n+1)\bigr) = \biggl( I+\frac {A}{u+nS}\biggr) U^{\mathit{DT}}(n), \end{equation}
where $I$ denotes the two-dimensional identity matrix and
\[ A := {}^t\hspace*{-2pt}R =\pmatrix{ a&c\cr b&d }. \]
\subsection{Asymptotics of the discrete-time process $U^{\mathit{DT}}(n)$}
As mentioned in Section \ref{intro} and briefly recalled hereunder, a discrete-time P\'olya--Eggenberger urn process has two different kinds of asymptotics depending on the ratio of the eigenvalues of its replacement matrix $R$. With our notation, these eigenvalues are $S$ and\vspace*{-1pt}
\[ m:=a-c=d-b. \]
Let us denote by $u_1$ and $u_2$ the two following linear eigenforms\footnote{An eigenform of an endomorphism $f$ is an eigenvector of $^t\hspace*{-2pt} f$; some authors call these linear forms left eigenvectors of $f$, referring to matrix operations.} of $A$, respectively associated with the eigenvalues $S$ and $m$, which means that $u_1\circ A=Su_1$ and $u_2\circ A=mu_2$:\vspace*{-1pt}
\begin{equation}\label{u} u_1(x,y) = \frac{1}{S}(x+y), \qquad u_2(x,y)= \frac{1}{S}(bx-cy), \end{equation}
and denote by $(v_1,v_2)$ the dual basis of $(u_1,u_2)$:\vspace*{-1pt}
\begin{equation}\label{v} v_1 = \frac S{(b+c)} \pmatrix{ c\cr b }, \qquad v_2 = \frac S{(b+c)} \pmatrix{ 1\cr -1 }. \end{equation}
The vectors $v_k$ are eigenvectors of $A$ and the projections on the eigenlines are $u_1v_1$ and $u_2v_2$.
For any positive real $x$ and any nonnegative integer $n$, if one denotes by $\gamma_{x,n}$ the polynomial\vspace*{-1pt}
\[ \gamma_{x,n} (t):= \prod_{k=0}^{n-1} \biggl(1 + \frac t{x+k}\biggr), \]
the matrix $\gamma_{\frac mS,n}(\frac AS)$ in nonsingular and it is immediate from (\ref{mart}) that\vspace*{-1pt} \[ \gamma_{ \frac mS,n}\biggl(\frac AS\biggr)^{-1} U^{\mathit{DT}}(n) \] is a (vector-valued) martingale.
As indicated in the \hyperref[intro]{Introduction}, the ratio of $R$ eigenvalues is denoted by\vspace*{-1pt}
\[ \sigma:=\frac mS\leq1. \]
The case of \emph{small} urn processes ({i.e.,} when $\sigma\leq1/2$) has been well studied; in this case, when $R$ is not triangular, the random vector admits a Gaussian central limit theorem (see Janson \cite{Jan}). Triangular replacement matrices impose a particular treatment and lead most often to a nonnormal second-order limit (see Janson \cite{JanTrig} or \cite{Sahnoun2}).
Our subject of interest is the case of \emph{large} urns, that is, when $\sigma>1/2$. In this case, $\frac1{S} U^{\mathit{DT}}(n)$ is a large P\'olya process with replacement matrix $\frac1{S} R$ in the sense of \cite{Pou08}.
As a matter of consequence, the projections of the above vector-valued martingale on the eigenlines of $A$, which are of course also martingales, converge in any ${\rm L}^p$, $p\geq1$ (and a.s.). In particular (second projection),
\[ M^{\mathit{DT}}(n):= \frac{1}{\gamma_{\frac uS,n}(\sigma) } u_2(U^{\mathit{DT}}(n)) \]
is a convergent martingale; since $\gamma_{u,n}(\sigma) = n^{\sigma} \frac{\Gamma(u)}{\Gamma (u+\sigma)}(1+o(1))$, denoting by
\begin{equation}\label{martingalediscrete} W^{\mathit{DT}} := \lim_{n\rightarrow+\infty} \frac{1}{n^{\sigma}} u_2(U^{\mathit{DT}}(n)), \end{equation}
a slight adaptation of \cite{Pou08} leads to the following theorem. Note that this theorem was essentially proven by Athreya and Karlin \cite{AK} and Janson \cite{Jan} for random replacement matrices. The convergence in $L^p$-spaces when $R$ is nonrandom is shown by the indicated adaptation of \cite{Pou08}.
\begin{Th} \label{asymptotiquediscrete} Suppose that $\sigma\in\,]1/2,1[$. Then, as $n$ tends to infinity,
\[ U^{\mathit{DT}}(n) = nv_1 + n^{\sigma} W^{\mathit{DT}} v_2 + o( n^{\sigma} ), \]
where $v_1$ and $v_2$, defined in \textup{(\ref{v})}, are eigenvectors associated with the eigenvalues $S$ and $m$; $W^{\mathit{DT}}$ is defined by (\ref{martingalediscrete}); $o( \cdot)$ means a.s. and in any $L^p, p\geq1$.
\end{Th}
\subsection{Parametrization and hypotheses} \label{parametrage}
The subject of the paper is $W^{\mathit{DT}}$ distribution in Theorem \ref {asymptotiquediscrete} so that the P\'olya urn process will be supposed large. Furthermore, the replacement matrix $R$ is supposed to be not triangular because this case has to be treated separately with regard to its limit law, as attested by Janson \cite{JanTrig}, Flajolet et al. \cite{FlajDP}, \cite{Sahnoun2} and the present paper.
Under these conditions, the assumptions on the replacement matrix $R=\left( {a\atop c}\enskip {b\atop d }
\right)$ are: $a+b=c+d=S$ (balance condition), $S/2<m=a-c=d-b<S$ (large urn) and $b,c\geq1$ (not triangular). Because of the balance condition, the parametrization of P\'olya urns is governed by three free parameters. The computation of Fourier transforms will show in Section \ref {calculdeF} that a natural choice of free parameters is $(m,S,b)$. The assumption ``large and nontriangular'' is equivalent, in terms of these data, to the following:
\[ R=\pmatrix{a&b\cr c&d }
=\pmatrix{S-b&b\cr S-m-b&m+b } \]
with
\begin{equation} \label{hypSmb} \cases{ m+2\leq S\leq2m-1, \cr 1\leq b\leq S-m-1. }
\end{equation}
Note that these inequalities imply $S\geq5$ and $m\geq3$ and that, for a given $m$, the point $(m,b)$ belongs to a triangle as represented in Figure~\ref{fig1}.
\begin{figure}
\caption{Parameters $(b,S)$ for a given $m$.}
\label{fig1}
\end{figure}
For small values of $S$, large urn processes have the following possible replacement matrices: for $S\in\{1,2\}$, only $R=S\operatorname{Id}_2$ defines a large urn; for $S\in\{3,4\}$, all large urns have triangular matrices. For $S\in\{5,6\}$, only $R= \left( { S-1\atop 1} \enskip {1\atop S-1 } \right) $ defines a nontriangular large urn. For $S=7$, apart from triangular or symmetric matrices, there are only two replacement matrices that define large urns: $ \left( { 6\atop 2}\enskip{1\atop 5 }
\right) $ and the other one derived from it by permutation of coordinates.
\section{Embedding in continuous time and martingale connection}\label{method}
\subsection{Embedding}\label{embeddingSection}
The idea of embedding discrete urn models in continuous-time branching processes goes back at least to Athreya and Karlin \cite{AK}. A description is given in Athreya and Ney \cite{AN}, Section 9. The method has been recently revisited and developed by Janson \cite{Jan}.
We define the continuous-time Markov branching process $(U^{\mathit{CT}}(t), t\geq0)$ as being the embedded process of $(U^{\mathit{DT}}(n), n\geq0)$. Following, for instance, Bertoin \cite{Bertoin}, Section 1.1, this means that it is defined as the continuous-time Markov chain having as jump rate, at any nonzero point $(x,y)\in\mathbb{N}^2$, the finite measure given by the transition probability of the discrete-time process [formula (\ref{probasTransition})]. One gets this way a branching process having the following dynamical description in terms of red and black balls. In the urn, at any moment, each ball is equipped with an $\mathit{\mathcal{E}xp}(1)$-distributed\footnote{ For any positive real $a$, $\operatorname{\mathit{\mathcal{E}xp}}(a)$ denotes the exponential distribution with parameter $a$.} random clock, all the clocks being independent. When the clock of a red ball rings, $a$ red balls and $b$ black balls are added in the urn; when the ringing clock belongs to a black ball, one adds $c$ red balls and $d$ black balls, so that the replacement rules are the same as in the discrete-time urn process.
The successive jumping times of $(U^{\mathit{CT}}(t), t\geq0)$, will be denoted by
\[ 0=\tau_0 < \tau_1 < \cdots< \tau_n < \cdots. \]
The $n$th jumping time is the time of the $n$th dislocation of the branching process. The process is thus constant on any interval $[\tau_n,\tau_{n+1}[$.
In the sequel,
\[ \bigl( U_{(\alpha,\beta)}^{\mathit{CT}}(t),t\geq0\bigr) \]
will denote the process starting from the nonzero vector $(\alpha ,\beta)$. Thus, for any initial condition $(\alpha, \beta)$, for any $t\geq0$,
\[ U_{(\alpha,\beta)}^{\mathit{CT}}(t)= U_{(\alpha,\beta)}^{\mathit{DT}}( a(t)), \]
where
\[ a(t):= \inf\{ n\geq0, \tau_n \geq t \}. \]
\begin{Lem} \label{embed} (i) for $n\geq0$, the distribution of $\tau_{n+1} - \tau_{n}$ is $\mathit{\mathcal{E} xp} (u+Sn)$ where $u$ denotes the total number of balls at time $0$;\vspace*{-6pt} \begin{longlist}[(iii)] \item[(ii)] the processes $(\tau_n)_{n\geq0}$ and $(U^{\mathit{CT}}(\tau_n))_{n\geq 0}$ are independent;
\item[(iii)] the processes $(U^{\mathit{CT}}(\tau_n))_{n\geq0}$ and $(U^{\mathit{DT}}(n))_{n\geq 0}$ have the same distribution. \end{longlist} \end{Lem}
\begin{pf} The total number of balls at time $t\in[\tau_n,\tau_{n+1}[$ is $u+Sn$. Therefore, (i) is a consequence of the fact that the minimum of $k$ independent random variables $\mathit{\mathcal{E} xp} (1)$-distributed is $\mathit{\mathcal{E} xp} (k)$-distributed. (ii) is the classical independence between the jump chain and the jump times in such Markov processes. The initial states and evolution rules of both Markov chains in discrete time and in continuous time are the same ones, so that (iii) holds. \end{pf}
\textit{Convention}: From now on, thanks to (iii) of Lemma \ref{embed}, we will classically consider that the discrete-time process and the continuous-time process are built on the \emph{same} probability space on which
\begin{equation} \label{embedding} (U^{\mathit{CT}}(\tau_n))_{n\geq0} = (U^{\mathit{DT}}(n))_{n\geq0} \qquad \mbox{a.s.} \end{equation}
\subsection{Asymptotics of the continuous-time process $U^{\mathit{CT}}(t)$} \label{asymptCT}
Let $v_1$ and $v_2$ the linearly independent eigenvectors of $A$ defined by (\ref{v}). In the case of large urns, the asymptotics of the continuous-time process $(U^{\mathit{CT}}(t))_{t\geq0}$ is given in the following theorem.
\begin{Th}[(Asymptotics of continuous-time process)]\label{continuousurn} When $t$ tends to infinity,
\begin{equation} \label{asymptotiqueCT} U^{\mathit{CT}}(t)=e^{St} \xi v_1\bigl( 1+o(1)\bigr) +e^{mt}W^{\mathit{CT}}v_2\bigl( 1+o(1)\bigr) , \end{equation}
where $\xi$ and $W^{\mathit{CT}}$ are real-valued random variables, the little $o$ being almost sure and in any ${\rm L}^p$-space, $p\geq1$. Furthermore, $\xi$ is $\operatorname{Gamma}(u /S)$-distributed, where $u =\alpha+\beta$ is the total number of balls at time $0$. \end{Th}
\begin{Rem} Another formulation of this Theorem is: in the basis $(v_1,v_2)$, the coordinate of $U^{\mathit{CT}}(t)$ along $v_1$ has $e^{St} \xi$ as its dominant term while the coordinate of $U^{\mathit{CT}}(t)$ along $v_2$ has $e^{mt} W^{\mathit{CT}}$ as its dominant term. \end{Rem}
\begin{pf*}{Proof of Theorem~\ref{continuousurn}} The embedding in continuous time has been studied in Athreya and Karlin \cite{AK} and in Janson \cite{Jan}. It has become classical that the process
\[ ( e^{-tA}U^{\mathit{CT}}(t)) _{t\geq0} \]
is a vector-valued martingale and that, in the case of large urns ($\sigma>1/2$), this martingale is bounded in ${\rm L}^2$, thus converges. Its projections on the eigenlines $\mathbb{R}v_1$ and $\mathbb {R}v_2$, \emph {\textup{that is},} respectively,
\[ e^{-St}u_1( U^{\mathit{CT}}(t)) \quad\mbox{and}\quad e^{-mt}u_2( U^{\mathit{CT}}(t)) \]
are also ${\rm L}^2$-convergent real valued martingales, thus converge almost surely. Their respective limits are named $\xi$ and $W^{\mathit{CT}}$. What still has to be proved is that these martingales converge in fact in any ${\rm L}^p$, $p\geq1$. The identification of $\xi$ distribution will be a consequence of this proof.
The infinitesimal generator of the Markov process $(U^{\mathit{CT}}(t))_t$ is the finite-difference operator
\[ \Phi(f)(x,y)=x\{ f(x+a,y+b)-f(x,y)\} +y\{ f(x+c,y+d)-f(x,y)\}, \]
defined for any (measurable) function $f$ and any $(x,y)\in\mathbb{R}^2$. For a very synthetic reference on semi-groups of Markov continuous-time processes, one can refer to Bertoin \cite{Bertoin}, Chapter 1. This operator $\Phi$ acts on two-variable polynomials. This action has been studied in detail in \cite{Pou08} in a more general framework. More precisely, for any integer $d\geq1$, the operator $\Phi$ acts on the finite-dimensional space of polynomials of degree less than $d$, so that, for any two-variable polynomial $P$ and for any $t\geq0$,
\begin{equation} \label{generateurInfinitesimal} \mathbb{E}( P(U^{\mathit{CT}}(t))) =\exp( t\Phi )( P) ( (U^{\mathit{CT}}(0))), \end{equation}
where, in this formula, $\Phi$ denotes the restriction of $\Phi$ itself on any finite-dimensional polynomials space containing $P$. The properties of $\Phi$ listed in the following lemma are proven in \cite{Pou08} and will be used here.
\begin{Lem} \label{lemmePhi} There exists a unique family of polynomials $Q_{p,q}\in\mathbb{R}[x,y]$, $p,q$ nonnegative integers, called \emph{reduced polynomials}, such that: \begin{longlist}[(1)] \item[(1)] $Q_{0,0}=1$, $Q_{1,0}=u_1$ and $Q_{0,1}=u_2$ [see (\ref{u}) for a definition of eigenforms $u_1$ and $u_2$];
\item[(2)] $\Phi(Q_{p,q})=(pS+qm)Q_{p,q}$ for all nonnegative integers $p,q$;
\item[(3)] $u_1^pu_2^q-Q_{p,q}\in\operatorname{Span}\{ Q_{p',q'}, p'S+q'm<pS+qm\} $ for all nonnegative integers $p,q$. \end{longlist} \end{Lem}
Note that the reduced polynomial $Q_{p,q}$ is in fact the projection of $u_1^pu_2^q$ on a suitable characteristic subspace of $\Phi$ restriction to some finite-dimensional polynomial space, and that this spectral decomposition of $\Phi$ on polynomials has a particularly simple form (it is diagonalizable) because the urn is large and two-dimensional. See \cite{Pou08} for more details.
Formula (\ref{generateurInfinitesimal}) and property {(ii)} of Lemma \ref{lemmePhi} lead to
\[ \forall(p,q)\in\mathbb{Z}_{\geq0}^2\qquad \mathbb{E}( Q_{p,q}( U^{\mathit{CT}}(t))) =e^{t(pS+qm)} Q_{p,q}(U^{\mathit{CT}}(0)). \]
This implies straightforwardly, with {(iii)} of Lemma \ref {lemmePhi}, that, for any $(p,q)$,
\begin{equation} \label{momentsReduits} \mathbb{E}( u_1^pu_2^q( U^{\mathit{CT}}(t))) =e^{t(pS+qm)} Q_{p,q}(U^{\mathit{CT}}(0)) +o\bigl( e^{t(pS+qm)}\bigr). \end{equation}
In particular, the martingales $e^{-St}u_1( U^{\mathit{CT}}(t))$ and $e^{-mt}u_2( U^{\mathit{CT}}(t))$ are ${\rm L}^p$-bounded for any $p\geq1$ and their respective limits, namely $\xi$ and $W^{\mathit{CT}}$ satisfy, for any nonnegative integer $p$,
\begin{equation} \label{momentsXiW} \mathbb{E}\xi^p=Q_{p,0}( U^{\mathit{CT}}(0))
\quad\mbox{and}\quad \mathbb{E}( W^{\mathit{CT}}) ^p=Q_{0,p}( U^{\mathit{CT}}(0)). \end{equation}
The convergence part of the theorem follows now from the spectral decomposition of $A$: for any $t\geq0$,
\[ U^{\mathit{CT}}(t)=u_1( U^{\mathit{CT}}(t))\cdot v_1+u_2( U^{\mathit{CT}}(t))\cdot v_2. \]
Besides, it is proven in \cite{Pou08}, or one can check it after an easy computation, that the reduced polynomials corresponding to the powers of $u_1$ have the following closed form expression
\[ Q_{p,0}=u_1( u_1+1)( u_1+2)\cdots( u_1+p-1). \]
Thanks to formula (\ref{momentsXiW}), this shows that the $p$th moment of $\xi$ is, for any integer $p\geq0$,
\[ \mathbb{E}\xi^p=\frac u S\biggl( \frac u S +1\biggr)\biggl( \frac u S +2\biggr)\cdots\biggl( \frac u S +p-1\biggr) =\frac{\Gamma( \frac{u}{ S}+p)}{\Gamma( \frac{u}{ S})}, \]
where $u$ is the total number of balls at time $0$ [remember that $u_1(U^{\mathit{CT}}(0))=u /S$, see (\ref{u})]. One identifies this way the required Gamma$(u /S)$ distribution, characterized by its moments. \end{pf*}
\begin{Rem} Notice that the distribution of $\xi$ has been given by Janson \cite {Jan} calculating first the distribution of $u_1(U^{\mathit{CT}}(t))$for every $t$:
\[ u_1(U^{\mathit{CT}}(t)) = \frac u S + Z(t), \]
where $Z(t)$ is a negative binomial distribution. \end{Rem}
\begin{Rem} Reduced polynomials $Q_{0,p}$ do not have a known closed form, so that reproducing the above method in order to compute the moments of $W^{\mathit{CT}}$ fails. \end{Rem}
\begin{Rem} It follows from the proof that the real-valued random variables $\xi$ and $W^{\mathit{CT}}$ are respective limits of the martingales
\begin{eqnarray*} \xi&=&\lim_{t\to+\infty}e^{-St}u_1( U^{\mathit{CT}}(t)), \\ W^{\mathit{CT}} &=&\lim_{t\to+\infty}e^{-mt}u_2( U^{\mathit{CT}}(t)). \end{eqnarray*}
They are not independent and their joint moments are computed from formula~(\ref{momentsReduits}): for any nonnegative integers $p,q$,
\[ E[ ( \xi) ^p( W^{\mathit{CT}}) ^q ]=Q_{p,q}( U^{\mathit{CT}}(0)). \]
For example, their respective means are $E\xi=u_1(U^{\mathit{CT}}(0))=\frac 1S(\alpha+\beta)$ and $EW^{\mathit{CT}}=u_2(U^{\mathit{CT}}(0))=\frac1S(b\alpha-c\beta)$, whereas
\[ E( \xi W^{\mathit{CT}}) =\frac{(\alpha+\beta+m)(b\alpha-c\beta)}{S^2} \]
as can be shown by computation of the reduced polynomial $Q_{1,1}=(u_1+\sigma)u_2$ (one can directly check that this polynomial is an eigenvector of $\Phi$, associated with the eigenvalue $S+m$).
\end{Rem}
\begin{Rem} When the urn is small ($\sigma<1/2$), the same method shows that the result on the first projection is still valid: the martingale\break $(e^{-St}u_1(U^{\mathit{CT}}(t)))_t$ converges in any ${\rm L}^p$ ($p\geq1$) to a $\operatorname{Gamma}(u /S)$ distributed random variable. On the contrary, the martingale $(e^{-mt}u_2(U^{\mathit{CT}}(t)))_t$ diverges and it is shown in Janson \cite{Jan} that the second projection satisfies a central limit theorem: when $\sigma=\frac mS<1/2$,
\[ e^{- \frac S2 t}u_2( U^{\mathit{CT}}(t)) \mathop{\hbox to 23pt{\rightarrowfill} }\limits_{t\to+\infty}\limits^{\mathcal{D}}\mathcal{N}, \]
where $\mathcal{N}$ is a normal distribution. In the case $\sigma=1/2$, the normalization must be modified and one gets the convergence in law of $\sqrt te^{-St/2}u_2(U^{\mathit{CT}}(t))$ to a normal distribution. \end{Rem}
\begin{Rem} The distributions of the $W^{\mathit{CT}}$ are infinitely divisible, because they are limits of infinitely divisible ones, obtained by scaling and projection of infinitely divisible ones. Indeed, in finite time, the distributions of the $U_{(\alpha, \beta)}^{\mathit{CT}}(t) $ are infinitely divisible. It has been said by Janson \cite{Jan}, proof of Theorem 3.9. With our notations, especially the one of (\ref{dislocationVectorielle}), it relies on the fact that
\[ U_{(\alpha, \beta)}^{\mathit{CT}}(t) \stackrel{\mathcal{L}}{=}[n] U_{( {\alpha}/n, {\beta}/n)}^{\mathit{CT}}(t), \]
where a continuous-time branching process (with the same branching dynamics as before), starting from real (nonnecessary integer) conditions, is suitably defined. \end{Rem}
\subsection{DT and CT connections} \label{connections}
Apply the first projection to the embedding principle (\ref{embedding}):
\[ u_1 ( U^{\mathit{CT}}(\tau_n)) = u_1 (U^{\mathit{DT}}(n) ) \qquad \mbox{a.s}. \]
By definition (\ref{u}) of $u_1$, this number is $\frac{1}S$ times the number of balls in the urn at time $n$, which equals $\frac{1}S (u +Sn) = n (1+o(1))$. Since stopping times $\tau_n$ tend to $+\infty$, renormalizing by $e^{-S\tau_n}$ and applying the convergence result of Section \ref{asymptCT} leads to
\begin{equation}\label{limxi} \xi= \lim_{ n\rightarrow+\infty} n e^{ -S\tau_n}. \end{equation}
Apply now the second projection to the embedding principle (\ref{embedding}):
\[ u_2 ( U^{\mathit{CT}}(\tau_n)) = u_2 (U^{\mathit{DT}}(n) ) \qquad \mbox{a.s}. \]
Renormalizing by $e^{-m\tau_n}$ implies that
\[ e^{ -m \tau_n} u_2 ( U^{\mathit{CT}}(\tau_n)) = W^{\mathit{CT}}(\tau_n ) = e^{ -m \tau _n} \gamma_{\frac{u}{S},n}(\sigma) M^{\mathit{DT}}(n)\qquad \mbox{a.s}. \]
which is a ``martingale connection'' in finite time.
Thanks to (\ref{limxi}) and Theorem \ref{asymptCT}, passing to the limit $n\to\infty$ leads to the following theorem, already mentioned in Janson \cite{Jan} in a more general framework.
Note that the independence between $\xi$ and $W^{\mathit{DT}}$ comes from Lemma \ref{embed}(ii).
\begin{Th}[(Martingale connection)] \label{thmartingaleconnection}
\begin{equation} \label{martingaleconnection} W^{\mathit{CT}} = \xi^{\sigma} W^{\mathit{DT}} \qquad \mbox{a.s}. \end{equation}
$\xi$ and $W^{\mathit{DT}}$ being independent. \end{Th}
The distribution of $\xi^{\sigma}$ is invertible (see footnote in the \hyperref[intro]{Introduction}), so that any information on $W^{\mathit{CT}}$ can be pulled back to $W^{\mathit{DT}}$ thanks to connection (\ref{martingaleconnection}).
\section{Dislocation equations for continuous urns}\label{dislocation}
\subsection{Vectorial finite time dislocation equations} \label{dislocationvect}
By embedding in continuous time, the previous section provided a branching process $(U_{(\alpha, \beta)}^{\mathit{CT}}(t), t\geq0)$. The independence properties of this process imply that it is equal to the sum of $\alpha$ copies of $U_{(1,0)}^{\mathit{CT}}(t)$ (the process starting from one red ball) and $\beta$ copies of $U_{(0,1)}^{\mathit{CT}}(t)$ (the process starting from one black ball). We are led to study these two $\mathbb{R}^2$-valued processes.
Let us now apply the strong Markov branching property to these processes: let us denote by $\tau$ the first splitting time for any of these processes (they have the same $\mathit{\mathcal{E} xp} (1)$ distribution). We get the following vectorial finite time dislocation equations:
\begin{equation} \label{dislocationVectorielle} \forall t>\tau \cases{ U_{(1,0)}^{\mathit{CT}}(t)\stackrel{\mathcal{L}}{=}[a+1]U_{(1,0)}^{\mathit{CT}}(t-\tau )+[b]U_{(0,1)}^{\mathit{CT}}(t-\tau),
\cr U_{(0,1)}^{\mathit{CT}}(t)\stackrel{\mathcal{L}}{=}[c]U_{(1,0)}^{\mathit{CT}}(t-\tau )+[d+1]U_{(0,1)}^{\mathit{CT}}(t-\tau), }
\end{equation}
where the notation $[n]X+[m]Y$ stands for the sum of $n$ copies of the random variable $X$ and $m$ copies of the random variable $Y$ ($n$ and $m$ are nonnegative integers).
\begin{Rem} The above equations could be written with a.s. equalities. Taking a probability space of trees is more convenient. The price to pay is just to write the different processes for each subtree with different indexes and to distinguish the two splitting times for the two starting situations. \end{Rem}
\subsection{Limit dislocation equations}\label{dislocationlimite}
Remember that $( e^{-mt}u_2( U_{(1,0)}^{\mathit{CT}}(t) ))_t$ and $( e^{-mt}u_2(U_{(0,1)}^{\mathit{CT}}(t) ))_t$ are martingales having, respectively, $u_2(U_{(1,0)}^{\mathit{CT}}(0) )=b/S$ and $u_2(U_{(0,1)}^{\mathit{CT}}(0))=-c/S$ as expectations. They converge in ${\rm L}^{p}$ for every nonnegative integer $p\geq1$. We are interested in the probability distributions of
\begin{equation}\label{defXY} X:= \lim_{t\rightarrow+\infty}e^{-mt}u_2\bigl(U_{(1,0)}^{\mathit{CT}}(t) \bigr) \quad\mbox{and}\quad Y:= \lim_{t\rightarrow+\infty }e^{-mt}u_2\bigl(U_{(0,1)}^{\mathit{CT}}(t) \bigr). \end{equation}
Projecting along the second eigenline, scaling and passing to the limit in system (\ref{dislocationVectorielle}) lead straightforwardly to the following proposition.
\begin{Prop} The limit random variables $X$ and $Y$ are solution of the following (scalar) limit dislocation equations:
\begin{equation} \label{dislocationProjetee} \cases{ X\stackrel{\mathcal{L}}{=}e^{-m\tau}( [a+1]X+[b]Y), \cr Y\stackrel{\mathcal{L}}{=}e^{-m\tau}([c]X+[d+1]Y), } \end{equation}
with
\begin{equation}\label{esperances} \mathbb{E}(X) = \frac b S, \qquad \mathbb{E}(Y) = - \frac{c} S, \end{equation}
where all the mentioned variables are independent. \end{Prop}
\begin{Rem} Janson \cite{Jan} in his Theorem 3.9 gets the same limit dislocation equations. He obtains the unicity of the solution in $L^2$ by a fixed point method. Hereunder in Section \ref{calculdeF}, calculating explicitly the solution of the fixed point system (\ref {dislocationProjetee}) together with conditions (\ref{esperances}), we give in passing another proof of the unicity in $L^2$. \end{Rem}
\section{Characteristic functions: fundamental differential system}
Let $\mathcal{F}$ and $\mathcal{G}$ be respectively, the characteristic functions of $X$ and $Y$:
\[ \forall x\in\mathbb{R}\qquad
\mathcal{F}(x)=\mathbb{E}(e^{ixX})=\int_{-\infty}^{+\infty }e^{ixt}\,d\mu_X(t) \]
with a similar formula for $\mathcal{G}$. Since $X$ and $Y$ admit moments of all orders, $\mathcal{F}$ and $\mathcal{G}$ are infinitely differentiable on $\mathbb{R}$.
\begin{Prop} The characteristic functions $\mathcal{F}$ and $\mathcal{G}$ are solutions of the differential system
\begin{equation} \label{systemeFourier} \cases{ \mathcal{F}(x)+mx\mathcal{F}'(x)=\mathcal{F}(x)^{a+1}\mathcal {G}(x)^b, \cr \mathcal{G}(x)+mx\mathcal{G}'(x)=\mathcal{F}(x)^{c}\mathcal{G}(x)^{d+1}, } \end{equation}
and satisfy the boundary conditions at the origin
\begin{equation} \label{conditionsBordSystemeFourier} \cases{ \mathcal{F}(x)=1+i\dfrac{b}Sx+O(x^2), \cr \mathcal{G}(x)=1-i\dfrac{c}Sx+O(x^2). } \end{equation}
\end{Prop}
\begin{pf} Conditioning on $\tau$ the distribution of which is exponential with mean~$1$, the first dislocation equation (\ref{dislocationProjetee}) implies successively that, for any $x\in \mathbb{R}$,
\begin{eqnarray*} \mathcal{F}(x)&=&\mathbb{E}\bigl(
\mathbb{E}\bigl( \exp\bigl( ixe^{-m\tau}([a+1]X+[b]Y)|\tau\bigr) \bigr)\bigr) \\ &=& \int_0^{+\infty}\mathcal{F}^{a+1}\bigl( xe^{-mt}\bigr)\mathcal {G}^{b} \bigl( xe^{-mt}\bigr)e^{-t}\,dt.
\end{eqnarray*}
After a change of variable under the integral, this functional equation can be written
\[ \forall x\neq0\qquad
\mathcal{F}(x)=\frac{x}{m|x|^{1+1/m}}\int_0^x\mathcal
{F}^{a+1}(t)\mathcal{G}^b(t) \frac{dt}{|t|^{1-1/m}}. \]
Differentiation of this equality and the similar one obtained from the second dislocation equation in (\ref{dislocationProjetee}) lead to the result. The boundary conditions come elementarily from the computation of the means of $X$ and $Y$ and from the existence of their second moment (Taylor expansion of $\mathcal{F}$ and $\mathcal{G}$ at $0$). \end{pf}
\begin{Rem} The differential system (\ref{systemeFourier}) is singular at $0$ so that the unicity of its solution that satisfies the boundary condition (\ref {conditionsBordSystemeFourier}) is not an elementary consequence of general theorems for ordinary differential equations. \end{Rem}
\section{Resolution of the fundamental differential system}\label{resolution}
\subsection{Change of functions: Heuristics}\label{heuristics}
Formally, without carefully checking which $m$th roots should be considered, if the variables $x\in\mathbb{R}$ and $w\in\mathbb{C}$ are related by $x^Sw^m=1$, the change of functions
\[ \cases{ f(w) =x^{1/m}\mathcal{F}(x),\cr g(w)=x^{1/m}\mathcal{G}(x) } \]
reduces the problems (\ref{systemeFourier}) and (\ref {conditionsBordSystemeFourier}) to the regular differential system
\begin{equation} \label{systemeMonomial} \cases{ f'=\dfrac{-1}{S}f^{a+1}g^b, \cr g'=\dfrac{-1}{S}f^cg^{d+1}, } \end{equation}
with boundary conditions at infinity
\begin{equation} \label{conditionsBordSystemeMonomial} \cases{
f(w)=w^{-1/S}+i\dfrac bSw^{-{(1+m)}/S}+O\bigl( |w|^{- {(1+2m)}/S}\bigr),
\vspace*{1pt}\cr g(w)=w^{-1/S}-i\dfrac cSw^{-({1+m})/S}+O\bigl( |w|^{- ({1+2m})/S}\bigr). } \end{equation}
The basic fact for the resolution of (\ref{systemeMonomial}) is that it admits $1/g^m-1/f^m$ as first integral: if $K$ is any complex number such that the constant function $1/g^m-1/f^m$ equals $1/K^m$, then $g^m$ can be straightforwardly expressed as a function of $f$ and (\ref{systemeMonomial}) implies that $f$ is solution of the ordinary differential equation
\begin{equation} \label{equaDiffAbelienne} f'\times\frac{( 1+( f/K)^m )^{b/m}}{( f/K)^{S+1}} =-\frac{K^{S+1}}{S} \end{equation}
with boundary conditions coming from (\ref{conditionsBordSystemeMonomial}).
This leads to consider a primitive of the function $z\mapsto (1+z^m)^{b/m}/z^{S+1}$ in the complex field.
\subsection{Abelian integral $I$ and its inverse $J$}
For all integers $m$, $S$ and $b$ that satisfy $S\geq5$, $S/2< m< S$, $1\leq b<S/2$, we denote by $I=I_{m,S,b}$ the function
\[ I(z)=\int_{[z,z\infty)}(1+u^m)^{ b/m}\frac {du}{u^{S+1}} =\frac1{z^S}\int_1^{+\infty}[ 1+(tz)^m] ^{ b/m}\frac{dt}{t^{S+1}},
\]
where $[z,z\infty)$ denotes the ray $\{ tz, t\geq1\}$ and where the power $1/m$ is used for the principal determination of the $m$th root. The function $I$ is an Abelian integral on the curve $x^m-y^m=1$ (which is isomorphic to the famous Fermat curve $x^m+y^m=1$ by a straightforward linear change of variables), defined on the open set
\[ \mathcal{O}_m=\mathbb{C}\bigm\backslash\bigcup_{p\in\{ 0,\dots,m-1\} }\mathbb{R}_{\geq 0}e^{({i\pi}/m)(1+2p)}. \]
Note that the integral is convergent because $S-b+1\geq3$. Let $\mathcal{S}_m$ be the open sector of the complex plane defined by
\[ \mathcal{S}_m=\biggl\{ z\in\mathbb{C}\setminus\{ 0\},\ -\frac\pi m<\arg (z)<\frac\pi m\biggr\}. \]
The open set $\mathcal{O}_m$ is the union of the images of $\mathcal{S}_m$ under all rotations of angles $2k\pi/m$ around the origin, $k\in\mathbb{Z}$.
In the following, the notation ${b/m\choose n}$ denotes the ordinary binomial coefficient, generalized for rational (or even complex) values of $b/m$ by Euler's Gamma function. As everywhere else in the paper, the positive integer $a$ is $a=S-b$.
\begin{Prop}[(Properties of $I$)]\label{developI} \begin{longlist}[(iii)] \item[(i)] $I$ is holomorphic on $\mathcal{O}_m$ and for any $z\in\mathcal{O}_m$,
\begin{equation} \label{primitiveI} I'(z)=-\frac{( 1+z^m) ^{ b/m}}{z^{S+1}}. \end{equation}
\item[(ii)] For any $m$th root of unity $\omega$ and for any $z\in\mathcal{O}_m$,
\begin{equation} \label{rotationsI} I(\omega z)=\omega^{-S}I(z). \end{equation}
\item[(iii)] The function $I$ admits a power series expansion in the neighborhood of infinity in any connected component of $\mathcal{O}_m$. On $\mathcal{S}_m$, this expansion is given by the formula
\begin{equation} \label{DSEinfinideI} \qquad I(z)=\sum_{n\geq0}\frac1{a+mn}\pmatrix{ b/m\cr n }z^{-a-mn} =\frac1{az^a}+\frac b{m(a+m)}\frac1{z^{a+m}}+\cdots, \end{equation}
valid for any $z\in\mathcal{S}_m$, $|z|\geq1$.
\item[(iv)] The function $I$ admits a Laurent series expansion in the neighborhood of the origin in any connected component of $\mathcal{O}_m$. On $\mathcal{S}_m$, this expansion is given by the formula
\begin{equation} \label{DSL0deI}
I(z)=\frac1{Sz^S}+\frac b{m(S-m)}\frac1{z^{S-m}} +C_0
-\sum_{n\geq2}\pmatrix{ b/m\cr n }\frac{z^{mn-S}}{mn-S}, \end{equation}
where $C_0$ is the constant
\begin{equation} \label{serieC_0} C_0=\sum_{n\geq0}\pmatrix{ b/m\cr n }\biggl( \frac1{a+mn}+\frac 1{mn-S}\biggr) . \end{equation}
Formula (\ref{DSL0deI}) is valid for any $z\in\mathcal{S}_m$,
$|z|\leq1$.
\item[(v)] $C_0<0$. \end{longlist} \end{Prop}
\begin{pf} (i) and (ii) are direct consequences of the definition of $I$. Expansion~(\ref{DSEinfinideI}) and its validity for $z\in\mathcal
{S}_m$, $|z|>1$ comes directly from the power series expansion of $\zeta\mapsto
(1+\zeta)^{b/m}$ in the definition of $I$. Its validity for $|z|=1$ is given by the convergence of the series at such a $z$ and application of Abel's theorem,\footnote{We refer to the following theorem of Abel: if a series $\sum_na_n$ is convergent, then the power series $\sum_na_nz^n$ converges uniformly on the segment $[0,1]$.} proving (iii). To prove expansion (\ref{DSL0deI}), notice first that $I$ is holomorphic on the simply connected domain $\mathcal{S}_m$ and $I'(z)$ tends to $0$ as $z$ tends to infinity, so that integration on the ray $[z,z\infty)$ is equivalent to integration on $[z,1]$ followed by integration on $[1,+\infty)$. Thus,
\[ I(z)=I(1)+\int_{[z,1]}(1+u^m)^{b/m}\frac{du}{u^{S+1}}. \]
Power series expansion of $u\mapsto(1+u)^{b/m}$ under this last integral leads then to~(\ref{DSL0deI}). The proof of (iv) is again made complete by application of Abel's theorem. Note that, since $S$ is not a multiple of $m$ because of our assumptions on the parameters, the denominators in Formula (\ref{DSL0deI}) do not vanish. Finally, if $\alpha_n$ denotes the general term of the series (\ref {serieC_0}), a straightforward computation shows that
\[ \alpha_0+\alpha_1= \frac{(S-a)(m^2+aS)(S-a-m)}{amS(a+m)(S-m)}<0, \]
the last inequality coming from $S-a-m<S-S/2-S/2=0$ and from the other hypotheses on the parameters. Furthermore, $\alpha_{2n}+\alpha_{2n+1}<0$ for any $n\geq1$, which concludes the proof. [Hint: compute $\alpha_{2n}+\alpha_{2n+1}$, factorize $ {{b/m} \choose {2n}}$ by ${{b/m} \choose {2n+1}}$, use the fact $(2n+1)/(2n-b/m)>1$, notice that ${{b/m}\choose {2n+1}}>0$ because $0<b/m=(S-a)/m<S/2m<1$.] \end{pf}
Let $\mathbb{H}$ denote Poincar\'e half-plane:
\[ \mathbb{H}=\{ z\in\mathbb{C}, \Im(z)>0\} \quad \mbox{and} \quad \overline{\mathbb{H}} = \{ \overline z, z \in\mathbb{H}\} . \]
\begin{Prop} \label{Iconforme} The analytic function $I\dvtx \mathcal{S}_m\bigcap\mathbb{H}\to\mathbb {C}$ is a conformal mapping onto the open subset
\[ \mathcal{U}=\biggl\{ z, -\frac{a\pi}{m}<\arg(z)<0\biggr\} \cup\biggl( I_1 +\biggl\{ z,\ -\frac{S\pi}m<\arg(z)<-\frac{a\pi}{m}\biggr\} \biggr) \]
(see Figure \ref{fig2}), where
\begin{equation} \label{I1} I_1:= \frac1m B\biggl(\frac am,\frac dm\biggr)e^{-{(ia\pi)/m}}, \end{equation}
and where $B$ denotes Euler's Beta function $B(x,y)=\Gamma(x)\Gamma (y)/\Gamma(x+y)$. \end{Prop}
\begin{figure}
\caption{Domain $\mathcal{S}_m\bigcap\mathbb{H}$ and its image by $I$.}
\label{fig2}
\end{figure}
\begin{pf} Let $\zeta_m = \exp(i\pi/m)$. We show hereunder that the restriction of $I$ to the sector $\mathcal {S}_m\cap Cl(\mathbb{H})$ (where $Cl(\mathbb{H})$ denotes the topological closure of $\mathbb{H}$) admits a continuous continuation to the ray $\{ t \zeta_m, t>0\}$
and that this continuation maps homeomorphically the boundary of the sector $\mathcal{S}_m\cap\mathbb{H}$ onto the boundary of $\mathcal{U}$. The result is then a consequence of elementary geometrical conformal theory (see, for example, Saks and Zygmund \cite{SaksZygmund}).
Let $h\in\mathbb{H}$, $r>0$, $t>1$ and $z=r(1-h)\zeta_m$. When $h$ tends to $0$, then $1+(tz)^m=1-r^mt^m+mr^mt^mh+O(h^2)$ so that the value of $m$th root principal determination of $1+(tz)^m$ according to the sign of $1-(rt)^m$ leads to the respective limits in terms of Beta incomplete functions:
\begin{itemize}[$\bullet$] \item[$\bullet$] if $r\geq1$, then
\begin{equation} \label{IBordInf} \lim_{z\to r\zeta_m, z\in\mathcal{S}_m}I(z)= \frac1m\zeta_m^{-a}\int_0^{1/r^m}(1-u)^{b/m}u^{c/m}\,du; \end{equation}
\item[$\bullet$] if $r\leq1$, then
\begin{equation} \label{IBordSup} \lim_{z\to r\zeta_m, z\in\mathcal{S}_m}I(z)= I_1+\frac1m\zeta_m^{-S}\int_1^{1/r^m}(u-1)^{b/m}u^{c/m}\,du. \end{equation} \end{itemize}
The complex number $I_1$ is simply
\[ I_1=\lim_{z\to\zeta_m, z\in\mathcal{S}_m}I(z); \]
formula (\ref{I1}) is a consequence of the integral representation of Euler Beta function $B(\alpha, \beta)=\int_0^1(1-u)^{\alpha-1}u^{\beta-1}\,du$. The monotonicity of real integrals (\ref{IBordInf}) and (\ref {IBordSup}) with respect to $r$ show that the continuous continuation of $I$ defined by these formulae maps decreasingly the ray $]0,+\infty[$ onto itself and respectively, the ray $]0,\zeta_m]$ onto the ray $\{ I_1+ t\zeta_m^{-S}, t\geq0 \}$ and the ray $[\zeta_m,\zeta_m\infty)$ onto $[I_1,0[$. \end{pf}
\begin{Rem} By computation in the realm of hypergeometric functions, one shows that the numbers $C_0$ defined by (\ref{serieC_0}) and $I_1$ defined by (\ref{I1}) are related by
\[
C_0=-\frac{\sin\pi(1+b/m)}{\sin\pi(1+S/m)}|I_1| =-\frac1m\frac{\sin\pi(1+b/m)}{\sin\pi(1+S/m)}B\biggl( \frac {S-b}m,\frac{m+b}m\biggr) . \]
\end{Rem}
\begin{Def} Let $J=J_{m,S,b}\dvtx \mathbb{C} \setminus]-\infty,0]\to\mathcal{S}_m$ the only continuous function defined by: \begin{itemize} \item[$\bullet$] $\forall z\in\overline{\mathbb{H}}$, $J(z)=I^{-1}(z)$ in the sense of Proposition \ref{Iconforme} ($\overline{\mathbb{H}}$ is an open subset of $\mathcal{U}$ so that this functional inverse exists);
\item[$\bullet$] $\forall z\in\mathbb{H}$, $J(z)=\overline{J(\overline z)}$ (complex conjugacy). \end{itemize} \end{Def}
The properties of $I$ shown in Propositions \ref{developI} and \ref {Iconforme} imply that $J$ is a conformal mapping between $\mathbb{C} \setminus]-\infty,0]$ and an open subset of $\mathcal{S}_m$ (use Schwarz reflection principle), that maps $\mathbb{H}$ into $\mathcal{S}_m\cap\overline{\mathbb{H}}$ and $\overline{\mathbb{H}}$ into $\mathcal{S}_m\cap{\mathbb{H}}$. If $\mathcal{C}$ denotes the inverse of the negative real axis by the restriction of $I$ to $\mathcal{S}_m\cap\mathbb{H}$, then the boundary of the image of $J$ is $\mathcal{C}\cup\overline
{\mathcal{C}}\cup\{ 0\}$ (see Figure \ref{fig3}). Furthermore, the restriction of $J$ to the positive real half-line is the inverse of $I$'s and $J$ is the unique analytic expansion of $(I_{|]0,+\infty[})^{-1}$ to the slit plane. Naturally, the formula $J(\overline z)=\overline{J(z)}$ is valid when $z$ is any nonnegative complex number.
\begin{Prop} \label{imageJ} For any negative real number $x$, both limits
\[ \lim_{z\to x, z\in\mathbb{H}}J(z)
\quad \mbox{and}\quad \lim_{z\to x, z\in\overline{\mathbb{H}}}J(z) \]
exist, are nonreal and conjugate (thus, different). \end{Prop}
\begin{pf} Direct consequence of the preceding properties of $J$ and Proposition~\ref{Iconforme} (see Figure \ref{fig3}). \end{pf}
We adopt the following notation:
\begin{equation} \label{Jlimite} \forall x<0\qquad \cases{ J(x-)=\displaystyle\lim_{z\to x, z\in\overline{\mathbb{H}}}J(z)\in\mathcal {S}_m\cap\mathbb{H}, \cr J(x+)=\displaystyle\lim_{z\to x, z\in\mathbb{H}}J(z)\in\mathcal{S}_m\cap \overline{\mathbb{H}}. }
\end{equation}
\begin{figure}
\caption{Action of $J$ on the slit plane $\mathbb{C}\setminus\mathbb {R}_{\leq0}$.}
\label{fig3}
\end{figure}
\begin{Prop} \label{DSPJ} The function $J$ admits, as $z$ tends to infinity in the slit plane $\mathbb{C}\setminus\mathbb{R}_-$, an asymptotic Puiseux series expansion at any order in the scale
\[ \biggl(\frac1z\biggr)^{1/S+p\sigma+q},\qquad (p,q)\in\mathbb{N}^2,
\]
where all fractional powers denote principal determination. The beginning of this asymptotic expansion is
\begin{eqnarray} \label{DSPinfiniDeJ} J(z)&=&\biggl( \frac1{Sz}\biggr)^{1/S} +\frac{b}{m(S-m)}\biggl( \frac1{Sz}\biggr)^{{(m+1)}/{S}}\nonumber \\[-8pt]\\[-8pt] &&{}+C_0\biggl( \frac1{Sz}\biggr)^{{(S+1)}/{S}} +o\biggl(\frac{1}{z}\biggr)^{{(S+1)}/{S}}.\nonumber \end{eqnarray}
\end{Prop}
\begin{pf} Expansion (\ref{DSL0deI}) leads to (\ref{DSPinfiniDeJ}) using the reversion formula $J\circ I=\break \operatorname{Id}$. \end{pf}
\subsection{Computation of characteristic functions} \label{calculdeF}
This section gives an explicit closed form of characteristic functions $\mathcal{F}$ and $\mathcal{G}$ for the elementary continuous-time urn processes $X$ and $Y$ [defined in (\ref{defXY})] associated with the replacement matrix $R=\left( {a\atop c}\enskip {b\atop d }
\right)$, in terms of the just defined functions $J$. Remember: the urn is supposed to be large and nontriangular so that $b>0$ and $c> 0$. Let $\kappa$ be the positive number defined by
\begin{equation} \label{kappa} \kappa = \sqrt[m\,]{\frac{S}{m(S-m)}}. \end{equation}
\begin{Th} \label{characteristicFunctions} The characteristic functions $\mathcal{F}$ and $\mathcal{G}$ are the unique solutions of the differential system (\ref{systemeFourier}) that satisfy boundary conditions (\ref {conditionsBordSystemeFourier}). They are given by the formulae
\begin{equation} \label{formulesFG} \hspace*{15pt} \forall x>0\hspace*{8pt} \cases{ \mathcal{F} (x)=\kappa e^{-{i\pi/(2m)}}x^{-1/m} J_{m,S,b}\biggl( C_0+\dfrac{\kappa^Se^{-{i\pi S/(2m)}}}{S}x^{-S/m}\biggr),
\cr \mathcal{G} (x)=\kappa e^{{i\pi/(2m)}}x^{-1/m} J_{m,S,c}\biggl( C_0+\dfrac{\kappa^Se^{{i\pi S/(2m)}}}{S}x^{-S/m}\biggr) }\hspace*{-12pt} \end{equation}
and
\begin{equation} \label{moitie} \forall x\in\mathbb{R}\qquad \mathcal{F}(-x)=\overline{\mathcal{F}(x)}, \qquad \mathcal{G}(-x)=\overline{\mathcal{G}(x)}. \end{equation}
\end{Th}
\begin{pf} (1) We first solve (\ref{systemeFourier}) on $\mathbb{R}_{>0}$. Let $F$ and $G$ be solutions of (\ref{systemeFourier}) that satisfy (\ref{conditionsBordSystemeFourier}). Lets do the change of variable $x\in\mathbb{R}_{>0}\to w=x^{-S/m}\in \mathbb{R}_{>0}$ and the change of functions
\[ f(w)=w^{-1/S}F( w^{-m/S}) \quad \mbox{and} \quad g(w)=w^{-1/S}G( w^{-m/S}) \]
that is straightforwardly reversed by the formula $F(x)=x^{-1/m}f(x^{-S/m})$ and a similar one for $G$ and $g$. Then $f$ and $g$ are solutions of (\ref{systemeMonomial}) on $\mathbb {R}_{>0}$ and satisfy boundary conditions (\ref{conditionsBordSystemeMonomial}) at $+\infty$. In particular, since (\ref{systemeMonomial}) is a nonsingular differential system, Cauchy--Lipschitz theorem guarantees that if $(f,g)$ is any solution, then $f$ ({resp.}, $g$) is identically zero or does not vanish. This implies that $f$ and $g$ do not vanish on $\mathbb{R}_{>0}$. Because of the balance conditions $a+b=c+d$, differentiation of $1/g^m-1/f^m$ leads to the fact that this function is constant on $\mathbb{R}_{>0}$ (first integral). Furthermore, boundary conditions at $+\infty$ (\ref {conditionsBordSystemeMonomial}) imply that this constant value is $i\frac mS(b+c)$. If $K$ denotes the complex number
\[ K=\kappa\exp\biggl(-\frac{i\pi}{2m}\biggr) \]
[$\kappa>0$ has been defined by formula (\ref{kappa})], this shows that
\begin{equation} \label{integralePremiere} \forall w>0\qquad \frac1{g^m(w)}-\frac1{f^m(w)}=\frac1{K^m}. \end{equation}
Since $f/g$ is continuous on $\mathbb{R}_{>0}$, does not vanish and tends to $1$ at $+\infty$ (\ref{conditionsBordSystemeMonomial}), relation $(f/g)^m=1+(f/K)^m$ implies that, on $\mathbb{R}_{>0}$,
\begin{equation} \label{gFonctionDef} g=\frac{f}{(1+( {f/K}) ^m )^{1/m}} \end{equation}
(principal determination of the $m$th root). Reporting in the first equation of (\ref{systemeMonomial}) shows that $f$ is necessarily a solution of equation (\ref{equaDiffAbelienne}) on $\mathbb{R}_{>0}$. Boundary conditions (\ref{conditionsBordSystemeMonomial}) imply that, when $w$ tends to $+\infty$, $\frac1Kf(w)\sim\frac1\kappa e^{i\pi /2m}w^{-1/S}\in\mathcal{S}_m$, so that equation (\ref{equaDiffAbelienne}) can be written
\[ \frac d{dw}I_{m,S,b}\biggl(\frac{f(w)}K\biggr)=\frac{K^S}S \]
in a neighborhood of $+\infty$. Integration of this equation shows that
\[ I_{m,S,b}\circ\biggl( \frac fK\biggr) (w)=\frac{K^S}Sw+C_1 \]
in a neighborhood of $w=+\infty$, for a suitable complex constant $C_1$. The determination of $C_1$ is made by means of local expansions: since $f$ tends to $0$ at $+\infty$, using (\ref{DSL0deI}) and previous equality leads to
\[ C_1+\frac{K^S}Sw =\frac{K^S}{Sf(w)^S} +\frac{b}{m(S-m)}\frac{K^{S-m}}{f(w)^{S-m}} +C_0 +o(1), \]
when $w$ tends to $+\infty$, so that boundary conditions (\ref {conditionsBordSystemeMonomial}) lead to the equality $C_1=C_0$. Note that this computation makes use of the big-O in (\ref {conditionsBordSystemeMonomial}), of the assumption $1-2m/S<0$ (large urn) and of the relation $S-m=b+c$. Thus, necessarily,
\begin{equation} \label{formulef} f(w)=KJ_{m,S,b}\biggl( C_0+\frac{K^S}Sw\biggr) \end{equation}
for any $w$ in a neighborhood of $+\infty$. The function $w\to KJ_{m,S,b}(C_0+K^Sw/S)$ is well defined on $\mathbb {R}_{>0}$ because $C_0<0$ [Proposition \ref{developI}(5)] and $-\pi <\operatorname{arg} (K^S)<-\pi/2$, so that it is the only maximal solution on $\mathbb{R}_{>0}$ of equation (\ref{equaDiffAbelienne}) that satisfies the first equation of (\ref{conditionsBordSystemeMonomial}). This shows finally that
\[ \forall x>0\qquad F (x)= Kx^{-1/m} J_{m,S,b}\biggl( C_0+\frac{K^S}Sx^{-S/m}\biggr). \]
Since $-K^m=\overline K^m$, the same arguments show that, for any $w>0$,
\[ g(w)=\overline KJ_{m,S,c}\biggl( C_0+\frac{\overline K^S}{S}w\biggr) , \]
which shows completely formula (\ref{formulesFG}).
(2) The resolution on $\mathbb{R}_{<0}$ is made the same way. To this effect, lets do the new change of variable
$x\in\mathbb{R}_{<0}\to w=|x|^{-S/m}e^{i\pi S/m}\in\mathbb {R}_{>0}e^{i\pi S/m}$.
Lets do as well the change of functions
\[
f(w)=e^{-i\pi/m}|w|^{-1/S}F( -|w|^{-m/S}) \]
and \[
g(w)=e^{-i\pi/m}|w|^{-1/S}G( -|w|^{-m/S}). \]
These changes of variable and functions are reversed by the formulae
$x=-|w|^{-m/S}$ and
$F(x)=e^{i\pi/m}|x|^{-1/m}f(e^{i\pi S/m}|x|^{-S/m})$ with a similar formula for $G$ and $g$. Functions $f$ and $g$ are still solutions of (\ref{systemeMonomial})
but boundary conditions become, as $|w|$ tends to infinity,
\begin{equation} \label{conditionsBordSystemeMonomialNegatif} \cases{
f(w)=e^{-i{(\pi/m)}}|w|^{-1/S}
\biggl( 1-i\dfrac bS|w|^{-{m}/S}+O( |w|^{-{2m}/S} )\biggr),
\cr g(w)=e^{-i({\pi}/{m})}|w|^{-1/S}
\biggl( 1+i\dfrac cS|w|^{-{m}/S}+O( |w|^{-{2m}/S} )\biggr) . }
\end{equation}
This implies that First integral (\ref{integralePremiere}) is still valid (same $K$) and, since $f$ and $g$ are still equivalent at infinity, relation (\ref{gFonctionDef}) is satisfied. Boundary conditions (\ref{conditionsBordSystemeMonomialNegatif}) imply that, when $w$ tends to $+\infty$, $\frac1Kf(w)\sim\frac1\kappa
|w|^{-1/S}e^{-i\pi/2m}\in\mathcal{S}_m$. Consequently, the same arguments as before show that formula (\ref {formulef}) remains valid (note that $C_0+wK^S/S\in\mathbb{H}$ so that this formula is well defined for any $w$). This shows that
\[
\forall x<0\qquad F (x)= Ke^{{i\pi}/{m}}|x|^{-1/m}
J_{m,S,b}\biggl( C_0+\frac{K^S}Se^{i\pi({S}/{m})}|x|^{- S/m}\biggr). \]
Since $Ke^{i\pi/m}=\overline K$, one gets finally $F(-x)=\overline {F(x)}$ for any real number $x$. The proof of the whole theorem is made complete by the same arguments for $G$. \end{pf}
\begin{Rem} Formula (\ref{moitie}) on characteristic functions comes directly from the fact that $X$ and $Y$ are real-valued random variables. \end{Rem}
We want to know more about the analyticity properties of $\mathcal{F}$ and $\mathcal{G}$ around $0$. Let $\varphi=\varphi_{m,S,b}$ be the function defined by the formula
\begin{equation} \label{defPhi} \varphi(z)=\kappa z^{-1/m}J_{m,S,b}\biggl( C_0+\frac{\kappa ^S}{S}( z^{-1/m}) ^S\biggr), \end{equation}
where the power $1/m$ denotes the principal determination of the $m$th root. Note that $\kappa$ and $C_0$, respectively, defined by formulas (\ref {kappa}) and (\ref{serieC_0}) are functions of $m$, $S$ and $b$ too. If $\rho$ denotes the positive number
\[
\rho= \biggl( \frac{S|C_0|}{\kappa^S}\biggr) ^{-m/S}
=\frac{S^{1-S/m}|C_0|^{-m/S}}{m(S-m)}, \]
it follows from the properties of $J$ that $\varphi$ is defined and holomorphic on the open set
\[ \mathcal{V}=\mathbb{C}\setminus\{ (-\infty,0]\cup[\rho ,+\infty) \}. \]
Furthermore, the characteristic functions $\mathcal{F}$ and $\mathcal {G}$ are restrictions of $\varphi$ functions on the imaginary axis: for any $x\in\mathbb{R}$,
\[ \mathcal{F}(x)=\varphi_{m,S,b}(ix) \quad \mbox{and}\quad \mathcal{G}(x)=\varphi_{m,S,c}(-ix). \]
Note that $\kappa$ is a function of $(m,S)$ so that the same $\kappa$ appears in both functions $\varphi_{m,S,b}$ and $\varphi_{m,S,c}$ (respective constants $C_0$ and $\rho$ are however different).
\begin{Prop}\label{analyticPhi} The function $\varphi$, holomorphic on $\mathcal{V}$, cannot be analytically extended on a larger subset of $\mathbb{C}$. However, setting $\varphi(0)=1$ defines a
continuously differentiable extension of $\varphi$ on $\mathcal {V}\cup\{ 0\}$. \end{Prop}
\begin{pf} The half-line $[\rho,+\infty)$ is the locus of complex $z$ such that $C_0+\frac{\kappa^S}{S}( z^{-1/m}) ^S$ is a real nonpositive number (remember that $m<S<2m$). Since the principal determination of the $m$th root is well defined and nonzero in a neighbourhood of this half-line, Proposition \ref{imageJ} implies that $\varphi$ cannot be continuously extended at any point of $[\rho,+\infty)$.
If $x$ is a negative number, definition of the principal determination of the $m$th root leads to the existence of both limits
\[ \cases{\displaystyle \lim_{z\to x, z\in\mathbb{H}}\varphi(z)=
\kappa e^{-i{(\pi/m)}}|x|^{-1/m}J\biggl( C_0+\dfrac{\kappa
^S}{S}e^{-i{(\pi S/m)}}|x|^{- S/m}\biggr) :=\varphi(x+), \cr \displaystyle\lim_{z\to x, z\in\overline{\mathbb{H}}}\varphi(z):=\varphi (x-)=\overline{\varphi(x+)}. }
\]
Since
the image of $J$ is included in $\mathcal{S}_m$, the limit $\varphi(x+)$ belongs to the open sector $e^{-i{(\pi/m)}}\mathcal{S}_m$ which contains no real number, so that $\varphi(x+)\neq\varphi(x-)$. This shows that $\varphi$ cannot be continuously extended at any point of $\mathbb{R}_{<0}$.
When $z$ tends to $0$ in the slit plane $\mathbb{C}\setminus\mathbb{R}_{<0}$, Proposition \ref{DSPJ} shows that $\varphi(z)$ tends to $1$. One step more, computing the derivative of $\varphi$ in terms of $J$ using the algebraic expression of $I'$ (\ref{primitiveI}) implies, with expansion (\ref {DSPinfiniDeJ}), that
\[ \lim_{z\to0, z\in\mathbb{C}\setminus\mathbb{R}_{\leq0}}\varphi '(z)=\frac bS. \]
\upqed\end{pf}
\begin{Cor}\label{rayonnul} The exponential moment generating series
\[ \sum\frac{\mathbb{E}(X^p)}{p!}T^p
\quad\mbox{and}\quad \sum\frac{\mathbb{E}(Y^p)}{p!}T^p \]
have a radius of convergence equal to $0$. \end{Cor}
\begin{pf} These series are the Taylor series of $\varphi_{m,S,b}$ and $\varphi _{m,S,c}$ at $0$. If these radii were positive, these functions could be analytically extended to a neighborhood of the origin. \end{pf}
\begin{Rem} The singularity of $\varphi$ at the origin is thus not due to ramification but to a divergent Taylor series phenomenon. Indeed, the apparent ramification coming from the $m$th root at the origin in formula (\ref{defPhi}) is compensated by both Puiseux expansion (\ref{DSPinfiniDeJ}) and the $S$th power of the $m$th root appearing in the argument of $J$ in formula (\ref{defPhi}). \end{Rem}
\section{Density of $W^{\mathit{CT}}$} \label{densiteWCT}
Notice, with the notation (\ref{Jlimite}) that
\begin{equation} \label{equiv} \mathcal{F}(x)\mathop{\sim}\limits_{x\to+\infty}\kappa J(C_0-)x^{-1/m}, \end{equation}
where the nonreal complex number $J(C_0-)$ is different from $0$ (see Figure 3).
A first consequence is that $\mathcal{F}(x)$ tends to $0$ when $x$ tends to $+\infty$. Hence, the probability distribution function of $W^{\mathit{CT}}$ is continuous so that the law of $W^{\mathit{CT}}$ has no point mass.
A second consequence is that $\mathcal{F}$ is not in ${\rm L}^1$ so that $W^{\mathit{CT}}$ distribution cannot be obtained by classical Fourier inversion. Nevertheless, we obtain in Section \ref{inversionFourier} an expression of this density using the derivative of the characteristic function $\mathcal{F}$. Before, we need firstly to ensure that the support of $W^{\mathit{CT}}$ is the whole real line ${\mathbb{R}}$ which is proven in Section \ref{support} and secondly to ensure that $W^{\mathit{CT}}$ admits a density which is proven in Section \ref{densityconnection} using the martingale connection (\ref {martingaleconnection}). As usually, this kind of connection induces a smoothing phenomenon between $W^{\mathit{DT}}$ and $W^{\mathit{CT}}$, allowing us to prove that $W^{\mathit{CT}}$ has a density, whatever $W^{\mathit{DT}}$ distribution is.
\subsection{Support of $W^{\mathit{CT}}$}\label{support}
\begin{Prop} \label{supportWCT} The support of $W^{\mathit{CT}}$ is ${\mathbb{R}}$. \end{Prop}
\begin{pf} As in (\ref{defXY}), let $X$ denote the random variable $W^{\mathit{CT}}$ starting from one red ball. Because of the branching property (see beginning of Section \ref {dislocationvect}), it suffices to prove that the support of $X$ is the whole real line $\mathbb{R}$. General results on infinite divisibility (see, for instance, Steutel and van Harn \cite{Steutel}, page 186) ensure that the support of an infinitely divisible random variable having a continuous probability distribution function is either a half-line or ${\mathbb{R}}$. Suppose that the support of $X$ is $[\alpha, +\infty[$ for a given real number $\alpha$. Then denoting $X$ distribution by $\mu_X$,
\[ \mathbb{E}(e^{-sX}) = \int_{\alpha}^{+\infty} e^{-st} \,d\mu_X(t) = e^{-s\alpha} \int_{\alpha}^{+\infty} e^{-s(t-\alpha)} \,d\mu_X(t) \]
exists for every real number $s\geq0$. Hence, the function $L\dvtx s \rightarrow\mathbb{E}(e^{-sX})$ is analytic on the half-plane $\{ \Re z >0\}$, continuous on the boundary of this half-plane and $\lim _{t\rightarrow\pm\infty} \mathbb{E}(e^{itX}) = 0$. By unicity of the analytic continuation, necessarily:
\[ L(s)
= \varphi(-s)\qquad \forall s, \Re(s) \geq0, \]
where $\varphi$ has been introduced in (\ref{defPhi}). But it has been proven in Proposition \ref{analyticPhi} that $\varphi $ cannot be analytically extended on the half-plane $\{ \Re z <0\}$. There is a contradiction: the support of $X$ cannot be a half-line $[\alpha, +\infty[$.
In the same way, if we suppose that the support of $W^{\mathit{CT}}$ is $]- \infty, \beta]$ for a given real number $\beta$, we are led to a contradiction, because $\varphi$ cannot be analytically extended on the whole half-plane $\{ \Re z >0\}$ (Proposition \ref{analyticPhi}). \end{pf}
\subsection{Connection between the distribution of $W^{\mathit{DT}}$ and the density of $W^{\mathit{CT}}$}\label{densityconnection}
\begin{Prop} \label{density1} Let $\mu$ be the distribution of $W^{\mathit{DT}}$ (it is a probability measure on ${\mathbb{R}}$).
(1) $W^{\mathit{CT}}$ admits a density $p$ on ${\mathbb{R}}$ given by
\[ \cases{ \forall w>0\qquad p(w)=\dfrac{1}{\sigma}\dfrac{1}{\Gamma( {1/S})} w^{- 1+ \frac 1m } \displaystyle\int_{]0,+\infty[}v^{-{1}/m} e^{-( w/v) ^{{1}/{\sigma}}} d\mu(v), \cr
\forall w<0\qquad p(w)=\dfrac{1}{\sigma}\dfrac{1}{\Gamma( {1}/{S})} |w|^{- 1 +\frac 1m}
\displaystyle\int_{]-\infty,0[}|v|^{-{1}/m} e^{-( w/v) ^{{1}/{\sigma}}} \,d\mu(v). }
\]
(2) The density $p$ is infinitely differentiable and increasing on ${\mathbb{R}} _{<0}$, infinitely differentiable and decreasing on ${\mathbb{R}}_{>0}$; it is not continuous at $0$: $\lim_{w\rightarrow0,w\neq0} p(w)= +\infty$.
In particular, the distribution is unimodal, the mode is $0$. \end{Prop}
\begin{pf} (1) To exhibit a density, let us take any real-valued bounded continuous function $h$ defined on $\mathbb{R}$ and, thanks to the martingale connection (\ref{martingaleconnection}), compute
\[ \mathbb{E}(h(W^{\mathit{CT}})) = \int_{{\mathbb{R}}}\int_0^{+\infty} h(uv) g(u) \,du \,d\mu(v), \]
where $g$ is the density of $\xi^{\sigma}$. After the change of variable $w=uv$, we get
\begin{eqnarray*} \mathbb{E}(h(W^{\mathit{CT}})) &=&
\int_{]-\infty, 0[}\frac{d\mu(v)}{|v|} \int_{-\infty }^0 h(w) g\biggl( \frac{w}v\biggr)\, dw \\ &&{} + \mu(\{0\})h(0) + \int_{]0, +\infty[}\frac{d\mu(v)}{v} \int_0^{+\infty} h(w) g\biggl( \frac{w}v\biggr)\, dw.
\end{eqnarray*}
Recall that $W^{\mathit{CT}}$ has no point mass (see Section \ref{densiteWCT}, introductory paragraph), so we get that $W^{\mathit{CT}}$ admits a density given by
\begin{eqnarray} \label{defp} \qquad p(w) = {\bf1}_{{\mathbb{R}}_{<0}} (w) \int_{]-\infty, 0[} g\biggl( \frac
{w}v\biggr) \frac{d\mu(v)}{|v|} + {\bf1}_{{\mathbb{R}}_{>0}} (w) \int_{]0,+\infty[} g\biggl( \frac {w}v \biggr)\frac{d\mu(v)}v . \end{eqnarray}
The only point to verify is that the integrals in formula (\ref{defp}) are well defined. The density $g$ is explicit. To simplify the notation, we consider the case when we start from one ball ($u = 1$). In this case,
\begin{equation}\label{densitedeG} g(x) = \frac{1}{\sigma}\frac{1}{\Gamma({1}/{S})} x^{ - 1+ \frac 1m}e^{-x^{ {1}/{\sigma}}} {\bf1}_{x>0}, \end{equation}
so that, for any nonzero $w$,
\[
\frac{1}{|v|}g\biggl( \frac{w}v\biggr)
= C |w|^{- 1+\frac{1}{m}} |v|^{-{1}/m} e^{-|w|^{{1}/{\sigma
}}|v|^{-{1}/{\sigma}}} \]
is bounded as a function of $v$.
(2) Let us prove that $\lim_{w\rightarrow0^+} p(w)= +\infty$, looking at
\[ \lim_{w\rightarrow0^+} w^{- 1+\frac{1}{m} }\int_{]0,+\infty[} v^{-{1}/m} e^{-( w /v)^{{1}/{\sigma}}}\,d\mu(v). \]
The last integral, for any $w< 1$, is greater than
\[ \int_{]0,+\infty[} v^{-{1}/m} e^{-(1 /v)^{{1}/{\sigma }}}\,d\mu(v), \]
so that it is sufficient to prove that this integral is a positive constant. If not, this integral would be equal to zero, and this happens only if the support of $\mu$ is included in ]$-\infty, 0$]. By the martingale connection (\ref{martingaleconnection}), this would imply that the support of $W^{\mathit{CT}}$ is included in ]$-\infty, 0$], which is not the case because of Proposition \ref{supportWCT}.
The result on the limit of $p$ at $0^-$ is proved the same way. Differentiability is immediate by dominated convergence and monotonicity comes from derivation of formula (\ref{defp}). \end{pf}
\begin{Rem} The distribution of $W^{\mathit{CT}}$ is not symmetric around $0$ (the expectation equals $\frac b S\not= 0$ when one starts with only one red ball). \end{Rem}
\subsection{Fourier inversion}\label{inversionFourier}
The characteristic function $\mathcal{F}$ is not integrable. Nevertheless, formulas (\ref{systemeFourier}) and (\ref{equiv}), imply straightforwardly that, for any real $x\not= 0$,\vspace*{-2pt}
\[ \mathcal{F}'(x) = \frac1{mx} \mathcal{F}(x) [ \mathcal {F}^a(x)\mathcal{G}^b(x) -1 ]\vspace*{-2pt} \]
and that $\mathcal{F}'$ is in ${\rm L}^1$. Theorem \ref{density2} gives an explicit expression of the density of $W^{\mathit{CT}}$ by means of inverse Fourier transform of $\mathcal{F}'$, completing Proposition \ref{density1}.
\begin{Th} \label{density2} The density $p$ on $\mathbb{R}$ of the random variable $W^{\mathit{CT}}$ is given, for any $x\not= 0$, by\vspace*{-2pt}
\begin{equation} \label{densite} p(x) = \frac1{2i\pi x} \int_{{\mathbb{R}}} e^{-itx} \mathcal{F}'(t) \, dt.\vspace*{-2pt} \end{equation}
\end{Th}
\begin{pf} Let $F$ be the probability distribution function of $W^{\mathit{CT}}$. We are going to show that $\forall x\not= 0$,\vspace*{-2pt}
\begin{equation} \label{sufficient} \lim_{h\rightarrow0} \frac{F(x+h) - F(x)}{h} = \frac1{2i\pi x} \int _{{\mathbb{R}}} e^{-itx} \mathcal{F}'(t) \, dt,\vspace*{-2pt} \end{equation}
which is sufficient to prove that $W^{\mathit{CT}}$ admits a continuous density given by (\ref{densite}).
For any $h\neq0$, let $d_h$ be the function defined on $\mathbb {R}\setminus \{ 0\}$ by
\[ d_h(t):= \frac{1-e^{-ith}}{ith} \]
and continuated by continuity at $0$. It follows from the general Fourier inversion theorem (see, for instance, Lukacs \cite{Lukacs}, Theorem 3.2.1, page~38) that $\forall x\in\mathbb{R}$, $\forall h\not= 0$, since $x$ and $x+h$ are continuity points of $F$ (remember that $F$ is continuous because its characteristic function tends to $0$ at infinity),\vspace*{-2pt}
\[ \frac{F(x+h) - F(x)}{h} = \lim_{T\rightarrow+\infty} I_{T,h}(x),\vspace*{-2pt} \]
where\vspace*{-2pt}
\[ I_{T,h} (x): = \frac1 {2\pi} \int_{-T}^T e^{-itx}d_h(t)\mathcal {F}(t) \, dt.\vspace*{-2pt} \]
Integrating by parts implies that, for any $x\neq0$,\vspace*{-2pt}
\[ I_{T,h}(x)= I_{T,h}^{(1)} (x) + I_{T,h}^{(2)}(x) + I_{T,h}^{(3)}(x)\vspace*{-2pt} \]
where\vspace*{-2pt}
\[ \cases{ I_{T,h}^{(1)} (x)= \dfrac1{2\pi} \biggl[ -\dfrac{e^{-iTx} }{ix} d_h(T)\mathcal{F}(T)+\dfrac{e^{iTx} }{ix} d_h(-T)\mathcal{F}(-T) \biggr] , \vspace*{2pt}\cr I_{T,h}^{(2)}(x)= \dfrac1{2i\pi x} \int_{-T}^T e^{-itx} d_h(t) \mathcal{F}'(t) \,dt, \vspace*{2pt}\cr I_{T,h}^{(3)}(x)=
\dfrac1{2i\pi x} \int_{-T}^T e^{-itx} d'_h(t) \mathcal{F}(t) \,dt.\vspace*{-1pt} }
\]
It is elementary to see that $d_h(t) $ has the following properties: $\forall h \not= 0, \forall t\not= 0$,\vspace*{-2pt}
\begin{eqnarray} \label{P1}
| d_h(t) | &=& \biggl| \frac{\sin{th}/ 2}{ {th}/ 2} \biggr|
\leq\min\biggl\{ 1, \frac2{|th|}\biggr\}, \\[-2pt] \label{P2}
| d_h'(t) | &\leq&\min\biggl\{ \frac{|h|} 2, \frac2 {|t|}\biggr\}. \end{eqnarray}
Since $\mathcal{F}$ is bounded (it is a characteristic function) and since $d_h$ tends to $0$ at infinity,\vspace*{-2pt}
\[ \lim_{T\rightarrow+\infty} I_{T,h}^{(1)} (x) = 0.\vspace*{-2pt} \]
Since $\mathcal{F}' \in{\rm L}^1$, (\ref{P1}) and Lebesgue dominated convergence theorem lead to\vspace*{-2pt}
\[ \lim_{T\rightarrow+\infty} I_{T,h}^{(2)} (x) = \frac1{2i\pi x} \int _{{\mathbb{R}}} e^{-itx} d_h(t) \mathcal{F}'(t)\, dt .\vspace*{-2pt} \]
At least, (\ref{P2}) implies that $d_h'\mathcal{F} \in{\rm L}^1$ so that, by dominated convergence,\vspace*{-2pt}
\[ \lim_{T\rightarrow+\infty} I_{T,h}^{(3)} (x)= \frac1{2i\pi x} \int _{{\mathbb{R}}}e^{-itx} d_h'(t) \mathcal{F}(t)\, dt .\vspace*{-2pt} \]
So, for any $x\neq0$ and $h\neq0$,\vspace*{-1pt}
\[ \frac{F(x+h) - F(x)}{h} = \frac1{2i\pi x} \int_{{\mathbb{R}}} e^{-itx}d_h(t) \mathcal{F}'(t)\, dt + \frac1{2i\pi x} \int_{{\mathbb{R}}} e^{-itx} d_h'(t) \mathcal{F}(t)\, dt .\vspace*{-1pt} \]
To get (\ref{sufficient}), it is now sufficient to take the limit when $h\rightarrow0$, using dominated convergence and (\ref{P2}). \end{pf}
\begin{Rem*} We have not found the following result in the literature but the arguments of this proof lead to the following proposition. \end{Rem*}
\begin{Prop} Let $\mathcal{F}$ be the characteristic function of a probability distribution function $F$.
Suppose that $\mathcal{F}$ is derivable, $\mathcal{F}' \in{\rm L}^1$ ($\mathcal{F}$ is not necessarily in~${\rm L}^1$) and $\frac{\mathcal {F}(t)} t \in {\rm L}^1 $. Then $F$ admits a density $p$ given for all $x\not=0$ by
\[ p(x) = \frac1{2i\pi x} \int_{{\mathbb{R}}} e^{-itx} \mathcal{F}'(t) \, dt. \]
\end{Prop}
\section{Concluding remarks} \label{conclusion}
\subsection{More colors}
The same questions arise naturally for limit laws of large urn processes with any finite number of colors. Embedding in continuous time, martingale connection, dislocation equations on elementary limit distributions and differential system (\ref{systemeFourier}) on Fourier transforms or on formal Laplace power series can be generalized. However, the resolution of (\ref{systemeFourier}) relies on the question of its integrability, even if an explicit closed form of its solutions may not be necessary to derive properties of the corresponding distributions.
The space requirements of an $m$-ary search tree is a special case of P\'olya--Eggenberger urn process with $m-1$ colors (see \cite{ChPo}, for example). Because of the negativeness of the diagonal entries $-1, -2, \ldots , -(m-1)$ of its replacement matrix, the corresponding continuous-time Markov process is not a branching process. However, the discrete-time node process of an $m$-ary search tree can be embedded into a branching process. When $m\geq27$, the corresponding limit laws can be studied with the same method as in the present paper. This is the subject of a forthcoming companion paper.
\subsection{Laplace series} \label{serieLaplace}
Remember from Section \ref{dislocationlimite} that $X$ ({resp.}, $Y$) is the martingale limit $W^{\mathit{CT}}$ of the continuous-time urn process starting from $(1,0)$ [{resp.}, from $(0,1)$]. For $n\geq0$, let
\[ a_n=\mathbb{E}(X^n)\quad \mbox{and}\quad b_n=\mathbb{E}(Y^n), \]
and let $F$ and $G$ be the Laplace series of $X$ and $Y$, {\it\textup {that is,}} the formal exponential series of the moments:
\[ F(T)=\sum_{n\geq0}\frac{a_n}{n!}T^n \quad \mbox{and}\quad G(T)=\sum_{n\geq0}\frac{b_n}{n!}T^n\in\mathbb{R}[[T]]. \]
From equations (\ref{dislocationProjetee}), we write recursion formulae relating $(a_k)_{0\leq k\leq n}$ and\break $(b_k)_{0\leq k\leq n}$. Thanks to the multinomial formula, they arrange themselves into the differential system with boundary conditions:
\begin{equation} \label{systemeFormel} \cases{ F(T)+mTF'(T)=F(T)^{a+1}G(T)^b, \cr G(T)+mTG'(T)=F(T)^{c}G(T)^{d+1}, \cr F(0)=G(0)=1, \cr
F'(0)=\dfrac b S \quad \mbox{and}\quad G'(0)=- \dfrac{c} S. }
\end{equation}
The fact that the urn is large implies that equations (\ref {systemeFormel}) characterize the moments of $X$ and $Y$. Indeed, proceed by recursion: for any $n\geq2$, $v_n=(a_n,b_n)$ is the solution of a linear system of the form $(R-nmI)(v_n)={}$[polynomial function of $v_1,\ldots,v_{n-1}$], $R$ being the replacement matrix of the process (\ref{matriceUrne}). Since the urn is large, $nm>nS/2\geq S$ so that $nm$ is not an eigenvalue of $R$.
A remarkable fact, which explains why we have worked with characteristic functions and not with Laplace transforms, is that, for nontriangular urns, {\it\textup{that is,}} when $bc\neq0$, series $F$ and $G$ have a radius of convergence equal to $0$ (Corollary~\ref {rayonnul}).
\subsection{Question}
The main theorem provides a family of distributions, those of the $W^{\mathit{CT}}$, indexed by the three parameters $S,m,b$ of the urn and by the initial condition $(\alpha, \beta)$. A challenging question is: can the physical relations between these distributions be translated into relations between the Abelian integrals? In otherwords, can the addition formulas between Abelian integrals be interpreted by a combinatorial/probabilistic approach using these distributions?
\section*{Acknowledgments}
The authors warmly thank Philippe Flajolet for stimulating discussions, Brigitte Chauvin being welcome in Project Algorithms at INRIA Rocquencourt.
\printaddresses
\end{document} | arXiv |
\begin{definition}[Definition:Differential Equation]
A '''differential equation''' is a mathematical equation for an unknown function of one or several variables relating:
:$(1): \quad$ The values of the function itself
:$(2): \quad$ Its derivatives of various orders.
\end{definition} | ProofWiki |
The purpose of the $\sf ZFC$ Axiom of Infinity
The purpose of the $\sf ZFC$ Axiom of Infinity seems to me to have no other purpose than to provide a set from which the natural numbers can be extracted. Is this correct? If so, do other $\sf ZFC$ axioms, e.g. Pairing, likewise have no other purpose than to enable this extraction process?
elementary-set-theory
Dan ChristensenDan Christensen
$\begingroup$ The purpose of the majority of the axioms of ZFC is to allow us to construct set-theoretic versions of the standard types of objects that we use in mathematics. "Unordered pair" is not directly useful, but in combination with other axioms it lets us construct a set-theoretic version of ordered pair, which is of great importance. $\endgroup$ – André Nicolas Aug 25 '13 at 20:49
$\begingroup$ ??? The purpose of the axioms taken as a whole is to describe the universe of sets as (we think) we understand it. $\endgroup$ – Brian M. Scott Aug 25 '13 at 20:49
$\begingroup$ @DanChristensen: We want a framework within which all standard mathematical constructions can be done. Definitely the ZFC construction of ordered pair is artificial, that's not what ordered pairs really are. And again, if we are introducing individual natural numbers, the convoluted construction starting from the empty set and using sickeningly many braces is not a nice one. But it works. More elaborately, the set of reals can be constructed within ZFC, by say formalizing the Dedekind cut approach. When we do analysis, we can largely forget about the set-theoretic background. (Cont.) $\endgroup$ – André Nicolas Aug 26 '13 at 2:32
$\begingroup$ As asked, this is very naive. Set theory is a theory of infinite sets. What is remarkable of the axiom of infinite is not that it provides us with a formal surrogate for the natural numbers, but rather that this suffices, when combined with the other axioms, to give us the rich landscape that follows. $\endgroup$ – Andrés E. Caicedo Aug 26 '13 at 3:53
$\begingroup$ Still makes no sense, but anyway, under what ought to be the natural interpretation of your question, you literally do not get anything new. These "axioms" are provable in $\mathsf{ZF}$ without the axiom of infinity. In this theory, you cannot prove that $\omega$ is a set, but you can prove that as a (perhaps proper) class, it satisfies both first and second order $\mathsf{PA}$. Yes, you want the redundancy here, because in this theory you cannot prove the existence of infinite sets, so the second order version may be vacuously true, while the first order version has content regardless. $\endgroup$ – Andrés E. Caicedo Aug 28 '13 at 5:17
Set theory is a theory of infinite sets, one could say that this is the point (that it serves us as a foundation for mathematics is extra, the cherry on top). What is remarkable about the axiom of infinity is not that it provides us with a formal surrogate for the natural numbers (I mean, we better do have something in our axioms that allows us to find such a surrogate, else, this would be a terrible theory of infinity, and an even worse foundation), but rather that this suffices, when combined with the other axioms, to give us the rich landscape that follows.
That said, the axiom of infinity is definitely used to prove many results beyond the construction of the naturals. "There are dense linear orders without end points" is an example. "There is an $\omega_1$-Aronszajn tree" is another. "Every Goodstein sequence terminates", etc. (Note that the last is an example of a statement about the natural numbers.)
Assuming the other axioms, the axiom of infinity is trivially equivalent to the statement "$\omega$ exists". In this sense, any use of the axiom of infinity is just using that there is a "set of natural numbers". But, as the examples above indicate, these uses can lead in many different directions, well beyond anything resembling the natural numbers (and, remarkably, also give us theorems about the natural numbers, as an additional bonus). One can wonder whether using the axiom is truly necessary for some of these results. The answer is yes and, not only that, but we in fact have a very good understanding of what results precisely need the use of the axiom of infinity. Namely, as Peter Smith's answer indicates, the theory resulting from replacing the axiom of infinity with its negation is just first order Peano arithmetic. Any theorem that goes beyond this framework needs the axiom of infinity. (This is not to say that any result which is not explicitly about natural numbers requires the axiom of infinity. We can code and discuss certain infinite objects in this setting, but not everything we would like. A precise formalization of this is carried out in the context of subsystems of second order arithmetic. In particular, the theory known as $\mathsf{ACA}_0$ allows us to prove some explicit results about some infinite sets, without requiring any commitments beyond the resources of first order Peano Arithmetic. See here for a brief introduction, and this book for details.)
In particular, the axiom of infinity goes well beyond the Peano axioms (and not simply in terms of consistency strength or expressive power). The Peano axioms are provable in $\mathsf{ZF}$ without the axiom of infinity. In this theory, you cannot prove that $\omega$ is a set, but you can prove that as a (perhaps proper) class, it satisfies both first and second order $\mathsf{PA}$. Typically, the second order formulation of the axioms subsumes the first order formulation, but not here, since in this theory one cannot prove the existence of infinite sets, so the second order version may be vacuously true, while the first order version still has content.
Andrés E. CaicedoAndrés E. Caicedo
$\begingroup$ Thanks Andres. I have accepted your answer, but I wonder what you think of an alternative to the AOI that I proposed a few days ago at math.stackexchange.com/questions/472045/… Do agree with posters there that it is equivalent to AOI? $\endgroup$ – Dan Christensen Aug 28 '13 at 19:26
$\begingroup$ I'll take a look later today, thanks for the link. $\endgroup$ – Andrés E. Caicedo Aug 28 '13 at 19:28
Here's a well-known bit of folklore. The theory "ZFC - the axiom of infinity + the negation of the axiom of infinity" is equivalent to Peano Arithmetic in the sense that each theory is interpretable in the other and the interpretations are inverse to each other. (Roughly speaking, anyway. The details involved in spelling this out accurately as a tight result are a bit tricky. See the paper by Kaye and Wong.)
Hence there's one good sense in which ZFC with the negation of the axiom of infinity gets you arithmetic -- so, in exactly what sense is the axiom of infinity "needed to get the natural numbers"?
Pedro Tamaroff♦
Peter SmithPeter Smith
$\begingroup$ Well, you could do arithmetic on natural numbers without the axiom. But you wouldn't have the SET of all natural numbers, which is needed to construct the other number systems such as the real numbers. Please correct me if I'm wrong. $\endgroup$ – Brusko651 Aug 25 '13 at 21:05
$\begingroup$ Sure, there won't be an infinite set! But as I recall the OP was originally asking about whether the axiom was "needed to get the natural numbers", though the question has since been edited (I think!). $\endgroup$ – Peter Smith Aug 25 '13 at 21:08
$\begingroup$ @PeterSmith I'm not saying the axiom of infinity is necessary for constructing the natural numbers, but that it seems to be used for this and only this purpose. But I stand to be corrected. Are you saying that the natural numbers can be constructed in ZFC without Infinity? That would be interesting. $\endgroup$ – Dan Christensen Aug 26 '13 at 2:29
$\begingroup$ @Dan: In a model of $\sf ZFC$ with the negation of the axiom of infinity instead, the natural numbers are just the ordinals of the universe. $\endgroup$ – Asaf Karagila♦ Aug 26 '13 at 3:26
$\begingroup$ See also here. $\endgroup$ – Andrés E. Caicedo Aug 26 '13 at 3:31
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Why is $\exists$ used in the ZFC Axiom of Power Set instead of $\exists !$ | CommonCrawl |
\begin{document}
\title{Substationarity in Spatial Point Processes} \def\eqalign#1{\null\,\vcenter{\openup\jot\ialign
{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$
\hfil\crcr#1\crcr}}\,}
\setcounter{page}{1} \begin{abstract} The goal of the article is to develop the approach of substationarity to spatial point processes (SPPs). Substationarity is a new concept, which has never been studied in the literature. It means that the distribution of SPPs can only be invariant under location shifts within a linear subspace of the domain. Theoretically, substationarity is a concept between stationariy and nonstationarity, but it belongs to nonstationarity. To formally propose the approach, the article provides the definition of substationarity and an estimation method for the first-order intensity function. As the linear subspace may be unknown, it recommends using a parametric way to estimate the linear subspace and a nonparametric way to estimate the first-order intensity function, indicating that it is a semiparametric approach. The simulation studies show that both the estimators of the linear subspace and the first-order intensity function are reliable. In an application to a forest wildfire data set, the article concludes that substationarity of wildfire occurrences may be assumed along the longitude, indicating that latitude is a more important factor than longitude in forest wildfire studies.
\end{abstract}
{\bf AMS 2000 subject classification:} 62M30, 62G05.
{\it Key Words:} Intensity Functions; Kernel Methods; Nonstationarity; Semiparametric Estimation; Spatial Point Processes (SPPs); Substationarity.
\section{Introduction} \label{sec:introduction}
The goal of the article is to develop the concept of {\it substationarity} for spatial point processes (SPPs). Substationarity a new concept, which has not been studied in the literature. Theoretically, substationarity can bridge stationarity and nonstationarity, two well-known concepts in the literature of spatial statistics. Substationarity means that the distribution of an SPP is only invariant under any location shift within a linear subspace of the domain. Stationarity means that the distribution is invariant under any location shift within the entire domain. Nonstationarity is the complementary concept of stationarity. It means that the distribution of the SPP can be affected by at least one location shift in the domain. If an SPP is substationary, then its distribution may still be affected by a location shift if it is outside the linear subspace. Therefore, the intersection of substationarity and nonstationarity is not empty. Substationarity provides a way to treat nonstationarity. It can make inferences on nonstationarity easy and convenient.
The idea of the research is motivated from our recent work on typical events in natural hazards \cite{zhang2014a}. According to its scientific definition, a natural hazard is a naturally occurring event that might have a negative effect on human or environments. Natural hazards include wildfires, tornados, and earthquakes. In our work on forest wildfires, we identified an inhomegenous wildfire pattern in Alberta (Canada) forests. The proportion of large wildfires in the north was higher than that in the south, but the frequency of wildfires in the south was higher than that in the north. Wildfire activities were not significantly affected by their longitude values. It seems that substationarity might be held along the longitude, indicating that it is an important concept in forest wildfire studies.
Statistical approaches to SPPs are important in many scientific disciplines such as forestry \cite{stoyan1994}, epidemiology \cite{benes2005,diggle2006}, wildfires \cite{peng2005,schoenberg2004}, or earthquakes \cite{ogata1988,zhang2014d}. In statistics, an SPP is treated as a pattern of random points developed in an Euclidean space. The number of points within a bounded subset of the Euclidean space is finite. Point distributions and dependence structures are modeled by intensity functions \cite{diggle2003}. The simplifying assumptions of stationarity and isotropy have been developed to make the analysis convenient. Various well-known tools have been proposed. Examples include the $K$-function \cite{ripley1976}, the $L$-function \cite{besag1977}, and the pair correlation function \cite{stoyan1996}. As stationarity is an important assumption, a few methods have been proposed to evaluate it \cite{guan2008,zhang2014c}. Becuase of the concern of the stationarity assumption, recent research often models SPPs under nonstationarity \cite{moller2007,waagepetersen2009}. An important concept called the second-order intensity-reweighted stationarity (SOIRS) has been proposed \cite{baddeley2000}. This concept is powerful in the joint analysis of the first-order and second-order intensity functions under nonstationarity. With the aid of SOIRS, a number of methods for nonstationarity have been proposed \cite{diggle2007,guan2009,guan2010,henrys2009,waagepetersen2007}. SOIRS only specifies the relationship between the first-order and the second-order intensity functions. It does not contain any assumptions related to substationarity, implying that statistical approaches to substationarity can be combined with SOIRs.
The purpose of the article is to develop a formal statistical approach to substationarity in SPPs, including the concept of substationarity and corresponding estimation methods. Since the linear subspace may still be unknown, estimation of the subspace must also be involved. In our approach, we want to estimate the subspace via a parametric way and intensity functions given the linear subspace via a nonparametric way. Therefore, we classify our estimation as a semiparametric approach. The nonparametric component provides the intensity functions given the linear subspace and the parametric component supplies the linear subspace. We evaluate the properties of our estimation methods by simulations and applications. In simulations, we evaluate the performance of the estimators of the linear subspace and the first-order intensity function by studying their mean square error (MSE) values. In applications, we implement our approach to forest wildfire data. We conclude that estimation under substationarity can provide more precise and reliable results than that under nonstationarity.
To the best of our knowledge, the article is the first one to formally discuss the concept of substationarity. As it has not been previously proposed, it is important to have a formal statistical definition of substationarity at the beginning. Although many research problems can be specified, we only focus on estimation of the first-order intensity functions under substationarity. Many nonparametric or semiparametric methods can be adopted, but we only study the kernel method since it is convenient.
The article is organized as follows. In Section \ref{sec:spatial point processes}, we review the concept of SPPs. In Section \ref{sec:substationarity}, we provide the definition of substationarity, including the evaluation of its theoretical properties. In Section \ref{sec:estimation}, we propose a method to estimate the first-order intensity function under substationarity. In Section \ref{sec:simulation}, we evaluate the performance of our estimators by Monte Carlo simulations. In Section \ref{sec:application}, we apply our approach to the Alberta forest wildfire data. The paper ends with some discussion in Section \ref{sec:discussion}.
\section{Spatial Point Processes} \label{sec:spatial point processes}
A spatial point process (SPP) ${\cal N}({\cal S})$ on ${\cal S}$ is composed of random points in a measurable ${\cal S}\subseteq\mathbb{R}^d$. It is treated as the restriction of ${\cal N}$, the SPP on the entire $\mathbb{R}^d$, with points only observed in ${\cal S}$. Therefore, points of ${\cal N}$ in ${\cal S}^c$ (the complementary set of ${\cal S}$) are not observed. Let $\mathscr{B}$ and $\mathscr{B}(A)$ be the collections of Borel sets of $\mathbb{R}^d$ and a measurable $A\subseteq\mathbb{R}^d$, respectively. Let $N(A)$ and $N$ be the numbers of points in $A$ and $\mathbb{R}^d$, respectively. Then, $N(A)$ is finite if $A$ is bounded and $P[N(A)=0]=1$ for any $A\in\mathscr{B}(\mathbb{R}^d)$ with $|A|=0$, where $|A|$ is the Lebesgue measure on $\mathbb{R}^d$.
An SPP ${\cal N}$ is $k$th-order stationary if \begin{equation} \label{eq:kth order stationary} P[N(A_1)=n_1,\cdots,N(A_l)=n_l]=P[N(A_1+{\bf h})=n_1,\cdots,N(A_l+{\bf h})=n_l] \end{equation} for any ${\bf h}\in\mathbb{R}^d$, $l\le k$, $A_1,\cdots,A_l\in\mathscr{B}(\mathbb{R}^d)$, and $n_1,\cdots,n_l\in\mathbb{N}$, where $A+{\bf h}=\{{\bf s}+{\bf h}:{\bf s}\in A\}$. It is strong stationary if (\ref{eq:kth order stationary}) holds for any $l\in\mathbb{N}$. We say ${\cal N}({\cal S})$ is $k$th-order stationary and strong stationary, respectively, if it can be derived by restricting a $k$th-order stationary or a strong stationary ${\cal N}$ on ${\cal S}$.
The $k$th-order intensity function of ${\cal N}$ is defined as $$
\lambda_k({\bf s}_1,\cdots, {\bf s}_k)=\lim_{\rho(U_{{\bf s}_i})\rightarrow 0,i=1,\ldots,k}{{\rm E}\{\prod_{i=1}^k N(U_{{\bf s}_i})\}\over \prod_{i=1}^k |U_{{\bf s}_i}|}, $$ where ${\bf s}_1,\cdots, {\bf s}_k\in\mathbb{R}^d$ are distinct, $U_{\bf s}$ is a neighbor of ${\bf s}$, and $\rho(U_{\bf s})$ is the diameter of $U_{\bf s}$, provided that it almost surely exists in the Lebesgue measure on $\mathbb{R}^d$. If ${\cal N}$ is $k$th-order and strong stationary, respectively, then $\lambda_{l}({\bf s}_1+{\bf h},\cdots,{\bf s}_l+{\bf h})$ is independent of ${\bf h}$ almost surely with respect to the Lebesgue measure on $\mathbb{R}^d$ for any positive $l\le k$ and any $l\in\mathbb{N}$, respectively.
The mean structure of ${\cal N}$ is $$ \mu(A)={\rm E}[N(A)] =\int_A\lambda({\bf s})d{\bf s},$$ where $\lambda({\bf s})=\lambda_1({\bf s})$ is the first-order intensity function. The covariance structure of ${\cal N}$ is \begin{equation} \label{eq:covariance of counts in two events} \eqalign{ {\rm Cov}[N(A_1),N(A_2)]=&\int_{A_1}\int_{A_2}\{ \lambda_2({\bf s}_1,{\bf s_2})-\lambda({\bf s}_1)\lambda({\bf s}_2)\} d{\bf s_2}d{\bf s_1}+\int_{A_1\cap A_2}\lambda({\bf s})d{\bf s}\cr =&\int_{A_1}\int_{A_2}\{ g({\bf s}_1,{\bf s_2})-1\} \lambda({\bf s}_1)\lambda({\bf s}_2) d{\bf s_2}d{\bf s_1}+\mu(A_1\cap A_2), } \end{equation}
where $ g({\bf s}_1,{\bf s}_2)=\lambda_2({\bf s}_1,{\bf s}_2)/\{ \lambda({\bf s}_1)\lambda({\bf s}_2)\}$ is the pair correlation function. The covariance function of ${\cal N}$ is $$ \Gamma({\bf s}_1,{\bf s}_2)=\{ g({\bf s_1},{\bf s}_2)-1\} \lambda({\bf s}_1)\lambda({\bf s}_2)+\lambda({\bf s}_1)\delta_{{\bf s}_1,{\bf s}_1}({\bf s}_2,{\bf s}_2),$$
where $\delta_{{\bf s},{\bf s}}$ represents the point measure at $({\bf s},{\bf s})\in\mathbb{R}^d\times\mathbb{R}^d$. By the covariance function, (\ref{eq:covariance of counts in two events}) becomes \begin{equation} \label{eq:covariance function} {\rm Cov}[N(A_1),N(A_2)]=\int_{A_1}\int_{A_2}\Gamma({\bf s}_1,{\bf s}_2)d{\bf s}_2d{\bf s}_1. \end{equation}
If $g({\bf s}_1,{\bf s}_2)$ only depends on ${\bf s}_1-{\bf s}_2$ or $\|{\bf s}_1-{\bf s}_2\|$ such that it can be expressed as $g({\bf s}_1-{\bf s}_2)$ or $g(\|{\bf s}_1-{\bf s}_2\|)$, then ${\cal N}$ is called a second-order intensity-reweighted stationary (SOIRS) or a second-order intensity-reweighted isotropic (SOIRI) SPP. SOIRS and SOIRI are important concepts for nonstationary SPPs as it can model the first-order and second-order intensity functions together \cite{baddeley2000}.
If ${\cal N}$ is first-order stationary, then $\lambda({\bf s})=c$ and $\mu(A)=c|A|$ for some $c>0$. If ${\cal N}$ is second-order stationary, then $\lambda({\bf s})=c$, $\mu(A)=c|A|$, $g({\bf s}_1,{\bf s}_2)=g({\bf s}_1-{\bf s}_2)$,
$${\rm Cov}[{\cal N}(A_1),{\cal N}(A_2)]=c^2\int_{A_1}\int_{A_2}\{g({\bf s}_1-{\bf s}_2)-1\}d{\bf s}_2d{\bf s}_1+c|A_1\cap A_2|$$ and
$${\rm V}[N(A)]=c^2\int_{A}\int_{A} \{g({\bf s}_1-{\bf s}_2)-1\}d{\bf s}_2d{\bf s}_1+c|A|.$$ If ${\cal N}$ is Poisson, then $g({\bf s}_1,{\bf s}_2)=1$, indicating that $V\{N(A)\}={\rm E}[N(A)]$ for any bounded $A\in\mathscr{B}(\mathbb{R}^d)$. Only the mean structure is important in Poisson SPPs. However, both the mean and variance structures are important in non-Poisson SPPs.
\section{Substationarity} \label{sec:substationarity}
The main purpose of this section is to provide the formal definition of substationarity as well as corresponding properties. As substationarity is a new concept which has not been studied in the literature before, it is also important to provide asymptotic theory under substationarity. The theory are useful in the evaluation of theoretical properties of estimators provided in the next section.
\begin{defn} \label{defn:definition of substationarity} We say ${\cal N}$ is $k$th-order substationary in a linear subspace ${\cal L}\subseteq\mathbb{R}^d$ if (\ref{eq:kth order stationary}) holds for any ${\bf h}\in{\cal L}$, $l\le k$, $A_1,\cdots,A_l\in\mathscr{B}(\mathbb{R}^d)$, and $n_1,\cdots,n_l\in\mathbb{N}$. We say ${\cal N}$ is strong substationary in ${\cal L}$ if it is $k$th-order substationary in ${\cal L}$ for any $l\in\mathbb{N}$. For any ${\cal S}\subseteq\mathbb{R}^d$, we say ${\cal N}({\cal S})$ is $k$th-order substationary or strong substationary in ${\cal L}$ or ${\cal L}\cap {\cal S}$ equivalently if ${\cal N}({\cal S})$ can be restricted by a $k$th-order substationary or strong substationary ${\cal N}$ in ${\cal L}$ on ${\cal S}$. \end{defn}
Obviously, if ${\cal N}$ is $k$th-order substationary and its $k$th-order intensity function almost surely exists, then \begin{equation} \label{eq:kth substationary} \lambda_l({\bf s}_1,\cdots,{\bf s}_l)=\lambda_l({\bf s}_1+{\bf h},\cdots,{\bf s}_l+{\bf h}) \end{equation}
almost surely with respect to the Lebesgue measure of $\mathbb{R}^d$ for any ${\bf h}\in {\cal L}$, $l\le k$, and distinct ${\bf s}_1,\cdots,{\bf s}_l\in\mathbb{R}^d$. If ${\cal N}$ is $k$th-order substationary in ${\cal L}$, then it is also $k$-th order substationary in any linear subspace ${\cal L}'\subseteq{\cal L}$. Therefore, the linear subspace ${\cal L}$ in Definition \ref{defn:definition of substationarity} is generally not unique.
\begin{defn} \label{defn:definition of intrinsic substationarity} We say ${\cal N}$ is $k$th-order intrinsically substationary or intrinsically strong substationary in ${\cal L}$ if it is substationary or strong substationary in ${\cal L}$ but not in any linear subspace ${\cal L}'$ of $\mathbb{R}^d$ satisfying ${\cal L}\subseteq{\cal L}'$ but ${\cal L}\not={\cal L}'$. We say ${\cal N}({\cal S})$ is $k$th-order intrinsically substationary or intrinsically strong substationary in ${\cal L}$ or ${\cal L}\cap{\cal S}$ equivalently if it can be restricted by a $k$th-order intrinsically substationary or intrinsically strong substationary in ${\cal L}$ on ${\cal S}$. \end{defn}
If ${\cal N}$ is substarionary in both ${\cal L}_1$ and ${\cal L}_2$, then (\ref{eq:kth order stationary}) holds for any ${\bf h}_1\in{\cal L}_1$ and ${\bf h}_2\in{\cal L}_2$. For any ${\bf h}\in{\rm span}\{{\cal L}_1,{\cal L}_2\}$, there exist ${\bf h}_1\in{\cal L}_1$ and ${\bf h}_2\in{\cal L}_2$ such that ${\bf h}={\bf h}_1+{\bf h}_2$. For any $l\le k$, we have $$\eqalign{ P[N(A_1+{\bf h})=n_1,\cdots,N(A_l+{\bf h})=n_l]=&P[N(A_1+{\bf h}_1+{\bf h}_2)=n_1,\cdots,N(A_l+{\bf h}_1+{\bf h}_2)=n_l]\cr =&P[N(A_1+{\bf h}_1)=n_1,\cdots,N(A_l+{\bf h}_1)=n_l]\cr =&P[N(A_1)=n_1,\cdots,N(A_l)=n_l],\cr }$$ implying that ${\cal N}$ is also substationary in ${\rm Span}\{{\cal L}_1,{\cal L}_2\}$. Thus, the linear subspace ${\cal L}$ in Definition \ref{defn:definition of intrinsic substationarity} is unique. A $k$th-order intrinsically substationary ${\cal N}$ in ${\cal L}$ is $k$th-order stationary if and only if ${\cal L}=\mathbb{R}^d$. If ${\cal N}$ is intrinsically substationary in ${\cal L}$, then it is substationary in any linear subspace ${\cal L'}$ of ${\cal L}$ but not in any linear subspace ${\cal L}'$ of $\mathbb{R}^d$ strictly covering ${\cal L}$.
\begin{figure}
\caption{ Equality of expected counts in two subsets under substationarity along the horizontal axis}
\label{fig:mean structures of two sets under substationarity}
\end{figure}
If ${\cal N}$ is substationary in ${\cal L}$, then for any ${\bf h}\in{\cal L}$ there is $\mu(A)=\mu(A+{\bf h})$. This statement can be true in a more general case. Suppose ${\cal N}$ is substationary in the horizontal axis of $\mathbb{R}^2$ (i.e., $d=2$) such that ${\cal L}=\{(x,0):x\in\mathbb{R}\}$. Then, the first-order intensity of ${\cal L}$ only depends on the vertical value of the point, indicating that we can express $\lambda({\bf s})=\lambda(y)$ for any ${\bf s}=(x,y)\in\mathbb{R}^2$. Let $\nu_r$ be the Lebesgue measure on $\mathbb{R}^r$. For any $A\in\mathbb{R}^2$, there is $$\mu(A)=\int_{-\infty}^\infty \lambda(y)\nu_1(A_y)dy,$$ where $A_y=\{{\bf s}=(x,y):(x,y)\in A\}$. For any measurable bounded $A,B\subseteq\mathbb{R}^2$, we may still have $\mu(A)=\mu(B)$ even if $B\not=A+{\bf h}$ for any ${\bf h}\in{\cal L}$ (e.g., the case displayed in Figure \ref{fig:mean structures of two sets under substationarity}). We summarize this issue into the following theorems.
\begin{thm} \label{thm:equality of two sets in a partition} Let ${\cal N}$ be substationary in ${\cal L}\subseteq \mathbb{R}^d$. For any measurable bounded $A,B\in\mathbb{R}^2$, if there exist a partition $\{A_1,A_2,\cdots\}$ of $A$ and a partition $\{B_1,B_2,\cdots\}$ of $B$ such that for every $i$ there exists ${\bf h}_i\in{\cal L}$ satisfying $B_i=A_i+{\bf h}_i$, then $\mu(A)=\mu(B)$. \end{thm}
\noindent {\bf Proof:} Straightforwardly, there is $$\mu(A)=\sum_{i=1}^\infty \mu(A_i)=\sum_{i=1}^\infty \mu(A_i+{\bf h}_i)=\sum_{i=1}^\infty \mu(B_i)=\mu(B).$$ Then, we draw the conclusion.
$\diamondsuit$
\begin{thm} \label{thm:equality of two sets} Let ${\cal N}$ be substationary in ${\cal L}\subseteq \mathbb{R}^d$. For any measurable bounded $A,B\in\mathbb{R}^2$, if $\nu_r(A_{\bf v})=\nu_r(B_{\bf v})$ almost surely for any ${\bf v}\in\mathbb{R}^d$, where $A_{\bf v}=\{{\bf s}\in A: {\bf s}-{\bf v}\in {\cal L}\}$ and $r$ is the dimension of ${\cal L}$, then ${\rm E}[N(A)]={\rm E}[N(B)]$. \end{thm}
\noindent {\bf Proof:} Let ${\bf u}_1,\cdots,{\bf u}_d$ be the orthogonal bases of $\mathbb{R}^d$, where the previous $r$ vectors form the orthogonal bases of ${\cal L}$. Let ${\cal L}^\perp=\{{\bf v}\in\mathbb{R}^d:{\bf v}=\sum_{i=r+1}^d x_i{\bf u}_i,x_i\in\mathbb{R}\}$ be the orthogonal space of ${\cal L}$ in $\mathbb{R}^d$ . Let ${\bf s}_{\cal L}$ and ${\bf s}_{{\cal L}^\perp}$ be the orthogonal projection of ${\bf s}$ on ${\cal L}$ and ${\cal L}^\perp$, respectively. Then, the first-order intensity function of ${\cal N}$ can be expressed as $\lambda({\bf s})=\lambda({\bf s}_{{\cal L}^\perp})$ for any ${\bf s}\in A$. We have $$\eqalign{ \mu(A)=&\int_{{\bf s}\in A}\lambda({\bf s})d{\bf s}\cr =&\int_{{\cal L}^\perp}\lambda({\bf s}_{{\cal L}^\perp})\nu_r(A_{{\bf s}_{{\cal L}^\perp}})d{\bf s}_{{\cal L}^\perp}\cr =&\int_{{\cal L}^\perp}\lambda({\bf s}_{{\cal L}^\perp})\nu_r(B_{{\bf s}_{{\cal L}^\perp}})d{\bf s}_{{\cal L}^\perp}\cr =&\int_{{\bf s}\in B}\lambda({\bf s})d{\bf s}\cr =&\mu(B). }$$ We draw the conclusion.
$\diamondsuit$
Theorems \ref{thm:equality of two sets in a partition} and \ref{thm:equality of two sets} can be used to study the relationship between expected numbers of counts between two regions. It is not enough to use them to study their joint distribution. As it depends on types of ${\cal N}$, we study the properties of the joint distribution under the framework of asymptotics. Let $A_{z,{\cal L}}=\{{\bf v}+z{\bf u}:{\bf v}+{\bf u}\in A,{\bf v}\in{{\cal L}}^{\perp},{\bf u}\in{\cal L}\}$ and $A_{{\bf v},z,{\cal L}}=\{{\bf s}\in A_{z,{\cal L}}:{\bf s}-{\bf v}\in{\cal L}\}$ for any $A\in\mathscr{B}(\mathbb{R}^d)$, where ${\cal L}$ is a linear subspace of $\mathbb{R}^d$. Then, $A_{{\bf v},z,{\cal L}}=\{{\bf v}+z{\bf u}:{\bf v}+{\bf u}\in A,{\bf v}\in{{\cal L}}^\perp,{\bf u}\in{\cal L}\}$ and $\nu_{r}(A_{{\bf v},z,{\cal L}})=z^r\nu_{r}(A_{{\bf v},1,{\cal L}})$. If ${\cal N}$ is substationary in ${\cal L}$ and $A$ is bounded, then $$\eqalign{ \mu(A_{z,{\cal L}})=&\int_{{\bf s}\in A_{z,{\cal L}}} \lambda({\bf s})d{\bf s}\cr =&\int_{{{\cal L}}^\perp}\lambda({\bf s}_{{\cal L}^\perp})\nu_{r}(A_{{\bf s}_{{{\cal L}}^\perp},z,{\cal L}})d{\bf s}_{{{\cal L}}^\perp}\cr =&z^{r}\int_{{{\cal L}}^\perp}\lambda({\bf s}_{{\cal L}^\perp})\nu_{r}(A_{{\bf s}_{{{\cal L}}^\perp},1,{\cal L}})d{\bf s}_{{{\cal L}}^\perp}\cr =&z^{r}\mu(A). }$$ If ${\cal N}$ is Poisson, then ${\rm V}[N(A_{z,{\cal L}})]=\mu(A_{z,{\cal L}})=z^r\mu(A)$ and $$M_{z,{\cal L}}(A)=z^{-{r\over 2}}[N(A_{z,{\cal L}})-\mu(A_{z,{\cal L}})]\stackrel{D}\rightarrow N[0,\mu(A)]$$
as $z\rightarrow\infty$.
Let ${\cal A}$ be a collection of Borel sets of $\mathbb{R}^d$. Let ${\cal A}_{z,{\cal L}}$, $N({\cal A}_{z,{\cal L}})$, $\mu({\cal A}_{z,{\cal L}})$ be vectors composed of $A_{z,{\cal L}}$, $N(A_{z,{\cal L}})$, and $\mu(A_{z,{\cal L}})$ for all $A\in{\cal A}$, respectively. If ${\cal A}$ is a finite collection of disjoint subsets such that it can be expressed as ${\cal A}=\{A_1,\cdots,A_m\}$ with disjoint $A_1,\cdots,A_m$, then \begin{equation} \label{eq:limiting distribution of many sets} M_{z,{\cal L}}({\cal A})\stackrel{D}\rightarrow N[0,{\rm diag}(\mu({\cal A}))], \end{equation} where $M_{z,{\cal L}}({\cal A})$ is the vector composed of $M_{z,{\cal L}}(A)$ for all $A\in{\cal A}$.
For any $V\in\mathscr{B}({{\cal L}}^\perp)$, let \begin{equation} \label{eq:defintion of collection in functional limit theorm} A_{{\bf t},V}=(0,t_1{\bf u}_1]\times\cdots\times(0,t_{r}{\bf u}_{r}]\times V, \end{equation} where $t_i>0$, $(0,t_i{\bf u}_i]=\{{\bf s}=x{\bf u}_i:0<x\le t_i\}$, and ${\bf u}_1,\cdots,{\bf u}_{r}$ are the orthogonal bases of ${\cal L}$. Then, \begin{equation} \label{eq:limiting distribution of bivariate sets}
\left(\begin{array}{c} M_{z,{\cal L}}(A_{{\bf t},V}) \cr M_{z,{\cal L}}(A_{{\bf t}',V})\end{array}\right)\stackrel{D}\rightarrow N\left[\left(\begin{array}{c} 0 \cr 0 \end{array}\right), \left(\begin{array}{cc} \mu(A_{{\bf t},V}) & \mu(A_{{\bf t}\wedge{\bf t}',V}) \cr \mu(A_{{\bf t}\wedge{\bf t}',V}) & \mu(A_{{\bf t}',V})\cr\end{array}\right)\right], \end{equation} as $z\rightarrow\infty$. The finite-dimensional central limit theorem of $N({\cal A}_{z,{\cal L}})$ can be derived by (\ref{eq:limiting distribution of many sets}) and (\ref{eq:limiting distribution of bivariate sets}), but it is not enough for us to study properties of the estimator of the first-order intensity proposed in the next section. To study the properties, we need the functional central limit theorem of $M_{z,{\cal L}}({\cal A})$ when ${\cal A}$ contains infinitely number of measurable subsets of $\mathbb{R}^d$. A typical way to show functional central limit theorem is to combine the finite-dimensional asymptotics with the tightness \cite{whitt2007}. A typical way to prove the tightness is the evaluation of the bracketing entropy number, which is used in the following theorem.
\begin{thm} \label{thm:functional central limit theorem for Poisson substationary SPP} Let ${\cal N}$ be a Poisson substationary SPP in ${\cal L}$. If ${\cal A}_V=\{A_{{\bf t},V}: {\bf t}=(t_1,\cdots,t_{r})\in[0,\infty)^{r}\}$ for some $V\subseteq\mathscr{B}({{\cal L}}^\perp)$, then $M_{z,{\cal L}}({\cal A}_V)$ weakly converges to a mean zero Gaussian random field on $[0,\infty)^{r}$ with the covariance structure given by the right side of (\ref{eq:limiting distribution of bivariate sets}). \end{thm}
\noindent {\bf Proof:} We show the conclusion by the standard empirical process approach. Let ${\cal A}_{V,{\bf a}}=\{A_{{\bf t},V}: {\bf t}=(t_1,\cdots,t_{r})\in[0,a_1]\times\dots\times\prod_{i=1}^{r} [0,a_i]\}$ for any ${\bf a}=(a_1,\cdots,a_{r})^\top\in (0,\infty)^{r}$. Let $F({\bf t})=\mu(A_{{\bf t},V})/\mu(A_{{\bf a},V})$ for any ${\bf t}\preceq {\bf a}$. Then, $F$ is an $r$-dimensional marginal uniformly distributed CDF on the $\sigma$-field generated by ${\cal A}_{{\bf a},V}$. Let $F_i$ be the $i$th CDF of $F$. For any $\epsilon\in(0,1)$, there is an integer $J$ such that ${r}/\epsilon^2\le J\le r/\epsilon^2+1$. Let $x_{ij}=ja_i/(J+1)$ for $j=0,1,\cdots,J+1$. Then, $\epsilon^2/(\epsilon^2+r)\le F_i(x_{i(j+1)})-F_i(x_{ij})\le \epsilon^2/r$. Let $X_{\epsilon}=\{{\bf x}=(x_1,\cdots,x_{r}): x_i=x_{ij}\ {\rm for\ some}\ j=0,1,\cdots,J+1\}$. Then, $\#X_\epsilon=(J+2)^{r}\le [(r+3)/\epsilon^2]^{r}$. For any $g_{\bf x}\in{\cal G}=\{I_{{\bf x}}: {\bf x}\in\prod_{i=1}^{r}[0,a_i]\}$, we can find ${\bf x}',{\bf x}''\in X_\epsilon$ such that ${\bf x}'\preceq {\bf y}\preceq{\bf x}''$ but there is no ${\bf x}^*\in X_\epsilon$ satisfying $x_i'<x_i^*< x_i''$ for some $i=1,\cdots,r$, where $x_i$, $x_i^*$, and $x_i''$ are the $i$th component of ${\bf x}$, ${\bf x}^*$, and ${\bf x}''$, respectively. Then, $g_{{\bf x}'}\le g_{\bf x}\le g_{{\bf x}''}$ and
$$\|g_{{\bf x}''}-g_{{\bf x}'}\|_{F}^2=\int_{\prod_{i=1}^{r}[0,a_i]}|g_{{\bf x}''}({\bf x})-g_{{\bf x}'}({\bf x})|^2F(d{\bf x})\le \sum_{i=1}^{r}[F_i(x_i'')-F_i(x_i')]\le\epsilon^2.$$ Because $$\int_0^1 \log^{1/2}(\#X_\epsilon)d\epsilon\le\int_0^1 \{r[\log(r+3)+2\log\epsilon]\}^{1/2}d\epsilon <\infty,$$ we conclude that ${\cal G}$ is $F$-Donsker \cite[P. 270]{vandervaart1998}, implying that the conclusion holds in $\prod_{i=1}^{r}[0,a_i]$ for any ${\bf a}\in(0,\infty)^{r}$. We draw the conclusion of the theorem by letting $a_i\rightarrow\infty$ for all $i$.
$\diamondsuit$
Theorem \ref{thm:functional central limit theorem for Poisson substationary SPP} supplies the functional central limit theorem of $M_{z,{\cal L}}({\cal A})$ if ${\cal N}$ is Poisson, but it does not provide any similar result of $M_{z,{\cal L}}({\cal A})$ if it is not. A critical issue in the case when ${\cal N}$ is non-Poison is the presence of dependence structures. In particular, for any disjoint $A$ and $B$, if ${\cal N}$ is Poisson, then $N(A)$ and $N(B)$ are independent Poisson random variables with expected values $\mu(A)$ and $\mu(B)$, respectively. If ${\cal N}$ is not Poisson, then the dependence between $N(A)$ and $N(B)$ must be addressed. This requires us to study the property of the second-order intensity function.
Let $A$ and $B$ be bounded measurable subsets of $\mathbb{R}^d$. For any ${\bf h}\in{\cal L}$, there is $$\eqalign{ {\rm Cov}[N(A+{\bf h}),N(B)]=&\int_{A+{\bf h}}\int_{B}\Gamma({\bf s}_1,{\bf s}_2)d{\bf s}_2d{\bf s}_1\cr
=&\int_{A}\int_{B}[g({\bf s}_1-{\bf s}_2-{\bf h})-1]\lambda({\bf s}_1)\lambda({\bf s}_2)d{\bf s}_2d{\bf s}_1+\int_{(A+{\bf h})\cap B}\lambda({\bf s})d{\bf s}.\cr } $$
If $\|{\bf h}\|$ is large such that $(A+{\bf h})\cap B=\phi$, then $${\rm Cov}[N(A+{\bf h}),N(B)]=\int_{A}\int_{B}[g({\bf s}_1-{\bf s}_2-{\bf h})-1]\lambda({\bf s}_1)\lambda({\bf s}_2)d{\bf s}_2d{\bf s}_1.$$
If $g({\bf s}_1-{\bf s}_2-{\bf h})\rightarrow 1$ as $\|{\bf h}\|\rightarrow\infty$, then ${\rm Cov}[N(A+{\bf h}),N(B)]\rightarrow 0$, indicating that $N(A+{\bf h})$ and $N(B)$ are almost independent. To theoretically address this issue, we need to assume that ${\cal N}$ satisfies the strong mixing condition. This approach was first introduced for dependent random variables by \cite{rosenblatt1956} and later extended to stationary SPPs by \cite{ivanoff1982}. Here we want to modify it to substationarity SPPs.
Suppose ${\cal N}$ is substationarity in ${\cal L}$. Let $\mathscr{B}(A)$ be the collection of Borel sets generated by $A$. Denote the diameter of $A$ by $\rho(A)$ and $\rho(A_1,A_2)$ as the minimum distance between $A_1$ and $A_2$, where $\rho(A)=\sup_{{\bf s},{\bf s}'\in A}\|{\bf s}-{\bf s}'\|$ and $\rho(A_1,A_2)=\min_{{\bf s}\in A_1,{\bf s}'\in A_2}\|{\bf s}-{\bf s}'\|$. Let $$\eqalign{
\alpha(u,v)=\sup\{|P&(U_1\cap U_2)-P(U_1)P(U_2)|:U_1\in\mathscr{B}(A_1),U_2\in\mathscr{B}(A_2),\cr & \rho(A_1,A_2)\ge u, \rho(A_1)\le v,\rho(A_2)\le v, A_1,A_2\in\mathscr{B}(\mathbb{R}^d)\} } $$ be the mixing coefficients, where $P(U)$ is the distribution of $N(U)$. We say ${\cal N}$ is {\it strongly mixing} if $\alpha(zu,zv)\rightarrow 0$ as $z\rightarrow\infty$.
We want to derive the functional central limit theorem of $M_{z,{\cal L}}({\cal A}_{{\bf t},V})$ for ${\bf t}\in[0,\infty)^{r}$ and $V\in\mathscr{B}({{\cal L}}^\perp)$. Our proof is based on a classical way. It was initially introduced by \cite{ibragimov1962} and later modified by \cite{herrndorf1984}. The main idea is to split $A_{z,{\cal L}}$ for $A\in{\cal A}_{{\bf t},V}$ into two components $B$ and $C$. Both $B$ and $C$ can be writing into the sum of blocks, where counts in blocks of $B$ are almost independent and counts in blocks of $C$ can be igrnored. This is a popular idea in the proof of the asymptotic normality for stationary time series, which can also be used to SPPs. Since the proof of our functional central limit theorm is just a simple usage of the popular idea, we decide to only briefly display it.
\begin{thm} \label{thm:functional central limit theorem strong mixing} Assume ${\cal N}$ is strongly mixing and substationary in ${\cal L}$. If the fourth intensity function of ${\cal N}$ is uniformly bounded and \begin{equation} \label{eq:mixing coefficient order} \int_0^\infty z^{d-{1\over 2}}\alpha(zu,zv)dz<\infty \end{equation} for any $u$ and $v$, then $M_{z,{\cal L}}({\cal A}_{V})$ weakly converges to a Gaussian process with independent increments. \end{thm}
\noindent {\bf Proof:} Let $A_i=U_i\times V$ For any disjoint $U_1,\cdots,U_m\in\mathscr{B}({\cal L})$. Define ${\cal A}=\{A_1,\cdots,A_m\}$. Using the method in Theorem 1.3 of \cite{ibragimov1962}, we can partition ${\cal A}$ into many small blocks, denoted by ${\cal B}=\{{\cal B}_{1},\cdots,{\cal B}_{k_1}\}$ and ${\cal C}=\{C_{1},\cdots,{\cal C}_{k_2}\}$, where $k_1,k_2\rightarrow\infty$ as $z\rightarrow\infty$, such that $$\min_{B\in{\cal B}_{j,n},B'\in{\cal B}_{j',n},j\not=j'}\rho(B,B')\ge u$$
and $N({\cal A}_{z,{\cal L}})=N({\cal B}_{z,{\cal L}})+N({\cal C}_{z,{\cal L}})$. By the method of Theorem 1.4 in \cite{ibragimov1962}, we can choose $k_1$ such that it is bounded by $z^{(1+u)/(2d)}$ for any positive $u$ if $z$ is sufficiently large. Then, there is
$$\left|{\rm E}e^{it\sum_{j=1}^m M_{z,{\cal L}}(A_j)}-\prod_{j=1}^{k_1}{\rm E}e^{it M_{z,{\cal L}}({\cal B}_{j})}\right|\le 4k_1\alpha(zu,zv),$$ where $v=\max(\rho(U_i))$. If (\ref{eq:mixing coefficient order}) holds, then the right side of the above goes to $0$ as $z\rightarrow\infty$. Since $\lambda_4$ is uniformly bounded, we conclude that the Lyapounov Condition \cite[P. 362]{billingsley1995} holds, implying that the asymptotic normality holds. We draw the conclusion about the central limit theorem of $M_{z,{\cal L}}({\cal A})$ for finite ${\cal A}$. By the same method in the proof of the tightness that we have displayed in Theorem \ref{thm:functional central limit theorem for Poisson substationary SPP}, we can show the tightness of the distribution of $M_{z,{\cal L}}({\cal A}_{V})$ for sufficiently large $z$. Then, we draw the functional central limit theorem for $M_{z,{\cal L}}({\cal A}_{V})$, implying the conclusion of the theorem.
$\diamondsuit$
\begin{cor} \label{cor:functional central limit theorem strong mixing} If all conditions of Theorem \ref{thm:functional central limit theorem strong mixing} hold, then there exists $C>0$ such that for any $A\in\mathscr{B}(\mathbb{R}^d)$ there is $M_{z,L}(A)\stackrel{D}\rightarrow N(0,C^2\mu(A))$. \end{cor}
\noindent {\bf Proof:} At the beginning, we assume that there exists ${\bf t}\in\mathbb{R}^r$ and $V\subseteq\mathscr{B}({\cal L}^\top)$ such that $A=A_{{\bf t},V}$. If we partition $(0,t_1{\bf u}_1]\times\cdots\times(0,t_r{\bf u}_r]$ into countable small rectangles, denoted by ${\cal A}=\{U_i: i\in\mathbb{N}\}$, then we can express $A_{{\bf t},V}=\bigcup_{i=1}^\infty U_i\times V$. By theorem \ref{thm:functional central limit theorem strong mixing}, $M_{z,{\cal L}}({\cal A})\stackrel{D}\rightarrow N(0,D_{\cal A})$, where $D_{\cal A}$ is a diagonal matrix determined by the property of ${\cal A}$ and it satisfies all of the assumptions of $\sigma$-finite measures in ${\cal L}$. Therefore, there exists a $\sigma$-finite measure $\tilde\mu$ on ${\cal L}$ such that $M_{z,{\cal L}}(A_{{\bf t},V})\stackrel{D}\rightarrow N(0,\tilde\mu(A_{{\bf t},V}))$. Note that ${\cal A}_{V}$ is a $\pi$-system \cite[P. 42]{billingsley1995}, we conclude that $\tilde\mu$ can be uniquely determined. Then, there is $M_{z,L}(A)\stackrel{D}\rightarrow N(0,\tilde\mu(A))$ for any $A\in\mathscr{B}(\mathbb{R}^d)$. By the expression of $V[N(A_{z,L})]$ given by (\ref{eq:covariance of counts in two events}), we conclude that $\tilde\mu(A)$ is proportional to $\mu(A)$, implying the conclusion.
$\diamondsuit$
A main interest in practice is to estimate the first-order intensity function $\lambda({\bf s})$ under substationarity. As $\lambda({\bf s})$ only varies in ${\cal L}^\perp$, it is equivalent to estimate $\lambda({\bf s}_{{\cal L}^\perp})$ and ${\cal L}$ together. Since it is generally inappropriate to model $\lambda({\bf s}_{{\cal L}^\perp})$ parametrically, we propose a nonparametric way to estimate it. Note that ${\cal L}$ can be formulated by a rotation of a linear subspace spanned by coordinates, we propose a parametric way to estimate it. Therefore, we classify our estimation as a semiparametric approach. The functional central limit theorems given by Theorems \ref{thm:functional central limit theorem for Poisson substationary SPP} and \ref{thm:functional central limit theorem strong mixing} provide the theoretical basis of the approach.
\section{Estimation} \label{sec:estimation}
Let ${\cal N}$ be substationary in ${\cal L}\subseteq\mathbb{R}^d$. Assume points of ${\cal N}$ are only collected in bounded ${\cal S}\in\mathscr{B}(\mathbb{R}^d)$ such that they can be represented by ${\cal N}({\cal S})$. Our main interest is to estimate $\lambda({\bf s}_{{\cal L}^\perp})$ and ${\cal L}$ simultaneously by ${\cal N}({\cal S})$. Since ${\cal L}$ is unknown, we propose a two-step method to estimate them. In the first step, we estimate $\lambda({\bf s}_{{\cal L}^\perp})$ with a given ${\cal L}$, where a nonparametric way is adopted. In the second step, we estimate ${\cal L}$, where a parametric way is adopted. The second step needs the formulation of the estimator in the first step.
We propose a kernel-based method to estimate $\lambda({\bf s})$ for a given ${\cal L}$. We investigate the usual kernel-based method without using substationarity \cite{diggle1985}. It provides an estimator of $\lambda({\bf s})$ as \begin{equation} \label{eq:general kernel estimator of the first-order} \hat\lambda_{h}({\bf s})=C_{h}^{-1}({\bf s})\int_{\cal S} K_{h}({\bf s}'-{\bf s}) N(d{\bf s}'), \end{equation} where $K_{h}({\bf s})=K({\bf s}/h)/h^d$ with bandwidth $h\in\mathbb{R}$ is a kernel density function on $\mathbb{R}^d$ and $C_{h}({\bf s})=\int_{\cal S}K_{h}({\bf s}'-{\bf s})d{\bf s}'$ is the Berman-Diggle boundary correction \cite{berman1989}. By Campbell's Theorem, we obtain \begin{equation} \label{eq:expected value of the general kernerl estimator} {\rm E}[\hat\lambda_{h}({\bf s})] =C_h^{-1}({\bf s})\int_{\cal S}K_h({\bf s}'-{\bf s})\lambda({\bf s}')d{\bf s}' \end{equation} and \begin{equation} \label{eq:variance of the general kernerl estimator} \eqalign{ {\rm V}[\hat\lambda_h({\bf s})]=&C_h^{-2}({\bf s})\int_{\cal S}\int_{\cal S}K_h({\bf s}'-{\bf s})K_h({\bf s}''-{\bf s})[g({\bf s}',{\bf s}'')-1]\lambda({\bf s}')\lambda({\bf s}'')d{\bf s}''d{\bf s}'\cr &+C_h^{-2}({\bf s})\int_{\cal S}K_h^2({\bf s}'-{\bf s})\lambda({\bf s})d{\bf s}. }\end{equation}
We modify (\ref{eq:general kernel estimator of the first-order}) for a substationary ${\cal N}$ in ${\cal L}$. We obtain an estimator of $\lambda({\bf s}_{{\cal L}^\perp})$ (or $\lambda({\bf s})$, equivalently) as \begin{equation} \label{eq:kernel estimator of the first-order substationarity} \hat\lambda_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp})=C_{h,{\cal L}^\perp}^{-1}({\bf s}_{{\cal L}^\perp})\int_{\cal S} K_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp}'-{\bf s}_{{\cal L}^\perp}) N(d{\bf s}'), \end{equation} where $K_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp})=K({\bf s}_{{\cal L}^\perp}/h)/h^r$ with $h\in\mathbb{R}$ is a kernel density function on ${\cal L}^\perp$ and $C_{h,{\cal L}^\perp}({\bf s})=\int_{\cal S} K_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp}'-{\bf s}_{{\cal L}^\perp})d{\bf s}'$ is still the boundary correction. Still by Campbell's Theorem, we obtain \begin{equation} \label{eq:expected value kernel estimator of the first-order substationarity} {\rm E}[\hat\lambda_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp})]=C_{h,{\cal L}^\perp}^{-1}({\bf s}_{{\cal L}^\perp})\int_{\cal S} K_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp}'-{\bf s}_{{\cal L}^\perp})\lambda({\bf s}_{{\cal L}^\perp}')d{\bf s}', \end{equation} and \begin{equation} \label{eq:variance kernel estimator of the first-order substationarity} \eqalign{ {\rm V}[\hat\lambda_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp})]=&C_{h,{\cal L}^\perp}^{-2}({\bf s}_{{\cal L}^\perp})\int_{\cal S}\int_{\cal S} K_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp}'-{\bf s}_{{\cal L}^\perp})K_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp}''-{\bf s}_{{\cal L}^\perp})[g({\bf s}',{\bf s}'')-1]\cr &\lambda({\bf s}_{{\cal L}^\perp}')\lambda({\bf s}_{{\cal L}^\perp}'')d{\bf s}''d{\bf s}'+C_{h,{\cal L}^\perp}^{-2}({\bf s})\int_{\cal S}K_{h,{\cal L}^\perp}^2({\bf s}_{{\cal L}^\perp}'-{\bf s}_{{\cal L}^\perp})\lambda({\bf s}_{{\cal L}^\perp}')d{\bf s}'.\cr } \end{equation} If $r=0$, then ${\cal L}=\{{\bf 0}\}$ and (\ref{eq:kernel estimator of the first-order substationarity}) becomes \begin{equation} \label{eq:estimator of intensity under stationarity}
\hat\lambda={n\over |{\cal S}|}. \end{equation} Since ${\cal N}$ is stationary in this case, the first-order intensity function is a constant, indicating that the estimator must be a constant.
We compare the MSEs (mean square errors) of $\hat\lambda_h({\bf s})$ and $\hat\lambda_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp})$ as $z\rightarrow\infty$ in the case when ${\cal S}=A_{z,{\cal L}}$ for a bounded $A\in\mathscr{B}(\mathbb{R}^d)$. We find that the bias of $\hat\lambda_h({\bf s})$, which is given by ${\rm Bias}[\hat\lambda_h({\bf s})]={\rm E}[\hat\lambda_h({\bf s})]-\lambda({\bf s})$, can go to $0$ as $h\rightarrow0$, but it can simultaneously cause ${\rm V}[\hat\lambda_h({\bf s})]\rightarrow\infty$. To make ${\rm V}[\hat\lambda_h({\bf s})]$ small, we need to choose a large $h$, which increases the value of ${\rm Bias}[\hat\lambda_h({\bf s})]$. Thus, ${\rm MSE}[\hat\lambda_h({\bf s})]=\{{\rm E}[\hat\lambda_h({\bf s})]-\lambda({\bf s})\}^2+{\rm V}[\hat\lambda_h({\bf s})]$ cannot go to $0$ as $z\rightarrow\infty$. However, by a way to select $h$, we can make ${\rm MSE}[\hat\lambda_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp})]\rightarrow 0$ as $z\rightarrow\infty$.
\begin{thm} \label{thm:unbiasedness of estimators if h is small}
Let ${\cal N}$ be substationary in ${\cal L}$ and ${\cal S}=A_{z,{\cal L}}$ for a bounded $A\in\mathscr{B}(\mathbb{R}^d)$ with $|\partial A|=0$. Suppose all of conditions of Theorem \ref{thm:functional central limit theorem strong mixing} hold. Assume $\lambda({\bf s}_{{\cal L}^\perp})$ is positive and continuous in the interior of ${\cal S}$ and $\nu_r(A_{\bf v})$ is almost surely continuous in any ${\bf v}\in{\cal A}^\perp$. For an interior point ${\bf s}$ of $A$, if $h\rightarrow 0$ and $hz\rightarrow\infty$, then ${\rm MSE}[\hat\lambda_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp})]\rightarrow 0$ as $z\rightarrow\infty$. \end{thm}
\noindent {\bf Proof:} For an interior point of ${\bf s}\in A$, there is $$\eqalign{ {\rm E}[\hat\lambda_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp})]=&\left\{\int_{A_{z,{\cal L}}}K_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp}'-{\bf s}_{{\cal L}^\perp})d{\bf s}'\right\}^{-1}\int_{A_{z,{\cal L}}} K_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp}'-{\bf s}_{{\cal L}^\perp})\lambda({\bf s}_{{\cal L}^\perp}')d{\bf s}'\cr =&\left\{\int_{A}K_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp}'-{\bf s}_{{\cal L}^\perp})d{\bf s}'\right\}^{-1}\int_A K_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp}'-{\bf s}_{{\cal L}^\perp})\lambda({\bf s}_{{\cal L}^\perp}')d{\bf s}'\cr =&\left\{\int_{{\cal L}^\perp}\nu_r(A_{{\bf s}_{{\cal L}^\perp}+h{\bf v}})K({\bf v})d{\bf v}\right\}^{-1}\int_{{\cal L}^\perp}\nu_r(A_{{\bf s}_{{\cal L}^\perp}+h{\bf v}})K({\bf v})\lambda({\bf s}_{{\cal L}^\perp}+h{\bf v})d{\bf v}. }$$ If $h\rightarrow 0$ as $z\rightarrow\infty$, then by the continuity of $\nu_r(A_{\bf v})$ and $\lambda({\bf s}_{{\cal L}^\perp})$ there is $$\eqalign{ \lim_{z\rightarrow\infty}{\rm E}[\hat\lambda_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp})] =&\lambda({\bf s}_{{\cal L}^\perp}).\cr }$$ By (\ref{eq:variance kernel estimator of the first-order substationarity}), there is $$\eqalign{ {\rm V}[\hat\lambda_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp})]=&\left\{\int_{A}K_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp}'-{\bf s}_{{\cal L}^\perp})d{\bf s}'\right\}^{-2}\int_{A}\int_{A} K_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp}'-{\bf s}_{{\cal L}^\perp})K_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp}''-{\bf s}_{{\cal L}^\perp})\cr &\times\{g[{\bf s}',{\bf s}''+z({\bf s}_{\cal L}''-{\bf s}_{\cal L}')]-1\}\lambda({\bf s}_{{\cal L}^\perp}')\lambda({\bf s}_{{\cal L}^\perp}'')d{\bf s}''d{\bf s}'\cr &+z^{-r}\left\{\int_{A}K_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp}'-{\bf s}_{{\cal L}^\perp})d{\bf s}'\right\}^{-2}\int_{A}K_{h,{\cal L}^\perp}^2({\bf s}_{{\cal L}^\perp}'-{\bf s}_{{\cal L}^\perp})\lambda({\bf s}_{{\cal L}^\perp}')d{\bf s}'. }$$ By Theorem \ref{thm:functional central limit theorem strong mixing}, we conclude that the first term of the above goes to $0$ as $z\rightarrow\infty$. Therefore, we only need to study the second term. It is $$\eqalign{ &{1\over h^rz^r}\left\{\int_{{\cal L}^\perp}\nu_r(A_{{\bf s}_{{\cal L}^\perp}+h{\bf v}})K({\bf v})d{\bf v}\right\}^{-2}\int_{{\cal L}^\perp}K^2({\bf v})\lambda({\bf s}_{{\cal L}^\perp}+h{\bf v})d{\bf v}, }$$ which goes to zero if $hz\rightarrow\infty$.
$\diamondsuit$
{\it Example 1:} We interpret Theorem \ref{thm:unbiasedness of estimators if h is small} in a special case. Assume that ${\cal N}$ is substationary in $\mathbb{R}^2$ and ${\cal L}=\{(x,0):x\in\mathbb{R}\}$ such that $d=2$, $r=1$, and the first-order intensity function can be expressed as $\lambda({\bf s})=\lambda(y)$, where ${\bf s}=(x,y)$. Suppose ${\cal S}=[0,z]\times[0,\omega]$ such that observations of ${\cal N}$ can be expressed by points within $[0,z]\times[0,\omega]$, denoted by ${\bf s}_1,\cdots,{\bf s}_n$, where $n={N({\cal S})}$ is the total number of observed points. If we choose $K({\bf s})=(2\pi)^{-1}e^{-(x^2+y^2)/2}$ for the case when substationarity is not accounted for, then $K_h({\bf s})=\phi(x/h)\phi(y/h)/h^2=(2\pi h^2)^{-1}e^{-(x^2+y^2)/(2h^2)}$, where $\phi$ is the PDF of $N(0,1)$. By (\ref{eq:general kernel estimator of the first-order}), there is $$\eqalign{ {\rm E}[\hat\lambda_h({\bf s})]=\left\{\int_0^z\int_0^\omega{1\over 2\pi h^2}e^{-{(x'-x)^2+(y'-y)^2\over 2h^2}}dy'dx'\right\}^{-1}\int_0^z\int_0^\omega{1\over 2\pi h^2}e^{-{(x'-x)^2+(y'-y)^2\over 2h^2}}\lambda({\bf s}')d{\bf s}'. }$$ Then, $\lim_{h\rightarrow 0}{\rm E}[\hat\lambda_h({\bf s})]=\lambda({\bf s})$, implying that the bias of $\hat\lambda_h({\bf s})$ can only disappear as $h\rightarrow\infty$ but this can make ${\rm V}[\hat\lambda_h({\bf s})]$ large. If we choose $K(y)=(2\pi)^{-1/2}e^{-y^2/2}$ for the case when substationarity is accounted for, then $K_{h,{\cal L}^\perp}(y)=\phi(y/h)/h=(\sqrt{2\pi}h)^{-1}e^{-y^2/(2h^2)}$. By (\ref{eq:kernel estimator of the first-order substationarity}), there is $${\rm E}[\hat\lambda_{h,{\cal L}^\perp}(y)]=\left\{\int_0^\omega {1\over\sqrt{2\pi}h}e^{-{(y'-y)^2\over 2h^2}}dy'\right\}^{-1}\int_0^\omega {1\over\sqrt{2\pi}h}e^{-{(y'-y)^2\over 2h^2}}\lambda(y)dy.$$ Then, $\lim_{h\rightarrow 0}{\rm E}[\hat\lambda_{h,{\cal S}^\perp}({\bf s})]=\lambda({\bf s})$, implying that the bias of $\hat\lambda_{h,{\cal L}^\perp}({\bf s})$ also disappears as $h\rightarrow\infty$. By (\ref{eq:variance kernel estimator of the first-order substationarity}), there is $$\eqalign{ {\rm V}[\hat\lambda_{h,{\cal L}^\perp}(y)]=&\left\{\int_0^\omega {1\over\sqrt{2\pi}h}e^{-{(y'-y)^2\over 2h^2}}dy'\right\}^{-2}\int_0^\omega\int_0^\omega {1\over 2\pi h^2}e^{-{(y'-y)^2+(y''-y)^2\over 2h^2}}\lambda(y')\lambda(y'')\cr &\times\left\{{1\over z}\int_0^z \{g[(0,y'),(x'',y'')]-1\}dx''\right\}dy'dy''\cr &+{1\over z}\left\{\int_0^\omega {1\over\sqrt{2\pi}h}e^{-{(y'-y)^2\over 2h^2}}dy'\right\}^{-2}\int_0^\omega {1\over\sqrt{2\pi}h}e^{-{(y'-y)^2\over 2h^2}}\lambda(y')dy'. }$$ If all conditions of Theorem \ref{thm:functional central limit theorem strong mixing} hod, then $\lim_{x''\rightarrow\infty}g[(0,y'),(x'',y'')]-1=0$. Thus, the first term of above goes to $0$ as $z\rightarrow\infty$. Further, we conclude the second term goes to zero if $zh\rightarrow\infty$. Thus, we have the conclusion of Theorem \ref{thm:unbiasedness of estimators if h is small}.
As ${\cal L}$ is also unknown, we should have a way to estimate ${\cal L}$ in the usage of $\hat\lambda_{h,{\cal L}^\perp}({\bf s})$. Let ${\cal L}={\rm span}\{{\bf u}_1,\cdots,{\bf u}_r\}$, where ${\bf u}_1,\cdots,{\bf u}_r$ are orthonormal vectors of ${\cal L}$. Then, it is enough to provide an estimator of $\{{\bf u}_1,\cdots,{\bf u}_r\}$ is our method. If $r=0$, then ${\cal N}$ is not substationary in any linear subspace of $\mathbb{R}^d$. If $r=d$, then ${\cal N}$ is stationary in the entire $\mathbb{R}^d$. Otherwise, ${\cal N}$ is substationary in ${\cal L}$ but nonstationary in $\mathbb{R}^d$. Note that ${\cal L}$ can be represented by an orthogonal projection ${\bf Q}$ in $\mathbb{R}^d$. Let $\mathscr{Q}$ be the collection of the orthogonal projections from $\mathbb{R}^d$ to an $r$-dimensional linear subspace. Estimation of ${\cal L}$ is equivalent to estimation of ${\bf Q}\in\mathscr{Q}$. Let \begin{equation} \label{eq:likelihood function} \ell[\lambda({\bf s})]=\sum_{i=1}^n \log\lambda({\bf s})-\int_{\cal S}\lambda({\bf s})d{\bf s} \end{equation} be the loglikelihood function of ${\cal N}({\cal S})$ if ${\cal N}$ is Poisson. Then, $\ell[\lambda({\bf s})]$ can be treated as the composite loglikelihood of ${\cal N}({\cal S})$ if ${\cal N}$ is non-Poisson \cite{guan2010}. Therefore, we can estimate ${\bf Q}$ by \begin{equation} \label{eq:estimator of the space} \hat{\bf Q}_h=\mathop{\arg\!\max}_{{\bf Q}\in\mathscr{Q}}\ell[\hat\lambda_{h,{\cal L}^\perp}({\bf s})]. \end{equation}
To apply (\ref{eq:estimator of the space}), we need to provide a way to determine the best $h$ in $\hat{\bf Q}_h$, where we recommend using the generalized cross validation (GCV) approach \cite{golub1979}.
\section{Simulation} \label{sec:simulation}
We carried out a simulation study to evaluate the performance of $\hat\lambda_{h,{\cal L}^\perp}({\bf s})=\hat\lambda_{h,{\cal L}^\perp}({\bf s}_{{\cal L}^\perp})$ given by (\ref{eq:kernel estimator of the first-order substationarity}). We simulated realizations from Poisson and Poisson cluster SPPs in a rectangle region ${\cal S}=[0,z]\times[0,\omega]$, the region used in Example 1. We chose $\omega=1$ in our simulation. We selected these processes because they are popular in modeling ecological, environmental, geographical data. In both processes, we chose the first-order intensity function as \begin{equation} \label{eq:first-order intensity function in simulation} \lambda({\bf s})={100\Gamma^2(a)\over\Gamma(2a)}y^{a-1}(1-y)^{a-1} \end{equation} for a selected $a\ge 1$ such that we always had $\kappa={\rm E}[N({\cal S})]=100z$. Note that $\lambda(s)/100$ is the PDF of $Beta(a,a)$ distribution. We chose $a=1.0,1.5,2.0,2.5,3.0$ in our simulations. If $a=1$, then ${\cal N}$ was stationary in the entire $\mathbb{R}^2$; otherwise, it was only substationary in ${\cal L}=\{(x,0): x\in\mathbb{R}\}$. Since ${\cal L}$ might be unknown, we also evaluated the performance of $\hat{\cal L}$, the estimator of ${\cal L}$ given by (\ref{eq:estimator of the space}).
To obtain a Poisson SPP, we first generated the number of points from the $Poisson(\kappa)$ distribution and then identically and independently generated the locations of these points. The horizontal values of these points were generated from the uniform distribution on $[0,z]$. The vertical values of these points were generated from the $Beta(a,a)$ distribution. To obtain a Poisson cluster SPP, we first generated their parent points from a Poisson SPP with its first-order intensify function equal to $\lambda({\bf s})/\gamma$ by the same method for the Poisson SPP. After parent points were derived, we generated offspring points. Each parent point generated $Poisson(\gamma)$ offspring points independently. The position of each offspring point relative to its parent point was defined as a radially symmetric Gaussian random variable with a standard deviation $\sigma$. We chose $\gamma=5$ and $\sigma=0.02$ in all the cases of Poisson cluster SPPs that we studied.
We studied two cases in the implementation of $\hat\lambda_{h,{\cal L}}({\bf s})$. In the first case, we assumed that ${\cal L}$ was known such that we could directly apply (\ref{eq:kernel estimator of the first-order substationarity}). We chose $K_{h,{\cal L}^\perp}(y)=\phi(y/h)/h$ as the density of $N(0,h^2)$. Then, we had $C_{h,{\cal L}^\perp}(y)=z\{\Phi[(\omega-y)/h]-\Phi(-y/h)\}$, where $\Phi$ is the CDF of $N(0,1)$, indicating that \begin{equation} \label{eq:estimator known L simulation} \hat\lambda_{h,{\cal L}^\perp}(y)=\left\{z\left[\Phi({\omega-y\over h})-\Phi(-{y\over h})\right]\right\}^{-1}\sum_{i=1}^n {1\over\sqrt{2\pi}h}e^{-{(y_i-y)^2\over 2h^2}}, 0<y<\omega. \end{equation} In the second case, we assumed that ${\cal L}$ was unknown. We also needed to estimate ${\cal L}$. Note that any one-dimensional linear subspace of $\mathbb{R}^2$ can be expressed as \begin{equation} \label{eq:linear subspace with angle} {\cal L}_{\theta}=\{(u\cos\theta,u\sin\theta):u\in\mathbb{R}\}, \theta\in[-{\pi\over 2},{\pi\over 2}), \end{equation} indicating that its vertical space is \begin{equation} \label{eq:vertical linear subspace with angle} {\cal L}_{\theta}^\perp=\{(-v\sin\theta,v\cos\theta):v\in\mathbb{R}\}, \theta\in[-{\pi\over 2},{\pi\over 2}). \end{equation} We chose $K_{h,{\cal L}_\theta^\perp}(v)=\phi(v/h)/h$ on ${\cal L}_{\theta}^\perp$.
To apply (\ref{eq:kernel estimator of the first-order substationarity}), we computed the analytic expression of $C_{h,{\cal L}_\theta^\perp}(v)$. If $\theta=0$, then $$C_{h,{\cal L}_\theta^\perp}(v)=z\left[\Phi({\omega-v\over h})-\Phi(-{v\over h})\right].$$ If $\theta=-\pi/2$, then $$C_{h,{\cal L}_\theta^\perp}(v)=\omega\left[\Phi({z-v\over h})-\Phi(-{v\over h})\right].$$ If $0<\theta<\pi/2$, then $$\eqalign{ C_{h,{\cal L}_\theta^\perp}(v)=&\left({z\over\cos\theta}+{v\over\sin\theta\cos\theta}\right)\left\{\Phi\left({(\omega\cos\theta-z\sin\theta)\wedge 0-v\over h}\right)-\Phi\left({-z\sin\theta-v\over h} \right)\right\}\cr &+{h\over\sin\theta\cos\theta}\left[\phi\left({-z\sin\theta-v\over h}\right)-\phi\left({(\omega\cos\theta-z\sin\theta)\wedge 0-v\over h}\right)\right]\cr &+\left({z\over\cos\theta}\wedge{\omega\over\sin\theta}\right)\left\{\Phi\left({(\omega\cos\theta-z\sin\theta)\vee 0-v\over h}\right)-\Phi\left({(\omega\cos\theta-z\sin\theta)\wedge 0-v\over h}\right)\right\}\cr &+\left({\omega\cos\theta-v\over\sin\theta\cos\theta}\right)\left\{\Phi\left({\omega\cos\theta-v\over h}\right)-\Phi\left[{(\omega\cos\theta-z\sin\theta)\vee 0-v\over h}\right]\right\}\cr &-{h\over\sin\theta\cos\theta}\left\{\phi\left({(\omega\cos\theta-z\sin\theta)\vee 0-v\over h}\right)-\phi\left({\omega\cos\theta-v\over h}\right) \right\}.\cr },$$ where $-z\sin\theta\le v\le \cos\theta$. If $-\pi/2<\theta<0$, then $$\eqalign{ C_{h,{\cal L}_\theta^\perp}(v)=&-{v\over\sin\theta\cos\theta}\left\{\Phi\left({(-z\sin\theta)\wedge(\omega\cos\theta)-v\over h}\right)-\Phi\left(-{v\over h}\right)\right\}\cr &-{h\over\sin\theta\cos\theta}\left\{\phi\left(-{v\over h}\right)-\phi\left({(-z\sin\theta)\wedge(\omega\cos\theta)-v\over h}\right)\right\}\cr &+\left[{z\over\cos\theta}\wedge\left(-{\omega\over\sin\theta}\right)\right]\left\{\Phi\left({(-z\sin\theta)\vee(\omega\cos\theta)-v\over h}\right)-\Phi\left({(-z\sin\theta)\wedge(\omega\cos\theta)-v\over h}\right)\right\}\cr &+{z\sin\theta-\omega\cos\theta+v\over\sin\theta\cos\theta}\left\{\Phi\left({-z\sin\theta+\omega\cos\theta-v\over h}\right)-\Phi\left({(-z\sin\theta)\vee(\omega\cos\theta)-v\over h}\right)\right\}\cr &+{h\over\sin\theta\cos\theta}\left\{\phi\left({(-z\sin\theta)\vee(\omega\cos\theta)-v\over h}\right)-\phi\left({-z\sin\theta+\omega\cos\theta-v\over h}\right)\right\},\cr }$$ where $0\le v\le -z\sin\theta+\cos\theta$. For a given $\theta\in[-\pi/2,\pi/2)$, we calculated $\hat\lambda_{h,{\cal L}_\theta^\perp}({\bf s}_{{\cal L}_\theta^\perp})$ by (\ref{eq:kernel estimator of the first-order substationarity}) as \begin{equation} \label{eq:estimate linear subspace with angle} \hat\lambda_{h,{\cal L}_\theta^\perp}(v)=C_{h,{\cal L}_\theta^\perp}^{-1}(v)\sum_{i=1}^n {1\over\sqrt{2\pi}h}e^{-{(y_i\cos\theta -x_i\sin\theta-v)^2\over 2h^2}} \end{equation} for $(-z\sin\theta)\wedge 0\le v\le \cos\theta+(-z\sin\theta)\vee0$, where points were given by ${\bf s}_i=(x_i,y_i)$ for $i=1,\cdots,n$. We calculated $\hat\theta_h$ by (\ref{eq:estimator of the space}) and (\ref{eq:estimate linear subspace with angle}). We defined $\mathscr{Q}=\{\theta:{\bf Q}_{\theta}\}$ in the implementation of (\ref{eq:estimator of the space}), where ${\bf Q}_{\theta}{\bf s}=y\cos\theta-x\sin\theta$ was an orthogonal project from $\mathbb{R}^2$ to ${\cal L}_{\theta}$. The estimator $\hat\theta_h$ was the value of $\theta$ corresponding to $\hat{\bf Q}_h$ given by (\ref{eq:estimator of the space}). With $\hat\theta_h$, we calculated the value of $\hat\lambda_{h,\hat{\cal L}^\perp}(v)$ with $\hat{\cal L}={\cal L}_{\hat\theta_h}$, which was treated as the estimator of $\lambda({\bf s})$ under substationarity with an unknown ${\cal L}$. It was compared with $\hat\lambda_{h,{\cal L}^\perp}(y)$, the estimator of $\lambda({\bf s})$ with a known ${\cal L}$.
We evaluated the performance of the MSE (mean squares error) of $\hat\theta_h$ and the MISE (mean integrated square error) of $\hat\lambda_{h,{\cal L}_\theta^\perp}(v)$ for selected $a$, $z$, and $h$. The performance of $\hat\lambda_{h,{\cal L}_\theta^\perp}(v)$ was compared with that of $\hat\lambda_h({\bf s})$ given by (\ref{eq:general kernel estimator of the first-order}) and $\hat\lambda$ given by (\ref{eq:estimator of intensity under stationarity}), where we chose $K({\bf s})$ as the density of the standard bivariate normal distribution in the computation of $\hat\lambda_h({\bf s})$.
\begin{table} \caption{\label{tab:performance of estimator of theta} Simulations (with 1,000 replications) for root MSEs of $\hat\theta_h$ (given by degrees) with respect to selected $a$, $z$, and $h$ in the Poisson and Poisson cluster processes.} \begin{center} \begin{tabular}{cccccccccc}\hline
& & \multicolumn{4}{c}{$h$ for Poisson} & \multicolumn{4}{c}{$h$ for Poisson Cluster}\\ $a$ & $z$ & $0.01$ & $0.02$ & $0.05$ & $0.1$ & $0.01$ & $0.02$ & $0.05$ & $0.1$ \\\hline $1.5$&$1$&$4.24$&$4.14$&$4.61$&$5.12$&$4.51$&$4.52$&$4.84$&$5.22$\\ &$2$&$4.18$&$3.87$&$3.85$&$3.23$&$4.07$&$4.19$&$4.77$&$4.73$\\ &$5$&$2.84$&$2.22$&$1.12$&$1.03$&$4.51$&$4.13$&$3.64$&$3.06$\\ &$10$&$1.55$&$0.45$&$0.32$&$0.32$&$4.83$&$4.25$&$3.07$&$1.59$\\ $2.0$&$1$&$3.75$&$4.28$&$4.66$&$4.54$&$4.24$&$4.50$&$4.92$&$5.04$\\ &$2$&$3.39$&$2.83$&$2.63$&$2.58$&$3.82$&$3.90$&$4.07$&$4.41$\\ &$5$&$1.27$&$0.82$&$0.56$&$0.50$&$3.20$&$2.98$&$2.06$&$1.73$\\ &$10$&$0.32$&$0.20$&$0.19$&$0.20$&$1.81$&$1.67$&$0.58$&$0.72$\\ $2.5$&$1$&$3.72$&$4.03$&$4.07$&$4.10$&$4.03$&$4.49$&$4.43$&$4.90$\\ &$2$&$2.78$&$2.78$&$2.05$&$1.93$&$3.73$&$3.90$&$3.66$&$3.90$\\ &$5$&$0.77$&$0.54$&$0.43$&$0.39$&$2.18$&$1.94$&$1.22$&$1.25$\\ &$10$&$0.22$&$0.19$&$0.13$&$0.16$&$0.89$&$0.62$&$0.39$&$0.42$\\ $3.0$&$1$&$3.78$&$4.04$&$4.05$&$3.73$&$4.08$&$4.30$&$4.84$&$4.86$\\ &$2$&$2.97$&$2.45$&$1.70$&$1.62$&$3.56$&$3.54$&$3.79$&$3.47$\\ &$5$&$0.69$&$0.48$&$0.37$&$0.37$&$1.57$&$1.52$&$0.92$&$0.88$\\ &$10$&$0.24$&$0.15$&$0.11$&$0.14$&$0.48$&$0.49$&$0.34$&$0.35$\\\hline \end{tabular} \end{center} \end{table}
We simulated $1000$ realizations for each selected cases. To evaluate the performance of $\hat\theta_h$, we computed its MSE value by $\sum_{i=1}^{1000}\hat\theta_{hi}^2/1000$, where $\hat\theta_{hi}$ was the value of $\hat\theta_h$ in the $i$th realization (Table \ref{tab:performance of estimator of theta}). We did not put the case when $a=1$ in the table as $\theta$ was not well-defined. The results showed that the root MSEs of $\hat\theta_h$ were all close to $0$, indicating that the estimator was accurate. The MSEs of $\hat\theta_h$ decreased as $z$ increased. This was interpreted by Theorem \ref{thm:unbiasedness of estimators if h is small}. The MSEs decreased as $a$ increased since the strength of nonstationarity increased as $a$ became large. For the same $a$ and $z$ values, the MSEs of $\hat\theta_h$ was also affected by the bandwidth $h$ in the kernel approach is always an important issue to be investigated. In all the cases that we studied, the MSEs of $\hat\theta_h$ in the Poisson SPPs was always lower than those in the Poisson cluster SPPs. This was expected as for the same $\kappa$ value the number of independent clusters in the Poisson cluster SPPs was lower than the number of independent points in the Poisson SPPs.
\begin{table} \caption{\label{tab:performance of estimator of intensity function}Simulations (with 1,000 replications) of root MSEs of $\hat\lambda_{h,{\cal L}^\perp}(y)$, $\hat\lambda_{h,{\cal L}_{\hat\theta_h}^\perp}(v)$, $\hat\lambda_h({\bf s})$, and $\hat\lambda$ with respected to selected $a$, $z$, and $h$ in the Poisson and Poisson cluster processes.} \begin{center} \begin{tabular}{ccccccccccc}\hline & & & \multicolumn{4}{c}{Poisson} & \multicolumn{4}{c}{Poisson Cluster}\\ $a$ & $z$ & $h$ & $\hat\lambda_{h,{\cal L}^\perp}(y)$ & $\hat\lambda_{h,\hat{\cal L}^\perp}(v)$ & $\hat\lambda_h({\bf s})$ & $\hat\lambda$ & $\hat\lambda_{h,{\cal L}^\perp}(y)$ & $\hat\lambda_{h,\hat{\cal L}^\perp}(v)$ & $\hat\lambda_h({\bf s})$ & $\hat\lambda$ \\\hline $1$&$1$&$0.05$&$25.13$&$36.95$&$61.17$&$10.33$&$54.43$&$60.22$&$133.01$&$23.24$\\
& &$0.10$&$17.31$&$17.95$&$32.19$&$9.20$&$39.23$&$40.85$&$70.90$&$21.27$\\
&$2$&$0.05$&$17.48$&$19.45$&$59.72$&$6.55$&$39.44$&$45.13$&$132.72$&$16.81$\\
& &$0.10$&$13.16$&$13.95$&$31.60$&$8.03$&$29.17$&$31.69$&$70.90$&$16.57$\\
&$5$&$0.05$&$11.21$&$13.56$&$59.10$&$4.87$&$24.21$&$31.22$&$130.30$&$9.50$\\
& &$0.10$&$8.25$&$9.82$&$31.17$&$4.83$&$18.45$&$22.35$&$68.69$&$10.81$\\
&$10$&$0.05$&$7.45$&$10.96$&$58.60$&$2.49$&$16.32$&$25.29$&$129.79$&$6.13$\\
& &$0.10$&$5.64$&$7.91$&$30.42$&$3.26$&$13.29$&$18.42$&$67.95$&$7.64$\\ $2$&$1$&$0.05$&$24.08$&$26.00$&$59.14$&$45.91$&$54.52$&$57.75$&$130.70$&$50.25$\\
& &$0.10$&$21.31$&$21.90$&$33.93$&$45.66$&$40.20$&$41.14$&$70.03$&$49.43$\\
&$2$&$0.05$&$17.58$&$19.94$&$58.17$&$45.18$&$38.16$&$43.02$&$128.43$&$47.38$\\
& &$0.10$&$17.23$&$19.44$&$32.98$&$45.25$&$28.95$&$32.67$&$69.97$&$47.12$\\
&$5$&$0.05$&$12.00$&$12.76$&$57.96$&$44.95$&$24.67$&$29.93$&$127.09$&$45.75$\\
& &$0.10$&$14.40$&$15.67$&$32.44$&$44.91$&$21.59$&$24.55$&$67.69$&$45.96$\\
&$10$&$0.05$&$8.92$&$9.33$&$57.41$&$44.82$&$17.15$&$19.44$&$126.20$&$45.24$\\
& &$0.10$&$13.64$&$14.12$&$32.28$&$44.83$&$16.93$&$18.46$&$66.67$&$45.21$\\ $3$&$1$&$0.05$&$23.95$&$26.08$&$59.25$&$66.17$&$50.81$&$55.22$&$129.91$&$68.65$\\
& &$0.10$&$21.88$&$22.67$&$34.36$&$66.12$&$41.83$&$43.57$&$71.43$&$69.77$\\
&$2$&$0.05$&$17.23$&$18.96$&$58.47$&$65.83$&$37.31$&$42.61$&$126.97$&$67.28$\\
& &$0.10$&$19.25$&$21.20$&$33.38$&$66.04$&$31.94$&$34.50$&$68.15$&$67.71$\\
&$5$&$0.05$&$11.32$&$12.00$&$56.88$&$65.61$&$24.30$&$27.19$&$125.90$&$66.37$\\
& &$0.10$&$15.21$&$15.92$&$32.49$&$65.61$&$20.87$&$22.39$&$66.81$&$66.07$\\
&$10$&$0.05$&$7.97$&$8.27$&$56.77$&$65.53$&$16.56$&$18.32$&$124.36$&$65.85$\\
& &$0.10$&$14.71$&$15.01$&$32.22$&$65.55$&$18.67$&$19.71$&$66.36$&$65.89$\\\hline \end{tabular} \end{center} \end{table}
We also evaluated the performance of four different estimators of the first-order intensity functions. Although we studied all of the selected cases in our simulations, we only put some of them in Table \ref{tab:performance of estimator of intensity function} to reduce the size of the table. We used $\hat\lambda_{h,{\cal L}^\perp}(y)$ to represent the case when $\theta$ was known. We used $\hat\theta_{h,\hat{\cal L}^\perp}(v)$ to represented the case when $\theta$ was unknown. We used $\hat\lambda_h({\bf s})$ to represent the case when substationarity was not taken into account. We used $\hat\lambda$ to represent the case when stationarity was assumed. All of the minimum MSEs were reached by $\hat\lambda$ when $a=1$ as the SPPs were stationary in this case. The MSEs of $\hat\lambda$ increased in $a$ since the strength of nonstationarity became large as $a$ increased. For the same $a$ and $h$ values, the MSEs of $\hat\lambda_{h,{\cal L}^\perp}(y)$ and $\hat\lambda_{h,\hat{\cal L}^\perp}(v)$ decreased in $z$. We interpreted this by Theorem \ref{thm:unbiasedness of estimators if h is small}. The MSEs of $\hat\lambda_h({\bf s})$ did not vary significantly as $z$ changed since the size of the region was not a critical issue in its computation. For all of the cases with $a>1$ that we studied, the MSEs of $\hat\lambda_{h,{\cal L}^\perp}(y)$ and $\hat\lambda_{h,\hat{\cal L}^\perp}(v)$ were lower than those of $\hat\lambda_{h}({\bf s})$ and $\hat\lambda$, indicating that efficiency was gained by accounting for substationarity.
\section{Application} \label{sec:application}
We applied our approach to the {\it Alberta Forest Wildfire} data. The {\it Alberta Forest Wildfire} data consisted of forest wildfire activities occurred in Alberta, Canada, from 1931 to 2012. The Canadian Alberta Forest Service initiated the modern era of wildfire record keeping in 1931. Since 1996, paper-based wildfire information was no long retained. The wildfire historical data were entered at the field level on the Fire Information Resource Evaluation System (FIREs), which can be freely downloaded from the internet. We collected the historical forest wildfire data from 1996 to 2010 within a rectangle spanned from $117$ longitude West to $110$ longitude West in the horizontal direction and from $54.7$ latitude North to $58$ latitude North in the vertical direction (Figure \ref{fig:fire locations in a rectangle region}(a)). We treated the rectangle as the study region in our approach. The region contained $8125$ wildfire occurrences with all of the three greatest wildfires occurred in Alberta forests during the $15$ years period. The greatest wildfire occurred in 2002 at $111.8$ longitude West and $55.5$ latitude North with area burned $2388.67{\rm km}^2$. The second greatest wildfire occurred in $1998$ at $116.5$ longitude West and $54.7$ latitude North with area burned $1631.38{\rm km}^2$. The third greatest wildfire occurred in $1998$ at $114.3$ longitude West and $47.5$ latitude West with area burned $1554.5{\rm km}^2$. The total burned area in the region was over $60\%$ of the total burned area in the entire region.
\begin{figure}
\caption{ Wildfires locations and estimates of intensity under nonstationarity in Alberta Forests from 1996 to 2010 in the selected region, where bandwidths were given by degrees.}
\label{fig:fire locations in a rectangle region}
\end{figure}
The study region contained a large portion of boreal forests in Alberta, which was dominated in plain areas. A small portion of boreal forests of Alberta was in the mountain areas, located in the southwestern region of Alberta. We focused our study on the plain areas since tree densities and topographic conditions were significantly different between the mountain and plain areas.
The geographical distribution of boreal forest wildfires is considered as a major dominant disturbance in the high latitude area of the North Hemisphere \cite{podur2002}. It has been pointed out that wildfire activities in boreal forest are significantly affected by latitude but not by longitude \cite{xiao2007}. It is expected to have low numbers of wildfire occurrences with high values of area burned in the north than those in the south \cite{zhang2014a}, indicating that substationarity might be assumed along the longitude. To confirm this, we calculated the estimates of $\lambda({\bf s})$ with the standard bivariate normal kernel via (\ref{eq:general kernel estimator of the first-order}) under nonstationarity. We used a few bandwidth values and found the results were not stable (Figure \ref{fig:fire locations in a rectangle region}(b), 2(c), and 2(d)). However, all of our results showed that the estimates of the intensity were high in the south but low in the north.
We assumed fire occurrences were substationary in a linear space of $\mathbb{R}^d$, where the linear space was ${\cal L}={\cal L}_{\theta}$ given by (\ref{eq:linear subspace with angle}). We calculated $\hat\theta_h$ with a normal kernel in (\ref{eq:estimator of the space}). We treated $\hat\theta_h$ as an estimator of $\theta$ for a given $h$. We compared values of $\hat\theta_h$ with various choices of $h$. We found that $\hat\theta_h$ was reliable. For instance, we got $\hat\theta_h=-0.002$ (given by arc degree, same as the following) if $h=0.01$, $\hat\theta_h=-0.001$ if $h=0.02$, $\hat\theta_h=-0.003$ if $h=0.05$, and $\hat\theta_h=-0.007$ if $h=0.1$. Therefore, we had $\hat\theta_h\approx0$, indicating that we might simply choose ${\cal L}={\cal L}_0$ in our estimation. To investigate this issue, we compared the values of $\ell[\hat\lambda_{h,\hat{\cal L}^\perp}({\bf s})]$ and $\ell[\hat\lambda_{h,{\cal L}_0^\perp}({\bf s})]$ with selected $h$ in (\ref{eq:likelihood function}). The values of $\ell[\hat\lambda_{h,\hat{\cal L}^\perp}({\bf s})]-\ell[\hat\lambda_{h,{\cal L}_0^\perp}({\bf s})]$ were $1.66$, $0.89$, $3.38$, and $5.3$ when $h$ were $0.01$, $0.02$, $0.05$, and $0.1$, respectively. Comparing these values with the differences of loglikelihood functions affected by $h$, which were often more than a few hundred, we concluded that the values of $\ell[\hat\lambda_{h,\hat{\cal L}^\perp}({\bf s})]-\ell[\hat\lambda_{h,{\cal L}_0^\perp}({\bf s})]$ could be ignored. Therefore, we could use $\theta=0$ in the computation of the estimates of the first-order intensity function.
\begin{figure}
\caption{ Estimates of the first-order intensity in the {\it Alberta Forest Wildfire} data under substationarity along the longitude.}
\label{fig:fire intensity under substationarity}
\end{figure}
Simply using $\theta=0$, we obtained ${\cal L}_0=\{(x,0):x\in\mathbb{R}\}$. We used ${\cal L}_0$ to estimate the first-order intensity function of wildfire occurrences under substationarity. We computed values of $\hat\lambda_{h,{\cal L}_0^\perp}(y)$ with various choices of $h$. All of the results were close (e.g. as those displayed by Figure \ref{fig:fire intensity under substationarity}), indicating that our approach was reliable. We found that the intensity of wildfire occurrences was almost maximized at $55.8$ latitude North. It decreased fast to the north but slowly to the south. The north part was consistent with our previous conclusion but the south part was a concern. We studied the reason by looking at the terrestrial ecozones. We found that ecozones in the south of the study region was dominated by grassland, which might affect the occurrences of forest wildfires \cite{pitman2007,xiao2007}.
\section{Discussion} \label{sec:discussion}
In this article, we propose the concept of substationarity and provide a semiparametric method to estimate the first-order intensity function of a spatial point process. The method is modified from the classical kernel density estimation for random variables. Classical kernel density estimation is formulated under the assumption that sampling data are collected identically and independently from a continuous distribution. This assumption is violated because the dependence structure is often present in spatial point data. A common way to account for dependence structures in SPPs is to use the second-order intensity functions. As specific relationship between the first-order and the second-order intensity functions can be formulated under the concept of SOIRs, it is possible to have methods to account for both the first-order and the second-order intensity functions simultaneously under the concept of SOIRs.
Although we have only discussed the kernel-based approach, two other nonparametric or semiparmetric approaches may also be considered. The {\it local polynomial} approach is modified from the kernel approach \cite{cleveland1988,fan1993}. It is based on the idea of the weighted localized polynomial regression, where the weights are determined by kernel functions of explanatory variables. The {\it smoothing spline} approach estimates a smooth function by minimizing a penalized likelihood function \cite{gu2013,kimeldorf1971,wahba1990}. The penalized likelihood function has two terms. The negative loglikelihood term controls the goodness-of-fit value. The penalty term controls the smoothness value. Both the local polynomial and the smooth spline approaches can be used to estimate the intensity functions of SPPs under substationarity.
As a relative concept, nonsubstationarity is also an important concept for spatial point data. A nonsubstationarity approach must be adopted if assumptions of substationarity are violated. Based on the concept of substationarity, a few possible ways may be proposed for nonsubstationarity. An easy way is to borrow the idea of additive models in nonparametric statistics \cite{friedman1981,hastie1990}. Assume intensity functions of a nonsubstationary SPP can be expressed by the sum of intensity functions of a few substationary SPPs. If the linear space of the substationary SPPs are different such that their intersection only contains the origin, then the additive model provides nonsubstatioary intensity functions. The structure of additive models for nonsubstationarity in SPPs is essentially different from the structure of additive models in nonparametric statistics. Additive models in SPPs attempt to model additivity by intensity functions. Additive model in nonparametric statistics attempt to model additivity by mean structures. Additive models in SPPs contain dependence structures but additive models in nonparametric statistics do not. This is an interesting research question to be studied in the future.
\end{document} | arXiv |
On the compensator of the default process in an information-based model
Matteo Ludovico Bedini ORCID: orcid.org/0000-0002-8735-95551,
Rainer Buckdahn2,3 &
Hans-Jürgen Engelbert4
Probability, Uncertainty and Quantitative Risk volume 2, Article number: 10 (2017) Cite this article
This paper provides sufficient conditions for the time of bankruptcy (of a company or a state) for being a totally inaccessible stopping time and provides the explicit computation of its compensator in a framework where the flow of market information on the default is modelled explicitly with a Brownian bridge between 0 and 0 on a random time interval.
One of the most important objects in a mathematical model for credit risk is the time τ (called default time) at which a certain company (or state) bankrupts. Modelling the flow of market information concerning a default time is crucial and in this paper we consider a process, β=(β t , t≥0), whose natural filtration \(\mathbb {F}^{\beta }\) describes the flow of information available for market agents about the time at which the default occurs. For this reason, the process β will be called the information process. In the present paper, we define β to be a Brownian bridge between 0 and 0 of random length τ:
$$\beta_{t}:=W_{t}-\frac{t}{\tau\vee t}W_{\tau\vee t},\quad t \geq 0, $$
where W=(W t , t≥0) is a Brownian motion independent of τ.
In this paper, the focus is on the classification of the default time with respect to the filtration \(\mathbb {F}^{\beta }\) and our main result is the following: If the distribution of the default time τ admits a continuous density f with respect to the Lebesgue measure, then τ is a totally inaccessible stopping time and its compensator K=(K t , t≥0) is given by
$$K_{t}=\int_{0}^{t\wedge\tau}\frac{f(s)}{\int_{s}^{\infty} v^{\frac{1}{2}}\,(2\pi s\,\left(v-s\right))^{-\frac{1}{2}}\,f(v)\mathrm{d} v}\,\mathrm{d} L^{\beta}\left(s,0\right), $$
where L β(t,0) is the local time of the information process β at level 0 up to time t.
Knowing whether the default time is a predictable, accessible, or totally inaccessible stopping time is very important in a mathematical credit risk model. A predictable default time is typical of structural credit risk models, while totally inaccessible default times are one of the most important features of reduced-form credit risk models. In the first framework, market agents know when the default is about to occur, while in the latter default occurs by surprise. The fact that financial markets cannot foresee the time of default of a company makes the reduced-form models well accepted by practitioners. In this sense, totally inaccessible default times seem to be the best candidates for modelling times of bankruptcy. We refer, among others, to the papers of Jarrow and Protter (2004) and of Giesecke (2006) on the relations between financial information and the properties of the default time, and also to the series of papers of Jeanblanc and Le Cam (2008,2009,2010). It is remarkable that in our setting the default time is a totally inaccessible stopping time under the common assumption that it admits a continuous density with respect to the Lebesgue measure. Both the hypothesis that the default time admits a continuous density and its consequence that the default occurs by surprise are standard in mathematical credit risk models, but in the information-based approach there is the additional feature of an explicit model for the flow of information which is more sophisticated than the standard approach. There, the available information on the default is modelled through \(\left (\mathbb {I}_{\left \{ \tau \leq t\right \} },\,t\geq 0\right)\), the single-jump process occurring at τ, meaning that people only know if the default has or has not occurred. Financial reality can be more complex and there are actually periods in which default is more likely to happen than in others. In the information-based approach, periods of fear of an imminent default correspond to situations where the information process is close to 0, while periods when investors are relatively sure that the default is not going to occur immediately correspond to situations where β t is far from 0.
The paper is organized as follows. In the section "The information process and its basic propertiesThe information process andits basic properties", we recall the definition and the main properties of the information process. In the section "The compensator of the default time", we state and prove Theorem 3.2, which is the main result of the paper. In Appendix Appendix A, we provide the properties of the local time associated with the information process. In Appendix Appendix B, we give the proofs of some auxiliary lemmas. Finally, in Appendix Appendix C, for the sake of easy reference, we recall the so-called Laplacian approach developed by Meyer (1966) (see, e.g., his book (Meyer 1966)) for computing the compensator of a right-continuous potential of class (D). It is an important ingredient of the approach adopted in this note to determine the compensator of the \(\mathbb {F}^{\beta }\)-submartingale \(\left (\mathbb {I}_{\left \{ \tau \leq t\right \} },\,t\geq 0\right)\).
The idea of modelling the information about the default time with a Brownian bridge defined on a stochastic interval was introduced in the thesis (Bedini 2012). The definition of the information process β, the study of its basic properties, and an application to the problem of pricing a Credit Default Swap (one of the most traded derivatives in the credit market) have also recently appeared in the paper (Bedini et al. 2016).
Non-trivial and sufficient conditions for making the default time a predictable stopping time will be considered in another paper, (Bedini and Hinz 2017). Other topics related to Brownian bridges on stochastic intervals (which will not be considered in this paper) are concerned with the problem of studying the progressive enlargement of a reference filtration \(\mathbb {F}\) by the filtration \(\mathbb {F}^{\beta }\) generated by the information process and further applications to Mathematical Finance.
The information process and its basic properties
We start by recalling the definition and the basic properties of a Brownian bridge between 0 and 0 of random length. The material in this section gives a résumé of some of the results obtained in the paper (Bedini et al. 2016), to which we shall refer for the proofs and more details on the basic properties of such a process.
If \(A\subseteq \mathbb {R}\) (where \(\mathbb {R}\) denotes the set of real numbers), then the set A + is defined as \(A_{+}:=A\cap \{x\in \mathbb {R}:x\geq 0\}\). If E is a topological space, then \(\mathcal {B}(E)\) denotes the Borel σ-algebra over E. The indicator function of a set A will be denoted by \(\mathbb {I}_{A}\). A function \(f:\mathbb {R}\rightarrow \mathbb {R}\) will be said to be càdlàg if it is right-continuous with limits from the left.
Let \(\left (\Omega,\mathcal {F},\mathbf {P}\right)\) be a complete probability space. We denote by \(\mathcal {N}_{P}\) the collection of P-null sets of \(\mathcal {F}\). If \(\mathcal {L}\) is the law of the random variable ξ we shall write \(\xi \sim \mathcal {L}\). Unless otherwise specified, all filtrations considered in the following are supposed to satisfy the usual conditions of right continuity and completeness.
Let τ:Ω→(0,+∞) be a strictly positive random time, whose distribution function is denoted by F: \(F(t):=\mathbf {P}\left (\tau \leq t\right),\;t\in \mathbb {R}_{+}\). The time τ models the random time at which some default occurs and, hereinafter, it will be called default time.
Let W=(W t , t≥0) be a Brownian motion defined on \(\left (\Omega,\mathcal {F},\mathbf {P}\right)\) and starting from 0. We shall always make use of the following assumption:
Assumption 2.1
The random time τ and the Brownian motion W are independent.
Given W and a strictly positive real number r, a standard Brownian bridge \(\beta ^{r}=\left (\beta _{t}^{r},\,t\geq 0\right)\) between 0 and 0 of length r is defined by
$$\beta^{r}_{t}=W_{t}-\frac{t}{r\vee t}W_{r\vee t},\quad t\geq 0\,. $$
For further references on Brownian bridges, see, e.g., Section 5.6.B of the book (Karatzas and Shreve 1991) by Karatzas and Shreve.
Now, we are going to introduce the definition of the Brownian bridge of random length (see (Bedini et al. 2016), Definition 3.1).
The process β=(β t , t≥0) given by
$$ \beta_{t}:=W_{t}-\frac{t}{\tau\vee t}W_{\tau\vee t},\quad t\geq 0\,, $$
will be called Brownian bridge of random length τ. We will often say that β=(β t , t≥0) is the information process (for the random time τ based on W).
The natural filtration of β will be denoted by \(\mathbb {F}^{\beta }=(\mathcal {F}_{t}^{\beta })_{t\geq 0}\):
$$\mathcal{F}_{t}^{\beta}:=\sigma\left(\beta_{s},\,0\leq s\leq t\right)\vee\mathcal{N}_{P}\,. $$
Note that according to (Bedini et al. 2016), Corollary 6.1, the filtration \(\mathbb {F}^{\beta }\) (denoted therein by \(\mathbb {F}^{P}\)) satisfies the usual conditions of right-continuity and completeness.
The law of β, conditional on τ=r, is the same as that of a standard Brownian bridge between 0 and 0 of length r (see (Bedini et al. 2016), Lemma 2.4 and Corollary 2.2). In particular, if 0<t<r, the law of β t , conditional on τ=r, is Gaussian with expectation zero and variance \(\frac {t\left (r-t\right)}{r}\):
$$\mathbf{P}\left(\beta_{t}\in\cdot\;\big|\tau=r\right)=\mathcal{N}\left(0,\frac{t\left(r-t\right)}{r}\right), $$
where \(\mathcal {N}\left (\mu,\sigma ^{2}\right)\) denotes the Gaussian law of mean μ and variance σ 2.
By p(t,·,y), we denote the density of a Gaussian random variable with mean \(y\in \mathbb {R}\) and variance t>0:
$$ p\left(t,x,y\right):=\frac{1}{\sqrt{2\pi t}}\exp\left[-\frac{\left(x-y\right)^{2}}{2t}\right],\,x\in\mathbb{R}. $$
For later use, we also introduce the functions φ t (t>0):
$$ \varphi_{t}\left(r,x\right):=\left\{ \begin{array}{ll} p\left(\frac{t\left(r-t\right)}{r},x,0\right), & 0<t<r,\ x\in\mathbb{R},\\ 0, & r\leq t,\ x\in\mathbb{R}. \end{array}\right. $$
We notice that for 0<t<r the conditional density of β t , conditional on τ=r, is just equal to the density φ t (r,·) of a standard Brownian bridge \(\beta ^{r}_{t}\) of length r at time t.
We proceed with the property that the default time τ is nonanticipating with respect to the filtration \(\mathbb {F}^{\beta }\) and the Markov property of the information process β.
Lemma 2.4
For all t>0, {β t =0}={τ≤t}, P-a.s. In particular, τ is an \(\mathbb {F}^{\beta }\)-stopping time.
See (Bedini et al. 2016), Proposition 3.1 and Corollary 3.1. □
The information process β is a Markov process with respect to the filtration \(\mathbb {F}^{\beta }\): For all 0≤t<u and measurable real functions g such that g(β u )is integrable,
$$\mathbf{E}[g(\beta_{u})|\mathcal{F}^{\beta}_{t}]=\mathbf{E}[g(\beta_{u})|\beta_{t}],\quad \mathbf{P}\text{-a.s.} $$
See Theorem 6.1 in (Bedini et al. 2016). □
As the following theorem combined with Theorem 2.5 shows, the function ϕ t defined by
$$ \phi_{t}\left(r,x\right):=\frac{\varphi_{t}\left(r,x\right)}{{\int_{\left(t,+\infty\right)}\varphi_{t}\left(v,x\right)\,\mathrm{d} F(v)}}, $$
\(\left (r,t\right)\in \left (0,+\infty \right)\times \mathbb {R}_{+}\), \(x\in \mathbb {R}\), is, for t<r, the a posteriori density function of τ on {τ>t}, conditional on β t =x.
Let t>0, \(g:\mathbb {R}^{+}\rightarrow \mathbb {R}\) be a Borel function such that E[|g(τ)|]<+∞. Then, P-a.s.
$$ \mathbf{E}\left[g(\tau)|\mathcal{F}_{t}^{\beta}\right]=g(\tau)\mathbb{I}_{\left\{ \tau\leq t\right\} }+{\int_{\left(t,+\infty\right)}g(r)}\,\phi_{t}\left(r,\beta_{t}\right)\,\mathrm{d} F(r)\mathbb{I}_{\left\{ t<\tau\right\} }. $$
See Theorem 4.1, Corollary 4.1 and Corollary 6.1 in (Bedini et al. 2016). □
Before stating the next result, which is concerned with the semimartingale decomposition of the information process, let us give the following definition:
Let B be a continuous process, \(\mathbb {F}\) a filtration, and T an \(\mathbb {F}\)-stopping time. Then, B is called an \(\mathbb {F}\)-Brownian motion stopped at T, if B is an \(\mathbb {F}\)-martingale with square variation process 〈B,B〉 t =t∧T, t≥0.
Now, we introduce the real-valued function u defined by
$$ u\left(s,x\right):=\mathbf{E}\left[\frac{\beta_{s}}{\tau-s}\mathbb{I}_{\left\{ s<\tau\right\} }\big|\beta_{s}=x\right],\quad s\in\mathbb{R}_{+},\;x\in\mathbb{R}. $$
The process b defined by
$$b_{t}:=\beta_{t}+\int_{0}^{t}u(s,\beta_{s})\,\mathrm{d} s,\quad t\geq 0\,, $$
is an \(\mathbb {F}^{\beta }\)-Brownian motion stopped at τ. The information process βis therefore an \(\mathbb {F}^{\beta }\)-semimartingale with decomposition
$$ \beta_{t}=b_{t}-\int_{0}^{t\wedge\tau}u\left(s,\beta_{s}\right)\,\mathrm{d} s,\quad t\geq 0\,. $$
The quadratic variation of the information process β is given by
$$ \left\langle \beta,\beta\right\rangle_{t}=\left\langle b,b\right\rangle_{t}=t\wedge\tau, \quad t\geq 0\,. $$
The compensator of the default time
In this section, we explicitly compute the compensator of the single-jump process with the jump occurring at τ, which will be denoted by H= (H t , t≥0):
$$ H_{t}:=\mathbb{I}_{\left\{ \tau\leq t\right\} },\quad t\geq 0. $$
The process H, called default process, is an \(\mathbb {F}^{\beta }\)-submartingale and its compensator is also known as the compensator of the \(\mathbb {F}^{\beta }\)-stopping time τ. Our main goal consists in providing a representation of the compensator of H. As we shall see below, this representation involves the local time L β(t,0) of the information process β (see Appendix Appendix A for properties of local times of continuous semimartingales and, in particular, of β). From its representation we immediately obtain that the compensator of the default process H is continuous. As a result, from the continuity of the compensator of H it follows that the default time τ is a totally inaccessible \(\mathbb {F}^{\beta }\)-stopping time.
In this section, the following assumption will always be in force.
(i) The distribution function F of τ admits a continuous density function f with respect to the Lebesgue measure λ + on \(\mathbb {R}_{+}\).
(ii) F(t)<1 for all t≥0.
The following theorem is the main result of this paper:
Suppose that Assumption 3.1 is satisfied.
(i) The process K=(K t , t≥0) defined by
$$ K_{t}:=\int_{0}^{t\wedge\tau}\frac{f(s)}{\int_{s}^{\infty}\varphi_{s}\left(v,0\right)f(v)dv}\,\mathrm{d} L^{\beta}(s,0),\quad t\geq 0\,, $$
is the compensator of the default process H. Here, L β(t,x) denotes the local time of the information process β up to t at level x.
(ii) The default time τ is a totally inaccessible stopping time with respect to the filtration \(\mathbb {F}^{\beta }\).
First, we verify statement (ii) under the supposition that (i) is true. Obviously, as L β(s,0) is continuous in s (see Lemma A.4), the process K given by (10) is continuous. Consequently, because of the well-known equivalence between this latter property and the continuity of the compensator (see, e.g., (Kallenberg 2002), Corollary 25.18), we can conclude that the default time τ is a totally inaccessible stopping time with respect to \(\mathbb {F}^{\beta }\).
Now, we prove statement (i) of the theorem. For every h>0 we define the process \(K^{h}=\left (K_{t}^{h},\,t\geq 0\right)\) by
$$\begin{array}{@{}rcl@{}} K_{t}^{h}&:=&\frac{1}{h}\int_{0}^{t}\left(\mathbb{I}_{\left\{ s<\tau\right\} }-\mathbf{E}\left[\mathbb{I}_{\left\{ s+h<\tau\right\} }|\mathcal{F}_{s}^{\beta}\right]\right)\,\mathrm{d} s\\ &=&\int_{0}^{t}\frac{1}{h}\mathbf{P}\left(s<\tau<s+h|\mathcal{F}_{s}^{\beta}\right)\,\mathrm{d} s\,,\quad\mathbf{P}\textrm{-a.s.} \end{array} $$
The proof is divided into two parts. In the first part, we prove that \(\phantom {\dot {i}\!}K_{t}-K_{t_{0}}\) is the P-a.s. limit of \(\phantom {\dot {i}\!}K_{t}^{h}-K^{h}_{t_{0}}\) as h ↓0, for every t 0,t≥0 such that 0<t 0<t. In the second part of the proof, we show that the process K is indistinguishable from the compensator of H. Auxiliary results used throughout the proof are postponed to Appendix Appendix B.
For the first part of the proof, we fix t 0,t such that 0<t 0<t and notice that
$$ \begin{aligned} K_{t}^{h}-K_{t_{0}}^{h} & =\int_{t_{0}}^{t}\frac{1}{h}\mathbf{P}\left(s<\tau<s+h|\mathcal{F}_{s}^{\beta}\right)\,\mathrm{d} s\\ &=\int_{t_{0}\wedge\tau}^{t\wedge\tau}\frac{1}{h}\left(\frac{\int_{s}^{s+h}\varphi_{s}\left(r,\beta_{s}\right)f(r)\,\mathrm{d} r}{\int_{s}^{\infty}\varphi_{s}\left(v,\beta_{s}\right)f(v)\,\mathrm{d} v}\right)\,\mathrm{d} s, \end{aligned} $$
where the last equality is a consequence of Theorem 2.6 and Definition (4) of the a posteriori density function of τ. Later, we shall verify that
$$ {\lim}_{h\downarrow 0} \int_{t_{0}\wedge\tau}^{t\wedge\tau}\frac{1}{h}\left(\frac{\int_{s}^{s+h}\varphi_{s}\left(r,\beta_{s}\right)\left[f(r)-f(s)\right]\,\mathrm{d} r}{\int_{s}^{\infty} \varphi_{s}\left(v,\beta_{s}\right)f(v)\,\mathrm{d} v}\right)\,\mathrm{d} s=0\quad \mathbf{P}\text{-a.s.} $$
So, we have to deal with the limit behaviour as h ↓0 of
$$\begin{array}{@{}rcl@{}} &&\int_{t_{0}\wedge\tau}^{t\wedge\tau}\frac{1}{h}\left(\frac{\int_{s}^{s+h}\varphi_{s}\left(r,\beta_{s}\right)\,\mathrm{d} r}{\int_{s}^{\infty} \varphi_{s}\left(v,\beta_{s}\right)f(v)\,\mathrm{d} v}\right)\,f(s)\,\mathrm{d} s\\ &&=\int_{t_{0}\wedge\tau}^{t\wedge\tau}\frac{1}{h}\left(\frac{\int_{0}^{h}\varphi_{s}\left(s+u,\beta_{s}\right)\,\mathrm{d} u}{\int_{s}^{\infty} \varphi_{s}\left(v,\beta_{s}\right)f(v)\,\mathrm{d} v}\right)\,f(s)\mathrm{d} s\\ &&=\int_{t_{0}\wedge\tau}^{t\wedge\tau}\frac{1}{h}\int_{0}^{h}p\left(\frac{su}{s+u},\beta_{s},0\right)\,\mathrm{d} u\, g(s,\beta_s)\,f(s)\,\mathrm{d} s, \end{array} $$
where we have introduced the function \(g:\left (0,+\infty \right)\times \mathbb {R}\rightarrow \mathbb {R}_{+}\) by
$$ g\left(s,x\right):=\left(\int_{s}^{\infty}\varphi_{s}\left(v,x\right)f(v)\,\mathrm{d} v\right)^{-1},\quad s>0,\,x\in\mathbb{R}\,. $$
In (14), we want to replace \(p\left (\frac {su}{s+u},\beta _{s},0\right)\) with p(u,β s ,0). To this end, we estimate the absolute value of the difference:
$$\begin{array}{@{}rcl@{}} &&\left\vert p\left(\frac{su}{s+u},x,0\right)-p(u,x,0)\right\vert\\ &&=p(u,x,0)\,\left\vert\left(\frac{s+u}{s}\right)^{\frac{1}{2}} \exp\left(-\frac{x^{2}}{2s}\right)-1\right\vert\\ &&\le{p}(u,x,0)\left[\left(\frac{s+u}{s}\right)^{\frac{1}{2}} \left\vert\exp\left(-\frac{x^{2}}{2s}\right)-1\right\vert +\left\vert\left(\frac{s+u}{s}\right)^{\frac{1}{2}} -1\right\vert\right]\\ &&\le\left((2\pi)^{\frac{1}{2}}|x|\right)^{-1}\exp\left(-\frac{1}{2}\right)\left(\frac{s+1}{s}\right)^{\frac{1}{2}}\left(\frac{x^{2}}{2s}\right)+(2\pi u)^{-\frac{1}{2}} \left(\frac{u}{2s}\right)\\ &&\le{c}_1|x|+c_{2}\,u^{\frac{1}{2}}\,, \end{array} $$
with some constants c 1 and c 2, for 0≤u≤h≤1 and s∈[t 0,t], where for the estimate of the first summand we have used that the function u↦p(u,x,0) has its unique maximum at u=x 2, the standard estimate 1−e −z≤z for all z≥0 and that \((s+1)s^{-1}\le 1+t_{0}^{-1}\) as well as for the estimate of the second summand the inequalities \(p(u,x,0)\le (2\pi u)^{-\frac {1}{2}}\) and \(|\sqrt {\frac {s+u}{s}}-1|\le \frac {u}{2s}\). Putting x=β s , and integrating from 0 to h, and dividing by h, for 0≤u≤h≤1 and s∈[t 0,t], from (16) we obtain
$$\begin{array}{@{}rcl@{}} &&\frac{1}{h}\int_{0}^{h}\left\vert p\left(\frac{su}{s+u},\beta_{s},0\right) -p\left(u,\beta_{s},0\right)\right\vert\,\mathrm{d} u\, g(s,\beta_s)\,f(s)\\ &&\quad\le\left(c_1|\beta_s|+c_{2}\,h^{\frac{1}{2}}\right)\,g(s,\beta_s)\,f(s)\\ &&\quad\le\left(c_1|\beta_s|+c_{2}\right)\,c_{3}\,C(t_0,t,\beta_s), \end{array} $$
where C(t 0,t,x) is an upper bound of g(s,x) on [t 0,t] continuous in x (see Lemma B.2) and c 3 is an upper bound for the continuous density function f on [t 0,t]. The right-hand side is integrable over [t 0,t] with respect to the Lebesgue measure λ +. On the other side, by the fundamental theorem of calculus, we have that, for every x≠0,
$$ {\lim}_{h\downarrow 0}\frac{1}{h}\int_{0}^h p(u,x,0)\,\mathrm{d} u=0,\quad {\lim}_{h\downarrow 0}\frac{1}{h}\int_{0}^h p\left(\frac{su}{s+u},x,0\right)\,\mathrm{d} u=0\,. $$
For this, we notice that p(0,x,0)=0 is a continuous extension of the function u↦p(u,x,0) if x≠0. By Corollary A.3, we have that the set {0≤s≤t∧τ: β s =0} has Lebesgue measure zero. Then, using Lebesgue's theorem on dominated convergence, we can conclude that P-a.s.
$$ {\lim}_{h\downarrow 0}\int_{t_{0}\wedge\tau}^{t\wedge\tau}\frac{1}{h}\int_{0}^{h}\left\vert p\left(\frac{su}{s+u},\beta_{s},0\right)-p(u,\beta_s,0)\right\vert\,\mathrm{d} u\, g(s,\beta_s)\,f(s)\,\mathrm{d} s=0\,. $$
This completes the first step of the proof of the first part, meaning that in (14) we can replace \(p\left (\frac {su}{s+u},\beta _{s},0\right)\) with p(u,β s ,0) for identifying the limit.
The second step of the first part is to prove that
$$\begin{array}{@{}rcl@{}} {\lim}_{h\downarrow 0} \int_{t_{0}\wedge\tau}^{t\wedge\tau}\frac{1}{h}\int_{0}^{h}p\left(u,\beta_{s},0\right)\,\mathrm{d} u\, g(s,\beta_s)\,f(s)\,\mathrm{d} s=K_t-K_{t_0}\,,\quad \mathbf{P}\text{-a.s.} \end{array} $$
$$ q(h,x):=\frac{1}{h}\int_{0}^h p(u,x,0)\,\mathrm{d} u,\quad 0<h\le 1, \ x\in\mathbb{R}\,, $$
an application of the occupation time formula (see Corollary A.3) yields
$$\begin{array}{@{}rcl@{}} &&\int_{t_{0}\wedge\tau}^{t\wedge\tau}\frac{1}{h}\int_{0}^{h}p\left(u,\beta_{s},0\right)\,\mathrm{d} u\, g(s,\beta_s)\,f(s)\,\mathrm{d} s\\ &&=\int_{t_{0}\wedge\tau}^{t\wedge\tau}\,q(h,\beta_s)\, g(s,\beta_s)\,f(s)\,\mathrm{d} s\\ &&=\int_{-\infty}^{+\infty}\left(\int_{t_0}^{t}g\left(s,x\right)f(s)\,\mathrm{d} L^\beta(s,x)\right)q\left(h,x\right)\,\mathrm{d} x\,,\quad \mathbf{P}\text{-a.s.} \end{array} $$
For every h>0, q(h,·) is a probability density function with respect to the Lebesgue measure on \(\mathbb {R}\). According to Lemma B.1, the probability measures Q h with density q(h,·) converge weakly to the Dirac measure δ 0 at 0. On the other hand, Lemma B.4 shows that the function \(x\mapsto \int _{t_{0}}^{t}g\left (s,x\right)f(s)\,\mathrm {d} L^{\beta }(s,x)\) is continuous and bounded. Hence, in (20) we can pass to the limit and obtain the following
$$\begin{array}{@{}rcl@{}} &&{\lim}_{h\downarrow0}\int_{t_{0}\wedge\tau}^{t\wedge\tau}\frac{1}{h}\int_{0}^{h}p\left(u,\beta_{s},0\right)\,\mathrm{d} u\, g(s,\beta_s)\,f(s)\,\mathrm{d} s\\ &&=\int_{t_0}^{t}g\left(s,0\right)f(s)\,\mathrm{d} L^\beta(s,0),\quad \mathbf{P}\text{-a.s.} \end{array} $$
In the third step of the proof of the first part, we must show that (13) holds. Note that the function f is uniformly continuous on [t 0,t+1]. We fix ε>0 and choose 0<δ≤1 such that |f(s+u)−f(s)|≤ε for every 0≤u<δ. Proceeding similarly as above, we obtain the following
$$\begin{array}{@{}rcl@{}} \limsup_{h\downarrow 0}\!\!\! &&\left\vert\int_{t_{0}\wedge\tau}^{t\wedge\tau}\frac{1}{h}\left(\frac{\int_{s}^{s+h}\varphi_{s}\left(r,\beta_{s}\right)\left[f(r)-f(s)\right]\,\mathrm{d} r}{\int_{s}^{\infty} \varphi_{s}\left(v,\beta_{s}\right)f(v)\,\mathrm{d} v}\right)\,\mathrm{d} s\right\vert\\ &\le&\limsup_{h\downarrow 0}\int_{t_{0}\wedge\tau}^{t\wedge\tau}\frac{1}{h}\int_{0}^{h}p\left(\frac{su}{s+u},\beta_{s},0\right)\,\big|f(s+u)-f(s)\big|\,\mathrm{d} u\, g(s,\beta_s)\,\mathrm{d} s\\ &\le&\varepsilon\,\limsup_{h\downarrow 0}\int_{t_{0}\wedge\tau}^{t\wedge\tau}\frac{1}{h}\int_{0}^{h}p\left(\frac{su}{s+u},\beta_{s},0\right)\,\mathrm{d} u\, g(s,\beta_s)\,\mathrm{d} s\\ &=&\varepsilon\,\limsup_{h\downarrow 0}\int_{t_{0}\wedge\tau}^{t\wedge\tau}\frac{1}{h}\int_{0}^{h}p\left(u,\beta_{s},0\right)\,\mathrm{d} u\, g(s,\beta_s)\,\mathrm{d} s\\ &=&\varepsilon\,\int_{t_0}^{t}g\left(s,0\right)\,\mathrm{d} L^\beta(s,0),\quad \mathbf{P}\text{-a.s.} \end{array} $$
Since ε>0 is choosen arbitrarily and the integral above is P-a.s. finite, we conclude that (13) holds.
The first part of the proof is complete.
The second part of the proof relies on the so-called Laplacian approach of P.-A. Meyer and, for the sake of easy reference, related results are recalled in Appendix Appendix C. Let us denote by K w the compensator of the default process H introduced in (9): \(H_{t}:=\mathbb {I}_{\left \{ \tau \leq t\right \}},\ t\geq 0\). We first show that \(K^{h}_{t}\) converges to \(K^{w}_{t}\) as h ↓0 in the sense of the weak topology σ(L 1,L ∞) (see Definition C.3), for every t≥0. We then prove that the process K is actually indistinguishable from K w.
For the sake of simplicity of the notation, if a sequence of integrable random variables \(\left (\xi _{n}\right)_{n\in \mathbb {N}}\) converges to an integrable random variable ξ in the sense of the weak topology σ(L 1,L ∞), we will write
$$\xi_{n}\xrightarrow[n\rightarrow+\infty]{{\!~\!}^{\sigma\left(L^{1},L^{\infty}\right)}}\xi. $$
Furthermore, we will denote by G the right-continuous potential of class (D) (cf. beginning of Appendix Appendix C) given by
$$ G_{t}:=1-H_{t}=\mathbb{I}_{\left\{ t<\tau\right\}},\quad t\geq 0\,. $$
By Corollary C.5, we know that there exists a unique integrable predictable increasing process \(K^{w}=\left (K_{t}^{w},\,t\geq 0\right)\) which generates, in the sense of Definition C.1, the potential G given by (22) and, for every \(\mathbb {F}^{\beta }\)-stopping time T, we have that
$$K_{T}^{h}\xrightarrow[h\downarrow0]{{\!~\!}^{\sigma\left(L^{1},L^{\infty}\right)}}K_{T}^{w}. $$
The process K w is actually the compensator of H. Indeed, it is a well-known fact that the process H admits a unique decomposition
$$ H=M+A $$
into the sum of a right-continuous martingale M and an adapted, natural, increasing, integrable process A. The process A is then called the compensator of H. On the other hand, from the definition of the potential generated by an increasing process (see Definition C.1), the process
$$ L:=G+K^{w} $$
is a martingale. By combining the definition (22) of the process G and (24), we obtain the following decomposition of H:
$$H=1-L+K^{w}. $$
However, by the uniqueness of the decomposition (23), we can identify the martingale M with 1−L and we have that A=K w, up to indistinguishability. Since the submartingale H and the martingale 1−L appearing in the above proof are right-continuous, the process K w is also right-continuous.
By applying Lemma C.8, we see that \(K_{t}-K_{t_{0}}\phantom {\dot {i}\!}\) is a modification of \(K^{w}_{t}-K^{w}_{t_{0}}\phantom {\dot {i}\!}\), for all t 0,t such that 0<t 0<t. Passing to the limit as t 0 ↓0, we get \(K_{t}=K_{t}^{w}\) P-a.s. for all t≥0. Since both processes have right-continuous sample paths they are indistinguishable.
The theorem is proved. □
We close this part of the present paper with the following observations.
(1) Note that \(\left (\mathbb {I}_{\left \{ \tau \leq t\right \} },\,t\geq 0\right)\) does not admit an intensity with respect to the filtration \(\mathbb {F}^{\beta }\) (hence, it is not possible to apply, for example, Aven's Lemma for computing the compensator (see, e.g., (Aven 1985))).
(2) Assumption 3.1(ii) on the distribution function F that F(t)<1 for all t≥0 ensures that the denominator of the integrand of the right-hand side of (10) is always strictly positive. However, it can be removed. Indeed, if the density function f of τ is continuous (as required by Assumption 3.1(i)), then exactly as above we can show that relation (10) is satisfied for all t≤t 1:= sup{t>0: F(t)<1}. On the other hand, it is obvious that τ≤t 1 P-a.s. (hence, the right-hand side of (10) is constant for t∈[t 1,∞)) and also that the compensator K=(K t , t≥0) of \(\left (\mathbb {I}_{\left \{ \tau \leq t\right \} },\,t\geq 0\right)\) is constant on [t 1,∞). Altogether, it follows that relation (10) is satisfied for all t≥0.
On the local time of the information process
In this section, we introduce and study the local time process associated with the information process.
For any continuous semimartingale X=(X t , t≥0) and for any real number x, it is possible to define the (right) local time L X(t,x) associated with X at level x up to time t using Tanaka's formula (see, e.g., (Revuz and Yor 1999), Theorem VI.(1.2)) as follows:
$$ L^{X}(t,x):=|X_{t}-x|-|X_{0}-x|-\int_{0}^{t}\text{sign}\left(X_{s}-x\right)\,\mathrm{d} X_{s},\quad t\geq 0, $$
where sign(x):=1 if x>0 and sign(x):=−1 if x≤0. The process L X(·,x)=(L X(t,x), t≥0) appearing in relation (25) is called the (right) local time of X at level x.
Now, we recall the occupation time formula for local times of continuous semimartingales which is given in a form convenient for our applications. By 〈X,X〉, we denote the square variation process of a continuous semimartingale X.
Lemma A.1
Let X=(X t , t≥0) be a continuous semimartingale. There is a P-negligible set outside of which
$$\int_{0}^{t}h\left(s,X_{s}\right)\mathrm{d} \left\langle X,X\right\rangle_{s}=\int_{-\infty}^{+\infty}\left(\int_{0}^{t}h\left(s,x\right)\,\mathrm{d} L^{X}\left(s,x\right)\right)\,\mathrm{d} x\,, $$
for every t≥0 and every non-negative Borel function h on \(\mathbb {R}_{+}\times \mathbb {R}\).
See Corollary VI.(1.6) from the book by Revuz and Yor (1999) for the case when h is a non-negative Borel function defined on \(\mathbb {R}\) (i.e., it does not depend on time). The statement of the lemma is then proved by first considering the case in which h has the form \( h\left (t,x\right)=\mathbb {I}_{\left [u,v\right ]}(t)\gamma (x) \) for 0≤u<v<∞ and a non-negative Borel function γ on \(\mathbb {R}\), and then using monotone class arguments (see Revuz and Yor (1999), Exercise VI.(1.15) or Rogers and Williams (2000), Theorem IV.(45.4)). □
Concerning continuity properties of local times, there is the following result.
Let X=(X t , t≥0) be a continuous semimartingale with canonical decomposition given by X=M+A, where M is a local martingale and A a finite variation process. Then, there exists a modification of the local time process \(\left (L^{X}\left (t,x\right),t\geq 0,\,x\in \mathbb {R}\right)\) of X such that the map (t,x)↦L X(t,x) is continuous in t and càdlàg in x, P-a.s. Moreover,
$$ L^{X}\left(t,x\right)-L^{X}\left(t,x-\right) =2\int_{0}^{t}\mathbb{I}_{\{x\}}(X_s)\, \mathrm{d} A_{s}, $$
for all \(t\geq 0,\,x\in \mathbb {R}\), P-a.s.
See, e.g., (Revuz and Yor 1999), Theorem VI.(1.7). □
The information process β is a continuous semimartingale (cf. Theorem 2.8), hence the local time L β(t,x) of β at level \(x\in \mathbb {R}\) up to time t≥0 is well defined. The occupation time formula takes the following form.
Corollary A.3
$$\intop_{0}^{t\wedge\tau}h\left(s,\beta_{s}\right)\mathrm{d} s=\intop_{0}^{t}h\left(s,\beta_{s}\right)\mathrm{d} \left\langle \beta,\beta\right\rangle_{s}=\intop_{-\infty}^{+\infty}\left(\intop_{0}^{t}h\left(s,x\right)\,\mathrm{d} L^{\beta}\left(s,x\right)\right)\,\mathrm{d} x\,, $$
for all t≥0 and all non-negative Borel functions h on \(\mathbb {R}_{+}\times \mathbb {R}\), P-a.s.
The first equality follows from relation (8) and the second is an application of Lemma A.1. □
An important property of the local time L β is the existence of a bicontinuous version.
There is a version of L β such that the map \((t,x)\in \mathbb {R}_{+}\times \mathbb {R}\mapsto L^{\beta }\left (t,x\right)\) is continuous, P-a.s.
We choose a version of the local time L β according to Lemma A.2. Using (26), we have that
$$L^{\beta}\left(t,x\right)-L^{\beta}\left(t,x-\right)=-2\int_{0}^{t\wedge\tau} \mathbb{I}_{\{x\}}(\beta_{s})\,u\left(s,\beta_{s}\right)\mathrm{d} s, $$
for all \(t\geq 0,\,x\in \mathbb {R}\), P-a.s., where u is the function defined by (6). Applying Corollary A.3 to the right-hand side of the last equality above, we see that
$$2\intop_{0}^{t\wedge\tau}\mathbb{I}_{\{x\}}(\beta_{s})\,u\left(s,\beta_{s}\right)\,\mathrm{d} s=2\intop_{-\infty}^{+\infty}\mathbb{I}_{\{x\}}(y)\left(\intop_{0}^{t}u\left(s,y\right)\,\mathrm{d} L^{\beta}\left(s,y\right)\right)\,\mathrm{d} y=0, $$
and hence L β(t,x)−L β(t,x−)=0, for all \(t\geq 0,\,x\in \mathbb {R}\), P-a.s., because {x} has Lebesgue measure zero. This completes the proof. □
We also make use of the boundedness of the local time with respect to the space variable.
The function x↦L β(t,x) is bounded for all \(t\in \mathbb {R}_{+}\) P-a.s. (the bound may depend on t and ω).
It follows from the occupation time formula (or from Revuz and Yor (1999), Corollary VI.(1.9)) that the local time L β(t,·) vanishes outside of the compact interval [−M t (ω),M t (ω)] where
$$ M_{t}(\omega) :=\sup_{s\in\left[0,t\right]}\left|\beta_{s}(\omega)\right|,\quad t\geq0,\;\omega\in\Omega\,, $$
which together with the continuity of L β(t,·) (see Lemma A.4) yields the boundedness of this function, P-a.s. □
Outside a negligible set, for fixed \(x\in \mathbb {R}\), the local time L β(·,x) is a positive continuous increasing function, and we can associate with it a random measure on \(\mathbb {R}_{+}\):
$$L^{\beta}\left(B,x\right):=\int_{B}\mathrm{d} L^{\beta}\left(s,x\right),\quad B\in\mathcal{B}\left(\mathbb{R}_{+}\right). $$
Outside a negligible set, for any sequence \(\left (x_{n}\right)_{n\in \mathbb {N}}\) in \(\mathbb {R}\) converging to \(x\in \mathbb {R}\), the sequence \(\left (L^{\beta }\left (\cdot,x_{n}\right)\right)_{n\in \mathbb {N}}\) converges weakly to L β(·,x), i.e.,
$$\int_{\mathbb{R}_{+}} g(s)L^{\beta}\left(\mathrm{d} s,x_{n}\right)\xrightarrow[n\rightarrow \infty]{}\int_{\mathbb{R}_{+}} g(s)L^{\beta}\left(\mathrm{d} s,x\right), $$
for all bounded and continuous functions \(g:\mathbb {R}_{+}\mapsto \mathbb {R}\).
We fix a negligible set outside of which L β is bicontinuous (cf. Lemma A.4) and outside of which we will be working now. The measures \(\left (L^{\beta }\left (\cdot,x_{n}\right)\right)_{n\in \mathbb {N}}\) are finite on \(\mathbb {R}\) and they are supported by [0,τ]. By the continuity of L β(t,·), we have that \(L^{\beta }\left (s,x_{n}\right)\xrightarrow [n\rightarrow \infty ]{}L^{\beta }\left (s,x\right),\,s\geq 0\), from which it follows that
$$ L^{\beta}\left(\left[0,s\right],x_{n}\right)\xrightarrow[n\rightarrow \infty]{}L^{\beta}\left(\left[0,s\right],x\right),\quad s\geq0\,. $$
We also have this convergence for the whole space \(\mathbb {R}_{+}\):
$$L^{\beta}\left(\mathbb{R}_{+},x_{n}\right)=L^{\beta}\left(\left[0,\tau\right],x_{n}\right) \xrightarrow[n\rightarrow\infty]{}L^{\beta}\left(\left[0,\tau\right],x\right)=L^{\beta}\left(\mathbb{R}_{+},x\right)\,. $$
From this, we can conclude that the measures L β(·,x n ) converge weakly to L β(·,x). □
Auxiliary results
In (19), we had introduced the function q by
$$q(h,x):=\frac{1}{h}\int_{0}^{h} p(u,x,0)\,\mathrm{d} u,\quad 0<h\le 1, \ x\in\mathbb{R}\,, $$
where p(t,·,y) is the density of the normal distribution with variance t and expectation y (see (2)).
Lemma B.1
The functions q(h,·) are probability density functions with respect to the Lebesgue measure on \(\mathbb {R}\). The probability measures \(\mathbb {Q}_{h}\) on \(\mathbb {R}\) associated with the density q h converge weakly as h ↓0 to the Dirac measure δ 0 at 0.
The first statement of the lemma is obvious. For verifying the second statement, let f be a bounded continuous function on \(\mathbb {R}\). Using Fubini's theorem, we obtain
$$\begin{array}{@{}rcl@{}} \int_{\mathbb{R}} f(x)\,\mathbb{Q}_{h}(\mathrm{d} x)&=&\int_{\mathbb{R}} f(x)\,q_{h}(x)\,\mathrm{d} x\\ &=&\int_{\mathbb{R}} f(x)\,\left(\frac{1}{h}\int_{0}^hp(u,x,0)\,\mathrm{d} u\right)\,\mathrm{d} x\\ &=&\frac{1}{h}\int_{0}^{h}\left(\int_{\mathbb{R}} f(x)\,p(u,x,0)\,\mathrm{d} x\right)\,\mathrm{d} u\\ &=&\frac{1}{h}\int_{0}^{h}\left(\int_{\mathbb{R}} f(x)\,\mathcal{N}(0,u)(\mathrm{d} x)\right)\,\mathrm{d} u\,. \end{array} $$
Since the function \(u\in [0,1]\mapsto \mathcal {N}(0,u)\), which associates to every u∈[0,1], the centered Gaussian law \(\mathcal {N}(0,u)\) is continuous with respect to weak convergence of probability measures (note that \(\mathcal {N}(0,0)=\delta _{0}\)), we observe that the function \(u\in [0,1]\mapsto \int _{\mathbb {R}} f(x)\,\mathcal {N}(0,u)(\mathrm {d} x)\) is continuous. An application of the fundamental theorem of calculus yields that the right-hand side converges to \(\int _{\mathbb {R}} f(x)\,\delta _{0}(\mathrm {d} x)\) as h ↓0 and hence
$${\lim}_{h\downarrow0}\int_{\mathbb{R}} f(x)\,\mathbb{Q}_{h}(\mathrm{d} x)=f(0)\,, $$
proving the second statement of the lemma. □
Now, we consider the function g introduced in (15):
$$g\left(s,x\right):=\Big(\int_{s}^{\infty}\varphi_{s}\left(v,x\right)f(v)\,\mathrm{d} v\Big)^{-1},\quad s>0,\,x\in\mathbb{R}\,. $$
(1) For all \(x\in \mathbb {R}\) and 0<t 0<t, the function \(g\left (\cdot,x\right): [t_{0},t]\mapsto \mathbb {R}\) is bounded, i.e., there exists a real constant C(t 0,t,x) such that
$$\sup_{s\in[t_{0},t]}g\left(s,x\right)\leq C(t_{0},t,x)\,. $$
(2) For all \(x\in \mathbb {R}\) and 0<t 0<t, the function \(g(\cdot,x): [t_{0},t]\mapsto \mathbb {R}\) is continuous, i.e., for all s n ,s∈[t 0,t] such that s n →s,
$${\lim}_{s_{n}\rightarrow s}g(s_{n},x)=g(s,x)\,. $$
(3) Let \(\left (x_{n}\right)_{n\in \mathbb {N}}\) be a sequence converging monotonically to \(x\in \mathbb {R}\). Then, for all 0<t 0<t,
$$\sup_{s\in\left[t_{0},t\right]}\left|g(s,x_{n})-g(s,x)\right| \xrightarrow[n\rightarrow \infty]{}0\,. $$
Let us define, for every s∈[t 0,t] and \(x\in \mathbb {R}\),
$$\begin{array}{@{}rcl@{}} D\left(s,x\right)&&\\ &:=&\int_{s}^{\infty} \sqrt{\frac{v}{2\pi s\,(v-s)}}\exp\left(-\frac{v\,x^{2}}{2s\,(v-s)}\right)f(v)\,\mathrm{d} v\,, \end{array} $$
and rewrite g as
$$ g(s,x)=\frac{1}{D\left(s,x\right)},\quad s\in\left[t_{0},t\right],\;x\in\mathbb{R}\,. $$
In order to prove statement (1), it suffices to verify that there exists a constant \(\tilde {C}\left (t_{0},t,x\right)\) such that
$$ 0<\tilde{C}\left(t_{0},t,x\right)\leq D\left(s,x\right),\quad s\in[t_{0},t],\; x\in\mathbb{R}\,. $$
Such a constant can be found by setting
$$ \tilde{C}\left(t_{0},t,x\right):=\int_{t}^{\infty}\sqrt{\frac{1}{2\pi t}}\exp\left(-\frac{v\,x^{2}}{2t_{0}(v-t)}\right)f(v)\,\mathrm{d} v\,, $$
proving the first statement of the lemma.
In order to prove statement (2) of the lemma, it suffices to verify that the function s↦D(s,x), s∈[t 0,t], is continuous, a fact that can be proved using Lebesgue's dominated convergence theorem. Indeed, let s n ,s∈[t 0,t] such that s n →s as n→∞. Rewriting (29), we get
$$\begin{array}{@{}rcl@{}} D\left(s_n,x\right)\\ &=&\int_{t_0}^{\infty} \mathbb{I}_{(s_n,+\infty)}(v)\sqrt{\frac{v}{2\pi s_{n}\,(v-s_n)}}\exp\left(-\frac{v\,x^{2}}{2s_{n}\,(v-s_n)}\right)f(v)\,\mathrm{d} v\,. \end{array} $$
First, we consider the integral from t to ∞: For v≥t, we can make an upper estimate of the integrand by \(\sqrt {\frac {v}{2\pi t_{0}\,(v-t)}}\, f(v)\) which is integrable over [t,+∞). For the second part of the integral from t 0 to t, we estimate the integrand by \(\mathbb {I}_{(s_{n},+\infty)}(v)\sqrt {\frac {t}{2\pi t_{0}\,(v-s_{n})}}\,c\), where c is an upper bound of f on [t 0,t], and by integrating we observe that
$${\lim}_{n\rightarrow\infty}\int_{t_{0}}^{t}\!\mathbb{I}_{(s_{n},+\infty)}(v) \sqrt{\frac{t}{2\pi t_{0}\,(v-s_{n})}}\,\mathrm{d} v \,=\,\int_{t_{0}}^{t}\!\mathbb{I}_{(s,+\infty)}(v)\sqrt{\frac{t}{2\pi t_{0}\,(v-s)}}\,\mathrm{d} v\,. $$
As the integrands are nonnegative, we get convergence in L 1([t 0,t]) and hence uniform integrability (cf. Theorem C.7). This means that the sequence
$$I_{(s_{n},+\infty)}(v)\sqrt{\frac{v}{2\pi s_{n}\,(v-s_{n})}}\exp\left(-\frac{v\,x^{2}}{2s_{n}\,(v-s_{n})}\right)f(v) $$
is uniformly integrable on [t 0,t] and we can apply Lebesgue's theorem (cf. Theorem C.7) to conclude
$$\begin{array}{@{}rcl@{}} \lefteqn{{\lim}_{n\rightarrow\infty}\int_{t_0}^t \mathbb{I}_{(s_n,+\infty)}(v)\sqrt{\frac{v}{2\pi s_{n}\,(v-s_n)}}\exp\left(-\frac{v\,x^{2}}{2s_{n}\,(v-s_n)}\right)f(v)\,\mathrm{d} v}\\ &=&\int_{t_0}^t \mathbb{I}_{(s,+\infty)}(v)\sqrt{\frac{v}{2\pi s\,(v-s)}}\exp\left(-\frac{v\,x^{2}}{2s\,(v-s)}\right)f(v)\,\mathrm{d} v\,. \end{array} $$
Summarizing, we get
$${\lim}_{n\rightarrow\infty}D\left(s_{n},x\right)=D\left(s,x\right) $$
and the proof of statement (2) of the lemma finished.
We turn to the proof of statement (3) of the lemma. Using relation (30), we see that
$$\left|g\left(s,x_{n}\right)-g\left(s,x\right)\right| =\frac{\left|D\left(s,x_{n}\right)-D\left(s,x\right)\right|} {D\left(s,x_{n}\right)D\left(s,x\right)} $$
and from inequality (31) we get that
$$\sup_{s\in\left[t_{0},t\right]}\left|g\left(s,x_{n}\right)-g\left(s,x\right)\right| \leq\frac{\sup_{s\in\left[t_{0},t\right]}\left|D\left(s,x_{n}\right)-D\left(s,x\right)\right|} {\tilde{C}\left(t_{0},t,x_{n}\right)\,\tilde{C}\left(t_{0},t,x\right)}, $$
where \(\tilde {C}\left (t_{0},t,x\right)\) is defined by (32). It is easy to see that
$${\lim}_{n\rightarrow \infty} \frac{1} {\tilde{C}\left(t_{0},t,x_{n}\right)\tilde{C}\left(t_{0},t,x\right)}=\frac{1}{\tilde{C}\left(t_{0},t,x\right)^{2}}<+\infty\,. $$
Hence, it remains to prove that
$$\sup_{s\in\left[0,t\right]}\left|D\left(s,x_{n}\right)-D\left(s,x\right)\right|\xrightarrow[n\rightarrow \infty]{}0. $$
By assumption, the sequence x n converges monotonically to x. In such a case, it is easy to see that the sequence of functions D(·,x n ) is monotone. Furthermore, using Lebesgue's dominated convergence theorem, we verify that D(s,x n ) converges to D(s,x), for all s∈[t 0,t]. Since the function s↦D(s,x) is also continuous on [t 0,t], according to Dini's theorem, D(·,x n ) converges uniformly to D(·,x) on [t 0,t]. This implies the third statement of the lemma and the proof is finished. □
Let h,h n be bounded and continuous functions on a metric space E, and μ,μ n be finite measures on \(\left (E,\mathcal {B}(E)\right)\). Suppose that the following two conditions are satisfied:
The sequence of functions h n converges uniformly to h.
The sequence of measures μ n converges weakly to μ.
Then, \({\lim }_{n\uparrow +\infty }\int _{E}h_{n}\,\mathrm {d}\mu _{n}=\int _{E}h \,\mathrm {d}\mu \).
It can immediately be verified that
$$\begin{aligned} &\left|\int_{E}h_{n}\,\mathrm{d}\mu_{n}-\int_{E}h \,\mathrm{d}\mu\right|\\ &\leq\sup_{x\in E}\left|h(x)-h_{n}(x)\Big|\int_{E}\,\mathrm{d} \mu_{n}+\Big|\int_{E}h \,\mathrm{d} \mu_{n}-\int_{E}h\, \mathrm{d} \mu\right|, \end{aligned} $$
which converges to 0 as n ↑+∞. □
Let 0<t 0<t. The function \(k:\,\mathbb {R}\rightarrow \mathbb {R}_{+}\) given by
$$k(x):=\int_{t_{0}}^{t}g\left(s,x\right)\,f(s)\,\mathrm{d} L^{\beta}(s,x),\quad x\in \mathbb{R}\,, $$
is bounded and continuous, where the function g is given by (15).
Let us first restrict to a compact subset E of \(\mathbb {R}\). First, we prove the right- and left-continuity, hence the continuity, of the function k. Let x n be a sequence from E converging monotonically to x∈E. From Lemma B.2, we know that the bounded and continuous functions \(g\left (\cdot,x_{n}\right):\,[t_{0},t]\rightarrow \mathbb {R}\) converge uniformly to the bounded and continuous function \(g\left (\cdot,x\right):\,[t_{0},t]\rightarrow \mathbb {R}\) as n→∞. From Lemma A.6, we obtain that the sequence of measures L β(·,x n ) converges weakly to L β(·,x) as n→∞. Applying Lemma B.3, we have that
$$\begin{array}{@{}rcl@{}} {\lim}_{n\rightarrow\infty}k\left(x_{n}\right)&=&{\lim}_{n\rightarrow\infty}\int_{t_0}^{t}g\left(s,x_{n}\right)f(s)\,\mathrm{d} L^{\beta} \left(s,x_{n}\right)\\ &=&\int_{t_0}^{t}g\left(s,x\right)f(s)\,\mathrm{d} L^{\beta} \left(s,x\right)=k(x). \end{array} $$
Consequently, the function k is continuous on E. The boundedness of k now follows from the compactness of E. In order to show that the statement also holds for \(\mathbb {R}\), let us choose E=[−M t −1,M t +1] (see (27) for notation). As L β(s,x)=0, s∈[0,t], x∉[−M t ,M t ] (see the proof of Lemma A.5), the statement follows. □
The Meyer approach to the compensator
Below, we briefly recall the approach developed by Meyer (1966) for computing the compensator of a right-continuous potential of class (D). In this section, \(\mathbb {F}=\left (\mathcal {F}_{t}\right)_{t\geq 0}\) denotes a filtration satisfying the usual hypothesis of right-continuity and completeness.
We begin with the definition of a right-continuous potential of class (D). Let X=(X t , t≥0) be a right-continuous \(\mathbb {F}\)-supermartingale and let \(\mathcal {T}\) be the collection of all finite \(\mathbb {F}\)-stopping times relative to this family. The process X is said to belong to the class(D) if the collection of random variables \(X_{T},\,T\in \mathcal {T}\) is uniformly integrable. We say that the right-continuous supermartingale X is a potential if the random variables X t are non-negative and if
$${\lim}_{t\rightarrow+\infty}\mathbf{E}\left[X_{t}\right]=0. $$
Definition C.1
Let C=(C t , t≥0) be an integrable \(\mathbb {F}\)-adapted right-continuous increasing process, and let L=(L t , t≥0) be a right-continuous modification of the martingale \(\left (\mathbf {E}\left [C_{\infty }|\mathcal {F}_{t}\right ],\,t\geq 0\right)\); the process Y=(Y t , t≥0) given by
$$Y_{t}:=L_{t}-C_{t} $$
is called the potential generated by C.
The following result establishes a connection between potentials generated by an increasing process and potentials of class (D). Let h be a strictly positive real number and X=(X t , t≥0) be a potential of class (D), and denote by (p h X t , t≥0) the right-continuous modification of the supermartingale \(\left (\mathbf {E}\left [X_{t+h}|\mathcal {F}_{t}\right ],\,t\geq 0\right)\).
Theorem C.2
Let X=(X t , t≥0) be a potential of class (D), let h>0 and \(A^{h}=\left (A_{t}^{h},\,t\geq 0\right)\) be the process defined by
$$ A_{t}^{h}:=\frac{1}{h}\intop_{0}^{t}\left(X_{s}-p_{h}X_{s}\right)\mathrm{d} s. $$
Then, A h is an integrable increasing process which generates a potential of class (D) \(X^{h}=\left (X_{t}^{h},\,t\geq 0\right)\) dominated by X, i.e., the process X−X h is a potential. It holds that
$$X_{t}^{h}=\frac{1}{h}\mathbf{E}\left[\int_{0}^{h}X_{t+s}\,\mathrm{d} s|\mathcal{F}_{t}\right], \quad \mathbf{P}\text{-a.s.},\; t\geq 0\,. $$
See, e.g., (Meyer 1966), VII.T28. □
An increasing process A=(A t , t≥0) is called natural (with respect to the filtration \(\mathbb {F}\)) if, for every bounded right-continuous \(\mathbb {F}\)-martingale M=(M t , t≥0), we have
$$\mathbf{E}\left[\intop_{\left(0,t\right]}M_{s}\,\mathrm{d} A_{s}\right]=\mathbf{E}\left[\intop_{\left(0,t\right]}M_{s-}\,\mathrm{d} A_{s}\right],\quad t>0\,. $$
It is well known that an increasing process A is natural with respect to \(\mathbb {F}\) if and only if it is \(\mathbb {F}\)-predictable.
For the following definition of convergence in the sense of the weak topology σ(L 1,L ∞), see (Meyer 1966), II.10.
Let \(\left (\xi _{n}\right)_{n\in \mathbb {N}}\) be a sequence of integrable real-valued random variables. The sequence \(\left (\xi _{n}\right)_{n\in \mathbb {N}}\) is said to converge to an integrable random variable ξ in the weak topology σ(L 1,L ∞) if
$${\lim}_{n\rightarrow+\infty}\mathbf{E}\left[\xi_{n}\eta\right]=\mathbf{E}\left[\xi\eta\right],\;\textrm{for all }\eta\in L^{\infty}\left(\mathbf{P}\right). $$
Let X=(X t , t≥0) be a right-continuous potential of class (D). Then, there exists an integrable natural increasing process A=(A t , t≥0) which generates X, and this process is unique. For every stopping time T we have
$$A_{T}^{h}\xrightarrow[h\downarrow0]{{\!~\!}^{\sigma\left(L^{1},L^{\infty}\right)}}A_{T}. $$
In the framework of the information-based approach, the process H=(H t , t≥0), given by (9), is a bounded increasing process which is \(\mathbb {F}^{\beta }\)-adapted. It is a submartingale and it can be immediately seen that the process G=(G t , t≥0), given by (22), is a right-continuous potential of class (D). By Theorem C.2, the processes K h, h>0, defined by (11), generate a family of potentials G h dominated by G.
Corollary C.5
There exists a unique integrable natural increasing process \(K^{w}=\left (K_{t}^{w},\,t\geq 0\right)\) which generates the process G, defined by (22) and, for every \(\mathbb {F}^{\beta }\)-stopping time T, we have that
$$K_{T}^{h}\xrightarrow[h\downarrow0]{{\!~\!}^{\sigma\left(L^{1},L^{\infty}\right)}}K_{T}^{w}, $$
where K h is the process defined by (11).
See Theorem C.4. □
Compactness Criterion of Dunford–Pettis Let \(\mathcal {A}\) be a subset of the space L 1(P). The following two properties are equivalent:
\(\mathcal {A}\) is uniformly integrable;
\(\mathcal {A}\) is relatively compact in the weak topology σ(L 1,L ∞).
See (Meyer 1966), II.T23. □
Let \(\left (\xi _{n}\right)_{n\in \mathbb {N}}\) be a sequence of integrable random variables converging in probability to a random variable ξ. Then, ξ n converges to ξ in L 1(P) if and only if \(\left (\xi _{n}\right)_{n\in \mathbb {N}}\) is uniformly integrable. If the random variables ξ n are non-negative, then \(\left (\xi _{n}\right)_{n\in \mathbb {N}}\) is uniformly integrable if and only if
$${\lim}_{n\rightarrow+\infty}\mathbf{E}\left[\xi_{n}\right]=\mathbf{E}[\xi]<+\infty\,. $$
Lemma C.8
Let \(\left (\xi _{n}\right)_{n\in \mathbb {N}}\) be a sequence of random variables and ξ,η∈L 1(P) such that:
\(\xi _{n}\xrightarrow [\:n\rightarrow +\infty ]{\sigma \left (L^{1},L^{\infty }\right)}\eta ;\)
ξ n →ξ, P-a.s.
Then, η=ξ, P-a.s.
From condition (1), we see that \(\left (\xi _{n}\right)_{n\in \mathbb {N}}\) is relatively compact in the weak-topology σ(L 1,L ∞). By Theorem C.6, it follows that the family \(\left (\xi _{n}\right)_{n\in \mathbb {N}}\) is uniformly integrable. We also know that ξ n →ξ P-a.s. Hence, by Theorem C.7, we see that ξ n →ξ in the L 1-norm and, consequently, \(\xi _{n}\xrightarrow [\:n\rightarrow +\infty ]{\sigma \left (L^{1},L^{\infty }\right)}\xi \). The statement of the lemma then follows by the uniqueness of the limit. □
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This work has been financially supported by the European Community's FP 7 Program under contract PITN-GA-2008-213841, and Marie Curie ITN ≪Controlled Systems ≫.
The three authors worked together on the manuscript and approved its final version.
Numerix, Milano, Italy
Matteo Ludovico Bedini
Université de Bretagne Occidentale, Brest, France
Rainer Buckdahn
School of Mathematics, Shandong University, Jinan, Shandong Province, People's Republic of China
Friedrich-Schiller-Universität, Fakultät für Mathematik und Informatik, Institut für Stochastik, Jena, Germany
Hans-Jürgen Engelbert
Correspondence to Matteo Ludovico Bedini.
Bedini, M., Buckdahn, R. & Engelbert, HJ. On the compensator of the default process in an information-based model. Probab Uncertain Quant Risk 2, 10 (2017). https://doi.org/10.1186/s41546-017-0017-4
Default time
Totally inaccessible stopping time
Brownian bridge on random intervals
Compensator process | CommonCrawl |
Polarization of an algebraic form
In mathematics, in particular in algebra, polarization is a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables. Specifically, given a homogeneous polynomial, polarization produces a unique symmetric multilinear form from which the original polynomial can be recovered by evaluating along a certain diagonal.
This article is about formulas for higher-degree polynomials. For formula that relates norms to inner products, see Polarization identity.
Although the technique is deceptively simple, it has applications in many areas of abstract mathematics: in particular to algebraic geometry, invariant theory, and representation theory. Polarization and related techniques form the foundations for Weyl's invariant theory.
The technique
The fundamental ideas are as follows. Let $f(\mathbf {u} )$ be a polynomial in $n$ variables $\mathbf {u} =\left(u_{1},u_{2},\ldots ,u_{n}\right).$ Suppose that $f$ is homogeneous of degree $d,$ which means that
$f(t\mathbf {u} )=t^{d}f(\mathbf {u} )\quad {\text{ for all }}t.$
Let $\mathbf {u} ^{(1)},\mathbf {u} ^{(2)},\ldots ,\mathbf {u} ^{(d)}$ be a collection of indeterminates with $\mathbf {u} ^{(i)}=\left(u_{1}^{(i)},u_{2}^{(i)},\ldots ,u_{n}^{(i)}\right),$ so that there are $dn$ variables altogether. The polar form of $f$ is a polynomial
$F\left(\mathbf {u} ^{(1)},\mathbf {u} ^{(2)},\ldots ,\mathbf {u} ^{(d)}\right)$
which is linear separately in each $\mathbf {u} ^{(i)}$ (that is, $F$ is multilinear), symmetric in the $\mathbf {u} ^{(i)},$ and such that
$F\left(\mathbf {u} ,\mathbf {u} ,\ldots ,\mathbf {u} \right)=f(\mathbf {u} ).$
The polar form of $f$ is given by the following construction
$F\left({\mathbf {u} }^{(1)},\dots ,{\mathbf {u} }^{(d)}\right)={\frac {1}{d!}}{\frac {\partial }{\partial \lambda _{1}}}\dots {\frac {\partial }{\partial \lambda _{d}}}f(\lambda _{1}{\mathbf {u} }^{(1)}+\dots +\lambda _{d}{\mathbf {u} }^{(d)})|_{\lambda =0}.$
In other words, $F$ is a constant multiple of the coefficient of $\lambda _{1}\lambda _{2}\ldots \lambda _{d}$ in the expansion of $f\left(\lambda _{1}\mathbf {u} ^{(1)}+\cdots +\lambda _{d}\mathbf {u} ^{(d)}\right).$
Examples
A quadratic example. Suppose that $\mathbf {x} =(x,y)$ and $f(\mathbf {x} )$ is the quadratic form
$f(\mathbf {x} )=x^{2}+3xy+2y^{2}.$
Then the polarization of $f$ is a function in $\mathbf {x} ^{(1)}=\left(x^{(1)},y^{(1)}\right)$ and $\mathbf {x} ^{(2)}=\left(x^{(2)},y^{(2)}\right)$ given by
$F\left(\mathbf {x} ^{(1)},\mathbf {x} ^{(2)}\right)=x^{(1)}x^{(2)}+{\frac {3}{2}}x^{(2)}y^{(1)}+{\frac {3}{2}}x^{(1)}y^{(2)}+2y^{(1)}y^{(2)}.$
More generally, if $f$ is any quadratic form then the polarization of $f$ agrees with the conclusion of the polarization identity.
A cubic example. Let $f(x,y)=x^{3}+2xy^{2}.$ Then the polarization of $f$ is given by
$F\left(x^{(1)},y^{(1)},x^{(2)},y^{(2)},x^{(3)},y^{(3)}\right)=x^{(1)}x^{(2)}x^{(3)}+{\frac {2}{3}}x^{(1)}y^{(2)}y^{(3)}+{\frac {2}{3}}x^{(3)}y^{(1)}y^{(2)}+{\frac {2}{3}}x^{(2)}y^{(3)}y^{(1)}.$
Mathematical details and consequences
The polarization of a homogeneous polynomial of degree $d$ is valid over any commutative ring in which $d!$ is a unit. In particular, it holds over any field of characteristic zero or whose characteristic is strictly greater than $d.$
The polarization isomorphism (by degree)
For simplicity, let $k$ be a field of characteristic zero and let $A=k[\mathbf {x} ]$ be the polynomial ring in $n$ variables over $k.$ Then $A$ is graded by degree, so that
$A=\bigoplus _{d}A_{d}.$
The polarization of algebraic forms then induces an isomorphism of vector spaces in each degree
$A_{d}\cong \operatorname {Sym} ^{d}k^{n}$
where $\operatorname {Sym} ^{d}$ is the $d$-th symmetric power of the $n$-dimensional space $k^{n}.$
These isomorphisms can be expressed independently of a basis as follows. If $V$ is a finite-dimensional vector space and $A$ is the ring of $k$-valued polynomial functions on $V$ graded by homogeneous degree, then polarization yields an isomorphism
$A_{d}\cong \operatorname {Sym} ^{d}V^{*}.$
The algebraic isomorphism
Furthermore, the polarization is compatible with the algebraic structure on $A,$so that
$A\cong \operatorname {Sym} ^{\cdot }V^{*}$
where $\operatorname {Sym} ^{\cdot }V^{*}$ is the full symmetric algebra over $V^{*}.$
Remarks
• For fields of positive characteristic $p,$ the foregoing isomorphisms apply if the graded algebras are truncated at degree $p-1.$
• There do exist generalizations when $V$ is an infinite dimensional topological vector space.
See also
• Homogeneous function – Function with a multiplicative scaling behaviour
References
• Claudio Procesi (2007) Lie Groups: an approach through invariants and representations, Springer, ISBN 9780387260402 .
| Wikipedia |
List of works by Petr Vaníček
This is the list of works by Petr Vaníček.
Remarks
• B Book
• TB Textbook
• LN Lecture Notes
• PR Paper in a Refereed Journal
• R Research Paper
• C Critique, Reference Paper
• IP Invited Paper to a Meeting
• NP Paper Read at a Meeting
• TH Thesis
• RT Report (non-technical)
• RW Review Paper (technical)
List of works
NrTypeAuthorsYearTitlePublisherLanguage
1TH, RVaníček, P.1959Use of triplets of stars in the method of equal altitudes.Czech Technical University, PragueCzech, Eng. thesis
2TBPleskot, V., Culik, J., Kafka, J. and Vaníček, P.1964Basic Programming for Ural I.SNTL, PragueCzech
3R, PRVaníček, P.1964Use of complex numbers for adjusting geodetic traverses. Aplikace Matematiky, No. 3, pp. 35–39,NCSAV, PragueCzech
4TBCulik, J., T. Hruskova, P. Vaníček1965Programming for Ural 2.SNTL, PragueCzech
5R, PRKabelac, J., P. Vaníček1965Computation of deviations of vertical from astronomical observations using the method of equal altitudes, Travaux de l'Institute de Geophysique 1964, No. 197, pp. 41–65NCSAV, Prague
6C, PRVaníček, P.1965Reference of J. Sutti's Paper, A Priori Exactitude in Reading on Nomograms.Referativnij Zhurnal-Mathematika, MoscowRussian
7PRVaníček, P.1965Teaching computer programming and numerical analysis in the University of Paris. Vysoka Skola, No. 7, pp. 7–12SPN, PragueCzech
8TH, RVaníček, P.1967Unharmonic analysis and its applications in geophysics. Ph.D. dissertationCzechoslovak Academy of Sciences, PragueCzech
9IP, RVaníček, P.1968Unharmonic analysis of the drift of horizontal pendulums.International Symposium Exchange of Experience with Tiltmeter Observations and a Critical Analysis of their Physical Significance, MoscowRussian
10R, PRVaníček, P.1969Approximate spectral analysis by least-squares fit.Astroph. and Sp. Sci., Vol. 4, pp. 387–39l
11R, PRVaníček, P.1969New analysis of the earth pole wobble.Studia Geoph. et Geod., Vol. 13, pp. 225–230.
12NP, RLennon, G. W., P. Vaníček1969Calibration tests and the comparative performance of horizontal pendulums at a single station.Proc. of 6th Int. Symp. on Earth Tides, Strasbourg, pp. 183–193.
13NP, RVaníček, P.1969Theory of motion of horizontal pendulum with a Zollner suspensionProc. 6th Int. Symp. on Earth Tides, Strasbourg, pp. 180–182
14NP, RVaníček, P.1969An analytical technique to minimize noise in a search for lines in the low frequency spectrumProc. of 6th Int. Symp. on Earth Tides, Strasbourg, pp. 170–173
15IP, RQuraishee, G. S., P. Vaníček1970A search for low frequencies in residual tide and mean sea level observations by means of the least-squares spectral analysisRep. on Symp. on Coastal Geodesy, Munich, pp. 485–493
16NP,RVaníček, P.1970Spectral analysis by least-squares fit51st An. Meet. of AGU, Washington
17LNVaníček, P.1971Introduction to adjustment calculusDepartment of Surveying Engineering Lecture Notes, University of New Brunswick, Fredericton
18R, PRVaníček, P.1971Further development and properties of the spectral analysis by least-squaresAstroph. and Sp. Scie., Vol. 12, pp. 10–33
19IP, RVaníček, P.1971Spectral analysis by least-squares fitIUGG meeting, Moscow
20LNVaníček, P.1971Physical Geodesy 1Department of Surveying Engineering Lecture Notes 21, University of New Brunswick, FrederictonAlso translated to Spanish as Geodesia Fisica Aplicada, tomo I. 1980, Secretaria de programacion y presupuesto, Detenal, Mexico, DF (several editions).
21NP, RGregerson, L. F., G. Symonds, P. Vaníček,1971Reports on experiments with a gyroscope equipped with electronic registrationIUGG meeting, Moscow
22NP,RGregerson, L. F., G. Symonds, P. Vaníček,1971Reports on experiments with a gyroscope equipped with electronic registrationFIG Meeting, Wiesbaden
23R, PRVaníček, P.1971An attempt to determine long-periodic variations in the drift of horizontal pendulumsStudia Geoph. et Geod., Vol. 15, pp. 416–420
24C, PRVaníček, P.1971Critique of M. Romanowski's "The Theory of Random Errors Based on the Concept of Modulated Normal Distribution."The Canadian Surveyor, Vol. 25, pp. 467–468
25R, PRVaníček, P., G. W. Lennon,1972The theory of motion of the horizontal pendulum with a Zollner suspensionStudia Geoph. et Geod., Vol. 16, pp. 30–50
26R, PRVaníček, P.1972Dynamical aspects of suspended gyro-compassThe Canadian Surveyor, Vol. 26, pp. 72–83
27RVaníček, P., J. D. Boal, T. A. Porter1972Proposed new system of heights for CanadaSurveys and Mapping Branch Report No. 72-3, Ottawa
28R, PRVaníček, P., A. C. Hamilton1972Further analysis of vertical crustal movement observations in the Lac St. Jean area, QuebecCan. J. of Earth Sci., Vol. 9, pp. 1139–1147
29LNVaníček, P.1972Brief outline of the Molodenskij theoryDepartment of Surveying Engineering Lecture Notes 23, University of New Brunswick, Fredericton
30LNVaníček, P.1972Physical geodesy IIDepartment of Surveying Engineering Lecture Notes 24, University of New Brunswick, FrederictonAlso translated into Spanish as Geodesia Fisica Applicade, tomo II, 1978. Secretaria de programacion y presupuesto, Detenal, Mexico, DF (several editions).
31LNVaníček, P.1972The earth-pole wobbleDepartment of Surveying Engineering Lecture Notes 25, University of New Brunswick, Fredericton
32LNVaníček, P., D. E. Wells1972The least-squares approximation and related topicsDepartment of Surveying Engineering Lecture Notes 22, University of New Brunswick, Fredericton
33LNVaníček, P.1972TensorsDepartment of Surveying Engineering Lecture Notes 27, University of New Brunswick, Fredericton
34R, PRMerry, C. L., P. Vaníček1973Horizontal control and the geoid in CanadaThe Canadian Surveyor, Vol. 27, pp. 23–31
35RVaníček, P., D. Woolnough1973A programme package for packing and generalising digital cartographic dataDepartment of Surveying Engineering Technical Report 23, University of New Brunswick, Fredericton
36NP,RMerry, C.L., P. Vaníček1973Computation of the geoid from deflections of vertical using a least-squares surface fitting technique54th Ann. Meeting of AGU, Washington
37NPVaníček, P., C. L. Merry1973The influence of geoid-ellipsoid separation on the Canadian horizontal controlCIS Annual Meeting, Ottawa
38R, PRVaníček, P., C. L. Merry1973Determination of the geoid from deflections of vertical using a least-squares surface fitting techniqueBulletin Géodésique, No. 109, pp. 261–279
39LNVaníček, P.1973Gravimetric satellite geodesyDepartment of Surveying Engineering Lecture Notes 32, University of New Brunswick, Fredericton
40LNVaníček, P.1973The earth tidesDepartment of Surveying Engineering Lecture Notes 36, University of New Brunswick, Fredericton
41PRKrakiwsky, E. J., P. Vaníček, L. A. Gale, A. C. Hamilton1973Objectives and philosophy of the International Symposium on Problems Related to the Redefinition of North American NetworksThe Canadian Surveyor, Vol. 27, p. 246
42IP, RVaníček, P.1973Use of relevelling in small regions for vertical crustal movement determination3rd GEOP Conference, Columbus, Ohio
43LNVaníček, P.1973Introduction of adjustment calculus (2nd rev. ed.)Department of Surveying Engineering Lecture Notes 35, University of New Brunswick, FrederictonAlso translated into Spanish.
44R, PRMerry, C. L., P. Vaníček,1974The geoid and translation componentsThe Canadian Surveyor, Vol. 28, pp. 56–62
45C, PRVaníček, P., E. J. Krakiwsky,1974Letter to the editorThe Canadian Surveyor, Vol. 28, pp. 91–92
46IP, RWKrakiwsky, E. J., P. Vaníček,1974Geodetic research needed for the redefinition of the size and shape of CanadaSymposium Geodesy for Canada, pp. A1-A17
47RMerry, C. L., P. Vaníček,1974A method for astrogravimetric geoid determinationDepartment of Surveying Engineering Technical Report 27, University of New Brunswick, Fredericton
48NP,R,PRVaníček, P., D. E. Wells,1974Positioning of horizontal geodetic datumsThe Canadian Surveyor, Vol. 28, No. 5, pp. 531–538
49NP, R, PRMerry, C. L., P. Vaníček,1974A technique for determining the geoid from a combination of astrogeodetic and gravimetric deflectionsThe Canadian Surveyor, Vol. 28, No. 5, pp. 549–554
50IP, RWVaníček, P.1974Present status and the future of geodesy in CanadaFirst CGU Meeting, St. John's
51R, PRVaníček, P., D. Christodulides1974A method for evaluating vertical crustal movements from scattered geodetic relevellingsCanad. J. of Earth Sci., Vol. 11(5): 605-610
52R, PRThomson, D.B., P. Vaníček1974Note on reduction of spatial distances to a reference ellipsoidThe Survey Review, XXII, pp, 1-4
53RChrzanowski, A., A.C. Hamilton, E.J. Krakiwsky, P. Vaníček1974An evaluation of the geodetic networks in OntarioResearch report prepared for the Ministry of Natural Resources, Province of Ontario, Toronto
54BVaníček, P. (ed)1974Proceedings of the International Symposium on Problems Related to the Redefinition of North American Geodetic NetworksFredericton, May, CIS, Ottawa
55RVaníček, P., A. Chrzanowski, E. J. Krakiwsky, A. C. Hamilton1974A critical review of alternatives with respect to the geodetic system of the Maritime ProvincesResearch report prepared for the Land Registration and Information Service, Fredericton
56RTVaníček, P.1975Contribution to Geosciences in Canada (Geodesy)Canadian Geoscience Council, 1974
57NP, R, PRVaníček, P.1975Vertical crustal movements in Nova Scotia as determined from scattered geodetic relevellingsTectonophysics, 29, pp. 183–189
58RVaníček, P.1975Report on geocentric and geodetic datumsDepartment of Surveying Engineering Technical Report 32, University of New Brunswick, Fredericton
59RNassar, M.M., P. Vaníček1975Levelling and gravityDepartment of Surveying Engineering Technical Report 33, University of New Brunswick, Fredericton
60RTKrakiwsky, E.J., P. Vaníček (eds)1975Geodesy in Canada 1971-1974Canadian national report to IAG, Canadian National Committee for IUGG
61PRWells, D. E., P. Vaníček1975Alignment of geodetic and satellite coordinate systems to the average terrestrial systemBulletin Géodésique, 117, pp. 241–257
62RHamilton, A. C., A. Chrzanowski, P. Vaníček1975A critical review of existing and possible map projection systems for the Maritime ProvincesResearch report prepared for the Land Registration and Information Service, Fredericton
63NPHamilton, A. C., A. Chrzanowski, P. Vaníček,R. Castonguay1975Map projections, grid coordinates, and geo-codesCommonwealth Survey Officers Conference, Cambridge
64RMasry, S. E., P. Vaníček1975Deviation of camera rotations from LTN-51 inertial navigation systemsDepartment of Surveying Engineering Technical Report 34, University of New Brunswick, Fredericton
65R, PRVaníček, P., D. F. Woolnough1975Reduction of linear cartographic data based on generation of pseudo-hyperbolaeThe Cartographic Journal, Vol. 12(2): 112-119
66LNVaníček, P.1976Physical geodesyDepartment of Surveying Engineering Lecture Notes 43, University of New Brunswick, Fredericton
67BVaníček, P. (ed)1976Proceedings of the 1975 CGU Symposium of Satellite Geodesy and GeodynamicsPublications of the Earth Physics Branch No. 45, 3, Ottawa
68R, PRVaníček, P.1976Vertical crustal movements pattern in Maritime CanadaCanad. J. of Earth Sci., 13(5): 661-667
69IP, RWVaníček, P.1976Papel de la geodesia en la sociedadProceedings of Congreso Nacional de Fotogrametria, Fotointerpretacion y Geodesia, Mexico City, May
70RTVaníček, P.1976Contribution to Geosciences in Canada (Geodesy) 1976Canadian Geoscience Council
71RVaníček, P.1977Vertical crustal movements in southern OntarioEarth Physics Branch, Open File Report K10-77-12, Ottawa
72IP, RVaníček, P., M. M. Nassar, F.W. Young1977Vertical crustal movements and sea-level recordsAnnual Meeting of CGU, Vancouver
73RVaníček, P.1977Sea level variations in Maritime CanadaResearch report prepared for the Geodetic Survey of Canada, Ottawa
74IP, RW,PRP. Vaníček1977Geodesy and geophysicsIN: Proceedings of Geophysics in the Americas, Ottawa, September 1976, Eds. Tanner and Dence, Publications of the Earth Physics Branch No. 46, 3, pp. 45–48Also translated into Spanish.
75R, PRWells, D.E., P. Vaníček1978Least squares spectral analysisBedford Institute of Oceanography Report No. BI-R-78-8
76R, PRVaníček, P.1978To the problem of noise reduction in sea level records used in vertical crustal movement detectionPhys. Earth Plan. Int. 17, pp. 265–280
77R, IP, RT, PRVaníček, P., E. J. Krakiwsky1978Geodesy rebornProceedings of Annual Meeting of ACSM-ASP, Washington, February, pp. 369–373,Also in Surveying and Mapping, XXXVIII(1): 23-26
78IP,RVaníček, P.1978Contemporary vertical crustal movements in southern Ontario from geodetic dataAGU Annual Meeting, Miami, April
79NP, RTNagy, D., P. Vaníček1978Preparation of contemporary vertical crustal movement map of Canada: A progress reportCGU Annual Meeting, London, Ontario, May
80RAnderson, E., P. Vaníček1978Suggestions relating to the classification, accuracy, and execution of vertical control surveysResearch report prepared for the Geodetic Survey of Canada, Ottawa
81IP, RWVaníček, P.1978Earthquake prediction: Some myths and factsIXth World Congress, S.V.U., Cleveland, Ohio, October
82IP, RWVaníček, P.1978Gravity needs in geodesyContribution to U.S. National Academy of Science Workshop on Gravity Satellites, Washington, D.C., November
83R, PRVaníček, P., M. Elliott, R. O. Castle1979Four dimensional modelling of recent vertical movements in the area of the southern California upliftTectonophysics, 52, pp. 287–300.
84R, RW, PRLambert, A., P. Vaníček1979Contemporary crustal movements in CanadaCanad. J. of Earth Sci., 16(3, part 2): 647–668.
85IP, RTVaníček, P.1979Vertical crustal movements—Terrestrial techniquesGEOP-9 Conference (Session Leader's Report), EOS, Transactions of AGU, 60(28): 524.
86PRVaníček, P.1979Tensor structure and the least squaresBulletin Géodésique, 53, pp. 221–225.
87NPNagy, D., P. Vaníček1979Map of contemporary vertical crustal movements in Canada6th Annual Meeting CGU, Fredericton.
88NPSteeves, R.R., P. Vaníček1979Earth tide tilt observations at UNB6th Annual Meeting CGU, Fredericton.
89RTVaníček, P.1979UNB contribution to Canadian National Report for IAGCanberra, Australia.
90PRVaníček, P.1979Sixth annual meeting of Canadian Geophysical UnionGeoscience Canada 6(4): 216–217.
91PRVaníček, P.1979Dr. E.J. Krakiwsky leaves the University of New Brunswick for the University of CalgaryThe Canadian Surveyor, 33(4): 396.
92NPAnderson, E. G., P. Vaníček1979Redefinition of the vertical geodetic network in CanadaIUGG General Assembly, Canberra.
93B, IP, RWVaníček, P.1980Inertial technology in surveyingIn: Notes for CIS Regional Geodetic Seminars, ed. G. Lachapelle, CIS, Ottawa, pp. 103–120.
94R, RTVaníček, P., D. Nagy1980Report on the compilation of the map of vertical crustal movements in CanadaEarth Physics Branch, Open File Report No. 80-2, Ottawa.
95R, PRVaníček, P., E. W. Grafarend1980On the weight estimation in levellingNOAA Technical Report, NOS 86 NGS 17.
96R, RW, PRVaníček, P., R. O. Castle, E. I. Balazs1980Geodetic levelling and its applicationsReviews of Geophysics and Space Physics, 18(2): 505–524, Reprinted in Advances in Geodesy, eds. E.W. Grafarend and R.M. Rapp, AGU, Washington, 1984.
97R, PRVaníček, P.1980Tidal corrections to geodetic quantitiesNOAA Technical Report, NOS 83 NGS 14, p. 30.
98R, PRVaníček, P., D. Nagy1980The map of contemporary vertical crustal movements in CanadaEOS, Transactions of AGU, 61(4): 145–147.
99C, PRVaníček, P.1980Review of "Proceedings of First International Conference on the Redefinition of the North American Geodetic Vertical Control Network"EOS, Transactions of AGU, 61(21): 467.
100NPVaníček, P.1980Heights based on observed gravityProceedings of the Second International Symposium on Problems Related to the Redefinition of North American Vertical Geodetic Networks, Ottawa, May, pp. 553–566.
101NPCastle, R. O. and P. Vaníček,1980Interdisciplinary considerations in the formulation of the new North American vertical datum.Proceedings of the Second International Symposium on Problems Related to the Redefinition of North American Vertical Geodetic Networks, Ottawa, May, pp. 285–300.
102IPVaníček, P.1980Vertical positioning—the state of the art.AGU/CGU Spring Meeting, Toronto.
103NPThapa, K., P. Vaníček1980A study of the effect of individual observations in horizontal geodetic networks.AGU/CGU Spring Meeting, Toronto.
104NPDelikaraoglou, D., D. E. Wells, P. Vaníček1980Analysis of GEOS-3 altimetry in Hudson Bay.AGU/CGU Spring Meeting, Toronto (abstract: EOS, Transactions of the American Geophysical Union, 61, p. 208).
105RVaníček, P.1980Investigation of some problems related to the redefinition of Canadian levelling networks.Department of Surveying Engineering Technical Report 72, University of New Brunswick, Fredericton.
106IPVaníček, P.1980How artificial satellites, the moon, and quasars help us learn more about the shape of the earth.Xth World Congress, S.V.U., Washington, October.
107NP, PRVaníček, P. and D. Nagy1981On the compilation of the map of contemporary vertical crustal movements in Canada.Tectonophysics, 71, pp. 75–87.
108NPWells, D. E., D. Delikaraoglou, P. Vaníček1981Navigating with the Global Positioning System, today and in the future.Presented at 74th Annual Meeting of CIS, St. John's, May.
109RWells, D. E., P. Vaníček, D. Delikaraoglou1981Pilot study of the application of NAVSTAR/GPS to geodesy in Canada.Research report for the Geodetic Survey of Canada. Department of Surveying Engineering Technical Report 76, University of New Brunswick, Fredericton.
110R, PRVaníček, P., K. Thapa, D. Schneider1981The use of strain to identify incompatible observations and constraints in horizontal geodetic networks.Manuscripta Geodaetica, 6(3): 257–281.
111C, PRVaníček, P.1981Review of "Fundamental Principles of General Relativity Theories".Manuscripta Geodaetica, 6(2): 245.
112C, PRVaníček, P.1981Review of "Earth History and Plate Tectonics."Manuscripta Geodaetica, 6(2): 245–246.
113RMerry, C. L., P. Vaníček1981The zero frequency response of sea level to meteorological influences.Department of Surveying Engineering Technical Report 82, University of New Brunswick, Fredericton, 83 pages.
114IPVaníček, P.1981Recent shifts in the geodetic perception of the role of the earth's gravity field.Fall Meeting of AGU, San Francisco, December.
115PRVaníček, P.1981Statement of intent for the election to presidency of AGU geodesy section.EOS, Transactions of AGU, 62(50): 1186.
116TBVaníček, P., E. J. Krakiwsky,1982Geodesy: The Concepts.North-Holland, Amsterdam, 691 pages.
117R, PRWells, D. E., D. Delikaraoglou, P. Vaníček1982Marine navigation with NAVSTAR/GPS today and in the future.The Canadian Surveyor, 36(1): 9-28.
118R, PRVaníček, P.1982To the problem of holonomity of height systems.Letter to the editor, The Canadian Surveyor, 36(1): 122–123.
119IP, PR, R, RTVaníček, P., A. C. Hamilton1982Do geodesy and digital cartography belong in the conceptual model for integrated surveying and mapping?Proceedings of the 42nd Annual Meeting of ACSM, Denver, March, pp. 289–296.
120R, NPDare P., P. Vaníček1982Strength analysis of horizontal networks using strain.Proceedings of Meeting of FIG Study Group 5B, (Survey Control Networks), Aalborg, July, pp. 181–196.
121NPCarrera, G., P. Vaníček1982Effect of reference ellipsoid misalignment on deflection components and geodetic azimuth.Presented to Congreso National de Fotogrametria, Fotointerpretacion y Geodesia, Mexico City, September.
122C, PRVaníček, P.1982.Review of "Multidisciplinary Approach to Earthquake Prediction".Manuscripta Geodaetica, 8(1): 80.
123RDavidson, D. D. Delikaraoglou, R. B. Langley, B. Nickerson, P. Vaníček and D. E. Wells1982Global Positioning System: Differential positioning simulations.Department of Surveying Engineering Technical Report 90, University of New Brunswick, Fredericton, 141 pages.
124IPVaníček, P.1982Is sea level really level?Congress S.V.U., Pittsburgh, October.
125IPVaníček, P.1982Geometrical strength analysis: A tool for assessment of geodetic networks.Presented to Fifth UN Regional Cartographic Conference for Africa, Cairo.
126B, IP, RWBlais, J. A. R. and P. Vaníček1983Overview of advanced estimation methods in surveying.In: Notes for CIS Adjustments Seminar, ed. E.J. Krakiwsky, CIS, Ottawa, pp. 308–350.
127PR, R, NPMerry, C. L., P. Vaníček1983Investigation of local variation of sea surface topography.IAG Symposium on Marine Geodesy, Tokyo, May. Marine Geodesy, 7(1-4): 101–126.
128PR, RDare, P., P. Vaníček,1983The use of strain in the design of horizontal networks.Geodeszja #79, Akademia Gorniczo-Hutnicza, Kraków, pp. 133–144.
129TBVaníček, P., M. Craymer1983Autocorrelation functions as a diagnostic tool in levelling.In H. Pelzer and W. Niemeier (editors), Precise Levelling, Dummler Verlag, Bonn, pp. 327–341.
130R, NPDelikaraoglou, D., D. Davidson, R. B. Langley, B. G. Nickerson, P. Vaníček, D. E. Wells1983Geodetic baseline accuracies from differential GPS observations.Spring Meeting of AGU, Baltimore, May–June (abstract: EOS, Transactions of the American Geophysical Union, 64, p. 210).
131RPLachapelle, G., P. Vaníček, (eds).1983Geodesy, gravity and geodynamics in Canada.Canadian Quadriannual Report to IAG, Hamburg, August, 87 pages.
132R, NPWells, D. E., D. A. Davidson, D. Delikaraoglou, R. B. Langley, B. G. Nickerson, P. Vaníček1983The comparative accuracies of Global Positioning System differential positioning modes.Proceedings of the IAG Symposium on The Future of Terrestrial and Space Methods for Positioning, Hamburg, August. The Ohio State University, Columbus, Vol. 2, pp. 192–221.
133IPVaníček, P.1983Geodetic symbiosis of the developed and developing countries.Proceedings of IAG Symposium on Strategy for Solving Geodetic Problems in Developing Countries, Hamburg, August, Vol. 2, pp. 597–605.
134IP, RVaníček, P., D. E. Wells, A. Chrzanowski, A. C. Hamilton, R. B. Langley, J. D. McLaughlin, B. G. Nickerson1983The future of geodetic networks.Proceedings of the IAG Symposium on the Future of Terrestrial and Space Methods for Positioning, Hamburg, August. The Ohio State University, Columbus, Oh, Vol. 2, pp. 372–379.Also translated into Polish as "Przyszlosc sieci geodezyjnych", Przeglad Geodezyjny, 1984, LVI(8-9): 5–7.
135NP, RVaníček, P., S. John,1983Evaluation of geoid solutions for Canada using different kinds of data.Proceedings of IAG Symposium on Improved Gravity Field Estimations on Global Basis, Hamburg, August, Vol. 2, pp. 609–624.
136IP, RVaníček, P.1983Diagrammatic approach to adjustment calculus.Proceedings of T. Banachiewic's Commemorative Conference on Development of Theory and Techniques of Astronomical and Geodetic Calculations, Kraków, May. Geodezja #86, Akademia Gorniczo-Hutnicza, 1986, pp. 28–39.
137LNVaníček, P.1983Transit satellite positioning.Lecture notes for UNDP graduate course in geodesy for East Africa. Regional Centre for Services in Surveying, Mapping and Remote Sensing, Nairobi, Kenya, July, p. 21.
138PR, RVaníček, P., M. Craymer1983Autocorrelation functions in the search for systematic errors in levelling.Manuscripta Geodaetica, 8(4): 321–341.
139PR, RVaníček, P., R. B. Langley, D. E. Wells, D. Delikaraoglou1984Geometrical aspects of differential GPS positioning.Bulletin Géodésique, 58, pp. 37–52.
140NP, RWCarrera, G., P. Vaníček1984Review of techniques for determining vertical crustal movements from levelling data.Proceedings of Third International Symposium on Land Subsidence, Venice, March, pp. 195–202.
141R, NPPagiatakis, S. D., R. B. Langley, P. Vaníček1984Ocean tide loading: A global model for the analysis of VLBI observations.Proceedings of the Third International Symposium on the Use of Artificial Satellites for Geodesy and Geodynamics, Ermioni, Greece, September, 1982, pp. 328–340.
142IPVaníček, P., G. Beutler, A. Chrzanowski, W. Faig, R. Langley, J. McLaughlin, D. E. Wells,1984Implications of new space techniques in land surveying.Presented at Annual Meeting of CIS, Quebec City, May.Also translated into Polish as "Wplyw rozwoju metod kosmicznych na pomiary terenowe". Przeglad Geodezyjny, 1984 LVI(11): 10–11.
143RLangley, R. B., G. Beutler, D. Delikaraoglou, B. G. Nickerson, R. Santerre, P. Vaníček, D. E. Wells1984Studies in the application of the Global Positioning System to differential positioning.Department of Surveying Engineering Technical Report 108, University of New Brunswick, Fredericton, 201 pages.
144NP, RLugoe, F., P. Vaníček1984Strain effect of an existing (densified) network on a densification network.Spring Meeting of AGU, Cincinnati, May.
145NP, RBeutler, G., D. A. Davidson, R. B. Langley, R. Santerre, H.D. Valliant, P. Vaníček, D.E. Wells1984The Ottawa Macrometer™ experiment: An independent analysis.Annual Meeting of CGU (CGU/CMOS), Halifax, May–June.
146IP,RWCarrera, G., P. Vaníček1984The wide use of tide gauge observations in geodesy.Annual Meeting of CGU (CGU/CMOS), Halifax, May–June.
147NP, RPagiatakis, D., P. Vaníček1984Analysis of tidal tilt and gravity measurements at the Fredericton earth tide station.Annual Meeting of CGU (CGU/CMOS), Halifax, May–June.
148RCarrera, G., R. C. Gunn, P. Tetrault, P. Vaníček1984Preliminary research for the geodetic monitoring of Ontario Hydro's Cornwall dyke.Survey Science Technical Report 4, University of Toronto/Erindale College, Mississauga, 175 pages.
149RBeutler, G., D. A. Davidson, R. B. Langley, R. Santerre, P. Vaníček, D. E. Wells1984Some theoretical aspects of geodetic positioning using carrier phase difference observations of GPS satellites.Department of Surveying Engineering Technical Report 109, University of New Brunswick, Fredericton. Also Satellite Station Zimmerwald, University of Bern Astronomical Institute Technical Report 14, Bern, Switzerland, 79 pages.
150IP, RKleusberg, A., G. Beutler, D. Delikaraoglou, R. Langley, R. Santerre, R. Steeves, H. Valliant, P. Vaníček, D. Wells1984Comparison of Macrometer™ V-1000 and Texas Instruments 4100 GPS survey results.AGU Fall Meeting, San Francisco, December (abstract: EOS Transactions of the American Geophysical Union, 65, p. 853).
151IP, RWVaníček, P.1984Sea level and vertical crustal movements.AGU Chapman Conference on Vertical Crustal Motion, Harpers Ferry, WV, October.
152R, NPCraymer, M., P. Vaníček1984Further analysis of the 1981 southern California field test for levelling refraction.AGU Chapman Conference on Vertical Crustal Motion, Harpers Ferry, WV, October.
153RW, PRVaníček, P., G. Beutler, A. Chrzanowski, W. Faig, R. B. Langley, J. D. McLaughlin, D. E. Wells1984Implications of new space techniques in land surveying.South African Journal, 19(6), pp. 32–36.
154PR,RVaníček, P., G. Carrera1985Reference ellipsoid misalignment, deflection components and geodetic azimuths.The Canadian Surveyor, 39(2): 123–130.
155RWCarrera, G., P. Vaníček1985The use of sea level tide gauge observations in geodesy.Lighthouse, Journal of the Canadian Hydrographic Service, Ed. No. 31, May, pp. 13–15.
156R, NPCarrera, G., P. Vaníček1985A temporal homogenization of the Canadian height network.Proceedings of NAVD '85 Symposium, Rockville, MD, April, pp. 217–226.
157R, NPCraymer, M., P. Vaníček1985An investigation of systematic errors in Canadian levelling lines.Proceedings of NAVD '85 Symposium, Rockville, MD, April, pp. 441–450.
158R,NPVaníček, P., A. Kleusberg, R. B. Langley, R. Santerre, D. E. Wells1985On the elimination of biases in processing differential GPS observations.Proceedings of the First International Symposium on Precise Positioning with GPS, Rockville, MD, April, Vol. I, pp. 315–324.
159R, NPKleusberg, A., R. B. Langley, R. Santerre, P. Vaníček, D. E. Wells, G. Beutler1985Comparison of survey results from different types of GPS receivers.Proceedings of the First International Symposium on Precise Positioning with GPS, Rockville, MD, April, Vol. II, pp. 579–592.
160R, NPKleusberg, A., R. B. Langley, S. H. Quek, R. Santerre, P. Vaníček, D. E. Wells1985Experience with GPS at the University of New Brunswick.12th Annual Meeting of CGU, Calgary, May (abstract: Geophysics, 50(8):1382-1383).
161RW, IPVaníček, P.1985Geodetic uses of and techniques to determine mean sea level.Annual meeting of Geological Association of Canada, Fredericton, May.
162R, NPCraymer, M., P. Vaníček, A. Tarvydas1985Rigorous updating of adjusted networks.CIS Annual Meeting, Edmonton, May.
163R,NPKleusberg, A., P. Vaníček1985The geoid and how to get it.CIS Annual Meeting, Edmonton, May.
164RVaníček, P., G. Carrera, M. Craymer1985Corrections for systematic errors in the Canadian levelling networks.Survey Science Technical Report 10, University of Toronto/Erindale College, Mississauga. Also reprinted as Geodetic Survey of Canada Contract Report 85–0001, 128 pages.
165RVaníček, P., G. Beutler, A. Kleusberg, R. B. Langley, R. Santerre, D. E. Wells1985DIPOP: Differential Positioning Program package for the Global Positioning System.Department of Surveying Engineering Technical Report 115, University of New Brunswick, Fredericton. Also reprinted as Geodetic Survey of Canada Contract Report 85-005.
166BU.S. National Research Council (P. Vaníček one of the authors)1985Geodesy: A Look to the Future.Committee on Geodesy, Commission on Physical Sciences, Mathematics and Resources; National Academy Press, Washington, 179 pages.
167R, NPInzinga,T., P.Vaníček1985A two-dimensional navigation algorithm using a probabilistic force field.Proceedings of the Third International Symposium on Inertial Technology for Surveying and Geodesy, Banff, Alberta, September, pp. 241–255.
168R, NPPagiatakis, S., P. Vaníček1985Atmospheric perturbations on tidal tilt and gravity measurements at the UNB earth tides station.Proceedings of the 10th International Symposium on Earth Tides, Madrid, September, pp. 905–922.
169R, NPLangley, R. B., A. Kleusberg, D. Parrot, R. Santerre, P. Vaníček, D. E. Wells1985(Plans to do) DIPOP processing of observations from the spring 1985 GPS "bake-off".Fall Annual Meeting of AGU, San Francisco, December (abstract: EOS, Transactions of the American Geophysical Union, 66(46) p. 844).
170RWells, D.E., P. Vaníček, S. Pagiatakis1985Least-squares spectral analysis revisited.Department of Surveying Engineering Technical Report 84, University of New Brunswick, Fredericton, 68 pages.
171R, PRVaníček, P. and A. Kleusberg1985What an external gravitational potential can really tell us about mass distribution.Bollettino di Geofisica Teorica ed Applicata, Vol. XXCII, No. 108, December, pp. 243–250.
172R, PRVaníček, P., F. N. Lugoe1986Rigorous densification of horizontal network.Journal of Surveying Engineering, Vol. 112, No. 1, pp. 18–29.
173TBVaníček, P. and E. J. Krakiwsky1986Geodesy: The Concepts.2nd rev. ed., North-Holland, Amsterdam, 697 pages.Reprinted in China and in Iran.
174R, NPLangley, R. B., A. Kleusberg, R. Santerre, D. E. Wells, P. Vaníček1986DIPOP: An interactive software package for precise positioning with GPS.ACSM/ISP Spring Annual Meeting, Washington.
175RVaníček, P., A. Kleusberg, R. G. Chang, H. Fashir, N. Christou, M. Hofman, T. Kling, T. Arsenault1986The Canadian Geoid.Geodetic Survey of Canada; Energy, Mines and Resources Canada, Ottawa, Technical Report No.???. Also Department of Surveying Engineering, University of New Brunswick, Fredericton, Technical Report No. 129, pp. 123.
176R, NPVaníček, P., A. Kleusberg1986Canadian experience with heterogeneous geoid data combinations.Presented to the IAG International Symposium on the Definition of the Geoid, Forence, Italy, May, Bolletino di Geodesia e Scienze Affini, XLV, No. 2, pp. 127–138.
177RVaníček, P., T. Arsenault, N. Christou, E. Derenyi, A. Kleusberg, S. Pagiatakis, D. E. Wells and R. Yazdani,1986Satellite altimetry applications for marine gravity.Department of Surveying Engineering Technical Report No. 128, University of New Brunswick, Fredericton, pp. 184.
178NPParrot, D., R. B. Langley, A. Kleusberg, R. Santerre, P. Vaníček, D. Wells1986The spring 1985 GPS High-Precision Baseline Test: Very preliminary results.Presented at the GPS Technology Workshop, Jet Propulsion Laboratory, Pasadena, CA, March.
179R, NPLangley, R. B., D. Parrot, R. Santerre, P. Vaníček, D. E. Wells1986The Spring 1985 GPS high-precision baseline test: Preliminary analyses with DIPOP.Proceedings of the Fourth International Geodetic Symposium on Satellite Positioning, University of Texas at Austin, Austin, TX, April, pp. 1073–1088.
180NP, RDoucet, K., H. Janes, D. Delikaraoglou, D. E. Wells, R. B. Langley, P. Vaníček1986Examples of geodetic GPS network design.Presented at joint Annual Meeting of Geological Association of Canada, Mineralological Association of Canada, and Canadian Geophysical Union, Ottawa, May.
181PRCraymer, M. and P. Vaníček1986Further analysis of the 1981 southern California field test for levelling refraction."Journal of Geophysical Research, Vol. 91, No. B9, August, pp. 9045-9055.
182RHamilton, A., D. Wells, A. Chrzanowski, W. Faig, R. Langley, P. Vaníček, J. McLaughlin1986Control survey study for LRIS.Department of Surveying Engineering Technical Report No. 124, University of New Brunswick, Fredericton, 110 pages.
183NP, RVaníček, P.1986Are geodetic networks going to survive the space age?Proceedings of Symposium on Geodetic Positioning for the Surveyor, University of Cape Town, Cape Town, South Africa, 8-9, September, pp. 63-70.
184NP, RVaníček, P.1986The accuracy of GPS-determined positions.Proceedings of Symposium on Geodetic Positioning for the Surveyor, University of Cape Town, Cape Town, South Africa, 8-9, September, pp. 133-142.
185Vaníček, P.1986Letter to the Editor.The Canadian Surveyor, Vol. 40, No. 1, p. 53.
186RJanes, H., K. Doucet, B. Roy, D. E. Wells, R. B. Langley, P. Vaníček, M. Craymer1986:"GPSNET": A program for the interactive design of geodetic GPS networks.Canadian Geodetic Survey Contract Report No. 0SZ85-00115, Canadian Engineering Surveys Co. Ltd., Edmonton, Alberta, 200 pages.
187TBGuide to GPS Positioning1986Prepared under the leadership of D.E. Wells by N. Beck, D. Delikaraoglou, A. Kleusberg, E.J. Krakiwsky, G. Lachapelle, R. B. Langley, M. Nakiboglu, K.-P. Schwarz, J.M. Tranquilla, P. Vaníček, D.E. Wells.Canadian GPS Associates 1986, 600 pages.
188IPVaníček, P.1986Gravimetric Geoid for Canada."Mathematical-geodetic methods for the determination of geoid and topography", Workshop organized by Geodetic Institute of University of Stuttgart, Lambrecht, FRG, October 1–3.
189R, PRVaníček, P. and A. Kleusberg,1987The Canadian geoid—Stokesian approach.Manuscripta Geodaetica, 12(2), pp. 86–98.
190PRVaníček, P., and L. E. Sjöberg,1987A note on vertical crustal movement determination techniques.Department of Geodesy, Royal Institute of Technology, Tech. Report No. 9, 15 pages.
191PRVaníček, P. (editor), P. A. Cross, J. Hannah, L. Hradilek, R. Kelm, J. Makinen, C. L. Merry, L. E. Sjoberg, R. R. Steeves, P. Vanicek, and D. B. Zilkoski1987Four-dimensional geodetic positioningReport of the IAG SSG 4.96, Manuscripta Geodaetica, Vol. 12(3), pp. 147–222.
192IPVaníček, P.1987New technology helps geodesy to become useful for other earth sciences.Presented at School of Geodesy "A. Marussi", Erice, Sicily, 15–25 June.
193IPVaníček, P.1987Satellite altimetry: Application to marine gravity determination.Presented at School of Geodesy "A. Marussi", Erice, Sicily, 15–25 June.
194IP, NPVaníček, P.1987Impact of Post-Glacial Rebound on Positions.Presented at the scientific meeting of IAG, Section V, Vancouver, August 12.
195IP, NPSanterre, R., M. R. Craymer, A. Kleusberg, R.B. Langley, D. Parrot, S.H. Quek, P. Vaníček, D.E. Wells, F. Wilkins1987Precise Relative GPS Positioning with DIPOP 2.0.Presented to: IAG Section II Scientific Meetings: "Advanced Space Technology", IUGG XIX General Assembly, Vancouver, 14 August.
196IP, NPCraymer, M. R. and P. Vaníček,1987NETAN: A Program for the Interactive Analysis of Geodetic Networks.Presented to: Session 2: Geodetic Networks, IAG Symposium GSI (Positioning), IUGG XIX General Assembly, Vancouver, B.C., 14 August.
197NPVaníček, P.1987Four-Dimensional Geodetic Positioning in Contributions to Geodetic Theory and Methodology,Editor: K.-P. Schwarz, IAG, Section IV, p. 221-226.
198RVaníček, P., P. Tetreault and M. Goadsby,1987Use of GPS for the maintenance of Ontario networksUofT, Survey Science, Tech. Rep. #12, 120 pages.
199IP, PR, RCarrera, G. and P. Vaníček,1988A comparison of present sea level linear trends from tide gauges, map of crustal movements and radiocarbon curves in Eastern Canada,Presented to XIIth Congress of INQUA, Ottawa, Aug. 7, Aug. 7, 1987. Palaeogeography, palaeoclimatology, palaeoecology, 68, pp. 127–134.
200BVaníček, P.1988Satellite Geodesy & GeodynamicsContribution to "The Encyclopedia of Field and General Geology", Vol. XIV, edited by C.W. Finkl, Jnr. Van Nostrand Reinhold Company Inc., pp. 737–744.
201BVaníček, P.1988Guide to GPS PositioningPrepared under the leadership of D. Wells by N. Beck, D. Delikaraoglou, A. Kleusberg, E.J. Krakiwsky, G. Lachapelle, R. B. Langley, M. Nakiboglu, K.-P Schwarz, J.M. Tranquilla, P. Vaníček, D.E. Wells. Canadian GPS Associates 1986, 600 pages. (second edition)
202NPChristou, N., P. Vaníček and C. Ware1988Can the geoid add anything to our knowledge of the lithosphere?Presented at 15th Annual Meeting CGU, Saskatoon, Sa, May.1988
203NPVaníček, P.1988Hiking and biking with GPS : The Canadian Perspective.International GPS Workshop, Darmstadt, April 10–13. GPS-Techniques Applied to Geodesy and Surveying, Groten, E. and R. Strauss (editors), Springer's Lecture Notes in Earth Science #19, pp. 225–229.
204RWCraymer, M. R. and P. Vaníček1988Sequential adjustment methods for the maintenance of geodetic networksCISM Seminar on the Impact of NAD 83, CISM, pp. 243–262.
205IPVaníček, P. and M. Kwimbere1988Displacement versus strain.Proceedings of 5th International Symposium on Deformation Surveys, Fredericton, June 6–9, pp. 557–562.
206NPCarrera, G. and P. Vaníček1988Compilation of a new recent crustal movements map for CanadaProceedings of 5th International Symposium on Deformation Surveys, Fredericton, June 6–9, pp. 113–118.
207RTVaníček, P.1988CGU Takes OffEOS, 69 (20), May 17, page 594.
208IPVaníček, P.1989Position Oriented SocietyAbstract for Quo Vadimus Symposium, IUGG, Vancouver, Aug. 9-22.
209B,RWVaníček, P.1989Adjustment methodsIn: Encyclopaedia of Geophysics, ed R. Fairbridge, Van Nostrand Reinhold, pp. 21–26.
210BCohen, S. and P. Vaníček (eds)1989Slow Deformation and Transmission of Stress in the EarthProceedings of Symposium on Slow Deformations and Transmission of Stress in the Earth, IUGG General Assembly, Vancouver, B.C., August 1987), American Geophysical Union, Washington, D.C., 138 pages.
211PRCraymer, M. R., P. Vaníček and A. Tarvydas1989NETAN-a computer program for the interactive analysis of geodetic networksCISM Journal. 43(1), pp. 25–37.
212PR, RChristou, N., P. Vaníček and C. Ware1989Geoid and density anomaliesEOS., 70 (22), pp. 625–631.
213C, PRCraymer, M. and P. Vaníček1989Comment on "Saugus-Palmdale, California, Field Test for Refraction Error in Historical Levelling Surveys" by R.S. Stein, C.T. Whalen, S.R. Holdahl, W.E. Strange, and W. Thatcher, and Reply to "Comment on 'Further Analysis of the 1981 Southern California Field Test for Levelling Refraction by M.R. Craymer and P. Vaníček' by R.S. Stein, C.T. Whalen, S.R. Holdahl, W.E. Strange, and W. Thatcher."JGR, 94 (B6), pp. 7667–7672.
214IPCraymer, M. R., D. E. Wells, P. Vaníček, P., Rapatz and R. Devlin1989Specifications and Procedures for the Evaluation of Urban GPS Surveys.Proceedings of 5th International Geodetic Symposium on Satellite Positioning, Las Cruces, N.M., March 1989, pp. 815–824.
215NPCraymer, M.R. and P. Vaníček1989Sequential Adjustment Methods for the Maintenance of Geodetic NetworksCISM Annual Meeting, Halifax, June.
216NPDevlin, R., P. Vaníček, D. Wells, M. Craymer, P. and C. Barnes1989Urban GPS SurveysCISM Annual Meeting, Halifax, June.
217RCraymer, M. R., D. E. Wells and P. Vaníček1989Report on urban GPS research project phase III-Evaluation Volume 3: Specifications and Guidelines. Geodetic Research Services Limited contract report for the City of Edmonton, Transportation Dept.Engineering Division, Edmonton, Alberta, May 1989, 37 pages.
218RCraymer, M. R., D. E. Wells, P. Vaníček and P. Rapatz1989Report on urban GPS research project phase III-Evaluation Volume 2: Evaluation of urban GPS surveys. Geodetic Research Services Limited contract report for the City of Edmonton, Transportation Dept.Engineering Division, Edmonton, Alberta, March 1989, 250 pages.
219RCraymer, M. R., A. Tarvydas and P. Vaníček1989NETAN: A program package for the interactive covariance, strain and strength analysis of networks.Geodetic Survey of Canada Contract Report, DSS Contract No. OSQa83-00102, Surveys and Mapping Branch, Energy, Mines and Resources Canada, Ottawa, May 1987, 177 pages.
220RCarrera, G. and P. Vaníček1989"Response": A System for the Determination of Float-Type Tide Gauge Response Functions.Geodetic Research Services Limited Contract Report for DSS, OSC 88-00292-(014), 100 pages.
221IPVaníček, P., M. Craymer, and G. Carrera1989Recompilation of map of recent vertical crustal movements in Canada: a progress report.Paper presented at Annual CGU Meeting, Montreal, May 17–19.
222NPZhang, C., L. E. Sjöberg and P. Vaníček1989Accuracy of the geoid computed from gravity disturbances.Paper presented at annual CGU meeting, Montreal, May 17–19.
223PRVaníček, P.1989Review of Proceedings of INSMAP 86PAGEOPH. 132(3), pp. 609–610.
224IPBlitzkow, D, P. Vaníček and R.B. Langley1989Processamento de observaçoes GPS com o DIPOP.Paper presented at GPS Workshop at Federal University of Paraná, Curitiba, October.
225NPVaníček, P. and L. E. Sjöberg1989Kernel modification in generalized Stokes's technique for geoid determination.Proceedings of General Meeting of IAG Edinburgh, Scotland, Aug. 3-12, 1989, Sea Surface Topography and the Geoid (Eds. H. Sünkel and T. Baker), Springer, 1990, pp. 31–38.
226NPCraymer, M. R., P. Vaníček and G. Carrera1989A report on the recompilation of the map of recent vertical crustal movements for Canada.Poster presented at the IAG General Meeting, Edinburgh, U.K., 2–12 August.
227NPVaníček, P., R. B. Langley, D. E. Wells, A. Kleusberg and J. McLaughlin1989Geographic position determination: a case for the Global Positioning System.Paper presented at URISA '89 conference, Boston, Mass., Aug. 7–9.
228RWVaníček, P.1990Some possible additional answers (reviewer's comments)Quo Vadimus (Geophysics for the Next Generation), Eds. G.D. Garland & J.R. Ajel, AGU, Wash. D.C., pp. 11–12.
229PRSjöberg, L. E., P. Vaníček and M. Kwimbere1990Estimates of present rates of geoid uplift in Eastern North AmericaManuscripta Geodaetica, Vol. 15, No. 5, pp. 261–272.
230NPCraymer, M. R. and P. Vaníček1990A comparison of various algorithms for the spectral analysis of unevenly spaced data series.Paper presented at CISM/CGU annual meeting, Ottawa, May 22–25.
231NPVaníček, P., G. H. Carrera and M.R. Craymer1990Map of recent crustal movements in CanadaPaper presented at CISM/CGU annual meeting, Ottawa, May 22–25.
232CVaníček, P.1990Review of "Gravimetry" by W. TorgePAGEOPH 134(2), pp. 475–476.
233IP,PRVaníček, P.1990Vertical datum and the "NAD'88"Paper presented at ACSM/ASPRS annual convention, Denver, March 18–24. Also printed in Surveying and Land Information Systems, Vol. 51, No. 2, 1991, pp. 83–86.
234RVaníček, P., C. Zhang and P. Ong1990Computation of a file of geoidal heights using Molodenskij's truncation methodUniversity of New Brunswick, Dept. Surveying Engineering, T.R. #147, 106 pp.
235RVaníček, P. and T. Hou1990Towards a Sequential Tidal Analysis and PredictionA contract report for Geometrix, Inc., Dartmouth, N.S., 27 pp.
236RCraymer, M. R. and P. Vaníček1990A Statistical Analysis of Rod Scale Errors in Historic Geodetic LevellingContract report for USGS, 36 pp.
237RVaníček, P., E. J. Krakiwsky, M. Craymer, Y. Gao, P. Ong1990"Robustness Analysis"Department of Surveying Engineering Technical Report No. 156, University of New Brunswick, 115 pp.
238PRCraymer, M. R., D. E. Wells, P. Vaníček and R. L. Devlin1990Specifications for Urban GPS SurveysSurveying and Land Information Systems 50(4), pp. 251–259.
239CVaníček, P.1991Review of "Gravity and Low-Frequency Geodynamics", edited by R. TeisseyrePAGEOPH 135(3), pp. 498–499.
240PRVaníček, P.1991Robustness of Geodetic NetworksJournal of the Association of Czechoslovakian Surveyors. (GAKO), Vol. 79, No. 6, pp. 111–113.Czech
241PRVaníček, P. and L. E. Sjöberg1991Reformulation of Stokes's Theory for Higher than Second-Degree Reference Field and Modification of Integration KernelsJGR, 96(B4),pp. 6529–6539.
242R,NPVaníček, P., P. Ong and Changyou Zhang1991New Gravimetric Geoid for Canada: the "UNB'90" SolutionProceedings of First International Geoid Commission Symposium, Milan, June 11–13, 1990 and printed in Determination of the Geoid. Present and Future, Springer-Verlay, New York, pp. 214–219.
243PRSchneider, D. and P. Vaníček1991A New Look at the USGS 1970-1980 Horizontal Crustal Deformation Data around Hollister (California)JGR 96 (B13), pp. 21641–21657.
244RCarrera, G. H., P. Vaníček and M. R. Craymer1991The compilation of a map of Recent Vertical Crustal Movements in CanadaUniversity of New Brunswick, Dept. Surveying Engineering, T.R. # 153; also published as Contract Report 91-001, File Number: 50SS.23244-7-4257, Energy, Mines and Resources Canada, 107 pp.
245NPVaníček, P., D. E. Wells and M. Kwimbere1991Towards the determination of continental slope footlineGALOS Technical Meeting, IUGG General Assembly, Vienna, August 22.
246PR,RWVaníček, P., R. B. Langley and A. Kleusberg1991"Geodesy: still the scientific backbone of surveying and mapping."Journal ACSGS, Vol. 45(4), pp. 383–4.
247IPVaníček, P.1991Geodetic modelling of superficial earth deformationsAGU Chapman Conference on crustal motions, Annapolis, Md., Sept. 22–25.
248PRVaníček, P., Zhang C., and L. E. Sjöberg1992Comparison of Stokes's and Hotine's approaches to geoid computationManuscripta Geodaetica, 17 (1), pp. 29–35.
249NPVajda, P. P. Ong, M. C. Santos, P. Vaníček and M. R. Craymer1992Comparison of geoidal deflections computed from UNB'91 geoid with observed astro-deflectionsAGU/CGU/MSA joint spring meeting, Montreal, May 12–14, 1992.
250NPOng, P. and P. Vaníček1992An investigation into the datum independence problem in robustness analysis.AGU/CGU/MSA joint spring meeting, Montreal, May 12–14, 1992.
251NPCraymer,M. R., S.Blackie, P. Vaníček, E. J. Krakiwsky and D.Szabo1992Robustness analysis of geodetic networks.AGU/CGU/MSA joint spring meeting, Montreal, May 12–14, 1992
252IPSideris, M. G., P. Vaníček and A Mainville1992The Canadian Geoid Committee and the geoid in Canada.AGU/CGU/MSA joint spring meeting, Montreal, May 12–14, 1992
253IPVaníček, P.1993The problem of a maritime boundary involving two horizontal geodetic datums.Presented at First International Conference on Geodetic Aspects of the Law of the Sea (GALOS), Bali, Indonesia, June 8–13, 1992
254RKrakiwsky, E. J., P. Vaníček and D. Szabo1993Further development and testing of robustness analysis.Final report to Geodetic Survey of Canada, DSS contract file # 39SS.23244-1-4482, March 1993. 77pp.
255NPEngels, J., E. Grafarend, W. Keller, Z. Martinec, F. Sansó and P. Vaníček1993The geoid as an inverse problem to be regularized.Proceedings of the International Conference "Inverse Problems: Principles and Applications in Geophysics, Technology and Medicine", Potsdam, Germany, Aug.30-Sept.3, 1993, Akademie Verlag GmbH, Berlin, pp. 122–166.
256PRMartinec,Z., C. Matyska, E. W. Grafarend and P. Vaníček1993On Helmert's 2nd condensation method.Manuscripta Geodaetica, 18 pp. 417–421.
257BVaníček, P. and N. Christou (editors)1993Geoid and its geophysical interpretations,CRC Press, Boca Raton, Fla., USA. 343 pp.
258NPCraymer, M. R., P. Vaníček, E. J. Krakiwsky and D.Szabo1993Robustness analysis: a new method of assessing the strength of geodetic networks.Annual meeting of CISM, Toronto.
259NPVaníček, P. and Z. Martinec1993Can the geoid be evaluated to a one-centimetre accuracy?-a look at the theory.CGU Annual Meeting, Banff, Alberta, May 9–12, 1993.
260NPVajda, P. and P. Vaníček1993Truncated geoid and its geophysical interpretation.CGU Annual meeting, Banff, Alberta, May 9–12, 1993.
261NPSzabo, D. J., M. R. Craymer, E. J. Krakiwsky, and P. Vaníček1993Robustness measures for geodetic networks.Proceedings of the 7th International FIG Symposium on Deformation Measurements, Banff, Alberta, May 3 to 7, 1993. pp 151–160.
262NPCraymer, M., P. Vaníček, E. J. Krakiwsky and D. Szabo1993Robustness Analysis.First International Symposium on Mathematical and Physical Foundations of geodesy, Stuttgart, Germany, September 7–9, 1993.
263PRVaníček, P. and Z. Martinec1994Stokes-Helmert scheme for the evaluation of a precise geoidManuscripta Geodaetica 19 pp. 119–128.
264PRHou T. and P. Vaníček1994Towards a real-time analysis of tides.International Hydrographic Review, LXXI (1), Monaco, pp. 29–52.
265PR, NPVaníček, P. and G. Carrera1994Treatment of sea level records in linear vertical crustal motion modelling.Proceedings of the 8-th International Symposium on Recent Crustal Movements, Kobe, Japan, December 6–11, 1993, special issue of Journal of Geodetic Society of Japan, pp. 305–309.
266NPCarrera, G. and P. Vaníček1994Compilation of a new map of recent vertical crustal movements in Canada.The 8-th International Symposium on Recent Crustal Movements, Kobe, Japan, December 6–11, 1993.
267PRMartinec, Z. and P. Vaníček1994The indirect effect of Stokes-Helmert's technique for a spherical approximation of the geoid.Manuscripta Geodaetica 19(2), pp. 213–219.
268PRMartinec, Z. and P. Vaníček1994Direct topographical effect of Helmert's condensation for a spherical geoid.Manuscripta Geodaeticaa 19(3), pp. 257–268.
269PRVaníček, P., D. E. Wells and T. Hou1994Determination of the Foot of the Continental Slope.DSS Contract # 23420-3-R207/01-OSC Report for Geological Survey of Canada, Atlantic Geoscience Centre, Bedford Institute of Oceanography, Dartmouth, N.S., 49 pp.
270CVaníček, P.1994New home for the Finnish Geodetic Institute,Geomatica, 48 (3), p. 243.
271IPVaníček, P., D. E. Wells and T. Hou1994Continental slope foot-line determination: Geometrical Aspects,International Workshop on LOS Article 76, UNB, Fredericton, N.B., April 14–15, Proceedings "Law of the Sea Article 76 Workshop", pp. 57–67.
272NPVaníček, P., D. E. Wells, T. Hou and Z. Ou1994First experiences with continental slope foot-line determination from real bathymetric dataProceedings of international symposium INSMAP 94, Hannover, Germany, September 19–23, pp. 385–397.
273IPVaníček, P.1994On the global vertical datum and its role in maritime boundary demarcation;Proceedings of international symposium INSMAP 94, Hannover, Germany, September 19–23, pp. 243–250.
274PRMartinec, Z., P. Vaníček, A. Mainville and M. Véronneau1995The effect of lake water on geoidal heightsManuscripta Geodaetica, 20, pp. 193–203.
275PRCraymer, M. R., P. Vaníček and R. O. Castle1995Estimation of Rod Scale Errors in Geodetic LevellingJGR, 100 (B8), pp. 15129–15146.
276RVaníček, P., A. Kleusberg, Z. Martinec, W. Sun, P. Ong, M. Najafi, P. Vajda, L. Harrie, P. Tomášek and B. ter Horst1995Compilation of a precise regional geoid,DSS Contract # 23244-1-4405/01-SS Report for Geodetic Survey Division, Ottawa, 45 pp.
277NPVaníček, P. and W. Sun1995Downward continuation of Helmert's gravityCGU annual meeting, Banff, May 22–25, 1995.
278NPSun, W., S. Okubo and P. Vaníček1995Surface displacements from dislocationsIUGG General Assembly, Boulder, Colo., July, 1995.
279NPOu, Z. and P. Vaníček1995Automatic tracing of the foot of the continental slopeIUGG General Assembly, Boulder, Colo., July, 1995.
280NPCraymer, M. R., P. Vaníček and E. J. Krakiwsky1995Application of Reliability and Robustness Analysis to Large Geodetic NetworksIUGG General Assembly, Boulder, Colo., July, 1995.
281NPSun, W. and P. Vaníček1995Downward continuation of Helmert's gravity disturbanceIUGG General Assembly, Boulder, Colo., July, 1995.
282NPSantos, M., P. Vaníček and R. B. Langley1995GPS real time orbit improvementIUGG General Assembly, Boulder, Colo., July, 1995.
283NPSantos, M. C., P. Vaníček and R. B. Langley1995An assessment of the effect of mathematical correlations on GPS network computation: a summary.XVII Congresso Brasileiro de Cartografia, Salvador, Bahia, Brazil, August, 1995.
284NPSantos, M. C., P. Vaníček and R. B. Langley1995Orbit improvement and generation of ephemerides for the global positionong system satellites: a summary.XVII Congresso Brasileiro de Cartografia, Salvador, Bahia, Brazil, August, 1995. Printed in Revista Brasileira de Cartografia, 46, October 1995, pp. 95–99.
285RVaníček, P., P. Ong, E. J. Krakiwsky, and M. R. Craymer1996Application of robustness analysis to large geodetic networks,DSS Contract # 23244-3-4363/01-SQ Report for Geodetic Survey Division, Ottawa, Technical Report #180, GGE, UNB, pp 82.
286RWells, D. E., A. Kleusberg and P. Vaníček1996A seamless vertical-reference surface for acquisition, management and display (ECDIS) of hydrographic data,CHS Contract # IIHS4-122 Report for Canadian Hydrographic Survey, Ottawa, Technical Report # 179, GGE, UNB, pp. 73.
287PRVaníček, P., M. Najafi, Z. Martinec, L. Harrie and L.E.Sjöberg1996Higher-order reference field in the generalized Stokes-Helmert scheme for geoid computation.Journal of Geodesy, 70 (3), pp. 176–182.
288PRMartinec, Z., P. Vaníček, A. Mainville and M. Véronneau1996Evaluation of topographical effects in precise geoid determination from densely sampled heights,Journal of Geodesy, 70(11), pp. 746–754.
289PRVaníček, P. and R. R. Steeves1996Transformation of coordinates between two horizontal geodetic datums.Journal of Geodesy, 70(11), pp. 740–745.
290BVaníček, P. (with contributions of GALOS members)1996Geodetic Commentary to TALOS Manual,Complement to Special Publication No. 51, International Hydrographic Bureau, Monaco. pp. 11.
291PROu, Z. and P. Vaníček1996Automatic tracing of the foot of the continental slope.Marine Geodesy 19, pp. 181–195.
292RWFeatherstone, W. E. and P. Vaníček1996The usage of Stokes in the possessive form,Bulletin of the International Geoid Service, No.5, International Geoid Service, Milan, Italy, December 1996, pp. 153–154.
293PRSun, W., S. Okubo and P. Vaníček1996Surface displacements caused by earthquake dislocations in realistic earth models.Journal of Geophysical Research, Vol.101, No.B4, pp. 8561–8578.
294CVaníček, P., A. M. Abolghasem and M. Najafi1996The need for precise geoid and how to get itNCC Scientific and Technical Quarterly Journal, Vol.7, No.1, Serial 25, pp. 16–22.Persian
295PRMartinec, Z. and P. Vaníček1996Formulation of the boundary-value problem for geoid determination with a higher-order reference field.Geophysical Journal International, 126, pp. 219–228.
296NPSun, W. and P. Vaníček1996On the discrete problem of downward Helmert's gravity continuation.Proceedings of Session G7 (Techniques for local geoid determination), Annual meeting of European Geophysical Society, The Hague, May 6–10, 1996, Reports of the Finnish Geodetic Institute, 96:2, pp. 29–34.
297PRVaníček, P., W. Sun, P. Ong, Z. Martinec, P. Vajda and B. ter Horst1996Downward continuation of Helmert's gravity,Journal of Geodesy 71 (1), pp. 21–34.
298PRSantos, M. C., P. Vaníček and R. B. Langley1996Principles of Orbit Improvement and Generation of Ephemerides for the Global Positioning System Satellites.Revista Brasileira de Geofisica (Brazilian Journal of Geophysics), Vol. 14 No. 3, pp. 253–262.
299PROu, Z. and P. Vaníček1996The effect of data density on the accuracy of foot-line determination through maximum curvature surface by automatic ridge-tracing algorithm.International Hydrographic Review, Vol. LXXIII (2), pp. 27–38.
300NPVaníček, P. and Z. Ou1997Automatic tracing of continental slope foot-line from bathymetric data.Proceedings of the Second International GALOS Conference, Bali, July 1 to 4, 1996, pp. 267–302.
301PRSantos, M. C., P. Vaníček and R. B. Langley1997Effect of Mathematical Correlations in GPS Network Computation Using Phase Double Difference Observation.Journal of Surveying Engineering, Vol. 123, No. 3, pp. 101–112.
302IPVaníček, P., P. Novák and J. Huang1997Construction of mean Helmert's anomalies on the geoid, presented at Geoid Workshop,Geodetic Survey Division, Ottawa, April 28–30.
303NPVaníček, P.1997Some technical aspects of the delimitation of maritime spaces defined by the LOS,Proceedings of "Curso de Derecho del Mar", organised by 'Comision Permanente del Pacifico Sur' and 'Academia Diplomatica del Peru', Lima, August 26–30, 1997Spanish
304NPVaníček, P., M. Veronneau and Z. Martinec1997Determination of mean Helmert's anomalies on the geoid,IAG General Assembly, Rio de Janeiro, Sept. 3 to 9.
305NPFeatherstone, W. E. and P. Vaníček1997To modify or not to modify?,IAG General Assembly, Rio de Janeiro, Sept. 3 to 9.
306PRVajda, P. and P. Vaníček1997On gravity inversion for point mass anomalies by means of the truncated geoid.Studia Geophysica et Geodaetica, 41, pp. 329–344.
307CVaníček, P.1998Review of "On Being the Head of a Department: a Personal View", by J. Conway.Journal of Geodesy, 72, 12, p. 709.
308PRSun, W. and P. Vaníček1998On some problems of the downward continuation of 5' x 5' mean Helmert's gravity disturbance.Journal of Geodesy, 72, 7–8, pp. 411–420.
309PRVajda, P. and P. Vaníček1998On the numerical evaluation of the truncated geoid.Contributions to Geophysics and Geodesy, Geophysical Institute of Slovak Academy of Sciences, Bratislava, Slovakia, Vol. 28, No.1, pp. 15–27.
310PR,CVaníček, P.1998The height of reason (a letter to the editor),GPS World, April 1998, p. 14.
311PRVaníček, P.1998On the errors in the delimitation of maritime spaces.International Hydrographic Review, LXXV(1), March, pp. 59–64.
312IPVaníček, P., P. Novák and J. Huang1998Geoid modelling at UNB,presented at Geoid Workshop, Geodetic Survey Division, Ottawa, May 14–15.
313NPNovák, P. and P. Vaníček1998Atmospherical Corrections for the Evaluation of Mean Helmert's Gravity Anomalies.CGU Annual Meeting, Quebec City, May 18–20, 1998.
314PRVaníček, P. and W. E. Featherstone1998Performance of three types of Stokes's kernel in the combined solution for the geoid,Journal of Geodesy, 72, 12, pp. 684–697.
315PRVajda, P. and P. Vaníček1998A note on spectral filtering of the truncated geoid.Contributions to Geophysics and Geodesy, Vol.28, No.4, pp. 253–262.
316RKrakiwsky, J. K., P. Vaníček, D. Szabo and M. R. Craymer1999Development and testing of in-context confidence regions for geodetic survey network.Report # 99-001, Geodetic Survey Division, Geomatics Canada, Ottawa, 26 p.
317PRVajda, P. and P. Vaníček1999Truncated geoid and gravity inversion for one point mass anomaly.Journal of Geodesy 73, pp. 58–66.
318PRVaníček, P., J. Huang, P. Novák, M. Véronneau, S. Pagiatakis, Z. Martinec and W. E. Featherstone1999Determination of boundary values for the Stokes-Helmert problem.Journal of Geodesy 73, pp. 180–192.
319RTAndersen, O. B., D. Fritsch and P. Vaníček1999Getting ready for the next century (International evaluation of Finnish Geodetic Institute),Finnish Ministry of Agriculture and Forestry, Helsinki, 68 p.
320IPNovák, P. and P. Vaníček1999Effect of distant topographical masses on geoid determination.CGU Annual Meeting, Banff, May 9–12, 1999.
321IPVaníček, P. and P. Novák1999Comparison between planar and spherical models of topography.CGU Annual Meeting, Banff, May 9–12, 1999.
322NPHuang, J. and P. Vaníček1999A faster algorithm for numerical Stokes's integration.CGU Annual Meeting, Banff, May 9–12, 1999.
323NPNovák, P., P. Vaníček, M. Véronneau, W. E. Featherstone and S.A. Holmes1999On the accuracy of Stokes's integration in the precise high-frequency geoid determination.AGU Spring Meeting, Boston, May 31 - June 3.
324NPHuang, J., P. Vaníček, W. Brink and S. Pagiatakis1999Effect of topographical mass density variation on gravity and the geoid in the Canadian Rocky mountains.AGU Spring Meeting, Boston, May 31 - June 3.
325NPVaníček, P. and J. Wong1999On the downward continuation of Helmert's gravity anomalies.AGU Spring Meeting, Boston, May 31 - June 3.
326NPSideris, M., P. Vaníček,J. Huang, and I.N. Tsiavos1999Comparison of downward continuation techniques of terrestrial gravity anomalies,
327NPFeatherstone, W. E., J. Evans and P. Vaníček1999Optimal selection of the degree of geopotential model and integration radius in regional gravimetric geoid computation.IUGG General Assembly, Birmingham, July 18 – 30.
328PRNajafi, M., P. Vaníček, P. Ong and M.R. Craymer1999Accuracy of a regional geoid,Geomatica 53,3, pp. 297–305.
329PRVaníček, P. and M. Omerbasic1999Does a navigation algorithm have to ue Kalman filter?Canadian Aeronautical and Space Institute Journal, 45, 3, pp. 292–296.
330PRFeatherstone, W. E. and P. Vaníček1999The role of coordinate systems, coordinates and heights in horizontal datum transformations,The Australian Surveyor, 44(2), pp. 143–150.
331PRVajda, P. and P. Vaníček1999The instant of the dimple onset for the high degree truncated geoid.Contributions to Geophysics and Geodesy,Vol. 29/3, pp. 193–204.
332IPVaníček, P.1999Propagation of errors from shore baselines seaward.Proceedings of ABLOS International Conference, Monaco, September 9 to 10, International Hydrographic Bureau, Monaco, pp. 110–119.
333RWVaníček, P.2000The detection of crustal movements by geodetic space techniques.Festschrift in honour of Adam Chrzanowski, Technical Report # 205, GGE, UNB, pp. 133–138.
334PRHuang, J., P. Vaníček and P. Novák2000An alternative algorithm to FFT for the numerical evaluation of Stokes's integral.Studia Geophysica et Geodaetica 44, pp. 374–380.
335IPSideris, M., K. R. Thompson and P. Vaníček2000Current status of precise geoid determination in Canada for geo-referencing and oceanography/hydrography applications,Geomatics 2000, Montreal, March 8.
336IPVaníček, P., J. Janák and M. Véronneau2000Impact of Digital Elevation Models on geoid modelling,Geomatics 2000, Montreal, March 8.
337IPVaníček, P. and J. Janák2000The UNB technique for precise geoid determination,CGU meeting, Banff, May 24–26.
338IPJanák, J. and P. Vaníček2000UNB North American geoid 2000 model: theory, intermediate and final results,GEOIDE annual meeting, Calgary, May 25–26.
339NPOmerbasic M. and P. Vaníček2000Least Squares Spectral Analysis of gravity data from the Canadian super-conducting gravimeter: an ongoing project report,poster presentation at GEOIDE annual meeting, Calgary, May 25–26.
340NPVaníček, P., J. Janák and J. Huang2000Mean Vertical Gradient of Gravity,Poster presentation at GGG2000 conference, Banff, July 31 – August 4. GGG2000 Proceedings (Ed. M.Sideris), pp259–262.
341IPVéronneau, M., S. D. Pagiatakis, P. Vaníček, P. Novák, J. Huang, J. Janák, M.G. Sideris and O. Esan2000Canadian Gravimetric Geoid Model 2000 (CGG2000): Preliminary results.GGG2000 conference, Banff, July 31 - August 4.
342NPVaníček, P. and J. Janák2000Truncation of 2D spherical convolution integration with an isotropic kernel, Algorithms 2000 conference,Tatranska Lomnica, Slovakia, September 15–18.
343PRXu, B. and P. Vaníček2001Navigation with position potential.Navigation 47(3), pp. 227–236.
344PRHuang, J., P. Vaníček, S. Pagiatakis and W. Brink2001Effect of topographical mass density variation on gravity and the geoid in the Canadian Rocky mountains.Journal of Geodesy 74 (11-12), pp. 805–815.
345PRNovák, P., P. Vaníček, M. Véronneau, W.E. Featherstone and S.A. Holmes2001On the accuracy of modified Stokes's integration in high-frequency gravimetric geoid determination: A comparison of two numerical techniques.Journal of Geodesy 74, pp. 644–654.
346PRVaníček, P., P. Novák and Z. Martinec2001Geoid, topography, and the Bouguer plate or shell.Journal of Geodesy 75 (4), pp. 210–215.
347PRVaníček, P., M. R. Craymer, and E.J.Krakiwsky2001Robustness analysis of geodetic networks,Journal of Geodesy 75 (4), pp. 199–209.
348BVaníček, P.2001Geodesy. Chapter in Encyclopedia of Science and Technology,Academic Press. 32 pages.
349PRVajda, P., L. Brimich and P. Vaníček2001Geodynamic applications of the truncation filtering methodology: A synthetic case study for a point source of heat: Progress report,Contributions to Geophysics and Geodesy 30/4, pp. 311– 322.
350PRNovák, P., P. Vaníček, Z. Martinec and M. Véronneau2001The effect of distant terrain on gravity and the geoid.Journal of Geodesy 75 (9-10), pp. 491–504.
351NPJanák, J. and P. Vaníček2001Systematic error of the geoid model in the Rocky Mountains,CGU annual conference, Ottawa, May 15–17, 2001.
352NPOmerbasic M. and P. Vaníček2001Accurate spectral analysis of very strong earthquakes’ signatures in superconducting gravimeter records,poster presentation at the Digital Earth conference, Fredericton, June 25–28.
353NPJanák, J., P. Vaníček and B. Alberts2001Point and mean values of topographical effects,the Digital Earth conference, Fredericton, June 25–28.
354IPVaníček, P. and J. Janák2001Refinement of the UNB geoid model: progress report for proj.#10,poster presentation at GEOIDE annual meeting, Fredericton, June 21–22.
355IPJanák, J. and P. Vaníček2001Improvement of the University of New Brunswick's gravimetric geoid model for Canada,poster presentation at IAG General Assembly, Budapest, Sept. 3 to 7.
356IPHuang, J., P. Vaníček and S. Pagiatakis2001On some numerical aspects of downward continuation of gravity anomalies,Proceedings of IAG General Assembly, Budapest, Sept. 3 to 7, Paper #58BD.
357PRVajda, P. and P. Vaníček2002The 3-D truncation filtering methodology defined for planar and spherical models: Interpreting gravity data generated by point masses.Studia Geophysica et Geodaetica 46, pp. 469–484.
358NPOmerbasic M. and P. Vaníček2002Last Squares Spectral Analysis of very-strong-earthquake-excited gravity variations recorded by the Canadian global superconducting gravimeter.Presented at the 74th annual meeting of the Eastern Section, Seismological Society of America, Boston, October 20–22.
359PRVaníček, P.,P. Novák, S. Pagiatakis and M. R. Craymer2002On the proper determination of transformation parameters of a horizontal geodetic datum,Geomatica 56 (4), pp. 329–340.
360PRHernandez, N. A., M. R. Gomez and P. Vaníček2002The far zone contribution in spherical Stokes's integration,Revista Cartografica 74–75, pp. 61–74.
361PRHuang, J., M. G. Sideris, P. Vaníček and I.N.Tsiavos2003Numerical investigation of downward continuation techniques for gravity anomalies.Bollettino di Geofisica Teorica ed Applicata LXII, No.1, pp. 34–48.
362NPBerber, M., P. Dare, and P. Vaníček2003"An Innovative Method for the Quality Assessment of Precise Geomatics Engineering Networks."The Mathematics of Information Technology and Complex Systems (MITACS) Atlantic Interchange, Dalhousie University, 24 March 2003, Halifax, NS, Canada.
363NPVaníček, P., R. Tenzer and J.Huang2003Role of "No Topography space" in the Stokes-Helmert scheme for geoid determination.CGU annual meeting, Banff, May 10–14.
364NPSantos, M, R. Tenzer and P. Vaníček2003Effect of terrain on orthometric height.CGU annual meeting, Banff, May 10–14.
365NPTenzer, R., and P. Vaníček2003New results for UNB geoid.CGU annual meeting, Banff, May 10–14.
366NPMartin, B.-A., C.MacPhee, R. Tenzer, P. Vaníček and M. Santos2003Mean gravity along plumbline.CGU annual meeting, Banff, May 10–14.
367NPVaníček, P., R. Tenzer and M. Santos2003New views of the spherical Bouguer gravity anomaly. Poster presentation,IUGG general assembly, Sapporo, Japan, June 27 - July 8.
368NPTenzer, R., P. Vaníček and M. Santos2003Corrections to be applied to Helmert's orthometric heights.Poster presentation, IUGG general assembly, Sapporo, Japan, June 27 - July 8.
369PRBerber, M., P. Dare and P. Vaníček.2003On the application of robustness analysis to geodetic networks,Meeting of Canadian Society for Civil Engineering, Moncton, June 4–7.
370PRTenzer, R., P. Vaníček and P. Novák2003Far-zone contribution to the topographical effects in the Stokes-Helmert method of geoid determination.Studia Geophysica et Geodaetica 47, pp. 467–480.
371PRTenzer, R. and P. Vaníček2003Geoid-quasigeoid correction in formulation of the fundamental formula of physical geodesy.Revista Brasileira de Cartografia. 55(1), pp. 57 – 61.
372PRVaníček, P., J. Janák and W.E. Featherstone2003Truncation of spherical convolution integration with an isotropic kernel,Studia Geophysica et Geodaetica, 47 (3), pp. 455–465.
373NPOmerbasic, M. and P. Vaníček2003Earth-model discrimination via terrestrial gravimetric spectroscopy.Abstract in AGU winter assembly, San Francisco, 2–6 December 2003
374PRTenzer, R. and P. Vaníček2003The correction to Helmert's orthometric height due to actual lateral variation of topographical density.Revista Brasileira de Cartografia, 55(2), pp. 44–47.
375PRTenzer, R., P. Vaníček and S. van Eck der Sluijs2003The far-zone contribution to upward continuation of gravity anomalies.Revista Brasileira de Cartografia, 55(2), pp. 48–54.
376PRVaníček, P., R.Tenzer, L.E. Sjöberg, Z. Martinec and W.E.Featherstone2004New views of the spherical Bouguer gravity anomaly.Journal of Geophysics International 159(2), pp. 460–472.
377NPBerber, M, P.Vaníček,and P. Dare2004Quality control of geodetic networks through robustness analysis.AGU/CGU Annual Meeting, Montreal, 17 to 21 May.
378NPKoohzare, A., P. Vaníček and M.Santos2004Glacial isostatic adjustment observed using historical tide gauge records and precise relevelling data in Eastern Canada.AGU/CGU Annual Meeting, Montreal, 17 to 21 May.
379NPSantos, M., Tenzer, R. and P. Vaníček2004Mean gravity along the plumbline.AGU/CGU Annual Meeting, Montreal, 17 to 21 May.
380NPKingdon, R., R. Tenzer, P. Vaníček and M. Santos2004Calculation of the Spherical Terrain Correction to Helmert's Orthometric Height.AGU/CGU Annual Meeting, Montreal, 17 to 21 May.
381NPYang, H., P. Vaníček and M. Santos2004Atmospheric effects in three-space scenario for the Stokes-Helmert method of geoid determination,AGU/CGU Annual Meeting, Montreal, 17 to 21 May.
382NPKoohzare, A., P. Vaníček and M.Santos2004Spatial analysis and treatment of tide gauge records using GIS.GEOIDE Annual meeting, Ottawa June 1 to 2.
383NPYang, H., P. Vaníček, M. Santos and R. Tenzer2004An introduction to Stokes-Helmert method for precise geoid determination.GEOIDE Annual meeting, Ottawa June 1 to 2.
384NPBaran, I.,S.J.Classens, W.E. Featherstone, S.A. Holmes, M. Kuhn and P Vaníček2004First Results of Australian Synthetic Earth Gravity Model (AUSSEGM),GGSM04, Porto, Aug.30 to Sept. 3.
385NPBaran, I., S. J. Claessens, W.E. Featherstone, S.A. Holmes, M. Kuhn, P. Vaníček2004Australian synthetic earth gravity field model (AUSSEGM) – a regional earth gravity model.Poster presentation at AGU Fall meeting, San Francisco, December 13 to 17.
386PRVaníček, P. and M. Najafi2004New cartographic mapping for Iran.Journal of Spatial Science, 49(2), pp. 33–44.
387PRVaníček, P., M. Santos, R.Tenzer and A.Hernandez2004Algunos aspectos sobre alturas ortométricas y normales.Revista Cartografica, Vol. 76–77, pp. 79–86.
388PRTenzer, R., P. Vaníček,S. van Eck der Sluijs and A.Hernandez2004On some numerical aspects of primary indirect topographical effect computation in the Stokes-Helmert theory of geoid determination.Revista Cartografica, Vol. 76–77, pp. 71–77.
389PRVajda, P., P.Vaníček, P. Novák, and B. Meurers2004On the evaluation of Newton integrals in geodetic coordinates: Exact formulation and spherical approximation.Contributions to Geophysics and Geodesy, 34 (4), pp. 289–314.
390PRVajda, P., P. Vaníček and B.Meurers2004On the removal of the effect of topography on gravity disturbance in gravity data inversion or interpretation,Contributions to Geophysics and Geodesy, 34 (4), pp. 339–369.
391PRJanák, J. and P. Vaníček2004Mean free-air gravity anomalies in the mountains.Studia Geophysica et Geodaetica, 49(1), pp. 31–42.
392PRHernandez, A.N., P.Vaníček, M. Santos and R. Tenzer2004Evaluación del Effecto Atmosférico Directo en el Área de Influencia de la Solución Geoidal Mexicana,Revista Cartografica, No.78-79, pp 7–12.
393IREllmann, A. and P. Vaníček2005UNB application of Stokes-Helmert's approach to geoid computation.Annual meeting of EGS, Vienna, April 24.
394PRTenzer, R, P. Vaníček, M. Santos, W. E. Featherstone, and M. Kuhn2005Rigorous orthometric heights.Journal of Geodesy 79, pp. 1432–1394. doi:10.1007/s001-005-0445-2.
395NPEllman, A., P.Vaníček, M. Santos2005No Topography approach to Stokes-Helmert's geoid modelling: results for a test area in the Canadian Rockies,Presented at the annual meeting of CGU, Banff, 8–11 May.
396NPKoohzare, A., P. Vaníček and M. Santos2005Compilation of a map of vertical crustal-movements in Eastern Canada using spline polynomials,Paper presented at the annual meeting of CGU, Banff, 8–11 May. Extended abstract printed in Elements, Vol 23, No. 2, pp. 30–34.
397NPKingdon, R., P. Vaníček, M. Santos, A. Ellmann, R. Tenzer2005Corrections for the improvement of the Canadian height system.Canadian Geophysical Union. Annual Meeting, Banff, May 8–11, 2005.
398IPEllman, A., P.Vaníček, M. Santos2005Precise geoid determination for geo-referencing and oceanography,Poster presented at the annual meeting of the GEOIDE, Quebec City, May 29–31.
399NPTenzer, R., A. Ellmann, P. Moore and P. Vaníček2005On the evaluation of gravity disturbances.Poster presentation, IAG General Assembly, Cairns, Australia, August 22–26.
400NPKoohzare, A., P. Vaníček and M. Santos2005The use of smooth piecewise algebraic approximation in the determination of vertical crustal movements in Eastern Canada.Poster presentation, IAG General Assembly, Cairns, Australia, August 22–26.
401PRKingdon, R., P. Vaníček, M. Santos, A. Ellmann and R. Tenzer2005Toward an improved height system for Canada.Geomatica, Vol. 59, No. 3, pp. 241 to 249.
402PRVajda, P., P.Vaníček, and B.Meurers2006A new physical foundation for anomalous gravity.Studia Geodaetica et Geofysica No. 50, pp. 189 to 216.
403IPVajda, P., P. Vaníček and B. Meurers2006On the relation between anomalous gravity and the attraction of earth's subsurface anomalous density.Presented at 2-nd Workshop on International Gravity Field Research, Smolenice castle, Slovak Republic, May 8–9, 2006
404NPVajda, P., P. Vaníček, P. Novák, R. Tenzer and A. Ellmann2006Secondary indirect effects in gravimetry.2-nd Workshop on International Gravity Field Research, Smolenice castle, Slovak Republic, May 8–9, 2006
405NPKoohzare, A., P. Vaníček and M. Santos2006The Contribution of Northern US Geodetic Data to the Study of Vertical Deformations of the Crust in Canada,Poster presentation at AGU Spring Assembly, Baltimore, MD, May 23–26.
406NPKingdon, R., A. Ellmann, P. Vaníček and M. Santos2006The cost of assuming a lateral density distribution in corrections to Helmert orthometric heights,Oral presentation at AGU Spring Assembly, Baltimore, MD, May 23–26.
407PRSanso, F. and P.Vaníček2006The orthometric height and the holonomity problem.Presented at the Grafarend symposium in Stuttgart, February 18;Journal of Geodesy 80 (5), pp. 225–232.
408NPEllmann, A., P. Vaníček, M. Santos and R. Kingdon2006The partnership of the precise geoid and orthometric heights.Oral presentation at Canadian Geophysical Union Annual Meeting, Banff, May 14–16.
409NPKingdon, R., A. Ellmann, P. Vaníček and M. Santos2006Estimating the cost to Helmert heights of lateral approximation of the topographical density distribution.Canadian Geophysical Union Annual Meeting, Banff, May 14–16.
410NPKoohzare, A., P. Vaníček and M. Santos2006Radial basis functions fitting methods as applied to determine postglacial tilt in the Canadian Prairies.Canadian Geophysical Union Annual Meeting, Banff, May 14–16.
411NPEllmann, A., P. Vaníček, M. Santos and R. Kingdon2006Symbiosis of orthometric heights with the geoid,The 8th GEOIDE Annual Scientific Conference – Poster presentation, Banff, May 31 – June 2.
412PRBerber, M., P. Dare and P. Vaníček2006Robustness analysis of 2D networks,Journal of Surveying Engineering, 132 (4), pp. 168–175.
413PRBaran, I., M. Kuhn, S.J. Claessens, W.E. Featherstone, S.A Holmes and P. Vaníček2006A synthetic Earth's gravity model designed specifically for testing regional gravimetric geoid determination algorithms,Journal of Geodesy 80(1):1-16.
414PRKoohzare, A., P. Vaníček and M. Santos2006Compilation of the map of recent vertical crustal movements in Eartern Canada using GIS,Journal of Surveying Engineering, 132 (4), pp. 160–167.
415IPKingdon, R. W., C. Hwang, U.-S. Hsiao, A. Ellmann, M. Santos and P. Vaníček2006Applications of satellite altimetry to evaluating effects of lake water on gravity and the geoid,IAG Workshop 2006: Coast and Land Applications of Satellite Altimetry, Beijing (oral presentation), July 20 – 22.
416NPFeatherstone, W. and P. Vaníček2006A new philosophy on the use of modified Stokes's kernels,First International Symposium of The International Gravity Field Service (IGFS), oral presentation, August 28 - September 1, 2006, Istanbul, Turkey.
417NPKingdon, R. W., M. Santos, A. Ellmann and P. Vaníček2006Estimating the shortcomings of 2D density models in calculating topograpfical effects on gravity and orthometric heights,First International Symposium of The International Gravity Field Service (IGFS), oral presentation, August 28 - September 1, 2006, Istanbul, Turkey.
418NPEllmann A., P. Vaníček and M. Santos2006Validation of the Stokes-Helmert geoid determination using Synthetic Earth Gravity Model,First International Symposium of The International Gravity Field Service (IGFS), oral presentation, August 28 - September 1, 2006, Istanbul, Turkey.
419PRSantos, M., P. Vaníček, W. E., Featherstone, R. Kingdon, B.-A. Martin, M.Kuhn and R.Tenzer2006Relation between the rigorous and Helmert's definitions of orthometric heights.Journal of Geodesy, Vol. 80, pp. 691–704.
420PRTenzer, R., P. Novak, P. Moore, M. Kuhn and P. Vaníček2006Explicit formula for the geoid-quasigeoid separation,Studia Geodaetica et Geofysica 50, pp. 607–618.
421PRKutoglu, H. and P. Vaníček2006Effect of common point selection on coordinate transformation parameter determination.Studia Geodaetica et Geofysica (50), pp. 525–536.
422IPEllmann, A. and P. Vaníček2006UNB application of Stokes-Helmert's approach to geoid computation,Journal of Geodynamics 43 (2), pp. 200–213, doi:10.1016/j.jog.2006.09.019.
423NPEllmann, A., P. Vaníček, M. Santos and R. Kingdon2007Interrelation between the geoid and orthometric heights,First International Symposium of The International Gravity Field Service (IGFS), oral presentation, August 28 - September 1, 2006, Istanbul, Turkey. Forsberg R.; Kilicoglu, A. (Eds.). Proceedings of the 1st International Symposium of the International Gravity Field Service "Gravity Field of the Earth", pp. 130 – 135. General Command of Mapping, Ankara, Turkey
424PRVajda, P., P. Vaníček, P. Novák, R. Tenzer and A. Ellmann2007Secondary indirect effect in gravity anomaly data inversion or interpretation,J. Geophys. Res., 112, B06411, doi:10.1029/2006JB004470.
425NPHatam, Y., R. Bayer, Y. Djamour, P. Vaníček, N. LeMoign, M. Mohammad Karim, A.M. Abolghasem, M. Karpychev, R. Sadat, S. Rafiey2007A new (tele cabin /land) national gravity calibration line for Iran,Poster presentation at General Assembly of European Geosciences Union, Vienna, April 15–20.
427NPCheraghi, H., Y. Hatam, P. Vaníček, M. Najafi Alamdari, Y. Djamour, J. Qarakhani, R. Saadat2007Effect of lateral topographical density variations on the geoid in Iran,Poster presentation at General Assembly of European Geosciences Union, Vienna, April 15–20.
428NPKoohzare, A., P. Vaníček, and M. Santos2007The pattern of Vertical Crustal movements in Canada using geodetic data,oral presentation Canadian Geophysical Union Annual Meeting, St. John's, May 30-June 2.
429NPAvalos, D., M. Santos and P. Vaníček2007The Mexican Gravimetric Geoid: state-of-the-art and future directionsoral presentation Canadian Geophysical Union Annual Meeting, St. John's, May 30-June 2.
430IPVajda P., A. Ellmann, B. Meurers, P. Vaníček, P. Novák, R. Tenzer2007On compiling and interpreting anomalous gravity data.Oral presentation at 7th Slovak Geophysical Conference, June 13–14, 2007, Bratislava.
431NPKoohzare, A., P. Vaníček., and M. Santos2007Geodetic modeling and geophysical interpretation of recent vertical crustal movements and gravity changes over Canada,Oral presentation to Twenty fourth assembly of IUGG, Perugia, July 2–13.
432NPHatam, Y., Y. Djamour, P. Vaníček, R. Bayer, M. Mohammad Karim, A. M. Abolghasem, M. Najafi Alamdari, R. Saadat, H. Cheraghi, S. Rafiey, S. Arabi, H. Nankali, S. Hoseini2007Designing and Implementation of the Multi-purpose Physical Geodesy and Geodynamics Network of (MPGGNI2005),Poster presentation, Twenty fourth assembly of IUGG, Perugia, July 2–13.
433NPTenzer, R., A. Ellmann, P. Novák, P. Vajda, P. Vaníček, P. Moore2007The Earth's gravity field components of the differences between gravity disturbances and gravity anomalies.Poster presentation, Twenty fourth assembly of IUGG, Perugia, July 2–13.
434NPBlitzkow, D., A.C.O.C. de Matos, I. De Oliveira Campos, A. Ellmann, P. Vaníček and M. Santos2007An attempt for an Amazon geoid model using Helmert gravity.Oral presentation to Twenty fourth assembly of IUGG, Perugia, July 2–13.
435NPVajda P., A. Ellmann, B. Meurers, P. Vaníček, P. Novák, R. Tenzer2007On a refined global topographic correction to gravity disturbances.Oral presentation to Twenty fourth assembly of IUGG, Perugia, July 2–13, 2007.
436NPKingdon, R., P. Vaníček and M. Santos2007Coping with topographical density in three dimensions: an application of forward modeling to orthometric height calculations.Oral presentation to Twenty fourth assembly of IUGG, Perugia, July 2–13.
437NPAvalos, D., M. C. Santos, P. Vaníček and A. Hernandez2007Insight into the Mexican Gravimetric Geoid (GGM05),Poster presentation to twenty fourth assembly of IUGG, Perugia, July 2–13, 2007. Appeared also in the Proceedings.
438PRVaníček, P., E. W. Grafarend and M. Berber2007Strain invariants,Journal of Geodesy Vol.82, Nos. 4–5, pp 263–268. doi:10.1007/s00190-007-0175-8.
Sources
• Santos, Marcelo, ed. (January 2003). Honoring The Academic Life of Petr Vanicek. Fredericton: Department of Geodesy and Geomatics Engineering, University of New Brunswick. pp. 184–219.
| Wikipedia |
Proper linear model
In statistics, a proper linear model is a linear regression model in which the weights given to the predictor variables are chosen in such a way as to optimize the relationship between the prediction and the criterion. Simple regression analysis is the most common example of a proper linear model. Unit-weighted regression is the most common example of an improper linear model.
Bibliography
• Dawes, R. M. (1979). "The robust beauty of improper linear models in decision making". American Psychologist. 34 (7): 571–582. doi:10.1037/0003-066X.34.7.571. S2CID 14428212.
| Wikipedia |
Talks are at noon on Monday in B660
Sept 9 everyone Open problem session
in E575
Sept 16 Farzad Aryan On distribution of squares modulo a composite number \(q\)
(University of Lethbridge)
A natural number \(s\) is said to be a square modulo a composite number \(q\) if it is a square modulo each of the prime numbers dividing \(q\). Let \(p\) be a prime number, then \[\textbf{Prob}(s \text{ is a square mod }p)=\displaystyle{\frac{p+1}{2p} \approx \frac{1}{2}}.\] Roughly speaking, the probability of a number to be a square modulo \(q\) is \(\displaystyle{ \frac{1}{2^{\omega(q)}}}\), where \(\omega(q)\) is the number of prime divisors of \(q\).
Fix \(h\) and let \(\mathcal{ X}: \{1, 2, \cdots , q\} \rightarrow \mathbb{N}\) be a random variable, given by \[\mathcal{ X}(i)= \#\{s \in [i, i+h] : s \text{ is a square modulo } q\}.\] For the mean, we have \({\rm \bf E}(\mathcal{ X})\approx h/ 2^{\omega(q)}\), and, in this talk, we show the following bound for the variance: \[{\rm \bf Var}(\mathcal{ X}) \leq {\rm \bf E}(\mathcal{ X}) \approx \frac{h}{2^{\omega(q)}} . \]
Sept 23 Nathan Ng Zhang's theorem on bounded gaps between primes
in B660
In April 2013, Yitang Zhang announced one of the great theorems in the history of number theory. He showed there exists an absolute constant C such that infinitely many consecutive primes differ by C. This theorem goes a long way towards proving the twin prime conjecture. In this talk I will give an overview of Zhang's theorem and some of the main ideas in the proof.
Sept 30 Nathan Ng Zhang's theorem on bounded gaps between primes, part 2
In this talk, I will focus on the Goldston-Pintz-Yildirim (GPY) method for detecting small gaps between primes. In particular, I will discuss the choice of weight function in their optimization argument and the role of primes in arithmetic progressions. Finally, we will consider the Motohashi-Pintz/Zhang variant of the GPY argument which yields bounded gaps between primes.
Oct 7 Jeff Bleaney Symmetries of an Elliptic Net
In 1948, Morgan Ward introduced the concept of an Elliptic Divisibility Sequence (EDS) as an integer sequence \((W_{n})\) which satisfies the recurrence relation $$W_{m+n}W_{m-n}W_{1}^{2} = W_{m+1}W_{m-1}W_{n}^{2} - W_{n+1}W_{n-1}W_{m}^{2},$$ and satisfies the additional property that \(W_{m}|W_{n}\) whenever \(m|n\). Of particular interest to Ward, were what he called symmetries of an EDS. Ward showed that if \((W_{n})\) is an EDS with \(W_{r} = 0\), then we have $$W_{r+i} = ab^{i}W_{i},$$ for some \(a\) and \(b\). In her Ph.D. thesis in 2008, Kate Stange generalized the concept of an EDS to an \(n\)-dimensional array called an Elliptic Net.
We will discuss the connections between EDS's, Elliptic Nets, and elliptic curves, and give a generalization of Ward's symmetry theorem for elliptic nets.
Oct 21 Adam Felix On the distribution of torsion points modulo primes
We will discuss a paper of Chen and Kuan, in which they study the distribution of torsion points modulo primes over several different commutative algebraic groups. They demonstrate that the average is related to some generalized divisor function for these groups.
Oct 28 Dave Morris What is a Coxeter group?
Coxeter groups arise in a wide variety of areas, so every mathematician should know some basic facts about them, including their connection to "Dynkin diagrams." Proofs about these "groups generated by reflections" mainly use group theory, geometry, and combinatorics.
Nov 4 Soroosh Yazdani Solving \(S\)-unit equations
Let \(S\) be a finite collection of prime numbers. We say a number is an \(S\)-unit if it is a product of powers of primes in \(S\). For instance \(-3/8\) is an example of a \(\{2,3\}\)-unit. Many interesting Diophantine equations are reduced to solving equations of the form \[ x+y=1 \] with \(x\) and \(y\) both being an \(S\)-unit. Using linear forms of logarithms, we can show that there only finitely many solutions to these \(S\)-unit equations. In this talk, I will explain an algorithm (due primarily to Smart and Wildanger) on how we can actually enumerate all these solutions.
Nov 15 Patrick Ingram The arithmetic of post-critically finite morphisms
(Colorado State University)
Let \(f\) be an endomorphism of \(N\)-dimensional projective space. In complex dynamics, it has been known for a century (at least when \(N = 1\)) that the orbits of the critical points determines much of the dynamics of \(f\). Morphisms for which all of these critical orbits are finite (so-called PCF maps) turn out to be an important class to understand. Thurston proved, when \(N = 1\), that there are no algebraic families of PCF maps, except for a small number of easy-to-understand examples. I will discuss some recent research into the arithmetic properties of these maps, as well as a partial extension of Thurston's result to arbitrary dimension.
Nov 18 James Parks Distribution conjectures for elliptic curves on average
Let \(E\) be an elliptic curve over \(\mathbb{Q}\). In this talk we consider several open conjectures about the distribution of local invariants associated with the reductions of \(E\) modulo \(p\) as \(p\) varies over the primes. In order to gain evidence for the conjectures, we consider them on average over a family of elliptic curves.
Nov 25 Habiba Kadiri Zero density and primes
In this talk we present some new Chebyshev bounds for the function \(\psi(x)\). In 1962, Rosser and Schoenfeld provided a method to estimate the error term in the approximation \(|\psi(x)-x|\). Since then, progress on the numerical verification of the Riemann Hypothesis and widening the zero-free region have allowed to improve numerically these bounds. In this talk we present a new method by introducing a smooth weight and by using the first explicit zero density estimate for the Riemann zeta function. We also present new results for primes in short intervals, based on this zero density estimate.
Dec 2 Darcy Best Biangular Lines
A set of unit vectors \(V \subset \mathbb{C}^n\) is called biangular if for any \(u,v \in V, u \neq v\), $$|\langle u,v\rangle| \in \left\{0,\alpha\right\}$$ for some \(0 < \alpha < 1\). There are well-known upper bounds on the size of these sets of vectors. We will discuss these upper bounds, and the implications when they are met, including the generation of combinatorial objects such as strongly regular graphs and association schemes.
Dec 9 Renate Scheidler Continued Fractions With Bounded Period Length
(University of Calgary)
It is well-known that the continued fraction expansion of a quadratic irrational is horizontally symmetric about its centre. However, an additional vertical symmetry is exhibited by the continued fraction expansions arising from a certain one-parameter family of positive integers known as Schinzel sleepers. This talk provides a method for generating any Schinzel sleeper and investigates their period lengths as well as both their horizontal and vertical symmetries.
This is joint work with Kell Cheng (Hongkong Institute of Education) as well as Richard Guy and Hugh Williams (University of Calgary). The talk is geared toward an audience with a background corresponding to no more than a first number theory course.
Past semesters: Fall 2007 Fall 2008 Fall 2009 Fall 2010 Fall 2011 Fall 2012
Spring 2008 Spring 2009 Spring 2010 Spring 2012 Spring 2013 | CommonCrawl |
\begin{document}
\title{Spectral order for contact manifolds with convex boundary}
\author{Andr\'as Juh\'asz} \address{Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK} \email{[email protected]}
\author{Sungkyung Kang} \address{Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK} \email{[email protected]}
\subjclass[2010]{57M27; 57R17; 57R58} \keywords{Contact structure; Spectral order; Heegaard Floer homology}
\date{}
\begin{abstract} We extend the Heegaard Floer homological definition of spectral order for closed contact 3-manifolds due to Kutluhan, Mati\'c, Van Horn-Morris, and Wand to contact 3-manifolds with convex boundary. We show that the order of a codimension zero contact submanifold bounds the order of the ambient manifold from above. As the neighborhood of an overtwisted disk has order zero, we obtain that overtwisted contact structures have order zero. We also prove that the order of a small perturbation of a $2\pi$ Giroux torsion domain has order at most two, hence any contact structure with positive Giroux torsion has order at most two (and, in particular, a vanishing contact invariant). \end{abstract}
\maketitle
\section{Introduction}
Algebraic torsion of closed contact $(2n-1)$-manifolds was defined by Latschev and Wendl~\cite{key-11} via symplectic field theory. It is an invariant with values in $\mathbb{N} \cup \{\infty\}$ whose finiteness gives obstructions to the existence of symplectic fillings and exact symplectic cobordisms. They also showed that the order of algebraic torsion is zero if and only if the contact homology is trivial -- in particular, if the contact structure is overtwisted -- and it has order at most one in the presence of positive Giroux torsion. Note that the analytical foundations of symplectic field theory are still under development. Hence, in the appendix, Hutchings provided a similar numerical invariant for contact 3-manifolds via embedded contact homology; however, it is currently unknown whether this is independent of the contact form.
Motivated by the isomorphism between embedded contact homology and Heegaard Floer homology, Kutluhan, Mati\'c, Van Horn-Morris, and Wand~\cite{key-10, key-14} defined a Heegaard Floer homological analogue of algebraic torsion for closed contact 3-manifolds called \emph{spectral order} (or \emph{order} in short), and denoted it by~$\mathbf{o}$. Their definition uses open book decompositions, and gives a refinement of the Ozsv\'ath-Szab\'o contact invariant~$c(\xi)$. Using the fact that an overtwisted contact structure is supported by an open book with non right-veering monodromy, they proved that $\mathbf{o}(M,\xi) = 0$ if $\xi$ is overtwisted.
In this paper, we extend $\mathbf{o}$ to contact manifolds with convex boundary, following the definition of Kutluhan et al.~in the closed case. The definition is in terms of a partial open book decomposition of the underlying sutured manifold supporting the contact structure, and a collection of arcs on the page, containing a basis. This data gives rise to a filtration of the sutured Floer boundary map, and the spectral order is the index of the first page of the associated spectral sequence where the distinguished generator representing the contact invariant vanishes, or $\infty$ otherwise. Then we take the minimum over all partial open books together with a collections of arcs containing a basis. (This extension of the definition of~$\mathbf{o}$ was also independently observed by Kutluhan et al.~\cite{key-14}.)
Our first main result is that the spectral order of a codimension zero contact submanifold gives an upper bound on the spectral order of the ambient manifold.
\begin{thm} \label{thm:ineq}
Let $(M,\xi)$ be a contact $3$-manifold with convex boundary. If $(N,\xi|_N)$ is a codimension zero submanifold of~$\text{Int}(M)$ with convex boundary,
then \[
\mathbf{o}(N,\xi|_N) \ge \mathbf{o}(M,\xi). \] \end{thm}
We will prove this result in Section~\ref{sec:ineq}. As a corollary, we show that if a contact manifold with convex boundary is overtwisted, then it has spectral order zero. This follows immediately from a simple computation that a neighborhood of an overtwisted disk has spectral order zero.
In Section~\ref{sec:gt}, we carry out a computation that shows that the spectral order of a slight enlargement of a Giroux $2\pi$-torsion $T^2 \times I$ has spectral order at most two. In particular, every contact manifold with positive Giroux torsion has vanishing Ozsv\'ath-Szab\'o invariant, which was proved in the closed case by Ghiggini, Honda, and Van Horn-Morris~\cite{key-5} (the sutured case also follows from their work when combined with \cite[Theorem~1.1]{key-2}). Together with Theorem~\ref{thm:ineq}, we obtain the following corollary.
\begin{thm} \label{thm:gt} If a contact $3$-manifold $(M,\xi)$ with convex boundary has Giroux $2\pi$-torsion, then \[ \mathbf{o}(M,\xi) \le 2. \] \end{thm}
The inequality $\mathit{AT} \le 1$ was shown in the closed case by Latschev and Wendl~\cite[Theorem~2]{key-11} via symplectic field theory, and conjectured in the Heegaard Floer setting in the closed case by Kutluhan et al.~\cite[Question~6.3]{key-14}. More generally, they asked whether the presence of planar $k$-torsion (see~\cite[Section~3.1]{key-11} for a definition) implies that the spectral order is at most~$k$.
\section{Spectral order for manifolds with boundary}
We first recall the definition of spectral order for closed contact $3$-manifolds due to Kutluhan, Mati\'c, Van Horn-Morris, and Wand~\cite{key-14}. Let $(M,\xi)$ be a closed contact 3-manifold. By the Giroux correspondence theorem \cite{key-9}, the contact structure $\xi$ is supported by some open book decomposition~$(S,\phi)$ of $M$, which is well-defined up to positive stabilizations. In particular, $M$ is identified with $S \times I/{\sim}$, where $(x,1) \sim (\phi(x),0)$ for every $x \in S$, and $(x,t) \sim (x,t')$ for every $x \in \partial S$ and $t$, $t' \in I$.
An arc basis on~$S$ is a set of pairwise disjoint properly embedded arcs that forms a basis of $H_{1}(S,\partial S)$. A collection of pairwise disjoint arcs $\underline{\mathbf{a}} = \{a_1,\dots,a_n\}$ on $S$ that contains a basis induces an ``overcomplete'' Heegaard diagram $(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta})$ of $M$, as follows. We obtain~$\underline{\mathbf{b}} = \{b_1,\dots,b_n\}$ by isotoping $\underline{\mathbf{a}}$
such that the endpoints of $\underline{\mathbf{a}}$ are moved in the positive direction along $\partial S$, and $|a_i \cap b_j| = \delta_{ij}$ for $i$, $j \in \{1, \dots n\}$. Then we set $\Sigma = (S\times\{1/2\}) \cup_{\partial S} (-S\times\{0\})$. Furthermore, we let $\boldsymbol{\alpha} = \{\alpha_1, \dots, \alpha_n\}$ and $\boldsymbol{\beta} = \{\beta_1,\dots,\beta_n\}$, where \[ \begin{split} \alpha_i &:= (a_i \times\{1/2\}) \cup (a_i \times \{0\}) \text{ and}\\ \beta_i &:= (b_i \times \{1/2\}) \cup (\phi(b_i) \times \{0\}) \end{split} \] for $i \in \{1, \dots, n \}$. We also choose a basepoint in each connected component of $S \setminus \underline{\mathbf{a}}$ away from the isotopy between $\underline{\mathbf{a}}$ and $\underline{\mathbf{b}}$, and denote the set of these on~$S \times \{1/2\} \subset \Sigma$ by~$\mathbf{z}$. Then $(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\mathbf{z})$ is a multi-pointed Heegaard diagram of~$M$.
We say that a domain $D$ in the diagram $(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\mathbf{z})$ connects $\mathbf{x}$, $\mathbf{y}\in\mathbb{T}_{\alpha}\cap\mathbb{T}_{\beta}$ if $\partial(\partial D\cap\boldsymbol{\alpha})=\mathbf{x}-\mathbf{y}$ and $\partial(\partial D\cap\boldsymbol{\beta})=\mathbf{y}-\mathbf{x}$, and $n_z(D) = 0$ for every $z \in \mathbf{z}$. We denote by $D(\mathbf{x},\mathbf{y})$ the set of such domains. If $\mathbf{x} = (x_1, \dots, x_n)$, then there is a unique permutation $\pi_\mathbf{x} \in S_n$ such that $x_i \in \alpha_i \cap \beta_{\pi_\mathbf{x}(i)}$ for every $i \in \{1, \dots, n\}$. Using the above Heegaard diagram, Kutluhan at al.~\cite{key-14} defined a function~$J_{+}$ that assigns an integer \[
J_{+}(D)=n_{\mathbf{x}}(D)+n_{\mathbf{y}}(D)-e(D)+|\mathbf{x}|-|\mathbf{y}| \] to every domain $D\in D(\mathbf{x},\mathbf{y})$. Here, $n_{\mathbf{x}}(D)$ is the sum over all $p\in\mathbf{x}$ of the averages of the coefficients of $D$ at the four regions around
$p$, the term $e(D)$ is the Euler measure of $D$, and $|\mathbf{x}|$,
$|\mathbf{y}|$ are the number of cycles in the permutations $\pi_\mathbf{x}$ and $\pi_\mathbf{y}$, respectively. When $D$ is a domain of Maslov index~$1$, the equality $e(D)=1-n_{\mathbf{x}}(D)-n_{\mathbf{y}}(D)$ holds by the work of Lipshitz~\cite{key-6}, so the formula becomes \[
J_{+}(D)=2(n_{\mathbf{x}}(D)+n_{\mathbf{y}}(D))-1+|\mathbf{x}|-|\mathbf{y}|. \] For any topological Whitney disk $\psi \in \pi_{2}(\mathbf{x},\mathbf{y})$, we can define $J_{+}(\psi)$ as the value $J_{+}(D(\psi))$, where $D(\psi)$ is the domain of $C$. The function $J_{+}$ is additive in the sense that \[ J_{+}(D_{1}+D_{2})=J_{+}(D_{1})+J_{+}(D_{2}) \] for every $D_{1} \in D(\mathbf{x}_1,\mathbf{x}_2)$ and $D_{2} \in D(\mathbf{x}_2, \mathbf{x}_3)$. Furthermore, $J_{+}(u)$ is always a nonnegative even integer for any J-holomorphic disk~$u$. Hence, we have a splitting \[ \widehat{\partial}_{\mathit{HF}}=\partial_{0}+\partial_{1}+\partial_{2}+\cdots \] of the Heegaard Floer differential $\widehat{\partial}_{\mathit{HF}}$, where $\partial_{i}$ is defined by counting all J-holomorphic disks $u$ satisfying $\mu(u)=1$ and $J_{+}(u)=2i$. As shown in~\cite{key-14}, this gives a spectral sequence \[ E^k(S,\phi,\underline{\mathbf{a}})= H_{\ast}\left(E^{k-1}(S,\phi,\underline{\mathbf{a}}), d^{k-1}\right), \] induced by the filtered complex \[ \left(C = \bigoplus_{i\in \mathbb{N}}\widehat{CF}(\Sigma,\boldsymbol{\beta},\boldsymbol{\alpha},\mathbf{z})_{i} \text{, }\widehat{\partial}\right). \] The $j$-th coordinate of the differential $\widehat{\partial} \underline{c}$ for $\underline{c} = (c_{i})_{i \in \mathbb{N}} \in C$ and $j \in \mathbb{N}$ is defined as \[ \left(\widehat{\partial}\underline{c}\right)_j = \sum_{i=0}^\infty \partial_{i} c_{i+j}, \] and the filtration is given by \[ \mathcal{F}_p C = \bigoplus_{i = 0}^p \widehat{CF}(\Sigma,\boldsymbol{\beta},\boldsymbol{\alpha},\mathbf{z})_i. \] Note that here we deviate slightly from the definition of Kutluhan et al.~\cite{key-14} in that we take the direct sum defining~$C$ over~$\mathbb{N}$ instead of~$\mathbb{Z}$, but as we shall see, the arising notion of spectral order is exactly the same.
Recall that a filtered complex \[ \dots \subseteq \mathcal{F}_{p-1}C \subseteq \mathcal{F}_p C \subseteq \mathcal{F}_{p+1} C \subseteq \dots \] induces a spectral sequence by setting \[ \begin{split} Z^k_p &= \{\, x \in \mathcal{F}_p C \,\colon\, \partial x \in \mathcal{F}_{p-k} C\,\} \text{ and} \\ B^k_p &= \mathcal{F}_p C \cap \partial\mathcal{F}_{p+k} C. \end{split} \] For $k \in \mathbb{N}$, the \emph{$k$-page} is the complex $\left(E^k = \bigoplus_{p \in \mathbb{Z}} E^k_p, d^k \right)$, where \[ E^k_p = \frac{Z^k_p}{Z^{k-1}_{p-1} + B^{k-1}_p}, \] and the differential $d^k \colon E^k_p \to E^k_{p-k}$ is induced by the differential $\partial$ on the complex~$C$.
For an open book decomposition $(S,\phi)$ supporting $\xi$, and a collection of arcs $\underline{\mathbf{a}}$ on~$S$ containing a basis, we denote the induced spectral sequence defined above by $E^{n}(S,\phi,\underline{\mathbf{a}})$. Then note that, for every $k \in \mathbb{Z}_{>0}$, \[ Z^k_0(S,\phi,\underline{\mathbf{a}}) = \{\, (c_i)_{i \in \mathbb{N}} \,\colon\, c_i = 0 \text{ for $i > 0$ and } \partial_0 c_0 = 0 \,\}. \]
Recall that the contact element $\mathit{EH}(\xi) \in \widehat{\HF}(-M)$ is defined as the homology class of the intersection point \[ \mathbf{x}_\xi := (\underline{\mathbf{b}} \cap \underline{\mathbf{a}}) \times \{1/2\} \in \mathbb{T}_{\beta} \cap \mathbb{T}_{\alpha}, \] where $\underline{\mathbf{a}} \times \{1/2\}$ and $\underline{\mathbf{b}} \times \{1/2\}$ are subsets of $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$, respectively, by definition. As there are no non-trivial pseudo-holomorphic disks emanating from $\mathbf{x}_\xi$ in $(\Sigma,\boldsymbol{\beta},\boldsymbol{\alpha})$ that contribute to $\widehat{\partial}_{\mathit{HF}}$, it follows that $\partial_i \mathbf{x}_\xi = 0$ for every $i \in \mathbb{N}$. We often view $\mathbf{x}_\xi$ as an element of~$C$ supported in degree zero; i.e., as a sequence $(d_i)_{i \in \mathbb{N}}$ such that $d_0 = \mathbf{x}_\xi$ and $d_i = 0$ for $i > 0$. As such, $\mathbf{x}_\xi \in Z^k_0(S,\phi,\underline{\mathbf{a}})$ for every $k \in \mathbb{N}$. The following is \cite[Definitions~2.1 and~2.2]{key-14}.
\begin{defn} Let $(M,\xi)$ be a closed contact $3$-manifold. We say that $\mathbf{o}(S,\phi,\underline{\mathbf{a}}) = k$ if the distinguished generator \[ \mathbf{x}_\xi \in \widehat{\mathit{CF}}(\Sigma,\boldsymbol{\beta},\boldsymbol{\alpha},\mathbf{z})_{0}, \] viewed in degree~0, is nonzero in $E^{k}(S,\phi,\underline{\mathbf{a}})$, and zero in~$E^{k+1}(S,\phi,\underline{\mathbf{a}})$. Then we define the \emph{spectral order of $(M,\xi)$} as \[ \mathbf{o}(M,\xi)=\min\left\{\, \mathbf{o}(S,\phi,\underline{\mathbf{a}})\,\colon\, (S,\phi) \mbox{ supports } \xi \mbox{ and } \underline{\mathbf{a}} \subset S \mbox{ contains an arc basis}\,\right\} . \] \end{defn}
Implicit in the above definition is the choice of an almost complex structure~$J$ on $\text{Sym}^g(\Sigma)$. Kutluhan et al.~\cite[Proposition~3.1]{key-14} showed that $\mathbf{o}(S,\phi,\underline{\mathbf{a}}, J)$ is independent of~$J$, hence we suppress it from our notation throughout.
\begin{rem*} The contact element $\mathbf{x}_{\xi}$, viewed in degree zero, vanishes in $E^{k+1}(S,\phi,\underline{\mathbf{a}})$ if and only if it is contained in \[ B_{0}^k(S,\phi,\underline{\mathbf{a}}) = \mathcal{F}_0 C \cap \widehat{\partial} \mathcal{F}_k C = \widehat{\mathit{CF}}(\Sigma,\boldsymbol{\beta},\boldsymbol{\alpha})_0 \cap \widehat{\partial} \left( \bigoplus_{i = 0}^k \widehat{\mathit{CF}}(\Sigma,\boldsymbol{\beta},\boldsymbol{\alpha})_i \right). \] This holds precisely if there exist elements $c_{i} \in \widehat{CF}(\Sigma,\boldsymbol{\beta},\boldsymbol{\alpha},\mathbf{z})$ for $i \in \{\, 0,\dots,k \,\}$ such that \begin{equation} \label{eqn:main} \begin{split} \sum_{i=0}^{k}\partial_{i} c_{i} &= \mathbf{x}_\xi, \text{ and} \\ \sum_{i=0}^{k-j}\partial_{i}c_{i+j} &= 0 \text{ for all } j > 0. \end{split} \end{equation} Indeed, if we set $c_i = 0$ for $i > k$, then the entries of $\widehat{\partial}(c_i)_{i \in \mathbb{N}}$ correspond to the left-hand side of equation~\eqref{eqn:main}, and so this equation translates to $\widehat{\partial}(c_i)_{i \in \mathbb{N}} = (d_j)_{j \in \mathbb{N}}$, where $d_0 = \mathbf{x}_{\xi}$ and $d_j = 0$ for $j > 0$. As equation~\eqref{eqn:main} coincides with the one defining $\mathcal{B}^k(S,\phi,\underline{\mathbf{a}})$ in~\cite[p.~5]{key-14}, it follows that it does not matter whether we take the direct sum over $\mathbb{N}$ or $\mathbb{Z}$ when we define~$\mathbf{o}$. \end{rem*}
Before extending this definition to manifolds with boundary, we first review the definition of partial open book decompositions, introduced by Honda, Kazez, and Mati\'{c}~\cite{key-2}. We follow the treatment of Etgu and Ozbagci~\cite{key-4}. An abstract partial open book decomposition is a triple $\mbox{\ensuremath{\mathcal{P}}}=(S,P,h)$, where \begin{itemize} \item $S$ is a compact, oriented, connected surface with nonempty boundary, \item $P=P_{1}\cup\dots\cup P_{r}$ is a proper subsurface of $S$ such that $S$ is obtained from $\overline{S\setminus P}$ by successively attaching $1$-handles $P_{1},\dots,P_{r}$,
\item $h:P\rightarrow S$ is an embedding such that $h|_{A}=\mbox{Id}_{A}$, where $A=\partial P\cap\partial S$. \end{itemize} Given a partial open book decomposition $(S,P,h)$, we associate to it a sutured $3$-manifold $(M,\Gamma)$, as follows. Let $H=S\times[-1,0]/{\sim}$, where $(x,t)\sim(x,t')$ for every $x \in \partial S$ and $t$, $t'\in[-1,0]$. Furthermore, let $N = P \times I/{\sim}$, where $(x,t)\sim(x,t')$ for every $x\in A$ and $t$, $t'\in I$. We obtain the manifold~$M$ by gluing $(x,0)\in\partial N$ to $(x,0)\in\partial H$ and $(x,1)\in\partial N$ to $(h(x),-1)\in\partial H$ for every $x\in P$. The sutures are defined as \[ \Gamma=(\overline{\partial S\setminus\partial P})\times\{0\}\cup-(\overline{\partial P\setminus\partial S})\times\{1/2\}. \] Then \[ \Sigma=(P\times\{0\}\cup-S\times\{-1\})/\sim \] is a Heegaard surface for $(M,\Gamma)$.
Let $\xi$ be a contact structure on $M$ such that $\partial M$ is convex with dividing set $\Gamma$. Similarly to the original Giroux correspondence, we say that~$\xi$ is compatible with the partial open book decomposition $(S,P,h)$ if \begin{itemize} \item $\xi$ is tight on the handlebodies $H$ and $N$, \item $\partial H$ is a convex surface with dividing set $\partial S\times\{0\}$, \item $\partial N$ is a convex surface with dividing set $\partial P\times\{1/2\}$. \end{itemize} Then the relative Giroux correspondence theorem says that $\xi$ is uniquely determined up to contact isotopy, and given such a contact structure $\xi$, any two partial open book decompositions compatible with $\xi$ are related by positive stabilizations.
We now extend the definition of spectral order to manifolds with boundary. Suppose that a contact $3$-manifold $(M,\xi)$ with convex boundary $\partial M$ and dividing set $\Gamma$ is given. Then $(M,\Gamma)$ is a balanced sutured manifold if $M$ has no closed components. Indeed, every convex surface has a non-empty dividing set. Furthermore, $\chi(R_{+}(\Gamma))=\chi(R_{-}(\Gamma))$ by \cite[Proposition~3.5]{polytope}. Then we have a compatible partial open book decomposition $\mathcal{P} = (S,P,h)$. An arc basis for $(S,P,h)$ is a set $\underline{\mathbf{a}}$ of properly embedded arcs in~$P$ with endpoints on~$A$ such that $S\setminus\underline{\mathbf{a}}$ deformation retracts onto $\overline{S\setminus P}$. Similarly to the closed case, a partial open book decomposition of $M$, together with a collection of pairwise disjoint arcs $\underline{\mathbf{a}}$ containing a basis and an appropriate choice of basepoints, gives a multipointed sutured Heegaard diagram $(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta},\mathbf{z})$ of $(M,\Gamma)$. Here, $\mathbf{z}$ consists of a basepoint in each component of $P \setminus \underline{\mathbf{a}}$ disjoint from $\partial P \setminus \partial S$.
The differential~$\widehat{\partial}_{\mathit{SFH}}$ of the sutured Floer chain complex counts the number of J-holomorphic curves $u$ with $\mu(u)=1$, modulo the $\mathbb{R}$-action, that do not intersect the suture $\Gamma=\partial\Sigma$ and the basepoints~$\mathbf{z}$. For any topological Whitney disk $\psi$ from $\mathbf{x}\in\mathbb{T}_{\alpha}\cap\mathbb{T}_{\beta}$ to $\mathbf{y}\in\mathbb{T}_{\alpha}\cap\mathbb{T}_{\beta}$ that does not intersect $\partial\Sigma$ and~$\mathbf{z}$, we define the number $J_{+}(\psi)$ as in the closed case by \[
J_{+}(\psi)=n_{\mathbf{x}}(D)+n_{\mathbf{y}}(D)-e(D)+|\mathbf{x}|-|\mathbf{y}|, \] where $D=D(\psi)$ is the domain of $\psi$. Since the equality $e(D)=1-n_{\mathbf{x}}(D)-n_{\mathbf{y}}(D)$ for $\mu(D)=1$ still holds in the sutured case, we get that \begin{equation}
J_{+}(\psi)=2(n_{\mathbf{x}}(D)+n_{\mathbf{y}}(D))-1+|\mathbf{x}|-|\mathbf{y}|\label{eqn:J} \end{equation} when $\mu(\psi)=1$. As in the closed case, the function $J_+$ is clearly additive, and the same argument as in~\cite[Section~2.2]{key-14} shows that $J_+(u)$ is a non-negative even integer for any $J$-holomorphic disk~$u$.
Hence, we can split the sutured Floer differential $\widehat{\partial}_{\mathit{SFH}}$ as \[ \widehat{\partial}_{\mathit{SFH}} = \partial_{0} + \partial_{1}+\cdots, \] where $\partial_{r}$ counts J-holomorphic disks~$u$ with $\mu(u)=1$ and $J_{+}(u)=2r$.
Just like in the closed case, the pair $\left(\bigoplus_{i \in \mathbb{N}} CF(\Sigma,\boldsymbol{\beta},\boldsymbol{\alpha},\mathbf{z})_i \text{, } \widehat{\partial}\right)$, where the map $\widehat{\partial}$ is defined as \[ \widehat{\partial}(c_{i})_{i \in \mathbb{N}} = \left(\sum_{i=0}^\infty \partial_{i} c_{i+j}\right)_{j \in \mathbb{N}}, \] is a filtered chain complex. Using its induced spectral sequence, we can define the spectral order of $(M,\xi)$ in the following way.
\begin{defn} For a contact $3$-manifold $(M,\xi)$ with convex boundary, a compatible partial open book decomposition $\mathcal{P}$, and a collection of pairwise disjoint arcs $\underline{\mathbf{a}}$ containing an arc basis, denote the induced spectral sequence by $E^k(\mathcal{P},\underline{\mathbf{a}})$. We say that $\mathbf{o}(\mathcal{P},\underline{\mathbf{a}})=k$ if the distinguished generator $\mathbf{x}_\xi \in \mathit{CF}(\Sigma,\boldsymbol{\beta},\boldsymbol{\alpha},\mathbf{z})_{0}$ in degree~0 remains nonzero in~$E^k(\mathcal{P},\underline{\mathbf{a}})$, but vanishes in $E^{k+1}(\mathcal{P},\underline{\mathbf{a}})$. Then we define the \emph{spectral order} \[ \mathbf{o}(M,\xi)= \min\left\{ \mathbf{o}(\mathcal{P},\underline{\mathbf{a}})\,\colon\,\mathcal{P}\mbox{ supports $\xi$ and }\underline{\mathbf{a}} \mbox{ contains a basis}\right\} . \] This is always a nonnegative integer. \end{defn}
We will need the following lemma for for the proof of Theorem~\ref{thm:ineq}.
\begin{lem}\label{lem:ball} Let $(M, \xi)$ be a contact $3$-manifold with (possibly empty) convex boundary, and suppose that $B \subset \text{Int}(M)$ is a tight contact ball. If $M_0 = M \setminus \text{Int}(B)$
and $\xi_0 = \xi|_{M_0}$, then \[ \mathbf{o}(M_0,\xi_0) \ge \mathbf{o}(M,\xi). \] Furthermore, we have equality if $M$ is closed. \end{lem}
\begin{proof}
We denote by $\Gamma_B$ the dividing set of $\xi$ on $\partial B$.
Let $\mathcal{P}_0 = (S_0,P_0,h_0)$ be a partial open book decomposition of $(M_0,\gamma_0)$
supporting $\xi_0$, together with a collection of arcs $\underline{\mathbf{a}}_0$ containing a basis,
and write $(\Sigma_0, \boldsymbol{\beta}_0, \boldsymbol{\alpha}_0, \mathbf{z}_0)$ for the corresponding based sutured diagram
of $(-M_0, -\Gamma_0)$.
There is a disk component $D_B$ of $\overline{S_0 \setminus P_0}$ corresponding
to $R_+(\Gamma_B)$, and a disk component $D_B'$ of $\overline{S \setminus h(P)}$
corresponding to $R_-(\Gamma_B)$. Then $h$ uniquely extends to a diffeomorphism
\[
h \colon P \cup D_B \to h(P) \cup D_B',
\]
up to isotopy. If we set $S = S_0$ and $P = P \cup D_B$,
then $\mathcal{P} := (S,P,h)$ is a partial open book of $(M,\xi)$. Furthermore, $\underline{\mathbf{a}} = \underline{\mathbf{a}}_0$
contains an arc basis for $\mathcal{P}$. As $D_B$ lies in a component
of $P \setminus \underline{\mathbf{a}}$ disjoint from $\partial P \setminus \partial S$,
we need to add a basepoint $z_B$ here. The based diagram $(\Sigma,\boldsymbol{\beta},\boldsymbol{\alpha},\mathbf{z})$ corresponding to $(\mathcal{P}, \underline{\mathbf{a}})$
is obtained by filling in a boundary component of $\Sigma_0$ with the disk $D_B \times \{0\}$,
and taking $\mathbf{z} = \mathbf{z}_0 \cup \{z_B\}$. Hence, $\mathbf{o}(\mathcal{P},\underline{\mathbf{a}}) = \mathbf{o}(\mathcal{P}_0,\underline{\mathbf{a}}_0)$,
as their defining filtered chain complexes agree.
It follows that $\mathbf{o}(M_0,\xi_0) \ge \mathbf{o}(M,\xi)$.
Now suppose that $M$ is closed. Let $(S,h)$ be an arbitrary open book
decomposition of $(M,\xi)$, and $\underline{\mathbf{a}}$ an arbitrary collection of arcs containing a basis.
Then each component of $S \setminus \underline{\mathbf{a}}$ is homeomorphic to a disk; let $D$ be one of them.
Consider the partial open book $\mathcal{P}_0 = (S_0, P_0, h_0)$, where $S_0 = S$,
$P_0 = S \setminus D$, and $h_0 = h|_{P_0}$. Then $\mathcal{P}_0$ supports $(M_0,\xi_0)$.
If $(\Sigma,\boldsymbol{\beta},\boldsymbol{\alpha},\mathbf{z})$ is the diagram arising from $(S,h,\underline{\mathbf{a}})$, and
$(\Sigma_0,\boldsymbol{\beta}_0,\boldsymbol{\alpha}_0,\mathbf{z}_0)$ is the diagram arising from $(\mathcal{P}_0,\underline{\mathbf{a}})$,
then $\Sigma = \Sigma_0 \cup (D \times \{0\})$, $\boldsymbol{\beta} = \boldsymbol{\beta}_0$, $\boldsymbol{\alpha} = \boldsymbol{\alpha}_0$,
and we obtain $\mathbf{z}_0$ by removing the unique point $\mathbf{z} \cap D$. Hence,
$\mathbf{o}(S,h,\underline{\mathbf{a}}) = \mathbf{o}(\mathcal{P}_0,\underline{\mathbf{a}})$. Since $(S,h,\underline{\mathbf{a}})$ was arbitrary,
we obtain that $\mathbf{o}(M,\xi) \ge \mathbf{o}(M_0,\xi_0)$. \end{proof}
Using Theorem~\ref{thm:ineq}, we will show that actually equality holds in the first part of Lemma~\ref{lem:ball}.
\section{Inequality of spectral orders} \label{sec:ineq}
The goal of this section is to prove Theorem~\ref{thm:ineq} from the introduction. We first briefly recall the construction of the contact gluing map $\Phi$ on sutured Floer homology, defined by Honda, Kazez, and Mati\'c~\cite{key-2}. Let $(M, \Gamma_M)$ be a sutured manifold, and let $(N, \Gamma_N)$ be a sutured submanifold of~$\text{Int}(M)$. Furthermore, let $\xi$ be a contact structure on $M \setminus \text{Int}(N)$ with convex boundary and dividing set~$\Gamma_M$ on $\partial M$ and dividing set~$\Gamma_N$ on $\partial N$. We can suppose that $M \setminus N$ has no isolated components; i.e., every component of $M \setminus N$ intersects~$\partial M$. Indeed, by Lemma~\ref{lem:ball}, if we remove a tight contact ball from each isolated component, $\mathbf{o}(M,\xi)$ does not decrease.
Choose a collar neighborhood $Z \simeq \partial N \times I$ of~$\partial N$
in~$M \setminus \text{Int}(N)$ such that $Z \cap N = \partial N \times \{0\}$, on which the contact structure $\xi$ is $I$-invariant, and write $N' = M \setminus \text{Int}(N \cup Z)$. Let $\Sigma_{N'}$ be a Heegaard surface compatible with $\xi|_{N'}$, and let~$\Sigma_Z$ be a Heegaard surface compatible with $\xi|_Z$. Then, for any sutured Heegaard diagram $\mathcal{H} = (\Sigma,\boldsymbol{\beta},\boldsymbol{\alpha})$ of $(N,\Gamma_N)$ that is contact-compatible near $\partial N$ in the sense of Honda, Kazez, and Mati\'c~\cite{key-2}, the union $\Sigma\cup\Sigma_Z \cup \Sigma_{N'}$
is a Heegaard surface for~$(M,\Gamma_M)$, and we can complete $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ to attaching sets of $(M,\Gamma_M)$ by adding $\boldsymbol{\alpha}'$ and $\boldsymbol{\beta}'$ compatible with~$\xi|_{N' \cup Z}$. We write \[ \mathcal{H}' = (\Sigma\cup\Sigma_Z \cup\Sigma_{N'},\boldsymbol{\beta}\cup\boldsymbol{\beta}',\boldsymbol{\alpha}\cup\boldsymbol{\alpha}'). \] Then the map \begin{eqnarray*} \Phi_\xi \colon \mathit{CF}(\mathcal{H}) & \rightarrow & \mathit{CF}(\mathcal{H}'),\\ \mathbf{y} & \mapsto & (\mathbf{y},\mathbf{x}') \end{eqnarray*} is a chain map, where $\mathbf{x}' \in \mathbb{T}_{\beta'} \cap \mathbb{T}_{\alpha'}$
is the canonical representative of the contact class $\mathit{EH}(\xi|_{N' \cup Z})$. Note that this construction makes sense even if we replace Heegaard diagrams with multipointed Heegaard diagrams.
\begin{proof}[Proof of Theorem~\ref{thm:ineq}]
As in the statement of Theorem~\ref{thm:ineq}, let $(M,\xi)$ be a contact $3$-manifold with convex boundary and dividing set~$\Gamma_M$, and let $N$ be a codimension zero submanifold of~$\text{Int}(M)$, also with convex boundary and dividing set~$\Gamma_N$. Then let $\mathcal{P}_{N} = (S_{N},P_{N},h_{N})$ be a partial open book decomposition of $(N,\xi|_{N},\Gamma_{N})$, together with a choice of an arc basis~$\underline{\mathbf{a}}_N$, and let $\mathcal{H} = (\Sigma,\boldsymbol{\beta},\boldsymbol{\alpha})$ be the corresponding diagram of $(-N,-\Gamma_N)$.
Let $\zeta$ be an $I$-invariant contact structure on $\partial N \times I$ such that $\partial N \times \{t\}$ is convex with dividing set $\Gamma_N$ for every $t \in I$. According to the Remark after \cite[Lemma~4.1]{key-2}, we can first extend $\mathcal{H}$ to a diagram of $(-N \cup (\partial N \times [0,2]), -\Gamma_N \times \{2\})$ that is contact compatible near $\partial N \times \{2\}$, by gluing two Heegaard surfaces arising from certain special partial open book decompositions of $(\partial N \times I, \zeta)$. We denote the resulting compact compatible diagram $(\Sigma \cup \Sigma_\zeta, \boldsymbol{\beta} \cup \boldsymbol{\beta}_\zeta, \boldsymbol{\alpha} \cup \boldsymbol{\alpha}_\zeta)$. Then, using Step~2 of \cite[Section~4]{key-2}, and as explained above, we can further extend this to a diagram \[ \mathcal{H}' = (\Sigma \cup \Sigma_\zeta \cup \Sigma_Z \cup \Sigma_{N'}, \boldsymbol{\beta} \cup \boldsymbol{\beta}_\zeta \cup \boldsymbol{\beta}', \boldsymbol{\alpha} \cup \boldsymbol{\alpha}_\zeta \cup \boldsymbol{\alpha}') \] of $(-M,-\Gamma_M)$. Analogously to the gluing map, we obtain a chain map \begin{eqnarray*} \Phi \colon \mathit{CF}(\mathcal{H}) & \rightarrow & \mathit{CF}(\mathcal{H}'),\\ \mathbf{y} & \mapsto & (\mathbf{y},\mathbf{x}') \end{eqnarray*} where $\mathbf{x}' \in \mathbb{T}_{\beta_\zeta \cup \beta'} \cap \mathbb{T}_{\alpha_\zeta \cup \alpha'}$
is the canonical representative of the contact class $\mathit{EH}(\xi|_{(\partial N \times [0,2]) \cup Z \cup N'})$. As $\mathcal{H}$ is not necessarily contact compatible, we do not claim that $\Phi$ is the contact gluing map under naturality, but this is not necessary for our purposes.
By construction, $\Phi$ maps the contact class~$\mathbf{x}_{\xi|_N}$ to the contact class~$\mathbf{x}_\xi$. Note that this construction of Honda, Kazez, and Mati\'c~\cite{key-2} actually gives a partial open book $\mathcal{P} = (S,P,h)$ supporting $(M,\xi)$ and an arc basis~$\underline{\mathbf{a}}$ that extend $\mathcal{P}_N$ and~$\underline{\mathbf{a}}_N$, respectively.
Now consider the case when $\underline{\mathbf{a}}_N$ is not an arc basis, but a collection of pairwise disjoint arcs that contains an arc basis. Then we need to choose basepoints $\mathbf{z}$ such that every connected component of $P_N \setminus \cup \underline{\mathbf{a}}_N$ that does not intersect $\partial P_N \setminus \partial S_N$ has exactly one basepoint. The gluing process can be applied to this case without modification, to get a collection of pairwise disjoint arcs~$\underline{\mathbf{a}}$ in~$P$. After gluing, every connected component of $P \setminus \cup \underline{\mathbf{a}}$ disjoint from $\partial P \setminus \partial S$ contains exactly one basepoint, since such a component must come from $P_N \setminus \cup \underline{\mathbf{a}}_N$, and other components do not contain a basepoint. Hence, the data $(\mathcal{P},\underline{\mathbf{a}},\mathbf{z})$ satisfies the conditions needed to define its order. The proof of the fact that the gluing map is a chain map between Floer chain complexes~\cite{key-2} also applies to this case without further modification, for the same reason.
\begin{lem} \label{lem:morphism} Let $\Phi$ be as above. Then the map \[ \overline{\Phi} \colon \bigoplus_{i \in \mathbb{N}} \mathit{CF}(\mathcal{H})_i \to \bigoplus_{i \in \mathbb{N}} \mathit{CF}(\mathcal{H}')_i \] defined by $\overline{\Phi}\left((c_i)_{i \in \mathbb{N}}\right) = (\Phi(c_i))_{i \in \mathbb{N}}$ is a filtered chain map, hence induces a morphism $(\Phi^k)_{k \in \mathbb{N}}$ of spectral sequences; i.e., $\Phi^0 = \overline{\Phi}$, and \[ \Phi^k \colon E^k(\mathcal{P}_{N},\underline{\mathbf{a}}_{N}) \rightarrow E^k(\mathcal{P},\underline{\mathbf{a}}) \] is a chain map for every $k \in \mathbb{N}$ such that the map induced on homology is $\Phi^{k+1}$. \end{lem}
\begin{proof} Let $\mathbf{x}$, $\mathbf{y} \in \mathbb{T}_\beta \cap \mathbb{T}_\alpha$. Any holomorphic disk~$u$ from $(\mathbf{x},\mathbf{x}^{\prime})$ to $(\mathbf{y},\mathbf{x}^{\prime})$ in $\mathit{CF}(\mathcal{H}')$ is actually a holomorphic disk from $\mathbf{x}$ to $\mathbf{y}$ in $\mathit{CF}(\mathcal{H})$; i.e., its domain $D := D(u)$ is zero outside $\Sigma$; see~\cite[p.~12]{key-2}. Since the Euler measure and the point measure of $D$ depend only on the non-zero coefficients, the Maslov index of $u$ in $\mathcal{H}$ and in $\mathcal{H}'$ are the same. Suppose that $\mu(u) = 1$. Then, in $\mathcal{H}'$, we have \begin{eqnarray*} J_{+}(u) & = & 2(n_{(\mathbf{x},\mathbf{x}^{\prime})}(D)+n_{(\mathbf{y},\mathbf{x}^{\prime})}(D)) -
1 + |(\mathbf{x},\mathbf{x}^{\prime})|-|(\mathbf{y},\mathbf{x}^{\prime})|\\
& = & 2(n_{\mathbf{x}}(D)+n_{\mathbf{y}}(D))-1+|\mathbf{x}|+|\mathbf{x}^{\prime}|-|\mathbf{y}|-|\mathbf{x}^{\prime}|\\
& = & 2(n_{\mathbf{x}}(D)+n_{\mathbf{y}}(D))-1+|\mathbf{x}|-|\mathbf{y}|. \end{eqnarray*} This is the same as the value of $J_{+}(u)$ in $\mathcal{H}$. Hence $\Phi$ preserves the $J_{+}$ filtration.
Now, by the definition of the differential $\partial_i$, the map $\Phi$ commutes with $\partial_i$ for all $i \in \mathbb{N}$. Hence, it commutes with the total differential $\widehat{\partial}$, and so $\overline{\Phi}$ is a filtered chain map. Therefore $\overline{\Phi}$ induces a morphism $(\Phi^k)_{k \in \mathbb{N}}$ between the corresponding spectral sequences. \end{proof}
Since $\Phi(\mathbf{x}_{\xi|_N}) = \mathbf{x}_{\xi}$, Lemma~\ref{lem:morphism} implies that if
$\mathbf{x}_{\xi|_N}$ vanishes in $E^k(\mathcal{P}_{N},\underline{\mathbf{a}}_{N})$, then it also vanishes in $E^k(\mathcal{P},\underline{\mathbf{a}})$. Hence, by the definition of the spectral order, \[ \mathbf{o}(\mathcal{P}_{N},\underline{\mathbf{a}}_{N})\ge \mathbf{o}(\mathcal{P},\underline{\mathbf{a}})\ge \mathbf{o}(M,\xi). \] Taking the minimum of over all possible choices of $(\mathcal{P}_{N},\underline{\mathbf{a}}_{N})$, we get that \[
\mathbf{o}(N,\xi|_{N})\ge \mathbf{o}(M,\xi), \] as required. This concludes the proof of Theorem~\ref{thm:ineq}. \end{proof}
We are now in a position to strengthen Lemma~\ref{lem:ball}.
\begin{cor} Let $(M, \xi)$ be a connected contact $3$-manifold with (possibly empty) convex boundary, and suppose that $B \subset \text{Int}(M)$ is a tight contact ball. If $M_0 = M \setminus \text{Int}(B)$
and $\xi_0 = \xi|_{M_0}$, then \[ \mathbf{o}(M_0,\xi_0) = \mathbf{o}(M,\xi). \] \end{cor}
\begin{proof}
We have already shown the closed case in Lemma~\ref{lem:ball},
so we can suppose that $\partial M \neq \emptyset$. Let $M' \subset \text{Int}(M)$
be a codimension zero submanifold of $M$ with convex boundary,
such that $(M',\xi')$ is contactomorphic to $(M,\xi)$, where $\xi' = \xi|_{M'}$.
Since $M$ is connected, we can assume that $B \subset \text{Int}(M) \setminus M'$.
If we apply Theorem~\ref{thm:ineq} to the sequence $M' \subset M_0 \subset M$,
we obtain that
\[
\mathbf{o}(M',\xi') \ge \mathbf{o}(M_0,\xi_0) \ge \mathbf{o}(M,\xi).
\]
As $(M,\xi)$ and $(M',\xi')$ are contactomorphic, $\mathbf{o}(M,\xi) = \mathbf{o}(M',\xi')$,
and the result follows. \end{proof}
\section{Calculation of upper bounds on some spectral orders} \label{sec:gt}
Let $(M,\xi)$ be a contact $3$-manifold with convex boundary. Suppose that $(M,\xi)$
is overtwisted. Then, by definition, it contains an embedded overtwisted disk $\Delta$. This has a standard neighborhood; i.e., there exists a neighborhood $U\supset\Delta$ such that $(U,\xi|_{U})$ is contactomorphic to a neighborhood of the disk $\Delta_{std}=\{z=0,\rho\le\pi\}$ inside the standard overtwisted contact structure on $\mathbb{R}^{3}$, which is defined as follows~\cite{key-8}: \[ \xi_{OT}=\ker(\cos\rho \, dz+\rho\sin\rho\, d\phi). \]
Inside $U$, we can perturb $\Delta$ to a convex surface $D$. Take a neighborhood $V=D\times [-1,1]$ such that $\xi|_{\text{Int}(M)}$ is $\mathbb{R}$-invariant. After rounding its edges, we obtain an open subset $V_{0}\simeq D^3$ such that the dividing set $\Gamma_{V_{0}}$ on $\partial V_{0}$ is given by three disjoint curves. Honda, Kazez, and Mati\'c~\cite[Example~1]{key-1} gave a partial open book decomposition of $N=\overline{V}_0$, and the corresponding Heegaard diagram is shown in Figure~\ref{fig:ot}. \begin{figure}
\caption{A sutured Heegaard diagram arising from a partial open book decomposition of a neighborhood of an overtwisted disk. We obtain the Heegaard surface by identifying the two bold horizontal arcs.}
\label{fig:ot}
\end{figure} This diagram can be used to show that $\mathbf{o}(M,\xi)=0$, which was proven by Kutluhan et al.~\cite{key-10} in the closed case using the fact that an overtwisted contact structure admits an open book whose monodromy is not right-veering.
\begin{rem} It is convenient and customary to present the sutured diagram $(\Sigma,\boldsymbol{\beta},\boldsymbol{\alpha})$ arising from a partial open book decomposition $(S,P,h)$ and arcs basis $\underline{\mathbf{a}}$ on the surface $-S \times \{0\} \subset \Sigma$. Instead of gluing in $P \times \{0\}$, for each $a \in \underline{\mathbf{a}}$, we identify the opposite edges of $N(a) \cap \partial S$ for a regular neighborhood $N(a)$ of $a$ in $S$. This is possible since $P = N(\underline{\mathbf{a}})$. \end{rem}
\begin{prop} \label{prop:ot} If~$N$ is the standard neighborhood of an overtiwsted disk in the contact manifold $(M,\xi)$ as above, then \[
\mathbf{o}(N,\xi|_{N})=0. \] \end{prop}
\begin{proof}
Honda, Kazez, and Mati\'c~\cite[Example~1]{key-1} computed that $c(N,\xi|_{N}) = 0$; we extend their proof. Consider the partial open book decomposition of $(N,\xi)$ shown in Figure~\ref{fig:ot}. The contact element $\mathit{EH}(N,\xi|_{N})$ is represented by the point~$x$, which is zero in homology because $\partial y = x$. The only J-holomophic curve from $y$ to $x$ is the bigon, which satisfies $J_{+}=0$. Hence $\mathbf{o}(N,\xi|_{N})\le0$. \end{proof}
\begin{thm} \label{thm:OT} If the contact manifold $(M,\xi)$ with convex boundary is overtwisted, then $\mathbf{o}(M,\xi)=0$. \end{thm}
\begin{proof}
We have $\mathbf{o}(M,\xi)\le \mathbf{o}(N,\xi|_{N})=0$ by Theorem~\ref{thm:ineq} and Proposition~\ref{prop:ot}. \end{proof}
We now consider the case when $(M,\xi)$ has Giroux $2\pi$-torsion. Recall that a contact manifold $(M,\xi)$ has $2\pi$-torsion if it admits an embedding \[ (M_{2\pi},\eta_{2\pi})=(T^{2}\times[0,1],\ker(\cos(2\pi t)\,dx-\sin(2\pi t)\,dy))\hookrightarrow(M,\xi). \] The boundary of $(M_{2\pi},\eta_{2\pi})$ is not convex. However, as in \cite[Lemma~5]{key-5}, if it embeds in $(M,\xi)$, then there exist small $\epsilon_0$, $\epsilon_1 >0$ such that the slightly extended domain \[ (M',\eta')= \left(T^{2}\times[-\epsilon_0,1+\epsilon_1],\ker(\cos(2\pi t)\,dx-\sin(2\pi t)\,dy) \right) \] also embeds inside $(M,\xi)$ such that $T^2 \times \{-\epsilon_0\}$ and $T^2 \times \{\epsilon_1\}$ are pre-Lagrangian tori with integer slopes $s_0$ and $s_1$ that form a basis of $H_1(T^2)$. By the work of Ghiggini~\cite{key-3}, we can perturb $\partial M'$ to get a new contact submanifold $\widetilde{M}$ such that $\partial\widetilde{M}$ is convex, and the slopes of the dividing sets are $s_0$ and $s_1$. After a change of coordinates in $\widetilde{M}$, we can assume these slopes are $0$ and $\infty$.
The contact manifold $\widetilde{M}$ is non-minimally-twisting and consists of five basic slices, which means that we can construct a partial open book decomposition of it by attaching four bypasses to a partial open book diagram of a basic slice, which can be found in Examples~4, 5, and~6 of~\cite{key-1}. The diagram we get is shown in Figure~\ref{fig:gt}. \begin{figure}
\caption{A sutured diagram arising from a partial open book decomposition of a neighborhood of a Giroux torsion domain. The opposite green arcs in the boundary are identified.}
\label{fig:gt}
\end{figure}
\begin{figure}
\caption{We apply the Sarkar-Wang algorithm by isotoping the red curves along the dashed arcs.}
\label{fig:dashed}
\end{figure}
Applying the Sarkar-Wang algorithm~\cite{key-12} to this diagram along the dotted arcs in Figure~\ref{fig:dashed}, we obtain the one in Figure~\ref{fig:SW}. It is easy to check that every region that does not intersect the boundary is either a bigon or a quadrilateral. In Figure~\ref{fig:SW}, the $\beta$-curves are shown in red and the $\alpha$-curves in blue, and the opposite green arcs in the boundary of the surface are identified. The intersection points between $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ are labeled $x_1, \dots, x_{18}$ from right-to-left along the horizontal blue arc, and along the four vertical blue arcs they are labeled from top-to-bottom $y_1, \dots, y_{15}$, $z_1, \dots, z_{10}$, $w_1, \dots, w_6$, and $v_1, \dots, v_3$, respectively. \begin{figure}
\caption{The diagram after applying the Sarkar-Wang algorithm.}
\label{fig:SW}
\end{figure}
\begin{figure}
\caption{The quadrilaterals and bigons relevant to the computation are shaded.}
\label{fig:color}
\end{figure}
The contact element $\mathit{EH}\left(\xi|_{\widetilde{M}}\right)$ is represented by the unordered tuple $(x_{1},y_{1},z_{1},w_{1}, v_1)$. We now directly prove that the contact invariant of $\widetilde{M}$ is zero and calculate its spectral order with respect to the given diagram, thus giving an upper bound on $\mathbf{o}(\widetilde{M})$.
If $Q$ is a quadrilateral component of $\Sigma \setminus (\boldsymbol{\alpha} \cup \boldsymbol{\beta})$ disjoint from $\partial \Sigma$ with corners $c_1$, $c_2$, $c_3$, $c_4 \in \boldsymbol{\alpha} \cap \boldsymbol{\beta}$, then we say that $c_{1}$, $c_{3}$ are its from-corners and $c_{2}$, $c_{4}$ are its to-corners if \[ \partial(\partial Q \cap \boldsymbol{\alpha}) = c_1 + c_3 - c_2 - c_4. \] For any generator $(c_1,c_3,\dots) \in \mathbb{T}_\alpha \cap \mathbb{T}_\beta$, the coefficient of $(c_{2},c_{4},\dots)$ in the boundary $\partial(c_{1},c_{3},\dots)$ is the number of such quadrilaterals.
Since the only quadrilateral whose to-corners are in $\{x_{1},y_{1},z_{1},w_{1},v_1\}$ is $y_{1}y_{2}z_{1}z_{2}$, we get that \[ \partial(x_{1},y_{2},z_{2},w_{1},v_1)=(x_{1},y_{1},z_{1},w_{1},v_1)+(x_{1},y_{3},z_{2},w_{1},v_1), \] where the last term comes from the bigon $y_{2}y_{3}$. This quadrilateral and bigon are shaded grey in Figure~\ref{fig:SW}.
The only quadrilateral whose to-corners are in $\{x_{1},y_{2},z_{2},w_{1},v_1\}$ is $x_{1}x_{2}y_{3}y_{4}$, and we have that \[ \partial(x_{2},y_{4},z_{2},w_{1},v_1)=(x_{1},y_{3},z_{2},w_{1},v_1)+(x_{3},y_{4},z_{2},w_{1},v_1), \] where the last term comes from the bigon $x_{2}x_{3}$. This quadrilateral and bigon are shaded pink in Figure~\ref{fig:SW}.
The only quadrilateral whose to-corners are in $\{x_{3},y_{4},z_{2},w_{1},v_1\}$ is $x_{3}x_{4}w_{1}w_{2}$, and we have that \[ \partial(x_{4},y_{4},z_{2},w_{2},v_1)=(x_{3},y_{4},z_{2},w_{1},v_1)+(x_{5},y_{4},z_{2},w_{2},v_1), \] where the last term comes from the bigon $x_{4}x_{5}$. This quadrilateral and bigon are shaded light blue in Figure~\ref{fig:SW}.
The only quadrilateral whose to-corners are in $\{x_{5},y_{4},z_{2},w_{2},v_1\}$ is $x_{5}x_{6}z_{2}z_{3}$, shaded green in Figure~\ref{fig:SW}, and we have that \[ \partial(x_{6},y_{4},z_{3},w_{2},v_1)=(x_{5},y_{4},z_{2},w_{2},v_1)+(x_9,y_1,z_3,w_2,v_1), \] where the last term comes from the quadrilateral $x_6x_9y_4y_1$.
The only quadrilateral whose to-corners are in $\{x_9,y_1,z_3,w_2,v_1\}$ is $y_1y_{15}z_{3}z_{2}$, and we have that \[ \partial(x_{9},y_{15},z_{2},w_{2},v_1)=(x_9,y_1,z_3,w_2,v_1)+(x_9,y_{14},z_2,w_2,v_1), \] where the last term comes from the bigon $y_{15}y_{14}$. This quadrilateral and bigon are shaded yellow in Figure~\ref{fig:SW}.
The only quadrilateral whose to-corners are in $\{x_9,y_{14},z_2,w_2,v_1\}$ is $y_{14}y_{13}v_1v_2$, and we have that \[ \partial(x_{9},y_{13},z_{2},w_{2},v_2)=(x_9,y_{14},z_2,w_2,v_1)+(x_9,y_{12},z_2,w_2,v_2), \] where the last term comes from the bigon $y_{13}y_{12}$. This quadrilateral and bigon are shaded blue in Figure~\ref{fig:SW}.
Finally, the only quadrilateral whose to-corners are in $\{x_9,y_{12},z_2,w_2,v_2\}$ is $y_{12}y_{11}w_2w_3$, shown in red, and we have that \[ \partial(x_{9},y_{11},z_{2},w_{3},v_2)=(x_9,y_{12},z_2,w_2,v_2). \]
Hence, over $\mathbb{F}_2$, \[ \begin{split} \partial((x_{1},y_{2},z_{2},w_{1},v_1)+(x_{2},y_{4},z_{2},w_{1},v_1)+(x_{4},y_{4},z_{2},w_{2},v_1)+(x_{6},y_{4},z_{3},w_{2},v_1)\\ + (x_{9},y_{15},z_{2},w_{2},v_1) + (x_{9},y_{13},z_{2},w_{2},v_2) + (x_{9},y_{11},z_{2},w_{3},v_2))= (x_{1},y_{1},z_{1},w_{1}), \end{split} \]
which is exactly $\mathbf{x}_{\xi|_{\widetilde{M}}}$. Thus $\mathit{EH}\left(\xi|_{\widetilde{M}}\right)=0$, so the spectral order of $\widetilde{M}$ is finite.
\begin{rem*} This result, together with the gluing map of \cite{key-2}, gives an explicit computational proof of the fact that the contact invariant of any contact $3$-manifold with Giroux $2\pi$-torsion vanishes, which was proven in the closed case by Ghiggini, Honda, and Van Horn-Morris~\cite{key-5}. \end{rem*}
\begin{rem*} In \cite[Example 6-(c)]{key-1}, Honda, Kazez, and Mati\'c showed that if we only attach four bypasses to a basic slice; i.e., if the contact structure is minimally twisting, then the contact invariant is non-zero because it embeds in the unique Stein fillable contact structure on~$T^{3}$, which already has non-zero contact invariant. This can also be shown explicitly using a computation analogous to, but simpler than the one above. Hence, it is necessary to enlarge the Giroux $2\pi$-torsion domain a bit to obtain vanishing of the contact element. \end{rem*}
\begin{prop} \label{prop:gt} For the perturbed Giroux $2\pi$-torsion domain $\widetilde{M}$, we have \[
\mathbf{o}\left(\widetilde{M}, \xi|_{\widetilde{M}} \right)\le 2. \] \end{prop}
\begin{proof} The complete list of the $J$-holomorphic disks used in the calculations above and the values used to compute their $J_{+}$ are given in the table below. If we label the $\alpha$- and $\beta$-curves such that $x_1 \in \alpha_1 \cap \beta_1$, $y_1 \in \alpha_2 \cap \beta_2$, $z_1 \in \alpha_3 \cap \beta_3$, $w_1 \in \alpha_4 \cap \beta_4$, and $v_1 \in \alpha_5 \cap \beta_5$, then \[ \begin{split} x_1, x_9, y_4, y_{16} \in \beta_1, \\ x_6, y_1, z_2 \in \beta_2, \\ z_1, x_2, x_3, y_2, y_3, y_{11}, w_2 \in \beta_3, \\ x_4, x_5, y_{14}, y_{15}, z_3, w_1, v_2 \in \beta_4, \\ y_{12}, y_{13}, w_3, v_1 \in \beta_5. \end{split} \] Furthermore, $x_i \in \alpha_1$, $y_i \in \alpha_2$, $z_i \in \alpha_3$, $w_i \in \alpha_4$, and $v_i \in \alpha_5$
for any~$i$. Note that if there is a bigon connecting $\mathbf{x}$, $\mathbf{y} \in \mathbb{T}_\alpha \cap \mathbb{T}_\beta$, then $|\mathbf{x}| = |\mathbf{y}|$. Using this, \[ \begin{split}
|(x_1,y_1,z_1,w_1,v_1)| = |(1)(2)(3)(4)(5)| = 5, \\
|(x_1,y_2,z_2,w_1,v_1)| = |(1)(23)(4)(5)| = 4 = |(x_1,y_3,z_2,w_1,v_1)|, \\
|(x_2,y_4,z_2,w_1,v_1)| = |(132)(4)(5)| = 3 = |(x_3,y_4,z_2,w_1,v_1)|, \\
|(x_4,y_4,z_2,w_2,v_1)| = |(1432)(5)| = 2 = |(x_5,y_4,z_2,w_2,v_1)|, \\
|(x_6,y_4,z_3,w_2,v_1)| = |(12)(34)(5)| = 3, \\
|(x_9,y_1,z_3,w_2,v_1)| = |(1)(2)(34)(5)| = 4, \\
|(x_9,y_{15},z_2,w_2,v_1)| = |(1)(243)(5)| = 3 = |(x_9,y_{14},z_2,w_2,v_1)| \\
|(x_9,y_{14},z_2,w_2,v_2)| = |(1)(2543)| = 2 = |(x_9,y_{12},z_2,w_2,v_2)|, \\
|(x_9,y_{11},z_2,w_3,v_2)| = |(1)(23)(45)| = 3. \end{split} \]
From the table below, we see that every $J$-holomorphic disk $u$ used to compute the differential satisfies $J_{+}(u) \le 2$; cf.~Equation~\ref{eqn:J}.
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline Type & Name & $2(n_{\mathbf{x}}+n_{\mathbf{y}})$ & $|\mathbf{x}|-|\mathbf{y}|$ & $J_{+}$\tabularnewline \hline \hline quadrilateral & $y_{1}y_{2}z_{1}z_{2}$ & 2 & -1 & 0\tabularnewline \hline quadrilateral & $x_{1}x_{2}y_{3}y_{4}$ & 2 & -1 & 0\tabularnewline \hline quadrilateral & $x_{3}x_{4}w_{1}w_{2}$ & 2 & -1 & 0\tabularnewline \hline quadrilateral & $x_{5}x_{6}z_{2}z_{3}$ & 2 & 1 & 2\tabularnewline \hline quadrilateral & $x_6 x_9 y_4 y_1$ & 2 & -1 & 0 \tabularnewline \hline quadrilateral & $y_1 y_{15} z_3 z_2$ & 2 & -1 & 0 \tabularnewline \hline quadrilateral & $y_{14} y_{13} v_1 v_2$ & 2 & -1 & 0 \tabularnewline \hline quadrilateral & $y_{12} y_{11} w_2 w_3$ & 2 & 1 & 2 \tabularnewline \hline bigon & $y_{2}y_{3}$ & 1 & 0 & 0\tabularnewline \hline bigon & $x_{2}x_{3}$ & 1 & 0 & 0\tabularnewline \hline bigon & $x_{4}x_{5}$ & 1 & 0 & 0\tabularnewline \hline bigon & $y_{15}y_{14}$ & 1 & 0 & 0\tabularnewline \hline bigon & $y_{13}y_{12}$ & 1 & 0 & 0\tabularnewline \hline \end{tabular} \par\end{center}
For simplicity, we will write $(i,j,k,l,m)$ for the generator $(x_i,y_j,z_k,w_l,v_m)$. Then let \[ \begin{split} b_0 &= (1,2,2,1,1) + (2,4,2,1,1) + (4,4,2,2,1) + (6,4,3,2,1) + (9,15,2,2,1) + (9,13,2,2,2), \\ b_1 &= (6,4,3,2,1) + (9,15,2,2,1) + (9,13,2,2,2) + (9,11,2,3,2), \text{ and} \\ b_2 &= (9,11,2,3,2), \end{split} \] considered as chains with $\mathbb{Z}_2$ coefficients. Using the table above, \[ \begin{split} \partial_0 b_0 &= (1,1,1,1,1) + (5,4,2,2,1) + (9,12,2,2,2), \\ \partial_0 b_1 &= (9,12,2,2,2) \text{ and } \partial_1 b_1 = (5,4,2,2,1) + (9,12,2,2,2), \\ \partial_0 b_2 &= 0 \text{, } \partial_1 b_2 = (9,12,2,2,2), \text { and } \partial_2 b_2 = 0. \end{split} \] Hence $\partial_0 b_0 + \partial_1 b_1 = (1,1,1,1,1)$, $\partial_0 b_1 + \partial_1 b_2 = 0$, and $\partial_0 b_2 = 0$. So, if we set $b_i = 0$ for every $i > 2$, then $\widehat{\partial}(b_i)_{i \in \mathbb{N}} = (c_i)_{i \in \mathbb{N}}$, where $c_0 = (1,1,1,1,1)$
represents the $\mathit{EH}$ class, and $c_i = 0$ for every $i > 0$. By equation~\eqref{eqn:main}, the element $\mathbf{x}_{\xi|_{\widetilde{M}}}$ lies in $B^2_0$, and hence vanishes in $E^3$; i.e., $\mathbf{o}\left(\widetilde{M}, \xi|_{\widetilde{M}}\right)\le 2$, as claimed. \end{proof}
As an immediate corollary, we obtain Theorem~\ref{thm:gt} from the introduction.
\begin{cor} If a contact $3$-manifold $(M,\xi)$ with convex boundary has Giroux $2\pi$-torsion, then \[ \mathbf{o}(M,\xi) \le 2. \] \end{cor}
\begin{proof} If the Giroux $2\pi$-torsion domain $M_{2\pi}$ embeds in $M$, then the perturbed domain $\widetilde{M}$ also embeds in $M$, by the argument outlined at the beginning of this section. Then Theorem~\ref{thm:ineq} and Proposition~\ref{prop:gt} imply that \[
\mathbf{o}(M,\xi)\le \mathbf{o}\left(\widetilde{M}, \xi|_{\widetilde{M}}\right) \le 2. \] \end{proof}
\section{Open questions}
We raise some questions that naturally arise from the discussions above. First, as in the case of closed contact $3$-manifolds, we would like to know how the spectral order $\mathbf{o}(\mathcal{P},\underline{\mathbf{a}})$ depends on the choice of partial open book decomposition~$\mathcal{P}$ and arc system~$\underline{\mathbf{a}}$.
\begin{rem*} Given two possible choices of partial open book decompositions $(\mathcal{P},\underline{\mathbf{a}})$ and $(\mathcal{P}',\underline{\mathbf{a}}')$ for a given contact $3$-manifold $(M,\xi)$ with convex boundary, it is natural to ask whether $\mathbf{o}(\mathcal{P},\underline{\mathbf{a}})=\mathbf{o}(\mathcal{P}',\underline{\mathbf{a}}')$. In the closed case, according to Kutluhan et al.~\cite{key-14}, the number $\mathbf{o}(S,\phi,\underline{\mathbf{a}})$ does not depend on the isotopy class of the arc basis $\underline{\mathbf{a}}$, but if two arc bases differ by an arc-slide, the corresponding values of $\mathbf{o}$ might not be the same. Since our definition of $\mathbf{o}$ is a direct generalization of the original one, the same holds in our case. \end{rem*}
Now, given the inequality $\mathbf{o}(N,\xi|_{N}) \ge \mathbf{o}(M,\xi)$, whenever $(N,\xi|_{N})$ is a compact codimension zero submanifold of $(M,\xi)$ with convex boundary, we are led to the following question.
\begin{qn} \label{qn:1} If a contact $3$-manifold $(N,\xi)$ with convex boundary satisfies $\mathbf{o}(M,\xi_{M}) \le k$ for every closed contact $3$-manifold $(M,\xi_{M})$ in which $(N,\xi)$ embeds, do we have $\mathbf{o}(N,\xi)\le k$? \end{qn}
An affirmative answer to Question~\ref{qn:1} would imply that the inequality
$\mathbf{o}(N,\xi|_{N}) \le \mathbf{o}(M,\xi)$ is sharp and cannot be improved without giving extra conditions even when~$M$ is assumed to be closed. We can ask the following question regarding the spectral order of planar torsion domains.
\begin{qn} \label{qn:2} Is there a way to prove that the order of a Giroux torsion domain is at most~$1$, instead of~$2$? \end{qn}
The upper bound to the spectral order of a Giroux torsion domain is predicted to be $1$ by \cite[Question~6.3]{key-14}, since a Giroux torsion domain is a planar torsion domain of order $1$. However, our computation only allows us to prove that it is at most $2$. If the above question has an affirmative answer, then we must be able to prove it via explicit computation by starting from a complete system of arcs, and then duplicating the arcs, one by one. The problem is that the resulting diagram is too large for practical computation by hand.
Finally, probably the most interesting question in this area is whether the converse of Theorem~\ref{thm:OT} holds, analogously to \cite[Question~6.1]{key-14}.
\begin{qn} If $\mathbf{o}(M,\xi) = 0$, then does this imply that $\xi$ is overtwisted? \end{qn}
\end{document} | arXiv |
Strang splitting
Strang splitting is a numerical method for solving differential equations that are decomposable into a sum of differential operators. It is named after Gilbert Strang. It is used to speed up calculation for problems involving operators on very different time scales, for example, chemical reactions in fluid dynamics, and to solve multidimensional partial differential equations by reducing them to a sum of one-dimensional problems.
Fractional step methods
As a precursor to Strang splitting, consider a differential equation of the form
${\frac {d{y}}{dt}}=L_{1}({y})+L_{2}({y})$
where $L_{1}$, $L_{2}$ are differential operators. If $L_{1}$ and $L_{2}$ were constant coefficient matrices, then the exact solution to the associated initial value problem would be
$y(t)=e^{(L_{1}+L_{2})t}y_{0}$.
If $L_{1}$ and $L_{2}$ commute, then by the exponential laws this is equivalent to
$y(t)=e^{L_{1}t}e^{L_{2}t}y_{0}$.
If they do not, then by the Baker–Campbell–Hausdorff formula it is still possible to replace the exponential of the sum by a product of exponentials at the cost of a first order error:
$e^{(L_{1}+L_{2})t}y_{0}=e^{L_{1}t}e^{L_{2}t}y_{0}+{\mathcal {O}}(t)$.
This gives rise to a numerical scheme where one, instead of solving the original initial problem, solves both subproblems alternating:
${\tilde {y}}_{1}=e^{L_{1}\Delta t}y_{0}$
$y_{1}=e^{L_{2}\Delta t}{\tilde {y}}_{1}$
${\tilde {y}}_{2}=e^{L_{1}\Delta t}y_{1}$
$y_{2}=e^{L_{2}\Delta t}{\tilde {y}}_{2}$
etc.
In this context, $e^{L_{1}\Delta t}$ is a numerical scheme solving the subproblem
${\frac {d{y}}{dt}}=L_{1}({y})$
to first order. The approach is not restricted to linear problems, that is, $L_{1}$ can be any differential operator.
Strang splitting
Strang splitting extends this approach to second order by choosing another order of operations. Instead of taking full time steps with each operator, instead, one performs time steps as follows:
${\tilde {y}}_{1}=e^{L_{1}{\frac {\Delta t}{2}}}y_{0}$
${\bar {y}}_{1}=e^{L_{2}\Delta t}{\tilde {y}}_{1}$
$y_{1}=e^{L_{1}{\frac {\Delta t}{2}}}{\bar {y}}_{1}$
${\tilde {y}}_{2}=e^{L_{1}{\frac {\Delta t}{2}}}y_{1}$
${\bar {y}}_{2}=e^{L_{2}\Delta t}{\tilde {y}}_{2}$
$y_{2}=e^{L_{1}{\frac {\Delta t}{2}}}{\bar {y}}_{2}$
etc.
One can prove that Strang splitting is second order by using either the Baker-Campbell-Hausdorff formula, Rooted tree analysis or a direct comparison of the error terms using Taylor expansion. For the scheme to be second order accurate, $e^{\cdots }$ must be a second order approximation to the solution operator as well.
See also
• List of operator splitting topics
• Matrix splitting
References
• Strang, Gilbert. On the construction and comparison of difference schemes. SIAM Journal on Numerical Analysis 5.3 (1968): 506–517. doi:10.1137/0705041
• McLachlan, Robert I., and G. Reinout W. Quispel. Splitting methods. Acta Numerica 11 (2002): 341–434. doi:10.1017/S0962492902000053
• LeVeque, Randall J., Finite volume methods for hyperbolic problems. Vol. 31. Cambridge University Press, 2002. (pbk ISBN 0-521-00924-3)
| Wikipedia |
Patrizia Gianni
Patrizia M. Gianni (born 1952)[1] is an Italian mathematician specializing in computer algebra. She is known for her early research on Gröbner bases including her discovery of the FGLM algorithm for changing monomial orderings in Gröbner bases,[2] and for her development of the components of the Axiom computer algebra system concerning polynomials and rational functions.[3]
Gianni is a professor of algebra in the mathematics department of the University of Pisa.[4] She earned a laurea from the University of Pisa,[3] and has worked for IBM Research as well as for the University of Pisa.[5]
References
1. Birth year from Library of Congress catalog entry, retrieved 2022-03-15
2. Mora, Teo (2016), Solving polynomial equation systems. Vol. IV. Buchberger theory and beyond, Encyclopedia of Mathematics and its Applications, vol. 158, Cambridge University Press, p. 469, doi:10.1017/CBO9781316271902, ISBN 978-1-107-10963-6, MR 3559383, at the Gröbner basis workshop held at Cornell University in October 1988 ... Patrizia Gianni and Daniel Lazard independently presented the FGLM algorithm
3. Jenks, Richard D.; Sutor, Robert S. (1992), "Contributors", Axiom: The Scientific Computation System, Springer, p. p. xxiii, doi:10.1007/978-1-4612-2940-7, ISBN 978-1-4612-2940-7
4. Patrizia Gianni, University of Pisa Mathematics Department, retrieved 2022-03-15
5. "Patrizia M. Gianni", ACM Digital Library, Association for Computing Machinery, retrieved 2022-03-15
External links
• Home page
• Patrizia Gianni publications indexed by Google Scholar
Authority control
International
• ISNI
• VIAF
National
• Norway
• Israel
• United States
• Netherlands
Academics
• CiNii
• Mathematics Genealogy Project
Other
• IdRef
| Wikipedia |
Swift–Hohenberg equation
The Swift–Hohenberg equation (named after Jack B. Swift and Pierre Hohenberg) is a partial differential equation noted for its pattern-forming behaviour. It takes the form
${\frac {\partial u}{\partial t}}=ru-(1+\nabla ^{2})^{2}u+N(u)$
where u = u(x, t) or u = u(x, y, t) is a scalar function defined on the line or the plane, r is a real bifurcation parameter, and N(u) is some smooth nonlinearity.
The equation is named after the authors of the paper,[1] where it was derived from the equations for thermal convection.
The webpage of Michael Cross[2] contains some numerical integrators which demonstrate the behaviour of several Swift–Hohenberg-like systems.
References
1. J. Swift,P.C. Hohenberg (1977). "Hydrodynamic fluctuations at the convective instability". Phys. Rev. A. 15 (1): 319–328. Bibcode:1977PhRvA..15..319S. doi:10.1103/PhysRevA.15.319.{{cite journal}}: CS1 maint: uses authors parameter (link)
2. Java applet demonstrations
| Wikipedia |
How would a pH curve look like for titration of diluted weak base compared to concentrated one?
Let's say I take $\pu{80 g}$ of a weak base, dilute it with $\pu{50 ml}$ of water and titrate it with a strong acid. I get a titration curve. Now I take again $\pu{80 g}$ of the same weak base but this time I dilute it with $\pu{200 ml}$ of water and titrate it with the strong acid. What will be the new titration curve compared to the first one?
And here is what confuses me:
On one hand, the diluted base should have lower $\mathrm{pH}$. But on the other hand, at half-equivalence point, $K_\mathrm{b}=\mathrm{pOH}$, and since it's the same base, $K_\mathrm{b}$ is the same, and thus $\mathrm{pOH}$ and $\mathrm{pH}$ at half equivalence point should be the same. I am also not sure about the endpoint.
acid-base titration
ShanyShany
$\begingroup$ Welcome to Chemistry.SE. We'd like you to take the Tour to familiarize with the site. $\endgroup$ – Mathew Mahindaratne Apr 15 '18 at 19:36
$\begingroup$ @Shany was last seen on July 19, 2018, and this was her/his only question. Maybe this question has out-lived its shelf life and should be flushed with success? Just a thought. $\endgroup$ – Ed V Jul 19 '19 at 1:51
I am first going to do the 'flip side' of the OP's scenario and then basically flip it back. So we start with a 0.1 M aqueous solution of benzoic acid and the initial solution volume is 25 mL. Benzoic acid is a monoprotic weak acid with $\mathrm{p}K_\mathrm{a} = 4.20$. The initial pH is 2.606. The strong base used for the titration is 0.1 M NaOH. At the equivalence point, the pH = 8.454 and the solution volume is 50 mL (= 25 mL benzoic acid solution + 25 mL NaOH solution). Thus, the pH is simply the pH of 0.05 M sodium benzoate solution. After all 50 mL of NaOH has been added, the pH = 12.523 and the final solution volume is 75 mL. At the half-equivalence point, 12.5 mL of NaOH has been added and pH = 4.185.
Now assume the initial benzoic acid solution is only 0.025 M, but the initial volume is 100 mL. The total number of moles of benzoic acid is the same as before. The initial pH is 2.912: the benzoic acid solution is only 25% as concentrated. At the equivalence point, the pH = 8.254 and the solution volume is 125 mL (= 100 mL benzoic acid solution + 25 mL NaOH solution). The equivalence point pH is closer to pH = 7 because the solution is simply a 0.02 M sodium benzoate solution, i.e., the number of moles of sodium benzoate is now in a volume that is 2.5 times larger, so the concentration is 40% of 0.05 M. After all 50 mL of NaOH has been added, the pH = 12.222 and the final solution volume is 150 mL. At the half-equivalence point, 12.5 mL of MaOH has been added and pH = 4.188, moving slightly up toward pH = 7. See the titration curves below:
Now flipping back: the OP's scenario involving a weak base titrated with strong acid. For reasons that will become clear, assume the hypothetical weak base has $\mathrm{p}K_\mathrm{b} = 9.80$, and that the strong acid is 0.1 M HCl solution. Note that the $\mathrm{p}K_\mathrm{b}$ is simply 14 (treated as exact throughout) - 4.20.
As before, start by assuming 25 mL of 0.1 M weak base. The initial pH = 14 – 2.606 = 11.394. At the equivalence point, the pH = 5.546 and the solution volume is 50 mL (= 25 mL weak base solution + 25 mL HCl solution). Thus, the pH is simply the pH of a 0.05 M weak base salt solution. After all 50 mL of HCl has been added, the pH = 1.477 and the final solution volume is 75 mL. At the half-equivalence point, 12.5 mL of HCl has been added and pH = 9.815.
Now assume the initial weak base solution is only 0.025 M, but the initial volume is 100 mL. The total number of moles of weak base is the same as before. The initial pH is 11.088: the weak base solution is only 25% as concentrated. At the equivalence point, the pH = 5.746 and the solution volume is 125 mL (= 100 mL weak base solution + 25 mL HCl solution). The equivalence point pH is closer to pH = 7 because the solution is simply 0.02 M weak base salt solution, i.e., the number of moles of weak base salt is now in a volume that is 2.5 times larger, so the concentration is 40% of 0.05 M. After all 50 mL of HCl has been added, the pH = 1.778 and the final solution volume is 150 mL. At the half-equivalence point, 12.5 mL of HCl has been added and pH = 9.812, moving slightly down toward pH = 7. For the titration curves, simply imagine vertically flipping the titration curves, in the figure above, about a horizontal line at pH = 7. Or just look at the plot:
One last thing: The pH values at half-equivalences were close to the respective $\mathrm{p}K$ values, as expected, but any buffer can be broken by dumping it into swimming pool full of water. So dilution may matter a great deal, or relatively little, depending on the specifics.
Ed VEd V
The equivalence is, by definition, the moment when $n_{base}=n_{acid}$. As you titrate the same mass of base (consequently the same quantity), your equivalence will be reached at the same point. The only difference, to me, between the two curves will be the fact that you will less see the equivalence when the solution is diluted. First curve is 80g in 50mL, second one is 80g in 200mL.
Thomas PrévostThomas Prévost
$\begingroup$ Thanks a lot! But what about the pH at the begining? is it the same? On one hand, the diluted base should have lower pH. But on the other hand, at half equivalence point, Kb=pOH, and since it's the same base, Kb is the same, and thus pOH and pH at half equivalence point should be the same. $\endgroup$ – Shany Apr 17 '18 at 19:25
$\begingroup$ your initial pH will change, as it will get closer of the one of water. But keep in mind that the equivalence depends only of $n_{acid}$. Hence, $n_{acid}=n_{base} \Leftrightarrow C_{acid}V_0=C_{base}V_{eq}$. So between your two titrations, pOH at equivalence and half-eq will remain the same, but volumes will change (as the quantity of base to reach the required pOH will change) $\endgroup$ – Thomas Prévost Apr 17 '18 at 19:44
$\begingroup$ Thank you, so I understand that except the lower initial ph of the diluted base, other points should be pretty much the same (half equivalence point, volume untill equivalence point, ph at equivalence point). Did I get it straight? $\endgroup$ – Shany Apr 25 '18 at 5:52
$\begingroup$ That's pretty much it, but your volumes won't be the same, as you start from a different pH, and concentration. The quantities will be the same, but not volumes (they will change, as you don't have the same concentrations of base). But all other values will remain unchanged. $\endgroup$ – Thomas Prévost Apr 25 '18 at 5:59
At the beginning of the titration, the solution is a weak base. At the equivalence point, the solution is a weak acid (with some more spectator ions). The pH of these solutions changes with dilution, both in the direction of neutral pH.
In contrast, at the half-equivalence point (or midpoint) of the titration, the solution is a 1:1 buffer. As long as the buffer concentration is large compared to the concentration of hydronium and hydroxide ions, the pH will be close to the pKa (and the pOH close to the pKb), irrespective of dilution.
So if you dilute the weak base with water before the titration, nothing will change along the x-axis (i.e. half-equivalence point and equivalence point will occur at the same volume of added strong acid). Along the y-axis (where you plot the pH), the curve will "shrink" toward the pH of the buffer (i.e. the starting pH will be lower - less basic - and the pH at the equivalence point will be higher - less acidic). Past the equivalence point, the pH is controlled by the excess of strong acid. Because the strong acid is diluted into a greater volume if you add water to the weak base before starting the titration, this part of the titration curve will also show higher - less acidic - pH values.
Karsten TheisKarsten Theis
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Using a climate-to-fishery model to simulate the influence of the 1976–1977 regime shift on anchovy and sardine in the California Current System
Haruka Nishikawa ORCID: orcid.org/0000-0002-0766-13411,
Enrique N. Curchitser2,
Jerome Fiechter3,
Kenneth A. Rose4 &
Kate Hedstrom5
Progress in Earth and Planetary Science volume 6, Article number: 9 (2019) Cite this article
The influence of the well-known 1976–1977 regime shift on the Northern anchovy (Engraulis mordax) and the Pacific sardine (Sardinops caeruleus) populations in the California Current System (CCS) is investigated using a climate-to-fishery model. This model consists of four coupled submodels (regional ocean circulation model; Eulerian nutrient-phytoplankton-zooplankton-detritus model; individual-based full life cycle anchovy and sardine model; agent-based fishery model). Analysis of a historical simulation (1958–1990) showed that survival fraction of age-0 anchovy was lower just after 1977, while survival fraction of age-0 sardine was relatively unaffected by the regime shift. The age-0 survival of both species was influenced by the growth in the larval stage. Simulated zooplankton densities in the historical simulation shifted from high to low in 1976–1977 in the CCS, with the shift being most drastic in winter in the coastal area. The model also shows that anchovy larvae feed extensively from winter to early spring in the coastal area, while sardine larvae were mainly distributed in the offshore area. The differential seasonal and spatial responses of zooplankton in the simulation caused anchovy survival to be more sensitive than sardine to the 1976–1977 regime shift. The model-generated zooplankton shift was a result of reduced phytoplankton production due to lowered nutrient concentrations after 1977 due to the weakening of both the coastal upwelling and mixed layer shoaling, which reduced the vertical nutrient flux from the bottom layer to the surface layer.
Anchovy (Engraulis sp.) and sardine (Sardinops sp.) are important harvested species in four regions of the world: California Current system (CCS), Humboldt Current System, Benguela Current System, and the Kuroshio-Oyashio Current System. Those stocks exhibit a low frequency (decadal) variability and, except for the Humboldt system, out-of-phase-like synchrony (to various degrees) of stock variation between anchovy and sardine (Lluch-Belda et al. 1989; Lluch-Belda et al. 1992; Schwartzlose et al. 1999). Kawasaki (1983) and Lluch-Belda et al. (1989) found synchronized stock variation patterns of anchovy and sardine world-wide and suggested that these stock variations are influenced by global-scale climate forcing. Many correlation-based studies suggest that there are many processes linking climate forcing with the resulting variability in fish stocks (e.g., Yatsu et al. 2008), but it is difficult to separate the influences of the various environmental and biological factors and harvest that all often covary together. Our approach here is to use a coupled modeling system and apply the model to sardine and anchovy in the CCS to elucidate the various processes that connect climate forcing to sardine and anchovy stock variation.
A reliable climate-to-fishery model is capable of demonstrating how certain modes of climate variability can impact fish stocks. The climate-to-fishery model used in this paper was originally implemented for anchovy and sardine in the CCS and, using a 1959 to 2008 historical simulation, was shown to adequately reproduce decadal scale stock variation of Northern anchovy (Engraulis mordax) and the Pacific sardine (Sardinops caeruleus) (Rose et al. 2015). Further analysis of the model output demonstrated the role of environmental variability and prey availability in setting up conditions favorable for the anchovy and sardine populations in the CCS (Fiechter et al. 2015). In the present study, we further investigate an updated historical simulation (using an expanded model grid) by focusing on identifying the environmental factors and life stages that influenced the sardine and anchovy population responses after a notable climate change event, namely the 1976–1977 North Pacific regime shift.
The North Pacific climate exhibited an abrupt change from 1976 to 1977 (Graham 1994; Miller et al. 1994), which was characterized by a decrease in sea surface temperature (SST) in the central Pacific and a rise of SST in the eastern Pacific including the California Current (Venrick et al. 1987; Cayan 1992; Roemmich 1992). This period also coincided with major changes in many North Pacific ecosystems that collectively have become recognized as an ecological regime shift or phase transition (McGowan et al. 2003). Examples of species that were influenced by the regime shift in the CCS include giant kelp (Parnell et al. 2010), zooplankton (Mackas et al. 2001), salmon (Beamish et al. 2000), soles (McFarlane et al., 2000), and marine mammals (Stone et al. 1997).
Considering the different stock variation patterns of anchovy and sardine in the CCS, we speculate that the regime shift should have opposite impacts on their populations. Consequently, some studies imply that the change of ocean environment associated with the 1976–1977 regime shift triggered a species replacement (Benson and Trites 2002; Chavez et al. 2003). Hare and Mantua (2000) reported that survival of age-0 anchovy on the US west coast decreased after the regime shift. They also describe the decrease of zooplankton abundance. Their suggestion that the regime shift lowered anchovy survival via reduced food availability can be tested using a coupled modeling framework. Fiechter et al. (2015), using the same model as used here, identified periods of enhanced adult anchovy growth associated with increased prey availability in the CCS leading to increased egg production. Anchovy and sardine are both plankton feeders that occur from British Columbia to Baja California and can form mixed schools (Radovitch 1979). If the poor feeding condition caused a low age-0 survival of anchovy, then we might expect that age-0 survival of sardine was also low during the same time period. Previous studies, including the model used here, suggested that age-0 survival exerts a significant control on adult sardine abundance via increased egg survival (i.e., reduced development time) during periods of warmer ocean conditions, such as those that occurred following the 1976–1977 regime shift (Fiechter et al. 2015). While sardine catch data in British Columbia did not show the apparent phase transition during the regime shift (McFarlane and Beamish, 1999), Barnes et al. (1992) suggested that age-0 sardine improved in California during the 1970s.
The sardine and anchovy ecological niches overlap to some extent. Due to differences in their spawning timing and locations, the overlap is relatively small in their early life stages. Peak spawning season of anchovy is from January to April and that of sardine is from February to June (Moser et al. 2001). Sardine spawning grounds extend more offshore than anchovy spawning grounds (Hernandez-Vazquez 1994). There are also differences in their feeding ecology. Sardine has finer mesh gill rakers than anchovy and therefore consumes smaller plankton as forage (Van der Lingen et al. 2006).
There are two important upwelling conditions that supply nutrients to the surface layer in the CCS. One is coastal upwelling due to alongshore wind stress (Ekman transport), and another is curl-driven upwelling due to wind stress curl (Ekman pumping) (Rykaczewski and Checkley 2008). While coastal upwelling is typically more intense and leads to primary production dominated by larger phytoplankton types, its effects are limited to the nearshore region. In contrast, weaker curl-driven upwelling extends farther offshore and supports primary production associated with smaller phytoplankton types. Based on species distribution and diet preferences, Rykaczewski and Checkley (2008) hypothesized that sardine survival in the CCS is related to the magnitude of curl-driven upwelling, while anchovy survival is related to the magnitude of coastal upwelling. The idea that anchovy and sardine survival are controlled by the different environmental factors related to differences in their habitat niche might explain the different response of age-0 dynamics to the 1976–1977 regime shift. Our analysis expands on modeling analyses to date (Deyle et al. 2013; Lindegren et al. 2013; Kaplan et al. 2017) by using coupled 3D ecosystem models to examine food availability and fish experiences on detailed spatial and temporal scales that are difficult from field data and spatially aggregated models. In addition, although the fish surveys have been conducted in the CCS for decades, reliable stock size estimates for sardine are not available around the regime shift due to the moratorium on fishing along the California coast from 1967 to 1985.
We used the climate-to-fishery model of Rose et al. (2015) with the only changes being the southward expansion of the model domain to better represent the natural range of the fish populations and increased horizontal and vertical resolution of the grid. Below, we briefly present the four submodels that comprise the climate-fishery model and describe the methods we used to analyze the model output for regime shifts and for estimating coastal and curl-driven upwelling. A complete description of the coupled climate-to-fishery model is given in Rose et al. (2015). An updated simulation of the period from 1958 to 1990 is analyzed here; our focus is on dynamics before and after the 1977 regime shift.
Climate-to-fishery model
Regional ocean circulation model
The ocean circulation model for the CCS is an implementation of the Regional Ocean Modeling System (ROMS; Haidvogel et al. 2008; Shchepetkin and McWilliams 2005) along the west coast of North America. The domain extends in latitude from the Baja California Peninsula (19° N) to British Columbia (50° N). The grid extends about 2300 km alongshore and about 1200 km offshore. The current domain is expanded from the original domain of Fiechter et al. (2015), which originally extended from 30 to 48° N, to allow for movement of sardine and anchovy farther south. The current horizontal grid resolution is 7 km, and the vertical resolution is set to 50 non-uniform terrain-following levels. The ROMS model is forced on all open boundaries by weekly averaged fields from the Simple Ocean Data Assimilation (SODA) reanalysis to force the model with realistic transport values and temperature profiles (Carton et al. 2000; Rose et al. 2015). Surface forcing is derived from the datasets for Common Ocean-Ice Reference Experiments (CORE2; Large and Yeager 2008), which consist of 6 hourly winds, air temperature, sea level pressure and specific humidity, and daily short-wave and downwelling long-wave radiation and monthly precipitation.
NPZ model
The nitrogen concentration-based NPZ food web model NEMURO (Kishi et al. 2007) used in the present study contains 11 compartments, which represent small phytoplankton (PS), large phytoplankton (PL), small zooplankton (ZS), large zooplankton (ZL), predatory zooplankton (ZP), nitrate (NO3), ammonium (NH4), silicate (Si(OH)4), particulate organic nitrogen (PON), biogenic silica (Opal), and dissolved organic nitrogen (DON). NEMURO is coupled to ocean circulation by solving a transport equation in ROMS for each NEMURO component at every time step. The initial and boundary conditions for nitrate and silicic acid are derived from monthly climatological concentrations (World Ocean Atlas 2001, 1° × 1° horizontal resolution; Conkright and Boyer 2002). This model can reproduce realistic chlorophyll-a, nitrate, and temperature distribution (Rose et al. 2015).
Individual-based fish model
The individual-based fish model (IBM) is parameterized to represent anchovy and sardine. The IBM tracks individuals in continuous (Lagrangian) space within the ROMS 3D grid. Anchovy and sardine life cycles are defined by the egg, yolk-sac larva, larva, juvenile, and adult life stages. Temperature determines egg and yolk-sac larval development; length determines metamorphosis and maturation. Individuals are aged by 1 year on January 1. The IBM also includes a migratory predator species (similar to albacore) that imposes dynamically varying predation mortality rates on anchovy and sardine. While the population dynamics of sardine and anchovy are simulated with a full life cycle approach (adults release young who grow up to be adults and release young, etc.), only the movement and consumption of the predator species individuals is simulated. The position of each individual in three-dimensional space is updated hourly and followed both in continuous (longitude, latitude, depth) space and grid cell location.
Growth of larval, juvenile, age-1, and older anchovy and sardine individuals is computed based on bioenergetics and a functional response relationship. The functional response determines the hourly consumption rate and uses zooplankton concentrations interpolated to its location from NEMURO-generated zooplankton concentration in nearby spatial cells. The diet preferences (weighting of different zooplankton groups) vary according to species and life stage.
Four sources of mortality of sardine and anchovy are represented: natural, starvation, predation by albacore, and a fishing fleet. Natural mortality has a constant rate specific to each life stage. Starvation occurs when the fish weight drops below a minimum value allowed for that length. The effects of albacore predation (on both species) and fishing (sardine only) mortality is limited to age-1 and older individuals. Mortality values are derived from Butler et al. (1993) and Lo et al. (1995). Since their estimation of mortality values were based on observation, they estimated the value with a width. We choose the average values that they assumed for simulation, but it is possible that the uncertainty of estimation causes the uncertainty of the simulation results.
Eggs, yolk-sac larvae, and larvae are moved horizontally using a passive transport (particle-tracking). Juvenile and adult anchovy and sardine individuals are moved every hour using a three-dimensional behavior (independent of physics) based on a kinesis approach (Humston et al. 2000, 2004; Watkins and Rose 2013). The movement cues are the temperature and prey availability at the individual's present location. A seasonal feeding migration is also imposed on sardine individuals by biasing the kinesis movement poleward in May and equatorward during the September spawning migration. Eggs and yolk-sac larvae were positioned in a vertical layer, while larvae, juveniles, and adults were moved vertically within the water column by first locating them horizontally with kinesis and then moving them to the cell within the water column with the highest prey availability. The simulated sardine egg and larval distributions are consistent with the observation data (Politikos et al. 2018).
Reproduction of mature fish (greater than a specific length on January 1) is represented as an allocation between somatic growth and egg production. Each batch of eggs is kept track of in an individual as the eggs develop or are resorbed; the model is configured to produce realistic numbers of egg per batch and number of batches per year for anchovy and sardine. The timing of batch releases depends mostly on the energy stores of the individual at the beginning of the spawning season (January 1 to May 1 for anchovy and February 1 to July 1 for sardine). A fecundity of batch depends on an individual female weight (Hunter et al. 1985). In the simulation therefore, eggs per individual is related to weight and lagged annual growth rates and weights of younger classes. A previous study investigated the simulation and concluded that the long-term anchovy abundance is associated with age-1 growth via age-2 egg production (Fiechter et al. 2015).
The IBM uses a version of the super-individual approach (Scheffer et al. 1995), where a fixed number of model individuals are followed, with each model individual worth some number of identical population individuals. Mortality from all sources (except starvation) acts to reduce the worth of each super-individual over time. Upon reaching a maximum age, super-individuals are removed and their place in arrays is available to place new super-individuals to represent the next year's production of eggs. Location, worth (abundance of the individual), weight, growth, diet, and reproduction (if mature) are recorded every first hour of the day for each super-individual.
Agent-based fishery model
Fishing occurs every day of the year, as long as boats have access to a grid cell where expected net revenue is positive. The daily location choice, and associated daily catch, of individual boats that fish sardine individuals is simulated using a simplified multinomial logit, agent-based approach where boats maximize their expected net revenue for each trip. Details of the agent-based fishery model are written in Rose et al. (2015).
Model outputs
Model outputs reported here are computed (summed, averaged) taking into account the worth of the super-individuals. For example, total adult biomass is the sum of worth over super-individuals and average length-at-age is a statistically weighted average over individuals using their worth as the weighting factors. The specific calculations for many of the outputs reported here are detailed in Table 6 of Rose et al. (2015) and Additional file 1: Table S1.
Regime shift detection
A regime shift index (RSI) (Rodionov 2004) is applied to model results to identify significant regime shifts. The basic idea of detecting regime shift is to compare the average values of variables before and after a regime shift. The difference diff, which depends on accuracy of detection and detection length is set.
$$ \mathrm{diff}=t\sqrt{2{\sigma}_l^2/l} $$
Here, t is the value of t distribution with 2l–2° of freedom at the given probability level p, \( {\sigma}_l^2 \) is the average variance for running l year intervals in the time series of variables, and l is the cutoff length of the regimes. We set the significant criterion at the probability level 0.05 by a t test with certain years of the cutoff length. Three, five, seven, and ten years are used in RSI detection for the recruitment, and 10 years are used for other cases. Then, the levels that should be reached in the subsequent l years to qualify for a shift from regime R1 to regime R2 are estimated.
$$ {\overline{x_{R2}}}^{\prime }=\overline{x_{R1}}\pm \mathrm{diff} $$
\( \overline{x_{R1}} \) is the average of variables during regime R1. Finally, RSI for variable x in year j is calculated.
$$ {\mathrm{RSI}}_{i,j}=\sum \limits_{i=j}^{j+m}\frac{x_i^{\ast }}{l{\sigma}_l},m=0,\dots, l-1 $$
When the regime shift is up, \( {x}_i^{\ast }={x}_i-\overline{{x_{R2}}^{\prime }} \) and when the shift is down, \( {x}_i^{\ast }=\overline{{x_{R2}}^{\prime }}-{x}_i \). If at any time from i = j + 1 to i = j + l – 1 the RSI value turns negative, the detection fails. We applied this regime shift detection method to the time series of model-generated stage survivals of anchovy and sardine, and the biological and physical variables (some are inputs and some are outputs) that affect the stage survival (e.g., zooplankton density, wind speed). The idea was to detect any responses of model results to the 1976–1977 regime shift.
We also use our computed RSI values to show a spatial and temporal distribution pattern of a regime shift for certain environmental variables (e.g., zooplankton density) for each month. The environmental variables are binned on the horizontal grids from 7 km × 7 km to 1° × 1° and averaged in the upper 30 m of the water column, which corresponds to the main habitat depth of anchovy and sardine larvae.
When the RSI is detected in the significant levels of 0.05, 0.1, and 0.2 in 1977, we draw the RSI distributions on the map of the model grid. The original RSI always returns a positive value if the regime shift is up or down. In this study, we assign negative value for detected regimes that result in a negative shift.
Upwelling
Upwelling in the CCS is produced by two different processes: coastal upwelling and curl-driven upwelling (Pickett and Schwing 2006). Here, we estimate the vertical transport associated with both types of upwelling using the results for the historical simulation. Coastal upwelling velocity, wcoast, can be calculated given the density of seawater, ρ0, alongshore wind stress within 10 km of the coastline, τa, and the local Rossby radius of deformation, Rd (Smith 1968). τa was obtained from interpolated CORE2 data that is used as boundary conditions for ROMS. Based on earlier studies in the region, a Rossby radius of 10 km is used in the calculation (Pickett and Schwing 2006; Rykaczewski and Checkley 2008).
$$ {w}_{\mathrm{coast}}=\frac{\tau_a}{\rho_0f}\bullet \frac{1}{R_d} $$
The curl-driven upwelling velocity, w, can be calculated from Smith (1968):
$$ w=\frac{1}{\rho_0f}\nabla \times \boldsymbol{\tau} $$
where ∇ × τ is the curl of the derived wind stress vector; ρ0 is the density of seawater, and f is the Coriolis parameter. Volumes of the coastal upwelling and curl-driven upwelling for a certain area during a certain period are calculated by these coastal upwelling and curl-driven upwelling velocities.
We divide the main sardine and anchovy habitats from 24° N to 40° N into a coastal region and an offshore region because observations and previous simulation reported that anchovy larvae have greater affinity for coastal waters than sardine larvae (Hernandez-Vazquez 1994; Rose et al. 2015). The coastal region is defined as nearshore and is dominated by upwelling. The vertical velocity is integrated from 2 m to 120 m depth (Song et al. 2011) and averaged in 2° × 2° grid cells from January to June for each year. We define the region where the averaged vertical velocity from 1967 to 1986 is upward as the predominant upwelling region. Also, we divided larvae based on the center of gravity location of larval stage distribution. When the center of gravity is located inside (outside) the coastal region, the larva is named as "coastal larvae" ("offshore larvae"). To avoid artificial effects from dynamics near the boundaries, we removed the larvae from offshore larvae that are distributed in the grid cells next to the boundary. This is resulted in about 1% of offshore sardine larvae being removed from analyses.
In the latter half of analysis, we focused on the physical and biological components averaged in the coastal region. Zooplankton, phytoplankton, nutrients, and mixed layer depth (MLD) data used there were from simulation results, and heat flux, wind stress, wind speed, and temperature data were from the model input derived from the CORE2 data. In this study, the MLD was operationally defined as the depth where the temperature is 0.8 °C lower than the surface (Kara et al. 2000).
Analysis of the mixed layer depth
The change in the MLD also had an impact on the surface ecosystem dynamics in the historical simulation. In the CCS, the heat flux from the atmosphere to the ocean, the intensity of turbulent wind mixing, and coastal upwelling erode stratification in the upper layer of the water column and deepen the mixed layer (Husby and Nelson 1982). Among these factors, the MLD variation is mainly due to wind (Jeronimo and Gomez-Valdes 2010). We therefore estimated the impact of the wind speed regime shift on the MLD regime shift in the simulation by using a bulk formulation of the entrainment velocity. The entrainment velocity is a measure of the evolution of the mixed layer, and therefore, we use the time integration of the entrainment velocity as a metric for mixed layer development. Qiu and Kelly (1993) provides a bulk model of the entrainment velocity, which contains the effects of wind mixing, absorption of the short-wave radiation, and potential energy changes from the net heat flux. We extracted the effect of wind mixing on the entrainment velocity from their bulk formulation as follows:
$$ {W}_e={m}_0{u}_{\ast}^3\bullet \frac{1}{\mathrm{MLD}}\bullet \frac{1}{\Delta T}\bullet \frac{2}{\alpha g} $$
In Eq. 6, the entrainment velocity We (m/s) is the objective variable. The parameter m0 (set to 0.5) is the coefficient for the vertical wind mixing, u* is the frictional velocity defined as u* = (τ/ρ0)1/2 for wind stress τ and sea water density ρ0 (= 1025 kg/m3), MLD is the mixed layer depth at that time, α (= 0.00025 °C−1) is thermal expansion coefficient, g is gravitational acceleration, and ΔT (set to 1.0 °C, Qiu and Kelly 1993) is the temperature difference between the mixed layer and entrained subsurface water. The entrainment velocity at each time step is then calculated according to Eq. 6. We integrate the estimated entrainment velocities from October to February for each year in order to estimate the evolution of the mixed layer depth due to changes in the winds throughout the season. Then, we compared the average of the mixed layer development by wind for the periods before (1967–1976) and after (1977–1986) the regime shift.
Comparison between model results and observations
Simulated long-term increasing trends of age-1 and older biomass of anchovy and sardine compared favorably to catch data from the Food and Agriculture Organization (Fig. 1). For anchovy, reported catch increased from 1958, with a maximum catch recorded in 1981 (Fig. 1a, gray line with crosses), until catch then sharply decreased after 1983. Simulated age-1 and older biomass of anchovy also increased from 1964 to 1982 and slightly decreased from 1982 to 1984, but then remained fairly constant after 1985 (Fig. 1a, black line with closed circles). Thus, simulated anchovy biomass exhibits a similar increasing trend as catch but levels off rather than sharply drops. Observations show that the El Niño in 1983 had a strongly negative influence on anchovy populations (Fielder 1984), which may be underestimated in the model simulation.
Comparison between model simulation results and observations for adult population biomass and catch. Model simulated a age-1 and older anchovy biomass and catch from FAO from 1958 to 1990 and b age-1 and older sardine biomass and catch from FAO from 1964 to 1990. Model simulation data is plotted in black line with close circles and FAO data is plotted in gray line with crossed through
Simulated adult biomass and catch of sardine both show an increasing trend (Fig. 1b). The two drops in catch (1984 and 1989, Fig. 1b, gray line with crosses) are not captured by the model.
In our analysis, we focus on the interannual variability of the survival of the early life stages. Previous studies used empirical data and estimated recruitment rates (age-0 biomass per spawning stock biomass) from 1964 to 1991 for anchovy (Jacobson et al. 1994) and from 1936 to 1964 and from 1987 to 1991 for sardine (Jacobson and MacCall 1995). We compare the normalized recruitment rate from our climate-to-fishery model with these reported estimates. In our calculation of recruitment rate, we used mature adult biomass instead of spawning stock biomass. Anchovy recruitment rate anomalies from previous study were negative from 1964 to late 1960s, positive from 1971 to 1976 and relatively negative after 1977 (Fig. 2a). In the simulation, the recruitment rate anomalies are negative before 1964, positive from 1965 to 1976, and negative after 1977 (Fig. 2b). The regime shift in 1977 is detected with a 7-year cutoff length for the previous study and also detected with a 10-year cutoff length for the current simulation. Although there is a difference between the previous study and the simulation around late 1960s, a regime shift occurred in 1977 after a regime of 7–10 years in both cases. For the sardine recruitment, it is difficult to compare the simulation with previous study, because the overlap of data period is very short. In the previous study, a regime was about 3 years occurred just prior to 1964, but the trend in 1970s was unknown (Fig. 2c). In the historical simulation, a shift was detected in 196 after an approximate 5-year regime period and there is no regime shift in 1977 (Fig. 2d). Mean and standard deviation of anchovy recruitment rate in previous study during 1964–1989 is 2.71 ± 2.39. For the historical simulation, mean and standard deviation for the same period is 1.41 ± 0.29. Thus, the stock level and amplitude of stock variation was underestimated in the historical simulation. Sardine recruitment shows the same tendency. Mean and standard deviation of sardine recruitment rate in previous study during 1936–1964 and 1987–1991 is 4.10 ± 4.03 compared to 0.75 ± 0.11 during 1959–1990 in the simulation. For the sardine, the comparison periods are different. But the sardine stock level and amplitude of stock variation was also underestimated in the simulation.
Comparison of recruitment rate anomaly between model simulation and estimates based on analysis of field data. Time series of anchovy a estimated by Jacobson et al. (1994) from 1964 to 1991, b estimated by the climate-to-fishery model from 1958 to 1990, normalized sardine rate c estimated by Jacobson and MacCall (1995) from 1936 to 1964 and from 1987 to 1991 and d estimated by the climate-to-fishery model from 1958 to 1990. Mean recruitment rate that was used for calculating anomaly is a 2.71, b 1.40, c 4.09, and d 0.75. Triangles represent years when the regime shift index detected by RSI. Cutoff lengths are a 7 years, b 10 years, c 3 years, and d 5 years
We further investigated the variation in recruitment rate by examining the survival of the larval stage. Larval stage dynamics is often a driver of recruitment success (Hjort 1914; Houde 1987; Cushing 1990). In the California Current System, the California Cooperative Oceanic Fisheries Investigations project (CalCOFI) (calcofi.org) is a long-term observation program focused on physical and biological ocean conditions related to larval stage dynamics. We compare larval abundance between the model and the CalCOFI data. Simulated larval abundance is defined by the cumulative number of larvae produced each year. Larvae abundance data from CalCOFI is the integrated counts of larvae per 10 m2 in oblique net tows from January to June for anchovy and from January to August for sardine.
CalCOFI data show that anchovy larvae appeared to increase from the end of 1960s to the early 1970s, abundance became low in late 1970s until a peak in the mid-1980s, after which larval abundance returned to relatively low levels (Fig. 3a). Larval abundance in the simulation reproduces a similar increasing trend from the end of 1960s to early 1970s and then relatively stable values after that (Fig. 3a). For sardine larvae, the simulated abundance shows a similar long-term increasing trend as the CalCOFI data from the end of the 1960s to the early 1980s, but it does not reproduce the short-term fluctuation in mid-1980s (Fig. 3b).
Comparison between model simulation results and field data for larval abundances. Time series of the larval abundance generated by the climate-to-fishery model (black line with closed circles) and from CALCOFI field data (gray line with open circles) for a anchovy and for b sardine
While simulated and observed recruitment rate of anchovy showed a clear response to the 1977 regime shift (Fig. 2a and b), simulated adult biomass (Fig. 1a) and larval abundance (Fig. 3a) did not. This is because the egg production increases during this time period in the simulation that more than offset the lowered recruitment rates (Fig. 4a). Thus, the regime shift did not have a negative influence on the anchovy biomass itself. The egg production depends on adult female biomass and an individual female weight, which is determined by lagged annual growth rates. If the adult biomass increase led the egg production increase, the age-0 abundance would increase prior to the egg increase. Given there is no lag between interannual variation of age-0 abundance and egg production in the simulation (Fig. 4a), this implicates that variation in egg production was due the average weights of age-1 and age-2 that are the main contributors to annual egg production. During the 1970s, age-1 growth rate showed sharp drops in 1972 and 1976 (Fig. 4b). As a result of lower age-2 weight, hence lower reproductive output, egg production in 1973 and 1977 were almost the same as the previous years (Fig. 4a) despite the continuous increase in annual egg production over time in the simulation.
Time series of simulated a anchovy age-0 abundance (black line with closed circles) and egg abundance (gray line with crosses) and b yearly growth rate of age-1
Simulated age-0 and stage-specific survivals
We focus our analysis on age-0 survival, which is the determinate of recruitment rate that showed a negative response in anchovy after the 1977 regime shift. We compare the age-0 survival (eggs to recruitment at age-1) with the survival fractions during each of the egg–yolk-sac, larva, and juvenile stages. Stage survival is the ratio of the sum of worth of individuals existing a stage to the sum of worth entering the same stage, expressed as a percentage. For anchovy, age-0 survival shows three negative drops: 1972 to 1973, 1976 to 1977, and 1982 to 1983 (Fig. 5a). 1972–1973 drop occurs only in egg–yolk-sac stage survival (Fig. 5b), and 1976–1977 and 1982–1983 drops only occur in larval stage survival (Fig. 5c). Caution is needed in interpreting simulated juvenile stage survival because the stage ends on December 31 of each year, so prolonged larval stage would cause a shortened juvenile stage regardless of the dynamics of growth and mortality during the juvenile stage. A better proxy for the interannual differences in conditions experienced by the juveniles is their average growth rate, which is not influenced by a forced ending day for the stage. Juvenile growth rate for anchovies (Fig. 5d) shows, at best, a weak correspondence to age-0 survival. While 1983 has low juvenile growth rate, the value for 1977 was intermediate and the lowest value was for 1972 that has among the highest age-0 survival.
Time series of simulated anchovy age-0 and stage-specific survivals. Age-0 survival is shown in a as black line and the other panels as a gray line. b Egg–yolk-sac stage survival (black line with closed circles), c larval stage survival (black line with closed circles), and d juvenile daily growth rate (black line with closed circles) from 1967 to 1986 of the historical simulation
The only significant positive correlation (r = 0.89, P < 0.001, d.f. = 18, t test) between overall age-0 survival and stage-specific survivals is for the larval stage. Average age-0 survival from 1967 to 1976 is 0.00106% and then a lower value of 0.00086% from 1977 to 1986. Thus, the age-0 survival drops by 19% from before to after the regime shift. The survival decline during the larval stage is from 0.061 to 0.046%, or a decrease of about 24%. The larval duration becomes longer after 1977 (Fig. 6a), which is caused by a lower growth rate that decreases stage survival. Our analysis shows that the 1976–1977 regime shift has a negative effect on the age-0 survival of anchovy through the slower growth causing a decrease in larval stage survival.
Time series of simulated a anchovy larval duration and b sardine larval duration for 1967 to 1986 of the historical simulation
For sardine, correlation analyses between age-0 survival and egg–yolk-sac stage survival, larval stage survival, or juvenile stage survival also suggest that the larval stage survival is most influential for causing interannual variation in the age-0 survival (Fig. 7). As with anchovy, larval stage survival (Fig. 7c) has a significant correlation (r = 0.88, P < 0.001, d.f. = 18, t test) with duration of the stage (Fig. 6b), and juvenile growth rate also did not show a strong relationship to age-0 survival (Fig. 7d). However, in contrast to anchovy, there are no significant changes in the sardine age-0 survival (Fig. 7a), and other stage survivals including the larval stage (Fig. 7c), corresponding to the 1977 (or 1983) drops seen in anchovy age-0 and larval stage survival. At least for the simulated dynamics, age-0 survival of sardine did not show any consistent differences between before versus after the 1977 regime shift.
a–d Same as Fig. 5 but for sardine
Coastal larvae and offshore larvae
While the simulated anchovy larvae distribution focuses on the coastal region, the simulated sardine larvae distribution extends from the coastal region to the offshore region (Fig. 8). On average over the historical simulation, 79% of the larvae are coastal for anchovy while only 31% are coastal for sardine. Simulated anchovy larvae are present from late January to early May and sardine larvae occur later, typically from March to July. Larval stage survival of coastal sardine larvae shows a slow decrease after 1977 (Fig. 9, black line). This decline is judged to be a "regime shift" by the regime shift detection test. The decline of the larval stage survival between 1967–1976 and 1977–1986 is 15.6%. The decline is smaller than the decline of anchovy larval stage survival (Fig. 5c). Survival of the offshore sardine larvae does not show such a regime shift like change (Fig. 9, gray line). However, these changes in larval stage survival did not propagate through to age-0 survival or recruitment rate, because relatively few sardine larvae are found in the coastal habitat.
Simulated horizontal spatial distributions of anchovy and sardine larvae averaged over the historical simulation. a Distribution of the vertical velocity and the boundary between the coastal region and offshore region denoted as black lines. Upward velocity was positive. b Distribution ratio during larval stage for anchovy and c for sardine. Distribution ratio was calculated based on the location of each super-individual and weighted by its abundance
Time series of simulated sardine larval stage survival. Offshore sardine larvae (gray line with crosses), coastal sardine larvae (black line with closed circles), and all sardine larvae (broken line) from 1967 to 1986 of the historical simulation
Factors controlling the interannual larval survival
Hereafter, we focus on the interannual variation of the larval stage. Larval mortality is determined by the growth, which is controlled by food availability and temperature. We compare the time series of larval stage survival with ambient zooplankton, which is calculated by daily zooplankton densities interpolated to the exact 3D position of each larva. Anchovies eat small and large zooplankton, while sardines only eat small zooplankton. The ambient small and large zooplankton densities correlate with the larval stage survival of anchovy (Fig. 10a and b). The correlation coefficients between stage survival and small and large zooplankton densities are 0.86 (P < 0.001, d.f. = 18) and 0.91 (P < 0.001, d.f. = 18). Ambient zooplankton densities exhibit significant changes after 1977. The declines in means from 1967–1976 to 1977–1986 are 0.026 mmol N/m3 (10.5%) for small zooplankton and 0.027 mmol N/m3 (12.3%) for large zooplankton. The ambient zooplankton density for sardine also has significant positive correlation with the stage survival of both coastal and offshore sardine larvae (Fig. 10c and d). Corresponding to the drop in survival of the coastal sardine larvae, the ambient zooplankton density on the coastal sardine feeding grounds decreases around mid-1970s (Fig. 10c). The decline in mean densities between 1967–1976 and 1977–1986 is 0.018 mmol N/m3 (9.0%), which is smaller than the declines simulated for anchovy. Though warmer ocean temperatures could also lead to higher growth, the model results show no significant correlations between ambient temperature (computed similarly as zooplankton densities) and larval stage survival for anchovy and coastal or offshore sardine from the late 1960s to early 1980s (not shown). These results suggest that food availability is a main factor controlling interannual variation of the larval stage survival for both anchovy and sardine. The decrease of the ambient zooplankton density in anchovy habitat after the regime shift causes lower survival of anchovy larvae. Ambient zooplankton density of sardine that feed in the same coastal region as anchovy also shows a weak regime shift, while the prime habitat for sardine larvae (offshore) did not show a significant zooplankton regime shift and so the larval stage survival of offshore sardine larvae did not show a regime shift signal.
Comparison between larval stage survival of anchovy and sardine and zooplankton densities. Time series of simulated a ambient small zooplankton (ZS) density (black line with closed circles) and larval stage survival of anchovy (gray line with crosses), b ambient large zooplankton (ZL) density and anchovy larval survival, c ambient ZS density and larval stage survival of coastal sardine, and d ambient ZS density and larval stage survival of offshore sardine. R denotes the correlation coefficient, and P denotes the probability value. Zooplankton density is the values recorded concurrently as what each larval individual experienced and is averaged accounting for the worth of the different super-individuals
Regime shift of the plankton and nutrients in the coastal region
There are two possible mechanisms in the model that changed the ambient zooplankton densities for anchovy and for coastal sardine larvae during 1976–1977. One is a change of zooplankton density itself and the other is a change of spatial or seasonal distribution of the larvae. We cannot find any larval distribution pattern changes that seemed likely to result in a decrease ambient zooplankton density experienced by the larvae around mid-1970s. On the other hand, simulated zooplankton density near the coastal region shows a decline in 1977 (Fig. 11). Thus, zooplankton density itself decreased in the feeding grounds of anchovy and coastal sardine larvae.
Spatial distribution of detections of regime shifts by the RSI for 1977 in the historical simulation. For February and June, the detections are shown for small plus large zooplankton densities, small plus large phytoplankton densities, and for nitrate concentration. Solid lines represent the boundary between the coastal region and offshore region, and broken lines represent a boundary of the model domain used for averaging values between the two regions
There is a notable feature in spatial distribution pattern in the coastal zone associated with this negative regime shift, which rarely occurs in the offshore region. The frequency of detection of negative regime shifts in the coastal area was exceptionally high for 1977. Negative regime shifts occur widely from January to May (typified by February in Fig. 11), but shows much lower occurrences beginning in June (Fig. 11). Detailed monthly changes in the distribution of regime shift detections are shown in Additional file 2: Figure S1, Additional file 3: Figure S2 and Additional file 4: Figure S3. We note that the location and season of the zooplankton regime shift, winter–early spring in the coastal region, overlap with main feeding grounds of the anchovy larvae. However, the feeding grounds of sardine larvae are only partially affected.
In the NPZ model, one of the controlling factors of zooplankton density is the flux of nitrate and silicate from nutrients to zooplankton through phytoplankton production. The same regime shift detection around 1977 in zooplankton density shows that both phytoplankton and nitrate also decreased in the coastal region from winter to early spring (Fig. 11). Temperature was a controlling factor for phytoplankton growth in the model, and the MLD affected phytoplankton growth via changes in primary controlling factors, such as temperature, light exposure, and nutrient availability. After 1977, temperature rose and MLD decreased in the coastal region during winter to early spring (Fig. 12a). The MLD averaged from January to April decreased by 2.4 m between 1967–1976 and 1977–1986, and the maximum depth decreased by 2.5 m. This decrease implies that temperature and light conditions, which are depending on averaged MLD, were favorable for phytoplankton growth after the 1977 regime shift. Averaged nitrate concentration in the coastal region from January to April, averaged over the upper 30 m of the water column, decreases by 0.88 mmol N/m3 (7%) between 1967–1976 and 1977–1986 mean (Fig. 12b) and ammonium and silicate concentrations also showed similar decreases after 1977 (Fig. 12c and d). The decline of nutrients resulted in an average decrease of 0.06 mmol N/m3 (16%) of phytoplankton (Fig. 12e) leading to an average decrease of 0.08 mmol N/m3 (21%) in zooplankton density (Fig. 12f).
Time series of selected physical and biological variables, a average and maximum mixed layer depth, b nitrate concentration, c silicate concentration, d ammonium concentration, e small plus large phytoplankton density, and f small plus large zooplankton density from the historical simulation. Values are averaged in the coastal region from January to April for a–f, and broken line represents the averaged value from 1967 to 1987. Nutrients and planktons are averaged in the upper 30 m of the water column, the main habitat of the anchovy and sardine larvae
Regime shift effects on nutrient supply
There are three main nutrient supply routes to the coastal surface layer: horizontal advection, vertical advection, and vertical diffusion. Horizontal diffusion is small enough to be negligible when compared to the other two factors. We compare the nitrate supply into the coastal upper 30 m of the water column across the lateral and bottom boundary by the three supply processes from before the regime shift (1967–1976) to the period after (1977–1986) between October and June. The supplied nutrients are not always immediately taken up for photosynthesis. The autumn nutrient 1 year prior affects the nutrient concentration in winter. The total nitrate supply from December to April decreases after the regime shift, but then increases during May to June (Fig. 13a). The significant difference between before and after the regime shift is shown from December to January and in April. This result is consistent with the former analysis that the negative nitrate regime shift disappears after May.
Simulated seasonal changes of nitrate supply for before and after the regime shift in 1977. a Total nitrate supply, b nitrate supply by vertical advection, c nitrate supply by horizontal advection, and d nitrate supply by vertical diffusion into the coastal upper 30 m of the water column. Black line with closed circles represents 1966–1975 mean for October–December and 1967–1976 mean for January–June. Gray line with crosses represents 1976–1985 mean for October–December and 1977–1986 mean for January–June. Error bar represents the standard error. The positive/negative value denotes that nitrate increases/decreases in the coastal upper 30 m of the water column
The nitrate supply by vertical advection shows a similar seasonal change as the total nitrate supply, and with a larger reduction between 1967–1976 and 1977–1986 (Fig. 13b). The significant difference between before and after the regime shift is shown from October to January and in April. In December, while the vertical advection is positive before 1977, the vertical advection is negative after 1978. The negative value implies that nitrate is removed from the coastal surface layer. The nitrate supply due to horizontal advection and by vertical diffusion increases after the regime shift (Fig. 13c and d). Figure 14 shows the time series of nitrate supply integrated from October to April. The total nitrate supply and the nitrate supply by vertical advection drop in 1977 (Fig. 14a and b), but there is no drop in 1977 for the nitrate supply due to horizontal advection and by vertical diffusion (Fig. 14c and d). Since the nitrate supply by vertical advection decreases 1.02 × 1014 mmol N from average 1967–1976 mean to 1977–1986 mean, the total nitrate at the same period decreases 6.96 × 1013 mmol N. Thus, the seasonal and interannual variation of the total nitrate supply mostly depends on the vertical advection, which after the regime shift, decreases during autumn to early spring in the coastal region. Given that silicate supply is also mostly controlled by the vertical advection and that the main source of ammonium is nitrification, the same vertical advection effects applies to all nutrients.
Simulated time series of nitrate supply for 1967 to 1986 of the historical simulation. a Total nitrate supply, b nitrate supply by vertical advection, c nitrate supply by horizontal advection, and d nitrate supply by vertical diffusion into the coastal upper 30 m of the water column integrated from October to April. Broken lines represent the values averaged over 1967 to 1986
Nutrient-rich subsurface water is introduced into the surface layer in the model by two processes: deepening of the seasonal mixed layer and the upward movement of a volume of water (upwelling). The maximum depth of winter mixed layer becomes shallow after the regime shift in the coastal region (Fig. 12a), causing a reduction in the nutrients available to phytoplankton in the mixed layer.
In the CCS, there are two types of upwelling: coastal upwelling and the curl-driven upwelling. The mean volume of the coastal upwelling (8.2 × 1012 m3) in the simulation is 13 times larger than that of the curl-driven upwelling (6.0 × 1011 m3) (Fig. 15). Coastal upwelling was reduced in the simulation in autumn of 1976 and winter of 1977. From October to April, the coastal upwelling decreased from 9.0 × 1012 m3 (1967–1976 mean) to 7.3 × 1012 m3 (1977–1986 mean) (Fig. 15a). Contrary to the coastal upwelling, there is not a significant change during the mid-1970s for the curl-driven upwelling (Fig. 15b). Our analyses show that both the change in MLD and coastal upwelling caused the reduced vertical nutrient supply in the regime shift.
Simulated time series of upwelling volumes for 1967 to 1986 of the historical simulation. a The coastal upwelling and b the curl-driven upwelling integrated from October to April. The values from October to December are grouped with the following year. Broken lines represent the values averaged over 1967 to 1986
Regime shift of the heat flux and the wind stress
We investigate the historical simulation for interannual variation of the heat flux and the wind speed in the coastal region between October and April. While we do not find a step-like regime shift in the upward net heat flux, there is a noticeable drop in 1977. Average net heat flux decreases by 26.7 W/m2 between 1967–1976 and 1977–1986 (Fig. 16a). Thus, it is possible that weakened cooling from autumn to early spring after 1977 enhanced the stratification.
Simulated time series of heat flux for 1967 to 1986 of the historical simulation. Values are averaged from October to April for a net heat flux, b sensible heat flux (black line with closed triangles) and the latent heat flux (gray line with open triangles), and c longwave radiation flux (black line with closed triangles) and short-wave radiation flux (gray line with open triangles) averaged from October to April. The values from October to December are grouped with the following year. Broken line in a represents the net heat flux averaged over 1967 to 1986
Net heat flux is composed of sensible heat flux, latent heat flux, short-wave radiation flux, and long-wave radiation flux. Time series of each flux is shown in Fig. 16b and c. Among them, only the sensible heat flux decreased after 1977 (Fig. 16b, black line with closed triangles). The reduction of the sensible heat flux (35.4 W/m2 from 1967–1976 to 1977–1986) resulted in a weak cooling. Sensible heat flux is mainly controlled by a sea surface absolute wind speed and the difference between sea surface temperature and atmospheric temperature. A regime shift from positive to negative phase in 1977 for the absolute wind speed in the simulation and the speed decreases from an average of 5.4 m/s during 1967–1976) to 5.2 m/s during 1977–1986 (Fig. 17a). On the other hand, the difference of temperature between sea surface and atmosphere does not have a regime shift or specific decrease in 1977 (Fig. 17b). These results suggest that the weakened absolute wind speed in the simulation causes a weak cooling, strong stratification, and shallower MLD. Using Eq. 6, this difference of 0.2 m/s would generate about a 4-m difference in MLD during February, which is comparable to the 2.5 m decrease of the maximum MLD during the historical simulation (Fig. 12a).
Time series of wind and difference of temperature used as input to the historical simulation. Values are averaged from October to April for a the absolute wind speed and b the difference of temperature between sea surface and atmosphere. The values from October to December are grouped with the following year. Broken lines represent average values computed over 1967 to 1986
The reduced coastal upwelling itself, together with the MLD shoaling, causes the nutrient regime shift in the historical simulation. Alongshore wind speed that controls the coastal upwelling shifts from strong phase to weak phase in 1977 (Fig. 18) the same as the absolute wind speed (Fig. 17a). The notable difference between the mean wind speed before regime shift and that after the regime shift appears clear from late November to April in both wind speeds (Fig. 19). This seasonal variation is consistent with the result that the regime shift reduced vertical nutrient flux (Fig. 13) and lowered surface nutrient concentrations occurred from winter to early spring (Fig. 11). Thus, as the background of the nutrient regime shifts, there is a decrease of the wind speed from late autumn to early spring in the coastal region including the near shore region.
Time series of alongshore wind speed used as input to the historical simulation. Values are averaged from October to April. The values from October to December are grouped with the following year. Broken lines represent average values computed over 1967 to 1986
Seasonal changes of wind used as input to the historical simulation. a Alongshore wind speed within 10 km of the coastline, b absolute wind speed in the coastal region, c difference of the alongshore wind speed from 1967–1976 mean to 1977–1986 mean, and d difference of the absolute wind speed in the coastal region from 1967–1976 mean to 1977–1986 mean. In a and b, red line denotes value averaged over 1967 to 1976 and blue line denotes value averaged over 1977 to 1986. The values from October to December are grouped with the following year
Possible impact of the 1976–1977 regime shift
We used our analysis of the historical simulation described above to formulate a conceptual diagram on how the 1977 North Pacific regime shift affected anchovy and sardine (Fig. 20). Wind speed, especially the alongshore component, decreased in the California coastal region from late autumn to early spring leading to reduced coastal upwelling. The weak upwelling is linked to the intensity of the stratification. In addition, the reduced absolute wind speed cannot sufficiently cool and mix the surface layer, which results in a relatively shallow mixed layer. Nutrient concentrations decreased due to the weak coastal upwelling and shallow mixed layer, which resulted in a low zooplankton density in the coastal region between winter and spring. The forage decline impacts the stage survival of anchovy larvae that feed mainly in the coastal region. However, while a part of the sardine larvae was under the influence of the coastal forage decline, most sardine larvae fed in the offshore region from spring to summer and were thus less exposed to reduced prey availability. This explains why the age-0 survival of anchovy had become low after the regime shift, while the regime shift had a little impact on the age-0 survival for sardine.
Conceptual summary of regime shift effects on sardine and anchovy based on the analysis of the historical simulation. Schematic diagram illustrating a chain reaction from the wind regime shift after 1977 to anchovy and sardine dynamics. The weakened wind speed from autumn to early spring links to the low food density in the coastal area, which affects many of the anchovy larvae. Consequently, age-0 anchovy survival is reduced after the regime shift. On the other hand, most of the sardine larvae that were distributed in the offshore region from spring to summer were unaffected by the regime shift
Based on our historical simulation using the climate-to-fishery model, we suggest that the influence of the regime shift had both temporal (seasonal) and spatial footprints. The causes emerge due to the different responses to the regime shift by anchovy and sardine in relation to their particular ecology. The possibility that anchovy is more sensitive to changes in coastal upwelling than sardine has been suggested by Rykaczewski and Checkley (2008). This finding is also supported by an earlier analysis using the same model as used here (Fiechter et al. 2015), which found that the primary factor controlling anchovy population abundance in the CCS was changes in egg production associated with varying adult growth conditions (i.e., prey availability) in the coastal upwelling region. Our results support this hypothesis by explaining the underlying process of how the changes in coastal upwelling affected anchovy and sardine. Furthermore, we find that considering temporal variability within the year is crucial for understanding the niche differences between sardine and anchovy (e.g., Fig. 20).
Our simulation is not perfect. The biomass does not reproduce the yearly variation completely, and recruitment rate is smaller than that of observation data. One of the causes of model uncertainty is the parameter value of the individual-based fish model. The biological parameter (e.g., mortality rate) is not a uniquely determined value in the real world. For some parameters, field survey data presented a value range and we chose a value in a range. The historical simulation did not fully reproduce the absolute values (biomass, abundance) and interannual variation of the sardine and anchovy dynamics. Our model is best viewed as a tool for exploring how environmental variation on decadal scales can affect anchovy and sardine recruitment and population dynamics. The model skill of ours is not yet sufficient for use for tactical fishery management (Collie et al. 2016). We used the historical simulation here to examine how different physical and biological processes related to the 1977 regime shift combined to affect anchovy and sardine recruitment.
Influence of coastal upwelling
There are two possible mechanisms for the simulated nutrient decline to the CCS: weakened coastal upwelling and shallowing of the winter mixed layer. While the coastal upwelling intensity has been regarded as a main controlling factor for productivity in the CCS (e.g., Ryther 1969), impact of stratification (MLD) change on nutrient supply is also significant (Traganza et al. 1987; Jacox and Edwards 2011). However, upwelling and stratification changes are not independent. Both weak coastal upwelling and strong stratification (shallow mixed layer) are caused by weakening of wind, but weak coastal upwelling itself also causes strong stratification. Thus, we considered that the weakening of coastal upwelling is important as direct and indirect factors driving the regime shift in nutrients and consequently in anchovy.
A study of zooplankton decline in the 1976–1977 regime shift by McGowan et al. (2003) did not find a significant correlation between the upwelling index, which is a measure of coastal upwelling, and zooplankton density in the CCS. Our results that the regime shift in the coastal upwelling is only significant between winter and early spring may help refine analyses like McGowan et al. (2003). The importance of upwelling phenology in the CCS for seasonal development of the ecosystem is also confirmed by krill and juvenile rockfish abundance (Schroeder et al. 2014).
Because the coastal upwelling is most active from spring to summer (Garcia-Reyes and Largier 2012), the interannual variation during the weak upwelling season (autumn and winter) may be difficult to detect in observational data. The alongshore wind speed that caused coastal upwelling dropped to near 0 in early winter and sometimes even reversed (Fig. 19a). The difference between upwelling and downwelling can be critical in determining nutrient conditions. Thus, even though the absolute value of alongshore wind speed in winter is not large, the interannual variation of the coastal upwelling had a substantial impact on the ecosystem through the nutrient supply.
The purpose of this study is to investigate the influence of the 1976–1977 regime shift on Northern anchovy and Pacific sardine in the CCS by using a climate-to-fishery model. In the model simulation, the recruitment rate of anchovy decreased after 1977 due to the low survival of larvae, whereas sardine recruitment rate and larval survival did not show such declines. The differences resulted from the seasonal and spatial differences of larval distribution between anchovy and sardine. The anchovy larvae feed extensively from winter to early spring in the coastal region, where the changes in the wind dynamics were strong. After 1977, the absolute wind speed in the coastal region and the alongshore wind speed near shore decreased from autumn to early spring. This led to a decline of zooplankton density from winter to early spring due to poor nutrient supply from the subsurface waters. This reduced prey availability caused low survival of anchovy larvae. On the other hand, since sardine larvae were distributed largely offshore from spring to summer, most of them were less exposed to reduced prey availability.
Our analysis of the historical simulation illustrates the utility of using climate-to-fishery models to identify possible cause-and-effect explanations for the population dynamics of sardine and anchovy and other species. The model generates a consistent and comprehensive dataset (model output) on the 3D dynamics of physics, nutrients, phytoplankton, zooplankton, and fish that would be impossible to obtain with field data. Of course, as with all complicated ecosystem-based models, there are many assumptions and caveats that need to be considered when interpreting model results. Further analyses should consider additional proposed regime shifts to look for consistency in explanations, investigation into how the 1977 regime shift may have affected the predators and fishery of sardine, and continued testing of the model by comparison to field data to better assess model skill and appropriate levels of confidence in critical model results.
CalCOFI:
California Cooperative Oceanic Fisheries Investigation project
CCS:
California Current System
CORE2:
Common Ocean-Ice Reference Experiments
Dissolved organic nitrogen
NEMURO:
North Pacific Ecosystem Model for Understanding Regional Oceanography
NPZ:
Nitrogen-phytoplankton-zooplankton
Opal:
Biogenic silica
PL:
Large phytoplankton
Particulate organic nitrogen
Small phytoplankton
ROMS:
Regional Ocean Modeling System
RSI:
Regime shift index
SODA:
Simple Ocean Data Assimilation
ZL:
Large zooplankton
ZP:
Predatory zooplankton
ZS:
Small zooplankton
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We would like to acknowledge Dr. David Checkley of Scripps Institution of Oceanography for his valuable comments and suggestions.
HN was supported by JSPS KAKENHI Grant-in-Aid for JSPS Fellows Number 15 J01506 and Grant-in-Aid for Early-Career Scientists Grant number 18 K14516.
Please contact the author for data requests.
Japan Agency for Marine-Earth Science and Technology, 3173-25 Showa-machi, Kanazawa-ku, Yokohama, Kanagawa, 236-0001, Japan
Haruka Nishikawa
Department of Environmental Sciences, Institute of Marine and Coastal Sciences, Rutgers University, 14 College Farm Rd, New Brunswick, NJ, 08901-8551, USA
Enrique N. Curchitser
Ocean Sciences Department, University of California Santa Cruz, 1156 High Street, Santa Cruz, CA, 95064, USA
Jerome Fiechter
University of Maryland Center for Environmental Science, Horn Point Laboratory, PO Box, 775, Cambridge, MD, 21613, USA
Kenneth A. Rose
College of Fisheries and Ocean Sciences, University of Alaska Fairbanks, 2150 Koyukuk Drive, Fairbanks, AK, 99775-7220, USA
Kate Hedstrom
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ENC, JF, KAR, and KH developed the climate-to-fishery model. HN analyzed the model output and wrote the first draft of the manuscript. ENC, JF, and KAR provided ideas for the study and helped in the interpretation of the model output. ENC and KAR collaborated with the corresponding author in the construction of the manuscript through several drafts. All co-authors participated in the discussions about the results and commented on the manuscript. All authors read and approved the final manuscript.
Correspondence to Haruka Nishikawa.
Table S1. Definition of variables in figures. (DOCX 43 kb)
Figure S1. Simulated spatial distribution of detections from the RSI in 1977 for small plus large zooplankton density. Solid lines represent the boundary between the coastal region and offshore region and broken lines represents a boundary of the model domain. This figure is supplemental figure for Fig. 11. (TIF 575 kb)
Figure S2. Same as Additional file 2: Figure S1, but for small plus large phytoplankton density. (TIF 17111 kb)
Figure S3. Same as Additional file 2: Figure S1, but for nitrate concentration. (TIF 17111 kb)
Nishikawa, H., Curchitser, E.N., Fiechter, J. et al. Using a climate-to-fishery model to simulate the influence of the 1976–1977 regime shift on anchovy and sardine in the California Current System. Prog Earth Planet Sci 6, 9 (2019) doi:10.1186/s40645-019-0257-2
Individual-based model
Regime shift
2. Atmospheric and hydrospheric sciences | CommonCrawl |
Is this decoherence?
I have a very basic understanting of decoherence (i.e. I,ve read the Wikipedia page), but I was recently reading Heisenberg's The Physical Principles of the Quantum Theory and I came across a thought experiment which, I think, is decoherence, as it shows loss of interference and seems analogous to optical decoherence. According to Wikipedia decoherence was first developed in 1952 and this book is from 1930, so I don't know.
This is, not exactly but paraphrased, the experiment.
To begin with, imagine a beam of atoms of width $d$ is sent through a field $F$ which is inhomogeneous in the $x$-direction, and which can separate atoms into states $n$ and $m$ with energies $E_n$ and $E_m$ respectively (Like the Stern-Gerlach experiment). The energy of state $m$ depends on the field so the force in the $x$-direction experienced by an atom in that state will be $$-\frac{\partial E_m}{\partial x} \, .$$ Thus the beam will be deflected by an angle $$\frac{\partial E_m}{\partial x}\frac{T}{p}$$ where $T$ is the amount of time the atom spends in the field and $p$ is the atom's momentum. The angular separation of the beams in state $m$ and state $n$ is therefore $$\alpha=\left(\frac{\partial E_m}{\partial x}-\frac{\partial E_n}{\partial x}\right)\frac{T}{p} \, .$$ For the beam to be split into separate beams the angular separation must be larger than the natural scattering of the beam by diffraction, so $$\alpha\ge\frac{\lambda}{d}=\frac{h}{pd}$$ where $d$ is the beam width and $\lambda$ is the wavelength of the particles. Since $E_m$ depends on field strength, there will be a change in the phase of the the atom's state $|m\rangle$ associated with the atoms passing through the field, with $\varphi_m$ being the phase change and $\varphi=\varphi_m-\varphi_n$. However, since the beam has a width and the Field is not uniform, the phase change will vary depending on the atom 'passing' through different parts of the field. This will introduce an uncertainty $\Delta\varphi$ in the phase change difference. Since the phase change is $$\varphi_m=\frac{2\pi}{h}E_m T$$ And the uncertainty in position is the width of the beam $d$ the uncertainty in the energy will be $$\frac{\partial E_m}{\partial x}d$$ So we have $$\Delta\varphi=2\pi\left(\frac{\partial E_m}{\partial x}-\frac{\partial E_n}{\partial x}\right)\frac{Td}{h}=2\pi \frac{pd}{h}\alpha$$ So $\Delta\varphi\ge 2\pi$, so the phase is entirely undetermined and random.
Now imagine a beam of atoms in state $n$ is sent to a detector which detects if an atom is in a state $l$. The probability of measuring an atom in the state $l$ when it is in the state $n$ is $$|\langle l | n \rangle|^2=|\sum_m\langle l | m \rangle\langle m | n \rangle|^2=\sum_{m'm''}\langle l | m' \rangle\langle m' | n \rangle\langle m'' | l \rangle\langle n | m'' \rangle$$ Now imagine before the atoms get to the detector, we introduce the field seen previously which separates the atoms in their states $m$ now to the probability we must necessarily introduce the phase factor, averaged out over all the possible phase changes introduced by the field $$\sum_{m'm''}\langle l | m' \rangle\langle m' | n \rangle\langle m'' | l \rangle\langle n | m'' \rangle\langle e^{\varphi_{m'}-\varphi_{m''}}\rangle$$ And (assuming the phase difference has the same probability for all possible values, reasonable given its large uncertainty) $$\langle e^{\varphi_{m'}-\varphi_{m''}}\rangle=\delta_{m'm''}$$ And so the probability becomes $$\sum_m|\langle l | m \rangle|^2|\langle m | n \rangle|^2$$ The interference is lost, and the classical probability is obtained.
quantum-mechanics probability measurement-problem decoherence quantum-measurements
Phineas Nicolson
Phineas NicolsonPhineas Nicolson
$\begingroup$ Hi. I think the basic answer to your question is "yes", but some of the details are puzzling. I marked one of the equations $(\star)$: can you add some explanation of where that equation comes from? $\endgroup$
– DanielSank
$\begingroup$ I've made the edits. Is that clear enough? $\endgroup$
– Phineas Nicolson
Formally this is decoherence - loss of coherence that results in the density matrix of the atom becoming mixed.
But assume that the field is static and all its fluctuations are only spatial in its nature. Let's replace the detector with some sort of mirror (that also changes the state of the atom in such a way that the field has the opposite effect) and put detector near the beam emitter. Then on the way back the state of the atom will evolve as if time was reversed. As result the original state of the atom will be restored to the initial one without loss of information.
The decoherence in the general sense can happen due to very different processes and sometimes the distinction is made. What is often called "true decoherence" happens due to the entanglement of the system under consideration with the uncontrollable environmental degrees of freedom. What you describe however is what is called "fake decoherence" which is much more classical in its nature - it is the loss of coherence that happens because of our lack of knowledge of the fundamentally unitary evolution of the system.
Both happen because of certain lack of knowledge however the latter is constrained to the system itself whereas the former involves the whole macroscopic environment. In the "fake decoherence" the information about the initial state is recoverable if only in principle from the knowledge just about the system itself. In contrast in the "true decoherence" the information about the initial state is lost in a much more severe way as to recover it even in principle you need to know the state of the whole macroscopic environment.
OONOON
$\begingroup$ So then, in interpretations without Wave function Collapse where it is only apparent due to decoherence, the 'wave function collapse' is in theory reversible, it just isn't in practice due to the randomness and complexity inherent in the enviroment, similar to entropy, is that right? $\endgroup$
$\begingroup$ @user140323 Well, I would like to know what do you mean when you say "interpretations without Wave function Collapse". People often misunderstand what "collapse" mean and often present rather stupid ways to "get rid of collapse" using decoherence. Decoherence theory indeed explains a lot of stuff about the macroscopic world that was rather poorly understood earlier and in many cases the logic follows what you've said. $\endgroup$
– OON
$\begingroup$ @user140323 Just a small example of stupid ways. Sometimes people say the following: we consider the system interacting with environment in a certain way and trace out all the environment and get diagonal density matrix of the system in ceratain basis. Then they say - that describes the measurement. It doesn't. What you've done is said "I don't care about all the environment, only about system". But in the measurent you care about certain part of environment - the state of the apparatus. That way you describe as if you turned the apparatus on but didn't care to look at results $\endgroup$
$\begingroup$ @user140323 What decoherence theory helps you to understand is how the purely quantum world starts behaving classically when you consider the macroscopic stuff. That's provide the basis for the Copenhagen interpretation idea that we not only can but have to formulate our experiments in the classical terms (because we ourselves are macroscopic) $\endgroup$
Wikipedia is mistaken if it says that decoherence was only developed in 1952. Decoherence is simply part of quantum theory, and it is there in many early discussions of the interactions of small systems with measuring apparatuses, such as the one you describe from Heisenberg. Heisenberg judged that these sorts of physical descriptions were valuable in understanding what is going on, but they do not resolve all the issues concerning measurement and observation. He was right about that.
Andrew SteaneAndrew Steane
$\begingroup$ Interesting. What are other early examples of decoherence that we can find? $\endgroup$
$\begingroup$ My remark just refers to the fact that whenever one either takes an average over a degree of freedom entangled with the state, or over a phase that cannot be controlled in the lab, then decoherence is simply part of the prediction of the dynamics as described by Schrodinger's equation. This can be found, and was found, by anyone who analyses a system of anything more than modest size of complexity (e.g. try Bohm). Later on people tried to imbue the word 'decoherence' with added significance, as if it settled all the issues concerning measurement. $\endgroup$
– Andrew Steane
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Cayleyan
In algebraic geometry, the Cayleyan is a variety associated to a hypersurface by Arthur Cayley (1844), who named it the pippian in (Cayley 1857) and also called it the Steiner–Hessian.
See also
• Quippian
References
• Cayley, Arthur (1844), "Mémoire sur les courbes du troisième ordre", Journal de Mathématiques Pures et Appliquées, 9: 285–293, Collected Papers, I, 183–189
• Cayley, Arthur (1857), "A Memoir on Curves of the Third Order", Philosophical Transactions of the Royal Society of London, The Royal Society, 147: 415–446, doi:10.1098/rstl.1857.0021, ISSN 0080-4614, JSTOR 108626
• Dolgachev, Igor V. (2012), Classical Algebraic Geometry: a modern view (PDF), Cambridge University Press, ISBN 978-1-107-01765-8, archived from the original (PDF) on 2014-05-31, retrieved 2012-04-06
| Wikipedia |
Applied Water Science
March 2013 , Volume 3, Issue 1, pp 133–144 | Cite as
Modeling sedimentation rates of Malilangwe reservoir in the south-eastern lowveld of Zimbabwe
Tatenda Dalu
Edwin Munyaradzi Tambara
Bruce Clegg
Lenin Dzibakwe Chari
Tamuka Nhiwatiwa
First Online: 28 December 2012
Modelling the sedimentation rates using the Wallingford (2004) equations with the aid of NDVI (remote sensing) to assess land degradation was carried out for Malilangwe reservoir catchment in the south eastern lowveld of Zimbabwe. Siltation life of the reservoir was determined from rate of incoming sediment, trap efficiency and reservoir capacity using the Wallingford method. The average rainfall of the study area was about 560 mm while runoff from the catchment ranged from 0.3 mm (minimum) to 199 mm (maximum) with an overall average runoff of 50.03 mm. Results showed that the overall mean annual sediment concentration was approximately 2,400 ppm. The reservoir capacity to inflow ratio was estimated at 0.8 with a sedimentation rate of 120.1 tkm−2 year−1. Calculated probability of the dam filling is 26.8 %. Results also showed that the siltation life of the reservoir was >100 years according to the Wallingford method. The Normalised Difference Vegetation Index (NDVI) showed progressive decline (p < 0.05) of the vegetation health from 2000 to 2009. While acknowledging the limitations of techniques used, this study demonstrates in part the effectiveness of sedimentation modelling and remote sensing as a tool for the production of baseline data for assessment and monitoring levels of land degradation in the Malilangwe reservoir catchment.
Sedimentation NDVI Catchment Reservoir Degradation
In Zimbabwe, sediment load has exceeded normal design limits in many reservoirs, thus reducing storage capacity and shortening their useful life for human benefit. According to van der Wall (1986) and Mambo and Archer (2007), Africa now stands for rapid land degradation, declining fertility, soil erosion and drought. Sedimentation of reservoirs, in the light of man accelerated erosion, is according to the Zimbabwean Government a major time bomb (van der Wall 1986; Mambo and Archer 2007). Within the framework of the development of a National Master Water Plan for Zimbabwe, a reconnaissance study in siltation and soil erosion was carried out in May 1984–January 1985 (van der Wall 1986). It has been reported that over 50 % of 132 small dams surveyed in Masvingo Province in Zimbabwe by Elwell in 1985 were silted (Khan et al. 2007).
Land use change is listed as the biggest threat to global biodiversity largely due to deforestation activities (Enters 1998). Land degradation in Zimbabwe has been caused mainly by the decline of forest areas through cutting down of trees for agriculture and fuel (van der Wall 1986; Mambo and Archer 2007). The widespread impacts of deforestation are also reflected at a national and regional level through vastly elevated soil erosion rates, sedimentation of major waterways and an increased frequency and severity of floods (Adger 1992; Ewers 2006). Of the major causes of soil degradation, deforestation and removal of natural vegetation account for 43 % with overgrazing, improper agricultural practices and over-exploitation of natural vegetation contributing 29, 24, and 4 %, respectively (Enters 1998). Land degradation is the most widespread and severe in communal areas which are characterized by deforested landscapes, poor quality pasture and soil infertility. The recent land resettlement programme that started in 2000 has left most of the country forests facing serious threat of deforestation increasing from 1.41 % (1990–2000) to 16.4 % (2000–2005). Degradation mostly manifests as gullies that render large tracts of land virtually unusable, threatening water supply and quality (Mambo and Archer 2007).
Recent ecological studies have highlighted the relevance of the Normalized Difference Vegetation Index (NDVI) as a tool for assessing changes in vegetation cover (Pettorelli et al. 2005). Land degradation is believed to be one of the most severe and widespread environmental problems in Zimbabwe and globally. It is, therefore, important to understand spatial and temporal distributions of vegetation in a region in order to assess changes in land cover. Remotely sensed NDVI may provide the basis for an early warning of land degradation (Scanlon et al. 2002; Wessels et al. 2004). However, the method is not without limitations and mis-registration of spectral images may lead to a considerable number of errors and unusable results (Lu et al. 2003). The calculation of NDVI values is influenced by a number of factors such as clouds, atmospheric, soil, anisotropic and spectral effects (Crippen 1990; Wessels et al. 2004). Modified indices such as Soil Adjusted Vegetation Index (SAVI) and Global Environment Monitoring Index (GEMI) have been developed indices to correct for some of the confounding factors that affect NDVI (Wessels et al. 2004). Despite its limitations, NDVI remains a valuable quantitative vegetation monitoring tool.
NDVI as a proxy for monitoring land degradation
Changes in vegetation features of terrestrial landscapes have long been used as indicators for susceptibility to degradation (Lambin and Ehrlich 1996; Mambo and Archer 2007). Vegetation cover is the commonly indicator in assessing susceptibility to degradation. Tucker et al. (1991) highlighted that vegetation cover is not a good indicator in long-term dynamics of land degradation in arid and semi-arid areas. Recent advances in remote sensing technologies have seen increased use of different spectral indices. One of the widely exploited spectral indices is the Normalised Difference Vegetation Index (NDVI) which measures "greenness" (chlorophyll content). The NDVI is a measurement of the balance between energy received and energy emitted by vegetation (Meneses-Tavor 2011). It has been observed that NDVI increases near-linearly with increasing leaf area index and then enters an asymptotic phase in which NDVI increases very slowly with increasing leaf area index (Roderick et al. 1996; Wessels et al. 2004; Jiang et al. 2006). The NDVI equation produces values in the range of −1 to +1. Higher values (0.8–0.9) are indicators of high photosynthetic activity linked to scrub land, temperate forest, rain forest and agricultural activity while values closer to zero means no vegetation (Crippen 1990; Weier and Herring 2000). Values in the range of −0.2 to 0.05 are indicative of snow, inland water bodies, deserts and exposed soils (Crippen 1990; Roderick et al. 1996; White et al. 1997; Bacour et al. 2006).
The use of NDVI as an indicator of degradation is based on the premise that NDVI values reflect the level of photosynthetic activity in a plant community which in turn indicate vegetation health (Barrow 1991; Mambo and Archer 2007; Meneses-Tavor 2011). Therefore, degradation of ecosystem vegetation, or a decrease in green, would be reflected in a decrease in NDVI values. The NDVI is also correlated with certain biophysical properties of the vegetation canopy, such as leaf area index, fractional vegetation cover, vegetation condition, and biomass (Meneses-Tavor 2011).
Impacts of sedimentation
The effect of sedimentation in a dam is that it reduces the dam's water holding capacity, with decline in capacity; the yield is reduced both in quantity and reliability. The relationships between reservoir yields under certain risk levels, storage ratios and the reliability of inflow, have been well established for Zimbabwe by Mitchell (1987). The deposition of eroded soil sediments in water bodies from either natural or anthropogenic impacts can result in the destruction of aquatic habitats and a reduction in the diversity and abundance of aquatic life. Diversity and population size of fish species such as Labeo altivelis and benthic macroinvertebrates associated with coarse substrates can be greatly reduced if the substrates are covered with sand and silt. Tomasson and Allanson (1983) showed that the growth rates of Barbus and Labeo sp. in Lake Le Roux, South Africa were greatly reduced when transparency of the water decreased due to increased sediment input in the lake. Moreover, increased turbidity decreases the water's aesthetic appeal, human enjoyment of lake and reservoir recreational activities and interferes with disinfection of the water prior to it being pumped to the end-users. If the river cross-section is sufficiently reduced by sediment build-up, sedimentation can increase downstream flooding. In addition, some metal ions, pesticides and nutrients may combine with sediment particles and be transported downstream.
Information on upstream land use activities and land cover change, sediment yield within a catchment is required for controlling sediment accumulation in reservoirs. Presently, there have been very few studies in Zimbabwe that have looked at the problem of reservoir siltation (ZINWA 2004). Therefore, there is not much data available to establish the correlation between changes in land use and land cover with sedimentation rates in reservoirs. If this is not addressed, sediment loads could exceed normal design parameters in some reservoirs, thus reducing storage capacity and a shortened useful lifespan. The main objective of this research was to assess changes in land use and model the impacts it would have on sedimentation rates in a small reservoir.
Malilangwe Wildlife Reserve is located in the Chiredzi District of the south-eastern lowveld of Zimbabwe (20°58′ 21°02′S, 31°47′32°01′E) (Fig. 1). Malilangwe reservoir is an impounded reservoir formed in 1964 and is used for water supply in the reserve. It is situated on the Nyamasikana River, a tributary of the Chiredzi River which in turn flows into the Runde River. It is a gravity section masonry dam with a surface area of 211 hectares and has a maximum volume of 1.2 × 107 m3 at full capacity, as well as a catchment of about 200 km2. The dam wall was initially built to a height of 10 m in 1963 and the dam was filled for the first time in 1965. In 1965, the dam wall was raised to a height of 19–22 m in 1984, and finally to a height of 24 m in 1988. The dam wall was raised by an additional 1.75 m in 1999 and has a current height of 25.75 m. Malilangwe reservoir last spilled in hot–wet season of 2000 after the Cyclone Eline induced floods in 2000.
Location of Malilangwe reservoir (shaded black area) and major water supplying rivers in the catchment (MC)
Normalised Difference Vegetation Index images from different years were captured during the dry season (September) and wet season (March) of the following years: 2000, 2002, 2005, 2007 and 2009. The images were captured at the same time of the year in order to minimise the expression of variations in such factors as light quality, geometry of the observation and variances in the state of a community over the course of a year (Mambo and Archer 2007). Images were downloaded from the United States Geological Survey (USGS), Global Visualisation Viewer (GloVis, website: www.glovis.usgs.gov). The images were processed using the Integrated Land and Water Information System (ILWIS) Version 3.3 GIS software which employed the map value function to extract NDVI values from the sample locations. Sampling points were randomly selected in the catchment area and in the reference sites with the aid of ArcView GIS 3.2a software. The points demarcated the positions of sites whose NDVI values were used for analysis. Coordinates (UTM) of each point were recorded.
NDVI expressed as a ratio between measured reflexivity in the red and the infra-red bands was calculated as:
$$ {\text{NDVI }} = \, \left( {\text{NIR} -{\text{ R}}} \right) \, / \, \left( {\text{NIR}} + {\text{R}} \right), $$
where NIR (in TM imagery) is near infra-red band 4; and R is red band 3 (N.B. living vegetation absorbs light in the frequency range of band 3 but shows almost no absorption in the range of band 4).
The NDVI data were first tested normally using the One-Sample Kolmogorov–Smirnov test. Since the data were normally distributed, a two-way ANOVA was carried out to test for differences in NDVI values. A Spectral Time Series analysis on SPSS 16.0 (SPSS 2007) was carried out on the NDVI data to find out whether there were any significant changes in vegetation for Malilangwe Wildlife Reserve (reference) and catchment. A two-way ANOVA was also carried out on the data set using the MYSTAT ver. 12 (Systat 2007).
In this study, Malilangwe Wildlife Reserve was used as the reference site which represents a natural functional ecosystem (Fig. 1). Reference sites are supposed to occur in similar biotic zones, in close proximity with the study site, and exposed to similar natural disturbances (Society for Ecological Restoration 2004).
Estimation of sedimentation rates
Mathematical equations from studies by van der Wall (1986), Wallingford (2004), Khan et al. (2007) and Mavima et al. (2010) were used in the estimation of sediment rates and the following procedure using several models was followed;
Gross mean annual reservoir inflow (MAI) (m3 year−1) was calculated by:
$$ {\text{MAI }} = {\text{ CA}}\; \times \;{\text{MAR}}, $$
where CA is the catchment area (km2), MAR the mean annual runoff (mm).
Annual runoff volume (ARV) (m3) was calculated by:
$$ {\text{ARV }} = \, P_{\text{a}} \; \times \;{\text{CA}}\; \times \;1,000, $$
where Pa is the annual precipitation (mm), CA the catchment area (km2).
Sediment trap efficiency (St) as a percentage (the trap efficiency is generally assumed to be 100 % for most reservoirs were the gross storage ratio >0.1) was calculated by:
$$ S_{t} = \, \left( {0.1 \, + \, 9\; \times \;{\text {SR}_{g}}} \right)\; \times \;100\;{\text{or}}\;S_{t} = 0.1116\; \times \;{\text{In}}\left( {C/ \, I} \right), $$
where C is the reservoir capacity at spillway crest level, I the inflow volume of water to the reservoir and the relationship predicts the annual sediment trapping efficiency of a dam from the ratio of the dam capacity to the annual inflow volume, SRg the gross storage ratio.
Sediment concentration was calculated according to Wallingford (2004) method, since catchment characterisation was not carried out. Wallingford studies were carried out in the same region, the lowveld as Malilangwe Reservoir, and the mean annual sedimentation concentration was estimated for reservoir as highlighted below based on observation data. The method uses the description which best fitted the catchment and the Malilangwe catchment fell between two descriptions; basin with low slopes and very well-developed conservation (1,200 ppm) and basin with moderate topography and well-developed conservation (3,600 ppm). The sediment concentrations were then averaged to give the mean annual sediment concentration of 2,400 ppm (2,400 mg l−1) for the catchment.
The predictive equation adopted from Wallingford (2004) and Khan et al. (2007) was used for estimation of sediment yield for the catchment of the Malilangwe reservoir with the mean annual Sy then calculated using the formula:
$$ S_{\text{y}} = \, X \, \left( {{\text{MAR }}/ \, 1000} \right), $$
where Sy is the mean annual sediment yield (tkm−2 year−1), X the sediment concentration/density, and MAR the mean annual runoff (mm).
We used the Ministry of Lands and Water of Zimbabwe (MoLWZ) (1984) methods; to correlate the coefficient of variation (CV) of mean annual runoff with MAR, we used a fitted relationship:
$$ {\text{CV }} = \, \left( {0.00139\;{\text{MAR}}} \right)^{2} - \, 0.7538\;{\text{MAR }} + \, 154.5 \, \left( {R^{2} = \, 0.87} \right), $$
where CV is the coefficient of variance (%), MAR the mean annual runoff (mm).
The probability of a dam filling can be estimated from the coefficient of variation of annual runoff and the dam capacity to annual inflow ratio, using a procedure developed for dams in Zimbabwe described in Mitchell (1987). Mitchell argues that given the relatively short records and other deficiencies in the available data, the use of complex statistical functions is not justified, and that the Wiebul distribution can be used to represent the distribution of annual inflows to a dam:
$$ P \, = \, e^{{ - {\text{km}}}} , $$
where P is the probability of a dam filling from empty, km the (c × V/I) n , V the Dam storage volume (m3), I the annual inflow (m3), c the constant related to CV (taken as 1.11), n the constant related to CV (taken as 0.84).
A Pearson correlation between sedimentation parameters (rainfall, runoff, sedimentation yield, Malilangwe catchment NDVI and Wildlife Reserve NDVI) was carried out using MYSTAT ver. 12 (Systat 2007) to test if year and season had an influence on parameters.
Storage capacity losses due to siltation
The proportion of the incoming sediment load that is trapped in a dam varies with the sizes of the sediments transported to the dam, the water velocities or retention time in the dam, and the proportion of the incoming flows that is passed over the spillway. The interrelationship between these parameters is too complex to be considered in the design of small dams (Wallingford 2004). Trap efficiency was assumed to remain constant at 100 % as Murwira et al. (2009) projected decreases in precipitation and runoff for the lowveld region. The loss in a dam's storage capacity over a specified time period is estimated using equation:
$$ C_{n} = \, 1 \, - \, \left[ {n\; \times \;S_{\text {y}} \; \times \;{\text{CA}}\; \times \;S_{\text {t}} /\left( {C\; \times \;{\text{Den}}} \right)} \right], $$
where C n is the proportion of original storage capacity left after n years of siltation, n the number of years, Sy the catchment sediment yield (tkm−2 year−1), CA the catchment area (km2), St the sediment trap efficiency, C the dam's original capacity at full supply level (m3), and Den the settled density of dam sediment deposits (taken as 1.2 tm−3).
NDVI analysis
Geographical Information Systems (GIS-NDVI) images were used for monitoring vegetation changes in the Malilangwe catchment (MC). Figures 2, 3 and 4 show vegetation change in the catchment and Malilangwe Wildlife Reserve (MR) from 2000 to 2009. The catchment showed progressive decline in NDVI values as shown by the decrease in cover especially along the Malilangwe Wildlife Reserve boundary meaning that vegetation was being cleared or greenness of plants was decreasing. The catchment showed a more extensive decrease in vegetation cover than the reserve in areas along major rivers supplying Malilangwe reserve. From the NDVI images and values, all sites in the wildlife reserve and the catchment between the years 2000 and 2009 consisted of sparse vegetation as a range of 0.1–0.5 represent sparse vegetation and >0.6 represent dense vegetation. In 2002 and 2007 vegetation had decreased in both the reserve and catchment as seen by low NDVI values (Figs. 2, 3, 4). In 2009, the catchment and reserve were still covered by sparse vegetation except for the southern tip of the reserve which had little patches of vegetation (Figs. 2, 3, 4).
Colour composite Landsat satellite images covering Malilangwe study area in the dry season for A 2000, B 2002, C 2005, D 2007 and E 2009 overlaid with degraded areas mapped by National Land Cover (NLC). Map units are in kilometres, Universal Transverse Mercator (UTM) zone 36 South based on WGS 1984 spheroid
Boxplots of NDVI values for Malilangwe catchment and Malilangwe Wildlife Reserve (2000, 2002, 2005, 2007 and 2009)
Spectral time series analysis for the dry (A, B) and wet (C, D) season for the Malilangwe catchment (A, C) and Wildlife Reserve (B, D)
The ranges of NDVI values are shown in Figs. 2 and 5. Mean NDVI values were generally higher for the catchment compared to the wildlife reserve. Mean NDVI values for Malilangwe Wildlife Reserve (MR) for 2000 were 0.4322, indicating that healthy vegetation mainly consisting of woodland as 2000 was a cyclone year. The other different colour tones identified, with lower NDVI values (from 0.3229 to 0.3973) indicate high moisture, given the low NDVI reflectance of moisture and also given the similarity of the tones to water bodies. These areas were characterized by open savanna grasslands. There was a decrease in 2002 (NDVI = 0.3229) and 2005 (NDVI = 0.3729) and 2007 (NDVI = 0.3973) before increasing in 2009 (NDVI = 0.4411) (Fig. 5). Tones indicating moisture patterns, as identified in 2002, 2005 and 2007, have completely disappeared in 2009. Overall mean NDVI value change for the wildlife reserve from 2000 to 2009 was 0.045. NDVI values decreased from the year 2000 up to 2007 for the catchment compared to the wildlife reserve. For the Malilangwe catchment (MC), mean NDVI values showed a generally declining trend although they fluctuated between the years. In 2000, mean NDVI was 0.4381; 2002 (NDVI = 0.3950); 2005 (NDVI = 0.4354) and 2007 (NDVI = 0.3855). The general reduction in NDVI values indicated a possible reduction in healthy vegetation across the catchment, but the area to the north of the catchment falling under woodland class shows high NDVI values. Similar to the wildlife reserve, there was an increase in NDVI values in 2009 (NDVI = 0.4812) (Figs. 2, 5). The overall mean NDVI value change for the catchment from 2000 to 2009 was 0.043.
Colour composite Landsat satellite images covering Malilangwe study area in the wet season for (A) 2000, (B) 2002, (C) 2005, (D) 2007 and (E) 2009 overlaid with degraded areas mapped by National Land Cover (NLC). Map units are in kilometres, Universal Transverse Mercator (UTM) zone 36 South based on WGS 1984 spheroid
The available NDVI data were not enough for a comprehensive time series analysis to be carried out on the data. It was necessary to identify if there had been any significant changes in vegetation cover over the years and also if there were any differences in terms of NDVI between the catchment and reference sites. ANOVA revealed significant differences between the years and sites (p < 0.05). However, for the years the trend implied by these differences could not be established. Two-way ANOVA test showed a significant (p < 0.05) effect of seasons on NDVI years with the interaction effect also present (Table 1). This showed higher NDVI values within the wet season, and the significant (p < 0.05) interaction also showed higher NDVI values in wet season than dry season (Fig. 6).
Two-way ANOVA test on the effect of seasons and years on NDVI values
F value
Year × season
Year × site
Season × site
Year × Season × site
Interaction plot showing the effect of seasons and year on NDVI values
Sedimentation rates and reservoir capacity–inflow ratios
The sedimentation rates, capacity inflow ratios are presented in Table 2. The reservoir capacity or volume was 1.2 × 107 m3 and inflows were determined from the long-term data collected over a 60-year period. The reservoir capacity to inflow ratio was estimated at 0.8. The calculated gross storage ratio was 4 for the reservoir, and the sediment trap efficiency was assumed to be 100 % as the reservoir has no outlet for water. The catchment sediment yield was estimated at 120.1 tkm−2 year−1 with a mean annual sediment concentration of 2,400 ppm (Table 2). Using the relationship developed by Mugabe et al. (2007), the Malilangwe catchment area is about 200 km2 and with mean annual rainfall of 562 mm, runoff was calculated at 50 mm and the coefficient of variance of mean annual runoff was 120.3 %. The runoff coefficient for the catchment was calculated as 0.1 (Table 2). The probability of the dam filling with water was calculated at 26.8 %.
Results of sedimentation rates for Malilangwe reservoir
Reservoir volume
1.2 × 107
Annual runoff volume
1 × 107
Mean annual runoff (MAR)
Sediment trap efficiency
Runoff coefficient
Mean annual sediment concentration
Catchment sediment yield
tkm−2 year−1
Probability of the dam filling
Coefficient of variance of MAR
Capacity–inflow ratio
Predicted capacity storage losses for the Malilangwe reservoir calculated using the Wallingford (2004) method is shown in Table 3. It is predicted that the dam will lose 16 % of its storage capacity over 100 years at current levels of sedimentation (120.1 tkm−2 year−1). The reduction in water yield over the same period is expected to be larger than 16 %. It is also projected that 32 % of the storage capacity will be lost over a 100-year period with double the sedimentation (Table 3).
Projected capacity storage losses (%) for Malilangwe reservoir for 0, 10, 25, 50, 80 and 100 years for different sediment yield rates
Sedimentation storage loss of reservoir in percentage
Sedimentation rate increases 120.1 tkm−2 year−1—0 %, 150.1 tkm−2 year−1—25 %, 180.1 tkm−2 year−1—50 %, 210.1 tkm−2 year−1—75 % and 240.2 tkm−2 year−1—100 %
A Pearson correlation of sedimentation parameters with season and year showed that season and year was significantly correlated with rainfall, runoff, sedimentation yield, catchment and Wildlife Reserve NDVI. Thus, season and year was significantly negatively correlated with rainfall (r = −0.754), runoff (r = −0.754), sedimentation yield (r = −0.754), catchment (r = −0.917) and Wildlife Reserve NDVI (r = −0.933).
Remote sensing was used to assess changes in vegetation cover in the study area in a period covering almost a decade. The interpretation of the NDVI data pointed toward a progressive decline in vegetation cover in the Malilangwe catchment particularly in areas close to the reserve boundary. We employed NDVI values as a surrogate for the assessment of vegetation (deforestation) change rates in the catchment. Vegetation change thus showed a general decline in the catchment for the period 2000–2009 with a mean yearly change of 0.043 NDVI values whilst the wildlife reserve showed a slight increase in NDVI values of relatively 0.045 NDVI units. The mean NDVI value, observed in 2002 and 2007, decrease in both the wildlife reserve and the catchment could be attributed to the severe droughts in those years that significantly reduced vegetation cover. In contrast, NDVI increases observed in 2005 and in 2009 in both the wildlife reserve and the catchment) could be attributed to good rainfall during the 2 years. Therefore, climatic factors have an important role in determining vegetation patterns. This variable is of great concern as this can be linked to protected climatic changes for the region where dry areas are expected to get less and less rainfall.
The use of statistical tools to analyse NDVI values for the 2000–2009 period, identified a large vegetation change in the catchment closer to the boundary with the communal areas. The catchment vegetation is facing serious anthropogenic impacts such as deforestation, veld fires and vegetation clearing for farmland and firewood which change the vegetation structure resulting in it being different from that of the wildlife reserve. There was a significant difference found between catchment and wildlife reserve with the latter having higher vegetation cover. It is evident that the catchment is slowly being degraded from the spectral time series analysis (p < 0.05) of 2000–2009, though influenced by stochastic setbacks such as cyclone, drought and human-induced anthropogenic events which are resulting in runoff and soil erosion changes in the catchment. Deforestation had a significant effect on the runoff and sediment discharge from Malilangwe catchment because it brought about a number of interferences such as increased surface runoff in streams and rivers and soil erosion which resulted in sedimentation of rivers and the reservoir as shown by significant correlations as suggested also by Dinor et al. (2007). The study also showed a strong relationship between sedimentation parameters as highlighted by Murwira et al. (2009) that found a strong positive and significant correlation between rainfall and runoff in the Save mega-basin. They noted that an increase in rainfall has a simultaneous increase in runoff and the inverse is also true and they also found a decreasing but not significant trend in the rainfall and runoff over the years. A deficit of over 5 × 104 mega litres of water in certain sub-catchments of the Save basin was projected given the worst case scenario of decreased rainfall that was observed (Murwira et al. 2009).
In Ivory Coast, deforestation increased surface runoff and sediment yield by 50–1,000 times compared to the forested areas. Similar effects of deforestation have been reported from East Africa, Kenya where sediment yield from agricultural and grazed catchments was significantly more than from partially or forested catchments. Li et al. (2007) found that deforestation increased both surface and subsurface runoff by about 20 % because some of the water that was formerly intercepted by vegetation and evaporated became overland flow. They also observed that about 80 % of the increase was due to an increase in subsurface drainage of soil moisture that would have been transpired by plants in the control experiment.
Field studies should be carried out to determine the actual amounts of sediments in the catchment and reservoir so as to provide the actual rate of sedimentation related to land use change in the catchment. While acknowledging the limitations of the techniques applied, this study demonstrates in part the effectiveness of remote sensing as a tool for the production of baseline data for assessment and monitoring of land degradation in the Malilangwe catchment. Field studies measuring sedimentation and erosion rates must also be carried out to aid remote sensing data.
The annual catchment sediment yield for Malilangwe reservoir was calculated at 120.1 tkm−2 year−1. The value of sediment yield (120.1 tkm−2 year−1) obtained is similar to studies done by van der Wall (1986) where about half of the basins observed in Zimbabwe yielded more than 100 tkm−2 year−1. The sediment yields for the Malilangwe reservoir of 120.1 tkm−2 year−1 were compared to that of Chikwedziwa (45 tkm−2 year−1) which is in the same geographic region and area (van der Wall 1986). This could be attributed to little land degradation and low population density per unit area (70 inhabitants' per km2) in Chikwedziwa at that time (1987), but the sediment rate for this area is expected to be higher than 45 tkm−2 year−1 at present. In the study area, high rates of degradation and population increase are estimated at 2.68 and 2.2 % per year, respectively (Lorup et al. 1998). Population growth and the poorly organised land resettlement since the year 2000 have been cited major factors contributing to the deterioration of the environment in the catchment (Mambo and Archer 2007).
The ratio of reservoir capacity to inflow indirectly provides an index of residence time of sediment laden water in the reservoir (Reddy 2005). Most sediment enters reservoirs during high inflow periods and ideally if the capacity–inflow ratio is small, much of it will be discharged over the spillway (Reddy 2005). If the capacity–inflow ratio is large, much of this water is retained in the reservoir resulting in high sediment trap efficiency (Reddy 2005). The capacity–inflow ratio for the dam was 0.8 which is higher than the recommended ratio of 0.3 for long economic life of the dam. In highly degraded catchments, where large sediment yield is expected, the capacity to inflow ratio of about 0.5 is mostly recommended (Wallingford 2004). With such a high capacity–inflow ratio for the dam, the reservoir is expected to have high siltation and shorter economic life.
The calculated gross storage ratio for Malilangwe reservoir was greater than 0.1 (calculated = 4): according to Khan et al. (2007), a gross storage ratio of greater than 0.1 means the sediment trap efficiency is 100 %. Thus, with a sediment trap efficiency of 100 %, all sediments that enter the reservoir are retained and trapped in the reservoir. A total depth of about 8.6 m or 33.1 % of storage capacity has been lost due to the sediment accumulation in the reservoir over a 48-year period (1963–2011) with the current estimated maximum depth at 15.4 m from a previous of 25.75 m. The Save River catchment in Zimbabwe loses an estimated 2.6 × 108 m3 year−1 of its storage capacity annually (Marshall and Maes 1994). At current sedimentation rates, the reservoir is projected to lose 16 % of its storage capacity in 100 years. With increase in land degradation in the Malilangwe reservoir catchment, the reservoir estimated life span of over 100 years will be drastically reduced unless drastic measures are taken to address the problem. Khan et al. (2007) showed that the bigger the reservoir and catchment, the better is the siltation life. All projections on storage capacity showed that the reservoir will not lose over 50 % of capacity in 100 years.
The coefficient of variation for mean annual runoff was calculated at 120.3 % with a capacity–inflow ratio of 0.8 which indicates a probability of filling of 26.8 %. This is less than the 80 % sometimes adopted as a target value for conventional water supply and irrigation systems (Wallingford 2004) as this would probably be acceptable as it is customary to retain some water in the dam at the end of the dry season (carry-over) to provide insurance against low rainfall in the following year. Since the probability of the dam filling (>26.8 %) is considerably less than the recommended levels, hence measures must be put in place to conserve much of the water in the reservoir. Other factors such as environmental factors (temperature, rainfall and evaporation rates), changes in catchment activities such deforestation, poor agricultural methods and water use and demand will be major determinants of the reservoir lifespan.
The annual runoff volume of 1 × 107 m3 indicates that a storage volume larger than 0.8 × 1 × 107 = 8 × 106 m3 is required, meaning that an 11 m dam height from the current 15.4 m dam height should be selected as this gives the best ratio of volume of water stored at any given year. O'Connor (2007) calculated runoff volume 6.2 × 106 m3 for Malilangwe reservoir which was lower than the calculated runoff volume of 1 × 107 m3 in this study. Magadza (2002) projected that mean annual runoff of some major rivers in southern Africa is estimated to decline by as much as 20–45 % within the next 50 years meaning that the rate of erosion and rate of sedimentation also is strongly related to runoff and rainfall.
Many benefits might be obtained from improving the management of the Malilangwe reservoir catchment so as to reduce siltation rates in the reservoir. These include addressing poor land practices leading to low soil productivity and the value of wood and non-wood products to be obtained from increased tree planting and improved management of natural forest areas. It is strongly recommended that Malilangwe must be geared for increased variability in the water availability in the reservoir with an increased frequency of droughts and floods as projected by Murwira et al. (2009). This calls for the use of more dams and groundwater sources (boreholes) within the wildlife reserve to reduce the negative effects of increased water fluctuations in the Malilangwe Reservoir. This sedimentation problem can only be solved through wider soil conservation technologies that are readily understood and implemented by locals and is within their financial reach. These measures require limited labour and require no foregone benefits but lead to substantially increased benefits within catchment populations.
The study provided an indication of degradation hot spots within Malilangwe catchment, where interventions to prevent further change and where practices for mitigating the associated degradation should be targeted. The generated data will be useful for managers of the Malilangwe Reservoir, environmental organisations, government and local communities in helping curb land degradation and aid in the coordination of mitigation.
This study was made possible through the financial support of the German Academic Exchange Services (DAAD—A/10/02914) and Malilangwe Trust Research Grant. Special thanks go to Godfrey Pachavo of the Department of Geography and Environmental Sciences, University of Zimbabwe who assisted during the NDVI analysis and Mr. Vincent G. Maposa a Research Analyst at Frost and Sullivan (South Africa) for his contribution and critical analysis of the manuscript. Our appreciation also goes to Elizabeth Munyoro and the technical staff of the Department of Biological Sciences, University of Zimbabwe, for all their technical support during the study.
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This article is published under license to BioMed Central Ltd. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
1.Department of Biological SciencesUniversity of ZimbabweMt. Pleasant HarareZimbabwe
2.Department of Zoology and EntomologyRhodes UniversityGrahamstownSouth Africa
3.Malilangwe Wildlife ReserveChiredziZimbabwe
Dalu, T., Tambara, E.M., Clegg, B. et al. Appl Water Sci (2013) 3: 133. https://doi.org/10.1007/s13201-012-0067-9
Accepted 19 November 2012
First Online 28 December 2012
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\begin{document}
\title[Non-hopfian relatively free groups] {Non-hopfian relatively free groups} \author{S.V. Ivanov} \address{Department of Mathematics\\ University of Illinois \\ Urbana, IL 61801, U.S.A.} \email{[email protected]} \thanks{The first author is supported in part by NSF grant DMS 00-99612}
\author{A.M. Storozhev} \address{Australian Mathematics Trust\\ University of Canberra \\ Belconnen, ACT 2601, Australia} \email{[email protected] }
\subjclass[2000]{Primary 20E10, 20F05, 20F06}
\begin{abstract} To solve problems of Gilbert Baumslag and Hanna Neumann, posed in the 1960's, we construct a nontrivial variety of groups all of whose noncyclic free groups are non-hopfian. \end{abstract} \maketitle
\section{Introduction}
Recall that a group $G$ is called {\em hopfian} if every epimorphism $G \to G$ is an automorphism. Let $F_m$ be a free group of finite rank $m >1$, $N$ be a normal subgroup of $F_m$ and $V(N)$ a verbal subgroup of $N$ defined by a set of words $V$. In \cite{B63}, Baumslag proved that if both quotients $F_m / N$, $N / V(N)$ are residually finite then the group $F_m / V(N)$ is also residually finite. In this connection, Baumslag \cite{B63} posed the following problem. Is $F_m / V(N)$ a hopfian group if the group $F_m/N$ is hopfian? In particular, if $F_m = N$ then $F_m/N$ is trivial and hopfian and so the Baumslag problem asks about the hopfian property of relatively free groups $F_m / V(F_m)$, where $V(F_m)$ is a verbal (or fully invariant) subgroup of $F_m $. The problem on the hopfian property of finitely generated relatively free groups was independently stated by H. Neumann \cite[Problem 15]{N67}. Recall that a finitely generated residually finite group is hopfian (e.g., see Corollary 41.44 \cite{N67}). Hence, if one could show that relatively free groups are residually finite then H. Neumann's problem would be solved in the affirmative. However, this is not the case in general and it follows from Novikov--Adian results \cite{NA68}, \cite{A75} (see also \cite{O82}, \cite{O89}, \cite{I94}, \cite{L96}, \cite{I98}) on the Burnside problem for odd $n \ge 665$ and Kostrikin--Zelmanov results \cite{K59}, \cite{K86}, \cite{Z91} on the restricted Burnside problem for $n = p^k$, where $p$ is prime, that the free $m$-generator Burnside group $B(m,n)= F_m /F_m^n$ of exponent $n$ is not residually finite if $m >1$ and $n$ is odd, $n = p^k >665$ (in fact, $B(m,n)$ is not residually finite for all $m >1$ and $n \gg 1$ as follows from results of \cite{K59}, \cite{K86}, \cite{Z91}, \cite{Z92}, \cite{HH56}, \cite{NA68}, \cite{I94} and the classification of finite simple groups). Whether the group $B(m,n)$ is hopfian (or, more generally, whether there is a hopfian non-residually finite relatively free group) is still unknown (the problem on the hopfian property of $B(m,n)$ for odd $n \gg 1$ is stated in \cite[Problem 11.36(c)]{KN92}).
In this article we construct a variety of groups of exponent 0 all of whose noncyclic free groups are non-hopfian providing thereby negative solutions to the foregoing problems of Baumslag and H. Neumann. To construct the identities that define such a variety of groups, we let $[a,b] = aba^{-1}b^{-1}$ be the commutator of $a$, $b$ and set \begin{equation} \label{v1v2} v_0(x,y) =x, \quad v_1(x,y) =[((x^dy^d)^dx^d)^d, x^d]^d y , \quad v_2(x,y) =[v_1(x,y)^d, x^d] . \end{equation}
Now we define the words $w_1(x,y)$, $w_2(x,y)$ by following formulas \begin{multline} \label{w1} w_1(x,y) = x^{{\varepsilon}_1} v_1(x,y)^{n} x^{{\varepsilon}_2} v_1(x,y)^{n+2} \dots
x^{{\varepsilon}_{h/2-2}} v_1(x,y)^{n+h-6} \\
x^{{\varepsilon}_{h/2-1}} v_1(x,y)^{n+h-4}
x^{{\varepsilon}_{h/2}} v_1(x,y)^{(n+h-2) + h/2} x^{{\varepsilon}_1} v_1(x,y)^{-(n+1)} \\ x^{{\varepsilon}_2} v_1(x,y)^{-(n+3)} \dots x^{{\varepsilon}_{h/2-1}} v_1(x,y)^{-(n+h-3)} x^{{\varepsilon}_{h/2}} v_1(x,y)^{-(n+h-1)} , \end{multline} \begin{multline} \label{w2} w_2(x,y) = y v_2(x,y)^{n^2+1} v_1(x,y)^{{\varepsilon}_2} v_2(x,y)^{n^2+2} v_1(x,y)^{{\varepsilon}_3} v_2(x,y)^{n^2+3} \dots \\ \dots v_1(x,y)^{{\varepsilon}_{h-1}} v_2(x,y)^{n^2+h-1} v_1(x,y)^{{\varepsilon}_{h}} v_2(x,y)^{n^2+h} , \end{multline} where $h \equiv 0 \pmod{20}$, \begin{gather*} {\varepsilon}_{10k+1}= {\varepsilon}_{10k+2}= {\varepsilon}_{10k+3}= {\varepsilon}_{10k+5}= {\varepsilon}_{10k+6}= 1, \\ {\varepsilon}_{10k+4}= {\varepsilon}_{10k+7}= {\varepsilon}_{10k+8}= {\varepsilon}_{10k+9}= {\varepsilon}_{10k+10}= -1 \end{gather*} for $k= 0,1,\dots ,h/10-1$, and $h, d, n$ are sufficiently large positive integers (with $n \gg d \gg h \gg 1$). Note that the sums of exponents on $x, y$ in both (\ref{w1}) and (\ref{w2}) are zeros.
Now we can state our main result (note that this result was announced in \cite[Theorem~5]{IO91}; for the sake of simplicity of proofs we changed the identities $w_1(x,y) \equiv 1$, $w_2(x,y) \equiv 1$ presented in \cite{IO91}).
\begin{Th} Let $\mathfrak{M}$ be the variety of groups defined by identities $w_1(x, y) \equiv 1$, $w_2(x, y) \equiv 1$, where the words $w_1(x, y)$, $w_2(x, y)$ are given by formulas $(\ref{w1})$, $(\ref{w2})$. Then any free group of rank $m > 1$ in the variety $\mathfrak{M}$ is not hopfian. \end{Th}
To prove this Theorem, in Sect.~2, we inductively construct a group presentation by means of generators and defining relations. In Sect.~3, we apply the geometric machinery of graded diagrams, developed by Ol'shanskii \cite{O85}, \cite{O89}, to study subpresentations of this presentation and to prove a number of technical lemmas. In particular, we use the notation and terminology of \cite{O89} and all notions that are not defined in this paper can be found in \cite{O89}. In Sect.~4, we show that the group presentation constructed in Sect.~2 defines a free group of rank $m > 1$ in the variety $\mathfrak{M}$ and this group is non-hopfian.
\section{Inductive Construction}
As in \cite{O89}, we will use numerical parameters $$ \alpha \succ \beta \succ \gamma \succ \delta \succ {\varepsilon} \succ \zeta \succ \eta \succ \iota $$ and $h = \delta^{-1}$, $d = \eta^{-1}$, $n = \iota^{-1}$ ($h, d, n$ were already used in (\ref{w1})--(\ref{w2})) and employ the least parameter principle (LPP) (according to LPP a small positive value for, say, $\zeta$ is chosen to satisfy all inequalities whose smallest (in terms of the relation $\succ$) parameter is $\zeta$).
Let $\mathcal{A} = \{ a_1, \dots, a_m \}$ be an alphabet, $m > 1$, and $F(\mathcal{A})$ be the free group in $\mathcal{A}$. Elements of $F(\mathcal{A})$ are referred to as {\em words in $\mathcal{A}^{\pm 1} = \mathcal{A} \cup \mathcal{A}^{-1}$} or just words. Denote $G(0) = F(\mathcal{A})$ and let the set $\R_0$ be empty. To define the group $G(i)$ by induction on $i \ge 1$, assume that the group $G(i-1)$ is already constructed by its presentation $$
G(i-1) = {\langle} \mathcal{A} \; \| \; R=1, R \in \R_{i-1} {\rangle} . $$ Let $\mathcal{X}_i$ be a set of words (in $\mathcal{A}^{\pm 1}$) of length $i$, called {\em periods of rank $i$}, which is maximal with respect to the following two properties: \begin{enumerate} \item[(A1)] If $A \in \mathcal{X}_i$ then $A$ (that is, the image of $A$
in $G(i-1)$) is not conjugate in $G(i-1)$ to a power of a word of length $< |A| = i$.
\item[(A2)] If $A$, $B$ are distinct elements of $\mathcal{X}_i$ then $A$ is not conjugate in $G(i-1)$ to $B$ or $B^{-1}$. \end{enumerate}
If the images of two words $X$, $Y$ are equal in the group $G(i-1)$, $i \ge 1$, then we will say that $X$ is {\em equal in rank} $i-1$ to $Y$ and will write $ X \overset {i-1} = Y$. Analogously, we will say that two words $X$, $Y$ are {\em conjugate in rank} $i-1$ if their images are conjugate in the group $G(i-1)$. As in \cite{O89}, a word $A$ is called {\em simple} in rank $i-1$, $i \ge 1$, if $A$ is conjugate in rank
$i-1$ neither to a power $B^k$, where $|B| <|A|$ nor to a power of period of some rank $\le i-1$. We will also say that two pairs $(X_1, X_2)$, $(Y_1, Y_2)$ of words are conjugate in rank $i-1$, $i \ge 1$, if there is a word $W$ such that $X_1 \overset {i-1} = W Y_1 W^{-1}$ and $X_2 \overset {i-1} = W Y_2 W^{-1}$.
Consider the set of all possible pairs $(X, Y)$ of words in $\mathcal{A}^{\pm 1}$ and pick $z^* \in \{1, 2\}$. This set is partitioned by equivalence $z^*$-classes $\mathcal{C}_\ell(z^*)$, $\ell =1,2, \dots$, of the equivalence relation $\overset { z^*} \sim$ defined as follows: $(X_1, Y_1) \overset { z^*} \sim (X_2, Y_2)$ if and only if the pairs $( v_{z^*}(X_1, Y_1), w_{z^*}(X_1, Y_1) )$ and $( v_{z^*}(X_2, Y_2), w_{z^*}(X_2, Y_2))$ are conjugate in rank $i-1$. It is convenient to enumerate (in some way) $$ \mathcal{C}_{A^f, 1}(z^*) , \mathcal{C}_{A^f, 2}(z^*), \dots $$ all $z^*$-classes of pairs $(X, Y)$ such that $w_{z^*}(X, Y) \overset {i-1} \neq 1$ and $v_{z^*}(X, Y)$ is conjugate in rank $i-1$ to some power $A^f$, where $A \in \mathcal{X}_i$ and $f$ are fixed.
It follows from definitions that every class $\mathcal{C}_{A^f, j}(z^*)$ contains a pair $$ (X_{A^f, j, z^*}, \bar Y_{A^f, j, z^*} ) $$ with the following properties. The word $X_{A^f, j, z^*}$ is graphically (that is, letter-by-letter) equal to a power of $ B_{A^f, j, z^*}$, where $ B_{A^f, j, z^*}$ is simple in rank $i-1$ or a period of rank $\le i-1$; $\bar Y_{A^f, j, z^*} \equiv Z_{A^f, j, z^*} Y_{A^f, j, z^*} Z_{A^f, j, z^*}^{-1}$, where the symbol \ '$\equiv$' \ means the graphical equality, $Y_{A^f, j, z^*}$ is graphically equal to a power of $C_{A^f, j, z^*}$, where $C_{A^f, j, z^*}$ is simple in rank $i-1$ or a period of rank $\le i-1$. We can also assume that if $D_1 \in \{ A, B_{A^f, j, z^*}, C_{A^f, j, z^*} \}$ is conjugate in rank $i-1$ to $D_2^{\pm 1}$, where $D_2 \in \{ A, B_{A^f, j, z^*}, C_{A^f, j, z^*} \}$, then $D_1 \equiv D_2$. Finally, the word $Z_{A^f, j, z^*}$ is picked for fixed $X_{A^f, j, z^*}$,
$Y_{A^f, j, z^*}$ so that the length $|Z_{A^f, j, z^*}|$ is minimal (and the pair $(X_{A^f, j, z^*}, Z_{A^f, j, z^*} Y_{A^f, j, z^*} Z_{A^f, j, z^*}^{-1})$ belongs to $\mathcal{C}_{A^f, j}(z^*)$). Similar to \cite{O85}, \cite{O89}, \cite{S94}, the triple $(X_{A^f, j, z^*}, Y_{A^f, j, z^*}, Z_{A^f, j, z^*} )$ is called an $(A^f, j, z^*)$-{\em triple} corresponding to the class $\mathcal{C}_{A^f, j}(z^*)$ (in rank $i-1$).
Now for every class $\mathcal{C}_{A^f, j}(z^*)$ we pick a corresponding $(A^f, j, z^*)$-{triple} $$ (X_{A^f, j, z^*}, Y_{A^f, j, z^*}, Z_{A^f, j, z^*} ) $$ in rank $i-1$ and construct a defining word $R_{A^f, j, z^*}$ of rank $i$ as follows. Pick a word $W_{A^f, j, z^*}$ of minimal length so that $$ v_{z^*}( X_{A^f, j, z^*}, \bar Y_{A^f, j, z^*} ) \overset {i-1} = W_{A^f, j, z^*} A^{f} W_{A^f, j, z^*}^{-1} . $$ Let $T_{A^f, j, z^*}$, $U_{A^f, j, z^*}$ be words of minimal length such that \begin{gather*} T_{A^f, j, z^*} \overset {i-1} = W_{A^f, j, z^*}^{-1} v_{z^*-1}( X_{A^f, j, z^*}, \bar Y_{A^f, j, z^*} ) W_{A^f, j, z^*} \\ U_{A^f, j, z^*} \overset {i-1} = W_{A^f, j, z^*}^{-1} \bar Y_{A^f, j, z^*} W_{A^f, j, z^*} \end{gather*} (recall that $v_0(x,y) = x$, see (\ref{v1v2})).
If $z^* =1$ then, in accordance with (\ref{w1}), we set \begin{multline} \label{R1} R_{A^f, j, 1} = T_{A^f, j, 1}^{{\varepsilon}_1} A^{ n f } T_{A^f, j, 1}^{{\varepsilon}_2} A^{ (n+2) f } \dots T_{A^f, j, 1}^{{\varepsilon}_{h/2-2}} A^{ (n+h-6) f } T_{A^f, j, 1}^{{\varepsilon}_{h/2-1}} A^{ (n+h-4) f } \\ T_{A^f, j, 1}^{{\varepsilon}_{h/2}} A^{ (n+h-2 + h/2) f } T_{A^f, j, 1}^{{\varepsilon}_1} A^{ -(n+1) f } T_{A^f, j, 1}^{{\varepsilon}_2} A^{ -(n+3) f } \dots \\ \dots T_{A^f, j, 1}^{{\varepsilon}_{h/2-1}} A^{ -(n+h-3) f } T_{A^f, j, 1}^{{\varepsilon}_{h/2}} A^{ -(n+h-1) f } , \end{multline} where ${\varepsilon}_1, \dots, {\varepsilon}_{h/2}$, $h, n$ are defined as in (\ref{w1}).
If $z^* =2$ then, in accordance with (\ref{w2}), we put \begin{multline} \label{R2} R_{A^f, j, 2} = U_{A^f, j, 2} A^{ (n^2+1) f } T_{A^f, j, 2}^{{\varepsilon}_2} A^{ (n^2+2) f } T_{A^f, j, 2}^{{\varepsilon}_3} A^{ (n^2+3) f } \dots \\ \dots T_{A^f, j, 2}^{{\varepsilon}_{h-1}} A^{ (n^2+h-1) f } T_{A^f, j, 2}^{{\varepsilon}_{h}} A^{ (n^2+h) f } , \end{multline} where ${\varepsilon}_2, \dots, {\varepsilon}_h$, $h, n$ are defined as in (\ref{w2}).
It follows from definitions that the word $R_{A^f, j, z^*}$ is conjugate in rank $i-1$ (by the word $W_{A^f, j, z^*}^{-1}$) to $w_{z^*}( X_{A^f, j, z^*}, \bar Y_{A^f, j, z^*}) \overset {i-1} \neq 1$.
The set $\Ss_i$ of defining words of rank $i$ consists of all possible words $R_{A^f, j, z^*}$ given by (\ref{R1})--(\ref{R2}) (over all equivalence classes $\mathcal{C}_{A^f, j}(z^*)$, $A \in \mathcal{X}_i$). Finally, we put $\R_i = \R_{i-1} \cup \Ss_i$ and set \begin{gather}
\label{Gi} G(i) = {\langle} \mathcal{A} \; \| \; R=1, R \in \R_i {\rangle} . \end{gather}
The inductive definition of groups $G(i)$, $i \ge 0$, is
now complete and we can consider the limit group $G(\infty)$ given by
defining words of all ranks $j =1,2, \dots$ \begin{gather}
\label{G8} G(\infty) = {\langle} \mathcal{A} \; \| \; R=1, R \in \cup_{j=0}^\infty \R_j {\rangle} . \end{gather}
\section{Several Lemmas}
We will prove (Lemma \ref{L11}) that the group $G(\infty)$, defined by \eqref{G8}, is the free group of the variety $\mathfrak{M}$ in the alphabet $\mathcal{A}$, that is, $G(\infty)$ is naturally isomorphic to the quotient $F(\mathcal{A}) / W_{1,2}(F(\mathcal{A}))$, where $W_{1,2}(F(\mathcal{A}))$ is the verbal subgroup of $F(\mathcal{A})$, defined by the set $W_{1,2} = \{ w_1(x,y), w_2(x,y) \}$. We will also show that $G(\infty)$ is not hopfian (Lemma \ref{L12}). But first we need to study the presentation (\ref{Gi}) of $G(i)$ and establish a number of technical lemmas.
As in Sects. 29--30 \cite{O89}, the following Lemmas \ref{L1}--\ref{L10} are proved by induction on $i \ge 0$ (whose base for $i =0$ is trivial). \begin{lemma}
\label{L1} The presentation $(\ref{Gi})$ of $G(i)$ satisfies the condition $R$ of \cite[Sect. 25]{O89}. \end{lemma}
\begin{proof} This proof is quite similar to the proof of Lemma 29.4 \cite{O89}. Inductive references to Lemmas 30.3, 30.4, 30.5 \cite{O89} (in rank $i-1$) are replaced by references to Lemma \ref{L6}. Note that, by Lemma \ref{L6}, we have that $$
n^2 > 100\zeta^{-1}(n+h) \ge |f|(n +h) $$ (LPP: $\delta = h^{-1} \succ \zeta \succ \iota = n^{-1}$) which implies that, repeating the arguments of Lemma 29.3 \cite{O89}, we can conclude that the defining relators $R$, $R'$ correspond to the same value of $z^*$. Therefore, Lemma \ref{L10} enables us to finish the proof of the analog of Lemma 29.3 as in \cite{O89} (note that we need Lemma \ref{L10} only when $z^* =2$). \end{proof}
\begin{lemma}
\label{L2A} Suppose that $X_1$, $X_2$ are some words and $k \ne 0$ is an integer. Then
$(a)$ If $[ X_1^k, X_2^k ] \overset i = 1$, then $[ X_1, X_2 ] \overset i = 1$;
$(b)$ If $X_1^{k} \overset i = X_2^k$, then $X_1 \overset i = X_2$. In particular, the group $G(i)$, defined by $(\ref{Gi})$, is torsion-free. \end{lemma}
\begin{proof} (a) It follows from definitions and Lemma 25.2 \cite{O89} that the group $G(i)$, defined by (\ref{Gi}), is torsion-free. Hence, we can apply Lemma 25.12 \cite{O89} to equality $[ X_1^k, X_2^k ] \overset i = 1$ and obtain that $[ X_1, X_2 ]\overset i = 1$, as required.
(b) The equality $X_1^{k} \overset i = X_2^k$ implies that $[ X_1^k, X_2^k ] \overset i = 1$ and, by part (a), we have $[ X_1, X_2 ]\overset i = 1$. Hence, $(X_1 X_2^{-1})^{k} \overset i = 1$ and, as above, it follows from definitions and Lemma 25.2 \cite{O89} that $X_1 X_2^{-1} \overset i = 1$. Lemma \ref{L2A} is proved. \end{proof}
Recall that a subgroup $H$ of a group $G$ is called {\em antinormal} if for every $g \in G$ the inequality $g H g ^{-1} \cap H \neq \{ 1\}$ implies that $g \in H$.
\begin{lemma}
\label{L2B} $(a)$ Every word is conjugate in rank $i$ to a power of either a simple in rank $i$ word or a period of rank $\le i$.
$(b)$ Suppose that each of $A$, $B$ is either a simple
in rank $i$ word or a period of rank $\le i$ and $A^k$ is conjugate
in rank $i$ to $B^\ell$, $\ell \neq 0$. Then $A$ is conjugate in rank $i$
to $B$ or to $B^{-1}$.
$(c)$ Let $A$ be a simple in rank $i$ word or a period of rank $\le i$.
Then the cyclic subgroup ${\langle} A{\rangle}$, generated by $A$, of the group
$G(i)$, defined by $(\ref{Gi})$, is antinormal. \end{lemma}
\begin{proof} Part (a) follows from definitions (see also Lemma 18.1 \cite{O89}). Since the group $G(i)$ is torsion-free by Lemma \ref{L2A}, we can argue as in the proof of Lemma 25.17 \cite{O89} (see also the proof of Theorem 19.4 \cite{O89}) to prove part (b) and obtain that an equality of the form $ZA^k Z^{-1} \overset i = A^\ell$, $\ell \neq 0$, implies that $Z \in {\langle} A {\rangle}$, as required in part (c). Lemma \ref{L2B} is proved. \end{proof}
In addition to $A$-, $B$-, $\dots$, $H$-maps which are introduced and investigated in \cite{O89}, we will need $I$-maps (cf. \cite{S94}) defined as follows. A $B$-map $\Delta$ is called an {\em $I$-map} if following properties (I1)--(I5) hold.
\begin{enumerate} \item[(I1)] $\Delta$ is a map on a sphere punctured at least once and at most thrice.
\item[(I2)] The cyclic sections of the boundary $\partial \Delta$ of $\Delta$ are products of sections (some of which or all can be cyclic) of two types: long sections and short sections.
\item[(I3)] If $s$ is a long section of $\partial \Delta$ then $s$ is smooth of rank
$r(s)$ and $|s| > 10 \zeta^{-1} r(s)$.
\item[(I4)] If $t$ is a short section of $\partial \Delta$ then $|t| < \zeta |s_0|$, where $s_0$ is a long section of minimal length.
\item[(I5)] If $L(\partial\Delta)$ and $S(\partial\Delta)$ are the numbers of long and short sections of $\partial \Delta$, respectively, then $S(\partial\Delta) \le 2 L(\partial\Delta)$ and $1 \le L(\partial\Delta) \le 10$. \end{enumerate}
\begin{lemma} \label{L2I} Suppose that $\Delta$ is an $I$-map. Then there is a system of pairwise disjoint regular contiguity submaps of long sections to long sections in $\Delta$ such that no two distinct contiguity submaps of a long section $s_1$ to a long section $s_2$ are contained in any larger contiguity submap of $s_1$ to $s_2$ and the sum of contiguity arcs of contiguity submaps of the system is greater than $(1-\alpha^{1/4})L_s$, where $L_s$ is the sum of lengths of all long sections of $\partial \Delta$. \end{lemma} \begin{proof} Without loss of generality, we can assume that every short section $t$ of $\partial \Delta$ is geodesic in $\Delta$ (and that if $t$ is cyclic then $t$ is cyclically geodesic; note that we can always replace $t$ by a homotopic to $t$ in $\Delta$ geodesic path).
This proof is analogous to the proof of Lemma 24.6 \cite{O89} (see also Lemmas 23.15, 24.2 \cite{O89} on $C$- and $D$-maps). Repeating arguments of the proof of Lemma 24.6 \cite{O89}, we can establish similar estimates for an $I$-map $\Delta$. Note that we need to make straightforward corrections of the number of distinguished contiguity submaps between $p$ and $q$, where $p$, $q$ are sections of $\partial \Delta$ or $\partial \Pi$ and $\Pi$ is a 2-cell of $\Delta$, and of the number of distinguished contiguity submaps between sections of $\partial \Delta$. As in the proof of Lemma 24.6 \cite{O89}, we obtain the estimate $M < \alpha \nu(\Delta)$, where $M$ is the sum of weights of all inner edges of $\Delta$ and $\nu(\Delta)$ is the total weight of $\Delta$.
Now we can argue as in the proof of Lemma 23.15 \cite{O89} to derive that the sum of lengths of outer arcs of long sections of $\partial \Delta$ is greater than $(1-\alpha^{1/4})L_s$. \end{proof}
\begin{lemma}
\label{L2C} Suppose that each of $A_1$, $A_2$, $B$ is a simple in rank $i$ word or a period of rank $\le i$, $[ A_1^{k_1} , Z A_2^{k_2} Z^{-1} ] \overset i \neq 1$ for some word $Z$, and $B^\ell$ is conjugate in rank $i$
either to $[ A_1^{d k_1} , Z A_2^{d k_2} Z^{-1} ]$ or to $ A_1^{dk_1} Z A_2^{dk_2} Z^{-1}$. Then $0 < |\ell | \le 100 \zeta^{-1}$ and either $\max (
|A_1^{d k_1}|, |A_2^{d k_2}| ) \le \zeta^{-1} |B^\ell |$ if $B^\ell$ is conjugate in rank $i$ to $[ A_1^{dk_1} , Z A_2^{dk_2} Z^{-1} ]$ or $\max (
|A_1^{dk_1}|, |A_2^{dk_2}| ) \le \zeta^{-2} |B^\ell |$ if $B^\ell$ is conjugate in rank $i$ to $ A_1^{dk_1} Z A_2^{dk_2} Z^{-1}$. \end{lemma}
\begin{proof} Without loss of generality (see also Lemma \ref{L2B}), we can assume that if $B_1 \in \{ A_1^{\pm 1}, A_2^{\pm 1}, B^{\pm 1} \}$ is conjugate in rank $i$ to $B_2 \in \{ A_1^{\pm 1}, A_2^{\pm 1}, B^{\pm 1} \}$, then $B_1 \equiv B_2$.
If $\ell = 0$, that is, either $[ A_1^{dk_1} , Z A_2^{dk_2} Z^{-1} ] \overset i = 1$ or $ A_1^{dk_1} Z A_2^{dk_2} Z^{-1} \overset i = 1$ then, by Lemma \ref{L2A}, we have that either $[ A_1^{k_1} , Z A_2^{k_2} Z^{-1} ] \overset i = 1$ or $ A_1^{k_1} Z A_2^{k_2} Z^{-1} \overset i = 1$, contrary to lemma's hypothesis $[ A_1^{k_1} , Z A_2^{k_2} Z^{-1} ] \overset i \neq 1$. Hence, $\ell \ne 0$.
First assume that \begin{equation} \label{L2C:1}
[ A_1^{dk_1} , Z A_2^{dk_2} Z^{-1} ] \overset i = W_B B^\ell W_B^{-1} \end{equation} for some word $W_B$. Then there is a reduced diagram $\Delta_1$ of rank $i$ on a thrice punctured sphere the labels of 3 cyclic sections of whose boundary $\partial \Delta_1$ are $A_1^{d k_1}$, $A_1^{-d k_1}$, $B^\ell$ ($\Delta_1$ can be constructed from a simply connected diagram $\Delta_0$ of rank $i$ for equality (\ref{L2C:1})
by identifying sections of $\partial \Delta_0$ labelled by $Z A_2^{dk_2} Z^{-1}$ and $Z A_2^{-dk_2} Z^{-1}$, $W_B$ and $W_B^{-1}$). If $|\ell | > 100 \zeta^{-1}$, then $\Delta_1$ is a $G$-map (see \cite[Sect. 24.2]{O89}) and, as in the proof of 25.19 \cite{O89}, it follows from Lemma 24.8 \cite{O89} that $B^\ell \overset i = 1$, contrary to Lemma \ref{L2A} and $\ell \ne 0$. Hence, $|\ell | \le 100 \zeta^{-1}$, as desired.
Suppose that $|B^\ell | < \zeta | A_1^{d k_1} |$. Then $\Delta_1$ is an $E$-map (see \cite[Sect. 24.2]{O89}) and it follows from Lemmas 24.6, 25.10 \cite{O89} that cyclic sections of $\partial \Delta_1$, labelled by $A_1^{d k_1}$, $A_1^{-d k_1}$, are $A_1$-compatible. Now it is easy to see that $B^\ell \overset i = 1$, whence $\ell = 0$ by Lemma \ref{L2A}. This contradiction to
$\ell \ne 0$ shows that $| A_1^{d k_1} | \le \zeta^{-1} |B^\ell|$, as desired. Analogously, $| A_2^{d k_2} | \le \zeta^{-1} |B^\ell|$ (using equality (\ref{L2C:1}), we can construct a similar diagram $\Delta_2$ the labels of 3 cyclic sections of whose boundary $\partial \Delta_2$ are $A_2^{d k_2}$, $A_2^{-d k_2}$, $B^\ell$, and then argue as before).
Now assume that \begin{equation} \label{L2C:2}
A_1^{dk_1} Z A_2^{dk_2} Z^{-1} \overset i = W_B B^\ell W_B^{-1} \end{equation} for some word $W_B$. Then there is a reduced diagram $\Delta$ of rank $i$ on a thrice punctured sphere the labels of 3 cyclic sections of whose boundary $\partial \Delta$ are $A_1^{d k_1}$, $A_2^{d k_2}$, $B^{-\ell}$ ($\Delta$ can be constructed from a simply connected diagram $\Delta_0$ of rank $i$ for equality (\ref{L2C:2}) by identifying the sections of $\partial \Delta_0$ labelled by $Z$ and $Z^{-1}$, $W_B$
and $W_B^{-1}$). If $|\ell | > 100 \zeta^{-1}$, then $\Delta$ is a $G$-map and, using Lemmas 24.8, 25.10 \cite{O89}, we can conclude that one of cyclic sections of $\partial \Delta$ is compatible with another cyclic section of $\partial \Delta$. Then it follows from Lemmas 3 and 24.9 \cite{O89} that $[ A_1^{dk_1} , Z A_2^{dk_2} Z^{-1} ] \overset i = 1$ and so, by Lemma \ref{L2A}, $[ A_1^{k_1}
, Z A_2^{k_2} Z^{-1} ] \overset i = 1$, contrary to lemma's hypothesis. This contradiction shows that $|\ell | \le 100 \zeta^{-1}$.
Suppose that $|B^\ell | < \zeta \min ( | A_1^{dk_1}|, |A_2^{dk_2}|)$. Then $\Delta$ is an $E$-map (see \cite[Sect. 24.2]{O89}) and, it follows from Lemmas 24.6, 25.10 \cite{O89} that two distinct cyclic sections of $\partial \Delta$ are compatible. As above, this implies that $[ A_1^{k_1} , Z A_2^{k_2} Z^{-1} ] \overset i = 1$, contrary to lemma's hypothesis. Hence, it is shown that
$|B^\ell | \ge \zeta \min ( | A_1^{dk_1}|, |A_2^{dk_2}|)$. For definiteness, let $|A_1^{dk_1}| \le |A_2^{dk_2}| $. Then \begin{equation}
\label{L2C:3} \zeta |A_1^{dk_1}| \le |B^\ell | . \end{equation}
If $ |A_2^{dk_2}| \le \zeta^{-2} |B^\ell | $ then our proof is obviously finished. So we may assume that \begin{equation} \label{L2C:4}
|A_2^{dk_2}| > \zeta^{-2} |B^\ell | . \end{equation}
It follows from inequalities (\ref{L2C:3})--(\ref{L2C:4}) that $|A_1^{dk_1}|,
|B^\ell | <\zeta |A_2^{dk_2}|$ and we can see that $\Delta$ is an $I$-map.
This, however, is impossible by Lemmas \ref{L2I} and 25.10 \cite{O89}.
This contradiction completes the proof of Lemma \ref{L2C}. \end{proof}
Now suppose that $X, Y$ are some words and \begin{equation} \label{XY1}
[X, Y] \overset {i} \neq 1 . \end{equation}
Conjugating the pair $(X, Y)$ in rank $i$ if necessary, we can assume that $X \equiv B^{k_B}$, $Y \equiv Z C^{k_C} Z^{-1}$, where each of $B, C$ is either simple in rank $i$ or a period of some rank $\le i$ and, when $B^{k_B}$, $C^{k_C}$ are fixed, the word $Z$ is picked to have minimal length. Furthermore, consider the following equalities \begin{gather*} X^dY^d \overset {i} = W_D D^{k_D} W_D^{-1} , \qquad (X^dY^d )^d X^d \overset {i} = W_E E^{k_E} W_E^{-1} , \\ [((X^dY^d )^d X^d)^d, X^d] \overset {i} = W_F F^{k_F} W_F^{-1} , \quad [((X^dY^d )^d X^d)^d, X^d]^dY \overset {i} = W_G G^{k_G} W_G^{-1} , \\ [([((X^dY^d )^d X^d)^d, X^d]^dY)^d,X^d] \overset {i} = W_H H^{k_H} W_H^{-1} , \end{gather*} where each of $ D, E, F, G, H$ is either simple in rank $i$ or a period of some rank $\le i$ and the conjugating words $W_D, W_E, W_F, W_G, W_H$ are picked (when $D, E, F, G, H$ are fixed) to have minimal length.
Without loss of generality, we can also assume that if $A_1 \in \{ B, C, D, E, F, G, H\}$ is conjugate in rank $i$ to $A_2^{\pm 1}$, where $A_2 \in \{ B, C, D, E, F, G, H\}$, then $A_1 \equiv A_2$.
\begin{lemma}
\label{L2}
In the foregoing notation, the following estimates hold \begin{gather}
0 < | k_D | \le 100 \zeta^{-1} , \quad \max(| B^{d k_B} |, | C^{d k_C} |) \le
\zeta^{-2} | D^{k_D} | , \label{L2:1} \\
|Z| < 3 \zeta^{-2} | D^{k_D} | , \quad
|W_D| < 5 \zeta^{-2} | D^{k_D} |, \label{L2:2} \\
0 < | k_E | \le 100 \zeta^{-1} , \quad
| D^{d k_D} | \le \zeta^{-2} | E^{k_E} | , \label{L2:3} \\
|W_E| < 2\zeta^{-2}| E^{k_E} |, \label{L2:4} \\
0 < | k_F | \le 100 \zeta^{-1} , \quad
| E^{d k_E} | \le \zeta^{-1} | F^{k_F} | , \label{L2:5} \\
|W_F| < 2 \zeta^{-1} | F^{k_F} |. \label{L2:6} \end{gather} \end{lemma}
\begin{proof} If $k_D = 0$, that is, $X^dY^d \overset {i} = 1$, then, by Lemma \ref{L2A}, $XY \overset {i} = 1$ and, therefore, $[X, Y ] \overset {i} = 1$, contrary to inequality (\ref{XY1}). Hence $k_D \neq 0$.
In view of inequality (\ref{XY1}), we can apply Lemma \ref{L2C} to the pair
$X \equiv B^{k_B}$, $Y \equiv Z C^{k_C} Z^{-1}$ which yields that $| k_D |
\le 100 \zeta^{-1}$ and $| B^{d k_B} |, | C^{d k_C} |\le \zeta^{-2} |
D^{k_D} |$. Inequalities (\ref{L2:1}) are proved.
In view of equality $B^{d k_B}Z C^{d k_C} Z^{-1} \overset {i} = W_D D^{k_D} W_D^{-1}$, there is a reduced diagram $\Delta$ of rank $i$ on a thrice punctured sphere the labels of three cyclic sections of whose boundary $\partial \Delta$ are $B^{d k_B}$, $C^{d k_C}$, $D^{-k_D}$. It follows from Lemmas 22.2, 24.9 \cite{O89} that $$
|Z| < (1+4\gamma) ( | B^{d k_B} |+ | C^{d k_C} | + | D^{k_D} | ) . $$ In view of inequalities (\ref{L2:1}), we have $$
|Z| < (1+4\gamma) ( 2\zeta^{-2} +1 ) | D^{k_D} | < 3 \zeta^{-2} |
D^{k_D}| , $$ as claimed in (\ref{L2:2}).
By estimates (\ref{L2:1}) and already proven inequality $|Z| < 3 \zeta^{-2}
|D^{k_D} |$, we have $$
|X^dY^d| = | B^{d k_B} |+ | C^{d k_C} | + 2 | Z | < 8\zeta^{-2} | D^{k_D} | . $$ Hence, it follows from Lemmas \ref{L1} and 22.1 \cite{O89} that $$
|W_D| < (\gamma + \tfrac 1 2)( | X^dY^d | + | D^{k_D} | ) < 5 \zeta^{-2} |
D^{k_D} | $$ and inequalities (\ref{L2:2}) are proved.
Now assume that $k_E=0$. Then $(X^dY^d)^dX^d \overset {i} = 1$ which implies that $$ [(X^dY^d)^d,X^d]\overset {i} = 1 . $$ Hence, by Lemma \ref{L2A}, we have $[X, Y ] \overset {i} = 1$, contrary to inequality (\ref{XY1}).
Consider the equality \begin{gather} \label{WDB} W_D D^{dk_D} W_D^{-1}B^{d k_B} \overset {i} = W_E E^{k_E} W_E^{-1} . \end{gather}
In view of (\ref{L2:1}), $$
| B|, |C| \le d^{-1} \zeta^{-2}| D^{k_D} | \le d^{-1} \zeta^{-2} \cdot 100
\zeta^{-1}| D | $$
and so $| B|, |C| < | D |$ (LPP: $\zeta \succ \eta = d^{-1}$). Hence, $ [W_D D^{dk_D} W_D^{-1}, B^{d k_B} ] \overset {i} \neq 1$ (otherwise, we would have a contradiction to Lemma \ref{L2B}). This last inequality enables us to apply Lemma \ref{L2C} to equality (\ref{WDB}) and conclude that $| k_E | \le 100
\zeta^{-1}$, $| D^{d k_D} | \le \zeta^{-2} | E^{k_E} |$. Inequalities (\ref{L2:3}) are proved.
By (\ref{L2:1}), (\ref{L2:2}), (\ref{L2:3}), we obtain $$
|W_D D^{dk_D} W_D^{-1}B^{d k_B}| < (1+11\zeta^{-2}d^{-1})| D^{dk_D} |
<2\zeta^{-2}| E^{k_E} | $$ (LPP: $\zeta \succ \eta = d^{-1}$). Therefore, it follows from Lemmas \ref{L1} and 22.1 \cite{O89} that $$
|W_E| < (\gamma + \tfrac 1 2)( | W_D D^{dk_D} W_D^{-1}B^{d k_B} | + |
E^{k_E} | )
< 2\zeta^{-2}| E^{k_E} | $$ as claimed in (\ref{L2:4}).
Next assume that $k_F=0$. Then $[((X^dY^d )^d X^d)^d, X^d]\overset {i} = 1$. Hence, by Lemma \ref{L2A}, we obtain $[X, Y ] \overset {i} = 1$, which contradicts inequality (\ref{XY1}).
Now consider the equality $[ W_E E^{dk_E} W_E^{-1} , B^{d k_B}] \overset {i}
= W_F F^{k_F} W_F^{-1}$. By Lemma \ref{L2C}, $| k_F | \le 100 \zeta^{-1}$, $|
E^{d k_E} | \le \zeta^{-1} | F^{k_F} |$, and estimates (\ref{L2:5}) are proved.
In view of equality $[ W_E E^{dk_E} W_E^{-1} , B^{d k_B}] \overset {i} = W_F F^{k_F} W_F^{-1}$, it follows from Lemma 22.1 \cite{O89} that $$
| W_F | < ( \gamma + \tfrac 12 )\cdot 2( 2| W_E | +
| E^{d k_E} | + | B^{d k_B } | + \tfrac 12 | F^{k_F} |) . $$ Hence, by estimates (\ref{L2:1}), (\ref{L2:3}), (\ref{L2:4}), (\ref{L2:5}), we get
\begin{multline*} | W_F | < (1 + 2 \gamma ) ( (2 \zeta^{-2} d^{-1} +1
)| E^{d k_E} | + \zeta^{-4} d^{-2}| E^{d k_E} | + \tfrac 12 | F^{k_F} |) < \\ < (1 + 2 \gamma ) ( ( (2 \zeta^{-2} d^{-1} +1 ) + \zeta^{-4}
d^{-2} ) \zeta^{-1} + \tfrac 12) | F^{k_F} |) < 2
\zeta^{-1}| F^{k_F} | \end{multline*} (LPP: $\gamma \succ \zeta \succ \eta = d^{-1}$), as required. Lemma \ref{L2} is proved. \end{proof}
\begin{lemma}
\label{L4}
In the foregoing notation, the following inequalities hold \begin{gather}
0 < | k_G | \le 10 \zeta^{-1} , \label{L4:1} \\
| F^{d k_F} | \le \zeta^{-1} | G^{k_G} | , \label{L4:2} \\
|W_G| < \zeta^{-1} | G^{k_G} | . \label{L4:3} \end{gather} \end{lemma}
\begin{proof} If $k_G =0$ then $[((X^dY^d )^d X^d)^d, X^d]^dY \overset i = 1$ which implies that $F^{dk_F}$ is conjugate in rank $i$ to $C^{-k_C}$. It follows from definitions and Lemma \ref{L2B} that $F\equiv C$. However, it follows from Lemma \ref{L2} that \begin{equation}
|C|\le \zeta^{-5}d^{-3}|F^{k_F}|\le 100\zeta^{-6}d^{-3}|F|<|F| \label{L4:4.0} \end{equation} (LPP: $\zeta \succ \eta = d^{-1}$). Hence $k_G \neq 0$.
By definitions, we have \begin{equation} W_F F^{d k_F} W_F^{-1} Z C^{ k_C} Z^{-1} \overset {i} = W_G G^{k_G} W_G^{-1} . \label{L4:4} \end{equation} By Lemma \ref{L2}, \begin{multline}
| W_F^{-1} Z C^{ k_C} Z^{-1} W_F | <
(4\zeta^{-1} + 6 \zeta^{-5}d^{-2} + \zeta^{-5}d^{-3}) | F^{k_F} | < \\
< 5 \zeta^{-1} d^{-1} | F^{d k_F} | < \zeta | F^{d k_F} | \label{L4:5} \end{multline} (LPP: $\zeta \succ \eta = d^{-1}$).
If $| G^{k_G} | < \zeta | F^{d k_F} | $ then a reduced annular diagram of rank $i$ for conjugacy of words $ F^{d k_F} W_F^{-1} Z C^{ k_C} Z^{-1} W_F$ \ and \ $G^{k_G}$ (see (\ref{L4:4})) is an $F$-map (see Sect. 24.2 \cite{O89}) whose existence contradicts Lemma \ref{L1} and Lemmas 24.7, 25.10 \cite{O89}. Therefore,
$ | G^{k_G} | \ge \zeta | F^{d k_F} | $
and (\ref{L4:2}) is proved.
In view of equality (\ref{L4:4}), there is a reduced diagram $\Delta$ of rank $i$ on a thrice punctured sphere the labels of whose cyclic sections are $F^{d k_F}$, $C^{k_C}$, $G^{-k_G}$. It follows from Lemma \ref{L2} that $$
| C^{k_C} | < \zeta^{-5}d^{-4}|F^{dk_F}| < \zeta^2 | F^{d k_F} | $$
(LPP: $\zeta \succ \eta = d^{-1}$). Hence, by (\ref{L4:2}), $| C^{k_C} | <
\zeta \min( | F^{d k_F} | , | G^{k_G}| )$. If $| k_G | > 10 \zeta^{-1}$, then $\Delta$ is an $E$-map (see Sect. 24.2 \cite{O89}) and we can argue as in the proof of Lemma 25.19 \cite{O89} to show that $F \equiv G$ and the cyclic sections of $\Delta$ labelled by $F^{d k_F}$, $G^{-k_E}$ are $F$-compatible. Then
$C^{k_C}$ is conjugate in rank $i$ to a power of $F$. This, however, is impossible by Lemma \ref{L2B} and inequality (\ref{L4:4.0}). Hence, $| k_G | \le 10 \zeta^{-1}$ and inequalities (\ref{L4:1}) are proved.
By Lemma \ref{L1}, we can apply Lemma 22.1 \cite{O89} to a reduced annular diagram of rank $i$ for conjugacy of words $W_F F^{d k_F} W_F^{-1} Z C^{ k_C} Z^{-1}$ and $G^{k_G}$ to obtain, using estimates (\ref{L4:2}), (\ref{L4:5}), that $$
| W_G | < ( \gamma + \tfrac 12 )( \zeta+1) (| F^{d k_F} | + | G^{k_G} | )
\le ( \gamma + \tfrac 12 )( \zeta+1) ( \zeta^{-1} +1)| G^{k_G} | < \zeta^{-1}
| G^{k_G} | , $$ as claimed in (\ref{L4:3}). Lemma \ref{L4} is proved. \end{proof}
\begin{lemma}
\label{L5} In the foregoing notation, the following inequalities hold \begin{gather}
0 < | k_H | \le 100 \zeta^{-1} , \quad
| G^{d k_G} | \le \zeta^{-1} | H^{k_H} | , \label{L5:1} \\
|W_H| < 2 \zeta^{-1} | H^{k_H} | . \label{L5:2} \end{gather} \end{lemma}
\begin{proof} Assume that $[([((X^dY^d )^d X^d)^d, X^d]^dY)^d,X^d] \overset {i} = 1$. Then, by Lemma \ref{L2A}, we also have $[ [((X^dY^d )^d X^d)^d, X^d]^dY , X ] \overset {i} = 1$. Since $X \equiv B^{k_B}$, it follows from Lemma \ref{L2B} that $[((X^dY^d )^d X^d)^d, X^d]^dY \overset {i} = B^\ell$ for some $\ell$ and so $G^{k_G}$ is conjugate in rank $i$ to $B^\ell$. By Lemmas \ref{L2} and \ref{L4}, $$
| B^{k_B} | \le \zeta^{-5}d^{-3}|F^{k_F}| \le
\zeta^{-6}d^{-4}|G^{k_G}| \le 10 \zeta^{-7}d^{-4} | G | < | G| $$
(LPP: $\zeta \succ \eta = d^{-1}$), whence $|B| < |G|$. This, however, contradicts Lemma \ref{L2B}. Hence, $[([((X^dY^d )^d X^d)^d, X^d]^dY)^d,X^d] \overset {i} \neq 1$ and so $k_H \neq 0$.
By definitions, $ [ W_G G^{d k_G } W_G^{-1}, B^{d k_B } ] \overset {i} =
W_H H^{k_H} W_H^{-1}$. By Lemma \ref{L2C}, $| k_H | \le 100 \zeta^{-1}$, $|
G^{d k_G} | \le \zeta^{-1} | H^{k_H} |$ and estimates (\ref{L5:1}) are proved.
As in proofs of Lemmas \ref{L2}, \ref{L4}, we have from Lemmas \ref{L1} and 22.1 \cite{O89} that $$
| W_H | < ( \gamma + \tfrac 12 )\cdot 2( 2| W_G | +
| G^{d k_G} | + | B^{d k_B } | + \tfrac 12 | H^{k_H} |) . $$ Hence, by Lemmas \ref{L2}, \ref{L4} and estimates (\ref{L5:1}), \begin{multline*}
| W_H | < (1 + 2 \gamma ) ( (2 \zeta^{-1} d^{-1} +1 )| G^{d k_G} | +
\zeta^{-6} d^{-4}| G^{d k_G} | + \tfrac 12 | H^{k_H} |)< \\ < (1 + 2
\gamma ) (( 2 \zeta^{-1} d^{-1} +1 + \zeta^{-6} d^{-4} )\zeta^{-1} + \tfrac 12 ) | H^{k_H} | < 2 \zeta^{-1}| H^{k_H} | \end{multline*} (LPP: $\gamma \succ \zeta \succ \eta = d^{-1}$), as required in (\ref{L5:2}). Lemma \ref{L5} is proved. \end{proof}
\begin{lemma}
\label{L6} Let $R_{A^f, j, z^*}$ be a defining word of rank $i+1$ defined by
$(\ref{R1})$ if $z^* = 1$ or by $(\ref{R2})$ if $z^* = 2$. Then $0 < | f|
\le 100 \zeta^{-1}$, $|A| > d$, the words $T_{A^f, j, z^*}$, $U_{A^f, j, z^*}$ do not belong to the cyclic subgroup ${\langle} A {\rangle}$, generated by $A$, of $G(i)$ and $$
\max(|T_{A^f, j, z^*} | , |U_{A^f, j, z^*} |) < d |A| . $$ \end{lemma}
\begin{proof} First we let $z^* = 1$. It follows from definitions that, in the foregoing notation, we can assume that $$ A \equiv G , \quad T_{A^f, j, 1} \overset {i} = W_G^{-1} B^{k_B} W_G , \quad U_{A^f, j, 1} \overset {i} = W_G^{-1} Z C^{k_C} Z^{-1} W_G , $$ and $f = f(A^f, j, 1)$ is $k_G$. Hence, in view of Lemmas \ref{L2}, \ref{L4}, \begin{gather*}
0 < | f| \le 100 \zeta^{-1} , \\
|A| \ge 10^{-1} \zeta^2 | F^{d k_F} | \ge 10^{-1} \zeta^7 d^3 | B^{d k_B} | \ge 10^{-1} \zeta^7 d^3 > d \end{gather*} (LPP: $\zeta \succ \eta = d^{-1}$), and \begin{multline*}
|T_{A^f, j, 1} | , |U_{A^f, j, 1} | \le 2( |W_G| +|Z|) + | B^{k_B}| + |
C^{k_C}| < \\
< ( 2( \zeta^{-1} + 3 \zeta^{-6} d^{-3}) + 2 \zeta^{-6} d^{-4}) | G^{k_G} |
< 3 \zeta^{-1} | G^{k_G} | \le 30 \zeta^{-2} | G|< d |A| \end{multline*} (LPP: $\zeta \succ \eta = d^{-1}$).
Assume that one of $T_{A^f, j, 1}$, $U_{A^f, j, 1}$ belongs to ${\langle} G {\rangle} \subseteq G(i)$. Then one of $B^{k_B}$, $C^{k_C}$ is conjugate in rank $i$ to a power of $G$. However, by Lemmas \ref{L2}, \ref{L4}, $$
|B^{k_B}|, |C^{k_C}| \le \zeta^{-6} d^{-4} | G^{k_G} | \le 10 \zeta^{-7}
d^{-4} | G | < |G| $$
(LPP: $\zeta \succ \eta = d^{-1}$), whence $|B|, |C| < |G|$ which is a contradiction to Lemma \ref{L2B}.
Now we let $z^* = 2$. It follows from definitions that, in the foregoing notation, we can assume that $$ A \equiv H, \quad T_{A^f, j, 2} \overset {i} = W_H^{-1} W_G G^{k_G} W_G^{-1} W_H , \quad U_{A^f, j, 2} \overset {i} = W_H^{-1} Z C^{k_C} Z^{-1} W_H , $$ and $f = f(A^f, j, 2)$ is $k_H$. Hence, in view of Lemmas \ref{L2}, \ref{L4}, \ref{L5}, we have \begin{gather*}
0 < | f | \le 100 \zeta^{-1} , \\
|A| \ge 10^{-2} \zeta^2 |G^{d k_G} | \ge
10^{-2} \zeta^3 d |F^{d k_F} | \ge 10^{-2} \zeta^3 d^2 > d \end{gather*} (LPP: $\zeta \succ \eta = d^{-1}$) and \begin{multline*}
|T_{A^f, j, 2} | , |U_{A^f, j, 2} | \le 2( |W_H| + |W_G| +|Z|) +
| G^{k_G}| + | C^{k_C}| < \\ < ( 2( 2\zeta^{-1} + \zeta^{-2} d^{-1} +3 \zeta^{-7} d^{-4} )
+ \zeta^{-1} d^{-1} + \zeta^{-7} d^{-5}) | H^{k_H} | < \\
< 5 \zeta^{-1} | H^{k_H} | \le 500 \zeta^{-2} | H | < d |A| \end{multline*} (LPP: $\zeta \succ \eta = d^{-1}$), as required.
Assume that one of $T_{A^f, j, 2}$, $U_{A^f, j, 2}$ belongs to ${\langle} H {\rangle} \subseteq G(i)$. Then one of $G^{k_G}$, $C^{k_C}$ is conjugate in rank $i$ to a power of $H$. However, we saw above that
$|B|, |C| < |G|$ and it follows from Lemma \ref{L5} that $$
|G| \le \zeta^{-1} d^{-1} \cdot 100 \zeta^{-1} |H| < |H| $$
(LPP: $\zeta \succ \eta = d^{-1}$). Therefore, $|G|, |C| < |H|$ and, as before, we have a contradiction to Lemma \ref{L2B}. Lemma \ref{L6} is proved. \end{proof}
\begin{lemma}
\label{L7} Suppose that $(X_1,Y_1)$ and $(X_2,Y_2)$ are two pairs of words such that $(X_1^dY_1^d )^d X_1^d$ is conjugate in rank $i$ to $(X_2^dY_2^d )^d X_2^d$ and $[X_1, Y_1] \overset {i} \neq 1$, $[X_2, Y_2] \overset {i} \neq 1$. Then the pairs $(X_1,Y_1)$ and $(X_2,Y_2)$ are conjugate in rank $i$. \end{lemma}
\begin{proof} Without loss of generality we can assume that \begin{gather*} X_1\equiv B_1^{k_{B_1}} , \quad X_2\equiv B_2^{k_{B_2}} , \quad \ Y_1 \equiv Z_1C_1^{k_{C_1}}Z_1^{-1} , \quad Y_2 \equiv Z_2C_2^{k_{C_2}}Z_2^{-1} , \\ X_1^dY_1^d \equiv W_{D_1}D_1^{k_{D_1}}W_{D_1}^{-1} , \quad X_2^dY_2^d \equiv W_{D_2}D_2^{k_{D_2}}W_{D_2}^{-1} , \end{gather*} where $B_1$, $B_2$, $C_1$, $C_2$, $D_1$, $D_2$ are words simple in rank $\,i\,$ or periods of rank $\le i$, the words $Z_1$, $Z_2$, $W_{D_1}$, $W_{D_2}$ have the minimal lengths among all words satisfying the corresponding equalities, and if $A_1 \in \{ B_1, B_2, C_1, C_2, D_1, D_2\}$ is conjugate in rank $i$ to $A_2^{\pm 1}$, where $A_2 \in \{ B_1, B_2, C_1, C_2, D_1, D_2\}$, then $A_1 \equiv A_2$.
Consider a reduced annular diagram $\Delta$ of rank $i$ for conjugacy of the words $W_{D_1}D_1^{dk_{D_1}}W_{D_1}^{-1}B_1^{dk_{B_1}}$ and $W_{D_2}D_2^{dk_{D_2}}W_{D_2}^{-1}B_2^{dk_{B_2}}$. Denote two cyclic sections of the boundary $\partial \Delta$ of $\Delta$ by $p_1q_1$ and $(p_2q_2)^{-1}$, where \begin{gather*} \varphi(p_1) \equiv D^{d k_{D_1}}, \quad \varphi(q_1) \equiv W_{D_1}^{-1}B_1^{dk_{B_1}}W_{D_1} , \\ \varphi(p_2) \equiv D^{d k_{D_2}}, \quad \varphi(q_2) \equiv W_{D_2}^{-1}B_2^{dk_{B_2}}W_{D_2} . \end{gather*}
It follows from Lemma \ref{L2} that \begin{gather*}
|W_{D_1}^{-1}B_1^{dk_{B_1}}W_{D_1}|<11\zeta^{-2}| D_1^{k_{D_1}}|\le 11\zeta^{-2} d^{-1}| D_1^{d k_{D_1}}| < \zeta^{3}| D_1^{d k_{D_1}}| \end{gather*} (LPP: $\zeta \succ \eta = d^{-1}$). Analogously,
$|W_{D_2}^{-1}B_2^{dk_{B_2}}W_{D_2}|< \zeta^{3}| D_2^{d k_{D_2}}|$. Hence, \begin{equation}
|q_1| < \zeta^3|p_1|, \quad |q_2| < \zeta^3|p_2| . \label{L7:1} \end{equation}
Assume that $|q_1|\ge \zeta |p_2|$. Then it follows from (\ref{L7:1}) that $$
|p_2q_2|<(1+\zeta^3)|p_2|\le
\zeta^{-1}(1+\zeta^3)|q_1|<\zeta^2(1+\zeta^3)|p_1|<\zeta|p_1|. $$
Hence, $|p_2q_2|<\zeta|p_1|$ and, in view of (\ref{L7:1}), $\Delta$ is an $F$-map whose existence contradicts Lemma \ref{L1} and Lemmas 24.7, 25.10 \cite{O89}.
Therefore, we can assume that $|q_1|< \zeta |p_2|$ and, similarly, $|q_2|<
\zeta |p_1|$. Now we can see that $\Delta$ is an $I$-map. It follows from Lemmas \ref{L2I} and 25.10 \cite{O89} that $D_1\equiv D_2\equiv D$ and the sections $p_1$, $p_2$ are $D$-compatible. By Lemma \ref{L2}, $|k_{D_1}|\le 100\zeta^{-1}$. Hence, using estimate (\ref{L7:1}), we get $$
|q_1|<\zeta^3|p_1|=\zeta^3|D_1^{dk_{D_1}}|\le 100\zeta^{2}|D_1^{d}|<\zeta|D^{d}_1| =\zeta|D^{d}| . $$
Analogously, $|q_2| <\zeta|D^{d}|$.
If $k_{D_1}\ne k_{D_2}$, then cutting $\Delta$ along a simple path, that makes $p_1$ and $p_2$ $D$-compatible, we could turn $\Delta$ into an $F$-map whose existence contradicts Lemma \ref{L1} and Lemmas 24.7, 25.10 \cite{O89}. Hence, it is shown that $k_{D_1}=k_{D_2}=k$. Cutting $\Delta$ along a simple path, that makes $p_1$ and $p_2$ $D$-compatible, we can see that $$ W_{D_2}D^{\ell_D} W_{D_1}^{-1}B_1^{dk_{B_1}}W_{D_1}D^{-\ell_D}W_{D_2}^{-1} \overset i = B_2^{dk_{B_2}} $$ for some integer $\ell_D$. By Lemma \ref{L2B}, $B_1$ and $(B_2)^{\pm 1}$ are conjugate in rank $i$. Hence, $B_1\equiv B_2\equiv B$. It also follows from Lemma \ref{L2B} that $k_{B_1}=k_{B_2}$ and $W_{D_2}D^{\ell_D}W_{D_1}^{-1} \overset i = B^{\ell_B}$ for some integer $\ell_B$. This implies that $W_{D_1} \overset i = B^{-\ell_B}W_{D_2}D^{\ell_D}$ and so \begin{multline*} X_1^dY_1^d \overset i = W_{D_1}D^k W_{D_1}^{-1} \overset i = B^{-\ell_B}W_{D_2}D^{\ell_D} D^k D^{-\ell_D}W_{D_2}^{-1}B^{\ell_B} \overset i = \\ \overset i = B^{-\ell_B}X_2^dY_2^dB^{\ell_B} \overset i = X_2^dB^{-\ell_B}Y_2^dB^{\ell_B}. \end{multline*} Since $X_1 \equiv X_2 \equiv B^{k_{B_1}}$, we obtain that $Y_1^d \overset i = B^{-\ell_B}Y_2^dB^{\ell_B}$. By Lemma \ref{L2A}, $Y_1 \overset i = B^{-\ell_B} Y_2 B^{\ell_B}$ and we see that the pair $(X_2, Y_2)$ is conjugate in rank $i$ to $(X_1, Y_1)$ by $B^{\ell_B}$. Lemma \ref{L7} is proved. \end{proof}
\begin{lemma}
\label{L9} Suppose that $(X_1,Y_1)$ and $(X_2,Y_2)$ are two pairs of words such that $(X_1^dY_1^d )^d X_1^d$ is conjugate in rank $i$ to $((X_2^dY_2^d )^d X_2^d)^{-1}$ and $[X_1, Y_1] \overset {i} \neq 1$, $[X_2, Y_2] \overset {i} \neq 1$. Then the pairs $(X_1, Y_1)$ and $(X_2^{-1}, Y_2^{-1})$ are conjugate in rank $i$. \end{lemma}
\begin{proof} Observe that $((X_2^dY_2^d )^d X_2^d)^{-1} \overset 0 = ((X_2^{-1})^d(Y_2^{-1})^d )^d (X_2^{-1})^d$. Therefore, if $(X_1^dY_1^d )^d X_1^d$ is conjugate in rank $i$ to $((X_2^dY_2^d )^d X_2^d)^{-1}$, then $(X_1^dY_1^d )^d X_1^d$ is conjugate in rank $i$ to $((X_2^{-1})^d(Y_2^{-1})^d )^d (X_2^{-1})^d$. Hence, our claim follows from Lemma~\ref{L7}. \end{proof}
\begin{lemma}
\label{L10} Suppose that $(X_1,Y_1)$ and $(X_2,Y_2)$ are two pairs of words such that $v_1(X_1,Y_1)$ is conjugate in rank $i$ to $v_1(X_2,Y_2)$ and $[X_1, Y_1] \overset {i} \neq 1$, $[X_2, Y_2] \overset {i} \neq 1$. Then the pairs $(X_1, Y_1)$ and $(X_2, Y_2)$ are conjugate in rank $i$. \end{lemma}
\begin{proof} Conjugating the pairs $(X_1, Y_1)$, $(X_2, Y_2)$ in rank $i$ if necessary, we can assume that $$ X_1 \equiv B_1^{k_{B_1}} , \quad Y_1 \equiv Z_1 C_1^{k_{C_1}} Z_1^{-1} , \quad X_2 \equiv B_2^{k_{B_2}} , \quad Y_2 \equiv Z_2 C_2^{k_{C_2}} Z_2^{-1} , $$ where each of $B_1$, $B_2$, $C_1$, $C_2$ is either simple in rank $i$ or a period of some rank $\le i$ and, when $B_1^{k_{B_1}}$, $B_2^{k_{B_2}}$, $C_1^{k_{C_1}}$, $C_2^{k_{C_2}}$ are fixed, the words $Z_1$, $Z_2$ are picked to have minimal length. Furthermore, consider the following equalities \begin{gather*} X_1^dY_1^d \overset {i} = W_{D_1} D_1^{k_{D_1}} W_{D_1}^{-1} , \quad X_2^dY_2^d \overset {i} = W_{D_2} D_2^{k_{D_2}} W_{D_2}^{-1} , \\ (X_1^dY_1^d )^d X_1^d \overset {i} = W_{E_1} E_1^{k_{E_1}} W_{E_1}^{-1} , \quad (X_2^dY_2^d )^d X_2^d \overset {i} = W_{E_2} E_2^{k_{E_2}} W_{E_2}^{-1} , \\ [((X_1^dY_1^d )^d X_1^d)^d, X_1^d] \overset {i} = W_{F_1} F_1^{k_{F_1}} W_{F_1}^{-1} , \quad [((X_2^dY_2^d )^d X_2^d)^d, X_2^d] \overset {i} = W_{F_2} F_2^{k_{F_2}} W_{F_2}^{-1} , \end{gather*} where each of $D_1$, $E_1$, $F_1$, $D_2$, $E_2$ $F_2$ is either a simple in rank $i$ word or a period of some rank $\le i$ and the conjugating words $W_{D_1}$, $W_{E_1}$, $W_{F_1}$, $W_{D_2}$, $W_{E_2}$, $W_{F_2}$ are picked (when $D_1$, $E_1$, $F_1$, $D_2$, $E_2$ $F_2$ are fixed) to have minimal lengths.
We can also assume that if $A_1 \in \{ B_1, C_1, D_1, E_1, F_1, B_2, C_2, D_2, E_2, F_2\}$ is conjugate in rank $i$ to $A_2^{\pm 1}$, where $A_2 \in \{ B_1, C_1, D_1, E_1, F_1, B_2, C_2, D_2, E_2, F_2\}$, then $A_1 \equiv A_2$.
Consider a reduced annular diagram $\Delta$ of rank $i$ for conjugacy of the words $W_{F_1} F_1^{dk_{F_1}} W_{F_1}^{-1}Z_1 C_1^{k_{C_1}} Z_1^{-1}$ and $W_{F_2} F_2^{dk_{F_2}} W_{F_2}^{-1}Z_2 C_2^{k_{C_2}} Z_2^{-1}$. Denote two cyclic sections of the boundary $\partial \Delta$ of $\Delta$ by $p_1q_1$ and $(p_2q_2)^{-1}$, where \begin{gather*} \varphi(p_1) \equiv F_1^{dk_{F_1}} , \quad
\varphi(q_1) \equiv W_{F_1}^{-1}Z_1 C_1^{k_{C_1}} Z_1^{-1}W_{F_1} , \\ \varphi(p_2) \equiv F_2^{dk_{F_2}} , \quad \varphi(q_2) \equiv W_{F_2}^{-1}Z_2 C_2^{k_{C_2}} Z_2^{-1}W_{F_2} . \end{gather*} It follows from Lemma \ref{L2} that \begin{multline*}
|W_{F_1}^{-1}Z_1 C_1^{k_{C_1}}
Z_1^{-1}W_{F_1}|<(4\zeta^{-1}+7\zeta^{-5}d^{-2})|F_1^{k_{F_1}}| \le \\ \le
(4\zeta^{-1}+7\zeta^{-5}d^{-2})d^{-1}|F_1^{d k_{F_1}}| < \zeta^3
|F_1^{dk_{F_1}}| \end{multline*}
(LPP: $\zeta \succ \eta = d^{-1}$). Analogously, $|W_{F_2}^{-1}Z_2 C_2^{k_{C_2}} Z_2^{-1}W_{F_2}|< \zeta^3 |F_2^{dk_{F_2}}|$. Hence, \begin{equation}
|q_1| < \zeta^3|p_1| , \quad |q_2| < \zeta^3|p_2|. \label{L10:1} \end{equation}
Assume that $|q_1|\ge \zeta |p_2|$. Then it follows from (\ref{L10:1}) that $$
|p_2q_2|<(1+\zeta^3)|p_2|\le
\zeta^{-1}(1+\zeta^3)|q_1|<\zeta^2(1+\zeta^3)|p_1|<\zeta|p_1|. $$
Hence, $|p_2q_2|<\zeta|p_1|$ and, in view of (\ref{L10:1}), $\Delta$ is an $F$-map whose existence contradicts Lemma \ref{L1} and Lemmas 24.7, 25.10 \cite{O89}.
Therefore, we can assume that $|q_1|< \zeta |p_2|$ and, similarly, $|q_2|<
\zeta |p_1|$. Now we can see that $\Delta$ is an $I$-map. By Lemmas \ref{L2I} and 25.10 \cite{O89}, $F_1\equiv F_2\equiv F$ and sections $p_1$
and $p_2$ are $F$-compatible. By Lemma \ref{L2}, $|k_{F_1}| \le 100\zeta^{-1}$. Hence, using estimate (\ref{L10:1}), we have $$
|q_1|<\zeta^3|p_1|=\zeta^3|F_1^{dk_{F_1}}|\le 100\zeta^{2}|F_1^{d}|<\zeta|F^{d}_1| = \zeta|F^{d}|. $$
Analogously, $|q_2|<\zeta|F^{d}|$.
If $k_{F_1}\ne k_{F_2}$, then, cutting $\Delta$ along a simple path, that makes $p_1$ and $p_2$ $F$-compatible, we could turn $\Delta$ into an $F$-map whose existence contradicts Lemma \ref{L1} and Lemmas 24.7, 25.10 \cite{O89}. Hence, $k_{F_1}=k_{F_2}$ and, cutting $\Delta$ along a simple path, that makes $p_1$ and $p_2$ $F$-compatible, we can see that $Y_1$ and $Y_2$ are conjugate in rank $i$.
Since $k_{F_1}=k_{F_2}$, it follows from definitions that the word $[W_{E_1} E_1^{dk_{E_1}} W_{E_1}^{-1}, B_1^{dk_{B_1}}]$ is conjugate in rank $i$ to $[W_{E_2} E_2^{dk_{E_2}} W_{E_2}^{-1}, B_2^{dk_{B_2}}]$. Let $\Delta$ be a reduced annular diagram of rank $i$ for conjugacy of these two words. Denote two cyclic sections of the boundary $\partial \Delta$ of $\Delta$ by $p_1q_1p_2q_2$ and $(p_3q_3p_4q_4)^{-1}$, where \begin{gather*} \varphi(p_1) \equiv \varphi(p_2)^{-1} \equiv E_1^{d k_{E_1}} , \quad \varphi(q_1) \equiv \varphi(q_2)^{-1} \equiv W_{E_1}^{-1} B_1^{dk_{B_1}} W_{E_1} \\ \varphi(p_3) \equiv \varphi(p_4)^{-1} \equiv E_2^{d k_{E_2}} , \quad \varphi(q_3) \equiv \varphi(q_4)^{-1} \equiv W_{E_2}^{-1} B_2^{dk_{B_2}} W_{E_2} . \end{gather*} It follows from Lemma \ref{L2} that \begin{multline*}
|W_{E_1}^{-1} B_1^{dk_{B_1}} W_{E_1}|<(4\zeta^{-2}+\zeta^{-4}d^{-1} ) |
E_1^{k_{E_1}}| \le \\ \le (4\zeta^{-2}+\zeta^{-4}d^{-1} ) d^{-1} | E_1^{d k_{E_1}}| <\zeta^3| E_1^{dk_{E_1}}| \end{multline*} (LPP: $\zeta \succ \eta = d^{-1}$). Therefore, \begin{equation}
|q_1|, |q_2| < \zeta^3|p_1| \quad \mbox{and} \quad |q_1|, |q_2| <
\zeta^3|p_2|. \label{L10:3} \end{equation} Similarly, \begin{equation}
|q_3|, |q_4| < \zeta^3|p_3| \quad \mbox{and} \quad |q_3|, |q_4| <
\zeta^3|p_4|. \label{L10:4} \end{equation}
Assume that $|q_1|\ge \zeta |p_3|$. Then, by (\ref{L10:3})--(\ref{L10:4}), we have $$
|p_3q_3p_4q_4|<2(1+\zeta^3)|p_3|\le 2\zeta^{-1}(1+\zeta^3)|q_1|<2\zeta^2(1+\zeta^3)|p_1|<\zeta|p_1|. $$ In view of estimates (\ref{L10:3})--(\ref{L10:4}), we can turn $\Delta$ into an $E$-map $\Delta'$ by pasting together $q_1$ and $q_2$. It follows from Lemmas 24.6, 25.10 \cite{O89} that the images of $p_1$ and $p_2$ in $\Delta'$ are $E_1$-compatible. This implies that $k_{F_2}=0$, contrary to Lemma \ref{L2}.
Therefore, we can assume that $|q_1|< \zeta |p_3|$ and, similarly, $|q_3|< \zeta |p_1|$. Now we can see that $\Delta$ is an $I$-map. If $p_1$ and $p_2$ are $E_1$-compatible in $\Delta$, then $k_{F_2}=0$, contrary to Lemma \ref{L2}. Hence, $p_1$ and $p_2$ may not be $E_1$-compatible in $\Delta$. Similarly, $p_3$ and $p_4$ are not $E_2$-compatible in $\Delta$. Therefore, it follows from Lemmas \ref{L2I} and 25.10 \cite{O89} that $E_1\equiv E_2\equiv E$ and either $p_1$ is $E$-compatible with $p_3^{-1}$ and $p_2$ is $E$-compatible with $p_4^{-1}$ or $p_1$ is $E$-compatible with $p_4^{-1}$ and $p_2$ is $E$-compatible with $p_3^{-1}$. Let $t_1$ and $t_2$ be simple disjoint paths in $\Delta$ that make corresponding pairs of paths $p_1$, $p_2$, $p_3^{-1}$, $p_4^{-1}$ $E$-compatible. Let us cut $\Delta$ along $t_1$ and $t_2$ and then paste the two resulting diagrams along the images of $q_3$ and $q_4^{-1}$ (recall that $\varphi(q_3) \equiv \varphi(q_4)^{-1}$). Let $\Delta_0$ denote the simply connected diagram of rank $i$ thus obtained from $\Delta$. Observe that \begin{equation} \varphi (\partial \Delta_0) \overset i = E_1^{\ell_E} W_{E_1}^{-1} B_1^{dk_{B_1}}W_{E_1} E_1^{-\ell_E} W_{E_1}^{-1} B_1^{-dk_{B_1}}W_{E_1} \overset i = 1 , \label{L10:5} \end{equation} where either $\ell_E = d(k_{E_1} - k_{E_2})$ in the case when $p_1$ is $E_1$-compatible in $\Delta$ with $p_3^{-1}$ and $p_2$ is $E_1$-compatible with $p_4^{-1}$ or $\ell_E = d(k_{E_1} + k_{E_2})$ in the case when $p_1$ is $E_1$-compatible in $\Delta$ with $p_4^{-1}$ and $p_2$ is $E_1$-compatible with $p_3^{-1}$. If $k_{E_1} \neq \pm k_{E_2}$ then it follows from Lemma \ref{L2B}, equality (\ref{L10:5}) and definitions that $B_1 \equiv E_1$. On the other hand, it follows from Lemma \ref{L2} that $$
| B_1| \le \zeta^{-4} d^{-2} |E_1^{k_{E_1}} | < 100 \zeta^{-5}
d^{-2} |E_1 | < |E_1 | $$ (LPP: $\zeta \succ \eta = d^{-1}$). This contradiction shows that $k_{E_1} = \pm k_{E_2}$ and so $E_1^{k_{E_1} } \equiv E_2^{\pm k_{E_2}}$. By definitions, this means that the word $(X_1^dY_1^d )^d X_1^d$ is conjugate in rank $i$ to $((X_2^dY_2^d )^d X_2^d)^{\pm 1}$.
Assume that $(X_1^dY_1^d )^d X_1^d$ is conjugate in rank $i$ to $((X_2^dY_2^d )^d X_2^d)^{-1}$. Then, by Lemma \ref{L9}, $Y_1$ is conjugate in rank $i$ to $Y_2^{-1}$. On the other hand, as we saw above, $Y_1$ is conjugate in rank $i$ to $Y_2$. Hence, $Y_2$ is conjugate in rank $i$ to $Y_2^{-1}$. This, however, by Lemmas \ref{L2B} and \ref{L2A}, implies that $Y_2 \overset i = 1$. This contradiction to $[X_1, Y_1] \overset {i} \neq 1$ proves that $(X_1^dY_1^d )^d X_1^d$ is conjugate in rank $i$ to $(X_2^dY_2^d )^d X_2^d$. Now a reference to Lemma \ref{L7} completes the proof of Lemma \ref{L10}. \end{proof}
\section{Proof of Theorem}
Our Theorem is immediate from the following below Lemmas \ref{L11}--\ref{L12}.
\begin{lemma}
\label{L11} The group $G(\infty)$, defined by presentation \eqref{G8}, is naturally isomorphic to the free group $F(\mathcal{A}) / W_{1,2}(F(\mathcal{A}))$ of the variety $\mathfrak{M}$ in the alphabet $\mathcal{A}$. \end{lemma}
\begin{proof} It follows from the definition of defining words of the group $G(\infty)$ that each of them is in $W_{1,2}(F(\mathcal{A}))$ and so there is a natural epimorphism $$ G(\infty) \to F(\mathcal{A}) / W_{1,2}(F(\mathcal{A})) . $$
Suppose that $\widetilde X$, $\widetilde Y$ are some words in $\mathcal{A}^{\pm 1}$ and \begin{equation} w_{z^*}(\widetilde X, \widetilde Y ) \neq 1 \label{L11:1} \end{equation} in the group $G(\infty)$, where $z^* \in \{ 1,2\}$.
Let $A$ be a period of some rank such that $A^f$ for some $f$ is conjugate in $G(\infty)$ to $v_{z^*}(\widetilde X, \widetilde Y )$. (The existence of such an $A$ follows from definitions and Lemma \ref{L2B}; see also Lemma 18.1 \cite{O89}.) Note that, in view of (\ref{L11:1}), $[\widetilde X, \widetilde Y ] \neq 1$ in $G(\infty)$. Hence, by Lemmas \ref{L2}, \ref{L4}, \ref{L5}, we can replace the pair $(\widetilde X, \widetilde Y )$ by a conjugate in the group $G(\infty)$ pair $(X, Y)$ such that $X \equiv B^{k_B}$, $Y \equiv Z C^{k_C} Z^{-1}$, and $v_{z^*}(X, Y ) = W_A A^f W_A^{-1}$ in $G(\infty)$, where $B$, $C$ are some periods,
$|f|>0$ and \begin{gather*}
|X^d | + |Y^d | = | B^{d k_B} | + |C^{d k_C} | +2 |Z| <
8\zeta^{-5}d^{-2} | F^{k_F} | \le 8 \zeta^{-6}d^{-3} |A^f| . \end{gather*}
Hence, the length $ | v_{z^*}( X, Y ) | $, which is either
$|[((X^dY^d )^d X^d)^d, X^d]^dY|$ if $z^* = 1$ or \newline
$|[([((X^dY^d )^d X^d)^d, X^d]^dY)^d,X^d]|$ if $z^* = 2$, can be estimated as follows \begin{multline*}
| v_{z^*}( X, Y ) | \le 2d(d(2d(d( |X^d| + |Y^d|) + |X^d|) + 2|X^d|)+|Y|) +
2|X^d| < \\
< 5 d^4( |X^d| + |Y^d|) < 40 d \zeta^{-6} |A^f| < 10^4 d \zeta^{-7} |A| \end{multline*}
for $0 < |f| \le 100 \zeta^{-1} $ by Lemmas \ref{L4}, \ref{L5}. Consider a reduced annular diagram $\Delta$ of some rank $i'$ for conjugacy of $v_{z^*}( X, Y )$ and $A^f$. By Lemmas
\ref{L1} and 22.1 \cite{O89}, $\Delta$ can be cut into a simply connected diagram $\Delta_1$ along a simple path $t$ which connects points on distinct components of $\partial \Delta$ with $| t| < \gamma
|\partial \Delta |$. Therefore, $$
|\partial \Delta_1 | < (1 + 2 \gamma) |\partial \Delta | < (1 + 2 \gamma)(10^4 d \zeta^{-7}+
100\zeta^{-1} )|A|
< \tfrac 12 n |A| $$ (LPP: $\gamma \succ \zeta \succ \eta = d^{-1} \succ \iota = n^{-1}$). Then, by Lemmas \ref{L1}, 20.4 and 23.16 \cite{O89}
applied to $\Delta_1$, the diagram $\Delta_1$ contains no 2-cells of rank $ \ge |A|$, whence $\Delta_1$, $\Delta$ are diagrams of rank $|A|
-1$. Since $A \in \mathcal{X}_{|A|}$, it follows from the construction of defining words of rank $|A|$ that there will be a defining word in $\Ss_{|A|}$ which guarantees that $w_{z^*}(X, Y)
\overset {|A|} = 1$. A contradiction to the assumption (\ref{L11:1}) proves that $G(\infty)$ is in $\mathfrak{M}$ and Lemma \ref{L11} is proved. \end{proof}
\begin{lemma}
\label{L12} The group $G(\infty)$, defined by presentation \eqref{G8}, is not hopfian. \end{lemma}
\begin{proof} By Lemma \ref{L11}, $G(\infty)$ is the free group of $\mathfrak{M}$ in $\mathcal{A}$ and, therefore, every map $a_1 \to U_1, \dots$, $a_m \to U_m$, where $U_1, \dots, U_m$ are words in $\mathcal{A}^{\pm 1}$, extends to a homomorphism $G(\infty) \to G(\infty)$.
Consider a homomorphism $\psi_{\infty} : G(\infty) \to G(\infty)$ defined by $\psi_{\infty}(a_j) = a_j$ if $j \neq 2$ and $\psi_{\infty}(a_2) = v_1(a_1, a_2)$. Observe that the relation $w_2(a_1, a_2) = 1$ in $G(\infty)$ and the definition (\ref{w2}) of the word $w_2(x, y)$ ensure that $a_2 \in {\langle} a_1, v_1(a_1, a_2) {\rangle} \subseteq G(\infty)$. Hence, $\psi_{\infty}$ is an epimorphism. Assume that $\psi_{\infty}$ is an automorphism. Then it follows from the relation $w_1(a_1, a_2) = 1$ in $G(\infty)$ that the word \begin{multline*} U \equiv a_1^{{\varepsilon}_1} a_2^{n} a_1^{{\varepsilon}_2} a_2^{n+2} \dots a_1^{{\varepsilon}_{h/2-2}} a_2^{n+h-6} a_1^{{\varepsilon}_{h/2-1}} a_2^{n+h-4} a_1^{{\varepsilon}_{h/2}} a_2^{(n+h-2) + h/2} \\ a_1^{{\varepsilon}_1} a_2^{-(n+1)} a_1^{{\varepsilon}_2} a_2^{-(n+3)} \dots a_1^{{\varepsilon}_{h/2-1}} a_2^{-(n+h-3)} a_1^{{\varepsilon}_{h/2}} a_2^{-(n+h-1)} \end{multline*}
is equal to 1 in $G(\infty)$. Note that $U$ is cyclically reduced and $|U| <(n+h)h$. Consequently, \begin{equation}
|U| < (n+h) h < (1-\alpha)(h-1) n d \label{T:3} \end{equation} (LPP: $ \alpha \succ \zeta = h^{-1} \succ \eta = d^{-1} \succ \iota = n^{-1}$).
On the other hand, since $U=1$ in $G(\infty)$ and $U \neq 1$ in the free group $F(\mathcal{A})$, it follows from Lemmas \ref{L1} and 23.16 \cite{O89} that \begin{equation}
|U| >(1-\alpha) |\partial \Pi | , \label{T:1} \end{equation} where $\Pi$ is a 2-cell of positive rank. Furthermore, it follows from Lemmas \ref{L6} and 20.4 \cite{O89} (together with condition $B$, see Sect. 20.4 \cite{O89}) that \begin{equation}
|\partial \Pi | > (h-1)n r(\Pi) > (h-1)n d \label{T:2} \end{equation} for an arbitrary 2-cell $\Pi$ of rank $r(\Pi) \ge 1$. However, inequalities (\ref{T:1})--(\ref{T:2}) contradict (\ref{T:3}). This contradiction shows that $U \neq 1$ in $G(\infty)$. Thus, $U$ is in the kernel of $\psi_{\infty}$ and $G(\infty)$ is not hopfian, as required. \end{proof}
In conclusion, we remark that, for every $i \ge 0$, the group $G(i)$, given by presentation (\ref{Gi}), is finitely presented (this follows from definitions and Lemmas \ref{L2}--\ref{L6}) and satisfies a linear isoperimetric inequality (this can be proved similar to Lemma 21.1 \cite{I94}). Therefore, $G(i)$ is a torsion-free Gromov hyperbolic group (see \cite{G87}) and so, by Sela's results \cite{Sl97}, $G(i)$ is a hopfian group. Therefore, the non-hopfian group $G(\infty)$, given by presentation (\ref{G8}), is a limit of hopfian groups $G(i)$, $i=0,1,\dots$.
\end{document} | arXiv |
The geometry of the Wigner caustic and adecomposition of a curve into parallel arcs
The geometry of the Wigner caustic and adecomposition of a curve into parallel arcs Domitrz, Wojciech; Zwierzyński, Michał 2022-03-01 00:00:00 In this paper we study global properties of the Wigner caustic of parameterized closed planar curves. We find new results on its geometry and singular points. In particular, we consider the Wigner caustic of rosettes, i.e. regular closed parameterized curves with non-vanishing curvature. We present a decomposition of a curve into parallel arcs to describe smooth branches of the Wigner caustic. By this construction we can find the number of smooth branches, the rotation number, the number of inflexion points and the parity of the number of cusp singularities of each branch. We also study the global properties of the Wigner caustic on shell (the branch of the Wigner caustic connecting two inflexion points of a curve). We apply our results to whorls—the important object to study the dynamics of a quantum particle in the optical lattice potential. Keywords Semiclassical dynamics · Affine equidistants · Wigner caustic · Singularities · Planar curves Mathematics Subject Classification 53A04 · 53A15 · 58K05 · 81Q20 1 Introduction In 1932 Eugene Wigner introduced the celebrated Wigner function to study quantum corrections to classical statistical mechanics ([31]).This function relates the wave- The work of W. Domitrz and M. Zwierzynski ´ was partially supported by NCN grant no. DEC-2013/11/B/ST1/03080. Wojciech Domitrz [email protected] Michał Zwierzynski ´ [email protected] Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warszawa, Poland 0123456789().: V,-vol 7 Page 2 of 35 W. Domitrz, M. Zwierzyński function that appears in Schrödinger's equation to a probability distribution in phase space. The Wigner function of a pure state is defined in the following way W (p, q) = ψ (q − ζ)ψ(q + ζ) exp (2ipζ/) dζ, −∞ 2 2 where (p, q) ∈ R are momentum and position, and ψ ∈ L (R) is the wavefunction. In [1] Berry studied the semiclassical limit of Wigner's phase-space representation of quantum states. He proved that for 1-dimensional systems, that correspond to smooth (Lagrangian) curves M in the phase space (R ,ω = d p ∧ dq), the semiclassical limit of the Wigner function of the classical correspondence M of a pure quantum state takes on high values at points in a neighborhood of M and also in a neighborhood of a singular closed curve, which is called the Wigner caustic of M or the Wigner catastrophe (see [1,4,10,23] for details). Geometrically the Wigner caustic of a planar curve M is the locus of midpoints of chords connecting points on M with parallel tangent lines ([1,10,11,23]). It is also the caustic of a certain Lagrangian map defined in the following way (see [10,11,23] for details). 2 2 For the canonical symplectic form ω = d p ∧ dq on R the map : T R ∗ 2 2 ∗ 2 v → ω(v, ·) ∈ T R is an isomorphism between the bundles T R and T R . Then ω ˙ = dα = d p ˙ ∧ dq + d p ∧ dq ˙ is a symplectic form on T R , where α is the ∗ 2 2 2 canonical Liouville 1-form on T R . The linear diffeomorphism : R × R → 2 2 2 T R = R × R , + + − − + − + − + − + − 1 (p , q , p , q ) = (p, q, p ˙, q ˙ ) = p + p , q + q , p − p , q − q 2 1 ∗ ∗ pulls the symplectic form ω ˙ on T R back to the canonical symplectic (π ω − π ω) + − 2 2 2 2 2 on the product R × R , where π ,π : R × R → R are the projections on the + − first and on the second component, respectively. If M is a smooth regular planar curve then M is an immersed Lagrangian submanifold of (R ,ω). Hence 1 (M × M ) is an 2 2 2 2 2 immersed Lagrangian submanifold of (T R , ω) ˙ .Let π ,π : T R = R × R → R 1 2 be the projections on the first and on the second component, respectively. Then π and π define Lagrangian fibre bundles with the symplectic structure ω ˙ . Then the caustic of the Lagrangian map (the set of its critical values) π ◦ 1 is the Wigner M ×M caustic [5,6,10,11,23]. On the other hand the Lagrangian map π ◦ 1 is the M ×M secant map of M [13]. If M is (locally) described as dS M = (p, q) ∈ R | p = (q) dq then the generating family of the Lagrangian submanifold 1 (M × M ) has the fol- lowing form F (p, q,β) = (S(q + β) − S(q − β)) − pβ. 2 The geometry of the Wigner caustic Page 3 of 35 7 Fig. 1 A non-convex curve together with its Wigner caustic Fig. 2 An improper affine sphere (with different opacities) generated from a curve in Fig. 1 The front of the Legendrian submanifold of the contact manifold (T R × R, dz + α) generated by F is a singular 2-dimensional improper affine sphere, where z is a coordinate on R. The caustic of this front is composed of the curve M and its Wigner caustic. Hence the geometry of the Wigner caustics provides information on singularities of improper affine spheres. In Fig. 1 we present a non-convex planar curve with its Wigner caustic and in Fig. 2 we show the improper affine sphere generated by M in the construction described above (see [5,6] for details). In [8](seealso[2–4]) the dynamics of a quantum particle in the optical lattice potential was investigated. The authors analyze the evolution of the Wigner function. The function undergoes a number of catastrophic changes. For a semiclassical approx- imation the Wigner caustic consists of the rainbow diagram (the original curve M) and a locus of midpoints of chords joining points on the rainbow diagram with parallel tan- gent lines. But the catastrophe set of the exact Wigner function, in addition, contains a locus of midpoints of chords joining points on neighboring rainbow diagrams with parallel tangents. Hence the Wigner caustic of the curve M should be investigated not only locally but globally too. It turns out that its global geometry is very important for understanding the quantum-classical correspondence breakdown. It allows to extract important information without using simplifying approximations. 7 Page 4 of 35 W. Domitrz, M. Zwierzyński Fig. 3 i A closed regular curve M, ii M and E (M ) Singularities of the Wigner caustic for ovals occur exactly from antipodal pairs (the tangent lines at the two points are parallel and the curvatures are equal). The well- known Blaschke-Süss theorem states that there are at least three pairs of antipodal points on an oval ([22,26]). The absolute value of the oriented area of the Wigner caustic gives the exact relation between the perimeter and the area of the region bounded by closed regular curves of constant width and improves the classical isoperimetric inequality for convex curves ([34,35,37–39]). Furthermore this oriented area improves the isoperimetric defect in the reverse isoperimetric inequality ([7]). Recently the properties of the middle hedgehog, which is a generalization of the Wigner caustic for non-smooth convex curves, were studied in [29,30]. The Wigner caustic in the literature regarding hedgehogs is known also as a projective hedgehog (see [27,28] and the literature cited therein). The Wigner caustic could be generalized to obtain an affine λ-equidistant, which is the locus of points of the above chords which divide the chord segments between base points with a fixed ratio λ. The singular points of affine equidistants create the Centre Symmetry Set, the natural generalization of the center of symmetry, which is widely studied in [11,16,18,20,24]. The geometry of an affine extended wave front, i.e. the set λ ∈[0, 1]{λ}× E (M ), where E (M ) is an affine λ λ λ-equidistant of a manifold M, was studied in [11,15]. Local properties of singularities of the Wigner caustic and affine equidistants were studied in many papers [5,9–12,19,23,25]. In this paper we study global properties of the Wigner caustic of a generic planar closed curve. In [1] Berry proved that if M is a convex curve, then generically the Wigner caustic is a parametrized connected curve with an odd number of cusp singularities and this number is not smaller than 3. It is not true in general for any closed planar curve. If M is a parametrized closed curve with self-intersections or inflexion points then the Wigner caustic has at least two branches (smoothly parametrized components). We present a decomposition of a curve into parallel arcs and thanks to this decomposition we are able to describe the geometry of branches of the Wigner caustic. In general the geometry of the Wigner caustic of a regular closed curve is quite complicated (see Fig. 3). In Sect. 2 we briefly sketch some of the known results on the Wigner caustic and affine equidistants. The geometry of the Wigner caustic Page 5 of 35 7 Section 3 contains the algorithm to describe branches of the Wigner caustic and affine equidistants of any generic regular parameterized closed curve. Subsection 3.1 provides an example of an application of this algorithm to a particular curve. In the beginning of Sect. 4 we present global propositions on the number of cusps and inflexion points of the Wigner caustic. We show that the procedure based on a decomposition presented in Sect. 3 can be applied to obtain the number of branches of the Wigner caustic, the number of inflexion points and the parity of the number of cusp singularities of each branch. After that we study global properties of the Wigner caustic on shell, i.e. the branch of the Wigner caustic which connects two inflexion points of a curve. We present the results on the parity of the number of cusp points of the branches of the Wigner caustic on shell. We also prove that each such branch has even number of inflexion points and there are even number of inflexion points on a path of the original curve between the endpoints of this branch. In Sect. 5 we use the decomposition introduced in Sect. 3 to study the geometry of the Wigner caustic of generic regular closed parameterized curves with non-vanishing curvature and of some generic regular closed parameterized curves with two inflexion points. Finally, in Sect. 6 we study the Wigner caustic of whorls. All the pictures of the Wigner caustic in this manuscript were made in the application created by the second author [36] and in Mathematica [32]. 2 Preliminaries Let M be a smooth parameterized curve in the affine plane R , i.e. the image of the ∞ 2 C smooth map from an interval to R . A smooth curve is closed if it is the image of ∞ 1 2 a C smooth map from S to R . A smooth curve is regular if its velocity does not vanish. A regular curve is simple if it has no self-intersection points. A regular simple closed curve is convex if its signed curvature has a constant sign. Let (s , s ) s → 1 2 2 k f (s) ∈ R be a parameterization of M. A point f (s ) is a C regular point of M for k = 1, 2,... or k =∞ if there exists ε> 0 such that f (s −ε, s +ε) is a C smooth 0 0 1-dimensional manifold. A point f (s ) is a singular point if it is not C regular for any k > 0. A curve is singular if it has at least one singular point. A singular point p is 2 3 2 called a cusp if M is locally diffeomorphic at p to a curve (−1, 1) t → (t , t ) ∈ R at 0. A point f (s ) is a cusp of M if and only if f (s ) = 0 and the vectors f (s ) and 0 0 0 f (s ) are linearly independent. A point f (s ) is an inflexion point of M if its signed 0 0 curvature changes sign. An inflexion point f (s ) is non-degenerate (or ordinary)if det f (s ), f (s ) = 0 which means that the order of contact of M with the tangent 0 0 line to M at f (s ) is equal to 2. If the curvature vanishes at f (s ) but does not change 0 0 sign, then this point is called an undulation point. Remark 2.1 Let (a, b) s → f (s) ∈ R be a parameterization of M, then the order of contact of M with the tangent line to M at p = f (t ) is k if and only if i k+1 d f d f d f d f det (t ), (t ) = 0for i = 1, 2,..., k and det (t ), (t ) = 0. i k+1 ds ds ds ds (2.1) 7 Page 6 of 35 W. Domitrz, M. Zwierzyński Definition 2.2 A pair of points a, b ∈ M (a = b) is called a parallel pair if the tangent lines to M at a and b are parallel. Definition 2.3 A chord passing through a pair a, b ∈ M, is the line: l(a, b) = λa + (1 − λ)b λ ∈ R . Let A be a subset of R , then clA denotes the closure of A. Definition 2.4 The Wigner caustic of M is the following set a + b E (M ) = cl a, b is a parallel pair of M . Remark 2.5 The Wigner caustic of M is an example of an affine λ-equidistant set of M, E (M ) = cl λa + (1 − λ)b a, b is a parallel pair of M , where λ = . Definition 2.4 is different from definitions in papers [5,10–12,18,19,25, 33,37,38]. The closure in the definition is needed to include inflexion points of M in E (M ). For details see Remark 2.18. Note that, for any given λ ∈ R,wehaveE (M ) = E (M ). Thus, the case λ = λ 1−λ is special. In particular we have E (M ) = E (M ) = M if M is closed. 0 1 Definition 2.6 The Centre Symmetry Set of M, denoted by CSS(M ), is the envelope of all chords passing through parallel pairs of M. Bitangent lines of M are parts of CSS(M ) ([11,16]). If M is a generic convex curve, then CSS(M ), the Wigner caustic and E (M ) for a generic λ are smooth closed curves with at most cusp singularities ([1,16,20,24]), cusp singularities of all E (M ) are on regular parts of CSS(M ) ([20]), the number of cusps of CSS(M ) and E (M ) is odd and not smaller than 3 ([1,16], see also [22]), the number of cusps of CSS(M ) is not smaller than the number of cusps of E (M ) ([11]). Let us denote by κ (p) the signed curvature of a smooth regular curve M at p.Let a, b be a parallel pair of M. Assume that κ (b) = 0. Let us fix local arc length parameterizations of M nearby the points a, b by f : (s , s ) → R and by 0 1 g : (t , t ) → R , respectively. Let us assume that the parameterizations at a and b 0 1 are in opposite directions, i.e. the velocities at a, b are opposite. Then there exists a function t : (s , s ) → (t , t ) such that 0 1 0 1 f (s) =−g (t (s)). (2.2) It is easy to see that by the implicit function theorem the function t is smooth and κ ( f (s)) t (s) = . (2.3) κ (g(t (s))) M The geometry of the Wigner caustic Page 7 of 35 7 Then by γ (s) = f (s) + g(t (s)) (2.4) 2 2 we will denote a local natural parameterization of the Wigner caustic. Whenever we will write about singular points of the Wigner caustic we will denote these points as the singular points of the parameterization given by (2.4). By direct calculations we get the following lemma. Lemma 2.7 Let M be a regular curve. Let a, b be a parallel pair of M, such that M a+b is parameterized at a and b in opposite directions and κ (b) = 0. Let p = be a regular point of E 1 (M ). Then (i) the tangent line to E 1 (M ) at p is parallel to the tangent lines to M at a and b. (ii) the curvature of E 1 (M ) at p is equal to 2κ (a)|κ (b)| M M κ (p) = . E (M ) |κ (b) − κ (a)| M M Lemma 2.7(ii) implies the following propositions. Proposition 2.8 [20] Let a, b be a parallel pair of a regular curve M, such that M is parameterized at a and b in opposite directions and one of a and b is not an inflexion a+b point. Then the point is a singular point of E (M ) if and only if κ (a) = κ (b). M M a+b Proposition 2.9 Let a, b be a parallel pair of a regular closed curve M. Then is an inflexion point of E (M ) if and only if one of the points a, b is an inflexion point of M. Let τ denote the translation by a vector p ∈ R . Definition 2.10 A curve M is curved to the same side at a and b (resp. curved to different sides), where a, b is a parallel pair of M, if the center of curvature of M at a and the center of curvature of τ (M ) at a = τ (b) lie on the same side (resp. on a−b a−b the different sides) of the tangent line to M at a. We illustrate above definition in Fig. 4. a+b Corollary 2.11 If M is curved to the same side at a parallel pair a, b, then is a regular point of the Wigner caustic of M. Proof Let us locally parameterize M at a and b in opposite directions. Then by Propo- a+b sition 2.8 a point is a singular point of E 1 (M ) if and only if κ (a) = 1. (2.5) κ (b) The right hand side of (2.5) is positive, then κ (a) and κ (b) have the same sign, M M therefore M is curved to different sides at a and b. 7 Page 8 of 35 W. Domitrz, M. Zwierzyński Fig. 4 i A curve curved to the same side at a parallel pair a, b, ii a curve curved to different sides at a parallel pair a, b ∞ 1 2 ∞ 1 2 We denote by C (S , R ) the set of C mappings from S to R ,i.e.the setof smooth closed parameterized planar curves, and by · the dot product in R . ∞ 1 2 Remark 2.12 Let f (s ), f (s ) be a parallel pair of f ∈ C (S , R ). A singular point 1 2 f (s )+ f (s ) 1 2 of the Wigner caustic of f (that is a point for which κ (s ) = κ (s ) and f 1 f 2 f (s ) · f (s )< 0) is a cusp if and only if κ (s ) = κ (s ), where κ denotes the 1 2 1 2 f f f signed curvature of f with respect to the parameterization of f , and κ denotes the derivative of the curvature with the respect to the arc length parameter ([10]). ∞ 1 2 Theorem 2.13 Let G be the subset of C (S , R ) such that each curve f in G satisfies the following conditions: (i) f is a regular curve with only non-degenerate inflexion points and no undulation points, (ii) f has only transverse self-crossings, (iii) if f (s ), f (s ) is a parallel pair of f , then f (s ) or f (s ) is not an inflexion point, 1 2 1 2 (iv) if f (s ), f (s ) is a parallel pair of f , the points f (s ), f (s ) are not inflexion 1 2 1 2 points of f , the dot product f (s ) · f (s ) is negative (respectively positive), and 1 2 κ (s ) = κ (s ) (respectively κ (s ) =−κ (s )), then κ (s ) = κ (s ), where f 1 f 2 f 1 f 2 1 2 f f κ denote the derivative of the curvature with respect to the arc length parameter. ∞ 1 2 ∞ Then G is a generic subset of C (S , R ) with Whitney C topology and the Wigner caustic of f ∈ G is the finite union of smooth curves with at most cusp singularities. Proof Since the intersection of two generic subsets is still a generic subset, it is enough to show that properties from each point are generic. The set of smooth regular closed ∞ 1 2 curves is an open and dense subset of C (S , R ) because the set of 1-jets of smooth 1 1 2 non-regular closed curves is a smooth submanifold of J (S , R ) of codimension 2. 1 2 Let f : S → R be smooth and regular. Having only non-degenerate inflexion points and no undulation points is equivalent to the following property: det( f (s), f (s)) = 0 ⇒ det( f (s), f (s)) = 0. (2.6) 3 1 3 1 2 Condition (2.6) means that the map j f : S → J (S , R ) is transversal to the 3 1 2 following submanifold of J (S , R ): 3 3 1 2 j g(s) ∈ J (S , R ) g (s) = 0, det g (s), g (s) = 0 . The geometry of the Wigner caustic Page 9 of 35 7 By the Thom Transversality Theorem (e.g. see Theorem 4.9 in [21]) Property (i) is generic. To prove genericity of the conditions (ii–iv) we will use the Thom Transversal- ity Theorem for multijets (e.g. see Theorem 4.13 in [21] for details). We denote by k 1 2 1 (2) 1 1 1 J (S , R ) the s-fold k-jet bundle and by (S ) the set S × S \{(s, s) | s ∈ S }. 1 1 (2) 1 1 2 Genericity of (ii) follows from transversality of j f : (S ) → J (S , R ) to the 2 2 1 1 2 following submanifold of J (S , R ): 1 1 1 1 2 ( j g(s ), j h(s )) ∈ J (S , R ) g(s ) = h(s ), g (s ) = 0, h (s ) = 0 . 1 2 1 2 1 2 Transversality means that if f (s ) = f (s ) for s = s , then det( f (s ), f (s )) = 0. 1 2 1 2 1 2 Therefore, Condition (ii) is generic. Genericity of Property (iii) follows from transversality of the second multijet j f : 1 (2) 2 1 2 (S ) → J (S , R ) to the submanifold 2 2 2 1 2 ( j g, j h) ∈ J (S , R ) det g (s ), h (s ) = 0, g (s ) = 0, h (s ) = 0 . 1 2 1 2 2 2 This means that if det f (s ), f (s ) = 0for s = s , then κ (s ) + κ (s ) = 0. 1 2 1 2 1 2 f f Hence, Property (iii) is generic. Now we assume that f satisfies (iii). Genericity of Property (iv) for f (s )· f (s )< 1 2 3 1 (2) 3 1 2 0 follows from the transversality of j f : (S ) → J (S , R ) to the submanifold 2 2 3 3 3 1 2 W := ( j g, j h) ∈ J (S , R ) g (s ) = 0, h (s ) = 0, det(g (s ), h (s )) = 0, 1 2 1 2 g (s ) · h (s )< 0,κ (s ) = κ (s ) . 1 2 g 1 h 2 By direct calculations one can show that this means that if j f (s , s ) ∈ W , then 1 2 κ (s ) = κ (s ), which is equivalent to the condition for a cusp singularity in a 1 2 f f singular point of the Wigner caustic (see Remark 2.12). The proof for the case f (s ) · f (s )> 0 is similar. From now one, when we will talk about generic curves, we will mean a curve from the set G. Furthermore, genericity of f implies the following geometric properties of f . ∞ 1 2 Proposition 2.14 [13]If f ∈ C (S , R ) has only non-degenerate inflexion points and has no undulation points, then the number of inflexion points of f and the rotation number of f are finite. Definition 2.15 The tangent line of E 1 (M ) at a cusp point p is the limit of a sequence of 1-dimensional vector spaces T M in RP for any sequence q of regular points q n of E (M ) converging to p. This definition does not depend on the choice of a converging sequence of regular points. By Lemma 2.7(i) we can see that the tangent line to E 1 (M ) at the cusp point a+b is parallel to tangent lines to M at a and b. 2 7 Page 10 of 35 W. Domitrz, M. Zwierzyński Remark 2.16 If M is an oval, then we have well defined the continuous normal vector a+b field on the double covering M of E 1 (M ), M a → ∈ E 1 (M ) by taking the 2 2 normal vector to M compatible with the parameterization of M at the point a, and a+b defining this vector as a normal vector to the Wigner caustic at . Let us notice that the continuous normal vector field to E 1 (M ) at regular and cusp points is perpendicular to the tangent line to E 1 (M ). Using this fact and the above definition we define the rotation number in the following way. Definition 2.17 The rotation number of the Wigner caustic of a generic curve M is the rotation number of the continuous normal vector field of the Wigner caustic. Remark 2.18 Let p be an inflexion point of M. Then the CSS(M ) is tangent to this inflexion point and has an endpoint there. The set E (M ) for λ = has an inflexion point at p (as the limit point) and is tangent to M at p. The Wigner caustic is tangent to M at p too and it has an endpoint there. The Wigner caustic and the Centre Symmetry Set approach p from opposite sides ([1,10,16,19]). This branch of the Wigner caustic is studied in Sect. 4. If M is a generic regular closed curve then E 1 (M ) is a union of smooth parametrized curves. Each of these curves we will call a smooth branch of the Wigner caustic of M. In Fig. 5 we illustrate a non-convex curve M,E 1 (M ), and different smooth branches of E (M ). 3 A decomposition of a curve into parallel arcs In this section we assume that M is a generic regular closed curve. We will present a decomposition of M into parallel arcs which will help us to study the geometry of the smooth branches of the Wigner caustic of M. 1 2 Definition 3.1 Let S s → f (s) ∈ R be a parameterization of a smooth closed curve M, such that f (0) is not an inflexion point. A function ϕ : S →[0,π ] is called an angle function of M if ϕ (s) is the oriented angle between f (s) and f (0) modulo π. We identify the set [0,π ] modulo π with S . 1 1 Definition 3.2 A point ϕ in S is a local extremum of ϕ if there exists s in S such that ϕ (s) = ϕ, ϕ (s) = 0, ϕ (s) = 0. The local extremum ϕ of ϕ is a local M M M M maximum (resp. minimum) if ϕ (s)< 0(resp. ϕ > 0). We denote by M(ϕ ) the M M set of local extrema of ϕ . The angle function has the following properties. Proposition 3.3 Let M be a generic regular closed curve. Let f be the arc length parameterization of M and let ϕ be the angle function of M. Then (i) f (s ), f (s ) is a parallel pair of M if and only if ϕ (s ) = ϕ (s ), 1 2 M 1 M 2 (ii) ϕ (s) is equal to the signed curvature of M with respect to the parameterization of M, The geometry of the Wigner caustic Page 11 of 35 7 Fig. 5 i A non-convex curve M with two inflexion points (the dashed line) and E (M ), ii–vi M and different smooth branches of E (M ) (iii) M has an inflexion point at f (s ) if and only if ϕ (s ) is a local extremum. 0 M 0 (iv) if ϕ (s ), ϕ (s ) are local extrema and there is no extremum on ϕ (s , s ) , M 1 M 2 M 1 2 then one of extrema ϕ (s ), ϕ (s ) is a local maximum and the other one is a M 1 M 2 local minimum. Lemma 3.4 Let ϕ be the angle function of a generic regular closed curve M. Then the function ϕ has an even number of local extrema, i.e. M has an even number of inflexion points . Proof It is a consequence of the fact that the number of local extrema of a generic 1 1 smooth function from S to S is even. Let ϕ be the angle function of M. Definition 3.5 The sequence of local extrema is the following sequence (ϕ ,ϕ ,..., 0 1 ϕ ) where {ϕ ,ϕ ,...,ϕ }= M(ϕ ) and the order is compatible with the 2n−1 0 1 2n−1 M 1 1 orientation of S = ϕ (S ). M 7 Page 12 of 35 W. Domitrz, M. Zwierzyński Fig. 6 i A closed regular curve M with points p = f (s ) tangent lines to M at these points, ii a graph of i i the angle function ϕ with ϕ and s values M i i Definition 3.6 The sequence of division points S is the following sequence −1 (s , s ,..., s ), where {s , s ,..., s }= ϕ (M(ϕ )) if M(ϕ ) is not empty, 0 1 k−1 0 1 k−1 M M −1 otherwise {s , s ,..., s }= ϕ ϕ (0) , and the order of S is compatible with ( ) 0 1 k−1 M M the orientation of M. Let M have inflexion points. If s belongs to the sequence of division points, then f (s ) is an inflexion point or a point which is parallel to an inflexion point. In the case when M has no inflexion points the sequence of division points consists of 0 and points s such that f (0), f (s ) are parallel pairs. k k In Fig. 6 we illustrate an example of a closed regular curve M, the angle function ϕ and the sequence of division points. Let us notice that the images of points in the sequence of division points divide the curve M into arcs. Some of these arcs (say A and A ) have the property that for any point a ∈ A there exists a point a ∈ A 2 i i j j such that a , a is a parallel pair for i = j ∈{1, 2}. Such arcs we will call parallel i j arcs. The set of arcs splits into subsets such that any two arcs in the same subset are parallel (see Definition 3.9). Proposition 3.7 If M is a generic regular closed curve and a ∈ M is an inflexion point then the number of points b ∈ M, such that b = a and a, b is a parallel pair, is even. Proof There are no inflexion points b ∈ M such that a, b is a parallel pair and the point a is not a self-intersection point of M, since the curve M is generic. The inflexion points of M correspond to local extrema of the angle function ϕ . We divide the graph of the angle function ϕ into continuous paths of the form (t,ϕ (t )) | t ∈[t , t ],ϕ (t ), ϕ (t ) ∈{0,π }, ∀t ∈ (t , t )ϕ (t)/∈{0,π } . M 1 2 M 1 M 2 1 2 M Let α belong to (0,π). First we assume that α is not equal to a local extremum of a path P. Then a line ϕ = α intersects the path P an even number of times if P is a path from 0 to 0 or from π to π, since both the beginning and the end of P areonthe same side of the line (see Fig. 7i). This line intersects P an odd number of times if P is a path from 0 to π or from π to 0, since the beginning and the end of P are on different sides of the line (see Fig. 7ii). Now we assume that α is equal to a local extremum of P. In this case the line ϕ = α intersects a path P an odd number of times if P is a path from 0 to 0 or from π to π The geometry of the Wigner caustic Page 13 of 35 7 Fig. 7 Continuous paths Fig. 8 Vertical perturbation nearby a local maximum (see Fig. 7iii) and this line intersects P an even number of times if P is a path from 0 to π or from π to 0 (see Fig. 7iv), since by a small local vertical perturbation around the extremum point we obtain the previous cases and the numbers of intersection points have a difference ±1(seeFig. 8). Let us note that a path from 0 to 0 or from π to π corresponds to an arc of a curve with the rotation number equalling 0 and a path from 0 to π or from π to 0 corresponds to an arc of a curve with the rotation number equalling ± . Since the rotation number of M is an integer, the number of paths from 0 to π or from π to 0 in the graph of ϕ is even. Each path of this type intersects every horizontal line ϕ = α at least once. −1 Thus the number of intersections of ϕ and the line ϕ = ϕ ( f (a)) is odd. But the M M number of points b = a such that a, b is a parallel pair is one less than the number of −1 intersection points of the graph of ϕ and the line ϕ = ϕ ( f (a)). M M The number of inflexion points of a generic regular closed curve is even. Thus by Proposition 3.7 we have the following corollary. Corollary 3.8 If M is a generic regular closed curve then #S is even. We recall that in this section we assume that M is a generic regular closed curve and let #S = 2m. The functions m , M :{0, 1,..., 2m − 1} →{0, 1,..., 2m − 1} are analogs 2m 2m of the minimum and the maximum functions modulo 2m, respectively. Namely, 2m − 1, if {k, l}={0, 2m − 1}, m (k, l) := 2m min(k, l), otherwise, 7 Page 14 of 35 W. Domitrz, M. Zwierzyński 0, if {k, l}={0, 2m − 1}, M (k, l) := 2m max(k, l), otherwise. We denote by (s , s ) an interval (s , L + s ), where L is the length of 2m−1 0 2m−1 M 0 M M. In the following definition indexes i in ϕ are computed modulo 2n, indexes j , j +1 in p p and p p are computed modulo 2m. j j +1 j +1 j Definition 3.9 If M(ϕ ) ={ϕ ,ϕ ,...,ϕ }, then for every i ∈{0, 1,..., 2n − M 0 1 2n−1 1},a set of parallel arcs is the following set = p p k − l =±1mod(2m), ϕ (s ) = ϕ ,ϕ (s ) = ϕ , i k l M k i M l i +1 ϕ (s , s ) = (ϕ ,ϕ ) , M m (k,l) M (k,l) i i +1 2m 2m where p = f (s ) and p p = f s , s . k k k l m (k,l) M (k,l) 2m 2m If M(ϕ ) is empty then we define only one set of parallel arcs as follows: = p p , p p ,..., p p , p p . 0 0 1 1 2 2m−2 2m−1 2m−1 0 The set of parallel arcs has the following property. 1 2 Proposition 3.10 Let f : S → R be the arc length parameterization of M. For every two arcs p p ,p p in the well defined map k l k l i p p p → P(p) ∈ p p , k l k l where the pair p, P(p) is a parallel pair of M, is a diffeomorphism. Definition 3.11 Let p p , p p belong to the same set of parallel arcs, then k k l l 1 2 1 2 p p k k 1 2 denotes the following set (the arc) p p l l 1 2 cl (a, b) ∈ M × M a ∈ p p , b ∈ p p , a, b is a parallel pair of M . k k l l 1 2 1 2 p ... p p p k k k k 1 n i i +1 In addition denotes i = 1n − 1 . We will call p ... p p p l l l l 1 n i i +1 this set a glueing scheme. Remark 3.12 If p p belongs to a set of parallel arcs, then there are neither inflexion k l points nor points with tangent lines parallel to tangent lines at inflexion points of M in p p \{ p , p }. k l k l Definition 3.13 The -point map ([11]) is the map a + b π 1 : M × M → R ,(a, b) → . 2 The geometry of the Wigner caustic Page 15 of 35 7 Fig. 9 Two arcs A and A of 1 2 M belonging to the same set of parallel arcs and E (A ∪ A ) 1 2 Let A = p p and A = p p be two arcs of M which belong to 1 k k 2 l l 1 2 1 2 the same set of parallel arcs. It is easy to see that E A ∪ A consists of one 1 2 p p k k 1 2 arc under π 1 (see Fig. 9). From this observation we get the following p p 2 l l 1 2 proposition. Proposition 3.14 The Wigner caustic E 1 (M ) is the image of the union of i different arcs under the -point map π . Proposition 3.15 Let M be a generic regular closed curve which is not convex. If a p p k k 1 2 glueing scheme is of the form , then this scheme can be prolonged in a p p l l 1 2 p p p k k k 1 2 3 unique way to such that (k , l ) = (k , l ). 1 1 3 3 p p p l l l 1 2 3 Proof Let us consider p p k k 1 2 . (3.1) p p l l 1 2 Let A = p p \{ p , p }, A = p p \{ p , p }. By Remark 3.12 A 1 k k k k 2 l l l l 1 1 2 1 2 1 2 1 2 and A must be curved to the same side or in the opposite sides at any parallel pair in A ∪ A (see Fig. 10i–ii). Let us consider the case in Fig. 10i, the other case is similar. 1 2 Then (3.1) can be prolonged in the following two ways. (1) Neither p nor p is an inflexion point of M. Then (3.1) can be prolonged to k l 2 2 p p p k k k 1 2 3 , where k = k and l = l (see Fig. 10iii). 1 3 1 3 p p p l l l 1 2 3 (2) One of points p , p is an inflexion point of M. Let us assume that this is p . k l k 2 2 2 p p p k k k 1 2 3 Then (3.1) can be prolonged to , where k = k (see Fig. 10iv). 1 3 p p p l l l 1 2 1 7 Page 16 of 35 W. Domitrz, M. Zwierzyński Fig. 10 Possible prolongations of an arc of a curve Since M is generic, at least one of the points p , p is not an inflexion point of M. k l 2 2 Remark 3.16 To avoid repetition in the union in Definition 3.11 we assume that no pair except the beginning and the end can appear twice in the glueing scheme. p p k l Furthermore, if the pair is in the glueing scheme than the pair does not appear p p l k unless they are the beginning and the end of the scheme. The image of a glueing scheme under the -point map π 1 represents parts of branches of the Wigner caustic. If we equip the set of all possible glueing schemes with the inclusion relation, then this set is partially ordered. There is only finite number of arcs from which we can construct branches of E (M ). Therefore we can define a maximal glueing scheme. Definition 3.17 A maximal glueing scheme is a glueing scheme which is a maximal element of the set of all glueing schemes equipped with the inclusion relation. Remark 3.18 If M is a generic regular convex curve, then the set of parallel arcs is equal p p 0 1 to ={ p p , p p }. Then the only maximal glueing scheme is 0 0 1 1 0 p p 1 0 Proposition 3.19 The set of all glueing schemes equipped with the inclusion relation is the disjoint union of totally ordered sets. Proof It follows from uniqueness of the prolongation of the glueing scheme (see Proposition 3.15). The geometry of the Wigner caustic Page 17 of 35 7 1 2 Lemma 3.20 Let f : S → R be the arc length parameterization of M. Then (i) for every two different arcs p p ,p p in there exists exactly one k k l l i 1 2 1 2 p p p p p p k k k k l l 1 2 2 1 1 2 maximal glueing scheme containing or or or p p p p p p l l l l k k 1 2 2 1 1 2 p p l l 2 1 p p k k 2 1 p ... p k k (ii) every maximal glueing scheme is in the following form , where p ... p l l { p , p }={ p , p } whenever p = p and p = p k l k l k l k l (iii) if p is an inflexion point of M, then there exists a maximal glueing scheme which is in the form p p ... p p k k k l 1 n p p ... p p k l l l 1 n where p is a different inflexion point of M and p = p for i = 1, 2,..., n. l k l i i Proof (i) is a consequence of the uniqueness of the prolongation of a glueing scheme (see Proposition 3.15). The proof of (ii) follows from (i) and the fact that the following equalities hold: p p p p p p p p k k k k k k l l 1 2 2 1 1 2 1 2 = and = . p p p p p p p p l l l l l l k k 1 2 2 1 1 2 1 2 p p k k To prove (iii) let us prolong to the maximal glueing scheme G.Any p p k l point p in the sequence of division points S belongs to exactly two arcs in all sets l M of parallel arcs. Then by (ii) this maximal glueing scheme is in the following form p p ... p p k k k l 1 n , (3.2) p p ... p p k l l l 1 n If (3.2) would contain some other inflexion point p in the middle, then (3.2) would contain the following part: p p r r r p p r r r which is impossible by (i). Theorem 3.21 The image of every maximal glueing scheme of M under the -point map π 1 is a branch of the Wigner caustic of M and all branches of the Wigner caustic can be obtained in this way. 1 2 Proof Let f : S → R be the arc length parameterization of M. It is easy to see that S = {s} (s , s ) m (k,l) M (k,l) 2m 2m s∈S i (k,l)∈ M i 7 Page 18 of 35 W. Domitrz, M. Zwierzyński and then M = { f (s)}∪ f (s , s ) , m (k,l) M (k,l) 2m 2m s∈S i (k,l)∈ M i where denotes the disjoint union. Then by Proposition 3.10 we obtain that E (M ) = E p p ∪ p . (3.3) 1 1 k l k l 2 2 i p p ,p k l i k l p p =p k l k l p p p k l k l Since E 1 p p ∪ p p = π 1 = π 1 and k l k l 2 2 p p 2 p p k l k l p p k l every arc is in exactly one maximal glueing scheme, then every branch of k l the Wigner caustic is the image of a maximal glueing scheme under the -point map π 1 . As a summary of this section we present an algorithm to find all maximal glueing schemes. Algorithm 1 (Finding all maximal glueing schemes of a generic regular closed curve 1 2 M parametrized by f : S → R ) (1) Find the set of local extrema of the angle function ϕ of M (see Definition 3.1 and Definition 3.2). (2) Find the sequence of local extrema (see Definition 3.5). (3) Find the sequence of division points (see Definition 3.6). (4) Find the sets of parallel arcs (see Definition 3.9). (5) Create the following set := p p , p p : p p = p p , k l k l k l k l 1 1 2 2 1 1 2 2 ∃ p p ∈ ∧ p p ∈ ∨ p p ∈ ∧ p p ∈ . i k l i k l i l k i l k i 1 1 2 2 1 1 2 2 (6) If there exists a number k such that p is an inflexion point of M and there exists the set of arcs p p , p p or p p , p p in , create a k l k l l k l k 1 2 1 2 p p k l glueing scheme , remove the used set of arcs from and go to step (7). p p k l Otherwise go to step (8). ... p (7) If the created glueing scheme is of the form and there exists the set of ... p arcs p p , p p or p p , p p in , then prolong the k k l l k k l l 1 2 1 2 2 1 2 1 ... p p k k 1 2 scheme to the following scheme , remove the used set of arcs ... p p l l 1 2 The geometry of the Wigner caustic Page 19 of 35 7 Fig. 11 Acurve M as in Fig. 6 and different branches of E (M ) from and go to step (7), otherwise the considered glueing scheme is a maximal glueing scheme and then go to step (6). (8) If is empty, then all maximal glueing schemes for E 1 (M ) were created, other- wise find any set of arcs p p , p p in , create a glueing scheme k l k l 1 1 2 2 p p k l 1 1 , remove the used set of arcs from and go to step (7). p p k l 2 2 3.1 An example of construction of branches of the Wigner caustic Let M be a curve illustrated in Fig. 6. Then the sets of parallel arcs are as follows = p p , p p , 0 0 1 4 5 = p p , p p , p p , p p . 1 1 2 3 2 3 4 5 0 Then there exist two maximal glueing schemes of M: p p p p p 0 1 2 3 4 , (3.4) p p p p p 4 5 0 5 0 p p p p 2 1 2 3 . (3.5) p p p p 2 3 4 3 By Proposition 4.3 the number of cusps of the branch which correspond to (3.4)is odd. By Corollary 4.4 in the glueing scheme (3.4) there are two parallel pairs containing an inflexion point of M – the pairs: ... p p ... 2 3 ... p p ... 0 5 Therefore this branch of the Wigner caustic has exactly two inflexion points—see Fig. 11ii. The same conclusion holds for the glueing scheme (3.5) and the branch in Fig. 11i. In this case we exclude the first and the last parallel pair. 7 Page 20 of 35 W. Domitrz, M. Zwierzyński Fig. 12 A continuous normal vector field at a cusp singularity 4 The geometry of the Wigner caustic of regular curves In this section we start with propositions on numbers of inflexion points and cusp singularities of the Wigner caustic which follows from properties of maximal glueing schemes introduced in Sect. 3. Proposition 4.1 Let M be a generic regular closed curve. If M has 2n inflexion points then there exist exactly n smooth branches of E 1 (M ) connecting pairs of inflexion points on M and every inflexion point of M is the end of exactly one branch of E 1 (M ). Other branches of E 1 (M ) are closed curves. Proof It is a consequence of Lemma 3.20 and Theorem 3.21. Lemma 4.2 Let C be a closed smooth curve with at most cusp singularities. If the rotation number of C is an integer, then the number of cusps of C is even and if the rotation number of C is a half-integer, then the number of C is odd. Proof A continuous normal vector field to the germ of a curve with a cusp singularity is directed outside the cusp on one of two connected regular components and is directed inside the cusp on the other component as it is illustrated in Fig. 12. That observation ends the proof. Proposition 4.3 Let M be a generic regular closed curve. Letn be a unit continuous normal vector field to M. Let C be a smooth branch of E 1 (M ) which does not connect inflexion points. Then the number of cusps of C is odd if and only if the maximal glueing p ... p k l scheme of C is in the following form and n (p ) =−n (p ). M l M k p ... p l k Proof If the normal vectors to M at p and p are opposite, then the rotation number k l of C is equal to , where r is an odd integer. By Lemma 4.2 the number of cusps in C is odd. Otherwise the rotation number of C is an integer, therefore the number of cusps of C is even. By Proposition 2.9, Corollary 3.8 and Proposition 4.1 we get the following corol- laries on inflexion points of branches of the Wigner caustic of M. Corollary 4.4 Let M be a generic regular closed curve. Let C be a smooth branch of the Wigner caustic of M. Then the number of inflexion points of C is equal to the The geometry of the Wigner caustic Page 21 of 35 7 number of parallel pairs containing an inflexion point of M in the maximal glueing scheme for C unless they are the beginning or the end of the maximal glueing scheme which connects the inflexion points of M. Corollary 4.5 Let M be a generic regular closed curve. Let 2n > 0 be the number of inflexion points of M and let #S = 2m. Then E 1 (M ) has 2m − 2n inflexion points. Now we study the properties of the Wigner caustic on shell, i.e. the branch of the Wigner caustic connecting two inflexion points, see Fig. 11i. We are interested in the parity of the number of cusps and the parity of the number of inflexion points on this branch. 1 2 Theorem 4.6 Let M be a generic regular closed curve. Let S s → f (s) ∈ R be a parameterization of M, let f (t ),f (t ) be inflexion points of M and let C be a branch 1 2 of the Wigner caustic of M which connects f (t ) and f (t ). Then the number of cusps 1 2 of C is odd if and only if exactly one of the inflexion points f (t ), f (t ) is a singular 1 2 point of the curve C ∪ f [t , t ] . 1 2 Proof By genericity of M the points f (t ) and f (t ) are ordinary inflexion points of 1 2 M. By Corollary 4.8. in [10] we know that the germ of the Wigner caustic at an inflexion point of a generic curve M together with M are locally diffeomorphic to the following germ at (0, 0): 2 2 2 (p, q) ∈ R : p = 0 ∪ (p, q) ∈ R : p =−q , q ≤ 0 . Let N = C ∪ f [t , t ] . Then N is a closed curve. The germ of N at f (t ) for 1 2 i i = 1, 2 is locally diffeomorphic to one of the following germs at (0, 0): 2 2 2 (p, q) ∈ R : p = 0, q ≤ 0 ∪ (p, q) ∈ R : p =−q , q ≤ 0 , (4.1) 2 2 2 (p, q) ∈ R : p = 0, q > 0 ∪ (p, q) ∈ R : p =−q , q ≤ 0 . (4.2) In other points N has at most cusp singularities. Note that the point (0, 0) is a singular point of the germ of type (4.1) and the point (0, 0) is a C -regular point of the germ of type (4.2) (see Fig. 13). Let M p → n (p) ∈ S be a continuous normal vector field to M. Let us assume that the maximal glueing scheme for C has the following form p p ... p p k k k k 1 2 n−1 n p p ... p p l l l l 1 2 n−1 n where k = l , k = l . Without loss of generality we can assume that k < k .Let 1 1 n n 1 n us define a normal vector field n to N as follows: • n (p) = n (p) for p ∈ f [t , t ] , N M 1 2 7 Page 22 of 35 W. Domitrz, M. Zwierzyński Fig. 13 A continuous normal vector field to the germs of type (4.1)and (4.2) a+b • n (p) = n (a) for p ∈ C, where p = , a, b is a parallel pair of M such that N M there exists i ∈{1, 2,..., n − 1} such that a ∈ p p , b ∈ p p . k k l l i i +1 i i +1 The vector field n is a continuous unit normal field to N . The normal vector field around the points of type (4.1) and (4.2) is described in Fig. 13. Thus by the same argument as in the proof of Lemma 4.2 we can get that the total number of cusps and singularities of type (4.1)in N is even, so the number of cusps of C is odd if and only if exactly one of the inflexion points f (t ), f (t ) is of type (4.1). 1 2 In Fig. 5iii there is exactly one point of type (4.1), in Fig. 11i there is an even number of points of type (4.1). Proposition 4.7 Let M be a regular curve. Let (a, b) s → f (s) ∈ R be a param- eterization of M and let f (s ) be an ordinary inflexion point of M. Let t be a smooth function-germ on R at s such that f (s), f (t (s)) is a parallel pair and lim t (s) = s . 0 0 s→s Let κ (s) be the curvature of M at a point f (s). Then κ (s) lim =−1. (4.3) s→s 0 κ (t (s)) Furthermore let C be a branch of the Wigner caustic which ends in f (s ).If d f d f det (s ), (s ) = 0, (4.4) 0 0 ds ds then C ∪ f [s , b) at f (s ) is of type (4.1)if 0 0 d κ (s) lim > 0 (4.5) s→s 0 ds κ (t (s)) M The geometry of the Wigner caustic Page 23 of 35 7 and C ∪ f [s , b) at f (s ) is of type (4.2)if 0 0 d κ (s) lim < 0. (4.6) s→s 0 ds κ (t (s)) Proof Without loss of generality we may assume that locally f (s) = (s, F (s)), (4.7) where F (s) = as + G(s), a = 0 and s = 0, where G(s) ∈ m , where m is the 0 n maximal ideal of smooth function-germs R → R vanishing at 0. Let us notice that (s, F (s)), (t , F (t )) is a parallel pair of M nearby f (0) if and only if s = t and F (s) − F (t ) = 0. This is equivalent to (s − t )(3as + 3at + H (s, t )) = 0, where H ∈ m and let P(s, t ) = 3as + 3at + H (s, t ).Let t : (R, 0) → (R, 0) be a function-germ at 0 such that P(s, t (s)) = 0. (4.8) By the implicit function theorem the function-germ t is well defined, because ∂ P (0, 0) = 3a = 0. By (4.8) we get that ∂t ∂ P (s, t ) ∂s t (s) =− . (4.9) ∂ P (s, t ) ∂t It implies that t (0) =−1. (4.10) Since F (s) = F (t (s)), then for s = 0 F (s) κ (s) t (s) = = . (4.11) F (t (s)) κ (t (s)) Thus (4.3) holds (Fig. 14). (4) The condition (4.4) means that F (0) = 0. It implies that M is not locally centrally symmetric around f (s ) = (0, 0). 0 7 Page 24 of 35 W. Domitrz, M. Zwierzyński Fig. 14 Acurve M with an inflexion point and the Wigner caustic of M (the dashed line) The branch of the Wigner caustic which contains f (0) has the following parame- terization x 1 (s) = s + t (s), F (s) + F (t (s)) . (4.12) Therefore x (s) = (1 + t (s)) 1, F (s) . (4.13) Since C ∪ f [t , t ] at f (t ) can be only of type (4.1)or(4.2), then C ∪ f [t , t ] 1 2 1 1 2 at f (t ) is of type (4.1) if and only if x (s) f (s)< 0 whenever s → t , therefore by (4.13) we get that 1 + t (s)< 0. By (4.10) we get that t (s)> 0 and by (4.11)we finish the proof. Remark 4.8 Under the assumptions of Theorem 4.7 if locally f (s) = (s, F (s)) then (4) d κ (s) 2F (s ) M 0 lim =− . (4.14) (3) s→s 0 ds κ (t (s)) 3F (s ) M 0 1 2 Theorem 4.9 Let M be a generic regular closed curve. Let S s → f (s) ∈ R be a parameterization of M, let f (s ),f (s ) be inflexion points of M and let C be a 1 2 branch of the Wigner caustic of M which connects f (s ) and f (s ). Then the number 1 2 of cusps of C is odd if and only if d κ (s) d κ (s) M M lim · lim > 0, (4.15) ± ∓ ds κ (t (s)) ds κ (t (s)) s→s M 1 s→s M 2 1 2 where κ (s) denotes the curvature of M at f (s), the pairs f (s), f (t (s)) and M 1 f (s), f (t (s)) are parallel pairs such that t (s) → s whenever s → s and s < t (s) 2 i i i i for the left-hand side neighborhood of s for i = 1, 2. i The geometry of the Wigner caustic Page 25 of 35 7 Proof By genericity of M we get that f (s ) and f (s ) are ordinary inflexion points. 1 2 Then the theorem is a consequence of Theorem 4.6 and Proposition 4.7. Now we study inflexion points on the Wigner caustic on shell. 1 2 Theorem 4.10 Let M be a generic regular closed curve. Let S s → f (s) ∈ R be a parameterization of M and let C be a branch of the Wigner caustic which connects two inflexion points f (t ) and f (t ) of M. Then the number of inflexion points of C 1 2 and the number of inflexion points of the arc f (t , t ) are even. 1 2 1 1 Proof Let ϕ : S → S be the angle function of M. By the genericity of M all local extrema of ϕ are different. Let 1 1 ψ ,ψ :[0, T]→ graph ϕ ⊂ S × S 1 2 M be the following continuous functions: ψ (0) = ψ (0) = t ,ϕ (t ) ,ψ (T ) = ψ (T ) = t ,ϕ (t ) , 1 2 1 M 1 1 2 2 M 2 ψ (t ) = s (t ), ϕ (s (t )) for i = 1, 2, i i M i where continuous functions s , s :[0, T]→ S satisfy ϕ s (t ) = ϕ s (t ) and 1 2 1 M 1 M 2 s (t ) = s (t ) for t ∈ (0, T ). 1 2 Since f (t ) is an inflexion point then ϕ (t ) is a local extremum. Without loss 1 M 1 of generality we assume that ϕ (t ) is a local minimum. To prove that the number M 1 of inflexion points in f (t , t ) is even it is enough to show that ϕ (t ) is a local 1 2 M 2 maximum. The numbers of local maxima and local minima of ϕ are equal. Thus the difference between the number of local maxima and local minima of ϕ is one. For small S −{t } ε> 0 the arcs ψ and ψ define the opposite orientations of the graph of ϕ 1 2 M [0,ε] [0,ε] and ϕ ◦ s increases. Let ϕ s (t ) for i = 1or i = 2 be a local extremum of M i M i [0,ε] ˜ ˜ ϕ such that there are no extrema on ϕ s (0, t ) and ϕ s (0, t ) . Since ϕ (t ) is M M 1 M 2 M 1 ˜ ˜ ˜ a local minimum then ϕ (s (t )) is a local maximum and ψ (t − ε, t + ε) for j = i M i j changes the orientation in t (see Fig. 15). The numbers of local maxima and local minima of ϕ are equal but the arcs S −[t ,t ] ψ and ψ define the same orientation of graph ϕ . Since the function 1 2 M [t˜,t˜+ε] [t˜,t˜+ε] ϕ increases then the next extremum is a local minimum. The number of local ˜ ˜ [t ,t +ε] minima decreases by 1 and the orientations are opposite after crossing the minimum. Thus the defined orientations are opposite if and only if the difference between the number of local maxima and local minima to cross is one. Since for small ε> 0the arcs ψ and ψ define the opposite orientations of the graph of ϕ , 1 2 M [T −ε,T ] [T −ε,T ] then ϕ (t ) must be a local maximum. M 2 A point (a + b) is an inflexion point of C if and only if one of the points of the parallel pair a, b is an inflexion point of M. The number of inflexion points of 7 Page 26 of 35 W. Domitrz, M. Zwierzyński Fig. 15 A change of the angle function ϕ Fig. 16 Acurve M with 8 inflexion points (the dashed line) and branches of the Wigner caustic between inflexion points of M C is equal to the sum of the number of changes of the orientations of ψ and ψ 1 2 ˜ ˜ ˜ ˜ because ψ (t − ε, t + ε) changes the orientation in t if and only if ϕ (s (t )) is a i M i local extremum. Since ϕ (t ) is a minimum and ϕ (t ) is a maximum, then the total M 1 M 2 number of changes of the orientations is even. In Fig. 16 we illustrate a closed curve M and branches of the Wigner caustic between inflexion points of M.InFig. 17 we illustrate a closed curve M such that the branch of the Wigner caustic which connects two inflexion points of M has no inflexion points. Lemma 4.11 Let C be a smooth closed curve with at most cusp singularities. Then the number of inflexion points of C is even. Proof If C is regular, i.e. has no cusp singularities, then by Lemma 3.4 we get that C has an even number of inflexion points. If C has cusp singularities, then we change C nearby each cusp in the way illustrated in Fig. 18 creating two more inflexion points. After this transformation of C we obtain a regular closed curve C such that the parity of the numbers of inflexion points of C and C are equal. Therefore the number of inflexion points of C is even. The geometry of the Wigner caustic Page 27 of 35 7 Fig. 17 Acurve M (the dashed line) and E (M ) Fig. 18 Acurve C with the cusp singularity at x and a curve C with inflexion points at p and q Proposition 4.12 Let M be a generic regular closed curve. Then the number of inflex- ion points of each smooth branch of the Wigner caustic of M is even. Proof Let us notice that all branches of E 1 (M ) except the branches of the Wigner caustic which connect two inflexion points of M are closed curves. So the result for these branches follows from Lemma 4.11. Otherwise it follows from Theorem 4.10. 5 The Wigner caustic of closed curves with at most 2 inflexion points In this section we study the geometry of the Wigner caustic of closed regular curves with non-vanishing curvature (rosettes) and of closed regular curves with exactly two inflexion points. Definition 5.1 A smooth curve γ : (s , s ) → R is called a loop if it is a simple 1 2 + − curve with non-vanishing curvature such that lim s → s γ(s) = lim s → s γ(s).A 1 2 loop γ is called convex if the absolute value of its rotation number is not greater than 1, otherwise it is called non-convex. We illustrate examples of loops in Fig. 19. Theorem 5.2 ([14]) The Wigner caustic of a loop has a singular point. Theorem 5.3 Let C be a generic regular closed parameterized curve with non- vanishing curvature with rotation number equal to n. Then 7 Page 28 of 35 W. Domitrz, M. Zwierzyński Fig. 19 i A convex loop L (the dashed line) and E (L), ii a non-convex loop L (the dashed line) and E (L) 1 1 2 2 (i) the number of smooth branches of E 1 (C ) is equal to n, (ii) at least branches of E 1 (C ) are regular closed parameterized curves with non-vanishing curvature, (iii) n − 1 branches of E 1 (C ) have a rotation number equal to n and one branch has a rotation number equal to , (iv) every smooth branch of E (C ) has an even number of cusps if n is even, (v) exactly one branch of E (C ) has an odd number of cusps if n is odd, (vi) cusps of E 1 (C ) created from loops of C are in the same smooth branch of n n E 1 (C ), (vii) the total number of cusps of E 1 (C ) is not smaller than 2, Proof Since the rotation number of C is n, for any point a in C there exist exactly n n 2n − 1 points b = a such that a, b is a parallel pair of C . Thus the set of parallel arcs has the following form = p p , p p ,..., p p , p p . 0 0 1 1 2 2n−2 2n−1 2n−1 0 Let E 1 (C ) be a smooth branch of E 1 (C ). We can create the following maximal n n ,k 2 2 glueing schemes. • A maximal glueing scheme of E 1 (C ) for k ∈{1, 2,..., n − 1}: ,k p p p ... p p p 0 1 2 2n−2 2n−1 0 p p p ... p p p k k+1 k+2 k−2 k−1 k • A maximal glueing scheme of E 1 (C ): ,n p p p . . . p p 0 1 2 n−1 n p p p . . . p p n n+1 n+2 2n−1 0 The total number of arcs of the glueing schemes for the Wigner caustic presented above is n(2n −1). By Proposition 3.14 the total number of different arcs of the Wigner The geometry of the Wigner caustic Page 29 of 35 7 caustic is equal to the same number. Thus there are no more maximal glueing schemes for the Wigner caustic of C . If (a , a ,..., a ) is a sequence of points in C with the order compat- 0 1 2n−1 n ible with the orientation of C such that a , a is a parallel pair, then C is n i j n curved to the same side at a and a if and only if i − j is even. Thus branches i j E 1 (C ), E 1 (C ),..., E 1 n (C ) are created from parallel pairs a, b in C n n n n ,2 ,4 ,2· 2 2 2 2 such that C is curved to the same side at a and b and all the other branches of the Wigner caustic of C are created from parallel pairs a, b in C such that n n C is curved to different sides at a and b. By Corollaries 2.11 and 4.5 branches E 1 (C ), E 1 (C ),..., E 1 n (C ) are regular closed parameterized curves with n n n ,2 ,4 ,2· 2 2 2 2 non-vanishing curvature. By Proposition 4.3 the branch E 1 (C ) is the only branch of the Wigner caustic ,n of C which has an odd number of cusps if n is odd. We can see that the part of the Wigner caustic created from loops of C are all in E 1 (C ).Every C for n > 1 has at least one loop, so E 1 (C ) has at least one cusp, n n n ,1 ,1 2 2 but because E 1 (C ) has an even number of cusps, then E 1 (C ) has at least two n n ,1 ,1 2 2 cusps. In Fig. 20i we illustrate a curve of the type C and E 1 (C ).InFig. 20iii–vi we 4 4 illustrate different smooth branches of E 1 (C ). Theorem 5.4 Let W be a generic closed curve with the rotation number n. Let W n n have exactly two inflexion points such that one of the arcs of W connecting inflexion points is an embedded curve with the absolute value of the rotation number smaller than . Then (i) the number of smooth branches of E 1 (W ) is equal to n + 1, (ii) n − 1 branches of E 1 (W ) have a rotation number equal to n and one branch has a rotation number equal to , (iii) n − 1 branches of E 1 (W ) have four inflexion points and two branches have two inflexion points, (iv) every smooth branch of E (W ), except a branch connecting inflexion points of W , has an even number of cusps if n is even, (v) exactly one smooth branch of E 1 (W ), except a branch connecting inflexion points of W , has an odd number of cusps if n is odd, (vi) cusps of E 1 (W ) created from convex loops of W are in the same smooth branch n n of E 1 (W ). Proof One can notice that the graph of the angle function ϕ has the form presented in Fig. 21. For that parameterization we get that f (s ) and f (s ) corresponds to inflexion 0 1 points of W and the sets of the parallel arcs are as follows: = p p , p p , p p , p p ,..., p p , p p , 0 2 3 4 5 6 7 8 9 4n−2 4n−1 4n 4n+1 = p p , p p , p p , p p ,..., p p , p p . 1 0 1 1 2 3 4 5 6 4n−1 4n 4n+1 0 7 Page 30 of 35 W. Domitrz, M. Zwierzyński Fig. 20 i Acurve C , ii E (C ),(iii-vi) C and different smooth branches of E (C ) 1 1 4 4 4 4 2 2 We proceed in the same way like in the proof of Theorem 5.3. An example of a curve W and its Wigner caustic are illustrated in Fig. 11. 6 The Wigner caustic of whorls In [3] waves with vacuum wavenumber k, travelling in the ξ direction, incident nor- mally on a medium that varies periodically and weakly in the η direction were studied. This problem describes the diffraction of light by ultrasound and diffraction of beams of atoms by beams of light and dynamics of a quantum particle in an optical lattice potential ([8]). The geometry of the Wigner caustic Page 31 of 35 7 Fig. 21 An angle function of W Fig. 22 The surface parameterized by (6.1) with different opacities 1 n In natural dimensionless variables y = qη, x = q ξ (for details see [3]) the 2 n rays regarded as curves η(ξ ) are described in the following way: −1 2 2 y(x , t ) = sin sin t sn x + K (sin t )| sin t , dy(x , t ) 2 2 p(x , t ) = = sin t cn x + K (sin t )| sin t , dx π π where 0 ≤ x < ∞ and − ≤ t ≤ , K (m) is the elliptic function, sn(n|m) and 2 2 cn(n|m) are Jacobi's elliptic sine and Jacobi's cosine functions, respectively. In Fig. 22 we illustrate a surface parameterized by 3π π π 0, × − , (x , t ) → x , y(x , t ), p(x , t ) ∈ R . (6.1) 2 2 2 For fixed values of x in (6.1) we obtain so-called whorls ([3]) or rainbow diagrams ([8])—see Fig. 23. Catastrophic manifolds of the semiclassical Wigner catastrophes are formed by the Wigner caustic of a fixed whorl and by the whorl by itself ([8]). It is worth mentioning that by its construction ([3]), whorls are π-periodic in the y-value (see Fig. 24). We illustrate the Wigner caustic of the periodic whorl from Fig. 24 in Fig. 25. Notice that every center of symmetry of the π-whorl belongs to the Wigner caustic. Now, we explain why the Wigner caustic of the whorl for x = π has singular points. We apply a result on existence of singular points of the Wigner caustic ([14]). 7 Page 32 of 35 W. Domitrz, M. Zwierzyński Fig. 23 Whorls/Rainbow diagrams Fig. 24 The periodic whorl for x = π Fig. 25 The periodic whorl for x = π and its Wigner caustic The geometry of the Wigner caustic Page 33 of 35 7 Fig. 26 The whorl for x = π with tangent lines and parallel arcs Fig. 27 Translated parallel arcs from Fig. 26 Proposition 6.1 (Proposition 3.7 in [14]) Let F and F be embedded regular curves 0 1 with endpoints p, q and p, q , respectively. Let be the line through q parallel to 0 1 0 1 T F and let be the line through q parallel to T F . Let c = ∩ ,b = ∩T F , p 0 1 0 p 1 0 1 0 0 p 1 b = ∩ T F . Let us assume that 1 1 p 0 (i) the line T F is parallel to T F , and the line T F is parallel to T F , p 0 q 1 q 0 p 1 1 0 (ii) the curvature of F for i = 0, 1 does not vanish at any point, (iii) absolute values of rotation numbers of F and F are the same and smaller than 0 1 (iv) for every point a in F there is exactly one point a in F such that a , a is a i i j j i j parallel pair for i = j, (v) F , F are curved to different sides at every parallel pair a , a such that a ∈ F 0 1 0 1 i i for i = 0, 1. Let ρ (respectively ρ ) be the maximum (respectively the minimum) of the set max min c − b c − b 1 0 , .If ρ < 1 or ρ > 1, then the Wigner caustic of F ∪ F max min 0 1 q − b q − b 1 1 0 0 has a singular point. In Fig. 26 we present a π-whorl with tangent lines for parameters: t =−0.125, t ≈−1.40562, t =−0.4, t ≈−1.4511, together with parallel arcs with endpoints at 7 Page 34 of 35 W. Domitrz, M. Zwierzyński these points. In Fig. 27 we illustrate translated parallel arcs from Fig. 26, which fulfil assumptions of Proposition 6.1. Therefore, the Wigner caustic created from parallel arcs in Fig. 26 has a singular point. This method can be applied for other whorls, too. Furthermore, notice that the tangent lines to the π-whorl at the points a = (0, 0), a = (0, 1), b = (0, −1) are horizontal, and a is an inflexion point of the π-whorl. a +a a +b 0 0 Hence, by Proposition 2.9 the points = (0, 0.5) and = (0, −0.5) are 2 2 inflexion points of the Wigner caustic of the π-whorl. These points are nearby singular points of the Wigner caustic of the π-whorl. For more figures of the whorls and its Wigner caustics see [8]. Acknowledgements The authors benefitted from the hospitality of the Faculty of Mathematics of the University of Valencia during the preparation of this manuscript. Special thanks to their host, M. Carmen Romero Fuster, for suggesting the subject of this paper and many useful comments. The authors also thank Zbigniew Szafraniec for fruitful discussions and suggestions. Data Availability We do not generate any data for or from this research. Declarations Conflict of interests The authors declare that there is no conflict of interests. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. 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Domitrz, Wojciech; Zwierzyński, Michał
Analysis and Mathematical Physics
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In this paper we study global properties of the Wigner caustic of parameterized closed planar curves. We find new results on its geometry and singular points. In particular, we consider the Wigner caustic of rosettes, i.e. regular closed parameterized curves with non-vanishing curvature. We present a decomposition of a curve into parallel arcs to describe smooth branches of the Wigner caustic. By this construction we can find the number of smooth branches, the rotation number, the number of inflexion points and the parity of the number of cusp singularities of each branch. We also study the global properties of the Wigner caustic on shell (the branch of the Wigner caustic connecting two inflexion points of a curve). We apply our results to whorls—the important object to study the dynamics of a quantum particle in the optical lattice potential. Keywords Semiclassical dynamics · Affine equidistants · Wigner caustic · Singularities · Planar curves Mathematics Subject Classification 53A04 · 53A15 · 58K05 · 81Q20 1 Introduction In 1932 Eugene Wigner introduced the celebrated Wigner function to study quantum corrections to classical statistical mechanics ([31]).This function relates the wave- The work of W. Domitrz and M. Zwierzynski ´ was partially supported by NCN grant no. DEC-2013/11/B/ST1/03080. Wojciech Domitrz [email protected] Michał Zwierzynski ´ [email protected] Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warszawa, Poland 0123456789().: V,-vol 7 Page 2 of 35 W. Domitrz, M. Zwierzyński function that appears in Schrödinger's equation to a probability distribution in phase space. The Wigner function of a pure state is defined in the following way W (p, q) = ψ (q − ζ)ψ(q + ζ) exp (2ipζ/) dζ, −∞ 2 2 where (p, q) ∈ R are momentum and position, and ψ ∈ L (R) is the wavefunction. In [1] Berry studied the semiclassical limit of Wigner's phase-space representation of quantum states. He proved that for 1-dimensional systems, that correspond to smooth (Lagrangian) curves M in the phase space (R ,ω = d p ∧ dq), the semiclassical limit of the Wigner function of the classical correspondence M of a pure quantum state takes on high values at points in a neighborhood of M and also in a neighborhood of a singular closed curve, which is called the Wigner caustic of M or the Wigner catastrophe (see [1,4,10,23] for details). Geometrically the Wigner caustic of a planar curve M is the locus of midpoints of chords connecting points on M with parallel tangent lines ([1,10,11,23]). It is also the caustic of a certain Lagrangian map defined in the following way (see [10,11,23] for details). 2 2 For the canonical symplectic form ω = d p ∧ dq on R the map : T R ∗ 2 2 ∗ 2 v → ω(v, ·) ∈ T R is an isomorphism between the bundles T R and T R . Then ω ˙ = dα = d p ˙ ∧ dq + d p ∧ dq ˙ is a symplectic form on T R , where α is the ∗ 2 2 2 canonical Liouville 1-form on T R . The linear diffeomorphism : R × R → 2 2 2 T R = R × R , + + − − + − + − + − + − 1 (p , q , p , q ) = (p, q, p ˙, q ˙ ) = p + p , q + q , p − p , q − q 2 1 ∗ ∗ pulls the symplectic form ω ˙ on T R back to the canonical symplectic (π ω − π ω) + − 2 2 2 2 2 on the product R × R , where π ,π : R × R → R are the projections on the + − first and on the second component, respectively. If M is a smooth regular planar curve then M is an immersed Lagrangian submanifold of (R ,ω). Hence 1 (M × M ) is an 2 2 2 2 2 immersed Lagrangian submanifold of (T R , ω) ˙ .Let π ,π : T R = R × R → R 1 2 be the projections on the first and on the second component, respectively. Then π and π define Lagrangian fibre bundles with the symplectic structure ω ˙ . Then the caustic of the Lagrangian map (the set of its critical values) π ◦ 1 is the Wigner M ×M caustic [5,6,10,11,23]. On the other hand the Lagrangian map π ◦ 1 is the M ×M secant map of M [13]. If M is (locally) described as dS M = (p, q) ∈ R | p = (q) dq then the generating family of the Lagrangian submanifold 1 (M × M ) has the fol- lowing form F (p, q,β) = (S(q + β) − S(q − β)) − pβ. 2 The geometry of the Wigner caustic Page 3 of 35 7 Fig. 1 A non-convex curve together with its Wigner caustic Fig. 2 An improper affine sphere (with different opacities) generated from a curve in Fig. 1 The front of the Legendrian submanifold of the contact manifold (T R × R, dz + α) generated by F is a singular 2-dimensional improper affine sphere, where z is a coordinate on R. The caustic of this front is composed of the curve M and its Wigner caustic. Hence the geometry of the Wigner caustics provides information on singularities of improper affine spheres. In Fig. 1 we present a non-convex planar curve with its Wigner caustic and in Fig. 2 we show the improper affine sphere generated by M in the construction described above (see [5,6] for details). In [8](seealso[2–4]) the dynamics of a quantum particle in the optical lattice potential was investigated. The authors analyze the evolution of the Wigner function. The function undergoes a number of catastrophic changes. For a semiclassical approx- imation the Wigner caustic consists of the rainbow diagram (the original curve M) and a locus of midpoints of chords joining points on the rainbow diagram with parallel tan- gent lines. But the catastrophe set of the exact Wigner function, in addition, contains a locus of midpoints of chords joining points on neighboring rainbow diagrams with parallel tangents. Hence the Wigner caustic of the curve M should be investigated not only locally but globally too. It turns out that its global geometry is very important for understanding the quantum-classical correspondence breakdown. It allows to extract important information without using simplifying approximations. 7 Page 4 of 35 W. Domitrz, M. Zwierzyński Fig. 3 i A closed regular curve M, ii M and E (M ) Singularities of the Wigner caustic for ovals occur exactly from antipodal pairs (the tangent lines at the two points are parallel and the curvatures are equal). The well- known Blaschke-Süss theorem states that there are at least three pairs of antipodal points on an oval ([22,26]). The absolute value of the oriented area of the Wigner caustic gives the exact relation between the perimeter and the area of the region bounded by closed regular curves of constant width and improves the classical isoperimetric inequality for convex curves ([34,35,37–39]). Furthermore this oriented area improves the isoperimetric defect in the reverse isoperimetric inequality ([7]). Recently the properties of the middle hedgehog, which is a generalization of the Wigner caustic for non-smooth convex curves, were studied in [29,30]. The Wigner caustic in the literature regarding hedgehogs is known also as a projective hedgehog (see [27,28] and the literature cited therein). The Wigner caustic could be generalized to obtain an affine λ-equidistant, which is the locus of points of the above chords which divide the chord segments between base points with a fixed ratio λ. The singular points of affine equidistants create the Centre Symmetry Set, the natural generalization of the center of symmetry, which is widely studied in [11,16,18,20,24]. The geometry of an affine extended wave front, i.e. the set λ ∈[0, 1]{λ}× E (M ), where E (M ) is an affine λ λ λ-equidistant of a manifold M, was studied in [11,15]. Local properties of singularities of the Wigner caustic and affine equidistants were studied in many papers [5,9–12,19,23,25]. In this paper we study global properties of the Wigner caustic of a generic planar closed curve. In [1] Berry proved that if M is a convex curve, then generically the Wigner caustic is a parametrized connected curve with an odd number of cusp singularities and this number is not smaller than 3. It is not true in general for any closed planar curve. If M is a parametrized closed curve with self-intersections or inflexion points then the Wigner caustic has at least two branches (smoothly parametrized components). We present a decomposition of a curve into parallel arcs and thanks to this decomposition we are able to describe the geometry of branches of the Wigner caustic. In general the geometry of the Wigner caustic of a regular closed curve is quite complicated (see Fig. 3). In Sect. 2 we briefly sketch some of the known results on the Wigner caustic and affine equidistants. The geometry of the Wigner caustic Page 5 of 35 7 Section 3 contains the algorithm to describe branches of the Wigner caustic and affine equidistants of any generic regular parameterized closed curve. Subsection 3.1 provides an example of an application of this algorithm to a particular curve. In the beginning of Sect. 4 we present global propositions on the number of cusps and inflexion points of the Wigner caustic. We show that the procedure based on a decomposition presented in Sect. 3 can be applied to obtain the number of branches of the Wigner caustic, the number of inflexion points and the parity of the number of cusp singularities of each branch. After that we study global properties of the Wigner caustic on shell, i.e. the branch of the Wigner caustic which connects two inflexion points of a curve. We present the results on the parity of the number of cusp points of the branches of the Wigner caustic on shell. We also prove that each such branch has even number of inflexion points and there are even number of inflexion points on a path of the original curve between the endpoints of this branch. In Sect. 5 we use the decomposition introduced in Sect. 3 to study the geometry of the Wigner caustic of generic regular closed parameterized curves with non-vanishing curvature and of some generic regular closed parameterized curves with two inflexion points. Finally, in Sect. 6 we study the Wigner caustic of whorls. All the pictures of the Wigner caustic in this manuscript were made in the application created by the second author [36] and in Mathematica [32]. 2 Preliminaries Let M be a smooth parameterized curve in the affine plane R , i.e. the image of the ∞ 2 C smooth map from an interval to R . A smooth curve is closed if it is the image of ∞ 1 2 a C smooth map from S to R . A smooth curve is regular if its velocity does not vanish. A regular curve is simple if it has no self-intersection points. A regular simple closed curve is convex if its signed curvature has a constant sign. Let (s , s ) s → 1 2 2 k f (s) ∈ R be a parameterization of M. A point f (s ) is a C regular point of M for k = 1, 2,... or k =∞ if there exists ε> 0 such that f (s −ε, s +ε) is a C smooth 0 0 1-dimensional manifold. A point f (s ) is a singular point if it is not C regular for any k > 0. A curve is singular if it has at least one singular point. A singular point p is 2 3 2 called a cusp if M is locally diffeomorphic at p to a curve (−1, 1) t → (t , t ) ∈ R at 0. A point f (s ) is a cusp of M if and only if f (s ) = 0 and the vectors f (s ) and 0 0 0 f (s ) are linearly independent. A point f (s ) is an inflexion point of M if its signed 0 0 curvature changes sign. An inflexion point f (s ) is non-degenerate (or ordinary)if det f (s ), f (s ) = 0 which means that the order of contact of M with the tangent 0 0 line to M at f (s ) is equal to 2. If the curvature vanishes at f (s ) but does not change 0 0 sign, then this point is called an undulation point. Remark 2.1 Let (a, b) s → f (s) ∈ R be a parameterization of M, then the order of contact of M with the tangent line to M at p = f (t ) is k if and only if i k+1 d f d f d f d f det (t ), (t ) = 0for i = 1, 2,..., k and det (t ), (t ) = 0. i k+1 ds ds ds ds (2.1) 7 Page 6 of 35 W. Domitrz, M. Zwierzyński Definition 2.2 A pair of points a, b ∈ M (a = b) is called a parallel pair if the tangent lines to M at a and b are parallel. Definition 2.3 A chord passing through a pair a, b ∈ M, is the line: l(a, b) = λa + (1 − λ)b λ ∈ R . Let A be a subset of R , then clA denotes the closure of A. Definition 2.4 The Wigner caustic of M is the following set a + b E (M ) = cl a, b is a parallel pair of M . Remark 2.5 The Wigner caustic of M is an example of an affine λ-equidistant set of M, E (M ) = cl λa + (1 − λ)b a, b is a parallel pair of M , where λ = . Definition 2.4 is different from definitions in papers [5,10–12,18,19,25, 33,37,38]. The closure in the definition is needed to include inflexion points of M in E (M ). For details see Remark 2.18. Note that, for any given λ ∈ R,wehaveE (M ) = E (M ). Thus, the case λ = λ 1−λ is special. In particular we have E (M ) = E (M ) = M if M is closed. 0 1 Definition 2.6 The Centre Symmetry Set of M, denoted by CSS(M ), is the envelope of all chords passing through parallel pairs of M. Bitangent lines of M are parts of CSS(M ) ([11,16]). If M is a generic convex curve, then CSS(M ), the Wigner caustic and E (M ) for a generic λ are smooth closed curves with at most cusp singularities ([1,16,20,24]), cusp singularities of all E (M ) are on regular parts of CSS(M ) ([20]), the number of cusps of CSS(M ) and E (M ) is odd and not smaller than 3 ([1,16], see also [22]), the number of cusps of CSS(M ) is not smaller than the number of cusps of E (M ) ([11]). Let us denote by κ (p) the signed curvature of a smooth regular curve M at p.Let a, b be a parallel pair of M. Assume that κ (b) = 0. Let us fix local arc length parameterizations of M nearby the points a, b by f : (s , s ) → R and by 0 1 g : (t , t ) → R , respectively. Let us assume that the parameterizations at a and b 0 1 are in opposite directions, i.e. the velocities at a, b are opposite. Then there exists a function t : (s , s ) → (t , t ) such that 0 1 0 1 f (s) =−g (t (s)). (2.2) It is easy to see that by the implicit function theorem the function t is smooth and κ ( f (s)) t (s) = . (2.3) κ (g(t (s))) M The geometry of the Wigner caustic Page 7 of 35 7 Then by γ (s) = f (s) + g(t (s)) (2.4) 2 2 we will denote a local natural parameterization of the Wigner caustic. Whenever we will write about singular points of the Wigner caustic we will denote these points as the singular points of the parameterization given by (2.4). By direct calculations we get the following lemma. Lemma 2.7 Let M be a regular curve. Let a, b be a parallel pair of M, such that M a+b is parameterized at a and b in opposite directions and κ (b) = 0. Let p = be a regular point of E 1 (M ). Then (i) the tangent line to E 1 (M ) at p is parallel to the tangent lines to M at a and b. (ii) the curvature of E 1 (M ) at p is equal to 2κ (a)|κ (b)| M M κ (p) = . E (M ) |κ (b) − κ (a)| M M Lemma 2.7(ii) implies the following propositions. Proposition 2.8 [20] Let a, b be a parallel pair of a regular curve M, such that M is parameterized at a and b in opposite directions and one of a and b is not an inflexion a+b point. Then the point is a singular point of E (M ) if and only if κ (a) = κ (b). M M a+b Proposition 2.9 Let a, b be a parallel pair of a regular closed curve M. Then is an inflexion point of E (M ) if and only if one of the points a, b is an inflexion point of M. Let τ denote the translation by a vector p ∈ R . Definition 2.10 A curve M is curved to the same side at a and b (resp. curved to different sides), where a, b is a parallel pair of M, if the center of curvature of M at a and the center of curvature of τ (M ) at a = τ (b) lie on the same side (resp. on a−b a−b the different sides) of the tangent line to M at a. We illustrate above definition in Fig. 4. a+b Corollary 2.11 If M is curved to the same side at a parallel pair a, b, then is a regular point of the Wigner caustic of M. Proof Let us locally parameterize M at a and b in opposite directions. Then by Propo- a+b sition 2.8 a point is a singular point of E 1 (M ) if and only if κ (a) = 1. (2.5) κ (b) The right hand side of (2.5) is positive, then κ (a) and κ (b) have the same sign, M M therefore M is curved to different sides at a and b. 7 Page 8 of 35 W. Domitrz, M. Zwierzyński Fig. 4 i A curve curved to the same side at a parallel pair a, b, ii a curve curved to different sides at a parallel pair a, b ∞ 1 2 ∞ 1 2 We denote by C (S , R ) the set of C mappings from S to R ,i.e.the setof smooth closed parameterized planar curves, and by · the dot product in R . ∞ 1 2 Remark 2.12 Let f (s ), f (s ) be a parallel pair of f ∈ C (S , R ). A singular point 1 2 f (s )+ f (s ) 1 2 of the Wigner caustic of f (that is a point for which κ (s ) = κ (s ) and f 1 f 2 f (s ) · f (s )< 0) is a cusp if and only if κ (s ) = κ (s ), where κ denotes the 1 2 1 2 f f f signed curvature of f with respect to the parameterization of f , and κ denotes the derivative of the curvature with the respect to the arc length parameter ([10]). ∞ 1 2 Theorem 2.13 Let G be the subset of C (S , R ) such that each curve f in G satisfies the following conditions: (i) f is a regular curve with only non-degenerate inflexion points and no undulation points, (ii) f has only transverse self-crossings, (iii) if f (s ), f (s ) is a parallel pair of f , then f (s ) or f (s ) is not an inflexion point, 1 2 1 2 (iv) if f (s ), f (s ) is a parallel pair of f , the points f (s ), f (s ) are not inflexion 1 2 1 2 points of f , the dot product f (s ) · f (s ) is negative (respectively positive), and 1 2 κ (s ) = κ (s ) (respectively κ (s ) =−κ (s )), then κ (s ) = κ (s ), where f 1 f 2 f 1 f 2 1 2 f f κ denote the derivative of the curvature with respect to the arc length parameter. ∞ 1 2 ∞ Then G is a generic subset of C (S , R ) with Whitney C topology and the Wigner caustic of f ∈ G is the finite union of smooth curves with at most cusp singularities. Proof Since the intersection of two generic subsets is still a generic subset, it is enough to show that properties from each point are generic. The set of smooth regular closed ∞ 1 2 curves is an open and dense subset of C (S , R ) because the set of 1-jets of smooth 1 1 2 non-regular closed curves is a smooth submanifold of J (S , R ) of codimension 2. 1 2 Let f : S → R be smooth and regular. Having only non-degenerate inflexion points and no undulation points is equivalent to the following property: det( f (s), f (s)) = 0 ⇒ det( f (s), f (s)) = 0. (2.6) 3 1 3 1 2 Condition (2.6) means that the map j f : S → J (S , R ) is transversal to the 3 1 2 following submanifold of J (S , R ): 3 3 1 2 j g(s) ∈ J (S , R ) g (s) = 0, det g (s), g (s) = 0 . The geometry of the Wigner caustic Page 9 of 35 7 By the Thom Transversality Theorem (e.g. see Theorem 4.9 in [21]) Property (i) is generic. To prove genericity of the conditions (ii–iv) we will use the Thom Transversal- ity Theorem for multijets (e.g. see Theorem 4.13 in [21] for details). We denote by k 1 2 1 (2) 1 1 1 J (S , R ) the s-fold k-jet bundle and by (S ) the set S × S \{(s, s) | s ∈ S }. 1 1 (2) 1 1 2 Genericity of (ii) follows from transversality of j f : (S ) → J (S , R ) to the 2 2 1 1 2 following submanifold of J (S , R ): 1 1 1 1 2 ( j g(s ), j h(s )) ∈ J (S , R ) g(s ) = h(s ), g (s ) = 0, h (s ) = 0 . 1 2 1 2 1 2 Transversality means that if f (s ) = f (s ) for s = s , then det( f (s ), f (s )) = 0. 1 2 1 2 1 2 Therefore, Condition (ii) is generic. Genericity of Property (iii) follows from transversality of the second multijet j f : 1 (2) 2 1 2 (S ) → J (S , R ) to the submanifold 2 2 2 1 2 ( j g, j h) ∈ J (S , R ) det g (s ), h (s ) = 0, g (s ) = 0, h (s ) = 0 . 1 2 1 2 2 2 This means that if det f (s ), f (s ) = 0for s = s , then κ (s ) + κ (s ) = 0. 1 2 1 2 1 2 f f Hence, Property (iii) is generic. Now we assume that f satisfies (iii). Genericity of Property (iv) for f (s )· f (s )< 1 2 3 1 (2) 3 1 2 0 follows from the transversality of j f : (S ) → J (S , R ) to the submanifold 2 2 3 3 3 1 2 W := ( j g, j h) ∈ J (S , R ) g (s ) = 0, h (s ) = 0, det(g (s ), h (s )) = 0, 1 2 1 2 g (s ) · h (s )< 0,κ (s ) = κ (s ) . 1 2 g 1 h 2 By direct calculations one can show that this means that if j f (s , s ) ∈ W , then 1 2 κ (s ) = κ (s ), which is equivalent to the condition for a cusp singularity in a 1 2 f f singular point of the Wigner caustic (see Remark 2.12). The proof for the case f (s ) · f (s )> 0 is similar. From now one, when we will talk about generic curves, we will mean a curve from the set G. Furthermore, genericity of f implies the following geometric properties of f . ∞ 1 2 Proposition 2.14 [13]If f ∈ C (S , R ) has only non-degenerate inflexion points and has no undulation points, then the number of inflexion points of f and the rotation number of f are finite. Definition 2.15 The tangent line of E 1 (M ) at a cusp point p is the limit of a sequence of 1-dimensional vector spaces T M in RP for any sequence q of regular points q n of E (M ) converging to p. This definition does not depend on the choice of a converging sequence of regular points. By Lemma 2.7(i) we can see that the tangent line to E 1 (M ) at the cusp point a+b is parallel to tangent lines to M at a and b. 2 7 Page 10 of 35 W. Domitrz, M. Zwierzyński Remark 2.16 If M is an oval, then we have well defined the continuous normal vector a+b field on the double covering M of E 1 (M ), M a → ∈ E 1 (M ) by taking the 2 2 normal vector to M compatible with the parameterization of M at the point a, and a+b defining this vector as a normal vector to the Wigner caustic at . Let us notice that the continuous normal vector field to E 1 (M ) at regular and cusp points is perpendicular to the tangent line to E 1 (M ). Using this fact and the above definition we define the rotation number in the following way. Definition 2.17 The rotation number of the Wigner caustic of a generic curve M is the rotation number of the continuous normal vector field of the Wigner caustic. Remark 2.18 Let p be an inflexion point of M. Then the CSS(M ) is tangent to this inflexion point and has an endpoint there. The set E (M ) for λ = has an inflexion point at p (as the limit point) and is tangent to M at p. The Wigner caustic is tangent to M at p too and it has an endpoint there. The Wigner caustic and the Centre Symmetry Set approach p from opposite sides ([1,10,16,19]). This branch of the Wigner caustic is studied in Sect. 4. If M is a generic regular closed curve then E 1 (M ) is a union of smooth parametrized curves. Each of these curves we will call a smooth branch of the Wigner caustic of M. In Fig. 5 we illustrate a non-convex curve M,E 1 (M ), and different smooth branches of E (M ). 3 A decomposition of a curve into parallel arcs In this section we assume that M is a generic regular closed curve. We will present a decomposition of M into parallel arcs which will help us to study the geometry of the smooth branches of the Wigner caustic of M. 1 2 Definition 3.1 Let S s → f (s) ∈ R be a parameterization of a smooth closed curve M, such that f (0) is not an inflexion point. A function ϕ : S →[0,π ] is called an angle function of M if ϕ (s) is the oriented angle between f (s) and f (0) modulo π. We identify the set [0,π ] modulo π with S . 1 1 Definition 3.2 A point ϕ in S is a local extremum of ϕ if there exists s in S such that ϕ (s) = ϕ, ϕ (s) = 0, ϕ (s) = 0. The local extremum ϕ of ϕ is a local M M M M maximum (resp. minimum) if ϕ (s)< 0(resp. ϕ > 0). We denote by M(ϕ ) the M M set of local extrema of ϕ . The angle function has the following properties. Proposition 3.3 Let M be a generic regular closed curve. Let f be the arc length parameterization of M and let ϕ be the angle function of M. Then (i) f (s ), f (s ) is a parallel pair of M if and only if ϕ (s ) = ϕ (s ), 1 2 M 1 M 2 (ii) ϕ (s) is equal to the signed curvature of M with respect to the parameterization of M, The geometry of the Wigner caustic Page 11 of 35 7 Fig. 5 i A non-convex curve M with two inflexion points (the dashed line) and E (M ), ii–vi M and different smooth branches of E (M ) (iii) M has an inflexion point at f (s ) if and only if ϕ (s ) is a local extremum. 0 M 0 (iv) if ϕ (s ), ϕ (s ) are local extrema and there is no extremum on ϕ (s , s ) , M 1 M 2 M 1 2 then one of extrema ϕ (s ), ϕ (s ) is a local maximum and the other one is a M 1 M 2 local minimum. Lemma 3.4 Let ϕ be the angle function of a generic regular closed curve M. Then the function ϕ has an even number of local extrema, i.e. M has an even number of inflexion points . Proof It is a consequence of the fact that the number of local extrema of a generic 1 1 smooth function from S to S is even. Let ϕ be the angle function of M. Definition 3.5 The sequence of local extrema is the following sequence (ϕ ,ϕ ,..., 0 1 ϕ ) where {ϕ ,ϕ ,...,ϕ }= M(ϕ ) and the order is compatible with the 2n−1 0 1 2n−1 M 1 1 orientation of S = ϕ (S ). M 7 Page 12 of 35 W. Domitrz, M. Zwierzyński Fig. 6 i A closed regular curve M with points p = f (s ) tangent lines to M at these points, ii a graph of i i the angle function ϕ with ϕ and s values M i i Definition 3.6 The sequence of division points S is the following sequence −1 (s , s ,..., s ), where {s , s ,..., s }= ϕ (M(ϕ )) if M(ϕ ) is not empty, 0 1 k−1 0 1 k−1 M M −1 otherwise {s , s ,..., s }= ϕ ϕ (0) , and the order of S is compatible with ( ) 0 1 k−1 M M the orientation of M. Let M have inflexion points. If s belongs to the sequence of division points, then f (s ) is an inflexion point or a point which is parallel to an inflexion point. In the case when M has no inflexion points the sequence of division points consists of 0 and points s such that f (0), f (s ) are parallel pairs. k k In Fig. 6 we illustrate an example of a closed regular curve M, the angle function ϕ and the sequence of division points. Let us notice that the images of points in the sequence of division points divide the curve M into arcs. Some of these arcs (say A and A ) have the property that for any point a ∈ A there exists a point a ∈ A 2 i i j j such that a , a is a parallel pair for i = j ∈{1, 2}. Such arcs we will call parallel i j arcs. The set of arcs splits into subsets such that any two arcs in the same subset are parallel (see Definition 3.9). Proposition 3.7 If M is a generic regular closed curve and a ∈ M is an inflexion point then the number of points b ∈ M, such that b = a and a, b is a parallel pair, is even. Proof There are no inflexion points b ∈ M such that a, b is a parallel pair and the point a is not a self-intersection point of M, since the curve M is generic. The inflexion points of M correspond to local extrema of the angle function ϕ . We divide the graph of the angle function ϕ into continuous paths of the form (t,ϕ (t )) | t ∈[t , t ],ϕ (t ), ϕ (t ) ∈{0,π }, ∀t ∈ (t , t )ϕ (t)/∈{0,π } . M 1 2 M 1 M 2 1 2 M Let α belong to (0,π). First we assume that α is not equal to a local extremum of a path P. Then a line ϕ = α intersects the path P an even number of times if P is a path from 0 to 0 or from π to π, since both the beginning and the end of P areonthe same side of the line (see Fig. 7i). This line intersects P an odd number of times if P is a path from 0 to π or from π to 0, since the beginning and the end of P are on different sides of the line (see Fig. 7ii). Now we assume that α is equal to a local extremum of P. In this case the line ϕ = α intersects a path P an odd number of times if P is a path from 0 to 0 or from π to π The geometry of the Wigner caustic Page 13 of 35 7 Fig. 7 Continuous paths Fig. 8 Vertical perturbation nearby a local maximum (see Fig. 7iii) and this line intersects P an even number of times if P is a path from 0 to π or from π to 0 (see Fig. 7iv), since by a small local vertical perturbation around the extremum point we obtain the previous cases and the numbers of intersection points have a difference ±1(seeFig. 8). Let us note that a path from 0 to 0 or from π to π corresponds to an arc of a curve with the rotation number equalling 0 and a path from 0 to π or from π to 0 corresponds to an arc of a curve with the rotation number equalling ± . Since the rotation number of M is an integer, the number of paths from 0 to π or from π to 0 in the graph of ϕ is even. Each path of this type intersects every horizontal line ϕ = α at least once. −1 Thus the number of intersections of ϕ and the line ϕ = ϕ ( f (a)) is odd. But the M M number of points b = a such that a, b is a parallel pair is one less than the number of −1 intersection points of the graph of ϕ and the line ϕ = ϕ ( f (a)). M M The number of inflexion points of a generic regular closed curve is even. Thus by Proposition 3.7 we have the following corollary. Corollary 3.8 If M is a generic regular closed curve then #S is even. We recall that in this section we assume that M is a generic regular closed curve and let #S = 2m. The functions m , M :{0, 1,..., 2m − 1} →{0, 1,..., 2m − 1} are analogs 2m 2m of the minimum and the maximum functions modulo 2m, respectively. Namely, 2m − 1, if {k, l}={0, 2m − 1}, m (k, l) := 2m min(k, l), otherwise, 7 Page 14 of 35 W. Domitrz, M. Zwierzyński 0, if {k, l}={0, 2m − 1}, M (k, l) := 2m max(k, l), otherwise. We denote by (s , s ) an interval (s , L + s ), where L is the length of 2m−1 0 2m−1 M 0 M M. In the following definition indexes i in ϕ are computed modulo 2n, indexes j , j +1 in p p and p p are computed modulo 2m. j j +1 j +1 j Definition 3.9 If M(ϕ ) ={ϕ ,ϕ ,...,ϕ }, then for every i ∈{0, 1,..., 2n − M 0 1 2n−1 1},a set of parallel arcs is the following set = p p k − l =±1mod(2m), ϕ (s ) = ϕ ,ϕ (s ) = ϕ , i k l M k i M l i +1 ϕ (s , s ) = (ϕ ,ϕ ) , M m (k,l) M (k,l) i i +1 2m 2m where p = f (s ) and p p = f s , s . k k k l m (k,l) M (k,l) 2m 2m If M(ϕ ) is empty then we define only one set of parallel arcs as follows: = p p , p p ,..., p p , p p . 0 0 1 1 2 2m−2 2m−1 2m−1 0 The set of parallel arcs has the following property. 1 2 Proposition 3.10 Let f : S → R be the arc length parameterization of M. For every two arcs p p ,p p in the well defined map k l k l i p p p → P(p) ∈ p p , k l k l where the pair p, P(p) is a parallel pair of M, is a diffeomorphism. Definition 3.11 Let p p , p p belong to the same set of parallel arcs, then k k l l 1 2 1 2 p p k k 1 2 denotes the following set (the arc) p p l l 1 2 cl (a, b) ∈ M × M a ∈ p p , b ∈ p p , a, b is a parallel pair of M . k k l l 1 2 1 2 p ... p p p k k k k 1 n i i +1 In addition denotes i = 1n − 1 . We will call p ... p p p l l l l 1 n i i +1 this set a glueing scheme. Remark 3.12 If p p belongs to a set of parallel arcs, then there are neither inflexion k l points nor points with tangent lines parallel to tangent lines at inflexion points of M in p p \{ p , p }. k l k l Definition 3.13 The -point map ([11]) is the map a + b π 1 : M × M → R ,(a, b) → . 2 The geometry of the Wigner caustic Page 15 of 35 7 Fig. 9 Two arcs A and A of 1 2 M belonging to the same set of parallel arcs and E (A ∪ A ) 1 2 Let A = p p and A = p p be two arcs of M which belong to 1 k k 2 l l 1 2 1 2 the same set of parallel arcs. It is easy to see that E A ∪ A consists of one 1 2 p p k k 1 2 arc under π 1 (see Fig. 9). From this observation we get the following p p 2 l l 1 2 proposition. Proposition 3.14 The Wigner caustic E 1 (M ) is the image of the union of i different arcs under the -point map π . Proposition 3.15 Let M be a generic regular closed curve which is not convex. If a p p k k 1 2 glueing scheme is of the form , then this scheme can be prolonged in a p p l l 1 2 p p p k k k 1 2 3 unique way to such that (k , l ) = (k , l ). 1 1 3 3 p p p l l l 1 2 3 Proof Let us consider p p k k 1 2 . (3.1) p p l l 1 2 Let A = p p \{ p , p }, A = p p \{ p , p }. By Remark 3.12 A 1 k k k k 2 l l l l 1 1 2 1 2 1 2 1 2 and A must be curved to the same side or in the opposite sides at any parallel pair in A ∪ A (see Fig. 10i–ii). Let us consider the case in Fig. 10i, the other case is similar. 1 2 Then (3.1) can be prolonged in the following two ways. (1) Neither p nor p is an inflexion point of M. Then (3.1) can be prolonged to k l 2 2 p p p k k k 1 2 3 , where k = k and l = l (see Fig. 10iii). 1 3 1 3 p p p l l l 1 2 3 (2) One of points p , p is an inflexion point of M. Let us assume that this is p . k l k 2 2 2 p p p k k k 1 2 3 Then (3.1) can be prolonged to , where k = k (see Fig. 10iv). 1 3 p p p l l l 1 2 1 7 Page 16 of 35 W. Domitrz, M. Zwierzyński Fig. 10 Possible prolongations of an arc of a curve Since M is generic, at least one of the points p , p is not an inflexion point of M. k l 2 2 Remark 3.16 To avoid repetition in the union in Definition 3.11 we assume that no pair except the beginning and the end can appear twice in the glueing scheme. p p k l Furthermore, if the pair is in the glueing scheme than the pair does not appear p p l k unless they are the beginning and the end of the scheme. The image of a glueing scheme under the -point map π 1 represents parts of branches of the Wigner caustic. If we equip the set of all possible glueing schemes with the inclusion relation, then this set is partially ordered. There is only finite number of arcs from which we can construct branches of E (M ). Therefore we can define a maximal glueing scheme. Definition 3.17 A maximal glueing scheme is a glueing scheme which is a maximal element of the set of all glueing schemes equipped with the inclusion relation. Remark 3.18 If M is a generic regular convex curve, then the set of parallel arcs is equal p p 0 1 to ={ p p , p p }. Then the only maximal glueing scheme is 0 0 1 1 0 p p 1 0 Proposition 3.19 The set of all glueing schemes equipped with the inclusion relation is the disjoint union of totally ordered sets. Proof It follows from uniqueness of the prolongation of the glueing scheme (see Proposition 3.15). The geometry of the Wigner caustic Page 17 of 35 7 1 2 Lemma 3.20 Let f : S → R be the arc length parameterization of M. Then (i) for every two different arcs p p ,p p in there exists exactly one k k l l i 1 2 1 2 p p p p p p k k k k l l 1 2 2 1 1 2 maximal glueing scheme containing or or or p p p p p p l l l l k k 1 2 2 1 1 2 p p l l 2 1 p p k k 2 1 p ... p k k (ii) every maximal glueing scheme is in the following form , where p ... p l l { p , p }={ p , p } whenever p = p and p = p k l k l k l k l (iii) if p is an inflexion point of M, then there exists a maximal glueing scheme which is in the form p p ... p p k k k l 1 n p p ... p p k l l l 1 n where p is a different inflexion point of M and p = p for i = 1, 2,..., n. l k l i i Proof (i) is a consequence of the uniqueness of the prolongation of a glueing scheme (see Proposition 3.15). The proof of (ii) follows from (i) and the fact that the following equalities hold: p p p p p p p p k k k k k k l l 1 2 2 1 1 2 1 2 = and = . p p p p p p p p l l l l l l k k 1 2 2 1 1 2 1 2 p p k k To prove (iii) let us prolong to the maximal glueing scheme G.Any p p k l point p in the sequence of division points S belongs to exactly two arcs in all sets l M of parallel arcs. Then by (ii) this maximal glueing scheme is in the following form p p ... p p k k k l 1 n , (3.2) p p ... p p k l l l 1 n If (3.2) would contain some other inflexion point p in the middle, then (3.2) would contain the following part: p p r r r p p r r r which is impossible by (i). Theorem 3.21 The image of every maximal glueing scheme of M under the -point map π 1 is a branch of the Wigner caustic of M and all branches of the Wigner caustic can be obtained in this way. 1 2 Proof Let f : S → R be the arc length parameterization of M. It is easy to see that S = {s} (s , s ) m (k,l) M (k,l) 2m 2m s∈S i (k,l)∈ M i 7 Page 18 of 35 W. Domitrz, M. Zwierzyński and then M = { f (s)}∪ f (s , s ) , m (k,l) M (k,l) 2m 2m s∈S i (k,l)∈ M i where denotes the disjoint union. Then by Proposition 3.10 we obtain that E (M ) = E p p ∪ p . (3.3) 1 1 k l k l 2 2 i p p ,p k l i k l p p =p k l k l p p p k l k l Since E 1 p p ∪ p p = π 1 = π 1 and k l k l 2 2 p p 2 p p k l k l p p k l every arc is in exactly one maximal glueing scheme, then every branch of k l the Wigner caustic is the image of a maximal glueing scheme under the -point map π 1 . As a summary of this section we present an algorithm to find all maximal glueing schemes. Algorithm 1 (Finding all maximal glueing schemes of a generic regular closed curve 1 2 M parametrized by f : S → R ) (1) Find the set of local extrema of the angle function ϕ of M (see Definition 3.1 and Definition 3.2). (2) Find the sequence of local extrema (see Definition 3.5). (3) Find the sequence of division points (see Definition 3.6). (4) Find the sets of parallel arcs (see Definition 3.9). (5) Create the following set := p p , p p : p p = p p , k l k l k l k l 1 1 2 2 1 1 2 2 ∃ p p ∈ ∧ p p ∈ ∨ p p ∈ ∧ p p ∈ . i k l i k l i l k i l k i 1 1 2 2 1 1 2 2 (6) If there exists a number k such that p is an inflexion point of M and there exists the set of arcs p p , p p or p p , p p in , create a k l k l l k l k 1 2 1 2 p p k l glueing scheme , remove the used set of arcs from and go to step (7). p p k l Otherwise go to step (8). ... p (7) If the created glueing scheme is of the form and there exists the set of ... p arcs p p , p p or p p , p p in , then prolong the k k l l k k l l 1 2 1 2 2 1 2 1 ... p p k k 1 2 scheme to the following scheme , remove the used set of arcs ... p p l l 1 2 The geometry of the Wigner caustic Page 19 of 35 7 Fig. 11 Acurve M as in Fig. 6 and different branches of E (M ) from and go to step (7), otherwise the considered glueing scheme is a maximal glueing scheme and then go to step (6). (8) If is empty, then all maximal glueing schemes for E 1 (M ) were created, other- wise find any set of arcs p p , p p in , create a glueing scheme k l k l 1 1 2 2 p p k l 1 1 , remove the used set of arcs from and go to step (7). p p k l 2 2 3.1 An example of construction of branches of the Wigner caustic Let M be a curve illustrated in Fig. 6. Then the sets of parallel arcs are as follows = p p , p p , 0 0 1 4 5 = p p , p p , p p , p p . 1 1 2 3 2 3 4 5 0 Then there exist two maximal glueing schemes of M: p p p p p 0 1 2 3 4 , (3.4) p p p p p 4 5 0 5 0 p p p p 2 1 2 3 . (3.5) p p p p 2 3 4 3 By Proposition 4.3 the number of cusps of the branch which correspond to (3.4)is odd. By Corollary 4.4 in the glueing scheme (3.4) there are two parallel pairs containing an inflexion point of M – the pairs: ... p p ... 2 3 ... p p ... 0 5 Therefore this branch of the Wigner caustic has exactly two inflexion points—see Fig. 11ii. The same conclusion holds for the glueing scheme (3.5) and the branch in Fig. 11i. In this case we exclude the first and the last parallel pair. 7 Page 20 of 35 W. Domitrz, M. Zwierzyński Fig. 12 A continuous normal vector field at a cusp singularity 4 The geometry of the Wigner caustic of regular curves In this section we start with propositions on numbers of inflexion points and cusp singularities of the Wigner caustic which follows from properties of maximal glueing schemes introduced in Sect. 3. Proposition 4.1 Let M be a generic regular closed curve. If M has 2n inflexion points then there exist exactly n smooth branches of E 1 (M ) connecting pairs of inflexion points on M and every inflexion point of M is the end of exactly one branch of E 1 (M ). Other branches of E 1 (M ) are closed curves. Proof It is a consequence of Lemma 3.20 and Theorem 3.21. Lemma 4.2 Let C be a closed smooth curve with at most cusp singularities. If the rotation number of C is an integer, then the number of cusps of C is even and if the rotation number of C is a half-integer, then the number of C is odd. Proof A continuous normal vector field to the germ of a curve with a cusp singularity is directed outside the cusp on one of two connected regular components and is directed inside the cusp on the other component as it is illustrated in Fig. 12. That observation ends the proof. Proposition 4.3 Let M be a generic regular closed curve. Letn be a unit continuous normal vector field to M. Let C be a smooth branch of E 1 (M ) which does not connect inflexion points. Then the number of cusps of C is odd if and only if the maximal glueing p ... p k l scheme of C is in the following form and n (p ) =−n (p ). M l M k p ... p l k Proof If the normal vectors to M at p and p are opposite, then the rotation number k l of C is equal to , where r is an odd integer. By Lemma 4.2 the number of cusps in C is odd. Otherwise the rotation number of C is an integer, therefore the number of cusps of C is even. By Proposition 2.9, Corollary 3.8 and Proposition 4.1 we get the following corol- laries on inflexion points of branches of the Wigner caustic of M. Corollary 4.4 Let M be a generic regular closed curve. Let C be a smooth branch of the Wigner caustic of M. Then the number of inflexion points of C is equal to the The geometry of the Wigner caustic Page 21 of 35 7 number of parallel pairs containing an inflexion point of M in the maximal glueing scheme for C unless they are the beginning or the end of the maximal glueing scheme which connects the inflexion points of M. Corollary 4.5 Let M be a generic regular closed curve. Let 2n > 0 be the number of inflexion points of M and let #S = 2m. Then E 1 (M ) has 2m − 2n inflexion points. Now we study the properties of the Wigner caustic on shell, i.e. the branch of the Wigner caustic connecting two inflexion points, see Fig. 11i. We are interested in the parity of the number of cusps and the parity of the number of inflexion points on this branch. 1 2 Theorem 4.6 Let M be a generic regular closed curve. Let S s → f (s) ∈ R be a parameterization of M, let f (t ),f (t ) be inflexion points of M and let C be a branch 1 2 of the Wigner caustic of M which connects f (t ) and f (t ). Then the number of cusps 1 2 of C is odd if and only if exactly one of the inflexion points f (t ), f (t ) is a singular 1 2 point of the curve C ∪ f [t , t ] . 1 2 Proof By genericity of M the points f (t ) and f (t ) are ordinary inflexion points of 1 2 M. By Corollary 4.8. in [10] we know that the germ of the Wigner caustic at an inflexion point of a generic curve M together with M are locally diffeomorphic to the following germ at (0, 0): 2 2 2 (p, q) ∈ R : p = 0 ∪ (p, q) ∈ R : p =−q , q ≤ 0 . Let N = C ∪ f [t , t ] . Then N is a closed curve. The germ of N at f (t ) for 1 2 i i = 1, 2 is locally diffeomorphic to one of the following germs at (0, 0): 2 2 2 (p, q) ∈ R : p = 0, q ≤ 0 ∪ (p, q) ∈ R : p =−q , q ≤ 0 , (4.1) 2 2 2 (p, q) ∈ R : p = 0, q > 0 ∪ (p, q) ∈ R : p =−q , q ≤ 0 . (4.2) In other points N has at most cusp singularities. Note that the point (0, 0) is a singular point of the germ of type (4.1) and the point (0, 0) is a C -regular point of the germ of type (4.2) (see Fig. 13). Let M p → n (p) ∈ S be a continuous normal vector field to M. Let us assume that the maximal glueing scheme for C has the following form p p ... p p k k k k 1 2 n−1 n p p ... p p l l l l 1 2 n−1 n where k = l , k = l . Without loss of generality we can assume that k < k .Let 1 1 n n 1 n us define a normal vector field n to N as follows: • n (p) = n (p) for p ∈ f [t , t ] , N M 1 2 7 Page 22 of 35 W. Domitrz, M. Zwierzyński Fig. 13 A continuous normal vector field to the germs of type (4.1)and (4.2) a+b • n (p) = n (a) for p ∈ C, where p = , a, b is a parallel pair of M such that N M there exists i ∈{1, 2,..., n − 1} such that a ∈ p p , b ∈ p p . k k l l i i +1 i i +1 The vector field n is a continuous unit normal field to N . The normal vector field around the points of type (4.1) and (4.2) is described in Fig. 13. Thus by the same argument as in the proof of Lemma 4.2 we can get that the total number of cusps and singularities of type (4.1)in N is even, so the number of cusps of C is odd if and only if exactly one of the inflexion points f (t ), f (t ) is of type (4.1). 1 2 In Fig. 5iii there is exactly one point of type (4.1), in Fig. 11i there is an even number of points of type (4.1). Proposition 4.7 Let M be a regular curve. Let (a, b) s → f (s) ∈ R be a param- eterization of M and let f (s ) be an ordinary inflexion point of M. Let t be a smooth function-germ on R at s such that f (s), f (t (s)) is a parallel pair and lim t (s) = s . 0 0 s→s Let κ (s) be the curvature of M at a point f (s). Then κ (s) lim =−1. (4.3) s→s 0 κ (t (s)) Furthermore let C be a branch of the Wigner caustic which ends in f (s ).If d f d f det (s ), (s ) = 0, (4.4) 0 0 ds ds then C ∪ f [s , b) at f (s ) is of type (4.1)if 0 0 d κ (s) lim > 0 (4.5) s→s 0 ds κ (t (s)) M The geometry of the Wigner caustic Page 23 of 35 7 and C ∪ f [s , b) at f (s ) is of type (4.2)if 0 0 d κ (s) lim < 0. (4.6) s→s 0 ds κ (t (s)) Proof Without loss of generality we may assume that locally f (s) = (s, F (s)), (4.7) where F (s) = as + G(s), a = 0 and s = 0, where G(s) ∈ m , where m is the 0 n maximal ideal of smooth function-germs R → R vanishing at 0. Let us notice that (s, F (s)), (t , F (t )) is a parallel pair of M nearby f (0) if and only if s = t and F (s) − F (t ) = 0. This is equivalent to (s − t )(3as + 3at + H (s, t )) = 0, where H ∈ m and let P(s, t ) = 3as + 3at + H (s, t ).Let t : (R, 0) → (R, 0) be a function-germ at 0 such that P(s, t (s)) = 0. (4.8) By the implicit function theorem the function-germ t is well defined, because ∂ P (0, 0) = 3a = 0. By (4.8) we get that ∂t ∂ P (s, t ) ∂s t (s) =− . (4.9) ∂ P (s, t ) ∂t It implies that t (0) =−1. (4.10) Since F (s) = F (t (s)), then for s = 0 F (s) κ (s) t (s) = = . (4.11) F (t (s)) κ (t (s)) Thus (4.3) holds (Fig. 14). (4) The condition (4.4) means that F (0) = 0. It implies that M is not locally centrally symmetric around f (s ) = (0, 0). 0 7 Page 24 of 35 W. Domitrz, M. Zwierzyński Fig. 14 Acurve M with an inflexion point and the Wigner caustic of M (the dashed line) The branch of the Wigner caustic which contains f (0) has the following parame- terization x 1 (s) = s + t (s), F (s) + F (t (s)) . (4.12) Therefore x (s) = (1 + t (s)) 1, F (s) . (4.13) Since C ∪ f [t , t ] at f (t ) can be only of type (4.1)or(4.2), then C ∪ f [t , t ] 1 2 1 1 2 at f (t ) is of type (4.1) if and only if x (s) f (s)< 0 whenever s → t , therefore by (4.13) we get that 1 + t (s)< 0. By (4.10) we get that t (s)> 0 and by (4.11)we finish the proof. Remark 4.8 Under the assumptions of Theorem 4.7 if locally f (s) = (s, F (s)) then (4) d κ (s) 2F (s ) M 0 lim =− . (4.14) (3) s→s 0 ds κ (t (s)) 3F (s ) M 0 1 2 Theorem 4.9 Let M be a generic regular closed curve. Let S s → f (s) ∈ R be a parameterization of M, let f (s ),f (s ) be inflexion points of M and let C be a 1 2 branch of the Wigner caustic of M which connects f (s ) and f (s ). Then the number 1 2 of cusps of C is odd if and only if d κ (s) d κ (s) M M lim · lim > 0, (4.15) ± ∓ ds κ (t (s)) ds κ (t (s)) s→s M 1 s→s M 2 1 2 where κ (s) denotes the curvature of M at f (s), the pairs f (s), f (t (s)) and M 1 f (s), f (t (s)) are parallel pairs such that t (s) → s whenever s → s and s < t (s) 2 i i i i for the left-hand side neighborhood of s for i = 1, 2. i The geometry of the Wigner caustic Page 25 of 35 7 Proof By genericity of M we get that f (s ) and f (s ) are ordinary inflexion points. 1 2 Then the theorem is a consequence of Theorem 4.6 and Proposition 4.7. Now we study inflexion points on the Wigner caustic on shell. 1 2 Theorem 4.10 Let M be a generic regular closed curve. Let S s → f (s) ∈ R be a parameterization of M and let C be a branch of the Wigner caustic which connects two inflexion points f (t ) and f (t ) of M. Then the number of inflexion points of C 1 2 and the number of inflexion points of the arc f (t , t ) are even. 1 2 1 1 Proof Let ϕ : S → S be the angle function of M. By the genericity of M all local extrema of ϕ are different. Let 1 1 ψ ,ψ :[0, T]→ graph ϕ ⊂ S × S 1 2 M be the following continuous functions: ψ (0) = ψ (0) = t ,ϕ (t ) ,ψ (T ) = ψ (T ) = t ,ϕ (t ) , 1 2 1 M 1 1 2 2 M 2 ψ (t ) = s (t ), ϕ (s (t )) for i = 1, 2, i i M i where continuous functions s , s :[0, T]→ S satisfy ϕ s (t ) = ϕ s (t ) and 1 2 1 M 1 M 2 s (t ) = s (t ) for t ∈ (0, T ). 1 2 Since f (t ) is an inflexion point then ϕ (t ) is a local extremum. Without loss 1 M 1 of generality we assume that ϕ (t ) is a local minimum. To prove that the number M 1 of inflexion points in f (t , t ) is even it is enough to show that ϕ (t ) is a local 1 2 M 2 maximum. The numbers of local maxima and local minima of ϕ are equal. Thus the difference between the number of local maxima and local minima of ϕ is one. For small S −{t } ε> 0 the arcs ψ and ψ define the opposite orientations of the graph of ϕ 1 2 M [0,ε] [0,ε] and ϕ ◦ s increases. Let ϕ s (t ) for i = 1or i = 2 be a local extremum of M i M i [0,ε] ˜ ˜ ϕ such that there are no extrema on ϕ s (0, t ) and ϕ s (0, t ) . Since ϕ (t ) is M M 1 M 2 M 1 ˜ ˜ ˜ a local minimum then ϕ (s (t )) is a local maximum and ψ (t − ε, t + ε) for j = i M i j changes the orientation in t (see Fig. 15). The numbers of local maxima and local minima of ϕ are equal but the arcs S −[t ,t ] ψ and ψ define the same orientation of graph ϕ . Since the function 1 2 M [t˜,t˜+ε] [t˜,t˜+ε] ϕ increases then the next extremum is a local minimum. The number of local ˜ ˜ [t ,t +ε] minima decreases by 1 and the orientations are opposite after crossing the minimum. Thus the defined orientations are opposite if and only if the difference between the number of local maxima and local minima to cross is one. Since for small ε> 0the arcs ψ and ψ define the opposite orientations of the graph of ϕ , 1 2 M [T −ε,T ] [T −ε,T ] then ϕ (t ) must be a local maximum. M 2 A point (a + b) is an inflexion point of C if and only if one of the points of the parallel pair a, b is an inflexion point of M. The number of inflexion points of 7 Page 26 of 35 W. Domitrz, M. Zwierzyński Fig. 15 A change of the angle function ϕ Fig. 16 Acurve M with 8 inflexion points (the dashed line) and branches of the Wigner caustic between inflexion points of M C is equal to the sum of the number of changes of the orientations of ψ and ψ 1 2 ˜ ˜ ˜ ˜ because ψ (t − ε, t + ε) changes the orientation in t if and only if ϕ (s (t )) is a i M i local extremum. Since ϕ (t ) is a minimum and ϕ (t ) is a maximum, then the total M 1 M 2 number of changes of the orientations is even. In Fig. 16 we illustrate a closed curve M and branches of the Wigner caustic between inflexion points of M.InFig. 17 we illustrate a closed curve M such that the branch of the Wigner caustic which connects two inflexion points of M has no inflexion points. Lemma 4.11 Let C be a smooth closed curve with at most cusp singularities. Then the number of inflexion points of C is even. Proof If C is regular, i.e. has no cusp singularities, then by Lemma 3.4 we get that C has an even number of inflexion points. If C has cusp singularities, then we change C nearby each cusp in the way illustrated in Fig. 18 creating two more inflexion points. After this transformation of C we obtain a regular closed curve C such that the parity of the numbers of inflexion points of C and C are equal. Therefore the number of inflexion points of C is even. The geometry of the Wigner caustic Page 27 of 35 7 Fig. 17 Acurve M (the dashed line) and E (M ) Fig. 18 Acurve C with the cusp singularity at x and a curve C with inflexion points at p and q Proposition 4.12 Let M be a generic regular closed curve. Then the number of inflex- ion points of each smooth branch of the Wigner caustic of M is even. Proof Let us notice that all branches of E 1 (M ) except the branches of the Wigner caustic which connect two inflexion points of M are closed curves. So the result for these branches follows from Lemma 4.11. Otherwise it follows from Theorem 4.10. 5 The Wigner caustic of closed curves with at most 2 inflexion points In this section we study the geometry of the Wigner caustic of closed regular curves with non-vanishing curvature (rosettes) and of closed regular curves with exactly two inflexion points. Definition 5.1 A smooth curve γ : (s , s ) → R is called a loop if it is a simple 1 2 + − curve with non-vanishing curvature such that lim s → s γ(s) = lim s → s γ(s).A 1 2 loop γ is called convex if the absolute value of its rotation number is not greater than 1, otherwise it is called non-convex. We illustrate examples of loops in Fig. 19. Theorem 5.2 ([14]) The Wigner caustic of a loop has a singular point. Theorem 5.3 Let C be a generic regular closed parameterized curve with non- vanishing curvature with rotation number equal to n. Then 7 Page 28 of 35 W. Domitrz, M. Zwierzyński Fig. 19 i A convex loop L (the dashed line) and E (L), ii a non-convex loop L (the dashed line) and E (L) 1 1 2 2 (i) the number of smooth branches of E 1 (C ) is equal to n, (ii) at least branches of E 1 (C ) are regular closed parameterized curves with non-vanishing curvature, (iii) n − 1 branches of E 1 (C ) have a rotation number equal to n and one branch has a rotation number equal to , (iv) every smooth branch of E (C ) has an even number of cusps if n is even, (v) exactly one branch of E (C ) has an odd number of cusps if n is odd, (vi) cusps of E 1 (C ) created from loops of C are in the same smooth branch of n n E 1 (C ), (vii) the total number of cusps of E 1 (C ) is not smaller than 2, Proof Since the rotation number of C is n, for any point a in C there exist exactly n n 2n − 1 points b = a such that a, b is a parallel pair of C . Thus the set of parallel arcs has the following form = p p , p p ,..., p p , p p . 0 0 1 1 2 2n−2 2n−1 2n−1 0 Let E 1 (C ) be a smooth branch of E 1 (C ). We can create the following maximal n n ,k 2 2 glueing schemes. • A maximal glueing scheme of E 1 (C ) for k ∈{1, 2,..., n − 1}: ,k p p p ... p p p 0 1 2 2n−2 2n−1 0 p p p ... p p p k k+1 k+2 k−2 k−1 k • A maximal glueing scheme of E 1 (C ): ,n p p p . . . p p 0 1 2 n−1 n p p p . . . p p n n+1 n+2 2n−1 0 The total number of arcs of the glueing schemes for the Wigner caustic presented above is n(2n −1). By Proposition 3.14 the total number of different arcs of the Wigner The geometry of the Wigner caustic Page 29 of 35 7 caustic is equal to the same number. Thus there are no more maximal glueing schemes for the Wigner caustic of C . If (a , a ,..., a ) is a sequence of points in C with the order compat- 0 1 2n−1 n ible with the orientation of C such that a , a is a parallel pair, then C is n i j n curved to the same side at a and a if and only if i − j is even. Thus branches i j E 1 (C ), E 1 (C ),..., E 1 n (C ) are created from parallel pairs a, b in C n n n n ,2 ,4 ,2· 2 2 2 2 such that C is curved to the same side at a and b and all the other branches of the Wigner caustic of C are created from parallel pairs a, b in C such that n n C is curved to different sides at a and b. By Corollaries 2.11 and 4.5 branches E 1 (C ), E 1 (C ),..., E 1 n (C ) are regular closed parameterized curves with n n n ,2 ,4 ,2· 2 2 2 2 non-vanishing curvature. By Proposition 4.3 the branch E 1 (C ) is the only branch of the Wigner caustic ,n of C which has an odd number of cusps if n is odd. We can see that the part of the Wigner caustic created from loops of C are all in E 1 (C ).Every C for n > 1 has at least one loop, so E 1 (C ) has at least one cusp, n n n ,1 ,1 2 2 but because E 1 (C ) has an even number of cusps, then E 1 (C ) has at least two n n ,1 ,1 2 2 cusps. In Fig. 20i we illustrate a curve of the type C and E 1 (C ).InFig. 20iii–vi we 4 4 illustrate different smooth branches of E 1 (C ). Theorem 5.4 Let W be a generic closed curve with the rotation number n. Let W n n have exactly two inflexion points such that one of the arcs of W connecting inflexion points is an embedded curve with the absolute value of the rotation number smaller than . Then (i) the number of smooth branches of E 1 (W ) is equal to n + 1, (ii) n − 1 branches of E 1 (W ) have a rotation number equal to n and one branch has a rotation number equal to , (iii) n − 1 branches of E 1 (W ) have four inflexion points and two branches have two inflexion points, (iv) every smooth branch of E (W ), except a branch connecting inflexion points of W , has an even number of cusps if n is even, (v) exactly one smooth branch of E 1 (W ), except a branch connecting inflexion points of W , has an odd number of cusps if n is odd, (vi) cusps of E 1 (W ) created from convex loops of W are in the same smooth branch n n of E 1 (W ). Proof One can notice that the graph of the angle function ϕ has the form presented in Fig. 21. For that parameterization we get that f (s ) and f (s ) corresponds to inflexion 0 1 points of W and the sets of the parallel arcs are as follows: = p p , p p , p p , p p ,..., p p , p p , 0 2 3 4 5 6 7 8 9 4n−2 4n−1 4n 4n+1 = p p , p p , p p , p p ,..., p p , p p . 1 0 1 1 2 3 4 5 6 4n−1 4n 4n+1 0 7 Page 30 of 35 W. Domitrz, M. Zwierzyński Fig. 20 i Acurve C , ii E (C ),(iii-vi) C and different smooth branches of E (C ) 1 1 4 4 4 4 2 2 We proceed in the same way like in the proof of Theorem 5.3. An example of a curve W and its Wigner caustic are illustrated in Fig. 11. 6 The Wigner caustic of whorls In [3] waves with vacuum wavenumber k, travelling in the ξ direction, incident nor- mally on a medium that varies periodically and weakly in the η direction were studied. This problem describes the diffraction of light by ultrasound and diffraction of beams of atoms by beams of light and dynamics of a quantum particle in an optical lattice potential ([8]). The geometry of the Wigner caustic Page 31 of 35 7 Fig. 21 An angle function of W Fig. 22 The surface parameterized by (6.1) with different opacities 1 n In natural dimensionless variables y = qη, x = q ξ (for details see [3]) the 2 n rays regarded as curves η(ξ ) are described in the following way: −1 2 2 y(x , t ) = sin sin t sn x + K (sin t )| sin t , dy(x , t ) 2 2 p(x , t ) = = sin t cn x + K (sin t )| sin t , dx π π where 0 ≤ x < ∞ and − ≤ t ≤ , K (m) is the elliptic function, sn(n|m) and 2 2 cn(n|m) are Jacobi's elliptic sine and Jacobi's cosine functions, respectively. In Fig. 22 we illustrate a surface parameterized by 3π π π 0, × − , (x , t ) → x , y(x , t ), p(x , t ) ∈ R . (6.1) 2 2 2 For fixed values of x in (6.1) we obtain so-called whorls ([3]) or rainbow diagrams ([8])—see Fig. 23. Catastrophic manifolds of the semiclassical Wigner catastrophes are formed by the Wigner caustic of a fixed whorl and by the whorl by itself ([8]). It is worth mentioning that by its construction ([3]), whorls are π-periodic in the y-value (see Fig. 24). We illustrate the Wigner caustic of the periodic whorl from Fig. 24 in Fig. 25. Notice that every center of symmetry of the π-whorl belongs to the Wigner caustic. Now, we explain why the Wigner caustic of the whorl for x = π has singular points. We apply a result on existence of singular points of the Wigner caustic ([14]). 7 Page 32 of 35 W. Domitrz, M. Zwierzyński Fig. 23 Whorls/Rainbow diagrams Fig. 24 The periodic whorl for x = π Fig. 25 The periodic whorl for x = π and its Wigner caustic The geometry of the Wigner caustic Page 33 of 35 7 Fig. 26 The whorl for x = π with tangent lines and parallel arcs Fig. 27 Translated parallel arcs from Fig. 26 Proposition 6.1 (Proposition 3.7 in [14]) Let F and F be embedded regular curves 0 1 with endpoints p, q and p, q , respectively. Let be the line through q parallel to 0 1 0 1 T F and let be the line through q parallel to T F . Let c = ∩ ,b = ∩T F , p 0 1 0 p 1 0 1 0 0 p 1 b = ∩ T F . Let us assume that 1 1 p 0 (i) the line T F is parallel to T F , and the line T F is parallel to T F , p 0 q 1 q 0 p 1 1 0 (ii) the curvature of F for i = 0, 1 does not vanish at any point, (iii) absolute values of rotation numbers of F and F are the same and smaller than 0 1 (iv) for every point a in F there is exactly one point a in F such that a , a is a i i j j i j parallel pair for i = j, (v) F , F are curved to different sides at every parallel pair a , a such that a ∈ F 0 1 0 1 i i for i = 0, 1. Let ρ (respectively ρ ) be the maximum (respectively the minimum) of the set max min c − b c − b 1 0 , .If ρ < 1 or ρ > 1, then the Wigner caustic of F ∪ F max min 0 1 q − b q − b 1 1 0 0 has a singular point. In Fig. 26 we present a π-whorl with tangent lines for parameters: t =−0.125, t ≈−1.40562, t =−0.4, t ≈−1.4511, together with parallel arcs with endpoints at 7 Page 34 of 35 W. Domitrz, M. Zwierzyński these points. In Fig. 27 we illustrate translated parallel arcs from Fig. 26, which fulfil assumptions of Proposition 6.1. Therefore, the Wigner caustic created from parallel arcs in Fig. 26 has a singular point. This method can be applied for other whorls, too. Furthermore, notice that the tangent lines to the π-whorl at the points a = (0, 0), a = (0, 1), b = (0, −1) are horizontal, and a is an inflexion point of the π-whorl. a +a a +b 0 0 Hence, by Proposition 2.9 the points = (0, 0.5) and = (0, −0.5) are 2 2 inflexion points of the Wigner caustic of the π-whorl. These points are nearby singular points of the Wigner caustic of the π-whorl. For more figures of the whorls and its Wigner caustics see [8]. Acknowledgements The authors benefitted from the hospitality of the Faculty of Mathematics of the University of Valencia during the preparation of this manuscript. Special thanks to their host, M. Carmen Romero Fuster, for suggesting the subject of this paper and many useful comments. The authors also thank Zbigniew Szafraniec for fruitful discussions and suggestions. Data Availability We do not generate any data for or from this research. Declarations Conflict of interests The authors declare that there is no conflict of interests. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Berry, M.V.: Semi-classical mechanics in phase space: a study of Wigners function. Philos. Trans. R. Soc. Lond. A 287, 237–271 (1977) 2. Berry, M.V., Balazs, N.L.: Evolution of semiclassical quantum states in phase space. J. Phys. A Math. Gen. 12, 625–642 (1979) 3. Berry, M.V., ODell, D.H.J.: Ergodicity in wave-wave diffraction. J. Phys. A Math. Gen. 32, 3571–3582 (1999) 4. Berry, M.V., Wright, F.J.: Phase-space projection identities for diffraction catastrophes. J. Phys. A Math. Gen. 13, 149–160 (1980) 5. Craizer, M., Domitrz, W., Rios, P.D.M.: Even dimensional improper affine spheres. J. Math. Anal. Appl. 421, 1803–1826 (2015) 6. 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Analysis and Mathematical Physics – Springer Journals
Published: Mar 1, 2022
Keywords: Semiclassical dynamics; Affine equidistants; Wigner caustic; Singularities; Planar curves; 53A04; 53A15; 58K05; 81Q20
Semi-classical mechanics in phase space: a study of Wigners function
Berry, MV
Evolution of semiclassical quantum states in phase space
Berry, MV; Balazs, NL
Ergodicity in wave-wave diffraction
Berry, MV; ODell, DHJ
Phase-space projection identities for diffraction catastrophes
Berry, MV; Wright, FJ
Even dimensional improper affine spheres
Craizer, M; Domitrz, W; Rios, PDM
Singular improper affine spheres from a given Lagrangian submanifold
A note on hurwitzs inequality
Cufi, J; Gallego, E; Reventós, A
On the phase-space catastrophes in dynamics of the quantum particle in an optical lattice potential
Ćosić, M; Petrović, S; Bellucci, S
Singularities of Affine Equidistants: extrinsic geometry of surfaces in 4-space
Domitrz, W; Janeczko, S; Rios, PDM; Ruas, MAS
The Wigner caustic on shell and singularities of odd functions
Domitrz, W; Manoel, M; Rios, PDM
Singularities of Equidistants and global centre symmetry sets of Lagrangian submanifolds
Domitrz, W; Rios, PDM
Singularities of affine equidistants: projections and contacts
Domitrz, W; Rios, PDM; Ruas, MAS
The geometry of the secant caustic of a planar curve
Domitrz, W; Romero Fuster, MC; Zwierzyński, M
Singular points the Wigner caustic and affine equidistants of planar curves
Domitrz, W; Zwierzyński, M
The Gauss-Bonnet theorem for coherent tangent bundles over surfaces with boundary and its applications
Local forms and transitions of the medial axis
Giblin, PJ; Kimia, BB; Siddiqi, K; Pizer, S
Centre symmetry sets of families of plane curves
Giblin, PJ; Reeve, GM
Bifurcations of affine equidistants
Giblin, PJ; Warder, JP; Zakalyukin, VM
Singularities of centre symmetry sets
Giblin, PJ; Zakalyukin, VM
An antipodal set of a periodic function
Góźdź, S
Geometry of two dimensional tori in phase space: projections
Ozorio de Almeida, AM; Hannay, J
Bifurcations of the center of symmetry
Janeczko, S
Symmetry defect of algebraic varieties
Janeczko, S; Jelonek, Z; Ruas, MAS
New notion of index for hedgehogs of R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}^3$$\end{document} and applications, rigidity and related topics in geometry
Martinez-Maure, Y
Uniqueness results for the Minkowski problem extended to hedgehogs
On the quantum correction for thermodynamic equilibrium
Wigner, E
Envelopes of families of wave fronts and control theory
Zakalyukin, VM
A mixed symmetric chernoff type inequality and its stability properties
Zhang, D
The lower bounds of the mixed isoperimetric deficit
The improved isoperimetric inequality and the Wigner caustic of planar ovals
Zwierzyński, M
Isoperimetric equalities for rosettes
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Domitrz, W., & Zwierzyński, M. (2022). The geometry of the Wigner caustic and adecomposition of a curve into parallel arcs. Analysis and Mathematical Physics, 12(1),
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The sum of three numbers $a$, $b$, and $c$ is 99. If we increase $a$ by 6, decrease $b$ by 6 and multiply $c$ by 5, the three resulting numbers are equal. What is the value of $b$?
We are given the equations $a+b+c=99$ and $a+6=b-6=5c$. Solve $b-6=5c$ for $b$ to find $b=5c+6$, and solve $5c=a+6$ for $a$ to find $a=5c-6$. Substituting both of these equations into $a+b+c=99$, we have $(5c-6)+(5c+6)+c=99$. Simplifying the left hand side, we get $11c=99$ which implies $c=9$. Substituting into $b=5c+6$, we have $b=5(9)+6=\boxed{51}$. | Math Dataset |
Logarithms and Age Counting
Amusingly, the age difference between a 45-year-old man and a 25-year-old woman doesn't seem as big as the age difference between them 20 years earlier, when the woman was a little 5-year-old girl. This remark was the insight the late science popularizer Albert Jacquart liked to give to his readers to explain logarithms. This article pays tribute to the great scientist by introducing age difference as he liked to tell it.
October 9, 2013 ArticleCalculus, MathematicsLê Nguyên Hoang 7493 views
I had just finished reading his book La Science à l'usage des non-scientifiques when, sadly, Albert Jacquard, aged 2.06, passed away, on September 11, 2013.
What? He was 2.06?
That's how he liked his age to be given! And, to pay tribute to this renowned geneticist and great science popularizer, I've decided to write this article to explain how and why he liked ages to be given that way!
Logarithms and Scales
Before getting to age counting, let me first introduce logarithms, which are the key mathematical objects to describe ages as Albert Jacquart liked to do it.
Logarithms? What the hell is that?
The logarithm is an operator to solve equations like $10^x = 100$.
The solution is 2, right?
Yes! That's why we write $\log_{10} (100) = 2$.
But what's the point in solving such an equation?
Mathematicians like to study the solutions of equations… But more practically, the logarithm is an amazing tool to write down in a readable way extremely huge or extremely small number. For instance, the logarithms of all the scales of the universe only range between -35 and 27! This is what's remarkable with logarithms! They enable to capture all the scales of the complex universe with two-digit numbers! These scales are obtained with logarithms are called logarithmic scales. For an awesome example, check this awesome animation by Cary Huang on htwins.net.
The fact that all scales of our universe lie within 2-digit logarithmic values actually highlights how small astronomical figures are. In comparison, mathematics and cryptography often deal with much larger numbers, whose logarithms can equal millions! Find out more about very big numbers with my talk in A Trek through 20th Century Mathematics.
So, logarithmic scales are great to talk about scales of the universe?
Yes! But that's not all. In fact, that's not where they are used the most! Many other measurements are made with such logarithm scales. This is the case, for instance, of decibels, used to measure the intensity of signals, like in acoustics or photography, as you can read it in my article on high dynamic range. In both cases, sensors like our ears, eyes, microphones or cameras have the amazing ability of capturing a very large range of sound or light intensities. For instance, our eyes can see simultaneously a dark spot and a spot a million times brighter, while our ears can hear sounds from 0 to 120 decibels, the latter being 10^12 times louder than the former! Making sense of these large scales is more doable with logarithmic scales!
Another important example is the measure of acid concentration in water (the pH), as explained in this great video by Steve Kelly on TedEducation:
So, if I get it right, logarithms aren't necessarily in base 10, right?
Yes. But let's talk about other bases later…
What do you want to talk about?
One area in which logarithms are essential is to describe of growth. Because logarithms capture huge numbers with small numbers, it means that it takes an extremely huge numbers for its logarithm to be big. Mathematically, we say that logarithms have smaller growth than any function $x^\alpha$ for $\alpha > 0$. Thus, we often use them as a benchmark to discuss small growths. A crucial example of that is the fundamental prime number theorem.
The what theorem?
The prime number theorem is a characterization of the distribution of primes. First observed by Carl Friedrich Gauss, it was later proved by Hadamard and de la Vallée-Poussin. It says that the average gap between consecutive primes grows as we consider bigger and bigger numbers, and this growth is the same as the growth of logarithms. But, rather than my explanations, listen to Marcus du Sautoy's:
What I find particularly puzzling with the prime number theorem is that the logarithm involved is actually the natural logarithm we'll talk about later! As you'll see, the definition of the natural logarithm has absolutely nothing to do with primes!
Other important examples of comparisons of growth appear in complexity theory. You can read about it, for instance, in my article on parallelization.
Age Counting
From a mathematical perspective, being able to write down huge numbers isn't a ground-breaking achievement. That's not what makes logarithms essential to mathematics. What does is their ability to transform multiplications into additions. And that's the key property of logarithms which led Albert Jacquart to propose to use them to define ages.
Albert Jacquart noticed that there was not as much age difference between a 25-year old girl and a 45-year old man as there is between a 5-year old girl and a 25-year old. This idea is illustrated by the following epic extract from Friends, where Monica tells her parents that she is dating their old friend Richard:
Humm.. That's funny! Age difference seems to decrease… with age…
Well that's not that surprising when you think about it. After all, a 25-year old man has lived 5 times longer than a 5-year old girl, but there's not such a big ratio between a 45-year old man and a 25-year old girl.
So you're saying that age difference should be counted in terms of age ratio rather than… age difference?
Well, sort of… According to Albert Jacquart, the reason why age difference doesn't mean what it should mean is because we're not using the right unit to measure age!
What? Our measure of time is wrong?
Our measure of time is fine. But what's misleading is our way of saying how old we are! In particular, a right way to express age difference should rather correspond to the ratio of the amounts of time lived.
The first step is to compare lifetime to a relevant characteristic amount of time which represents life. Albert Jacquart liked to choose the human gestation duration (9 months). Then, he proposed to write the number of gestation duration we have lived as a power of 10. And this power would then be defined as the age. In other words, the age is now defined as the logarithm of the number of gestation durations one has lived. This corresponds to the following formulas:
Recall, that the Age in the formula is how Albert Jacquart defines it! It's not what we usually call age!
So, how old are Monica and Richard?
25-year-old Monica has lived 300 months. That's $300/9 = 33$ gestation durations. Thus, her age is $\log_{10}(33) \approx 1.52$. Meanwhile, 45-year-old Richard has age $\log_{10}(45 \times 12/9) \approx 1.78$.
This all sounds complicated… Why should we do that?
Well, we said that what matters is the ratio of the amount of times people have been alive, right? So, to compare the ages of Richard and Monica, we would have to divide the respective Richard's lifetime by Monica's lifetime… And, magically, we obtain the following equation:
So the ratio of lifetimes now corresponds to… an actual age difference! That's what we wanted!
So what are the age differences between Monica and Richard at the time of the episode and 20 years earlier?
The age difference between 25-year-old Monica and 45-year-old Richard is $1.78-1.52 \approx 0.26$. Twenty years before that, the age difference was $\log_{10}(25 \times 12/9) – \log_{10}(5 \times 12/9) = 0.70$. Compared to twenty years earlier, Monica and Richard are now nearly the same age! Plus, wait another 20 years, and their age difference would then be 0.16…
I know most people don't like decimal numbers. To fix this, we can choose another base for the logarithm, or, more simply, multiply all ages by 100! And before you go around and say this is nuts, it's actually the core of Weber-Fechner law!
Products Become Sums
This ability logarithms have to transform multiplication into addition is the core of its potency in mathematics. Before computers were invented, this yielded a powerful way to quickly compute huge multiplications, as explained in the following video by Numberphile:
But with the invention of computers, this has become completely useless, right?
Pretty much! But that's not the only application of the ability of logarithms to transform products into sums. In statistics, to adjust models, one classical technics consists in searching for parameters which make the observations the most likely. This is known as the maximum likelihood estimation method. It's the one I used to estimate the levels of national football teams to simulate world cups!
What do logarithms have to do with that?
The likelihood is then a probability of a great number of events occurring. Assuming these events independent, the likelihood then equals the multiplication of the probabilities of the events. Yet, to maximize the likelihood, the classical approach consists in differentiating it. And, as you've probably learned it, differentiating a product is quite hard. The awesomeness of logarithms is to transform the product into a sum, which is infinitely easier to differentiate!
The moral is that, whenever you must differentiate a complex product, try to differentiate its logarithm instead!
I'll keep that in mind!
Another area where logarithms are essential is Shannon's information theory and entropy in thermodynamics. In particular, to express the amount of information a system can contain, Shannon had the brilliant idea to consider it to be the logarithm of the number of states it can be in.
Why is that such a great idea?
When you have two hard drives, the number of states they can be in is the product of the number of states each can be in. By using Shannon's quantification of information, called entropy, the amount of information two hard drives can contain is now the sum of the amounts of information of each of them! That's what we really mean when we say that 1 Gigabytes plus 1 Gigabytes equals 2 Gigabytes! Behind this simple sentence lies the omnipotence of logarithms!
From a pure mathematical viewpoint, this ability of logarithms to transform products into additions is a fundamental connection between the two operations. We say that logarithms induce an equivalence between products of positive numbers and sums of real numbers.
More precisely, the set of logarithms is exactly the set of continuous group isomorphisms from $(\mathbb R^*_+, \times)$ to $(\mathbb R, +)$. In fact, the logarithm is even differentiable, and its inverse is too, making the logarithm a Lie group diffeomorphism!
Does that mean that we can go the other way around?
Yes! The other ways around are known as exponentials. Exponentials transform sums into products. Just like logarithms, exponentials are defined by a base. If an exponential and a logarithm are defined with the same base, then the exponential of the logarithm and the logarithm of the exponential get us back to our initial point. For instance, $\log_{10} (10^x) = 10^{\log_{10}(x)} = x$. Geometrically, this beautiful property means that the main diagonal is an axis of symmetry between the graphs of exponentials and the graphs of logarithms, as displayed in the figure on the right.
Calculus and Natural Logarithm
The area where logarithms have strived the most is calculus, especially in differential and integral calculus.
That's because the primitive of $1/x$ is… a logarithm!
If you don't know what a primitive is, don't be scared! I'll explain everything!
A logarithm? Why on earth would that be?
Hehe!!! Let's prove it! Let's show that the primitive of $1/x$ transforms multiplications into additions. Since only logarithms can do that continuously, this will prove that the primitive must be a logarithm.
Wait… What's a primitive?
A primitive is a measure of the area below the curve. In our case, the primitive we will be focusing on equals the area below the curve $1/x$ between $1$ and $X$, as described below. Let's call it $Area(1,X)$, instead of its usual complicated notation $\int_1^X dx/x$.
Note that if $X$ is actually in the left of 1, then $Area(1,X)$ is the opposite of the area under the curve between $X$ in $1$. In other words, $Area(1,X) = – Area(X,1)$.
Now, I want you to prove that $Area(1,X)$ is actually a logarithm of $X$!
What? I thought I was only supposed to read!
Come on! It's a cool exercise!
I have no idea where to start!
Read what I've just said earlier!
You said something about proving that the primitive transforms multiplications into additions…
Yes! What does that mean?
I guess it means that I have to prove that $Area(1,X \times Y) = Area(1,X) + Area(1,Y)$…
But how on earth can I prove that?
When I'm stuck, I like to doodle…
Good idea! Let me draw the three areas!
Here, let me help you out:
So, to prove $Area(1,X \times Y) = Area(1,X) + Area(1,Y)$, what you really need to prove is that…
The green area is the same as the blue one!
Exactly! Technically, what you've just used is Chasles relation $Area(1, X \times Y) = Area(1, Y) + Area(Y, XY)$. By then subtracting $Area(1,Y)$ in both sides of the equation above, the equation to prove then becomes $Area(Y, XY) = Area(1,X)$. That's the equality of the green and blue areas!
But you're not done yet…
I know… But how can I prove that the green and blue areas are equal?
Compare them!
Humm… I know! For one thing, the blue area is a horizontal stretching of the green one by a factor $Y$!
Exactly! The blue area is horizontally $Y$ times longer! What about vertically?
I know! If we then contract the green area vertically by a factor $Y$, its area won't have changed!
Bingo! Here's a figure of the operations you are talking about!
To be fair, you'd still need to prove that the horizontally stretched and vertically contracted green area perfectly matches the original blue area. But I'll leave that as an homework exercise!
That's why the green and blue areas area equal… And $Area(1, XY) = Area(1,X) + Area(1,Y)$! That's brilliant!
I know! That's why the primitive of $1/x$ is a logarithm! It's known as the natural logarithm, and is commonly denoted $\ln x = Area(1,x)$. The base of this logarithm is a weird number though, called Euler's number in reference to the great mathematician Leonhard Euler. It is commonly denoted $e$ and is approximately $e \approx 2.7$. It stands for the solution to the equation $Area(1,x) = 1$.
But how does $\ln$ compare to $\log_{10}$?
You should try to figure it out yourself!
Come on! I've just proved a difficult theorem!
To find out how to change base, you can simply play around with formulas. Eventually, you'll obtain $\log_c x = \log_b x / \log_b c$. Thus, in particular, $\ln x = \log_{10} x / \log_{10} e$.
This last section is going to be more technical… A read of my article on differential calculus and infinite series is advised. If you're not familiar with these important topics of mathematics, you should still be able to follow the main ideas though.
Historically, the invention of logarithms was accompanied with the first studies of infinite sums, also known as infinite series. In particular, power series were to provide deep insights into common diverse functions, including logarithms.
What's a power series?
A power series is an infinite sum of terms $a_n$ multiplied by $x^n$. We write it $\sum a_n x^n$, and it sort of means $a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \ldots$, and so on to infinity. But, as you can read it in my article on infinite series where I explain why $1+2+4+8+16+… = -1$, infinite sums can be tricky!
What do they have to do with logarithm?
Amusingly, any of the usual functions we use can be written as a power series. The function is then uniquely identified by the terms $a_n$ of the series of $\sum a_n x^n$. And this is the case of logarithms! Sort of…
Well, actually, the function we should try to write as series is rather $\ln(1+x)$… So, without further ado, let's find the terms $a_n$ corresponding to $\ln(1+x)$!
How can we do that?
First, we write $\ln(1+x) = \sum a_n x^n$. Then, we'll use the differential properties of the natural logarithm we have found out earlier.
Are you referring to the fact that the natural logarithm is the primitive of $1/x$?
Yes! This means that if we differentiate the natural logarithm, we should obtain is $1/x$. Now, if you remember your courses of computation of derivatives, you should now be able to compute the derivative of $\ln(1+x)$!
If you've forgotten, note that the derivative of $f(g(x))$ is $g'(x) f'(g(x))$…
I've found $1/(1+x)$!
Excellent! Now, let's find the power series of $1/(1+x)$! The key is for you to remember (or learn about) how to calculate sums of geometric series…
I don't remember it!
Let me redo the calculation then… We have $\sum x^n = 1 + x^1 + x^2 + x^3 + \ldots = 1 + x(1+x+x^2+ \ldots) = 1 + x \sum x^n$. Therefore, $(1-x) \sum x^n = 1$, and $\sum x^n = 1/(1-x)$.
I remember now! But that's not $1/(1+x)$…
To get from $1/(1-x)$ to $1/(1+x)$, we just need to replace $x$ by $-x$! This gives us $1/(1+x) = 1/(1-(-x)) = \sum (-x)^n = \sum (-1)^n x^n$. That's a power series!
But how can we now retrieve the power series of $\ln(1+x)$?
We have $\ln(1+x) = \sum a_n x^n$. If we differentiate both sides, we now have $1/(1+x) = \sum n a_n x^{n-1}$. Arranging both sides then yields $\sum (-1)^n x^n = \sum (n+1) a_{n+1} x^n$. Thus…
$a_{n+1} = (-1)^n/(n+1)$, right?
Yes, or, by replacing $n+1$ by $n$, we have $a_n = (-1)^{n+1}/n$.
Each term $a_n$ could have also been computed by differentiating $\ln(1+x)$ $n$ times, taking the values of the derivatives for $x=0$ and dividing that by $n!$. Such a technic yields the Taylor and Maclaurin series.
We can now write $\ln(1+x)$ as a power series!
Well, sort of…
What do you mean? Is it not right?
Sadly, you won't be able to compute $\ln 3$ with this formula, as the power series does not converge for $x=2$! It just gets bigger towards plus infinity and minus infinity alternatively! In fact, as displayed in the animation on the right where terms in the power series are added sequentially to get closer to the actual value of $\sum a_n x^n$, the equality will only hold for logarithms of values between 0 and 2! That's the horribly everlasting trouble of power series!
Technically, power series always have a convergence radius $R$, possibly infinite. This means that the equality of a power series with the function it stands for will only yield for $-R < x < R$. In our case, the convergence radius equals 1, which means that, for $-1 < x < 1$, we have $\ln(1+x) = \sum (-1)^{n+1} x^n/n$. But this no longer holds if $|x| > 1$.
Complex Calculus
This last section is going to be even more technical. A read of my article on complex numbers is greatly advised here.
Still, an amazing empowering of the expansion of $\ln(1+x)$ in infinite series is the possibility we now have to define logarithms of complex numbers! Indeed, for any complex number $z$ whose module is smaller than 1, the series $\sum (-1)^{n+1} z^n/n$ converges, and defines a value for $\ln(1+z)$.
In fact, this formula can be used for any algebraic ring. For instance, you can use it to define $\ln(I_n+M)$ when $M$ is a linear endomorphism or a square matrix! The awesome thing is that, providing power series are well-defined, if $MN = NM$, we would always have $\ln(MN) = \ln M + \ln N$, as this property is encoded in the power series expansion!
OK… but this limits us to merely a small area of the complex plane…
Indeed, the values $1+z$ for $|z| <1$ is a disk centered on $1$ and of radius 1. But, amazingly, we can then write the power series of $\ln(c+z)$, for any $c$ such that $\ln c$ has been defined! In another disk now centered on $c$, this will define new values for the complex logarithm $\ln z$! By doing so, we will have expanded the domain of definition of the natural logarithm in the complex plane.
The power series expansion of $\ln(c+z)$ can be deduced, for instance, with the equality $\ln(c+z) = \ln (c(1+z/c)) = \ln c + \ln (1+z/c) = \ln c + \sum (-1)^{n+1} z^n/(nc^n)$. This proves that the radius of convergence is then $|c|$, which means that the origin is at the edge of the disk of convergence.
An example of three first steps of expansions is pictured below:
By continuing this on to infinity, we can now define the logarithms for nearly all points in the complex plane! This amazing technic is known as analytic continuation.
This idea of analytic continuation is a critical step in Riemann hypothesis, one of the Millenium prize problems and the greatest open problem in number theory.
Will we reach all points in the complex plane?
Some points will be unreachable no matter how hard we try to expand the analytic continuation. But in the case of the logarithm, the only unreachable point is $0$! We say that $0$ is a pole of the natural logarithm.
Can't we have some contradictory values for the logarithm between two expansions?
Unfortunately, yes we can… The thing is that each expansion is valid locally. Each expansion is in agreement with the expansions of its neighbors. However, as we turn around the origin, we have some expansions which have been built from a clockwise expansion of the original expansion around the origin, while others have been built anti-clockwise. These two kinds of expansions won't agree. This fundamental result says that the natural logarithm cannot be uniquely expanded to the whole complex plane!
Does it have to do with $e^{2ik\pi} = 1$ for all integers $k$?
Exactly! The natural logarithm is actually defined up to $2i\pi$! That's why Bernhard Riemann had the brilliant idea of defining the natural logarithm on a sort of infinite helicoidal staircase rather than on a complex plane. This staircase is known as the Riemann surface of the natural logarithm, and is explained by Jason Ross in the video extract below:
To provide a nearly natural well-defined natural logarithm in the complex plane, mathematicians often choose to cut it along a forbidden half line starting at the origin. Typically, the half line of negative number is chosen, and we choose the determination of the logarithm which yields $\ln 1 = 0$. Then, we apply the analytic continuation, but we forbid an analytic continuation to cross the forbidden half line. These restrictions ensure that the expansion of the natural logarithm to the complex plane minus the forbidden half line is well-defined and unique. This is what's pictured below:
The natural logarithm we obtain by doing so is such that $\ln z$ always has an imaginary part in $]-\pi, \pi[$. It is known as the principal value of the logarithm.
Let's Conclude
The take-away message of this article is that logarithms are a hidden structure between multiplications and additions. This is the fundamental property of logarithms, and it has many direct applications in computations, calculus and information theory. And age counting… An important implication of that property is the fact that logarithms can capture the size of huge numbers by small ones. This has plenty of applications to measurements in physics and chemistry. It is also essential to describe growths, like in the prime number theorem.
Finally, since we have defined logarithms for complex numbers, let's mention what happens if we try to define logarithms for other sorts of numbers. In particular, in modular arithmetic, the logarithm modulo $p$ base $b$ of $n$ is naturally defined as the power $x$ such that $b^x$ is congruent to a certain number $n$ modulo $p$. If $p$ is prime, then this logarithm is well-defined. However, computing $\log_{b,p}(n)$ is considered as a difficult problem, and is thus an interesting property for cryptography. This is what's explained in this great video by ArtOfTheProblem:
To find out more about cryptography, read Scott's article on cryptography!
Finally, I can't resist showing you a surprising connections between logarithms and newspaper digits. More precisely, between logarithms and first digits of newspaper numbers. This mind-blowing connection is known as Benford's laws… Well, I'll just let James Grime explain it to you!
What Benford's law hints at is that age isn't the only measure which should rather be quantified with logarithmic scales…
© 2016 par Lê Nguyên Hoang, avec Wordpress. | CommonCrawl |
Complex numbers and algebra
Exploring the beauty of complex numbers, their origins and why they are important
Exploring the landscape of the Riemann Zeta function, adapted from Flickr, user parameter_bond
Jamie Handitye
Imaginary and complex numbers. Perhaps they are incorrectly named, and Gauss even called for the 'imaginary' numbers to be renamed lateral numbers. However, most young mathematicians are not concerned with their naming but instead might be asking questions such as:
Do they even have a meaning?
Do they relate to any other area of mathematics?
These are justified questions. One might start with their discovery. Though complex numbers are commonly taught to students through considering the roots of the function $f(x)=x^2+1$, their origin is in fact from investigating cubic curves in the 16th century.
We start just before the 16th century. Cubic equations (equations of the form $ax^3+bx^2+cx+d=0$) had been reduced to simpler cubics in the form $x^3+px+q=0$. These cubics without an $x^2$ term were known as depressed cubics and this made solving cubic equations much easier.
Niccolò Fontana Tartaglia, the self-taught student that defeated Antonio Fiore. Public Domain
We now skip ahead to the 16th century. An Italian professor, Scipione del Ferro, at Bologna University had found a formula to solve the depressed cubics with $p$ positive and $q$ negative. On his death bed in 1526, he confided his proof and formula to his student Antonio Fiore. Fiore was at the university Bologna, where they regularly held mathematical competitions. Fiore now had the formula that mathematicians had been searching for and with the proof in hand, challenged a self-taught Niccolò Tartaglia. In an amazing feat, Tartaglia was able to derive the formula for the cubic before the competition and won against Fiore. The formula (without proof) was again passed on from a Tartaglia to Gerolamo Cardano. Cardano simply worked backwards and reconstructed the proof.
'L'Algebra', a central figure in the understanding of imaginary numbers. Public Domain
One problem was that part of the formula could have led to taking the root of a negative number and Cardano was the first to really consider it as a possibility. However, he never included this in any of his writing. On the other hand, Rafael Bombelli, in his writing 'l'Algebra' explicitly made note of this. It took time for complex numbers to be accepted but eventually, their usefulness outweighed the difficulty some mathematicians had in understanding them.
The cubic history of the complex numbers is fascinating. However, looking at quadratic functions might be easier. Let us reconsider the curve $f(x)=x^2+1$. According to an algebraic principle named the fundamental theorem of algebra, every polynomial with degree $n$ must have exactly $n$ roots. Therefore, since this curve has degree 2, it should have exactly two roots. An attempt to find the roots of $f(x)$ might look like this:
$$x^2+1=0 $$
$$x^2=-1 \, \, \, \therefore \, \, \, \text{"No solutions"}$$
This doesn't make sense though. The fundamental theorem of algebra states that there should be 2 roots, yet this answer suggests there are none. To correct the mistake, the phrase "No solutions" should read "No real solutions". Taking the square root yields the result that $x=\pm \sqrt{-1}$. This produces exactly two roots for the function. $\sqrt{-1}$ was special and was given its own symbol, $\mathrm{i}$, and become the imaginary unit.
Creating beauty
Mathematicians' belief in the existence of $\mathrm{i}$ as a plausible mathematical concept was vital for what is considered by some to be the most beautiful equations in all of mathematics. The first was discovered by Abraham De Moivre. We will slowly work towards it by first noticing that:
$$(\cos \theta+\mathrm{i}\sin \theta)^2 =\cos^2\theta – \sin^2\theta +(2\cos \theta \sin \theta)\mathrm{i}=\cos2\theta+\mathrm{i}\sin2\theta \, \, \text{(1)}.$$
Maybe we can start to see a pattern and we might even conjecture that:
$$(\cos\theta+\mathrm{i}\sin\theta)^n= \cos(n\theta)+\mathrm{i}\sin(n\theta).$$
Let $P(n)$ be the mathematical statement that equation (1) is true for all real values of $\theta$. We will prove this by induction on $n$ starting from the `base case', $n=1$. We clearly have:
$$(\cos\theta+\mathrm{i}\sin\theta)^1=\cos1\theta+\mathrm{i}\sin1\theta$$
so $P(1)$ is true. Now, for the inductive step, we assume the `Inductive Hypothesis' that $P(k)$ is true for some positive integer $k$. That means:
$$(\cos\theta+\mathrm{i}\sin\theta)^k=\cos k\theta+\mathrm{i}\sin k\theta.$$
We will use this to show that $P(k + 1)$ is correct.:
\begin{align*}
(\cos\theta+\mathrm{i}\sin\theta)^{k+1}&=(\cos\theta+\mathrm{i}\sin\theta)^k (\cos\theta+\mathrm{i}\sin\theta)\\
&=(\cos k\theta+\mathrm{i}\sin k\theta)(\cos\theta+\mathrm{i}\sin\theta)\\
&=\cos k\theta\cos\theta+\mathrm{i}\cos k\theta\sin\theta-\sin k\theta\sin\theta+\mathrm{i}\sin k\theta\cos\theta\\
&=\cos k\theta\cos\theta-\sin k\theta\sin\theta+\mathrm{i}(\sin k\theta\cos\theta+\cos k\theta\sin\theta)\\
&=\cos (k\theta+\theta)+\mathrm{i}\sin(k\theta+\theta)\\
&=\cos((k+1)\theta)+\mathrm{i}\sin((k+1)\theta)
\end{align*}
We conclude that, if $P(k)$ is true, then $P(k+1)$ is true. Therefore, since $P(1)$ is true $P(n)$ is true for all positive integers $n$ by the principle of mathematical induction.
The second formula I want to mention follows from the first and is perhaps even more aesthetically pleasing. Although discovered by Roger Coates, the equation is named Euler's identity due to how much Euler was able to manipulate and further this identity. It is given as;
$$e^{i\theta} = \cos\theta+\mathrm{i}\sin\theta.$$
This proof we are going to give is one of three proofs that Euler gave to try and prove this beauty. To get to this mathematical gem, we start by considering the expansion of $e^x$:
$$e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots.$$
Next, consider the case where the index is an imaginary number, say $\mathrm{i}x$:
\begin{align*}e^{\mathrm{i}x}&=1+\frac{\mathrm{i}x}{1!}+\frac{(\mathrm{i}x)^2}{2!}+\frac{(\mathrm{i}x)^3}{3!}+\frac{(\mathrm{i}x)^4}{4!}+\frac{(\mathrm{i}x)^5}{5!}+\frac{(\mathrm{i}x)^6}{6!}+\frac{(\mathrm{i}x)^7}{7!}+\frac{(\mathrm{i}x)^8}{8!}+\cdots \\
&=1+\mathrm{i}\frac{x}{1!}-\frac{x^2}{2!}-\mathrm{i}\frac{x^3}{3!}+\frac{x^4}{4!}+\mathrm{i}\frac{x^5}{5!}-\frac{x^6}{6!}-\mathrm{i}\frac{x^7}{7!}+\frac{x^8}{8!}+\cdots
This can be separated into real parts and imaginary parts:
$$e^{\mathrm{i}x}=(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}+ \cdots ) + \mathrm{i}(\frac{x}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!} +\cdots)$$
If you are familiar with the Maclaurin series expansion, you will quickly realise that the infinite sums in parenthesis are the expansions for $\cos x$ and $\sin x$ respectively. Ergo:
$$e^{\mathrm{i}x}=\cos x+\mathrm{i}\sin x$$
However, to show that $x$ here is an angle measured in radians, the formula is most commonly seen as:
$e^{\mathrm{i}\theta}=\cos\theta+\mathrm{i}\sin\theta$,
and therefore $e^{i\pi} = -1$. Wow! The elegance of Euler's identity is that it yields more wonderful results. For example, we can continue exploring by taking $\theta$ to be the sum of two angles. Let these angles be A and B, where A and B are measured in radians. This produces the following identities which we used in the proof of De Moivre's formula:
$$(\cos A+\mathrm{i}\sin A)(\cos B+\mathrm{i}\sin B) =e^{\mathrm{i}A} e^{\mathrm{i}B} = e^{\mathrm{i}(A+B)}=\cos (A+B)+\mathrm{i}\sin(A+B)$$
but also:
$$(\cos A+\mathrm{i}\sin A)(\cos B+\mathrm{i}\sin B) = \cos A\cos B-\sin A\sin B+\mathrm{i}(\cos A\sin B+\sin A\cos B).$$
Equating real and imaginary parts gives:
$$\cos A\cos B-\sin A\sin B= \cos(A+B)$$
$$\cos A\sin B+\sin A\cos B= \sin(A+B)$$
In the same way we could replace $\theta$ with $A-B$ to derive expressions for $\cos(A-B)$ and for $\sin(A-B)$. In addition, all the other trigonometric identities used in A-level mathematics can be derived just from these identities.
Many students might be aware of the geometric proof of these identities and might still be struggling to memorise these sometimes-confusing identities. However, I hope that after seeing this you might gain a deeper understanding of your learning, truly enjoying the hidden beauty of mathematics instead of rote learning mathematics. Perhaps the elegance of using Euler's formula is that it highlights something quite interesting: there is a clear link between algebra and geometry, namely complex numbers. Furthermore, it provides evidence that mathematics is a collection of unified and collected ideas.
Why do they matter?
The discovery of complex numbers was extremely important in algebra. To fully appreciate this, we must consider our number system, something that many students are never explicitly taught. Initially, humans made use of natural numbers $\{1,2,3…\}$ to count naturally occurring objects- for example, 20 cows or 3 apples. Integers were then discovered to understand concepts such as debt. Integers extended the number of system and the natural numbers were a subset of the integers as shown by the image below. The next extension of the numbers system was the discovery of the rational numbers, which are defined as the ratio of two integers. The number system was yet again extended through the introduction of the real numbers, a set which contained the rational and irrational numbers. Mathematicians thought that the number system was complete, and that algebra was now understood. However, the discovery of the complex numbers exposed the incompleteness of the number system. In fact, the complex numbers were the final extension of the number system and are what is known as the algebraic closure of the number system. They completed traditional algebra.
Let's reconsider the function curve $f(x)=x^2+1$ we were working with at the beginning. When we attempted to find the roots of this function, we arrived at the equation $x^2=-1$. Now consider the case $z^n=\pm1$, where $n$ is a real number. The first equation, $z^n=1$, is a classic equation in complex analysis. Its solutions are named the roots of unity as they are the roots of one (unity). They are very interesting, but I was more intrigued with a variation of the roots of unity. Replacing 1 with $i$ to get:
$$z^n=\mathrm{i}.$$
I came across this idea whilst considering if I could take the square root of $\sqrt{-1}$. Since:
$$e^{\mathrm{i}\theta}=\cos\theta+\mathrm{i}\sin\theta,$$
we could find a value of $\theta$ such that $\cos\theta=0$ and $\sin\theta=1$ to obtain an expression for $\mathrm{i}$. For example, $\theta=\pi/2$. Note: there are infinitely many values of $\theta$ that satisfy these requirements since $\sin\theta$ and $\cos\theta$ are periodic functions. However, working with values in the range $-\pi < \theta \leq \pi$ is easier. So we have $e^{\mathrm{i} \pi/2}=\cos \pi/2+\mathrm{i}\sin \pi/2$, or:
$$e^{\mathrm{i} \pi/2}=\mathrm{i}.$$
As a result, taking roots becomes simpler. For example:
$$\sqrt{\mathrm{i}}=e^{\mathrm{i}\pi/4}$$
$$e^{\mathrm{i} \pi/4}=\cos \frac{\pi}{4}+\mathrm{i}\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}+\mathrm{i}\frac{\sqrt{2}}{2}.$$
Think about the shape that all the roots of $\mathrm{i}$ would make on the complex plane. Try and plot these points on some graphing software such as Desmos. Could you explain why they form this shape by considering Euler's identity and your knowledge of trigonometry.
We can continue exploring. What about taking the $\mathrm{i}$-th root of $\mathrm{i}$? Recall that $e^{\mathrm{i}\pi/2}=\mathrm{i}$, so maybe:
$$\sqrt[\mathrm{i}]{\mathrm{i}}= (e^{\mathrm{i} \pi/2})^{1/\mathrm{i}}$$
$$\sqrt[\mathrm{i}]{\mathrm{i}}=e^{\pi/2}.$$
Since $e$ and $\pi$ are real numbers, we can deduce that $\sqrt[\mathrm{i}]{\mathrm{i}}$ is a real number. This is a stunning result but there is a slight problem. In the same way that the principal value of $\sin^{-1}(1/2)$ is $\pi/6$ rads, $e^{\pi/2}$ can be considered to be the principal value of $\sqrt[\mathrm{i}]{\mathrm{i}}$. There are actually infinitely many values we could assign to the expression $\sqrt[i]{i}$.
In conclusion, complex numbers are something of beauty. They are just as real as real numbers, just as tangible and just as necessary. As I said before, complex numbers show that mathematics is not a collection of separated ideas but instead consists of linked and connected fields. In this case, we have seen the intrinsic link between complex analysis, geometry and algebra.
A challenge
But the fun doesn't end there. As you might have seen, investigating complex numbers really is fascinating. Complex numbers can even be applied to situations that don't seem to require them. We leave it is a challenge to the reader to use complex numbers to evaluate the following integral:
$$\int_{0}^{1}\frac{\sin(\log x)}{\log x}dx.$$
A solution will be uploaded to this blog in the near future.
A solution has now been uploaded! See this article.
Jamie is a second year mathematics student at Christ's College Cambridge. His main interests are in group theory, number theory, and a touch of algebraic geometry.
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"How to Feed a Brain is an important book. It's the book I've been looking for since sustaining multiple concussions in the fall of 2013. I've dabbled in and out of gluten, dairy, and (processed) sugar free diets the past few years, but I have never eaten enough nutritious foods. This book has a simple-to-follow guide on daily consumption of produce, meat, and water.
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Phenylpiracetam (Phenotropil) is one of the best smart drugs in the racetam family. It has the highest potency and bioavailability among racetam nootropics. This substance is almost the same as Piracetam; only it contains a phenyl group molecule. The addition to its chemical structure improves blood-brain barrier permeability. This modification allows Phenylpiracetam to work faster than other racetams. Its cognitive enhancing effects can last longer as well.
Take at 11 AM; distractions ensue and the Christmas tree-cutting also takes up much of the day. By 7 PM, I am exhausted and in a bad mood. While I don't expect day-time modafinil to buoy me up, I do expect it to at least buffer me against being tired, and so I conclude placebo this time, and with more confidence than yesterday (65%). I check before bed, and it was placebo.
For obvious reasons, it's difficult for researchers to know just how common the "smart drug" or "neuro-enhancing" lifestyle is. However, a few recent studies suggest cognition hacking is appealing to a growing number of people. A survey conducted in 2016 found that 15% of University of Oxford students were popping pills to stay competitive, a rate that mirrored findings from other national surveys of UK university students. In the US, a 2014 study found that 18% of sophomores, juniors, and seniors at Ivy League colleges had knowingly used a stimulant at least once during their academic career, and among those who had ever used uppers, 24% said they had popped a little helper on eight or more occasions. Anecdotal evidence suggests that pharmacological enhancement is also on the rise within the workplace, where modafinil, which treats sleep disorders, has become particularly popular.
Methylphenidate, commonly known as Ritalin, is a stimulant first synthesised in the 1940s. More accurately, it's a psychostimulant - often prescribed for ADHD - that is intended as a drug to help focus and concentration. It also reduces fatigue and (potentially) enhances cognition. Similar to Modafinil, Ritalin is believed to reduce dissipation of dopamine to help focus. Ritalin is a Class B drug in the UK, and possession without a prescription can result in a 5 year prison sentence. Please note: Side Effects Possible. See this article for more on Ritalin.
Critics will often highlight ethical issues and the lack of scientific evidence for these drugs. Ethical arguments typically take the form of "tampering with nature." Alena Buyx discusses this argument in a neuroethics project called Smart Drugs: Ethical Issues. She says that critics typically ask if it is ethically superior to accept what is "given" instead of striving for what is "made". My response to this is simple. Just because it is natural does not mean it is superior.
But though it's relatively new on the scene with ambitious young professionals, creatine has a long history with bodybuilders, who have been taking it for decades to improve their muscle #gains. In the US, sports supplements are a multibillion-dollar industry – and the majority contain creatine. According to a survey conducted by Ipsos Public Affairs last year, 22% of adults said they had taken a sports supplement in the last year. If creatine was going to have a major impact in the workplace, surely we would have seen some signs of this already.
The first night I was eating some coconut oil, I did my n-backing past 11 PM; normally that damages my scores, but instead I got 66/66/75/88/77% (▁▁▂▇▃) on D4B and did not feel mentally exhausted by the end. The next day, I performed well on the Cambridge mental rotations test. An anecdote, of course, and it may be due to the vitamin D I simultaneously started. Or another day, I was slumped under apathy after a promising start to the day; a dose of fish & coconut oil, and 1 last vitamin D, and I was back to feeling chipper and optimist. Unfortunately I haven't been testing out coconut oil & vitamin D separately, so who knows which is to thank. But still interesting.
I have also tried to get in contact with senior executives who have experience with these drugs (either themselves or in their firms), but without success. I have to wonder: Are they completely unaware of the drugs' existence? Or are they actively suppressing the issue? For now, companies can ignore the use of smart drugs. And executives can pretend as if these drugs don't exist in their workplaces. But they can't do it forever.
My first impression of ~1g around 12:30PM was that while I do not feel like running around, within an hour I did feel like the brain fog was lighter than before. The effect wasn't dramatic, so I can't be very confident. Operationalizing brain fog for an experiment might be hard: it doesn't necessarily feel like I would do better on dual n-back. I took 2 smaller doses 3 and 6 hours later, to no further effect. Over the following weeks and months, I continued to randomly alternate between potassium & non-potassium days. I noticed no effects other than sleep problems.
One often-cited study published in the British Journal of Pharmacology looked at cognitive function in the elderly and showed that racetam helped to improve their brain function.19 Another study, which was published in Psychopharmacology, looked at adult volunteers (including those who are generally healthy) and found that piracetam helped improve their memory.20
Nootropics. You might have heard of them. The "limitless pill" that keeps Billionaires rich. The 'smart drugs' that students are taking to help boost their hyperfocus. The cognitive enhancers that give corporate executives an advantage. All very exciting. But as always, the media are way behind the curve. Yes, for the past few decades, cognitive enhancers were largely sketchy substances that people used to grasp at a short term edge at the expense of their health and well being. But the days of taking prescription pills to pull an all-nighter are so 2010. The better, safer path isn't with these stimulants but with nootropics. Nootropics consist of dietary supplements and substances which enhance your cognition, in particular when it comes to motivation, creativity, memory, and other executive functions. They play an important role in supporting memory and promoting optimal brain function.
Oxiracetam is one of the 3 most popular -racetams; less popular than piracetam but seems to be more popular than aniracetam. Prices have come down substantially since the early 2000s, and stand at around 1.2g/$ or roughly 50 cents a dose, which was low enough to experiment with; key question, does it stack with piracetam or is it redundant for me? (Oxiracetam can't compete on price with my piracetam pile stockpile: the latter is now a sunk cost and hence free.)
…It is without activity in man! Certainly not for the lack of trying, as some of the dosage trials that are tucked away in the literature (as abstracted in the Qualitative Comments given above) are pretty heavy duty. Actually, I truly doubt that all of the experimenters used exactly that phrase, No effects, but it is patently obvious that no effects were found. It happened to be the phrase I had used in my own notes.
Running low on gum (even using it weekly or less, it still runs out), I decided to try patches. Reading through various discussions, I couldn't find any clear verdict on what patch brands might be safer (in terms of nicotine evaporation through a cut or edge) than others, so I went with the cheapest Habitrol I could find as a first try of patches (Nicotine Transdermal System Patch, Stop Smoking Aid, 21 mg, Step 1, 14 patches) in May 2013. I am curious to what extent nicotine might improve a long time period like several hours or a whole day, compared to the shorter-acting nicotine gum which feels like it helps for an hour at most and then tapers off (which is very useful in its own right for kicking me into starting something I have been procrastinating on). I have not decided whether to try another self-experiment.
I largely ignored this since the discussions were of sub-RDA doses, and my experience has usually been that RDAs are a poor benchmark and frequently far too low (consider the RDA for vitamin D). This time, I checked the actual RDA - and was immediately shocked and sure I was looking at a bad reference: there was no way the RDA for potassium was seriously 3700-4700mg or 4-5 grams daily, was there? Just as an American, that implied that I was getting less than half my RDA. (How would I get 4g of potassium in the first place? Eat a dozen bananas a day⸮) I am not a vegetarian, nor is my diet that fantastic: I figured I was getting some potassium from the ~2 fresh tomatoes I was eating daily, but otherwise my diet was not rich in potassium sources. I have no blood tests demonstrating deficiency, but given the figures, I cannot see how I could not be deficient.
Similar to the way in which some athletes used anabolic steroids (muscle-building hormones) to artificially enhance their physique, some students turned to smart drugs, particularly Ritalin and Adderall, to heighten their intellectual abilities. A 2005 study reported that, at some universities in the United States, as many as 7 percent of respondents had used smart drugs at least once in their lifetime and 2.1 percent had used smart drugs in the past month. Modafinil was used increasingly by persons who sought to recover quickly from jet lag and who were under heavy work demands. Military personnel were given the same drug when sent on missions with extended flight times.
The placebos can be the usual pills filled with olive oil. The Nature's Answer fish oil is lemon-flavored; it may be worth mixing in some lemon juice. In Kiecolt-Glaser et al 2011, anxiety was measured via the Beck Anxiety scale; the placebo mean was 1.2 on a standard deviation of 0.075, and the experimental mean was 0.93 on a standard deviation of 0.076. (These are all log-transformed covariates or something; I don't know what that means, but if I naively plug those numbers into Cohen's d, I get a very large effect: \frac{1.2 - 0.93}{0.076}=3.55.)
I have a needle phobia, so injections are right out; but from the images I have found, it looks like testosterone enanthate gels using DMSO resemble other gels like Vaseline. This suggests an easy experimental procedure: spoon an appropriate dose of testosterone gel into one opaque jar, spoon some Vaseline gel into another, and pick one randomly to apply while not looking. If one gel evaporates but the other doesn't, or they have some other difference in behavior, the procedure can be expanded to something like and then half an hour later, take a shower to remove all visible traces of the gel. Testosterone itself has a fairly short half-life of 2-4 hours, but the gel or effects might linger. (Injections apparently operate on a time-scale of weeks; I'm not clear on whether this is because the oil takes that long to be absorbed by surrounding materials or something else.) Experimental design will depend on the specifics of the obtained substance. As a controlled substance (Schedule III in the US), supplies will be hard to obtain; I may have to resort to the Silk Road.
These are quite abstract concepts, though. There is a large gap, a grey area in between these concepts and our knowledge of how the brain functions physiologically – and it's in this grey area that cognitive enhancer development has to operate. Amy Arnsten, Professor of Neurobiology at Yale Medical School, is investigating how the cells in the brain work together to produce our higher cognition and executive function, which she describes as "being able to think about things that aren't currently stimulating your senses, the fundamentals of abstraction. This involves mental representations of our goals for the future, even if it's the future in just a few seconds."
(In particular, I don't think it's because there's a sudden new surge of drugs. FDA drug approval has been decreasing over the past few decades, so this is unlikely a priori. More specifically, many of the major or hot drugs go back a long time. Bacopa goes back millennia, melatonin I don't even know, piracetam was the '60s, modafinil was '70s or '80s, ALCAR was '80s AFAIK, Noopept & coluracetam were '90s, and so on.)
One thing to notice is that the default case matters a lot. This asymmetry is because you switch decisions in different possible worlds - when you would take Adderall but stop you're in the world where Adderall doesn't work, and when you wouldn't take Adderall but do you're in the world where Adderall does work (in the perfect information case, at least). One of the ways you can visualize this is that you don't penalize tests for giving you true negative information, and you reward them for giving you true positive information. (This might be worth a post by itself, and is very Litany of Gendlin.)
(As I was doing this, I reflected how modafinil is such a pure example of the money-time tradeoff. It's not that you pay someone else to do something for you, which necessarily they will do in a way different from you; nor is it that you have exchanged money to free yourself of a burden of some future time-investment; nor have you paid money for a speculative return of time later in life like with many medical expenses or supplements. Rather, you have paid for 8 hours today of your own time.)
"In the hospital and ICU struggles, this book and Cavin's experience are golden, and if we'd have had this book's special attention to feeding tube nutrition, my son would be alive today sitting right here along with me saying it was the cod liver oil, the fish oil, and other nutrients able to be fed to him instead of the junk in the pharmacy tubes, that got him past the liver-test results, past the internal bleeding, past the brain difficulties controlling so many response-obstacles back then. Back then, the 'experts' in rural hospitals were unwilling to listen, ignored my son's unexpected turnaround when we used codliver oil transdermally on his sore skin, threatened instead to throw me out, but Cavin has his own proof and his accumulated experience in others' journeys. Cavin's boxed areas of notes throughout the book on applying the brain nutrient concepts in feeding tubes are powerful stuff, details to grab onto and run with… hammer them!
The main concern with pharmaceutical drugs is adverse effects, which also apply to nootropics with undefined effects. Long-term safety evidence is typically unavailable for nootropics.[13] Racetams — piracetam and other compounds that are structurally related to piracetam — have few serious adverse effects and low toxicity, but there is little evidence that they enhance cognition in people having no cognitive impairments.[19] | CommonCrawl |
\begin{definition}[Definition:Parallelepiped/Height]
:400px
The '''height''' of a parallelepiped is the length of the perpendicular from the plane of the base to the plane opposite.
In the above diagram, $h$ is the '''height''' of the parallelepiped whose base is $AB$.
Category:Definitions/Parallelepipeds
\end{definition} | ProofWiki |
\begin{document}
\title{Implications of a deeper level explanation of the deBroglie--Bohm version of quantum mechanics\vspace*{
amount} }
\author{Gerhard \surname{Grössing}\textsuperscript{}}
\email[E-mail: ]{[email protected]}
\homepage[Visit: ]{http://www.nonlinearstudies.at/}
\author{Siegfried \surname{Fussy}\textsuperscript{}}
\email[E-mail: ]{[email protected]}
\homepage[Visit: ]{http://www.nonlinearstudies.at/}
\author{Johannes \surname{Mesa Pascasio}\textsuperscript{}}
\email[E-mail: ]{[email protected]}
\homepage[Visit: ]{http://www.nonlinearstudies.at/}
\author{Herbert \surname{Schwabl}\textsuperscript{}}
\email[E-mail: ]{[email protected]}
\homepage[Visit: ]{http://www.nonlinearstudies.at/}
\affiliation{\textsuperscript{}Austrian Institute for Nonlinear Studies, Akademiehof\\
Friedrichstr.~10, 1010 Vienna, Austria\\ \vspace*{1cm} }
\date{\today} \begin{abstract} Elements of a ``deeper level'' explanation of the deBroglie--Bohm (dBB) version of quantum mechanics are presented. Our explanation is based on an analogy of quantum wave-particle duality with bouncing droplets in an oscillating medium, the latter being identified as the vacuum's zero-point field. A hydrodynamic analogy of a similar type has recently come under criticism by Richardson \emph{et~al}.~\cite{Richardson.2014analogy}, because despite striking similarities at a phenomenological level the governing equations related to the force on the particle are evidently different for the hydrodynamic and the quantum descriptions, respectively. However, said differences are not relevant if a radically different use of said analogy is being made, thereby essentially referring to emergent processes in our model. If the latter are taken into account, one can show that the forces on the particles are identical in both the dBB and our model. In particular, this identity results from an exact matching of our emergent velocity field with the Bohmian ``guiding equation''. One thus arrives at an explanation involving a deeper, i.e.\ subquantum, level of the dBB version of quantum mechanics. We show in particular how the classically-local approach of the usual hydrodynamical modeling can be overcome and how, as a consequence, the configuration-space version of dBB theory for \emph{N} particles can be completely substituted by a ``superclassical'' emergent dynamics of \emph{N} particles in real 3-dimensional space. \begin{lyxgreyedout} \noindent \global\long\def\VEC#1{\mathbf{#1}} \global\long\def\,\mathrm{d}{\,\mathrm{d}} \global\long\def{\rm e}{{\rm e}} \global\long\def\meant#1{\left<#1\right>} \global\long\def\meanx#1{\overline{#1}} \global\long\def\ensuremath{\genfrac{}{}{0pt}{1}{-}{\scriptstyle (\kern-1pt +\kern-1pt )}}{\ensuremath{\genfrac{}{}{0pt}{1}{-}{\scriptstyle (\kern-1pt +\kern-1pt )}}} \global\long\def\ensuremath{\genfrac{}{}{0pt}{1}{+}{\scriptstyle (\kern-1pt -\kern-1pt )}}{\ensuremath{\genfrac{}{}{0pt}{1}{+}{\scriptstyle (\kern-1pt -\kern-1pt )}}} \global\long\def\partial{\partial} \end{lyxgreyedout}
\end{abstract}
\keywords{quantum mechanics, hydrodynamics, deBroglie--Bohm theory, guiding equation, configuration space, zero-point field}
\maketitle
\section{Introduction\label{sec:intro}}
The Schrödinger equation for $N>1$ particles does not describe a wave function in ordinary 3-dimensional space, but instead in an abstract $3N$-dimensional space. For quantum realists, including Schrödinger and Einstein, for example, this has always been considered as ``indigestible''. This holds even more so for a realist, causal approach to quantum phenomena such as the deBroglie--Bohm (dBB) version of quantum mechanics. David Bohm himself has admitted this, calling it a ``serious problem'': ``While our theory can be extended formally in a logically consistent way by introducing the concept of a wave in a $3N$-dimensional space, it is evident that this procedure is not really acceptable in a physical theory, and should at least be regarded as an artifice that one uses provisionally until one obtains a better theory in which everything is expressed once more in ordinary $3$-dimensional space.''~\cite{Bohm.1997causality} (For more detailed accounts of this discussion already in the early years of quantum mechanics, see~\cite{Norsen.2010theory} and \cite{Norsen.2014can}.)
In the present paper, we shall refer to our attempt towards such a ``better theory'' in terms of a deeper level, i.e.\ subquantum, approach to the dBB theory, and thus to quantum theory in general. In fact, with our model we have in a series of papers already obtained several essential elements of nonrelativistic quantum theory~\cite{Groessing.2008vacuum,Groessing.2009origin,Groessing.2012doubleslit,Groessing.2013dice}. They derive from the assumption that a particle of energy $E=\hbar\omega$ is actually an oscillator of angular frequency $\omega$ phase-locked with the zero-point oscillations of the surrounding environment, the latter containing both regular and fluctuating components and being constrained by the boundary conditions of the experimental setup via the buildup and maintenance of standing waves. The particle in this approach is an off-equilibrium steady state oscillation maintained by a constant throughput of energy provided by the (\textquotedblleft classical\textquotedblleft ) zero-point energy field. We have, for example, applied the model to the case of interference at a double slit, thereby obtaining the exact quantum mechanical probability density distributions on a screen behind the double slit, the average trajectories (which because of the averaging are shown to be identical to the Bohmian ones), and the involved probability density currents. Our whole model is constructed in close analogy to the bouncing/walking droplets above the surface of a vibrated liquid in the experiments first performed by Yves Couder, Emmanuel Fort and co-workers~\cite{Couder.2006single-particle,Couder.2012probabilities,Fort.2010path-memory}, which in many respects can serve as a classical prototype guiding our intuition for the modeling of quantum systems.
However, there are also obvious differences between the mentioned physics of classical bouncers/walkers on the one hand, and the hydrodynamic-like models for quantum systems like our own model or the dBB one on the other hand. In a recent paper, Richardson \emph{et~al}.~\cite{Richardson.2014analogy} have probed more thoroughly into the hydrodynamic analogy of dBB-type quantum wave-particle duality with that of the classical bouncing droplets. Apart from the obvious difference in that Bohmian theory is distinctly nonlocal, whereas droplet-surface interactions are rooted in classical hydrodynamics and thus in a manifestly local theory, Richardson \emph{et~al}.\ focus on the following observation: the evidently different nature of the Bohmian force upon a quantum particle as compared to the force that a surface wave exerts upon a droplet. In fact, wherever the probability density in the dBB picture is close to zero, the quantum force becomes singular and will very quickly push any particle away from that area. Conversely, the hydrodynamic force directs the droplet into the trough of the wave! So, the probability of finding a droplet in the minima never reaches zero as it does for a quantum particle. The authors conclude that these discrepancies between the two models highlight ``a major difference between the hydrodynamic force and the quantum force''.\ \cite{Richardson.2014analogy}
Although these authors generally recover in numerical hydrodynamic simulations the results of the Paris group (later confirmed also by the group of John Bush at MIT~\cite{Bush.2015pilot-wave}) on single-slit diffraction and double-slit interference, they also point out the (already known) striking contrast between the trajectory behaviors for the bouncing droplet systems and dBB-type quantum mechanics, respectively. Whereas the latter exhibits the well-known no-crossing property, the trajectories of the former do to a large extent cross each other. So, again, the physics in the two models is apparently fundamentally different, despite some striking similarities on a phenomenological level. As to the differences, one may very well expect that they will even become more severe when moving from one-particle to $N$-particle systems.
So, all in all, the paper by Richardson \emph{et~al}.~\cite{Richardson.2014analogy} cautions against the assumption of too close a resemblance of bouncer/walker systems and the hydrodynamic-like modeling of quantum systems like the dBB one, with their main argument being that the hydrodynamic force on a droplet strikingly contrasts with the quantum force on a particle in the dBB theory. However, we shall here argue against the possible conclusion that one has thus reached the limits of applicability of the hydrodynamic bouncer analogy for quantum modeling. On the contrary, as we have already pointed out in previous papers, it is a more detailed model inspired by the bouncer/walker experiments that can show the fertility of said analogy. It enables us to show that our model, being of the type of an ``emergent quantum mechanics''~\cite{Groessing.2012emerqum11-book,Groessing.2014emqm13-book}, can provide a deeper-level explanation of the dBB version of quantum mechanics (Chapter~2). Moreover, as we shall also show, it turns out to provide an identity of an emergent force on the bouncer in our hydrodynamic-like model with the quantum force in Bohmian theory (Chapter~3). Finally, in Chapter~4 we shall discuss the ``price'' to be paid in order to arrive at our explanation of dBB theory in that some kind of nonlocality, or a certain ``systemic nonlocality'', has to be admitted in the model from the start. However, the simplicity and elegance of our derived formalism, combined with arguments about the reasonableness of a corresponding hydrodynamic-like modeling, will show that our approach may be a viable one w.r.t.\ understanding the emergence of quantum phenomena from the interactions and contextualities provided by the combined levels of classical boundary conditions and those of a subquantum domain.
\section{Identity of the emergent kinematics of \emph{$N$} bouncers in real $3$-dimensional space with the configuration-space version of deBroglie--Bohm theory for \emph{$N$} particles \label{sec:config}}
Consider one particle in an $n$-slit system. In quantum mechanics, as well as in our quantum-like modeling via an emergent quantum mechanics approach, one can write down a formula for the total intensity distribution $P$ which is very similar to the classical formula. For the general case of $n$ slits, it holds with phase differences $\varphi_{ij}=\varphi_{i}-\varphi_{j}$ that \begin{equation} P=\sum_{i=1}^{n}\left(P_{i}+\sum_{j=i+1}^{n}2R_{i}R_{j}\cos\varphi_{ij}\right),\label{eq:Sup2.1} \end{equation} where the phase differences are defined over the whole domain of the experimental setup. Apart from the role of the relative phase with important implications for the discussions on nonlocality~\cite{Groessing.2013dice}, there is one additional ingredient that distinguishes~(\ref{eq:Sup2.1}) from its classical counterpart, namely the ``dispersion of the wavepacket''. As in our model the ``particle'' is actually a bouncer in a fluctuating wave-like environment, i.e.~analogously to the bouncers of Couder and Fort's group, one does have some (e.g.\ Gaussian) distribution, with its center following the Ehrenfest trajectory in the free case, but one also has a diffusion to the right and to the left of the mean path which is just due to that stochastic bouncing. Thus the total velocity field of our bouncer in its fluctuating environment is given by the sum of the forward velocity $\VEC v$ and the respective diffusive velocities $\VEC u_{\mathrm{L}}$ and $\VEC u_{\mathrm{R}}$ to the left and the right. As for any direction $i$ the diffusion velocity $\VEC u_{\mathrm{i}}=D\frac{\nabla_{i}P}{P}$ does not necessarily fall off with the distance, one has long effective tails of the distributions which contribute to the nonlocal nature of the interference phenomena~\cite{Groessing.2013dice}. In sum, one has three, distinct velocity (or current) channels per slit in an $n$-slit system.
We have previously shown~\cite{Fussy.2014multislit,Groessing.2014relational} how one can derive the Bohmian guidance formula from our bouncer/walker model. To recapitulate, we recall the basics of that derivation here. Introducing classical wave amplitudes $R(\VEC w_{i})$ and generalized velocity field vectors $\VEC w_{i}$, which stand for either a forward velocity $\VEC v_{i}$ or a diffusive velocity $\VEC u_{i}$ in the direction transversal to $\VEC v_{i}$, we account for the phase-dependent amplitude contributions of the total system's wave field projected on one channel's amplitude $R(\VEC w_{i})$ at the point $(\VEC x,t)$ in the following way: We define a \emph{conditional probability density} $P(\VEC w_{i})$ as the local wave intensity $P(\VEC w_{i})$ in one channel (i.e.~$\VEC w_{i}$) upon the condition that the totality of the superposing waves is given by the ``rest'' of the $3n-1$ channels (recalling that there are 3 velocity channels per slit). The expression for $P(\VEC w_{i})$ represents what we have termed ``relational causality'': any change in the local intensity affects the total field, and \emph{vice versa}, any change in the total field affects the local one. In an $n$-slit system, we thus obtain for the conditional probability densities and the corresponding currents, respectively, i.e.\ for each channel component $\mathit{i}$, \begin{align} P(\VEC w_{i}) & =R(\VEC w_{i})\VEC{\hat{w}}_{i}\cdot{\displaystyle \sum_{j=1}^{3n}}\VEC{\hat{w}}_{j}R(\VEC w_{j})\label{eq:Proj-1}\\ \VEC J\mathrm{(}\VEC w_{i}\mathrm{)} & =\VEC w_{i}P(\VEC w_{i}),\qquad i=1,\ldots,3n, \end{align} with \begin{equation} \cos\varphi_{i,j}:=\VEC{\hat{w}}_{i}\cdot\VEC{\hat{w}}_{j}. \end{equation} Consequently, the total intensity and current of our field read as \begin{align} P_{\mathrm{tot}}= & {\displaystyle \sum_{i=1}^{3n}}P(\VEC w_{i})=\left({\displaystyle \sum_{i=1}^{3n}}\VEC{\hat{w}}_{i}R(\VEC w_{i})\right)^{2}\label{eq:Ptot6-1}\\ \VEC J_{\mathrm{tot}}= & \sum_{i=1}^{3n}\VEC J(\VEC w_{i})={\displaystyle \sum_{i=1}^{3n}}\VEC w_{i}P(\VEC w_{i}),\label{eq:Jtot6-1} \end{align}
leading to the \textit{emergent total velocity} \begin{equation} \VEC v_{\mathrm{tot}}=\frac{\VEC J_{\mathrm{tot}}}{P_{\mathrm{tot}}}=\frac{{\displaystyle \sum_{i=1}^{3n}}\VEC w_{i}P(\VEC w_{i})}{{\displaystyle \sum_{i=1}^{3n}}P(\VEC w_{i})}\,.\label{eq:vtot_fin-1} \end{equation}
In~\cite{Fussy.2014multislit,Groessing.2014relational} we have shown with the example of $n=2,$ i.e.\ a double slit system, that Eq.~(\ref{eq:vtot_fin-1}) can equivalently be written in the form \begin{equation} \VEC v_{\mathrm{tot}}=\frac{R_{1}^{2}\VEC v_{\mathrm{1}}+R_{2}^{2}\VEC v_{\mathrm{2}}+R_{1}R_{2}\left(\VEC v_{\mathrm{1}}+\VEC v_{2}\right)\cos\varphi+R_{1}R_{2}\left(\VEC u_{1}-\VEC u_{2}\right)\sin\varphi}{R_{1}^{2}+R_{2}^{2}+2R_{1}R_{2}\cos\varphi}\,.\label{eq:vtot-1} \end{equation}
The trajectories or streamlines, respectively, are obtained according to $\VEC{\dot{x}}=\VEC v_{\mathrm{tot}}$ in the usual way by integration. As first shown in~\cite{Groessing.2012doubleslit}, by re-inserting the expressions for convective and diffusive velocities, respectively, i.e.\ $\VEC v_{i}=\frac{\nabla S_{i}}{m}$, $\VEC u_{i}=-\frac{\hbar}{m}$$\frac{\nabla R_{i}}{R_{i}}$, one immediately identifies Eq.~(\ref{eq:vtot-1}) with the Bohmian guidance formula. Naturally, employing the Madelung transformation for each path $j$ ($j=1$ or $2$), \begin{equation} \psi_{j}=R_{j}{\rm e}^{\mathrm{i}S_{j}/\hbar},\label{eq:3.14-1} \end{equation}
and thus $P_{j}=R_{j}^{2}=|\psi_{j}|^{2}=\psi_{j}^{*}\psi_{j}$, with $\varphi=(S_{1}-S_{2})/\hbar$, and recalling the usual trigonometric identities such as $\cos\varphi=\frac{1}{2}\left({\rm e}^{\mathrm{i}\varphi}+{\rm e}^{-\mathrm{i}\varphi}\right)$, one can rewrite the total average current immediately in the usual quantum mechanical form as \begin{equation} \begin{array}{rl} {\displaystyle \mathbf{J}_{{\rm tot}}} & =P_{{\rm tot}}\mathbf{v}_{{\rm tot}}\\[3ex]
& ={\displaystyle (\psi_{1}+\psi_{2})^{*}(\psi_{1}+\psi_{2})\frac{1}{2}\left[\frac{1}{m}\left(-\mathrm{i}\hbar\frac{\nabla(\psi_{1}+\psi_{2})}{(\psi_{1}+\psi_{2})}\right)+\frac{1}{m}\left(\mathrm{i}\hbar\frac{\nabla(\psi_{1}+\psi_{2})^{*}}{(\psi_{1}+\psi_{2})^{*}}\right)\right]}\\[3ex]
& ={\displaystyle -\frac{\mathrm{i}\hbar}{2m}\left[\Psi^{*}\nabla\Psi-\Psi\nabla\Psi^{*}\right]={\displaystyle \frac{1}{m}{\rm Re}\left\{ \Psi^{*}(-\mathrm{i}\hbar\nabla)\Psi\right\} ,}} \end{array}\label{eq:3.18-1} \end{equation}
where $P_{{\rm tot}}=|\psi_{1}+\psi_{2}|^{2}=:|\Psi|^{2}$.
Eq.~(\ref{eq:vtot_fin-1}) has been derived for one particle in an $n$-slit system. However, it is straightforward to extend this derivation to the many-particle case. Due to the purely additive terms in the expressions for the total current and total probability density, respectively, also for \emph{N} particles, the only difference now is that the currents' nabla operators have to be applied at all of the locations of the respective \emph{N} particles, thus providing the quantum mechanical formula \begin{equation} {\displaystyle \mathbf{J}_{{\rm tot}}}\left(N\right)={\displaystyle \sum_{i=1}^{N}}\frac{1}{m_{i}}{\rm Re}\left\{ \Psi\left(t\right)^{*}(-\mathrm{i}\hbar\nabla_{i})\Psi\left(t\right)\right\} , \end{equation} where $\Psi\left(t\right)$ now is the total $N$-particle wave function, whereas the total velocity fields are given by \begin{equation} \VEC v_{i}\left(t\right)=\frac{\hbar}{m_{i}}\mathrm{Im}\frac{\nabla_{i}\Psi\left(t\right)}{\Psi\left(t\right)}\;\forall i=1,...,N. \end{equation}
Note that this result is similar in spirit to that of Norsen \emph{et~al.~}\cite{Norsen.2010theory,Norsen.2014can} who with the introduction of a \emph{conditional wave function} $\tilde{\psi}_{i}$, as opposed to the configuration-space wave function $\Psi$, rewrite the guidance formula, for each particle, in terms of the $\tilde{\psi}_{i}$:
\begin{equation}
\frac{\,\mathrm{d} X_{i}\left(t\right)}{\,\mathrm{d} t}=\frac{\hbar}{m_{i}}\mathrm{Im}\left.\frac{\nabla\Psi}{\Psi}\right|_{\mathbf{\boldsymbol{x}=\boldsymbol{X}\left(t\right)}}\equiv\frac{\hbar}{m_{i}}\mathrm{Im}\left.\frac{\nabla\tilde{\psi}_{i}}{\tilde{\psi}_{i}}\right|_{x=X_{i}\left(t\right)}, \end{equation} where the $X_{i}$ denote the location of one specific particle and $\mathbf{X}\left(t\right)=\left\{ X_{1}\left(t\right),...,X_{N}\left(t\right)\right\} $ the actual configuration point. Thus, in this approach, each $\tilde{\psi}_{i}$ can be regarded as a wave propagating in physical 3-dimensional space.
In sum, with our introduction of a conditional probability $P(\VEC w_{i})$ for channels $\VEC w_{i}$, which include subquantum velocity fields, we obtain the guidance formula also for $N$-particle systems. \textsl{Therefore, what looks like the necessity in the dBB theory to superpose wave functions in configuration space in order to provide an ``indigestible'' guiding wave, can equally be obtained by superpositions of all relational amplitude configurations of waves in real 3-dimensional space. }\textsl{\emph{The central ingredient for this to be possible is to consider the }}\textsl{emergence}\textsl{\emph{ of the velocity field from the interplay of the totality of all of the system's velocity channels. We have termed the framework of our approach a ``superclassical'' one, because in it are combined classical levels at vastly different scales, i.e.\ at the subquantum and the macroscopic levels, respectively.
{}}}
\section{Identity of the emergent force on a particle modeled by a bouncer system and the quantum force of the deBroglie--Bohm theory\label{sec:force}}
With the results of the foregoing Chapter, we can now return to and resolve the problem discussed in Chapter~1 of the apparent incompatibility between the Bohmian force upon a quantum particle and the force exerted on a bouncing droplet as formulated by Richardson \emph{et~al}.~\cite{Richardson.2014analogy}. In fact, already a first look at the bouncer/walker model of our group provides a clear difference as compared to the hydrodynamical force studied by Richardson \emph{et~al}. For, whereas the latter investigate the effects of essentially a single bounce on the fluid surface and the acceleration of the bouncer as a consequence of this interaction, our bouncer/walker model for quantum particles involves a much more complex dynamical scenario: We consider the effects of a huge number of bounces, i.e.\ typically of the order of $\nicefrac{1}{\omega}$, like approximately $10^{21}$ bounces per second of an electron, which constitute effectively a ``heating up'' of the bouncer's surrounding, i.e.\ the subquantum medium related to the zero-point energy field.
Note that as soon as a microdynamics is assumed, the development of heat fluxes is a logical necessity if the microdynamics is constrained by some macroscopic boundaries like that of a slit system, for example. As we have shown in some detail~\cite{Groessing.2010emergence}, the thermal field created by such a huge number of bounces in a slit system leads to an emergent average behavior of particle trajectories which is identified as anomalous, and more specifically as ballistic, diffusion. As such, the particle trajectories exiting from, say, a Gaussian slit behave exactly as if they were subject to a Bohmian quantum force. We were also able to show that this applies also to $n$-slit systems, such that one arrives at a subquantum modeling of the emergent interference effects at $n$ slits whose predicted average behavior is identical to that provided by the dBB theory.
It is then easily shown that the average force acting on a particle in our model is the same as the Bohmian quantum force. For, due to the identity of our emerging velocity field with the guidance formula, and because they essentially differ only via the notations due to different forms of bookkeeping, their respective time derivatives must also be identical. Thus, from Eq.~(\ref{eq:vtot_fin-1}) one obtains the particle acceleration field (using a one-particle scenario for simplicity) in an $n$-slit system as \begin{align} a_{\mathrm{tot}}\left(t\right) & =\frac{\,\mathrm{d}\mathbf{v}_{{\rm tot}}}{\,\mathrm{d} t}=\frac{\,\mathrm{d}}{\,\mathrm{d} t}\left(\frac{{\displaystyle \sum_{i=1}^{3n}}\VEC w_{i}P(\VEC w_{i})}{{\displaystyle \sum_{i=1}^{3n}}P(\VEC w_{i})}\right)\nonumber \\
& =\frac{1}{\left({\displaystyle \sum_{i=1}^{3n}}P(\VEC w_{i})\right)^{2}}\left\{ \vphantom{\frac{{\displaystyle \sum^{3n}}}{{\displaystyle \sum^{3n}}}}\sum_{i=1}^{3n}\left[P(\VEC w_{i})\frac{\,\mathrm{d}\VEC w_{i}}{\,\mathrm{d} t}+\VEC w_{i}\frac{\,\mathrm{d} P(\VEC w_{i})}{\,\mathrm{d} t}\right]\left({\displaystyle \sum_{i=1}^{3n}}P(\VEC w_{i})\right)\right.\label{eq:3.1}\\
& \qquad\qquad\qquad\qquad\qquad\qquad\left.-\left({\displaystyle \sum_{i=1}^{3n}}\VEC w_{i}P(\VEC w_{i})\right)\left({\displaystyle \sum_{i=1}^{3n}\frac{\,\mathrm{d} P(\VEC w_{i})}{\,\mathrm{d} t}}\right)\vphantom{\frac{{\displaystyle \sum^{3n}}}{{\displaystyle \sum^{3n}}}}\right\} .\nonumber \end{align} Note in particular that~(\ref{eq:3.1}) typically becomes infinite for regions $\left(\mathbf{x},t\right)$ where $P_{\mathrm{tot}}={\displaystyle \sum_{i=1}^{3n}}P(\VEC w_{i})\rightarrow0$, in accordance with the Bohmian picture.
From~(\ref{eq:3.1}) we see that even the acceleration of one particle in an $n$-slit system is a highly complex affair, as it nonlocally depends on all other accelerations and temporal changes in the probability densities across the whole experimental setup! In other words, this force is truly emergent, resulting from a huge amount of bouncer-medium interactions, both locally and nonlocally. This is of course radically different from the scenario studied by Richardson \emph{et~al}.\ where the effect of only a single local bounce is compared with the quantum force. From our new perspective, it is then hardly a surprise that the comparison of the two respective forces provides distinctive differences. However, as we just showed, with the emergent scenario proposed in our model, complete agreement with the Bohmian quantum force is established.
\section{Choose your Poison: How to introduce Nonlocality in a Hydrodynamic-like model for quantum systems?\label{sec:poison}}
As already mentioned in the introduction of this paper, purely classical hydrodynamical models are manifestly local and thus inadequate tools to explain quantum mechanical nonlocality. Although nonlocal correlations may also be obtainable within hydrodynamical modeling~\cite{Brady.2013violation}, there is no way to also account for dynamical nonlocality~\cite{Tollaksen.2010quantum} in this manner. So, as correctly observed by Richardson \emph{et~al.}~\cite{Richardson.2014analogy}, droplet-surface wave interaction scenarios are not enough to serve as a full-fledged analogy of the distinctly nonlocal dBB theory, for example.
The question thus arises how in our much more complex, but still ``hydrodynamic-like'' bouncer/walker model nonlocal, or nonlocal-like, effects can come about. To answer this, one needs to consider in more detail how the elements of our model are constructed, which finally provide an elegant formula, Eq.~(\ref{eq:vtot_fin-1}), identical with the guidance formula in a (for simplicity: one-particle) system with $n$ slits. (As shown above, the extension to $N$ particles is straightforward.) As we consider, without restriction of generality, the typical example of Gaussian slits, we introduce the Gaussians in the usual way, with $\sigma$ related to the slit width, for the probability density distributions (which in our model coincide with ``heat'' distributions due to the bouncers' stirring up of the vacuum) just behind the slit. The important feature of these Gaussians is that we do not implement any cutoff for the distributions, but maintain the long tails which actually then extend across the whole experimental setup, even if these are only very small and practically often negligible amplitudes in the regions far away from the slit proper. As the emerging probability density current is given by the denominator of Eq.~(\ref{eq:vtot-1}), we see that in fact the product $R_{1}R_{2}$ may be negligibly small for regions where only a long tail of one Gaussian overlaps with another Gaussian, nevertheless the last term in~(\ref{eq:vtot-1}) can be very large despite the smallness of $R_{1}$ or $R_{2}$. It is this latter part which is responsible for the genuinely quantum-like nature of the average momentum, i.e.\ for its nonlocal nature. This is similar in the Bohmian picture, but here given a more direct physical meaning in that this last term refers to a difference in diffusive currents as explicitly formulated in the last term of~(\ref{eq:vtot-1}). Because of the mixing of diffusion currents from both channels, we call this decisive term in $\mathbf{J_{\mathrm{\mathit{\mathrm{tot}}}}=P_{{\rm tot}}\mathbf{v}_{{\rm tot}}}$ the ``entangling current''.\ \cite{Mesa.2013variable}
Thus, one sees that formally one obtains genuine quantum mechanical nonlocality in a hydrodynamic-like model with one particular ``unusual'' characteristic: the extremely feeble but long tails of (Gaussian or other) distribution functions for probability densities exiting from a slit extend nonlocally across the whole experimental setup. So, we have nonlocality by explicitly putting it into our model. After all, if the world is nonlocal, it would not make much sense to attempt its reconstruction with purely local means. Still, so far we have just stated a formal reason for how nonlocality may come about. Somewhere in any theory, so it seems, one has to ``choose one's poison'' that would provide nonlocality in the end. But what would be a truly ``digestible'' physical explanation? Here is where at present only some speculative clues can be given.
For one thing, strict nonlocality in the sense of absolute simultaneity of space-like separated events can never be proven in any realistic experiment, because infinite precision is not attainable. This means, however, that very short time lapses must be admitted in any operational scenario, with two basic options remaining: i) either there is a time lapse due to the finitely fast ``switching'' of experimental arrangements in combination with instantaneous information transfer (but not signaling; see Walleczek and Grössing, {[}forthcoming{]}), or ii) the information transfer itself is not instantaneous, but happens at very high speeds $v\ggg c$.
How, then, can the implementation of nonlocal or nonlocal-like processes with speeds $v\ggg c$ be argued for in the context of a hydrodynamic-like bouncer/walker model? We briefly mention two options here. Firstly, one can imagine that the ``medium'' we employ in our model is characterized by oscillations of the zero-point energy throughout space, i.e.\ between any two or more macroscopic boundaries as given by experimental setups. Between these boundaries standing wave configurations may emerge (similar to the Paris group's experiments, but now explicitly extending synchronously over nonlocal distances). Here it might be helpful to remind ourselves that we deal with solutions of the diffusion (heat conduction) equation. At least (but perhaps only) formally, any change of the boundary conditions is effective ``instantaneously'' across the whole setup. Alternatively, if the experimental setup is changed such that old boundary conditions are substituted by new ones, due to the all-space-pervading zero-point energy oscillations, one ``immediately'' (i.e.\ after a very short time of the order $t\sim\frac{1}{\omega}$) obtains a new standing wave configuration that now effectively implies an almost instantaneous change of probability density distributions, or relative phase changes, for example. The latter would then become ``immediately'' effective in that changed phase information is available across the whole domain of the extended probability density distribution. We have referred to this state of affairs as ``systemic nonlocality''~\cite{Groessing.2013dice}. So, one may speculate that it is something like ``eigenvalues'' of the universe's network of zero-point fluctuations that may be responsible for quantum mechanical nonlocality-eigenvalues which (almost?) instantaneously change whenever the boundary conditions are changed.
A second option even more explicitly refers to the universe as a whole, or, more particularly, to spacetime itself. If spacetime is an emergent phenomenon as some recent work suggests~\cite{Padmanabhan.2014general}, this would very likely have strong implications for the modeling and understanding of quantum phenomena. Just as in our model of an emergent quantum mechanics we consider quantum theory as a possible limiting case of a deeper level theory, present-day relativity and concepts of spacetime may be approximations of, and emergent from a superclassical, deeper level theory of gravity and/or spacetime. It is thus a potentially fruitful task to bring both attempts together in the near future. \begin{acknowledgments} We thank Jan Walleczek for many enlightening discussions, and the Fetzer Franklin Fund for partial support of the current work. \end{acknowledgments}
\providecommand{\href}[2]{#2}\begingroup\raggedright
\endgroup
\end{document} | arXiv |
Annals of Actuarial Science
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Published on behalf of Institute and Faculty of Actuaries
FirstView articles
Volume 12 - Issue 1 - March 2018
A stochastic Expectation–Maximisation (EM) algorithm for construction of mortality tables
Luz Judith R. Esparza, Fernando Baltazar-Larios
Published online by Cambridge University Press: 04 May 2017, pp. 1-22
In this paper, we present an extension of the model proposed by Lin & Liu that uses the concept of physiological age to model the ageing process by using phase-type distributions to calculate the probability of death. We propose a finite-state Markov jump process to model the hypothetical ageing process in which it is possible the transition rates between non-consecutive physiological ages. Since the Markov process has only a single absorbing state, the death time follows a phase-type distribution. Thus, to build a mortality table the challenge is to estimate this matrix based on the records of the ageing process. Considering the nature of the data, we consider two cases: having continuous time information of the ageing process, and the more interesting and realistic case, having reports of the process just in determined times. If the ageing process is only observed at discrete time points we have a missing data problem, thus, we use a stochastic Expectation–Maximisation (SEM) algorithm to find the maximum likelihood estimator of the intensity matrix. And in order to do that, we build Markov bridges which are sampled using the Bisection method. The theory is illustrated by a simulation study and used to fit real data.
Ruin problems in Markov-modulated risk models
David C.M. Dickson, Marjan Qazvini
Published online by Cambridge University Press: 30 May 2017, pp. 23-48
Chen et al. (2014), studied a discrete semi-Markov risk model that covers existing risk models such as the compound binomial model and the compound Markov binomial model. We consider their model and build numerical algorithms that provide approximations to the probability of ultimate ruin and the probability and severity of ruin in a continuous time two-state Markov-modulated risk model. We then study the finite time ruin probability for a discrete m-state model and show how we can approximate the density of the time of ruin in a continuous time Markov-modulated model with more than two states.
The cost and value of UK pensions
Paul J. Sweeting
Published online by Cambridge University Press: 13 June 2017, pp. 49-66
Over the last 20 years, the extent of defined benefit provision has declined substantially in the United Kingdom. Whilst most of the focus has been on deficits relating to past benefit accrual, the increasing cost of future benefit accrual is also important. There are two reasons for this. First, the change in the cost of defined benefit accrual represents the difference in the earnings for employees with membership of a defined benefit scheme and those with membership of a defined contribution scheme. Second, the current cost of defined benefit accrual gives an indication of the cost of an adequate pension. As such, it can be compared with levels of contribution to defined contribution schemes to determine whether these are adequate. I therefore look at how the cost of pensions has changed relative to the cost of non-pensions earnings. I also look at the main components of the change in pensions cost – those relating to benefits payable, discount rates and longevity – to analyse their relative importance. I find that the cost of employing a member of defined benefit pension scheme has consistently outpaced the cost of employing someone in a defined contribution arrangement. I also find that the current cost of accrual is significantly higher than the average level of payments to defined contribution schemes.
Yet more on a stochastic economic model: Part 4: a model for share earnings, dividends, and prices
A. D. Wilkie, Şule Şahin
Published online by Cambridge University Press: 19 June 2017, pp. 67-105
In this paper, we develop an extension to the Wilkie model, introducing share earnings and cover (earnings/dividends) as new variables, and deriving share dividends from them. Earnings are available from April 1962, but only for the Non-Financial index, and for the All-Share one only from 1992. We construct a Composite Earnings Index from these series. We then find a suitable annual time series model for changes in earnings, and then for cover, which is mean-reverting. We compare this new model with the original model, in which changes in dividends were modelled directly. We also investigate monthly data to give parameters for stochastic interpolation. We observe an unusual change in earnings over 2015–2016, consider the implications of this and show specimen simulations.
An actuarial investigation into maternal hospital cost risk factors for public patients
Jananie William, Michael A. Martin, Catherine Chojenta, Deborah Loxton
We investigate an actuarial approach to identifying the factors impacting government-funded maternal hospital costs in Australia, with a focus on women who experience adverse birth outcomes. We propose a two-phase modelling methodology that adopts actuarial methods from typical insurance claim cost modelling and extends to other statistical techniques to account for the large volume of covariates available for modelling. Specifically, Classification and Regression Trees and generalised linear mixed models are employed to analyse a data set that links longitudinal survey and administrative data from a large sample of women. The results show that adverse births are a statistically significant risk factor affecting maternal hospital costs in the antenatal and delivery periods. Other significant cost risk factors in the delivery period include mode of delivery, private health insurance status, diabetes, smoking status, area of residence and onset of labour. We demonstrate the efficacy of using actuarial techniques in non-traditional areas and highlight how the results can be used to inform public policy.
An optimal multi-layer reinsurance policy under conditional tail expectation
Amir T. Payandeh Najafabadi, Ali Panahi Bazaz
An usual reinsurance policy for insurance companies admits one or two layers of the payment deductions. Under optimality criterion of minimising the Conditional Tail Expectation (CTE) risk measure of the insurer's total risk, this article generalises an optimal stop-loss reinsurance policy to an optimal multi-layer reinsurance policy. To achieve such optimal multi-layer reinsurance policy, this article starts from a given optimal stop-loss reinsurance policy f(⋅). In the first step, it cuts down the interval [0, ∞) into intervals [0, M1) and [M1, ∞). By shifting the origin of Cartesian coordinate system to (M1, f(M1)), and showing that under the CTE criteria $$f\left( x \right)I_{{[0,M_{{\rm 1}} )}} \left( x \right){\plus}\left( {f\left( {M_{{\rm 1}} } \right){\plus}f\left( {x{\minus}M_{{\rm 1}} } \right)} \right)I_{{[M_{{\rm 1}} ,{\rm }\infty)}} \left( x \right)$$ is, again, an optimal policy. This extension procedure can be repeated to obtain an optimal k-layer reinsurance policy. Finally, unknown parameters of the optimal multi-layer reinsurance policy are estimated using some additional appropriate criteria. Three simulation-based studies have been conducted to demonstrate: (1) the practical applications of our findings and (2) how one may employ other appropriate criteria to estimate unknown parameters of an optimal multi-layer contract. The multi-layer reinsurance policy, similar to the original stop-loss reinsurance policy is optimal, in a same sense. Moreover, it has some other optimal criteria which the original policy does not have. Under optimality criterion of minimising a general translative and monotone risk measure ρ(⋅) of either the insurer's total risk or both the insurer's and the reinsurer's total risks, this article (in its discussion) also extends a given optimal reinsurance contract f(⋅) to a multi-layer and continuous reinsurance policy.
Optimal reinsurance: a reinsurer's perspective
Fei Huang, Honglin Yu
In this paper, the optimal safety loading that the reinsurer should set in the reinsurance pricing is studied, which is novel in the literature. It is first assumed that the insurer will choose the form of the reinsurance contract by following the results derived in Cai et al. Different optimality criteria from the reinsurer's perspective are then studied, such as maximising the expectation of the profit, maximising the utility of the profit and minimising the value-at-risk of the reinsurer's total loss. By applying the concept of comonotonicity, the problem in which the reinsurer is facing two risks with unknown dependency structure is also solved. Closed-form solutions are obtained when the underlying losses are zero-modified exponentially distributed. Finally, numerical examples are provided to illustrate the results derived.
Projection models for health expenses
Marcus Christiansen, Michel Denuit, Nathalie Lucas, Jan-Philipp Schmidt
Published online by Cambridge University Press: 18 December 2017, pp. 185-203
This note proposes a practical way for modelling and projecting health insurance expenditures over short time horizons, based on observed historical data. The present study is motivated by a similar age structure generally observed for health insurance claim frequencies and yearly aggregate losses on the one hand and mortality on the other hand. As an application, the approach is illustrated for German historical inpatient costs provided by the Federal Financial Supervisory Authority. In particular, similarities and differences to mortality modelling are addressed.
A History of British Actuarial Thought, Turnbull Craig, Palgrave Macmillan, 2017, Cham, Switzerland, 345pp, ISBN: 978-3-319-33182-9 | CommonCrawl |
# Introductory Algebra
Anne Gloag<br>Andrew Gloag<br>Melissa Kramer
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## flexbook
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## Chapter Outline
1.1 Integers and RATIONAL NUmbers
1.2 Addition and Subtraction of Rational Numbers
1.3 MUltiplication and Division of Rational Numbers
1.4 ORDER OF OPERATIONS
1.5 Chapter 1 ReVieW
Real numbers are all around us. The majority of numbers we use in calculations are considered real numbers. This chapter defines a real number and explains important properties and rules that apply to real numbers.
### Integers and Rational Numbers
Integers and rational numbers are important in daily life. The price per square yard of carpet is a rational number. The number of frogs in a pond is expressed using an integer. The organization of real numbers can be drawn as a hierarchy. Look at the hierarchy below.
The most generic number is the real number; it can be a combination of negative, positive, decimal, fraction, or non-repeating decimal values. Real numbers have two major categories: rational numbers and irrational numbers.
Definition: Irrational numbers are numbers that can be written as non-repeating, non-terminating decimals such as $\pi$ or $\sqrt{2}$.
Definition: Rational numbers are numbers that can be written in the form $\frac{a}{b}$, where $a$ and $b$ are integers $a$ and $b \neq 0$. All Irrational Numbers and Rational Numbers are Real Numbers
Definition: Integers are all the whole numbers, zero and the negatives of the whole numbers. i.e. $\{\ldots-5,-4,-3,-2$, $-1,0,1,2,3,4,5 \ldots\}$ All Integers are Rational Numbers.
Definition: Whole Numbers are all Counting Numbers and the number zero. i.e. $\{0,1,2,3,4,5 \ldots\}$ All Whole Numbers are Integer.
Definition: Counting Numbers are the the natural numbers from 1 to infinity. i.e. $\{1,2,3,4,5 \ldots\}$ All Counting Numbers are Whole Numbers.
## A Review of Fractions
You can think of a rational number as a fraction of a cake. If you cut the cake into $b$ slices, your share is $a$ of those slices. For example, when we see the rational number $\frac{1}{2}$, we imagine cutting the cake into two parts. Our share is one of those parts. Visually, the rational number $\frac{1}{2}$ looks like this.
There are two main types of fractions: - Proper fractions are rational numbers where the numerator is less than the denominator. A proper fraction represents a number less than one. With a proper fraction you always end up with less than a whole cake!
- Improper fractions are rational numbers where the numerator is greater than or equal to the denominator. Improper fractions can be rewritten as a mixed number - an integer plus a proper fraction. An improper fraction represents a number greater than or equal to one.
When evaluating fractions, it is possible for two fractions to give the same numerical value. These fractions are called equivalent fractions. For example, look at a visual representation of the rational number $\frac{2}{4}$.
The visual of $\frac{1}{2}$ is equivalent to the visual of $\frac{2}{4}$. We can write out the prime factors of both the numerator and the denominator and cancel matching factors that appear in both the numerator and denominator.
$\left(\frac{2}{4}\right)=\left(\frac{\not 2 \cdot 1}{2 \cdot 2 \cdot 1}\right)$ We then re-multiply the remaining factors. $\left(\frac{2}{4}\right)=\left(\frac{1}{2}\right)$
Therefore, $\frac{1}{2}=\frac{2}{4}$. This process is called reducing the fraction, or writing the fraction in lowest terms. Reducing a fraction does not change the value of the fraction; it simplifies the way we write it. When we have canceled all common factors, we have a fraction in its simplest form.
Example 1: Classify and simplify the following rational numbers.
a) $\left(\frac{3}{7}\right)$
b) $\left(\frac{9}{3}\right)$
c) $\left(\frac{50}{60}\right)$
## Solution:
a) Because both 3 and 7 are prime numbers, $\frac{3}{7}$ is a proper fraction written in its simplest form.
b) The numerator is larger than the denominator, therefore, this is an improper fraction.
$$
\frac{9}{3}=\frac{3 \times 3}{3}=\frac{3}{1}=3
$$
c) This is a proper fraction; $\frac{50}{60}=\frac{5 \times 2 \times 5}{6 \times 2 \times 5}=\frac{5}{6}$
## Ordering Rational Numbers
To order rational numbers is to arrange them according to a set of directions, such as ascending (lowest to highest) or descending (highest to lowest). Ordering rational numbers is useful when determining which unit cost is the cheapest.
Example 2: Cans of tomato sauce come in three sizes: 8 ounces, 16 ounces, and 32 ounces. The costs for each size are $\$ 0.59, \$ 0.99$, and $\$ 1.29$, respectively. Find the unit cost and order the rational numbers in ascending order.
Solution: Use proportions to find the cost per ounce. $\frac{\$ 0.59}{8}=\frac{\$ 0.07375}{\text { ounce }} ; \frac{\$ 0.99}{16}=\frac{\$ 0.061875}{\text { ounce }} ; \frac{\$ 1.29}{32}=\frac{\$ 0.040}{\text { ounce }}$. Arranging the rational numbers in ascending order: $0.040,0.061875,0.07375$ Example 3: Which is greater $\frac{3}{7}$ or $\frac{4}{9}$ ?
Solution: Begin by creating a common denominator for these two fractions. Which number is evenly divisible by 7 and 9 ? $7 \times 9=63$, therefore the common denominator is 63 .
$$
\frac{3 \times 9}{7 \times 9}=\frac{27}{63} \quad \frac{4 \times 7}{9 \times 7}=\frac{28}{63}
$$
Because $28>27, \frac{4}{9}>\frac{3}{7}$
For more information regarding how to order fractions, watch this YouTube video.
KhanAcademy: Ordering Fractions
## Graph and Compare Integers
More specific than the rational numbers are the integers. Integers are whole numbers and their negatives. When comparing integers, you will use the math verbs such as less than, greater than, approximately equal to, and equal to. To graph an integer on a number line, place a dot above the number you want to represent.
Example 4: Compare the numbers 2 and -5.
Solution: First, we will plot the two numbers on a number line.
We can compare integers by noting which is the greatest and which is the least. The greatest number is farthest to the right, and the least is farthest to the left.
In the diagram above, we can see that 2 is farther to the right on the number line than -5 , so we say that 2 is greater than $\mathbf{- 5}$. We use the symbol >to mean "greater than".
Therefore, $2>-5$.
This can also be read as $\mathbf{- 5}$ is less than $\mathbf{2}$, because the -5 is farther to the left on the number line. To write this, we use the symbol <to mean "less than".
Therefore, $-5<2$
## Numbers and Their Opposites
Every number has an opposite, which represents the same distance from zero but in the other direction.
A special situation arises when adding a number to its opposite. The sum is zero. This is summarized in the following property.
The Additive Inverse Property: For any real number $a, a+-a=0$.
## Absolute Value
Absolute value represents the distance from zero when graphed on a number line. For example, the number 7 is 7 units away from zero. The number -7 is also 7 units away from zero. Therefore, the absolute value of 7 and the absolute value of -7 are both 7 .
We write the absolute value of -7 like this: $|-7|$.
We read the expression $|x|$ like this: "the absolute value of $x$."
- Treat absolute value expressions like parentheses. If there is an operation inside the absolute value symbols evaluate that operation first.
- The absolute value of a number or an expression is always positive or zero. It cannot be negative. With absolute value, we are only interested in how far a number is from zero, not the direction.
Example 5: Evaluate the following absolute value expressions.
a) $|5+4|$
b) $3-|4-9|$
c) $|-5-11|$
d) $-|7-22|$
Solution:
a)
$$
\begin{aligned}
|5+4| & =|9| \\
& =9
\end{aligned}
$$
b)
$$
\begin{aligned}
3-|4-9| & =3-|-5| \\
& =3-5 \\
& =-2
\end{aligned}
$$
c)
$$
\begin{aligned}
|-5-11| & =|-16| \\
& =16
\end{aligned}
$$
d)
$$
\begin{aligned}
-|7-22| & =-|-15| \\
& =-(15) \\
& =-15
\end{aligned}
$$
## Practice Set: Integers and Rational Numbers
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. Integersand Rational Numbers (13:00)
1. Define absolute value.
2. What are the two types of fractions?
3. Give an example of a real number that is not an integer.
4. What standards separate a rational number from an irrational number?
5. The tick-marks on the number line represent evenly spaced integers. Find the values of $a, b, c, d$ and $e$.
In $6-8$, determine what fraction of the whole each shaded region represents.
In $9-12$, place the following sets of rational numbers in order from least to greatest.
9. $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$
10. $\frac{11}{12}, \frac{12}{11}, \frac{13}{10}$
11. $39 \quad 49 \quad 59$
12. $\frac{7}{11}, \frac{8}{13}, \frac{12}{19}$
In $13-18$, find the simplest form of the following rational numbers.
13. $\frac{22}{44}$
14. $\frac{9}{27}$
15. $\frac{12}{18}$
16. $\frac{315}{420}$
17. $\frac{19}{10}$
18. $\frac{99}{11}$
In $19-27$, simplify.
19. $|-98.4|$
20. $|123.567|$
21. $-|16-98|$
22. $11-|-4|$
23. $|4-9|-|-5|$
24. $|-5-11|$
25. $-|-7|$
26. $|-2-88|-|88+2|$
27. $|-5-99|-|16-7|$
In 28 - 33, compare the two real numbers using the symbol <or $>$.
28. 8 and 7.99999
29. -4.25 and $\frac{-17}{4}$
30. 65 and -1
31. 10 units left of zero and 9 units right of zero
32. A frog is sitting perfectly on top of number 7 on a number line. The frog jumps randomly to the left or right, but always jumps a distance of exactly 2. Describe the set of numbers that the frog may land on, and list all the possibilities for the frog's position after exactly 5 jumps.
33. Will a real number always have an additive inverse? Explain your reasoning.
## Addition and Subtraction of Rational Num- bers
## Addition of Rational Numbers
A football team gains 11 yards on one play then loses 5 yards on another play and loses 2 yards on the third play. What is the total yardage loss or gain?
A loss can be expressed as a negative integer. A gain can be expressed as a positive integer. To find the net gain or loss, the individual values must be added together. Therefore, the sum is $11+-5+-2=4$. The team has a net gain of 4 yards.
Addition can also be shown using a number line. If you need to add $2+3$, start by making a point at the value of 2 and move three integers to the right. The ending value represents the sum of the values.
Example 1: Find the sum of $-2+3$ using a number line.
Solution: Begin by making a point at -2 and moving three units to the right. The final value is 1 , so $-2+3=1$
When the value that is being added is positive, we "jump" to the right. If the value is negative, we jump to the left (in a negative direction).
Example 2: Find 2-3 using a number line.
Solution: Begin by making a point at 2. The expression represents subtraction, so we will count three jumps to the left.
The solution is: $2-3=-1$
## Algebraic Properties of Addition
In a previous lesson you learned the Additive Inverse Property. This property states that the sum of a number and its opposite is zero. Algebra has many other properties that help you manipulate and organize information.
The Commutative Property of Addition: For all real numbers $a$, and $b, a+b=b+a$.
To commute means to change locations, so the Commutative Property of Addition allows you to rearrange the objects in an addition problem.
The Associative Property of Addition: For all real numbers $a, b$, and $c,(a+b)+c=a+(b+c)$
To associate means to group together, so the Associative Property of Addition allows you to regroup the objects in an addition problem.
The Identity Property of Addition: For any real number $a, a+0=a$
This property allows you to use the fact that the sum of any number and zero is the original value. For this reason, we call zero the additive identity.
Example 3: Simplify the following using the properties of addition:
a) $9+(1+22)$
b) $4211+0$
Solution:
a) It is easier to regroup $9+1$, so by applying the Associative Property of Addition, $(9+1)+22=10+22=32$
b) The Additive Identity Property states the sum of a number and zero is itself, therefore $4211+0=4211$
Nadia and Peter are building sand castles on the beach. Nadia built a castle two feet tall, stopped for ice-cream and then added one more foot to her castle. Peter built a castle one foot tall before stopping for a sandwich. After his sandwich, he built up his castle by two more feet. Whose castle is the taller?
Nadia's castle is $(2+1)$ feet tall. Peter's castle is $(1+2)$ feet tall. According to the Commutative Property of Addition, the two castles are the same height.
## Adding Rational Numbers
To add rational numbers, we must first remember how to rewrite mixed numbers as improper fractions. Begin by multiplying the denominator of the mixed number to the whole value. Add the numerator to this product. This value is the numerator of the improper fraction. The denominator is the original.
Example 4: Write $11 \frac{2}{3}$ as an improper fraction: Solution: $3 \times 11=33+2=35$. This is the numerator of the improper fraction.
$$
11 \frac{2}{3}=\frac{35}{3}
$$
Now that we know how to rewrite a mixed number as an improper fraction, we can begin to add rational numbers. There is one thing to remember when finding the sum or difference of rational numbers: the denominators must be equivalent.
The Addition Property of Fractions: For all real numbers $a, b$, and $c, \frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}$
Watch this video for further explanation on adding fractions with unlike denominators. This video shows how to add fractions using a visual model.
http://www.teachertube.com/viewVideo.php?video_id=103926\\&title=Adding_Fractions_with_Unlike_Denominators
## Subtraction of Rational Numbers
In the previous two lessons you have learned how to find the opposite of a rational number and to add rational numbers. You can use these two concepts to subtract rational numbers. Suppose you want to find the difference of 9 and 12. Symbolically, it would be $9-12$. Begin by placing a dot at nine and move to the left 12 units.
$$
9-12=-3
$$
Rule: To subtract a number, add its opposite.
$$
3-5=3+(-5)=-2 \quad 9-16=9+(-16)=-7
$$
A special case of this rule can be written when trying to subtract a negative number.
The Opposite-Opposite Property: Since taking the opposite of a number changes its sign, we can say that $-(-b)=$ $b$. So it is also true that for any real numbers $a$ and $b, a-(-b)=a+b$.
Example 1: Simplify $-6-(-13)$
Solution: Using the Opposite-Opposite Property, the double negative is rewritten as a positive.
$$
-6-(-13)=-6+13=7
$$
Example 2: Simplify $\frac{5}{6}-\left(-\frac{1}{18}\right)$ :
Solution: Begin by using the Opposite-Opposite Property
$$
\frac{5}{6}+\frac{1}{18}
$$
Next, create a common denominator: $\frac{5 \times 3}{6 \times 3}+\frac{1}{18}=\frac{15}{18}+\frac{1}{18}$
Add the fractions: $\frac{16}{18}$
Reduce: $\frac{2 \times 2 \times 2 \times 2}{3 \times 3 \times 2}=\frac{8}{9}$
## Practice Set: Addition of Integers
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. Additionof RationalNumbers (7:40)
In exercises 1 and 2, write the sum represented by the moves on the number line.
Find the sum. Write fractions in their simplest form. Convert improper fractions to mixed numbers.
3. $\frac{3}{7}+\frac{2}{7}$
4. $\frac{3}{10}+\frac{1}{5}$
5. $\frac{5}{16}+\frac{5}{12}$
6. $\frac{3}{8}+\frac{9}{16}$
7. $\frac{8}{25}+\frac{7}{10}$
8. $\frac{1}{6}+\frac{1}{4}$
9. $\frac{7}{15}+\frac{2}{9}$
10. $\frac{5}{19}+\frac{2}{27}$
11. $-2.6+11.19$
12. $-8+13$
13. $-7.1+(-5.63)$
14. $9.99+(-0.01)$
15. $4 \frac{7}{8}+1 \frac{1}{2}$
16. $-3 \frac{1}{3}+\left(-2 \frac{3}{4}\right)$
In $17-20$, which property of addition does each situation involve?
17. Whichever order your groceries are scanned at the store, the total will be the same.
18. Suppose you go buy a DVD for $\$ 8.00$, another for $\$ 29.99$, and a third for $\$ 14.99$. You can add $(8+29.99)+$ 14.99 or you can add $8+(29.99+14.99)$ to obtain the total.
19. Shari's age minus the negative of Jerry's age equals the sum of the two ages.
20. Kerri has 16 apples and has added zero additional apples. Her current total is 16 apples.
21. Nadia, Peter and Ian are pooling their money to buy a gallon of ice cream. Nadia is the oldest and gets the greatest allowance. She contributes half of the cost. Ian is next oldest and contributes one third of the cost. Peter, the youngest, gets the smallest allowance and contributes one fourth of the cost. They figure that this will be enough money. When they get to the check-out, they realize that they forgot about sales tax and worry there will not be enough money. Amazingly, they have exactly the right amount of money. What fraction of the cost of the ice cream was added as tax?
22. A blue whale dives 160 feet below the surface then rises 8 feet. Write the addition problem and find the sum.
23. The temperature in Chicago, Illinois one morning was $-8^{\circ} \mathrm{F}$. Over the next six hours the temperature rose 25 degrees Fahrenheit. What was the new temperature?
## Practice Set: Subtraction of Integers
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. Subtraction of Rational Numbers (10:22)
## MEDIA
Click image to the left for more content. In $1-20$, subtract the following rational numbers. Be sure that your answer is in the simplest form.
1. $9-14$
2. $2-7$
3. $21-8$
4. $8-(-14)$
5. $-11-(-50)$
6. $\frac{5}{12}-\frac{9}{18}$
7. $5.4-1.01$
8. $\frac{2}{3}-\frac{1}{4}$
9. $\frac{3}{4}-\frac{1}{3}$
10. $\frac{1}{4}-\left(-\frac{2}{3}\right)$
11. $\frac{15}{11}-\frac{9}{7}$
12. $\frac{2}{13}-\frac{1}{11}$
13. $-\frac{7}{8}-\left(-\frac{8}{3}\right)$
14. $\frac{7}{27}-\frac{9}{39}$
15. $\frac{6}{11}-\frac{3}{22}$
16. $-3.1-21.49$
17. $\frac{13}{64}-\frac{7}{40}$
18. $\frac{11}{70}-\frac{11}{30}$
19. $-68-(-22)$
20. $\frac{1}{3}-\frac{1}{2}$
21. KMN stock began the day with a price of $\$ 4.83$ per share. At the closing bell, the price dropped $\$ 0.97$ per share. What was the closing price of KMN stock?
## Multiplication and Division of Rational Numbers
When you began learning how to multiply whole numbers, you replaced repeated addition with the multiplication sign $(\times)$. For example,
$$
6+6+6+6+6=5 \times 6=30
$$
Multiplying rational numbers is performed the same way. We will start with the Multiplication Property of -1 .
## Properties of Multiplication
The Multiplication Property of -1: For any real number $a,(-1) \times a=-a$.
This can be summarized by saying "a number times a negative 1 is the opposite of the number".
Example 1: Evaluate $(-1) \cdot 9,876$.
Solution: Using the Multiplication Property of $-1:(-1) \cdot 9,876=-9,876$.
This property can also be used when the values are negative, as shown in example 2.
Example 2: Evaluate $(-1) \cdot-322$.
Solution: Using the Multiplication Property of $-1:(-1) \cdot-322=322$.
A basic algebraic property is the Multiplicative Identity. Similar to the Additive Identity, this property states that any value multiplied by 1 will result in the original value.
The Identity Property of Multiplication: For any real numbers $a,(1) \times a=a$.
A third property of multiplication is the Multiplication Property of Zero. This property states that any value multiplied by zero will result in zero.
The Zero Property of Multiplication: For any real numbers $a,(0) \times a=0$.
## Multiplying Rational Numbers
You've decided to make cookies for a party. The recipe you've chosen makes 6 dozen cookies, but you only need 2 dozen. How do you reduce the recipe?
In this case, you should not use subtraction to find the new values. Subtraction means to make less by taking away. You haven't made any cookies, therefore cannot take any away. Instead, you need to make $\frac{2}{6}$ or $\frac{1}{3}$ of the original recipe. This process involves multiplying fractions.
For any real numbers $a, b, c, d$ where $b \neq 0$ and $d \neq 0$,
$$
\frac{a}{b} \cdot \frac{c}{d}=\frac{a c}{b d}
$$
Example 3: The original cookie recipe calls for 8 cups flour. How much is needed for the reduced recipe?
Solution: Begin by writing the multiplication situation. $8 \cdot \frac{1}{3}$. We need to rewrite this product in the form of the property above. In order to perform this multiplication, you need to rewrite 8 as the fraction $\frac{8}{1}$.
$$
8 \times \frac{1}{3}=\frac{8}{1} \times \frac{1}{3}=\frac{8 \cdot 1}{1 \cdot 3}=\frac{8}{3}=2 \frac{2}{3}
$$
You will need $2 \frac{2}{3}$ cups flour.
Multiplication of fractions can also be shown visually. For example, to multiply $\frac{1}{3} \cdot \frac{2}{5}$, draw one model to represent the first fraction and a second model to represent the second fraction.
By placing one model (divided in thirds horizontally) on top of the other (divided in fifths vertically) you divide one whole rectangle into $b d$ smaller parts. Shade $a c$ smaller regions.
The product of the two fractions is the $\frac{\text { shaded regions }}{\text { total regions }}$
$$
\frac{1}{3} \cdot \frac{2}{5}=\frac{2}{15}
$$
Example 4: Simplify $\frac{3}{7} \cdot \frac{4}{5}$
Solution: By drawing visual representations, you can see
$$
\frac{3}{7} \cdot \frac{4}{5}=\frac{12}{35}
$$
## More Properties of Multiplication
Properties that hold true for addition such as the Associative Property and Commutative Property also hold true for multiplication. They are summarized below.
The Associative Property of Multiplication: For any real numbers $a, b$, and $c$,
$$
(a \cdot b) \cdot c=a \cdot(b \cdot c)
$$
The Commutative Property of Multiplication: For any real numbers $a$ and $b$,
$$
a(b)=b(a)
$$
The Same Sign Multiplication Rule: The product of two positive or two negative numbers is positive.
The Different Sign Multiplication Rule: The product of a positive number and a negative number is a negative number.
## Solving Real-World Problems Using Multiplication
Example 5: Anne has a bar of chocolate and she offers Bill a piece. Bill quickly breaks off $\frac{1}{4}$ of the bar and eats it. Another friend, Cindy, takes $\frac{1}{3}$ of what was left. Anne splits the remaining candy bar into two equal pieces which she shares with a third friend, Dora. How much of the candy bar does each person get?
Solution: Think of the bar as one whole.
$1-\frac{1}{4}=\frac{3}{4}$. This is the amount remaining after Bill takes his piece.
$\frac{1}{3} \times \frac{3}{4}=\frac{1}{4}$. This is the fraction Cindy receives.
$\frac{3}{4}-\frac{1}{4}=\frac{2}{4}=\frac{1}{2}$. This is the amount remaining after Cindy takes her piece.
Anne divides the remaining bar into two equal pieces. Every person receives $\frac{1}{4}$ of the bar.
Example 6: Doris' truck gets $10 \frac{2}{3}$ miles per gallon. Her tank is empty so she fills it with $5 \frac{1}{2}$ gallons.
How far can she travel?
Solution: Begin by writing each mixed number as an improper fraction.
$$
10 \frac{2}{3}=\frac{32}{3} \quad 5 \frac{1}{2}=\frac{11}{2}
$$
Now multiply the two values together.
$$
\frac{32}{3} \cdot \frac{11}{2}=\frac{352}{6}=58 \frac{4}{6}=58 \frac{2}{3}
$$
Doris can travel $58 \frac{2}{3}$ miles on $5 \frac{1}{2}$ gallons of gas.
## Division of Rational Numbers
So far in this chapter you have added, subtracted, and multiplied rational numbers. It now makes sense to learn how to divide rational numbers. We will begin with a definition of inverse operations.
Inverse operations "undo" each other.
For example, addition and subtraction are inverse operations because addition cancels subtraction and vice versa. The additive identity results in a sum of zero. In the same sense multiplication and division are inverse operations. This leads into the next property: The Inverse Property of Multiplication.
For every nonzero number $a$, there is a multiplicative inverse $\frac{1}{a}$ such that $a\left(\frac{1}{a}\right)=1$.
The values of $a$ and $\frac{1}{a}$ are called reciprocals. In general, two nonzero numbers whose product is 1 are reciprocals.
Reciprocal: The reciprocal of a nonzero rational number $\frac{a}{b}$ is $\frac{b}{a}$.
Note: The number zero does not have a reciprocal.
## Using Reciprocals to Divide Rational Numbers
When dividing rational numbers, use the following rule:
## "When dividing rational numbers, multiply by the 'right' reciprocal."
In this case, the "right" reciprocal means to take the reciprocal of the fraction on the right-hand side of the division operator.
Example 7: Simplify $\frac{2}{9} \div \frac{3}{7}$.
Solution: Begin by multiplying by the "right" reciprocal
$$
\frac{2}{9} \times \frac{7}{3}=\frac{14}{27}
$$
Example 8: Simplify $\frac{7}{3} \div \frac{2}{3}$.
Solution: Begin by multiplying by the "right" reciprocal.
$$
\frac{7}{3} \div \frac{2}{3}=\frac{7}{3} \times \frac{3}{2}=\frac{7 \cdot 3}{2 \cdot 3}=\frac{7}{2}
$$
Instead of the division symbol $\div$, you may see a large fraction bar. This is seen in the next example.
Example 9: Simplify $\frac{\frac{2}{3}}{\frac{7}{8}}$.
Solution: The fraction bar separating $\frac{2}{3}$ and $\frac{7}{8}$ indicates division.
$$
\frac{2}{3} \div \frac{7}{8}
$$
Simplify as in example 2:
$$
\frac{2}{3} \times \frac{8}{7}=\frac{16}{21}
$$
## Using Reciprocals to Solve Real-World Problems
The need to divide rational numbers is necessary for solving problems in physics, chemistry, and manufacturing. The following example illustrates the need to divide fractions in physics.
Example 10: Newton's Second Law relates acceleration to the force of an object and its mass: $a=\frac{F}{m}$. Suppose $F=7 \frac{1}{3}$ and. Find a, the acceleration.
Solution: Before beginning the division, the mixed number of force must be rewritten as an improper fraction.
Replace the fraction bar with a division symbol and simplify: $a=\frac{22}{3} \div \frac{1}{5}$
$\frac{22}{3} \times \frac{5}{1}=\frac{110}{3}=36 \frac{2}{3}$. Therefore, the acceleration is $36 \frac{2}{3} \mathrm{~m} / \mathrm{s}^{2}$
Example 11: Anne runs a mile and a half in one-quarter hour. What is her speed in miles per hour?
Solution: Use the formula speed $=\frac{\text { distance }}{\text { time }}$.
$$
s=1.5 \div \frac{1}{4}
$$
Rewrite the expression and simplify: $s=\frac{3}{2} \cdot \frac{4}{1}=\frac{4 \cdot 3}{2 \cdot 1}=\frac{12}{2}=6 \mathrm{mi} / \mathrm{hr}$
## Practice Set: Multiplication of Integers
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. Multiplicati on of Rational Numbers (8:56)
Multiply the following rational numbers.
1. $\frac{1}{2} \cdot \frac{3}{4}$
2. $-7.85 \cdot-2.3$ 3. $\frac{2}{5} \cdot \frac{5}{9}$
3. $\frac{1}{3} \cdot \frac{2}{7} \cdot \frac{2}{5}$
4. $4.5 \cdot-3$
5. $\frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5}$
6. $\frac{5}{12} \times \frac{9}{10}$
7. $\frac{27}{5} \cdot 0$
8. $\frac{2}{3} \times \frac{1}{4}$
9. $-11.1(4.1)$
10. $\frac{3}{4} \times \frac{1}{3}$
11. $\frac{15}{11} \times \frac{9}{7}$
12. $\frac{2}{7} \cdot-3.5$
13. $\frac{1}{13} \times \frac{1}{11}$
14. $\frac{7}{27} \times \frac{9}{14}$
15. $\left(\frac{3}{5}\right)^{2}$
16. $\frac{1}{11} \times \frac{22}{21} \times \frac{7}{10}$
17. $5.75 \cdot 0$
In $19-21$, state the property that applies to each of the following situations.
19. A gardener is planting vegetables for the coming growing season. He wishes to plant potatoes and has a choice of a single 8 by 7 meter plot, or two smaller plots of 3 by 7 meters and 5 by 7 meters. Which option gives him the largest area for his potatoes?
20. Andrew is counting his money. He puts all his money into $\$ 10$ piles. He has one pile. How much money does Andrew have?
21. Nadia and Peter are raising money by washing cars. Nadia is charging $\$ 3$ per car, and she washes five cars in the first morning. Peter charges $\$ 5$ per car (including a wax). In the first morning, he washes and waxes three cars. Who has raised the most money?
In 22 - 30, find the multiplicative inverse of each of the following.
22. 100
23. $\frac{2}{8}$
24. $-\frac{19}{21}$
25. 7
26. $-\frac{z^{3}}{2 x y^{2}}$
27. 0
28. $\frac{1}{3}$
29. $\frac{-19}{18}$
30. $\frac{3 x y}{8 z}$
## Practice Set: Division of Integers
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. Divisionof RationalNumbers (8:20)
In 1 - 9, divide the following rational numbers. Be sure that your answer in the simplest form.
1. $\frac{5}{2} \div \frac{1}{4}$
2. $\frac{5}{11} \div \frac{6}{7}$
3. $\frac{1}{2} \div \frac{1}{2}$
4. $-\frac{x}{2} \div \frac{5}{7}$
5. $\frac{1}{2} \div \frac{x}{4 y}$
6. $\left(-\frac{1}{3}\right) \div\left(-\frac{3}{5}\right)$
7. $\frac{7}{2} \div \frac{7}{4}$
8. $11 \div\left(-\frac{x}{4}\right)$
In $10-12$, evaluate the expression.
10. $\frac{x}{y}$ for $x=\frac{3}{8}$ and $y=\frac{4}{3}$
11. $4 z \div u$ for $u=0.5$ and $z=10$
12. $\frac{-6}{m}$ for $m=\frac{2}{5}$
In 13 - 16, answer the questions.
13. The label on a can of paint states that it will cover 50 square feet per pint. If I buy a $\frac{1}{8}$ pint sample, it will cover a square two feet long by three feet high. Is the coverage I get more, less or the same as that stated on the label?
14. The world's largest trench digger, "Bagger 288," moves at $\frac{3}{8} \mathrm{mph}$. How long will it take to dig a trench $\frac{2}{3}$ mile long?
15. Explain why the reciprocal of a nonzero rational number is not the same as the opposite of that number.
16. Explain why zero does not have a reciprocal.
### Order of Operations
## The Aviary Dilemma
Keisha loves the birds in the aviary at the city zoo. Her favorite part of the aviary is the bird rescue. Here the zoo staff rescues injured birds, helps them to heal and then releases them again. Currently, they have 256 birds in the rescue. Today, Keisha has a special visit planned with Ms. Thompson who is in charge of the bird rescue.
When Keisha arrives, Ms. Thompson is already hard at work. She tells Keisha that there are new baby birds in the rescue. Three of the birds have each given birth to five baby birds. Keisha can't help grinning as she walks around. She can hear the babies chirping. In fact, it sounds like they are everywhere.
"It certainly sounds like a lot more babies," Keisha says.
"Yes," Ms. Thompson agrees. "We also released two birds yesterday."
"That is great news," Keisha says smiling.
"Yes, but we also found three new injured birds. Our population has changed again."
"I see," Keisha adds, “That is $256+3 \times 5-2+3$ that equals 1296 birds, I think. I'm not sure, that doesn't seem right."
Is Keisha's math correct?
How many birds are there now?
Can you figure it out?
This is a bit of a tricky question. You will need to learn some new skills to help Keisha determine the number of birds in the aviary.
Pay attention. By the end of the lesson, you will know all about the order of operations. Then you will be able to help Keisha with the bird count.
## Evaluating Numerical Expressions with the Four Operations
This lesson begins with evaluating numerical expressions. But before we can do that we need to answer one key question, "What is an expression?"
To understand what an expression is, let's compare it with an equation. An equation is a number sentence that can be solved. It has an equal sign where one side of the equals sign is equal to the other side of the equals sign.
Example
$$
3+4=7
$$
This is an equation. It has an equals sign and can be solved.
What is an expression then?
An expression is a number sentence without an equals sign. It can be simplified and/or evaluated.
Example
$$
4+3 \times 5
$$
Now this expression can be confusing because it has both addition and multiplication in it.
Do we need to add or multiply first?
To figure this out, we are going to learn something called the Order of Operations.
The Order of Operation is a way of evaluating expressions. It lets you know what order to complete each operation in.
Order of Operations
$P$ - parentheses
E - exponents
MD - multiplication or division in order from left to right
AS - addition or subtraction in order from left to right
Take a few minutes to write these down in a notebook.
Now that you know the order of operations, let's go back to our example.
Example
$$
4+3 \times 5
$$
Here we have an expression with addition and multiplication.
We can look at the order of operations and see that multiplication comes before addition. We need to complete that operation first.
$$
\begin{aligned}
& 4+3 \times 5 \\
& 4+15 \\
& =19
\end{aligned}
$$
When we evaluate this expression using order of operations, our answer is 20.
What would have happened if we had NOT followed the order of operations?
Example
$$
4+3 \times 5
$$
We probably would have solved the problem in order from left to right.
$$
\begin{aligned}
& 4+3 \times 5 \\
& 7 \times 5 \\
& =35
\end{aligned}
$$
This would have given us an incorrect answer. It is important to always follow the order of operations.
Here are few for you to try on your own.
1. $8-1 \times 4+3=$
2. $2 \times 6+8 \div 2=$
3. $5+9 \times 3-6+2=$
## Evaluating Numerical Expressions Using Powers and Grouping Symbols
We can also use the order of operations when we have exponent powers and grouping symbols like parentheses.
Let's review where exponents and parentheses fall in the order of operations.
## Order of Operations
$P$ - parentheses
E - exponents
MD - multiplication or division in order from left to right
## AS - addition or subtraction in order from left to right
Wow! You can see that according to the order of operations parentheses comes first. We always do the work in parentheses first. Then we evaluate exponents.
Let's see how this works with a new example.
Example
$$
2+(3-1) \times 2
$$
In this example, we can see that we have four things to look at.
We have 1 set of parentheses, addition, subtraction in the parentheses and multiplication.
We can evaluate this expression using the order of operations.
Example
$$
\begin{aligned}
& 2+(3-1) \times 2 \\
& 2+2 \times 2 \\
& 2+4 \\
& =6
\end{aligned}
$$
## Our answer is 6.
## What about when we have parentheses and exponents?
Example
$$
35+3^{2}-(3 \times 2) \times 7
$$
We start by using the order of operations. It says we evaluate parentheses first.
$$
\begin{aligned}
& 3 \times 2=6 \\
& 35+3^{2}-6 \times 7
\end{aligned}
$$
Next, we evaluate exponents
$$
\begin{aligned}
& 3^{2}=3 \times 3=9 \\
& 35+9-6 \times 7
\end{aligned}
$$
Next, we complete multiplication or division in order from left to right. We have multiplication.
$$
\begin{aligned}
& 6 \times 7=42 \\
& 35+9-42
\end{aligned}
$$
Next, we complete addition and/or subtraction in order from left to right.
$$
\begin{aligned}
35+9 & =44 \\
44-42 & =2
\end{aligned}
$$
## Our answer is 2.
Now, consider the expression $8+(5)(4)-(6+10 \div 2)+44$. Simply using order of operations. Then watch the vide o explanation for this problem.
Here are a few for you to try on your own.
1. $16+2^{3}-5+(3 \times 4)$
2. $9^{2}+2^{2}-5 \times(2+3)$
3. $8^{2} \div 2+4-1 \times 6$
## Use the Order of Operations to Determine if an Answer is True
We just finished using the order of operations to evaluate different expressions.
We can also use the order of operations to "check" our work.
In this section, you will get to be a "Math Detective."
As a math detective, you will be using the order of operations to determine whether or not someone else's work is correct.
Here is a worksheet that has been completed by Joaquin.
Your task is to check Joaquin's work and determine whether or not his work is correct.
Use your notebook to take notes.
If the expression has been evaluated correctly, then please make a note of it. If it is incorrect, then re-evaluate the expression correctly.
Here are the problems that are on Joaquin's worksheet.
$$
\begin{aligned}
& \text { 1. } 7 \times(4+1)-7 \times 2=21 \\
& \text { 2. } 3^{2}+4^{2}-9+(3 \times 2)=22 \\
& \text { 3. } 6+3 \times 2-5+(7-1)=19 \\
& \text { 4. }(8 \times 2)-3^{2}+(5 \times 2)=17 \\
& \text { 5. } 18-2 \times 3+(6 \times 3)=66
\end{aligned}
$$
## Did you check Joaquin's work?
## Let's see how you did with your answers. Take your notebook and check your work with these correct answers.
## Let's begin with problem number 1.
We start by adding $4+1$ which is 5 . Then we multiply $7 \times 5$ and $7 \times 2$. Since multiplication comes next in our order of operations. Finally we subtract $35-14=21$.
Joaquin's work is correct.
Problem Number 2
We start by evaluating the exponents. 3 squared is 9 and 4 squared is 16 . Next we multiply $3 \times 2=6$. Finally we can complete the addition and subtraction in order from left to right. Our final answer is 22.
Joaquin's work is correct.
Problem Number 3
We start with the multiplication and multiply $3 \times 2$ which is 6 . Then we complete the parentheses $7-1=6$. Now we can complete the addition and subtraction in order from left to right. The answer correct is 13.
Uh Oh, Joaquin's answer is incorrect. How did Joaquin get 19 as an answer?
Well, if you look, Joaquin did not follow the order of operations. He just did the operations in order from left to right. If you don't multiply $3 \times 2$ first, then you get 19 as an answer instead of 16 .
Problem Number 4
Let's complete the work in parentheses first, $8 \times 2=16$ and $5 \times 2=10$. Next we evaluate the exponent, 3 squared is 9. Now we can complete the addition and subtraction in order from left to right. The answer is 17.
Joaquin's work is correct.
Problem Number 5
First, we need to complete the work in parentheses, $6 \times 3=18$. Next, we complete the multiplication $2 \times 3=6$. Now we can evaluate the addition and subtraction in order from left to right. Our answer is 30 .
Uh Oh, Joaquin got mixed up again. How did he get 66? Let's look at the problem. Oh, Joaquin subtracted $18-2$ before multiplying. You can't do that. He needed to multiply $2 \times 3$ first then he needed to subtract. Because of this, Joaquin's work is not accurate.
## How did you do?
## Remember, a Math Detective can check any answer by following the order of operations.
## Insert Grouping Symbols to Make a Given Answer True
Sometimes a grouping symbol can help us to make an answer true. By putting a grouping symbol, like parentheses, in the correct spot, we can change an answer.
## Let's try this out.
Example
$$
5+3 \times 2+7-1=22
$$
Now if we just solve this problem without parentheses, we get the following answer.
$$
5+3 \times 2+7-1=17
$$
How did we get this answer?
Well, we began by completing the multiplication, $3 \times 2=6$. Then we completed the addition and subtraction in order from left to right. That gives us an answer of 17 .
However, we want an answer of 22.
Where can we put the parentheses so that our answer is 22 ?
This can take a little practice and you may have to try more than one spot too.
Let's try to put the parentheses around $5+3$.
Example
$$
(5+3) \times 2+7-1=22
$$
Is this a true statement?
Well, we begin by completing the addition in parentheses, $5+3=8$. Next we complete the multiplication, $8 \times 2=16$. Here is our problem now.
$$
16+7-1=22
$$
Next, we complete the addition and subtraction in order from left to right.
## Our answer is 22.
Here are a few for you to try on your own. Insert a set of parentheses to make each a true statement.
1. $6-3+4 \times 2+7=39$
2. $8 \times 7+3 \times 8-5=65$
3. $2+5 \times 2+18-4=28$
## The Aviary Dilemma
## Let's look back at Keisha and Ms. Thompson and the bird dilemma at the zoo.
## Here is the original problem.
Keisha loves the birds in the aviary at the city zoo. Her favorite part of the aviary is the bird rescue. Here the zoo staff rescues injured birds, helps them to heal and then releases them again. Currently, they have 256 birds in the rescue. Today, Keisha is has a special visit planned with Ms. Thompson who is in charge of the bird rescue. When Keisha arrives, Ms. Thompson is already hard at work. She tells Keisha that there are new baby birds in the rescue. Three of the birds have each given birth to five baby birds. Keisha can't help grinning as she walks around. She can hear the babies chirping. In fact, it sounds like they are everywhere.
"It certainly sounds like a lot more babies," Keisha says.
"Yes," Ms. Thompson agrees. "We also released two birds yesterday."
"That is great news," Keisha says smiling.
"Yes, but we also found three new injured birds. Our population has changed again."
"I see," Keisha adds, “That is $256+3 \times 5-2+3$ that equals 1296 birds, I think. I'm not sure, that doesn't seem right."
We have an equation that Keisha wrote to represent the comings and goings of the birds in the aviary.
Before we figure out if Keisha's math is correct, let's underline any important information in the problem.
Wow, there is a lot going on. Here is what we have to work with.
256 birds
$3 \times 5$ - three birds each gave birth to five baby birds
1. birds were released
2. injured birds were found.
Since we started with 256 birds, that begins our equation. Then we can add in all of the pieces of the problem. $256+3 \times 5-2+3=$
This is the same equation that Keisha came up with. Let's look at her math.
Keisha says, "That is $256+3 \times 5-2+3$ that equals 1296 birds, I think. I'm not sure, that doesn't seem right." It isn't correct. Keisha forgot to use the order of operations.
According to the order of operations, Keisha needed to multiply $3 \times 5$ BEFORE completing any of the other operations.
Let's look at that.
$$
\begin{array}{r}
256+3 \times 5-2+3= \\
256+15-2+3= \\
256+13+3= \\
269+3= \\
272=
\end{array}
$$
Now we can complete the addition and subtraction in order from left to right.
$256+15-2+3=272$
The new bird count in the aviary is 272 birds.
## Practice Set
Evaluate each expression according to the order of operations.
1. $2+3 \times 4+7=$
2. $4+5 \times 2+9-1=$
3. $6 \times 7+2 \times 3=$
4. $4 \times 5+3 \times 1-9=$
5. $5 \times 3 \times 2+5-1=$
6. $4+7 \times 3+8 \times 2=$
7. $9-3 \times 1+4-7=$
8. $10+3 \times 4+2-8=$
9. $11 \times 3+2 \times 4-3=$
10. $6+7 \times 8-9 \times 2=$
11. $3+4^{2}-5 \times 2+9=$
12. $2^{2}+5 \times 2+6^{2}-11=$
13. $3^{2} \times 2+4-9=$
14. $6+3 \times 2^{2}+7-1=$
15. $7+2 \times 4+3^{2}-5=$
16. $3+(2+7)-3+5=$
17. $2+(5-3)+7^{2}-11=$
18. $4 \times 2+(6-4)-9+5=$
19. $8^{2}-4+(9-3)+12=$
20. $7^{3}-100+(3+4)-9=$
Check each answer using order of operations. Write whether the answer is true or false.
21. $4+5 \times 2+8-7=15$
22. $4+3 \times 9+6-10=104$
23. $6+2^{2} \times 4+3 \times 6=150$
24. $3+6 \times 3+9 \times 7-18=66$
25. $7 \times 2^{3}+4-9 \times 3-8=25$ Directions: Insert grouping symbols to make each a true statement.
26. $4+5-2+3-2=8$
27. $2+3 \times 2-4=6$
28. $1+9 \times 4+3+2-1=121$
29. $7+4 \times 3-5 \times 2=23$
30. $2^{2}+5 \times 8-3+4=33$
### Chapter 1 Review
Compare the real numbers. Which real number is the largest?
1. 7 and -11
2. $\frac{4}{5}$ and $\frac{11}{16}$
3. $\frac{10}{15}$ and $\frac{2}{3}$
4. 0.985 and $\frac{31}{32}$
5. -16.12 and $\frac{-300}{9}$
Order the real numbers from least to greatest.
6. $\frac{8}{11}, \frac{7}{10}, \frac{5}{9}$
7. $\frac{2}{7}, \frac{1}{11}, \frac{8}{13}, \frac{4}{7}, \frac{8}{9}$
Graph these values on the same number line.
8. $3 \frac{1}{3}$
9. -1.875
10. $\frac{7}{8}$
11. $0.1 \overline{6}$
12. $\frac{-55}{5}$
Simplify by performing the operation(s).
13. $\frac{8}{5}-\frac{4}{3}$
14. $\frac{4}{3}-\frac{1}{2}$
15. $\frac{1}{6}+1 \frac{5}{6}$
16. $\frac{-5}{4} \times \frac{1}{3}$
17. $\frac{4}{9} \times \frac{7}{4}$
18. $-1 \frac{5}{7} \times-2 \frac{1}{2}$
19. $\frac{1}{9} \div-1 \frac{1}{3}$
20. $\frac{-3}{2} \div \frac{-10}{7}$
21. $-3 \frac{7}{10} \div 2 \frac{1}{4}$
22. $1 \frac{1}{5}-\left(-3 \frac{3}{4}\right)$
23. $4 \frac{2}{3}+3 \frac{2}{3}$
24. $5.4+(-9.7)$
25. $(-7.1)+(-0.4)$
26. $(-4.79)+(-3.63)$
27. $(-8.1)-(-8.9)$
28. $1.58-(-13.6)$
29. $(-13.6)+12-(-15.5)$ 30. $(-5.6)-(-12.6)+(-6.6)$
30. $19.4+24.2$
31. $8.7+3.8+12.3$
32. $9.8-9.4$
33. $2.2-7.3$
Which property has been applied?
35. $6.78+(-6.78)=0$
36. $9.8+11.2+1.2=9.8+1.2+11.2$
37. $\frac{4}{3}-\left(-\frac{5}{6}\right)=\frac{4}{3}+\frac{5}{6}$
38. $8(11)\left(\frac{1}{8}\right)=8\left(\frac{1}{8}\right)(11)$
Solve the real-world situation.
39. Carol has 18 feet of fencing and purchased an addition 132 inches. How much fencing does Carol have?
40. Ulrich is making cookies for a fundraiser. Each cookie requires $\frac{3}{8}$ pound of dough. He has 12 pounds of cookie dough. How many cookies can Ulrich make?
41. Bagger 288 is a trench digger, which moves at $\frac{3}{8}$ miles/hour. How long will it take to dig a trench 14 miles long?
42. Georgia started with a given amount of money, $a$. She spent $\$ 4.80$ on a large latte, $\$ 1.20$ on an English muffin, $\$ 68.48$ on a new shirt, and $\$ 32.45$ for a present. She now has $\$ 0.16$. How much money, $a$, did Georgia have in the beginning?
43. The formula for an area of a square is $A=s^{2}$. A square garden has an area of 145 meters ${ }^{2}$. Find the length of the garden exactly.
## CHAPTER
## Introduction to Variables
## Chapter Outline
### VARIABLE EXPRESSIONS
2.2 PATterns AND EXPRESSIONS
2.3 Combining LiKe Terms
### The Distributive Property
### Addition and Subtraction of Polynomials
In order to represent real life situations mathematically, we often use symbols to represent unknown quantities. We call these symbols variables. Each mathematical subject requires knowledge of manipulating expressions and equations to solve for a variable. Careers such as automobile accident investigators, quality control engineers, and insurance originators use equations to determine the value of variables.
Throughout this chapter, you will learn how variables can be used to represent unknown quantities in various situations, and how to simplify algebraic expressions with variables using the Order of Operations.
### Variable Expressions
## Who Speaks Math, Anyway?
When someone is having trouble with algebra, they may say, "I don't speak math!" While this may seem weird to you, it is a true statement. Math, like English, French, Spanish, or Arabic, is a like a language that you must learn in order to be successful. In order to understand math, you must practice the language.
Action words, like run, jump, or drive, are called verbs. In mathematics, operations are like verbs because they involve doing something. Some operations are familiar, such as addition, multiplication, subtraction, or division. Operations can also be much more complex like an raising to an exponent or taking a square root.
Example 1: Suppose you have a job earning $\$ 8.15$ per hour. What could you do to quickly find out how much money you would earn for different hours of work?
Solution: You could make a list of all the possible hours, but that would take forever! So instead, you let the "hours you work" be replaced with a symbol, like $h$ for hours, and write an equation such as:
amount of money $=8.15 \mathrm{~h}$
In mathematics, variable is a symbol, usually an English letter, written to replace an unknown or changing quantity.
Example 2: What variable symbol would be a good choice for the following situations?
a. the number of cars on a road
b. time in minutes of a ball bounce
c. distance from an object
Solution: There are many options, but here are a few to think about.
a. Cars is the changing value, so $c$ is a good choice.
b. Time is the changing value, so $t$ is a good choice.
c. Distance is the varying quantity, so $d$ is a good choice.
## Why Do They Do That?
Just like in the English language, mathematics uses several words to describe one thing. For example, sum, addition, more than, and plus all mean to add numbers together. The following definition shows an example of this.
Definition: To evaluate means to complete the operations in the math sentence. Evaluate can also be called simplify or answer.
To begin to evaluate a mathematical expression, you must first substitute a number for the variable.
Definition: To substitute means to replace the variable in the sentence with a value.
Now try out your new vocabulary.
Example 3: EVALUATE $7 y-11$, when $y=4$.
Solution: Evaluate means to follow the directions, which is to take 7 times $y$ and subtract 11 . Because $y$ is the number 4,
$$
\begin{aligned}
& 7 \times 4-11 \\
& 28-11
\end{aligned}
$$
17
The solution is 17 . We have "substituted" the number 4 for $y$.
Multiplying 7 and 4
Subtracting 11 from 28
Because algebra uses variables to represent the unknown quantities, the multiplication symbol $\times$ is often confused with the variable $x$. To help avoid confusion, mathematicians replace the multiplication symbol with parentheses ( ), the multiplication dot $\cdot$, or by writing the expressions side by side.
Example 4: Rewrite $P=2 \times l+2 \times w$ with alternative multiplication symbols.
Solution: $P=2 \times l+2 \times w$ can be written as $P=2 \cdot l+2 \cdot w$
It can also be written as $P=2 l+2 w$.
The following is a real-life example that shows the importance of evaluating a mathematical variable.
Example 5: To prevent major accidents or injuries, horses must be fenced in a rectangular pasture. If the dimensions of the pasture are 300 feet by 225 feet, how much fencing should the ranch hand purchase to enclose the pasture?
Solution: Begin by drawing a diagram of the pasture and labeling what you know.
To find the amount of fencing needed, you must add all the sides together;
$$
L+L+W+W
$$
By substituting the dimensions of the pasture for the variables $L$ and $W$, the expression becomes
$$
300+300+225+225
$$
Now we must evaluate by adding the values together. The ranch hand must purchase 1,050 feet of fencing.
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:Variable Expressions (12:26)
In $1-5$, evaluate the expression.
1. $5 m+7$ when $m=3$.
2. $\frac{1}{3}(c)$ when $c=63$.
3. $\$ 8.15(h)$ when $h=40$.
4. $(k-11) \div 8$ when $k=43$.
5. Evaluate $(-2)^{2}+3(j)$ when $j=-3$.
In $6-13$, evaluate the expressions. Let $a=-3, b=2, c=5$, and $d=-4$.
6. $2 a+3 b$
7. $4 c+d$ 8. $5 a c-2 b$
8. $\frac{2 a}{c-d}$
9. $\frac{3 b}{d}$
10. $\frac{a-4 b}{3 c+2 d}$
11. $\frac{1}{a+b}$
12. $\frac{a b}{c d}$
In $14-21$, evaluate the expressions. Let $x=-1, y=2, z=-3$, and $w=4$.
14. $8 x^{3}$
15. $\frac{5 x^{2}}{6 z^{3}}$
16. $3 z^{2}-5 w^{2}$
17. $x^{2}-y^{2}$
18. $\frac{z^{3}+w^{3}}{z^{3}-w^{3}}$
19. $2 x^{2}-3 x^{2}+5 x-4$
20. $4 w^{3}+3 w^{2}-w+2$
21. $3+\frac{1}{z^{2}}$
In 22 - 26, choose an appropriate variable to describe each situation.
22. The number of hours you work in a week
23. The distance you travel
24. The height of an object over time
25. The area of a square
26. The number of steps you take in a minute In 27 - 31, evaluate the real-life problems.
27. The measurement around the widest part of these holiday bulbs is called their circumference. The formula for circumference is $2 \pi r$, where $\pi \approx 3.14$ and $r$ is the radius of the circle. Suppose the radius is 1.25 inches. Find the circumference.
29. The dimensions of a piece of notebook paper are 8.5 inches by 11 inches. Evaluate the writing area of the paper. The formula for area is length $\times$ width.
30. Sonya purchases 16 cans of soda at $\$ 0.99$ each. What is the amount Sonya spent on soda?
31. Mia works at a job earning $\$ 4.75$ per hour. How many hours should she work to earn $\$ 124.00$ ?
32. The area of a square is the side length squared. Evaluate the area of a square with side length 10.5 miles.
### Patterns and Expressions
In mathematics, especially in algebra, we look for patterns in the numbers that we see. Using mathematical verbs and variables studied in previous lessons, expressions can be written to describe a pattern.
Definition: An algebraic expression is a mathematical phrase combining numbers and/or variables using mathematical operations.
Consider a theme park charging admission of $\$ 28$ per person. In this case, the total amount of money collected can be described by the phrase, "Twenty-eight times the number of people who enter the park."
The English phrase above can be translated (to write in another language) into an algebraic expression.
Example 1: Write an expression to describe the amount of revenue of the theme park.
Solution: An appropriate variable to describe the number of people could be $p$. Rewriting the English phrase into a mathematical phrase, it becomes $28 \times p$.
Example 2: Write an algebraic expression for the following phrase.
The product of $c$ and 4.
Solution: The verb is product, meaning "to multiply." Therefore, the phrase is asking for the answer found by multiplying $c$ and 4 . The nouns are the number 4 and the variable $c$. The expression becomes $4 \times c, 4(c)$, or using shorthand, $4 c$.
Example 3: Write an algebraic expression for the following phrase.
3 times the sum of $c$ and 4 .
Solution: In this example, the phrase consists of " 3 times" followed by the phrase "the sum of c and 4 ". If we put the second phrase in parenthesis, we get $3 \times($ the sum of $\mathrm{c}$ and 4 ). This can be shortened to $3(c+4)$.
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:Patterns and Equations (13:18)
For exercises $1-15$, translate the English phrase into an algebraic expression. For the exercises without a stated variable, choose a letter to represent the unknown quantity.
1. Sixteen more than a number
2. The quotient of $h$ and 8
3. Forty-two less than $y$
4. The product of $k$ and three
5. The sum of $g$ and -7
6. $r$ minus 5.8
7. 6 more than 5 times a number
8. 6 divided by a number minus 12
9. A number divided by -11
10. 27 less than a number times four
11. The quotient of 9.6 and $m$
12. 2 less than 10 times a number
13. The quotient of $d$ and five times $s$
14. 35 less than $x$
15. The product of $6,-9$, and $u$
In exercises 16 - 24, write an English phrase for each algebraic expression
16. $J-9$
17. $\frac{n}{14}$
18. $17-a$
19. $3 l-16$
20. $\frac{1}{2}(h)(b)$
21. $\frac{b}{3}+\frac{z}{2}$
22. $4.7-2 f$
23. $5.8+k$
24. $2 l+2 w$
In exercises 25 - 28, define a variable to represent the unknown quantity and write an expression to describe the situation.
25. The unit cost represents the quotient of the total cost and number of items purchased. Write an expression to represent the unit cost of the following: The total cost is $\$ 14.50$ for $n$ objects.
26. The area of a square is the side length squared.
27. The total length of ribbon needed to make dance outfits is 15 times the number of outfits.
28. What is the remaining amount of chocolate squares if you started with 16 and have eaten some? 29. Describe a real-world situation that can be represented by $h+9$.
29. What is the difference between $\frac{7}{m}$ and $\frac{m}{7}$ ?
### Combining Like Terms
In the last lesson you learned how to write single-variable expressions and single variable equations. Now you are going to learn to work with single-variable expressions. The first thing that you are going to learn is how to simplify an expression.
## What does it mean to simplify?
To simplify means to make smaller or to make simpler. When we simplify in mathematics, we aren't solving anything, we are just making it smaller.
How do we simplify expressions?
Sometimes, you will be given an expression using variables where there is more than one term. A term can be either a number or or a variable or it can be a number multiplied by a variable. As an example, take a look at the following expression:
$4 x+3$
This expression consists of two terms. The first term is $4 x$. It is a number and a variable. The second term is the 3 . It does not have a variable in it, but it is still a term.
We haven't been given a value for $x$, so there isn't anything else we can do with this expression. It stays the same. If we have been given a value for $x$, then we could evaluate the expression. You have already worked on evaluating expressions.
When there is more than one LIKE TERM in an expression, we can simplify the expression.
What is a like term?
A like term means that the terms in question use the same variable, raised to the same power.
$4 x$ and $5 x$ are like terms. They both have $x$ as the variable. They are alike.
$6 x$ and $2 y$ are not like terms. One has an $x$ and one has a $y$. They are not alike.
When expressions have like terms, they can be simplified. We can simplify the sums and differences of expressions with like terms. Let's start with sums. Here is an example:
Example 1
$$
5 x+7 x
$$
First, we look to see if these terms are alike. Both of them have an $x$ so they are alike.
Next, we can simplify them by adding the numerical part of the terms together. The $x$ stays the same.
$$
\begin{gathered}
5 x+7 x \\
=12 x
\end{gathered}
$$
You can think of the $x$ as a label that lets you know that the terms are alike.
Let's look at another example.
Example 2
$$
7 x+2 x+5 y
$$
First, we look to see if the terms are alike. Two of the terms have $x$ 's and one has a $y$. The two with the $x$ 's are alike. The one with the $y$ is not alike. We can simplify the ones with the $x$ 's.
Next, we simplify the like terms.
$$
7 x+2 x=9 x
$$
We can't simplify the $5 y$ so it stays the same.
$$
9 x+5 y
$$
This is our answer.
We can also simplify expressions with differences and like terms. Let's look at an example.
Example 3
$$
9 y-2 y
$$
First, you can see that these terms are alike because they both have $y$ 's. We simplify the expression by subtracting the numerical part of the terms.
$9-2=7$
Our answer is $7 y$.
Sometimes you can combine like terms that have both sums and differences in the same example.
Example 4
$$
8 x-3 x+2 y+4 y
$$
We begin with the like terms.
$$
\begin{aligned}
& 8 x-3 x=5 x \\
& 2 y+4 y=6 y
\end{aligned}
$$
Next, we put it all together.
$$
5 x+6 y
$$
This is our answer.
Remember that you can only combine terms that are alike!!!
For further explanation, watch the video on simplifying expressions and combininglike terms:
## Practice Set
Simplify the expressions by combining like terms.
1. $7 z+2 z+4 z$
2. $25 y-13 y$
3. $7 x+2 x+4 a$
4. $45 y-15 y+13 y$
5. $-32 m+12 m$
6. $-6 x+7 x-12 x$
7. $14 a+18 b-5 a-8 b$
8. $-32 m+12 m$
9. $-11 t-12 t-7 t$
10. $15 w+7 h-15 w+21 h$
11. $-7 x+39 x$
12. $3 x^{2}+21 x+5 x+10 x^{2}$
13. $6 x y+7 y+5 x+9 x y$
14. $10 a b+9-2 a b$
15. $-7 m n-2 m n^{2}-2 m n+8$
### The Distributive Property
When we multiply an algebraic expression by a number or by another algebraic expression, we need to use the distributive property.
The Distributive Property: For any real numbers or expressions $\boldsymbol{A}, \boldsymbol{B}$ and $\mathbf{C}$ :
$A(B+C)=A B+A C$
$A(B-C)=A B-A C$
Example 1: Determine the value of 11(2+6) using both order of operations and the Distributive Property.
Solution: Using the order of operations: $11(2+6)=11(8)=88$
Using the Distributive Property: $11(2+6)=11(2)+11(6)=22+66=88$
Regardless of the method, the answer is the same.
Example 2: Simplify $7(3 x-5)$
Solution 1: Think of this expression as seven groups of $(3 x-5)$. You could write this expression seven times and add all the like terms. $(3 x-5)+(3 x-5)+(3 x-5)+(3 x-5)+(3 x-5)+(3 x-5)+(3 x-5)=21 x-35$
Solution 2: Apply the Distributive Property. $7(3 x-5)=7(3 x)+7(-5)=21 x-35$
Example 3: Simplify $\frac{2}{7}\left(3 y^{2}-11\right)$
Solution: Apply the Distributive Property.
$$
\begin{aligned}
& \frac{2}{7}\left(3 y^{2}+-11\right)=\frac{2}{7}\left(3 y^{2}\right)+\frac{2}{7}(-11) \\
& \frac{6 y^{2}}{7}-\frac{22}{7}
\end{aligned}
$$
## Identifying Expressions Involving the Distributive Property
The Distributive Property often appears in expressions, and many times it does not involve parentheses as grouping symbols. Previously, we saw how the fraction bar acts as a grouping symbol. The following example involves using the Distributive Property with fractions.
Example 4: Simplify $\frac{2 x+4}{8}$ Solution: Think of the denominator as: $\frac{2 x+4}{8}=\frac{1}{8}(2 x+4)$
Now apply the Distributive Property: $\frac{1}{8}(2 x)+\frac{1}{8}(4)=\frac{2 x}{8}+\frac{4}{8}$
Simplify: $\frac{x}{4}+\frac{1}{2}$
## Solve Real-World Problems Using the Distributive Property
The Distributive Property is one of the most common mathematical properties seen in everyday life. It crops up in business and in geometry. Anytime we have two or more groups of objects, the Distributive Property can help us solve for an unknown.
Example 5: An octagonal gazebo is to be built as shown below. Building code requires five-foot long steel supports to be added along the base and four-foot long steel supports to be added to the roof-line of the gazebo. What length of steel will be required to complete the project?
Solution: Each side will require two lengths, one of five and four feet respectively. There are eight sides, so here is our equation.
Steel required $=8(4+5)$ feet.
We can use the distributive property to find the total amount of steel:
Steel required $=8 \times 4+8 \times 5=32+40$ feet.
A total of 72 feet of steel is needed for this project.
## Practice Set: Distributive Property
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:Distributive Property (5:39)
Use the Distributive Property to simplify the following expressions.
1. $(x+4)-2(x+5)$
2. $\frac{1}{2}(4 z+6)$
3. $(4+5)-(5+2)$
4. $(x+2+7)$
5. $0.25(6 q+32)$
6. $y(x+7)$
7. $-4.2(h-11)$
8. $13 x(3 y+z)$
9. $\frac{1}{2}(x-y)-4$
10. $0.6(0.2 x+0.7)$
11. $(2-j)(-6)$
12. $(r+3)(-5)$
13. $6+(x-5)+7$
14. $6-(x-5)+7$
15. $4(m+7)-6(4-m)$
16. $-5(y-11)+2 y$
Use the Distributive Property to simplify the following fractions.
17. $\frac{8 x+12}{4}$
19. $\frac{11 x+12}{2}$
20. $\frac{3 y+2}{6}$
21. $-\frac{6 z-2}{3}$
22. $\frac{7-6 p}{3}$
In 23 - 25, write an expression for each phrase.
23. $\frac{2}{3}$ times the quantity of $n$ plus 16
24. Twice the quantity of $m$ minus 3
25. $-4 x$ times the quantity of $x$ plus 2
26. A bookshelf has five shelves, and each shelf contains seven poetry books and eleven novels. How many of each type of book does the bookcase contain?
27. Use the Distributive Property to show how to simplify 6(19.99) in your head.
28. A student rewrote $4(9 x+10)$ as $36 x+10$. Explain the student's error.
29. Use the Distributive Property to simplify 9(5998) in your head. 30. Amar is making giant holiday cookies for his friends at school. He makes each cookie with 6 oz of cookie dough and decorates them with macadamia nuts. If Amar has $5 \mathrm{lbs}$ of cookie dough $(1 \mathrm{lb}=16 \mathrm{oz})$ and 60 macadamia nuts, calculate the following.
(a) How many (full) cookies can he make?
(b) How many macadamia nuts can he put on each cookie, if each is supposed to be identical?
### Addition and Subtraction of Polynomials
Some algebraic expressions are called polynomials. A polynomial is made up of different terms that contain positive integer powers of the variables. Here is an example of a polynomial:
$$
4 x^{3}+2 x^{2}-3 x+1
$$
The example above is a polynomial with four terms.
The numbers appearing in each term in front of the variable are called the coefficients. The number appearing all by itself without a variable is called a constant term.
In this case the coefficient of $x^{3}$ is $\mathbf{4}$, the coefficient of $x^{2}$ is $\mathbf{2}$, the coefficient of $x$ is $\mathbf{- 3}$ and the constant term is $\mathbf{1}$.
## Degrees of Polynomials and Standard Form
Each term in the polynomial has a different degree. The degree of the term is the power of the variable in that term. A constant term is said to have a degree of 0 .
$$
\begin{array}{ll}
4 x^{3} & \text { has degree } 3 \text { or } 3^{\text {rd }} \text { order term. } \\
2 x^{2} & \text { has degree } 2 \text { or } 2^{\text {nd }} \text { order term. } \\
-3 x & \text { has degree } 1 \text { or } 1^{\text {st }} \text { order term. } \\
1 & \text { has degree } 0 \text { and is called the constant. }
\end{array}
$$
By definition, the degree of the polynomial is the same as the term with the highest degree. This example is a polynomial of degree 3 .
## Example 1
For the following polynomials, identify the coefficient of each term, the constant, the degree of each term and the degree of the polynomial.
a) $x^{5}-3 x^{3}+4 x^{2}-5 x+7$
## Solution
a) $x^{5}-3 x^{3}+4 x^{2}-5 x+7$
The coefficients of each term in order are $1,-3,4$, and -5 and the constant is 7 .
The degrees of each term are 5,3,2,1, and 0 . Therefore the degree of the polynomial is 5 .
Often, we arrange the terms in a polynomial in order of decreasing power. This is called standard form. The following polynomials are in standard form:
$$
\begin{aligned}
& 4 x^{4}-3 x^{3}+2 x^{2}-x+1 \\
& a^{4}-2 a^{3}+3 a^{2}-5 a+2
\end{aligned}
$$
The first term of a polynomial in standard form is called the leading term, and the coefficient of the leading term is called the leading coefficient.
The first polynomial above has the leading term $4 x^{4}$, and the leading coefficient is 4 .
The second polynomial above has the leading term $a^{4}$, and the leading coefficient is 1 .
## Example 2
Rearrange the terms in the following polynomials so that they are in standard form. Indicate the leading term and leading coefficient of each polynomial.
a) $7-3 x^{3}+4 x$
b) $a-a^{3}+2$
c) $-4 b+4+b^{2}$
## Solution
a) $7-3 x^{3}+4 x$ becomes $-3 x^{3}+4 x+7$. Leading term is $-3 x^{3}$; leading coefficient is -3 .
b) $a-a^{3}+2$ becomes $-a^{3}+a+2$. Leading term is $-a^{3}$; leading coefficient is -1 .
c) $-4 b+4+b^{2}$ becomes $b^{2}-4 b+4$. Leading term is $b^{2}$; leading coefficient is 1 .
## Simplifying Polynomials
A polynomial is simplified if it has no terms that are alike. Recall that Like terms are terms in the polynomial that have the same variable(s) with the same exponents, whether they have the same or different coefficients.
When a polynomial has like terms, we can simplify it by combining those terms.
$$
x^{2}+\underline{6 x}-\underline{4 x}+2
$$
Like terms
We can simplify this polynomial by combining the like terms $6 x$ and $-4 x$ into $(6-4) x$, or $2 x$. The new polynomial is $x^{2}+2 x+2$.
## Example 3
Simplify the following polynomial by collecting like terms and combining them.
$2 x-4 x^{2}+6+x^{2}-4+4 x$
## Solution
Rearrange the terms so that like terms are grouped together: $\left(-4 x^{2}+x^{2}\right)+(2 x+4 x)+(6-4)$
Combine each set of like terms: $-3 x^{2}+6 x+2$
## Simplifying Polynomials using the Distributive Property
We need to employ the Distributive Property to simplify expressions that involve multiplying a polynomial by a leading coefficient. This is also often required when we add or subtract polynomial expressions. In the last section we learned the distributive property:
Distributive Property: $A(B+C)=A B+A C$
When multiplying a polynomial by a leading coefficient, we multiply the coefficient by each term of the polynomial. For example, let's multiply the polynomial $2 x+6$ by 4 . First we write this expression as $4(2 x+6)$. Then we distribute by multiplying each term of the polynomial by 4 .
$4(5 \mathrm{x}+6)=4(5 \mathrm{x})+4(6)=20 \mathrm{x}+24$
## Example 4
Distributing a leading coefficient through a polynomial
a) Simplify $3(2 x+6)$
b) Simplify $-4(3 x-2)$
c) Simplify $2+4(6 x-4)$
d) Simplify $3-(2 x-3)$
## Solution
a) Distribute the 3 through the polynomial by multiplying it by each term
$3(2 x+6)=3(2 x)+3(6)=6 x+18$
b) Distribute the -4 through the polynomial by multiplying it by each term
$-4(2 x+6)=(-4)(2 x)+(-4)(6)=-8 x-24$
c) In this example, we need to make sure we distribute the 4 through the polynomial before we add the 2 . Then we combine like terms for our final answer.
$2+4(6 x-4)=2+4(6 x)+4(-4)=2+24 x-16=24 x-14$
d) In the last example, we are presented with a number minus a polynomial. In this case, we have to distribute the negative sign through the polynomial and then combine like terms. To do this, we convert the expression to an addition problem.
$$
\begin{aligned}
3-(2 x-4) & =3+(-1)(2 x-4) \\
\text { Distribute: } & =3+(-1)(2 x)-(-1)(4) \\
\text { Combine Like Terms: } & =3-2 x+4 \\
& =-2 x+7
\end{aligned}
$$
## Simplifying when Adding and Subtracting Polynomials
To add two or more polynomials, write their sum and then simplify by combining like terms.
## Example 6a
Add and simplify the resulting polynomial.
Add $3 x^{2}-4 x+7$ and $2 x^{3}-4 x^{2}-6 x+5$
Solution
$$
\begin{aligned}
& \left(3 x^{2}-4 x+7\right)+\left(2 x^{3}-4 x^{2}-6 x+5\right) \\
\text { Group like terms: } & =2 x^{3}+\left(3 x^{2}-4 x^{2}\right)+(-4 x-6 x)+(7+5) \\
\text { Simplify: } & =2 x^{3}-x^{2}-10 x+12
\end{aligned}
$$
To subtract one polynomial from another, distribute a -1 to each term of the polynomial you are subtracting.
## Example 7
a) Subtract $x^{3}-3 x^{2}+8 x+12$ from $4 x^{2}+5 x-9$
b) Subtract $5 b^{2}-2 a^{2}$ from $4 a^{2}-8 a b-9 b^{2}$
## Solution
a)
$$
\begin{aligned}
\left(4 x^{2}+5 x-9\right)-1\left(x^{3}-3 x^{2}+8 x+12\right) & =4 x^{2}+5 x-9-x^{3}+3 x^{2}-8 x-12 \\
\text { Group like terms: } & =-x^{3}+\left(4 x^{2}+3 x^{2}\right)+(5 x-8 x)+(-9-12) \\
\text { Simplify: } & =-x^{3}+7 x^{2}-3 x-21
\end{aligned}
$$
b)
$$
\begin{aligned}
\left(4 a^{2}-8 a b-9 b^{2}\right)-1\left(5 b^{2}-2 a^{2}\right) & =4 a^{2}-8 a b-9 b^{2}-5 b^{2}+2 a^{2} \\
\text { Group like terms: } & =\left(4 a^{2}+2 a^{2}\right)+\left(-9 b^{2}-5 b^{2}\right)-8 a b \\
\text { Simplify: } & =6 a^{2}-14 b^{2}-8 a b
\end{aligned}
$$
Note: An easy way to check your work after adding or subtracting polynomials is to substitute a convenient value in for the variable, and check that your answer and the problem both give the same value. For example, in part (b) above, if we let $a=2$ and $b=3$, then we can check as follows:
$$
\begin{aligned}
& \text { Given } \\
& \left(4 a^{2}-8 a b-9 b^{2}\right)-\left(5 b^{2}-2 a^{2}\right) \\
& \left(4(2)^{2}-8(2)(3)-9(3)^{2}\right)-\left(5(3)^{2}-2(2)^{2}\right) \\
& (4(4)-8(2)(3)-9(9))-(5(9)-2(4)) \\
& (-113)-37 \\
& -150
\end{aligned}
$$
Solution
$$
\begin{aligned}
& 6 a^{2}-14 b^{2}-8 a b \\
& 6(2)^{2}-14(3)^{2}-8(2)(3) \\
& 6(4)-14(9)-8(2)(3) \\
& 24-126-48 \\
& -150
\end{aligned}
$$
Since both expressions evaluate to the same number when we substitute in arbitrary values for the variables, we can be reasonably sure that our answer is correct.
Note: When you use this method, do not choose 0 or 1 for checking since these can lead to common problems.
## Problem Solving Using Addition or Subtraction of Polynomials
One application that uses polynomials is finding the area of a geometric figure.
## Example 7
Write a polynomial that represents the area of each figure shown.
a)
b)
c)
d)
## Solution
a) This shape is formed by two squares and two rectangles.
The blue square has area $y \times y=y^{2}$.
The yellow square has area $x \times x=x^{2}$.
The pink rectangles each have area $x \times y=x y$.
To find the total area of the figure we add all the separate areas:
$$
\begin{aligned}
\text { Total area } & =y^{2}+x^{2}+x y+x y \\
& =y^{2}+x^{2}+2 x y
\end{aligned}
$$
b) This shape is formed by two squares and one rectangle. The yellow squares each have area $a \times a=a^{2}$.
The orange rectangle has area $2 a \times b=2 a b$.
To find the total area of the figure we add all the separate areas:
$$
\begin{aligned}
\text { Total area } & =a^{2}+a^{2}+2 a b \\
& =2 a^{2}+2 a b
\end{aligned}
$$
c) To find the area of the green region we find the area of the big square and subtract the area of the little square.
The big square has area $: y \times y=y^{2}$.
The little square has area : $x \times x=x^{2}$.
$$
\text { Area of the green region }=y^{2}-x^{2}
$$
d) To find the area of the figure we can find the area of the big rectangle and add the areas of the pink squares.
The pink squares each have area $a \times a=a^{2}$.
The blue rectangle has area $3 a \times a=3 a^{2}$.
To find the total area of the figure we add all the separate areas:
$$
\text { Total area }=a^{2}+a^{2}+a^{2}+3 a^{2}=6 a^{2}
$$
Another way to find this area is to find the area of the big square and subtract the areas of the three yellow squares:
The big square has area $3 a \times 3 a=9 a^{2}$.
The yellow squares each have area $a \times a=a^{2}$.
To find the total area of the figure we subtract:
$$
\begin{aligned}
\text { Area } & =9 a^{2}-\left(a^{2}+a^{2}+a^{2}\right) \\
& =9 a^{2}-3 a^{2} \\
& =6 a^{2}
\end{aligned}
$$
## Further Practice
For more practice adding and subtracting polynomials, try playing the Battleship game at http://www.quia.com/ba/2 8820.html. (The problems get harder as you play; watch out for trick questions!)
## Practice Set
Express each polynomial in standard form. Give the degree of each polynomial.
1. $3-2 x$
2. $8-4 x+3 x^{3}$
3. $-5+2 x-5 x^{2}+8 x^{3}$
4. $x^{2}-9 x^{4}+12$
5. $5 x+2 x^{2}-3 x$ Add and simplify.
6. $(x+8)+(-3 x-5)$
7. $\left(-2 x^{2}+4 x-12\right)+\left(7 x+x^{2}\right)$
8. $\left(2 a^{2} b-2 a+9\right)+\left(5 a^{2} b-4 b+5\right)$
9. $\left(6.9 a^{2}-2.3 b^{2}+2 a b\right)+\left(3.1 a-2.5 b^{2}+b\right)$
10. $\left(\frac{3}{5} x^{2}-\frac{1}{4} x+4\right)+\left(\frac{1}{10} x^{2}+\frac{1}{2} x-2 \frac{1}{5}\right)$ Subtract and simplify.
11. $\left(-t+5 t^{2}\right)-\left(5 t^{2}+2 t-9\right)$
12. $\left(-y^{2}+4 y-5\right)-\left(5 y^{2}+2 y+7\right)$
13. $\left(-5 m^{2}-m\right)-\left(3 m^{2}+4 m-5\right)$
14. $\left(2 a^{2} b-3 a b^{2}+5 a^{2} b^{2}\right)-\left(2 a^{2} b^{2}+4 a^{2} b-5 b^{2}\right)$
15. $\left(3.5 x^{2} y-6 x y+4 x\right)-\left(1.2 x^{2} y-x y+2 y-3\right)$
16. Find the area of the following figures.
17.
18.
19.
## Chapter 3
Polynomials and Exponents, Part 1
## Chapter Outline
3.1 Exponential Properties involving Products
3.2 Multiplying Two Polynomials
3.3 Special Products of Polynomials
As we saw in the last section, polynomial expressions often involve variables with exponents. We learned how to add and subtract and simplify these expressions in the last chapter. This chapter focuses on multiplying expressions. We will learn the properties of exponents which will will be important to simplify our expressions, and we will learn how to multiply polynomials.
Definition: An exponent is a power of a number which shows how many times that number is multiplied by itself. An example would be $2^{3}$. You would multiply 2 by itself 3 times, $2 \times 2 \times 2$. The number 2 is the base and the number 3 is the exponent. The value $2^{3}$ is called the power.
Example 1: Write in exponential form: $a \cdot a \cdot a \cdot a$
Solution: You must count the number of times the base, $a$ is being multiplied by itself. It's being multiplied 4 times so the solution is $a^{4}$
Note: When you raise negative numbers to a power, you need to keep track of the negatives. Recall that
(negative number $) \times($ positive number $)=$ negative number
$($ negative number $) \times($ negative number $)=$ positive number For even powers of negative numbers, the answer will always be positive. Pairs can be made with each number and the negatives will be cancelled out.
$$
(-2)^{4}=(-2)(-2)(-2)(-2)=(-2)(-2) \cdot(-2)(-2)=+16
$$
For odd powers of negative numbers, the answer is always negative. Paris can be made but there will still be one negative number unpaired making the answer negative.
$$
(-2)^{5}=(-2)(-2)(-2)(-2)(-2)=(-2)(-2) \cdot(-2)(-2) \cdot(-2)=-32
$$
### Exponential Properties Involving Products
When we multiply the same numbers, each with different powers, it is easier to combine them before solving. This is why we use the Product of Powers Property.
Product of Powers Property: for all real numbers $x, x^{n} \cdot x^{m}=x^{n+m}$
Example 1: Multiply $x^{4} \cdot x^{5}$
Solution: We could write this in expanded form to see that:
$$
\left(x^{4}\right)\left(x^{5}\right)=\underbrace{(x \cdot x \cdot x \cdot x}_{x^{4}}) \cdot \underbrace{(x \cdot x \cdot x \cdot x \cdot x}_{x^{5}})=\underbrace{(x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x)}_{x^{9}}
$$
Or we could use the product of powers property to find the same result:
$x^{4} \cdot x^{5}=x^{4+5}=x^{9}$
Now let us consider the situation where we are raising an exponent to a power:
$$
\begin{aligned}
& \left(x^{4}\right)^{3}=x^{4} \cdot x^{4} \cdot x^{4} \quad 3 \text { factors of } x \text { to the power } 4 . \\
& \underbrace{(x \cdot x \cdot x \cdot x}_{x^{4}}) \cdot \underbrace{(x \cdot x \cdot x \cdot x)}_{x^{4}} \cdot \underbrace{(x \cdot x \cdot x \cdot x}_{x^{4}})=\underbrace{(x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x)}_{x^{12}}
\end{aligned}
$$
This situation is summarized below.
Power of a Product Property: for all real numbers $x$,
$$
\left(x^{n}\right)^{m}=x^{n \cdot m}
$$
The Power of a Product Property is similar to the Distributive Property. Everything inside the parentheses must be taken to the power outside. For example, $\left(x^{2} y\right)^{4}=\left(x^{2}\right)^{4} \cdot(y)^{4}=x^{8} y^{4}$. Watch how it works the long way.
$$
\underbrace{(x \cdot x \cdot y)}_{x^{2} y} \cdot \underbrace{(x \cdot x \cdot y)}_{x^{2} y} \cdot \underbrace{(x \cdot x \cdot y)}_{x^{2} y} \cdot \underbrace{(x \cdot x \cdot y)}_{x^{2} y}=\underbrace{(x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y)}_{x^{8} y 4}
$$
The Power of a Product Property does not work if you have a sum or difference inside the parenthesis. For example, $(x+y)^{2} \neq x^{2}+y^{2}$. Because it is an addition equation, it should look like $(x+y)(x+y)$.
Example 2: Simplify $\left(x^{2}\right)^{3}$
Solution: $\left(x^{2}\right)^{3}=x^{2 \cdot 3}=x^{6}$
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. Exponent Properties Involving Products (14:00)
## Consider $a^{5}$.
a. What is the base?
b. What is the exponent?
c. What is the power?
d. How can this power be written using repeated multiplication?
Determine whether the answer will be positive or negative. You do not have to provide the answer.
2. $-\left(3^{4}\right)$
3. $-8^{2}$
4. $10 \times(-4)^{3}$
5. What is the difference between $-5^{2}$ and $(-5)^{2}$ ? Enter both of these expressions, exactly as they appear, in your calculator. Why does the calculator give you different answers for each expression?
Write in exponential notation.
6. $2 \cdot 2$
7. $(-3)(-3)(-3)$
8. $y \cdot y \cdot y \cdot y \cdot y$
9. $(3 a)(3 a)(3 a)(3 a)$
10. $4 \cdot 4 \cdot 4 \cdot 4 \cdot 4$
11. $3 x \cdot 3 x \cdot 3 x$
12. $(-2 a)(-2 a)(-2 a)(-2 a)$
13. $6 \cdot 6 \cdot 6 \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y$
Find each number:
14. $1^{10}$
15. $0^{3}$
16. $7^{3}$
17. $-6^{2}$
18. $5^{4}$ 19. $3^{4} \cdot 3^{7}$
19. $2^{6} \cdot 2$
20. $\left(4^{2}\right)^{3}$
21. $(-2)^{6}$
22. $(0.1)^{5}$
23. $(-0.6)^{3}$
Multiply and simplify.
25. $6^{3} \cdot 6^{6}$
26. $2^{2} \cdot 2^{4} \cdot 2^{6}$
27. $3^{2} \cdot 4^{3}$
28. $x^{2} \cdot x^{4}$
29. $x^{2} \cdot x^{7}$
30. $\left(y^{3}\right)^{5}$
31. $\left(-2 y^{4}\right)(-3 y)$
32. $\left(4 a^{2}\right)(-3 a)\left(-5 a^{4}\right)$
Simplify.
33. $\left(a^{3}\right)^{4}$
34. $(x y)^{2}$
35. $\left(3 a^{2} b^{3}\right)^{4}$
36. $\left(-2 x y^{4} z^{2}\right)^{5}$
37. $\left(3 x^{2} y^{3}\right) \cdot\left(4 x y^{2}\right)$
38. $(4 x y z) \cdot\left(x^{2} y^{3}\right) \cdot\left(2 y z^{4}\right)$
39. $\left(2 a^{3} b^{3}\right)^{2}$
40. $(-8 x)^{3}(5 x)^{2}$
41. $\left(4 a^{2}\right)\left(-2 a^{3}\right)^{4}$
42. $(12 x y)(12 x y)^{2}$
43. $\left(2 x y^{2}\right)\left(-x^{2} y\right)^{2}\left(3 x^{2} y^{2}\right)$
### Multiplying Two Polynomials
Let's start by multiplying two binomials together. A binomial is a polynomial with two terms, so a product of two binomials will take the form $(a+b)(c+d)$.
We can still use the Distributive Property here if we do it cleverly. First, let's think of the first set of parentheses as one term. The Distributive Property says that we can multiply that term by c, multiply it by $\mathrm{d}$, and then add those two products together: $(a+b)(c+d)=(a+b) \cdot c+(a+b) \cdot d$.
We can rewrite this expression as $c(a+b)+d(a+b)$. Now let's look at each half separately. We can apply the distributive property again to each set of parentheses in turn, and that gives us $c(a+b)+d(a+b)=c a+c b+d a+d b$.
What you should notice is that when multiplying any two polynomials, every term in one polynomial is multiplied by every term in the other polynomial.
## Example 1
Multiply and simplify: $(2 x+1)(x+3)$
## Solution
We must multiply each term in the first polynomial by each term in the second polynomial. Let's try to be systematic to make sure that we get all the products.
First, multiply the first term in the first set of parentheses by all the terms in the second set of parentheses.
## FIGURE 3.1
Now we're done with the first term. Next we multiply the second term in the first parenthesis by all terms in the second parenthesis and add them to the previous terms.
FIGURE 3.2
$(2 x+1)(x+3)=(2 x)(x)+(2 x)(3)+(1)(x)+(1)(3)$
Now we're done with the multiplication and we can simplify:
$(2 x)(x)+(2 x)(3)+(1)(x)+(1)(3)=2 x^{2}+6 x+x+3=2 x^{2}+7 x+3$
Example 2
Multiply and simplify:
a) $(4 x-5)(x-20)$
b) $(3 x-2)(3 x+2)$
c) $\left(3 x^{2}+2 x-5\right)(2 x-3)$
Solution
a) This would be
$$
(4 x-5)(x-20)=(4 x)(x)+(4 x)(-20)+(-5)(x)+(-5)(-20)=4 x^{2}-80 x-5 x+100=4 x^{2}-85 x+100
$$
b)
$$
(3 x-2)(3 x+2)=(3 x)(3 x)+(3 x)(2)+(-2)(3 x)+(-2)(2)=9 x^{2}+6 x-6 x-4=9 x^{2}-4
$$
c)
$$
\left(3 x^{2}+2 x-5\right)(2 x-3)=\left(3 x^{2}\right)(2 x)+\left(3 x^{2}\right)(-3)+(2 x)(2 x)+(2 x)(-3)+(-5)(2 x)+(-5)(-3)=6 x^{3}-9 x^{2}+4 x^{2}-6 x-10 x+
$$
Solve Real-World Problems Using Multiplication of Polynomials
In this section, we'll see how multiplication of polynomials is applied to finding the areas and volumes of geometric shapes.
## Example 3
Find the areas of the following figures:
a)
b)
Find the volumes of the following figures:
c)
d)
## Solution
a) We use the formula for the area of a rectangle: Area $=$ length $\times$ width.
For the big rectangle:
$$
\begin{aligned}
\text { Length } & =b+3, \text { Width }=b+2 \\
\text { Area } & =(b+3)(b+2) \\
& =b^{2}+2 b+3 b+6 \\
& =b^{2}+5 b+6
\end{aligned}
$$
b) We could add up the areas of the blue and orange rectangles, but it's easier to just find the area of the whole big rectangle and subtract the area of the yellow rectangle.
$$
\begin{aligned}
\text { Areaofbigrectangle } & =20(12)=240 \\
\text { Area of yellow rectangle } & =(12-x)(20-2 x) \\
& =240-24 x-20 x+2 x^{2} \\
& =240-44 x+2 x^{2} \\
& =2 x^{2}-44 x+240
\end{aligned}
$$
The desired area is the difference between the two:
$$
\begin{aligned}
\text { Area } & =240-\left(2 x^{2}-44 x+240\right) \\
& =240+\left(-2 x^{2}+44 x-240\right) \\
& =240-2 x^{2}+44 x-240 \\
& =-2 x^{2}+44 x
\end{aligned}
$$
c) The volume of this shape $=($ area of the base $)($ height $)$.
$$
\begin{aligned}
\text { Areaofthebase } & =x(x+2) \\
& =x^{2}+2 x \\
\text { Height } & =2 x+1 \\
\text { Volume } & =\left(x^{2}+2 x\right)(2 x+1) \\
& =2 x^{3}+x^{2}+4 x^{2}+2 x \\
& =2 x^{3}+5 x^{2}+2 x
\end{aligned}
$$
d) The volume of this shape $=($ area of the base $)($ height $)$.
$$
\begin{aligned}
\text { Areaofthebase } & =(4 a-3)(2 a+1) \\
& =8 a^{2}+4 a-6 a-3 \\
& =8 a^{2}-2 a-3 \\
\text { Height } & =a+4 \\
\text { Volume } & =\left(8 a^{2}-2 a-3\right)(a+4)
\end{aligned}
$$
so the volume is $8 a^{3}+30 a^{2}-11 a-12$.
## Practice Set
Multiply the following monomials.
1. $(2 x)(-7 x)$
2. $(10 x)(3 x y)$
3. $(4 m n)\left(0.5 n m^{2}\right)$
4. $\left(-5 a^{2} b\right)\left(-12 a^{3} b^{3}\right)$
5. $\left(3 x y^{2} z^{2}\right)\left(15 x^{2} y z^{3}\right)$
Multiply and simplify.
6. $17(8 x-10)$
7. $2 x(4 x-5)$
8. $9 x^{3}\left(3 x^{2}-2 x+7\right)$
9. $3 x\left(2 y^{2}+y-5\right)$
10. $10 q\left(3 q^{2} r+5 r\right)$
11. $-3 a^{2} b\left(9 a^{2}-4 b^{2}\right)$
12. $(x-3)(x+2)$
13. $(a+b)(a-5)$
14. $(x+2)\left(x^{2}-3\right)$
15. $\left(a^{2}+2\right)\left(3 a^{2}-4\right)$
16. $(7 x-2)(9 x-5)$
17. $(2 x-1)\left(2 x^{2}-x+3\right)$
18. $(3 x+2)\left(9 x^{2}-6 x+4\right)$
19. $\left(a^{2}+2 a-3\right)\left(a^{2}-3 a+4\right)$ 20. $3(x-5)(2 x+7)$
20. $5 x(x+4)(2 x-3)$
Find the areas of the following figures.
Find the volumes of the following figures.
### Special Products of Polynomials
We saw that when we multiply two binomials we need to make sure to multiply each term in the first binomial with each term in the second binomial. Let's look at another example.
Multiply two linear binomials (binomials whose degree is 1):
$$
(2 x+3)(x+4)
$$
When we multiply, we obtain a quadratic polynomial (one with degree 2) with four terms:
$$
2 x^{2}+8 x+3 x+12
$$
The middle terms are like terms and we can combine them. We simplify and get $2 x^{2}+11 x+12$. This is a quadratic, or second-degree, trinomial (polynomial with three terms).
You can see that every time we multiply two linear binomials with one variable, we will obtain a quadratic polynomial. In this section we'll talk about some special products of binomials.
## Find the Square of a Binomial
One special binomial product is the square of a binomial. Consider the product $(x+4)(x+4)$.
Since we are multiplying the same expression by itself, that means we are squaring the expression. $(x+4)(x+4)$ is the same as $(x+4)^{2}$.
When we multiply it out, we get $x^{2}+4 x+4 x+16$, which simplifies to $x^{2}+8 x+16$.
Notice that the two middle terms - the ones we added together to get $8 x$-were the same. Is this a coincidence? In order to find that out, let's square a general linear binomial.
$$
\begin{aligned}
(a+b)^{2} & =(a+b)(a+b)=a^{2}+a b+a b+b^{2} \\
& =a^{2}+2 a b+b^{2}
\end{aligned}
$$
Sure enough, the middle terms are the same. How about if the expression we square is a difference instead of a sum?
$$
\begin{aligned}
(a-b)^{2} & =(a-b)(a-b)=a^{2}-a b-a b+b^{2} \\
& =a^{2}-2 a b+b^{2}
\end{aligned}
$$
It looks like the middle two terms are the same in general whenever we square a binomial. The general pattern is: to square a binomial, take the square of the first term, add or subtract twice the product of the terms, and add the square of the second term. You should remember these formulas:
$$
\begin{gathered}
(a+b)^{2}=a^{2}+2 a b+b^{2} \\
\text { and } \\
(a-b)^{2}=a^{2}-2 a b+b^{2}
\end{gathered}
$$
Remember! Raising a polynomial to a power means that we multiply the polynomial by itself however many times the exponent indicates. For instance, $(a+b)^{2}=(a+b)(a+b)$. Don't make the common mistake of thinking that $(a+b)^{2}=a^{2}+b^{2}$ ! To see why that's not true, try substituting numbers for $a$ and $b$ into the equation (for example, $a=4$ and $b=3$ ), and you will see that it is not a true statement. The middle term, $2 a b$, is needed to make the equation work.
We can apply the formulas for squaring binomials to any number of problems.
## Example 1
Square each binomial and simplify.
a) $(x+10)^{2}$
b) $(2 x-3)^{2}$
c) $\left(x^{2}+4\right)^{2}$
d) $(5 x-2 y)^{2}$
## Solution
Let's use the square of a binomial formula to multiply each expression.
a) $(x+10)^{2}=$
$$
(x+10)(x+10)=(x)(x)+(x)(10)+(10)(x)+(10)(10)=x^{2}+10 x+10 x+100
$$
which simplifies to $x^{2}+20 x+100$.
b) $(2 x-3)^{2}=$
$$
(2 x-3)(2 x-3)=(2 x)(2 x)+(2 x)(-3)+(-3)(2 x)+(-3)(-3)=4 x^{2}-6 x-6 x+9
$$
which simplifies to $4 x^{2}-12 x+9$.
c) $\left(x^{2}+4\right)^{2}$
we can multiply this out, or use the formula. If we let $a=x^{2}$ and $b=4$, then
$$
\begin{aligned}
\left(x^{2}+4\right)^{2} & =\left(x^{2}\right)^{2}+2\left(x^{2}\right)(4)+(4)^{2} \\
& =x^{4}+8 x^{2}+16
\end{aligned}
$$
d) $(5 x-2 y)^{2}$
If we let $a=5 x$ and $b=2 y$, then
$$
\begin{aligned}
(5 x-2 y)^{2} & =(5 x)^{2}-2(5 x)(2 y)+(2 y)^{2} \\
& =25 x^{2}-20 x y+4 y^{2}
\end{aligned}
$$
## Find the Product of Binomials Using Sum and Difference Patterns
Another special binomial product is the product of a sum and a difference of terms. For example, let's multiply the following binomials.
$$
\begin{aligned}
(x+4)(x-4) & =x^{2}-4 x+4 x-16 \\
& =x^{2}-16
\end{aligned}
$$
Notice that the middle terms are opposites of each other, so they cancel out when we collect like terms. This is not a coincidence. This always happens when we multiply a sum and difference of the same terms. In general,
$$
\begin{aligned}
(a+b)(a-b) & =a^{2}-a b+a b-b^{2} \\
& =a^{2}-b^{2}
\end{aligned}
$$
When multiplying a sum and difference of the same two terms, the middle terms cancel out. We get the square of the first term minus the square of the second term. You should remember this formula.
Sum and Difference Formula: $(a+b)(a-b)=a^{2}-b^{2}$
Let's apply this formula to a few examples.
## Example 2
Multiply the following binomials and simplify.
a) $(x+3)(x-3)$
b) $(5 x+9)(5 x-9)$
c) $\left(2 x^{3}+7\right)\left(2 x^{3}-7\right)$
d) $(4 x+5 y)(4 x-5 y)$
## Solution
a) Let $a=x$ and $b=3$, then:
$$
\begin{aligned}
(a+b)(a-b) & =a^{2}-b^{2} \\
(x+3)(x-3) & =x^{2}-3^{2} \\
& =x^{2}-9
\end{aligned}
$$
b) Let $a=5 x$ and $b=9$, then:
$$
\begin{aligned}
(a+b)(a-b) & =a^{2}-b^{2} \\
(5 x+9)(5 x-9) & =(5 x)^{2}-9^{2} \\
& =25 x^{2}-81
\end{aligned}
$$
c) Let $a=2 x^{3}$ and $b=7$, then:
$$
\begin{aligned}
\left(2 x^{3}+7\right)\left(2 x^{3}-7\right) & =\left(2 x^{3}\right)^{2}-(7)^{2} \\
& =4 x^{6}-49
\end{aligned}
$$
d) Let $a=4 x$ and $b=5 y$, then:
$$
\begin{aligned}
(4 x+5 y)(4 x-5 y) & =(4 x)^{2}-(5 y)^{2} \\
& =16 x^{2}-25 y^{2}
\end{aligned}
$$
For additional explanation, watch the video on special products of polynomials.
## Solve Real-World Problems Using Special Products of Polynomials
Now let's see how special products of polynomials apply to geometry problems and to mental arithmetic.
## Example 3
Find the area of the following square:
## Solution
The length of each side is $(a+b)$, so the area is $(a+b)(a+b)$.
Notice that this gives a visual explanation of the square of a binomial. The blue square has area $a^{2}$, the red square has area $b^{2}$, and each rectangle has area $a b$, so added all together, the area $(a+b)(a+b)$ is equal to $a^{2}+2 a b+b^{2}$. The next example shows how you can use the special products to do fast mental calculations.
## Example 4
Use the difference of squares and the binomial square formulas to find the products of the following numbers without using a calculator.
a) $43 \times 57$
b) $112 \times 88$
c) $45^{2}$
d) $481 \times 319$
## Solution
The key to these mental "tricks" is to rewrite each number as a sum or difference of numbers you know how to square easily.
a) Rewrite 43 as $(50-7)$ and 57 as $(50+7)$.
Then $43 \times 57=(50-7)(50+7)=(50)^{2}-(7)^{2}=2500-49=2451$
b) Rewrite 112 as $(100+12)$ and 88 as $(100-12)$.
Then $112 \times 88=(100+12)(100-12)=(100)^{2}-(12)^{2}=10000-144=9856$
c) $45^{2}=(40+5)^{2}=(40)^{2}+2(40)(5)+(5)^{2}=1600+400+25=2025$
d) Rewrite 481 as $(400+81)$ and 319 as $(400-81)$.
Then $481 \times 319=(400+81)(400-81)=(400)^{2}-(81)^{2}$
$(400)^{2}$ is easy - it equals 160000.
$(81)^{2}$ is not easy to do mentally, so let's rewrite 81 as $80+1$.
$(81)^{2}=(80+1)^{2}=(80)^{2}+2(80)(1)+(1)^{2}=6400+160+1=6561$
Then $481 \times 319=(400)^{2}-(81)^{2}=160000-6561=153439$
## Practice Set - Using Special Products of Polynomials
Use the special product rule for squaring binomials to multiply these expressions.
1. $(x+9)^{2}$
2. $(3 x-7)^{2}$
3. $(5 x-y)^{2}$
4. $\left(2 x^{3}-3\right)^{2}$
5. $\left(4 x^{2}+y^{2}\right)^{2}$
6. $(8 x-3)^{2}$
7. $(2 x+5)(5+2 x)$
8. $(x y-y)^{2}$
Use the special product of a sum and difference to multiply these expressions.
9. $(2 x-1)(2 x+1)$
10. $(x-12)(x+12)$
11. $(5 a-2 b)(5 a+2 b)$
12. $(a b-1)(a b+1)$
13. $\left(z^{2}+y\right)\left(z^{2}-y\right)$
14. $\left(2 q^{3}+r^{2}\right)\left(2 q^{3}-r^{2}\right)$
15. $(7 s-t)(t+7 s)$
16. $\left(x^{2} y+x y^{2}\right)\left(x^{2} y-x y^{2}\right)$
Find the area of the lower right square in the following figure. 17.
## CHAPTER 4 Polynomials and Exponents, Part 2
## Chapter Outline
### Exponential Properties Involving Quotients
4.2 Exponential Properties Involving Zero and Negative Exponents
4.3 Division OF POLYNOMIALS
In this chapter, we learn about the rules that apply to dividing terms with exponents, and we learn how to divide simple polynomials.
### Exponential Properties Involving Quo- tients
In this lesson you will learn how to simplify quotients of numbers and variables.
Quotient of Powers Property: When the exponent in the numerator is larger than the exponent in the denominator, for all real numbers $x, \frac{x^{n}}{x^{m}}=x^{n-m}$
When dividing expressions with the same base, keep the base and subtract the exponent in the denominator (bottom) from the exponent in the numerator (top). When we have problems with different bases, we apply the rule separately for each base. To simplify $\frac{x^{7}}{x^{4}}$, repeated multiplication can be used.
$$
\begin{aligned}
\frac{x^{7}}{x^{4}} & =\frac{\not x \cdot x \cdot \not x \cdot \not x \cdot x \cdot x \cdot x}{\not x \cdot \not x \cdot \not x \cdot x}=\frac{x \cdot x \cdot x}{1}=x^{3} \\
\frac{x^{5} y^{3}}{x^{3} y^{2}} & =\frac{\not x \cdot x \cdot \not x \cdot x \cdot x}{\not x \cdot \not x \cdot x x} \cdot \frac{y \cdot y \cdot y}{y \cdot y}=\frac{x \cdot x}{1} \cdot \frac{y}{1}=x^{2} y \text { OR } \frac{x^{5} y^{3}}{x^{3} y^{2}}=x^{5-3} \cdot y^{3-2}=x^{2} y
\end{aligned}
$$
Example 1: Simplify each of the following expressions using the quotient of powers property.
(a) $\frac{x^{10}}{x^{5}}$
(b) $\frac{x^{5} \gamma^{4}}{x^{3} \gamma^{2}}$
## Solution:
(a) $\frac{x^{10}}{x^{5}}=\chi^{10-5}=\chi^{5}$
(b) $\frac{x^{5} y^{4}}{x^{3} y^{2}}=x^{5-3} \cdot y^{4-2}=x^{2} y^{2}$
Power of a Quotient Property: $\left(\frac{x^{n}}{y^{m}}\right)^{p}=\frac{x^{n \cdot p}}{y^{m \cdot p}}$
The power inside the parenthesis for the numerator and the denominator multiplies with the power outside the parenthesis. The situation below shows why this property is true.
$$
\left(\frac{x^{3}}{y^{2}}\right)^{4}=\left(\frac{x^{3}}{y^{2}}\right) \cdot\left(\frac{x^{3}}{y^{2}}\right) \cdot\left(\frac{x^{3}}{y^{2}}\right) \cdot\left(\frac{x^{3}}{y^{2}}\right)=\frac{(x \cdot x \cdot x) \cdot(x \cdot x \cdot x) \cdot(x \cdot x \cdot x) \cdot(x \cdot x \cdot x)}{(y \cdot y) \cdot(y \cdot y) \cdot(y \cdot y) \cdot(y \cdot y)}=\frac{x^{12}}{y^{8}}
$$
Example 2: Simplify the following expression.
$$
\left(\frac{x^{10}}{y^{5}}\right)^{3}
$$
Solution: $\left(\frac{x^{10}}{y^{5}}\right)^{3}=\frac{x^{10 \cdot 3}}{y^{5 \cdot 3}}=\frac{x^{30}}{y^{15}}$
## Practice Set: Exponents Involving Quotients
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. Exponent Properties Involving Quotients (9:22)
Evaluate the following expressions.
1. $\frac{5^{6}}{5^{2}}$
2. $\frac{67}{6^{3}}$
3. $\frac{3^{10}}{3^{4}}$
4. $\left(\frac{2^{2}}{3^{3}}\right)^{3}$
Simplify the following expressions.
1. $\frac{a^{3}}{a^{2}}$
2. $\frac{x^{9}}{x^{5}}$
3. $\frac{x^{10}}{x^{5}}$
4. $\frac{a^{6}}{a}$
5. $\frac{a^{5} b^{4}}{a^{3} b^{2}}$
6. $\frac{4}{4}$
7. $\left(\frac{3^{4}}{5^{2}}\right)^{2}$
8. $\left(\frac{a^{3} b^{4}}{a^{2} b}\right)^{3}$
9. $\frac{x^{6} y^{5}}{x^{2} y^{3}}$
10. $\frac{6 x^{2} y^{3}}{2 x y^{2}}$
11. $\left(\frac{2 a^{3} b^{3}}{8 a^{7} b}\right)^{2}$
12. $\left(x^{2}\right)^{2} \cdot \frac{x^{6}}{x^{4}}$
13. $\left(\frac{16 a^{2}}{4 b^{5}}\right)^{3} \cdot \frac{b^{2}}{a^{16}}$
14. $\frac{6 a^{3}}{2 a^{2}}$
15. $\frac{15 x^{5}}{5 x}$
16. $\left(\frac{18 a^{10}}{15 a^{4}}\right)^{4}$
17. $\frac{25 y x^{6}}{20 y^{5} x^{2}}$ 18. $\left(\frac{x^{6} y^{2}}{x^{4} y^{4}}\right)^{3}$
18. $\left(\frac{6 a^{2}}{4 b^{4}}\right)^{2} \cdot \frac{5 b}{3 a}$
19. $\frac{(3 a b)^{2}\left(4 a^{3} b^{4}\right)^{3}}{\left(6 a^{2} b\right)^{4}}$
20. $\frac{\left(2 a^{2} b c^{2}\right)\left(6 a b c^{3}\right)}{4 a b^{2} c}$
### Exponential Properties Involving Zero and Negative Exponents
In the previous lessons we have dealt with powers that are positive whole numbers. In this lesson, you will learn how to simplify expressions when the exponent is zero, or negative.
Exponents of Zero: for all real numbers of $x, x \neq 0, x^{0}=1$
Example: $1=\frac{x^{4}}{x^{4}}=x^{4-4}=x^{0}$. This example is simplified using the Quotient of Powers Property. We can also see why we make the restriction that $x \neq 0$, since $\frac{0^{4}}{0^{4}}$ would require us to divide by zero which is never allowed. Thus, $0^{0}$ is not defined.
## Simplifying Expressions with Negative Exponents
The next objective is negative exponents. When we use the quotient rule and we subtract a greater number from a smaller number, the answer will become negative. The variable and the power will be moved to the denominator of a fraction. You will learn how to write this in an expression.
Example: $\frac{x^{4}}{x^{6}}=x^{4-6}=x^{-2}$. If we write the expression in expanded form, we have $\frac{\chi \cdot x \cdot x \cdot \chi}{\chi \cdot x \cdot x \cdot \chi \cdot x \cdot \chi}$. The four $x$ 's on top will cancel out with four $x$ 's on bottom. This will leave $2 x$ 's remaining on the bottom which makes your answer look like $\frac{1}{x^{2}}$. So $x^{-2}=\frac{1}{x^{2}}$. In general:
Negative Power rule for Exponents: $\frac{1}{x^{n}}=x^{-n}$ where $\chi \neq 0$
Example: $x^{-6} y^{-2}=\frac{1}{x^{6}} \cdot \frac{1}{y^{2}}=\frac{1}{x^{6} y^{2}}$. The negative power rule for exponents is applied to both variables separately in this example.
Multimedia Link: For more help with these types of exponents, watch this http://tinyurl.com/7n3ae2l.
Example 1: Write the following expressions without fractions.
(a) $\frac{2}{x^{2}}$
(b) $\frac{x^{2}}{y^{3}}$
Solution:
(a) $\frac{2}{x^{2}}=2 x^{-2}$
(b) $\frac{x^{2}}{y^{3}}=x^{2} y^{-3}$
Notice example $1 \mathrm{a}$, the number 2 is in the numerator. This number is multiplied to $x^{-2}$. It could also look like this, $2 \cdot \frac{1}{x^{2}}$ to be better understood.
It is important when evaluating expressions you remember the order of operations. Evaluate what is inside the parentheses or other grouping symbols, then evaluate the exponents, then perform multiplication/division from left to right, then perform addition/subtraction from left to right.
Example 2: Evaluate the following expression
(a) $3 \cdot 5^{2}-10 \cdot 5+1$
Solution: $3 \cdot 5^{2}-10 \cdot 5+1=3 \cdot 25-10 \cdot 5+1=75-50+1=26$
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. Zeroand Negative Exponents (14:04)
Simplify the following expressions. Be sure the final answer includes only positive exponents.
1. $x^{-1} \cdot y^{2}$
2. $x^{-4}$
3. $\frac{x^{-3}}{x^{-7}}$
4. $\frac{1}{x}$
5. $\frac{2}{x^{2}}$
6. $\frac{x^{2}}{y^{3}}$
7. $\frac{3}{x y}$
8. $3 x^{-3}$
9. $a^{2} b^{-3} c^{-1}$
10. $4 x^{-1} y^{3}$
11. $\frac{2 x^{-2}}{y^{-3}}$
12. $\frac{x^{-3} y^{-5}}{z^{-7}}$
13. $\left(\frac{a}{b}\right)^{-2}$
14. $\left(3 a^{-2} b^{2} c^{3}\right)^{3}$
15. $x^{-3} \cdot x^{3}$
Simplify the following expressions without any fractions in the answer.
16. $\frac{a^{-3}\left(a^{5}\right)}{a^{-6}}$
17. $\frac{5 x^{6} y^{2}}{x^{8} y}$
18. $\frac{\left(4 a b^{6}\right)^{3}}{(a b)^{5}}$
19. $\frac{4 a^{2} b^{3}}{2 a^{5} b}$
20. $\left(\frac{x}{3 y^{2}}\right)^{3} \cdot \frac{x^{2} y}{4}$
21. $\left(\frac{a b^{-2}}{b^{3}}\right)^{2}$
22. $\frac{x^{-3} y^{2}}{x^{2} y^{-2}}$
23. $\frac{\left(3 x^{3}\right)\left(4 x^{4}\right)}{(2 y)^{2}}$ 24. $\frac{a^{-2} b^{-3}}{c^{-1}}$
Evaluate the following expressions to a single number.
25. $3^{-2}$
26. $(6.2)^{0}$
27. $8^{-4} \cdot 8^{6}$
28. $5^{0}$
29. $7^{2}$
30. $\left(\frac{2}{3}\right)^{3}$
31. $3^{-3}$
Evaluate the following expressions for $x=2, y=-1, z=3$.
32. $2 x^{2}-3 y^{3}+4 z$
33. $\left(x^{2}-y^{2}\right)^{2}$
34. $\left(\frac{3 x^{2} y^{5}}{4 z}\right)^{-2}$
35. $x^{2} 4 x^{3} y^{4} 4 y^{2}$ if $x=2$ and $y=-1$
36. $a^{4}\left(b^{2}\right)^{3}+2 a b$ if $a=-2$ and $b=1$
37. $5 x^{2}-2 y^{3}+3 z$ if $x=3, y=2$, and $z=4$
38. $\left(\frac{a^{2}}{b^{3}}\right)^{-2}$ if $a=5$ and $b=3$
39. $3 \cdot 5^{5}-10 \cdot 5+1$
40. $\frac{2 \cdot 4^{2}-3 \cdot 5^{2}}{3^{2}}$
### Division of Polynomials
When we take the ratio of two polynomials, we call the result a rational expression. We can also think of this as division of polynomials.
Some examples of rational expressions are
$$
\frac{x^{2}-x}{x} \quad \frac{4 x^{2}-3 x+4}{2 x}
$$
Just as with rational numbers, the expression on the top is called the numerator and the expression on the bottom is called the denominator. In special cases we can simplify a rational expression by dividing the numerator by the denominator.
## Divide a Polynomial by a Monomial
We'll start by dividing a polynomial by a monomial. To do this, we divide each term of the polynomial by the monomial. When the numerator has more than one term, the monomial on the bottom of the fraction serves as the common denominator to all the terms in the numerator.
## Example 1
Divide.
a) $\frac{8 x^{2}-4 x+16}{2}$
b) $\frac{3 x^{2}+6 x-1}{x}$
c) $\frac{-3 x^{2}-18 x+6}{9 x}$
## Solution
a) $\frac{8 x^{2}-4 x+16}{2}=\frac{8 x^{2}}{2}-\frac{4 x}{2}+\frac{16}{2}=4 x^{2}-2 x+8$
b) $\frac{3 x^{3}+6 x-1}{x}=\frac{3 x^{3}}{x}+\frac{6 x}{x}-\frac{1}{x}=3 x^{2}+6-\frac{1}{x}$
c) $\frac{-3 x^{2}-18 x+6}{9 x}=-\frac{3 x^{2}}{9 x}-\frac{18 x}{9 x}+\frac{6}{9 x}=-\frac{x}{3}-2+\frac{2}{3 x}$
A common error is to cancel the denominator with just one term in the numerator.
Consider the quotient $\frac{3 x+4}{4}$.
Remember that the denominator of 4 is common to both the terms in the numerator. In other words we are dividing both of the terms in the numerator by the number 4.
The correct way to simplify is:
$$
\frac{3 x+4}{4}=\frac{3 x}{4}+\frac{4}{4}=\frac{3 x}{4}+1
$$
A common mistake is to cross out the number 4 from the numerator and the denominator, leaving just $3 x$. This is incorrect, because the entire numerator needs to be divided by 4 .
## Example 2
Divide $\frac{5 x^{3}-10 x^{2}+x-25}{-5 x^{2}}$.
Solution
$$
\frac{5 x^{3}-10 x^{2}+x-25}{-5 x^{2}}=\frac{5 x^{3}}{-5 x^{2}}-\frac{10 x^{2}}{-5 x^{2}}+\frac{x}{-5 x^{2}}-\frac{25}{-5 x^{2}}
$$
The negative sign in the denominator changes all the signs of the fractions:
$$
-\frac{5 x^{3}}{5 x^{2}}+\frac{10 x^{2}}{5 x^{2}}-\frac{x}{5 x^{2}}+\frac{25}{5 x^{2}}=-x+2-\frac{1}{5 x}+\frac{5}{x^{2}}
$$
Simplify the expression:
$$
\frac{18 x^{3}-3 x^{2}+6 x-4}{6 x}
$$
Watch the video explanation toseethis example workedout.
## Practice Set - Division of Polynomials
Divide the following polynomials:
1. $\frac{2 x+4}{2}$
2. $\frac{x-4}{x}$
3. $\frac{5 x-35}{5 x}$
4. $\frac{x^{2}+2 x-5}{x}$
5. $\frac{4 x^{2}+12 x-36}{-4 x}$
6. $\frac{2 x^{2}+10 x+7}{2 x^{2}}$
7. $\frac{x^{3}-x}{-2 x^{2}}$
8. $\frac{5 x^{4}-9}{3 x}$
9. $\frac{x^{3}-12 x^{2}+3 x-4}{12 x^{2}}$
10. $\frac{3-6 x+x^{3}}{-9 x^{3}}$
11. $\frac{x^{2}-6 x-12}{5 x^{4}}$
## Chapter Outline
5.1 ThE SOLUTION OF AN EQUATION
5.2 ONE-STEP EqUATIONS
5.3 Two-STEP EquATIONS
5.4 Multi-Step Equations
5.5 Equations With VARIABLES ON BOTH SideS
Mathematical equations are used in many different career fields. Medical researchers use equations to determine the length of time it takes for a drug to circulate throughout the body, botanists use equations to determine the amount of time it takes a Sequoia tree to reach a particular height, and environmental scientists can use equations to approximate the number of years it will take to repopulate the bison species.
In this chapter, you will learn how to manipulate linear equations to solve for the unknown quantity represented by the variable. You already have some experience solving equations. This chapter is designed to help formalize the mental math you use to answer questions in daily life.
### The Solution of an Equation
When an algebraic expression is set equal to another value, variable, or expression, a new mathematical sentence is created. This sentence is called an equation.
Definition: An algebraic equation is a mathematical sentence connecting an expression to a value, variable, or another expression with an equal sign (=).
Definition: The solution to an equation is the value (or multiple values) that make the equation true..
What is the value of $m$ in the following equation?
$$
\frac{1}{4} m=20.00
$$
Think: One-quarter can also be thought of as divide by four. What divided by 4 equals 20.00 ?
The solution is 80 . So, the money was $\$ 80.00$.
Checking an answer to an equation is almost as important as the equation itself. By substituting the value for the variable, you are making sure both sides of the equation balance.
Example 1: Check that $x=5$ is the solution to the equation $3 x+2=-2 x+27$.
Solution: To check that $x=5$ is the solution to the equation, substitute the value of 5 for the variable, $x$ :
$$
\begin{aligned}
3 x+2 & =-2 x+27 \\
3 \cdot x+2 & =-2 \cdot x+27 \\
3 \cdot 5+2 & =-2 \cdot 5+27 \\
15+2 & =-10+27 \\
17 & =17
\end{aligned}
$$
Because $17=17$ is a true statement, we can conclude that $x=5$ is a solution to $3 x+2=-2 x+27$.
Example 2: Is $z=3$ a solution to $z^{2}+2 z=8$ ?
Solution: Begin by substituting the value of 3 for $z$.
$$
\begin{aligned}
3^{2}+2(3) & =8 \\
9+6 & =8 \\
15 & =8
\end{aligned}
$$
Because $15=8$ is NOT a true statement, we can conclude that $z=3$ is not a solution to $z^{2}+2 z=8$.
## Practice Problems
Check that the given number is a solution to the corresponding equation.
1. $a=-3 ; 4 a+3=-9$ 2. $x=\frac{4}{3} ; \frac{3}{4} x+\frac{1}{2}=\frac{3}{2}$
2. $y=2 ; 2.5 y-10.0=-5.0$
3. $z=-5 ; 2(5-2 z)=20-2(z-1)$
### One-Step Equations
You have been solving equations since the beginning of this textbook although you may not have recognized it. For example, in a previous lesson, you determined the answer to the pizza problem below.
$\$ 20.00$ was one-quarter of the money spent on pizza.
$\frac{1}{4} m=20.00$ What divided by 4 equals $20.00 ?$
The solution is 80 . So, the amount of money before buying the pizza was $\$ 80.00$.
By working through this question mentally, you were applying mathematical rules and solving for the variable $m$.
Definition: To solve an equation means to write an equivalent equation that has the variable by itself on one side. This is also known as isolating the variable.
In order to begin solving equations, you must understand three basic concepts of algebra: inverse operations, equivalent equations, and the Addition Property of Equality.
## Inverse Operations and Equivalent Equations
In lesson 1, you learned how to simplify an expression using the order of operations: Parentheses, Exponents, Multiplication and Division completed in order from left to right, and Addition and Subtraction (also completed from left to right). Each of these operations has an inverse. Inverse operations "undo" each other when combined.
For example, the inverse of addition is subtraction. The inverse of an exponent is a root.
Example 1: Determine the inverse of division.
Solution: To undo dividing something, you would multiply.
By applying the same inverse operations to each side of an equation, you create an equivalent equation.
Definition: Equivalent equations are two or more equations having the same solution.
The Addition Property of Equality
Just like Spanish, Chemistry, or even music, mathematics has a set of rules you must follow in order to be successful. These are called properties, theorems, or axioms. These have been proven or agreed upon years ago so you can apply them to many different situations.
For example, The Addition Property of Equality allows you to apply the same operation to each side of the equation, or what you do to one side of an equation you can do to the other.
## The Addition Property of Equality
For all real numbers $a, b$, and $c$ :
If $a=b$, then $a+c=b+c$.
## Solving One-Step Equations Using Addition or Subtraction
Because subtraction can be considered adding a negative, the Addition Property of Equality also works if you need to subtract the same value from each side of an equation.
## Example 2
Solve for $y: 16=y-11$.
Solution: When asked to solve for $y$, your goal is to write an equivalent equation with the variable $y$ isolated on one side.
Write the original equation $16=y-11$.
Apply the Addition Property of Equality $16+11=y-11+11$
Simplify by adding like terms $27=y$.
The solution is $y=27$.
Example 3: One method to weigh a horse is to load it into an empty trailer with a known weight and reweigh the trailer. A Shetland pony is loaded onto a trailer that weighs 2,200 pounds empty and re-weighed. The new weight is 2,550 pounds. How much does the pony weigh?
Solution: Choose a variable to represent the weight of the pony, say $p$.
Write an equation $2550=2200+p$.
Apply the Addition Property of Equality $2550-2200=2200+p-2200$
Simplify $350=p$.
The Shetland pony weighs 350 pounds.
Equations that take one step to isolate the variable are called one-step equations. Such equations can also involve multiplication or division.
## The Multiplication Property of Equality
The Multiplication Property of Equality says that if you multiply one side of an equation by a number, you will maintain the equality if you also multiply the other side of that equation by the number.
For all real numbers $a, b$, and $c$ :
If $a=b$, then $a(c)=b(c)$
Since division is like multiplying by the reciprocal, the Multiplication Property of Equality also works if you divide each side of the equation by the same number.
## Solving One-Step Equations Using Multiplication or Division
## Example 4
Solve for $k:-8 k=-96$
Solution: Because $-8 k=-8 \times k$, the inverse operation of multiplication is division. Therefore, we must cancel multiplication by applying the Multiplication Property of Equality.
Write the original equation $-8 k=-96$.
Apply the Multiplication Property of Equality $-8 k \div-8=-96 \div-8$
The solution is $k=12$.
When working with fractions, you must remember: $\frac{a}{b} \times \frac{b}{a}=1$. In other words, in order to cancel a fraction using division, you really must multiply by its reciprocal.
Example 5: Solve $\frac{1}{8} \cdot x=1.5$.
The variable $x$ is being multiplied by one-eighth. Instead of dividing two fractions, we multiply by the reciprocal of $\frac{1}{8}$, which is 8 .
$$
\begin{aligned}
8\left(\frac{1}{8} \cdot x\right) & =8(1.5) \\
x & =12
\end{aligned}
$$
## Solving Real World Problems Using Equations
As was mentioned in the chapter opener, many careers base their work on manipulating linear equations. Consider the botanist studying bamboo as a renewable resource. She knows bamboo can grow up to 60 centimeters per day. If the specimen she measured was 1 meter tall, how long would it take to reach 5 meters in height? By writing and solving this equation, she will know exactly how long it should take for the bamboo to reach the desired height.
Example 6: In good weather, tomato seeds can grow into plants and bear ripe fruit in as little as 19 weeks. Lorna planted her seeds 11 weeks ago. How long must she wait before her tomatoes are ready to be picked?
Solution: The variable in question is the number of weeks until the tomatoes are ready. Call this variable $w$.
Write an equation $w+11=19$
Solve for $w$ by using the Addition Property of Equality.
$$
\begin{aligned}
w+11-11 & =19-11 \\
w & =8
\end{aligned}
$$
It will take as little as 8 weeks for the plant to bear ripe fruit.
Example 7: In 2004, Takeru Kobayashi, of Nagano, Japan, ate $53 \frac{1}{2}$ hot dogs in 12 minutes. He broke his previous world record, set in 2002, by three more hot dogs. Calculate how many minutes it took him to eat one hot dog.
## Solution:
Letting $m$ represent the number of minutes to eat one hot dog. Then, $53.5 m=12$
Applying the Multiplication Property of Equality,
$$
\begin{aligned}
\frac{53.5 m}{53.5} & =\frac{12}{53.5} \\
m & =0.224 \text { minutes }
\end{aligned}
$$
It took approximately 0.224 minutes or 13.5 seconds to eat one hot dog.
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:One-Step Equations (12:30)
Solve for the given variable.
1. $x+11=7$
2. $x-1.1=3.2$
3. $7 x=21$
4. $4 x=1$
5. $\frac{5 x}{12}=\frac{2}{3}$
6. $x+\frac{5}{2}=\frac{2}{3}$
7. $x-\frac{5}{6}=\frac{3}{8}$
8. $0.01 x=11$
9. $q-13=-13$
10. $z+1.1=3.0001$
11. $21 s=3$
12. $t+\frac{1}{2}=\frac{1}{3}$
13. $\frac{7 f}{11}=\frac{7}{11}$
14. $\frac{3}{4}=-\frac{1}{2} \cdot y$
15. $6 r=\frac{3}{8}$
16. $\frac{9 b}{16}=\frac{3}{8}$
17. Peter is collecting tokens on breakfast cereal packets in order to get a model boat. In eight weeks he has collected 10 tokens. He needs 25 tokens for the boat. Write an equation and determine the following information.
(a) How many more tokens he needs to collect, $n$.
(b) How many tokens he collects per week, $w$.
(c) How many more weeks remain until he can send off for his boat, $r$.
18. Juan has baked a cake and wants to sell it in his bakery. He is going to cut it into 12 slices and sell them individually. He wants to sell it for three times the cost of making it. The ingredients cost him $\$ 8.50$, and he allowed $\$ 1.25$ to cover the cost of electricity to bake it. Write equations that describe the following statements.
(a) The amount of money that he sells the cake for $(u)$.
(b) The amount of money he charges for each slice $(c)$. (c) The total profit he makes on the cake $(w)$.
19. Solve the remaining two questions regarding Takeru Kobayashi.
### Two-Step Equations
Suppose Shaun weighs 146 pounds and wants to lose enough weight to wrestle in the 130-pound class. His nutritionist designed a diet for Shaun so he will lose about 2 pounds per week. How many weeks will it take Shaun to weigh enough to wrestle in his class?
This is an example which can be solved by working backward. In fact, you may have already found the answer by using this method. The solution is 8 weeks.
By translating this situation into an algebraic sentence, we can begin the process of solving equations. To solve an equation means to undo all the operations of the sentence, leaving a value for the variable.
Translate Shauns situation into an equation:
$$
-2 w+146=130
$$
This sentence has two operations: addition and multiplication. To find the value of the variable, we must use both properties of Equality: The Addition Property of Equality and the Multiplication Property of Equality.
Procedure to Solve Equations of the Form $a x+b=$ some number:
1. Use the Addition Property of Equality to get the variable term ax alone on one side of the equation:
$$
\text { ax = some number }
$$
2. Use the Multiplication Property of Equality to get the variable $x$ alone on one side of the equation:
$$
x=\text { some number }
$$
Example 1: Solve Shauns problem.
Solution: $-2 w+146=130$
Apply the Addition Property of Equality: $-2 w+146-146=130-146$
Simplify: $-2 w=-16$
Apply the Multiplication Property of Equality: $-2 w \div-2=-16 \div-2$
The solution is $w=8$.
It will take 8 weeks for Shaun to weigh 130 pounds.
## Solving Equations by Combining Like Terms
Michigan has a $6 \%$ sales tax. Suppose you made a purchase and paid \\$95.12, including tax. How much was the purchase before tax?
Begin by determining the noun that is unknown and choose a letter as its representation.
The purchase price is unknown so this is our variable. Call it $p$. Now translate the sentence into an algebraic equation.
$$
\begin{aligned}
\text { price }+(0.06) \text { price } & =\text { total amount } \\
p+0.06 p & =95.12
\end{aligned}
$$
To solve this equation, you must know how to combine like terms.
Like terms are expressions that have identical variable parts.
According to this definition, you can only combine like terms if they are identical. Combining like terms only applies to addition and subtraction! This is not a true statement when referring to multiplication and division.
The numerical part of an algebraic term is called the coefficient. To combine like terms, you add (or subtract) the coefficients of the identical variable parts.
Combine the like terms: $p+0.06 p=1.06 p$, since $p=1 p$
Simplify: $1.06 p=95.12$
Apply the Multiplication Property of Equality: $1.06 p \div 1.06=95.12 \div 1.06$
Simplify: $p=89.74$
The price before tax was $\$ 89.74$.
The next several examples show how algebraic equations can be created to solve real world situations.
Example 1: An emergency plumber charges $\$ 65$ as a call-out fee plus an additional $\$ 75$ per hour. He arrives at a house at 9:30 and works to repair a water tank. If the total repair bill is $\$ 196.25$, at what time was the repair completed?
Solution: Translate the sentence into an equation. The number of hours it took to complete the job is unknown, so call it $h$.
Write the equation: $65+75(h)=196.25$
Apply the Addition Property and simplify:
$$
\begin{aligned}
65+75(h)-65 & =196.25-65 \\
75(h) & =131.25
\end{aligned}
$$
Simplify: $h=1.75$
The plumber worked for 1.75 hours, or 1 hour, 45 minutes. Since he started at 9:30, the repair was completed at 11:15.
Example 2: To determine the temperature in Fahrenheit, multiply the Celsius temperature by 1.8 then add 32. Determine the Celsius temperature if it is $89^{\circ} \mathrm{F}$. Solution: Translate the sentence into an equation. The temperature in Celsius is unknown; call it $C$.
Write the equation: $1.8 C+32=89$
Apply the Addition Property and simplify:
$$
\begin{aligned}
1.8 C+32-32 & =89-32 \\
1.8 C & =57
\end{aligned}
$$
Apply Multiplication Property of Equality: $1.8 C \div 1.8=57 \div 1.8$
Simplify: $C=31.67$
If the temperature is $89^{\circ} \mathrm{F}$, then it is $31.67^{\circ} \mathrm{C}$.
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:Two-Step Equations (13:50)
Solve and check your solution.
1. $1.3 x-0.7 x=12$
2. $6 x-1.3=3.2$
3. $5 x-(3 x+2)=1$
4. $4(x+3)=1$
5. $5 q-7=\frac{2}{3}$
6. $\frac{3}{5} x+\frac{5}{2}=\frac{2}{3}$
7. $s-\frac{3 s}{8}=\frac{5}{6}$
8. $0.1 y+11=0$
9. $\frac{5 q-7}{12}=\frac{2}{3}$
10. $\frac{5(q-7)}{12}=\frac{2}{3}$
11. $33 t-99=0$
12. $5 p-2=32$
13. $14 x+9 x=161$ 14. $3 m-1+4 m=5$
14. $8 x+3=11$
15. $24=2 x+6$
16. $66=\frac{2}{3} k$
17. $\frac{5}{8}=\frac{1}{2}(a+2)$
18. $16=-3 d-5$
19. Jayden purchased a new pair of shoes. Including a $7 \%$ sales tax, he paid $\$ 84.68$. How much did his shoes cost before sales tax?
20. A mechanic charges $\$ 98$ for parts and $\$ 60$ per hour for labor. Your bill totals $\$ 498.00$, including parts and labor. How many hours did the mechanic work?
21. An electric guitar and amp set costs $\$ 1195.00$. You are going to pay $\$ 250$ as a down payment and pay the rest in 5 equal installments. How much should you pay each month?
22. Jade is stranded downtown with only $\$ 10$ to get home. Taxis cost $\$ 0.75$ per mile, but there is an additional $\$ 2.35$ hire charge. Write a formula and use it to calculate how many miles she can travel with her money. Determine how many miles she can ride.
23. Jasmins Dad is planning a surprise birthday party for her. He will hire a bouncy castle, and will provide party food for all the guests. The bouncy castle costs $\$ 150$ dollars for the afternoon, and the food will cost $\$ 3.00$ per person. Andrew, Jasmins Dad, has a budget of $\$ 300$. Write an equation to help him determine the maximum number of guests he can invite.
### Multi-Step Equations
So far in this chapter you have learned how to solve one-step equations of the form $y=a x$ and two-step equations of the form $y=a x+b$. This lesson will expand upon solving equations to include solving multi-step equations and equations involving the Distributive Property.
## Solving Multi-Step Equations by Combining Like Terms
In the last lesson, you learned the definition of like terms and how to combine such terms. We will use the following situation to further demonstrate solving equations involving like terms.
You are hosting a Halloween party. You will need to provide 3 cans of soda per person, 4 slices of pizza per person, and 37 party favors. You have a total of 79 items. How many people are coming to your party?
This situation has several pieces of information: soda cans, slices of pizza, and party favors. Translate this into an algebraic equation.
$$
3 p+4 p+37=79
$$
This equation requires three steps to solve. In general, to solve any equation you should follow this procedure.
## Procedure to Solve Equations:
1. Remove any parentheses by using the Distributive Property or the Multiplication Property of Equality:
2. Simplify each side of the equation by combining like terms.
3. Isolate the $a x$ term. Use the Addition Property of Equality to get the variable on one side of the equal sign and the numerical values on the other.
4. Isolate the variable. Use the Multiplication Property of Equality to get the variable alone on one side of the equation.
## Check your solution.
Example 1: Determine the number of party-goers in the opening example.
Solution: $3 p+4 p+37=79$
Combine like terms: $7 p+37=79$
Apply the Addition Property of Equality: $7 p+37-37=79-37$
Simplify: $7 p=42$
Multiplication Property of Equality: $7 p \div 7=42 \div 7$
The solution is $p=6$.
There are six people coming to the party.
## Solving Multi-Step Equations by Using the Distributive Property
When faced with an equation such as $2(5 x+9)=78$, the first step is to remove the parentheses. There are two options to remove the parentheses. You can apply the Distributive Property or you can apply the Multiplication Property of Equality. This lesson will show you how to use the Distributive Property to solve multi-step equations. Example 2: Solve for $x$ : $2(5 x+9)=78$
Solution: Apply the Distributive Property: $10 x+18=78$
Apply the Addition Property of Equality: $10 x+18-18=78-18$
Simplify: $10 x=60$
Multiplication Property of Equality: $10 x \div 10=60 \div 10$
The solution is $x=6$.
Check: Does $10(6)+18=78$ ? Yes, so the answer is correct.
Example 3: Kashmir needs to fence in his puppy. He will fence in three sides, connecting it to his back porch. He wants the run to be 12 feet long and he has 40 feet of fencing. How wide can Kashmir make his puppy enclosure?
Solution: Translate the sentence into an algebraic equation. Let $w$ represent the width of the enclosure.
$$
w+w+12=40
$$
Solve for $w$ :
$$
\begin{aligned}
2 w+12 & =40 \\
2 w+12-12 & =40-12 \\
2 w & =28 \\
2 w \div 2 & =28 \div 2 \\
w & =14
\end{aligned}
$$
The dimensions of the enclosure are 14 feet wide by 12 feet long.
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:Multi-Step Equations (15:01)
Solve for the given variable. 1. $3(x-1)-2(x+3)=0$
2. $7(w+20)-w=5$
3. $9(x-2)=3 x+3$
4. $2\left(5 a-\frac{1}{3}\right)=\frac{2}{7}$
5. $\frac{2}{9}\left(i+\frac{2}{3}\right)=\frac{2}{5}$
6. $4\left(v+\frac{1}{4}\right)=\frac{35}{2}$
7. $22=2(p+2)$
8. $-(m+4)=-5$
9. $48=4(n+4)$
10. $\frac{6}{5}\left(v-\frac{3}{5}\right)=\frac{6}{25}$
11. $-10(b-3)=-100$
12. $6 v+6(4 v+1)=-6$
13. $-46=-4(3 s+4)-6$
14. $8(1+7 m)+6=14$
15. $0=-7(6+3 k)$
16. $35=-7(2-x)$
17. $-3(3 a+1)-7 a=-35$
18. $-2\left(n+\frac{7}{3}\right)=-\frac{14}{3}$
19. $-\frac{59}{60}=\frac{1}{6}\left(-\frac{4}{3} r-5\right)$
20. $\frac{4 y+3}{7}=9$
21. $(c+3)-2 c-(1-3 c)=2$
22. $5 m-3[7-(1-2 m)]=0$
23. $f-1+2 f+f-3=-4$
24. Find four consecutive even integers whose sum is 244.
25. Four more than two-thirds of a number is 22 . What is the number?
26. The total cost of lunch is $\$ 3.50$, consisting of a juice, a sandwich, and a pear. The juice cost 1.5 times as much as the pear. The sandwich costs $\$ 1.40$ more than the pear. What is the price of the pear?
27. Camden High has five times as many desktop computers as laptops. The school has 65 desktop computers. How many laptops does it have?
28. A realtor receives a commission of $\$ 7.00$ for every $\$ 100$ of a homes selling price. How much was the selling price of a home if the realtor earned $\$ 5,389.12$ in commission?
### Equations with Variables on Both Sides
As you may now notice, equations come in all sizes and styles. There are single-step, double-step, and multi-step equations. In this lesson, you will learn how to solve equations with a variable appearing on each side of the equation. The process you need to solve this type of equation is similar to solving a multi-step equation. The procedure is repeated here.
## Procedure to Solve Equations:
1. Remove any parentheses by using the Distributive Property or the Multiplication Property of Equality:
2. Simplify each side of the equation by combining like terms.
3. Isolate the ax term. Use the Addition Property of Equality to get the variable on one side of the equal sign and the numerical values on the other.
4. Isolate the variable. Use the Multiplication Property of Equality to get the variable alone on one side of the equation.
5. Check your solution.
Karen and Sarah have bank accounts. Karen has a starting balance of $\$ 125.00$ and is depositing \\$20 each week. Sarah has a starting balance of $\$ 43$ and is depositing $\$ 37$ each week. When will the girls have the same amount of money?
To solve this problem, you could use the guess and check method. You are looking for a particular week in which the bank accounts are equal. This could take a long time! You could also translate the sentence into an equation. The number of weeks is unknown so this is our variable, call it $w$. Now translate this situation into an algebraic equation:
$$
125+20 w=43+37 w
$$
This is a situation in which the variable $w$ appears on both sides of the equation. To begin to solve for the unknown, we must use the Addition Property of Equality to gather the variables on one side of the equation.
Example 1: Determine when Sarah and Karen will have the same amount of money.
Solution: Using the Addition Property of Equality, move the variables to one side of the equation:
$$
125+20 w-20 w=43+37 w-20 w
$$
Simplify: $125=43+17 w$
Solve:
$$
\begin{aligned}
125-43 & =43-43+17 w \\
82 & =17 w \\
82 \div 17 & =17 w \div 17 \\
w & \approx 4.82
\end{aligned}
$$
It will take about 4.8 weeks for Sarah and Karen to have equal amounts of money.
Example 2: Solve for $h: 3(h+1)=11 h-23$
Solution: First you must remove the parentheses by using the Distributive Property:
$$
3 h+3=11 h-23
$$
Gather the variables on one side:
$$
3 h-3 h+3=11 h-3 h-23
$$
Simplify:
$$
3=8 h-23
$$
Solve using the steps from lesson 3.3:
$$
\begin{aligned}
3+23 & =8 h-23+23 \\
26 & =8 h \\
26 \div 8 & =8 h \div 8 \\
h & =\frac{13}{4}=3.25
\end{aligned}
$$
Multimedia Link: Watch this video - http://www.teachertube.com/viewVideo.php?video_id=55491\\&title=Solvin g_equations_with_variables_on_both_sides for further information on how to solve an equation with a variable on each side of the equation.
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. Equations with Variableson BothSides (9:28)
Solve for the given variable.
1. $3(x-1)=2(x+3)$
2. $7(x+20)=x+5$
3. $9(x-2)=3 x+3$
4. $2\left(a-\frac{1}{3}\right)=\frac{2}{5}\left(a+\frac{2}{3}\right)$
5. $\frac{2}{7}\left(t+\frac{2}{3}\right)=\frac{1}{5}\left(t-\frac{2}{3}\right)$
6. $\frac{1}{7}\left(v+\frac{1}{4}\right)=2\left(\frac{3 v}{2}-\frac{5}{2}\right)$
7. $\frac{y-4}{11}=\frac{2}{5} \cdot \frac{2 y+1}{3}$
8. $\frac{z}{16}=\frac{2(3 z+1)}{9}$
9. $\frac{q}{16}+\frac{q}{6}=\frac{(3 q+1)}{9}+\frac{3}{2}$
10. $21+3 b=6-6(1-4 b)$
11. $-2 x+8=8(1-4 x)$
12. $3(-5 v-4)=-6 v-39$
13. $-5(5 k+7)=25+5 k$
14. Manoj and Tamar are arguing about how a number trick they heard goes. Tamar tells Andrew to think of a number, multiply it by five and subtract three from the result. Then Manoj tells Andrew to think of a number add five and multiply the result by three. Andrew says that whichever way he does the trick he gets the same answer. What was Andrew's number?
15. I have enough money to buy five regular priced CDs and have $\$ 6$ left over. However all CDs are on sale today, for $\$ 4$ less than usual. If I borrow \\$2, I can afford nine of them. How much are CDs on sale for today?
16. Jaime has a bank account with a balance of $\$ 412$ and is saving $\$ 18$ each week. George has a bank account with a balance of $\$ 874$ and is spending $\$ 44$ dollars each week. When will the two have the same amount of money?
17. Cell phone plan A charges $\$ 75.00$ each month and $\$ 0.05$ per text. Cell phone plan B charges $\$ 109$ dollars and $\$ 0.00$ per text.
(a) At how many texts will the two plans charge the same?
(b) Suppose you plan to text 3,000 times per month. Which plan should you choose? Why?
18. To rent a dunk tank Modern Rental charges $\$ 150$ per day. To rent the same tank, Budgetwise charges $\$ 7.75$ per hour.
(a) When will the two companies charge the same?
(b) You will need the tank for a 24-hour fund raise-a-thon. Which company should you choose?
## CHAPTER
Linear Equations: Real World Applications
## Chapter Outline
### WRITING EQUATIONS
6.2 Ratios and Proportions
6.3 Scale AND INDIRECT MEASUREMENT
6.4 Percent Problems
Algebra can be used to solve for the unknown quantity represented by a variable, but to solve real world applications, you need to be able to translate the situation into a mathematical equation. We have had some practice writing expressions from situations in Chapter 2. Now we will practice writing equations.
### Writing Equations
Suppose there is a concession stand selling burgers and French fries. Each burger costs $\$ 2.50$ and each order of French fries costs $\$ 1.75$. You and your family will spend exactly $\$ 25.00$ on food. How many burgers can be purchased? How many orders of fries? How many of each type can be purchased if your family plans to buy a combination of burgers and fries?
The underlined word exactly lends a clue to the type of mathematical sentence you will need to write to model this situation.
These words can be used to symbolize the equal sign:
Exactly, equivalent, the same as, identical, is
The word exactly is synonymous with equal, so this word is directing us to write an equation. Using the methods previously learned, read every word in the sentence and translate each into mathematical symbols.
Example 1: Your family is planning to only purchase burgers. How many can be purchased with $\$ 25.00$ ?
## Solution:
Step 1: Choose a variable to represent the unknown quantity, say $b$ for burgers
Step 2: Write an equation to represent the situation: $2.50 b=25.00$
Step 3: Think. What number multiplied by 2.50 equals 25.00
The solution is 10 , so your family can purchase exactly ten burgers.
Example 2: Translate the following into equations:
a) 9 less than twice a number is 33 .
b) Five more than four times a number is 21 .
c) $\$ 20.00$ was one-quarter of the money spent on pizza.
## Solutions:
a) Let a number be $n$. So, twice a number is $2 n$.
Nine less than that is $2 n-9$.
The word is means the equal sign, so $2 n-9=33$ b) Let a number be $x$. So five more than four times a number is 21 can be written as: $4 x+5=21$
c) Let of the money be $m$. The equation could be written as $\frac{1}{4} m=20.00$
## Practice Set
Define the variables and translate the following statements into algebraic equations.
1. Peters Lawn Mowing Service charges $\$ 10$ per job and $\$ 0.20$ per square yard. Peter earns $\$ 25$ for a job.
2. Renting the ice-skating rink for a birthday party costs $\$ 200$ plus $\$ 4$ per person. The rental costs $\$ 324$ in total.
3. Renting a car costs $\$ 55$ per day plus $\$ 0.45$ per mile. The cost of the rental is $\$ 100$.
4. Nadia gave Peter 4 more blocks than he already had. He already had 7 blocks.
5. An amount of money is invested at $5 \%$ annual interest. The interest earned at the end of the year is equal to $\$ 250$.
6. You buy hamburgers at a fast food restaurant. A hamburger costs $\$ 0.49$. You have at $\$ 3$ to spend. Write an equation for the number of hamburgers you can buy.
### Ratios and Proportions
There are many situations that can be represented with Algebraic equations. We will focus now on situations that involve proportions.
Ratios and proportions have a fundamental place in mathematics. They are used in geometry, size changes, and trigonometry. This lesson expands upon the idea of fractions to include ratios and proportions.
A ratio is a fraction comparing two things.
A rate is a fraction comparing two things with different units.
You have experienced rates many times: 65 mi/hour, \\$1.99/pound, \\$3.79/yd ${ }^{2}$. You have also experienced ratios. A student to teacher ratio shows approximately how many students one teacher is responsible for in a school.
Example 1: The State Dining Room in the White House measures approximately 48 feet long by 36 feet wide. Compare the length of the room to the width, and express your answer as a ratio.
Solution:
$$
\frac{48 \text { feet }}{36 \text { feet }}=\frac{4}{3}
$$
The length of the State Dining Room is $\frac{4}{3}$ the width.
A proportion is a statement that two fractions are equal: $\frac{a}{b}=\frac{c}{d}$.
Example 2: Is $\frac{2}{3}=\frac{6}{12}$ a proportion?
Solution: Find the least common multiple of 3 and 12 to create a common denominator.
$$
\frac{2}{3}=\frac{8}{12} \neq \frac{6}{12}
$$
This is NOT a proportion because these two fractions are not equal.
A ratio can also be written using a colon instead of the fraction bar.
$\frac{a}{b}=\frac{c}{d}$ can also be read, $a$ is to $b$ as $c$ is to $d$ or $a: b=c: d$
You can use the Multiplication Property of Equality to find that:
If $\frac{a}{b}=\frac{c}{d}$, then $a d=b c$. This is true because we can multiply both sides of the equation by $b$ which gives us $a=\frac{b c}{d}$, then we can multiply both sides of the equation by $d$, which gives us $a d=b c$.
$a d$ is the product of the numerator of the first fraction and the denominator of the second fraction, and $b c$ is the product of the denominator of the first fraction and the numerator of the second fraction. The are called crossproducts.
Example 3: Solve $\frac{a}{9}=\frac{7}{6}$
Solution: Use the cross-products to solve.
$$
\begin{aligned}
& 6 a=7(9) \\
& 6 a=63
\end{aligned}
$$
Solve for $a$ :
$$
a=10.5
$$
Consider the following situation: A train travels at a steady speed. It covers 15 miles in 20 minutes. How far will it travel in 7 hours, assuming it continues at the same rate? This is an example of a problem that can be solved using several methods, including proportions.
To solve using a proportion, you need to translate the statement into an algebraic sentence. The key to writing correct proportions is to keep the units the same in each fraction.
$$
\frac{\text { miles }}{\text { time }}=\frac{\text { miles }}{\text { time }} \quad \frac{\text { miles }}{\text { time }} \neq \frac{\text { time }}{\text { miles }}
$$
You will be asked to solve this in the practice set.
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:Ratio and Proportion (10:25)
Write the following comparisons as ratios. Simplify fractions where possible.
1. $\$ 150$ to $\$ 3$
2. 150 boys to 175 girls
3. 200 minutes to 1 hour
4. 10 days to 2 weeks
Write the following ratios as a unit rate.
5. 54 hotdogs to 12 minutes
6. $5000 \mathrm{lbs}$ to $250 \mathrm{in}^{2}$
7. 20 computers to 80 students
8. 180 students to 6 teachers
9. 12 meters to 4 floors
10. 18 minutes to 15 appointments
11. Give an example of a proportion that uses the numbers $5,1,6$, and 30
12. In the following proportion, identify the means and the extremes: $\frac{5}{12}=\frac{35}{84}$
Solve the following proportions.
13. $\frac{13}{6}=\frac{5}{x}$
14. $\frac{1.25}{7}=\frac{3.6}{x}$
15. $\frac{6}{19}=\frac{x}{11}$
16. $\frac{1}{x}=\frac{0.01}{5}$
17. $\frac{300}{4}=\frac{x}{99}$
18. $\frac{2.75}{9}=\frac{x}{\left(\frac{2}{9}\right)}$
19. $\frac{1.3}{4}=\frac{x}{1.3}$
20. $\frac{0.1}{1.01}=\frac{1.9}{x}$
21. $\frac{5 p}{12}=\frac{3}{11}$
22. $-\frac{9}{x}=\frac{4}{11}$
23. $\frac{n+1}{11}=-2$
24. A restaurant serves 100 people per day and takes in $\$ 908$. If the restaurant were to serve 250 people per day, what might the cash collected be?
25. The highest mountain in Canada is Mount Yukon. It is $\frac{298}{67}$ the size of Ben Nevis, the highest peak in Scotland. Mount Elbert in Colorado is the highest peak in the Rocky Mountains. Mount Elbert is $\frac{220}{67}$ the height of Ben Nevis and $\frac{44}{48}$ the size of Mont Blanc in France. Mont Blanc is 4800 meters high. How high is Mount Yukon?
26. At a large high school it is estimated that two out of every three students have a cell phone, and one in five of all students have a cell phone that is one year old or less. Out of the students who own a cell phone, what proportion own a phone that is more than one year old?
27. The price of a Harry Potter Book on Amazon.com is $\$ 10.00$. The same book is also available used for $\$ 6.50$. Find two ways to compare these prices.
28. To prepare for school, you purchased 10 notebooks for $\$ 8.79$. How many notebooks can you buy for $\$ 5.80$ ?
29. It takes 1 cup mix and $\frac{3}{4}$ cup water to make 6 pancakes. How much water and mix is needed to make 21 pancakes?
30. Ammonia is a compound consisting of a 1:3 ratio of nitrogen and hydrogen atoms. If a sample contains 1,983 hydrogen atoms, how many nitrogen atoms are present?
31. The Eagles have won 5 out of their last 9 games. If this continues, how many games will they have won in the 63-game season?
32. Solve the train situation described earlier in this lesson.
### Scale and Indirect Measurement
We are occasionally faced with having to make measurements of things that would be difficult to measure directly: the height of a tall tree, the width of a wide river, the height of the moons craters, even the distance between two cities separated by mountainous terrain. In such circumstances, measurements can be made indirectly, using proportions and similar triangles. Such indirect methods link measurement with geometry and numbers. In this lesson, we will examine some of the methods for making indirect measurements.
A map is a two-dimensional, geometrically accurate representation of a section of the Earths surface. Maps are used to show, pictorially, how various geographical features are arranged in a particular area. The scale of the map describes the relationship between distances on a map and the corresponding distances on the earth's surface. These measurements are expressed as a fraction or a ratio.
In the last lesson, you learned the different ways to write a fraction: using the fraction bar, a colon, and in words. Outside of mathematics books, ratios are often written as two numbers separated by a colon (:). Here is a table that compares ratios written in two different ways.
## TABLE 6.1:
| Ratio | Is Read As | Equivalent To |
| :--- | :--- | :--- |
| $1: 20$ | one to twenty | $\left(\frac{1}{20}\right)$ |
| $2: 3$ | two to three | $\left(\frac{2}{3}\right)$ |
| $1: 1000$ | one to one-thousand | $\left(\frac{1}{1000}\right)$ |
If a map had a scale of 1:1000 (one to one-thousand), one unit of measurement on the map (1 inch or 1 centimeter for example) would represent 1000 of the same units on the ground. A 1:1 (one to one) map would be a map as large as the area it shows!
Example : Anne is visiting a friend in London, and is using the map above to navigate from Fleet Street to Borough Road. She is using a 1:100,000 scale map, where $1 \mathrm{~cm}$ on the map represents $1 \mathrm{~km}$ in real life. Using a ruler, she measures the distance on the map as $8.8 \mathrm{~cm}$. How far is the real distance from the start of her journey to the end?
The scale is the ratio of distance on the map to the corresponding distance in real life and can be written as a proportion.
$$
\frac{\text { dist.on map }}{\text { real dist. }}=\frac{1}{100,000}
$$
By substituting known values, the proportion becomes:
$$
\begin{aligned}
\frac{8.8 \mathrm{~cm}}{\text { real dist. }(x)} & =\frac{1}{100,000} & & \text { Cross multiply. } \\
880000 \mathrm{~cm} & =x & & 100 \mathrm{~cm}=1 \mathrm{~m} . \\
x & =8800 \mathrm{~m} & & 1000 \mathrm{~m}=1 \mathrm{~km} .
\end{aligned}
$$
The distance from Fleet Street to Borough Road is $8800 \mathrm{~m}$ or $8.8 \mathrm{~km}$.
We could, in this case, use our intuition: the $1 \mathrm{~cm}=1 \mathrm{~km}$ scale indicates that we could simply use our reading in centimeters to give us our reading in $\mathrm{km}$. Not all maps have a scale this simple. In general, you will need to refer to the map scale to convert between measurements on the map and distances in real life!
Example 1: Oscar is trying to make a scale drawing of the Titanic, which he knows was 883 feet long. He would like his drawing to be 1:500 scale. How long, in inches, must his sheet of paper be?
Solution: We can reason that since the scale is $1: 500$ that the paper must be $\frac{883}{500}=1.766$ feet long. Converting to inches gives the length at 12(1.766) in $=21.192 \mathrm{in}$.
The paper should be at least 22 inches long.
### Percent Problems
Percent problems can be written as Algebraic equations. This section focuses on solving percept problems Algebraically. A percent is a ratio whose denominator is 100 . Before we can use percents to solve problems, let's review how to convert percents to decimals and fractions and vice versa.
To convert a decimal to a percent, multiply the decimal by 100 and add a \% sign.
Example 1: Convert 0.3786 to a percent.
$$
0.3786 \times 100=37.86 \%
$$
To convert a percentage to a decimal, divide the percentage by 100 and drop the $\%$ sign.
Example 2: Convert 98.6\% into a decimal.
$$
98.6 \div 100=0.986
$$
When converting fractions to percents, we can substitute $\frac{x}{100}$ for $x \%$, where $x$ is the unknown.
Example 3: Express $\frac{3}{5}$ as a percent.
We start by representing the unknown as $x \%$ or $\frac{x}{100}$.
$$
\begin{array}{rlrl}
\left(\frac{3}{5}\right) & =\frac{x}{100} & & \text { Cross multiply. } \\
5 x & =100 \cdot 3 & \\
5 x & =300 & \\
x & =\frac{300}{5}=60 & & \text { Divide both sides by } 5 \text { to solve for } x . \\
\left(\frac{3}{5}\right) & =60 \% & &
\end{array}
$$
Now that you remember how to convert between decimals and percents, you are ready for The Percent Equation.
## The Percent Equation
The key words in a percent equation will help you translate it into a correct algebraic equation. Remember the equal sign symbolizes the word is and the multiplication symbol symbolizes the word of.
Example 4: What is $30 \%$ of 85 .
Solution: First, translate into an equation.
$$
n=30 \% \times 85
$$
Convert the percent to a decimal and simplify.
$$
\begin{aligned}
& n=0.30 \times 85 \\
& n=25.5
\end{aligned}
$$
Example 5: 50 is $15 \%$ of what number?
Solution: Translate into an equation.
$$
50=15 \% \times w
$$
Rewrite the percent as a decimal and solve.
$$
\begin{aligned}
50 & =0.15 \times w \\
\frac{50}{.15} & =\frac{0.15 \times w}{.15} \\
333 \frac{1}{3} & =w
\end{aligned}
$$
For more help with the percent equation, watch this 4-minute video recorded by Kens MathWorld. How to SolvePer centEquations (4:10)
## Finding the Percent of Change
A useful way to express changes in quantities is through percents. You have probably seen signs such as $20 \%$ more free, or save $35 \%$ today. When we use percents to represent a change, we generally use the formula:
$$
\text { Percent change }=\left(\frac{\text { final amount }- \text { original amount }}{\text { original amount }}\right) \times 100 \%
$$
A positive percent change would thus be an increase, while a negative change would be a decrease.
Example 6: A school of 500 students is expecting a $20 \%$ increase in students next year. How many students will the school have? Solution: Using the percent of change equation, translate the situation into an equation. Because the $20 \%$ is an increase, it is written as a positive value.
$$
\begin{aligned}
& \text { Percent change }=\left(\frac{\text { final amount }- \text { original amount }}{\text { original amount }}\right) \times 100 \% \\
& 20 \%=\left(\frac{\text { final amount }-500}{500}\right) \times 100 \% \\
& \text { Divide both sides by } 100 \% \text {. } \\
& 0.2=\frac{x-500}{500} \\
& \text { Let } x=\text { final amount. } \\
& 100=x-500 \\
& \text { Multiply both sides by } 500 \text {. } \\
& 600=x \\
& \text { Add } 500 \text { to both sides. }
\end{aligned}
$$
The school will have 600 students next year.
Example 7: A \\$150 mp3 player is on sale for $30 \%$ off. What is the price of the player?
Solution: Using the percent of change equation, translate the situation into an equation. Because the $30 \%$ is a discount, it is written as a negative.
$$
\begin{array}{rlrl}
\text { Percent change } & =\left(\frac{\text { final amount }- \text { original amount }}{\text { original amount }}\right) \times 100 \% \\
\left(\frac{x-150}{150}\right) \cdot 100 \% & =-30 \% & & \text { Divide both sides by } 100 \% . \\
\left(\frac{x-150}{150}\right) & =-0.3 \% & & \text { Multiply both sides by } 150 . \\
x-150=150(-0.3) & =-45 & & \text { Add } 150 \text { to both sides. } \\
x & =-45+150 & & \\
x & =105 & &
\end{array}
$$
The mp3 player will cost $\$ 105$.
Many real situations involve percents. Consider the following.
## Example 8
In 2004, the US Department of Agriculture had 112,071 employees, of which 87,846 were Caucasian. Of the remaining minorities, African-American and Hispanic employees had the two largest demographic groups, with 11,754 and 6,899 employees respectively.*
a) Calculate the total percentage of minority (non-Caucasian) employees at the USDA.
b) Calculate the percentage of African-American employees at the USDA.
c) Calculate the percentage of minority employees at the USDA who were neither African-American nor Hispanic.
## Solution:
a) The total number of employees is 112,071 . We know that the number of Caucasian employees is 87,846, which means that there must be $(112,071-87,846)=24,225$ non-Caucasian employees.
$$
\begin{aligned}
\text { Rate } \times 112,071 & =24,225 \\
\text { Rate } & \approx 0.216 \\
\text { Rate } & \approx 21.6 \%
\end{aligned}
$$
Divide both sides by 112,071 .
Multiply by 100 to obtain percent :
Approximately $21.6 \%$ of USDA employees in 2004 were from minority groups.
b) Total $=112071$ Part $=11754$
$$
\begin{aligned}
\text { Rate } \times 112071 & =11754 \\
\text { Rate } & \approx 0.105 \\
\text { Rate } & \approx 10.5 \%
\end{aligned}
$$
Divide both sides by 112071
Multiply by 100 to obtain percent :
Approximately $10.5 \%$ of USDA employees in 2004 were African-American.
c) We now know there are 24225 non-Caucasian employees. That means there must be $(24225-11754-6899)=$ 5572 minority employees who are neither African-American nor Hispanic. The part is 5572.
$$
\begin{aligned}
\text { Rate } \times 112071 & =5572 \\
\text { Rate } & \approx 0.05 \\
\text { Rate } & \approx 5 \%
\end{aligned}
$$
Divide both sides by 112071 .
Multiply by 100 to obtain percent.
Approximately 5\% of USDA minority employees in 2004 were neither African-American nor Hispanic.
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:PercentProblems (14:15)
Express the following decimals as percents.
1. 0.011 2. 0.001
2. 0.91
3. 1.75
4. 20
Express the following fractions as a percent (round to two decimal places when necessary).
6. $\frac{1}{6}$
7. $\frac{5}{24}$
8. $\frac{6}{7}$
9. $\frac{11}{7}$
10. $\frac{13}{97}$
Express the following percentages as reduced fractions.
11. $11 \%$
12. $65 \%$
13. $16 \%$
14. $12.5 \%$
15. $87.5 \%$
Find the following.
16. $32 \%$ of 600 is what number?
17. $\frac{3}{4} \%$ of 16 is what number?
18. $9.2 \%$ of 500 is what number
19. 8 is $20 \%$ of what number?
20. 99 is $180 \%$ of what number?
21. What percent of 7.2 is 45 ?
22. What percent of 150 is 5 ?
23. What percent of 50 is 2500 ?
24. A sweatshirt costs $\$ 35$. Find the total cost if the sales tax is $7.75 \%$.
25 . This year you got a $5 \%$ raise. If your new salary is $\$ 45,000$, what was your salary before the raise?
26. It costs $\$ 250$ to carpet a room that is $14 \mathrm{ft} \times 18 \mathrm{ft}$. How much does it cost to carpet a room that is $9 \mathrm{ft} \times 10 \mathrm{ft}$ ?
27. A department store has a $15 \%$ discount for employees. Suppose an employee has a coupon worth $\$ 10$ off any item and she wants to buy a $\$ 65$ purse. What is the final cost of the purse if the employee discount is applied before the coupon is subtracted?
28. A realtor earns $7.5 \%$ commission on the sale of a home. How much commission does the realtor make if the home sells for $\$ 215,000$ ?
29. The fire department hopes to raise $\$ 30,000$ to repair a fire house. So far the department has raised $\$ 1,750.00$. What percent is this of their goal?
30. A $\$ 49.99$ shirt goes on sale for $\$ 29.99$. By what percent was the shirt discounted?
31. A TV is advertised on sale. It is $35 \%$ off and has a new price of $\$ 195$. What was the pre-sale price?
32. An employee at a store is currently paid $\$ 9.50$ per hour. If she works a full year, she gets a $12 \%$ pay rise. What will her new hourly rate be after the raise?
33. Store A and Store B both sell bikes, and both buy bikes from the same supplier at the same prices. Store A has a $40 \%$ mark-up for their prices, while store B has a $90 \%$ mark-up. Store B has a permanent sale and will always sell at $60 \%$ off those prices. Which store offers the better deal?
34. 788 students were surveyed about their favorite type of television show. $18 \%$ stated that their favorite show was reality-based. How many students said their favorite show was reality-based?
## CHAPTER 7 Literal Equations and Inequalities
## Chapter Outline
7.1 LITERAL EQUATIONS
7.2 INEQUALITIES
7.3 INEQUALITIES USING MULTIPLICATION AND Division
7.4 MULTI-STEP InEQUaLities
### Literal Equations
In the previous section, we looked at problem solving strategies using formulas. Formulas are examples of Literal Equations and in this section, we are going to look at Literal Equations in more depth.
Officially, a Literal Equation is an equation with several variables (more than one). Some of the more common Literal Equations that we see all the time are:
## TABLE 7.1:
| $A=L W$ | Area $=$ Length $*$ Width | Area of a Rectangle |
| :---: | :---: | :---: |
| $\mathrm{P}=2 \mathrm{~L}+2 \mathrm{~W}$ | $\begin{array}{l}\text { Perimeter }=\text { Twice the Length }+ \\ \text { Twice the Width }\end{array}$ | Perimiter of a Rectangle |
| $\mathrm{A}=\pi \mathrm{r}^{2}$ | Area $=\mathrm{Pi} *$ the Square of the Radius | Area of a Circle |
| $F=1.8 C+32$ | $\begin{array}{l}\text { Temperature in Fahrenheit }=\text { the } \\ \text { temperature in Celsius } * 1.8+32\end{array}$ | $\begin{array}{l}\text { Conversion from Celsius to Fahren- } \\ \text { heit }\end{array}$ |
| $\mathrm{D}=\mathrm{RT}$ | Distance $=$ Rate $*$ Time | Distance Formula |
| $A=P+P r t$ | $\begin{array}{l}\text { Accrued Value }=\text { Principal }+ \text { Prin- } \\ \text { cial } * \text { Interest Rate } * \text { Time }\end{array}$ | Simple Interest Formula |
Sometimes we need to rearrange a formula to solve for a different variable. For example, we may know the Area and the Length of a rectangle, and we want to find the width. Or we may know the time that weve been travelling and the total distance that weve travelled, and we want to find our average speed (rate). To do this, we need to isolate the variable of interest on one side of the equal sign, with all the other terms on the opposite side of the equal sign. Luckily, we can use the same techniques that we have been using to solve for the value of a variable.
Example 1: Find the Width of a Rectangle given the Area and the Length
Solution: The Area of a rectangle is given by the formula $A=l w$. We need to solve this equation for $w$.
$$
\begin{array}{ll}
A=l w & \text { Area = Length * Width } \\
\frac{A}{l}=\frac{l w}{l} & \text { Divide both sides by Length } \\
\frac{A}{l}=w & \text { Equivalent form solved for Width }
\end{array}
$$
Example 2: Find the Width of a Rectangle when given the Perimeter and Length
Solution: The perimeter of a rectangle is given by the formula $P=2 l+2 w$. We need to solve for $w$.
This one is actually a little more complicated. The formula has one term on the left hand side and two terms on the right. It's important to remember that we have to move one term to the left first.
$$
\begin{aligned}
P & =2 l+2 w & & \text { Perimeter }=\text { twice the Length }+ \text { twice the Width } \\
P-2 l & =2 l+2 w-2 l & & \text { Subtract } 2 l \text { from both sides } \\
P-2 l & =2 w & & \text { This leaves us with one term on the right } \\
\frac{P-2 l}{2} & =\frac{2 w}{2} & & \text { Divide both sides by } 2 \\
\frac{P-2 l}{2} & =w & & \text { Equivalent form solved for Width }
\end{aligned}
$$
Example 3: Find the length of one side of a triangle, given the Perimeter and the other two sides.
Solution: The Perimeter of a triangle is the sum of the lengths of the three sides, so $P=a+b+c$. We need to solve for $a, b$ or $c$.
This one seems to give people problems, so let's take a look at it.
$$
\begin{aligned}
P & =a+b+c & & \text { Perimeter }=\text { sum of the three sides. Let's solve for c } \\
P-a & =a+b+c-a & & \text { Subtract a from both sides } \\
P-a & =b+c & & \\
P-a-b & =b+c-b & & \text { Subtract } \mathrm{b} \text { from both sides } \\
P-a-b & =c & & \text { Equivalent form solved for c }
\end{aligned}
$$
Example 4: Find the voltage given the current and the power.
Solution: A formula that relates power with current and voltage is $I=\frac{P}{E}$, where $I$ is the current, $P$ is the power, and $E$ is the voltage.
In this example, the variable you want to solve for is part of the denominator of a fraction.
$$
\begin{aligned}
I & =\frac{P}{E} & & \text { Amps }=\text { Watts/Voltage } \\
\left(\frac{E}{1}\right) I & =\frac{P}{E}\left(\frac{E}{1}\right) & & \text { Multiply both sides E/1 } \\
E I & =P & & \text { Now } \mathrm{E} \text { is in the Numerator on the left } \\
\frac{E I}{I} & =\frac{P}{I} & & \text { Divide both sides by } \mathrm{I} \\
E & =\frac{P}{I} & & \text { Equivalent form solved for E }
\end{aligned}
$$
Example 5: Problem Solving - Building a planter
Dave is building a planter for his wife. It's going to be two feet wide and four feet long. He wants the Planter to hold 20 cubic feet of soil to ensure the plants have room to grow. How high should the planter be?
Dave knows that the volume of a box (Rectangular Prism) is $\mathrm{V}=\mathrm{l}^{*} \mathrm{~W}^{*} \mathrm{~h}$ so he decides to solve the equation for $\mathrm{h}$ and plug it into his calculator
$$
\begin{aligned}
V & =l w h \\
\frac{V}{l w} & =\frac{l w h}{l w} \\
\frac{V}{l w} & =h
\end{aligned}
$$
Divide both sides by lw
Equivalent form solved for $\mathrm{h}$
Given the new formula, Dave plugs $20 /(2 * 4)$ into his calculator and gets 2.5 . Therefore, Dave should build his planter 2.5 feet high.
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both.CK-12 Ba sic Algebra:Solvingfora variable (2:41)
For each of the following Equations, solve for the indicated variable
1. The formula for converting from Fahrenheit to Celsius is
Convert this formula so it solves for $F$
$$
C=\frac{5}{9}(F-32)
$$
2. $\mathrm{P}=$ IRT solve for $T$
3. $C=2 \pi r$ solve for $r$
4. $\mathrm{y}=5 \mathrm{x}-6$ solve for $x$
5. $4 \mathrm{x}-3 \mathrm{y}=6$ solve for $y$
6. $\mathrm{y}=\mathrm{mx}+\mathrm{b}$ solve for $b$
7. $\mathrm{ax}+\mathrm{by}=\mathrm{c}$ solve for $y$
8. $\mathrm{A}=\mathrm{P}+\mathrm{Prt}$ solve for $t$
9. $\mathrm{V}=\mathrm{LWH}$ solve for $L$
10. $\mathrm{A}=4 \mathrm{pr}^{2}$ solve for $r^{2}$
11. $\mathrm{V}=\mathrm{pr}^{2} \mathrm{~h}$ solve for $h$
12. $5 \mathrm{x}-\mathrm{y}=10$ solve for $x$
13. $A=(x+y) / 2$ solve for $y$
14. $12 \mathrm{x} 4 \mathrm{y}=20$ solve for $y$
## 5. $A=1 / 2 h(b+c)$ solve for $b$
16. John knows that the formula to calculate how far he has traveled is average speed $*$ time traveled. Write a literal equation that represents this formula using $\mathrm{D}$ for distance, $\mathrm{r}$ for average rate and $\mathrm{t}$ for time traveled. Then rewrite the formula, solving for average speed, $\mathrm{r}$.
17. Mary paid $\$ 100$ to set up a face painting booth at the local fair. She is going to charge customers $\$ 5$ each. She figures out that her profits will be $\$ 5$ per customer minus her $\$ 100$ rental fee and comes up with the formula $\mathrm{P}=5 \mathrm{c}$ 100 where $\mathrm{P}$ is profit and $c$ is the number of customers. She wants to earn at least $\$ 400$ at the fair. Help Mary rewrite her formula so that she can calculate c. How many customers will she need?
### Inequalities
## Sometimes Things Are Not Equal
In some cases there are multiple answers to a problem or the situation requires something that is not exactly equal to another value. When a mathematical sentence involves something other than an equal sign, an inequality is formed.
Definition: An algebraic inequality is a mathematical sentence connecting an expression to a value, variable, or another expression with an inequality sign.
Listed below are the most common inequality signs.
$>$ greater than
$\geq$ greater than or equal to
$\leq$ less than or equal to
$<$ less than
$\neq$ not equal to
Below are several examples of inequalities.
$$
3 x<5 \quad \frac{3 x}{4} \geq \frac{x}{2}-3 \quad 4-x \leq 2 x
$$
Example 1: Translate the following into an inequality: Avocados cost $\$ 1.59$ per pound. How many pounds of avocados can be purchased for less than $\$ 7.00$ ?
Solution: Choose a variable to represent the number of pounds of avocados purchased, say $a$.
$$
1.59(a)<7
$$
You will be asked to solve this inequality in the exercises
## Checking the Solution to an Inequality
Unlike equations, inequalities typically have more than one solution. Checking solutions to inequalities are more complex than checking solutions to equations. The key to checking a solution to an inequality is to choose a number that occurs within the solution set. Example 2: Check $m \leq 10$ is a solution to $4 m+30 \leq 70$.
Solution: If the solution set is true, any value less than or equal to 10 should make the original inequality true.
Choose a value less than 10 , say 4 . Substitute this value for the variable $m$.
$$
\begin{aligned}
& 4(4)+30 \\
& 16+30 \\
& 46 \leq 70
\end{aligned}
$$
The value found when $m=4$ is less than 70 . Therefore, the solution set is true.
Why was the value 10 not chosen? Endpoints are not chosen when checking an inequality because the direction of the inequality needs to be tested. Special care needs to be taken when checking the solutions to an inequality.
Solutions to one-variable inequalities can be graphed on a number line or in a coordinate plane.
Example 3: Graph the solutions to $t>3$ on a number line.
Solution: The inequality is asking for all real numbers 3 or larger.
You can also write inequalities given a number line of solutions.
Example 4: Write the inequality pictured below.
Solution: The value of four is colored in, meaning that four is a solution to the inequality. The red arrow indicates values less than four. Therefore, the inequality is:
$$
x \leq 4
$$
Inequalities that include the value are shown as $\leq$ or $\geq$. The line underneath the inequality stands for or equal to. We show this relationship by coloring in the circle above this value, as in the previous example. For inequalities without the or equal to the circle above the value remains unfilled.
## Three Ways to Express Solutions to Inequalities
1. Inequality notation: The answer is expressed as an algebraic inequality, such as $d \leq \frac{1}{2}$.
2. Interval notation: This notation uses brackets to denote the range of values in an inequality.
- Square or closed brackets [ ] indicate that the number is included in the solution
- Round or open brackets ( ) indicate that the number is not included in the solution.
Interval notation also uses the concept of infinity $\infty$ and negative infinity $-\infty$. For example, for all values of $d$ that are less than or equal to $\frac{1}{2}$, you could use set notation as follows: $\left(-\infty, \frac{1}{2}\right\}$ 3. As a graphed sentence on a number line.
Example 5: $(8,24)$ states that the solution is all numbers between 8 and 24 but does not include the numbers 8 and 24.
$[3,12)$ states that the solution is all numbers between 3 and 12 , including 3 but not including 12 .
## Inequalities Using Addition or Subtraction
To solve inequalities, you need some properties.
Addition Property of Inequality: For all real numbers $a, b$, and $c$ :
If $x<a$, then $x+b<a+b$
If $x<a$, then $x-c<a-c$
The two properties above are also true for $\leq$ or $\geq$.
Because subtraction can also be thought of as add the opposite, these properties also work for subtraction situations.
Just like one-step equations, the goal is to isolate the variable, meaning to get the variable alone on one side of the inequality symbol. To do this, you will cancel the operations using inverses.
Example 6: Solve for $x: x-3<10$
Solution: To isolate the variable $x$, you must cancel subtract 3 using its inverse operation, addition.
$$
\begin{aligned}
x-3+3 & <10+3 \\
x & <13
\end{aligned}
$$
Now, check your answer. Choose a number less than 13 and substitute it into your original inequality. If you choose 0 , and substitute it you get:
$$
0-3<10=-3<10
$$
What happens at 13 ? What happens with numbers greater than 13 ?
Example 7: Solve for $x: x+4>13$
Solution:
Writing Real Life Inequalities
As described in the chapter opener, inequalities appear frequently in real life. Solving inequalities is an important part of algebra. Example 8: Write the following statement as an algebraic inequality. You must maintain a balance of at least $\$ 2,500$ in your checking account to avoid a finance charge.
Solution: The key phrase in this statement is at least. This means you can have $\$ 2,500$ or more in your account to avoid a finance charge.
Choose the variable to describe the money in your account, say $m$.
Write the inequality $m \geq 2500$
Graph the solutions using a number line.
Example 9: Translate into an algebraic inequality: The speed limit is 65 miles per hour.
Solution: To avoid a ticket, you must drive 65 or less. Choose a variable to describe your possible speed, say $s$. Write the inequality $s \leq 65$.
Graph the solutions to the inequality using a number line.
In theory, you cannot drive a negative number of miles per hour. This concept will be a focus later in this chapter.
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:Inequalities Using AdditionandSubtraction (7:48)
MEDIA
Click image to the left for more content.
1. What are the three methods of writing the solution to an inequality?
Graph the solutions to the following inequalities using a number line.
2. $x<-3$
3. $x \geq 6$
4. $x>0$
5. $x \leq 8$
6. $x<-35$
7. $x>-17$
8. $x \geq 20$
9. $x \leq 3$
Write the inequality that is represented by each graph.
Write the inequality given by the statement. Choose an appropriate letter to describe the unknown quantity.
18. You must be at least 48 inches tall to ride the Thunderbolt Rollercoaster.
19. You must be younger than 3 years old to get free admission at the San Diego Zoo.
20. Charlie needs more than $\$ 1,800$ to purchase a car.
21. Cheryl can have no more than six pets at her house.
22. The shelter can house no more than 16 rabbits.
Solve each inequality and graph the solution on the number line.
23. $x-1>-10$
24. $x-1 \leq-5$
25. $-20+a \geq 14$
26. $x+2<7$
27. $x+8 \leq-7$
28. $5+t \geq \frac{3}{4}$
29. $x-5<35$
30. $15+g \geq-60$
31. $x-2 \leq 1$
32. $x-8>-20$
33. $11+q>13$
34. $x+65<100$
35. $x-32 \leq 0$
36. $x+68 \geq 75$
37. $16+y \leq 0$
### Inequalities Using Multiplication and Divi- sion
Equations are mathematical sentences in which the two sides have the same weight. By adding, subtracting, multiplying or dividing the same value to both sides of the equation, the balance stays in check. However, inequalities begin off-balance. When you perform inverse operations, the inequality will remain off-balance. This is true with inequalities involving both multiplication and division.
Before we can begin to solve inequalities involving multiplication or division, you need to know two properties, the Multiplication Property of Inequalities and the Division Property of Inequalities.
Multiplication Property of Inequality: For all real positive numbers $a, b$, and $c$ :
If $x<a$, then $x(c)<a(c)$.
If $x>a$, then $x(c)>a(c)$.
Division Property of Inequality: For all real positive numbers $a, b$, and $c$ :
If $x<a$, then $x \div(c)<a \div(c)$.
If $x>a$, then $x \div(c)>a \div(c)$.
Consider the inequality $2 x \geq 12$. To find the solutions to this inequality, we must isolate the variable $x$ by using the inverse operation of multiply by 2 , which is dividing by 2 .
$$
\begin{aligned}
2 x & \geq 12 \\
\frac{2 x}{2} & \geq \frac{12}{2} \\
x & \geq 6
\end{aligned}
$$
This solution can be expressed in four ways. One way is already written, $x \geq 6$. Below are the three remaining ways to express this solution:
- $\{x \mid x \geq 6\}$
- $[6, \infty)$
- Using a number line:
Example 1: Solve for $y: \frac{y}{5} \leq 3$. Express the solution using all four methods.
Solution: The inequality above is read, $y$ divided by 5 is less than or equal to 3 . To isolate the variable $y$, you must cancel division using its inverse operation, multiplication.
$$
\begin{aligned}
\frac{y}{5} \cdot \frac{5}{1} & \leq 3 \cdot \frac{5}{1} \\
y & \leq 15
\end{aligned}
$$
One method of writing the solution is $y \leq 15$.
The other three are: - $(-\infty, 15]$
- $\{y \mid y \leq 15\}$
Multiplying and Dividing an Inequality by a Negative Number
Notice the two properties in this lesson focused on $c$ being only positive. This is because those particular properties of multiplication and division do not apply when the number being multiplied (or divided) is negative.
Think of it this way. When you multiply a value by -1 , the number you get is the negative of the original.
$$
6(-1)=-6
$$
Multiplying each side of a sentence by -1 results in the opposite of both values.
$$
\begin{gathered}
5 x(-1)=4(-1) \\
-5 x=-4
\end{gathered}
$$
When multiplying by a negative, you are doing the opposite of everything in the sentence, including the verb.
$$
\begin{aligned}
x & >4 \\
x(-1) & >4(-1) \\
-x & <-4
\end{aligned}
$$
This concept is summarized below.
Multiplication/Division Rule of Inequality: For any real number $a$, and any negative number $c$,
If $x<a$, then $x \cdot c>a \cdot c$
If $x<a$, then $\frac{x}{c}>\frac{a}{c}$
As with the other properties of inequalities, these also hold true for $\leq$ or $\geq$.
Example 2: Solve for $r:-3 r<9$
Solution: To isolate the variable $r$, we must cancel multiply by -3 using its inverse operation, dividing by -3 .
$$
\frac{-3 r}{-3}<\frac{9}{-3}
$$
Since you are dividing by -3 , everything becomes opposite, including the inequality sign.
$$
r>-3
$$
Example 3: Solve for $p: 12 p<-30$
Solution: To isolate the variable $p$, we must cancel multiply by 12 using its inverse operation, dividing by 12 .
$$
\frac{12 p}{12}<\frac{-30}{12}
$$
Because 12 is not negative, you do not switch the inequality sign.
$$
p<\frac{-5}{2}
$$
In set notation, the solution would be: $\left(-\infty, \frac{-5}{2}\right)$
Multimedia Link: For more help with solving inequalities involving multiplication and division, visit Khan Academy's website. http://khanexercises.appspot.com/video?v=PNXozoJWsWc
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:Inequalities Using Multiplication and Division (10:27)
1. In which cases do you change the inequality sign?
Solve each inequality. Give the solution using inequality notation and with a solution graph.
2. $3 x \leq 6$
3. $\frac{x}{5}>-\frac{3}{10}$
4. $-10 x>250$
5. $\frac{x}{-7} \geq-5$
6. $9 x>-\frac{3}{4}$
7. $\frac{x}{-15} \leq 5$
8. $620 x>2400$ 9. $\frac{x}{20} \geq-\frac{7}{40}$
9. $-0.5 x \leq 7.5$
10. $75 x \geq 125$
11. $\frac{x}{-3}>-\frac{10}{9}$
12. $\frac{k}{-14} \leq 1$
13. $\frac{x}{-15}<8$
14. $\frac{x}{2}>40$
15. $\frac{x}{-3} \leq-12$
16. $\frac{x}{25}<\frac{3}{2}$
17. $\frac{x}{-7} \geq 9$
18. $4 x<24$
19. $238<14 d$
20. $-19 m \leq-285$
21. $-9 x \geq-\frac{3}{5}$
22. $-5 x \leq 21$
23. The width of a rectangle is 16 inches. Its area is greater than 180 square inches.
- - Write an inequality to represent this situation.
- Graph the possible lengths of the rectangle.
24. Ninety percent of some number is at most 45 .
- - Write an inequality to represent the situation.
- Write the solutions as an algebraic sentence.
25. Doubling Marthas jam recipe yields at least 22 pints.
- $\quad$ - Write an inequality to represent the situation.
- Write the solutions using interval notation.
### Multi-Step Inequalities
The previous two lessons focused on one-step inequalities. Inequalities, like equations, can require several steps in order to isolate the variable. These inequalities are called multi-step inequalities. With the exception of the Multiplication/Division Rule of Inequality, the process of solving multi-step inequalities is identical to solving multistep equations.
## Procedure to Solve an Inequality:
1. Remove any parentheses by using the Distributive Property.
2. Simplify each side of the inequality by combining like terms.
3. Isolate the $a x$ term. Use the Addition Property of Inequality to get the variable on one side of the equal sign and the numerical values on the other.
4. Isolate the variable. Use the Multiplication Property of Inequality to get the variable alone on one side of the equation.
(a) Remember to reverse the inequality sign if you are multiplying or dividing by a negative.
5. Check your solution.
Example 1: Solve for $w$ : $6 x-5<10$
Solution: Begin by using the checklist above.
1. Parentheses? No
2. Like terms on the same side of inequality? No
3. Isolate the $a x$ term using the Addition Property.
$$
6 x-5+5<10+5
$$
Simplify:
$$
6 x<15
$$
4. Isolate the variable using the Multiplication or Division Property.
$$
\frac{6 x}{6}<\frac{15}{6}=x<\frac{5}{2}
$$
5. Check your solution. Choose a number less than 2.5 , say 0 and check using the original inequality.
$$
\begin{aligned}
6(0)-5 & <10 ? \\
-5 & <10
\end{aligned}
$$
Yes, the answer checks. $x<2.5$ Example 2: Solve for $x$ : $-9 x<-5 x-15$
Solution: Begin by using the checklist above.
1. Parentheses? No
2. Like terms on the same side of inequality? No
3. Isolate the $a x$ term using the Addition Property.
$$
-9 x+5 x<-5 x+5 x-15
$$
Simplify:
$$
-4 x<-15
$$
4. Isolate the variable using the Multiplication or Division Property.
$$
\frac{-4 x}{-4}<\frac{-15}{-4}
$$
Because the number you are dividing by is negative, you must reverse the inequality sign.
$$
x>\frac{15}{4} \rightarrow x>3 \frac{3}{4}
$$
5. Check your solution by choosing a number larger than 3.75, say 10 .
$$
\begin{aligned}
& -9(10)<-5(10)-15 ? \\
& \checkmark-90<-65
\end{aligned}
$$
Example 3: Solve for $x$ : $4 x-2(3 x-9) \leq-4(2 x-9)$
Solution: Begin by using the previous checklist.
1. Parentheses? Yes. Use the Distributive Property to clear the parentheses.
$$
4 x+(-2)(3 x)+(-2)(-9) \leq-4(2 x)+(-4)(-9)
$$
Simplify:
$$
4 x-6 x+18 \leq-8 x+36
$$
2. Like terms on the same side of inequality? Yes. Combine these
$$
-2 x+18 \leq-8 x+36
$$
3. Isolate the ax term using the Addition Property.
$$
-2 x+8 x+18 \leq-8 x+8 x+36
$$
Simplify:
$$
\begin{aligned}
6 x+18 & \leq 36 \\
6 x+18-18 & \leq 36-18
\end{aligned}
$$
4. Isolate the variable using the Multiplication or Division Property.
$$
\frac{6 x}{6} \leq \frac{18}{6} \rightarrow x \leq 3
$$
5. Check your solution by choosing a number less than 3, say -5 .
$$
\begin{aligned}
4(-5)-2(3 \cdot-5-9) & \leq-4(2 \cdot-5-9) \\
\checkmark 28 & <76
\end{aligned}
$$
Identifying the Number of Solutions to an Inequality
Inequalities can have infinitely many solutions, no solutions, or a finite set of solutions. Most of the inequalities you have solved to this point have an infinite amount of solutions. By solving inequalities and using the context of a problem, you can determine the number of solutions an inequality may have.
Example 4: Find the solutions to $x-5>x+6$
Solution: Begin by isolating the variable using the Addition Property of Inequality.
$$
x-x-5>x-x+6
$$
Simplify.
$$
-5>6
$$
This is an untrue inequality. Negative five is never greater than six. Therefore, the inequality $x-5>x+6$ has no solutions.
Previously we looked at the following sentence. The speed limit is 65 miles per hour.
The algebraic sentence is: $s \leq 65$.
Example 5: Find the solutions to $s \leq 65$.
Solution: The speed at which you drive cannot be negative. Therefore, the set of possibilities using interval notation is $[0,65]$.
## Solving Real-World Inequalities
Example 6: In order to get a bonus this month, Leon must sell at least 120 newspaper subscriptions. He sold 85 subscriptions in the first three weeks of the month. How many subscriptions must Leon sell in the last week of the month?
Solution: The amount of subscriptions Leon needs is at least 120. Choose a variable to represent the varying quantitythe number of subscriptions, say $n$. The inequality that represents the situation is $n+85 \geq 120$.
Solve by isolating the variable $n . n \geq 35$
Leon must sell 35 or more subscriptions in order to receive his bonus.
Example 7: The width of a rectangle is 12 inches. What must the length be if the perimeter is at least 180 inches? Diagram not drawn to scale.
Solution: The perimeter is the sum of all the sides.
$$
12+12+x+x \geq 180
$$
Simplify and solve for the variable $x$ :
$$
\begin{aligned}
12+12+x+x & \geq 180 \rightarrow 24+2 x \geq 180 \\
2 x & \geq 156 \\
x & \geq 78
\end{aligned}
$$
The length of the rectangle must be 78 inches or larger.
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:Multi-Step Inequalities (8:02)
Solve each of the following inequalities and graph the solution set.
1. $6 x-5<10$
2. $-9 x<-5 x-15$
3. $-\frac{9 x}{5} \leq 24$
4. $\frac{9 x}{5}-7 \geq-3 x+12$
5. $\frac{5 x-1}{4}>-2(x+5)$
6. $4 x+3<-1$
7. $2 x<7 x-36$
8. $5 x>8 x+27$
9. $5-x<9+x$
10. $4-6 x \leq 2(2 x+3)$
11. $5(4 x+3) \geq 9(x-2)-x$
12. $2(2 x-1)+3<5(x+3)-2 x$
13. $8 x-5(4 x+1) \geq-1+2(4 x-3)$
14. $2(7 x-2)-3(x+2)<4 x-(3 x+4)$
15. $\frac{2}{3} x-\frac{1}{2}(4 x-1) \geq x+2(x-3)$
16. At the San Diego Zoo, you can either pay $\$ 22.75$ for the entrance fee or $\$ 71$ for the yearly pass which entitles you to unlimited admission. At most how many times can you enter the zoo for the $\$ 22.75$ entrance fee before spending more than the cost of a yearly membership?
17. Proteeks scores for four tests were $82,95,86$ and 88 . What will he have to score on his last test to average at least 90 for the term?
18. Raul is buying ties and he wants to spend $\$ 200$ or less on his purchase. The ties he likes the best cost $\$ 50$. How many ties could he purchase?
19. Virena's Scout Troup is trying to raise at least $\$ 650$ this spring. How many boxes of cookies must they sell at $\$ 4.50$ per box in order to reach their goal?
## CHAPTER
## Graphs and Graphing Linear Equations
## Chapter Outline
### The Coordinate Plane
8.2 WHAT MAKES A GOOD GRAPH
8.3 GRAPH A LINEAR EQUATION
8.4 GRAPHING USING INTERCEPTS
The ability to graph linear equations is important in mathematics. In fact, graphing equations and solving equations are two of the most important concepts in mathematics. If you master these, all mathematical subjects will be much easier, even Calculus!
This chapter focuses on the visual representations of data. You will learn how graphs are created using the Coordinate Plane and how to create and interpret graphs using sets of data. Finally, you will learn what defines a good graph and what mistakes to avoid when creating a graph.
### The Coordinate Plane
## Introduction
Lydia lives 2 blocks north and one block east of school; Travis lives three blocks south and two blocks west of school. What's the shortest line connecting their houses?
## The Coordinate Plane
We've seen how to represent numbers using number lines; now we'll see how to represent sets of numbers using a coordinate plane. The coordinate plane can be thought of as two number lines that meet at right angles. The horizontal line is called the $x$-axis and the vertical line is the $y$-axis. Together the lines are called the axes, and the point at which they cross is called the origin. The axes split the coordinate plane into four quadrants, which are numbered sequentially (I, II, III, IV) moving counter-clockwise from the upper right.
## Identify Coordinates of Points
When given a point on a coordinate plane, it's easy to determine its coordinates. The coordinates of a point are two numbers - written together they are called an ordered pair. The numbers describe how far along the $x$-axis and $y$-axis the point is. The ordered pair is written in parentheses, with the $x$-coordinate (also called the abscissa) first and the $y$-coordinate (or the ordinate) second. $(-107.2,-0.005)$ An ordered pair with an $x$ - value of one and a $y$ - value of seven
An ordered pair with an $x$ - value of zero and a $y$-value of five
An ordered pair with an $x$ - value of -2.5 and a $y$ - value of four
An ordered pair with an $x$-value of -107.2 and a $y$-value of -0.005
Identifying coordinates is just like reading points on a number line, except that now the points do not actually lie on the number line! Look at the following example.
## Example 1
Find the coordinates of the point labeled $P$ in the diagram above
## Solution
Imagine you are standing at the origin (the point where the $x$-axis meets the $y$-axis). In order to move to a position where $P$ was directly above you, you would move 3 units to the right (we say this is in the positive $x$-direction). The $x$-coordinate of $P$ is +3 .
Now if you were standing at the 3 marker on the $x$-axis, point $P$ would be 7 units above you (above the axis means it is in the positive $y$ direction).
The $y$-coordinate of $P$ is +7 .
## The coordinates of point $P$ are $(3,7)$.
## Example 2
Find the coordinates of the points labeled $Q$ and $R$ in the diagram to the right.
## Solution
In order to get to $Q$ we move three units to the right, in the positive $x$-direction, then two units down. This time we are moving in the negative $y$-direction. The $x$-coordinate of $Q$ is +3 , the $y$-coordinate of $Q$ is -2 .
The coordinates of $R$ are found in a similar way. The $x$-coordinate is +5 (five units in the positive $x$-direction) and the $y$-coordinate is again -2 .
The coordinates of $Q$ are $(3,-2)$. The coordinates of $R$ are $(5,-2)$.
## Example 3
Triangle $A B C$ is shown in the diagram to the right. Find the coordinates of the vertices $A, B$ and $C$.
Point $A$ :
$x-$ coordinate $=-2$
$y$ - coordinate $=+5$
Point $B$ :
$x$ - coordinate $=+3$
$y$ - coordinate $=-3$
Point $C$ :
$x-$ coordinate $=-4$
$y-$ coordinate $=-1$
Solution
A(-2,5)
$B(3,-3)$
$C(-4,-1)$
Plot Points in a Coordinate Plane
Plotting points is simple, once you understand how to read coordinates and read the scale on a graph. As a note on scale, in the next two examples pay close attention to the labels on the axes.
## Example 4
Plot the following points on the coordinate plane.
$A(2,7) \quad B(-4,6) \quad D(-3,-3) \quad E(0,2) \quad F(7,-5)$
Point $A(2,7)$ is 2 units right, 7 units up. It is in Quadrant $\mathrm{I}$.
Point $B(-4,6)$ is 4 units left, 6 units up. It is in Quadrant II.
Point $D(-3,-3)$ is 3 units left, 3 units down. It is in Quadrant III.
Point $E(0,2)$ is 2 units up from the origin. It is right on the $y$-axis, between Quadrants I and II.
Point $F(7,-5)$ is 7 units right, 5 units down. It is in Quadrant IV.
## Example 5
Plot the following points on the coordinate plane.
$A(2.5,0.5) \quad B(\pi, 1.2) \quad C(2,1.75) \quad D(0.1,1.2) \quad E(0,0)$
Here we see the importance of choosing the right scale and range for the graph. In Example 4, our points were scattered throughout the four quadrants. In this case, all the coordinates are positive, so we don't need to show the negative values of $x$ or $y$. Also, there are no $x$-values bigger than about 3.14, and 1.75 is the largest value of $y$. We can therefore show just the part of the coordinate plane where $0 \leq x \leq 3.5$ and $0 \leq y \leq 2$.
Here are some other important things to notice about this graph: - The tick marks on the axes don't correspond to unit increments (i.e. the numbers do not go up by one each time). This is so that we can plot the points more precisely.
- The scale on the $x$-axis is different than the scale on the $y$-axis, so distances that look the same on both axes are actually greater in the $x$-direction. Stretching or shrinking the scale in one direction can be useful when the points we want to plot are farther apart in one direction than the other.
- The tick marks are equally spaced on each axis. The axes are like number lines - it is important that each tick mark represents the same increment.
For more practice locating and naming points on the coordinate plane, try playing the Coordinate Plane Game at http://tinyurl.com/72myodv.
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:TheCoordinate Plane (6:50)
In questions 1-6, identify the coordinate of the given letter.
Graph the following ordered pairs on one Cartesian plane. Identify the quadrant where each ordered pair is located.
7. $(4,2)$
8. $(-3,5.5)$
9. $(4,-4)$
10. $(-2,-3)$
11. $\left(\frac{1}{2},-\frac{3}{4}\right)$
12. $(-0.75,1)$
13. $\left(-2 \frac{1}{2},-6\right)$
14. $(1.60,4.25)$
Using the directions given in each problem, find and graph the coordinates on a Cartesian plane.
15. Six left, four down
16. One-half right, one-half up
17. Three right, five down
18. Nine left, seven up
19. Four and one-half left, three up
20. Eight right, two up
21. One left, one down
22. One right, three-quarter down
23. Plot the vertices of triangle $A B C:(0,0),(4,-3),(6,2)$
24. The following three points are three vertices of square $A B C D$. Plot them on a coordinate plane then determine what the coordinates of the fourth point, $D$, would be. Plot that point and label it. $A(-4,-4) B(3,-4) C(3,3)$
25. Does the ordered pair $(2,0)$ lie in a quadrant? Explain your thinking.
26. Why do you think $(0,0)$ is called the origin?
27. Ian has the following collection of data. Graph the ordered pairs and make a conclusion from the graph.
## TABLE 8.1:
Year
1973
1980
1986 \% of Men Employed in the United States
75.5
72.0
71.0 TABLE 8.1: (continued)
| Year | \% of Men Employed in the United States |
| :--- | :--- |
| 1992 | 69.8 |
| 1997 | 71.3 |
| 2002 | 69.7 |
| 2005 | 69.6 |
| 2007 | 69.8 |
| 2009 | 64.5 |
Words of Wisdom from the Graphing Plane
Not all axes will be labeled for you. There will be many times you are required to label your own axes. Some problems may require you to graph only the first quadrant. Others need two or all four quadrants. The tic marks do not always count by ones. They can be marked in increments of 2,5 , or even $\frac{1}{2}$. The axes do not even need to have the same increments! The Cartesian plane below shows an example of this.
The increments by which you count your axes should MAXIMIZE the clarity of the graph.
### What makes a good graph
When creating a graph, there are guidelines to follow to ensure that it is easy to read and easy to interpret.
1. The horizontal axis should be properly labeled with the name and units of the input variable.2. The vertical axis should be properly labeled with the name and units of the output variable.3. Use an appropriate scale.
- Start at or just below the lowest replacement value.
- End at or just above the highest replacement value.
- Scale the graph so the adjacent tick marks are equal distance apart.
- Use numbers that make sense for the given data set.
- The axes meet at $(0,0)$ Use a "/r" between the origin and the first tick mark if the scale does not begin at 0 .
## Example 1
The graph shown in Example 3 is a very good example of how a graph should be created. It has all the elements of a good graph as noted above.
1. Both the horizontal axis and vertical axis are labeled with the name of the variable and the unit. The horizontal axis has the variable name Weight and the unit Pounds (lbs) The vertical axis is labeled with the variable name Length and the unit Inches. If the units were not included, there would be no way to know what unit of weight or length was being used
2. The horizontal and vertical scales both start with 0 and are consistent. Note that they are not the same scale. Weight changes by a scale of 1 and Length changes by a scale of 5 .
3. The numbers for both scales make sense for the data and make good use of the space available.
4. The data points are labeled with the $\mathrm{x}$ and $\mathrm{y}$ value
## Example 2
The last example that we are going to look at in this lesson is an example of a "bad" graph. It has examples of everything you should "not" do when creating a graph.
1. The Horizontal Axis does not have the units. It is impossible to tell if Time is being measured in Minutes, Seconds or something else.
2. The Vertical Axis does not have the Variable Name. What is being measured in Feet?
3. The horizontal scale is not consistent and the vertical scale does not start at 0
4. The graph makes very poor use of the available space
Remember as you create your graphs that labeling is critical. It is important that the graph be both readable and accurate. Follow the guidelines above at all times to ensure good practice.
## Analyzing Graphs
We often use graphs to represent relationships between two linked quantities. It's useful to be able to interpret the information that graphs convey. For example, the chart below shows a fluctuating stock price over ten weeks. You can read that the index closed the first week at about \\$68, and at the end of the third week it was at about \\$62. You may also see that in the first five weeks it lost about $20 \%$ of its value, and that it made about $20 \%$ gain between weeks seven and ten. Notice that this relationship is discrete, although the dots are connected to make the graph easier to interpret.
Analyzing graphs is a part of life - whether you are trying to decide to buy stock, figure out if your blog readership is increasing, or predict the temperature from a weather report. Many graphs are very complicated, so for now we'll start off with some simple linear conversion graphs. Algebra starts with basic relationships and builds to more complicated tasks, like reading the graph above.
## Example 3
Below is a graph for converting marked prices in a downtown store into prices that include sales tax. Use the graph to determine the cost including sales tax for a $\$ 6.00$ pen in the store.
To find the relevant price with tax, first find the correct pre-tax price on the $x$-axis. This is the point $x=6$.
Draw the line $x=6$ up until it meets the function, then draw a horizontal line to the $y$-axis. This line hits at $y \approx 6.75$ (about three fourths of the way from $y=6$ to $y=7$ ).
The approximate cost including tax is $\$ 6.75$.
## Example 4
The chart for converting temperature from Fahrenheit to Celsius is shown to the right. Use the graph to convert the following:
a) $70^{\circ}$ Fahrenheit to Celsius
b) $0^{\circ}$ Fahrenheit to Celsius
c) $30^{\circ}$ Celsius to Fahrenheit
d) $0^{\circ}$ Celsius to Fahrenheit
## Solution
a) To find $70^{\circ}$ Fahrenheit, we look along the Fahrenheit-axis (in other words the $x$-axis) and draw the line $x=70$ up to the function. Then we draw a horizontal line to the Celsius-axis ( $y$-axis). The horizontal line hits the axis at a little over 20 (21 or 22).
## $70^{\circ}$ Fahrenheit is approximately equivalent to $21^{\circ}$ Celsius.
b) To find $0^{\circ}$ Fahrenheit, we just look at the $y$-axis. (Don't forget that this axis is simply the line $x=0$.) The line hits the $y$-axis just below the half way point between -15 and -20 .
$0^{\circ}$ Fahrenheit is approximately equivalent to $-18^{\circ}$ Celsius.
c) To find $30^{\circ}$ Celsius, we look up the Celsius-axis and draw the line $y=30$ along to the function. When this horizontal line hits the function, we draw a line straight down to the Fahrenheit-axis. The line hits the axis at approximately 85.
## $30^{\circ}$ Celsius is approximately equivalent to $85^{\circ}$ Fahrenheit.
d) To find $0^{\circ}$ Celsius, we look at the Fahrenheit-axis (the line $y=0$ ). The function hits the $x$-axis just right of 30 .
## $0^{\circ}$ Celsius is equivalent to $32^{\circ}$ Fahrenheit.
## Example 5
Here is an example of a graph you might see reported in the news. Most mainstream scientists believe that increased emissions of greenhouse cases, particularly carbon dioxide, are contributing to the warming of the planet. The graph below illustrates how carbon dioxide levels have increased as the world has industrialized.
## Global concentration of Co2 in the atmosphere Parts per million (ppm)
From this graph, we can find the concentration of carbon dioxide found in the atmosphere in different years.
1900 - 285 parts per million
1930 - 300 parts per million
1950 - 310 parts per million
1990 - 350 parts per million
## Practice Set
1. The students at a local high school took The Youth Risk Behavior Survey. The graph below shows the percentage of high school students who reported that they were current smokers. A person qualifies as a current smoker if he/she has smoked one or more cigarettes in the past 30 days. What percentage of high-school students were current smokers in the following years?
2. 1991
3. 1996
4. 2004
5. 2005
2. The graph below shows the average life-span of people based on the year in which they were born. This information comes from the National Vital Statistics Report from the Center for Disease Control. What is the average life-span of a person born in the following years?
3. 1940
4. 1955
5. 1980
6. 1995
3. The graph below shows the median income of an individual based on his/her number of years of education. The top curve shows the median income for males and the bottom curve shows the median income for females. (Source: US Census, 2003.) What is the median income of a male who has the following years of education?
4. 10 years of education
5. 17 years of education
6. What is the median income of a female who has the same years of education?
7. 10 years of education
8. 17 years of education
4. The graph below shows a conversion chart for converting between weight in kilograms and weight in pounds. Use it to convert the following measurements.
1. (a) 4 kilograms into weight in pounds
(b) 9 kilograms into weight in pounds
(c) 12 pounds into weight in kilograms
(d) 17 pounds into weight in kilograms
5. Use the graph from problem 4 to answer the following questions.
a. An employee at a sporting goods store is packing 3-pound weights into a box that can hold 8 kilograms. How many weights can she place in the box?b. After packing those weights, there is some extra space in the box that she wants to fill with one-pound weights. How many of those can she add?c. After packing those, she realizes she misread the label and the box can actually hold 9 kilograms. How many more one-pound weights can she add?
### Graph a Linear Equation
In mathematics, we tend to use the words formula and equation to describe the rules we get when we express relationships algebraically. Interpreting and graphing these equations is an important skill that youll use frequently in math.
## Example 1
A taxi costs more the further you travel. Taxis usually charge a fee on top of the per-mile charge to cover hire of the vehicle. In this case, the taxi charges $\$ 3$ as a set fee and $\$ 0.80$ per mile traveled. Here is the equation linking the cost in dollars $(y)$ to hire a taxi and the distance traveled in miles $(x)$.
$$
y=0.8 x+3
$$
Graph the equation and use your graph to estimate the cost of a seven-mile taxi ride.
## Solution
Well start by making a table of values. We will take a few values for $x(0,1,2,3$, and 4$)$, find the corresponding $y$-values, and then plot them. Since the question asks us to find the cost for a seven-mile journey, we need to choose a scale that can accommodate this.
First, heres our table of values:
## TABLE 8.2:
$x$
0
1
2
3
4 $y$
3
3.8
4.6
5.4
6.2
To find the cost of a seven-mile journey, first we find $x=7$ on the horizontal axis and draw a line up to our graph. Next, we draw a horizontal line across to the $y$-axis and read where it hits. It appears to hit around half way between $y=8$ and $y=9$. Let's call it 8.5.
## A seven mile taxi ride would cost approximately $\$ 8.50$ (\\$8.60 exactly).
Here are some things you should notice about this graph and the formula that generated it:
- The graph is a straight line (this means that the equation is linear).
- The graph crosses the $y$-axis at $y=3$ (notice that theres $a+3$ in the equationthats not a coincidence!). This is the base cost of the taxi.
- Every time we move over by one square we move up by 0.8 squares (notice that thats also the coefficient of $x$ in the equation). This is the rate of charge of the taxi (cost per mile).
- If we move over by three squares, we move up by $3 \times 0.8$ squares.
## Example 2
A small business has a debt of $\$ 500,000$ incurred from start-up costs. It predicts that it can pay off the debt at a rate of $\$ 85,000$ per year according to the following equation governing years in business $(x)$ and debt measured in thousands of dollars $(y)$.
$$
y=-85 x+500
$$
Graph the above equation and use your graph to predict when the debt will be fully paid.
## Solution
First, we start with our table of values:
## TABLE 8.3: (continued)
| $x$ | $y$ |
| :--- | :--- |
| 2 | 330 |
| 3 | 245 |
| 4 | 160 |
Then we plot our points and draw the line that goes through them:
Notice the scale weve chosen here. Theres no need to include any points above $y=500$, but its still wise to allow a little extra.
Next we need to determine how many years it takes the debt to reach zero, or in other words, what $x$-value will make the $y$-value equal 0 . We know its greater than four (since at $x=4$ the $y$-value is still positive), so we need an $x$-scale that goes well past $x=4$. Here weve chosen to show the $x$-values from 0 to 12 , though there are many other places we could have chosen to stop.
To read the time that the debt is paid off, we simply read the point where the line hits $y=0$ (the $x$-axis). It looks as if the line hits pretty close to $x=6$. So the debt will definitely be paid off in six years.
To see more simple examples of graphing linear equations by hand, see the Khan Academy video on graphing lines at http://tinyurl.com/7m2o2ya. The narrator shows how to graph several linear equations, using a table of values to plot points and then connecting the points with a line.
## Graphs and Equations of Horizontal and Vertical Lines
## Example 3
Mad-cabs have an unusual offer going on. They are charging $\$ 7.50$ for a taxi ride of any length within the city limits. Graph the function that relates the cost of hiring the taxi $(y)$ to the length of the journey in miles $(x)$.
To proceed, the first thing we need is an equation. You can see from the problem that the cost of a journey doesnt depend on the length of the journey. It should come as no surprise that the equation then, does not have $x$ in it. Since any value of $x$ results in the same value of $y(7.5)$, the value you choose for $x$ doesnt matter, so it isnt included in the equation. Here is the equation:
$$
y=7.5
$$
The graph of this function is shown below. You can see that its simply a horizontal line.
Any time you see an equation of the form $y=$ constant, the graph is a horizontal line that intercepts the $y$-axis at the value of the constant.
Similarly, when you see an equation of the form $x=$ constant, then the graph is a vertical line that intercepts the $x$-axis at the value of the constant. (Notice that that kind of equation is a relation, and not a function, because each $x$-value (theres only one in this case) corresponds to many (actually an infinite number) $y$-values.)
## Example 4
Plot the following graphs.
(a) $y=4$
(b) $y=-4$
(c) $x=4$
(d) $x=-4$
(a) $y=4$ is a horizontal line that crosses the $y$-axis at 4 .
(b) $y=-4$ is a horizontal line that crosses the $y$-axis at 4 .
(c) $x=4$ is a vertical line that crosses the $x$-axis at 4 .
(d) $x=-4$ is a vertical line that crosses the $x$-axis at 4 .
## Example 5
Find an equation for the $x$-axis and the $y$-axis.
Look at the axes on any of the graphs from previous examples. We have already said that they intersect at the origin (the point where $x=0$ and $y=0$ ). The following definition could easily work for each axis.
$x$-axis: A horizontal line crossing the $y$-axis at zero.
$y$-axis: A vertical line crossing the $x$-axis at zero.
So using example 3 as our guide, we could define the $x$-axis as the line $y=0$ and the $y$-axis as the line $x=0$.
## Lesson Summary
- Equations with the variables $y$ and $x$ can be graphed by making a chart of values that fit the equation and then plotting the values on a coordinate plane. This graph is simply another representation of the equation and can be analyzed to solve problems.
- Horizontal lines are defined by the equation $y=$ constant and vertical lines are defined by the equation $x=$ constant.
- Be aware that although we graph the function as a line to make it easier to interpret, the function may actually be discrete.
## Practice Set
1. Make a table of values for the following equations and then graph them.
(a) $y=2 x+7$
(b) $y=0.7 x-4$
(c) $y=6-1.25 x$
2. Think of a number. Multiply it by 20, divide the answer by 9, and then subtract seven from the result. (a) Make a table of values and plot the function that represents this sentence.
(b) If you picked 0 as your starting number, what number would you end up with?
(c) To end up with 12, what number would you have to start out with?
3. Write the equations for the five lines ( $A$ through $E$ ) plotted in the graph below.
4. In the graph above, at what points do the following lines intersect?
(a) $A$ and $E$
(b) $A$ and $D$
(c) $C$ and $D$
(d) $B$ and the $y$-axis
(e) $E$ and the $x$-axis
(f) $C$ and the line $y=x$
(g) $E$ and the line $y=\frac{1}{2} x$
(h) $A$ and the line $y=x+3$
5. At the airport, you can change your money from dollars into euros. The service costs $\$ 5$, and for every additional dollar you get 0.7 euros.
(a) Make a table for this and plot the function on a graph.
(b) Use your graph to determine how many euros you would get if you give the office $\$ 50$.
(c) To get 35 euros, how many dollars would you have to pay?
(d) The exchange rate drops so that you can only get 0.5 euros per additional dollar. Now how many dollars do you have to pay for 35 euros?
### Graphing Using Intercepts
## Introduction
Sanjits office is 25 miles from home, and in traffic he expects the trip home to take him an hour if he starts at 5 PM. Today he hopes to stop at the post office along the way. If the post office is 6 miles from his office, when will Sanjit get there?
If you know just one of the points on a line, youll find that isnt enough information to plot the line on a graph. As you can see in the graph above, there are many linesin fact, infinitely many linesthat pass through a single point. But what if you know two points that are both on the line? Then theres only one way to graph that line; all you need to do is plot the two points and use a ruler to draw the line that passes through both of them.
There are a lot of options for choosing which two points on the line you use to plot it. In this lesson, well focus on two points that are rather convenient for graphing: the points where our line crosses the $x$-and $y$-axes, or intercepts. Well see how to find intercepts algebraically and use them to quickly plot graphs.
Look at the graph above. The $y$-intercept occurs at the point where the graph crosses the $y$-axis. The $y$-value at this point is 8 , and the $x$-value is 0 .
Similarly, the $x$-intercept occurs at the point where the graph crosses the $x$-axis. The $x$-value at this point is 6 , and the $y$-value is 0 .
So we know the coordinates of two points on the graph: $(0,8)$ and $(6,0)$. If wed just been given those two coordinates out of the blue, we could quickly plot those points and join them with a line to recreate the above graph.
Note: Not all lines will have both an $x$-and a $y$-intercept, but most do. However, horizontal lines never cross the $x$-axis and vertical lines never cross the $y$-axis.
For examples of these special cases, see the graph below.
Finding Intercepts
Example 1 Find the intercepts of the line $y=13-x$ and use them to graph the function.
## Solution
The first intercept is easy to find. The $y$-intercept occurs when $x=0$. Substituting gives us $y=13-0=13$, so the $y$-intercept is $(0,13)$.
Similarly, the $x$-intercept occurs when $y=0$. Plugging in 0 for $y$ gives us $0=13-x$, solving this equation for $\mathrm{x}$ gives us $x=13$. So $(13,0)$ is the $x$-intercept.
To draw the graph, simply plot these points and join them with a line.
## Example 2
Graph the following equations by finding intercepts.
a) $y=2 x+3$
b) $y=7-2 x$
c) $4 x-2 y=8$
d) $2 x+3 y=-6$
## Solution
a) Find the $y$-intercept by plugging in $x=0$ :
$$
y=2 \cdot 0+3=3 \quad-\text { the } y \text {-intercept is }(0,3)
$$
Find the $x$-intercept by plugging in $y=0$ :
$$
\begin{aligned}
0 & =2 x+3 & & - \text { subtract } 3 \text { from both sides : } \\
-3 & =2 x & & - \text { divide by } 2: \\
-\frac{3}{2} & =x & & - \text { the } x \text {-intercept is }(-1.5,0)
\end{aligned}
$$
b) Find the $y$-intercept by plugging in $x=0$ :
$$
y=7-2 \cdot 0=7 \quad-\text { the } y-\text { intercept is }(0,7)
$$
Find the $x$-intercept by plugging in $y=0$ :
$$
\begin{aligned}
0 & =7-2 x & & - \text { subtract } 7 \text { from both sides: } \\
-7 & =-2 x & & - \text { divide by }-2: \\
\frac{7}{2} & =x & & - \text { the } x \text { - intercept is }(3.5,0)
\end{aligned}
$$
c) Find the $y$-intercept by plugging in $x=0$ :
$$
\begin{aligned}
4 \cdot 0-2 y & =8 & & \\
-2 y & =8 & & - \text { divide by }-2 \\
y & =-4 & & - \text { the } y-\text { intercept is }(0,-4)
\end{aligned}
$$
Find the $x$-intercept by plugging in $y=0$ :
$$
\begin{aligned}
4 x-2 \cdot 0 & =8 & & \\
4 x & =8 & & - \text { divide by } 4: \\
x & =2 & & - \text { the } x \text {-intercept is }(2,0)
\end{aligned}
$$
d) Find the $y$-intercept by plugging in $x=0$ :
$$
\begin{aligned}
2 \cdot 0+3 y & =-6 & & \\
3 y & =-6 & & - \text { divide by } 3: \\
y & =-2 & & - \text { the } y-\text { intercept is }(0,-2)
\end{aligned}
$$
Find the $x$-intercept by plugging in $y=0$ :
$$
\begin{aligned}
2 x+3 \cdot 0 & =-6 & & \\
2 x & =-6 & & - \text { divide by } 2: \\
x & =-3 & & - \text { the } x \text {-intercept is }(-3,0)
\end{aligned}
$$
Finding Intercepts for General (Standard) Form Equations Using the Cover-Up Method
Look at the last two equations in the previous example. These equations are written in general form. General form equations are always written coefficient times $x$ plus (or minus) coefficient times $y$ equals value. In other words, they look like this:
$$
a x+b y=c
$$
There is a neat method for finding intercepts in standard form, often referred to as the cover-up method.
## Example 3
Find the intercepts of the following equations:
a) $7 x-3 y=21$
b) $12 x-10 y=-15$
c) $x+3 y=6$
## Solution
To solve for each intercept, we realize that at the intercepts the value of either $x$ or $y$ is zero, and so any terms that contain that variable effectively drop out of the equation. To make a term disappear, simply cover it (a finger is an excellent way to cover up terms) and solve the resulting equation.
a) To solve for the $y$-intercept we set $x=0$ and cover up the $x$-term:
$$
\text { (D) }-3 y=21
$$
$-3 y=21$
$y=-7 \quad(0,-7)$ is the $y-$ intercept.
Now we solve for the $x$-intercept:
$$
7 x-O=21
$$
$7 x=21$
$x=3 \quad(3,0)$ is the $x$-intercept.
b) To solve for the $y$-intercept $(x=0)$, cover up the $x$-term:
$$
\text { (2) }-10 y=-15
$$
$-10 y=-15$
$y=1.5 \quad(0,1.5)$ is the $y-$ intercept.
Now solve for the $x$-intercept $(y=0)$ :
$12 x=-15$
$x=-\frac{5}{4} \quad(-1.25,0)$ is the $x$-intercept.
c) To solve for the $y$-intercept $(x=0)$, cover up the $x$-term:
$$
\text { (B) } 3 y
$$
$3 y=6$
$y=2 \quad(0,2)$ is the $y$-intercept.
Solve for the $y$-intercept:
$$
x=6
$$
$x=6 \quad(6,0)$ is the $x$-intercept.
The graph of these functions and the intercepts is below:
To learn more about equations in standard form, try the Java applet at http://www.analyzemath.com/line/line.htm (scroll down and click the click here to start button.) You can use the sliders to change the values of $a, b$, and $c$ and see how that affects the graph.
## Solving Real-World Problems Using Intercepts of a Graph
## Example 4
Jessie has $\$ 30$ to spend on food for a class barbecue. Hot dogs cost \\$0.75 each (including the bun) and burgers cost $\$ 1.25$ (including the bun). Plot a graph that shows all the combinations of hot dogs and burgers he could buy for the barbecue, without spending more than $\$ 30$.
This time we will find an equation first, and then we can think logically about finding the intercepts.
If the number of burgers that Jessie buys is $x$, then the money he spends on burgers is $1.25 x$
If the number of hot dogs he buys is $y$, then the money he spends on hot dogs is $0.75 y$
So the total cost of the food is $1.25 x+0.75 y$.
The total amount of money he has to spend is $\$ 30$, so if he is to spend it ALL, we can use the following equation:
$$
1.25 x+0.75 y=30
$$
We can solve for the intercepts using the cover-up method. First the $y$-intercept
$$
0+0.75 y=30
$$
$0.75 y=30$
$y=40 \quad y$-intercept: $(0,40)$
Then the $x$-intercept $(y=0)$ :
$$
1.25 x+0=30
$$
$1.25 x=30$
$x=24 \quad x$-intercept: $(24,0)$
Now we plot those two points and join them to create our graph, shown here:
We could also have created this graph without needing to come up with an equation. We know that if John were to spend ALL the money on hot dogs, he could buy $\frac{30}{.75}=40$ hot dogs. And if he were to buy only burgers he could buy $\frac{30}{1.25}=24$ burgers. From those numbers, we can get 2 intercepts: ( 0 burgers, 40 hot dogs) and ( 24 burgers, 0 hot dogs). We could plot these just as we did above and obtain our graph that way.
As a final note, we should realize that Jesus problem is really an example of an inequality. He can, in fact, spend any amount up to $\$ 30$. The only thing he cannot do is spend more than $\$ 30$. The graph above reflects this: the line is the set of solutions that involve spending exactly $\$ 30$, and the shaded region shows solutions that involve spending less than $\$ 30$. Well work with inequalities some more in Chapter 6.
## Lesson Summary
- A $y$-intercept occurs at the point where a graph crosses the $y$-axis (where $x=0$ ) and an $x$-intercept occurs at the point where a graph crosses the $x$-axis (where $y=0$ ).
- The $y$-intercept can be found by substituting $x=0$ into the equation and solving for $y$. Likewise, the $x$-intercept can be found by substituting $y=0$ into the equation and solving for $x$.
- Equations in general form can be solved for the intercepts by covering up the $x$ (or $y$ ) term and solving the equation that remains.
## Practice Set
1. Find the intercepts for the following equations using substitution.
(a) $y=3 x-6$
(b) $y=-2 x+4$
(c) $y=14 x-21$
(d) $y=7-3 x$
(e) $y=2.5 x-4$
(f) $y=1.1 x+2.2$
(g) $y=\frac{3}{8} x+7$
(h) $y=\frac{5}{9}-\frac{2}{7} x$
2. Find the intercepts of the following equations using the cover-up method.
(a) $5 x-6 y=15$
(b) $3 x-4 y=-5$
(c) $2 x+7 y=-11$
(d) $5 x+10 y=25$
(e) $5 x-1.3 y=12$
(f) $1.4 x-3.5 y=7$
(g) $\frac{3}{5} x+2 y=\frac{2}{5}$
(h) $\frac{3}{4} x-\frac{2}{3} y=\frac{1}{5}$
3. Use any method to find the intercepts and then graph the following equations.
(a) $y=2 x+3$
(b) $6(x-1)=2(y+3)$
(c) $x-y=5$
(d) $x+y=8$
4. At the local grocery store strawberries cost $\$ 3.00$ per pound and bananas cost $\$ 1.00$ per pound.
(a) If I have $\$ 10$ to spend on strawberries and bananas, draw a graph to show what combinations of each I can buy and spend exactly $\$ 10$.
(b) Plot the point representing 3 pounds of strawberries and 2 pounds of bananas. Will that cost more or less than $\$ 10$ ?
(c) Do the same for the point representing 1 pound of strawberries and 5 pounds of bananas.
5. A movie theater charges $\$ 7.50$ for adult tickets and $\$ 4.50$ for children. If the theater takes in $\$ 900$ in ticket sales for a particular screening, draw a graph which depicts the possibilities for the number of adult tickets and the number of child tickets sold.
6. Why can't we use the intercept method to graph the following equation? $3(x+2)=2(y+3)$
7. Name two more equations that we cant use the intercept method to graph.
## CHAPTER
## Introduction to Functions
## Chapter Outline
### FUnCTIONS AND FUnCTION Notation
### Functions AS GRAPHS
### USING FUnCTION NOTATION
Joseph decided to spend the day at the Theme Park. He could buy an all day pass, but he plans to ride only a few rides, so instead, he is going to pay by ride. Each ride costs $\$ 2.00$. To describe the amount of money Joseph will spend, several mathematical concepts can be used.
First, an expression can be written to describe the relationship between the cost per ride and the number of rides, $r$. An equation can also be written if the total amount he wants to spend is known. An inequality can be used if Joseph wanted to spend less than a certain amount.
### Functions and Function Notation
Example 1: Using Joseph's situation, write the following:
a. An expression representing his total amount spent
b. An equation that shows Joseph wants to spend exactly $\$ 22.00$ on rides
c. An inequality that describes the fact that Joseph will not spend more than $\$ 26.00$ on rides
Solution: The variable in this situation is the number of rides Joseph will pay for. Call this $r$.
a. $2(r)$
b. $2(r)=22$
c. $2(r) \leq 26$
In addition to an expression, equation, or inequality, Joseph's situation can be expressed in the form of a function.
Definition: A function is a relationship between two variables such that the input value has ONLY one output value.
## Writing Equations as Functions
A function is a set of ordered pairs in which the first coordinate, the input, matches with exactly one second coordinate, the output. Equations that follow this definition can be written in function notation. The $y$ coordinate represents the dependent variable, meaning the answers of this variable depend upon what is substituted for the other variable.
Consider Joseph's equation $m=2 r$. Using function notation, the value of the equation (the money spent $m$ ), is replaced with $f(r) . f$ represents the function name and $(r)$ represents the variable. In this case the parentheses do not mean multiplication - they separate the function name from the independent variable.
Example 2: Rewrite the following equations in function notation.
a. $y=7 x-3$
b. $d=65 t$
c. $F=1.8 C+32$
Solution:
a. According to the definition of a function, $y=f(x)$, so $f(x)=7 x-3$.
b. This time the dependent variable is $d$. Function notation replaces the independent variable, so $d=f(t)=65 t$.
c. $F=f(C)=1.8 C+32$
## Why Use Function Notation?
Why is it necessary to use function notation? The necessity stems from using multiple equations. Function notation allows one to easily decipher between the equations. Suppose Joseph, Lacy, Kevin, and Alfred all went to the theme park together and chose to pay $\$ 2.00$ for each ride. Each person would have the same equation $m=2 r$. Without asking each friend, we could not tell which equation belonged to whom. By substituting function notation for the dependent variable, it is easy to tell which function belongs to whom. By using function notation, it will be much easier to graph multiple lines.
Example 3: Write functions to represent the total each friend spent at the park.
Solution: $J(r)=2 r$ represents Joseph's total, $L(r)=2 r$ represents Lacy, $K(r)=2 r$ represents Kevin, and $A(r)=2 r$ represents Alfred's total.
## Using a Function to Generate a Table
A function really is a type of equation. Therefore, a table of values can be created by choosing values to represent the independent variable. The answers to each substitution represent $f(x)$.
Use Joseph's function to generate a table of values. Because the variable represents the number of rides Joseph will pay for, negative values do not make sense and are not included in the value of the independent variable.
| | TABLE 9.1: |
| :--- | :--- |
| $r$ | $J(r)=2 r$ |
| 0 | $2(0)=0$ |
| 1 | $2(1)=2$ |
| 2 | $2(2)=4$ |
| 3 | $2(3)=6$ |
| 4 | $2(4)=8$ |
| 5 | $2(5)=10$ |
| 6 | $2(6)=12$ |
As you can see the list cannot include every possibility. A table allows for precise organization of data. It also provides an easy reference for looking up data and offers a set of coordinate points that can be plotted to create a graphical representation of the function. A table does have limitations; namely it cannot represent infinite amounts of data and does not always show the possibility of fractional values for the independent variable.
## Domain and Range of a Function
The set of all possible input values for the independent variable is called the domain. The domain can be expressed in words, as a set, or as an inequality. The values resulting from the substitution of the domain represents the range of a function.
The domain of Joseph's situation will not include negative numbers because it does not make sense to ride negative rides. He also cannot ride a fraction of a ride, so decimals and fractional values do not make sense as input values. Therefore, the values of the independent variable $r$ will be whole numbers beginning at zero.
Domain: All whole numbers
The values resulting from the substitution of whole numbers are whole numbers times two. Therefore, the range of Joseph's situation is still whole numbers just twice as large.
Range: All whole numbers Example 4: A tennis ball is bounced from a height and bounces back to $75 \%$ of its previous height. Write its function and determine its domain and range.
Solution: The function of this situation is $h(b)=0.75 b$, where $b$ represents the previous bounce height.
Domain: The previous bounce height can be any positive number, so $b \geq 0$.
Range: The new height is $75 \%$ of the previous height, and therefore will also be any positive number (decimal or whole number), so the range is all positive real numbers.
Multimedia Link For another look at the domain of a function, see the following video where the narrator solves a sample problem from the California Standards Test about finding the domain of an unusual function. KhanAcademy CA Algebra I Functions (6:34)
## Write a Function Rule
In many situations, data is collected by conducting a survey or an experiment. To visualize the data it is arranged into a table. Most often, a function rule is needed to predict additional values of the independent variable.
Example 5: Write a function rule for the table.
$\begin{array}{llllll}\text { Number of CDs } & 2 & 4 & 6 & 8 & 10 \\ \text { Cost }(\$) & 24 & 48 & 72 & 96 & 120\end{array}$
Solution: You pay $\$ 24$ for 2 CDs, $\$ 48$ for 4 CDs, $\$ 120$ for 10 CDs. That means that each CD costs $\$ 12$.
We can write the function rule.
Cost $=\$ 12 \times$ number of CDs or $f(x)=12 x$ Example 6: Write a function rule for the table.
$\begin{array}{lrrrrrrr}\text { amp; } x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \text { amp; } y & 3 & 2 & 1 & 0 & 1 & 2 & 3\end{array}$
Solution: The values of the dependent variable are always the positive outcomes of the input values. This relationship has a special name, the absolute value. The function rule looks like this: $f(x)=|x|$.
## Represent a Real-World Situation with a Function
Let's look at a real-world situation that can be represented by a function.
Example 7: Maya has an internet service that currently has a monthly access fee of $\$ 11.95$ and a connection fee of $\$ 0.50$ per hour. Represent her monthly cost as a function of connection time.
Solution: Let $x=$ the number of hours Maya spends on the internet in one month and let $y=$ Maya's monthly cost. The monthly fee is $\$ 11.95$ with an hourly charge of $\$ 0.50$.
The total cost $=$ flat fee + hourly fee $\times$ number of hours. The function is $y=f(x)=11.95+0.50 x$
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:Domain and Range ofa Function (12:52)
1. Rewrite using function notation: $y=\frac{5}{6} x-2$.
2. What is one benefit of using function notation?
3. Define domain.
4. True or false. Range is the set of all possible inputs for the independent variable.
5. Generate a table from $-5 \leq x \leq 5$ for $f(x)=-(x)^{2}-2$
6. Use the following situation for question 6: Sheri is saving for her first car. She currently has $\$ 515.85$ and is savings \\$62 each week.
(a) Write a function rule for the following situation:
(b) Can the domain be "all real numbers?" Explain your thinking.
(c) How many weeks would it take Sheri to save $\$ 1,795.00$ ? Identify the domain and range of the following functions.
7. Dustin charges $\$ 10$ per hour for mowing lawns.
8. Maria charges $\$ 25$ per hour for tutoring math, with a minimum charge of $\$ 15$.
9. $f(x)=15 x-12$
10. $f(x)=2 x^{2}+5$
11. $f(x)=\frac{1}{x}$
12. What is the range of the function $y=x^{2}-5$ when the domain is $-2,-1,0,1,2$ ?
13. What is the range of the function $y=2 x-\frac{3}{4}$ when the domain is $-2.5,1.5,5$ ?
14. Angie makes $\$ 6.50$ per hour working as a cashier at the grocery store. Make a table of values that shows her earning for input values $5,10,15,20,25,30$.
15. The area of a triangle is given by: $A=\frac{1}{2} b h$. If the height of the triangle is 8 centimeters, make a table of values that shows the area of the triangle for heights $1,2,3,4,5$, and 6 centimeters.
16. Make a table of values for the function $f(x)=\sqrt{2 x+3}$ for input values $-1,0,1,2,3,4,5$.
17. Write a function rule for the table
$\begin{array}{rrrrr}\text { amp; input } & 3 & 4 & 5 & 6 \\ \text { amp; output } & 9 & 16 & 25 & 36\end{array}$
18. Write a function rule for the table
$\begin{array}{lllll}\text { hours } & 0 & 1 & 2 & 3 \\ \text { cost } & 15 & 20 & 25 & 30\end{array}$
19. Write a function rule for the table
$\begin{array}{rrrrr}\text { amp; } \text { input } & 0 & 1 & 2 & 3 \\ \text { amp; output } & 24 & 12 & 6 & 3\end{array}$
20. Write a function that represents the number of cuts you need to cut a ribbon in $x$ number of pieces.
21. Solomon charges a $\$ 40$ flat rate and $\$ 25$ per hour to repair a leaky pipe. Write a function that represents the total fee charge as a function of hours worked. How much does Solomon earn for a 3 hour job?
22. Rochelle has invested $\$ 2500$ in a jewelry making kit. She makes bracelets that she can sell for $\$ 12.50$ each. How many bracelets does Rochelle need to make before she breaks even?
23. Make up a situation in which the domain is all real numbers but the range is all whole numbers.
### Functions as Graphs
Once a table has been created for a function, the next step is to visualize the relationship by graphing the coordinates (independent value, dependent value). In previous courses, you have learned how to plot ordered pairs on a coordinate plane. The first coordinate represents the horizontal distance from the origin (the point where the axes intersect). The second coordinate represents the vertical distance from the origin.
To graph a coordinate point such as $(4,2)$ we start at the origin.
Because the first coordinate is positive four, we move 4 units to the right.
From this location, since the second coordinate is positive two, we move 2 units up.
Example 1: Plot the following coordinate points on the Cartesian plane.
(a) $(5,3)$
(b) $(-2,6)$
(c) $(3,-4)$
(d) $(-5,-7)$ Solution: We show all the coordinate points on the same plot.
Notice that:
For a positive $x$ value we move to the right.
For a negative $x$ value we move to the left.
For a positive $y$ value we move up.
For a negative $y$ value we move down.
When referring to a coordinate plane, also called a Cartesian plane, the four sections are called quadrants. The first quadrant is the upper right section, the second quadrant is the upper left, the third quadrant is the lower left and the fourth quadrant is the lower right.
Suppose we wanted to visualize Joseph's total cost of riding at the park. Using the table below, the graph can be constructed as (number of rides, total cost) TABLE 9.2:
| $r$ | $J(r)=2 r$ |
| :--- | :--- |
| 0 | $2(0)=0$ |
| 1 | $2(1)=2$ |
| 2 | $2(2)=4$ |
| 3 | $2(3)=6$ |
| 4 | $2(4)=8$ |
| 5 | $2(5)=10$ |
| 6 | $2(6)=12$ |
The green dots represent the combination of $(r, J(r))$. The dots are not connected because the domain of this function is all whole numbers. By connecting the points we are indicating that all values between the ordered pairs are also solutions to this function. Can Joseph ride $2 \frac{1}{2}$ rides? Of course not! Therefore, we leave this situation as a scatter plot.
Example 2: Graph the function that has the following table of values.
$\begin{array}{llllll}\text { Side of the Square } & 0 & 1 & 2 & 3 & 4 \\ \text { Area of the Square } & 0 & 1 & 4 & 9 & 16\end{array}$
Solution: The table gives us five sets of coordinate points:
$(0,0),(1,1),(2,4),(3,9),(4,16)$. To graph the function, we plot all the coordinate points. Because the length of a square can be fractional values, but not negative, the domain of this function is all positive real numbers, or $x \geq 0$. This means the ordered pairs can be connected with a smooth curve. It will continue forever in the positive direction, shown by an arrow.
## Writing a Function Rule Using a Graph
In many cases you are given a graph and asked to determine its function. From a graph, you can read pairs of coordinate points that are on the curve of the function. The coordinate points give values of dependent and independent variables. These variables are related to each other by a rule. It is important we make sure this rule works for all the points on the curve.
In this course you will learn to recognize different kinds of functions. There will be specific methods that you can use for each type of function that will help you find the function rule. For now, we will look at some basic examples and find patterns that will help us figure out the relationship between the dependent and independent variables.
Example 3: The graph below shows the distance that an inchworm covers over time. Find the function rule that shows how distance and time are related to each other.
Solution: Make table of values of several coordinate points to identify a pattern.
$\begin{array}{llllllll}\text { Time } & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \text { Distance } & 0 & 1.5 & 3 & 4.5 & 6 & 7.5 & 9\end{array}$
We can see that for every second the distance increases by 1.5 feet. We can write the function rule as:
Distance $=1.5 \times$ time
The equation of the function is $f(x)=1.5 x$
## Determining Whether a Relation is a Function
You saw that a function is a relation between the independent and the dependent variables. It is a rule that uses the values of the independent variable to give the values of the dependent variable. A function rule can be expressed in words, as an equation, as a table of values and as a graph. All representations are useful and necessary in understanding the relation between the variables.
Definition: A relation is a set of ordered pairs. Mathematically, a function is a special kind of relation.
Definition: A function is a relation between two variables such that each input value has EXACTLY one output value.
This usually means that each $x$-value has only one $y$-value assigned to it. But, not all functions involve $x$ and $y$.
Consider the relation that shows the heights of all students in a class. The domain is the set of people in the class and the range is the set of heights. Each person in the class cannot be more than one height at the same time. This relation is a function because for each person there is exactly one height that belongs to him or her.
Notice that in a function, a value in the range can belong to more than one element in the domain, so more than one person in the class can have the same height. The opposite is not possible, one person cannot have multiple heights.
Example 4: Determine if the relation is a function.
a) $(1,3),(-1,-2),(3,5),(2,5),(3,4)$
b) $(-3,20),(-5,25),(-1,5),(7,12),(9,2)$
## Solution:
a) To determine whether this relation is a function, we must follow the definition of a function. Each $x$-coordinate can have ONLY one $y$-coordinate. However, since the $x$-coordinate of 3 has two $y$-coordinates, 4 and 5 , this relation is NOT a function.
b) Applying the definition of a function, each $x$-coordinate has only one $y$-coordinate. Therefore, this relation is a function.
## Determining Whether a Graph is a Function
One way to determine whether a relation is a function is to construct a flow chart linking each dependent value to its matching independent value. Suppose, however, all you are given is the graph of the relation. How can you determine whether it is a function?
You could organize the ordered pairs into a table or a flow chart, similar to the student and height situation. This could be a lengthy process, but it is one possible way. A second way is to use the Vertical Line Test. Applying this test gives a quick and effective visual to decide if the graph is a function.
Theorem: Part A) A relation is a function if there are no vertical lines that intersect the graphed relation in more than one point.
Part B) If a graphed relation does not intersect a vertical line in more than one point, then that relation is a function. Is this graphed relation a function?
By drawing a vertical line (the red line) through the graph, we can see that the vertical line intersects the circle more than once. Therefore, this graph is NOT a function.
Here is a second example:
No matter where a vertical line is drawn through the graph, there will be only one intersection. Therefore, this graph is a function.
Example 4: Determine if the relation is a function.
Solution: Using the Vertical Line Test, we can conclude the relation is a function.
For more information:
Watch this YouTube video giving step-by-step instructions of the Vertical Line Test. CK-12 Basic Algebra:Ver tical Line Test (3:11)
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:Functions as Graphs (9:34)
In $1-5$, plot the coordinate points on the Cartesian plane.
1. $(4,-4)$
2. $(2,7)$
3. $(-3,-5)$
4. $(6,3)$
5. $(-4,3)$
Using the coordinate plane below, give the coordinates for a - e.
In $7-9$, graph the relation on a coordinate plane. According to the situation, determine whether to connect the ordered pairs with a smooth curve or leave as a scatterplot. 7.
$\begin{array}{rrrrrr}X & -10 & -5 & 0 & 5 & 10 \\ Y & -3 & -0.5 & 2 & 4.5 & 7\end{array}$
## Side of cube (in inches)
0
1
2
3
4
## TABLE 9.3:
Volume of cube (in inches ${ }^{3}$ )
0
1
8
27
64
## TABLE 9.4:
## Time (in hours)
$-2$
$-1$
0
1
2 Distance (in miles)
$-50$
25
0
5
50
In 10 - 12, graph the function.
10. Brandon is a member of a movie club. He pays a $\$ 50$ annual membership and $\$ 8$ per movie.
11. $f(x)=(x-2)^{2}$
12. $f(x)=3 \cdot 2^{x}$
In $13-16$, determine if the relation is a function.
13. $(1,7),(2,7),(3,8),(4,8),(5,9)$
14. $(1,1),(1,-1),(4,2),(4,-2),(9,3),(9,-3)$
15.
$\begin{array}{llllll}\text { Age } & 20 & 25 & 25 & 30 & 35 \\ \text { Number of jobs by that age } & 3 & 4 & 7 & 4 & 2\end{array}$
16.
$\begin{array}{lrrrrr}\text { amp; } x & -4 & -3 & -2 & -1 & 0 \\ \text { amp;y } & 16 & 9 & 4 & 1 & 0\end{array}$
In 17 and 18, write a function rule for the graphed relation.
In $22-23$, determine whether the graphed relation is a function.
### Using Function Notation
Function notation allows you to easily see the input value for the independent variable inside the parentheses.
Example: Consider the function $f(x)=-\frac{1}{2} x^{2}$
Evaluate $f(4)$.
Solution: The value inside the parentheses is the value of the variable $x$. Use the Substitution Property to evaluate the function for $x=4$.
$$
\begin{aligned}
& f(4)=-\frac{1}{2}\left(4^{2}\right) \\
& f(4)=-\frac{1}{2} \cdot 16 \\
& f(4)=-8
\end{aligned}
$$
To use function notation, the equation must be written in terms of $x$. This means that the $y$-variable must be isolated on one side of the equal sign.
Example: Rewrite $9 x+3 y=6$ using function notation.
Solution: The goal is to rearrange this equation so the equation looks like $y=$. Then replace $y=$ with $f(x)=$.
$$
\begin{aligned}
9 x+3 y & =6 & & \text { Subtract } 9 x \text { from both sides. } \\
3 y & =6-9 x & & \text { Divide by } 3 . \\
y & =\frac{6-9 x}{3}=2-3 x & & \\
f(x) & =2-3 x & &
\end{aligned}
$$
## Functions as Machines
You can think of a function as a machine. You start with an input (some value), the machine performs the operations (it does the work), and your output is the answer. For example, $f(x)=3 x+2$ takes some number, $x$, multiplies it by 3 and adds two. As a machine, it would look like this:
When you use the function machine to evaluate $f(2)$, the solution is $f(2)=8$.
Example 1: A function is defined as $f(x)=6 x-36$. Determine the following:
a) $f(2)$
b) $f(p)$
Solution: a) Substitute $x=2$ into the function $f(x): f(2)=6 \cdot 2-36=12-36=-24$
b) Substitute $x=p$ into the function $f(x): f(p)=6 p+36$
## Practice Set
For each of the following functions evaluate $f(-3) ; f(7) ; f(0)$, and $f(z)$ :
1. $f(x)=-2 x+3$
2. $f(x)=0.7 x+3.2$
3. $f(x)=\frac{5(2-x)}{11}$
4. $f(t)=\frac{1}{2} t^{2}+4$
5. $f(x)=3-\frac{1}{2} x$
6. The roasting guide for a turkey suggests cooking for 100 minutes plus an additional 8 minutes per pound.
(a) Write a function for the roasting time, given the turkey weight in pounds $(x)$.
(b) Determine the time needed to roast a 10-lb turkey.
(c) Determine the time needed to roast a 27-lb turkey.
(d) Determine the maximum size turkey you could roast in $4 \frac{1}{2}$ hours.
7. $F(C)=1.8 C+32$ is the function used to convert Celsius to Fahrenheit. Find $F(100)$ and explain what it represents.
8. A prepaid phone card comes with $\$ 20$ worth of calls. Calls cost a flat rate of $\$ 0.16$ per minute. Write the value of the card as a function of minutes per calls. Use a function to determine the number of minutes of phone calls you can make with the card.
9. You can burn 330 calories during one hour of bicycling. Write this situation using $b(h)$ as the function notation. Evaluate $b(0.75)$ and explain its meaning.
10. Sadie has a bank account with a balance of $\$ 650.00$. She plans to spend $\$ 55$ per week.
(a) Write this using function notation.
(b) Evaluate her account after 10 weeks. What can you conclude?
## CHAPTER 10
Linear Functions
## Chapter Outline
10.1 Slope and RATE OF Change
10.2 GRAPHS USING SLOPE-INTERCEPT FoRM
10.3 Problem-Solving Strategies - Creating and Interpreting Graphs
## Slope and Rate of Change
## Introduction
Wheelchair ramps at building entrances must have a slope between $\frac{1}{16}$ and $\frac{1}{20}$. If the entrance to a new office building is 28 inches off the ground, how long does the wheelchair ramp need to be?
We come across many examples of slope in everyday life. For example, a slope is in the pitch of a roof, the grade or incline of a road, or the slant of a ladder leaning on a wall. In math, we use the word slope to define steepness in a particular way.
$$
\text { Slope }=\frac{\text { distance moved vertically }}{\text { distance moved horizontally }}
$$
To make it easier to remember, we often word it like this:
$$
\text { Slope }=\frac{\text { rise }}{\text { run }}
$$
In the picture above, the slope would be the ratio of the height of the hill to the horizontal length of the hill. In other words, it would be $\frac{3}{4}$, or 0.75 .
If the car were driving to the right it would climb the hill - we say this is a positive slope. Any time you see the graph of a line that goes up as you move to the right, the slope is positive.
If the car kept driving after it reached the top of the hill, it might go down the other side. If the car is driving to the right and descending, then we would say that the slope is negative.
Heres where it gets tricky: If the car turned around instead and drove back down the left side of the hill, the slope of that side would still be positive. This is because the rise would be -3 , but the run would be -4 (think of the $x$-axis - if you move from right to left you are moving in the negative $x$-direction). That means our slope ratio would be $\frac{-3}{-4}$, and the negatives cancel out to leave 0.75 , the same slope as before. In other words, the slope of a line is the same no matter which direction you travel along it.
## Find the Slope of a Line
A simple way to find a value for the slope of a line is to draw a right triangle whose hypotenuse runs along the line. Then we just need to measure the distances on the triangle that correspond to the rise (the vertical dimension) and the run (the horizontal dimension).
## Example 1
Find the slopes for the three graphs shown.
## Solution
There are already right triangles drawn for each of the lines - in future problems youll do this part yourself. Note that it is easiest to make triangles whose vertices are lattice points (i.e. points whose coordinates are all integers).
a) The rise shown in this triangle is 4 units; the run is 2 units. The slope is $\frac{4}{2}=2$.
b) The rise shown in this triangle is 4 units, and the run is also 4 units. The slope is $\frac{4}{4}=1$.
c) The rise shown in this triangle is 2 units, and the run is 4 units. The slope is $\frac{2}{4}=\frac{1}{2}$.
## Example 2
Find the slope of the line that passes through the points $(1,2)$ and $(4,7)$.
## Solution
We already know how to graph a line if were given two points: we simply plot the points and connect them with a line. Heres the graph:
Since we already have coordinates for the vertices of our right triangle, we can quickly work out that the rise is $7-2=5$ and the run is $4-1=3$ (see diagram). So the slope is $\frac{7-2}{4-1}=\frac{5}{3}$.
If you look again at the calculations for the slope, youll notice that the 7 and 2 are the $y$-coordinates of the two points and the 4 and 1 are the $x$-coordinates. This suggests a pattern we can follow to get a general formula for the slope between two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ :
Slope between $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$
or $m=\frac{\Delta y}{\Delta x}$
In the second equation the letter $m$ denotes the slope (this is a mathematical convention youll see often) and the Greek letter delta $(\Delta)$ means change. So another way to express slope is change in $y$ divided by change in $x$. In the next section, youll see that it doesnt matter which point you choose as point 1 and which you choose as point 2.
## Example 3
Find the slopes of the lines on the graph below.
## Solution
Look at the lines - they both slant down (or decrease) as we move from left to right. Both these lines have negative slope.
The lines dont pass through very many convenient lattice points, but by looking carefully you can see a few points that look to have integer coordinates. These points have been circled on the graph, and well use them to determine the slope. Well also do our calculations twice, to show that we get the same slope whichever way we choose point 1 and point 2.
For Line A:
$$
\begin{array}{lll}
\left(x_{1}, y_{1}\right)=(-6,3) & \left(x_{2}, y_{2}\right)=(5,-1) & \left(x_{1}, y_{1}\right)=(5,-1) \quad\left(x_{2}, y_{2}\right)=(-6,3) \\
\text { amp; } m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{(-1)-(3)}{(5)-(-6)}=\frac{-4}{11} \approx-0.364 & m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{(3)-(-1)}{(-6)-(5)}=\frac{4}{-11} \approx-0.364
\end{array}
$$
For Line $B$
$$
\begin{array}{ll}
\left(x_{1}, y_{1}\right)=(-4,6) \quad\left(x_{2}, y_{2}\right)=(4,-5) & \left(x_{1}, y_{1}\right)=(4,-5) \quad\left(x_{2}, y_{2}\right)=(-4,6) \\
m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{(-5)-(6)}{(4)-(-4)}=\frac{-11}{8}=-1.375 & m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{(6)-(-5)}{(-4)-(4)}=\frac{11}{-8}=-1.375
\end{array}
$$
You can see that whichever way round you pick the points, the answers are the same. Either way, Line $A$ has slope $\mathbf{- 0 . 3 6 4}$, and Line $B$ has slope $-\mathbf{1 . 3 7 5}$.
Khan Academy has a series of videos on finding the slope of a line, starting at http://tinyurl.com/7jklqx7.
Find the Slopes of Horizontal and Vertical lines
## Example 4
Determine the slopes of the two lines on the graph below.
## Solution
There are 2 lines on the graph: $A(y=3)$ and $B(x=5)$.
Lets pick 2 points on line Asay, $\left(x_{1}, y_{1}\right)=(-4,3)$ and $\left(x_{2}, y_{2}\right)=(5,3)$ and use our equation for slope:
$$
m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{(3)-(3)}{(5)-(-4)}=\frac{0}{9}=0
$$
If you think about it, this makes sense - if $y$ doesnt change as $x$ increases then there is no slope, or rather, the slope is zero. You can see that this must be true for all horizontal lines.
Horizontal lines $(y=$ constant $)$ all have a slope of 0 .
Now lets consider line $B$. If we pick the points $\left(x_{1}, y_{1}\right)=(5,-3)$ and $\left(x_{2}, y_{2}\right)=(5,4)$, our slope equation is $m=$ $\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{(4)-(-3)}{(5)-(5)}=\frac{7}{0}$. But dividing by zero isnt allowed!
In math we often say that a term which involves division by zero is undefined. (Technically, the answer can also be said to be infinitely largeor infinitely small, depending on the problem.)
Vertical lines $(x=$ constant $)$ all have an infinite (or undefined) slope.
## Find a Rate of Change
The slope of a function that describes real, measurable quantities is often called a rate of change. In that case the slope refers to a change in one quantity $(y)$ per unit change in another quantity $(x)$. (This is where the equation $m=\frac{\Delta y}{\Delta x}$ comes inremember that $\Delta y$ and $\Delta x$ represent the change in $y$ and $x$ respectively.)
## Example 5
A candle has a starting length of 10 inches. 30 minutes after lighting it, the length is 7 inches. Determine the rate of change in length of the candle as it burns. Determine how long the candle takes to completely burn to nothing.
## Solution
First well graph the function to visualize what is happening. We have 2 points to start with: we know that at the moment the candle is lit $($ time $=0)$ the length of the candle is 10 inches, and after 30 minutes $($ time $=30)$ the length is 7 inches. Since the candle length depends on the time, well plot time on the horizontal axis, and candle length on the vertical axis.
The rate of change of the candles length is simply the slope of the line. Since we have our 2 points $\left(x_{1}, y_{1}\right)=(0,10)$ and $\left(x_{2}, y_{2}\right)=(30,7)$, we can use the familiar version of the slope formula:
$$
\text { Rate of change }=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{(7 \text { inches })-(10 \text { inches })}{(30 \text { minutes })-(0 \text { minutes })}=\frac{-3 \text { inches }}{30 \text { minutes }}=-0.1 \text { inches per minute }
$$
Note that the slope is negative. A negative rate of change means that the quantity is decreasing with timejust as we would expect the length of a burning candle to do.
To find the point when the candle reaches zero length, we can simply read the $x$-intercept off the graph (100 minutes). We can use the rate equation to verify this algebraically:
$$
\begin{aligned}
\text { Length burned } & =\text { rate } \times \text { time } \\
10 & =0.1 \times 100
\end{aligned}
$$
Since the candle length was originally 10 inches, our equation confirms that 100 minutes is the time taken.
## Example 6
The population of fish in a certain lake increased from 370 to 420 over the months of March and April. At what rate is the population increasing?
## Solution
Here we dont have two points from which we can get $x$ - and $y$-coordinates for the slope formula. Instead, well need to use the alternate formula, $m=\frac{\Delta y}{\Delta x}$.
The change in $y$-values, or $\Delta y$, is the change in the number of fish, which is $420-370=50$. The change in $x$-values, $\Delta x$, is the amount of time over which this change took place: two months. So $\frac{\Delta y}{\Delta x}=\frac{50 \text { fish }}{2 \text { months }}$, or 25 fish per month.
## Interpret a Graph to Compare Rates of Change
## Example 7
The graph below represents a trip made by a large delivery truck on a particular day. During the day the truck made two deliveries, one taking an hour and the other taking two hours. Identify what is happening at each stage of the trip (stages A through E).
## Solution
Here are the stages of the trip:
a) The truck sets off and travels 80 miles in 2 hours.
b) The truck covers no distance for 2 hours.
c) The truck covers $(120-80)=40$ miles in 1 hour.
d) The truck covers no distance for 1 hour.
e) The truck covers -120 miles in 2 hours.
Lets look at each section more closely.
A. Rate of change $=\frac{\Delta y}{\Delta x}=\frac{80 \text { miles }}{2 \text { hours }}=40$ miles per hour
Notice that the rate of change is a speedor rather, a velocity. (The difference between the two is that velocity has a direction, and speed does not. In other words, velocity can be either positive or negative, with negative velocity representing travel in the opposite direction. Youll see the difference more clearly in part E.)
Since velocity equals distance divided by time, the slope (or rate of change) of a distance-time graph is always a velocity.
So during the first part of the trip, the truck travels at a constant speed of $40 \mathrm{mph}$ for 2 hours, covering a distance of 80 miles.
B. The slope here is 0 , so the rate of change is $0 \mathrm{mph}$. The truck is stationary for one hour. This is the first delivery stop.
C. Rate of change $=\frac{\Delta y}{\Delta x}=\frac{(120-80) \text { miles }}{(4-3) \text { hours }}=40$ miles per hour. The truck is traveling at $40 \mathrm{mph}$.
D. Once again the slope is 0 , so the rate of change is $0 \mathrm{mph}$. The truck is stationary for two hours. This is the second delivery stop. At this point the truck is 120 miles from the start position.
E. Rate of change $=\frac{\Delta y}{\Delta x}=\frac{(0-120) \text { miles }}{(8-6) \text { hours }}=\frac{-120 \text { miles }}{2 \text { hours }}=-60$ miles per hour. The truck is traveling at negative 60 mph.
Wait a negative speed? Does that mean that the truck is reversing? Well, probably not. Its actually the velocity and not the speed that is negative, and a negative velocity simply means that the distance from the starting position is decreasing with time. The truck is driving in the opposite direction back to where it started from. Since it no longer has 2 heavy loads, it travels faster (60 mph instead of $40 \mathrm{mph}$ ), covering the 120 mile return trip in 2 hours. Its speed is $60 \mathrm{mph}$, and its velocity is $-60 \mathrm{mph}$, because it is traveling in the opposite direction from when it started out.
## Lesson Summary
- Slope is a measure of change in the vertical direction for each step in the horizontal direction. Slope is often represented as $m$.
- Slope can be expressed as $\frac{\text { rise }}{\text { run }}$, or $\frac{\Delta y}{\Delta x}$.
- The slope between two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ is equal to $\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$.
- Horizontal lines (where $y=a$ constant) all have a slope of 0 .
- Vertical lines (where $x=a$ constant) all have an infinite (or undefined) slope.
- The slope (or rate of change) of a distance-time graph is a velocity.
## Practice Set
1. Use the slope formula to find the slope of the line that passes through each pair of points.
(a) $(-5,7)$ and $(0,0)$
(b) $(-3,-5)$ and $(3,11)$
(c) $(3,-5)$ and $(-2,9)$
(d) $(-5,7)$ and $(-5,11)$
(e) $(9,9)$ and $(-9,-9)$
(f) $(3,5)$ and $(-2,7)$
(g) $(2.5,3)$ and $(8,3.5)$
2. For each line in the graphs below, use the points indicated to determine the slope.
3. For each line in the graphs above, imagine another line with the same slope that passes through the point (1, $1)$, and name one more point on that line.
### Graphs Using Slope-Intercept Form
## Learning Objectives
- Identify the slope and $y$-intercept of equations and graphs.
- Graph an equation in slope-intercept form.
- Understand what happens when you change the slope or intercept of a line.
- Identify parallel lines from their equations.
## Introduction
The total profit of a business is described by the equation $y=15000 x-80000$, where $x$ is the number of months the business has been running. How much profit is the business making per month, and what were its start-up costs? How much profit will it have made in a year?
## Identify Slope and intercept
So far, weve been writing a lot of our equations in slope-intercept formthat is, weve been writing them in the form $y=m x+b$, where $m$ and $b$ are both constants. It just so happens that $m$ is the slope and the point $(0, b)$ is the $y$-intercept of the graph of the equation, which gives us enough information to draw the graph quickly.
## Example 1
Identify the slope and $y$-intercept of the following equations.
a) $y=3 x+2$
b) $y=0.5 x-3$
c) $y=-7 x$
d) $y=-4$
## Solution
a) Comparing
, we can see that $m=3$ and $b=2$. So $y=3 x+2$ has a slope of 3 and a $y$-intercept of $(\mathbf{0 , 2})$.
b)
has a slope of 0.5 and a $y$-intercept of $(0,-3)$. Notice that the intercept is negative. The $b$-term includes the sign of the operator (plus or minus) in front of the numberfor example, $y=0.5 x-3$ is identical to $y=0.5 x+(-3)$, and that means that $b$ is -3 , not just 3 .
c) At first glance, this equation doesnt look like its in slope-intercept form. But we can rewrite it as $y=-7 x+0$, and that means it has a slope of -7 and a $y$-intercept of $(\mathbf{0 , 0})$. Notice that the slope is negative and the line passes through the origin.
d) We can rewrite this one as $y=0 x-4$, giving us a slope of $\mathbf{0}$ and a $y$-intercept of $(\mathbf{0}, \mathbf{- 4})$. This is a horizontal line.
## Example 2
Identify the slope and y-intercept of the lines on the graph shown below.
The intercepts have been marked, as well as some convenient lattice points that the lines pass through.
## Solution
a) The $y$-intercept is $(\mathbf{0 , 5})$. The line also passes through $(2,3)$, so the slope is $\frac{\Delta y}{\Delta x}=\frac{-2}{2}=-1$.
b) The $y$-intercept is $(\mathbf{0 , 2})$. The line also passes through $(1,5)$, so the slope is $\frac{\Delta y}{\Delta x}=\frac{3}{1}=3$.
c) The $y$-intercept is $(\mathbf{0}, \mathbf{- 1})$. The line also passes through $(2,3)$, so the slope is $\frac{\Delta y}{\Delta x}=\frac{4}{2}=2$.
d) The $y$-intercept is $(\mathbf{0}, \mathbf{- 3})$. The line also passes through $(4,-4)$, so the slope is $\frac{\Delta y}{\Delta x}=\frac{-1}{4}=-\frac{1}{4}$ or -0.25 .
## Graph an Equation in Slope-Intercept Form
Once we know the slope and intercept of a line, its easy to graph it. Just remember what slope means. Let's look back at this example from a previous lesson.
Ali is trying to work out a trick that his friend showed him. His friend started by asking him to think of a number, then double it, then add five to the result. Ali has written down a rule to describe the first part of the trick. He is using the letter $x$ to stand for the number he thought of and the letter $y$ to represent the final result of applying the rule. He wrote his rule in the form of an equation: $y=2 x+5$.
Help him visualize what is going on by graphing the function that this rule describes.
In that example, we constructed a table of values, and used that table to plot some points to create our graph.
We also saw another way to graph this equation. Just by looking at the equation, we could see that the $y$-intercept was $(0,5)$, so we could start by plotting that point. Then we could also see that the slope was 2 , so we could find another point on the graph by going over 1 unit and up 2 units. The graph would then be the line between those two points.
Heres another problem where we can use the same method.
## Example 3
Graph the following function: $y=-3 x+5$
## Solution
To graph the function without making a table, follow these steps:
1. Identify the $y$-intercept: $b=5$
2. Plot the intercept: $(0,5)$
3. Identify the slope: $m=-3$. (This is equal to $\frac{-3}{1}$, so the rise is -3 and the run is 1 .)
4. Move over 1 unit and down 3 units to find another point on the line: $(1,2)$
5. Draw the line through the points $(0,5)$ and $(1,2)$.
Notice that to graph this equation based on its slope, we had to find the rise and runand it was easiest to do that when the slope was expressed as a fraction. Thats true in general: to graph a line with a particular slope, its easiest to first express the slope as a fraction in simplest form, and then read off the numerator and the denominator of the fraction to get the rise and run of the graph.
## Example 4
Find integer values for the rise and run of the following slopes, then graph lines with corresponding slopes.
a) $m=3$
b) $m=-2$
c) $m=0.75$
d) $m=-0.375$
## Solution
a)
b)
c)
d)
Changing the Slope or Intercept of a Line
The following graph shows a number of lines with different slopes, but all with the same $y$-intercept: $(0,3)$.
You can see that all the functions with positive slopes increase as we move from left to right, while all functions with negative slopes decrease as we move from left to right. Another thing to notice is that the greater the slope, the steeper the graph.
This graph shows a number of lines with the same slope, but different $y$-intercepts.
Notice that changing the intercept simply translates (shifts) the graph up or down. Take a point on the graph of $y=2 x$, such as $(1,2)$. The corresponding point on $y=2 x+3$ would be $(1,5)$. Adding 3 to the $y$-intercept means we also add 3 to every other $y$-value on the graph. Similarly, the corresponding point on the $y=2 x-3$ line would be $(1,-1)$; we would subtract 3 from the $y$-value and from every other $y$-value.
Notice also that these lines all appear to be parallel. Are they truly parallel?
To answer that question, well use a technique that youll learn more about in a later chapter. Well take 2 of the equationssay, $y=2 x$ and $y=2 x+3$ and solve for values of $x$ and $y$ that satisfy both equations. That will tell us at what point those two lines intersect, if any. (Remember that parallel lines, by definition, are lines that dont intersect.) So what values would satisfy both $y=2 x$ and $y=2 x+3$ ? Well, if both of those equations were true, then $y$ would be equal to both $2 x$ and $2 x+3$, which means those two expressions would also be equal to each other. So we can get our answer by solving the equation $2 x=2 x+3$.
But what happens when we try to solve that equation? If we subtract $2 x$ from both sides, we end up with $0=3$. That cant be true no matter what $x$ equals. And that means that there just isnt any value for $x$ that will make both of the equations we started out with true. In other words, there isnt any point where those two lines intersect. They are parallel, just as we thought.
And wed find out the same thing no matter which two lines wed chosen. In general, since changing the intercept of a line just results in shifting the graph up or down, the new line will always be parallel to the old line as long as the slope stays the same.
Any two lines with identical slopes are parallel.
## Further Practice
To get a better understanding of what happens when you change the slope or the $y$-intercept of a linear equation, try playing with the Java applet athttp://standards.nctm.org/document/eexamples/chap7/7.5/index.htm. http://tinyu rl.com/7laf6oh.
## Lesson Summary
- A common form of a line (linear equation) is slope-intercept form: $y=m x+b$, where $m$ is the slope and the point $(0, b)$ is the $y$-intercept
- Graphing a line in slope-intercept form is a matter of first plotting the $y$-intercept $(0, b)$, then finding a second point based on the slope, and using those two points to graph the line.
- Any two lines with identical slopes are parallel.
## Practice Set
1. Identify the slope and $y$-intercept for the following equations.
(a) $y=2 x+5$
(b) $y=-0.2 x+7$
(c) $y=x$
(d) $y=3.75$
2. Identify the slope of the following lines.
3. Identify the slope and $y$-intercept for the following functions.
4. Plot the following functions on a graph.
(a) $y=2 x+5$
(b) $y=-0.2 x+7$
(c) $y=x$
(d) $y=3.75$
5. Which two of the following lines are parallel?
(a) $y=2 x+5$
(b) $y=-0.2 x+7$
(c) $y=x$
(d) $y=3.75$
(e) $y=-\frac{1}{5} x-11$
(f) $y=-5 x+5$
(g) $y=-3 x+11$
(h) $y=3 x+3.5$
6. What is the $y$-intercept of the line passing through $(1,-4)$ and $(3,2)$ ?
7. What is the $y$-intercept of the line with slope -2 that passes through $(3,1)$ ?
### Problem-Solving Strategies - Creating and Interpreting Graphs
## Introduction
In this chapter, weve been solving problems where quantities are linearly related to each other. In this section, well look at a few examples of linear relationships that occur in real-world problems, and see how we can solve them using graphs. Remember back to our Problem Solving Plan:
1. Understand the Problem
2. Devise a PlanTranslate
3. Carry Out the PlanSolve
4. LookCheck and Interpret
## Example 1
A cell phone company is offering its costumers the following deal: You can buy a new cell phone for \\$60 and pay a monthly flat rate of $\$ 40$ per month for unlimited calls. How much money will this deal cost you after 9 months?
## Solution
Lets follow the problem solving plan.
Step 1: The phone costs $\$ 60$; the calling plan costs $\$ 40$ per month.
Let $x=$ number of months.
Let $y=$ total cost.
Step 2: We can solve this problem by making a graph that shows the number of months on the horizontal axis and the cost on the vertical axis.
Since you pay $\$ 60$ when you get the phone, the $y$-intercept is $(0,60)$.
You pay $\$ 40$ for each month, so the cost rises by $\$ 40$ for 1 month, so the slope is 40 .
We can use this information to graph the situation.
Step 3: The question was How much will this deal cost after 9 months? We can now read the answer from the graph. We draw a vertical line from 9 months until it meets the graph, and then draw a horizontal line until it meets the vertical axis.
We see that after 9 months you pay approximately $\$ \mathbf{\$ 2 0}$.
Step 4: To check if this is correct, lets think of the deal again.
Originally, you pay $\$ 60$ and then $\$ 40$ a month for 9 months.
$$
\begin{aligned}
\text { Phone } & =\$ 60 \\
\text { Calling plan } & =\$ 40 \times 9=\$ 360 \\
\text { Total cost } & =\$ 420 .
\end{aligned}
$$
The answer checks out.
## Example 2
A stretched spring has a length of 12 inches when a weight of $2 \mathrm{lbs}$ is attached to the spring. The same spring has a length of 18 inches when a weight of $5 \mathrm{lbs}$ is attached to the spring. What is the length of the spring when no weights are attached?
## Solution
Step 1: We know: the length of the spring $=12$ inches when weight $=2 \mathrm{lbs}$
the length of the spring $=18$ inches when weight $=5 \mathrm{lbs}$
We want: the length of the spring when weight $=0 \mathrm{lbs}$
Let $x=$ the weight attached to the spring.
Let $y=$ the length of the spring.
Step 2: We can solve this problem by making a graph that shows the weight on the horizontal axis and the length of the spring on the vertical axis.
We have two points we can graph:
When the weight is $2 \mathrm{lbs}$, the length of the spring is 12 inches. This gives point $(2,12)$.
When the weight is $5 \mathrm{lbs}$, the length of the spring is 18 inches. This gives point $(5,18)$.
Graphing those two points and connecting them gives us our line.
Step 3: The question was: What is the length of the spring when no weights are attached?
We can answer this question by reading the graph we just made. When there is no weight on the spring, the $x$-value equals zero, so we are just looking for the $y$-intercept of the graph. On the graph, the $y$-intercept appears to be approximately 8 inches.
Step 4: To check if this correct, lets think of the problem again.
You can see that the length of the spring goes up by 6 inches when the weight is increased by $3 \mathrm{lbs}$.
To find the length of the spring when there is no weight attached, we can look at the spring when there are $2 \mathrm{lbs}$ attached. For each pound we take off, the spring will shorten by 2 inches. If we take off $2 \mathrm{lbs}$, the spring will be shorter by 4 inches. So, the length of the spring with no weights is 12 inches -4 inches $=8$ inches.
## The answer checks out.
Example 3 Christine took 1 hour to read 22 pages of Harry Potter. She has 100 pages left to read in order to finish the book. How much time should she expect to spend reading in order to finish the book?
## Solution
Step 1: We know - Christine takes 1 hour to read 22 pages.
We want - How much time it takes to read 100 pages.
Let $x=$ the time expressed in hours.
Let $y=$ the number of pages.
Step 2: We can solve this problem by making a graph that shows the number of hours spent reading on the horizontal axis and the number of pages on the vertical axis.
We have two points we can graph:
Christine takes 1 hour to read 22 pages. This gives point $(1,22)$.
A second point is not given, but we know that Christine would take 0 hours to read 0 pages. This gives point $(0,0)$. Graphing those two points and connecting them gives us our line.
Number of Pages Read by Time
Step 3: The question was: How much time should Christine expect to spend reading 100 pages? We can find the answer from reading the graph - we draw a horizontal line from 100 pages until it meets the graph and then we draw the vertical until it meets the horizontal axis. We see that it takes approximately 4.5 hours to read the remaining 100 pages.
Step 4: To check if this correct, lets think of the problem again.
We know that Christine reads 22 pages per hour - this is the rate at which she is reading. To find how many hours it takes her to read 100 pages, we divide the number of pages by the rate. In this case, $\frac{100 \text { pages }}{22 \text { pages } / \text { hour }}=4.54$ hours. This is very close to the answer we got from reading the graph.
The answer checks out.
## Example 4
Aatif wants to buy a surfboard that costs \\$249. He was given a birthday present of $\$ 50$ and he has a summer job that pays him $\$ 6.50$ per hour. To be able to buy the surfboard, how many hours does he need to work?
## Solution
Step 1: We know - The surfboard costs $\$ 249$.
Aatif has $\$ 50$.
His job pays $\$ 6.50$ per hour.
We want - How many hours Aatif needs to work to buy the surfboard.
Let $x=$ the time expressed in hours
Let $y=$ Aatifs earnings
Step 2: We can solve this problem by making a graph that shows the number of hours spent working on the horizontal axis and Aatifs earnings on the vertical axis.
Aatif has $\$ 50$ at the beginning. This is the $y$-intercept: $(0,50)$.
He earns $\$ 6.50$ per hour.
We graph the $y$-intercept of $(0,50)$, and we know that for each unit in the horizontal direction, the line rises by 6.5 units in the vertical direction. Here is the line that describes this situation.
Step 3: The question was: How many hours does Aatif need to work to buy the surfboard?
We find the answer from reading the graph - since the surfboard costs $\$ 249$, we draw a horizontal line from $\$ 249$ on the vertical axis until it meets the graph and then we draw a vertical line downwards until it meets the horizontal axis. We see that it takes approximately 31 hours to earn the money.
Step 4: To check if this correct, lets think of the problem again.
We know that Aatif has $\$ 50$ and needs $\$ 249$ to buy the surfboard. So, he needs to earn $\$ 249-\$ 50=\$ 199$ from his job.
His job pays $\$ 6.50$ per hour. To find how many hours he need to work we divide: $\frac{\$ 199}{\$ 6.50 / \text { hour }}=30.6$ hours. This is very close to the answer we got from reading the graph.
The answer checks out.
## Lesson Summary
The four steps of the problem solving plan when using graphs are:
## Understand the Problem
2. Devise a PlanTranslate: Make a graph.
3. Carry Out the PlanSolve: Use the graph to answer the question asked.
## LookCheck and Interpret
## Practice Set
Solve the following problems by making a graph and reading it.
1. A gym is offering a deal to new members. Customers can sign up by paying a registration fee of $\$ 200$ and a monthly fee of $\$ 39$.
(a) How much will this membership cost a member by the end of the year?
(b) The old membership rate was $\$ 49$ a month with a registration fee of $\$ 100$. How much more would a years membership cost at that rate?
(c) Bonus: For what number of months would the two membership rates be the same?
2. A candle is burning at a linear rate. The candle measures five inches two minutes after it was lit. It measures three inches eight minutes after it was lit.
(a) What was the original length of the candle?
(b) How long will it take to burn down to a half-inch stub?
(c) Six half-inch stubs of candle can be melted together to make a new candle measuring $2 \frac{5}{6}$ inches (a little wax gets lost in the process). How many stubs would it take to make three candles the size of the original candle?
3. A dipped candle is made by taking a wick and dipping it repeatedly in melted wax. The candle gets a little bit thicker with each added layer of wax. After it has been dipped three times, the candle is $6.5 \mathrm{~mm}$ thick. After it has been dipped six times, it is $11 \mathrm{~mm}$ thick.
(a) How thick is the wick before the wax is added?
(b) How many times does the wick need to be dipped to create a candle $2 \mathrm{~cm}$ thick?
4. Tali is trying to find the thickness of a page of his telephone book. In order to do this, he takes a measurement and finds out that 55 pages measures $\frac{1}{8}$ inch. What is the thickness of one page of the phone book?
5. Bobby and Petra are running a lemonade stand and they charge 45 cents for each glass of lemonade. In order to break even they must make $\$ 25$.
(a) How many glasses of lemonade must they sell to break even?
(b) When theyve sold $\$ 18$ worth of lemonade, they realize that they only have enough lemons left to make 10 more glasses. To break even now, theyll need to sell those last 10 glasses at a higher price. What does the new price need to be?
6. Dale is making cookies using a recipe that calls for 2.5 cups of flour for two dozen cookies. How many cups of flour does he need to make five dozen cookies?
7. To buy a car, Jason makes a down payment of $\$ 1500$ and pays $\$ 350$ per month in installments.
(a) How much money has Jason paid at the end of one year?
(b) If the total cost of the car is $\$ 8500$, how long will it take Jason to finish paying it off?
(c) The resale value of the car decreases by $\$ 100$ each month from the original purchase price. If Jason sells the car as soon as he finishes paying it off, how much will he get for it? 8. Anne transplants a rose seedling in her garden. She wants to track the growth of the rose so she measures its height every week. On the third week, she finds that the rose is 10 inches tall and on the eleventh week she finds that the rose is 14 inches tall. Assuming the rose grows linearly with time, what was the height of the rose when Anne planted it?
9. Ravi hangs from a giant spring whose length is $5 \mathrm{~m}$. When his child Nimi hangs from the spring its length is 2 $\mathrm{m}$. Ravi weighs $160 \mathrm{lbs}$ and Nimi weighs $40 \mathrm{lbs}$. Write the equation for this problem in slope-intercept form. What should we expect the length of the spring to be when his wife Amardeep, who weighs $140 \mathrm{lbs}$, hangs from it?
10. Nadia is placing different weights on a spring and measuring the length of the stretched spring. She finds that for a 100 gram weight the length of the stretched spring is $20 \mathrm{~cm}$ and for a 300 gram weight the length of the stretched spring is $25 \mathrm{~cm}$.
(a) What is the unstretched length of the spring?
(b) If the spring can only stretch to twice its unstretched length before it breaks, how much weight can it hold?
11. Andrew is a submarine commander. He decides to surface his submarine to periscope depth. It takes him 20 minutes to get from a depth of 400 feet to a depth of 50 feet.
(a) What was the submarines depth five minutes after it started surfacing?
(b) How much longer would it take at that rate to get all the way to the surface?
12. Kierstas phone has completely run out of battery power when she puts it on the charger. Ten minutes later, when the phone is $40 \%$ recharged, Kierstas friend Danielle calls and Kiersta takes the phone off the charger to talk to her. When she hangs up 45 minutes later, her phone has $10 \%$ of its charge left. Then she gets another call from her friend Kwan.
(a) How long can she spend talking to Kwan before the battery runs out again?
(b) If she puts the phone back on the charger afterward, how long will it take to recharge completely?
13. Marji is painting a 75-foot fence. She starts applying the first coat of paint at 2 PM, and by 2:10 she has painted 30 feet of the fence. At 2:15, her husband, who paints about $\frac{2}{3}$ as fast as she does, comes to join her.
(a) How much of the fence has Marji painted when her husband joins in?
(b) When will they have painted the whole fence?
(c) How long will it take them to apply the second coat of paint if they work together the whole time?
## СHAРTER 11 More on Linear Functions
## Chapter Outline
11.1 Writing LINEAR EQUATIONS IN SLOPE-INTERCEPT FoRM
11.2 Writing LINEAR Equations in STANDARd Form
11.3 LINEAR INEQUALITIES IN TWO VARIABLES
### Writing Linear Equations in Slope- Intercept Form
Previously, you learned how to graph solutions to two-variable equations in slope-intercept form. This lesson focuses on how to write an equation for a graphed line. There are two things you will need from the graph to write the equation in slope-intercept form:
1. The $y$-intercept of the graph and
2. The slope of the line.
Having these two things will allow you to make the appropriate substitutions in the slope-intercept formula. Recall from the last chapter,
Slope-intercept form: $y=($ slope $) x+(y$-intercept $)$ or $y=m x+b$
Example 1: Write the equation for a line with a slope of 4 and a $y$-intercept $(0,-3)$.
Solution: Slope-intercept form needs two things: the slope and $y$-intercept. To write the equation, you substitute the values into the formula.
$$
\begin{aligned}
& y=(\text { slope }) x+(y-\text { intercept }) \\
& y=4 x+-3 \\
& y=4 x-3
\end{aligned}
$$
You can also use a graphed line to determine the slope and $y$-intercept.
Example 2: Use the graph below to write its equation in slope-intercept form.
Solution: The $y$-intercept is $(0,2)$. Using the slope triangle, you can determine the slope is $\frac{\text { rise }}{\text { run }}=\frac{+3}{+1}=\frac{3}{1}$. Substituting the value 2 for $b$ and the value 3 for $m$, the equation for this line is $y=3 x+2$.
## Writing an Equation Given the Slope and a Point
Sometimes it may be difficult to determine the $y$-intercept. Perhaps the $y$-intercept is rational instead of an integer. Maybe you dont know the $y$-intercept. All you have is the slope and an ordered pair. You can use this information to write the equation in slope-intercept form. To do so, you will need to follow several steps. Step 1: Begin by writing the formula for slope-intercept form $y=m x+b$.
Step 2: Substitute the given slope for $m$.
Step 3: Use the ordered pair you are given $(x, y)$, substitute these values for the variables $x$ and $y$ in the equation.
Step 4: Solve for $b$ (the $y$-intercept of the graph).
Step 5: Rewrite the original equation in step 1, substituting the slope for $m$ and the $y$-intercept for $b$.
Example 3: Write an equation for a line with slope of 4 that contains the ordered pair $(-1,5)$.
Solution:
Step 1: Begin by writing the formula for slope-intercept form.
$$
y=m x+b
$$
Step 2: Substitute the given slope for $m$.
$$
y=4 x+b
$$
Step 3: Use the ordered pair you are given $(-1,5)$, substitute these values for the variables $x$ and $y$ in the equation.
$$
5=(4)(-1)+b
$$
Step 4: Solve for $b$ (the $y$-intercept of the graph).
$$
\begin{aligned}
5 & =-4+b \\
5+4 & =-4+4+b \\
9 & =b
\end{aligned}
$$
Step 5: Rewrite $y=m x+b$, substituting the slope for $m$ and the $y$-intercept for $b$
$$
y=4 x+9
$$
Example 4: Write the equation for a line with a slope of -3 containing the point $(3,-5)$.
Solution: Using the five-steps from above:
$$
\begin{aligned}
y & =(\text { slope }) x+(y-\text { intercept }) \\
y & =-3 x+b \\
-5 & =-3(3)+b \\
-5 & =-9+b \\
4 & =b \\
y & =-3 x+4
\end{aligned}
$$
## Writing an Equation Given Two Points
In many cases, especially real-world situations, you are not given the slope nor are you given the $y$-intercept. You might only have two points to use to determine the equation of the line.
To find an equation for a line between two points, you need two things:
1. The $y$-intercept of the graph; and
2. The slope of the line.
Previously, you learned how to determine the slope between two points. Lets repeat the formula here:
The slope between any two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ is: slope $=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$
The procedure for determining a line given two points is the same five-step process as writing an equation given the slope and a point.
Example 5: Write the equation for the line containing the points $(3,2)$ and $(-2,4)$.
Solution: You need the slope of the line. Find the line's slope by using the formula. Choose one ordered pair to represent $\left(x_{1}, y_{1}\right)$ and the other ordered pair to represent $\left(x_{2}, y_{2}\right)$.
$$
\text { slope }=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{4-2}{-2-3}=\frac{-2}{5}=-\frac{2}{5}
$$
Now use the five-step process to find the equation for this line.
Step 1: Begin by writing the formula for slope-intercept form.
$$
y=m x+b
$$
Step 2: Substitute the given slope for $m$.
$$
y=-\frac{2}{5} x+b
$$
Step 3: Use one of the ordered pairs you are given $(-2,4)$, substitute these values for the variables $x$ and $y$ in the equation.
$$
4=\left(-\frac{2}{5}\right)(-2)+b
$$
Step 4: Solve for $b$ (the $y$-intercept of the graph).
$$
\begin{aligned}
4 & =\frac{4}{5}+b \\
4-\frac{4}{5} & =\frac{4}{5}-\frac{4}{5}+b \\
\frac{16}{5} & =b
\end{aligned}
$$
Step 5: Rewrite $y=m x+b$, substituting the slope for $m$ and the $y$-intercept for $b$.
$$
y=-\frac{2}{5} x+\frac{16}{5}
$$
Example 6: Write the equation for a line containing the points $(-4,1)$ and $(-2,3)$.
## Solution:
1. Start with the slopeintercept form of the line $y=m x+b$.
2. Find the slope of the line. $m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{3-1}{-2-(-4)}=\frac{2}{2}=1$
3. Substitute the value of slope for $m: y=(1) x+b$
4. Substitute the coordinate $(-2,3)$ into the equation for the variables $x$ and $y: 3=-2+b \Rightarrow b=5$
5. Rewrite the equation, substituting the slope for $m$ and the $y$-intercept for $b . y=x+5$
## Writing a Function in Slope-Intercept Form
Remember that a linear function has the form $f(x)=m x+b$. Here $f(x)$ represents the $y$ values of the equation or the graph. So $y=f(x)$ and they are often used interchangeably. Using the functional notation in an equation often provides you with more information.
For instance, the expression $f(x)=m x+b$ shows clearly that $x$ is the independent variable because you substitute values of $x$ into the function and perform a series of operations on the value of $x$ in order to calculate the values of the dependent variable, $y$.
In this case when you substitute $x$ into the function, the function tells you to multiply it by $m$ and then add $b$ to the result. This process generates all the values of $y$ you need.
Example 7: Consider the function $f(x)=3 x-4$. Find $f(2), f(0)$, and $f(-1)$.
Solution: Each number in parentheses is a value of $x$ that you need to substitute into the equation of the function.
$$
f(2)=2 ; f(0)=-4 ; \text { and } f(-1)=-7
$$
Function notation tells you much more than the value of the independent variable. It also indicates a point on the graph. For example, in the above example, $f(-1)=-7$. This means the ordered pair $(-1,-7)$ is a solution to $f(x)=3 x-4$ and appears on the graphed line. You can use this information to write an equation for a function.
Example 8: Write an equation for a line with $m=3.5$ and $f(-2)=1$.
Solution: You know the slope and you know a point on the graph $(-2,1)$. Using the methods presented in this lesson, write the equation for the line.
Begin with slope-intercept form Substitute the value for the slope.
Use the ordered pair to solve for $b$.
Rewrite the equation.
or
$$
\begin{aligned}
y & =m x+b \\
y & =3.5 x+b \\
1 & =3.5(-2)+b \\
b & =8 \\
y & =3.5 x+8 \\
f(x) & =3.5 x+8
\end{aligned}
$$
## Solve Real-World Problems Using Linear Models
Lets apply the methods we just learned to a few application problems that can be modeled using a linear relationship.
Example 9: Nadia has $\$ 200$ in her savings account. She gets a job that pays $\$ 7.50$ per hour and she deposits all her earnings in her savings account. Write the equation describing this problem in slopeintercept form. How many hours would Nadia need to work to have $\$ 500$ in her account?
Solution: Begin by defining the variables:
$y=$ amount of money in Nadias savings account
$x=$ number of hours
The problem gives the $y$-intercept and the slope of the equation.
We are told that Nadia has $\$ 200$ in her savings account, so $b=200$.
We are told that Nadia has a job that pays $\$ 7.50$ per hour, so $m=7.50$.
By substituting these values in slopeintercept form $y=m x+b$, we obtain $y=7.5 x+200$.
To answer the question, substitute $\$ 500$ for the value of $y$ and solve.
$$
500=7.5 x+200 \Rightarrow 7.5 x=300 \Rightarrow x=40
$$
Nadia must work 40 hours if she is to have $\$ 500$ in her account.
Example 10: A stalk of bamboo of the family Phyllostachys nigra grows at steady rate of 12 inches per day and achieves its full height of 720 inches in 60 days. Write the equation describing this problem in slopeintercept form. How tall is the bamboo 12 days after it started growing?
Solution: Define the variables.
$y=$ the height of the bamboo plant in inches
$x=$ number of days The problem gives the slope of the equation and a point on the line.
The bamboo grows at a rate of 12 inches per day, so $m=12$.
We are told that the plant grows to 720 inches in 60 days, so we have the point $(60,720)$.
Start with the slope-intercept form of the line.
Substitute 12 for the slope.
Substitute the point $(60,720)$.
Substitute the value of $b$ back into the equation.
$$
\begin{aligned}
y & =m x+b \\
y & =12 x+b \\
720 & =12(60)+b \Rightarrow b=0 \\
y & =12 x
\end{aligned}
$$
To answer the question, substitute the value $x=12$ to obtain $y=12(12)=144$ inches.
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:Linear Equations inSlope InterceptForm (14:58)
1. What is the formula for slope-intercept form? What do the variables $m$ and $b$ represent?
2. What are the five steps needed to determine the equation of a line given the slope and a point on the graph (not the $y$-intercept)?
3. What is the first step of finding the equation of a line given two points?
Find the equation of the line in slopeintercept form.
4. The line has slope of 7 and $y$-intercept of -2 .
5. The line has slope of -5 and $y$-intercept of 6 .
6. The line has slope $=-2$ and $a y$-intercept $=7$.
7. The line has slope $=\frac{2}{3}$ and $a y$-intercept $=\frac{4}{5}$.
8. The line has slope of $-\frac{1}{4}$ and contains point $(4,-1)$.
9. The line has slope of $\frac{2}{3}$ and contains point $\left(\frac{1}{2}, 1\right)$.
10. The line has slope of -1 and contains point $\left(\frac{4}{5}, 0\right)$.
11. The line contains points $(2,6)$ and $(5,0)$.
12. The line contains points $(5,-2)$ and $(8,4)$. 13. The line contains points $(3,5)$ and $(-3,0)$.
13. The slope of the line is $-\frac{2}{3}$ and the line contains point $(2,-2)$.
14. The slope of the line is 3 and the line contains point $(3,-5)$.
15. The line contains points $(10,15)$ and $(12,20)$
17.
19.
20.
Find the equation of the linear function in slopeintercept form.
21. $m=5, f(0)=-3$
22. $m=-2$ and $f(0)=5$
23. $m=-7, f(2)=-1$
24. $m=\frac{1}{3}, f(-1)=\frac{2}{3}$
25. $m=4.2, f(-3)=7.1$
26. $f\left(\frac{1}{4}\right)=\frac{3}{4}, f(0)=\frac{5}{4}$
27. $f(1.5)=-3, f(-1)=2$
28. $f(-1)=1$ and $f(1)=-1$
29. To buy a car, Andrew puts a down payment of $\$ 1500$ and pays $\$ 350$ per month in installments. Write an equation describing this problem in slope-intercept form. How much money has Andrew paid at the end of one year?
30. Anne transplants a rose seedling in her garden. She wants to track the growth of the rose so she measures its height every week. On the third week, she finds that the rose is 10 inches tall and on the eleventh week she finds that the rose is 14 inches tall. Assuming the rose grows linearly with time, write an equation describing this problem in slope-intercept form. What was the height of the rose when Anne planted it?
31. Ravi hangs from a giant exercise spring whose length is $5 \mathrm{~m}$. When his child Nimi hangs from the spring its length is $2 \mathrm{~m}$. Ravi weighs $160 \mathrm{lbs}$. and Nimi weighs $40 \mathrm{lbs}$. Write the equation for this problem in slopeintercept form. What should we expect the length of the spring to be when his wife Amardeep, who weighs 140 lbs., hangs from it?
32. Petra is testing a bungee cord. She ties one end of the bungee cord to the top of a bridge and to the other end she ties different weights and measures how far the bungee stretches. She finds that for a weight of $100 \mathrm{lbs}$., the bungee stretches to 265 feet and for a weight of $120 \mathrm{lbs}$., the bungee stretches to 275 feet. Physics tells us that in a certain range of values, including the ones given here, the amount of stretch is a linear function of the weight. Write the equation describing this problem in slopeintercept form. What should we expect the stretched length of the cord to be for a weight of $150 \mathrm{lbs}$ ? As the past few lessons of this chapter have shown, there are several ways to write a linear equation. This lesson introduces another method: standard form. You have already seen examples of standard form equations in a previous lesson. For example, here are some equations written in standard form.
$$
\begin{aligned}
0.75(h)+1.25(b) & =30 \\
7 x-3 y & =21 \\
2 x+3 y & =-6
\end{aligned}
$$
The standard form of a linear equation has the form $A x+B y=C$, where $A, B$, and $C$ are integers and $A$ and $B$ are not both zero.
Example 1: Rewrite $y-5=3(x-2)$ in standard form.
Solution: Use the Distributive Property to simplify the right side of the equation
$$
y-5=3 x-6
$$
Rewrite this equation so the variables $x$ and $y$ are on the same side of the equation.
$$
\begin{aligned}
y-5+6 & =3 x-6+6 \\
y-y+1 & =3 x-y \\
1 & =3 x-y,
\end{aligned}
$$$$
\text { where } \mathrm{A}=3, \mathrm{~B}=-1 \text {, and } \mathrm{C}=1 \text {. }
$$
Example 2: Rewrite $5 x-7=y$ in standard form.
Solution: Rewrite this equation so the variables $x$ and $y$ are on the same side of the equation.
$$
\begin{aligned}
5 x-7+7 & =y+7 \\
5 x-y & =y-y+7 \\
5 x-y & =7
\end{aligned}
$$
where $\mathrm{A}=5, \mathrm{~B}=-1$, and $\mathrm{C}=7$.
## Finding Slope and Intercept of a Standard Form Equation
Slope-intercept form of a linear equation contains the slope of the equation explicitly, but the standard form does not. Since the slope is such an important feature of a line, it is useful to figure out how you would find the slope if you were given the equation of the line in standard form.
Begin with standard form: $A x+B y=C$.
If you rewrite this equation in slope-intercept form, it becomes
$$
\begin{aligned}
A x-A x+B y & =C-A x \\
\frac{B y}{B} & =\frac{-A x+C}{B} \\
y & =\frac{-A}{B} x+\frac{C}{B}
\end{aligned}
$$
When you compare this form to slope-intercept form, $y=m x+b$, you can see that the slope of a standard form equation is $\frac{-A}{B}$ and the $y$-intercept is $\frac{C}{B}$.
The standard form of a linear equation $A x+B y=C$ has the following:
slope $=\frac{-A}{B}$ and $y$-intercept $=\frac{C}{B}$.
Example 3: Find the slope and $y$-intercept of $2 x-3 y=-8$.
Solution: Using the definition of standard form, $A=2, B=-3$, and $C=-8$.
$$
\begin{array}{r}
\text { slope }=\frac{-A}{B}=\frac{-2}{-3} \rightarrow \frac{2}{3} \\
y-\text { intercept }=\frac{C}{B}=\frac{-8}{-3} \rightarrow \frac{8}{3}
\end{array}
$$
The slope is $\frac{2}{3}$ and the $y$-intercept is $\frac{8}{3}$.
## Applying Standard Form to Real-World Situations
Example 4: Nimitha buys fruit at her local farmers market. This Saturday, oranges cost $\$ 2$ per pound and cherries cost $\$ 3$ per pound. She has $\$ 12$ to spend on fruit. Write an equation in standard form that describes this situation. If she buys 4 pounds of oranges, how many pounds of cherries can she buy?
Solution: Define the variables: $x=$ pounds of oranges and $y=$ pounds of cherries
The equation that describes this situation is: $2 x+3 y=12$
If she buys 4 pounds of oranges, we substitute $x=4$ in the equation and solve for $y$. $2(4)+3 y=12 \Rightarrow 3 y=12-8 \Rightarrow 3 y=4 \Rightarrow y=\frac{4}{3}$. Nimitha can buy $1 \frac{1}{3}$ pounds of cherries.
Example 5: Jethro skateboards part of the way to school and walks for the rest of the way. He can skateboard at 7 miles per hour and he can walk at 3 miles per hour. The distance to school is 6 miles. Write an equation in standard form that describes this situation. If Jethro skateboards for $\frac{1}{2}$ an hour, how long does he need to walk to get to school?
Solution: Define the variables: $x=$ hours Jethro skateboards and $y=$ hours Jethro walks
The equation that describes this situation is $7 x+3 y=6$.
If Jethro skateboards $\frac{1}{2}$ hour, we substitute $x=0.5$ in the equation and solve for $y$.
$7(0.5)+3 y=6 \Rightarrow 3 y=6-3.5 \Rightarrow 3 y=2.5 \Rightarrow y=\frac{5}{6}$. Jethro must walk $\frac{5}{6}$ of an hour.
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:Linear Equations inStandard Form (10:08)
1. What is the standard form of a linear equation? What do $A, B$, and $C$ represent?
2. What is the meaning of clear the fractions? How would you go about doing so? 3. Consider the equation $A x+B y=C$. What are the slope and $y$-intercept of this equation?
Rewrite the following equations in standard form.
4. $y=3 x-8$
5. $y=-x-6$
6. $y=\frac{5}{3} x-4$
7. $0.30 x+0.70 y=15$
8. $5=\frac{1}{6} x-y$
9. $y-7=-5(x-12)$
10. $2 y=6 x+9$
11. $y=\frac{9}{4} x+\frac{1}{4}$
12. $y+\frac{3}{5}=\frac{2}{3}(x-2)$
13. $3 y+5=4(x-9)$
Find the slope and $y$-intercept of the following lines.
14. $5 x-2 y=15$
15. $3 x+6 y=25$
16. $x-8 y=12$
17. $3 x-7 y=20$
18. $9 x-9 y=4$
19. $6 x+y=3$
20. $x-y=9$
21. $8 x+3 y=15$
22. $4 x+9 y=1$
Write each equation in standard form.
23. The farmers market also sells tomatoes and corn. Tomatoes are selling for $\$ 1.29$ per pound and corn is selling for $\$ 3.25$ per pound. If you buy 6 pounds of tomatoes, how many pounds of corn can you buy if your total spending cash is $\$ 11.61$ ?
24. The local church is hosting a Friday night fish fry for Lent. They sell a fried fish dinner for $\$ 7.50$ and a baked fish dinner for $\$ 8.25$. The church sold 130 fried fish dinners and took in $\$ 2,336.25$. How many baked fish dinners were sold?
25. Andrew has two part time jobs. One pays $\$ 6$ per hour and the other pays $\$ 10$ per hour. He wants to make $\$ 366$ per week. Write an equation in standard form that describes this situation. If he is only allowed to work 15 hours per week at the $\$ 10$ per hour job, how many hours does he need to work per week at his $\$ 6$ per hour job in order to achieve his goal?
26. Anne invests money in two accounts. One account returns $5 \%$ annual interest and the other returns $7 \%$ annual interest. In order not to incur a tax penalty, she can make no more than $\$ 400$ in interest per year. Write an equation in standard form that describes this problem. If she invests $\$ 5000$ in the $5 \%$ interest account, how much money does she need to invest in the other account?
### Linear Inequalities in Two Variables
When a linear equation is graphed in a coordinate plane, the line splits the plane into two pieces. Each piece is called a half-plane. The diagram below shows how the half-planes are formed when graphing a linear equation.
A linear inequality in two variables can also be graphed. Instead of only graphing the boundary line $(y=m x+b)$, you must also include all the other ordered pairs that could be solutions to the inequality. This is called the solution set and is shown by shading, or coloring the half-plane that includes the appropriate solutions.
When graphing inequalities in two variables, you must be remember when the value is included $\leq$ or $\geq$ or not included $\langle$ or $\rangle$. To represent these inequalities on a coordinate plane, instead of shaded or unshaded circles, we use solid and dashed lines. We can tell which half of the plane the solution is by looking at the inequality sign.
- $>$ The solution is the half plane above the line.
- $\geq$ The solution is the half plane above the line and also all the points on the line.
- $<$ The solution is the half plane below the line.
- $\leq$ The solution is the half plane below the line and also all the points on the line.
The solution of $y>m x+b$ is the half plane above the line. The dashed line shows that the points on the line are not part of the solution.
The solution of $y \geq m x+b$ is the half plane above the line and all the points on the line.
The solution of $y<m x+b$ is the half plane below the line.
The solution of $y \leq m x+b$ is the half plane below the line and all the points on the line.
Example 1: Graph the inequality $y \geq 2 x-3$. Solution: This inequality is in a slope-intercept form. Begin by graphing the line. Then determine the half-plane to color.
- The inequality is $\geq$, so the line is solid.
- The inequality states to shade the half-plane above the boundary line.
In general, the process used to graph a linear inequality in two variables is:
Step 1: Graph the equation using the most appropriate method.
- Slope-intercept form uses the $y$-intercept and slope to find the line
- Standard form uses the intercepts to graph the line
- Point-slope uses a point and the slope to graph the line
Step 2: If the equal sign is not included draw a dashed line. Draw a solid line if the equal sign is included.
Step 3: Shade the half plane above the line if the inequality is greater than. Shade the half plane under the line if the inequality is less than.
Example: A pound of coffee blend is made by mixing two types of coffee beans. One type costs $\$ 9.00$ per pound and another type costs $\$ 7.00$ per pound. Find all the possible mixtures of weights of the two different coffee beans for which the blend costs $\$ 8.50$ per pound or less.
Solution: Begin by determining the appropriate letters to represent the varying quantities.
Let $x=$ weight of $\$ 9.00$ per pound coffee beans in pounds and let $y=$ weight of $\$ 7.00$ per pound coffee beans in pounds
Translate the information into an inequality. $9 x+7 y \leq 8.50$.
Because the inequality is in standard form, it will be easier to graph using its intercepts.
weight of $\$ 9$ coffee beans
When $x=0, y=1.21$. When $y=0, x=0.944$.
Graph the inequality. The line will be solid. We shade below the line.
We only graphed the first quadrant of the coordinate plane because neither bag should have a negative weight.
The blue-shaded region tells you all the possibilities of the two bean mixtures that will give a total less than or equal to $\$ 8.50$.
Example 2: Julian has a job as an appliance salesman. He earns a commission of $\$ 60$ for each washing machine he sells and \\$130 for each refrigerator he sells. How many washing machines and refrigerators must Julian sell in order to make $\$ 1,000$ or more in commission?
Solution: Determine the appropriate variables for the unknown quantities. Let $x=$ number of washing machines Julian sells and let $y=$ number of refrigerators Julian sells
Now translate the situation into an inequality. $60 x+130 y \geq 1000$.
Graph the standard form inequality using its intercepts. When $x=0, y=16.667$. When $y=0, x=7.692$. The line will be solid.
We want the ordered pairs that are solutions to Julian making more than $\$ 10,000$, so we shade the half-plane above the boundary line.
## Graphing Horizontal and Vertical Linear Inequalities
Linear inequalities in one variable can also be graphed in the coordinate plane. They take the form of horizontal and vertical lines, however the process is identical to graphing oblique, or slanted, lines.
Example: Graph the inequality $x>4$ on 1) a number line and 2) the coordinate plane.
Solution: Remember what the solution to $x>4$ looks like on a number line.
The solution to this inequality is the set of all real numbers $x$ that are bigger than four but not including four.
On a coordinate plane, the line $x=4$ is a vertical line four units to the right of the origin. The inequality does not equal four, so the vertical line is dashed this shows the reader the ordered pairs on the vertical line $x=4$ are not solutions to the inequality.
The inequality is looking for all $x$-coordinates larger than four. We then color the half-plane to the right, symbolizing $x>4$.
Graphing absolute value inequalities can also be done in the coordinate plane. To graph the inequality $|x| \geq 2$, we remember lesson six of this chapter and rewrite the absolute value inequality.
$x \leq-2$ or $x \geq 2$
Then graph each inequality on a coordinate plane.
In other words, the solution is all the coordinate points for which the value of $x$ is smaller than or equal to -2 and greater than or equal to 2 . The solution is represented by the plane to the left of the vertical line $x=-2$ and the plane to the right of line $x=2$.
Both vertical lines are solid because points on the line are included in the solution.
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:Graphing Inequalities (8:03)
1. Define half-plane.
2. In which cases would the boundary line be represented by a dashed line?
3. In which cases would the boundary line be represented by a solid line?
4. What is a method to help you determine which half-plane to color?
Graph each inequality in a coordinate plane. 5. $x<20$
6. $y \geq-5$
7. $y \leq 6$
8. $|x|>10$
9. $|y| \leq 7$
10. $y \leq 4 x+3$
11. $y>-\frac{x}{2}-6$
12. $y \leq-\frac{1}{2} x+5$
13. $3 x-4 y \geq 12$
14. $x+7 y<5$
15. $y<-4 x+4$
16. $y>\frac{7}{2} x+3$
17. $6 x+5 y>1$
18. $6 x-5 y \leq 15$
19. $2 x-y<5$
20. $y+5 \leq-4 x$
21. $x-\frac{1}{2} y \geq 5$
22. $y \leq-\frac{x}{3}+5$
23. $5 x-2 y>4$
24. $30 x+5 y<100$
25. $y \geq-x$
26. $6 x-y<4$
27. Lili can make yarn ankle bracelets and wrist bracelets. She has 600
28. yards of yarn available. It takes 6 yards to make one wrist bracelet and 8 yards to make one anklet. Find all the possible combinations of anklets and bracelets she can make without going over her available yarn.
29. An ounce of gold costs $\$ 670$ and an ounce of silver costs $\$ 13$. Find all possible weights of silver and gold that makes an alloy that costs less than $\$ 600$ per ounce.
30. A phone company charges 50 cents per minute during the daytime and 10 cents per minute at night. How many daytime minutes and night time minutes would you have to use to pay more than $\$ 20.00$ over a 24 -hour period?
31. Jessie has $\$ 30$ to spend on food for a class barbeque. Hot dogs cost $\$ 0.75$ each (including the bun) and burgers cost $\$ 1.25$ (including bun and salad). Plot a graph that shows all the combinations of hot dogs and burgers he could buy for the barbecue, spending less than $\$ 30.00$.
32. At the local grocery store strawberries cost $\$ 3.00$ per pound and bananas cost $\$ 1.00$ per pound. If I have $\$ 10$ to spend between strawberries and bananas, draw a graph to show what combinations of each I can buy and spend at most $\$ 10$.
## CHAPTER 12 Systems of Linear Equations
## Chapter Outline
12.1 SOLVING LINEAR SYSTEMS BY GRAPHING
12.2 SOLVING LINEAR SYSTEMS BY SUBSTITUTION
12.3 SOlVING LINEAR SYSTEMS BY Elimination
12.4 Special Types OF Linear Systems
## Solving Linear Systems by Graphing
## Introduction
A linear system of equations is a set of equations that must be solved together to find the one solution that fits them both.
In this lesson, well discover methods to determine if an ordered pair is a solution to a system of two equations. Then well learn to solve the two equations graphically and confirm that the solution is the point where the two lines intersect. Finally, well look at real-world problems that can be solved using the methods described in this chapter.
## Determine Whether an Ordered Pair is a Solution to a System of Equations
Consider this system of equations:
$$
\begin{aligned}
& y=x+2 \\
& y=-2 x+1
\end{aligned}
$$
Since the two lines are in a system, we deal with them together by graphing them on the same coordinate axes. We can use any method to graph them; lets do it by making a table of values for each line.
Line 1: $y=x+2$
## TABLE 12.1:
| $x$ | $y$ |
| :--- | :--- |
| 0 | 2 |
| 1 | 3 |
Line 2: $y=-2 x+1$
## TABLE 12.2 :
We already know that any point that lies on a line is a solution to the equation for that line. That means that any point that lies on both lines in a system is a solution to both equations.
So in this system:
- Point $A$ is not a solution to the system because it does not lie on either of the lines.
- Point $B$ is not a solution to the system because it lies only on the blue line but not on the red line.
- Point $C$ is a solution to the system because it lies on both lines at the same time.
In fact, point $C$ is the only solution to the system, because it is the only point that lies on both lines. For a system of equations, the geometrical solution is the intersection of the two lines in the system. The algebraic solution is the ordered pair that solves both equationsin other words, the coordinates of that intersection point.
You can confirm the solution by plugging it into the system of equations, and checking that the solution works in each equation.
## Example 1
Determine which of the points $(1,3),(0,2)$, or $(2,7)$ is a solution to the following system of equations:
$$
\begin{aligned}
& y=4 x-1 \\
& y=2 x+3
\end{aligned}
$$
## Solution
To check if a coordinate point is a solution to the system of equations, we plug each of the $x$ and $y$ values into the equations to see if they work.
Point $(1,3)$ :
$$
\begin{aligned}
y & =4 x-1 \\
3^{?} & =? 4(1)-1 \\
3 & =3 \text { solution checks } \\
y & =2 x+3 \\
3^{?} & =? 2(1)+3 \\
3 & \neq 5 \text { solution does not check }
\end{aligned}
$$
Point $(1,3)$ is on the line $y=4 x-1$, but it is not on the line $y=2 x+3$, so it is not a solution to the system.
Point $(0,2)$ :
$$
\begin{aligned}
y & =4 x-1 \\
2^{?} & =? 4(0)-1 \\
2 & \neq-1 \text { solution does not check }
\end{aligned}
$$
Point $(0,2)$ is not on the line $y=4 x-1$, so it is not a solution to the system. Note that it is not necessary to check the second equation because the point needs to be on both lines for it to be a solution to the system.
Point $(2,7)$ :
$$
\begin{aligned}
y & =4 x-1 \\
7^{?} & =? 4(2)-1 \\
7 & =7 \text { solution checks } \\
y & =2 x+3 \\
7^{?} & =? 2(2)+3 \\
7 & =7 \text { solution checks }
\end{aligned}
$$
Point $(2,7)$ is a solution to the system since it lies on both lines.
## The solution to the system is the point $(2,7)$.
## Determine the Solution to a Linear System by Graphing
The solution to a linear system of equations is the point, (if there is one) that lies on both lines. In other words, the solution is the point where the two lines intersect.
We can solve a system of equations by graphing the lines on the same coordinate plane and reading the intersection point from the graph.
This method most often offers only approximate solutions, so its not sufficient when you need an exact answer. However, graphing the system of equations can be a good way to get a sense of whats really going on in the problem youre trying to solve, especially when its a real-world problem.
## Example 2
Solve the following system of equations by graphing:
$$
\begin{aligned}
& y=3 x-5 \\
& y=-2 x+5
\end{aligned}
$$
## Solution
Graph both lines on the same coordinate axis using any method you like.
In this case, lets make a table of values for each line.
Line 1: $y=3 x-5$
## TABLE 12.3:
| $x$ | $y$ |
| :--- | :--- |
| 1 | -2 |
| 2 | 1 |
Line 2: $y=-2 x+5$
## TABLE 12.4:
| $x$ | $y$ |
| :--- | :--- |
| 1 | 3 |
| 2 | 1 |
The solution to the system is given by the intersection point of the two lines. The graph shows that the lines intersect at point $(2,1)$. So the solution is $x=2, y=1$ or $(2,1)$.
## Example 3
Solve the following system of equations by graphing:
$$
\begin{aligned}
2 x+3 y & =6 \\
4 x-y & =-2
\end{aligned}
$$
## Solution
Since the equations are in standard form, this time well graph them by finding the $x$ - and $y$-intercepts of each of the lines.
Line 1: $2 x+3 y=6$
$x$-intercept: set $y=0 \Rightarrow 2 x=6 \Rightarrow x=3$ so the intercept is $(3,0)$
$y$-intercept: set $x=0 \Rightarrow 3 y=6 \Rightarrow y=2$ so the intercept is $(0,2)$
Line 2: $-4 x+y=2$
$x$-intercept: set $y=0 \Rightarrow-4 x=2 \Rightarrow x=-\frac{1}{2}$ so the intercept is $\left(-\frac{1}{2}, 0\right)$
$y$-intercept: set $x=0 \Rightarrow y=2$ so the intercept is $(0,2)$
The graph shows that the lines intersect at $(0,2)$. Therefore, the solution to the system of equations is $x=0, y=2$.
## Solving a System of Equations Using a Graphing Calculator
As an alternative to graphing by hand, you can use a graphing calculator to find or check solutions to a system of equations.
## Example 4
Solve the following system of equations using a graphing calculator.
$$
\begin{array}{r}
x-3 y=4 \\
2 x+5 y=8
\end{array}
$$
To input the equations into the calculator, you need to rewrite them in slope-intercept form (that is, $y=m x+b$ form).
$x-3 y=4$
$\Rightarrow \quad y=\frac{1}{3} x-\frac{4}{3}$
$2 x+5 y=8$
$y=-\frac{2}{5} x+\frac{8}{5}$
Press the $[\mathbf{y}=]$ button on the graphing calculator and enter the two functions as:
$$
\begin{aligned}
& Y_{1}=\frac{x}{3}-\frac{4}{3} \\
& Y_{2}=\frac{-2 x}{5}+\frac{8}{5}
\end{aligned}
$$
Now press [GRAPH]. Heres what the graph should look like on a TI-83 family graphing calculator with the window set to $-5 \leq x \leq 10$ and $-5 \leq y \leq 5$.
There are a few different ways to find the intersection point.
Option 1: Use [TRACE] and move the cursor with the arrows until it is on top of the intersection point. The values of the coordinate point will be shown on the bottom of the screen. The second screen above shows the values to be $X=4.0957447$ and $Y=0.03191489$. Use the [ZOOM] function to zoom into the intersection point and find a more accurate result. The third screen above shows the system of equations after zooming in several times. A more accurate solution appears to be $X=4$ and $Y=0$.
Option 2 Look at the table of values by pressing [2nd] [GRAPH]. The first screen below shows a table of values for this system of equations. Scroll down until the $Y$-values for the two functions are the same. In this case this occurs at $X=4$ and $Y=0$.
(Use the [TBLSET] function to change the starting value for your table of values so that it is close to the intersection point and you dont have to scroll too long. You can also improve the accuracy of the solution by setting the value of $\Delta$ Table smaller.)
Option 3 Using the [2nd] [TRACE] function gives the second screen shown above.
Scroll down and select intersect.
The calculator will display the graph with the question [FIRSTCURVE]? Move the cursor along the first curve until it is close to the intersection and press [ENTER].
The calculator now shows [SECONDCURVE]?
Move the cursor to the second line (if necessary) and press [ENTER].
The calculator displays [GUESS]?
Press [ENTER] and the calculator displays the solution at the bottom of the screen (see the third screen above).
The point of intersection is $X=4$ and $Y=0$. Note that with this method, the calculator works out the intersection point for you, which is generally more accurate than your own visual estimate.
## Solve Real-World Problems Using Graphs of Linear Systems
Consider the following problem:
Peter and Nadia like to race each other. Peter can run at a speed of 5 feet per second and Nadia can run at a speed of 6 feet per second. To be a good sport, Nadia likes to give Peter a head start of 20 feet. How long does Nadia take to catch up with Peter? At what distance from the start does Nadia catch up with Peter?
Lets start by drawing a sketch. Heres what the race looks like when Nadia starts running; well call this time $t=0$.
Now lets define two variables in this problem:
$t=$ the time from when Nadia starts running
$d=$ the distance of the runners from the starting point.
Since there are two runners, we need to write equations for each of them. That will be the system of equations for this problem.
For each equation, we use the formula: distance $=$ speed $\times$ time Nadias equation: $d=6 t$
Peters equation: $d=5 t+20$
(Remember that Peter was already 20 feet from the starting point when Nadia started running.)
Lets graph these two equations on the same coordinate axes.
Time should be on the horizontal axis since it is the independent variable. Distance should be on the vertical axis since it is the dependent variable.
We can use any method for graphing the lines, but in this case well use the slopeintercept method since it makes more sense physically.
To graph the line that describes Nadias run, start by graphing the $y$-intercept: $(0,0)$. (If you dont see that this is the $y$-intercept, try plugging in the test-value of $x=0$.)
The slope tells us that Nadia runs 6 feet every one second, so another point on the line is $(1,6)$. Connecting these points gives us Nadias line:
To graph the line that describes Peters run, again start with the $y$-intercept. In this case this is the point $(0,20)$.
The slope tells us that Peter runs 5 feet every one second, so another point on the line is $(1,25)$. Connecting these points gives us Peters line:
Time (seconds)
In order to find when and where Nadia and Peter meet, well graph both lines on the same graph and extend the lines until they cross. The crossing point is the solution to this problem.
The graph shows that Nadia and Peter meet 20 seconds after Nadia starts running, and 120 feet from the starting point.
These examples are great at demonstrating that the solution to a system of linear equations means the point at which the lines intersect. This is, in fact, the greatest strength of the graphing method because it offers a very visual representation of system of equations and its solution. You can also see, though, that finding the solution from a graph requires very careful graphing of the lines, and is really only practical when youre sure that the solution gives integer values for $x$ and $y$. Usually, this method can only offer approximate solutions to systems of equations, so we need to use other methods to get an exact solution.
## Practice Set
Determine which ordered pair satisfies the system of linear equations.
1.
$$
\begin{aligned}
& y=3 x-2 \\
& y=-x
\end{aligned}
$$
(a) $(1,4)$
(b) $(2,9)$
(c) $\left(\frac{1}{2}, \frac{-1}{2}\right)$ 2.
$$
\begin{aligned}
& y=2 x-3 \\
& y=x+5
\end{aligned}
$$
(a) $(8,13)$
(b) $(-7,6)$
(c) $(0,4)$
3.
$$
\begin{gathered}
2 x+y=8 \\
5 x+2 y=10
\end{gathered}
$$
(a) $(-9,1)$
(b) $(-6,20)$
(c) $(14,2)$
4.
$$
\begin{aligned}
3 x+2 y & =6 \\
y & =\frac{1}{2} x-3
\end{aligned}
$$
(a) $\left(3, \frac{-3}{2}\right)$
(b) $(-4,3)$
(c) $\left(\frac{1}{2}, 4\right)$
5.
$$
\begin{aligned}
& 2 x-y=10 \\
& 3 x+y=-5
\end{aligned}
$$
(a) $(4,-2)$
(b) $(1,-8)$
(c) $(-2,5)$
Solve the following systems using the graphing method.
6.
$$
\begin{aligned}
& y=x+3 \\
& y=-x+3
\end{aligned}
$$
7.
$$
\begin{aligned}
& y=3 x-6 \\
& y=-x+6
\end{aligned}
$$
8.
$$
\begin{aligned}
2 x & =4 \\
y & =-3
\end{aligned}
$$
9.
$$
\begin{aligned}
y & =-x+5 \\
-x+y & =1
\end{aligned}
$$
10.
$$
\begin{array}{r}
x+2 y=8 \\
5 x+2 y=0
\end{array}
$$
11.
$$
\begin{gathered}
3 x+2 y=12 \\
4 x-y=5
\end{gathered}
$$
12. $5 x+2 y=-4$
$x-y=2$
13. $2 x+4=3 y$
$x-2 y+4=0$
14.
$$
\begin{aligned}
y & =\frac{1}{2} x-3 \\
2 x-5 y & =5
\end{aligned}
$$
15.
$$
\begin{aligned}
& y=4 \\
& x=8-3 y
\end{aligned}
$$
16. Try to solve the following system using the graphing method:
$y=\frac{3}{5} x+5$
$y=-2 x-\frac{1}{2}$.
(a) What does it look like the $x$-coordinate of the solution should be?
(b) Does that coordinate really give the same $y$-value when you plug it into both equations?
(c) Why is it difficult to find the real solution to this system?
17. Try to solve the following system using the graphing
method:
$y=4 x+8$
$y=5 x+1$.
Use a grid with $x$-values and $y$-values ranging from -10 to 10 .
(a) Do these lines appear to intersect?
(b) Based on their equations, are they parallel?
(c) What would we have to do to find their intersection point?
18. Try to solve the following system using the graphing method:
$y=\frac{1}{2} x+4$
$y=\frac{4}{9} x+\frac{9}{2}$.
Use the same grid as before.
(a) Can you tell exactly where the lines cross? (b) What would we have to do to make it clearer?
Solve the following problems by using the graphing method.
19. Marys car has broken down and it will cost her $\$ 1200$ to get it fixedor, for $\$ 4500$, she can buy a new, more efficient car instead. Her present car uses about $\$ 2000$ worth of gas per year, while gas for the new car would cost about $\$ 1500$ per year. After how many years would the total cost of fixing the car equal the total cost of replacing it?
20. Juan is considering two cell phone plans. The first company charges $\$ 120$ for the phone and $\$ 30$ per month for the calling plan that Juan wants. The second company charges $\$ 40$ for the same phone but charges $\$ 45$ per month for the calling plan that Juan wants. After how many months would the total cost of the two plans be the same?
21. A tortoise and hare decide to race 30 feet. The hare, being much faster, decides to give the tortoise a 20 foot head start. The tortoise runs at $0.5 \mathrm{feet} / \mathrm{sec}$ and the hare runs at 5.5 feet per second. How long until the hare catches the tortoise?
### Solving Linear Systems by Substitution
## Introduction
In this lesson, well learn to solve a system of two equations using the method of substitution.
## Solving Linear Systems Using Substitution of Variable Expressions
Lets look again at the problem about Peter and Nadia racing.
Peter and Nadia like to race each other. Peter can run at a speed of 5 feet per second and Nadia can run at a speed of 6 feet per second. To be a good sport, Nadia likes to give Peter a head start of 20 feet. How long does Nadia take to catch up with Peter? At what distance from the start does Nadia catch up with Peter?
In that example we came up with two equations:
Nadias equation: $d=6 t$
Peters equation: $d=5 t+20$
Each equation produced its own line on a graph, and to solve the system we found the point at which the lines intersectedthe point where the values for $d$ and $t$ satisfied both relationships. When the values for $d$ and $t$ are equal, that means that Peter and Nadia are at the same place at the same time.
But theres a faster way than graphing to solve this system of equations. Since we want the value of $d$ to be the same in both equations, we could just set the two right-hand sides of the equations equal to each other to solve for $t$. That is, if $d=6 t$ and $d=5 t+20$, and the two $d$ s are equal to each other, then by the transitive property we have $6 t=5 t+20$. We can solve this for $t$ :
$$
\begin{aligned}
6 t & =5 t+20 \\
t & =20 \\
d & =6 \cdot 20=120
\end{aligned}
$$
subtract 5 t from both sides:
substitute this value for $t$ into Nadia's equation:
Even if the equations werent so obvious, we could use simple algebraic manipulation to find an expression for one variable in terms of the other. If we rearrange Peters equation to isolate $t$ :
$$
\begin{aligned}
d & =5 t+20 \\
d-20 & =5 t \\
\frac{d-20}{5} & =t
\end{aligned}
$$
We can now substitute this expression for $t$ into Nadias equation $(d=6 t)$ to solve:
$$
\begin{aligned}
d & =6\left(\frac{d-20}{5}\right) & & \text { multiply both sides by } 5: \\
5 d & =6(d-20) & & \text { distribute the } 6: \\
5 d & =6 d-120 & & \text { subtract } 6 d \text { from both sides : } \\
-d & =-120 & & \text { divide by }-1: \\
d & =120 & & \text { substitute value for } d \text { into our expression for } t: \\
t & =\frac{120-20}{5}=\frac{100}{5}=20 & &
\end{aligned}
$$
So we find that Nadia and Peter meet 20 seconds after they start racing, at a distance of 120 feet away.
The method we just used is called the Substitution Method. In this lesson youll learn several techniques for isolating variables in a system of equations, and for using those expressions to solve systems of equations that describe situations like this one.
## Example 1
Lets look at an example where the equations are written in standard form.
Solve the system
$$
\begin{array}{r}
2 x+3 y=6 \\
-4 x+y=2
\end{array}
$$
Again, we start by looking to isolate one variable in either equation. If you look at the second equation, you should see that the coefficient of $y$ is 1 . So the easiest way to start is to use this equation to solve for $y$.
Solve the second equation for $y$ :
$$
\begin{aligned}
-4 x+y & =2 \\
y & =2+4 x
\end{aligned}
$$
add $4 x$ to both sides :
Substitute this expression into the first equation:
$$
\begin{aligned}
2 x+3(2+4 x) & =6 & & \text { distribute the } 3: \\
2 x+6+12 x & =6 & & \text { collect like terms : } \\
14 x+6 & =6 & & \text { subtract } 6 \text { from both sides : } \\
14 x & =0 & & \text { and hence }: \\
x & =0 & &
\end{aligned}
$$
Substitute back into our expression for $y$ :
$$
y=2+4 \cdot 0=2
$$
As you can see, we end up with the same solution $(x=0, y=2)$ that we found when we graphed these functions back in a previous lesson. So long as you are careful with the algebra, the substitution method can be a very efficient way to solve systems. Next, lets look at a more complicated example. Here, the values of $x$ and $y$ we end up with arent whole numbers, so they would be difficult to read off a graph!
## Example 2
Solve the system
$$
\begin{aligned}
& 2 x+3 y=3 \\
& 2 x-3 y=-1
\end{aligned}
$$
Again, we start by looking to isolate one variable in either equation. In this case it doesnt matter which equation we useall the variables look about equally easy to solve for.
So lets solve the first equation for $x$ :
$$
\begin{aligned}
2 x+3 y & =3 & & \text { subtract } 3 y \text { from both sides : } \\
2 x & =3-3 y & & \text { divide both sides by } 2: \\
x & =\frac{1}{2}(3-3 y) & &
\end{aligned}
$$
Substitute this expression into the second equation:
$$
\begin{aligned}
2 \cdot \frac{1}{2}(3-3 y)-3 y & =-1 & & \text { cancel the fraction and re }- \text { write terms : } \\
3-3 y-3 y & =-1 & & \text { collect like terms : } \\
3-6 y & =-1 & & \text { subtract } 3 \text { from both sides : } \\
-6 y & =-4 & & \text { divide by }-6: \\
y & =\frac{2}{3} & &
\end{aligned}
$$
Substitute into the expression we got for $x$ :
$$
\begin{aligned}
& x=\frac{1}{2}\left(3-\not \supset\left(\frac{2}{\not z}\right)\right) \\
& x=\frac{1}{2}
\end{aligned}
$$
So our solution is $x=\frac{1}{2}, y=\frac{2}{3}$. You can see how the graphical solution $\left(\frac{1}{2}, \frac{2}{3}\right)$ might have been difficult to read accurately off a graph!
## Solving Real-World Problems Using Linear Systems
Simultaneous equations can help us solve many real-world problems. We may be considering a purchasefor example, trying to decide whether its cheaper to buy an item online where you pay shipping or at the store where you do not. Or you may wish to join a CD music club, but arent sure if you would really save any money by buying a new CD every month in that way. Or you might be considering two different phone contracts. Lets look at an example of that now.
## Example 3
Anne is trying to choose between two phone plans. The first plan, with Vendafone, costs $\$ 20$ per month, with calls costing an additional 25 cents per minute. The second company, Sellnet, charges $\$ 40$ per month, but calls cost only 8 cents per minute. Which should she choose? You should see that Annes choice will depend upon how many minutes of calls she expects to use each month. We start by writing two equations for the cost in dollars in terms of the minutes used. Since the number of minutes is the independent variable, it will be our $x$. Cost is dependent on minutes the cost per month is the dependent variable and will be assigned $y$.
For Vendafone: $y=0.25 x+20$
For Sellnet: $y=0.08 x+40$
By writing the equations in slope-intercept form $(y=m x+b)$, you can sketch a graph to visualize the situation:
The line for Vendafone has an intercept of 20 and a slope of 0.25 . The Sellnet line has an intercept of 40 and a slope of 0.08 (which is roughly a third of the Vendafone lines slope). In order to help Anne decide which to choose, well find where the two lines cross, by solving the two equations as a system.
Since equation 1 gives us an expression for $y(0.25 x+20)$, we can substitute this expression directly into equation 2 :
$$
\begin{aligned}
0.25 x+20 & =0.08 x+40 \\
0.25 x & =0.08 x+20 \\
0.17 x & =20 \\
x & =117.65 \text { minutes }
\end{aligned}
$$
subtract 20 from both sides:
subtract $0.08 x$ from both sides :
divide both sides by 0.17 :
rounded to 2 decimal places.
So if Anne uses 117.65 minutes a month (although she cant really do exactly that, because phone plans only count whole numbers of minutes), the phone plans will cost the same. Now we need to look at the graph to see which plan is better if she uses more minutes than that, and which plan is better if she uses fewer. You can see that the Vendafone plan costs more when she uses more minutes, and the Sellnet plan costs more with fewer minutes.
So, if Anne will use 117 minutes or less every month she should choose Vendafone. If she plans on using 118 or more minutes she should choose Sellnet.
## Mixture Problems
Systems of equations crop up frequently in problems that deal with mixtures of two thingschemicals in a solution, nuts and raisins, or even the change in your pocket! Lets look at some examples of these.
## Example 4
Janine empties her purse and finds that it contains only nickels (worth 5 cents each) and dimes (worth 10 cents each). If she has a total of 7 coins and they have a combined value of 45 cents, how many of each coin does she have? Since we have 2 types of coins, lets call the number of nickels $x$ and the number of dimes $y$. We are given two key pieces of information to make our equations: the number of coins and their value.
$$
\begin{array}{lll}
\text { of coins equation: } & x+y=7 & \text { (number of nickels })+(\text { number of dimes }) \\
\text { value equation: } & 5 x+10 y=55 & (\text { since nickels are worth } 5 c \text { and dimes } 10 c)
\end{array}
$$
We can quickly rearrange the first equation to isolate $x$ :
$$
\begin{aligned}
& a m p ; x=7-y \\
& 5(7-y)+10 y=55 \\
& 35-5 y+10 y=55 \\
& 35+5 y=55 \\
& 5 y=20 \\
& \underline{y=4} \\
& a m p ; x+4=7 \\
& \underline{x=3}
\end{aligned}
$$
now substitute into equation 2 :
distribute the 5 :
collect like terms :
subtract 35 from both sides:
divide by 5 :
substitute back into equation 1 :
subtract 4 from both sides:
## Janine has 3 nickels and 4 dimes.
Sometimes a question asks you to determine (from concentrations) how much of a particular substance to use. The substance in question could be something like coins as above, or it could be a chemical in solution, or even heat. In such a case, you need to know the amount of whatever substance is in each part. There are several common situations where to get one equation you simply add two given quantities, but to get the second equation you need to use a product. Three examples are below.
## TABLE 12.5:
## Type of mixture
Coins (items with $\$$ value)
Chemical solutions
Density of two substances
## First equation
total number of items $\left(n_{1}+n_{2}\right)$
total solution volume $\left(V_{1}+V_{2}\right)$
total amount or volume of mix
## Second equation
total value (item value $\times$ no. of items)
amount of solute $(\mathrm{vol} \times$ concentration)
total mass (volume $\times$ density)
For example, when considering mixing chemical solutions, we will most likely need to consider the total amount of solute in the individual parts and in the final mixture. (A solute is the chemical that is dissolved in a solution. An example of a solute is salt when added to water to make a brine.) To find the total amount, simply multiply the amount of the mixture by the fractional concentration. To illustrate, lets look at an example where you are given amounts relative to the whole.
## Example 5
A chemist needs to prepare $500 \mathrm{ml}$ of copper-sulfate solution with a $15 \%$ concentration. She wishes to use a high concentration solution (60\%) and dilute it with a low concentration solution (5\%) in order to do this. How much of each solution should she use?
## Solution
To set this problem up, we first need to define our variables. Our unknowns are the amount of concentrated solution $(x)$ and the amount of dilute solution $(y)$. We will also convert the percentages $(60 \%, 15 \%$ and $5 \%)$ into decimals (0.6, 0.15 and 0.05$)$. The two pieces of critical information are the final volume $(500 \mathrm{ml})$ and the final amount of solute $(15 \%$ of $500 \mathrm{ml}=75 \mathrm{ml})$. Our equations will look like this:
Volume equation: $x+y=500$
Solute equation: $0.6 x+0.05 y=75$
To isolate a variable for substitution, we can see its easier to start with equation 1 :
$$
\begin{aligned}
& x+y=500 \\
& x=500-y \\
& 0.6(500-y)+0.05 y=75 \\
& 300-0.6 y+0.05 y=75 \\
& 300-0.55 y=75 \\
& -0.55 y=-225 \\
& \underline{y=409 m l} \\
& x=500-409=\underline{91 \mathrm{ml}}
\end{aligned}
$$
subtract y from both sides:
now substitute into equation 2 :
distribute the 0.6 :
collect like terms :
subtract 300 from both sides :
divide both sides by -0.55 :
substitute back into equation for $x$ :
## So the chemist should mix $91 \mathrm{ml}$ of the $60 \%$ solution with $409 \mathrm{ml}$ of the $5 \%$ solution.
## Further Practice
For lots more practice solving linear systems, check out this web page: http://www.algebra.com/algebra/homework/ coordinate/practice-linear-system.epl
After clicking to see the solution to a problem, you can click the back button and then click Try Another Practice Linear System to see another problem.
## Practice Set
1. Solve the
system:
$$
\begin{aligned}
& x+2 y=9 \\
& 3 x+5 y=20
\end{aligned}
$$
2. Solve the system:
$$
\begin{aligned}
& x-3 y=10 \\
& 2 x+y=13
\end{aligned}
$$
3. Solve the
system:
$$
\begin{aligned}
& 2 x+0.5 y=-10 \\
& x-y=-10
\end{aligned}
$$
4. Solve the
system:
$$
2 x+0.5 y=3
$$$$
x+2 y=8.5
$$
5. Solve the
system:
$$
\begin{aligned}
& 3 x+5 y=-1 \\
& x+2 y=-1
\end{aligned}
$$
6. Solve the
system:
$3 x+5 y=-3$
$x+2 y=-\frac{4}{3}$
7. Solve the
system:
$x-y=-\frac{12}{5}$
$2 x+5 y=-2$
8. Of the two non-right angles in a right angled triangle, one measures twice as many degrees as the other. What are the angles?
9. The sum of two numbers is 70 . They differ by 11 . What are the numbers?
10. A number plus half of another number equals 6 ; twice the first number minus three times the second number equals 4 . What are the numbers?
11. A rectangular field is enclosed by a fence on three sides and a wall on the fourth side. The total length of the fence is 320 yards. If the field has a total perimeter of 400 yards, what are the dimensions of the field?
12. A ray cuts a line forming two angles. The difference between the two angles is $18^{\circ}$. What does each angle measure?
13. I have $\$ 15$ and wish to buy five pounds of mixed nuts for a party. Peanuts cost $\$ 2.20$ per pound. Cashews cost $\$ 4.70$ per pound.
(a) How many pounds of each should I buy?
(b) If I suddenly realize I need to set aside $\$ 5$ to buy chips, can I still buy 5 pounds of nuts with the remaining $\$ 10$ ?
(c) Whats the greatest amount of nuts I can buy?
14. A chemistry experiment calls for one liter of sulfuric acid at a $15 \%$ concentration, but the supply room only stocks sulfuric acid in concentrations of $10 \%$ and $35 \%$.
(a) How many liters of each should be mixed to give the acid needed for the experiment?
(b) How many liters should be mixed to give two liters at a $15 \%$ concentration? 15. Bachelle wants to know the density of her bracelet, which is a mix of gold and silver. Density is total mass divided by total volume. The density of gold is $19.3 \mathrm{~g} / \mathrm{cc}$ and the density of silver is $10.5 \mathrm{~g} / \mathrm{cc}$. The jeweler told her that the volume of silver in the bracelet was $10 \mathrm{cc}$ and the volume of gold was $20 \mathrm{cc}$. Find the combined density of her bracelet.
16. Jason is five years older than Becky, and the sum of their ages is 23 . What are their ages?
17. Tickets to a show cost $\$ 10$ in advance and $\$ 15$ at the door. If 120 tickets are sold for a total of $\$ 1390$, how many of the tickets were bought in advance?
18. The multiple-choice questions on a test are worth 2 points each, and the short-answer questions are worth 5 points each.
(a) If the whole test is worth 100 points and has 35 questions, how many of the questions are multiple-choice and how many are short-answer?
(b) If Kwan gets 31 questions right and ends up with a score of 86 on the test, how many questions of each type did she get right? (Assume there is no partial credit.)
(c) If Ashok gets 5 questions wrong and ends up with a score of 87 on the test, how many questions of each type did he get wrong? (Careful!)
(d) What are two ways you could have set up the equations for part c?
(e) How could you have set up part b differently?
## Solving Linear Systems by Elimination
## Introduction
In this lesson, well see how to use simple addition and subtraction to simplify our system of equations to a single equation involving a single variable. Because we go from two unknowns ( $x$ and $y$ ) to a single unknown (either $x$ or $y$ ), this method is often referred to by solving by elimination. We eliminate one variable in order to make our equations solvable! To illustrate this idea, lets look at the simple example of buying apples and bananas.
## Example 1
If one apple plus one banana costs $\$ 1.25$ and one apple plus 2 bananas costs $\$ 2.00$, how much does one banana cost? One apple?
It shouldnt take too long to discover that each banana costs $\$ 0.75$. After all, the second purchase just contains 1 more banana than the first, and costs $\$ 0.75$ more, so that one banana must cost $\$ 0.75$.
Heres what we get when we describe this situation with algebra:
$$
\begin{aligned}
a+b & =1.25 \\
a+2 b & =2.00
\end{aligned}
$$
Now we can subtract the number of apples and bananas in the first equation from the number in the second equation, and also subtract the cost in the first equation from the cost in the second equation, to get the difference in cost that corresponds to the difference in items purchased.
$$
(a+2 b)-(a+b)=2.00-1.25 \rightarrow b=0.75
$$
That gives us the cost of one banana. To find out how much one apple costs, we subtract $\$ 0.75$ from the total cost of one apple and one banana.
$$
a+0.75=1.25 \rightarrow a=1.25-0.75 \rightarrow a=0.50
$$
So an apple costs 50 cents.
To solve systems using addition and subtraction, well be using exactly this idea by looking at the sum or difference of the two equations we can determine a value for one of the unknowns.
## Solving Linear Systems Using Addition of Equations
Often considered the easiest method of solving systems of equations, the addition (or elimination) method lets us combine two equations in such a way that the resulting equation has only one variable. We can then use simple algebra to solve for that variable. Then, if we need to, we can substitute the value we get for that variable back into either one of the original equations to solve for the other variable.
## Example 2
Solve this system by addition:
$$
\begin{aligned}
& 3 x+2 y=11 \\
& 5 x-2 y=13
\end{aligned}
$$
## Solution
We will add everything on the left of the equals sign from both equations, and this will be equal to the sum of everything on the right:
$$
(3 x+2 y)+(5 x-2 y)=11+13 \rightarrow 8 x=24 \rightarrow x=3
$$
A simpler way to visualize this is to keep the equations as they appear above, and to add them together vertically, going down the columns. However, just like when you add units, tens and hundreds, you MUST be sure to keep the $x^{\prime}$ s and $y^{\prime}$ in their own columns. You may also wish to use terms like " $0 y$ " as a placeholder!
$$
\begin{gathered}
3 x+2 y=11 \\
+\quad(5 x-2 y)=13 \\
\hline 8 x+0 y=24
\end{gathered}
$$
Again we get $8 x=24$, or $x=3$. To find a value for $y$, we simply substitute our value for $x$ back in.
Substitute $x=3$ into the second equation:
$$
\begin{aligned}
5 \cdot 3-2 y & =13 & & \text { since } 5 \times 3=15, \text { we subtract } 15 \text { from both sides : } \\
-2 y & =-2 & & \text { divide by }-2 \text { to get }: \\
y & =1 & &
\end{aligned}
$$
The reason this method worked is that the $y$-coefficients of the two equations were opposites of each other: 2 and -2. Because they were opposites, they canceled each other out when we added the two equations together, so our final equation had no $y$-term in it and we could just solve it for $x$.
In a little while well see how to use the addition method when the coefficients are not opposites, but for now lets look at another example where they are.
## Example 3
Andrew is paddling his canoe down a fast-moving river. Paddling downstream he travels at 7 miles per hour, relative to the river bank. Paddling upstream, he moves slower, traveling at 1.5 miles per hour. If he paddles equally hard in both directions, how fast is the current? How fast would Andrew travel in calm water?
## Solution
First we convert our problem into equations. We have two unknowns to solve for, so well call the speed that Andrew paddles at $x$, and the speed of the river $y$. When traveling downstream, Andrew speed is boosted by the river current, so his total speed is his paddling speed plus the speed of the river $(x+y)$. Traveling upstream, the river is working against him, so his total speed is his paddling speed minus the speed of the river $(x-y)$.
Downstream Equation: $x+y=7$ Upstream Equation: $x-y=1.5$
Next well eliminate one of the variables. If you look at the two equations, you can see that the coefficient of $y$ is +1 in the first equation and -1 in the second. Clearly $(+1)+(-1)=0$, so this is the variable we will eliminate. To do this we simply add equation 1 to equation 2 . We must be careful to collect like terms, and make sure that everything on the left of the equals sign stays on the left, and everything on the right stays on the right:
$$
(x+y)+(x-y)=7+1.5 \Rightarrow 2 x=8.5 \Rightarrow x=4.25
$$
Or, using the column method we used in example 2:
$$
\begin{gathered}
x+y=7 \\
+x-y=1.5 \\
\hline 2 x+0 y=8.5
\end{gathered}
$$
Again we get $2 x=8.5$, or $x=4.25$. To find a corresponding value for $y$, we plug our value for $x$ into either equation and isolate our unknown. In this example, well plug it into the first equation:
$$
\begin{aligned}
4.25+y & =7 \quad \text { subtract } 4.25 \text { from both sides : } \\
y & =2.75
\end{aligned}
$$
## Andrew paddles at 4.25 miles per hour. The river moves at 2.75 miles per hour.
## Solving Linear Systems Using Multiplication
So far, weve seen that the elimination method works well when the coefficient of one variable happens to be the same (or opposite) in the two equations. But what if the two equations dont have any coefficients the same?
It turns out that we can still use the elimination method; we just have to make one of the coefficients match. We can accomplish this by multiplying one or both of the equations by a constant.
Heres a quick review of how to do that. Consider the following questions:
1. If 10 apples cost $\$ 5$, how much would 30 apples cost?
2. If 3 bananas plus 2 carrots cost $\$ 4$, how mush would 6 bananas plus 4 carrots cost?
If you look at the first equation, it should be obvious that each apple costs \\$0.50. So 30 apples should cost \\$15.00.
The second equation is trickier; it isnt obvious what the individual price for either bananas or carrots is. Yet we know that the answer to question 2 is $\$ 8.00$. How?
If we look again at question 1, we see that we can write an equation: $10 a=5$ ( $a$ being the cost of 1 apple). So to find the cost of 30 apples, we could solve for $a$ and then multiply by 30 but we could also just multiply both sides of the equation by 3. We would get $30 a=15$, and that tells us that 30 apples cost $\$ 15$.
And we can do the same thing with the second question. The equation for this situation is $3 b+2 c=4$, and we can see that we need to solve for $(6 b+4 c)$, which is simply 2 times $(3 b+2 c)$ ! So algebraically, we are simply multiplying the entire equation by 2 :
$$
\begin{aligned}
2(3 b+2 c) & =2 \cdot 4 \quad \text { distribute and multiply }: \\
6 b+4 c & =8
\end{aligned}
$$
So when we multiply an equation, all we are doing is multiplying every term in the equation by a fixed amount.
## Solving a Linear System by Multiplying One Equation
If we can multiply every term in an equation by a fixed number (a scalar), that means we can use the addition method on a whole new set of linear systems. We can manipulate the equations in a system to ensure that the coefficients of one of the variables match.
This is easiest to do when the coefficient as a variable in one equation is a multiple of the coefficient in the other equation.
## Example 4
Solve the system:
$$
\begin{aligned}
& 7 x+4 y=17 \\
& 5 x-2 y=11
\end{aligned}
$$
## Solution
You can easily see that if we multiply the second equation by 2 , the coefficients of $y$ will be +4 and -4 , allowing us to solve the system by addition:
2 times equation 2 :
$$
\begin{array}{r}
10 x-4 y=22 \quad \text { now add to equation one }: \\
\frac{+(7 x+4 y)=17}{17 x=34} \\
\text { divide by } 17 \text { to get }: \quad x=2
\end{array}
$$
Now simply substitute this value for $x$ back into equation 1 :
$$
\begin{aligned}
7 \cdot 2+4 y & =17 & & \text { since } 7 \times 2=14, \text { subtract } 14 \text { from both sides }: \\
4 y & =3 & & \text { divide by } 4: \\
y & =0.75 & &
\end{aligned}
$$
## Example 5
Anne is rowing her boat along a river. Rowing downstream, it takes her 2 minutes to cover 400 yards. Rowing upstream, it takes her 8 minutes to travel the same 400 yards. If she was rowing equally hard in both directions, calculate, in yards per minute, the speed of the river and the speed Anne would travel in calm water.
## Solution
Step one: first we convert our problem into equations. We know that distance traveled is equal to speed $\times$ time. We have two unknowns, so well call the speed of the river $x$, and the speed that Anne rows at $y$. When traveling downstream, her total speed is her rowing speed plus the speed of the river, or $(x+y)$. Going upstream, her speed is hindered by the speed of the river, so her speed upstream is $(x-y)$.
Downstream Equation: $2(x+y)=400$ Upstream Equation: $8(x-y)=400$
Distributing gives us the following system:
$$
\begin{aligned}
& 2 x+2 y=400 \\
& 8 x-8 y=400
\end{aligned}
$$
Right now, we cant use the method of elimination because none of the coefficients match. But if we multiplied the top equation by 4 , the coefficients of $y$ would be +8 and -8 . Lets do that:
$$
\begin{gathered}
8 x+8 y=1,600 \\
+\quad(8 x-8 y)=400 \\
\hline 16 x=2,000
\end{gathered}
$$
Now we divide by 16 to obtain $x=125$.
Substitute this value back into the first equation:
$$
\begin{aligned}
2(125+y) & =400 & & \text { divide both sides by } 2: \\
125+y & =200 & & \text { subtract } 125 \text { from both sides }: \\
y & =75 & &
\end{aligned}
$$
## Anne rows at 125 yards per minute, and the river flows at 75 yards per minute.
## Solving a Linear System by Multiplying Both Equations
So what do we do if none of the coefficients match and none of them are simple multiples of each other? We do the same thing we do when were adding fractions whose denominators arent simple multiples of each other. Remember that when we add fractions, we have to find a lowest common denominatorthat is, the lowest common multiple of the two denominatorsand sometimes we have to rewrite not just one, but both fractions to get them to have a common denominator. Similarly, sometimes we have to multiply both equations by different constants in order to get one of the coefficients to match.
## Example 6
Andrew and Anne both use the I-Haul truck rental company to move their belongings from home to the dorm rooms on the University of Chicago campus. I-Haul has a charge per day and an additional charge per mile. Andrew travels from San Diego, California, a distance of 2060 miles in five days. Anne travels 880 miles from Norfolk, Virginia, and it takes her three days. If Anne pays $\$ 840$ and Andrew pays $\$ 1845$, what does I-Haul charge
a) per day?
b) per mile traveled?
## Solution
First, well set up our equations. Again we have 2 unknowns: the daily rate (well call this $x$ ), and the per-mile rate (well call this $y$ ).
Annes equation: $3 x+880 y=840$
Andrews Equation: $5 x+2060 y=1845$ We cant just multiply a single equation by an integer number in order to arrive at matching coefficients. But if we look at the coefficients of $x$ (as they are easier to deal with than the coefficients of $y$ ), we see that they both have a common multiple of 15 (in fact 15 is the lowest common multiple). So we can multiply both equations.
Multiply the top equation by 5 :
$$
15 x+4400 y=4200
$$
Multiply the lower equation by -3 :
$$
-15 x-6180 y=-5535
$$
Add:
$$
\begin{gathered}
15 x+4400 y=4200 \\
+\quad(-15 x-6180 y)=-5535 \\
\hline-1780 y=-1335
\end{gathered}
$$
Divide by $-1780: y=0.75$
Substitute this back into the top equation:
$$
\begin{aligned}
& 3 x+880(0.75)=840 \quad \text { since } 880 \times 0.75=660, \text { subtract } 660 \text { from both sides : } \\
& 3 x=180 \quad \text { divide both sides by } 3 \\
& x=60
\end{aligned}
$$
## I-Haul charges $\$ 60$ per day plus $\$ 0.75$ per mile.
## Comparing Methods for Solving Linear Systems
Now that weve covered the major methods for solving linear equations, lets review them. For simplicity, well look at them in table form. This should help you decide which method would be best for a given situation.
## TABLE 12.6:
Method:
Graphing
#### Abstract
Best used when you... ...dont need an accurate answer.
## Advantages:
Often easier to see number and quality of intersections on a graph. With a graphing calculator, it can be the fastest method since you dont have to do any computation.
## Comment:
Can lead to imprecise answers with non-integer solutions.
## TABLE 12.6: (continued)
Method:
Substitution
## Best used when you...
... have an explicit equa- tion for one variable (e.g. $y=14 x+2)$
Advantages:
Works on all systems. Reduces the system to one variable, making it easier to solve. Elimination by Addition ... have matching coeffior Subtraction
Elimination by Multiplication and then Addition and Subtraction cients for one variable in both equations.
... do not have any variables defined explicitly or any matching coefficients. Easy to combine equations to eliminate one variable. Quick to solve. Works on all systems. Makes it possible to combine equations to eliminate one variable.
## Comment:
You are not often given explicit functions in systems problems, so you may have to do extra work to get one of the equations into that form.
It is not very likely that a given system will have matching coefficients.
Often more algebraic manipulation is needed to prepare the equations.
The table above is only a guide. You might prefer to use the graphical method for every system in order to better understand what is happening, or you might prefer to use the multiplication method even when a substitution would work just as well.
## Example 7
Two angles are complementary when the sum of their angles is $90^{\circ}$. Angles $A$ and $B$ are complementary angles, and twice the measure of angle $A$ is $9^{\circ}$ more than three times the measure of angle B. Find the measure of each angle.
## Solution
First we write out our 2 equations. We will use $x$ to be the measure of angle $A$ and $y$ to be the measure of angle $B$. We get the following system:
$$
\begin{aligned}
x+y & =90 \\
2 x & =3 y+9
\end{aligned}
$$
First, well solve this system with the graphical method. For this, we need to convert the two equations to $y=m x+b$ form:
$$
\begin{array}{ll}
x+y=90 & \Rightarrow y=-x+90 \\
2 x=3 y+9 & \Rightarrow y=\frac{2}{3} x-3
\end{array}
$$
The first line has a slope of -1 and a $y$-intercept of 90 , and the second line has a slope of $\frac{2}{3}$ and a $y$-intercept of -3 . The graph looks like this:
In the graph, it appears that the lines cross at around $x=55, y=35$, but it is difficult to tell exactly! Graphing by hand is not the best method in this case!
Next, well try solving by substitution. Lets look again at the system:
$$
\begin{aligned}
x+y & =90 \\
2 x & =3 y+9
\end{aligned}
$$
Weve already seen that we can start by solving either equation for $y$, so lets start with the first one:
$$
y=90-x
$$
Substitute into the second equation:
$$
\begin{array}{ll}
2 x=3(90-x)+9 & \text { distribute the } 3: \\
2 x=270-3 x+9 & \text { add } 3 x \text { to both sides }: \\
5 x=270+9=279 & \text { divide by } 5: \\
x=55.8^{\circ} &
\end{array}
$$
Substitute back into our expression for $y$ :
$$
y=90-55.8=34.2^{\circ}
$$
Angle $A$ measures $55.8^{\circ}$; angle $B$ measures $34.2^{\circ}$.
Finally, well try solving by elimination (with multiplication):
Rearrange equation one to standard form:
$$
a m p ; x+y=90 \quad \Rightarrow 2 x+2 y=180
$$
Multiply equation two by 2 :
$$
2 x=3 y+9 \quad \Rightarrow 2 x-3 y=9
$$
Subtract:
$$
\begin{array}{r}
2 x+2 y=180 \\
-\quad(2 x-3 y)=-9 \\
\hline 5 y=171
\end{array}
$$
Divide by 5 to obtain $y=34.2^{\circ}$
Substitute this value into the very first equation:
$$
\begin{aligned}
x+34.2 & =90 \\
x & =55.8^{\circ}
\end{aligned}
$$
subtract 34.2 from both sides:
## Angle $A$ measures $55.8^{\circ}$; angle $B$ measures $34.2^{\circ}$.
Even though this system looked ideal for substitution, the method of elimination worked well too. Once the equations were rearranged properly, the solution was quick to find. Youll need to decide yourself which method to use in each case you see from now on. Try to master all the techniques, and recognize which one will be most efficient for each system you are asked to solve.
The following Khan Academy video contains three examples of solving systems of equations using addition and subtraction as well as multiplication (which is the next topic): http://www.youtube.com/watch?v=nok99JOhcjo (9:57). (Note that the narrator is not always careful about showing his work, and you should try to be neater in your mathematical writing.)
For even more practice, we have this video. One common type of problem involving systems of equations (especially on standardized tests) is age problems." In the following video the narrator shows two examples of age problems, one involving a single person and one involving two people. Khan Academy Age Problems (7:13)
## Practice Set
1. Solve the
system:
$3 x+4 y=2.5$
$5 x-4 y=25.5$
2. Solve the
system:
$5 x+7 y=-31$
$5 x-9 y=17$ 3. Solve the
system:
$3 y-4 x=-33$
$5 x-3 y=40.5$
4. Nadia and Peter visit the candy store. Nadia buys three candy bars and four fruit roll-ups for $\$ 2.84$. Peter also buys three candy bars, but can only afford one additional fruit roll-up. His purchase costs $\$ 1.79$. What is the cost of a candy bar and a fruit roll-up individually?
5. A small plane flies from Los Angeles to Denver with a tail wind (the wind blows in the same direction as the plane) and an air-traffic controller reads its ground-speed (speed measured relative to the ground) at 275 miles per hour. Another, identical plane, moving in the opposite direction has a ground-speed of 227 miles per hour. Assuming both planes are flying with identical air-speeds, calculate the speed of the wind.
6. An airport taxi firm charges a pick-up fee, plus an additional per-mile fee for any rides taken. If a 12-mile journey costs $\$ 14.29$ and a 17 -mile journey costs $\$ 19.91$, calculate:
(a) the pick-up fee
(b) the per-mile rate
(c) the cost of a seven mile trip
7. Calls from a call-box are charged per minute at one rate for the first five minutes, then a different rate for each additional minute. If a 7 -minute call costs $\$ 4.25$ and a 12 -minute call costs $\$ 5.50$, find each rate.
8. A plumber and a builder were employed to fit a new bath, each working a different number of hours. The plumber earns $\$ 35$ per hour, and the builder earns $\$ 28$ per hour. Together they were paid $\$ 330.75$, but the plumber earned $\$ 106.75$ more than the builder. How many hours did each work?
9. Paul has a part time job selling computers at a local electronics store. He earns a fixed hourly wage, but can earn a bonus by selling warranties for the computers he sells. He works 20 hours per week. In his first week, he sold eight warranties and earned \\$220. In his second week, he managed to sell 13 warranties and earned $\$ 280$. What is Pauls hourly rate, and how much extra does he get for selling each warranty?
Solve the following systems using multiplication.
10. $5 x-10 y=15$
$3 x-2 y=3$
11. $5 x-y=10$
$3 x-2 y=-1$
12. $5 x+7 y=15$
$7 x-3 y=5$
13. $9 x+5 y=9$
$12 x+8 y=12.8$
14. $4 x-3 y=1$
$3 x-4 y=4$
15. $7 x-3 y=-3$
$6 x+4 y=3$
Solve the following systems using any method.
16. $x=3 y$
$x-2 y=-3$
17. $y=3 x+2$
$y=-2 x+7$
18. $5 x-5 y=5$
$5 x+5 y=35$
19. $y=-3 x-3$
$3 x-2 y+12=0$ 20. $3 x-4 y=3$
$4 y+5 x=10$
21. $9 x-2 y=-4$
$2 x-6 y=1$
22. Supplementary angles are two angles whose sum is $180^{\circ}$. Angles $A$ and $B$ are supplementary angles. The measure of Angle $A$ is $18^{\circ}$ less than twice the measure of Angle $B$. Find the measure of each angle.
23. A farmer has fertilizer in $5 \%$ and $15 \%$ solutions. How much of each type should he mix to obtain 100 liters of fertilizer in a $12 \%$ solution?
24. A 150-yard pipe is cut to provide drainage for two fields. If the length of one piece is three yards less that twice the length of the second piece, what are the lengths of the two pieces?
25. Mr. Stein invested a total of $\$ 100,000$ in two companies for a year. Company As stock showed a $13 \%$ annual gain, while Company B showed a 3\% loss for the year. Mr. Stein made an $8 \%$ return on his investment over the year. How much money did he invest in each company?
26. A baker sells plain cakes for $\$ 7$ and decorated cakes for $\$ 11$. On a busy Saturday the baker started with 120 cakes, and sold all but three. His takings for the day were $\$ 991$. How many plain cakes did he sell that day, and how many were decorated before they were sold?
27. Twice Johns age plus five times Claires age is 204. Nine times Johns age minus three times Claires age is also 204. How old are John and Claire?
### Special Types of Linear Systems
## Introduction
As we saw previously, a system of linear equations is a set of linear equations which must be solved together. The lines in the system can be graphed together on the same coordinate graph and the solution to the system is the point at which the two lines intersect.
Or at least thats what usually happens. But what if the lines turn out to be parallel when we graph them?
If the lines are parallel, they wont ever intersect. That means that the system of equations they represent has no solution. A system with no solutions is called an inconsistent system.
And what if the lines turn out to be identical?
If the two lines are the same, then every point on one line is also on the other line, so every point on the line is a solution to the system. The system has an infinite number of solutions, and the two equations are really just different forms of the same equation. Such a system is called a dependent system.
But usually, two lines cross at exactly one point and the system has exactly one solution:
A system with exactly one solution is called a consistent system.
To identify a system as consistent, inconsistent, or dependent, we can graph the two lines on the same graph and see if they intersect, are parallel, or are the same line. But sometimes it is hard to tell whether two lines are parallel just by looking at a roughly sketched graph.
Another option is to write each line in slope-intercept form and compare the slopes and $y$ - intercepts of the two lines. To do this we must remember that:
- Lines with different slopes always intersect.
- Lines with the same slope but different $y$-intercepts are parallel.
- Lines with the same slope and the same $y$-intercepts are identical.
## Example 1
Determine whether the following system has exactly one solution, no solutions, or an infinite number of solutions.
$$
\begin{array}{r}
2 x-5 y=2 \\
4 x+y=5
\end{array}
$$
## Solution
We must rewrite the equations so they are in slope-intercept form
$2 x-5 y=2$
$-5 y=-2 x+2$
$y=\frac{2}{5} x-\frac{2}{5}$
$x+y=5$
$y=-4 x+5$
$y=-4 x+5$
The slopes of the two equations are different; therefore the lines must cross at a single point and the system has exactly one solution. This is a consistent system.
## Example 2
Determine whether the following system has exactly one solution, no solutions, or an infinite number of solutions.
$$
\begin{aligned}
3 x & =5-4 y \\
6 x+8 y & =7
\end{aligned}
$$
## Solution
We must rewrite the equations so they are in slope-intercept form
$3 x=5-4 y$
$4 y=-3 x+5$
$y=-\frac{3}{4} x+\frac{5}{4}$
$6 x+8 y=7$
$8 y=-6 x+7$
$y=-\frac{3}{4} x+\frac{7}{8}$ The slopes of the two equations are the same but the $y$-intercepts are different; therefore the lines are parallel and the system has no solutions. This is an inconsistent system.
## Example 3
Determine whether the following system has exactly one solution, no solutions, or an infinite number of solutions.
$$
\begin{array}{r}
x+y=3 \\
3 x+3 y=9
\end{array}
$$
## Solution
We must rewrite the equations so they are in slope-intercept form
$x+y=3$
$\Rightarrow \quad y=-x+3$
$y=-x+3$
$x+3 y=9$
$3 y=-3 x+9$
$\Rightarrow$
$$
y=-x+3
$$
The lines are identical; therefore the system has an infinite number of solutions. It is a dependent system.
## Determining the Type of System Algebraically
A third option for identifying systems as consistent, inconsistent or dependent is to just solve the system and use the result as a guide.
## Example 4
Solve the following system of equations. Identify the system as consistent, inconsistent or dependent.
$$
\begin{array}{r}
10 x-3 y=3 \\
2 x+y=9
\end{array}
$$
## Solution
Lets solve this system using the substitution method.
Solve the second equation for $y$ :
$$
2 x+y=9 \Rightarrow y=-2 x+9
$$
Substitute that expression for $y$ in the first equation:
$$
\begin{aligned}
10 x-3 y & =3 \\
10 x-3(-2 x+9) & =3 \\
10 x+6 x-27 & =3 \\
16 x & =30 \\
x & =\frac{15}{8}
\end{aligned}
$$
Substitute the value of $x$ back into the second equation and solve for $y$ :
$$
2 x+y=9 \Rightarrow y=-2 x+9 \Rightarrow y=-2 \cdot \frac{15}{8}+9 \Rightarrow y=\frac{21}{4}
$$
The solution to the system is $\left(\frac{15}{8}, \frac{21}{4}\right)$. The system is consistent since it has only one solution.
## Example 5
Solve the following system of equations. Identify the system as consistent, inconsistent or dependent.
$$
\begin{aligned}
& 3 x-2 y=4 \\
& 9 x-6 y=1
\end{aligned}
$$
## Solution
Lets solve this system by the method of multiplication.
Multiply the first equation by 3 :
$3(3 x-2 y=4)$
$9 x-6 y=12$
$9 x-6 y=1$
$9 x-6 y=1$
Add the two equations:
$$
\begin{aligned}
9 x-6 y & =4 \\
9 x-6 y & =1 \\
\hline 0 & =13 \quad \text { This statement is not true. }
\end{aligned}
$$
If our solution to a system turns out to be a statement that is not true, then the system doesnt really have a solution; it is inconsistent.
## Example 6
Solve the following system of equations. Identify the system as consistent, inconsistent or dependent.
$$
\begin{array}{r}
4 x+y=3 \\
12 x+3 y=9
\end{array}
$$
## Solution
Lets solve this system by substitution.
Solve the first equation for $y$ :
$$
4 x+y=3 \Rightarrow y=-4 x+3
$$
Substitute this expression for $y$ in the second equation:
$$
\begin{array}{r}
12 x+3 y=9 \\
12 x+3(-4 x+3)=9 \\
12 x-12 x+9=9 \\
9=9
\end{array}
$$
This statement is always true.
If our solution to a system turns out to be a statement that is always true, then the system is dependent.
A second glance at the system in this example reveals that the second equation is three times the first equation, so the two lines are identical. The system has an infinite number of solutions because they are really the same equation and trace out the same line.
Lets clarify this statement. An infinite number of solutions does not mean that any ordered pair $(x, y)$ satisfies the system of equations. Only ordered pairs that solve the equation in the system (either one of the equations) are also solutions to the system. There are infinitely many of these solutions to the system because there are infinitely many points on any one line.
For example, $(1,-1)$ is a solution to the system in this example, and so is $(-1,7)$. Each of them fits both the equations because both equations are really the same equation. But $(3,5)$ doesnt fit either equation and is not a solution to the system.
In fact, for every $x$-value there is just one $y$-value that fits both equations, and for every $y$-value there is exactly one $x$-valuejust as there is for a single line.
Lets summarize how to determine the type of system we are dealing with algebraically.
- A consistent system will always give exactly one solution.
- An inconsistent system will yield a statement that is always false (like $0=13$ ).
- A dependent system will yield a statement that is always true (like 9=9).
## Applications
In this section, well see how consistent, inconsistent and dependent systems might arise in real life.
## Example 7
The movie rental store CineStar offers customers two choices. Customers can pay a yearly membership of \\$45 and then rent each movie for $\$ 2$ or they can choose not to pay the membership fee and rent each movie for \\$3.50. How many movies would you have to rent before the membership becomes the cheaper option?
## Solution
Lets translate this problem into algebra. Since there are two different options to consider, we can write two different equations and form a system.
The choices are membership and no membership. Well call the number of movies you rent $x$ and the total cost of renting movies for a year $y$.
TABLE 12.7:
| | flat fee | rental fee | total |
| :--- | :--- | :--- | :--- |
| membership | $\$ 45$ | $2 x$ | $y=45+2 x$ |
| no membership | $\$ 0$ | $3.50 x$ | $y=3.5 x$ |
The flat fee is the dollar amount you pay per year and the rental fee is the dollar amount you pay when you rent a movie. For the membership option the rental fee is $2 x$, since you would pay $\$ 2$ for each movie you rented; for the no membership option the rental fee is 3.50x, since you would pay $\$ 3.50$ for each movie you rented.
Our system of equations is:
$y=45+2 x$
$y=3.50 x$ Heres a graph of the system:
Now we need to find the exact intersection point. Since each equation is already solved for $y$, we can easily solve the system with substitution. Substitute the second equation into the first one:
$y=45+2 x$
$$
\Rightarrow 3.50 x=45+2 x \Rightarrow 1.50 x=45 \Rightarrow x=30 \text { movies }
$$
$y=3.50 x$
You would have to rent 30 movies per year before the membership becomes the better option.
This example shows a real situation where a consistent system of equations is useful in finding a solution. Remember that for a consistent system, the lines that make up the system intersect at single point. In other words, the lines are not parallel or the slopes are different.
In this case, the slopes of the lines represent the price of a rental per movie. The lines cross because the price of rental per movie is different for the two options in the problem
Now lets look at a situation where the system is inconsistent. From the previous explanation, we can conclude that the lines will not intersect if the slopes are the same (and the $y$-intercept is different). Lets change the previous problem so that this is the case.
## Example 8
Two movie rental stores are in competition. Movie House charges an annual membership of $\$ 30$ and charges \\$3 per movie rental. Flicks for Cheap charges an annual membership of \\$15 and charges \\$3 per movie rental. After how many movie rentals would Movie House become the better option?
## Solution
It should already be clear to see that Movie House will never become the better option, since its membership is more expensive and it charges the same amount per movie as Flicks for Cheap.
The lines on a graph that describe each option have different $y$-interceptsnamely 30 for Movie House and 15 for Flicks for Cheapbut the same slope: 3 dollars per movie. This means that the lines are parallel and so the system is inconsistent.
Now lets see how this works algebraically. Once again, well call the number of movies you rent $x$ and the total cost of renting movies for a year $y$.
## TABLE 12.8:
| | flat fee | rental fee | total |
| :--- | :--- | :--- | :--- |
| Movie House | $\$ 30$ | $3 x$ | $y=30+3 x$ |
| Flicks for Cheap | $\$ 15$ | $3 x$ | $y=15+3 x$ |
The system of equations that describes this problem is:
$y=30+3 x$
$y=15+3 x$
Lets solve this system by substituting the second equation into the first equation:
$15+3 x=30+3 x \Rightarrow 15=30 \quad$ This statement is always false.
This means that the system is inconsistent.
## Example 9
Peter buys two apples and three bananas for $\$ 4$. Nadia buys four apples and six bananas for $\$ 8$ from the same store. How much does one banana and one apple costs?
## Solution
We must write two equations: one for Peters purchase and one for Nadias purchase.
Lets say $a$ is the cost of one apple and $b$ is the cost of one banana.
| | cost of apples | cost of bananas | total cost |
| :--- | :--- | :--- | :--- |
| Peter | $2 a$ | $3 b$ | $2 a+3 b=4$ |
| Nadia | $4 a$ | $6 b$ | $4 a+6 b=8$ |
The system of equations that describes this problem is:
$2 a+3 b=4$
$4 a+6 b=8$
Lets solve this system by multiplying the first equation by -2 and adding the two equations:
$-2(2 a+3 b=4) \quad \Rightarrow \quad-4 a-6 b=-8$
$4 a+6 b=8 \quad \frac{4 a+6 b=8}{0+0=0}$
This statement is always true. This means that the system is dependent.
Looking at the problem again, we can see that we were given exactly the same information in both statements. If Peter buys two apples and three bananas for $\$ 4$, it makes sense that if Nadia buys twice as many apples (four apples) and twice as many bananas (six bananas) she will pay twice the price (\\$8). Since the second equation doesnt give us any new information, it doesnt make it possible to find out the price of each fruit.
## Practice Set
Express each equation in slope-intercept form. Without graphing, state whether the system of equations is consistent, inconsistent or dependent.
1.
$$
\begin{aligned}
& 3 x-4 y=13 \\
& y=-3 x-7
\end{aligned}
$$
2.
$$
\begin{array}{r}
\frac{3}{5} x+y=3 \\
1.2 x+2 y=6
\end{array}
$$
3.
$$
\begin{aligned}
& 3 x-4 y=13 \\
& y=-3 x-7
\end{aligned}
$$
4.
$$
\begin{aligned}
3 x-3 y & =3 \\
x-y & =1
\end{aligned}
$$
5.
$$
\begin{array}{r}
0.5 x-y=30 \\
0.5 x-y=-30
\end{array}
$$
6.
$$
\begin{array}{r}
4 x-2 y=-2 \\
3 x+2 y=-12
\end{array}
$$
7. $3 x+y=4$
$y=5-3 x$
8. $x-2 y=7$
$4 y-2 x=14$
Find the solution of each system of equations using the method of your choice. State if the system is inconsistent or dependent.
9. $3 x+2 y=4$
$-2 x+2 y=24$
10. $5 x-2 y=3$
$2 x-3 y=10$
11. $3 x-4 y=13$
$y=-3 x-7$
12. $5 x-4 y=1$
$-10 x+8 y=-30$
13. $4 x+5 y=0$
$3 x=6 y+4.5$
14. $-2 y+4 x=8$
$y-2 x=-4$
15. $x-\frac{1}{2} y=\frac{3}{2}$
$3 x+y=6$
16. $0.05 x+0.25 y=6$
$x+y=24$
17. $x+\frac{2}{3} y=6$
$3 x+2 y=2$
18. A movie theater charges $\$ 4.50$ for children and $\$ 8.00$ for adults. (a) On a certain day, 1200 people enter the theater and $\$ 8375$ is collected. How many children and how many adults attended?
(b) The next day, the manager announces that she wants to see them take in $\$ 10000$ in tickets. If there are 240 seats in the house and only five movie showings planned that day, is it possible to meet that goal?
(c) At the same theater, a 16-ounce soda costs $\$ 3$ and a 32-ounce soda costs $\$ 5$. If the theater sells 12,480 ounces of soda for $\$ 2100$, how many people bought soda? (Note: Be careful in setting up this problem!)
19. Jamal placed two orders with an internet clothing store. The first order was for 13 ties and 4 pairs of suspenders, and totaled \\$487. The second order was for 6 ties and 2 pairs of suspenders, and totaled \\$232. The bill does not list the per-item price, but all ties have the same price and all suspenders have the same price. What is the cost of one tie and of one pair of suspenders?
20. An airplane took four hours to fly 2400 miles in the direction of the jet-stream. The return trip against the jet-stream took five hours. What were the airplanes speed in still air and the jet-stream's speed?
21. Nadia told Peter that she went to the farmers market and bought two apples and one banana, and that it cost her \\$2.50. She thought that Peter might like some fruit, so she went back to the seller and bought four more apples and two more bananas. Peter thanked Nadia, but told her that he did not like bananas, so he would only pay her for four apples. Nadia told him that the second time she paid $\$ 6.00$ for the fruit.
(a) What did Peter find when he tried to figure out the price of four apples?
(b) Nadia then told Peter she had made a mistake, and she actually paid $\$ 5.00$ on her second trip. Now what answer did Peter get when he tried to figure out how much to pay her?
(c) Alicia then showed up and told them she had just bought 3 apples and 2 bananas from the same seller for $\$ 4.25$. Now how much should Peter pay Nadia for four apples?
## CHAPTER
Problem Solving
## Chapter Outline
13.1 A Problem-Solving Plan
13.2 Problem-Solving Strategies: Make a Table; Look for a Pattern
13.3 Problem-Solving Strategies: Guess and Check and Work BackWARDS
13.4 Problem-Solving Strategies - Using Graphs to solve a problem
13.5 Problem-Solving Strategies: Use a Formula
This chapter is an appendix to the text. You will find a number of different problem solving activities that may or may not be used in the class. The problems can be great learning tools - so you may want to try them on your own!
### A Problem-Solving Plan
Much of mathematics apply to real-world situations. To think critically and to problem solve are mathematical abilities. Although these capabilities may be the most challenging, they are also the most rewarding.
To be successful in applying mathematics in real-life situations, you must have a "toolbox" of strategies to assist you. The last few lessons of many chapters in this FlexBook are devoted to filling this toolbox so you to become a better problem solver and tackle mathematics in the real world.
## Step \\#1: Read and Understand the Given Problem
Every problem you encounter gives you clues needed to solve it successfully. Here is a checklist you can use to help you understand the problem.
$\sqrt{ }$ Read the problem carefully. Make sure you read all the sentences. Many mistakes have been made by failing to fully read the situation.
$\sqrt{ }$ Underline or highlight key words. These include mathematical operations such as sum, difference, product, and mathematical verbs such as equal, more than, less than, is. Key words also include the nouns the situation is describing such as time, distance, people, etc.
$\checkmark \checkmark$ Ask yourself if you have seen a problem like this before. Even though the nouns and verbs may be different, the general situation may be similar to something else you've seen.
$\sqrt{ }$ What are you being asked to do? What is the question you are supposed to answer?
$\sqrt{ }$ What facts are you given? These typically include numbers or other pieces of information.
Once you have discovered what the problem is about, the next step is to declare what variables will represent the nouns in the problem. Remember to use letters that make sense!
## Step \\#2: Make a Plan to Solve the Problem
The next step in the problem-solving plan is to make a plan or develop a strategy. How can the information you know assist you in figuring out the unknown quantities?
Here are some common strategies that you will learn.
- Drawing a diagram
- Making a table
- Looking for a pattern
- Using guess and check
- Working backwards - Using a formula
- Reading and making graphs
- Writing equations
- Using linear models
- Using dimensional analysis
- Using the right type of function for the situation
In most problems, you will use a combination of strategies. For example, drawing a diagram and looking for patterns are good strategies for most problems. Also, making a table and drawing a graph are often used together. The "writing an equation" strategy is the one you will work with the most frequently in your study of algebra.
## Step \\#3: Solve the Problem and Check the Results
Once you develop a plan, you can use it to solve the problem.
The last step in solving any problem should always be to check and interpret the answer. Here are some questions to help you to do that.
- Does the answer make sense?
- If you substitute the solution into the original problem, does it make the sentence true?
- Can you use another method to arrive at the same answer?
## Step \\#4: Compare Alternative Approaches
Sometimes a certain problem is best solved by using a specific method. Most of the time, however, it can be solved by using several different strategies. When you are familiar with all of the problem-solving strategies, it is up to you to choose the methods that you are most comfortable with and that make sense to you. In this book, we will often use more than one method to solve a problem. This way we can demonstrate the strengths and weaknesses of different strategies when applied to different types of problems.
Regardless of the strategy you are using, you should always implement the problem-solving plan when you are solving word problems. Here is a summary of the problem-solving plan.
Step 1: Understand the problem.
Step 2: Devise a plan - Translate. Come up with a way to solve the problem. Set up an equation, draw a diagram, make a chart, or construct a table as a start to begin your problem-solving plan.
Step 3: Carry out the plan - Solve.
Step 4: Check and Interpret: Check to see if you have used all your information. Then look to see if the answer makes sense.
## Solve Real-World Problems Using a Plan
Example 1: Jeff is 10 years old. His younger brother, Ben, is 4 years old. How old will Jeff be when he is twice as old as Ben?
Solution: Begin by understanding the problem. Highlight the key words.
Jeff is $\mathbf{1 0}$ years old. His younger brother, Ben, is $\mathbf{4}$ years old. How old will Jeff be when he is twice as old as Ben? The question we need to answer is. "What is Jeff's age when he is twice as old as Ben?"
You could guess and check, use a formula, make a table, or look for a pattern. The key is "twice as old." This clue means two times, or double Ben's age. Begin by doubling possible ages. Let's look for a pattern.
$4 \times 2=8$. Jeff is already older than 8.
$5 \times 2=10$. This doesn't make sense because Jeff is already 10 .
$6 \times 2=12$. In two years, Jeff will be 12 and Ben will be 6 . Jeff will be twice as old.
Jeff will be 12 years old.
Example 2: Matthew is planning to harvest his corn crop this fall. The field has 660 rows of corn with 300 ears per row. Matthew estimates his crew will have the crop harvested in 20 hours. How many ears of corn will his crew harvest per hour?
Solution: Begin by highlighting the key information.
Matthew is planning to harvest his corn crop this fall. The field has 660 rows of corn with 300 ears per row. Matthew estimates his crew will have the crop harvested in 20 hours. How many ears of corn will his crew harvest per hour?
You could draw a picture (it may take a while), write an equation, look for a pattern, or make a table. Let's try to use reasoning.
We need to figure out how many ears of corn are in the field. $660(300)=198,000$. This is how many ears are in the field. It will take 20 hours to harvest the entire field, so we need to divide 198,000 by 20 to get the number of ears picked per hour.
$$
\frac{198,000}{20}=9,900
$$
The crew can harvest 9,900 ears per hour.
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:Word Problem-Solving Plan 1 (10:12)
MEDIA
Click image to the left for more content.
1. What are the four steps to solving a problem?
2. Name three strategies you can use to help make a plan. Which one(s) are you most familiar with already?
3. Which types of strategies work well together? Why?
4. Suppose Matthew's crew takes 36 hours to harvest the field. How many ears per hour will they harvest?
5. Why is it difficult to solve Ben and Jeff's age problem by drawing a diagram?
6. How do you check a solution to a problem? What is the purpose of checking the solution?
7. There were 12 people on a jury, with four more women than men. How many women were there?
8. A rope 14 feet long is cut into two pieces. One piece is 2.25 feet longer than the other. What are the lengths of the two pieces?
9. A sweatshirt costs $\$ 35$. Find the total cost if the sales tax is $7.75 \%$.
10. This year you got a $5 \%$ raise. If your new salary is $\$ 45,000$, what was your salary before the raise?
11. It costs $\$ 250$ to carpet a room that is $14 \mathrm{ft} \times 18 \mathrm{ft}$. How much does it cost to carpet a room that is $9 \mathrm{ft} \times 10 \mathrm{ft}$ ?
12. A department store has a $15 \%$ discount for employees. Suppose an employee has a coupon worth $\$ 10$ off any item and she wants to buy a $\$ 65$ purse. What is the final cost of the purse if the employee discount is applied before the coupon is subtracted?
13. To host a dance at a hotel, you must pay $\$ 250$ plus $\$ 20$ per guest. How much money would you have to pay for 25 guests?
14. It costs $\$ 12$ to get into the San Diego County Fair and $\$ 1.50$ per ride. If Rena spent $\$ 24$ in total, how many rides did she go on?
15. An ice cream shop sells a small cone for $\$ 2.92$, a medium cone for $\$ 3.50$, and a large cone for $\$ 4.25$. Last Saturday, the shop sold 22 small cones, 26 medium cones, and 15 large cones. How much money did the store earn?
16. The sum of angles in a triangle is 180 degrees. If the second angle is twice the size of the first angle and the third angle is three times the size of the first angle, what are the measures of the angles in the triangle?
## Mixed Review
17. Choose an appropriate variable for the following situation: It takes Lily 45 minutes to bathe and groom a dog. How many dogs can she groom in an 9-hour day?
18. Translate the following into an algebraic inequality: Fourteen less than twice a number is greater than or equal to 16.
19. Write the pattern of the table below in words and using an algebraic equation.
$\begin{array}{lllll}a m p ; x & -2 & -1 & 0 & 1 \\ a m p ; y & -8 & -4 & 0 & 4\end{array}$
20. Check that $m=4$ is a solution to $3 y-11 \geq-3$.
21. What is the domain and range of the graph below?
### Problem-Solving Strategies: Make a Table; Look for a Pattern
This lesson focuses on two of the strategies introduced in the previous chapter: making a table and looking for a pattern. These are the most common strategies you have used before algebra. Let's review the four-step problemsolving plan from Lesson 1.7.
Step 1: Understand the problem.
Step 2: Devise a plan - Translate. Come up with a way to solve the problem. Set up an equation, draw a diagram, make a chart, or construct a table as a start to begin your problem-solving plan.
Step 3: Carry out the plan - Solve.
Step 4: Check and Interpret: Check to see if you used all your information. Then look to see if the answer makes sense.
## Using a Table to Solve a Problem
When a problem has data that needs to be organized, a table is a highly effective problem-solving strategy. A table is also helpful when the problem asks you to record a large amount of information. Patterns and numerical relationships are easier to see when data are organized in a table.
Example 1: Josie takes up jogging. In the first week she jogs for 10 minutes per day, in the second week she jogs for 12 minutes per day. Each week, she wants to increase her jogging time by 2 minutes per day. If she jogs six days per week each week, what will be her total jogging time in the sixth week?
Solution: Organize the information in a table
## TABLE 13.1:
| Week 1 | Week 2 | Week 3 | Week 4 |
| :--- | :--- | :--- | :--- |
| 10 minutes | 12 minutes | 14 minutes | 16 minutes |
| 60 min/week | 72 min/week | 84 min/week | 96 min/week |
We can see the pattern that the number of minutes is increasing by 12 each week. Continuing this pattern, Josie will run 120 minutes in the sixth week.
Don't forget to check the solution! The pattern starts at 60 and adds 12 each week after the first week. The equation to represent this situation is $t=60+12(w-1)$. By substituting 6 for the variable of $w$, the equation becomes $t=60+12(6-1)=60+60=120$
## Solve a Problem by Looking for a Pattern
Some situations have a readily apparent pattern, which means that the pattern is easy to see. In this case, you may not need to organize the information into a table. Instead, you can use the pattern to arrive at your solution.
Example 2: You arrange tennis balls in triangular shapes as shown. How many balls will there be in a triangle that has 8 layers?
1 layer
2 layer
3 layer
One layer: It is simple to see that a triangle with one layer has only one ball.
Two layers: For a triangle with two layers we add the balls from the top layer to the balls of the bottom layer. It is useful to make a sketch of the different layers in the triangle.
Three layers: we add the balls from the top triangle to the balls from the bottom layer.
We can fill the first three rows of the table.
1
2
3
4
$1 \quad 3$
6
$6+4=10$
To find the number of tennis balls in 8 layers, continue the pattern.
5
6
7
8
$10+5=15$
$15+6=21$
$21+7=28$
$28+8=36$
There will be 36 tennis balls in the 8 layers. Check: Each layer of the triangle has one more ball than the previous one. In a triangle with 8 layers, each layer has the smae number of balls as its position. When we add these we get:
$1+2+3+4+5+6+7+8=36$ balls
The answer checks out.
## Comparing Alternative Approaches to Solving Problems
In this section, we will compare the methods of "Making a Table" and "Looking for a Pattern" by using each method in turn to solve a problem.
Example 3: Andrew cashes a \\$180 check and wants the money in $\$ 10$ and $\$ 20$ bills. The bank teller gives him 12 bills. How many of each kind of bill does he receive?
## Solution: Method 1: Making a Table
| Tens | 0 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Twenties | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
The combination that has a sum of 12 is six $\$ 10$ bills and six $\$ 20$ bills.
## Method 2: Using a Pattern
The pattern is that for every pair of $\$ 10$ bills, the number of $\$ 20$ bills reduces by one. Begin with the most number of $\$ 20$ bills. For every $\$ 20$ bill lost, add two $\$ 10$ bills.
$$
6(\$ 10)+6(\$ 20)=\$ 180
$$
Check: Six $\$ 10$ bills and six $\$ 20$ bills $=6(\$ 10)+6(\$ 20)=\$ 60+\$ 120=\$ 180$.
## Using These Strategies to Solve Problems
Example 4: Students are going to march in a homecoming parade. There will be one kindergartener, two firstgraders, three second-graders, and so on through $12^{\text {th }}$ grade. How many students will be walking in the homecoming parade?
Could you make a table? Absolutely. Could you look for a pattern? Absolutely.
Solution 1: Make a table:
$$
\begin{array}{rrrrrrrrrrrrr}
a m p ; K & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13
\end{array}
$$
The solution is the sum of all the numbers, 91 . There will be 91 students walking in the homecoming parade.
Solution 2: Look for a pattern.
The pattern is: The number of students is one more than their grade level. Therefore, the solution is the sum of numbers from 1 (kindergarten) through 13 (12 grade). The solution is 91.
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:Word Problem-Solving Strategies (12:51)
1. Go back and find the solution to the problem in Example 1.
2. Britt has $\$ 2.25$ in nickels and dimes. If she has 40 coins in total how many of each coin does she have?
3. A pattern of squares is placed together as shown. How many squares are in the $12^{\text {th }}$ diagram?
4. Oswald is trying to cut down on drinking coffee. His goal is to cut down to 6 cups per week. If he starts with 24 cups the first week, cuts down to 21 cups the second week, and drops to 18 cups the third week, how many weeks will it take him to reach his goal?
5. Taylor checked out a book from the library and it is now 5 days late. The late fee is 10 cents per day. How much is the fine?
6. How many hours will a car traveling at 75 miles per hour take to catch up to a car traveling at 55 miles per hour if the slower car starts two hours before the faster car?
7. Grace starts biking at 12 miles per hour. One hour later, Dan starts biking at 15 miles per hour, following the same route. How long would it take him to catch up with Grace?
8. Lemuel wants to enclose a rectangular plot of land with a fence. He has 24 feet of fencing. What is the largest possible area that he could enclose with the fence?
## Mixed Review
9. Determine if the relation is a function: $\{(2,6),(-9,0),(7,7),(3,5),(5,3)\}$.
10. Roy works construction during the summer and earns $\$ 78$ per job. Create a table relating the number of jobs he could work, $j$, and the total amount of money he can earn, $m$.
11. Graph the following order pairs: $(4,4) ;(-5,6),(-1,-1),(-7,-9),(2,-5)$
12. Evaluate the following expression: $-4(4 z-x+5)$; use $x=-10$, and $z=-8$.
13. The area of a circle is given by the formula $A=\pi r^{2}$. Determine the area of a circle with radius $6 \mathrm{~mm}$.
14. Louie bought 9 packs of gum at $\$ 1.19$ each. How much money did he spend?
15. Write the following without the multiplication symbol: $16 \times \frac{1}{8} c$.
### Problem-Solving Strategies: Guess and Check and Work Backwards
This lesson will expand your toolbox of problem-solving strategies to include guess and check and work backwards. Let's begin by reviewing the four-step problem-solving plan.
Step 1: Understand the problem.
Step 2: Devise a plan - Translate.
Step 3: Carry out the plan - Solve.
Step 4: Look - Check and Interpret.
## Develop and Use the Strategy: Guess and Check
The strategy for the "guess and check" method is to guess a solution and use that guess in the problem to see if you get the correct answer. If the answer is too big or too small, then make another guess that will get you closer to the goal. You continue guessing until you arrive at the correct solution. The process might sound like a long one; however, the guessing process will often lead you to patterns that you can use to make better guesses along the way.
Here is an example of how this strategy is used in practice.
Example 1: Nadia takes a ribbon that is 48 inches long and cuts it in two pieces. One piece is three times as long as the other. How long is each piece?
Solution: We need to find two numbers that add to 48 . One number is three times the other number.
$\begin{array}{llll}\text { Guess } & 5 \text { and } 15 & \text { the sum is } 5+15=20 & \text { which is too small } \\ \text { Guess bigger numbers } & 6 \text { and } 18 & \text { the sum is } 6+18=24 & \text { which is too small }\end{array}$
However, you can see that the previous answer is exactly half of 48 .
Multiply 6 and 18 by two.
$\begin{array}{lll}\text { Our next guess is } \quad 12 \text { and } 36 & \text { The sum is } 12+36=48\end{array}$
## Develop and Use the Strategy: Work Backwards
The "work backwards" method works well for problems in which a series of operations is applied to an unknown quantity and you are given the resulting value. The strategy in these problems is to start with the result and apply the operations in reverse order until you find the unknown. Let's see how this method works by solving the following problem.
Example 2: Anne has a certain amount of money in her bank account on Friday morning. During the day she writes a check for \\$24.50, makes an ATM withdrawal of \\$80, and deposits a check for \\$235. At the end of the day, she sees that her balance is \\$451.25. How much money did she have in the bank at the beginning of the day?
Solution: We need to find the money in Anne's bank account at the beginning of the day on Friday. From the unknown amount, we subtract $\$ 24.50$ and $\$ 80$ and we add $\$ 235$. We end up with $\$ 451.25$. We need to start with the result and apply the operations in reverse. Start with $\$ 451.25$. Subtract $\$ 235$, add $\$ 80$, and then add $\$ 24.50$.
$$
451.25-235+80+24.50=320.75
$$
Anne had $\$ 320.75$ in her account at the beginning of the day on Friday.
## Plan and Compare Alternative Approaches to Solving Problems
Most word problems can be solved in more than one way. Often one method is more straightforward than others. In this section, you will see how different problem-solving approaches compare for solving different kinds of problems.
Example 3: Nadia's father is 36. He is 16 years older than four times Nadia's age. How old is Nadia?
Solution: This problem can be solved with either of the strategies you learned in this section. Let's solve the problem using both strategies.
## Guess and Check Method:
We need to find Nadia's age.
We know that her father is 16 years older than four times her age, or $4 \times$ (Nadia's age) +16 .
We know her father is 36 years old.
## Work Backwards Method:
Nadia's father is 36 years old.
To get from Nadia's age to her father's age, we multiply Nadia's age by four and add 16.
Working backward means we start with the father's age, subtract 16, and divide by 4 .
## Solve Real-World Problems Using Selected Strategies as Part of a Plan
Example 4: Hana rents a car for a day. Her car rental company charges $\$ 50$ per day and $\$ 0.40$ per mile. Peter rents a car from a different company that charges $\$ 70$ per day and $\$ 0.30$ per mile. How many miles do they have to drive before Hana and Peter pay the same price for the rental for the same number of miles?
Solution: Hana's total cost is $\$ 50$ plus $\$ 0.40$ times the number of miles.
Peter's total cost is $\$ 70$ plus $\$ 0.30$ times the number of miles.
Guess the number of miles and use this guess to calculate Hana's and Peter's total cost.
Keep guessing until their total cost is the same.
$$
\begin{array}{lll}
\text { Guess } & 50 \text { miles } & \\
\text { Check } & \$ 50+\$ 0.40(50)=\$ 70 & \$ 70+\$ 0.30(50)=\$ 85 \\
\text { Guess } & 60 \text { miles } & \\
\text { Check } & \$ 50+\$ 0.40(60)=\$ 74 & \$ 70+\$ 0.30(60)=\$ 88
\end{array}
$$
Notice that for an increase of 10 miles, the difference between total costs fell from $\$ 15$ to $\$ 14$. To get the difference to zero, we should try increasing the mileage by 140 miles. Guess $\quad 200$ miles
Check $\quad \$ 50+\$ 0.40(200)=\$ 130 \quad \$ 70+\$ 0.30(200)=\$ 130 \quad$ correct
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:ProblemSolvingWord Problems 2 (12:20)
1. Finish the problem we started in Example 3.
2. Nadia is at home and Peter is at school, which is 6 miles away from home. They start traveling toward each other at the same time. Nadia is walking at 3.5 miles per hour and Peter is skateboarding at 6 miles per hour. When will they meet and how far from home is their meeting place?
3. Peter bought several notebooks at Staples for $\$ 2.25$ each and he bought a few more notebooks at Rite-Aid for $\$ 2$ each. He spent the same amount of money in both places and he bought 17 notebooks in total. How many notebooks did Peter buy in each store?
4. Andrew took a handful of change out of his pocket and noticed that he was holding only dimes and quarters in his hand. He counted that he had 22 coins that amounted to $\$ 4$. How many quarters and how many dimes does Andrew have?
5. Anne wants to put a fence around her rose bed that is one and a half times as long as it is wide. She uses 50 feet of fencing. What are the dimensions of the garden?
6. Peter is outside looking at the pigs and chickens in the yard. Nadia is indoors and cannot see the animals. Peter gives her a puzzle. He tells her that he counts 13 heads and 36 feet and asks her how many pigs and how many chickens are in the yard. Help Nadia find the answer.
7. Andrew invests $\$ 8000$ in two types of accounts: a savings account that pays $5.25 \%$ interest per year and a more risky account that pays $9 \%$ interest per year. At the end of the year, he has $\$ 450$ in interest from the two accounts. Find the amount of money invested in each account.
8. There is a bowl of candy sitting on our kitchen table. This morning Nadia takes one-sixth of the candy. Later that morning Peter takes one-fourth of the candy that's left. This afternoon, Andrew takes one-fifth of what's left in the bowl and finally Anne takes one-third of what is left in the bowl. If there are 16 candies left in the bowl at the end of the day, how much candy was there at the beginning of the day?
9. Nadia can completely mow the lawn by herself in 30 minutes. Peter can completely mow the lawn by himself in 45 minutes. How long does it take both of them to mow the lawn together?
### Problem-Solving Strategies - Using Graphs to solve a problem
In this section, we will look at a few examples of linear relationships that occur in real-world problems and see how we can solve them using grpahs. Remember back to our Problem Solving Plan:
## Understand the problem
2. Devise a plan - Translate
3. Carry out the plan - Solve
4. Look - Check and Interpret
## Example 1
A cell phone company is offering its costumers the following deal. You can buy a new cell phone for $\$ 60$ and pay a monthly flat rate of $\$ 40$ per month for unlimited calls. How much money will this deal cost you after 9 months?
## Solution
Let's follow the problem solving plan.
Step 1: The cell phone costs $\$ 60$, the calling plan costs $\$ 40$ per month
Let $x=$ number of months
Let $y=$ total cost in dollars
Step 2: Let's solve this problem by making a graph that shows the number of months on the horizontal axis and the cost on the vertical axis.
Since you pay $\$ 60$ for the phone when you get the phone, then the $y$-intercept is $(0,60)$.
You pay $\$ 40$ for each month, so the cost rises by $\$ 40$ for one month, so the slope $=40$.
We can graph this line using the slope-intercept method.
Step 3: The question was: "How much will this deal cost after 9 months?"
We can now read the answer from the graph. We draw a vertical line from 9 months until it meets the graph, and then draw a horizontal line until it meets the vertical axis.
We see that after 9 months you pay approximately $\$ 420$.
Step 4: To check if this is correct, let's think of the deal again. Originally, you pay $\$ 60$ and then $\$ 40$ for 9 months.
$$
\begin{aligned}
\text { Phone } & =\$ 60 \\
\text { Calling plan } & =\$ 40 \times 9=\$ 360 \\
\text { Total cost } & =\$ 420 .
\end{aligned}
$$
## The answer checks out.
## Example 2
A stretched spring has a length of 12 inches when a weight of $2 \mathrm{lbs}$ is attached to the spring. The same spring has a length of 18 inches when a weight of $5 \mathrm{lbs}$ is attached to the spring. It is known from physics that within certain weight limits, the function that describes how much a spring stretches with different weights is a linear function. What is the length of the spring when no weights are attached?
## Solution
Let's apply problem solving techniques to our problem.
## Step 1:
We know: the length of the spring $=12$ inches when weight $=2 \mathrm{lbs}$.
the length of the spring $=18$ inches when weight $=5 \mathrm{lbs}$.
We want: the length of the spring when the weight $=0 \mathrm{lbs}$.
Let $x=$ the weight attached to the spring.
Let $y=$ the length of the spring
## Step 2
Let's solve this problem by making a graph that shows the weight on the horizontal axis and the length of the spring on the vertical axis.
We have two points we can graph.
When the weight is $2 \mathrm{lbs}$, the length of the spring is 12 inches. This gives point $(2,12)$.
When the weight is $5 \mathrm{lbs}$, the length of the spring is 18 inches. This gives point $(5,18)$.
If we join these two points by a line and extend it in both directions we get the relationship between weight and length of the spring.
## Step 3
The question was: "What is the length of the spring when no weights are attached?"
We can answer this question by reading the graph we just made. When there is no weight on the spring, the $x$ value equals to zero, so we are just looking for the $y$-intercept of the graph. Looking at the graph we see that the $y$-intercept is approximately 8 inches.
## Step 4
To check if this correct, let's think of the problem again.
You can see that the length of the spring goes up by 6 inches when the weight is increased by $3 \mathrm{lbs}$, so the slope of the line is $\frac{6 \text { inches }}{3 \text { lbs }}=2$ inches $/$ lb. To find the length of the spring when there is no weight attached, we look at the spring when there are $2 \mathrm{lbs}$ attached. For each pound we take off, the spring will shorten by 2 inches. Since we take off $2 \mathrm{lbs}$, the spring will be shorter by 4 inches. So, the length of the spring with no weights is 12 inches -4 inches $=8$ inches.
## The answer checks out.
## Example 3
Christine took one hour to read 22 pages of Harry Potter and the Order of the Phoenix. She has 100 pages left to read in order to finish the book. Assuming that she reads at a constant rate of pages per hour, how much time should she expect to spend reading in order to finish the book?
Solution: Let's apply the problem solving techniques:
Step 1: We know that it takes Christine takes 1 hour to read 22 pages.
We want to know how much time it takes her to read 100 pages.
Let $x=$ the time expressed in hours.
Let $y=$ the number of pages.
Step 2: Let's solve this problem by making a graph that shows the number of hours spent reading on the horizontal axis and the number of pages on the vertical axis.
We have two points we can graph.
Christine takes one hour to read 22 pages. This gives point $(1,22)$.
A second point is not given but we know that Christine takes 0 hours to read 0 pages. This gives the point $(0,0)$.
If we join these two points by a line and extend it in both directions we get the relationship between the amount of time spent reading and the number of pages read.
Step 3: The question was: "How much time should Christine expect to spend reading 100 pages?"
We find the answer from reading the graph - we draw a horizontal line from 100 pages until it meets the graph and then we draw the vertical until it meets the horizontal axis. We see that it takes approximately $\mathbf{4 . 5}$ hours to read the remaining 100 pages.
Step 4: To check if this correct, let's think of the problem again. We know that Christine reads 22 pages per hour. This is the slope of the line or the rate at which she is reading. To find how many hours it takes her to read 100 pages, we divide the number of pages by the rate. In this case, $\frac{100 \text { pages }}{22 \text { pages per hour }}=4.54$ hours. This is very close to what we gathered from reading the graph.
## The answer checks out.
## Example 4
Aatif wants to buy a surfboard that costs $\$ 249$. He was given a birthday present of $\$ 50$ and he has a summer job that pays him $\$ 6.50$ per hour. To be able to buy the surfboard, how many hours does he need to work?
## Solution
Let's apply the problem solving techniques.
## Step 1
We know - Surfboard costs $\$ 249$.
He has $\$ 50$.
His job pays $\$ 6.50$ per hour.
We want - How many hours does Aatif need to work to buy the surfboard?
Let $x=$ the time expressed in hours
Let $y=$ Aatif's earnings
## Step 2
Let's solve this problem by making a graph that shows the number of hours spent working on the horizontal axis and Aatif's earnings on the vertical axis.
Peter has $\$ 50$ at the beginning. This is the $y$-intercept of $(0,50)$.
He earns $\$ 6.50$ per hour. This is the slope of the line.
We can graph this line using the slope-intercept method. We graph the $y$-intercept of $(0,50)$ and we know that for each unit in the horizontal direction the line rises by 6.5 units in the vertical direction. Here is the line that describes this situation.
## Step 3
The question was "How many hours does Aatif need to work in order to buy the surfboard?"
We find the answer from reading the graph. Since the surfboard costs $\$ 249$, we draw a horizontal line from $\$ 249$ on the vertical axis until it meets the graph and then we draw a vertical line downwards until it meets the horizontal axis. We see that it takesapproximately $\mathbf{3 1}$ hours to earn the money.
## Step 4
To check if this correct, let's think of the problem again.
We know that Aatif has $\$ 50$ and needs $\$ 249$ to buy the surfboard. So, he needs to earn $\$ 249-\$ 50=\$ 199$ from his job.
His job pays $\$ 6.50$ per hour. To find how many hours he need to work we divide $\frac{\$ 199}{\$ 6.50 \text { per hour }}=30.6$ hours. This is very close to the result we obtained from reading the graph.
## The answer checks out.
## Review Questions
Solve the following problems by making a graph and reading a graph.
1. A gym is offering a deal to new members. Customers can sign up by paying a registration fee of $\$ 200$ and a monthly fee of $\$ 39$. How much will this membership cost a member by the end of the year?
2. A candle is burning at a linear rate. The candle measures five inches two minutes after it was lit. It measures three inches eight minutes after it was lit. What was the original length of the candle?
3. Tali is trying to find the width of a page of his telephone book. In order to do this, he takes a measurement and finds out that 550 pages measures 1.25 inches. What is the width of one page of the phone book?
4. Bobby and Petra are running a lemonade stand and they charge 45 cents for each glass of lemonade. In order to break even they must make $\$ 25$. How many glasses of lemonade must they sell to break even?
### Problem-Solving Strategies: Use a For- mula
Some problems are easily solved by applying a formula, such as The Percent Equation or the area of a circle. In this lesson, you will include using formulas in your toolbox of problem-solving strategies.
An architect is designing a room that is going to be twice as long as it is wide. The total square footage of the room is going to be 722square feet. What are the dimensions in feet of the room?
This situation applies very well to a formula. The formula for the area of a rectangle is: $A=l(w)$, where $l=$ length and $w=$ width. From the situation, we know the length is twice as long as the width. Translating this into an algebraic equation, we get:
$$
A=(2 w) w
$$
Simplifying the equation: $A=2 w^{2}$
Substituting the known value for $A$ : $722=2 w^{2}$
$$
\begin{aligned}
2 x^{2} & =722 \\
x^{2} & =361 \\
x & =\sqrt{361}=19 \\
2 x & =2 \times 19=38 \\
x & =19
\end{aligned}
$$
Divide both sides by 2 .
Take the square root of both sides.
The width is 19 feet and the length is 38 feet.
## Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:Word Problem Solving 3 (11:06)
MEDIA
Click image to the left for more content.
1. Patricia is building a sandbox for her daughter. It's going to be five feet wide and eight feet long. She wants the height of the sandbox to be four inches above the height of the sand. She has 30 cubic feet of sand. How high should the sandbox be?
2. A 500-sheet stack of copy paper is 1.75 inches high. The paper tray on a commercial copy machine holds a two foot high stack of paper. Approximately how many sheets is this?
3. It was sale day at Macy's and everything was $20 \%$ less than the regular price. Peter bought a pair of shoes, and using a coupon, got an additional 10\% off the discounted price. The price he paid for the shoes was $\$ 36$. How much did the shoes cost originally?
4. Peter is planning to show a video file to the school at graduation, but he's worried that the distance that the audience sits from the speakers will cause the sound and the picture to be out of sync. If the audience sits 20 meters from the speakers, what is the delay between the picture and the sound? (The speed of sound in air is 340 meters per second).
5. Rosa has saved all year and wishes to spend the money she has on new clothes and a vacation. She will spend $30 \%$ more on the vacation than on clothes. If she saved $\$ 1000$ in total, how much money (to the nearest whole dollar) can she spend on the vacation?
6. On a DVD, data is stored between a radius of $2.3 \mathrm{~cm}$ and $5.7 \mathrm{~cm}$. Calculate the total area available for data storage in square $\mathrm{cm}$.
7. If a Blu-ray ${ }^{T M}$ DVD stores 25 gigabytes (GB), what is the storage density, in GB per square $\mathrm{cm}$ ?
8. The volume of a cone is given by the formula Volume $=\frac{\pi r^{2}(h)}{3}$, where $r=$ radius, and $h=$ height of cone. Determine the amount of liquid a paper cone can hold with a 1.5 inches diameter and a 5 inches height.
9. Consider the conversion 1 meter $=39.37$ inches. How many inches are in a kilometer? (Hint: A kilometer is equal to 1,000 meters)
10. Yanni's motorcycle travels 108 miles/hour. $1 \mathrm{mph}=0.44704$ meters/second. How many meters did Yanni travel in 45 seconds?
11. The area of a rectangle is given by the formula $A=l(w)$. A rectangle has an area of 132 square centimeters and a length of 11 centimeters. What is the perimeter of the rectangle?
12. The surface area of a cube is given by the formula: SurfaceArea $=6 x^{2}$, where $x=$ side of the cube. Determine the surface area of a die with a 1 inch side length.
## Lesson Summary
The four steps of the problem solving plan when using graphs are:
## Understand the Problem
2. Devise a Plan-Translate: Make a graph.
3. Carry Out the Plan-Solve: Use the graph to answer the question asked.
4. Look-Check and Interpret
## Problem Set
Solve the following problems by making a graph and reading it.
1. A gym is offering a deal to new members. Customers can sign up by paying a registration fee of $\$ 200$ and a monthly fee of $\$ 39$.
(a) How much will this membership cost a member by the end of the year?
(b) The old membership rate was $\$ 49$ a month with a registration fee of $\$ 100$. How much more would a year's membership cost at that rate?
(c) Bonus: For what number of months would the two membership rates be the same?
2. A candle is burning at a linear rate. The candle measures five inches two minutes after it was lit. It measures three inches eight minutes after it was lit.
(a) What was the original length of the candle?
(b) How long will it take to burn down to a half-inch stub?
(c) Six half-inch stubs of candle can be melted together to make a new candle measuring $2 \frac{5}{6}$ inches (a little wax gets lost in the process). How many stubs would it take to make three candles the size of the original candle?
3. A dipped candle is made by taking a wick and dipping it repeatedly in melted wax. The candle gets a little bit thicker with each added layer of wax. After it has been dipped three times, the candle is $6.5 \mathrm{~mm}$ thick. After it has been dipped six times, it is $11 \mathrm{~mm}$ thick.
(a) How thick is the wick before the wax is added?
(b) How many times does the wick need to be dipped to create a candle $2 \mathrm{~cm}$ thick?
4. Tali is trying to find the thickness of a page of his telephone book. In order to do this, he takes a measurement and finds out that 55 pages measures $\frac{1}{8}$ inch. What is the thickness of one page of the phone book?
5. Bobby and Petra are running a lemonade stand and they charge 45 cents for each glass of lemonade. In order to break even they must make $\$ 25$.
(a) How many glasses of lemonade must they sell to break even?
(b) When they've sold $\$ 18$ worth of lemonade, they realize that they only have enough lemons left to make 10 more glasses. To break even now, they'll need to sell those last 10 glasses at a higher price. What does the new price need to be?
6. Dale is making cookies using a recipe that calls for 2.5 cups of flour for two dozen cookies. How many cups of flour does he need to make five dozen cookies?
7. To buy a car, Jason makes a down payment of $\$ 1500$ and pays $\$ 350$ per month in installments.
(a) How much money has Jason paid at the end of one year?
(b) If the total cost of the car is $\$ 8500$, how long will it take Jason to finish paying it off?
(c) The resale value of the car decreases by $\$ 100$ each month from the original purchase price. If Jason sells the car as soon as he finishes paying it off, how much will he get for it? 8. Anne transplants a rose seedling in her garden. She wants to track the growth of the rose so she measures its height every week. On the third week, she finds that the rose is 10 inches tall and on the eleventh week she finds that the rose is 14 inches tall. Assuming the rose grows linearly with time, what was the height of the rose when Anne planted it?
9. Ravi hangs from a giant spring whose length is $5 \mathrm{~m}$. When his child Nimi hangs from the spring its length is 2 $\mathrm{m}$. Ravi weighs $160 \mathrm{lbs}$ and Nimi weighs $40 \mathrm{lbs}$. Write the equation for this problem in slope-intercept form. What should we expect the length of the spring to be when his wife Amardeep, who weighs 140 lbs, hangs from it?
10. Nadia is placing different weights on a spring and measuring the length of the stretched spring. She finds that for a 100 gram weight the length of the stretched spring is $20 \mathrm{~cm}$ and for a 300 gram weight the length of the stretched spring is $25 \mathrm{~cm}$.
(a) What is the unstretched length of the spring?
(b) If the spring can only stretch to twice its unstretched length before it breaks, how much weight can it hold?
11. Andrew is a submarine commander. He decides to surface his submarine to periscope depth. It takes him 20 minutes to get from a depth of 400 feet to a depth of 50 feet.
(a) What was the submarine's depth five minutes after it started surfacing?
(b) How much longer would it take at that rate to get all the way to the surface?
12. Kiersta's phone has completely run out of battery power when she puts it on the charger. Ten minutes later, when the phone is $40 \%$ recharged, Kiersta's friend Danielle calls and Kiersta takes the phone off the charger to talk to her. When she hangs up 45 minutes later, her phone has $10 \%$ of its charge left. Then she gets another call from her friend Kwan.
(a) How long can she spend talking to Kwan before the battery runs out again?
(b) If she puts the phone back on the charger afterward, how long will it take to recharge completely?
13. Marji is painting a 75-foot fence. She starts applying the first coat of paint at 2 PM, and by 2:10 she has painted 30 feet of the fence. At 2:15, her husband, who paints about $\frac{2}{3}$ as fast as she does, comes to join her.
(a) How much of the fence has Marji painted when her husband joins in?
(b) When will they have painted the whole fence?
(c) How long will it take them to apply the second coat of paint if they work together the whole time?
| Textbooks |
\begin{document}
\title[Even and odd Kauffman bracket ideals]{Even and odd Kauffman bracket ideals for genus-1 tangles}
\author{Susan M. Abernathy} \address{Department of Mathematics\\ Angelo State University\\ ASU Station \#10900\\ San Angelo, TX 76909\\ USA} \email{[email protected]} \thanks{The first author was supported as a research assistant by NSF-DMS-1311911} \urladdr{http://www.angelo.edu/faculty/sabernathy/}
\author{Patrick M. Gilmer} \address{Department of Mathematics\\ Louisiana State University\\ Baton Rouge, LA 70803\\ USA} \email{[email protected]} \thanks{The second author was partially supported by NSF-DMS-1311911} \urladdr{www.math.lsu.edu/\textasciitilde gilmer/}
\subjclass[2010]{57M25} \date{August 22, 2016}
\begin{abstract}
This paper refines previous work by the first author. We study the question of which links in the 3-sphere can be obtained as closures of a given 1-manifold in an unknotted solid torus in the 3-sphere (or genus-1 tangle) by adjoining another 1-manifold in the complementary solid torus. We distinguish between even and odd closures, and define even and odd versions of the Kauffman bracket ideal. These even and odd Kauffman bracket ideals are used to obstruct even and odd tangle closures. Using a basis of Habiro's for the even Kauffman bracket skein module of the solid torus, we define bases for the even and odd skein module of the solid torus relative to two points. These even and odd bases allow us to compute a finite list of generators for the even and odd Kauffman bracket ideals of a genus-1 tangle. We do this explicitly for three examples. Furthermore, we use the even and odd Kauffman bracket ideals to conclude in some cases that the determinants of all even/odd closures of a genus-1 tangle possess a certain divisibility.
\end{abstract}
\keywords{tangles, tangle embedding, determinants, Kauffman bracket skein module}
\maketitle
\tableofcontents
\section{Introduction}\label{intro}
Let $M\subseteq S^3$ be a compact, oriented 3-manifold with boundary. Then an $(M,2n)$-tangle is 1-manifold with $2n$ boundary components properly embedded in $M$. We refer to $(S^1 \times D^2,2)$-tangles where $S^1 \times D^2$ is a unknotted solid torus in $S^3$ as genus-1 tangles.
An $(M,2n)$-tangle $\mathcal{T}$ embeds in a link $L\subseteq S^3$ if there exists a complementary 1-manifold $\mathcal{T}^\prime$ with $2n$ boundary components in $S^3-Int(M)$ such that upon gluing $\mathcal{T}^\prime$ to $\mathcal{T}$ along their boundaries, we obtain a link isotopic to $L$. Such a link is called a closure of $\mathcal{T}$. We refer to $\mathcal{T}^\prime$ as the complementary 1-manifold of the closure. Note that if $\mathcal{T}$ is a genus-1 tangle, then $\mathcal{T}^\prime$ is also a genus-1 tangle. The focus of this paper is genus-1 tangle embedding.
In \cite{ab, thesis}, the first author defined the notion of even and odd closures for any genus-1 tangle $\mathcal{G}$ with respect to a longitude $l$ on the boundary of the solid torus which misses the boundary points of $\mathcal{T}$.
If we choose $l$ to be the longitude pictured in Figure \ref{fig:krebes}
and assume that the boundary points of $\mathcal{T}$ are in the complement of $l$, then we may think of even and odd closures
intuitively as follows. Even (respectively, odd) closures are those whose complementary 1-manifold passes through the hole of the solid torus containing $\mathcal{G}$ an even (respectively, odd) number of times. For the remainder of this paper, when we discuss even and odd closures, we mean even and odd with respect to the longitude $l$.
In \cite{abkb,thesis}, the first author defined the Kauffman bracket ideal of an $(M,2n)$-tangle $\mathcal{T}$ to be the ideal $I_\mathcal{T}$ of $\mathbb{Z}[A,A^{-1}]$ generated by the reduced Kauffman bracket polynomials of all closures of $\mathcal{T}$.
This ideal gave an obstruction to embedding. In the case $(M,2n)= (B^3,4)$, this ideal was first studied by Przytycki, Silver and Williams \cite{psw}.
The first author outlined a method for computing this ideal in the case of genus-1 tangles
using skein theory techniques. In this paper, we define an even and odd version of the Kauffman bracket ideal for genus-1 tangles. The even Kauffman bracket ideal of a genus-1 tangle $\mathcal{G}$ is the ideal $\ev{\mathcal{G}}$ generated by the reduced Kauffman bracket polynomials of all even closures of $\mathcal{G}$. The odd Kauffman bracket ideal $\od{\mathcal{G}}$ is defined similarly. If an ideal is equal to $\mathbb{Z}[A,A^{-1}]$, we refer to that ideal as trivial.
The following proposition is an immediate consequence of these definitions.
\begin{proposition} Let $\mathcal{G}$ be a genus-1 tangle. If $\ev{\mathcal{G}}$ (respectively, $\od{\mathcal{G}}$) is non-trivial, then the unknot is not an even (respectively, odd) closure of $\mathcal{G}$. More generally, if $L$ is an even (respectively, odd) closure of $\mathcal{G}$, then the reduced Kauffman bracket polynomial of $L$ must lie in $\ev{\mathcal{G}}$ (respectively, $\od{\mathcal{G}}$). Finally, we have that $I_\mathcal{G} = \ev{\mathcal{G}} + \od{\mathcal{G}}$. \end{proposition}
In \S\ref{section:kbsm}, we recall the basics of Kauffman bracket skein modules. In \S\ref{section:evenodd}, we define bases for the even and odd Kauffman bracket skein modules of $S^1 \times D^2$ relative to two points. These even and odd bases are defined in terms of a basis for the even skein module of the solid torus due to Habiro \cite{hab}.
In \S\ref{section:graphbasis}, we recall the graph basis defined in \cite{abkb,thesis}. In \S\ref{section:method}, we outline a method for computing a finite list of generators for the even and odd Kauffman bracket ideals
of genus-1 tangles with two boundary points.
We note that if the ordinary Kauffman bracket ideal is non-trivial, then both the even and odd Kauffman bracket ideals must be non-trivial.
However, the converse is not true. In \S\ref{section:examples},
we examine some specific examples. We
show that Krebes's, tangle $\mathcal{A}$ \cite{kr}, pictured in Figure \ref{fig:krebes}, has trivial even ideal and non-trivial odd ideal. In \cite{abkb,thesis}, we showed that the ordinary Kauffman bracket ideal of Krebes's tangle $\mathcal{A}$ is trivial.
We give an example of a rather simple genus-1 tangle, $\mathcal{D}$ in Figure \ref{fig:smallex} which has non-trivial even but trivial odd Kauffman bracket ideals. See Figure \ref{fig:smalltangletrivialclosure} for an odd closure
of $\mathcal{D}$
which is trivial.
We
also consider a
particularly interesting tangle $\mathcal{H}$ (in Figure \ref{fig:85ex}).
This tangle has non-trivial even ideal and non-trivial odd ideal. Thus it does not posses a a trivial closure, but
the ordinary Kauffman bracket ideal of $\mathcal{H}$ is trivial.
The determinant $\det(L)$ of a link $L$ is a classical link with well-known alternative definitions. On the one hand, this invariant is the absolute value of the determinant of a Seifert matrix for $L$ symmetrized. It can also be described as the order of the first homology group of the double branched cover of $S^3$ along $L$ (this is interpreted to be zero if this homology group is infinite). In \cite{ab}, the first author used the homology of double branched covers to show that any odd closure of Krebes's tangle has determinant divisible by $3$. Here we can reach this result as a consequence of our calculation of the odd Kauffman
bracket ideal Krebes's tangle. We also obtain similar results for other tangles in the same way. Ultimately this approach to the determinants of closures rests
on Jones' observation \cite[Corollary 13]{J} that his polynomial evaluated at $t=-1$ is the determinant (up to sign), and Kauffman's bracket
polynomial
description \cite{K} of the Jones polynomial. In \S\ref{section:determinant}, we relate the even and odd ideals of an genus-1 tangle to the determinants of even and odd closures of that tangle.
\begin{figure}
\caption{Krebes's tangle $\mathcal{A}$ and a longitude $l$.}
\label{fig:krebes}
\end{figure}
\section{Kauffman bracket skein modules}\label{section:kbsm}
The Kauffman bracket polynomial of a framed link $D$, denoted by $\langle D \rangle$, is an element of $\mathbb{Z}[A,A^{-1}]$ given by the following three relations, where $\delta = -A^2-A^{-2}$: \begin{enumerate}[(i)] \item $\displaystyle \langle\begin{minipage}{.5in}\begin{center}\includegraphics[width=.4in]{unsmoothed.pdf}\end{center}\end{minipage} \rangle= A \langle\begin{minipage}{.5in}\begin{center}\includegraphics[width=.4in]{a-smoothing.pdf}\end{center}\end{minipage} \rangle+ A^{-1}\langle\begin{minipage}{.5in}\begin{center}\includegraphics[width=.4in]{b-smoothing.pdf}\end{center}\end{minipage} \rangle $ \item $ \langle D^\prime \coprod \begin{minipage}{.4in}\begin{center}\includegraphics[width=.3in]{circle.pdf}\end{center}\end{minipage} \rangle= \delta \langle D^\prime\rangle.$ \item $ \langle \begin{minipage}{.1in}\begin{center}\end{center}\end{minipage} \rangle= 1.$ \end{enumerate} We let $\langle D \rangle ^\prime$ denote the reduced Kauffman bracket polynomial of $D$; that is, where $\langle D \rangle ^\prime = \langle D\rangle/ \delta$.
The Kauffman bracket skein module of a 3-manifold $M$ is the $\mathbb{Z}[A,A^{-1}]$-module $K(M)$ generated by isotopy classes of framed links in $M$ modulo the Kauffman bracket relations above.
Of particular concern to us is the relative Kauffman bracket skein module. Let $M$ be a compact oriented 3-manifold with boundary and a set of $m$ specified marked framed
points on $\partial M$. Then the Kauffman bracket skein module of $M$ relative to the $m$ marked points is the $\mathbb{Z}[A,A^{-1}]$-module $K(M,m)$ generated by isotopy classes of framed 1-manifolds with boundary the marked framed
points modulo the above Kauffman bracket relations. We can view any genus-1 tangle (equipped with the blackboard framing) as a skein element in $K(S^1 \times D^2,2)$.
As in \cite{abkb, thesis}, we generalize the Hopf pairing on $K(S^1 \times D^2)$ defined in \cite{bhmv92} to obtain the relative Hopf pairing $\langle\text{ , }\rangle: K(S^1 \times D^2,2) \times K(S^1 \times D^2,2) \rightarrow K(S^3) = \mathbb{Z}[A,A^{-1}]$. Given $a$ and $b$ in $K(S^1 \times D^2,2)$, we let $$\langle a, b \rangle = \left\langle\hskip.05in\begin{minipage}{1in}\includegraphics[width=1in]{relativepairing.pdf}\end{minipage}\hspace{.04in}\right\rangle$$ where $a$ and $b$ lie in regular neighborhoods of the trivalent graphs.
If a genus-1 tangle $\mathcal{G}$ embeds in a link $L\subseteq S^3$, then we can describe this tangle embedding using the relative Hopf pairing. We have that $\langle L \rangle = \langle\mathcal{G},\mathcal{G}^\prime\rangle$ for some $\mathcal{G}^\prime \in K(S^1 \times D^2,2)$.
\section{Even and odd relative skein modules}\label{section:evenodd}
As in \cite{bhmv92}, we let $z$ denote a standard banded core of $S^1 \times D^2$ and the element this core represents in $K(S^1 \times D^2)$. A basis for $K(S^1 \times D^2)$ is given by $\{z^n\}_{n \ge 0}$. As described in \cite{hab}, one can obtain a $\mathbb{Z}_2$-graded algebra structure on the Kauffman bracket skein module $K(S^1 \times D^2)$ by letting $K^\text{even}(S^1 \times D^2)$ be the subalgebra of $K(S^1 \times D^2)$ generated by $z^2$ and $K^\text{odd}(S^1 \times D^2)$ be $zK^\text{even}(S^1 \times D^2)$. Then, one has that $K(S^1 \times D^2) = K^\text{even}(S^1 \times D^2) \oplus K^\text{odd}(S^1 \times D^2)$. This is because the Kauffman skein relations respect $\mathbb{Z}_2$-homology classes \cite[p.105]{gh}.
Suppose now that $S^1 \times D^2$ is equipped with two marked framed points in $\partial(S^1 \times D^2)$ and an essential curve $l$ in $\partial(S^1 \times D^2)$ missing the marked points and which bounds a disk $\mathfrak D$ in $S^1 \times D^2$.
Let $u$ be a framed 1-manifold in $S^1 \times D^2$ with the two given marked points as boundary. Then we say that $u$ is even (respectively, odd), if $u$ intersects $\mathfrak D$ an even (respectively, odd) number of times. Let $K^\text{even}(S^1 \times D^2,2)$ and $K^\text{odd}(S^1 \times D^2,2)$ be the submodules of $K(S^1 \times D^2,2)$ generated by all even and odd 1-manifolds, respectively. Then, we have that $K(S^1 \times D^2,2) = K^\text{even}(S^1 \times D^2,2) \oplus K^\text{odd}(S^1 \times D^2,2)$.
Note that if $L$ is an even closure of a genus-1 tangle $\mathcal{G}$, then the Kauffman bracket polynomial of $L$ can be written as $\langle L \rangle = \langle \mathcal{G},\mathcal{G}^\prime\rangle$ where
$\mathcal{G} \in K(S^1 \times D^2,2)$ and
$\mathcal{G}^\prime \in K^\text{even}(S^1 \times D^2,2)$. Here $l$ is a ``longitude'' for the first copy of $S^1 \times D^2$ and a ``meridian'' for the second copy of
$S^1 \times D^2$.The analogous statement is true for odd closures.
\begin{figure}
\caption{An even element of $K(S^1 \times D^2,2)$, where $\mathfrak D$ is the shaded disk}
\label{fig:evenskeinelt}
\end{figure}
In \cite{bhmv92}, a basis $\{Q_n\}_{n \ge 0}$ for $K(S^1 \times D^2)$ is given. It is orthogonal with respect to the Hopf pairing. Here $Q_n =\displaystyle \prod_{i=0}^{n-1} (z-\phi_i)$, where $\phi_i = -A^{2i+2}-A^{-2i-2}$ (in \cite{bhmv92} and elsewhere this is denoted $\lambda_i$). In the case $n=0$, we interpret the empty product as the identity which is represented by the empty link. In \cite{hab}, Habiro modifies the definition of $Q_n$
to obtain a new basis for the even submodule $K^\text{even}(S^1 \times D^2)$ given by $S_n = \displaystyle\prod_{i=0}^{n-1} (z^2-\phi_i^2)$ for $n\geq 0$. Note that $S_n = Q_n\displaystyle \prod_{i=0}^{n-1}(z + \phi_i)$.
We adapt Habiro's basis to obtain bases for the $K^\text{even}(S^1 \times D^2,2)$ and $K^\text{odd}(S^1 \times D^2,2)$. We refer to them as the even basis and odd basis, respectively, and define them as follows
(using the same $\mathfrak D$ as pictured in Figure \ref{fig:evenskeinelt}). The even basis consists of the following elements, where $n\geq 0$: $$x_n^\text{even} = \begin{minipage}{1in}\includegraphics[width=1in]{evenshort.pdf}\end{minipage}\text{ and }y_n^\text{even} = \begin{minipage}{1.03in}\includegraphics[width=1in]{evenlong.pdf}\end{minipage}.$$ Similarly, the odd basis consists of the following elements, where $n\geq 0$: $$x_n^\text{odd} = \begin{minipage}{1in}\includegraphics[width=1in]{oddshort.pdf}\end{minipage}\text{ and }y_n^\text{odd} = \begin{minipage}{1.03in}\includegraphics[width=1in]{oddlong.pdf}\end{minipage}.$$
That these are bases follows ultimately from the basis for $K(S^1 \times D^2,2)$ consisting of framed links described by isotopy classes of diagrams without crossings and without contractible loops in $S^1 \times D^2$. One also uses the fact that there is a triangular unimodular change of basis matrix over $\mathbb{Z}[A,A^{-1}]$ relating the bases $\{S_n\}$ and \{$z^{2m} \}$ for $K^\text{even}(S^1 \times D^2)$.
\section{Graph basis of $K_R(S^1 \times D^2,2)$}\label{section:graphbasis}
Trivalent graphs will be interpreted as in \cite{abkb, thesis, gh, kl, mv,l}.
Any unlabelled edge is assumed to be colored one. The colors of the three edges incident to a single vertex must form an admissible triple. Given non-negative integers $a$, $b$, and $c$, the triple $(a,b,c)$ is admissible if $|a-b| \leq c\leq a + b$ and $a + b + c \equiv 0 (\text{mod } 2)$. We use the notation of \cite{kl}: $\Delta_n$, $\theta (a,b,c)$, $\Tet \begin{bmatrix} a & b & e\\ c & d & f \end{bmatrix}$, and $\lambda^{a \text{ }b}_c$.
We use the graph basis defined in \cite{abkb,thesis}. Given a pair of non-negative integers $(i,\varepsilon)$ such that $\varepsilon = i+1$ or $\varepsilon = i-1$, let \[g_{i,\varepsilon} =\hspace{.05in}\begin{minipage}{1.25in}\includegraphics[width=1.25in]{graphbasiselt2.pdf}\end{minipage}.\]
Let $R$ denote the ring $\mathbb{Z}[A,A^{-1}]$ localized by inverting $A^{k}-1$ for all natural numbers $k$, and let $K_R(M,m)$ denote the Kauffman bracket skein module of $M$ relative to $m$ points with coefficients in $R$. According to \cite[Theorem 2.3]{pr}, we have that $K_R(M,m) = K(M,m) \otimes R$, so we can essentially view $K(M,m)$ as a subset of $K_R(M,m)$. We make this distinction because when computing a finite list of generators for the even and odd Kauffman bracket ideals, we pass through
$K_R(S^1 \times D^2,2)$ when using the doubling pairing defined in \cite[\S2.3]{abkb}. However, each of the generators we obtain is in fact an element of $K(S^1 \times D^2,2)$.
Recall, according to \cite{hp}, $K_R(S^1\times S^2)/\text{torsion}$ is isomorphic to $R$. Let $\psi: K_R(S^1\times S^2) \rightarrow R$ be the epimorphism that sends the empty link to $1 \in R.$ The doubling pairing is defined to be the symmetric pairing $\langle \text{ , }\rangle_D: K_R(S^1 \times D^2,2) \times K_R(S^1 \times D^2,2) \rightarrow R$ obtained by gluing two solid tori containing skein elements together via a certain orientation-reversing homeomorphism to obtain a skein element in $S^1\times S^2$, and evaluating this skein element under $\psi.$ Figure \ref{fig:doublingpairing} illustrates the doubling pairing of two graph basis elements. The thick dark colored loop indicates where a $0$-framed surgery is to be performed, converting $S^3$ to $S^1\times S^2$. According to \cite[Theorem 2.4]{abkb}, the graph basis is orthogonal with respect to the doubling pairing.
\begin{figure}
\caption{The doubling pairing of two graph basis elements. The bold loop indicates a 0-framed surgery.}
\label{fig:doublingpairing}
\end{figure}
\section{Applications to genus-1 tangle embedding}\label{section:method}
Let $\mathcal{G}$ be a genus-1 tangle. The Kauffman bracket polynomial of any even closure $L$ of $\mathcal{G}$ can be written as $\langle L \rangle = \langle \mathcal{G},\mathcal{G}^\prime\rangle$ where $\mathcal{G}^\prime \in K^\text{even}(S^1 \times D^2,2)$. So, $\langle \mathcal{G},x^\text{even}_n\rangle/\delta$ and $\langle \mathcal{G},y^\text{even}_n\rangle/\delta$ form a generating set for $I_\mathcal{G}^\text{even}$. Similarly, $\langle \mathcal{G},x^\text{odd}_n\rangle/\delta$ and $\langle \mathcal{G},y^\text{odd}_n\rangle/\delta$ form a generating set for $I_\mathcal{G}^\text{odd}$.
We will see in this section that these generating sets are finite.
We follow the same basic procedure as in \cite{abkb, thesis} to obtain finite lists of generators for $I_\mathcal{G}^\text{even}$ and $I_\mathcal{G}^\text{odd}$. First, we write $\mathcal{G}$ as a linear combination of graph basis elements $\mathcal{G} = \sum c_{i,\varepsilon}g_{i,\varepsilon}$. Since the graph basis is orthogonal, we have that $c_{i,\varepsilon} = \langle \mathcal{G}, g_{i,\varepsilon} \rangle_D / \langle g_{i,\varepsilon} , g_{i,\varepsilon} \rangle_D$ and only finitely many $c_{i,\varepsilon}$ are non-zero.
We then use this linear combination to compute the relative Hopf pairing of $\mathcal{G}$ with the even (respectively, odd) basis to obtain a generating set for $I_\mathcal{G}^\text{even}$ (respectively, $I_\mathcal{G}^\text{odd}$). The following results allow us to compute the relative Hopf pairing of the graph basis with the even and odd bases. The proof of Lemma \ref{lemma:removings} below is similar to that of \cite[Lemma 4.1]{abkb}.
\begin{lemma}\label{lemma:removings} $$\begin{minipage}{.75in}\includegraphics[height=.8in]{removings.pdf}\end{minipage} = \displaystyle\prod_{k=0}^{n-1}(\phi_i^2 - \phi_k^2) \hspace{.04in}\begin{minipage}{.3in}\includegraphics[height=.65in]{strandcoloredi.pdf}\end{minipage}$$ If $n=0$, we interpret $\displaystyle\prod_{k=0}^{n-1}(\phi_i^2 - \phi_k^2)$ as $1$. \end{lemma}
Notice that $\prod_{k=0}^{n-1}(\phi_i^2 - \phi_k^2)$ is zero if $n > i$.
\begin{proposition}\label{prop:pairing}
\begin{enumerate}[(i)] \item $\langle g_{i,\varepsilon},x^\text{even}_n\rangle = \theta(1,i,\varepsilon) \displaystyle\prod_{k=0}^{n-1}(\phi_i^2 - \phi_k^2)$.\\ \item $\langle g_{i,\varepsilon},y^\text{even}_n\rangle = \phi_i (\lambda^{i\text{ } 1}_\varepsilon)^{-2} \theta(1,i,\varepsilon) \displaystyle\prod_{k=0}^{n-1}(\phi_i^2 - \phi_k^2)$.\\ \item $\langle g_{i,\varepsilon},x^\text{odd}_n\rangle = \phi_i \theta(1,i,\varepsilon) \displaystyle\prod_{k=0}^{n-1}(\phi_i^2 - \phi_k^2)$.\\ \item $\langle g_{i,\varepsilon},y^\text{odd}_n\rangle = (\lambda^{i\text{ } 1}_\varepsilon)^{-2} \theta(1,i,\varepsilon) \displaystyle\prod_{k=0}^{n-1}(\phi_i^2 - \phi_k^2)$.\\ \end{enumerate} \noindent Each of these is zero if $n>i$. \end{proposition}
\begin{proof} \begin{enumerate}[(i)] \item We have from Lemma \ref{lemma:removings} that $$ \langle g_{i,\varepsilon}, x^\text{even}_n\rangle = \hspace{.05in}\begin{minipage}{1in}\includegraphics[height=.75in]{pairinggraphxeven.pdf}\end{minipage} = \displaystyle\prod_{k=0}^{n-1}(\phi^2_i - \phi^2_k)\hspace{.05in} \begin{minipage}{.65in}\includegraphics[height=.5in]{pairinggraphx2.pdf}\end{minipage} = \displaystyle\theta(1,i,\varepsilon)\prod_{k=0}^{n-1}(\phi^2_i - \phi^2_k).$$\\
\item Using Lemma \ref{lemma:removings}, \cite[ equations 2.2, 2.5]{abkb} and $\lambda^{i\text{ } j}_k= \lambda^{j\text{ }i}_k,$ $$\begin{array}{r c l} \langle g_{i,\varepsilon}, y^\text{even}_n\rangle & = & \begin{minipage}{.75in}\includegraphics[height=.85in]{pairinggraphyeven.pdf}\end{minipage} = \hspace{.05in} \displaystyle\phi_i\prod_{k=0}^{n-1}(\phi^2_i - \phi^2_k)\hspace{.05in} \begin{minipage}{.75in}\includegraphics[height=.85in]{pairinggraphy2.pdf}\end{minipage}\\ & & \\ & = & \displaystyle\phi_i(\lambda^{i\text{ } 1}_\varepsilon)^{-1}(\lambda^{1\text{ } i}_\varepsilon)^{-1}\prod_{k=0}^{n-1}(\phi^2_i - \phi^2_k)\hspace{.05in} \begin{minipage}{.6in}\includegraphics[height=.5in]{pairinggraphy3.pdf}\end{minipage}\\ & = & \displaystyle\phi_i(\lambda^{i\text{ } 1}_\varepsilon)^{-2}\theta(1,i,\varepsilon)\prod_{k=0}^{n-1}(\phi^2_i - \phi^2_k). \end{array}$$
\item \begin{align*}\langle g_{i,\varepsilon}, x^\text{odd}_n\rangle & = \hspace{.05in}\begin{minipage}{1.1in}\includegraphics[height=.75in]{pairinggraphxodd.pdf}\end{minipage} = \displaystyle\phi_i\prod_{k=0}^{n-1}(\phi^2_i - \phi^2_k)\hspace{.05in} \begin{minipage}{.65in}\includegraphics[height=.5in]{pairinggraphx2.pdf}\end{minipage}\\ & = \displaystyle\phi_i\theta(1,i,\varepsilon)\prod_{k=0}^{n-1}(\phi^2_i - \phi^2_k).\end{align*}\\
\item $$\begin{array}{r c l} \langle g_{i,\varepsilon}, y^\text{odd}_n\rangle & = & \begin{minipage}{.75in}\includegraphics[height=.85in]{pairinggraphyodd.pdf}\end{minipage} = \hspace{.05in} \displaystyle\prod_{k=0}^{n-1}(\phi^2_i - \phi^2_k)\hspace{.05in} \begin{minipage}{.75in}\includegraphics[height=.85in]{pairinggraphy2.pdf}\end{minipage}\\ & & \\ & = & \displaystyle(\lambda^{i\text{ } 1}_\varepsilon)^{-1}(\lambda^{1\text{ } i}_\varepsilon)^{-1}\prod_{k=0}^{n-1}(\phi^2_i - \phi^2_k)\hspace{.05in} \begin{minipage}{.6in}\includegraphics[height=.5in]{pairinggraphy3.pdf}\end{minipage}\\ & = & \displaystyle\displaystyle(\lambda^{i\text{ } 1}_\varepsilon)^{-2} \theta(1,i,\varepsilon)\prod_{k=0}^{n-1}(\phi^2_i - \phi^2_k). \end{array}$$ \end{enumerate} \end{proof}
Proposition \ref{prop:pairing} implies that only finitely many of $\langle \mathcal{G},x^\text{even}_n\rangle/\delta$ and $\langle \mathcal{G},y^\text{even}_n\rangle/\delta$ will be non-zero.
Similarly only finitely many of $\langle \mathcal{G},x^\text{odd}_n\rangle/\delta$ and $\langle \mathcal{G},y^\text{odd}_n\rangle/\delta$ will be non-zero. This is why we choose to define the even/odd bases for $K^\text{even/odd}(S^1 \times D^2,2)$ as we did above.
If, for instance, we replace $S_n$ by $z^{2n}$ in these definitions, then we would not have this finiteness.
\section{Examples}\label{section:examples}
We compute the even and odd Kauffman bracket ideals for three tangles $\mathcal{A}$, $\mathcal{D}$, and $\mathcal{H}$. In each of these computations, our first step is to compute the doubling pairing of the tangle in question with the graph basis. We leave out the full computation for the sake of brevity, but we follow the same procedure as in \cite[Appendix A]{abkb}. We then write each tangle as a linear combination of graph basis elements. It turns out that due to admissibility conditions, $\mathcal{A}$, $\mathcal{D}$, and $\mathcal{H}$ may all be written as $c_{0,1}g_{0,1} + c_{2,1}g_{2,1} + c_{2,3}g_{2,3}$ for some coefficients $c_{i,\varepsilon} \in R$. Thus we have using Proposition \ref{prop:pairing}:
\begin{lemma}\label{i012} If $\mathcal{G}$= $\mathcal{A}$, $\mathcal{D}$, or $\mathcal{H}$, $I_\mathcal{G}^\text{even}$ is generated by $\langle \mathcal{G}, x^\text{even}_i\rangle/\delta$ and $\langle \mathcal{G}, y^\text{even}_i\rangle/\delta$ where $0\leq i \leq 2$. Similarly, $I_\mathcal{G}^\text{odd}$ is generated by $\langle \mathcal{G}, x^\text{odd}_i\rangle/\delta$ and $\langle \mathcal{G}, y^\text{odd}_i\rangle/\delta$ where $0\leq i \leq 2$. \end{lemma}
For $\mathcal{G}$= $\mathcal{A}$, $\mathcal{D}$ or $\mathcal{H}$, we followed the same procedure as in \cite{abkb,thesis}, to find $I_\mathcal{A}^\text{even}$ and $I_\mathcal{A}^\text{odd}$, using Proposition \ref{prop:pairing}, Lemma \ref{i012}, and Mathematica. One can verify directly that the claimed ideals are indeed non-trivial using the computations in \S\ref{section:determinant}.
\subsection{Krebes's tangle $\mathcal{A}$ }\label{subsection:krebes}
We consider the genus-1 tangle given by Krebes \cite{kr} pictured in Figure \ref{fig:krebes}. We have that $$ \langle \mathcal{A}, g_{i,\varepsilon}\rangle_D = \begin{minipage}{.11in}\includegraphics[width=1.1in]{dpairingkrebes}\end{minipage} \hskip1in \quad \text{is the sum of } $$
\noindent $$
\frac{ \lambda^{1 \text{ }1}_i (\lambda^{1\text{ }1}_j)^{-1} (\lambda^{1\text{ }1}_k)^{-1} (\lambda^{1\text{ }1}_l)^{-1} \Delta_j \Delta_k \Delta_l
\Tet \begin{bmatrix} 1 & i & \varepsilon\\ 1 & j & 1 \end{bmatrix}
\Tet \begin{bmatrix} l & 1 & j \\ 1 & k & 1 \end{bmatrix}
\Tet \begin{bmatrix} 1 & \varepsilon & 1 \\ k & l & j \end{bmatrix}}
{\theta(1, 1, i) \theta(1,1,j) \theta(1,1,k) \theta(1,1,l) \theta(1,j,\varepsilon) \theta(\varepsilon, k, 1) \theta(l, k, j)} $$
\noindent over all $j$, $k$, and $l$ such that the following triples are admissible: $(1,1,i)$, $(1,1,j)$, $(1,j,\varepsilon)$, $(1,1,k)$, $(\varepsilon,k,1)$, $(1,1,l)$, and $(l,k,j)$. Admissibility conditions imply that 0 and 2 are the only possible admissible values for $j$, $k$, and $l$. Note that, if $i \ne 0, 2$, there are no such $j$,$k$, $l$, and the given sum is over an empty index set. Thus, the value of the sum is zero. So $\langle \mathcal{A}, g_{i,\varepsilon}\rangle_D =0$ unless $i=0$ or $i=2$.
The coefficients for $\mathcal{A}$ as a linear combination of the graph basis are \begin{equation*} c_{0,1} = \frac{-1-A^8+A^{12}}{1+A^4}, \quad c_{2,1} = \frac{-1+A^4+A^{12}}{A^6+A^{10}+A^{14}}, \text{ and } c_{2,3} = 1.\end{equation*}
After further computation, we obtain the generating sets given in the following result.
\begin{proposition} The even Kauffman bracket ideal $I_\mathcal{A}^\text{even}$ of Krebes's tangle $\mathcal{A}$ is trivial. The odd Kauffman bracket ideal of $\mathcal{A}$ is $I_\mathcal{A}^\text{odd} = \langle 9, 4+A^4\rangle$ which is non-trivial. \end{proposition}
\subsection{A small tangle, $\mathcal{D}$}\label{subsection:smallex}
We now consider the genus-1 tangle $\mathcal{D}$ pictured in Figure \ref{fig:smallex}. In contrast to Krebes's example, $\mathcal{D}$ has a non-trivial even Kauffman bracket ideal and a trivial odd Kauffman bracket ideal. We remark that we could also obtain a tangle with these properties from Krebes's tangle by sliding one endpoint of Krebes's tangle across the longitude $\ell$ and dragging the rest of tangle along behind this endpoint.
\begin{figure}
\caption{A genus-1 tangle, denoted by $\mathcal{D}$.}
\label{fig:smallex}
\end{figure}
We have that $$ \langle \mathcal{D}, g_{i,\varepsilon}\rangle_D = \begin{minipage}{1.1in}\includegraphics[width=1.1in]{dpairingd}\end{minipage} \quad \text{is the sum of } $$ $$\frac{ \lambda^{1 \text{ }1}_i (\lambda^{1\text{ }1}_j)^{-3}\Delta_j \Tet \begin{bmatrix} 1 & 1 & j \\ 1 & \varepsilon & i \end{bmatrix} \Tet \begin{bmatrix} 1 & i & \varepsilon \\ 1 & j & 1 \end{bmatrix}} {\theta(1, 1, i)\theta(1,1,j)\theta(1,\varepsilon,j)} $$
\noindent over all integers $j$ such that the following are admissible triples: $(1,1,i)$, $(1,1,j)$, and $(1, \varepsilon,j)$. Admissibility conditions imply that 0 and 2 are the only possible admissible values for $j$, and $\langle \mathcal{D}, g_{i,\varepsilon}\rangle_D =0$ unless $i=0$ or $i=2$.
The coefficients for $\mathcal{D}$ as a linear combination of the graph basis are \begin{equation*}
c_{0,1} = \frac{1-A^4-A^{12}}{A^2+A^6}, \quad c_{2,1} = \frac{1+A^8-A^{12}}{A^8+A^{12}+A^{16}} ,
\text{ and } c_{2,3} = A^2 .\end{equation*}
We obtain the following generating sets after further computation.
\begin{proposition} The even Kauffman bracket ideal of $\mathcal{D}$ is $I_\mathcal{D}^\text{even} = \langle 9,-2+A^4\rangle$ which is non-trivial. The odd Kauffman bracket ideal $I_\mathcal{D}^\text{odd}$ of $\mathcal{D}$ is trivial. \end{proposition}
Indeed, one can see that the odd closure shown in Figure \ref{fig:smalltangletrivialclosure} is the unknot, so $I_\mathcal{D}^\text{odd}$ must be trivial.
\begin{figure}
\caption{A trivial odd closure of the tangle $\mathcal{D}$.}
\label{fig:smalltangletrivialclosure}
\end{figure}
\subsection{A particularly interesting tangle, $\mathcal{H}$} \label{subsection:85ex}
We consider the genus-1 tangle $\mathcal{H}$ pictured in Figure \ref{fig:85ex}. \begin{figure}
\caption{A genus-1 tangle, denoted by $\mathcal{H}$.}
\label{fig:85ex}
\end{figure}
We have that $$ \langle \mathcal{H}, g_{i,\varepsilon}\rangle_D = \begin{minipage}{1.15in}\includegraphics[width=1.1in]{dpairing85}\end{minipage} \quad \text{is the sum of } $$ \noindent $
\frac{ \lambda^{1 \text{ }1}_i (\lambda^{1\text{ }1}_j)^{-3} (\lambda^{1\text{ }1}_k)^{-3}
\lambda^{1\text{ }1}_l
\Delta_j \Delta_k \Delta_l
\Tet \begin{bmatrix} 1 & i & \varepsilon\\ 1 & j & 1 \end{bmatrix}
\Tet \begin{bmatrix} \varepsilon & i & 1 \\ 1 & k & 1 \end{bmatrix}
\Tet \begin{bmatrix} 1 & 1 & l \\ 1 & \varepsilon & j \end{bmatrix}
\Tet \begin{bmatrix} 1 & k & \varepsilon \\ 1 & l & 1 \end{bmatrix}}
{\theta(1, 1, i) \theta(1,1,j) \theta(1,1,k) \theta(1,1,l) \theta(1,j,\varepsilon) \theta(1, k, \varepsilon) \theta(1,l,\varepsilon)}
$
\noindent over all $j$, $k$, and $l$ such that the following triples are admissible: $(1,1,i)$, $(1,1,j)$, $(1,j,\varepsilon)$, $(1,1,k)$, $(1,k, \varepsilon)$, $(1,1,l)$, and $(1,l,\varepsilon)$. Admissibility conditions imply that 0 and 2 are the only possible admissible values for $j$, $k$, and $l$. So, $\langle \mathcal{H}, g_{i,\varepsilon}\rangle_D =0$ unless $i=0$ or $i=2$.
The coefficients for $\mathcal{H}$ as a linear combination of the graph basis are \begin{align*}c_{0,1} =& \frac{-1+2A^4-3A^8+2A^{12}-3A^{16}+2A^{20}-A^{24}+A^{28}}{A^{12}+A^{16}} \\c_{2,1} = & \frac{-1+A^4-2A^8+3A^{12}-2A^{16}+3A^{20}-2A^{24}+A^{28}}{A^{18}+A^{22}+A^{26}} \text{ and } c_{2,3} = A^4.\end{align*}
The following generating sets are obtained after further computation.
\begin{proposition} The even Kauffman bracket ideal of $\mathcal{H}$ is $I_\mathcal{H}^\text{even} = \langle 5,1+A^4\rangle$ which is non-trivial. The odd Kauffman bracket ideal of $\mathcal{H}$ is $I_\mathcal{H}^\text{odd} = \langle 9,4+A^4\rangle$ which is also non-trivial. \end{proposition}
These corollaries follow immediately.
\begin{corollary} The Kauffman bracket ideal $I_\mathcal{H}$ of the genus-1 tangle $\mathcal{H}$ is trivial. \end{corollary}
\begin{corollary} The genus-1 tangle $\mathcal{H}$ does not embed in the unknot. \end{corollary}
Although $\mathcal{H}$ is not obstructed from embedding in the unknot by the ordinary Kauffman bracket ideal, the even and odd Kauffman bracket ideals, working together, do provide an obstruction.
\section{Relation to Determinants}\label{section:determinant}
Let $\omega$ denote $e^{\frac{\pi i}4}$, and $\Omega:\mathbb{Z}[A,A^{-1}] \rightarrow \mathbb{Z}[\omega]$ be the ring epimorphism sending $A$ to $\omega$. According to \cite[Prop. 1 on p. 329; \S 11]{kr}, \begin{equation}\det(L)= \omega^j \Omega({\langle L \rangle'}) \label{kre}\end{equation} for an integer $j$, chosen so that $\omega^j \Omega({\langle L \rangle'})$ is a non-negative integer. In fact, \ref{kre} follows easily from \cite[Corollary 3]{J} and \cite[Thm 2.8]{K} without consideration of the ``monocyclic states'' used in \cite{kr}.
\begin{proposition}\label{omeg} If $L$ is a closure of tangle $\mathcal{G}$, then $\det(L) \in \Omega(I_\mathcal{G}) \cap \mathbb{Z}$. If $L$ is an even closure, then $\det(L) \in \Omega(\ev{\mathcal{G}}) \cap \mathbb{Z}.$ If $L$ is an odd closure, then $\det(L) \in \Omega(\od{\mathcal{G}}) \cap \mathbb{Z}.$ \end{proposition}
\begin{proof} If $L$ is
a closure of $\mathcal{G}$, $A^j \langle L \rangle' \in I_\mathcal{G}$ for all $j \in \mathbb{Z}$. So for some $j$,
$$\det(L)= \omega^j \Omega (\langle L \rangle') =\Omega (A^j \langle L \rangle') \in \Omega(I_\mathcal{G}).$$
As $\det(L) \in \mathbb{Z}$, $\det(L) \in \Omega(I_\mathcal{G}) \cap \mathbb{Z}$. The other statements are proved similarly. \end{proof}
Let $\langle n \rangle _\mathbb{Z}$
denote the $\mathbb{Z}$-ideal generated by $n$. For the ideals computed in the examples above, noting that $\Omega(A^4)=-1$, we have:
\begin{equation} \Omega(\od{\mathcal{A}}) \cap \mathbb{Z} = \Omega(\langle 9,4+A^4 \rangle) \cap \mathbb{Z} =\Omega(\langle 3 \rangle) \cap \mathbb{Z}=\langle 3 \rangle_\mathbb{Z}.\label{3}\end{equation} \begin{equation*}\Omega(\ev{\mathcal{H}}) \cap \mathbb{Z}= \Omega(\langle 9,4+A^4 \rangle) \cap \mathbb{Z}=\Omega(\langle 3 \rangle) \cap \mathbb{Z}=\langle 3 \rangle_\mathbb{Z}.\end{equation*} \begin{equation*} \Omega(\ev{\mathcal{D}}) \cap \mathbb{Z}= \Omega(\langle 9,-2+A^4 \rangle) \cap \mathbb{Z}=\Omega(\langle 3 \rangle) \cap \mathbb{Z}= \langle 3 \rangle_\mathbb{Z}.\end{equation*} \begin{equation*} \Omega(\od{\mathcal{H}}) \cap \mathbb{Z}= \Omega(\langle 5,1+A^4 \rangle) \cap \mathbb{Z}=\Omega(\langle 5 \rangle) \cap \mathbb{Z}= \langle 5 \rangle_\mathbb{Z}.\end{equation*}
The first sentence in Proposition \ref{omeg2} has the same content as \cite[Theorem 1.3]{ab}.
\begin{proposition}\label{omeg2} Let $L$ be
an odd closure of $\mathcal{A}$, then $\det(L) \equiv 0 \pmod{3}$. Let $L$ be
an even closure of $\mathcal{D}$, then $\det(L) \equiv 0 \pmod{3}$. If $L$ is an odd closure of $\mathcal{H}$, $\det(L) \equiv 0 \pmod{5}$. If $L$ is an even closure of $\mathcal{H}$, $\det(L) \equiv 0 \pmod{3}$. \end{proposition}
\begin{proof} If $L$ is
an odd closure of $\mathcal{A}$, by Proposition \ref{omeg}, $\det(L) \in \Omega(\od{\mathcal{A}}) \cap \mathbb{Z}$. By (\ref{3}), $\det(L) \equiv 0 \pmod{3}$. The other statements are proved similarly. \end{proof}
Note the tangle $\mathcal{F}$ (pictured below)
was shown in \cite{abkb} to have $I_\mathcal{F}= \langle 11,4-A^4 \rangle$. Thus $\Omega (I_\mathcal{F}) \cap Z= \langle 11,5 \rangle_\mathbb{Z} =\langle1\rangle_\mathbb{Z} =\mathbb{Z}$.
\begin{figure}
\caption{The genus-1 tangle $\mathcal{F}$.}
\label{fig:tanglef}
\end{figure}
\end{document} | arXiv |
\begin{document}
\centerline{\large{\bf {Scale-free property for degrees and weights}}}
\centerline{\large{\bf {in an $N$-interactions random graph model}}}
\centerline{ {\sc Istv\'an Fazekas} and {\sc Bettina Porv\'azsnyik} }
Faculty of Informatics, University of Debrecen, P.O. Box 12, 4010 Debrecen, Hungary, \centerline{ e-mail: [email protected], [email protected].}
\begin{abstract} A general random graph evolution mechanism is defined. The evolution is a combination of the preferential attachment model and the interaction of $N$ vertices $\left( N \geq 3\right)$. A vertex in the graph is characterized by its degree and its weight. The weight of a given vertex is the number of the interactions of the vertex. The asymptotic behaviour of the graph is studied. Scale-free properties both for the degrees and the weights are proved. It turns out that any exponent in $(2,\infty)$ can be achieved. The proofs are based on discrete time martingale theory. \end{abstract}
\renewcommand{\thefootnote}{} {\footnotetext{ {\bf Key words and phrases:} Random graph, preferential attachment, scale-free, power law, submartingale, Doob-Meyer decomposition.
{\bf Mathematics Subject Classification:} 05C80, 60G42.
The publication was supported by the T\'AMOP-4.2.2.C-11/1/KONV-2012-0001 project. The project has been supported by the European Union, co-financed by the European Social Fund. }
\section{Introduction} Network theory became a popular field during the last $15$ years. Several real-world networks were investigated such as the WWW, the Internet, social and biological networks (see \cite{durrett} for an overview). It turned out that a main common characteristic of such networks is their scale-free nature, in other words the asymptotic power law degree distribution, i.e. $p_k \sim Ck^{-\gamma}$, as $k\to\infty$. Using large data sets, it was shown that for the WWW the in-degree and the out-degree of web pages follow power law with $\gamma_{\rm{in}}=2.1$ and $\gamma_{\rm{out}}=2.7$, for the Internet $\gamma=2.3$, for the movie actor network $\gamma=2.3$, for the collaboration graph of mathematicians $\gamma=2.4$ (see \cite{durrett} for details). To describe the phenomenon, in \cite{barabasi} the preferential attachment model was suggested. In the preferential attachment model the growing mechanism of the random graph is the following. At every time $t=2,3,\dots$ a new vertex with $m$ edges is added so that the edges link the new vertex to $m$ old vertices. The probability $\pi_i$ that the new vertex will be connected to the old vertex $i$ depends on the degree $d_i$ of vertex $i$, so that $\pi_i= d_i/\sum_j d_j$. The power law degree distribution in the preferential attachment model was proved by a couple of methods (see, e.g. \cite{bollobas}).
There are several modifications of the preferential attachment model (see \cite{cooper}, \cite{SGWN}). It is also known that besides the degrees of the vertices other characteristics of the graph can be important (see \cite{SGWN}). In \cite{BaMo1} a model based on the interaction of three vertices was introduced. The power law degree distribution in that model was proved in \cite{BaMo2}. In \cite{FIPB}, instead of the three-interactions model, interactions of four vertices were studied. It turned out that in the seemingly complicated four-interactions model the asymptotic behaviour is as simple as in the three-interactions model. Therefore it is hopeful that the overburdening formulae of the $N$-interactions model lead to tractable asymptotic results.
In this paper, we extend the model and the results of \cite{BaMo1}, \cite{BaMo2} and \cite{FIPB} to interactions of $N$ vertices. Our model is the following. A complete graph with $m$ vertices we call an $m$-clique, for short. We denote an $m$-clique by the symbol $K_m$. In our model at time $n=0$ we start with a $K_N$. The initial weight of this graph is one. This graph contains $N$ vertices, $ \binom{N}{2}$ edges, \dots , $\binom{N}{M}$ $M$-cliques $\left( M \leq N \right)$. Each of these objects has initial weight $1$. After the initial step we start to increase the size of the graph. At each step, the evolution of the graph is based on the interaction of $N$ vertices. More precisely, at each step $n=1,2,\dots$ we consider $N$ vertices and draw all non-existing edges between these vertices. So we obtain a $K_N$. The weight of this graph $K_N$ and the weights of all cliques in $K_N$ are increased by $1$. (That is we increase the weights of $N$ vertices, $ \binom{N}{2}$ edges, \dots , $N$ different $\left(N-1\right)$-cliques and the $N$-clique $K_N$ itself.) The choice of the $N$ interacting vertices is the following.
There are two possibilities at each step. With probability $p$ we add a new vertex that interacts with $N-1$ old vertices, on the other hand, with probability $\left( 1-p \right)$, $N$ old vertices interact. Here $0 < p \leq 1$ is fixed.
When we add a new vertex, then we choose $N-1$ old vertices and they together will form an $N$-clique. However, to choose the $N-1$ old vertices we have two possibilities. With probability $r$ we choose an $\left(N-1\right)$-clique from the existing $\left(N-1\right)$-cliques according to the weights of the $\left(N-1\right)$-cliques. It means that an $\left(N-1\right)$-clique of weight $w_t$ is chosen with probability $w_t/\sum_h w_h$. On the other hand, with probability $1-r$, we choose among the existing vertices uniformly, that is all $N-1$ vertices have the same chance.
At a step when we do not add a new vertex, then $N$ old vertices interact. As in the previous case, we have two possibilities. With probability $q$, we choose one $K_N$ of the existing $N$-cliques according to their weights. It means that an $N$-clique of weight $w_t$ is chosen with probability $w_t/\sum_h w_h$. On the other hand, with probability $1-q$, we choose among the existing vertices uniformly, that is all subsets consisting of $N$ vertices have the same chance.
In this paper we show that the above mechanism results in a scale-free graph. To prove our results we follow the lines of \cite{BaMo1}, \cite{BaMo2}. Let $X(n,d,w)$ denote the number of vertices of weight $w$ and degree $d$ after the $n$th step. Let $V_n$ denote the number of vertices after the $n$th step. Let ${\mathcal F}} \def\GD{{\cal G}_{n-1}$ denote the $\sigma$-algebra of observable events after the $(n-1)$th step. First we calculate the conditional expectation ${{\mathbb {E}}} \def\D{{\mathbb {D}}} \{ X(n,d,w)|{\mathcal F}} \def\GD{{\cal G}_{n-1} \}$, see Lemma \ref{EX/F}. Then we prove (Theorem \ref{limX/V}) that the ratio $\frac{X \left( n,d,w \right)}{V_n}$ converges to $x_{d,w}$ almost surely (a.s.) as $n \rightarrow \infty $, where the limits $x_{d,w}$ are fixed non-negative numbers. The main tool of the proof is the Doob-Meyer decomposition of submartingales. We remark that in the $3$-interactions model the limits $x_{d,w}$ are always positive (see \cite{BaMo2}). However, in the $N$-interactions model the limits $x_{d,w}$ can be zero unless $N$ is equal to $3$. It is an important phenomenon, because the appearance of zero limits simplifies the seemingly intractable formulae.
We show that $x_{d,w}$, $d=N-1,N,\dots, \left(N-1\right)w$, $w=1,2,\dots$, is a proper two-dimensional discrete probability distribution (Lemma \ref{xdw}). Then we turn to the scale-free property for the weights. Let $X \left( n,w \right)$ denote the number of vertices of weight $w$ after the $n$th step. Then for all $w=1,2,\dots$ we have $$ \dfrac{X \left( n,w \right)}{V_n} \rightarrow x_{w} = x_{N-1,w} + x_{N,w} + \dots + x_{\left(N-1\right)w,w} $$ almost surely. Moreover, $x_{w} \sim C w^{- \left( 1 + \frac{1}{\alpha} \right)}$, as $w\to\infty$ (Theorem \ref{theorem:scalefreeWeights}), that is the distribution $x_{w}$ has a tail which decays as a power-law with exponent $ 1 + \frac{1}{\alpha}$. To derive the above results from Theorem \ref{limX/V}, we need only some known facts about the $\varGamma$-function, see the proofs of Lemma \ref{xdw} and Theorem \ref{theorem:scalefreeWeights}. Finally, we obtain the scale-free property for the degrees. Let us denote by $U\left( n,d \right)$ the number of vertices of degree $d$ after the $n$th step. For any $d \geq N-1$ we have \begin{equation*} \dfrac{U\left( n,d \right)}{V_n} \to u_d \end{equation*} a.s. as $n \to \infty$, where $u_d$, $d=N-1,N, N+1,\dots$, are positive numbers. Furthermore, \begin{equation*} u_d \sim \dfrac{\varGamma \left( 1 + \frac{\beta + 1}{\alpha} \right)}{\alpha_2 \varGamma \left( 1 + \frac{\beta}{\alpha} \right)}\left( \dfrac{\alpha d}{\alpha_2} \right)^{- \left( 1 + \frac{1}{\alpha} \right)}, \end{equation*} as $d \to \infty$, where $\alpha$, $\beta$ and $\alpha_2$ are appropriate constants (see Theorem \ref{ThmScaleFreeDegree}). We see that in both cases the exponent is $1 + \frac{1}{\alpha}$.
\section{The evolution of the graph}
Throughout the paper $0<p \leq 1$, $0 \leq r \leq 1$, $0 \leq q \leq 1$ are fixed numbers. Let $X(n,d,w)$ denote the number of vertices of weight $w$ and degree $d$ after the $n$th step. Let $V_n$ denote the number of vertices after the $n$th step.
\begin{rem} Each vertex has initial weight $1$ and initial degree $N-1$. When a vertex takes part in an interaction, the weight of this vertex is increased by $1$ and the degree of this vertex may increase by $0, 1, 2, \dots $ or $N-1$. Therefore $X(n,d,w)$ can be positive only for $1 \leq w \leq n+1$ and $N-1 \leq d \leq \left(N-1\right)w$. \end{rem}
Let ${\mathcal F}} \def\GD{{\cal G}_{n-1}$ denote the $\sigma$-algebra of observable events after the $(n-1)$th step. We compute the conditional expectation of $X(n,d,w)$ with respect to ${\mathcal F}} \def\GD{{\cal G}_{n-1}$ for $w \geq 1$. The results of this paper will be based on it. The particular cases $N=3$ and $N=4$ are included in \cite{BaMo2} and \cite{FIPB}, respectively.
Let \begin{equation*} \alpha_1 = \left(1-p\right) q, \quad \alpha_2 = \dfrac{N-1}{N}pr, \quad \alpha = \alpha_1 + \alpha_2, \quad \beta = \left( N-1 \right)\left( 1-r \right) + \dfrac{N\left( 1-p \right)\left( 1-q \right)}{p}. \end{equation*}
\begin{lem} \label{EX/F}
One has \begin{equation*}
{{\mathbb {E}}} \def\D{{\mathbb {D}}} \{ X(n,d,w)|{\mathcal F}} \def\GD{{\cal G}_{n-1} \} = X(n-1,d,w) \left[ 1-\left(\dfrac{w}{n}\alpha + \dfrac{p}{V_{n-1}}\beta \right) \right] + \end{equation*} \begin{equation*} +X(n-1,d,w-1) \left[ \left( 1-p \right) \left( q\dfrac{w-1}{n} + \left( 1-q \right)\dfrac {\binom{d}{N-1}} {\binom{V_{n-1}}{N}} \right) \right] + \end{equation*} \begin{equation*} + X(n-1,d-1,w-1) \left[ p \left(r \dfrac{\left(N-1\right)\left( w-1 \right)}{Nn} + \left( 1-r \right) \dfrac{\binom{d-1}{N-2}}{\binom{V_{n-1}}{N-1}} \right) + \right. \end{equation*} \begin{equation*} \left. + \left( 1-p \right)\left( 1-q \right)\dfrac {\binom{d-1}{N-2} \left( V_{n-1}-d \right)} {\binom{V_{n-1}}{N}} \right] + \dots + \end{equation*} \begin{equation*} + X(n-1,d-m,w-1) \left[ p \left( 1-r \right)\dfrac { \binom{d-m}{N-m-1} \binom{V_{n-1}-d+m-1}{m-1}} {\binom{V_{n-1}}{N-1}} + \right. \end{equation*} \begin{equation*} \left. + \left( 1-p \right)\left( 1-q \right)\dfrac {\binom{d-m}{N-m-1} \binom{ V_{n-1}-d+m-1}{m} } {\binom{V_{n-1}}{N}} \right] + \dots + \end{equation*} \begin{equation*} +X(n-1,d-\left(N-1\right),w-1) \left[ p\left( 1-r \right) \dfrac{\binom{V_{n-1}-d+N-2}{N-2}}{\binom{V_{n-1}}{N-1}} \right. \end{equation*} \begin{equation} \label{CondExp} \left. + \left( 1-p \right) \left( 1-q \right)\dfrac{\binom{V_{n-1}-d+N-2}{N-1}}{\binom{V_{n-1}}{N}} \right] + p\delta_{d,N-1}\delta_{w,1} \end{equation} for $w\ge 1$ and $N-1\le d \le \left( N-1 \right)w$, $1 < m < N-1$. Here $\delta_{k,l}$ denotes the Dirac-delta. \end{lem}
\begin{pf} The total weight of $N$-cliques after $(n-1)$ steps is $n$. The total weight of $\left(N-1\right)$-cliques after $(n-1)$ steps is $Nn$. The total weight of $\left(N-1\right)$-cliques having a fixed common vertex of weight $w$ is $\left( N-1 \right)w$. Moreover, after $(n-1)$ steps, we have the following. When we choose $\left(N-1\right)$ vertices randomly, then the probability that a given vertex is chosen is $$ \frac{\binom{V_{n-1}-1}{N-2}}{\binom{V_{n-1}}{N-1}} = \frac{N-1}{V_{n-1}}. $$ When we choose $N$ vertices randomly, then the probability that a given vertex is chosen is $$ \frac{\binom{V_{n-1}-1}{N-1}}{\binom{V_{n-1}}{N}} = \frac{N}{V_{n-1}}. $$ Therefore the probability that an old vertex of weight $w$ takes part in the interaction at step $n$ is $$ p \left(r \dfrac{\left(N-1\right)w}{Nn} + (1-r) \dfrac{N-1}{V_{n-1}}\right) + \left(1-p\right) \left(q \dfrac{w}{n} + (1-q) \dfrac{N}{V_{n-1}}\right) = \dfrac{w}{n}\alpha + \dfrac{p}{V_{n-1}}\beta, $$ where \begin{equation*} \alpha = (1-p)q + \dfrac{\left(N-1\right)pr}{N}, \quad \beta = \dfrac{1}{p} \left\{\left(N-1\right)p(1-r) + N(1-p)(1-q) \right\}. \end{equation*}
A new vertex always takes part in the interaction. At each step with probability $p$ a new vertex with weight $1$ and with degree $\left(N-1\right)$ is born.
Consider a fixed vertex with weight $w$ and degree $d$. The probability that in the $n$th step \begin{itemize}
\item neither its degree $d$ nor its weight $w$ change is
$$ 1-\left(\dfrac{w}{n}\alpha + \dfrac{p}{V_{n-1}}\beta \right)\,;
$$
\item its degree does not change but its weight is increased by 1 is
$$ \left( 1-p \right) \left( q\dfrac{w-1}{n} + \left( 1-q \right)\dfrac {\binom{d}{N-1}} {\binom{V_{n-1}}{N}} \right)\,;
$$
\item both its degree and its weight are increased by 1 is
$$ p \left(r \dfrac{\left(N-1\right)\left( w-1 \right)}{Nn} + \left( 1-r \right) \dfrac{\binom{d-1}{N-2}}{\binom{V_{n-1}}{N-1}} \right) + \left( 1-p \right)\left( 1-q \right)\dfrac {\binom{d-1}{N-2} \left( V_{n-1}-d \right)} {\binom{V_{n-1}}{N}}\,;
$$
\item its degree is increased by $m$ ($1<m<N-1$) and its weight is increased by 1 is
$$ p \left( 1-r \right)\dfrac { \binom{d-m}{N-m-1} \binom{V_{n-1}-d+m-1}{m-1}} {\binom{V_{n-1}}{N-1}} + \left( 1-p \right)\left( 1-q \right)\dfrac {\binom{d-m}{N-m-1} \binom{ V_{n-1}-d+m-1}{m} } {\binom{V_{n-1}}{N}}\,;
$$
\item its degree is increased by $N-1$ and its weight is increased by 1 is
$$ p \left( 1-r \right) \dfrac{\binom{V_{n-1}-d+N-2}{N-2}}{\binom{V_{n-1}}{N-1}} + \left( 1-p \right) \left( 1-q \right)\dfrac{\binom{V_{n-1}-d+N-2}{N-1}}{\binom{V_{n-1}}{N}}\,.
$$ \end{itemize} Using the above formulae, we obtain equation \eqref{CondExp}.
$\Box$ \end{pf}
The following theorem is an extension of \textit{Theorem 3.1} of \cite{BaMo2}, see also \textit{Theorem 2.1} of \cite{FIPB}. For $N>3$ we shall see, that several terms are asymptotically negligible, therefore the final expressions are as simple as in the case of $N=3$.
\begin{thm} \label{limX/V} Let $0<p<1$, $q>0$, $r>0$ and $(1-r)(1-q)>0$. For any fixed $w$ and $d$ with $1 \leq w$ and $N-1 \leq d \leq w\left(N-1 \right)$ we have \begin{equation} \label{X/Vtox} \dfrac{X \left( n,d,w \right)}{V_n} \rightarrow x_{d,w} \end{equation} almost surely as $n \rightarrow \infty $, where $x_{d,w}$ are fixed non-negative numbers. Furthermore, the numbers $x_{d,w}$ satisfy the following recurrence: $$ x_{N-1,1} = \dfrac{1}{\alpha + \beta +1} > 0, \quad \quad \quad x_{d,1}=0, \, \text{ for } \, d\ne N-1, $$ \begin{equation} \label{rekurzio_x(d,w)} x_{d,w} = \dfrac{1}{\alpha w + \beta +1} \left[ \alpha_1 \left( w-1 \right) x_{d,w-1} + \alpha_2 \left( w-1\right)x_{d-1,w-1} +\beta x_{d-\left( N-1 \right),w-1} \right], \end{equation} for $w\ge 2$, $N-1\le d\le w\left(N-1 \right)$, where \begin{equation*} \alpha_1 = \left(1-p\right) q, \quad \alpha_2 = \dfrac{N-1}{N}pr, \quad \alpha = \alpha_1 + \alpha_2, \quad \beta = \left( N-1 \right)\left( 1-r \right) + \dfrac{N\left( 1-p \right)\left( 1-q \right)}{p}. \end{equation*}
If $w \geq 1$ is fixed then there exists $d$ with $N-1\le d \le w\left(N-1\right)$ such that $x_{d,w}$ is positive and if $w \geq 1$ and $N \geq 4 $ then there exists $d$ with $N-1\le d \le w\left(N-1\right)$ such that $x_{d,w}$ is equal to zero. Moreover, in the cases when $x_{d,w} = 0$ we have \begin{equation*} \dfrac{X \left( n,d,w \right)}{V_n} = {\rm {o}} \left( n^{-a} \right), \end{equation*} where $a$ is a positive number which may depend on $w$ and $d$.
If $N-1\le d \le w\left(N-1\right)$ does not satisfied, then $x_{d,w}=0$. \end{thm}
\begin{pf} We follow the lines of \cite{BaMo2}. Introduce notation \begin{equation} \label{c_def} c(n,w) = \prod_{i=w-1}^{n} \left( 1-\dfrac{\alpha w}{i}-\dfrac{\beta p}{V_{i-1}} \right)^{-1}, \quad n \geq w-1, w \geq 1. \end{equation}
$c(n,w)$ is an ${\mathcal F}} \def\GD{{\cal G}_{n-1}$-measurable random variable. Applying the Marcinkiewicz strong law of large numbers to the number of vertices, we have \begin{equation} \label{Markinkiewicz} V_n = pn + {\rm {o}} \left( n^{1/2 + \varepsilon} \right) \end{equation} almost surely, for any $\varepsilon >0$.
Using \eqref{Markinkiewicz} and the Taylor expansion for $\log(1+x)$, we obtain \begin{equation*} \log c\left( n,w \right) = -\sum_{i=w-1}^{n} \log \left( 1-\dfrac{\alpha w}{i}-\dfrac{\beta }{i + {\rm {o}} \left( i^{1/2 + \varepsilon} \right)} \right) = \left( \alpha w + \beta \right) \sum_{i=w-1}^{n} \dfrac{1}{i} + {\rm {O}} \left( 1 \right), \end{equation*} where the error term is convergent as $n \to \infty$. Therefore \begin{equation} \label{c(n,w)_asz.} c(n,w) \sim a_{w} n^{\alpha w + \beta} \end{equation} almost surely, as $n \to \infty$, where $a_{w}$ is a positive random variable.
Let \begin{equation*} \label{Z_def} Z \left( n,d,w \right) = c\left( n,w \right) X \left( n,d,w \right) \quad \text{for} \quad 1 \leq w, \, N-1 \leq d \leq w\left(N-1 \right). \end{equation*}
Using \eqref{CondExp}, we can see that $ \left\{ Z \left( n,d,w \right) , {\mathcal F}} \def\GD{{\cal G}_{n} , n=w-1,w,w+1,\dots \right\}$ is a non-negative submartingale for any fixed $1 \leq w$, $N-1 \leq d \leq \left( N-1 \right)w$. Define $ Z \left( n,d,w \right) = 0$ for $n=1,2,\dots,w-2$. Applying the Doob-Meyer decomposition to $ Z \left( n,d,w \right)$, we can write \begin{equation*} \label{Doob_Meyer} Z \left( n,d,w \right) = M \left( n,d,w \right) + A \left( n,d,w \right), \end{equation*} where $M \left( n,d,w \right)$ is a martingale and $A \left( n,d,w \right)$ is a predictable increasing process. The general form of $M \left( n,d,w \right)$ and $A \left( n,d,w \right)$ are the following: \begin{equation} \label{M_alt}
M \left( n,d,w \right) = \sum_{i=1}^{n} \left[ Z \left( i,d,w \right) - {{\mathbb {E}}} \def\D{{\mathbb {D}}} \left( Z \left( i,d,w \right) | {\mathcal F}} \def\GD{{\cal G}_{i-1} \right) \right], \end{equation} \begin{equation} \label{A_alt}
A \left( n,d,w \right) = {{\mathbb {E}}} \def\D{{\mathbb {D}}} Z \left( 1,d,w \right) + \sum_{i=2}^{n} \left[ {{\mathbb {E}}} \def\D{{\mathbb {D}}} \left( \left( Z \left( i,d,w \right) | {\mathcal F}} \def\GD{{\cal G}_{i-1} \right) - Z \left( i-1,d,w \right) \right) \right], \end{equation} where ${\mathcal F}} \def\GD{{\cal G}_{0}$ is the trivial $\sigma$-algebra. Using \eqref{A_alt} and \eqref{CondExp}, we have \begin{multline} \label{A(n,d,w)} \begin{split} A & \left( n,d,w \right) = {{\mathbb {E}}} \def\D{{\mathbb {D}}} Z \left( 1,d,w \right)+\\ & +\sum_{i=2}^{n} \left[ c \left( i,w \right) X \left( i-1,d,w-1 \right) \left( 1-p \right) \left( q \dfrac{w-1}{i} + \left( 1-q \right) \dfrac{\binom{d}{N-1}}{\binom{V_{i-1}}{N}} \right) \right.+ \\
& +c \left( i,w \right) X(i-1,d-1,w-1) \times \\ & \times \left( p \left(r \dfrac{\left(N-1\right)\left( w-1 \right)}{Ni} + \left( 1-r \right) \dfrac{\binom{d-1}{N-2}}{\binom{V_{i-1}}{N-1}} \right) \right. +
\left( 1-p \right) \left( 1-q \right) \left. \dfrac{\binom{d-1}{N-2} \left( V_{i-1}-d \right)}{\binom{V_{i-1}}{N}} \right) +\\
& + \dots +c\left( i,w \right)X(i-1,d-m,w-1) \times \\ & \times \left( p \left( 1-r \right)\dfrac { \binom{d-m}{N-m-1} \binom{V_{i-1}-d+m-1}{m-1} } {\binom{V_{i-1}}{N-1}} \right.
+ \left. \left( 1-p \right)\left( 1-q \right)\dfrac {\binom{d-m}{N-m-1} \binom{ V_{i-1}-d+m-1}{m} } {\binom{V_{i-1}}{N}} \right) +\\
& + \dots +c\left( i,w \right) X(i-1,d-\left(N-1\right),w-1) \times \\ & \times \left( p \left( 1-r \right) \dfrac{\binom{V_{i-1}-d+N-2}{N-2}}{\binom{V_{i-1}}{N-1}} + \left( 1-p \right) \left( 1-q \right) \left. \dfrac {\binom{V_{i-1}-d+N-2}{N-1}} {\binom{V_{i-1}}{N}} \right) + c \left( i,w \right) p\delta_{d,N-1}\delta_{w,1} \right]. \end{split} \end{multline}
Let $B \left( n,d,w \right)$ the sum of the conditional variances of $Z \left( n,d,w \right)$. Now we give an upper bound of $B \left( n,d,w \right)$.
\begin{equation*}
B \left( n,d,w \right) = \sum_{i=2}^{n} {\D}^2 \left( Z \left( i,d,w \right) | {\mathcal F}} \def\GD{{\cal G}_{i-1} \right) =
\sum_{i=2}^{n} {{\mathbb {E}}} \def\D{{\mathbb {D}}} \{ \left( Z \left( i,d,w \right) - {{\mathbb {E}}} \def\D{{\mathbb {D}}} \left( Z \left( i,d,w \right) |{\mathcal F}} \def\GD{{\cal G}_{i-1}\right) \right)^2 | {\mathcal F}} \def\GD{{\cal G}_{i-1} \} = \end{equation*}
\begin{equation*} =
\sum_{i=2}^{n} c\left( i,w \right)^2 {{\mathbb {E}}} \def\D{{\mathbb {D}}} \{ \left( X \left( i,d,w \right) - {{\mathbb {E}}} \def\D{{\mathbb {D}}} \left( X \left( i,d,w \right) |{\mathcal F}} \def\GD{{\cal G}_{i-1}\right) \right)^2 | {\mathcal F}} \def\GD{{\cal G}_{i-1}
\} \leq \end{equation*}
\begin{equation*} \leq
\sum_{i=2}^{n} c\left( i,w \right)^2 {{\mathbb {E}}} \def\D{{\mathbb {D}}} \{ \left( X \left( i,d,w \right) - X \left( i-1,d,w \right) \right)^2 | {\mathcal F}} \def\GD{{\cal G}_{i-1}
\} \leq \end{equation*}
\begin{equation}\label{Bndw} \leq
N^2 \sum_{i=2}^{n} c\left( i,w \right)^2 = {\rm {O}}\left( n^{2 \left( \alpha w+ \beta \right) +1} \right). \end{equation}
Above we used that $c\left( i,w \right)$ is ${\mathcal F}} \def\GD{{\cal G}_{i-1}$-measurable and at each step $N$ vertices can interact. Finally, we applied \eqref{c(n,w)_asz.}. Jensen's inequality implies that $M^2 \left( n,d,w \right)$ is a (non-negative) submartingale if $M \left( n,d,w \right)$ is a martingale. Now we can apply the Doob-Meyer decomposition to $M^2 \left( n,d,w \right)$. It is known that $B(n, d,w)$, that is the sum of the conditional variances of terms $Z(n,d,w)$ from formula \eqref{Bndw}, is the same (up to an additive constant) as the increasing predictable process in the Doob-Meyer decomposition of the non-negative submartingale $M^2(n,d,w)$. Therefore the Doob-Meyer decomposition is $$ M^2 \left( n,d,w \right) = Y \left( n,d,w \right) + B \left( n,d,w \right), $$ where $Y \left( n,d,w \right)$ is a martingale and the predictable increasing process $B \left( n,d,w \right)$ is given by \eqref{Bndw}.
We use induction on $w$. Let $w=1$. We can see that a vertex of weight $1$ could take part in an interaction when it was born. Therefore its degree must be equal to $N-1$. By \eqref{A(n,d,w)}, \begin{equation} \label{Teljes indukcio:w=1} A \left( n,N-1,1 \right) \sim p \sum_{i=2}^{n} c\left( i,1 \right) \sim p \sum_{i=2}^{n} a_{1} i^{\alpha + \beta} \sim p a_{1} \dfrac{n^{\alpha + \beta + 1}}{\alpha + \beta + 1} \end{equation} a.s. as $n \to \infty$. \\ By \eqref{Bndw}, $B \left( n,N-1,1 \right) = {\rm {O}} \left( n^{2\left( \alpha + \beta \right) + 1} \right)$ and therefore $ \left(B \left( n,N-1,1 \right)\right)^{\frac{1}{2}} \log B \left( n,N-1,1 \right) = \,\,\, = {\rm {O}} \left( A \left( n,N-1,1 \right)\right)$. Therefore, by Proposition VII-2-4 of \cite{neveu}, \begin{equation} \label{Z(n,3,1)_A} Z \left( n,N-1,1 \right) \sim A \left( n,N-1,1 \right) \quad \text{a.s. on the event } \quad \{A \left( n,N-1,1 \right) \to \infty\} \quad \text{as}\quad n \to \infty. \end{equation} As, by \eqref{Teljes indukcio:w=1}, $A(n,N-1,1) \to \infty$ a.s., therefore using \eqref{Teljes indukcio:w=1}, \eqref{Markinkiewicz} and \eqref{c(n,w)_asz.}, relation \eqref{Z(n,3,1)_A} implies \begin{equation} \dfrac{X \left( n,N-1,1 \right)}{V_n} = \dfrac{Z \left( n,N-1,1 \right)}{c\left( n,1 \right)V_n} \sim \dfrac{A \left( n,N-1,1 \right)}{c\left( n,1 \right)V_n} \sim \dfrac{p a_{1} \dfrac{n^{\alpha + \beta +1}}{\alpha +\beta +1}}{a_{1} n^{\alpha + \beta }p n} = \dfrac{1}{\alpha + \beta +1} = x_{N-1,1} > 0 \end{equation} almost surely. So \eqref{X/Vtox} is valid for $w=1$.
Now let $w=2$. In this case the degree of the vertex must be $N-1 \leq d \leq 2\left(N-1\right)$. If $w=2$ and $d=N-1,N \,\,\text{or}\,\, 2\left(N-1\right)$, then we shall show $x_{d,w} > 0$. By \eqref{Markinkiewicz}, \eqref{c(n,w)_asz.} and \eqref{A(n,d,w)}, we can compute the asymptotic behaviour of $A \left( n,d,2 \right)$ as follows $$ A \left( n,N-1,2 \right) \sim p a_{2} \dfrac{n^{2\alpha + \beta +1}}{2\alpha + \beta +1} \left( 1-p \right) q x_{N-1,1} \to\infty, $$ $$ A \left( n,N,2 \right) \sim p a_{2} \dfrac{n^{2\alpha + \beta +1}}{2\alpha + \beta +1} \dfrac{N-1}{N} p r x_{N-1,1} \to\infty, $$ $$ A \left( n,2\left(N-1\right),2 \right) \sim p a_{2} \dfrac{n^{2\alpha + \beta +1}}{2\alpha + \beta +1} \left[ \left(N-1\right) \left( 1-r \right) + \dfrac{N \left( 1-p \right) \left( 1-q \right) }{p} \right] x_{N-1,1} \to\infty. $$ Moreover, $$ B \left( n,d,2 \right) = {\rm {O}}\left( n^{2 \left( \alpha 2+ \beta \right) +1} \right), \text{thus} \left(B \left( n,d,2 \right)\right)^{\frac{1}{2}} \log B \left( n,d,2 \right) = {\rm {O}} \left( A \left( n,d,2 \right)\right) $$ for $d=N-1,N \text{ and } 2\left(N-1\right). $ Therefore in these cases $Z \left( n,d,2 \right) \sim A \left( n,d,2 \right)$ a.s. on $\{A \left( n,d,2 \right) \to \infty\}$ as $n \to \infty$. It implies that \begin{equation} \dfrac{X \left( n,d,2 \right)}{V_n} = \dfrac{Z \left( n,d,2 \right)}{c\left( n,2 \right)V_n} \sim \dfrac{A \left( n,d,2 \right)}{c\left( n,2 \right)V_n} \sim \dfrac{p a_{2} \dfrac{n^{2\alpha + \beta +1}}{2\alpha +\beta +1}}{a_{2} n^{2\alpha + \beta }p n} T_{d,2} = \dfrac{T_{d,2}}{2\alpha + \beta +1} \end{equation} with appropriate $T_{d,2}$. Therefore we have \begin{equation*} \dfrac{X \left( n,N-1,2 \right)}{V_n} \to \dfrac{\left( 1-p \right) q x_{N-1,1}}{2\alpha + \beta +1} = x_{N-1,2} > 0, \end{equation*} \begin{equation*} \dfrac{X \left( n,N,2 \right)}{V_n} \to \dfrac{ \dfrac{N-1}{N} p r x_{N-1,1}}{2\alpha + \beta +1} = x_{N,2} > 0, \end{equation*} \begin{equation*} \dfrac{X \left( n,2\left(N-1\right),2 \right)}{V_n} \to \left[ \left( N-1 \right) \left( 1-r \right) + \dfrac{N \left( 1-p \right) \left( 1-q \right)}{p} \right] \dfrac{x_{N-1,1}}{2\alpha + \beta +1} = x_{2\left(N-1\right),2} > 0, \end{equation*} as $n\to\infty$. So \eqref{X/Vtox} is valid for $w=2$, $d=N-1, N, 2\left(N-1\right)$.
Consider the cases when $N+1 \leq d \leq 2N-3$ and $w=2$. These cases are different from the previous ones. By \eqref{A(n,d,w)} and using \textit{Remark 1}, we have \begin{multline} \begin{split} A & \left( n,d,2 \right) = {{\mathbb {E}}} \def\D{{\mathbb {D}}} Z \left( 1,d,2 \right)+\\
&+\sum_{i=2}^{n} c\left( i,w \right)X(i-1,d-m,1) \times \\ & \times \left( p \left( 1-r \right)\dfrac { \binom{d-m}{N-m-1} \binom{V_{i-1}-d+m-1}{m-1}} {\binom{V_{i-1}}{N-1}} \right.
+ \left. \left( 1-p \right)\left( 1-q \right)\dfrac {\binom{d-m}{N-m-1} \binom{ V_{i-1}-d+m-1}{m} } {\binom{V_{i-1}}{N}} \right), \end{split} \end{multline} where $d-m = N-1$.
Using this and the limit of $X(i-1,N-1,1)/V_{i-1}$, we obtain the asymptotic behaviour of $A \left( n,d,2 \right)$ as follows $$ A \left( n,d,2 \right) \sim \sum_{i=2}^{n} a_{2} i^{2\alpha + \beta} x_{N-1,1} i p \left[ \dfrac{\binom{d-m}{N-m-1} \left( N-1\right)! }{\left(m-1\right)! } \dfrac{p \left( 1-r \right)}{\left( pi \right)^{N-m}} + \dfrac{\binom{d-m}{N-m-1} N! }{m!} \dfrac{ \left( 1-p \right) \left( 1-q \right)}{ \left( pi \right)^{N-m}} \right] \sim $$ $$ \sim a_{2} x_{N-1,1} C \sum_{i=2}^{n} i^{2\alpha + \beta +1 +m -N} \sim $$ \begin{equation} \label{A(n,5,2)_as} \sim a_{2} x_{N-1,1} C \dfrac{n^{2\alpha + \beta +2 +m -N}}{2\alpha + \beta +2 +m -N} = {\rm {O}}\left( n^{2\alpha + \beta } \right), \end{equation} because $N-m\geq 2$. Here $C$ denotes an appropriate constant.
Now we have \begin{equation*} \dfrac{X \left( n,d,2 \right)}{V_n} = \dfrac{Z \left( n,d,2 \right)}{c\left( n,2 \right)V_n} = \dfrac{M \left( n,d,2 \right) + A \left( n,d,2 \right)}{c\left( n,2 \right)V_n}. \end{equation*} The behaviour of $A \left( n,d,2 \right)$ is given by \eqref{A(n,5,2)_as}. We denoted by $B \left( n,d,w \right)$ the increasing predictable process in the Doob-Meyer decomposition of $M^2 \left( n,d,w \right)$. We know that $B \left( n,d,2 \right) = {\rm {O}}\left( n^{ 4\alpha+ 2\beta +1} \right)$ and so $ \left(B \left( n,d,2 \right)\right)^{\frac{1}{2}} \log B \left( n,d,2 \right) = {\rm {O}} \left( n^{2\alpha + \beta + \frac{1}{2} + \varepsilon}\right)$ with arbitrary small positive $\varepsilon$.
Applying Proposition VII-2-4 of \cite{neveu}, we have $$ M \left( n,d,2 \right) = {\rm {o}} \left( \left(B \left( n,d,2 \right)\right)^{\frac{1}{2}} \log B \left( n,d,2 \right) \right) = {\rm {o}} \left( n^{2\alpha + \beta + \frac{1}{2} + \varepsilon} \right)\,\,\, \text{a.e. on}\,\,\,\{ B \left( n,d,2 \right) \to \infty \}. $$ Moreover, on the set $\{ B \left( \infty,d,2 \right) < \infty \}$, the sequence $M( n,d,2)$ is a.s. convergent. So $M( n,d,2) = {\rm {o}} \left( n^{2\alpha + \beta + \frac{1}{2} + \varepsilon} \right)$ a.s.\,\,. Therefore, using \eqref{Markinkiewicz} and \eqref{c(n,w)_asz.}, we have \begin{equation} \dfrac{X \left( n,d,2 \right)}{V_n} = \dfrac{M \left( n,d,2 \right) + A \left( n,d,2 \right)}{c\left( n,2 \right)V_n} \leq C \dfrac{n^{2\alpha + \beta + \frac{1}{2} + \varepsilon}}{n^{2\alpha + \beta}n} = C \dfrac{1}{n^a} \to 0, \end{equation} where $n \to \infty$ and $ \dfrac{1}{4} < a < \dfrac{1}{2}$. So the proposition is valid for $w=1$ and $w=2$.
Suppose that the statement is true for all weights less than $w$ and for all possible degrees. First we study the positive limits. Consider $A(n,d,w)$ in \eqref{A(n,d,w)} and assume that at least one of the coefficients $x_{d,w-1}$, $x_{d-1,w-1}$, $x_{d-\left(N-1\right),w-1}$ is positive. Then by \eqref{Markinkiewicz}, \eqref{c(n,w)_asz.}, \eqref{A(n,d,w)} and using the induction hypothesis, we have \begin{multline*} \begin{split} A \left( n,d,w \right) \sim & \sum_{i=2}^{n} c \left( i,w \right) \left[ X\left(i-1,d,w-1\right) \left( 1-p \right) q \dfrac{w-1}{i} + \right. \\ &\left. + X\left(i-1,d-1,w-1\right) p r \dfrac{\left(N-1\right)\left( w-1 \right)}{Ni} \right. +\\ & + X\left(i-1,d-\left(N-1\right),w-1\right) \left( \dfrac{p \left( 1-r \right) \left( N-1 \right)}{pi} +
\left. \dfrac{ \left( 1-p \right) \left( 1-q \right) N}{pi} \right) \right] \sim
\end{split} \end{multline*} \begin{multline*} \begin{split} \sim \sum_{i=2}^{n} \left[ c \left( i,w \right) x_{d,w-1} pi \left( 1-p \right) q \dfrac{w-1}{i}
+c \left( i,w \right) x_{d-1,w-1} pi p r \dfrac{\left(N-1\right)\left( w-1 \right)}{Ni} \right. +\\
+ c\left( i,w \right) x_{d-\left(N-1\right),w-1} pi \left( \dfrac{ \left( 1-r \right) \left( N-1 \right)}{i} +
\left. \dfrac{ \left( 1-p \right) \left( 1-q \right) N}{pi} \right) \right] \sim \end{split} \end{multline*} \begin{multline*} \begin{split} \sim \sum_{i=2}^{n} a_{w} i^{\alpha w + \beta} & \left[ x_{d,w-1} p \alpha_{1} \left(w-1\right)
+ x_{d-1,w-1} p \alpha_{2} \left( w-1 \right) \right. +\\ & + x_{d-\left(N-1\right),w-1} p\beta \left.\left. \vphantom{\dfrac{1}{2}} \right) \vphantom{\dfrac{1}{2}} \right] \sim \end{split} \end{multline*} \begin{equation} \label{A_as} \begin{split} \sim p a_{w} \dfrac{n^{\alpha w + \beta +1}}{\alpha w + \beta +1} & \left[ \alpha_1 \left(w-1\right) x_{d,w-1}
+\alpha_2 \left( w-1 \right)x_{d-1,w-1} + \beta x_{d-\left( N-1 \right),w-1}
\right]. \end{split} \end{equation} During the above computation we deleted all terms having asymptotically smaller degree than the others.
First, we examine the case when the limits are positive. Suppose that there is at least one positive term in \eqref{A_as}. Therefore $A(n,d,w) \sim p a_w n^{\alpha w +\beta +1} x_{d,w} \to \infty $ (because $x_{d,w}>0$). In this case $\left(B \left( n,d,w \right)\right)^{\frac{1}{2}} \log B \left( n,d,w \right) = {\rm {O}} \left( A \left( n,d,w \right)\right)$. So, using Proposition VII-2-4 of \cite{neveu}, we have $Z \left( n,d,w \right) \sim A \left( n,d,w \right)$. Therefore \begin{equation} \dfrac{X \left( n,d,w \right)}{V_n} = \dfrac{Z \left( n,d,w \right)}{c\left( n,w \right)V_n} \sim \dfrac{A \left( n,d,w \right)}{c\left( n,w \right)V_n} \sim \dfrac{p a_{w} n^{\alpha w + \beta +1} x_{d,w}}{a_{w} n^{\alpha w + \beta }p n} = x_{d,w} \quad \text{a.s.} \quad \text{as} \quad n \to \infty, \end{equation} where, by \eqref{A_as}, \begin{equation*} x_{d,w} = \dfrac{1}{\alpha w + \beta +1} \left[ \alpha_{1} \left( w-1 \right) x_{d,w-1} + \alpha_{2}\left( w-1\right)x_{d-1,w-1} + \beta x_{d-\left(N-1\right),w-1} \right]. \end{equation*}
To handle the case when the limit is $0$, we argue as follows. Consider the case when the coefficients $x_{d,w-1}$, $x_{d-1,w-1}$, $x_{d-\left(N-1\right),w-1}$ are equal to zero.
By \eqref{A(n,d,w)} and using the induction hypothesis, we have \begin{multline*} \begin{split} A & \left( n,d,w \right) \sim \sum_{i=2}^{n} \left[ \vphantom{\dfrac{1}{2}} \right. a_{w} i^{\alpha w + \beta} \left( {\rm {O}} \left( \dfrac{1}{i^a} \right) + \sum_{m=2}^{N-2} X(i-1,d-m,w-1) \right. \times \\ & \times \left( p \left( 1-r \right)\dfrac { \binom{d-m}{N-m-1} \left(N-1\right)!} {\left(m-1\right)!} \dfrac{1}{\left(pi\right)^{N-m}} \right.
+ \left. \left. \left( 1-p \right)\left( 1-q \right)\dfrac {\binom{d-m}{N-m-1} N! } {m!} \dfrac{1}{\left(pi\right)^{N-m}} \right) \right) \sim \end{split} \end{multline*} \begin{equation*} \sim C_1 \sum_{i=2}^{n} a_{w} i^{\alpha w + \beta -a} +
C_2 \sum_{i=2}^{n} \sum_{m=2}^{N-2} i^{\alpha w + \beta} x_{d-m,w-1} \dfrac{1}{\left(pi\right)^{N-m-1}} \end{equation*} \begin{equation*} \leq C \dfrac{n^{\alpha w + \beta + 1 - a}}{\alpha w + \beta + 1 - a} + C \dfrac{n^{\alpha w + \beta }}{\alpha w + \beta} = \end{equation*} \begin{equation} \label{as_A(n,3w-1,w)} = {\rm {O}} \left( n^{\alpha w + \beta +1 -a} \right), \end{equation} where $C$ denotes an appropriate constant. So in this case the asymptotic behaviour of $A \left( n,d,w \right)$ is given by \eqref{as_A(n,3w-1,w)}. On the other hand, $B \left( n,d,w \right) = {\rm {O}} \left( n^{2 \left(\alpha w + \beta \right)+1} \right)$.
Using \eqref{Markinkiewicz} and \eqref{c(n,w)_asz.}, Proposition VII-2-4 of \cite{neveu}
implies \begin{equation*} \dfrac{X \left( n,d,w \right)}{V_n} = \dfrac{Z \left( n,d,w \right)}{c\left( n,w \right)V_n} = \dfrac{M \left( n,d,w \right) + A \left( n,d,w \right)}{c\left( n,w \right)V_n} = \end{equation*} \begin{equation} =\dfrac{{\rm {O}} \left( n^{\alpha w + \beta +1 -a} \right)}{n^{\alpha w + \beta}pn} = {\rm {O}} \left( n^{-a} \right) \to 0 = x_{d,w} \,, \quad {\text{a.s.}} \end{equation} So we have obtained the desired result for the case $0$ limit as well.
$\Box$ \end{pf}
\begin{rem} We can see that for each $d$ with $d \geq N$ there exists $w$ such that $x_{d,w} > 0$. \end{rem}
\section{The scale-free property for the weights and degrees}
\begin{lem} \label{xdw}
Let $p>0$ and define
$$
x_{w} = x_{N-1,w} + x_{N,w} + \dots + x_{\left(N-1\right)w,w}
$$
for $w=1,2,\dots$\,.
Then $x_{w}$, $w=1,2,\dots$\,, are positive numbers satisfying the following recurrence: $$ x_{1} = \dfrac{1}{\alpha + \beta +1}, $$ \begin{equation} \label{rekurziox(w)-re} x_{w} = \dfrac{\alpha \left( w-1 \right) + \beta}{\alpha w + \beta +1}x_{w-1}, \quad \text{if} \quad w > 1, \end{equation} where \begin{equation*} \alpha = \left(1-p\right) q + \dfrac{N-1}{N}pr, \quad \beta = \left( N-1 \right)\left( 1-r \right) + \dfrac{N\left( 1-p \right)\left( 1-q \right)}{p}. \end{equation*}
$x_{w}$, $w=1,2,\dots$\,, is a discrete probability distribution. Moreover, $x_{d,w}$, $d=N-1,N,\dots, \left(N-1 \right) w$, $w=1,2,\dots$\,, is a two-dimensional discrete probability distribution. \end{lem}
\begin{pf} If $\alpha=0$, then the statement is an obvious consequence of \eqref{rekurzio_x(d,w)}. Now assume $\alpha\ne 0$. As $x_{d,w}$ is defined as $x_{d,w}=0$ for $d\notin\{N-1,N,\dots, \left( N-1 \right)w\}$, therefore $x_{w} = \sum_{d} x_{d,w}$. From the recurrence \eqref{rekurzio_x(d,w)} for $x_{d,w}$, we obtain \begin{multline*} x_{w} = \sum_{d=N-1}^{\left( N-1 \right)w} x_{d,w} = \sum_{d} x_{d,w} = \\
=\dfrac{1}{\alpha w + \beta +1} \left[ \alpha_1 \left( w-1 \right) \sum_{d} x_{d,w-1} + \alpha_2 \left( w-1\right) \sum_{d} x_{d-1,w-1} +\beta \sum_{d} x_{d-\left(N-1\right),w-1} \right] = \end{multline*} $$
= \dfrac{\alpha \left( w-1 \right) + \beta}{\alpha w + \beta +1}x_{w-1}\,. $$ Using this recursive formula for $x_w$, we obtain $$ x_w = x_1 \prod_{j=2}^w \dfrac{\alpha \left( j-1 \right) + \beta}{\alpha j + \beta +1} = \dfrac{1}{\alpha + \beta +1} \dfrac{\alpha + \beta}{2 \alpha + \beta +1} \dfrac{2 \alpha + \beta}{3 \alpha + \beta +1} \dots \dfrac{\left( w-1 \right) \alpha + \beta}{w \alpha + \beta +1} = $$ $$
= \dfrac{1}{\alpha w + \beta +1} \prod_{j=1}^{w-1} \dfrac{ \frac{\beta}{\alpha} + j}{\frac{\beta +1}{\alpha} + j}
= \dfrac{1}{\alpha w + \beta +1} \dfrac{\varGamma \left( w + \frac{\beta}{\alpha} \right)}{\varGamma \left( 1 + \frac{\beta}{\alpha} \right)}
\dfrac{\varGamma \left( 1 + \frac{\beta +1}{\alpha} \right)}{\varGamma \left( w + \frac{\beta+1}{\alpha} \right)} = $$ \begin{equation} \label{recformx_w} = \dfrac{\varGamma \left( 1 + \frac{\beta +1}{\alpha} \right)}{\alpha \varGamma \left( 1 + \frac{\beta}{\alpha} \right)} \dfrac{\varGamma \left( w + \frac{\beta}{\alpha} \right)}{\varGamma \left( w + \frac{\beta+1}{\alpha}+ 1 \right)}. \end{equation} Moreover, by \cite{prudnikov}, we have the following formula: $$ \sum_{k=0}^{n} \dfrac{\varGamma \left( k+a \right)}{\varGamma \left( k+b \right)} = \dfrac{1}{a-b+1} \left[ \dfrac{\varGamma \left( n+a+1 \right)}{\varGamma \left( n+b \right)} - \dfrac{\varGamma \left( a \right)}{\varGamma \left( b-1 \right)} \right]. $$ Therefore, by some calculation, we obtain $ \sum_{k=1}^{n} x_k \to 1$, as $n \to \infty$. So $\sum_{w=1}^{\infty} x_w =1$. As $\sum_d x_{d,w}=x_w$, so $\sum_{w=1}^{\infty}\sum_{d=N-1}^{\left(N-1\right)w} x_{d,w} =1$ and therefore $x_{d,w}$, $d=N-1,N,\dots, \left(N-1\right)w$, $w=1,2,\dots$\,, is a two-dimensional discrete probability distribution.
$\Box$ \end{pf}
Let $X \left( n,w \right)$ denote the number of vertices of weight $w$ after $n$ steps. Next theorem is the scale-free property for the weights. It is an extension of \textit{Theorem 3.1} in \cite{BaMo1}, see also \textit{Theorem 3.1} of \cite{FIPB}.
\begin{thm} \label{theorem:scalefreeWeights}
Let $0<p<1$, $q>0$, $r>0$ and $(1-r)(1-q)>0$. Then for all $w=1,2,\dots$ we have \begin{equation} \dfrac{X \left( n,w \right)}{V_n} \rightarrow x_{w} \end{equation} almost surely, as $n \rightarrow \infty $, where $x_{w}$, $w=1,2, \dots$\,, are positive numbers satisfying the recurrence \eqref{rekurziox(w)-re}. Moreover, \begin{equation} \label{x_{w}_asz.} x_{w} \sim C w^{- \left( 1 + \frac{1}{\alpha} \right)}, \end{equation}
as $w \to \infty$, with $C =\varGamma \left( 1 + \frac{\beta +1}{\alpha} \right) \big/ \left({\alpha \varGamma \left( 1 + \frac{\beta}{\alpha} \right)}\right) $. \end{thm}
\begin{pf} We have $$ X \left( n,w \right) = X \left( n,N-1,w \right) + X \left( n,N ,w \right) + \dots + X \left( n,\left(N-1\right)w,w \right). $$ By Theorem \ref{limX/V}, \begin{equation*} \dfrac{X \left( n,w \right)}{V_n} \rightarrow x_{w} = x_{N-1,w}+ \dots + x_{\left(N-1\right)w,w} \end{equation*} almost surely, as $n \rightarrow \infty $. Here each $x_{w}$ is positive.
Using formula \eqref{recformx_w} and the Stirling-formula for the Gamma function, we have $$ x_w = \dfrac{\varGamma \left( 1 + \frac{\beta +1}{\alpha} \right)}{\alpha \varGamma \left( 1 + \frac{\beta}{\alpha} \right)} \dfrac{\varGamma \left( w + \frac{\beta}{\alpha} \right)}{\varGamma \left( w + \frac{\beta+1}{\alpha}+ 1 \right)}
\sim C_0
\dfrac{\left(w + \frac{\beta}{\alpha}\right)^{\left(w + \frac{\beta}{\alpha}\right)}}
{\left(w + \frac{\beta}{\alpha} + \frac{1}{\alpha} + 1\right)^{\left(w + \frac{\beta}{\alpha} + \frac{1}{\alpha} + 1\right)}} = $$ $$ = C_0 \left( \dfrac{\left(w + \frac{\beta}{\alpha}\right)}{\left(w + \frac{\beta}{\alpha} + \frac{1}{\alpha} + 1\right)}\right)^{\left(w + \frac{\beta}{\alpha}\right)} \dfrac{1}{\left(w +\frac{\beta}{\alpha} + \frac{1}{\alpha} + 1\right)^{\frac{1}{\alpha} + 1}} \sim C w^{- \left( 1 + \frac{1}{\alpha} \right)}, $$ where $C_0 =\dfrac{\varGamma \left( 1 + \frac{\beta +1}{\alpha} \right)}{\alpha \varGamma \left( 1 + \frac{\beta}{\alpha} \right) } \dfrac{1}{\left(\frac{1}{e}\right)^{1 + \frac{1}{\alpha}}} $ and $C=\dfrac{\varGamma \left( 1 + \frac{\beta +1}{\alpha} \right)}{\alpha \varGamma \left( 1 + \frac{\beta}{\alpha} \right) }$.
$\Box$ \end{pf}
Now we construct a representation of the limiting joint distribution of degrees and weights.
Let $W$ be a random variable with distribution ${{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left( W = w\right) = x_w\,\,, w=1,2,\dots $\,\,. Let $\xi_1 \equiv N-1$ and $\xi_2,\xi_3,\,\dots$ be independent random variables being independent of $W$, too. For $w \geq 2$ let $\xi_w$ have the following distribution: \begin{equation*} {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left( \xi_w = 0\right) = \dfrac{\alpha_1 \left( w-1 \right)}{\alpha \left( w-1 \right) + \beta}, \quad {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left( \xi_w = 1\right) = \dfrac{\alpha_2 \left( w-1 \right)}{\alpha \left( w-1 \right) + \beta}, \quad {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left( \xi_w = N-1\right) = \dfrac{\beta}{\alpha \left( w-1 \right) + \beta}\,. \end{equation*}
Introduce notation $ S_w = \xi_1 + \xi_2 + \dots + \xi_w $\,.
The following representation of the joint distribution of degrees and weights is useful to obtain scale-free property for degrees.
\begin{thm} \label{jointDistr} ${{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_W=d,W=w\right) = x_{d,w}$ \,for all\quad $w=1,2,\dots$\,,\,\,\,$d=N-1,N,\dots,\left( N-1 \right)w$. \end{thm}
\begin{pf} If $w=1$ and $d=N-1$ we have \begin{equation*} {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_W=N-1,W=1\right)
= {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(\xi_1=N-1,W=1\right) = {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(W=1\right) = x_1 = x_{N-1,1}\,. \end{equation*} If $w=1$ and $d \ne N-1$, then ${{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_W=d,W=1\right) = 0 = x_{d,1}$. \\ If $w=2$ and $d \not \in \{ N-1,N,2\left( N-1 \right) \}$ then we have \begin{equation*} {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_W=d,W=2\right) = {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_2=d,W=2\right) = {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(\xi_1=N-1, \xi_2 = d-\left( N-1 \right), W=2\right) = 0 = x_{d,2}\,. \end{equation*} Using the recursion \eqref{rekurziox(w)-re} and the assumption that $\xi_1, \xi_2, \xi_3, \dots$ are independent random variables which are independent of $W$, we have for $w \geq 2$ \begin{equation*} {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_W=d,W=w\right) = {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_w=d,W=w\right) = {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_w=d \right){{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(W=w\right)= \end{equation*} \begin{eqnarray*}
= \left[{{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_{w-1}=d\right){{\mathbb {P}}} \def\Q{{\mathbb {Q}}}\left( \xi_w=0 \right) + {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_{w-1}=d-1\right){{\mathbb {P}}} \def\Q{{\mathbb {Q}}}\left( \xi_w=1 \right)+{{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_{w-1}=d-\left( N-1 \right) \right){{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(\xi_w=\left( N-1 \right)\right)\right] \times \end{eqnarray*} \begin{eqnarray*}
\times {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(W=w-1\right)\dfrac{x_w}{x_{w-1}} = \end{eqnarray*} \begin{eqnarray*}
= \left[{{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_{w-1}=d\right){{\mathbb {P}}} \def\Q{{\mathbb {Q}}}\left( W = w-1 \right)\alpha_1 \left(w-1\right) + {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_{w-1}=d-1\right){{\mathbb {P}}} \def\Q{{\mathbb {Q}}}\left( W = w-1 \right)\alpha_2 \left(w-1\right)+ \right.
\end{eqnarray*} \begin{eqnarray*} + \left. {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_{w-1}=d-\left( N-1 \right) \right){{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(W = w-1\right)\beta \right] \dfrac{1}{\alpha w + \beta +1} = \end{eqnarray*} \begin{eqnarray*}
= \dfrac{1}{\alpha w + \beta +1} \left[ \alpha_1 \left(w-1\right) {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_W=d,W=w-1\right) +
\alpha_2 \left(w-1\right) \left(S_W=d-1,W=w-1\right) + \right.
\end{eqnarray*}
\begin{eqnarray*}
\left. + \beta {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_W=d-\left( N-1 \right),W=w-1\right) \right].
\end{eqnarray*} Now, we can see that the sequence ${{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_W=d,W=w\right)$ satisfies the same recursion \eqref{rekurzio_x(d,w)} as $x_{d,w}$\,.
$\Box$ \end{pf}
\begin{thm} \label{Theorem-x(d,w)_as}
Suppose that $\alpha_1 > 0$ and $\alpha_2 > 0$\,.
Then \begin{equation} \label{x_dw=} x_{d,w} = x_w \dfrac{\alpha}{\sqrt{2 \pi \alpha_1 \alpha_2 w}} \left[ \exp \left( - \dfrac{\left(d - {{\mathbb {E}}} \def\D{{\mathbb {D}}} S_w\right)^2}{2 {\D}^2 S_w} \right) + {\rm {O}} \left( w^{-\frac{1}{2}} \right)\right] \,,\,\,\,\text{as}\,\,\,w \to \infty\,, \end{equation}
where the error term ${\rm {O}} \left( w^{-\frac{1}{2}} \right)$ does not depend on $d$\,. \end{thm} \begin{pf} We can follow the ideas of the proof of \textit{Theorem 4.2} in \cite{BaMo2}.
Let $w \geq 1$. By the definition of the expected value, we have \begin{equation*} {{\mathbb {E}}} \def\D{{\mathbb {D}}} \xi_w = \dfrac{\alpha_2 \left( w-1 \right)}{\alpha \left( w-1 \right) + \beta} + \left( N-1\right) \dfrac{\beta}{\alpha \left( w-1 \right) + \beta} = \dfrac{\alpha_2}{\alpha} + \dfrac{\left(\left( N-1\right) \alpha -\alpha_2\right) \beta}{\alpha \left(\alpha \left( w-1 \right) + \beta \right)}\,, \end{equation*} if $w \geq 2$, hence \begin{equation*} {{\mathbb {E}}} \def\D{{\mathbb {D}}} S_w = {{\mathbb {E}}} \def\D{{\mathbb {D}}} \xi_1 + \dots + {{\mathbb {E}}} \def\D{{\mathbb {D}}} \xi_w = w \dfrac{\alpha_2}{\alpha} + {\rm {O}} \left(\log w\right)\,, \end{equation*} as $w \to \infty$. Similarly, by simple computation, we have \begin{equation} \label{D} {\D}^2 \xi_w = \dfrac{\alpha_1 \alpha_2}{\alpha^2} + {\rm {O}} \left( \dfrac{1}{w} \right)\,,\,\,\,\,\ \ {\D}^2 S_w = \dfrac{\alpha_1 \alpha_2}{\alpha^2} w + {\rm {O}} \left( \log w \right)\,, \end{equation} as $w \to \infty$. \\ Now, we can apply Theorem VII.2.5 in \cite{petrov} for $S_w$. The conditions of that theorem are satisfied, therefore we have \begin{equation} \label{Petrov}
\sup_{d \in {{\mathbb {Z}}} } \left| {\D} S_w {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left( S_w = d \right) - \dfrac{1}{\sqrt{2\pi}} \exp \left( - \dfrac{\left( d- {{\mathbb {E}}} \def\D{{\mathbb {D}}} S_w \right)^2}{2 {\D}^2 S_w} \right)\right| = {\rm {O}} \left( \dfrac{1}{\sqrt{w}} \right)\,. \end{equation} Using \eqref{D} and \eqref{Petrov}, we obtain
$\left| {\D} S_w - \dfrac{\sqrt{\alpha_1 \alpha_2 w}}{\alpha} \right| {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left( S_w = d \right) = {\rm {O}} \left( w^{-\frac{1}{2}} \right)$. Therefore, it follows from \eqref{Petrov}, that \begin{equation} \label{Alk_Petrov}
\sup_{d \in {{\mathbb {Z}}} } \left| \dfrac{\sqrt{\alpha_1 \alpha_2 w}}{\alpha}\,\, {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left( S_w = d \right) -
\dfrac{1}{\sqrt{2\pi}} \exp \left( - \dfrac{\left( d- {{\mathbb {E}}} \def\D{{\mathbb {D}}} S_w \right)^2}{2 {\D}^2 S_w} \right)\right| = {\rm {O}} \left( \dfrac{1}{\sqrt{w}} \right)\,. \end{equation} The independence of $W$ and $\xi_i$ implies that \begin{equation*} x_{d,w} = {{\mathbb {P}}} \def\Q{{\mathbb {Q}}}\left( S_W = d, W = w \right) = {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left( S_w = d \right)x_w\,. \end{equation*} Using this in \eqref{Alk_Petrov}, we can obtain the desired result.
$\Box$ \end{pf}
Our last theorem is an extension of \textit{Theorem 4.3} in \cite{BaMo2} (see also \textit{Theorem 3.4} of \cite{FIPB}) to the case of $N$ interactions. The theorem shows the scale-free property for the degrees.
\begin{thm} \label{ThmScaleFreeDegree} Let $0<p<1$, $q>0$, $r>0$ and $(1-r)(1-q)>0$. Let us denote by $U\left( n,d \right)$ the number of vertices of degree $d$ after $n$ steps, that is $U\left( n,d \right) = \sum_{w : \frac{d}{N-1} \leq w \leq n+1} X \left( n,d,w \right)$\,. Then, for any $d \geq N-1$ we have \begin{equation} \label{u_d} \dfrac{U\left( n,d \right)}{V_n} \to u_d = \sum_w x_{d,w} \end{equation} a.s. as $n \to \infty$\,, where $u_d$, $d = N-1, N, \dots$, are positive numbers. Furthermore, \begin{equation} \label{u_d as} u_d \sim \dfrac{\varGamma \left( 1 + \frac{\beta + 1}{\alpha} \right)}{\alpha_2 \varGamma \left( 1 + \frac{\beta}{\alpha} \right)}\left( \dfrac{\alpha d}{\alpha_2} \right)^{- \left( 1 + \frac{1}{\alpha} \right)}\,, \end{equation} as $d \to \infty$\,. \end{thm}
\begin{pf} By Theorems \ref{limX/V} and \ref{jointDistr}, $\dfrac{X \left( n,d,w \right)}{V_n}$ converges almost surely to the distribution $x_{d,w}= {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_W=d,W=w\right)$. But the cardinalities of terms in $\sum_{w : \frac{d}{N-1} \leq w \leq n+1} X \left( n,d,w \right)$ are not bounded when $n \to \infty$. However, using that $x_{d,w}$, $d=N-1,N,\dots, \left(N-1 \right)w$, $w=1,2,\dots$ is a proper two-dimensional discrete distribution, therefore the convergence of the marginal distributions is a consequence of the convergence of the two-dimensional distributions. So we obtain \eqref{u_d}.
To obtain \eqref{u_d as}, we follow the lines of \cite{BaMo2}.
Let $$ f = \dfrac{\alpha}{\alpha_2} d\,, \quad H =H_d = \left\lbrace w : f-f^{\frac{1}{2}+ \varepsilon} \leq w \leq f+f^{\frac{1}{2}+ \varepsilon} \right\rbrace\,, $$ $$ H^{-} = H_d^{-} = \left\lbrace w : w < f-f^{\frac{1}{2}+ \varepsilon } \right\rbrace \,, \quad H^{+} = H_d^{+} = \left\lbrace w : w > f+f^{\frac{1}{2}+ \varepsilon} \right\rbrace\, $$ with some fixed $0 < \varepsilon < \dfrac{1}{6}$.
Using Hoeffding's exponential inequality (Theorem 2 in \cite{HO}) for $w \in H^{-}$ we have \begin{equation*} {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_w = d \right) \leq {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_w \geq d \right) \leq {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_w - {{\mathbb {E}}} \def\D{{\mathbb {D}}} S_w \geq d - \dfrac{\alpha_2}{\alpha}w - {\rm {O}} \left( \log w \right) \right) \leq \end{equation*} \begin{equation*} \leq \exp \left\lbrace - \dfrac{2}{\left(N-1\right)^2w} \left( d - \dfrac{\alpha_2}{\alpha}w - {\rm {O}} \left( \log w \right) \right)^2 \right\rbrace = \exp \left\lbrace -\dfrac{2}{\left(N-1\right)^2} \left( \dfrac{\alpha_2}{\alpha} \right)^2 \dfrac{\left( f-w - {\rm {O}} \left( \log w \right) \right)^2}{w}\right\rbrace. \end{equation*} Here $w \in H^{-}$ implies that $$ \left( f-w - {\rm {O}} \left( \log w \right) \right)^2 = \left( f-w \right)^2 - 2 \left( f-w \right) {\rm {O}} \left( \log w \right) + \left({\rm {O}} \left( \log w \right)\right)^2 \geq f^{1+2\varepsilon} - {\rm {O}} \left( f \log f \right). $$ Therefore in the case when $w \in H^{-}$ we have $$ {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_w = d \right) \leq \exp \left\lbrace -\dfrac{2}{\left(N-1\right)^2} \left( \dfrac{\alpha_2}{\alpha} \right)^2 \dfrac{ f^{1+2\varepsilon} -{\rm {O}} \left( f \log f \right)}{f}\right\rbrace = \exp \left\lbrace - \dfrac{2}{\left(N-1\right)^2}\left( \dfrac{\alpha_2}{\alpha} \right)^2 f^{2\varepsilon} + {\rm {O}} \left(\log f \right) \right\rbrace. $$ Using this, we can obtain that $$ {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_W=d,W \in H^{-}\right) = \sum_{w \in H^{-}} {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_w=d,W = w\right) \leq \sum_{w \in H^{-}} {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_w=d\right) \leq $$ \begin{equation} \label{H-} \leq f \exp \left\lbrace - \dfrac{2}{\left(N-1\right)^2}\left( \dfrac{\alpha_2}{\alpha} \right)^2 f^{2\varepsilon} + {\rm {O}} \left(\log f \right) \right\rbrace = {\rm {o}} \left( f^{- \left( 1 + \frac{1}{\alpha} \right)} \right). \end{equation} Similarly, if $w \in H^{+}$, again by Hoeffding's inequality, we have \begin{multline*} {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_w = d \right) \leq {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_w \leq d \right) \leq {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_w - {{\mathbb {E}}} \def\D{{\mathbb {D}}} S_w \leq d - \dfrac{\alpha_2}{\alpha}w\right) \leq \\ \leq \exp \left\lbrace -\dfrac{2}{\left(N-1\right)^2w} \left(d- \dfrac{\alpha_2}{\alpha}w \right)^2 \right\rbrace =
\exp \left\lbrace - \dfrac{2}{\left(N-1\right)^2}\left( \dfrac{\alpha_2}{\alpha} \right)^2 \dfrac{\left(f-w\right)^2}{w}\right\rbrace. \end{multline*} Using that $w\in H^+$ and $\frac{1}{2} + \varepsilon <1$, we obtain $2 \left( w-f \right) \geq f^{\frac{1}{2} + \varepsilon} + w - f \geq f^{\frac{1}{2} + \varepsilon} + \left(w-f\right)^{\frac{1}{2} + \varepsilon} \geq w^{\frac{1}{2} + \varepsilon}$ for $d$ large enough. Therefore $$ {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_w = d \right) \leq \exp \left\lbrace -\dfrac{2}{\left(N-1\right)^2} \left( \dfrac{\alpha_2}{\alpha} \right)^2 \dfrac{w^{1+2\varepsilon}}{4w}\right\rbrace = \exp \left\lbrace - \dfrac{1}{2\left(N-1\right)^2}\left( \dfrac{\alpha_2}{\alpha} \right)^2 w^{2\varepsilon} \right\rbrace. $$ Hence \begin{equation} \label{H+} {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_W=d,W \in H^{+}\right) \leq \sum_{\{ w \, : \, f < w \}} \exp \left\lbrace - \dfrac{1}{2\left(N-1\right)^2} \left( \dfrac{\alpha_2}{\alpha} \right)^2 w^{2\varepsilon} \right\rbrace = {\rm {o}} \left( f^{\left(-1+\frac{1}{\alpha}\right)} \right) \end{equation} for $f$ large enough.
Now consider the case when $w \in H$. First we need some general facts. Consider the set $$ B= \left\{ (d,w) \, : \, w\ge 1, d\ge N-1, w\in H_d \right\}. $$ It is easy to see that when $(d,w)\in B$ then $d\to \infty$ if and only if $w\to \infty$. More precisely, $$ \frac{w}{d} \to 1, \quad {\text{ if}} \quad d\to \infty \quad {\text{and}} \quad (d,w)\in B. $$ We have $w =f + {\rm {O}} \left( f^{\frac{1}{2} + \varepsilon}\right)$. Then (with $\varepsilon_1 >0$ arbitrarily small) \begin{equation} \label{H++} - \dfrac{\left(d - {{\mathbb {E}}} \def\D{{\mathbb {D}}} S_w\right)^2}{2 {\D}^2 S_w} =
- \dfrac{\left(d - w \dfrac{\alpha_2}{\alpha} - {\rm {O}} \left(\log w\right)\right)^2}{2 \dfrac{\alpha_1 \alpha_2}{\alpha^2} w + {\rm {O}} \left( \log w \right)} = - \dfrac{\alpha_2}{\alpha_1} \dfrac{\left( f-w -{\rm {O}} \left(\log w\right) \right)^2}{2w + {\rm {O}} \left( \log w \right)}= \end{equation} \begin{equation*} = - \dfrac{\alpha_2}{\alpha_1} \dfrac{\left( f-w \right)^2 + {\rm {O}} \left( f^{\frac{1}{2} + \varepsilon + \varepsilon_1} \right)}{2w + {\rm {O}} \left( \log w \right)}= - \dfrac{\alpha_2}{\alpha_1} \dfrac{\left( f-w \right)^2 + {\rm {O}} \left( f^{\frac{1}{2} + \varepsilon + \varepsilon_1} \right)}{2f} \dfrac{2f}{2f + {\rm {O}} \left( f^{\frac{1}{2} + \varepsilon} \right)} = \end{equation*} \begin{equation*} = - \dfrac{\alpha_2}{\alpha_1} \dfrac{\left( f-w \right)^2 + {\rm {O}} \left( f^{\frac{1}{2} + \varepsilon + \varepsilon_1} \right)}{2f} \left[ 1 - \dfrac{{\rm {O}} \left( f^{\frac{1}{2} + \varepsilon}\right)}{2f + {\rm {O}} \left( f^{\frac{1}{2} + \varepsilon} \right)} \right] = - \dfrac{\alpha_2}{\alpha_1} \dfrac{\left( f-w \right)^2}{2f} + {\rm {O}} \left( f^{- \frac{1}{2} + 3\varepsilon} \right), \end{equation*} as $d \to \infty $. Here the error term does not depend on $w$. We shall apply Theorem \ref{Theorem-x(d,w)_as} that is formula \eqref{x_dw=}. The asymptotic behaviour of $x_{w}$ is known from \eqref{x_{w}_asz.}. Using these facts and \eqref{H++}, we obtain $$ x_{d,w} \sim C w^{- \left( 1 + \frac{1}{\alpha} \right)} \dfrac{\alpha}{\sqrt{2 \pi \alpha_1 \alpha_2 w}} \left[ \exp \left\lbrace - \dfrac{\alpha_2}{\alpha_1} \dfrac{\left( f-w \right)^2}{2f} + {\rm {O}} \left( f^{- \frac{1}{2} + 3\varepsilon} \right) \right\rbrace + {\rm {O}} \left( w^{-\frac{1}{2}} \right) \right] \sim $$ $$ \sim C f^{- \left( 1 + \frac{1}{\alpha} \right)} \dfrac{\alpha}{\alpha_2} \dfrac{1}{\sqrt{2 \pi \frac{\alpha_1}{\alpha_2} f}} \exp \left\lbrace -\dfrac{\left( f-w \right)^2}{2 \frac{\alpha_1}{\alpha_2}f} \right\rbrace $$ as $d \to \infty$ and $w \in H$, where $C= \varGamma \left( 1 + \frac{\beta + 1}{\alpha} \right)/ \left(\alpha \varGamma \left( 1 + \frac{\beta}{\alpha} \right)\right)$. Therefore $$ \sum_{w \in H} x_{d,w} \sim \sum_{f-f^{\frac{1}{2} + \varepsilon} < w < f+f^{\frac{1}{2} + \varepsilon}} C f^{- \left( 1 + \frac{1}{\alpha} \right)} \dfrac{\alpha}{\alpha_2} \dfrac{1}{\sqrt{2 \pi \frac{\alpha_1}{ \alpha_2}f}} \exp \left\lbrace - \dfrac{\left( f-w \right)^2}{2 \frac{\alpha_1}{ \alpha_2}f} \right\rbrace \sim $$ $$ \sim C f^{- \left( 1 + \frac{1}{\alpha} \right)} \dfrac{\alpha}{\alpha_2} \sum_{-f^{\frac{1}{2}+\varepsilon} < k < f^{\frac{1}{2}+\varepsilon}} \dfrac{1}{\sqrt{2 \pi \frac{\alpha_1}{ \alpha_2}f}} \exp \left\lbrace - \dfrac{k^2}{2 \frac{\alpha_1}{ \alpha_2}f} \right\rbrace = $$ $$ = A \sum_{-f^{\varepsilon} < \frac{k}{\sqrt{f}} < f^{\varepsilon}} \dfrac{1}{\sqrt{f}} \dfrac{1}{\sqrt{2 \pi \frac{\alpha_1}{ \alpha_2}}} \exp \left\lbrace - \dfrac{\left(\frac{k}{\sqrt{f}}\right)^2}{2 \frac{\alpha_1}{ \alpha_2}} \right\rbrace \to A \int_{-\infty}^{+\infty} \dfrac{1}{\sqrt{2 \pi \frac{\alpha_1}{ \alpha_2}}} \exp \left\lbrace - \dfrac{x^2}{2 \frac{\alpha_1}{ \alpha_2}} \right\rbrace dx = A. $$ Thus we have \begin{equation} \label{H} {{\mathbb {P}}} \def\Q{{\mathbb {Q}}} \left(S_W=d,W \in H\right) \sim A = \dfrac{\varGamma \left( 1 + \frac{\beta + 1}{\alpha} \right)}{\alpha_2 \varGamma \left( 1 + \frac{\beta}{\alpha} \right)} \left( \dfrac{\alpha d}{\alpha_2} \right)^{- \left( 1 + \frac{1}{\alpha} \right)}, \end{equation} as $d \to \infty$. Finally, from \eqref{H-}, \eqref{H+} and \eqref{H}, we obtain $$ u_d = \sum_w x_{d,w} = \sum_{w \in H^{-}} x_{d,w} + \sum_{w \in H} x_{d,w} + \sum_{w \in H^{+}} x_{d,w} \sim {\rm {o}} \left( f^{- \left( 1+\frac{1}{\alpha}\right)} \right) + C \dfrac{\alpha}{\alpha_2} f^{- \left( 1+\frac{1}{\alpha}\right)} + {\rm {o}} \left( f^{- \left( 1+\frac{1}{\alpha}\right)} \right) \sim $$ $$ \sim \dfrac{\varGamma \left( 1 + \frac{\beta +1}{\alpha} \right)}{\alpha_2 \varGamma \left( 1 + \frac{\beta}{\alpha} \right)} \left(\dfrac{\alpha}{\alpha_2}d\right)^{-\left(1 + \frac{1}{\alpha}\right)}, $$ as $d \to \infty $. The proof is complete.
$\Box$ \end{pf}
\end{document} | arXiv |
QRHFIRR The symmetries of the orbitals which are involved in the QRHF orbital occupation alteration. A minus sign indicates that a $\beta$ orbital of the associated symmetry will be depopulated, while a positive sign indicates that an $\alpha$ orbital will be populated.
Data type: floating point. Dimension: QRHFTOT. Written by: xjoda.
This page has been visited 731 times since December 2010. | CommonCrawl |
Singleton (mathematics)
In mathematics, a singleton, also known as a unit set[1] or one-point set, is a set with exactly one element. For example, the set $\{0\}$ is a singleton whose single element is $0$.
Part of a series on statistics
Probability theory
• Probability
• Axioms
• Determinism
• System
• Indeterminism
• Randomness
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• Collectively exhaustive events
• Elementary event
• Mutual exclusivity
• Outcome
• Singleton
• Experiment
• Bernoulli trial
• Probability distribution
• Bernoulli distribution
• Binomial distribution
• Normal distribution
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• Continuous or discrete
• Expected value
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• Conditional independence
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For a sequence with one member, see 1-tuple.
Properties
Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity guarantees that no set is an element of itself. This implies that a singleton is necessarily distinct from the element it contains,[1] thus 1 and {1} are not the same thing, and the empty set is distinct from the set containing only the empty set. A set such as $\{\{1,2,3\}\}$ is a singleton as it contains a single element (which itself is a set, however, not a singleton).
A set is a singleton if and only if its cardinality is 1. In von Neumann's set-theoretic construction of the natural numbers, the number 1 is defined as the singleton $\{0\}.$
In axiomatic set theory, the existence of singletons is a consequence of the axiom of pairing: for any set A, the axiom applied to A and A asserts the existence of $\{A,A\},$ which is the same as the singleton $\{A\}$ (since it contains A, and no other set, as an element).
If A is any set and S is any singleton, then there exists precisely one function from A to S, the function sending every element of A to the single element of S. Thus every singleton is a terminal object in the category of sets.
A singleton has the property that every function from it to any arbitrary set is injective. The only non-singleton set with this property is the empty set.
Every singleton set is an ultra prefilter. If $X$ is a set and $x\in X$ then the upward of $\{x\}$ in $X,$ which is the set $\{S\subseteq X:x\in S\},$ is a principal ultrafilter on $X.$[2] Moreover, every principal ultrafilter on $X$ is necessarily of this form.[2] The ultrafilter lemma implies that non-principal ultrafilters exist on every infinite set (these are called free ultrafilters). Every net valued in a singleton subset $X$ of is an ultranet in $X.$
The Bell number integer sequence counts the number of partitions of a set (OEIS: A000110), if singletons are excluded then the numbers are smaller (OEIS: A000296).
In category theory
Structures built on singletons often serve as terminal objects or zero objects of various categories:
• The statement above shows that the singleton sets are precisely the terminal objects in the category Set of sets. No other sets are terminal.
• Any singleton admits a unique topological space structure (both subsets are open). These singleton topological spaces are terminal objects in the category of topological spaces and continuous functions. No other spaces are terminal in that category.
• Any singleton admits a unique group structure (the unique element serving as identity element). These singleton groups are zero objects in the category of groups and group homomorphisms. No other groups are terminal in that category.
Definition by indicator functions
Let S be a class defined by an indicator function
$b:X\to \{0,1\}.$
Then S is called a singleton if and only if there is some $y\in X$ such that for all $x\in X,$
$b(x)=(x=y).$
Definition in Principia Mathematica
The following definition was introduced by Whitehead and Russell[3]
$\iota $‘$x={\hat {y}}(y=x)$ Df.
The symbol $\iota $‘$x$ denotes the singleton $\{x\}$ and ${\hat {y}}(y=x)$ denotes the class of objects identical with $x$ aka $\{y:y=x\}$. This occurs as a definition in the introduction, which, in places, simplifies the argument in the main text, where it occurs as proposition 51.01 (p.357 ibid.). The proposition is subsequently used to define the cardinal number 1 as
$1={\hat {\alpha }}((\exists x)\alpha =\iota $‘$x)$ Df.
That is, 1 is the class of singletons. This is definition 52.01 (p.363 ibid.)
See also
• Class (set theory) – Collection of sets in mathematics that can be defined based on a property of its members
• Uniqueness quantification – Logical property of being the one and only object satisfying a condition
References
1. Stoll, Robert (1961). Sets, Logic and Axiomatic Theories. W. H. Freeman and Company. pp. 5–6.
2. Dolecki & Mynard 2016, pp. 27–54.
3. Whitehead, Alfred North; Bertrand Russell (1910). Principia Mathematica. Vol. I. p. 37.
• Dolecki, Szymon; Mynard, Frederic (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.
Set theory
Overview
• Set (mathematics)
Axioms
• Adjunction
• Choice
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• dependent
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• Determinacy
• Extensionality
• Infinity
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Operations
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Set types
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Set theorists
• Paul Bernays
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• Kurt Gödel
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| Wikipedia |
Proceedings of the 13th Annual MCBIOS conference
Improving sensitivity of linear regression-based cell type-specific differential expression deconvolution with per-gene vs. global significance threshold
Edmund R. Glass1 &
Mikhail G. Dozmorov1
The goal of many human disease-oriented studies is to detect molecular mechanisms different between healthy controls and patients. Yet, commonly used gene expression measurements from blood samples suffer from variability of cell composition. This variability hinders the detection of differentially expressed genes and is often ignored. Combined with cell counts, heterogeneous gene expression may provide deeper insights into the gene expression differences on the cell type-specific level.
Published computational methods use linear regression to estimate cell type-specific differential expression, and a global cutoff to judge significance, such as False Discovery Rate (FDR). Yet, they do not consider many artifacts hidden in high-dimensional gene expression data that may negatively affect linear regression. In this paper we quantify the parameter space affecting the performance of linear regression (sensitivity of cell type-specific differential expression detection) on a per-gene basis.
We evaluated the effect of sample sizes, cell type-specific proportion variability, and mean squared error on sensitivity of cell type-specific differential expression detection using linear regression. Each parameter affected variability of cell type-specific expression estimates and, subsequently, the sensitivity of differential expression detection. We provide the R package, LRCDE, which performs linear regression-based cell type-specific differential expression (deconvolution) detection on a gene-by-gene basis. Accounting for variability around cell type-specific gene expression estimates, it computes per-gene t-statistics of differential detection, p-values, t-statistic-based sensitivity, group-specific mean squared error, and several gene-specific diagnostic metrics.
The sensitivity of linear regression-based cell type-specific differential expression detection differed for each gene as a function of mean squared error, per group sample sizes, and variability of the proportions of target cell (cell type being analyzed). We demonstrate that LRCDE, which uses Welch's t-test to compare per-gene cell type-specific gene expression estimates, is more sensitive in detecting cell type-specific differential expression at α < 0.05 missed by the global false discovery rate threshold FDR < 0.3.
Detection of differential gene expression at the cell type-specific level (deconvolution) aims to provide deeper insight into underlying biological causes of a given pathology. Investigators studying disease mechanisms benefit from knowing which genes in which cell types are differentially expressed. Yet, deconvolution is complicated by the prohibitive cost of extraction of pure cell type specimens, and non-linearity of amplified pure samples [1]. Statistical methods of quantifying cell type-specific differential gene expression (CDE) are a viable alternative to deconvolve heterogeneous gene expression signal into the cell type-specific measures that can be compared for significant differences [2–4].
There are two rationales behind CDE. One is that group-wise differential expression analysis on heterogeneous measures provides no information about which cell types are the source of any detected differences [2]. The other is that differential expression detection analysis applied only to heterogeneous tissue may miss the true cell type-specific expression differences. Thus, CDE analysis may uncover cell type-specific signal not seen at the heterogeneous level [5] (Additional file 1: section 1.1).
Previous efforts primarily focused on quantifying cell proportions from heterogeneous tissue is by using a priori known cell signatures as predictors in a linear regression model [3, 4, 6–10]. The other, less developed approach focuses on cell specific gene expression detection. It relies on linear regression to deconvolve heterogeneous gene expression measures using cell proportions as predictors. The coefficient estimates in this setup represent average cell type-specific expression levels, comparable if two groups are analyzed [5, 10–13]. Both approaches require two pieces of information: 1) the heterogeneous gene expression measures, and 2) either the cell signatures (first approach), or cell proportions (second approach).
Two algorithms addressing the second approach have been published (csSAM, DSection) [5, 11]. The csSAM approach uses heterogeneous observations as outcomes in a linear regression model, and the measured cell proportions as predictors. Two regressions, one per study group (e.g., case–control groups), are performed and the difference between coefficient estimates represents the cell type-specific differential expression estimates. Group label permutations are performed and false discovery rates (FDR) are estimated across the range of effect sizes per cell type. The csSAM authors acknowledge that increasing sample variability will improve cell type-specific expression accuracy, and we quantified the effect of such variability.
DSection assumes that cell proportion measures are imprecise and that this imprecision must be accounted for. DSection uses a Bayesian approach to "de-noise" cell proportion measures prior to linear regression deconvolution. The authors of DSection contrast their method to a "gold standard" of using linear regression when cell proportions are precisely known. The DSection authors correctly point out that, in real settings, no exact knowledge of cell proportions is known and that measurements are presumed to be estimates. They also acknowledge that the choice of prior information to use with their Bayesian approach has a subjective component.
In this study, we investigated the sensitivity of linear regression to detect cell type-specific gene expression differences on per-gene basis. Parameters affecting the variability of cell type-specific expression estimates (Fig. 1), and the sensitivity of cell type-specific differential expression, include, 1) sample size per study group, 2) average spread of heterogeneous measures around a linear regression prediction fit (size of residuals, quantified by mean squared error – MSE), and 3) variability of cell type-specific proportions across samples. We tested the effect of each parameter in simulation settings, while controlling other parameters (Fig. 2). For fixed values of sample size and cell proportion variability, any modification of MSE or cell type-specific differential expression affected the sensitivity of LRCDE. Since MSE and cell type-specific differential expression are gene-dependent, we conclude that any evaluation of sensitivity of cell type-specific differential expression detection must be assessed on per-gene basis, instead of a global significance threshold. We implement our approach in the LRCDE R package that utilizes variability of the per-gene cell type-specific expression estimates, and is more sensitive in detecting true cell type-specific differentially expressed genes as compared with the global significance cutoff.
Differential expression detection sensitivity is primarily affected by two factors: cell type-specific expression estimate (point estimate) variability and cell type-specific differential expression (a). A two-sample t-statistic is computed using the observed effect size (cell type-specific differential expression (Additional file 1: section 1.3). If the t-statistic does not exceed the t-critical value, which is based on the alpha significance threshold, then we cannot conclude that a significant difference has been observed between the two groups. Given an observed difference which is determined to be significant, then we may reject the null hypothesis of no difference between controls and cases and calculate sensitivity for this observed difference, based upon the distance from the case group expression estimate to the t-critical value (b). Bell curves represent distribution of cell type-specific expression estimates (point estimates - vertical dashed lines). The cell type-specific differential expression estimate (effect size) corresponds to the distance between vertical dashed lines for cases and controls (blue/purple bell curves, respectively)
Parameters affecting sensitivity of cell type-specific differential expression detection. The effect of (a) per-group samples sizes (10, 14, and 18); (b) cell proportion SD (0.05, 0.1, and 0.15); (c) MSE (0.5, 1.5, and 2.5); (d) log2-transformation vs. as-is data. Unless specified otherwise, log2-transformed data was used, and the following parameters were held constant: per-group sample size - 14, condition number - 100, cell proportion SD - 0.1 (0.6 for C), MSE - 1.5
Modeling heterogeneous gene expression measures using linear regression (LR) given sample specific cell proportions (deconvolution)
We model heterogeneous gene expression measures across samples as cumulative contributions of cell type-specific gene expression measures weighted by the corresponding cell proportions of P cell types. A biologically meaningful constraint of this model is that cell proportions for any given sample should sum up to 1, or 100 % [5]. As proposed, heterogeneous gene expression measures (y mn , where n is gene index, m is sample index) are modeled using a linear regression approach:
$$ {y}_{mn}={\displaystyle {\sum}_{k=1}^p{\beta}_{kn}{x}_{km}+{\varepsilon}_{mn}} $$
where β kn is the average theoretical cell type-specific gene expression for the k th of p total cell types, x km is the cell proportion (predictor), and ε mn is a normally distributed random error defined as the difference between the observed values y mn and values predicted by the linear regression ŷ mn , (y mn − ŷ mn ). This allows obtaining linear regression coefficient estimates \( {\widehat{\beta}}_{kn} \), interpreted as cell type-specific gene expression estimates. Intuitively, eq. 1 describes a linear relationship between heterogeneous gene expression level y mn and contribution of cell type-specific gene expression estimates \( {\widehat{\beta}}_{kn} \) weighted by the corresponding cell proportions x km . The model in eq. 1 contains no intercept term since we assume zero heterogeneous expression (y mn = 0) in the absence of individual cell contributions. Thus, for each gene we have a total of P cell type-specific gene expression estimates (regression coefficients, one per cell type) in the model. Model in eq. 1 is more compactly represented in matrix form, for a single gene j:
$$ {\mathbf{y}}_j=\mathbf{X}{\boldsymbol{\upbeta}}_j+\boldsymbol{\upvarepsilon} $$
The matrix form eq. 2 suggests the form in which β j is estimated:
$$ {\widehat{\boldsymbol{\upbeta}}}_j={\left(\mathbf{X}\mathbf{\hbox{'}}\mathbf{X}\right)}^{-1}\mathbf{X}\mathbf{\hbox{'}}{\mathbf{y}}_j $$
Fitted regression estimates are then given by:
$$ {\widehat{\mathbf{y}}}_j=\mathbf{X}{\widehat{\boldsymbol{\upbeta}}}_j $$
which are required in order to calculate residual values needed to estimate the variance of \( {\widehat{\boldsymbol{\upbeta}}}_j \).
Obtaining cell type-specific gene expression estimates carries a quantifiable level of uncertainty. This uncertainty can be expressed as a function of sample size, number of cell types, the size of the residuals, and the variability of cell type proportions. The formula for the theoretical variance of the linear regression coefficient for simple linear regression (single predictor vector X) provides an intuitive illustration of how various parameters affect the variance:
$$ \operatorname{var}\left({\widehat{\beta}}_1\right)=\frac{\sigma^2}{{\displaystyle {\sum}_{i=1}^m{\left({x}_i-\overline{x}\right)}^2}} $$
In practice, the estimated variance of \( {\widehat{\beta}}_1 \) in eq. 5 uses the mean squared error (MSE) as an estimate of σ 2, represented as s 2:
$$ {s}^2=MSE=\frac{{\displaystyle {\sum}_{i=1}^m{\left({y}_i-{\widehat{y}}_i\right)}^2}}{\left(M-P\right)} $$
In this simple linear regression context of eq. 5 and eq. 6, M is the sample size and P is typically equal to 2, since there are two parameters being estimated: an intercept term \( {\widehat{\beta}}_0 \) and the coefficient of the predictor variable: \( {\widehat{\beta}}_1 \). Thus, the estimated variance of \( {\widehat{\beta}}_1 \) in simple linear regression is represented as:
$$ \widehat{\operatorname{var}{\widehat{\beta}}_1}=\frac{{\displaystyle {\sum}_{i=1}^m{\left({y}_i-{\widehat{y}}_i\right)}^2}/\left(M-P\right)}{{\displaystyle {\sum}_{i=1}^m{\left({x}_i-\overline{x}\right)}^2}} $$
where y i − ŷ i is the residual for sample i, and \( {x}_i-\overline{x} \) is the difference between the predictor for sample i and the mean of x across all M samples. In this way, predictor variability is captured in the denominator of eq. 7, as is sample size M. Residual variability is captured in the numerator of eq. 7. Each component of eq. 7 affects the estimated variance of \( {\widehat{\beta}}_1 \) (Additional file 1: section 1.2).
In multivariate linear regression, matrix notation simplifies representation of the variances of all P regression coefficients. The theoretical variance-covariance matrix Σ of linear regression coefficients is represented as:
$$ \sum ={\sigma}^2\left(\mathbf{X}\mathbf{\hbox{'}}{\mathbf{X}}^{-1}\right) $$
where the variances of the k th individual \( {\widehat{\beta}}_k \) regression coefficients are found on the diagonal of Σ. In eq. 8 it is less intuitive to see the way in which individual parameters affect the variances of the individual \( {\widehat{\beta}}_k \) in matrix form, yet the principles are the same as in eq. 5. Predictor variability is captured in the inverse of the design matrix: (X ' X)− 1, analogous to the denominator of eq. 5.
As with simple single variable regression, σ 2 is estimated by MSE, represented by s 2 providing the estimated covariance matrix:
$$ \widehat{\sum}={s}^2\left(\mathbf{X}\mathbf{\hbox{'}}{\mathbf{X}}^{-1}\right) $$
where s 2 is:
$$ {s}^2=\frac{\left(\mathbf{y}-\mathbf{X}\widehat{\boldsymbol{\upbeta}}\right)\mathbf{\hbox{'}}\left(\mathbf{y}-\mathbf{X}\widehat{\boldsymbol{\upbeta}}\right)}{M-P}=\frac{\mathbf{y}\mathbf{\hbox{'}}\mathbf{y}-\widehat{\boldsymbol{\upbeta}}\hbox{'}\mathbf{X}\mathbf{\hbox{'}}\mathbf{y}}{M-P}=\frac{\mathbf{y}\mathbf{\hbox{'}}\mathbf{y}-\mathbf{y}\hbox{'}\mathbf{X}{\left(\mathbf{X}\mathbf{\hbox{'}}\mathbf{X}\right)}^{-\mathbf{1}}\mathbf{X}\mathbf{\hbox{'}}\mathbf{y}}{M-P} $$
The primary focus of this paper is to evaluate the effects of sample size, residual variability, and MSE on the estimated variances \( \widehat{\sum} \) of the cell type-specific expression estimates \( {\widehat{\beta}}_k \) and the effect this has upon differential expression detection sensitivity.
(Matrix notation for equations 2, 3, 4, 8, 9 and 10 is attributed to Graybill [14]).
Linear regression-based estimation of differential expression at the cell type-specific level
Differential expression analysis implies comparison of two or more groups for detectable gene expression differences. For simplicity, we consider two-group design, such as a case–control study.
To obtain group specific cell type-specific gene expression estimates \( \left({\widehat{\beta}}_{kn}\right) \), we apply linear regression separately to each group of heterogeneous gene expression measures (two regressions). The linear regression coefficient estimates are taken as surrogates for estimated cell type-specific average gene expressions. A difference between these cell type-specific estimates represents the level of gene expression change between the two groups in a given cell type:
$$ {\widehat{\delta}}_{kn}={\left(\widehat{\beta}\right)}_{kn}^{cases}-{\left(\widehat{\beta}\right)}_{kn}^{controls} $$
where \( {\widehat{\delta}}_{kn} \) is estimated effect size, k is the specific cell type and n is the genomic site.
Measuring the cell type-specific gene expression differences between groups using linear regression (LR) requires accurate cell type-specific gene expression estimates. Any factors affecting the variability of cell type-specific gene expression estimates per group will affect the sensitivity to detect cell type-specific differences between groups (Fig. 1) (Additional file 1: section 1.2).
Testing for significant differential expression using two-sample t-test
We used Welch's two-sample t-test to determine if the observed effect size (11) is significant (Additional file 1: section 1.3). A per-gene t-statistic is compared against the 1-α critical t-value (α = 0.05). A t-statistic exceeding the t-critical value is determined to be evidence to reject the null hypothesis of no difference between groups. If a significant difference is determined, sensitivity is then calculated as the upper tail probability beyond the critical t-value. If no significant difference is determined, then sensitivity is the alpha level threshold (Fig. 1). LRCDE sensitivity is based upon calculated t-statistic.
Simulation of cell type-specific expressions with known differential expression
Simulated data was used to assess performance of LRCDE under controlled conditions in which the cell type-specific differential expression was known. To establish a "gold-standard" of known cell type-specific differential expressions to benchmark LRCDE estimates of cell type-specific differential expressions, synthetic data with controlled changes [15] was constructed in three steps.
First, we created synthetic P cell proportions with known standard deviation across M samples per group for the target cell type p. For the sake of comparable per-group regressions, we simulate the condition where both groups have identical cell proportions (Additional file 1: Section 1.4). Second, we created synthetic matrixes of cell type-specific gene expression estimates for both control and case groups. We applied a uniform range of effect sizes (from 0.001 to 1.0) to half of the "genes" in the target cell type p of the case group ("true changes", Additional file 1: Section 1.5). Finally, the cross-product of both synthetic cell type expression and synthetic cell proportion matrices was taken for each group to produce simulated matrices of heterogeneous "fitted values" analogous to the predicted values obtained from linear regression. Normally distributed "noise" was added to the "fitted values" to simulate residual values obtained from a linear regression (Additional file 1: section 1.6). Having these a priori known cell type-specific expressions and differential expressions provided us with a benchmark against which to compare the results of LRCDE analysis.
Synthetic data is assembled by joining the two heterogeneous gene expression matrices ("cases" and "controls") into one 2 M by J heterogeneous gene expression matrix with a vector of group labels. The two cell proportion matrices, identical for groups of "cases" and "controls" were joined to obtain one 2 M by P cell proportion matrix.
Assessing LRCDE sensitivity from simulated data
Simulations over the parameter space were compared using sensitivity based upon simulated (and therefore known) cell type-specific differential expression. To quantify sensitivity, the total number of detected differentially expressed genes was divided by the total number of a priori known differentially expressed genes. The sensitivity from simulation using various levels of parameters was compared in order to illustrate the effects of sample size, MSE, and cell proportion variability.
Parameters affecting cell type-specific expression estimate variance
Parameters that directly impact the variability around cell type-specific expression estimates (linear regression coefficient estimates) include sample size, MSE, and cell type-specific proportion variability across sample (cell proportion SD). For simulations, all but one parameter is held constant. A single simulation is performed for each of a series of discrete values of the parameter of interest. Sensitivity is assessed at each level of the parameter of interest and sensitivity curves are plotted against significance level thresholds for each simulation.
Assessing the effect of high condition number of cell proportion matrix
A source of variability of cell type specific expression estimates is the "conditioning" or invertibility of the dot-product of the cell proportions predictor matrix. Multivariate linear regression relies upon the dot-product of the predictor matrix, which must be invertible:
$$ \mathrm{dot}\ \mathrm{product}=\mathbf{X}\mathbf{\hbox{'}}\mathbf{X} $$
where X is the M by P matrix of cell proportion predictors and X' is the X matrix transpose. The inverted matrix is denoted:
$$ \mathrm{inverted}\ \mathbf{X}\ \mathrm{matrix}={\left(\mathbf{X}\mathbf{\hbox{'}}\mathbf{X}\right)}^{-1} $$
A non-invertible matrix is referred to as "singular". A least squares linear regression solution cannot be obtained when the predictor matrix is singular and thus non-invertible. The condition number (CD) of a matrix X ' X is the ratio of the absolute values of the largest to smallest eigenvalues:
$$ CD=\left|\frac{ \max \left( eigen\left(\mathbf{X}\mathbf{\hbox{'}}\mathbf{X}\right)\right)}{ \min \left( eigen\left(\mathbf{X}\mathbf{\hbox{'}}\mathbf{X}\right)\right)}\right| $$
and X ' X can be factored as:
$$ \mathbf{X}\mathbf{\hbox{'}}\mathbf{X}=\mathbf{A}\boldsymbol{\Lambda } \mathbf{A}\mathbf{\hbox{'}} $$
where Λ is a diagonal matrix with eigenvalues of X ' X on the diagonal. Thus, a singular matrix, which has at least one zero eigenvalue, has an undefined condition number. In the case of cell type-specific differential expression detection, the linear regression predictor matrix is the matrix of cell proportions. It is near-singular squared cell proportion predictor matrices which result in unreliable cell expression estimates, and thus unreliable differential expression estimates [16]. Thus, a near-singular cell proportions predictor matrix is a source of cell expression estimate variability [6, 17]. The instability of cell type-specific expression and subsequent differential expression estimates cannot be observed from a single linear regression based upon a single cell proportions predictor matrix. The instability becomes apparent when observing estimates based upon different cell proportions predictor matrices, each with identical standard deviations across samples of the target cell type and identical condition numbers of the squared predictor matrix. It is the exact values comprising the matrices, which vary slightly between matrices (small perturbations) resulting in increasingly greater fluctuations of cell type-specific expression estimates with increasing condition numbers. We aimed at investigating the effect of the condition number for cell proportion on the sensitivity of LRCDE analysis.
Cell proportions were simulated with the target cell standard deviation of 0.2 over samples and a condition number of 100 with 5 total cell types. Group samples sizes were fixed at 10. MSE was fixed at 0.1. Effect size was fixed at 0.2. Allowing the random seed generator to float, this same set of parameters was simulated over 100 iterations. Note that the random seed set prior to the initial iteration in order to allow for replicable results. Thus, a cell proportions predictor matrix with identical target cell standard deviation and condition number of approximately 100 was re-created once for each of the 100 iterations. Letting the random seed float between iterations allowed small perturbations of the cell proportions across all iterations. Each calculated sensitivity observation was collected into a vector and stored for plotting.
Condition numbers of 100, 200, 500, 1000, 5000, 10000, 25000, 50000, and 75000 were simulated and sensitivity was recorded using the same 100-iteration method. The resulting vectors of sensitivity observations for each of the 100 iterations at distinct condition numbers were plotted in adjacent boxplots and variability of sensitivity at each level of condition number was visually compared.
Dropping cell proportion predictors to address high condition number
As the inherent biological restriction of cell proportions to sum up to 1 leads to high multicollinearity and, consequently, high condition number, reducing multicollinearity may improve the sensitivity of linear regression. We tested the effect of dropping at least one cell proportion predictor with the lowest mean cell in order to reduce multicollinearity. Using target cell proportion standard deviation of 0.1, we tested a model with 5 cell proportions which sum identically to 1 across each sample. These properties of the cell proportion matrix are identical to the model in [5], and result in a condition number of ~75000.
Using simulated conditions, we dropped cell proportions of cell types without introduced differential expression. As dropping a cell type will ultimately affect biological interpretation, we dropped cell types with the lowest mean cell proportions. Intuitively, this may be considered as ignoring potential "noise" in cell proportion measurements.
Comparing sensitivity of cell type-specific differential expression detection using LRCDE vs. false discovery rate-based methods
False discovery rates (FDR) calculations implemented in the csSAM method [5] rely upon repeated permutation of group membership labels and subsequent repeats of the linear regression cell type-specific differential expression step. A series of 100 cut points is constructed between zero and the greatest cell type-specific differential expression effect size. Differences associated with all genes in a given cell type are then compared against each successive cut point, and the total number of gene differences larger than the cut point is the number of "calls". The average number of permutation differences across all genes is also compared to this sequence of cut points. For each cut point in each cell type, a potential FDR is calculated by dividing the average number of permutation differences greater than the cut point by the number of calls at the cut point. FDRs are subsequently assigned per gene per cell type by comparing differential expression estimates against cut points and assigning the FDR associated with the greatest cut point smaller than the estimated expression difference.
We contrasted the cell type-specific FDRs with t-statistics p-values calculated by the LRCDE approach, which performs analysis on a cell type by cell type gene-by-gene basis. Significance thresholds of 0.0 to 0.3 for FDR and 0.0 to 0.1 (1-α for a two-sided test) for t-statistic p-value were used to compare true positive rates (TPRs) for both methods tested on the same simulated data over a range differential expression from 0.001 to 1.0. TPRs versus threshold values were then compared graphically for both FDR and t-statistic calculated p-values.
Functional enrichment analysis
Lists of cell type-specific differentially expressed gene names were analyzed using ToppFun module of the ToppGene Suite [18] using default settings.
Software used for analysis
RStudio [19] v.0.99.491. R packages: GEOquery [20] v.2.36.0, pROC [21] v.1.8, CellMix [22] 1.6.2. CsSAM [5] version 1.2.4, Computer specifications used: Hardware: Intel i7-6700 K 4-core 4.0 GHz, 32 Gb RAM. Operating System: Ubuntu v.15.10, Linux kernel v.4.2.0-35-generic.
Parameters affecting sensitivity of linear regression for cell type-specific differential expression detection
Sensitivity (true positive rates – TPR) of linear regression cell type-specific differential expression (LRCDE) detection is affected in two ways. Either, 1) the size of the true differential expression between study groups is changed (effect size), or 2) the variability of cell type-specific expression estimates is changed in one or both study groups.
Variability around cell type-specific expression estimates is affected by three main parameters: sample sizes, residuals sizes (quantified by mean squared error – MSE), and cell type-specific proportion variability across samples. The latter is also dependent on the total number of cell types, five in our study. We found that small changes around this total number of cell types included as predictors had a negligible effect on LRCDE sensitivity (data not shown). Yet, increasing the number of cell types increases the number of predictors in the linear model, making it less parsimonious with respect to the sample size. Furthermore, larger number of cell types decreases cell type-specific proportion variability, and should be avoided. In summary, the variability of cell type-specific expression estimates relative to the size of actual cell type-specific differential expression (eq. 9) drives the sensitivity of differential expression detection (Additional file 1: section 1.2).
Increased group sample sizes increases sensitivity of LRCDE
Increasing the number of samples in one or both study groups resulted in overall increased sensitivity of differential expression detection, as quantified by TPR curves. As before, other parameters were held fixed. Increasing sample size increases LRCDE sensitivity by reducing variability around cell type-specific expression estimates. Since sample size M is in the denominator of the MSE (eq. 10), this result is not surprising. Figure 2a depicts typical increases in TPR observed as sample sizes are increased (Additional file 2: Table S2).
Increased cell proportion variability across samples increased LRCDE sensitivity
Each cell type used as a predictor in LRCDE always exhibits some degree of variability in its relative proportions across samples. Variability of the proportions of any particular cell type can be quantified by standard deviation. Smaller standard deviation indicates lower variability across samples and conversely larger standard deviation indicates higher variability. In the biologically improbable case of all proportions of a given cell being identical across all samples, then there would be zero variability making linear regression unfeasible.
Cell type-specific differential expression detection sensitivity increases with increased variability of cell type-specific proportions across samples. We simulated cell proportions and tested LRCDE detection sensitivity for several levels of cell proportion variability (Fig. 2b) while holding other parameters fixed (Additional file 3: Table S3). As cell proportion variability across samples is increased, sensitivity of cell type-specific differential expression detection for all genes in that particular cell type improved.
Reducing mean squared error (MSE) increases sensitivity of LRCDE
Residuals are the differences between actual heterogeneous expression measures (observations) and those same measures as predicted by a regression line (fitted values). Each gene will have a unique set of residuals from linear regression. Sums of squared residuals divided by degrees of freedom (sample size minus the number of cell types) is mean squared error (MSE - numerator of eq. 9). Overall size of residuals as quantified by MSE is one measure of "goodness of fit", i.e., how well the observed data is predicted by the regression. When other parameters are held fixed (sample sizes, effect size, and cell proportion variability) and MSE is decreased (decreased overall size of residuals), the result is increased differential expression detection sensitivity (Fig. 2c).
Increase in sensitivity is due to the fact that cell type-specific expression estimate variances are decreased proportionately as MSE is decreased (eq. 9). We confirmed this relationship between MSE and variability around cell type-specific expression estimates by simulations (Additional file 4: Table S4). As variability around group-wise cell type-specific expression estimates cover less of a significantly detected difference between these estimates, the sensitivity of differential expression detection increases.
Changes in cell proportion variability for cell type p affect the variance (eq. 9) of cell type-specific expression estimates. Since cell proportion variability is captured in the inverse of the design matrix (eq. 9), any increase in cell proportion variability results in a decrease in cell type-specific expression estimate variability. This decrease in cell type-specific expression estimate variability improves sensitivity of LRCDE.
High condition number of cell proportions predictor matrix results in inconsistent sensitivity
Comparing sensitivity over 100 iterations for each of cell proportions dot product condition numbers of 100, 200, 500, 1000, 5000, 10000, 25000, 50000, and 75000 resulted in fluctuations of t-statistic based sensitivity plotted in Fig. 3. With a condition number of 100, sensitivity remains within a consistent range of values between 0.977 and 0.999. When condition number reaches 1000, we noticed fluctuations of sensitivity from a high of 0.999 to a low of 0.939. At condition number of 5000, the range from maximum sensitivity to minimum had broadened with maximum of 0.999 to minimum of 0.805. When condition number is 10000 maximum sensitivity remains at 0.999 while minimum is 0.364. This loss of consistency of sensitivity with increasing condition number illustrates the instability of an "ill-conditioned" cell proportion matrix.
Large condition number of the cell proportion matrix negatively affects stability of sensitivity. Cell proportion matrixes were simulated to obtain condition numbers 100, 200, 500, 1000, 5000, 10000, 25000, 50000, and 75000. Each condition number was simulated 100 times by small perturbations of the cell proportion values. The following parameters were held constant: SD for the target cell proportion – 0.2, per-group sample size – 10, MSE – 0.1, cell type-specific effect size – 0.1
Dropping cell proportions with the lowest mean reduces multicollinearity and improves sensitivity
Dropping a single cell type with the lowest proportion mean while retaining ~98 % of total proportions across all samples resulted in reduction of condition number (CD) from ~75000 down to ~31500. This resulted in noticeable improvement in sensitivity of t-statistic p-values (Fig. 4a). However, dropping 3 cells while retaining ~94 % of total proportions produced CD of ~56, and, consequently, further improved sensitivity of both FDR and t-statistic p-values (Fig. 4b). Although dropping cell types with the lowest proportion mean appears a viable statistical method to improve sensitivity of linear regression-based cell type-specific differential expression analysis, it warrants further investigation of how biological interpretation of the cell proportion estimates is altered.
Dropping cell proportions reduces condition number and improves sensitivity. Effect of dropping (a) 1 cell type; (b) 3 cell proportions with the lowest proportion mean. The following parameters were used: log2-transformed data, per-group sample size – 10, MSE – 1.5, cell type-specific effect ranged from 0.001 to 1.0 over 500 genes out of 1000 simulated genes, SD of the cell type with introduced changes – 0.1, condition number (5 cells total) – 75000
Summary of the parameters affecting sensitivity of cell type-specific differential expression
The principal metric driving the sensitivity of LRCDE is the relationship between cell type-specific expression estimate variance and estimated group-wise differential expression. The only way to increase sensitivity of linear regression differential expression detection is to either 1) reduce variances around cell type-specific expression estimates or 2) increase size of cell type-specific differential expression (effect size). While the latter is an experimentally given parameter, the former is affected by several parameters: sample size, sizes of residuals per linear regression as quantified by mean squared error (MSE), or cell type proportion (predictor) variability across samples. These three parameters are the components of variance eq. 9.
LRCDE accounts for cell type-specific variability of differential gene expression estimate, missed by FDR-based analysis
Figure 2 shows the difference in sensitivity between the FDR-based approach vs. the t-statistic p-value approach. Using simulated data, we tested cell type-specific differential expression detection over a range of known effect sizes ranging from 0.001 to 1.0 spread over the 500 changed genes in the target cell. At lower sample sizes, FDR fails to detect any of the known differences below a 0.3 threshold. With sample size of 10 per group (smallest tested), at all but the 0.001 effect size, the t-statistic p-value indicates significant changes with sensitivity greater than 0.68 at an alpha significance threshold of 0.025 (for a two-sided test). At 28 samples per group, FDR sensitivity increases from 0 up to 0.848 at a 0.3 threshold. Under the same conditions, t-statistic p-value has maximum sensitivity of 0.968, indicating the per-gene significance testing using two-sample t-test improves sensitivity of cell type-specific differential expression detection.
All panels in Fig. 2 are representative of the increased sensitivity of the per-gene t-statistics p-values vs. FDR. In all cases, the t-statistic p-value is more sensitive than FDR to differential expression, particularly in smaller sample sizes. Our results demonstrate that FDR is insensitive to gene-specific variability of cell type-specific expression estimates, leading to higher overall FDRs and thus decreased sensitivity. In contrast, the t-statistic incorporates the variability captured in eq. 2 on a cell-by-cell and gene-by-gene basis.
Ignoring significance thresholds for FDR and t-statistic p-values gives the illusion of perfect discrimination or a 100 % true positive rate and 0 % false positive rate (FPR) when analyzing simulated data (Additional file 1: section 1.7). In simulated data, we created the situation in which normally distributed residuals are mean centered around zero. Since linear regression coefficient estimates are unbiased estimates, both regression-based methods will precisely target the known differential expression values, regardless of coefficient variability. For this reason, any known differential expression will always have a lower FDR and a lower t-statistic p-value than genes with no differential expression. Thus, in order to truly measure the merits of either method, any measure of differential expression detection performance must be viewed in light of significance thresholds for both FDR and t-statistic p-value.
Biological significance of cell type-specific differentially expressed genes
We compared the performance of per-gene LRCDE analysis with the global FDR threshold-based analysis (csSAM) by analyzing the human whole-blood gene expression measures from 24 kidney transplant patients used by the authors of csSAM [5]. We used liberal FDR < 0.3 threshold for the csSAM method, and the Bonferroni-corrected α = 0.05 level for the p-value cutoff in the LRCDE method (Table 1).
Table 1 The number of cell type-specific differentially expressed probes (genes) identified in kidney transplant gene expression data from [5]
We identified 59 (10 iterations: 0 to 169) upregulated genes in monocytes at an FDR 0.15, and zero upregulated genes in the other four cell types. Using FDR <0.3, 1203 (10 iterations: 902 to 1696) monocyte-specific genes were detected. In contrast, LRCDE analysis was able to identify significant differentially expressed genes in all five cell types (Table 1, Additional file 5: Table S5, Fig. 5).
Venn diagram of overlaps among cell type-specific differentially expressed unique gene names in the kidney transplant data set
Notably, genes detected as differentially expressed in neutrophils were enriched in two functional categories: Ion channel activity/membrane and extracellular matrix/adhesion (Additional file 6: Table S6). Genes detected in lymphocytes and monocytes were enriched in RNA binding/transcription factor activity. Despite being measured in blood, these genes were also enriched in genes involved in kidney regenerative processes ("Human Kidney_Sallustio10_2134genes_DiscriminatedARPCsFromRPTEC/MSC" co-expression category, Additional file 6: Table S6).
Despite many genes were identified as differentially expressed in eosinophils and basophils, they were marginally enriched in processes without obvious biological scheme. This may be attributed to the fact that both cell types had very low mean and SD, making detection of cell type-specific differentially expressed genes less reliable. It remains to be further investigated how the mean/SD of the cell types affect biological outcome of cell type-specific deconvolution.
This paper addresses several key issues of cell type-specific differential expression detection methods based on linear regression (LRCDE). One is the fact that there is a level of uncertainty attached to any detected differential expression at the cell type-specific level, which will change depending upon values of several parameters. Furthermore, this level of uncertainty is different for each gene, and should be accounted for on a per-cell type per-gene basis. The other is the severe multicolinearity of predictors quantified in the condition number of the cell proportions matrix (design matrix).
One of the primary goals of our work has been to quantify the parameter space affecting the sensitivity of LRCDE. One source of variability, the cell proportion measures used as predictor values, affects detection sensitivity within that particular cell type for all genes in the heterogeneous data set. The other source of variability, the mean squared error (MSE - size of residuals) for a given regression, affects the sensitivity for each gene individually.
Characteristics of the cell proportion measures across samples also affect LRCDE detection sensitivity for each gene in the data set. There are two measures of interest. One is the degree of variability for any single cell type across samples. Greater variability of proportion measures across samples will result in higher differential expression detection sensitivity for that specific cell type. Thus, sensitivity will vary from cell type to cell type. This need for variability across samples is acknowledged by the authors of csSAM [5]: "…accurate estimates of rare cell types may be aided by sample enrichment or inclusion of highly variable samples". In this work we have demonstrated the effect of cell type-specific proportion variability upon sensitivity of cell type-specific differential expression detection.
The other measure attached to the cell proportion matrix is a condition number. The condition number of the cell proportion matrix is a global measure of multi-collinearity across all samples and all cell types. It is a function of the ratio of the maximum and minimum eigenvalues of the design matrix. Lower values of the condition number closer to 1 (minimum possible condition number) indicate lower multicollinearity resulting in stable cell type-specific expression estimates (regression coefficients). As the condition number and the variability around cell type-specific estimates are increased, variability of sensitivity also increased. When the condition number associated with the cell proportion predictor matrix is in the tens of thousands then the confidence intervals on the observed sensitivity will increase to the point that conclusive differential expression detection is questionable (Fig. 4). Conditioning of the predictor's design matrix is therefore a non-trivial source of instability of cell type-specific expression estimates (coefficients estimates from linear regression) and should not be dismissed. Caution is urged when evaluating results of any analysis of linear regression cell type-specific differential expression detection when the cell proportions predictor matrix has a "high condition number" (above 1000). Such "ill-conditioned" cell proportions produce cell type-specific expression estimates which may be unreliable [16], resulting in untrustworthy differential expression estimates. Future use of linear regression techniques for estimation of biological values should take into consideration the condition number of the cell proportion matrix as a source of estimate variability.
Our preliminary investigations of a high multicollinearity and condition number problem suggest that the solution may be simply to drop at least one cell type with the lowest proportion mean. Our results show decreasing condition number and increasing sensitivity of the linear regression-based cell type-specific differential expression analysis. Yet, we have not investigated the biological implications of dropping cell types on cell type-specific expression estimates. We expect that the heterogeneous signal previously allocated to the dropped cell types will then be distributed across the remaining cell type-specific expression estimates. Furthermore, we have only tested the effects of dropping cell types under simulated conditions, when only one cell type contains a priori known differentially expressed genes, which is an exception in real biological data. We aim at further investigating the effect of dropping cell types as a means of effectively handling the multicollinearity and large condition number issues without harming biological relevance or cell type-specific differential expression detection.
The practice of log2 transformation of heterogeneous microarray gene measures prior to linear regression deconvolution has been criticized on the basis that log2 transformation of the outcome variable breaks the linear relationship between outcomes and predictors (cell type-specific expression estimates - linear regression coefficients). It has been shown that without applying some back-transformation after performing LRCDE to log2 transformed heterogeneous observations, the cell type-specific expressions will be underestimated [23]. Furthermore, linear regression coefficient estimates may be difficult to interpret in the absence of a linear relationship between the outcome heterogeneous observations and the cell proportion predictors. We tested the effect of log2 transformation on the sensitivity of both FDR-based and LRCDE analyses, but did not identify any measurable sensitivity increase in either method (Fig. 2d). We aim to further investigate the effect of log2 transformation post hoc linear regression by quantifying normality of the residuals and other diagnostic parameters of linear modeling.
The primary drawback of the LRCDE sensitivity calculation is that is relies upon a single linear regression step per group to compute t-statistic based on the standard error estimates of the cell type-specific expressions. This approach carries the implicit assumption of a known distribution of linear regression coefficient estimates. Other algorithms rely upon multiple iterations of the linear regression step in which permutations of group membership labels provides an estimated "null distribution" against which to compare initial estimates. The advantage of these permutation methods is that they do not rely upon assumptions as to the distribution of coefficient estimates. A way to overcome this limitation may be to include facility for a permutation method in which two-sample t-statistics may then be based upon a similar comparison of initial observation versus permuted null observations.
We demonstrated the greater sensitivity to detection of known cell type-specific differential expression using a per-gene two-sample t-test approach to differential expression detection. We have drawn attention to parameters affecting gene-specific variability of cell type-specific expression estimates and thus the sensitivity of cell type-specific differential expression detection. Our approach is implemented in an R package, LRCDE, available on GitHub (https://github.com/ERGlass/lrcde.dev), which performs cell type-specific differential expression analysis on a cell type-by-cell type, and gene-by-gene basis. LRCDE estimates cell type-specific differential expression, calculates two-sample t-statistic, t-statistic p-value, and, given a significantly detected difference, outputs sensitivity based upon t-statistic (Fig. 1).
Linear regression-based cell type-specific differential expression (LRCDE) is more sensitive to detect significant cell type specific differential expression than FDR approach. The gene-specific LRCDE sensitivity is a function of sample size, cell type-specific proportion variability, and mean squared error of linear regression for that gene. Larger sample sizes, lower MSE for a linear regression for a given gene, and greater cell type specific proportion variability are needed to achieve greater sensitivity in detecting smaller significant cell type-specific differential expression (smaller effect sizes). The greater the cell proportion variability across samples results in the greater the sensitivity of LRCDE. The magnitude of cell type-specific expression estimate variability relative to size of actual cell type-specific differential expression (eq. 9) drives the sensitivity of differential expression detection. Finally, when the cell proportion matrix has a condition number greater than 1000, then results of LRCDE may be unreliable given the instability introduced by the "ill-conditioned" cell proportions. Preliminary investigation suggests that dropping cell types with low proportion mean increases the sensitivity of linear regression-based cell type-specific differential expression, and should be further investigated.
CDE:
Cell type-specific differential expression
csSAM:
Cell-specific significance analysis of microarrays
FC:
Fold change
FPR:
False positive rate
LRCDE:
Linear regression cell type-specific differential expression
MSE:
Mean squared error
Receiver operator characteristic (curve)
TPR:
True positive rate
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This article has been published as part of BMC Bioinformatics Volume 17 Supplement 13, 2016: Proceedings of the 13th Annual MCBIOS conference. The full contents of the supplement are available online at http://bmcbioinformatics.biomedcentral.com/articles/supplements/volume-17-supplement-13.
This work and its publication was partially supported by the Virginia Commonwealth University start-up fund (to MGD).
All data, software and material are available at https://github.com/ERGlass/lrcde.dev.
Conceived the study: MGD. Performed experiments: ERG, MGD. Wrote the paper: ERG, MGD. Both authors read and approved the final manuscript.
Edmund R. Glass is a graduate student in the Department of Biostatistics of Virginia Commonwealth University, Richmond, Virginia. Mikhail G. Dozmorov is assistant professor in the Department of Biostatistics of Virginia Commonwealth University, Richmond, Virginia.
Department of Biostatistics, Virginia Commonwealth University, School of Medicine, PO Box 980032, Richmond, VA, 23298, USA
Edmund R. Glass & Mikhail G. Dozmorov
Edmund R. Glass
Mikhail G. Dozmorov
Correspondence to Mikhail G. Dozmorov.
Supplementary Methods and Notes. (DOCX 154 kb)
An example of LRCDE and csSAM analysis results of simulated data with 18 samples per group. Target condition number (kappa) – 100, target mean squared error – 1.5, target cell proportion SD – 0.1. Range of the effect sizes introduced in the first 500 genes – 0.001 to 1.0. "site" – Probe ID; "Gene" – Gene label; "base" – estimated non-negative control group cell type-specific expression level; "case" - estimated non-negative case group cell type-specific expression level; "diff.est" – estimated cell type-specific differential expression between case and base; "mse.control" – Observed mean squared error of the control group regression; "mse.case" – Observed mean squared error of the case group regression; "cell" – Cell type name; "cell.sd" – Cell type-specific standard deviation of cell proportions across samples; "kappa.1/2" – Observed condition number of squared cell proportions matrix for the case/control group, respectively; "t.crit" – Critical t-value against which observed t-statistic is compared; "t.stat" – Observed t-statistic; "p.val.t" – Unadjusted p-value for t-statistic; "se1/2"– Standard error of the control/case group cell type-specific expression (regression coefficient) estimate, respectively; "se.p" – Welch's combined standard error calculated for t-statistic; "t.power" – Observed power calculated from t-statistic and critical t-value; "FDR" – False discovery rate as reported by csSAM. (XLSX 136 kb)
An example of LRCDE and csSAM analysis results of simulated data with target cell proportion standard deviation of 0.15. Sample size – 14, target condition number (kappa) – 100, target mean squared error – 1.5. Range of the effect sizes introduced in the first 500 genes – 0.001 to 1.0. Column names legend as in Additional file 1. (XLSX 137 kb)
An example of LRCDE and csSAM analysis results of simulated data with target MSE of 0.5 per regression. Sample size – 14, target condition number (kappa) – 100, target mean squared error – 2.5, target cell proportion SD – 0.06. Range of the effect sizes introduced in the first 500 genes – 0.001 to 1.0. Column names legend as in Additional file 1. (XLSX 138 kb)
Additional file 5: Table 5.
LRCDE and csSAM analysis results of the kidney transplant dataset. Column names legend as in Additional file 1. (XLSX 2842 kb)
Functional enrichment analysis of LRCDE significant differentially expressed genes in kidney transplant dataset. "Category" – General functional category; "ID" - unique identifier of the functional category; "Name" – Category name; "Source" – Database reference; "p-value" - non-adjusted p-value; "q-value FDR B&H" – FDR-adjusted p-value; "Hit Count in Query List/Genome" – Number of differential genes/genes in the whole genome annotated with a functional category, respectively; "Hit in Query List" – Names of differentially expressed genes annotated with a functional category. (XLSX 246 kb)
Glass, E.R., Dozmorov, M.G. Improving sensitivity of linear regression-based cell type-specific differential expression deconvolution with per-gene vs. global significance threshold. BMC Bioinformatics 17, 334 (2016). https://doi.org/10.1186/s12859-016-1226-z
Cell type-specific | CommonCrawl |
\begin{document}
\begin{frontmatter}
\title{Entropic properties of $D$-dimensional Rydberg systems}
\author[label1]{I. V. Toranzo} \address[label1]{Departamento de F\'{\i}sica At\'{o}mica, Molecular y Nuclear, Universidad de Granada, Granada 18071, Spain\\ Instituto Carlos I de F\'{\i}sica Te\'orica y Computacional, Universidad de Granada, Granada 18071, Spain}
\author[label1]{D. Puertas-Centeno}
\author[label1]{J. S. Dehesa} \ead{[email protected]}
\begin{abstract} The fundamental information-theoretic measures (the Rényi $R_{p}[\rho]$ and Tsallis $T_{p}[\rho]$ entropies, $p>0$) of the highly-excited (Rydberg) quantum states of the $D$-dimensional ($D>1$) hydrogenic systems, which include the Shannon entropy ($p \to 1$) and the disequilibrium ($p = 2$), are analytically determined by use of the strong asymptotics of the Laguerre orthogonal polynomials which control the wavefunctions of these states. We first realize that these quantities are derived from the entropic moments of the quantum-mechanical probability $\rho(\vec{r})$ densities associated to the Rydberg hydrogenic wavefunctions $\Psi_{n,l,\{\mu\}}(\vec{r})$, which are closely connected to the $\mathfrak{L}_{p}$-norms of the associated Laguerre polynomials. Then, we determine the ($n\to\infty$)-asymptotics of these norms in terms of the basic parameters of our system (the dimensionality $D$, the nuclear charge and the hyperquantum numbers $(n,l,\{\mu\}$) of the state) by use of recent techniques of approximation theory. Finally, these three entropic quantities are analytically and numerically discussed in terms of the basic parameters of the system for various particular states. \end{abstract}
\begin{keyword} Entropic uncertainty measures of Shannon \sep Rényi and Tsallis types \sep $D$-dimensional hydrogenic systems \sep $D$-dimensional quantum physics \sep Rydberg states. \end{keyword}
\end{frontmatter}
\section{Introduction}\label{s1}
Rydberg systems are ballooned-up atoms which can be made by exciting the outermost electron in certain elements, so that all the inner electrons can lumped together and regarded, along with the atom's nucleus, as a unified core, with the lone remaining electron lying outside \cite{lundee,gallagher}. Thus, they are as if the atom were a hydrogenic system, a heavy version of hydrogen. Due to their extraordinary properties (high magnetic susceptibility, \textcolor{red}{relatively} long lifetime, high kinetic energy,…) these systems, where the outermost electrons are highly excited but not ionized, have been used in multiple scientific areas ranging from plasmas and diamagnetism to astrophysics, quantum chaos and strongly interacting systems. Recently it has been argued that they might be just the basic elements for processing quantum information (see e.g., \cite{shiell,saffman}). Indeed, these outsized atoms can be sustained for a long time in a quantum superposition condition (what is very convenient for creating qubits) and they can interact strongly with other such atoms; this property makes them very useful for devising the kind of logic gates needed to process information.\\
The $D$-dimensional hydrogenic system (i.e. an electron or a negatively-charged particle moving around a nucleus or a positively-charged core which electromagnetically binds it in its orbit), with $D>1$, is the main prototype to model the behavior of most multidimensional quantum many-body systems with standard ($D = 3$) and non-standard ($D\neq 3$) dimensionalities \cite{witten,herschbach_93, burgbacher:jmp99, aquilanti:cp97, aquilanti_96, andrew:ajp90}. It embraces a large variety of three-dimensional physical systems (e.g., hydrogenic atoms and ions, exotic atoms, antimatter atoms, Rydberg atoms,…) and a number of nanotechnological objects which have been shown to be very useful in semiconductor nanostructures (e.g., quantum wells, wires and dots) \cite{harrison_05, li:pla07} and quantum computation (e.g., qubits) \cite{nieto:pra00, dykman:prb03}. Moreover, it plays a crucial role for the interpretation of numerous phenomena of quantum cosmology \cite{amelinocamelia_05} and quantum field theory \cite{witten, itzykson_06,dong}. As well, the $D$-dimensional hydrogenic wavefunctions have been used as complete orthonormal sets for many-body atomic and molecular problems \cite{aquilanti:aqc01, aquilanti:irpc01,coletti} in both position and momentum spaces. Finally, the existence of non-standard hydrogenic systems has been proved for $D<3$ \cite{li:pla07,caruso} and suggested for $D>3$ \cite{burgbacher:jmp99}.\\
The multidimensional extension of Rydberg hydrogenic states (i.e. states where the electron has a large principal quantum number $n$, so being highly excited), with standard and non-standard dimensionalities, has been investigated (see section 5 of \cite{dehesa_2010}, and \cite{tarasov,lopez_2009,aptekarev_2010,dehesa_sen12}) up until now by means of the following spreading measures: central moments, variances, logarithmic expectation values, Shannon entropy and Fisher information. These measures were found to be expressed in terms of the principal and orbital hyperquantum numbers and the space dimensionality $D$. In this work we go much beyond this study by calculating the Rényi \cite{renyi1} and Tsallis \cite{tsallis} entropies (also called by \textit{information generating functionals} \cite{golomb}) of the Rydberg states defined by \begin{eqnarray} \label{eq:renentrop} R_{p}[\rho] &=& \frac{1}{1-p}\ln W_{p}[\rho]; \quad 0<p<\infty,\\ \label{eq:tsalentrop} T_{p}[\rho] &=& \frac{1}{p-1}(1-W_{p}[\rho]); \quad 0<p<\infty, \end{eqnarray} where the symbol $W_{p}[\rho]$ denotes the entropic moments of $\rho(\vec{r})$ defined as \begin{equation} \label{eq:entropmom}
W_{p}[\rho] = \int_{\mathbb{R}^D} [\rho(\vec{r})]^{p}\, d\vec{r} =\| \rho\|_p^p;\quad p > 0. \end{equation}
The symbol $\|\cdot\|_p$ denotes the $\mathfrak{L}_{p}$-norm for functions:
$\|\Phi\|_p=\left(\int_{\mathbb{R}^D} |\Phi(\vec{r})|^p d\vec{r}\right)^{1/p}$. Note that both Rényi and Tsallis measures include the Shannon entropy, $S[\rho] = \lim_{p\rightarrow 1} R_{p}[\rho] = \lim_{p\rightarrow 1} T_{p}[\rho]$, and the disequilibrium, $\langle\rho\rangle = \exp(R_{2}[\rho])$, as two important particular cases. Moreover, they are interconnected as indicated later on. Their properties have been recently reviewed \cite{dehesa_sen12,bialynicki3,jizba}; see also \cite{aczel,dehesa_88,dehesa_89,romera_01,leonenko,guerrero}. Moreover, the R\'enyi entropies and their associated uncertainty relations have been widely used to investigate a great deal of quantum-mechanical properties and phenomena of physical systems and processes \cite{bialynicki2,dehesa_sen12,bialynicki3}, ranging from the quantum-classical correspondence \cite{sanchezmoreno} and quantum entanglement \cite{bovino} to pattern formation and Brown processes \cite{cybulski1,cybulski2}, quantum phase transition \cite{calixto} and disordered systems \cite{varga}.
The structure of this work is the following. First, in Section II, we give the wavefunctions of the stationary $D$-dimensional hydrogenic states in position space and their squares, the quantum probability densities $\rho\left(\vec{r} \right)$. Then we define the entropic moments and the Rényi entropy of this density, and we show that for the very excited states the calculation of the latter quantity essentially converts into the determination of the asymptotics of the $\mathfrak{L}_{p}$-norm of the Laguerre polynomials which control the states' wavefunctions. In Section III we use a powerful technique of approximation theory recently developed by Aptekarev et al \cite{aptekarev_2012,aptekarev_2010,aptekarev_2016} to determine these Laguerre norms in terms of $D$ and the hyperquantum numbers of the system. In Section IV the Shannon, Rényi and Tsallis entropies are studied both analytically and numerically for the $D$-dimensional hydrogenic states by means of $D$, the hyperquantum numbers and the nuclear charge $Z$ of the system. Finally, some concluding remarks are given.
\section{The $D$-dimensional Rydberg problem: entropic formulation}
In this section we briefly describe the quantum probability density of the stationary states of the $D$-dimensional hydrogenic system in position space. Then, we pose the determination of the entropic moments and the Rényi entropies of this density in the most appropriate mathematical manner for our purposes. Atomic units will be used throughout.\\ The time-independent Schr\"{o}dinger equation of a $D$-dimensional ($D \geqslant 1$) hydrogenic system (i.e., an electron moving under the action of the $D$-dimensional Coulomb potential $\displaystyle{V(\vec{r})=-\frac{Z}{r}}$) is given by \begin{equation}\label{eqI_cap1:ec_schrodinger} \left( -\frac{1}{2} \vec{\nabla}^{2}_{D} - \frac{Z}{r} \right) \Psi \left( \vec{r} \right) = E \Psi \left(\vec{r} \right), \end{equation} where $\vec{\nabla}_{D}$ denotes the $D$-dimensional gradient operator, $Z$ is the nuclear charge, and the electronic position vector $\vec{r} = (x_1 , \ldots , x_D)$ in hyperspherical units is given as $(r,\theta_1,\theta_2,\ldots,\theta_{D-1}) \equiv
(r,\Omega_{D-1})$, $\Omega_{D-1}\in S^{D-1}$, where $r \equiv |\vec{r}| = \sqrt{\sum_{i=1}^D x_i^2} \in [0 \: ; \: +\infty)$ and $x_i = r \left(\prod_{k=1}^{i-1} \sin \theta_k \right) \cos \theta_i$ for $1 \le i \le D$
and with $\theta_i \in [0 \: ; \: \pi), i < D-1$, $\theta_{D-1} \equiv \phi \in [0 \: ; \: 2 \pi)$. It is assumed that the nucleus is located at the origin and, by convention, $\theta_D = 0$ and the empty product is the unity. . \\ It is known \cite{nieto,yanez_1994,dehesa_2010,dong} that the energies belonging to the discrete spectrum are given by \begin{equation} \label{eqI_cap1:energia} E= -\frac{Z^2}{2\eta^2},\hspace{0.5cm} \hspace{0.5cm} \eta=n+\frac{D-3}{2}; \hspace{5mm} n=1,2,3,..., \end{equation} and the associated eigenfunction can be expressed as \begin{equation}\label{eqI_cap1:FunOnda_P} \Psi_{\eta,l, \left\lbrace \mu \right\rbrace }(\vec{r})=\mathcal{R}_{\eta,l}(r)\,\, {\cal{Y}}_{l,\{\mu\}}(\Omega_{D-1}). \end{equation}
Then, the quantum probability density of a $D$-dimensional hydrogenic stationary state $(n,l,\{\mu\})$ is given by the squared modulus of the position eigenfunction as \begin{equation} \label{eq:denspos}
\rho_{n,l,\{\mu\}}(\vec{r}) = \rho_{n,l}(\tilde{r})\,\, |\mathcal{Y}_{l,\{\mu\}}(\Omega_{D-1})|^{2}, \end{equation} where the radial part of the density is the univariate function \begin{eqnarray} \label{eq:radensity} \rho_{n,l}(\tilde{r}) &=& [\mathcal{R}_{n,l}(r)]^2 = \frac{\lambda^{-d}}{2 \eta} \,\, \frac{\omega_{2L+1}(\tilde{r})}{\tilde{r}^{d-2}}\,\,[{\widehat{\mathcal{L}}}_{\eta-L-1}^{(2L+1)}(\tilde{r})]^{2} \end{eqnarray} with $L$, defined as the ``grand orbital angular momentum quantum number'', and the adimensional parameter $\tilde{r}$ given by \begin{align} \label{eqI_cap1:Lyr} L&=l+\frac{D-3}{2}, \hspace{0.5cm} l=0, 1, 2, \ldots \\ \label{rtilde} \tilde{r}&=\frac{r}{\lambda},\hspace{0.5cm} \hspace{0.5cm}\lambda=\frac{\eta}{2Z}. \end{align}
The symbols $\mathcal{L}_{n}^{(\alpha)}(x)$ and $\widehat{\mathcal{L}}_{n}^{(\alpha)}(x)$ denote the orthogonal and orthonormal, respectively, Laguerre polynomials with respect to the weight $\omega_\alpha(x)=x^{\alpha} e^{-x}$ on the interval $\left[0,\infty \right) $, so that \begin{equation}\label{eqI_cap1:laguerre_orto_ortogo} {\widehat{\mathcal{L}}}^{(\alpha)}_{m}(x)= \left( \frac{m!}{\Gamma(m+\alpha+1)}\right)^{1/2} {\mathcal{L}}^{(\alpha)}_{m}(x), \end{equation} and finally \begin{equation} K_{\eta,L} = \lambda^{-\frac{D}{2}}\left\{\frac{(\eta-L-1)!}{2\eta(\eta+L)!}\right\}^{\frac{1}{2}}=\left\{\left(\frac{2Z}{n+\frac{D-3}{2}}\right)^{D}\frac{(n-l-1)!}{2\left(n+\frac{D-3}{2}\right)(n+l+D-3)!} \right\}^{\frac{1}{2}} \equiv K_{n,l} \label{eq:4} \end{equation}
represents the normalization constant which ensures that $\int \left| \Psi_{\eta,l, \left\lbrace \mu \right\rbrace }(\vec{r}) \right|^2 d\vec{r} =1$.
The angular eigenfunctions are the hyperspherical harmonics, $\mathcal{Y}_{l,\{\mu\}}(\Omega_{D-1})$, defined as \begin{equation} \label{eq:hyperspherarm} \mathcal{Y}_{l,\{\mu\}}(\Omega_{D-1}) = \mathcal{N}_{l,\{\mu\}}e^{im\phi}\nonumber\times \prod_{j=1}^{D-2}\mathcal{C}^{(\alpha_{j}+\mu_{j+1})}_{\mu_{j}-\mu_{j+1}}(\cos\theta_{j})(\sin\theta_{j})^{\mu_{j+1}} \end{equation} with the normalization constant \begin{equation} \label{eq:normhypersphar} \mathcal{N}_{l,\{\mu\}}^{2} = \frac{1}{2\pi}\times\nonumber\\ \prod_{j=1}^{D-2} \frac{(\alpha_{j}+\mu_{j})(\mu_{j}-\mu_{j+1})![\Gamma(\alpha_{j}+\mu_{j+1})]^{2}}{\pi \, 2^{1-2\alpha_{j}-2\mu_{j+1}}\Gamma(2\alpha_{j}+\mu_{j}+\mu_{j+1})},\nonumber\\ \end{equation} where the symbol $\mathcal{C}^{\lambda}_{n}(t)$ denotes the Gegenbauer polynomial of degree $n$ and parameter $\lambda$.\\
Now we can calculate any spreading measure of the $D$-dimensional hydrogenic system beyond the known the variance and the ordinary moments (or radial expectation values) of its density, which are already known \cite{dehesa_2010}. The most relevant spreading quantities are the entropic moments $W_{p}[\rho_{n,l,\{\mu\}}]$, because they characterize the density and moreover we can obtain from them the main entropic measures of the system such as the Rényi, Shannon and Tsallis entropies. They are given as \begin{eqnarray} \label{eq:entropmom2} W_{p}[\rho_{n,l,\{\mu\}}] &=& \int_{\mathbb{R}^D} [\rho_{n,l,\{\mu\}}(\vec{r})]^{p}\, d\,\vec{r}\nonumber\\ &=& \int\limits_{0}^{\infty}[\rho_{n,l}(r)]^{p}\,r^{D-1}\,dr\times \Lambda_{l,\{\mu\}}(\Omega_{D-1}), \end{eqnarray} where we have used that the volume element is \[ d\vec{r} =r^{D-1}dr\,d\Omega_{D-1} , \quad d\Omega_{D-1} = \left(\prod_{j=1}^{D-2}\sin^{2\alpha_{j}}\theta_{j}\right)d\phi, \] (with $2\alpha_{j}= D-j-1$) and the angular part is given by \begin{equation} \label{eq:angpart}
\Lambda_{l,\{\mu\}}(\Omega_{D-1}) = \int_{S^{D-1}}|\mathcal{Y}_{l,\{\mu\}}(\Omega_{D-1})|^{2p}\, d\Omega_{D-1}. \end{equation} Then, from Eqs. (\ref{eq:renentrop}) and (\ref{eq:entropmom2}) we can obtain the Rényi entropies of the $D$-dimensional hydrogenic state $(n,l,\{\mu\})$ as follows \begin{equation} \label{eq:renyihyd1} R_{p}[\rho_{n,l,\{\mu\}}] = R_{p}[\rho_{n,l}]+R_{p}[\mathcal{Y}_{l,\{\mu\}}], \end{equation} where $R_{p}[\rho_{n,l}]$ denotes the radial part \begin{equation} \label{eq:renyihyd2} R_{p}[\rho_{n,l}] = \frac{1}{1-p}\ln \int_{0}^{\infty} [\rho_{n,l}]^{p} r^{D-1}\, dr, \end{equation} and $R_{p}[\mathcal{Y}_{l,\{\mu\}}]$ denotes the angular part \begin{equation} \label{eq:renyihyd3} R_{p}[\mathcal{Y}_{l,\{\mu\}}] = \frac{1}{1-p}\ln \Lambda_{l,\{\mu\}}(\Omega_{D-1}). \end{equation} In this work we are interested in the entropic properties of the high extreme region of the system, embracing the highly and very highly excited (Rydberg) states (recently shown to be experimentally accessible \cite{lundee}) where these properties are most difficult to compute because they possess large and very large values of $n$. Since the dependence of both the entropic moments and the Rényi entropies on the principal hyperquantum number $n$ is concentrated in their radial parts according to Eqs. (\ref{eq:entropmom2})-(\ref{eq:renyihyd3}), the computation of $R_{p}[\rho_{n,l,\{\mu\}}]$ for the Rydberg states of $D$-dimensional hydrogenics systems practically reduces to determine the value of the radial Rényi entropy, $R_{p}[\rho_{n,l}]$, in the limiting case $n\rightarrow \infty$. Moreover, by taking into account the expression (\ref{eq:radensity}) of the radial density and Eqs. (\ref{eq:entropmom2}) and (\ref{eq:renyihyd2}), this problem converts into the study of the asymptotics ($n\to\infty$) of the $\mathfrak{L}_{p}$-norm of the Laguerre polynomials \begin{equation}\label{eq:c1.2} N_{n}(\alpha,p,\beta)=\int\limits_{0}^{\infty}\left(\left[\widehat{\mathcal{L}}_{n}^{(\alpha)}(x)\right]^{2}\,w_{\alpha}(x)\right)^{p}\,x^{\beta}\,dx\;,\quad \alpha >-1\, \, , p>0\,\, , \beta +p\alpha >-1. \end{equation} Indeed, from Eq. (\ref{eq:renyihyd2}) one has that the radial Rényi entropy can be expressed as \begin{equation} \label{eq:renyihyd4} R_{p}[\rho_{n,l}] = \frac{1}{1-p}\ln\left[\frac{\eta^{D(1-p)-p}}{2^{D(1-p)+p}Z^{D(1-p)}}N_{n,l}(D,p) \right], \end{equation} where the norm $N_{n,l}(\alpha,p,\beta) \equiv N_{n,l}(D,p)$ is given by \begin{equation}\label{eq:c1.21} N_{n,l}(D,p)=\int\limits_{0}^{\infty}\left(\left[\widehat{\mathcal{L}}_{n-l-1}^{(\alpha)}(x)\right]^{2}\,w_{\alpha}(x)\right)^{p}\,x^{\beta}\,dx, \end{equation} with \begin{equation} \label{eq:condition} \alpha=2L+1=2l+D-2\,,\;l=0,1,2,\ldots,n-1,\, p>0\,\, \text{and}\,\, \beta=(2-D)p+D-1 \end{equation} We note that (\ref{eq:condition}) guarantees the convergence of integral (\ref{eq:c1.21}); i.e. the condition $\beta+p\alpha= 2lp+D-1 > -1$ is always satisfied for physically meaningful values of the parameters.\\
\section{Laguerre $\mathfrak{L}_{p}$-norms and radial entropies: Asymptotics }
Let us now study the asymptotics at large $n$ of the Laguerre hydrogenic norms $N_{n,l}(D,p)$ given by Eq. (\ref{eq:c1.21}) in terms of all possible values of the involved parameters $(D,p)$. It controls the asymptotic values of the radial Rényi entropy $R_{p}[\rho_{n,l}]$ given by Eq. (\ref{eq:renyihyd4}) and, because of Eq. (\ref{eq:renyihyd1}), the total Rényi entropy of the Rydberg $D$-dimensional hydrogenic states. Since the exact evaluation of these norm-like functionals is a very difficult task, not yet solved, we will tackle this problem by means of the determination of the asymptotical behavior (i.e., at large $n$) of the general functional $N_{n}(\alpha,p,\beta)$ by extensive use of the strong asymptotics of Laguerre polynomials. The results obtained are contained in the following theorem.\\
\textbf{Theorem.} The asymptotics ($n\to\infty$) of the Laguerre hydrogenic functionals $N_{n,l}(D,p)$ defined by Eq. (\ref{eq:c1.21}) are given by the following expressions for all possible values of $D$ and $p>0$: \begin{enumerate} \item If $\beta > 0$, there are two subcases:
\begin{enumerate}
\item If $D>2$, then
\begin{equation}\label{13}
N_{n,l}(D, p)= C(\beta,p)\,(2(n-l-1))^{1+\beta-p}\,(1+\bar{\bar{o}}(1)),\; \text{for}\,\, p\in \left(0,\frac{D-1}{D-2}\right)
\end{equation}
\item If $D=2$ (so, $\beta=1$), then
\begin{equation}\label{14}
N_{n,l}(D, p)=\left\{
\begin{array}{ll}
C(1,p)\,(2(n-l-1))^{2-p}\,(1+\bar{\bar{o}}(1))\;, & p\in (0,2)
\\
\displaystyle\frac{\ln (n-l-1)+\underline{\underline{O}}(1)}{\pi^{2}}\;, & p=2(*)
\\
\displaystyle\frac{C_{A}(p)}{\pi^p}\,(4(n-l-1))^{\frac{2}{3}(2-p)}(1+\bar{\bar{o}}(1))\, & p\in(2,5)
\\
\left(\displaystyle\frac{C_{A}(p)}{\pi^p 4^{2}}+C_{B}(\alpha,1,p)\right)\,(n-l-1)^{-2},
& p=5
\\
C_{B}(\alpha,1,p)\,(n-l-1)^{-2}\;, & p\in(5,\infty) .
\end{array}
\right.
\end{equation} (*) Cosine-Airy regime
\end{enumerate} \item If $\beta = 0$ (so, $D\neq 2$ and $p=\frac{D-1}{D-2}$), then \begin{equation}\label{9} N_{n,l}(D, p)=\left\{ \begin{array}{ll} C(0,p)\,(2(n-l-1))^{(1-p)}\,(1+\bar{\bar{o}}(1))\;, & p=\frac{D-1}{D-2} \\ \displaystyle\frac{\ln (n-l-1)+\underline{\underline{O}}(1)}{\pi^{2}(n-l-1)}\;, & p=2,\, (D=3)(*)
\end{array} \right. \end{equation} (*)Cosine-Airy regime \item If $\beta < 0$ (so, either $p<\frac{D-1}{D-2}$ and $D<2$ or $p>\frac{D-1}{D-2}$ and $D>2$), then \begin{equation}\label{8} N_{n,l}(D, p)=\left\{ \begin{array}{ll} C(\beta,p)\,(2(n-l-1))^{1+\beta-p}\,(1+\bar{\bar{o}}(1)),\quad & p\in \left( \frac{D-1}{D-2}, \frac{2 D}{2 D-3}\right)
\\ \displaystyle\frac{2\,\Gamma(p+1/2)\,\,\,(\ln n+\underline{\underline{O}}(1))}{\pi^{p+1/2}\,\Gamma(p+1)\,(4(n-l-1))^{1+\beta}}\,,\quad & p=2+2\,\beta = \frac{2 D}{2 D-3}(*), \\ C_{B}(\alpha,\beta,p)\,(n-l-1)^{-(1+\beta)}\,(1+\bar{\bar{o}}(1)),\quad & p>2+2\,\beta = \frac{2 D}{2 D-3} \end{array} \right. \end{equation}
(*)Cosine-Bessel regime\\ where the constants $C, C_{B}, C_{A}$ are given by \begin{equation}\label{2.6} C(\beta,p):=\displaystyle\frac{2^{\beta+1}}{\pi^{p+1/2}}\,\displaystyle\frac{\Gamma(\beta+1-p/2)\,\Gamma(1-p/2)\,\Gamma(p+1/2)} {\Gamma(\beta+2-p)\,\Gamma(1+p)}\;, \end{equation} \begin{equation}\label{2.5} C_{A}(p):= \int_{-\infty}^{+\infty} \left[ \frac{2\pi}{\sqrt[3]{2}}\,\, {\rm Ai}^2\left( -\frac{t \sqrt[3]{2}}{2} \right) \right]^p dt\,, \end{equation} and \begin{equation}\label{2.4}
C_{B}(\alpha,\beta,p):=2\int\limits_{0}^{\infty}t^{2\beta+1}|J_{\alpha}|^{2p}(2t)\,dt\;, \end{equation} respectively, $\alpha=2l+D-2$ and $\beta=(2-D)p+D-1$.
The symbols $Ai(t)$ and $J_{\alpha}(z)$ denote the Airy and the Bessel functions (see \cite{szego_75}) defined by $$ Ai(y)=\frac{\sqrt[3]{3}}{\pi}\,A(-3\sqrt{3}y),\quad A(t)=\frac{\pi}{3}\,\sqrt{\frac{t}{3}}\, \left[J_{-1/3}\left(2\left(\frac{t}{3}\right)^{\frac{3}{2}}\right)+J_{1/3}\left(2\left(\frac{t}{3}\right)^{\frac{3}{2}}\right)\right]. $$ and $$ J_{\alpha}(z)=\sum_{\nu=0}^{\infty}\frac{(-1)^{\nu}}{\nu!\,\Gamma(\nu+\alpha+1)}\,\left(\frac{z}{2}\right)^{\alpha+2\nu}\;. $$ respectively.
\textit{Comment}: For $D=3$ it happens that $\frac{2D}{2D-3} =\frac{D-1}{D-2} = 2$, and then the quantities $N_{n,l}(D,p)$ are given by the third asymptotical expression in (\ref{8}). For higher dimensions one has $\frac{2D}{2D-3} >\frac{D-1}{D-2}$, and the three expressions in (\ref{8}) hold. \end{enumerate}
\textit{Hints}. We use the effective Aptekarev et al's technique \cite{aptekarev_2012,aptekarev_2010,aptekarev_2016} recently applied for oscillator-like systems. This technique determines the ($n\to\infty$)-asymptotics of the Laguerre hydrogenic norms $N_{n}(\alpha,p,\beta)$ by taking care of the values of the parameters $\alpha,\beta$ and $p$. It turns out that the dominant contribution to the asymptotical value of the integral (\ref{eq:c1.2}) comes from different regions of integration defined according to the values of the involved parameters, which characterize various asymptotic regimes. Thus, we have to use various asymptotical representations for the Laguerre polynomials at the different scales.
Altogether there are five asymptotical regimes which can give (depending on $\alpha,\beta$ and $p$) the dominant contribution in the asymptotics of $N_{n}(\alpha, p, \beta)$. Three of them exhibit the growth of $N_{n}(\alpha, p, \beta)$ as some $n$th-power law with an exponent which depends on $\alpha,\beta$ and $p$. We call them by Bessel, Airy and cosine (or oscillatory) regimes, which are characterized by the constants $C_{B}$, $C_{A}$ and $C$, respectively, mentioned above. The Bessel regime corresponds to the neighborhood of zero (i.e. the left end point of the interval of orthogonality), where the Laguerre polynomials can asymptotically be expressed by means of Bessel functions (taken for expanding scale of the variable). Then, at the right of zero (cosine regime) the oscillatory behavior of the polynomials (in the bulk region of zeros location) is modeled asymptotically by means of the trigonometric functions; and at the neighborhood of the extreme right of zeros (Airy regime) the asymptotics of the Laguerre polynomials is controlled by Airy functions. Finally, in the neighborhood of the infinity point of the orthogonality interval the polynomials have growing asymptotics. Moreover, there are regions where these asymptotics match each other. Namely, asymptotics of the Bessel functions for big arguments match the trigonometric function, as well as the asymptotics of the Airy functions do the same.
In addition, there are two transition regimes: cosine-Bessel and cosine-Airy. If the contributions of these regimes dominate in the integral (\ref{eq:c1.2}), then the asymptotics of $N_{n}(\alpha, p, \beta)$ besides the degree on $n$ have the factor $\ln n$. Note also that if these regimes dominate, then the gamma factors in constant $C(\beta,p)$ in (\ref{2.6}) for the oscillatory cosine regime blow up. For the cosine-Bessel regime this happens when $\beta+1-p/2=0$, and for the cosine-Airy regime when $1-p/2=0$.
\section{Information entropies of the $D$-dimensional Rydberg states}
In this section we determine the Rényi, Shannon and Tsallis entropies of the $D$-dimensional Rydberg hydrogenic states in terms of the spatial dimension $D$, the order parameter $p$, the hyperquantum numbers $(n,l,\{\mu\})$ and the nuclear charge $Z$. First, attention is focussed on the Rényi and Shannon entropies since the Tsallis entropy can be obtained from the Rényi one by means of the relation \begin{equation} \label{eq:tsalren} T_{p}[\rho] = \frac{1}{1-p}[e^{(1-p)R_{p}[\rho]}-1]. \end{equation} Then, for illustration, we numerically discuss the Rényi entropy $R_{p}[\rho_{n,0,0}]$ of some hydrogenic Rydberg $(ns)$-states in terms of $D$, $p$, $n$ and $Z$.\\
Let us start by pointing out that the total Rényi entropy $R_{p}[\rho_{n,l,\{\mu\}}]$ of the Rydberg states can be obtained in a straightforward manner by taking into account Eq. (\ref{eq:renyihyd1}), the values of the radial Rényi entropy $R_{p}[\rho_{n,l}]$ derived from Eq. (\ref{eq:renyihyd4}), the asymptotical ($n \to \infty$) values of the Laguerre norms $N_{n,l}(D,p)$ given by the previous theorem, and the angular Rényi entropy $R_{p}[\mathcal{Y}_{l,\{\mu\}}]$ given by Eqs. (\ref{eq:angpart}) and (\ref{eq:renyihyd3}); keep in mind that the angular part of the Rényi entropy does not depend on the principal quantum number $n$.\\
What about the Shannon entropy $S[\rho_{n,l,\{\mu\}}]$ of the Rydberg hydrogenic states?. To calculate its value for any stationary state $(n,l,\{\mu\})$, we take into account (a) that $\lim_{p \to +1} R_p[\rho] = S[\rho]$ for any probability density $\rho$, (b) the expression (\ref{eq:renyihyd1}), (c) the following limiting value of the radial Rényi entropy $R_{p}[\rho_{n,l,\{\mu\}}]$ of the Rydberg hydrogenic states obtained for a fixed dimension $D$ from (\ref{eq:renyihyd4})-(\ref{eq:c1.21}) and the previous theorem, \begin{eqnarray} \label{eq:radshan} \lim_{p \to +1} R_p[\rho_{n,l}] &=& \lim_{p \to +1} \frac{1}{1-p} \ln \left[\frac{\eta^{D(1-p)-p}}{2^{D(1-p)+p}Z^{D(1-p)}} C(\beta,p)\,(2n)^{1+\beta-p}\right]\nonumber\\ &=& 2 D \ln n +(2-D) \ln 2 +\ln\pi -D\ln Z+D-3, \end{eqnarray} (d) the condition $n>>l$ and (e) that \begin{equation} \label{eq:angshan} \lim_{p \to +1} R_{p}[\mathcal{Y}_{l,\{\mu\}}] = \lim_{p \to +1} \frac{1}{1-p} \ln \Lambda_{l,m}(\Omega_{D-1}) = S[\mathcal{Y}_{l,\{\mu\}}], \end{equation} (remember (\ref{eq:angpart}) and (\ref{eq:renyihyd1}) for the first equality) where $S[\mathcal{Y}_{l,\{\mu\}}]$ is the Shannon-entropy functional of the spherical harmonics \cite{dehesa2} given by \begin{equation} S[\mathcal{Y}_{l,\{\mu\}}] = -\int_{S^{D-1}}[\mathcal{Y}_{l,\{\mu\}}]^{2}\, \ln \,[\mathcal{Y}_{l,\{\mu\}}]^{2}\,d\Omega_{D-1}. \end{equation} which does not depend on $n$ and can be calculated as indicated in \cite{lopez_2009}. In particular, from the previous indications we find the following values \begin{equation} \label{eq:nsrenyi} R_{p}[\rho_{n,0,0}] = R_p[\rho_{n,0}] + R_{p}[\mathcal{Y}_{0,0}] = R_p[\rho_{n,0}] + \frac{1}{1-p}\ln f(p,D) \end{equation} for $p \neq 1$, and \begin{equation} \label{eq:sn00} S[\rho_{n,0,0}] = 2 D \ln n +(2-D) \ln 2 +\ln\pi -D\ln Z+D-3 + S(\mathcal{Y}_{0,0}) + o(1) \end{equation} for the Rényi and Shannon entropy of the ($n$\textit{s})-Rydberg hydrogenic states, respectively, where $f(p,D)$ and $S(\mathcal{Y}_{0,0})$ have the values (see Appendix A): \begin{eqnarray}
f(p,D) &=& \int_{\Omega_{D-1}} |\mathcal{Y}_{0,0}(\Omega_{D-1})|^{2p}\, d\Omega_{D-1}\nonumber \\ &=& 2^{D(1-p)}\pi^{\frac{1}{2} (-Dp+D+p-1)}\left[\frac{\Gamma (D)}{\Gamma \left(\frac{D+1}{2}\right)}\right]^{p-1} \end{eqnarray} and \begin{eqnarray} \label{eq:shany00} S(\mathcal{Y}_{0,0}) &=& -\ln \mathcal{N}_{0,0}^{2}\nonumber \\ &=& D\ln 2 + \frac{D-1}{2}\ln\pi +\ln \frac{\Gamma \left(\frac{D+1}{2}\right)}{\Gamma (D)},\nonumber\\ \label{eq:harmorenyi} \end{eqnarray} respectively. Note that $\lim_{p \to +1} \frac{1}{1-p}\ln f(D,p) = S(\mathcal{Y}_{0,0})$. In the particular case $D=3$ (i.e., for real hydrogenic systems), one has that $\lim_{p \to +1} \frac{1}{1-p}\ln f(3,p) = S(Y_{0,0}) = \ln (4\pi)$, as expected.\\
\textcolor{red}{To conclude and} for illustrative purposes, we first numerically investigate the dependence of the Rényi entropy $R_{p}[\rho_{n,0,0}]$ for some Rydberg $(ns)$-states on the quantum number $n$, the order parameter $p$ and the nuclear charge $Z$. \textcolor{red}{To start with}, we study the variation of the $p$-th order Rényi entropy of these states in terms of $n$, within the interval $n= 50-200$, when $p$ is fixed. As an example, the cases $p=\frac{5}{4},\frac{10}{7},\frac{3}{4},3$ for the $D$-dimensional hydrogenic Rydberg $(ns)$-states with $D=6,5,4$, and $2$, respectively, are plotted in Fig. \ref{fig.1}. We observe that the behavior of the Rényi entropy has always an increasing character for any dimensionality $D>2$. \\
\noindent
\textcolor{red}{Then}, in Fig. \ref{fig.2}, we analyze the dependence of the Rényi entropy, $R_{p}[\rho]$, on the order $p$ for the Rydberg hydrogenic state $(n=100, l=1, D=4)$. We observe that the Rényi entropy decreases monotonically as the integer order $p$ is increasing. This behavior indicates that the quantities with the lowest orders (particularly the cases $p=1$ and $p=2$, which correspond to the Shannon entropy and the disequilibrium or second-order Rényi entropy) are most significant for the quantification of the spreading of the electron distribution of the system. In fact, this behavior occurs for all the $D$-dimensional states; we should expect it since the Rényi entropy is defined by (\ref{eq:renentrop}) as a continuous and non-increasing function in $p$.
\noindent \textcolor{red}{Later}, in Fig. \ref{fig.3}, we illustrate the behavior of the Rényi entropy, $R_{p}[\rho]$, as a function of the atomic number $Z$ of the Rydberg hydrogenic states $(n=100,l=0)$ with $(p=3, D=2)$ and $(p=\frac{3}{4},D=4)$, where $Z$ goes from $1$ (hydrogen) to $103$ (lawrencium). We observe that the Rényi entropy decreases monotonically as $Z$ increases, which points out the fact that the probability distribution of the system tends to separate out from equiprobability more and more as the number of electrons in the nucleus of the atom increases; so, quantifying the greater complexity of the system as the atomic number grows.\\
\noindent Finally, we investigate the behavior of the Rényi entropy, $R_{p}[\rho]$, of the Rydberg hydrogenic state as a function of the dimensionality $D$. We show it in Fig. \ref{fig.4} for the Rydberg state $(n=100, l=0)$ with $p=\frac{1}{2}$ and $4$ of the hydrogen atom with various integer values of the dimensionality $D \in [50,200]$. We observe that in both cases the Rényi entropy has a monotonically increasing behavior as $D$ grows, which indicates that the larger the dimension, more classically the system behaves (or in other words, the closer is the system to its classical counterpart).
\begin{figure}
\caption{Variation of the Rényi entropy, $R_{p}[\rho]$, with respect to $n$ for the Rydberg hydrogenic $(ns)$-states with $(p=\frac{5}{4}, D=6)(\square)$, $(p=\frac{10}{7}, D=5)(\bullet))$, $(p=\frac{3}{4}, D=4)(\circ))$ and $(p=3, D=2)(\blacksquare))$. }
\label{fig.1}
\end{figure}
\begin{figure}
\caption{Variation of the Rényi entropy $R_{p}[\rho_{n,1,0}]$, respectively, with respect to $p$ for the Rydberg state $(n=100,l=1)$ of the hydrogen atom ($Z=1$) with $D=4$.}
\label{fig.2}
\end{figure}
\begin{figure}
\caption{Variation of the Rényi entropy, $R_{p}[\rho]$, with respect to the nuclear charge $Z$ for the Rydberg hydrogenic state $(n=100, l=0)$ of the $D$-dimensional hydrogen atom with $D=2(\blacksquare)$ and $D=4(\bullet)$.}
\label{fig.3}
\end{figure}
\begin{figure}
\caption{Variation of the Rényi entropy, $R_{p}[\rho]$, with respect to the dimensionality $D$ for the Rydberg hydrogenic state $(n=100, l=0)$ of the $D$-dimensional hydrogen atom with $p=\frac{1}{2}(\blacksquare)$ and $p=4(\bullet)$.}
\label{fig.4}
\end{figure}
\section{Conclusions}
In this work we determine the three main entropic measures (Rényi, Shannon, Tsallis) of the quantum probability density of stationary $D$-dimensional Rydberg ($n >> 1$) hydrogenic states in terms of the basic parameters which characterize them; namely, the dimensionality $D$, the hyperquantum numbers $(n,l,\{\mu\})$ and the nuclear charge $Z$ of the system. In fact, the Shannon and Tsallis entropies can be obtained from the Rényi one. The Rényi entropy has been calculated by first decomposing it into two parts of radial and angular types, and realizing that the angular part does not depend on $n$, so that the true problem to be solved is the calculation of the radial Rényi entropy in the limit of large $n$. The radial Rényi entropy has been shown to be expressed in terms of the $\mathfrak{L}_{p}$-norms of the Laguerre polynomials which control the Rydberg states we are interested in. Then, the remaining asymptotics of these Laguerre norms is determined by means of a recent technique of approximation theory. Finally, we numerically apply this theoretical methodology to some particular Rydberg hydrogenic states of $s$ and $p$ types. We find that the Rényi entropy monotonically decreases (increases) when the nuclear charge (the dimensionality) is decreasing (increasing) for some $s$ and $p$ states. \section*{Acknowledgments} This work has been partially supported by the Projects FQM-7276 and FQM-207 of the Junta de Andaluc\'ia and the MINECO-FEDER grants FIS2014-54497P and FIS2014-59311-P. I. V. Toranzo acknowledges the support of ME under the program FPU. We acknowledge useful discussions with Alexander I. Aptekarev.
\appendix
\section{Calculation of $f(p,D)$ and $S(\mathcal{Y}_{0,0})$}
Let us first calculate the factor $f(p,D)$ which appears in Eq. (\ref{eq:nsrenyi}):
\begin{eqnarray}
f(p,D) &=& \int_{\Omega_{D-1}} |\mathcal{Y}_{0,0}(\Omega_{D-1})|^{2p}\, d\Omega_{D-1}\nonumber \\ &=& \int_{\Omega_{D-1}} (\mathcal{N}_{0,0}^{2})^{p}\, d\Omega_{D-1}\nonumber \\ &=& (\mathcal{N}_{0,0}^{2})^{p}\,2 \pi \prod _{j=1}^{D-2} \frac{\sqrt{\pi } \,\Gamma \left(\frac{D-j}{2}\right)}{\Gamma \left(\frac{1}{2} (D-j+1)\right)}\nonumber \\ &=& (2 \pi )^{1- p}\left[\prod _{j=1}^{D-2} \frac{(D-j-1) \Gamma \left(\frac{1}{2} (D-j-1)\right)^2}{\pi \, 2^{-D+j+3} \Gamma (D-j-1)}\right]^{p}\prod _{j=1}^{D-2} \frac{\sqrt{\pi }\, \Gamma \left(\frac{D-j}{2}\right)}{\Gamma \left(\frac{1}{2} (D-j+1)\right)},\nonumber\\
&=& 2^{1- p}\pi^{\frac{D}{2}(1-p)}\left[\prod _{j=1}^{D-2} \frac{\Gamma \left(\frac{1}{2} (D-j+1)\right)}{\Gamma \left(\frac{D-j}{2}\right)}\right]^{p}\prod _{j=1}^{D-2}\frac{\Gamma \left(\frac{D-j}{2}\right)}{\Gamma \left(\frac{1}{2} (D-j+1)\right)}\nonumber \\
&=&2^{1- p}\pi^{\frac{D}{2}(1-p)} \left[ \Gamma\left(\frac{D}{2}\right) \right]^{p-1}\nonumber\\
\end{eqnarray}
Let us now compute the factor $S(\mathcal{Y}_{0,0})$ which appears in Eq. (\ref{eq:sn00}): \begin{eqnarray} \label{eq:shany00} S(\mathcal{Y}_{0,0}) &=& -\ln \mathcal{N}_{0,0}^{2}\nonumber \\ &=& -\ln\left[\frac{1}{2 \pi }\prod _{j=1}^{D-2} \frac{(D-j-1) \Gamma \left(\frac{1}{2} (D-j-1)\right)^2}{\pi\, 2^{-D+j+3}\Gamma (D-j-1)}\right],\nonumber \\ &=& \ln 2 \pi - \ln \prod _{j=1}^{D-2} \frac{\Gamma \left(\frac{1}{2} (D-j+1)\right)}{\sqrt{\pi } \,\Gamma \left(\frac{D-j}{2}\right)}\nonumber \\ &=& \ln 2 \pi - \left(1-\frac{D}{2}\right)\ln \pi - \ln\prod_{j=1}^{D-2}\frac{\Gamma \left(\frac{1}{2} (D-j+1)\right)}{\Gamma \left(\frac{D-j}{2}\right)}\nonumber\\
&=& \ln 2 \pi - \left(1-\frac{D}{2}\right)\ln \pi - \ln \Gamma\left( \frac{D}{2} \right)\nonumber \\ &=& \ln 2 +\frac{D}{2}\ln \pi - \ln \frac{2^{1-D}\pi^{1/2}(D-1)!}{\left(\frac{D-1}{2}\right)!} \nonumber\\ &=& D\ln 2 + \frac{D-1}{2}\ln\pi +\ln \frac{\Gamma \left(\frac{D+1}{2}\right)}{\Gamma (D)}.\nonumber\\ \label{eq:harmorenyi} \end{eqnarray}
\end{document} | arXiv |
Consumer stated preferences for dairy products with carbon footprint labels in Italy
Maurizio Canavari1 &
Silvia Coderoni ORCID: orcid.org/0000-0001-8751-73762
Agricultural and Food Economics volume 8, Article number: 4 (2020) Cite this article
Carbon footprint (CF) labels on agri-food products represent one of the most important tools to convey information to consumers about the greenhouse gases emissions associated with their purchase behaviour.
Together with the growing interest of consumers in CF labels, the subject has gained attention also in the scientific literature, and formal evaluations of consumer response to carbon labelling have been published. Studies in this area aim at analysing consumers' preferences for buying products with a lower CF label or their willingness to pay (WTP) for these products.
The objective of this paper is twofold. First, the study proposes a review of the literature that so far has analysed consumer WTP for CF label, focusing on Italian consumers. Second, it uses the results of two surveys of consumers' attitudes towards dairy products with a lower CF label to analyse the factors determining a positive stated WTP. Results point out that a positive WTP for lower CF products is more likely to be declared by respondents who believe that buying products with less environmental impact can combat climate change. Conversely, highly price-sensitive consumers are less likely to be willing to pay more for CF-labelled products.
Climate change mitigation is one of the key environmental goals of agricultural production worldwide (Gerber et al. 2013). Moreover, in Europe, climate change mitigation objectives and the contribution that agriculture is expected to provide have reached the top of the political agenda (European Commission 2016). Climate action is one of the main priorities of the Common Agricultural Policy (CAP) and agricultural greenhouse gases (GHG) emissions' mitigation has become both an objective of the new architecture of the first pillar payments and a focus area of the actual Rural Development Policy programming period (Council of the European Union 2013a, 2013b).
According to many studies in this field, however, supply-side options alone, i.e. options that tackle production aspects of GHG mitigation, are not sufficient to reach the ambitious mitigation targets set by European and international climate policy agenda (European Commission 2011, 2016). In addition, though the most cost-effective ways to reduce GHG are carbon taxes and cap and trade systems (Nordhaus 2013; Stern 2007), these economic instruments are unlikely to be implemented in the near future in the agricultural sector, both in the EU (Coderoni and Esposti 2018) and in the United States (Shewmake et al. 2015). Thus, demand-side solutions to climate change, which consist of more sustainable consumption patterns, are becoming important tools to curb agricultural GHG emissions (Garnett 2011; Bajželj et al. 2014; Armel et al. 2011; Brunelle et al. 2017; Creutzig et al. 2016; de Boer et al. 2016).
In this respect, the so-called "sustainable labels", i.e. types of labels that are designed to convey to the consumer concepts related to all the facets of sustainability, are the most common tools supporting changes in consumption patterns (Vermeir and Verbeke 2006; Zander and Hamm 2010). When sustainable labels try to show to consumers the overall impact of the product converting it into a standardised measure of carbon dioxide emissions, they are referred to as "carbon footprint" (CF) labels. CF labels in practice indicate the quantity (in grams) of carbon dioxide equivalent (CO2e)Footnote 1 emitted into the atmosphere throughout all the life cycle of a product or service, which comprises production, transport, transformation, distribution and purchase (Sander et al. 2016). The rationale for these labels, when applied to food products, is that they may help to orient the consumer towards buying more GHG saving agricultural products and thus mitigating agriculture's contribution to global warming.
Despite the potentially relevant role of demand-side options in tackling climate change, there have been a few consumer studies on WTP for carbon footprint labels (Hoek et al. 2017), especially for Italian agriculture, where the bulk of the empirical literature has focussed on the potential and effectual role of the production processes to mitigate agricultural GHG emissions.Footnote 2 Thus, it would be essential to analyse consumers' preferences for purchasing products with a label showing a lower CF, to understand what drives their choices, and to recognise to what extent there is a mitigation potential deriving from Italian consumers' choices for the Italian agricultural sector.
In this context, the objective of this paper is twofold. First, we review the literature that until 2018 has analysed consumer preferences and WTP for CF label, with a focus on Italian consumers. Second, we illustrate some of the results of two separate pilot surveys aimed at detecting whether consumers state a positive WTP for dairy products with a CF label.
The remainder of the paper is structured as follows: Section 2 introduces some definition of sustainable labels, specifically referring to CF; Section 3 presents the literature review; Section 4 shows the case studies analysed, while Section 5 presents and discusses the results of the analysis; Section 6 finally proposes some concluding policy remarks and future research guidelines.
Carbon footprint labels in the agri-food sector
According to Miranda-Ackerman and Azzaro-Pantel (2017), "New consumer awareness is shifting industry towards more sustainable practices, creating a virtuous cycle between producers and consumers enabled by eco-labelling".
This consumer awareness is the foundation of sustainable consumption, which is grounded in a decision-making process that takes into account not only individual needs and wants, but also their social responsibility. In fact, as De Pelsmacker et al. (2005) have found, when dealing with sustainability concerns, an important driver for change is the inclination of the "ethical consumer [that] feels responsible towards society and expresses these feelings by means of his purchase behaviour".Footnote 3
The concept of sustainability has deeply evolved from the primer environmentalist approach (Kumar et al. 2012) and now it comprises, in its most widespread use, three different aspects: the economic, the environmental and the social one (Vermeir and Verbeke 2006). Sustainable products are those products whose characteristics respect one or more of these aspects (Vackier et al. 2002).
Eco-labelling, or environmentally sustainable labels, are a means to inform consumers of the environmental performance of either the products or the production systems they come from, and they can also inform the consumer on measures taken by the producers to minimise the product's environmental impact.
One particular type of sustainable label is the so-called CF label, which is an indicator of the total amount of CO2, or the equivalent of CO2 in the case of the emission of other GHG (usually expressed in grams), emitted into the atmosphere along the whole "life cycle" of a particular product or service. Thus, the calculation comprises not only production but also transport, transformation, distribution, use and disposal.
In the agri-food sector, the European Commission has identified 129 (both public and private) information plans concerning the concept of sustainability (Grunert et al. 2014). Among these labels, the organic brand (referred to also as "bio") is the most widely used in the Italian market. Local production, however, is gaining popularity among Italian consumers, even though a universal label for the definition of such products has not yet been established (Bazzani and Canavari 2013, 2017).
CF labels are rarely present in the agri-food marketFootnote 4, and only recently, consumers have occasionally had access to information about the CF of products, both in Italy and in most European countries. Tesco experience is exemplary in this field: the retailer, together with the Carbon Trust, has started introducing the first CF label in food retailing in 2009, claiming that they would have labelled all the 50,000 own-brand products (The Economist 2011). However, in 2012, when they only have been able to label 500 products, they had to give up the project. The reasons for this failure were that: consumers found the labels complicated and difficult to understand (so the company was looking for alternatives to replace the CF); the process of labelling the products revealed much more time consuming than planned and other retailers were slow in adopting CF labelling. Thus, the uptake of the label could not reach the desired critical mass (Financial Times 2012).
Nowadays, there are only a few CF labels that have continued in the marketplace. However, as mentioned by Peschel et al. (2016) and Grebitus et al. (2015), Eurobarometer survey's results have found 72% of EU citizens agreeing that CF information on products should be mandatory (European Commission 2009). More recently, about 90% of EU citizens have declared that buying environmentally friendly products can bring real benefits to the environment (European Commission 2012).
In this context, it should be of much interest to investigate the drivers and the socio-economic characteristics of respondents that can influence a positive WTP of consumers towards CF-labelled products.
A literature review of WTP studies on CF for food products
Consumers' preferences for lower CF label products have not yet been widely explored in the literature (Vanclay et al. 2011), also because of scarce market presence and uptake, and only recently, there has been a growing body of literature proposing formal evaluations of consumer response to carbon labelling.Footnote 5
We performed a literature review to examine the works available in the Italian and international scientific literature that so far have analysed consumer preferences and WTP for CF label.Footnote 6
Articles were selected by checking against pre-determined criteria for eligibility and relevance. Firstly, the following keywords have been identified: "footprint" (and its possible variation "foot-print"), "consumer", "food", together with their Italian translations. Secondly, a search for the abstracts of the articles has been done based on these keywords in the primary databases for scientific relevant literature (Scopus, Web of Knowledge, AgEconSearch, EconPapers), and thus pertinent articles have been selected.Footnote 7
Approximately 300 articles have been consulted (including 150 references from Scopus and 130 from the Web of Knowledge, largely overlapping). Those papers went through a screening process that made emerge only 27 of them for an in-depth analysis, as they were in line with the specific goals of the review. These low figures reinforce the argument that the topic has not been widely explored in the empirical literature so far, in particular for the Italian consumers. Table 1 summarises the main aspects of the selected studies: country, products of reference, the methodology used and the main findings.
Table 1 Published articles regarding WTP evaluations for CF labels: country, products of reference, methodology used and main findings
Most of these studies show that in general consumers are responsive to CF on different products indicating lower emissions, than conventional ones. However, as Vanclay et al. (2011) found, CF labels are most effective when combined with lower prices.
Moreover, Akaichi et al. (2013) and Onozaka and Mcfadden (2011) highlighted that consumers have been particularly likely to buying low-CO2 products in case they were also labelled with local origin, and according to Hoek et al. (2017), the combination of a health and carbon logo has a more positive effect than the logos separately or no logo. An interesting result is that from Shewmake et al. (2015) that have shown how even if CF labels can lower GHG emissions, they can also have the potential to incur in the opposite effect if their implementation does not account for consumer beliefs as well as complementary and substitute relationships among different products.
Among the sorted articles, only the works by Caputo and co-authors (Caputo et al. 2012, 2013b), Vecchio (2013), Vecchio and Annunziata (2015), Lombardi et al. (2017) and Colantuoni et al. (2016) focused specifically on the Italian market.
Caputo et al. (2012) provided information on the presence of food miles' labels and the level of GHG emissions related to transport, finding a positive influence of both information on consumers' utility. Caputo et al. (2013b) found that Neapolitan consumers interviewed have shown a greater WTP for transport distance information label (food miles) rather than for the more comprehensive CF label. This finding suggests that the local origin of the product might have an impact on the Italian consumers' purchasing choice. Italian consumers could thus be more concerned with labels related to a concept of sustainability together with the local origin.
Vecchio (2013) and Vecchio and Annunziata (2015) evaluate young consumers' attitude towards sustainability labels. Vecchio (2013) found a positive young adult wine drinkers' WTP for CF-labelled wine. Vecchio and Annunziata (2015) found a positive (1.41€) WTP for CF-labelled chocolate bars and identified some factors positively affecting WTP for CF label: age (younger individuals express a higher WTP); gender (female respondents); intensity of trust in the specific labels and the preference for food obtained in an environmentally friendly way.
Lombardi et al. (2017) analyse consumers' preferences when buying fresh milk and find an average premium price of 0.55€ per litre.
On the contrary, Colantuoni et al. (2016) explore the market potential of domestic early potato and find that Italian (and German) respondents were unwilling to pay more for CF certification. Marginal WTP estimated was, in fact, negative and higher for Italians than for Germans.
As regards the type of product, the preference for low CF product has been found for both livestock and vegetable foodstuff. Echeverría et al. (2014) have analysed the WTP of Chilean consumers for both a product of vegetable origin (bread) and an animal product (milk) and found that respondents showed greater sensitivity when evaluating animal products as they were more likely to pay for lower CF for milk than for bread. To this respect, product origin (animal or vegetal) can be acknowledged as an additional aspect that could potentially affect the preferences of Italian consumers for CF labels: e.g. Canavari and Nayga (2009) have shown that Italian consumers exhibit differentiated behaviours when consumer choice is related to GMO products of vegetable origin rather than of animal origin.
As regards the methodological aspect, the WTP for low CF products was primarily estimated by hypothetical choice experiments. Only five studies out of 27 have used non-hypothetical methods (i.e. experimental auctions in four cases and a real choice experiment in one case).
The two case studies in the dairy sector: data and method
The two case studies carried out dealt with consumers' habits related to dairy foodstuffs purchasing and were performed through two different surveys aimed at evaluating consumer understanding, knowledge, and preference for low CF products. Besides, they aimed at identifying the products' characteristics that influence purchasing behaviour and the consumers' WTP for the purchase of 1 litre of fresh milk with a lower CF label in comparison to a conventional one. The focus on dairy foodstuff was driven by the importance of livestock products in the debate at international level for their higher contribution to climate change with respect to vegetable foodstuff production (Gerber et al. 2013; GRAIN and IATP 2018).
The two studies were conducted among Italian consumers from December 2016 to February 2017, in both cases using an online questionnaire gathering information on consumption choices and socio-economic characteristics for 393 consumers interviewed (215 in case study A and 178 in case study B, respectively). The questionnaires were similar but not identical, and they were composed of four sections: the first on consumers' habits, the second on their environmental awareness, the third about their knowledge of environmental and CF labels, and a section dealing with personal socio-demographic information (Author1 et al. 2018).
Though the use of web instruments to administer the questionnaire has facilitated reaching a high number of respondents, it raises the issue of representativeness of the sample, because this sampling method tends to gather self-selected respondents. Consequently, it usually generates a biased sample, in which younger people with a higher level of education or web literacy are overrepresented (Canavari et al. 2005). Therefore, the samples cannot be considered representative of Italian consumers; nevertheless, they allow obtaining quite interesting information about the relationship among the variables analysed. Though WTP estimations based on a non-representative sample cannot be used to extend WTP results to the population analysed, as figures would be biased, the relationships among the socio-economic characteristics of respondents the positive WTP eventually expressed, remain valid.
Table 2 and Table 3 show the descriptive statistics of the surveys analysed. As mentioned, the questionnaires were similar, but not identical. Thus, Table 2 shows the replies to the question that differed among surveys, while Table 3 shows only the shared items analysed.
Table 2 Descriptive statistics of the questions differing between the two surveys
Table 3 Descriptive statistics of the questions asked in both surveys
As regards methodological aspects, WTP analysis was conducted with different approaches, both relying on the contingent evaluation (CV) method. CV is a method of estimating the value that a respondent gives to a specific good or a good attribute, by asking to directly report his WTP, rather than inferring it from observed behaviours in market transactions. For this reason, CV is often referred to as a "stated preference" method, on opposite to "revealed preference" method based on existing prices. There are different CV approaches that depend on the way consumers are asked about their preferences. In the present analysis, in the case of study A, the survey used the open-ended contingent valuation method that relies upon asking directly to consumers to state their WTP for the product considered. An open-ended question is a question that is designed to encourage a full, meaningful answer, using the subject's personal knowledge or feelings. In the case of study B, we relied upon a dichotomous choice contingent valuation: a hypothetical purchase situation has been proposed to estimate the WTP, by comparing product 1 (milk bottle with 200 g of CO2e emissions) at the fixed price of 1.30€ with product 2 (bottle of milk with 150 g of CO2e emissions). Respondents were asked to state their preference between the two products according to a price variation of 0.10€ of product 2, up to a maximum value of 2.00€. Given these different approaches for the elicitation of the WTP in the two studies, the consumer's preference has been evaluated considering whether the choice of the respondent (individual outcome variable) was to state a positive WTP for lower CF products or not. The dichotomous choice contingent valuation, proposing such a take-it-or-leave-it survey valuation question, is more likely to reflect real individual purchase decisions. Moreover, such elicitation format has shown to be less susceptible to strategic bidding behaviour than the open-ended one.
A conditional logit model has been estimated to investigate the determinants of the probability for consumers to declare a positive WTP for products with lower CF, based on explanatory variables, as responses shared to both surveys, expressing some socio-demographic characteristics and attitudes of the consumers interviewed. The general equation for the conditional logit model estimated is:
$$ {P}_i\left({y}_i\ne 0\ |{X}_i\right)=\frac{\exp \left({X}_{i\kern0.5em }\beta \right)}{1+\exp \left({X}_{i\kern0.5em }\beta \right)\ } $$
where i indicates the generic individual; Pi is the predicted probability of individual i to make a specific choice; β is a vector of unknown parameters and X a vector of explanatory variables expressing the characteristics and choices of the individual expected to influence the respective choice. By including among these variables some features of the individuals interviewed, we assume heterogeneous respondents' preferences.
For the sake of brevity, we only report the variables that were significant (Table 4).
Table 4 Estimations results
Results discussion
Table 2 shows the answers to both the surveys to the different questions analysed. As regards survey A, interestingly, almost all the respondents declare to know the climate change phenomenon, are (on average) interested in it and think that the consumption of products with an environmental label helps contrasting climate change. This survey also reported a set of questions on which tools could be used to promote the knowledge and dissemination of CF labels and the web instruments and education were judged the most important, followed by campaigns, advertising, the label itself and newspapers.
As regards survey B, the majority of respondents consider it important to have a CF label to inform purchase decisions in an environmental sense and thinks that buying organic food helps to reduce GHG emissions. However, it seems that the majority of the respondents do not read the label but gives importance to the sensory quality or expiration date when buying food.
Table 3 summarises some descriptive statistics of the pooled sample analysed, made by the common questions. As mentioned, being the sample self-selected and based on an online survey, some demographics reflect the nature of the data source. The respondents are 64% female. Almost all the respondents have at least a high school diploma, and 52% have a university degree (or higher). Despite the low presence of CF label in the Italian market, a majority of subjects declared to know the concept of CF labels.
As regards WTP, results indicate that only 24% of the total sample states not to be willing to pay more for a litre of milk with lower CF. This figure is likely underestimated since it is based on a stated preference survey, and the sample is self-selected.
For respondents that declared a positive WTP, in case study A, an average 9% premium price for lower CF milk has emerged, with maximum values of 50%. The premium price was on average 0.19€, assuming an average price of 2€. In case study B, the average WTP was more than 30%. The consumer is likely to pay € 1.68 per bottle of low CF milk and therefore, compared to the high-impact product proposed at the price of € 1.30, the surplus difference is € 0.38 (Author1 et al. 2018).
The results of the logit model estimation summarised in Table 4 allow identifying the drivers behind this positive WTP.Footnote 8
Results indicate that in the sample analysed, if a respondent gives high importance to low impact products to tackle climate change, this trait positively affects the probability to be willing to pay more for CF-labelled milk. Also, the format of the different surveys may matter: survey B respondents are more likely to show a positive WTP compared to survey A respondents.
As regards socio-demographic variables, respondents who are more sensitive to price when buying products (about 40% of the sample) are less likely to be willing to pay more for products with a lower CF label; this result is consistent with what other authors in this field have found (see among others Vanclay et al. 2011).
Instead, the only knowledge of the CF concept does not seem to be relevant in determining the stated perception of value. Also, age and education do not affect the WTP of consumers, similarly to what was found from the detailed analysis of case study A published in another article (Author1 and Author2 2019).
As regards gender, females show a slightly higher WTP than males, but this result is significant only at the 0.10 level. Hence, even if this result goes in the direction of what found in previous work (e.g. Steiner et al. 2017; Vecchio and Annunziata 2015), its statistical significance is quite poor.
CF labels represent one of the most important tools to help to tackle climate change through consumers' informed purchases behaviour.
Despite their relevance for demand-driven mitigation options, their presence is still scarce in the Italian food sector, and so it is also for studies aimed at investigating Italian consumers' WTP for products with lower CF. From the literature review, a positive WTP for lower CF products seems to emerge, though not for all products and respondents' socio-economic characteristics.
The two pilot case studies presented, focused on Italian consumers' habits when purchasing milk, allowed us to make a rough evaluation of their preferences for low-carbon-labelled dairy products.
Results, though based on convenience and probably biased samples and stated preferences, suggest that the interest of consumers in CF labels may exist. Findings are generally in line with previous studies indicating that respondents that give high importance to foodstuff produced with low environmental impact to mitigate GHG emissions have shown to be more willing to attribute a positive premium price to CF-labelled products. Also, the data confirm that CF labels could be most effective when combined with prices lower than (or at least equal to) conventional products (Vanclay et al. 2011), as more price-sensitive consumers are less prone to perceive a higher value for lower CF products.
Those results, if confirmed by larger and representative samples, may have interesting policy implications. In fact, they would suggest that a policy framework aiming at promoting demand-side mitigation options in the agricultural sector should tackle both the consumers' side, informing consumers about the environmental impact of food production and the potential of environmental label in reducing it, and the producers' side, helping the food supply chain reducing its GHG emission in a cost-effective way.
About the consumers' side, policies should aim at both enhancing consumers' awareness about climate change challenge and ensure that the system of certification is reliable and easily interpretable by consumers. To this respect, the initiative of the European Commission (2013) on "building the single market for green products facilitating better information on the environmental performance of products and organisations", is of utmost importance.
As regards the production side, results would suggest to producers that a lower CF would be appealing if offered at the same or a lower price. Indeed, the possibility to couple lower prices with lower GHG emissions in the agricultural sector is not rare, because technical studies on the mitigation potential of agri-food productions have found many of the so-called "win-win solution" to climate change, i.e. strategies that allow saving both GHG emissions and production costs (Coderoni et al. 2015). When a win-win solution is adopted, thus, lower CF products can be produced at lower costsFootnote 9 that could, in turn, be translated into lower selling prices, as entrepreneurs participating in the CF labelling scheme have declared to be willing to do (Coderoni and Pontrandolfi 2016). If these solutions are applied, thus, CF product uptake could be easily enabled.
Also, Rural Development Programmes funds could be used to reduce farmers' costs of adopting GHG saving techniques, as they provide incentives for both GHG calculation and certification and farms' investments to implement mitigation strategies identified. This should be made taking into account the likely evolution of the food systems as a whole (Macombe 2018).
Given the limitations of this study, a more in-depth analysis is needed to estimate Italian consumers' WTP for CF labels accurately. Future research should on one side, rely upon a larger and nation-wide representative sample to avoid the problems linked to self-selected and biased samples; on the other side, it should focus on non-hypothetical techniques, such as experimental auctions to obtain reliable estimations of WTP (Lusk and Shogren 2007). In fact, the studies based on hypothetical choices, generally, tend to overestimate the WTP and the experience of Tesco with CF-labelled products seems to confirm this gap between stated and real behaviour. A further research avenue could be the consideration of a more comprehensive framework for the analysis of the environmental impact of food consumption, covering not only the GHG emissions generated, but also the use of resources such as water and land, and the generation of waste (Candy et al. 2018).
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
CO2e is a term that describes different greenhouse gases in a common unit. A quantity of non-CO2 GHG (i.e. methane or nitrous oxide) can be expressed as CO2e by multiplying the amount of the GHG by its global warming potential (GWP).
For Italian agriculture case study, both micro and macro level have been explored (see among others: Rete Rurale Nazionale 2012; Coderoni and Esposti 2014; Baldoni et al. 2017, 2018).
Nevertheless, studies have found that convenience, value for money, habit, personal health concerns, hedonism and individual responses to social and institutional norms are still relevant aspects driving everyday consumption practices (SDC 2003).
Instead, for other products (like home appliances, paper products, detergents, etc.), there is abundance of eco-labelling initiatives.
The most recent literature on the evaluation of consumer preferences for sustainable food, is focusing on a more comprehensive evaluation of the environmental impact of production that encompasses different resources exploited by agricultural activities (Steiner et al. 2017; Grebitus et al. 2016; Grunert et al. 2014). Some of these recent works have been included in the present analyses, however, only the results regarding consumers' preference for CF labels are reported, in line with the objectives of the study.
This literature review is an update until early 2018 of the review performed by Canavari and Bazzani (2016), which covers articles published until 2014. For more details on methodological aspects related to some of the cited papers, the reader can refer to the aforementioned work.
We acknowledge that the criteria used for the selection of the papers might have caused the exclusion of some important works on the topic analysed. Thus, the literature review has also considered papers that were cited by the ones selected, even if they did not contain the chosen keywords, to allow a more comprehensive analysis of the phenomenon.
As regards to the goodness-of-fit of the model, the value of the Pseudo R2 is typical of fairly fitting models (McFadden 1979: 307).
Looking at the discrimination ability of the model (i.e. the capacity of correctly distinguishing between positive and negative replies), the area under the receiver operating characteristic (ROC) curve is reported. This value gives the probability that the model correctly ranks a randomly chosen pairs of observations. In the model, the area under the ROC curve is 0.735 which is an acceptable value as this figure should be higher than 0.5 (but lower than 1) to indicate a satisfactorily fitting model.
For example, because the product certification procedures allow highlighting hot spot in energy consumption or emission intensive packaging that can be reduced.
Akaichi F, de Grauw S, Darmon P, Revoredo-Giha C (2016) Does fair trade compete with carbon footprint and organic attributes in the eyes of consumers? Results from a Pilot Study in Scotland, the Netherlands and France. J Agric Environ Ethics 29(6):969–984. https://doi.org/10.1007/s10806-016-9642-7
Akaichi F, Nayga RM, Gil JM. (2013) Do consumers make tradeoffs with respect to GHG emissions, local, and food miles attributes? Evidence from Experimental Auctions of US Rice. INRA (Ed.), 1-28. Paris: INRA.
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Canavari M, Coderoni S, Giuliodori L, Visi E (2018) Consumer stated preferences for environmental labels: two case studies in the dairy sector, Proceedings of the 54th SIDEA Conference-25th SIEA Conference Cooperative Strategies and value creation in sustainable food supply chain, Bisceglie/Trani, September 13-16 2017, ISBN 9788891786883, FrancoAngeli Edizioni, Milan.
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Baldoni E, Coderoni S, Esposti R (2018) The complex farm-level relationship between environmental performance and productivity. The Case of Carbon Footprint of Lombardy farms. Environ Sci Policy 89C:73–82
Bazzani C, Canavari M (2013) Alternative agri-food networks and short food supply chains: a review of the literature. Economia Agro-Alimentare 15(2):11–34. https://doi.org/10.3280/ECAG2013-002002
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Candy S, Turner GM, Sheridan J, Carey R (2018) Quantifying Melbourne's "Foodprint": a scenario modelling methodology to determine the environmental impact of feeding a city. Economia Agro-Alimentare / Food Economy 20(3):371–399. doi: https://doi.org/10.3280/ECAG2018-003007
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Caputo V, Nayga RM, Scarpa R (2013a) Food miles or carbon emissions? Exploring labelling preference for food transport footprint with a stated choice study. Austr J Agric Resour Econ 57(4):465–482. https://doi.org/10.1111/1467-8489.12014
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The authors would like to thank Elena Visi and Elena Visi for providing the data they collected for their bachelor's thesis. This paper has been selected as one of the best papers of the First joint SIDEA-SIEA Conference in Bisceglie/Trani (13-16 September 2017). It has been accepted for publication in this journal following the usual review process.
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Department of Agricultural and Food Sciences, Alma Mater Studiorum-Università di Bologna, viale Giuseppe Fanin, 50, Bologna, 40127, Italy
Maurizio Canavari
Department of Economics and Social Sciences, Università Politecnica delle Marche, Piazzale Martelli, 8, 60121, Ancona, Italy
Silvia Coderoni
Authors are listed in alphabetical order. This paper was developed jointly by the authors; nevertheless, the individual contribution may be identified as follows: Section 1, 2 and 3 to MC; Section 4, 5 and 6 to SC. Both the authors have approved the manuscript for submission.
Correspondence to Silvia Coderoni.
Canavari, M., Coderoni, S. Consumer stated preferences for dairy products with carbon footprint labels in Italy. Agric Econ 8, 4 (2020). https://doi.org/10.1186/s40100-019-0149-1
Carbon footprint label
Environmental labels
Consumer preferences
JEL codes | CommonCrawl |
\begin{definition}[Definition:Jacobi's Equation of Functional/Dependent on N Functions]
Let:
:$\ds \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
be a (real) functional, where $\map {\mathbf y} a = A$ and $\map {\mathbf y} b = B$.
Let:
:$\ds \int_a^b \paren {\mathbf h' \mathbf P \mathbf h' + \mathbf h \mathbf Q \mathbf h} \rd x$
be a quadratic functional, where:
:$P_{ij} = \dfrac 1 2 F_{y_i'y_j'}$
:$Q_{ij} = \dfrac 1 2 \paren {F_{y_i y_j} - \dfrac \d {\d x} F_{y_i y_j'} }$
Then the Euler's equation of the latter functional:
:$-\map {\dfrac \d {\d x} } {\mathbf P \mathbf h'} + \mathbf Q \mathbf h = \mathbf 0$
is called '''Jacobi's Equation''' of the former functional.
{{NamedforDef|Carl Gustav Jacob Jacobi|cat = Jacobi}}
\end{definition} | ProofWiki |
6-simplex honeycomb
In six-dimensional Euclidean geometry, the 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets. These facet types occur in proportions of 1:1:1 respectively in the whole honeycomb.
6-simplex honeycomb
(No image)
TypeUniform 6-honeycomb
FamilySimplectic honeycomb
Schläfli symbol{3[7]}
Coxeter diagram
6-face types{35} , t1{35}
t2{35}
5-face types{34} , t1{34}
t2{34}
4-face types{33} , t1{33}
Cell types{3,3} , t1{3,3}
Face types{3}
Vertex figuret0,5{35}
Symmetry${\tilde {A}}_{6}$×2, [[3[7]]]
Propertiesvertex-transitive
A6 lattice
This vertex arrangement is called the A6 lattice or 6-simplex lattice. The 42 vertices of the expanded 6-simplex vertex figure represent the 42 roots of the ${\tilde {A}}_{6}$ Coxeter group.[1] It is the 6-dimensional case of a simplectic honeycomb. Around each vertex figure are 126 facets: 7+7 6-simplex, 21+21 rectified 6-simplex, 35+35 birectified 6-simplex, with the count distribution from the 8th row of Pascal's triangle.
The A*
6
lattice (also called A7
6
) is the union of seven A6 lattices, and has the vertex arrangement of the dual to the omnitruncated 6-simplex honeycomb, and therefore the Voronoi cell of this lattice is the omnitruncated 6-simplex.
∪ ∪ ∪ ∪ ∪ ∪ = dual of
Related polytopes and honeycombs
This honeycomb is one of 17 unique uniform honeycombs[2] constructed by the ${\tilde {A}}_{6}$ Coxeter group, grouped by their extended symmetry of the Coxeter–Dynkin diagrams:
A6 honeycombs
Heptagon
symmetry
Extended
symmetry
Extended
diagram
Extended
group
Honeycombs
a1 [3[7]] ${\tilde {A}}_{6}$
i2 [[3[7]]] ${\tilde {A}}_{6}$×2
1
2
r14 [7[3[7]]] ${\tilde {A}}_{6}$×14
3
Projection by folding
The 6-simplex honeycomb can be projected into the 3-dimensional cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
${\tilde {A}}_{6}$
${\tilde {C}}_{3}$
See also
Regular and uniform honeycombs in 6-space:
• 6-cubic honeycomb
• 6-demicubic honeycomb
• Truncated 6-simplex honeycomb
• Omnitruncated 6-simplex honeycomb
• 222 honeycomb
Notes
1. "The Lattice A6". Archived from the original on 2012-01-19. Retrieved 2011-05-11.
• Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence A000029 18-1 cases, skipping one with zero marks
References
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings)
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
| Wikipedia |
Nonlinear Dynamics (journal)
Nonlinear Dynamics, An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems is a monthly peer-reviewed scientific journal covering all nonlinear dynamic phenomena associated with mechanical, structural, civil, aeronautical, ocean, electrical, and control systems. It is published by Springer Nature and the editor-in-chief of the journal is Walter Lacarbonara (Sapienza University of Rome).
Nonlinear Dynamics
Cover of Volume 84, Number 3 from May 2016
DisciplineChaos theory, engineering
LanguageEnglish
Edited byWalter Lacarbonara
Publication details
History1990-present
Publisher
Springer Nature
FrequencyMonthly
Impact factor
5.741 (2021)
Standard abbreviations
ISO 4 (alt) · Bluebook (alt1 · alt2)
NLM (alt) · MathSciNet (alt )
ISO 4Nonlinear Dyn.
Indexing
CODEN (alt · alt2) · JSTOR (alt) · LCCN (alt)
MIAR · NLM (alt) · Scopus
CODENNODYES
ISSN0924-090X (print)
1573-269X (web)
LCCN94660702
OCLC no.634426729
Links
• Journal homepage
• Online archive
It should not be confused with the similarly named Russian journal Nelineinaya Dinamika (or the Russian Journal of Nonlinear Dynamics).
Abstracting and indexing
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\begin{document}
\begin{frontmatter}
\title{A complete convergence theorem for voter model perturbations} \runtitle{Complete convergence theorem}\vspace*{6pt}
\begin{aug} \author[A]{\fnms{J. Theodore} \snm{Cox}\corref{}\thanksref{t1}\ead[label=e1]{[email protected]}} \and \author[B]{\fnms{Edwin A.} \snm{Perkins}\thanksref{t2}\ead[label=e3]{[email protected]}} \thankstext{t1}{Supported in part by grants from the NSA and NSF.} \thankstext{t2}{Supported in part by an NSERC Discovery grant.}\vspace*{3pt} \runauthor{J.~T. Cox and E. A. Perkins} \affiliation{Syracuse University and University of British Columbia}\vspace*{3pt} \address[A]{Department of Mathematics\\ Syracuse University\\ Syracuse, New York 13244\\ USA\\ \printead{e1}}
\address[B]{Department of Mathematics\\ University of British Columbia\\ 1984 Mathematics Road\\ Vancouver, British Columbia V6T 1Z2\\ Canada\\ \printead{e3}} \end{aug}
\received{\smonth{8} \syear{2012}} \revised{\smonth{12} \syear{2012}}
\begin{abstract} We prove a complete convergence theorem for a class of symmetric voter model perturbations with annihilating duals. A special case of interest covered by our results is the stochastic spatial Lotka--Volterra model introduced by Neuhauser and Pacala [\textit{Ann. Appl. Probab.} \textbf {9} (1999) 1226--1259]. We also treat two additional models, the ``affine'' and ``geometric'' voter models. \end{abstract}
\begin{keyword}[class=AMS] \kwd[Primary ]{60K35} \kwd{82C22} \kwd[; secondary ]{60F99} \end{keyword}
\begin{keyword} \kwd{Complete convergence theorem} \kwd{Lotka--Volterra} \kwd{interacting particle system} \kwd{voter model perturbation} \kwd{annihilating dual} \end{keyword}
\vspace*{6pt} \end{frontmatter}
\section{Introduction}\label{secintro} In our earlier study of voter model perturbations \mbox{\cite{CDP11,CMP,CP05,CP07,CP08}} we found conditions for survival, extinction and coexistence for these interacting particle systems. Our goal here is to show that under additional conditions it is possible to determine all stationary distributions and their domains of attraction. We start by introducing the primary example of this work, a competition model from~\cite{NP}.
The state of the system at time $t$ is represented by a spin-flip process $\xi_t$ taking values in $\{0,1\}^{{\mathbb{Z}}^d}$. The dynamics will in part be determined by a fixed probability kernel $p\dvtx{\mathbb{Z}}^d\to[0,1]$. We assume throughout that
\begin{equation} \label{passump} \begin{tabular}{p{280pt}@{}}
$p(0)=0, p(x)$ is symmetric, irreducible, and has covariance matrix $\sigma^2$I for some $\sigma^2\in(0, \infty).$ \end{tabular}
\end{equation}
For most of our results we will need to assume that $p(x)$ has exponential tails, that is,
\begin{equation}
\label{exptail} \exists\kappa>0, C<\infty\mbox{ such that } p(x) \le Ce^{-\kappa|x|}\ \forall x\in{\mathbb{Z}^d}. \end{equation}
\eject\noindent
Here $|(x_1,\ldots,x_d)|=\max_i|x_i|$. We define the \emph{local density} $f_i=f_i(x,\xi)$ of type $i$ near $x\in{\mathbb{Z}}^d$ by
\[ f_i(x,\xi)=\sum_{y\in{\mathbb{Z}^d}}p(y-x)1 \bigl\{ \xi(y)=i \bigr\},\qquad i=0,1. \]
Given $p(x)$ satisfying \eqref{passump} and nonnegative parameters $(\alpha_0,\alpha_1)$, the sto\-chastic Lotka--Volterra model of \cite{NP}, $\mathrm{LV}(\alpha_0,\alpha_1)$, is the spin-flip process $\xi_t$ with rate function $c_{\mathrm{LV}}(x,\xi)$ given by
\begin{equation} \label{LVrates} c_{\mathrm{LV}}(x,\xi) = \cases{ f_1(x,\xi) \bigl(f_0(x,\xi) + \alpha_0f_1(x,\xi) \bigr),&\quad $\mbox{if }\xi(x)=0$,\vspace*{2pt} \cr f_0(x,\xi) \bigl(f_1(x,\xi) + \alpha_1f_0(x,\xi) \bigr),&\quad $ \mbox{if }\xi(x)=1.$} \end{equation}
All the spin-flip rate functions we will consider, including $c_{\mathrm{LV}}$, will satisfy the hypothesis of Theorem B.3 in \cite{Lig99}. By that result, for such a rate function $c(x,\xi)$, there is a unique $\{0,1\}^{{\mathbb{Z}}^d}$-valued Feller process $\xi_t$ with generator equal to the closure of $\Omega f(\xi)=\sum_{x\in{\mathbb{Z}}^d}c(x,\xi)(f(\xi^x)-f(\xi))$ on the space of functions $f$ depending on finitely many coordinates of $\xi$. Here $\xi^x$ is $\xi$ but with the coordinate at~$x$ flipped.
One goal of \cite{NP} was to establish coexistence for $\mathrm{LV}(\alpha_0,\alpha_1)$ for some $\alpha_i$. If we let
$|\xi|=\sum_{x\in{\mathbb{Z}^d}}\xi(x)$ and $\hat\xi(x)=1-\xi (x)$ for all $x\in{\mathbb{Z}^d}$, then coexistence for a spin-flip process $\xi_t$ means that there is a stationary distribution $\mu$ for $\xi_t$ such that
\begin{equation}
\label{coexist} \mu\bigl(| \xi| = |\widehat\xi|=\infty\bigr) = 1. \end{equation}
In \cite{NP}, coexistence was proved for
\begin{equation} \label{diag} \alpha=\alpha_0=\alpha_1\in[0,1) \end{equation}
close enough to 0 and $p(x)=1_{{\mathcal{N}}}(x)/|{\mathcal{N}}|$, where
\begin{equation}
\label{NPN} {\mathcal{N}}=\bigl\{x\in{\mathbb{Z}^d}\dvtx0<|x|\le L\bigr\}, \qquad L \ge1, \end{equation}
excluding only the case $d=L=1$.
A special case of $\mathrm{LV}(\alpha_0,\alpha_1)$ is the voter model. If we set $\alpha_0=\alpha_1=1$ and use $f_0+f_1=1$, then $c_{\mathrm{LV}}(x,\xi)$ reduces to the rate function of the voter model,
\begin{equation} \label{VMrates} c_{\mathrm{VM}}(x,\xi) = \bigl(1-\xi(x) \bigr)f_1(x, \xi) + \xi(x) f_0(x,\xi). \end{equation}
It is well known (see Chapter~V of \cite{Lig}, Theorems~V.1.8 and V.1.9 in particular) that coexistence for the voter model is dimension dependent. Let $\mathbf{0}$ (resp., $\mathbf{1}$) be the element of $\{0,1\}^{\mathbb{Z}^d}$ which is identically 0 (resp., 1), and let $\delta_\mathbf{0}$, $\delta_\mathbf{1}$ be the corresponding unit point masses. If $d\le2$, then there are exactly two extremal stationary distributions, $\delta_\mathbf{0}$ and $\delta_\mathbf{1}$, and hence no coexistence. If $d\ge3$, then there is a one-parameter family $\{P_u,u\in[0,1]\}$ of translation invariant extremal stationary distributions, where $P_u$ has density $u$, that is, $P_u(\xi(x)=1)=u$. For $u\ne0,1$, each $P_u$ satisfies \eqref{coexist}, so there is coexistence.
Returning to the general Lotka--Volterra model, coexistence for $\mathrm{LV}(\alpha_0,\alpha_1)$ for certain $(\alpha_0,\alpha_1)$ near $(1,1)$ (including $\alpha_0=\alpha_1<1$, $1-\alpha_i$ small enough) was obtained in \cite{CP07} for $d\ge3$ and in \cite{CMP} for $d=2$. The methods used in this work require symmetry in the dynamics between $0$'s and $1$'s, that is, condition \eqref{diag}. Under this assumption, Theorem~4 of \cite{CP07} and Theorem~1.2 of \cite{CMP} reduce to the following, with $\mathrm{LV}(\alpha)$ denoting the Lotka--Volterra model when \eqref{diag} holds.
\renewcommand{\Alph{theorem}}{\Alph{theorem}} \begin{theorem}\label{thmA} Assume $d\ge2$ and \eqref{diag} holds. If $d=2$, assume also that
$\sum_{x\in{\mathbb{Z}}^2}|x|^3p(x)<\infty$. Then there exists $\alpha_c=\alpha_c(d)<1$ such that coexistence holds for $\mathrm{LV}(\alpha)$ for $\alpha\in(\alpha_c,1)$. \end{theorem}
Given coexistence, one would like to know if there is more than one stationary distribution satisfying the coexistence condition \eqref{coexist}, if so, what are all stationary distributions and from what initial states is there weak convergence to a given stationary distribution. To state our answers to these questions for $\mathrm{LV}(\alpha)$ we need some additional notation. Define the hitting times
\[ \tau_\mathbf{0}=\inf\{t\ge0\dvtx\xi_t=\mathbf{0}\},\qquad \tau_\mathbf{1}=\inf\{ t\ge0\dvtx\xi_t=\mathbf{1}\}, \]
and the probabilities, for $\xi\in\{0,1\}^{\mathbb{Z}^d}$,
\begin{eqnarray*} \beta_0(\xi)& =& P_\xi(\tau_{\mathbf{0}}<\infty),\qquad \beta_1(\xi) =P_\xi(\tau_{\mathbf{1}}<\infty),\\ \beta_\infty(\xi) &=& P_\xi(\tau_{\mathbf{0}}= \tau_{\mathbf{1}}=\infty), \end{eqnarray*}
where $P_\xi$ is the law of our process starting at $\xi$. The point masses $\delta_\mathbf{0},\delta_\mathbf{1}$ are clearly stationary distributions for $\mathrm{LV}(\alpha)$. We write $\xi_t\Rightarrow\mu$ to mean that the law of $\xi_t$ converges weakly to the probability measure $\mu$. A law $\mu$ on $\{0,1\}^{{\mathbb{Z}}^d}$ is symmetric if and only if $\mu(\xi\in\cdot)=\mu(\hat\xi\in \cdot)$.
We note here that for any translation invariant spin-flip system $\xi_t$ satisfying the hypothesis (B4) of Theorem~B.3 in \cite{Lig99},
\begin{equation}
\label{betanonzero} \beta_0(\xi)=0\qquad\mbox{if }|\xi|=\infty\quad\mbox{and}\quad \beta_1(\xi)=0\qquad\mbox{if } |\hat\xi|=\infty. \end{equation}
To see this for $\beta_0$, assume $\xi_0$ satisfies $|\xi_0|=\infty$. By assumption, there is a uniform maximum flip rate $M$ at all sites in all configurations. So for $\xi_0$ and $x$ such that $\xi_0(x)=1$,
$P(\xi_t(x)=1)\ge e^{-Mt}$. Since $|\xi_0|=\infty$, we may choose $A_n\subset{\mathbb{Z}^d}$ satisfying
$|A_n|= n$, $\min\{|x-y|\dvtx x,y\in A_n, x\ne y\}\to\infty$ as $n\to\infty$, and $\xi_0(x)=1\ \forall x\in A_n$. Our hypotheses and translation invariance allow us to apply Theorem~I.4.6 of \cite{Lig} and conclude that for any fixed $t>0$, $E (\prod_{x\in A_n}\hat\xi_t(x) ) -\prod_{x\in A_n}E(\hat\xi_t(x))\to0$.\vadjust{\goodbreak} It follows that for any $n$, there are $\{\varepsilon_n\}$ approaching $0$ so that
\begin{eqnarray*} P \bigl(\xi_t(x)=0\ \forall x\in\xi_0 \bigr) &\le& P \bigl(\xi_t(x)=0\ \forall x\in A_n \bigr) \\ &\le&\varepsilon_n+\prod_{x\in A_n}P \bigl( \xi_t(x)=0 \bigr) \\ &\le&\varepsilon_n+ \bigl(1-e^{-Mt} \bigr)^n \rightarrow0\qquad \mbox{as }n\to\infty. \end{eqnarray*}
Recall (see Corollary~V.1.13 of \cite{Lig}) that for the voter model itself and $\xi_0$ translation invariant with $P(\xi_0(x)=1)=u$, we have $\xi_t\Rightarrow u\delta_\mathbf{1}+(1-u)\delta_\mathbf{0}$ if $d\le2$, and $\xi _t\Rightarrow P_u$ if $d\ge3$ and $\xi_0$ is ergodic.
\begin{thmm}\label{thmLVCCT} Assume $d\ge2$, and \eqref{exptail}. There exists $\alpha_c<1$ such that for all $\alpha\in(\alpha_c,1)$, $\mathrm{LV}(\alpha)$ has a unique translation invariant symmetric stationary distribution $\nu_{1/2}$ satisfying the coexistence property \eqref{coexist}, such that for all $\xi_0\in \{0,1\}^{{\mathbb{Z}^d}}$,
\begin{equation} \label{eqLVCCT} \xi_t \Rightarrow\beta_0(\xi_0) \delta_\mathbf{0}+ \beta_\infty(\xi_0) \nu_{1/2} +\beta_1(\xi_0)\delta_{\mathbf{1}}\qquad \mbox{as } t\to\infty. \end{equation}
\end{thmm}
Theorem~\ref{thmLVCCT} is a \emph{complete convergence theorem}, it gives complete answers to the questions raised above. The first theorem of this type for infinite particle systems was proved for the contact process in \cite{G78}, where $\beta_1(\xi)=0$ for $\xi\ne \mathbf{1}$ and $\delta_\mathbf{1}$ is not a stationary distribution. Our result is closely akin to the complete convergence theorem proved in \cite{H99} for the threshold voter model. (Indeed, we make use of a number of ideas from \cite{H99}.) A more recent example is Theorem 4 in \cite{SS} for the $d=1$ ``rebellious voter model.'' For
$p(x)=1_{{\mathcal{N}}}(x)/|{\mathcal{N}}|$, ${\mathcal{N}}$ as in \eqref{NPN}, the existence and uniqueness of $\nu_{1/2}$ in the above context follows from results in~\cite{SS} and Theorem \ref{thmA}. The relationship between Theorem~\ref{thmLVCCT} and results in \cite{SS} is discussed further in Remarks~\ref{LVAV} and \ref{SScomp} below.
For $\mathrm{LV}(\alpha)$, we note that if $0<|\xi_0|<\infty$, then $0<\beta_0(\xi_0)<1$, where the upper bound is valid for $\alpha$ close enough to $1$, and if $|\xi_0|=\infty$, then $\beta_0(\xi_0)=0$. By the symmetry condition \eqref{diag}, this implies that the obvious symmetric statements with $(\widehat\xi_0,\beta_1)$ in place of $(\xi_0,\beta_0)$ also hold by \eqref{diag}. To see the above, note first that
$|\xi_0|<\infty$ trivially implies $\beta_0>0$ since one can prescribe a finite sequence of flips that leads to the trap $\mathbf{0}$. The fact that $\beta_0<1$ for $\alpha<1$ close enough to $1$ follows from the survival results in \cite{CP07} for $d\ge3$ (see Theorem~1 there), and in
\cite{CMP} for $d=2$ (see Theorem~1.4 there). Finally, $\beta_0(\xi _0)=0$ if $|\xi_0|=\infty$ holds by \eqref{betanonzero}.
As our earlier comments on the ergodic theory of the voter model show, the situation is quite different for $\alpha=1$ as \eqref{eqLVCCT} does not hold. Moreover, by constructing blocks of alternating $0$'s and $1$'s on larger and larger annuli one can construct an initial $\xi_0\in\{0,1\}^{\mathbb{Z}^d}$ for which the law of $\xi_t$ does not converge as $t\to\infty$. This suggests that the above theorem is rather delicate. Nonetheless we make the following conjecture:
\begin{conj} For $\alpha_i<1$, close enough to $1$ and with $\alpha =(\alpha_0,\alpha_1)$ in the coexistence region of Theorem~1.10 of \cite {CDP11}, the complete convergence theorem holds with a unique nontrivial stationary distribution $\nu_\alpha$ in place of~$\nu_{1/2}$. \end{conj}
If $\alpha$ approaches $(1,1)$ so that $\frac{1-\alpha_1}{1-\alpha_0}\to m$, then by Theorem~1.10 of \cite{CDP11}, the limiting particle density of $\nu_\alpha$ must approach $u^*(m)$, where $u^*$ is as in (1.50) of~\cite{CDP11}. Hence one obtains the one-parameter family of invariant laws for the voter model in the limit along different slopes approaching $(1,1)$.
The $d\ge3$ case of Theorem~\ref{thmLVCCT} is a special case of a general result for certain \emph{voter model perturbations}. We will define this class following the formulation in~\cite{CDP11} (instead of \cite{CP05}), and then give the additional required definitions needed for our general result. A \emph{voter model perturbation} is a family of spin-flip systems $\xi^\varepsilon_t$, $0<\varepsilon\le\varepsilon_0$ for some $\varepsilon_0>0$, with rate functions
\begin{equation} \label{vmpert} c_\varepsilon(x,\xi) = c_{\mathrm{VM}}(x,\xi) + \varepsilon^2 c^*_\varepsilon(x,\xi) \ge0,\qquad x\in{\mathbb{Z}}^d, \xi\in \{0,1\}^{{\mathbb{Z}}^d}, \end{equation}
where $c^*_\varepsilon(x,\xi)$ is a translation invariant, signed perturbation of the form
\begin{equation} \label{vmpert2} c^*_\varepsilon(x,\xi)= \bigl(1-\xi(x) \bigr)h_1^\varepsilon(x, \xi) + \xi(x)h_0^\varepsilon(x,\xi). \end{equation}
Here we assume \eqref{passump} and \eqref{exptail} hold, and for some finite $N_0$ there is a law $q_Z$ of $(Z^1,\ldots,Z^{N_0})\in{\mathbb{Z}}^{dN_0}$, functions $g^\varepsilon_i$ on $\{0,1\}^{N_0}$, $i=0,1$ and $\varepsilon_1\in(0,\infty]$ so that $g_i^\varepsilon\ge0$, and for $i=0,1$, $\xi\in\{0,1\}^{{\mathbb{Z}}^d}$, $x\in{\mathbb{Z}}^d$ and $\varepsilon\in (0,\varepsilon_1)$,
\begin{equation} \label{hgrepn}h_i^\varepsilon(x,\xi) = -\varepsilon_1^{-2}f_i(x, \xi)+E_Z \bigl(g_i^\varepsilon\bigl(\xi\bigl(x+ Z^1 \bigr), \ldots,\xi\bigl(x+ Z^{N_0} \bigr) \bigr) \bigr). \end{equation}
Here $E_Z$ is expectation with respect to $q_Z$. We also suppose that (decrease $\kappa>0$ if necessary)
\begin{equation} \label{expbd2} P \bigl( Z^*\ge x \bigr) \le C e^{-\kappa x}\qquad\mbox{for }x>0, \end{equation}
where $Z^* = \max\{ |Z^1|, \ldots,|Z^{N_0}| \}$, and there are limiting maps $g_i\dvtx\{0,1\}^{N_0}\to{\mathbb{R}}_+$ such that for some $c_g,r_0>0$,
\begin{equation} \label{gcvgce} \bigl\Vert g_i^\varepsilon-g_i \bigr\Vert_\infty\le c_g\varepsilon^{r_0},\qquad i=0,1. \end{equation}
In addition, we will always assume that for $0<\varepsilon\le\varepsilon_0$,
\begin{equation} \label{traps} \mathbf{0}\mbox{ is a trap for }\xi^\varepsilon_t, \mbox{ that is, }c_\varepsilon(x,\mathbf{0})=0. \end{equation}
In adding \eqref{gcvgce} and \eqref{traps} to the definition of \textit{voter model perturbation} we have taken some liberty with the definition in \cite{CDP11}, but these conditions do appear later in that work for all the results to hold.
It is easy to check that $\mathrm{LV}(\alpha_0,\alpha_1)$ is a voter model perturbation, as is done in Section~1.3 of~\cite{CDP11}. We will just note here that if $\alpha_i=\alpha_i^\varepsilon=1+\varepsilon^2\theta_i $, $\theta_i\in{\mathbb{R}}$ and $h^\varepsilon_i(x,\xi)=\theta_{1-i}f_i(x,\xi)^2$, $i=0,1$, then $c_{\mathrm{LV}}(x,\xi)$ has the form given in \eqref{vmpert} and~\eqref{vmpert2}. Additional examples of voter model perturbations are given in Section~1 of \cite{CDP11}.\vadjust{\goodbreak} In fact, many interesting models from the life sciences and social sciences reduce to the voter model for a specific choice of parameters, and thus in many cases can be viewed as voter model perturbations.
Coexistence results for voter model perturbations are given in \cite{CDP11} and \cite{CP07} for $d\ge3$ (and for the two-dimensional Lotka--Volterra model in \cite{CMP}). Here we will additionally require that our voter model perturbations be \emph{cancellative processes}, which we now define following Section~III.4 of \cite{Lig}; see also Chapter~III of \cite{G79}. Let $Y$ be the collection of finite subsets of ${\mathbb{Z}^d}$ and for $x\in{\mathbb{Z}^d}$, $\xi\in\{0,1\} ^{\mathbb{Z}^d}$ and $A\in Y$, let $H(\xi,A)=\prod_{a\in A}(2\xi(a)-1)$ (an empty product is 1). We will call a translation invariant flip rate function $c(x,\xi)$ (not necessarily a voter model perturbation) cancellative if there is a positive constant $k_0$ and a map $q_0\dvtx Y\to[0,1]$ such that
\begin{equation} \label{q00} c(x,\xi) = \frac{k_0}{2} \biggl( 1- \bigl(2\xi(x)-1 \bigr) \sum _{A\in Y}q_0(A-x)H(\xi,A) \biggr), \end{equation}
where $A-x=\{a-x\dvtx a\in A\}$, $q_0(\varnothing)=0$,
\begin{eqnarray} \sum_{A\in Y} q_0(A)&=& 1\quad \mbox{and } \label{q01} \\
\sum_{A\in Y} |A|q_0(A)&<&\infty. \label{q02} \end{eqnarray}
This is a subclass of the corresponding processes defined in \cite{Lig}. It follows from~\eqref{q01} that $c(x,\mathbf{1})=0$ and so $\mathbf{1}$ is a trap for $\xi$. The above rate will satisfy the hypothesis of Theorem B.3 in \cite{Lig99} and so, as discussed above, determines a unique $\{0,1\}^{{\mathbb{Z}}^d}$-valued Feller process; see the discussion in Section III.4 of \cite{Lig} leading to (4.8) there. (One can also check easily that the same is true of our voter model perturbations but at times we will only assume the above cancellative property.)
Given $c(x,\xi), k_0,q_0$ as above, we can define a continuous time Markov chain taking values in $Y$ by the following. For $F,G\in Y$, $F\neq G$, define
\begin{equation} \label{qmatrix} Q(F,G) = k_0\sum_{x\in F} \sum_{A\in Y} q_0(A-x) 1 \bigl\{ \bigl(F \setminus\{x\} \bigr)\Delta A=G \bigr\}, \end{equation}
where $\Delta$ is the symmetric difference operator. As noted in \cite{Lig}, $Q$ is the $Q$-matrix of a nonexplosive Markov chain $\zeta_t$ taking values in $Y$; see also \cite{G79}. If we think of~$\zeta_t$ as the set of sites occupied by a system of particles at time $t$, then the interpretation of \eqref{qmatrix} is this. If the current state of the chain is $F$, then at rate $k_0$ for each $x\in F$:
\begin{longlist}[(1)]
\item[(1)] $x$ is removed from $F$, and
\item[(2)] with probability $q_0(A-x)$, particles are sent from $x$ to $A$, with the proviso that a particle landing on an occupied site $y$ annihilates itself and the particle at $y$.\vadjust{\goodbreak} \end{longlist}
Perhaps the simplest example of a cancellative/annihilative pair $(\xi_t,\zeta_t$) is the voter model and its dual annihilating random walk system. Here $c_{\mathrm{VM}}(x,\xi)$ satisfies~\eqref{q00}
with $k_0=1$, $q_0(\{y\})=p(y)$, $q_0(A)=0$ if $|A|>1$; again, see \cite {G79} and \cite{Lig}. A second example, as shown in \cite{NP}, is the Lotka--Volterra process, assuming
\eqref{diag} and $p(x)=1_{\mathcal{N}}(x)/|{\mathcal{N}}|$, ${\mathcal{N}}$ satisfies \eqref{NPN} [this will be extended to our general $p(\cdot)$'s in Section~\ref{secalmostlast}].
The Markov chain $\zeta_t$ is the \emph{annihilating dual} of $\xi_t$. The general duality equation of Theorem~III.4.13 of \cite{Lig} (see also Theorem~III.1.5 of \cite{G79}) and \cite{Lig}, simplifies in the current setting to the following \textit{annihilating duality} equation:
\begin{equation} \label{eqduality0} E \bigl(H(\xi_t,\zeta_0) \bigr) = E \bigl(H(\xi_0,\zeta_t) \bigr)\qquad \forall \xi_0\in\{0,1\}^{\mathbb {Z}^d}, \zeta_0\in Y. \end{equation}
In Section~\ref{secH} we will recall from \cite{G79} and \cite{Lig} several implications of this duality equation for the ergodic theory of $\xi_t$.
Let $Y_e$ (resp., $Y_o$) denote the set of
$A\in Y$ with $|A|$ even (resp., odd). We call $\zeta_t$ (or $Q$) \emph{parity preserving} if
\begin{equation} \label{parity} Q(F,G)=0 \qquad\mbox{unless }F,G\in Y_e \mbox{ or } F,G\in Y_o. \end{equation}
Clearly $\zeta_t$ is parity preserving if and only if $q_0(A)=0$ for all $A\in Y_e$. If $\zeta_t$ is parity preserving we will call $\zeta_t$ \emph{irreducible} if $\zeta_t$ is irreducible on $Y_o$ and also on $Y_e\setminus\{\varnothing\}$, and $Q(A,\varnothing)>0$ for some $A\ne\varnothing$.
One fact we need now is Corollary~{III.1.8} of \cite{G79}. Let $\mu_{1/2}$ be Bernoulli product measure on $\{0,1\}^{\mathbb{Z}^d}$ with density $1/2$. Then under \eqref{traps} there is a translation invariant distribution $\nu_{1/2}$ with density $1/2$ such that
\begin{equation} \label{nuhalf1} \mbox{if the law of $\xi_0$ is $ \mu_{1/2}$ then }\xi_t\Rightarrow\nu_{1/2} \mbox{ as }t \to\infty; \end{equation}
see \eqref{zetaconv} below for a proof. For a cancellative process, $\nu_{1/2}$ will always denote this measure. We note that $\nu_{1/2}$ might be $\frac12 (\delta_\mathbf{0}+\delta_\mathbf{1})$ and hence not have the coexistence property \eqref{coexist}.
Theorem~1.15 of \cite{CDP11} gives conditions which guarantee coexistence for $\xi^\varepsilon_t$ for small positive $\varepsilon$. One assumption of that result, which we will need here, requires a function $f$ defined in terms of the voter model equilibria $P_u$ previously introduced. For bounded functions $g$ on $\{0,1\}^{\mathbb{Z}^d}$ write $\langle g\rangle _u=\int g(\xi)\,dP_u(\xi)$, and note that $\langle g(\xi)\rangle_u =\langle g(\widehat\xi)\rangle_{1-u}$. As in \cite{CDP11}, define
\begin{equation} \label{f} f(u) = \bigl\langle\bigl(1-\xi(0) \bigr)c^*(0,\xi) - \xi (0)c^*(0,\xi) \bigr\rangle_u,\qquad u\in[0,1], \end{equation}
where $c^*$ is as in \eqref{vmpert2} but with $g_i$ in place of $g_i^\varepsilon$. As noted in Section 1 of \cite{CDP11}, $f$ is a polynomial of degree at most $N_0+1$, and is a cubic for $\mathrm{LV}(\alpha_0,\alpha_1)$.
We extend our earlier definitions of $\beta_i$ and $\tau_i$ to general spin-flip processes~$\xi$.
\begin{definition*}[(Complete convergence)] We say that \textit{the complete convergence theorem} holds for a given cancellative process $\xi_t$ if \eqref{eqLVCCT} holds for all initial states $\xi_0\in\{0,1\}^{{\mathbb{Z}}^d}$, where $\nu_{1/2}$ is given in \eqref{nuhalf1}, and that it holds \textit{with coexistence} if, in addition, $\nu _{1/2}$ satisfies \eqref{coexist}. \end{definition*}
\begin{thmm}\label{thmCCTpert} Assume $d\ge3$, $c_\varepsilon(x,\xi)$ is a voter model perturbation satisfying \eqref{exptail}, \eqref{q00}--\eqref{q02} and $f'(0)>0$.
Then there exists $\varepsilon_1>0$ such that if $0<\varepsilon<\varepsilon_1$ the complete convergence theorem with coexistence holds for $\xi^\varepsilon_t$. \end{thmm}
\begin{rem}\label{remexpvsattractive} As can be seen in our proof of Theorem~\ref{thmCCTpert}, it is possible to drop the exponential tail condition \eqref{exptail} if the voter model perturbations are attractive, as is the case for $\mathrm{LV}(\alpha)$; see, for example, (8.5) with $C_{8.3}=1$ in \cite{CP07} for the latter. To do this one uses the coexistence result in Section~6 of \cite{CP07} rather than that in Section~6 of \cite{CDP11}. In particular it follows that in Theorem~\ref{thmLVCCT} the complete convergence result holds for the Lotka--Volterra models considered there for $d\ge3$ without the exponential tail condition \eqref{exptail}. For $\mathrm{LV}(\alpha)$ with $d=2$ we will have to use coexistence results in \cite{CMP} to derive the complete convergence results, and instead of \eqref{exptail} these results only require
\[
\sum_{x\in{\mathbb{Z}}^2}|x|^3p(x)<\infty. \]
See Remark~\ref{noexptail} in Section~\ref{secalmostlast}.
\end{rem}
Theorem~1.3 of \cite{CDP11} states that if the ``initial rescaled approximate densities of~$1$'s'' approach a continuous function $v$ in a certain sense, then the rescaled approximate densities of $\xi_t$ converge to the unique solution of the reaction diffusion equation
\[ \frac{\partial u}{\partial t}=\frac{\sigma^2}{2}\Delta u+f(u),\qquad u_0=v. \]
Hence the condition $f'(0)>0$ means there is a positive drift for the local density of $1$'s when the density of 1's is very small and so by symmetry a negative drift when the density of 1's is close to $1$. In this way we see that this condition promotes coexistence. It also excludes voter models themselves for which the complete convergence theorem fails.
We present two additional applications of Theorem~\ref{thmCCTpert}.
\begin{example}[(Affine voter model)]\label{ex1} Suppose
\begin{equation} \label{N}\qquad {\mathcal{N}}\in Y\mbox{ is nonempty, symmetric and does not contain the origin.} \end{equation}
The corresponding \textit{threshold voter model} rate function, introduced in \cite{CD91}, is
\[ c_{\mathrm{TV}}(x,\xi) = 1 \bigl\{\xi(x+y)\ne\xi(x)\mbox{ for some } y\in{\mathcal{N}} \bigr\}. \]
See Chapter~II of \cite{Lig99} for a general treatment of threshold voter models, and \cite{H99} for a complete convergence theorem. The affine voter model with parameter $\alpha\in[0,1]$, $\mathrm{AV}(\alpha)$, is the spin-flip system with rate function
\begin{equation} \label{AVrates} c_{\mathrm{AV}}(x,\xi)=\alpha c_{\mathrm{VM}}(x,\xi)+(1- \alpha)c_{\mathrm{TV}}(x,\xi), \end{equation}
where $c_{\mathrm{VM}}$ is as in \eqref{VMrates}. This model is studied in \cite{SS} with voter kernel $p(x)=1_{{\mathcal{N}}
}(x)/|{\mathcal{N}}|$, as an example of a competition model where locally rare types have a competitive advantage. \end{example}
\begin{thmm}\label{thmtvm} Assume $d\ge3$, \eqref{exptail} holds and ${\mathcal{N}}$ satisfies \eqref{N}. There is an $\alpha_c\in(0,1)$ so that for all $\alpha\in(\alpha_c,1)$, the complete convergence theorem with coexistence holds for $\mathrm{AV}(\alpha)$. \end{thmm}
\begin{rem}\label{LVAV} It was shown in Theorem~3(a) of \cite{SS} that, excluding the case $d=1$ and ${\mathcal{N}}=\{-1,1\}$, if
$p(x)=1_{{\mathcal{N}}}(x)/|{\mathcal{N}}|$, ${\mathcal{N}}$ as in \eqref{NPN}, and coexistence holds for $\mathrm{LV}(\alpha)$, respectively, $\mathrm{AV}(\alpha)$, for a given $\alpha<1$, then there is a unique translation invariant stationary distribution $\nu_{1/2}$ satisfying \eqref{coexist}. Hence this is true for $\mathrm{LV}(\alpha)$ in $d\ge2$ for $\alpha<1$, and sufficiently close to $1$, by Theorem~\ref{thmA}, and for $\alpha$ sufficiently small by \cite {NP}. It is also true for $\mathrm{AV}(\alpha)$ for $\alpha=0$ by results in \cite{CD91} and \cite{Lig94}. The same result in \cite{SS} also shows that if, in addition, the dual satisfies a certain ``nonstability'' condition, then $\xi_t\Rightarrow\nu_{1/2}$ if the law of $\xi_0$ is translation invariant and satisfies \eqref{coexist}. The complete convergence results in Theorems~\ref {thmLVCCT} and \ref{thmtvm} above (which are special cases of Theorem~\ref{thmCCTpert} if $d\ge3$) assert a stronger and unconditional conclusion for both models for $\alpha$ near $1$. \end{rem}
\begin{example}[(Geometric voter model)]\label{ex2} Let ${\mathcal{N}}$ satisfy \eqref{N}. The geometric voter model with parameter $\theta\in[0,1]$, $\operatorname{GV}(\theta)$, is the spin-flip system with rate function
\begin{equation}
\label{geom1} c_{\mathrm{GV}}(x,\xi) = \frac{1-\theta^{j}}{1-\theta^{|{\mathcal{N}}
|}}\qquad \mbox{if } \sum _{y\in {\mathcal{N}}}1 \bigl\{\xi(x+y)\ne\xi(x) \bigr\}=j, \end{equation}
where the ratio is interpreted as $j/|{\mathcal{N}}|$ if $\theta=1$. This \emph{geometric} rate function was introduced in \cite{CD91}, where it was shown to be cancellative. As $\theta$ ranges from 0 to 1 these dynamics range from the threshold voter model to the voter model. It turns out that the geometric voter model is a voter model perturbation for $\theta$ near 1, and the following result is another consequence of Theorem~\ref{thmCCTpert}. \end{example}
\begin{thmm}\label{thmgeom} Assume $d\ge3$ and ${\mathcal{N}}$ satisfies \eqref{N}. There is a $\theta_c\in(0,1)$ so that for all $\theta\in(\theta_c,1)$, the complete convergence theorem with coexistence holds for $\operatorname{GV}(\theta)$. \end{thmm}
\begin{rem}[(Comparison with \cite{H99} and \cite{SS})]\label{SScomp} The emphasis in \cite{SS} was on the use of the annihilating dual to study the invariant laws and the long time behavior of cancellative systems. A general result (Theorem 6 of \cite{SS}) gave conditions on the dual to ensure the existence of a unique translation invariant stationary law $\nu_{1/2}$ which satisfies the coexistence property \eqref{coexist} and a stronger local nonsingularity property. It also gives stronger conditions on the dual under which $\xi_t\Rightarrow\nu_{1/2}$ providing the initial law is translation invariant and satisfies the above local nonsingularity condition. The general nature of these interesting results make them potentially useful in a variety of settings if the hypotheses can be verified.
In our work we focus on cancellative systems which are also voter perturbations. A non-annihilating dual particle system was constructed in \cite{CDP11} to analyze the latter, and it is by using both dual processes that we are able
to obtain a complete convergence theorem in Theorem~\ref{thmCCTpert} for small perturbations and $d\ge3$ ($d\ge2$ for LV in Theorem~\ref{thmLVCCT}).
Theorem~1.1 of \cite{H99} gives a complete convergence theorem for the threshold voter model, the spin-flip system with rate function $c_{\mathrm{TV}}$ given in Example~\ref{ex1} above, and a complete convergence result \textup{is} established in \cite{SS} for the one-dimensional ``rebellious voter model'' for a sufficiently small parameter value. In both of these works, one fundamental step is to show that the annihilating dual $\zeta_t$ grows when it survives, a result we will adapt for use here; see Lemma~\ref{lemH} and the discussion following Remark~\ref{remHaltcond} below. Both \cite{H99} and \cite{SS} then use special properties of the particle systems being studied to complete the proof. In Proposition~\ref{propCCT} below we give general conditions under which a cancellative spin-flip system will satisfy a complete convergence theorem with coexistence. We then verify the required conditions for the voter model perturbations arising in Theorems~\ref{thmCCTpert} and~\ref{thmLVCCT}. \end{rem}
We conclude this section with a ``flow chart'' of the proof of the main results, including an outline of the paper. First, the rather natural condition we impose that~$\mathbf{0}$ is a trap for our cancellative systems $\xi_t$ will imply that $\xi_t$ is in fact symmetric with respect to interchange of $0$'s and $1$'s; see Lemma~\ref {elemann} in Section~\ref{secH}. This helps explain the asymmetric \textit{looking} condition $f'(0)>0$ in Theorem~\ref{thmCCTpert} and the restriction of our results to $\mathrm{LV}$ with $\alpha_1=\alpha_2$. Section~\ref{secH} also reviews the ergodic theory of cancellative and annihilating systems.
As noted above, the core of our proof, Proposition~\ref{propCCT}, establishes a complete convergence theorem for cancellative particle systems (where $\mathbf{0}$ is a trap), assuming three conditions: (i) growth of the dual system when it survives, that is, \eqref{eqH}, (ii) a condition \eqref {flip2} ensuring a large number of $0$--$1$ pairs at locations separated by a fixed vector $x_0$ for large $t$ with high probability (ruling out clustering which clearly is an obstruction to any complete convergence theorem) and (iii) a condition \eqref{oddgoal} which says if the initial condition $\xi_0$ contains a large number of $0$--$1$ pairs with $1$'s in a set $A$, then at time $1$ the probability of an odd number of $1$'s in $A$ will be close to $1/2$. With these inputs, the proof of Propostion~\ref{propCCT} in Section~\ref{secflip} is a reasonably straightforward duality argument. This result requires no voter perturbation assumptions and may therefore have wider applicability.
We then verify the three conditions for voter model perturbations. The dual growth condition \eqref{eqH} is established in Lemma~\ref{lemH} and Remark~\ref{remHaltcond} in Section~\ref{secH}, assuming the dual is irreducible and the cancellative system itself satisfies $\limsup _{t\to\infty} P(\xi_t(0)=1)>0$ when $\xi_0=\delta_0$. The latter condition will be an easy by-product of our percolation arguments in Section~\ref{secthmproof}. The irreducibility of the annihilating dual is proved for cancellative systems which are voter model perturbations in Section~\ref{secirred}; see Corollary~\ref{vpirred}. Condition \eqref{oddgoal} is verified for voter model perturbations in Lemma~\ref {lemodd2} of Section~\ref{secflip}, following ideas in \cite{BDD}. In Section~\ref{secthmproof} (see Lemma~\ref{lemflip2}) condition \eqref{flip2} is derived for the voter model perturbations in Theorem~\ref{thmCCTpert} using a comparison with oriented percolation which in turn relies on input from \cite{CDP11} (see Lemma~\ref {vmpertspercolate}) and our condition $f'(0)>0$. Another key in this argument is the use of certain irreducibility properties of voter perturbations to help set up the appropriate block events. More specifically, with positive probability it allows us to transform a $0$--$1$ pair at a couple of input sites at one time into a mixed configuration which has a ``positive density'' of both $0$'s and $1$'s at a later time; see Lemma~\ref{vmirred}. The percolation comparison will provide a large number of the inputs, and the mixed configuration will be chosen to ensure a $0$--$1$ pair at sites with the prescribed separation by $x_0$. In Section~\ref{secthmproof} we finally prove Theorem~\ref{thmCCTpert}. Theorem~\ref{thmLVCCT} is proved in Section~\ref{secalmostlast}, and the proofs of Theorems~\ref{thmtvm} and \ref{thmgeom} are given in Section~\ref{seclast}. All of these latter results are proved as corollaries to Theorem~\ref{thmCCTpert}, except for the two-dimensional case of Theorem~\ref{thmLVCCT}, where the input for the percolation argument is derived from \cite{CP07} instead of \cite{CDP11}.
\section{Cancellative and annihilating processes: Growth of the annihilating dual}\label{secH} Our main objective in this section (Lemma~\ref{lemH} below) is to show the dual growth condition: under appropriate hypotheses, the annihilating dual process $\zeta_t$ will either die out or grow without bound as $t\to\infty$.
We begin by pointing out the consequences of the assumption that $\mathbf{0}$ is a trap for $\xi_t$. We assume here that $c(x,\xi)$ is a translation invariant cancellative flip rate function satisfying \eqref{q00}--\eqref{q02}, $\xi_t$ is the corresponding cancellative process and $\zeta_t$ the corresponding annihilating process [the Markov chain on $Y$ with $Q$-matrix defined in \eqref{qmatrix}]. In part (iv) below we identify $\xi_t$ with the set of sites of type $1$. Recall that $H(\xi,A)=\prod_{a\in A}(2\xi(x)-1)$.
\begin{lem}\label{elemann} If $\xi_t$ and $\zeta_t$ are as above, then the following are equivalent: \begin{longlist}[(iii)] \item[(i)] $\mathbf{0}$ is a trap for $\xi_t$.
\item[(ii)] $q_0(A)=0$ for all $A\in Y_e$, that is, $\zeta_t$ is parity-preserving.\vadjust{\goodbreak}
\item[(iii)] $\xi_t$ is symmetric, that is, $c(x,\xi)=c(x,\widehat\xi)$.
\item[(iv)] The simplified duality equation holds
\begin{equation}
\label{eqduality2} P\bigl(|\xi_t \cap\zeta_0| \mbox{ is odd}\bigr) = P\bigl(|\xi_0 \cap\zeta_t| \mbox{ is odd}\bigr)\qquad \forall \xi_0\in\{ 0,1\}^{\mathbb{Z}^d}, \zeta_0\in Y. \end{equation}
\end{longlist} \end{lem}
\begin{pf} Note that $H(\mathbf{0}, A)=(-1)^{|A|}$, which by \eqref{q00} implies
\[ c(0,\mathbf{0})=\frac{k_0}{2} \biggl(1+\sum_{A\in Y}q_0(A)
(-1)^{|A|} \biggr). \]
Thus $\mathbf{0}$ is a trap for $\xi_t$ if and only if
$ \sum_{A\in Y}q_0(A) (-1)^{|A|}=-1$. Using \eqref{q01}, we see that
\[
\sum_{A\in Y}q_0(A) (-1)^{|A|}= \sum_{A\in Y_e}q_0(A) -\sum _{A\in Y_o}q_0(A) \ge\sum _{A\in Y_e}q_0(A)-1, \]
so (i) and (ii) are equivalent.
Using $H(\widehat\xi,A)=(-1)^{|A|}H(\xi,A)$ and \eqref{q00}, (ii) implies (iii) because
\begin{eqnarray*}
c(x,\widehat\xi)&=&\frac{k_0}{2} \biggl(1- \bigl(1-2\xi(x) \bigr)\sum _{A\in Y_o } q_0(A-x) (-1)^{|A|}H( \xi,A) \biggr) \\ &=&\frac{k_0}{2} \biggl(1- \bigl(2\xi(x)-1 \bigr)\sum _{A\in Y} q_0(A-x)H(\xi,A) \biggr) \\ &=&c(x,\xi). \end{eqnarray*}
Conversely, if $c(0,\xi)=c(0,\widehat\xi)$ for all $\xi$, the previous calculation shows that
\[
\sum_{A\in Y}q_0(A)H(\xi,A) = \sum _{A\in Y}q_0(A) (-1)^{|A|+1}H(\xi,A). \]
Plug in $\xi=\mathbf{1}$ to get
\[ \sum_{A\in Y}q_0(A) = \sum _{A\in Y_o}q_0(A) - \sum_{A\in Y_e}q_0(A), \]
which implies $q_0(A)=0$ if $|A|$ is even. We now have that conditions (i)--(iii) are equivalent.
The duality equation \eqref{eqduality0} is easily seen to be equivalent to
\[
P\bigl(|\zeta_0|-|\xi_t\cap\zeta_0|\mbox{ is odd}\bigr) =P\bigl(|\zeta_t|-|\xi_0\cap\zeta_t| \mbox{ is odd}\bigr)\qquad \forall\xi_0\in\{0,1\}^{\mathbb{Z}^d}, \zeta_0\in Y. \]
If $\zeta_t$ is parity preserving, then this is equivalent to (iv). Conversely, if (iv) holds, and we apply it with $\xi_0=\mathbf{0}$ and $\zeta_0=\{x\}$, we get $P(\xi_t(x)=1)=0$ for all $t>0$. Since this holds for all $x\in{\mathbb{Z}^d}$, $\mathbf{0}$ must be a trap for $\xi_t$. \end{pf}
We give a brief review (cf. \cite{G79,Lig}) of the application of annihilating duality to the ergodic theory of $\xi_t$. Recall that $\mu_{1/2}$ is Bernoulli product measure with density $1/2$ on $\{0,1\}^{\mathbb{Z}^d}$.\vadjust{\goodbreak} Let $\zeta^A_t$ denote the Markov chain $\zeta_t$ with initial state $A$, and let $\xi_0$ have law $\mu_{1/2}$. It is easy to see by integrating \eqref{eqduality2} with respect to the law of $\xi_0$ that
\begin{eqnarray}
\label{zetaconv} P\bigl(|\xi_t\cap A| \mbox{ is odd}\bigr) &= & E \bigl(P
\bigl(\bigl| \zeta^A_t\cap\xi_0\bigr|\mbox{ is odd }| \zeta^A_t \bigr)1 \bigl(\zeta^A_t \neq\varnothing\bigr) \bigr) \nonumber \\[-8pt] \\[-8pt] \nonumber &=& \tfrac12 P \bigl(\zeta^A_t\ne\varnothing \bigr) \qquad\mbox{for all }A\in Y. \end{eqnarray}
The right-hand side above is monotone in $t$ ($\varnothing$ is a trap for $\zeta_t$), and so the left-hand side above converges as $t\to\infty$. By inclusion--exclusion arguments the class of functions
\begin{eqnarray}
\label{odddet}\mbox{$ \bigl\{\xi\to1\bigl(|\xi\cap A| \mbox{ is odd}\bigr)\dvtx A\in Y \bigr\}$ is a determining class,} \end{eqnarray}
and hence also a convergence determining class since the state space is compact. Therefore the above convergence not only implies \eqref{nuhalf1}, it characterizes $\nu_{1/2}$ via: for all $A\in Y$,
\begin{equation}
\label{nuhalf2} \nu_{1/2}\bigl(\xi\dvtx|\xi\cap A| \mbox{ is odd}\bigr) =\tfrac12 P \bigl(\zeta^A_t\ne\varnothing\ \forall t\ge0 \bigr). \end{equation}
The measure $\nu_{1/2}$ is necessarily a translation invariant stationary distribution for $\xi_t$ with density $1/2$, and a consequence of \eqref{nuhalf2} is that $\nu_{1/2}\ne\frac12(\delta_{0} + \delta_{1})$ if and only if for some $x\ne y\in{\mathbb{Z}^d}$,
\begin{equation} \label{dualsurvival} P \bigl(\zeta^{\{x,y\}}_t\ne\varnothing \ \forall t\ge0 \bigr)>0. \end{equation}
Thus, a sufficient condition for coexistence for $\xi_t$ is
\eqref{dualsurvival}. Indeed, if \eqref{dualsurvival} holds, then $\nu_{1/2}(\xi\in\cdot|\xi\notin\{\mathbf{0},\mathbf{1}\})$ is a translation invariant stationary distribution for
$\xi_t$ which must satisfy \eqref{coexist}. (There are countably many configurations $\xi$ with $|\xi|<\infty$, none of which can have positive probability because there are countably many distinct translates of each one.)
Establishing \eqref{dualsurvival} directly is a difficult problem for most annihilating systems. [Not so for the annihilating dual of the voter model, since \eqref{dualsurvival} follows trivially from transience if $d\ge3$ but fails if $d\le2$.] To use annihilating duality to go beyond~\eqref{nuhalf1} requires more information about the behavior of
$\zeta_t$. In particular, one needs that either $|\zeta_t|\to0$ or
$|\zeta_t|\to\infty$ as $t\to\infty$; see \cite{BDD}, for instance. The following general result gives a condition for this which we can check for certain voter model perturbations. It is a key ingredient in the proofs of Theorems~\ref{thmLVCCT} and~\ref{thmCCTpert}.
We now assume that $\mathbf{0}$ is a trap for $\xi_t$, and so all the properties listed in Lemma~\ref{elemann} will hold.
\begin{lem}[(Handjani \cite{H99}, Sturm and Swart \cite{SS})]\label{lemH} Let $\zeta_t$ be a translation invariant annihilating process with $Q$-matrix given in \eqref{qmatrix} satisfying \eqref{q01} and~\eqref{q02}. If $\zeta_t$ is irreducible, parity-preserving, and satisfies
\begin{equation} \label{eqH0} \limsup_{t\to\infty}P \bigl(0\in\zeta_t^{\{0\}} \bigr) >0, \end{equation}
then
\begin{equation}
\label{eqH} \lim_{t\to\infty} P \bigl(0<\bigl|\zeta^B_t\bigr| \le K \bigr) = 0\qquad \mbox{for all nonempty }B\in Y\mbox{ and } K\ge1.\vadjust{\goodbreak} \end{equation}
\end{lem}
\begin{rem}\label{remHaltcond} If $\zeta_t$ has associated cancellative process $\xi_t$ which has $\mathbf{0}$ as a trap, then the parity-preserving hypothesis in the above result follows by Lemma~\ref{elemann}. If we let $\xi^{\{0\}}_t$ denote this process with initial state $\xi_0^{\{0\}}=\{0\}$, then by the duality equation \eqref{eqduality2}, \eqref{eqH0} is equivalent to
\begin{eqnarray} \label{eqH3} \limsup_{t\to\infty}P \bigl(\xi^{\{0\}}_t(0)=1 \bigr) > 0. \end{eqnarray} \end{rem}
The limit \eqref{eqH} was proved in \cite{H99} (see Proposition~2.6 there) for the annihilating dual of the threshold voter model. The arguments in that work are in fact quite general, and with some work can be extended to establish Lemma~\ref{lemH} as stated above. Rather than provide the necessary details, we appeal instead to Theorem~12 of \cite{SS}, which is proved using a related but somewhat different approach. To apply this result, and hence establish Lemma~\ref{lemH}, we must do two things. The first is to show that (3.54) in \cite{SS} [see \eqref{354} below] holds; the second is to show that our condition \eqref{eqH0} implies the nonstability condition in Theorem~12 of \cite{SS}. The latter is nonpositive recurrence of $\zeta$ ``modulo translations''; see the conclusion of Lemma~\ref {lemstability} below.
In preparation for these tasks we give a ``graphical construction'' (as in \cite{G78} or~\cite{CD91}) of $\zeta_t$. For $x\in{\mathbb{Z}}^d$, let $\{(S_n^x,A_n^x,)\dvtx n\in{\mathbb{N}}\}$ be the points of independent Poisson point processes $\{\Gamma^x(ds,dA)\dvtx x\in{\mathbb{Z}}^d\}$ on ${\mathbb{R}} _+\times Y$ with rates $k_0\,dsq_0(dA)$. For $R\subset{\mathbb{R}}^d$ and $0\le t_1\le t_2$ we let
\[ {\mathcal{F}}\bigl(R\times[t_1,t_2] \bigr)=\sigma\bigl(
\Gamma^x |_{{\mathbb{Z}}^d\times [t_1,t_2]}\dvtx x\in R \bigr). \]
Then for $S_i=R_i\times I_i$ as above ($i=1,2$), ${\mathcal{F}}(S_1)$ and ${\mathcal{F}}(S_2)$ are independent if $S_1\cap S_2=\varnothing$. At time $S^x_n$ draw arrows from $x$ to $x+y$ for each $y\in A^x_n\setminus\{0\}$. If $0\notin A^x_n$ put a $\delta$ at $x$ (at time $S^x_n$). For $x,y\in{\mathbb{Z}^d}$ and $s<t$ we say that $(x,s)\to(y,t)$ if there is a path from $(x,s)$ to $(y,t)$ that goes across arrows, or up but not through $\delta$'s. That is, $(x,s)\to(y,t)$ if there are sequences $x_0=x,x_1,\ldots,x_n=y$ and $s_0=s<s_1<\cdots<s_n <s_{n+1}=t$ such that:
\begin{longlist}[(ii)]
\item[(i)] for $1\le m\le n$, there is an arrow from $x_{m-1}$ to $x_{m}$ at time $s_m$;
\item[(ii)] for $1\le m\le n+1$, there are no $\delta$'s in $\{x_{m-1}\}\times(s_{m-1},s_m)$, \end{longlist}
and no $\delta$ at $(y,t)$. For $0\le s<t$, $x,y\in{\mathbb{Z}^d}$ and $B\in Y$ define
\begin{eqnarray*} N^{(x,s)}_t(y) &=& \mbox{ the number of paths up from }(x,s) \mbox{ to }(y,t), \\ \zeta^{B,s}_t &= &\biggl\{y\dvtx\sum _{x\in B} N^{(x,s)}_t(y) \mbox{ is odd} \biggr\}, \\ \bar\zeta^{B,s}_t &=& \biggl\{y\dvtx\sum _{x\in B} N^{(x,s)}_t(y)\ge1 \biggr\}, \end{eqnarray*}
and write $\zeta^B_t$ for $\zeta^{B,0}_t$ and $\bar\zeta^B_t$ for $\zeta^{B,0}_t$.\vadjust{\goodbreak}
The process $\zeta_t$ is the annihilating Markov chain on $Y$ with $Q$-matrix as in \eqref{qmatrix}. The process $\bar\zeta_t$ is additive, meaning $\bar\zeta^{B,s}_t = \bigcup_{x\in B}\bar\zeta^{(x,s)}_t $. Both $\zeta_t$ and $\bar\zeta_t$ are nonexplosive Markov chains on $Y$. Also, it is clear that for every $B\in Y$,
\begin{equation} \label{contained} \zeta^{B,s}_t\subset\bar \zeta^{B,s}_t \qquad \forall0\le s\le t<\infty, \end{equation}
and also that for any fixed $t>0$,
\begin{equation} \label{boundedgrowth} \lim_{K\to\infty}P \bigl(\bar \zeta^{\{0\}}_u\subset[-K,K]^d\ \forall0\le u \le t \bigr) = 1. \end{equation}
Furthermore if $A,B\in Y$ satisfy $\min_{a\in A,b\in B}|a-b|>2K$ and $t>s\ge0$, then
\begin{eqnarray} \label{separated} \zeta_t^{A\cup B,s}= \zeta_t^{A,s}\cup\zeta_t^{B,s} \nonumber \\[-8pt] \\[-8pt]
\eqntext{\mbox{on the event }\bigl\{\bar\zeta^{A,s}_u\subset A+[-K,K]^d, \bar\zeta^{B,s}_u\subset B+[-K,K]^d \ \forall s\le u\le t \bigr\}} \end{eqnarray}
(where $A+B=\{x+y\dvtx x\in A, y\in B\}$).
The following result is key to verifying condition (3.54) of \cite{SS}.
\begin{lem}\label{graphrep} Let $A\in Y$, $r\in{\mathbb{N}}$ and $B_m=\{y^m_1,\ldots,y^m_r\}\in Y$ be such that
$\lim_{m\to\infty} \min_i|y^m_i|=\infty$. If $\zeta^A$ and $\zeta ^{B_m}$ are independent copies of $\zeta$ with the given initial conditions, then for each $t\ge0$ and $n\in{\mathbb{N}}$,
\[
\lim_{m\to\infty}P \bigl(\bigl|\zeta_t^{A\cup B_m}\bigr|=n
\bigr)-P \bigl(\bigl|\zeta_t^A\bigr|+\bigl|\zeta_t^{B_m}\bigr|=n \bigr)=0. \]
\end{lem}
\begin{pf} Assume $(\zeta^B_t)$ are constructed as above for $B\in Y$ and $t\ge0$. For $K\in{\mathbb{N}}$ define $\tilde\zeta_t^{B,(K)}$ as $\zeta^B_t$ but now only count paths which are contained in $B+[-K,K]^d$. This implies that
\begin{equation} \label{tildemeas} \tilde\zeta_t^{B,(K)}\mbox{ is }{\mathcal{F}} \bigl( \bigl(B+[-K,K]^d \bigr)\times[0,t] \bigr)\mbox{-measurable}. \end{equation}
Fix $\varepsilon>0$. By \eqref{boundedgrowth} and the additivity of $\bar \zeta _t$ we may choose $K(\varepsilon)\in{\mathbb{N}}$ so that if $K\ge K(\varepsilon)$, then
\begin{eqnarray} \label{contain}\qquad && P \bigl(\bar\zeta^A_u\subset A+[-K,K]^d\mbox{ and }\bar\zeta^{B_m}_u \subset B_m+[-K,K]^d\mbox{ for all }u\in[0,t] \bigr) \nonumber \\[-8pt] \\[-8pt] \nonumber &&\qquad >1-\varepsilon\qquad \mbox{for all }m\in{\mathbb{N}}. \end{eqnarray}
Write $\tilde\zeta_t^B$ for $\tilde\zeta^{B,(K(\varepsilon))}_t$. Choose
$m(\varepsilon)\in{\mathbb{N}}$ so that $\min_{a\in A,b\in B_m}|a-b|>2K(\varepsilon)$ for $m\ge m(\varepsilon)$. It follows from \eqref{separated} and \eqref{contained} that on the set in \eqref{separated} with $K=K(\varepsilon)$, for $m\ge m(\varepsilon)$,
\begin{equation}
\label{zetasum}\bigl|\zeta^{A\cup B_m}_t\bigr|=\bigl|\zeta^A_t\bigr|+\bigl|
\zeta_t^{B_m}\bigr| \end{equation}
and
\begin{equation} \label{tildeeq} \zeta^A_t=\tilde\zeta^A_t \quad\mbox{and}\quad\zeta_t^{B_m}=\tilde\zeta^{B_m}_t. \end{equation}
[The latter is an easy check using \eqref{separated}.] We conclude from the last two results that
\begin{equation}
\label{tildesum} P \bigl(\bigl|\zeta^{A\cup B_m}_t\bigr|\neq\bigl|\tilde
\zeta^A_t\bigr|+\bigl|\tilde\zeta^{B_m}_t\bigr| \bigr)<\varepsilon\qquad\mbox{for }m\ge m(\varepsilon). \end{equation}
By \eqref{tildemeas} and the choice of $m(\varepsilon)$ we see that $\tilde \zeta^A_t$ and $\tilde\zeta^{B_m}_t$ are independent for $m\ge m(\varepsilon )$. Using this independence and then \eqref{tildesum} we conclude that
\begin{eqnarray*}
\hspace*{-4pt}&&\Biggl|P \bigl(\bigl|\zeta_t^{A\cup B_m}\bigr|=n \bigr)- \Biggl(\sum _{k=0}^nP \bigl(\bigl|\zeta^A_t\bigr|=k
\bigr)P \bigl(\bigl|\zeta_t^{B_m}\bigr|=n-k \bigr) \Biggr) \Biggr| \\
\hspace*{-4pt}&&\qquad\le\varepsilon+ \Biggl|P \bigl(\bigl|\tilde\zeta^A_t\bigr|+\bigl|\tilde\zeta
^{B_m}_t\bigr|=n \bigr)- \Biggl(\sum _{k=0}^nP \bigl(\bigl|\zeta^A_t\bigr|=k
\bigr)P \bigl(|\zeta_t^{B_m}|=n-k \bigr) \Biggr) \Biggr| \\
\hspace*{-4pt}&&\qquad\le\varepsilon+ \Biggl|\sum_{k=0}^n \bigl[P \bigl(\bigl|
\tilde\zeta^A_t\bigr|=k \bigr)P \bigl(\bigl|\tilde\zeta
^{B_m}_t\bigr|=n-k \bigr)-P \bigl(\bigl|\zeta^A_t\bigr|=k
\bigr)P \bigl(\bigl|\zeta_t^{B_m}\bigr|=n-k \bigr) \bigr] \Biggr| \\ \hspace*{-4pt}&&\qquad\le3\varepsilon. \end{eqnarray*}
In the last line we have used \eqref{tildeeq}. The result follows. \end{pf}
Say that $A,B\in Y_o$ are equivalent if they are translates of each other, let $\tilde Y_o$ denote the set of equivalence classes, and (abusing notation slightly) let $\tilde A$ denote the equivalence class containing $A\in Y$. Since the dynamics of $\zeta$ are translation invariant, for parity-preserving $\zeta$ we may define $\tilde\zeta_t$ as the $\tilde Y_o$-valued Markov process obtained by taking the equivalence class of $\zeta_t$. The nonstability requirement of Theorem~12 of \cite{SS} is that $\tilde\zeta_t$ not be positive recurrent on $\tilde Y_o$.
\begin{lem}\label{lemstability} If $\zeta$ is parity-preserving, irreducible and satisfies \eqref{eqH0}, then $\tilde\zeta_t$ is not positive recurrent. \end{lem}
\begin{pf} We use the same arguments as in the proof of Lemma~2.4 of \cite{H99}. First, $\zeta_t$ cannot be positive recurrent on $Y_o$. To check this, we first note that translation invariance implies
\[ P \bigl(\zeta_t^{\{x\}}=\{x\} \bigr)=P \bigl( \zeta_t^{\{0\}}=\{0\} \bigr)\qquad \mbox{for all }t\ge0, x\in{ \mathbb{Z}^d}. \]
If $\zeta_t$ is positive recurrent on $Y_o$, then the limit $\mu(A)=\lim_{t\to\infty}P(\zeta^B_t=A)$ exists and is positive for all $A,B\in Y_o$. Letting $t\to\infty$ above, this implies $\mu(\{ 0\})= \mu(\{ x\})$ for all $x$ which is impossible, so $\zeta_t$ is not positive recurrent on $Y_o$. A consequence of this is that for any fixed $k>0$,
\[ \lim_{t\to\infty} P \bigl(\zeta^{\{0\}}_t \subset[-k,k]^d \bigr) = 0. \]
Next, suppose $\tilde\zeta_t$ is positive recurrent on $\tilde Y_o$, with some stationary distribution~$\tilde\mu$, which must satisfy
\[ \tilde\mu\bigl(A\in\tilde Y_o\dvtx\operatorname{diam}(A) \le k \bigr) \to1 \qquad\mbox{as }k\to\infty,\vadjust{\goodbreak} \]
where $\operatorname{diam}(A)=\max\{|x-y|\dvtx x,y\in A\}$ is well defined for $A\in \tilde Y_o$. For any $k,t$, since $\operatorname{diam}(\tilde\zeta^{\{0\}}_t) =\operatorname{diam}(\zeta^{\{0\}}_t)$, we have
\[ P \bigl(0\in\zeta^{\{0\}}_t \bigr) \le P \bigl( \zeta^{\{0\}}_t\subset[-k,k]^d \bigr) +P \bigl( \operatorname{diam} \bigl(\tilde\zeta^{\{0\}}_t \bigr) > k \bigr). \]
Letting $t\to\infty$ gives
\[ \limsup_{t\to\infty}P \bigl(0\in\zeta^{\{0\}}_t \bigr) \le\tilde\mu\bigl(A\in\tilde Y_o\dvtx\operatorname{diam}(A)>k \bigr). \]
The right-hand side above tends to 0 as $k\to\infty$, so we have a contradiction to the assumption \eqref{eqH0}. \end{pf}
\begin{pf*}{Proof of Lemma~\ref{lemH}} Thanks to the above lemma we have verified all the hypotheses of Theorem~12 of \cite{SS} except for their (3.54) which we now state in our notation: for each $n\in{\mathbb{Z}}_+$, $L\ge1$ and $t>0$,
\begin{eqnarray}
\label{354}\qquad &&\inf\bigl\{P \bigl(\bigl|\zeta_t^A\bigr|=n \bigr)
\dvtx|A|=n+2 \mbox{ and } 0<|i-j|\le L \mbox{ for some }i,j\in A \bigr \} \nonumber \\[-8pt] \\[-8pt] \nonumber &&\qquad>0. \end{eqnarray}
Assume \eqref{354} fails. Then for some $n$, $L$ and $t$ as above, by translation invariance and compactness of $Y$ (with the subspace topology it inherits from $\{0,1\}^{{\mathbb{Z}}^d}$), there are $\{A_m\}\subset Y$ so that for some integer $2\le s\le n+2$ and $x_2\in[-L,L]^d$,
$A_m=\{0,x_2,\ldots,x_s\}\cup\{x_{s+1}^m,\ldots,x^m_{n+2}\}\equiv A\cup B_m$, where $\lim_{m\to\infty}|x_i^m|=\infty$ for each $i\in\{s+1,\ldots,n+2\}$ and
\begin{equation}
\label{fail354}\lim_{m\to\infty}P \bigl(\bigl|\xi_t^{A_m}\bigr|=n \bigr)=0. \end{equation}
By the irreducibility of $\zeta$, $P(|\zeta_t^{A}|=s-2)=p>0$. If $\zeta_t^{A}$ and $\zeta_t^{B_m}$ are as in Lemma~\ref{graphrep}, then by that result,
\begin{eqnarray*}
\lim_{m\to\infty}P \bigl(\bigl|\zeta_t^{A_m}\bigr|=n
\bigr)&=&\lim_{m\to\infty}P \bigl(\bigl|\zeta^A_t\bigr|+\bigl|
\zeta^{B_m}_t\bigr|=n \bigr) \\
&\ge &P \bigl(\bigl|\zeta^A_t\bigr|=s-2 \bigr)\liminf _{m\to\infty}P \bigl(\bigl|\zeta_t^{B_m}\bigr|=|B_m| \bigr) \\ &\ge& p\exp\bigl\{-k_0(n+2-s)T \bigr\}>0. \end{eqnarray*}
In the last line we use the fact that by its graphical construction,
$\zeta^{B_m}$ will remain constant up to time $t$ if none of the $|B_m|$ independent rate $k_0$ Poisson processes attached to each of the sites in $B_m$ fire by time $t$. This contradicts \eqref{fail354} and so~\eqref{354} must hold. We now may apply Theorem~12 of \cite{SS} to obtain the required conclusion. \end{pf*}
\section{Irreducibility}\label{secirred} In addition to the explicit irreducibility requirement for $\zeta_t$ in Lemma~\ref{lemH}, some arguments in Section~\ref{secthmproof} will require irreducibility type conditions for the voter model perturbations $\xi^\varepsilon_t$. We collect and prove the necessary results for both processes in this section.\vadjust{\goodbreak}
Assuming $\sum_{y\in{\mathbb{Z}^d}} q_0(\{y\})>0$, define the step distribution of a random walk associated with $q_0$ by
\[ q(x)=q_0 \bigl(\{x\} \bigr) \Big/\sum_{y\in{\mathbb{Z}^d}} q_0 \bigl(\{y\} \bigr). \]
\begin{lem}\label{suffindecomp} Let $\zeta_t$ be a parity-preserving annihilating process with $Q$-matrix given in \eqref{qmatrix}. Assume $q_0(A_0)>0$ for some
$A_0\in Y$ with $|A_0|\ge3$, and for some symmetric, irreducible random walk kernel $r$ on ${\mathbb{Z}^d}$, $q(x)>0$ whenever $r(x)>0$. Then $\zeta_t$ is irreducible. \end{lem}
\begin{pf} The proof is elementary but awkward, so we will only sketch the argument. Note that if $x\in A$ and $y\notin A$, then
\[ Q \bigl(A, \bigl(A\setminus\{x\} \bigr)\cup\{y\} \bigr)\ge q_0 \bigl( \{y-x\} \bigr)=cq(y-x). \]
So by using only the $q_0(\{x\})$ ``clocks'' with $r(x)>0$, $\zeta_t$ can with positive probability execute exactly any finite sequence of transitions that the annihilating random walk system with step distribution $r$ can. We will refer to ``$r$-random walks'' below in describing such transitions.
We first check that the assumptions on $q_0$ imply that
$\zeta_t$ can reach any set $B$ with $|B|=|\zeta_0|$ with positive probability. To see this, we first construct a set $B'$ by starting $r$-random walks at each site of $B$ and then moving them apart, one at a time, avoiding collisions, to widely separated locations, resulting in $B'$. Note that by reversing this entire sequence of steps, it is possible to move $r$-random walks starting at the sites of $B'$ to $B$ without collisions. This uses the symmetry of $r$. Now, to move $r$-walks from $\zeta_0$ to $B$ we first move walks from $\zeta_0$ to some $\zeta_0'$, avoiding collisions, where the sites of $\zeta_0'$ are widely separated. Pair off points from $\zeta_0'$ and $B'$ and move $r$-walks one at a time from $\zeta_0'$ to $B'$ without collisions. This is possible if $\zeta_0'$ and $B'$ are sufficiently spread out since $r$ is irreducible. Finally, move the walks from $B'$ to $B$ without collisions as discussed above.
It should be clear that if $\zeta_0\ne\varnothing$, then
$\zeta_t$ can reach a set $B$ such that $|B|=|\zeta_0|-2$, since this is the case for annihilating random walks. Finally, if $\zeta_0\ne\varnothing$, then
$\zeta_t$ can reach a set $B$ with $|B|\ge|\zeta_0|+2$. Choose $x_1$ far from $\zeta_0$ so that $\zeta_0$ and $x_1+A_0$ are disjoint, and such that for some $x_0\in\zeta_0$, an $r$-walk starting at $x_0$ can reach $x_1$ by a sequence of steps avoiding $\zeta_0$. Now using the ``$A_0$ clock'' at $x_1$ we get a transition from $(\zeta_0\setminus\{x_0\})\cup\{x_1\}$ to $\zeta_0'=(\zeta_0\setminus\{x_0\})\Delta(x_1+A_0)$, and
$|\zeta_0'|\ge|\zeta_0|+2$. \end{pf}
The next result will allow us to apply the above lemma to voter model perturbations. Recall $p(x)$ satisfies \eqref{passump}, and $c_{\mathrm{VM}}(x,\xi)$ is the corresponding voter model flip rate function.
\begin{lem}\label{suffirred} There is an $\varepsilon_2=\varepsilon_2(p(\cdot))>0$ and $R_1=R_1(p(\cdot))$ such that: \begin{longlist}[(ii)]
\item[(i)] $p(\cdot| |x|<R_1)$ is irreducible, and\vadjust{\goodbreak}
\item[(ii)] if $c(x,\xi)=c_{\mathrm{VM}}(x,\xi)+\tilde c(x,\xi)$ is a translation invariant, cancellative flip rate function with $\mathbf{0}$ as a trap such that
\begin{equation}
\label{c*cond}\Vert\tilde c\Vert_\infty<\varepsilon_2,\qquad \sum _{x\neq 0}\bigl|\tilde c(0,\delta_x)\bigr|< \varepsilon_2, \end{equation}
then the dual kernel $q_0$ satisfies
\begin{equation}
\label{qdomp} q_0 \bigl(\{x\} \bigr)> (k_0 3)^{-1}p(x) \qquad\mbox{for all }0<|x|<R_1. \end{equation} \end{longlist}
\end{lem}
\begin{pf}
Since $p$ is irreducible, we may choose $R_1$ so that $p(\cdot|
|x|<R_1)$ is also irreducible. Assume \eqref{c*cond} holds for an appropriate $\varepsilon_2$ which will be chosen below. We will write $\hat \xi(B)$ for $\sum_{x\in B}\hat\xi(x)$. Also, \emph{in this proof only}, we will let $A$ denote a \emph{random set} with probability mass function $q_0$, and write $E_0(g(A))=\int g \,dP_0$ for $\sum_{B\in Y}g(B)q_0(B)$. With this notation, by our hypotheses we have
\begin{equation} \label{tworep} c_{\mathrm{VM}}(0,\xi)+\tilde c(0,\xi)=\frac{k_0}{2} \bigl[1+(-1)^{\xi (0)}E_0 \bigl((-1)^{\hat\xi(A)} \bigr) \bigr]. \end{equation}
Recall by Lemma~\ref{elemann} that $P_0(|A|\mbox{ is odd})=1$. Therefore if we set $\xi=\delta_x$ for $x\neq0$ in \eqref{tworep}, we get
\[ p(x)+\tilde c(0,\delta_x)=\frac{k_0}{2} \bigl[1+E_0
\bigl((-1)^{|A\setminus\{x\}|} \bigr) \bigr]=k_0P_0(x\in A), \]
and so
\begin{equation} \label{P0x}P_0(x\in A)= \bigl(p(x)+\tilde c(0,\delta_x) \bigr)k_0^{-1}. \end{equation}
If we take $\xi=\delta_{\{x_0,x_1\}}$ in \eqref{tworep}, where $x_0$, $x_1$ are two distinct nonzero points, then we get
\begin{eqnarray*} p(x_0)+p(x_1)+\tilde c(0,\delta_{\{x_0,x_1\}})& =&
\frac{k_0}{2} \bigl[1+E_0 \bigl((-1)^{|A\setminus\{x_0,x_1\}|} \bigr) \bigr] \\ &=&k_0P \bigl(1_A(x_0)\neq1_A(x_1) \bigr), \end{eqnarray*}
and so
\begin{equation} \label{P0x1x2} P_0 \bigl(1_A(x_0) \neq1_A(x_1) \bigr)= \bigl(p(x_0)+p(x_1)+ \tilde c(0,\delta_{\{ x_0,x_1\}}) \bigr)k_0^{-1}. \end{equation}
For any two distinct nonzero points, $x_0$ and $x_1$, we have
\[ P_0 \bigl(1_A(x_0)\ne1_A(x_1) \bigr) = P_0(x_0\in A) + P_0(x_1 \in A) -2 P_0(x_0\in A,x_1\in A). \]
Therefore, we see that \eqref{P0x} and \eqref{P0x1x2} imply
\begin{eqnarray*} P_0 \bigl(\{x_0,x_1\}\subset A \bigr) &=& \tfrac{1}{2} \bigl[P_0(x_0\in A)+P_0(x_1 \in A)-P_0 \bigl(1_A(x_0)\neq 1_A(x_1) \bigr) \bigr] \\ &=& \bigl[\tilde c(0,\delta_{x_0})+\tilde c(0,\delta_{x_1})- \tilde c(0,\delta_{\{x_0,x_1\}}) \bigr](2k_0)^{-1}, \end{eqnarray*}
which gives the simple bound
\[ P_0 \bigl(\{x_0,x_1\}\subset A \bigr)\le
\tfrac{3}{2}\|\tilde c\|_\infty k_0^{-1}. \]
Note that if $0\neq x\in A$ but $A\neq\{x\}$, then $P_0$-a.s. $A$ must contain $x$ and another nonzero point as $|A|$ is a.s. odd, and so for $0<|x|<R_1$ and $R_2>R_1$,
\begin{eqnarray*} &&P_0 \bigl(A=\{x\} \bigr)\\ &&\qquad\ge P_0(x\in A)-\sum _{x_1\notin\{0, x\}}P_0 \bigl(\{x,x_1\} \subset A \bigr) \\ &&\qquad\ge k_0^{-1} \biggl[p(x)+\tilde c(0, \delta_x)-(3/2)\Vert\tilde c\Vert_\infty(2R_2+1)^d-
\sum_{|x_1|>R_2}P_0(x_1\in A) \biggr] \\ &&\qquad\ge k_0^{-1} \biggl[p(x)- \bigl(1+2(2R_2+1)^d
\bigr)\Vert\tilde c\Vert_\infty-\sum_{|x_1|>R_2} \bigl(p(x_1)+\tilde c(0,\delta_{x_1}) \bigr) \biggr]. \end{eqnarray*}
We have used the previous displays and \eqref{P0x} in the above. Recalling the bounds in our assumption \eqref{c*cond} on $\tilde c$, we conclude that
\begin{equation} \label{qlb} P_0 \bigl(A=\{x\} \bigr)\ge k_0^{-1}
\biggl[p(x)-\sum_{|x_1|>R_2}p(x_1)-2 \bigl(1+(2R_2+1)^d \bigr)\varepsilon_2 \biggr]. \end{equation}
Now let
\[
\eta=\eta\bigl(p(\cdot) \bigr)=\inf\bigl\{p(x)\dvtx|x|<R_1\mbox{ and }p(x)>0 \bigr\}>0, \]
choose $R_2=R_2(p(\cdot))>R_1$ so that $\sum_{|x_1|>R_2}p(x_1)<\eta/3$ and define
\[ \varepsilon_2=\frac{\eta}{6((2R_1+1)^d+1)}. \]
Then by \eqref{qlb}
\[ P_0 \bigl(A=\{x\} \bigr)\ge(3k_0)^{-1}p(x)
\qquad\mbox{for all }0<|x|<R_1, \]
and we are done. \end{pf}
For the rest of this section we assume $\{\xi^\varepsilon\dvtx0<\varepsilon\le \varepsilon _0\}$ is a voter model perturbation with rate function $c_\varepsilon$ [so that \eqref{vmpert}--\eqref{traps} are valid] which is also cancellative for each $\varepsilon$ as above with dual kernels $q_0^\varepsilon$ satisfying \eqref{q00}--\eqref{q02}. In particular the $\tilde c$ in Lemma~\ref{suffirred} is now $\varepsilon^2c^*_\varepsilon$. By Lemma~\ref{elemann}, all the conclusions of that result hold.
\begin{cor}\label{vpirred} Assume that
\begin{equation} \label{create} \mbox{for small enough $\varepsilon$, } q_0^\varepsilon(A)>0
\mbox{ for some $A\in Y$ with }|A|>1. \end{equation}
Then there is an $\varepsilon_3>0$ depending on $p$, $\varepsilon_1$, $\{g_i^\varepsilon\}$ and the $\varepsilon$ required in \eqref{create} so that if $0<\varepsilon<\varepsilon_3$, then the annihilating dual with kernel $q_0^\varepsilon$ is irreducible. \end{cor}
\begin{pf} Let $R_1$ be as in Lemma~\ref{suffirred}. An easy calculation shows that
\[
\Vert\tilde c\Vert_\infty\vee\biggl(\sum_x\bigl|
\tilde c(0,\delta_x)\bigr| \biggr)\le\varepsilon^2 \Biggl[ \varepsilon_1^{-2}+\bigvee_{i=0}^1\bigl\Vert g_i^\varepsilon\bigr\Vert_\infty\Biggr]\le \varepsilon^2C \]
for some constant $C$, independent of $\varepsilon$. Therefore for $\varepsilon <\varepsilon _3$ ($\varepsilon_3$ as claimed) we have the hypotheses, and hence conclusion, of Lemma~\ref{suffirred}. This allows us to apply Lemma~\ref
{suffindecomp} with $r(\cdot)=p(\cdot| |x|<R_1)$ and hence conclude that the annihilating dual $\zeta$ is irreducible for such $\varepsilon$. \end{pf}
\begin{rem}\label{nvm} Clearly \eqref{create} is a necessary condition for the conclusion to hold. In fact if it fails, it is easy to check that $c_\varepsilon(x,\xi)$ is a multiple of the voter model rates with random walk kernel $q^\varepsilon_0(\{x\})$. Hence this condition just eliminates voter models for which the conclusions of Corollary~\ref{vpirred}, as well as Lemma~\ref{lemH} and Proposition~\ref{propCCT} below, will also fail in general.
Note that if \eqref{create} fails, then for some $\varepsilon_n\downarrow0$,
\[ c^*_{\varepsilon_n}(0,\xi)=\varepsilon_n^{-2}c_{\varepsilon_n}(0, \xi)-\varepsilon_n^{-2}c_{\mathrm{VM} }(0,\xi)= \lambda_n\tilde c^n_{\mathrm{VM}}(0,\xi)- \varepsilon_n^{-2}c_{\mathrm{VM} }(0,\xi), \]
where $\tilde c^n_{\mathrm{VM}}(0,\xi)$ is the rate function for the voter model with kernel $q_0^{\varepsilon_n}(\{\cdot\})$. From this it is easy to check that if $\langle\cdot\rangle_u$ is expectation with respect to the voter model equilibrium for $c_{\mathrm{VM}}$ with density $u$, then
\[ \bigl\langle(1-\xi)c^*_{\varepsilon_n}(0,\xi)-\xi c^*_{\varepsilon_n}(0,\xi) \bigr\rangle_u=0, \]
and so by \eqref{gcvgce} and \eqref{f}, $f(u)\equiv0$. Therefore, the condition $f'(0)>0$ in Theorem~\ref{thmCCTpert} implies \eqref{create}. \end{rem}
Next we prove an irreducibility property for the voter model perturbations $\xi^\varepsilon_t$ themselves. To do so we introduce the (unscaled) graphical representation for $\xi^\varepsilon_t$ used in \cite{CDP11}. First put
\[
\bar c= \sup_{\varepsilon<\varepsilon_0} \bigl(\bigl\|g^\varepsilon_1
\bigr\|_\infty+\bigl\|g^\varepsilon_0\bigr\|_\infty+1 \bigr)< \infty. \]
For $x\in{\mathbb{Z}^d}$, introduce independent Poisson point processes on ${\mathbb{R}}_+$, $\{ T^{x}_n, n \ge1 \}$ and $\{ T^{*,x}_n, n \ge1\}$, with rates $1$ and $\varepsilon^2\bar c$, respectively. For $x\in{\mathbb{Z}^d}$ and $n\ge1$, define independent random variables $X_{x,n}$ with distribution $p(\cdot)$, $Z_{x,n}=(Z^1_{x,n}, \ldots, Z^{N_0}_{x,n})$ with distribution $q_Z(\cdot)$, and $U_{x,n}$ uniform on $(0,1)$. These random variables are independent of the Poisson processes, and all are independent of any initial condition $\xi^\varepsilon_0\in\{0,1\}^{{\mathbb{Z}}^d}$. For all $x\in{\mathbb{Z}^d}$ we allow $\xi^\varepsilon_t(x)$ to change only at times $t\in\{T^x_n,T^{*,x}_n,n\ge1\}$. At the voter times $T^x_n, n \ge1$ we draw a voter arrow from $(x,T^x_n)$ to $(x+X_{x,n},T^x_n)$ and set $\xi^\varepsilon_{T^x_n}(x)=\xi^\varepsilon_{T^x_n-}(x+X_{x,n})$. At the times $T^{*,x}_n$, $n\ge1$ we draw ``*-arrows'' from $(x,T^{*,x}_n)$\vadjust{\goodbreak} to each $(x+Z^i_{x,n},T^{*,x}_n)$, $1\le i\le N_0$, and if $\xi^\varepsilon_{T^{*,x}_n-}(x)=i$ we set $\xi^\varepsilon_{T^{*,x}_N}(x)=1-i$ if
\[ U_{x,n} < g^\varepsilon_{1-i} \bigl( \xi^\varepsilon_{T^{*,x}_n-} \bigl(x+Z^1_{x,n} \bigr), \ldots, \xi^\varepsilon_{T^{*,x}_n-} \bigl(x+Z^{N_0}_{x,n} \bigr) \bigr)/\bar c. \]
As noted in Section 2 of \cite{CDP11}, this recipe defines a pathwise unique process $\xi^\varepsilon_t$ whose law is specified by the flip rates in \eqref{vmpert}. We refer to this as the graphical construction of $\xi^\varepsilon_t$. For $x\in{\mathbb{Z}}^d$, $\{(X_{(x,n)},T_n^x)\dvtx n\in{\mathbb{N}}\}$ and $\{ (Z_{x,n},T^{*,x}_n,U_{x,n})\dvtx n\in{\mathbb{N}}\}$ are the points of independent collections of independent Poisson point processes, $(\Lambda^x_w(dy,dt),x\in{\mathbb{Z}}^d)$ and $(\Lambda^x_r(dy,dt,du),\break x\in{\mathbb{Z}}^d)$, on ${\mathbb{Z}}^d\times{\mathbb{R}}_+$ with rate $dt p(\cdot)$, and on ${\mathbb{Z}}^d\times{\mathbb{R}}_+\times[0,1]$ with rate\break $\varepsilon^2\bar c\,dt q_Z(\cdot) \,du$, respectively. For $R\subset{\mathbb{R}}^d$ and $0\le t_1\le t_2$ we let
\[ \mathcal{G}\bigl(R\times[t_1,t_2] \bigr)=\sigma\bigl(
\Lambda^x_w|_{{\mathbb{Z}}^d\times[t_1,t_2]}, \Lambda^{x'}_r|_{{\mathbb{Z}}^d\times [t_1,t_2]\times[0,1]} \dvtx x,x'\in R \bigr), \]
that is, the $\sigma$-field generated by the points of the graphical construction in $R\times[t_1,t_2]$.
A coalescing branching random walk dual for $\xi^\varepsilon_t$ is constructed in \cite{CDP11}. We give here only the part of that dual which we need. Using only the Poisson processes $T^x_n,x\in{\mathbb{Z}^d}$, define a coalescing random walk system as follows. Fix $t>0$. For each $y\in{\mathbb{Z}^d}$ define $B^{y,t}_u, u\in[0,t]$ by putting $B^{y,t}_0=y$ and then proceeding ``down'' in the graphical construction and using the voter arrows to jump. More precisely, if $T^{y}_1>t$ put $B^{y,t}_u=y$ for all $u\in[0,t]$. Otherwise, choose the largest $T^y_j=s<t$, and put $B^{y,t}_u=y$ for $u\in[0,t-s)$ and $B^{y,t}_{t-s}=x+X_{x,j}$. Continue in this fashion to complete the construction of $B^{y}_u,u\in[0,t]$. Note that each $B^{y,t}_u$ is a rate one random walk with step distribution $p(\cdot)$ and that the walks coalesce when they meet: if $B^{x,t}_u= B^{y,t}_u$ for some $u\in[0,t]$, then $B^{x,t}_s=B^{y,t}_s$ for all $u\le s\le t$. On the event that no $*$-arrow is encountered along the path $B^{x,t}_\cdot$, that is, $(z,T^{*, z}_n)\ne (B^{x,t}_{t-u},t-u)$ for all $z,n$ and $0\le u\le t$, then
\begin{equation} \label{vmdual} \xi^\varepsilon_t(x) = \xi^\varepsilon_0 \bigl(B^{x,t}_t \bigr) \qquad \forall\xi^\varepsilon_0 \in\{0,1\}^{\mathbb{Z}^d}. \end{equation}
\begin{lem}\label{vmirred} Fix $t>0$, distinct $y_0,y_1\in{\mathbb{Z}^d}$ and finite disjoint $B_0,B_1\subset{\mathbb{Z}^d}$. Then there exists a finite $\Lambda=\Lambda(y_0,y_1,B_0,B_1)\subset{\mathbb{Z}^d}$ and a $\mathcal{G}(\Lambda\times[0,t])$-measurable event $G=G(t,y_0,y_1,B_0,B_1)$ such that $P(G)>0$ and on $G$: \begin{longlist}[(iii)] \item[(i)] $T^{*,z}_1>t$ for all $z\in\Lambda$;
\item[(ii)] $B^{x,t}_u\in\Lambda$ for all $x\in B_0\cup B_1$, $u\in[0,t]$;
\item[(iii)] $B^{x,t}_t=y_i$ for all $x\in B_i$, $i=0,1$. \end{longlist} If $\xi^\varepsilon_0(y_i)=i$, $i=0,1$, then on the event $G$, $\xi^\varepsilon_t(x)=i$ for all $x\in B_i$, $i=0,1$. \end{lem}
\begin{pf} We reason as in the proof of Lemma~\ref{suffindecomp}, but now working with the dual of $\xi ^\varepsilon $, using the fact that the $B^{y,t}_u$ are independent, irreducible random walks as long as they do not meet. There are sets $B'_0,B'_1$ which are far apart, each with widely separated points, such that a sequence of walk steps can move the walks from $B_0$ to $B'_0$ and $B_1$ to $B'_1$ without collisions. If $B_0'$ and $B_1'$ are sufficiently far apart, then by irreducibility there is a sequence of steps resulting in the walks from $B_0'$ coalescing at some site $y'_0$, the walks from $B_1'$ coalescing at some site $y'_1$, all without collisions between the two collections of walks, and with $y'_0$ and $y'_1$ far apart. Now by moving one walk at a time it is possible to prescribe a set of walk steps which take the two walks from $y_0$ and $y_1$ to $y'_0$ and $y'_1$, respectively, without collisions between the two walks. By reversing these steps (recall $p$ is symmetric) we can therefore have the above walks follow steps which will take them from $y'_0$ and $y'_1$ to $y_0$ and $y_1$, respectively, without collisions. In this way we can prescribe walk steps which occur with positive probability and ensure that $B_t^{x,t}=y_i$ for all $x\in B_i$. Let $\Lambda$ be a finite set large enough to contain all the positions of the walks in this process, and let $G$ be the event that $T^{*,x}_1>t$ for all $x\in\Lambda$, and such that the $T^{x}_n$ and $X_{x,n}$, $x\in \Lambda$, allow for the above prescribed sequence of walk steps to occur by time $t$. Then $G$ has the desired properties, and on this event, $\xi^\varepsilon_t(x)=\xi^\varepsilon_0(B^{x,t}_t)$ for all $x\in B_0\cup B_1$ by \eqref{vmdual}. Now the fact that (iii) holds on $G$, implies the final conclusion by the choice of $y_i$. \end{pf}
In addition to Lemma~\ref{vmirred} we will need the simpler fact that for any fixed $t>0$ and $z\in{\mathbb{Z}^d}$,
\begin{equation} \label{02z} \inf_{\xi^\varepsilon_0\dvtx\xi^\varepsilon_0(0)=1}P \bigl(\xi^\varepsilon_t(z)=1 \bigr)>0. \end{equation}
This is clear because there is a sequence of random walk steps leading from $0$ to $z$, and there is positive probability that the walk makes these steps before time $t$ and that no other transitions occur at any site in the sequence.
\begin{rem} It is clear that the above holds equally well for voter model perturbations in $d=2$. \end{rem}
\section{A complete convergence theorem for cancellative systems}\label{secflip}
To make effective use of annihilating duality we will need to know that for large $t$, if $\xi_t\ne\mathbf{0},\mathbf{1}$ and finite $A\subset{\mathbb{Z}^d}$ is large, then there will be many sites in $\xi_t\cap A$ which can flip values in a fixed time interval, and that the probability there will be an odd number of these flips is close to $1/2$. For $x\in{\mathbb{Z}^d}$ and $A\subset{\mathbb{Z}^d}$ define
\[ A(x,\xi) = \bigl\{y\in A\dvtx\xi(y)=1 \mbox{ and }\xi(y+x)=0 \bigr\}. \]
The conditions we will use are: there exists $x_0\in{\mathbb{Z}^d}$ such that
\begin{equation} \label{flip2}\qquad \lim_{K\to\infty}\mathop{\sup_{A\subset{\mathbb
{Z}^d}}}_{|A|\ge K} \limsup_{t\to
\infty} P \bigl(|\xi_{t}|>0 \mbox{ and }
A(x_0,\xi_t) = \varnothing\bigr)=0\qquad \mbox{if }|
\widehat\xi_0|=\infty \end{equation}
and
\begin{equation} \label{oddgoal} \lim_{K\to\infty}\mathop{\sup_{A\in Y,\xi_0\in\{ 0,1\}^{\mathbb{Z}^d}\dvtx}}_{
|A(x_0,\xi _0)|\ge K}
\bigl|P\bigl(|\xi_1\cap A|\mbox{ is odd}\bigr)-\tfrac12 \bigr| =0. \end{equation}
We will verify in Lemmas~\ref{lemodd2} and \ref{lemflip2} below that our voter model perturbations have these properties for all sufficiently small $\varepsilon$, but first we will show how they are used along with \eqref{eqH} to obtain complete convergence of $\xi_t$. Recall $\nu_{1/2}$ is the translation invariant stationary measure in \eqref{nuhalf1}.
\begin{prop}\label{propCCT} Let $\xi_t$ be a translation invariant cancellative spin-flip system with rate function $c(x,\xi)$ satisfying \eqref{traps}--\eqref{q02}, \eqref{flip2}, and \eqref{oddgoal}. Let $\zeta_t$ be the annihilating dual with $Q$-matrix given in \eqref{qmatrix} and assume that \eqref{eqH} holds. Then $\nu_{1/2}$
satisfies \eqref{coexist}, and if $|\widehat\xi_0|=\infty$ then
\begin{equation} \label{eqgoal} \xi_{t} \Rightarrow\beta_0(\xi_0) \delta_0 + \bigl(1-\beta_0(\xi_0) \bigr) \nu_{1/2} \qquad\mbox{as }t\to\infty. \end{equation}
\end{prop}
\begin{pf} We start with some preliminary facts. First, \eqref{flip2} implies that for any $m<\infty$ and $\xi_0\in\{0,1\}^{\mathbb{Z}^d}$ with
$|\widehat\xi_0|=\infty$,
\begin{equation} \label{flip2a} \lim_{K\to\infty}\mathop{\sup_{A\subset{\mathbb
{Z}^d}}}_{|A|\ge K} \limsup_{t\to
\infty} P \bigl(|\xi_{t}|>0 \mbox{ and }\bigl |
A(x_0,\xi_t)\bigr|<m \bigr)=0. \end{equation}
This is because
$|A|\ge mK$ implies $A$ can be written as the disjoint union of sets $A_1,\ldots,A_m$ with each $|A_i|\ge K$, and
\[
\bigl\{|\xi_{t}|>0\mbox{ and } \bigl| A(x_0,
\xi_t)\bigr|<m \bigr\} \subset\bigcup_{i=1}^m
\bigl\{|\xi_{t}|>0\mbox{ and } \bigl| A_i(x_0,
\xi_t)\bigr|=0 \bigr\}. \]
Applying \eqref{flip2} we obtain \eqref{flip2a}.
Next, we need a slight upgrade of the basic duality equation. As shown in \cite{G79}, \eqref{eqduality2} can be extended by applying the Markov property of $\xi_t$ at a time $v<t$. If the processes $\xi_t$ and $\zeta_t$ are independent, then for all $u,v\ge0$,
\begin{equation}
\label{eqduality3} P\bigl(|\xi_{v+u} \cap\zeta_0| \mbox{ is odd }\bigr)= P\bigl(|\xi_v \cap\zeta_u| \mbox{ is odd }\bigr). \end{equation}
Let $\nu_{1/2}$ be defined by \eqref{nuhalf2}. Since we are assuming
$|\widehat\xi_0|=\infty$, we have $\beta_1(\xi_0)=0$ by \eqref{betanonzero}. In view of \eqref{nuhalf2} and $\delta_{\mathbf{0}}(|\xi\cap A|\mbox{ is odd})=0$, to prove \eqref{eqgoal} it suffices to prove [recall \eqref{odddet}] that for fixed $A\in Y$,
\begin{equation}
\label{goalflip} \lim_{t\to\infty} P \bigl(|\xi_{t}\cap A| \mbox{ is odd} \bigr) =\frac12 \beta_\infty(\xi_0)P \bigl( \zeta^A_t\ne\varnothing\ \forall t\ge0 \bigr). \end{equation}
Fix $\varepsilon>0$. By \eqref{oddgoal} there exists $K_1<\infty$ such that if $B\in Y$
and $|B(x_0,\xi_0)|\ge K_1$, then
\begin{equation}
\label{odd1} \bigl| P\bigl(|\xi_1\cap B|\mbox{ is odd}\bigr)-\tfrac12 \bigr| <\varepsilon. \end{equation}
By \eqref{flip2a}, there exists $K_2<\infty$ and $s_0<\infty$ such that if
$|B|\ge K_2$ and $s\ge s_0$, then
\begin{equation}
\label{flip3} P \bigl(\xi_s\ne\varnothing\mbox{ and } \bigl|
B(x_0,\xi_s)\bigr|<K_1 \bigr)< \varepsilon. \end{equation}
By \eqref{eqH} we can choose $T=T(A,K_2)<\infty$ large enough so that
\begin{equation}
\label{dualbig1} P \bigl(0<\bigl|\zeta^A_T\bigr|\le K_2 \bigr) < \varepsilon. \end{equation}
\begin{figure}
\caption{$P(|\xi_{s+1}\cap\zeta^A_T|\mbox{ is odd})\approx \frac12P(\xi_s\ne\varnothing)P(\zeta^A_T\ne\varnothing)$.}
\label{fig1}
\end{figure}
For $t>1+T+s_0$ let $s=t-(1+T)$ and put $u=T$ and
$v=s+1$ in \eqref{eqduality3}. Then $P(|\xi_{t}\cap A|\mbox{ is odd}) = P(|\xi_{s+1}\cap\zeta^A_T|\mbox{ is odd})$, where $\xi_t$ and $\zeta^A_t$ are independent. (At this point the reader may want to consult Figure \ref{fig1} and Remark~\ref{Figdesc} below.) Making use of the Markov property of $\xi_t$, we obtain
\begin{eqnarray*}
&&P\bigl(|\xi_{t}\cap A|\mbox{ is odd}\bigr)- \frac12 P(\xi_s\ne \varnothing)P \bigl(\zeta^A_T\ne\varnothing\bigr) \\ &&\qquad= \sum_{B\ne\varnothing}P \bigl(\zeta^A_T=B
\bigr) \biggl[ P\bigl(|\xi_{s+1}\cap B|\mbox{ is odd}\bigr)-\frac12 P( \xi_s\ne\varnothing) \biggr] \\ &&\qquad = \sum_{B\ne\varnothing}P \bigl(\zeta^A_T=B
\bigr) E \biggl[ \biggl(E_{\xi_s}\bigl(|\xi_1\cap B|\mbox{ is odd}\bigr)-\frac12 \biggr) 1\{\xi_s\ne\varnothing\} \biggr]. \end{eqnarray*}
By \eqref{dualbig1},
\begin{eqnarray}
\label{decomp2}\qquad &&\biggl|P\bigl(|\xi_{t}\cap A|\mbox{ is odd}\bigr)- \frac12 P_\xi(\xi_s\ne\varnothing)P \bigl(\zeta^A_T
\ne\varnothing\bigr)\biggr| \nonumber \\[-8pt] \\[-8pt] \nonumber
&&\qquad< \varepsilon+ \sum_{|B|> K_2}P \bigl(\zeta^A_T=B
\bigr) E \biggl[ \biggl|E_{\xi_s}\bigl(|\xi_1\cap B|\mbox{ is odd}\bigr)-
\frac12 \biggr| 1\{\xi_s\ne\varnothing\} \biggr]. \end{eqnarray}
By \eqref{flip3}, since $s>s_0$, each expectation in the last sum is bounded above by
\begin{equation}
\label{decomp3} \varepsilon+ E \bigl[ \bigl| E_{\xi_s}\bigl(|\xi_1\cap B|
\mbox{ is odd}\bigr)-\tfrac12 \bigr| 1 \bigl\{B(x_0,\xi_s)\ge K_1 \bigr\} \bigr]. \end{equation}
Applying the bound \eqref{odd1} in this last expression, and then combining \eqref{decomp2} and~\eqref{decomp3} we obtain
\[
\bigl|P\bigl(|\xi_{t}\cap A|\mbox{ is odd}\bigr)- \tfrac12 P(\xi_s\ne
\varnothing)P \bigl(\zeta^A_T\ne\varnothing\bigr)\bigr| < 3 \varepsilon. \]
Let $t$ (and hence $s$) tend to infinity, and then $T$ tend to infinity above to complete the proof of \eqref{goalflip} and hence \eqref{eqgoal}.
Finally, let $|\xi_0|=|\widehat\xi_0|=\infty$. Then $\beta_0(\xi_0)=0$ by \eqref{betanonzero}, so \eqref{eqgoal} and \eqref{flip2a}
imply that for any finite $m$, $\nu_{1/2}(|\xi|\ge m,
|\widehat\xi|\ge m) =1$, and this implies coexistence. \end{pf}
\begin{rem}\label{Figdesc} Figure~\ref{fig1} above gives a graphical view of the above argument. Time runs up for $\xi$ and down for $\zeta$. Conditional on $\xi_s\ne\varnothing$ and $\zeta^A_T\ne\varnothing$, \eqref{eqH} guarantees that $B=\zeta^A_T$ is large, and \eqref{eqduality3} guarantees $B(x_0,\xi_s)$ is large. In the dashed boxes in Figure \ref{fig1}, the $(\bullet\circ)$ pairs indicate the locations $(x,x+x_0)$, $x\in B(x_0,\xi_s)$. Finally,
\eqref{oddgoal} now guarantees that $|\xi_{s+1}\cap B|$ will be odd with probability approximately $\frac12$. \end{rem}
Verification of \eqref{flip2} for our voter model perturbations requires a comparison with oriented percolation which we will save for the next section. Here we present a proof of \eqref{oddgoal}, based heavily on ideas from \cite{BDD}. See Lemma~7 of \cite{SS} for a purely cancellative version of this result.
\begin{lem}\label{lemodd2} If $\xi^\varepsilon_\cdot$ is a voter model perturbation, then there exists $\varepsilon_1>0$ and $x_0\in{\mathbb{Z}^d}$ such that \eqref {oddgoal} holds for $\xi^\varepsilon_\cdot$ if $\varepsilon<\varepsilon_1$. \end{lem}
\begin{pf} Fix any $x_0$ with $p(x_0)>0$. We will prove that if $\delta>0$, then there exists $K$ such that if
$|A(x_0,\xi_0)|\ge K$, then
\begin{equation}
\label{oddgoalep} \bigl| P \bigl(\bigl|\xi^\varepsilon_1\cap A\bigr|\mbox{ is odd} \bigr) -\tfrac12 \bigr| <\delta. \end{equation}
Using the graphical construction of $\xi^\varepsilon_t$ described in Section~\ref{secirred}, we define a version of the ``almost isolated sites'' of \cite{BDD}. First we give the informal definition. For $x\in{\mathbb{Z}^d}$, let $U(x)$ be the indicator of the event that during the time period $[0,1]$, no change can occur at site $x+x_0$ and no change can occur at $x$ except possibly due to a (first) voter arrow directed from $x$ to $x+x_0$. Let $V(x)$ be the indicator of the event that during the time period $[0,1]$ no site
$y$ outside $\{x,x+x_0\}$ can change due to the value at $x$. More formally, for $y\in{\mathbb{Z}^d}$ and $A\in Y$ with $|A|\le N_0$, define
\begin{eqnarray*} \tau(y,A)& =& \min\bigl\{T^y_n\dvtx A= \{X_{y,n} \}, n\in{\mathbb{N}}\bigr\} \\ &&{}\wedge\min\bigl\{T^{*,y}_n\dvtx A= \bigl \{Z^1_{y,n},\ldots,Z^{N_0}_{y,n} \bigr\}, n \in{\mathbb{N}}\bigr\}, \end{eqnarray*}
and $\tau(y)=\min\{\tau(y,A)\dvtx A\in Y$\}. We can now define
\begin{eqnarray*} U(x) &= &1 \bigl\{ \tau(x+x_0)>1, X_{x,1}=x_0, T^x_2>1\mbox{ and }T^{*,x}_1>1 \bigr \}\quad \mbox{ and} \\ V(x) &= &1 \bigl\{ \tau(y,A)>1 \ \forall y\in{\mathbb{Z}^d}\setminus \{x,x+x_0 \}\mbox{ and } A\in Y\colon x\in y+A \bigr\}, \end{eqnarray*}
and call $x$ almost isolated if $U(x)V(x)=1$.
By standard properties of Poisson processes,
\begin{equation} \label{Poissonprop}\tau(x,A)\mbox{ and } \tau(y,B)\mbox{ are independent whenever }x\ne y\mbox{ or }A\ne B. \end{equation}
We also define
\[ \nu(A) = P \bigl(\{X_{0,1}\}=A \bigr) + P \bigl( \bigl \{Z^1_{0,1},\ldots,Z^{N_0}_{0,1} \bigr\}=A \bigr), \]
and observe that $\nu(A)=0$ if $|A|>N_0$, and $\sum_{A\in Y}\nu(A)= 2$. Use the fact that $\{T^0_n\dvtx\{X_{0,n}\}=A\}$ are the points of a Poisson point process with rate $P(\{X_{0,1}\}=A)$, and $\{T^{*,0}_n\dvtx\{ Z^1_{0,n},\ldots,Z_{0,n}^{N_0}\}=A\}$ are the points of an independent Poisson point process with rate $\bar c\varepsilon^2P(A=\{Z^1_{0,n},\ldots,Z_{0,n}^{N_0}\})$ to conclude that
\begin{eqnarray} \label{rangeest}\qquad P \bigl(\tau(y,A)>1 \bigr)&=&\exp\bigl(-P \bigl( \{X_{0,1} \}=A \bigr)-\bar c\varepsilon^2P \bigl( \bigl\{ Z^1_{0,n}, \ldots,Z_{0,n}^{N_0} \bigr\}=A \bigr) \bigr) \nonumber \\[-8pt] \\[-8pt] \nonumber & \ge& e^{-(1+\bar c)\nu(A)}. \end{eqnarray}
For each $x\in{\mathbb{Z}^d}$, the variables $U(x),V(x)$ are independent [this much is clear from \eqref{Poissonprop}], and we claim that $u_0=E(U(x))$ and $v_0=E(V(x))$ are positive uniformly in $\varepsilon$. To check this for $v_0$ we apply \eqref{rangeest} to get
\begin{eqnarray*} v_0 &\ge&\exp\biggl(-(1+\bar c)\sum_{A\in Y} \sum_{y\in{\mathbb{Z}^d}\setminus \{x\}} \nu(A) 1\{x-y\in A\} \biggr) \\
& \ge&\exp\biggl(-(1+\bar c)\sum_{A\in Y}|A|\nu(A) \biggr), \end{eqnarray*}
which is positive, uniformly in $\varepsilon\le\varepsilon_0$. For $u_0$, we have by the choice of $x_0$,
\[ u_0\ge\exp\biggl\{-(1+\bar c)\sum_{A\in Y} \nu(A) \biggr\} p(x_0)P \bigl(T^0_2>1 \bigr)P \bigl(T_1^{*,0}>1 \bigr)>0. \]
If $w_0=w_0(\varepsilon)=u_0v_0$, then we have verified that
\begin{equation} \label{gammalbnd}\gamma=\min\bigl\{w_0(\varepsilon)\dvtx0<\varepsilon\le\varepsilon _0 \bigr\}>0. \end{equation}
Now suppose ${\mathcal{Y}}=\{y_1,\ldots,y_J\}\subset{\mathbb{Z}^d}$ and
$|y_i-y_j|>2|x_0|$ for $i\ne j$. Then $U(y_1),\ldots,U(y_J)$ are independent, but
$V(y_1),\ldots,V(y_J)$ are not. Nevertheless, we claim they are almost independent if all $|y_i-y_j|$, $i\ne j$, are large, and hence if we let $W(y_i)=U(y_i)V(y_i)$, then $W(y_1),\ldots, W(y_J)$ are almost independent. More precisely, we claim that for any $J\ge2$ and $a_i\in\{0,1\}$, $1\le i\le J$,
\begin{eqnarray}
\label{ai}\qquad && \lim_{n\to\infty} \mathop{\sup_{{\mathcal{Y}}=\{y_1,\ldots,y_J\},}}_{|y_i-y_j|\ge n \ \forall i\ne j}
\Biggl|P \bigl( W(y_i)=a_i \ \forall1\le i\le J \bigr) -\prod _{i=1}^JP \bigl(W(y_i)=a_i
\bigr) \Biggr| \nonumber \\[-8pt] \\[-8pt] \nonumber &&\qquad = 0. \end{eqnarray}
For the time being, let us suppose this fact.
Given $J$ and ${\mathcal{Y}}=\{y_1,\ldots,y_J\}$, let $S(J,{\mathcal{Y}})=\sum_{y\in{\mathcal{Y}}}W(y)$. Then \eqref{ai} implies $S(J,{\mathcal{Y}})$ is approximately binomial if the $y_i$ are well separated. That is, if ${\mathcal{B}}(J,w_0)$ is a binomial random variable with parameters $J$ and $w_0$, and
\[
\Delta(J,{\mathcal{Y}},k)= \bigl|P \bigl(S(J,{\mathcal{Y}})=k \bigr) - P \bigl({\mathcal{B}}(J,w_0)=k
\bigr) \bigr|, \]
then \eqref{ai} implies that for $k=0,\ldots,J$,
\begin{equation} \label{spreadout} \lim_{n\to\infty}\mathop{\sup_{{\mathcal{Y}}=(y_1,\ldots,y_J)}}_{
|y_i-y_j|\ge n, i\ne j} \Delta(J,{\mathcal{Y}},k) =0. \end{equation}
Now fix $\delta>0$. A short calculation shows that
\begin{equation}
\label{mireq} p_0=P \bigl(T^x_1>1|U(x)=1
\bigr)=P \bigl(T_1^x>1|T^x_2>1 \bigr)=\tfrac{1}{2}. \end{equation}
By \eqref{gammalbnd} and \eqref{spreadout}, we may choose $J=J(\delta)$ such that
\begin{equation} \label{oddgamma} (1-\gamma)^J <\delta, \end{equation}
and then $n=n(J,\delta)$ so that for all ${\mathcal{Y}}=\{y_1,\ldots,y_J\}$ with $|y_i-y_j|\ge n$ for $i\ne j$,
\begin{equation} \label{binomialapprox} \Delta(J,{\mathcal{Y}},0) <\delta. \end{equation}
Given $J$ and $n$, it is easy to see that there exists $K=K(J,n)$ such that if $B\subset{\mathbb{Z}^d}$ and $|B|\ge K$, then $B$ must contain some ${\mathcal{Y}}=\{y_1,\ldots,y_J\}$ such that $|y_i-y_j|\ge n$ for $i\ne j$.
Now suppose that $|A(x_0,\xi^\varepsilon_0)|\ge K$ and ${\mathcal{Y}}=\{y_1,\ldots, y_J\}\subset A(x_0,\xi^\varepsilon_0)$ with
$|y_i-y_j|\ge n$ for all $i\neq j$. Let ${\mathcal{I}}$ be the set of $y_j$ with $W(y_j)=1$, so that
$|{\mathcal{I}}|=S(J,{\mathcal{Y}})$. Let $\mathcal{G}$ be the $\sigma$-field generated by
\begin{eqnarray} \label{list} &&\bigl\{1(y_j\in{\mathcal{I}})\dvtx j=1,\ldots,J \bigr\} \nonumber \\[-8pt] \\[-8pt] \nonumber &&\qquad{}\cup\bigl\{1 \bigl\{x\in{\mathcal{I}}^c \bigr\} \bigl (T_n^x,T^{*,x}_n,X_{x,n},Z_{x,n},U_{x,n} \dvtx x\in{\mathbb{Z}^d},n\ge1 \bigr) \bigr\}. \end{eqnarray}
If $g_j=1\{T^{y_j}_1>1\}$, then conditional on $\mathcal{G}$,
\begin{equation} \label{Bin2} \{g_j\dvtx y_j\in{\mathcal{I}}\}\mbox{ are i.i.d. Bernoulli rv's with mean $p_0=\tfrac{1}{2}$}, \end{equation}
and $X=\sum_j 1\{y_j\in{\mathcal{I}}\}g_j$ is binomial with parameters $(|{\mathcal{I}}
|,p_0=\frac{1}{2})$. This is easily checked by conditioning on the $\mathcal{G}$-measurable set ${\mathcal{I}} $ and using \eqref{mireq}.
Let $h=\sum_{x\in A}1\{x\notin{\mathcal{I}})\xi^\varepsilon_1(x)$. Then at time 1 we have the decomposition
\begin{equation}
\label{decomp1} \bigl|\xi^\varepsilon_1\cap A\bigr|=h + X, \end{equation}
where we have used the fact that for $y_j\in{\mathcal{I}}$, $\xi^\varepsilon_s(y_j)$ will flip from a $1=\xi^\varepsilon_0(y_j)$ to a $0=\xi^\varepsilon_0(y_j+x_0)$ during the time interval $[0,1]$ if and only if $g_j=0$. Since $h$ is $\mathcal{G}$-measurable,
\begin{eqnarray}
P \bigl( \bigl|\xi^\varepsilon_1\cap A\bigr| \mbox{ is odd}\mid\mathcal{G}\bigr) ( \omega) = P \bigl( X =1-h(\omega) \bmod2 \mid\mathcal{G}\bigr) (\omega)= \tfrac{1}{2} \nonumber\\
\eqntext{\mbox{a.s. on }\bigl\{|{\mathcal{I}}|>0\bigr\},} \end{eqnarray}
the last by an elementary binomial calculation and the fact that conditional on $\mathcal{G}$,
$X$ is binomial with parameters $(|{\mathcal{I}}|,\frac{1}{2})$. Take expectations in the above and use~\eqref{binomialapprox} and then \eqref{oddgamma} to conclude that
\begin{eqnarray}
\label{odd2} \bigl|P \bigl(\bigl| \xi^\varepsilon_1\cap A\bigr| \mbox{ is odd}
\bigr) - \tfrac12\bigr| &\le& P\bigl(|{\mathcal{I}}|=0\bigr) \nonumber\\ &\le&\delta+P \bigl({\mathcal{B}}(J,\gamma)=0 \bigr) \\ \nonumber &\le&\delta+(1-\gamma)^J\le2\delta. \end{eqnarray}
This yields inequality \eqref{oddgoalep}.
It remains to verify \eqref{ai}. This is easy if $p$ and $q_Z$ have finite support; see Remark~\ref{finsuppeasy} below. In general, the idea is to write
\begin{equation} \label{Vprod} V(y_i)=V_1(y_i)V_2(y_i) V_3(y_i), \end{equation}
where $V_1(y_1),\ldots, V_1(y_J)$ are independent and independent of $U(y_1),\ldots,U(y_J)$, and
$V_2(y_1),V_3(y_1),\ldots,V_2(y_J),V_3(y_J)$ are all one with high probability if the $y_i$ are sufficiently spread out. Let $n\ge2|x_0|$ and ${\mathcal{Y}}=\{y_1,\ldots,y_J\}$ be given with
$|y_i-y_j|\ge n$, and let ${\mathcal{Y}}_0=\{y_1+x_0,\ldots,y_J+x_0\}$. Note that $y_1,y_1+x_0,\ldots,y_J,y_J+x_0$ are distinct. Define
\begin{eqnarray*} V_1(y_i)&=& 1 \bigl\{ \tau(z,A)>1 \ \forall z\notin{\mathcal{Y}} \cup{\mathcal{Y}}_0, A\in Y\dvtx(z+A)\cap{\mathcal{Y}}=\{y_i\} \bigr\}, \\ V_2(y_i)&=&1 \bigl\{\tau(z,A)>1 \ \forall z\notin{\mathcal{Y}}\cup {\mathcal{Y}}_0, A\in Y\dvtx z+A\supset\{y_i,y_j\}\mbox{ for some }j\ne i \bigr\}, \\ V_3(y_i)&= &1 \bigl\{\tau(z,A)>1 \ \forall z\in({\mathcal{Y}}\cup {\mathcal{Y}}_0)\setminus\{y_i,y_i+x_0 \}, A\in Y\dvtx y_i\in z+A \bigr\}. \end{eqnarray*}
A bit of elementary logic shows that \eqref{Vprod} holds. If a pair $(z,A)$ occurs in the definition of some $V_1(y_i)$, then it cannot occur in any $V_1(y_j), j\ne i$, and hence $V_1(y_1),\ldots,V_1(y_J)$ are independent, and also independent of $U(y_1),\ldots,U(y_J)$. Therefore, to prove \eqref{ai} it suffices to prove that
\begin{equation}
\label{ai2} \lim_{n\to\infty} \mathop{\sup_{{\mathcal{Y}}=\{y_1,\ldots,y_J\},}}_{|y_i-y_j|\ge n \ \forall i\ne j}P \bigl(V_2(y_i)V_3(y_i)\ne1 \bigr) = 0. \end{equation}
We treat $V_3(y_i)$ first. By \eqref{rangeest},
\begin{eqnarray*} P \bigl(V_3(y_i)=1 \bigr) &\ge&\exp\biggl(-(1+\bar c) \sum_{z\in({\mathcal{Y}}\cup{\mathcal{Y}}_0)\setminus\{ y_i,y_i+x_0\}} \sum_{A\in Y} \nu(A)1\{y_i\in z+A\} \biggr) \\ &\ge&\exp\biggl(-2J(1+\bar c) \sum_{A\in Y}\nu(A)1 \bigl\{\operatorname{diam}(A)>n \bigr\} \biggr) \\ &\to& 1\qquad \mbox{as }n\to\infty. \end{eqnarray*}
To treat $V_2(y_i)$ we note that if a pair $(z,A)$ occurs in the definition of $V_2(y)$, then $\operatorname{diam}(A)\ge n$, so
\begin{eqnarray*} \hspace*{-4pt}&&P \bigl(V_2(y_i)=1 \bigr) \\ \hspace*{-4pt}&&\quad\ge\exp\biggl(-(1+\bar c)\sum_{B\subset{\mathcal{Y}}}\sum _{A\in Y}\sum_{z\ne y_i} 1 \bigl \{y_i\in B= (z+A)\cap{\mathcal{Y}},\operatorname{diam}(A)\ge n \bigr\}\nu(A) \biggr). \end{eqnarray*}
In the sum above, given $B\ni y_i$ there at most $|A|$ choices for $z$ such that $(z+A)\cap{\mathcal{Y}}=B$. In fact, there are at most $|A|$ choices of $z$ such that $y_i\in z+A$ as this implies $z\in y_i-A$. Thus
\begin{eqnarray*} P \bigl(V_2(y_i)=1 \bigr) &\ge&\exp\biggl(-(1+\bar c) \sum_{B\subset{\mathcal{Y}}}\sum_{A\in Y} 1
\bigl\{\operatorname{diam}(A)\ge n \bigr\}|A|\nu(A) \biggr) \\ &\ge&\exp\biggl(-(1+\bar c) 2^J\sum_{A\in Y}
1 \bigl\{\operatorname{diam}(A)\ge n \bigr\}|A|\nu(A) \biggr) \\ &\to&1 \qquad\mbox{as }n\to\infty. \end{eqnarray*}
This proves \eqref{ai2} and hence \eqref{ai}. \end{pf}
\begin{rem}\label{finsuppeasy}Note that if $p(\cdot)$ and $q_Z(\cdot )$ have finite support, then the proof simplifies somewhat because for large enough $n$ the left-hand side of \eqref{ai} is zero. This is because the $A$'s arising in the definition of $V(x)$ will have uniformly bounded diameter which will show that for
$|y_i-y_j|$ large, $V(y_1),\ldots,V(y_J)$ will be independent. \end{rem}
\section{\texorpdfstring{Proof of Theorem~\protect\ref{thmCCTpert}} {Proof of Theorem 1.2}} \label{secthmproof} To prove Theorem~\ref{thmCCTpert} it will suffice, in view of Proposition~\ref{propCCT}, Lemmas~\ref{lemH} and \ref{lemodd2} and Remark~\ref{remHaltcond}, to prove that for small enough~$\varepsilon$, both conditions \eqref{eqH3} and \eqref{flip2} hold for $\xi^\varepsilon_t$. The proof of \eqref{flip2} is given in Lemma~\ref{lemflip2} below after first developing the necessary oriented percolation machinery. With \eqref{flip2} in hand the proof of \eqref{eqH3} is then straightforward.
We suppose now that $\xi^\varepsilon$ is a voter model perturbation with rate function $c_\varepsilon(x,\xi)$ and that all the assumptions of Theorem~\ref{thmCCTpert} are in force. We also assume that $\xi^{\varepsilon}$ is constructed using the Poisson processes $T^x_n,T^{*,x}_n$ and the variables $X_{x,n},Z^i_{x,n},U_{x,n}$ as in Section~\ref{secirred}. We assume
$|{\hat\xi}^\varepsilon_0|=\infty$, so that by \eqref{betanonzero} $\beta_1(\xi^\varepsilon_0)=0$. By the results of \cite{CDP11} for small $\varepsilon$ we expect that when $\xi
^\varepsilon_t$ survives, there will be blocks in space--time, in the graphical construction, containing both 0's and 1's, which dominate a super-critical oriented percolation. The percolation process necessarily spreads out. So if $A\subset{\mathbb{Z}^d}$ is large, eventually there will be many blocks containing 0's and 1's near the sites of $A$ at times just before $t$, allowing for many independent tries to force $|A(\xi_t,x_0)|\ge1$.
Let ${\mathbb{Z}^d_e}$ be the set of $x\in{\mathbb{Z}^d}$ such that $\sum_iz_i$ is even. Let ${\mathcal{L}}=\{(x,n)\subset{\mathbb{Z}^d}\times{\mathbb{Z}}^+\dvtx\sum_i x_i +n\mbox{ is even}\}$. We equip ${\mathcal{L}}$ with edges from $(x,n)$ to $(x+e,n+1)$ and $(x-e,n+1)$ for all $e\in\{e_1,\ldots,e_d\}$, where $e_i$ is the $i$th unit basis vector. Given a family of Bernoulli random variables $\theta(x,n), (x,n)\in{\mathcal{L}}$, we define open paths in ${\mathcal{L}}$ using the $\theta(y,n)$ and the edges in ${\mathcal{L}}$ in the usual way. That is, a sequence of points $z_0,\ldots,z_n$ in ${\mathcal{L}}$ is an open path from $z_0$ to $z_n$ if and only if there is an edge from $z_i$ to $z_{i+1}$ and $\theta(z_i)=1$ (in which case we say site $z_i$ is open) for $i=0,\ldots,n-1$. We will write $(x,n)\to(y,m)$ to indicate there is an open path in ${\mathcal{L}}$ from $(x,n)$ to $(y,m)$. Define the open cluster starting at $(x,n)\in{\mathcal{L}}$,
\[ {\mathcal{C}}(x,n) = \bigl\{ (y,m)\in{\mathcal{L}}\dvtx m\ge n \mbox{ and } (x,n)\to(y,m) \mbox{ in } {\mathcal{L}}\bigr\}. \]
For $(x,n)\in{\mathcal{L}}$ let $W^{(x,n)}_{m}=\{y\dvtx(x,n)\to(y,m)\}$, $m\ge n$. We will write $W^0_n$ for $W^{(0,0)}_n$. For $k=1,\ldots,d$, say that $(x,n)\to_k (y,m)$ if there is an open path from $(x,n)$ to $(y,m)$ using only edges of the form $(x,n)\to(x+e_k,n+1)$ or $(x,n)\to(x-e_k,n+1)$. We define the corresponding ``slab'' clusters ${\mathcal{C}}_k(x,n)$ and processes $W^{(x,n)}_{k,m}$ using these paths. Clearly ${\mathcal{C}}_k(x,n)\subset{\mathcal{C}}(x,n)$ and $W^{(x,n)}_{k,m}\subset W^{(x,n)}_m$. If $W_0\subset{\mathbb{Z}}^d$, let $W_m=\bigcup_{x\in W_0}W_m^{(x,0)}$.
\begin{lem}\label{perclem} Suppose the $\{\theta(z,n)\}$ are i.i.d., and $1-\gamma=P(\theta(x,n)=1)\ge1-6^{-4}$. Then
\begin{equation}
\label{percolate} \rho_\infty=P \bigl(\bigl|{\mathcal{C}}_1(0,0)\bigr|=\infty \bigr)>0 \end{equation}
and
\begin{equation}
\label{0occupied} \lim_{K\to\infty} \mathop{\sup _{A\subset 2{\mathbb{Z}^d}}}_{|A|\ge K} \limsup_{n\to\infty}P \bigl(W^0_{2n}\ne\varnothing\mbox{ and } W^0_{2n}\cap A=\varnothing\bigr) =0. \end{equation}
\end{lem}
\begin{pf} For \eqref{percolate}, see Theorem~A.1 (with $M=0$) in \cite{Dur95}. The limit \eqref{0occupied} is known for $d=1$, while the $d>1$ case is an immediate consequence of the ``shape theorem'' for $W^0_{2n}$, the discrete time analogue of the shape theorem for the contact process in \cite{DG82}. Since this discrete time result does not appear in the literature, we will give a direct proof of \eqref{0occupied}, but for the sake of simplicity will restrict ourselves to the $d=2$ case. We need the following $d=1$ results, which we state using our ``slab'' notation,
\begin{equation} \label{percdens} \exists\rho_1>0 \mbox{ such that}\qquad\liminf _{n\to\infty} P \bigl((x,0)\in W^{0}_{1,2n} \bigr) \ge\rho_1\qquad \mbox{for all } x\in2{\mathbb{Z}},\hspace*{-35pt} \end{equation}
and for fixed $K_0\in{\mathbb{N}}$,
\begin{equation} \label{K0occupied1d1} \lim_{K\to\infty}\sup_{A\subset2{\mathbb{Z}}\times\{
0\}, |A|\ge K} \limsup_{n\to \infty} P \bigl(W^0_{1,2n}\ne
\varnothing,\bigl|W^0_{1,2n}\cap A \bigr|<K_0\bigr) = 0. \end{equation}
These facts are easily derived using the methods in \cite{Dur84}; see also Lemma~3.5 in~\cite{DN}, the Appendix in \cite{DN91} and Section~2 of \cite{BN94}.
The idea of the proof of the $d=2$ case of \eqref{0occupied} is the following. If $n$ is large, then on the event $W^0_{2n}\ne\varnothing$ we can find, with high probability, a point $z\in W^0_{2k}$ for some small $k$ such that $W^{(z,2k)}_{1,2m}\ne\varnothing$ for some large $m<n$. With high probability $W^{(z,2k)}_{1,2m}$ will contain many points $z'$ from which we can start independent ``$e_2$'' slab processes $W^{(z',2m)}_{2,2n}$. Many of these will be large, providing many independent chances for $W^{(z',2m)}_{2,2n}\cap A\ne\varnothing$, forcing $W^0_{2n} \cap A\ne\varnothing$.
Here are the details. We may assume without loss of generality that all sets $A$ considered here are finite. Fix $\delta>0$, and choose positive integers $J_0,K_0$ satisfying $(1-\rho_\infty)^{J_0}<\delta$ and $(1-\rho_1)^{K_0}<\delta$. By \eqref{K0occupied1d1} we can choose a positive integer $K_1$ such that for all
$A\subset2{\mathbb{Z}}\times\{0\}$, $|A|\ge K_1$,
\begin{equation} \label{K0occupied1d2} \limsup_{n\to\infty}P \bigl(W^0_{1,2n}
\ne\varnothing, \bigl|W^0_{1,2n}\cap A\bigr|<K_0 \bigr) < \delta. \end{equation}
For $x=(x_1,x_2)\in{\mathbb{Z}}^2$ and $A\subset{\mathbb{Z}}^2$ let $\pi_1x=(x_1,0)$,
$\pi_2x=(0,x_2)$ and $\pi_iA=\{\pi_ia\dvtx a\in A\}$, $i=1,2$. Observe that at least one of the $|\pi_iA|\ge\sqrt{|A|}$. We now fix any $A\subset2{\mathbb{Z}}^2$ with $|A|\ge K_1^2$, and suppose
$|\pi_1A|\ge K_1$. For convenience later in the argument, fix any $A'\subset A$ such that $\pi_1A'=\pi_1A$ and $\pi_1$ is one-to-one on $A'$. By \eqref{K0occupied1d2} we may choose a positive integer $n_1=n_1(A)$ such that
\begin{equation} \label{K0occupied1d3} P \bigl( W^0_{1,2n}\ne\varnothing,
\bigl|W^0_{1,2n}\cap\pi_1 A'\bigr|<K_0 \bigr) < \delta\qquad\mbox{for all }n\ge n_1. \end{equation}
We may increase $n_1$ if necessary so that
$P(|{\mathcal{C}}_1(0,0)|<\infty, W^0_{1,2n_1}\ne\varnothing)<\delta$, which implies that
\begin{equation} \label{dieaftern1} P \bigl(W^0_{1,2n_1}\ne\varnothing, W^0_{1,2n}=\varnothing\bigr) < \delta\qquad\mbox{for all }n\ge n_1. \end{equation}
Let $m(j)=2(j-1)n_1, j=1,2,\ldots,$ and define a random sequence of points $z_1,z_2,\ldots$ as follows. If $W^{0}_{m(j)}\ne\varnothing$, let $z_j$ be the point in $W^{0}_{m(j)}$ closest to the origin, with some convention in the case of ties. If $W^{0}_{m(j)}=\varnothing$, put $z_j=0$. Define
\[ N = \inf\bigl\{j \dvtx z_j\in W^{0}_{m(j)} \mbox{ and } W^{(z_j,m(j))}_{1,m(j+1)}\ne\varnothing\bigr\}. \]
Since $P(W^0_{1,2n_1}=\varnothing)\le1- \rho_\infty$, the Markov property implies
\begin{eqnarray*} P \bigl(W^0_{m(j)}\ne\varnothing, N>j \bigr) &=& P \bigl(W^0_{m(j)}\ne\varnothing, N>j-1, W^{(z_j,m(j))}_{1,m(j+1)}= \varnothing\bigr) \\ &\le&(1- \rho_{\infty}) P \bigl(W^0_{m(j)}\ne \varnothing, N>j-1 \bigr). \end{eqnarray*}
The above is at most $(1- \rho_{\infty}) P(W^0_{m(j-1)}\ne \varnothing, N>j-1)$, so iterating this, we get
\begin{equation} \label{geombnd}P \bigl(W^0_{m(j)}\ne\varnothing, N>j \bigr) \le(1-\rho_\infty)^j, \end{equation}
and if $n>J_0n_1$, then
\begin{equation} \label{geombound}\quad P \bigl(W^0_{2n}\ne\varnothing, N>J_0 \bigr) \le P \bigl(W^0_{m(J_0)}\ne \varnothing, N>J_0 \bigr) \le(1-\rho_{\infty})^{J_0}< \delta. \end{equation}
We need a final preparatory inequality. Using \eqref{K0occupied1d3} and the Markov property, for $n>n_1$ we have
\begin{eqnarray*} &&P \bigl(W^0_{1,2n}\ne\varnothing, W^0_{2n} \cap A=\varnothing\bigr) \\
&&\qquad\le\delta+\sum_{B\subset A', |B|\ge K_0}P \bigl(W^0_{1,2n_1} \cap\pi_1A'=\pi_1B \bigr) P \bigl(x\notin W^{(\pi_1x,2n_1)}_{2,2n}\ \forall x\in B \bigr) \\
&&\qquad\le\delta+\sum_{B\subset A', |B|\ge K_0}P \bigl(W^0_{1,2n_1} \cap\pi_1A'=\pi_1B \bigr) \prod _{x\in B}P \bigl(x\notin W^{(\pi_1x,2n_1)}_{2,2n} \bigr), \end{eqnarray*}
the last step by independence of the slab processes. Thus, employing \eqref{percdens},
\begin{equation} \label{prelim} \limsup_{n\to\infty}P \bigl(W^0_{1,2n} \ne\varnothing, W^0_{2n}\cap A=\varnothing\bigr) \\ \le\delta+(1-\rho_1)^{K_0} <2\delta. \end{equation}
We are ready for the final steps. For each $j\le J_0$ and $n\ge J_0n_1$, by the Markov property and \eqref{dieaftern1},
\begin{eqnarray*} &&P \bigl(W^0_{2n}\ne\varnothing,W^0_{2n} \cap A=\varnothing,N=j \bigr) \\ &&\qquad\le P \bigl(W^0_{m(j)}\ne\varnothing,N>j-1, W^{(z_j,m(j))}_{1,m(j+1)}\ne\varnothing, W^{(z_j,m(j))}_{2n} \cap A=\varnothing\bigr) \\ &&\qquad=\sum_{z}P \bigl(W^0_{m(j)} \ne\varnothing, N>j-1,z_j=z \bigr) \\ &&\qquad\qquad{} \times P \bigl(W^{(z,m(j))}_{1,m(j+1)}\ne\varnothing, W^{(z,m(j))}_{2n} \cap A=\varnothing\bigr) \\ &&\qquad\le\sum_{z}P \bigl(W^0_{m(j)} \ne\varnothing, N>j-1,z_j=z \bigr) \\ &&\hspace*{14pt}\qquad\quad{} \times\bigl(\delta+ P \bigl(W^{(z,m(j))}_{1,2n}\ne\varnothing, W^{(z,m(j))}_{2n}\cap A=\varnothing\bigr) \bigr). \end{eqnarray*}
Applying \eqref{prelim} and then \eqref{geombnd}, we obtain
\begin{eqnarray*} \limsup_{n\to\infty} P \bigl(W^0_{2n}\ne \varnothing,W^0_{2n}\cap A=\varnothing,N=j \bigr) &\le&3 \delta P \bigl(W^0_{m(j)}\ne\varnothing,N>j-1 \bigr) \\ &\le&3\delta(1-\rho_\infty)^{j-1}. \end{eqnarray*}
It follows that
\begin{eqnarray*} &&\limsup_{n\to\infty} P \bigl(W^0_{2n}\ne \varnothing,W^0_{2n}\cap A=\varnothing\bigr) \\ &&\qquad \le\limsup_{n\to\infty} P \bigl(W^0_{2n} \ne\varnothing,N>J_0 \bigr)+ 3\delta\sum _{j=1}^{J_0}(1- \rho_\infty)^{j-1} \\ &&\qquad\le\delta+ 3\delta/\rho_\infty \end{eqnarray*}
by using \eqref{geombound} and summing the series. This completes the proof. \end{pf}
Now we follow \cite{Dur95} and Section~6 of \cite{CDP11} in describing a setup which connects our spin-flip systems with the percolation process defined above. Let $K,L,T$ be finite positive constants with $K, L\in {\mathbb{N}} $, let $r=\frac{1}{16d}$, $Q_\varepsilon=[0,\lceil\varepsilon^{r-1}\rceil]^d\cap{\mathbb{Z}}^d$ and $Q(L)=[-L,L]^d$. We define a set $H$ of configurations in $\{0,1\}^{{\mathbb{Z}}^d}$ to be an unscaled version of the set of configurations in $\{0,1\}^{\varepsilon{\mathbb{Z}}^d}$ of the same name in Section~6 of \cite{CDP11}, that is,
\begin{eqnarray*}
&&H= \biggl\{\xi\in\{0,1\}^{{\mathbb{Z}}^d}\dvtx|Q_\varepsilon|^{-1} \sum_{y\in Q_\varepsilon}\xi(x+y)\in I^*\\ &&\hspace*{33pt}\mbox{for all }x\in Q(L) \cap\bigl( \bigl[0, \bigl\lceil\varepsilon^{r-1} \bigr\rceil \bigr]^d \cap{\mathbb{Z}}^d \bigr) \biggr\}. \end{eqnarray*}
Here $I^*$ is a particular closed subinterval of $(0,1)$; it is $I^*_\eta$ in the notation of Section~6 in \cite{CDP11}. The key property we will need of $H$ is
\begin{equation} \label{Hprop} \mbox{for each $\xi\in H$ there are $y_0,y_1 \in Q(L)\cap{\mathbb{Z}}^d$ s.t. $\xi(y_i)=i$ for $i=0,1$.}\hspace*{-35pt} \end{equation}
This is immediate from the definition and the fact that $I^*$ is a closed subinterval of $(0,1)$. For $z\in{\mathbb{Z}}^d$, let $\sigma_z\dvtx\{0,1\}^{{\mathbb{Z}}^d}\to\{0,1\}^{{\mathbb{Z}} ^d}$ be the translation map, $\sigma_z(\xi)(x)=\xi(x+z)$, and let $0<\gamma'<1$. Recall from Section~\ref{secirred} that for $R\subset{\mathbb{R}}^d$, $\mathcal{G} (R\times[0,T])$ is the $\sigma$-field generated by the points of the graphical construction in the space--time region $R\times[0,T]$. For each $\xi\in H$, $G_\xi$ will denote an event such that:
\begin{longlist}[(iii)]
\item[(i)]$G_\xi$ is $\mathcal{G}([-KL,KL]^d\times[0,T])$-measurable;
\item[(ii)] if $\xi_0=\xi\in H$, then on $G_\xi$, $\xi_T\in\sigma_{Le}H$ for all $e\in\{e_1,-e_1,\ldots,e_d,-e_{d}\}$;
\item[(iii)]$P(G_\xi)\ge1-\gamma'$ for all $\xi\in H$. \end{longlist}
We are now in a position to quote the facts we need from Section~6 of \cite{CDP11}, which depend heavily on our assumption $f'(0)>0$ [and by symmetry $f'(1)=f'(0)>0$]. This allows us to use Proposition~1.6 of \cite{CDP11} to show that Assumption 1 of that reference is in force and so by a minor modification of Lemma~6.3 of \cite{CDP11} we have the following.
\begin{lem}\label{vmpertspercolate} For any $\gamma'\in(0,1)$ there exists $\varepsilon_1>0$ and finite $K\in{\mathbb{N}}$ such that for all $0<\varepsilon<\varepsilon_1$ there exist $L,T,\{G_\xi,\xi\in H\}$, all depending on $\varepsilon$, satisfying the basic setup given above. \end{lem}
Lemma 6.3 of \cite{CDP11} deals with a rescaled process on the scaled lattice $\varepsilon{\mathbb{Z}}^d$ but here we have absorbed the scaling parameters into our constants $T$ and $L$ and then shifted $L$ slightly so that it is a natural number. In fact $L$ will be of the form $\lceil c \varepsilon^{-1}\log(1/\varepsilon)\rceil$.
Given $\xi=\xi_0\in\{0,1\}^{\mathbb{Z}^d}$ we define
\begin{equation} \label{Vndef} V_n= \bigl\{x\dvtx(x,n)\in{\mathcal{L}}\mbox{ and } \sigma_{-Lx} \xi_{nT}\in H \bigr\}. \end{equation}
Note that $V_n=\varnothing$ and $V_{n+1}\ne\varnothing$ is possible. Theorem~A.4 of \cite{Dur95} and its proof imply that there are $\{0,1\}$-valued random variables $\{\theta'(z,n)\dvtx (z,n)\in {\mathcal{L}} \}$ so that if $\{{W_m'}^{(x,n)}\dvtx m\in{\mathbb{Z}}_+, (x,n)\in{\mathcal{L}}\}$ and $\{{\mathcal{C}} '(z,n)\dvtx(z,n)\in {\mathcal{L}}\}$ are constructed from $\{\theta'(z,n)\}$ as above, then
\begin{equation} \label{VWprime} \mbox{if $x\in V_n$, then ${W'_m}^{(x,n)} \subset V_m$ for all $m\ge n$,} \end{equation}
and $\{W'_n\}$ is a $2K$-dependent oriented percolation process, that is,
\begin{eqnarray} \label{modMdep} P \bigl(\theta'(z_k,n_k) = 1 \mid\theta'(z_j,n_j),j< k \bigr) \ge1- \gamma' \end{eqnarray}
whenever $(z_j,n_j), 1\le j\le k$, satisfy $n_j<n_k$, or $n_j=n_k$ and $|z'_j-z'_k|> 2K$, for all $j<k$. The Markov property of $\xi^\varepsilon$ allows us to only require $n_j<n_k$ as opposed to $n_k-n_j>2K$ in the above, as in Section~6 of \cite{CDP11}.
Let $\Delta=(2K+1)^{d+1}$. By Theorem~B26 of \cite{Lig99}, modified as in Lemma~5.1 of~\cite{CDP11}, if $\gamma'$ (in Lemma~\ref{percolate}) is taken small enough so that $1-\gamma= (1-(\gamma')^{1/\Delta})^2 \ge1/4$, then the $\theta'(z,n)$ can be coupled with i.i.d. Bernoulli variables $\theta(z,n)$ such that
\begin{eqnarray} \label{thetathetaprime} \theta(z,n) &\le&\theta'(z,n) \qquad\mbox{for all }(z,n)\in{\mathcal{L}}\quad \mbox{and} \nonumber \\[-8pt] \\[-8pt] \nonumber P \bigl(\theta(z,n)=1 \bigr)&=&1-\gamma. \end{eqnarray}
(The simpler condition on $\gamma$ and $\gamma'$ in Theorem B26 of \cite{Lig99} and above in fact follows from that in \cite{LSS} and Lemma~5.1 of \cite {CDP11} by some arithmetic, and the explicit value of $\Delta$ comes from the fact that we are now working on ${\mathbb{Z}}^d$.) If the coupling part of \eqref{thetathetaprime} holds, then $W_n\subset V_n$ for all $n$, and \eqref{VWprime} implies
\begin{equation} \label{percext} x\in V_n \mbox{ implies } W^{(x,n)}_m \subset V_m \mbox{ for all } m\ge n. \end{equation}
Now choose $\gamma'$ small enough in Lemma~\ref{vmpertspercolate} so that
\begin{equation} \label{LSSbnd} 1-\gamma= \bigl(1- \bigl(\gamma' \bigr)^{1/\Delta} \bigr)^2>1-6^{-4}. \end{equation}
We can now verify condition \eqref{flip2}.
\begin{lem}\label{lemflip2} If $\xi^\varepsilon$ is a voter model perturbation satisfying the hypotheses of Theorem~\ref{thmCCTpert}, then there exists $\varepsilon_1>0$ and $x_0\in{\mathbb{Z}^d}$ such that \eqref{flip2} holds for $\xi^\varepsilon$ if $\varepsilon<\varepsilon_1$. \end{lem}
\begin{pf} For $\gamma'$ as above, let $\varepsilon_1$ be as in Lemma~\ref{vmpertspercolate}, so that for $0<\varepsilon<\varepsilon_1$ all the conclusions of that lemma hold, as well as the setup \eqref{Hprop}--\eqref{percext}, with $\rho_\infty>0$. There are two main steps in the proof. In the first, we show that if $A\subset2{\mathbb{Z}^d}$ is large, then for all large $n$, $\xi^\varepsilon_{2nT}\ne\varnothing$ will imply $V_{2n}\cap A$ is also large; see~\eqref{K0occupied} below. To do this, we argue that there is a uniform positive lower bound on $P_\xi(\exists z\in V_2\mbox{ with } W^{(z,2)}_m\ne\varnothing\ \forall m\ge2)$, $\xi\notin\{\mathbf{0},\mathbf{1}\}$. Iteration leads to \eqref {K0occupied}. In the second step, we consider $A\subset{\mathbb{Z}^d}$ large, and for $a\in A$ choose points $\ell(a)\in{\mathbb{Z}^d}$ such that $a\in 2L\ell(a)+Q(L)$. If $A$ is sufficiently large, there will be many points $a_i\in A$ which are widely separated. By the first step, for large $n$, there will be many points $2\ell (a_i)\in V_{2n}$, and for each of these there will be points $y^0_i,y^1_i\in2L\ell(a_i)+Q(L)$ such that $\xi^\varepsilon_{2nT}(y^0_i)=0$ and $\xi^\varepsilon_{2nT}(y^1_i)=1$. Given these points, it will follow from Lemma~\ref{vmirred} that there is a uniform positive lower bound on the probabilities of independent events on which $\xi^\varepsilon_t(a_i)=1$ and $\xi^\varepsilon_t(a_i+x_0)=0$ for all $t\in[(2n+1)T, (2n+3)T]$. Many of these events will occur, forcing $A(\xi_t,x_0)$ to be large; see \eqref{emptyA} below. Condition \eqref{flip2} now follows easily.
It is convenient to start with two estimates which depend only on the process $\xi^\varepsilon$ (and not on the percolation construction). We claim that by Lemma~\ref{vmirred} with $t=2T$,
\begin{equation} \min_{x\in Q(L), k=1,\ldots,d}\mathop{\inf_{\xi\in\{0,1\}^{\mathbb {Z}^d}\dvtx}}_{ \xi(x)=1,\xi(x+e_k)=0} P_\xi\bigl(\xi^\varepsilon_{2T}\in H \bigr)>0. \end{equation}
To see this, note that for small $\varepsilon$, $\xi\in H$ depends only on the coordinates $\xi(x)$, $x\in Q(L+1)$. This means there are disjoint sets $B_0,B_1\subset Q(L+1)$ so that $\xi^\varepsilon_{2T}(x)=i$ for all $x\in B_i$, $i=0,1$, implies $\xi^\varepsilon_{2T}\in H$. If $G(2T,y_0,y_1,B_0,B_1)$ and $\Lambda(y_0,y_1,B_0,B_1)$ are as in Lemma~\ref{vmirred} with $(y_0,y_1)=(x+e_k,x)$, then for $x\in Q(L)$ the above infimum is bounded below by $P(G(2T,x+e_k,x,B_0,B_1))>0$. If $\xi\notin\{\mathbf{0},\mathbf{1}\}$, there must exist $k\in\{1,\ldots,d\}$ and $x,z\in{\mathbb{Z}^d}$ with $x\in2Lz+Q(L)$ and $\xi(x)=1$, $\xi(x+e_k)=0$. It now follows from translation invariance that
\begin{equation} \label{rho1} \rho_1= \inf_{\xi\notin\{\mathbf{0},\mathbf{1}\}}P_\xi \bigl( \exists z\in{\mathbb{Z}^d}\mbox{ such that } \xi^\varepsilon_{2T}\circ\sigma_{2Lz}\in H \bigr)>0. \end{equation}
Let $\rho_2=\rho_1\rho_\infty>0$.
Next, suppose $y_0,y_1,y\in Q(L)$, $B_{1}=\{y\}$, $B_0=\{y+x_0\}$ and $G(T,y_0,y_1,\break B_0, B_1)$ be as in Lemma~\ref{vmirred}. To also require that $\xi^\varepsilon_u$ be constant at $y,y+x_0$ for $u\in[T,3T]$, we let $\tilde G(T,y_0,y_1,B_0,B_1)$ be the event
\[ G(T,y_0,y_1,B_0,B_1)\cap\bigl \{\mbox{$T^z_m,T^{*,z}_m \notin[T,3T]$ for $z=y,y+x_0$ and all $m\ge1$} \bigr\}. \]
Note that each $\tilde G$ is an intersection of two independent events each with positive probability, and so $P(\tilde G)>0$. Making use of the notation of Lemma~\ref{vmirred}, choose $\tilde M<\infty$ such that
\[ \Lambda=\bigcup_{y_0,y_1,y\in Q(L)}\Lambda(y_0,y_1,B_0,B_1) \subset[-\tilde M,\tilde M]^d, \]
and put
\begin{equation} \label{deltaprime} \tilde\delta=\min_{y_0,y_1,y\in Q(L)}P \bigl(\tilde G(T,y_0,y_1,B_0,B_1) \bigr)>0. \end{equation}
If $\xi^\varepsilon_0(y_i)=i,i=0,1$, then
\begin{equation} \begin{tabular}{p{260pt}@{}} \label{Gimpl}$\tilde G(T,y_0,y_1,B_0,B_1)$ implies $\xi^\varepsilon_t(y)=1,\xi^\varepsilon_t(y+x_0)=0$ for all $t\in[T,3T]$. \end{tabular} \end{equation}
We now start the proof of
\begin{equation}
\label{0occupied2} \lim_{K\to\infty} \mathop{\sup _{A\subset2{\mathbb{Z}^d}}}_{|A|\ge K} \lim_{n\to\infty}P_\xi \bigl(\xi^\varepsilon_{2nT}\ne\varnothing\mbox{ and } V_{2n}\cap A=\varnothing\bigr) =0. \end{equation}
Fix $\delta>0$. By \eqref{0occupied} there exists $K_1=K_1(\delta)<\infty$
such that if $A\subset2{\mathbb{Z}^d}$ with $|A|\ge K_1$, then there exists $n_1=n_1(A)<\infty$ such that
\begin{equation} \label{afternA} P \bigl(W^0_{2n}\ne\varnothing, W^0_{2n}\cap A=\varnothing\bigr) < \delta\qquad\mbox{for all }n\ge n_1. \end{equation}
We may increase $n_1$ if necessary so that
$P(|{\mathcal{C}}(0,0)|<\infty, W^0_{2n_1}\ne\varnothing)<\delta$, which implies that
\begin{equation} \label{dieaftern12} P \bigl(W^0_{2n_1}\ne\varnothing, W^0_{2n}=\varnothing\bigr) < \delta\qquad\mbox{for all }n\ge n_1. \end{equation}
For $j=1,2,\ldots,$ let $m(j)=(j-1)(2n_1+2)$, and define a random sequence of sites $z_j$, as follows. If $V_{m(j)+2}=\varnothing$, put $z_j=0$. If not, choose $z\in V_{m(j)+2}$ with minimal norm (with some convention for ties), and put $z_j=z$. By the Markov property and \eqref{rho1},
\begin{equation} \label{N1} \inf_{\xi\notin\{\mathbf{0},\mathbf{1}\}}P_\xi\bigl(z_1
\in V_2, \bigl|{\mathcal{C}}(z_1,2)\bigr|=\infty\bigr) \ge\rho_2. \end{equation}
Let
\[ N=\inf\bigl\{j\dvtx z_j\in V_{m(j)+2}\mbox{ and } W^{z_j,m(j)+2}_{m(j+1)}\ne\varnothing\bigr\} \]
and ${\mathcal{F}}_n$ be the $\sigma$-algebra generated by $\mathcal{G}({\mathbb{R}}^d\times[0,nT])$ and the $\theta(z,k)$ for $z\in{\mathbb{Z}}^d, k<n$. It follows from our construction and \eqref{N1} that almost surely on the event $\{\xi^\varepsilon_{m(j)T}\ne\varnothing\}$,
\begin{eqnarray*} &&P_\xi\bigl(z_j\in V_{m(j)+2} \mbox{ and } W^{(z_j,m(j)+2)}_{m(j+1)}=\varnothing\mid{\mathcal{F}}_{m(j)} \bigr) \\ &&\qquad=P_{\xi^\varepsilon_{m(j)T}} \bigl(z_1\in V_{2} \mbox{ and } W^{(z_1,2)}_{2n_1+2}=\varnothing\bigr) \\
&&\qquad\le P_{\xi^\varepsilon_{m(j)T}} \bigl(z_1\in V_2, \bigl|
{\mathcal{C}}(z_1,2)\bigr|<\infty\bigr) \\ &&\qquad\le1-\rho_2. \end{eqnarray*}
In the last line note that by \eqref{Hprop} if the initial state is $\mathbf{1}$, the probability is zero as~$\mathbf{1}$ is a trap. Since the event on the LHS is ${\mathcal{F}}_{m(j+1)}$-measurable, we may iterate this inequality to obtain
\begin{equation} \label{Ntailbound}P_\xi\bigl(\xi^{\varepsilon}_{m(j)T}\ne \varnothing, N>j \bigr)\le(1-\rho_2)^{j}. \end{equation}
Taking $J_0>2$ large enough so that $(1-\rho_2)^{J_0}<\delta$, and then $2n> m(J_0+1)$,
\begin{eqnarray} \label{decompJ0}&& P_\xi\bigl(\xi^\varepsilon_{2nT}\ne \varnothing, V_{2n}\cap A=\varnothing\bigr) \nonumber \\[-8pt] \\[-8pt] \nonumber &&\qquad\le\delta+ \sum _{j=1}^{J_0}P_\xi\bigl( \xi^\varepsilon_{m(j)T}\ne\varnothing, V_{2n}\cap A= \varnothing, N=j \bigr). \end{eqnarray}
For $j\le J_0$, almost surely on the event $\{\xi^\varepsilon_{m(j)T}\ne\varnothing, N>j-1\}$,
\begin{eqnarray*} &&P_\xi\bigl(z_j\in V_{m(j)+2}, W^{(z_j,m(j)+2)}_{m(j+1)}\ne\varnothing, V_{2n}\cap A= \varnothing\mid{\mathcal{F}}_{m(j)} \bigr) \\ &&\qquad= P_{\xi^\varepsilon_{m(j)}} \bigl(z_1\in V_{2}, W^{(z_1,2)}_{2n_1}\ne\varnothing, V_{2n-2n_1}\cap A= \varnothing\bigr) \\ &&\qquad\le P_{\xi^\varepsilon_{m(j)}} \bigl(z_1\in V_{2}, W^{(z_1,2)}_{2n_1}\ne\varnothing, W^{(z_1,2)}_{2n-2n_1} \cap A=\varnothing\bigr) \\ &&\qquad\le\delta+P_{\xi^\varepsilon_{m(j)}} \bigl(z_1\in V_{2}, W^{(z_1,2)}_{2n-2n_1}\ne\varnothing, W^{(z_1,2)}_{2n-2n_1} \cap A=\varnothing\bigr) \\ &&\qquad\le2\delta, \end{eqnarray*}
where the last three inequalities follow from \eqref{percext}, \eqref{dieaftern12}, \eqref{afternA} and the fact that $n\ge2n_1$ by our choice of $n$ above. Combining this bound with \eqref{decompJ0} and then using \eqref {Ntailbound}, we obtain
\begin{eqnarray*} P_\xi\bigl(\xi^\varepsilon_{2nT}\ne\varnothing, V_{2n}\cap A=\varnothing\bigr) &\le&\delta+ 2\delta\sum _{j=1}^{J_0}P_\xi\bigl( \xi^\varepsilon_{m(j)T}\ne\varnothing, N> j-1 \bigr) \\ &\le&\delta+ 2\delta\sum_{j=1}^{J_0}(1- \rho_2)^{j-1} \\ &\le&\delta+2\delta/\rho_2. \end{eqnarray*}
This establishes \eqref{0occupied2}, which along with the argument proving \eqref{flip2a}, implies that for any $K_0<\infty$,
\begin{equation}
\label{K0occupied} \lim_{K\to\infty}\mathop{\sup _{A\subset 2{\mathbb{Z}^d}}}_{|A|\ge K} \limsup_{n\to\infty}P_\xi
\bigl(\xi^\varepsilon_{2nT} \ne\varnothing, |V_{2n}\cap A|\le K_0 \bigr)=0. \end{equation}
Now fix $K_0<\infty$ so that $(1-\tilde\delta)^{K_0}<\delta$. By \eqref {K0occupied} there exists
$K_1<\infty$ such that for $A'\subset2{\mathbb{Z}}^d$ satisfying $|A'|\ge K_1$, there exists $n_1(A')$ so that
\begin{equation} \label{end2} P_\xi\bigl(\xi^\varepsilon_{2nT}\ne
\varnothing\mbox{ and }\bigl|V_{2n}\cap A'\bigr|\le K_0 \bigr) <\delta\qquad\mbox{if } n\ge n_1 \bigl(A' \bigr). \end{equation}
For $a\in{\mathbb{Z}^d}$ let $\ell(a)$ be the minimal point in some ordering of ${\mathbb{Z}}^d$ such that $a\in2L\ell(a)+Q(L)$. For $A\subset{\mathbb{Z}^d}$ let
$\ell(A)=\{\ell(a),a\in A\}$. With $K_0,K_1$ as above, choose $K_2<\infty$ so that if $A\subset{\mathbb{Z}^d}$ and $|A|\ge K_2$, then $\ell(A)$ contains $K_1$ points,
$\ell(a_1),\ldots,\ell(a_{K_1})$, such that $|\ell(a_i)-\ell(a_{j})|2L\ge4\tilde M$ for $i\ne j$. The regions $2L\ell(a_1)+[-\tilde M,\tilde M]^d,\ldots, 2L\ell(a_{K_1})+[-\tilde M,\tilde M]^d$ are pairwise disjoint. Let $A'=\{2\ell(a_1),\ldots,2\ell(a_{K_1})\}\subset2{\mathbb{Z}}^d$.
Now suppose $t\in[(2n+1)T,(2n+3)T]$ for some integer $n\ge n_1(A')$.\break By~\eqref{end2}, on the event
$\{|\xi^\varepsilon_{2nT}|>0\}$, except for a set of probability at most $\delta$, $V_{2n}$ will contain at least $K_0$ points of $A'$. If $2\ell(a_i)$ is such a point, then by the definitions of $V_{2n}$ and $H$, there will exist points $y^i_0,y^i_1\in2L\ell(a_i)+Q(L)$ such that $\xi^\varepsilon_{2nT}(y^i_0)=0,\xi^\varepsilon_{2nT}(y^i_1)=1$. Conditional on this, by \eqref{deltaprime} and \eqref{Gimpl}, the probability that $\xi^\varepsilon_t(a_i)=1,\xi^\varepsilon_t(a_i+x_0)=0$ is at least $\tilde\delta$. By independence of the Poisson point process on disjoint space--time regions, it follows that
\begin{equation} \label{emptyA} P \bigl(\xi^\varepsilon_{2nT}\ne\varnothing\mbox{ and } A \bigl(x_0,\xi^\varepsilon_t \bigr)= \varnothing\bigr) <\delta+ (1-\tilde\delta)^{K_0}, \end{equation}
and therefore since $t>2nT$,
\[ P \bigl(\xi^\varepsilon_{t}\ne\varnothing\mbox{ and } A \bigl(x_0,\xi^\varepsilon_t \bigr)=\varnothing \bigr) <\delta+ (1-\tilde\delta)^{K_0}<2\delta, \]
the last by our choice of $K_0$. This proves \eqref{flip2}.
Finally, \eqref{N1} implies by \eqref{percext}, \eqref{Hprop}, the definition of $V_n$ and the fact that ${\mathbf{1}}$ is a trap by Lemma~\ref{elemann}, that
\begin{equation} \label{survival} \inf_{\xi\neq\mathbf{0}}P_\xi\bigl( \xi^\varepsilon_t\ne\varnothing\ \forall t\ge0 \bigr)\ge \rho_2. \end{equation}
This will be used below. \end{pf}
\begin{pf*}{Proof of Theorem~\ref{thmCCTpert}} We verify the assumptions of Proposition~\ref{propCCT}. It follows from Lemma~\ref{elemann} and \eqref{traps} that $c_\varepsilon(x,\xi)$ is symmetric and $\zeta^\varepsilon_t$, the annihilating dual of $\xi^\varepsilon_t$, is parity preserving. By Corollary~\ref{vpirred} (which applies by Remark~\ref{nvm}) there exists $\varepsilon_3>0$ such that if $0<\varepsilon<\varepsilon_3$, then $\zeta^\varepsilon_t$ is irreducible. By Lemmas~\ref{lemodd2} and \ref{lemflip2} (and the proof of the latter), there exists $0<\varepsilon_4<\varepsilon_3$ such that if $0<\varepsilon<\varepsilon_4$, then \eqref{flip2}, \eqref{oddgoal} and \eqref{survival} hold for $\xi^\varepsilon_t$.
Assume now that $0<\varepsilon<\varepsilon_4$. It remains to check that the dual growth condition~\eqref{eqH} (the conclusion of Lemma~\ref{lemH}) holds, and to do this it suffices by Remark~\ref{remHaltcond} to show that \eqref{eqH3} for $\xi^\varepsilon$ holds. By \eqref{flip2} and \eqref{survival}, there is a $\delta_1>0$, $t_0<\infty$ and $A\in Y$ so that for all $t\ge t_0$ (with $\xi^\varepsilon_0=1_{\{0\}}$),
\[ P \bigl(\xi^\varepsilon_t(a)=1\mbox{ for some }a\in A \bigr) \ge P \bigl(A \bigl(x_0,\xi_t^\varepsilon\bigr) \neq\varnothing\bigr)\ge\delta_1. \]
Next apply \eqref{02z}, translation invariance and the Markov property to conclude that for $t$ as above,
\begin{eqnarray*} P \bigl(\xi^\varepsilon_{t+1}(0)=1 \bigr)&\ge& E \bigl(1 \bigl( \xi^\varepsilon_t(a)=1\mbox{ for some }a\in A \bigr)P_{\xi^\varepsilon_t} \bigl(\xi_1^\varepsilon(0)=1 \bigr) \bigr) \\ &\ge&\delta_1\min_{a\in A}\inf_{\xi^\varepsilon_0\dvtx\xi^\varepsilon _0(0)=1}P_{\xi_0^\varepsilon } \bigl(\xi_1^\varepsilon(-a)=1 \bigr)\ge\delta_2>0. \end{eqnarray*}
This proves \eqref{eqH3}, and all the assumptions of Proposition~\ref{propCCT} have now been verified for $\xi^\varepsilon_t$ if $0<\varepsilon<\varepsilon_4$, and thus the weak limit \eqref{eqgoal} also holds. Finally, by~\eqref{betanonzero}
this result implies the full complete convergence\vspace*{1pt} theorem with coexistence if $|\hat\xi_0^\varepsilon|=\infty$. If $|\hat
\xi_0^\varepsilon|<\infty$, then $|\xi^\varepsilon_0|=\infty$, and the result now follows by the symmetry of $\xi^\varepsilon$; recall Lemma~\ref{elemann}. \end{pf*}
\section{\texorpdfstring{Proof of Theorem~\protect\ref{thmLVCCT}} {Proof of Theorem 1.1}}\label{secalmostlast} Let us check that $\mathrm{LV}(\alpha)$, $\alpha\in(0,1)$, is cancellative.
(This was done in \cite{NP} for the case $p(x)=1_{{\mathcal{N}}}(x)/|{\mathcal{N}}|$ for ${\mathcal{N}}$ satisfying \eqref{N}.) For the more general setting here, we assume $p(x)$ satisfies \eqref{passump}, and allow any $d\ge1$. We first observe that if $c(x,\xi)$ has the form given in \eqref{q00}, then it follows from \eqref{q01} that
\[ c(0,\xi) = k_0\sum_{A\in Y} q_0(A) \frac12 \bigl[1 - \bigl(2\xi(0)-1 \bigr)H(\xi,A) \bigr]. \]
From this it is clear that the sum of two positive multiples of cancellative rate functions is cancellative. It follows from a bit of arithmetic that if \eqref{diag} holds, then $\mathrm{LV}(\alpha)$ with $\varepsilon^2=1-\alpha>0$ has flip rates
\[ c_{\mathrm{LV}}(x,\xi)=\alpha c_{\mathrm{VM}}(x,\xi)+\varepsilon^2f_0f_1(x, \xi). \]
We have already noted that $c_{\mathrm{VM}}$ is cancellative, and so by the above we need only check that $c^*(x,\xi)=f_0(x,\xi)f_1(x,\xi)$ is cancellative.
To do this we let $p^{(2)}(0)=\sum_{x\in{\mathbb{Z}^d}}(p(x))^2$,
$k_0 = (1-p^{(2)}(0))/2$, $q_0(A)=0$ if $|A|\ne3$ and
\[ q_0 \bigl(\{0,x,y\} \bigr)= k_0^{-1}p(x)p(y)\qquad \mbox{if $0,x,y$ are distinct}. \]
Note that $\sum_{A\in Y}q_0(A)=1$ because [recall that $p(0)=0$]
\[ \sum_{\{x,y\}}q_0 \bigl(\{0,x,y\} \bigr) = \frac{1}{2k_0}\sum_{x\ne y}p(x)p(y) = \frac{1}{2k_0} \bigl(1-p^{(2)}(0) \bigr) = 1. \]
Also, for $0,x,y$ distinct,
\[ \tfrac12 \bigl[1- \bigl(2\xi(0)-1 \bigr)H \bigl(\xi,\{0,x,y\} \bigr) \bigr] = 1 \bigl\{ \xi(x)\ne\xi(y) \bigr\}. \]
With these facts it is easy to see that
\begin{eqnarray*} &&k_0\sum_{A\in Y} q_0(A) \frac12 \bigl[1 - \bigl(2\xi(0)-1 \bigr)H(\xi,A) \bigr] \\ &&\qquad= \sum_{x,y} p(x)p(y)1 \bigl\{\xi(x)\ne\xi(y) \bigr \} = f_0(0,\xi)f_1(0,\xi), \end{eqnarray*}
proving $c^*(x,\xi)=f_0(x,\xi)f_1(x,\xi)$ is cancellative and hence so is $\mathrm{LV}(\alpha)$.
Although we won't need it, we calculate the parameters of the branching annihilating dual. Adding in the voter model, we see that they are
\begin{eqnarray*} k_0&=& \alpha+ (1-\alpha)\frac{1-p^2(0)}{2},\qquad q_0 \bigl( \{y\} \bigr) = \frac{\alpha}{k_0} p(y),\\
q_0 \bigl(\{0,y,z\} \bigr)& =& \frac{1-\alpha}{k_0} p(y)p(z), \end{eqnarray*}
and $q_0(A)=0$ otherwise. One can see from this that $\zeta_t$, the dual of $\mathrm{LV}(\alpha)$, describes a system of particles evolving according to the following rules: (i) a particle at $x$ jumps to $y$ at rate $\alpha p(y-x)$; (ii) a particle at $x$ creates two particles and sends them to $y,z$ at rate $(1-\alpha) p(y-x)p(z-x)$; (iii) if a particle attempts to land on another particle, then the two particles annihilate each other.
Assume $d\ge3$. The function $f(u)$ as shown in Section~1.3 of \cite{CDP11} is a cubic, and under the assumption \eqref{diag} reduces to $f(u)=2p_3(1-\alpha)u(1-u)(1-2u)$, where $p_3$ is a certain (positive) coalescing random walk probability. Thus $f'(0)>0$, so the complete convergence theorem with coexistence for $\mathrm{LV}(\alpha)$ for $\alpha$ sufficiently close to one follows from Theorem~\ref{thmCCTpert}.
Now suppose $d=2$. It suffices to prove an analogue of Lemma~\ref{vmpertspercolate} as the above results will then allow us to apply the proof of Theorem~\ref{thmCCTpert} in the previous section to give the result. As the results of \cite{CDP11} do not apply, we will use results from \cite{CMP} instead and proceed as in Section~4 of \cite{CP07}. Instead of \eqref{exptail}, we only require (as was the case in \cite{CMP})
\begin{equation}
\label{3mom} \sum_{x\in{\mathbb{Z}}^2}|x|^3p(x)< \infty. \end{equation}
We will need some notation from \cite{CMP}. For $N>1$, let $\xi^{(N)}$ be the $\mathrm{LV}(\alpha_N)$ process where
\[ \alpha_N=1-\frac{(\log N)^3}{N}, \]
and consider the rescaled process, $\xi^N_t(x)=\xi^{(N)}_{Nt}(x\sqrt N)$, for $x\in S_N={\mathbb{Z}}^2/\sqrt N$. The associated process taking values in $M_F({\mathbb{R}}^2)$ (the space of finite\vadjust{\goodbreak} measures on the plane with the weak topology) is
\begin{equation} \label{MVP} X_t^N=\frac{\log N}{N}\sum _{x\in S_N}\xi_t^N(x) \delta_x. \end{equation}
For parameters $K,L'\in{\mathbb{N}}$, $K>2$ and $L'>3$, which will be chosen below, we let $\underline\xi^N_t(x)\le\xi^N_t(x)$, $x\in S_N$ be a coupled particle system where particles are ``killed'' when they exit $(-KL',KL')^2$, as described in Proposition~2.1 of \cite{CP07}. (Here a particle corresponds to a $1$.) In particular $\underline\xi^N_t(x)=0$
for all $|x|\ge KL'$. $\underline X^N_t$ is defined as in \eqref{MVP} with $\underline\xi^N$ in place of $\xi^N$.
We will need to keep track of some of the dependencies in the constant $C_{8.1}$ in Lemma~8.1 of \cite{CMP}. As in that result, $B^N$ is a rate $N\alpha_N=N-(\log N)^3$ random walk on $S_N$ with step distribution $p_N(x)=p(x\sqrt N)$, $x\in S_N$, starting at the origin.
\begin{lem}\label{lem81mod} There are positive constants $c_0$ and $\delta_0$ and a nondecreasing function $C_0(\cdot)$, so that if $t>0$, $K,L'\in{\mathbb{N}}$, $K>2$ and $L'>3$, and $X_0^N=\underline X_0^N$ is supported on $[-L',L']^2$, then
\begin{eqnarray} \label{killbnd} &&E \bigl(X_t^N(1)-\underline X_t^N(1) \bigr) \nonumber \\ &&\qquad\le X_0^N(1) \Bigl[c_0e^{c_0 t}P\Bigl(\sup_{s\le t}
\bigl|B^{N}_s\bigr|>(K-1)L'-3 \Bigr)\\ &&\hspace*{52pt}\qquad\quad{}+C_0(t) \bigl(1\vee X_0^N(1) \bigr) (\log N)^{-\delta_0} \Bigr].\nonumber \end{eqnarray}
\end{lem}
\begin{pf} This is a simple matter of keeping track of the $t$-dependency in some of the constants arising in the proof of Lemma~8.1 in \cite{CMP}. \end{pf}
Recall from Theorem~1.5 of \cite{CMP} that if $X_0^N\to X_0$ in $M_F({\mathbb{R}} ^2)$, then $\{X^N\}$ converges weakly in $D({\mathbb{R}}_+,M_F({\mathbb{R}}^2))$ to a two-dimensional super-Brownian motion, $X$, with branching rate $4\pi \sigma^2$, diffusion coefficient $\sigma^2$ and drift $\eta>0$ [write $X$ is $\mathrm{SBM}(4\pi\sigma^2,\sigma^2,\eta)$], where $\eta$ is the constant $K$ in (6) of \cite{CMP} (not to be confused with our parameter $K$). See (MP) in Section~1 of \cite{CMP} for a precise definition of SBM. The important point for us is that the positivity of $\eta$ will mean that the supercritical $X$ will survive with positive probability, and on this set will grow exponentially fast.
We next prove a version of Proposition~4.2 of \cite{CP07} which when symmetrized is essentially a scaled version of the required Lemma~\ref {vmpertspercolate}. To be able to choose $\gamma'$ as in \eqref{LSSbnd}, so that we may apply Lemma~5.1 of \cite{CDP11}, we will have to be more careful with the selection of constants in the proof of Proposition~4.2 in the above reference. We start by choosing $c_1>0$ so that
\begin{equation} \label{c1choice} \bigl(1-e^{-c_1} \bigr)^2>1-6^{-4}, \end{equation}
and then setting
\[ \gamma'_K=e^{-c_1(2K+1)^3}.\vadjust{\goodbreak} \]
\begin{lem}\label{lemprop42} There are $T'>1$, $L',K,J'\in{\mathbb{N}}$ with $K>2$, $L'>3$, and $\varepsilon_1\in(0,\frac{1}{2})$ such that if $0<1-\alpha <\varepsilon_1$, $N>1$ is chosen so that $\alpha=1-\frac{(\log N)^3}{N}$, and $I_{\pm e_i}=\pm2L'e_i+[-L',L']^2$, then
\[ \underline X_0^N \bigl( \bigl[-L',L' \bigr]^2 \bigr)\ge J'\mbox{ implies }P \bigl(\underline X_{T'}^N(I_{\pm e_i})\ge J' \mbox{ for }i=1,2 \bigr)\ge1-\gamma'_K. \]
\end{lem}
\begin{pf} By the monotonicity of $\underline X^N$ in its initial condition (Proposition~2.1(b) of \cite{CP07} and the monotonicity of $\mathrm{LV}(\alpha)$ discussed, e.g., in Section 1 of \cite{CP07}), we may assume that $\underline X_0^N({\mathbb{R}}^2\setminus[-L',L']^2)=0$ and $\underline X_0^N([-L',L']^2)\in[J',2J']$, where $L'$ and $J'$ are chosen below.
We will choose a number of constants which depend on an integer $K>2$ and will then choose $K$ large enough near the end of the proof. Assume $B=(B^1,B^2)$ is a $2$-dimensional Brownian motion with diffusion parameter $\sigma^2$, starting at $x$ under $P_x$ and fix $p>\frac{1}{2}$. Set
\begin{equation} \label{Tdef} T'=c_2K^{2p}, \end{equation}
where a short calculation shows that if $c_2$ is chosen large enough, depending on $\sigma^2$ and $\eta$, then for any $K>2$,
\begin{equation}
\label{Tdef2} e^{\eta T'/2}\inf_{|x|\le K^p} P_x \bigl(B_1\in\bigl[K^p,3K^p \bigr]^2 \bigr)\ge5. \end{equation}
Now put $L'=K^p\sqrt{T'}$, increasing $c_2$ slightly so that $L'\in{\mathbb{N}}$. If $I=[-L',L']^2$ and $X$ is the limiting super-Brownian motion described above, then as in Lemma~12.1(b) of \cite{DP99}, there is a $c_3(K)$ so that
\begin{equation} \label{SBMbnd}\qquad \forall J'\in{\mathbb{N}}\mbox{ and }i\le2,\mbox{ if }X_0(I)\ge J',\mbox{ then }P \bigl(X_{T'}(I_{\pm e_i})<4J' \bigr)\le c_3/J'. \end{equation}
Next choose $J'=J'(K)\in{\mathbb{N}}$ so that
\[ \frac{c_3}{J'}\le\frac{\gamma'_K}{100}. \]
As in Lemma~4.4 of \cite{CP07}, the weak convergence of $X^N$ to $X$ and \eqref{SBMbnd} show that for $N\ge N_1(K)$,
\begin{equation} \label{massbnd} \forall i\le2\mbox{ if }X_0^N(I)\ge J', \mbox{ then }P \bigl(X_{T'}^N(I_{\pm e_i})<4J' \bigr)\le\frac{\gamma'_K}{50}. \end{equation}
Next use Lemma~\ref{lem81mod}, the fact that $X^N_{T'}-\underline X^N_{T'}$ is a nonnegative measure and Donsker's theorem to see that there is a $c_4>0$ and an $\varepsilon_N=\varepsilon_N(K)\to0$ as $N\to\infty$, so that for any $i\le2$,
\begin{eqnarray*} &&P \bigl(X_{T'}^N(I_{\pm e_i})-\underline X_{T'}^N(I_{\pm e_i})\ge2J' \bigr) \\ &&\qquad\le\frac{X_0^N(1)}{2J'} \Bigl[c_0e^{c_0T'}
\Bigl(P_0 \Bigl(\sup_{s\le T'}|B_s|>(K-1)L'-3 \Bigr)+\varepsilon_N \Bigr) \\ &&\hspace*{96pt}\qquad\quad{} +C_0 \bigl(T' \bigr) \bigl(1\vee X_0^N(1) \bigr) (\log N)^{-\delta_0} \Bigr] \\ &&\qquad\le\bigl[c'_0e^{c_0T'} \bigl(\exp \bigl(-c_4K^{2+2p} \bigr)+\varepsilon_N \bigr)+C_0 \bigl(T' \bigr)2J'(\log N)^{-\delta_0} \bigr], \end{eqnarray*}
where the fact that $X_0^N(1)\le2J'$ and the definition of $L'$ are used in the last line. It follows that for $K\ge K_0$ and $N\ge N_2(K)$, the above is bounded by
\begin{equation} \label{survbnd} 2c'_0e^{c_0T'}\exp \bigl(-c_4K^{2+2p} \bigr)\le2c'_0e^{c_5K^{2p}-c_4K^{2+2p}} \le\frac{\gamma'_K}{50}. \end{equation}
The fact that $p>\frac{1}{2}$ is used in the last inequality. We finally choose $K\in{\mathbb{N}}^{>2}$, $K\ge K_0$. Therefore the bounds in \eqref{massbnd} and \eqref{survbnd} show that for $N\ge N_1(K)\vee N_2(K)$ and $i\le2$,
\begin{eqnarray*} P \bigl(\underline X_{T'}^N(I_{\pm e_i})<2J' \bigr)&\le& P \bigl(X^N_{T'}(I_{\pm e_i}) \le4J' \bigr)+P \bigl(X_{T'}^N(I_{\pm e_i})- \underline X_{T'}^N(I_{\pm e_i}) \ge2J' \bigr) \\ &\le&\frac{\gamma'_K}{25}. \end{eqnarray*}
Sum over the $4$ choices of $\pm e_i$ to prove the required result because the condition on $N$ is implied by taking $1-\alpha=(\log N)^3/N$ small enough. \end{pf}
\begin{pf*}{Completion of proof of Theorem~\ref{thmLVCCT}} By symmetry we have an analogue of the above lemma with $0$'s in place of $1$'s. Let $\alpha$ and $N$ be as in Lemma~\ref{lemprop42}. Now undo the scaling and set $L=\sqrt N L'$, $J=\frac{N}{\log N}J'$ and $T=T' N$. Slightly abusing our earlier notation we let $\underline{\xi }_t\le \xi^{(N)}_t$ be the unscaled coupled particle system where particles are killed upon exiting $(-KL,KL)^2$ and let $\tilde I_{\pm e_i}=\pm e_iL+[-L,L]^2$. We define
\[ G_\xi= \bigl\{\underline{\xi}_T(\tilde I_{\pm e_i})\ge J, \hat{\underline\xi}_T(\tilde I_{\pm e_i})\ge J\mbox{ for }i=1,2 \bigr\}, \]
where ${\underline\xi}_0=\xi$. Lemma~\ref{lemprop42} gives the conclusion of Lemma~\ref {vmpertspercolate} with $\varepsilon=1-\alpha$, $\gamma'=\gamma'_K$ and now with
\[ H= \bigl\{\xi\in\{0,1\}^{{\mathbb{Z}^d}}\dvtx\xi\bigl([-L,L]^d \bigr) \ge J, \hat\xi\bigl([-L,L]^d \bigr)\ge J \bigr\}. \]
Note that by \eqref{c1choice} and the definition of $\gamma'_K$, we have $1-\gamma>1-6^{-4}$ where $\gamma$ is as in \eqref{LSSbnd}. The definition of $\underline\xi$ gives the required measurability of $G_\xi$. Note that $H$ depends only on $\{\xi(x)\dvtx x\in[-L,L]^d\} $, and $\xi\in H$ implies $\xi(x)=1$ and $\xi(x')=0$ for some $x,x'\in [-L,L]^d$. These are the only properties of $H$ used in the previous proof. Finally it is easy to adjust the parameters so that $L\in{\mathbb{N}}$ as in Lemma~\ref{vmpertspercolate}. One way to do this is to modify \eqref {SBMbnd} so the conclusion of Lemma 6.2 becomes
\[ \underline X_0^N \bigl(I' \bigr)\ge J'\mbox{ implies }P \bigl(\underline X_T^N \bigl(I'_{\pm e_i} \bigr)\ge J' \mbox{ for }i=1,2 \bigr)\ge1-\gamma'_K, \]
where $I'=[-L'-1,L'+1]^2$ and $I'_{\pm e_i}=\pm2L'e_i+[-L'+1,L'-1]^2$. Then for $N$ large (in addition to the constraints above, $N\ge9$ will do) one can easily check that the above argument is valid with $L=\lfloor\sqrt{N}L'\rfloor\in{\mathbb{N}}$. Therefore, with the conclusion of this version of Lemma~\ref{vmpertspercolate} in hand, the result for $d=2$ now follows as in the proof of Theorem~\ref {thmCCTpert}. \end{pf*}
\begin{rem}\label{noexptail} The above argument works equally well for $\mathrm{LV}(\alpha)$ for $d\ge3$ even without assuming \eqref{3mom}. Only a few constants need to be altered, for example, $p=(d-1)/2$ and $\gamma '_K=e^{c_1(2K+1)^{d+1}}$. More generally the argument is easily adjusted to give the result for the general voter model perturbations in Theorem~\ref{thmCCTpert} (for $d\ge3$) without assuming \eqref{exptail}, provided the particle systems are also attractive. This last condition is needed to use the results in \cite{CP07}. \end{rem}
\section{\texorpdfstring{Proofs of Theorems~\protect\ref{thmtvm} and \protect\ref{thmgeom}} {Proofs of Theorems 1.3 and 1.4}}\label{seclast} \mbox{} \begin{pf*}{Proof of Theorem~\ref{thmtvm}} Let $\xi_t$ be the affine voter model with parameter $\alpha\in(0,1)$, and $d\ge3$. If $\varepsilon^2=1-\alpha$, then the rate function of $\xi$ is of the form in~\eqref{vmpert} and \eqref{vmpert2} where
\begin{equation} \label{avh} h_i(x,\xi)=-f_i(x,\xi)+1 \bigl(\xi(y)=i \mbox{ for some }y\in{\mathcal{N}}\bigr). \end{equation}
Taking $Z_1,\ldots,Z_{N_0}$ to be the distinct points in ${\mathcal{N}}$ we see that $\mathrm{AV}(\alpha)$ is a voter model perturbation. The fact that $c_{\mathrm{TV} }(x,\xi)$ is cancellative was established in Section~2 of \cite{CD91}, and so, as for $\mathrm{LV}(\alpha)$, we may conclude that $\mathrm{AV}(\alpha)$ is a cancellative process. It is easy to check that $c_{\mathrm{AV}}(x,\xi)$ is not a pure voter model rate function, so the only remaining condition of Theorem~\ref {thmCCTpert} to check is $f'(0)>0$.
To compute $f(u)$, let $\{B^x_u,u\ge0, x\in{\mathbb{Z}^d}\}$ be a system of coalescing random walks with step distribution $p(x)$, and put $A^F_t=\{B^x_t,x\in F\}$, $F\in Y$. The slight abuse of notation
$|A^F_\infty|=\lim_{t\to\infty}|A^F_t|$ is convenient. If $\xi_0(x)$ are i.i.d. Bernoulli with $E(\xi_0(x))=u$, and $F_0,F_1\in Y$ are disjoint, then (see (1.26) in~\cite{CDP11})
\begin{eqnarray} \label{coaldual2} &&\bigl\langle\xi(y)= 0\ \forall y\in F_0, \xi(x)= 1\ \forall x\in F_1 \bigr\rangle_u \nonumber \\[-8pt] \\[-8pt] \nonumber &&\qquad= \sum_{i,j}(1-u)^iu^j P
\bigl(\bigl|A^{F_0}_\infty\bigr|=i, \bigl|A^{F_1}_\infty\bigr|=j,
\bigl|A^{F_0\cup F_1}_\infty\bigr|=i+j \bigr). \end{eqnarray}
From \eqref{f} and \eqref{avh} we have $f(u)=G_0(u)-G_1(u)$, where
\begin{eqnarray*} G_0(u) &=& \bigl\langle1 \bigl\{\xi(0)=0 \bigr\} \bigl(-f_1(0, \xi)+ 1 \bigl\{\xi(y)\ne0\mbox{ for some }y\in{\mathcal{N}}\bigr \} \bigr) \bigr\rangle_u, \\ G_1(u) &=& \bigl\langle1 \bigl\{\xi(0)=1 \bigr\} \bigl(-f_0(0, \xi)+ 1 \bigl\{\xi(y)\ne1\mbox{ for some }y\in{\mathcal{N}}\bigr \} \bigr) \bigr\rangle_u. \end{eqnarray*}
If $c_0=\sum_ep(e)P(|A_\infty^{\{0,e\}}|=2)$, then the assumption that $0\notin{\mathcal{N}}$ and \eqref{coaldual2} imply
\begin{eqnarray*} G_0(u)&=&-c_0u(1-u)+ \bigl\langle1 \bigl\{\xi(0)=0 \bigr\} \bigr\rangle_u - \bigl\langle1 \bigl\{\xi(0)=\xi(y)=0\mbox{ for all }y\in{\mathcal{N}}\bigr\} \bigr\rangle_u \\
&=&-c_0u(1-u)+1-u - \sum_{j=1}^{|{\mathcal{N}}|+1}
(1-u)^jP \bigl(\bigl|A_\infty^{{\mathcal{N}}\cup\{0\}}\bigr|=j \bigr). \end{eqnarray*}
Similarly,
\[
G_1(u) = -c_0u(1-u)+ u - \sum _{j=1}^{|{\mathcal{N}}|+1} u^jP \bigl(\bigl|A_\infty^{{\mathcal{N}}\cup\{0\}}\bigr|=j \bigr). \]
Therefore if $A=|A_\infty^{{\mathcal{N}}\cup\{0\}}|$, we obtain
\begin{eqnarray*} f'(0) &=& G_0'(0)-G_1'(0)=-1+
\sum_{j=1}^{|{\mathcal{N}}|+1}jP(A=j)-1+P(A=1) \\ &=&E \bigl(A-1-1(A>1) \bigr). \end{eqnarray*}
Note that since $A$ is ${\mathbb{N}}$-valued, we have $A-1-1(A>1)\ge 0$ with equality holding if and only if $A\in\{1,2\}$. Hence to show
$f'(0)>0$ it suffices to establish that $P(A>2)>0$. But since $|{\mathcal{N}}\cup\{0\}|\ge3$ by the symmetry assumption on ${\mathcal{N}}$, the required inequality is easy to see by the transience of the random walks $B^x_u$. The complete convergence theorem with coexistence holds if $\varepsilon>0$ is small enough, depending on ${\mathcal{N}}$, by Theorem~\ref{thmCCTpert}. \end{pf*}
\begin{pf*}{Proof of Theorem~\ref{thmgeom}} Let $\eta^\theta_t$ be the geometric voter model with rate function given in~\eqref{geom1}. Then $\eta^\theta_t$ is cancellative for all $\theta\in[0,1]$ (see Section~2 of~\cite{CD91}), and it is clear that $\eta^\theta_t$ is not a pure voter model for $\theta<1$. [The latter follows from the fact that $q_0(A)>0$ for any odd subset of ${\mathcal{N}}\cup\{0\}$.] The next step is to check that $\eta^\theta_t$ is a voter model perturbation. Clearly $\mathbf{0}$ is a trap. If we set $\varepsilon^2=1-\theta$ and $a_j=c(0,\xi)$ for $\xi(0)=0$ and $\sum_{x\in{\mathcal{N}}}\xi(x)=j$, then
\begin{eqnarray*} a_j&=& \Biggl[\sum_{k=1}^j \pmatrix{j\cr k} \bigl(-\varepsilon^2 \bigr)^k \Biggr] \Bigg/ \Biggl[
\sum_{k=1}^{|{\mathcal{N}}|} \pmatrix{|{\mathcal{N}}|\cr k} \bigl(- \varepsilon^2 \bigr)^k \Biggr] \\
&=&\frac{j\varepsilon^2-{j\choose2}\varepsilon^4+O(\varepsilon^6)}{|{\mathcal{N}}|\varepsilon^2-{|{\mathcal{N}}
|\choose2}\varepsilon^4+O(\varepsilon^6)}, \end{eqnarray*}
where ${j\choose2}=0$ if $j=1$. A straightforward calculation [we emphasize that $c_{\mathrm{VM}}$ and $f_0,f_1$ are defined using
$p(x)=1_{{{\mathcal{N}}}}(x)/|{\mathcal{N}}|$ which satisfies \eqref{passump} and \eqref{exptail}] now shows that
\begin{equation} \label{geompert}\qquad c_{\mathrm{GV}}(x,\xi) = c_{\mathrm{VM}} (x,\xi) +
\varepsilon^2\frac{|{\mathcal{N}}|}{2} f_0(x,\xi)f_1(x,\xi) +O \bigl(\varepsilon^4 \bigr)\qquad\mbox{as }\varepsilon\to0, \end{equation}
where the $O(\varepsilon^4)$ term is uniform in $\xi$ and may be written as a function of $f_1(0,\xi)$. It follows from Proposition 1.1 of \cite{CDP11} and symmetry that $\xi ^\varepsilon$ is a voter model perturbation.
To apply Theorem~\ref{thmCCTpert} it only remains to check that $f'(0)>0$, where
\begin{eqnarray*} f(u) &=& \bigl\langle \bigl(1-2\xi(0) \bigr)f_1(0, \xi)f_0(0,\xi) \bigr\rangle_u \\ &=&\sum_{x,y}p(x)p(y) \bigl\langle \bigl(1-2\xi(0) \bigr)\xi(x) \bigl(1-\xi(y) \bigr) \bigr\rangle_u. \end{eqnarray*}
Using \eqref{coaldual2} it is easy to see that for $x,y,0$ distinct,
\begin{eqnarray*} \bigl\langle\xi(x) \bigl(1-\xi(y) \bigr) \bigr\rangle_u &=&u(1-u)P
\bigl(\bigl|A^{\{x,y\}}_\infty\bigr|=2 \bigr), \\ \bigl\langle\xi(0)\xi(x) \bigl(1-\xi(y) \bigr) \bigr\rangle_u
&=&u(1-u)P \bigl(\bigl|A^{\{0,x\}}_\infty\bigr|=1,\bigl|A^{\{0,x,y\}}_\infty\bigr|=2 \bigr) \\
&&{} +u^2(1-u)P \bigl(\bigl|A^{\{0,x,y\}}_\infty\bigr|=3 \bigr). \end{eqnarray*}
If we plug the decomposition ($x,y,0$ still distinct)
\begin{eqnarray*}
P \bigl(\bigl|A^{\{x,y\}}_\infty\bigr|&=&2 \bigr)= P \bigl(\bigl|A^{\{0,x\}}_\infty
\bigr|=1,\bigl|A^{\{x,y\}}_\infty\bigr|=2 \bigr) \\
&&{}+ P \bigl(\bigl|A^{\{0,y\}}_\infty\bigr|=1,\bigl|A^{\{x,y\}}_\infty\bigr|=2
\bigr) + P \bigl(\bigl|A^{\{0,x,y\}}_\infty\bigr|=3 \bigr) \end{eqnarray*}
into the above we find that
\[ f(u) = u(1-u) (1-2u)\sum_{x,y}p(x)p(y) P
\bigl(\bigl|A^{\{0,x,y\}}_\infty\bigr|=3 \bigr), \]
and thus $f'(0)=\sum_{x,y}p(x)p(y)P(|A^{\{0,x,y\}}_\infty|=3)>0$ as required. \end{pf*}
\section*{Acknowledgment} We thank an anonymous referee for a careful reading of the paper and making a number of helpful suggestions which have improved the presentation.
\printaddresses
\end{document} | arXiv |
TMF:
TMF, 1971, Volume 9, Number 3, Pages 440–444 (Mi tmf4517)
This article is cited in 2 scientific papers (total in 2 papers)
Perturbation theory for one-dimensional Schrodinger equations that can be used in a region where the wave function is small
V. S. Pekar
Abstract: The usual perturbation theory series converges badly in the region where the wavefunction $\psi$ is small and the relative correction to $\psi$ is great. The new simple perturbation method is proposed, which is valid, in particular, in the region where $\psi$ is small. The method is based on expanding in the perturbation theory series not the function $\psi$ itself, but its logarithmic derivative,$\frac{d}{dx}\ln\psi$. Corrections of any order to eigen-functions and eigen-values are expressed in quadratures instead of infinite seria. The examples are considered which demonstrate the rapid convergence of the method proposed in cases when the series of the usual theory converges badly.
Theoretical and Mathematical Physics, 1971, 9:3, 1256–1258
Citation: V. S. Pekar, "Perturbation theory for one-dimensional Schrodinger equations that can be used in a region where the wave function is small", TMF, 9:3 (1971), 440–444; Theoret. and Math. Phys., 9:3 (1971), 1256–1258
\Bibitem{Pek71}
\by V.~S.~Pekar
\paper Perturbation theory for one-dimensional Schrodinger equations that can be used in a~region where the wave function is small
\jour TMF
\vol 9
\mathnet{http://mi.mathnet.ru/tmf4517}
\jour Theoret. and Math. Phys.
\pages 1256--1258
\crossref{https://doi.org/10.1007/BF01043417}
http://mi.mathnet.ru/eng/tmf4517
http://mi.mathnet.ru/eng/tmf/v9/i3/p440
This publication is cited in the following articles:
V. S. Polikanov, "On a rapidly converging perturbation theory for a discrete spectrum", Theoret. and Math. Phys., 24:2 (1975), 794–798
G. V. Vikhnina, V. S. Pekar, "Excited states in logarithmic perturbation theory", Theoret. and Math. Phys., 68:1 (1986), 740–743
First page: 1 | CommonCrawl |
\begin{document}
\title{KAM theorem for reversible mapping of low smoothness with application ootnote{
E-mail:[email protected](Li); [email protected](Qi); [email protected](Yuan) }
\begin{abstract} Assume the mapping $$A:\left\{
\begin{array}{ll}
x_{1}=x+\omega+y+f(x,y),\\
y_{1}=y+g(x,y),
\end{array}
\right. (x, y)\in \mathbb{T}^{d}\times B(r_{0})
$$ is reversible with respect to $G: (x, y)\mapsto (-x, y),$ and $| f | _{C^{\ell}(\mathbb{T}^{d}\times B(r_{0}))}\leq \varepsilon_{0}, | g |_{C^{\ell+d}(\mathbb{T}^{d}\times B(r_{0}))}\leq \varepsilon_{0},$
where $B(r_{0}):=\{|y|\le r_0:\; y\in\mathbb R^d\},$ $\ell=2d+1+\mu$ with $0<\mu\ll 1.$ Then when $\varepsilon_{0}=\varepsilon_{0}(d)>0$ is small enough and $\omega$ is Diophantine, the map $A$ possesses an invariant
torus with rotational frequency $\omega.$ As an application of the obtained theorem, the Lagrange stability is proved for a class of reversible Duffing equation with finite smooth perturbation.
\end{abstract}
\section{Introduction and Main Results}\label{s1}
Kolmogorov \cite{Kol}, Arnold \cite{Arnold1963} and Moser \cite{Moser1962} established the well-known KAM theory after their names. Let us begin with a nearly integrable Hamiltonian $H=H_0(y)+\varepsilon\, R(x,y)$ where $x\in\mathbb T^d$ is angle variable and $y$ is action variable in some compact set of $\mathbb R^d$. Endow $H$ the symplectic structure $d\, y\wedge d\, x$. Assume $H_0$ is non-degenerate in the Kolmogorov's sense: $\text{det}\,\left( \frac{\partial^2}{\partial y^2}\, H_0(y) \right)\neq 0$. In 1954's ICM, Kolmogorov announced that for any Diophantine vector $\omega:= \frac{\partial}{\partial y}\, H_0(y)$, the Hamiltonian $H$ possesses an invariant torus which carries quasi-periodic motion with rotational frequency vector $\omega$ provided that $H$ is analytic and $\varepsilon$ is small enough. This result is called Kolmogorov's invariant-tori-theorem. Kolmogorov himself gave an outline of proof in \cite{Kol}. Arnold \cite{Arnold1963} gave a detail proof for the Kolmogorov's theorem. Arnold's proof is a little bit different from Kolmogorov's outline. Recently one has found that Kolmogorov's proof is valid and of more merits. Kolmogorov's basic idea is to overcome the difficulty arising from resonances (small divisors) by Newton iteration method. The main contribution of Moser \cite{Moser1962} to the KAM theory was to extend Kolmogorov's invariant-tori-theorem to smooth category. Moser exploited smooth approximation technique closely related to idea of Nash \cite{Nash1956} to overcome the loss of regularity due to the inversion of certain (non-elliptic) differential operators at each Newton iteration step. In the original work of Moser \cite{Moser1962}, which deals with twist area-preserving maps (corresponding to the Hamiltonian system case in ¡°one and a half¡± degrees of freedom), the perturbation was assumed to be
$C^{333}$. The smoothness assumption (in the twist map case) was later relaxed to five by R\"ussmann \cite{Russmann1970}. The Moser's theorem with improvement by R\"ussmann is usually called Moser's twist theorem. For the Hamiltonian case we refer to \cite{Moser1969, Zehnder1976}, and, especially, \cite{Poschel1982}, where Kolmogorov¡¯s theorem is proved under the hypothesis that the perturbation is $C^{\ell}$ with $\ell> 2d$.
It is well-known that a center ( phase space is foliated by $1$-dimensional invariant tori) of planar linear system can changed into a focus by a non-Hamiltonian nonlinear perturbation so that all invariant tori are broken down. From this one sees that the Hamiltonian structure plays an important role in preserving the invariant tori undergoing perturbations. Besides the Hamiltonian structure (or symplectic structure for mappings), there is so-called reversible structure for differential equations or mappings on which KAM theory can be constructed. Moser \cite{Moser1973} and Arnold \cite{Arnold1984} initiated the study of reversible differential equations or reversible mappings. In 1973, Moser \cite{Moser1973} constructed a KAM theorem for \[
\dot x=\omega+y+f(x,y),\quad \dot y =g(x,y),\]
where $f$ and $g$ are analytic in their arguments and reversible with respect to the involution $(x,y)\mapsto (-x,y)$, that is, \[f(-x,y)=f(x,y),\quad g(-x,y)=-g(x,y).\] The KAM theory for analytic reversible equations (vector-fields) of more general form was deeply investigated in Sevryuk \cite{Sevryuk1986, Sevryuk95, Sevryuk98, Sevryuk2011, Sevryuk2012} and Broer \cite{Broer2009}. Zhang \cite{Zhang2008} constructed a KAM theorem for a class of reversible equations which are assume to be $C^\ell$ smooth where the low bound of $\ell<\infty$ is not specified. Sevryuk \cite{Sevryuk1986} also studied deeply the KAM theory for reversible mappings. For example, Sevryuk \cite{Sevryuk1986} constructed a KAM theorem for a reversible mapping $A$ with respect to $G:$ $$A: \left(
\begin{array}{c}
x \\
y \\
z \\
\end{array}
\right)\mapsto\left(
\begin{array}{c}
x+\lambda y+f^{1}(x, y, z) \\
y+f^{2} (x, y, z)\\
z+f^{3}(x, y, z) \\
\end{array}
\right),\;G: \left(
\begin{array}{c}
x \\
y \\
z \\
\end{array}
\right)\mapsto \left(
\begin{array}{c}
-x+\alpha^{1}(x, y, z) \\
y+\alpha^{2}(x, y, z) \\
z+\alpha^{3}(x, y, z) \\
\end{array}
\right), $$ where $(x, y, z)$ is in some domain in $\mathbb{T}^{n}\times \mathbb{R}^{p}\times\mathbb{R}^{q},$ constant $\lambda\in (0, 1]$,
$f^{j}$ $(j=1, 2, 3)$ and $\alpha^{j} (j=1, 2, 3)$ are real analytic in some domain. Liu \cite{Liu2005} established a KAM theorem for analytic and reversible mapping which is quasi-periodic in $x$. In those works the mappings are required to be analytic.
Naturally one hopes to construct KAM theory for reversible mapping of finite smoothness. Especially one can ask what the lowest smoothness assumption is for reversible mapping.
Actually, KAM theorem for reversible mapping of finite smoothness is useful in the study of some ordinary differential equations. Dieckerhoff-Zehnder \cite{Dieckerhoff-Zehnder1987} showed the Lagrange stability for Duffing equation
\[\ddot x+x^{2n+1}+\sum_{j=0}^{2n}a_{j}(t)x^{j}=0,\;a_{j}(t)\in C^{\infty}(\mathbb{T}^{1})\]
using Moser's twist theorem. See \cite{Laederich-Levi1991, Liu1989, Liu1992, Yuan1995,Yuan1998,Yuan2000} for more details. Levi \cite{Levi1991} generalizes the polynomial $ x^{2n+1}+\sum_{j=0}^{2n}a_{j}(t)x^{j}$ to any finite smooth function of $g(x,t)$ with some suitable conditions, by using the facts that the mapping is required to be finite smooth rather than analytic in Moser's twist theorem and that the Duffing equation is a Hamiltonian system. Liu \cite{Liu1991, Liu1998, Liu2005}, Piao \cite{Piao2008} and Yuan-Yuan \cite{Rong2001} proved the Lagrange stability for the Duffing equation
\begin{equation}\label{1.1} \ddot{x}+\left(\sum_{j=0}^{[(n-1)/2]}{b_{j}}(t)x^{2j+1}\right)\dot{x}+x^{2n+1}+\sum_{j=0}^{n}a_{j}(t)x^{2j+1}=p(t), \end{equation} which is reversible with respect to $G: (x, \dot{x}, t) \mapsto (-x, \dot{x}, -t),$ where either $p(t)=0$ or $p(t)$ is odd. If we want to generalize \eqref{1.1} to a general reversible system $\ddot{x}+g(x,\dot x, t)=0$ where $g$ is finite smooth in each variable, then we need to construct a KAM theorem for reversible mapping of finite smoothness. This is one of aims that we write the present paper.
To that end, let $\mathbb{T}^{d}=(\mathbb{R}/2 \pi \mathbb{Z})^{d}, B(r)=\{y\in \mathbb{R}^{d}\mid |y|<r\}$ with $r>0,$ and let us consider a twist mapping $$A_{0}:\;\;\left\{
\begin{array}{ll}
x_{1}=x+ \omega+ y, \\
y_{1}=y,
\end{array}
\right.$$ where $(x, y)\in \mathbb{T}^{d}\times B(r_{0})$ with some $r_{0}>0$ is a constant, as well as $\omega\in \mathbb{R}^{d}$ is called frequency of $A_{0}.$ It is clear that $A_{0}$ possesses an invariant torus $$\mathcal{J}_{0}:=\{x_{1}=x+\omega: x\in \mathbb{T}^{d}\} \times \{y_{1}=0\}.$$ We will prove that the invariant torus $\mathcal{J}_{0}$ is preserved undergoing a small perturbation of finite smoothness, provided that $\omega$ is Diophantine. More exactly, we have the following theorem:
\begin{theorem}\label{thm1} Consider a mapping $A$ which is the perturbation of $A_{0}:$ $$A:\left\{
\begin{array}{ll}
x_{1}=x+\omega+y+f(x,y),\\
y_{1}=y+g(x,y),
\end{array}
\right. (x, y)\in \mathbb{T}^{d}\times B(r_{0}). $$ Suppose that \begin{itemize}
\item[(A1)] $\omega\in DC(\kappa, \tau)$ with $0<\kappa <1,$ $\tau> d,$ that is,
there exist constants $1>\kappa>0$ and $\tau>d$ such that
\begin{equation}\label{eq1}
\mid \langle k, \omega\rangle+ j\mid\geq \frac{\kappa}{|k|^{\tau}},\;\;\forall \;(k, j)\in \mathbb{Z}^{d}\times \mathbb{Z},\;\;k\neq 0. \end{equation} (In order to the smoothness of perturbations $f$ and $g$ is sharp, we take $\tau=d+\frac{\mu}{100}$ with $0<\mu\ll1$.)
\item[(A2)] Given $\ell=2d+1+\mu$ with $0<\mu\ll 1,$ and $f, g: \mathbb{T}^{d}\times B(r_{0})\rightarrow \mathbb{R}^{d}$ are $C^{\ell}$ and $C^{\ell+d},$ respectively, and
$$ |f|_{C^{\ell}(\mathbb{T}^{d}\times B(r_{0}))}\leq \varepsilon,\;\;|g|_{C^{\ell+d}(\mathbb{T}^{d}\times B(r_{0}))}\leq \varepsilon.$$
\item[(A3)] The mapping $A$ is reversible with respect to the involution $G: (x, y)\mapsto (-x, y),$ that is, $$AGA=G\;\;\mbox{on} \;\;\mathbb{T}^{d}\times B(r_{0}).$$ \end{itemize} Then there exists $\varepsilon_{0}=\varepsilon_{0}(\tau, d, r_{0})>0$ such that for any $0<\varepsilon<\varepsilon_{0},$ the mapping $A$ has an invariant torus $\Gamma$ and the restriction of $A$ on $\Gamma$ is expressed by $$A\mid _{\Gamma}: x\mapsto x+\omega.$$
\end{theorem}
\begin{theorem}\label{1.2} Consider a system of non-autonomous differential equations \begin{equation} \label{a} (a):\;\;\left\{
\begin{array}{ll}
\dot{x}=\omega+y+f(x, y, t), \\
\dot{y}=g(x, y, t),
\end{array} \right.\;\;(x, y, t)\in \mathbb{T}^{d}\times B(r_{0})\times \mathbb{T}:= D. \end{equation} Suppose that \begin{itemize}
\item [(a1)] $\omega\in DC(\kappa, \tau)$ with $0<\kappa <1,$ $\tau>d.$
\iffalse that is, there exist constants $0<\kappa <1,$ $\tau>d$ such that
$$|\langle k, \omega\rangle+ j|\geq \frac{\kappa}{|k|^{\tau}},\;\;(k, j)\in \mathbb{Z}^{d}\times \mathbb{Z},\;\;k\neq 0,$$\fi
\item [(a2)] $f$, $g: \mathbb{T}^{d}\times B(r_{0})\times \mathbb{T}\rightarrow \mathbb{R}^{d}$ are $C^{\ell}$ and $C^{\ell+d},$ respectively, and
$$|f|_{C^{\ell}(D)}\leq \varepsilon,\;\;|g|_{C^{\ell+d}(D)}\leq \varepsilon.$$
\item [(a3)] The system \eqref{a} of differential equations is reversible with the involution $G: (x, y, t)\mapsto (-x, y, -t)$, that is,
for any $(x, y, t)\in D,$
\begin{eqnarray*}
&f(-x, y, -t)=f(x, y, t),\\
&g(-x, y, -t)=-g(x, y, t).
\end{eqnarray*} \end{itemize} Then there exists $\varepsilon_{0}=\varepsilon_{0}(\tau, d, r_{0})>0$ such that for any $0<\varepsilon <\varepsilon_{0}$ there exists a coordinate changes \begin{eqnarray*} \psi: \left\{
\begin{array}{ll}
x=\xi+u(\xi, \eta, t) \\
y=\eta+v(\xi, \eta, t)
\end{array}
\right. \end{eqnarray*} such that the map $\psi$ restricted to $\{\xi=\omega t: t\in \mathbb{R}\}\times \{\eta=0\}\times \{t: t\in \mathbb{T}\}$ is a real $C^{0}$ embedding into $\mathbb{T}^{d}\times \mathbb{R}^{d}$ of a rotational torus with frequency $\omega$ for the system \eqref{a}. \end{theorem}
\begin{rem}\label{rem0} The first equation in \eqref{a} can be replaced by $\dot{x}=\omega+\alpha(y)+f(x, y, t)$ with $\partial_{y}\alpha(y)\geq C_{0}>0$ in the sense of positive definite matrix. In Theorem \ref{1.2}, the condition that $f$ and $g$ are $2\pi$ periodic in time $t$ can be generalized to that $f$ and $g$ are quasi-periodic with frequency $\tilde\omega\in \mathbb R^{\tilde d}$ ($\tilde d\ge 1$) in time $t$, if we take $\ell=2(d+\tilde d)-1+\mu$ and replace the condition $(a1)$ by $(\omega,\tilde\omega)\in\mathbb R^{d+\tilde d}\bigcap DC(\kappa,\tau)$ with $0<\kappa<1, \; \tau>d+\tilde d$. \end{rem}
\begin{rem}\label{rem1} For exact and area-preserving twist maps on annulus, it was proved by Herman in \cite{Her83} that unperturbed invariant curves can be destructed by $C^{3-\delta}$ $(0<\delta <1)$ arbitrarily small perturbation, and the unperturbed invariant curves can be preserved by $C^{3+\delta}$ sufficiently small perturbation for some special Diophantine frequency $\omega$ of zero Lebesgue measure but infinitely many numbers.
Recently, Cheng-Wang \cite{Cheng} showed that for an integrable Hamiltonian $H_{0}=\frac{1}{2}\sum_{i}^{d}y_{i}^{2}$ $(d\geq 2),$ any Lagrangian torus with a given unique rotation vector can be destructed by arbitrarily $C^{2d-\delta}$ small Hamiltonian perturbations. Checking those counter-examples above, it was seen that the results on destructed invariant tori hold still true if an additional reversible condition is imposed on the symplectic mapping or Hamiltonian vector. So the optimal smoothness of the reversible mapping $A$ depending periodically on time $t$ should be larger or equal to $2 d+1+\mu$. It is not clear whether the smoothness $3 d+1+\mu$ is optimal or this requirement is a shortcoming of our proof. It is worth to investigate further. \end{rem}
\begin{rem} The proof of Theorem \ref{thm1} is different from that of Moser's twist theorem. We recall that Moser invoked the smooth approximation technique from Nash's idea to overcome the loss of regularity during Newton iteration. More exactly, Moser used a smoothness operator, say $S_{s}$, to decompose the perturbation vector field $(f, g)$ into $(f, g)=(S_{s}f+(1-S_{s})f, S_{s}g+(1-S_{s})g),$ where $S_{s}f$ and $S_{s}g$ are more smooth than $f$ and $g,$ and $(1-S_{s})f$ and $(1-S_{s})g$ are smaller than $f$
and $g.$ Then he eliminated the perturbations $(S_{s}f, S_{s}g).$ An important fact is that $(S_{s}f, S_{s}g)$ is still symplectic if $(f, g)$ is symplectic. Unfortunately,
when $(f, g)$ is reversible with respect to the involution $G,$ we do not know if $(S_{s}f, S_{s}g)$ is, too, reversible with respect to $G.$ So we could not transplant Moser's
trick to deal with the reversible mapping of finite smoothness. In the present paper, we regard the reversible mapping $A$ in Theorem
\ref{thm1} as the Poincare map of a reversible differential equation. And then we construct a KAM theorem (Theorem \ref{1.2}) for a reversible differential equation which
is periodic in time. Then we proved Theorem \ref{thm1} by using Theorem \ref{1.2}. \end{rem}
This paper is organized as follows: Section 2 we give out the approximation theorem of Jackson-Moser-Zhender. In Section 3, we give out iterative constants and iterative domains in Newton iteration. In Section 4, we give out the key iterative lemma (See Lemma \ref{lem5.1}). Using Lemma \ref{lem5.1} further, we give proof of Theorems \ref{1.2} and \ref{thm1} . In Section 5, we derive the homological equations and give out the estimates of the solutions of the homological equations to eliminates perturbations. In Section 6, we make the estimate of new perturbations. In Section 7, we give an application of the obtained Theorem \ref{thm1} to the Lagrange stability for reversible Duffing equation with finite smooth nonlinear perturbation.
\iffalse \section{Diophantine conditions } We prove that almost any $\omega\in \mathbb{R}^{d}$ satisfies the Diophantine conditions \eqref{eq1}.
\begin{lemma} Let $\Pi_{0}=[0, 1]^{d}.$ Then there exists a subset $\widetilde{\Pi}_{0}\subset \Pi_{0}$ with $$Leb \widetilde{\Pi}_{0}\leq C \kappa\;\;(0<\kappa\ll 1)$$ such that for any $\omega\in \Pi_{0}\backslash \widetilde{\Pi}_{0},$ $\omega\in DC(\kappa, \tau),$ where $Leb \widetilde{\Pi}_{0}$ denotes Lebesgue measure for set $\widetilde{\Pi}_{0}.$ \end{lemma}
\begin{proof}
Let $$\Pi_{k,j}=\{\omega\in \Pi_{0}\mid |\<k, \omega\rangle+ j|<\frac{\kappa}{|k|^{\tau}}\},\;\;(k, j)\in \mathbb{Z}^{d}\times \mathbb{Z},\;\;k\neq 0.$$
Note that $|\langle k, \omega \rangle|\leq |k|,$ where $|k|=|k_{1}|+|k_{2}|+\cdots+|k_{d}|$ for $k=(k_{1}, \cdots, k_{d}).$ Thus if $|j|> |k|+1,$ then
$$|\<k, \omega\rangle+ j|\geq |k|+1-|\<k, \omega\rangle|\geq 1 >\frac{ \kappa}{|k|^{\tau}}.$$
It follows that $\Pi_{k, j}=\phi.$ Now we assume $|j|\leq |k|+1.$ Let $f(\omega)=\<k, \omega\rangle+2\pi j.$ Since $k\in \mathbb{Z}^{d}\setminus \{0\},$ there exists a unit vector $\nu\in \mathbb{Z}^{d}$ such that
$$\frac{d f(\omega)}{d \nu}\geq \frac{|k|}{d}.$$ It follows that
$$Leb \Pi_{k,j}\leq \frac{2d(\sqrt{2})^{d-1}\kappa}{|k|^{\tau+1}}.$$ So
$$Leb \left(\bigcup_{j\in \mathbb{Z}}\Pi_{k,j}\right)\leq \frac{2d(\sqrt{2})^{d-1} \kappa}{ |k|^{\tau+1}}\sum_{|j|\leq |k|+1} 1\leq \frac{2d(\sqrt{2})^{d-1}\kappa}{ |k|^{\tau+1}}2(|k|+2)
\leq \frac{C \kappa}{|k|^{\tau}}.$$ Moreover,
$$ Leb \left(\bigcup_{k\in\mathbb{Z}^{d}\setminus \{0\}}\bigcup_{j\in \mathbb{Z}}\Pi_{k,j}\right)\leq \sum_{k\in\mathbb{Z}^{d}\setminus \{0\}} \frac{C\kappa}{|k|^{\tau}}\leq C\kappa,$$ where the last constant $C=C(\tau)$ depends on $\tau.$ Let $\widetilde{\Pi}_{0}=\bigcup_{k\in\mathbb{Z}^{d}\setminus \{0\}}\bigcup_{j\in \mathbb{Z}}\Pi_{k,j}.$ Then $Leb \widetilde{\Pi}_{0}\leq C\kappa.$ \end{proof}
\fi
\section{Approximation Lemma }
First we denote by $|\cdot|$ the norm of any finite dimensional Euclidean space. Let $C^\mu(\mathbb R^m)$ for $0<\mu<1$ denote the space of bounded H\"older continuous functions $f:\; \mathbb R^m\to \mathbb R^n$ with the norm
\[|f|_{C^\mu}=\sup_{0<|x-y|<1}\frac{|f(x)-f(y)|}{|x-y|^\mu}+\sup_{x\in\mathbb R^m}|f(x)|.\]
If $\mu=0$ the $|f|_{C^\mu}$ denotes the sup-norm. For $\ell=k+\mu$ with $k\in\mathbb N$ and $0\le \mu<1$ we denote by $C^\ell(\mathbb R^m)$ the space of functions $f:\; \mathbb R^m\to \mathbb R^n$ with H\"older continuous partial derivatives $\partial^\alpha\, f
\in\, C^\mu(\mathbb R^m)$ for all multi-indices $\alpha=(\alpha_1,...,\alpha_m)\in\mathbb N^m$ with $|\alpha|=\alpha_1+...+\alpha_m\le k $. We define the norm
\[|f|_{C^\ell}:=\sum_{|\alpha|\le \ell} |\partial^\alpha f|_{C^\mu}\] for $\mu=\ell-[\ell]<1$. In order to give an approximate lemma, we define the kernel function \[K(x)=\frac{1}{(2\pi)^m}\int_{\mathbb R^m}\widehat{K}(\xi)e^{\mathbf{i}\, \<x,\xi\rangle}\, d\xi,\; x\in\mathbb C^m,\]
where $\widehat{K}(\xi)$ is a $C^\infty$ function with compact support, contained in the ball $|\xi|\le a$ with a constant $a>0$, that satisfies \[\partial^\alpha\,\widehat{K}(0)=\begin{cases} 1, & \text{if}\, \alpha=0,\\ 0, & \text{if}\, \alpha\neq 0.\end{cases} \]
Then $K:\, \mathbb C^m\to \mathbb R^n$ is a real analytic function with the property that for every $j>0$ and every $p>0$, there exists a constant $c_1=c_1(j,p)>0$ such that for all $\beta\in\mathbb N^m$ with $|\beta|\le j$, \begin{equation}
\left|\partial^\beta\, K(x+\mathbf{i} y) \right|\le c_{1} (1+|x|)^{-p}e^{a|y|},\;\; x,y\in\mathbb R^m.\label{y1} \end{equation}
\begin{lemma}\label{lem3}(Jackson-Moser-Zehnder) There is a family of convolution operators
\begin{equation} (S_{s}F)(x)=s^{-m}\int_{\mathbb{R}^{m}}K(s^{-1}(x-y))F(y)dy,\;\;0<s\leq 1,\;\;\forall\;F\in C^{0}(\mathbb{R}^{m})\label{y2}\end{equation} from $C^{0}(\mathbb{R}^{m})$ into the linear space of entire (vector) functions on $\mathbb{C}^{m}$ such that for every $\ell>0$ there exist a constant $c=c(\ell)>0$ with the following properties: If $F\in C^{\ell}(\mathbb{R}^{m}),$
then for $|\alpha|\leq \ell$ and $|\mathrm{Im} x|\leq s,$
\begin{equation}\label{2.11}
|\partial^{\alpha}(S_{s}F)(x)-\sum_{|\beta|\leq \ell-|\alpha|}\partial^{\alpha+\beta}F(\mathrm{Re} x)({\bf{i}}\, \mathrm{Im} x)^{\beta}/\beta!|\leq c\,|F|_{C^{\ell}}s^{\ell-|\alpha|}. \end{equation}
Moreover, in the real case\begin{eqnarray}
&& |S_{s}F-F|_{C^{p}}\leq c |F|_{C^{\ell}}s^{\ell-p},\;\;p\leq \ell,\label{2.13-1}\\
&& |S_{s}F|_{C^{p}}\leq c|F|_{C^{\ell}}s^{\ell-p},\;\;p\leq \ell.\label{2.13} \end{eqnarray} Finally, if $F$ is periodic in some variables then so are the approximating functions $S_{s}F$ in the same variables. \end{lemma} \iffalse \begin{rem}\label{rem*} Usually, $K(z)=\widehat{\varphi}(z)=\int_{\mathbb{R}^{n}}e^{{\bf i}z\cdot y}\varphi(y)dy,$ $z\in \mathbb{R}^{n},$ where $\varphi: \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ is $C^{\infty}$ and has compact support. In order to fulfill our aim, when $n=2d,$ we furthermore impose conditions on $\varphi: \varphi(-x, y)=(-1)^{d}\varphi(x, y),$ $\forall (x,y)\in \mathbb{R}^{2d}.$ Thus, $K(-x, y)=(-1)^{d}K(x, y).$ And $\ell$ can be replaced by any $\ell\in\mathbb{R_{+}}\setminus \mathbb{Z}.$ \end{rem} \fi \begin{rem}\label{rem**}
Moreover we point out that from \eqref{2.13} one can easily deduce the following well-known convexity estimates which will be used later on \begin{eqnarray}
&&|f|_{C^{\alpha}}^{l-k}\leq c|f|_{C^{k}}^{l-\alpha}|f|_{C^{l}}^{\alpha-k},\;\;k\leq \alpha\leq l,\label{5.4-1}\\
&& |f\cdot g|_{C^{s}}\leq c(|f|_{C^{s}}|g|_{C^{0}}+|f|_{C^{0}}|g|_{C^{s}}),\;\;s\geq 0.\label{5.5-1} \end{eqnarray} See \cite{Salamon2004, Zehnder1976} for the proofs of Lemma \ref{lem3} and the inequalities \eqref{5.4-1} and \eqref{5.5-1}. \end{rem}
\begin{rem}\label{rem**y} From the definition of the operator $S_s$, we clearly have
\begin{equation} \sup_{x,y\in\mathbb R^m,|y|\le s}\, \left|S_s\, F(x+\mathbf{i}\, y) \right|\le C |F|_{C^0}.\label{3.x}\end{equation} In fact, by the definition of $S_s$, we have that for any $x,y\in\mathbb R^m$ with $|y|\le s$,
\begin{eqnarray}\left|S_s\, F(x+\mathbf{i}\, y) \right|&=&\left|s^{-m}\int_{\mathbb R^m}K(s^{-1}(x+\mathbf{i}\, y-z))F(z)\, d\,z \right| \\
&=& \left|\int_{\mathbb R^m}K(\mathbf{i}\,s^{-1} y+\xi)F(x-s \xi)\, d\,\xi\right| \\
&\le & |F|_{C^0}\, \int_{\mathbb R^m}\left|K(\mathbf{i}\,s^{-1} y+\xi)\right|\, d\,\xi\\
&\le & C \, |F|_{C^0}, \end{eqnarray} where we used \eqref{y1} in the last inequality. The following lemma shows that the operator $S_{s}$ commutes with the involution map $G: (x, y, t)\mapsto (-x, y, -t).$ \end{rem}
\begin{lemma}\label{lem3.2} Let $m=2d+1$ in Lemma \ref{lem3}. Then
(1)when $F(-x, y, -t)=-F(x, y, t),$ for $\forall (x, y, t)\in\mathbb{T}^{d}\times B(r)\times \mathbb{T}$ with $r>0,$ we have
$$S_{s} F(-x, y, -t)=- S_{s}F(x, y, t).$$
(2) when $F(-x, y, -t)=F(x, y, t),$ for $\forall (x, y, t)\in\mathbb{T}^{d}\times B(r)\times \mathbb{T}$ with $r>0,$ we have
$$S_{s} F(-x, y, -t)=S_{s}F(x, y, t).$$
\end{lemma}
\begin{proof} Let $\xi=(\xi_{1}, \xi_{2}, t)\in \mathbb{R}^{d}\times \mathbb{R}^{d}\times\mathbb{R}=\mathbb{R}^{m}.$ Clearly we can choose the kernel function $\widehat{K}(\xi)$ such that $$\widehat{K}(-\xi_{1}, \xi_{2}, -t)=\widehat{K}(\xi_{1}, \xi_{2}, t),\;\;\forall \xi\in\mathbb{R}^{m}.$$ It follows that $$K(-x, y, -t)=(-1)^{d+1}K(x, y, t),\;\;\forall (x, y, t)\in \mathbb{R}^{m}.$$ By the definition of $S_{s},$ $$S_{s}F(x, y, t)=s^{-(2d+1)}\int_{\mathbb{R}^{d}\times \mathbb{R}^{d}\times \mathbb{R}} K(s^{-1}(x-\tilde{x}, y-\tilde{y}, t-\tilde{t}))F(\tilde{x}, \tilde{y}, \tilde{t})d \tilde{x} d \tilde{y} d \tilde{t}.$$ So \begin{eqnarray*} S_{s}F(-x, y, -t)&=&s^{-(2d+1)}\int_{\mathbb{R}^{d}\times \mathbb{R}^{d}\times \mathbb{R}} K(s^{-1}(-x-\tilde{x}, y-\tilde{y}, -t-\tilde{t}))F(\tilde{x}, \tilde{y}, \tilde{t})d \tilde{x} d \tilde{y} d \tilde{t}\\ &=& s^{-(2d+1)}(-1)^{d+1}\int_{\mathbb{R}^{d}\times \mathbb{R}^{d}\times \mathbb{R}} K(s^{-1}(x+\tilde{x}, y-\tilde{y}, t+\tilde{t}))F(\tilde{x}, \tilde{y}, \tilde{t})d \tilde{x} d \tilde{y} d \tilde{t}\\ &=&s^{-(2d+1)}\int_{\mathbb{R}^{d}\times \mathbb{R}^{d}\times \mathbb{R}} K(s^{-1}(x-{x}^{*}, y-\tilde{y}, t-{t}^{*}))F(-{x}^{*}, \tilde{y}, -{t}^{*})d {x}^{*} d \tilde{y} d {t}^{*}\\ &=& \mp s^{-(2d+1)}\int_{\mathbb{R}^{d}\times \mathbb{R}^{d}\times \mathbb{R}} K(s^{-1}(x-\tilde{x}, y-\tilde{y}, t-\tilde{t}))F(\tilde{x}, \tilde{y}, \tilde{t})d \tilde{x} d \tilde{y} d \tilde{t}\\ &=& \mp S_{s}F(x, y, t), \end{eqnarray*} where $\mp=-$ for case (1), $\mp=+$ for case (2). \end{proof}
Consider a $\mathbb{R}^{n}-$ valued function $F: D\rightarrow \mathbb{R}^{n}$ with
$$|F|_{C}^{\ell}(D)\leq \varepsilon.$$ Recall $D=\mathbb{T}^{d}\times B(r_{0})\times \mathbb{T}.$ By Whitney's extension theorem, we can find a $\mathbb{R}^{n}-$ valued function $\widetilde{F}: \mathbb{T}^{d}\times \mathbb{R}^{d}\times \mathbb{T}\rightarrow \mathbb{R}^{n}$ such that $\widetilde{F}\mid_{D}=F$ \;\;(i. e. $\widetilde{F}$ is the extension of $F$) and
$$|\widetilde{F}|_{C^{|\alpha|}(\mathbb{T}^{d}\times \mathbb{R}^{d}\times \mathbb{T})}\leq C_{\alpha}\mid F\mid _{C^{|\alpha|}(D)},
\;\;\forall \alpha\in \mathbb{Z}^{d}_{+},\;|\alpha|\leq \ell,$$ where $C_{\alpha}$ is a constant depends only $\ell$ and $d.$
Let $z=(x, y, t)$ for brevity, define, for $\forall s>0,$ $$(S_{s} \widetilde{F})(z)=s^{-(2d+1)}\int_{\mathbb{T}^{d}\times \mathbb{R}^{d}\times \mathbb{T}}K(s^{-1}(z-\tilde{z}))\widetilde{F}(\tilde{z})d\tilde{z}.$$
Let $\mathbb{T}^{d}_{s}=\{\phi\in (\mathbb{C}/ 2\pi \mathbb{Z})^{d}: |Im \phi|<s\},$ $\mathbb{R}^{d}_{s}=\{x\in \mathbb{C}^{d}\mid |Im x|<s\}.$ Fix a sequence of fast decreasing numbers $s_{\nu} \downarrow 0,$ $\nu\in \mathbb{Z}_{+}$ and $s_{0}\leq 1/2.$ Let $$F^{(\nu)}(z)=(S_{s_{\nu}}\widetilde{F})(z),\;\;\nu\geq 0.$$ Then $F^{(\nu)}$'s $(\nu\geq 0)$ are entire functions in $\mathbb{C}^{2d+1},$ in particular, which obey the following properties.
\begin{itemize}
\item [(1)] $F^{(\nu)}$'s $(\nu\geq 0)$ are real analytic \footnote{that is, $F^{\nu}(z)$ is analytic in
$\mathbb{T}^{d}_{s_{\nu}}\times\mathbb{R}^{d}_{s_{\nu}}\times\mathbb{T}_{s_{\nu}}$, and is real when $z$ is real.} on the complex domain $\mathbb{T}^{d}_{s_{\nu}}\times \mathbb{R}^{d}_{s_{\nu}}\times \mathbb{T}_{s_{\nu}}:=D_{s_{\nu}};$
\item [(2)] The sequence of functions $F^{(\nu)}(z)$ satisfies the bounds \begin{eqnarray}
&&\sup_{z\in D_{s_{\nu}}}|F^{(\nu)}(z)-F(z)|\leq C |F|_{C^{\ell}(D)}s_{\nu}^{\ell},\label{09.1}\\
&& \sup_{z\in D_{s_{\nu+1}}}|F^{(\nu+1)}(z)-F^{(\nu)}(z)|\leq C |F|_{C^{\ell}(D)}s_{\nu}^{\ell},\label{09.2} \end{eqnarray} where constants $C=C(d, \ell)$ depend on only $d$ and $\ell;$
\item [(3)] The first approximate $F^{(0)}(z)=(S_{s_{0}}\widetilde{F})(z)$ is ``small" with respect to $F.$ Precisely,
\begin{equation}\label{09.3}
|F^{(0)}(z)|\leq C|F|_{C^{\ell}(D)},\;\;\forall z\in D_{s_{0}},
\end{equation} where constant $C=C(d, \ell)$ is independent of $s_{0}.$
\item [(4)] From Lemma \ref{lem3}, we have that
\begin{equation}\label{09.4}F(z)=F^{(0)}(z)+\sum_{\nu=0}^{\infty}(F^{(\nu+1)}(z)-F^{(\nu)}(z)),\;\;\forall z\in D.\end{equation}
Let
\begin{equation} \label{09.5}F_{0}(z)=F^{(0)}(z),\;\;F_{\nu+1}(z)=F^{\nu+1}(z)-F^{\nu}(z).\end{equation}
Then
\begin{equation}\label{09.6}F(z)=\sum_{\nu=0}^{\infty}F_{\nu}(z),\;\;\forall z\in D.\end{equation} \end{itemize}
By Lemma\ref{lem3.2}, we have \begin{eqnarray} && F_{\nu}(-x, y, -t)=-F_{\nu}(x, y, t),\;\;\text{if}\;\;F(-x, y, -t)=-F(x, y, t),\label{09.7}\\ && F_{\nu}(-x, y, -t)=F_{\nu}(x, y, t),\;\;\text{if}\;\;F(-x, y, -t)=F(x, y, t).\label{09.8} \end{eqnarray}
\section{Iterative constants} \begin{itemize} \item Given constant $\mu$ with $0<\mu\ll 1$, and let $\tau=d+\frac{\mu}{100}$, $\widetilde{\mu}=\frac{\mu}{100(2\tau+1+\mu)};$
\item $\ell=2 d+1+\mu\;;$
\item $\varepsilon_{0}=\varepsilon,$ \footnote{We hope that the readers are able to distinguish this $\varepsilon_{0}$ with that in Theorems 1.1 and 1.2.} $\varepsilon_{\nu}=\varepsilon^{(1+{\widetilde{\mu}})^{\nu}},$
$\nu=0, 1, 2, \cdots,$ which measures the size of perturbation at $\nu-$th step of Newton iteration;
\item $s_{\nu}=\varepsilon_{\nu}^{1/\ell},\;\;\nu=0, 1, 2, \cdots$, which measures the width of angle variable in analytic approximation; \item $r_{\nu}=s_{\nu}^{d+1+\frac{\mu}{10}},\;\;\nu=0, 1, 2, \cdots$, which measures the size of action variable in analytic approximation;
\item $s^{(j)}_{\nu}=s_{\nu}-\frac{j}{100\, \ell}(s_{\nu}-s_{\nu+1}),\; j=1,2,...,100\, \ell$, which are bridges between $s_{\nu}$ and $s_{\nu+1}$;
\item $r^{(j)}_{\nu}=r_{\nu}-\frac{j}{100\, \ell}(r_{\nu}-r_{\nu+1}),\; j=1,2,...,100\, \ell$, which are bridges between $r_{\nu}$ and $r_{\nu+1}$; \item
$B_{\mathbb{C}}(r)=\{y\in \mathbb{C}^{d}: |y|\leq r\},$ for $\forall r\geq 0;$
\item $D(s, r)=\mathbb{T}^{d}_{s}\times B_{\mathbb{C}}(r)\times \mathbb{T}_{s},\;\;\forall r\geq 0, s\geq 0.$
\item
For a $\mathbb{C}^{n}-$ valued function $F(x, y, t)$ analytic in $D(s, r),$ denote
$$||F||_{s, r}=\sup_{z=(x, y, t)\in D(s, r)}|F(z)|,$$ here (and other places) $|\cdot|$ is Euclidean norm. \end{itemize}
\section{Iterative Lemma}\label{lemma4.1}
Let us return to function $f=f(x, y, t),$ $g=g(x, y, t)$ in Theorem \ref{1.2}. Let $z=(x, y,t)$ for brevity. With the above preparation, we can rewrite equation \eqref{a} in Theorem \ref{1.2} as follows:
\begin{equation} \label{92.0}\dot{x}=\omega+y+\sum_{\nu=0}^{\infty}f_{\nu}(z),\;\dot{y}=\sum_{\nu=0}^{\infty}g_{\nu}(z),\end{equation}
where \begin{equation}\label{92.1} f_{\nu}, g_{\nu}: \mathbb{T}^{d}_{s_{\nu}}\times \mathbb{R}^{d}_{s_{\nu}}\times\mathbb{T}_{s_{\nu}}\rightarrow\mathbb{C}^{n}\end{equation} are real analytic, and \begin{equation}\label{92.2}\parallel f_{\nu}\parallel_{s_{\nu}, r_{\nu}}\leq C \varepsilon_{\nu},\;\;\parallel g_{\nu}\parallel_{s_{\nu},r_{\nu}}\leq C\varepsilon_{\nu}s_{\nu}^{d}\end{equation} and for any $(x,y,t)\in D(s_{\nu}, r_{\nu})$ \begin{equation}\label{92.3} f_{\nu}(-x, y, -t)=f_{\nu}(x, y, t),\;\;g_{\nu}(-x, y, -t)=-g_{\nu}(x, y, t).\end{equation} The basic idea in KAM theory is to kill the perturbations $f$ and $g$ by Newton iteration. The procedure of the iteration is as follows:
$1^{st}$ step: to search for a involution map $\Phi_{0}$ (which keeps the involution $G: (x, y, t)\mapsto (-x, y, -t)$ unchanged) such that the analytic vector-field $${(a)_{0}:\;\; (\omega+y+f_{0}, g_{0})}$$ is changed by $\Phi_{0}$ into \begin{eqnarray*} (a)_{0}\circ \Phi_{0}&=&(\omega+y+f_{0}, g_{0})\circ \Phi_{0}\\ &=&(\omega+y+f^{0}_{1}, g_{1}^{0}):=(a)_{1}, \end{eqnarray*} where $f^{0}_{1}=O (\varepsilon_{1})$ and $g^{1}_{0}=O(\varepsilon_{1}s^{d}_{1}).$
$2^{nd}$ step: to search for a involution map $\Phi_{1}$ such that $(a)_{1}+(f_{1}, g_{1})$ is changed into $(\omega+y+f_{2}^{0}, g_{2}^{0}),$ where $f_{2}^{0}=O(\varepsilon_{2},)$ $g_{2}^{0}=O(\varepsilon_{2}s_{2}^{d})$. The combination of steps 1 and 2 implies that $(\omega+y+f_{0}+f_{1}, g_{0}+g_{1})\circ \Phi_{0}\circ\Phi_{1}=(\omega+y+f^{0}_{2}, g^{0}_{2}).$ Repeating the above procedure, at $m+1^{th}$ step, we have that $(a)_{m}+(f_{m}, g_{m})$ is changed by $\Phi_{m}$ into $(\omega+y+f^{0}_{m+1}, g_{m+1}^{0}),$ where $f_{m+1}^{0}=O(\varepsilon_{m+1}),$ $g_{m+1}^{0}=O(\varepsilon_{m+1}s_{m+1}^{d}).$ That is, $(\omega+y+\sum_{j=0}^{m}f_{j}, \sum_{j=0}^{m}g_{j})\circ \Phi^{(m)}=(\omega+y+f^{0}_{m+1}, g_{m+1}^{0}),$ where $\Phi^{(m)}=\Phi_{0}\circ\Phi_{1}\circ\cdots \circ\Phi_{m}.$ Finally letting $m\rightarrow \infty,$ and letting $$\Phi^{\infty}:=\lim_{m\rightarrow \infty}\Phi^{(m)},$$ we have \begin{eqnarray*} &&(\omega+y+f,g)\circ \Phi^{\infty}\\ &&=\lim_{m\rightarrow \infty} (\omega+y+\sum_{j=0}^{m}f_{j}, \sum_{j=0}^{m}g_{j})\circ \Phi^{(m)}\\ &&=\lim_{m\rightarrow \infty} (\omega+y+f_{m+1}^{0}, g_{m+1}^{0})\\ &&=\lim_{m\rightarrow \infty} (\omega+y+O(\varepsilon_{m+1}), O(\varepsilon_{m+1}s_{m+1}^{d}))\\ &&=(\omega+y, 0). \end{eqnarray*} From this, we see that $(\Phi^{\infty})^{-1}(\{x=\omega t\}\times \{y=0\}\times \{t: t\in \mathbb{T}\})$ is an invariant torus of the original vector-field. This iterative procedure can be found in \cite{Chierchia-Qian}. The following iterative Lemma is a materialization of the above iterative procedure.
\begin{lemma}\label{lem5.1}(Iterative Lemma) Let $\omega\in DC(\kappa, \tau).$ Assume that we have $m$ coordinate changes $\Phi_{0}=\Psi_{0}^{-1},\cdots, \Phi_{m-1}=\Psi_{m-1}^{-1},$ which obey $$\Psi_{j}: D(s_{j}, r_{j})\rightarrow D(s_{j-1}, r_{j-1})\;\;(j=0, 1, \cdots, m-1)$$ of the form $$\Psi_{j}: x=\tilde{x}+u_{j}(\tilde{x}, \tilde{y}, t),\;y=\tilde{y}+v_{j}(\tilde{x}, \tilde{y}, t),\;(j=0, 1, \cdots, m-1)$$ and $u_{j},$ $v_{j}$ are real for real arguments and analytic in each argument with estimates \begin{equation}\label{y5.1}\parallel u_{j}\parallel_{s_{j}, r_{j}}\leq C\varepsilon_{j}s_{j}^{-d}, \parallel v_{j}\parallel_{s_{j}, r_{j}}\leq C\varepsilon_{j}\;\;(j=0, 1, \cdots, m-1)\end{equation} such that the system of equations $$(a)^{(m-1)}: \left\{
\begin{array}{ll}
\dot{\tilde{x}}=\omega+\tilde{y}+\sum_{j=0}^{m-1}f_{j}(\tilde{x}, \tilde{y}, t),\\
\dot{\tilde{y}}=\sum_{j=0}^{m-1}g_{j}(\tilde{x}, \tilde{y}, t)
\end{array}
\right. $$ is changed by $\Phi^{(m-1)}=\Phi_{0}\circ \cdots\circ\Phi_{m-1}$ into $$(a)_{*}^{(m-1)}: \left\{
\begin{array}{ll}
\dot{{x}}=\omega+{y}+f^{0}_{m}({x}, {y}, t),\\
\dot{{y}}=g^{0}_{m}({x}, {y}, t),
\end{array}
\right. $$ where $f_{m}^{0},$ $g_{m}^{0}$ obey
\begin{itemize}
\item [$(1)_{m}$] The functions $f_{m}^{0}$, $g_{m}^{0}$ are real for real arguments;
\item [$(2)_{m}$] The functions $f_{m}^{0}$, $g_{m}^{0}$ are analytic in $D(s_{m}, r_{m})$ with estimates \begin{equation}\label{y5.1} \parallel f_{m}^{0}\parallel_{s_{m}, r_{m}}\leq C \varepsilon_{m},\;\;\parallel g_{m}^{0}\parallel_{s_{m}, r_{m}}\leq C \varepsilon_{m}s_{m}^{d};\end{equation}
\item [$(3)_{m}$] The functions $f_{m}^{0}$, $g_{m}^{0}$ is reversible with respect to involution map: $G: (x, y, t)\mapsto (-x, y, -t),$ that is, \begin{equation}\label{y5.2} f_{m}^{0}(-x, y, -t)=f_{m}^{0}(x, y, t),\;\;g_{m}^{0}(-x, y, -t)=-g_{m}(x, y, t).\end{equation} \end{itemize} Then there is a coordinate change $\Phi_{m}=\Psi^{-1}_{m}:$ $$\Psi_{m}: D(s_{m+1}, r_{m+1})\rightarrow D(s_{m}, r_{m})$$ of the form \begin{equation}\label{y5.3} \Psi_{m}: \xi=x+u_{m}(x, y, t),\;\;\eta=y+v_{m}(x, y, t)\end{equation} and \begin{equation}\label{y5.4} \parallel u_{m}\parallel_{s_{m}, r_{m}}\leq C \varepsilon_{m}s_{m}^{-d},\;\;\parallel v_{m}\parallel_{s_{m}, r_{m}}\leq C\varepsilon_{m}\end{equation} such that $\Psi_{m},$ which is reversible with respect to $G: (x, y, t)\mapsto (-x, y, -t),$ changes the modified equations: \begin{equation}\label{y5.5} (a_{j})^{*}: \left\{
\begin{array}{ll}
\dot{x}=\omega+y+f_{m}^{0}(x, y, t)+f_{m}(x, y, t)\\
\dot{y}=g_{m}^{0}(x, y, t)+g_{m}(x, y, t)
\end{array}
\right. \end{equation} into \begin{equation} \label{eq4.11}(a)^{(m)}_{*}: \left\{
\begin{array}{ll}
\dot{\xi}=\omega+\eta+f_{m+1}^{0}(\xi, \eta, t),\\
\dot{\eta}=g_{m+1}^{0}(\xi, \eta, t),
\end{array}
\right. \end{equation} where $f_{m+1}^{0}$ and $g_{m+1}^{0}$ obey the conditions $(1)_{m},$ $(2)_{m}$ and $(3)_{m}$ by replacing $m$ by $m+1.$ In other words, $\Phi^{(m)}:=\Phi^{(m-1)}\circ\Phi_{m}$ changes \begin{equation}\label{eq5.10} \left\{
\begin{array}{ll}
\dot{x}=\omega+y+\sum_{j=0}^{m}f_{j}(x, y, t)\\
\dot{y}=\sum_{j=0}^{m}g_{j}(x, y, t)
\end{array} \right. \end{equation} into $(a)^{(m)}_{*}.$
\end{lemma}
{\bf Proof of Theorem 1.2} We see that $$\Psi^{\infty}=\lim_{m\rightarrow \infty}\Psi_{1}\circ \cdots\circ\Psi_{m}: D(0, 0)\rightarrow D(s_{0}, r_{0})\subset D.$$ The proof for the existence of limit $\Psi^{\infty}$ is now standard. We omit the detail. See Moser \cite{Moser1962} for example. Let $\Phi^{\infty}=\lim_{m\rightarrow \infty} \Phi^{(m)}.$ Thus $\Psi^{\infty}=(\Phi^{\infty})^{-1}$ and $\Psi^{\infty}(\{\omega t\}\times \{y=0\}\times \{t: t\in \mathbb{T}\})$ is a $C^{0}$ embedding torus of the original equations \eqref{a}.
{\bf Proof of Theorem 1.1} By Proposition 4.5 of Sevryuk \cite{Sevryuk1986}, we know that the map $A$ in Theorem 1.1 can be regarded as the time-1 map ( Poincare map ) of equation \eqref{a} in Theorem 1.2. The proof is completed by Theorem 1.2.
\section{Derivation of homological equation}
Let us recall \eqref{y5.5}. Let \begin{eqnarray} &&f_{m}^{(m)}(x, y, t)=f_{m}^{0}(x, y, t)+f_{m}(x, y, t),\label{18.1}\\ &&g_{m}^{(m)}(x, y, t)=g_{m}^{0}(x, y, t)+g_{m}(x, y, t).\label{18.2} \end{eqnarray} By the conditons $(1)_{m},$ $(2)_{m}$ and $(3)_{m}$ in the iterative lemma, and observing \eqref{92.1}, \eqref{92.2} and \eqref{92.3}, we have \begin{itemize}
\item [$(i)_{m}$] the functions $f_{m}^{(m)},$ $g_{m}^{(m)}$ are real for real arguments;
\item [$(ii)_{m}$] the functions $f_{m}^{(m)},$ $g_{m}^{(m)}$ are analytic in $D(s_{m}, r_{m}):$
\begin{equation}\label{eq*}\parallel f_{m}^{(m)}\parallel_{s_{m}, r_{m}}\leq C \varepsilon_{m},\;\;\parallel g_{m}^{(m)}\parallel_{s_{m}, r_{m}}\leq C \varepsilon_{m}s_{m}^{d};\end{equation}
\item [$(iii)_{m}$] the functions $f_{m}^{(m)},$ $g_{m}^{(m)}$ are reversible with respect to $G: (x, y, t)\mapsto (-x, y, -t),$ that is
\begin{eqnarray} &&f_{m}^{(m)}(-x, y, -t)=f_{m}^{(m)}(x, y, t),\label{eq6.17}\\ &&g_{m}^{(m)}(-x, y, -t)=-g_{m}^{(m)}(x, y, t).\label{eq6.18} \end{eqnarray} \end{itemize}
Consider a map $\Phi=\Phi_{m}$ of the form \begin{equation}\label{eq6.1} \Phi=\Phi_{m}: \left\{
\begin{array}{ll}
x=\xi+U(\xi, \eta, t)\\
y=\eta+V(\xi, \eta, t)
\end{array}
\right. \end{equation} and its inverse $\Phi^{-1}$ is of the form \begin{equation}\label{eq6.2}\Psi=\Phi^{-1}: \left\{
\begin{array}{ll}
\xi=x+u(x, y, t),\\
\eta=y+v(x,y,t),
\end{array}
\right. \end{equation} where $u, v, U, V$ will be specified. Inserting \eqref{eq6.2} into equation $(a_{j})^{*}$ with $j=m-1$ (i.e. \eqref{y5.5}), and noting \eqref{18.1} and \eqref{18.2}, we have that \begin{eqnarray} \dot{\xi}&=&\omega+\eta\\ &+& \omega\cdot \partial_{x}u+\partial_{t} u+f_{m}^{(m)}(x, y, t)-v\label{eq6.3}\\ &+& \partial_{y} u\cdot g_{m}^{(m)}(x, y, t)+\partial_{x}u\cdot f_{m}^{(m)}(x, y, t)\label{eq6.4}\\ &+& \partial_{x}u\cdot y,\label{eq6.6} \end{eqnarray}
\begin{eqnarray} \dot{\eta}&=&\omega\cdot\partial_{x}v+\partial_{t}v+g_{m}^{(m)}(x,y,t)\label{eq6.8}\\ &+& \partial_{y}v\cdot g_{m}^{(m)}(x, y, t)+\partial_{x}v\cdot f_{m}^{(m)}(x, y, t)\label{eq6.9}\\ &+& \partial_{x}v\cdot y \label{eq6.11} \end{eqnarray} and where $u=u(x, y, t),$ $v=v(x, y, t),$ $\omega\cdot\partial_{x}=\Sigma_{j=1}^{d}\omega_{j}\partial_{x_{j}},$ and $x=\xi+U(\xi, \eta, t),$ $y=\eta+V(\xi, \eta, t)$ will be implicity defined by \eqref{eq6.2}.
Letting \eqref{eq6.3}=0 and \eqref{eq6.8}=0, we derive homological equations: \begin{equation}\label{eq6.15} \omega\cdot \partial_{x}u+\partial_{t}u-v+f_{m}^{(m)}(x, y, t)=0 \end{equation} and \begin{equation}\label{eq6.16} \omega\cdot \partial_{x}v+\partial_{t}v+g_{m}^{(m)}(x, y, t)=0. \end{equation}
Let $\widehat{g}_{m}^{(m)}(k, l, y)$ is the $(k, l)-$ Fourier coefficient of $g_{m}^{(m)}(x, y, t),$ with respect to variable $(x, t)$, that is, $$\widehat{g}_{m}^{(m)}(k, l, y)=\frac{1}{(2\pi)^{d+1}}\int_{0}^{2\pi}\cdots\int_{0}^{2\pi} g_{m}^{(m)}(x, y, t)e^{-\sqrt{-1}(\langle k, x\rangle+l t)}dx dt,$$ where $k\in \mathbb{Z}^{d}, l\in \mathbb{Z}.$ Similarly, we can define $\widehat{f}_{m}^{(m)}(k, l, y),$ etc.
By \eqref{eq6.18}, we have \begin{equation}\label{eq6.19} \widehat{g}_{m}^{(m)}(0, 0, y)=0,\;\;y\in {B}_{\mathbb{C}}(r_{m}).\end{equation} By passing to Fourier coefficients, homological equation \eqref{eq6.16} reads \begin{equation}\label{eq6.20} \sqrt{-1}(\langle k, \omega\rangle+l)\,\widehat{v}(k, l, y)=-\widehat{g}^{(m)}_{m}(k, l, y),\end{equation} where $(k, l)\in \mathbb{Z}^{d}\times \mathbb{Z}\setminus \{(0, 0)\},$ $y\in B_{\mathbb{C}}(r_{m}).$ So we have \begin{equation}\label{eq6.21}\widehat{v}(k,l, y)=\sqrt{-1}\frac{\widehat{g}_{m}^{(m)}(k, l, y)}{\langle k, \omega\rangle+l}.\end{equation} We notice that when $(k, l)=(0, 0),$ $\langle k, \omega\rangle+l=0$ and $\widehat{g}_{m}^{(m)}(0, 0, y)=0.$ Thus we have a freedom to choose $\widehat{v}(0, 0, y)$ in \eqref{eq6.20}.
\begin{lemma}\label{ly5.1} \cite{Russmann1975, Russmann1983} Assume $\omega$ satisfies
$$|\langle k, \omega\rangle+j|\geq \kappa/|k|^{\tau},\;\;\forall (k, j)\in\mathbb{Z}^{d}\times\mathbb{Z}\setminus\{(0,0)\}.$$ Then the inequalities
$$\sum_{|k|\leq n}|\langle k, \omega\rangle+j|^{-2}\leq C\kappa^{-2}n^{2\tau},\;\;|k|=|k_{1}|+\cdots+|k_{d}|$$ hold for $n=1, 2, \cdots.$ \end{lemma}
\begin{proof} The proof for $d=1$ is given in \cite{Russmann1975,Russmann1983}. For $d>1,$ the proof goes well. For the convenience of the readers, we copy the proof from \cite{Russmann1975,Russmann1983} with a minor modification. If we numerate the numbers of the set
$$\{\<k, \omega\rangle+l\mid |k|\leq n, l\in \mathbb{Z}\},$$ according to their natural order \begin{eqnarray*} &&\cdots < d_{-2}< d_{-1}< 0 < d_{1}< d_{2}<\cdots,\\ && d_{j}=\<k_{j}, \omega\rangle- l_{j}, j=\pm 1, \pm2, \cdots. \end{eqnarray*} According to $\omega\in D(\kappa, \tau),$ $d_{1}\geq \kappa/ n^{\tau}$, we obtain
$$d_{j+1}-d_{j}=|\<k_{j+1}-k_{j}, \omega\rangle+ (l_{j}-l_{j+1})|\geq \kappa/2^{\tau}n^{\tau},\;\;j=1, 2, \cdots.$$ Thus $$d_{j}\geq j\kappa (2n)^{-\tau},\;\;j=1, 2, \cdots.$$ It follows that $$\sum_{j=1}^{\infty}d_{j}^{-2}\leq \kappa^{-2}(2n)^{2\tau}\sum_{j=1}^{\infty}\frac{1}{j^{2}}\leq C\kappa^{-2}n^{2\tau}.$$ In the same way, we have $$\sum_{j=-\infty}^{-1}d_{j}^{-2}\leq C\kappa^{-2}n^{2\tau}.$$ Consequently, we have
$$\sum_{|k|\leq n}|\langle k, \omega\rangle+l|^{-2}\leq C \kappa^{-2}n^{2\tau}.$$\end{proof}
We are now in position to estimate $v(x, y ,t)$. First, by Parseval's identity
\begin{equation}\label{li-1}\sup_{y\in B_{\mathbb{C}}(r_{m})}\sum_{(k, l)\in \mathbb{Z}^{d+1}\setminus \{0\}}\mid \widehat{g}_{m}^{(m)}(k, l, y)\mid^{2}e^{2(|k|+|l|)s_{m}} \leq C\parallel g_{m}^{(m)}\parallel^{2}_{s_{m}, r_{m}},\end{equation} where $C=C(\kappa,\tau)$ depends on $\kappa$ and $ \tau.$ Following \cite{Nash1956,Piao2008}, let $$G_{n}(y)=\sum_{\begin{array}{c}
(k, l)\in \mathbb{Z}^{d+1} \\
1\leq |k|+|l|\leq n \\
\end{array}}\mid\widehat{g}^{(m)}_{m}(k, l, y)\mid\frac{e^{(|k|+|l|)s_{m}}}{|\langle k, \omega\rangle+l|},\;\;(n=1, 2, \cdots).$$ Then by Cauchy-Schwarz inequality and Lemma \ref{ly5.1}, we get \begin{eqnarray*}
G_{n}(y)&\leq & C\sqrt{\sum_{(k,l)\in\mathbb{Z}^{d+1}}\mid\widehat{g}_{m}^{(m)}(k, l,y)\mid^{2}e^{2(|k|+|l|)s_{m}}}\sqrt{\sum_{1\leq |k|+|l|\leq n} \frac{1}{\mid \langle k, \omega\rangle+l\mid^{2}}}\\
&\leq & C\parallel g_{m}^{(m)}\parallel_{s_{m}, r_{m}}\frac{|n|^{\tau}}{\kappa}. \end{eqnarray*} Letting $\tilde b_m=\frac{1}{200\ell}(s_m-s_{m+1})\ge \frac{1}{400\ell} s_m$ and letting $G_{0}(y)=0,$ we obtain by means of Abel's partial summation, for any $N\gg 1,$ \begin{eqnarray*}
&&\sum_{0<|k|+|l|\leq N}\mid \widehat{g}_{m}^{(m)}(k, l, y)\mid\frac{e^{(|k|+|l|)(s_{m}-\tilde b_m)}}{\mid \langle k, \omega\rangle+l\mid}\\ &=&(1-e^{-\tilde b_m})\sum_{n=1}^{N}G_{n}(y)e^{-n \tilde b_m}+G_{N}(y)e^{-(N+1) \tilde b_m}\\ &\leq &C\parallel g_{m}^{(m)}\parallel_{s_{m}, r_{m}}\sum_{n=1}^{\infty}n^{\tau}(e^{-n\tilde b_m}-e^{-(n+1)\tilde b_m})\\ &\leq & C\parallel g_{m}^{(m)}\parallel_{s_{m}, r_{m}}(\tilde b_m)^{-\tau}\\ & \leq &(400\ell)^\tau C \parallel g_{m}^{(m)}\parallel_{s_{m}, r_{m}} s_m^{-\tau}. \end{eqnarray*}
It follows \begin{eqnarray}\label{6.?} \parallel v\parallel_{s _{m}^{(1)}, r_{m}}&\leq &C\parallel g_{m}^{(m)}\parallel_{s_{m},r_{m}}s_{m}^{-\tau}\nonumber\\ &\leq & C\varepsilon_{m}s_{m}^{-\tau+d}=C\varepsilon_{m}s_{m}^{-\frac{\mu}{100}}\nonumber\\ &\ll& r_{m}. \end{eqnarray} Recall that we have a freedom to choose $\widehat{v}(0,0,y)$ such that \begin{equation} \label{li.3} -\widehat{v}(0,0,y)+\widehat{f}_{m}^{(m)}(0,0,y)=0,\;\;\forall y\in B_{\mathbb{C}}(r_{m}).\end{equation} In view of \eqref{eq*} and \eqref{li.3}, by applying the same method to \eqref{eq6.15}, we have \begin{eqnarray} \label{li.4} \parallel u\parallel_{s_{m}^{(2)}, r_{m}}&\leq &
C\parallel v\parallel_{s_{m}^{(1)},r_{m}}s_{m}^{-\tau}+C\parallel f_{m}^{(m)}\parallel_{s_{m},r_{m}}s_{m}^{-\tau}\nonumber\\
&\leq &C\varepsilon_{m}s_{m}^{-(\tau+\frac{\mu}{100})}.\end{eqnarray}
By Cauchy estimate, we have that for $0\leq p,$ $0\leq q$ and $p+q=1,$
\begin{eqnarray*}
&&\parallel \partial_{x}^{p}\partial_{y}^{q}v\parallel_{s_{m}^{(2)}, r_{m}^{(1)}}\leq C\varepsilon_{m}s_{m}^{-\frac{\mu}{100}}\max\{s_{m}^{-1}, r_{m}^{-1}\}
\leq C s_{m}^{l-\frac{\mu}{100}-(d+1+\frac{\mu}{100})}
\leq Cs_{m}^{d-\frac{\mu}{100}}\ll 1,\\
&&\parallel \partial_{x}^{p}\partial_{y}^{q} u\parallel_{s^{(3)}_{m},r^{(1)}_{m}}\leq C \varepsilon_{m}s_{m}^{-(\tau+\frac{\mu}{100})}\max\{s_{m}^{-1}, r_{m}^{-1}\}
\leq Cs_{m}^{l-\tau-\frac{\mu}{100}-(d+1+\frac{\mu}{100})}\leq Cs_{m}^{\frac{\mu}{100}}\ll 1.
\end{eqnarray*}
By \eqref{eq6.2}, \eqref{6.?} and \eqref{li.4} and by means of implicit theorem, we have that $\Phi=\Psi^{-1}=\Phi_{m}:$
\begin{eqnarray}\label{eq6.30}
\Phi:\left\{
\begin{array}{ll}
x=\xi+U(\xi, \eta, t),\\
y=\eta+V(\xi, \eta, t),
\end{array}
\right.
\end{eqnarray} where $(\xi, \eta, t)\in D(s_{m}^{(4)}, r_{m}^{(2)})$ and $U, V$ are real analytic in $D(s_{m}^{(4)}, r^{(2)}_{m}),$ and satisfy \begin{eqnarray} && \parallel U\parallel_{s_{m}^{(4)},r_{m}^{(2)}}\leq C\varepsilon_{m}s_{m}^{-2\tau},\label{eq6.31}\\ && \parallel V\parallel_{s_{m}^{(4)},r_{m}^{(2)}}\leq C\varepsilon_{m}s_{m}^{-\tau},\label{eq6.32}\\ &&\Psi(D(s_{m+1}, r_{m+1})) \subset\Psi(D(s_{m}^{(4)}, r_{m}^{(2)}))\subset D(s_{m}, r_{m})\label{eq6.33} \end{eqnarray} Now let us consider the reversibility of the changed system. First of all, by applying \eqref{eq6.17}, \eqref{eq6.18} to \eqref{eq6.15} and \eqref{eq6.16}, we have \begin{equation}\label{eq6.34}u(-x, y,-t)=-u(x, y, t),\;\;v(-x, y, -t)=v(x,y, t).\end{equation} Then by $\Phi \circ \Psi=id,$ we get \begin{eqnarray} && u(x,y, t)+U(x+u(x, y, t), y+V(x, y, t), t)=0,\label{eq6.35}\\ && v(x,y, t)+V(x+u(x, y, t), y+V(x, y, t), t)=0.\label{eq6.36} \end{eqnarray} By \eqref{eq6.34}, \eqref{eq6.35} and \eqref{eq6.36}, we have \begin{equation}\label{eq6.37}U(-\xi, \eta, -t)=-U(\xi, \eta, t),\;\;V(-\xi, \eta, -t)=V(\xi, \eta, t),\end{equation} where $(\xi, \eta, t)\in D(s_{m}^{(4)},r_{m}^{(2)}).$ It follows that the changed system are still reversible with respect to $G: (x, y, t)\mapsto (-x, y, -t).$
\section{Estimates of new perturbations} \begin{itemize}
\item Estimate of \eqref{eq6.4}. By \eqref{6.?}, \eqref{li.3}, \eqref{li.4} and \eqref{eq*}, and regarding \eqref{eq6.4} as a function of $(x, y, t),$ we have $$\parallel \eqref{eq6.4}(x,y,t)\parallel_{s_{m}^{(5)}, r_{m}^{(3)}}\leq \varepsilon_{m}^{\tilde{\mu}}\varepsilon_{m}=\varepsilon_{m+1}.$$
\iffalse
\item Estimate of \eqref{eq6.5}. Regarding \eqref{eq6.5} as a function of $(x, y, t),$ using \eqref{eq*}, \eqref{6.?} and \eqref{li.4}, we have \begin{eqnarray*} \parallel \eqref{eq6.5}(x,y,t)\parallel_{s_{m}^{(7)}, r_{m}^{(3)}}&\leq & \parallel \int_{0}^{1}\partial_{x}f^{(m)}_{m}(x+zu, y+zv, t)udz\parallel_{s_{m}^{(7)}, r_{m}^{(3)}}\\ &+& \parallel \int_{0}^{1}\partial_{y}f^{(m)}_{m}(x+zu, y+zv, t)vdz\parallel_{s_{m}^{(7)}, r_{m}^{(3)}}\\ &\leq & C\varepsilon_{m}s_{m}^{-1}\varepsilon_{m}s_{m}^{-(\tau+\frac{\mu}{100})}+C\varepsilon_{m}r_{m}^{-1}\varepsilon_{m}s_{m}^{-\frac{\mu}{100}}\\ &\leq & \varepsilon_{m+1}. \end{eqnarray*} \fi
\item Estimate of \eqref{eq6.6}. By Cauchy estimate, and in view of \eqref{li.4}, we have \begin{eqnarray} \parallel \eqref{eq6.6}(x, y, t)\parallel_{s_{m}^{(3)}, r_{m}^{(2)}}&\leq &\varepsilon_{m}s_{m}^{-(\tau+\frac{\mu}{100}+1)}r_{m}\\ &=& s_{m}^{\frac{4}{50}\mu}\varepsilon_{m}<\varepsilon_{m+1}. \end{eqnarray} \end{itemize} Let $$f_{m+1}^{0}=\eqref{eq6.4}+\eqref{eq6.6}.$$ Moreover, by \eqref{eq6.31} and \eqref{eq6.32}, we have \begin{equation}\label{eq6.50} \parallel f_{m+1}^{0}(\xi, \eta, t)\parallel_{s_{m+1}, r_{m+1}}\leq \varepsilon_{m+1}.\end{equation} Let $g_{m+1}^{0}=\eqref{eq6.9}+\eqref{eq6.11}.$ Similarly, we can prove $$\parallel g_{m}^{0}(\xi, \eta, t)\parallel_{s_{m+1}, r_{m+1}}\leq \varepsilon_{m+1}s_{m+1}^{d}.$$ Thus we have proved the claim $(2)_{m}$ in the iterative Lemma with replacing $m$ by $m+1.$
Now we are in position to prove $(3)_{m}$ with replacing $m$ by $m+1.$ Let us return to the homological equation \eqref{eq6.15} and \eqref{eq6.16}. Replacing $(x, y, t)$ by $(-x, y, -t),$ and in view of $$f_{m}^{(m)}(-x, y, -t)=f_{m}^{(m)}(x,y ,t),\;\;g_{m}^{(m)}(-x, y, -t)=-g_{m}^{(m)}(x,y ,t).$$ We see that $u(-x, y, -t), -v(-x, y, -t)$ are still a pair of the solutions of \eqref{eq6.15} and \eqref{eq6.16}. Noting that the solutions of \eqref{eq6.15} and \eqref{eq6.16} are unique. So $$u(-x, y, -t)=u(x, y, t), v(-x, y, -t)=-v(x, y, t).$$ This implies that \eqref{eq4.11} is reversible with respect to $G: (x, y, t)\mapsto (-x, y, -t).$ Thus $(3)_{m}$ holds true with replacing $m$ by $m+1.$
Again returning to \eqref{eq6.15} and \eqref{eq6.16}. Note that $f_{m}^{(m)}$ and $g_{m}^{(m)}$ are real for real arguments. It follows that so are $u$ and $v$. Moreover, $f_{m+1}^{0}$ and $g_{m+1}^{0}$ are real for real arguments. We omit the detail here. This completes the proof of the iterative Lemma.
\ \ \section{Application: Lagrange stability for a class of Li\'{e}nard equation} Let $C>0$ be a universal constant which maybe different in different places. Consider the Li\'{e}nard equation \begin{equation}\label{1} \ddot{x}+x^{2n+1}+g(x, t)+f(x, t)\dot{x}=0, \end{equation} where $f$ and $g$ satisfy \begin{itemize}
\item [$(f)_{1}$] $f(x, t)$ is odd in $x,$ even in $t,$ and of period 1 in $t$,
\item [$(f)_{2}$] $\exists \;0\leq p\leq n-1$, and $\exists$ integer $N>0$ such that
$$ |x^{k}\partial_{x}^{k}\partial_{t}^{\ell}f(x, t)|\leq C|x|^{p},\;\;|x|\gg 1,\;\;0\leq k\leq N,\;\;0\leq\ell\leq 2,$$
\item [$(g)_{1}$] $g(x, t)$ is odd in $x,$ even in $t,$ and of period 1 in $t$,
\item [$(g)_{2}$] $\exists\; q$ with $0\leq q\leq 2n-1$ such that
$$|x^{k}\partial_{x}^{k}\partial_{t}^{\ell}g(x, t)|\leq C|x|^{q},\;\;|x|\gg 1, \;\;0\leq k\leq N,\;\;0\leq \ell \leq 2. $$ \end{itemize}
\ \
{Note that Eq. \eqref{1} is equivalent to the plane system \begin{equation}\label{6} \dot{x}=y,\;\;\dot{y}=-x^{2n+1}-g(x, t)-f(x, t)y. \end{equation} First of all, we consider a special Hamiltonian system \begin{equation}\label{7} \dot{x}=y, \;\;\dot{y}=-x^{2n+1} \end{equation} with Hamiltonian \begin{equation}\label{8} h(x,y)=\frac{y^{2}}{2}+\frac{x^{2(n+1)}}{2(n+1)}. \end{equation} Suppose that $(x_{0}(t), y_{0}(t))$ is the solution of Eq. \eqref{7}, with the initial conditions $(x_{0}(0), y_{0}(0))=(0, 1).$ Clearly it is periodic. Let $T_{0}$ be its minimal positive period. It follows from \eqref{7} that $x_{0}(t)$ and $y_{0}(t)$ possess the following properties: \begin{itemize}
\item [(a)] $x_{0}(t+T_{0})=x_{0}(t)$ and $y_{0}(t+T_{0})=y_{0}(t)$;
\item [(b)] $x_{0}'(t)=y_{0}(t)$ and $y_{0}'(t)=-(x_{0}(t))^{2n+1}$;
\item [(c)] $(n+1)((y_{0}(t))^{2}+(x_{0}(t))^{2n+2}=n+1$;
\item [(d)]$x_{0}(-t)=-x_{0}(t)$ and $y_{0}(-t)=y_{0}(t).$ \end{itemize} Following \cite{Dieckerhoff-Zehnder1987,Liu1998} we construct transformation $\psi: \mathbb{R}^{+}\times \mathbb{T} \rightarrow \mathbb{R}^{2}/\{0\},$ where $(x, y)=\psi(\lambda, \theta)$ with $\lambda>0$ and $\theta$ (mod 1) being given by the formula \begin{equation}\label{9} \psi: x=c^{\alpha}\rho^{\alpha}x_{0} (\frac{\theta T_{0}}{2\pi}),\;\;y=c^{\beta}\rho^{\beta}y_{0} (\frac{\theta T_{0}}{2\pi}), \end{equation} where $\alpha=\frac{1}{n+2},$ $\beta=1-\alpha$ and $c=\frac{2\pi}{\beta T_{0}}.$ By using the transformation $\psi,$ and noting properties (b) and (d), Eq. \eqref{6} is transformed into}
\iffalse \begin{equation}\label{11} \left\{
\begin{array}{ll}
c^{\alpha}\alpha\rho^{\alpha-1}x_{0}\dot{\rho}+c^{\alpha}\rho^{\alpha}y_{0}T_{0}\dot{\theta}=c^{\beta}\rho^{\beta}y_{0},\\
c^{\beta}\beta\rho^{\beta-1}y_{0}\dot{\rho}+c^{\beta}\rho^{\beta}y_{0}(-x_{0}^{2n+1})T_{0}\dot{\theta} =-x^{2n+1}-g(c^{\alpha}\rho^{\alpha}x_{0}, t)-f(c^{\alpha}\rho^{\alpha}x_{0}, t)c^{\beta}\rho^{\beta}y_{0},
\end{array} \right. \end{equation} where $x_{0}=x_{0}(\theta T_{0}),$ $y_{0}=y_{0}(\theta T_{0}).$ Noting $$
\left|
\begin{array}{cc}
c^{\alpha}\alpha\rho^{\alpha-1}x_{0} & c^{\alpha}\rho^{\alpha}y_{0}T_{0} \\
c^{\beta}\beta\rho^{\beta-1}y_{0} & c^{\beta}\rho^{\beta}(-x_{0}^{2n+1})T_{0} \\
\end{array}
\right|=1. $$ It follows from \eqref{11} that
$$\dot{\rho}=\left|
\begin{array}{cc}
c^{\beta}\rho^{\beta}y_{0} & c^{\alpha}\rho^{\alpha}y_{0}T_{0} \\
-c^{\alpha(2n+1)}\rho^{\alpha(2n+1)}x_{0}^{2n+1}-g(c^{\alpha}\rho^{\alpha}x_{0}, t)-f(c^{\alpha}\rho^{\alpha}x_{0}, t)c^{\beta}\rho^{\beta}y_{0} & c^{\beta}\rho^{\beta}(-x_{0}^{2n+1}T_{0})\\
\end{array}
\right|, $$
$$\dot{\theta}=\left|
\begin{array}{cc}
c^{\alpha}\alpha \rho^{\alpha-1}x_{0} & c^{\beta}\rho^{\beta}y_{0} \\
c^{\beta}\beta\rho^{\beta-1}y_{0} & -c^{\alpha(2n+1)}\rho^{\alpha(2n+1)}x_{0}^{2n+1}-g(c^{\alpha}\rho^{\alpha}x_{0}, t)-f(c^{\alpha}\rho^{\alpha}x_{0}, t)c^{\beta}\rho^{\beta}y_{0}\\
\end{array}
\right|. $$ That is ,\fi
\begin{equation}\label{12} \left\{
\begin{array}{ll}
\dot{\rho}=\frac{-1}{2\pi}(c\rho T_{0}y_{0}^{2}f(c^{\alpha}\rho^{\alpha}x_{0}, t)+c^{\alpha}\rho^{\alpha}y_{0}T_{0}g(c^{\alpha}\rho^{\alpha}x_{0}, t))\triangleq F_{1}( \theta, \rho, t),\\
\dot{\theta}=c_{0}\rho^{2\beta-1}+c\alpha x_{0}y_{0}f(c^{\alpha}\rho^{\alpha}x_{0}, t)+c^{\alpha}{\alpha}\rho^{\alpha-1}x_{0}g(c^{\alpha}\rho^{\alpha}x_{0}, t)\triangleq c_{0}\rho^{2\beta-1}+F_{2}(\theta, \rho, t),
\end{array} \right. \end{equation} where $c_{0}=\beta c^{2\beta},$ $x_{0}=x_{0}(\frac{\theta T_{0}}{2\pi}),$ $y_{0}=y_{0}(\frac{\theta T_{0}}{2\pi}),$ ${2\beta-1=\frac{n}{n+2}}.$
\begin{dfn} Given $\rho_{*}\gg 1.$ Consider $\rho\geq \rho_{*},$ $\theta\in \mathbb{T}.$ For $\gamma\in\mathbb R,$ $q\geq 0,$ we call $y=y(\theta, \rho, t)\in P_{q,p}(\gamma)$ if
$$\sup_{(\theta, \rho, t)\in \mathbb{T}\times [\rho_{*}, +\infty]\times
\mathbb{T}}|\rho^{q-\gamma}\!\!\sum_{
k+\ell\leq q, \; k, \ell\geq 0
}\partial_{\theta}^{k}\partial_{\rho}^{\ell}\partial_{t}^{p}y(\theta, \rho, t)|\leq C<\infty.$$ In light of $(f)_{1}$, $(f)_{2}$ and $(g)_{1}$, $(g)_{2},$ we have \begin{eqnarray} && F_{1}\in P_{N, 2}(\frac{2n+1}{n+2}), F_{1}(-\theta, \rho, t)=-F_{1}(\theta, \rho, t), F_{1}(\theta, \rho, -t)=F_{1}(\theta, \rho, t),\\ &&\forall (\theta, \rho, t)\in \mathbb{T}\times [\rho_{*}, +\infty)\times \mathbb{T},\nonumber\\ && F_{2}\in P_{N, 2}(\frac{n-1}{n+2}),F_{2}(-\theta, \rho, t)=F_{2}(\theta, \rho, t), F_{2}(\theta, \rho, -t)=F_{2}(\theta, \rho, t),\\ &&\forall (\theta, \rho, t)\in \mathbb{T}\times [\rho_{*}, +\infty)\times \mathbb{T}.\nonumber \end{eqnarray} For $\forall C>0,$ we define the domain $$D_{C}=\{(\theta, \lambda, t)\mid \theta\in\mathbb{T}, t\in \mathbb{T}, \lambda\geq C\}.$$ \end{dfn}
\begin{lemma}\label{lem100} There exists a diffeomorphism depending periodically on $t,$ $$\Psi: \rho=\mu+U(\phi, \mu, t),\;\;\theta=\phi+V(\phi, \mu, t)$$ such that $$D_{c_{+}}\subset \Psi(D_{c_{0}})\subset D_{c_{-}}\;\;\mbox{for}\;\;1\ll c_{+}<c_{0}<c_{-},$$ and \eqref{12} is changed by $\Psi$ into \begin{equation*}\label{*} \left\{
\begin{array}{ll}
\dot{\mu}=\widetilde{F}_{1}(\phi, \mu, t),\\
\dot{\phi}=c_{0}\mu^{2\beta-1}+h(\mu, t)+\widetilde{F}_2(\phi, \mu, t),
\end{array} \right. \end{equation*} and \begin{eqnarray} && \widetilde{F}_{1}\in P_{5,0}\left(\frac{-1}{n+2}\right),\;\;\widetilde{F}_{1}(-\phi, \mu, -t)=-\widetilde{F}_{1}(\phi, \mu, t),\label{30}\\ && \widetilde{F}_{2}\in P_{5,0}\left(\frac{-1}{n+2}\right),\;\;\widetilde{F}_{2}(-\phi, \mu, -t)={\widetilde{F}_{2}(\phi, \mu, t)},\label{31}\\ && h\in P_{5,0}\left(\frac{n-1}{n+2}\right),\;\;\mbox{which is independant of}\;\;\phi\; \mbox{and}\;h(\mu, -t)=-h(\mu, t). \end{eqnarray} \end{lemma}
\begin{proof} The proof is similar to that in Propositions 3.2 and 3.3 in \cite{Liu1998}. \end{proof} Let $\lambda=c_{0}\mu^{2\beta-1}.$ Then \eqref{*} reads \begin{equation}\label{**} \left\{
\begin{array}{ll}
\dot{\lambda}=F_{1}^{*}(\phi, \lambda, t),\\
\dot{\phi}=\lambda+\widetilde{h}(\lambda, t)+F_{2}^{*}(\phi, \lambda, t),
\end{array} \right. \end{equation} where \begin{eqnarray} && F_{1}^{*}\in P_{5,0}\left(\frac{-1}{n}\right),\;\;F_{1}^{*}(-\phi, \lambda, -t)=-F_{1}^{*}(\phi, \lambda, t),\label{35}\\ && F_{2}^{*}\in P_{5,0}\left(\frac{-1}{n}\right),\;\;F_{2}^{*}(-\phi, \lambda, -t)=F_{2}^{*}(\phi, \lambda, t),\label{36}\\ && \widetilde{h}(\lambda, t)\in P_{5,0}\left(\frac{n-1}{n}\right),\;\; h(\lambda, -t)=-h(\lambda, t).\label{37} \end{eqnarray} Following Lemma 4.1 in \cite{Liu1998}, we get the Poinc\'{a}re map of \eqref{**} is of the form $$P: \;\theta_{1}=\theta+r(\lambda)+\widetilde{f}(\theta, \lambda),\;\lambda_{1}=\lambda+\widetilde{g}(\theta, \lambda),\;\lambda\geq \lambda_{*}\gg 1,$$ where
$r(\lambda)=\lambda+\int_{0}^{1}\widetilde{h}(\lambda, t)dt,$ $$\left|\sum_{k+\ell\leq 5}D_{\lambda}^{k}\partial_{\theta}^{\ell}\widetilde{f}\;\right|\leq C \lambda^{-\frac{1}{n}},\;\;\left|\sum_{k+\ell\leq 5}D_{\lambda}^{k}\partial_{\theta}^{\ell}\widetilde{g}\;\right|\leq C \lambda^{-\frac{1}{n}}.$$ Let $\rho=r(\lambda).$ By \eqref{37} and the implicit function theorem, we have that $P$ is of the form $$P: \theta_{1}=\theta+\rho+f^{*}(\theta, \rho),\;\;\rho_{1}=\rho+g^{*}(\theta, \rho),$$ where
$$\left|\sum_{k+\ell\leq 5}D_{\rho}^{k}\partial_{\theta}^{\ell}f^{*}\;\right|\leq \rho^{-\frac{1}{n}}<\varepsilon_{0},\;\;
\left|\sum_{k+\ell\leq 5}D_{\rho}^{k}\partial_{\theta}^{\ell}g^{*}\;\right|\leq \rho^{-\frac{1}{n}}<\varepsilon_{0},\;\;\rho\in [m, m+1],\;\;m\gg 1.$$ Let $\Phi= \psi\circ\Psi$. It is easy to see that equation \eqref{**} is reversible with respect to $G:\; (\phi, \lambda, t)\mapsto (-\phi, \lambda, -t)$. By Lemma 2.2 in \cite{Liu1998}, we have that $P$ is reversible with respect to $G$.
Using Theorem \ref{thm1}, we get that $P$ has an invariant carve $\Gamma_{m}\subset \mathbb{T}\times [m, m+1]$ $(\forall m\gg 1).$ It follows that the original equation has a family of invariant curves which are around the infinity. Thus, we have that
$$\sup_{t\in \mathbb{R}}|x(t)|+|\dot{x}(t)|\leq C,$$ where $(x, \dot{x})$ is the solution of \eqref{6}, and $C$ depends on initial $(x(0),\dot{x}(0)).$
\end{document} | arXiv |
(Redirected from Brassiere measurements)
Measuring for bra size: around the torso at the inframammary fold and over the bust
Bra size (also known as brassiere measurement or bust size) indicates the size characteristics of a bra. While there are a number of bra sizing systems in use around the world, the bra sizes usually consist of a number, indicating the size of the band around the woman's torso, and one or more letters that indicate the breast cup size. Bra cup sizes were first invented in 1932 while band sizes became popular in the 1940s. For convenience, because of the impracticality of determining the size dimensions of each breast, the volume of the bra cup, or cup size, is based on the difference between band length and over-the-bust measurement.
Manufacturers try to design and manufacture bras that correctly fit the majority of women, while individual women try to identify correctly fitting bras among different styles and sizing systems.[1]
The shape, size, position, symmetry, spacing, firmness, and sag of individual women's breasts vary considerably. Manufacturers' bra size labelling systems vary from country to country because no international standards exist. Even within a country, one study found that the bra size label was consistently different from the measured size.[2] As a result of all these factors, about 25% of women have a difficult time finding a properly fitted bra,[3] and some women choose to buy custom-made bras due to the unique shape of their breasts.
1 Measurement method origins
1.1 Cup design origins
1.2 Band measurement origins
1.3 Other innovations
2 Consumer fitting
2.1 Larger breasts and bra fit
2.2 Bad bra-fit symptoms
2.3 Obtaining best fit
2.4 Confirming bra fit
3 Consumer measurement difficulties
3.1 Asymmetric breasts
3.2 Breast volume variation
3.3 Increases in average bra size
4 Consumer measurement methods
4.1 Band measurement methods
4.1.1 Underbust +0
4.1.3 Sizing chart
4.1.4 Underarm/upper bust
4.2 Cup measurement methods
4.3 The meaning of cup sizes varies
4.4 Plastic Surgeon Measuring System
4.5 Consumer fit research
5 Manufacturer design standards
5.2 Australia/New Zealand
5.3 United States
5.4 Europe / International
5.5 South Korea/Japan
5.6 France/Belgium/Spain
5.7 Italy
5.8 Advertising and retail influence
5.9 Engineered Alternative to traditional bras
5.10 Calculating cup volume and breast weight
Measurement method origins[edit]
1932 advertisement by S.H. Camp and Company, the first to correlate A-to-D cup size with the volume of the breast
On 21 November 1911, Parisienne Madeleine Gabeau received a United States patent for a brassiere with soft cups and a metal band that supported and separated the breasts. To avoid the prevailing fashion that created a single "monobosom"[citation needed], her design provided: "...that the edges of the material d may be carried close along the inner and under contours of the breasts, so as to preserve their form, I employ an outlining band of metal b which is bent to conform to the lower curves of the breast."[4]
Cup design origins[edit]
The term "cup" was not used to describe bras until 1916[5] when two patents were filed.[6]
In October 1932, S.H. Camp and Company were the first to measure cup size by the letters of the alphabet, A, B, C and D, although the letters represented how pendulous the breasts were and not their volume. Camp's advertising in the February 1933 issue of Corset and Underwear Review featured letter-labeled profiles of breasts. Cup sizes A to D were not intended to be used for larger-breasted women.[7]
Warner's 1944 advertisement for its Alphabet Bras in cup sizes A to D
In 1937, Warner's introduced its Alphabet Bra with four cup sizes (A, B, C and D) to its product descriptions. Before long, these cup sizes got nicknames: egg cup, tea cup, coffee cup and challenge cup, respectively.[8] Two other companies, Model and Fay-Miss (renamed in 1935 as the Bali Brassiere Company), followed, offering A, B, C and D cup sizes in the late 1930s. Catalogue companies continued to use the designations Small, Medium and Large through the 1940s.[9] Britain did not adopt the American cups in 1933, and resisted using cup sizes for its products until 1948. The Sears Company finally applied cup sizes to bras in its catalogue in the 1950s.[10]
Band measurement origins[edit]
Adjustable bands were introduced using multiple hook and eye closures in the 1930s. Prior to the widespread use of bras, the undergarment of choice for Western women was a corset. To help women meet the perceived ideal female body shape, corset and girdle manufacturers used a calculation called hip spring, the difference between waist and hip measurement (usually 10–12 inches (25–30 cm)).[11]
The band measurement system was created by U.S. bra manufacturers just after World War II.[citation needed]
Other innovations[edit]
The underwire was first added to a strapless bra in 1937 by André, a custom-bra firm.[12] Patents for underwire-type devices in bras were issued in 1931 and 1932, but were not widely adopted by manufacturers until after World War II when metal shortages eased.[13]
In the 1930s, Dunlop chemists were able to reliably transform rubber latex into elastic thread. After 1940, "whirlpool", or concentric stitching, was used to shape the cup structure of some designs.[14] The synthetic fibres were quickly adopted by the industry because of their easy-care properties. Since a brassiere must be laundered frequently, easy-care fabric was in great demand.[citation needed]
Consumer fitting[edit]
For best results, the breasts should be measured twice: once when standing upright, once bending over at the waist with the breasts hanging down.[citation needed] If the difference between these two measurements is more than 10 cm, then the average is chosen for calculating the cup size.[citation needed] A number of reports, surveys and studies in different countries have found that between 80% to 85% of women wear incorrectly fitted bras.[15][unreliable source?]
In November 2005, Oprah Winfrey produced a show devoted to bras and bra sizes, during which she talked about research that eight out of ten women wear the wrong size bra.[16]
Larger breasts and bra fit[edit]
A woman wearing an Australian brand, Elle Macpherson, with bra size 30E exhibits correct band fit (30) with a cup fit (E) that is too small for her. Her bra band is parallel to the ground and her breasts do not spill out under her arm, but there is major spilling of breast tissue over the top of the cup.
Studies have revealed that the most common mistake made by women when selecting a bra was to choose too large a back band and too small a cup, for example, 38C instead of 34E, or 34B instead of 30D.[17][18]
The heavier a person's build, the more difficult it is to obtain accurate measurements, as measuring tape sinks into the flesh more easily.[19]
In a study conducted in the United Kingdom of 103 women seeking mammoplasty, researchers found a strong link between obesity and inaccurate back measurement. They concluded that "obesity, breast hypertrophy, fashion and bra-fitting practices combine to make those women who most need supportive bras the least likely to get accurately fitted bras."[19]
One issue that complicates finding a correctly fitting bra is that band and cup sizes are not standardized, but vary considerably from one manufacturer to another, resulting in sizes that only provide an approximate fit.[20][21] Women cannot rely on labeled bra sizes to identify a bra that fits properly.[22][23] Scientific studies show that the current system of bra sizing may be inaccurate.[24]
Manufacturers cut their bras differently, so, for example, two 34B bras from two companies may not fit the same person.[25] Customers should pay attention to which sizing system is used by the manufacturer. The main difference is in how cup sizes increase, by 2 cm or 1 inch (= 2.54 cm, see below). Some French manufacturers also increase cup sizes by 3 cm.[26] Unlike dress sizes, manufacturers do not agree on a single standard.
British bras currently range from A to LL cup size (with Rigby&Peller recently introducing bras by Elila which go up to US-N-Cup), while most Americans can find bras with cup sizes ranging from A to G. Some brands (Goddess, Elila) go as high as N, a size roughly equal to a British JJ-Cup. In continental Europe, Milena Lingerie from Poland produces up to cup R. Larger sizes are usually harder to find in retail outlets. As the cup size increases, the labeled cup size of different manufacturers' bras tend to vary more widely in actual volume.[23] One study found that the label size was consistently different from the measured size.[27]
Even medical studies have attested to the difficulty of getting a correct fit.[28] Research by plastic surgeons has suggested that bra size is imprecise because breast volume is not calculated accurately:
The current popular system of determining bra size is inaccurate so often as to be useless. Add to this the many different styles of bras and the lack of standardization between brands, and one can see why finding a comfortable, well-fitting bra is more a matter of educated guesswork, trial, and error than of precise measurements.[29]
The use of the cup sizing and band measurement systems has evolved over time and continues to change. Experts recommend that women get fitted by an experienced person[30] at a retailer offering the widest possible selection of bra sizes and brands.
Bad bra-fit symptoms[edit]
Back view of a properly fitted bra band worn parallel to the floor
If the straps dig into the shoulder, leaving red marks or causing shoulder or neck pain, the bra band is not offering enough support.[31] If breast tissue overflows the bottom of the bra, under the armpit, or over the top edge of the bra cup, the cup size is too small.[31] Loose fabric in the bra cup indicates the cup size is too big.[32] If the underwires poke the breast under the armpit or if the bra's center panel does not lie flat against the sternum,[33] the cup size is too small.[31] If the band rides up the torso at the back, the band size is too big. If it digs into the flesh, causing the flesh to spill over the edges of the band, the band is too small.[31] If the band feels tight, this may be due to the cups being too small; instead of going up in band size a person should try going up in cup size. Similarly a band might feel too loose if the cup is too big. It is possible to test whether a bra band is too tight or too loose by reversing the bra on her torso so that the cups are at the back and then check for fit and comfort. Generally, if the wearer must continually adjust the bra or experiences general discomfort, the bra is a poor fit and she should get a new fitting.[34]
Obtaining best fit[edit]
Bra extension for the band
Bra experts recommend that women, especially those whose cup sizes are D or larger, get a professional bra fitting from the lingerie department of a clothing store or a specialty lingerie store.[30] However, even professional bra fitters in different countries including New Zealand[35] and the United Kingdom[36] produce inconsistent measurements of the same person. There is significant heterogeneity in breast shape, density, and volume. As such, current methods of bra fitting may be insufficient for this range of chest morphology.
A 2004 study by Consumers Reports in New Zealand found that 80% of department store bra fittings resulted in a poor fit.[37] However, because manufacturer's standards widely vary,[38] women cannot rely on their own measurements to obtain a satisfactory fit.[citation needed] Some bra manufacturers and distributors state that trying on and learning to recognize a properly fitting bra is the best way to determine a correct bra size, much like shoes.
A correctly fitting bra should meet the following criteria:[39][40]
When viewed from the side, the edge of the chest band should be horizontal, should not ride up the back and should be firm but comfortable.
Each cup's underwire at the front should lie flat against the sternum (not the breast), along the inframammary fold, and should not dig into the chest or the breasts, rub or poke out at the front.
The breasts should be enclosed by the cups and there should be a smooth line where the fabric at the top of the cup ends.
The apex of the breast, the nipple, must be in the center of the cup.
The breast should not bulge over the top or out the sides of the cups, even with a low-cut style such as the balconette bra.
The straps of a correctly fitted bra should not dig into or slip off the shoulder, which suggests a too-large band.
The back of the bra should not ride up and the chest band should remain parallel to the floor when viewed from the back.
The breasts should be supported primarily by the band around the rib cage, rather than by the shoulder straps.
The woman should be able to breathe and move easily without the bra slipping around.
Confirming bra fit[edit]
One method to confirm that the bra is the best fit has been nicknamed the Swoop and Scoop. After identifying a well-fitting bra, the woman bends forward (the swoop), allowing her breasts to fall into the bra, filling the cup naturally, and then fastening the bra on the outermost set of hooks.[41][42] When the woman stands up, she uses the opposite hand to place each breast gently into the cup (the scoop), and she then runs her index finger along the inside top edge of the bra cup to make sure her breast tissue does not spill over the edges.[31][43]
Experts suggest that women choose a bra band that fits well on the outermost hooks.[41][42] This allows the wearer to use the tighter hooks on the bra strap as it stretches during its lifetime of about eight months.[42] The band should be tight enough to support the bust, but the straps should not provide the primary support.[44]
Consumer measurement difficulties[edit]
A bra is one of the most complicated articles of clothing to make. A typical bra design has between 20 and 48 parts, including the band, hooks, cups, lining, and straps. Major retailers place orders from manufacturers in batches of 10,000. Orders of this size require a large-scale operation to manage the cutting, sewing and packing required.[45]
Constructing a properly fitting brassiere is difficult. Adelle Kirk, formerly a manager at the global Kurt Salmon management consulting firm that specializes in the apparel and retail businesses, said that making bras is complex:
Bras are one of the most complex pieces of apparel. There are lots of different styles, and each style has a dozen different sizes, and within that there are a lot of colors. Furthermore, there is a lot of product engineering. You've got hooks, you've got straps, there are usually two parts to every cup, and each requires a heavy amount of sewing. It is very component intensive.[46]
Asymmetric breasts[edit]
Obtaining the correct size is complicated by the fact that up to 25% of women's breasts display a persistent, visible breast asymmetry,[47] which is defined as differing in size by at least one cup size. For about 5% to 10% of women, their breasts are severely different, with the left breast being larger in 62% of cases.[48] Minor asymmetry may be resolved by wearing a padded bra, but in severe cases of developmental breast deformity—commonly called "Amazon's Syndrome" by physicians, may require corrective surgery due to morphological alterations caused by variations in shape, volume, position of the breasts relative to the inframammary fold, the position of the nipple-areola complex on the chest, or both.[49]
Breast volume variation[edit]
Obtaining the correct size is further complicated by the fact that the size and shape of women's breasts change, if they experience menstrual cycles, during the cycle[15] and can experience unusual or unexpectedly rapid growth in size due to pregnancy, weight gain or loss, or medical conditions.[50] Even breathing can substantially alter the measurements.[28]
Some women's breasts can change shape by as much as 20% per month:
"Breasts change shape quite consistently on a month-to-month basis, but they will individually change their volume by a different amount ... Some girls will change less than 10% and other girls can change by as much as 20%." Would it be better not to wear a bra at all then? "... In fact there are very few advantages in wearing existing bras. Having a bra that's generally supportive would have significant improvement particularly in terms of stopping them going south ... The skin is what gives the breasts their support"[50]
Increases in average bra size[edit]
In 2010, the most common bra size sold in the UK was 36D.[17][51] In 2004, market research company Mintel reported that bust sizes in the United Kingdom had increased from 1998 to 2004 in younger as well as older consumers, while a more recent study showed that the most often sold bra size in the US in 2008 was 36D.[52]
Researchers ruled out increases in population weight as the explanation and suggested it was instead likely due to more women wearing the correct, larger size.[citation needed]
Consumer measurement methods[edit]
Bra retailers recommend several methods for measuring band and cup size. These are based on two primary methods, either under the bust or over the bust, and sometimes both. Calculating the correct bra band size is complicated by a variety of factors. The American National Standards Institute states that while a voluntary consensus of sizes exists, there is much confusion to the 'true' size of clothing.[38] As a result, bra measurement can be considered an art and a science.[1] Online shopping and in-person bra shopping experiences may differ because online recommendations are based on averages and in-person shopping can be completely personalized so the shopper may easily try on band sizes above and below her between measured band size. For the woman with a large cup size and a between band size, they may find their cup size is not available in local stores so may have to shop online where most large cup sizes are readily available on certain sites. Others recommend rounding to the nearest whole number.[53]
Band measurement methods[edit]
There are several possible methods for measuring the bust.
Underbust +0[edit]
A measuring tape is pulled around the torso at the inframammary fold. The tape is then pulled tight while remaining horizontal and parallel to the floor. The measurement in inches is then rounded to the nearest even number for the band size.[54][55] As of March 2018[update], Kohl's uses this method for its online fitting guide.[56]
This method begins the same way as the underbust +0 method, where a measuring tape is pulled tight around the torso under the bust while remaining horizontal. If the measurement is even, 4 is added to calculate the band size. If it is odd, 5 is added. Kohl's used this method in 2013.[57] The "war on plus four" was a name given to a campaign (circa 2011) against this method, with underbust +0 supporters claiming that the then-ubiquitous +4 method fails to fit a majority of women.[58] Underbust +4 method generally only applies to the US and UK sizes.
Sizing chart[edit]
Currently, many large U.S. department stores determine band size by starting with the measurement taken underneath the bust similar to the aforementioned underbust +0 and underbust +4 methods. A sizing chart or calculator then uses this measurement to determine the band size.[59] Band sizes calculated using this method vary between manufacturers.
Underarm/upper bust[edit]
A measuring tape is pulled around the torso under the armpit and above the bust. Because band sizes are most commonly manufactured in even numbers, the wearer must round to the closest even number.[60]
Cup measurement methods[edit]
Pictogram for the European bra size 70B using EN 13402-1
Bra-wearers can calculate their cup size by finding the difference between their bust size and their band size.[59][61] The bust size, bust line measure, or over-bust measure is the measurement around the torso over the fullest part of the breasts, with the crest of the breast halfway between the elbow and shoulder,[62] usually over the nipples,[63] ideally while standing straight with arms to the side and wearing a properly fitted bra.[53] This practice assumes the current bra fits correctly. The measurements are made in the same units as the band size, either inches or centimetres. The cup size is calculated by subtracting the band size from the over-the-bust measurement.[64][65]
The meaning of cup sizes varies[edit]
Cup sizes vary from one country to another. For example, a U.S. H-cup does not have the same size as an Australian, even though both are based on measurements in inches. The larger the cup size, the bigger the variation.[66]
Over the bust/band measurement difference and cup size [67][68]
Difference (inches) <1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Cupsize U.S. AA A B C D DD/E DDD/F DDDD/G H I J K L M N O P Q R
Cupsize Austr. AA A B C D DD E F G H I J K L M N O P
Cupsize UK AA A B C D DD E F FF G GG H HH J JJ K KK L LL M
Surveys of bra sizes tend to be very dependent on the population studied and how it was obtained. For instance, one U.S. study reported that the most common size was 34B, followed by 34C, that 63% were size 34 and 39% cup size B. However, the survey sample was drawn from 103 Caucasian student volunteers at a Midwest U.S. university aged 18–25, and excluded pregnant and nursing women.[69]
Triumph Survey
UK 57% 18% 19% 6%
Denmark 50% 19% 24% 7%
Netherlands 36% 27% 29% 8%
Belgium 28% 28% 35% 9%
France 26% 29% 38% 7%
Sweden 24% 30% 33% 14%
Greece 23% 28% 40% 9%
Switzerland 19% 24% 43% 14%
Austria 11% 27% 51% 10%
Italy 10% 21% 68% 1%
Plastic Surgeon Measuring System[edit]
Measuring cup size
7.0 17.8 A
8.0 20.3 B
9.0 22.9 C
10.0 25.4 D
11.0 27.9 DD
Bra-wearers who have difficulty calculating a correct cup size may be able to find a correct fit using a method adopted by plastic surgeons. Using a flexible tape measure, position the tape at the outside of the chest, under the arm, where the breast tissue begins.[70] Measure across the fullest part of the breast, usually across the nipple, to where the breast tissue stops at the breast bone.[71]
Conversion of the measurement to cup size is shown in the "Measuring cup size" table.[71]
Note that, in general, countries that employ metric cup sizing (like in § Continental Europe) have their own system of 2 cm (0.79 in) increments that result in cup sizes which differ from those using inches, since 1 inch (2.5 cm) does not equal 2 centimetres (0.79 in).[citation needed]
These cup measurements are only correct for converting cup sizes for a 34-inch (86 cm) band to cm using this particular method, because cup size is relative to band size.[citation needed] This principle means that bras of differing band size can have the same volume. For example, the cup volume is the same for 30D, 32C, 34B, and 36A. These related bra sizes of the same cup volume are called sister sizes.[citation needed] For a list of such sizes, refer to § Calculating cup volume and breast weight.
Consumer fit research[edit]
A 2012 study by White and Scurr University of Portsmouth compared method that adds 4 to the band size over-the-bust method used in many United Kingdom lingerie shops with and compared that to measurements obtained using a professional method.[citation needed] The study relied on the professional bra-fitting method described by McGhee and Steele (2010).[72] The study [73] utilized a five-step approach to obtain the best fitting bra size for an individual. The study measured 45 women using the traditional selection method that adds 4 to the band size over-the-bust method. Women tried bras on until they obtained the best fit based on professional bra fitting criteria. The researchers found that 76% of women overestimated their band and 84% underestimated their cup size. When women wear bras with too big a band, breast support is reduced. Too small a cup size may cause skin irritation. They noted that "ill-fitting bras and insufficient breast support can lead to the development of musculoskeletal pain and inhibit women participating in physical activity.".[74] The study recommended that women should be educated about the criteria for finding a well-fitting bra.[73] They recommended that women measure under their bust to determine their band size rather than the traditional over the bust measurement method.[73]
Manufacturer design standards[edit]
Bra-labeling systems used around the world are at times misleading and confusing. Cup and band sizes vary around the world.[75] In countries that have adopted the European EN 13402 dress-size standard, the torso is measured in centimetres and rounded to the nearest multiple of 5 cm. Bra-fitting experts in the United Kingdom state that many women who buy off the rack without professional assistance wear up to two sizes too small.[citation needed]
Manufacturer Fruit of the Loom attempted to solve the problem of finding a well-fitting bra for asymmetrical breasts by introducing Pick Your Perfect Bra, which allow women to choose a bra with two different cup sizes, although it is only available in A through D cup sizes.[76]
Approximate (band) size equivalents between various systems
Under bust (cm)
Under bust (in)
EU, Iran
FR, BE, ES
US, UK
UK, RoI
AU, NZ
6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
UK dress
4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
One very prominent discrepancy between the sizing systems is the fact that the US band sizes, based on inches, does not correspond to its centimeter based EU counterpart. E.g. 30in equals 76 cm which would suggest that US band size 30 is equivalent EU band size 75. However, this is not correct. Instead, US band size 30 corresponds to EU band size 65 and UK band size 34. This discrepancy stems from the fact that US band sizes were originally based on above bust and under armpit measurement while EU and UK band sizes are based on under bust measurement. This causes confusion and is what led to the Underbust +4 measuring method.[77]
There are several sizing systems in different countries.
Cup size is determined by one of two methods: in the US and UK, increasing cup size every inch method; and in all other systems by increasing cup size for every two centimeters. Since one inch equals 2.54 centimeters, there is considerable discrepancy between the systems, which becomes more exaggerated as cup sizes increase. Many bras are only available in 36 sizes.[30]
UK[edit]
These are equivalent UK cup volumes
The UK and US use the inch system. The difference in chest circumference between the cup sizes is always one inch, or 2.54 cm. The difference between 2 band sizes is 2 inches or 5.08 cm.
Leading brands and manufacturers including Panache, Bestform, Gossard, Freya, Curvy Kate, Bravissimo and Fantasie, which use the British standard band sizes (where underbust measurement equals band size) 28-30-32-34-36-38-40-42-44, and so on. Cup sizes are designated by AA-A-B-C-D-DD-E-F-FF-G-GG-H-HH-J-JJ-K-KK-L.[78]
However, some clothing retailers and mail order companies have their own house brands and use a custom sizing system. Marks and Spencers uses AA-A-B-C-D-DD-E-F-G-GG-H-J, leaving out FF and HH, in addition to following the US band sizing convention. As a result, their J-Cup is equal to a British standard H-cup.[citation needed] Evans and ASDA sell bras (ASDA as part of their George clothing range) whose sizing runs A-B-C-D-DD-E-F-G-H. Their H-Cup is roughly equal to a British standard G-cup.[citation needed]
Some retailers reserve AA for young teens, and use AAA[79][80][81] for women.
Australia/New Zealand[edit]
Australia and New Zealand cup and band sizes are in metric increases of 2 cm per cup similar to many European brands. Cup labelling methods and sizing schemes are inconsistent and there is great variability between brands. In general, cup sizes AA-DD follow UK labels but thereafter split off from this system and employ European labels (no double letters with cups progressing from F-G-H etc. for every 2 cm increase).[82] However, a great many local manufacturers employ unique labelling systems[83] Australia and New Zealand bra band sizes are labelled in dress size, although they are obtained by under bust measurement whilst dress sizes utilise bust-waist-hip.[84] In practice very few of the leading Australian manufacturers produce sizes F+ and many disseminate sizing misinformation.[85][86] The Australian demand for DD+ is largely met by various UK, US and European major brands. This has introduced further sizing scheme confusion that is poorly understood even by specialist retailers.[1]
United States[edit]
Bra-sizing in the United States is very similar to the United Kingdom. Band sizes use the same designation in inches and the cups also increase by 1-inch-steps. However, some manufacturers use conflicting sizing methods. Some label bras beyond a C cup as D-DD-DDD-DDDD-E-EE-EEE-EEEE-F..., some use the variation: D1, D2, D3, D4, D5..... but many use the following system: A, B, C, D, DD, DDD, G, H, I, J, K, L, M, N, O. and others label them like the British system D-DD-E-F-FF... Comparing the larger cup sizes between different manufacturers can be difficult.[citation needed]
In 2013, underwear maker Jockey International offered a new way to measure bra and cup size. It introduced a system with ten cup sizes per band size that are numbered and not lettered, designated as 1–36, 2–36 etc. The company developed the system over eight years, during which they scanned and measured the breasts and torsos of 800 women. Researchers also tracked the women's use of their bras at home.[87] To implement the system, women must purchase a set of plastic cups from the company to find their Jockey cup size. Some analysts were critical of the requirement to buy the measurement kit, since women must pay about US$20 to adopt Jockey's proprietary system, in addition to the cost of the bras themselves.[87]
Europe / International[edit]
Underbust circumference
Underbust size
FR/BE/ES
63–67 80 65 1 I
68–72 85 70 2 II
73–77 90 75 3 III
78–82 95 80 4 IV, IIII
83–87 100 85 5 V
88–92 105 90 6 VI
93–97 110 95 7 VII
98–102 115 100 8 VIII
103–107 120 105 9 IX, VIIII
108–112 125 110 10 X
European bra sizes are based on centimeters. They are also known as International. Abbreviations such as EU, Intl and Int are all referring to the same European bra size convention. These sizes are used in most of Europe and large parts of the world.
Difference [cm]
The underbust measurement is rounded to the nearest multiple of 5 cm. Band sizes run 65, 70, 75, 80 etc., increasing in steps of 5 cm, similar to the English double inch. A person with a measured underbust circumference of 78–82 cm should wear a band size 80. The tightness or snugness of the measurement (e.g. a tape measure or similar) depends on the adipose tissue softness. Softer tissue require tightening when measuring, this to ensure that the bra band will fit snugly on the body and stay in place. A loose measurement can, and often do, vary from the tighter measurement. This causes some confusion as a person with a loose measurement of 84 cm would think they have band size 85 but due to a lot of soft tissue the same person might have a snugger and tighter and of 79 cm and should choose the more appropriate band size of 80 or even smaller band size.
The cup labels begin normally with "A" for an 11±1 cm difference between bust and underbust circumference measurement measured loosely (i.e. not tightly as for bra band size), i.e. the not between bust circumference and band size (that normally require some tightening when measured).[77] To clarify the important difference in measuring: Underbust measuring for bra band is done snugly and tight while measuring underbust for determining bra cups is done loosely. For people with much soft adipose tissue these two measurements will not be identical. In this sense the method to determine European sizes differ compared to English systems where the cup sizes are determined by bust measurement compared to bra band size. European cups increase for every additional 2 cm in difference between bust and underbust measurement, instead of 2.5 cm or 1-inch, and except for the initial cup size letters are neither doubled nor skipped. In very large cup sizes this causes smaller cups than their English counterparts.
This system has been standardized in the European dress size standard EN 13402 introduced in 2006, but was in use in many European countries before that date.
South Korea/Japan[edit]
In South Korea and Japan the torso is measured in centimetres and rounded to the nearest multiple of 5 cm. Band sizes run 65-70-75-80..., increasing in steps of 5 cm, similar to the English double inch. A person with a loosely measured underbust circumference of 78–82 cm should wear a band size 80.
The cup labels begin with "AAA" for a 5±1.25 cm difference between bust and underbust circumference, i.e. similar bust circumference and band size as in the English systems. They increase in steps of 2.5 cm, and except for the initial cup size letters are neither doubled nor skipped.
Japanese sizes are the same as Korean ones, but the cup labels begin with "AA" for a 7.5±1.25 cm difference and usually precedes the bust designation, i.e. "B75" instead of "75B".
This system has been standardized in the Korea dress size standard KS K9404 introduced in 1999 and in Japan dress size standard JIS L4006 introduced in 1998.
France/Belgium/Spain[edit]
The French and Spanish system is a permutation of the Continental European sizing system. While cup sizes are the same, band sizes are exactly 15 cm larger than the European band size.
Italy[edit]
The Italian band size uses small consecutive integers instead of the underbust circumference rounded to the nearest multiple of 5 cm. Since it starts with size 0 for European size 60, the conversion consists of a division by 5 and then a subtraction of 12. The size designations are often given in Roman numerals.
Cup sizes have traditionally used a step size of 2.5 cm, which is close to the English inch of 2.54 cm, and featured some double letters for large cups, but in recent years some Italian manufacturers have switched over to the European 2-cm system.
Here is a conversion table for bra sizes in Italy with respect other countries:
UK & USA
Advertising and retail influence[edit]
Manufacturers' marketing and advertising often appeals to fashion and image over fit, comfort, and function.[28][88] Since about 1994, manufacturers have re-focused their advertising, moving from advertising functional brassieres that emphasize support and foundation, to selling lingerie that emphasize fashion while sacrificing basic fit and function, like linings under scratchy lace.[89]
Engineered Alternative to traditional bras[edit]
English mechanical engineer and professor John Tyrer from Loughborough University has devised a solution to problematic bra fit by re-engineering bra design. He started investigating the problem of bra design while on an assignment from the British government after his wife returned disheartened from an unsuccessful shopping trip.[90][91] His initial research into the extent of fitting problems soon revealed that 80% of women wear the wrong size of bra.[citation needed]. He theorised that this widespread practice of purchasing the wrong size was due to the measurement system recommended by bra manufacturers. This sizing system employs a combination of maximum chest diameter (under bust) and maximum bust diameter (bust) rather than the actual breast volume which is to be accommodated by the bra. According to Tyrer, "to get the most supportive and fitted bra it's infinitely better if you know the volume of the breast and the size of the back.".[90] He says the A, B, C, D cup measurement system is flawed. "It's like measuring a motor car by the diameter of the gas cap." "The whole design is fundamentally flawed. It's an instrument of torture."[91] Tyrer has developed a bra design with crossed straps in the back.[citation needed] These use the weight of one breast to lift the other using counterbalance.[citation needed] Standard designs constrict chest movement during breathing.[citation needed] One of the tools used in the development of Tyrer's design has been a projective differential shape body analyzer for 40,000 GBP.[citation needed]
Breasts weigh up to ~1 kg and not ~0.2 .. 0.3 kg.[90][92][93][94] Tyrer said, "By measuring the diameter of the chest and breasts current measurements are supposed to tell you something about the size and volume of each breast, but in fact it doesn't".[90] Bra companies remain reluctant to manufacture Tyrer's prototype,[91] which is a front closing bra with more vertical orientation and adjustable cups.[91]
Calculating cup volume and breast weight[edit]
The average breast weighs about 0.5 kilograms (1.1 lb).[95] Each breast contributes to about 4–5% of the body fat.[citation needed] The density of fatty tissue is more or less equal to 0.9 kg/l for all women.[citation needed]
If a cup is a hemisphere, its volume V is given by the following formula:[96]
V = 2 π r 3 3 {\displaystyle V={2\pi r^{3} \over 3}} V = π D 3 12 {\displaystyle V={\pi D^{3} \over 12}}
where r is the radius of the cup, and D is its diameter.
If the cup is an hemi-ellipsoid, its volume is given by the formula :
V = 2 π a b c 3 {\displaystyle V={2\pi abc \over 3}} V ≈ π × c w × c d × w l 12 {\displaystyle V\approx {\pi \times cw\times cd\times wl \over 12}}
where a, b and c are the three semi-axes of the hemi-ellipsoid, and cw cd and wl are respectively the cup width, the cup depth and the length of the wire.
Cups give a hemi-spherical shape to breasts and underwires give shape to cups.[citation needed] So the curvature radius of the underwire is the key parameter to determine volume and weight of the breast.[citation needed] The same underwires are used for the cups of sizes 36A, 34B, 32C, 30D etc. ... so those cups have the same volume.[citation needed] The reference numbers of underwire sizes are based on a B cup bra,[97] for example underwire size 32 is for 32B cup (and 34A, 30C...). An underwire size 30 width has a curvature diameter of 3-inch 5/6 ≈ 9.7 cm and this diameter increases by 1⁄3 inch ≈ 0.847 cm by size.[97] The table below shows volume calculations for some cups that can be found in a ready-to-wear large size shop.[98]
Underwire size
Bra size (US system)
Bra size (UK system)
Cup diameter[97]
Volume of one cup
Weight of both breasts
30 32A 30B 28C 32A 30B 28C 9.7 cm (3 in 5/6) 240 cm3 (0.51 US pt) 0.43 kg (0.95 lb)
32 34A 32B 30C 28D 34A 32B 30C 28D 10.6 cm (4 in 1/6) 310 cm3 (0.66 US pt) 0.56 kg (1.2 lb)
34 36A 34B 32C 30D 28E 36A 34B 32C 30D 28DD 11.4 cm (4 in 1/2) 390 cm3 (0.82 US pt) 0.70 kg (1.5 lb)
36 38A 36B 34C 32D 30E 28F 38A 36B 34C 32D 30DD 28E 12.3 cm (4 in 5/6) 480 cm3 (1.0 US pt) 0.86 kg (1.9 lb)
38 40A 38B 36C 34D 32E 30F 28G 40A 38B 36C 34D 32DD 30E 28F 13.1 cm (5 in 1/6) 590 cm3 (1.2 US pt) 1.1 kg (2.4 lb)
40 42A 40B 38C 36D 34E 32F 30G 28H 42A 40B 38C 36D 34DD 32E 30F 28FF 14.0 cm (5 in 1/2) 710 cm3 (1.5 US pt) 1.3 kg (2.9 lb)
42 44A 42B 40C 38D 36E 34F 32G 30H 28I 44A 42B 40C 38D 36DD 34E 32F 30FF 28G 14.8 cm (5 in 5/6) 850 cm3 (1.8 US pt) 1.5 kg (3.3 lb)
44 44B 42C 40D 38E 36F 34G 32H 30I 28J 44B 42C 40D 38DD 36E 34F 32FF 30G 28GG 15.7 cm (6 in 1/6) 1,000 cm3 (2.1 US pt) 1.8 kg (4.0 lb)
46 44C 42D 40E 38F 36G 34H 32I 30J 28K 44C 42D 40DD 38E 36F 34FF 32G 30GG 28H 16.5 cm (6 in 1/2) 1,180 cm3 (2.5 US pt) 2.1 kg (4.6 lb)
48 44D 42E 40F 38G 36H 34I 32J 30K 28L 44D 42DD 40E 38F 36FF 34G 32GG 30H 28HH 17.4 cm (6 in 5/6) 1,370 cm3 (2.9 US pt) 2.5 kg (5.5 lb)
50 44E 42F 40G 38H 36I 34J 32K 30L 28M 44DD 42E 40F 38FF 36G 34GG 32H 30HH 28J 18.2 cm (7 in 1/6) 1,580 cm3 (3.3 US pt) 2.8 kg (6.2 lb)
52 44F 42G 40H 38I 36J 34K 32L 30M 28N 44E 42F 40FF 38G 36GG 34H 32HH 30J 28JJ 19.0 cm (7 in 1/2) 1,810 cm3 (3.8 US pt) 3.3 kg (7.3 lb)
54 44G 42H 40I 38J 36K 34L 32M 30N 28O 44F 42FF 40G 38GG 36H 34HH 32J 30JJ 28K 19.9 cm (7 in 5/6) 2,060 cm3 (4.4 US pt) 3.7 kg (8.2 lb)
56 44H 42I 40J 38K 36L 34M 32N 30O 28P 44FF 42G 40GG 38H 36HH 34J 32JJ 30K 28KK 20.7 cm (8 in 1/6) 2,340 cm3 (4.9 US pt) 4.2 kg (9.3 lb)
58 44I 42J 40K 38L 36M 34N 32O 30P 44G 42GG 40H 38HH 36J 34JJ 32K 30KK 21.6 cm (8 in 1/2) 2,640 cm3 (5.6 US pt) 4.8 kg (11 lb)
60 44J 42K 40L 38M 36N 34O 32P 44GG 42H 40HH 38J 36JJ 34K 32KK 22.4 cm (8 in 5/6) 3,000 cm3 (6.3 US pt) 5.3 kg (12 lb)
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Journal of Inequalities and Applications
The obstacle problem for non-coercive equations with lower order term and \(L^{1}\)-data
Jun Zheng ORCID: orcid.org/0000-0003-4868-32451
Journal of Inequalities and Applications volume 2019, Article number: 205 (2019) Cite this article
The aim of this paper is to study the obstacle problem associated with an elliptic operator having degenerate coercivity, a low order term, and \(L^{1}\)-data. We prove the existence of an entropy solution to the obstacle problem and show its continuous dependence on the \(L^{1}\)-data in \(W^{1,q}(\varOmega )\) with some \(q>1\).
1.1 Problem setting and main result
Let Ω be a bounded domain in \(\mathbb{R}^{N}\) (\(N\geq 2\)), \(1< p<+\infty \), and \(\theta \geq 0\). Given functions \(g, \psi \in W ^{1,p}(\varOmega )\cap L^{\infty }(\varOmega )\) and data \(f\in L^{1}(\varOmega )\), the aim of this paper is to study the obstacle problem for nonlinear non-coercive elliptic equations with lower order term, governed by the operator
$$\begin{aligned} Au=-\operatorname{div} \frac{a(x, \nabla u)}{(1+ \vert u \vert )^{\theta (p-1)}}+b \vert u \vert ^{r-2}u, \end{aligned}$$
where \(b>0\) is a constant, and \(a:\varOmega \times \mathbb{R}^{N} \rightarrow \mathbb{R}^{N}\) is a Carathéodory function, satisfying the following conditions:
$$\begin{aligned} &a(x,\xi )\cdot \xi \geq \alpha \vert \xi \vert ^{p}, \end{aligned}$$
$$\begin{aligned} & \bigl\vert a(x,\xi ) \bigr\vert \leq \beta \bigl(j(x)+ \vert \xi \vert ^{p-1}\bigr), \end{aligned}$$
$$\begin{aligned} &\bigl(a(x, \xi )-a(x, \eta )\bigr) (\xi -\eta )>0, \end{aligned}$$
$$\begin{aligned} & \bigl\vert a(x, \xi )-a(x, \zeta ) \bigr\vert \leq \gamma \textstyle\begin{cases} \vert \xi -\zeta \vert ^{p-1}, & \text{if } 1< p< 2, \\ (1+ \vert \xi \vert + \vert \zeta \vert )^{p-2} \vert \xi -\zeta \vert , & \text{if } p\geq 2 \end{cases}\displaystyle \end{aligned}$$
for almost every x in Ω and for every ξ, η, ζ in \(\mathbb{R}^{N}\) with \(\xi \neq \eta \), where \(\alpha ,\beta ,\gamma >0\) are constants, and j is a nonnegative function in \(L^{p'}( \varOmega )\).
If f has a fine regularity, e.g., \(f\in W^{-1,p'}(\varOmega )\), the obstacle problem corresponding to \((f,\psi ,g) \) can be formulated in terms of the inequality
$$\begin{aligned} & \int _{\varOmega }\frac{a(x, \nabla u)}{(1+ \vert u \vert )^{\theta (p-1)}}\cdot \nabla (u-v)\, \mathrm{d}x+ \int _{\varOmega }b \vert u \vert ^{r-2}u(u-v)\, \mathrm{d}x \\ & \quad \leq \int _{\varOmega }f(u-v)\,\mathrm{d}x, \quad \forall v\in K_{g,\psi }\cap L^{\infty }(\varOmega ), \end{aligned}$$
whenever \(1\leq r< p\) and the convex subset
$$\begin{aligned} K_{g,\psi }= \bigl\{ v\in W^{1,p}(\varOmega ); v-g \in W^{1,p}_{0}(\varOmega ), v\geq \psi , \text{a.e. in } \varOmega \bigr\} \end{aligned}$$
is nonempty. However, if \(f\in L^{1}(\varOmega )\), (6) is not well-defined. Following [1, 3, 5, 19] etc., we are led to the more general definition of a solution to the obstacle problem, using the truncation function
$$\begin{aligned} T_{s}(t)=\max \bigl\{ -s,\min \{s,t\}\bigr\} , \quad s,t\in \mathbb{R}. \end{aligned}$$
Definition 1
An entropy solution of the obstacle problem associated with operator A and functions \((f,\psi ,g)\) with \(f\in L^{1}(\varOmega )\) is a measurable function u such that \(u\geq \psi \) a.e. in Ω, \(\frac{a(x, \nabla u)}{(1+ \vert u \vert )^{\theta (p-1)}}\in (L^{1}( \varOmega ))^{N}\), \(\vert u \vert ^{r-1}\in L^{1}(\varOmega )\), and, for every \(s>0\), \(T_{s}(u)-T_{s}(g)\in W_{0}^{1,p}(\varOmega )\) and
$$\begin{aligned} & \int _{\varOmega }\frac{a(x, \nabla u)}{(1+ \vert u \vert )^{\theta (p-1)}}\cdot \nabla \bigl(T_{s}(u-v)\bigr)\,\mathrm{d}x+ \int _{\varOmega }b \vert u \vert ^{r-2}uT_{s}(u-v) \,\mathrm{d}x \\ & \quad \leq \int _{\varOmega }fT_{s}(u-v)\,\mathrm{d}x, \quad \forall v\in K_{g,\psi }\cap L^{\infty }(\varOmega ). \end{aligned}$$
Observe that no global integrability condition is required on u nor on its gradient in the definition. As pointed out in [3, 8], if \(T_{s} (u) \in W^{1,p}(\varOmega )\) for all \(s > 0\), then there exists a unique measurable vector field \(U:\varOmega \rightarrow \mathbb{R}^{N}\) such that \(\nabla (T_{s} (u)) = \chi _{\{ \vert u \vert < s\}}U\) a.e. in Ω, \(s > 0\), which, in fact, coincides with the standard distributional gradient of ∇u whenever \(u \in W^{1,1}(\varOmega )\).
Before stating the main result, we make some basic assumptions throughout this paper, i.e., without special statements, we always assume that
$$\begin{aligned} 2-\frac{1}{N}< p< N, \quad\quad 1\leq r< p, \quad\quad 0\leq \theta < \min \biggl\{ \frac{N}{N-1}-\frac{1}{p-1},\frac{p-r}{p-1} \biggr\} ,\quad\quad b>0, \end{aligned}$$
and \(\psi ,g\in W^{1,p}(\varOmega )\cap L^{\infty }(\varOmega )\) with \((\psi -g)^{+}\in W^{1,p}_{0}(\varOmega )\) such that \(K_{g,\psi }\neq \emptyset \). The following theorem is the main result obtained in this paper.
Let \(f\in L^{1}(\varOmega )\). Then there exists at least one entropy solution u of the obstacle problem associated with \((f,\psi ,g) \). In addition, u depends continuously on f, i.e., if \(f_{n}\rightarrow f\) in \(L^{1}(\varOmega )\) and \(u_{n}\) is a solution to the obstacle problem associated with \((f_{n},\psi ,g)\), then
$$\begin{aligned} u_{n}\rightarrow u \quad \textit{in } W^{1,q}(\varOmega ),\forall q \in \textstyle\begin{cases} ( \frac{N(r-1)}{N+r-1}, \frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)} ), & \textit{if }\frac{2N-1}{N-1}\leq r< p, \\ (1,\frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)} ), & \textit{if } 1\leq r< \min \{\frac{2N-1}{N-1},p\}. \end{cases}\displaystyle \end{aligned}$$
1.2 Some comments and remarks
Consider the Dirichlet boundary value problem having a form
$$\begin{aligned} \textstyle\begin{cases} -\operatorname{div} \frac{ \vert \nabla u \vert ^{p-2}\nabla u}{(1+ \vert u \vert )^{\theta (p-1)}}+bu=f, & \text{in } \varOmega , \\ u=0, & \text{on } \partial \varOmega , \end{cases}\displaystyle \end{aligned}$$
with \(p>1\), \(\theta \in (0,1]\), \(b\geq 0\), \(f\in L^{1}(\varOmega )\). The item \(-\operatorname{div}\frac{ \vert \nabla u \vert ^{p-2}\nabla u}{(1+ \vert u \vert )^{\theta (p-1)}}\) may not be coercive when u tends to infinity. Due to this fact, the classical methods used to prove the existence of a solution for elliptic equations, e.g., [14], cannot be applied even if \(b=0\) and the data f is regular. Moreover, \(\frac{ \vert \nabla u \vert ^{p-2}\nabla u}{(1+ \vert u \vert )^{\theta (p-1)}}\), u and f are only in \(L^{1}(\varOmega )\), not in \(W^{-1,p'}(\varOmega )\). All these characteristics prevent us from employing the duality argument [17] or nonlinear monotone operator theory [18] directly.
To overcome these difficulties, "cutting" the non-coercivity term and using the technique of approximation, a pseudomonotone and coercive differential operator on \(W^{1,p}_{0}(\varOmega )\) can be applied to establish a priori estimates on approximating solutions. As a result, existence of solutions, or entropy solutions, can be obtained by taking limitation for \(f\in L^{m}(\varOmega )\), \(m\geq 1\), and \(b> 0\) due to the almost everywhere convergence of gradients of the approximating solutions, see, e.g., [4, 6, 9,10,11, 15] (see also [1, 2, 7, 12, 13, 16] for \(b=0\)). However, there is little literature that considers regularities for entropy solutions of obstacle problems governed by (1) and functions \((f,\psi ,g) \) with \(f\in L^{1}(\varOmega )\), except [19], in which the authors considered the obstacle problem (7) with \(b=0\) and \(L^{1}\)-data.
Motivated by the study on the non-coercive elliptic equations (9) and the problem considered in [19], in this paper, we consider the obstacle problem governed by (1) and functions \((f,\psi ,g) \) with \(f\in L^{1}(\varOmega )\). By the truncation method used in [8] and [19], we prove the existence of an entropy solution and show its continuous dependence on the \(L^{1}\)-data in \(W^{1,q}(\varOmega )\) with some \(q\in (1,p)\).
In the following, we give some remarks on our main result and inequalities that will be needed in the proofs. Some notations are provided at the end of this subsection.
\(0\leq \theta < \min \{\frac{N}{N-1}-\frac{1}{p-1}, \frac{p-r}{p-1} \}\Rightarrow r-1<(1-\theta )(p-1)< \frac{N(p-1)(1- \theta )}{N-1-\theta (p-1)}\). Therefore Theorem 1 guarantees \(\vert u \vert ^{r-1}\in L^{1}(\varOmega )\), and the second integration in (7) makes sense.
We will show that \(\frac{a(x, \nabla u)}{(1+ \vert u \vert )^{\theta (p-1)}}\in (L^{1}(\varOmega ))^{N} \) in Proposition 4. Therefore, the first integration in (7) makes sense.
\(( \frac{N(r-1)}{N+r-1}, \frac{N(p-1)(1-\theta )}{N-1- \theta (p-1)} )\subset (1,\frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)} )\) if \(\frac{2N-1}{N-1}\leq r< p\). Indeed, \(\theta < \frac{p-r}{p-1}+\frac{p(r-1)}{N(p-1)}\Leftrightarrow \frac{N(p-1)(1- \theta )}{N-1-\theta (p-1)}>\frac{N(r-1)}{N+r-1}\), while \(\frac{2N-1}{N-1}\leq r \) gives \(\frac{N(r-1)}{N+r-1}\geq 1 \). Thus \(u_{n}\rightarrow u\) in \(W^{1,q}(\varOmega )\) for all \(q\in (1,\frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)} )\).
\(r-1<\frac{Nq}{N-q}\). Indeed, by \(1\leq r<\frac{2N-1}{N-1}\), there holds \(r-1<\frac{N}{N-1}< \frac{Nq}{N-q}\) for any \(q>1\), particularly, for \(q\in (1, \frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)} )\). For \(r\geq \frac{2N-1}{N-1}\), it suffices to note that \(q> \frac{N(r-1)}{N+r-1} \Leftrightarrow r-1< \frac{Nq}{N-q}\).
\(q< p\). Indeed, \(0\leq \theta <\frac{N}{N-1}-\frac{1}{p-1}< \frac{N-1}{p-1} \Rightarrow \frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)}<p \).
Checking proofs in this paper (e.g., setting \(r=1\)), one may find that, for \(b=0\), (8) holds with
$$\begin{aligned} u_{n}\rightarrow u \quad \text{in } W^{1,q}(\varOmega ),\forall q \in \biggl(1,\frac{N(p-1)(1- \theta )}{N-1-\theta (p-1)} \biggr), \end{aligned}$$
which is the same as [19, Theorem 1]. Thus, Theorem 1 can be seen as an extension of [19, Theorem 1].
\(\Vert u \Vert _{p}:= \Vert u \Vert _{L^{p}(\varOmega )}\) is the norm of u in \(L^{p}(\varOmega )\), where \(1\leq p\leq \infty \). \(\Vert u \Vert _{1,p}:= \Vert u \Vert _{W^{1,p}(\varOmega )}\) is the norm of u in \(W^{1,p}(\varOmega )\), where \(1< p<\infty \). \(p':=\frac{p}{p-1}\) with \(1< p<\infty \). \(\{u>s\}:=\{x\in \varOmega ;u(x)>s \}\). \(\{u\leq s\}:=\varOmega \setminus \{u>s\}\). \(\{u< s\}:=\{x\in \varOmega ;u(x)< s\}\). \(\{u\geq s\}:=\varOmega \setminus \{u< s\}\). \(\{u=s\}:=\{x \in \varOmega ;u(x)=s\}\). \(\{t\leq u < s\}:=\{u\geq t\}\cap \{u<s\} \). For a measurable set E in \(\mathbb{R}^{N}\), \(\vert E \vert :=\mathcal{L}^{N}(E)\), where \(\mathcal{L}^{N}\) is the Lebesgue measure of \(\mathbb{R}^{N}\). For a real-valued function u, \(u^{+}=\max \{u,0\}\), \(u^{-}=(-u)^{+}\). Without special statements, positive integers are denoted by n, h, k, \(k_{0}\), K. C is a positive constant, which may be different from each other.
2 Lemmas on entropy solutions
It is worthy to note that, for any smooth function \(f_{n}\), there exists at least one solution to the obstacle problem (6). Indeed, one can proceed exactly as in [1, 11] to obtain \(W^{1,p}\)-solutions due to assumptions (2)–(4) on a and \(r-1< p\). These solutions, in particular, are also entropy solutions. In this section, using the method of [8] and [19], we establish several auxiliary results on convergence of sequences of entropy solutions when \(f_{n}\rightarrow f\) in \(L^{1}(\varOmega )\).
Lemma 2
Let \(v_{0}\in K_{g,\psi }\cap L^{\infty }(\varOmega )\), and let u be an entropy solution of the obstacle problem associated with \((f,\psi ,g)\). Then we have
$$\begin{aligned} \int _{\{ \vert u \vert < t\}}\frac{ \vert \nabla u \vert ^{p}}{(1+ \vert u \vert )^{\theta (p-1)}}\,\mathrm{d}x \leq C \bigl(1+t^{r}\bigr), \quad \forall t>0, \end{aligned}$$
where C is a positive constant depending only on α, β, p, r, b, \(\Vert j \Vert _{p'}\), \(\Vert \nabla v_{0} \Vert _{p}\), \(\Vert v_{0} \Vert _{\infty }\), and \(\Vert f \Vert _{1}\).
Take \(v_{0}\) as a test function in (7). For t large enough such that \(t- \Vert v_{0} \Vert _{\infty }>0\), we get
$$\begin{aligned} \int _{\{ \vert v_{0}-u \vert < t\}}\frac{a(x, \nabla u)\cdot \nabla u}{(1+ \vert u \vert )^{ \theta (p-1)}}\,\mathrm{d}x\leq{}& \int _{\{ \vert v_{0}-u \vert < t\}}\frac{a(x, \nabla u)\cdot \nabla v_{0}}{(1+ \vert u \vert )^{\theta (p-1)}}\,\mathrm{d}x \\ & {} + \int _{\varOmega }\bigl(f-b \vert u \vert ^{r-2}u\bigr) T_{t}(u-v_{0})\,\mathrm{d}x. \end{aligned}$$
We estimate each integration in the right-hand side of (11). It follows from (3) and Young's inequality with \(\varepsilon >0\) that
$$\begin{aligned}& \int _{\{ \vert v_{0}-u \vert < t\}}\frac{a(x, \nabla u)\cdot \nabla v_{0}}{(1+ \vert u \vert )^{ \theta (p-1)}}\,\mathrm{d}x\leq \int _{\{ \vert v_{0}-u \vert < t\}}\frac{\beta ( \vert j \vert + \vert \nabla u \vert ^{p-1})\cdot \vert \nabla v_{0} \vert }{(1+ \vert u \vert )^{\theta (p-1)}}\,\mathrm{d}x \\& \hphantom{\int _{\{ \vert v_{0}-u \vert < t\}}\frac{a(x, \nabla u)\cdot \nabla v_{0}}{(1+ \vert u \vert )^{ \theta (p-1)}}\,\mathrm{d}x}\leq \int _{\{ \vert v_{0}-u \vert < t\}}\frac{\beta \varepsilon ( \vert j \vert ^{p'}+ \vert \nabla u \vert ^{p})}{(1+ \vert u \vert )^{\theta (p-1)}}\,\mathrm{d}x \\& \hphantom{\int _{\{ \vert v_{0}-u \vert < t\}}\frac{a(x, \nabla u)\cdot \nabla v_{0}}{(1+ \vert u \vert )^{ \theta (p-1)}}\,\mathrm{d}x}\quad {} + \int _{\{ \vert v_{0}-u \vert < t\}}\frac{\beta C(\varepsilon ) \vert \nabla v_{0} \vert ^{p}}{(1+ \vert u \vert )^{ \theta (p-1)}}\,\mathrm{d}x \\& \hphantom{\int _{\{ \vert v_{0}-u \vert < t\}}\frac{a(x, \nabla u)\cdot \nabla v_{0}}{(1+ \vert u \vert )^{ \theta (p-1)}}\,\mathrm{d}x} \leq \varepsilon \int _{\{ \vert v_{0}-u \vert < t\}}\frac{ \vert \nabla u \vert ^{p}}{(1+ \vert u \vert )^{ \theta (p-1)}}\,\mathrm{d}x \\& \hphantom{\int _{\{ \vert v_{0}-u \vert < t\}}\frac{a(x, \nabla u)\cdot \nabla v_{0}}{(1+ \vert u \vert )^{ \theta (p-1)}}\,\mathrm{d}x}\quad {} +C\bigl( \Vert j \Vert _{p'}^{p'}+ \Vert \nabla v_{0} \Vert _{p}^{p}\bigr), \end{aligned}$$
$$\begin{aligned}& \begin{aligned}[b] - \int _{\varOmega }b \vert u \vert ^{r-2}uT_{t}(u-v_{0}) \,\mathrm{d}x={}&- \int _{\{ \vert u-v_{0} \vert \leq t\}} b \vert u \vert ^{r-2}uT_{t}(u-v_{0}) \,\mathrm{d}x \\ & {} - \int _{\{ \vert u-v_{0} \vert > t\}}b \vert u \vert ^{r-2}uT_{t}(u-v_{0}) \,\mathrm{d}x. \end{aligned} \end{aligned}$$
Note that on the set \(\{ \vert u-v_{0} \vert \leq t\}\),
$$\begin{aligned} \bigl\vert \vert u \vert ^{r-2}uT_{t}(u-v_{0}) \bigr\vert \leq t \bigl\vert t+ \Vert v_{0} \Vert _{\infty } \bigr\vert ^{r-1}\leq C\bigl(1+t ^{r} \bigr), \end{aligned}$$
where C is a constant depending only on r, \(\Vert v_{0} \Vert _{\infty }\).
On the set \(\{ \vert u-v_{0} \vert > t\}\), we have \(\vert u \vert \geq t- \Vert v_{0} \Vert _{\infty } >0\), thus u and \(T_{t}(u-v_{0})\) have the same sign. It follows
$$\begin{aligned} - \int _{\{ \vert u-v_{0} \vert > t\}} b \vert u \vert ^{r-2}uT_{t}(u-v_{0}) \,\mathrm{d}x\leq 0. \end{aligned}$$
Combining (13)–(15), we get
$$\begin{aligned}& - \int _{\varOmega }b \vert u \vert ^{r-2}uT_{t}(u-v_{0}) \,\mathrm{d}x\leq C\bigl(1+t^{r}\bigr), \end{aligned}$$
$$\begin{aligned}& \begin{aligned}[b] \int _{\{ \vert v_{0}-u \vert < t\}}\frac{ \vert \nabla u \vert ^{p}}{(1+ \vert u \vert )^{\theta (p-1)}} \,\mathrm{d}x\leq{}& C\bigl( \Vert j \Vert _{p'}^{p'}+ \Vert \nabla v_{0} \Vert _{p}^{p}+t \Vert f \Vert _{1}+1+t ^{r}\bigr) \\ \leq{}& C\bigl(1+t^{r}\bigr). \end{aligned} \end{aligned}$$
Replacing t with \(t+ \Vert v_{0} \Vert _{\infty }\) in (17) and noting that \(\{ \vert u \vert < t\}\subset \{ \vert v_{0}-u \vert < t+ \Vert v_{0} \Vert _{\infty }\}\), one may obtain the desired result. □
In the rest of this section, let \(\{u_{n}\}\) be a sequence of entropy solutions of the obstacle problem associated with \((f_{n},\psi ,g)\) and assume that
$$\begin{aligned} f_{n}\rightarrow f \quad \text{in } L^{1}(\varOmega ) \quad \text{and} \quad \Vert f_{n} \Vert _{1}\leq \Vert f \Vert _{1}+1. \end{aligned}$$
There exists a measurable function u such that \(u_{n}\rightarrow u \) in measure, and \(T_{k}(u_{n})\rightharpoonup T_{k}(u)\) weakly in \(W^{1,p}(\varOmega )\) for any \(k>0\). Thus \(T_{k}(u_{n})\rightarrow T _{k}(u) \) strongly in \(L^{p}(\varOmega )\) and a.e. in Ω.
Let s, t, and ε be positive numbers. One may verify that, for every \(m,n\geq 1\),
$$\begin{aligned} \mathcal{L}^{N}\bigl(\bigl\{ \vert u_{n}-u_{m} \vert >s\bigr\} \bigr)\leq{}& \mathcal{L}^{N}\bigl(\bigl\{ \vert u_{n} \vert >t \bigr\} \bigr)+ \mathcal{L}^{N}\bigl( \bigl\{ \vert u_{m} \vert >t\bigr\} \bigr) \\ & {} +\mathcal{L}^{N}\bigl(\bigl\{ \bigl\vert T_{k}(u_{n})-T_{k}(u_{m}) \bigr\vert >s\bigr\} \bigr), \end{aligned}$$
$$\begin{aligned} \mathcal{L}^{N}\bigl(\bigl\{ \vert u_{n} \vert >t \bigr\} \bigr)=\frac{1}{t^{p}} \int _{\{ \vert u_{n} \vert >t\}}t ^{p}\,\mathrm{d}x\leq \frac{1}{t^{p}} \int _{\varOmega } \bigl\vert T_{t}(u_{n}) \bigr\vert ^{p} \,\mathrm{d}x. \end{aligned}$$
Due to \(v_{0}=g+(\psi -g)^{+}\in K_{g,\psi }\cap L^{\infty }(\varOmega )\), by Lemma 2, one has
$$\begin{aligned} \int _{\varOmega } \bigl\vert \nabla T_{t}(u_{n}) \bigr\vert ^{p}\,\mathrm{d}x= \int _{\{ \vert u_{n} \vert < t\}} \vert \nabla u_{n} \vert ^{p}\,\mathrm{d}x \leq C(1+t)^{\theta (p-1)}\bigl(1+t^{r} \bigr). \end{aligned}$$
Note that \(T_{t}(u_{n})-T_{t}(g)\in W^{1,p}_{0}(\varOmega )\). By (19), (20), and Poincaré's inequality, for every \(t> \Vert g \Vert _{\infty }\) and for some positive constant C independent of n and t, there holds
$$\begin{aligned} \mathcal{L}^{N}\bigl(\bigl\{ \vert u_{n} \vert >t \bigr\} \bigr)\leq{}& \frac{1}{t^{p}} \int _{\varOmega } \bigl\vert T _{t}(u_{n}) \bigr\vert ^{p}\,\mathrm{d}x \\ \leq{}& \frac{2^{p-1}}{t^{p}} \int _{\varOmega } \bigl\vert T_{t}(u_{n})-T_{t}(g) \bigr\vert ^{p} \,\mathrm{d}x+\frac{2^{p-1}}{t^{p}} \Vert g \Vert _{p}^{p} \\ \leq{}& \frac{C}{t^{p}} \int _{\varOmega } \bigl\vert \nabla T_{t}(u_{n})- \nabla T _{t}(g) \bigr\vert ^{p}\,\mathrm{d}x+ \frac{2^{p-1}}{t^{p}} \Vert g \Vert _{p}^{p} \\ \leq{}&\frac{C}{t^{p}} \int _{\varOmega } \bigl\vert \nabla T_{t}(u_{n}) \bigr\vert ^{p}\,\mathrm{d}x+\frac{C}{t ^{p}} \Vert g \Vert _{1,p}^{p} \\ \leq{}& \frac{C(1+t^{r+\theta (p-1)})}{t^{p}}. \end{aligned}$$
Since \(0\leq \theta <\frac{p-r}{p-1}\), there exists \(t_{\varepsilon }>0\) such that
$$\begin{aligned} \mathcal{L}^{N}\bigl(\bigl\{ \vert u_{n} \vert >t \bigr\} \bigr)< \frac{\varepsilon }{3}, \quad \forall t\geq t_{\varepsilon }, \forall n\geq 1. \end{aligned}$$
Now we have as in (19)
$$\begin{aligned} \mathcal{L}^{N}\bigl(\bigl\{ \bigl\vert T_{t_{\varepsilon }}(u_{n})-T_{t_{\varepsilon }}(u _{m}) \bigr\vert >s\bigr\} \bigr) &=\frac{1}{s^{p}} \int _{\{ \vert T_{t_{\varepsilon }} (u_{n})-T_{t_{\varepsilon }}(u_{m}) \vert >s \}} s^{p}\,\mathrm{d}x \\ &\leq \frac{1}{s^{p}} \int _{\varOmega } \bigl\vert T_{t_{\varepsilon }}(u_{n})-T _{t_{\varepsilon }}(u_{m}) \bigr\vert ^{p}\,\mathrm{d}x. \end{aligned}$$
Using (20) and the fact that \(T_{t}(u_{n})-T_{t}(g)\in W^{1,p} _{0}(\varOmega )\) again, we see that \(\{T_{t_{\varepsilon }}(u_{n})\}\) is a bounded sequence in \(W^{1,p}(\varOmega )\). Thus, up to a subsequence, \(\{T_{t_{\varepsilon }}(u_{n})\}\) converges strongly in \(L^{p}( \varOmega )\). Taking into account (22), there exists \(n_{0}=n_{0}(t _{\varepsilon },s)\geq 1\) such that
$$\begin{aligned} \mathcal{L}^{N}\bigl(\bigl\{ \bigl\vert T_{t_{\varepsilon }}(u_{n})-T_{t_{\varepsilon }}(u _{m}) \bigr\vert >s\bigr\} \bigr)< \frac{\varepsilon }{3}, \quad \forall n,m\geq n_{0}. \end{aligned}$$
Combining (18), (21), and (23), we obtain
$$\begin{aligned} \mathcal{L}^{N}\bigl(\bigl\{ \vert u_{n}-u_{m} \vert >s\bigr\} \bigr)< \varepsilon , \quad \forall n,m\geq n_{0}. \end{aligned}$$
Hence \(\{u_{n}\}\) is a Cauchy sequence in measure, and therefore there exists a measurable function u such that \(u_{n}\rightarrow u\) in measure. The remainder of the lemma is a consequence of the fact that \(\{T_{k}(u_{n})\}\) is a bounded sequence in \(W^{1,p}(\varOmega )\). □
Proposition 4
There exist a subsequence of \(\{u_{n}\}\) and a measurable function u such that, for each q given in (8), we have
$$\begin{aligned} u_{n}\rightarrow u \quad \textit{strongly in } W^{1,q}( \varOmega ). \end{aligned}$$
Furthermore, if \(0\leq \theta < \min \{ \frac{1}{N-p+1},\frac{N}{N-1}-\frac{1}{p-1},\frac{p-r}{p-1} \}\), then
$$\begin{aligned} \frac{a(x, \nabla u_{n})}{(1+ \vert u_{n} \vert )^{\theta (p-1)}} \rightarrow \frac{a(x, \nabla u)}{(1+ \vert u \vert )^{\theta (p-1)}} \quad \textit{strongly in } \bigl(L^{1}(\varOmega )\bigr)^{N}. \end{aligned}$$
To prove Proposition 4, we need two preliminary lemmas.
There exist a subsequence of \(\{u_{n}\}\) and a measurable function u such that, for each q given in (8), we have \(u_{n}\rightharpoonup u \) weakly in \(W^{1,q}(\varOmega )\), and \(u_{n}\rightarrow u \) strongly in \(L^{q}(\varOmega ) \).
Let \(k>0\) and \(n\geq 1\). Define \(D_{k}=\{ \vert u_{n} \vert \leq k\}\) and \(B_{k}=\{k\leq \vert u_{n} \vert < k+1\}\). Using Lemma 2 with \(v_{0}=g+(\psi -g)^{+}\), we get
$$\begin{aligned} \int _{D_{k}}\frac{ \vert \nabla u_{n} \vert ^{p}}{(1+ \vert u_{n} \vert )^{\theta (p-1)}} \,\mathrm{d}x\leq C \bigl(1+k^{r}\bigr), \end{aligned}$$
where C is a positive constant depending only on α, β, b, p, r, \(\Vert j \Vert _{p'}\), \(\Vert f \Vert _{1}\), \(\Vert \nabla v_{0} \Vert _{p}\), and \(\Vert v_{0} \Vert _{\infty }\).
Using the function \(T_{k}(u_{n})\) for \(k> \{ \Vert g \Vert _{\infty }, \Vert \psi \Vert _{\infty }\}\), as a test function for the problem associated with \((f_{n},\psi ,g)\), we obtain
$$\begin{aligned} & \int _{\varOmega }\frac{a(x,\nabla u_{n})\cdot \nabla (T_{1}(u_{n}-T_{k}(u _{n})))}{(1+ \vert u_{n} \vert )^{\theta (p-1)}}\,\mathrm{d}x+ \int _{\varOmega }b \vert u_{n} \vert ^{r-2}u _{n}T_{1}\bigl(u_{n}-T_{k}(u_{n}) \bigr)\,\mathrm{d}x \\ & \quad \leq \int _{\varOmega }f_{n}T_{1} \bigl(u_{n}-T_{k}(u_{n})\bigr)\, \mathrm{d}x, \end{aligned}$$
which and (2) give
$$\begin{aligned} \int _{B_{k}} \frac{\alpha \vert \nabla u_{n} \vert ^{p}}{(1+ \vert u_{n} \vert )^{\theta (p-1)}}\,\mathrm{d}x+ \int _{\varOmega }b \vert u_{n} \vert ^{r-2} u_{n}T_{1}\bigl(u_{n}-T_{k}(u_{n}) \bigr)\,\mathrm{d}x \leq \Vert f_{n} \Vert _{1}\leq \Vert f \Vert _{1}+1. \end{aligned}$$
Note that on the set \(\{ \vert u_{n} \vert \geq k+1\} \), \(u_{n}\) and \(T_{1}(u_{n}-T _{k}(u_{n}))\) have the same sign. Then
$$\begin{aligned} \int _{\varOmega } \vert u_{n} \vert ^{r-2}u_{n}T_{1}\bigl(u_{n}-T_{k}(u_{n}) \bigr)\,\mathrm{d}x ={}& \int _{D_{k}} \vert u_{n} \vert ^{r-2}u_{n}T_{1}\bigl(u_{n}-T_{k}(u_{n}) \bigr)\,\mathrm{d}x \\ & {} + \int _{B_{k}} \vert u_{n} \vert ^{r-2}u_{n}T_{1}\bigl(u_{n}-T_{k}(u_{n}) \bigr)\,\mathrm{d}x \\ & {} + \int _{\{ \vert u_{n} \vert \geq k+1\}} \vert u_{n} \vert ^{r-2}u_{n}T_{1}\bigl(u_{n}-T_{k}(u _{n})\bigr)\,\mathrm{d}x \\ \geq{}& \int _{B_{k}} \vert u_{n} \vert ^{r-2}u_{n}T_{1}\bigl(u_{n}-T_{k}(u_{n}) \bigr) \,\mathrm{d}x. \end{aligned}$$
Thus we have
$$\begin{aligned} \int _{B_{k}} \frac{\alpha \vert \nabla u_{n} \vert ^{p}}{(1+ \vert u_{n} \vert )^{\theta (p-1)}}\,\mathrm{d}x+ \leq{}& \Vert f \Vert _{1}+1- \int _{B_{k}}b \vert u_{n} \vert ^{r-2}u_{n}T_{1}\bigl(u_{n}-T_{k}(u _{n})\bigr)\,\mathrm{d}x \\ \leq{}& \Vert f \Vert _{1}+1+ \int _{B_{k}}b \vert u_{n} \vert ^{r-1}\,\mathrm{d}x \\ \leq{}& C \biggl(1+ \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q^{*}}\,\mathrm{d}x \biggr)^{\frac{r-1}{q ^{*}}} \vert B_{k} \vert ^{1-\frac{r-1}{q^{*}}} \biggr) , \end{aligned}$$
where q is given in (8) and \(q^{*}=\frac{Nq}{N-q} \).
Let \(s=\frac{q\theta (p-1)}{p}\). Note that \(q< p\) and \(\frac{ps}{p-q}< q ^{*}\). For \(\forall k>0\), we estimate \(\int _{B_{k}} \vert \nabla u_{n} \vert ^{q} \,\mathrm{d}x\) as follows:
$$\begin{aligned} \int _{B_{k}} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x={}& \int _{B_{k}}\frac{ \vert \nabla u _{n} \vert ^{q}}{(1+ \vert u_{n} \vert )^{s}}\cdot \bigl(1+ \vert u_{n} \vert \bigr)^{s}\,\mathrm{d}x \\ \leq{}& \biggl( \int _{B_{k}}\frac{ \vert \nabla u_{n} \vert ^{p}}{(1+ \vert u_{n} \vert )^{ \theta (p-1)}}\,\mathrm{d}x \biggr)^{\frac{q}{p}} \biggl( \int _{B_{k}}\bigl(1+ \vert u _{n} \vert \bigr)^{\frac{ps}{p-q}}\,\mathrm{d}x \biggr)^{\frac{p-q}{p}} \\ \leq{}& C \biggl( \int _{B_{k}}\frac{ \vert \nabla u_{n} \vert ^{p}}{(1+ \vert u_{n} \vert )^{ \theta (p-1)}}\,\mathrm{d}x \biggr)^{\frac{q}{p}} \biggl( \vert B_{k} \vert ^{ \frac{p-q}{p}}+ \biggl( \int _{B_{k}} \vert u_{n} \vert ^{\frac{ps}{p-q}}\,\mathrm{d}x \biggr)^{\frac{p-q}{p}} \biggr) \\ \leq{}& C \biggl( \int _{B_{k}}\frac{ \vert \nabla u_{n} \vert ^{p}}{(1+ \vert u_{n} \vert )^{ \theta (p-1)}}\,\mathrm{d}x \biggr)^{\frac{q}{p}} \biggl( \vert B_{k} \vert ^{ \frac{p-q}{p}} + \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q^{*}}\,\mathrm{d}x \biggr)^{\frac{s}{q ^{*}}} \vert B_{k} \vert ^{\frac{p-q}{p}-\frac{s}{q^{*}}} \biggr) \\ \leq{}& C \biggl( \vert B_{k} \vert ^{\frac{p-q}{p}}+ \vert B_{k} \vert ^{\frac{p-q}{p}-\frac{s}{q ^{*}}} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q^{*}}\,\mathrm{d}x \biggr)^{ \frac{s}{q^{*}}}+ \vert B_{k} \vert ^{1-p_{1}} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \biggr)^{p_{1}} \\ & {} + \vert B_{k} \vert ^{1-p_{2}} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q^{*}}\,\mathrm{d}x \biggr)^{p _{2}} \biggr) \quad \text{by (25)} \\ ={}& C \biggl( \vert B_{k} \vert ^{\frac{p-q}{p}}+ \vert B_{k} \vert ^{\frac{p-q}{p}-\frac{s}{q ^{*}}} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q^{*}}\,\mathrm{d}x \biggr)^{\frac{s}{q^{*}}} \\ & {} + \vert B_{k} \vert ^{1-p_{1}-C_{1}} \vert B_{k} \vert ^{C_{1}} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q ^{*}}\,\mathrm{d}x \biggr)^{p_{1}} \\ & {} + \vert B_{k} \vert ^{1-p_{2}-C_{2}} \vert B_{k} \vert ^{C_{2}} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q ^{*}}\,\mathrm{d}x \biggr)^{p_{2}} \biggr), \end{aligned}$$
where \(p_{1}=\frac{q}{p}\frac{r-1}{q^{*}}\), \(p_{2}=\frac{s}{q^{*}}+ \frac{q}{p}\frac{r-1}{q^{*}}\), \(C_{1}\) and \(C_{2}\) are positive constants to be chosen later.
Note that \(\theta <\frac{p-r}{p-1}\), it follows
$$\begin{aligned} \frac{\theta (p-1)}{p}+\frac{r-1}{p}< \frac{p-1}{p}< 1- \frac{1}{N}=1- \frac{1}{q}+\frac{1}{q^{*}}. \end{aligned}$$
$$\begin{aligned} \frac{\theta q(p-1)}{p}+\frac{q(r-1)}{p}+1< q+\frac{q}{q^{*}} \quad &\Leftrightarrow\quad s+\frac{q(r-1)}{p}+1< q+\frac{q}{q^{*}} \\ & \Leftrightarrow\quad p_{2}+\frac{1-p _{2}}{q^{*}+1}< \frac{q}{q^{*}}. \end{aligned}$$
Note that \(p_{1}< p_{2}<1\). Then, for \(i=1,2\), we always have
$$\begin{aligned} p_{i}+\frac{1-p_{i}}{q^{*}+1}< \frac{q}{q^{*}}< 1. \end{aligned}$$
From this, we may find positive \(C_{i}\) (\(i=1,2\)) such that
$$\begin{aligned} p_{i}+\frac{1-p_{i}}{q^{*}+1}< p_{i}+C_{i}< \frac{q}{q^{*}}< 1, \quad i=1,2. \end{aligned}$$
$$\begin{aligned} \frac{1-p_{i}}{q^{*}+1}< C_{i}\quad \Leftrightarrow \quad 1-p_{i}-C_{i}< C_{i}q^{*}, \quad i=1,2, \end{aligned}$$
which implies
$$\begin{aligned} C_{i}\alpha _{i}q^{*}= \frac{C_{i}q^{*}}{1-p_{i}-C_{i}}>1, \quad i=1,2, \end{aligned}$$
with \(\alpha _{i}=\frac{1}{1-p_{i}-C_{i}}>1\), \(i=1,2\). Let \(\beta _{i}=\frac{1}{p _{i}+C_{i}}>1\), \(i=1,2\). Then we have \(\frac{1}{\alpha _{i}}+\frac{1}{ \beta _{i}}=1\) (\(i=1,2\)).
Since \(\vert B_{k} \vert \leq \frac{1}{k^{q^{*}}}\int _{B_{k}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \), \(\vert B_{k} \vert ^{1-p_{1}-C_{1}} \leq \vert \varOmega \vert ^{1-p_{1}-C_{1}}\), and \(\vert B_{k} \vert ^{1-p_{2}-C_{2}} \leq \vert \varOmega \vert ^{1-p_{2}-C_{2}}\), we have, for \(k\geq k_{0}\geq 1\),
$$\begin{aligned} \int _{B_{k}} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x\leq{}&\frac{C}{k^{q^{*} ( \frac{p-q}{p}-\frac{s}{q^{*}} )}} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \biggr)^{\frac{p-q}{p}} \\ & {} +\frac{C}{k^{q^{*}C_{1}}} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q^{*}}\,\mathrm{d}x \biggr)^{p_{1}+C_{1}}+\frac{C}{k^{q^{*}C_{2}}} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q ^{*}}\,\mathrm{d}x \biggr)^{p_{2}+C_{2}}. \end{aligned}$$
Summing up from \(k=k_{0}\) to \(k=K\) and using Hölder's inequality, one has
$$\begin{aligned} \sum_{k=k_{0}}^{K} \int _{B_{k}} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x\leq{}& C \Biggl(\sum_{k=k_{0}}^{K} \frac{1}{k^{q^{*}(\frac{p-q}{p}-\frac{s}{q ^{*}})\frac{p}{q}}} \Biggr)^{\frac{q}{p}}\cdot \Biggl(\sum _{k=k_{0}} ^{K} \int _{B_{k}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \Biggr)^{\frac{p-q}{p}} \\ & {} +C \Biggl(\sum_{k=k_{0}}^{K} \frac{1}{k^{q^{*}C_{1}\alpha _{1}}} \Biggr)^{\frac{1}{ \alpha _{1}}} \cdot \Biggl(\sum _{k=k_{0}}^{K} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q ^{*}} \,\mathrm{d}x \biggr)^{\beta _{1}(p_{1}+C_{1})} \Biggr)^{\frac{1}{\beta _{1}}} \\ & {} +C \Biggl(\sum_{k=k_{0}}^{K} \frac{1}{k^{q^{*}C_{2}\alpha _{2}}} \Biggr)^{\frac{1}{ \alpha _{2}}} \cdot \Biggl(\sum _{k=k_{0}}^{K} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q ^{*}} \,\mathrm{d}x \biggr)^{\beta _{2}(p_{2}+C_{2})} \Biggr)^{\frac{1}{\beta _{2}}} \\ ={}& C \Biggl(\sum_{k=k_{0}}^{K} \frac{1}{k^{q^{*}(\frac{p-q}{p}-\frac{s}{q ^{*}})\frac{p}{q}}} \Biggr)^{\frac{q}{p}}\cdot \Biggl(\sum _{k=k_{0}} ^{K} \int _{B_{k}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \Biggr)^{\frac{p-q}{p}} \\ & {} +C \Biggl(\sum_{k=k_{0}}^{K} \frac{1}{k^{q^{*}C_{1}\alpha _{1}}} \Biggr)^{\frac{1}{ \alpha _{1}}} \cdot \Biggl(\sum _{k=k_{0}}^{K} \int _{B_{k}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \Biggr)^{p_{1}+C_{1}} \\ & {} +C \Biggl(\sum_{k=k_{0}}^{K} \frac{1}{k^{q^{*}C_{2}\alpha _{2}}} \Biggr)^{\frac{1}{ \alpha _{2}}} \cdot \Biggl(\sum _{k=k_{0}}^{K} \int _{B_{k}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \Biggr)^{p_{2}+C_{2}}. \end{aligned}$$
$$\begin{aligned} \int _{\{ \vert u_{n} \vert \leq K\}} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x= \int _{D_{k_{0}}} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x+\sum_{k=k_{0}}^{K} \int _{B_{k}} \vert \nabla u _{n} \vert ^{q} \,\mathrm{d}x. \end{aligned}$$
To estimate the first integral in the right-hand side of (29), we compute by using Hölder's inequality and (24), obtaining
$$\begin{aligned} \int _{D_{k_{0}}} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x \leq{}& \biggl( \int _{D_{k _{0}}}\frac{ \vert \nabla u_{n} \vert ^{p}}{(1+ \vert u_{n} \vert )^{\theta (p-1)}}\,\mathrm{d}x \biggr)^{\frac{q}{p}} \biggl( \int _{D_{k_{0}}}\bigl(1+ \vert u_{n} \vert \bigr)^{ \frac{ps}{p-q}} \,\mathrm{d}x \biggr)^{\frac{p-q}{p}} \\ \leq{}& C, \end{aligned}$$
where C depends only on α, β, b, p, θ, \(\Vert j \Vert _{p'}\), \(\Vert f \Vert _{1}\), \(\Vert \nabla v_{0} \Vert _{p}\), \(\Vert v_{0} \Vert _{\infty }\), and \(k_{0}\).
Note that \(\sum_{k=k_{0}}^{K} \frac{1}{k^{q^{*}(\frac{p-q}{p}-\frac{s}{q ^{*}})\frac{p}{q}}}\) and \(\sum_{k=k_{0}}^{K} \frac{1}{k^{q^{*}C _{i}\alpha _{i}}}\) converge as \(K\rightarrow \infty \) due to the fact that \(q^{*}(\frac{p-q}{p}-\frac{s}{q^{*}})\frac{p}{q}>1\) and \(q^{*}C_{i} \alpha _{i}>1\) by (27), respectively. Combining (28)–(30), we get for \(k_{0}\) large enough
$$\begin{aligned} \int _{\{ \vert u_{n} \vert \leq K\}} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x\leq{}& C+C \biggl( \int _{\{ \vert u_{n} \vert \leq K\}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \biggr)^{ \frac{p-q}{p}} \\ & {} + C \biggl( \int _{\{ \vert u_{n} \vert \leq K\}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \biggr)^{p _{1}+C_{1}} \\ & {} +C \biggl( \int _{\{ \vert u_{n} \vert \leq K\}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \biggr)^{p _{2}+C_{2}}. \end{aligned}$$
Since \(p>q\), \(T_{K}(u_{n})\in W^{1,q}(\varOmega )\), \(T_{K}(g)=g\in W^{1,q}( \varOmega )\) for \(K> \Vert g \Vert _{\infty }\). Hence \(T_{K}(u_{n})-g\in W^{1,q} _{0}(\varOmega )\). Using the Sobolev embedding \(W^{1,q}_{0}(\varOmega ) \subset L^{q^{*}}(\varOmega )\) and Poincaré's inequality, we obtain
$$\begin{aligned} \bigl\Vert T_{K}(u_{n}) \bigr\Vert ^{q}_{q^{*}}\leq{}& 2^{q-1}\bigl( \bigl\Vert T_{K}(u_{n})-g \bigr\Vert ^{q}_{q ^{*}}+ \Vert g \Vert ^{q}_{q^{*}}\bigr) \\ \leq{}& C\bigl( \bigl\Vert \nabla \bigl(T_{K}(u_{n})-g \bigr) \bigr\Vert ^{q}_{q}+ \Vert g \Vert ^{q}_{q^{*}}\bigr) \\ \leq{}& C \bigl( \bigl\Vert \nabla T_{K}(u_{n}) \bigr\Vert ^{q}_{q}+ \Vert \nabla g \Vert ^{q}_{q}+ \Vert g \Vert ^{q}_{q^{*}} \bigr) \\ \leq{}& C \biggl(1+ \int _{\{ \vert u_{n} \vert \leq K\}} \vert \nabla u_{n} \vert ^{q} \,\mathrm{d}x \biggr). \end{aligned}$$
Using the fact that
$$\begin{aligned} \int _{\{ \vert u_{n} \vert \leq K\}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x\leq \int _{\{ \vert u_{n} \vert \leq K\}} \bigl\vert T_{K}( u_{n}) \bigr\vert ^{q^{*}} \,\mathrm{d}x\leq \bigl\Vert T _{K}( u_{n}) \bigr\Vert ^{q^{*}}_{q^{*}}, \end{aligned}$$
we obtain from (31)–(32)
$$\begin{aligned} \int _{\{ \vert u_{n} \vert \leq K\}} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x\leq{}& C+C \biggl(1+ \int _{\{ \vert u_{n} \vert \leq K\}} \vert \nabla u_{n} \vert ^{q} \,\mathrm{d}x \biggr)^{\frac{q ^{*}}{q}\frac{p-q}{p}} \\ & {} +C \biggl(1+ \int _{\{ \vert u_{n} \vert \leq K\}} \vert \nabla u_{n} \vert ^{q} \,\mathrm{d}x \biggr)^{(p_{1}+C_{1})\frac{q^{*}}{q}} \\ & {} +C \biggl(1+ \int _{\{ \vert u_{n} \vert \leq K\}} \vert \nabla u_{n} \vert ^{q} \,\mathrm{d}x \biggr)^{(p_{2}+C_{2})\frac{q^{*}}{q}} . \end{aligned}$$
Note that \(p< N\Leftrightarrow \frac{q^{*}}{q}\frac{p-q}{p}<1\) and \((p_{i}+C_{i})\frac{q^{*}}{q}<1 \) by (26). It follows from (34) that, for \(k_{0}\) large enough, \(\int _{\{ \vert u_{n} \vert \leq K\}} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x \) is bounded independently of n and K. Using (32) and (33), we deduce that \(\int _{\{ \vert u_{n} \vert \leq K\}} \vert u_{n} \vert ^{q^{*}}\,\mathrm{d}x\) is also bounded independently of n and K. Letting \(K\rightarrow \infty \), we deduce that \(\Vert \nabla u_{n} \Vert _{q}\) and \(\Vert u_{n} \Vert _{q^{*}}\) are uniformly bounded independently of n. Particularly, \(u_{n}\) is bounded in \(W^{1,q}(\varOmega )\). Therefore, there exist a subsequence of \(\{u_{n}\}\) and a function \(v\in W^{1,q}(\varOmega )\) such that \(u_{n}\rightharpoonup v\) weakly in \(W^{1,q}(\varOmega )\), \(u_{n}\rightarrow v\) strongly in \(L^{q}(\varOmega )\) and a.e. in Ω. By Lemma 3, \(u_{n}\rightarrow u\) in measure in Ω, we conclude that \(u=v\) and \(u\in W^{1,q}(\varOmega )\). □
There exist a subsequence of \(\{u_{n}\}\) and a measurable function u such that \(\nabla u_{n}\) converges almost everywhere in Ω to ∇u.
Define \(A(x,u,\xi )=\frac{a(x,\xi )}{(1+ \vert u \vert )^{\theta (p-1)}}\) (for the sake of simplicity, we omit the dependence of \(A(x,u,\xi )\) on x). Let \(h>0\), \(k> \max \{ \Vert g \Vert _{\infty }, \Vert \psi \Vert _{\infty }\}\), and \(n\geq h+k\). Take \(T_{k}(u)\) as a test function for (7), obtaining
$$\begin{aligned} I_{7}(n,k,h)\leq \int _{\varOmega }f_{n}T_{h} \bigl(u_{n}-T_{k}(u)\bigr)\,\mathrm{d}x+ \int _{\varOmega }b \vert u_{n} \vert ^{r-2}u_{n}T_{h}\bigl(u_{n}-T_{k}(u) \bigr)\,\mathrm{d}x, \end{aligned}$$
$$\begin{aligned} I_{7}(n,k,h)= \int _{\varOmega }A(u_{n},\nabla u_{n}) \cdot \nabla T_{h}\bigl(u_{n}-T_{k}(u) \bigr)\,\mathrm{d}x. \end{aligned}$$
Note that \(r-1< q^{*}\), and \(\int _{\varOmega } \vert u_{n} \vert ^{q^{*}}\,\mathrm{d}x\) is uniformly bounded (see the proof of Lemma 5), thus \(\vert u_{n} \vert \) converges strongly in \(L^{1}(\varOmega )\). Therefore we have
$$\begin{aligned} \lim_{n\rightarrow \infty } \int _{\varOmega } \vert u_{n} \vert ^{r-2}u_{n}T_{h}\bigl(u_{n}-T_{k}(u) \bigr)\,\mathrm{d}x= \int _{\varOmega } \vert u \vert ^{r-2}uT_{h} \bigl(u-T_{k}(u)\bigr)\,\mathrm{d}x. \end{aligned}$$
Then, using the strong convergence of \(f_{n}\) in \(L^{1}(\varOmega )\), one has
$$\begin{aligned} \lim_{n\rightarrow \infty }I_{7}(n,k,h) \leq \int _{\varOmega }-fT_{h}\bigl(u-T_{k}(u) \bigr)\,\mathrm{d}x+ \int _{\varOmega }b \vert u \vert ^{r-2}uT_{h} \bigl(u-T_{k}(u)\bigr)\,\mathrm{d}x. \end{aligned}$$
$$\begin{aligned} \lim_{k\rightarrow \infty }\lim_{n\rightarrow \infty }I_{7}(n,k,h) \leq 0. \end{aligned}$$
Thanks to Lemma 3 and Lemma 5, we can proceed exactly as [19, Lemma 6] to conclude that, up to subsequence, \(\nabla u_{n}\rightarrow \nabla u\) a.e. □
Proof of Proposition 4
We shall prove that \(\nabla u_{n}\) converges strongly to ∇u in \(L^{q}(\varOmega )\) for each q being given by (8). To do that,we will apply Vitali's theorem, using the fact that by Lemma 5, \(\nabla u_{n}\) is bounded in \(L^{q}(\varOmega )\) for each q given by (8). So let \(s\in (q,\frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)})\) and \(E\subset \varOmega \) be a measurable set. Then we have by Hölder's inequality
$$\begin{aligned} \int _{E} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x\leq \biggl( \int _{E} \vert \nabla u_{n} \vert ^{r} \,\mathrm{d}x \biggr)^{\frac{q}{s}}\cdot \vert E \vert ^{\frac{s-q}{s}}\leq C \vert E \vert ^{ \frac{s-q}{s}}\rightarrow 0 \end{aligned}$$
uniformly in n, as \(\vert E \vert \rightarrow 0\). From this and from Lemma 6, we deduce that \(\nabla u_{n}\) converges strongly to ∇u in \(L^{q}(\varOmega )\).
Now assume that \(0\leq \theta < \min \{\frac{1}{N-p+1},\frac{N}{N-1}- \frac{1}{p-1},\frac{p-r}{p-1}\}\). Note that since \(\nabla u_{n}\) converges to ∇u a.e. in Ω, to prove the convergence
$$\begin{aligned} \frac{a(x, \nabla u_{n})}{(1+ \vert u_{n} \vert )^{\theta (p-1)}} \rightarrow \frac{a(x, \nabla u)}{(1+ \vert u \vert )^{\theta (p-1)}} \quad \text{strongly in } \bigl(L^{1}(\varOmega )\bigr)^{N}, \end{aligned}$$
it suffices, thanks to Vitali's theorem, to show that, for every measurable subset \(E\subset \varOmega \), \(\int _{E} \vert \frac{a(x, \nabla u_{n})}{(1+ \vert u_{n} \vert )^{\theta (p-1)}} \vert \,\mathrm{d}x\) converges to 0 uniformly in n, as \(\vert E \vert \rightarrow 0\). Note that \(p-1<\frac{N(p-1)(1- \theta )}{N-1-\theta (p-1)})\) by assumptions. For any \(q\in (p-1, \frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)} )\), we deduce by Hölder's inequality
$$\begin{aligned} \int _{E} \biggl\vert \frac{a(x, \nabla u_{n})}{(1+ \vert u_{n} \vert )^{\theta (p-1)}} \biggr\vert \,\mathrm{d}x\leq{}& \beta \int _{E}\bigl(j+ \vert \nabla u_{n} \vert ^{p-1}\bigr)\,\mathrm{d}x \\ \leq{}& \beta \Vert j \Vert _{p'} \vert E \vert ^{\frac{1}{p}}+\beta \biggl( \int _{E} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x \biggr)^{\frac{p-1}{q}} \vert E \vert ^{\frac{q-p+1}{q}} \\ \rightarrow &0 \quad \text{uniformly in } n \text{ as } \vert E \vert \rightarrow 0. \end{aligned}$$
There exists a subsequence of \(\{u_{n}\}\) such that, for all \(k>0\),
$$\begin{aligned} \frac{a(x, \nabla T_{k}(u_{n}))}{(1+ \vert T_{k}(u_{n}) \vert )^{\theta (p-1)}} \rightarrow \frac{a(x, \nabla T_{k}(u))}{(1+ \vert T_{k}(u) \vert )^{\theta (p-1)}} \quad \textit{strongly in } \bigl(L^{1}(\varOmega )\bigr)^{N}. \end{aligned}$$
See the proof of [19, Lemma 7]. □
3 Proof of the main result
Now we have gathered all the lemmas needed to prove the existence of an entropy solution to the obstacle problem associated with \((f,\psi ,g)\). In this part, let \(f_{n}\) be a sequence of smooth functions converging strongly to f in \(L^{1}(\varOmega )\), with \(\Vert f_{n} \Vert _{1}\leq \Vert f \Vert _{1}+1\). We consider the sequence of approximated obstacle problems associated with \((f_{n},\psi ,g)\). The proof can be proceeded in the same way as in [8] and [19]. We provide details for readers' convenience.
Proof of Theorem 1
Let \(v\in K_{g, \psi }\cap L^{\infty }(\varOmega )\). Taking v as a test function in (7) associated with \((f_{n},\psi ,g)\), we get
$$\begin{aligned} & \int _{\varOmega }\frac{a(x, \nabla u_{n})}{(1+ \vert u_{n} \vert )^{\theta (p-1)}} \cdot \nabla \bigl(T_{t}(u_{n}-v)\bigr)\,\mathrm{d}x+ \int _{\varOmega }b \vert u_{n} \vert ^{r-2}u _{n}T_{t}(u_{n}-v)\, \mathrm{d}x \\ & \quad \leq \int _{\varOmega }f_{n}T_{t}(u_{n}-v) \,\mathrm{d}x. \end{aligned}$$
Since \(\{ \vert u_{n}-v \vert < t\}\subset \{ \vert u_{n} \vert < s\}\) with \(s=t+ \Vert v \Vert _{\infty }\), the previous inequality can be written as
$$\begin{aligned} \int _{\varOmega }\chi _{n}\nabla _{A} T_{s}(u_{n})\cdot \nabla v\,\mathrm{d}x \geq{}& \int _{\varOmega }-f_{n}T_{t}(u_{n}-v) \,\mathrm{d}x+ \int _{\varOmega }b \vert u _{n} \vert ^{r-2}u_{n}T_{t}(u_{n}-v)\, \mathrm{d}x \\ & {} + \int _{\varOmega }\chi _{n}\nabla _{A} T_{s}(u_{n})\cdot \nabla T_{s}(u _{n})\,\mathrm{d}x, \end{aligned}$$
where \(\chi _{n}=\chi _{\{ \vert u_{n}-v \vert < t\}}\) and \(\nabla _{A}u=\frac{a(x, \nabla u)}{(1+ \vert u \vert )^{\theta (p-1)}}\). It is clear that \(\chi _{n} \rightharpoonup \chi \) weakly* in \(L^{\infty }(\varOmega )\). Moreover, \(\chi _{n}\) converges a.e. to \(\chi _{\{ \vert u-v \vert < t\}}\) in \(\varOmega \setminus \{ \vert u-v \vert =t\}\). It follows that
$$\begin{aligned} \chi =\textstyle\begin{cases} 1, & \text{in } \{ \vert u-v \vert < t\}, \\ 0, & \text{in } \{ \vert u-v \vert >t\}. \end{cases}\displaystyle \end{aligned}$$
Note that we have \(\mathcal{L}^{N}(\{ \vert u-v \vert =t\})=0\) for a.e. \(t\in (0,\infty )\). So there exists a measurable set \(\mathcal{O} \subset (0,\infty )\) such that \(\mathcal{L}^{N}(\{ \vert u-v \vert =t\})=0\) for all \(t\in (0,\infty )\setminus \mathcal{O}\). Assume that \(t\in (0,\infty )\setminus \mathcal{O}\). Then \(\chi _{n}\) converges weakly* in \(L^{\infty }(\varOmega )\) and a.e. in Ω to \(\chi =\chi _{\{ \vert u-v \vert < t \}}\). Since \(\nabla T_{s}(u_{n})\) converges a.e. to \(\nabla T_{s}(u)\) in Ω (Proposition 4), we obtain by Fatou's lemma
$$\begin{aligned} \liminf_{n\rightarrow \infty } \int _{\varOmega }\chi _{n}\nabla _{A} T_{s}(u _{n})\cdot \nabla T_{s}(u_{n}) \,\mathrm{d}x\geq \int _{\varOmega }\chi \nabla _{A} T_{s}(u) \cdot \nabla T_{s}(u)\,\mathrm{d}x. \end{aligned}$$
Using the strong convergence of \(\nabla _{A} T_{s}(u_{n})\) to \(\nabla _{A} T_{s}(u)\) in \(L^{1}(\varOmega )\) (Lemma 7) and the weak* convergence of \(\chi _{n}\) to χ in \(L^{\infty }(\varOmega )\), we obtain
$$\begin{aligned} \lim_{n\rightarrow \infty } \int _{\varOmega }\chi _{n}\nabla _{A} T_{s}(u _{n})\cdot \nabla v\,\mathrm{d}x= \int _{\varOmega }\chi \nabla _{A} T_{s}(u) \cdot \nabla v\,\mathrm{d}x. \end{aligned}$$
Moreover, due to the strong convergence of \(f_{n}\) to f and \(\vert u_{n} \vert ^{r-2}u_{n} \) to \(\vert u \vert ^{r-2}u\) (by \(r-1< q^{*}\) and the boundedness of \(\Vert u_{n} \Vert _{q^{*}}\)) in \(L^{1}(\varOmega )\), and the weak* convergence of \(T_{t}(u_{n}-v)\) to \(T_{t}(u-v)\) in \(L^{\infty }( \varOmega )\), by passing to the limit in (35) and taking into account (36)–(37), we obtain
$$\begin{aligned} \int _{\varOmega }\chi \nabla _{A} T_{s}(u) \cdot \nabla v\,\mathrm{d}x- \int _{\varOmega }\chi \nabla _{A} T_{s}(u) \cdot \nabla T_{s}(u)\,\mathrm{d}x \geq{}& \int _{\varOmega }-fT_{t}(u-v)\,\mathrm{d}x \\ & {} + \int _{\varOmega }b \vert u \vert ^{r-2}uT_{t}(u-v) \,\mathrm{d}x, \end{aligned}$$
which can be written as
$$\begin{aligned} \int _{\{ \vert v-u \vert \leq t\}}\chi \nabla _{A} T_{s}(u)\cdot (\nabla v-\nabla u)\,\mathrm{d}x\geq{}& \int _{\varOmega }-fT_{t}(u-v)\,\mathrm{d}x \\ & {} + \int _{\varOmega }b \vert u \vert ^{r-2}uT_{t}(u-v) \,\mathrm{d}x, \end{aligned}$$
or since \(\chi =\chi _{\{ \vert u-v \vert < t\}}\) and \(\nabla (T_{t}(u-v))= \chi _{\{ \vert u-v \vert < t\}}\nabla (u-v)\)
$$\begin{aligned} & \int _{\varOmega }\nabla _{A} u\cdot \nabla T_{t}( u-v)\,\mathrm{d}x+ \int _{\varOmega }b \vert u \vert ^{r-2}uT_{t}(u-v) \,\mathrm{d}x \\ & \quad \leq \int _{\varOmega }fT_{t}(u-v)\,\mathrm{d}x,\forall t\in (0,\infty ) \setminus \mathcal{O}. \end{aligned}$$
For \(t\in \mathcal{O}\), we know that there exists a sequence \(\{t_{k}\}\) of numbers in \((0,\infty )\setminus \mathcal{O}\) such that \(t_{k}\rightarrow t\) due to \(\vert \mathcal{O} \vert =0\). Therefore, we have
$$\begin{aligned} \int _{\varOmega }\nabla _{A} u\cdot \nabla T_{t_{k}}( u-v)\,\mathrm{d}x+ \int _{\varOmega }b \vert u \vert ^{r-2}uT_{t_{k}}(u-v) \,\mathrm{d}x\leq \int _{\varOmega }fT _{t_{k}}(u-v)\,\mathrm{d}x. \end{aligned}$$
Since \(\nabla (u-v)=0\) a.e. in \(\{ \vert u-v \vert =t\} \), the left-hand side of (38) can be written as
$$\begin{aligned} \int _{\varOmega }\nabla _{A} u\cdot \nabla T_{t_{k}}( u-v)\,\mathrm{d}x= \int _{\varOmega \setminus \{ \vert u-v \vert =t\}}\chi _{\{ \vert u-v \vert < t_{k}\}}\nabla _{A} u \cdot \nabla ( u-v)\,\mathrm{d}x. \end{aligned}$$
The sequence \(\chi _{\{ \vert u-v \vert < t_{k}\}}\) converges to \(\chi _{\{ \vert u-v \vert < t \}}\) a.e. in \(\varOmega \setminus \{ \vert u-v \vert =t\} \) and therefore converges weakly* in \(L^{\infty }(\varOmega \setminus \{ \vert u-v \vert =t\})\). We obtain
$$\begin{aligned} \lim_{k\rightarrow \infty } \int _{\varOmega }\nabla _{A} u\cdot \nabla T _{t_{k}}( u-v)\,\mathrm{d}x={}& \int _{\varOmega \setminus \{ \vert u-v \vert =t\}} \chi _{\{ \vert u-v \vert < t\}}\nabla _{A} u\cdot \nabla ( u-v)\,\mathrm{d}x \\ ={}& \int _{\varOmega }\chi _{\{ \vert u-v \vert < t\}}\nabla _{A} u \cdot \nabla ( u-v) \,\mathrm{d}x \\ ={}& \int _{\varOmega }\nabla _{A} u\cdot \nabla T_{t}(u-v)\,\mathrm{d}x. \end{aligned}$$
For the right-hand side of (38), we have
$$\begin{aligned} \biggl\vert \int _{\varOmega }fT_{t_{k}}(u-v)\,\mathrm{d}x- \int _{\varOmega }fT_{t}(u-v) \,\mathrm{d}x \biggr\vert \leq \vert t_{k}-t \vert \cdot \Vert f \Vert _{1}\rightarrow 0 \quad \text{as }k\rightarrow \infty . \end{aligned}$$
Similarly, we have
$$\begin{aligned} \biggl\vert \int _{\varOmega } \vert u \vert ^{r-2}uT_{t_{k}}(u-v) \,\mathrm{d}x- \int _{\varOmega } \vert u \vert ^{r-2}uT _{t}(u-v)\,\mathrm{d}x \biggr\vert &\leq \vert t_{k}-t \vert \cdot \bigl\Vert \vert u \vert ^{r-1} \bigr\Vert _{1} \\ &\rightarrow 0 \quad \text{as } k\rightarrow \infty . \end{aligned}$$
It follows from (38)–(41) that we have the inequality
$$\begin{aligned} & \int _{\varOmega }\nabla _{A} u\cdot \nabla T_{t}( u-v)\,\mathrm{d}x+ \int _{\varOmega }b \vert u \vert ^{r-2}uT_{t}(u-v) \,\mathrm{d}x \\ & \quad \leq \int _{\varOmega }fT_{t}(u-v)\,\mathrm{d}x,\quad \forall t\in (0,\infty ). \end{aligned}$$
Hence, u is an entropy solution of the obstacle problem associated with \((f,\psi ,g)\). The dependence of the entropy solution on the data \(f\in L^{1}(\varOmega )\) is guaranteed by Proposition 4. □
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The author would like to thank the reviewers for their valuable suggestions and comments to improve the quality of this paper.
This research was partially supported by the Fundamental Research Funds for the Central Universities: No. 2682019LK01.
School of Mathematics, Southwest Jiaotong University, Chengdu, China
Jun Zheng
This paper was completed by JZ independently. All authors read and approved the final manuscript.
Correspondence to Jun Zheng.
The author declares that he has no competing interests.
Zheng, J. The obstacle problem for non-coercive equations with lower order term and \(L^{1}\)-data. J Inequal Appl 2019, 205 (2019). https://doi.org/10.1186/s13660-019-2157-9
35J87
Obstacle problem
Non-coercive equation
Entropy solution
\(L^{1}\)-data
Lower order term | CommonCrawl |
\begin{document}
\begin{abstract}
Let $X$ be a smooth projective curve of genus $g\geq 2$ over the
complex numbers. A holomorphic triple $(E_1,E_2,\phi)$ on $X$ consists of two
holomorphic vector bundles $E_{1}$ and $E_{2}$ over $X$ and a
holomorphic map $\phi \colon E_{2} \to E_{1}$.
There is a concept of stability for triples which depends on a
real parameter $\sigma$.
In this paper, we determine the Hodge polynomials of the moduli
spaces of $\sigma$-stable triples with $\rk(E_1)=\rk(E_2)=2$, using the theory of mixed Hodge
structures (in the cases that they are smooth and compact).
This gives in particular the Poincar{\'e} polynomials
of these moduli spaces.
As a byproduct, we also give the Hodge
polynomial of the moduli space of even degree rank $2$ stable
vector bundles. \end{abstract}
\maketitle
\section{Introduction} \label{sec:introduction}
Let $X$ be a smooth projective curve of genus $g\geq 2$ over the field of complex numbers. A holomorphic triple $T = (E_{1},E_{2},\phi)$ on $X$ of rank $(n_1,n_2)$ consists of two holomorphic vector bundles $E_{1}$ and $E_{2}$ over $X$ (of ranks $n_1$ and $n_2$, and degrees $d_1$ and $d_2$, respectively) and a holomorphic map $\phi \colon E_{2} \to E_{1}$. There is a concept of stability for a triple which depends on the choice of a parameter $\sigma \in \mathbb{R}$. Let $\cN_\s$ and $\cN_\s^s$ denote the moduli spaces of $\sigma$-semistable and $\sigma$-stable triples, respectively. These have been widely studied in \cite{BGP,BGPG,GPGM,MOV}.
The range of the parameter $\sigma$ is an interval $I\subset \mathbb{R}$ split by a finite number of \textit{critical values} $\sigma_c$ in such a way that, when $\sigma$ moves without crossing a critical value, then $\cN_\s$ remains unchanged, but when $\sigma$ crosses a critical value, $\cN_\s$ undergoes a transformation which we call \textit{flip}. The study of this process allows us to obtain information on the topology of all moduli spaces $\cN_\s$, for any $\sigma$, once we know such information for one particular $\cN_\s$ (usually the one corresponding to the minimum or maximum possible value of the parameter).
One of the main motivations to study the topology of the moduli spaces of triples is that they appear when looking at the topology of the moduli spaces of Higgs bundles \cite{Hi,Go,GPGM} via Morse theory techniques. Higgs bundles are pairs $(E,\Phi)$, formed by a holomorphic vector bundle $E$ of rank $r$ and a holomorphic map $\Phi:E\to E\otimes K$, where $K$ is the canonical bundle of the curve, and they are intimately related to the representation varieties of the fundamental group of the surface underlying the complex curve into the general Lie group $\GL(r,\mathbb{C})$. The moduli spaces of triples, and the more general moduli spaces of chains \cite{AG,GK,AGS} appear as critical sets of a natural Morse-Bott function on the moduli space of Higgs bundles \cite{Hi,Go}.
When the rank of $E_2$ is one, we have the so-called \textit{pairs} \cite{BD,GP,MOV}. The moduli spaces of pairs are smooth for any rank $n_1$, and in the case of rank $n_1=2$ and fixed determinant, they are very well-understood thanks to the work of Thaddeus \cite{Th}. In this case, the flips have a very nice geometrical interpretation, consisting of blowing up an embedded subvariety and then blowing-down the exceptional divisor in a different way. Moreover, there are also very explicit descriptions of the moduli spaces of pairs for the minimum and maximum possible values of $\sigma$.
The flips do not have such a nice behavior for moduli spaces of triples of rank $(n_1,n_2)$ with \mbox{$n_1+n_2>3$}. The flip locus may have singularities, it may consist of several irreducible components intersecting in a non-transverse way, the moduli spaces themselves may have singularities for \mbox{$n_1,n_2\geq 2$}, and the moduli spaces for $\sigma$ large are difficult to handle in the situation when $n_1=n_2$, since then they are described in terms of Quot schemes.
These difficulties can be overcome in two different ways. The first way is to introduce parabolic structures with generic weights. The moduli spaces of parabolic triples have been studied in \cite{GPGM}, where the Poincar{\'e} polynomials have been given for the moduli of parabolic triples of ranks $(2,1)$. The parabolic weights tend to prevent the singularities of the moduli spaces and flip loci. However, for obtaining information on the moduli space of non-parabolic triples, one should relate the parabolic and the non-parabolic situations.
The second route to compute the Poincar{\'e} polynomials of the moduli spaces of triples was introduced in \cite{MOV}. It consists of using the theory of mixed Hodge structures of Deligne \cite{De} to compute the Hodge polynomials of the moduli space. The Hodge polynomials recover the usual Poincar{\'e} polynomial when we deal with a smooth compact algebraic variety, but they can be defined for non-smooth and non-compact algebraic varieties as well. This allows to compute the Poincar{\'e} polynomials of the moduli spaces of triples which are smooth and compact, no matter if the flip loci have singularities.
In this paper, we use mixed Hodge theory to compute the Hodge polynomials of some of the moduli spaces of triples of rank $(2,2)$. By the results of \cite{BGPG}, if $d_1-d_2 > 4g-4$ then $\mathcal{N}_\sigma^s$ is smooth. Moreover, when $d_1+d_2$ is odd, the moduli spaces $\mathcal{N}_\sigma$ only consist of $\sigma$-stable triples for non-critical values of $\sigma$, therefore $\mathcal{N}_\sigma^s$ are projective varieties. Because of this, we shall compute the Hodge polynomials of the moduli spaces of triples of rank $(2,2)$ in the case $d_1-d_2 > 4g-4$ and $d_1+d_2$ odd. This gives in particular the Poincar{\'e} polynomials of these moduli spaces.
We start by reviewing the rudiments of mixed Hodge theory and the standard results on triples that we shall use throughout the paper, in Sections \ref{sec:virtual} and \ref{sec:stable-triples}. Then Section \ref{sec:Poincare(2,1)} recalls the computations of the Hodge polynomials of the moduli spaces of triples of ranks $(2,1)$ and $(1,2)$, from \cite{MOV}. In Section~\ref{sec:poly-bundles-rk-two} we use the Hodge polynomial of the moduli spaces of triples to deduce the Hodge polynomials of the moduli spaces of rank $2$ stable vector bundles. The case of odd degree rank $2$ bundles is already known \cite{BaR,EK,MOV}, but we do the case of even degree rank $2$ stable bundles, proving the following result (see~Theorem \ref{thm:rank2even}).
\noindent \textbf{Theorem A.}\ {\em Let $M^s(2,d)$ denote the moduli space of rank $2$, degree $d$ stable vector bundles on $X$. If $d$ is even then the Hodge polynomial of $M^s(2,d)$ is
\begin{align*}
e(M^s(2,d))=\, &
\frac{1}{2(1-uv)(1-(uv)^2)} \bigg(
2(1+u)^{g}(1+v)^g(1+u^2v)^{g}(1+uv^2)^g \\ & -
(1+u)^{2g}(1+v)^{2g} (1+ 2 u^{g+1} v^{g+1} -u^2v^2) -
(1-u^2)^g(1-v^2)^g(1-uv)^2 \bigg) \, .
\end{align*} }
Note that the moduli space $M^s(2,d)$ is smooth but non-compact.
Next we move to the study of the moduli spaces of triples of rank $(2,2)$, which are the main focus of the paper. The critical values are computed in Section \ref{sec:critical(2,2)}.
In Section \ref{sec:small} we compute the Hodge polynomial of the moduli space of stable triples of rank $(2,2)$ for the smallest allowable values of the parameter $\sigma$, proving the following result (see Theorem \ref{thm:(2,2,odd,even)} and Corollary \ref{cor:(2,2,odd,even)-2}).
\noindent \textbf{Theorem B.}\
{\em Let $\mathcal{N}_\sigma=\mathcal{N}_\sigma(2,2,d_1,d_2)$ be the moduli space
of $\sigma$-stable triples of rank $(2,2)$. Assume that $d_1-d_2 > 4g-4$ and
$d_1+d_2$ is odd. Let $\sigma_m=\frac{d_1}2 - \frac{d_2}2$ be the minimum
value of the parameter $\sigma$ and ${\s_m^+}=\sigma_m + \epsilon$ for
$\epsilon>0$ small. Then $\mathcal{N}_{\s_m^+}$ is smooth and projective, it only
consists of stable triples, and its Hodge polynomial is
$$
\begin{aligned}
e(\cN_{\smp}) = & \ \frac{(1+u)^{2g}(1+v)^{2g}(1-(uv)^N)
(u^gv^g(1+u)^g(1+v)^g-(1+u^2v)^g(1+uv^2)^g)}{(1-uv)^3(1-(uv)^2)^2}
\cdot \\ & \bigg(
(1+u)^{g} (1+v)^g (u^{g+1}v^{g+1} + u^{N+g-1}v^{N+g-1}) -
(1+u^2v)^{g}(1+uv^2)^g (1 + u^Nv^{N})\bigg) \, ,
\end{aligned}
$$ where $N=d_1-d_2-2g+2$. }
\enlargethispage{1\baselineskip}
Under the condition $d_1-d_2>4g-4$, the Hodge polynomial of $\mathcal{N}_{\s_m^+}(2,2,d_1,d_2)$ when both $d_1,d_2$ are odd is easily given (see Theorem \ref{thm:(2,2,odd,odd)}). When both $d_1,d_2$ are even, it may be computed with similar techniques to those of Theorem \ref{thm:(2,2,odd,even)}. However, to remove the condition $d_1-d_2>4g-4$ is not possible with the current techniques.
The contribution of the flips to the Hodge polynomials of the moduli spaces of $\sigma$-stables triples of rank $(2,2)$ is computed in Section \ref{sec:simple}. This is added up to the information for the Hodge polynomial of the small parameter moduli space to get the Hodge polynomial of the moduli space of $\sigma$-stable triples of rank $(2,2)$ for the largest values of $\sigma$ in Section \ref{sec:large}. We get the following result (see Theorem \ref{thm:finally} and Corollary \ref{cor:finally}).
\noindent \textbf{Theorem C.}\ {\em Let $\mathcal{N}_\sigma=\mathcal{N}_\sigma(2,2,d_1,d_2)$ be the moduli space
of $\sigma$-stable triples of rank $(2,2)$. Assume that $d_1-d_2 > 4g-4$ and
$d_1+d_2$ is odd. Let $\sigma_M=d_1 - d_2$. Then all the moduli spaces
$\mathcal{N}_\sigma$ are isomorphic for $\sigma>\sigma_M$. Let
$\sigma_M^+=\sigma_M + \epsilon$ for
$\epsilon>0$. Then
$\mathcal{N}_{\sigma_M^+}$ is smooth and projective, it only
consists of stable triples, and
$$
\begin{aligned}
e(\mathcal{N}_{\sigma_M^+})=& \ \frac{(1+u)^{2g}(1+v)^{2g}}{(1-uv)^3(1-(uv)^2)^2}
\Bigg[
(1+u^2v)^{2g}(1+uv^2)^{2g}(1-(uv)^{2N})\\
&
-N \, (1+u^2v)^g(1+uv^2)^g(1+u)^g(1+v)^g (uv)^{N+g-1}(1-(uv)^2)
\\
&
+
(1+u)^{2g}(1+v)^{2g}(1+uv)^2(uv)^{2g -2 +(N+1)/2 }
\Big((1-(uv)^{N+1}) - \frac{N+1}{2} \, (1-uv)(1+(uv)^{N})\Big)\\
&
-g(1+u)^{2g-1}(1+v)^{2g-1}
(1-(uv)^2)^2(uv)^{2g -2 +(N+1)/2}(1-(uv)^{N})
\Bigg] \, , \end{aligned} $$ where $N=d_1-d_2-2g+2$. }
The computation of the contribution of the flips to the Hodge polynomials of the moduli spaces of $\sigma$-stables triples of rank $(2,2)$ is done under the assumptions $d_1+d_2$ odd and $d_1-d_2>2g-2$. This~can be extended to the case $d_1+d_2$ even, keeping in mind that in this case we will find the Hodge polynomials of the moduli spaces $\mathcal{N}_\sigma^s$ which are non-compact and of the moduli spaces $\mathcal{N}_\sigma$ which have singularities at non-stable points. However the assumption $d_1-d_2>2g-2$ cannot be removed with the current techniques.
The Poincar\'e polynomials of the moduli spaces $\mathcal{N}_{\sigma_m^+}$ and $\mathcal{N}_{\sigma_M^+}$ are obtained from the Hodge polynomials, for $d_1-d_2 > 4g-4$ and $d_1+d_2$ odd (see Corollaries \ref{cor:(2,2,odd,even)} and \ref{cor:finally-n}), since they are smooth projective varieties.
{\bf Acknowledgements:} First and third authors partially supported through grant MCyT (Spain) MTM2004-07090-C03-01/02.
\section{Hodge Polynomials} \label{sec:virtual}
\subsection{Hodge-Deligne theory} \label{subsec:Hodge-Deligne}
Let us start by recalling the Hodge-Deligne theory of algebraic varieties over $\mathbb{C}$. Let $H$ be a finite-dimensional complex vector space. A {\em pure Hodge structure of weight $k$} on $H$ is a decomposition
$$
H=\bigoplus\limits_{p+q=k} H^{p,q}
$$ such that $H^{q,p}=\overline{H}^{p,q}$, the bar denoting complex conjugation in $H$. We denote
$$
h^{p,q}(H)=\dim H^{p,q}\ ,
$$ which is called the Hogde number of type $(p,q)$. A Hodge structure of weight $k$ on $H$ gives rise to the so-called {\em Hodge filtration} $F$ on $H$, where
$$
F^p= \bigoplus\limits_{s\geq p} H^{s,p-s}\ ,
$$ which is a descending filtration. Note that $\Gr_F^p H = F^p/F^{p+1}= H^{p,q}$.
Let $H$ be a finite-dimensional complex vector space. A {\em (mixed) Hodge structure} over $H$ consists of an ascending weight filtration $W$ on $H$ and a descending Hodge filtration $F$ on $H$ such that $F$ induces a pure Hodge filtration of weight $k$ on each $\Gr^W_k H= W_k/W_{k-1}$. Again we define
$$
h^{p,q}(H)=\dim \, H^{p,q}\, , \qquad \text{where}\quad H^{p,q}
=\Gr_F^p\Gr^W_{p+q} H\, .
$$
Deligne has shown \cite{De} that, for each complex algebraic variety $Z$, the cohomology $H^k(Z)$ and the cohomology with compact support $H_c^k(Z)$ both carry natural Hodge structures. If $Z$ is a compact smooth projective variety (hence compact K{\"a}hler) then the Hodge structure $H^k(Z)$ is pure of weight $k$ and coincides with the classical Hodge structure given by the Hodge decomposition of harmonic forms into $(p,q)$ types.
\begin{definition} \label{def:Hodge-poly} For \emph{any} complex algebraic variety $Z$ (not necessarily smooth, compact or irreducible), we define the Hodge numbers as
$$
h^{k,p,q}_c(Z)=h^{p,q}(H^k_c(Z))=\dim \Gr^p_F \Gr_{p+q}^W H^k_c(Z)\, .
$$ Introduce the Euler characteristic
$$
\chi^{p,q}_c(Z) = \sum_k (-1)^k h^{k,p,q}_c(Z)
$$ The \emph{Hodge polynomial} of $Z$ is defined \cite{DK} as
$$
e(Z)=e(Z)(u,v)= \sum_{p,q} (-1)^{p+q}\chi_c^{p,q}(Z) u^p v^q\, .
$$ \end{definition}
If $Z$ is smooth and projective then the mixed Hodge structure on $H^k_c(Z)$ is pure of weight $k$, so $\Gr_k^W H_c^k(Z) =H_c^k(Z)=H^k(Z)$ and the other pieces $\Gr_m^W H_c^k(Z)=0$, $m\neq k$. So
$$
\chi_c^{p,q}(Z)=(-1)^{p+q} h^{p,q}(Z),
$$ where $h^{p,q}(Z)$ is the usual Hodge number of $Z$. In this case,
$$
e(Z)(u,v)= \sum_{p,q} h^{p,q}(Z) u^p v^q\,
$$ is the (usual) Hodge polynomial of $Z$. Note that in this case, the Poincar{\'e} polynomial of $Z$ is
\begin{equation}\label{eqn:Poinca}
P_Z(t)=\sum_k b^k(Z) t^k= \sum_k \left(
\sum_{p+q=k} h^{p,q}(Z) \right) t^k= e(Z)(t,t).
\end{equation} where $b^k(Z)$ is the $k$-th Betti number of $Z$.
\begin{theorem}[{\cite[Theorem 2.2]{MOV}}] \label{thm:Du}
Let $Z$ be a complex algebraic variety. Suppose that $Z$ is
a finite disjoint union $Z=Z_1\cup \cdots \cup Z_n$, where the
$Z_i$ are algebraic subvarieties. Then
$$
e(Z)= \sum_i e (Z_i).
$$
$\Box$ \end{theorem}
Note that we can assign to \textit{any} complex algebraic variety $Z$ (not necessarily smooth, compact or irreducible) a polynomial
$$
P_Z(t) = e(Z)(t,t)= \sum_m (-1)^m \chi^m_c(Z) \, t^m=\sum_{k,m} (-1)^{k+m}
\dim \Gr_m^W H_c^k(Z) \, t^m ,
$$ where
$$
\chi^m_c (Z)= \sum_{p+q=m} \chi^{p,q}_c(Z).
$$ This is called the {\em virtual Poincar{\'e} polynomial} of $Z$ (see \cite{FM,Du}). It satisfies an additive property analogous to that of Theorem \ref{thm:Du} and it recovers the usual Poincar{\'e} polynomial when $Z$ is a smooth projective variety.
The following Hodge polynomials will be needed later: \begin{itemize}
\item Let $Z=\mathbb{P}^{n}$, then $e(Z)= 1+uv+(uv)^2+\cdots +
(uv)^{n}=(1-(uv)^{n+1})/(1-uv)$. For future reference,
we shall denote
\begin{equation}\label{eqn:Pn}
e_n := e
(\mathbb{P}^{n-1})=e
(\mathbb{P}(\mathbb{C}^n))
=\frac{1-(uv)^{n}}{1-uv}\ .
\end{equation}
\item Let $\Jac^d X$ be the Jacobian of (any) degree $d$ of a
(smooth, projective) complex curve $X$ of genus $g$. Then
\begin{equation}\label{eqn:Jac}
e(\Jac^d X)=(1+u)^g(1+v)^g.
\end{equation}
\end{itemize}
\begin{lemma}[{\cite[Lemma 2.3]{MOV}}]\label{lem:vb}
Suppose that $\pi:Z\to Y$ is an algebraic fiber bundle with fiber $F$ which is
locally trivial in the Zariski topology, then $e(Z)=e(F)\,e(Y)$.
(In particular this is true for $Z=F\times Y$.)
$\Box$ \end{lemma}
\begin{lemma}\label{lem:Gri}
Suppose that $\pi:Z\to Y$ is a map between quasi-projective varieties which is
a locally trivial fiber bundle in the usual topology, with fibers projective
spaces $F=\mathbb{P}^N$ for some $N>0$. Then $e(Z)=e(F)\,e(Y)$. \end{lemma}
\begin{proof} This follows from \cite{Gri,Del}. For completeness we provide a proof. Let $H$ be a hyperplane section of $Z$. We have a morphism of Hodge structures:
\begin{equation}\label{eqn:Leray}
\begin{array}{ccc}
L: H^*(\mathbb{P}^N)\otimes H^*_c(Y) &\to& H^*_c(Z) \\
h^i \otimes \alpha &\mapsto & H^i\cap \pi^*(\alpha)\, ,
\end{array}
\end{equation} where $h$ is the hyperplane class of the projective space. Note that $L$ is not multiplicative. Let us see that $L$ is injective. If $x=\sum H^i \cap \pi^*(\alpha_i)=0$, let $i_0$ be the maximum $i$ for which $\alpha_{i}\neq 0$. Then
$$
0 = \pi_*(H^{N-i_0}\cap x)=\alpha_{i_0}\, .
$$ So $L$ must be injective. On the other hand, the Leray spectral sequence of the fibration $\pi$ has $E_2$-term isomorphic to $H^*(\mathbb{P}^N)\otimes H^*_c(Y)$ and converges to $H^*_c(Z)$. So $\dim H^*(\mathbb{P}^N)\otimes H^*_c(Y) \geq \dim H^*_c(Z)$ and $L$ must be bijective. Therefore $L$ is an isomorphism of Hodge structures, and the result follows. \end{proof}
\begin{lemma}\label{lem:Gr} The Hodge polynomial of the Grassmannian $\Gr(k,N)$ is
$$
e(\Gr(k,N))=
\frac{(1- (uv)^{N-k+1}) \cdots (1- (uv)^{N-1})
(1- (uv)^{N})}{(1- uv)\cdots(1-(uv)^{k-1}) (1-(uv)^{k})}\, .
$$ \end{lemma}
\begin{proof} This is well-known, but we provide a proof for completeness.
Let us review first the case of the projective space $\mathbb{P}^{N-1}=(\mathbb{C}^N -\{0\}) /(\mathbb{C} -\{0\})$. Then $\mathbb{C}^N -\{0\} \to \mathbb{P}^{N-1}$ is a locally trivial fibration, since it is the restriction of the universal line bundle $U \to \mathbb{P}^{N-1}$ to the complement of the zero section. Using either Lemma \ref{lem:vb} or Lemma \ref{lem:Gri}, we have $e(\mathbb{C}^N-\{0\})=e(\mathbb{C}-\{0\})\, e(\mathbb{P}^{N-1})$, i.e.\ $(uv)^{N}-1=(uv-1) e(\mathbb{P}^{N-1})$, from where (\ref{eqn:Pn}) is recovered. Now in the case of $k>1$, denote
$$
F(k,n)= \{ (v_1,\ldots,v_k) \, | \, \text{$v_i$ are linearly
independent vectors of $\mathbb{C}^n$}\}\ .
$$ Then $\Gr(k,N)=F(k,N)/\GL(k,\mathbb{C})$ and there is a locally trivial fibration $F(k,N) \to \Gr(k,N)$ with fiber $\GL(k,\mathbb{C})\cong F(k,k)$ (again it is the principal bundle associated to the universal bundle $U\to \Gr(k,N)$). So by Lemma \ref{lem:vb}, $e(\Gr(k,N))=e(F(k,N))/e(F(k,k))$. Now we use that the map
$$
F(k,n) \longrightarrow F(k-1,n),
$$ given by forgetting the last vector, is a locally trivial fibration, with fiber $\mathbb{C}^n-\mathbb{C}^{k-1}$. Using Lemma \ref{lem:vb} and Theorem \ref{thm:Du}, we have $e(F(k,n))=e(F(k-1,n))\, e(\mathbb{C}^n-\mathbb{C}^{k-1})= e(F(k-1,n))\, ((uv)^{n}-(uv)^{k-1})$. By recursion this gives
$$
e(F(k,n))=((uv)^{n}-(uv)^{k-1}) \cdots ((uv)^{n}-uv) ((uv)^{n}-1)\, .
$$ So
\begin{eqnarray*}
e(\Gr(k,N)) &= & \frac{((uv)^{N}-(uv)^{k-1}) \cdots ((uv)^{N}-uv)((uv)^{N}-1)}{((uv)^{k}-(uv)^{k-1})
\cdots ((uv)^{k}-uv)((uv)^{k}-1)} \\ &=&
\frac{(1- (uv)^{N-k+1}) \cdots (1- (uv)^{N-1}) (1- (uv)^{N})}{(1- uv)\cdots(1-(uv)^{k-1}) (1-(uv)^{k})
} \, .
\end{eqnarray*} \end{proof}
\begin{lemma} \label{lem:Z2}
Let $M$ be a smooth projective variety. Consider the algebraic variety
$Z=(M\times M)/\mathbb{Z}_2$, where $\mathbb{Z}_2$ acts as $(x,y)\mapsto (y,x)$. The Hodge
polynomial of $Z$ is
$$
e(Z)= \frac12 \Big(e(M)(u,v)^2 + e(M)(-u^2,-v^2)\Big) \, .
$$ \end{lemma}
\begin{proof} The cohomology of $Z$ is
$$
H^*(Z)=H^*(M\times M)^{\mathbb{Z}_2}=(H^*(M)\otimes H^*(M))^{\mathbb{Z}_2}\, .
$$ This is an equality of Hodge structures. The Hodge structure of $M$ is of pure type, therefore the Hodge structure of $Z$ is also of pure type. Moreover,
$$
H^{p,q}(Z) = \left( \bigoplus_{p_1+p_2=p \atop q_1+q_2=q} H^{p_1,q_1}(M)\otimes H^{p_2,q_2}(M) \right)^{\mathbb{Z}_2}\, .
$$ Therefore we have
$$
h^{p,q}(Z) =\frac12 \hspace{-4mm} \sum_{{p_1+p_2=p \atop q_1+q_2=q} \atop (p_1,q_1)\neq
(p_2,q_2)} \hspace{-5mm}
h^{p_1,q_1}(M)h^{p_2,q_2}(M) \quad + \ \epsilon_{p,q}\, ,
$$ where
$$
\epsilon_{p,q}=
\left\{\begin{array}{ll} 0, & \text{$p$ or $q$ odd}, \\
\dim \left(\mathrm{Sym}^2 H^{p_1,q_1}(M)\right) , \qquad & p=2p_1, q=2q_1, p_1+q_1 \text{ even}, \\
\dim \left(\bigwedge^2 H^{p_1,q_1}(M)\right), & p=2p_1, q=2q_1, p_1+q_1 \text{
odd}.
\end{array}\right.
$$ If $V$ is a vector space of dimension $n$, then $\dim \left(\mathrm{Sym}^2 V\right)=\frac12 (n^2+n)$ and $\dim \left(\bigwedge^2 V\right)=\frac12 (n^2-n)$, so
$$
\epsilon_{p,q}=\left\{\begin{array}{ll} 0, & \text{$p$ or $q$ odd}, \\
\frac12 ( h^{p_1,q_1}(M)^2 + (-1)^{p_1+q_1} h^{p_1,q_1}(M) ),\qquad
& p=2p_1, \, q=2q_1\, . \end{array}\right.
$$
This yields
\begin{eqnarray*}
e(Z) &=& \sum h^{p,q}(Z)u^pv^q \\ &=& \frac12 \sum
h^{p_1,q_1}(M)h^{p_2,q_2}(M)u^{p_1+p_2}v^{q_1+q_2} + \frac12
\sum (-1)^{p_1+q_1} h^{p_1,q_1}(M) u^{2p_1}v^{2q_1} \\ &=& \frac12 \, e(M)\cdot
e(M) + \frac12 \, e(M)(-u^2,-v^2)\, .
\end{eqnarray*} \end{proof}
\section{Moduli spaces of triples} \label{sec:stable-triples}
\subsection{Holomorphic triples} \label{subsec:triples-definitions}
Let $X$ be a smooth projective curve of genus $g\geq 2$ over $\mathbb{C}$. A \emph{holomorphic triple} $T = (E_{1},E_{2},\phi)$ on $X$ consists of two holomorphic vector bundles $E_{1}$ and $E_{2}$ over $X$, of ranks $n_1$ and $n_2$ and degrees $d_1$ and $d_2$, respectively, and a holomorphic map $\phi \colon E_{2} \to E_{1}$. We refer to $(n_1,n_2,d_1,d_2)$ as the \emph{type} of $T$, to $(n_1,n_2)$ as the \emph{rank} of $T$, and to $(d_1,d_2)$ as the \emph{degree} of $T$.
A homomorphism from $T' = (E_1',E_2',\phi')$ to $T = (E_1,E_2,\phi)$ is a commutative diagram
\begin{displaymath}
\begin{CD}
E_2' @>\phi'>> E_1' \\
@VVV @VVV \\
E_2 @>\phi>> E_1,
\end{CD}
\end{displaymath} where the vertical arrows are holomorphic maps. A triple $T'=(E_1',E_2',\phi')$ is a subtriple of $T = (E_1,E_2,\phi)$ if $E_1'\subset E_1$ and $E_2'\subset E_2$ are subbundles,
$\phi(E_2')\subset E_1'$ and $\phi'=\phi|_{E_2'}$. A subtriple $T'\subset T$ is called \emph{proper} if $T'\neq 0 $ and $T'\neq T$. The quotient triple $T''=T/T'$ is given by $E_1''=E_1/E_1'$, $E_2''=E_2/E_2'$ and $\phi'' \colon E_2''\to E_1''$ being the map induced by $\phi$. We usually denote by $(n_1',n_2',d_1',d_2')$ and $(n_1'',n_2'',d_1'',d_2'')$, the types of the subtriple $T'$ and the quotient triple $T''$.
\begin{definition} \label{def:s-slope} For any $\sigma \in \mathbb{R}$ the \emph{$\sigma$-slope} of $T$ is defined by
$$
\mu_{\sigma}(T) =
\frac{d_1+d_2}{n_1+n_2} + \sigma \frac{n_{2}}{n_{1}+n_{2}}\ .
$$ To shorten the notation, we define the \emph{$\mu$-slope} and \emph{$\lambda$-slope} of the triple $T$ as $\mu=\mu(E_{1} \oplus E_{2})= \frac{d_1+d_2}{n_1+n_2}$ and $\lambda=\frac{n_{2}}{n_{1}+n_{2}}$, so that $\mu_{\sigma}(T)=\mu+\sigma \lambda$. \end{definition}
\begin{definition}\label{def:sigma-stable} We say that a triple $T = (E_{1},E_{2},\phi)$ is \emph{$\sigma$-stable} if
$$
\mu_{\sigma}(T') < \mu_{\sigma}(T) ,
$$ for any proper subtriple $T' = (E_{1}',E_{2}',\phi')$. We define \emph{$\sigma$-semistability} by replacing the above strict inequality with a weak inequality. A triple is called \emph{$\sigma$-polystable} if it is the direct sum of $\sigma$-stable triples of the same $\sigma$-slope. It is \emph{$\sigma$-unstable} if it is not $\sigma$-semistable, and \emph{strictly $\sigma$-semistable} if it is $\sigma$-semistable but not $\sigma$-stable. A $\sigma$-destabilizing subtriple $T'\subset T$ is a proper subtriple satisfying $\mu_{\sigma}(T') \geq \mu_{\sigma}(T)$. \end{definition}
We denote by
$$
\mathcal{N}_\sigma = \mathcal{N}_\sigma(n_1,n_2,d_1,d_2)
$$ the moduli space of $\sigma$-polystable triples $T = (E_{1},E_{2},\phi)$ of type $(n_1,n_2,d_1,d_2)$, and drop the type from the notation when it is clear from the context. The open subset of $\sigma$-stable triples is denoted by $\mathcal{N}_\sigma^s = \mathcal{N}_\sigma^s(n_1,n_2,d_1,d_2)$. This moduli space is constructed in \cite{BGP} by using dimensional reduction. A direct construction is given by Schmitt \cite{Sch} using geometric invariant theory.
There are certain necessary conditions in order for $\sigma$-semistable triples to exist. Let $\mu_i=\mu(E_i)=d_i/n_i$ stand for the slope of $E_i$, for $i=1,2$. We write
\begin{align*}
\sigma_m = &\mu_1-\mu_2\ , \\[5pt]
\sigma_M = & \left(1+ \frac{n_1+n_2}{|n_1 - n_2|}\right)(\mu_1 - \mu_2)\ ,
\qquad \mbox{if $n_1\neq n_2$\ .}
\end{align*}
\begin{proposition}\cite{BGPG} \label{prop:alpha-range} The moduli space $\mathcal{N}_\sigma(n_1,n_2,d_1,d_2)$ is a complex projective variety. For $n_1, n_2>0$, let $I$ denote the interval $I=[\sigma_m,\sigma_M]$ if $n_1\neq n_2$, or $I=[\sigma_m,\infty)$ if $n_1=n_2$. A necessary condition for $\mathcal{N}_\sigma(n_1,n_2,d_1,d_2)$ to be non-empty is that $\sigma\in I$.
$\Box$ \end{proposition}
\subsection{Critical values}\label{subsec:critical-values}
To study the dependence of the moduli spaces $\mathcal{N}_\sigma$ on the parameter, we need to introduce the concept of critical value \cite{BGP,MOV}.
\begin{definition}\label{def:critical} The values of $\sigma_c\in I$ for which there exist $0 \le n'_1 \leq n_1$, $0 \le n'_2 \leq n_2$, $d'_1$ and $d'_2$, with $n_1'n_2\neq n_1n_2'$, such that
\begin{equation}\label{eqn:sigmac}
\sigma_c=\frac{(n_1+n_2)(d_1'+d_2')-(n_1'+n_2')(d_1+d_2)}{n_1'n_2-n_1n_2'},
\end{equation} are called \emph{critical values}. \end{definition}
Given a triple $T=(E_1,E_2,\phi)$, the condition of $\sigma$-(semi)stability for $T$ can only change when $\sigma$ crosses a critical value. If $\sigma=\sigma_c$ as in (\ref{eqn:sigmac}) and if $T$ has a subtriple $T'\subset T$ of type $(n_1',n_2',d_1',d_2')$, then $\mu_{\sigma_c}(T')=\mu_{\sigma_c}(T)$ and
\begin{enumerate}\itemsep=5pt
\item if $\lambda'>\lambda$ (where $\lambda'$ is the $\lambda$-slope of
$T'$), then $T$ is not $\sigma$-stable for
$\sigma>\sigma_c$, \item if $\lambda'<\lambda$, then $T$ is not $\sigma$-stable for
$\sigma<\sigma_c$.
\end{enumerate} Note that $n_1'n_2\neq n_1n_2'$ is equivalent to $\lambda'\neq \lambda$.
Of course, it may happen that there is no triple $T$ as above and hence that the moduli spaces $\mathcal{N}_\sigma$ and $\mathcal{N}_\sigma^s$ do not change when crossing $\sigma_c$ (see Remark \ref{rem:virtual-not-all}).
\begin{proposition}[{\cite[Proposition 2.6]{BGPG}}] \label{prop:triples-critical-range} Fix $(n_1,n_2,d_1,d_2)$. Then
\begin{enumerate}
\item[(1)] The critical values are a finite number of values $\sigma_c \in I$.
\item[(2)] The stability and semistability criteria for two values of $\sigma$
lying between two consecutive critical values are equivalent; thus
the corresponding moduli spaces are isomorphic.
\item[(3)] If $\sigma$ is not a critical value and $\mathrm{gcd}(n_1,n_2,d_1+d_2) = 1$,
then $\sigma$-semistability is equivalent to $\sigma$-stability, i.e.,\
$\mathcal{N}_\sigma=\mathcal{N}_\sigma^s$.
\end{enumerate}
$\Box$ \end{proposition}
Note that if $\mathrm{gcd}(n_1,n_2,d_1+d_2)\neq 1$ then it may happen that there exists triples $T$ which are strictly $\sigma$-semistable for non-critical values of $\sigma$.
\subsection{Extensions and deformations of triples} \label{subsec:extensions-of-triples}
The homological algebra of triples is controlled by the hypercohomology of a certain complex of sheaves which appears when studying infinitesimal deformations \cite[Section 3]{BGPG}. Let $T'=(E'_1,E'_2,\phi')$ and $T''=(E''_1,E''_2,\phi'')$ be two triples of types $(n_{1}',n_{2}',d_{1}',d_{2}')$ and $(n_{1}'',n_{2}'',d_{1}'',d_{2}'')$, respectively. Let $\Hom(T'',T')$ denote the linear space of homomorphisms from $T''$ to $T'$, and let $\Ext^1(T'',T')$ denote the linear space of equivalence classes of extensions of the form
$$
0 \longrightarrow T' \longrightarrow T \longrightarrow T'' \longrightarrow 0,
$$ where by this we mean a commutative diagram
$$
\begin{CD}
0@>>>E_1'@>>>E_1@>>> E_1''@>>>0\\
@.@A\phi' AA@A \phi AA@A \phi'' AA\\
0@>>>E'_2@>>>E_2@>>>E_2''@>>>0.
\end{CD}
$$ To analyze $\Ext^1(T'',T')$ one considers the complex of sheaves
\begin{equation} \label{eqn:extension-complex}
C^{\bullet}(T'',T') \colon ({E_{1}''}^{*} \otimes E_{1}') \oplus
({E_{2}''}^{*} \otimes E_{2}')
\overset{c}{\longrightarrow}
{E_{2}''}^{*} \otimes E_{1}',
\end{equation} where the map $c$ is defined by
$$
c(\psi_{1},\psi_{2}) = \phi'\psi_{2} - \psi_{1}\phi''.
$$
\begin{proposition}[{\cite[Proposition 3.1]{BGPG}}]
\label{prop:hyper-equals-hom}
There are natural isomorphisms
\begin{align*}
\Hom(T'',T') &\cong \mathbb{H}^{0}(C^{\bullet}(T'',T')), \\
\Ext^{1}(T'',T') &\cong \mathbb{H}^{1}(C^{\bullet}(T'',T')),
\end{align*} and a long exact sequence associated to the complex $C^{\bullet}(T'',T')$:
$$
\begin{array}{c@{\,}c@{\,}c@{\,}l@{\,}c@{\,}c@{\,}c}
0 &\longrightarrow \mathbb{H}^0(C^{\bullet}(T'',T')) &
\longrightarrow & H^0(({E_{1}''}^{*} \otimes E_{1}') \oplus ({E_{2}''}^{*} \otimes
E_{2}'))
& \longrightarrow & H^0({E_{2}''}^{*} \otimes E_{1}') \\[3pt]
& \longrightarrow \mathbb{H}^1(C^{\bullet}(T'',T')) &
\longrightarrow & H^1(({E_{1}''}^{*} \otimes E_{1}') \oplus ({E_{2}''}^{*} \otimes
E_{2}'))
& \longrightarrow & H^1({E_{2}''}^{*} \otimes E_{1}') \\[3pt]
& \longrightarrow \mathbb{H}^2(C^{\bullet}(T'',T')) & \longrightarrow & 0. & &
\end{array}
$$
$\Box$ \end{proposition}
We introduce the following notation: \begin{align*}
h^{i}(T'',T') &= \dim\mathbb{H}^{i}(C^{\bullet}(T'',T')), \\
\chi(T'',T') &= h^0(T'',T') - h^1(T'',T') + h^2(T'',T'). \end{align*}
\begin{proposition}[{\cite[Proposition 3.2]{BGPG}}]
\label{prop:chi(T'',T')}
For any holomorphic triples $T'$ and $T''$ we have
\begin{align*}
\chi(T'',T') &= \chi({E_{1}''}^{*} \otimes E_{1}')
+ \chi({E_{2}''}^{*} \otimes E_{2}')
- \chi({E_{2}''}^{*} \otimes E_{1}') \\[5pt]
&= (1-g)(n''_1 n'_1 + n''_2 n'_2 - n''_2 n'_1) + n''_1 d'_1 - n'_1 d''_1
+ n''_2 d'_2 - n'_2 d''_2
- n''_2 d'_1 + n'_1 d''_2,
\end{align*} where $\chi(E)=\dim H^0(E) - \dim H^1(E)$ is the Euler characteristic of $E$.
$\Box$ \end{proposition}
Since the space of infinitesimal deformations of $T$ is isomorphic to $\mathbb{H}^{1}(C^{\bullet}(T,T))$, the previous results also apply to studying deformations of a holomorphic triple $T$.
\begin{theorem}[{\cite[Theorem 3.8]{BGPG}}]\label{thm:smoothdim} Let $T=(E_1,E_2,\phi)$ be an $\sigma$-stable triple of type $(n_1,n_2,d_1,d_2)$.
\begin{enumerate}
\item[(1)] The Zariski tangent space at the point defined by $T$
in the moduli space of stable triples is isomorphic to
$\mathbb{H}^{1}(C^{\bullet}(T,T))$.
\item[(2)] If\/ $\mathbb{H}^{2}(C^{\bullet}(T,T))= 0$, then the moduli space of
$\sigma$-stable triples is smooth in a neighbourhood of the point
defined by $T$.
\item[(3)] At a smooth point $T\in \mathcal{N}^s_\sigma(n_1,n_2,d_1,d_2)$ the
dimension of the moduli space of $\sigma$-stable triples is
\begin{align*}
\dim \mathcal{N}^s_\sigma(n_1,n_2,d_1,d_2)
&= h^{1}(T,T) = 1 - \chi(T,T) \\
&= (g-1)(n_1^2 + n_2^2 - n_1 n_2) - n_1 d_2 + n_2 d_1 + 1.
\end{align*}
\item[(4)] Let $T=(E_1,E_2,\phi)$ be a $\sigma$-stable triple. If $T$
is injective or surjective (meaning that $\phi:E_2\to E_1$ is
injective or surjective) then the moduli space is smooth at $T$.
\end{enumerate}
$\Box$ \end{theorem}
\subsection{Crossing critical values} \label{subsec:crossing-critical-values}
Fix the type $(n_1,n_2,d_1,d_2)$ for the moduli spaces of holomorphic triples. We want to describe the differences between two spaces $\mathcal{N}^s_{\sigma_1}$ and $\mathcal{N}^s_{\sigma_2}$ when $\sigma_1$ and $\sigma_2$ are separated by a critical value. Let $\sigma_c\in I$ be a critical value and set
$$
{\s_c^+} = \sigma_c + \epsilon,\quad {\s_c^-} = \sigma_c -
\epsilon,
$$ where $\epsilon > 0$ is small enough so that $\sigma_c$ is the only critical value in the interval $({\s_c^-},{\s_c^+})$.
\begin{definition}\label{def:flip-loci} We define the \textit{flip loci} as
\begin{align*}
\mathcal{S}_{{\s_c^+}} &= \{ T\in\mathcal{N}_{{\s_c^+}} \ |
\ \text{$T$ is ${\s_c^-}$-unstable}\} \subset\mathcal{N}_{{\s_c^+}} \ ,\\
\mathcal{S}_{{\s_c^-}} &= \{ T\in\mathcal{N}_{{\s_c^-}} \ |
\ \text{$T$ is ${\s_c^+}$-unstable}\}
\subset\mathcal{N}_{{\s_c^-}} \ .
\end{align*} and $\mathcal{S}_{{\s_c^\pm}}^s=\mathcal{S}_{\s_c^\pm} \cap \mathcal{N}_{\s_c^\pm}^s$ for the stable part of the flip loci. \end{definition}
Note that for $\sigma_c=\sigma_m$, $\mathcal{N}_{\sigma_m^-}$ is empty, hence $\mathcal{N}_{{\s_m^+}}= \mathcal{S}_{{\s_m^+}}$. Analogously, when $n_1\neq n_2$, $\mathcal{N}_{\sigma_M^+}$ is empty and $\mathcal{N}_{\sigma_M^-}= \mathcal{S}_{\sigma_M^-}$.
\begin{lemma}\label{lem:fliploci}
Let $\sigma_c$ be a critical value. Then
\begin{itemize}\itemsep=5pt
\item[(1)] $\mathcal{N}_{{\s_c^+}}-\mathcal{S}_{{\s_c^+}}=\mathcal{N}_{{\s_c^-}}-\mathcal{S}_{{\s_c^-}}$.
\item[(2)] $\mathcal{N}^s_{{\s_c^+}}-\mathcal{S}_{{\s_c^+}}^s=
\mathcal{N}^s_{{\s_c^-}}-\mathcal{S}_{{\s_c^-}}^s=\mathcal{N}^s_{\sigma_c}$.
\end{itemize} \end{lemma}
\begin{proof} Item (1) is an easy consequence of the definition of flip loci. Item (2) is the content of \cite[Lemma 5.3]{BGPG}. \end{proof}
Let us describe the flip loci $\mathcal{S}_{\sigma_c^{\pm}}$. Let $\sigma_c$ be a critical value, and let $(n_1',n_2',d_1',d_2')$ such that $\lambda' \neq \lambda$ and (\ref{eqn:sigmac}) holds. Put $(n_1'',n_2'',d_1'',d_2'')=(n_1-n_1',n_2-n_2',d_1-d_1',d_2-d_2')$. Denote $\mathcal{N}_\sigma'=\mathcal{N}_\sigma(n_1',n_2',d_1',d_2')$ and $\mathcal{N}_\sigma''=\mathcal{N}_\sigma(n_1'',n_2'',d_1'',d_2'')$.
\begin{lemma}[{\cite[Lemma 4.7]{MOV}}] \label{lem:semistable} Let $T\in \mathcal{S}_{\s_c^+}$ (resp.\ $T\in \mathcal{S}_{\s_c^-}$). Then $T$ sits in a non-split exact sequence
\begin{equation}\label{eqn:extension-triples}
0\to T'\to T\to T''\to 0,
\end{equation} where $\mu_{\sigma_c}(T')=\mu_{\sigma_c}(T)=\mu_{\sigma_c}(T'')$, $\lambda'<\lambda$ (resp.\ $\lambda'
>\lambda$) and $T'$ and $T''$ are both $\sigma_c$-semistable.
Conversely, if $T'\in \mathcal{N}_{\sigma_c}'$ and $T''\in \mathcal{N}_{\sigma_c}''$ are both $\sigma_c$-stable, and $\lambda'<\lambda$ (resp.\ $\lambda' >\lambda$). Then for any non-trivial extension {\rm (\ref{eqn:extension-triples})}, $T$ lies in $\mathcal{S}_{\s_c^+}^s$ (resp.\ in $\mathcal{S}_{\s_c^-}^s$). Moreover, such $T$ can be written uniquely as an extension {\rm (\ref{eqn:extension-triples})} with $\mu_{\sigma_c}(T')=\mu_{\sigma_c}(T)$.
In particular, suppose $\sigma_c$ is not a critical value for the moduli spaces of triples of types $(n_1',n_2',d_1',d_2')$ and
$(n_1'',n_2'',d_1'',d_2'')$, $\mathrm{gcd}(n_1',n_2',d_1'+d_2')=1$ and
$\mathrm{gcd}(n_1'',n_2'',d_1''+d_2'')=1$. Then if $\lambda'<\lambda$ (resp.\
$\lambda'>\lambda$), there is a bijective correspondence
between non-trivial extensions {\rm (\ref{eqn:extension-triples})}, with
$T'\in \mathcal{N}_{\sigma_c}'$ and $T''\in \mathcal{N}_{\sigma_c}''$ and triples
$T\in \mathcal{S}_{\s_c^+}$ (resp.\ $\cS_{\scm}$).
$\Box$ \end{lemma}
\begin{theorem} \label{thm:Smas}
Let $\sigma_c$ be a critical value with $\lambda'<\lambda$
(resp.\ $\lambda'>\lambda$). Assume
\begin{itemize}
\item[(i)] $\sigma_c$ is not a critical value for the moduli spaces
of triples of types $(n_1',n_2',d_1',d_2')$ and
$(n_1'',n_2'',d_1'',d_2'')$, $\mathrm{gcd}(n_1',n_2',d_1'+d_2')=1$ and
$\mathrm{gcd}(n_1'',n_2'',d_1''+d_2'')=1$.
\item[(ii)] $\mathbb{H}^0(C^\bullet(T'',T'))=\mathbb{H}^2(C^\bullet(T'',T'))
=0$, for every $(T',T'')\in \mathcal{N}_{\sigma_c}'\times \mathcal{N}_{\sigma_c}''$.
\end{itemize}
Then $\cS_{\scp}$ (resp.\ $\cS_{\scm}$) is the projectivization of a
bundle of
rank $-\chi(T'',T')$ over $\mathcal{N}_{\sigma_c}' \times \mathcal{N}_{\sigma_c}''$. \end{theorem}
\begin{proof} This is the content of \cite[Theorem 4.8]{MOV}. Note that by \cite{Sch}, the moduli spaces $\mathcal{N}_{\sigma_c}'$ and $\mathcal{N}_{\sigma_c}''$ are fine moduli spaces (since $\mathrm{gcd}(n_1',n_2',d_1'+d_2')=1$ and $\mathrm{gcd}(n_1'',n_2'',d_1''+d_2'')=1$), so the hypothesis (iii) in \cite[Theorem 4.8]{MOV} is satisfied. \end{proof}
The construction of the flip loci can be used for the critical value $\sigma_c=\sigma_m$, which allows to describe the moduli space $\mathcal{N}_{{\s_m^+}}$. We refer to the value of $\sigma$ given by $\sigma={\s_m^+}=\sigma_m+\epsilon$ as \textit{small}.
Let $M(n,d)$ denote the moduli space of polystable vector bundles of rank $n$ and degree $d$ over $X$. This moduli space is projective. We also denote by $M^s(n,d)$ the open subset of stable bundles, which is smooth of dimension $n^2(g-1)+1$. If $\mathrm{gcd}(n,d)=1$, then $M(n,d)=M^s(n,d)$.
\begin{proposition}[{\cite[Proposition 4.10]{MOV}}] \label{prop:moduli-small} There is a map
$$
\pi:\mathcal{N}_{{\s_m^+}}=\mathcal{N}_{{\s_m^+}}(n_1,n_2,d_1,d_2) \to M(n_1,d_1) \times M(n_2,d_2)
$$ which sends $T=(E_1,E_2,\phi)$ to $(E_1,E_2)$.
\begin{enumerate}
\item[(i)] If $\mathrm{gcd}(n_1,d_1)=1$, $\mathrm{gcd}(n_2,d_2)=1$ and $\mu_1-\mu_2>2g-2$, then $\mathcal{N}_{{\s_m^+}}^s=\mathcal{N}_{{\s_m^+}}$ is a projective bundle over $M(n_1,d_1) \times M(n_2,d_2)$, whose fibers are projective spaces of dimension $n_2d_1-n_1d_2- n_1n_2(g-1)-1$.
\item[(ii)] In general, if $\mu_1-\mu_2>2g-2$, then the open subset
$$
\pi^{-1}(M^s(n_1,d_1)\times M^s(n_2,d_2)) \subset \mathcal{N}_{{\s_m^+}}
$$ is a projective bundle over $M^s(n_1,d_1) \times M^s(n_2,d_2)$, whose fibers are projective spaces of dimension $n_2d_1-n_1d_2- n_1n_2(g-1)-1$.
\end{enumerate}
$\Box$ \end{proposition}
\section{Hodge polynomials of the moduli spaces of triples of ranks $(2,1)$ and $(1,2)$} \label{sec:Poincare(2,1)}
\subsection{Moduli of triples of rank $(2,1)$} \label{subsec:triples(2,1)}
In this section we recall the main results of \cite{MOV}. Let $\cN_\s=\cN_\s(2,1,d_1,d_2)$ denote the moduli space of $\sigma$-polystable triples $T=(E_1,E_2,\phi)$ where $E_1$ is a vector bundle of degree $d_1$ and rank $2$ and $E_2$ is a line bundle of degree $d_2$. By Proposition \ref{prop:alpha-range}, $\sigma$ is in the interval
$$
I=[\sigma_m,\sigma_M]=
[\mu_1-\mu_2\,,\,4(\mu_1-\mu_2)]=[d_1/2-d_2,2d_1-4d_2], \qquad\mbox{ where }
\mu_1-\mu_2\ge0\, .
$$ Otherwise $\mathcal{N}_\sigma$ is empty.
\begin{theorem}[{\cite[Theorem 5.1]{MOV}}]\label{thm:moduli(2,1)} For $\sigma\in I$, $\cN_\s$ is a projective variety. It is smooth and of (complex) dimension $3g-2 + d_1 - 2 d_2$ at the stable points $\cN_\s^s$. Moreover, for non-critical values of $\sigma$, $\cN_\s=\cN_\s^s$ (hence it is smooth and projective).
$\Box$ \end{theorem}
The critical values corresponding to $n_1=2$, $n_2=1$ are given by Definition \ref{def:critical}\ :
\begin{enumerate}
\item[(1)] $n'_1=1$, $n'_2=0$. The corresponding $\sigma_c$-destabilizing
subtriple is of the form $0\to E_1'$, where $E_1'=M$ is a line bundle of
degree $\deg(M)=d_M$. The critical value is
$$
\sigma_c= 3d_M-d_1-d_2\, .
$$
\item[(2)] $n'_1=1$, $n'_2=1$. The corresponding
$\sigma_c$-destabilizing subtriple $T'$ is of the form $E_2\to
E_1'$, where $E_1'$ is a line bundle. Let $T''=T/T'$ be the
quotient bundle, which is of the form $0\to E_1''$, where
$E_1''=M$ is a line bundle, and let $d_M=\deg(M)$ be its degree.
Then $d_2'=d_2$, $d_1'=d_1-d_M$ and
$$
\sigma_c=-\big( 3(d_1-d_M+d_2)-2(d_1+d_2)\big)= 3d_M-d_1-d_2\, .
$$
\item[(3)] $n'_1=2$, $n'_2=0$. In this case, the only possible subtriple
is $0\to E_1$. This produces the critical value
$$
\sigma_c=\frac{d_1-2d_2}{2}= \mu_1-\mu_2=\sigma_m\, ,
$$
i.e.,\ the minimum of the interval $I$ for
$\sigma$.
\item[(4)] $n'_1=0$, $n'_2=1$. The subtriple $T'$ must be of the form $E_2\to
0$. This forces $\phi=0$ in $T=(E_1,E_2,\phi)$. So $T$ is decomposable,
of the form $T'\oplus T''=(0,E_2,0)\oplus (E_1,0,0)$, and $T$ is
$\sigma$-unstable for any $\sigma\neq \sigma_c$, where
$$
\sigma_c=\frac{2d_2-d_1}{-2}= \mu_1-\mu_2=\sigma_m\, .
$$ \end{enumerate}
\begin{lemma}[{\cite[Lemma 5.3]{MOV}}] \label{lem:dM} Let $\sigma_c=3d_M-d_1-d_2$ be a critical value. Then
\begin{equation} \label{eqn:dM-bounds}
\mu_1\leq d_M\leq d_1-d_2 \, ,
\end{equation} and $\sigma_c=\sigma_m \Leftrightarrow d_M=\mu_1$.
$\Box$ \end{lemma}
The Hodge polynomials of the moduli spaces $\mathcal{N}_\sigma$ for non-critical values of $\sigma$ are given in \cite[Theorem 6.2]{MOV}. As this moduli space is projective and smooth, we may recover the Poincar{\'e} polynomial from the Hodge polynomial via the formula (\ref{eqn:Poinca}).
\begin{theorem}[{\cite[Theorem 6.2]{MOV}}] \label{thm:polinomiono(2,1)no-critico} Suppose that $\sigma>\sigma_m$ is not a critical value. Set $d_0=\Big[\frac13(\sigma+d_1+d_2)\Big]+1$. Then the Hodge polynomial of $\cN_\s=\cN_\s (2,1,d_1,d_2)$ is
$$
e(\cN_\s)= \mathop{\mathrm{coeff}}_{x^0}
\left[\frac{(1+u)^{2g}(1+v)^{2g}(1+ux)^{g}(1+vx)^{g}}{(1-uv)(1-x)(1-uvx)x^{d_1-d_2-d_0}}
\Bigg(\frac{(uv)^{d_1-d_2-d_0}}{1-(uv)^{-1}x}-
\frac{(uv)^{-d_1+g-1+2d_0}}{1-(uv)^2x}\Bigg)\right] .
$$
$\Box$ \end{theorem}
\subsection{Moduli space of triples of rank $(1,2)$} \label{sec:Poincare(1,2)}
Triples of rank $(1,2)$ are of the form $\phi: E_2\to E_1$, where $E_2$ is a rank $2$ bundle and $E_1$ is a line bundle. By Proposition \ref{prop:alpha-range}, $\sigma$ is in the interval
$$
I=[\sigma_m,\sigma_M]=
[\mu_1-\mu_2\,,\,4(\mu_1-\mu_2)]=[d_1-d_2/2,4d_1-2d_2], \qquad\mbox{ where }
\mu_1-\mu_2\ge0\, .
$$
\begin{theorem}\label{thm:moduli(1,2)} For $\sigma\in I$, $\cN_\s$ is a projective variety. It is smooth and of (complex) dimension $3g -2 + 2 d_1 - d_2$ at the stable points $\cN_\s^s$. Moreover, for non-critical $\sigma$, $\cN_\s=\cN_\s^s$ (hence it is smooth and projective). \end{theorem}
\begin{proof} Given a triple $T=(E_1,E_2,\phi)$ one has the dual triple $T^*=(E_2^*,E_1^*,\phi^*)$, where $E_i^*$ is the dual of $E_i$ and $\phi^*$ is the transpose of $\phi$. The map $T\mapsto T^*$ defines an isomorphism
$$
\mathcal{N}_\sigma(1,2,d_1,d_2) \cong \mathcal{N}_\sigma(2,1,-d_2,-d_1)\, .
$$ The result now follows from Theorem \ref{thm:moduli(2,1)}. \end{proof}
Also from Lemma \ref{lem:dM}, we get
\begin{lemma}[{\cite[Lemma 7.2]{MOV}}] \label{lem:dM-2} The critical values for $\cN_\s(1,2,d_1,d_2)$ are the numbers $\sigma_c=3d_M+d_1+d_2$, where $-\mu_2\leq d_M \leq d_1-d_2$. Also $\sigma_c=\sigma_m \Leftrightarrow d_M=-\mu_2$.
$\Box$ \end{lemma}
\begin{theorem}[{\cite[Theorem 7.3]{MOV}}] \label{thm:polinomiono(1,2)no-critico} Consider $\mathcal{N}_\sigma=\mathcal{N}_\sigma(1,2,d_1,d_2)$. Let $\sigma>\sigma_m$ be a non-critical value. Set $d_0=\Big[\frac13(\sigma-d_1-d_2)\Big]+1$. Then the Hodge polynomial of $\cN_\s$ is
$$
e(\cN_\s)= \mathop{\mathrm{coeff}}_{x^0}
\left[\frac{(1+u)^{2g}(1+v)^{2g}(1+ux)^{g}(1+vx)^{g}}{(1-uv)(1-x)(1-uvx)x^{d_1-d_2-d_0}}
\Bigg(\frac{(uv)^{d_1-d_2-d_0}}{1-(uv)^{-1}x}-
\frac{(uv)^{d_2+g-1+2d_0}}{1-(uv)^2x}\Bigg)\right] .
$$ \end{theorem}
\begin{proof} We use that $e(\cN_\s(1,2,d_1,d_2))=e(\cN_\s(2,1,-d_2,-d_1))$ and the formula in Theorem \ref{thm:polinomiono(2,1)no-critico}, where $d_1$ and $d_2$ are substituted by $-d_2$, $-d_1$ and
$$
d_0= \left[ \frac13 (\sigma -d_2-d_1)\right] +1 \, .
$$ \end{proof}
\section{Hodge polynomial of the moduli space of rank $2$ even degree stable bundles} \label{sec:poly-bundles-rk-two}
Let $M(2,d)$ denote the moduli space of polystable vector bundles of rank $2$ and degree $d$ over $X$. As $M(2,d)\cong M(2,d+2k)$, for any integer $k$, there are two moduli spaces, depending on whether the degree is even or odd. We are going to apply the results of the Section \ref{sec:Poincare(2,1)} to compute the Hodge polynomials of these moduli spaces.
We first recall the Hodge polynomial of the moduli space of rank $2$ odd degree stable bundles from \cite{BaR,EK,MOV}
\begin{theorem}[{\cite[Proposition 8.1]{MOV}}]\label{thm:rank2odd} The Hodge polynomial of $M(2,d)$ with odd degree $d$, is
$$
e(M(2,d)) = \frac{(1+u)^{g}(1+v)^g(1+u^2v)^{g}(1+uv^2)^g
-(uv)^{g}(1+u)^{2g}(1+v)^{2g}}
{(1-uv)(1-(uv)^2)}\,.
$$
$\Box$ \end{theorem}
Now we compute the Hodge polynomial of the moduli space of rank $2$ even degree stable bundles. Note that this moduli space is smooth but non-compact. It is irreducible and of dimension $4g-3$.
\begin{theorem}\label{thm:rank2even} The Hodge polynomial of $M^s(2,d)$ with even degree $d$, is
\begin{align*}
e(M^s(2,d))=\, &
\frac{1}{2(1-uv)(1-(uv)^2)} \bigg( 2(1+u)^{g}(1+v)^g(1+u^2v)^{g}(1+uv^2)^g \\
&-
(1+u)^{2g}(1+v)^{2g} (1+ 2 u^{g+1} v^{g+1} -u^2v^2) -
(1-u^2)^g(1-v^2)^g(1-uv)^2 \bigg) \, .
\end{align*} \end{theorem}
\begin{proof} We compute this by relating $M^s(2,d)$ with the moduli space $\mathcal{N}_{{\s_m^+}}=\mathcal{N}_{{\s_m^+}}(2,1,d,d_2)$ of triples of rank $(2,1)$ for small $\sigma$. Choose $(n_1,d_1)=(2,d)$ and $(n_2,d_2)=(1,d_2)$. If $d_2$ is very negative so that $\mu_1-\mu_2=d/2-d_2>2g-2$ then Proposition \ref{prop:moduli-small} (ii) applies. We shall choose the maximum possible value of $d_2$ for this condition to hold, i.e.\ $d-2d_2=4g-2$.
There is a decomposition $\cN_{\smp}= X_0 \sqcup X_1 \sqcup X_2 \sqcup X_3 \sqcup X_4$ into locally closed algebraic subsets, defined by the following strata:
\begin{itemize} \item[(1)] The open subset $X_0\subset \mathcal{N}_{{\s_m^+}}$ consists of those triples of the form $\phi: L\to E$, where $E$ is a stable rank $2$ bundle of degree $d$, $L$ is a line bundle of degree $d_2$, and $\phi$ is a non-zero map (defined up to multiplication by non-zero scalars). Actually, by Proposition \ref{prop:moduli-small} there is a map
$$
\pi: \mathcal{N}_{{\s_m^+}} \to M(2,d)\times \Jac^{d_2} X,
$$ and $X_0=\pi^{-1}(M^s(2,d)\times \Jac^{d_2}X)$. Proposition \ref{prop:moduli-small} (ii) says that $X_0$ is a projective bundle over $M^s(2,d)\times \Jac^{d_2}X$ with fibers isomorphic to $\mathbb{P}^{d-2d_2-2g+2-1}=\mathbb{P}^{2g-1}$. By Lemma \ref{lem:Gri},
$$
e(X_0)=e(M^s(2,d)) e(\Jac\, X) e_{2g}\, ,
$$ where $e_{2g}=e(\mathbb{P}^{2g-1})$ following the notation in (\ref{eqn:Pn}).
\item[(2)] The subset $X_1$ parametrizes triples $\phi:L\to E$ where $E$ is a strictly semistable bundle of degree $d$ which sits as a non-trivial extension
\begin{equation}\label{eqn:111}
0\to L_1\to E\to L_2\to 0,
\end{equation} with $L_1\not\cong L_2$, $L_1,L_2\in \Jac^{d/2} X$ and $L\in \Jac^{d_2}X$.
Let $Y_1$ be the family which parametrizes such bundles $E$. For fixed $L_1,L_2$ with $L_1\not\cong L_2$, the extensions (\ref{eqn:111}) are determined by $\mathbb{P} \Ext^1(L_2,L_1)$. As $L_1,L_2$ are non-isomorphic, $\dim \Ext^1(L_2,L_1)=\dim H^1(L_1\otimes L_2^*)=g-1$, so $\mathbb{P} \Ext^1(L_2,L_1) \cong \mathbb{P}^{g-2}$. Therefore $Y_1$ is a fiber bundle over $\Jac^{d/2}X\times \Jac^{d/2}X
- \Delta$, where $\Delta$ is the diagonal, with fibers isomorphic to $\mathbb{P}^{g-2}$. Thus using Theorem \ref{thm:Du} and Lemma \ref{lem:Gri},
\begin{equation}\label{eqn:Y1}
{}\qquad e(Y_1)= \big(e(\Jac X)^2 - e(\Jac X)\big) e_{g-1} \, .
\end{equation}
Now we want to describe $X_1$. For each fixed $E\in Y_1$ as in (\ref{eqn:111}), and $L\in \Jac^{d_2}X$, there is an exact sequence
$$
0\to \Hom(L,L_1) \to \Hom (L,E)\to \Hom(L,L_2) \to 0.
$$ Here $\Ext^1(L,L_1)=0$ since $\deg(L_1)-\deg(L)=d/2-d_2>2g-2$. So we may write $\Hom(L,E) \cong \Hom(L,L_1)\oplus \Hom(L,L_2)$, non-canonically. Let us see when $\phi\in \Hom(L,E)$ gives rise to a ${\s_m^+}$-stable triple $T=(E,L,\phi)$. First note that $T$ is $\sigma_m$-semistable, since by Section \ref{subsec:triples(2,1)}, the only possibility for not being $\sigma_m$-semistable is to have a subtriple of rank $(0,1)$, i.e., a line subbundle $M\subset E$, which by Lemma \ref{lem:dM} should have degree $d_M>\mu_1$, contradicting the semistability of $E$. If $T$ is not ${\s_m^+}$-stable then it must have a $\sigma_m$-destabilizing subtriple $T'$ of rank $(1,1)$ by Section \ref{subsec:triples(2,1)}. Such subtriple is of the form $\phi:L\to L'$, with $L'\subset E$. As $\mu_{\sigma_m}(T')=\mu_{\sigma_m}(T) \implies \mu(L')=\mu(E)$, $L'$ is a destabilizing subbundle of $E$. But the only destabilizing subbundle of $E$ is $L_1$, so $\phi$ satisfies $\phi(L)\subset L_1$. Equivalently, $\phi=(\phi_1,0)\in \Hom(L,E)$ gives rise to ${\s_m^+}$-unstable triples.
This discussion implies that given $(E,L)\in Y_1 \times \Jac^{d_2} X$, the morphisms $\phi$ giving rise to ${\s_m^+}$-stable triples $(E,L,\phi)$ are those in
\begin{equation}\label{eqn:X1}
\Hom(L,E) - \Hom (L,L_1).
\end{equation} By Riemann-Roch, $\dim\Hom(L,E)=d-2d_2-2g+2=2g$ and $\dim \Hom (L,L_1)=d/2-d_2-g+1=g$. So the space (\ref{eqn:X1}) is isomorphic to $\mathbb{C}^{2g}-\mathbb{C}^{g}$.
The isomorphism class of the triple $T=(E,L,\phi)$ is determined up to multiplication by non-zero scalar $(E,L,\phi)\mapsto (E,L, \lambda\phi)$, since $\Aut(T)=\mathbb{C}^*$. This follows from the fact that $\Aut(E)=\mathbb{C}^*$ (since $E$ is a non-trivial extension (\ref{eqn:111})) and $\Aut(L)=\mathbb{C}^*$. Taking into account the $\mathbb{C}^*$-action by automorphisms, the fibers of the map $\pi: X_1\to Y_1 \times \Jac^{d_2} X$ are isomorphic to the projectivization of (\ref{eqn:X1}), i.e.\ $\mathbb{P}^{2g-1}-\mathbb{P}^{g-1}$. Hence
$$
{}\qquad e(X_1)= e(\Jac X) e(Y_1) (e_{2g} -e_{g})
= e(\Jac X)^2 (e (\Jac X)-1) e_{g-1} (e_{2g}-e_{g})
\, .
$$ (For this, write $X_1=X_1'-X_1''$, where $X_1'$ is a $\mathbb{P}^{2g-1}$-bundle over $Y_1$ and $X_1''$ is a $\mathbb{P}^{g-1}$-bundle over $Y_1$. By Theorem \ref{thm:Du}, $e(X_1)=e(X_1')-e(X_1'')$. Now use Lemma \ref{lem:Gri} to compute $e(X_1')$ and $e(X_1'')$.)
\item[(3)] The subset $X_2$ parametrizes triples $\phi:L\to E$ where $E$ is a strictly semistable bundle of degree $d$ which sits as a non-trivial extension
$$
0\to L_1\to E\to L_1\to 0
$$ with $L_1 \in \Jac^{d/2} X$ and $L\in \Jac^{d_2}X$.
The family $Y_2$ parametrizing such bundles $E$ is a fiber bundle over $\Jac^{d/2}X$ with fibers $\mathbb{P} \Ext^1(L_1,L_1) =\mathbb{P} H^1(\mathcal{O})=\mathbb{P}^{g-1}$ (actually, this fiber bundle is trivial, so $Y_2=\Jac^{d/2} X\times \mathbb{P}^{g-1}$). Thus by Lemma \ref{lem:vb},
\begin{equation}\label{eqn:Y2}
{}\qquad e(Y_2)= e(\Jac X) e_{g} \, .
\end{equation}
For each $L_1\in \Jac^{d/2} X$, there is an exact sequence
$$
0\to \Hom(L,L_1) \to \Hom (L,E)\to \Hom(L,L_1) \to 0.
$$ So we may write $\Hom(L,E)\cong \Hom(L,L_1)\oplus \Hom(L,L_1)$, non-canonically. In order to describe $X_2$, let us see when a triple $T=(E,L,\phi)$, with $E\in Y_2$, is ${\s_m^+}$-stable. As before, the morphisms $\phi$ giving rise to ${\s_m^+}$-stable triples $(E,L,\phi)$ are those in
\begin{equation}\label{eqn:X2}
\Hom(L,E) - \Hom (L,L_1) = \Hom(L,L_1) \times (\Hom(L,L_1)- \{0\}) \,
.
\end{equation}
For a bundle $E$ in $Y_2$, the automorphism group of $E$ is $\mathbb{C}\times \mathbb{C}^*$, where $\mathbb{C}\times \mathbb{C}^*$ acts on $\Hom(L,E)$ by
$$
(a, \lambda)\cdot (\phi_1,\phi_2) =(\lambda \phi_1 + a \phi_2,
\lambda \phi_2).
$$ Thus for any $(E,L)\in Y_2 \times \Jac^{d_2} X$, the morphisms $\phi$ giving rise to ${\s_m^+}$-stable triples $(E,L,\phi)$ are parametrized by
\begin{equation}\label{eqn:X22}
(\Hom(L,L_1) \times (\Hom(L,L_1)- \{0\}))/\mathbb{C}\times \mathbb{C}^* \, .
\end{equation} This is a fiber bundle over $\mathbb{P}\Hom(L,L_1)=(\Hom(L,L_1)- \{0\})/\mathbb{C}^*$ with fibers isomorphic to $\Hom(L,L_1)/\mathbb{C}\phi_2$ for every $[\phi_2]\in \mathbb{P}\Hom(L,L_1)$. As $\dim\Hom(L,E)=d-2d_2-2g+2=2g$ and $\dim \Hom (L,L_1)=d/2-d_2-g+1=g$, the space (\ref{eqn:X22}) is a $\mathbb{C}^{g-1}$-bundle over $\mathbb{P}^{g-1}$.
Therefore $X_2\to Y_2 \times \Jac^{d_2} X$ is $\mathbb{C}^{g-1}$-bundle over a $\mathbb{P}^{g-1}$-bundle over $Y_2 \times \Jac^{d_2} X$. So
$$
{}\qquad e(X_2)= e(\Jac X)e(Y_2) e_g (e_g-e_{g-1}) =
e(\Jac X)^2 e_g^2 (e_g-e_{g-1})
\, .
$$ (To apply Lemma \ref{lem:Gri}, we write $X_2\to P$, where $P$ is the $\mathbb{P}^{g-1}$-bundle over $Y_2 \times \Jac^{d_2} X$. Then $X_2=X_2'-X_2''$, where $X_2'$ is a $\mathbb{P}^{g-1}$-bundle over $P$ and $X_2''$ is a $\mathbb{P}^{g-2}$-bundle over $P$.)
\item[(4)] The subset $X_3$ parametrizes triples $\phi:L\to E$ where $E$ is a decomposable bundle of the form $E=L_1\oplus L_2$, $L_1\not\cong L_2$, $L_1,L_2\in \Jac^{d/2}X$ and $L\in \Jac^{d_2}X$. The space parametrizing such bundles $E$ is
\begin{equation}\label{eqn:Y3}
Y_3=\tilde{Y}_3/\mathbb{Z}_2, \qquad \text{where } \
\tilde{Y}_3 = \Jac^{d/2}X\times \Jac^{d/2}X -\Delta\, ,
\end{equation} with $\mathbb{Z}_2$ acting by permuting the two factors.
As before, the condition for $\phi\in \Hom(L,E)$ to give rise to a ${\s_m^+}$-unstable triple is that there is a subtriple $\phi:L\to L'$ where $\mu(L')=\mu(E)$. There are only two possible such choices for $L'$, namely $L_1$ and $L_2$. So given $(E,L)\in Y_3\times \Jac^{d_2} X$, the morphisms $\phi\in \Hom(L,E)=\Hom(L,L_1)\oplus \Hom(L,L_2)$ giving rise to ${\s_m^+}$-stable triples $(E,L,\phi)$ are those with both components non-zero, i.e., lying in
$$
(\Hom (L,L_1)-\{0\})\times (\Hom(L,L_2)-\{0\}) .
$$
The automorphisms of $E$ are $\Aut(E)=\mathbb{C}^*\times \mathbb{C}^*$, therefore the map $\phi\in \Hom(L,E)=\Hom(L,L_1)\oplus \Hom(L,L_2)$ is determined up to the action of $\mathbb{C}^*\times \mathbb{C}^*$ on both factors. So $\phi$ are parametrized by
$$
\mathbb{P}\Hom (L,L_1) \times \mathbb{P}\Hom(L,L_2).
$$
Let $\tilde{X}_3\to \tilde{Y}_3 \times \Jac^{d_2}X$ be the fiber bundle with fiber over $(L_1,L_2,L)$ equal to $\mathbb{P}\Hom (L,L_1) \times \mathbb{P}\Hom(L,L_2)$. Then $X_3=\tilde X_3/\mathbb{Z}_2$, where $\mathbb{Z}_2$ acts by permuting $(\phi_1,\phi_2)\mapsto (\phi_2,\phi_1)$. This covers the action of $\mathbb{Z}_2$ on $\tilde{Y}_3$. Now $\tilde X_3=X_3'-X_3''$, where $\pi: X_3'\to \Jac^{d/2}X\times \Jac^{d/2}X\times \Jac^{d_2}X$ is a $\mathbb{P}^{g-1}\times \mathbb{P}^{g-1}$-bundle and $X_3''=\pi^{-1}(\Delta\times \Jac^{d_2}X)$. If $A\to \Jac^{d/2} X\times \Jac^{d_2}X$ is the $\mathbb{P}^{g-1}$-bundle with fiber over $L_1$ equal to $\mathbb{P}\Hom (L,L_1)$, then $X_3'=A\times_{\Jac^{d_2}X} A$. We apply Lemma \ref{lem:Z2} fiberwise: $A\to \Jac^{d_2}X$ is a fibration whose fiber is $A_L$, which in turn is a fibration over $\Jac^{d/2}X$ with fibers $\mathbb{P}\Hom (L,L_1)$. Then $X_3'$ fibers over $\Jac^{d_2}X$ with fibers $(A_L\times A_L)/\mathbb{Z}_2$. Now
$$
\begin{aligned}
H^*((A\times_{\Jac^{d_2}X} A)/\mathbb{Z}_2) &= H^*(A\times_{\Jac^{d_2}X}
A)^{\mathbb{Z}_2} \\ &= (H^*(A_L \times A_L)\otimes H^*(\Jac^{d_2}X))^{\mathbb{Z}_2} \\ &=
(H^*(A_L \times A_L))^{\mathbb{Z}_2}\otimes H^*(\Jac^{d_2}X)\, .
\end{aligned}
$$ So
$$
\begin{aligned}
e(X_3'/\mathbb{Z}_2) & = e((A_L\times A_L)/\mathbb{Z}_2) e(\Jac X) \\
&= \frac12 \Big( e(\Jac X)^2e_g^2 + (1-u^2)^g(1-v^2)^g
\frac{1-(uv)^{2g}}{1-u^2v^2}\Big) e(\Jac X)\, .
\end{aligned}
$$ On the other hand, $X_3''$ is a $\mathbb{P}^{g-1}\times \mathbb{P}^{g-1}$-bundle over $\Delta\times \Jac^{d_2}X$, the action of $\mathbb{Z}_2$ is trivial on the base, and acts by permutation on the fibers. So $X_3''/\mathbb{Z}_2$ is a bundle over $\Delta\times \Jac^{d_2}X$ with fibers
$$
(\mathbb{P} \Hom (L,L_1)\times \mathbb{P}\Hom(L,L_1))/\mathbb{Z}_2
=(\mathbb{P}^{g-1}\times \mathbb{P}^{g-1})/\mathbb{Z}_2.
$$ This fibration is locally trivial in the Zariski topology, since it is associated to a locally trivial (in the Zariski topology) vector bundle over $\Delta\times \Jac^{d_2}X$. Hence by Lemma \ref{lem:vb} and Lemma \ref{lem:Gri},
$$
e(X_3''/\mathbb{Z}_2)= e(\Jac X)^2 e(\mathbb{P}^{g-1}\times \mathbb{P}^{g-1}/\mathbb{Z}_2) =
\frac12e(\Jac X)^2 \bigg( e_g^2 + \frac{1-(uv)^{2g}}{1-u^2v^2}\bigg)\, .
$$ Finally using Theorem \ref{thm:Du},
$$
\begin{aligned}
e(X_3)= & \ e(\tilde{X}_3/\mathbb{Z}_2)= e(X_3'/\mathbb{Z}_2)-e(X_3''/\mathbb{Z}_2) \\
= & \ \frac12 \bigg( e(\Jac X)^2e_g^2 + (1-u^2)^g(1-v^2)^g
\frac{1-(uv)^{2g}}{1-u^2v^2}\bigg) e(\Jac X)
-\frac12e(\Jac X)^2 \bigg( e_g^2 +
\frac{1-(uv)^{2g}}{1-u^2v^2}\bigg)\, .
\end{aligned}
$$
\item[(5)] The subset $X_4$ parametrizes triples $\phi:L\to E$, where $E$ is a decomposable bundle of the form $E=L_1\oplus L_1$, $L_1\in \Jac^{d/2}X$ and $L\in \Jac^{d_2}X$. Such bundles $E$ are parametrized by $Y_4=\Jac^{d/2}X$. The morphism $\phi$ lives in
\begin{equation}\label{eqn:Y4}
\Hom(L,E)=\Hom(L,L_1)\oplus \Hom(L,L_1)=\Hom(L,L_1)\otimes \mathbb{C}^2\, .
\end{equation}
The condition for a triple $T=(E,L,\phi)$ to be ${\s_m^+}$-unstable is that there is a destabilizing subbundle $L' \subset E$. A destabilizing subbundle of $E$ is necessarily isomorphic to $L_1$ and there exists $(a,b)\neq (0,0)$ such that $L'\cong L_1 \hookrightarrow E$ is given by $x\mapsto (ax, bx)$. This means that $\phi=(a\psi,b\psi) \in \Hom(L,L_1)\otimes \mathbb{C}^2$, for some $\psi\in \Hom(L,L_1)$. All this discussion implies that the set of $\phi$ giving rise to ${\s_m^+}$-stable triples are those of the form $\phi=(\phi_1,\phi_2)\in\Hom(L,L_1)\otimes \mathbb{C}^2$, with $\phi_1,\phi_2$ linearly independent.
The automorphisms of $T=(E,L,\phi)$ are $\Aut(T)\cong \Aut(E)=GL(2,\mathbb{C})$. This acts on (\ref{eqn:Y4}) via the standard representation of $GL(2,\mathbb{C})$ on $\mathbb{C}^2$. So the morphisms $\phi$ are parametrized by the grassmannian $\Gr(2, \Hom(L,L_1))$. As $\dim \Hom(L,L_1)=g$, we have that $\Gr(2, \Hom(L,L_1))\cong \Gr(2,g)$.
Moreover $X_4\to Y_4\times \Jac^{d_2}X$ is a locally trivial fibration in the Zariski topology since it is associated to the (locally trivial in the Zariski topology) vector bundle over $Y_4\times \Jac^{d_2}X$ with fibers $\Hom(L,L_1)$. Using Lemma \ref{lem:Gr},
$$
e(X_4)= e (\Jac X)^2 e(\Gr(2, g))
= e (\Jac X)^2
\frac{(1-(uv)^{g-1})(1-(uv)^{g})}{(1-(uv)^2)(1-uv)}
\, .
$$ \end{itemize}
Putting all together,
\begin{equation}\label{eqn:M(2,even)}
\begin{aligned}
e(\cN_{\smp})=& \
e(X_0)+e(X_1)+ e(X_2)+e(X_3)+e(X_4)\\
=& \ e(M^s(2,d)) e(\Jac X)e_{2g} + e(\Jac X)^2 (e (\Jac X)-1) e_{g-1} (e_{2g}-e_{g})
+ e(\Jac X)^2 e_g^2 (e_g-e_{g-1}) \\
& +\frac12 \bigg( e(\Jac X)^2e_g^2 + (1-u^2)^g(1-v^2)^g
\frac{1-(uv)^{2g}}{1-u^2v^2}\bigg) e(\Jac X)
-\frac12e(\Jac X)^2 \bigg( e_g^2 +
\frac{1-(uv)^{2g}}{1-u^2v^2}\bigg)\\
&+e (\Jac X)^2 e(\Gr(2, g))\,.
\end{aligned}
\end{equation} To compute the left hand side, we use Theorem \ref{thm:polinomiono(2,1)no-critico} for $\sigma={\s_m^+}=\mu_1-\mu_2+\epsilon$, $\epsilon>0$ small. It gives
$$
d_0=\big[\mbox{$\frac13$}(\mu_1-\mu_2+\varepsilon+2\mu_1+\mu_2)\big]+1=
[\mu_1]+1=\frac{d}2+1 \,.
$$ Substuting into the formula for $e(\cN_\s)$ with $d_1=d/2$ and $d-2d_2=4g-2$, the Hodge polynomial of $\cN_{\smp}$ equals
$$
e(\cN_{\smp})=
\mathop{\mathrm{coeff}}_{x^0}\Bigg[
\frac{(1+u)^{2g}(1+v)^{2g}(1+ux)^{g}(1+vx)^g}
{(1-uv)(1-x)(1-uvx)x^{2g-2}}
\Bigg( \frac{(uv)^{2g-2}}{1-(uv)^{-1}x} -
\frac{(uv)^{g+1}}{1-(uv)^2x}\Bigg)\Bigg] \, .
$$ Using the following equality (see the proof of \cite[Proposition 8.1]{MOV})
$$
\mathop{\mathrm{coeff}}_{x^{0}}
\frac{(1+ux)^{g}(1+vx)^g}{(1-ax)(1-bx)(1-cx)x^{2g-2}}=
\frac{(a+u)^{g}(a+v)^{g}}{(a-b)(a-c)}+
\frac{(b+u)^{g}(b+v)^{g}}{(b-a)(b-c)}+
\frac{(c+u)^{g}(c+v)^{g}}{(c-a)(c-b)}\, ,
$$ one gets the following expression
$$ \begin{aligned}
e(\cN_{\smp}) = & \
\frac{(1+u)^{2g}(1+v)^{2g}}{(1-uv)^2(1-(uv)^2)}
\big[(1+u^2v)^{g}(1+v^2u)^g(1-(uv)^{2g})+ \\ & \ +(1+u)^{g} (1+v)^g
\big((uv)^{3g-1}+(uv)^{2g+1}- (uv)^{2g-1}-(uv)^{g+1}\big)
\big]\, .
\end{aligned}
$$
Finally we substitute this into (\ref{eqn:M(2,even)}) to get the Hodge polynomial $e(M^s(2,d))$ as in the statement. \end{proof}
\begin{corollary} \label{cor:rank2even-ss} The Hodge polynomial of the moduli space of polystable rank $2$ even degree $d$ vector bundles is
\begin{align*}
e(M(2,d))=\, &
\frac{1}{2(1-uv)(1-(uv)^2)} \bigg( 2(1+u)^{g}(1+v)^g(1+u^2v)^{g}(1+uv^2)^g \\
&-
(1+u)^{2g}(1+v)^{2g} (1+ 2 u^{g+1} v^{g+1} -u^2v^2) -
(1-u^2)^g(1-v^2)^g(1-uv)^2 \bigg) \\ & +
\frac12 \Big((1+u)^{2g}(1+v)^{2g}
+ (1-u^2)^{g}(1-v^2)^g \Big) \, .
\end{align*} \end{corollary}
\begin{proof}
We only need to compute $e(M^{ss}(2,d))$, where
$M^{ss}(2,d)=M(2,d)-M^s(2,d)$ is the locus of non-stable and
polystable rank $2$ bundles of degree $d$. Such bundles are of
the form $L_1\oplus L_2$, where $L_1,L_2\in \Jac^{d/2} X$.
Therefore $M^{ss}(2,d) \cong (\Jac\ X\times \Jac\ X )/\mathbb{Z}_2$. By Lemma \ref{lem:Z2} and (\ref{eqn:Jac}),
$$
e((\Jac\ X\times \Jac\ X )/\mathbb{Z}_2)= \frac12 \Big((1+u)^{2g}(1+v)^{2g}
+ (1-u^2)^{g}(1-v^2)^g \Big)\, .
$$ Adding this to $e(M^s(2,d))$ in Theorem \ref{thm:rank2even} we get the result. \end{proof}
For instance, the formula of Corollary \ref{cor:rank2even-ss} for $g=2$ gives
$$
e(M(2,0))=(1+u)^2(1+v)^2(1+uv+u^2v^2+u^3v^3)\, .
$$ This formula agrees with \cite[Remark 4.11]{Ki}. Note that the moduli space $M(2,0)$ is smooth for $g=2$ (see \cite{NR2}).
\section{Critical values for triples of rank $(2,2)$}\label{sec:critical(2,2)}
Now we move to the analysis of the moduli spaces of $\sigma$-polystable triples of rank $(2,2)$. Let $\mathcal{N}_\sigma= \mathcal{N}_\sigma(2,2,d_1,d_2)$. By Proposition \ref{prop:alpha-range}, $\sigma$ takes values in the interval
$$
I=[\sigma_m,\infty)=[\mu_1-\mu_2,\infty)\, , \quad \hbox{ where $d_1-d_2\geq 0$}.
$$ Otherwise $\mathcal{N}_\sigma$ is empty.
\begin{theorem} \label{thm:moduli(2,2)} For $\sigma\in I$, $\mathcal{N}_\sigma$ is a projective variety. It is smooth of dimension $4g+2d_1-2d_2-3$ at any $\sigma$-stable point for $\sigma\geq 2g-2$, or at any $\sigma$-stable injective triple. Moreover, if $d_1+d_2$ is odd then $\mathcal{N}_\sigma=\mathcal{N}_\sigma^s$ for non-critical $\sigma$. \end{theorem}
\begin{proof} Projectiveness follows from Proposition \ref{prop:alpha-range}. The smoothness at injective triples follows from Theorem \ref{thm:smoothdim}(4); the dimension follows from Theorem \ref{thm:smoothdim}(3); the smoothness result for $\sigma\geq 2g-2$ comes from \cite[Theorem 3.8(6)]{BGPG}. If $d_1+d_2$ is odd then $\mathrm{gcd}(2,2,d_1+d_2)=1$ and so, for non-critical $\sigma$, $\mathcal{N}_\sigma=\mathcal{N}_\sigma^s$, by Proposition \ref{prop:triples-critical-range} (3). On the other hand, if $d_1+d_2$ is even, then it may happen that there are strictly $\sigma$-semistable triples for non-critical values of $\sigma$. \end{proof}
Let us now compute the critical values for $\mathcal{N}_\sigma(2,2,d_1,d_2)$. According to (\ref{eqn:sigmac}) we have the following possibilities for $n_1=2$, $n_2=2$:
\begin{enumerate}
\item[(1)] $n'_1=1$, $n'_2=0$. The corresponding $\sigma_c$-destabilizing
subtriple is of the form $0\to E_1'$ where $E_1'=L$ is a line
bundle of degree $d_L$. The critical value is
$$
\sigma_c= \frac{4d_L-(d_1+d_2)}{2}=2d_L-\mu_1-\mu_2 \, .
$$
\item[(2)] $n'_1=1$, $n'_2=2$. The $\sigma_c$-destabilizing subtriple $T'$ is of the form $E_2 \to E_1'$ where $E_1'$ is a line bundle. The quotient triple $T''=T/T'$ is of the form $0\to E_1''$, where $E_1''=L$ is a line bundle of degree $d_L$, and $d_1'=d_1-d_L$. Note that $\phi:E_2\to E_1$ is not injective. The critical value is
$$
\sigma_c= \frac{4(d_1-d_L+d_2)-3(d_1+d_2)}{-2}= 2d_L-\mu_1-\mu_2 \, .
$$
\item[(3)] $n'_1=2$, $n'_2=1$. The $\sigma_c$-destabilizing subtriple $T'$
is of the form $E_2'\to E_1$, where $E_2'$ is a line bundle. Then
the quotient triple $T''=T/T'$ is of the form $E_2''\to 0$, where
$E_2''=F$ is a line bundle of degree $d_F$, and $d_2'=d_2-d_F$.
$$
\sigma_c =\frac{4(d_1+d_2-d_F)-3(d_1+d_2)}{2}=\mu_1+\mu_2-2d_F \, .
$$
\item[(4)] $n'_1=0$, $n'_2=1$. The $\sigma_c$-destabilizing subtriple is
of the form $E_2'\to 0$, where $E_2'=F$ is a line bundle of
degree $d_F$. Again in this case $\phi$ is not injective. The
corresponding critical value is
$$
\sigma_c= \frac{4d_F-(d_1+d_2)}{-2}=\mu_1+\mu_2-2d_F\, .
$$
\item[(5)] $n'_1=2$, $n'_2=0$. The subtriple is of the form $0\to E_1$. The corresponding critical value is $\sigma_c=\mu_1-\mu_2=\sigma_m$.
\item[(6)] $n'_1=0$, $n'_2=2$. The subtriple is of the form $E_2\to 0$. This only happens if $\phi=0$, and so $T=(0,E_2,0)\oplus (E_1,0,0)$. The critical value is $\sigma_c=\mu_1-\mu_2=\sigma_m$, and the triple is $\sigma$-unstable for any $\sigma\neq \sigma_m$.
\end{enumerate}
Note that the case $n'_1=1$, $n'_2=1$ does not appear, since $\lambda'=\lambda$ and therefore this does not give a critical value. In the Cases (1), (3) and (5), we have $\lambda'<\lambda$, so the corresponding triples are $\sigma$-unstable for $\sigma<\sigma_c$. In the Cases (2), (4) and (6), we have $\lambda'>\lambda$, so the corresponding triples are $\sigma$-unstable for $\sigma>\sigma_c$.
\begin{proposition}\label{prop:bounds-for-sc}
\begin{itemize}
\item[(i)] Let $\sigma_c=2d_L-\mu_1-\mu_2$ be a critical value corresponding to the
Cases {\rm (1)} or {\rm (3)}. Then $\mu_1 \leq d_L\leq (3\mu_1-\mu_2)/2$.
Also $d_L =\mu_1 \iff \sigma_c=\sigma_m$.
\item[(ii)] Let $\sigma_c=\mu_1+\mu_2-2d_F$ be a critical value corresponding to
the Cases {\rm (2)} or {\rm (4)}. Then $(3\mu_2-\mu_1)/2 \leq d_F\leq
\mu_2$. Also $d_F =\mu_2 \iff \sigma_c=\sigma_m$.
\end{itemize} \end{proposition}
\begin{proof} We shall do the first item, since the second is analogous. Fix the critical value $\sigma_c=2d_L-\mu_1-\mu_2$ and suppose that there is a strictly $\sigma_c$-semistable triple $T$ in either Case (1) or (3) above. Then the subtriple $T'$ and quotient triple $T''$ are both $\sigma_c$-semistable by Lemma \ref{lem:semistable}. In either case, there exists a $\sigma_c$-semistable triple of type $(1,2,d_1-d_L,d_2)$. By Proposition \ref{prop:alpha-range} applied to this situation, we get
$$
d_1-d_L-\frac{d_2}2 \leq \sigma_c= 2d_L-\frac{d_1}2-\frac{d_2}2 \leq
4\left(d_1-d_L-\frac{d_2}2 \right)\, .
$$ We can write this inequality in the equivalent form
$$
\frac{d_1}2 \leq d_L \leq \frac{3d_1-d_2}4 \,.
$$ \end{proof}
\begin{theorem} \label{thm:stabilize}
Let $\sigma_M= 2(\mu_1-\mu_2)$. For
$\sigma>\sigma_M$ the moduli spaces of $\sigma$-(semi)stable triples do not change,
and all $\sigma$-semistable triples
$T=(E_1,E_2,\phi)$ are injective, i.e., $T$ defines an
exact sequence of the form
$$
0 \to E_2 \overset{\phi}{\longrightarrow} E_1 \to S \to 0,
$$
where $S$ is a torsion sheaf of degree $d_1 -d_2$. \end{theorem}
\begin{proof}
If we are in the first situation in Proposition
\ref{prop:bounds-for-sc}, then $\sigma_c =2d_L-\mu_1-\mu_2 \leq
3\mu_1-\mu_2-\mu_1-\mu_2=2(\mu_1-\mu_2)$. In the second
situation, $\sigma_c=\mu_1+\mu_2-2d_F\leq \mu_1+\mu_2-
(3\mu_2-\mu_1)=2(\mu_1-\mu_2)$.
Now let $T$ be a $\sigma$-semistable triple for $\sigma>2(\mu_1-\mu_2)$.
If $\phi:E_2\to E_1$ were not injective, then $T$ has a
subtriple $T'=(0,\ker\phi,0)$ with $\lambda'>\lambda$. This
forces $\mu_\sigma(T')>\mu_\sigma(T)$ for $\sigma$ large, and hence for $\sigma$
bigger than the last critical value. \end{proof}
\begin{remark}\label{rem:stabilize}
Note that for any critical value $\sigma_c$, all the triples in
$\cS_{\scm}$ are not injective. \end{remark}
\begin{remark}\label{rem:stabilization}
By \cite[Proposition 6.5]{BGPG} there is a value $\sigma_0$ such that
all $\sigma$-semistable triples for $\sigma>\sigma_0$ are injective. By
\cite[Theorem 8.6]{BGPG} there is a value $\sigma_L$ such that the
moduli spaces $\mathcal{N}_\sigma$ are isomorphic for all $\sigma>\sigma_L$.
In our case, $n_1=n_2=2$,
both numbers are $2(\mu_1-\mu_2)$. \end{remark}
\begin{remark} \label{rem:virtual-not-all}
In Proposition \ref{prop:bounds-for-sc} we see that, for the
triples of rank $(2,2)$, there are critical values for which
the moduli spaces do not change (those corresponding to
$d_L>(3\mu_1-\mu_2)/2$ and those corresponding to
$d_F<(3\mu_2-\mu_1)/2$). \end{remark}
\begin{remark}\label{rem:simple-double}
If we have simultaneously $\sigma_c=2d_L-\mu_1-\mu_2$ and
$\sigma_c=\mu_1+\mu_2-2d_F$, then $2d_L-\mu_1-\mu_2=\mu_1+\mu_2-2d_F
\implies d_1+d_2 =2d_L+2d_F$ is an even number.
Therefore, if
$d_1+d_2\notin 2\mathbb{Z}$, then Cases (1) and (3) (resp.\ Cases (2) and
(4)) do not happen
simultaneously (for the same critical value). So the flip locus $\cS_{\scp}$
(resp.\ $\cS_{\scm}$) will consist only of triples of one type for
any $\sigma_c>\sigma_m$. In this situation
the critical values $\sigma_c\in (\mu_1+\mu_2+ \mathbb{Z}) \cap
[\mu_1-\mu_2, 2(\mu_1-\mu_2)]$.
If $d_1+d_2\in 2\mathbb{Z}$, then Cases (1) and (3) (resp.\ Cases (2) and
(4)) do happen simultaneously. The flip locus $\cS_{\scp}$ (resp.\ $\cS_{\scm}$)
consists of two types of triples, which yields two components
that must be considered independently. In this situation
the critical values $\sigma_c\in (\mu_1+\mu_2+ 2\mathbb{Z}) \cap
[\mu_1-\mu_2, 2(\mu_1-\mu_2)]$. \end{remark}
In the next section, it will be useful to have a vanishing result for the hypercohomology $\mathbb{H}^2$ to find the flip loci $\mathcal{S}_{\s_c^\pm}$ for the moduli spaces of triples of type $(2,2,d_1,d_2)$.
\begin{proposition} \label{prop:h2-vanishing}
Let $T=(E_1,E_2,\phi)$ be a strictly $\sigma_c$-semistable triple of
type $(2,2,d_1,d_2)$ with $\sigma_c>\sigma_m$, $T'=(E_1',E_2',\phi')$ a destabilizing
subtriple and $T''=T/T'=(E_1'',E_2'',\phi'')$ the corresponding
quotient triple.
\begin{itemize}
\item[(1)] If $T \in \cS_{\scm}$ then $\mathbb{H}^{2}(C^{\bullet}(T'',T')) = 0$.
\item[(2)] If $T \in \cS_{\scp}$ then $\mathbb{H}^{2}(C^{\bullet}(T'',T')) = 0$,
if $d_1-d_2 > 2g-2$.
\end{itemize} \end{proposition}
\begin{proof} By Proposition \ref{prop:hyper-equals-hom} and Serre duality, the vanishing $\mathbb{H}^{2}(C^{\bullet}(T'',T'))=0$ is equivalent to the injectivity of the map
$$
\begin{array}{ccc}
H^{0}({E_{1}'}^* \otimes E_2'' \otimes K) & \overset{P}{\longrightarrow} &
H^{0}({E_{1}'}^* \otimes E_1'' \otimes
K)\oplus H^{0}({E_{2}'}^* \otimes E_2'' \otimes K) \\
\psi & \longmapsto & ((\phi'' \otimes Id) \circ \psi, \, \psi
\circ \phi').
\end{array}
$$
\begin{itemize} \item[(1)] If $T \in \cS_{\scm}$, then $H^{0}({E_{1}'}^* \otimes E_2'' \otimes K)$ is trivial because either we are in Case (4) and so $E_{1}'=0$ or we are in Case (2) and so $E_{2}''=0$.
\item[(2)] If $T \in \cS_{\scp}$, we may have two cases: \begin{itemize} \item[(a)] If we are in Case (3), then $E_1'=E_1$ and $E_1''=0$. The map $P$ is
$$
\begin{array}{ccc}
H^{0}(E_{1}^* \otimes E_2'' \otimes K) & \overset{P}{\longrightarrow} &
H^{0}({E_{2}'}^* \otimes E_2'' \otimes K) \\
\psi & \longmapsto & \psi \circ \phi'.
\end{array}
$$ If $P$ is not injective, let $\psi: E_{1} \rightarrow E_2'' \otimes K$ be a non-trivial homomorphism in $\ker P$. Then, as $\phi': E_2' \rightarrow E_1$, $\psi$ must factor through the quotient $E_1/E_2'$. Both $E_1/E_2'$ and $E_2'' \otimes K$ are line bundles, hence $\deg(E_1/E_2')= d_1-d_2' \leq \deg(E_2'' \otimes K) =d_2''+2g-2$. This yields $d_1-d_2 \leq 2g-2$.
\item[(b)] If we are in Case (1), then $E_2'=0$ and $E_2''=E_2$. Then the map $P$ is
$$
\begin{array}{ccc}
H^{0}({E_{1}'}^* \otimes E_2'' \otimes K) & \overset{P}{\longrightarrow} &
H^{0}({E_{1}'}^* \otimes E_1'' \otimes K) \\
\psi & \longmapsto & (\phi'' \otimes Id) \circ \psi.
\end{array}
$$ If $P$ is not injective, let $\psi: E_{1}' \rightarrow E_2 \otimes K$ be a non-trivial homomorphism in $\ker P$. Denote by $Q$ the kernel of $\phi'': E_2 \rightarrow E_1''$, so $\psi$ must factor through $Q \otimes K$. As $E_1'$ and $Q\otimes K$ are line bundles, we have $\deg({E_1'})=d_1' \leq \deg(Q \otimes K) = d_2-d_1''+2g-2$, which is rewritten as $d_1-d_2 \leq 2g-2$.
\end{itemize} In both cases, if $P$ is not injective then $d_1-d_2 \leq 2g-2$. Therefore, if $d_1-d_2> 2g-2$, then $P$ must be injective. \end{itemize} \end{proof}
\begin{remark}\label{rem:compareBGPG} This result is a sort of improvement of \cite[Proposition 3.6]{BGPG} for the case of triples of rank $(2,2)$. Here we prove the vanishing of $\mathbb{H}^2$ for \textit{any} critical value $\sigma_c$ under the condition $\sigma_m=\mu_1-\mu_2> g-1$, whereas in \cite[Proposition 3.6]{BGPG} it is proved the vanishing of $\mathbb{H}^2$ only for critical values $\sigma_c>2g-2$ (but without condition in $\sigma_m$). \end{remark}
\section{Hodge polynomial of the moduli of triples of rank $(2,2)$ and small $\sigma$} \label{sec:small}
In this section we want to compute the Hodge polynomial of the moduli space
$$
\mathcal{N}_{{\s_m^+}}=\mathcal{N}_{{\s_m^+}}(2,2,d_1,d_2)
$$ of $\sigma$-stable triples of types $(2,2,d_1,d_2)$ for $\sigma$ small, under the assumption $\mu_1-\mu_2>2g-2$. The study of $\mathcal{N}_{{\s_m^+}}$ is simpler when both $d_1$ and $d_2$ are odd, since in this case the bundles are automatically stable. However in this case $d_1+d_2$ is even and hence $\gcd(2,2,d_1+d_2)\neq 1$. So there may be strictly $\sigma$-semistable triples in $\mathcal{N}_\sigma$ for non-critical values of $\sigma$, making the moduli space $\mathcal{N}_\sigma^s$ non-compact and the moduli space $\mathcal{N}_\sigma$ singular (this does not happen for $\sigma={\s_m^+}$; see Theorem \ref{thm:(2,2,odd,odd)}).
\begin{theorem}\label{thm:(2,2,odd,odd)} Suppose that $d_1$ and $d_2$ are odd and that $\mu_1-\mu_2>2g-2$. Then $\mathcal{N}_{\s_m^+}=\mathcal{N}_{\s_m^+}^s$, it is smooth, compact and
$$
e(\mathcal{N}_{\s_m^+}) = \left(\frac{(1+u)^{g}(1+v)^g(1+u^2v)^{g}(1+uv^2)^g
-(uv)^{g}(1+u)^{2g}(1+v)^{2g}}
{(1-uv)(1-(uv)^2)}\right)^2
\frac{1-(uv)^{2d_1-2d_2-4g+4}}{1-uv} \, .
$$ \end{theorem}
\begin{proof}
The equality $\mathcal{N}_{\s_m^+}=\mathcal{N}_{\s_m^+}^s$ is a consequence of Proposition
\ref{prop:moduli-small} (i).
Next, since $\sigma_m=\mu_1-\mu_2>2g-2$, Theorem
\ref{thm:moduli(2,2)} implies that
the moduli $\cN_{\smp}$ is smooth and compact.
By Proposition \ref{prop:moduli-small} (i), it is
the projectivization of a fiber bundle over $M(2,d_1) \times
M(2,d_2)$ of rank $2d_1-2d_2-4g+4$. Therefore
$$
e(\mathcal{N}_{\s_m^+}) = e(M(2,d_1)) e(M(2,d_2)) e_{2d_1-2d_2-4g+4} \, .
$$
The result follows now applying Theorem \ref{thm:rank2odd}. \end{proof}
The case where $d_1$ is odd and $d_2$ is even is more involved, since we have to deal with the presence of strictly semistable bundles in $M(2,d_2)$.
\begin{theorem}\label{thm:(2,2,odd,even)} Suppose that $d_1$ is odd and $d_2$ is even and that $\mu_1-\mu_2>2g-2$. Then $\mathcal{N}_{\s_m^+}=\mathcal{N}_{\s_m^+}^s$, it is smooth and compact and
$$
\begin{aligned}
e(\cN_{\smp}) = & \ \frac{(1+u)^{2g}(1+v)^{2g}(1-(uv)^N)
(u^gv^g(1+u)^g(1+v)^g-(1+u^2v)^g(1+uv^2)^g)}{(1-uv)^3(1-(uv)^2)^2}
\cdot \\ &
\bigg(
(1+u)^{g} (1+v)^g (u^{g+1}v^{g+1} + u^{N+g-1}v^{N+g-1}) -
(1+u^2v)^{g}(1+uv^2)^g (1 + u^Nv^{N})\bigg) \, ,
\end{aligned}
$$ where $N=d_1-d_2-2g+2$. \end{theorem}
\begin{proof} As $d_1+d_2$ is odd, Theorem \ref{thm:moduli(2,2)} implies that $\cN_{\smp}=\cN_{\smp}^s$, and it is smooth and compact, since $\sigma_m=\mu_1-\mu_2>2g-2$. To compute $e(\mathcal{N}_{\s_m^+})$ we decompose $\mathcal{N}_{\s_m^+}=X_0\sqcup X_1\sqcup X_2\sqcup X_3\sqcup X_4$, where:
\begin{enumerate} \item[(1)] The open subset $X_0\subset \cN_{\smp}$ consists of those triples of the form $\phi:E_2\to E_1$, where $E_1$ and $E_2$ are both stable bundles, and $\phi$ is a non-zero map defined up to multiplication by scalars. By Proposition \ref{prop:moduli-small} (ii), $X_0 \to M(2,d_1)\times M^s(2,d_2)$ is a projective fibration whose fibers are projective spaces of dimension $2d_1-2d_2-4g+4-1=2N-1$. Therefore, and using the notation (\ref{eqn:Pn}),
$$
e(X_0)= e(M(2,d_1)) e(M^s(2,d_2))e_{2N} \, .
$$
\item[(2)] The subset $X_1$ parametrizes ${\s_m^+}$-stable triples of the form $\phi:E_2\to E_1$ where $E_2$ is a strictly semistable bundle of degree $d_2$ which is a non-split extension
$$
0 \to L_1 \to E_2\to L_2\to 0,
$$ where $L_1, L_2\in \Jac^{d_2/2} X$ are non-isomorphic and $E_1$ is a stable bundle. The space $Y_1$ parametrizing such bundles $E_2$ was described in (2) of the proof of Theorem \ref{thm:rank2even} and its Hodge polynomial is given in (\ref{eqn:Y1}).
Now in order to describe $X_1$, we must characterize when a triple $T=(E_1,E_2,\phi)$, with $E_2\in Y_1$, is ${\s_m^+}$-stable. As $T$ is $\sigma_m$-semistable, then the only possibility for $T$ being ${\s_m^+}$-unstable is that it has a subtriple $T'$ of rank $(1,2)$ or $(0,1)$, corresponding to Cases (2) or (4) of Section \ref{sec:critical(2,2)}, respectively. If $T'$ is of rank $(1,2)$, then it is of the form $E_2\to L$, where $L$ is a line bundle of degree $d_L=\mu_1$, by Proposition \ref{prop:bounds-for-sc}. But this is impossible, since $d_1$ is odd. If $T'$ is of rank $(0,1)$, then it is of the form $F \to 0$, where $F$ is a line bundle of degree $d_F=\mu_2$, by Proposition \ref{prop:bounds-for-sc}. Therefore $F$ is a destabilizing subbundle for $E_2$. Since the only destabilizing subbundle of $E_2$ is $L_1$, we have $F=L_1$. So it must be $\phi(L_1)=0$. Any such $\phi$ lies in the image of the inclusion $\Hom(L_2,E_1)\hookrightarrow \Hom(E_2,E_1)$, under the natural projection $E_2\to L_2$. This discussion implies that given $(E_1,E_2)\in M(2,d_1)\times Y_1$, the morphisms $\phi$ giving rise to ${\s_m^+}$-stable triples $(E_1,E_2,\phi)$ are those in
$$
\Hom (E_2,E_1) - \Hom (L_2,E_1)\ .
$$ Note that since the group of automorphisms of $E_1$ and $E_2$ are both equal to $\mathbb{C}^*$, $\phi$ is defined up to multiplication by non-zero scalars. So the map $\pi:X_1\to M(2,d_1)\times Y_1$ is a fibration with fiber over $(E_1,E_2)$ equal to
\begin{equation}\label{eqn:333}
\mathbb{P}\Hom (E_2,E_1) - \mathbb{P} \Hom (L_2,E_1)\ .
\end{equation} By Riemman-Roch, $\dim \Hom(E_2,E_1)=2d_1-2d_2-4g+4=2N$, since $\mu_1-\mu_2>2g-2$ implies that $H^1(E_2^*\otimes E_1)= H^0(E_1\otimes E_2^*\otimes K)=0$, $E_1$ and $E_2$ being both semistable bundles. Also $\dim \Hom(L_2,E_1)=d_1-2(d_2/2)-2g+2=d_1-d_2-2g+2=N$, since $\mu_1-\deg L_2=\mu_1-d_2/2>2g-2$. Hence (\ref{eqn:333}) is isomorphic to $\mathbb{P}^{2N-1}-\mathbb{P}^{N-1}$. Therefore as in (2) of the proof of Theorem \ref{thm:rank2even},
\begin{align*}
e(X_1)= & \ e(M(2,d_1)) e(Y_1) (e_{2N}-e_N) \\
=& \ e(M(2,d_1))e(\Jac X)(e(\Jac X)-1)e_{g-1} (e_{2N}-e_{N}) \, .
\end{align*}
\item[(3)] The subset $X_2$ parametrizes ${\s_m^+}$-stable triples of the form $\phi:E_2\to E_1$ where $E_2$ is a strictly semistable bundle of degree $d_2$ which is non-split extension
$$
0 \to L_1 \to E_2\to L_1\to 0,
$$ where $L_1 \in \Jac^{d_2/2} X$ and $E_1$ is a stable bundle. The space $Y_2$ parametrizing such bundles $E_2$ was described in (3) of the proof of Theorem \ref{thm:rank2even} and its Hodge polynomial is given in (\ref{eqn:Y2}).
To describe $X_2$, we must characterize when a triple $T=(E_1,E_2,\phi)$, with $E_2\in Y_2$, is ${\s_m^+}$-stable. As before, given $(E_1,E_2)\in M(2,d_1)\times Y_2$, the morphisms $\phi$ giving rise to ${\s_m^+}$-stable triples $(E_1,E_2,\phi)$ are those in
$$
\Hom (E_2,E_1) - \Hom (L_1,E_1)\ .
$$
For a triple $T=(E_1,E_2,\phi)\in X_2$, $\Aut(E_1)=\mathbb{C}^*$, so $\Aut(T)\cong \Aut(E_2)=\mathbb{C}\times\mathbb{C}^*$. There is an exact sequence
$$
0\to \Hom(L_1,E_1) \to \Hom(E_2,E_1)\to \Hom(L_1,E_1)\to 0
$$ Under the (non-canonical) decomposition $\Hom(E_2,E_1)\cong \Hom(L_1,E_1)\oplus \Hom(L_1,E_1)$, $\Aut(E_2)$ acts as $(a,\lambda) (x,y)\mapsto (\lambda x+a y, \lambda y)$. So the fiber of $\pi:X_2\to M(2,d_1)\times Y_2$ is
$$
(\Hom (E_2,E_1) - \Hom (L_1,E_1))/\mathbb{C}\times \mathbb{C}^* \cong
(\mathbb{C}^{2N}-\mathbb{C}^N)/\mathbb{C}\times\mathbb{C}^*\, ,
$$ which is a $\mathbb{C}^{N-1}$-bundle over $\mathbb{P}^{N-1}$. Therefore as in (3) of the proof of Theorem \ref{thm:rank2even},
\begin{align*}
e(X_2)= & \ e(M(2,d_1)) e(Y_2)
(e_{N}-e_{N-1})e_N \\ =& \ e(M(2,d_1)) e(\Jac X) e_g
(e_{N}-e_{N-1})e_N \, .
\end{align*}
\item[(4)] The subset $X_3$ parametrizes ${\s_m^+}$-stable triples of the form $\phi:E_2\to E_1$ where $E_1$ is a stable bundle and $E_2=L_1\oplus L_2$, $L_1\not\cong L_2$, are two line bundles of degree $d_2/2$. The space $Y_3$ parametrizing such bundles is described in (\ref{eqn:Y3}).
As above, the condition for $\phi\in \Hom(E_2,E_1)$ to give rise to a ${\s_m^+}$-unstable triple is that there is a subtriple $T'$ of the form $F\to 0$, with $F$ a line bundle of degree $d_F=\mu_2$. Then it must be either $F=L_1$ or $F=L_2$. This means that $\phi\in (\Hom(L_1,E_1)\oplus \{0\} )\cup (\{0\}\oplus\Hom(L_2,E_1)) \subset \Hom(E_2,E_1)$. Therefore, given $(E_1,E_2)\in M(2,d_1)\times Y_3$, the morphisms $\phi$ giving rise to ${\s_m^+}$-stable triples $(E_1,E_2,\phi)$ are those in
$$
(\Hom (L_1,E_1)-\{0\})\times (\Hom(L_2,E_1)-\{0\}).
$$
The group of automorphisms of $E_2$ is $\mathbb{C}^*\times \mathbb{C}^*$ acting on $L_1\oplus L_2$ by diagonal matrices. Therefore $\phi\in (\Hom (L_1,E_1)-\{0\})\times (\Hom(L_2,E_1)-\{0\})$ is defined up to the action of $\mathbb{C}^*\times \mathbb{C}^*$, where each $\mathbb{C}^*$ acts by multiplication on each of the two summands. So the map $\pi:X_3\to M(2,d_1)\times Y_3$ has fiber
\begin{equation}\label{eqn:444}
\mathbb{P} \Hom (L_1,E_1)\times\mathbb{P}\Hom(L_2,E_1).
\end{equation} By Riemann-Roch, $\dim\Hom(L_1,E_1)=\dim\Hom(L_2,E_1)= d_1-d_2-2g+2$. Therefore (\ref{eqn:444}) is isomorphic $\mathbb{P}^{N-1}\times \mathbb{P}^{N-1}$. To compute $e(X_3)$ we work as in (4) of the proof of Theorem \ref{thm:rank2even}. Write $X_3=\tilde{X}_3/\mathbb{Z}_2=X_3'/\mathbb{Z}_2-X_3''/\mathbb{Z}_2$, where $X_3'$ is a fibration over $M(2,d_1)$ with fiber $(A_{E_1}\times A_{E_1})/\mathbb{Z}_2$, where $A_{E_1}$ is a projective bundle over $\Jac^{d_2/2}X$ with fibers $\mathbb{P} \Hom (L,E_1)\cong \mathbb{P}^{N-1}$, and $\mathbb{Z}_2$ acts by permutation. $X_3''$ is a fibration over $M(2,d_1)\times \Jac^{d_2/2}X$ with fibers $(\mathbb{P}^{N-1}\times \mathbb{P}^{N-1})/\mathbb{Z}_2$. So using Theorem \ref{thm:Du},
$$
\begin{aligned}
e(X_3)= & \ e(\tilde{X}_3/\mathbb{Z}_2)= e(X_3'/\mathbb{Z}_2)-e(X_3''/\mathbb{Z}_2) \\
= & \ \frac12 e(M(2,d_1)\bigg(\bigg( e(\Jac X)^2e_N^2 + (1-u^2)^g(1-v^2)^g
\frac{1-(uv)^{2N}}{1-u^2v^2}\bigg) \\ &
-e(\Jac X) \bigg( e_N^2 +
\frac{1-(uv)^{2N}}{1-u^2v^2}\bigg)\bigg)\, .
\end{aligned}
$$
\item[(5)] The subset $X_4$ parametrizes triples $\phi:E_2\to E_1$, where $E_1$ is a stable bundle and $E_2=L_1\oplus L_1$, $L_1\in \Jac^{d_2/2}X$. Such bundles $E_2$ are parametrized by $Y_4=\Jac^{d_2/2} X$. The map $\phi$ lies in
\begin{equation}\label{eqn:555}
\Hom(E_2,E_1)=\Hom(L_1,E_1)\oplus \Hom(L_1,E_1) \cong
\Hom(L_1,E_1)\otimes \mathbb{C}^2\, .
\end{equation}
The condition for a triple $T=(E_1,E_2,\phi)$ to be ${\s_m^+}$-unstable is that there is a line subbundle $F\subset E_2$ of degree $d_F=\mu_2$ such that $\phi(F)=0$. A destabilizing subbundle of $E_2$ is necessarily isomorphic to $L$ and there exists $(a,b)\neq (0,0)$ such that $F\cong L\hookrightarrow E_2$ is given by $x\mapsto (ax,bx)$. So $\phi=(a\psi,b\psi)\in\Hom(L_1,E_1)\otimes \mathbb{C}^2$, for some $\psi\in \Hom(L,E_1)$. Therefore $T=(E_1,E_2,\phi)$ is ${\s_m^+}$-stable if $\phi=(\phi_1,\phi_2)\in\Hom(L_1,E_1)\otimes \mathbb{C}^2$ satisfies that $\phi_1,\phi_2$ are linearly independent.
On the other hand, a triple $(E_1,E_2,\phi)\in X_4$ is determined up to the action of $\Aut(E_2)=GL(2,\mathbb{C})$. This acts on (\ref{eqn:555}) via the standard representation on $\mathbb{C}^2$. Thus for $(E_1,E_2)\in M(2,d_1)\times Y_4$, the morphisms $\phi$ giving rise to ${\s_m^+}$-stable triples $(E_1,E_2,\phi)$ are parametrized by $\Gr(2,\Hom(L_1,E_1))$. But $\dim \Hom(L_1,E_1)=d_1-d_2-2g+2=N$, so this fiber is isomorphic to $\Gr(2,N)$. So
$$
e(X_4)=e(M(2,d_1)) e(Y_4) e(\Gr(2,N))=e(M(2,d_1)) e(\Jac X) e(\Gr(2,N)) \ .
$$
\end{enumerate}
Adding up all contributions together we get
\begin{align*}
e(\mathcal{N}_{{\s_m^+}})= & \ e(X_0)+ e(X_1)+e(X_2)+e(X_3)+e(X_4) \\
=& \ e(M(2,d_1)) \Bigg( e(M^s(2,d_2))e_{2N} + e(\Jac X) (e(\Jac X)-1) e_{g-1}
(e_{2N}-e_{N})
\\
&\qquad+ e(\Jac X) e_g (e_{N}-e_{N-1})e_N
+ \frac12 \bigg( e(\Jac X)^2e_N^2 + (1-u^2)^g(1-v^2)^g
\frac{1-(uv)^{2N}}{1-u^2v^2}\bigg)
\\
&\qquad
-\frac12 e(\Jac X) \bigg( e_N^2 +
\frac{1-(uv)^{2N}}{1-u^2v^2}\bigg)
+ e(\Jac X) e(\Gr(2,N)) \Bigg) \\
=& \ \frac{(1+u)^{2g}(1+v)^{2g}(1-(uv)^N)
(u^gv^g(1+u)^g(1+v)^g-(1+u^2v)^g(1+uv^2)^g)}{(1-uv)^3(1-(uv)^2)^2} \cdot
\\ &
\bigg(
(1+u)^{g} (1+v)^g (u^{g+1}v^{g+1} + u^{N+g-1}v^{N+g-1}) -
(1+u^2v)^{g}(1+uv^2)^g (1 + u^Nv^{N})\bigg) \, .
\end{align*} \end{proof}
\begin{corollary}\label{cor:(2,2,odd,even)-2} Suppose that $d_1$ is even and $d_2$ is odd and that $\mu_1-\mu_2>2g-2$. Then $\cN_{\smp}=\cN_{\smp}^s$, it is smooth and compact and its Hodge polynomial has the same formula as that of Theorem \ref{thm:(2,2,odd,even)}, where $N=d_1-d_2-2g+2$. \end{corollary}
\begin{proof} We use the isomorphism $\mathcal{N}_{\sigma}(2,2,d_1,d_2)\cong \mathcal{N}_{\sigma}(2,2,-d_2,-d_1)$. Note that
$$
d_1-d_2= (-d_2) - (-d_1)\, ,
$$ so that the small value ${\s_m^+}=\mu_1-\mu_2$ and the condition on the slopes $\mu_1-\mu_2>2g-2$ is the same for both moduli spaces $\mathcal{N}_{\sigma}(2,2,d_1,d_2)$ and $\mathcal{N}_{\sigma}(2,2,-d_2,-d_1)$. Now we apply Theorem \ref{thm:(2,2,odd,even)} to get the stated formula where $N=-d_2-(-d_1)-2g+2$. \end{proof}
\begin{corollary}\label{cor:(2,2,odd,even)} Suppose that $d_1+d_2$ is odd and $\mu_1-\mu_2>2g-2$. Then the Poincar{\'e} polynomial of $\mathcal{N}_{\s_m^+}$ is
$$
P_t(\cN_{\smp}) = \frac{(1\!+\!t)^{4g} (1\!-\!t^{2N})(t^{2g}(1\!+\!t)^{2g} \!-\! (1\!+\!t^3)^{2g})
((1\!+\!t)^{2g} (t^{2g\!+\!2}\!+\! t^{2N\!+\!2g\!-\!2}) \!-\! (1\!+\!t^3)^{2g}
(1 \!+\! t^{2N}))}{(1\!-\!t^2)^3(1\!-\!t^4)^2} \, ,
$$ where $N=d_1-d_2-2g+2$.
$\Box$ \end{corollary}
\begin{proof}
$\mathcal{N}_{{\s_m^+}}$ is smooth and projective, so $P_t(\mathcal{N}_{{\s_m^+}})=e(\mathcal{N}_{{\s_m^+}})(t,t)$.
The result follows from Theorem \ref{thm:(2,2,odd,even)} and
Corollary \ref{cor:(2,2,odd,even)-2}. \end{proof}
We could deal also with the case when $d_1$ and $d_2$ are both even and $d_1-d_2>4g-4$. This is similar to the case just treated in Theorem \ref{thm:(2,2,odd,even)}, with the further complication that there are semistable loci for both $E_1$ and $E_2$.
However, dealing with the case $d_1-d_2\leq 4g-4$ is more complicated, since Proposition \ref{prop:moduli-small} does not apply as there is a Brill-Noether problem consisting on determining the loci of those $(E_1,E_2)$ where $\dim \Hom(E_2,E_1\otimes K)$ is constant.
\section{Contribution of the flips to the Hodge polynomials} \label{sec:simple}
In this section, we shall compute the change in the Hodge polynomial of $\mathcal{N}_\sigma(2,2,d_1,d_2)$ when we cross a critical value $\sigma_c$. We restrict to the case $d_1+d_2$ is odd, since in the case $d_1+d_2$ even there may be strictly $\sigma$-semistable triples for non-critical values of $\sigma$ (and in this case $\mathcal{N}_\sigma^s$ is non-compact and $\mathcal{N}_\sigma$ is non-smooth). For $d_1+d_2$ odd, Theorem \ref{thm:moduli(2,2)} guarantees that $\mathcal{N}_\sigma$ is compact and smooth for any non-critical $\sigma\geq 2g-2$. The critical values are given in Proposition \ref{prop:bounds-for-sc}. These are of two types. The following two propositions treat them separately.
\begin{proposition}\label{prop:diff-poly-simple-2} Let $\sigma_c=2d_L-\mu_1-\mu_2$ be a critical value for triples of type $(2,2,d_1,d_2)$ with $d_1+d_2$ odd, such that $\sigma_c>\sigma_m$. Suppose that $\mu_1-\mu_2>g-1$. Then
$$
\begin{aligned}
e(\cN_{\s_c^+})-e(\cN_{\s_c^-})=
\mathop{\mathrm{coeff}}_{x^0}\Bigg[&
\frac{(1+u)^{3g}(1+v)^{3g}(1+ux)^{g}(1+vx)^g\big((uv)^{g-1-d_1+2d_L}-(uv)^{1-g+d_1-d_2}\big)}
{(1-uv)^2(1-x)(1-uvx)x^{[3\mu_1-\mu_2]-2d_L}}\\
&
\Bigg(\frac{(uv)^{(3d_1-d_2-1)/2-2d_L}}{1-(uv)^{-1}x}
-\, \frac{(uv)^{2d_L-d_1+g}}{1-(uv)^2x}\Bigg) \Bigg]\,.
\end{aligned}
$$ \end{proposition}
\begin{proof} Theorem \ref{thm:moduli(2,2)} implies that $\mathcal{N}_{\s_c^\pm}=\mathcal{N}_{\s_c^\pm}^s$. Then Lemma \ref{lem:fliploci} and the properties of the Hodge polynomials give
$$
e(\mathcal{N}_{{\s_c^+}})-e( \mathcal{N}_{{\s_c^-}}) = e(\mathcal{S}_{{\s_c^+}}) -e(\mathcal{S}_{{\s_c^-}}).
$$
Let us start by studying $\cS_{\scp}$. By Lemma \ref{lem:semistable}, any $T\in \cS_{\scp}$ sits in a non-split extension
\begin{equation}\label{eqn:sec8}
0 \rightarrow T' \rightarrow T \rightarrow T'' \rightarrow 0
\end{equation} in which $T'$ and $T''$ are $\sigma_c$-semistable, $\lambda'<\lambda$ and $\mu_{\sigma_c}(T')=\mu_{\sigma_c}(T)=\mu_{\sigma_c}(T'')$. Since $T$ corresponds to Case (1) in Section \ref{sec:critical(2,2)}, we have $T'\in \mathcal{N}_{\sigma_c}'$ and $T''\in \mathcal{N}_{\sigma_c}''$, where
\begin{align*}
\mathcal{N}_{\sigma_c}' &= \mathcal{N}_{\sigma_c}(1,0,d_L,0) \cong \Jac^{d_L} X, \\
\mathcal{N}_{\sigma_c}''&= \mathcal{N}_{\sigma_c}(1,2,d_1-d_L,d_2).
\end{align*} The moduli space of triples of rank $(1,0)$ has no critical values; and for the moduli space of triples of rank $(1,2)$, the critical values are of the form $3d_M + d_1''+d_2''$, by Lemma \ref{lem:dM-2}, and are in particular integers. But $\sigma_c=2d_L-\frac{d_1+d_2}{2} \notin\mathbb{Z}$, so $\sigma_c$ is not a critical value for $\mathcal{N}_{\sigma_c}''$.
By \cite[Proposition 3.5]{BGPG}, $\mathbb{H}^0(T'',T')=0$ and by Proposition \ref{prop:h2-vanishing} (2), $\mathbb{H}^2(T'',T')=0$ . So Theorem \ref{thm:Smas} implies that $\cS_{\scp}$ is the projectivization of a bundle over $\mathcal{N}_{\sigma_c}'\times \mathcal{N}_{\sigma_c}''$ of rank
$$
-\chi(T'',T')= 1 -g+d_1-d_2\, .
$$ Therefore
$$
e(\cS_{\scp})= e(\Jac^{d_L} X)\, e(\mathcal{N}_{\sigma_c}(1,2,d_1-d_L,d_2))\, e_{1 -g+d_1-d_2}\, .
$$
The case of $\cS_{\scm}$ is similar. Any $T\in\cS_{\scm}$ sits in an exact sequence (\ref{eqn:sec8}) with $T'\in \mathcal{N}_{\sigma_c}'$ and $T''\in \mathcal{N}_{\sigma_c}''$, where
\begin{align*}
\mathcal{N}_{\sigma_c}' &= \mathcal{N}_{\sigma_c}(1,2,d_1-d_L,d_2), \\
\mathcal{N}_{\sigma_c}''&= \mathcal{N}_{\sigma_c}(1,0,d_L,0)\cong \Jac^{d_L} X,
\end{align*} corresponding to the Case (2) in Section \ref{sec:critical(2,2)}. The hypothesis of Theorem \ref{thm:Smas} are satisfied and so $\cS_{\scm}$ is the projectivization of a bundle over $\mathcal{N}_{\sigma_c}'\times \mathcal{N}_{\sigma_c}''$ of rank
$$
-\chi(T'',T')= g-1 -d_1+2d_L\, .
$$ Therefore
$$
e(\cS_{\scm})= e(\Jac^{d_L} X) \, e(\mathcal{N}_{\sigma_c}(1,2,d_1-d_L,d_2)) \, e_{g-1 -d_1+2d_L}\, .
$$
Substracting, we get
\begin{align*}
e(\cS_{\scp})-e(\cS_{\scm}) &= (e_{1
-g+d_1-d_2}- e_{g-1 -d_1+2d_L}) (1+u)^{g}(1+v)^g
e(\mathcal{N}_{\sigma_c}(1,2,d_1-d_L,d_2)) = \\
&= \frac{(uv)^{g-1-d_1+2d_L}-(uv)^{1-g+d_1-d_2}}{1-uv}
(1+u)^{g}(1+v)^g e(\mathcal{N}_{\sigma_c}(1,2,d_1-d_L,d_2))\, .
\end{align*}
Being $\sigma_c$ a non-critical value for the moduli of triples of rank $(1,2)$, we can apply Theorem \ref{thm:polinomiono(1,2)no-critico} to compute the Hodge polynomial of $\mathcal{N}_\sigma(1,2,d_1-d_L,d_2)$. First,
$$
\begin{aligned}
d_0 &=\left[\frac13 (2d_L-\mu_1-\mu_2-(d_1-d_L)-d_2)\right]+1\\
&=d_L +[-\mu_1-\mu_2] +1 \, .
\end{aligned}
$$ So $e(\mathcal{N}_\sigma(1,2,d_1-d_L,d_2))$ equals
$$
\mathop{\mathrm{coeff}}_{x^0} \Bigg[
\frac{(1+u)^{2g}(1+v)^{2g}(1+ux)^{g}(1+vx)^{g}}{(1-uv)(1-x)(1-uv x)x^{d_1-d_2-d_L-d_0}}
\Bigg(\frac{(uv)^{d_1-d_2-d_L-d_0}}{1-(uv)^{-1}x}-\,
\frac{(uv)^{d_2+g-1+2d_0}}{1-(uv)^2x}\Bigg) \Bigg]\, ,
$$ where $d_1-d_2-d_L-d_0= [3\mu_1-\mu_2]-2d_L=(3d_1-d_2-1)/2-2d_L$ and $d_2+2d_0=2d_L-d_1+1$. The result follows from this. \end{proof}
\begin{proposition}\label{prop:diff-poly-simple-1} Let $\sigma_c=\mu_1+\mu_2 - 2d_F$ be a critical value for triples of type $(2,2,d_1,d_2)$ with $d_1+d_2$ odd, such that $\sigma_c>\sigma_m$. Suppose that $\mu_1-\mu_2>g-1$. Then
$$
\begin{aligned}
e(\cN_{\s_c^+})-e(\cN_{\s_c^-})=
\mathop{\mathrm{coeff}}_{x^0}\Bigg[&
\frac{(1+u)^{3g}(1+v)^{3g}(1+ux)^{g}(1+vx)^g\big((uv)^{g-1+d_2-2d_F}-(uv)^{1-g+d_1-d_2}\big)}
{(1-uv)^2(1-x)(1-uvx)x^{2d_F - [3\mu_2-\mu_1]-1}}\\
&
\Bigg(\frac{(uv)^{2d_F+(d_1-3d_2-1)/2}}{1-(uv)^{-1}x}
-\, \frac{(uv)^{d_2-2d_F+g}}{1-(uv)^2x}\Bigg) \Bigg] \,.
\end{aligned}
$$ \end{proposition}
\begin{proof} This is very similar to the proof of Proposition \ref{prop:diff-poly-simple-2}. Again
$$
e(\mathcal{N}_{{\s_c^+}})-e( \mathcal{N}_{{\s_c^-}}) = e(\mathcal{S}_{{\s_c^+}})
-e(\mathcal{S}_{{\s_c^-}}).
$$ We start with $\cS_{\scp}$. Any $T\in \cS_{\scp}$ sits in a non-split extension like (\ref{eqn:sec8}), with $\mu_{\sigma_c}(T')=\mu_{\sigma_c}(T)=\mu_{\sigma_c}(T'')$, $T'\in \mathcal{N}_{\sigma_c}'$ and $T''\in \mathcal{N}_{\sigma_c}''$, where
\begin{align*}
\mathcal{N}_{\sigma_c}' &= \mathcal{N}_{\sigma_c}(2,1,d_1,d_2-d_F), \\
\mathcal{N}_{\sigma_c}''&= \mathcal{N}_{\sigma_c}(0,1,0,d_F)\cong \Jac^{d_F} X,
\end{align*} corresponding to the Case (3) in Section \ref{sec:critical(2,2)}. The moduli space of triples of rank $(0,1)$ has no critical values; and for the moduli space of triples of rank $(2,1)$, the critical values are of the form $3d_M -d_1' -d_2' \in\mathbb{Z}$, whilst $\sigma_c=\frac{d_1+d_2}{2} - 2d_F \notin\mathbb{Z}$, so $\sigma_c$ is not a critical value for $\mathcal{N}_{\sigma_c}'$. The other conditions of Theorem \ref{thm:Smas} are checked as before. So $\cS_{\scp}$ is the projectivization of a bundle over $\mathcal{N}_{\sigma_c}'\times \mathcal{N}_{\sigma_c}''$ of rank
$$
-\chi(T'',T')= 1 -g+d_1-d_2\, .
$$ Therefore
$$
e(\cS_{\scp})= e(\Jac^{d_F} X)\, e(\mathcal{N}_{\sigma_c}(2,1,d_1,d_2-d_F))\, e_{1 -g+d_1-d_2}\, .
$$
Moving to $\cS_{\scm}$, any $T\in\cS_{\scm}$ sits in an exact sequence (\ref{eqn:sec8}) with $T'\in \mathcal{N}_{\sigma_c}'$ and $T''\in \mathcal{N}_{\sigma_c}''$, where
\begin{align*}
\mathcal{N}_{\sigma_c}' &= \mathcal{N}_{\sigma_c}(0,1,0,d_F)\cong \Jac^{d_F} X, \\
\mathcal{N}_{\sigma_c}''&= \mathcal{N}_{\sigma_c}(2,1,d_1,d_2-d_F),
\end{align*} corresponding to the Case (4) in Section \ref{sec:critical(2,2)}. Arguing as before, we have that $\cS_{\scm}$ is the projectivization of a bundle over $\mathcal{N}_{\sigma_c}'\times \mathcal{N}_{\sigma_c}''$ of rank
$$
-\chi(T'',T')= g-1 +d_2-2d_F\, .
$$ Therefore
$$
e(\cS_{\scm})= e(\Jac^{d_F} X) \, e(\mathcal{N}_{\sigma_c}(2,1,d_1,d_2-d_F)) \,
e_{g-1 +d_2-2d_F}\, .
$$
Substracting, we get
\begin{align*}
e(\cS_{\scp})-e(\cS_{\scm}) &= (e_{1
-g+d_1-d_2}-e_{g-1 +d_2-2d_F}) (1+u)^{g}(1+v)^g
e(\mathcal{N}_{\sigma_c}(2,1,d_1,d_2-d_F)) = \\
&= \frac{(uv)^{g-1+d_2-2d_F}-t^{1-g+d_1-d_2}}{1-uv}
(1+u)^{g}(1+v)^g e(\mathcal{N}_{\sigma_c}(2,1,d_1,d_2-d_F))\, .
\end{align*}
Being $\sigma_c$ a non-critical value for the moduli of triples of rank $(2,1)$, we can apply Theorem \ref{thm:polinomiono(2,1)no-critico} to compute the Hodge polynomial of $\mathcal{N}_\sigma(2,1,d_1,d_2-d_F)$. First,
$$
\begin{aligned}
d_0 &=\left[\frac13 (\mu_1+\mu_2-2d_F
+d_1+d_2-d_F\right]+1\\
&=[\mu_1+\mu_2] -d_F +1
\end{aligned}
$$ So $e(\mathcal{N}_\sigma(2,1,d_1,d_2-d_F))$ equals
$$
\mathop{\mathrm{coeff}}_{x^0} \Bigg[
\frac{(1+u)^{2g}(1+v)^{2g}(1+ux)^{g}(1+vx)^{g}}{(1-uv)(1-x)(1-uv x)x^{d_1-d_2+d_F-d_0}}
\Bigg(\frac{(uv)^{d_1-d_2+d_F-d_0}}{1-(uv)^{-1}x}-\,
\frac{(uv)^{-d_1+g-1+2d_0}}{1-(uv)^2x}\Bigg) \Bigg]\, ,
$$ where $d_1-d_2+d_F-d_0=2d_F - [3\mu_2-\mu_1]-1= 2d_F+(d_1-3d_2-1)/2$ and $-d_1+2d_0=d_2-2d_F+1$. The result follows from this. \end{proof}
We gather together Propositions \ref{prop:diff-poly-simple-2} and \ref{prop:diff-poly-simple-1} in a single result.
\begin{corollary}\label{cor:diff-poly-simple}
The critical values $\sigma_c>\sigma_m$ for triples of type $(2,2,d_1,d_2)$ with
$d_1+d_2$ odd are of the form $\sigma_c=\mu_1-\mu_2 +n$,
$1\leq n \leq [\mu_1-\mu_2]$, $n\in \mathbb{Z}$. Suppose that $\mu_1-\mu_2>g-1$.
Then
$$
\begin{aligned}
e(\cN_{\s_c^+})-e(\cN_{\s_c^-})=
\mathop{\mathrm{coeff}}_{x^0}\Bigg[&
\frac{(1+u)^{3g}(1+v)^{3g}(1+ux)^{g}(1+vx)^g\big((uv)^{g-1+n}-(uv)^{1-g+d_1-d_2}\big)}
{(1-uv)^2(1-x)(1-uvx)x^{[\mu_1-\mu_2]-n}}\\
&
\Bigg(\frac{(uv)^{(d_1-d_2-1)/2-n}}{1-(uv)^{-1}x}
-\, \frac{(uv)^{g+n}}{1-(uv)^2x}\Bigg) \Bigg]\,.
\end{aligned}
$$ \end{corollary}
\begin{proof}
For simplicity let us assume that $d_1$ is odd and $d_2$ is even
(the other case is analogous). We~have the
following possibilities:
\begin{enumerate}
\item[(a)] If
$\sigma_c=2d_L-\mu_1-\mu_2$, write $d_L =\mu_1+\frac12+m$ with $m$
integer. Then $\sigma_c=\mu_1-\mu_2+2m+1$. As~$\mu_1<d_L\leq
\frac{3\mu_1-\mu_2}2$ by Proposition \ref{prop:bounds-for-sc} (i),
we have $0\leq m\leq (\mu_1-\mu_2-1)/2$. Substituting the
values $3d_1-d_2-1-4d_L= d_1-d_2-1-4m-2$, $2d_L-d_1+g= g+2m+1$,
$[3\mu_1-\mu_2]-2d_L= [\mu_1-\mu_2]-2m-1$ and $g-1-d_1+2d_L= g+2m$
into the formula of Proposition \ref{prop:diff-poly-simple-2}, one
gets
$$
\begin{aligned}
e(\cN_{\s_c^+})-e(\cN_{\s_c^-})=
\mathop{\mathrm{coeff}}_{x^0}\Bigg[&
\frac{(1+u)^{3g}(1+v)^{3g}(1+ux)^{g}(1+vx)^g\big((uv)^{g+2m}-(uv)^{1-g+d_1-d_2}\big)}
{(1-uv)^2(1-x)(1-uvx)(1-(uv)^{-1}x)x^{[\mu_1-\mu_2]-2m-1}}\\
&
\Bigg(\frac{(uv)^{(d_1-d_2-1)/2-2m-1}}{1-(uv)^{-1}x}
-\, \frac{(uv)^{g+2m+1}}{1-(uv)^2x}\Bigg) \Bigg]\,.
\end{aligned}
$$
\item[(b)] If $\sigma_c=\mu_1+\mu_2-2d_F$, write $d_F =\mu_2-m-1$ with $m$ an integer. Then $\sigma_c=\mu_1-\mu_2+2m+2$. As~$\frac{3\mu_2-\mu_1}2 \leq d_F< \mu_2$ by Proposition \ref{prop:bounds-for-sc} (i), we have $0\leq m\leq (\mu_1-\mu_2)/2-1$. Substituting the values $4d_F+d_1-3d_2-1=d_1-d_2-1-4m-4$, $d_2-2d_F+g=g+2m+2$, $2d_F - [3\mu_2-\mu_1]-1=[\mu_1-\mu_2] -2m-2$ and $g-1+d_2-2d_F= g+2m+1$ into the formula of Proposition \ref{prop:diff-poly-simple-1}, we~have
$$
\begin{aligned}
e(\cN_{\s_c^+})-e(\cN_{\s_c^-})=
\mathop{\mathrm{coeff}}_{x^0}\Bigg[&
\frac{(1+u)^{3g}(1+v)^{3g}(1+ux)^{g}(1+vx)^g\big((uv)^{g+2m+1}-(uv)^{1-g+d_1-d_2}\big)}
{(1-uv)^2(1-x)(1-uvx)(1-(uv)^{-1}x)x^{[\mu_1-\mu_2]-2m-2}}\\
&
\Bigg(\frac{(uv)^{(d_1-d_2-1)/2-2m-2}}{1-(uv)^{-1}x}
-\, \frac{(uv)^{g+2m+2}}{1-(uv)^2x}\Bigg) \Bigg]\,.
\end{aligned}
$$ \end{enumerate} Case (a) corresponds to $n=2m+1$ odd, and Case (b) to $n=2m+2$ even in the formula in the statement. The range for $n$ is $1\leq n\leq \mu_1-\mu_2$. But, since $\mu_1-\mu_2$ is not an integer, this range is actually $1\leq n \leq [\mu_1-\mu_2]$. \end{proof}
\section{Hodge polynomial of the moduli of triples of rank $(2,2)$ and large $\sigma$} \label{sec:large}
Now we use all the information in Sections \ref{sec:critical(2,2)}--\ref{sec:simple}\ to compute the Hodge polynomial of the $\cN_\s(2,2,d_1,d_2)$, for any non-critical $\sigma>\sigma_m$. Recall that by Theorem \ref{thm:stabilize}, there is a value $\sigma_M= 2(\mu_1-\mu_2)$ such that for $\sigma>\sigma_M$ all the moduli spaces $\mathcal{N}_\sigma$ are isomorphic. We refer to
$$
\mathcal{N}_{\sigma_M^+}=\mathcal{N}_{\sigma_M^+}(2,2,d_1,d_2)
$$ as the \emph{large $\sigma$} moduli space.
\begin{proposition}\label{prop:finally}
Suppose that $d_1$ is even and $d_2$ is odd and that $\mu_1-\mu_2>g-1$.
Let $\sigma>\sigma_m$ be a non-critical value. Set
$n_0=\min\{[\sigma-\mu_1+\mu_2], [\mu_1-\mu_2]\}$. Then
$$
\begin{aligned}
e(\mathcal{N}_{\sigma}) & - e(\mathcal{N}_{{\s_m^+}}) = \mathop{\mathrm{coeff}}_{x^0}\Bigg[
\frac{(1+u)^{3g}(1+v)^{3g}(1+ux)^{g}(1+vx)^g}
{(1-uv)^2(1-x)(1-uvx)x^{[\mu_1-\mu_2]}}\\
& \quad
\Bigg(\frac{(uv)^{g-1+(d_1-d_2-1)/2}x(1-x^{n_0})}{(1-(uv)^{-1}x)(1-x)}
-\, \frac{(uv)^{(3d_1-3d_2-1)/2-g}x (1- (uv)^{-n_0}
x^{n_0})}{(1-(uv)^{-1}x)^2}\\
&\quad -\, \frac{(uv)^{2g+1} x (1-(uv)^{2n_0}x^{n_0})}{(1-(uv)^2x)^2}
+\, \frac{(uv)^{d_1-d_2+2}x (1-(uv)^{n_0} x^{n_0})}{(1-(uv)^2x)(1-uvx)}\Bigg) \Bigg] \, .
\end{aligned}
$$ \end{proposition}
\begin{proof} By Corollary \ref{cor:diff-poly-simple}, the critical values are of the form $\sigma_c=\mu_1-\mu_2 +n$ with $1\leq n \leq [\mu_1-\mu_2]$. Now $\sigma_m<\sigma_c<\sigma$ is equivalent to $n\leq [\sigma-\mu_1+\mu_2]$ (note that $\sigma-\mu_1+\mu_2\not\in \mathbb{Z}$ since $\sigma$ is not critical). Therefore,
$$
\begin{aligned}
e(\mathcal{N}_{\sigma})- & e(\mathcal{N}_{{\s_m^+}}) = \sum_{\sigma_m<\sigma_c<\sigma}
e(\cN_{\s_c^+})-e(\cN_{\s_c^-})= \\ = &\sum_{n=1}^{n_0}
\mathop{\mathrm{coeff}}_{x^0}\Bigg[
\frac{(1+u)^{3g}(1+v)^{3g}(1+ux)^{g}(1+vx)^g\big((uv)^{g-1+n}-(uv)^{1-g+d_1-d_2}\big)}
{(1-uv)^2(1-x)(1-uvx)x^{[\mu_1-\mu_2]-n}}\\
& \quad
\Bigg(\frac{(uv)^{(d_1-d_2-1)/2-n}}{1-(uv)^{-1}x}
-\, \frac{(uv)^{g+n}}{1-(uv)^2x}\Bigg) \Bigg] = \\ &= \mathop{\mathrm{coeff}}_{x^0}\Bigg[
\frac{(1+u)^{3g}(1+v)^{3g}(1+ux)^{g}(1+vx)^g}
{(1-uv)^2(1-x)(1-uvx)x^{[\mu_1-\mu_2]}}\\
& \quad
\Bigg(\frac{1}{1-(uv)^{-1}x} \sum_{n=1}^{n_0}(uv)^{g-1+(d_1-d_2-1)/2}x^n-
\frac{1}{1-(uv)^{-1}x} \sum_{n=1}^{n_0}(uv)^{1-g+(3d_1-3d_2-1)/2-n}
x^n \\ & \quad
-\, \frac{1}{1-(uv)^2x} \sum_{n=1}^{n_0}(uv)^{2g-1+2n}x^n +
\frac{1}{1-(uv)^2x} \sum_{n=1}^{n_0}(uv)^{1+d_1-d_2+n} x^n\Bigg) \Bigg] =
\end{aligned}
$$
$$
\begin{aligned}
\qquad &=
\mathop{\mathrm{coeff}}_{x^0}\Bigg[
\frac{(1+u)^{3g}(1+v)^{3g}(1+ux)^{g}(1+vx)^g}
{(1-uv)^2(1-x)(1-uvx)x^{[\mu_1-\mu_2]}}\\
& \quad
\Bigg(\frac{(uv)^{g-1+(d_1-d_2-1)/2}x(1-x^{n_0})}{(1-(uv)^{-1}x)(1-x)}
-\, \frac{(uv)^{1-g+(3d_1-3d_2-1)/2-1}x (1- (uv)^{-n_0}
x^{n_0})}{(1-(uv)^{-1}x)^2}\\
&\quad -\, \frac{(uv)^{2g-1+2} x (1-(uv)^{2n_0}x^{n_0})}{(1-(uv)^2x)^2}
+\, \frac{(uv)^{1+d_1-d_2+1}x (1-(uv)^{n_0} x^{n_0})}{(1-(uv)^2x)(1-uvx)}\Bigg) \Bigg]\ .
\end{aligned}
$$ \end{proof}
\begin{theorem} \label{thm:finally}
Suppose that $d_1$ is odd and $d_2$ is even.
Then the large $\sigma$ moduli space $\mathcal{N}_{\sigma_M^+}=\mathcal{N}^s_{\sigma_M^+}$ is smooth
and compact. If $\mu_1-\mu_2>2g-2$, its Hodge polynomial is
$$
\begin{aligned}
e(\mathcal{N}_{\sigma_M^+})=& \ \frac{(1+u)^{2g}(1+v)^{2g}}{(1-uv)^3(1-(uv)^2)^2}
\Bigg[
(1+u^2v)^{2g}(1+uv^2)^{2g}(1-(uv)^{2N})\\
&
-N \, (1+u^2v)^g(1+uv^2)^g(1+u)^g(1+v)^g (uv)^{N+g-1}(1-(uv)^2)
\\
&
+
(1+u)^{2g}(1+v)^{2g}(1+uv)^2(uv)^{2g -2 +(N+1)/2 }
\Big((1-(uv)^{N+1}) - \frac{N+1}{2} \, (1-uv)(1+(uv)^{N})\Big)\\
&
-g(1+u)^{2g-1}(1+v)^{2g-1}
(1-(uv)^2)^2(uv)^{2g -2 +(N+1)/2}(1-(uv)^{N})
\Bigg] \, , \end{aligned} $$ where $N=d_1-d_2-2g+2$. \end{theorem}
\begin{proof} The first statement follows from Theorem \ref{thm:moduli(2,2)}. To compute $e(\mathcal{N}_{\sigma_M^+})-e(\mathcal{N}_{\sigma_m^+})$ we use
Proposition~\ref{prop:finally} for $\sigma=\sigma_M^+$. Note that in this case
$n_0=[\mu_1-\mu_2]$. All the terms in the formula of
Proposition \ref{prop:finally} involving $x^{n_0}$ yield positive
powers of $x$, so they can be disregarded for computing
$\mathop{\mathrm{coeff}}_{x^0}$. Hence
$$
\begin{aligned}
e&(\mathcal{N}_{\sigma_M^+}) = e(\mathcal{N}_{\sigma_m^+}) +\mathop{\mathrm{coeff}}_{x^0}\Bigg[
\frac{(1+u)^{3g}(1+v)^{3g}(1+ux)^{g}(1+vx)^g}
{(1-uv)^2(1-x)(1-uvx)x^{[\mu_1-\mu_2]}} \cdot \\
&
\Bigg(\frac{(uv)^{g-1+(d_1-d_2-1)/2}x}{(1-(uv)^{-1}x)(1-x)}
-\, \frac{(uv)^{(3d_1-3d_2-1)/2-g}x }{(1-(uv)^{-1}x)^2} -\, \frac{(uv)^{2g+1} x }{(1-(uv)^2x)^2}
+\, \frac{(uv)^{d_1-d_2+2}x }{(1-(uv)^2x)(1-uvx)}\Bigg)
\Bigg]\, .
\end{aligned}
$$
As $\mu_1-\mu_2>2g-2$, let $m\geq 0$ such that $[\mu_1-\mu_2]=2g-2+m$. Introduce the following function
$$
F(a,b,c)=\mathop{\mathrm{coeff}}_{x^0}\Bigg( \frac{(1+ux)^{g}(1+vx)^gx^{3-2g-m}}{(1-ax)^2(1-bx)(1-cx)}
\Bigg)=\mathrm{Res}_{x=0}\Bigg(\frac{(1+ux)^{g}(1+vx)^gx^{2-2g-m}}{(1-ax)^2(1-bx)(1-cx)}\Bigg)\
,
$$ where $a,b,c\neq 0$. So
\begin{equation} \label{eqn:acabando}
\begin{aligned}
e(\mathcal{N}_{\sigma_M^+})=& \ e(\mathcal{N}_{{\s_m^+}})+
\frac{(1+u)^{3g}(1+v)^{3g}}{(1-uv)^2} \bigg( (uv)^{3g-3+m} F(1,uv,(uv)^{-1})
- (uv)^{5g-5+3m} F((uv)^{-1},1,uv) \\
& - (uv)^{2g+1} F((uv)^2,1,uv)
+ (uv)^{4g-1+2m} F(uv,1,(uv)^2) \bigg)
\end{aligned}
\end{equation} using $d_1-d_2=4g-3+2m$.
The function
$$
G(x)=\frac{(1+ux)^{g}(1+vx)^g x^{2-2g-m}}{(1-ax)^2(1-bx)(1-cx)}
$$ is a meromorphic function on $\mathbb{C}\cup \{\infty\}$ with poles at $x=0$, $x=1/a$, $x=1/b$ and $x=1/c$. Note that there is no pole at $\infty$. So
$$
F(a,b,c)=-\mathrm{Res}_{x=1/a} G(x) -\mathrm{Res}_{x=1/b} G(x) -\mathrm{Res}_{x=1/c}
G(x)\, .
$$ An easy calculation, using that
$$
\begin{aligned}
&\mathrm{Res}_{x=1/a} G(x) = \frac{d}{dx}\bigg|_{x=1/a} \big( G(x)(x-1/a)^2\big)\,
. \\
&\mathrm{Res}_{x=1/b} G(x) = G(x)(x-1/b) |_{x=1/b}\, , \\
&\mathrm{Res}_{x=1/c} G(x) = G(x)(x-1/c) |_{x=1/c}\, ,
\end{aligned}
$$ yields
$$
\begin{aligned}
F(a,b,c)&=
\frac{a^{m-1}b(a+u)^g(a+v)^g}{(a-b)^2(c-a)}
+
\frac{a^{m-1}c(a+u)^g(a+v)^g}{(b-a)(c-a)^2}\\
&+
\frac{b^m(b+u)^g(b+v)^g}{(a-b)^2(b-c)}
+
\frac{c^m(c+u)^g(c+v)^g}{(c-a)^2(c-b)}\\
&+
\frac{a^{m-1}(a+u)^{g-1}(a+v)^{g-1}}{(a-b)(a-c)}
\Big(
g \, a(2a+u+v)+(m-2)(a+u)(a+v)
\Big) \, .
\end{aligned}
$$
Using this into (\ref{eqn:acabando}) and Theorem \ref{thm:(2,2,odd,even)}, we have
$$
\begin{aligned}
e(\mathcal{N}_{\sigma_M^+})
\!=\!& \ \frac{(1+u)^{2g}(1+v)^{2g}}{(1-uv)^3(1-(uv)^2)^2}
\Bigg[
(1+u^2v)^{2g}(1+uv^2)^{2g}(1-(uv)^{4g+4m-2})\\
&
+(1-2m-2g)(1+u^2v)^g(1+uv^2)^g(1+u)^g(1+v)^g (uv)^{3g+2m-2}(1-(uv)^2)
\\
&
+\!
(1\!+\!u)^{2g}(1\!+\!v)^{2g}(1\!+\!uv)^2(uv)^{3g+m-2}
\Big((1\!-\!(uv)^{2g+2m}) \!-\! (m\!+\!g)(1\!-\!uv)(1\!+\!(uv)^{2g+2m-1})\Big)\\
&
-g(1+u)^{2g-1}(1+v)^{2g-1}
(1-(uv)^2)^2(uv)^{3g+m-2}(1-(uv)^{2g+2m-1})
\Bigg] \, . \end{aligned} $$ As $N=d_1-d_2 -2g+2 =2m+2g-1$, we get the formula in the statement. \end{proof}
\begin{corollary}\label{cor:finally}
Suppose that $d_1$ is even and $d_2$ is odd.
Then the large $\sigma$ moduli space $\mathcal{N}_{\sigma_M^+}=\mathcal{N}^s_{\sigma_M^+}$ is smooth
and compact. If $\mu_1-\mu_2>2g-2$ its Hodge polynomial has the same formula
as that of Theorem \ref{thm:finally}. \end{corollary}
\begin{proof} Use the isomorphism $\mathcal{N}_\sigma(2,2,d_1,d_2)\cong \mathcal{N}_\sigma(2,2,-d_2,-d_1)$ together with Theorem \ref{thm:finally}. \end{proof}
\begin{corollary}\label{cor:finally-n}
Suppose that $d_1+d_2$ is odd and $\mu_1-\mu_2>2g-2$. Then the
Poincar{\'e} polynomial of $\mathcal{N}_{\sigma_M^+}$ is
$$
\begin{aligned}
P_t(\mathcal{N}_{\sigma_M^+})=& \ \frac{(1+t)^{4g}}{(1-t^2)^3(1-t^4)^2}
\Bigg[
(1+t^3)^{4g}(1-t^{4N})
- N \, (1+t^3)^{2g}(1+t)^{2g} t^{2N+2g-2} (1-t^4) \\
& + (1+t)^{4g}(1+t^2)^2 t^{N+4g-3}
\Big((1-t^{2N+2}) - \frac{N+1}{2} \, (1-t^2)(1+t^{2N})\Big)\\
&
-g \, (1+t)^{4g-2}
(1-t^4)^2 t^{N +4g-3}(1-t^{2N})
\Bigg]\, , \end{aligned} $$ where $N=d_1-d_2 -2g+2$.
$\Box$ \end{corollary}
\end{document} | arXiv |
\begin{document}
\begin{abstract} Let $S = K[x_1, \ldots, x_n ]$ be a polynomial ring over a field $K$, and $E = \bigwedge {\langle} y_1, \ldots, y_n {\rangle}$ an exterior algebra. The {\it linearity defect} $\operatorname{ld}_E(N)$ of a finitely generated graded $E$-module $N$ measures how far $N$ departs from ``componentwise linear". It is known that $\operatorname{ld}_E(N) < \infty$ for all $N$. But the value can be arbitrary large, while the similar invariant $\operatorname{ld}_S(M)$ for an $S$-module $M$ is always at most $n$. We will show that if $I_\Delta$ (resp. $J_\Delta$) is the squarefree monomial ideal of $S$ (resp. $E$) corresponding to a simplicial complex $\Delta \subset 2^{\{1, \ldots, n \}}$, then $\operatorname{ld}_E(E/J_\Delta) = \operatorname{ld}_S(S/I_\Delta)$. Moreover, except some extremal cases, $\operatorname{ld}_E(E/J_\Delta)$ is a topological invariant of the geometric realization
$|\Delta^\vee|$ of the Alexander dual $\Delta^\vee$ of $\Delta$. We also show that, when $n \geq 4$, $\operatorname{ld}_E(E/J_\Delta) = n-2$ (this is the largest possible value) if and only if $\Delta$ is an $n$-gon. \end{abstract}
\title{Linearity Defects of Face Rings}
\section{Introduction} Let $A = \bigoplus_{i \in {\mathbb N}} A_i$ be a graded (not necessarily commutative) noetherian algebra over a field $K \, (\cong A_0)$. Let $M$ be a finitely generated graded left $A$-module, and $P_\bullet$ its minimal free resolution. Eisenbud et al. \cite{EFS} defined the {\it linear part} $\operatorname{lin}(P_\bullet)$ of $P_\bullet$, which is the complex obtained by erasing all terms of degree $\geq 2$ from the matrices representing the differential maps of $P_\bullet$ (hence $\operatorname{lin}(P_\bullet)_i = P_i$ for all $i$). Following Herzog and Iyengar \cite{HI}, we call $\operatorname{ld}_A(M) = \sup\{ \, i \mid H_i(\operatorname{lin}(P_\bullet)) \ne 0 \, \}$ the {\it linearity defect} of $M$. This invariant and related concepts have been studied by several authors (e.g., \cite{EFS, HI, MZ, R02, Y7}). We say a finitely generated graded $A$-module $M$ is {\it componentwise linear} (or, {\it (weakly) Koszul} in some literature) if $M_{{\langle} i {\rangle}}$ has a linear free resolution for all $i$. Here $M_{{\langle} i {\rangle}}$ is the submodule of $M$ generated by its degree $i$ part $M_i$. Then we have $$\operatorname{ld}_A(M) = \min \{ \, i \mid \text{the $i^{\rm th}$ syzygy of $M$ is componentwise linear} \, \}.$$
For this invariant, a remarkable result holds over an exterior algebra $E = \bigwedge {\langle} y_1, \ldots, y_n {\rangle}$. In \cite[Theorem~3.1]{EFS}, Eisenbud et al. showed that any finitely generated graded $E$-module $N$ satisfies $\operatorname{ld}_E(N) < \infty$ while $\operatorname{proj.dim}_E(N) = \infty$ in most cases. (We also remark that Martinez-Villa and Zacharia~\cite{MZ} proved the same result for many selfinjective Koszul algebras). If $n \geq 2$, then we have $\sup \{ \, \operatorname{ld}_E(N) \mid \text{$N$ a finitely generated graded $E$-module} \, \} = \infty$. But Herzog and R\"omer proved that if $J \subset E$ is a {\it monomial} ideal then $\operatorname{ld}_E(E/J) \leq n-1$ (c.f. \cite{R02}).
A monomial ideal of $E = \bigwedge {\langle} y_1, \ldots, y_n {\rangle}$ is always of the form $J_\Delta := ( \, \prod_{i \in F} y_i \mid F \not \in \Delta \, )$ for a simplicial complex $\Delta \subset 2^{\{1, \ldots, n \}}$. Similarly, we have the {\it Stanley-Reisner ideal} $I_\Delta := ( \, \prod_{i \in F} x_i \mid F \not \in \Delta \, )$ of a polynomial ring $S=K[x_1, \ldots, x_n ]$. In this paper, we will show the following.
\begin{thm} With the above notation, we have $\operatorname{ld}_E(E/J_\Delta) = \operatorname{ld}_S (S/I_\Delta)$. Moreover, if $\operatorname{ld}_E(E/J_\Delta) > 0$ (equivalently, $\Delta \ne 2^{T}$ for any $T \subset [n]$), then $\operatorname{ld}_E(E/J_\Delta)$ is a topological invariant of the geometric realization
$|\Delta^\vee|$ of the Alexander dual $\Delta^\vee$. (But $\operatorname{ld}(E/J_\Delta)$ may depend on $\operatorname{char}(K)$.) \end{thm}
By virtue of the above theorem, we can put $\operatorname{ld}(\Delta):= \operatorname{ld}_E(E/J_\Delta) = \operatorname{ld}_S (S/I_\Delta)$. If we set $d:= \min \{ \,i \mid [I_\Delta]_i \ne 0 \, \} = \min \{ \,i \mid [J_\Delta]_i \ne 0 \, \}$, then $\operatorname{ld}(\Delta) \leq \max \{1, n- d\}$. But, if $d=1$ (i.e., $\{ i \} \not \in \Delta$ for some $1 \leq i \leq n$), then $\operatorname{ld}(\Delta) \leq \max \{1, n-3\}$. Hence, if $n \geq 3$, we have $\operatorname{ld}(\Delta) \leq n-2$ for all $\Delta$.
\begin{thm} Assume that $n \geq 4$. Then $\operatorname{ld}(\Delta) = n-2$ if and only if $\Delta$ is an $n$-gon. \end{thm}
While we treat $S$ and $E$ in most part of the paper, some results on $S$ can be generalized to a normal semigroup ring, and this generalization makes the topological meaning of $\operatorname{ld}(\Delta)$ clear. So \S2 concerns a normal semigroup ring. But, in this case, we use an irreducible resolution (something analogous to an injective resolution), not a projective resolution.
\section{Linearity Defects for Irreducible Resolutions} Let $C \subset {\mathbb Z}^n \subset {\mathbb R}^n$ be an affine semigroup (i.e., $C$ is a finitely generated additive submonoid of ${\mathbb Z}^n$), and $R := K[{\bf x}^{\bf c} \mid {\bf c} \in C] \subset K[x_1^{\pm 1}, \ldots, x_n^{\pm 1}]$ the semigroup ring of $C$ over the field $K$. Here ${\bf x}^{\bf c}$ for ${\bf c} = (c_1, \ldots, c_n) \in C$ denotes the monomial $\prod_{i=1}^n x_i^{c_i}$. Let $ {\mathbf P} := {\mathbb R}_{\geq 0} C \subset {\mathbb R}^n$ be the polyhedral cone spanned by $C$. We always assume that ${\mathbb Z} C = {\mathbb Z}^n$, ${\mathbb Z}^n \cap {\mathbf P} = C$ and $C \cap (-C) = \{ 0 \}$. Thus $R$ is a normal Cohen-Macaulay integral domain of dimension $n$ with a maximal ideal ${\mathfrak m} := ({\bf x}^{\bf c} \mid 0 \ne {\bf c} \in C)$.
Clearly, $R = \bigoplus_{{\bf c} \in C} K {\bf x}^{\bf c}$ is a ${\mathbb Z}^n$-graded ring. We say a ${\mathbb Z}^n$-graded ideal of $R$ is a {\it monomial ideal}. Let $\operatorname{*mod}R$ be the category of finitely generated ${\mathbb Z}^n$-graded $R$-modules and degree preserving $R$-homomorphisms. As usual, for $M \in \operatorname{*mod}R$ and ${\bf a} \in {\mathbb Z}^n$, $M_{\bf a}$ denotes the degree ${\bf a}$ component of $M$, and $M({\bf a})$ denotes the shifted module of $M$ with $M({\bf a})_{\bf b} = M_{{\bf a} + {\bf b}}$.
Let ${\mathbf L}$ be the set of non-empty faces of the polyhedral cone $ {\mathbf P}$. Note that $\{ 0\}$ and $ {\mathbf P}$ itself belong to ${\mathbf L}$. For $F \in {\mathbf L}$, $P_F := ( \, {\bf x}^{\bf c} \mid {\bf c} \in C \setminus F \, )$ is a prime ideal of $R$. Conversely, any monomial prime ideal is of the form $P_F$ for some $F \in {\mathbf L}$. Note that $P_{\{ 0\}} = {\mathfrak m}$ and $P_ {\mathbf P} = (0)$. Set $K[F] := R/P_F \cong K[ \, {\bf x}^{\bf c} \mid {\bf c} \in C \cap F]$ for $F \in {\mathbf L}$. The Krull dimension of $K[F]$ equals the dimension $\dim F$ of the polyhedral cone $F$.
For a point $u \in {\mathbf P}$, we always have a unique face $F \in {\mathbf L}$ whose relative interior contains $u$. Here we denote $s(u) = F$.
\begin{dfn}[\cite{Y2}]\label{sq} We say a module $M \in \operatorname{*mod}R$ is {\it squarefree}, if it is $C$-graded (i.e., $M_{\bf a} = 0$ for all ${\bf a} \not \in C$), and the multiplication map $M_{\bf a} \ni y \mapsto {\bf x}^{\bf b} y \in M_{{\bf a} + {\bf b}}$ is bijective for all ${\bf a}, {\bf b} \in C$ with $s({\bf a}+{\bf b}) = s({\bf a})$. \end{dfn}
For a monomial ideal $I$, $R/I$ is a squarefree $R$-module if and only if $I$ is a radical ideal (i.e., $\sqrt{I} = I$). Regarding ${\mathbf L}$ as a partially ordered set by inclusion, we say $\Delta \subset {\mathbf L}$ is an {\it order ideal}, if $\Delta \ni F \supset F' \in {\mathbf L}$ implies $F' \in \Delta$. If $\Delta$ is an order ideal, then $I_\Delta := ( \, {\bf x}^{\bf c} \mid {\bf c} \in C, \, s({\bf c}) \not \in \Delta \, ) \subset R$ is a radical ideal. Conversely, any radical monomial ideal is of the form $I_\Delta$ for some $\Delta$. Set $K[\Delta] := R/I_\Delta$. Clearly, $$ K[\Delta]_{\bf a} \cong \begin{cases} K & \text{if ${\bf a} \in C$ and $s({\bf a}) \in \Delta$,}\\ 0 & \text{otherwise.} \end{cases} $$ In particular, if $\Delta = {\mathbf L}$ (resp. $\Delta = \{ \, \{ 0 \} \, \}$), then $I_\Delta = 0$ (resp. $I_\Delta = {\mathfrak m}$) and $K[\Delta] = R$ (resp. $K[\Delta] = K$). When $R$ is a polynomial ring, $K[\Delta]$ is nothing else than the Stanley-Reisner ring of a simplicial complex $\Delta$. (If $R$ is a polynomial ring, then the partially ordered set ${\mathbf L}$ is isomorphic to the power set $2^{\{1, \ldots, n\}}$, and $\Delta$ can be seen as a simplicial complex.)
For each $F \in {\mathbf L}$, take some ${\bf c}(F) \in C \cap \operatorname{rel-int}(F)$ (i.e., $s({\bf c}(F)) =F$). For a squarefree $R$-module $M$ and $F, G \in {\mathbf L}$ with $G \supset F$, \cite[Theorem~3.3]{Y2} gives a $K$-linear map $\varphi^M_{G, F}: M_{{\bf c}(F)} \to M_{{\bf c}(G)}$. They satisfy $\varphi^M_{F,F} = \operatorname{Id}$ and $\varphi^M_{H, G} \circ \varphi^M_{G, F} = \varphi^M_{H,F}$ for all $H \supset G \supset F$. We have $M_{\bf c} \cong M_{{\bf c}'}$ for ${\bf c}, {\bf c}' \in C$ with $s({\bf c}) = s({\bf c}')$. Under these isomorphisms, the maps $\varphi^M_{G, F}$ do not depend on the particular choice of ${\bf c}(F)$'s.
Let $\operatorname{Sq}(R)$ be the full subcategory of $\operatorname{*mod}R$ consisting of squarefree modules. As shown in \cite{Y2}, $\operatorname{Sq}(R)$ is an abelian category with enough injectives. For an indecomposable squarefree module $M$, it is injective in $\operatorname{Sq}(R)$ if and only if $M \cong K[F]$ for some $F \in {\mathbf L}$. Each $M \in \operatorname{Sq}(R)$ has a minimal injective resolution in $\operatorname{Sq}(R)$, and we call it a {\it minimal irreducible resolution} (see \cite{Y8} for further information). A minimal irreducible resolution is unique up to isomorphism, and its length is at most $n$.
Let $\omega_R$ be the ${\mathbb Z}^n$-graded canonical module of $R$. It is well-known that $\omega_R$ is isomorphic to the radical monomial ideal $(\, {\bf x}^{\bf c} \mid {\bf c} \in C, \, s(c) = {\mathbf P} \, )$. Since we have $\operatorname{Ext}^i_R(M^\bullet,\omega_R) \in \operatorname{Sq}(R)$ for all $M^\bullet \in \operatorname{Sq}(R)$, ${\mathbf D}(-) := {\rm R}{\Hom}_R(-, \omega_R)$ gives a duality functor from the derived category $D^b(\operatorname{Sq}(R)) \, (\cong D^b_{\operatorname{Sq}(R)} (\operatorname{*mod}R))$ to itself.
In the sequel, for a $K$-vector space $V$, $V^*$ denotes its dual space. But, even if $V = M_{\bf a}$ for some $M \in \operatorname{*mod}R$ and ${\bf a} \in {\mathbb Z}^n$, we set the degree of $V^*$ to be 0.
\begin{lem}[{\cite[Lemma~3.8]{Y8}}]\label{explicit form} If $M \in \operatorname{Sq}(R)$, then ${\mathbf D}(M)$ is quasi-isomorphic to the complex $D^{\bullet} : 0 \to D^0 \to D^1 \to \cdots \to D^n \to 0$ with $$D^i = \bigoplus_{\substack{F \in {\mathbf L} \\ \dim F = n-i}} (M_{{\bf c}(F)})^* \otimes_K K[F].$$ Here the differential is the sum of the maps $$(\pm\varphi_{F,F'}^M)^* \otimes \operatorname{nat} : (M_{{\bf c}(F)})^* \otimes_K K[F] \to (M_{{\bf c}(F')})^* \otimes_K K[F']$$ for $F,F' \in {\mathbf L}$ with $F \supset F'$ and $\dim F = \dim F'+1$, and $\operatorname{nat}$ denotes the natural surjection $K[F] \to K[F']$. We can also describe ${\mathbf D}(M^\bullet)$ for a complex $M^\bullet \in D^b(\operatorname{Sq}(R))$ in a similar way. \end{lem}
\noindent{\bf Convention.} In the sequel, as an explicit complex, ${\mathbf D}(M^\bullet)$ for $M^\bullet \in D^b(\operatorname{Sq}(R))$ means the complex described in Lemma~\ref{explicit form}.
Since ${\mathbf D} \circ {\mathbf D} \cong \operatorname{Id}_{D^b(\operatorname{Sq}(R))}$, ${\mathbf D} \circ {\mathbf D} (M)$ is an irreducible resolution of $M$, but it is far from being minimal. Let $(I^\bullet, \partial^\bullet)$ be a minimal irreducible resolution of $M$. For each $i \in {\mathbb N}$ and $F \in {\mathbf L}$, we have a natural number $\nu_i(F, M)$ such that $$I^i \cong \bigoplus_{F \in {\mathbf L}} K[F]^{\nu_i(F,M)}.$$ Since $I^\bullet$ is minimal, $z \in K[F] \subset I^i$ with $\dim F = d$ is sent to $$\partial^i(z) \in \bigoplus_{\substack{G \in {\mathbf L} \\ \dim G < d}} K[G]^{\nu_{i+1}(G,M)} \subset I^{i+1}.$$ The above observation on ${\mathbf D} \circ {\mathbf D}(M)$ gives the formula (\cite[Theorem~4.15]{Y2}) $$\nu_i(F,M) = \dim_K [\operatorname{Ext}^{n-i-\dim F}_R (M, \omega_R)]_{{\bf c}(F)}.$$
For each $l \in {\mathbb N}$ with $0 \leq l \leq n$, we define the $l$-{\it linear strand} $\operatorname{lin}_l(I^\bullet)$ of $I^\bullet$ as follows: The term $\operatorname{lin}_l(I^\bullet)^i$ of cohomological degree $i$ is $$\bigoplus_{\dim F = l-i} K[F]^{\nu_i(F,M)},$$ which is a direct summand of $I^i$, and the differential $\operatorname{lin}_l(I^\bullet)^i \to \operatorname{lin}_l(I^\bullet)^{i+1}$ is the corresponding component of the differential $\partial^i: I^i \to I^{i+1}$ of $I^\bullet$. By the minimality of $I^\bullet$, we can see that $\operatorname{lin} _l \ep{I^\bullet}$ are cochain complexes. Set $\operatorname{lin}(I^\bullet) := \bigoplus_{0 \leq l \leq n} \operatorname{lin}_l(I^\bullet)$. Then we have the following. For a complex $M^\bullet$ and an integer $p$, let $M^\bullet[p]$ be the $p^{\rm th}$ translation of $M^\bullet$. That is, $M^\bullet[p]$ is a complex with $M^i[p] = M^{i+p}$.
\begin{thm}[{\cite[Theorem~3.9]{Y8}}]\label{lin(M)} With the above notation, we have $$\operatorname{lin}_l(I^\bullet) \cong {\mathbf D}(\operatorname{Ext}_R^{n-l}(M,\omega_R))[n-l].$$ Hence $$\operatorname{lin}(I^\bullet) \cong \bigoplus_{i \in {\mathbb Z}} {\mathbf D}(\operatorname{Ext}_R^i(M,\omega_R))[i].$$ \end{thm}
\begin{dfn} Let $I^\bullet$ be a minimal irreducible resolution of $M \in \operatorname{Sq}(R)$. We call $\max \{ \, i \mid H^i(\operatorname{lin}(I^\bullet)) \ne 0 \, \}$ the {\it linearity defect of the minimal irreducible resolution} of $M$, and denote it by $\operatorname{ld.irr}_R(M)$. \end{dfn}
\begin{cor}\label{ld for irr} With the above notation, we have $$\max\{ \, i \mid H^i(\operatorname{lin}_l(I^\bullet)) \ne 0 \, \} = l - \operatorname{depth}_R ( \, \operatorname{Ext}_R^{n-l}( M, \omega_R ) \,),$$ and hence $$\operatorname{ld.irr}_R(M) = \max \{ \, i - \operatorname{depth}_R ( \, \operatorname{Ext}_R^{n-i}( M, \omega_R ) \, ) \mid 0 \leq i \leq n \, \}.$$ Here we set the depth of the 0 module to be $+ \infty$. \end{cor}
\begin{proof} By Theorem~\ref{lin(M)}, we have $H^i(\operatorname{lin}_l(I^\bullet)) = \operatorname{Ext}^{i+l}_R(\operatorname{Ext}^l_R(M,\omega_R), \omega_R)$. Since $\operatorname{depth}_R N = \min \{ \, i \mid \operatorname{Ext}^{n-i}_R(N, \omega_R) \ne 0 \, \}$ for a finitely generated graded $R$-module $N$, the assertion follows. \end{proof}
\begin{dfn}[Stanley \cite{St}] Let $M \in \operatorname{*mod}R$. We say $M$ is {\it sequentially Cohen-Macaulay} if there is a finite filtration $$0 = M_0 \subset M_1 \subset \cdots \subset M_r = M$$ of $M$ by graded submodules $M_i$ satisfying the following conditions. \begin{itemize} \item[(a)] Each quotient $M_i/M_{i-1}$ is Cohen-Macaulay. \item[(b)] $\dim(M_i/M_{i-1}) < \dim (M_{i+1}/M_i)$ for all $i$. \end{itemize} \end{dfn}
Remark that the notion of sequentially Cohen-Macaulay module is also studied under the name of a ``Cohen-Macaulay filtered module" (\cite{Sc}).
Sequentially Cohen-Macaulay property is getting important in the theory of Stanley-Reisner rings. It is known that $M \in \operatorname{*mod}R$ is sequentially Cohen-Macaulay if and only if $\operatorname{Ext}^{n-i}_R(M,\omega_R)$ is a zero module or a Cohen-Macaulay module of dimension $i$ for all $i$ (c.f. \cite[III. Theorem~2.11]{St}). Let us go back to Corollary~\ref{ld for irr}. If $N := \operatorname{Ext}_R^{n-i}( M, \omega_R) \ne 0$, then $\operatorname{depth}_R N \leq \dim_R N \leq i$. Hence $\operatorname{depth}_R N = i$ if and only if $N$ is a Cohen-Macaulay module of dimension $i$. Thus, as stated in \cite[Corollary~3.11]{Y8}, $\operatorname{ld.irr}_R(M)=0$ if and only if $M$ is sequentially Cohen-Macaulay.
Let $I^\bullet: 0 \to I^0 \stackrel{\partial^0}{\to} I^1 \stackrel{\partial^1}{\to} I^2 \to \cdots$ be an irreducible resolution of $M \in \operatorname{Sq}(R)$. Then it is easy to see that $\ker(\partial^i)$ is sequentially Cohen-Macaulay if and only if $i \geq \operatorname{ld.irr}_R(M)$. In particular, $$\operatorname{ld.irr}_R(M) = \min\{ \, i \mid \text{$\ker(\partial^i)$ is sequentially Cohen-Macaulay}\}.$$
We have a hyperplane $H \subset {\mathbb R}^n$ such that $B:= H \cap {\mathbf P}$ is an $(n-1)$-dimensional polytope. Clearly, $B$ is homeomorphic to a closed ball of dimension $n-1$.
For a face $F \in {\mathbf L}$, set $|F|$ to be the relative interior of $F \cap H$. If $\Delta \subset {\mathbf L}$ is an order ideal,
then $|\Delta| := \bigcup_{F \in \Delta} |F|$ is a closed subset of $B$, and
$\bigcup_{F \in \Delta} |F|$ is a {\it regular cell decomposition}
(c.f. \cite[\S 6.2]{BH}) of $|\Delta|$.
Up to homeomorphism, (the regular cell decomposition of) $|\Delta|$ does not depend on the particular choice of the hyperplane $H$.
The dimension $\dim |\Delta|$ of $|\Delta|$ is given by
$ \max \{ \, \dim |F| \mid F \in \Delta \, \}$. Here $\dim |F|$ denotes
the dimension of $|F|$ as a cell (we set $\dim \emptyset = -1$),
that is, $\dim |F| = \dim F -1 = \dim K[F]-1$.
Hence we have $\dim K[\Delta] = \dim |\Delta| +1$.
If $F \in \Delta$, then $U_F := \bigcup_{F' \supset F} |F'|$ is an open set of $B$. Note that $\{ \, U_F \mid \{ 0 \} \ne F \in {\mathbf L} \, \}$ is an open covering of $B$. In \cite{Y6}, from $M \in \operatorname{Sq}(R)$, we constructed a sheaf $M^+$ on $B$. More precisely, the assignment $$\Gamma(U_F, M^+) = M_{{\bf c}(F)}$$ for each $F \ne \{ 0 \}$ and the map $$\varphi_{F,F'}^M:\Gamma(U_{F'},M^+) = M_{{\bf c}(F')} \to M_{{\bf c}(F)} = \Gamma(U_F,M^+)$$ for $F, F' \ne \{ 0 \}$ with $F \supset F'$ (equivalently, $U_{F'} \supset U_{F}$) defines a sheaf. Note that $M^+$ is a {\it constructible sheaf}
with respect to the cell decomposition $B=\bigcup_{F \in {\mathbf L}} |F|$. In fact, for all $\{ 0 \} \ne F \in {\mathbf L}$,
the restriction $M^+|_{|F|}$ of $M^+$ to $|F| \subset B$ is a constant sheaf with coefficients in $M_{{\bf c}(F)}$. Note that $M_{\bf 0}$ is ``irrelevant" to $M^+$, where ${\bf 0}$ denotes $(0,0, \ldots, 0) \in {\mathbb Z}^n$.
It is easy to see that $K[\Delta]^+ \cong j_* \underline{K}_{|\Delta|}$,
where $\underline{K}_{|\Delta|}$ is the constant sheaf on $|\Delta|$ with coefficients in $K$,
and $j$ denotes the embedding map $|\Delta| \hookrightarrow B$. Similarly, we have that $(\omega_R)^+ \cong h_! \underline{K}_{B^\circ}$, where $\underline{K}_{B^\circ}$ is the constant sheaf on the relative interior $B^\circ$ of $B$, and $h$ denotes the embedding map $B^\circ \hookrightarrow B$. Note that $(\omega_R)^+$ is the orientation sheaf of $B$ over $K$.
\begin{thm}[{\cite[Theorem~3.3]{Y6}}]\label{Hoch} For $M \in \operatorname{Sq}(R)$, we have an isomorphism $$H^i(B; M^+) \cong [H_{\mathfrak m}^{i+1}(M)]_{{\bf 0}} \quad \text{for all $i \geq 1$},$$ and an exact sequence $$0 \to [H_{\mathfrak m}^0(M)]_{\bf 0} \to M_{\bf 0} \to H^0( B; M^+) \to [H_{\mathfrak m}^1(M)]_{\bf 0} \to 0.$$ In particular, we have
$[H_{\mathfrak m}^{i+1}(K[\Delta])]_{\bf 0} \cong \tilde{H}^i(|\Delta| ; K)$ for all $i \geq 0$,
where $\tilde{H}^i(|\Delta| ; K)$ denotes the $i^{ th}$ reduced cohomology
of $|\Delta|$ with coefficients in $K$. \end{thm}
Let $\Delta \subset {\mathbf L}$ be an order ideal and $X := |\Delta|$. Then $X$ admits Verdier's dualizing complex $\mathcal{D}^\bullet_X$, which is a complex of sheaves of $K$-vector spaces. For example, $\mathcal{D}^\bullet_B$ is quasi-isomorphic to $(\omega_R)^+[n-1]$.
\begin{thm}[{\cite[Theorem~4.2]{Y6}}]\label{Verdier} With the above notation, if $\operatorname{ann}(M) \supset I_\Delta$ (equivalently, $\operatorname{supp} (M^+) := \{ x \in B \mid (M^+)_x \ne 0 \} \subset X$), then we have $$\operatorname{supp} (\operatorname{Ext}_R^i(M, \omega_R)^+) \subset X \quad \text{and} \quad
\operatorname{Ext}_R^i(M, \omega_R)^+ |_X \cong {\mathcal Ext}^{i-n+1}(M^+|_X, \mathcal{D}^\bullet_X).$$ \end{thm}
\begin{thm}\label{ldirr sheaf} Let $M$ be a squarefree $R$-module with $M \ne 0$ and $[H_{\mathfrak m}^1(M)]_{\bf 0} = 0$, and $X$ the closure of $\operatorname{supp} (M^+)$. Then $\operatorname{ld.irr}_R(M)$ only depends on
the sheaf $M^+|_X$ (also independent from $R$). \end{thm}
\begin{proof} We use Corollary~\ref{ld for irr}. In the notation there, the case when $i = 0$ is always unnecessary to check. Moreover, by the present assumption, we have $\operatorname{depth}_R (\, \operatorname{Ext}_R^{n-1}(M, \omega_R) \, ) \geq 1$ (in fact, $\operatorname{Ext}_R^{n-1}(M, \omega_R)$ is either the 0 module, or a 1-dimensional Cohen-Macaulay module). So we may assume that $i > 1$.
Recall that $$\operatorname{depth}_R ( \, \operatorname{Ext}_R ^{n-i}(M, \omega_R) \, ) = \min \{ \, j \mid \operatorname{Ext}^{n-j}_R( \, \operatorname{Ext}_R^{n-i}(M, \omega_R), \omega_R \, ) \ne 0 \, \}.$$ By Theorem~\ref{Verdier}, $[\operatorname{Ext}^{n-j}_R( \, \operatorname{Ext}_R^{n-i}(M, \omega_R), \omega_R \, )]_{\bf a}$ can be determined by
$M^+|_X$ for all $i,j$ and all ${\bf a} \ne 0$. If $j > 1$, then $[\operatorname{Ext}^{n-j}_R(\, \operatorname{Ext}_R^{n-i}(M, \omega_R), \omega_R \, )]_{\bf 0}$ is isomorphic to \begin{eqnarray*} [H_{\mathfrak m}^j(\operatorname{Ext}_R^{n-i}(M, \omega_R))]_{\bf 0}^* &\cong& H^{j-1}( \, B; \, \operatorname{Ext}_R^{n-i}(M, \omega_R)^+ \, )^* \\ &\cong&
H^{j-1}( \, X ; \, {\mathcal Ext}^{-i-1}(M^+ |_X; \mathcal{D}^\bullet_X) \,)^* \end{eqnarray*} (the first and the second isomorphisms follow from Theorem~\ref{Hoch} and Theorem~\ref{Verdier}, respectively),
and determined by $M^+|_X$. So only $[\operatorname{Ext}^{n-j}_R(\operatorname{Ext}_R^{n-i}(M, \omega_R), \omega_R)]_{\bf 0}$ for $j= 0,1$ remain. As above, they are isomorphic to $[H_{\mathfrak m}^j(\operatorname{Ext}_R^{n-i}(M, \omega_R))]_{\bf 0}^*$. But, by \cite[Lemma~5.11]{Y8}, we can compute $[H_{\mathfrak m}^j(\operatorname{Ext}_R^{n-i}(M, \omega_R))]_{\bf 0}$
for $i > 1$ and $j= 0,1$ from the sheaf $M^+|_X$. So we are done. \end{proof}
\begin{thm}\label{topological} For an order ideal $\Delta \subset {\mathbf L}$ with $\Delta \ne \emptyset$, $\operatorname{ld.irr}_R(K[\Delta])$ depends
only on the topological space $|\Delta|$. \end{thm}
Note that $\operatorname{ld.irr}_R(K[\Delta])$ may depend on $\operatorname{char}(K)$.
For example, if $|\Delta|$ is homeomorphic to a real projective plane, then $\operatorname{ld.irr}_R(K[\Delta])=0$ if $\operatorname{char}(K) \ne 2$, but $\operatorname{ld.irr}_R(K[\Delta])=2$ if $\operatorname{char}(K) =2$.
Similarly, some other invariants and conditions (e.g., the Cohen-Macaulay property of $K[\Delta]$) studied in this paper depend on $\operatorname{char}(K)$. But, since we fix the base field $K$, we always omit the phrase ``over $K$".
\begin{proof}
If $|\Delta|$ is not connected, then $[H_{\mathfrak m}^1(K[\Delta])]_{\bf 0} \ne 0$ by Theorem~\ref{Hoch}, and we cannot use Theorem~\ref{ldirr sheaf} directly. But even in this case, $\operatorname{depth}_R (\, \operatorname{Ext}^{n-i}_R(K[\Delta], \omega_R) \, )$ can be computed for all $i \ne 1$
by the same way as in Theorem~\ref{ldirr sheaf}. In particular, they only depend on $|\Delta|$. So the assertion follows from the next lemma. \end{proof}
\begin{lem} We have $\operatorname{depth}_R (\, \operatorname{Ext}^{n-1}_R(K[\Delta], \omega_R) \, ) \in \{ 0,1, + \infty \}$, and $$\operatorname{depth}_R ( \, \operatorname{Ext}^{n-1}_R(K[\Delta], \omega_R) \, )= 0 \ \, \text{if and only if} \ \,
\text{$|\Delta'|$ is not connected.}$$ Here $\Delta' := \Delta \setminus \{ \, F \mid
\text{$F$ is a maximal element of $\Delta$ and $\dim |F| = 0 $} \, \}.$ \end{lem}
\begin{proof} Since $\dim_R \operatorname{Ext}^{n-1}_R(K[\Delta], \omega_R) \leq 1$,
the first statement is clear. If $\dim |\Delta| \leq 0$, then
$|\Delta'| = \emptyset$ and $\operatorname{depth}_R (\, \operatorname{Ext}^{n-1}_R(K[\Delta], \omega_R) \,) \geq 1$. So, to see the second statement,
we may assume that $\dim |\Delta| > 1$. Set $J := I_{\Delta'}/I_\Delta$ to be an ideal of $K[\Delta]$. Note that either $J$ is a 1-dimensional Cohen-Macaulay module or $J=0$. From the short exact sequence $0 \to J \to K[\Delta] \to K[\Delta'] \to 0$, we have an exact sequence $$0 \to \operatorname{Ext}_R^{n-1}(K[\Delta'],\omega_R) \to \operatorname{Ext}^{n-1}_R(K[\Delta],\omega_R) \to \operatorname{Ext}_R^{n-1}(J,\omega_R)\to 0.$$ Since $\operatorname{Ext}_R^{n-1}(J,\omega_R)$ has positive depth, $\operatorname{depth}_R (\, \operatorname{Ext}_R^{n-1}(K[\Delta'],\omega_R) \, )=0$ if and only if $\operatorname{depth}_R (\, \operatorname{Ext}^{n-1}_R(K[\Delta],\omega_R) \, ) =0$. But, since $K[\Delta']$ does not have 1-dimensional associated primes, $\operatorname{Ext}^{n-1}_R(K[\Delta'],\omega_R)$ is an artinian module. Hence we have the following. \begin{eqnarray*} \operatorname{depth}_R (\, \operatorname{Ext}^{n-1}_R(K[\Delta'],\omega_R) \, ) =0 &\Longleftrightarrow& [\operatorname{Ext}^{n-1}_R(K[\Delta'],\omega_R)]_{\bf 0} \ne 0 \\ &\Longleftrightarrow&
[H_{\mathfrak m}^1(K[\Delta'])]_{\bf 0} = \tilde{H}^0(|\Delta'|;K) \ne 0 \\ &\Longleftrightarrow&
\text{$|\Delta'|$ is not connected.} \end{eqnarray*} \end{proof}
\section{Linearity Defects of Symmetric and Exterior Face Rings} Let $S := K[x_1, \ldots, x_n]$ be a polynomial ring, and consider its natural ${\mathbb Z}^n$-grading. Since $S = K[{\mathbb N}^n]$ is a normal semigroup ring, we can use the notation and the results in the previous section.
Now we introduce some conventions which are compatible with the previous notation. Let ${\bf e}_i := (0, \ldots, 0,1,0, \ldots 0) \in {\mathbb R}^n$ be the $i^{\rm th}$ unit vector, and $ {\mathbf P}$ the cone spanned by ${\bf e}_1, \ldots, {\bf e}_n$. We identify a face $F$ of $ {\mathbf P}$ with the subset $\{ \, i \mid {\bf e}_i \in F \, \}$ of $[n] := \{ 1,2, \ldots, n \}$. Hence the set ${\mathbf L}$ of nonempty faces of $ {\mathbf P}$ can be identified with the power set $2^{[n]}$ of $[n]$. We say ${\bf a} = (a_1, \ldots, a_n)\in {\mathbb N}^n$ is {\it squarefree}, if $a_i = 0,1$ for all $i$. A squarefree vector ${\bf a} \in {\mathbb N}^n$ will be identified with the subset $\{ \, i \mid a_i = 1 \, \}$ of $[n]$. Recall that we took a vector ${\bf c}(F) \in C$ for each $F \in {\mathbf L}$ in the previous section. Here we assume that ${\bf c}(F)$ is the squarefree vector corresponding to $F \in {\mathbf L} \cong 2^{[n]}$. So, for a ${\mathbb Z}^n$-graded $S$-module $M$, we simply denote $M_{{\bf c}(F)}$ by $M_F$. In the first principle, we regard $F$ as a subset of $[n]$, or a squarefree vector in ${\mathbb N}^n$, rather than the corresponding face of $ {\mathbf P}$. For example, we write $P_F = (x_i \mid i \not \in F)$, $K[F] \cong K[x_i \mid i \in F]$. And $S(-F)$ denotes the rank 1 free $S$-module $S(-{\bf a})$, where ${\bf a} \in {\mathbb N}^n$ is the squarefree vector corresponding to $F$.
Squarefree $S$-modules are defined by the same way as Definition~\ref{sq}. Note that the free module $S(-{\bf a})$, ${\bf a} \in {\mathbb Z}^n$, is squarefree if and only if ${\bf a}$ is squarefree. Let $\operatorname{*mod}S$ (resp. $\operatorname{Sq}(S)$) be the category of finitely generated ${\mathbb Z}^n$-graded $S$-modules (resp. squarefree $S$-modules). Let $P_\bullet$ be a ${\mathbb Z}^n$-graded minimal free resolution of $M \in \operatorname{*mod}S$. Then $M$ is squarefree if and only if each $P_i$ is a direct sum of copies of $S(-F)$ for various $F \subset [n]$. In the present case, an order ideal $\Delta$ of ${\mathbf L} \, (\cong 2^{[n]})$ is essentially a simplicial complex, and the ring $K[\Delta]$ defined in the previous section is nothing other than the {\it Stanley-Reisner ring} (c.f. \cite{BH,St}) of $\Delta$.
Let $E = \bigwedge {\langle} y_1, \ldots, y_n{\rangle}$ be the exterior algebra over $K$. Under the {\it Bernstein-Gel'fand-Gel'fand correspondence} (c.f. \cite{EFS}), $E$ is the counter part of $S$. We regard $E$ as a ${\mathbb Z}^n$-graded ring by $\deg y_i = {\bf e}_i = \deg x_i$ for each $i$. Clearly, any monomial ideal of $E$ is ``squarefree", and of the form $J_\Delta := ( \, \prod_{i \in F} y_i \mid F \subset [n], \, F \not \in \Delta \, )$ for a simplicial complex $\Delta \subset 2^{[n]}$. We say $K{\langle} \Delta{\rangle} := E/J_\Delta$ is the {\it exterior face ring} of $\Delta$.
Let $\operatorname{*mod}E$ be the category of finitely generated ${\mathbb Z}^n$-graded $E$-modules and degree preserving $E$-homomorphisms. Note that, for graded $E$-modules, we do not have to distinguish left modules from right ones. Hence $${\bf D}_E(-) := \bigoplus_{{\bf a} \in {\mathbb Z}^n} \operatorname{Hom}_{\operatorname{*mod}E}(-, E({\bf a}))$$ gives an exact contravariant functor from $\operatorname{*mod}E$ to itself satisfying ${\bf D}_E \circ {\bf D}_E = \operatorname{Id}$.
\begin{dfn}[R\"omer~\cite{R0}]\label{sqE} We say $N \in \operatorname{*mod}E$ is {\it squarefree}, if $N = \bigoplus_{F \subset [n]} N_F$ (i.e., if ${\bf a} \in {\mathbb Z}^n$ is not squarefree, then $N_{\bf a} = 0$). \end{dfn}
An exterior face ring $K{\langle} \Delta {\rangle}$ is a squarefree $E$-module. But, since a free module $E({\bf a})$ is not squarefree for ${\bf a} \ne 0$, the syzygies of a squarefree $E$-module are {\it not} squarefree. Let $\operatorname{Sq}(E)$ be the full subcategory of $\operatorname{*mod}E$ consisting of squarefree modules. If $N$ is a squarefree $E$-module, then so is ${\bf D}_E(N)$. That is, ${\bf D}_E$ gives a contravariant functor from $\operatorname{Sq}(E)$ to itself.
We have functors ${\mathcal S}: \operatorname{Sq}(E) \to \operatorname{Sq}(S)$ and ${\mathcal E}: \operatorname{Sq}(S) \to \operatorname{Sq}(E)$ giving an equivalence $\operatorname{Sq}(S) \cong \operatorname{Sq}(E)$. Here ${\mathcal S}(N)_F = N_F$ for $N \in \operatorname{Sq}(E)$ and $F \subset [n]$, and the multiplication map ${\mathcal S}(N)_F \ni z \mapsto x_i z \in {\mathcal S}(N)_{F \cup \{ i \}}$ for $i \not \in F$ is given by ${\mathcal S}(N)_F =N_F \ni z \mapsto (-1)^{\alpha(i, F)} y_i z \in N_{F \cup \{ i \}}= {\mathcal S}(N)_{F \cup \{ i \}}$, where $\alpha(i,F) = \# \{ \, j \in F \mid j < i \, \}$. For example. ${\mathcal S}(K{\langle} \Delta{\rangle}) \cong K[\Delta]$. See \cite{R0} for detail.
Note that ${\bf A} := {\mathcal S} \circ {\bf D}_E \circ {\mathcal E}$ is an exact contravariant functor from $\operatorname{Sq}(S)$ to itself satisfying ${\bf A} \circ {\bf A} = \operatorname{Id}$. It is easy to see that ${\bf A}(K[F]) \cong S(-F^{\sf c})$, where $F^{\sf c} := [n] \setminus F$. We also have ${\bf A}(K[\Delta]) \cong I_{\Delta^\vee}$, where $$\Delta^\vee := \{ \, F \subset [n] \mid F^{\sf c} \not \in \Delta \, \}$$ is the {\it Alexander dual} complex of $\Delta$. Since ${\bf A}$ is exact, it exchanges a (minimal) free resolution with a (minimal) irreducible resolution.
Eisenbud et al.\ (\cite{Ei, EFS}) introduced the notion of the {\it linear strands} and the {\it linear part} of a minimal free resolution of a graded $S$-module. Let $P_\bullet : \cdots \to P_1 \to P_0 \to 0$ be a ${\mathbb Z}^n$-graded minimal $S$-free resolution of $M \in \operatorname{*mod}S$. We have natural numbers $\beta_{i,{\bf a}}(M)$ for $i \in {\mathbb N}$ and ${\bf a} \in {\mathbb Z}^n$ such that $P_i = \bigoplus_{{\bf a} \in {\mathbb Z}^n} S(-{\bf a})^{\beta_{i,{\bf a}}(M)}$. We call $\beta_{i,{\bf a}}(M)$ the {\it graded Betti numbers} of $M$.
Set $|{\bf a}| = \sum_{i=1}^na_i$ for ${\bf a} = (a_1, \ldots, a_n) \in {\mathbb N}^n$. For each $l \in {\mathbb Z}$, we define the $l$-{\it linear strand} $\operatorname{lin}_l(P_\bullet)$ of $P_\bullet$ as follows: The term $\operatorname{lin}_l(P_\bullet)_{i}$ of homological degree $i$
is $$\bigoplus_{|{\bf a}|=l+i} S(-{\bf a})^{\beta_{i, {\bf a}}(M)},$$ which is a direct summand of $P_i$, and the differential $\operatorname{lin}_l(P_\bullet)_i \to \operatorname{lin}_l(P_\bullet)_{i-1}$ is the corresponding component of the differential $P_i \to P_{i-1}$ of $P_\bullet$. By the minimality of $P_\bullet$, we can easily verify that $\operatorname{lin} _l \ep{P_\bullet}$ are chain complexes (see also \cite[\S 7A]{Ei}). We call
$\operatorname{lin}(P_\bullet) := \bigoplus_{l \in {\mathbb Z}} \operatorname{lin}_l(P_\bullet)$ the {\it linear part} of $P_\bullet$. Note that the differential maps of $\operatorname{lin}(P_\bullet)$ are represented by matrices of linear forms. We call $$\operatorname{ld}_S(M):= \max \{ i \mid H_i(\operatorname{lin}(P_\bullet)) \ne 0 \}$$ the {\it linearity defect} of $M$.
Sometimes, we regard $M \in \operatorname{*mod}S$ as a ${\mathbb Z}$-graded module by
$M_j = \bigoplus_{|{\bf a}|=j} M_{\bf a}$.
In this case, we set $\beta_{i,j}(M):= \bigoplus_{|{\bf a}|=j} \beta_{i,{\bf a}}(M)$. Then $\operatorname{lin}_l(P_\bullet)_i = S(-l-i)^{\beta_{i,l+i}(M)}$.
\begin{rem} For $M \in \operatorname{*mod}S$, it is clear that $\operatorname{ld}_S(M) \leq \operatorname{proj.dim}_S(M) \leq n$, and there are many examples attaining the equalities. In fact, $\operatorname{ld}_S(S/(x_1^2, \ldots, x_n^2)) = n$. But if $M \in \operatorname{Sq}(S)$, then we always have $\operatorname{ld}_S(M) \leq n-1$. In fact, for a squarefree module $M$, $\operatorname{proj.dim}_S(M) = n$, if and only if $\operatorname{depth}_S M = 0$, if and only if $M \cong K \oplus M'$ for some $M' \in \operatorname{Sq}(S)$. But $\operatorname{ld}_S(K) = 0$ and $\operatorname{ld}_S(M' \oplus K) = \operatorname{ld}_S(M')$. So we may assume that $\operatorname{proj.dim}_S M' \leq n-1$. \end{rem}
\begin{prop}\label{ld_S} Let $M \in \operatorname{Sq}(S)$, and $P_\bullet$ its minimal graded free resolution. We have $$\max \{ \, i \mid H_i(\operatorname{lin}_l(P_\bullet))\not= 0\, \} = n-l - \operatorname{depth}_S ( \, \operatorname{Ext}_S^l( {\bf A}(M), S) \, ), $$ and hence $$\operatorname{ld}_S (M) = \max \{ \, i - \operatorname{depth}_S ( \, \operatorname{Ext}_S^{n-i}( {\bf A}(M), S) \, ) \mid 0 \leq i \leq n \, \}.$$ \end{prop}
\begin{proof} Note that $I^\bullet := {\bf A}(P_\bullet)$ is a minimal irreducible resolution of ${\bf A}(M)$. Moreover, we have ${\bf A}(\operatorname{lin}_l(P_\bullet)) \cong \operatorname{lin}_{n-l}(I^\bullet)$. Since ${\bf A}$ is exact, $$\max \{ \, i \mid H_i(\operatorname{lin}_l(P_\bullet)) \ne 0 \, \} = \max \{ \, i \mid H^i(\operatorname{lin}_{n-l}(I^\bullet)) \ne 0 \, \},$$ and hence \begin{equation}\label{ldirr = ldA} \operatorname{ld}_S(M) = \operatorname{ld.irr}_S({\bf A}(M)). \end{equation} Hence the assertions follow from Corollary~\ref{ld for irr} (note that $S \cong \omega_S$ as underlying modules). \end{proof}
For $N \in \operatorname{*mod}E$, we have a ${\mathbb Z}^n$-graded minimal $E$-free resolution $P_\bullet$ of $N$. By the similar way to the $S$-module case, we can define the linear part $\operatorname{lin}(P_\bullet)$ of $P_\bullet$, and set $\operatorname{ld}_E(N) := \max \{ \, i \mid H_i(\operatorname{lin}(P_\bullet)) \ne 0 \, \}$. (In \cite{R02,Y7}, $\operatorname{ld}_E(N)$ is denoted by $\operatorname{lpd}(N)$. ``lpd" is an abbreviation for ``linear part dominate".) In \cite[Theorem~3.1]{EFS}, Eisenbud et al. showed that $\operatorname{ld}_E (N) < \infty$ for all $N \in \operatorname{*mod}E$. Since $\operatorname{proj.dim}_E (N) = \infty$ in most cases, this is a strong result. If $n \geq 2$, then we have $\sup \{ \, \operatorname{ld}_E(N) \mid N \in \operatorname{*mod}E \, \} = \infty$. In fact, since $E$ is selfinjective, we can take ``cosyzygies". But, if $N \in \operatorname{Sq}(E)$, then $\operatorname{ld}_E(N)$ behaves quite nicely.
\begin{thm}\label{S & E} For $N \in \operatorname{Sq}(E)$, we have $\operatorname{ld}_E(N) = \operatorname{ld}_S({\mathcal S}(N)) \leq n-1$. In particular, for a simplicial complex $\Delta \subset 2^{[n]}$, we have $\operatorname{ld}_E(K{\langle} \Delta {\rangle}) = \operatorname{ld}_S(K[\Delta])$. \end{thm}
\begin{proof} Using the Bernstein-Gel'fand-Gel'fand correspondence, the second author described $\operatorname{ld}_E (N)$ in \cite[Lemma~4.12]{Y7}. This description is the first equality of the following computation, which proves the assertion. \begin{eqnarray*} \operatorname{ld}_E (N) &=& \max \{ \, i - \operatorname{depth}_S ( \, \operatorname{Ext}_S^{n-i}( {\mathcal S} \circ {\bf D}_E(N), S) \, ) \mid 0 \leq i \leq n \, \} \quad \text{(by \cite{Y7})} \\ &=& \max \{ \, i - \operatorname{depth}_S ( \, \operatorname{Ext}_S^{n-i}( {\bf A} \circ {\mathcal S}(N), S) \, ) \mid 0 \leq i \leq n \, \} \quad \text{(see below)} \\ &=& \operatorname{ld}_S ({\mathcal S}(N)) \quad \text{(by Proposition~\ref{ld_S}).} \end{eqnarray*} Here the second equality follows from the isomorphisms ${\mathcal S} \circ {\bf D}_E(N) \cong {\mathcal S} \circ {\bf D}_E \circ {\mathcal E} \circ {\mathcal S} (N) \cong {\bf A} \circ {\mathcal S}(N)$. \end{proof}
\begin{rem} Herzog and R\"omer showed that $\operatorname{ld}_E(N) \leq \operatorname{proj.dim}_S ({\mathcal S}(N))$ for $N \in \operatorname{Sq}(E)$ (\cite[Corollary~3.3.5]{R02}). Since $\operatorname{ld}_S({\mathcal S}(N)) \leq \operatorname{proj.dim}_S ({\mathcal S}(N))$ (the inequality is strict quite often), Theorem~\ref{S & E} refines their result. Our equality might follow from the argument in \cite{R02}, which constructs a minimal $E$-free resolution of $N$ from a minimal $S$-free resolution of ${\mathcal S}(N)$. But it seems that certain amount of computation will be required. \end{rem}
Theorem \ref{S & E} suggests that we may set $$ \operatorname{ld} (\Delta) := \operatorname{ld}_S (K[\Delta]) = \operatorname{ld} _E (K{\langle} \Delta {\rangle}). $$
\begin{thm}\label{ld topological} If $I_\Delta \ne (0)$ (equivalently, $\Delta \ne 2^{[n]}$), then $\operatorname{ld}_S (I_\Delta)$ is a topological
invariant of the geometric realization $|\Delta^\vee|$ of the Alexander dual $\Delta^\vee$ of $\Delta$. If $\Delta \ne 2^{T}$ for any $T \subset [n]$, then
$\operatorname{ld}( \Delta )$ is also a topological invariant of $|\Delta^\vee|$ (also independent from the number $n= \dim S$). \end{thm}
\begin{proof} Since ${\bf A}(I_\Delta) = K[\Delta^\vee]$ and $\Delta^\vee \ne \emptyset$, the first assertion follows from Theorem~\ref{topological} and the equality \eqref{ldirr = ldA} in the proof of Proposition~\ref{ld_S}.
It is easy to see that $\Delta \ne 2^T$ for any $T$ if and only if $\operatorname{ld}( \Delta ) \geq 1$. If this is the case, $\operatorname{ld} ( \Delta) = \operatorname{ld}_S(I_\Delta)+1$, and the second assertion follows from the first. \end{proof}
\begin{rem} (1) For the first statement of Theorem~\ref{ld topological}, the assumption that $I_\Delta \ne (0)$ is necessary. In fact, if $I_\Delta =(0)$, then $\Delta = 2^{[n]}$ and $\Delta^\vee = \emptyset$. On the other hand, if we set $\Gamma :=2^{[n]} \setminus [n]$,
then $\Gamma^\vee = \{ \emptyset \}$ and $|\Gamma^\vee| =
\emptyset = |\Delta^\vee|$. In view of Proposition~\ref{ld_S}, it might be natural to set $\operatorname{ld}_S(I_\Delta) = \operatorname{ld}_S( \, (0) \, ) = -\infty$. But, $I_\Gamma = \omega_S$ and hence $\operatorname{ld}_S(I_\Gamma) = 0$. One might think it is better to set $\operatorname{ld}_S( \, (0) \, ) = 0$ to avoid the problem. But this convention does not help so much, if we consider $K[\Delta]$ and $K[\Gamma]$. In fact, $\operatorname{ld}_S(K[\Delta]) = \operatorname{ld}_S (S) = 0$ and $\operatorname{ld}_S(K[\Gamma]) = \operatorname{ld}_S(S/\omega_S) =1$.
(2) Let us think about the second statement of the theorem. Even if we forget the assumption that $\Delta \ne 2^T$, $\operatorname{ld}(\Delta)$ is almost a topological invariant. Under the assumption that $I_\Delta \ne 0$, we have the following. \begin{itemize} \item $\operatorname{ld}(\Delta) \leq 1$ if and only if $K[\Delta^\vee]$ is sequentially Cohen-Macaulay. Hence we can determine whether $\operatorname{ld}(\Delta) \leq 1$
from the topological space $|\Delta^\vee|$. \item $\operatorname{ld}(\Delta) = 0$, if and only if all facets of $\Delta^\vee$ have dimension $n-2$, if and only if
$|\Delta^\vee|$ is Cohen-Macaulay and has dimension $n-2$. \end{itemize} Hence, if we forget the number ``$n$", we can not determine
whether $\operatorname{ld}(\Delta) = 0$ from $|\Delta^\vee|$. \end{rem}
\section{An upper bound of linearity defects.} In the previous section, we have seen that $\operatorname{ld}_E(N) = \operatorname{ld}_S({\mathcal S}(N))$ for $N\in \operatorname{Sq}(E)$, in particular $\operatorname{ld} _E \ep{\exfacering{\d}} = \operatorname{ld} _S \ep{\symfacering{\d}}$ for a simplicial complex $\d$. In this section, we will give an upper bound of them, and see that the bound is sharp.\\ \quad For $0 \not= N\in \operatorname{*mod}E$, regarding $N$ as a ${\mathbb Z} $-graded module, we set $\operatorname{indeg} _E\ep{N} := \min\sbra{\, i \mid N_i \not= 0\, }$, which is called the {\it initial degree} of $N$, and $\operatorname{indeg} _S\ep{M}$ is similarly defined as $\operatorname{indeg} _S\ep{M} := \min\sbra{\, i\mid M_i \not= 0\, }$ for $0\not= M\in \operatorname{*mod}S$. If $\d \ne 2^{\left[ n\right] }$ (equivalently $\stideali \d \not= 0$ or $\stidealj \d \not= 0$), then we have $\operatorname{indeg} _S\ep{\stideali{\d}} = \operatorname{indeg} _E\ep{\stidealj{\d}} = \min \sbra{\, \sharp F \mid F\subset \left[ n\right] ,F\not\in \d \, }$, where $\sharp F$ denotes the cardinal number of $F$. So we set $$ \operatorname{indeg} \ep{\d} := \operatorname{indeg} _S\ep{\stideali{\d}} = \operatorname{indeg} _E\ep{\stidealj{\d}}. $$ Since $\operatorname{ld} \ep{2^{\left[ n\right] }} = \operatorname{ld} _S \ep{S} = \operatorname{ld} _E \ep{E} = 0$ holds, we henceforth exclude this trivial case; we assume that $\d \not= 2^{\left[ n\right] }$.\\ \indent We often make use of the following facts:
\begin{lem}\label{sec:res lemma} Let $0\not= M \in \operatorname{*mod}S$ and let $P_\bullet$ be a minimal graded free resolution of $M$. Then \begin{enumerate} \item $\operatorname{lin} _i \ep{P_\bullet} = 0$ for all $i < \operatorname{indeg} _S\ep{M}$, i.e., there are only $l$-linear strands with $l\ge \operatorname{indeg} _S\ep{M}$ in $P_\bullet${\rm ;} \item $\operatorname{lin} _{\operatorname{indeg} _S\ep{M}} \ep{P_\bullet}$ is a subcomplex of $P_\bullet${\rm ;} \item if $M\in \operatorname{Sq}(S)$, then $\operatorname{lin} \ep{P_\bullet} = \dsum{}{0\le l \le n} \operatorname{lin} _l\ep{P_\bullet}$, and $\operatorname{lin} _l\ep{P_\bullet}_i = 0$ for all $i > n-l$ and all $0\le l \le n$, where the subscript $i$ is a homological degree. \end{enumerate} \end{lem} \begin{proof} (1) and (2) are clear. (3) holds from the fact that $P_i \cong \dsum{}{F\subset \left[ n\right]}S\ep{-F}^{\betti iF}$. \end{proof}
\begin{thm}\label{sec:bound of ld} For $0\not= N\in \operatorname{Sq}(E)$, it follows that $$ \operatorname{ld} _E \ep{N}\le \max \sbra{ 0, n-\operatorname{indeg} _E\ep{N} -1}. $$ \\ \quad By Theorem \ref{S & E}. this is equivalent to say that for $M \in \operatorname{Sq}(S)$, $$ \operatorname{ld} _S \ep{M}\le \max \sbra{0, n-\operatorname{indeg}_S \ep{M} - 1}. $$ \end{thm} \begin{proof} It suffices to show the assertion for $M\in \operatorname{Sq}(S)$. Set $\operatorname{indeg} _S\ep{M} = d$ and let $P_\bullet$ be a minimal graded free resolution of $M$. The case $d=n$ is trivial by Lemma \ref{sec:res lemma} (1), (3). Assume that $d\le n-1$. Observing that $\operatorname{lin} _l \ep{P_\bullet}_i = S \ep{-l-i}^{\betti i{i+l}}$, where $\betti i{i+l}$ are ${\mathbb Z}$-graded Betti numbers of $M$, Lemma \ref{sec:res lemma} (1), (3) implies that the last few steps of $P_\bullet$ are of the form \begin{align*} \begin{CD} 0@> >> S\ep{-n}^{\betti {n-d}n} @> >>S\ep{-n}^{\betti{n-d-1}n} \oplus S\ep{-n+1}^{\betti{n-d-1}{n-1}} @> >>\cdots . \end{CD} \end{align*} Hence $\operatorname{lin} _d\ep{P_\bullet}_{n-d} = S\ep{-n}^{\betti {n-d}n} = P_{n-d}$. Since $\operatorname{lin} _d\ep{P_\bullet}$ is a subcomplex of the acyclic complex $P_\bullet$ by Lemma \ref{sec:res lemma} (2), we have $\H{n-d}{\operatorname{lin} _d\ep{P_\bullet}} = 0$, so that $\operatorname{ld} _S\ep{M} \le n-d -1$. \end{proof}
Note that $\stidealj{\d}\in \operatorname{Sq}(E)$ (resp. $\stideali{\d}\in \operatorname{Sq}(S)$). Since $\operatorname{ld} \ep{\d} \le \operatorname{ld} _E\ep{\stidealj{\d}} + 1$ (resp. $\operatorname{ld} \ep{\d} \le \operatorname{ld} _S\ep{\stideali{\d}} + 1$) holds, we have a bound for $\operatorname{ld} \ep{\d}$, applying Theorem \ref{sec:bound of ld} to $\stidealj{\d}$ (resp. $\stideali{\d}$).
\begin{cor}\label{sec:bound of d} For a simplicial complex $\d$ on $\left[ n\right] $, we have $$ \operatorname{ld} \ep{\d} \le \max \sbra{1, n-\operatorname{indeg} \ep{\d}}. $$ \end{cor}
Let $\d ,\Gamma$ be simplicial complexes on $\left[ n\right] $. We denote $\d *\Gamma $ for the join $$ \sbra{\, F\cup G \mid F\in \d , G\in \Gamma \, } $$ of $\d$ and $\Gamma $, and for our convenience, set $$ \operatorname{ver} \ep{\d} := \sbra{\ v\in \left[ n\right]\mid \sbra{v} \in \d \ }. $$
\begin{lem}\label{sec:lem of ld} Let $\d$ be a simplicial complex on $\left[ n\right]$. Assume that $\operatorname{indeg} \ep{\d} = 1$, or equivalently $\operatorname{ver} \ep{\d} \not= \left[ n\right]$. Then we have $$ \operatorname{ld} \ep{\d} = \operatorname{ld} \ep{\d *\sbra{v}} $$ for $v \in \left[ n\right] \setminus \operatorname{ver} \ep{\d}$. \end{lem}
\begin{proof} We may assume that $v=1$. Let $P_\bullet$ be a minimal graded free resolution of $\symfacering{\d *\sbra{1}}$ and $\mathcal{K} \ep{x_1}$ the Koszul complex $$ \begin{CD} 0@> >>S\ep{-1} @> x_1 >> S @> >>0 \end{CD} $$ with respect to $x_1$. Consider the mapping cone $P_\bullet \tensor{}{S} \mathcal{K} \ep{x_1}$ of the map $P_\bullet \ep{-1} \overset{x_1}{\longrightarrow} P_\bullet$. There is the short exact sequence $$ \begin{CD} 0@> >>P_\bullet @> >>P_\bullet \tensor{}{S} \mathcal{K} \ep{x_1} @> >>P_\bullet\ep{-1}\bra{-1}@> >>0, \end{CD} $$ whence we have $\H i{P_\bullet\tensor{}{S} \mathcal{K} \ep{x_1}} = 0$ for all $i\ge2$ and the exact sequence $$ \begin{CD} 0@> >>\H 1{P_\bullet\tensor{}{S} \mathcal{K} \ep{x_1}} @> >> \H 0{P_\bullet\ep{-1}} @> x_1>> \H 0{P_\bullet}. \end{CD} $$ But since $\H 0{P_\bullet} = \symfacering{\d*\sbra{1}}$ and $x_1$ is regular on it, we have $\H 1{P_\bullet \tensor{}{S} \mathcal{K} \ep{x_1}} = 0$. Thus $P_\bullet \tensor{}{S} \mathcal{K}\ep{x_1}$ is acyclic and hence a minimal graded free resolution of $\symfacering{\d}$. Note that $\operatorname{lin} \ep{P_\bullet\tensor{}{S} \mathcal{K}\ep{x_1}} = \operatorname{lin} \ep{P_\bullet} \tensor{}{S} \mathcal{K}\ep{x_1}$: in fact, we have \begin{align*} \operatorname{lin} _l\ep{P_\bullet \tensor{}{S} \mathcal{K} \ep{x_1}}_i &= \operatorname{lin} _l\ep{P_\bullet \tensor{}{S} S}_i \oplus \operatorname{lin}_l\ep{P_\bullet \bra{-1}\tensor{}{S} S\ep{-1}}_i \\
&= \ep{\operatorname{lin} _l\ep{P_\bullet }_i \tensor{}{S} S} \oplus \ep{\operatorname{lin} _l\ep{P_\bullet}_{i-1} \tensor{}{S}S\ep{-1}} \\
&= \ep{\operatorname{lin} _l\ep{P_\bullet } \tensor{}{S} \mathcal{K} \ep{x_1}}_i, \end{align*} where the subscripts $i$ denote homological degrees, and the differential map $$ \operatorname{lin} _l\ep{P_\bullet\tensor{}{S} \mathcal{K} \ep{x_1}}_i \longrightarrow \operatorname{lin} _l\ep{P_\bullet\tensor{}{S} \mathcal{K} \ep{x_1}}_{i-1} $$ is composed by $\partial ^{{\langle} l{\rangle} }_i$, $-\partial ^{{\langle} l{\rangle} }_{i-1}$, and the multiplication map by $x_1$, where $\partial ^{{\langle} l{\rangle} }_i$ (resp. $\partial ^{{\langle} l{\rangle} }_{i-1}$) is the $i^{\text{th}}$(resp. $\ep{i-1}^{\text{st}}$) differential map of the $l$-linear strand of $P_\bullet$. Hence there is the short exact sequence
\begin{align*} \begin{CD} 0@> >>\operatorname{lin} \ep{P_\bullet} @> >> \operatorname{lin} \ep{P_\bullet \tensor{}{S} \mathcal{K} \ep{x_1}} @> >> \operatorname{lin} \ep{P_\bullet}\ep{-1}\bra{-1}@> >>0, \end{CD} \end{align*} which yields that $\H i{\operatorname{lin} \ep{P_\bullet \tensor{}{S} \mathcal{K} \ep{x_1}}} = 0$ for all $i\ge \operatorname{ld} \ep{\d *\sbra{1}} +2$, and the exact sequence \begin{multline*} 0 \longrightarrow \H {\operatorname{ld} \ep{\d *\sbra{1}} +1}{\operatorname{lin} \ep{P_\bullet \tensor{}{S} \mathcal{K} \ep{x_1}}} \longrightarrow \H {\operatorname{ld} \ep{\d *\sbra{1}}}{\operatorname{lin} \ep{P_\bullet}\ep{-1}} \\ \overset{x_1}{\longrightarrow} \H {\operatorname{ld} \ep{\d *\sbra{1}}}{\operatorname{lin} \ep{P_\bullet}} \longrightarrow \H {\operatorname{ld} \ep{\d *\sbra{1}}}{\operatorname{lin} \ep{P_\bullet \tensor{}{S} \mathcal{K} \ep{x_1}}}. \end{multline*} Since $x_1$ does not appear in any entry of the matrices representing the differentials of $\operatorname{lin} \ep{P_\bullet}$, it is regular on $\H {\bullet}{\operatorname{lin} \ep{P_\bullet}}$, and hence we have $$ \H {\operatorname{ld} \ep{\d *\sbra{1}}+1}{\operatorname{lin} \ep{P_\bullet \tensor{}{S} \mathcal{K} \ep{x_1}}} = 0 $$ and $$ \H {\operatorname{ld} \ep{\d *\sbra{1}}}{\operatorname{lin} \ep{P_\bullet \tensor{}{S} \mathcal{K} \ep{x_1}}} \not= 0, $$ since $\H {\operatorname{ld} \ep{\d *\sbra{1}}}{\operatorname{lin} \ep{P_\bullet}} \not= 0$. Therefore $\operatorname{ld} \ep{\d} = \operatorname{ld} \ep{\d *\sbra{1}}$. \end{proof}
Let $\d$ be a simplicial complex on $\left[ n\right] $. For $F\subset \left[ n\right] $, we set $$ \d _F := \sbra{\, G\in \d \mid G\subset F\, }. $$ \quad The following fact, due to Hochster, is well known, but because of our frequent use, we mention it. \begin{prop}[c.f. \cite{BH,St}]\label{sec:Hoc's formula a} For a simplicial complex $\d$ on $\left[ n\right] $, we have $$ \betti ij\ep{\symfacering{\d}}=\sum _{F\subset \left[ n\right], \sharp F =j}\dim _K\redhom{j-i-1}{\d _F}K , $$ where $\betti ij\ep{\symfacering{\d}}$ are the ${\mathbb Z} $-graded Betti numbers of $\symfacering{\d}$. \end{prop}
Now we can give a new proof of \cite[Proposition 4.15]{Y7}, which is the latter part of the next result.
\begin{prop}[cf. {\cite[Proposition 4.15]{Y7}}]\label{sec:indeg is 1} Let $\d$ be a simplicial complex on $\left[ n\right] $. If $\operatorname{indeg} \d = 1$, then we have $$ \operatorname{ld} \ep{\d} \le \max \sbra{1, n-3}. $$ Hence, for any $\d$, we have $$ \operatorname{ld} \ep{\d} \le \max \sbra{1, n-2}. $$ \end{prop}
\begin{proof} The second inequality follows from the first one and Corollary \ref{sec:bound of d}. So it suffices to show the first. We set $\mathcal{V} := \left[ n\right] \setminus \operatorname{ver} \ep{\d}$. Our hypothesis $\operatorname{indeg} \d = 1$ implies that $\mathcal{V} \not= \varnothing$. By Lemma \ref{sec:lem of ld}, the proof can be reduced to the case $\sharp \mathcal{V}=1$. We may then assume that $\mathcal{V} = \sbra{1}$. Thus we have only to show that $\operatorname{ld} \ep{\d *\sbra{1}} \le \max \sbra{1,n-3}$. Since we have $\operatorname{indeg} \ep{\d *\sbra{1}} \ge 2$, we may assume $n\ge 4$ by Corollary \ref{sec:bound of d}. The length of the $0$-linear strand of $\symfacering{\d *\sbra{1}}$ is $0$, and hence we concentrate on the $l$-linear strands with $l\ge1$. Let $P_\bullet$ be a minimal graded free resolution of $\symfacering{\d *\sbra{1}}$. Since, as is well known, the cone of a simplicial complex, i.e., the join with a point, is acyclic, we have $$ \betti in \ep{\symfacering{\d *\sbra{1}}} = \dim _K\redhom{n-i-1}{\d *\sbra{1}}K = 0 $$ by Proposition \ref{sec:Hoc's formula a}. Thus $\operatorname{lin} _l\ep{P_\bullet}_{n-l} = 0$ for all $l\ge 1$. Now applying the same argument as the last part of the proof of Theorem \ref{sec:bound of ld} (but we need to replace $n$ by $n-1$), we have $$ \H{n-2}{\operatorname{lin} \ep{P_\bullet}} = 0, $$ and so $\operatorname{ld} \ep{\d *\sbra{1}} \le n-3$. \end{proof}
According to \cite[Proposition 4.14]{Y7}, we can construct a squarefree module $N\in \operatorname{Sq}(E)$ with $\operatorname{ld} _E \ep{N} = \operatorname{proj.dim} _S\ep{{\mathcal S}\ep{N}} = n-1$. By Theorems~\ref{S & E} and \ref{sec:bound of ld}, $M:={\mathcal S}\ep{N}$ satisfies that $\operatorname{indeg} _S\ep{M} = 0$ and $\operatorname{ld} _S\ep{M} = n-1$. For $0\le i \le n-1$, let $\Omega _i\ep{M}$ be the $i^{\text{th}}$ syzygy of $M$. Then $\Omega _i\ep{M}$ is squarefree, and we have that $\operatorname{ld} _S\ep{\Omega _i\ep{M}} = \operatorname{ld} _S\ep{M} - i = n-i-1$ and $\operatorname{indeg} _S\ep{\Omega _i\ep{M}} \ge \operatorname{indeg} _S\ep{M} + i = i$. Thus by Theorem \ref{sec:bound of ld}, we know that $\operatorname{indeg} _S\ep{\Omega _i\ep{M}} = i$ and $\operatorname{ld} _S\ep{\Omega _i\ep{M}} = n - \operatorname{indeg} _S\ep{\Omega _i\ep{M}} -1$. So the bound in Theorem \ref{sec:bound of ld} is optimal.\\ \quad In the following, we will give an example of a simplicial complex $\d$ with $\operatorname{ld} \ep{\d} = n-\operatorname{indeg} \ep{\d}$ for $2\le \operatorname{indeg} \ep{\d} \le n-2$, and so we know the bound in Proposition~\ref{sec:bound of d} is optimal if $\operatorname{indeg} \ep{\d} \ge 2$, that is, $\operatorname{ver} \ep{\d} = \left[ n\right]$. \\
Given a simplicial complex $\d$ on $\left[ n\right] $, we denote $\sk {\d}i$ for the $i^{\text{th}}$ skeleton of $\d$, which is defined as $$ \sk {\d}i := \sbra{\, F\in \d \mid \# F \le i+1\, }. $$ \begin{exmp}\label{sec:sharp}
Set $\Sigma := 2^{\left[ n\right]}$, and let $\Gamma $ be a simplicial complex on $\left[ n\right] $ whose geometric realization $|\Gamma |$ is homeomorphic to the $\ep{d-1}$-dimensional sphere with $2\le d<n-1$, which we denote by $S^{d-1}$. (For $m>d$ there exists a triangulation of $S^{d-1}$ with $m$ vertices. See, for example, \cite[Proposition 5.2.10]{BH}). Consider the simplicial complex $\d :=\Gamma \cup \sk{\Sigma }{d-2}$. We will verify that $\d$ is a desired complex, that is, $\operatorname{ld} \ep{\d} = n - \operatorname{indeg} \ep{\d}$. For brief notation, we put $t:= \operatorname{indeg} \d$ and $l:=\operatorname{ld} \ep{\d }$. \\ \quad First, from our definition, it is clear that $t \ge d$. Thus it is enough to show that $n-d\le l$: in fact we have that $l\le n-t \le n-d \le l$ by Corollary \ref{sec:bound of d}, and hence that $t=d$ and $l = n-d$. Our aim is to prove that $$ \betti {n-d}n\ep{\symfacering{\d}} \not= 0 \quad \text{and} \quad \betti{n-d -1}{n-1}\ep{\symfacering{\d}} = 0, $$ since, in this case, we have $\H {n-d}{\operatorname{lin} _d\ep{P_\bullet}}\not= 0$, and hence $n-d\le l$. \\ \quad Now, let $F\subset \left[ n\right]$, and $\redcomp{\bullet }{\d _F}$, $\redcomp{\bullet }{\Gamma _F}$ be the augmented chain complexes of $\d _F$ and $\Gamma _F$, respectively.
Since $\sk{\Sigma}{d-2}$ have no faces of dimension $\ge d-1$, we have $\redcomp{d-1}{\d _F} = \redcomp{d-1}{\Gamma _F}$ and hence $\redhom{d-1}{\d _F}K = \redhom{d-1}{\Gamma _F}K$. On the other hand, our assumption that $|\Gamma |\approx S^{d-1}$ implies that $\Gamma $ is Gorenstein, and hence that \begin{align*} \redhom{d-1}{\Gamma _F}K = \begin{cases} K & \text{if $F=\left[ n\right] $;} \\ 0 & \text{otherwise.} \end{cases} \end{align*} Therefore, by Proposition \ref{sec:Hoc's formula a}, we have that \begin{align*} \betti {n-d}n\ep{\symfacering{\d}} &= \dim _K\redhom{d-1}{\Gamma}K = 1\not= 0;\\ \betti{n-d -1}{n-1}\ep{\symfacering{\d}} &= \sum _{F\subset \left[ n\right], \sharp F=n-1}\dim _K\redhom{d-1}{\Gamma _F}K = 0. \end{align*} \end{exmp}
\section{A simplicial complex $\d$ with $\operatorname{ld} \ep{\d} = n-2$ is an $n$-gon} Following the previous section, we assume that $\d \not= \left[ n\right]$, throughout this section. We say a simplicial complex on $\left[ n\right] $ is an $n$-{\it gon} if its facets are $\sbra{1,2},\sbra{2,3},\cdots ,\sbra{n-1,n},$ and $\sbra{n,1}$ after a suitable permutation of vertices. Consider the simplicial complex $\d$ on $\left[ n\right]$ given in Example \ref{sec:sharp}. If we set $d = 2$, then $\d$ is an $n$-gon. Thus if a simplicial complex $\d$ on $\left[ n\right] $ is an $n$-gon, we have $\operatorname{ld} \ep{\d} = n-2$. Actually, the inverse holds, that is, if $\operatorname{ld} \ep{\d} = n-2$ with $n\ge 4$, $\d$ is nothing but an $n$-gon.
\begin{thm} \label{sec:the case any dim} Let $\d$ be a simplicial complex on $\left[ n\right]$ with $n \ge 4$. Then $\operatorname{ld} \ep{\d}=n-2$ if and only if $\d$ is an $n$-gon. \end{thm}
In the previous section, we introduced Hochster's formula (Proposition \ref{sec:Hoc's formula a}), but in this section, we need explicit correspondence between $\bra{\tor{\bullet}S{\symfacering{\d}}K}_F$ and reduced cohomologies of $\d _F$, and so we will give it as follows. \\ \quad Set $V:=\angle{x_1,\dots ,x_n}=S_1$ and let $\lbull{\mathcal{K}}:=S\tensor{}{K} \bigwedge V$ be the Koszul complex of $S$ with respect to $x_1,\dots ,x_n$. Then we have $$ \ebra{\tor iS{\symfacering{\d}}K}_F=\H i{\bra{\symfacering{\d}\tensor {}{S}\lbull{\mathcal{K} }}_F}=\H i{\ebra{\symfacering{\d}\tensor {}{K}\bigwedge V}_F} $$ for $F\subset \left[ n\right]$. Furthermore, the basis of the $K$-vector space $\ebra{\symfacering{\d}\tensor {}{K}\bigwedge V}_F$ is of the form ${\bf x} ^G\tensor{}{}\wedge ^{F\setminus G}{\bf x}$ with $G \in \d _F$, where ${\bf x} ^G = \prod _{i\in G}x_i$ and $\wedge ^{F\setminus G}{\bf x} = x_{i_1} \wedge \cdots \wedge x_{i_k}$ for $\sbra{i_1,\cdots ,i_k} = F\setminus G$ with $i_1<\cdots <i_k$. Thus the assignment $$ \varphi ^i:\redcocomp{i-1}{\d _F}\ni e_G^{*} \longmapsto \ep{-1}^{\sign GF}{\bf x} ^G\tensor{}{}\wedge ^{F\setminus G}{\bf x}\in \ebra{\symfacering{\d}\tensor {}{K}\bigwedge V}_F $$ with $G\in \d _F$ gives the isomorphism $\ubull{\varphi}:\redcocomp {\bullet}{\d _F}\ebra{-1} \longrightarrow \ebra{\symfacering{\d}\tensor {}{K}\bigwedge V}_F$ of chain complexes, where $\redcocomp {i-1}{\d _F}$ (resp. $\redcomp{i-1}{\d _F}$) is the $\ep{i-1}^{\text{st}}$ term of the augmented cochain (resp. chain) complex of $\d _F$ over $K$, $e_G$ is the basis element of $\redcomp{i-1}{\d _F}$ corresponding to $G$, and $e_G^{*}$ is the $K$-dual base of $e_G$. Here we set $$
\alpha \ep{A,B}:=\sharp \sbra{\ep{a,b}\ |\ a>b, a\in A, b\in B} $$ for $A,B \subset \left[ n\right]$. Thus we have the isomorphism \begin{equation}\label{eq:tor=redhom} \bar{\varphi}:\redcohom{i-1}{\d _F}K \longrightarrow \ebra{\tor {\sharp F -i}S{\symfacering{\d}}K}_F. \end{equation}
\begin{lem} \label{sec:position of ld} Let $\d$ be a simplicial complex on $\left[ n\right]$ with $\operatorname{indeg} \ep{\d} \ge 2$, and $P_\bullet$ a minimal graded free resolution of $\symfacering{\d}$. We denote $\lbull Q$ for the subcomplex of $P_\bullet$ such that $Q_i := \bigoplus _{j \le i + 1} S\ep{-j}^{\betti ij} \subset \bigoplus _{j\in {\mathbb Z}}S\ep{-j}^{\betti ij} = P_i$. Assume $n\ge 4$. Then the following are equivalent. \begin{enumerate} \item $\operatorname{ld} \ep{\d} = n-2$; \item $\H{n-2}{\operatorname{lin} _2\ep{P_\bullet}}\not= 0$; \item $\H{n-3}{\lbull Q}\not= 0$. \end{enumerate} \end{lem}
In the case $n \ge 5$, the condition ($3$) is equivalent to $\H{n-3}{\operatorname{lin} _1\ep{P_\bullet}} \not= 0$.
\begin{comment} \begin{proof} First of all, we note $\operatorname{lin} _0\ep{P_\bullet}_i = 0$ for $i \ge 1$ holds, since $\operatorname{indeg} \ep{\d} \ge 2$.\\ \quad ($1$) $\Leftarrow $ ($2$). Choose $z \in \operatorname{lin}_2\ep{P_\bullet}$ whose image in $\H{n-2}{\operatorname{lin} _2\ep{P_\bullet}}$ is not $0$. Then $\partial _{n-2}\ep{z} \not= 0$; otherwise $z = 0$ by Lemma \ref{sec:res lemma} and the acyclicity of $P_\bullet$, which is a contradiction. Since $\partial ^{\angle{2}}_{n-2} \ep{z} = 0$, it follows $\partial _{n-2}\ep{\z} \in Q_{n-3}$. \quad ($2$) $\Leftarrow $ ($2$). Since $\operatorname{ld} \ep{\d} = n-2 \ge 2 \not= 0$, $\operatorname{ld} \ep{\d}$ is attained by $1$- or $2$-linear strands of $P_\bullet$. Thus it suffices to show that $\H i{\operatorname{lin} _1\ep{P_\bullet}} = 0$ for $i\ge n-2$. Note that since now $\operatorname{indeg} \ep{\d} = 2$, it follows that $\operatorname{proj.dim} _S\ep{\symfacering{\d}}\le n-1$. The last few steps of $P_\bullet$ and $\operatorname{lin} _1\ep{P_\bullet}$ is of the form: $$ \begin{CD} 0@> >>P_{n-1}@> \partial _{n-1}>>P _{n-2}@>\partial _{n-2}>>P _{n-3}; \\ 0@> >>\operatorname{lin} _1\ep{P_\bullet}_{n-1}@> \partial ^{{\langle} 1{\rangle} }_{n-1}>>\operatorname{lin} _1\ep{P_\bullet}_{n-2}@>\partial ^{{\langle} 1{\rangle} }_{n-2}>>\operatorname{lin} _1\ep{P_\bullet}_{n-3}. \end{CD} $$ Since by Lemma \ref{sec:res lemma} the latter complex is a subcomplex of the former, $P_{n-1} = S\ep{-n}^{\betti {n-1}n} = \operatorname{lin} _1\ep{P_\bullet}_{n-1}$ holds, and $P_\bullet$ is acyclic, we conclude that $$ \H {n-1}{\operatorname{lin} _1\ep{P_\bullet}} = \H {n-2}{\operatorname{lin} _1\ep{P_\bullet}} = 0. $$ \quad ($2$) $\Leftrightarrow$ ($3$). We denote for $\partial _i$, $\partial ^{\angle{l}}_i$, and $\tilde{\partial }_i$, the $i^{\text{th}}$ differential maps of $P_\bullet$, $\operatorname{lin} _l\ep{P_\bullet}$, and $\lbull Q$, respectively. By Lemma \ref{sec:res lemma} and a routine, we can easily verify the implication ``$\Rightarrow$". We shall show the inverse implication. Assume that $\H{n-3}{\lbull Q} \not= 0$. Then there is a non-zero element $z \in Q_{n-3}$ such that $\partial _{n-3}\ep{z} = \tilde{\partial }_{n-3}\ep{z} = 0$. Hence by acyclicity of $P_\bullet$, $z = \partial _{n-2}\ep{z_1 + z_2}$ for some $z_1 \in Q_{n-2}$ and $0 \not = z_2 \in \operatorname{lin} _2\ep{P_\bullet}_{n-2}$, since $\operatorname{lin} _l\ep{P_\bullet}_{n-2} = 0$ for $l\ge 2$ by Lemma \ref{sec:res lemma} ($3$). Then $z_2$ satisfies $\partial ^{\angle{2}}_{n-2}\ep{z_2} = 0$, and the image of $\partial ^{\angle{2}}_{n-2}\ep{z_2}$ in $\H{n-2}{\operatorname{lin}_2\ep{P_\bullet}}$ is not $0$. \end{proof} \end{comment}
\begin{proof} Since $\operatorname{indeg} \ep{\d} \ge 2$, $\operatorname{lin} _0\ep{P_\bullet}_i = 0$ holds for $i \ge 1$. Clearly, $\H i{\lbull Q} = \H i{\operatorname{lin} _1\ep{P_\bullet}}$ for $i \geq 2$. Since $\operatorname{lin} _l\ep{P_\bullet}_i = 0$ for $i\ge n-2$ and $l\ge 3$ by Lemma \ref{sec:res lemma} and that $\operatorname{ld} \ep{\d} \le n-2$ by Proposition~\ref{sec:indeg is 1}, it suffices to show the following. \begin{align}\label{eq:what to show} \H{n-2}{\operatorname{lin}_2\ep{P_\bullet}} \cong \H{n-3}{\lbull Q} \quad \text{and} \quad \H i{\lbull Q} = 0 \quad \text{for $i \ge n-2$}. \end{align} Since $\lbull Q$ is a subcomplex of $P_\bullet$, there exists the following short exact sequence of complexes. $$ 0 \longrightarrow \lbull Q \longrightarrow P_\bullet \longrightarrow \tilde{P_\bullet} := P_\bullet /\lbull Q \longrightarrow 0, $$ which induces the exact sequence of homology groups $$ \H i{P_\bullet} \longrightarrow \H i{\tilde{P_\bullet}} \longrightarrow \H{i-1}{\lbull Q} \longrightarrow \H{i-1}{P_\bullet}. $$ Hence the acyclicity of $P_\bullet$ implies that $\H i{\tilde{P_\bullet}} \cong \H{i-1}{\lbull Q}$ for all $i \ge 2$. Now $\H i{\tilde{P_\bullet}} = 0$ for $i \ge n-1$ by Lemma \ref{sec:res lemma} and the fact that $\tilde{P}_i = \oplus _{l \ge 2}\operatorname{lin} _l\ep{P_\bullet}_i$. So the latter assertion of \eqref{eq:what to show} holds, since $n-2 \ge 2$. The former follows from the equality $\H{n-2}{\tilde{P_\bullet}} = \H{n-2}{\operatorname{lin}_2\ep{P_\bullet}}$, which is a direct consequence of the fact that $\operatorname{lin}_2\ep{P_\bullet}$ is a subcomplex of $\tilde{P_\bullet}$, that $\tilde{P}_{n-2} = \operatorname{lin} _2\ep{P_\bullet}_{n-2}$, and that $\tilde{P}_{n-1} = 0$. \end{proof}
Let $\d$ be a $1$-dimensional simplicial complex on $\left[ n\right]$ (i.e., $\d$ is essentially a simple graph). A {\it cycle} $C$ in $\d$ of length $t$ ($\ge 3$) is a sequence of edges of $\d$ of the form ($v_1$, $v_2$), ($v_2$, $v_3$)$,\dots ,$($v_t$, $v_1$) joining distinct vertices $v_1,\dots v_t$. \\
Now we are ready for the proof of Theorem \ref{sec:the case any dim}.
\renewcommand{\textit{Proof of Theorem \ref{sec:the case any dim}}}{\textit{Proof of Theorem \ref{sec:the case any dim}}} \begin{proof} The implication ``$\Leftarrow$" has been already done in the beginning of this section. So we shall show the inverse. By Proposition \ref{sec:indeg is 1}, we may assume that $\operatorname{indeg} \ep{\d} \ge 2$. Let $P_\bullet$ be a minimal graded free resolution of $\symfacering{\d}$ and $\lbull Q$ as in Lemma \ref{sec:position of ld}. Note that $\lbull Q$ is determined only by $\ebra{\stideali{\d}}_2$ and that it follows $\ebra{\stideali{\d}}_2 = \ebra{\stideali{\sk{\d}1}}_2$. If the $1$-skeleton $\sk{\d}1$ of $\d$ is an $n$-gon, then so is $\d$ itself. Thus by Lemma \ref{sec:position of ld}, we may assume that $\dim \d = 1$. Since $\operatorname{ld} \ep{\d} = n-2$, by Lemma \ref{sec:position of ld} we have $$ \redhom 1{\d}K \cong \redcohom 1{\d}K \cong \ebra{\tor{n-2}S{\symfacering{\d}}K}_{\left[ n\right]} \not= 0, $$ and hence $\d$ contains at least one cycle as a subcomplex. So it suffices to show that $\d$ has no cycles of length $\le n - 1$. Suppose not, i.e., $\d$ has some cycles of length $\le n-1$. To give a contradiction, we shall show \begin{align}\label{eq:contra} 0 \longrightarrow \operatorname{lin} _2\ep{P_\bullet}_{n-2} \longrightarrow \operatorname{lin} _{2}\ep{P_\bullet}_{n-3} \end{align} is exact; in fact it follows $\H{n-2}{\operatorname{lin} _2\ep{P_\bullet}} = 0$, which contradicts to Lemma \ref{sec:position of ld}. For that, we need some observations (this is a similar argument to that done in Theorem 4.1 of \cite{Y}). Consider the chain complex $\symfacering{\d}\tensor{}{K}\bigwedge V\tensor{}{K}S$ where $V$ is the $K$-vector space with the basis $x_1,\dots ,x_n$. We can define two differential map $\vartheta ,\partial $ on it as follows: \begin{align*} \vartheta \ep{f\tensor{}{}\wedge ^G{\bf x}\tensor{}{} g} = \sum _{i\in G}\ep{-1}^{\sign iG}\ep{x_if\tensor{}{}\wedge ^{G\setminus \sbra{i}}{\bf x}\tensor{}{}g}; \\ \partial \ep{f\tensor{}{}\wedge ^G{\bf x}\tensor{}{} g} = \sum _{i\in G}\ep{-1}^{\sign iG}\ep{f\tensor{}{}\wedge ^{G\setminus \sbra{i}}{\bf x}\tensor{}{}x_ig}. \end{align*} By a routine, we have that $\partial \vartheta +\vartheta \partial = 0$, and easily we can check that the $i^{\text{th}}$ homology group of the chain complex $\ep{\symfacering{\d}\tensor{}{K}\bigwedge V\tensor{}{K}S,\vartheta}$ is isomorphic to the $i^{\text{th}}$ graded free module of a minimal free resolution $P_\bullet$ of $\symfacering{\d}$. Since, moreover, the differential maps of $\operatorname{lin} \ep{P_\bullet}$ is induced by $\partial$ due to Eisenbud-Goto \cite{EG} and Herzog-Simis-Vasconcelos \cite{HSV}, $\operatorname{lin} _l\ep{P_\bullet}_i \longrightarrow \operatorname{lin} _l\ep{P_\bullet}_{i-1}$ can be identified with \begin{align*} \dsum{}{F\subset \left[ n\right] ,\sharp F=i+l}\bra{\tor iS{\symfacering{\d}}K}_F\tensor{}{K}S \overset{\bar{\partial}}{\longrightarrow } \dsum{}{F\subset \left[ n\right] ,\sharp F =i-1+l}\bra{\tor {i-1}S{\symfacering{\d}}K}_F\tensor{}{K}S, \end{align*} where $\bar{\partial}$ is induced by $\partial $. In the sequel, $-\{i\}$ denotes the subset $\left[ n\right] \setminus \{i\}$ of $\left[ n\right]$. Then we may identify the sequence \eqref{eq:contra} with $$ 0 \longrightarrow \ebra{\tor {n-2}S{\symfacering{\d}}K}_{\left[ n\right]}\tensor{}{K}S \overset{\bar{\partial}}{\longrightarrow } \dsum{}{i \in \left[ n\right] }\bra{\tor {n-3}S{\symfacering{\d}}K}_{-\{i\}}\tensor{}{K}S $$ and hence, by the isomorphism \eref{tor=redhom}, with \begin{align}\label{eq:lin_1} \begin{CD} 0 \longrightarrow \redcohom 1{\d}K \tensor{}{K}S @> \bar{\varepsilon }>> {\displaystyle \dsum{}{i \in \left[ n\right]}\redcohom 1{\d _{-\{i\}}}K \tensor{}{K}S}. \end{CD} \end{align} Here $\bar{\varepsilon }$ is composed by $\bar{\varepsilon}_i: \redcohom 1{\d}K \tensor{}{K}S \to
\redcohom 1{\d_{-\{i\}}}K \tensor{}{K}S$
which is induced by the chain map $$\varepsilon_i: \redcocomp \bullet{\d}\otimes _K S \longrightarrow \redcocomp \bullet{\d _{- \{i\}}}\otimes _K S, $$ $$ \varepsilon _i\ep{e_G^{*}\otimes 1} =\begin{cases} \ep{-1}^{\sign iG}e_G^{*} \otimes x_i & \text{if $i \not \in G$}; \\ 0 & \text{otherwise}. \end{cases} $$ \quad Well, let $C$ be a cycle in $\d$ of the form ($v_1$, $v_2$), ($v_2$, $v_3$)$,\dots ,$($v_t$, $v_1$) with distinct vertices $v_1, \cdots v_t$. We say $C$ has a {\it chord} if there exists an edge ($v_i$, $v_j$) of $G$ such that $j \not\equiv i+1$ ($\operatorname{mod}\ t$), and $C$ is said to be {\it minimal} if it has no chord. It is easy to see that the $1^{\text{st}}$ homology of $\d$ is generated by those of minimal cycles contained in $\d$, that is, we have the surjective map: $$ \dsum{}{\substack{C\subset \d \\ C : \text{minimal cycle}}} \redhom 1{C}K \longrightarrow \redhom 1{\d}K. $$
Now by our assumption that $\d$ contains a cycle of length $\le n-1$ (that is, $\d$ itself is not a minimal cycle), we have the surjective map \begin{equation}\label{eta} \begin{CD} {\displaystyle \dsum{}{i \in \left[ n\right]}\redhom 1{\d _{- \{ i \}}}K} @> \bar{\eta} >>\redhom 1{\d}K \end{CD} \end{equation} where $\bar{\eta}$ is induced by the chain map $\eta :\dsum{}{}\redcomp \bullet{\d _{- \{ i \}}} \longrightarrow \redcomp \bullet{\d}$, and $\eta$ is the sum of $$\eta _i:\redcomp \bullet{\d _{- \{ i \}}} \ni e_G \mapsto \ep{-1}^{\sign iG}e_G \in \redcomp \bullet{\d}.$$ Taking the $K$-dual of \eqref{eta}, we have the injective map \begin{align*} \begin{CD} \redcohom 1{\d}K @>\bar{\eta}^{*}>> {\displaystyle \dsum{}{i \in \left[ n\right]}\redcohom 1{\d _{- \{ i \}}}K}, \end{CD} \end{align*} where $\bar{\eta}^{*}$ is the $K$-dual map of $\bar{\eta}$, and composed by the $K$-dual $$\bar{\eta}_i^*:\redcohom 1{\d}K \to \redcohom 1{\d_{-\{i\}}}K$$ of $\bar{\eta}_i$. Then for all $0\not= z \in \redcohom 1{\d}K$, we have $\bar{\eta}^{*}_i \ep{z} \not= 0$ for some $i$. Recalling the map $\bar{\varepsilon }: \redcohom 1{\d}K \tensor{}{K}S \to \dsum{}{} \redcohom 1{\d _{-\{i\}}}K \tensor{}{K}S$ in \eqref{eq:lin_1} and its construction, we know for $z\in \redcohom 1{\d}K$, $$ \bar{\varepsilon }\ep{z\otimes y} = \sum ^n_{i=1}\bar{\eta}_i^{*}\ep{z}\otimes x_iy, $$ and hence $\bar{\varepsilon }$ is injective. \end{proof}
\begin{rem} (1) If $\Delta$ is an $n$-gon, then $\Delta^\vee$ is an $(n-3)$-dimensional Buchsbaum complex with $\tilde{H}_{n-4}(\Delta^\vee;K) = K$. If $n=5$, then $\Delta^\vee$ is a triangulation of the M\"obius band. But, for $n \geq 6$, $\Delta^\vee$ is not a homology manifold. In fact, let $\sbra{1,2},\sbra{2,3},\cdots ,\sbra{n-1,n}, \sbra{n,1}$ be the facets of $\d$, then if $F = [n] \setminus \{1,3,5 \}$, easy computation shows that $\lk{\dual\Delta}F$ is a 0-dimensional complex with 3 vertices, and hence $\redhom 0{\lk{\dual\Delta}F}K = K^2$.\\ \quad (2) If $\operatorname{indeg} \Delta \geq 3$, then the simplicial complexes given in Example \ref{sec:sharp} are not the only examples which attain the equality $\operatorname{ld}(\Delta) = n -\operatorname{indeg} \ep{\Delta}$. We shall give two examples of such complexes. \\ \quad Let $\Delta$ be the triangulation of the real projective plane $\mathbb{P}^2\mathbb{R}$ with 6 vertices which is given in \cite[figure 5.8, p.236]{BH}. Since $\mathbb{P}^2\mathbb{R}$ is a manifold, $K[\Delta]$ is Buchsbaum. Hence we have $H_{\mathfrak m}^2(K[\Delta]) = [H_{\mathfrak m}^2(K[\Delta])]_0 \cong \tilde{H}_1(\Delta;K)$. So, if $\operatorname{char}(K) =2$, then we have $\operatorname{depth}_S (\operatorname{Ext}^4_S(K[\Delta], \omega_S))=0$. Note that we have $\d = \dual\d$ in this case. Therefore, easy computation shows that $$ \operatorname{ld} \ep{\dual \d} = \operatorname{ld}(\Delta ) = 3 = 6-3 = 6 - \operatorname{indeg} \ep{\Delta}. $$ \quad Next, as is well known, there is a triangulation of the torus with $7$ vertices. Let $\d$ be the triangulation. Since $\dim \d = 2$, we have $\operatorname{indeg} \ep{\dual \d} = 7 - \dim \d - 1= 4$. Observing that $\symfacering{\d}$ is Buchsbaum, we have, by easy computation, that $$ \operatorname{ld} \ep{\dual\d} = 3 = 7 -4 = 7 - \operatorname{indeg} \ep{\dual\d}. $$ Thus $\dual\d$ attains the equality, but is not a simplicial complex given in Example \ref{sec:sharp}, since it follows, from Alexander's duality, that \begin{align*} \dim _K \redhom i{\dual\d}K = \dim _K \redhom{4-i}{\d}K = \begin{cases} 2\not= 1 & \text{for $i=3$;} \\ 0 & \text{for $i\ge 4$.} \end{cases} \end{align*} \quad More generally, the dual complexes of $d$-dimensional Buchsbaum complexes $\d$ with $\redhom{d-1}{\d}K \not= 0$ satisfy the equality $$ \operatorname{ld}\ep{\dual\d} = n - \operatorname{indeg}\ep{\dual\d}, $$ but many of them differ from the examples in Example \ref{sec:sharp}, and we can construct such complexes more easily as $\operatorname{indeg} \ep{\dual\d}$ is larger. \end{rem}
\end{document} | arXiv |
Written by ika, posted on TheWizardBay.
Elements of Set Theory : The Note
This is my note during the reading of the book Elements of Set Theory by Herbert B. Enderton .
The book is great.
Baby Set Theory
Sets - An Informal View
In a naive approach, here is some definition.
A set is a collection of things(called its members or elements), and the collection being regarded as a singel object.
\(t\in A\) means \(t\) is a member of \(A\), and \(t\notin A\) means \(t\) is not a member of \(A\).
\(A==B\) could be translated that \(A\) has the exact same members as \(B\), in which case, \(A\) could be \(\{2,3,5,7\}\) and \(B\) could be the set of all solutions to the equation \(x^4-17x^3+101x^2-247x+210=0\)
to which, we say:
Principle of Extensionality \(\quad\) If two sets have exactly the same members, then they are equal.
In a less naive approach, we could say:
Principle of Extensionality \(\quad\) If \(A\) and \(B\) are sets s.t. for \(\forall t\) \[ t\in A \iff t\in B \] then \(A=B\).
Empty set, noted as \(\emptyset\), has no members at all. and it's the only set with no members.
\(\emptyset \in \{\emptyset\}\) and \(\emptyset \notin \emptyset\), so \(\emptyset \neq \{\emptyset\}\)
Union and Intersection \(\quad\) noted respectively as \(\cup\) and \(\cap\), means all elements of \(A\) and/or \(B\) and all elements of \(A\) and \(B\)
The set of all subsets of \(A\) is noted as power set \(\mathscr{P} A\) of \(A\), also could be noted as \(2^{A}\)
Method of abstraction \(\quad\) is a very flexible way of naming a set. Notation used for the set of all objects \(x\) s.t. the condition P(x) holds is \[ \{x|P(x)\} \]
This method causes two disastrous paradoxes in set theory, which is
Berry's Paradox \(\quad\) an expression which is self-referenced negatively
Russell's Paradox \(\quad\) exemplified by \[\{x|x\notin x\}\]
None of the later work will depend on this informal description.
Skip if you want.
First we gather around all the things that are not sets but we want to have as members of our sets, call such things as atoms. Atoms are not sets of any kind. We put all of those atoms in a set which could be noted as \(A\), it is the first set in out description.
And now imagine a hierarchy \[ V_0 \subseteq V_1 \subseteq V_2 \subseteq \cdot\cdot\cdot \] of sets, and we take \(V_0 = A\), the set of all atoms, and we difine the rest of the sets recursively, for example: \[ V_1 = V_0 \cup \mathscr{P}V_0 = A \cup \mathscr{P}A \] so, the general formula for constructing this hierarchy of sets would be: \[ V_{n+1} = V_n \cup \mathscr{P}V_n \] Thus we obtain $V0, V1, V2,….$, but this infinite hierarchy does not include enough sets. For example, the infinite set \[ \{\emptyset,\{\emptyset\},\{\{\emptyset\}\},...\} \] but we have \(\emptyset \in V_1\),$\{ ∅ \} ∈ V2 $, etc. To remedy this lack, we contrust an infinite union \[ V_{\omega} = V_0\cup V_1 \cup \cdot\cdot\cdot \] and then let \(V_{\omega+1} = V_{\omega} \cup \mathscr{P}V_{\omega}\), and in general for any \(\alpha\), \[ V_{\alpha+1} = V_{\alpha} \cup \mathscr{P} V_{\alpha} \] so we could conclude that for each set \(S\), \(\exists \alpha\) s.t. \(S \in V_{\alpha}\)
And for now, if we banish the atoms, instead take \(A=\emptyset\), and here is the ordinals.
There is no "set of all sets", which is later presented as a theorem, provable from the axioms.
Would like to comment? Start a discussion in my public inbox by sending an email to ~ika/[email protected] [mailing list etiquette] | CommonCrawl |
Affluent Millennial Investing Survey
How to Invest with Confidence
Financial Technology & Automated Investing
Investing Portfolio Management
Bet Smarter With the Monte Carlo Simulation
By Tzveta Iordanova
In finance, there is a fair amount of uncertainty and risk involved with estimating the future value of figures or amounts due to the wide variety of potential outcomes. Monte Carlo simulation (MCS) is one technique that helps to reduce the uncertainty involved in estimating future outcomes. MCS can be applied to complex, non-linear models or used to evaluate the accuracy and performance of other models. It can also be implemented in risk management, portfolio management, pricing derivatives, strategic planning, project planning, cost modeling and other fields.
MCS is a technique that converts uncertainties in input variables of a model into probability distributions. By combining the distributions and randomly selecting values from them, it recalculates the simulated model many times and brings out the probability of the output.
Basic Characteristics
MCS allows several inputs to be used at the same time to create the probability distribution of one or more outputs.
Different types of probability distributions can be assigned to the inputs of the model. When the distribution is unknown, the one that represents the best fit could be chosen.
The use of random numbers characterizes MCS as a stochastic method. The random numbers have to be independent; no correlation should exist between them.
MCS generates the output as a range instead of a fixed value and shows how likely the output value is to occur in the range.
Some Frequently Used Probability Distributions in MCS
Normal/Gaussian Distribution – Continuous distribution applied in situations where the mean and the standard deviation are given and the mean represents the most probable value of the variable. It is symmetrical around the mean and is not bounded.
Lognormal Distribution – Continuous distribution specified by mean and standard deviation. This is appropriate for a variable ranging from zero to infinity, with positive skewness and with normally distributed natural logarithm.
Triangular Distribution – Continuous distribution with fixed minimum and maximum values. It is bounded by the minimum and maximum values and can be either symmetrical (the most probable value = mean = median) or asymmetrical.
Uniform Distribution – Continuous distribution bounded by known minimum and maximum values. In contrast to the triangular distribution, the likelihood of occurrence of the values between the minimum and maximum is the same.
Exponential Distribution – Continuous distribution used to illustrate the time between independent occurrences, provided the rate of occurrences is known.
The Math Behind MCS
Consider that we have a real-valued function g(X) with probability frequency function P(x) (if X is discrete), or probability density function f(x) (if X is continuous). Then we can define the expected value of g(X) in discrete and continuous terms respectively:
E(g(X))=∑−∞+∞g(x)P(x), where P(x)>0 and∑−∞+∞P(x)=1E(g(X))=∫−∞+∞g(x)f(x)dx, where f(x)>0 and ∫−∞+∞f(x)dx=1Next, make random drawings of , callednX(x1,…,xn)trial runs or simulation runs, calculate g(x1),…,g(xn)\begin{aligned}&E(g(X))=\sum^{+\infty}_{-\infty}g(x)P(x),\\&\qquad\qquad\qquad\qquad\qquad\text{ where }P(x)>0\text{ and} \sum^{+\infty}_{-\infty}P(x)=1\\&E(g(X))=\int^{+\infty}_{-\infty}g(x)f(x)\,dx,\\&\qquad\qquad\qquad\qquad\text{ where }f(x)>0\text{ and }\int^{+\infty}_{-\infty}f(x)\,dx=1\\&\text{Next, make $n$ random drawings of $X (x_1,\ldots,x_n)$, called}\\&\text{trial runs or simulation runs, calculate $g(x_1),\ldots,g(x_n)$}\\&\text{and find the mean of $g(x)$ of the sample:}\end{aligned}E(g(X))=−∞∑+∞g(x)P(x), where P(x)>0 and−∞∑+∞P(x)=1E(g(X))=∫−∞+∞g(x)f(x)dx, where f(x)>0 and ∫−∞+∞f(x)dx=1Next, make n random drawings of X(x1,…,xn), calledtrial runs or simulation runs, calculate g(x1),…,g(xn)
gnμ(x)=1n∑i=1ng(xi), which represents the final simulatedvalue of E(g(X)).Therefore gnμ(X)=1n∑i=1ng(X) will be the Monte Carloestimator of E(g(X)).As n→∞,gnμ(X)→E(g(X)),thus we are now able tocompute the dispersion around the estimated mean withthe unbiased variance of gnμ(X):\begin{aligned}&g^\mu_n(x)=\frac{1}{n}\sum^n_{i=1}g(x_i),\text{ which represents the final simulated}\\&\text{value of }E(g(X)).\\\\&\text{Therefore }g^\mu_n(X)=\frac{1}{n}\sum^n_{i=1}g(X)\text{ will be the Monte Carlo}\\&\text{estimator of }E(g(X)).\\\\&\text{As }n\to\infty, g^\mu_n(X)\to E(g(X)), \text{thus we are now able to}\\&\text{compute the dispersion around the estimated mean with}\\&\text{the unbiased variance of }g^\mu_n(X)\text{:}\\&Var(g^\mu_n(X))=\frac{1}{n-1}\sum^n_{i=1}(g(x_i)-g^\mu_n(x))^2.\end{aligned}gnμ(x)=n1i=1∑ng(xi), which represents the final simulatedvalue of E(g(X)).Therefore gnμ(X)=n1i=1∑ng(X) will be the Monte Carloestimator of E(g(X)).As n→∞,gnμ(X)→E(g(X)),thus we are now able tocompute the dispersion around the estimated mean withthe unbiased variance of gnμ(X):
Simple Example
How will the uncertainty in unit price, unit sales and variable costs affect the EBITD?
Copyright Unit Sales)-(Variable Costs + Fixed Costs)
Let us explain the uncertainty in the inputs – unit price, unit sales and variable costs – using triangular distribution, specified by the respective minimum and maximum values of the inputs from the table.
Sensitivity Chart
A sensitivity chart can be very useful when it comes to analyzing the effect of the inputs on the output. What it says is that unit sales account for 62% of the variance in the simulated EBITD, variable costs for 28.6% and unit price for 9.4%. The correlation between unit sales and EBITD and between unit price and EBITD is positive or an increase in unit sales or unit price will lead to an increase in EBITD. Variable costs and EBITD, on the other hand, are negatively correlated, and by decreasing variable costs we will increase EBITD.
Beware that defining the uncertainty of an input value by a probability distribution that does not correspond to the real one and sampling from it will give incorrect results. In addition, the assumption that the input variables are independent might not be valid. Misleading results might come from inputs that are mutually exclusive or if significant correlation is found between two or more input distributions.
The MCS technique is straightforward and flexible. It cannot wipe out uncertainty and risk, but it can make them easier to understand by ascribing probabilistic characteristics to the inputs and outputs of a model. It can be very useful for determining different risks and factors that affect forecasted variables and, therefore, it can lead to more accurate predictions. Also note that the number of trials should not be too small, as it might not be sufficient to simulate the model, causing clustering of values to occur.
What Can The Monte Carlo Simulation Do For Your Portfolio?
How to use Monte Carlo simulation with GBM
Tools for Fundamental Analysis
Learn How to Create a Monte Carlo Simulation Using Excel
The Uses And Limits Of Volatility
How to Use Monte Carlo Analysis to Estimate Risk
Using Common Stock Probability Distribution Methods
How Risk Analysis Works
Risk analysis is the process of assessing the likelihood of an adverse event occurring within the corporate, government, or environmental sector.
Monte Carlo Simulation
Monte Carlo simulations are used to model the probability of different outcomes in a process that cannot easily be predicted due to the intervention of random variables.
How Discrete Distribution Works
A discrete distribution is a statistical distribution that shows the probabilities of outcomes with finite values.
What Are the Odds? How Probability Distribution Works
A probability distribution is a statistical function that describes possible values and likelihoods that a random variable can take within a given range.
How the Least Squares Criterion Method Works
The least-squares criterion is a method of measuring the accuracy of a line in depicting the data that was used to generate it. That is, the formula determines the line of best fit.
Sensitivity analysis determines how different values of an independent variable affect a particular dependent variable under a given set of assumptions. | CommonCrawl |
StreAM-\(T_g\): algorithms for analyzing coarse grained RNA dynamics based on Markov models of connectivity-graphs
Sven Jager ORCID: orcid.org/0000-0003-1410-91141,
Benjamin Schiller2,
Philipp Babel1,
Malte Blumenroth1,
Thorsten Strufe2 &
Kay Hamacher3
Algorithms for Molecular Biology volume 12, Article number: 15 (2017) Cite this article
In this work, we present a new coarse grained representation of RNA dynamics. It is based on adjacency matrices and their interactions patterns obtained from molecular dynamics simulations. RNA molecules are well-suited for this representation due to their composition which is mainly modular and assessable by the secondary structure alone. These interactions can be represented as adjacency matrices of k nucleotides. Based on those, we define transitions between states as changes in the adjacency matrices which form Markovian dynamics. The intense computational demand for deriving the transition probability matrices prompted us to develop StreAM-\(T_g\), a stream-based algorithm for generating such Markov models of k-vertex adjacency matrices representing the RNA.
We benchmark StreAM-\(T_g\) (a) for random and RNA unit sphere dynamic graphs (b) for the robustness of our method against different parameters. Moreover, we address a riboswitch design problem by applying StreAM-\(T_g\) on six long term molecular dynamics simulation of a synthetic tetracycline dependent riboswitch (500 ns) in combination with five different antibiotics.
The proposed algorithm performs well on large simulated as well as real world dynamic graphs. Additionally, StreAM-\(T_g\) provides insights into nucleotide based RNA dynamics in comparison to conventional metrics like the root-mean square fluctuation. In the light of experimental data our results show important design opportunities for the riboswitch.
The computational design of switchable and catalytic ribonucleic acids (RNA) becomes a major challenge for synthetic biology [1]. So far, available models and simulation tools to design and analyze functionally complex RNA based devices are very limited [2]. Although several tools are available to assess secondary as well as tertiary RNA structure [3], current capabilities to simulate dynamics are still underdeveloped [4] and rely heavily on atomistic molecular dynamics (MD) techniques [5]. RNA structure is largely modular and composed of repetitive motifs [4] that form structural elements such as hairpins and stems based on hydrogen-bonding patterns [6]. Such structural modules play an important role for nano design [1, 7].
In order to understand RNA dynamics [8, 14] we develop a new method to quantify all possible structural transitions, based on a coarse grained, transferable representation of different module sizes. The computation of Markov State Models (MSM) have recently become practical to reproduce long-time conformational dynamics of biomolecules using data from MD simulations [15].
To this end, we convert MD trajectories into dynamic graphs and derive the Markovian dynamics in the space of adjacency matrices. Aggregated matrices for each nucleotide represent RNA coarse grained dynamics. However, a full investigation of all transitions is computationally expensive.
To address this challenge we extend StreaM—a stream-based algorithm for counting 4-vertex motifs in dynamic graphs with an outstanding performance for analyzing (bio)molecular trajectories [16]. The extension StreAM computes one transition matrix for a single set of vertices or a full set for combinatorial many matrices. To gain insight into global folding and stability of an RNA molecule, we propose StreAM-\(T_g\): It combines all adjacency-based Markov models for a nucleotide into one global weighted stochastic transition matrix \(T_g(a)\). However, deriving Markovian dynamics from MD simulations of RNA is an emerging method to describe folding pathways [13] or to elucidate the kinetics of stacking interactions [11]. Especially MSM of atomistic aptamer simulations like the theophylline [12] and thrombin aptamer could help to understand structure-function relationships as well as the folding process [18]. Nonetheless, all the methods mentioned above rely on Root Mean Square Deviation (RMSD) computations in combination with clustering in order to identify relevant transition states. For StreAM-\(T_g\), the transition states are given by small adjacency matrices representing structural motifs.
The remainder of this paper is structured as follows: In "Our approach for coarse grained analysis", we introduce the concept of StreAM-\(T_g\) as well as our biological test setup. We describe details of the algorithm in "Algorithm". We present runtime evaluations as well as application scenario of our algorithm in "Evaluation" for a synthetic tetracycline (TC) dependent riboswitch (TC-Aptamer). Furthermore, we investigate the influence upon ligand binding of four different TC derivates and compare them with a conventional method. Finally, we summarize our work in "Summary, conclusion, and future work".
Our approach for coarse grained analysis
Structural representation of RNA
Predicting the function of complex RNA molecules depends critically on understanding both, their structure as well as their conformational dynamics [17, 19]. To achieve the latter we propose a new coarse grained RNA representation. For our approach, we start with an MD simulation to obtain a trajectory of the RNA. We reduce these simulated trajectories to nucleotides represented by their (\(C3'\)) atoms. From there, we represent RNA structure as an undirected graph [20] using each \(C3'\) as a vertex and distance dependent interactions as edges [3]. It is well known that nucleotide-based molecular interactions take place between more than one partner [21]. For this reason interactions exist for several edges observable in the adjacency matrix (obtained via a Euclidean distance cut-off) of \(C3'\) coordinates at a given time-step. The resulting edges represent, e.g., strong local interactions such as Watson-Crick pairing, Hoogsteen, or \(\pi{-}\pi\)-stacking.
Our algorithm estimates adjacency matrix transition rates of a given set of vertices (nucleotides) and builds a Markov model. Moreover, by deriving all Markov models of all possible combinations of vertices, we can reduce them afterwards into a global weighted transition matrix for each vertex representing the ensemble that the nucleotide modeled as a vertex is immersed in.
Dynamic graphs, their analysis, and Markovian dynamics
A graph \(G = (V,E)\) is an ordered pair of vertices \(V = \{v_1, v_2, \dots v_{|V|}\}\) and edges E. We refer to a single vertex of V as a. Here, we only consider undirected graphs without self-loops, i.e., \(E \subseteq \{\{v,w\}: v,w \in V, v \ne w\}\). We define a self-loop as an edge that connects a vertex to itself. For a subset \(V'\) of the vertex set V, we refer to \(G[V'] = (V', E'), \; E' := \{\{v,w\} \in E : v, w \in V'\}\) as the \(V'\) -induced subgraph of G. We refer to the powerset of V as \(\mathbb {P}(V)\). The adjacency matrix \(A(G) = A_{i,j}\) (Eq. 1) of a graph G is a \(|V| \times |V|\) matrix, defined as follows:
$$\begin{aligned} A_{i,j} := \left\{ \begin{array}{rl} 0 &{} : i< j \wedge \{v_i, v_j\} \notin E \\ 1 &{} : i < j \wedge \{v_i, v_j\} \in E \\ \Diamond &{} : \text {otherwise} \end{array} \right. \end{aligned}$$
Dynamic graph example. Example of a dynamic graph and induced subgraphs for \(V' = \{a, b, c, d\}\). The first row shows the dynamic graph \(G_t\) and the second the induced subgraph \(V'\) with its respective adjacency matrix. At the bottom is a short example of how to compute the adjacency id for the displayed subgraphs
Here, the symbol \(\Diamond\) denotes for an undefined matrix entry. We denote the set of all adjacency matrices of size k as \(\mathcal{A}_k\), with \(|\mathcal{A}_k| = 2^{\frac{k \cdot (k-1)}{2}}\). In our current implementation k can takes values in \(\{2,3,4,5,6,7,8,9,10\}\). With concat(A), we denote the row-by-row concatenation of all defined values of an adjacency matrix A. We define the adjacency id of a matrix A as the numerical value of the binary interpretation of its concatenation, i.e., \(id(A) = concat(A)_2 \in \mathbb {N}\). We refer to \(id(V') := id(A(G[V']))\) as the adjacency id of the \(V'\)-induced subgraph of G. For example, the concatenation of the adjacency matrix of graph \(G_1[V']\) (shown in Fig. 1) is \(concat(A(G_1[V'])) = \text {011011}\) and its adjacency id is \(id(V') = 011011_2 = 27_{10}\).
As a dynamic graph \(G_t = (V, E_t)\), we consider a graph whose edge set changes over time. For each point in time \(t \in [1,\tau ]\), we consider \(G_t\) as the snapshot or state of the dynamic graph at that time. The transition of a dynamic graph \(G_{t-1}\) to the next state \(G_{t}\) is described by a pair of edge sets which contain the edges added to and removed from \(G_{t-1}\), i.e., \((E^+_{t}, E^-_{t})\). We refer to these changes as a batch, defined as follows: \(E^+_{t} := E_{t} \backslash E_{t-1}\) and \(E^-_{t} := E_{t-1} \backslash E_{t}\). The batch size is referred as \(\delta _t=|E^+_t|+|E^-_t|\) and the average batch size is refered as \(\delta _{avg}\) and is defined as \(\frac{\sum _t \delta _t}{\tau }\).
The analysis of dynamic graphs is commonly performed using stream- or batch-based algorithms. Both output the desired result for each snapshot \(G_t\). Stream-based algorithms take a single update to the graph as input, i.e., the addition or removal of an edge e. Batch-based algorithms take a pair \((E^+_{t+1},E^-_{t+1})\) as input. They can always be implemented by executing a stream-based algorithm for each edge addition \(e \in E^+_{t+1}\) and removal \(e \in E^-_{t+1}\). We refer to \(id_t(V')\) as the adjacency id of the \(V'\)-induced subgraph of each snapshot of \(G_t\). The result of analyzing the adjacency id of \(V'\) for a dynamic graph \(G_t\) is a list \((id_t(V'): t \in [1,\tau ])\). We consider each pair \((id_t(V'), id_{t+1}(V'))\) as an adjacency transition of \(V'\) and denote the set of all transitions as \(\mathcal{T}(V')\). Then, we define the local transition matrix \(T(V')\) of \(V'\) as a \(|\mathcal{A}_k| \times |\mathcal{A}_k|\) matrix, which contains the number of transitions between any two adjacency ids over time, i.e., \(T_{i,j}(V') := |(i+1,j+1) \in \mathcal{T}(V')|\) for an adjacency size k. From \(T(V')\), we can derive a Markov model to describe these transitions.
By combining all possible \(T(V')\) where \(V' \in \mathbb {P}(V): |V'| = k\) and \(a \in V'\), we derive a transition tensor \(C_{a}(V)\). Thus \(C_{a}(V)\) has the dimensions of \(|\mathcal{A}_k| \times |\mathcal{A}_k| \times (k-1)! \left( {\begin{array}{c}|V|\\ k-1\end{array}}\right)\).
We define the weighting matrix \(W(V')\) with the dimensions of \(|\mathcal{A}_k| \times (k-1)! \left( {\begin{array}{c}|V|\\ k-1\end{array}}\right)\). \(W(V')\) contains the weighting for every subset \(V' \in C_{a}(V)\). It is defined as \(W(V'):= \frac{S(V')}{\sum _{V' \in C_a(V)} S(V')}\). Here, \(S(V')\) is a matrix containing the sum of every transition between adjacency \(id(V')\) and every other \(id(V')\) of the same matrix \(T(V')\) for all \(V' \in C_a(V)\). Hence \(S(V')\) has the dimensions \(|\mathcal{A}_k| \times (k-1)! \left( {\begin{array}{c}|V|\\ k-1\end{array}}\right)\). Thus \(W(V')\) is considered as the local distribution weighted by its global distribution of transitions matrices of \(V'\). Finaly, we define a global transition matrix, a vertex a is immeresd in, as \(T_g(a)=\sum _{V' \in C_{a}(V)} W(V') \times T(V')\) with the dimensions \(|\mathcal{A}_k| \times |\mathcal{A}_k|\).
For a local or global transition matrix the respective dominant eigenvectorFootnote 1 is called \(\pi\) and represents the stationary distribution attained for infinite (or very long) times. The corresponding conformational entropy of the ensemble of motifs is \(H:=-\sum _i \pi _i \cdot \log \pi _i\). The change in conformational entropy upon, e.g., binding a ligand is then given as \(\Delta H = H_{wt}-H_{complex}\).
MD simulation setup
TC-derivates. TC-derivates illustrated as chemical structures. Here we show the structure of Tetracycline (left top), Anhydrotetracycline (right top), Doxycycline (left bottom) and 6-deoxy-6-demethyl-Tetracycline (right bottom). The illustrated derivates share the characteristic 4-ring-structure and functional groups
We use a structure of a synthetic tetracycline binding riboswitch (PDB: 3EGZ, chain B, resolution: 2.2 Å, Fig. 2) [23] and perform six simulations: the TC-Aptamer with five different tetracycline types in complex and one without tetracycline. As tetracycline binding alters the structural entropy of the molecule [24] our proposed method should be able to detect changes in (local) dynamics due the presence of tetracycline. All simulations were performed using the GROMACS software package (version 2016). For water molecules, we used the TIP3P model, the RNA interact through the CHARMM force field, while the tetracycline analogs interact through a modified CHARMM force field from Aleksandrov and Simonson [25, 26]. The systems were first energy minimized and equilibrated for 1 ns in the NVT-ensemble at a temperature of 300 K and for 5 ns in the NpT-ensemble at a temperature of 300 K and a pressure of 1 bar. During the equilibration, temperature was controlled using the velocity-rescale thermostat [27] (\(\tau _{\text {T}} = {0.1}~{\mathrm{ps}}\)) and pressure was controlled using the Berendsen barostat [28] (\(\tau _{\text {P}}={0.5}~{\mathrm{ps}}\)). Isothermal compressibility was set to \({4.5}\times 10^{-5}\,\mathrm{bar}^{-1}\), which is the corresponding value for water. Production runs were performed for 500 ns. The temperature was controlled using the Nosé-Hoover thermostat [29, 30] (\(\tau _{\text {T}} = {1}~{\mathrm{ps}}\)) and pressure was controlled using the Parrinello-Rahman barostat [31] (\(\tau _{\text {P}}={1}~{\mathrm{ps}}\)) during the production runs. Bond lengths were constrained using the LINCS [32] algorithm. The Lennard-Jones nonbonded interactions were evaluated using a cutoff distance of 1.2 nm. The electrostatic interactions were evaluated using the particle mesh Ewald method with a real space cutoff 1.2 nm and a grid-spacing 0.12 nm. Long-range corrections to energy and pressure due to the truncation of Lennard-Jones potential were accounted for. The equations of motion were integrated using a 2 fs time step.
Tetracycline derivates
Structural representation of TC-Aptamer. a Crystal structure of TC-Aptamer with a cut-off of 13 Å and using \(C3'\) atom for coarse graining reveals edges for dominant WC base-pairings. Important structural parts are annotated according to [23]. b Secondary structure representation of TC-Aptamer. Nucleotides are displayed as vertices and connections are based on hydrogen-bonding patterns. Nucleotides participating in TC-binding are colored in red. Graphics were created using Pymol and R [39, 47]
For the comparison of TC derivates we use tetracycline (tc), doxycycline (dc), anhydrotetracycline (atc) and 6-deoxy-6-demythyltetracycline (ddtc) in our MD simulation. These four analogs share the characteristic 4-ring-structure and functional groups of all tetracyclines. Still, the possibility and the mode of interaction with the RNA is an open question. The first ring of tetracycline carries a dimethylamino group, while the third ring carries a hydroxy and a methyl group facing towards the same direction away from the 4-ring-system. The detailed chemical structures are shown in Fig. 3. In comparison to these two rings the fourth, aromatic ring has an especially small steric volume on this side of the molecule. From tc over dc and atc to ddtc this steric volume is further reduced by shifting the aforementioned hydroxy and methyl group away from the fourth ring or eliminating some of them entirely. Note, that our graph-based approach is capable to easily distinguish between different modes of interaction upon changes in the, e.g., the side-chains of the rings. The molecular data of tc, dc, atc and ddtc was created using the Avogadro software [33]. Structures were manually constructed and moved into the extended conformation described to be 3 kcal/mol more stable than its twisted alternative by Alexandrov et al. [24]. The molecules were then fitted to the position of 7-chlorotetracycline (7-cl-tc) bound in the TC-Aptamer structure used for simulation. Note, that the geometry of 7-cl-tc was already present in the crystal structure of the TC-Aptamer. All considered antibiotics show different properties upon ligand binding. They range from high activity (tc, 7-cl-tc) to weak activity (dc, ddtc, atc) based on in vivo experiments [34].
RNA trajectory and contact probability
An RNA trajectory X is represented as a list of T frames \(X = (\vec {x}_{t_0},\vec {x}_{t_1}, \ldots )\). Each frame \(\vec {x}_t\in \mathbb {R}^{3n}\) contains the three-dimensional coordinates of the simulated system of the n atoms at the respective point in time t. We define a binary contact matrix B(t) with dimensions \(|V| \times |V|\). Its entries scan range between \(\{0,1\}\). A single contact \(B_{i,j}(t)\) between one pair of atom coordinates \(\vec {r}_{i}(t)\) and \(\vec {r}_{j}(t)\) is generated if their Euclidean distance [L2-norm, \(L2(\ldots )\)] is shorter than d. Thus B(t) entries are defined as follows:
$$\begin{aligned} B_{i,j}(t) := \left\{ \begin{array}{rl} 0 &{} : d < L2(\vec {r}_{i}(t)-\vec {r}_{j}(t))\\ 1 &{} : d > L2(\vec {r}_{i}(t)-\vec {r}_{j}(t))\\ \end{array} \right. \end{aligned}$$
The contact probability of one pair of atom coordinates \(\vec {r}_{i}\) and \(\vec {r}_{j}\) is defined as:
$$\begin{aligned} P(X,\vec {r}_{i},\vec {r}_{j}) = \frac{\sum _{t=1}^T B_{ij}(t)}{T}. \end{aligned}$$
Graph transformation
All considered MD simulations have a total length of 500 ns using an integration stepsize of 2 fs. We created snapshots every 250 ps resulting in 100,000 frames. We generated dynamic graphs \(G_t = (V, E_t)\) containing \(|V|=65\) vertices (Table 1), each modelling a nucleic \(3C'\) (Fig. 2). This resolution is sufficient to represent both small secondary structure elements as well as large quaternary RNA complexes [35, 36]. We create undirected edges between two vertices in case their Euclidean cut-off (d) is shorter than \(\{ d \in N | 10 \le d \le 15 \}\) Å (cmp. Table 1).
Markov state models (MSM) of local adjacency and global transition matrix
StreAM counts adjacency transitions (e.g. as a set \(\mathcal{T}(V')\)) of an induced subgraph for a given adjacency size. Now the transition matrix \(T(V')\) can be derived from \(\mathcal{T}(V')\) but not all possible states are necessarily visited in a given, finite simulation, although a "missing state" potentially might occur in longer simulations. In order to allow for this, we introduce a minimal pseudo-count [37] of \(P_k=\frac{1}{|\mathcal{A}_k|}\). All models that fullfill \(\{V' \in \mathbb {P}(V) : |V'| = k, a \in V'\}\) have the same matrix dimension and thus can be envisioned to be combined in a tensor \(C_a(V)\). Now, \(C_{a~i,j,l}(V)\) is one entry of the tensor of transitions between adjacency id i and j in the l th transition matrix \(T(V')\) with \(|l|=\left( {\begin{array}{c}|V|\\ k-1\end{array}}\right) \times k-1\). Thus \(C_a(V)\) contains all \(T(V')\) a specific vertex is immersed in and due to this it contains all possible information of local markovian dynamics. To derive \(T_g(a)\) every entry \(C_{a~i,j,l}(V)\) is normalized by the count of all transitions of i in all matrices \(S(V)_{j,l} = \sum _{i} C_{a~i,j,l}(V)\). For a given set of l transition matrices \(T(V')\) we can combine them into a global model with respect to their probability:
$$\begin{aligned} T_{g~i,j}(a) = \sum _{l} \frac{S(V)_{jl}}{\sum _{l} S(V)_{jl}} \cdot C_{a~i,j,l}(V) . \end{aligned}$$
Stationary distribution and entropy
As \(T_g(a)\) (Eq. 4) is a row stochastic matrix we can compute its dominant eigenvector from a spectral decomposition. It represents a basic quantity of interest: the stationary probability \(\vec {\pi }:=\left( \pi _1, \ldots , \pi _{i},\ldots \right)\) of micro-states i [37]. To this end we used the markovchain library in R [38, 39]. For measuring the changes in conformational entropy \(H := -\sum _{i=1}^{|\mathcal{A}_k|}{\pi _i \cdot \log \pi _i}\) upon binding a ligand, we define \(\Delta H = H_{wt}-H_{complex}\), form a stationary distribution.
Conventional analysis: root mean square fluctuation (RMSF)
The flexibility of an atom can be quantitatively assessed by its Root-mean-square fluctuation (RMSF). This measure is the time average L2-norm \(L2(\ldots )\) of one particular atom's position \(\vec {r}_{i}(t)\) to its time-averaged position \(\bar{\vec {r}_{i}}\). The RMSF of an nucleotide i (represented by its respective \(C3'\) atom) is defined as:
$$\begin{aligned} RMSF(X,r_{i}) := \sqrt{\frac{1}{T} \cdot \sum _{t=1}^T L2(\vec {r}_{i}(t),\bar{\vec {r}_{i}}~)^2} \end{aligned}$$
In this section, we introduce the required algorithms to compute \(T_g(a)\). First, we describe StreAM, a stream-based algorithm for computing the adjacency \(id(V')\) for a given \(V'\). Afterwards we describe, the batch-based computation using StreAM \(_B\) to derive \(id_t(V')\). By computing the adjacency id of a dynamic graph \(G_t[V']\) we derive a list \((id_t(V'): t \in [1,\tau ])\) where each pair \([id_t(V'), id_{t+1}(V')]\) represents an adjacency transition. The respective transitions are than stored in \(\mathcal{T}(V')\). Now, a single \(T(V')\) can be derived by counting the transitions in \(\mathcal{T}(V')\). At last we introduce StreAM-\(T_g\), an algorithm for the computation of a global transition matrix \(T_g(a)\) for a given vertex a from a dynamic graph \(G_t[V]\). To this end, StreAM-\(T_g\) computes the tensor \(C_a(V)\) which includes every single matrix \(T(V')\) where \(V' \in \mathbb {P}(V)\) and \(|V'| = k\) with vertex \(a \in V'\). Finally, StreAM-\(T_g\) computes \(T_g(a)\) from \(C_a(V)\).
StreAM and StreAM\(_B\).
We compute the adjacency id \(id(V')\) for vertices \(V' \subseteq V\) in the dynamic graph \(G_t\) using the stream-based algorithm StreAM, as described in Algorithm 1. Here, \(id(V') \in [0,|\mathcal {A}_{|V'|}|)\) is the unique identifier of the adjacency matrix of the subgraph \(G[V']\). Each change to \(G_t\) consists of the edge \(\{a,b\}\) and a type to mark it as addition or removal (abbreviated to add,rem). In addition to edge and type, StreAM takes as input the ordered list of vertices \(V'\) and their current adjacency id.
An edge \(\{a,b\}\) is only processed by StreAM in case both a and b are contained in \(V'\). Otherwise, its addition or removal has clearly no impact on \(id(V')\).
Assume \(pos(V',a), pos(V',b) \in [1,k]\) to be the positions of vertices a and b in \(V'\). Then, \(i = min(pos(V',a), pos(V',b))\) and \(j = max(pos(V',a), pos(V',b))\) are the row and column of adjacency matrix \(A(G[V'])\) that represent the edge \(\{a,b\}\). In the bit representation of its adjacency id \(id(V')\), this edge is represented by the bit \((i-1) \cdot k + j - i \cdot (i+1)/2\). When interpreting this bit representation as a number, an addition or removal of the respective edge corresponds to the addition or subtraction of \(2^{k \cdot (k-1) / 2 - ((i-1) \cdot k + j - i \cdot (i+1)/2)}\). This operation is performed to update \(id(V')\) for each edge removal or addition. In the following, we refer to this position as \(e(a,b,V') := \frac{|V'| \cdot (|V'|-1)}{2} - [(i-1) \cdot |V'| + j - \frac{i \cdot (i+1)}{2}]\).
Furthermore, in Algorithm 2 we show StreAM \(_B\) for the batch-based computation of the adjacency id for vertices \(V'\)
StreAM-\(T_g\)
For the design or redesign of aptamers it is crucial to provide experimental researchers informations about e.g. dynamics at the nulceotide level. To this end, StreAM-\(T_g\) combines every adajcency-based transition matrix, one nucleotide participates in, into a global model \(T_g(a)\). This model can be derived for every nucleotide of the regarded RNA structure and contains all the structural transition of a nuclotide between the complete ensemble of remaining nucleotides. In order to do this, we present StreAM-\(T_g\), an algorithm for the computation of global transition matrices, one particular vertex is participating in, given in Algorithm 3. A full computation with StreAM-\(T_g\) can be divided into the following steps. The first step is the computation of all possible Markov models that fulfill \(V' \in \mathbb {P}(V) : |V'| = k\) with StreAM for a given k with \(k \in [2,10]\). This results in \(\left( {\begin{array}{c}|V|\\ k\end{array}}\right) \cdot k!=\frac{|V|!}{\left( |V|-k\right) !}\) combinations. Afterwards, StreAM-\(T_g\) sorts the matrices by vertex id into different sets, each with the size of \(\left( {\begin{array}{c}|V|\\ k-1\end{array}}\right) \cdot (k-1)!\). For each vertex a, StreAM-\(T_g\) combines the obtained \(T(V')\) that fulfill \(a \in V'\) in a transition tensor \(C_a(V)\), which is normalized by \(W(V')\) the global distribution of transition states a vertex is immersing in, taking the whole ensemble into account. \(W(V')\) can be directly computed from \(C_a(V)\) (e.g. "Dynamic graphs, their analysis, and Markovian dynamics")
StreAM-\(T_g\) optimization using precomputed contact probability
The large computational demands for a full computation of the \(\left( {\begin{array}{c}|V|\\ k\end{array}}\right) \cdot k!=\frac{|V|!}{\left( |V|-k\right) !}\) transition matrices to derive a set of \(T_g(a)\), motivated us to implement an optimization: The number of Markov models can be reduced by considering only adjacencies including possible contacts between at least two vertices of \(G_t = (V, E_t)\). This can be precomputed before the full computation by considering the contact probability \(P(X,\vec {r}_{i},\vec {r}_{j})\) between vertices. To this end we only compute transition matrices forming a contact within the dynamic graph with \(P(X,\vec {r}_{i},\vec {r}_{j}) > 0\).
As StreAM-\(T_g\) is intended to analyze large MD trajectories we first measure the speed of StreAM for computing a single \(\mathcal{T}(V')\) to estimate overall computational resources. With this in mind, we benchmark different \(G_t\) with increasing adjacency size k (Table 1). Furthermore, we need to quantify the dependence of computational speed with respect to \(\delta _{t}\). Note, \(\delta _{t}\) represents changes in conformations within \(G_t\). For the full computation of \(T_g(a)\), we want to measure computing time in order to benchmark StreAM-\(T_g\) by increasing network size |V| and k for a given system due to exponentially increasing matrix dimensions \(|\mathcal{A}_k| = 2^{\frac{k \cdot (k-1)}{2}}\) (\(k=3\) 8, \(k=4\) 64, \(k=5\) 1,024, \(k=6\) 32,768, \(k=7\) 2,097,152 size of matrix dimensions). We expect due to combinatorial complexity of matrix computation a linear relation between |V| and speed and an exponential relation between increasing k and speed. To access robustness of influence of d robustness regarding the computation of \(T_g(a)\) stationary distribution \(\vec {\pi }\). We expect a strong linear correlation between derived stationary distributions. Details are shown in "Robustness against threshold". We compare Markovian dynamics between the native TC-Aptamer and the structure in complex with 7-cl-tc with experimental data. We discuss the details in "Workflow" and "Application to molecular synthetic biology". Furthermore, we want to illustrate the biological relevance by applying it to a riboswitch design problem; this is shown in detail in "Application to molecular synthetic biology". For the last part, we investigate the ligand binding of four different TC derivates using StreAM-\(T_g\) and compare them with a classical metric (e.g. RMSF) in "Comparison of tetracycline derivates".
Evaluation setup
All benchmarks were performed on a machine with four Intel(R) Xeon(R) CPU E5-2687W v2 processors with 3.4GHz running a Debian operating system. We implemented StreAM in Java; all sources are available in a GitHub repository.Footnote 2 The final implementation StreAM-\(T_g\) is integrated in a Julia repository.Footnote 3 We created plots using the AssayToolbox library for R [39, 40]. We generate all random graphs using a generator for dynamic graphsFootnote 4 derived for vertex combination.
Table 1 Details of the dynamic graphs obtained from MD simulation trajectories
Runtime dependencies of StreAM on adjacency size
For every dynamic graph \(G_t(V,E_t)\), we selected a total number of 100,000 snapshots to measure StreAM runtime performance. In order to perform benchmarks with increasing k, we chose randomly nodes \(k \in [3, 10]\) and repeated this 500 times for different numbers of snapshots (every 10,000 steps). We determined the slope (speed \(\frac{frames}{ms}\)) of compute time vs. k for random and MD graphs with different parameters (Table 1).
Runtime dependence of StreAM on batch size
We measured runtime performance of StreAM for the computation of a set of all transitions \(\mathcal{T}(V')\) with different adjacency sizes k as well as dynamic networks with increasing batch sizes. To test StreAM batch size dependencies, 35 random graphs were drawn with increasing batch size and constant numbers of vertex and edges. All graphs contained 100,000 snapshots and k is calculated from 500 random combinations of vertices.
Runtime dependencies of StreAM-\(T_g\) on network size
We benchmarked the full computation of \(T_g(a)\) with different \(k \in [3, 5]\) for increasing network sizes |V|. Therefore we performed a full computation with StreAM. StreAM-\(T_g\) sorts the obtained transition list, converts them into transition matrices and combines them into a global Markov model for each vertex.
Runtime evaluation
Runtime performance of StreAM-\(T_g\). a Speed of computing a set of \(\mathcal{T}(V')\) using StreAM. b Performance of \(T_g(a)\) full computation with increasing network size |V| and different adjacency sizes \(k=3,4,5\). c Speed of StreAM with increasing batch size for \(k=3,10\)
Figure 4b shows computational speeds for each dynamic graph. Speed decreases linearly with a small slope (Fig. 4a). While this is encouraging the computation of transition matrices for \(k > 5\) is still prohibitively expensive due to the exponential increase of the matrix dimensions with \(2^{\frac{k \cdot (k-1)}{2}}\). For \(G_t\) obtained from MD simulations, we observe fast speeds due to small batch sizes (Table 1).
Figure 4b reveals that \(T_{cpu}\) increases linearly with increasing |V| and with k exponentially. We restrict the \(T_g(a)\) full computation to \(k<5\). In Fig. 4c, speed decreases linearly with \(\delta _{t}\). As \(\delta _{t}\) represents the changes between snapshots our observation has implications for the choice of MD integration step lengths as well as trajectory granularity.
Performance enhancing by precomputed contact probability
The exponential increase of transition matrix dimensions with \(2^{\frac{k \cdot (k-1)}{2}}\) is an obvious disadvantage of the proposed method. However, there exist several \(T(V')\) where every vertex is never in contact with another vertex from the set. These adjacencies remain only in one state during the whole simulation. To avoid the computation of the respective Markov models we precomputed \(P(X,\vec {r}_{i},\vec {r}_{j})\) of all vertices. Thus only combinations are considered with \(P(X,\vec {r}_{i},\vec {r}_{j}) > 0\). This procedure leads to a large reduction of \(T_{cpu}\) due to fewer number of matrices to be computed to derive \(T_g(a)\). To illustrate this reduction, we compute the number of adjacencies left after a precomputation of \(P(X,\vec {r}_{i},\vec {r}_{j})\) as a function of d for the TC-Aptamer simulation without TC. The remaining number of transition matrices for adjacency sizes \(k=3,4,5\) are shown in Fig. 5b. For further illustration we show the graph of the RNA molecule obtained for a cut-off of \(d=15\) Å in Fig. 5a.
Precomputation with different cut-offs. a Illustration of the the first frame of the TC-Aptamer simulation without TC th created with a cut-off of \(d=15 \) Å. Vertices (representing nucleotides) are colored in black and edges (representing interactions) in red. The edges belonging to the backbone are furthermore highlighted in black. Graphics were created using Pymol and R [39, 47]. b Number of \(\mathcal{T}(V')\) for a full computation of \(T_g(a)\) after selection with contact probability as function of cut-off d for three different adjacency sizes (\(k=3,4,5\)). The dashed lines show the number of matrices normally required for a full computation [\(k=3\), 262,080 matrices (green); \(k=4\) , 16,248,960 matrices (black); \(k=5\), 991,186,560 matrices (blue)]
We can observe that using a precomputation of \(P(X,\vec {r}_{i},\vec {r}_{j})\) to a full computation of \(T_g(a)\) hardly depends on the Euclidean cut-off (d) for all considered adjacencies. The reduced computational costs in case of a full computation can be expressed by a significant smaller number of transition matrices left to compute for all considered adjacency sizes \(k=3,4,5\). For example if we use \(k=4\) and \(d=13\) Å we have to compute 16,248,960 transition matrices, if we use a precomputation of \(P(X,\vec {r}_{i},\vec {r}_{j})\) we can reduce this value to 2,063,100, this roughly eightfold. Furthermore, in case of new contact formation due to an increased d the number of transition matrices can increase.
Robustness against threshold
Here, we investigate the influence of threshold d for the full computation of \(T_g(a)\). To this end, we created dynamic graphs with different \(d \in [11, 15]\) Å of the TC-Aptamer simulation without TC. Here, we focus on a simple model with an adjacency size of \(k=3\), thus with eight states. In particular, we focus on the local adjacency matrix of combination 52, 54 and 51 because these nucleotides are important for TC binding and stabilization of intermediates.
Robustness for \(T_g(a)\) of the native riboswitch. a Scatter plot matrix of computed \(\vec {\pi }\) for each \(T_g(a)\) at different d. The lower triangle includes the scatterplots obtained at different d. The diagonal includes the histogram of all 65 \(\vec {\pi }\) and the upper triangle includes the Pearson product moment correlation of the corresonding scatterplots. b Illustration of single \(T(V')\) derived for vertex combination 52, 54 and 51 for \(d \in [11, 15]\) Å as heat maps
To access the overall robustness of a full computation of \(T_g(a)\) we compute the stationary distribution for every \(T_g(a)\) and afterwards we compare them with each other. For the comparison we use the Pearson product moment correlation (Pearson's r). Figure 6 illustrates the comparison of stationary distributions obtained from 65 \(T_g(a)\) for unit sphere dynamic graphs with different d.
The obtained Pearson correlations r are also shown in Fig. 6 (a, upper triangle). We observed a high robustness expressed by an overall high correlation (\(r= 0.938\) to \(r = 0.98\)) of the dynamic graphs created with different d. However transient states disappear with increasing threshold d (Fig. 6b). This observation stems from the fact that the obtained graph becomes more and more densely connected. One consequence of a high threshold d is that the adjacency remain in the same state.
Accuracy of StreAM
In this section we discuss the accuracy of StreAM for the computation of a set of all transitions \(\mathcal{T}(V')\) on finite data samples. Our approach estimates the transition probabilities from a trajectory as frequencies of occurrences. It could be shown that uncertainties derived from a transition matrix (e.g derived from a molecular dynamics simulation) decreases with increasing simulation time [22]. Thus the error and bias in our estimator are driven by the available data set size to derive \(\mathcal{T}(V')\). Additionally, there is an implicit influence of k on the accuracy since the number of k determines the transition matrix dimensions. Consequently, the available trajectory (system) data must be at least larger than the number of entries in the transition matrix to be estimated in order to use StreAM.
Application to molecular synthetic biology
This section is devoted to investigate possible changes in Markovian dynamics of the TC-Aptamer upon binding of 7-cl-tc. This particular antibiotic is part of the crystal structure of the TC-Aptamer thus structure of 7-cl-tc has the correct geometry and orientation of functional groups.
For both simulations of "Workflow", we computed 16,248,960 transition matrices and combined them into 65 global models (one for each vertex of the riboswitch). To account for both the pair-interactions and potential stacking effects we focus on \(k=4\)-vertex adjacencies and use dynamic RNA graphs with \(d=13\) Å. One global transition matrix contains all the transitions a single nucleotide participates in. The stationary distribution and the implied entropy (changes) help to understand the effects of ligand binding and potential improvements on this (the design problem at hand). The \(\Delta H\) obtained are shown in Fig. 7.
\(\Delta H\) (in bit) comparison for 7-cl-tc. \(\Delta H\) for \(T_g(a)\) of the native riboswitch and the one in complex with 7-cl-tc. Nucleotides with 7-cl-tc in complex are colored in red. At the top, we annotate the nucleotides with secondary structure information. A positive value of \(\Delta H\) indicates a loss and a negative a gain of conformational entropy
A positive value of \(\Delta H\) in Fig. 7 indicates a loss of conformational entropy upon ligand binding. Interestingly, the binding loop as well as complexing nucleotides gain entropy. This is due to the fact of rearrangements between the nucleotides in spatial proximity to the ligand because 70% of the accessible surface area of TC is buried within the binding pocket L3 [23]. Experiments confirmed that local rearrangement of the binding pocket are necessary to prevent a possible release of the ligand [41]. Furthermore crystallographic studies have revealed that the largest changes occur in L3 upon TC binding [23]. Furthermore, we observe the highest entropy difference for nucleotide G51. Experimental data reveals that G51 crosslinks to tetracycline when the complex is subjected to UV irradiation [42]. These findings suggest a strong interaction with TC and thus a dramatic, positive change in \(\Delta H\). Nucleotides A52 and U54 show a positive entropy difference inside L3. Interestingly, molecular probing experiments show that G51, A52, and U54 of L3 are—in the absence of the antibiotic—the most modified nucleotides [23, 34]. Clearly, they change their conformational flexibility upon ligand binding due they direct interaction with the solvent. U54 further interacts with A51,A52,A53 and A55 building the core of the riboswitch [23]. Taken together, these observations reveal that U54 is necessary for the stabilization of L3. A more flexible dynamics (\(\Delta H\)) will change the configuration of the binding pocket and promotes TC release.
Comparison of tetracycline derivates
In this section, we want to investigate possible changes in configuration entropy by binding of different TC derivates. Moreover, we want to contrast StreAM-\(T_g\) to conventional metrics like RMSF (Eq. 5) using the entropy of the stationary distributions obtained from \(T_g(a)\). Therefore, we simulated a set consisting of four different antibiotics (atc, dc, ddtc, tc) in complex with the riboswitch of "Workflow". The structures of all derivates, each with different functional groups and different chemical properties, are shown in Fig. 3. For this approach we use a precomputation of \(P(X,\vec {r_{i}},\vec {r_{j}})\) to reduce the number of transition matrices for a full computation of \(T_g(a)\). Hence for all four simulations of TC derivates, we computed 1,763,208 (for tc), 1,534,488 (for atc), 2,685,816 (for dc) and 2,699,280 (for ddtc) transition matrices and combined them into 65 global models \(T_g(a)\) each. Similar to "Application to molecular synthetic biology", we compute \(\Delta H = H_{wt}-H_{complex}\) from the stationary distribution as well as \(\Delta RMSF = RMSF_{wt}-RMSF_{complex}\) from individual RMSF computations. The results are shown in Fig. 8.
Comparison of \(\Delta H\) and \(\Delta RMSF\). a \(\Delta H\) for \(T_g(a)\) between the native riboswitch and the complex with four different TC derivates. \(\Delta H\) is plotted against nucleotide position as a bar plot. A positive value of \(\Delta H\) indicates a loss and a negative a gain of conformational entropy. b \(\Delta RMSF\) between the native riboswitch and the complex with four different TC derivates (antibiotic). A positive value of \(\Delta RMSF\) indicates a loss and a negative an increase in fluctuations
The \(\Delta RMSF\) in Fig. 8b and in \(\Delta H\) Fig. 8a shows a similar picture in terms of nucleotide dynamics. If we focus on atc we can observe a loss of conformational entropy upon ligand binding for almost every nucleotide. Considering this example the RMSF only detects a significant loss of nucleotide-based dynamics ranging from nucleotide 37–46. However, for dc, we observe the same effects like for dc. Contrary to this observation we detect, for ddtc, an increase in dynamic upon ligand binding as well as negative \(\Delta RMSF\) values. For tc, we observe a similar picture as for 7-cl-tc ("Comparison of tetracycline derivates"). In a next step, we want to compare the obtained differences in stationary distribution with experimental values. To this end,we use an experimental metric: xfold values. A xfold value describes the efficiency of regulation in vivo and is given as the ratio of fluorescence without and with antibiotic in the experimental setup [43]. Unfortunately, atc reveals no experimental dynamics due to growth inhibition caused by the toxicity of the respective tc derivative [43]. In contrast to atc, dc and ddtc show only a weak performance (xfold = 1.1) in comparison to tc (xfold = 5.8) and 7-cl-tc (xfold = 3.8) [43]. On the one hand, atc and dc appear overall too rigid and on the other hand ddtc too flexible to obtain a stable bound structure, implying insufficient riboswitch performance. For our design criterion of high xfold, we conclude that only certain nucleotides are allowed to be affected upon ligand binding. In particular, we need flexible nucleotides for the process of induced ligand binding (like nucleotide G51 Fig. 7) and stabilization of the complex intermediates ("Application to molecular synthetic biology"). Additionally, the switch needs rigidity for nucleotides building the stem region of the TC-Aptamer upon ligand binding (like nucleotides A51, A52 and A53 Fig. 7).
Summary, conclusion, and future work
Simulation tools to design and analyze functionally RNA based devices are nowadays very limited. In this study, we developed a new method StreAM-\(T_g\) to analyze structural transitions, based on a coarse grained representation of RNA MD simulations, in order to gain insights into RNA dynamics. We demonstrate that StreAM-\(T_g\) fulfills our demands for a method to extract the coarse-grained Markovian dynamics of motifs of a complex RNA molecule. Moreover StreAM-\(T_g\) provides valuable insights into nucleotide based RNA dynamics in comparison to conventional metrics like the RMSF.
The effects observed in a designable riboswitch can be related to known experimental facts, such as conformational altering caused by ligand binding. Hence StreAM-\(T_g\) derived Markov models in an abstract space of motif creation and destruction. This allows for the efficient analysis of large MD trajectories.
Thus we hope to elucidate molecular relaxation timescales, spectral analysis in relation to single-molecule studies, as well as transition path theory in the future. At present, we use it for the design of switchable synthetic RNA based circuits in living cells [2, 44].
To broaden the application areas of StreAM-\(T_g\) we will extend it to proteins as well as evolutionary graphs mimicking the dynamics of molecular evolution in sequence space [45].
Guaranteed to exist due to the Perron-Frobenius theorem with an eigenvalue of \(\lambda = 1\).
https://github.com/BenjaminSchiller/Stream.
http://www.cbs.tu-darmstadt.de/streAM-Tg.tar.gz.
https://github.com/BenjaminSchiller/DNA.datasets
RMSF:
root-mean-square fluctuation
TC:
anhydrotetracycline
ddtc:
6-deoxy-6-demythyltetracycline
7-cl-tc:
7-chlorotetracycline
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Conceptualization of research work by SJ and KH. Data preparation by SJ, PB and MB. Implementation by SJ and BS. Benchmarks and Analysis of algorithm by SJ. Graphics by SJ, BS and PB. Writing of the manuscript by SJ, BS, KH, TS, MB and PB. Valuable suggestions to improve the manuscript by SJ, TS and KH. This Paper is an extended version of the research article: StreAM-\(T_g\) : Algorithms for Analyzing Coarse Grained RNA Dynamics Based on Markov Models of Connectivity-Graphs [46]. All authors read and approved the final manuscript.
The authors declares that they have no competing interests.
Generator for dynamic graphs: https://github.com/BenjaminSchiller/DNA.datasets Implementation of StreAM-\(T_g\): http://www.cbs.tu-darmstadt.de/streAM-Tg.tar.gz Implementation of StreAM and StreAM \(_B\): https://github.com/BenjaminSchiller/Stream
The Authors gratefully acknowledge financial support by the LOEWE project CompuGene of the Hessen State Ministry of Higher Education, Research and the Arts. Parts of this work have also been supported by the DFG, through the Cluster of Excellence cfaed as well as the CRC HAEC.
Department of Biology, TU Darmstadt, Schnittspahnstr. 2, 64283, Darmstadt, Germany
Sven Jager, Philipp Babel & Malte Blumenroth
Department of Computer Science, TU Dresden, Nöthnitzer Str. 46, 01187, Dresden, Germany
Benjamin Schiller & Thorsten Strufe
Department of Biology, Department of Computer Science, Department of Physics, TU Darmstadt, Schnittspahnstr. 2, 64283, Darmstadt, Germany
Kay Hamacher
Sven Jager
Benjamin Schiller
Philipp Babel
Malte Blumenroth
Thorsten Strufe
Correspondence to Sven Jager.
Jager, S., Schiller, B., Babel, P. et al. StreAM-\(T_g\): algorithms for analyzing coarse grained RNA dynamics based on Markov models of connectivity-graphs. Algorithms Mol Biol 12, 15 (2017). https://doi.org/10.1186/s13015-017-0105-0
Accepted: 16 May 2017
Markovian dynamics
Coarse graining
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November 2019, 12(7): 2085-2095. doi: 10.3934/dcdss.2019134
Periodic and subharmonic solutions for a 2$n$th-order $\phi_c$-Laplacian difference equation containing both advances and retardations
Peng Mei , Zhan Zhou , and Genghong Lin
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
* Corresponding author: Zhan Zhou
Received December 2017 Revised May 2018 Published December 2018
We consider a 2$n$th-order nonlinear difference equation containing both many advances and retardations with $\phi_c$-Laplacian. Using the critical point theory, we obtain some new and concrete criteria for the existence and multiplicity of periodic and subharmonic solutions in the more general case of the nonlinearity.
Keywords: Periodic and subharmonic solution, 2$n$th-order, nonlinear difference equation, $\phi_c$-Laplacian, critical point theory.
Mathematics Subject Classification: 39A23.
Citation: Peng Mei, Zhan Zhou, Genghong Lin. Periodic and subharmonic solutions for a 2$n$th-order $\phi_c$-Laplacian difference equation containing both advances and retardations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2085-2095. doi: 10.3934/dcdss.2019134
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Peng Mei Zhan Zhou Genghong Lin
\begin{document}$n$\end{document}th-order \begin{document}$\phi_c$\end{document}-Laplacian difference equation containing both advances and retardations" readonly="readonly"> | CommonCrawl |
Yakov Sinai
Yakov Grigorevich Sinai (Russian: Я́ков Григо́рьевич Сина́й; born September 21, 1935) is a Russian–American mathematician known for his work on dynamical systems. He contributed to the modern metric theory of dynamical systems and connected the world of deterministic (dynamical) systems with the world of probabilistic (stochastic) systems.[1] He has also worked on mathematical physics and probability theory.[2] His efforts have provided the groundwork for advances in the physical sciences.[1]
Yakov Sinai
Yakov G. Sinai
Born
Yakov Grigorevich Sinai
(1935-09-21) September 21, 1935
Moscow, Russian SFSR, Soviet Union
NationalityRussian / American
Alma materMoscow State University
Known forMeasure-preserving dynamical systems, various works on dynamical systems, mathematical and statistical physics, probability theory, mathematical fluid dynamics
SpouseElena B. Vul
AwardsBoltzmann Medal (1986)
Dannie Heineman Prize (1990)
Dirac Prize (1992)
Wolf Prize (1997)
Nemmers Prize (2002)
Lagrange Prize (2008)
Henri Poincaré Prize (2009)
Foreign Member of the Royal Society (2009)
Leroy P. Steele Prize (2013)
Abel Prize (2014)
Marcel Grossmann Award (2015)
Scientific career
FieldsMathematics
InstitutionsMoscow State University, Landau Institute for Theoretical Physics, Princeton University
Doctoral advisorAndrey Kolmogorov
Doctoral studentsLeonid Bunimovich
Nikolai Chernov
Dmitry Dolgopyat
Svetlana Jitomirskaya
Anatole Katok
Konstantin Khanin
Grigory Margulis
Leonid Polterovich
Marina Ratner
Corinna Ulcigrai
Sinai has won several awards, including the Nemmers Prize, the Wolf Prize in Mathematics and the Abel Prize. He serves as the professor of mathematics at Princeton University since 1993 and holds the position of Senior Researcher at the Landau Institute for Theoretical Physics in Moscow, Russia.
Biography
Yakov Grigorevich Sinai was born into a Russian Jewish academic family on September 21, 1935, in Moscow, Soviet Union (now Russia).[3] His parents, Nadezda Kagan and Gregory Sinai, were both microbiologists. His grandfather, Veniamin Kagan, headed the Department of Differential Geometry at Moscow State University and was a major influence on Sinai's life.[3]
Sinai received his bachelor's and master's degrees from Moscow State University.[2] In 1960, he earned his Ph.D., also from Moscow State; his adviser was Andrey Kolmogorov. Together with Kolmogorov, he showed that even for "unpredictable" dynamic systems, the level of unpredictability of motion can be described mathematically. In their idea, which became known as Kolmogorov–Sinai entropy, a system with zero entropy is entirely predictable, while a system with non-zero entropy has an unpredictability factor directly related to the amount of entropy.[1]
In 1963, Sinai introduced the idea of dynamical billiards, also known as "Sinai Billiards". In this idealized system, a particle bounces around inside a square boundary without loss of energy. Inside the square is a circular wall, of which the particle also bounces off. He then proved that for most initial trajectories of the ball, this system is ergodic, that is, after a long time, the amount of that time the ball will have spent in any given region on the surface of the table is approximately proportional to the area of that region. It was the first time anyone proved a dynamic system was ergodic.[1]
Also in 1963, Sinai announced a proof of the ergodic hypothesis for a gas consisting of n hard spheres confined to a box. The complete proof, however, was never published, and in 1987 Sinai declared that the announcement was premature. The problem remains open to this day.[4]
Other contributions in mathematics and mathematical physics include the rigorous foundations of Kenneth Wilson's renormalization group-method, which led to Wilson's Nobel Prize for Physics in 1982, Gibbs measures in ergodic theory, hyperbolic Markov partitions, proof of the existence of Hamiltonian dynamics for infinite particle systems by the idea of "cluster dynamics", description of the discrete Schrödinger operators by the localization of eigenfunctions, Markov partitions for billiards and Lorenz map (with Bunimovich and Chernov), a rigorous treatment of subdiffusions in dynamics, verification of asymptotic Poisson distribution of energy level gaps for a class of integrable dynamical systems, and his version of the Navier–Stokes equations together with Khanin, Mattingly and Li.
From 1960 to 1971, Sinai was a researcher in the Laboratory of Probabilistic and Statistical Methods at Moscow State University. In 1971 he accepted a position as senior researcher at the Landau Institute for Theoretical Physics in Russia, while continuing to teach at Moscow State. He had to wait until 1981 to become a professor at Moscow State, likely because he had supported the dissident poet, mathematician and human rights activist Alexander Esenin-Volpin in 1968.[5]
Since 1993, Sinai has been a professor of mathematics at Princeton University, while maintaining his position at the Landau Institute. For the 1997–98 academic year, he was the Thomas Jones Professor at Princeton, and in 2005, the Moore Distinguished Scholar at the California Institute of Technology.[3]
In 2002, Sinai won the Nemmers Prize for his "revolutionizing" work on dynamical systems, statistical mechanics, probability theory, and statistical physics.[2] In 2005, the Moscow Mathematical Journal dedicated an issue to Sinai writing "Yakov Sinai is one of the greatest mathematicians of our time ... his exceptional scientific enthusiasm inspire[d] several generations of scientists all over the world."[3]
In 2013, Sinai received the Leroy P. Steele Prize for Lifetime Achievement.[3] In 2014, the Norwegian Academy of Science and Letters awarded him the Abel Prize, for his contributions to dynamical systems, ergodic theory, and mathematical physics.[6] Presenting the award, Jordan Ellenberg said Sinai had solved real world physical problems "with the soul of a mathematician".[1] He praised the tools developed by Sinai which demonstrate how systems that look different may in fact have fundamental similarities. The prize comes with 6 million Norwegian krone,[1] equivalent at the time to $US 1 million or £600,000. He was also inducted into the Norwegian Academy of Science and Letters.[7]
Other awards won by Sinai include the Boltzmann Medal (1986), the Dannie Heineman Prize for Mathematical Physics (1990), the Dirac Prize (1992), the Wolf Prize in Mathematics (1997), the Lagrange Prize (2008) and the Henri Poincaré Prize (2009).[2][3] He is a member of the United States National Academy of Sciences, the Russian Academy of Sciences, and the Hungarian Academy of Sciences.[2] He is an honorary member of the London Mathematical Society (1992) and, in 2012, he became a fellow of the American Mathematical Society.[2][8] Sinai has been selected an honorary member of the American Academy of Arts and Sciences (1983), Brazilian Academy of Sciences (2000), the Academia Europaea, the Polish Academy of Sciences, and the Royal Society of London. He holds honorary degrees from the Budapest University of Technology and Economics, the Hebrew University of Jerusalem, Warwick University, and Warsaw University.[3]
Sinai has authored more than 250 papers and books. Concepts in mathematics named after him include Minlos–Sinai theory of phase separation, Sinai's random walk, Sinai–Ruelle–Bowen measures, and Pirogov–Sinai theory, Bleher–Sinai renormalization theory. Sinai has overseen more than 50 PhD candidates.[3] He has spoken at the International Congress of Mathematicians four times.[2] In 2000, he was a plenary speaker at the First Latin American Congress in Mathematics.[3]
Sinai is married to mathematician and physicist Elena B. Vul. The couple have written several joint papers.[3]
Selected works
• Introduction to Ergodic Theory. Princeton 1976.[9]
• Topics in Ergodic Theory. Princeton 1977, 1994.[10]
• Probability Theory – an Introductory Course. Springer, 1992.[10]
• Theory of probability and Random Processes (with Koralov). 2nd edition, Springer, 2007.[10]
• Theory of Phase Transitions – Rigorous Results. Pergamon, Oxford 1982.[10]
• Ergodic Theory (with Isaac Kornfeld and Sergei Fomin). Springer, Grundlehren der mathematischen Wissenschaften 1982.[10]
• "What is a Billiard?", Notices AMS 2004.[10]
• "Mathematicians and physicists = Cats and Dogs?" in Bulletin of the AMS. 2006, vol. 4.[10]
• "How mathematicians and physicists found each other in the theory of dynamical systems and in statistical mechanics", in Mathematical Events of the Twentieth Century (editors: Bolibruch, Osipov, & Sinai). Springer 2006, p. 399.[10]
References
1. Ball, Philip (March 26, 2014). "Chaos-theory pioneer nabs Abel Prize". Nature. Retrieved March 29, 2014.
2. "2002 Frederic Esser Nemmers Mathematics Prize Recipient". Northwestern University. Retrieved March 30, 2014.
3. "Yakov G. Sinai" (PDF). Abel Prize. Retrieved August 2, 2022.{{cite web}}: CS1 maint: url-status (link)
4. Uffink, Jos (2006). Compendium of the foundations of classical statistical physics (PDF). p. 91.
5. "Sinai biography". www-history.mcs.st-andrews.ac.uk. Retrieved June 28, 2017.
6. "2014: Yakov G. Sinai". www.abelprize.no. Retrieved August 2, 2022.{{cite web}}: CS1 maint: url-status (link)
7. "Gruppe 1: Matematiske fag" (in Norwegian). Norwegian Academy of Science and Letters. Retrieved March 30, 2016.
8. "List of Fellows of the American Mathematical Society". Retrieved July 20, 2013.
9. Chacon, R. V. (1978). "Review: Introduction to ergodic theory, by Ya. G. Sinai" (PDF). Bull. Amer. Math. Soc. 84 (4): 656–660. doi:10.1090/s0002-9904-1978-14515-7.
10. "Yakov Bibliography" (PDF). Princeton University. Retrieved March 30, 2014.
External links
Wikimedia Commons has media related to Yakov Grigorevich Sinai.
• Sinai on scholarpedia
• O'Connor, John J.; Robertson, Edmund F., "Yakov Sinai", MacTutor History of Mathematics Archive, University of St Andrews
• Yakov Sinai at the Mathematics Genealogy Project
• List of publications on the website of the Landau Institute for Theoretical Physics
Laureates of the Wolf Prize in Mathematics
1970s
• Israel Gelfand / Carl L. Siegel (1978)
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1980s
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• 2003 Jean-Pierre Serre
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| Wikipedia |
\begin{document}
\title{A survey of Heffter arrays}
\author{ Anita Pasotti\thanks{Partially supported by INdAM - GNSAGA} \\ DICATAM - Sez. Matematica\\ Università degli Studi di Brescia\\ Via Branze 43\\ I-25123 Brescia\ \ Italy\\ {\tt [email protected]} \\ \and Jeffrey H. Dinitz \\ Dept of Math. and Stat.\\ University of Vermont\\ Burlington, VT 05405, USA\\ {\tt [email protected]} }
\maketitle
\begin{center} \em Dedicated to Doug Stinson on the occasion of his 66th birthday. \end{center}
\begin{abstract} Heffter arrays were introduced by Archdeacon in 2015 as an interesting link between combinatorial designs and topological graph theory. Since the initial paper on this topic, there has been a good deal of interest in Heffter arrays as well as in related topics such as the sequencing of subsets of a group, biembeddings of cycle systems on a surface, and orthogonal cycle systems. This survey presents an overview of the current state of the art of this topic. We begin with an introduction to Heffter arrays for the reader who is unfamiliar with the subject, then we give a unified and comprehensive presentation of the major results, showing some proof methods also. This survey also includes sections on the connections of Heffter arrays to several other combinatorial objects, such as problems on partial sums and sequenceability, biembedding graphs on surfaces, difference families and orthogonal graph decompositions. These sections are followed by a section discussing the variants and generalizations of Heffter arrays which have been proposed. The survey itself is complemented by a list of unsolved problems as well as an updated and complete bibliography. \end{abstract}
\section{Introduction and Background} The notion of a Heffter array was introduced by Archdeacon \cite{A} in 2015. The original motivation was to aid in an embedding problem in topological graph theory, but they were also introduced as a very interesting combinatorial design object. The name is derived from their connection to Heffter's first difference problem, a problem related
to the construction of Steiner triple systems \cite{CR}. Specifically, in 1896 Heffter \cite{H} introduced his famous \emph{First Difference Problem} which asks if, given $v\equiv 1 \pmod 6$, the set $\left\{1,2,\ldots,\frac{v-1}{2}\right\}$ can be partitioned into $\frac{v-1}{6}$ triples $\{x, y, z\}$ such that either $x+y = z$ or $x+y+z=v$. The problem was eventually solved in 1939 by Peltesohn \cite{P}. We will show later in this survey that this problem is essentially solved (with an even stronger property) by a Heffter array with 3 rows and $\frac{v-1}{6}$ columns.
We begin with some definitions. Let $\mathbb{Z}_v$ be the cyclic group of odd order $v$ whose elements are denoted by $0$ and $\pm i$ where $i = 1, \ldots, \frac{v-1}{2}$. A \emph{half-set} $L$ of $\mathbb{Z}_v$ is a subset of $\mathbb{Z}_v\setminus \{0\}$ of size $\frac{v-1}{2}$
that contains exactly one of each pair $\{x,-x\}$. A \emph{Heffter system} $D(v, k)$, on a half-set $L$ of $\mathbb{Z}_v$, is a partition of $L$ into parts of size $k$ such that the elements in each part sum to $0$ modulo $v$, see \cite{A,MR}. Clearly a necessary condition for the existence of such a system is $v \equiv 1 \pmod{2k}$.
\begin{example}\label{ex:D315} A Heffter system $D(31,5)$: $$\{\{6,7,-10,-4,1\}, \{-9,5,2,-11,13\}, \{3,-12,8,15,-14\}\}.$$ \end{example} It is easy to see that starting from a Heffter system $D(v,3)$ one immediately gets a solution to the Heffter's First Difference Problem, as shown in the following example.
\begin{example}\label{ex:D313} A Heffter system $D(31,3)$: $$\{\{6,3,-9\}, \{7,5,-12\}, \{2,8,-10\},\{-4,-11,15\},\{1,13,-14\}\}.$$ \end{example} Then $\{\{6,3,9\}, \{7,5,12\}, \{2,8,10\},\{4,11,15\},\{1,13,14\}\}$ is a solution to the Heffter's First Difference Problem with $v=31$. Using this Heffter system we get the following base blocks $\{\{0,6,9\}, \{0 ,7, 12\}, \{0, 2,10\},\{0,4,15\},\{0,1,14\}\}$ which can then be developed in $\mathbb{Z}_{31}$ to construct a Steiner triple system of order 31. A detailed discussion of the construction of cycle systems from Heffter systems can be found in Section \ref{sec:DF}.
Two Heffter systems $D_m = D(2mn + 1,m)$ and $D_n = D(2mn + 1, n)$ on the same half-set $L$ of $\mathbb{Z}_{2mn+1}$ are \emph{orthogonal} if each part of $D_m$ intersects each part of $D_n$ in exactly one element, see \cite{A}. By a direct check one can see that the Heffter systems of Examples \ref{ex:D315} and \ref{ex:D313} are orthogonal.
In \cite{A} Archdeacon defined a \emph{Heffter array} $H(m, n)$ as an $m \times n$ array whose rows form a $D(2mn + 1, n)$ and whose columns form a $D(2mn + 1,m)$; these are called the row and column Heffter systems, respectively. Note that a Heffter array $H(m, n)$ is equivalent to a pair of orthogonal Heffter systems. An $H(m,n)$ gives two orthogonal Heffter systems by definition; its associated row system and its associated column system. Conversely, two orthogonal Heffter systems $D(2mn + 1,m)$ and $D(2mn + 1, n)$ give rise to an $H(m,n)$ whose cell of indices $i,j$ contains the only element which the $i$-th part of the first Heffter system shares with the $j$-th part of the second one.
\begin{example}\label{ex:35} Starting from the orthogonal Heffter systems of Examples \ref{ex:D315} and \ref{ex:D313} we obtain the following Heffter array $H(3,5)$ whose elements belong to $\mathbb{Z}_{31}$: $$
\begin{array}{|r|r|r|r|r|} \hline 6 & 7 & -10 & -4 & 1 \\ \hline -9 & 5 & 2 & -11 & 13 \\ \hline 3 & -12 & 8 & 15 & -14 \\ \hline \end{array} $$ \end{example}
In the same paper Archdeacon proposed a variation of Heffter arrays which allows for some empty cells. Two Heffter systems $D_h=D(2mh+1, h)$ and $D_k=D(2nk+1, k)$ on the same half-set of size $mh = nk$ are \emph{sub-orthogonal} if each part of $D_h$ intersects each part of $D_k$ in {\em at most} one element (see \cite{AHL} for information on sub-orthogonal factorizations). We note here that the term {\em sub-orthogonal} has not been widely adopted, hence for the remainder of this survey we will
use {\em orthogonal} to denote either orthogonal or sub-orthogonal. As before, we can form an $m \times n$ array, denoted by $H(m, n; h,k)$, whose cell of indices $i,j$ either contains the common element in the $i$-th part of $D_h$ and the $j$-th part of $D_k$, if any, or is empty otherwise. The $H(m,n)$ defined above is nothing but a $H(m,n;n,m)$.
In other words, when empty cells are allowed, a Heffter array can be defined as follows (this is now the standard definition): \begin{definition}\label{def:Heffter}
A \emph{Heffter array} $H(m,n;h,k)$ is an $m \times n$ matrix with entries from $\mathbb{Z}_{2nk+1}$ such that: \begin{itemize}
\item[{\rm (a)}] each row contains $h$ filled cells and each column contains $k$ filled cells; \item[{\rm (b)}] for every $x \in \mathbb{Z}_{2nk+1} \setminus \{0\}$, either $x$ or $-x$ appears in the array; \item[{\rm (c)}] the elements in every row and column sum to $0$ in $\mathbb{Z}_{2nk+1}$. \end{itemize} \end{definition}
\begin{example}\label{ex61284} An $H(6, 12; 8, 4)$ over $\mathbb{Z}_{97}$: $$
\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|} \hline && -1 & 2 & 5 & -6 &&& -25 & 26 & 29 & -30 \\ \hline && 3 & -4 & -7 & 8 &&& 27 & -28 & -31 & 32 \\ \hline 9 & -10 &&& -13 & 14 & 33 & -34 &&& -37 & 38 \\ \hline -11 & 12&&& 15 & -16 & -35 & 36 &&& 39 & -40 \\ \hline -17 & 18 & 21 & -22 &&& -41 & 42 & 45 & -46 && \\ \hline
19 & -20 & -23 & 24 &&& 43 &-44 & -47 & 48 && \\ \hline \end{array} $$ \end{example} Note that by the pigeon-hole principle zero cannot appear in the array, hence
Condition {\rm (b)} of Definition \ref{def:Heffter} is equivalent to requiring that the elements appearing in the array form a half-set of $\mathbb{Z}_{2nk+1}$. It is easy to see that trivial necessary conditions for the existence of an $H(m, n; h,k)$ are $mh = nk$, $3\leq k \leq m$, and $3\leq h \leq n$.
If a Heffter array $H(m, n; h,k)$ is square, that is $m = n$, then necessarily $h=k$. In this case it is a $H(n, n; k,k)$ and is denoted by $H(n; k)$. And just to clarify, a $H(n, k)$ (note the comma in place of the semicolon) is a Heffter array with no empty cells and with $n$ rows and $k$ columns, or a $H(n, k; k,n)$.
\begin{example}\label{ex:54} An $H(5; 4)$ over $\mathbb{Z}_{41}$:
$$\begin{array}{|r|r|r|r|r|} \hline
& 17 & -8 & -14&5 \\ \hline 1& & 18& -9 &-10\\ \hline -6 & 2 & &19 &-15\\ \hline -11&-12&3&&20 \\ \hline 16&-7&-13&4& \\ \hline \end{array} $$ \end{example}
\begin{definition}\label{def:Heffter.integer} A Heffter array is an \emph{integer Heffter array} if condition {\rm (c)} of Definition \ref{def:Heffter}
is strengthened so that the elements in every row and every column, seen as integers in $\pm\{1,2,\ldots,nk\}$, sum to zero in $\mathbb{Z}$. \end{definition} Note that the arrays of Examples \ref{ex:35}, \ref{ex61284} and \ref{ex:54} are integer Heffter arrays. In the following example we present a non-integer Heffter array.
\begin{example} A non-integer $H(7;3)$ over $\mathbb{Z}_{43}$:
$$\begin{array}{|r|r|r|r|r|r|r|} \hline 15& -13& -2&&&& \\\hline -11& 14& &-3&&&\\\hline -4& &-8& 12&&&\\\hline
&-1& 10& -9&&& \\\hline &&&& 5& 21& 17\\\hline &&&& 18& 6& 19\\\hline &&&& 20& 16& 7\\\hline \end{array}$$ \end{example}
A class of Heffter arrays of fundamental importance and which are particularly useful in recursive constructions of are so called \emph{shiftable} arrays, see \cite{ADDY}. An $n \times n$ array $A$ (possibly with empty cells) whose elements are integers is shiftable if each row and each column of $A$ contains the same number of positive and negative numbers. Given a shiftable array $A$ and a nonnegative integer $x$, let $A \pm x$ denote the array where $x$ is added to all the positive entries in $A$ and $-x$ is added to all the negative entries. Define the {\em support} of $A$ as the set containing the absolute value of the elements appearing in $A$. Note that if $A$ is shiftable with support $S$ and $x$ is a nonnegative integer, then $A\pm x$ has the same row and column sums as $A$ and has support $S + x$. In the case of a shiftable integer Heffter array $H(m,n;h,k)$, the array $H(m,n;h,k)\pm x$ has row and column sums equal to zero and its support is $\{1+x,2+x, \ldots ,nk+x\} $. The arrays of Examples \ref{ex61284} and \ref{ex:54} are shiftable. The existence of shiftable square Heffter arrays was established by constructive proofs in \cite{ADDY}.
\begin{theorem}{\rm\cite{ADDY}}\label{shift} There exists a shiftable integer $H(n;k)$ if and only if $k$ is even and $nk\equiv 0 \pmod 4$. \end{theorem}
Many of the known square Heffter arrays have a diagonal structure which has been shown to be useful for recursive constructions and for the applications to biembeddings. In addition, this diagonal structure has been extremely useful in computer searches for Heffter arrays. Given a square array of order $n$, for $i=1,\ldots,n$, the $i$-th diagonal is defined by $D_i=\{(i,1),(i+1,2),\ldots, (i-1,n)\}$. All the arithmetic on row and column indices is performed modulo $n$, where the set of reduced residues is $\{1,2,\ldots,n\}$. The diagonals $D_i,D_{i+1},\ldots, D_{i+k-1}$ are $k$ {\em consecutive diagonals}. Given $n\geq k\geq 1$, a partially filled array $A$ of order $n$ is $k$-\emph{diagonal} if its nonempty cells are exactly those of $k$ diagonals. Moreover, if these diagonals are consecutive, $A$ is said to be \emph{cyclically} $k$-\emph{diagonal}. The array of Example \ref{ex:54} is cyclically $4$-diagonal.
Shiftable cyclically $4$-diagonal $H(n;4)$ are constructed in \cite{ADDY} for all $n \ge 4$. In view of their particular structure, these arrays can be used to add four filled cells per row and column to an existing Heffter array $H$ if $H$ contains four consecutive diagonals of empty cells. This is described in the following lemma and example.
\begin{lemma}{\rm\cite{ADDY}}\label{lemma:shift} If there exists an integer Heffter array $H(n;k)$ which has $s$ disjoint sets of four consecutive empty diagonals, then there exists an integer $H(n;k+4s)$. Furthermore, if the $H(n;k)$ is shiftable, then the $H(n;k+4s)$ is shiftable too. \end{lemma}
\begin{example}\label{ex:H87} The following are a cyclically $3$-diagonal $H(8;3)$, say $H$, and a shiftable $H(8;4)$, say $A$: \begin{center} \begin{footnotesize}
$H=\begin{array}{|r|r|r|r|r|r|r|r|}\hline 8 & 16 & & & & & & -24 \\ \hline -17 & -6 & 23 & & & & & \\ \hline & -10 & -5 & 15 & & & & \\ \hline & & -18 & 7 & 11 & & & \\ \hline & & & -22 & 3 & 19 & & \\ \hline & & & & -14 & 2 & 12 & \\ \hline & & & & & -21 & 1 & 20 \\ \hline 9 & & & & & & -13 & 4 \\\hline \end{array}$ \end{footnotesize}
\begin{footnotesize}
A= $\begin{array}{|r|r|r|r|r|r|r|r|}\hline & & & 1 & -3 & -5 & 7 & \\ \hline & & & -4 & 2 & 8 & -6 & \\ \hline 15 & & & & & 9 & -11 & -13 \\ \hline -14 & & & & & -12 & 10 & 16 \\ \hline -19 & -21 & 23 & & & & & 17 \\ \hline 18 & 24 & -22 & & & & & -20 \\ \hline & 25 & -27 & -29 & 31 & & & \\ \hline & -28 & 26 & 32 & -30 & & & \\\hline \end{array}$ \end{footnotesize} \end{center}
Using $H$ and $A\pm 24$ one can construct the following $H(8;7)$ over $\mathbb{Z}_{113}$: \begin{center} \begin{footnotesize}
$\begin{array}{|r|r|r|r|r|r|r|r|}\hline 8 & 16 & & 25 & -27 & -29 & 31 & -24 \\ \hline -17 & -6 & 23 & -28 & 26 & 32 & -30 & \\ \hline 39 & -10 & -5 & 15 & & 33 & -35 & -37 \\ \hline -38 & & -18 & 7 & 11 & -36 & 34 & 40 \\ \hline -43 & -45 & 47 & -22 & 3 & 19 & & 41 \\ \hline 42 & 48 & -46 & & -14 & 2 & 12 & -44 \\ \hline & 49 & -51 & -53 & 55 & -21 & 1 & 20 \\ \hline 9 & -52 & 50 & 56 & -54 & & -13 & 4 \\\hline \end{array}$ \end{footnotesize} \end{center} \end{example}
Concerning the search for Heffter arrays constructed from diagonals, a useful notation was introduced in \cite{DW} and has been used in several subsequent papers including \cite{weak,RelH,CPEJC,CPPBiembeddings}. All the constructions in \cite{DW} were based on filling in the cells of a fixed collection of diagonals. To aid in the constructions the authors defined the following procedure for filling a sequence of cells on a diagonal. It is termed {\em diag} and it has six parameters, as follows.
In an $n \times n$ array $A$ the procedure $diag(r,c,s,\Delta_1,\Delta_2, \ell)$ installs the entries of an array $A$ as follows: \begin{center} $A[r+i\Delta_1,c+i\Delta_1]=s+i\Delta_2\mbox{ \ for \ }i=0,1,\dots,\ell-1.$\end{center} Here all arithmetic on the row and column indices is performed modulo $n$, where the set of reduced residues is $\{1,2, \ldots, n\}$. The following summarizes the parameters used in the $diag$ procedure:
\begin{tabular}{cl} $\bullet$ & $r$ denotes the starting row,\\ $\bullet$ &$c$ denotes the starting column,\\ $\bullet$ & $s$ denotes the starting symbol,\\ $\bullet$ & $\Delta_1$ denotes how much the row and column indices change at each step,\\ $\bullet$ & $\Delta_2$ denotes how much the symbol changes at each step, and\\ $\bullet$ & $\ell$ is the length of the chain.\\ \end{tabular}
\noindent The following example shows the use of the above definition.
\begin{example}\label{ex1} {\rm\cite{DW}} The array below is a shiftable integer Heffter array $H(11;4)$ where the filled cells are contained in two sets of consecutive diagonals and it is constructed via the following procedures:
\noindent
\begin{center} $\begin{array}{ll}
diag(4,1,1,1,2,11);\quad\quad & diag(5,1,-2,1,-2,11); \\
diag(4,7,-23,1,-2,11);\quad \quad & diag(5,7,24,1,2,11). \end{array}$ \end{center}
\renewcommand{6pt}{1.5pt} \begin{footnotesize} \[
\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|}\hline
& &38&-39& & & &-16&17& & \\ \hline
&&&40&-41&&&&-18&19& \\ \hline
&&&&42&-43&&&&-20&21 \\ \hline 1&& &&&44&-23&&&&-22 \\ \hline -2&3&&&&&24&-25&&& \\ \hline
&-4&5&&&&&26&-27&& \\ \hline
&&-6&7&&&&&28&-29& \\ \hline
&&&-8&9&&&&&30&-31 \\ \hline -33&&&&-10&11&&&&&32 \\ \hline 34&-35&&&&-12&13&&&& \\ \hline
&36&-37& & & &-14&15& && \\ \hline \end{array} \] \end{footnotesize} \renewcommand{6pt}{6pt} \end{example}
Concerning the existence problem, the first classes of Heffter arrays which were systematically studied were (1) square arrays and (2) Heffter arrays with no empty cells (termed {\em tight} Heffter arrays). Square integer Heffter arrays were considered in
\cite{ADDY}. In that paper the authors proved the following almost complete result.
In their proofs Lemma \ref{lemma:shift} plays a fundamental role.
\begin{theorem}{\rm\cite{ADDY}} If an integer $H(n;k)$ exists, then necessarily $3 \leq k \leq n$ and $nk \equiv 0,3\pmod 4$. Furthermore, these conditions are sufficient except possibly when $n\equiv 0\ \mbox{or} \ 3\pmod 4$ and $k\equiv 1\pmod 4$. \end{theorem}
It should be noted that \cite{ADDY} also contains partial results when $n\equiv 0\ \mbox{or} \ 3\pmod 4$ and $k\equiv 1$ (mod $4$). These missing cases were entirely solved in \cite{DW} using the $diag$ method described above. Thus we have a complete constructive solution for the integer square case. This result is summarized
in the following theorem.
\begin{theorem}\label{thm:intergerHeffter}{\rm \cite{ADDY,DW}} There exists an integer $H(n; k)$ if and only if $3 \leq k \leq n$ and $nk \equiv 0, 3\pmod4$. \end{theorem}
For all congruences classes of $n$ and $k$ for which an integer Heffter array necessarily cannot exist a non-integer Heffter array $H(n;k)$ was proven to exist in \cite{CDDY}. Hence we can state the following. \begin{theorem}\label{thm:Heffter}{\rm \cite{ADDY,CDDY,DW}} There exists an $H(n; k)$ if and only if $3 \leq k \leq n$. \end{theorem}
Table \ref{table} details the papers where existence is proven for each congruence class of $n$ and $k$ modulo 4. \begin{table}[h]
\begin{center}
\begin{tabular} {|c||c|c|c|c|} \hline $n\backslash k$& 0& 1& 2&3 \\ \hline\hline 0 &\cite{ADDY}&\cite{ADDY,DW}&\cite{ADDY}&\cite{ADDY} \\ \hline 1 &\cite{ADDY}& \cite{CDDY}& \cite{CDDY} &\cite{ADDY} \\ \hline 2 &\cite{ADDY}& \cite{CDDY} &\cite{ADDY}& \cite{CDDY} \\ \hline 3 &\cite{ADDY}&\cite{ADDY,DW}& \cite{CDDY}& \cite{CDDY} \\ \hline \end{tabular} \end{center} \label{table} \caption{Articles which prove the existence of square Heffter arrays $H(n;k)$.} \end{table}
We now turn to the case of rectangular Heffter arrays without empty cells $H(m,n)$, or so-called tight Heffter arrays. A complete constructive solution is presented in \cite{ABD}. Historically speaking, it was this paper that introduced the notion of shiftable Heffter arrays. The main theorem of that paper is given next.
\begin{theorem}{\rm \cite{ABD}} There exists an $H(m,n)$ if and only if $m,n\geq 3$. Moreover, these arrays are integer if and only if $mn\equiv 0,3 \pmod 4$. \end{theorem}
It has not been proven that integer rectangular arrays with empty cells exist for all possible orders. We note here that the necesary conditions for the existence of an integer $H ( m, n ; h, k )$ are that
$3 \leq h \leq n$, $3 \leq k \leq m$, $mh = nk$ and $nk \equiv 0,3 \pmod 4$.
Recent partial results for integer rectangular arrays with empty cells can be found in \cite{MP3}. We summarize these results in the following two theorems.
\begin{theorem}{\rm \cite{MP3}}\label{Fiore1} Let $m, n, h, k$ be such that $3 \leq h \leq n$, $3 \leq k \leq m$ and $mh = nk$. Set $d = \gcd ( h , k )$. There exists an integer $H ( m, n ; h, k )$ in each of the following cases: \begin{itemize} \item[$(\rm{1})$] $d \equiv 0 \pmod 4$; \item [$(\rm{2})$] $d \equiv 1 \pmod 4$ with $d \geq 5$ and $nk \equiv 3 \pmod 4$; \item[$(\rm{3})$] $d \equiv 2 \pmod 4$ and $nk \equiv 0 \pmod 4$; \item[$(\rm{4})$] $d \equiv 3 \pmod 4$ and $nk \equiv 0, 3 \pmod 4$. \end{itemize} \end{theorem}
\begin{theorem}{\rm \cite{MP3}}\label{Fiore2}
Let $m, n, h, k$ be such that $3 \leq h \leq n$, $3 \leq k \leq m$ and $mh = nk$.
If $h\equiv 0 \pmod 4$ and $k \neq 5$ is odd, then there exists an integer $H ( m, n ; h, k )$. \end{theorem}
This survey is organized as follows. In Section \ref{sec:simple} we discuss results on simple and globally simple Heffter arrays. Then in Section \ref{sec:biemb} we give the application of Heffter arrays to biembedding complete graphs on topological surfaces. In Section \ref{sec:sums} we consider the problem of sequencing subsets of a group. Section \ref{sec:DF} gives results concerning the connection between Heffter arrays, difference families and graph decompositions and in Section \ref{sec:variants} we survey results concerning variants and generalizations of Heffter arrays. Finally in Section \ref{sec:conclusions} we give some open problems related to Heffter arrays and the other objects covered in this survey.
\section{Simple Heffter Arrays}\label{sec:simple}
In \cite{A}, Archdeacon showed that Heffter arrays can be used to construct (simple) cycle decompositions of the complete graph if they satisfy an additional condition, called \emph{simplicity}. This concept is also related to several open problems on partial sums and sequenceability, these questions will be discussed in the Section \ref{sec:sums}.
Let $T$ be a finite subset of an additive group $G$. Given an ordering $\omega=(t_1,t_2,\ldots,t_k)$ of the elements in $T$, let $s_i=\sum_{j=1}^i t_j$ be the $i$-th partial sum of $\omega$. The ordering $\omega$ is said to be \emph{simple} if its partial sums $s_1,\ldots,s_k$ are pairwise distinct. We point out that, if $\sum_{t \in T}t=0$, then an ordering $\omega$ of the elements in $T$ is simple if and only if it does not contain any proper subsequence that sums to $0$. We also note that if $\omega=(t_1,t_2,\ldots,t_k)$ is a simple ordering so is $\omega^{-1}=(t_k,t_{k-1},\ldots,t_1)$.
In the following, with a little abuse of notation, we will identify each row and column of an $H(m,n;h,k)$ with the subset of $\mathbb{Z}_{2nk+1}$ of size $h$ and $k$, respectively, whose elements are the entries of the nonempty cells of such a row or column. For example we can view the first row of the $H(8;7)$ of Example \ref{ex:H87} as the subset $R_1=\{8,16,25,-27,-29,31,-24\}$ of $\mathbb{Z}_{113}$.
\begin{definition} An $H(m,n;h,k)$ is \emph{simple} if each row and each column admits a simple ordering. \end{definition}
It is easy to see that since each row and each column of an $H(m,n;h,k)$ does not contain $0$ or $2$-subsets of the form $\{x,-x\}$, and is such that $s_h=0$ and $s_k=0$, respectively, then every $H(m,n;h,k)$ with $h,k\leq 5$ is simple (indeed {\em any} subset of size $k\leq 10$ with these properties is simple \cite{AL}).
\begin{example}\label{ex:simple} The $H(8;7)$ of {\rm Example \ref{ex:H87}} is simple. To verify this property we need to provide a simple ordering for each row and each column. One can check that the $\omega_i$'s are simple orderings of the rows and the $\nu_i$'s are simple orderings of the columns: \begin{center} \begin{footnotesize} $\begin{array}{lcl} \omega_1= ( 8, 25, 16, -27, -29, 31, -24); & \quad &\nu_1= (8,39,-17,-38,-43,42,9 );\\ \omega_2= (-17, -6, -28, 23, 26, 32, -30); & \quad &\nu_2= ( 16,-6,-45,-10,48,49,-52 );\\ \omega_3= (39, -10, -5, 33, 15, -35, -37); & \quad & \nu_3= ( 23,-5,47,-18,-46,-51,50 );\\ \omega_4= (-38, -18, 7, -36, 11, 34, 40); & \quad &\nu_4= (25,-28,15,7,-53,-22,56 );\\ \omega_5= (-43, -45, 47, -22, 3, 41, 19); & \quad &\nu_5= (-27,26,11,3,55,-14,-54 );\\ \omega_6= ( 42, 48, -46, -14, 2, -44, 12); & \quad &\nu_6= ( -21,32,33,-36,19,2,-29);\\ \omega_7= (20, -51, -53, 55, -21, 1, 49);& \quad &\nu_7= (-13,-30,-35,34,12,1,31);\\ \omega_8= (-52, 9, 50, 56, -54, -13, 4); & \quad &\nu_8= (-37,-24,40,41,-44,20,4). \end{array}$ \end{footnotesize} \end{center} \end{example}
By \emph{natural ordering} of a row (column, respectively) one means the ordering from left to right (from top to bottom, respectively). Since the natural ordering of each row and each column of the $H(8;7)$ of Example \ref{ex:H87} contains a proper subsequence that sums to zero (this immediately follows by its construction), the natural ordering cannot be simple for any row and any column. Clearly, for larger $n$ and $k$ it is more difficult and tedious to provide explicit simple orderings for rows and columns of an $H(m,n;h,k)$. For this reason, in \cite{CMPPHeffter} the authors considered Heffter arrays which are simple with respect to the natural ordering of rows and columns and proposed the following definition.
\begin{definition}{\rm \cite{CMPPHeffter}} A Heffter array $H(m, n;h, k)$ is \emph{globally simple} if the natural ordering of each row and each column is simple. \end{definition}
A globally simple $H(m,n; h,k)$ is denoted by $SH(m,n; h,k)$ and a square globally simple $H(n;k)$ is denoted by $SH(n;k)$.
By above arguments, we have that the $H(8;7)$ of Example \ref{ex:H87} is simple but not globally simple. However a globally simple $H(8;7)$ does indeed exist and is presented in the following example.
\begin{example}\label{ex:SH87} A globally simple $H(8;7)$: \begin{center} \begin{footnotesize}
\begin{tabular}{|r|r|r|r|r|r|r|r|} \hline $4$ & $35$ & $-45$ & $46$ & $ $ & $20$ & $-36$ & $-24$ \\ \hline $48$ & $-5$ & $23$ & $-47$ & $-18$ & $ $ & $37$ & $-38$ \\ \hline $-32$ & $-10$ & $-6$ & $31$ & $-41$ & $42$ & $ $ & $16$ \\ \hline $33$ & $-34$ & $44$ & $3$ & $11$ & $-43$ & $-14$ & $ $ \\ \hline $ $ & $15$ & $-28$ & $-22$ & $7$ & $27$ & $-53$ & $54$ \\ \hline $-13$ & $ $ & $29$ & $-30$ & $56$ & $1$ & $12$ & $-55$ \\ \hline $-49$ & $50$ & $ $ & $19$ & $-40$ & $-21$ & $2$ & $39$ \\ \hline $9$ & $-51$ & $-17$ & $ $ & $25$ & $-26$ & $52$ & $8$ \\\hline \end{tabular} \end{footnotesize} \end{center} \end{example}
The Heffter arrays constructed in {\rm \cite{ADDY,DW}}, in general, are not globally simple and
easy modifications of the existing constructions seem not to produce globally simple arrays. One exception to this are the $SH(n;6)$'s constructed in {\rm \cite[Proposition 4.1]{CMPPHeffter}} which were obtained by switching the first two columns of the matrices given in {\rm \cite[Theorem 2.1]{ADDY}}. Globally simple Heffter arrays are more elegant than simple ones but also more difficult to construct, in fact there are few papers on these arrays, see \cite{BCDY,CDY,CMPPHeffter,DM}. The following two theorems summarize the results on square globally simple integer Heffter arrays. The first deals with the case when $k\leq 10$ and the second considers the existence of several of the congruence classes.
\begin{theorem}{\rm \cite{CMPPHeffter}}\label{thm:SH} Let $3\leq k\leq 10$. There exists an integer globally simple $H(n;k)$ if and only if $n \geq k$ and $nk\equiv 0,3 \pmod4$. \end{theorem}
\ \begin{theorem}{\rm \cite{BCDY}} \label{thm:globally_simple2} Let $n\geq k\geq 3$. There exists an integer globally simple $H(n;k)$ in each of the following cases: \begin{itemize}
\item[$(\rm{1})$ ] $k\equiv 0\pmod 4$;
\item[$(\rm{2})$ ] $n\equiv 1\pmod4$ and $k\equiv3\pmod 4$;
\item[$(\rm{3})$ ] $n\equiv 0\pmod4$, $k\equiv3\pmod 4$ and $n>>k$. \end{itemize} \end{theorem}
In the rectangular case we have the following theorem for Heffter arrays with either three or five rows. \begin{theorem}{\rm \cite{DM}} \label{thm:SH1}
There exists a globally simple $H(3,n)$ for every $n\geq 3$ and a globally simple $H(5,n)$ for all $3\leq n\leq 100$. \end{theorem}
\section{Applications to biembeddings}\label{sec:biemb}
One of the main reasons that Archdeacon introduced the notion of Heffter arrays is their connection with biembedding cycle systems on a topological surface. Several papers focus on this application, see, for instance \cite{CDY,CMPPHeffter,CPArxiv,DM}. We emphasize that there has been previous work done in the area of biembedding cycle systems, especially triple systems, see \cite{DGLM,DGLM2,GG,GK}. It is interesting to note that some of the earliest examples of Heffter arrays are derived from direct constructions of biembeddings. In particular, in \cite{ADDY} the authors used current graphs (developed by Youngs \cite{Y}) based on M\"{o}bius ladders with $n=4m+1$ rungs and on cylindrical ladders with $n=4m$ rungs to construct diagonal Heffter arrays $H(n;3)$. So not only do Heffter arrays yield biembeddings (as will be detailed in this section), but biembeddings have been used to make Heffter arrays.
We recall some basic definitions, see \cite{Moh} and also note that \cite{GT} and \cite{MT} are excellent references for graphs on surfaces. An \emph{embedding} of a graph (or multigraph) $\Gamma$ in a surface $\Sigma$ is a continuous injective mapping $\psi: \Gamma \to \Sigma$, where $\Gamma$ is viewed with the usual topology as $1$-dimensional simplicial complex. The connected components of $\Sigma \setminus \psi(\Gamma)$ are called $\psi$-\emph{faces}. If each $\psi$-face is homeomorphic to an open disc, then the embedding $\psi$ is said to be \emph{cellular}. In this context, we say that two embeddings $\psi:\Gamma\rightarrow\Sigma$ and $\psi':\Gamma'\rightarrow\Sigma'$ are isomorphic if and only if there is a graph isomorphism $\sigma:\Gamma\rightarrow\Gamma'$ such that $\sigma(F)$ is a $\psi'$-face if and only if $F$ is a $\psi$-face. A circuit decomposition of a graph $\Gamma$ is a collection of edge-disjoint subgraphs of $\Gamma$ such that every edge of $\Gamma$ belongs to exactly one circuit.
\begin{definition} A \emph{biembedding} of two circuit decompositions $\mathcal{D}$ and $\mathcal{D}'$ of a simple graph $\Gamma$ is a face $2$-colorable embedding
of $\Gamma$ in which one color class is comprised of the circuits in $\mathcal{D}$ and
the other class contains exactly the circuits in $\mathcal{D}'$. \end{definition}
If the faces are cycles the biembedding is said to be \emph{simple}.
When one is not interested in the circuits, we simply speak of a biembedding of the graph $\Gamma$.
Equivalently, a biembedding of a simple graph $\Gamma$ is a face $2$-colorable embedding of $\Gamma$.
\begin{example}\label{emb.fig} A biembedding of the complete graph $K_7$ on the torus. Note that the black cells are a Steiner triple system (a circuit decomposition of $K_7$ into triples) and the white cells are also a Steiner triple system.
\centerline{\hbox{ \psfig{figure=emb2.eps,height=2in,width=3in}}} \end{example}
In order to state our main theorem connecting Heffter arrays to biembeddings (Theorem \ref{Gustin}), we need the notion of compatible orderings of the rows and columns of the Heffter array.
\begin{definition} Let $A$ be a partially filled array with $F$ filled cells whose elements are pairwise distinct. Let $\omega_r$ ($\omega_c$, respectively) be any ordering of the rows (columns, respectively) of $A$. Then $\omega_r$ and $\omega_c$ are \emph{compatible} orderings if $\omega_r\circ \omega_c$ is a cycle of order $F$. \end{definition}
\begin{example} Given the $H(3,5)$ from Example \ref{ex:35}, use the natural ordering of the rows and columns to get row cycles
$\omega_r= (6, 7, {-10}, {-4}, 1)(-9, 5, 2, {-11}, 13)(3, {-12}, 8, 15,{-14})$
and the column cycles
$\omega_c= (6,{-9},3)(7,5,{-12})({-10},2,8)({-4},{-11},15)(1,13,{-14}) $. Then we see that $\omega_r\circ \omega_c = ( 6,5,8, {-4}, 13, 3,7, 2,15,1,{-9},{-12},{-10},{-11},{-14}) $ is a single cycle (of order 15) and hence $\omega_r$ and $\omega_c$ are compatible orderings. \end{example}
We will discuss the connection between compatible orderings and biembeddings below. It is because of this connection that the existence of compatible orderings for a given array has been considered in several papers, see \cite{CDY,CDP,CMPPHeffter,DM}. The following theorem details the necessary conditions for there to exist
orderings $\omega_r$ and $\omega_c$ of the row and column Heffter systems of a Heffter array that are compatible.
\begin{theorem}{\rm \cite{CDY,DM}}\label{thm:compatible} If there exist compatible orderings $\omega_r$ and $\omega_c$ for a Heffter array $H (m, n; h, k)$, then either: \begin{itemize} \item[{\rm (1)}] $m, n, h$ and $k$ are all odd; \item[{\rm (2)}] $m$ is odd, $n$ is even and $h$ is even; or \item[{\rm (3)}] $m$ is even, $n$ is odd and $k$ is even. \end{itemize} \end{theorem}
Hence, if a square Heffter array $H(n;k)$ admits compatible orderings, then $nk$ is odd. This condition is also sufficient when $k$ is small and the array is cyclically $k$-diagonal. In fact the following holds:
\begin{proposition}{\rm \cite{CDP}} Let $3\leq k< 200$ be an integer and let $A$ be a cyclically $k$-diagonal Heffter array of size $n\geq k$. Then there exist two compatible orderings $\omega_r$ and $\omega_c$ of the rows and the columns of $A$ if and only if $n$ and $k$ are both odd. \end{proposition}
In view of Theorem \ref{thm:compatible}, in order to obtain results on biembeddings, several authors have focused on square Heffter arrays with $n$ and $k$ odd. The following results pertain to the existence of compatible orderings in the case of cyclically $k$-diagonal Heffter arrays $H(n;k)$.
\begin{proposition}{\rm \cite{CDP,CMPPHeffter}}
Let $k\geq 3$ be an odd integer and let $A$ be a cyclically $k$-diagonal Heffter
array of size $n\geq k$. If $\gcd(n, k-1) = 1$, then there exist two compatible orderings $\omega_r$ and $\omega_c$ of the rows and the columns of $A$. \end{proposition}
\begin{proposition}{\rm \cite{CDP}} Let $k\geq3$ and let $A$ be a cyclically $k$-diagonal Heffter array of size $n\geq (k-2)(k-1)$. Then there exist two compatible orderings $\omega_r$ and $\omega_c$ of the rows and the columns of $A$ if and only if $n$ and $k$ are both odd. \end{proposition}
In the case of an $H(m,n;h,k)$ with no empty cells (an $H(m,n)$) the following theorem was proven in \cite{DM}. In the proof, an explicit compatible ordering of the rows and columns is given.
\begin{proposition}{\rm \cite{DM}} \label{compat} Let $A$ be a $m\times n$ Heffter array with no empty cells where at least one of $m$ and $n$ is odd. Then there exist compatible orderings, $\omega_r$ and $\omega_c$ on the rows and columns of $A$. \end{proposition}
So from Theorem \ref{thm:compatible} and Proposition \ref{compat} we have necesary and sufficient condition for an $H(m,n)$ to have compatible orderings. Specifically we have the following.
\begin{theorem} Let $A$ be a $m\times n$ Heffter array with no empty cells (an $H(m,n))$. Then there exist compatible orderings, $\omega_r$ and $\omega_c$ on the rows and columns of $A$ if and only if at least one of $m$ and $n$ is odd. \end{theorem}
Compatible orderings are the link between Heffter arrays and biembeddings, while the simplicity of a Heffter array will guarantee the biembedding is simple also. The following fundamental theorem from \cite{A} gives the connection between Heffter arrays with compatible orderings and biembeddings. The interested reader is referred to that paper for the full details of this connection, but briefly, using the compatible orderings of the Heffter array one obtains a current graph which in turn gives the rotation of the edges around each vertex. The rows and columns give the cycles in the two cycle decompositions.
\begin{theorem}\label{Gustin}{\rm \cite{A}} Given a Heffter array $H(m,n;h,k)$ with compatible orderings $\omega_r$ on the rows and $\omega_c$ on the columns, there exists a biembedding of the complete graph $K_{2nk+1}$
on an orientable surface such that every edge is on a face of size $h$ and a face of size $k$. Moreover, if $\omega_r$ and $\omega_c$ are both simple, then all faces are simple cycles. \end{theorem}
A $k$-cycle system of order $n$ is a set of cycles of length $k$, denoted by $C_k$, whose edges partition the edges of $K_n$, one speaks also of a $C_k$-decomposition of $K_n$. Restating the previous theorem in terms of biembeddings of cycle systems we have the following. \begin{theorem}\label{biembed2}{\rm \cite{A}} Given a Heffter array $H(m,n;h,k)$ with simple compatible orderings $\omega_r$ on the rows and $\omega_c$ on the columns, there exists an orientable biembedding of a $C_h$-decomposition and a $C_k$-decomposition both of $K_{2nk+1}$, or equivalently, there is an orientable biembedding of an $h$-cycle system and a $k$-cycle system both in $K_{2nk+1}$. \end{theorem}
We note again that a discussion of the construction of cycle systems from Heffter arrays will be given in Section \ref{sec:DF}. As an example of the application of Proposition \ref{compat} and Theorems \ref{Gustin} and \ref{biembed2} simple $H(3,n)$ were explicitly constructed in \cite{DM} for all $n\geq 3$ with an eye towards the biembedding problem. The following is the main result in that paper.
\begin{theorem}{\rm \cite{DM}} There exists a globally simple $H(3,n)$ for every $n\geq 3$. Thus for every $n \geq 3$ there exists a biembedding of $K_{6n+1}$ on an orientable surface such that each edge is on a $3$-cycle and an $n$-cycle and hence there is a biembedding of a Steiner triple system and an $n$-cycle system. \end{theorem}
The following problem was introduced in \cite{CDP} in view of its importance in looking for compatible orderings for Heffter arrays. However, it can also be considered as a tour problem on a toroidal board which is interesting \emph{per se}.
Let $A$ be an $m\times n$ toroidal partially filled array. By $r_i$ we denote the orientation of the $i$-th row of $A$, precisely $r_i=1$ if it is from left to right and $r_i=-1$ if it is from right to left. Analogously, for the $j$-th column, if its orientation $c_j$ is from top to bottom then $c_j=1$ otherwise $c_j=-1$. Assume that an orientation $\mathcal{R}=(r_1,\dots,r_m)$
and $\mathcal{C}=(c_1,\dots,c_n)$ is fixed. Given an initial filled cell $(i_1,j_1)$ consider the sequence $ L_{\mathcal{R},\mathcal{C}}(i_1,j_1)=((i_1,j_1),(i_2,j_2),\ldots,(i_\ell,j_\ell),$ $(i_{\ell+1},j_{\ell+1}),\ldots)$ where $j_{\ell+1}$ is the column index of the filled cell $(i_\ell,j_{\ell+1})$ of the row $R_{i_\ell}$ next to $(i_\ell,j_\ell)$ in the orientation $r_{i_\ell}$, and where $i_{\ell+1}$ is the row index of the filled cell of the column $C_{j_{\ell+1}}$ next to $(i_\ell,j_{\ell+1})$ in the orientation $c_{j_{\ell+1}}$. The problem is the following:\\
\textbf{Crazy Knight's Tour Problem.}{\rm \cite{CDP}} Given a toroidal partially filled array $A$, do there exist $\mathcal{R}$ and $\mathcal{C}$ such that the list $L_{\mathcal{R},\mathcal{C}}$ covers all the filled cells of $A$?\\
The Crazy Knight's Tour Problem for a given array $A$ is denoted by $P(A)$. Also, given a filled cell $(i,j)$, if $L_{\mathcal{R},\mathcal{C}}(i,j)$ covers all the filled positions of $A$, then
$(\mathcal{R},\mathcal{C})$ is said to be a solution of $P(A)$.
\begin{example}\label{ex0} Let $A$ be the following array, where the bullets ``$\bullet$'' denote the filled cells:
$$\begin{array}{|c|c|c|c|}\hline
\bullet & & & \bullet \\\hline
& \bullet & \bullet & \\\hline
& \bullet & \bullet & \bullet \\\hline
\bullet & & \bullet & \\\hline
\end{array}$$
Choosing $\mathcal{R}:=(-1,1,1,-1)$ and $\mathcal{C}:=(1,-1,1,1)$ we can cover all the filled positions of $A$, as shown in the table below where
in each filled cell we write $j$ if this position is the $j$-th element of the list $L_{\mathcal{R},\mathcal{C}}(1,1)$. The elements of $\mathcal{R}$ and $\mathcal{C}$ are represented by an arrow.
$$\begin{array}{r|r|r|r|r|}
& \downarrow & \uparrow & \downarrow & \downarrow \\\hline \leftarrow & 1 & & & 5 \\\hline \rightarrow & & 3 & 7 & \\\hline \rightarrow & & 8 & 4 & 2 \\\hline \leftarrow & 6 & & 9 & \\\hline
\end{array}$$ \end{example}
\begin{remark}
If $A$ is an $H(m, n; h, k)$ such that $P(A)$ admits a solution $(\mathcal{R}, \mathcal{C})$, then $A$ admits two compatible orderings $\omega_r$ and $\omega_c$ that can be determined as follows. For each row $R$ (column $C$, respectively) of $A$, let $\omega_R$ be the natural ordering if $r_i = 1$ ($c_i = 1$, respectively) and its inverse otherwise. Setting $\omega_r = \omega_{R_1} \circ \ldots \circ \omega_{R_m}$ and $\omega_c = \omega_{C_1} \circ \ldots \circ \omega_{C_n}$, since $(\mathcal{R}, \mathcal{C})$ is a solution of $P(A)$, then $\omega_r$ and $\omega_c$ are compatible. \end{remark}
The relationship between globally simple Heffter arrays, Crazy Knight's Tour Problem and biembeddings is explained in the following result. It is a reformulation of Theorem \ref{Gustin} in the case of square globally simple Heffter arrays in terms of $P(A)$. Clearly, a similar theorem holds in the rectangular case. \begin{theorem} Let $A$ be a globally simple Heffter array $H(n;k).$ If there exists a solution of $P(A)$, then there exists a biembedding of
$K_{2nk + 1}$ on an orientable surface whose face boundaries are $k$-cycles. \end{theorem}
The existence of integer globally simple Heffter arrays $H(n;k)$ with compatible orderings for $n\equiv 1\pmod 4$ and $k\equiv 3 \pmod 4$ has been proved (with infinite sporadic exceptions) in \cite{CDY}.
\begin{theorem}\label{HeffterBiembeddingsCavenagh}{\rm \cite{CDY}}
Let $n \equiv 1\pmod 4$, $t > 0$ and $n > 4t + 3$. If there exists $\alpha$ such that
$2t + 2\leq \alpha \leq n-2-2t$, $\gcd(n, \alpha) = 1$, $\gcd(n, \alpha-2t-1) = 1$ and $\gcd(n, n-1-\alpha-2t) = 1$, then there exists a globally simple integer Heffter array $H (n; 4t + 3)$ with row and column orderings that are both simple and compatible. \end{theorem}
\begin{theorem}{\rm \cite{CDY}}
Let $n \equiv 1\pmod 4$, $n > k\equiv 3 \pmod 4$ and either: $(a)$ $n$ is prime; $(b)$ $n = k + 2$; or $(c)$ $n\geq 7(k + 1)/3$ and if $n \equiv 3 \pmod 6$ then $k \equiv 7\pmod {12}$. Then there exists a globally simple integer Heffter array $H (n; k)$ with an ordering that is both simple and compatible. Furthermore [by Theorem \ref{biembed2}], there exists a face $2$-colorable embedding $\mathcal{G}$ of $K_{2nk+1}$ on an orientable surface such that the faces of each color are cycles of length $k$. Moreover, $\mathbb{Z}_{2nk+1}$ has a sharply vertex-transitive action on $\mathcal{G}$. \end{theorem}
\begin{definition}Two Heffter arrays $H$ and $H'$ are said to be {\em equivalent} if one can be obtained from the other by (i) rearranging rows or columns; (ii) replacing every entry $x$ in $H$ with $-x$; and/or (iii) taking the transpose. \end{definition}
The following theorem gives a lower bound on the number of inequivalent Heffter arrays $H(n; 4t + 3)$ that satisfy Theorem \ref{HeffterBiembeddingsCavenagh}. Let $\mathcal{H}(n)$ represent the number of derangements on $\{0,1,\ldots,n-1\}$. \begin{theorem}{\rm \cite{CDY}}\label{BoundCavenagh}
Let $n \equiv 1\pmod 4$, $t \geq 2$ and $n > 4t + 3$. If there exists $\alpha$ such that $2t + 2\leq\alpha \leq n-2-2t$, $\gcd(n, \alpha) = 1$, $\gcd(n, \alpha-2t-1) = 1$ and $\gcd(n, n-1-\alpha-2t) = 1$, then there exist at least $(n -2)(\mathcal{H}(t-2))^2\simeq (n-2)[(t-2)!/e]^2$ inequivalent globally simple integer Heffter arrays $H(n; 4t + 3)$, each with an ordering that is both simple and compatible. \end{theorem}
The other case, that is $n\equiv 3\pmod 4$ and $k\equiv 1 \pmod 4$, remains unsolved in general. In the same paper the authors also investigated the number of biembeddings of $K_v$ obtained from certain classes of Heffter arrays.
The following result is a corollary of Theorem \ref{BoundCavenagh}.
\begin{theorem}{\rm \cite{CDY}}
Let $n \equiv 1\pmod 4$ and $k \equiv 3\pmod 4$, $k \geq 11$ and either $n$ is prime or $n \geq (7k + 1)/3$. Further if $n \equiv 3\pmod 6$ assume $k \equiv 7\pmod {12}$. Then there exists at least $(n-2)[((k-11)/4)!/e]^2$ distinct face $2$-colorable embeddings of the complete graph $K_{2nk+1}$ onto an orientable surface where each face is a cycle of fixed length $k$, and the vertices can be labeled with the elements of $\mathbb{Z}_{2nk+1}$ in such a way that this group $(\mathbb{Z}_{2nk+1}, +)$ has a sharply vertex-transitive action on the embedding. \end{theorem}
Starting from the results of \cite{CDY}, in \cite{CPArxiv} the authors consider the number of non-isomorphic biembeddings of $K_v$ obtained from certain classes of Heffter arrays. Set the binary entropy function by $H(p):=-p\log_2{p}-(1-p)\log_2(1-p)$.
\begin{theorem}\label{GeneralBound}{\rm \cite{CPArxiv}} Let $v=2nk+1$, $k=4t+3$ and let $n\equiv 1 \pmod{4}$ be either a prime or $n\geq (7k+1)/3$. Moreover, if $n\equiv 3\pmod{6}$ assume $k\equiv 7\pmod{12}$. Then, the number of non-isomorphic face $2$-colorable embeddings of the complete graph $K_{2nk+1}$ onto an orientable surface where each face is a cycle of fixed length $k$
is at least $$\frac{(n-2)[\mathcal{H}(t-2)]^2}{2(2nk)^2}\approx \frac{\pi(t-2)^{2t-5}}{64e^{2t-2}n}\approx k^{k/2+o(k)}/v.$$ \end{theorem}
\begin{theorem}{\rm \cite{CPArxiv}}
Let $k\geq 3$ be odd, $n\geq 4k-3$ and $nk\equiv 3 \pmod{4}$. Assume also that $\gcd(n,k-1)=1$. Then, set $v=2nk+1$, the number of non-isomorphic face $2$-colorable embeddings of the complete graph $K_{2nk+1}$ onto an orientable surface where each face has fixed length $k$
is at least: $$ \frac{\binom{\lceil n/(k-1)\rceil}{ \lceil n/(4k-4)\rceil}}{(nk)^2}\gtrsim \frac{\sqrt{\frac{ 2(k-1)}{3n\pi}}2^{\frac{n}{k-1}\cdot H(1/4)+1}}{(nk)^2}\approx 2^{v\cdot \frac{H(1/4)}{2k(k-1)}+o(v,k)}.$$ Moreover, if: \begin{itemize} \item[$(\rm{1})$ ] $k=3$ the number of \emph{simple} non-isomorphic face $2$-colorable embeddings of $K_{6n+1}$ is at least $\frac{2^{n/2}}{9n^2} \approx 2^{v/12+o(v)}$;\\ \item[$(\rm{2})$ ] $k=5,7,9$ the number of \emph{simple} non-isomorphic face $2$-colorable embeddings of $K_{2nk+1}$ is at least $ \frac{\binom{\lfloor n/(k-1)\rfloor}{ \lfloor n/(4k-4)\rfloor}}{(nk)^2}\gtrsim \frac{\sqrt{\frac{ 2(k-1)}{3n\pi}}2^{\lfloor\frac{n}{k-1}\rfloor\cdot H(1/4)+1}}{(nk)^2}\approx 2^{v\cdot \frac{H(1/4)}{2k(k-1)}+o(v,k)}.$ \end{itemize} \end{theorem}
One additional embedding result follows directly by Theorem \ref{thm:SH}. \begin{theorem}{\rm \cite{CMPPHeffter}} There exists a biembedding of the complete graph of order $2nk + 1$ and one of the cocktail graph of order $2nk +2$ on orientable surfaces such that every face is a $k$-cycle, whenever $k \in \{3, 5, 7, 9\}$, $nk \equiv 3 \pmod 4$ and $n > k$. \end{theorem}
\section{Sequencing Subsets of a Group}\label{sec:sums}
We know from Theorem \ref{Gustin} that if there is a simple Heffter array $H(m,n;h,k)$ (with compatible orderings), then there is a biembedding of $K_{2nk+1}$
on an orientable surface such that every edge is on a face of size $h$ and a face of size $k$ and all faces are simple cycles. It is thus a natural question to ask just when the rows and columns of a Heffter array have a simple ordering, or more generally, when subsets of a group have a simple ordering. Indeed, if this is always the case, then Heffter arrays would always be simple and the condition for simplicity of the array would be unnecessary. It has indeed been an active area of research to show this, but at this time it is still not known for all subsets (and all groups). In this section we give some conjectures relating to the sequencing of subsets of groups as well as some of the results in this area.
We begin with the more classical notion of a sequenceable group. Let $G$ be a group of order $n$ (written additively). Given an ordering $\omega=(0,g_1,g_2,\ldots,g_{n-1})$ of the elements in $G$, let $s_i=\sum_{j=1}^i g_j$ be the $i$-th partial sum of $\omega$. The group $G$ is {\em sequenceable} if there exists some ordering of the elements of $G$ where all of these partial sums $s_0=0,s_1,\ldots,s_{n-1}$ are distinct (equivalently, if $G$ admits a simple ordering), while it is {\em R-sequenceable} if $s_0=0,s_1,\ldots,s_{n-2}$ are distinct and in addition $s_{n-1} = 0$. The concept of sequenceable group was introduced in 1961 by Gordon \cite{G}, even if similar ideas for cyclic groups were already presented in 1892 by Lucas \cite{L}. Alspach et al. \cite{AKP} proved that any finite abelian group is either sequenceable or R-sequenceable, confirming the Friedlander-Gordon-Miller conjecture.
Here is a short summary of the results concerning the sequencing and R-sequencing of groups. For an extensive survey on this topic the reader can refer to \cite{Osurvey}.
\begin{proposition}Results on sequencing of groups.
\begin{enumerate}
\item[$(\rm{1})$ ] An abelian group is sequenceable if and only if it has a unique element of order $2$.
\item[$(\rm{2})$ ] No non-abelian group of order less than $10$ is sequenceable. But Keedwell conjectures that all non-abelian groups of order greater than $10$ are sequenceable.
\item[$(\rm{3})$ ] Keedwell's conjecture has been proved true for the following classes of groups: \begin{enumerate} \item[$(\rm{a})$ ] All non-abelian groups of order $n$, $10\le n\le 32$; \item[$(\rm{b})$ ] The dihedral groups $D_n$; \item[$(\rm{c})$ ] $A_5$ and $S_5$; \item[$(\rm{d})$ ] Solvable groups with a unique element of order $2$; \item[$(\rm{e})$ ] Some non-solvable groups with a unique element of order $2$; \item[$(\rm{f})$ ] Some groups of order $pq$, $p$ and $q$ odd primes. \end{enumerate} \end{enumerate}
\end{proposition}
\begin{proposition} Results on R-sequencing of groups.
\begin{enumerate} \item[$(\rm{1})$ ] If $G$ is an R-sequenceable group, then its Sylow $2$-subgroup is either trivial or non-cyclic.
\item[$(\rm{2})$ ] The following groups are R-sequenceable: \begin{enumerate} \item[$(\rm{a})$ ] $\mathbb{Z}_n$, $n$ odd; \item[$(\rm{b})$ ] Abelian groups of order $n$, $gcd(n,6)=1$; \item[$(\rm{c})$ ] $D_n$, $n$ even; \item[$(\rm{d})$ ] $Q_{2n}$ if $n+1$ is a prime of the form $4k+1$, for which $-2$ is a primitive root; \item[$(\rm{e})$ ] Non-abelian groups of order $pq$, $p<q$ odd primes, with $2$ a primitive root of $p$. \end{enumerate} \end{enumerate} \end{proposition}
The problem of ordering the elements of a {\em subset} of a given group in such a way that all the partial sums are pairwise distinct is more recent than the sequenceable group question. Nonetheless it has also been extensively studied, most recently because of its connection to Heffter arrays, but not only for this reason. Indeed there are several open conjectures on partial sums of a finite group.
For example, several years ago Alspach made
the following conjecture, whose validity would shorten some cases of known proofs about the existence of cycle decompositions.
\begin{conjecture}\label{Conj:als} Let $T\subseteq \mathbb{Z}_v\setminus\{0\}$ such that $\sum_{t\in T}t\neq0$. Then there exists an ordering of the elements of $T$ such that the partial sums are all distinct and non-zero. \end{conjecture}
The first published paper on this problem is by Bode and Harborth \cite{BH} in which the authors proved that Conjecture \ref{Conj:als}
is valid if $|T|=v-1$ or $|T|=v-2$ or if $v\leq 16$ (the latter was obtained by computer verification). Recently, several other papers on this topic have been published, see \cite{AL,CDF,CDFORF,CP,HOS,O}.
Another conjecture, very close to the Alspach's conjecture, was originally proposed in 1971 by Graham \cite{Gr71} for cyclic groups of prime order and then extended in 2015 by
Archdeacon et al. \cite{ADMS} to every cyclic group as follows. \begin{conjecture}\label{Conj:ADMS}{\rm \cite{ADMS}} Let $T\subseteq \mathbb{Z}_v\setminus\{0\}$. Then $T$ admits a simple ordering. \end{conjecture}
Clearly Conjecture \ref{Conj:ADMS} implies Conjecture \ref{Conj:als}. In \cite{ADMS} the authors proved that Conjecture \ref{Conj:als} implies Conjecture \ref{Conj:ADMS}. Then they proved that Conjecture \ref{Conj:ADMS} is valid if $|T|\leq 6$ and (by computer verification) for $v\leq 25$.
In \cite{CMPPSums}, Costa et al. proposed the following conjecture on partial sums which was also motivated by the study of Heffter arrays.
\begin{conjecture}\label{Conj:nostra}{\rm \cite{CMPPSums}} Let $(G,+)$ be an abelian group. Let $T$ be a finite subset of $G\setminus\{0\}$ such that no $2$-subset $\{x,-x\}$ is contained in $T$ and with the property that $\sum_{t\in T} t=0$. Then $T$ admits a simple ordering. \end{conjecture}
If $G=\mathbb{Z}_v$, then every row and column of a Heffter array can be viewed as a subset $T$ considered in Conjecture \ref{Conj:nostra}. Hence if this conjecture were true for cyclic groups, then every Heffter array would be simple. In \cite{CMPPSums} the validity of Conjecture \ref{Conj:nostra} is proved theoretically for subsets $T$ of size less than $10$
and, with the aid of a computer, for every abelian group $G$ with $|G| \leq 27$.
Clearly if $G=\mathbb{Z}_v$, Conjecture \ref{Conj:nostra} immediately follows from Conjecture \ref{Conj:ADMS}. Indeed in \cite{CMPPSums} the authors extended Conjecture \ref{Conj:ADMS} to every abelian group and proved by computer that this extended conjecture is valid for every abelian group of order at most $23$. Also, they pointed out that in \cite{ADMS}, when proving the conjecture true for $|T|\leq 6$ the authors do not use the hypothesis that $T$ is a subset of a cyclic group. In fact, these proofs for $|T|\leq 6$ work in general for an arbitrary abelian group.
Some results are known about Conjectures \ref{Conj:als} and \ref{Conj:ADMS} on an arbitrary group. For instance the validity of Conjecture \ref{Conj:als} has been verified by computer for \emph{all abelian groups} of order at most $21$ in \cite{CMPPSums}. It has also been verified in \cite{AL} for any subset $T$ of size at most $9$ of an \emph{arbitrary abelian group}. On the other hand, Conjecture \ref{Conj:als} cannot be generalized to nonabelian groups as shown in \cite{An} by Anderson. Also, if we consider Conjectures \ref{Conj:als} and \ref{Conj:ADMS} in the case of abelian groups then, again, Conjecture \ref{Conj:als} implies Conjecture \ref{Conj:ADMS}. Hence it follows that Conjecture \ref{Conj:ADMS} holds, in the abelian case, for any subset $T$ of size at most $9$ and when $\sum_{t\in T}t=0$ for $T$ of size at most $10$, for details see \cite{ADMS}. This implies that also Conjecture \ref{Conj:nostra} holds for
subsets of size at most $10$. For Conjecture \ref{Conj:ADMS} it is also natural to investigate the nonabelian case. In \cite{CMPPSums} this conjecture is proved theoretically for subsets $T$ of size at most $5$ of \emph{every arbitrary group} and a computer verification is made for \emph{all groups} of order not exceeding $19$.
Finally, we recall that a subset of an (additive) abelian group is said to be \emph{sequenceable} if there is an ordering $(t_1,\ldots,t_k)$ of its elements such that the partial sums $(s_0,s_1,\ldots,s_k)$, where $s_0 = 0$, are distinct, with the possible exception that we may have $s_k=s_0=0$. The following theorem summarizes what is known about sequencing subsets of the cyclic group. We should note that most of the recent progress on this problem has been made by use of the {\em polynomial method}. This method relies on the non-vanishing corollary to the Combinatorial Nullstellensatz (see \cite{Alon,CDF,CDFORF,HOS,O}).
\begin{theorem}
Let $T \subseteq \mathbb{Z}_v\setminus \{0\}$ with $|T|=k$. Then $T$ is sequenceable in the following cases: \begin{itemize} \item[$(\rm{1})$ ] $k\leq 9$\ {\rm \cite{AL}}; \item[$(\rm{2})$ ] $k \leq 12$ when $v=pq$ where $p$ is prime and $q\leq 4$ \ {\rm \cite{CDFORF,HOS}}; \item[$(\rm{3})$ ] $k \leq 12$ when $v=mq$ where all the prime factors of $m$ are bigger than $\frac{k!}{2}$ and $q \leq 4$ \ {\rm \cite{CDFORF}}; \item[$(\rm{4})$ ] $k=v-3$ when $v$ is prime and $\sum_{t \in T}t=0$\ {\rm \cite{HOS}}; \item[$(\rm{5})$ ] $k=v-2$ when $\sum_{t \in T}t\neq 0$ \ {\rm \cite{BH}}; \item[$(\rm{6})$ ] $k=v-1$ \ {\rm \cite{AKP,G}}; \item[$(\rm{7})$ ] $v\leq 21$ and $v\leq 25$ when $\sum_{t \in T}t=0$ \ \rm{ \cite{ADMS,CMPPSums}}. \end{itemize} \end{theorem}
\section{Connections with difference families and graph decompositions}\label{sec:DF}
In this section we discuss the connection between Heffter arrays and cycle decompositions of the complete graph. We have already discussed the connection between Heffter arrays and biembeddings of two cycle systems (cycle decompositions). In this section we will explicitly show how one can produce the two cycle decompositions $\mathcal{D}$ and $\mathcal{D'}$ and that these cycle decompositions have the property that if two edges are in the same cycle of $\mathcal{D}$, then they will be in different cycles in $\mathcal{D}'$.
First of all we recall some basic definitions about graphs and graph decompositions. Good references about these concepts are \cite{W} and \cite{BEZ}, respectively. Given a graph $\Gamma$, by $V(\Gamma)$ and $E(\Gamma)$ we mean the vertex set and the edge set of $\Gamma$, respectively, and by $^\lambda \Gamma$ the multigraph obtained from $\Gamma$ by repeating each edge $\lambda$ times. We denote by $K_{q \times r}$ the complete multipartite graph with $q$ parts each of size $r$, obviously $K_{q \times 1}$ is nothing but the complete graph $K_q$. Finally, by $P_k$ we mean the path of length $k$.
Let $\Gamma$ be a subgraph of a graph $K$. A $\Gamma$-\emph{decomposition} of $K$ is a set $\mathcal{D}$ of subgraphs of $K$ isomorphic to $\Gamma$, called \emph{blocks}, whose edges partition $E(K)$. As already remarked in the previous section, a $C_k$-decomposition of $K_v$ is also called a $k$-\emph{cycle system of order} $v$.
\begin{definition} Two graph decompositions $\mathcal{D}$ and $\mathcal{D}'$ of the same graph $K$ are \emph{orthogonal} if for every $B \in \mathcal{D}$ and every $B' \in \mathcal{D}'$, $B$ intersects $B'$ in at most one edge. \end{definition}
We note the following obvious fact connecting the notions of biembedded cycle systems to orthogonal cycle systems. From this fact, many of our earlier results on biembeddings of cycle systems give results on orthogonal cycle systems.
\begin{proposition}\label{biembed.implies.orthog}
If there is a simple biembedding of a cycle system $ \mathcal{D}$ and a cycle system $ \mathcal{D}'$ in $K_v$, then $ \mathcal{D}$ and $ \mathcal{D'}$ are orthogonal cycle systems of order $v$. \end{proposition}
If the vertices of $\Gamma$ belong to a group $G$, given $g \in G$, by $\Gamma +g$ one means the graph whose vertex set is $V(\Gamma)+g$ and whose edge set is $\{\{x+g,y+g\}\mid \{x,y\}\in E(\Gamma)\}$. An \emph{automorphism group} of a $\Gamma$-decomposition $\mathcal{D}$ of $K$ is a group of bijections on $V(K)$ leaving $\mathcal{D}$ invariant. A $\Gamma$-decomposition of $K$ is said to be \emph{regular under a group} $G$ or $G$-\emph{regular} if it admits $G$ as an automorphism group acting sharply transitively on $V(K)$. Heffter arrays are related to \emph{cyclic} decompositions, namely decompositions regular under a cyclic group. We recall the following result.
\begin{proposition} Let $K$ be a graph with $v$ vertices. A $\Gamma$-decomposition $\mathcal{D}$ of $K$ is cyclic if and only if, up to isomorphisms, the following conditions hold: \begin{itemize} \item[$(\rm{1})$ ] the set of vertices of $K$ is $\mathbb{Z}_v$; \item[$(\rm{2})$ ] for all $B\in \mathcal{D}$, $B+1\in \mathcal{D}$. \end{itemize} \end{proposition}
Clearly, to describe a cyclic decomposition it is sufficient to exhibit a complete system $\mathcal{B}$ of representatives for the orbits of $\mathcal{D}$ under the action of $\mathbb{Z}_v$. The elements of $\mathcal{B}$ are called \emph{base blocks} of $\mathcal{D}$.
One of most efficient tools applied for constructing regular decompositions is the \emph{difference method}, see \cite{AB}. In particular, difference families over graphs have been introduced by Buratti \cite{B}, see also \cite{BP}. Given a graph $\Gamma$ with vertices in an additive group $G$, the \emph{list of differences} of $\Gamma$ is the multiset $\Delta\Gamma$ of all possible differences $x-y$ between two adjacent vertices of $\Gamma$. So, denoting by $D(\Gamma)$ the set of \emph{di-edges} of $\Gamma$, i.e., the set of all \underline{ordered} pairs $(x,y)$ with $\{x,y\} \in E(\Gamma)$, one can write: $$\Delta \Gamma = \{x-y \mid (x,y)\in D(\Gamma)\}.$$
More generally, if $\mathcal{F}=\{\Gamma_1,\Gamma_2,\ldots,\Gamma_{\ell}\}$ is a collection of graphs with vertices in $G$, one sets $\Delta \mathcal{F}=\Delta \Gamma_1 \cup \Delta \Gamma_2 \cup \ldots \cup \Delta \Gamma_{\ell}$, where in the union each element has to be counted with its multiplicity.
\begin{definition} Let $J$ be a subgroup of a group $G$ and let $\Gamma$ be a graph. A collection $\mathcal{F}$ of graphs isomorphic to $\Gamma$ and with vertices in $G$ is said to be a $(G,J,\Gamma,\lambda)$-\emph{difference family} (briefly DF) \emph{over $G$ relative to $J$} if each element of $G\setminus J$ appears exactly $\lambda$ times in $\Delta \mathcal{F}$ while no element of $J$ appears there. \end{definition}
In the case of $J=\{0\}$ one simply says that $\mathcal{F}$ is a $(G,\Gamma,\lambda)$-DF. Also, if $t$ is a divisor of $v$, by writing $(v,t,\Gamma,\lambda)$-DF one means a $(\mathbb{Z}_v,\frac{v}{t}\mathbb{Z}_v,\Gamma,\lambda)$-DF where $\frac{v}{t}\mathbb{Z}_v$ denotes the subgroup of $\mathbb{Z}_v$ of order $t$. Analogously, $(v,\Gamma,\lambda)$-DF means $(\mathbb{Z}_v,\Gamma,\lambda)$-DF. The connection between relative difference families and decompositions of a complete multipartite multigraph is given by the following result.
\begin{proposition}{\rm \cite{BP}}\label{DF-GD}
If $\mathcal{F}=\{\Gamma_1,\Gamma_2,\ldots,\Gamma_{\ell}\}$ is a $(G,J,\Gamma,\lambda)$-DF, then $\mathcal{D}=\{\Gamma_i+g \mid i=1,\ldots, \ell; g \in G\}$
is a $G$-regular $\Gamma$-decomposition of $^\lambda K_{q \times r}$, where $q=|G:J|$ and $r=|J|$. In particular, if $\mathcal{F}$ is $(v,\Gamma,1)$-DF then $\mathcal{D}$ is a cyclic $\Gamma$-decomposition of $K_{v}$. \end{proposition}
Now we show how starting from a simple Heffter array one can construct a pair of orthogonal cyclic cycle decompositions of the complete graph $K_v$ or of the cocktail party graph $K_{2\ell}-I$, namely the complete graph $K_{2\ell}$ minus the $1$-factor $I$ whose edges are $[0,\ell],[1,\ell+1],[2,\ell+2],\ldots,[\ell-1,2\ell-1]$. It is well known that the problem of finding necessary and sufficient conditions for cyclic cycle decompositions of $K_v$ and $K_{2\ell}-I$ has attracted much attention (see, for instance, \cite{BDF,V,WF} and \cite{BGL,BR,JM2008,JM2017}, respectively).
First of all, from a simple Heffter array one can construct two cyclic difference families whose blocks are cycles, as follows. Let $H$ be a simple $H(m,n;h,k)$, hence every row and every column of $H$ admits a simple ordering. Starting from a row of $H$ one can construct a cycle of length $h$ whose vertices are the partial sums of the simple ordering of such a row. Denoted by $\Gamma_{i}$ the cycle obtained from the $i$-th row $R_i$ of $H$, it is immediate to see that $\Delta\Gamma_i =\pm R_i$. Now, let $\mathcal{F}_R$ be the set of the $m$ cycles of length $h$ constructed from the $m$ rows of $H$ with this procedure. By Condition (\rm{b}) of Definition \ref{def:Heffter}, we have $\Delta{\mathcal{F}_R}=\mathbb{Z}_{2mh+1}\setminus\{0\}$. Hence $\mathcal{F}_R$ is a $(2mh+1, C_h,1)$-DF. Clearly, with the same technique, one can construct $n$ cycles of length $k$ starting from the columns of $H$. Let $\Gamma'_j$ be the cycle obtained from the $j$-th column and let $\mathcal{F}_C$ be the set of these cycles. Again by Condition (\rm{b}) of Definition \ref{def:Heffter}, we have $\Delta{\mathcal{F}_C}=\mathbb{Z}_{2nk+1}\setminus\{0\}$. Hence $\mathcal{F}_C$ is a $(2nk+1, C_k,1)$-DF.
\begin{remark}\label{rem:ortho}
For every $\Gamma_i \in \mathcal{F}_R$ and every $\Gamma'_j \in \mathcal{F}_C$, we see that $\Gamma_i$ and $\Gamma'_j$ contain at most one edge in common and hence $|\Delta \Gamma_i \cap \Delta \Gamma'_j|\in\{0,2\}$. In particular, the intersection is empty if the cell on the $i$-th row and the $j$-th column of $H$ is empty, while the intersection has size two otherwise. \end{remark}
By Proposition \ref{DF-GD}, the cycles of $\mathcal{F}_R$ are the base blocks of a cyclic $C_h$-decomposition $\mathcal{D}_R$ of $K_{2mh+1}$, while the cycles of $\mathcal{F}_C$ are the base blocks of a cyclic $C_k$-decomposition $\mathcal{D}_C$ of $K_{2nk+1}$. By Remark \ref{rem:ortho}, the
decompositions $\mathcal{D}_R$ and $\mathcal{D}_C$ are orthogonal.
\begin{example} Let $H$ be the $H(8;7)$ of {\rm Example \ref{ex:H87}} and let $\omega_i$'s and $\nu_i$'s be
the simple orderings given in {\rm Example \ref{ex:simple}}. Considering the partial sums of the $\omega_i$'s ($\nu_i$'s, respectively) in $\mathbb{Z}_{113}$ we obtain the cycles $\Gamma_i$'s ($\Gamma'_i$'s, respectively): \begin{center} \begin{footnotesize} $\begin{array}{lcl} \Gamma_1= ( 8, 33, 49, 22, -7, 24, 0); & \quad &\Gamma'_1= (8,47,30,-8,-51,-9,0);\\ \Gamma_2= (-17, -23, -51, -28, -2, 30, 0); & \quad &\Gamma'_2= ( 16,10,-35,-45,3,52,0 );\\ \Gamma_3= (39, 29, 24, 57, 72, 37, 0); & \quad &\Gamma'_3= ( 23,18,65,47,1,-50,0);\\ \Gamma_4= (-38, -56, -49, -85, -74, -40, 0); & \quad &\Gamma'_4= (25,-3,12,19,-34,-56,0);\\ \Gamma_5= (-43, -88, -41, -63, -60, -19, 0); & \quad &\Gamma'_5= (-27,-1,10,13,68,54,0);\\ \Gamma_6= ( 42, 90, 44, 30, 32, -12, 0); & \quad &\Gamma'_6= ( -21,11,44,8,27,29,0);\\ \Gamma_7= (20, -31, -84, -29, -50, -49, 0);& \quad &\Gamma'_7= (-13,-43,-78,-44,-32,-31,0);\\ \Gamma_8= (-52, -43, 7, 63, 9, -4, 0); & \quad &\Gamma'_8= (-37,-61,-21,20,-24,-4,0). \end{array}$ \end{footnotesize}
\end{center} Then $\mathcal{F}_R=\{\Gamma_1,\ldots,\Gamma_8\}$ and $\mathcal{F}_C=\{\Gamma'_1,\ldots,\Gamma'_8\}$ are two sets of base cycles of a pair of orthogonal cyclic $C_7$-decompositions of $K_{113}$. \end{example}
From the arguments in this section we get the following results. Note the similarity of Proposition \ref{orth.systems} to Theorem \ref{biembed2}, however we see that for the application of Heffter arrays to orthogonal cycle decompositions it is not required that the orderings of the rows and columns of the Heffter array be compatible. For results on orthogonal cycle decompositions see \cite{BCP} and the references therein.
\begin{proposition} If there exists a simple $H(m,n;h,k)$, then there exist a $(2mh+1,C_h,1)$-DF and a $(2nk+1, C_k,1)$-DF. \end{proposition}
\begin{proposition}\label{orth.systems} If there exists a simple $H(m,n;h,k)$, then there exist
a cyclic $C_h$-decomposition and a cyclic $C_k$-decomposition both of $K_{2nk+1}$, and these decompositions are orthogonal. \end{proposition}
Obviously if the array is globally simple one can immediately construct the cycles starting from the rows and from the columns, without providing the simple orderings. Also note that the elements of an $H(m,n;h,k)$ belong to $\mathbb{Z}_{2nk+1}$, hence when we say that an ordering of a row or of a column is simple, we mean that its partial sums are pairwise distinct modulo $2nk+1$. In \cite{CMPPHeffter}, the authors showed that it is useful to find orderings whose partial sums are pairwise distinct both modulo $2nk+1$ and modulo $2nk+2$. To this end they give the following definition.
\begin{definition}An $SH^*(m,n;h,k)$ is a globally simple $H(m,n;h,k)$ such that the natural ordering of each row and each column is simple also modulo $2nk+2$. \end{definition} The arrays of Theorem \ref{thm:SH} have this additional property. The usefulness of these arrays is explained by the following proposition.
\begin{proposition}\label{pr:ccp}{\rm \cite{CMPPHeffter}}
If there exists an $SH^*(m,n;h,k)$, then there exist: \begin{itemize} \item[{\rm (1)}] a cyclic $C_h$-decomposition and a cyclic $C_k$-decomposition both of $K_{2nk+1}$ and these decompositions are orthogonal; \item[{\rm (2)}] a cyclic $C_h$-decomposition and a cyclic $C_k$-decomposition both of $K_{2nk+2}-I$ and these decompositions are orthogonal. \end{itemize} \end{proposition}
\begin{example}\label{10.8} The following is an $SH^*(10;8)$ over $\mathbb{Z}_{161}$: \begin{center} \begin{footnotesize}
$\begin{array}{|r|r|r|r|r|r|r|r|r|r|}\hline 77 & 80 & -78 & -71 & -70 & -79 & & & 69 & 72 \\\hline
& & -17 & -20 & -25 & -28 & 26 & 19 & 18 & 27 \\\hline 5 & 8 & 13 & 16 & -14 & -7 & -6 & -15 & & \\\hline 34 & 43 & & & -33 & -36 & -41 & -44 & 42 & 35 \\\hline
& & 21 & 24 & 29 & 32 & -30 & -23 & -22 & -31 \\\hline 58 & 51 & 50 & 59 & & & -49 & -52 & -57 & -60 \\\hline -38 & -47 & & & 37 & 40 & 45 & 48 & -46 & -39 \\\hline -73 & -76 & 74 & 67 & 66 & 75 & & & -65 & -68 \\\hline -62 & -55 & -54 & -63 & & & 53 & 56 & 61 & 64 \\\hline -1 & -4 & -9 & -12 & 10 & 3 & 2 & 11 & & \\\hline \end{array}$ \end{footnotesize} \end{center} By the partial sums in $\mathbb{Z}_{161}$ of the natural orderings of the rows (columns, respectively) we obtain the cycles $\Gamma_i$'s ($\Gamma'_i$'s, respectively): \begin{center} \begin{footnotesize} $\begin{array}{lcl} \Gamma_1= ( 77, 157, 79, 8, -62, -141, -72, 0 ); & \;\; &\Gamma'_1= (77, 82, 116, 13, 136, 63, 1, 0) ;\\ \Gamma_2= (-17, -37, -62, -90, -64, -45, -27, 0 ); & \;\; &\Gamma'_2= (80, 88,131,21, 135, 59, 4, 0 );\\ \Gamma_3= (5, 13, 26, 42, 28, 21, 15, 0 ); & \;\; &\Gamma'_3= ( -78, -95, -82, -61, -11, 63, 9, 0 ); \\ \Gamma_4= (34, 77, 44, 8, -33, -77, -35, 0); & \;\; &\Gamma'_4= (-71, -91, -75, -51, 8, 75, 12, 0);\\ \Gamma_5= (21, 45, 74, 106, 76, 53, 31,0); & \;\;&\Gamma'_5= (-70, -95, -109, -142, -113, -76, -10, 0 );\\ \Gamma_6=(58, 109, 159, 57, 8, 117, 60, 0); & \;\; &\Gamma'_6= (-79, -107, -114, -150, -118, -78, -3, 0 );\\ \Gamma_7= (-38, -85, -48, -8, 37, 85, 39, 0 );& \;\; &\Gamma'_7= ( 26, 20, -21, -51, -100, -55, -2, 0 );\\ \Gamma_8= ( -73, -149, -75, -8, 58, 133, 68, 0 ); & \;\; &\Gamma'_8= (19, 4, -40, -63, -115, -67, -11, 0 );\\ \Gamma_9= (-62, -117, -10, -73, -20, -125, -64, 0 ); &\;\;&\Gamma'_9=( 69, 87, 129, 107, 50, 4, -61, 0 );\\ \Gamma_{10}=(-1, -5, -14, -26, -16, -13, -11, 0); & \;\; &\Gamma'_{10}=(72, 99, 134, 103, 43, 4, -64, 0 ). \end{array}$ \end{footnotesize} \end{center} Then $\mathcal{F}_R=\{\Gamma_1,\ldots,\Gamma_{10}\}$ and $\mathcal{F}_C=\{\Gamma'_1,\ldots,\Gamma'_{10}\}$ are two sets of base cycles of a pair of orthogonal cyclic $C_8$-decompositions of $K_{161}$.\\ Analogously, if we consider the partial sums of each row (column, respectively) in $\mathbb{Z}_{162}$, we obtain the cycles $\widetilde{\Gamma}_i$'s ($\widetilde{\Gamma}'_j$'s, respectively) where: \begin{center} \begin{footnotesize} $\begin{array}{lcl} \widetilde{\Gamma}_i= \Gamma_i, \quad i\ne 6,9; &\quad& \widetilde{\Gamma}'_j=\Gamma_j, \quad j\ne 1,2;\\ \widetilde{\Gamma}_6= (58, 109, 159, 56, 7, 117, 60, 0);& \quad &\widetilde{\Gamma}'_1= (77, 82, 116, 12, 136, 63, 1, 0);\\ \widetilde{\Gamma}_9=(-62, -117, -9, -72, -19, -125, -64, 0 ); & \quad &\widetilde{\Gamma}'_2= (80, 88, 131, 20, 135, 59, 4, 0 ). \end{array}$ \end{footnotesize} \end{center}
Now $\widetilde{\mathcal{F}_R}=\{\widetilde{\Gamma}_1,\ldots,\widetilde{\Gamma}_{10}\}$ and $\widetilde{\mathcal{F}_C}=\{\widetilde{\Gamma}'_1,\ldots,\widetilde{\Gamma}'_{10}\}$ are two sets of base cycles of a pair of orthogonal cyclic $C_8$-decompositions of the cocktail party graph $K_{162}-I$. \end{example}
\section{Variants and generalizations}\label{sec:variants}
In this section we will discuss several of the variants and generalizations of Heffter arrays that have been researched in earlier publications.
\subsection{Weak Heffter arrays}
In the original paper on Heffter arrays \cite{A}, Archdeacon also proposed the following weaker type of Heffter array. These weaker Heffter arrays again have applications to biembedding cycle systems as well as to orthogonal cycle systems. Two Heffter systems $D_h=D(2mh+1,h)$ and $D_k=D(2nk+1,k)$ with $mh = nk$ are {\em weakly sub-orthogonal} if the $i$-th part of $D_h$ has at most one element $a_{i,j}$ such that either $a_{i,j}$ or $-a_{i,j}$ is in the $j$-th part of $D_k$. Form a {\em weak Heffter array} $H(m,n;h,k)$ by placing $a_{i,j}$ in the cell on the $i$-th row and the $j$-th column. The upper sign on $\pm$ or $\mp$ is the {\em row sign} corresponding to its sign on $a_{i,j}$ in $D_h$, the lower sign is the {\em column sign} used in $D_k$. Using the row signs we get row sums 0 and the column signs give column sums 0.
The following is a formal definition of a weak Heffter array. \begin{definition}\label{def:weak}
A \emph{weak Heffter array} $H(m,n;h,k)$ is an $m \times n$ matrix $A$ such that: \begin{itemize}
\item[$(\rm{a_1})$] each row contains $h$ filled cells and each column contains $k$ filled cells; \item[$(\rm{b_1})$] for every $x \in \mathbb{Z}_{2nk+1} \setminus \{0\}$, there is exactly one cell of $A$ whose element is one of the following: $x,-x,\pm x,\mp x$, where the upper sign on $\pm$ or $\mp$ is the row sign and the lower sign is the column sign; \item[$(\rm{c_1})$] the elements in every row and column (with the corresponding sign) sum to $0$ in $\mathbb{Z}_{2nk+1}$. \end{itemize} \end{definition}
\begin{example} Consider the following Heffter systems on a half-set of $\mathbb{Z}_{25}$: $$D_4=\{\{1,-7-6,12\},\{2,-4,10,-8\},\{-3,-11,9,5\}\}$$ and $$D_3=\{\{1,2,-3\},\{-7,-4,11\},\{-6,-10,-9\},\{12,8,5\}\}.$$ Note that they are weakly sub-orthogonal and starting from these two systems one gets the following $3 \times 4$ weak Heffter array over $\mathbb{Z}_{25}$ without empty cells : $$
\begin{array}{|r|r|r|r|} \hline 1 & -7 & -6 & 12 \\ \hline 2 & -4 & \pm 10 & \mp 8 \\ \hline -3 & \mp 11 & \pm 9 & 5 \\ \hline \end{array} $$ \end{example}
To date there are no published papers specifically on weak Heffter arrays, but one is in preparation, see \cite{weak}. In \cite{A} Archdeacon showed how these arrays are related to current graphs and hence to biembeddings (in this case on a {\em non-orientable} surface). Also, by reasoning analogous to that in Section \ref{sec:DF}, it is easy to see that the following holds.
\begin{proposition}\label{weak} If there exists a simple weak $H(m,n;h,k)$, then there exist a $(2mh+1,C_h,1)$-DF and a $(2nk+1, C_k,1)$-DF. \end{proposition}
\begin{proposition} If there exists a simple weak $H(m,n;h,k)$, then there exist a cyclic $C_h$-decomposition and
a cyclic $C_k$-decomposition both of $K_{2nk+1}$. These decompositions are orthogonal. \end{proposition}
\subsection{$\lambda$-fold relative Heffter arrays}
In \cite{RelH,CPEJC} the authors proposed a natural generalization of Heffter arrays, which is related to signed magic arrays, difference families, graph decompositions and biembeddings. In this generalization, the array is missing the elements of a subgroup $J\subset \mathbb{Z}_v$ and each element of $ \mathbb{Z}_v \setminus J$ occurs exactly $\lambda$ times in the array. We give the following definition.
\begin{definition}\label{def:lambdaRelative} Let $v=\frac{2nk}{\lambda}+t$ be a positive integer, where $t$ divides $\frac{2nk}{\lambda}$, and let $J$ be the subgroup of $\mathbb{Z}_{v}$ of order $t$.
A $\lambda$-\emph{fold Heffter array $A$ over $\mathbb{Z}_{v}$ relative to $J$}, denoted by $^\lambda H_t(m,n; h,k)$, is an $m\times n$ array
with elements in $\mathbb{Z}_{v}\setminus J$ such that: \begin{itemize} \item[$(\rm{a_2})$] each row contains $h$ filled cells and each column contains $k$ filled cells; \item[$(\rm{b_2})$] the multiset $\{\pm x \mid x \in A\}$ contains each element of $\mathbb{Z}_v\setminus J$ exactly $\lambda$ times; \item[$(\rm{c_2})$] the elements in every row and column sum to $0$ in $\mathbb{Z}_v$. \end{itemize} \end{definition}
The terminology is due to their connection with \emph{relative} difference families. Trivial necessary conditions for the existence of a $^\lambda H_t(m,n; h,k)$ are $mh=nk$, $2\leq h \leq n$ and $2\leq k \leq m$. It is easy to see that condition $(\rm{b_2})$ of Definition \ref{def:lambdaRelative} asks that for every $x\in \mathbb{Z}_{v}\setminus J$ with $x$ different from the involution the number of occurrences of $x$ and $-x$ in the array is $\lambda$, while if the involution exists and it does not belong to $J$, then it has to appear exactly $\frac{\lambda}{2}$ times, also no element of $J$ appears in the array.
If $^\lambda H_t(m,n; h,k)$ is a square array, then it is denoted by $^\lambda H_t(n;k)$. If $\lambda=t=1$ one finds again the classical concept of Heffter array, i.e. an $^1 H_1(m,n; h,k)$ is just an $H(m,n; h,k)$. In general if $\lambda=1$, then it is omitted; the same holds for $t$.
Note also that if $\lambda=1$, then $3\leq h \leq n$ and $3\leq k \leq m$.
\begin{example}\label{ex:1} A $^3H_2(4;3)$ whose elements belong to $\mathbb{Z}_{10}$ and a $^4H_4(4;2)$ whose elements belong to $\mathbb{Z}_8$:
\center{
$\begin{array}{|r|r|r|r|} \hline & 1 & 2 & -3 \\ \hline 4 & & 4 & 2 \\ \hline -3 & 2 & & 1 \\ \hline -1 & -3 & 4 & \\ \hline \end{array}$ \quad\quad\quad\quad
$\begin{array}{|r|r|r|r|} \hline 1 & -1 & & \\ \hline -1 & 1 & & \\ \hline & & 3 & -3 \\ \hline & & -3 & 3 \\ \hline \end{array}$ } \end{example} As was the case for Heffter arrays, a $\lambda$-fold relative Heffter array is called \emph{integer} if condition ($\rm{c_2}$) in Definition \ref{def:lambdaRelative} is strengthened so that the elements in every row and every column, seen as integer in $\pm\left\{1,2,\ldots, \lfloor\frac{v}{2}\rfloor\right\}$, sum to $0$ in $\mathbb{Z}$. The $^3H_2(4;3)$ of Example \ref{ex:1} is not an integer Heffter array, while the $^4H_4(4;2)$ in the same example is integer.
\begin{example}\label{ex:inv}
An integer $^2H(5;3)$ over $\mathbb{Z}_{16}$ (note that the involution
appears exactly once in the array) and an integer $^2 H(6;4)$ over $\mathbb{Z}_{25}$:
\center{
$\begin{array}{|r|r|r|r|r|} \hline 3 & 5 & -8 & & \\ \hline & 2 & 5 & -7 & \\ \hline & & 3 & 1 & -4 \\ \hline -4 & & & 6 & -2 \\ \hline 1 & -7 & & & 6 \\ \hline \end{array}$ \hspace{1 cm}
$\begin{array}{|r|r|r|r|r|r|} \hline & & 1 & -2 & -5 & 6 \\ \hline & & -3 & 4 & 7 & -8 \\ \hline -9 & 10 & & & 1 & -2 \\ \hline 11 & -12 & & & -3 & 4 \\ \hline 5 & -6 &-9 & 10 & & \\ \hline -7 & 8 & 11 & -12 & & \\ \hline \end{array}$
}
\end{example}
\subsubsection{Necessary conditions}
In \cite{CPEJC} the authors established some necessary conditions for the existence of a $^\lambda H_t(m,n;h,k)$. Firstly, it is trivial to note that Conditions ($\rm{a_2})$ and ($\rm{b_2})$ of Definition \ref{def:lambdaRelative} imply that if there exists a $^{\lambda} H_t(m,n;h,k)$ with either $h=2$ or $k=2$, then $\lambda$ has to be even.
\begin{proposition}\label{prop:necctrivial}{\rm \cite{CPEJC}} Suppose that there exists a $^\lambda H_t(m,n;h,k)$ and set $v=\frac{2nk}{\lambda}+t$.
If either $v$ is odd or $v$ and $t$ are even, then $\lambda$ has to be a divisor of $nk$. \end{proposition}
\begin{proposition}\label{prop:nonexistence}{\rm \cite{CPEJC}} If $\lambda\equiv 2 \pmod 4$, $v=\frac{2nk}{\lambda}+t\equiv2\pmod 4$ and $t$ is odd, then a $^{\lambda} H_t(m,n;h,k)$ cannot exist. \end{proposition}
About the integer case there are the following more restrictive conditions. \begin{proposition}\label{prop:necc}{\rm \cite{RelH,CPEJC}} Suppose that there exists an integer $^\lambda H_t(m,n;h,k)$ with $\lambda$ odd. \begin{itemize} \item[$(\rm{1})$] If $t$ divides $\frac{nk}{\lambda}$, then $\frac{nk}{\lambda}\equiv 0 \pmod 4$ or $\frac{nk}{\lambda}\equiv -t \equiv \pm 1\pmod 4.$ \item[$(\rm{2})$] If $t=\frac{2nk}{\lambda}$, then $h$ and $k$ must be even. \item[$(\rm{3})$] If $t\neq \frac{2nk}{\lambda}$ does not divide $\frac{nk}{\lambda}$, then $\frac{2nk}{\lambda}+t\equiv 0 \pmod 8.$ \end{itemize} \end{proposition}
It is known that these conditions are not sufficient. For instance in \cite{RelH} it is proved that there is no integer $H_{3n}(n;3)$ for $n\geq 3$ and no integer $H_8(4;3)$, moreover in \cite{weak} it is shown the nonexistence of an integer $H_t(4;3)$ for $t=4,6$.
\subsubsection{Integer relative Heffter arrays with $\lambda=1$}
Turning to the existence question now, the first class investigated is that of integer relative Heffter arrays with $\lambda=1$.
\begin{theorem}{\rm \cite{RelH}} Let $3 \leq k \leq n$ with $k \neq 5$. There exists an integer $H_k (n; k)$ if and only if one of the following holds: \begin{itemize}
\item [$(\rm{1})$] $k$ is odd and $n \equiv 0, 3 \pmod 4$;
\item [$(\rm{2})$] $k \equiv 2 \pmod 4$ and $n$ is even;
\item [$(\rm{3})$] $k \equiv 0 \pmod 4$. \end{itemize} Furthermore, there exists an integer $H_5(n; 5)$ if $n \equiv 3 \pmod 4$ and it does not exist if $n \equiv 1, 2 \pmod 4$. \end{theorem} The case $k=5$ and $n\equiv0\pmod 4$ is still open. For these values of the parameters integer weak Heffter arrays have been constructed in \cite{weak}.
\begin{theorem} {\rm \cite{CPPBiembeddings}} There exists an integer globally simple $H_t(n;k)$ in the following cases: \begin{itemize}
\item [$(\rm{1})$] $k=3$, $t=n,2n$ for all odd $n$;
\item [$(\rm{2})$] $t=k=3,5,7,9$ for all $n\equiv 3 \pmod4$. \end{itemize} \end{theorem}
\begin{theorem} {\rm \cite{MP}} Let $m, n, h, k$ be such that $4 \leq h \leq n$, $4 \leq k \leq m$ and $mh = nk$. Let $t$ be a divisor of $2nk$. \begin{itemize}
\item [$(\rm{1})$] If $h, k \equiv 0 \pmod 4$, then there exists an integer $H_t(m, n; h, k)$.
\item [$(\rm{2})$] If $h \equiv 2 \pmod 4$ and $k \equiv 0 \pmod 4$, then there exists an integer $H_t(m, n; h, k)$ if and only if $m$ is even.
\item [$(\rm{3})$] If $h \equiv 0 \pmod 4$ and $k \equiv 2 \pmod 4$, then there exists an integer $H_t(m, n; h, k)$ if and only if $n$ is even.
\item [$(\rm{4})$] Suppose that $m$ and $n$ are both even. If $h, k \equiv 2 \pmod 4$, then there exists an integer $H_t(m, n; h, k)$. \end{itemize} \end{theorem}
\begin{theorem} {\rm \cite{MP3}} Let $m, n, h, k$ be such that $3 \leq h \leq n$, $3 \leq k \leq m$ and $mh = nk$. Suppose $d = \gcd(h, k) \geq 3$. \begin{itemize}
\item[$(\rm{1})$] If $d \equiv 1 \pmod 4$ and $nk \equiv 3 \pmod 4$, then there exists an integer $H_d(m, n; h, k)$.
\item[$(\rm{2})$] If $d \equiv 3 \pmod 4$ and $nk \equiv 0, 1 \pmod 4$, then there exists an integer $H_d(m, n; h, k)$.
\item[$(\rm{3})$] If $d = 3$ and $nk$ is odd, then there exists an integer $H_\frac{nk}{3}(m, n; h, k)$ and an integer $H_\frac{2nk}{3}(m, n; h, k)$. \end{itemize} \end{theorem}
\subsubsection{ $\lambda$-fold relative Heffter arrays} The following are the known existence results on $\lambda$-fold relative Heffter arrays.
\begin{theorem} {\rm \cite{MP3}} Let $m, n, h, k$ be such that $3 \leq h \leq n$, $3 \leq k \leq m$ and $mh = nk$. Suppose $d = \gcd(h, k) \geq 3$. \begin{itemize}
\item[$(\rm{1})$] If $d = 3$ and $nk \equiv 3 \pmod 4$, then there exists a $^2 H(m, n; h, k)$.
\item[$(\rm{2})$] If $d = 3$ and $nk \equiv 1 \pmod 4$, then there exists a $^3 H(m, n; h, k).$
\item[$(\rm{3})$] If $d = 3$, $nk$ is odd, and $\lambda$ divides $n$, then there exists a $^\lambda H_\frac{n}{\lambda}(m, n; h, k).$
\item[$(\rm{4})$] If $d = 3$, $nk$ is odd, and $\lambda$ divides $2n$, then there exists a $^\lambda H_\frac{2n}{\lambda}(m, n; h, k).$
\item[$(\rm{5})$] If $d = 5$ and $nk \equiv 3 \pmod 4$, then there exists a $^2 H(m, n; h, k).$ \end{itemize} \end{theorem}
\begin{theorem} {\rm \cite{MP2}} Let $m, n, h, k$ be such that $4 \leq h \leq n$, $4 \leq k \leq m$ and $mh = nk$. Let $\lambda$ be a divisor of $2nk$ and let $t$ be a divisor of $\frac{2nk}{\lambda}$.
There exists an integer $^\lambda H_t(m, n; h, k)$ in each of the following cases:
\begin{itemize}
\item[$(\rm{1})$] $h, k \equiv 0 \pmod 4$;
\item[$(\rm{2})$] $h \equiv 2 \pmod 4$ and $k \equiv 0 \pmod 4$;
\item[$(\rm{3})$] $h \equiv 0 \pmod 4$ and $k \equiv 2 \pmod 4$;
\item[$(\rm{4})$] $h, k \equiv 2 \pmod 4$ and $m, n$ both even.
\end{itemize} \end{theorem}
\begin{theorem}\label{lambda2}{\rm \cite{CPEJC}} There exists a $^2 H(n;k)$ if and only if $n\geq k\geq 3$ with $nk\not\equiv 1 \pmod 4$. \end{theorem}
\begin{theorem} \label{thm:noemptycell}{\rm \cite{CPEJC}} There exists a $^2 H(m,n;n,m)$ if and only if $mn\not\equiv 1 \pmod 4$ and $m,n$ are not both equal to $2$. \end{theorem}
\begin{theorem}\label{thm:skeven}{\rm \cite{CPEJC}} Let $h,k$ be even positive integers. There exists a $^2 H(m,n;h,k)$ if and only if $mh=nk$, $2\leq h\leq n$, $2\leq k \leq m$ and $h,k$ are not both equal to $2$. Moreover these arrays are integer when: \begin{itemize} \item[$(\rm{1})$] $h,k\neq 2$; \item[$(\rm{2})$] $h=2$ and either $n=2$ and $m=k\equiv 0\pmod 4$ or $n,k\geq3$; \item[$(\rm{3})$] $k=2$ and either $m=2$ and $n=h\equiv 0\pmod 4$ or $m,h\geq 3$. \end{itemize} \end{theorem}
\begin{proposition}\label{lambdak}{\rm \cite{CPEJC}} Let $n,k,\lambda$ be positive integers with $n\geq k\geq 3$ and $\lambda$ divisor of $k$.
Then there exists a $^\lambda H_{\frac{k}{\lambda}}(n;k)$ if one of the following is satisfied:
\begin{itemize}
\item[$(\rm{1})$] $k=5$ and $n\equiv 3 \pmod 4$;
\item[$(\rm{2})$] $k\neq 5$ odd and $n\equiv 0,3 \pmod 4$.
\end{itemize} \end{proposition}
\begin{proposition}\label{prop:n}{\rm \cite{CPEJC}}
For any odd integer $n\geq 3$ and for any divisor $\lambda$ of $n$ there exists a $^\lambda H_\frac{n}{\lambda}(n;3)$. \end{proposition}
\begin{proposition}\label{prop:2n}{\rm \cite{CPEJC}}
For any odd integer $n\geq 3$ and for any divisor $\lambda$ of $2n$ there exists a $^\lambda H_\frac{2n}{\lambda}(n;3)$. \end{proposition}
\begin{proposition}{\rm \cite{CPEJC}} Let $m,n,h,k,t,\lambda$ be such that $mh=nk$, $4\leq h\leq n$, $4\leq k\leq m$, $t$ is a divisor of $2nk$ and $\lambda$ is a divisor of $t$. A $^\lambda H_{\frac{t}{\lambda}}(m,n;h,k)$ exists in each of the following cases: \begin{itemize}
\item[$(\rm{1})$] $h\equiv k\equiv 0\pmod 4$;
\item[$(\rm{2})$] either $h\equiv 2\pmod 4$ and $k\equiv 0\pmod 4$ or
$h\equiv 0\pmod 4$ and $k\equiv 2\pmod 4$;
\item[$(\rm{3})$] $h\equiv k \equiv 2\pmod 4$ and $m$ and $n$ are even. \end{itemize} \end{proposition}
Some of previous results have been obtained thanks to one of the following two theorems which allow one to obtain new arrays from existing ones.
\begin{theorem}\label{thm:proiezione}{\rm \cite{CPEJC}} If there exists an $^\alpha H_t(m,n;h,k)$ then, for any divisor $\lambda$ of $t$, there exists a $^{\lambda\alpha} H_\frac{t}{\lambda}(m,n;h,k)$. \end{theorem}
\begin{theorem}{\rm \cite{MP3}} Let $m, n, h, k$ be such that $2 \leq h \leq n$, $2 \leq k \leq m$ and $mh = nk$. Let $\lambda$ be a divisor of $2nk$ and let $t$ be a divisor of $\frac{2nk}{\lambda}$.
Set $d = \gcd(h, k)$. If there exists a (integer) diagonal $^\lambda H_t(\frac{nk}{d};d)$,
then there exists a (integer) $^\lambda H_t(m, n; h, k)$. \end{theorem}
Other existence results can be found using the following recursive construction. \begin{proposition}\label{prop:recursive}{\rm \cite{CPEJC}} If there exists an (integer) $H_t(m,n;h,k)$ then there exists an (integer) $^{\alpha_1\lambda_1}H_t(\lambda_1m,\lambda_2n;\alpha_1h,\alpha_2k)$ for all positive integers $\alpha_1\leq \lambda_2$, $\alpha_2\leq\lambda_1$ such that $\alpha_1\lambda_1=\alpha_2\lambda_2$, $\alpha_1h\leq \lambda_2 n$ and $\alpha_2k\leq \lambda_1 m$. \end{proposition}
\subsubsection{Connection between relative Heffter arrays and biembeddings and graph decompositions}
In \cite{RelH,CPEJC} the authors investigated the connection of these generalized arrays with relative difference families, graph decompositions and biembeddings. To illustrate the results we need some definitions. Given a partially filled array $A$ by \emph{skeleton} of $A$, denoted by $skel(A)$, one means the set of the filled
positions of $A$, while by $\mathcal{E}(A)$ one denotes the multiset of the elements of $skel(A)$.
Analogously, by $\mathcal{E}(R_i)$ and $\mathcal{E}(C_j)$ we mean the multisets of elements
of the $i$-th row and of the $j$-th column, respectively, of $A$. Now we have to define a simple ordering of a \emph{multiset}. We consider the elements of $\mathcal{E}(A)$ indexed by the set $skel(A)$ (analogously for the rows and the columns). For example let $A$ be the $^4 H_4(4;2)$ constructed in Example \ref{ex:1}. Here $\mathcal{E}(A)=\{-1,-1,1,1,-3,-3,3,3\}$, we can view this multiset as the set $\{t_b \mid b\in skel(A)\}=\{t_{(1,1)},t_{(1,2)},t_{(2,1)},t_{(2,2)},t_{(3,3)},t_{(3,4)},t_{(4,3)},t_{(4,4)}\}$, where $t_{(1,1)}=1$, $t_{(1,2)}=-1$, $t_{(2,1)}=-1$, $t_{(2,2)}=1$, $t_{(3,3)}=3$, $t_{(3,4)}=-3$, $t_{(4,3)}=-3$ and $t_{(4,4)}=3$.
Given a finite multiset $T=[t_{b_1},\dots,t_{b_k}]$ whose (not necessarily distinct) elements are indexed by a set $B=\{b_1,\dots,b_k\}$ and a cyclic permutation $\alpha$ of $B$ we say that the list $\omega_{\alpha}=(t_{\alpha(b_1)},t_{\alpha(b_2)},\ldots,t_{\alpha(b_k)})$ is the ordering of the elements of $T$ associated to $\alpha$. In case the elements of $T$ belong to an abelian group $G$, we define $s_i=\sum_{j=1}^i t_{\alpha(b_j)}$, for any $i\in\{1,\ldots,k\}$, to be the $i$-th partial sum of $\omega_{\alpha}$ and we set $\mathcal{S}(\omega_{\alpha})=(s_1,\ldots,s_k)$. The ordering $\omega_{\alpha}$ is said to be \emph{simple} if $s_b\neq s_c$ for all $1\leq b < c\leq k$.
\begin{proposition} If there exists a simple $^\lambda H_t(m,n;h,k)$, then there exist a $(2mh+t,t,C_h,\lambda)$-DF and a $(2nk+t,t, C_k,\lambda)$-DF. \end{proposition}
\begin{proposition} If there exists a simple $^\lambda H_t(m,n;h,k)$, then there exist a cyclic $C_h$-decomposition $\mathcal{D}$ and a cyclic $C_k$-decomposition $\mathcal{D}'$ both of $^\lambda K_{\frac{2nk+t}{t}\times t}$. If $\lambda=1$ the decompositions $\mathcal{D}$ and $\mathcal{D}'$ are orthogonal. \end{proposition}
Also for these generalized arrays to get biembeddings one has to find compatible orderings. Note that in Section \ref{sec:biemb} we have defined compatible orderings only when $\mathcal{E}(A)$ is a set. Given an $m \times n$ partially filled array $A$, by $\alpha_{R_i}$ we will denote a cyclic permutation of the nonempty cells of the $i$-th row $R_i$ of $A$ and by $\omega_{R_i}$ (omitting the dependence on $\alpha_{R_i}$) the associated ordering of $\mathcal{E}(R_i)$. Similarly, for the $j$-th column $C_j$ of $A$, we define $\alpha_{C_j}$ and $\omega_{C_j}$.
If for any $i\in\{1,\ldots, m\}$ and for any $j\in\{1,\ldots,n\}$, the orderings $\omega_{R_i}$ and $\omega_{C_j}$ are simple, we define the permutation $\alpha_r$ of $skel(A)$
by $\alpha_{R_1}\circ \ldots \circ\alpha_{R_m}$ and we say that the associated action $\omega_r$ on $\mathcal{E}(A)$ is the simple ordering for the rows.
Similarly we define $\alpha_c=\alpha_{C_1}\circ \ldots \circ\alpha_{C_n}$ and the simple ordering $\omega_c$ for the columns. Now, given a $\lambda$-fold relative Heffter array $^\lambda H_t(m,n; h,k)$, say $A$, then the orderings $\omega_r$ (associated to the permutation $\alpha_r$) and $\omega_c$ (associated to the permutation $\alpha_c$) are said to be
\emph{compatible} if $\alpha_c \circ \alpha_r$ is a cycle of length $|skel(A)|$. In \cite{CPEJC} the authors showed that the necessary conditions of Theorem \ref{thm:compatible} hold more in general for a $^\lambda H_t(m,n; h,k)$.
\begin{theorem}\label{thm:biembedding}{\rm \cite{CPEJC}} Let $A$ be a $\lambda$-fold relative Heffter array $^\lambda H_t(m,n; h,k)$ that is simple with respect to the compatible orderings $\omega_r$ and $\omega_c$. Then there exists a cellular biembedding of a cyclic $C_h$-decomposition and a cyclic $C_k$-decomposition both of $^\lambda K_{(\frac{2nk}{\lambda t}+1)\times t}$ into an orientable surface. \end{theorem}
\subsection{Signed Magic Arrays}
The notion of a signed magic array was introduced by Khodkar, Schulz and Wagner \cite{KSW} in 2017. These are intimately related to (relative) Heffter arrays. We begin with the definition.
\begin{definition}\label{def:SMR} A \emph{signed magic array} $SMA(m,n;h,k)$ is an $m\times n$ array with entries from $X$, where $X=\left\{0,\pm1,\pm2,\ldots, \pm \frac{nk-1}{2}\right\}$ if $nk$ is odd and $X=\left\{\pm1,\pm2,\ldots,\pm \frac{nk}{2}\right\}$ if $nk$ is even, such that: \begin{itemize} \item[$(\rm{a_3})$] each row contains $h$ filled cells and each column contains $k$ filled cells; \item[$(\rm{b_3})$] every integer from the set $X$ appears exactly once in the array; \item[$(\rm{c_3})$] the sum of each row and of each column is $0$ in $\mathbb{Z}$. \end{itemize} \end{definition}
\begin{remark}\label{rem:magicHeffter} As was shown in \cite{CPEJC}, a signed magic array $SMA(m,n;h,k)$ with $nk$ even is an integer $^2 H(m,n;h,k)$, while in general the converse is not true. For instance, the $^2 H(6;4)$ of Example \ref{ex:inv} is not a signed magic array. On the other hand, in the particular case in which either $h=2$ or $k=2$ an integer $^2 H(m,n;h,k)$ is precisely an $SMA(m,n;h,k)$, see \cite{CPEJC} for details. \end{remark}
Existence results on signed magic arrays can be found in \cite{KE,KL20,KL,KLE,KSW,MP2,MP3}. We note that nearly all the results and techniques used on signed magic arrays are analogous to those for Heffter arrays. In particular, the existence of an $SMA(m, n; h, k)$ has been determined in the square case and when the array has no empty cells, as shown in the following two theorems.
\begin{theorem}{\rm \cite{KSW}} There exists an $SMA(n, n; k, k)$ if and only if either $n = k = 1$ or $3 \leq k \leq n$. \end{theorem}
\begin{theorem} {\rm \cite{KSW}} There exists an $SMA(m, n; n, m)$ if and only if one of the following cases occurs: \begin{itemize} \item[$(1)$] $m = n = 1$; \item[$(2)$] $m = 2$ and $n \equiv 0, 3 \pmod 4$; \item[$(3)$] $n = 2$ and $m \equiv 0, 3 \pmod 4$; \item[$(4)$] $m,n > 2$. \end{itemize} \end{theorem}
Also the cases where each column contains 2 or 3 filled cells have been solved. \begin{theorem}{\rm \cite{KE}} There exists an $SMA(m, n; h, 2)$ if and only if one of the following cases occurs: \begin{itemize} \item[$(1)$] $m = 2$ and $n = h \equiv 0, 3 \pmod 4$; \item[$(2)$] $m, h> 2$ and $mh = 2n$. \end{itemize} \end{theorem}
\begin{theorem}{\rm \cite{KLE}} There exists an $SMA(m, n; h, 3)$ if and only if $3 \leq m$, $h\leq n$ and $mh = 3n$. \end{theorem}
Necessary and sufficient conditions have been established also in the case when $h$ and $k$ are both even.
\begin{theorem}{\rm \cite{MP2}} Let $h, k$ be two even integers with $h, k \geq 4$. There exists an $SMA(m, n; h, k)$ if and only if $4\leq h \leq n$, $4 \leq k \leq m$ and $mh = nk$. \end{theorem}
Recent partial results for the general case can be found in \cite{MP3}. We summarize them in the following two theorems.
\begin{theorem}{\rm \cite{MP3}} Let $m, n, h, k$ be four integers such that $3 \leq h \leq n$, $3 \leq k \leq m$ and $mh = nk$. There exists an $SMA(m, n; h, k)$ whenever $\gcd(h, k) \geq 2$. \end{theorem}
\begin{theorem}{\rm \cite{MP3}} Let $m, n, h, k$ be four integers such that $3 \leq h \leq n$, $3 \leq k \leq m$ and $mh = nk$. If $h\equiv 0 \pmod 4$, then there exists an $SMA(m, n; h, k)$. \end{theorem}
\subsection{Non-zero sum Heffter arrays}
While studying the conjecture of Alspach (Conjecture \ref{Conj:als}), the authors of \cite{CDFP} proposed the following variant of a Heffter array which was termed a {\em non-zero sum Heffter array}. The formal definition follows.
\begin{definition}\label{def:NZS} Let $v=\frac{2nk}{\lambda}+t$ be a positive integer, where $t$ divides $\frac{2nk}{\lambda}$, and let $J$ be the subgroup of $\mathbb{Z}_{v}$ of order $t$.
A $\lambda$-\emph{fold non-zero sum Heffter array $A$ over $\mathbb{Z}_{v}$ relative to $J$}, denoted by $^\lambda NH_t(m,n; h,k)$, is an $m\times n$ array
with elements in $\mathbb{Z}_{v}$ such that: \begin{itemize} \item[$(\rm{a_4})$] each row contains $h$ filled cells and each column contains $k$ filled cells; \item[$(\rm{b_4})$] the multiset $\{\pm x \mid x \in A\}$ contains each element of $\mathbb{Z}_v\setminus J$ exactly $\lambda$ times; \item[$(\rm{c_4})$] the sum of the elements in every row and column {\underline {is different from $0$ }} in $\mathbb{Z}_v$. \end{itemize} \end{definition} If the array is square, then $h=k\geq 1$, while if $m\neq n$ then at least one of $h$ and $k$ has to be greater than $1$. A square non-zero sum Heffter array is denoted by $^\lambda NH_t(n;k)$. If $\lambda=1$ or $t=1$, then it is omitted.
\begin{example}\label{ex:NH75} An $NH(7;5)$ over $\mathbb{Z}_{71}$: \begin{center} \begin{footnotesize}
$\begin{array}{|r|r|r|r|r|r|r|}\hline 10 & & 16 & -1 & -2 & -3 & \\ \hline & 4 & & -6 & -7 & -5 & 22 \\ \hline -30 & 29 & 9 & -8 & & & 18 \\ \hline -11 & & -12 & -28 & -31 & 26 & \\ \hline & -14 & -15 & -13 & & -17 & 25 \\ \hline 27 & -34 & 20 & & -19 & & -32 \\ \hline 24 & 23 & & & 21 & -35 & 33 \\\hline \end{array}$ \end{footnotesize} \end{center} \end{example}
In \cite{CDFP} the authors focused on the case in which $\lambda=1$ and they proved that the trivial necessary conditions are also sufficient in the very general case in which the array is rectangular, empty cells are allowed and for every arbitrary subgroup $J$. We would like to note, however, that in the {\em square case} the proof of existence is quite easy when there exists an $H_t(n; k)$. Begin with the $H_t(n; k)$ and find a transversal of filled cells $T$ (this must exist). Then change the sign of the element in each of the cells of $T$ and note that now no row or column can add to $0$. Hence we have constructed an $NH_t(n; k)$. The general case is more involved and we emphasize that the proof of the general result is not constructive.
\begin{theorem}{\rm \cite{CDFP}}\ There exists an $NH_t(m,n;h,k)$ if and only if $mh=nk$, $m\geq k \geq 1$, $n \geq h \geq 1$ and $t$ divides $2nk$. \end{theorem}
These arrays have been introduced since, as with classical ones, they are related to open problems on partial sums, difference families, graph decompositions and biembeddings. The interested reader is referred to \cite{CDFP} where all the following results can be found.
By reasoning similar to what was done in Section \ref{sec:DF}, it is easy to see that if a row (column, respectively) of a $NH_t(m,n;h,k)$ admits an ordering whose partial sums are pairwise distinct and non-zero then one can construct a path of length $h$ ($k$, respectively). For example, starting from the first row of the $NH(7;5)$ of Example \ref{ex:NH75} and considering its natural ordering one obtains the path $\Gamma=[0,10,26,25,23,20]$ of length $5$, clearly $\Delta \Gamma=\pm R_1$. A non-zero sum Heffter array is said to be {\em simple} if every row and every column admits an ordering such that the partial sums are pairwise distinct and non-zero. Hence, since every row and column of a non-zero sum Heffter array can be viewed as a set $T$ considered in Conjecture \ref{Conj:als}, we have that if Conjecture \ref{Conj:als} were true, then every non-zero sum Heffter array would be simple. Investigating arrays having this additional property is important in view of the following results.
\begin{proposition}{\rm \cite{CDFP}} If there exists a simple $^\lambda NH_t(m,n;h,k)$, then there exist a $(2mh+t,t,P_h,\lambda)$-DF and a $(2nk+t,t, P_k,\lambda)$-DF. \end{proposition}
\begin{proposition}{\rm \cite{CDFP}} If there exists a simple $^\lambda NH_t(m,n;h,k)$, then there exist
a cyclic $P_h$-decomposition $\mathcal{D}$ and
a cyclic $P_k$-decomposition $\mathcal{D}'$ both of $^\lambda K_{\frac{2nk+t}{t}\times t}$. If $\lambda=1$ the decompositions $\mathcal{D}$ and $\mathcal{D}'$ are orthogonal. \end{proposition}
In view of previous propositions, in \cite{CDFP,MePa} the authors focused on globally simple non-zero sum Heffter arrays, obtaining the following existence results. All the proofs are constructive.
\begin{theorem}{\rm \cite{CDFP}}\label{NH1} For every $n\geq k\geq 1$ there exists a globally simple cyclically $k$-diagonal $NH(n;k)$. \end{theorem}
\begin{theorem}{\rm \cite{CDFP}} For every $m,n\geq 1$ there exists a globally simple $NH(m,n;n,m)$. \end{theorem}
\begin{theorem}{\rm \cite{MePa}}\label{NH:t} For every odd integer $n \geq 1$ and for every divisor $t$ of $n$, there exists a globally simple $NH_t(n; n)$. \end{theorem}
\begin{proposition}{\rm \cite{MePa}}\label{NH:t2} For every odd integer $n \geq 1$ and for every $t\in\{2,2n,n^2,2n^2\}$, there exists a globally simple $NH_t(n; n)$. \end{proposition}
By Theorems \ref{NH1} and \ref{NH:t} and Proposition \ref{NH:t2} one gets the following result (the case $n=2$ is trivial).
\begin{corollary}{\rm \cite{MePa}} Let $n$ be a prime. There exists a globally simple $NH_t(n; n)$ for every admissible $t$. \end{corollary}
The case of $\lambda$-fold non-zero sum Heffter arrays with $\lambda>1$ has been investigated in \cite{CDFlambda}. Here the authors consider a generic finite group $G$ and prove that there exists a non-zero sum $\lambda$-fold Heffter array over $G$ relative to $J$
whenever the trivial necessary conditions are satisfied and $|G|\geq 41$. Moreover, this value can be decreased to $29$ when the array does not contain empty cells.
We conclude this subsection with the theorem that gives the connection to biembeddings.
\begin{theorem}{\rm \cite{CDFP}} Let $A$ be a non-zero sum Heffter array $NH(m,n;h,k)$ with compatible orderings $\omega_r$ and $\omega_c$. Then there exists a cellular biembedding of two circuit decompositions
of $K_{2nk+1}$ into an orientable surface, such that the faces are multiples of $h$ and $k$ strictly larger than $h$ and $k$, respectively. \end{theorem}
Details on the faces of the biembeddings can be given considering the order of the row and column sums in the group, as done in \cite{C,MePa}.
\subsection{Heffter configurations}
Heffter arrays are generalized to {\it Heffter configurations} in \cite{BP22}. We recall that a $(v,b,k,r)$-configuration is a pair $(V,{\cal B})$ where $V$ is a set of $v$ {\it points} and $\cal B$ is a $b$-set of $k$-subsets ({\it blocks}) of $V$ such that any two distinct points are contained together in at most one block and any point occurs in exactly $r$ blocks. Such a configuration is {\it resolvable} if there exists a partition of $\cal B$ into $r$ {\it parallel classes} each of which is a partition of $V$, see \cite{BS21,Ge}. Also, it is $\mathbb{Z}_v$-{\it additive} if its point set is a subset of $\mathbb{Z}_v$ and each block sums to zero.
\begin{definition} A $(nk,nr,k,r)$ \emph{Heffter configuration} (briefly HC) is a $\mathbb{Z}_{2nk+1}$-additive resolvable $(nk,nr,k,r)$-configuration where the point set is a half-set of $\mathbb{Z}_{2nk+1}$. It is {\it proper} if $r\geq3$ and it is {\it simple} if each block admits a simple ordering. \end{definition}
The above terminology is justified by the fact that a $(nk,nr,k,r)$-HC is equivalent to $r$ mutually sub-orthogonal Heffter systems on the same half-set of $\mathbb{Z}_{2nk+1}$. Indeed each parallel class is nothing but a Heffter system D$(nk,k)$. Hence, a Heffter system D$(nk,k)$ is a $(nk,n,k,1)$-HC and an
Heffter array $H(n;k)$, say $A$, is a $(nk,2n,k,2)$-HC whose two parallel classes are the rows of $A$ and the columns of $A$.
The construction of proper HCs appears to be very hard. On the other hand some infinite series have been obtained recursively by means of small examples such as the following.
\begin{example} The integer shiftable $H(5;4)$ below gives a $(20,15,4,3)$-HC whose three parallel classes are:
the rows;
the columns; and the diagonals $D_1, \ldots, D_5$. $$
\begin{array}{|r|r|r|r|r|} \hline -1& 2 & 17 & -18 & \\ \hline 6 & 10 & -13 & & -3\\ \hline 9 & -5 & & -8 & 4\\ \hline -14 & & -16 & 11 & 19\\ \hline
& -7 & 12 & 15 &-20 \\ \hline \end{array} $$ \end{example}
Generalizing Proposition \ref{orth.systems}, we can state the following.
\begin{theorem}
A simple $(nk,nr,k,r)$-HC gives rise to $r$ mutually orthogonal cyclic $C_k$-decompositions of $K_{2nk+1}$.
\end{theorem}
\subsection{Further generalizations}
Another natural generalization is to work in an arbitrary additive group $G$ and not necessarily in the cyclic group. Some work in this direction has been done in \cite{CPPBiembeddings} where the authors proposed the very general concept of an \emph{Archdeacon array}.
\begin{definition}{\rm \cite{CPPBiembeddings}}
An \emph{Archdeacon array} over an abelian group $(G, +)$ is an $m \times n$ array $A$ with elements in $G$, such that: \begin{itemize}
\item[$(\rm{b_5})$] the multiset $\{\pm x \mid x \in A\}$ contains each element of $G \setminus \{0\}$ at most once;
\item[$(\rm{c_5})$] the elements in every row and column sum to $0$ in $G$. \end{itemize} \end{definition}
\begin{example} An Archdeacon array on $\mathbb{Z}_{41}\oplus \mathbb{Z}_d$, for $d\geq 3$: $$
\begin{array}{|r|r|r|r|r|} \hline (-1,1) & (0,-1) & (-14,0) & (9,0) & (6,0) \\ \hline (2,-1) & (-7,1) & & (-5,0) & (10,0)\\ \hline (17,0) & (12,0) & (-16,0) & & (-13,0)\\ \hline (-18,0) & (15,0) & (11,0) & (-8,0) & \\ \hline
& (-20,0) & (19,0) & (4,0) & (-3,0) \\ \hline \end{array} $$ \end{example}
Clearly, Heffter arrays are a particular kind of Archdeacon arrays. Examples, existence results and applications of these arrays can be found in Section $5$ of \cite{CPPBiembeddings}.
Another general concept termed a {\em quasi-Heffter array} has recently been proposed in \cite{C}. In that paper the reader can find all the details on the applications of these arrays to biembeddings and their full automorphism group.
\begin{definition}{\rm \cite{C}} Let $v = 2nk + t$ be a positive integer, where $t$ divides $2nk$, and let $J$ be the subgroup of $\mathbb{Z}_v$ of order $t$. A \emph{quasi-Heffter array} $A$ \emph{over $\mathbb{Z}_v$ relative to $J$}, denoted by $QH_t(m, n; h, k)$, is an $m \times n$ array with elements in $\mathbb{Z}_v$ such that: \begin{itemize}
\item[$(\rm{a_6})$] each row contains $h$ filled cells and each column contains $k$ filled cells;
\item[$(\rm{b_6})$] the multiset $\{\pm x \mid x \in A\}$ contains each element of $\mathbb{Z}_v \setminus J$ exactly once.
\end{itemize} \end{definition}
So in this case the definition does not require any property on the row and column sums. In general, given $S \subseteq \mathbb{Z}_v$ one may require that the sums of the rows and of the columns belong to $S$. Clearly, if $S=\{0\}$ we find the classical concept of a Heffter array and if $S=\mathbb{Z}_v\setminus\{0\}$ we get the notion of a non-zero sum Heffter array. For other choices of $S$ we have further variants of the classical concept having again applications to difference families, graph decompositions and biembeddings.
\section{Conclusions and Open Problems}\label{sec:conclusions}
Heffter arrays are a relatively new type of combinatorial design with original paper on this topic published in 2015. Since that time there have been a number of papers published on this subject as well as on topics that are related to Heffter arrays. Here we have surveyed all the major results of these papers. We believe that there are many more interesting questions relating to Heffter arrays and related subjects and we fully expect researchers to make good progress on these in the years to come. We conclude this survey with a short list of open problems. \begin{itemize} \item Complete the construction of classical Heffter arrays in the rectangular case with empty cells, for known results see Theorems \ref{Fiore1} and \ref{Fiore2}. \item Construct new infinite classes of globally simple Heffter arrays, for known results see Theorems \ref{thm:SH}, \ref{thm:globally_simple2} and \ref{thm:SH1}. \item Determine new solutions to the Crazy Knight's Tour Problem, see Section \ref{sec:biemb}. \item Determine new results concerning Conjectures \ref{Conj:als}, \ref{Conj:ADMS} and \ref{Conj:nostra}. \item Establish more restrictive necessary conditions for integer $^\lambda H_t(n;k)$, in order to improve Proposition \ref{prop:necc}. \end{itemize}
\end{document} | arXiv |
The family of unidirectional continuous fiber reinforced polymeric composites are currently used in automotive bumper beams and load floors. The material properties and mechanical characteristics of the compression molded parts are determined by the curing behavior, fiber orientation and formation of knit lines, which are in turn determined by the mold filling parameters. In this paper, a new model is presented which can be used to predict the 3-dimensional flow under consideration of the slip of mold-composites and anisotropic viscosity of composites during compression molding of unidirectional fiber reinforced thermoplastics for isothermal state. The composites is treated as an incompressible Newtonian fluid. The effects of longitudinal/transverse viscosity ratio A and slip parameter $\alpha$ on the buldging phenomenon and mold filling patterns are also discussed. | CommonCrawl |
Musings on a Youthful Planet
by Paul Gilster on January 3, 2008
People seem to be getting younger all the time. I'm told this is a common perception as you get older. In any case, it wasn't so long ago that I met the son of an acquaintance at an informal gathering. He looked to me to be about fourteen years old, but something warned me not to assume this. I said "What do you do? Are you in school?" His reply: "No, I've got my own dental practice downtown." I don't know how old you have to be to become a dentist, but I do know it's a lot older than fourteen!
Exoplanets and the stars they circle, on the other hand, seem to be mostly of a certain age, the denizens of relatively mature systems. Which is why TW Hydrae is so interesting. It's an infant in stellar terms, at eight to ten million years old only a fraction of the Sun's age. Like other stars in its age group, it is surrounded by a circumstellar disk of gas and dust, the sort of place where planets can form. And indeed, what seems to be the youngest planet yet detected has now been located within a gap in that disk. TW Hydrae b is about ten times as massive as Jupiter, orbiting in 3.56 days at a distance of some six million kilometers, or 0.04 AU.
Image: The newly discovered giant planet orbits around its young and active host star inside the inner hole of a dusty circumstellar disk (artist view). Credit: Max Planck Institute for Astronomy.
This work comes out of the Max Planck Institute for Astronomy (MPIA) in Heidelberg, where a team led by Johny Setiawan used European Southern Observatory equipment at La Silla (Chile) to make the find. TW Hydrae b turns out to be quite a catch. Starspots analogous to the sunspots on our own star can distort radial velocity readings, a particular problem with young stars whose surface is still relatively unstable. But MPIA's Ralf Launhardt seems sure of the result, saying:
"To exclude any misinterpretation of our data, we have investigated all activity indicators of TW Hydrae in detail. But their characteristics are significantly different from those of the main radial velocity variation. They are less regular and have shorter periods."
Bear in mind that none of the known extrasolar planets have, until now, been found around stars young enough to still have their circumstellar disks. Here again we're looking at a limitation in our methods, younger stars having been excluded from many searches because of the difficulty of measurement caused by the above mentioned solar activity. Now we're seeing some constraints on planetary formation, learning that a planet can form within a ten million year timeframe and, presumably, migrate inward as it interacts with the circumstellar disk to its present position.
Discoveries like this one add to our knowledge of planetary formation. Is this how all 'hot Jupiters' form? Things we need to learn more about include the average lifetime of a circumstellar disk, now thought to be somewhere between ten and thirty million years. And the core accretion model we're talking about here is still challenged by the gravitational instability alternative, which theoretically allows much faster formation of such giant worlds. The new planet becomes a helpful test case in which to simulate both scenarios as we look for still younger planets.
The paper is Setiawan, Henning et al., "A young massive planet in a star–disk system," Nature 451 (3 January 2008), pp. 38-41 (abstract). New Scientist also offers an article on TW Hydrae b.
andy January 3, 2008, 17:16
This system offers some very interesting insights, not least giving information about how far the planet has migrated (as I understand it from secondhand information about the paper's contents, the inner disc out to 0.5-4 AU from the star contains less material than the outer part, which could indicate the region through which the planet has travelled), and also how much material is left in the solar system to form planets from. This could have a significant bearing on whether hot Jupiter systems can form habitable planets or not.
In addition, TW Hydrae b seems to be rather more massive than the typical hot Jupiter. Such "hot superjupiters" seem to be quite rare. I wonder if the TW Hydrae environment will shed any light on why this may be.
Dylan January 3, 2008, 18:11
It sounds like this might also be how binary stellar systems form – at ten time the mass of jupiter it sounds like this could be a candidate for stellar evolution of it's own, however it's proximity of the main star sounds extremely close for a binary star system.
Can anybody comment regarding how this model relates to the formation of secondary stars in a system?
Administrator January 3, 2008, 19:39
I'll let someone else comment on the formation of binaries in this scenario, but I want to ask both andy and Dylan, and anyone else who may know, whether there is a simple breakdown somewhere showing estimated stellar ages for the exoplanet host stars we now know about? I realize that calculating stellar ages is itself a subject in flux, but it would be a useful link nonetheless if anyone has it.
And if I'm remembering the recent comment here on Struve's work correctly, wasn't the formation of close binaries the impetus for his original 'hot Jupiter' theory?
andy January 4, 2008, 7:41
A good starting point for exoplanet host star ages is Saffe, Gómez and Chavero (2005) "On the ages of exoplanet host stars" – it isn't an exhaustive list for exoplanet host stars (even of the ones known at the time), but it has a fair few in there.
ljk January 7, 2008, 12:21
Tidal Evolution of Close-in Extra-Solar Planets
Authors: Brian Jackson, Richard Greenberg, Rory Barnes
(Submitted on 4 Jan 2008)
Abstract: The distribution of eccentricities e of extra-solar planets with semi-major axes a greater than 0.2 AU is very uniform, and values for e are relatively large, averaging 0.3 and broadly distributed up to near 1. For a less than 0.2 AU, eccentricities are much smaller (most e less than 0.2), a characteristic widely attributed to damping by tides after the planets formed and the protoplanetary gas disk dissipated. Most previous estimates of the tidal damping considered the tides raised on the planets, but ignored the tides raised on the stars. Most also assumed specific values for the planets' poorly constrained tidal dissipation parameter Qp.
Perhaps most important, in many studies, the strongly coupled evolution between e and a was ignored. We have now integrated the coupled tidal evolution equations for e and a over the estimated age of each planet, and confirmed that the distribution of initial e values of close-in planets matches that of the general population for reasonable Q values, with the best fits for stellar and planetary Q being ~10^5.5 and ~10^6.5, respectively. The accompanying evolution of a values shows most close-in planets had significantly larger a at the start of tidal migration. The earlier gas disk migration did not bring all planets to their current orbits. The current small values of a were only reached gradually due to tides over the lifetimes of the planets. These results may have important implications for planet formation models, atmospheric models of "hot Jupiters", and the success of transit surveys.
Comments: accepted to ApJ
From: Brian Jackson [view email]
[v1] Fri, 4 Jan 2008 19:05:18 GMT (846kb)
ljk January 15, 2008, 9:14
Two Unusual Older Stars Giving Birth To Second
Wave Of Planets
ScienceDaily (Jan. 15, 2008) — Hundreds of millions —
or even billions — of years after planets would have
initially formed around two unusual stars, a second
wave of planetesimal and planet formation appears
to be taking place, UCLA astronomers and colleagues
"This is a new class of stars, ones that display conditions
now ripe for formation of a second generation of planets,
long, long after the stars themselves formed," said UCLA
astronomy graduate student Carl Melis, who reported the
findings today at the American Astronomical Society
meeting in Austin, Texas.
Grain Sedimentation in a Giant Gaseous Protoplanet
Authors: Ravit Helled, Morris Podolak, Attay Kovetz
(Submitted on 16 Jan 2008)
Abstract: We present a calculation of the sedimentation of grains in a giant gaseous protoplanet such as that resulting from a disk instability of the type envisioned by Boss (1998). Boss (1998) has suggested that such protoplanets would form cores through the settling of small grains. We have tested this suggestion by following the sedimentation of small silicate grains as the protoplanet contracts and evolves. We find that during the course of the initial contraction of the protoplanet, which lasts some $4\times 10^5$ years, even very small (greater than 1 micron) silicate grains can sediment to create a core both for convective and non-convective envelopes, although the sedimentation time is substantially longer if the envelope is convective, and grains are allowed to be carried back up into the envelope by convection. Grains composed of organic material will mostly be evaporated before they get to the core region, while water ice grains will be completely evaporated.
These results suggest that if giant planets are formed via the gravitational instability mechanism, a small heavy element core can be formed due to sedimentation of grains, but it will be composed almost entirely of refractory material. Including planetesimal capture, we find core masses between 1 and 10 M$_{\oplus}$, and a total high-Z enhancement of ~40 M$_{\oplus}$. The refractories in the envelope will be mostly water vapor and organic residuals.
Comments: accepted for publication in Icarus
From: Ravit Helled [view email]
[v1] Wed, 16 Jan 2008 06:10:49 GMT (22kb)
ljk January 23, 2008, 10:03
Angular Momentum Accretion onto a Gas Giant Planet
Authors: Masahiro N. Machida, Eiichiro Kokubo, Shu-ichiro Inutsuka, Tomoaki Matsumoto
Abstract: We investigate the accretion of angular momentum onto a protoplanet system using three-dimensional hydrodynamical simulations. We consider a local region around a protoplanet in a protoplanetary disk with sufficient spatial resolution. We describe the structure of the gas flow onto and around the protoplanet in detail. We find that the gas flows onto the protoplanet system in the vertical direction crossing the shock front near the Hill radius of the protoplanet, which is qualitatively different from the picture established by two-dimensional simulations. The specific angular momentum of the gas accreted by the protoplanet system increases with the protoplanet mass. At Jovian orbit, when the protoplanet mass M_p is M_p less than 1 M_J, where M_J is Jovian mass, the specific angular momentum increases as j \propto M_p. On the other hand, it increases as j \propto M_p^2/3 when the protoplanet mass is M_p greater than 1 M_J. The stronger dependence of the specific angular momentum on the protoplanet mass for M_p less than 1 M_J is due to thermal pressure of the gas. The estimated total angular momentum of a system of a gas giant planet and a circumplanetary disk is two-orders of magnitude larger than those of the present gas giant planets in the solar system. A large fraction of the total angular momentum contributes to the formation of the circumplanetary disk. We also discuss the satellite formation from the circumplanetary disk.
Comments: 39 pages,13 figures, Submitted to ApJ, For high resolution figures see this http URL
From: Masahiro Machida N [view email]
[v1] Tue, 22 Jan 2008 04:24:58 GMT (1354kb)
Disks, young stars, and radio waves: the quest for forming planetary systems
Authors: Claire J. Chandler, Debra S. Shepherd (NRAO)
Abstract: Kant and Laplace suggested the Solar System formed from a rotating gaseous disk in the 18th century, but convincing evidence that young stars are indeed surrounded by such disks was not presented for another 200 years.
As we move into the 21st century the emphasis is now on disk formation, the role of disks in star formation, and on how planets form in those disks. Radio wavelengths play a key role in these studies, currently providing some of the highest spatial resolution images of disks, along with evidence of the growth of dust grains into planetesimals.
The future capabilities of EVLA and ALMA provide extremely exciting prospects for resolving disk structure and kinematics, studying disk chemistry, directly detecting proto-planets, and imaging disks in formation.
Comments: 10 pages, 6 figures, to appear in the proceedings of the NRAO 50th Anniversary Science Symposium "Frontiers of Astrophysics", ASP Conf. Series
From: Claire J. Chandler [view email]
[v1] Thu, 24 Jan 2008 00:01:11 GMT (346kb)
ljk April 2, 2008, 11:12
Astronomers see 'youngest planet'
By Paul Rincon
Science reporter, BBC News, Belfast
An embryonic planet detected outside our Solar System
could be less than 2,000 years old, astronomers say.
The ball of dust and gas, which is in the process of turning
into a Jupiter-like giant, was detected around the star HL Tau,
by a UK team.
Research leader Dr Jane Greaves said the planet's growth
may have been kickstarted when another young star passed
the system 1,600 years ago.
Details were presented at the UK National Astronomy Meeting
in Belfast.
The scientists studied a disc of gas and rocky particles around
HL Tau, which is 520 light-years away in the constellation of
Taurus and thought to be less than 100,000 years old.
The disc is unusually massive and bright, making it an excellent
place to search for signs of planets in the process of formation.
The researchers say their picture is one of a proto-planet still
embedded in its birth material.
Dr Greaves, from the University of St Andrews, Scotland, said
the discovery of a forming planet around such a young star was
a major surprise.
"It wasn't really what we were looking for. And we were amazed
when we found it," she told BBC News.
"The next youngest planet confirmed is 10 million years old."
If the proto-planet is assumed to be the same age as the star it
orbits, this would be some one hundred times younger than the
previous record holder.
http://news.bbc.co.uk/2/hi/science/nature/7326318.stm
andy August 19, 2008, 6:01
It has been suggested the RV variations are actually caused by starspots:
arXiv: TW Hydrae: evidence of stellar spots instead of a Hot Jupiter
Next post: Suggestive Red Dust in Protoplanetary Disk
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Automotive Innovation
June 2019 , Volume 2, Issue 2, pp 93–101 | Cite as
Experimental and Numerical Study of Cervical Muscle Contraction in Frontal Impact
Zhenhai Gao
Zhao Li
Hongyu Hu
Fei Gao
In a crash situation, drivers typically make evasive maneuvers before an upcoming impact, which can affect the kinematics and injury during impact. The purpose of the current study was to investigate the response and effect of drivers' cervical muscles in a frontal impact. A crash scenario was developed using a vehicle driving simulator, and 10 volunteers were employed to drive the simulator at 20 km/h, 50 km/h, 80 km/h and 100 km/h. Electromyography (EMG) was recorded from the sternocleidomastoideus (SCM), splenius cervicis (SPL) and trapezium (TRP) muscles using a data acquisition system, and the level of muscle activation was calculated. A numerical study was conducted using data collected in the experiment. The results revealed that the cervical muscles were activated during drivers' protective action. EMG activity of cervical muscles before impact was greater than that during normal driving. EMG activity increased with driving speed, with the SCM and TRP exhibiting larger increases than the SPL. The kinematics and load of the driver were influenced by muscle activation. Before the collision, the head of an active model stretched backward, while the passive model kept the head upright. In low-speed impact, the torque and shear of the cervical muscle in the active model were much lower than those in the passive model, while the tension of the cervical muscle was higher in the active model compared with the passive model. The results indicated that the incidence of cervical injury in high-speed impact is complex.
Cervical injury Frontal impact Active muscle force Driving simulator
Electromyography
Sternocleidomastoideus
Splenius cervicis
Max voluntary contraction
Root mean square
New Car Assessment Programme
Traffic accidents are one of the leading causes of death worldwide. In 2015, there were 35,092 fatalities and 2,443,000 injuries related to road accidents in the USA [1], and 26,134 and 1,090,042 fatalities and injuries, respectively, in the European Union [2]. In cases of deaths and injuries in road accidents, the head and neck are the most vulnerable parts of the body [3, 4]. Although cervical injuries caused by rear-end collisions have been extensively studied, almost one-third of all neck injuries occur in frontal impacts [5]. Thus, neck injuries in frontal impacts warrant detailed research.
Crash tests are important for reducing injuries in accidents, but the testing method has limitations. A 50th percentile adult male dummy in a normal sitting posture exposed to standardized crash scenarios is commonly used in automobile crash safety tests. Even in the same crash situation, there are substantial differences between impact tests and a real accident. The dummies used in vehicle crash simulations are often stiffer than real human bodies and respond differently [6, 7]. Particularly in the soft and flexible cervical and spinal regions, dummies are typically unrealistically stiff and unlikely to have the same compliant response to human bodies [8]. In the real world, drivers tend to act on the brake pedal and/or steering wheel to avoid or prepare for an upcoming impact [9, 10, 11, 12]. However, a dummy cannot perform evasive maneuvers before impact. Thus, cars that achieve the highest scores in crash tests are optimized for protecting dummies, rather than human occupants.
With increasing computational power and progression in human body biomechanics, human body modeling (HBM) is widely utilized in the field of crash simulation. HBM involves computational modeling of human physiological structure and can be applied to study the influence of muscular activity and post-fracture response.
Studies of the effect of braced action on vehicle impact have been undertaken using a combination of experimental testing and numerical analysis with active HBM. In these tests, volunteers are typically subjected to low acceleration in sled devices while physiological signals are recorded. In numerical analysis, HBM with experimental parameters is employed in crash injury research. The investigation of muscular effects on cervical injuries can be conducted using the same research paradigms.
Choi et al. [13] employed volunteers in sled tests, measuring reaction forces on the steering wheel and brake pedal, as well as the activities of muscles in the upper and lower extremities. Computational analysis using a human finite element (FE) model with muscle activities obtained from an experiment in a similar environment was conducted and validated by experimental results [13]. Using a similar method, Ejima et al. [14] compared the behavior of relaxed and braced occupants in a pre-crash scenario. It was found that the neck and abdominal muscles were the most highly activated and kinematics of the head–neck–torso were strongly influenced by the muscle activity. To extend understanding of whiplash injury, Kumar et al. conducted sled tests using volunteers to determine the recruitment of cervical muscles with varied magnitudes of low velocity, different sitting positions under conditions of awareness and unawareness of upcoming frontal and rear-end impacts. Electromyography (EMG) signals of cervical muscles were found to exceed the maximum voluntary contraction (MVC) level, and excessive contraction of the muscles could contribute to injury [14, 15, 16]. To protect volunteers from injury, the sled test is limited to low-speed impact, meaning that responses of occupants in complicated high-speed accidents cannot be elicited using this method.
Reconstructing a virtual collision scenario in a driving simulator provides an alternative method. Virtual simulations could be an effective tool for examining realistic avoidance behavior of occupants in an imminent crash. Autrey et al. [17] designed a driving task involving a sudden frontal crash in a driving simulator, while the driver's actions were recorded by cameras throughout the driving period. The driving action was used in a numerical simulation of a frontal crash. Thus, a driving simulator method was used to investigate driving behavior and evasive maneuvers in an emergency situation. However, muscle activity, which could reveal additional information about human action, was not recorded in the study mentioned above.
The current study sought to investigate contraction of drivers' cervical muscles in a vehicle crash scenario and to examine the effect of neck muscle strength on collision injury, at low and high speeds. The study was composed of two parts. One part reported a driving simulator study designed to determine the cervical muscle activation in an unavoidable impact. Frontal impacts at different velocities were reconstructed in a driving simulator. EMG signals from selected cervical muscles during impact were recorded and analyzed. The other part reported a numerical simulation to study the effect of muscle force on injury outcome. The averaged muscle activations were applied to an active cervical HBM using MADYMO software.
2 Experimental Study of Cervical Muscle Activity During Emergency Braking
The experiments were performed in a driving simulator at the State Key Laboratory of Automobile Simulation and Control, Jilin University, China (Fig. 1). A sudden frontal impact was replicated in the simulator.
Open image in new window
2.1 Experimental Setup
The driving simulator had the capacity to perform yaw, pitch and roll motion and was equipped with a realistic control operation system. The steering wheel and brake pedal induced a sensation similar to that of normal real-world driving.
The driving scenario is shown in Fig. 2. The volunteer drove the host car in one lane at a specified speed. One obstacle car, which crossed into the other lane, suddenly cut into the participant's lane and stopped 10 meters in front of the host car.
Simulated crash scenario
2.2 Selection of Volunteers
The selection of volunteers was based on the following criteria: similar body size to 50 percent American male and no history of cervical spine pain in the past 12 months. The mean age, height and weight of the sample were 28 (SD 3.2) years, 174.1 (SD 4.1) cm and 67.4 (SD 3.8) kg. (Table 1)
Volunteer data
Sitting weight (cm)
BMI (kg/m2)
The protocol of the experiments was reviewed and approved by the Jilin University Ethics Committee, and all volunteers submitted their informed consent in a document that complied with the Helsinki Declaration.
2.3 Selection of Muscles of Interest
There are numerous muscles on the cervical spine, including small, intersegmental muscles and larger muscles that originate from the skull and insert to the thorax. The sternocleidomastoideus (SCM), trapezium (TRP) and splenius cervicis (SPL) were selected as muscles of interest. In the neural position, the SCM, SPL and TRP have the largest moment-generating capacity in flexion, extension and axial rotation [18]. These large surface muscles were selected as muscles of interest because of their large force-generating capacity and the ease of measurement with EMG. The main actions of the SCM include forward flexion of the head on the trunk when acting bilaterally. The TRP is capable of extending the head. The SPL extends the head and neck as well as assists in rotation.
2.4 Test Procedure
The corresponding magnitude of EMG in maximum voluntary contraction (MVC) root mean square (RMS) was measured and used in the normalization of EMG signals. EMG data collected during MVC and impact simulation were subjected to quantitative and statistical analysis.
The EMG was recorded using a MP150 physiology recorder produced by BioPac Systems, Inc (California, USA). The sample rate was set to 1 kHz. Proper electrode placement and locations were suggested by the SENIAM project [19]. The skin was shaved, wiped with 50/50 alcohol/distilled water and coated with electrode gel. The electrodes were placed bilaterally on the most prominent part of the SCM and the SPL at the C4 level. A ground electrode was placed above the right acromion.
The MVC was conducted approximately 30 min before the formal tests. The volunteers were asked to sit on the chair with arms perpendicular to the trunk to reduce the involvement of the biceps and deltoid, feet suspension to avoid lower extremity muscle force. The methods to elicit the MVC of the muscles were as follows: For the TRP, the volunteer shrugged the shoulder with maximum force; for the SCM, the volunteer performed left and right lateral flexion; and for the SPL, the volunteer performed flexion and extension of the head. The EMG amplitudes recorded during the simulation trials were normalized against these maximal values.
In the driving simulator, volunteers drove the simulated vehicle at speeds of 20 km/h, 50 km/h, 80 km/h and 100 km/h. The EMG signals of selected muscles were recorded.
EMG signals in the MVC test and simulation were subjected to band-pass filtering of 10–350 Hz, full wave rectified with a linear envelope.
RMS reflects the average level of muscle discharges over a period of time and is considered to be related to the number of motor units recruited, expressed as
$$ {\text{RMS}} = \sqrt {\frac{1}{T} \cdot \int_{0}^{T} {{\text{EMG}}^{2} (t){\text{d}}t} } $$
where EMG(t) is amplitude of signal and T denotes the length of the signal. RMS can be used to estimate the muscle strength produced by a muscle. The EMG amplitude corresponding to MVC for each muscle was given a value of 100%. The EMG amplitudes recorded in the simulation test were normalized against these maximal values.
In addition, the slope of EMG is expressed as follows:
$${\text{Slope}}_{\text{EMG}} = ({\text{EMG}}_{\text{max}} - {\text{EMG}}_{\text{act}})/T$$
where EMGmax is the peak value of EMG, EMGact is the activation of EMG and T is the time of EMG signal raising from activation state to peak value. The time of muscle activation was determined by an EMG increase of 5% of MVC over the baseline.
Statistical analysis was carried out using the SPSS statistical package to calculate descriptive statistics, correlation analysis between EMG and driving speed.
3 Numerical Study of Frontal Impact
In a numerical study of impact injury, muscular activation values were based on the data collected in the experimental study.
3.1 Human Body Model
A MADYMO 50th percentile male facet occupant model with active neck muscles was used in the simulation.
The facet occupant model is composed of chains of rigid and flexible bodies, which represent the inertial properties of the segments, connected by kinematic joints. The outer surface is described with meshes of shell-type contact element, representing the deformation of soft tissues defined by stress-based contact characteristics. The model allows a more accurate geometric representation and efficient computation.
The facet neck model designed for muscle tensing simulation is defined with relevant muscles addition to anatomical structures, as shown in Fig. 3. In the figure, the SCM, TRP and SPL are shown in blue, black and red. Most muscles are represented by more than one Hill-type muscle element to account for different attachment points of the muscle group. In this model, 16 pairs of cervical muscles are modeled by 68 symmetrical pairs of muscle elements. The muscle elements are divided into segments that curve around the vertebrae. The muscle activation can be defined by a time-varying function.
50th percentile male facet neck model (a) front view; (b) side view; (c) rear view
3.2 Simulation of Braced Occupant
The steering wheel and airbag typically make forceful contact with the occupant during a secondary collision. Thus, their settings and arrangement can have a significant effect on injury. As such, these factors were excluded in the simulation to reduce the interference of external factors.
The occupant model in driving posture was positioned in a sled system equipped with a seat, floor panel and belt. Sliding contact interfaces were defined between the seat and the skin of the model. The three-point belt system was set with standard belt stiffness and retractor properties.
The activity of muscles was set by a time-dependent curve, where the experimental slope was also considered. The simulations were conducted in two groups for each speed. The deceleration pulses were acquired from frontal 100% rigid barrier impacts at speeds of 20, 50, 80 and 100 km/h. In the passive occupant model, all activations of cervical muscles were set as 0.005 for model stability. In contrast to the passive model, activations of the SCM, TRP and SPL muscles in the active model were set according to the recorded experimental activation values. The whole simulation process took 250 ms, and the muscle activities and impact pulse were released at 0 ms and 100 ms, respectively. This left 100 ms for the muscles to react to the upcoming impact, because it takes time for muscles to generate muscular forces that can be transformed to joint movements after contraction.
In accord with the Euro NCAP requirements, neck shear, tension and extension were used as criteria to assess neck injuries. These parameters during impact were selected for injury analysis.
4 Results of the Experimental Study and Numerical Study
Nine of ten volunteers pressed the brake pedal with their right foot, gripped the steering wheel with both hands, stretched their elbows and pushed their trunk to the seatback when the obstacle vehicle cut in front of the host vehicle. The head had a slight backward tendency due to this movement, consistent with the evasive action of passengers described in previous studies [14]. One volunteer had a habitual evasive action of pressing the right foot to the brake pedal and rotating the steering wheel to avoid the obstacle at the same time. The data for this volunteer were removed from the analysis.
4.1 EMG Amplitude of Cervical Muscles
The normalized RMS of EMG (%RMS) was used to characterize the muscle force produced during the collision. The mean RMS of the three muscles of the SPL, SCM and TRP was calculated for nine volunteers. The levels of muscle activation are shown in Fig. 4. The SPL had the lowest magnitude of normalized EMG (less than 13%), while the TRP had the maximum activity (ranging from 18 to 26%). The SCM and TRP exhibited similar amplitude patterns. With the increase in driving speed, the %RMS tended to increase. This occurred because increased driving speed led to greater stimulation among volunteers, causing an increase in the level of muscle contraction. However, no regularity was found in the SPL.
Activation of cervical muscles
The slope of EMG activity in the SCM and TRP increased with driving speed. The slopes of the EMG increase in the SPL did not follow the pattern exhibited in the SCM and TRP, as shown in Fig. 5.
\({\text{Slope}}_{\text{EMG}}\) of cervical muscles
4.2 Statistical Analysis
In a multivariate analysis of variance, the driving speed and the muscle examined had a significant effect (P < 0.01) on the EMG values. Increasing acceleration was associated with greater peak EMG. A least significant difference of the EMG demonstrated that the SPL was significantly different from the SCM and TRP (P < 0.001), while the SCM and TRP were similar to each other.
Statistical analysis of the slope of EMG exhibited a similar pattern of results to the EMG values. The driving speed and muscle examined also had a significant effect on the slope of the EMG (P < 0.05). The slopes varied for the examined muscles, and the SPL significantly differed from the other two muscles.
4.3 Occupant Motion
The drivers' motion during frontal impact at speeds of 20 km/h and 50 km/h is shown in Fig. 6. The start of the impact pulse was set as the beginning of collision. The sled was stopped by the impact pulse, while the occupant kept moving forward, and the movement was then restricted by the seat belt. The motion of the neck can be divided into protraction and flexion. From 0 to 60 ms, the head exhibited horizontal translational displacement relative to the torso, which was called the phase of neck protraction motion. The neck was exposed to a high load when the terminal point of neck protraction was reached. This led to the phase of neck flexion motion, which was observed from 90 ms. The head, restrained by the neck, was moving downward. Comparing the motion of active and passive cervical occupants in frontal impacts of the same speed revealed differences in the trajectories of the head. In the phase of neck protraction motion, the head extended backward due to muscle force in the braced occupant, while the head remained upright in the relaxed occupant. In the phase of neck flexion motion, the head was retracted by the torso through the cervical spine. At 90 ms, the head of the active occupant model had a slightly more forward position than the passive model. At 120 ms and 150 ms, there were no differences between the two models. In addition, the results revealed that there were no differences in the trajectory of other segments.
Motion during collisions in different muscle activation states and impact speed: a passive, 20 km/h; b active, 20 km/h; c passive, 50 km/h; d active, 50 km/h
4.4 Injury Parameters
The injury parameters of the cervical muscles, with muscles in an activated or relaxed condition, are shown in Fig. 7.
Injury parameters in frontal impact: a 20 km/h; b 50 km/h; c 80 km/h; d 100 km/h
The main peak value of the loads was observed in both active and passive models. The peak value for all the parameters increased with impact speed. The load reached its peak at approximately 70 ms, 65 ms, 60 ms and 50 ms at speeds from 20 to 100 km/h, respectively. The active model was less stressed than the passive model in the 20 and 50 km/h impact conditions. In high-speed impacts, some parameters for the active model had high values. At the same time, because all of the loads started earlier in the active model, it can be assumed that the active model had small change rates of loads.
In the 20 km/h impact condition, the maximum torque reached 16.8 N·m and 27 N·m in the active model and passive model, respectively, and the maximum shear values were 1583.4 N and 1941 N. The active model exhibited greater tension in the neck than the passive model, by 34 N, although the incline of tension was smaller.
For the 50 km/h simulation, the torque reached 20.3 N·m and 33.4 N·m for the active and passive model, respectively. The shear reached 1952 N and 2420 N, respectively. The peak tension in the active and passive model was 921 N and 839 N. A second peak was observed at approximately 130 ms, and the peak values decreased compared with their counterpart in various degrees.
When exposed to high-impact pulses, the soft tissue performed poorly in terms of computational stability. The curves of tension and shear were completely different from those in the lower-speed condition. The active model exhibited greater tension force and shear force, while the passive model exhibited greater torque. In the 80 km/h impact condition, the second peak of tension and shear was greater than the first peak. In the 100 km/h impact condition, three peaks of a similar magnitude were observed in the tension curve. The first peak of the shear curve was much higher than the second peak.
Considering the differences between crash tests and real-word traffic accidents, it is critical to understand how closely simulated injuries conform to real-world injuries. The key issue relates to the investigation of occupant-initiated muscular activation and its effects on impact responses and occupant behavior during vehicle collisions.
Investigation of the reaction of occupants in low- to high-speed realistic impacts is challenging, because the sled test using human volunteers is typically limited to low-speed impacts for ethical reasons, constraining the realism of the driving environment. Therefore, reconstructing impact scenarios using a dynamic driving simulator could provide a powerful tool for investigating occupants' reactions in traffic accidents. HBMs that consider muscle activity are widely employed in studying muscle activity in crash scenarios.
In the current study, volunteers participated in driving simulators to investigate muscular responses of drivers' cervical muscles in a simulated frontal impact. A scenario of sudden unavoidable frontal impact was used, in which volunteers undertook simulated driving at speeds of 20, 50, 80 and 100 km/h.
The normalized EMG revealed that the maximum activity for the TRP ranged from 18 to 26%, while the activation of SCM varied from 12 to 22%. The SPL exhibited the lowest level of normalized EMG activity, which was less than 13%. The activation of the cervical muscles increased with driving speed. The SCM and TRP exhibited similar activation and differed from the SPL.
According to the anatomy of the human neck, the main actions of the SCM include forward flexion of the head on the trunk when acting bilaterally. The TRP is capable of extending the head. The SPL extends the head and neck as well as assisting in rotation. The SPL and TRP are agonist muscles for the extension of the head, while the SCM is the antagonist muscle. During the collision, the activated muscles change the kinematics of the occupant, typically causing the head to stretch backward. The SPL, which acts as an agonist muscle for the extension, is less activated than the SCM, which is the antagonist muscle. The introduction of the steering wheel provides an important constraint for the driver besides the seat and seat belt. With the anticipation of an unavoidable upcoming crash, the driver tends to extend the elbow and knee joints, pushes the trunk to the seatback and moves the head backward with the trunk [14]. The whole body muscles, especially in upper extremities and trunk, share the role of the SPL. Thus, the extension of the head is not only caused by the cervical muscles. At the same time, the brake pedal provides support to the leg, which also affects the whole body kinematics.
In the second part of the current study, a numerical study was carried out with an active cervical muscle model. Two key points should be noted. First, the muscle force was relatively small compared with the impact pulse; second, the duration of impact pulse was approximately 150 ms, and the movement of the segment was 50–150 ms after the activation of the muscle [20]. Thus, the active muscle force has traditionally been ignored in crash safety. The current study concentrated on the pre-crash reaction, including the muscle activation released before the impact pulse in the simulation. The muscle force had sufficient time to act and influence the kinematics of the occupant.
In terms of the trajectory of the models in different impact pulses, the head of the active model stretched backward before the pulse was transmitted to the human model, while no change was observed in the passive model. The head of the active model reached the end of the natural range of protraction earlier than the passive model. The motion of the other segments was not affected.
In terms of the neck loads for different speeds of frontal impact, the tension, shear and torque were relevant to the neck injury criteria. Activation of the cervical muscles appears to be a positive factor in reducing neck injury in lower-speed frontal impacts. The peak torques and shears in the active model were much lower than those in the passive model. The active model exhibited slightly greater neck tension than the passive model, but lower slope. In high-speed frontal impact, the inclusion of neck muscle activation reduced the torque but enhanced the shear, which could increase the risk of shear injury.
The muscle itself is one of the neck injury sites. The mechanism of cervical muscle injury involves the lengthening of the muscle by the inertia of the head during active contraction. The procedure is called eccentric contraction. In frontal impact, the posterior located muscles are activated and lengthened in the phase of protraction motion and flexion motion. Thus, the TRP and SPL would be likely to sustain more serious injury in the active model than the passive model.
One limitation of the current study is related to the absence of activation of other cervical muscles. The human neck has complex musculoskeletal anatomy with redundancy. All muscles are involved in movement, in accord with muscle recruitment strategies. With EMG measurement alone, it is not practical to acquire the activation scale for each muscle. In future, estimation of activation of cervical muscles based on measured activation should be studied.
Driving simulators provide an effective tool for studying the evasive behavior of occupants in real traffic accidents.
Drivers often perform evasive maneuvers when they are aware of an impending collision. Evasive behaviors typically cause a significant increase in EMG activity compared with normal driving. The SCM and TRP muscles exhibit a similar pattern of activation.
Contraction of the cervical muscles can positively reduce the risk of neck injury in lower-speed frontal impact. However, the risk of shear injury may increase in high-speed impacts.
The posterior muscles experience eccentric contraction in frontal impact, potentially leading to muscle injury.
This work is supported by National Natural Science Foundation of China (Nos. 51775236, 51675224, U1564214) and National Key R&D Program of China (Nos. 2017YFB0102600, 2018YFB0105205).
On behalf of all the authors, the corresponding author states that there is no conflict of interest.
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1.State Key Laboratory of Automotive Simulation and ControlJilin UniversityChangchunChina
2.School of Public HealthJilin UniversityChangchunChina
3.College of Biological and Agricultural EngineeringJilin UniversityChangchunChina
Gao, Z., Li, Z., Hu, H. et al. Automot. Innov. (2019) 2: 93. https://doi.org/10.1007/s42154-019-00060-6
Publisher Name Springer Singapore | CommonCrawl |
On the elliptic equation $\Delta u+ku-Ku^p=0$ on complete Riemannian manifolds and their geometric applications
Authors: Peter Li, Luen-fai Tam and DaGang Yang
Journal: Trans. Amer. Math. Soc. 350 (1998), 1045-1078
MSC (1991): Primary 58G03; Secondary 53C21
DOI: https://doi.org/10.1090/S0002-9947-98-01886-8
Abstract | References | Similar Articles | Additional Information
Abstract: We study the elliptic equation $\Delta u + ku - Ku^{p} = 0$ on complete noncompact Riemannian manifolds with $K$ nonnegative. Three fundamental theorems for this equation are proved in this paper. Complete analyses of this equation on the Euclidean space ${\mathbf {R}}^{n}$ and the hyperbolic space ${\mathbf {H}}^{n}$ are carried out when $k$ is a constant. Its application to the problem of conformal deformation of nonpositive scalar curvature will be done in the second part of this paper.
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J. Bland and M. Kalka, Complete metrics conformal to the hyperbolic disc, Proc. Amer. Math. Soc. 97 (1986), 128-132.
K.S. Cheng and J.T. Lin, On the elliptic equations $\Delta u=K(x)u^{\sigma }$ and $\Delta u=K(x)e^{2u}$, Trans. Amer. Math. Soc. 304 (1987), 639-668.
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K.S. Cheng and W.M. Ni, On the structure of the conformal scalar curvature equation on ${\mathbf {R}}^{n}$, Indiana Univ. Math. J. 41 (1992), 261-278.
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W.Y. Ding and W.M. Ni, On the elliptic equation $\Delta u +Ku^{\frac {n+2}{n-2}} =0$ and related topics, Duke Math. J. 52 (1985), 485-506.
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J. Lee and T. Parker, The Yamabe problem, Bull. Amer. Math. Soc. 17 (1987), 37-91.
R. McOwen, On the equation $\Delta u + K(x)e^{2u} = f$ and prescribed negative curvature on $\mathbf {R}^{2}$, J. Math. Anal. Appl. 103 (1984), 365-370.
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W.M. Ni, On the elliptic equation $\Delta u + K(x)u^{(n+2)/(n-2)} = 0$, its generalizations, and applications in geometry, Indiana Univ. Math. J. 31 (1982), 495-529.
W.M. Ni, On the elliptic equation $\Delta u + Ke^{2u}= 0$ and conformal metrics with prescribed Gaussian curvatures, Invent. Math. 66 (1982), 343-352.
E.S. Noussair, On the existence of solutions of nonlinear elliptic boundary value problems, J. Diff. Equations 34 (1979), 482-495.
R. Osserman, On the inequality $\Delta u \ge f(u)$, Pacific J. Math. 7 (1957), 1641-1647.
D.H. Sattinger, Conformal metrics in $\mathbf {R}^{2}$ with prescribed Gaussian curvature, Indiana Univ. Math. J. 22 (1972), 1-4.
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R. Schoen and S.T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), 47-71.
M. Troyanov, The Schwarz lemma for nonpositively curved Riemannian surfaces, Man. Math. 72 (1991), 251-256.
N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 265-274.
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Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 58G03, 53C21
Retrieve articles in all journals with MSC (1991): 58G03, 53C21
Affiliation: Department of Mathematics, University of California, Irvine, California 92697-3875
Email: [email protected]
Luen-fai Tam
Affiliation: Department of Mathematics, Chinese University of Hong Kong, Shatin, NT, Hong Kong
MR Author ID: 170445
Email: [email protected]
DaGang Yang
Affiliation: Department of Mathematics, Tulane University, New Orleans, Louisiana 70118
Email: [email protected]
Keywords: Conformal deformation, prescribing scalar curvature, complete Riemannian manifolds, semi-linear elliptic PDE, generalized maximum principle, analysis on manifolds
Received by editor(s): May 23, 1995
Additional Notes: The first two authors are partially supported by NSF grant DMS 9300422. The third author is partially supported by NSF grant DMS 9209330
Article copyright: © Copyright 1998 American Mathematical Society | CommonCrawl |
\begin{document}
\draft \twocolumn[\hsize\textwidth\columnwidth\hsize\csname @twocolumnfalse\endcsname
\title{Observation of three-photon Greenberger-Horne-Zeilinger entanglement} \author{Dik Bouwmeester, Jian-Wei Pan, Matthew Daniell, Harald Weinfurter \& Anton Zeilinger} \address{Institut f\"{u}r Experimentalphysik, Universit\"{a}t Innsbruck, Technikerstr. 25, A-6020 Innsbruck, Austria} \date{submitted to PRL} \maketitle \begin{abstract} We present the experimental observation of polarization entanglement for three spatially separated photons. Such states of more than two entangled particles, known as GHZ states, play a crucial role in fundamental tests of quantum mechanics versus local realism and in many quantum information and quantum computation schemes. Our experimental arrangement is such that we start with two pairs of entangled photons and register one photon in a way that any information as to which pair it belongs to is erased. The registered events at the detectors for the remaining three photons then exhibit the desired GHZ correlations. \end{abstract}
\vskip2pc ]
\narrowtext
Ever since the seminal work of Einstein, Podolsky and Rosen \cite{EINSTEIN} there has been a quest for generating entanglement between quantum particles. Although two-particle entanglements have long been demonstrated experimentally \cite{WU,BBO}, the preparation of entanglement between three or more particles remains an experimental challenge. Proposals have been made for experiments with photons \cite{ZEIL97} and atoms \cite{PARIS}, and three nuclear spins within a single molecule have been prepared such that they locally exhibit three-particle correlations \cite{NMR1}. However, until now there has been no experiment which demonstrates the existence of entanglement of more than two spatially separated particles. Here we report the experimental observation of polarization entanglement of three spatially separated photons.
The original motivation to prepare three-particle entanglements stems from the observation by Greenberger, Horne and Zeilinger (GHZ) that three-particle entanglement leads to a conflict with local realism for non-statistical predictions of quantum mechanics \cite{GHZ}. This is in contrast to the case of Einstein, Podolsky and Rosen experiments with two entangled particles testing Bell´s inequalities, where the conflict only arises for statistical predictions \cite{BELL}. We will experimentally address this issue in a forthcoming paper.
The incentive to produce GHZ states has been significantly increased by the advance of the field of quantum communication and quantum information processing. Entanglement between several particles is the most important feature of many such quantum communication and computation protocols \cite{A,B}.
We now describe the experimental arrangements for the observation of three-photon GHZ entanglement. The experimental techniques are similar to those that have been developed for our previous experiments on quantum teleportation \cite{BOU97} and entanglement swapping \cite{PAN98}. In fact, one of the main complications in the previous experiments, namely the creation of two pairs of photons by a single source, is here turned into a virtue.
The main idea, as was put forward in Ref.\cite{ZEIL97}, is to transform two pairs of polarization entangled photons into three entangled photons and a fourth independent photon. Figure 1 is a schematic drawing of our experimental setup. Pairs of polarization entangled photons are generated by a short (approx. 200 fs) pulse of UV-light of wavelength $\lambda=788nm$ from a mode-locked Ti-Sapphire laser, which passes through an optically nonlinear crystal (here Beta-Barium-Borate, BBO). The probability per pulse to create a single pair in the desired modes is rather low and of the order of a few $10^{-4}$. The pair creation is such that the following polarization entangled state is obtained \cite{BBO}. \begin{equation} \label{polstate} \frac{1}{\sqrt{2}} \left( \ket{H}_a\ket{V}_b - \ket{V}_a\ket{H}_b \right) \,. \end{equation} This state indicates that there is a superposition of the possibility that the photon in arm $a$ is horizontally polarized and the one in arm $b$ vertically polarized ($\ket{H}_a\ket{V}_b$), and the opposite possibility, i.e., $\ket{V}_a\ket{H}_b$. The minus sign indicates that there is a fixed phase difference of $\pi$ between the two possibilities. For our GHZ experiment this phase factor is actually allowed to have any value, as long as it is fixed for all pair creations.
The setup is such that arm $a$ continues towards a polarizing beamsplitter, where $H$ photons are transmitted towards detector T and $V$ photons are reflected, and arm $b$ continues towards a 50/50 polarization-independent beamsplitter. From each beamsplitter one output is directed to a second polarizing beamsplitter. In between the two polarizing beamsplitters there is a $\lambda/2$ retardation plate at an angle of $22.5^{\circ}$ which rotates the vertical polarization of the photons reflected by the first polarizing beamsplitter into a $45^{\circ}$ polarization, i.e. a superposition of $\ket{H}$ and $\ket{V}$ with equal amplitudes. We use three more detectors, $D_1$, $D_2$, and $D_3$, in the remaining output arms. Narrow band-width interference filters are placed in front of the four detectors ($\delta\lambda=4.5\,$ nm for the detector T and $\delta\lambda=3.6\,$ nm for the other three). Including filter losses, coupling into single-mode fibers, and the Si-avalanche detector efficiency, the total collection and detection probability of a photon is about 10\%.
Consider now the case that {\em two} pairs are generated by a single UV-pulse, and that the four photons are all detected, one by each detector T, $D_1$, $D_2$, and $D_3$. Our claim is that by the coincident detection of four photons and because of the brief duration of the UV pulse and the narrowness of the filters, one can conclude that a three-photon GHZ state has been recorded by detectors D$_1$, D$_2$, and D$_3$. The reasoning is as follows. When a four-fold coincidence recording is obtained, one photon in path $a$ must have been horizontally polarized and detected by the trigger detector T. Its companion photon in path $b$ must then be vertically polarized, and it has 50\% chance to be transmitted by the beamsplitter (see Figure 1) towards detector D$_3$ and 50\% chance to be reflected by the beamsplitter towards the final polarizing beamsplitter where it will be reflected to D$_2$. Consider the first possibility, i.e. the companion of the photon detected at T is detected by D$_3$ and necessarily carried polarization $V$. Then the counts at detectors D$_1$ and D$_2$ were due to a second pair, one photon travelling via path $a$ and the other one via path $b$. The photon travelling via path $a$ must necessarily be $V$ polarized in order to be reflected by the polarizing beamsplitter in path $a$; thus its companion, taking path $b$, must be $H$ polarized and after reflection at the beamspliter in path $b$ (with a 50\% probability) it will be transmitted by the final polarizing beamsplitter and arrive at detector D$_1$. The photon detected by D$_2$ therefore must be $H$ polarized since it came via path $a$ and had to transit the last polarizing beamsplitter. Note that this latter photon was $V$ polarized but after passing the $\lambda/2$ plate it became polarized at $45^{\circ}$ which gave it 50\% chance to arrive as an $H$ polarized photon at detector D$_2$. Thus we conclude that if the photon detected by D$_3$ is the companion of the T photon, then the coincidence detection by D$_1$, D$_2$, and D$_3$ corresponds to the detection of the state \begin{equation} \label{term1} \ket{H}_1 \ket{H}_2 \ket{V}_3\,. \end{equation} By a similar argument one can show that if the photon detected by D$_2$ is the companion of the T photon, the coincidence detection by D$_1$, D$_2$, and D$_3$ corresponds to the detection of the state \begin{equation} \label{term2} \ket{V}_1 \ket{V}_2 \ket{H}_3\,. \end{equation}
\begin{figure}
\caption{Schematic drawing of the experimental setup for the demonstration of Greenberger-Horne-Zeilinger entanglement for spatially separated photons. Conditioned on the registration of one photon at the trigger detector T, the three photons registered at D$_1$, D$_2$, and D$_3$ exhibit the desired GHZ correlations.}
\label{setup}
\end{figure}
In general the two possible states (\ref{term1}) and (\ref{term2}) corresponding to a four-fold coincidence recording will not form a coherent superposition, i.e. a GHZ state, because they could, in principle, be distinguishable. Besides possible lack of mode overlap at the detectors, the exact detection time of each photon can reveal which state is present. For example, state (\ref{term1}) is identified by noting that T and D$_3$, or D$_1$ and D$_2$, fire nearly simultaneously. To erase this information it is necessary that the coherence time of the photons is substantially longer than the duration of the UV pulse (approx. 200 fs) \cite{MAREK}. We achieved this by detecting the photons behind narrow band-width filters which yield a coherence time of approx. 500 fs. Thus, the possibility to distinguish between states (\ref{term1}) and (\ref{term2}) is no longer present, and, by a basic rule of quantum mechanics, the state detected by a coincidence recording of D$_1$, D$_2$, and D$_3$, conditioned on the trigger T, is the quantum superposition \begin{equation} \label{expGHZ} \frac{1}{\sqrt{2}} \left( \ket{H}_1 \ket{H}_2 \ket{V}_3 + \ket{V}_1 \ket{V}_2 \ket{H}_3 \right) \,, \end{equation} which is a GHZ state \cite{REM}.
The plus sign in Eq.~(\ref{expGHZ}) follows from the following more formal derivation. Consider two down-conversions producing the product state \begin{equation} \label{productstate} \frac{1}{2} \left( \ket{H}_a\ket{V}_b-\ket{V}_a\ket{H}_b \right) \left( \ket{H}^{\prime}_a\ket{V}^{\prime}_b- \ket{V}^{\prime}_a \ket{H}^{\prime}_b \right) \,. \end{equation} Initially we assume that the components $\ket{H}_{a,b}$ and $\ket{V}_{a,b}$ created in one down-conversion might be distinguishable from the components $\ket{H}^{\prime}_{a,b}$ and $\ket{V}^{\prime}_{a,b}$ created in the other one. The evolution of the individual components of state (\ref{productstate}) through the apparatus towards the detectors T, D$_1$, D$_2$, and D$_3$ is given by \begin{eqnarray} \ket{H}_a &\rightarrow& \ket{H}_T \,, \\ \ket{V}_b &\rightarrow& \frac{1}{\sqrt{2}}(\ket{V}_2+\ket{V}_3) \,,\\ \ket{V}_a &\rightarrow& \frac{1}{\sqrt{2}}(\ket{V}_1+\ket{H}_2) \,, \\ \ket{H}_b &\rightarrow& \frac{1}{\sqrt{2}}(\ket{H}_1+\ket{H}_3) \,. \end{eqnarray} Identical expressions hold for the primed components. Inserting these expressions into state (\ref{productstate}) and restricting ourselves to those terms where only one photon is found in each output we obtain \begin{eqnarray} \label{stateC} - \frac{1}{4\sqrt{2}} \left\{ \ket{H}_T \left( \ket{V}^{\prime}_1\ket{V}_2\ket{H}^{\prime}_3+ \ket{H}^{\prime}_1\ket{H}^{\prime}_2\ket{V}_3 \right) \right. \nonumber \\ \left. + \ket{H}^{\prime}_T \left( \ket{V}_1\ket{V}^{\prime}_2\ket{H}_3+\ket{H}_1\ket{H}_2\ket{V}^{\prime}_3 \right) \right\} \,. \end{eqnarray} If now the experiment is performed such that the photon states from the two down-conversions are indistinguishable, we finally obtain the desired state (up to an overall minus sign) \begin{equation} \label{expGHZfinal} \frac{1}{\sqrt{2}}\ket{H}_T \left( \ket{H}_1 \ket{H}_2 \ket{V}_3 + \ket{V}_1 \ket{V}_2 \ket{H}_3 \right) \,. \end{equation}
Note that the total photon state produced by our setup, i.e., the state before detection, also contains terms in which, for example, two photons enter the same detector. In addition, the total state contains contributions from single down-conversions. The four-fold coincidence detection acts as a projection measurement onto the desired GHZ state (\ref{expGHZfinal}) and filters out these undesireable terms. The efficiency for one UV pump pulse to yield such a four-fold coincidence detection is very low (of the order of $10^{-10}$). Fortunately, $7.6\times 10^{7}$ UV-pulses are generated per second, which yields about one double pair creation and detection per 150 seconds, which is just enough to perform our experiments. Triple pair creations can be completely neglected since they can give rise to a four-fold coincidence detection only about once every day.
To experimentally demonstrate that a GHZ state has been obtained by the method described above, we first verified that, conditioned on a photon detection by the trigger T, both the H$_1$H$_2$V$_3$ and the V$_1$V$_2$H$_3$ component are present and no others. This was done by comparing the count rates of the eight possible combinations of polarization measurements, H$_1$H$_2$H$_3$, H$_1$H$_2$V$_3$, ..., V$_1$V$_2$V$_3$. The observed intensity ratio between the desired and undesired states was 12:1. Existence of the two terms as just demonstrated is a necessary but not yet sufficient condition for demonstrating GHZ entanglement. In fact, there could in principle be just a statistical mixture of those two states. Therefore, one has to prove that the two terms coherently superpose. This we did by a measurement of linear polarization of photon 1 along $+45^{\circ}$, bisecting the H and V direction. Such a measurement projects photon 1 into the superposition \begin{equation} \label{photon1} \ket{+45^{\circ}}_1=\frac{1}{\sqrt{2}}(\ket{H}_1 + \ket{V}_1)\,, \end{equation} what implies that the state (\ref{expGHZfinal}) is projected into \begin{equation} \label{totalstate} \frac{1}{\sqrt{2}} \ket{H}_T\ket{+45^{\circ}}_1(\ket{H}_2 \ket{V}_3 + \ket{V}_2\ket{H}_3)\,. \end{equation}
Thus photon 2 and 3 end up entangled as predicted under the notion of "entangled entanglement" \cite{entent}. We conclude that demonstrating the entanglement between photon 2 and 3 confirms the coherent superposition in state (\ref{expGHZfinal}) and thus existence of the GHZ entanglement. In order to proceed to our experimental demonstration we represent the entangled state (2-3) in a linear basis rotated by $45^{\circ}$. The state then becomes \begin{equation} \label{45basis} \frac{1}{\sqrt{2}} \left( \ket{+45^{\circ}}_2\ket{+45^{\circ}}_3 - \ket{-45^{\circ}}_2 \ket{-45^{\circ}}_3 \right) \,, \end{equation} which implies that if photon 2 is found to be polarized along -$45^{\circ}$, photon 3 is also polarized along the same direction. We test this prediction in our experiment. The absence of the terms $\ket{+45^{\circ}}_2\ket{-45^{\circ}}_3$ and $\ket{-45^{\circ}}_2\ket{+45^{\circ}}_3$ is due to destructive interference and thus indicates the desired coherent superposition of the terms in the GHZ state (\ref{expGHZfinal}). The experiment therefore consisted of measuring four-fold coincidences between the detector T, detector 1 behind a +$45^{\circ}$ polarizer, detector 2 behind a -$45^{\circ}$ polarizer, and measuring photon 3 behind either a +$45^{\circ}$ polarizer or a -$45^{\circ}$ polarizer. In the experiment, the difference of arrival time of the photons at the final polarizer, or more specifically, at the detectors D1 and D2, was varied.
\begin{figure}
\caption{Experimental confirmation of GHZ entanglement. Graph (a) shows the results obtained for polarization analysis of the photon at D$_3$, conditioned on the trigger and the detection of one photon at D$_1$ polarized at $45^{\circ}$ and one photon at detector D$_2$ polarized $-45^{\circ}$. The two curves show the four-fold coincidences for a polarizer oriented at $-45^{\circ}$ and $45^{\circ}$ respectively in front of detector D$_3$ as function of the spatial delay in path $a$. The difference between the two curves at zero delay confirms the GHZ entanglement. By comparison (graph (b)) no such intensity difference is predicted if the polarizer in front of detector D$_1$ is set at 0$^{\circ}$. }
\label{result}
\end{figure}
The data points in Fig.2(a) are the experimental results obtained for the polarization analysis of the photon at D$_3$, conditioned on the trigger and the detection of two photons polarized at $45^{\circ}$ and $-45^{\circ}$ by the two detectors D$_1$ and D$_2$, respectively. The two curves show the four-fold coincidences for a polarizer oriented at $-45^{\circ}$ (squares) and $+45^{\circ}$ (circles) in front of detector D$_3$ as function of the spatial delay in path $a$. From the two curves it follows that for zero delay the polarization of the photon at D$_3$ is oriented along $-45^{\circ}$, in accordance with the quantum-mechanical predictions for the GHZ state. For non-zero delay, the photons travelling via path $a$ towards the second polarizing beamsplitter and those travelling via path $b$ become distinguishable. Therefore increasing the delay gradually destroys the quantum superposition in the three-particle state.
Note that one can equally well conclude from the data that at zero delay, the photons at D$_1$ and D$_3$ have been projected onto a two-particle entangled state by the projection of the photon at D$_2$ onto $-45^{\circ}$. The two conclusions are only compatible for a genuine GHZ state. We note that the observed visibility was as high as 75\%.
For an additional confirmation of state (\ref{expGHZfinal}) we performed measurements conditioned on the detection of the photon at D$_1$ under $0^{\circ}$ polarization (i.e. $V$ polarization). For the GHZ state (1/$\sqrt{2}$)(H$_1$H$_2$V$_3$ + V$_1$V$_2$H$_3$) this implies that the remaining two photons should be in the state V$_2$H$_3$ which cannot give rise to any correlation between these two photons in the $45^{\circ}$ detection basis. The experimental results of these measurement are presented in Fig.2(b). The data clearly indicate the absence of two-photon correlations and thereby confirm our claim of the observation of GHZ entanglement between three spatially separated photons.
Although the extension from two to three entangled particles might seem to be only a modest step forward, the implications are rather profound. First of all, GHZ entanglements allow for novel tests of quantum mechanics versus local realistic models. Secondly, three-particle GHZ states might find a direct application, for example, in third-man quantum cryptography. And thirdly, the method developed to obtain three-particle entanglement from a source of pairs of entangled particles can be extended to obtain entanglement between many more particles \cite{KNIGHT}, which are at the basis of many quantum communication and computation protocols. Most applications of GHZ states imply that the three particles have to be detected. Therefore, even as our setup only produces GHZ entanglement upon the condition that the three photons and the trigger photon are actually detected, our scheme can readily be used for many applications. The detection plays the double role of projecting onto the GHZ state and of performing a specific measurement on this state. Finally, we note that our experiment, together with our earlier realization of quantum teleportation \cite{BOU97} and entanglement swapping \cite{PAN98} provides necessary to tools to implement a number of novel entanglement distribution and network ideas as recently proposed \cite{E,F}.
We are very grateful to M.A.~Horne and D.M.~Greenberger for their useful criticism and detailed suggestions for improvements of our initial manuscript. This work was supported by the Austrian Science Foundation FWF (Project No. S6502), the Austrian Academy of Sciences, the US National Science Foundation NSF (Grant No. PHY 97-22614) and the TMR program of the European Union (Network Contract No. ERBFMRXCT96-0087).
\end{document} | arXiv |
Find the value of $x$ such that $\sqrt{x - 2} = 8$.
Squaring both sides of the equation $\sqrt{x - 2} = 8$, we get $x - 2 = 8^2 = 64$, so $x = 64 + 2 = \boxed{66}$. | Math Dataset |
\begin{document}
\title[AR \& HW Conj over quasi-fiber product rings]
{Auslander-Reiten and Huneke-Wiegand conjectures over quasi-fiber product rings}
\author{T. H. Freitas} \address{Universidade Tecnol\'ogica Federal do Paran\'a, 85053-525, Guarapuava-PR, Brazil} \email{[email protected]}
\author{V. H. Jorge P\'erez} \address{Universidade de S{\~a}o Paulo - ICMC, Caixa Postal 668, 13560-970, S{\~a}o Carlos-SP, Brazil} \email{[email protected]}
\author{R. Wiegand} \address{University of Nebraska-Lincoln} \email{[email protected] }
\author{S. Wiegand} \address{University of Nebraska-Lincoln}
\thanks{All four authors were partially supported by FAPESP-Brazil 2018/05271-6, 2018/05268-5 and CNPq-Brazil 421440/2016-3. RW was partially supported by Simons Collaboration Grant 426885.}
\keywords{Auslander-Reiten Conjecture, Huneke-Wiegand Conjecture, vanishing of Ext, fiber product rings, Tor-rigid modules} \subjclass[2010]{ 13D07, 13H10, 13C15.}
\begin{abstract}
In this paper we explore consequences of the vanishing of $\Ext$ for finitely generated modules over a quasi-fiber product ring $R$; that is, $R$ is a
local ring
such that $R/(\underline x)$ is a non-trivial fiber product ring, for some regular sequence $\underline x$ of $R$.
Equivalently,
the maximal ideal of $R/(\underline x)$ decomposes as a direct sum of two nonzero ideals.
Gorenstein quasi-fiber product rings
are AB-rings and are Ext-bounded. We show in Theorem~\ref{thm:arcminimal} that quasi-fiber product rings
satisfy a sharpened form of the Auslander-Reiten Conjecture. We also make some observations related to the Huneke-Wiegand conjecture for quasi-fiber product rings. \end{abstract}
\maketitle
\vskip-40pt
\begin{dedication}
{This article is dedicated to the memory of Nicholas Baeth} \end{dedication}
\section{Introduction}
This article is motivated
by the celebrated Auslander-Reiten Conjecture (ARC)
and the Huneke-Wiegand Conjecture for integral domains (HWC$_d$); see \cite[p. 70]{AR}, \cite{HL}, and
\cite[pp. 473--474]{HW}: \begin{definition}\label{defarchwc} Let $R$ be a commutative Noetherian local ring.
\begin{enumerate}
\item[] (ARC)\ \ {\bf Auslander-Reiten Conjecture.}
If $M$ is a finitely generated $R$-module such that
$\Ext^i_R(M,M\oplus R)=0$, for all $i\geq 1$, then $M$ is free.\\
\item[] (HWC$_d$)\ \ {\bf Huneke-Wiegand Conjecture (for domains).} If $R$ is a Gorenstein local domain and
$M$ is a finitely generated torsion-free $R$-module $M$ such that
$M\otimes_RM^\ast$ is
reflexive, then $M$ is free. \end{enumerate} \end{definition}
\noindent Here $M^*$ denotes the algebraic dual of $M$, namely, $\Hom_R(M,R)$. Recall that an $R$-module $M$ is t{\em orsion-free} provided every non-zerodivisor of $R$ is a non-zerodivisor on $M$.
Several positive cases for (ARC) are known; see, for instance, work of Huneke, Leuschke, Goto, Takahashi, Nasseh, Sather-Wagstaff, Christensen, Holm, Avramov, and Iyengar in \cite{HL}, \cite{GT}, \cite{CeT}, \cite{NS}, \cite{CH} and \cite{AINS}. Huneke and R.~Wiegand \cite{HW} established (HWC$_d$) over hypersurfaces (see Remark \ref{hw-main}), but (HWC$_d$) is still open for Gorenstein domains, even if $M$ is assumed to be an ideal of the ring; see the article of Huneke, R.~Wiegand,
and Iyengar \cite{HIW} or Celikbas \cite{Ce1}.
At the other extreme, we know of no counterexample to the following general form of the conjecture: \begin{enumerate} \item[] (G-HWC$_d$)\ \ {\bf Huneke-Wiegand Conjecture (generalized, domain).} Let $R$ be a local domain, and let $M$ be an $R$-module. If $M\otimes_RM^*$ is maximal Cohen-Macaulay (henceforth abbreviated ``MCM''), then $M$ is free. \end{enumerate} Any attempt to solve (G-HWC$_d$) is likely to involve knowing what properties (weaker than being free) one can conclude about $M$ from the assumption that $M\otimes_RM^*$ is MCM. For example, we might conjecture that $M$ is forced to be torsion-free:
\begin{enumerate} \item[] (G-HWC$_{tf}$)\ \ Let $R$ be a local domain, and let $M$ be an $R$-module. If $M\otimes_RM^*$ is maximal Cohen-Macaulay, then $M$ is torsion-free. \end{enumerate} For convenience we usually assume $M$ is torsion-free for our discussion here, as in (HWC$_d$).
(Assuming that $M$ is torsion-free in (HWC$_d$) avoids the trivial case where $M$ is torsion, and hence $M^*=0=M\otimes_RM^*$.)
Some partial results concerning these conjectures appear in articles by Huneke, Iyengar, and Wiegand \cite{HIW}; Celikbas \cite{Ce0}, \cite{Ce1}; Goto, Takahashi, Taniguchi and Truong \cite{GTTT}; and Garcia-Sanchez and Leamer \cite{P}.
By
a result from Celikbas and R. Wiegand \cite{CRW}, reproduced here as Proposition~\ref{ACT}.
the truth of (HWC$_d$) in the one-dimensional case would imply the general case.
(On the other hand, the truth of
a Huneke-Wiegand conjecture for Gorenstein quasi-fiber product rings---the focus of this article---does {\it not} seem to
reduce to the one-dimensional case; see Remark~\ref{ACTr}).
By Proposition~\ref{ACT}, the truth of (HWC$_d$) also would
imply (ARC)
for Gorenstein domains of
arbitrary dimension.
It is not known whether or not (ARC) implies (HWC$_d$).
Concerning (ARC), there are several situations in which the vanishing of $\Ext^i_R(M,M\oplus R)$ for a specific finite set of values of $i$ is enough
to deduce that $M$ is free, or, perhaps, that $M$ has finite projective dimension;
see for example, results of Huneke and Leuschke \cite[Main Theorem]{HL}; Araya \cite[Corollary 10]{Araya}; and Goto and Takahashi \cite[Theorem 1.5]{GT}.
Our Proposition~\ref{prop:aardvark} and Theorems~\ref{prop:jor} and \ref{thm:arcminimal} are results
along these lines. In addition, we consider a general version of (ARC) that involves two modules:
\begin{comment}
\begin{question} \label{question1} Is there a minimal number of vanishing of $\Ext^i_R(M,M\oplus R)$ that guarantees a bound for the projective dimension of $M$? \end{question}
\end{comment}
\begin{question}\label{question1} For $M$ and $N$ finitely generated $R$-modules, can one find integers $s$ and $t$, with $1\le s \le t$, such that the vanishing of
$\Ext^i_R(M,N\oplus R)$, for all $i$ with $ s\leq i \leq t$, ensures that
$M$ or $N$ has finite projective dimension? \end{question}
The main body of this paper is an investigation of Question~\ref{question1} and of these conjectures over
a quasi-fiber product ring.
(See Setting~\ref{fpset} for conventions and definitions.) Nasseh and Takahashi \cite{NT} introduced the notion of a ``local ring with quasi-decomposable maximal ideal" as an extension of the notion of fiber product ring; we call it a ``quasi-fiber product ring" here.
The class of quasi-fiber product rings includes, for instance, every regular local ring of dimension $d\ge 2$ and every non-hypersurface Cohen-Macaulay ring with minimal multiplicity and with infinite residue field,
as well as every two-dimensional non-Gorenstein normal domain with a rational singularity (\cite[Examples 4.7 and 4.8]{NT}) and,
of course, every fiber product ring.
Recently the
study of fiber product rings has become an active research topic, as is evident in articles by
Nasseh, Sather-Wagstaff, Takahashi and VandeBogart \cite{NS}, \cite{NSTV}, \cite{NT}, \cite{NTV}, \cite{T}, and the current authors \cite{TVRS}.
\begin{comment}
Our main results are the following (see Theorem \ref{thm:arcminimal} and Theorem \ref{lem:HWCQ}).
\begin{thm} Let $R$ be a quasi-fiber product ring
and let $M$ be an $R$-module. If $\Ext^5_R(M,M\oplus R)=\Ext^6_R(M,M\oplus R)=0$, then $\pd_RM \leq 1$. \end{thm}
\begin{thm} Let $R$ be a $d$-dimensional Cohen-Macaulay quasi-fiber product ring
with a canonical module $\omega_R$.
Let $M$ and $N$ be nonzero $R$-modules such that: \begin{itemize}
\item[(i)] $M$ is maximal Cohen-Macaulay.
\item[(ii)] M is locally free on $ {\rm Spec}R- \{\mathfrak{m}\}$.
\item[(iii)] $N$ is maximal Cohen-Macaulay and Tor-rigid.
\item[(iv)] $M\otimes N^\vee$ is maximal Cohen-Macaulay. \end{itemize}
Then $M$ or $N$ is free. \end{thm}
\end{comment}
We briefly describe
the contents of the paper. Section 2 gives the main definitions and basic facts for the rest of this work. Section 3 concerns the vanishing of $\Ext$ over a quasi-fiber product ring.
In Notation and Remarks~\ref{uabnot}, we define and discuss
``AB rings'' and ``Ext-bounded rings'', introduced in Huneke's and Jorgensen's article \cite{HJ}; we
prove, in Theorem~\ref{thm:uac} and Proposition~\ref{cor:class}, that a Gorenstein quasi-fiber product ring
has both of these properties. In Corollary \ref{cor:arc} we verify (ARC) for quasi-fiber product rings. Theorems~\ref{thm:Ext0} and ~\ref{evodthm00} and their corollaries
give implications of the vanishing of finitely many $\Ext_R^i(M,N)$
over a quasi-fiber product ring $R$ under an additional assumption
of Tor-rigidity for $N$.
The remaining theorems of Section 3 concern implications of the vanishing of finitely many $\Ext^i_R(M,M)$
over quasi-fiber product rings without the additional assumption
of Tor-rigidity.
We show in Theorem~\ref{thm:arcminimal} that quasi-fiber product rings satisfy a sharper version of (ARC): For $M$ a finitely generated module over a quasi-fiber product ring, there is a positive integer $b$ such that, if $\Ext^i_R(M,M\oplus R)=0$, for every $i$ with $1\le i\le b$, then $M$ is free. Moreover
Corollary \ref{cor:arcminimal} states that, if $M$ is a finitely generated module over a fiber product ring and $\Ext^i_R(M,M\oplus R)=0$, for every $i$ such that $1\le i\le 6$, then $M$ is free. This improves Nasseh and Sather-Wagstaff's result that fiber product rings satisfy (ARC) \cite{NS}.
In Section 4 we apply the results of Section 3 to obtain some positive results related to (HWC$_d$) and we consider
a more general condition involving two modules.
\section{Setup and background} This section gives basic definitions and properties that are used in later sections.
\begin{Setting}\label{fpset} Throughout this paper, $(R,\mathfrak{m},k)$, or simply $(R,\mathfrak{m})$, denotes a local ring with maximal ideal $\mathfrak{m}$ and residue field $k$. Local rings are always assumed to be commutative and Noetherian, and modules are always assumed to be finitely generated. \begin{itemize}
\item[(i)] $(R,\mathfrak{m},k)$ is the {\it fiber product ring}
$S\times_kT$ of two local rings $S$ and $T$, with the same residue field $k$, if $R$ is the subring of $S\times T$ consisting of pairs $(s,t)$ such that $s\in S$, $t\in T$ and $\pi_S(s) = \pi_T(t)$, where $\pi_S$ and $\pi_T$ denote reduction modulo the maximal ideals $\mathfrak{m}_S$ and $\mathfrak{m}_T$. We always assume fiber product rings are
{\em non-trivial}; that is, neither $S$ nor $T$ is equal to $k$.
\item[(i$'$)] $(R,\mathfrak{m},k)$ has {\it decomposable maximal ideal} if $\mathfrak{m}= I\oplus J$, where $I$ and $J$ are
nonzero ideals of $R$.
\item[(ii)] $(R,\mathfrak{m},k)$ is a {\it quasi-fiber product ring} if there exists an
$R$-sequence $\underline{x}:=x_1,\ldots,x_n$, of length $n\geq 0$, such that
$R/(\underline{x})$ is a non-trivial fiber product ring.
\item[(ii$'$)] $(R,\mathfrak{m},k)$ has {\it quasi-decomposable maximal ideal} if there exists an
$R$-sequence $\underline{x}:=x_1,\ldots,x_n$, of length $n\geq 0$, such that $\mathfrak{m}/(\underline{x})$ is decomposable. \end{itemize}
Ogoma \cite[Lemma 3.1]{Og} observed the following:
\begin{fact}\label{formulafiber} \noindent A local ring $(R,\mathfrak{m},k)$ has decomposable
maximal ideal if and only if $R$ can be realized as a non-trivial fiber product. In fact, if $\mathfrak{m}= I\oplus J$, the map $R\to S\times_{k}T$, given by $r\mapsto (r+I,r+J)$ is an isomorphism, where $S=R/I$ and $T=R/J$. For the converse,
if $R=S\times_k T$, then $\mathfrak{m}= \mathfrak{m}_S\oplus \mathfrak{m}_T$, where $\mathfrak{m}_S$ and $\mathfrak{m}_T$ are the maximal ideals of $S$ and $T$, respectively. \end{fact} Similarly, items (ii) and (ii$'$) are equivalent. We often say that $(R,\mathfrak{m},k)$ is a {\it quasi-fiber product ring with respect to} the regular sequence $\underline{x}$, or $\mathfrak{m}$ is {\it quasi-decomposable with respect to} $\underline{x}$. The case $n=0$ is the case of a fiber product ring, equivalently, a local ring with decomposable maximal ideal. In this article all fiber product rings and quasi-fiber product rings are assumed to be non-trivial. \end{Setting}
Examples of quasi-fiber rings abound. For example, every regular local ring of dimension $d\ge 2$ is a quasi-fiber ring. (If $x_1,\dots,x_d$ generate the maximal ideal, then $R/(x_1x_2,x_3,x_4,\dots,x_d)$ has decomposable maximal ideal.) Many interesting examples of quasi-fiber product rings can be found in the paper \cite[\S4]{NT} by Nasseh and Takahashi.
\begin{fact}\label{pdfin1} Let $(R,\mathfrak{m},k)$ be a fiber product ring and let $M$ be a finitely generated $R$-module. Then: \begin{enumerate}\item $\depth R\leq 1$ (\cite{L81} or \cite[Remark 1.9]{TVRS}). \item
If $\pd_RM <\infty$, then $\pd_RM \le 1$. \end{enumerate} {\rm For (2), use (1) and
Remark~\ref{rem:ABF}, the Auslander-Buchsbaum Formula:}
\end{fact}
\begin{remark}\label{rem:ABF} {\it Auslander-Buchsbaum Formula} \cite[A.5. Theorem, p. 310]{LW} Let $M$ be a nonzero module of finite projective dimension (pd) over a local ring~$R$. Then $\depth M+\pd_RM=\depth R$.
Thus $\pd_RM \le \depth R$. \end{remark}
\begin{definitionsremarks} \label{rem:rank} Recall that a finitely generated module $M$ over a local ring $R$ {\em has rank} provided there is an integer $r$ such that $M_P$ is $R_P$-free of rank $r$ for every $P\in\Ass(R)$. Equivalently, $M\otimes_RK$ is free as a $K$-module, where $K$ is the total quotient ring of $R$,
namely $K=\{\text{non-zerodivisors of}\ R\}^{-1}R$. If $R$ is an integral domain, $M$ {\em always} has rank. It is probably better, for moving about from one ring to another, not to assume that $R$ is a domain, but to invoke the weaker hypothesis that $M$ have rank. (Example~\ref{example1} shows why some such hypothesis is needed.)
A {\it hypersurface ring} is a local ring $(R,\mathfrak{m})$ whose $\mathfrak{m}$-adic completion $\widehat R$ has the form $\widehat R=S/fS$, where $(S,\mathfrak{m}_S)$ is a complete regular local ring and $f\in \mathfrak{m}_S$. More generally a local ring $R$ with maximal ideal $\mathfrak{m}$ is a {\it complete intersection} if the $\mathfrak{m}$-adic completion $\widehat R$ has the form $S/(f)$, where $f$ is a regular sequence and $S$ is a complete regular local ring. (By Cohen's Structure Theorem, the ring $S$ is a ring of formal power series over a field or over a discrete valuation ring.)
An $R$-module is {\em torsion-free} provided the natural map $M \to M\otimes_RK$ is injective. Equivalently, every non-zerodivisor in $R$ is a non-zerodivisor on $M$. This leads
to the following version of (HWC$_d$): \end{definitionsremarks}
\begin{definition} \label{defhwc'} Let $R$ be a local ring (not necessarily an integral domain).
\begin{enumerate} \item[] (HWC) {\bf Huneke-Wiegand Conjecture.} Assume $R$ is Gorenstein, and $M$ is a torsion-free $R$-module with rank. If $M\otimes_RM^*$ is maximal Cohen-Macaulay, then $M$ is free.
\end{enumerate}
\end{definition}
Following \cite{CeT}, we consider conditions, labeled (AR) and (HW) here, on a local ring $(R,\mathfrak{m})$: \begin{definition} \label{defarhw} Let $R$ be a local ring. \begin{enumerate} \item[] (AR)\ \ {\bf Artin-Reiten Condition.} For every finitely generated torsion-free $R$-module $M$, $$\Ext^i_R(M,M\oplus R)=0\text{ for every }i\geq 1\implies M\text{ is free.}$$
\item[] (HW) \ \ {\bf Huneke-Wiegand Condition.} For every finitely generated torsion-free module $M$ with rank,
$$M\otimes_R M^*\text{ is MCM }\implies M
\text{ is free.}$$
\end{enumerate}
Thus (ARC) says that {\em every} local ring $(R,\mathfrak{m})$ satisfies (AR), and (HWC) says that
{\em every} local Gorenstein ring satisfies (HW).
\end{definition}
Proposition~\ref{ACT} gives the connection between the Huneke-Wiegand Conjecture and the commutative version of the Auslander-Reiten Conjecture. The proof of Proposition~\ref{ACT} in \cite{CRW} is more explicit than in \cite{Araya} and \cite{CeT}.
\begin{proposition} \label{ACT} \cite[Proposition 8.6]{CRW}. Let $R$ be a local Gorenstein ring. Consider the following statements regarding the conditions of Definition~\ref{defarhw}:
$(i) ~R$ satisfies $($HW$)$.
$(ii) ~R_\mathfrak{p}$ satisfies $($HW$)$, for every prime ideal $\mathfrak{p}$ with height $\mathfrak{p} \le 1$.
$(iii) ~R$ satisfies $($AR$)$.
$(iv) ~R_\mathfrak{p}$ satisfies $($AR$)$, for every prime ideal $\mathfrak{p}$ with height $\mathfrak{p} \le 1$.
\noindent Then $(ii) \implies (i) \implies (iii) \Longleftarrow (iv)$. \end{proposition}
The main ideas in the proof of (ii)$\implies $(i) are \begin{enumerate} \item[(a)] \cite[A.1]{AuGol} A module $M$ is free $\iff$ the natural map $$ M\otimes_RM^*\to \Hom_R(M,M)\,, $$
taking $x\otimes f$ to the
homomorphism $y\mapsto (f(x))y$, for $x,y\in M$ and $f\in M^*$, is an
isomorphism; and \item[(b)] A map from a reflexive module to a torsion-free module is an isomorphism if and only if it is an isomorphism at each height-one prime ideal. \end{enumerate} Both (a) and (b) were used by Auslander in his proof of \cite[Proposition 3.3]{aus}. The implication (iv) $\implies$ (iii) is due to Araya \cite{Araya}. Two obvious questions: Does (i) $\implies$ (ii)? Does (iii) $\implies$ (iv)?
\begin{remark} \label{ACTr}By Proposition \ref{ACT}, the condition (HW) being
satisfied for one-dimensional local Gorenstein rings would ensure that it holds for every local Gorenstein ring. Attempts to prove (HW) for Gorenstein quasi-fiber rings, however, do not immediately reduce to the one-dimensional case, since localizations of two-dimensional
quasi-fiber product rings at height-one primes are {\it not} necessarily quasi-fiber rings. For example, a two-dimensional regular local ring is a quasi-fiber ring,
but its localizations at height-one primes are discrete valuation rings, which are {\em not} quasi-fiber rings. \end{remark}
\begin{fact}\label{factGor} Let $(R,\mathfrak{m})$ be a fiber product ring, say, $R= S\times_kT$.
The following statements are equivalent:
\begin{itemize}
\item[(i)] $R$ is Gorenstein.
\item[(ii)] $R$ is a $1$-dimensional hypersurface, as in Remark~\ref{rem:rank}.
\item[(iii)] $S$ and $T$ are discrete valuation rings. \end{itemize} \end{fact} The implications (i) $\iff$ (ii) $\implies$ (iii) constitute the Main Theorem of \cite{NTV}, while the implication (iii) $\implies$ (i) is Part (3) of \cite[Proposition 2.2]{EGI}.
\begin{fact}\label{factGorqfd1} Let $R$ be a Gorenstein quasi-fiber product ring of dimension 1. Then $R$ is
a Gorenstein fiber product ring.
To see this, let $\underline{x}=x_1,\ldots,x_n$ be an $R$-sequence such that $R/(\underline x)$ is a fiber product ring.
By
Fact~\ref{factGor}, the dimension of $R/(\underline x)$ is $1$. Thus $n=0$ and $R$ is
a Gorenstein fiber product ring. \end{fact}
Combining Fact \ref{factGor}
with Fact \ref{pdfin1}, we have:
\begin{rem}\label{rem:size} Let $R$ be a quasi-fiber product ring, and
let $n$ be the length of a regular sequence $\underline{x}$ such that $R/(\underline{x})$ is a fiber product ring.
Then $n$ is equal to either $\depth R-1$ or
$\depth R$. Moreover, if $R$ is Gorenstein, then $n=\depth R -1$.
\end{rem}
\begin{nota} \label{syzbnot} For an $R$-module $M$, let $\Omega_R^iM$ denote the $i^{\text{th}}$ syzygy of $M$ with respect to a minimal $R$-free resolution. We often write $\Omega_RM$ for $\Omega^1_RM$. \end{nota}
\begin{lem}\label{lem:syz} \cite[Lemma 5.1]{NT} Let $R$ be a local ring and $M$ an $R$-module. Let
$\underline{x}=x_1,\ldots,x_n$ be an $R$-sequence. Then $\underline{x}$ is a regular sequence on $\Omega_R^nM$.
\end{lem}
\begin{comment}
\begin{lem}\label{lem:change} \cite[11.65]{Rot} (Change of Rings long exact sequences of $\Ext$). Let $x$ be a nonzero divisor of an ring $R$, and set $S:= R/(x)$. Let $M$ and $N$ be $R$-modules. Then we have the following long exact sequence $$ 0 \to \Ext_S^1(M,N)\to \Ext_R^1(M,N)\to \Ext_S^0(M,N) \to \cdots $$ $$ \to \Ext_S^n(M,N) \to \Ext_R^n(M,N) \to \Ext_S^{n-1}(M,N) \to \cdots. $$
\end{lem}
\end{comment}
\begin{definitionsremarks} \label{defn:A-trans} (i) The {\em Auslander transpose}, written as $\D M$ or $\D_1M$, of a finitely generated module $M$
over a local ring $R$ is defined to be the cokernel of the map $F_0^*\to F_1^*$, where $$ F_1\to F_0\to M\to 0 $$ is a minimal resolution of $M$, with the $F_i$ free $R$-modules. Thus one has the exact sequence \begin{equation}\label{eq:A-trans} 0 \to M^* \to F_0^*\to F_1^* \to \D M \to 0\,.\tag{\ref{defn:A-trans}.0} \end{equation}
(ii) More generally, for $n\ge 1$ and a minimal free resolution $F$ of $M$ over~$R$, $$ F: \cdots F_n\to\cdots \to F_2\to F_1\to F_0\to M\to 0, $$ define the $n^{\text {th}}$ Auslander transpose $\D_nM$ by $\D_nM:=\coker(F_{n-1}^*\to F_n^*)$.
(iii) In \cite[p.4462]{Jor}, Jorgensen uses the notation ``$\D^0M$" to mean the same as our ``$\D M$", and ``$\D^n$'' to mean the same as our ``$\D_{n+1}$''. Note that, for every $i$ with $0\le i\le n$, $\D_{n}M= \D_{n-i}\Omega^i_RM$
\cite[p. 4462]{Jor}.
\end{definitionsremarks}
After one adjusts the notation as in Definitions and Remarks~\ref{defn:A-trans}(iii) above, \cite[Proposition 3.1(1)]{Jor} states:
\begin{proposition} \label{Jorprop} Let $R$ be a commutative Noetherian ring, $M$ be a finitely generated $R$-module, and $n\ge 1.$ If $\ext_R^i(M,M\oplus R)=0$, for every $i$ with $1\le i\le n$,
then:
$ (1) \Tor^R_i(\D_{n+1}M,M)=0$, for every $i$ with $1\le i\le n$.
$(2)$ The following sequence is exact
$$\aligned 0&\to \Tor_{n+2}^R(\D_{n+1}M,M)\to \Hom_R(M,R)\otimes_R M \\
&{\to} \Hom_R(M, M)\to \Tor_{n+1}^R(\D_{n+1}M,M)\to 0,
\endaligned$$
where the middle homomorphism $(\Hom_R(M,R)\otimes_R M
{\to} \Hom_R(M, M))$ is the natural one. \end{proposition}
\begin{facts}\label{fact:torsion} Here are some well-known facts concerning reflexive, maximal Cohen-Macaulay (MCM), and torsion-free $R$-modules. Let $R$ be a local ring and $M$ a non-zero $R$-module. \begin{enumerate}[(i)]
\item If $R$ is Gorenstein and $M$ is MCM, then $M$ is reflexive, and the dual module $M^*$ is also MCM.
(These follow from the fact that $R$ is its own canonical module. See \cite[Theorems 3.3.7 and 3.3.10(d)]{BH}.)
\item If $R$ is $1$-dimensional and Cohen-Macaulay, then $M$ is MCM if and only if $M$ is torsion-free.
\item If $R$ is Cohen-Macaulay and $n \ge \mathop{\rm dim} R$, then $\Omega^n_RM$ is MCM, by the Depth Lemma \cite[Lemma A.4]{LW}.
\item Suppose $R$ is Cohen-Macaulay and $\mathop{\rm dim} R \leq 2$. If $M$ is reflexive, then $M$ is MCM, since, by
Equation \eqref{eq:A-trans}, $M =M^{**}$ is the second syzygy of $\D(M^*)$. \end{enumerate}
\end{facts}
Since we have not found a proof of (ii) in the literature, we include one here: Since $R$ is CM, there is a non-zerodivisor $f\in \mathfrak{m}$. If $M$ is torsion-free, then $f$ is a non-zerodivisor on $M$, and hence $\depth M \ge1$, that is, $M$ is MCM. Conversely, suppose $M$ is MCM, and let $r$ be a zero-divisor on $M$. Then $r\in \mathfrak{p}$ for some $\mathfrak{p}\in \Ass M$. Now $\mathfrak{p}\ne \mathfrak{m}$ since $M$ is MCM, and hence $\mathfrak{p}$ is a minimal prime ideal of $R$. Therefore $r$ is a zero-divisor of $R$; this shows that $M$ is torsion-free.
\begin{remark}\label{hwcimp'} The truth of (HWC), the local ring version of the Huneke-Wiegand Conjecture in Definition~\ref{defhwc'},
would imply the truth of (HWC$_d$), the integral domain version in Definition~\ref{defarchwc}: Assume (HWC). By Proposition~\ref{ACT}, (HWC$_d$) reduces to the one-dimensional case.
Fact~\ref{fact:torsion}, parts (i) and (iv),
implies that the non-zero reflexive modules over a one-dimensional Gorenstein local ring are exactly the MCM modules. Thus a Gorenstein local domain satisfying (HW) of Definition~\ref{defarhw} also satisfies (HW$_d$), the domain form of the condition: \begin{itemize} \item[(HW$_d$)] If $M$ is a finitely generated torsion-free $R$-module such that $M\otimes_R M^*$ is reflexive, then $M$ is a free module. \end{itemize} The original conjecture (HWC$_d$) in Definition~\ref{defarchwc} is that every Gorenstein local domain satisfies (HW$_d$). \end{remark}
\section{Vanishing of Ext and the Auslander-Reiten Conjecture\label{SecVE}}
The results in this section can be compared with those in Section 6 of \cite{NT}. We begin by recalling the Auslander Condition (AC) and the
Uniform Auslander Condition (UAC) on a ring $R$:
\begin{itemize} \item[(AC)] For each $R$-module $M$, there is a non-negative integer $b = b_M$ such that, for every $R$-module $N$, one has \begin{equation}\label{eq:bound} \Ext^i_R(M,N) = 0\ \forall i \gg 0 \implies \Ext^i_R(M,N)=0,\ ~\forall i \ge b\,.
\tag{\ref{SecVE}.0.0} \end{equation} \item[(UAC)] There is an integer $b\ge0$ such that \eqref{eq:bound} holds for every pair $M,N$ of $R$-modules. \end{itemize}
\begin{nota/rem} \label{uabnot} \noindent (i) A number $b$ with the property required in (UAC) is called a {\em uniform Auslander bound}.
(ii) The smallest number $b$ with this property is called the {\em Ext-index} of $R$ \cite{HJ}.
(iii) To our knowledge, it is unknown whether every local ring satisfying (AC) actually satisfies the stronger condition (UAC).
(iv) A Gorenstein local ring satisfying (UAC) is called an {\em AB ring} \cite{HJ}. For a local AB ring, the Ext-index is known to be equal to $\mathop{\rm dim} R$
\cite[Proposition 3.1]{HJ}.
(v) Modules $M$ and $N$ over an AB ring $R$ satisfy the
following symmetry \cite[Theorem 4.1]{HJ}:
\begin{equation}\label{eq:symm}
\Ext_R^i(M,N) = 0, \ \forall i\gg 0 \iff \Ext_R^i(N,M) = 0, \ ~\forall i\gg 0\,.
\tag{\ref{uabnot}.0}
\end{equation}
(vi) Auslander conjectured \cite[p. 795]{aus-coll} that finite-dimensional modules over a finite-dimensional $k$-algebra satisfy (AC). This conjecture was disproved by Jorgensen and \c Sega \cite{JS}. Their counterexample is a finite-dimen-\\ sional {\em commutative} Gorenstein $k$-algebra, where $k$ can be taken to be any field that is not algebraic over a finite field. \end{nota/rem}
\begin{lemma}\label{lem:pdid} Let $R$ be a quasi-fiber product ring with respect to an $R$-sequence of length $n$. If $M$ and $N$ are $R$-modules such that $\pd_RM<\infty$ or $\id_RN<\infty$,
then $\Ext_R^i(M,N)=0$, for every $i>n+1$.
\end{lemma}
\begin{proof}
By
Remark~\ref{rem:size}, $\depth R\leq n+1$ . The Auslander-Buchsbaum Formula (Remark \ref{rem:ABF}) and Bass Formula \cite[Theorem 3.1.17]{BH}) show that $\pd_RM\leq \depth R \leq n+1$ or $\id_R N=\depth R\leq n+1$. In either case, $\Ext_R^i(M,N)=0$, for every $i$ with $i> n+1$. \end{proof}
\begin{thm}\label{thm:uac} If $R$ is a quasi-fiber product ring with respect to an $R$-sequence of length $n$, then the Ext-index of $R$ is at most $n+2$. In particular, every quasi-fiber product ring satisfies (UAC). If, further, $R$ is Gorenstein, then $R$ is an AB ring. \end{thm} \begin{proof} Assume that $\Ext_R^i(M,N)=0$ for all $i\gg 0$. Then \cite[Corollary 6.8]{NT} asserts that
$\pd_RM<\infty$ or $\id_RN<\infty$.
By Lemma~\ref{lem:pdid}, $\Ext_R^i(M,N)=0$, for every $i$ with $i> n+1$.
Therefore $R$ satisfies (UAC).
The last statement is clear by definition; an AB ring is a Gorenstein local ring satisfying (UAC). \end{proof}
\begin{definition}\label{gapdef} Let $M$ and $N$ be $R$-modules, and let $g$ and $m$ be positive integers. We say that $\Ext_R(M,N)$ has a {\it gap of length} $g$ with {\it lower bound} $m$ if $$ \aligned \Ext^i_R(M,N)&\neq 0,\text{ for }i=m-1\text{ and for }i=m+g;\text{ and }\\ \Ext^i_R(M,N) &=0,\text{ whenever }m\leq i\leq m+g-1.\endaligned $$ Then $(m,g)$ is called a { \it gap pair} for $\Ext_R(M,N)$. Set
\begin{equation*}\aligned \Ext\text{-}\operatorname{gap}_R(,N)&:={\rm sup}\{g \mid \Ext(M,N)\ {\rm has \ a \ gap \ of \ length} \ g\}; \text{ and}\\
\Ext\text{-}\operatorname{gap}(R)&:={\rm sup}\{\Ext\text{-}\operatorname{gap}_R(M,N)~|~M\text{ and }N\text{ are} \ R\text{-modules}\}. \endaligned \end{equation*} \end{definition}
\begin{example} \label{gapexmp} In Example~\ref{example1}, where $k$ is a field, $R$ is the fiber product ring $R=k[[x,y]]/(xy)$, and $M=R/(y)$, we show $\ext^i_R(M,M)=0$ for every odd $i>0$ and $\ext^i_R(M,M)=0$, for every even $i>0$. Thus every odd positive integer $m$ is a lower bound for a gap of length $1$; the set of gap pairs for $\ext_R(M,M)$ is $\{(m,1)~|~m$ is an odd integer\}. \end{example} \begin{rems}\label{rem:classNT} Let $R$ be a quasi-fiber product ring, let $M$ and $N$ be finitely generated $R$-modules, and let $n$ be the length of an $R$-sequence $\underline x$ such that $R/(\underline x)$ is a fiber product ring. With this setting, Nasseh and Takahashi give these interesting and useful results related to ``Tor-gaps", Question~\ref{question1} and Ext-gaps in \cite{NT}:
(1) \cite[Corollary 6.5]{NT} If there exists an integer $t$ with $t\ge \max\{5, n+1\}$ such that $ \Tor^R_i(M,N) =0$, for every $i$ with $t+n\le i \le t+n+\depth R$, then
$\pd_RM<\infty$ or $\pd_RN<\infty$.
(2) \cite[Corollary 6.6]{NT} Assume $R$ is a $d$-dimensional Cohen-Macaulay ring, and set $s:=d-\depth M$. If there exists an integer $t$ such that $t\ge 5$ and $ \Ext^i_R(M,N) =0$, for every $i$ with $t+s\le i \le t+s+d$, then
$\pd_RM<\infty$ or $\id_RN<\infty$.
(2$'$) (Restating (2)) If $R$ is a $d$-dimensional Cohen-Macaulay ring
such that both $\pd_RM$ and $\id_RN$ are infinite, if $s:=d-\depth M$, and if $t\ge 5$,
then there exists an integer $j$ with $ \Ext^j_R(M,N) \ne 0$ and $t+s\le j \le t+s+d$.
\end{rems}
In \cite{HJ}, a local ring is called
{\it Ext-bounded} if its Ext-gap is finite. It seems to be unknown whether every AB ring
is Ext-bounded \cite[\S6, Question 4]{HJ}. Proposition~\ref{cor:class} shows
that quasi-fiber product rings would not be a good place to look for a counterexample. If $\pd_RM<\infty$ or $\id_RN<\infty$, the gap is at most the dimension.
For convenience we give bounds on the size of the gaps associated with
various values of the lower bound $m$ if $\pd_RM$ and $\id_RN$ are both infinite. That is,
we give restrictions on the possible gap pairs having various values for $m$.
\begin{prop}\label{cor:class} Let $R$ be a Cohen-Macaulay quasi-fiber product ring of dimension $d$, let $m$ and $g$ be positive integers, and let $M$ and $N$ be finitely generated R-modules such that $ \Ext_R(M,N)$ has a gap of length $g$ with lower bound $m$. Set $s: = d-\depth M$. Then:
\begin{enumerate} \item [$(1)$] $g\le 2d+4$. Thus the {$\Extgap$} of $R$ is at most $2d+4$, and hence $R$ is $\Ext$-bounded. \item [$(2)$]If $\pd_RM$ or $\id_RN$ is finite, then $g\le d$ and $\Extgap_R(M,N)\le d$. \item [$(3)$]If $m\ge s+5$, then $g\le d$. \item [$(4)$]If $s\le m\le s+4$, then $g\le d+5-(m-s)$. (Thus, for example, $m=s+4\implies g\le d+1$, and $m=s\implies g\le d+5$.) \item [$(5)$]If $m\ge 5$, then $g\le s+d$, so $g\le 2d$. \item [$(6)$]If $0< m\le 4$, then $g\le s+d+(5-m)$, so $g\le 2d+(5-m)$; e.g. $m=1\implies g\le 2d+4.$ \item [$(7)$] $(i)$ If $s=0$ (that is, $\mathop{\rm dim} R=\depth M$) and $m\ge 5$, then $g\le d$. \\ $(ii)$ If $s=0$ and $0<m< 5$, then $g\le d+(5-m)$. \item [$(8)$] If $d=0$, then $m< 5$. \end{enumerate}
\end{prop}
\begin{proof} By Definition~\ref{gapdef}, we have, for every $i$ with $m \le i \le m+g-1, $ \begin{equation}\label{eq:Ext-van} \Ext_R^i(M,N) = 0 \text{ and } \Ext^{m+g}_R(M,N)\ne 0\,.\tag{\ref{cor:class}.a} \end{equation} Let $n$ be the length of a regular sequence $\underline{x}$ such that $R/(\underline{x})$ is a fiber product ring.
Remark~\ref{rem:size} and the fact that $R$ is Cohen-Macaulay yield \begin{equation}\label{eq:depth-bound} d = \depth R \in \{n,n+1\}\,. \tag{\ref{cor:class}.b} \end {equation}
Part (1) follows from parts (2), (5) and (6).
For part (2), Lemma~\ref{lem:pdid}
implies that
$\Ext^i_R(M,N) = 0$ for every $i> n+1$, and hence $m+g\le n+1$; that is, $g\le n - (m-1) \le n$. Since $n\le d$ by \eqref{eq:depth-bound}, we have $g\le d$. This proves item (2).
For the remainder of the proof, assume $\pd_RM= \infty$ and $\id_RN = \infty$. Then, by Remark~\ref{rem:classNT} (2$'$), for every $t\ge 5$, there is an integer $j$ satisfying \begin{equation}\label{eq:short-gap} t+s\le j \le t+s+d \text{ and }\Ext^j_R(M,N) \ne 0\,.\tag{\ref{cor:class}.c} \end{equation}
To prove (3),
assume that $m\ge 5+s$. Let $t:= m-s \ge 5$. By \eqref{eq:short-gap}, there exists $j$ with \begin{equation*} m=t+s \le j\le t+s+d = m+d\, \text{ and }\Ext^j_R(M,N) \ne 0. \end{equation*} By ({\ref{cor:class}.a}), $g-1<d$. Thus $g\le d$.
To prove (4), assume that $m= a+s$ for $0\le a\le 4$. Let $t:= 5$. By \eqref{eq:short-gap}, there exists $j$ with \begin{equation*} a+s=m<5+s \le j\le 5+s+d = (5-a) +(a+s)+d=m+(5-a)+d\, \end{equation*} { and }$\Ext^j_R(M,N) \ne 0.$ By ({\ref{cor:class}.a}), $g-1<(5-a)+d$. Thus $g\le d+5-a$.
For (5), let $t:=m$. By \eqref{eq:short-gap}, there exists a positive integer $j$ with
\begin{equation*} m+s \le j\le m+s+d, \text{ and }\Ext^j_R(M,N) \ne 0\,.
\end{equation*} Definition~\ref{gapdef} implies that $m+g-1 < m+s+d$. Now $g\le s+d\le 2d$, since $0\le s\le d$.
For (6), let $t:=5$ and $a:=5-m$. By \eqref{eq:short-gap}, there exists $j$ with \begin{equation*} a+m+s=5+s \le j\le 5+s+d=a+m+s+d, \text{ and }\Ext^j_R(M,N) \ne 0\,. \end{equation*} By Definition~\ref{gapdef}, $j> m+g-1$. Therefore $m+g-1 < a+m+s+d$, and hence $g\le a +s+d \le a+2d$.
Observe that $g\le 2d+4$ by parts (2), (5) and (6). Thus (1) holds.
For (7)(i), use (5), $ g\le s+d=d$, and, for (7)(ii), use (6), $$g\le ( 5-m)+s+d=(5-m) +d.$$
For (8), if $d=0$, then also $s=0$. If $m\ge 5$, then, by \eqref{eq:short-gap}, there exists $j$ with \begin{equation*} \aligned m+s \le j\le m+s+d, \text{ and }\Ext^j_R(M,N) \ne 0.
\endaligned\end{equation*} That is, $m\le j \le m\implies j=m\implies \Ext^m_R(M,N) \ne 0$. This contradicts ({\ref{cor:class}.a}). Thus $m<5$. \end{proof}
The Auslander-Reiten Conjecture (ARC) is the case $n=1$ of the Generalized Auslander-Reiten Conjecture (GARC):
\begin{enumerate}
\item[] (GARC)\ \
$ \text{If}\ \Ext^i_R(M,M\oplus R)=0 \text{ for all } i\geq n, \text{ then } \pd_RM< n\,.$ \end{enumerate}
\begin{cor}\label{cor:arc} Every quasi-fiber product ring satisfies {\rm (GARC)}, and so also {\rm (ARC)}. \end{cor} \begin{proof} Diveris \cite{diveris} introduced the notion of ``finitistic extension degree'' fed$(A)$, for a left Noetherian ring $A$: \begin{equation*}
\text{fed}(A):=\sup\{\sup\{i~|~\ext_A^i(M,M) \ne 0\}\},\end{equation*} where the outside sup is over all finitely generated left $A$-modules $M$. Diveris proved \cite[Corollary 2.12]{diveris} that (GARC) holds for $A$ if fed$(A) < \infty$. By Theorem \ref{thm:uac}, every quasi-fiber product ring $R$ satisfies (UAC), which certainly implies fed$(R)$ is finite. Thus (GARC) holds for quasi-fiber product rings. \end{proof}
It also follows from Nasseh and Sather-Wagstaff's theorem \cite[Theorem 4.5]{NS} that every quasi-fiber product ring $R$ satisfies (ARC) -- the case $n=1$ of (GARC).
They show
that fiber product rings (i.e. with {\em decomposable} maximal ideal) satisfy (ARC). By Celikbas' theorem \cite[Theorem 4.5 (1)]{CeT}, the (AR) property lifts modulo a regular sequence.
\begin{discussion}\label{AsTr}{\bf Auslander sequence and Tor-rigidity.} We introduce an important tool concerning the Auslander transpose (see Definition \ref{defn:A-trans}).
(i) From \cite[Theorem 2.8 (b)]{AuBr} or \cite[(1.1.1)]{Jor}, we have, for each $i\ge0$, the following
exact
sequence: \begin{equation*}\label{key}\begin{gathered} {\Tor_2^R(\D\Omega_R^iM,N) \to \Ext_R^i(M,R)\otimes_R N}\\ {\phantom{extext}\to \Ext_R^i(M,N) \to \Tor_1^R(\D\Omega_R^iM,N) \to 0.}\end{gathered} \tag{\ref{AsTr}.1}\end{equation*} The sequence ({\ref{AsTr}.1}) is known as the \emph{Auslander sequence}.
(ii) An $R$-module $M$ is said to be {\it Tor-rigid} provided that the following holds for every $R$-module $N$ and every $n\ge 1$ (see \cite{aus}): $$ \Tor_n^R(M,N)=0 \implies \Tor_{n+1}^R(M,N)=0. $$
(iii) Let $M$ be an $R$-module and let $h:M\to M^{**}$ be the canonical map. Recall that $M$ is {\em torsionless} provided $h$ is injective, and {\em reflexive} if $h$ is bijective. By mapping a free module onto $M^*$ and dualizing, we see that $M^{**}$ embeds in a free module. It follows that torsionless modules are torsion-free.
From \cite[Exercise 1.4.21]{BH} we obtain an exact sequence $$ 0\to \Ext^1_R(\D M,R) \to M\overset{h}{\to}M^{**} \to \Ext^2_R(\D M,R) \to 0\,. $$ Thus $M$ is torsionless if and only if $\Ext^1_R(\D M,R) = 0$, and reflexive
if and only if $\Ext^1_R(\D M,R)$ and $\Ext^2_R(\D M,R)$ are both zero.
\end{discussion}
We use Auslander's Theorem~\ref{austordep} and Lemma~\ref{jdextdep}, due to Jothilingam and Duraivel: \begin{theorem} \label{austordep} \cite[Theorem 1.2]{aus} Let $R$ be a local ring, and let $M$ and $N$ be nonzero $R$-modules such that $\pd M<\infty$. Let $q$ be the largest integer such that $\tor_q^R(M,N)\ne 0$. If $\depth (\tor_q^R(M,N))\le 1$ or $q=0$, then $\depth N=\depth(\tor_q^R(M,N)) +\pd M-q$. \end{theorem}
\begin{lemma} \label{jdextdep} \cite[Lemma, p. 2763]{JD} Let $R$ be a local ring, and let $M$ and $N$ be $R$-modules such that $N\ne 0$ and $$\Ext^i_R(M,N)=0, \ {\rm for} \ i=1,\ldots, {\rm max}\{1, \depth_RN-2\}.$$ Then $\depth N\le \depth (\Hom_R(M,N))$.\end{lemma}
Our next result, Theorem~\ref{thm:Ext0}, can be compared to the following result, due to Jothilingam and Duraivel:
\begin{theorem} \cite[Theorem 1]{JD}: Let $M$ be a module over a regular local ring $R$. If $$\Ext^i_R(M,N)=0, \ {\rm for} \ i=1,\ldots, {\rm max}\{1, \depth_RN-2\},$$ for some nonzero $R$-module $N$, then $M^\ast$ is free. \end{theorem}
\begin{thm}\label{thm:Ext0} Let $R$ be a quasi-fiber product ring.
Let $M$ and $N$ be nonzero $R$-modules. If $N$ is Tor-rigid and $$\Ext^i_R(M,N)=0 \ {\rm for} \ i=1,\ldots, {\rm max}\{1, \depth_RN-2\},$$ then $M^\ast$ is free or $\pd_RN <\infty$. \end{thm} \begin{proof} Since $\Ext^1_R(M,N)=0,$ the Auslander sequence \eqref{key} shows that $$ \Tor_1^R(\D\Omega_R^1M,N)=0\,. $$
The Tor-rigidity of $N$ implies that \begin{equation}\label{1}\Tor_i^R(\D\Omega_R^1M,N)=0 \ \ {\rm for} \ \ {\rm all} \ \ i\geq 1.\tag{\ref{thm:Ext0}.1} \end{equation}
The Auslander sequence \eqref{key} implies that $\Ext^1_R(M,R)\otimes_R N=0.$ Since $N$ is nonzero, Nakayama's lemma implies that $\Ext^1_R(M,R)=0$.
Using the minimal free resolution $$ F_2\to F_1 \to F_0 \to M \to 0 $$ to compute $\Ext_R(M,R)$, we have $$ 0 = \Ext^1_R(M,R) = \frac{\ker(F_1^\ast \to F_2^\ast)}{\text{im}(F_0^\ast \to F_1^\ast)}\,. $$ Therefore $$ \D M = \coker(F_0^\ast \to F_1^\ast) = \frac{F_1^\ast}{\text{im}(F_0^\ast \to F_1^\ast)} = \frac{F_1^\ast}{\ker(F_1^\ast \to F_2^\ast)}\,, $$ and hence the map $F_1^*\to F_2^*$ factors as $F_1^*\twoheadrightarrow \D M \hookrightarrow F_2^*$\,. This shows that $\D M \cong \text{im}(F_1^*\to F_2^*)$. Since $\D\Omega^1_RM = \coker(F_1^*\to F_2^*)$, we get a short exact sequence \begin{equation}\label{gleep} 0\to \D M \to F_2^\ast \to \D\Omega_R^1M\to 0\,.\tag{\ref{thm:Ext0}.2} \end{equation} Together \eqref{1} and \eqref{gleep} imply that \begin{equation}\label{glarb} \Tor_j^R(DM,N)=0 \text{ for all } j\ge 1\,.\tag{\ref{thm:Ext0}.3} \end{equation} Using \eqref{glarb} and \eqref{eq:A-trans}, we see that \begin{equation}\label{barf} \Tor_j^R(M^*,N) = 0 \text{ for all } j\ge 1\,.\tag{\ref{thm:Ext0}.4} \end{equation}
By \eqref{barf} and Remark~\ref{rem:classNT}(1), \begin{equation*} \pd_R M^\ast <\infty \text{ or } \pd_RN<\infty\,. \end{equation*}
If $\pd_RN<\infty\,$, we are done.
To complete the proof of Theorem~\ref{thm:Ext0}, assume $\pd_RM^\ast<\infty$. We show that $M^\ast$ is free. By \eqref{glarb} and the Auslander sequence \eqref{key} in the case $i=0$, we have the isomorphism \begin{equation}\label{eq:gorpse} M^\ast\otimes_RN =\Hom_R(M,R)\otimes_RN\cong \Hom_R(M,N). \tag{\ref{thm:Ext0}.5} \end{equation} Since $\pd_RM^\ast<\infty$, Theorem~\ref{austordep} with $q=0$ implies \begin{equation}\label{eq:gorpse2} \depth_R N=\depth_R(M^\ast\otimes_RN) +\pd_R M^*. \tag{\ref{thm:Ext0}.6} \end{equation} By \eqref{eq:gorpse} and Lemma~\ref{jdextdep}, \begin{equation}\label{eq:gorpse1} \depth_R(M^\ast\otimes_RN)= \depth_R (\Hom_R(M,N)) \ge \depth_RN.
\tag{\ref{thm:Ext0}.7}
\end{equation}
Putting together \eqref{eq:gorpse1} and \eqref{eq:gorpse2} yields that $\pd_R M^*=0$; that is, $M^\ast$ is free, as desired. \ \end{proof}
\begin{comment}
\begin{equation}\label{8} M^\ast\otimes_RN \cong \Hom_R(M,N). \tag{\ref{thm:Ext0}.8} \end{equation}
and the vanishing of $\Ext^1_R(M,R)$ yield the exact sequence \begin{equation}\label{eq:snort} 0\to M^\ast \to F_0^\ast \to (\Omega_R^1M)^\ast\to 0.\tag{\ref{thm:Ext0}.4} \end{equation} end{equation} The presentation $$ F_2\to F_1\to \Omega_R^1M \to 0 $$
provides the following exact sequence (analogous to \eqref{eq:A-trans}): \begin{equation}\label{5} 0\to (\Omega_R^1M)^\ast\to F_1^\ast \to F_2^\ast \to \D\Omega^1_RM \to 0.\tag{\ref{thm:Ext0}.5} \end{equation}
Now \eqref{eq:snort} and \eqref{5} yield the exact sequence \begin{equation}\label{6} 0\to M^\ast \to F_0^\ast \to F_1^\ast\to \D M\to 0\,.\tag{\ref{thm:Ext0}.6} \end{equation} Since $F^*_1\to F^*_2\to F_3^*$ is a complex, $\D M$ is the image of $F_1^*\to F^*_2$, and so we get an exact sequence \begin{equation}\label{7} 0\to \D M \to F_2^\ast \to \D\Omega_R^1M\to 0\,.\tag{\ref{thm:Ext0}.7} \end{equation} From \eqref{1} and the exact sequences \eqref{6} and \eqref{7}, we get the following: \begin{itemize}
\item[(i)] $\Tor^R_j(\D M,N)=0$ for all $j\geq 1$;
\item[(ii)] $\Tor_j^R(M^\ast,N)=0$ for all $j\geq 1$. \end{itemize}
Also we get the following statements: \begin{itemize}
\item[(iii)] $\pd_R M^\ast <\infty$ or $\pd_RN<\infty$.
If $\pd_RN<\infty$, we are done.
Thus to complete the proof of Theorem~\ref{thm:Ext0}, we assume $\pd_RM^\ast<\infty$ and we show that $M^\ast$ is free. By (i) and the Auslander sequence \eqref{key} in the case $i=0$, we have the isomorphism \begin{equation}\label{8} M^\ast\otimes_RN =\Hom(M,R)\otimes_RN\cong \Hom_R(M,N). \tag{\ref{thm:Ext0}.8'} \end{equation} Since $\pd_RM^\ast<\infty$, Theorem~\ref{austordep} with $q=0$ implies $$\depth N=\depth(M^\ast\otimes_RN) +\pd M^*.$$ By equation \eqref{8} and Lemma~\ref{jdextdep}, $$ \depth_R(M^\ast\otimes_RN)= \depth_R (\Hom_R(M,N)) \le \depth_RN.$$ Thus $\pd M^*=0$; that is, $M^\ast$ is free, as desired. \ \end{proof} \begin{comment} Using this exact sequence and Theorem \ref{thm:Ext0}, we get \begin{cor}\lab In either case, we can apply Theorem~\ref{austordep} and rearrange the conclusion using the Auslander-Buchsbaum Formula (Remark~\ref{rem:ABF}), to obtain the equation \begin{equation}\label{eq:burp1} \depth_RM^\ast+\depth_RN=\depth R +\depth_R(M^\ast\otimes_RN).\tag{\ref{thm:Ext0}.7} \end{equation}
To complete the proof, we may assume that $\pd_R(N)= \infty$. Then we know that $\pd_RM^* <\infty$ by \eqref{eq:gorpse}, and our goal is to show that $M$ is free.
el{cor:Trump} Let $R$ be a ring with quasi-decomposable maximal ideal. Let $M$ and $N$ be nonzero $R$-modules. Assume that $N$ is Tor-rigid, that $\pd_RN = \infty$, and
that. $$\Ext^i_R(M,N)=0, \ {\rm for } \ i=1,\ldots, {\rm max}\{1, \depth_RN-2\}.$$ \begin{itemize}
\item[(i)] If $M$ satisfies $(S_1)$, then M is torsion-free.
\item[(ii)] If $M$ satisfies $(S_2)$, then M is reflexive. \end{itemize} \end{cor} \begin{proof} It's true because we said so. \end{proof} \end{comment}
\begin{cor}\label{coro:Ext} Let $R$ be a quasi-fiber product ring with $\mathop{\rm dim} R\leq 3$ .
Let $M$ and $N$ be nonzero $R$-modules such that $N$ is Tor-rigid, $\Ext^1_R(M,N)=0$, and $\pd_RN = \infty$. Then $M^\ast$ is free. \end{cor} \begin{proof} Since $\depth_R N\le \depth R\le 3$, Theorem~\ref{thm:Ext0} applies. \end{proof}
The next result provides an answer for Question \ref{question1}.
\begin{cor}\label{GAR} Let $R$ be a quasi-fiber product ring.
Let $M$ and $N$ be nonzero $R$-modules, with $\pd_RN = \infty$. Suppose that \begin{itemize}
\item[(i)] $N$ is Tor-rigid.
\item[(ii)] $\Ext^i_R(M,N)=0$, for all $i$ with $1\le i\le\ldots, \le {\rm max}\{1, \depth_RN-2\}$.
\item[(iii)] $\Ext^j_R(M,R)=0$, for every $j$ with $1\le\ldots, \le\depth R.$ \end{itemize} Then $M$ is free. \end{cor} \begin{proof} By Theorem \ref{thm:Ext0}, $M^*$ is free. By \cite[Lemma 3.3]{DEL}, $M^*$ free and $\Ext^j_R(M,R)=0$, for all $j=1,\ldots, \depth R$, together imply that $M$ is free. \end{proof}
\begin{cor}\label{CMGAR} Let $R$ be a Cohen-Macaulay fiber product ring.
Let $M$ and $N$ be nonzero $R$-modules. If $N$ is Tor-rigid and $\Ext^1_R(M,N)=0,$ then $M$ is free or $\pd_RN \leq 1$. \end{cor} \begin{proof} If $\pd_RN < \infty$, then $\pd_RN \leq 1$, by Fact~\ref{pdfin1}(2). Thus we assume $\pd_RN=\infty$. By Fact~\ref{pdfin1}(1), $\depth R\leq 1$, and so $\mathop{\rm dim} R\le 1$.
Since $\Ext^1_R(M,N)=0,$ and $N$ is Tor-rigid, the Auslander sequence (\ref{key}) implies
$$\Tor_i^R(D\Omega_R^1M,N)=0,\text{ for all }i\geq 1.$$
Again from the Auslander sequence we deduce $\Ext^1_R(M,R)\otimes_RN=0$. Since $N$ is nonzero,
Nakayama's Lemma implies $\Ext^1_R(M,R)=0$. Now apply Corollary \ref{GAR}. \end{proof}
\begin{cor}\label{cor:CMEXT} Let $R$ be a Cohen-Macaulay fiber product ring.
Let $M$ and $N$ be nonzero $R$-modules. If $N$ is Tor-rigid and $\Ext^i_R(M,N)=0$ for some $i>1$, then $\pd_RM\leq 1$ or $\pd_RN \leq 1$. \end{cor} \begin{proof} Assume that $\pd_RN >1$. Since $\Ext_R^1(\Omega_R^{i-1}M,N) \cong \Ext_R^i(M,N) = 0$, Corollary \ref{CMGAR} implies that $\Omega_R^{i-1}M$ is free. Therefore $\pd_RM <\infty$, and now Fact \ref{pdfin1} implies that $\pd_RM\le 1$. \end{proof}
Corollary~\ref{cor:CMEXT} is somewhat relevant to a question raised by Jorgensen \cite[Question 2.7]{Jor}: If $\Ext^n_R(M,M) = 0$, over a complete intersection $R$, must $M$ have projective dimension at most $n-1$?
\begin{ex} \label{example1} \cite[Remark 2.6]{Jor}, \cite[Example 4.3]{AvBu} Let $k$ be a field, and let $R = k[[x,y]]/(xy)$. Let $M=R/(y) \cong Rx$. It is easy to compute $\Ext_R(M,M)$ and $\Tor^R(M,M)$ using the periodic free resolution $$ \dots \overset{x}{\to} R \overset{y}{\to} R \overset{x}{\to} R \overset{y}{\to} R \to 0 $$ of $M$. Then $\Ext_R^i(M,M)=0$, for every odd $i>0$, and $\Ext_R^i(M,M) \cong k$, for every even $i>0$. Therefore $\pd_RM=\infty$. Also, $\Tor^R_i(M,M)$ alternates between $0$ and $k$, but with $k$ at the {\em odd} indices. Thus $M$ is not Tor-rigid. This example justifies the rigidity hypothesis in Corollary \ref{cor:CMEXT}.
In addition, this example shows why one should insist that $R$ be a domain (or at least that $M$ have rank) in (HWC$_d$). One checks that $M^* \cong M$, and
hence $M\otimes_RM^* \cong (R/(y))\otimes_R(R/(y)) \cong R/(y) \cong M$, which is torsion-free, hence reflexive (as $R$ is one-dimensional and Gorenstein --- see Facts~\ref{fact:torsion}). But of course $M$ is not free. \end{ex}
\begin{comment}
\begin{thm} \label{evodthm00}
Let $R$ be a Cohen-Macaulay quasi-fiber product ring
with the sequence $\underline{x}=x_1,\ldots,x_n$. Let $M$ and $N$ be nonzero $R$-modules. If $N$ is Tor-rigid and, for some $t>n$,
$$\Ext^i_R(M,N)=0 \ \ \ {\rm for} \ \ {\rm all} \ \ t\leq i \leq t+n,$$ then $\pd_R M <\infty$ or $\pd_R N <\infty$. \end{thm} \begin{proof} The case $\mathop{\rm dim} R=0,1$ follows by Corollary \ref{CMGAR} and Corollary \ref{cor:CMEXT}. So it remains to consider the case $\mathop{\rm dim} R>1$. Note that, if $X:=\Omega_R^n(M)$ and $Y:=\Omega_R^n(N)$ we obtain $$\Ext^i_R(X,Y)=0 \ \ \ {\rm for} \ \ {\rm all} \ \ t\leq i \leq t+n.$$
By the short exact sequences $$0\to X/(x_1,\ldots,x_{j-1})X\stackrel{x_j}\to X/(x_1,\ldots,x_{j-1})X \to X/(x_1,\ldots,x_{j})X \to 0$$ for $1\leq j\leq n$, we deduce from the long exact sequence of Ext that $$\Ext^i_R(X/\underline{x}X,Y)=0 \ \ \ {\rm for} \ \ i = t+n.$$
Since $\underline{x}$ is $Y$-regular (Lemma \ref{lem:syz}), \cite[Lemma 2 (i), Section 18]{Mat} provides $$\Ext^i_{R/(\underline{x})}(X/\underline{x}X,Y/\underline{x}Y)=0 \ \ \ {\rm for} \ \ i=t.$$
Since $N$ is Tor-rigid, $Y$ is Tor-rigid and so $Y/\underline{x}Y$ is Tor-rigid $R/\underline{x}$-module. Further, since $R/(\underline{x})$ is a Cohen-Macaulay ring with decomposable maximal ideal, then we have $n=\mathop{\rm dim} R-1$ or $n=\mathop{\rm dim} R$ by Remark~\ref{rem:size}. Since $t>n$, Corollary \ref{cor:CMEXT} asserts $\pd_{R/(\underline{x})}X/\underline{x}X \leq 1$ or $\pd_{R/(\underline{x})}Y/\underline{x}Y \leq 1.$ Therefore \cite[Lemma 1.3.5]{BH} provides $ \pd_R X \leq 1$ or $\pd_R Y \leq 1,$ and so the desired conclusion follows.
\end{proof}
\end{comment}
In the proofs of the next theorems we use the change-of-rings isomorphisms (i) and (iii) of Lemma 2 in Chapter 18 of Matsumura's book \cite{Mat}. We record those formulas here, adjusting the notation to suit our situation:
\begin{rem}\label{Mats-Formulas} Let $R$ be a local ring, and let $N$ and $Z$ be $R$-modules. Let $\underline x = x_1,\dots, x_n$ be an $R$-sequence that is also an $N$-sequence, and assume that $({\underline x})Z=0$. Put $\overline R = R/(\underline x)$ and $\overline N = N/({\underline x})N$. The following formulas hold for every non-negative integer $i$: \begin{enumerate}[(a)] \item $\Ext_R^{i+n}(Z,N) \cong \Ext_{\overline R}^i(Z,{\overline N})$\,. \item $\Tor^R_i(Z,N) \cong \Tor^{\overline R}_i(Z,{\overline N})$\,. \end{enumerate} \end{rem}
\begin{thm} \label{evodthm00}
Let $R$ be a quasi-fiber product ring
with respect to the regular sequence $\underline{x}=x_1,\ldots,x_n$. Let $M$ and $N$ be nonzero $R$-modules. If $N$ is a Tor-rigid maximal
Cohen-Macaulay module and, for some $t>n$,
$$
\Ext^i_R(M,N)=0 \ \ {\rm whenever} \ \ t\leq i \leq t+n\,,
$$
then $\pd_R M <\infty$ or $N$ is free. \end{thm} \begin{proof}
First observe that $R$ is Cohen-Macaulay, since it admits a
Tor-rigid maximal Cohen-Macaulay module (\cite[Theorem 4.3]{aus} or \cite[Corollary 4.7]{CZGS}).
If $n=0$, $R$ is a fiber product ring. In this case, $\pd_R N \le 1$, by Corollaries \ref{CMGAR} and \ref{cor:CMEXT}. Since $N$ is maximal Cohen-Macaulay, $N$ is actually free, by the Auslander Buchsbaum Formula (Remark \ref{rem:ABF}).
Assume from now on that $n\ge 1$. Putting $X:=\Omega_R^n( M)$, noting
that $t-n\ge 1$, and shifting merrily, we obtain \begin{equation}\label{eq:burp} \Ext^i_R(X,N)=0 \ \ \ {\rm whenever} \ \ t-n\leq i \leq t\,. \tag{\ref{evodthm00}.0} \end{equation}
Put $Y_j = X/(x_1,\dots,x_j)X$, for $0\le j \le n$. Now $\underline x$ is $X$-regular by Lemma \ref{lem:syz}, and hence we have short exact sequences \begin{equation}\label{eq:gulp} 0\longrightarrow Y_{j-1}\stackrel{x_{j}}\longrightarrow Y_{j-1} \longrightarrow Y_j\longrightarrow 0,\ \ 1\le j \le n\,. \tag{\ref{evodthm00}.1} \end{equation}
We want to deduce, from the long exact sequence of Ext, that \begin{equation}\label{eq:aargh} \Ext^t_R(Y_n,N)=0\,.\tag{\ref{evodthm00}.2} \end{equation} To do this, let $T$ be the isosceles right-triangular region $$T:= \{(i,j) \mid 0 \le j \le i +n-t \le n\},$$ with vertices $(t-n,0)$, $(t,0)$ and $(t,n)$ in the $(i,j)$ plane, and let $$S = \{(i,j)\in T \mid \Ext_R^i(Y_j,N) = 0\}.$$ We claim that $S=T$. The bottom leg of $T$, namely, $\{(i,0) \mid t-n\le i \le t\}$, is contained in $S$, by \eqref{eq:burp}. Therefore, to prove the claim it suffices to show that \begin{equation}\label{eq:bleat} (i,j-1)\in S, \ (i+1,j-1)\in S, \text{ and } j \le n \implies (i+1,j) \in S\,. \tag{\ref{evodthm00}.3} \end{equation} But \eqref{eq:bleat} follows immediately from the following snippet of the long exact sequence of Ext stemming from the $j^{\text{th}}$ short exact sequence \eqref{eq:gulp}: $$ \to \Ext^i_R(Y_{j-1},N) \to \Ext^{i+1}_R(Y_j,N) \to \Ext_R^{i+1}(Y_{j-1},N)\to $$ This verifies the claim that $S=T$. In particular, $(t,n)$ belongs to $S$, and \eqref{eq:aargh} is verified.
Since $N$ is maximal Cohen-Macaulay, the $R$-regular sequence $\underline{x}$ is also $N$-regular, and so
Remark \ref{Mats-Formulas} (a) shows that \begin{equation}\label{eq:gasp} \Ext^{t-n}_{R/(\underline{x})}(X/\underline{x}X,N/\underline{x}N)=0\,. \end{equation} Since $N$ is Tor-rigid, we see from Remark \ref{Mats-Formulas} (b) that $N/{\underline x} N$ is a Tor-rigid $R/\underline{x}$-module. Now $\overline R$ is a Cohen-Macaulay fiber product ring,
so Corollaries \ref{cor:CMEXT} and \ref{CMGAR} yield $\pd_{R/(\underline{x})}X/\underline{x}X \leq 1$ or $\pd_{R/(\underline{x})}N/\underline{x}N \leq 1.$ By \cite[Lemma 1.3.5]{BH}, $ \pd_R X \leq 1$ or $\pd_R N \leq 1.$
If $ \pd_R X \leq 1$, then
$\pd_RM< \infty$, and we are done. Assume $\pd_RN \leq 1.$ The Auslander-Buchsbaum Formula (Remark \ref{rem:ABF}) and the fact that $N$ is maximal Cohen-Macaulay now imply $\pd_R N =0$, and so $N$ is free.
\end{proof}
If $R$ is assumed to be Gorenstein, we can get by without the assumption that $N$ is maximal Cohen-Macaulay. For this we need Remark~\ref{JorExtsyz}.
\begin{remark} \label{JorExtsyz} \cite[Formula {\bf (1.2)}, p. 164]{HJ} Let $R$ be a local Gorenstein ring, let $M$ be a finitely generated maximal Cohen-Macaulay $R$-module, and let $N$ be a
finitely generated $R$-module. Then $ \Ext^i_R(M,R)= 0$, for every $i\ge 1$. It follows that $ \Ext^i_R(M,N)= \Ext^{i+j}_R(M,\Omega^j_RN)$, for every $i\ge 1$ and $j\ge 0$. \end{remark} \begin{cor} \label{evodthm001}
Let $R$ be a $d$-dimensional Gorenstein quasi-fiber product ring
with respect to
the regular sequence $\underline{x}=x_1,\ldots,x_n$. Let $M$ and $N$ be nonzero $R$-modules. If $N$ is Tor-rigid, and there is an integer $t > n$ such that
$$
\Ext^i_R(M,N)=0 \ \ \ {\rm whenever } \ \ t\leq i \leq t+n\,,
$$
then $\pd_R M <\infty$ or $\pd_R N <\infty$. \end{cor} \begin{proof} By Remark~\ref{rem:size}, $n=d-1$. By Fact~\ref{fact:torsion}(iii), $X:=\Omega_R^d(M)$ is Cohen-Macaulay. We have $$ \Ext^i_R(X,N)=0 \ \ \ {\rm whenever} \ \ t-d\leq i \leq t+n-d.$$
If $Y:=\Omega_R^d(N)$, the hypothesis that $R$ is Gorenstein and the fact that $X$ is maximal Cohen-Macaulay imply, by Remark~\ref{JorExtsyz}, that $$ \Ext^i_R(X,Y)=0 \ \ \ {\rm whenever} \ \ t\leq i \leq t+n\,. $$
For $R$-modules $U$ and $V$, the isomorphisms $\Tor^R_m(U,\Omega V) \cong \Tor^R_{m+1}(U,V)$, for all $m\ge 1$, imply that a syzygy of a Tor-rigid module is Tor-rigid. Thus $Y$ is Tor-rigid and maximal Cohen-Macaulay. It follows from Theorem \ref{evodthm00} that at least one of $X$ and $Y$ has finite projective dimension, and, of course, the same must hold for $M$ and $N$. \end{proof}
\begin{comment} \begin{thm} \label{evodthm001}
Let $(R,\mathfrak{m}) $ be a $d$-dimensional Gorenstein quasi-fiber product ring
with respect to
the regular sequence $\underline{x}=x_1,\ldots,x_n$. Let $M$ and $N$ be nonzero $R$-modules. If $N$ is Tor-rigid and, for some $t>d$,
$$\Ext^i_R(M,N)=0 \ \ \ {\rm whenever } \ \ t\leq i \leq t+n,$$ then $\pd_R M <\infty$ or $\pd_R N <\infty$. \end{thm} \begin{proof} If $\mathop{\rm dim} R=0$, Remark~\ref{rem:size} yields $n\leq \mathop{\rm dim} R$, and then $R$ is a fiber product ring.
In this case, the result follows by Corollaries \ref{CMGAR} and \ref{cor:CMEXT}. So it remains to consider the case $\mathop{\rm dim} R\geq 1$.
Note that, if $X:=\Omega_R^d(M)$ one obtains $$\Ext^i_R(X,N)=0 \ \ \ {\rm for} \ \ {\rm all} \ \ t-d\leq i \leq t+n-d.$$
Now, if $Y:=\Omega_R^d(N)$, the hypothesis that $R$ is Gorenstein and using the fact that $X$ is maximal Cohen-Macaulay, \cite[1.2]{HJ} implies $$\Ext^i_R(X,Y)=0 \ \ \ {\rm for} \ \ {\rm all} \ \ t\leq i \leq t+n.$$
Since $\underline{x}$ is $X$-regular, by the short exact sequences $$0\to X/(x_1,\ldots,x_{j-1})X\stackrel{x_j}\to X/(x_1,\ldots,x_{j-1})X \to X/(x_1,\ldots,x_{j})X \to 0$$ for $1\leq j\leq n$, we deduce from the long exact sequence of Ext that $$\Ext^i_R(X/\underline{x}X,Y)=0 \ \ \ {\rm for} \ \ i = t+n.$$
Since $\underline{x}$ is $Y$-regular, \cite[Lemma 2 (i), Section 18]{Mat} provides \begin{equation}\label{extevothm00} \Ext^i_{R/(\underline{x})}(X/\underline{x}X,Y/\underline{x}Y)=0 \ \ \ {\rm for} \ \ i=t. \end{equation}
Since $N$ is Tor-rigid, $Y$ is Tor-rigid and so $Y/\underline{x}Y$ is Tor-rigid $R/\underline{x}$-module. Further, $t>d \geq 1$ and $R/(\underline{x})$ is a Gorenstein fiber product ring.
Therefore (\ref{extevothm00}) and Corollary \ref{cor:CMEXT} give $\pd_{R/(\underline{x})}X/\underline{x}X \leq 1$ or $\pd_{R/(\underline{x})}Y/\underline{x}Y \leq 1.$ Finally, \cite[Lemma 1.3.5]{BH} provides $ \pd_R X \leq 1$ or $\pd_R Y \leq 1,$ and this establishes our fact. \end{proof}
\end{comment}
Recall that an $R$-module $N$ is said to be a {\it rigid-test} module provided the following holds for all $R$-modules $Z$ (see \cite[Definition 2.3]{CZGS}): $$ \Tor_n^R(Z,N)=0 \ \text{ for some }
n\geq 1 \ \implies \ \Tor_{n+1}^R(Z,N)=0 \ {\rm and} \ \pd_RZ<\infty.
$$
\begin{cor}\label{cor:rigid-test}
Let $R$ be a Gorenstein quasi-fiber product ring
with respect to
the sequence $\underline{x}=x_1,\ldots,x_n$. Let $M$ and $N$ be nonzero $R$-modules. Assume $N$ is a rigid-test module and, for some $t>n$,
$$
\Ext^i_R(M,N)=0 \ \text{ whenever } \\ t\leq i \leq t+n\,.
$$
Then $\pd_R M <\infty$ or $R$ is regular. \end{cor} \begin{proof} Corollary \ref{evodthm001} implies $\pd_R M <\infty$ or $\pd_R N <\infty$, since $N$ is Tor-rigid. If $\pd_R M <\infty$, we are done.
Suppose $\pd_R N <\infty$. By \cite[Theorem 1.1]{CZGS}, a local ring having a rigid test module of finite projective (or injective) dimension must be regular. Therefore the desired conclusion follows.
\end{proof}
Perhaps it is worth stating the case $n = 0$ of Corollary \ref{cor:rigid-test}:
\begin{cor} Let $R$ be a Gorenstein fiber product ring.
Let $M$ and $N$ be nonzero $R$-modules such that $N$ is a rigid-test module. If $\Ext^i_R(M,N)=0$ for some $i>1$, then $\pd_RM \leq 1$.\qed \end{cor} \begin{proof} By Corollary \ref{cor:rigid-test}, one has $\pd_RM> \infty$ or $R$ is regular. Since $R$ is a not a domain, it cannot be regular. Therefore $\pd_R M <\infty$, and now Fact \ref{pdfin1} shows that $\pd_RM\le 1$. \end{proof}
\noindent {\bf Some additional results.} \ \ For the rest of this section, we investigate the vanishing of Ext without the assumption of Tor-rigidity.
Recall that an $R$-module $M$ satisfies {\it Serre's condition $(S_n)$} if \begin{equation*}\depth_{R_{\mathfrak{p}}}M_{\mathfrak{p}}\geq {\rm min}\{n, \mathop{\rm dim} R_{\mathfrak{p}}\},\tag{$\ast$} \end{equation*}
for all $\mathfrak{p} \in {\rm Supp}_R(M)$.
(Warning: Some sources define $(S_n)$ by the inequality \begin{equation}\depth_{R_{\mathfrak{p}}}M_{\mathfrak{p}}\geq {\rm min}\{n, \mathop{\rm dim} M_{\mathfrak{p}}\}.\tag{$\ast'$} \end{equation}
The two definitions are not equivalent. For example, using the definition with the
inequality ($*'$), every module of finite length satisfies $(S_n)$ for every $n$, whereas, by the definition using inequality
($*$), a nonzero module of finite length over a ring of positive dimension does not even satisfy $(S_1)$.)
For Gorenstein quasi-fiber product rings,
we obtain the following result.
\begin{thm}\label{prop:jor} Let $R$ be a $d$-dimensional Gorenstein quasi-fiber product ring
with respect to the regular sequence ${\underline x} = x_1,\dots, x_n$. Suppose that
$$
\Ext^i_R(M,M)=0 \ \ {\rm for } \ \ 2\leq i \leq d+1\,.
$$
\begin{itemize}
\item[(i)] If $M$ is a maximal Cohen-Macaulay $R$-module, then $M$ is free.
\item[(ii)] If $M$ satisfies $(S_k)$ for some $k\ge n$, then $\pd_RM \leq 1$.
\end{itemize} \end{thm} \begin{proof} (i) Since $M$ is maximal Cohen-Macaulay over a Gorenstein ring, $M$ is a $d^{\text{th}}$ syzygy, by \cite[Corollary A.15]{LW}. Then Lemma \ref{lem:syz} implies that $\underline{x}$ is an $M$-sequence. As in the proof of Theorem \ref{evodthm00}, apply the long exact sequence of Ext to the short exact sequences $$ 0\to M/(x_1,\ldots,x_{j-1})M\stackrel{x_j}\to M/(x_1,\ldots,x_{j-1})M \to M/(x_1,\ldots,x_{j})M \to 0\,, $$ for $1\leq j\leq n$. We deduce that $$ \Ext^i_R(M/\underline{x}M,M)=0 \ \ {\rm for} \ \ 2+n\leq i \leq d+1. $$
Now \ref{Mats-Formulas} (a) shows that \begin{equation}\label{eq:wheeze} \Ext^i_{R/(\underline{x})}(M/\underline{x}M,M/\underline{x}M)=0 \ \ {\rm for} \ \ 2\leq i \leq 1+d-n. \tag{\ref{prop:jor}.0} \end{equation} Since $R/(\underline{x})$ is a Gorenstein fiber product ring, Fact \ref{factGor} yields that $R/(\underline{x})$ is a $1$-dimensional hypersurface. Hence $n=d-1$ by Remark~\ref{rem:size}, and hence \eqref{eq:wheeze} just says that $\Ext^2_{R/(\underline{x})}(M/\underline{x}M,M/\underline{x}M)=0$. Now \cite[Proposition 2.5]{Jor} yields $\pd_{R/(\underline{x})} M/\underline{x}M \leq 1$.
Therefore $\pd_R M\leq 1$. Since $M$ is maximal Cohen-Macaulay, $M$ is free by the Auslander-Buchsbaum Formula.
(ii) Since $M$ satisfies $(S_k)$ for some $k\ge n$ and $R$ is Gorenstein, $M$ is $k^{\text{th}}$ syzygy by \cite[Corollary A.15]{LW}, and Lemma \ref{lem:syz} says that $\underline{x}$ is also an $M$-sequence. The proof is now similar to the proof of part (i). \end{proof}
The next result is a consequence of Theorem \ref{prop:jor}, since Gorenstein fiber product rings
are $1$-dimensional (see Fact \ref{factGor}).
\begin{cor}\label{cor:jor1} Let $R$ be a Gorenstein fiber product ring.
Let $M$ be a maximal Cohen-Macaulay $R$-module. If $\Ext^2_R(M,M)=0$, then $M$ is free. \end{cor}
Since a Gorenstein fiber product ring is a hypersurface (Fact \ref{factGor}), and hence a complete intersection, Corollary \ref{cor:jor1} agrees with \cite[Proposition 2.5]{Jor}.
Remarks~\ref{AvrBuc} contains a summary of related terminology and results from Avramov's and Buchweitz' article \cite{AvBu}; Theorem~\ref{TAvrBuc} and Corollary~\ref{cAvBu} follow from these remarks.
\begin{remarks} \label{AvrBuc} \cite{AvBu} Let $R$ be a a Noetherian ring.
(1) For $R$ local, a {\it quasi-deformation} (of codimension c) is a diagram of local homomorphisms $\CD R @>>> R' @<<< Q \endCD$,
the first being faithfully flat and the second surjective with kernel generated by a $Q$-regular sequence
(of length c). If $(R,\mathfrak{m})$ is a complete intersection (Remark~\ref{rem:rank}), then there exists a quasi-deformation $\CD R @>>> \widehat R @<<< Q \endCD.$
(2) Let $M\ne 0$ be a finitely generated module over $R$.
If R is local,
\noindent the {\it complete intersection dimension} over R is defined by $$\aligned\text{CI-}{\mathop{\rm dim}}_RM :=\inf\{&\pd(M\otimes_R R')- \pd_RR' , \text{ such that } \\ &\CD R@>>>R'@<<<Q\endCD\text { is a quasi-deformation}\}. \endaligned $$
\noindent For $R$ not local, the {\it complete intersection dimension} over R is defined by
$$\text{CI-}{\mathop{\rm dim}}_R M := \sup\{\text{CI-}{\mathop{\rm dim}}_R \mathfrak{m} ~| ~\mathfrak{m} \in \Max(R)\}; \quad \text{ CI-}{\mathop{\rm dim}}_R 0 = 0.$$
(3) \cite[4.1,4.1.5]{AvBu} Let R be a local ring, let $M$ be a finitely generated $R$-module with CI-${\mathop{\rm dim}}_R M<\infty$, and let
$\CD R@>>>R' @<<<Q\endCD$ be a quasi-deformation of codimension $c$ such that the module $M' = M \otimes_R R'$ has finite projective dimension over $Q$. By \cite[(1.4)]{AGP}, the Auslander-Buchsbaum Formula extends to:
$\text{CI-}{\mathop{\rm dim}}_R M =\depth R - \depth_R M\le \depth R \le \mathop{\rm dim} R.$ Thus $\text{CI-}{\mathop{\rm dim}}_R M <\infty$.
(4) \cite[5.1]{AvBu} If R is a local complete intersection and $M$ is a finitely generated $R$-module, then $\CD \pd_Q(M \otimes_R \widehat R) <\infty, \endCD $ for any quasi-deformation
$\phantom{local}\CD R @>>> \widehat R @<<< Q \endCD$, with $Q$ a regular ring.
(5) \cite[Theorem 4.2]{AvBu} Let $M$ be a finitely generated module of finite CI-dimension over $R$.
Then $M$ has finite projective dimension if and only if $\Ext^{2i}_ R (M, M) = 0$ for some $i \ge 1.$
\end{remarks} Remarks~\ref{AvrBuc} yield Theorem~\ref{TAvrBuc}, due to Avramov and Buchweitz.
\begin{theorem} \label{TAvrBuc} \cite[Theorem 4.2, (5.1), p. 24]{AvBu} If $R$ is a local complete intersection and $M$ is a finitely generated module such that $\Ext^{2i}_R(M,M)=~0$, for some $i\ge1$, then $M$ has finite projective dimension. \end{theorem} \begin{proof} Let $\CD R @>>> \widehat R @<<< Q \endCD$ be a quasi-deformation with $Q$ a complete regular local
ring. By (4) of Remarks~\ref{AvrBuc}, $\CD \pd_Q(M \otimes_R \widehat R) <\infty. \endCD $ By (3) of Remarks~\ref{AvrBuc}, $\text{CI-}{\mathop{\rm dim}}_R M <\infty$. Now (5) implies $\pd(M)<\infty$. \end{proof}
By
Fact~\ref{pdfin1}, we have a generalization of Corollary~\ref{cor:jor1}: \begin{corollary}\label{cAvBu}
If $R$ is a fiber product complete intersection ring and $M$ is a finitely generated module such that $\Ext^{2i}_R(M,M)=0$ for some $i\ge1$, ring, then $\pd_RM \le1$. \end{corollary}
Using Avramov's and Buchweitz' Theorem~\ref{TAvrBuc} and arguments similar to those in the proof of Theorem \ref{evodthm00}, we deduce the folloiwng:
\begin{prop}\label{prop:aardvark}
Let $R$ be a Gorenstein quasi-fiber product ring
with respect to the sequence $\underline{x}=x_1,\ldots,x_n$. Let $M$ be an $R$-module. If
for some even number $t>n$
$$
\Ext^i_R(M,M)=0 \ \ \ {\rm whenever} \ \ t\leq i \leq t+n,
$$
then $\pd_R M <\infty$. \end{prop} \begin{proof} Put $X=\Omega_R^nM$. By Lemma~\ref{lem:syz}, $\underline{x}$ is $X$-regular. Then $$ \Ext^i_R(X,M)=0\ \text{for}\ t-n\le i \le t \,. $$ With $Y_n = X/\underline{x} X$ and $N=X$, we obtain Equation~\eqref{eq:aargh}, exactly as in the proof of Theorem \ref{evodthm00}. Thus $$ \Ext^{t-n}_{R/\underline{x}R}(X/\underline{x}X,X/\underline{x}X) = 0. $$ Now Remark~\ref{Mats-Formulas} shows that $\Ext_R^t(X,X)=0$. Avramov's and Buchweitz's Theorem~\ref{TAvrBuc} implies $\pd_R X < \infty$, since $t$ is an even positive integer. Since $X$ is a syzygy of $M$, it follows that $\pd_RM<\infty$. \end{proof}
The next result provides a bound $b$ such that if $\Ext^i_R(M,M\oplus R)=0, $ for every $i$ with $1\leq i \leq b,$ then $M$ is free. This gives a positive answer to Question \ref{question1} in case $N=M\oplus R$ and improves Corollary \ref{cor:arc}.
\begin{thm}\label{thm:arcminimal} Let $R$ be a quasi-fiber product ring
with respect to the sequence
$\underline{x}=x_1,\ldots,x_n$. Let $M$ be a finitely generated $R$-module. If $t\geq {\rm max}\{5,n+1\}$ and
$$\Ext^i_R(M,M\oplus R)=0, \text{ for every }i\text{ with }1\leq i \leq t+\depth R+n,$$ then $M$ is free. Thus
$$\aligned n>4,& \Ext^i_R(M,M\oplus R)=0,\text{ for }1\leq i \leq 3n+2 \implies M\text{ is free.}\\
n\le 4,& \Ext^i_R(M,M\oplus R)=0,\text{ for }1\leq i \leq 5+2n+1 \implies M\text{ is free.}\\ n\le 4,& \Ext^i_R(M,M\oplus R)=0,\text{ for }1\leq i \leq 14 \implies M\text{ is free.} \endaligned $$ \end{thm} \begin{proof} Set $\alpha:= t+\depth R+n$. By Proposition~\ref{Jorprop}(1), \begin{equation}
\Tor_i^R(\D_{\alpha+1}M,M)=0, \ {\rm for \ every }~i~ {\rm with}~\ 1 \leq i \leq \alpha. \tag{\ref{thm:arcminimal}.1} \end{equation} Since $R$ is a quasi-fiber product ring, Equation~(\ref{thm:arcminimal}.1) and Remark~\ref{rem:classNT}(1) imply $\pd_R M <\infty$ or $\pd_R (\D_{\alpha+1}M) < \infty$. By the Auslander-Buchsbaum Formula, $\pd_R M \leq \depth R$ or $\pd_R (\D_{\alpha+1}M) \leq \depth R$. Thus \begin{equation}\label{VanTor}\Tor_i^R(\D_{\alpha +1}M,M)=0, \ {\rm for \ every }~i \ {\rm with }~ i>\depth R.\tag{\ref{thm:arcminimal}.2} \end{equation}
Now Proposition~\ref{Jorprop}(2) provides the exact sequence $$ \Hom_R(M,R)\otimes_R M \to \Hom_R(M, M)\to \Tor_{\alpha+1}^R(\D_{\alpha+1}M,M)\to 0,$$ where the homomorphism $ \Hom_R(M,R)\otimes_R M \to \Hom_R(M, M)$ is the natural one.
Since $\alpha+1> \depth R$, $\Tor_{\alpha+1}^R(\D_{\alpha+1}M,M)=0$, by (\ref{VanTor}). Thus the previous exact sequence yields a surjective map $$\Hom_R(M,R)\otimes_R M \to \Hom_R(M, M).$$
The freeness of $M$ follows by Proposition~\ref{ACT}(a), (\cite[A.1]{AuGol}).
The ``Thus" statement follows by setting $ t=n+1$ for the first line and setting $t=5$ in the second line, and using that $\depth R\le n+1$ by Remark~\ref{rem:size}. \end{proof}
A natural consequence of Theorem~\ref{thm:arcminimal} of particular interest is the case that $R$ is a fiber product ring; that is, $n=0$.
{Corollary}~\ref{cor:arcminimal} improves Nasseh and Sather-Wagstaff's result \cite[Theorem 4.5 (b)]{NS} that (ARC) holds for fiber product rings.
\begin{cor}\label{cor:arcminimal} Let $R$ be a fiber product ring
and let $M$ be an $R$-module. If $\Ext^i_R(M,M\oplus R)=0$, for $1\leq i \leq 6$, then $M$ is free. \end{cor}
\section{The Huneke-Wiegand Conjecture}
Conjecture {\rm(HWC)} is related to another condition on local rings we call (HW$^{2}$), also considered by Huneke and R.~Wiegand in \cite{HW}.
\begin{definition} Let $(R,\mathfrak{m})$ be a local ring; define (HW$^{2}$) on $R$ as follows: \begin{enumerate}
\item[] (HW$^{2}$)\ \ For every pair $M$ and $N$ of finitely generated $R$-modules such that $M$ or $N$ has rank,
if $M\otimes_R N$ is a maximal Cohen-Macaulay module,
then $M$ or $N$ is free.\end{enumerate} \end{definition}
Theorem~\ref{hw-main}, one of the main results of \cite{HW}, yields that every hypersurface satisfies {\rm (HW$^{2}$)}. \begin{thm}\label{hw-main} ~\cite[Theorem 3.1]{HW} Let $(R,\mathfrak{m},k)$ be an abstract hypersurface, and let $M$ and $N$ be finitely generated $R$-modules such that $M$ or $N$ has rank.
If $M\otimes_R N$ is a maximal Cohen-Macaulay module,
then both $M$ and $N$ are maximal Cohen-Macaulay modules and one of $M$ or $N$ is free.\end{thm} They also give Example~\ref{cinotHW''} below to show that even complete intersection rings may not satisfy (HW$^{2}$). \begin{example} \label{cinotHW''} \cite[Example 4.3]{HW}. Let $R:=k[[T^4, T^5, T^6]],~ I:=(T^4, T^5),$ and $J:=(T^4, T^6).$ Then $R$ is a complete intersection and $I\otimes_RJ$ is torsion-free, and so Cohen-Macaulay by Facts~\ref{fact:torsion}(ii), yet neither $I$ nor $J$ is free. For the proof that $I\otimes_RJ$ is torsion-free, see \cite{HW}. \end{example}
\begin{proposition}\label{HWfiber} Let $R$ be a Gorenstein fiber product ring and let $M$ be a torsion-free $R$-module with rank. If $M\otimes_RM^*$ is torsion-free, then $M$ is free.
\end{proposition} \begin{proof} Since $R$ is a Gorenstein fiber product ring, $R$ is a $1$-dimensional hypersurface (Fact \ref{factGor}). By \cite[Theorem 3.7]{HW}, either $M$ or $M^*$ is free. If $M^*$ is free, then so is $M^{**}$. Using (ii), and then (i), of Facts \ref{fact:torsion}, we see that $M$ is reflexive, and hence is free in either case. \end{proof}
\begin{cor} \label{HWCfpr} If $R$ is a Gorenstein fiber product ring, then $R$ satisfies {\rm(HW)}. That is, if $M$ is a torsion-free $R$-module with rank and $M\otimes_R M^*$ is MCM,
then $M$ is free. \end{cor} \begin{cor} \label{HWCqfpr1} If $R$ is a one-dimensional Gorenstein quasi-fiber product ring, then $R$ satisfies {\rm(HW)}.\end{cor}
To see Corollary \ref{HWCqfpr1}, apply Fact~\ref{factGorqfd1}.
Next we give an example showing why for higher dimensional rings simply assuming in (HWC) (see Definition~\ref{defhwc'}) that $M\otimes_RM^*$ is torsion-free (rather than reflexive) over a Gorenstein domain $R$ is not enough to conclude that $M$ is free. \begin{example}\label{guppybarf} Let $(R,\mathfrak{m})= k[[x,y]]$, a Gorenstein local domain with maximal ideal $\mathfrak{m}=Rx+Ry$. Then $\mathfrak{m}^{-1} = \{\alpha\in K \mid \alpha\mathfrak{m}\subseteq R\},$ where $K$ is the quotient field of $R$.
We claim that $\mathfrak{m}^{-1} = R$. Of course $\mathfrak{m}^{-1}$ is naturally isomorphic to $\mathfrak{m}^*$. Since $\mathfrak{m}^{-1} \supseteq R$, we prove the reverse inclusion. Let $\alpha\in \mathfrak{m}^{-1}$, and write $\alpha = \frac{a}{b}$ in lowest terms (recall that $R$ is a UFD). Suppose, by way of contradiction, that $b$ is not a unit of $R$, and let $p$ be a prime divisor of $b$. Since $\frac{a}{b}x\in R$ and $\frac{a}{b}y\in R$, we have $ax\in Rb$ and $ay\in Rb$. Therefore $p\mid ax$ and $p\mid ay$. The assumption about lowest terms means that $p\nmid a$, and hence $p\mid x$ and $p\mid y$. That's impossible, and the claim is proved. Now $\mathfrak{m}\otimes_R\mathfrak{m}^* \cong \mathfrak{m}\otimes_RR \cong \mathfrak{m}$, which is torsion-free. But, of course, $\mathfrak{m}$ is not free. \end{example}
\begin{conj}\label{HWregular} Let $(R,\mathfrak{m})$ be a Gorenstein local ring and let $x\in \mathfrak{m}$ be a non-zerodivisor. If $R/(x)$ satisfies {\rm(HW)}, then $R$ satisfies {\rm(HW}). \end{conj}
Conjecture \ref{HWregular} is related to Conjecture~\ref{HWGorqf}:
\begin{conj} \label{HWGorqf} Gorenstein quasi-fiber product rings
satisfy {\rm(HW)}. \end{conj}
In particular, we are interested in the case when $R$ has dimension two:
\begin{question} \label{HWGorqf2d} If $M$ is a finitely generated torsion-free module with rank over a two-dimensional Gorenstein quasi-fiber product ring $R$ and $M\otimes_RM^*$ is maximal Cohen-Macaulay, must $M$ be free?
\end{question}
We have some results related to the conjectures with additional hypotheses. First we make a remark and prove a lemma.
\begin{remark} \label{0dGor} If $R$ is a $0$-dimensional local ring, then $R$ satisfies {\rm(HW)}. This follows trivially since every $R$-module with rank is free. \end{remark}
One stumbling block to proving Conjectures~\ref{HWregular} and \ref{HWGorqf} and answering Question~\ref{HWGorqf2d} is that ``torsion-free with rank" for $M$ as an $R$-module is not necessarily
inherited by $M/IM$ as an $R/I$-module for an ideal $I$ of $R$, even if $I$ is a principal ideal generated by a non-zerodivisor on both $R$ and $M$:
\begin{example}\label{ex:platypus} Let $k$ be a field, let $R=k[[x,y]]$, and let $M$ be the maximal ideal $Rx+Ry$. Then $M$ is a torsion-free $R$-module. However $M/xM$ is not torsion-free
as a module over $\overline R=R/(x)$, since $x\in M\setminus xM$ implies $\overline x\ne \overline 0$ in $\overline M$, but $yx\in xM$, and so $\overline{y}\cdot \overline x=\overline 0$.
(Note that $\overline y$ is a non-zerodivisor of $\overline R = k[[y]]$.)
Also, for $N=R/(y)$
and $f=xy$, the module
$N$ has rank (because $R$ is a domain), but $\overline N:=N/fN$ does not have rank as an $R/(f)$-module. To see this: The ring $S:=R/(f)$ has two associated primes, namely
$P := S\overline x$ and $Q:= S\overline y$. Now $yN=0\implies \overline y\cdot\overline N=0\implies \overline N_P=0$,
since $\overline y$ is a unit of $S_P$. To see that $\overline N_Q\ne 0$, consider the element $\overline 1\in\overline N$, that is, the coset $1+(y)+xy N$. If $\frac{\overline 1}{\overline 1}=0$ in $\overline N_Q$, then,
for some $t\in S\setminus S\overline y$, we would have that $t\cdot\overline 1$ is
the coset $0+(y)+xy N$. Now $t\in S\setminus S\overline y\implies t$ is a coset of form $ g_1(x)+yg_2(y)+xyR,$ where $0\ne g_1(x)\in k[[x]]$ and $g_2(y)\in k[[y]]$. Then $$t\cdot\overline 1= (g_1(x)+yg_2(y)+xyR)(1+(y)+xy N)= g_1(x)+(y)+xy N\ne 0,$$ a contradiction. Thus $\overline N_Q\ne 0$.\end{example}
\begin{comment}
\begin{remark}\label{tf-MCM} Let $(R,\mathfrak{m})$ be a one-dimensional local CM ring and $M$ a finitely generated $R$-module. Then $M$ is torsion-free if and only if $M$ is MCM. \end{remark}\begin{proof}Since $R$ is CM, there is a non-zerodivisor $f\in \mathfrak{m}$. If $M$ is torsion-free, then $f$ is a non-zerodivisor on $M$, and hence $\depth M \ge1$, that is, $M$ is MCM. Conversely, suppose $M$ is MCM, and let $r$ be a zero-divisor on $M$. Then $r\in P$ for some $P\in \Ass M$. Now $P\ne \mathfrak{m}$ since $M$ is MCM, and hence $P$ is a minimal prime ideal of $R$. Therefore $r$ is a zero-divisor of $R$; this shows that $M$ is torsion-free. \end{proof}
\end{comment}
On the other hand, we do have Lemma~\ref{freemodregseq}: \begin{lemma} \label{freemodregseq} Let $N$ be a finitely generated module over a local ring $(R, \mathfrak{m})$. Let $\underline x=\{x_1,\ldots,x_n\}$ be a regular sequence of $R$ such that $\underline x
$ is a regular sequence on $N$ and $N/(\underline x)N$ is free. Then $N$ is free. \end{lemma} \begin{proof} First consider the case $n=1$, that is, $\underline x=\{x\}$, where $x\in\mathfrak{m}$ and $x$ is a non-zerodivisor on $R$ and $N$. By \cite[Lemma 4.9]{BH}, $\pd N=\pd (N/xN)$. Since $N/xN$ is free,
$\pd N=\pd (N/xN)=0$, and so $N$ is free.
For $n>1$, use induction and the equation $$N/(x_1,\ldots,x_{n})N=\frac{N/(x_1,\ldots,x_{n-1})N}{x_n(N/(x_1,\ldots,x_{n-1})N)}.$$ \vskip-20pt \end{proof}
By applying Lemma~\ref{freemodregseq}, we have Proposition~\ref{Gorqfp}.
\begin{proposition} \label{Gorqfp} Let $R$ be a Gorenstein quasi-fiber product ring, and let $(\underline x) $ be a regular sequence such that $R/(\underline x)$ is a fiber product ring.
Let $M$ be a finitely generated $R$-module such that \begin{enumerate} \item [$(1)$] $\underline x$ is a regular sequence on $M$,
\item [$(2)$] $M/(\underline x)M$ is torsion-free and has rank as an $(R/(\underline x))$-module, \item [$(3)$] $(M/(\underline x)M) \otimes_{R/(\underline{x})} \Hom_{R/(\underline x)}(M/(\underline{x})M,R/({\underline{x}}))$ is torsion-free as an $(R/(\underline x))$-module. \end{enumerate} \noindent Then $M$ is free. \end{proposition} \begin{proof} By Proposition~\ref{HWfiber}, $M/(\underline x)M$ is free as an $(R/(\underline x))$-module. Now Lemma~\ref{freemodregseq} implies that $M$ is free. \end{proof}
Proposition~\ref{Gorqfp} in the case $\mathop{\rm dim} R=2$ yields a partial affirmative answer to Question~\ref{HWGorqf2d} in Corollary~\ref{HWGorqf2dTry}. We need to replace the conditions on $M$ in (\ref{HWGorqf2d}) by conditions on $M/xM$, where $x$ is a regular element of $R$ such that $R/(x)$ is a fiber product ring and $x$ is regular on $M$. \begin{corollary} \label{HWGorqf2dTry} Let $(R, \mathfrak{m})$ be a two-dimensional
Gorenstein quasi-fiber product ring,
let $x$ be a non-zerodivisor of $R$ such that $x\in\mathfrak{m}$ and $R/(x)$ is a fiber product ring,
and let $M$ be a finitely generated $R$-module such that $x$ is regular on $M$. If
$M/xM$ is a torsion-free $(R/(x))$-module with rank, and
$(M/xM)\otimes_{R/(x)}\Hom_{R/(x)}(M/xM, R/(x))$ is maximal Cohen-Macaulay
as an $(R/(x))$-module, then $M$ is free. \end{corollary}
\begin{proof} By (ii) of Fact`\ref{fact:torsion} and Proposition~\ref{HWfiber}, one has $M/xM$ is free as an $(R/(x))$-module; now Lemma~\ref{freemodregseq} says that $M$ is free over $R$. \end{proof}
\begin{remark}\label{st2} Another stumbling block: It is easily seen that $M \otimes_R M^*$ MCM implies that tensoring the terms with $R/(\underline x)$ preserves MCM, so \begin{equation*} (M/(\underline{x})M) \otimes_{R/(\underline{x})} (M^*/(\underline x)M^*)\tag{\ref{st2}.0} \end{equation*} is MCM as an $(R/(\underline x))$-module (equivalently torsion-free
as an $(R/(\underline x))$-module, by (ii) of Facts~\ref{fact:torsion}). But it does not necessarily follow that \begin{equation*}(M/(\underline x)M) \otimes_{R/(\underline{x})}
\Hom_{R/(\underline x)}(M/(\underline{x})M,R/(\underline{x}))
\tag{\ref{st2}.1} \end{equation*}
is MCM as an $(R/(\underline x))$-module. If $ M^*/(\underline x)M^*= \Hom_{R/(\underline x)}(M/(\underline{x})M,R/(\underline{x}))$, then Expresssion~(\ref{st2}.1) would be MCM.
\end{remark}
\begin{comment} \begin{remark} \label{trylocht} We would like to do something like this:
Let $M$ be a finitely generated torsion-free module with rank over a two-dimensional Gorenstein quasi-fiber product ring $R$ such that $M\otimes_RM^*$ is maximal Cohen-Macaulay, and let $x$ be a non-zerodivisor of $R$ such that $x\in\mathfrak{m}$ and $R/xR$ is a Gorenstein fiber product ring. Also assume that $x$ is a non-zerodivisor on $M$.
Let $\mathfrak{p}$ be a height-one prime ideal of $R$. Then $R_\mathfrak{p}$ is a one-dimensional Gorenstein ring, $M_\mathfrak{p}$ is a finitely generated torsion-free module with rank over $R_\mathfrak{p}$ and
$M_\mathfrak{p}\otimes_{R_\mathfrak{p}} M^*_\mathfrak{p}= M\otimes_RM^*\otimes_R{R_\mathfrak{p}}$ is maximal Cohen-Macaulay as an $R_\mathfrak{p}$-module. However, $R_\mathfrak{p}$ is not necessarily a quasi-fiber product ring, nor is $R_\mathfrak{p}$ necessarily a hypersurface.
If $x\in \mathfrak{p}$ and $x$ is a non-zerodivisor, then $R_\mathfrak{p}/xR_\mathfrak{p}\cong (R/xR)_{(\mathfrak{p} /xR)}$ is a zero-dimensional Gorenstein ring and satisfies (HW). If $M_\mathfrak{p}/xM_\mathfrak{p}$ is torsion-free with rank as an $(R_\mathfrak{p}/ xR_\mathfrak{p})$-module, then $M_\mathfrak{p}/xM_\mathfrak{p}$ is free as an $(R_p/ xR_\mathfrak{p})$-module by Remark~\ref{0dGor}. Thus, by Lemma~\ref{freemodregseq}, $M_\mathfrak{p}$ would be free as an $R_\mathfrak{p}$-module. If $x\notin \mathfrak{p}$, then what is $R_\mathfrak{p}$?
Can we show that every $R_\mathfrak{p}$ that arises in this way satisfies (HW)? If so, then by Proposition~\ref{ACT} $M$ is free.\end{remark}
\end{comment}
Lemma~\ref{lem:bleat} gives a condition that implies $\overline {M^*}=({\overline M})^*$, for certain $R$-modules $M$ and
$x\in R$, where $I:=xR$, $\overline {M^*} :=M^*/IM^*$ , $\overline M:=M/IM$ and
$$
(\overline M)^*=(M/IM)^*:=\Hom_{R/I}(M/IM, R/I)\,.
$$ \begin{lem}\label{lem:bleat}
Let $(R,\mathfrak{m})$ be a local ring and $M$ and $N$ be $R$-modules. Let $x\in \mathfrak{m}$ be a NZD in $R$ and also a NZD on $N$.
For any $R$-module $V$, denote $V/xV$ by $\overline{V}$.
The natural map $\Hom_R(M,N) \to \Hom_{\overline{R}}(\overline{M},\overline{N})$
induces an injective homomorphism $\overline{\Hom_R(M,N)} \hookrightarrow
\Hom_{\overline{R}}(\overline{M},\overline{N})$. If, in addition,
$\Ext^1_R(M,N) = 0$, then the injective homomorphism is in fact an isomorphism.
\end{lem}
\begin{proof} Let $\pi_M:M \twoheadrightarrow \overline M$ and $\pi_N:N\twoheadrightarrow \overline N$ be the natural homomorphisms. Given an $R$-homomorphsim $f:M\to N$, there is a unique $\overline R$-homomorphsim
$\overline f:\overline M \to \overline{N}$ making the following diagram commute:
$$
\begin{CD}
M @>f>> N\\
@V\pi_MVV @VV\pi_NV\\
\overline M @>\overline{f}>> \overline N.
\end{CD}
$$
The kernel of the resulting homomorphism
$\Hom_R(M,N) \to \Hom_{\overline{R}}(\overline{M},\overline{N})$
taking $f$ to $\overline f$ is $K:= \{f\in\Hom_R(M,N) \mid f(M)\subseteq xN\}$.
We claim that $K = x\Hom_R(M,N)$. Obviously $x\Hom_R(M,N) \subseteq K$.
For the reverse inclusion, suppose $f\in K$. Then, for each $m\in M$, we have
$f(m) = xn$ for some $n \in M$. Moreover, the element $n$ is unique, since $x$ is a NZD
on $N$. The correspondence $g:M\to N$ taking $m$ to $n$ is easily seen to be an
$R$-homomorphism. Since $f = xg$, this proves the claim and provides a natural injection $\overline{\Hom_R(M,N)} \hookrightarrow \Hom_{\overline{R}}(\overline{M},\overline{N})$.
For the last statement, just apply $\Hom_R(M, -)$ to the short exact sequence $$ 0 \to N \stackrel x\to N \to N/xN \to 0\,. $$ (See \cite[Proposition 3.3.3]{BH} for the details.)
\end{proof}
Corollary~\ref{Gorqfp'} now follows from Proposition~\ref{Gorqfp}.
\begin{corollary} \label{Gorqfp'} Let $R$ be a Gorenstein quasi-fiber product ring, and let $(\underline x) $ be a regular sequence such that $R/(\underline x)$ is a fiber product ring.
Let $M$ be a finitely generated $R$-module such that \begin{enumerate} \item[$(1)$] $\underline x$ is a regular sequence on $M$,
\item[$(2)$] $M/(\underline x)M$ is torsion-free and has rank as an $(R/(\underline x))$-module, \item[$(3)$] $M \otimes M^*$ is MCM as an $R$-module. \item[$(4)$] $\ext^1_R(M,R)=0$.
\end{enumerate} Then $M$ is free. \end{corollary} \begin{proof} The new condition (4) implies $\overline{M}^*=\overline{M^*}$ by Lemma~\ref{lem:bleat}, and so by Remark~\ref{st2} and Proposition~\ref{Gorqfp}, the corollary holds. \end{proof}
Using Theorem~\ref{hw-main} and Lemma \ref{freemodregseq}, we obtain the next corollary, which may be useful for Question~\ref{HWGorqf2d}.
\begin{cor} \label{HWMNhypreg} Let $(R,\mathfrak{m})$ be a local ring, let $M$ and $N$ be finitely generated $R$-modules and let $\underline x\in \mathfrak{m}$ be a regular sequence on $M$ and $N$ such that \begin{enumerate} \item [$(1)$] $R/(\underline x)$ is a hypersurface, \item [$(2)$] $M/(\underline x)M$ has rank as an $(R/(\underline x))$-module, and \item [$(3)$]
$M \otimes_R N$ is MCM as an $R$-module.\end{enumerate} Then $M$ or $N$ is free. \end{cor}
\begin{proof} By Remark~\ref{hw-main}, $R/(x)$ satisfies (HW$^{2}$). Now $M \otimes_R N$ a MCM $R$-module implies $$(M/(\underline x)M) \otimes_{R/(\underline x)} (N/(\underline x)N)=(M \otimes_R N)\otimes_R (R/(\underline x))$$ is MCM as an $(R/(\underline x))$-module. Therefore $M/(\underline x)M$ or
$N/(\underline x)N$ is free as an $(R/(\underline x))$-module. By Lemma~\ref{freemodregseq}, $M$ or $N$ is free.
\end{proof}
\begin{comment} \section{Wishful Thinking} This is not intended to be a real section of the paper --- just some commentary; and this is the easiest way to make such comments.
Since Conjecture \ref{HWregular} is not a theorem until it has been proven, it is really a conjecture about a conjecture,
and hence a metaconjecture. There is still a problem with what (HWC$_d$) should say when $(R,\mathfrak{m})$ is not a domain.
Example~\ref{guppybarf} makes a convincing argument for insisting, in (HWC$_d$), that $M\otimes_RM^*$ be reflexive, rather than simply torsion-free. (In the example, $\mathfrak{m} \cong \mathfrak{m}\otimes_R\mathfrak{m}^*$ is definitely not reflexive, since $\mathfrak{m}^{**} \cong R^*\cong R$.) Also, in the Freitas Metaconjecture, it is important that $M$ be torsion-free, since we definitely want to know that $x$ is a NZD on $M$. (But, for (HWC$_d$) itself, it's not clear that $M$ being torsion-free is essential. In fact, see (G-HWC) in the introduction.)
Clearly, we need {\em some} extra condition on $M$ though, as the example $R$ above and the ring $R/(xy)$ indicate.
Without deciding on what this extra condition should be, let us call the condition ``glorpy'' for now. The ``Glorpy Huneke-Wiegand Conjecture is then
(GHWC) Over a local ring $R$, if $M$ is a glorpy $R$-module and $M\otimes M^*$ is reflexive, then $M$ is free.
Now let us outline the main steps in what should be a proof of the Freitas Metaconjecture: \begin{enumerate} \item Prove that if $M$ is a glorpy $R$-module, then $M/xM$ is a glorpy $(R/(x))$-module. \item Prove that if $M\otimes_RM^*$ is a reflexive $R$-module, then $\overline M\otimes_{\overline R}{\overline M}^*$ is a reflexive $\overline R$-module, where the dual is computed over $\overline R$, and where the bars represent reductions modulo $x$. \item Prove, for a reflexive $R$-module $N$, if $N/xN$ is $(R/(x))$-free, then $N$ is $R$-free. \end{enumerate}
Actually, we can prove (3); all we need is that $x$ is a NZD on $R$ and on $N$. This is a standard argument: Let $$ 0 \to K \to F \to N\to 0 $$ be exact, where $F$ is free and minimal. Tensor with $R/(x)$, getting $$ 0\to \Tor_1^R(N,R/(x)) \to \overline K \to \overline F \to \overline N \to 0 $$ If we can show that $\Tor_1^R(N,R/(x) = 0$, we'll have $\overline K = 0$, since $\overline F\twoheadrightarrow \overline N$ is minimal and $\overline N$ is assumed to be free. then $K = 0$ by NAK, and hence $N$ is free. To show that $\overline K = 0$, tensor $$ 0\to R \stackrel{x}\to R \to R/(x) \to 0 $$ with $N$, getting $$ 0\to \Tor_1^R(N,R/(x)) \to N\stackrel x\to N \to N/xN \to 0 $$ Thus $\Tor_1^R(N,R/(x))$ is the kernel of $N\stackrel x\to N$, which is injective by assumption, and hence $\Tor_1^R(N,R/(x))=0$.
Lemma~\ref{lem:bleat} might be useful in attempting to verify (1) and (2):
\begin{lem}\label{lem:bleatly}
Let $(R,\mathfrak{m})$ be a local ring and $M$ and $N$ be $R$-modules. Let $x\in \mathfrak{m}$ be a NZD in $R$ and also a NZD on $N$.
For any $R$-module $V$, denote $V/xV$ by $\overline{V}$.
The natural map $\Hom_R(M,N) \to \Hom_{\overline{R}}(\overline{M},\overline{N})$
induces an injective homomorphism $\overline{\Hom_R(M,N)} \hookrightarrow
\Hom_{\overline{R}}(\overline{M},\overline{N})$. If, in addition,
$\Ext^1_R(M,N) = 0$, then the injective homomorphism is in fact an isomorphism.
\end{lem}
\begin{proof} Let $\pi_M:M \twoheadrightarrow \overline M$ and $\pi_N:N\twoheadrightarrow \overline N$ be the natural homomorphisms. Given an $R$-homomorphsim $f:M\to N$, there is a unique $\overline R$-homomorphsim
$\overline f:\overline M \to \overline{N}$ making the following diagram commute:
$$
\begin{CD}
M @>f>> N\\
@V\pi_MVV @VV\pi_NV\\
\overline M @>\overline{f}>> \overline N
\end{CD}
$$
The kernel of the resulting homomorphism
$\Hom_R(M,N) \to \Hom_{\overline{R}}(\overline{M},\overline{N})$
taking $f$ to $\overline f$ is $K:= \{f\in\Hom_R(M,N) \mid f(M)\subseteq xN\}$.
We claim that $K = x\Hom_R(M,N)$. Obviously $x\Hom_R(M,N) \subseteq K$.
For the reverse inclusion, suppose $f\in K$. Then, for each $m\in M$, we have
$f(m) = xn$ for some $n \in M$. Moreover, the element $n$ is unique, since $x$ is a NZD
on $N$. The correspondence $g:M\to N$ taking $m$ to $n$ is easily seen to be an
$R$-homomorphism. Since $f = xg$, this proves the claim and provides a natural injection $\overline{\Hom_R(M,N)} \hookrightarrow \Hom_{\overline{R}}(\overline{M},\overline{N})$.
For the last statement, just apply $\Hom_R(M, -)$ to the short exact sequence $$ 0 \to N \stackrel x\to N \to N/xN \to 0\,. $$ (See \cite[Proposition 3.3.3]{BH} for the details.)
\end{proof}
{\bf Barf!} I had intended to prove that the injection above is actually bijective, but I failed. I still do not know whether or not it is always surjective.
\end{comment}
\end{document} | arXiv |
\begin{document}
\title{Galois actions on models of curves}
\author{Lars Halvard Halle}\email{[email protected]}
\address{Department of Mathematics, KTH, S--100 44 Stockholm,
Sweden}
\begin{abstract} We study group actions on regular models of curves. If $X$ is a smooth curve defined over the fraction field $K$ of a complete discrete valuation ring $R$, every tamely ramified extension $K'/K$ with Galois group $G$ induces a $G$-action on the extension $X_{K'}$ of $X$ to $K'$. In this paper we study the extension of this $G$-action to certain regular models of $X_{K'}$. In particular, we are interested in the induced action on the cohomology groups of the structure sheaf of the special fiber of such a regular model. We obtain a formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups. Inspired by this global study, we also consider similar group actions on the cohomology of the structure sheaf of the exceptional locus of a tame cyclic quotient singularity, and obtain an explicit polynomial formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups.
We apply these results to study a natural filtration of the special fiber of the N\'eron model of the Jacobian of $X$ by closed, unipotent subgroup schemes. We show that the jumps in this filtration only depend on the fiber type of the special fiber of the minimal regular model with strict normal crossings for $X$ over $\mathrm{Spec}(R)$, and in particular are independent of the residue characteristic. Furthermore, we obtain information about where these jumps occur. We also compute the jumps for each of the finitely many possible fiber type for curves of genus $1$ and $2$. \end{abstract}
\keywords{Models of curves, tame cyclic quotient singularities, group actions on cohomology, N\'eron models}
\maketitle
\section{Introduction}
\subsection{Stable reduction of curves and Jacobians} Let $X$ be a smooth, projective and geometrically connected curve of genus $ g(X) > 0 $, defined over the fraction field $K$ of a complete discrete valuation ring $R$, with algebraically closed residue field $k$. By a \emph{model} for $X$ over $R$, we mean an integral and normal scheme $ \mathcal{X} $ that is flat and projective over $ S = \mathrm{Spec}(R) $, and with generic fiber $ \mathcal{X}_K \cong X $.
Recall the semi-stable reduction theorem, due to Deligne and Mumford (\cite{DelMum}, Corollary 2.7), which states that there exists a finite, separable field extension $ L/K$ such that $X_L$ admits a semi-stable model over the integral closure $R_L$ of $R$ in $L$.
It can often be useful to work with the Jacobian $J/K$ of $X$. The question whether $X$ has semi-stable reduction over $S = \mathrm{Spec}(R)$ is reflected in the structure of the \emph{N\'eron model} $ \mathcal{J}/S$ (cf. ~\cite{Ner}) of $J$. In fact, $X$ has semi-stable reduction over $S$ if and only if $ \mathcal{J}_k^0 $, the identity component of the special fiber, has no \emph{unipotent radical} (\cite{DelMum}, Proposition 2.3).
In general, it is necessary to make ramified base extensions in order for $X$ to obtain semi-stable reduction. If the residue characteristic is positive, it can often be difficult to find explicit extensions over which $X$ obtains stable reduction. In the case where a tamely ramified extension suffices one can do this by considering the geometry of suitable regular models for $X$ over $ S $ (cf. ~\cite{Tame}). In this paper we study, among other things, how the geometry of the N\'eron model contains information that is relevant for obtaining semi-stable reduction for $X$.
\subsection{N\'eron models and tame base change} Let $ K'/K $ be a finite, separable and tamely ramified extension of fields, and let $R'$ be the integral closure of $R$ in $ K' $. Then $R'$ is a complete discrete valuation ring, with residue field $k$. Furthermore, $ K'/K $ is Galois, with group $ G = \boldsymbol{\mu}_n $, where $ n = \mathrm{deg}(K'/K) $.
Let $ \mathcal{J}'/S' $ be the N\'eron model of the Jacobian of $ X_{K'} $, where $ S' = \mathrm{Spec}(R') $. Due to a result by B. Edixhoven (\cite{Edix}, Theorem 4.2), it is possible to describe $ \mathcal{J}/S $ in terms of $ \mathcal{J}'/S' $, together with the induced $G$-action on $ \mathcal{J}' $. Namely, if $W$ denotes the \emph{Weil restriction} of $ \mathcal{J}'/S' $ to $S$ (cf. ~\cite{Ner}, Chapter 7), one can let $G$ act on $W$ in such a way that $ \mathcal{J} \cong W^G $, where $ W^G $ denotes the scheme of invariant points. In particular, one gets an isomorphism $ \mathcal{J}_k \cong W_k^G $. By \cite{Edix}, Theorem 5.3, this decription of $ \mathcal{J}_k $ induces a descending filtration $$ \mathcal{J}_k = F_n^0 \supseteq \ldots \supseteq F_n^i \supseteq \ldots \supseteq F_n^n = 0 $$ of $ \mathcal{J}_k $ by closed subgroup schemes.
In \cite{Edix}, Remark 5.4.5, a generalization of this setup is suggested. If we define $ \mathcal{F}^{i/n} = F_n^i $, where $ F_n^i $ is the $i$-th step in the filtration induced by the extension of degree $n$, one can consider the filtration $$ \mathcal{J}_k = \mathcal{F}^0 \supseteq \ldots \supseteq \mathcal{F}^a \supseteq \ldots \supseteq \mathcal{F}^1 = 0, $$ with indices in $ \mathbb{Z}_{(p)} \cap [0,1] $. In order for this to make sense, it is necessary that the construction of $ \mathcal{F}^a $ is independent of the choice of representatives $n$ and $i$ for $a$, and that $ \mathcal{F}^a $ descends with increasing $a$. We give a proof for these properties in Section \ref{Neron}.
The filtration $ \{ \mathcal{F}^a \} $ contains significant information about $\mathcal{J}$. For instance, one can show that the subgroup schemes $ \mathcal{F}^a $ are \emph{unipotent} for $ a > 0 $. So in a natural way, this filtration gives a measure on how far $ \mathcal{J}/S $ is from being \emph{semi-abelian}.
One way to study the filtration $ \{ \mathcal{F}^a \} $ is to determine where it \emph{jumps}. This will occupy a considerable part of this paper. The jumps in the filtration often give explicit numerical information about $X$. For instance, if $X$ obtains stable reduction after a tamely ramified extension, we show that the jumps occur at indices of the form $i/\tilde{n}$, where $\tilde{n}$ is the degree of the minimal extension that realizes stable reduction for $X$.
It follows from Edixhoven's theory that to determine the jumps in the filtration $ \{ F^i_n \} $ induced by an extension of degree $n$, one needs to compute the irreducible characters for the representation of $ \boldsymbol{\mu}_n $ on the tangent space $ T_{ \mathcal{J}'_k, 0 } $. We shall use such computations for infinitely many integers $n$ to describe the jumps of the filtration $ \{ \mathcal{F}^a \} $ with rational indices.
\subsection{} Contrary to the case of general abelian varieties, N\'eron models for Jacobians can be constructed in a fairly concrete way, using the theory of the relative Picard functor (cf. ~\cite{Ner}, Chapter 9). The following property will be of particular importance to us: If $ \mathcal{Z}/S' $ is a regular model for $ X_{K'}/K' $, then there is a canonical isomorphism $$ \mathrm{Pic}_{\mathcal{Z}/S'}^0 \cong (\mathcal{J}')^0, $$ where $ \mathrm{Pic}_{\mathcal{Z}/S'}^0 $ (resp. $(\mathcal{J}')^0$) is the identity component of $ \mathrm{Pic}_{\mathcal{Z}/S'} $ (resp. $\mathcal{J}'$). It follows that there is a canonical isomorphism $$ H^1(\mathcal{Z}_k, \mathcal{O}_{\mathcal{Z}_k}) \cong T_{\mathcal{J}'_k, 0}. $$
We shall work with regular models $ \mathcal{Z} $ of $ X_{K'} $ that admit $G$-actions that are compatible with the $G$-action on $ \mathcal{J}' $. It will then follow that the representation of $G$ on $ T_{\mathcal{J}'_k, 0} $ can be described in terms of the representation of $G$ on $ H^1(\mathcal{Z}_k, \mathcal{O}_{\mathcal{Z}_k}) $.
\subsection{} In order to find an $S'$-model for $X_{K'} $ with a compatible $G$-action, we take a model $ \mathcal{X} $ of $X$ over $S$, and consider its pullback $ \mathcal{X}_{S'} $ to $ S' $. This is in general not a model of $ X_{K'} $, but its normalization $ \mathcal{X}' $ will be a model. Furthermore, there exists a \emph{minimal} desingularization $ \mathcal{Y} \rightarrow \mathcal{X}' $ which is an isomorphism on generic fibers, thus producing in a natural way a regular model for $ X_{K'} $ over $S'$.
There is a natural $G$-action on $ \mathcal{X}_{S'} $ via the action on the second factor. This action lifts uniquely to the normalization $ \mathcal{X}' $, and to the minimal desingularization $ \mathcal{Y} $. The $G$-action restricts to the special fiber $ \mathcal{Y}_k $, and in particular, $G$ will act on the cohomology groups $ H^i(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $, for $ i = 0, 1 $.
In order to understand the $G$-action on $ H^i(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $, it is important that we have a good description of the geometry of $\mathcal{Y}$ and of the $G$-action on $ \mathcal{Y} $, and this is studied in Section \ref{extensions and actions} and Section \ref{lifting the action}. For this purpose, we demand that the model $ \mathcal{X} $ has good properties. To begin with, we shall require that $ \mathcal{X} $ is regular, and that the special fiber is a divisor with strict normal crossings, i.e., that $ \mathcal{X} $ is an SNC-model. However, we have to impose some restrictions on the geometry of $ \mathcal{X}_k $. In fact, we shall always require that any two irreducible components of $ \mathcal{X}_k $, whose multiplicities are both divisible by the residue characteristic, have empty intersection. This condition is automatically fulfilled if $X$ obtains stable reduction after a tamely ramified extension, but holds also for a larger class of curves.
Under these assumptions, it turns out that the normalization $ \mathcal{X}' $ of $ \mathcal{X}_{S'} $ has rather well behaved singularities, known as \emph{tame cyclic quotient singularities} (cf. ~\cite{CED}, Definition 2.3.6 and \cite{Tame}, Proposition 4.3). Furthermore, these singularities can be resolved explicitly, and it can be seen that the minimal desingularization $ \mathcal{Y} $ is an SNC-model for $ X_{K'} $.
We shall also make the assumption that $ n = \mathrm{deg}(K'/K) $ is relatively prime to the multiplicities of all the irreducible components of $ \mathcal{X}_k $. With this additional hypothesis, it turns out that we can describe the combinatorial structure of the special fiber $ \mathcal{Y}_k $ (i.e., the intersection graph of the irreducible components, their genera and multiplicities), in terms of the corresponding data for $ \mathcal{X}_k $.
If all the assumptions above are satisfied, it follows that all irreducible components of $ \mathcal{Y}_k $ are stable under the $G$-action on $ \mathcal{Y} $, and that all intersection points in $ \mathcal{Y}_k $ are fixed points. We can explicitly describe the action on the cotangent space of $ \mathcal{Y} $ at these intersection points, and the restriction of the $G$-action to each irreducible component of $ \mathcal{Y}_k $.
\subsection{Action on cohomology} Next, we study the representation of $ G = \boldsymbol{\mu}_n $ on $ H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $. In particular, we would like to compute the irreducible characters for this representation. So for every $ g \in G $, we want to compute the trace of the endomorphism of $ H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $ induced by $g$, and then use this information to find the characters.
There are some technical problems that need to be overcome in order to do this. First, since we allow the residue characteristic to be positive, just knowing the trace for each $g \in G$ may not give sufficient information to compute the characters. Instead, we have to compute the so called \emph{Brauer trace} for every $g \in G $ (cf. \cite{SerreLin}, Chapter 18). This means that we have to lift the eigenvalues and traces from characteristic $p$ to characteristic $0$. From knowing the Brauer trace for every $ g \in G $ we can compute the irreducible Brauer characters, and then the ordinary characters are obtained by reducing the Brauer characters modulo $p$. Second, the special fiber $ \mathcal{Y}_k $ will in general be singular, and even non-reduced. This complicates trace computations considerably.
To deal with these problems, we introduce in Section \ref{section 6} a certain filtration of the special fiber $ \mathcal{Y}_k $ by effective subdivisors, where the difference at the $i$-th step is an irreducible component $ C_i $ of $ \mathcal{Y}_k $. Since $ \mathcal{Y} $ is an SNC-model, each $ C_i $ is a smooth and projective curve, and with our assumption on $n$, the $G$-action restricts to each $ C_i $. Furthermore, to each step in this filtration, one can in a natural way associate an invertible $ G $-sheaf $ \mathcal{L}_i $, supported on $C_i$.
We apply the so called Lefschetz-Riemann-Roch formula (\cite{Don}, Corollary 5.5), in order to get a formula for the Brauer trace of the endomorphism induced by each $ g \in G $ on the formal difference $ H^0(C_i,\mathcal{L}_i) - H^1(C_i,\mathcal{L}_i) $. An important step is to show that our description of the action on $ \mathcal{Y} $ is precisely the data that is needed to obtain these formulas. Then we show that these traces add up to give the Brauer trace for the endomorphism induced by each $ g \in G $ on the formal difference $ H^0(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) - H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $. In particular, we give in Theorem \ref{thm. 9.13}, which is the first main result in this work, a formula for this Brauer trace, and show that it only depends on the combinatorial structure of $ \mathcal{X}_k $.
Let us also remark that in our applications, we already know the character for $ H^0(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $, and hence we will be able to compute the irreducible characters for $ H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $ in this way.
\subsection{Trace formulas for singularities} If $ x' \in \mathcal{X}' $ is a singular point, the $G$-action on $ \mathcal{Y} $ restricts to the exceptional fiber $ \mathcal{E}_{x'} := \rho^{-1}(x') \subset \mathcal{Y}_k $, where $ \rho : \mathcal{Y} \rightarrow \mathcal{X}' $ is the minimal desingularization. Hence $ G = \boldsymbol{\mu}_n $ acts on the cohomology groups $ H^i(\mathcal{E}_{x'}, \mathcal{O}_{\mathcal{E}_{x'}}) $, for $ i = 0,1 $. This situation is studied in Section \ref{trace formula}. We observe that the methods developed earlier in the paper also apply to this situation, and enable us to compute the Brauer trace of the endomorphism induced by $ g \in G $ on the formal difference $ H^0(\mathcal{E}_{x'}, \mathcal{O}_{\mathcal{E}_{x'}}) - H^1(\mathcal{E}_{x'}, \mathcal{O}_{\mathcal{E}_{x'}}) $.
The singularity $ x' \in \mathcal{X}' $ is determined by \emph{parameters} $ n $, $m_1$ and $m_2$, where $ n $ is the order of $G$, and $ m_1$ and $ m_2 $ are the multiplicities of the components of $ \mathcal{X}_k' $ intersecting at $x'$. If $n$ is large enough compared to $m_1$ and $m_2$, we obtain in Theorem \ref{Formula} an explicit closed polynomial formula for the Brauer trace in terms of the parameters of the singularity. It turns out that there is precisely one polynomial for each element in $ (\mathbb{Z}/M)^* $, where $M = \mathrm{lcm}(m_1,m_2) $.
Apart from the fact that we find this to be an interesting problem in its own right, these formulas are used later on in the paper, both for theoretical issues as well as for the explicit computations in Section \ref{computations and jumps}. In particular, by combining Theorem \ref{Formula} and Theorem \ref{thm. 9.13}, we obtain our main result Theorem \ref{improved formula}, which gives an explicit effective formula for the Brauer trace of the automorphism induced by $\xi \in \boldsymbol{\mu}_n $ on the alternating sum $ H^0(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) - H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $.
\subsection{} If now $ \mathcal{X}/S $ is the minimal SNC-model for $ X/K $, we prove in Theorem \ref{main character theorem} that the irreducible characters for the representation of $ G = \boldsymbol{\mu}_n $ on $ H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $ only depend on the combinatorial structure of the special fiber $ \mathcal{X}_k $, as long as $n$ is relatively prime to $l$, where $l$ is the least common multiple of the multiplicities of the irreducible components of $ \mathcal{X}_k $.
If $ \{ \mathcal{F}^a \} $ denotes the filtration of $ \mathcal{J}_k $, where $ \mathcal{J} $ is the N\'eron model of the Jacobian of $X$, we prove as a corollary that the jumps in the filtration $ \{ \mathcal{F}^a \} $ only depend on the combinatorial structure of $ \mathcal{X}_k $ (Corollary \ref{main jump corollary}). This is due to the fact that $ \mathbb{Z}_{ (p l) } \cap [0,1] $ is ``dense'' in $ \mathbb{Z}_{ (p) } \cap [0,1] $. Furthermore, in Corollary \ref{specific jump corollary}, we draw the conclusion that the jumps are actually independent of ~$p$, and that the jumps can only occur at finitely many rational numbers of a certain kind, depending on the combinatorial structure of $ \mathcal{X}_k $.
It is known that for a fixed genus $ g \geq 1 $, there are only a finite number of possible combinatorial structures for $ \mathcal{X}_k $, modulo a certain equivalence relation. Furthermore, in case $ g = 1 $ or $ g = 2 $, one has complete classifications (cf. ~\cite{Kod} for $g=1$ and \cite{Ueno}, \cite{Ogg} for $g=2$). In Section \ref{computations and jumps} we compute the jumps for each possible fiber type for $g=1$ (which were also computed by Schoof in \cite{Edix}) and for $g=2$.
\section{N\'eron models and tamely ramified extensions}\label{Neron}
\subsection{N\'eron models} Let $ R $ be a discrete valuation ring, with fraction field $K$ and residue field $k$, and let $A$ be an abelian variety over $K$. There exists a canonical extension of $A$ to a smooth group scheme $ \mathcal{A} $ over $ S = \mathrm{Spec}(R) $, known as the \emph{N\'eron model} (\cite{Ner}, Theorem 1.4/3). The N\'eron model is characterized by the following universal property: for every smooth morphism $ T \rightarrow S $, the induced map $ \mathcal{A}(T) \rightarrow A(T_K) $ is \emph{bijective}.
\subsection{N\'eron models and base change}\label{neronbase} We assume from now on that $R$ is strictly henselian. Let $ K'/K $ be a finite, separable extension of fields, and let $R'$ be the integral closure of $R$ in $K'$. Let $ \mathcal{A}'/S' $ denote the N\'eron model of the abelian variety $ A_{K'}/K' $, where $ S' = \mathrm{Spec}(R') $. In general, it is not so easy to describe how N\'eron models change under ramified base changes. However, in the case where $K'/K$ is tamely ramified, one can relate $ \mathcal{A}'/S' $ to $ \mathcal{A}/S $ in a nice way, due to a result by B. Edixhoven (\cite{Edix}, Theorem 4.2). We will in this section explain this relation, following the treatment in \cite{Edix}. We refer to this paper for further details.
From now on, we will assume that $ K'/K $ is tamely ramified. Then $ K'/K $ is Galois with group $ G = \boldsymbol{\mu}_n $, where $n$ is the degree of the extension. Let $G$ act on $ A_{K'} = A \times_{\mathrm{Spec}(K)} \mathrm{Spec}(K') $ (from the right), via the action on the right factor. By the universal property of $ \mathcal{A}' $, this $G$-action on $ A_{K'} $ extends uniquely to a right action on $ \mathcal{A}' $, such that the morphism $ \mathcal{A}' \rightarrow S' $ is equivariant. The idea is now to reconstruct $ \mathcal{A} $ as an invariant scheme for this action. However, since $ \mathcal{A} $ is an $S$-scheme, one first needs to ``push down'' from $S'$ to $S$.
The \emph{Weil restriction} of $ \mathcal{A}' $ to $S$ is a contravariant functor $$ \Pi_{S'/S}(\mathcal{A}'/S') : (\mathrm{Sch}/S)^0 \rightarrow (\mathrm{Sets}), $$ defined by assigning, for any $S$-scheme $T$, $ \Pi_{S'/S}(\mathcal{A}'/S')(T) = \mathcal{A}'(T') $, where $ T' = T \times_S S' $ (cf. ~\cite{Ner}, Chapter 7). This functor is representable by an $S$-scheme, which we will denote by $X$ (\cite{Edix}, Remark 2.1).
In \cite{Edix}, an equivariant $G$-action on $ X \rightarrow S $ is defined in the following way: Let $ T $ be an $S$-scheme, and let $ P \in X(T) = \mathcal{A}'(T') $ be a $T$-point, where $ T' = T \times_S S' $. For any $ g \in G $, let $ \rho_{\mathcal{A}'}(g) $ and $ \rho_{S'}(g) $ be the automorphisms of $ \mathcal{A}' $ and $S'$ induced by $g$. Finally, let $ \rho_{T'}(g) = 1_T \times \rho_{S'}(g) $. The action is then given by $$ P \cdot g = \rho_{\mathcal{A}'}(g) \circ P \circ \rho_{T'}(g)^{-1}. $$
\subsection{Relating $\mathcal{A}'$ and $\mathcal{A}$} In \cite{Edix}, Proposition 4.1, it is shown that the generic fiber of $ \Pi_{S'/S}(\mathcal{A}'/S') $ is an abelian variety, having $ \Pi_{S'/S}(\mathcal{A}'/S') $ as its N\'eron model. Furthermore, Construction 2.3 in \cite{Edix} gives a closed immersion $$ A \hookrightarrow (\Pi_{S'/S}(\mathcal{A}'/S'))_K, $$ which extends uniquely to a morphism $$ \mathcal{A} \rightarrow \Pi_{S'/S}(\mathcal{A}'/S'), $$ since $ \mathcal{A}/S $ is smooth and $ \Pi_{S'/S}(\mathcal{A}'/S') $ is a N\'eron model.
According to \cite{Edix}, Theorem 4.2, this morphism is a closed immersion, and induces an isomorphism \begin{equation}\label{Bastheorem} \mathcal{A} \cong (\Pi_{S'/S}(\mathcal{A}'/S'))^G, \end{equation} where $(\Pi_{S'/S}(\mathcal{A}'/S'))^G$ is the \emph{scheme of invariant points} for the $ G $-action defined above (\cite{Edix}, Chapter 3).
\subsection{Filtration of $\mathcal{A}_k$} One can use the isomorphism in \ref{Bastheorem} to study the special fiber $ \mathcal{A}_k $ in terms of $ \mathcal{A}'_k $, together with the $G$-action. Indeed, let $ R \subset R' = R[\pi']/(\pi'^n - \pi) $ be a tame extension, where $ \pi $ is a uniformizing parameter for $R$. Then we have that $ R'/\pi R' = k[\pi']/(\pi'^n) $. For any $ k $-algebra $C$, it follows that $$ \mathcal{A}_k(C) \cong X_k^G(C) \cong X_k(C)^G \cong \mathcal{A}'(C[\pi']/(\pi'^n))^G. $$
In \cite{Edix}, Chapter 5, this observation is used to construct a filtration of $\mathcal{A}_k$. To do this, let us first consider an $R$-algebra $ C $. The Weil restriction induces a map $$ \mathcal{A}(C) \rightarrow \mathcal{A}'(C \otimes_R R'), $$ which gives a map $$ \mathcal{A}(C) \rightarrow \mathcal{A}'(C \otimes_R R') \rightarrow \mathcal{A}'(C \otimes_R R'/(\pi'^i)), $$ for any integer $i$ such that $ 0 \leq i \leq n $. Define functors $ F^i\mathcal{A}_k $ by $$ F^i\mathcal{A}_k(C) = \mathrm{Ker}(\mathcal{A}(C) \rightarrow \mathcal{A}'(C \otimes_R R'/(\pi'^i))), $$ for any $ k = R/(\pi)$-algebra $C$. The functors $ F^i\mathcal{A}_k $ are represented by closed subgroup schemes of $ \mathcal{A}_k $, and give rise to a descending filtration
$$ \mathcal{A}_k = F^0\mathcal{A}_k \supseteq F^1\mathcal{A}_k \supseteq \ldots \supseteq F^n\mathcal{A}_k = 0. $$ Let us also remark that the group schemes $ F^i\mathcal{A}_k $ are \emph{unipotent} for $ i > 0 $.
One can describe the successive quotients of this filtration quite accurately: Let $ Gr^i \mathcal{A}_k $ denote the quotient $ F^i\mathcal{A}_k/F^{i+1}\mathcal{A}_k $, for $ i \in \{ 0, \ldots, n-1 \} $. Then, by Theorem 5.3 in \cite{Edix}, we have that $ Gr^0(\mathcal{A}_k) = (\mathcal{A}'_k)^{\boldsymbol{\mu}_n} $, and for $ 0 < i < n $, we have that $$ Gr^i \mathcal{A}_k \cong T_{\mathcal{A}'_k,0}[i] \otimes_k (m/m^2)^{\otimes i}, $$ where $ m \subset R'$ is the maximal ideal, and where $ T_{\mathcal{A}'_k,0}[i] $ denotes the subspace of $ T_{\mathcal{A}'_k,0} $ where $ \xi \in \boldsymbol{\mu}_n $ acts by multiplication by $ \xi^i $.
The filtration \emph{jumps} at the index $ i \in \{ 0, \ldots, n-1 \} $ if $ Gr^i \mathcal{A}_k \neq 0 $. Since $$ T_{\mathcal{A}'_k,0}[0] = (T_{\mathcal{A}'_k,0})^{\boldsymbol{\mu}_n} = T_{(\mathcal{A}'_k)^{\boldsymbol{\mu}_n},0} $$ (use \cite{Edix}, Proposition 3.2), it follows that the jumps are completely determined by the representation of $ \boldsymbol{\mu}_n $ on $ T_{\mathcal{A}'_k,0} $. In particular, it follows that there are at most $ \mathrm{dim}(A) $ jumps, since $ \mathrm{dim}_k T_{\mathcal{A}'_k,0} = \mathrm{dim}(A) $.
\subsection{Compositions of tame extensions} In Remark 5.4.5 \cite{Edix}, a generalization of the filtration of $ \mathcal{A}_k $ discussed above is suggested. Let $ \{ F^i_n \mathcal{A}_k \} $ denote the filtration induced by the tame extension of degree $n$. The idea is to put all the $ F^i_n \mathcal{A}_k $, for all positive integers not divisible by the residue characteristic $p$, in a common filtration of $ \mathcal{A}_k $. We shall later see that this filtration gives interesting information about $ \mathcal{A}_k $.
In order to set this up, it is necessary to understand how two filtrations $ \{ F^i_{n_1} \mathcal{A}_k \} $ and $ \{ F^i_{n_2} \mathcal{A}_k \} $ are related in the case where $ n_1 $ divides $n_2$. We will now explain this in some detail, since this is not done in \cite{Edix}.
Let $ R_1 $ and $ R_2 $ be tame extensions of $R$, where $ R_1 = R[t_1]/(t_1^{n_1} - \pi) $, $ R_2 = R[t_2]/(t_2^{n_2} - \pi) $ and where $ R_2 = R_1[t_2]/(t_2^{m} - t_1) $. Let $ \mathcal{A}_i $ denote the N\'eron model of $ A_{K_i} $ over $ R_i $ for $ i \in {1,2} $, where $ K_i $ is the fraction field of $R_i$. By Remark 2.1 in \cite{Edix}, we have that $$ \Pi_{S_i/S} (\mathcal{A}_i/S_i) \times_S \mathrm{Spec}(K) = \Pi_{K_i/K} (A_{K_i}/K_i). $$
Furthermore, Construction 2.3 in \cite{Edix} gives, for every $ T \rightarrow \mathrm{Spec}(K) $, a commutative diagram $$ \xymatrix{ \mathrm{Hom}_K(T,A) \ar[r] \ar[dr] & \mathrm{Hom}_{K_1}(T_{K_1}, A_{K_1}) \ar[d] \\
& \mathrm{Hom}_{K_2}(T_{K_2}, A_{K_2}), } $$ where all arrows are injections. So we get a commutative diagram $$ \xymatrix{ A \ar[rr]^-{\phi_1} \ar[drr]_-{\phi_2} & & \Pi_{K_1/K} (A_{K_1}/K_1) \ar[d]^{\phi_{1,2}} \\
& & \Pi_{K_2/K} (A_{K_2}/K_2). } $$ We now claim that all maps in this diagram are closed immersions. To see this, let $ G $ denote the Galois group of $ K_2/K $, and let $ H \subseteq G $ be the subgroup that fixes $K_1 $. Then $ H $ is the Galois group of $ K_2/K_1 $, and $ G/H $ is the Galois group of $K_1/K $.
We have that $ G $ acts on $ A_{K_2} = A \times_{\mathrm{Spec}(K)} \mathrm{Spec}(K_2) $ by its action on the right factor. Therefore, $G$ acts on $ \Pi_{K_2/K} (A_{K_2}/K_2) $ as defined in Section \ref{neronbase}, with invariant scheme $A$ (see the proof of \cite{Edix}, Theorem 4.2), and $ \phi_2 $ can be identified with the inclusion. By Proposition 3.1 in \cite{Edix}, it follows that $\phi_2$ is a closed immersion. The same argument for the $G/H$-action on $ \Pi_{K_1/K} (A_{K_1}/K_1) $ shows that $\phi_1$ is a closed immersion.
\begin{lemma} The morphism $ \phi_{1,2} $ above is a closed immersion. \end{lemma} \begin{pf} We need to understand how $ H $ acts on $ \Pi_{K_2/K} (A_{K_2}/K_2) $. For any $ g \in G $, let $ \rho_{K_2}(g) : \mathrm{Spec}(K_2) \rightarrow \mathrm{Spec}(K_2) $ be the corresponding automorphism. For any $ K $-scheme $T$, there is an induced automorphism given as $ \rho_{T_{K_2}}(g) = \mathrm{id}_T \times \rho_{K_2}(g) $ on $ T_{K_2} $, and similarly, there is an induced automorphism $ \rho_{A_{K_2}}(g) = \mathrm{id}_{A} \times \rho_{K_2}(g) $ on $ A_{K_2} $. Then $G$ acts, for any $ P \in \Pi_{K_2/K} (A_{K_2}/K_2)(T) $, by $ P \cdot g = \rho_{A_{K_2}}(g) \circ P \circ \rho_{T_{K_2}}(g)^{-1} $.
However, if $ h \in H $, the morphism $ \rho_{K_2}(h) $ is relative to $ \mathrm{Spec}(K_1) $. So we can in fact write $ \rho_{T_{K_2}}(h) = \mathrm{id}_{T_{K_1}} \times \rho_{K_2}(h) $ and $ \rho_{A_{K_2}}(g) = \mathrm{id}_{A_{K_1}} \times \rho_{K_2}(g) $.
Observe that we have the identity $$ \Pi_{K_2/K} (A_{K_2}/K_2)(T) = \mathrm{Hom}_{K_2}(T_{K_2}, A_{K_2}) = \Pi_{K_2/K_1} (A_{K_2}/K_2)(T_{K_1}), $$ and an $H$-action on $ \Pi_{K_2/K_1} (A_{K_2}/K_2) $ as defined in Section \ref{neronbase}. It follows immediately that the $H$-action on $ \Pi_{K_2/K} (A_{K_2}/K_2)(T) $ and $ \Pi_{K_2/K_1} (A_{K_2}/K_2)(T_{K_1}) $ can be identified.
But since $A_{K_1} $ is the invariant scheme of $ \Pi_{K_2/K_1} (A_{K_2}/K_2) $ under this $H$-action, we have that $$ \Pi_{K_2/K_1} (A_{K_2}/K_2)(T_{K_1})^H = \mathrm{Hom}_{K_1}(T_{K_1}, A_{K_1}) = \Pi_{K_1/K} (A_{K_1}/K_1)(T), $$ and it therefore follows that $$ \Pi_{K_2/K} (A_{K_2}/K_2)(T)^H = \mathrm{Hom}_{K_1}(T_{K_1}, A_{K_1}) = \Pi_{K_1/K} (A_{K_1}/K_1)(T). $$ Therefore, since $ \Pi_{K_1/K} (A_{K_1}/K_1) $ is the invariant scheme of $ \Pi_{K_2/K} (A_{K_2}/K_2) $ under the action of $H$, and $ \phi_{1,2} $ can be identified with the inclusion, it follows that $ \phi_{1,2} $ is a closed immersion. \end{pf}
By the N\'eronian property, the maps $ \phi_1 $, $ \phi_2 $ and $ \phi_{1,2} $ lift uniquely so that we have a commutative diagram \begin{equation}\label{GH diagram}
\xymatrix{ \mathcal{A} \ar[r]^-{\Phi_1} \ar[dr]_-{\Phi_2} & \Pi_{S_1/S} (\mathcal{A}_1/S_1) \ar[d]^{\Phi_{1,2}} \\ & \Pi_{S_2/S} (\mathcal{A}_2/S_2). } \end{equation}
\begin{lemma} The morphisms $ \Phi_1 $, $ \Phi_2 $ and $ \Phi_{1,2} $ are closed immersions. Furthermore, via these maps, we have that $ \mathcal{A} = \Pi_{S_2/S} (\mathcal{A}_2/S_2)^G $, $ \mathcal{A} = \Pi_{S_1/S} (\mathcal{A}_1/S_1)^{G/H} $ and $ \Pi_{S_1/S} (\mathcal{A}_1/S_1) = \Pi_{S_2/S} (\mathcal{A}_2/S_2)^H $. \end{lemma} \begin{pf} Except for the statement regarding $ \Phi_{1,2} $, this is Theorem 4.2 in \cite{Edix}. But the same proof can be applied to $ \Phi_{1,2} $. \end{pf}
Having established this relationship between the Weil restrictions, we get the following result for the filtrations:
\begin{lemma}\label{12extensions} Let $ \{ F_{n_j}^i \mathcal{A}_k \} $ be the filtration of $ \mathcal{A}_k $ induced by the extension $ R_j/R $, for $ j \in \{ 1,2 \} $. Then we have that $ F_{n_1}^i \mathcal{A}_k = F_{n_2}^{im} \mathcal{A}_k $, for any $ i \in \{ 0, \ldots, n_1 \} $. \end{lemma} \begin{pf} Let $C$ be an $R$-algebra. As $ C \otimes_R R_2 = (C \otimes_R R_1) \otimes_{R_1} R_2 $, Diagram \ref{GH diagram} above induces a commutative diagram \begin{equation}\label{diagram 14.4}
\xymatrix{ \mathcal{A}(C) \ar[r]^-{\Phi_1} \ar[dr]_-{\Phi_2} & \mathcal{A}_1(C \otimes_R R_1) \ar[d]^{\Phi_{1,2}} \\
& \mathcal{A}_2((C \otimes_R R_1) \otimes_{R_1} R_2). } \end{equation} Note that $ C \otimes_R R_j = C [t_j]/(t_j^{n_j} - \pi) $, and in the case where $ C $ is also an $ R/(\pi) $-algebra, we get that $ C \otimes_R R_j = C [t_j]/(t_j^{n_j}) $. In the following, we shall only consider the latter case. Let $i$ be an integer such that $ 0 \leq i \leq n_1 $. Then we have that $ C[t_1]/(t_1^i) \otimes_{R_1} R_2 = C[t_2]/(t_2^{im}) $, and that there is a commutative diagram \begin{equation}\label{diagram 14.5} \xymatrix{ \mathcal{A}_1(C[t_1]/(t_1^{n_1})) \ar[r]^{\beta_1} \ar[d]_{\Phi_{1,2}} & \mathcal{A}_1(C[t_1]/(t_1^i)) \ar[d]^{\Phi_{1,2}} \\ \mathcal{A}_2(C[t_2]/(t_2^{n_2})) \ar[r]_{\beta_2} & \mathcal{A}_2(C[t_2]/(t_2^{im})). } \end{equation}
Combining Diagram \ref{diagram 14.4} with Diagram \ref{diagram 14.5} gives a commutative diagram \begin{equation}\label{diagram 14.6} \xymatrix{ \mathcal{A}_k(C) \ar[r]^-{\alpha_1} \ar[dr]_{\alpha_2} & \mathcal{A}_1(C[t_1]/(t_1^i)) \ar[d]^-{\Phi_{1,2}} \\
& \mathcal{A}_2(C[t_2]/(t_2^{im})). } \end{equation} Since $\Phi_{1,2}$ is \emph{injective}, it follows that $ \mathrm{Ker}(\alpha_1) = \mathrm{Ker}(\alpha_2) $ for any $k$-algebra $C$, and hence $ F_1^i\mathcal{A}_k = F_2^{im}\mathcal{A}_k $ for all $i$ such that $ 0 \leq i \leq n_1 $. \end{pf}
\subsection{Filtration with rational indices}\label{ratfil}
Let $ a \in \mathbb{Z}_{(p)} \cap [0,1] $. If $ a = i/n $, then we define $ \mathcal{F}^a\mathcal{A}_k = F^i_n \mathcal{A}_k $, where $ F^i_n \mathcal{A}_k $ denotes the $i$-th step in the filtration induced by the tame extension of degree $n$.
\begin{prop}\label{ratfil} The construction above gives a descending filtration $$ \mathcal{A}_k = \mathcal{F}^0 \mathcal{A}_k \supseteq \ldots \supseteq \mathcal{F}^a\mathcal{A}_k \supseteq \ldots \supseteq \mathcal{F}^1 \mathcal{A}_k = 0 $$ of $ \mathcal{A}_k $ by closed subgroup schemes, where $ a \in \mathbb{Z}_{(p)} \cap [0,1] $. \end{prop} \begin{pf} It follows from Lemma \ref{12extensions} that the definition of $ \mathcal{F}^a\mathcal{A}_k $ does not depend on the choice of representatives for $a$. Hence the $ \mathcal{F}^a\mathcal{A}_k $ are well defined.
In order to show that the filtration is descending, take $a_1, a_2 \in \mathbb{Z}_{(p)} \cap [0,1] $ such that $ a_1 \leq a_2 $. Let $ a_1 = i_1/n_1 $ and $ a_2 = i_2/n_2 $. Since $ \mathcal{F}^a \mathcal{A}_k $ does not depend on the choice of representatives for $a$, we write $ a_1 = i_1 n_2/n_1 n_2 $ and $ a_2 = i_2 n_1/n_1 n_2 $. In particular, $ i_1 n_2 \leq i_2 n_1 $. But then we get that $$ \mathcal{F}^{a_1}\mathcal{A}_k = F^{i_1 n_2}_{n_1 n_2} \mathcal{A}_k \supseteq F^{i_2 n_1}_{n_1 n_2} \mathcal{A}_k = \mathcal{F}^{a_2}\mathcal{A}_k. $$ \end{pf}
Let $ x \in [0,1] $ be a real number, and let $ (x^j)_j $ (resp. $ (x_k)_k $) be a sequence of numbers in $ \mathbb{Z}_{(p)} \cap [0,1] $ converging to $x$ from above (resp. from below). We will say that $ \{ \mathcal{F}^a \mathcal{A}_k \} $ jumps at $x$ if $ \mathcal{F}^{x_k} \mathcal{A}_k \supsetneq \mathcal{F}^{x^j} \mathcal{A}_k $ for all $j$ and $k$. It is natural to ask how many jumps there are, and \emph{where} they occur. It is easily seen that since every filtration $ \{ F^i_n \mathcal{A}_k \} $ jumps at most $ g = \mathrm{dim}(A) $ times, it follows that the filtration $ \{ \mathcal{F}^a \mathcal{A}_k \} $ can have at most $g$ jumps.
Consider a positive integer $n$ that is not divisible by $p$, and let $ \{ F^i_n \mathcal{A}_k \} $ be the filtration induced by the extension of degree $n$. Let us assume that this filtration has a jump at $ i \in \{ 0, \ldots, n-1 \} $. Then we can say that $ \{ \mathcal{F}^a \mathcal{A}_k \} $ has a jump in the interval $ [i/n, (i+1)/n] $. By computing jumps in this way for increasing $n$, we get finer partitions of the interval $ [0,1] $, and increasingly better approximations of the jumps in $ \{ \mathcal{F}^a \mathcal{A}_k \} $.
It follows that one can compute the jumps of $ \{ \mathcal{F}^a \mathcal{A}_k \} $ by computing the jumps for the filtrations $ \{ F^i_n \mathcal{A}_k \} $ for ``sufficiently'' many $n$ that are not divisible by $p$. This would for instance be the case for a multiplicatively closed subset $ \mathcal{U} \subset \mathbb{N} $ such that $ \mathbb{Z}[\mathcal{U}^{-1}] \cap [0,1] $ is dense in $ \mathbb{Z}_{(p)} \cap [0,1] $.
\subsection{Jumps in the tamely ramified case} In the case where $ A/K $ obtains semi-abelian reduction over a tamely ramified extension $K'$ of $K$, the jumps of $ \{ \mathcal{F}^a \mathcal{A}_k \} $ have an interesting interpretation, which we will now explain. Let $ \widetilde{K} $ be the minimal extension over which $A$ has semi-abelian reduction (cf. ~\cite{Deschamps}, Th\'eor\`eme 5.15), and let $ \tilde{n} = \mathrm{deg}(\widetilde{K}/K) $. Then the jumps occur at rational numbers of the form $ k/\tilde{n} $, where $ k \in \{0, \ldots, \tilde{n} - 1 \} $. This is essentially due to the following observation:
\begin{lemma}\label{tame jumps} Let $ \widetilde{K}/K $ be the minimal extension over which $A/K$ obtains semi-abelian reduction, and let $ \tilde{n} = \mathrm{deg}(\widetilde{K}/K) $. Consider a tame extension $ K'/K $ of degree $ n $, factoring via $ \widetilde{K} $, and let $ m = n/\tilde{n} $. Let $\mathcal{A}'/S' $ be the N\'eron model of $A_{K'}$.
Then we have that the jumps in the filtration $ \{ F^i_n \mathcal{A}_k \} $ induced by $ S'/S $ occur at indices $ i = k n / \tilde{n} $, where $ 0 \leq k \leq \tilde{n} - 1$. \end{lemma} \begin{pf} Let $ \widetilde{\mathcal{A}}/\widetilde{S} $ be the N\'eron model of $ A_{\widetilde{K}} $. By assumption, we have that both $\mathcal{A}'$ and $ \widetilde{\mathcal{A}} $ are semi-abelian. Since $ \widetilde{\mathcal{A}}_{S'} $ is smooth, and $\mathcal{A}'$ has the N\'eronian property, we get a canonical morphism $ \widetilde{\mathcal{A}}_{S'} \rightarrow \mathcal{A}' $, lifting the identity map on the generic fibers. Since $ \widetilde{\mathcal{A}}_{S'} $ is semi-abelian, it follows from Proposition 7.4/3 in \cite{Ner} that this morphism induces an isomorphism $ (\widetilde{\mathcal{A}}_k)^0 \cong (\mathcal{A}'_k)^0 $. In particular, we get that $ T_{\widetilde{\mathcal{A}}_k,0} = T_{\mathcal{A}'_k, 0} $.
Consider now the filtration $ \{ F^i_m \widetilde{\mathcal{A}}_k \} $ of $ \widetilde{\mathcal{A}}_k $ induced by the extension $ S'/\widetilde{S} $. Since $ \widetilde{\mathcal{A}} $ is semi-abelian, we have that $ F^i_m \widetilde{\mathcal{A}}_k = 0 $ for all $ i > 0 $. Therefore, we get that $$ \widetilde{\mathcal{A}}_k = F^0_m \widetilde{\mathcal{A}}_k = Gr^0_m \widetilde{\mathcal{A}}_k = (\mathcal{A}'_k)^{\boldsymbol{\mu}_m}. $$ But now $$ (T_{\mathcal{A}'_k, 0})^{\boldsymbol{\mu}_m} = T_{(\mathcal{A}'_k)^{\boldsymbol{\mu}_m}, 0} = T_{\mathcal{A}'_k, 0}, $$ and so it follows that $ \boldsymbol{\mu}_m $ acts trivially on $ T_{\mathcal{A}'_k, 0} $.
Let us now consider the filtration $ \{ F^i_n \mathcal{A}_k \} $ induced by the extension $ S'/S $. The jumps in this filtration are determined by the $ \boldsymbol{\mu}_n $-action on $ T_{\mathcal{A}'_k, 0} $. Assume that $ T_{\mathcal{A}'_k, 0}[i] \neq 0 $, for some $ i \in \{0, \ldots, n -1 \} $. On this subspace, every $ \xi \in \boldsymbol{\mu}_n $ acts by multiplication by $ \xi^i $. We can identify $ \boldsymbol{\mu}_m $ with the $\tilde{n}$-th powers in $ \boldsymbol{\mu}_n $, and since we established above that $ \boldsymbol{\mu}_m $ acts trivially, it follows that $ \xi^{\tilde{n} i} = 1 $. So therefore $ \tilde{n} i = k n $ for some $ k \in \{0, \ldots, \tilde{n} - 1 \} $, and we get that $ i = k n/\tilde{n} $. \end{pf}
We can now formulate the following result:
\begin{prop}\label{tamejumpprop} If $ A/K $ obtains semi-abelian reduction over a tamely ramified extension of $K$, then the jumps in the filtration $ \{ \mathcal{F}^a \mathcal{A}_k \} $ occur at indices $ k/\tilde{n} $, where $ k \in \{0, \ldots, \tilde{n} - 1 \} $, and where $ \tilde{n} $ is the degree of the minimal extension $ \widetilde{K}/K $ that realizes semi-abelian reduction for $ A $. \end{prop} \begin{pf} Let us consider the sequence of integers $ (\tilde{n} m)_m $, where $m$ runs over the positive integers that are not divisibe by $p$. For the extension of degree $n = \tilde{n} m$, Lemma \ref{tame jumps} gives that the jumps of $ \{ F^i_n \mathcal{A}_k \} $ occur at indices $ i = k n / \tilde{n} $, where $ 0 \leq k \leq \tilde{n} - 1$. It follows that the jumps of $ \{ \mathcal{F}^a \mathcal{A}_k \} $ will be among the limits of the expressions $ i/n = k/ \tilde{n} $, as $m$ goes to infinity, and the result follows. \end{pf}
\subsection{The case of Jacobians}\label{jacobiancase} Let $X/K$ be a smooth, projective and geometrically connected curve of genus $ g > 0 $. We shall also make the assumption that $ X(K) \neq \emptyset $. If $K'/K $ is a tame extension, it follows that $ X_{K'} $ hase the same properties. Let $ J' = J_{K'} $ denote the Jacobian of $X_{K'}$, and let $ \mathcal{J}'/S' $ be the N\'eron model of $J'$ over $S'$.
We can let $G$ act on $ X_{K'} $ via the action on the second factor. Let $ \mathcal{Y}/S' $ be a regular model of $ X_{K'} $ such that the $G$-action on $ X_{K'} $ extends to $ \mathcal{Y} $. According to \cite{Ner}, Theorem 9.5/4, there is a canonical isomorphism $$ \mathrm{Pic}^0_{\mathcal{Y}/S'} \cong \mathcal{J}'^0, $$ where $ \mathcal{J}'^0 $ is the identity component of $ \mathcal{J}' $, and where $ \mathrm{Pic}^0_{\mathcal{Y}/S'} $ is the identity component of the relative Picard functor $ \mathrm{Pic}_{\mathcal{Y}/S'} $. Hence, on the special fibers, we get an isomorphism $$ \mathrm{Pic}^0_{\mathcal{Y}_k/k} \cong \mathcal{J}'^0_k. $$ By \cite{Ner}, Theorem 8.4/1, it follows that we can canonically identify \begin{equation}\label{H^1=T} H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) \cong T_{\mathcal{J}'_k,0}. \end{equation}
We are interested in computing the irreducible characters for the representation of $\boldsymbol{\mu}_n$ on $T_{\mathcal{J}'_k,0}$. With the identification in \ref{H^1=T} above, we see that this can be done by computing the irreducible characters for the representation of $\boldsymbol{\mu}_n$ on $ H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $.
By combining the discussion in this section with properties of the representation of $\boldsymbol{\mu}_n$ on $ H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $, we obtain in Corollary \ref{specific jump corollary} a quite precise description of the jumps of the filtration $ \{ \mathcal{F}^a \mathcal{J}_k \} $.
\section{Tame extensions and Galois actions}\label{extensions and actions}
\subsection{Some definitions.} Throughout this paper, $R$ will denote a complete discrete valuation ring, with fraction field $K$, and algebraically closed residue field $k$.
$X/K$ will be a smooth, projective, geometrically connected curve over $K$, of genus $g(X)>0$. For simplicity, we will say that $X$ is a \emph{curve} of genus $g$ over $K$. \begin{dfn} A scheme $ \mathcal{X} $ is called a \emph{model} of $X$ over $ S = \mathrm{Spec}(R) $ if $ \mathcal{X} $ is integral and normal, projective and flat over $S$, and with generic fibre $ \mathcal{X}_K \cong X $. \end{dfn}
It is easily seen that one can always find a model for $X$ (cf. ~\cite{Liubook}, Proposition 10.1.8). Let us now mention a few types of models that will frequently occur in this paper.
Since a model $ \mathcal{X} $ is normal, with smooth generic fiber, it follows that the singular locus consists of a finite set of closed points in the special fiber. Furthermore, there exists a \emph{strong} desingularization $ \phi : \mathcal{Z} \rightarrow \mathcal{X} $, i.e., $ \phi $ is an isomorphism over the regular locus of $ \mathcal{X} $ (\cite{Liubook}, Corollary 8.3.51). In fact, there even exists a \emph{minimal} (strong) desingularization $ \rho : \mathcal{Y} \rightarrow \mathcal{X} $, characterized by the property that any other desingularization of $ \mathcal{X} $ factors via $ \rho $ (\cite{Liubook}, Proposition 9.3.32).
It follows that we can always find a regular model $ \mathcal{X}/S $ for $X/K$. By blowing up points in the special fiber, we can even ensure that the irreducible components of the special fiber are smooth, and intersect transversally. Such a model will be called a \emph{strict normal crossings} model for $X/K$, or for short, an SNC-model.
\subsection{Construction}\label{2.2} Let $X/K$ be a curve of genus $g > 0 $, and let $ \mathcal{X}/S $ be an SNC-model for $X/K$.
Let $ K \subset K' $ be a finite, separable field extension, and let $R'$ be the integral closure of $R$ in $K'$. Since $R$ is complete, we have that $R'$ is a complete discrete valuation ring (\cite{Serre}, Proposition II.3). Making the finite base extension $S' = \mathrm{Spec}(R') \rightarrow S = \mathrm{Spec}(R) $, we obtain a commutative diagram $$ \xymatrix{
\mathcal{Y} \ar[d] \ar[r]^{\rho} & \mathcal{X}' \ar[d] \ar[r]^{f} & \mathcal{X} \ar[d]\\
S' \ar[r]^{\mathrm{id}} & S' \ar[r] & S, } $$ where $ \mathcal{X}' $ is the normalization of the pullback $ \mathcal{X}_{S'} = \mathcal{X} \times_S S' $ ($ \mathcal{X}_{S'} $ is integral by Lemma \ref{lemma construction} below), and $ \rho : \mathcal{Y} \rightarrow \mathcal{X}' $ is the minimal desingularization. The map $ f : \mathcal{X}' \rightarrow \mathcal{X} $ is the composition of the projection $ \mathcal{X}_{S'} \rightarrow \mathcal{X} $ with the normalization $ \mathcal{X}' \rightarrow \mathcal{X}_{S'} $.
\begin{lemma}\label{lemma construction} With the hypotheses above, the following statements hold: \begin{enumerate} \item The pullback $ \mathcal{X}_{S'} $ is integral. \item $ f : \mathcal{X}' \rightarrow \mathcal{X} $ is a finite morphism. \end{enumerate} \end{lemma} \begin{pf} (i) Let us first note that the generic fiber of $ \mathcal{X}_{S'} $ is the pullback $\mathcal{X}_K \otimes_K K' $, where $ \mathcal{X}_K $ is the generic fiber of $ \mathcal{X} $. By assumption $ \mathcal{X}_K $ is smooth and geometrically connected over $K$, hence the generic fiber of $ \mathcal{X}_{S'} $ is in particular integral. Now, since $ \mathcal{X}_{S'} \rightarrow S' $ is flat, it follows from \cite{Liubook}, Proposition 4.3.8, that $ \mathcal{X}_{S'} $ is integral as well.
(ii) Since $ R' $ is a \emph{complete} discrete valuation ring it is excellent (\cite{Liubook}, Theorem 8.2.39). As $ \mathcal{X}_{S'} $ is of finite type over $S'$, it follows that $ \mathcal{X}_{S'} $ is an excellent scheme, and hence the normalization morphism $ \mathcal{X}' \rightarrow \mathcal{X}_{S'} $ is finite (\cite{Liubook}, Theorem 8.2.39). The projection $ \mathcal{X}_{S'} \rightarrow \mathcal{X} $ is finite, since it is the pullback of the finite morphism $ S' \rightarrow S $. So the composition $f$ of these two morphisms is indeed finite. \end{pf}
\subsection{Galois actions} Let us now assume that the field extension $ K \subset K' $ is Galois with group $G$. Every $ \sigma \in G $ induces an automorphism of $R'$ that fixes $R$, and we have furthermore that $ R'^{G} = R $. So there is an injective group homomorphism $ G \rightarrow \mathrm{Aut}(S') $, and we may view $ S' \rightarrow S $ as the quotient map.
We can lift the $G$-action to $ \mathcal{X}_{S'} $, via the action on the second factor. So there is a group homomorphism $ G \rightarrow \mathrm{Aut}(\mathcal{X}_{S'}) $. For any element $ \sigma \in G $, we shall still denote the image in $ \mathrm{Aut}(\mathcal{X}_{S'}) $ by $ \sigma $. Proposition \ref{prop. 2.3} below shows that this action can be lifted uniquely both to the normalization $ \mathcal{X}' $ and to the minimal desingularization $ \mathcal{Y} $ of $ \mathcal{X}' $.
Recall the universal property of the normalization, saying that if $ g : \mathcal{Z} \rightarrow \mathcal{X}_{S'} $ is a \emph{dominant} morphism where $ \mathcal{Z} $ is a normal scheme, then $g$ factors uniquely via $ \mathcal{X}' \rightarrow \mathcal{X}_{S'} $.
\begin{prop}\label{prop. 2.3} With the hypotheses above, the following statements hold: \begin{enumerate} \item The $G$-action on $ \mathcal{X}_{S'} $ lifts uniquely to the normalization $ \mathcal{X}' $.
\item The $ G $-action on $ \mathcal{X}' $ lifts uniquely to the minimal desingularization $ \mathcal{Y} $.
\item For any $ \sigma \in G $, let $ \sigma $ denote the induced automorphism of $ \mathcal{X}' $, and let $ \tau $ be the unique lift of $ \sigma $ to $ \mathrm{Aut}(\mathcal{Y}) $. Then we have that $ \tau(\rho^{-1}(\mathrm{Sing}(\mathcal{X}'))) = \rho^{-1}(\mathrm{Sing}(\mathcal{X}')) $. That is, the exceptional locus is mapped into itself under the $G$-action on $ \mathcal{Y} $. \end{enumerate} \end{prop} \begin{pf} (i) The proofs of the liftings of the $G$-actions to $ \mathcal{X}' $ and $ \mathcal{Y} $ are similar, so we do not write out the details for $ \mathcal{X}' $.
(ii) Let $ \sigma \in G $. Then $ \sigma $ acts as an automorphism $ \sigma : \mathcal{X}' \rightarrow \mathcal{X}' $. Consider the diagram $$ \xymatrix{
\mathcal{Y} \ar[d]_{\rho} & \mathcal{Y} \ar[d]^{\rho} \\
\mathcal{X}' \ar[r]^{\sigma} & \mathcal{X}'. } $$
The composition $ \sigma \circ \rho : \mathcal{Y} \rightarrow \mathcal{X}' $ is a desingularization, so by the minimality of $ \rho : \mathcal{Y} \rightarrow \mathcal{X}' $, there exists a unique morphism $ \tau : \mathcal{Y} \rightarrow \mathcal{Y} $ such that the diagram $$ \xymatrix{
\mathcal{Y} \ar[d]_{\rho} \ar[r]^{\tau} & \mathcal{Y} \ar[d]^{\rho} \\
\mathcal{X}' \ar[r]^{\sigma} & \mathcal{X}' } $$ commutes. Let us first show that $ \tau $ is an automorphism. Let $ \sigma' = \sigma^{-1} $. Since $ \sigma $ is an automorphism, also $ \sigma' \circ \rho $ is a desingularization, hence there exists a unique $ \tau' : \mathcal{Y} \rightarrow \mathcal{Y} $ such that $ \rho \circ \tau' = \sigma' \circ \rho $.
It now follows that $$ \rho \circ \tau' \circ \tau = \sigma' \circ \rho \circ \tau = \sigma' \circ \sigma \circ \rho = \rho. $$ Hence the diagram $$ \xymatrix{
\mathcal{Y} \ar[dr]_{\rho} \ar[rr]^{\tau' \circ \tau} & & \mathcal{Y} \ar[dl]^{\rho} \\
& \mathcal{X}' & } $$ commutes. But also $ \mathrm{id}_{\mathcal{Y}} $ makes this diagram commute, and by the universal property of $ \rho $, we get that $ \tau' \circ \tau = \mathrm{id}_{\mathcal{Y}} $. By symmetry we also have that $ \tau \circ \tau' = \mathrm{id}_{\mathcal{Y}} $, hence $ \tau' = \tau^{-1} $, so in particular $ \tau $ is an automorphism.
This shows the existence of the map $ G \rightarrow \mathrm{Aut}(\mathcal{Y}) $. The proof that this map is indeed a group homomorphism is straightforward, and is omitted here.
(iii) The desingularization $ \rho : \mathcal{Y} \rightarrow \mathcal{X}' $ is \emph{strong}, i.e., it is an isomorphism over $ \mathrm{Reg}(\mathcal{X}') $. Since $ \sigma $ is an automorphism, we have that $ \sigma(\mathrm{Sing}(\mathcal{X}')) = \mathrm{Sing}(\mathcal{X}') $. But then $ \tau(\rho^{-1}(\mathrm{Reg}(\mathcal{X}'))) = \rho^{-1}(\mathrm{Reg}(\mathcal{X}')) $, so the claim follows immediately. \end{pf}
\subsection{Tame extensions}\label{assumption on degree} For the rest of this paper, we shall always make the assumption that the degree $ n = [K':K] $ is not divisible by the residue characteristic $p$. In other words, $ S' \rightarrow S $ is a tamely ramified extension. Since $k$ is algebraically closed, $k$ has a full set $ \boldsymbol{\mu}_n $ of $n$-th roots of unity. Furthermore, since $R$ is complete, and in particular henselian, we may lift all $n$-th roots of unity to $R$. We can choose a uniformizing parameter $ \pi \in R $ such that $ K' = K[\pi']/(\pi'^n - \pi) $. The extension $ K \subset K' $ is Galois, with group $ G = \boldsymbol{\mu}_n $. Then $ R' := R[\pi']/(\pi'^n - \pi) $ is the integral closure of $R$ in $K'$, and $ \pi'$ is a uniformizing parameter for $R'$.
\subsection{Assumptions on $\mathcal{X}$}\label{assumption on surface} In the rest of this paper, we shall make two assumptions in the situation in Section \ref{2.2}:
\begin{ass}\label{ass. 2.4} Let $ x \in \mathcal{X} $ be a closed point in the special fiber such that two irreducible components $C_1$ and $C_2$ of $ \mathcal{X}_k $ meet at $x$, and let $ m_i = \mathrm{mult}(C_i) $. We will always assume that \emph{at least} one of the $m_i$ is not divisible by $p$. \end{ass}
With this assumption, we can find an isomorphism $$ \widehat{\mathcal{O}}_{\mathcal{X},x} \cong R[[u_1,u_2]]/(\pi - u_1^{m_1} u_2^{m_2}) $$ (cf. \cite{CED}, proof of Lemma 2.3.2).
\begin{ass}\label{ass. 2.5} Let $l$ be the least common multiple of the multiplicities of the irreducible components of $\mathcal{X}_k$. Then we assume that $ \mathrm{gcd}(l,n) = 1 $. \end{ass}
Let us now recall a few facts developed in \cite{Tame}:
$\bullet$ Let $ x \in \mathcal{X} $ be a closed point in the special fiber. Because of Assumption \ref{ass. 2.5}, there is a unique point $ x' \in \mathcal{X}'_k $ that maps to $x$. The local analytic structure of $ \mathcal{X}' $ at $x'$ depends only on $n$ and the local analytic structure of $ \mathcal{X} $ at $x$. If $x$ belongs to a unique component of $ \mathcal{X}_k $, then $x'$ belongs to a unique component of $ \mathcal{X}'_k $, and $ \mathcal{X}' $ is regular at $x'$. If $x$ is an intersection point of two distinct components, then the same is true for $x'$, and $ \mathcal{X}' $ will have a \emph{tame cyclic quotient singularity} at $x'$.
$\bullet$ The minimal desingularization $ \mathcal{Y} $ of $ \mathcal{X}' $ is an SNC-model. Furthermore, the structure of $ \mathcal{Y} $ locally above a tame cyclic quotient singularity $x' \in \mathcal{X}'$ is completely determined by the structure locally at $ x = f(x') $ and the degree $n$ of the extension. The inverse image of $x'$ consists of a chain of smooth and rational curves whose multiplicities and self intersection numbers may be computed from the integers $n, m_1$ and $m_2$.
$\bullet$ For every irreducible component $C$ of $ \mathcal{X}_k $, there is precisely one component $C'$ of $ \mathcal{X}'_k $ that dominates $C$. The component $C'$ is isomorphic to $C$, and we have that $ \mathrm{mult}_{\mathcal{X}'_k}(C') = \mathrm{mult}_{\mathcal{X}_k}(C) $. It follows that the combinatorial structure of $ \mathcal{Y}_k $ is completely determined by the combinatorial structure of $ \mathcal{X}_k $ and the degree of $S'/S$.
\subsection{} We will now begin to describe the $G$-action on $ \mathcal{X}' $ and $ \mathcal{Y} $ in more detail. Assumptions \ref{ass. 2.4} and \ref{ass. 2.5} will impose some restrictions on this action.
\begin{prop}\label{action on desing} Let $ \rho : \mathcal{Y} \rightarrow \mathcal{X'} $ be the minimal desingularization. Then the following properties hold: \begin{enumerate} \item Let $D$ be an irreducible component of $ \mathcal{Y}_k $ that dominates a component of $ \mathcal{X}_k $. Then $D$ is stable under the $G$-action, and $G$ acts \emph{trivially} on $D$. \item Let $ x' \in \mathcal{X}' $ be a singular point, and let $E_1, \ldots, E_{l}$ be the exceptional components mapping to $x'$ under $ \rho $. Then every $E_i$ is stable under the $G$-action, and every node in the chain $ \rho^{-1}(x') $ is fixed under the $G$-action. \end{enumerate} \end{prop} \begin{pf} Let us first note that the map $ \mathcal{X}_{S'} \rightarrow \mathcal{X} $ is an isomorphism on the special fibers. Moreover, the action on the special fiber of $ \mathcal{X}_{S'} $ is easily seen to be trivial, so every closed point in the special fiber is fixed. Since the action on $ \mathcal{X}' $ commutes with the action on $ \mathcal{X}_{S'} $, it follows that every point in the special fiber of $ \mathcal{X}' $ is fixed. In particular, every irreducible component $C'$ of $ \mathcal{X}'_k $ is stable under the $G$-action, and the restriction of this action to $C'$ is trivial. Since the action on $ \mathcal{Y} $ commutes with the action on $ \mathcal{X}' $, it follows that the same is true for the strict transform $D$ of $C'$ in $ \mathcal{Y} $. This proves (i).
For (ii), we observe that since $x'$ is fixed, we have that $\rho^{-1}(x')$ is stable under the $G$-action. But also the two branches meeting at $x'$ are fixed. Let $D$ be the strict transform of any of these two branches. From part (i), it follows that the point where it meets the exceptional chain $\rho^{-1}(x')$ must be fixed. So if $E_1$ is the component in the chain meeting $\widetilde{D}$, then $E_1$ must be mapped into itself. Let $E_2$ be the next component in the chain. Then the point where $E_1$ and $E_2$ meet must also be fixed, so $E_2$ must also be mapped to itself. Continuing in this way, it is easy to see that all of the exceptional components are stable under the $G$-action, and that all nodes in $\rho^{-1}(x')$ are fixed points. \end{pf}
\begin{cor}\label{g^{-1}(Z) = Z} Let $ 0 \leq Z \leq \mathcal{Y}_k $ be an effective divisor. Then we have that the $G$-action restricts to $Z$. \end{cor} \begin{pf}
Since $Z$ is an effective Weil divisor, we can write $ Z = \sum_{C} r_C C $, where $C$ runs over the irreducible components of $ \mathcal{Y}_k $, and $r_C$ is a non-negative integer for all $C$. But Proposition \ref{action on desing} states that all irreducible components $C$ of $ \mathcal{Y}_k $ are stable under the $G$-action, and hence we get that the same holds for $ Z $. In other words, the action restricts to $Z$. \end{pf}
From Proposition \ref{action on desing} above, it follows that every node $y$ in $ \mathcal{Y}_k $ is a fixed point for the $G$-action on $ \mathcal{Y} $. Hence there is an induced action on $ \mathcal{O}_{\mathcal{Y},y} $ and on the cotangent space $ m_y/m_y^2 $, where $ m_y \subset \mathcal{O}_{\mathcal{Y},y} $ is the maximal ideal. In order to get a precise description of the action on the cotangent space, we will first describe the action on the completion $ \widehat{\mathcal{O}}_{\mathcal{Y},y} $.
Since, by Proposition \ref{action on desing}, every irreducible component $D$ of $ \mathcal{Y}_k $ is mapped to itself under the $G$-action, it follows that the $G$-action restricts to $D$ and that the points where $D$ meets the rest of the special fiber are fixed. In the case where $G$ acts non-trivially on $D$, we will see in Proposition \ref{prop. 3.3} that the fixed points for the $G$-action on $D$ are precisely the points where $D$ meets the rest of the special fiber. In particular, we wish to describe the action on $D$ locally at the fixed points.
\section{Desingularizations and actions}\label{lifting the action} In this section, we study how one can explicitly describe the action on the minimal desingularization $ \rho : \mathcal{Y} \rightarrow \mathcal{X}' $. Since we are only interested in this action locally at fixed points or stable components in the exceptional locus of $\rho$, we will begin with showing that we can reduce to studying the minimal desingularization locally at a singular point $ x' \in \mathcal{X}' $. This is an important step, since we have a good description of the complete local ring $ \widehat{\mathcal{O}}_{\mathcal{X}',x'} $. In particular, we can find a nice algebraization of this ring, with a compatible $G$-action. It turns out that it suffices for our purposes to study the minimal desingularization of this ring, and the lifted $G$-action.
In the second part of this section, we study the desingularization of the algebraization of $ \widehat{\mathcal{O}}_{\mathcal{X}',x'} $. We will use the explicit blow up procedure given in \cite{CED} for the resolution of tame cyclic quotient singularities, which will allow us to describe precisely how the $G$-action lifts. In particular, we describe the action on the completion of the local rings at the nodes in the exceptional locus, and the action on the exceptional components. These results are gathered in Proposition \ref{prop. 3.3}.
\subsection{Reduction to the complete local rings} If $ x' \in \mathcal{X}' $ is a singular point, we need to understand how $G$ acts on $ \widehat{\mathcal{O}}_{\mathcal{X}',x'} $. In order to do this, we consider $ f(x') = x \in \mathcal{X} $, where $ f : \mathcal{X}' \rightarrow \mathcal{X} $. Then $x$ is a closed point in the special fiber, and we have that $$ \widehat{\mathcal{O}}_{\mathcal{X},x} \cong R[[v_1,v_2]]/(\pi - v_1^{m_1} v_2^{m_2}), $$ where $m_1$ and $m_2$ are positive integers. Let $n$ be the degree of $ R'/R $. By assumption, we have that $ n $ is relatively prime to $m_1$ and $m_2$. Before we proceed, we would like to remark that we will in the discussion that follows use some properties that were proved in Section 2 of \cite{Tame}.
Consider now the pullback $ \mathcal{X}_{S'} $. We let $ G = \boldsymbol{\mu}_n $ act on $ \mathcal{X}_{S'} $ via its action on the second factor. We point out that we here choose the action given by $ [\xi](\pi') = \xi \pi' $ for any $ \xi \in \boldsymbol{\mu}_n $. Choosing this action is notationally convenient when we work with rings. However, the natural right action on $ \mathcal{X}_{S'} $ is the inverse to the one we use here. In particular, the irreducible characters for the representation of $ \boldsymbol{\mu}_n $ on $ H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $ induced by the action chosen here on $\mathcal{X}_{S'}$ will be the inverse characters to those induced by the right action.
By abuse of notation, let $ x \in \mathcal{X}_{S'} $ be the point mapping to $ x \in \mathcal{X} $. Then we have that the map $ \mathcal{O}_{\mathcal{X},x} \rightarrow \mathcal{O}_{\mathcal{X}_{S'},x} $ induced by the projection $ \mathcal{X}_{S'} \rightarrow \mathcal{X} $ can be described by the tensorization $$ \mathcal{O}_{\mathcal{X},x} \rightarrow \mathcal{O}_{\mathcal{X},x} \otimes_R R', $$ and that the $G$-action on $ \mathcal{O}_{\mathcal{X}_{S'},x} = \mathcal{O}_{\mathcal{X},x} \otimes_R R' $ is given by the action on $R'$.
Since $ \mathcal{O}_{\mathcal{X},x} \rightarrow \mathcal{O}_{\mathcal{X},x} \otimes_R R' $ is finite, completion commutes with tensoring with $R'$, so we get that $$ \widehat{\mathcal{O}}_{\mathcal{X}_{S'},x} = \widehat{\mathcal{O}}_{\mathcal{X},x} \otimes_R R', $$ and hence the $G$-action on $ \widehat{\mathcal{O}}_{\mathcal{X}_{S'},x} $ is induced from the action on $R'$ in the second factor. It follows that $$ \widehat{\mathcal{O}}_{\mathcal{X}_{S'},x} \cong R'[[v_1,v_2]]/(\pi'^n - v_1^{m_1} v_2^{m_2}), $$ and that the $G$-action is given by $ [\xi](\pi') = \xi \pi' $ and $ [\xi](v_i) = v_i $, for any $ \xi \in \boldsymbol{\mu}_n $.
Let $ \mathcal{X}' \rightarrow \mathcal{X}_{S'} $ be the normalization. Then $x'$ is the unique point mapping to $x$, and the induced map $ \widehat{\mathcal{O}}_{\mathcal{X}_{S'},x} \rightarrow \widehat{\mathcal{O}}_{\mathcal{X}',x'} $ is the normalization of $ \widehat{\mathcal{O}}_{\mathcal{X}_{S'},x} $. Furthermore, it follows also that the $G$-action on $ \widehat{\mathcal{O}}_{\mathcal{X}',x'} $ induced by the action on $ \mathcal{X}' $ is the unique lifting of the $G$-action on $ \widehat{\mathcal{O}}_{\mathcal{X}_{S'},x} $ to the normalization $ \widehat{\mathcal{O}}_{\mathcal{X}',x'} $.
Let $ \rho : \mathcal{Y} \rightarrow \mathcal{X}' $ be the minimal desingularization, and consider the fiber diagram $$ \xymatrix{ \widehat{\mathcal{Y}} \ar[d]_{\hat{\rho}} \ar[r]^f & \mathcal{Y} \ar[d]^{\rho} \\ \mathrm{Spec}(\widehat{\mathcal{O}}_{\mathcal{X}',x'}) \ar[r] & \mathcal{X}' .} $$ Then $ \hat{\rho} $ is the minimal desingularization of $ \mathrm{Spec}(\widehat{\mathcal{O}}_{\mathcal{X}',x'}) $ (cf. ~\cite{Lip}, Lemma 16.1, and use the fact that $ \mathcal{Y} $ is minimal), and hence the $G$-action on $ \mathrm{Spec}(\widehat{\mathcal{O}}_{\mathcal{X}',x'}) $ lifts uniquely to $ \widehat{\mathcal{Y}} $.
We have that $f$ induces an isomorphism of the exceptional loci $ \hat{\rho}^{-1}(x') $ and $ \rho^{-1}(x') $. Let $E$ be an exceptional component. Then the $G$-action restricts to $E$, and it is easily seen that $f$, when restricted to $E$, is equivariant.
Furthermore, for any closed point $ y \in \rho^{-1}(x') $, we have that $f$ induces an isomorphism $ \widehat{\mathcal{O}}_{\mathcal{Y},y} \cong \widehat{\mathcal{O}}_{\widehat{\mathcal{Y}},y} $ (one can argue in a similar way as in the proof of \cite{Liubook}, Lemma 8.3.49). If $y$ is a fixed point, it is easily seen that this isomorphism is equivariant. We therefore conclude that in order to describe the action on $\mathcal{Y}$ locally at the exceptional locus over $x'$, it suffices to consider the minimal desingularization of $ \mathrm{Spec}(\widehat{\mathcal{O}}_{\mathcal{X}',x'}) $.
\subsection{Equivariant algebraization} In order to find an algebraization of $ \widehat{\mathcal{O}}_{\mathcal{X}',x'} $, we consider first the polynomial ring $ V = R'[v_1,v_2]/(\pi'^n - v_1^{m_1} v_2^{m_2}) $. We let $G$ act on $ V $ by $ [ \xi ](\pi') = \xi \pi' $ and $ [ \xi ](v_i) = v_i $ for $ i = 1,2 $, for any $ \xi \in G $. Note that the maximal ideal $ P = (\pi', v_1,v_2) $ is fixed, and hence there is an induced action on the completion $ \widehat{V}_P = R'[[v_1,v_2]]/(\pi'^n - v_1^{m_1} v_2^{m_2}) $, given as above. This gives a $G$-equivariant algebraization of $ \widehat{\mathcal{O}}_{\mathcal{X}_{S'},x} $.
Let us consider now the $R'$-homomorphism $$ V = R'[v_1,v_2]/(\pi'^n - v_1^{m_1} v_2^{m_2}) \rightarrow T = R'[t_1,t_2]/(\pi' - t_1^{m_1} t_2^{m_2}), $$ given by $ v_i \mapsto t_i^n $. We let $ \boldsymbol{\mu}_n $ act on $T$, relatively to $R'$, by $ [ \eta ](t_1) = \eta t_1 $, $ [ \eta ](t_2) = \eta^r t_2 $, where $ r $ is the unique integer $ 0 < r < n $ such that $ m_1 + r m_2 \equiv_n 0 $. Note that this is an ad hoc action introduced to compute the normalization, which must not be confused with the natural $G$-action. Arguing as in Section 2 of \cite{Tame}, we find that the induced map $ V \rightarrow U := T^{\boldsymbol{\mu}_n} $ is the normalization of $V$. Furthermore, it is easily seen that there is a unique maximal ideal $Q \subset U $ mapping to $P$, corresponding to the origin $ (\pi, t_1,t_2) $ in $T$.
We will need the following lemma:
\begin{lemma}\label{lemma 4.1} Consider the fiber diagrams $$ \xymatrix{ V \ar[d] \ar[r] & V_P \ar[d] \ar[r] & \widehat{V}_P \ar[d]\\ U \ar[r] & U_P \ar[r] & \widehat{U}_P, } $$ where $ U_P := U \otimes_V V_P $, $ \widehat{U}_P := U_P \otimes_{V_P} \widehat{V}_P $, and where $ V \rightarrow U $ is the normalization.
Then we have that $ U_P = U_Q $, where $U_Q $ is the localization in the maximal ideal $ Q \subset U $. Furthermore, the maps $ V_P \rightarrow U_P $ and $ \widehat{V}_P \rightarrow \widehat{U}_P $ are the normalizations. \end{lemma} \begin{pf} Since $ V \rightarrow U $ is finite, it follows that $ V_P \rightarrow U_P $ is finite, and coincides with the normalization of $V_P$, since normalization commutes with localization. Furthermore, $ U_P $ is semi-local, with maximal ideals corresponding exactly to the maximal ideals of $U$ restricting to $P$. It follows that $ U_P = U_Q $, and that the map $ U \rightarrow U_Q $ above is the localization map.
Since $ V_P \rightarrow U_P $ is finite, we can identify the map $ U_P \rightarrow \widehat{U}_P $ with the completion of $ U_P $ in the radical ideal. We have that $V_P$ is reduced and excellent, since it is a localization of $V$. But then normalization commutes with completion, that is, $ \widehat{U}_Q = \widehat{U}_P = (\widehat{V}_P)' $, and the map $ \widehat{V}_P \rightarrow \widehat{U}_P = \widehat{U}_Q $ is the normalization (\cite{Liubook}, Proposition 8.2.41). \end{pf}
It follows from this lemma that $U$ is an equivariant algebraization of $ \widehat{\mathcal{O}}_{\mathcal{X}',x'} $. Let $ \rho_U : \mathcal{Z} \rightarrow \mathrm{Spec}(U) $ be the minimal desingularization. Then it follows that we have a fiber diagram $$ \xymatrix{ \widehat{\mathcal{Y}} \ar[d]_{\hat{\rho}} \ar[r] & \mathcal{Z} \ar[d]^{\rho_U} \\ \mathrm{Spec}(\widehat{\mathcal{O}}_{\mathcal{X}',x'}) \ar[r] & \mathrm{Spec}(U),} $$ where all maps commute with the various $G$-actions. In particular, we can now conclude that to describe the $G$-action on $ \widehat{\mathcal{Y}} $ locally at fixed points or components in the exceptional locus, it suffices to compute the corresponding data for $ \mathcal{Z} $.
\subsection{Lifting the action to the normalization}
We will continue to work with the rings $V$, $U$ and $T$. We saw above that there was a $G$-action on $V$. Since $ U $ is the normalization of $V$, this action will lift uniquely, and we want to give a precise description of that action. Let us first remark that by arguing exactly as in Section 2 in \cite{Tame}, it follows that $ U = T^{\boldsymbol{\mu}_n} $ is generated as a $V$-module by the monomials $t_1^i t_2^j $ such that $ 0 \leq i,j < n $, where $ i + r j \equiv 0 $ modulo $n$.
In order to describe the $G$-action on $U = T^{\boldsymbol{\mu}_n}$, it suffices to give the action on each of the $V$-module generators. This is done in the following lemma:
\begin{lemma}\label{lemma 3.1} Let $ t_1^i t_2^j \in T $ be such that $ i + rj \equiv 0 $ modulo $n$. Then we have, for any $ \xi \in G $, that $ [ \xi ] (t_1^i t_2^j) = \xi^{i \alpha_1} t_1^i t_2^j $, where $ \alpha_1 $ is an inverse to $m_1$ modulo $n$. \end{lemma} \begin{pf} In $T$ we have the relations $ t_1^{m_1} t_2^{m_2} = \pi' $, $ t_1^{n} = v_1 $ and $ t_2^{n} = v_2 $, hence all these elements lie in the image of $V$, and we therefore know how $ G $ acts on them. We have that $[ \xi ] (t_1^{m_1} t_2^{m_2}) = \xi t_1^{m_1} t_2^{m_2} $ and $ [ \xi ] (t_i^{n}) = t_i^{n} $, for any $ \xi \in G $.
For any $i,j$ as above, let $k(i,j)$ be the integer satisfying $ i + r j = k(i,j) n $. Furthermore, define $ K(i,j) := m_2 k(i,j) - k(m_1,m_2) j $. One computes easily that $ i m_2 = K(i,j) n + j m_1 $. From this, we get the relation $$ (t_1^{m_1} t_2^{m_2})^i = (t_1^i t_2^j)^{m_1} (t_2^{n})^{K(i,j)} $$ in $\mathrm{Frac}(V)$, and hence $$ (t_1^i t_2^j)^{m_1} = (t_1^{m_1} t_2^{m_2})^i (t_2^{n})^{ - K(i,j)}. $$ As $ \mathrm{gcd}(m_1,n) = 1 $, we can find integers $ \alpha_1, \beta_1 $ such that $ \alpha_1 m_1 + \beta_1 n = 1 $. So we get $$ t_1^i t_2^j = (t_1^i t_2^j)^{\alpha_1 m_1} (t_1^i t_2^j)^{\beta_1 n} = ((t_1^i t_2^j)^{m_1})^{\alpha_1} (t_1^{n})^{i \beta_1} (t_2^{n})^{j \beta_1} = $$ $$ (t_1^{m_1} t_2^{m_2})^{i \alpha_1} (t_2^{n})^{ - \alpha_1 K(i,j)} (t_1^{n})^{i \beta_1} (t_2^{n})^{j \beta_1}. $$ From this relation, it follows that $ [ \xi ] (t_1^i t_2^j) = \xi^{i \alpha_1} t_1^i t_2^j $. In particular, we note that every $ \boldsymbol{\mu}_n $-invariant monomial $ t_1^i t_2^j $ in $T$ is an ``eigenvector'' for the $G$-action, and that we can compute explicitly the eigenvalue. \end{pf}
\subsection{The action on the minimal desingularization}\label{section 3.4} Now that we know how $G$ acts on $U$, we will use this description to lift the $G$-action to the minimal desingularization of $ \mathrm{Spec}(U) $. For this, we shall follow closely the inductive construction of the desingularization given in \cite{CED}.
Before we begin with that, let us first note that $ t_1^{i} t_2^{j} $ is $\boldsymbol{\mu_n}$-invariant if and only if $ i + r j = n k(i,j) $, for some integer $ k(i,j) $. From Lemma \ref{lemma 3.1}, it follows that $ [ \xi ] (t_1^i t_2^j) = \xi^{i \alpha_1} t_1^{i} t_2^{j} $, for any $ \xi \in G $, where $ m_1 \alpha_1 \equiv_n 1 $.
\subsection{Changing coordinates}\label{subs. 3.5} Define elements $ z = t_1^{n} $ and $ w = t_2/t_1^r $. These are $\boldsymbol{\mu}_n$-invariant elements in the fraction field of $T$. Understanding how $G$ acts on these elements will be a key step in describing the action on the desingularization, so we work that out now.
Note that $ z = t_1^{n} = v_1 $, so $ [ \xi ](z) = z $. Moreover, since $ zw = t_1^{n-r} t_2 $ is an invariant monomial, we get that $ [ \xi ](zw) = [ \xi ](t_1^{n-r} t_2) = \xi^{(n-r) \alpha_1} t_1^{n-r} t_2 = \xi^{ -r \alpha_1} zw $, and it follows that $ [ \xi ](w) = [ \xi ](zw/z) = \xi^{- r \alpha_1} w $.
\subsection{Blowing up} In the following, we shall describe the minimal desingularization of $ Z = \mathrm{Spec}(U) $, which will be denoted $ \varrho : \widetilde{Z} \rightarrow Z $. This map will consist of a series of blow-ups $$ \ldots \rightarrow Z_{i+1} \rightarrow Z_i \rightarrow \ldots \rightarrow Z_1 \rightarrow Z_0 = Z, $$ where each step $ \varrho_{i+1} : Z_{i+1} \rightarrow Z_i $ is the blow-up in a certain ideal supported in the special fiber. Furthermore, each step will produce a regular, affine open chart of $ \widetilde{Z} $, that is stable under the $G$-action. For each of these charts we will be able to describe the $G$-action explicitly.
To do this, we shall use the new coordinates $z$ and $w$ of $U$. Let $i,j$ be any pair of integers such that $ 0 \leq j \leq (n/r)i $. Then we have that $ z^i w^j \in R'[t_1,t_2] $, and furthermore $$ R'[t_1,t_2]^{\boldsymbol{\mu}_n} = \oplus_{0 \leq j \leq (n/r)i} R'z^iw^j. $$ A trivial computation shows that $ t_1^{m_1} t_2^{m_2} = z^{\mu_1}w^{m_2} $, where $ m_1 + r m_2 = n \mu_1 $, so we can write $$ U = \frac{\oplus_{0 \leq j \leq (n/r)i} R'z^iw^j}{(z^{\mu_1}w^{m_2} - \pi')}. $$
The first map $ \varrho_1 : Z_1 \rightarrow Z $ in the sequence above is obtained by blowing up in the ideal generated by $z$ and $zw$. There are two affine charts covering $Z_1$, namely $D_+(z)$ and $ D_+(zw) $, where we adjoin the quotients $ zw/z = w $ and $ z/zw = 1/w $ respectively. We shall treat these two charts separately.
\subsection{The chart $D_+(z)$.} We have $$ D_+(z) = \mathrm{Spec}(U[w]) = \mathrm{Spec}(R'[z,w]/(z^{\mu_1}w^{m_2} - \pi')). $$ This affine piece is already regular, and the special fiber has two components. The component $ w = 0 $ is the strict transform of the branch with multiplicity $m_2$ in $Z$. The component $ z = 0 $ is the exceptional curve $E_1$, which has multiplicity $ \mu_1 $ in the special fiber.
$ D_+(z) $ is birational to $Z$ via $ \varrho_1 $. The coordinates $z$ and $w$ are elements of the function field of $Z$, and we computed above that $ [ \xi ] (z) = z $ and $ [ \xi ] (w) = \xi^{-r \alpha_1} w $, for any $ \xi \in G $. So the same is true for the action on the function field of $ D_+(z) $. We also have that $ z^{\mu_1} w^{m_2} = t_1^{m_1} t_2^{m_2} $, hence $ [ \xi ] (z^{\mu_1} w^{m_2}) = \xi z^{\mu_1} w^{m_2} $, and so it follows that $ D_+(z) $ is stable under the $G$-action.
On the chart $ D_+(z) $, the exceptional curve $E_1$ has affine ring $k[w]$, and the $G$-action restricted to $E_1$ is given by $ [ \xi ] (w) = \xi^{-r \alpha_1} w $.
\subsection{The chart $D_+(zw)$.} We have that $ D_+(zw) = \mathrm{Spec}(U[1/w]) $, where $$ U[1/w] = \frac{\oplus_{j \leq (n/r)i, 0 \leq i} R'z^iw^j}{(z^{\mu_1}w^{m_2} - \pi')}. $$
We will now change coordinates. We start by performing the first step of the Jung-Hirzebruch continued fraction expansion. So we write $ n = b_1 r - r_1 $, where $ b_1 = \left \lceil n/r \right \rceil \geq 2 $. There are now two possibilities; either \begin{enumerate} \item $ r = 1 $ and $ r_1 = 0 $, or \item $ r_1 > 0 $ and $ \mathrm{gcd}(r,r_1) = 1 $. \end{enumerate}
Consider first case (i), where $ r = 1 $ and $ r_1 = 0 $. Then we have $ b_1 = n $, so $ b_1 \mu_1 - m_2 = m_1 $. Let $ i_1 = b_1 i - j $ and $ j_1 = i $, and define new coordinates $ z_1 = 1/w $ and $ w_1 = z w^{b_1} $. By an easy computation, we see that $ z^i w^j = z_1^{i_1} w_1^{j_1} $. But then it follows that $$ U[1/w] = R'[z_1,w_1]/(z_1^{b_1 \mu_1 - m_2} w_1^{\mu_1} - \pi') = R'[z_1,w_1]/(z_1^{m_1} w_1^{\mu_1} - \pi'), $$ so in particular $ \mathrm{Spec}(U[1/w]) $ is regular. The special fiber consists of $ z_1 = 0 $, which is the strict transform of the branch of $Z_k$ with multiplicity $m_1$, and the component $ w_1 = 0 $, the exceptional curve $ E_1 $.
Since $ \mathrm{Spec}(U[1/w]) $ is birational to $Z$, and since we know the action on the elements $z$ and $w$, the action on $z_1$ and $w_1$ follows immediately. It is $ [ \xi ](z_1) = \xi^{ r \alpha_1 } z_1 $ and $ [ \xi ](w_1) = \xi^{ - b_1 r \alpha_1 } w_1 = w_1 $, since $ b_1 = n $. As $ z^{\mu_1} w^{m_2} = z_1^{m_1} w_1^{\mu_1} $, we get that $ [ \xi ] (z_1^{m_1} w_1^{\mu_1}) = \xi z_1^{m_1} w_1^{\mu_1} $, so $ \mathrm{Spec}(U[1/w]) $ is stable under the $G$-action. Note also that $E_1$ has affine ring $k[z_1]$ on this chart, and therefore the $G$-action restricted to $E_1$ is given by $ [ \xi ](z_1) = \xi^{ r \alpha_1 } z_1 $.
Let us now consider case (ii). Then we have that $ r_1 > 0 $ and $ \mathrm{gcd}(r,r_1) = 1 $. We shall still do the coordinate change $ z_1 = 1/w $, $ w_1 = z w^{b_1} $. With the equation $ n = b_1 r - r_1 $ in mind, it follows that the conditions $ j \leq (n/r)i $ and $ 0 \leq i $ can be rewritten as $ 0 \leq i \leq (r/r_1)(b_1 i-j) $, hence $ 0 \leq j_1 \leq (r/r_1) i_1 $. So we may write $$ U_1 := U[1/w] = \frac{\oplus_{0 \leq j_1 \leq (r/r_1)i_1} R'z_1^{i_1} w_1^{j_1}}{(z_1^{b_1 \mu_1 - m_2} w_1^{\mu_1} - \pi')}.$$ Notice that this is a ring essentially of the same type that we started with, where in addition to the coordinate change, the parameters are changed by $$ (n, r, m_1, m_2, \mu_1) \mapsto (r, r_1, m_1, \mu_1, b_1 \mu_1 - m_2). $$
We will now perform a new blow up, in the ideal $(z_1, z_1 w_1)$. Similarly to the case in Section \ref{subs. 3.5}, we get two affine charts. The chart $ D_+(z_1) $ is the spectrum of the ring $$ R'[z_1,w_1]/(z_1^{\mu_2} w_1^{\mu_1} - \pi'), $$ where we have put $ \mu_2 = b_1 \mu_1 - m_2 $. $G$ acts on this ring by $ [ \xi ](z_1) = \xi^{r \alpha_1} z_1 $ and $ [ \xi ](w_1) = \xi^{ - b_1 r \alpha_1} w_1 $. But as $ b_1 r = n + r_1 $, we get in fact that $ [ \xi ](w_1) = \xi^{ - r_1 \alpha_1} w_1 $, for any $ \xi \in G $. The new exceptional curve $E_2$ has affine ring $k[w_1]$ on this chart, and hence the restricted $G$-action is given by $ [ \xi ](w_1) = \xi^{ - r_1 \alpha_1} w_1 $.
The nature of the chart $ D_+(z_1w_1) $ depends entirely on our new set of parameters. That is, one computes the next step in the Jung-Hirzebruch continued fraction expansion $ r = b_2 r_1 - r_2 $. If $r_1 = 1$ (and hence $ r_2 = 0 $), also $ D_+(z_1w_1) $ is regular. In this case, we change coordinates in a similar fashion as above. Since $ z_1, w_1 $ are ``eigenelements'' under the $G$-action, the same will be true for the new coordinates $ z_2, w_2 $. In case $ r_2 > 0$, we change coordinates, and get a new ring of the same type as above, after changing our set of parameters.
\subsection{Induction step} As a preparation for the induction step, recall that when calculating the Jung-Hirzebruch continued fraction expansion of $ n/r $, we have a series of equations \begin{equation} r_{l-1} = b_{l+1} r_l - r_{l+1}, \end{equation} for all $ l \geq 0 $, where $ b_{l+1} = \left \lceil r_{l-1}/r_l \right \rceil $, and where we define $ r_{-1} = n $ and $ r_0 = r $. At each stage, we always have either $ r_l = 1 $ and $ r_{l+1} = 0 $, or $ r_{l+1} > 0 $ and $ \mathrm{gcd}(r_l, r_{l+1}) = 1 $. Furthermore, $ r_{l-1} > r_l $, so this process will eventually stop after a finite number of steps. Let $L$ denote the \emph{length} of the resolution, by which we mean that $ r_{L-1} = 1 $, and $ r_L = 0 $.
We also have a system of equations \begin{equation} \mu_{l+1} = b_l \mu_l - \mu_{l-1} \end{equation} for $ l \in \{ 1, \ldots, L \} $, where we define $ \mu_{0} = m_2 $, $ \mu_{1} = \mu = (m_1 + r m_2)/n $ and $ \mu_{L+1} = m_1 $.
Assume that we have reached the $(l-1)$-st step $$ Z_{l-1} \rightarrow \ldots \rightarrow Z_1 \rightarrow Z $$ in the blow-up procedure, and that $ Z_{l-1} $ is regular outside the open affine chart $\mathrm{Spec}(U_{l-1})$, where $$ U_{l-1} = \frac{\oplus_{0 \leq j_{l-1} \leq (r_{l-2}/r_{l-1}) i_{l-1}} R' z_{l-1}^{i_{l-1}} w_{l-1}^{j_{l-1}}}{(z_{l-1}^{\mu_l} w_{l-1}^{\mu_{l-1}} - \pi')}. $$ Assume furthermore that the $G$-action is given by $ [ \xi ](z_{l-1}) = \xi^{\alpha_1 r_{l-2}} z_{l-1} $, $ [ \xi ](w_{l-1}) = \xi^{- \alpha_1 r_{l-1}} w_{l-1} $ and $ [ \xi ](z_{l-1}^{\mu_l} w_{l-1}^{\mu_{l-1}}) = \xi z_{l-1}^{\mu_l} w_{l-1}^{\mu_{l-1}} $.
We will now blow up in the ideal $ (z_{l-1}, z_{l-1} w_{l-1}) $. Then we get two affine charts $ D_+(z_{l-1}) $ and $ D_+(z_{l-1}w_{l-1}) $. The affine ring for the chart $ D_+(z_{l-1}) $ is $$ U_{l-1}[w_{l-1}] = R'[z_{l-1}, w_{l-1}]/( z_{l-1}^{\mu_l} w_{l-1}^{\mu_{l-1}} - \pi' ). $$ Notice that this chart is regular, and that by assumption, we already know the $G$-action here.
The affine ring for the chart $ D_+(z_{l-1}w_{l-1}) $ is $$ U_{l-1}[1/w_{l-1}] = \frac{\oplus_{j_{l-1} \leq (r_{l-2}/r_{l-1}) i_{l-1}, 0 \leq i_{l-1}} R' z_{l-1}^{i_{l-1}} w_{l-1}^{j_{l-1}}}{(z_{l-1}^{\mu_l} w_{l-1}^{\mu_{l-1}} - \pi')}. $$ Let us introduce the new indexing $ i_l = b_l i_{l-1} - j_{l-1} $, $ j_l = i_{l-1} $, and define new coordinates $ z_l = 1/w_{l-1} $, $ w_l = z_{l-1} w_{l-1}^{b_l} $. There are two possibilities, that we will treat separately.
We shall first treat the case where $ r_{l-1} = 1 $ and $ r_l = 0 $. Then it is easy to compute that $ z_{l-1}^{\mu_l} w_{l-1}^{\mu_{l-1}} = z_{l}^{\mu_{l+1}} w_{l}^{\mu_{l}} $. We now have that $ b_l = r_{l-2} $, so the inequality $ j_{l-1} \leq (r_{l-2}/r_{l-1}) i_{l-1} $ can be written as $ i_l = b_l i_{l-1} - j_{l-1} \geq 0 $. It follows that $ i_l, j_l $ run over non-negative integers, and hence we get that $$ U_{l-1}[1/w_{l-1}] = R'[z_{l}, w_{l}]/( z_{l}^{\mu_{l+1}} w_{l}^{\mu_{l}} - \pi' ), $$ which is regular. It follows that $ [ \xi ](z_{l}) = \xi^{\alpha_1 r_{l-1}} z_l $ and $ [ \xi ](w_{l}) = \xi^{\alpha_1 r_{l-2} - \alpha_1 b_l r_{l-1}} w_{l} = \xi^{- \alpha_1 r_l } w_{l} = w_{l} $, since $ r_l = 0 $. Furthermore, we see that $ \mathrm{Spec}(U_{l-1}[1/w_{l-1}]) $ is stable under the $ G $-action.
In the second case we have $ r_l > 0 $ and $ \mathrm{gcd}(r_{l-1}, r_l) = 1 $. Since $ r_{l-2} = b_l r_{l-1} - r_l $, the inequalities $ j_{l-1} \leq (r_{l-2}/r_{l-1}) i_{l-1} $ and $ 0 \leq i_{l-1} $ can be written on the form $$ 0 \leq j_l \leq (r_{l-1}/r_l) i_l. $$ It follows that we can write $$ U_l := U_{l-1}[1/w_{l-1}] = \frac{\oplus_{0 \leq j_l \leq (r_{l-1}/r_l) i_l} R' z_l^{i_l} w_l^{j_l}}{(z_{l}^{\mu_{l+1}} w_{l}^{\mu_{l}} - \pi' )}. $$ Furthermore, $ [ \xi ](z_{l}) = \xi^{\alpha_1 r_{l-1}} z_l $ and $ [ \xi ](w_{l}) = \xi^{- \alpha_1 r_l } w_{l} $ for any $ \xi \in G $.
By induction, it now follows that we may cover $ \widetilde{Z} $ with a finite number of affine open charts $ \mathrm{Spec}(R'[z_l,w_l]/(z_{l}^{\mu_{l+1}} w_{l}^{\mu_{l}} - \pi')) $. These charts are stable under the $G$-action, and we have that $ [ \xi ] (z_l) = \xi^{\alpha_1 r_{l-1}} z_l $ and $ [ \xi ] (w_l) = \xi^{- \alpha_1 r_{l}} w_l $.
\subsection{}\label{3.11}
We sum up these results in Proposition \ref{prop. 3.3} below. This proposition will be important in later sections, when we consider the $G$-action on the cohomology groups $ H^i(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $.
\begin{prop}\label{prop. 3.3} The minimal desingularization $ \widetilde{Z} $ of $ Z = \mathrm{Spec}(U) $ can be covered by the affine charts $$ U_{l-1}[w_{l-1}] = R'[z_{l-1},w_{l-1}]/(z_{l-1}^{\mu_{l}} w_{l-1}^{\mu_{l-1}} - \pi'), $$ for $ l \in \{ 1, \ldots, L \} $, together with the affine chart $$ U_{L-1}[1/w_{L-1}] = R'[z_L,w_L]/(z_L^{\mu_{L+1}} w_L^{\mu_{L}} - \pi'). $$
These charts are $ G $-stable, and the $ G $-action is given by $ [ \xi ] (\pi') = \xi \pi' $, $ [ \xi ] (z_{l-1}) = \xi^{ \alpha_1 r_{l-2} } z_{l-1} $ and $ [ \xi ] (w_{l-1}) = \xi^{ - \alpha_1 r_{l-1} } w_{l-1} $, for all $ l \in \{ 1, \ldots, L+1 \} $, for any $ \xi \in G $.
Let $ E_l $ be the $ l $-th exceptional component. On the chart $U_{l-1}[w_{l-1}]$, we have that the affine ring for $ E_l $ is $ k[w_{l-1}] $, and $ G $ acts by $ [ \xi ] (w_{l-1}) = \xi^{ - \alpha_1 r_{l-1} } w_{l-1} $, for any $ \xi \in G $. On the affine chart $U_{l}[w_{l}]$, the affine ring for $ E_l $ is $ k[z_l] $, and $ G $ acts by $ [ \xi ] (z_{l}) = \xi^{ \alpha_1 r_{l-1} } z_{l} $. \end{prop}
Let us finally remark that the cotangent space to $ \widetilde{Z} $ at the fixed point that is the intersection point of $ E_l $ and $ E_{l+1} $ is generated by (the classes) of the local equations $ z_l $ and $ w_l $ for the curves. Therefore, Proposition \ref{prop. 3.3} gives a complete description of the action on the cotangent space. Furthermore, we can also read off the eigenvalues for the elements of this basis. Hence we immediately get an explicit description of the action on the cotangent space to the minimal desingularization of $\mathcal{X}'$ at the corresponding fixed point.
\section{Action on cohomology} In the previous two sections, we have described the $G$-action on $ \mathcal{Y}/S' $. We will now begin the study of the induced $G$-action on the cohomology groups $ H^i(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $.
We begin this section with recalling some generalities about coherent sheaf cohomology, and establish notation and terminology that will be used throughout the rest of the paper.
Let $X$ be a scheme, and let $ \mathcal{F} $ be a coherent sheaf on $X$. Assume that we have an automorphism $ g : X \rightarrow X $, and an isomorphism $ u : g^* \mathcal{F} \rightarrow \mathcal{F} $. This data will, as we explain below, give an automorphism of the cohomology groups $ H^i(X,\mathcal{F}) $. If $X$ is projective over a field $k$, these groups are finite dimensional vector spaces over $k$, and hence the traces of these automorphisms are defined. However, we will typically work over fields of positive characteristic, and then traces can behave badly, in particular with respect to representation theoretical questions. In our case, we shall see below that this can be solved by lifting to characteristic $0$, using the theory of \emph{Brauer characters}.
We end this section with considering the case where $X$ is a smooth curve, and where $ \mathcal{F} $ is an invertible sheaf. We give, in Proposition \ref{prop. 5.5} and Proposition \ref{prop. 5.6}, formulas for the alternating sum of the Brauer traces of automorphisms as mentioned above of the $ H^i(X,\mathcal{F}) $, in terms of certain data related to the fixed point locus. These formulas will be used numerous times in later sections.
\subsection{The general set-up}\label{inverse} Let $ f : X \rightarrow Y $ be a morphism of schemes, and let $ \mathcal{G} $ be a sheaf on $Y$. Let us first recall the construction of the inverse image sheaf $ f^{-1} \mathcal{G} $ on $X$. It is obtained as the sheafification of the presheaf $$ U \mapsto \underset{\rightarrow}{\mathrm{lim}} ~ \mathcal{G}(V), $$ where the direct limit is taken over all open sets $ V \subseteq Y $ such that $ U \subseteq f^{-1}(V) $.
In particular, we have a canonical homomorphism of global sections \begin{equation}\label{equation 5.1} \Gamma (Y, \mathcal{G}) \rightarrow \Gamma(X, f^{-1} \mathcal{G}). \end{equation}
\begin{rmk}We shall only use this construction when $f$ is an isomorphism, in which case the functor $ f^{-1} $ coincides with the direct image functor $ \varphi_* $, where $ \varphi $ denotes the inverse morphism to $ f $. Hence there is no need to sheafify. \end{rmk}
\begin{lemma}\label{lemma 5.2} The morphism $ f : X \rightarrow Y $ induces a natural and canonical homomorphism $$ H^p(f) : H^p(Y,\mathcal{G}) \rightarrow H^p(X, f^* \mathcal{G}), $$ for all $p \geq 0$. \end{lemma} \begin{pf} Let $$ 0 \rightarrow \mathcal{G} \rightarrow \mathcal{I}^{\bullet} $$ be an injective resolution in $ \mathcal{A}b(Y) $. Since the functor $ f^{-1} $ is exact, we get that $$ 0 \rightarrow f^{-1} \mathcal{G} \rightarrow f^{-1} \mathcal{I}^{\bullet} $$ is a resolution. Let us now choose an injective resolution $$ 0 \rightarrow f^{-1} \mathcal{G} \rightarrow \mathcal{J}^{\bullet} $$ in $ \mathcal{A}b(X) $. Then there is a map $$ \xymatrix{0 \ar[r] & f^{-1} \mathcal{G} \ar[r] \ar[d] & f^{-1} \mathcal{I}^{\bullet} \ar[d] \\
0 \ar[r] & f^{-1} \mathcal{G} \ar[r] & \mathcal{J}^{\bullet}, } $$ unique up to homotopy. Hence, on global sections, we get $$ \Gamma(Y, \mathcal{I}^{\bullet}) \rightarrow \Gamma(X, f^{-1} \mathcal{I}^{\bullet}) \rightarrow \Gamma(X, \mathcal{J}^{\bullet}), $$ which induces a map \begin{equation}\label{eq. 5.2} H^p(Y,\mathcal{G}) \rightarrow H^p(X, f^{-1} \mathcal{G}), \end{equation} for every $ p \geq 0 $. This map is independent of the choice of injective resolutions.
The canonical homomorphism $$ f^{-1} \mathcal{G} \rightarrow f^* \mathcal{G} = f^{-1} \mathcal{G} \otimes_{f^{-1} \mathcal{O}_Y} \mathcal{O}_X $$ of sheaves on $X$ induces by functoriality a map \begin{equation}\label{eq. 5.3} H^p(X, f^{-1} \mathcal{G}) \rightarrow H^p(X, f^* \mathcal{G}), \end{equation} for every $ p \geq 0 $. The composition of the maps \ref{eq. 5.2} and \ref{eq. 5.3} is then the desired $H^p(f)$. \end{pf}
The inverse image $ f^{-1} : \mathcal{A}b(Y) \rightarrow \mathcal{A}b(X) $ is an exact functor. Hence, the assignment $ \mathcal{G} \mapsto H^{\bullet}(X, f^{-1} \mathcal{G}) $ makes up a $ \delta $-functor from $ \mathcal{A}b(Y) $ to $ \mathcal{A}b $. Furthermore, the canonical map $$ \phi^0 : \Gamma (Y, \mathcal{G}) \rightarrow \Gamma(X, f^{-1} \mathcal{G}), $$ induces a unique sequence $ \phi^p : H^p(Y,\mathcal{G}) \rightarrow H^p(X, f^{-1} \mathcal{G}) $ of morphisms, for every $ p \geq 0 $. Given any sequence $$ 0 \rightarrow \mathcal{G}_1 \rightarrow \mathcal{G}_2 \rightarrow \mathcal{G}_3 \rightarrow 0 $$ in $ \mathcal{A}b(Y) $, the maps $ \phi^p $ commute with the differentials in the long exact sequences induced by $ H^p(Y,-) $ and $ H^p(X, f^{-1} (-)) $ respectively.
Consider now the case where $ f : X \rightarrow Y $ is an \emph{isomorphism}. Then the pullback $ f^* $ is an exact functor. Consequently, any exact sequence $$ 0 \rightarrow \mathcal{G}_1 \rightarrow \mathcal{G}_2 \rightarrow \mathcal{G}_3 \rightarrow 0 $$ of $ \mathcal{O}_Y $-modules gives an exact sequence $$ 0 \rightarrow f^* \mathcal{G}_1 \rightarrow f^* \mathcal{G}_2 \rightarrow f^* \mathcal{G}_3 \rightarrow 0 $$ of $ \mathcal{O}_X $-modules, and a commutative diagram
\begin{equation} \xymatrix{ 0 \ar[r] & f^{-1} \mathcal{G}_1 \ar[r] \ar[d] & f^{-1} \mathcal{G}_2 \ar[r] \ar[d] & f^{-1} \mathcal{G}_3 \ar[r] \ar[d] & 0 \\ 0 \ar[r] & f^* \mathcal{G}_1 \ar[r] & f^* \mathcal{G}_2 \ar[r] & f^* \mathcal{G}_2 \ar[r] & 0, } \end{equation} in $ \mathcal{A}b(X) $, where the vertical arrows are the canonical maps discussed in the proof of Lemma \ref{lemma 5.2}. Passing to cohomology, there are then induced maps $$ \psi^p : H^p(X, f^{-1} \mathcal{G}_i) \rightarrow H^p(X, f^* \mathcal{G}_i), $$ commuting with the differentials $$ H^p(X, f^{-1} \mathcal{G}_3) \rightarrow H^{p+1}(X, f^{-1} \mathcal{G}_1) $$ and $$ H^p(X, f^* \mathcal{G}_3) \rightarrow H^{p+1}(X, f^* \mathcal{G}_1), $$ for any $ p \geq 0 $ in the two long exact sequences.
Thus, if $ f : X \rightarrow Y $ is an isomorphism, and if $$ 0 \rightarrow \mathcal{G}_1 \rightarrow \mathcal{G}_2 \rightarrow \mathcal{G}_3 \rightarrow 0 $$ is a short exact sequence of $ \mathcal{O}_Y $-modules, the maps $$ H^p(f) : H^p(Y,\mathcal{G}_i) \rightarrow H^p(X, f^* \mathcal{G}_i) $$ constructed in Lemma \ref{lemma 5.2} commute with the differentials of the two long exact sequences.
\subsection{Automorphisms of schemes} Let $X$ be a scheme, let $ g : X \rightarrow X $ be a morphism, and let $ \mathcal{F} $ be a sheaf of $ \mathcal{O}_X $-modules. Assume that we are also given a homomorphism $$ u : g^* \mathcal{F} \rightarrow \mathcal{F} $$ of $ \mathcal{O}_X $-modules. By functoriality, $u$ induces a homomorphism on the cohomology groups $$ H^p(u) : H^p(X, g^* \mathcal{F}) \rightarrow H^p(X,\mathcal{F}), $$ for all $p \geq 0$.
In particular, if the map $ u : g^* \mathcal{F} \rightarrow \mathcal{F} $ is an isomorphism, then the induced map $ H^p(u) $ is also an isomorphism for all $p$.
\begin{dfn}\label{dfn 5.4} Let $ g : X \rightarrow X $ be an automorphism of a scheme $X$, $ \mathcal{F} $ a sheaf of $ \mathcal{O}_X $-modules, and $ u : g^* \mathcal{F} \rightarrow \mathcal{F} $ a homomorphism. The endomorphism $$ H^p(g,u) : H^p(X,\mathcal{F}) \rightarrow H^p(X,\mathcal{F}) $$ \emph{induced} by the couple $ (g,u) $ is defined as the composition of the maps $ H^p(g) $ and $ H^p(u) $. \end{dfn}
In case $ \mathcal{F} = \mathcal{O}_X $, there is a canonical isomorphism $ g^* \mathcal{O}_X \cong \mathcal{O}_X $, associated to the morphism $g$. So we get naturally an endomorphism of the cohomology groups
$$ H^p(g) : H^p(X,\mathcal{O}_X) \rightarrow H^p(X,\mathcal{O}_X), $$ for all $p \geq 0$.
\subsection{Short exact sequences}\label{5.4} Let $ g : X \rightarrow X $ be an automorphism, and consider a commutative diagram $$ \xymatrix{
0 \ar[r] & g^* \mathcal{F}_1 \ar[r] \ar[d]_{u_1} & g^* \mathcal{F}_2 \ar[r] \ar[d]_{u_2} & g^* \mathcal{F}_3 \ar[r] \ar[d]_{u_3} & 0 \\
0 \ar[r] & \mathcal{F}_1 \ar[r] & \mathcal{F}_2 \ar[r] & \mathcal{F}_3 \ar[r] & 0,} $$ where the horizontal lines are exact, and the maps $ u_i $ are homomorphisms of $ \mathcal{O}_X $-modules. The induced maps $ H^p(u_i) : H^p(X,g^* \mathcal{F}_i) \rightarrow H^p(X, \mathcal{F}_i) $ commute with the differentials of the induced long exact sequences in cohomology.
We may now combine the maps $ H^p(u_i) $ with the maps $ H^p(g) $ constructed earlier, and obtain a commutative diagram
\begin{equation} \xymatrix{ \ldots \ar[r] & H^p(X, \mathcal{F}_2) \ar[r] \ar[d]^{H^p(g,u_2)} & H^p(X, \mathcal{F}_3) \ar[r]^{\delta} \ar[d]^{H^p(g,u_3)} & H^{p+1}(X, \mathcal{F}_1) \ar[r] \ar[d]^{H^{p+1}(g,u_1)} & \ldots \\ \ldots \ar[r] & H^p(X, \mathcal{F}_2) \ar[r] & H^p(X, \mathcal{F}_3) \ar[r]^{\delta} & H^{p+1}(X, \mathcal{F}_1) \ar[r] & \ldots .} \end{equation}
\subsection{} If our scheme $X$ in addition is projective over a field $k$, and $ \mathcal{F} $ is a coherent sheaf of $ \mathcal{O}_X $-modules, the cohomology groups $ H^p(X,\mathcal{F}) $ are finite dimensional $k$-vector spaces. We can then define the trace $ \mathrm{Tr}(H^p(g,u)) $ of the endomorphism $ H^p(g,u) $. We will use the notation $$ \mathrm{Tr}(e(H^{\bullet}(g,u))) = \sum_{p=0}^{dim(X)} (-1)^p ~ \mathrm{Tr}(H^p(g,u)) $$ for the alternating sum of the traces.
\subsection{Witt vectors and Teichm\"uller liftings} In the case where $ p = \mathrm{char}(k) > 0 $, we let $W(k)$ denote the ring of \emph{Witt vectors} for $k$. Recall that $W(k)$ is a complete discrete valuation ring, $p$ is a uniformizing parameter in $W(k)$ and the residue field is $k$ (\cite{Serre}, Chap. II, par. 5).
Let $ FW(k) $ denote the fraction field of $W(k)$. An important feature of the Witt vectors is the fact that $ \mathrm{char}(FW(k)) = 0 $. Furthermore, there exists a unique multiplicative map $ w : k \rightarrow W(k) $ that sections the reduction map $ W(k) \rightarrow k $. The map $w$ is often referred to as the \emph{Teichm\"uller lifting} of $k$ to $ W(k) $. The existence of this map often makes it possible to lift computations from characteristic $p$ to characteristic $0$.
Since $k$ is assumed to be algebraically closed, it follows that $k$ has a full set of $n$-th roots of unity, for any $n$ not divisible by $p$. As $W(k)$ is complete, these lift uniquely to $W(k)$. Furthermore, reduction modulo $p$ induces an isomorphism of $ \boldsymbol{\mu}_n(W(k)) $ onto $ \boldsymbol{\mu}_n(k) $. Hence, if $ \lambda \in \boldsymbol{\mu}_n(k) $ is any $n$-th root of unity, it follows that $ w(\lambda) \in W(k) $ is an $n$-th root of unity, of the same order as $ \lambda $.
\subsection{Brauer characters} A few facts regarding \emph{modular characters} are needed, and are stated here in the case where $ G = \boldsymbol{\mu}_n $. We refer to \cite{SerreLin}, Chap. 18 for details. It should be mentioned that most of the constructions and properties below are actually valid for any finite group.
If $ E $ is a $ k[G] $-module, we let $ g_E $ denote the endomorphism of $ E $ induced by $ g \in G $. Since the order of $g$ divides $n$, and $n$ is relatively prime to $p$, it follows that $g_E$ is diagonalizable, and that all the eigenvalues $ \lambda_1, \ldots, \lambda_{e=\mathrm{dim} E} $ are $n$-th roots of unity. The \emph{Brauer character} is defined by assigning $$ \phi_E(g) = \sum_{i=1}^{e} w(\lambda_i), $$ where $ w(\lambda_i) $ is the Teichm\"uller lift of $ \lambda_i $ to $ W(k) $. It can be seen that the function $$ \phi_E : G \rightarrow W(k) $$ thus obtained is a class function on $G$. We shall call the element $ \phi_E(g) \in W(k) $ the \emph{Brauer trace} of $g_E$.
Two properties of the Brauer character are of particular importance to us. First, we have that $$ \overline{\phi_E(g)} = \mathrm{Tr}(g_E). $$ That is, we obtain the ordinary trace from the Brauer trace by reduction modulo $p$.
Another important property is that the Brauer character is additive on short exact sequences. That is, if $$ 0 \rightarrow E' \rightarrow E \rightarrow E'' \rightarrow 0 $$ is an exact sequence of $ k[G] $-modules, then $ \phi_E = \phi_{E'} + \phi_{E''} $. A useful consequence is that if $$ 0 \rightarrow E_0 \rightarrow \ldots \rightarrow E_i \rightarrow \ldots \rightarrow E_l \rightarrow 0 $$ is an exact sequence of $k[G]$-modules, we get that $ \sum_{i=0}^l (-1)^i \phi_{E_i}(g) = 0 $. That is, the alternating sum of the Brauer traces equals zero.
\begin{ntn} If $ V $ is a finite dimensional vector space over $k$, and $ \psi : V \rightarrow V $ is an automorphism, we will use the notation $ \mathrm{Tr}_{\beta}(\psi) $ for the Brauer trace of $\psi$. \end{ntn}
\subsection{Automorphisms of curves and invertible sheaves} We will now consider the situation where $ X $ is a smooth, connected and projective curve $C$ over an algebraically closed field $k$, and where $\mathcal{F} = \mathcal{L}$ is an invertible sheaf on $C$.
Let $ g : C \rightarrow C $ be an automorphism, and let $ u : g^*\mathcal{L} \rightarrow \mathcal{L} $ be an isomorphism, so that there are induced automorphisms $ H^p(g,u) $ for $ p = 0, 1 $. We would like to compute the alternating sum $ \sum_{p=0}^1 (-1)^p ~\mathrm{Tr}_{\beta} (H^p(g,u)) $ of the Brauer traces.
\subsection{Computing the trace when $g$ is trivial} Let us first consider the case when the automorphism $ g : C \rightarrow C $ is trivial, i.e., $ g = \mathrm{id}_C $. Then $ H^p(g) $ is the identity, so we need only consider the isomorphism $ u : \mathcal{L} = g^*(\mathcal{L}) \rightarrow \mathcal{L} $. Hence $ H^p(g,u) = H^p(u) : H^p(C,\mathcal{L}) \rightarrow H^p(C,\mathcal{L}) $, where $ u \in \mathrm{Aut}_{\mathcal{O}_C}(\mathcal{L}) $. The results we shall need for this situation are listed in Proposition \ref{prop. 5.5} below.
\begin{prop}\label{prop. 5.5} Let $ C $ be a smooth, projective and irreducible curve over an algebraically closed field $k$, and let $ \mathcal{L} $ be an invertible sheaf on $C$. Then we have that \begin{enumerate} \item $ \mathrm{Aut}_{\mathcal{O}_C}(\mathcal{L}) = k^* $. \item Let $ \lambda \in \mathrm{Aut}_{\mathcal{O}_C}(\mathcal{L}) $. Then the induced automorphism $$ H^p(\lambda) : H^p(C,\mathcal{L}) \rightarrow H^p(C,\mathcal{L}) $$ is given by multiplication by $ \lambda $ for all $ p \geq 0 $. \item The following equality holds in $W(k)$: $$ \mathrm{Tr}_{\beta}(e(H^{\bullet}(\lambda))) = w(\lambda) \cdot (\mathrm{deg}_C(\mathcal{L}) + 1 - p_a(C)). $$ \end{enumerate} \end{prop} \begin{pf} The statement in (i) is standard, and follows essentially from the fact that $ \mathcal{O}_C^*(C) = k^* $, and that any automorphism of $ \mathcal{O}_C $ is determined by $ \mathcal{O}_C^*(C) $.
Part (ii) can be seen using \v{C}ech cohomology.
Finally, (iii) follows easily, since according to (ii), all eigenvalues of $ H^p(\lambda) $ equal $ \lambda $, for each $p$. So we get that $$ \mathrm{Tr}_{\beta}(H^p(\lambda)) = w(\lambda) \cdot h^p(C,\mathcal{L}), $$ and hence, by the Riemann-Roch theorem, $$ \mathrm{Tr}_{\beta}(e(H^{\bullet}(\lambda))) = w(\lambda) \cdot (h^0(C,\mathcal{L}) - h^1(C,\mathcal{L}) ) = w(\lambda) \cdot (\mathrm{deg}_C(\mathcal{L}) + 1 - p_a(C)). $$ \end{pf}
\subsection{Computing the trace when $g$ is nontrivial} If the automorphism $ g : C \rightarrow C $ is non-trivial, things become substantially harder. In this case, we will need the \emph{Lefschetz-Riemann-Roch} formula. This formula is presented in Section \ref{LRR} below, and we also discuss the ingredients in the formula. We will follow the paper \cite{Don} by P. Donovan closely.
\subsection{The Lefschetz-Riemann-Roch formula}\label{LRR} Let $ Y $ be a smooth, projective variety over an algebraically closed field $k$. An automorphism $ y \in \mathrm{Aut}_k(Y) $ is said to be \emph{periodic} if $ y^n = \mathrm{id}_Y $ for some $ n \in \mathbb{N} $. The smallest such $n$ will be denoted the period of $y$. We shall always make the assumption that the characteristic of $k$ does not divide $n$. It is a basic fact (\cite{Don}, Lemma 4.1) that any connected component $ Z $ of the fixed point set of $y$ is non-singular. We shall say that $Z$ is a \emph{fixed} component.
Consider the set of isomorphism classes of homomorphisms $ \eta : y^* \mathcal{F} \rightarrow \mathcal{F} $, where $ \mathcal{F} $ is a coherent sheaf on $ Y $. Let $ M(y) $ denote the quotient of the free abelian group on this set, modulo certain relations that will not be written out explicitly here. In the special case where $ y = \mathrm{id}_Y $, we write $ M(Y) = M(\mathrm{id}_Y) $.
\begin{rmk} $ M(y) $ can be seen to be generated by homomorphisms $ \eta : y^* \mathcal{F} \rightarrow \mathcal{F} $, where $ \mathcal{F} $ is a locally free sheaf. Also, $ M(y) $ has the structure of a $ \mathbb{Z}[k] $-algebra in a natural way. So one may define $ M'(y) := M(y) \otimes_{\mathbb{Z}[k]} W(k) $. We refer to \cite{Don} for details. \end{rmk}
Recall from Definition \ref{dfn 5.4}, that a couple $ y, \eta $ induces endomorphisms $ H^p(y, \eta) $ on the cohomology groups $ H^p(Y,\mathcal{F}) $, for every $ p \geq 0 $. Let $ c_{!} \eta := \sum_p (-1)^p ~[H^p(y, \eta)] $ denote the alternating formal sum of these endomorphisms. The \emph{Lefschetz-Riemann-Roch} formula (\cite{Don}, Theorem 5.4, Corollary 5.5) computes the alternating sum of the Brauer traces of the endomorphisms $ H^p(y, \eta) $ as a sum of certain contributions over the fixed components of $y$.
\begin{thm}[The Lefschetz-Riemann-Roch formula, \cite{Don}, Corollary 5.5]\label{thm 5.7} Let $ (Y,y) $ be as above. If $Z$ is a fixed component of $y$, write $ i_Z : Z \rightarrow Y $ for the injection. Furthermore, write $ c_Z : Z \rightarrow \mathrm{Spec}(k) $ and $ c : Y \rightarrow \mathrm{Spec}(k) $ for the structure morphisms. Then the following equality holds in the ring $W(k)$ for $ \eta \in M(y) $ : $$ \mathrm{Tr}_{\beta}(c_{!} \eta) = \sum_Z c_{Z*}(Td(Z) \cdot ct (\lambda_Z)^{-1} \cdot ct(i_Z^{!} \eta)). $$ \end{thm}
We shall now discuss the terms appearing on the right hand side of this formula. First we need to explain the concept of the \emph{Chern trace}, as defined in \cite{Don}, Chapter 2. Let $ X/k $ be a smooth, projective and irreducible variety. Let $ \mathcal{E} $ be a locally free sheaf on $X$, and consider a homomorphism $ \psi : \mathcal{E} \rightarrow \mathcal{E} $. The characteristic equation of $ \psi $ has coefficients in $ k = \mathcal{O}_X(X) $, and hence all roots are in $k$. Let $ t $ be one of the roots, and define $$ \mathcal{E}_t := \mathrm{Ker}(\psi - t)^N, $$ for $ N > \mathrm{rk}(\mathcal{E}) $. This is a locally free subsheaf of $ \mathcal{E} $, and independent of the chosen $N$. Furthermore, $ \mathcal{E} = \oplus_t \mathcal{E}_t $, and $ \psi = \oplus_t \psi_t $.
It can now be shown that $ M(X) $ is generated by the images of the constant homomorphisms $ t : \mathcal{E} \rightarrow \mathcal{E} $, where $ t \in k $ and $ \mathcal{E} $ is locally free (\cite{Don}, Lemma 2.7). Let $K(X)$ be the Grothendieck group of locally free sheaves. Then the map $$ \alpha : M(X) \rightarrow K(X) \otimes \mathbb{Z}[k] $$ defined by $ \alpha([\psi]) = \sum_t [\mathcal{E}_t] \otimes [t] $ is a natural isomorphism (\cite{Don}, Proposition 2.8).
Recall the construction of the \emph{Chern character} $$ ch : K(X) \rightarrow A^*(X)_{\mathbb{Q}} $$ (\cite{Fulton}, p. 282). This is a homomorphism of rings, and for a line bundle $L$ on $X$ (which is the only case where we will need an explicit description), it is given by $$ ch [L] = \sum_{ i \geq 0 } (1/i!) c_1(L)^i. $$ The Chern trace $$ ct : M(X) \rightarrow A^*(X) \otimes FW(k) $$ is then defined as the composition of $ \alpha $ and the induced map $ ch \otimes w $. If $X$ is a reducible variety, then all constructions above can be done on each separate component of $X$, so the Chern trace may still be defined.
If $ Z $ is a fixed component for the periodic automorphism $ y : Y \rightarrow Y $, $ \lambda_Z $ is defined as follows: Let $ i_Z : Z \rightarrow Y $ be the inclusion. Since both $Z$ and $ Y $ are smooth, this is a regular immersion, and hence the conormal sheaf $ \mathcal{C}_{Z/Y} := \mathcal{J}/\mathcal{J}^2 $ is a locally free sheaf on $ Z $, of rank equal to the codimension $ q $ of $ Z $ in $ Y $. Furthermore, $y$ induces a canonical endomorphism $ \Phi : \mathcal{C}_{Z/Y} \rightarrow \mathcal{C}_{Z/Y} $. Hence there are induced endomorphisms $ \Lambda^t \Phi $ on $ \Lambda^t \mathcal{C}_{Z/Y} $ for every $ t \in \{ 0, \ldots, q \} $. Then one defines $$ \lambda_Z = \sum_{t=0}^q (-1)^t [\Lambda^t \Phi] \in M(Z). $$ In particular, $ ct (\lambda_Z) $ is now defined, and Lemma 4.3 in \cite{Don} asserts that this is a unit in $ A^*(Z) \otimes FW(k) $.
Let $ \eta : y^* \mathcal{F} \rightarrow \mathcal{F} $ be an element of $M(y) $. Then $ i_Z^{!} \eta $ is defined as (the class of) the restriction $$ i_Z^* \eta : i_Z^* y^* \mathcal{F} = i_Z^* \mathcal{F} \rightarrow i_Z^* \mathcal{F}, $$ and is an element of $ M(Z) $. Hence the Chern trace $ ct(i_Z^{!} \eta) \in A^*(Z) \otimes FW(k) $ is defined.
Finally, $ Td(Z) $ is the \emph{Todd class} of $Z$ in $ A^*(Z) \otimes_{\mathbb{Z}} FW(k) $ (\cite{Fulton}, p. 354).
\subsection{Formula in the case where $g$ is non-trivial} Let $C/k$ be a smooth, connected and projective curve, and let $ g \in \mathrm{Aut}_k(C) $ be a non-trivial automorphism of finite period $n$, where $n$ is not divisible by the characteristic of $k$. Furthermore, assume that a homomorphism $ u : g^* \mathcal{L} \rightarrow \mathcal{L} $ is given, where $ \mathcal{L} $ is an invertible sheaf on $C$. Denote by $ H^p(g,u) : H^p(C,\mathcal{L}) \rightarrow H^p(C,\mathcal{L}) $ the induced endomorphism, for any $ p \geq 0 $.
Theorem \ref{thm 5.7} above gives a formula for the alternating sum of the Brauer traces of the endomorphisms $ H^p(g,u) $. In order for this formula to be useful, we need to be able to compute the contributions from the fixed point locus. Proposition \ref{prop. 5.6} below gives an explicit description of these terms.
\begin{prop}\label{prop. 5.6} Let $C$, $ \mathcal{L} $, $g$ and $u$ be as above, and denote by $ C^g $ the (finite) set of fixed points of $g$. Then the following equality holds in $W(k)$: $$ \mathrm{Tr}_{\beta}(e(H^{\bullet}(g,u))) = \sum_{p=0}^1 (-1)^p ~\mathrm{Tr}_{\beta}(H^p(g,u)) = \sum_{z \in C^g} w(\lambda_u(z))/(1- w(\lambda_{dg}(z))), $$ where $ \lambda_u(z) $ is the unique eigenvalue of the fiber $ u(z) $ of the map $ u $ at $z$, and $ \lambda_{dg}(z) $ is the unique eigenvalue of the cotangent map on the fiber at $z$. \end{prop} \begin{pf} Let us first note that if $ z \in C^g $, then $ z = \mathrm{Spec}(k) $, and the structure map $ c_{ z } : \{ z \} \rightarrow \mathrm{Spec}(k) $ is the identity. Furthermore, the Chern trace $$ ct : M(\{z\}) = M(\mathrm{Spec}(k)) \rightarrow A^*(\mathrm{Spec}(k)) \otimes FW(k) $$ reduces in this case to the Brauer trace defined for endomorphisms of finite dimensional $k$-vector spaces (\cite{Don}, p. 264).
The Todd class $ Td(\{z\}) $ is just the class of a point in $ A^*(\mathrm{Spec}(k)) $ (\cite{Fulton}, Theorem 18.3 (5)), hence it is the identity element of $ A^*(\mathrm{Spec}(k)) $.
The conormal bundle $ \mathcal{C} := \mathcal{C}_{\{z\}/C} $ can be computed as $ m_z/m_z^2 $, where $ m_z \in \mathcal{O}_{C,z} $ is the maximal ideal corresponding to the point $z$. The endomorphism of $ \Lambda^0 \mathcal{C} $ is trivial, and $ dg : \Lambda^1 \mathcal{C} \rightarrow \Lambda^1 \mathcal{C} $ is just an endomorphism of $ 1 $-dimensional vectorspaces, hence determined by its unique eigenvalue $ \lambda_{dg}(z) \in k $. Let $ w(1) = 1 $ and $ w(\lambda_{dg}(z)) $ be the Teichm\"uller lifts. Then we get that $$ ct(\lambda_{\{z\}}) = 1 - w(\lambda_{dg}(z)) \in A^*(\mathrm{Spec}(k)) \otimes FW(k). $$
It remains to compute $ ct(i_z^! u) $. But $ i_z^! u $ is (the class of) the pullback $ i_z^*u $, which is $$ i_z^*u : i_z^* \mathcal{L} = i_z^* g^* \mathcal{L} \rightarrow i_z^* \mathcal{L}. $$ But this is just the fiber of $u$ at $z$, and therefore $$ ct(i_z^!u) = \mathrm{Tr}_{\beta}(u(z)) = w(\lambda_u(z)). $$ \end{pf}
\begin{rmk} The reader might want to compare Proposition \ref{prop. 5.6} with the \emph{Woods-Hole}-formula (\cite{SGA5}, Exp. III, Cor. 6.12), that gives a formula for the ordinary trace, instead of the Brauer trace. \end{rmk}
\begin{rmk} Throughout the rest of the text we will, when no confusion can arise, continue to write $ \lambda $ instead of $w(\lambda) $ for the Teichm\"uller lift of a root of unity $\lambda$. \end{rmk}
\section{Action on the minimal desingularization}\label{section 6} Recall the set-up in Section \ref{extensions and actions}. We considered an SNC-model $ \mathcal{X}/S $, and a tamely ramified extension $ S'/S $ of degree $n$ that is prime to the least common multiple of the multiplicities of the irreducible components of $ \mathcal{X}_k $. The minimal desingularization of the pullback $ \mathcal{X}_{S'}/S' $ is an SNC-model $ \mathcal{Y}/S' $, and the Galois group $ G = \boldsymbol{\mu_n} $ of the extension $ S'/S $ acts on $ \mathcal{Y} $.
Let $ g \in G $ be any element, and consider the induced automorphism $ g : \mathcal{Y} \rightarrow \mathcal{Y} $. In Corollary \ref{g^{-1}(Z) = Z}, we saw that $ g^{-1}(Z) = Z $ for any effective divisor $ 0 \leq Z \leq \mathcal{Y}_k $, and so the $G$-action restricts to $Z$. In particular, there is an induced action by $G$ on $ \mathcal{Y}_k $.
In the previous section, we saw that for each $ g \in G $, the induced automorphism $ g : \mathcal{Y}_k \rightarrow \mathcal{Y}_k $ gave an automorphism of the cohomology groups $ H^i(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $, and hence $G$ acts on $ H^i(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $ for all $ i \geq 0 $. Our ultimate goal is to compute the irreducible characters for this representation on $ H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $. To do this, we would ideally compute the Brauer trace of the automorphism of $ H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $ induced by $g$, for every group element $ g \in G $. This information would then be used to compute the Brauer character. However, we can not do this directly. Instead we will compute the Brauer trace of the automorphism induced by $g$ on the formal difference $ H^0(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) - H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $, for any $ g \in G $. In our applications, we know the character for $ H^0(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $, so this would suffice in order to compute the character for $ H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $.
The fact that $ \mathcal{Y}_k $ is not in general smooth, prevents us from using Proposition \ref{prop. 5.5} and Proposition \ref{prop. 5.6} directly. On the other hand, the irreducible components of $ \mathcal{Y}_k $ are smooth and proper curves. So we shall in fact show that it is possible to reduce to computing Brauer traces on each individual component of $ \mathcal{Y}_k $, where Proposition \ref{prop. 5.5} and Proposition \ref{prop. 5.6} do apply. The key step in obtaining this is to introduce a certain filtration of the special fiber $\mathcal{Y}_k$.
\subsection{Filtration of the special fiber} Let $ \{ C_{\alpha} \}_{\alpha \in \mathcal{A}} $ denote the set of irreducible components of $ \mathcal{Y}_k $, and let $ m_{\alpha} $ denote the multiplicity of $ C_{\alpha} $ in $ \mathcal{Y}_k $. Then $ \mathcal{Y}_k $ can be written in Weil divisor form as $$ \mathcal{Y}_k = \sum_{\alpha} m_{\alpha} C_{\alpha}. $$
\begin{dfn}\label{complete} A \emph{complete} filtration of $\mathcal{Y}_k$ is a sequence $$ 0 < Z_m < \ldots < Z_j < \ldots < Z_1 = \mathcal{Y}_k $$ of effective divisors $Z_j$ supported on $\mathcal{Y}_k$, such that for each $1 \leq j \leq m - 1 $ there exists an $ \alpha_j \in \mathcal{A} $ with $ Z_j - Z_{j+1} = C_{\alpha_j} $. So $ m = \sum_{\alpha} m_{\alpha} $. \end{dfn}
Loosely speaking, such a filtration of $\mathcal{Y}_k$ is obtained by removing the irreducible components of the special fiber one at the time (counted with multiplicity, of course).
\subsection{The steps in the filtration} At each step of a complete filtration, we can construct an exact sequence of sheaves.
\begin{lemma}\label{lemma 6.1} Let $ 0 \leq Z' < Z \leq \mathcal{Y}_k $ be divisors such that $ Z - Z' = C $, for some irreducible component $C$ of $ \mathcal{Y}_k $. Denote by $ \mathcal{I}_Z $ and $ \mathcal{I}_{Z'} $ the corresponding ideal sheaves in $ \mathcal{O}_{\mathcal{Y}} $. Let $ i_{Z} $, $ i_{Z'} $ and $ i_C $ be the canonical inclusions of $ Z $, $Z'$ and $ C $ in $\mathcal{Y}$. Furthermore, let $ \mathcal{L} = i_C^*(\mathcal{I}_{Z'}) $.
We then have an exact sequence $$ 0 \rightarrow (i_C)_* \mathcal{L} \rightarrow (i_{Z})_* \mathcal{O}_{Z} \rightarrow (i_{Z'})_* \mathcal{O}_{Z'} \rightarrow 0 $$ of $ \mathcal{O}_{\mathcal{Y}} $-modules. \end{lemma}
\begin{pf} The inclusions of the ideal sheaves $ \mathcal{I}_Z \subset \mathcal{I}_{Z'} \subset \mathcal{O}_{\mathcal{Y}} $ give an exact sequence $$ 0 \rightarrow \mathcal{K} \rightarrow \mathcal{O}_{\mathcal{Y}}/\mathcal{I}_Z \rightarrow \mathcal{O}_{\mathcal{Y}}/\mathcal{I}_{Z'} \rightarrow 0, $$ where $ \mathcal{K} $ denotes the kernel. We can identify $ \mathcal{O}_{\mathcal{Y}}/\mathcal{I}_Z = (i_{Z})_* \mathcal{O}_{Z} $ and $ \mathcal{O}_{\mathcal{Y}}/\mathcal{I}_{Z'} = (i_{Z'})_* \mathcal{O}_{Z'} $, so it remains to consider $ \mathcal{K} $.
Let $ U = \mathrm{Spec}(A) \subset \mathcal{Y} $ be an open affine set. Then $A$ is a regular domain, and the ideal sheaves $ \mathcal{I}_C $, $ \mathcal{I}_Z $ and $ \mathcal{I}_{Z'} $ restricted to $U$ correspond to invertible modules $ I_C $, $ I_{Z} $ and $ I_{Z'} $ in $\mathrm{Frac}(A)$ (and in fact in $A$). Since $ Z = Z' + C $, we have that $ I_{Z} = I_C I_{Z'} $.
Restricted to $U$, the exact sequence above is associated to the sequence $$ 0 \rightarrow I_{Z'}/I_{Z} \rightarrow A/I_{Z} \rightarrow A/I_{Z'} \rightarrow 0. $$ We can now make the identification $ I_{Z'}/I_{Z} = I_{Z'}/I_C I_{Z'} = I_{Z'} \otimes_A A/I_C $. But $ I_{Z'} \otimes_A A/I_C $ is exactly the pullback of the ideal sheaf of $Z'$ to $C$. It is easy to see that these isomorphisms glue to give the desired isomorphism of the sheaves. \end{pf}
\subsection{$G$-sheaves}\label{6.4} Let $ \mathcal{F} $ be an $ \mathcal{O}_{\mathcal{Y}} $-module. We will say that $ \mathcal{F} $ is a $G$-sheaf if there are isomorphisms $ g^*\mathcal{F} \rightarrow \mathcal{F} $, for every $g \in G$, satisfying certain natural compatibility properties (cf. \cite{Kock}, Chapter 1).
Let $ \mathcal{F} $ and $ \mathcal{G} $ be $G$-sheaves. A homomorphism of $G$-sheaves $ \mathcal{F} \rightarrow \mathcal{G} $ is an $ \mathcal{O}_{\mathcal{Y}} $-homomorphism that commutes with the $G$-sheaf structures of $ \mathcal{F} $ and $ \mathcal{G} $ respectively.
\begin{lemma}\label{G-sheaf} Let $ 0 \leq Z \leq \mathcal{Y}_k $ be an effective divisor, with ideal sheaf $ \mathcal{I}_Z $. Then we have that $ \mathcal{I}_Z $ is a $G$-sheaf. \end{lemma} \begin{pf} Let $ g \in G $ be any group element, and consider the corresponding automorphism $ g : \mathcal{Y} \rightarrow \mathcal{Y} $. Applying the exact functor $ g^{-1} $ to the inclusion $ \mathcal{I}_Z \subset \mathcal{O}_{\mathcal{Y}} $ gives an inclusion \begin{equation}\label{equation 6.1} g^{-1} \mathcal{I}_Z \subset g^{-1} \mathcal{O}_{\mathcal{Y}}. \end{equation} Composing this inclusion with the canonical map $ g^{\sharp} : g^{-1} \mathcal{O}_{\mathcal{Y}} \rightarrow \mathcal{O}_{\mathcal{Y}} $, we obtain a map $$ g^{-1} \mathcal{I}_Z \rightarrow \mathcal{O}_{\mathcal{Y}}. $$ Now, let $ \mathcal{J} $ be the sheaf of ideals generated by the image of $ g^{-1} \mathcal{I}_Z $ in $ \mathcal{O}_{\mathcal{Y}} $. We have that $ \mathcal{J} $ is the ideal sheaf of $ g^{-1}(Z) $. But in our case $ g^{-1}(Z) = Z $, and therefore $ \mathcal{J} = \mathcal{I}_Z $.
The inclusion in \ref{equation 6.1} above induces an injective map $$ g^*\mathcal{I}_Z \rightarrow g^* \mathcal{O}_{\mathcal{Y}} \cong \mathcal{O}_{\mathcal{Y}} $$ of $ \mathcal{O}_{\mathcal{Y}} $-modules, whose image is $ \mathcal{I}_Z = g^{-1} \mathcal{I}_Z \cdot \mathcal{O}_{\mathcal{Y}} $ (\cite{Hart}, II.7.12.2). Hence we obtain a homomorphism \begin{equation}\label{equation 6.2} u_Z : g^*\mathcal{I}_Z \rightarrow \mathcal{I}_Z \end{equation} of $ \mathcal{O}_{\mathcal{Y}} $-modules, which is necessarily an isomorphism.
It remains to check that the maps $ g^*\mathcal{I}_Z \rightarrow \mathcal{I}_Z $ for various elements $ g \in G $ satisfy the compatibility conditions, but this is immediate since the maps are derived from the action on the surface. \end{pf}
\begin{rmk} The isomorphism $ u_Z : g^* \mathcal{I}_Z \rightarrow \mathcal{I}_Z $ constructed in the proof of Lemma \ref{G-sheaf} may be specified on stalks. Indeed, for any $ y \in \mathcal{Y} $, we have the map $$ g^{\sharp}_y : (g^{-1} \mathcal{O}_{\mathcal{Y}})_y \cong \mathcal{O}_{\mathcal{Y},g(y)} \rightarrow \mathcal{O}_{\mathcal{Y},y}, $$ which is an isomorphism of rings, and the image of $ \mathcal{I}_{Z,g(y)} \subset \mathcal{O}_{\mathcal{Y},g(y)} $ under this map is precisely $ \mathcal{I}_{Z,y} $. The inclusion $ \mathcal{I}_{Z,g(y)} \subset \mathcal{O}_{\mathcal{Y},g(y)} $ gives, after tensoring with $ \mathcal{O}_{\mathcal{Y},y} $ via $ g^{\sharp}_y $, an injective map $$ \mathcal{I}_{Z,g(y)} \otimes_{\mathcal{O}_{\mathcal{Y},g(y)}} \mathcal{O}_{\mathcal{Y},y} \rightarrow \mathcal{O}_{\mathcal{Y},g(y)} \otimes_{\mathcal{O}_{\mathcal{Y},g(y)}} \mathcal{O}_{\mathcal{Y},y} \cong \mathcal{O}_{\mathcal{Y},y}. $$ Let $ f \in \mathcal{I}_{Z,g(y)} $ be a generator. Then $ f \otimes 1 $ generates $ \mathcal{I}_{Z,g(y)} \otimes_{\mathcal{O}_{\mathcal{Y},g(y)}} \mathcal{O}_{\mathcal{Y},y} $ as an $ \mathcal{O}_{\mathcal{Y},y} $-module, and the image of this element under the map above is $ g^{\sharp}_y(f) $, which generates $ \mathcal{I}_{Z,y} $. It follows that $ u_{Z,y}(f \otimes 1) = g^{\sharp}_y(f) $. \end{rmk}
\subsection{Exact sequence}\label{6.6} The construction above gives, for every $ g \in G $, a commutative diagram $$ \xymatrix{ g^* \mathcal{I}_Z \ar[d]_{u_{Z}} \ar@{^{(}->}[r] & g^* \mathcal{O}_{\mathcal{Y}} \ar[d]^{\cong}\\ \mathcal{I}_{Z} \ar@{^{(}->}[r] & \mathcal{O}_{\mathcal{Y}}. } $$ This induces a commutative diagram $$ \xymatrix{ 0 \ar[r] & g^* \mathcal{I}_Z \ar[r] \ar[d]_{u_{Z}} & g^* \mathcal{O}_{\mathcal{Y}} \ar[r] \ar[d]_{\cong} & g^*(i_Z)_* \mathcal{O}_{Z} \ar[r] \ar[d]_{v_Z} & 0\\ 0 \ar[r] & \mathcal{I}_Z \ar[r] & \mathcal{O}_{\mathcal{Y}} \ar[r] & (i_Z)_* \mathcal{O}_{Z} \ar[r] & 0,} $$ where the horizontal sequences are exact. Here we have identified $ (i_Z)_* \mathcal{O}_{Z} \cong \mathcal{O}_{\mathcal{Y}}/\mathcal{I}_Z $, and $ v_Z $ denotes the induced isomorphism on the cokernel.
\subsection{Relative version}\label{6.7} \begin{prop}\label{prop. 6.5} Let us keep the hypotheses and notation from Lemma \ref{lemma 6.1}. The sequence $$ 0 \rightarrow (i_C)_* \mathcal{L} \rightarrow (i_{Z})_* \mathcal{O}_{Z} \rightarrow (i_{Z'})_* \mathcal{O}_{Z'} \rightarrow 0 $$ is an exact sequence of $ G $-sheaves. \end{prop} \begin{pf} Let $ \mathcal{I}_{Z} \subset \mathcal{I}_{Z'} \subset \mathcal{O}_{\mathcal{Y}} $ be the inclusions of the ideal sheaves. From Lemma \ref{G-sheaf}, it follows that these maps are maps of $G$-sheaves. The result now follows from the fact that the category of $G$-modules on $\mathcal{Y}$ is an abelian category (\cite{Kock}, Lemma 1.3). \end{pf}
\begin{rmk} We would like to point out the following fact, that will be useful later: Let $ u_{Z'/Z} : g^*(\mathcal{I}_{Z'}/\mathcal{I}_{Z}) \rightarrow \mathcal{I}_{Z'}/\mathcal{I}_{Z} $ be the map that is the kernel of the diagram $$ \xymatrix{ g^*(\mathcal{O}_{\mathcal{Y}}/\mathcal{I}_{Z}) \ar[r] \ar[d]_{v_Z} & g^*(\mathcal{O}_{\mathcal{Y}}/\mathcal{I}_{Z'}) \ar[d]^{v_{Z'}} \\ \mathcal{O}_{\mathcal{Y}}/\mathcal{I}_{Z} \ar[r] & \mathcal{O}_{\mathcal{Y}}/\mathcal{I}_{Z'}.} $$ Then $ u_{Z'/Z} $ can also be described as the \emph{cokernel} of the diagram $$ \xymatrix{ g^*\mathcal{I}_{Z} \ar[r] \ar[d]_{u_Z} & g^*\mathcal{I}_{Z'} \ar[d]^{u_{Z'}}\\ \mathcal{I}_{Z} \ar[r] & \mathcal{I}_{Z'}.} $$ \end{rmk}
\subsection{}\label{6.8} Proposition \ref{prop. 6.5} implies that for any $g \in G$, we have a commutative diagram \begin{equation}\label{equation 6.3}
\xymatrix{ 0 \ar[r] & g^* (i_C)_* \mathcal{L} \ar[d]_{u_{Z'/Z}} \ar[r] & g^* (i_Z)_* \mathcal{O}_{Z} \ar[r] \ar[d]_{v_Z} & g^* (i_{Z'})_* \mathcal{O}_{Z'} \ar[r] \ar[d]_{v_{Z'}} & 0\\ 0 \ar[r] & (i_C)_* \mathcal{L} \ar[r] & (i_Z)_* \mathcal{O}_{Z} \ar[r] & (i_{Z'})_* \mathcal{O}_{Z'} \ar[r] & 0,} \end{equation} where the horizontal lines are exact sequences. To avoid unnecessarily complicated notation, we will write $ u $, $ v $ and $ v' $ for the vertical maps above, if no confusion can arise.
Consider now the long exact sequence in cohomology induced by the lower exact sequence in Diagram \ref{equation 6.3} \begin{equation} 0 \rightarrow H^0(\mathcal{Y},(i_C)_* \mathcal{L}) \rightarrow H^0(\mathcal{Y},(i_Z)_* \mathcal{O}_{Z}) \rightarrow H^0(\mathcal{Y},(i_{Z'})_* \mathcal{O}_{Z'}) \rightarrow \ldots \end{equation} $$ \ldots \rightarrow H^p(\mathcal{Y},(i_C)_* \mathcal{L})\rightarrow H^p(\mathcal{Y}, (i_Z)_* \mathcal{O}_{Z}) \rightarrow H^p(\mathcal{Y}, (i_{Z'})_* \mathcal{O}_{Z'}) \rightarrow \ldots. $$
The maps $ u $, $ v $ and $ v' $, together with the automorphism $ g : \mathcal{Y} \rightarrow \mathcal{Y} $ induce, for every $ p \geq 0 $, automorphisms $ H^p(g,u) $, $ H^p(g,v) $ and $ H^p(g,v') $ that commute with the differentials in the long exact sequence. That is, we obtain, for every $ g \in G $, an \emph{automorphism} of this long exact sequence.
Note that all the sheaves appearing in the exact sequence above are supported on the special fiber of $ \mathcal{Y} $. We will now explain how we can ``restrict'' the endomorphisms $ H^p(g,u) $, $ H^p(g,v) $ and $ H^p(g,v') $ to the support of the various sheaves.
\subsection{Restriction}\label{6.9} Let $X$ and $Y$ be schemes, and let $ i : X \rightarrow Y $ be a closed immersion. For any (quasi-coherent) $ \mathcal{O}_X $-module $ \mathcal{F} $, the canonical adjunction map $ \alpha_{\mathcal{F}} : i^* i_* \mathcal{F} \rightarrow \mathcal{F} $ is an isomorphism, since $ i $ is a closed immersion. Let $ \mathcal{G} $ be a quasi-coherent $ \mathcal{O}_Y $-module. The canonical adjunction map $ \beta_{\mathcal{G}} : \mathcal{G} \rightarrow i_* i^* \mathcal{G} $ is not necessarily an isomorphism. However, in the case where $ \mathcal{G} $ has support on $X$, $ \beta_{\mathcal{G}} $ is indeed an isomorphism.
Recall also that $ i^* $ and $ i_* $ are adjoint functors, the maps $ \alpha $ and $ \beta $ induce a natural bijection $$ \mathrm{Hom}_{\mathcal{O}_X}(i^* \mathcal{G}, \mathcal{F}) \cong \mathrm{Hom}_{\mathcal{O}_Y}(\mathcal{G}, i_* \mathcal{F}), $$ for any $ \mathcal{O}_X $-module $ \mathcal{F} $ and any $ \mathcal{O}_Y $-module $ \mathcal{G} $ on $Y$.
Consider now the case where an automorphism $ g : Y \rightarrow Y $ is given, that restricts to an automorphism $ f = g|_X : X \rightarrow X $. Hence there is a commutative diagram \begin{equation}\label{diagram 6.1} \xymatrix{ X \ar[r]^i \ar[d]_f & Y \ar[d]^g \\
X \ar[r]^i & Y. } \end{equation}
We will now apply these considerations to cohomology. Let us first note that if $ \mathcal{F} $ is a quasi-coherent sheaf on $X$, then the push-forward $ i_*(\mathcal{F}) $ is a quasi-coherent sheaf on $ Y $, since $i$ is a closed immersion.
\begin{prop}\label{prop. 6.6} Let $ \mathcal{F} $ be a quasi-coherent sheaf on $X$, and let $ u : g^* i_* \mathcal{F} \rightarrow i_* \mathcal{F} $ be a homomorphism of $ \mathcal{O}_Y $-modules. Then there is induced, for every $ p \geq 0 $, a commutative diagram $$ \xymatrix{ H^p(Y, i_* \mathcal{F}) \ar[rr]^{H^p(g,u)} \ar[d]_{H^p(i)} & & H^p(Y, i_* \mathcal{F}) \ar[d]^{H^p(i)} \\ H^p(X, i^* i_*\mathcal{F}) \ar[rr]^{H^p(f,i^*u)} & & H^p(X, i^* i_*\mathcal{F}),} $$ where the vertical arrows are isomorphisms. \end{prop} \begin{pf} Since $ g \circ i = i \circ f $, we may identify the induced homomorphisms \begin{equation}\label{equation 6.7} H^p(Y, i_*\mathcal{F}) \rightarrow H^p(X, (g \circ i)^* i_* \mathcal{F}) \end{equation} and \begin{equation}\label{equation 6.8} H^p(Y, i_*\mathcal{F}) \rightarrow H^p(X, (i \circ f)^* i_* \mathcal{F}). \end{equation} However, homomorphism \ref{equation 6.7} factors as $$ H^p(Y, i_*\mathcal{F}) \rightarrow H^p(Y, g^* i_*\mathcal{F}) \rightarrow H^p(X, i^* g^* i_* \mathcal{F}) = H^p(X, f^* i^* i_* \mathcal{F}), $$ and homomorphism \ref{equation 6.8} factors as $$H^p(Y, i_*\mathcal{F}) \rightarrow H^p(X, i^* i_* \mathcal{F}) \rightarrow H^p(X, f^* i^* i_* \mathcal{F}) $$ (\cite{EGAIII}, Chap. 0, 12.1.3.5). Consequently, there is a commutative diagram \begin{equation} \xymatrix{ H^p(Y, i_*\mathcal{F}) \ar[r]^{H^p(g)} \ar[d]_{H^p(i)} & H^p(Y, g^* i_*\mathcal{F}) \ar[d]^{H^p(i)} \\ H^p(X, i^* i_* \mathcal{F}) \ar[r]^{H^p(f)} & H^p(X, f^* i^* i_* \mathcal{F}). } \end{equation}
If we pull back the map $ u : g^* i_* \mathcal{F} \rightarrow i_* \mathcal{F} $ with $ i $, we get the map $$ i^*(u) : i^* g^* i_* \mathcal{F} = f^* i^* i_* \mathcal{F} \rightarrow i^* i_* \mathcal{F}. $$ By functoriality, we obtain a commutative diagram
\begin{equation} \xymatrix{ H^p(Y, g^* i_*\mathcal{F}) \ar[r]^{H^p(u)} \ar[d]_{H^p(i)} & H^p(Y, i_*\mathcal{F}) \ar[d]^{H^p(i)} \\ H^p(X, f^* i^* i_* \mathcal{F}) \ar[r]^{H^p(i^*u)} & H^p(X, i^* i_* \mathcal{F}). } \end{equation}
To complete the proof, we need to check that the maps $ H^p(i) $ are isomorphisms. In order to see this, we first note that the adjunction map $ \alpha_{\mathcal{G}} : \mathcal{G} \rightarrow i_* i^* \mathcal{G} $, where we put $ \mathcal{G} := i_* \mathcal{F} $, is an isomorphism. For any open $ V \subset Y $, there is induced a map $$ \Gamma(V, \mathcal{G}) \rightarrow \Gamma(V, i_* i^* \mathcal{G}) = \Gamma(i^{-1}(V), i^* \mathcal{G}), $$ that is also an isomorphism. Choose now an open affine cover $ \mathcal{V} = (V_j)_{j \in J} $ of $Y$. As $i$ is a closed immersion, the inverse image $ \mathcal{U} := i^{-1}(\mathcal{V}) $ is an open affine cover of $X$. The map on sections above induces a map of Cech-complexes $$ C^{\bullet}(\mathcal{V},\mathcal{G}) \rightarrow C^{\bullet}(\mathcal{U},i^*\mathcal{G}), $$ that is also an isomorphism. Passing on to cohomology, we therefore get isomorphisms $$ H^p(\mathcal{V},\mathcal{G}) \rightarrow H^p(\mathcal{U},i^* \mathcal{G}), $$ for every $ p \geq 0 $.
The natural maps $$ H^p(\mathcal{V},\mathcal{G}) \rightarrow H^p(Y,\mathcal{G}) $$ and $$ H^p(\mathcal{U},i^* \mathcal{G}) \rightarrow H^p(X,i^* \mathcal{G}) $$ are isomorphisms, by \cite{Hart}, Theorem III 4.5. By \cite{EGAIII}, Chap. 0, 12.1.4.2, the diagram $$ \xymatrix{ H^p(\mathcal{V},\mathcal{G}) \ar[r] \ar[d] & H^p(\mathcal{U},i^* \mathcal{G}) \ar[d] \\
H^p(Y,\mathcal{G}) \ar[r] & H^p(X,i^* \mathcal{G})} $$ commutes. Therefore, we can conclude that $$ H^p(Y,\mathcal{G}) \rightarrow H^p(X,i^* \mathcal{G}) $$ is an isomorphism. \end{pf}
\subsection{}\label{6.11}
We are now going to apply the technical results in Section \ref{6.9} to the setup in Section \ref{6.8}. To begin with, we note that the closed immersions $ i_C $, $ i_Z $ and $ i_{Z'} $ induce isomorphisms $$ H^p(\mathcal{Y}, (i_C)_* \mathcal{L}) \cong H^p(C,\mathcal{L}), $$ $$ H^p(\mathcal{Y}, (i_Z)_* \mathcal{O}_Z) \cong H^p(Z, \mathcal{O}_Z) $$ and $$ H^p(\mathcal{Y}, (i_{Z'})_* \mathcal{O}_{Z'}) \cong H^p(Z', \mathcal{O}_{Z'}), $$ for all $ p \geq 0 $. Here we have identified $ \mathcal{L} $ with $ (i_C)^* (i_C)_* \mathcal{L} $, and likewise for $ \mathcal{O}_Z $ and $ \mathcal{O}_{Z'} $.
Since $ C $, $ Z $ and $ Z' $ are projective curves over $k$, and since $ \mathcal{L} $, $ \mathcal{O}_Z $ and $ \mathcal{O}_{Z'} $ are coherent sheaves on the respective curves, the cohomology groups above are finite dimensional $k$-vector spaces, and nonzero only for $ p = 0 $ and $ p = 1 $. So the long exact sequence in cohomology associated to the short exact sequence in Section \ref{6.8} is simply \begin{equation}\label{sequence 6.7} 0 \rightarrow H^0(C,\mathcal{L}) \rightarrow H^0(Z, \mathcal{O}_Z) \rightarrow H^0(Z', \mathcal{O}_{Z'}) \rightarrow \end{equation} $$ H^1(C,\mathcal{L}) \rightarrow H^1(Z, \mathcal{O}_Z) \rightarrow H^1(Z', \mathcal{O}_{Z'}) \rightarrow 0. $$
Denote by $g_C$ the restriction of $g$ to $C$. By Proposition \ref{prop. 6.6}, restriction to $C$ gives a commutative diagram $$ \xymatrix{ H^p(\mathcal{Y},(i_C)_* \mathcal{L}) \ar[d]_{H^p(g,u)} \ar[rr]^{\cong} & & H^p(C,\mathcal{L}) \ar[d]^{H^p(g_C, i_C^*u)} \\ H^p(\mathcal{Y},(i_C)_* \mathcal{L}) \ar[rr]^{\cong} & & H^p(C,\mathcal{L}). } $$ Also, we get similar diagrams for $ H^p(g_Z, i_Z^* v) $ and $ H^p(g_{Z'}, i_{Z'}^* v') $.
Having made these identifications, we see that the automorphisms $ H^p(g_C, i_C^* u)$, $ H^p(g_Z, i_Z^* v) $ and $ H^p(g_{Z'}, i_{Z'}^* v') $, for $ p = 0, 1 $, fit together to give an automorphism of Sequence \ref{sequence 6.7} above. In particular, \ref{sequence 6.7} is an exact sequence of $ k[G] $-modules.
\begin{lemma}\label{lemma 6.7} Let us keep the notation and constructions from the discussion above. The following equality holds in $W(k)$: $$ \sum_{p=0}^1 (-1)^p ~\mathrm{Tr}_{\beta} (H^p(g_Z, i_Z^* v)) = $$ $$ \sum_{p=0}^1 (-1)^p ~\mathrm{Tr}_{\beta} (H^p(g_C,i_C^* u)) + \sum_{p=0}^1 (-1)^p ~\mathrm{Tr}_{\beta} (H^p(g_{Z'}, i_{Z'}^* v')). $$ \end{lemma} \begin{pf} The long exact sequence \ref{sequence 6.7} is an exact sequence of $k[G]$-modules. Therefore, for any $ g \in G $, the alternating sum of the Brauer traces equals zero. \end{pf}
\subsection{} We will now generalize the formula in Lemma \ref{lemma 6.7}. Let us write $ \mathcal{Y}_k = \sum_{\alpha} m_{\alpha} C_{\alpha} $, where $ \alpha \in \mathcal{A} $, and put $ m = \sum_{\alpha} m_{\alpha} $. Fix a complete filtration $$ 0 < Z_m < \ldots < Z_j < \ldots < Z_2 < Z_1 = \mathcal{Y}_k, $$ where $ Z_j - Z_{j+1} = C_j $ for some $ C_j \in \{ C_{\alpha} \}_{\alpha \in \mathcal{A} } $, for each $ j \in \{ 1, \ldots, m-1 \} $. At each step of this filtration, Lemma \ref{lemma 6.1} asserts that there is a short exact sequence $$ 0 \rightarrow (i_{C_j})_* \mathcal{L}_j \rightarrow (i_{Z_j})_* \mathcal{O}_{Z_j} \rightarrow (i_{Z_{j+1}})_* \mathcal{O}_{Z_{j+1}} \rightarrow 0. $$ Note in particular that $ Z_m = C_m $, for some $ C_m \in \{ C_{\alpha} \}_{\alpha \in \mathcal{A} } $. So we write $ \mathcal{O}_{Z_m} = \mathcal{L}_m $, for the sake of coherence.
For every $ j \in \{ 1, \ldots , m \} $, we have isomorphisms $$ u_j : g^* (i_{C_j})_* \mathcal{L}_j \rightarrow (i_{C_j})_* \mathcal{L}_j, $$ and $$ v_j : g^* (i_{Z_j})_* \mathcal{O}_{Z_j} \rightarrow (i_{Z_j})_* \mathcal{O}_{Z_j}. $$
\begin{prop}\label{prop. 6.6} The alternating sum of the Brauer traces of the automorphisms induced by $ g \in G $ on $ H^p(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $ can be computed by the formula $$ \sum_{p=0}^1 (-1)^p ~\mathrm{Tr}_{\beta} (H^p(g_{\mathcal{Y}_k})) = \sum_{j=1}^m (\sum_{p=0}^1 (-1)^p ~\mathrm{Tr}_{\beta} (H^p(g_{C_j}, i_{C_j}^* u_j))). $$ \end{prop} \begin{pf} Let us first note that $ (i_{Z_j})^*v_j $ can be identified with the canonical isomorphism $ g_{Z_j}^* \mathcal{O}_{Z_j} \cong \mathcal{O}_{Z_j} $, for every $j$. So the $v_j$ will be dropped from the notation.
For $ j = 1 $, we have that $ Z_1 = \mathcal{Y}_k $, and so Lemma \ref{lemma 6.7} gives that $$ \sum_{p=0}^1 (-1)^p ~\mathrm{Tr}_{\beta} (H^p(g_{\mathcal{Y}_k})) = $$ $$ \sum_{p=0}^1 (-1)^p ~\mathrm{Tr}_{\beta} (H^p(g_{C_1}, i_{C_1}^* u_1)) + \sum_{p=0}^1 (-1)^p \mathrm{Tr}_{\beta} ~(H^p(g_{Z_2})). $$
However, for any $ j \in \{ 1, \ldots , m-1 \} $, Lemma \ref{lemma 6.7} again gives that $$ \sum_{p=0}^1 (-1)^p ~\mathrm{Tr}_{\beta} (H^p(g_{Z_j})) = $$ $$ \sum_{p=0}^1 (-1)^p ~\mathrm{Tr}_{\beta} (H^p(g_{C_j}, i_{C_j}^* u_j)) + \sum_{p=0}^1 (-1)^p ~\mathrm{Tr}_{\beta} (H^p(g_{Z_{j+1}})). $$ So the result follows by induction. \end{pf}
\begin{rmk} The importance of Proposition \ref{prop. 6.6} is that it reduces the problem of computing $$ \sum_{p=0}^1 (-1)^p ~\mathrm{Tr}_{\beta} (H^p(g_{\mathcal{Y}_k})) $$ to instead computing the contributions $$ \sum_{p=0}^1 (-1)^p ~\mathrm{Tr}_{\beta} (H^p(g_{C_j}, i_{C_j}^* u_j)), $$ for certain invertible sheaves $ \mathcal{L}_j $, supported on the smooth irreducible components $ C_j $ of $ \mathcal{Y}_k $. For such computations, we may apply Proposition \ref{prop. 5.5} and Proposition \ref{prop. 5.6}. In order to apply these formulas, we need to understand the action on each irreducible component $C_j$ of $ \mathcal{Y}_k $. Furthermore, we need to understand the action on the fiber of the invertible sheaf $ \mathcal{L}_j $ at any fixed point on $C_j$. \end{rmk}
\subsection{Fibers at fixed points and cotangent spaces}\label{7.1} Let $ g \in G $, and consider the corresponding automorphism $$ g : \mathcal{Y} \rightarrow \mathcal{Y}. $$ Let $ y \in \mathcal{Y}_k $ be an intersection point of two irreducible components $ C $ and $ C' $ of $ \mathcal{Y}_k $. Then $ y $ is a fixed point for $ g $, so there is an induced automorphism $$ g^{\sharp}_y : \mathcal{O}_{\mathcal{Y},y} \rightarrow \mathcal{O}_{\mathcal{Y},y}. $$ As $ g^{\sharp}_y $ is a local homomorphism, it induces an automorphism of the cotangent space at $y$: $$ dg(y) : m_y/m_y^2 \rightarrow m_y/m_y^2. $$
Let $ f $ and $ f' $ be local equations for $ C $ and $ C' $ at $y$. That is, we have $ \mathcal{I}_{C,y} = (f) \subset \mathcal{O}_{\mathcal{Y},y} $ and similarly $ \mathcal{I}_{C',y} = (f') \subset \mathcal{O}_{\mathcal{Y},y} $. Since $ \mathcal{Y} $ has strict normal crossings, it follows that $ m_y = \mathcal{I}_{C,y} + \mathcal{I}_{C',y} = (f, f') $. In particular, the images of $f$ and $f'$ form a basis for $ m_y/m_y^2 $.
Since $C$ and $C'$ are stable under the $ G $-action, we have that $ \mathcal{I}_{C,y} $ and $ \mathcal{I}_{C',y} $ map to themselves via $ g^{\sharp}_y $. Hence, the images of $f$ and $f'$ are \emph{eigenvectors} for $ dg(y) $. Let $ \lambda $ and $ \lambda' $ be the corresponding eigenvalues.
\subsection{}\label{section 7.2}
Consider the inclusion $ (f) = \mathcal{I}_{C,y} \subset m_y $. By tensoring with $ \mathcal{O}_{\mathcal{Y},y}/m_y $, we get a map $$ \mathcal{I}_{C}(y) = \mathcal{I}_{C,y} \otimes_{\mathcal{O}_{\mathcal{Y},y}} \mathcal{O}_{\mathcal{Y},y}/m_y \rightarrow m_y \otimes_{\mathcal{O}_{\mathcal{Y},y}} \mathcal{O}_{\mathcal{Y},y}/m_y = m_y/m_y^2, $$ mapping $ f \otimes 1 $ to the image of $f$ in $ m_y/m_y^2 $. This is a non-degenerate map of $ k $-vector spaces, so we may identify $$ \mathcal{I}_{C}(y) = <f> \subset m_y/m_y^2. $$ Similarly, we may identify $$ \mathcal{I}_{C'}(y) = <f'> \subset m_y/m_y^2. $$
\begin{lemma}\label{Lemma 7.1} Let $ u_C : g^* \mathcal{I}_C \rightarrow \mathcal{I}_C $ be the map constructed in Lemma \ref{G-sheaf}. Then the eigenvalue of the induced map $ u_C(y) : (g^* \mathcal{I}_C)(y) \rightarrow \mathcal{I}_C(y) $ at the fiber in the fixed point $y$ is precisely the eigenvalue $\lambda$ of $f$ under the cotangent map $dg(y)$ at $y$.
Similarly, the eigenvalue of the map $ u_{C'}(y) : (g^* \mathcal{I}_{C'})(y) \rightarrow \mathcal{I}_{C'}(y) $ at the fiber in $y$ is precisely the eigenvalue $ \lambda'$ of $f'$ under $dg(y)$. \end{lemma} \begin{pf} Let $ g^{\sharp}_y : \mathcal{O}_{\mathcal{Y},y} \rightarrow \mathcal{O}_{\mathcal{Y},y} $ be the map of local rings induced from $ g : \mathcal{Y} \rightarrow \mathcal{Y} $. We can describe the stalk of $ u_C $ at $y$ in the following way: Tensoring the inclusion $ \mathcal{I}_{C,y} \subset \mathcal{O}_{\mathcal{Y},y} $ with $ g^{\sharp}_y $ gives $$ \mathcal{I}_{C,y} \otimes^g \mathcal{O}_{\mathcal{Y},y} \rightarrow \mathcal{O}_{\mathcal{Y},y} \otimes^g \mathcal{O}_{\mathcal{Y},y}, $$ where $ M \otimes^g \mathcal{O}_{\mathcal{Y},y} $ denotes, for any $ \mathcal{O}_{\mathcal{Y},y} $-module $M$, tensorization with $ \mathcal{O}_{\mathcal{Y},y} $ via the homomorphism $ g^{\sharp}_y $.
By composing with the canonical isomorphism $ \mathcal{O}_{\mathcal{Y},y} \otimes^g \mathcal{O}_{\mathcal{Y},y} \cong \mathcal{O}_{\mathcal{Y},y} $, we get an injective map $$ \mathcal{I}_{C,y} \otimes^g \mathcal{O}_{\mathcal{Y},y} \rightarrow \mathcal{O}_{\mathcal{Y},y} $$ with image $ \mathcal{I}_{C,y} $. The induced map $$ (g^* \mathcal{I}_C)_y = \mathcal{I}_{C,y} \otimes^g \mathcal{O}_{\mathcal{Y},y} \rightarrow \mathcal{I}_{C,y} $$ is the stalk $ (u_C)_y $. Let $ a \in \mathcal{I}_{C,y} $ be any element. Then we have that $$ (u_C)_y(a \otimes 1) = g^{\sharp}_y(a) \in \mathcal{I}_{C,y}. $$
We will now consider the fiber $ (u_C)(y) $ of $ (u_C)_y $ at $y$. Let us first point out that $ g^{\sharp}_y $ induces the identity on the residue field of $y$. That is, we have a commutative diagram $$ \xymatrix{ \mathcal{O}_{\mathcal{Y},y} \ar[r]^{g^{\sharp}_y} \ar[d] & \mathcal{O}_{\mathcal{Y},y} \ar[d] \\ \mathcal{O}_{\mathcal{Y},y}/m_y \ar[r]^{=} & \mathcal{O}_{\mathcal{Y},y}/m_y.} $$ It therefore follows that $$ (g^* \mathcal{I}_{C})(y) = (g^* \mathcal{I}_C)_y \otimes_{\mathcal{O}_{\mathcal{Y},y}} \mathcal{O}_{\mathcal{Y},y}/m_y = $$ $$ \mathcal{I}_{C,y} \otimes^g \mathcal{O}_{\mathcal{Y},y} \otimes_{\mathcal{O}_{\mathcal{Y},y}} \mathcal{O}_{\mathcal{Y},y}/m_y = \mathcal{I}_{C,y} \otimes_{\mathcal{O}_{\mathcal{Y},y}} \mathcal{O}_{\mathcal{Y},y}/m_y = \mathcal{I}_{C}(y). $$ With this identification, we see that the map $ (u_C)(y) $ on the fiber is $$ (u_C)(y) : \mathcal{I}_{C,y} \otimes_{\mathcal{O}_{\mathcal{Y},y}} \mathcal{O}_{\mathcal{Y},y}/m_y \rightarrow \mathcal{I}_{C,y} \otimes_{\mathcal{O}_{\mathcal{Y},y}} \mathcal{O}_{\mathcal{Y},y}/m_y, $$ where, for any $ a \in \mathcal{I}_{C,y} $, we have that $ (u_C)(y)(a \otimes 1) = g^{\sharp}_y(a) \otimes 1 $.
If we replace $ \mathcal{I}_{C,y} $ with the maximal ideal, we can argue in exactly the same way, and get a map $$ m_y \otimes_{\mathcal{O}_{\mathcal{Y},y}} \mathcal{O}_{\mathcal{Y},y}/m_y \rightarrow m_y \otimes_{\mathcal{O}_{\mathcal{Y},y}} \mathcal{O}_{\mathcal{Y},y}/m_y, $$ given by $ a \otimes 1 \mapsto g^{\sharp}_y(a) \otimes 1 $, for any $ a \in m_y $. Hence this is precisely the cotangent map $ dg(y) : m_y/m_y^2 \rightarrow m_y/m_y^2 $.
The inclusion $ \mathcal{I}_{C,y} \subset m_y $ now gives a commutative diagram
$$ \xymatrix{ \mathcal{I}_C(y) \ar[d]_{u_{C}(y)} \ar[r] & m_y/m_y^2 \ar[d]^{dg(y)}\\ \mathcal{I}_{C}(y) \ar[r] & m_y/m_y^2. } $$ Using the identifications above, we see that the eigenvalue of $ u_{C}(y) $ is precisely the eigenvalue $ \lambda $ of $f$ under the cotangent map $ dg(y) $.
In a similar way, we can show that the eigenvalue of $ u_{C'}(y) $ is precisely the eigenvalue $ \lambda' $ of $f'$ under the cotangent map $ dg(y) $. \end{pf}
\subsection{} Let $ 0 \leq Z \leq \mathcal{Y}_k $ be an effective divisor on the form $$ Z = a C + a' C' + Z_0, $$ where the effective divisor $ Z_0 $ does not contain $ C $ or $C'$. Then the ideal sheaf of $Z$ can be written as $$ \mathcal{I}_Z = \mathcal{I}_C^{\otimes a} \otimes \mathcal{I}_{C'}^{\otimes a'} \otimes \mathcal{I}_0, $$ where $ \mathcal{I}_0 $ is the ideal sheaf of $Z_0$.
\begin{lemma}\label{lemma 7.2} Let $ u_Z : g^* \mathcal{I}_Z \rightarrow \mathcal{I}_Z $ be the map constructed in Lemma \ref{G-sheaf}. Then the eigenvalue of $ u_Z $ at the fiber in $y$ is given by $$ \mathrm{Tr}(u_Z(y)) = \lambda^a \lambda'^{a'}. $$ \end{lemma} \begin{pf} One proves this in a similar way as Lemma \ref{Lemma 7.1}. \end{pf}
\subsection{} Finally, we consider divisors $ 0 \leq Z' < Z \leq \mathcal{Y}_k $, where $ Z - Z' = C $ for some irreducible component $ C $ of $ \mathcal{Y}_k $.
\begin{lemma}\label{lemma 7.3} Let $ u_{Z'/Z} : g^* (i_C)_* \mathcal{L} \rightarrow (i_C)_* \mathcal{L} $ be the map constructed in Section \ref{6.8}. Then the eigenvalue of the pullback $ i_C^* (u_{Z'/Z}) $ in the fiber at the fixed point $y$ is $$ \mathrm{Tr}(i_C^* (u_{Z'/Z})(y)) = \mathrm{Tr}(u_{Z'}(y)), $$ where $ u_{Z'} : g^* \mathcal{I}_{Z'} \rightarrow \mathcal{I}_{Z'} $ is the corresponding map for $ \mathcal{I}_{Z'} $. \end{lemma} \begin{pf} In the proof of Proposition \ref{prop. 6.5} we saw that $ u_{Z'/Z} $ fitted into a commutative diagram $$ \xymatrix{ g^*\mathcal{I}_{Z'} \ar[r]^-{g^* \pi} \ar[d]_{u_{Z'}} & g^*(\mathcal{I}_{Z'}/\mathcal{I}_{Z}) = g^* (i_C)_* \mathcal{L} \ar[d]^{u_{Z'/Z}} \\ \mathcal{I}_{Z'} \ar[r]^-{\pi} & \mathcal{I}_{Z'}/\mathcal{I}_{Z} = (i_C)_* \mathcal{L},} $$ where $ \pi $ is the quotient surjection. Let $ i_y : \{y\} \hookrightarrow \mathcal{Y} $ be the inclusion. Since $y$ is a fixed point, we have that $ g \circ i_y = i_y $, and therefore $ i_y^* g^* = i_y^* $. It follows that $ i_y^* \pi = i_y^* g^* \pi $. Moreover, $i_y$ factors via the inclusion $ i_C $, so pulling back with $i_y $, we get the diagram $$ \xymatrix{ \mathcal{L}(y) \ar[r]^-{i_y^* \pi} \ar[d]_{u_{Z'}(y)} & \mathcal{L}(y) \ar[d]^{u_{Z'/Z}(y)} \\ \mathcal{L}(y) \ar[r]^-{i_y^* \pi} & \mathcal{L}(y), } $$ and the result follows.
\end{pf}
\section{Resolution fibers of tame cyclic quotient singularities} In this section we study certain properties of tame cyclic quotient singularities, which will be important in the following sections. As a motivation for the discussion below, let us consider the following set-up: Let $ \mathcal{X} $ be an SNC-model, and let $ S'/S $ be a tame extension of degree $n$ that is prime to the multiplicities of the irreducible components of $ \mathcal{X}_k $. The normalization $ \mathcal{X}' $ of $ \mathcal{X}_{S'} $ has tame cyclic quotient singularities, depending on the combinatorial structure of $ \mathcal{X}_k $ and on $n$. We are interested in how the exceptional locus of the desingularization $ \rho : \mathcal{Y} \rightarrow \mathcal{X}' $ varies as the degree $n$ grows.
This is actually a local problem, so we will start by formalizing the situation, and identify singularities with certain parameters $m_1$, $m_2$ and $n$. We conclude in Proposition \ref{prop. 8.9}, that if $n$ is sufficiently large, the combinatorial structure of the exceptional locus in the minimal desingularization belongs, modulo chains of $(-2)$-curves, to only a finite number of types, depending only on $m_1$ and $m_2$.
\subsection{Construction}\label{8.1} Let $ m_1 $ and $ m_2 $ be positive integers, and let $ n $ be a positive integer that is not divisible by $p$, and that is relatively prime to $ \mathrm{lcm}(m_1,m_2) $. The integers $m_1$, $m_2$ and $n$ define a tame cyclic quotient singularity in the following way:
Consider the ring $ A = R[[u_1,u_2]]/( \pi - u_1^{m_1} u_2^{m_2} ) $, and let $ R \rightarrow R' $ be a tamely ramified extension of degree $n$. Let $ B = A \otimes_R R' = R'[[u_1,u_2]]/( \pi'^n - u_1^{m_1} u_2^{m_2} ) $, and let $ \widetilde{B} $ be the normalization of $B$. Then $ Z = \mathrm{Spec}(\widetilde{B}) $ is a tame cyclic quotient singularity.
Note also that $B$ can be equipped with the obvious $ G = \boldsymbol{\mu}_n $-action $ [ \xi ](\pi') = \xi \pi' $, for every $ \xi \in \boldsymbol{\mu}_n $.
\begin{dfn}\label{Def. 8.1} A tame cyclic quotient singularity arising as in \ref{8.1} above will be denoted as \emph{the} singularity $(m_1,m_2,n)$. That is, we identify the singularity with its parameters $ m_1 $, $ m_2 $ and $ n $. \end{dfn}
In this section, we will make a detailed study of the behaviour of the minimal resolution of a singularity with parameters $ (m_1,m_2,n) $, where we keep $m_1$ and $m_2$ fixed, but where $n$ varies.
Let $ \rho : \widetilde{Z} \rightarrow Z $ be the minimal desingularization. We saw in Section \ref{3.11} that the action of the Galois group $G$ on the special fiber $ \widetilde{Z}_k $ was completely determined by the parameters $ (m_1,m_2,n) $ of the singularity.
\subsection{Data associated to the singularity} Let us consider a singularity with parameters $ (m_1,m_2,n) $. Let $ m := \mathrm{gcd}(m_1,m_2) $ and $ M := \mathrm{lcm}(m_1,m_2) $, and let furthermore $r$ be the unique integer with $ 0<r<n $ such that $ m_1 + rm_2 = 0 $ modulo $n$. Write $ \frac{n}{r} = [b_1, \ldots , b_l, \ldots , b_L]_{JH} $ for the Jung-Hirzebruch expansion. As $L$ is the length of the expansion, there are $L$ exceptional components $ C_1, \ldots, C_L $, with self intersection numbers $ C_l^2 = - b_l $, for all $ l \in \{1, \ldots, L \} $. We let $ \mu_l $ denote the multiplicity of $C_l $, for all $l$. There are two series of numerical equations associated to the singularity. We have \begin{equation}\label{equation 8.1} r_{l-1} = b_{l+1} r_l - r_{l+1}, \end{equation} for $ 0 \leq l \leq L-1 $, where we put $ r_{-1} = n $ and $ r_0 = r $. Furthermore, we have
\begin{equation}\label{equation 8.2} \mu_{l+1} = b_l \mu_l - \mu_{l-1}, \end{equation} which is valid for $ 1 \leq l \leq L $. Here we define $ \mu_0 = m_2 $ and $ \mu_{L+1} = m_1 $. Note also that $ m_1 + r m_2 = n \mu_1 $ (see \cite{CED}, Corollary 2.4.3).
Let us also define $ C_0 $ to be the (formal) branch with multiplicity $ \mu_0 $, and $ C_{L+1} $ the (formal) branch with multiplicity $ \mu_{L+1} $. Let $ y_l $ be the node in the special fiber which is the intersection point of $ C_{l+1} $ and $ C_l $. Then we saw in Section \ref{3.11} that for any $ 0 \leq l \leq L $, we have $$ \widehat{\mathcal{O}}_{\widetilde{Z},y_l} = R'[[z_l,w_l]]/(z_l^{\mu_{l+1}} w_l^{\mu_l} - \pi'). $$ By Proposition \ref{prop. 3.3}, the Galois group $G$ acts on this ring by $ [ \xi ]( \pi' ) = \xi \pi' $, $ [ \xi ](z_l) = \xi^{ \alpha_1 r_{l-1} } z_l $ and $ [ \xi ](w_l) = \xi^{ - \alpha_1 r_l } w_l $, where $ \alpha_1 $ is an inverse to $ m_1 $ modulo $n$.
Since we have that $ I_{C_l} = (w_l) $ and $ I_{C_{l+1}} = (z_l) $, it follows that $ \widehat{\mathcal{O}}_{C_l, y_{l-1}} = k[[w_{l-1}]] $, and that $ \widehat{\mathcal{O}}_{C_l, y_{l}} = k[[z_l]] $. The cotangent space of $C_l$ at $y_{l-1}$ is generated by $ w_{l-1} $, and the eigenvalue is therefore $ \xi^{ - \alpha_1 r_{l-1} } $, and at $ y_{l} $ the cotangent space is generated by $ z_l $ with eigenvalue $ \xi^{ \alpha_1 r_{l-1} } $.
\subsection{Some general properties of the minimal resolution of quotient singularities} The lemma below lists some properties of the exceptional locus of the minimal resolution of a tame cyclic quotient singularity. These properties will be used numerous times in the rest of this paper.
\begin{lemma}\label{lemma 8.1} Let $ m_1 $ and $ m_2 $ be positive integers, and let $n$ be a positive integer not divisible by $p$ such that $ \mathrm{gcd}(n,m_1) = \mathrm{gcd}(n,m_2) = 1 $. Denote by $ (m_1,m_2,n) $ the associated singularity. Then the following properties hold: \begin{enumerate} \item If $d$ divides both $ \mu_k $ and $ \mu_{k+1} $ for some $ 0 \leq k \leq L $, then $d$ divides $ \mu_l $ for all $ 0 \leq l \leq L+1 $. \item Let $ m = \mathrm{gcd}(m_1,m_2) $. Then $m$ divides $ \mu_l $ for all $ 0 \leq l \leq L+1 $. \item Let $ l $ be an integer such that $ 1 \leq l \leq L $. Then the pairs of inequalities $ \mu_{l-1} < \mu_l $ and $ \mu_{l+1} \leq \mu_l $, or $ \mu_{l-1} \leq \mu_l $ and $ \mu_{l+1} < \mu_l $, can not occur. \end{enumerate} \end{lemma} \begin{pf} (i) Equation \ref{equation 8.2} gives that $ \mu_{k-1} = \mu_k b_k - \mu_{k+1} $. So if $d$ divides $ \mu_k $ and $ \mu_{k+1} $, it will also divide $ \mu_{k-1} $. Continuing inductively, we get that $d$ divides $ \mu_l $ for all $ 0 \leq l \leq k+1 $. The same argument shows that we also get that $d$ divides $ \mu_l $ for all $ k+1 \leq l \leq L+1 $ if we do induction for increasing $l$ instead.
(ii) Recall that we have the equation $ m_1 + r_0 m_2 = n \mu_1 $. So $m$ divides $ n \mu_1 $. But $n$ is relatively prime to both $m_1$ and $m_2$, and in particular to $m$, so $m$ must therefore divide $\mu_1$. But then we have that $m$ divides both $\mu_0$ and $\mu_1$, so by (i), we can conclude that $m$ divides $ \mu_l $ for all $ 0 \leq l \leq L+1 $.
(iii) Assume that $ \mu_{l-1} < \mu_l $ and $ \mu_{l+1} \leq \mu_l $ for some $l$. Using Equation \ref{equation 8.2}, we get that $$ 0 < b_l = (\mu_{l-1} + \mu_{l+1})/\mu_l < (\mu_l + \mu_l) /\mu_l = 2, $$ which implies that $ C_l^2 = - 1 $. Hence we get that $ C_l $ is a $(-1)$-curve, which contradicts the minimality of the resolution. \end{pf}
\begin{cor}\label{m_1=m_2} If $ m_1 = m_2 $, then $ \mu_l = m_1 = m_2 $ for all $ l $. \end{cor}
Let us also give a result about the action on the components in the exceptional locus.
\begin{lemma}\label{Lemma 8.4} Let $ \sigma = (m_1,m_2,n) $ be a singularity. Then the following property holds. Let $ \xi \in G $ be a primitive $n$-th root of unity, and denote by $g_{\xi}$ the induced automorphism on the minimal desingularization $ \widetilde{Z} $. Then the restriction $ g_{\xi}|_{C_l} $ is a non-trivial automorphism of each exceptional curve $C_l$, for all $ l = 1, \ldots , L $. Furthermore, the fixed points of $ g_{\xi}|_{C_l} $ are exactly the two points where $ C_l $ meets the rest of the special fiber.
\end{lemma} \begin{pf} Let $C_l$ be any of the exceptional components. Then we saw in Proposition \ref{prop. 3.3} that we may cover $C_l$ with the affine charts $ \mathrm{Spec}(k[w_{l-1}]) $ and $ \mathrm{Spec}(k[z_l]) $, where we glue along $ z_l = 1/w_{l-1} $. In these coordinates, the action is $ [\xi](z_l) = \xi^{\alpha_1 r_{l-1}} z_l $ and $ [\xi](w_{l-1}) = \xi^{ - \alpha_1 r_{l-1}} w_{l-1} $. Since $ \xi $ is a primitive root, it suffices to show that $ \alpha_1 r_{l} \not\equiv_n 0 $ for any $ l \in \{ 0, \ldots, L - 1 \} $. As $\alpha_1$ is invertible modulo $n$, this is the same as showing that $ r_{l} \not\equiv_n 0 $. But the $r_l$ satisfy the inequalities $$ n = r_{-1} > r_0 > \ldots > r_{L-1} = 1, $$ which proves the first statement.
The last statement follows from the explicit description of the action on the affine charts of the exceptional components. \end{pf}
\subsection{} We will now study the minimal resolution of a singularity $ (m_1,m_2,n) $ in more detail. In particular, we are interested in what happens if we keep $m_1$ and $ m_2 $ fixed, but let $n$ grow to infinity. We shall see that, modulo a certain equivalence relation, there are finitely many possibilities, corresponding to the elements in $ (\mathbb{Z}/M)^* $.
\begin{lemma}\label{lemma 8.4} Let $m_1, m_2$ be positive integers, and let $ n $ be a positive integer not divisible by $p$ such that $ \mathrm{gcd}(n,M) = 1 $. Assume that the equation $ m_1 + r m_2 = n \mu $ holds, where $ r $ is an integer such that $ 0 < r < n $, and where $ \mu $ is some positive integer. Recall that $ \mu $ is the multiplicity $ \mu_1 $ of the first exceptional component in the resolution of the singularity $(m_1,m_2,n)$. Then the following properties hold: \begin{enumerate} \item Assume that $ \mu < m_2 $. Let $ n' = n + k M $ and let $ r' = r + k \mu \frac{M}{m_2} $, for any $ k \in \mathbb{N} $. Then we have that $ \mathrm{gcd}(n',M) = 1 $, that $ 0 < r' < n' $, and that the equation $ m_1 + r' m_2 = n' \mu $ holds. \item If $ m_2 > m_1 $, we automatically have that $ \mu < m_2 $. \end{enumerate} \end{lemma} \begin{pf} (i) Assume that $t$ is a common divisor of $ n' $ and $M$. Since $ n = n' - kM $, it follows that $t$ also divides $n$, and hence $t=1$. In particular, $m_1$ and $m_2$ are invertible modulo $n'$. Furthermore, we see that $$ n' \mu = (n + kM) \mu = n \mu + k \mu \frac{M}{m_2} m_2 = m_1 + (r + k \mu \frac{M}{m_2}) m_2 = m_1 + r' m_2.$$ It is clear from this equation that $r'$ is invertible modulo $n'$. We now compute that $$ 0 < r' = r + k \mu \frac{M}{m_2} < n + k m_2 \frac{M}{m_2} = n + k M = n', $$ so it follows that $ 0 < r' < n' $.
(ii) Assume that $ \mu \geq m_2 $, so that $ n m_2 \leq n \mu $. Then we get $$ m_1 + r m_2 < m_2 + r m_2 = (r+1) m_2 \leq n m_2 \leq n \mu, $$ a contradiction. \end{pf}
We shall now consider the case where $ m_2 < m_1 $. Then we do not necessarily have that $ \mu \leq m_2 $, but we shall see that this is ``stably'' true. Note that the case $ m_1 = m_2 $ is treated in Corollary \ref{m_1=m_2}.
\begin{lemma} Let us keep the hypotheses from Lemma \ref{lemma 8.4}. Assume in addition that $ m_2 \leq \mu $. Then there exists a positive integer $ K $ such that $ m_1 + r' m_2 = n' \mu' $, where $ n' = n + K M $, $ 0 < r' < n' $ and where $ \mu' \leq m_2 $. The equality occurs only for $ \mu' = m_2 = \mathrm{gcd}(m_1,m_2) $. \end{lemma} \begin{pf} In case $ \mu = m_2 $, we take $K = 0 $ and hence $ n' = n $, so we may assume that $ \mu > m_2 $.
Observe that if $ n' = n + kM $, and $ R' = r + k \mu \frac{M}{m_2} $, we get that $$ m_1 + R' m_2 = m_1 + (r + k \mu \frac{M}{m_2}) m_2 = m_1 + r m_2 + k \mu M = n \mu + k \mu M = n' \mu. $$ But $ n' - R' = (n-r) + k (M - \mu \frac{M}{m_2}) $, so if $k \gg 0$, this is negative. Let $k_0$ be the smallest integer such that $ (n-r) + k_0 (M - \mu \frac{M}{m_2}) $ is negative, and put $ n' = n + k_0 M $. Arguing as in Lemma \ref{lemma 8.4}, we see that $m_1$, $m_2$ and hence $R'$ are relatively prime to $n'$. Let then $l_0$ be the unique integer such that $ 0 < R' - l_0 n' < n' $, and define $ r' = R' - l_0 n' $. It is easily computed that $$ m_1 + r' m_2 = n' (\mu - l_0 m_2). $$ We now claim that $ 0 < \mu' := \mu - l_0 m_2 \leq m_2 $. For if this was not the case, then $ m_2 < \mu - l_0 m_2 $, and we could find an integer $ l > l_0 $ such that $ 0 < \mu - l m_2 \leq m_2 $, giving the equation $$ m_1 + r'' m_2 = n' (\mu - l m_2), $$ where $ r'' = R' - l n' $, and where we necessarily have $ r'' < 0 $, by assumption on $l_0$. But then $ m_1 + r'' m_2 < m_1 < n' $, and as $ n' (\mu - l m_2) \geq n' $, we obtain a contradiction.
Finally, if $ \mu' = m_2 $, then $ \mu' $ divides $m_1$, and hence $ \mu' = m_2 = \mathrm{gcd}(m_1,m_2) $. \end{pf}
\begin{lemma}\label{lemma 8.6} Let $ \sigma = (m_1,m_2,n) $ be a singularity, and let $ \mu_l $, $ b_l $ and $r_l$ be the numerical data associated to the minimal resolution of $ \sigma $. Assume that there exists an integer $ \lambda > 0 $ such that $$ \mu_0 > \mu_1 > \ldots > \mu_{\lambda}. $$
Let $ n' = n + k M $ for any positive integer $k$, and denote by $ \mu_l', b_l' $ and $ r_l' $ the numerical data associated to the resolution of the singularity $ \sigma' = (m_1,m_2,n') $.
Then we have that $ \mu_l' = \mu_l $ for all $ 0 \leq l \leq \lambda $, that $ b_l' = b_l $ for all $ 1 \leq l \leq \lambda - 1 $ and that $ r_l' = r_l + k \frac{M}{m_2} \mu_{l+1} $ for all $ -1 \leq l \leq \lambda - 1 $. \end{lemma} \begin{pf} We shall prove this by induction. Notice that if $ \lambda = 1 $, this is just Lemma \ref{lemma 8.4}. So we may assume that $ \lambda > 1 $. Then the statement for $ \mu_1' $ and $ r_0' $ again follows from Lemma \ref{lemma 8.4}.
We have now that $ \mu_2 < \mu_1 $. With $ n' = n + kM $ and $ r_0' = r_0 + k \frac{M}{m_2} \mu_1 $, we need to show that $ \mu_2' = \mu_2 $, $ b_1' = b_1 $ and $ r_1' = r_1 + k \frac{M}{m_2} \mu_2 $. Let us define $ R_1' = r_1 + k \frac{M}{m_2} \mu_2 $. We now compute $$ b_1 r_0' - n' = b_1 (r_0 + k \frac{M}{m_2} \mu_1) - (n + k M) = (b_1 r_0 - n) + k \frac{M}{m_2}(b_1 \mu_1 - \mu_0) $$ $$ = r_1 + k \frac{M}{m_2} \mu_2 = R_1'. $$ But by assumption $ 0 \leq r_1 < r_0 $ and $ 1 \leq \mu_2 < \mu_1 $, so we get that $$ 0 < R_1' = r_1 + k \frac{M}{m_2} \mu_2 < r_0 + k \frac{M}{m_2} \mu_1 = r_0'. $$ In other words, $b_1$ is the unique integer such that $ 0 < b_1 r_0' - n' < r_0' $, and therefore $ b_1' = b_1 $ and $ r_1' = R_1' = r_1 + k \frac{M}{m_2} \mu_2 $. Furthermore, it follows that $$ \mu_2' = b_1' \mu_1' - \mu_0' = b_1 \mu_1 - \mu_0 = \mu_2. $$
Assume now that we have established that $ \mu_i' = \mu_i $ for all $ 0 \leq i \leq l $, that $ b_i' = b_i $ for all $ 1 \leq i \leq l-1 $ and that $ r_i' = r_i + k \frac{M}{m_2} \mu_{i+1} $ for all $ -1 \leq i \leq l-1 $, where $ 2 \leq l < \lambda $. We need to prove that $ \mu_{l+1}' = \mu_{l+1} $, that $ b_l' = b_l$ and that $ r_l' = r_l + k \frac{M}{m_2} \mu_{l+1} $. Define $ R_l' = r_l + k \frac{M}{m_2} \mu_{l+1} $. We then compute that $$ b_l r_{l-1}' - r_{l-2}' = b_l(r_{l-1} + k \frac{M}{m_2} \mu_l) - (r_{l-2} + k \frac{M}{m_2} \mu_{l-1}) = $$ $$ (b_l r_{l-1} - r_{l-2}) + k \frac{M}{m_2} (b_l \mu_l - \mu_{l-1}) = r_l + k \frac{M}{m_2} \mu_{l+1} = R_l'. $$ Since $ 0 \leq r_l < r_{l-1} $ and $ 1 \leq \mu_{l+1} < \mu_l $, we get that $$ 0 < R_l' = r_l + k \frac{M}{m_2} \mu_{l+1} < r_{l-1} + k \frac{M}{m_2} \mu_l = r_{l-1}', $$ and so it follows that $ b_l' = b_l $ and that $ r_l' = r_l + k \frac{M}{m_2} \mu_{l+1} $. Finally, we also see that $$ \mu_{l+1}' = b_l' \mu_l' - \mu_{l-1}' = b_l \mu_l - \mu_{l-1} = \mu_{l+1}, $$ which completes the proof. \end{pf}
\begin{lemma}\label{lemma 8.7} Consider again a singularity $ \sigma = (m_1,m_2,n) $. Assume that there exists a positive integer $ \lambda $ such that the multiplicities of the exceptional components in the minimal resolution of $ \sigma $ satisfy the inequalities $$ \mu_0 > \mu_1 > \ldots > \mu_{\lambda} < \mu_{\lambda + 1}. $$
Then there exists a positive integer $k_0$ such that the multiplicities of the components in the minimal resolution of the singularity $ \sigma' = (m_1,m_2,n') $, where $ n' = n + k_0 M $, satisfy $ \mu_0' = \mu_0, \ldots, \mu_{\lambda}' = \mu_{\lambda} $, and $ \mu_{\lambda + 1}' < \mu_{\lambda + 1} $. \end{lemma} \begin{pf} We will use the notation and computations from the proof of Lemma \ref{lemma 8.6}.
We have that \begin{equation}\label{equation 8.3} R_{\lambda}' := r_{\lambda} + k \frac{M}{m_2} \mu_{\lambda + 1} = b_{\lambda} r_{\lambda - 1}' - r_{\lambda -2 }', \end{equation} and $ r_{\lambda - 1}' = r_{\lambda - 1} + k \frac{M}{m_2} \mu_{\lambda} $. Observe that $$ r_{\lambda - 1}' - R_{\lambda}' = (r_{\lambda - 1} + k \frac{M}{m_2} \mu_{\lambda}) - (r_{\lambda} + k \frac{M}{m_2} \mu_{\lambda + 1}) = (r_{\lambda - 1} - r_{\lambda}) + k \frac{M}{m_2} (\mu_{\lambda} - \mu_{\lambda + 1}). $$ By assumption, $ r_{\lambda - 1} - r_{\lambda} > 0 $ and $ \mu_{\lambda} - \mu_{\lambda + 1} < 0 $, so for $ k \gg 0 $, we get that $ r_{\lambda - 1}' < R_{\lambda}' $.
Let $ k_0 $ be the smallest integer such that $ r_{\lambda - 1}' < R_{\lambda}' $. Since $ r_{\lambda}' < r_{\lambda - 1}' $, it then follows from Equation \ref{equation 8.3} that $ b_{\lambda}' < b_{\lambda} $. Furthermore, we easily compute from Equation \ref{equation 8.3} that $ (b_{\lambda} - l) r_{{\lambda} - 1}' - r_{\lambda - 2}' = R_{\lambda}' - l r_{\lambda - 1}' $. Now, let $ l_0 $ be the biggest integer such that $ 0 \leq R_{\lambda}' - l_0 r_{\lambda - 1}' $. We get that $ 0 \leq (b_{\lambda} - l_0) r_{\lambda - 1}' - r_{\lambda - 2}' < r_{\lambda - 1}' $, so it follows that $ b_{\lambda}' = b_{\lambda} - l_0 $ and $ r_{\lambda}' = R_{\lambda}' - l_0 r_{\lambda - 1}' $. But then $$ \mu_{\lambda + 1}' = b_{\lambda}' \mu_{\lambda}' - \mu_{\lambda - 1}' = (b_{\lambda} - l_0) \mu_{\lambda} - \mu_{\lambda - 1} = \mu_{\lambda + 1} - l_0 \mu_{\lambda} < \mu_{\lambda + 1}. $$ The rest of the statement follows immediately from Lemma \ref{lemma 8.6}. \end{pf}
\begin{cor}\label{cor. 8.8} Let $ \sigma = (m_1,m_2,n) $ be a singularity, and assume that there exists a positive integer $\lambda$ such that the multiplicities of the components in the minimal resolution of $\sigma$ satisfy the inequalities $$ \mu_0 > \mu_1 > \ldots > \mu_{\lambda} < \mu_{\lambda + 1}. $$
Then there exists a positive integer $K$ such that the multiplicities of the components in the minimal resolution of the singularity $ \sigma' = (m_1,m_2,n') $, where $ n' = n + K M $, satisfy $ \mu_0' = \mu_0, \ldots, \mu_{\lambda}' = \mu_{\lambda} $, and $ \mu_{\lambda + 1}' \leq \mu_{\lambda}' $. The equality occurs only in the case where $ \mu_{\lambda + 1}' = \mu_{\lambda}' = \mathrm{gcd}(m_1,m_2) $. \end{cor} \begin{pf} According to Lemma \ref{lemma 8.7}, we can find an integer $k_0$ such that for the extension of degree $n + k_0 M$, the associated multiplicities satisfy the equalities $ \mu_0' = \mu_0, \ldots, \mu_{\lambda}' = \mu_{\lambda} $, and the inequality $ \mu_{\lambda + 1}' < \mu_{\lambda + 1} $. If we still have that $ \mu_{\lambda}' < \mu_{\lambda + 1}' $, we can apply the same lemma again, by replacing $n$ with $n + k_0 M$, and looking at the multiplicities of the components in the exceptional fibre of the minimal resolution of this singularity instead. By repeated use of this procedure, the multiplicity of the $ (\lambda + 1) $-st component in the exceptional fiber will strictly decrease each time, so eventually we will have $ \mu_{\lambda + 1}' \leq \mu_{\lambda}' $. By then, we have made an extension of degree $n + KM$, which is the sought after extension.
If the inequality above is actually an equality, i.e., $ \mu_{\lambda + 1}' = \mu_{\lambda}' $, it follows from Lemma \ref{lemma 8.1} that $ \mu_{\lambda}' $ divides \emph{all} the multiplicities, and in particular it divides $m = \mathrm{gcd}(m_1,m_2) $. On the other hand, $ m $ divides all multiplicities in the chain, so we get that $ \mu_{\lambda + 1}' = \mu_{\lambda}' = \mathrm{gcd}(m_1,m_2) $. \end{pf}
Combining the results developed in this section, we get the following result:
\begin{prop}\label{prop. 8.9} Let $m_1, m_2$ be positive integers, let $ m = \mathrm{gcd}(m_1,m_2) $, and let $ M = \mathrm{lcm}(m_1,m_2) $. Let us furthermore fix a positive integer $ n_0 $ that is not divisible by $p$ and that is relatively prime to $M$. Then the following properties hold: \begin{enumerate} \item There exists an integer $ K \gg 0 $, such that the multiplicities of the components in the minimal resolution of the singularity $ \sigma = (m_1,m_2,n) $, where $ n = n_0 + K M $, satisfy $$ \mu_0 > \mu_1 > \ldots > \mu_{l_0} = \ldots = m = \ldots = \mu_{L + 1 - l_1} < \ldots < \mu_{L} < \mu_{L + 1}, $$ where $L$ denotes the \emph{length} of the singularity $ \sigma $. \item The integers $ \mu_2 , \ldots , \mu_{l_0} $ are uniquely determined by $ \mu_0 $ and $ \mu_1 $, and similarly $ \mu_{L + 1 - l_1}, \ldots , \mu_{L-1} $ are uniquely determined by $ \mu_{L} $ and $ \mu_{L+1} $. \item For any extension of degree $ n' = n + k M $, where $ k > 0 $, we have that the multiplicities $ \mu_l' $ of the components in the minimal resolution of the singularity $ \sigma' = (m_1,m_2,n') $ will only differ from the sequence of multiplicities associated to $ \sigma $ by inserting $m$'s ``in the middle''. \end{enumerate} \end{prop} \begin{pf} (i) From Corollary \ref{cor. 8.8}, it follows that we can find a positive integer $K \gg 0$ such that the multiplicities associated to the singularity $ \sigma = (m_1,m_2,n) $, where $ n = n_0 + KM $, satisfy the inequalities $$ \mu_0 > \mu_1 > \ldots > \mu_{l_0} = m $$ and $$ \mu_{L+1} > \mu_{L} > \ldots > \mu_{L + 1 - l_1} = m, $$ for some integers $ l_0 \geq 0 $ and $ l_1 \geq 0 $.
For any $ \mu_l $, where $ l_0 < l < L + 1 - l_1 $, we have that $ \mu_l = m $. Indeed, by part (ii) of Lemma \ref{lemma 8.1}, we have that $ m \geq \mu_l $, and part (iii) of the same lemma implies that we actually have equalities.
(ii) In case $ l_0 = 0 $ or $ l_0 = 1 $, the statement is empty. So we can assume that $ l_0 > 1 $. Let us first see that $ \mu_0 $ and $ \mu_1 $ determine $ \mu_2 $. By assumption, we have that $ \mu_0 > \mu_1 > \mu_2 \geq m $. These integers must satisfy the equation $ \mu_2 = b_1 \mu_1 - \mu_0 $, and since there is only one such integer $ \mu_2 $ satisfying these properties, it follows that $ \mu_2 $ is uniquely determined by $ \mu_0 $ and $ \mu_1 $.
Now let $ 1 < l < l_0 $, and assume that $ \mu_0 > \mu_1 > \ldots > \mu_{l} $ are given. Then $ \mu_{l+1} = b_l \mu_l - \mu_{l-1} $, and we have that $ \mu_{l+1} < \mu_l $. But then $ \mu_{l+1} $ is uniquely determined by $ \mu_l $ and $ \mu_{l-1} $, which are in turn uniquely determined by $ \mu_0 $ and $ \mu_1 $ by the induction hypothesis, so the result follows. The same argument applies to the other end of the chain.
(iii) This is a straightforward consequence of Lemma \ref{lemma 8.6}. \end{pf}
\begin{rmk} Proposition \ref{prop. 8.9} states that the minimal resolution of a singularity $(m_1,m_2,n)$ essentially only depends on $m_1$, $m_2$ and the residue class of $ n $ modulo $ M = \mathrm{lcm}(m_1,m_2) $. That is, if $ n' \gg 0 $, and $ n'' \gg 0 $, and if $ n' \equiv_M n'' $, then the exceptional locus of the minimal desingularization of $(m_1,m_2,n')$ is equal to the exceptional locus of the minimal desingularization of $(m_1,m_2,n'')$, modulo chains of $(-2)$-curves ``in the middle''. \end{rmk}
\section{Special filtrations for trace computations}\label{special filtrations} Let $ \mathcal{X}/S $ be an SNC-model, and let $ S' \rightarrow S $ be a tame extension of degree $n$, where $n$ is prime to the least common multiple of the multiplicities of the irreducible components of $ \mathcal{X}_k $. Let $ \mathcal{X}' $ be the normalization of $ \mathcal{X}_{S'} $, and let $ \mathcal{Y} $ be the minimal desingularization of $ \mathcal{X}' $.
This section is devoted to computing, for any $ g \in G = \boldsymbol{\mu}_n $, the Brauer trace of the automorphism induced by $g$ on the formal difference $ H^0(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) - H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $. Hence a lot of our previous work will come together in this section.
Our assumption on the degree of $ S'/S $ makes it possible to describe $ \mathcal{Y}_k $ in terms of $ \mathcal{X}_k $. In particular, since every component of $ \mathcal{Y}_k $ either is an exceptional curve, or dominates a component of $ \mathcal{X}_k $, it is natural to stratify the combinatorial structure of $ \mathcal{Y}_k $ according to the combinatorial structure of $ \mathcal{X}_k $.
This stratification proves to be very convenient for our trace computations. The section concludes with Theorem \ref{thm. 9.13}, wich gives a formula for the trace mentioned above as a sum of contributions associated in a natural way to the combinatorial structure of $ \mathcal{X}_k $.
\subsection{The graph $ \Gamma(\mathcal{X}_k) $}\label{9.1} We will associate a graph $ \Gamma(\mathcal{X}_k) $ to $ \mathcal{X}_k $ in the following way: The set of vertices, $ \mathcal{V} $, consists of the irreducible components of $ \mathcal{X}_k $. The set of edges, $ \mathcal{E} $, consists of the intersection points of $ \mathcal{X}_k $, and two distinct vertices $\upsilon$ and $\upsilon'$ are connected by $ \mathrm{Card} (\{ D_{\upsilon} \cap D_{\upsilon'} \}) $ edges, where $D_{\upsilon}$ is the irreducible component corresponding to $\upsilon$.
We define two natural functions on the set of vertices $ \mathcal{V} $. First, define the \emph{genus} $$ \mathfrak{g} : \mathcal{V} \rightarrow \mathbb{N}_0, $$ by $ \mathfrak{g}(\upsilon) = p_a(D_{\upsilon}) $. We also define the \emph{multiplicity} $$ \mathfrak{m} : \mathcal{V} \rightarrow \mathbb{N}, $$ by $ \mathfrak{m}(\upsilon) = \mathrm{mult}_{\mathcal{X}_k}(D_{\upsilon}) $.
The graph $ \Gamma(\mathcal{X}_k) $, together with the functions $\mathfrak{g}$ and $\mathfrak{m}$, encode all the combinatorial and numerical properties of $ \mathcal{X}_k $.
\subsection{A partition of the set of irreducible components of $ \mathcal{Y}_k $}\label{9.2}
Let $ \mathcal{S} $ denote the set of irreducible components of $ \mathcal{Y}_k $. If $ C \in \mathcal{S} $, then we have either: \begin{enumerate} \item $ C $ dominates a component $ D_{\upsilon} $ of $ \mathcal{X}_k $, or
\item $ C $ is a component of the exceptional locus of the minimal desingularization $ \rho : \mathcal{Y} \rightarrow \mathcal{X}' $. \end{enumerate}
In the first case, we have that $ p_a(C) = \mathfrak{g}(\upsilon) $, and $ \mathrm{mult}_{\mathcal{Y}_k}(C) = \mathfrak{m}(\upsilon) $. Furthermore, $G$ acts trivially on $C$. Since $C$ is the unique component of $ \mathcal{Y}_k $ corresponding to $ \upsilon $, we write $C = C_{\upsilon}$.
In the second case, we have that $ C $ is part of a chain of exceptional curves, corresponding uniquely to an edge $ \varepsilon \in \mathcal{E} $. Hence $ p_a(C) = 0 $. By choosing an ordering (or direction) of this chain, we can index the components in the chain by $ l $, for $ 1 \leq l \leq L(\varepsilon) $, where $ L(\varepsilon) $ is the length of the chain. So we can write $ C = C_{\varepsilon,l} $, for some $l \in \{1, \ldots, L(\varepsilon) \} $. By Lemma \ref{Lemma 8.4}, $G$ acts nontrivially on $ C $, with fixed points exactly at the two points where $C$ meets the rest of the special fiber.
The special fiber $ \mathcal{Y}_k $ can now be written, as an effective divisor on $ \mathcal{Y} $, in the form $$ \mathcal{Y}_k = \sum_{\varepsilon \in \mathcal{E}} \sum_{l = 1}^{L(\varepsilon)} \mu_{\varepsilon,l} C_{\varepsilon,l} + \sum_{\upsilon \in \mathcal{V}} m_{\upsilon} C_{\upsilon}, $$ where $ \mu_{\varepsilon,l} $ denotes the multiplicity of the component $ C_{\varepsilon,l} $, and $m_{\upsilon}$ is the multiplicity of $C_{\upsilon}$.
\subsection{Special filtrations}\label{9.3} We will now consider \emph{special} filtrations of $ \mathcal{Y}_k $, inspired by the partition of the set of irreducible components of $ \mathcal{Y}_k $ introduced above.
Let us to begin with choosing an ordering of the elements in $ \mathcal{V} $. We can then define the following sequence:
$$ 0 < \ldots < Z_{\mathcal{E}} =: Z_{\upsilon_{|\mathcal{V}| + 1}} < Z_{\upsilon_{|\mathcal{V}|}} < \ldots < Z_{\upsilon_i} < \ldots < Z_{\upsilon_1} = \mathcal{Y}_k, $$
where $ Z_{\mathcal{E}} := \mathcal{Y}_k - \sum_{\upsilon \in \mathcal{V}} m_{\upsilon} C_{\upsilon} $. The $ Z_{\upsilon_i} $ are defined inductively, for every $ i \in \{ 1, \ldots, |\mathcal{V}| \} $, by the refinements $$ Z_{\upsilon_{i+1}} = Z_{\upsilon_i}^{m_{\upsilon_i} + 1} < \ldots < Z_{\upsilon_i}^j < \ldots < Z_{\upsilon_i}^1 = Z_{\upsilon_i}, $$ where $ Z_{\upsilon_i}^{j + 1} = Z_{\upsilon_i} - j C_{\upsilon_i} $ for every $ j \in \{ 0, \ldots, m_{\upsilon_i} \} $.
Next, we choose an ordering of the elements in $ \mathcal{E} $. We can then define the following sequence:
$$ 0 =: Z_{\varepsilon_{|\mathcal{E}|+1}} < Z_{\varepsilon_{|\mathcal{E}|}} < \ldots < Z_{\varepsilon_i} < \ldots < Z_{\varepsilon_1} := Z_{\mathcal{E}}. $$
The $ Z_{\varepsilon_i} $ are defined inductively, for any $ i \in \{ 1, \ldots, |\mathcal{E}| \} $, by the refinements $$ Z_{\varepsilon_{i+1}} := Z_{\varepsilon_i, L(\varepsilon_i) + 1} < \ldots < Z_{\varepsilon_i,l} < \ldots < Z_{\varepsilon_i,1} := Z_{\varepsilon_i}, $$ which in turn are defined inductively, for every $ l \in \{ 1, \ldots, L(\varepsilon_i) \} $, by the further refinements $$ Z_{\varepsilon_i, l+1} := Z_{\varepsilon_i, l}^{\mu_l + 1} < \ldots < Z_{\varepsilon_i, l}^j < \ldots < Z_{\varepsilon_i, l}^1 := Z_{\varepsilon_i, l}, $$ where $ Z_{\varepsilon_i, l}^{j+1} := Z_{\varepsilon_i, l} - j C_{\varepsilon_i, l} $, for every $ j \in \{ 0, \ldots, \mu_l \} $.
\subsection{} In the rest of this paper, we shall always choose complete filtrations of $ \mathcal{Y}_k $ that are of the form \begin{equation} 0 < \ldots < Z_{\varepsilon_i} < \ldots < Z_{\varepsilon_1} = Z_{\mathcal{E}} < \ldots < Z_{\upsilon_i} < \ldots < Z_{\upsilon_1} = \mathcal{Y}_k, \end{equation} where $ Z_{\upsilon_{i+1}} < Z_{\upsilon_i} $ and $ Z_{\varepsilon_{i+1}} < Z_{\varepsilon_i} $ are subfiltrations as described above.
We shall soon see that the chosen orderings of the sets $ \mathcal{E} $ and $ \mathcal{V} $ are irrelevant. The nice feature of working with filtrations like this becomes evident when one wants to do trace computations \`a la Section \ref{section 6}. Then we may actually reduce to considering subfiltrations $ Z_{\upsilon_{i+1}} < Z_{\upsilon_i} $, which we interpret as contributions from the vertices of $\Gamma$, and subfiltrations $ Z_{\varepsilon_{i+1}} < Z_{\varepsilon_i} $, which we interpret as contributions from the edges.
\subsection{Contribution to the trace from vertices in $ \mathcal{V}$} Let us fix a vertex $\upsilon \in \mathcal{V}$. We shall now define and calculate the \emph{contribution} to the trace from $\upsilon$. To do this, we choose a filtration of $ \mathcal{Y}_k $ as in Section \ref{9.3}. Then there will be a subfiltration of the form:
$$ Z_{\mathcal{E}} \leq Z_{\upsilon}^{m_{\upsilon}+1} < \ldots < Z_{\upsilon}^k < \ldots < Z_{\upsilon}^1 = Z_{\upsilon} \leq \mathcal{Y}_k, $$ where $ Z_{\upsilon}^{k} - Z_{\upsilon}^{k + 1} = C_{\upsilon} $, for all $ 1 \leq k \leq m_{\upsilon} $. The invertible sheaf associated to the $k$-th step in this filtration is $ \mathcal{L}_{\upsilon}^k := j_{\upsilon}^*(\mathcal{I}_{Z_{\upsilon}^{k+1}}) $, where $ j_{\upsilon} : C_{\upsilon} \hookrightarrow \mathcal{Y} $ is the canonical inclusion.
We will use the following easy lemma, whose proof is omitted. \begin{lemma}\label{lemma 9.1} Assume that $ S'/S $ is a nontrivial extension. If $C_1$ and $C_2$ are two distinct components of $ \mathcal{Y}_k $, corresponding to elements in $ \mathcal{V} $, then they have empty intersection. \end{lemma}
Let $ D_1, \ldots, D_f $ be the components of $ Z_{\upsilon} $ that intersect $ C_{\upsilon} $ non-trivially, and that are not equal to $ C_{\upsilon} $. Let $ a_i $ denote the multiplicity of $D_i$. It follows from Lemma \ref{lemma 9.1} that the $D_i$ are exceptional components. Moreover, it follows from the way we constructed the filtration that the $D_i$ are precisely the components of $ \mathcal{Y}_k $ different from $ C_{\upsilon} $ that have non-empty intersection with $ C_{\upsilon} $. We can then write $$ Z_{\upsilon}^{k + 1} = (m_{\upsilon} - k) C_{\upsilon} + a_1 D_1 + \ldots + a_f D_f + Z_0, $$ where all components of $ Z_0 $ have empty intersection with $ C_{\upsilon} $. So we get that \begin{equation}\label{equation 9.2}
\mathcal{L}_{\upsilon}^k = j_{\upsilon}^*(\mathcal{I}_{Z_{\upsilon}^{k+1}}) = (\mathcal{I}_{C_{\upsilon}}|_{C_{\upsilon}})^{ \otimes m_{\upsilon} - k } \otimes (\mathcal{I}_{D_1}|_{C_{\upsilon}})^{\otimes a_1} \otimes \ldots \otimes (\mathcal{I}_{D_f}|_{C_{\upsilon}})^{\otimes a_f}. \end{equation}
Let $ g $ be an element of $ G = \boldsymbol{\mu}_n $, corresponding to a root of unity $ \xi $. Note that the restriction of the automorphism $ g $ to $C_{\upsilon}$ is $ \mathrm{id}_{C_{\upsilon}}$. Let $$ u_{\upsilon}^k : g^* (i_{C_{\upsilon}})_* \mathcal{L}_{\upsilon}^k \rightarrow (i_{C_{\upsilon}})_* \mathcal{L}_{\upsilon}^k $$ be the map constructed in Section \ref{6.8}, and let $ j_{\upsilon}^* (u_{\upsilon}^k) : \mathcal{L}_{\upsilon}^k \rightarrow \mathcal{L}_{\upsilon}^k $ be the pullback of this map to $C_{\upsilon}$.
\begin{dfn}\label{contribution from vertex} We define \emph{the contribution to the trace} from the vertex $ \upsilon \in \mathcal{V} $ as the sum \begin{equation} \mathrm{Tr}_{\upsilon}(\xi) = \sum_{k=1}^{m_{\upsilon}} \mathrm{Tr}_{\beta}(e(H^{\bullet}(\mathrm{id}_{C_\upsilon},j_{\upsilon}^*(u_{\upsilon}^k)))) . \end{equation} \end{dfn}
\begin{prop}\label{prop. 9.7} The contribution to the trace from the vertex $ \upsilon $ is $$ \mathrm{Tr}_{\upsilon}(\xi) = \sum_{k=1}^{m_{\upsilon}} (\xi^{\alpha_{m_{\upsilon}}})^{m_{\upsilon} - k} (k C_{\upsilon}^2 + 1 - p_a(C_{\upsilon})) $$ $$ = \sum_{k=0}^{m_{\upsilon}-1} (\xi^{\alpha_{m_{\upsilon}}})^{k} ((m_{\upsilon} - k) C_{\upsilon}^2 + 1 - p_a(C_{\upsilon})), $$ where $ \alpha_{m_{\upsilon}} $ is an inverse to $ m_{\upsilon} $ modulo $n$. \end{prop} \begin{pf} By Proposition \ref{prop. 5.6}, we have that $$ \mathrm{Tr}_{\beta}(e(H^{\bullet}(\mathrm{id}_{\upsilon}, j_{\upsilon}^*(u_{\upsilon}^k)))) = \lambda_k(\mathrm{deg}_{C_{\upsilon}}(\mathcal{L}_{\upsilon}^k) + 1 - p_a(C_{\upsilon})), $$ where $ \lambda_k = \mathrm{Tr}(j_{\upsilon}^*(u_{\upsilon}^k)(y)) $, for any point $ y \in C_{\upsilon} $. The proof will consist of specifying precisely the terms appearing in this formula.
Let us first compute $ \mathrm{deg}_{C_{\upsilon}}(\mathcal{L}_{\upsilon}^k) $. Since $ \mathcal{I}_{C_{\upsilon}} = \mathcal{O}_{\mathcal{Y}}(- C_{\upsilon}) $, it follows that
$$ \mathrm{deg}_{C_{\upsilon}}( \mathcal{I}_{C_{\upsilon}}|_{C_{\upsilon}}) = \mathrm{deg}_{C_{\upsilon}}(\mathcal{O}_{\mathcal{Y}}(- C_{\upsilon})|_{C_{\upsilon}}) = - \mathrm{deg}_{C_{\upsilon}}(\mathcal{O}_{\mathcal{Y}}(C_{\upsilon})|_{C_{\upsilon}}) = - C_{\upsilon}^2.$$ Furthermore, for any $ i \in \{ 1, \ldots, f \} $, we have that $ \mathcal{I}_{D_i} = \mathcal{O}_{\mathcal{Y}}(- D_i) $, and hence
$$ \mathrm{deg}_{C_{\upsilon}}(\mathcal{I}_{D_i}|_{C_{\upsilon}}) = \mathrm{deg}_{C_{\upsilon}}(\mathcal{O}_{\mathcal{Y}}(- D_i)|_{C_{\upsilon}}) = - 1, $$ since $ C_{\upsilon} $ and $ D_i $ intersect transversally at exactly one point. Since $ \mathrm{deg}_{C_{\upsilon}}(-) $ is additive on tensor products, it follows from Equation \ref{equation 9.2} that $$ \mathrm{deg}_{C_{\upsilon}}(\mathcal{L}_{\upsilon}^k) = - (m_{\upsilon} - k) C_{\upsilon}^2 - (a_1 + \ldots + a_f). $$ On the other hand, we have that $ - C_{\upsilon}^2 = (a_1 + \ldots + a_f)/m_{\upsilon} $, and therefore $ \mathrm{deg}_{C_{\upsilon}}(\mathcal{L}_{\upsilon}^k) = k C_{\upsilon}^2 $.
To see that $ \lambda_k = (\xi^{\alpha_{m_{\upsilon}}})^{m_{\upsilon} - k} $, let $D$ be one of the components of $ \mathcal{Y}_k $ meeting $ C_{\upsilon} $, and denote by $y$ the unique point where they intersect. Then $D$ is part of a chain of exceptional curves. Let $ C_{\upsilon'} $ be the component at ``the other end'' of this chain, and denote by $L$ the length of this chain. Then, using the notation and computations in Proposition \ref{prop. 3.3}, with $ C_{\upsilon} = C_{L+1} $ and $ D = C_L $, we can identify the fiber of $ \mathcal{I}_{C_{\upsilon}} $ at $ y = y_L $ with $ <z_{L+1}> $. And the eigenvalue of $ z_{L+1} $ for the automorphism induced by $\xi$ was precisely equal to $ \xi^{\alpha_{m_{\upsilon}}} $. It follows that $ \lambda_k = (\xi^{\alpha_{m_{\upsilon}}})^{m_{\upsilon} - k} $. \end{pf}
\begin{rmk} In particular, it is clear that this formula is independent of how we have chosen to order the elements in $ \mathcal{V} $. \end{rmk}
\subsection{Contribution to the trace from edges in $ \mathcal{E} $}\label{9.6} Let us now choose an edge $ \varepsilon \in \mathcal{E} $. In the filtration of $ \mathcal{Y}_k $, we can find a subfiltration $ 0 < Z_{\varepsilon} \leq Z_{\mathcal{E}} < \mathcal{Y}_k $, with the refinements $$ Z_{\varepsilon, L(\varepsilon) + 1} < \ldots < Z_{\varepsilon,l} < \ldots < Z_{\varepsilon,1} = Z_{\varepsilon}, $$ for any $ l \in \{ 1, \ldots, L(\varepsilon) \} $, and further refinements $$ Z_{\varepsilon,l+1} = Z_{\varepsilon,l}^{\mu_l + 1} < \ldots < Z_{\varepsilon,l}^{k} < \ldots < Z_{\varepsilon,l}^{1} = Z_{\varepsilon,l}, $$ where $ Z_{\varepsilon,l}^{k} - Z_{\varepsilon,l}^{k+1} = C_{\varepsilon,l} $, for any $ k \in \{ 1, \ldots, \mu_l \} $.
As we are working with a fixed $ \varepsilon $, we will for the rest of this section suppress the index $ \varepsilon $, to simplify the notation. Take now an integer $ l \in \{1, \ldots, L-1 \} $, and let $ j_l : C_l \hookrightarrow \mathcal{Y} $ be the canonical inclusion. Consider then the subfiltration involving the component $C_l$: $$ \ldots < Z^{\mu_l + 1}_l < \ldots < Z^k_l < \ldots < Z^1_l < \ldots . $$ At the $k$-th step in this filtration, we have $ Z_l^k - Z_l^{k+1} = C_l $ for all $ 1 \leq k \leq \mu_l $. The associated invertible sheaf at the $k$-th step is
$$ \mathcal{L}_l^k := j_l^*( \mathcal{I}_{Z_l^{k+1}}) = (\mathcal{I}_{C_l}|_{C_l})^{\otimes (\mu_l - k)} \otimes (\mathcal{I}_{C_{l+1}}|_{C_l})^{ \otimes \mu_{l+1} }. $$
For $ l = L $, we have $$ \ldots < Z^{\mu_L + 1}_L < \ldots < Z^k_L < \ldots < Z^1_L < \ldots, $$ and at the $ k $-th step, we have $ Z_L^k - Z_L^{k+1} = C_L $. As all components in $ Z_{L} $ other than $ C_L $ have empty intersection with $ C_L $, we get that
$$ \mathcal{L}_L^k := j_L^*( \mathcal{I}_{Z_L^{k+1}}) = (\mathcal{I}_{C_L}|_{C_L})^{\otimes (\mu_L - k)}. $$
Let $ g \in G $ be a group element corresponding to a root of unity $ \xi $. The restriction $ g|_{C_l} $ is either the identity on $C_l$, or has fixed points exactly at the two points $ y_l $ and $ y_{l-1} $ where $C_l$ meets the rest of the special fiber. We need to compute the fibers at $ y_l $ and $ y_{l-1} $ of $ \mathcal{L}_l^k $, and the corresponding eigenvalues for the automorphisms induced by $ g $ at these fibers.
Let $ u_l^k : g^* (j_l)_* \mathcal{L}_l^k \rightarrow (j_l)_* \mathcal{L}_l^k $ be the map defined in Section \ref{6.8}. We will now compute $ \mathrm{Tr}_{\beta}(e(H^{\bullet}(g|_{C_l},(j_l)^*(u_k^l)))) $.
\begin{lemma}\label{fiberlemma} For any $ l \in \{1, \ldots, L \} $, we have that $$ \mathrm{Tr}((j_l)^*(u_l^k)(y_{l-1})) = (\xi^{\alpha_1 r_{l-2}})^{\mu_l - k}. $$ For any $ l \in \{1, \ldots, L-1 \} $, we have that $$ \mathrm{Tr}((j_l)^*(u_l^k)(y_{l})) = (\xi^{ - \alpha_1 r_l})^{\mu_l - k} (\xi^{ \alpha_1 r_{l-1}})^{\mu_{l+1}}, $$ and for $ l = L $, we have that $$ \mathrm{Tr}((j_L)^*(u_L^k)(y_{L})) = (\xi^{- \alpha_1 r_L})^{\mu_L - k}. $$ \end{lemma} \begin{pf} In order to compute the eigenvalue of the map $ (j_l)^*(u_k^l)(y) $ on the fiber at $y \in C_l$, it suffices, by Lemma \ref{lemma 7.3}, to compute the eigenvalue of $ v_l^k(y) $, where $$ v_l^k : g^* \mathcal{I}_{Z_l^{k+1}} \rightarrow \mathcal{I}_{Z_l^{k+1}} $$ is the map constructed as in Section \ref{6.4}.
If $ l \in \{1, \ldots, L-1\} $, we can write $$ \mathcal{I}_{Z_l^{k+1}} = \mathcal{I}_{C_l}^{\otimes(\mu_l - k)} \otimes \mathcal{I}_{C_{l+1}}^{\otimes \mu_{l+1}} \otimes \mathcal{I}_0, $$ where $ \mathcal{I}_0 $ has support away from $ C_l $. If $l = L $, we have $$ \mathcal{I}_{Z_L^{k+1}} = \mathcal{I}_{C_L}^{\otimes(\mu_L - k)} \otimes \mathcal{I}_0. $$ Recall the notation from Proposition \ref{prop. 3.3}. For any $ l \in \{1, \ldots, L \} $, the fiber at $ y_{l-1} $ is $$ \mathcal{I}_{Z_l^{k+1}}(y_{l-1}) = \mathcal{I}_{C_l}^{\otimes(\mu_l - k)}(y_{l-1}) = <z_{l-1}>^{\otimes (\mu_l - k)}, $$ and the fiber at $y_l$ is, for any $ l \in \{1, \ldots, L-1 \} $, $$ \mathcal{I}_{Z_l^{k+1}}(y_l) = \mathcal{I}_{C_l}^{\otimes(\mu_l - k)}(y_l) \otimes \mathcal{I}_{C_{l+1}}^{\otimes \mu_{l+1}}(y_l) = <w_l>^{\otimes (\mu_l - k)} \otimes <z_l>^{ \otimes \mu_{l+1} }. $$ In the case $ l = L $, we get $$ \mathcal{I}_{Z_L^{k+1}}(y_L) = \mathcal{I}_{C_L}^{\otimes(\mu_L - k)}(y_L) = <w_L>^{\otimes (\mu_L - k)}. $$
By Lemma \ref{lemma 7.2}, $ \mathrm{Tr}(v_l^k(y_{l-1})) = \lambda^{\mu_l - k} $, where $ \lambda $ is the eigenvalue of the eigenvector $z_{l-1}$ under the automorphism induced by $ g $ on the cotangent space of $ \mathcal{Y} $ at $y_{l-1}$. Proposition \ref{prop. 3.3} then shows that $ \lambda = \xi^{\alpha_1 r_{l-2}} $, where $ \alpha_1 m_1 \equiv_n 1 $, so therefore \begin{equation}\label{equation 9.4} \mathrm{Tr}((j_l)^*(u_l^k)(y_{l-1})) = \mathrm{Tr}(v_l^k(y_{l-1})) = (\xi^{\alpha_1 r_{l-2}})^{\mu_l - k}. \end{equation}
For $ l \in \{1, \ldots, L-1 \} $, Lemma \ref{lemma 7.2} shows again that $ \mathrm{Tr}(v_l^k(y_{l})) = \lambda^{\mu_l - k} \lambda'^{\mu_{l+1}} $, where $ \lambda $ and $ \lambda' $ are the eigenvalues of the eigenvectors $ w_l $ and $ z_l $ under the automorphism induced by $ g $ on the cotangent space of $ \mathcal{Y} $ at $y_{l}$. Using Proposition \ref{prop. 3.3}, it is then easy to see that \begin{equation}\label{equation 9.5} \mathrm{Tr}((j_l)^*(u_l^k)(y_{l})) = \mathrm{Tr}(v_l^k(y_{l})) = (\xi^{ - \alpha_1 r_l})^{\mu_l - k} (\xi^{ \alpha_1 r_{l-1}})^{\mu_{l+1}}. \end{equation} In the case $ l = L $, we get by the same methods $$ \mathrm{Tr}((j_L)^*(u_L^k)(y_{L})) = \mathrm{Tr}(v_L^k(y_{L})) = (\xi^{- \alpha_1 r_L})^{\mu_L - k}. $$ \end{pf}
\begin{ntn} We will write $ \chi = \xi^{\alpha_1} $. \end{ntn}
\begin{prop}\label{lemma 9.8}
Let $ g \in G $ be a group element corresponding to a root of unity $ \xi $. If $ g|_{C_l} = \mathrm{id}_{C_l} $, then we have, for any $ l \in \{1, \ldots, L - 1 \} $, that
$$ \mathrm{Tr}_{\beta}(e(H^{\bullet}(g|_{C_l},(j_l)^*(u_l^k)))) = \chi^{ r_{l - 2} (\mu_l - k)} ((\mu_{l} - k) b_l - \mu_{l+1}+ 1). $$ If $l=L$, we have that
$$ \mathrm{Tr}_{\beta}(e(H^{\bullet}(g|_{C_L},(j_L)^*(u_L^k)))) = \chi^{ r_{L - 2} (\mu_L - k)} ((\mu_{L} - k) b_L + 1). $$ \end{prop} \begin{pf}
Since $ g|_{C_l} = \mathrm{id}_{C_l} $, we can apply Proposition \ref{prop. 5.5} (iii), which states that
$$ \mathrm{Tr}_{\beta}(e(H^{\bullet}(g|_{C_l},(j_l)^*(u_k^l)))) = \lambda(\mathrm{deg}_{C_l}(\mathcal{L}_l^k) + 1 - p_a(C_l)). $$
From Lemma \ref{fiberlemma}, it follows that $ \lambda = \chi^{r_{l - 2} (\mu_l - k)}$. Furthermore, we have that $ p_a(C_l) = 0 $. So it remains to compute the degree of $ \mathcal{L}_l^k $. For $ l \neq L $, we have that $ \mathcal{L}_l^k = (\mathcal{I}_{C_l}|_{C_l})^{\otimes (\mu_l - k)} \otimes (\mathcal{I}_{C_{l+1}}|_{C_l})^{ \otimes (\mu_{l+1})} $, and we need therefore to compute $ \mathrm{deg}_{C_l}(\mathcal{I}_{C_l}|_{C_l}) $ and $ \mathrm{deg}_{C_l}(\mathcal{I}_{C_{l+1}}|_{C_l}) $. But it is easily seen that $ \mathrm{deg}_{C_l}(\mathcal{I}_{C_l}|_{C_l}) = - C_l^2 = b_l $, and that $ \mathrm{deg}_{C_l}(\mathcal{I}_{C_{l+1}}|_{C_l}) = - 1 $. Hence we get that $ \mathrm{deg}_{C_l}(\mathcal{L}_l^k) = (\mu_l - k) b_l - \mu_{l+1} $. If $l=L$, we get instead that $ \mathrm{deg}_{C_L}(\mathcal{L}_L^k) = (\mu_L - k) b_L $. \end{pf}
\begin{prop}\label{lemma 9.10}
Assume that $ g|_{C_l} $ is not the identity on $C_l$. \begin{enumerate} \item If $ l \in \{2, \ldots, L - 1 \} $, we have that
$$ \mathrm{Tr}_{\beta}(e(H^{\bullet}(g|_{C_l},(j_l)^*(u_k^l)))) = \frac{\chi^{r_{l-2}(\mu_l - k)}}{1 - \chi^{- r_{l-1}}} + \frac{\chi^{ - r_l (\mu_l - k) + r_{l-1} \mu_{l+1} }}{1 - \chi^{r_{l-1}}}, $$ for any $ k = 1, \ldots , \mu_l $.
\item If $ l = 1 $, we get
$$ \mathrm{Tr}_{\beta}(e(H^{\bullet}(g|_{C_1},(j_1)^*(u_k^1)))) = \frac{1}{1 - \chi^{ - r_0 }} + \frac{ \chi^{ - r_1 (\mu_1 - k) + r_0 \mu_2}}{1 - \chi^{r_0}} $$ for any $ k = 1, \ldots , \mu_1 $.
\item Finally, if $ l = L $, we get
$$ \mathrm{Tr}_{\beta}(e(H^{\bullet}(g|_{C_L},(j_L)^*(u_k^L)))) = \frac{\chi^{ r_{L-2}(\mu_L - k)}}{1 - \chi^{ - r_{L-1}}} + \frac{1}{1 - \chi^{r_{L-1}}} $$ for any $ k = 1, \ldots , \mu_L $. \end{enumerate} \end{prop} \begin{pf}
Let us first assume that $ l \in \{2, \ldots, L - 1 \} $. We are going to apply Proposition \ref{prop. 5.6}. The only fixed points of the automorphism $ g|_{C_l} : C_l \rightarrow C_l $ are the two points where $C_l$ meets the other components, denoted as usual by $ y_{l-1}$ and $y_l$. From Lemma \ref{fiberlemma}, we have that $$ \mathrm{Tr}((j_l)^*(u_k^l)(y_{l-1})) = \chi^{r_{l-2}(\mu_l - k)}, $$ and that $$ \mathrm{Tr}((j_l)^*(u_k^l)(y_{l})) = \chi^{ - r_l (\mu_l - k) + r_{l-1} \mu_{l+1}}. $$
Let $ dg(y) : \Omega(y) \rightarrow \Omega(y) $ denote the induced automorphism of the cotangent space at $y$ on $ C_l $. It follows immediately from Proposition \ref{prop. 3.3} that $ \mathrm{Tr}(dg(y_{l-1})) = \chi^{-r_{l-1}} $, and that $ \mathrm{Tr}(dg(y_{l})) = \chi^{r_{l-1}} $.
Proposition \ref{prop. 5.6} now gives that
$$ \mathrm{Tr}_{\beta}(e(H^{\bullet}(g|_{C_l},(j_l)^*(u_l^k)))) = \frac{\chi^{r_{l-2}(\mu_l - k)}}{1 - \chi^{- r_{l-1}}} + \frac{\chi^{ - r_l (\mu_l - k) + r_{l-1} \mu_{l+1} }}{1 - \chi^{r_{l-1}}},$$
which is precisely the sought after formula.
The cases $ l = 1 $ and $ l = L $ are treated in the same fashion, when we recall that $ r_{-1} \equiv_n 0 $, and $ r_L := 0 $. \end{pf}
\begin{ntn}
We will write $ \mathrm{Tr}_{\varepsilon}^{l , k}(\xi) := \mathrm{Tr}_{\beta}(e(H^{\bullet}(g|_{C_l},(j_l)^*(u_l^k)))) $. \end{ntn}
We can then make the following definition:
\begin{dfn}\label{contribution from edge} Let
\begin{equation} \mathrm{Tr}_{\varepsilon}(\xi) : = \sum_{l = 1}^{L(\varepsilon)} \sum_{k = 1}^{\mu_l} \mathrm{Tr}_{\varepsilon}^{l , k}(\xi). \end{equation}
We say that $ \mathrm{Tr}_{\varepsilon}(\xi) $ is the \emph{contribution} from $ \varepsilon \in \mathcal{E} $ to the trace
$$ \mathrm{Tr}_{\beta}(e(H^{\bullet}(g|_{\mathcal{Y}_k})). $$ \end{dfn}
\begin{rmk} Let us note that $ \mathrm{Tr}_{\varepsilon}(\xi) $ is defined entirely in terms of the intrinsic data of the singularity associated to $ \varepsilon $. Furthermore, this expression does not depend on the order in which we chose $ \varepsilon $. It is also clear that this expression does not depend on the chosen subfiltration of the divisor $ \sum_{i = 1}^{L(\varepsilon)} m_{\varepsilon_i} C_{\varepsilon_i} $. In the next section we shall define this trace \emph{intrinsically} for the singularity. \end{rmk}
We will now show that we obtain a formula for the Brauer trace $ \mathrm{Tr}_{\beta}(e(H^{\bullet}(g|_{\mathcal{Y}_k}))) $, in terms of the vertex and edge contributions discussed above.
\begin{thm}\label{thm. 9.13} Let $ g \in G $ be a group element corresponding to a root of unity $\xi \in \boldsymbol{\mu}_n$. Then we have that
$$ \mathrm{Tr}_{\beta}(e(H^{\bullet}(g|_{\mathcal{Y}_k}))) = \sum_{\upsilon \in \mathcal{V}} \mathrm{Tr}_{\upsilon}(\xi) + \sum_{\varepsilon \in \mathcal{E}} \mathrm{Tr}_{\varepsilon}(\xi). $$ Furthermore, this expression depends only on $ \Gamma(\mathcal{X}_k) $ and the functions $ \mathfrak{g} $ and $ \mathfrak{m} $ defined in Section \ref{9.1}.
\end{thm} \begin{pf} We begin with choosing a special filtration $$ 0 < \ldots < Z_{\varepsilon_i} < \ldots < Z_{\varepsilon_1} = Z_{\mathcal{E}} < \ldots < Z_{\upsilon_i} < \ldots < Z_{\upsilon_1} = \mathcal{Y}_k. $$ It then follows from Proposition \ref{prop. 6.6} that
$$ \mathrm{Tr}_{\beta}(e(H^{\bullet}(g|_{\mathcal{Y}_k}))) = \sum_{\upsilon \in \mathcal{V}} \mathrm{Tr}_{\upsilon}(\xi) + \sum_{\varepsilon \in \mathcal{E}} \mathrm{Tr}_{\varepsilon}(\xi), $$ where $ \mathrm{Tr}_{\upsilon}(\xi) $ is the expression defined in Definition \ref{contribution from vertex} and $ \mathrm{Tr}_{\varepsilon}(\xi) $ is the expression defined in Definition \ref{contribution from edge}.
It follows from Proposition \ref{prop. 9.7} that $ \mathrm{Tr}_{\upsilon}(\xi) $ only depends on the combinatorial structure of $ \mathcal{X}_k $. Likewise, Proposition \ref{lemma 9.8} and Proposition \ref{lemma 9.10} give that $ \mathrm{Tr}_{\varepsilon}(\xi) $ only depends on the combinatorial structure of $ \mathcal{X}_k $. Therefore, the same is true for the sum of these expressions. \end{pf}
\section{Trace formulas for singularities}\label{trace formula}
\subsection{} In Section \ref{special filtrations}, given $ \epsilon \in \mathcal{E}(\mathcal{X}_k) $ and a tame extension $S'/S$ of degree $n$, we defined the expression $ \mathrm{Tr}_{\epsilon}(\xi) $ for any $ \xi \in \boldsymbol{\mu}_n $, and we were able to compute it using the global geometry of the surface $ \mathcal{Y} $, combined with the local description of the $ \boldsymbol{\mu}_n $-action on $ \mathcal{Y} $. However, the formula we obtain is expressed in terms of intrinsic data for the unique singularity of $ \mathcal{Y} $ associated to $ \epsilon $. So it makes sense to try to define this expression for any tame cyclic quotient singularity, without thinking of the global situation. Moreover, we will investigate the properties of the trace formulas, and in particular express them in a closed, polynomial form.
\subsection{} Let $ \sigma = (m_1,m_2,n) $ be a singularity, as defined in Definition \ref{Def. 8.1}, and let $$ \rho_{\sigma} : \widetilde{\mathcal{Z}} \rightarrow \mathcal{Z} $$
be the minimal desingularization. We denote by $ Z $ the exceptional fiber $ \rho_{\sigma}^{-1}(z) $, where $ z \in \mathcal{Z} $ is the unique singular point of $\mathcal{Z}$. Then $ G = \boldsymbol{\mu}_n $ acts on $ \widetilde{\mathcal{Z}} $, and in particular on $ Z $. So every $ g \in G $ induces an automorphism $ e(H^{\bullet}(g|_Z)) $ of $ e(H^{\bullet}(Z, \mathcal{O}_Z)) $.
In order to compute the trace of $ e(H^{\bullet}(g|_Z)) $, it is easily seen that we can apply the same methods as in Section \ref{special filtrations}. Let $ L = L(\sigma) $ denote the length of the resolution chain, and let the irreducible components of $ \rho_{\sigma}^{-1}(z) $ be denoted $C_l$, and let ~$ \mu_l = \mathrm{mult}(C_l) $. So again, we can choose a filtration $$ 0 < Z_m < \ldots < Z_i < \ldots < Z_2 < Z_1 = Z, $$ where $ m = \sum_{l=1}^{L} \mu_l $, and $ Z_i - Z_{i+1} = C_{l_i} $, for every $i$. At the $i$-th step in this filtration, there is as usual associated an invertible sheaf $ \mathcal{L}_i $, supported on $ C_{l_i} $, and for every $ g \in G $, an isomorphism $$ u_i : g^* (j_{C_{l_i}})_* \mathcal{L}_i \rightarrow (j_{C_{l_i}})_* \mathcal{L}_i, $$
where $ j_{C_{l_i}} : C_{l_i} \hookrightarrow \widetilde{\mathcal{Z}} $ is the canonical inclusion. This set of data gives, for every $i$, an automorphism $ e(H^{\bullet}(g|_{C_{l_j}}, (j_{C_{l_i}})^*u_i)) $ such that
$$ \mathrm{Tr}(e(H^{\bullet}(g|_Z))) = \sum_{j=1}^m \mathrm{Tr}(e(H^{\bullet}(g|_{C_{l_j}}, (j_{C_{l_i}})^*u_i))). $$
The trace is independent of which filtration we choose. So in particular, we can use the \emph{special} filtration $$ 0 =: Z_{L+1} < Z_L < \ldots < Z_l < \ldots < Z_1 := Z, $$ where the $Z_l$ are defined by the refinements $$ Z_{l+1} := Z_l^{\mu_l + 1} < \ldots < Z_l^k < \ldots < Z_l^1 := Z_l, $$ and $ Z_l^{k} - Z_l^{k+1} = C_l $, for every $ k \in \{ 1, \ldots, \mu_l \} $.
We immediately get the following result:
\begin{prop}Let $ g \in G $ be a group element corresponding to a root of unity $ \xi $. Let us write $ \mathrm{Tr}_{\sigma}(\xi) := \mathrm{Tr}(e(H^{\bullet}(g|_Z))) $ and $ \mathrm{Tr}_{\sigma}^{l,k} (\xi) := \mathrm{Tr}(e(H^{\bullet}(g|_{C_l}, (j_{C_l})^* u_l^k))) $. Then we have that $$ \mathrm{Tr}_{\sigma}(\xi) = \sum_{l=1}^L \sum_{k=1}^{\mu_l} \mathrm{Tr}_{\sigma}^{l,k} (\xi). $$ \end{prop}
\begin{rmk} The connection to Section \ref{special filtrations} is as follows: Let $ \varepsilon \in \mathcal{E}(\mathcal{X}_k) $ be an edge, and let $ \sigma = (m_1,m_2,n) $ be the singularity of $ \mathcal{Y} $ corresponding to $ \varepsilon $, after a base change $S'/S$ of degree $n$. Let $ \xi \in \boldsymbol{\mu}_n $. Then we have that $ \mathrm{Tr}_{\sigma}(\xi) = \mathrm{Tr}_{\varepsilon}(\xi) $. \end{rmk}
We are now going to investigate closer the terms in the formula $$ \mathrm{Tr}_{\sigma}(\xi) = \sum_{l=1}^L \sum_{k=1}^{\mu_l} \mathrm{Tr}_{\sigma}^{l,k} (\xi). $$
\begin{lemma}\label{sigmasing} Consider a singularity $ \sigma = (m_1,m_2,n) $, and let $ \xi \in \boldsymbol{\mu}_n $ be a primitive root of unity. Let $ \chi = \xi^{\alpha_1} $, where $ \alpha_1 m_1 \equiv_n 1 $. Then we have that \begin{enumerate} \item $$ \sum_{k=1}^{\mu_l} \mathrm{Tr}_{\sigma}^{l,k}(\xi) = \sum_{k=1}^{\mu_l} \frac{ \chi^{r_{l-2}(\mu_l - k)}}{1 - \chi^{- r_{l-1}}} + \sum_{k=1}^{\mu_l} \frac{ \chi^{ - r_l (\mu_l - k) + r_{l-1} \mu_{l+1} }}{1 - \chi^{r_{l-1}}}, $$ for any $ l \in \{2, \ldots, L - 1\} $, that
\item $$ \sum_{k=1}^{\mu_1} \mathrm{Tr}_{\sigma}^{1,k}(\xi) = \sum_{k=1}^{\mu_1} \frac{1}{1 - \chi^{ - r_0 }} + \sum_{k=1}^{\mu_1} \frac{ \chi^{ - r_1 (\mu_1 - k) + r_0 \mu_2}}{1 - \chi^{r_0}}, $$ and that \item $$ \sum_{k=1}^{\mu_L} \mathrm{Tr}_{\sigma}^{L,k}(\xi) = \sum_{k=1}^{\mu_L} \frac{\chi^{ r_{L-2}(\mu_L - k)}}{1 - \chi^{ - r_{L-1}}} + \sum_{k=1}^{\mu_L} \frac{1}{1 - \chi^{r_{L-1}}}. $$ \end{enumerate} \end{lemma} \begin{pf} The proof is similar to the proof of Lemma \ref{lemma 9.10}. \end{pf}
\subsection{Formal manipulations} Consider a singularity $ \sigma = (m_1,m_2,n) $. We have, for any $ \xi \in \boldsymbol{\mu}_n $, a trace $ \mathrm{Tr}_{\sigma}(\xi) = \sum_{l=1}^L \sum_{k=1}^{\mu_l} \mathrm{Tr}_{\sigma}^{l,k} (\xi) $. We would like to have an explicit formula for this sum. In particular, we would like to know if it is a polynomial in $\xi$, and in which way it depends on the parameters of the singularity. In order to deal with these questions, and to obtain a closed formula, some formal manipulations of the expressions in Lemma \ref{sigmasing} will be necessary. Unless otherwise mentioned, $ \xi $ will be a primitive root of unity.
The following lemma is an easy computation, whose proof is omitted. \begin{lemma}\label{Lemma 10.3} We have that $$ \chi^{ r_{l-1} \mu_{l+1} } \sum_{k=1}^{\mu_l} \frac{ \chi^{ - r_l (\mu_l - k) }}{1 - \chi^{r_{l-1}}} + \sum_{k=1}^{\mu_{l+1}} \frac{ \chi^{r_{l-1} (\mu_{l+1} - k)}}{1 - \chi^{- r_l}} = \frac{1 - \chi^{ r_{l-1} \mu_{l+1} - r_l \mu_l}}{(1 - \chi^{r_{l-1}})(1 - \chi^{- r_l})}, $$ for $ l \in \{ 1, \ldots, L-1 \}$. \end{lemma}
Let us make the following definition:
\begin{dfn} We define $$ \mathrm{Tr}_{y_l}(\xi) := \frac{1 - \chi^{ r_{l-1} \mu_{l+1} - r_l \mu_l}}{(1 - \chi^{r_{l-1}})(1 - \chi^{- r_l})}, $$ for every $ l \in \{ 1, \ldots, L-1 \}$. We also define $$ \mathrm{Tr}_{y_0}(\xi) := \frac{\mu_1}{1 - \chi^{-r_0}} $$ and $$ \mathrm{Tr}_{y_L}(\xi) := \frac{\mu_L}{1 - \chi^{r_{L-1}}}. $$ \end{dfn}
Note that it follows that $ \mathrm{Tr}_{\sigma}(\xi) = \sum_{l=0}^L \mathrm{Tr}_{y_l}(\xi) $. We shall find a more convenient way to write $ \mathrm{Tr}_{y_l} $. The first step is Lemma \ref{lemma 10.5} below, whose proof is omitted.
\begin{lemma}\label{lemma 10.5} For every $ l \in \{ 1, \ldots, L-1 \}$, we have $$ 1 - \chi^{ r_{l-1} \mu_{l+1} - r_l \mu_l} = $$ $$ (1 - \chi^{- r_l \mu_l})(1 - \chi^{r_{l-1} \mu_{l+1}}) + \chi^{r_{l-1} \mu_{l+1}} (1 - \chi^{- r_l \mu_l}) + \chi^{- r_l \mu_l}(1 - \chi^{r_{l-1} \mu_{l+1}}). $$ \end{lemma}
\begin{cor}\label{cor. 10.6} For every $ l \in \{ 1, \ldots, L-1 \}$, we can write $ \mathrm{Tr}_{y_l} $ in the following form: $$ \mathrm{Tr}_{y_l}(\xi) = \sum_{k=0}^{\mu_l - 1} (\chi^{- r_l})^k \sum_{k=0}^{\mu_{l+1} - 1} (\chi^{r_{l-1}})^k + $$ $$ \frac{\chi^{ r_{l-1} \mu_{l+1}}}{1 - \chi^{ r_{l-1}}} \sum_{k=0}^{\mu_l - 1} (\chi^{- r_l})^k + \frac{\chi^{- r_l \mu_l}}{1 - \chi^{- r_l}} \sum_{k=0}^{\mu_{l+1} - 1} (\chi^{r_{l-1}})^k. $$ \end{cor}
We introduce some notation for the terms appearing in $ \mathrm{Tr}_{y_l}(\xi) $. \begin{dfn}\label{dfn 10.7} We define \begin{enumerate} \item $$ \mathrm{Tr}_{\mu_l, \mu_{l+1}}(\xi) := \sum_{k=0}^{\mu_l - 1} (\chi^{- r_l})^k \sum_{k=0}^{\mu_{l+1} - 1} (\chi^{r_{l-1}})^k, $$ for all $ l \in \{ 0, \ldots, L \} $, \item $$ \mathrm{Tr}_{y_l}^{\mu_l}(\xi) := \frac{\chi^{ r_{l-1} \mu_{l+1}}}{1 - \chi^{ r_{l-1}}} \sum_{k=0}^{\mu_l - 1} (\chi^{- r_l})^k, $$ for all $ l \in \{ 1, \ldots, L \} $, and \item $$ \mathrm{Tr}_{y_l}^{\mu_{l+1}}(\xi) := \frac{\chi^{- r_l \mu_l}}{1 - \chi^{- r_l}} \sum_{k=0}^{\mu_{l+1} - 1} (\chi^{r_{l-1}})^k, $$ for all $ l \in \{ 0, \ldots, L-1 \} $. \end{enumerate} \end{dfn}
\begin{lemma}\label{lemma blabla} For any $ l $ such that $ 0 \leq l \leq L - 1 $, we have that $$ \mathrm{Tr}_{y_l}^{\mu_{l+1}}(\xi) + \mathrm{Tr}_{y_{l + 1}}^{\mu_{l+1}}(\xi) = - \sum_{k=0}^{\mu_{l+1} - 1} \chi^{r_{l-1} k - r_l (\mu_l - 1)} (1 + (\chi^{r_{l}}) + \ldots + (\chi^{r_l})^{b_{l+1}(\mu_{l+1} - k) - 2}). $$
Furthermore, we may also write this sum as $$ \sum_{k=0}^{\mu_{l+1} - 1} \chi^{r_{l-1} k} ((\chi^{ - r_{l}})^{ \mu_l - 1 } + (\chi^{ - r_{l}})^{ \mu_l - 2 } + \ldots + (\chi^{ - r_{l}})^{b_{l+1} k - ( \mu_{l + 2} - 1 )}). $$ \end{lemma} \begin{pf} Observe that we have $$ (\sum_{k=0}^{\mu_{l+1} - 1} (\chi^{r_{l-1}})^k) \frac{\chi^{- r_l \mu_l}}{1 - \chi^{- r_l}} + (\sum_{k=0}^{\mu_{l+1} - 1} (\chi^{ - r_{l+1}})^k) \frac{\chi^{ r_{l} \mu_{l+2}}}{1 - \chi^{ r_{l}}} $$ $$ = \frac{1}{1 - \chi^{ r_{l}}} \sum_{k=0}^{\mu_{l+1} - 1} (\chi^{ - r_{l+1} k + r_{l} \mu_{l+2} } - \chi^{r_{l-1} k - r_l (\mu_l - 1 ) } ). $$ We can rewrite $ \chi^{ - r_{l+1} k + r_{l} \mu_{l+2} } - \chi^{r_{l-1} k - r_l (\mu_l - 1 ) } $ as $$ - \chi^{r_{l-1} k - r_l (\mu_l - 1 ) } ( 1 - \chi^{ - r_{l+1} k + r_{l} \mu_{l+2} - r_{l-1} k + r_l (\mu_l - 1 ) }). $$ We then compute that $$ - r_{l+1} k + r_{l} \mu_{l+2} - r_{l-1} k + r_l (\mu_l - 1 ) = r_l (b_{l+1} ( \mu_{l+1} - k ) - 1), $$ since $ \mu_l + \mu_{l+2} = b_{l+1} \mu_{l+1} $ and $ r_{l-1} = b_{l+1} r_{l} - r_{l+1} $.
This gives that $$ \chi^{ - r_{l+1} k + r_{l} \mu_{l+2} } - \chi^{r_{l-1} k - r_l (\mu_l - 1 ) } = - \chi^{r_{l-1} k - r_l (\mu_l - 1 ) } ( 1 - (\chi^{r_l})^{b_{l+1} ( \mu_{l+1} - k ) - 1}). $$ Recall that $ b_{l+1} \geq 2 $, so for all $ 0 \leq k \leq \mu_{l+1} - 1 $, we have that $$ b_{l+1} ( \mu_{l+1} - k ) - 1 \geq 1. $$ But then it follows that $$ \frac{1 - (\chi^{r_l})^{b_{l+1} ( \mu_{l+1} - k ) - 1}}{1 - \chi^{ r_{l}}} = 1 + (\chi^{r_l}) + \ldots + (\chi^{r_l})^{b_{l+1} ( \mu_{l+1} - k ) - 2}. $$
The last statement follows from observing that $$ (\mu_l - 1) - (b_{l+1} ( \mu_{l+1} - k ) - 2) = b_{l+1} k - ( \mu_{l + 2} - 1 ). $$ \end{pf}
It is convenient to introduce some notation for the terms of this form.
\begin{dfn} For any $l$ such that $ 0 \leq l \leq L-1 $, we define $$ \mathrm{Tr}_{\mu_{l+1}}(\xi) := \mathrm{Tr}_{y_l}^{\mu_{l+1}}(\xi) + \mathrm{Tr}_{y_{l + 1}}^{\mu_{l+1}}(\xi). $$ \end{dfn}
It is now possible to express $ \mathrm{Tr}_{\sigma}(\xi) $ in the following form:
\begin{prop}\label{prop. 10.14} Let $ \sigma = (m_1, m_2, n) $ be a singularity, and let $ \xi \in \boldsymbol{\mu}_n $ be a primitive root. Then we have that $$ \mathrm{Tr}_{\sigma}(\xi) = \sum_{l=0}^{L} \mathrm{Tr}_{\mu_l, \mu_{l+1}}(\xi) + \sum_{l=1}^{L} \mathrm{Tr}_{\mu_{l}}(\xi). $$ \end{prop} \begin{pf} This is rather immediate, with the exception of the appearance of the terms $ \mathrm{Tr}_{\mu_0, \mu_1} $, $ \mathrm{Tr}_{\mu_1} $, $ \mathrm{Tr}_{\mu_L} $ and $ \mathrm{Tr}_{\mu_L,\mu_{L+1} } $ in the formula. But an easy computation shows that $$ \sum_{k=0}^{\mu_1 - 1} \frac{1}{1 - \chi^{ - r_0 }} + \mathrm{Tr}_{y_1}^{\mu_1} = (\sum_{k=0}^{\mu_1 - 1} \frac{1}{1 - \chi^{ - r_0 }} - \mathrm{Tr}_{y_0}^{\mu_1}) + (\mathrm{Tr}_{y_0}^{\mu_1} + \mathrm{Tr}_{y_1}^{\mu_1}) = \mathrm{Tr}_{\mu_0, \mu_1} + \mathrm{Tr}_{\mu_1}. $$ In a similar way, we compute $$ \mathrm{Tr}_{y_{L-1}}^{\mu_L} + \sum_{k=1}^{\mu_L} \frac{1}{1 - \chi^{r_{L-1}}} = \mathrm{Tr}_{\mu_L} + \mathrm{Tr}_{\mu_L,\mu_{L+1} }. $$ \end{pf}
Furthermore, the following is now automatic from our description:
\begin{cor}\label{nonprimformula} Let $ \xi \in G $ be any root of unity. Then we have that $$ \mathrm{Tr}_{\sigma}(\xi) = \sum_{l=0}^{L} \mathrm{Tr}_{\mu_l, \mu_{l+1}}(\xi) + \sum_{l=1}^{L} \mathrm{Tr}_{\mu_{l}}(\xi). $$ \end{cor} \begin{pf} Since $p$ does not divide the order of $ \boldsymbol{\mu}_n $, it follows that the action of $ \boldsymbol{\mu}_n $ on $ H^i(Z, \mathcal{O}_Z) $, where $ i \in \{0,1\} $, is diagonalizable. In particular, the irreducible Brauer characters are all of the form $ \xi \mapsto \xi^j $, for some $ j \geq 0 $. Therefore, the Brauer trace is given by the same polynomial for any $ \xi \in \boldsymbol{\mu}_n $. \end{pf}
\section{Trace formula}\label{Trace formula} Let $ \sigma = (m_1,m_2,n) $ be a singularity. In this section, we will prove an explicit formula for $\mathrm{Tr}_{\sigma}(\xi)$, under the assumption that $ n \gg 0 $. We will see that the ``shape'' of this formula is closely related to the properties of the exceptional locus of the minimal desingularization of $ \sigma $.
In Proposition \ref{prop. 8.9}, we saw that the exceptional fiber of the minimal desingularization of $ \sigma = (m_1, m_2, n) $, for $ n \gg 0 $, only depended on the residue class $ [n]_{M} $ of $ n $ modulo $ M = \mathrm{lcm}(m_1,m_2)$, modulo chains of curves with multiplicity $ m = \mathrm{gcd}(m_1,m_2) $. More precisely, the multiplicities satisfy: $$ m_2 = \mu_0 > \mu_1 > \ldots > \mu_{l_0} = \ldots = m = \ldots = \mu_{L+1-l_1} < \ldots < \mu_L < \mu_{L+1} = m_1. $$ The integer $l_0$ and the sequence of multiplicities $ \mu_1, \ldots, \mu_{l_0} $, as well as the integer $l_1$ and the sequence of multiplicities $ \mu_{L+1-l_1}, \ldots, \mu_L $ depend only on $ [n]_{M} $.
In fact, we shall see that $\mathrm{Tr}_{\sigma}(\xi)$ can be written as a polynomial, where the degree and the coefficients only depend on $ \mu_0 $, $ \mu_1 $, $ \mu_L $ and $ \mu_{L+1} $. Since $ \mu_0 $ and $ \mu_{L+1} $ are fixed, and $ \mu_1 $ and $ \mu_L $ only depend on the residue class of $n$ modulo $M$, it will follow that this also holds for $\mathrm{Tr}_{\sigma}(\xi)$.
As a further motivation for what we will do in this section, we remark that as $n$ goes to infinity (but with a fixed residue class modulo $M$), any $ \xi \in \boldsymbol{\mu}_n $ behaves ``less'' like a root of unity, and more like an independent variable. This suggests that the cancellations occuring in our formulas are of a formal nature, and that we should substitute $ \xi $ with a variable.
\subsection{Formal substitution} For any $ n \gg 0 $ with a fixed residue class $ [n]_{M} $, the self intersection numbers $ b_1, \ldots, b_{l_0} $ are constant, since they are computed in terms of the multiplicities via the formula $ b_l \mu_l = \mu_{l-1} + \mu_{l+1} $. However, the integers $ r_l $ may vary as $n$ varies. But on the other hand, these integers are related in terms of the $b_l$ via the equations $ r_{l-1} = b_{l+1} r_l - r_{l+1} $.
We will now define universal polynomials $ P_l $ inductively, by the following procedure: Put $ P_{-1} = 0 $, and $ P_0 = 1 $. Then we define $ P_l = b_l P_{l-1} - P_{l-2} $, for $ l \geq 1 $. Note that $ P_l = P_l(b_1, \ldots, b_l) $, when $ l \geq 1 $. For instance, we have $ P_1 = b_1 $, $ P_2 = b_2 b_1 - 1 $, $ P_3 = b_3(b_2b_1 - 1) - b_1 $, and so on.
The importance of these polynomials is that $ r_l \equiv_n P_l r_0 $. So for any $n$-th root of unity $\eta$, we have that $ \eta^{r_l} = \eta^{P_l r_0} $. Recall that $ \chi := \xi^{\alpha_1} $, and that $ - \alpha_1 r_0 \equiv_n \alpha_2 $. It follows that we can write \begin{equation}\label{equation 10.5} \chi^{r_l} = \xi^{\alpha_1 r_l} = \xi^{ \alpha_1 r_0 P_l} = (\xi^{ \alpha_2})^{ - P_l}. \end{equation}
\subsection{}\label{lemmas}
Recall from Proposition \ref{prop. 10.14} that we had $$ \mathrm{Tr}_{\sigma}(\xi) = \sum_{l=1}^{L+1} \mathrm{Tr}_{\mu_{l-1}, \mu_{l}}(\xi) + \sum_{l=1}^{L} \mathrm{Tr}_{\mu_l}(\xi), $$ for any $ \xi \in \boldsymbol{\mu}_n $. It turns out that a small reformulation of the expressions $ \mathrm{Tr}_{\mu_{l-1}, \mu_{l}}(\xi) $ and $ \mathrm{Tr}_{\mu_l}(\xi) $ is suitable for our computations later on. Let us also here remark that we will drop the reference to $ \xi $ from the notation.
Let $ q \in \{ 1, \ldots, L \} $ be an integer such that $ \mu_q = m $. For all $ l \in \{1, \ldots, q \} $ such that $ \mu_{l+1} > 1 $, we then define: \begin{equation} \mathrm{Tr}_{l-1,l} := \sum_{k=1}^{\mu_{l} - 1} (\chi^{r_{l-2}})^k ( (\chi^{-r_{l-1}})^{b_l k -1} + \ldots + 1), \end{equation} and \begin{equation} \mathrm{Tr}_l := - \sum_{k=0}^{\mu_{l} - 1} (\chi^{r_{l-2}})^k ( (\chi^{-r_{l-1}})^{b_l k -1} + \ldots + (\chi^{-r_{l-1}})^{b_l k - (\mu_{l+1} - 1)}). \end{equation}
We can replace $ \mathrm{Tr}_{\mu_{l-1}, \mu_{l}} + \mathrm{Tr}_{\mu_l} $ with $ \mathrm{Tr}_{l-1,l} + \mathrm{Tr}_l $ in the expression for $ \mathrm{Tr}_{\sigma}(\xi) $, as the lemma below shows.
\begin{lemma}\label{Lemma 11.3} For all $ l \in \{1, \ldots, q \} $ such that $ \mu_{l+1} > 1 $, we have that $$ \mathrm{Tr}_{\mu_{l-1}, \mu_{l}} + \mathrm{Tr}_{\mu_l} = \mathrm{Tr}_{l-1,l} + \mathrm{Tr}_l. $$ \end{lemma} \begin{pf} We will write $ \mathrm{Tr}_{\mu_l}^{(k)} $ for the $k$-th summand in the expression for $ \mathrm{Tr}_{\mu_l} $, and likewise for $ \mathrm{Tr}_{\mu_{l-1}, \mu_{l}}^{(k)} $, $ \mathrm{Tr}_l^{(k)} $ and $ \mathrm{Tr}_{l-1,l}^{(k)} $.
Let us first consider the case where $ b_l k - 1 > \mu_{l-1} - 1 $. Then we have that $$ \mathrm{Tr}_{l-1,l}^{(k)} = (\chi^{r_{l-2}})^k ( (\chi^{-r_{l-1}})^{b_l k -1} + \ldots + 1) $$ and $$ \mathrm{Tr}_l^{(k)} = - (\chi^{r_{l-2}})^k ( (\chi^{-r_{l-1}})^{b_l k -1} + \ldots + (\chi^{-r_{l-1}})^{b_l k - (\mu_{l+1} - 1)}), $$ that is, we add some monomials to $ \mathrm{Tr}_{\mu_{l-1}, \mu_{l}}^{(k)} $ and subtract exactly the same monomials from $ \mathrm{Tr}_{\mu_l}^{(k)} $. It follows easily that $$ \mathrm{Tr}_{\mu_{l-1}, \mu_{l}}^{(k)} + \mathrm{Tr}_{\mu_l}^{(k)} = \mathrm{Tr}_{l-1,l}^{(k)} + \mathrm{Tr}_l^{(k)}. $$
For $ k = 0 $, one easily computes that $$ \mathrm{Tr}_{\mu_l}^{(0)} = - \mathrm{Tr}_{\mu_{l-1}, \mu_{l}}^{(0)} + \mathrm{Tr}_l^{(0)}, $$ and for all $ k \geq 1 $ such that $ b_l k \leq \mu_{l-1} - 1 $, it is trivial to see that $$ \mathrm{Tr}_{\mu_{l-1}, \mu_{l}}^{(k)} + \mathrm{Tr}_{\mu_l}^{(k)} = \mathrm{Tr}_{l-1,l}^{(k)} + \mathrm{Tr}_l^{(k)}. $$
\end{pf}
We will now replace $ \chi^{-r_0} $ in the expressions $ \mathrm{Tr}_{l-1,l} $ and $ \mathrm{Tr}_l $ above with a formal variable $y$. There are two reasons for doing this. First, we can write these expressions in a compact form as polynomials in $y$. Second, when we consider various sums of these expressions, it is manageable to keep track of the formal cancellations that occur.
\begin{lemma}\label{Omskriving} Let $ l \in \{ 1, \ldots, q \} $ be such that $ \mu_{l+1} > 1 $. Put $ y = \chi^{-r_0} $. Then we can write \begin{enumerate} \item $$ \mathrm{Tr}_{l-1,l} = \sum_{k=1}^{\mu_l - 1} \sum_{m=1}^{b_l k} y^{k P_l - m P_{l-1}}, $$ and \item $$ \mathrm{Tr}_l = - \sum_{k=0}^{\mu_l - 1} \sum_{m=1}^{\mu_{l+1} - 1} y^{k P_l - m P_{l-1}}. $$ \end{enumerate} \end{lemma} \begin{pf} We have in case (i) that $$ \mathrm{Tr}_{l-1,l} = \sum_{k=1}^{\mu_l - 1} (\chi^{r_{l-2}})^k ( (\chi^{-r_{l-1}})^{b_l k - 1} + \ldots + 1) = $$ $$ \sum_{k=1}^{\mu_l - 1} y^{-P_{l-2} k} (y^{b_l P_{l-1} k - P_{l-1}} + \ldots + y^{b_l P_{l-1} k - b_l k P_{l-1}}) =$$ $$ \sum_{k=1}^{\mu_l - 1} \sum_{m=1}^{b_l k} y^{(b_l P_{l-1} - P_{l-2})k - m P_{l-1}} = \sum_{k=1}^{\mu_l - 1} \sum_{m=1}^{b_l k} y^{k P_l - m P_{l-1}}. $$
The proof of (ii) is similar, and is omitted.
\end{pf}
\subsection{} This section consists of three lemmas that we will use when proving the trace formula in Theorem \ref{Formula}. The reader might want to skip this section for now, and refer back to these results when needed in Section \ref{star} and Section \ref{explicit formula}.
\begin{lemma}\label{Lemma 11.4} Let $ \sigma = (m_1,m_2,n) $ be a singularity where $ n \gg 0 $, and let $ L $ be the length of the resolution of $\sigma$. Let $ q \in \{1, \ldots, L\} $ be such that $ \mu_q = m $ and choose an integer $ l \in \{1, \ldots, L\} $ such that $ l + 1 \leq q $. Consider the inequality $$ (*) ~ b_l t - \left \lceil t \frac{\mu_{l+1}}{\mu_l} \right \rceil \geq s, $$ where $ 1 \leq s \leq \mu_{l-1} $, and where $ t \geq 0 $ is an integer. Then we have that $ t(s) := \left \lceil s \frac{\mu_{l}}{\mu_{l-1}} \right \rceil $ is the smallest positive integer that satisfies the inequality $(*)$. Note in particular that $ 1 \leq t(s) \leq \mu_l $.
\end{lemma} \begin{pf} Let us first show that $ t(s) $ satisfies the inequality. By definition we have that $ s \frac{\mu_{l}}{\mu_{l-1}} \leq t(s) $, so it follows that $ s \leq t(s) \frac{\mu_{l-1}}{\mu_{l}} $. From the equality $ b_l \mu_l = \mu_{l+1} + \mu_{l-1} $ it follows that $ \frac{\mu_{l-1}}{\mu_{l}} = b_l - \frac{\mu_{l+1}}{\mu_{l}} $. So we get that $$ s \leq t(s)(b_l - \frac{\mu_{l+1}}{\mu_{l}}) = t(s) b_l - t(s) \frac{\mu_{l+1}}{\mu_{l}}. $$ From this, it follows that $ t(s) \frac{\mu_{l+1}}{\mu_{l}} \leq t(s) b_l - s $, and since $ t(s) $, $ b_l $ and $s$ are integers, we actually have that $ \left \lceil t(s) \frac{\mu_{l+1}}{\mu_{l}} \right \rceil \leq t(s) b_l - s $, and hence $t(s)$ satisfies $ (*) $.
Assume now that $ t \geq 0 $ is an integer such that $ b_l t - \left \lceil t \frac{\mu_{l+1}}{\mu_l} \right \rceil \geq s $, and that $ t < t(s) $. This implies that $ t < s \frac{\mu_{l}}{\mu_{l-1}} $, and so it follows that $$ t \mu_{l-1} < s \mu_{l} \leq \mu_{l} (b_l t - \left \lceil t \frac{\mu_{l+1}}{\mu_l} \right \rceil) = t \mu_{l-1} + t \mu_{l+1} - \mu_{l} \left \lceil t \frac{\mu_{l+1}}{\mu_l} \right \rceil. $$
So in fact, we get that $ \mu_{l} \left \lceil t \frac{\mu_{l+1}}{\mu_l} \right \rceil < t \mu_{l+1} $, and hence $ \left \lceil t \frac{\mu_{l+1}}{\mu_l} \right \rceil < t \frac{\mu_{l+1}}{\mu_l} $, a contradiction. \end{pf}
\begin{lemma}\label{Lemma Key2} Let $ \sigma = (m_1,m_2,n) $ be a singularity where $ n \gg 0 $. Let $ q \in \{1, \ldots, L\} $ be such that $ \mu_q = m $, and choose $ l \in \{ 0, \ldots, L-1 \} $ such that $ l + 2 \leq q $. Let us assume that $ \mu_{l+1} \geq 2 $. For all integers $ s $ and $ k $ such that $ 1 \leq s < k \leq \mu_{l+1} $, the strict inequality $$ b_{l+1} k - \left \lceil k \frac{\mu_{l+2}}{\mu_{l+1}} \right \rceil > b_{l+1} s - \left \lceil s \frac{\mu_{l+2}}{\mu_{l+1}} \right \rceil $$ holds. In particular, by taking $ k = \mu_{l+1} $, we get that $$ \mu_l > b_{l+1} s - \left \lceil s \frac{\mu_{l+2}}{\mu_{l+1}} \right \rceil, $$ for all $ s \in \{ 1, \ldots, \mu_{l+1} - 1 \} $. \end{lemma} \begin{pf} We first note that $ \mu_{l+1} \geq \mu_{l+2} $. Observe that $$ b_{l+1} k - b_{l+1} s = b_{l+1} (k - s) \geq 2 (k - s), $$ and likewise $$ k \frac{\mu_{l+2}}{\mu_{l+1}} - s \frac{\mu_{l+2}}{\mu_{l+1}} = (k-s) \frac{\mu_{l+2}}{\mu_{l+1}} \leq (k-s). $$ It follows that $$ k \frac{\mu_{l+2}}{\mu_{l+1}} \leq (k-s) + s \frac{\mu_{l+2}}{\mu_{l+1}} \leq (k-s) + \left \lceil s \frac{\mu_{l+2}}{\mu_{l+1}} \right \rceil. $$ Hence $$ \left \lceil k \frac{\mu_{l+2}}{\mu_{l+1}} \right \rceil - \left \lceil s \frac{\mu_{l+2}}{\mu_{l+1}} \right \rceil \leq (k-s). $$ By combining these inequalities, we get $$ \left \lceil k \frac{\mu_{l+2}}{\mu_{l+1}} \right \rceil - \left \lceil s \frac{\mu_{l+2}}{\mu_{l+1}} \right \rceil \leq (k-s) < 2 (k-s) \leq b_{l+1} k - b_{l+1} s, $$ and therefore $$ b_{l+1} k - \left \lceil k \frac{\mu_{l+2}}{\mu_{l+1}} \right \rceil > b_{l+1} s - \left \lceil s \frac{\mu_{l+2}}{\mu_{l+1}} \right \rceil. $$ \end{pf}
\begin{lemma}\label{Lemma 11 Key} Let $ \sigma = (m_1,m_2,n) $ be a singularity where $ n \gg 0 $. Let $ q \in \{1, \ldots, L\} $ be such that $ \mu_q = m $, and choose $ l \in \{ 0, \ldots, L-1 \} $ such that $ l + 2 \leq q $. Consider the inequalities $ \mu_l \geq \mu_{l+1} \geq \mu_{l+2} $. Let us assume that either \begin{enumerate} \item $ \mu_{l+1} > \mu_{l+2} $, or that \item $ \mu_{l+1} = \mu_{l+2} \geq 3 $. \end{enumerate} Then the inequality $ \mu_l - 1 \geq b_{l+1} $ holds. \end{lemma} \begin{pf} The equality $ \mu_l = b_{l+1} \mu_{l+1} - \mu_{l+2} $ can be written as $$ \mu_l = b_{l+1} + b_{l+1} (\mu_{l+1} - 1) - \mu_{l+2}. $$ Let us first consider case (i), where $ \mu_{l+1} > \mu_{l+2} $. Then $ \mu_{l+1} - 1 \geq \mu_{l+2} $, and $ b_{l+1} \geq 2 $, so $$ b_{l+1} (\mu_{l+1} - 1) - \mu_{l+2} \geq 2 \mu_{l+2} - \mu_{l+2} = \mu_{l+2} \geq 1, $$ and therefore $ \mu_l \geq b_{l+1} + 1 $, as desired.
In case (ii), where $ \mu_{l+1} = \mu_{l+2} \geq 3 $, we see that $$ b_{l+1} (\mu_{l+1} - 1) - \mu_{l+2} \geq 2 (\mu_{l+1} - 1) - \mu_{l+1} = \mu_{l+1} - 2 \geq 1, $$ and so again, it follows that $ \mu_l \geq b_{l+1} + 1 $. \end{pf}
\subsection{Formal cancellation}\label{star} We will now consider sums of the expressions $ \mathrm{Tr}_{l-1,l} $ and $ \mathrm{Tr}_{l} $. It turns out that there will be cancellations occurring in these sums, following a certain pattern. We will eventually, in Proposition \ref{Computation}, end up with a polynomial in $y$ derived from $ \mathrm{Tr}_{0,1} $, which we can compute precisely.
To set this up, let $ q \in \{ 1, \ldots , L-1 \} $ be an integer such that $ \mu_q = m $, and assume that $ \lambda \in \{ 0, \ldots , q \} $ is an index such that either $ \mu_{\lambda + 1} > \mu_{\lambda + 2} $ holds or that $ \mu_{\lambda + 1} = \mu_{\lambda + 2} \geq 3 $ holds. From Lemma \ref{Lemma 11 Key} above, it follows that $ \mu_l - 1 \geq b_{l+1} $ for all $ l \in \{ 0, \ldots, \lambda \} $, and hence it makes sense to define \begin{equation} \mathrm{Tr}_{l}^* := - \sum_{k = b_{l+1}}^{\mu_{l} - 1} \sum_{m=1}^{\left \lceil k \frac{\mu_{l+1}}{\mu_{l}} \right \rceil - 1} y^{k P_{l} - m P_{l-1} }, \end{equation} and \begin{equation} \mathrm{Tr}_{l-1,l}^* := \sum_{k = 1}^{\mu_{l} - 1} \sum_{m = \left \lceil k \frac{\mu_{l+1}}{\mu_{l}} \right \rceil}^{b_l k} y^{k P_{l} - m P_{l-1}. } \end{equation}
\begin{prop}\label{Prop Key} Let us keep the assumptions above. For any index $l$ such that $ 0 \leq l < \lambda $, the following identities hold: \begin{enumerate} \item $ \mathrm{Tr}_l^* = \mathrm{Tr}_l + \mathrm{Tr}_{l,l+1}^* $, and \item $ \mathrm{Tr}_{l-1,l}^* = \mathrm{Tr}_{l-1,l} + \mathrm{Tr}_l^* $. \end{enumerate} \end{prop} \begin{pf} By definition, we have that $$ \mathrm{Tr}_{l, l + 1}^* = \sum_{s = 1}^{\mu_{l + 1} - 1} \sum_{t = \left \lceil s \frac{\mu_{l + 2}}{\mu_{l + 1}} \right \rceil}^{b_{l + 1} s } y^{s P_{l + 1} - t P_{l}}. $$ Furthermore, it follows from Lemma \ref{Omskriving} that $$ \mathrm{Tr}_{l} = - \sum_{k = 0}^{\mu_{l} - 1} \sum_{m=1}^{\mu_{l + 1} - 1} y^{k P_{l} - m P_{l - 1} }, $$ and it is easily seen by re-indexing the expression for $ \mathrm{Tr}_{l, l + 1}^* $ that we may write $$ \mathrm{Tr}_{l, l + 1}^* = \sum_{m=1}^{\mu_{l + 1} - 1} \sum_{k = 0}^{b_{l + 1} m - \left \lceil m \frac{\mu_{l + 2}}{\mu_{l + 1}} \right \rceil} y^{k P_{l} - m P_{l - 1} }. $$
Let us put $$ Q_k := - \sum_{m=1}^{\mu_{l + 1} - 1} y^{k P_{l} - m P_{l - 1} }, $$ and $$ S_m := \sum_{k = 0}^{b_{l + 1} m - \left \lceil m \frac{\mu_{l + 2}}{\mu_{l + 1}} \right \rceil} y^{k P_{l} - m P_{l - 1} }. $$
Consider now $S_m$ for a fixed $ m \in \{1, \ldots, \mu_{l+1} - 1 \} $. We let the monomials in $S_m$ cancel monomials in $Q_k$ in the following systematic way: We let the $k$-th term $ y^{k P_{l} - m P_{l - 1} } $ in $S_m$, where $ 0 \leq k \leq b_{l + 1} m - \left \lceil m \frac{\mu_{l + 2}}{\mu_{l + 1}} \right \rceil $, cancel the $m$-th term $ y^{k P_{l} - m P_{l - 1} } $ in $Q_k$. Note that all terms in $S_m$ are cancelled in this way, since $$ \mu_l - 1 \geq b_{l + 1} m - \left \lceil m \frac{\mu_{l + 2}}{\mu_{l + 1}} \right \rceil, $$ for all $ m \in \{ 1, \ldots, \mu_{l+1} - 1 \} $, by Lemma \ref{Lemma Key2}. On the other hand, if we now fix $ k \in \{ 0, \ldots, \mu_l - 1 \} $, we see that $S_m$ will annihilate the $m$-th term in $Q_k$ precisely when $$ S_m(1) = b_{l + 1} m - \left \lceil m \frac{\mu_{l + 2}}{\mu_{l + 1}} \right \rceil + 1 \geq k + 1. $$
Let now $ m_k $ be the smallest positive integer such that $$ b_{l + 1} m_k - \left \lceil m_k \frac{\mu_{l + 2}}{\mu_{l + 1}} \right \rceil \geq k. $$ From Lemma \ref{Lemma 11.4}, we know that $ m_k = \left \lceil k \frac{\mu_{l + 1}}{\mu_{l}} \right \rceil $. Then it follows that the monomials $ y^{k P_{l} - m P_{l - 1} } $, with $ m_k \leq m \leq \mu_{l + 1} - 1 $, are cancelled in $Q_k $, provided that $ m_k \leq \mu_{l + 1} - 1 $. In the extremal case where $ m_k = \left \lceil k \frac{\mu_{l + 1}}{\mu_{l}} \right \rceil = \mu_{l + 1} $ nothing gets cancelled.
Note in particular that all monomials in $ Q_1, \ldots, Q_{b_{l+1} - 1} $ are cancelled, since in all these cases we have $ m_k = 1 $. We also immediately see that all monomials in $Q_0$ are cancelled.
So we put $ Q_k^* := 0 $, for all $ k \in \{0, \ldots, b_{l+1} - 1 \} $, and $$ Q_k^* := - \sum_{m=1}^{\left \lceil k \frac{\mu_{l + 1}}{\mu_{l}} \right \rceil - 1} y^{k P_{l} - m P_{l - 1} }, $$ for $ k \in \{ b_{l+1}, \ldots, \mu_l - 1 \} $. Note that from Lemma \ref{Lemma 11 Key} we have that $ \mu_l - 1 \geq b_{l+1} $, and note also that $ Q_k = Q_k^* $ for all $k$ such that $ \left \lceil k \frac{\mu_{l + 1}}{\mu_{l}} \right \rceil = \mu_{l + 1} $. It follows that $$ \mathrm{Tr}_l + \mathrm{Tr}_{l,l+1}^* = - \sum_{k = b_{l + 1}}^{\mu_{l} - 1} \sum_{m=1}^{\left \lceil k \frac{\mu_{l + 1}}{\mu_{l}} \right \rceil - 1} y^{k P_{l} - m P_{l - 1} } = \mathrm{Tr}_{l}^*. $$
It remains to prove the statement for $ \mathrm{Tr}_{l - 1, l}^* $. We have that $$ \mathrm{Tr}_{l - 1, l} = \sum_{k = 1}^{\mu_{l} - 1} \sum_{m = 1}^{b_{l} k } y^{k P_{l} - m P_{l - 1}}. $$ Let us put $ R_k := \sum_{m = 1}^{b_{l} k } y^{k P_{l} - m P_{l - 1}} $. If $ k \in \{1, \ldots, b_{l+1} - 1 \} $, one computes that $$ R_k^* := R_k + Q_k^* = R_k - 0 = \sum_{m = 1}^{b_{l} k } y^{k P_{l} - m P_{l - 1}} $$ $$ = \sum_{m = \left \lceil k \frac{\mu_{l + 1}}{\mu_{l}} \right \rceil}^{b_{l} k } y^{k P_{l} - m P_{l - 1}}, $$ since $ \left \lceil k \frac{\mu_{l + 1}}{\mu_{l}} \right \rceil = 1 $ in these cases. As $ b_l \mu_l = \mu_{l + 1} + \mu_{l - 1} > \mu_{l + 1} $, it follows that $ b_l k > k \frac{\mu_{l + 1}}{\mu_{l}} $ for any $ k \in \{ 1, \ldots, \mu_l - 1 \} $, and therefore, the inequality $$ b_l k \geq \left \lceil k \frac{\mu_{l + 1}}{\mu_{l}} \right \rceil $$ holds. Therefore, $$ R_k^* := R_k + Q_k^* = \sum_{m = 1}^{b_{l} k } y^{k P_{l} - m P_{l - 1}} - \sum_{m=1}^{\left \lceil k \frac{\mu_{l + 1}}{\mu_{l}} \right \rceil - 1} y^{k P_{l} - m P_{l - 1} } $$ $$ = \sum_{m = \left \lceil k \frac{\mu_{l + 1}}{\mu_{l}} \right \rceil}^{b_{l} k} y^{k P_{l} - m P_{l - 1} }. $$
Consequently, we get that $$ \mathrm{Tr}_{l - 1, l} + \mathrm{Tr}_l^* = \sum_{k = 1}^{\mu_{l} - 1} R_k^* = \sum_{k = 1}^{\mu_{l} - 1} \sum_{m = \left \lceil k \frac{\mu_{l + 1}}{\mu_{l}} \right \rceil}^{b_{l} k} y^{k P_{l} - m P_{l - 1} } = \mathrm{Tr}_{l - 1, l}^*, $$ and the proof is complete. \end{pf}
By induction, we get the following result:
\begin{cor}\label{Cor Key} $$ \mathrm{Tr}_{0,1}^* = \mathrm{Tr}_{0,1} + \mathrm{Tr}_1 + \ldots + \mathrm{Tr}_{\lambda - 1, \lambda}^*. $$ \end{cor}
We can compute the polynomial $ \mathrm{Tr}_{0,1}^* $ explicitly:
\begin{prop}\label{Computation} We have that $$ \mathrm{Tr}_{0,1}^* = \sum_{r=0}^{\mu_0 - 1} c_r y^r, $$ where $ c_0 = \mu_1 - 1 $, and $ c_r = \mu_1 - \left \lceil r \frac{\mu_{1}}{\mu_{0}} \right \rceil $ for all $ r \in \{ 1, \ldots, \mu_0 - 1\} $. \end{prop} \begin{pf} We have that $$ \mathrm{Tr}_{0,1}^* = \sum_{k = 1}^{\mu_{1} - 1} \sum_{m = \left \lceil k \frac{\mu_{2}}{\mu_{1}} \right \rceil}^{b_{1} k} y^{k P_{1} - m P_{0} }. $$ Since $ P_1 = b_1 $ and $ P_0 = 1 $, we have $$ \sum_{m = \left \lceil k \frac{\mu_{2}}{\mu_{1}} \right \rceil}^{b_{1} k} y^{k P_{1} - m P_{0} } = y^{b_{1} k - \left \lceil k \frac{\mu_{2}}{\mu_{1}} \right \rceil} + y^{b_{1} k - \left \lceil k \frac{\mu_{2}}{\mu_{1}} \right \rceil - 1} + \ldots + 1. $$ Let us denote this expression by $F_k$. From Lemma \ref{Lemma Key2}, we know that if $ l,k$ are integers such that $ 1 \leq l < k \leq \mu_1 - 1 $, then $$ b_1 k - \left \lceil k \frac{\mu_{2}}{\mu_{1}} \right \rceil > b_1 l - \left \lceil l \frac{\mu_{2}}{\mu_{1}} \right \rceil. $$ Therefore $ F_k(1) > F_l(1) $. That is, the number of monomials in $F_k$ strictly increases with $k$. This fact makes it easy to calculate the coefficients of the polynomial $ \sum_{k = 1}^{\mu_{1} - 1} F_k $. Let $ r \in \{ 1, \ldots, \mu_0-1 \} $. As the monomial $y^r$ appears at most once in $ F_k $, we see that its coefficient $c_r$ will equal the number of $F_k$ in which $y^r$ appears. And $ y^r $ will appear in $F_k$ exactly when $ F_k(1) \geq r + 1 $. Let $K_r$ denote the smallest positive integer such that the inequality $$ b_1 K_r - \left \lceil K_r \frac{\mu_{2}}{\mu_{1}} \right \rceil \geq r $$ holds. Then $y^r$ does not appear in any of the polynomials $ F_1, \ldots, F_{K_r - 1} $, but it does appear in all the polynomials $ F_{K_r}, \ldots, F_{\mu_1 - 1} $. Hence $ c_r = \mu_1 - K_r $. By Lemma \ref{Lemma 11.4}, it then follows that $ c_r = \mu_1 - \left \lceil r \frac{\mu_{1}}{\mu_{0}} \right \rceil $. Finally, it is easy to see that $c_0 = \mu_1 - 1 $, which concludes the proof. \end{pf}
\subsection{Explicit trace formula}\label{explicit formula} We are now ready to prove the formula for $ \mathrm{Tr}_{\sigma} $. The idea of the proof is as follows: We consider the resolution chain for $ \sigma = (m_1,m_2,n) $. Since we assume that $ n \gg 0 $, we have that the multiplicities $ \mu_l $ descend strictly from $ l = 0 $ to some $ l_0 $. After that, the multiplicities are constant equal to $ m = \mathrm{gcd}(m_1,m_2) $, and will then strictly increase from $ l = L + 1 - l_1 $ up to $ l = L + 1 $.
The idea is to ``cut'' the resolution chain somewhere in the constant locus, and treat the two halves independently. For each of the two parts of the chain, we have trace expressions $ \mathrm{Tr}_{l-1,l} $ and $ \mathrm{Tr}_l $ that we can sum up as in Section \ref{star}. The rest of the proof consists of computing the correction term.
Let us remark that in order for the results in Section \ref{star} to apply to both parts of the chain, we will need to perform a certain coordinate change, corresponding to switching the order of the formal branches of the singularity. This process is explained in Section \ref{coordinate change}, and the reader might want to consult this section while reading the proof of Theorem \ref{Formula}.
\begin{thm}\label{Formula} Let $ \sigma = (m_1,m_2,n) $ be a singularity as in Definition \ref{Def. 8.1}, where $ n \gg 0 $. Let $ m = \mathrm{gcd}(m_1,m_2) $, and let $ \alpha $ resp.~$ \alpha_1 $, resp.~$ \alpha_2 $ be inverses to $m$ resp.~$m_1$, resp.~$m_2$. For any root of unity $ \xi \in \boldsymbol{\mu}_n $, we let $ y = \xi^{\alpha_2} $, $ z = \xi^{\alpha_1} $ and $ w = \xi^{\alpha_1 (m_1/m)} = \xi^{\alpha_2 (m_2/m)} = \xi^{\alpha } $.
Then we have that $$ \mathrm{Tr}_{\sigma}(\xi) = \sum_{r=0}^{\mu_0 - 1} (\mu_1 - \left \lceil r \frac{\mu_{1}}{\mu_{0}} \right \rceil) y^r + \sum_{r=0}^{\mu_{L+1} - 1} (\mu_L - \left \lceil r \frac{\mu_{L}}{\mu_{L+1}} \right \rceil) z^r - \sum_{r=0}^{m-1} w^r. $$ The coefficients in this expression depend only on the residue class of $n$ modulo $ \mathrm{lcm}(m_1,m_2) $. \end{thm} \begin{pf} It suffices to give the proof for a \emph{primitive} root of unity (see the discussion in the proof of Corollary \ref{nonprimformula}). So throughout the proof, $ \xi $ will denote a primitive $n$-th root of unity.
Let $ q \in \{1, \ldots, L \} $ be an index such that $ \mu_{q-2} = \mu_{q-1} = \mu_{q} = \mu_{q+1} = \mu_{q+2} = m $. Since $ \xi $ is primitive, we have that $$ \mathrm{Tr}_{\sigma} = (\mathrm{Tr}_{y_0} + \ldots + \mathrm{Tr}_{y_q}) + (\mathrm{Tr}_{y_{q+1}} + \ldots + \mathrm{Tr}_{y_L}). $$ Recall from Corollary \ref{cor. 10.6} that $ \mathrm{Tr}_{y_l} = \mathrm{Tr}_{\mu_l, \mu_{l+1}} + \mathrm{Tr}_{y_l}^{\mu_l} + \mathrm{Tr}_{y_l}^{\mu_{l+1}} $, for $ 1 \leq l \leq L-1 $, and from Lemma \ref{lemma blabla} that $ \mathrm{Tr}_{y_l}^{\mu_{l+1}} + \mathrm{Tr}_{y_{l+1}}^{\mu_{l+1}} = \mathrm{Tr}_{\mu_{l+1}} $.
We will begin with assuming that $ m = \mathrm{gcd}(m_1,m_2) \geq 3 $. Then we have that $ \mathrm{Tr}_{\mu_{l}, \mu_{l+1}} + \mathrm{Tr}_{\mu_{l+1}} = \mathrm{Tr}_{l,l+1} + \mathrm{Tr}_{l+1} $ for all $ l \leq q $, and we get the equality $$ \mathrm{Tr}_{y_0} + \ldots + \mathrm{Tr}_{y_q} = \mathrm{Tr}_{0,1} + \mathrm{Tr}_1 + \ldots + \mathrm{Tr}_q + \mathrm{Tr}_{\mu_q, \mu_{q+1}} + \mathrm{Tr}_{y_q}^{\mu_{q+1}}. $$
We will now use the calculations and the notation from Section \ref{coordinate change}. Using Lemma \ref{lemma 10.25}, one computes easily that $ \mathrm{Tr}_{y_q}^{\mu_{q+1}} = \mathrm{Tr}_{y'_{L-q}}^{\mu'_{L-q}} $. Furthermore, from Corollary \ref{cor. 10.26}, it follows that $$ \mathrm{Tr}_{q+1} + \ldots + \mathrm{Tr}_{y_L} = \mathrm{Tr}_{y'_0} + \ldots + \mathrm{Tr}_{y'_{L-1-q}}. $$ Since $ \mathrm{Tr}_{y'_{L-1-q}} = \mathrm{Tr}_{\mu'_{L-1-q}, \mu'_{L-q}} + \mathrm{Tr}_{y'_{L-1-q}}^{\mu'_{L-1-q}} + \mathrm{Tr}_{y'_{L-1-q}}^{\mu'_{L-q}} $, we can write $$ \mathrm{Tr}_{y'_0} + \ldots + \mathrm{Tr}_{y'_{L-1-q}} = \mathrm{Tr}_{0,1}' + \ldots + \mathrm{Tr}_{L-1-q}' + \mathrm{Tr}_{\mu'_{L-1-q}, \mu'_{L-q}} + \mathrm{Tr}_{y'_{L-1-q}}^{\mu'_{L-q}}. $$ So all in all, we get that $$ \mathrm{Tr} = \mathrm{Tr}_{0,1} + \mathrm{Tr}_1 + \ldots + \mathrm{Tr}_q + \mathrm{Tr}_{\mu_q, \mu_{q+1}} + \mathrm{Tr}_{y'_{L-q}}^{\mu'_{L-q}} + $$ $$ \mathrm{Tr}_{0,1}' + \ldots + \mathrm{Tr}_{L-1-q}' + \mathrm{Tr}_{\mu'_{L-1-q}, \mu'_{L-q}} + \mathrm{Tr}_{y'_{L-1-q}}^{\mu'_{L-q}} = $$ $$ \mathrm{Tr}_{0,1} + \ldots + \mathrm{Tr}_q + \mathrm{Tr}_{\mu_q, \mu_{q+1}} + \mathrm{Tr}_{0,1}' + \ldots + \mathrm{Tr}_{L-q-1}' + \mathrm{Tr}_{L-1-q,L-q}' + \mathrm{Tr}_{L-q}'. $$ Corollary \ref{Cor Key} gives that $$ \mathrm{Tr}_{0,1} + \mathrm{Tr}_1 + \ldots + \mathrm{Tr}_q^* = \mathrm{Tr}_{0,1}^*, $$ and that $$ \mathrm{Tr}_{0,1}' + \ldots + \mathrm{Tr}_{L-q-1}' + \mathrm{Tr}_{L-1-q,L-q}' + (\mathrm{Tr}_{L-q}')^* = (\mathrm{Tr}_{0,1}')^*, $$ where the notation is the same as in Section \ref{star}. Moreover, the terms $ \mathrm{Tr}_{0,1}^* $ and $ (\mathrm{Tr}_{0,1}')^* $ can be explicitly computed using Proposition \ref{Computation}, and these two sums are indeed the first two terms in the theorem.
It remains to compute the correction term $ \mathrm{Tr}_q^0 + \mathrm{Tr}_{\mu_q, \mu_{q+1}} + (\mathrm{Tr}_{L-q}')^0 $, where $ \mathrm{Tr}_q^0 := \mathrm{Tr}_q - \mathrm{Tr}_q^* $, and $ (\mathrm{Tr}_{L-q}')^0 := \mathrm{Tr}_{L-q}' - (\mathrm{Tr}_{L-q}')^* $. It follows from the way we chose $q$, that $ \mu_q = \mu_{q+1} $ and that $ b_{q+1} = 2 $. With our explicit description of $ \mathrm{Tr}_q $ and $ \mathrm{Tr}_q^* $, it is then easy to compute that $$ \mathrm{Tr}_q^0 = - \sum_{k=0}^{\mu_q - 1} \sum_{n=k}^{\mu_{q+1} - 1} y^{k P_q - n P_{q-1}} + 1. $$ Likewise, we can write $$ \mathrm{Tr}_{\mu_q, \mu_{q+1}} = \sum_{k=0}^{\mu_q - 1} \sum_{n=0}^{\mu_{q+1} - 1} y^{k P_q - n P_{q-1}}. $$ An easy calculation now shows that $$ \mathrm{Tr}_q^0 + \mathrm{Tr}_{\mu_q, \mu_{q+1}} = 1 + \sum_{k=1}^{\mu_q - 1} \sum_{n=0}^{k - 1} y^{k P_q - n P_{q-1}}. $$
After ``changing coordinates'' as in Section \ref{coordinate change} below, we get that $$ (\mathrm{Tr}_{L-q}')^0 = - \sum_{s=0}^{\mu_{q+1} - 1} \sum_{t=s}^{\mu_q - 1} y^{t P_q - s P_{q-1}} + 1. $$ We would now like to calculate $$ \sum_{k=1}^{\mu_q - 1} \sum_{n=0}^{k - 1} y^{k P_q - n P_{q-1}} - \sum_{s=0}^{\mu_{q+1} - 1} \sum_{t=s}^{\mu_q - 1} y^{t P_q - s P_{q-1}}. $$ Before we do that, note that $ \mu_q = \mu_{q+1} = m $, and that the sum above may be written $$ \sum_{k=1}^{m - 1} \sum_{l=0}^{k - 1} y^{k P_q - l P_{q-1}} - \sum_{l=0}^{m - 1} \sum_{k=l}^{m - 1} y^{k P_q - l P_{q-1}}. $$ However, it is easily seen that $$ \sum_{k=1}^{m - 1} \sum_{l=0}^{k - 1} y^{k P_q - l P_{q-1}} = \sum_{l=0}^{m - 2} \sum_{k=l+1}^{m - 1} y^{k P_q - l P_{q-1}}, $$ and that $$ \sum_{l=0}^{m - 2} \sum_{k=l+1}^{m - 1} y^{k P_q - l P_{q-1}} - \sum_{l=0}^{m - 1} \sum_{k=l}^{m - 1} y^{k P_q - l P_{q-1}} $$ $$ = - \sum_{l=0}^{m - 1} y^{l P_q - l P_{q-1}} = - \sum_{l=0}^{m - 1} (y^{P_q - P_{q-1}})^l, $$ so it follows that $$ \mathrm{Tr}_q^0 + \mathrm{Tr}_{\mu_q, \mu_{q+1}} + (\mathrm{Tr}_{L-q}')^0 = 2 - \sum_{l=0}^{m - 1} (y^{P_q - P_{q-1}})^l. $$
Note now that $ y^{P_q - P_{q-1}} = \chi^{-r_q + r_{q+1}} $. It is easily seen that $$ - \mu_l r_l + \mu_{l+1} r_{l-1} = - \mu_{l+1} r_{l+1} + \mu_{l+2} r_{l} $$ for all $l$, so by induction, we get that $$ - \mu_q r_q + \mu_{q+1} r_{q-1} = \mu_{L+1} r_{L-1}, $$ since $ r_L = 0 $. As $ \mu_q = \mu_{q+1} = m $, we get, after dividing by $m$, that $$ -r_q + r_{q+1} = \mu_{L+1}/m = m_1/m, $$ remembering that $ r_{L-1} = 1 $. So it follows that $$ y^{P_q - P_{q-1}} = \chi^{\mu_{L+1}/m} = \xi^{\alpha_1 (m_1/m)}, $$ and the proof of the formula is complete in the case $m \geq 3$.
The two cases $ m = 1 $ and $ m = 2 $ remain. The proof in these cases is very similar to the one above, and is therefore omitted here. The main difference is that we have to cut the chain in three pieces: The part where the multiplicities descend, the part where the multiplicities are constant, and the part where the multiplicities ascend. On each of these parts, we have to compute certain sums of trace expressions, which can be done along the same lines as in the case where $ m \geq 3 $.
\end{pf}
We would like to end this section with the remark that now that we have obtained the explicit formula in Theorem \ref{Formula}, Theorem \ref{thm. 9.13} gives an effective formula for computing the trace $ \mathrm{Tr}_{\beta}(e(H^{\bullet}(g|_{\mathcal{Y}_k}))) $, where the notation is the same as in Section \ref{special filtrations}. We formulate this in Theorem \ref{improved formula} below. Recall the standard assumptions in Sections \ref{assumption on degree} and \ref{assumption on surface}.
\begin{thm}\label{improved formula} Let $ g \in G $ be a group element corresponding to a root of unity $\xi \in \boldsymbol{\mu}_n$, where $ n \gg 0 $. Then we have that
$$ \mathrm{Tr}_{\beta}(e(H^{\bullet}(g|_{\mathcal{Y}_k}))) = \sum_{\upsilon \in \mathcal{V}} \mathrm{Tr}_{\upsilon}(\xi) + \sum_{\varepsilon \in \mathcal{E}} \mathrm{Tr}_{\sigma(\varepsilon)}(\xi), $$ where $ \sigma(\varepsilon) $ is the unique singularity of $ \mathcal{X}' $ corresponding to $ \varepsilon $. The contributions $ \mathrm{Tr}_{\upsilon}(\xi) $ are given by Proposition \ref{prop. 9.7}, and the contributions $ \mathrm{Tr}_{\sigma(\varepsilon)}(\xi) $ are given by Theorem \ref{Formula}.
Furthermore, this expression depends only on $ \Gamma(\mathcal{X}_k) $ and the functions $ \mathfrak{g} $ and $ \mathfrak{m} $ defined in Section \ref{9.1}.
\end{thm}
\subsection{Coordinate change}\label{coordinate change} We consider now positive integers $ m_1 $, $m_2$ such that $ \mathrm{gcd}(m_1,m_2) = 1 $. Let $ M = \mathrm{lcm}(m_1,m_2) $, and let $ n $ be a positive integer such that $ \mathrm{gcd}(n,M) = 1 $. Let $ \alpha_i $ denote the inverse to $ m_i $ modulo $n$. We denote by $ \mu_l $, $ b_l $ and $ r_l $ the numerical data associated to the singularity $ \sigma = (m_1, m_2, n) $ as usual.
Let us now define $ m_1' := m_2 $ and $ m_2' := m_1 $, and consider the singularity $ \sigma' = (m_1', m_2', n) $, which is the same singularity as $(m_1,m_2,n)$, but with reverse ordering of the branches. Let $ \mu'_j = \mu_{L+1-j} $, $c_j = b_{L+1-j} $. Then the numerical data for $ (m_1', m_2', n) $ consists of $ \mu'_j $, $ c_j $ and $ s_j $, where the $c_j$ and $s_j$ satisfy the equations $ s_{j-1} = c_{j+1} s_j - s_{j+1} $.
The intersection points of the components in the exceptional locus are $ y_l = C_l \cap C_{l+1} $, so we let $ y'_j = y_{L-j} $, where $ l $ and $ j $ run from $ 0 $ to $L$.
Let $ \alpha'_1 $ denote the inverse to $m_1' $ modulo $n$. Notice then that the equation $ m_1 + r_0 m_2 = n \mu_1 $ gives that $ \alpha'_1 \equiv_n - \alpha_1 r_0 $, and $ m_1' + s_0 m_2' = n \mu_1' $ gives $ \alpha_1 \equiv_n - \alpha'_1 s_0 $.
\begin{lemma}\label{lemma 10.25} Let $ \chi = \xi^{\alpha_1} $ and $ \chi' = \xi^{\alpha_1'} $. Then we have that $ \chi^{r_{L-1-j}} = \chi'^{-s_j} $ for all $ 0 \leq j \leq L-1 $. \end{lemma} \begin{pf} As $ r_{L-1} = 1 $, we get that $ \chi^{r_{L-1}} = \xi^{\alpha_1} = \xi^{- \alpha'_1 s_0} = \chi'^{ - s_0 } $. Assume now that $ j > 0 $. By induction it then follows that $$ \chi^{r_{L-1-j}} = \xi^{\alpha_1 r_{L-1-j}} = \xi^{\alpha_1 b_{L+1-j} r_{L-j}} \xi^{ - \alpha_1 r_{L + 1 -j}} = \xi^{ - \alpha'_1 s_{j-1} c_j} \xi^{ \alpha'_1 s_{j-2}}$$ $$ = \xi^{ - \alpha'_1(c_j s_{j-1} - s_{j-2})} = \xi^{ - \alpha'_1 s_j } = \chi'^{-s_j}. $$ \end{pf}
Let us now assume that $ \xi \in \boldsymbol{\mu}_n $ is a primitive root. Then we can write $$ \mathrm{Tr}_{\sigma}(\xi) = \mathrm{Tr}_{y_0} + \ldots + \mathrm{Tr}_{y_l} + \ldots + \mathrm{Tr}_{y_L}, $$ where $ \mathrm{Tr}_{y_l} = \frac{1 - \xi}{(1 - \chi^{r_{l-1}})(1 - \chi^{- r_{l}})} $, $ \mathrm{Tr}_{y_0} = \frac{\mu_1}{1 - \chi^{- r_0}} $ and $ \mathrm{Tr}_{y_L} = \frac{\mu_L}{1 - \chi^{r_{L-1}}} $.
We similarly have that $$ \mathrm{Tr}_{\sigma'}(\xi) = \mathrm{Tr}_{y'_0} + \ldots + \mathrm{Tr}_{y'_j} + \ldots + \mathrm{Tr}_{y'_L}, $$ where $ \mathrm{Tr}_{y'_j} = \frac{1 - \xi}{(1 - \chi'^{s_{j-1}})(1 - \chi'^{- s_{j}})} $, $ \mathrm{Tr}_{y'_0} = \frac{\mu_1'}{1 - \chi'^{- s_0}} $ and $ \mathrm{Tr}_{y'_L} = \frac{\mu'_L}{1 - \chi'^{s_{L-1}}} $.
Using Lemma \ref{lemma 10.25}, we can draw the following conclusion: \begin{cor}\label{cor. 10.26} We have that $ \mathrm{Tr}_{y'_j} = \mathrm{Tr}_{y_{L-j}} $ for all $ j $, where $ 0 \leq j \leq L $. \end{cor}
Let us also remark that we will use the notation $ Tr_{\mu_{j-1}',\mu_j'} $, $ Tr_{\mu_j'} $ etc., for the expressions defined as in Section \ref{trace formula}, and the notation $ Tr'_{j-1,j} $ and $ Tr'_j $ for the expressions defined as in Section \ref{lemmas}.
\section{Character computations and jumps}\label{computations and jumps} Let $X/K$ be a smooth, projective and geometrically irreducible curve such that $X(K) \neq \emptyset $, and let $ \mathcal{X}/S $ be the minimal SNC-model of $X$. We have in previous sections studied properties of the action of $ \boldsymbol{\mu}_n $ on the cohomology groups $ H^i(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $, where $ \mathcal{Y} $ is the minimal desingularization of the pullback $ \mathcal{X}_{S'} $ for some tame extension $S'/S$ of degree $n$.
Let $\mathcal{J}/S$ be the N\'eron model of the Jacobian of $X$. We will in this section apply our results to the study of the filtration $ \{ \mathcal{F}^a \mathcal{J}_k \} $, where $ a \in \mathbb{Z}_{(p)} \cap [0,1] $, that we defined in Section \ref{ratfil}. We will first prove some general properties for these filtrations, and then present some computations for curves of genus $g = 1$ and $g = 2$.
We would at this point like to remark that in order to make the $ \boldsymbol{\mu}_n $-action on $ H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $ compatible with the action on $ T_{\mathcal{J}'_k, 0} $, we have to let $ \boldsymbol{\mu}_n $ act on $ R' $ by $ [\xi](\pi') = \xi^{-1} \pi' $, for any $ \xi \in \boldsymbol{\mu}_n $. We made the choice in previous sections, when working with surfaces, to let $ \boldsymbol{\mu}_n $ act by $ [\xi](\pi') = \xi \pi' $, in order to get simpler notation. This means that the irreducible characters for the representation on $ T_{\mathcal{J}'_k, 0} $ are the \emph{inverse} characters to those we compute when using our formulas for the representation on $ H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $.
\subsection{Filtrations for N\'eron models of Jacobians} Theorem \ref{thm. 9.13} states that the Brauer trace of the automorphism induced by any group element $ \xi \in \boldsymbol{\mu}_n $ on the formal difference $ H^0(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) - H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $ only depends on the combinatorial structure of $ \mathcal{X}_k $. With the assumption that $X(K) \neq \emptyset$, we can actually improve this result, and get a similar result for the character of the representation of $ \boldsymbol{\mu}_n $ on $ H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $:
\begin{thm}\label{main character theorem} Let $ X/K $ be a smooth, projective and geometrically connected curve having genus $ g(X) > 0 $, and assume that $ X(K) \neq \emptyset $. Let $ \mathcal{X} $ be the minimal SNC-model of $ X $ over $S$. Furthermore, let $ S'/S $ be a tame extension of degree $n$, where $ n $ is relatively prime to the least common multiple of the multiplicities of the irreducible components of $ \mathcal{X}_k $, and let $ \mathcal{Y}/S' $ be the minimal desingularization of $ \mathcal{X}_{S'} $.
Then the irreducible characters for the representation of $ \boldsymbol{\mu}_n $ on $ H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $ only depend on the intersection graph $ \Gamma(\mathcal{X}_k) $, together with the functions $ \mathfrak{g} $ and $ \mathfrak{m} $. \end{thm} \begin{pf} For any $ g \in G $, corresponding to a root $ \xi \in \boldsymbol{\mu}_n $, we have that
$$ \mathrm{Tr}_{\beta}(e(H^{\bullet}(g|_{\mathcal{Y}_k}))) = \sum_{\upsilon \in \mathcal{V}} \mathrm{Tr}_{\upsilon}(\xi) + \sum_{\varepsilon \in \mathcal{E}} \mathrm{Tr}_{\varepsilon}(\xi), $$ by Theorem \ref{thm. 9.13}. The contributions $ \mathrm{Tr}_{\upsilon}(\xi) $ can be computed using Proposition \ref{prop. 9.7}, and for the contributions $ \mathrm{Tr}_{\varepsilon}(\xi) $, we use Proposition \ref{lemma 9.8} and Proposition \ref{lemma 9.10}. In this way, we obtain a formula for the Brauer trace of the automorphism induced by any $ \xi \in \boldsymbol{\mu}_n $ on the formal difference $$ H^0(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) - H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}). $$
Since $X$, and hence $X_{K'}$, has a rational point, it follows from \cite{Liubook}, Corollary 9.1.32 that at least one of the irreducible components of $ \mathcal{Y}_k $ has multiplicity $1$. We can therefore conclude that $ H^0(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) = k $ (\cite{Arwin}, Lemma 2.6). Furthermore, the $ \boldsymbol{\mu}_n $-action on $ \mathcal{Y}_k $ is relative to the ground field $k$, so it follows that the eigenvalue for the automorphism induced by $ \xi $ on $ H^0(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $ equals $1$.
We therefore obtain the formula
$$ \mathrm{Tr}_{\beta}(H^1(g|_{\mathcal{Y}_k})) = 1 - (\sum_{\upsilon \in \mathcal{V}} \mathrm{Tr}_{\upsilon}(\xi) + \sum_{\varepsilon \in \mathcal{E}} \mathrm{Tr}_{\varepsilon}(\xi)). $$
Since the expressions $ \mathrm{Tr}_{\upsilon}(\xi) $ and $ \mathrm{Tr}_{\varepsilon}(\xi) $ only depend on the combinatorial structure of $ \mathcal{X}_k $, the same is true for $ \mathrm{Tr}_{\beta}(H^1(g|_{\mathcal{Y}_k})) $. This completes the proof, since the Brauer character for the representation of $ \boldsymbol{\mu}_n $ on $ H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $ is determined by the Brauer trace for the group elements $ \xi \in \boldsymbol{\mu}_n $. \end{pf}
Let $ \mathcal{J}/S $ be the N\'eron model of the Jacobian of $X/K$, and let $ \{ \mathcal{F}^a \mathcal{J}_k \} $, where $ a \in \mathbb{Z}_{(p)} \cap [0,1] $, be the filtration of $ \mathcal{J}_k $ defined in Section \ref{ratfil}. Then Theorem \ref{main character theorem} has the following consequence:
\begin{cor}\label{main jump corollary} The jumps in the filtration $ \{ \mathcal{F}^a \mathcal{J}_k \} $ with indices in $ \mathbb{Z}_{(p)} \cap [0,1] $ depend only on the intersection graph $ \Gamma(\mathcal{X}_k) $, together with the functions $ \mathfrak{g} $ and $ \mathfrak{m} $. In particular, they don't depend on $p$. \end{cor} \begin{pf} Let $S'/S$ be a tame extension of degree $n$, where $n$ is prime to $l$, the least common multiple of the multiplicities of the irreducible components of $ \mathcal{X}_k $. Let $ \mathcal{J}'/S' $ be the N\'eron model of the Jacobian of $X_{K'}$. Recall from Section \ref{jacobiancase} that we could make the identification $ H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) \cong T_{\mathcal{J}'_k,0} $.
The jumps in the filtration of $ \mathcal{J}_k $ induced by the extension $S'/S$ are determined by the irreducible characters for the representation of $ \boldsymbol{\mu}_n $ on $ T_{\mathcal{J}'_k,0} $. However, this representation is precisely the representation of $ \boldsymbol{\mu}_n $ on $ H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $, if we let $ \boldsymbol{\mu}_n $ act on $R'$ by $ [\xi](\pi') = \xi^{-1} \pi' $, for every $\xi$. By Theorem \ref{main character theorem}, the character for this representation only depends on $ \Gamma(\mathcal{X}_k) $, $ \mathfrak{g} $ and $ \mathfrak{m} $.
Since $ \mathbb{Z}_{(lp)} \cap [0,1] $ is \emph{dense} in $ \mathbb{Z}_{(p)} \cap [0,1] $, we conclude that the jumps of the filtration $ \{ \mathcal{F}^a \mathcal{J}_k \} $ with indices in $ \mathbb{Z}_{(p)} \cap [0,1] $ only depend on $ \Gamma(\mathcal{X}_k) $, $ \mathfrak{g} $ and $ \mathfrak{m} $. \end{pf}
With the two results above at hand, we can draw some conclusions about \emph{where} the jumps occur in the case of Jacobians. Let us first recall the following terminology from \cite{Tame}: An irreducible component $C$ of $ \mathcal{X}_k $ is called \emph{principal} if either $ P_a(C) > 0 $, or if $ C $ is smooth and rational and meets the rest of the components of $ \mathcal{X}_k $ in at least three points.
\begin{cor}\label{specific jump corollary} Let $ \tilde{n} $ be the least common multiple of the multiplicities of the principal components of $ \mathcal{X}_k $. Then the jumps in the filtration $ \{ \mathcal{F}^a \mathcal{J}_k \} $ occur at indices of the form $ i/\tilde{n} $, where $ 0 \leq i < \tilde{n} $. \end{cor} \begin{pf} Let us first note that if $X$ obtains semi-stable reduction over a tame extension $K'/K$, then the Jacobian of $X$ obtains semi-abelian reduction over the same extension. Furthermore, the minimal extension that gives semi-abelian reduction is the tame extension $ \widetilde{K}/K $ of degree $ \tilde{n} $ (\cite{Tame}, Theorem 7.1). So in this case, the statement follows from Proposition \ref{tamejumpprop}.
Let us now assume that $X$ needs a wildly ramified extension to obtain semi-stable reduction. Consider the combinatorial data $ (\Gamma(\mathcal{X}_k), \mathfrak{g}, \mathfrak{m}) $. It follows from \cite{Winters}, Corollary 4.3, that we can find an SNC-model $ \mathcal{Z}/\mathrm{Spec}(\mathbb{C}[[t]]) $, where the generic fiber of $ \mathcal{Z} $ is smooth, projective and geometrically connected, and where the special fiber of $ \mathcal{Z} $ has the \emph{same} combinatorial data as $ \mathcal{X}_k $.
Let $ \mathcal{J}_{\mathcal{Z}} $ be the N\'eron model of the Jacobian of the generic fiber of $ \mathcal{Z} $. Then the jumps of the filtration $ \{ \mathcal{F}^a \mathcal{J}_{\mathcal{Z},\mathbb{C}} \} $ occur at indices of the form $ i/\tilde{n} $, where $ 0 \leq i < \tilde{n} $. The result follows now from Corollary \ref{main jump corollary}. \end{pf}
\subsection{} Let $X/K$ be a smooth, projective and geometrically connected curve, and assume that $ X(K) \neq \emptyset $. Let $ \mathcal{X}/S $ be the minimal SNC-model of $X/K$. It is known that for a fixed genus $ g \geq 2 $, there are only finitely many possibilities for the combinatorial structure of the special fiber of $ \mathcal{X}/S$, modulo chains of $(-2)$-curves (\cite{Arwin}, Theorem 1.6). The same statement is, as we shall see below, also true for elliptic curves.
Let $ \mathcal{J}/S $ be the N\'eron model of the Jacobian of $X$. Since, by Corollary \ref{main jump corollary}, the jumps of the filtration $ \{ \mathcal{F}^a \mathcal{J}_k \} $ only depend on the combinatorial structure of $ \mathcal{X}_k $, one can, for each $g > 0$, classify these jumps. In the next sections, we will give the jumps for every fiber type of genus $1$ and $2$.
\begin{rmk} It is easy to see that chains of $(-2)$-curves do not affect the jumps. \end{rmk}
\subsection{Computations of jumps for $ g =1 $} Let $X/K$ be an \emph{elliptic} curve, and let $ \mathcal{E} $ be the minimal regular model of $ X $. It is a well known fact that there are only finitely many possibilities for the combinatorial structure of the special fiber $ \mathcal{E}_k $, modulo chains of $(-2)$-curves. The various possibilities were first classified in \cite{Kod}, and this is commonly referred to as the \emph{Kodaira classification}. For another treatment of this theory, we refer to \cite{Liubook}, Chapter 10.2. If now $ \mathcal{X}/S $ denotes the minimal SNC-model of $X$, it follows that there are only finitely many possibilities for the combinatorial structure of $ \mathcal{X}_k $, each one derived from the Kodaira classification. The symbols $ I, II, \ldots $ appearing in Table \ref{table 1} below are known as the \emph{Kodaira symbols} and refer to the fiber types in the Kodaira classification.
Let $ \mathcal{J}/S $ be the N\'eron model of $ J(X) = X $. It follows from Corollary \ref{main jump corollary} and Corollary \ref{specific jump corollary} that the (unique) jump in the filtration $ \{ \mathcal{F}^a \mathcal{J}_k \} $ only depends on the fiber type of $ \mathcal{X}/S $, and can only occur at finitely many \emph{rational} numbers. In Table \ref{table 1} below, we list the jumps for the various Kodaira types. Note that we obtain the same list as the one computed in \cite{Edix} by R. ~Schoof.
We would like to say a few words about how these computations are done. For each fiber type, we consider an infinite sequence $ (n_j)_{j \in \mathbb{N}} $, depending on the fiber type, where $ n_j \rightarrow \infty $ as $ j \rightarrow \infty $. For each $n_j$ in this sequence, let $ R_j/R $ be the tame extension of degree $n_j$, and let $ \pi_j $ be the uniformizing parameter of $ R_j $. Furthermore, let $ \boldsymbol{\mu}_{n_j} $ act on $ R_j $ by $ [\xi](\pi_j) = \xi \pi_j $. We can then use Theorem \ref{thm. 9.13} to compute the character for the induced representation of $ \boldsymbol{\mu}_{n_j} $ on $ H^1(\mathcal{Y}^j_k, \mathcal{O}_{\mathcal{Y}^j_k}) $, where $ \mathcal{Y}^j $ denotes the minimal desingularization of $ \mathcal{X}_{S_j} $, and where $ S_j = \mathrm{Spec}(R_j) $. This character is on the form $ \chi(\xi) = \xi^{i(j)} $.
The character for the representation of $ \boldsymbol{\mu}_{n_j} $ on $ T_{\mathcal{J}^j_k, 0} $ is the inverse of this character, $ \chi^{-1}(\xi) = \xi^{- i(j)} $. The jump of $ \{ \mathcal{F}^a \mathcal{J}_k \} $ will then be given by the limit of the expression $ [- i(j)]_{n_j}/n_j $ as $ j \rightarrow \infty $, where $ [- i(j)]_{n_j} \equiv_{n_j} - i(j) $, and $ 0 \leq [- i(j)]_{n_j} < n_j $.
In Example \ref{example genus 1} below, we explain in detail how these computations are done for fiber type $IV$ in the Kodaira classification.
\begin{ex}\label{example genus 1} Let $ \mathcal{X}/S $ have fibertype $IV$. In this case, the combinatorial data of $ \mathcal{X}_k $ consists of the set of vertices $ \mathcal{V} = \{ \upsilon_1, \ldots, \upsilon_4 \} $, where $ \mathfrak{m}(\upsilon_i) = 1 $ for $ i \in\{ 1,2,3 \}$, and $ \mathfrak{m}(\upsilon_4) = 3 $. Furthermore, we have that $ \mathfrak{g}(\upsilon_i) = 0 $ for all $i$. The set of edges consists of $ \mathcal{E} = \{ \varepsilon_1, \varepsilon_2, \varepsilon_3 \} $, where $ \varepsilon_i $ corresponds to the unique intersection point of the components $ \upsilon_i $ and $ \upsilon_4 $, for $ i = 1,2,3 $. Let us choose the ordering $ (\upsilon_i,\upsilon_4) $ for all $i$.
Let now $ n \gg 0 $ be a positive integer relatively prime to $ p $ and to $ \mathrm{lcm}( \{ \mathfrak{m}(\upsilon_i) \} ) = 3 $, and let $R'/R$ be a tame extension of degree $n$. Let $ \boldsymbol{\mu}_n $ act on $R'$ by $ [\xi](\pi') = \xi \pi' $ for any $ \xi \in \boldsymbol{\mu}_n$, where $ \pi' $ is a uniformizing parameter for $R'$.
For any $ g \in G $, corresponding to a root of unity $ \xi \in \boldsymbol{\mu}_n $, Theorem \ref{thm. 9.13} states that
$$ \mathrm{Tr}_{\beta}(e(H^{\bullet}(g|_{ \mathcal{Y}_k}))) = \sum_{\upsilon \in \mathcal{V}} \mathrm{Tr}_{\upsilon}(\xi) + \sum_{\varepsilon \in \mathcal{E}} \mathrm{Tr}_{\varepsilon}(\xi). $$
Let $ \sigma $ be the singularity $ (1,3,n) $. Then we have that $ \mathrm{Tr}_{\varepsilon_i}(\xi) = \mathrm{Tr}_{\sigma}(\xi) $ for all $ i \in \{ 1,2,3 \} $. It suffices to consider the case where $ n \equiv_3 1 $. One computes easily that $ \mu_l = 1 $ for all $ l \in \{ 1, \ldots, L(\sigma) \} $. From Theorem \ref{Formula}, we immediately get that $ \mathrm{Tr}_{\varepsilon_i}(\xi) = 1 $, for all $i$.
Proposition \ref{prop. 9.7} states that $$ \mathrm{Tr}_{\upsilon}(\xi) = \sum_{k=0}^{m_{\upsilon}-1} (\xi^{\alpha_{m_{\upsilon}}})^{k} ((m_{\upsilon} - k)C_{\upsilon}^2 + 1 - p_a(C_{\upsilon})), $$ for any $ \upsilon \in \mathcal{V} $, where $ \alpha_{m_{\upsilon}} m_{\upsilon} \equiv_n 1 $. As $ C_{\upsilon_i}^2 = - 1 $ for $ i \in \{ 1,2,3 \} $, we see that $ \mathrm{Tr}_{\upsilon_i}(\xi) = 0 $ for these vertices, and since $ C_{\upsilon_4}^2 = - 1 $, it follows that $ \mathrm{Tr}_{\upsilon_4}(\xi) = - 2 - \xi^{\alpha_3} $. In total, we get
$$ \mathrm{Tr}_{\beta}(e(H^{\bullet}(g|_{ \mathcal{Y}_k}))) = 3 + (- 2 - \xi^{\alpha_3}) = 1 - \xi^{\alpha_3}. $$ We can therefore conclude that the character for the representation of $ \boldsymbol{\mu}_n $ on $ H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $ is $ \chi(\xi) = \xi^{\alpha_3} $.
In order to compute the jump of the filtration $ \{ \mathcal{F}^a \mathcal{J}_k \} $, where $ \mathcal{J} $ is the N\'eron model of $J(X) = X$, we have to use the \emph{inverse} character, which is $ \chi^{-1}(\xi) = \xi^{[- \alpha_3]_n} $, where $ [- \alpha_3]_n = - \alpha_3 $ modulo $n$, and $ 0 \leq [- \alpha_3]_n < n $. The jump will be given by the limit of the expression $ ([- \alpha_3]_n)/n $ as $n$ goes to infinity over integers $n$ that are equivalent to $1$ modulo $3$.
Since $ n = 1 + 3 \cdot h $, for some integer $h$, we get that $ \alpha_3 = \frac{1 + 2 n}{3} $, where $ 0 < \alpha_3 < n $. Therefore, the jump occurs at the limit of $ ([- \alpha_3]_n)/n = \frac{n - 1}{3 n} $ which is $ 1/3 $.
\end{ex}
\begin{table}[htb]\caption{Genus $1$}\label{table 1}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|} \hline Fibertype & $(I)$ & $(I)^*$ & $(I_n)$ & $(I_n)^*$ & $(II)$ & $(II)^*$ & $(III)$ & $(III)^*$ & $(IV)$ & $(IV)^*$\\ \hline Jumps & $0$ & $1/2$ & $0$ & $1/2$ & $ 1/6 $ & $ 5/6 $ & $ 1/4 $ & $ 3/4 $ & $ 1/3 $ & $ 2/3 $ \\\hline \end{tabular} \end{table}
\subsection{Computations of jumps for $g=2$}\label{genus 2} Let $ X/K $ be a curve having genus equal to $2$. Like in the case for elliptic curves, there are finitely many possibilities, modulo chains of $(-2)$-curves, for the combinatorial structure of the special fiber of the minimal regular model of $X$. Moreover, there exists a complete classification of the various possible fiber types. This classification is mainly due to A.P. Ogg (\cite{Ogg}), with the exception of a few missing cases which were filled in by Y. Namikawa and K. ~Ueno in \cite{Ueno}. We will refer to the list of possible fiber types as the \emph{Ogg-classification}, and we will use the indexing from \cite{Ogg}, with the addition of the types $41_a$, $41_b$ and $41_c$ from \cite{Ueno} that were missing in \cite{Ogg}.
Let $ \mathcal{X}/S $ be the minimal SNC-model of $X$. Then there are only finitely many possibilities for the combinatorial structure of $ \mathcal{X}_k $, each derived from the fiber types in the Ogg-classification. Let $ \mathcal{J}/S $ be the N\'eron model of the Jacobian of $X$. The jumps in the filtration $ \{ \mathcal{F}^a \mathcal{J}_k \} $ depend only on the combinatorial structure of $ \mathcal{X}_k $, and can occur only at a finite set of rational numbers.
In order to compute the jumps for each fibertype, we proceed more or less in the same manner as we did in the case of elliptic curves. In Example \ref{example 13.1}, we explain in detail how this is done for fiber type $4$ in the Ogg-classification.
Most of the results for genus $2$ curves are gathered in Table \ref{table 2} below. However, some cases are treated separately in Section \ref{special types}.
\begin{ex}\label{example 13.1} We consider fiber type $4$ in the Ogg-classification. In this case, the set of vertices of $ \Gamma(\mathcal{X}_k) $ is $ \mathcal{V} = \{ \upsilon_1, \ldots, \upsilon_7 \} $, where $ \mathfrak{g}(\upsilon_i) = 0 $ for all $i$. Furthermore, we have that $ \mathfrak{m}(\upsilon_i) = 1 $ for $ i = 1,7 $, $ \mathfrak{m}(\upsilon_i) = 2 $ for $ i = 2,5,6 $, $ \mathfrak{m}(\upsilon_3) = 3 $ and $ \mathfrak{m}(\upsilon_4) = 4 $. The set of edges is $ \mathcal{E} = \{ \varepsilon_1, \varepsilon_2, \varepsilon_3, \varepsilon_4, \varepsilon_5, \varepsilon_6 \} $, where $ \varepsilon_1 = (\upsilon_1,\upsilon_2) $, $ \varepsilon_2 = (\upsilon_2,\upsilon_3) $, $ \varepsilon_3 = (\upsilon_3,\upsilon_4) $, $ \varepsilon_4 = (\upsilon_5,\upsilon_4) $, $ \varepsilon_5 = (\upsilon_6,\upsilon_4) $ and $ \varepsilon_6 = (\upsilon_7,\upsilon_4) $.
We have that $ \mathrm{lcm}( \{\mathfrak{m}(\upsilon_i)\}) = 12 $. Let $ n \gg 0$ be any integer not divisible by $p$, and such that $ n \equiv_{12} 1 $. Let $ R'/R $ be the extension of degree $n$, and let $ \pi'$ be the uniformizing parameter of $R'$. Let $ \mathcal{Y} $ be the minimal desingularization of $ \mathcal{X}_{S'} $. We let $ \boldsymbol{\mu}_n $ act on $ R'$ by $ [\xi](\pi') = \xi \pi' $, for any $ \xi \in \boldsymbol{\mu}_n $.
Now, let $ \xi \in \boldsymbol{\mu}_n$ be a root of unity. For any $ \upsilon \in \mathcal{V} $, Proposition \ref{prop. 9.7} gives that $$ \mathrm{Tr}_{\upsilon}(\xi) = \sum_{k=0}^{m_{\upsilon}-1} (\xi^{\alpha_{m_{\upsilon}}})^{k} ((m_{\upsilon} - k)C_{\upsilon}^2 + 1 - p_a(C_{\upsilon})). $$ As the computations are similar for all $ \upsilon \in \mathcal{V} $, we only do this explicitly for $ \upsilon_3 $. We have that $ p_a(C_{\upsilon_3}) = \mathfrak{g}(\upsilon_3) = 0 $, so it remains only to compute $ C_{\upsilon_3}^2 $. The edge $ \varepsilon_2 $ corresponds to the singularity $ \sigma_2 = (2,3,n) $ and the edge $ \varepsilon_3 $ corresponds to the singularity $ \sigma_3 = (3,4,n) $. Denote by $ C_l^{\sigma_2} $ the exceptional components in the resolution of $ \sigma_2 $, and by $ C_l^{\sigma_3} $ the components in the resolution of $ \sigma_3 $. Then $ C_1^{\sigma_2} $ and $ C_L^{\sigma_3} $ are the only two components of $ \mathcal{Y}_k $ that meet $ C_{\upsilon_3} $ (note the ordering of the formal branches in $ \sigma_2 $ and $ \sigma_3 $). It is easily computed that $ \mu_1^{\sigma_2} = 2 $ and that $ \mu_L^{\sigma_3} = 1 $. So it follows that $ C_{\upsilon_3}^2 = - 1 $, and therefore $$ \mathrm{Tr}_{ \upsilon_3 }(\xi) = - 2 - \xi^{\alpha_3}. $$ For the other vertices, we compute that $$ \mathrm{Tr}_{ \upsilon_1 }(\xi) = \mathrm{Tr}_{ \upsilon_7 }(\xi) = 0, $$ $$ \mathrm{Tr}_{ \upsilon_2 }(\xi) = \mathrm{Tr}_{ \upsilon_5 }(\xi) = \mathrm{Tr}_{ \upsilon_6 }(\xi) = - 1, $$ and $$ \mathrm{Tr}_{ \upsilon_4 }(\xi) = - 7 - 5 \xi^{\alpha_4} - 3 (\xi^{\alpha_4})^2 - (\xi^{\alpha_4})^3. $$
Next, we must compute the contributions from the singularities. We will only write out the details for $ \varepsilon_3 = (\upsilon_3,\upsilon_4) $. In this case, we need to compute $ \mathrm{Tr}_{\sigma_3}(\xi) $. It is easily computed that $ \mu_1^{\sigma_3} = 3 $ and $ \mu_L^{\sigma_3} = 1 $. Theorem \ref{Formula} then gives that $$ \mathrm{Tr}_{\varepsilon_3}(\xi) = \mathrm{Tr}_{\sigma_3}(\xi) = 3 + 2 \xi^{\alpha_4} + (\xi^{\alpha_4})^2. $$ For the contributions from the other edges, we compute in a similar fashion that $$ \mathrm{Tr}_{\varepsilon_1}(\xi) = \mathrm{Tr}_{\varepsilon_6}(\xi) = 1, $$ $$ \mathrm{Tr}_{\varepsilon_2}(\xi) = 2 + \xi^{\alpha_3}, $$ and $$ \mathrm{Tr}_{\varepsilon_4}(\xi) = \mathrm{Tr}_{\varepsilon_5}(\xi) = 3 + \xi^{\alpha_4} + (\xi^{\alpha_4})^2. $$
Summing up, we get $$ \sum_{i=1}^7 \mathrm{Tr}_{\upsilon_i }(\xi) + \sum_{i=1}^6 \mathrm{Tr}_{\varepsilon_i}(\xi) = 1 - \xi^{\alpha_4} - (\xi^{\alpha_4})^3. $$ We can therefore conclude that the irreducible characters for the induced representation of $ \boldsymbol{\mu}_n $ on $ H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $ are $ \chi_1(\xi) = \xi^{\alpha_4} $ and $ \chi_2(\xi) = \xi^{3 \alpha_4} $.
The irreducible characters for the representation of $ \boldsymbol{\mu}_n $ on $ T_{\mathcal{J}'_k,0} $ induced by the action $ [\xi](\pi') = \xi^{-1} \pi' $ on $R'$ are the inverse characters of these, $ \chi_1^{-1}(\xi) = \xi^{ - \alpha_4} $ and $ \chi_2^{-1}(\xi) = \xi^{ - 3 \alpha_4} $. It is easily seen that $ [- \alpha_4]_n = (n-1)/4 $, and that $ [- 3 \alpha_4]_n = (3n-3)/4 $. Hence the jumps occur at the limits $1/4$ and $ 3/4 $ of these expressions $ (n-1)/4n $ and $ (3n-3)/4n $ as $ n $ goes to infinity.
\end{ex}
\begin{table}[htb]\caption{Genus $2$}\label{table 2}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline Fibertype & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ \\ \hline Jumps & $ 1/6 $, $ 3/6 $ & $ 1/4 $, $ 3/4 $ & $ 1/4 $, $ 3/4 $ & $ 3/12 $, $ 10/12 $ & $ 3/10 $, $ 9/10 $ & $ 1/5 $, $ 3/5 $ \\ \hline \end{tabular}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline $9$ & $10$ & $11$ & $12$ & $ 13 $ & $ 15 $ & $ 16 $ & $ 17 $ & $18$ \\ \hline $ 0 $, $ 1/2 $ & $ 0 $, $ 2/3 $ & $ 0 $, $ 3/4 $ & $ 0 $, $ 1/2 $ & $ 0 $, $ 0 $ & $ 0 $, $ 1/2 $ & $ 0 $, $ 2/3 $ & $ 1/6 $, $ 5/6 $ & $1/6$, $ 5/6 $ \\ \hline \end{tabular}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline $ 19 $ & $20$ & $21$ & $22$ & $ 23 $ & $ 24a $ & $ 24 $\\ \hline
$ 4/6 $, $ 5/6 $ & $ 7/10 $, $9/10$ & $ 3/5 $, $ 4/5 $ & $5/8$, $7/8$ & $ 2/4 $, $ 3/4 $ & $ 1/4 $, $ 3/4 $ & $ 1/4 $, $ 3/4 $ \\ \hline \end{tabular}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline $25$ & $26$ & $27$ & $28$ & $30$ & $31$ & $32$ & $33$ \\ \hline $ 3/6 $, $ 5/6 $ & $ 2/6 $, $ 5/6 $ & $ 3/8 $, $ 7/8 $ & $ 5/12 $, $ 11/12 $ & $ 0 $, $ 3/4 $ & $ 1/6 $, $ 4/6 $ & $ 0 $, $ 1/2 $ & $ 1/2 $, $ 1/2 $ \\ \hline \end{tabular}
\begin{tabular}{|c|c|c|c|c|c|c|c|} \hline $34$ & $35$ & $36$ & $37$ & $38$ & $40$ & $41$ & $41_a$ \\ \hline $ 1/6 $, $ 2/6 $ & $ 0 $, $ 1/4 $ & $ 1/5 $, $ 2/5 $ & $ 0 $, $ 1/3 $ & $ 1/4 $, $ 2/4 $ & $ 0 $, $ 0 $ & $ 0 $, $ 1/2 $ & $ 0 $, $1/2 $\\ \hline \end{tabular}
\begin{tabular}{|c|c|c|c|} \hline $41_b$ & $41_c$ & $43$ & $44$\\ \hline $0$, $ 2/3 $ & $0$, $3/4$ & $ 1/3 $, $ 2/3 $ & $ 2/5 $, $ 4/5 $\\ \hline \end{tabular} \end{table}
\begin{table}[htb]\caption{Genus $2$, $1_{KOD}$ and $14_{KOD}$}\label{table 3}
\begin{tabular}{|c|c|c|c|c|c|c|c|} \hline $KOD$ & $(I_n)$ & $(I_n^*)$ & $(II)^*$ & $(III)$ & $(III)^*$ & $(IV)$ & $(IV)^*$ \\ \hline Jumps & 0, 0 & 0, 1/2 & 0, 5/6 & 0, 1/4 & 0, 3/4 & 0, 1/3 & 0, 2/3 \\ \hline \end{tabular} \end{table}
\subsection{Special fibertypes for $g=2$}\label{special types} In the Ogg-classification, there are some fiber types that are built up from $g=1$ fiber types. We treat these separately in this section.
Let $KOD$ denote any of the seven specific fiber types $ I_n $, $ I_n^* $, $ II^* $, $ III $, $ III^* $, $ IV $ and $ IV^* $ from the Kodaira-classification. The fiber types $1$ and $14$ in the Ogg-classification are constructed in a certain way from choosing a type $KOD$. The various fibers obtained in this way are denoted by $1_{KOD}$ and $ 14_{KOD} $, and the associated jumps are listed in Table \ref{table 3} (they are actually the same for $1_{KOD}$ and $ 14_{KOD} $).
The fiber types $2$ and $39$ in the Ogg-classification are built up in a certain way by choosing \emph{two} types $ KOD_a $ and $ KOD_b $ from the seven specific Kodaira types $ I_n $, $ I_n^* $, $ II^* $, $ III $, $ III^* $, $ IV $ and $ IV^* $. For each of the 28 possible unordered pairs $ (KOD_a,KOD_b) $ made from this list, we get a fiber type denoted here by $ 2_{KOD_a,KOD_b} $ and $ 39_{KOD_a,KOD_b} $, respectively. Let $j(KOD)$ denote the jump associated to fiber type $KOD$ in Table \ref{table 1}. Then we have that the jumps for the fiber type $ 2_{(KOD_a,KOD_b)} $ are $ j(KOD_a) $ and $ j(KOD_b) $, and likewise, we find that the jumps for type $ 39_{KOD_a,KOD_b} $ are $ j(KOD_a) $ and $ j(KOD_b) $.
The fiber types $ 29 $, $ 29_a $ and $ 42 $ also break up into several cases in a similar way as in the cases mentioned above. Let $KOD' $ denote any of the three specific fiber types $ II^* $, $ III^* $ and $ IV^* $. The jumps for $ 29_{KOD'_a,KOD'_b} $ are listed in Table \ref{table 5}, the jumps for $ 29a_{KOD'} $ are listed in Table \ref{table 6} and the jumps for $ 42_{KOD'} $ are listed in Table \ref{table 7}.
We refer to \cite{Ogg} for precise details regarding the constructions mentioned here.
\begin{table}[htb]\caption{Genus $2$, $29_{KOD'_a, KOD'_b}$}\label{table 5}
\begin{tabular}{|c|c|c|c|} \hline $KOD'_a, KOD'_b$ & $(II)^*,(II)^*$ & $(II)^*,(III)^*$ & $(II)^*,(IV)^*$ \\ \hline Jumps & 5/6, 5/6 & 9/12, 10/12 & 4/6, 5/6 \\ \hline \end{tabular} \end{table}
\begin{table}[htb]
\begin{tabular}{|c|c|c|} \hline $ (III)^*, (III)^* $ & $ (III)^*, (IV)^*$ & $ (IV)^*, (IV)^* $\\ \hline 3/4, 3/4 & 8/12, 9/12 & 2/3, 2/3 \\ \hline \end{tabular} \end{table}
\begin{table}[htb]\caption{Genus $2$, $29a_{KOD'}$}\label{table 6}
\begin{tabular}{|c|c|c|c|} \hline $KOD'$ & $(II)^*$ & $(III)^*$ & $(IV)^*$ \\ \hline Jumps & $ 3/6 $, $ 5/6 $ & $ 2/4 $, $ 3/4 $ & $ 3/6 $, $ 4/6 $\\ \hline \end{tabular} \end{table}
\begin{table}[htb]\caption{Genus $2$, $42_{KOD'}$}\label{table 7}
\begin{tabular}{|c|c|c|c|} \hline $KOD'$ & $(II)^*$ & $(III)^*$ & $(IV)^*$\\ \hline Jumps & $ 2/6 $, $ 5/6 $ & $ 4/12 $, $ 9/12 $ & $ 1/3 $, $ 2/3 $\\ \hline \end{tabular} \end{table}
\subsection{Final remarks and comments} It would be interesting to know, for a curve $X/K$, the significance of the numerators and the denominators of the jumps in the filtration $ \{ \mathcal{F}^a \} $ of $ \mathcal{J}_k $, where $ \mathcal{J} $ is the N\'eron model of the Jacobian of $X$.
One could also try to obtain a closed formula for the irreducible characters of the representation of $ \boldsymbol{\mu}_n $ on $ H^1(\mathcal{Y}_k, \mathcal{O}_{\mathcal{Y}_k}) $, where $ \mathcal{Y} $ is the minimal desingularization of $ \mathcal{X}_{S'} $, and $ S'/S $ is tamely ramified of degree prime to the least common multiple of the multiplicities of the irreducible components of $ \mathcal{X} $. It seems clear that such a formula would reflect combinatorial properties of the intersection graph of $ \mathcal{X}_k $.
We do not know if our results remain true in the case where the minimal SNC-model $ \mathcal{X} $ of $X/K$ does not fulfill our initial assumptions, that is, if distinct components of $ \mathcal{X}_k $ with multiplicities divisible by $p$ intersect nontrivially. The main problem is the lack of a good description of the minimal desingularization of $ \mathcal{X}_{S'} $, where $ S'/S $ is a tame extension.
Finally, we think it would be interesting to study these filtrations for N\'eron models of abelian varieties that are not Jacobians. In that case, it is not so clear what kind of data would suffice in order to determine the jumps.
\end{document} | arXiv |
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
Geometric Series (using graphics calculators)
Infinite sum for GP's
Applications of Geometric Progressions
Applications of Geometric Series
Sequences and Saving Money (Investigation) LIVE
Fibonacci Sequence
First Order Linear Recurrences Introduction
Graphs and Tables - Recurrence Relations
Solutions to Recurrence Relations
Steady state solutions to recurrence relations
Applications of Recurrence Relations
Think about the generating rule for a particular arithmetic progression given by $t_n=2n+3$tn=2n+3. We can cleverly rearrange this equation to $t_n=5+\left(2n-2\right)$tn=5+(2n−2) . By taking out common factors on the last two terms, the expression becomes:
$t_n=5+\left(n-1\right)\times2$tn=5+(n−1)×2
Comparing this to the general formula for the $n$nth term of an AP, given by $t_n=a+\left(n-1\right)d$tn=a+(n−1)d , we can immediately see that the first term is $5$5 and the common difference is $2$2.
Thus any generating rule of the form $t_n=dn+k$tn=dn+k where $d$d and $k$k are constants can be shown to be an arithmetic sequence. Just like the equation $y=mx+c$y=mx+c is drawn as a straight line, so the arithmetic progression given by $t_n=dn+k$tn=dn+k is plotted as a series of points that are all in a straight line.
The first term is represented by the left-most point shown $\left(t_1=5\right)$(t1=5) . The gradient of the marked points measured, as the vertical distance between the points, is the common difference. In our example, the line of points is rising, so this indicates a positive common difference. In other instances, the line might be falling and this would indicate a negative common difference.
In the example above, where , the common difference is immediately recognizable as the coefficient of . A simple way of finding the first term is to evaluate as . We could place these values in a table as follows:
tn 5 7 9 11 13
Arithmetic sequences are said to grow linearly literally meaning 'in a straight line'. We find applications of linear growth in many areas of life, including simple interest earnings, straight-line depreciation, monthly rental accumulation and many others. Whenever something grows or diminishes in constant quantities over equal time periods, then that growth or fall is said to be linear.
The $n$nth term of an arithmetic progression is given by the equation $T_n=12+4\left(n-1\right)$Tn=12+4(n−1).
Complete the table of values.
$n$n
$1$1 $2$2 $3$3 $4$4 $10$10
$T_n$Tn
By how much are consecutive terms in the sequence increasing?
Plot the points in the table on the graph.
If the points on the graph were joined, they would form:
a straight line
a curved line
The plotted points represent terms in an arithmetic sequence:
Complete the table of values for the given points.
$1$1 $2$2 $3$3 $4$4
Identify $d$d, the common difference between consecutive terms.
Write a simplified expression for the general $n$nth term of the sequence, $T_n$Tn.
Find the $15$15th term of the sequence.
A racing car starts the race with $150$150 litres of fuel. From there, it uses fuel at a rate of $5$5 litres per minute.
Complete the table of values:
Number of minutes passed ($x$x)
$0$0 $5$5 $10$10 $15$15 $20$20
Amount of fuel left in tank ($y$y)
Write an algebraic relationship linking the number of minutes passed ($x$x) and the amount of fuel left in the tank ($y$y).
After how many minutes, $x$x, will the car need to refuel (i.e. when there is no fuel left) ?
Use arithmetic and geometric sequences and series
Apply sequences and series in solving problems | CommonCrawl |
Mendelian randomization study of maternal influences on birthweight and future cardiometabolic risk in the HUNT cohort
Gunn-Helen Moen ORCID: orcid.org/0000-0002-8768-09041,2,3,4,
Ben Brumpton ORCID: orcid.org/0000-0002-3058-10593,5,6,
Cristen Willer ORCID: orcid.org/0000-0001-5645-49667,8,9,
Bjørn Olav Åsvold ORCID: orcid.org/0000-0003-3837-21013,10,
Kåre I. Birkeland1,
Geng Wang ORCID: orcid.org/0000-0003-2478-29192,
Michael C. Neale11,
Rachel M. Freathy ORCID: orcid.org/0000-0003-4152-223812,
George Davey Smith ORCID: orcid.org/0000-0002-1407-83144,6,13,
Deborah A. Lawlor ORCID: orcid.org/0000-0002-6793-22624,6,13,
Robert M. Kirkpatrick11,
Nicole M. Warrington ORCID: orcid.org/0000-0003-4195-775X2,3,6 na1 &
David M. Evans ORCID: orcid.org/0000-0003-0663-46212,6 na1
Genetic association study
Genetics research
There is a robust observational relationship between lower birthweight and higher risk of cardiometabolic disease in later life. The Developmental Origins of Health and Disease (DOHaD) hypothesis posits that adverse environmental factors in utero increase future risk of cardiometabolic disease. Here, we explore if a genetic risk score (GRS) of maternal SNPs associated with offspring birthweight is also associated with offspring cardiometabolic risk factors, after controlling for offspring GRS, in up to 26,057 mother–offspring pairs (and 19,792 father–offspring pairs) from the Nord-Trøndelag Health (HUNT) Study. We find little evidence for a maternal (or paternal) genetic effect of birthweight associated variants on offspring cardiometabolic risk factors after adjusting for offspring GRS. In contrast, offspring GRS is strongly related to many cardiometabolic risk factors, even after conditioning on maternal GRS. Our results suggest that the maternal intrauterine environment, as proxied by maternal SNPs that influence offspring birthweight, is unlikely to be a major determinant of adverse cardiometabolic outcomes in population based samples of individuals.
There is a robust and well-documented observational relationship between lower birthweight and higher risk of cardiometabolic diseases in later life, including cardiovascular disease (CVD) and type 2 diabetes (T2D). The Developmental Origins of Health and Disease (DOHaD) hypothesis posits that adverse environmental factors in utero or in the early years of life result in increased future risk of cardiometabolic disease1,2,3,4,5,6,7. Evidence in favor of DOHaD has primarily come from observational1,2,8 and animal studies9; however, definitive causal evidence from human studies is lacking.
Mendelian randomization (MR) is an epidemiological method used to investigate whether an observational association between an exposure and an outcome represents a causal relationship10. Several studies have recently attempted to use MR to investigate the relationship between lower birthweight and cardiometabolic disease to inform on the validity of DOHaD11,12,13. However, these MR studies have used sub-optimal methodologies in which only offspring genotypes are considered as genetic instruments to proxy offspring birthweight14. This limitation contrasts strikingly with the argument that many DOHaD proponents would make, i.e. that an adverse maternal environment during pregnancy, results in low birthweight and increased risk of future cardiometabolic disease1,4,6. This hypothesis is entirely distinct from postulating that birthweight itself has a direct causal effect on risk of cardiometabolic disease14. Thus, these early MR studies have ignored the potential contribution of the maternal genome (correlated 0.5 with the offspring genome15,16), meaning that any association between offspring SNPs and offspring cardiometabolic risk may in fact be due to maternal genotypes, violating core assumptions underlying MR17, and complicating interpretation of the results. Indeed, Smith and Ebrahim10 in their initial description of the MR methodology, noted that the appropriate way of using MR to investigate the effects of the intrauterine environment on offspring outcomes (in their example maternal folate intake and offspring neural tube defects), was to use maternal genotypes to proxy the intrauterine environment10.
MR principles can be harnessed to test aspects of DOHaD using maternal SNPs that are related to offspring birthweight and/or adverse maternal environmental exposures during pregnancy14,16,18,19,20. For example, one possibility is to test whether SNPs in the mother that are directly related to offspring birthweight are also associated with offspring cardiometabolic risk factors, after conditioning on offspring genotypes at the same loci. To understand why this analysis would be informative, consider Fig. 1, which illustrates four credible ways in which maternal SNPs can simultaneously be related to offspring birthweight and future offspring cardiometabolic risk factors. In panel (a), maternal birthweight associated SNPs produce an in utero environment that leads to reduced fetal growth and subsequently low offspring birthweight and developmental compensations that produce increased risk of offspring cardiometabolic disease in later life. In panel (b), low offspring birthweight itself is causal for increased risk of offspring cardiometabolic disease. Under panels (a) and (b), the existence of a relationship between maternal alleles associated with lower birthweight and higher cardiometabolic risk in the offspring (after conditioning on offspring genotype at the same loci) argues strongly in favor of a DOHaD mechanism, where developmental compensations to reduced fetal growth impact on future health. In panel (c), the inverse genetic correlation between offspring birthweight and offspring cardiometabolic disease is driven entirely by genetic pleiotropy in the offspring genome, and importantly, not via DOHaD mechanisms. Under this model, maternal genotypes related to lower offspring birthweight will not be associated with increased offspring cardiometabolic risk after conditioning on offspring genotype. Finally, in panel (d), SNPs that exert maternal effects on offspring birthweight also pleiotropically influence offspring cardiometabolic disease through the postnatal environment. If genotyped father–offspring pairs are also available, then paternal SNPs at the same loci can be tested for association with offspring cardiometabolic risk factors (conditional on offspring genotype). The existence of such associations would suggest that the postnatal environment (i.e. early life DOHaD influences such as via genetic nurture or dynastic effects rather than the intrauterine environment) may be responsible for the correlation between maternal genotypes and offspring cardiometabolic risk factors.
Fig. 1: Four credible ways in which maternal single nucleotide polymorphism (SNP)s can be related to offspring birthweight and offspring cardiometabolic risk factors.
a Maternal SNPs produce an adverse in utero environment that leads to fetal growth restriction and subsequently low offspring birthweight and developmental compensations that produce increased risk of offspring cardiometabolic disease in later life. b Maternal SNPs produce an adverse in utero environment that leads to fetal growth restriction and low offspring birthweight. Low offspring birthweight in turn is causal for increased risk of offspring cardiometabolic disease. c Maternal SNPs produce an adverse in utero environment that leads to fetal growth restriction and reduced birthweight. The same SNPs are transmitted to the offspring and pleiotropically influence offspring cardiometabolic risk through the offspring genome. d Maternal SNPs produce an adverse in utero environment that leads to fetal growth restriction and reduced offspring birthweight. SNPs that exert maternal effects on offspring birthweight also pleiotropically influence offspring cardiometabolic disease through the postnatal environment. The star on the arrows denotes the act of conditioning on maternal or offspring genotype blocking the association between maternal and offspring variables. The dotted paths indicate paths in which the maternal genotype can be related to offspring phenotype that are not to do with intrauterine growth restriction. Finally, we note that some offspring SNPs may also exert direct effects on offspring birthweight (these not shown). The presence of direct effects from offspring genotype on offspring birthweight is inconsequential so long as the relevant analyses are conditional on offspring genotype.
In other words, the presence of correlation between maternal genotypes and offspring cardiometabolic risk factors, after conditioning on offspring genotypes at the same loci, is highly suggestive of DOHaD mechanisms related to lower birthweight (providing these associations are not replicated in father–offspring pairs also). We emphasize that the paradigm illustrated in Fig. 1, which we use in our study, only tests one aspect of DOHaD (i.e. that maternal exposures that affect offspring birthweight are also causal for increased offspring cardiometabolic risk). It is possible that there are other maternal exposures that affect the offspring prenatal or postnatal environment, but do not influence offspring birthweight, and still affect future offspring cardiometabolic risk. We do not test for the influence of these exposures on offspring cardiometabolic risk in this study, but limit our attention to those that exert an effect on offspring birthweight (a distinction we explore further in the discussion).
We have previously used this paradigm to examine the association between maternal birthweight related SNPs and offspring blood pressure in the UK Biobank study as a preliminary test of the validity of this possible DOHaD mechanism18. Interestingly, this showed that maternal SNPs related to low offspring birthweight were actually associated with lower offspring systolic blood pressure after conditioning on offspring genotype at the same loci (i.e. the opposite of what would be expected if maternal intrauterine effects that reduce fetal growth result in higher later-life cardiometabolic risk). However, the number of mother–offspring pairs used in this previous study was small (N = 3,886) and systolic blood pressure was the only cardiometabolic risk factor investigated. Therefore, the results from this preliminary study need to be replicated and further cardiometabolic risk factors examined. The Norwegian based HUNT Study21, which contains approximately 70,000 genotyped individuals, including 45,849 parent–offspring pairs, is one of the few cohorts where such analyses can be conducted. The average age of the HUNT offspring is approximately 40 years, rendering this cohort not only one of the largest cohorts in the world with genotyped mother–offspring pairs (and father–offspring pairs) with birthweight information, but also one of the few with offspring old enough to have developed adverse cardiometabolic profiles.
In this work, we perform genetic association analyses in up to 26,057 genotyped mother–offspring pairs from the Norwegian HUNT Study in order to investigate whether there is evidence for a causal effect of the intrauterine environment (proxied by maternal SNPs that influence offspring birthweight) on offspring cardiometabolic risk factors. We investigate whether maternal genotypes associated with lower offspring birthweight are also associated with later life offspring cardiometabolic risk factors such as blood pressure, non-fasting glucose levels, body mass index (BMI), and lipid levels, after conditioning on offspring genotype at the same loci. We also perform similar analyses in up to 19,792 father–offspring pairs to investigate whether there is evidence for a postnatal environmental effect (genetic nurture or dynastic effects), rather than an intrauterine environmental effect. In the course of executing these analyses, we implement a computationally efficient genetic linear mixed model that not only enables the investigation of causal questions relevant to the specific DOHaD mechanism that is the focus of this paper, but also simultaneously accounts for the non-independence between siblings and the considerable cryptic relatedness within the HUNT Study. We show no evidence for a causal effect of the intrauterine environment (as proxied by maternal genetic effects on offspring birthweight) on offspring cardiometabolic risk factors. We do, however, find evidence that offspring SNPs pleiotropically influence both birthweight and future cardiometabolic risk factors, which helps explain the robust observational relationships between the variables.
Phenotypic correlations
HUNT offspring with recorded values for birthweight were on average 30.1 years old, with a minimum age of 19, and a maximum age of 41 at the time of measurement used in this study. Descriptive statistics on the mother–offspring and father–offspring pairs are presented in Table 1. It is important to note that only offspring born after 1967 had birthweight recorded and were included in this part of the analysis. Table 2 shows the phenotypic association between own birthweight and SBP, DBP, non-fasting glucose, non-fasting total, LDL and HDL cholesterol, non-fasting triglycerides, and BMI. Consistent with many previous observational epidemiological studies22,23,24,25, linear regression yielded negative point estimates of the observational relationship between birthweight and blood pressure, LDL, total cholesterol, and BMI. We also found evidence for positive quadratic terms in the model between birthweight and both BMI and glucose, suggesting U-shaped/J-shaped relationships between these variables. Finally, we found evidence for a positive linear relationship between HDL cholesterol and birthweight with additional evidence for a convex quadratic term indicating small and large babies are likely to have slightly reduced HDL levels in later life.
Table 1 Descriptive statistics for offspring cardiometabolic risk factors in the phenotypic association analyses.
Table 2 Association between offspring birthweight, offspring birthweight squared, and offspring cardiometabolic risk factor in mother–offspring and father–offspring pairs.
Analysis of fetal growth and cardiometabolic risk factors in the HUNT offspring
We first checked whether the GRSs of birthweight associated SNPs from the latest GWAS of birthweight18 were also related to offspring birthweight in HUNT. The full results are presented in Supplementary Table 1. In short, we found that maternal GRSs were strongly associated with increased offspring birthweight after conditioning on offspring GRS in HUNT. Offspring GRS was related to offspring birthweight, but this relationship attenuated after controlling for maternal GRS. In the case of the GRS consisting of SNPs that only had a maternal effect from the Warrington et al18 birthweight GWAS, offspring GRS was not strongly related to offspring birthweight after controlling for maternal GRS. As expected, paternal GRS was not associated with offspring birthweight after conditioning on offspring GRS. The effect size of the offspring GRS was similar in mother–offspring and father–offspring pairs, and did not attenuate after adjusting for paternal GRS.
For the primary analyses investigating the effect of GRS on offspring cardiometabolic traits, we had a total of 26,057 mother–offspring pairs and 19,792 father–offspring pairs. HUNT offspring were on average 40 years old, with a minimum age of 19, and a maximum age of 85 at the time of measurement used in this study. Descriptive statistics on all of the outcome variables in the two samples are presented in Table 3. Our asymptotic power calculations indicated that we had (≥80%) power to detect a maternal genetic effect that explained as little as 0.04% of the variance in offspring outcome (N = 26,057) (two tailed α = 0.05) and slightly lower power (>68%) (N = 19,792) to detect a paternal genetic effect responsible for a similar proportion of the offspring phenotypic variance. Due to some missing data in the offspring's cardiometabolic risk factors, the number of mother–offspring and father–offspring pairs differed slightly across the outcomes (Table 3). Although the sample size for some of the analyses is slightly lower (lowest being 25,461 mother–offspring pairs and 19,339 father–offspring pairs) we retain statistical power to detect an association of maternal GRS with offspring cardiometabolic risk factors (79% and 67%, respectively) using the same parameters as above.
Table 3 Descriptive statistics for offspring cardiometabolic risk factors in the primary analyses.
We found little evidence for an association between maternal (or paternal) GRS and any of the offspring cardiometabolic risk factors in later life, after adjusting for offspring GRS (Tables 4, 5; Supplementary Data 1). These tables show the estimated expected change in offspring cardiometabolic outcome per one unit (i.e. allele) increase in maternal/paternal genetic risk score after conditioning on offspring (or maternal/paternal) genetic risk score. These results hold for systolic blood pressure, which had previously been found to associate with maternal GRS in the Warrington et al GWAS of birthweight18. In contrast, there was strong evidence for a relationship between offspring GRS and some of the offspring phenotypes after conditioning on maternal GRS (Table 6). Specifically, there was evidence for a positive association between offspring GRS and both offspring glucose and LDL, and evidence for a negative relationship between offspring GRS and both systolic blood pressure and triglycerides. It is important to note that the blood samples used to measure lipids and glucose were non-fasting samples, which could influence these results.
Table 4 Results of regressing offspring cardiometabolic risk factors on maternal GRSa after conditioning on offspring GRSa in mother–offspring pairs.
Table 5 Results of regressing offspring cardiometabolic risk factors on paternal GRSa after conditioning on offspring GRSa in father–offspring pairs.
Table 6 Results of regressing offspring cardiometabolic risk factors on offspring GRSa after conditioning on maternal GRSa in mother–offspring pairs.
Cardiometabolic pathology becomes more apparent with increasing age. Indeed, it is possible that younger individuals within the HUNT Study do not show observable compensatory changes in cardiometabolic risk factors, reducing the power of our analyses to detect evidence for the observational associations between birthweight and cardiometabolic risk factors to be causal. We therefore divided our dataset into two strata based on age of the offspring (i.e. offspring under 40 years of age and offspring between 40 and 60 years of age). Our asymptotic power calculations indicated that we had (≥80%) power to detect a maternal genetic effect that explained as little as 0.09% of the variance in offspring SBP (N = 12,037 and N = 11,849) (α = 0.05) and slightly lower power (>66%) (N = 10,393 and N = 8402) to detect a paternal genetic effect responsible for a similar proportion of the offspring phenotypic variance. Table 7 (and Supplementary Table 2) shows the main results of the stratified analyses compared with those previously reported in the UK BioBank by Warrington et al in their GWAS of birthweight18. Whereas Warrington and colleagues found a significant positive effect of maternal GRS on offspring SBP when adjusting for offspring GRS, we find no effect in the stratified analyses.
Table 7 Association between maternal or paternal GRSa influencing offspring birthweight and offspring SBP after conditioning on offspring GRSa in different age strata compared with previous results from Warrington et al. 18.
The Developmental Origins of Health and Disease (DOHaD) hypothesis posits that adverse environmental factors in utero or in the early years of life result in increased future risk of cardiometabolic disease1,4,6. In this study, we used an MR paradigm to provide evidence for or against the existence of DOHaD mechanisms that are related to fetal growth and lower birthweight for a range of cardiometabolic risk factors16,18. Specifically, we tested whether a genetic risk score in mothers intended to proxy for maternal intrauterine influences on offspring birthweight was also associated with offspring cardiometabolic risk factors, whilst simultaneously conditioning on offspring GRS constructed from the same birthweight associated loci. There was no strong evidence of association in a sample of over 25,000 mother–offspring pairs from the Norwegian HUNT study, implying that if such an effect on cardiometabolic risk factors exists, it may be small compared to other sources of inter-individual variation, or only affects a few individuals.
Our study is, to the best of our knowledge, the largest parent–offspring MR study of DOHaD performed to date. The HUNT Study contains over 25,000 genotyped mother–offspring pairs where the majority of the offspring are middle-aged adults, and are therefore old enough to have begun developing observable signs of cardiometabolic disease. Our asymptotic calculations indicated that we had strong (≥80%) power to detect a maternal genetic effect that explained as little as 0.04% of the variance in offspring outcome (two tailed α = 0.05). In contrast, our previous study in the UK Biobank18 (where we first used this MR paradigm to investigate DOHaD), involved only 3886 mother–offspring pairs, and was likely underpowered. Interestingly, Warrington and colleagues found evidence for a positive relationship between maternal birthweight lowering SNPs and reduced offspring SBP (i.e. the opposite of what DOHaD would predict); however, this result did not replicate in our sample. Possible reasons for the discrepancy include the differences in sample ascertainment across the studies, or that the younger offspring in HUNT did not manifest a large enough effect18. When stratifying our analysis by age, we did find effects in the same direction as our original study for the 40-60 years age group; however, the statistical support for the effect was weak. Taken together, the UK BioBank and HUNT results provide converging evidence that maternal genetic effects that predispose to low offspring birthweight are not associated with increased systolic blood pressure in later life.
In contrast, we did find evidence for association between offspring GRS and a number of offspring cardiometabolic risk factors, even after conditioning on maternal GRS. These results are broadly consistent with the Fetal Insulin hypothesis26,27,28,29 and previous studies that have used LD score regression and G-REML approaches to suggest that much of the phenotypic correlation between birthweight and cardiometabolic risk is driven by genetic pleiotropy in the offspring genome rather than DOHAD mechanisms18,30. We note that the direction of the associations involving the offspring GRS and offspring phenotypes are a little difficult to interpret, since the GRS were defined on the basis of maternal genotypic effects on offspring birthweight, whereas these reported associations involve offspring GRS. Offspring genotypes at some of the same loci are known to have quantitatively and qualitatively different effects on offspring birthweight (including the direction of association) compared to the maternal effects. Also important to take into account is the fact that the lipid and glucose measurements were performed in non-fasting samples, which could influence these results, particularly as it is known that mean blood glucose levels and triglycerides are higher in the first three hours after calorie intake31. Nevertheless, our results show clearly that maternal SNPs that influence offspring birthweight have pleiotropic effects on offspring cardiometabolic traits when these same SNPs are transmitted to their offspring.
Another novel facet of our study was the use of the OpenMx software package to model the complicated data structure within the HUNT Study. Using traditional formulations of FIML to model the relatedness structure using a genetic relationship matrix would be computationally prohibitive within the HUNT sample, as maximizing the likelihood would involve an inversion of a matrix of order N. In contrast, our implementation permits complicated tests of association to be performed in the fixed effects part of the model, whilst simultaneously modeling cryptic relatedness in the random effects part of the model in a computationally efficient manner32. We hope that our implementation will prove useful in complicated genetic analyses of other large scale population-based cohorts where cryptic relatedness/population stratification is likely to be an issue. We have included an example R script in Supplementary Note 1 of the manuscript that can be used as a template by interested researchers. We caution users, however, that specification of the covariance part of the model is more rigid using our speed up in that only two variance components can be fitted simultaneously, one being a residual variance component that is uncorrelated across individuals.
Our approach has a number of limitations which we discuss in the remaining paragraphs. First, we assume that the maternal SNPs that affect offspring birthweight do so via fetal growth (as reflected in birthweight). This is important, because as many others have noted, it may not be fetal growth/birthweight itself that is relevant for the validity of DOHaD. Rather it could be poor development of different key organs, in key stages of the pregnancy or a particular adverse maternal environment due to famine, disease or a range of other factors. Indeed, it would likely be profitable to use the same framework to investigate the association between offspring cardiometabolic disease and other adverse maternal exposures, such as maternal BMI, maternal alcohol consumption, preeclampsia, and gestational diabetes. Their effect may be qualitatively and quantitatively different from the maternal effect on birthweight within healthy subjects delivering babies within the normal range. However, even though the mechanisms through which our maternal SNPs influence offspring birthweight are largely unknown (and therefore our genetic risk score is largely unspecific), we know that they play an important part in fetal growth of the offspring. Further MR studies on different maternal exposures are warranted including on those that do not necessarily exert observable effects on offspring birthweight, but proxy other more specific maternal environments. Moreover, we used unweighted GRS of birthweight associated SNPs in our MR framework. Using a weighted maternal GRS and conditioning on a weighted offspring GRS does not completely block the path through the offspring's genome, increasing type 1 error rate for the maternal effect on offspring cardiometabolic phenotype. To avoid the inflation in type 1 error, we use an unweighted maternal GRS and condition on offspring unweighted GRS, which is sufficient to block this path. However, the main reason for using an unweighted GRS is that weighting SNPs by the strength of association between maternal genotype and offspring birthweight would only be appropriate if the effect of the maternal SNP on the offspring's cardiometabolic phenotype was mediated through offspring birthweight (i.e. panel B of Fig. 1). However, we believe it is more likely that offspring birthweight is a marker of several latent processes, which may then affect the offspring's cardiometabolic phenotype (i.e. more akin to panel A of Fig. 1). Using weights derived from a maternal GWAS of birthweight may not accurately reflect SNP associations with these underlying latent processes, particularly if there are many such processes that are relevant for later life disease risk.
Second, our example here, and MR approaches in general, typically test small changes in an exposure. However, it may be that DOHaD mechanisms are important in the genesis of cardiometabolic risk, but only in the case of severe exposures (e.g. famine or obesity) at the extreme ends of the spectrum. These effects may be qualitatively different from small perturbations in the environment that produce relatively subtle variations in the normal healthy population. If DOHaD is only relevant in the case of extreme environmental effects, then MR approaches applied to population data may not be well suited to testing the hypothesis.
Third, although our methods rely on MR principles to inform on the validity of DOHaD (i.e. we use genetic variants to increase our study's robustness to environmental confounding), we did not perform formal instrumental variables analyses in this manuscript. The reason is that we do not have appropriate estimates of the effect of maternal genotypes on the intrauterine environment. We only have estimates of the relationship between SNPs and offspring birthweight, which is an imperfect proxy of fetal growth restriction. Therefore, it does not make sense to estimate causal effect sizes in our study as in typical MR analyses. However, we note that it may be possible to estimate the effect of a putative latent variable indexing growth restriction using, for example, latent variable models; this is an area of future research for our group.
Fourth, our power calculations show that we were well powered (>80% at α = 0.05) to detect an association between maternal genetic risk score and offspring cardiometabolic risk factors responsible for as little as 0.04% of the phenotypic variance. However, whilst our study, to the best of our knowledge, is the largest and most powerful genetic investigation into DOHaD to date, the actual variance in the offspring cardiometabolic risk factor explained by the maternal GRS, depends critically upon the underlying genetic model, and could be even smaller than 0.04%. In an attempt to make this clear, Fig. 2 is a path diagram that illustrates the relationship between maternal GRS, offspring GRS, an intrauterine environment that reduces fetal growth (modeled as a single latent unobserved variable), offspring birthweight and an offspring cardiometabolic risk factor. In this diagram, and consistent with most formulations of DOHaD, we assume that (i) there is no direct causal effect of birthweight on cardiometabolic risk (i.e. no arrow from birthweight to the cardiometabolic risk factor), and (ii) no effect of maternal GRS on the offspring cardiometabolic risk factor that goes through paths other than fetal growth restriction (e.g. no postnatal mechanisms). To make calculations and explication easier, we assume that all variables have been standardized to unit variance. Under this model, the correlation between birthweight and the cardiometabolic risk factor is a function of two processes. One is the effect of the intrauterine environment on birthweight and the cardiometabolic risk factors (i.e. the product of path coefficients λ1 and λ2). The second is the residual covariance between birthweight and the cardiometabolic risk factors. This latter pathway includes both environmental factors other than fetal growth restriction that affect both phenotypes and the effect of polygenes that are not modeled in the experiment whose joint effects are quantified by the parameter Θ. These correlations could be positive or negative individually, but when combined produce a very small (|r | <= 0.05) negative phenotypic correlation between birthweight and most of the cardiometabolic risk factors. The point is that, unless the residual covariance between birthweight and the cardiometabolic risk factor is positive, the values for path coefficients λ1 and λ2 are likely to be very small in order to be consistent with the observed phenotypic correlations.
Fig. 2: Path diagram of the relationship between maternal Genetic Risk Score (GRS), offspring GRS, the intrauterine environment, offspring birthweight and an offspring cardiometabolic risk factor.
Variables within square boxes represent observed variables, whereas variables in circles represent latent unobserved variables. Unidirectional arrows represent causal relationships from tail to head, whilst two headed arrows represent correlational relationships. Greek letters on one headed arrows represent path coefficients which quantify the expected causal effect of one variable on the other. Greek letters on two headed arrows represent covariances between variables. The two epsilon variables represent residual latent factors (both environmental and genetic) that are not modeled in the study. The coefficient Θ represents the covariance between the residual terms. We assume that all variables are standardized to have unit variance. Consequently, the residual variance of the offspring GRS is set to 0.75 since ¼ of the variance comes from the maternal genotype. For the purposes of the power calculation described in the discussion, we assume that maternal single nucleotide polymorphism (SNP)s that affect offspring birthweight do so through a single latent intrauterine factor, and that this factor also exerts long term effects on the offspring cardiometabolic risk factor of interest.
The variance in birthweight explained by the maternal GRS is a function of the direct association between the SNPs and the intrauterine environment (the path coefficient γ), and the effect of the intrauterine environment on birthweight (the path coefficient λ1—the precise formula being: γ2λ12). The variance explained in the cardiometabolic risk factor by the maternal GRS is equal to the product of the SNPs' direct effect on the intrauterine environment (path coefficient γ in Fig. 2), multiplied by the effect of the intrauterine environment on the cardiometabolic risk factor (path coefficient λ2 in Fig. 2) all squared. There are an infinite number of ways these parameters can vary to make the underlying model consistent with the pattern of observed correlations and the proportion of variance explained in birthweight by the maternal GRS. To give the reader an idea of the potentially small numbers involved, we assume that the correlation between birthweight and the cardiometabolic risk factor is completely explained by the intrauterine environment and λ1 = −0.5 and λ2 = 0.1 (so that the observed correlation r = λ1 λ2 = −0.05). In order for the underlying model to also be consistent with the maternal GRS explaining a small percentage of the variance in birthweight (say 0.5% of the variance), then the path coefficient between the maternal GRS and the latent intrauterine variable γ would equal \(\sqrt {\frac{{0.005}}{{\lambda ^2}}}\) = 0.1414. These values in turn would imply that the variance explained in the cardiometabolic risk factor by the maternal genetic risk score would be 0.14142 × 0.12 = 0.02%, which is a small proportion of the variance, and one that we are only moderately well powered to detect (>50%) in our study. Our point, however, is that the proportion of variance in the outcome explained by the maternal GRS may be very small, and so power may only be moderate despite the very large sample size of HUNT. The corollary to this though is that we are very well powered to detect larger effects of the intrauterine environment influencing offspring birthweight on cardiometabolic risk factors, and the fact that we do not detect these suggests that if such an effect is present, it is likely to be small.
Finally, we recognize that our act of conditioning on offspring GRS, may have induced a (spurious) correlation between maternal GRS and paternal GRS due to conditioning on a collider variable, potentially biasing the results of our maternal GRS analyses. However, any such bias is likely to be small in magnitude as it relies on the existence of (and is proportional to the size of) direct paternal genetic effects from the same SNPs on the offspring phenotype. As sizeable paternal genetic effects on offspring cardiometabolic risk are unlikely at these loci, we doubt that collider bias is a serious impediment to the validity of our study33.
In conclusion, we did not find evidence for a causal effect of the intrauterine environment (as proxied by maternal genetic effects on offspring birthweight) on offspring cardiometabolic risk factors in a population-based sample of individuals. We did, however, find evidence of genetic pleiotropy between offspring birthweight and offspring cardiometabolic risk factors which helps explain the robust observational relationships between the variables.
HUNT study
The Nord-Trøndelag Health Study (HUNT) is a large population-based health study of the inhabitants of Nord-Trøndelag County in central Norway that commenced in 1984. A comprehensive description of the study population has been previously reported21. Approximately every 10 years the entire adult population of Nord-Trøndelag (~90,000 adults in 1995) is invited to attend a health survey which includes comprehensive questionnaires, an interview, clinical examination, and detailed phenotypic measurements (HUNT1 (1984 to 1986); HUNT2 (1995 to 1997); HUNT3 (2006 to 2008) and HUNT4 (2017 to 2019)). These surveys have high participation, with 89%, 69%, 54%, and 54% of invited adults participating in HUNT1, 2, 3, and 4, respectively21,34. Additional phenotypic information is collected by integrating national registers. Approximately 90% of participants from HUNT2 and HUNT3 were genotyped in 201535, and the genotype and phenotype data used in the subsequent analysis are exclusively from these two surveys.
The HUNT Study was approved by the Regional Committee for Medical and Health Research Ethics, Norway and all participants gave informed written consent (REK Central application number 2018/2488).
Genotyping, quality control, and imputation
DNA from 71,860 HUNT samples was genotyped using one of three different Illumina HumanCoreExome arrays (HumanCoreExome12 v1.0, HumanCoreExome12 v1.1, and UM HUNT Biobank v1.0)35. Genomic position, strand orientation, and the reference allele of genotyped variants were determined by aligning their probe sequences against the human genome (Genome Reference Consortium Human genome build 37 and revised Cambridge Reference Sequence of the human mitochondrial DNA; http://genome.ucsc.edu) using BLAT36. Ancestry of all samples was inferred by projecting all genotyped samples into the space of the principal components of the Human Genome Diversity Project (HGDP) reference panel (938 unrelated individuals; downloaded from http://csg.sph.umich.edu/chaolong/LASER/)37,38, using PLINK v1.9039. The resulting genotype data were phased using Eagle2 v2.340. Imputation was performed on the 69,716 samples of recent European ancestry using Minimac3 (v2.0.1, http://genome.sph.umich.edu/wiki/Minimac3)41 with default settings (2.5 Mb reference based chunking with 500 kb windows) and a customized Haplotype Reference consortium release 1.1 (HRC v1.1) for autosomal variants and HRC v1.1 for chromosome X variants42.
Identifying genotyped parent–offspring pairs
Before the kinship analysis, the plink files with genotyped SNPs underwent a second stage of cleaning. Any individuals whose inferred sex contradicted their reported gender (N = 348) as well as individuals showing high or low heterozygosity (±5 SD from the mean) (N = 412) were removed (760 individuals in total). In addition, variants with minor allele frequency <0.005 or more than 5% missing rate were removed. Parent–offspring pairs were identified by kinship analysis using the KING software version 2.2.443. Only genotyped SNPs shared across the arrays on autosomal chromosomes were used for the analysis – a total of 257,488 SNPs.
From the analysis, 46,428 parent–offspring relationships were identified, in addition to 35,373 full siblings, 128,334 second degree relationships and 386,619 third degree relationships based on the kinship analysis performed using the KING software and recommended thresholds for relatedness implemented as part of this package43. Any parent–offspring pair with 15 years or fewer difference in birth year was removed from further analyses. After removing these pairs, a total of 26,057 mother–offspring pairs and 19,792 father–offspring pairs of European ancestry with genotype information passing QC were identified. Each parent had between one and eight offspring available for analysis. Supplementary Table 3 shows the number of offspring per mother/father available for analysis.
Genetic risk scores
SNPs previously associated with own or offspring birthweight at genome-wide levels of significance in the Early Growth Genetics (EGG) Consortium paper18 were extracted from the HUNT imputed genotype data in dosage format using plink239. Dosages were coded so that increasing dosages reflected maternal alleles associated with increased offspring birthweight based on conditional genome-wide association study (GWAS) results previously published18. Unweighted genetic risk scores (GRS) were constructed by simply adding the expected number of increasing birthweight alleles together for each individual. We used unweighted scores because we do not know the extent to which each allele influences growth restriction, and so weighting the scores by e.g. their observed effect on birthweight would be less appropriate. Three GRS were constructed—one using all autosomal SNPs shown to have an effect on birthweight (N = 204) from the recent EGG Consortium GWAS paper of birthweight18 that found 205 autosomal SNPs, but rs9267812 was not available in the HUNT data), one using SNPs shown to have a maternal effect (N = 71; i.e. some of these SNPs also had a fetal effect on birthweight), and one using SNPs that only had a significant maternal effect (N = 31) (Supplementary Data 2).
Outcome variables
During the health surveys (HUNT1-4)21 clinical examination, and detailed phenotypic measurements were performed on all participants, data from HUNT3 or HUNT2 were used in the subsequent work. For all cardiometabolic risk factors in the offspring (BMI, systolic blood pressure (SBP), diastolic blood pressure (DBP), non-fasting glucose (Glucose), total cholesterol, high density lipoprotein (HDL) cholesterol, low density lipoprotein (LDL) cholesterol, and triglycerides), the most recent value (e.g. values measured in HUNT3) was used if available. If the individuals were not a part of HUNT3, measurements from HUNT2 were used. Age at participation was calculated to correspond with the health survey chosen. Blood pressure was taken three times during the clinical examination, and SBP and DBP measurements were calculated as the average of the second and third measurement. For individuals who only had two blood pressure measurements taken (12% of offspring in the mother–offspring pairs and 9% of offspring in the father–offspring pairs), the second measurement was used.
For the blood measurements, samples were taken from non-fasting participants. In HUNT3, participants' total cholesterol was measured by enzymatic cholesterol esterase methodology; HDL cholesterol was measured by accelerator selective detergent methodology; triglycerides were measured by glycerol phosphate oxidase methodology; and glucose was measured by Hexokinase/G-6-PDH methodology (Abbott, Clinical Chemistry, USA). In HUNT2, participants' total and HDL cholesterol and triglycerides were measured by applying enzymatic colorimetric cholesterol esterase methods (Boeheringer Mannheim, Mannheim, Germany) and glucose was measured by an enzymatic hexokinase method. The measurements are shown in millimole per liter. Weight and height were measured in light clothes and BMI was calculated as weight (kilograms) divided by the squared value of height (in meters).
We adjusted the blood pressure measurements of individuals who self-reported using blood pressure lowering medication by adding 15 mmHg to their SBP and 10 mmHg to their DBP. We chose this procedure over including medication use as a covariate to avoid introduction of possible collider biases into the analyses44. Non-fasting LDL cholesterol was calculated using the Friedewald formula45. All values more than 4 standard deviations from the mean were removed. If the variable was not normally distributed (non-fasting triglycerides, BMI and non-fasting glucose) the values were natural log transformed before removing outlying values.
Phenotypic relationship between birthweight and cardiometabolic risk factors
Own birthweight was available for individuals in HUNT after linking with the Medical Birth Registry of Norway (MBRN)46 using the unique 11-digit identification numbers assigned to all Norwegian residents. This was performed by a third party and the researchers only had access to de-identified data. The registry commenced in 1967, when health authorities began reporting pregnancy-related data; therefore, birthweight measurements were only available for HUNT participants born in 1967 or later. The validity of information on birthweight in the MBRN has previously been reported as very good47. Individuals in HUNT with own birthweight who were part of a multiple birth (210 twins and 4 triplets) were excluded from the analysis. Additionally, we excluded individuals with a known congenital malformation (N = 317), if their birth was induced or performed via a cesarean section (N = 2,488), if their birthweight was under 1000 g (N = 1), or if they were born before 258 days of gestation (N = 451). To investigate if we could replicate the previously reported phenotypic associations between birthweight and cardiometabolic risk factors, we fitted a linear mixed model to N = 7,825 mother–offspring pairs and then N = 6,875 father–offspring pairs using the software package OpenMx48 using the procedure described below. We modeled offspring cardiometabolic risk factor as the outcome and included offspring birthweight, offspring birthweight squared, offspring age, offspring sex and measurement occasion (HUNT2 or HUNT3) as fixed effects. Offspring birthweight squared was included as a fixed effect to capture a possible non-linear relationship between birthweight and cardiometabiolic risk factors, as has been observed in some studies previously49,50,51. The other covariates were included to reduce error variance in the cardiometabolic risk factor and consequently increase the power of the analyses. The non-independence between siblings and the cryptic relatedness between offspring was modeled using a genetic relatedness matrix in the random effects part of the model as described below.
Analysis of fetal growth and later life outcomes in the offspring
Cryptic relatedness is a problem for genetic studies of large population-based cohorts like HUNT. Whilst point estimates from genetic association analyses will often be unbiased in the presence of cryptic relatedness, standard errors can be too small, meaning that statistical tests of association may have inflated Type 1 error rates. Dropping one person from each pair of putatively related individuals is inefficient and requires an arbitrary threshold to be specified in order to declare a pair of individuals related (e.g. first-order relatives). Thus, dropping individuals is unlikely to remove the non-independence of the error terms completely. In the case of single SNPs, this problem can be solved by using custom-written software packages. These software packages allow users to fit linear mixed models where a dataset is analyzed as one large set of related individuals and the similarity between individuals is parameterized by a genome-wide genetic relationship matrix. However, these software packages are designed for GWAS analysis and may not have the flexibility to enable users to fit more complicated statistical models such as those involving genetic risk scores and conditional association analyses, as we wish to do here.
We therefore parameterized our statistical model using the OpenMx package48 in the R statistics software. OpenMx allows users to model multivariate normal data flexibly in terms of fixed and random effects, and to estimate parameters simultaneously using full information maximum likelihood (FIML). We used the fixed effects part of the model to test for genetic association between the maternal GRS and offspring phenotype, and modeled the similarity between individuals in the random effects part of the model. The model for the fixed effects included terms for the genetic risk score of the mother (father), the genetic risk score of the offspring, age, sex and the measurement occasion (HUNT2 or HUNT3). Again, the inclusion of age, sex and measurement occasion in the fixed effects part of the model was to reduce error variance in the cardiometabolic risk factor of interest and hence increase power. In the random effects part of the model we modeled the similarity between individuals in terms of a genetic relationship matrix (GRM) and an identity matrix for residual effects:
$${\mathbf{{\Sigma}}} = {\mathbf{A}}\sigma _g^2 + {\mathbf{I}}\sigma _e^2,$$
where Σ is the expected N x N phenotypic covariance matrix, A is an N x N genetic relationship matrix calculated using the GCTA software version 1.9352, I is an N x N identity matrix, \(\sigma _A^2\) and \(\sigma _E^2\) are variance components due to additive genetic and residual sources of variation respectively, and N is the number of individuals in the analysis. When creating the GRM it was important to exclude the SNPs (and the SNPs in linkage disequilibrium around them) used in the genetic risk scores. Otherwise we would risk some of our association signal in the fixed effects part of the model being attenuated because they would also be modeled in the random effects part of the model. The GRM was therefore calculated after excluding the known birthweight SNPs and any SNPs 1 Mb away from them.
Estimating these parameters using FIML in OpenMx is computationally intensive in that it involves inverting a matrix of order N. Indeed, our initial attempts to do this suggested that fitting the model to the HUNT data this way may not be possible given the limitations of our computing hardware. We therefore reparametrized the statistical model using a factor rotation which converted the problem from one involving an N x N matrix, to one involving a 1 ×1 matrix (see Supplementary Note 2 for details). Our implementation involved first performing a spectral decomposition of the genetic relationship matrix (A), and then pre-multiplying the matrices of outcomes and fixed effects respectively by the matrix of eigenvectors32. This pre-multiplication has the effect of "rotating away" the dependence between outcome trait values, leaving the random effects uncorrelated. The problem then reduces from N correlated observations (modeled by an N x N matrix), to N independent observations, greatly facilitating computation. An R script with code illustrating our method is included in the Supplementary Note 1.
We first tested the relationship between maternal GRS and offspring birthweight in N = 7,825 mother–offspring pairs and N = 6,875 father–offspring pairs to confirm that our GRS explained some of the variance in offspring birthweight. We then performed our primary analyses testing the relationship between maternal GRS and each of the offspring cardiovascular risk factors, whilst conditioning on offspring GRS. We performed the same analyses in father–offspring pairs to assess whether there was evidence for a postnatal effect from either parent (Fig. 1d). In addition to analyzing all of the offspring together, we stratified the data into two groups; one group with offspring under age 40 at the time of measurement and one for offspring between 40 and 60 years of age. This was done to obtain a sample that would be easier to compare with the previous analysis of SBP in the UK Biobank Study by Warrington and colleagues18. Age strata for individuals over 60 is not presented due to the low number of individuals.
A flowchart of the sample selection is presented in Fig. 3.
Fig. 3: Flowchart showing the number of individuals (N) participating in the analysis and the exclusion criteria for each analysis.
A total of 69,716 genotyped HUNT participants were recruited from either HUNT2 or HUNT 3. Of these 46,428 parent–offspring relationships were identified using the King software. Parent–offspring pairs with ≤15 years difference in birth year were removed leaving 26,058 mother–offspring pairs and 19,792 father–offspring pairs for the main analysis. Birthweight was only available for offspring born after 1967. Additionally, for analyses involving birthweight as an outcome, offspring were excluded if they were part of a multiple birth, had congenital malformation, were born where the birth was induced or performed with C-section, their birthweight ≤1000 g, or were born before 258 days of gestation.
Power calculations
We were interested in the statistical power of our approach to detect maternal genetic effects on offspring cardiometabolic risk factors. We therefore used the Maternal and Offspring Genetic Effects Power Calculator (https://evansgroup.di.uq.edu.au/MGPC/) to calculate power to detect association53. We assumed N = 26,057 complete mother–offspring pairs, the absence of offspring genetic effects, and a Type 1 error rate of α = 0.05 (the presence/absence of offspring genetic effects has little influence on power to detect maternal genetic effects so long as the proportion of variance explained is small53).
The empirical datasets used with the HUNT study will be archived with the study and will be made available to individuals who obtain the necessary permissions from the study's Data Access Committee. Due to privacy issues, access to individual-level data requires permission from the HUNT Study, the Medical Birth Registry of Norway and the regional committee for medical research ethics. Requirements for access to data from the HUNT Study are described at www.ntnu.edu/hunt.
Code availability
Example code is provided in Supplementary Note 1.
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The authors would like to thank the research participants of the HUNT study. The Nord-Trøndelag Health Study (The HUNT Study) is a collaboration between HUNT Research Centre (Faculty of Medicine and Health Sciences, NTNU, Norwegian University of Science and Technology), Nord-Trøndelag County Council, Central Norway Regional Health Authority, and the Norwegian Institute of Public Health. The genotyping in HUNT was financed by the National Institutes of Health (NIH); University of Michigan; The Research Council of Norway; The Liaison Committee for Education, Research and Innovation in Central Norway; and the Joint Research Committee between St. Olavs hospital and the Faculty of Medicine and Health Sciences, NTNU. The genotype quality control and imputation has been conducted by the K.G. Jebsen Center for Genetic Epidemiology, Department of Public Health and Nursing, Faculty of Medicine and Health Sciences, NTNU. This research was carried out at the Translational Research Institute, Woolloongabba, QLD 4102, Australia. The Translational Research Institute is supported by a grant from the Australian Government. Support for this research have been given by the Norwegian Diabetes Association and Nils Normans minnegave. G.H.M. is supported by the Norwegian Research Council (Post doctorial mobility research grant 287198). N.M.W. is supported by an Australian National Health and Medical Research Council Early Career Fellowship (APP1104818). M.C.N. and OpenMx development were funded by NIH grant DA-018673. R.M.F. is supported by a Wellcome Trust and Royal Society Sir Henry Dale Fellowship (104150/Z/14/Z). D.A.L. is supported by the British Heart Foundation (AA/18/7/34219) and European Research Council (669545). D.A.L., G.D.S., D.M.E. are affiliated with a Unit that receives support from the University of Bristol and the UK Medical Research Council (MC_UU_00011/1 and MC_UU_00011/6). D.M.E. is funded by an Australian National Health and Medical Research Council Senior Research Fellowship (APP1137714) and NHMRC project grants (GNT1125200, GNT1157714, GNT1183074).
These authors jointly supervised this work: Nicole M. Warrington, David M. Evans.
Institute of Clinical Medicine, Faculty of Medicine, University of Oslo, Oslo, Norway
Gunn-Helen Moen & Kåre I. Birkeland
The University of Queensland Diamantina Institute, The University of Queensland, Woolloongabba, QLD, 4102, Australia
Gunn-Helen Moen, Geng Wang, Nicole M. Warrington & David M. Evans
K.G. Jebsen Center for Genetic Epidemiology, Department of Public Health and Nursing, NTNU, Norwegian University of Science and Technology, Trondheim, Norway
Gunn-Helen Moen, Ben Brumpton, Bjørn Olav Åsvold & Nicole M. Warrington
Population Health Science, Bristol Medical School, University of Bristol, Bristol, UK
Gunn-Helen Moen, George Davey Smith & Deborah A. Lawlor
Department of Thoracic and Occupational Medicine, St. Olavs Hospital, Trondheim University Hospital, Trondheim, Norway
Ben Brumpton
Medical Research Council Integrative Epidemiology Unit at the University of Bristol, Bristol, UK
Ben Brumpton, George Davey Smith, Deborah A. Lawlor, Nicole M. Warrington & David M. Evans
Department of Biostatistics and Center for Statistical Genetics, University of Michigan, Ann Arbor, USA
Cristen Willer
Department of Internal Medicine, University of Michigan, Ann Arbor, MI, USA
Department of Human Genetics, University of Michigan, Ann Arbor, MI, USA
Department of Endocrinology, St. Olavs Hospital, Trondheim University Hospital, Trondheim, Norway
Bjørn Olav Åsvold
Department of Psychiatry, Virginia Institute for Psychiatric and Behavioral Genetics, Virginia Commonwealth University, Richmond, VA, USA
Michael C. Neale & Robert M. Kirkpatrick
Institute of Biomedical and Clinical Science, College of Medicine and Health, University of Exeter, Exeter, UK
Rachel M. Freathy
Bristol NIHR Biomedical Research Centre, Bristol, UK
George Davey Smith & Deborah A. Lawlor
Gunn-Helen Moen
Kåre I. Birkeland
Geng Wang
Michael C. Neale
George Davey Smith
Deborah A. Lawlor
Robert M. Kirkpatrick
Nicole M. Warrington
David M. Evans
G.H.M. contributed to the design of the work, the data analysis, interpretation of the data and the draft of this manuscript. B.B. contributed to the acquisition and interpretation of the data and substantively revised the manuscript. C.W. contributed to the acquisition and analysis of the data. B.O.Å. contributed to the acquisition and interpretation of the data and substantively revised the manuscript. K.I.B. contributed to the interpretation of the data and substantively revised the manuscript. G.W. contributed to the design of the work and interpretation of the data. M.C.N. contributed to the analysis and substantively revised the manuscript. R.M.F. contributed to the interpretation of data and substantively revised the manuscript. G.D.S. contributed to the interpretation of data and substantively revised the manuscript. D.A.L. contributed to the interpretation of data and substantively revised the manuscript. R.M.K. contributed to the analysis and substantively revised the manuscript. N.M.W. contributed to conception, design of the work, interpretation of the data and substantively revised the manuscript. D.M.E. contributed to conception, design of the work, interpretation of the data and the draft of this manuscript. All authors approved the submitted version.
Correspondence to Gunn-Helen Moen or David M. Evans.
D.A.L. has received support from Roche Diagnostics and Medtronic Ltd for work unrelated to that presented here. All other authors declare no conflict of interests.
Peer review information Nature Communications thanks Marjo-Riitta Jarvelin and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Description of Additional Supplementary Files
Supplementary Data 1
Moen, GH., Brumpton, B., Willer, C. et al. Mendelian randomization study of maternal influences on birthweight and future cardiometabolic risk in the HUNT cohort. Nat Commun 11, 5404 (2020). https://doi.org/10.1038/s41467-020-19257-z
Two decades since the fetal insulin hypothesis: what have we learned from genetics?
Alice E. Hughes
Andrew T. Hattersley
Diabetologia (2021)
Association of diabetes-related variants in ADCY5 and CDKAL1 with neonatal insulin, C-peptide, and birth weight
Ivette-Guadalupe Aguilera-Venegas
Julia-del-Socorro Mora-Peña
Maria-Luisa Lazo-de-la-Vega-Monroy
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Introduction to the Special Issue on Statistical Genetic Methods for Human Complex Traits
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Behavior Genetics (2021) | CommonCrawl |
Insulin Resistance is Associated with Cognitive Decline Among Older Koreans with Normal Baseline Cognitive Function: A Prospective Community-Based Cohort Study
Sung Hye Kong ORCID: orcid.org/0000-0002-8791-09091,
Young Joo Park1,
Jun-Young Lee ORCID: orcid.org/0000-0002-5893-31242,
Nam H. Cho3 &
Min Kyong Moon ORCID: orcid.org/0000-0002-5460-28461,4
Scientific Reports volume 8, Article number: 650 (2018) Cite this article
Diabetes complications
We evaluated whether metabolic factors were associated with cognitive decline, compared to baseline cognitive function, among geriatric population. The present study evaluated data from an ongoing prospective community-based Korean cohort study. Among 1,387 participants who were >65 years old, 422 participants were evaluated using the Korean mini-mental status examination (K-MMSE) at the baseline and follow-up examinations. The mean age at the baseline was 69.3 ± 2.9 years, and 222 participants (52.6%) were men. The mean duration of education was 7.1 ± 3.6 years. During a mean follow-up of 5.9 ± 0.1 years, the K-MMSE score significantly decreased (−1.1 ± 2.7 scores), although no significant change was observed in the homeostasis model assessment of insulin resistance (HOMA-IR) value. Participants with more decreased percent changes in K-MMSE scores had a shorter duration of education (p = 0.001), older age (p = 0.022), higher baseline K-MMSE score (p < 0.001), and increased insulin resistance (∆HOMA-IR, p = 0.002). The correlation between the percent changes in K-MMSE and ∆HOMA-IR values remained significant after multivariable adjustment (B = −0.201, p = 0.002). During a 6-year follow-up of older Koreans with normal baseline cognitive function, increased insulin resistance was significantly correlated with decreased cognitive function.
The increasing geriatric population is associated with increasing prevalence of dementia and cognitive dysfunction in many countries, including Korea1,2,3. The World Health Organization has reported that 47.5 million people had dementia in 2016, with global totals projected to reach 75.6 million people in 2030 and 135.5 million people in 20504. Dementia in geriatric populations affects the individuals' and their families' quality of life, which creates a societal burden, and the global estimated economic burden was approximately 604 billion US dollars in 20104. Therefore, it would be helpful to identify and address modifiable factors that can help to reduce the global burden of dementia.
Epidemiological studies have revealed associations of cognitive impairment and/or dementia with type 2 diabetes mellitus5,6. Insulin resistance, hyperglycemia, and obesity are probable mechanistic links in this association. Few longitudinal studies have evaluated this issue, although many cross-sectional studies have confirmed the relationship between insulin resistance and cognitive decline7,8,9,10. One longitudinal study evaluated middle-aged adults (45–64 years old at baseline in the Atherosclerosis Risk in Communities cohort), and revealed that baseline hyperinsulinemia was associated with a rapid decline in cognitive function7. Another longitudinal nationwide population-based survey revealed that higher baseline HOMA-IR and fasting insulin levels were associated with a greater decline in verbal fluency8. However, to the best of our knowledge, no studies have evaluated the effects of longitudinal changes in insulin resistance on cognitive function.
Although clinical studies of patients with diabetes have demonstrated that hyperglycemia is associated with cognitive dysfunction, it remains controversial whether hyperglycemia is associated with cognitive dysfunction among older individuals or among individuals with normal glucose tolerance or prediabetes9,10,11,12. Among patients with impaired glucose tolerance, there is very little evidence regarding a relationship between impaired glucose tolerance and cognitive impairment12,13. However, among participants with normal glucose tolerance, poor glucose tolerance was associated with cognitive impairment12,13. The Leiden 85-plus Study prospectively evaluated 599 individuals from the age of 85 years and revealed that HbA1c concentrations were not associated with cognitive dysfunction9.
It has also been suggested that obesity is a risk factor for cognitive dysfunction, although this relationship remains unclear, as the few longitudinal studies have provided conflicting results14,15,16.
As geriatric populations are consistently growing, it is important to determine whether metabolic factors are related to cognitive dysfunction among older individuals, and whether addressing these metabolic factors can help prevent cognitive dysfunction. In the present prospective community-based cohort study, we aimed to identify metabolic factors that were associated with cognitive decline among older individuals with normal baseline cognitive function.
The present study evaluated data from the Ansung cohort study, which is a prospective community-based study that began in 2001 and is supported by National Genome Research Institute (Korean Centers for Disease Control and Prevention, Cheongju, Korea). That study is a part of the Korean Genome Epidemiology Study, which is a community-based epidemiological survey of Korean individuals who are 40–69 years old. The Ansung study recruited residents of Ansung who had lived in the surveyed region for ≥6 months. According to the 2000 census, Ansung is a rural community with 132,906 residents. Detailed information regarding the Ansung study's selection criteria and sampling plan has been published previously17,18, and the study's protocol was approved by the institutional review board of the Korean Centers for Disease Control and Prevention and the study was carried out in accordance with the protocol. Informed consent was obtained from all participants and/or their legal guardians. All methods were performed in accordance with the relevant guidelines and regulations.
The Ansung cohort study evaluated 1,387 participants who were >65 years old, completed a baseline examination in 2001, and were surveyed biennially until 2014. Data from these participants were considered for inclusion in the present study. However, we excluded 391 participants because they did not complete the baseline or follow-up Korean mini-mental status examination (K-MMSE), the Korean geriatric depression score tool (GDS-K), or the Korean dementia screening questionnaire (KDSQ). In addition, we excluded 557 participants with a K-MMSE score of <23, a KDSQ score of >5, or a GDS-K score of >10. Participants with impaired baseline cognitive function (low K-MMSE score or high KDSQ score) were excluded because we only intended to evaluate individuals with normal baseline cognitive function. Participants with high GDS scores were excluded to minimize the influence of depressiveness on the MMSE results, as depression can be associated with low MMSE scores19. Furthermore, we excluded 17 participants who had been diagnosed with stroke, dementia, depression, or head trauma. Thus, data from 422 eligible participants were included in the final analyses (Fig. 1). The mean follow-up duration was 5.9 ± 0.1 years.
Flow chart showing selection of the study population from Ansung cohort study.
Assessing cognitive impairment and depression
Cognitive impairment and depression were measured at the baseline and follow-up examinations. Cognitive impairment was evaluated using the K-MMSE and KDSQ tools. The 30-item K-MMSE was specifically developed and validated for assessing the general cognitive function of older Korean individuals20. The results are scored from 0 to 30 points, with scores of ≥23 points indicating normal cognition, scores of 17–22 points indicating mild cognitive impairment, and scores of <17 points indicating moderate-to-severe impairment. The 15-item KDSQ is a sensitive test for early dementia screening, and the results are not influenced by age or educational level21. The results are scored from 0 to 15 points, with scores of >5 points considered suggestive of cognitive impairment. Depressive symptoms were assessed using the 15-item GDS-K22. The results are scored from 0 to 15 points, with scores of >10 points considered suggestive of depressive mood. Changes in the K-MMSE, GDS-K, and KDSQ scores were calculated as:
$${\rm{\Delta }}K \mbox{-} \mathrm{MMSE}=\mathrm{follow} \mbox{-} \mathrm{up}\,K \mbox{-} \mathrm{MMSE}-{\rm{baseline}}\,K \mbox{-} \mathrm{MMSE}$$
$${\rm{Percent}}\,{\rm{changes}}\,{\rm{in}}\,K \mbox{-} \mathrm{MMSE}={\rm{\Delta }}K \mbox{-} \mathrm{MMSE}/{\rm{baseline}}\,K \mbox{-} \mathrm{MMSE}$$
$${\rm{\Delta }}\mathrm{GDS} \mbox{-} K=\mathrm{follow} \mbox{-} \mathrm{up}\,\mathrm{GDS} \mbox{-} K-{\rm{baseline}}\,\mathrm{GDS} \mbox{-} K$$
$${\rm{\Delta }}\mathrm{KDSQ}=\mathrm{follow} \mbox{-} \mathrm{up}\,{\rm{KDSQ}}-{\rm{baseline}}\,{\rm{KDSQ}}$$
Measuring anthropometric parameters
Face-to-face or telephone interviews were used to obtain data regarding the participants' age, sex, duration of education, medical history, alcohol consumption, and smoking status. Former smokers were defined as individuals who had smoked >5 packs of cigarettes during their lifetime, and participants were defined as having quit smoking if they had stopped smoking ≥6 months before the baseline examination. Former drinkers were defined as individuals who had consumed 5 g of ethanol/day, and participants were defined as having quit drinking if they had stopped consuming alcohol ≥6 months before the baseline examination. Medical histories of diabetes, hypertension, stroke, dementia, head trauma, depression, and other mental illness were identified based on self-reported diagnoses.
The participants' height and body weight were measured using the standard methods (scale and wall-mounted extensometer) while the participants were wearing light-weight clothes. Body mass index (BMI) was calculated as weight divided by height squared (kg/m2). Changes in BMI (∆BMI) were calculated as:
$${\rm{Equation}}\,4:{\rm{\Delta }}{\rm{BMI}}=\mathrm{follow} \mbox{-} \mathrm{up}\,{\rm{BMI}}-{\rm{baseline}}\,{\rm{BMI}}$$
Laboratory testing and homeostasis model assessment of insulin resistance calculation
All participants fasted for at least 14 h before undergoing the blood sampling. Plasma specimens were separated using centrifugation (2,000 rpm for 20 min at 4 °C) and tested immediately. Plasma glucose concentrations were measured using the hexokinase method (ADVIA 1650 Auto Analyzer; Bayer, Leverkusen, Germany), and plasma insulin concentrations were measured using the IRMA test kit (bioSource Europe S.A., Niverlles, Belgium). Fasting concentrations of total cholesterol, high-density lipoprotein cholesterol (HDL-C), low-density lipoprotein cholesterol (LDL-C), and triglycerides were measured enzymatically using the Hitachi 747 chemistry analyzer (Hitachi, Tokyo, Japan). Concentrations of HbA1c were evaluated using high-performance liquid chromatography with a Bio-Rad Variant II HbA1c analyzer (Bio-Rad, Montreal, Quebec, Canada). Consenting participants underwent apolipoprotein E genotyping using the methods of Hixson and Vernier23, and the results were categorized based on the presence or absence of the ε4 allele. HOMA-IR24, ∆HOMA-IR, ∆HbA1c, and ∆fasting insulin were calculated as:
$$\mathrm{HOMA} \mbox{-} \mathrm{IR}=({\rm{fasting}}\,{\rm{plasma}}\,{\rm{insulin}}\,[{\rm{\mu }}\mathrm{IU}/{\rm{mL}}]\times {\rm{fasting}}\,{\rm{plasma}}\,{\rm{glucose}}[{\rm{mg}}/{\rm{dL}}]\times 0.0555)/22.5$$
$${\rm{\Delta }}\mathrm{HOMA} \mbox{-} \mathrm{IR}=\mathrm{follow} \mbox{-} \mathrm{up}\,\mathrm{HOMA} \mbox{-} \mathrm{IR}-{\rm{baseline}}\,\mathrm{HOMA} \mbox{-} \mathrm{IR}$$
$${\rm{\Delta }}\mathrm{HbA}1{\rm{c}}=\mathrm{follow} \mbox{-} \mathrm{up}\,{\rm{HbA}}1{\rm{c}}-{\rm{baseline}}\,{\rm{HbA}}1{\rm{c}}$$
$${\rm{\Delta }}\mathrm{fasting}\,{\rm{insulin}}=\mathrm{follow} \mbox{-} \mathrm{up}\,{\rm{fasting}}\,{\rm{insulin}}-{\rm{baseline}}\,{\rm{fasting}}\,{\rm{insulin}}$$
Normally distributed data were presented as mean ± standard deviation, non-normally distributed data were reported as median (interquartile range [IQR]), and categorical data were reported as number (%). The participants' characteristics at the baseline and 6-year follow-up examinations were compared using the paired t-test, Wilcoxon signed-rank test, and Mann-Whitney U test, as appropriate. Pearson's correlation coefficient was used to estimate the relationship between cognitive function and the related parameters. B refers to standardized beta value. Associations of cognitive function with the other factors were analyzed using multivariable linear regression models. Model 1 was adjusted for age, sex, baseline K-MMSE score, education duration, and baseline GDS-K score. Model 2 was adjusted for the factors in model 1 plus smoking status, history of diabetes, history of hypertension, and BMI. Model 3 was adjusted for the factors in model 2 plus the apolipoprotein E ε4 genotype status. Differences were considered statistically significant at p-values of <0.05, and all analyses were performed using IBM SPSS software (version 22.0; IBM Corp., Armonk, NY, USA).
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
The participants' characteristics at the baseline and follow-up examinations are shown in Table 1. The mean follow-up duration was 5.9 ± 0.1 years, the mean age at baseline was 69.3 ± 2.9 years, and 222 participants (52.6%) were men. The mean education duration was 7.1 ± 3.6 years. At baseline, 115 participants (27.3%) had hypertension, compared to 207 participants (49.1%) at the follow-up. At baseline, 89 participants (21.1%) had diabetes, compared to 113 participants (26.8%) at the follow-up. The K-MMSE scores decreased significantly from 26.5 ± 1.9 at baseline to 25.4 ± 2.9 at the follow-up (∆K-MMSE, −1.1 ± 2.7, percent changes in K-MMSE, −4.1 ± 10.3%). The KDSQ scores increased significantly (median increase: 1.0, IQR: –1.0 to 4.0). The GDS-K scores also increased significantly at the follow-up. The laboratory test results from the follow-up revealed a significant decrease in the LDL-C concentration and significant increases in the HDL-C and creatinine concentrations. The laboratory results for lipid profile, including LDL-C, HDL-C, and triglycerides, were only analyzed for 371 participants who were not receiving dyslipidemia treatment. No significant differences were observed in the values for HOMA-IR, fasting insulin, and HbA1c. Seventy-two participants (17.1%) were found to have the apolipoprotein E ε4 genotype (Table 1).
Table 1 Characteristics of the participants at baseline and follow-up.
Correlations of K-MMSE with factors at baseline
Pearson's correlation analyses revealed that lower K-MMSE values were correlated with shorter education durations (r = 0.393, p < 0.001), higher KDSQ scores (r = −0.129, p = 0.008), and higher GDS-K scores (r = −0.128, p = 0.008). Baseline K-MMSE values were not significantly correlated with age, baseline BMI, ∆BMI, ∆KDSQ score, ∆GDS-K score, baseline HOMA-IR, ∆HOMA-IR, baseline fasting insulin, ∆fasting insulin, baseline HbA1c, ∆HbA1c, baseline lipid profiles, or creatinine (Table 2).
Table 2 Correlations between baseline K-MMSE and related factors.
Correlations of percent changes in K-MMSE with related factors
Pearson's correlation analyses revealed that a greater decrease in K-MMSE between baseline and follow-up was correlated with older age (r = –0.102, p = 0.037), shorter education duration (r = 0.168, p = 0.001), higher baseline K-MMSE score (r = –0.187, p < 0.001), and increases in HOMA-IR values (r = –0.155, p = 0.001) and fasting insulin levels between baseline and follow-up (r = –0.160, p = 0.001). The percent changes in K-MMSE values were not significantly correlated with the values for baseline BMI, ∆BMI, baseline KDSQ score, ∆KDSQ score, GDS-K score, ∆GDS-K score, baseline HOMA-IR, baseline fasting insulin, baseline HbA1c, ∆HbA1c, lipid profiles, or creatinine (Table 3).
Table 3 Correlations of percent changes in K-MMSE between baseline and follow-up and related factors.
Multivariable linear regression models for percent changes in K-MMSE and the hyperinsulinemia variables
After adjusting model 1 for age, sex, baseline K-MMSE score, education duration, and baseline GDS-K score, we observed that an increase in HOMA-IR was correlated with a reduction in K-MMSE between baseline and follow-up (B = −0.139, p = 0.003). After adjusting for the variables in model 2 (model 1 plus smoking status, distort of diabetes, history of hypertension, and BMI), we observed that the negative correlation between change of HOMA-IR and K-MMSE remained significant (standardized beta [B] = −0.137, p = 0.004). After adjusting for the variables in model 3 (model 2 plus apolipoprotein E ε4 genotype status), the correlation between change of HOMA-IR and K-MMSE remained significant (B = −0.138, p = 0.004). Percent changes in K-MMSE values were not correlated with ∆BMI, ∆GDS-K scores, and ∆HbA1c before and after adjusting covariates before and after adjusting covariates (Table 4).
Table 4 Multivariable linear regression models of percent changes in K-MMSE between baseline and follow-up and their correlations with metabolic factors.
This community-based prospective cohort study of older individuals with normal cognitive function revealed that cognitive decline was associated with a 6-year increase in insulin resistance, after adjusting for age, sex, baseline K-MMSE, education duration, baseline GDS-K, smoking status, history of diabetes, history of hypertension, BMI, and apolipoprotein E ε4 genotype status. Among the various metabolic factors, cognitive dysfunction was associated with changes in insulin resistance, but not with existing or changing hyperglycemia or obesity. Furthermore, greater cognitive declines were observed for participants with greater increases in insulin resistance.
Most previous studies investigating the association of insulin resistance with cognitive dysfunction have used a cross-sectional design, and few longitudinal studies have been performed7,8,9,10. During a 6-year follow-up, baseline hyperinsulinemia among middle-aged adults (45–64 years old at baseline) in the Atherosclerosis Risk in Communities cohort was associated with greater cognitive decline7. In addition, a recent 11-year Finnish nation-wide population-based survey revealed that higher baseline HOMA-IR and fasting insulin values independently predicted a greater decline in verbal fluency8. Furthermore, our findings are consistent with previous findings that insulin resistance is related to cognitive decline7,10,25,26,27,28. Moreover, we evaluated the relationship between changes in insulin resistance and cognitive decline, and the results suggest that controlling insulin resistance could help prevent cognitive decline in the older population.
Insulin receptors are widely distributed throughout the brain, which suggests that the brain is a major insulin target29. Furthermore, insulin resistance is observed in the neuronal cells of patients with Alzheimer's disease (AD), which suggests that insulin resistance may cause neuronal cell dysfunction and lead to cognitive dysfunction and dementia30,31. Interestingly, mice receiving a high-fat diet exhibited impaired binding to brain insulin receptors32, which agrees with the findings from the brains of patients with early-stage AD. Therefore, neurons in the brain might be damaged by insulin receptor dysfunction. Moreover, peripheral insulin resistance could induce neuronal damage though amyloid beta and cytokines, as peripheral hyperinsulinemia can increase amyloid beta concentrations33 and plasma and cerebrospinal fluid concentrations of interleukin-6 and tumor necrosis factor alpha34. The increased concentrations of amyloid beta and inflammatory cytokines could induce neuronal loss, amyloid beta plaques, and neurofibrillary tangles35.
The present study did not detect any significant relationship between the baseline HOMA-IR and K-MMSE values. Although many previous studies have identified a positive association between insulin resistance and cognitive dysfunction7,25,27,28, other studies did not detect any significant association36,37. These discrepancies may be related to differences in the study populations and cognitive function assessment methods. For example, studies with positive results generally evaluated individuals who did not have baseline cognitive dysfunction28, were relatively young7 or were women38,39. In addition, insulin resistance may mainly affect executive dysfunction, which is best evaluated using the Trail Making Test39,40 and/or verbal fluency38,41, rather than tools that consider all brain function domains. However, the present study did not detect any significant correlation between cognitive function and insulin resistance when we only analyzed women. Therefore, it appears that methodology differences are more important, rather than sex-based differences.
It is suggested that hyperglycemia is associated with poor cognitive outcomes. It has been shown in both cross-sectional studies42 and prospective studies43, although there are conflicting results. The present study failed to detect a significant correlation between hyperglycemia and cognitive function or change in cognitive function. This may be because the HbA1c range was relatively narrow (caused by the community-based design), while most studies with positive results evaluated patients with diabetes. Moreover, hyperglycemia mainly affects processing speed, attention, and visual-spatial processing44, which may not be accurately assessed using only the MMSE tool.
Although many studies have revealed an association between obesity and the risk of dementia12,14,15,16,43,44, the association appears to be complex. For example, some reports have indicated that higher BMI was associated with less cognitive decline in a cognitively unimpaired community-dwelling population, and there are reports of increased dementia risk for both obese and underweight people12,15,45. In addition, studies have demonstrated that midlife obesity is more strongly related to dementia, compared to obesity among older people12,14. We did not detect any significant association between BMI and K-MMSE or the change in K-MMSE. This may be related to the participants' relatively normal BMI (mean 23.9 kg/m2), or midlife BMI having a greater effect on the cognitive function of older individuals, compared to current BMI.
The present study evaluated the participants' serum cholesterol concentrations after excluding individuals who were receiving dyslipidemia medication. However, the follow-up examinations revealed lower concentrations of triglycerides and LDL-C, and higher concentrations of HDL-C. These changes may be partially explained by aging-related changes46, and enrollment in the cohort may also have improved the participants' lifestyle, as it was accompanied by an increase in HDL-C concentrations. Furthermore, there is the possibility of recall bias regarding medication histories, as these data were obtained using questionnaire responses, rather than prescription records.
The present study has several strengths. First, to the best of our knowledge, ours is the first study to evaluate the association between changes in insulin resistance and cognitive function. Thus, our findings may help elucidate the importance of controlling insulin resistance to prevent cognitive decline among older individuals. Second, we analyzed data from the Ansung cohort study, which provides a large study sample, a community-based prospective design, and long-term follow-up data with information regarding potential confounding factors. Third, we adjusted for important covariates, including GDS-K score, education duration, history of diabetes, history of hypertension, and apolipoprotein E ε4 genotype status. Fourth, we excluded patients with high GDS scores to minimize the influence of depressiveness on the MMSE results, as depression can be associated with low MMSE scores18. Fifth, we excluded participants with conditions that could influence cognitive function, such as stroke, dementia, depression, and head trauma.
The study also has potential limitations. First, although the current study used data from a large-scale prospective community-based cohort study, only 422 subjects were assessed for cognitive function, which may have limited the assessment of metabolic factors' effect on cognitive change. Second, the follow-up period may not be sufficient to detect a decrease in cognitive function among individuals with normal baseline cognitive function, and the 6-year change in the MMSE value was relatively small. Third, cognitive function was only evaluated using MMSE, although insulin resistance and hyperglycemia are reportedly more related to verbal performance or executive function. Fourth, increased insulin resistance was associated with a decrease in MMSE during the study, although we did not evaluate whether improvements in insulin resistance prevented cognitive decline. In addition, although the MMSE is a widely used tool for cognitive dysfunction screening, it cannot identify small changes in cognitive function (e.g., cognitive aging) because of a ceiling effect. Therefore, more accurate evaluation of cognitive function must be performed using detailed neurocognitive assessments without ceiling effects.
In conclusion, increased insulin resistance was associated with decreased cognitive function during a 6-year follow-up of older individuals with normal baseline cognitive function. However, baseline HbA1c, baseline BMI, ∆HbA1c, and ∆BMI values were not associated with changes in cognitive function. These relationships were independent of apolipoprotein E ε4 genotype status. Based on our results, further interventional studies are needed to evaluate the effect of controlling insulin resistance on cognitive dysfunction among older individuals.
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Department of Internal Medicine, Seoul National University College of Medicine, Seoul, Republic of Korea
Sung Hye Kong
, Young Joo Park
& Min Kyong Moon
Department of Psychiatry and Behavioral Science, Seoul National University Boramae Medical Center, Seoul, Republic of Korea
Jun-Young Lee
Department of Preventive Medicine, Ajou University School of Medicine, Suwon, Republic of Korea
Nam H. Cho
Department of Internal Medicine, Seoul National University Boramae Medical Center, Seoul, Republic of Korea
Min Kyong Moon
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S.H.K. and M.K.M researched data and wrote the manuscript. M.K.M., Y.J.P., and J.Y.L. contributed to the discussion and reviewed/edited the manuscript. N.H.C. contributed to the data research and discussion.
Correspondence to Nam H. Cho or Min Kyong Moon.
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Kong, S.H., Park, Y.J., Lee, J. et al. Insulin Resistance is Associated with Cognitive Decline Among Older Koreans with Normal Baseline Cognitive Function: A Prospective Community-Based Cohort Study. Sci Rep 8, 650 (2018). https://doi.org/10.1038/s41598-017-18998-0
DOI: https://doi.org/10.1038/s41598-017-18998-0
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# Lecture Notes on <br> GRAPH THEORY
Tero Harju<br>Department of Mathematics
University of Turku
FIN-20014 Turku, Finland
e-mail: [email protected]
1994 - 2011
## Introduction
Graph theory may be said to have its beginning in 1736 when EULER considered the (general case of the) Königsberg bridge problem: Does there exist a walk crossing each of the seven bridges of Königsberg exactly once? (Solutio Problematis ad geometriam situs pertinentis, Commentarii Academiae Scientiarum Imperialis Petropolitanae 8 (1736), pp. 128-140.)
It took 200 years before the first book on graph theory was written. This was "Theorie der endlichen und unendlichen Graphen" ( Teubner, Leipzig, 1936) by KÖNIG in 1936. Since then graph theory has developed into an extensive and popular branch of mathematics, which has been applied to many problems in mathematics, computer science, and other scientific and not-so-scientific areas. For the history of early graph theory, see
N.L. Biggs, R.J. Lloyd AND R.J. Wilson, “Graph Theory 1736 - 1936”, Clarendon Press, 1986.
There are no standard notations for graph theoretical objects. This is natural, because the names one uses for the objects reflect the applications. Thus, for instance, if we consider a communications network (say, for email) as a graph, then the computers taking part in this network, are called nodes rather than vertices or points. On the other hand, other names are used for molecular structures in chemistry, flow charts in programming, human relations in social sciences, and so on.
These lectures study finite graphs and majority of the topics is included in J.A. BONDY, U.S.R. MurTy, "Graph Theory with Applications", Macmillan, 1978.
R. DiESTEL, "Graph Theory", Springer-Verlag, 1997.
F. Harary, "Graph Theory", Addison-Wesley, 1969.
D.B. WEST, "Introduction to Graph Theory", Prentice Hall, 1996.
R.J. WiLSON, "Introduction to Graph Theory", Longman, (3rd ed.) 1985.
In these lectures we study combinatorial aspects of graphs. For more algebraic topics and methods, see
N. Biggs, "Algebraic Graph Theory", Cambridge University Press, (2nd ed.) 1993.
C. GoDsiL, G.F. RoYLE, "Algebraic Graph Theory”, Springer, 2001.
and for computational aspects, see
S. EveN, "Graph Algorithms", Computer Science Press, 1979. In these lecture notes we mention several open problems that have gained respect among the researchers. Indeed, graph theory has the advantage that it contains easily formulated open problems that can be stated early in the theory. Finding a solution to any one of these problems is another matter.
Sections with a star $(*)$ in their heading are optional.
## Notations and notions
- For a finite set $X,|X|$ denotes its size (cardinality, the number of its elements).
- Let
$$
[1, n]=\{1,2, \ldots, n\}
$$
and in general,
$$
[i, n]=\{i, i+1, \ldots, n\}
$$
for integers $i \leq n$.
- For a real number $x$, the floor and the ceiling of $x$ are the integers
$$
\lfloor x\rfloor=\max \{k \in \mathbb{Z} \mid k \leq x\} \text { and }\lceil x\rceil=\min \{k \in \mathbb{Z} \mid x \leq k\} .
$$
- A family $\left\{X_{1}, X_{2}, \ldots, X_{k}\right\}$ of subsets $X_{i} \subseteq X$ of a set $X$ is a partition of $X$, if
$$
X=\bigcup_{i \in[1, k]} X_{i} \quad \text { and } \quad X_{i} \cap X_{j}=\varnothing \text { for all different } i \text { and } j .
$$
- For two sets $X$ and $Y$,
$$
X \times Y=\{(x, y) \mid x \in X, y \in Y\}
$$
is their Cartesian product, and
$$
X \triangle Y=(X \backslash Y) \cup(Y \backslash X)
$$
is their symmetric difference. Here $X \backslash Y=\{x \mid x \in X, x \notin Y\}$.
- Two integers $n, k \in \mathbb{N}$ (often $n=|X|$ and $k=|Y|$ for sets $X$ and $Y$ ) have the same parity, if both are even, or both are odd, that is, if $n \equiv k(\bmod 2)$. Otherwise, they have opposite parity.
Graph theory has abundant examples of NP-complete problems. Intuitively, a problem is in $\mathrm{P}^{1}$ if there is an efficient (practical) algorithm to find a solution to it. On the other hand, a problem is in NP ${ }^{2}$, if it is first efficient to guess a solution and then efficient to check that this solution is correct. It is conjectured (and not known) that $\mathrm{P} \neq \mathrm{NP}$. This is one of the great problems in modern mathematics and theoretical computer science. If the guessing in NP-problems can be replaced by an efficient systematic search for a solution, then $\mathrm{P}=\mathrm{NP}$. For any one NP-complete problem, if it is in $\mathrm{P}$, then necessarily $\mathrm{P}=\mathrm{NP}$.
${ }^{1}$ Solvable - by an algorithm - in polynomially many steps on the size of the problem instances.
2 Solvable nondeterministically in polynomially many steps on the size of the problem instances.
### Graphs and their plane figures
Let $V$ be a finite set, and denote by
$$
E(V)=\{\{u, v\} \mid u, v \in V, u \neq v\} .
$$
the 2-sets of $V$, i.e., subsets of two distinct elements.
DEFINition. A pair $G=(V, E)$ with $E \subseteq E(V)$ is called a graph (on $V)$. The elements of $V$ are the vertices of $G$, and those of $E$ the edges of $G$. The vertex set of a graph $G$ is denoted by $V_{G}$ and its edge set by $E_{G}$. Therefore $G=\left(V_{G}, E_{G}\right)$.
In literature, graphs are also called simple graphs; vertices are called nodes or points; edges are called lines or links. The list of alternatives is long (but still finite).
A pair $\{u, v\}$ is usually written simply as $u v$. Notice that then $u v=v u$. In order to simplify notations, we also write $v \in G$ and $e \in G$ instead of $v \in V_{G}$ and $e \in E_{G}$.
DEFINition. For a graph $G$, we denote
$$
v_{G}=\left|V_{G}\right| \text { and } \varepsilon_{G}=\left|E_{G}\right| .
$$
The number $v_{G}$ of the vertices is called the order of $G$, and $\varepsilon_{G}$ is the size of $G$. For an edge $e=u v \in G$, the vertices $u$ and $v$ are its ends. Vertices $u$ and $v$ are adjacent or neighbours, if $u v \in G$. Two edges $e_{1}=u v$ and $e_{2}=u w$ having a common end, are adjacent with each other.
A graph $G$ can be represented as a plane figure by drawing a line (or a curve) between the points $u$ and $v$ (representing vertices) if $e=u v$ is an edge of $G$. The figure on the right is a geometric representation of the graph $G$ with $V_{G}=\left\{v_{1}, v_{2}, v_{3}, v_{4}, v_{5}, v_{6}\right\}$ and $E_{G}=\left\{v_{1} v_{2}, v_{1} v_{3}, v_{2} v_{3}, v_{2} v_{4}, v_{5} v_{6}\right\}$.
Often we shall omit the identities (names $v$ ) of the vertices in our figures, in which case the vertices are drawn as anonymous circles.
Graphs can be generalized by allowing loops $v v$ and parallel (or multiple) edges between vertices to obtain a multigraph $G=(V, E, \psi)$, where $E=\left\{e_{1}, e_{2}, \ldots, e_{m}\right\}$ is a set (of symbols), and $\psi: E \rightarrow E(V) \cup\{v v \mid v \in V\}$ is a function that attaches an unordered pair of vertices to each $e \in E: \psi(e)=u v$.
Note that we can have $\psi\left(e_{1}\right)=\psi\left(e_{2}\right)$. This is drawn in the figure of $G$ by placing two (parallel) edges that connect the common ends. On the right there is (a drawing of) a multigraph $G$ with vertices $V=\{a, b, c\}$ and edges $\psi\left(e_{1}\right)=a a, \psi\left(e_{2}\right)=a b, \psi\left(e_{3}\right)=b c$, and $\psi\left(e_{4}\right)=b c$.
Later we concentrate on (simple) graphs.
DEFINITION. We also study directed graphs or digraphs $D=(V, E)$, where the edges have a direction, that is, the edges are ordered: $E \subseteq V \times V$. In this case, $u v \neq v u$.
The directed graphs have representations, where the edges are drawn as arrows. A digraph can contain edges $u v$ and $v u$ of opposite directions.
Graphs and digraphs can also be coloured, labelled, and weighted:
DEFInition. A function $\alpha: V_{G} \rightarrow K$ is a vertex colouring of $G$ by a set $K$ of colours. A function $\alpha: E_{G} \rightarrow K$ is an edge colouring of $G$. Usually, $K=[1, k]$ for some $k \geq 1$.
If $K \subseteq \mathbb{R}$ (often $K \subseteq \mathbb{N}$ ), then $\alpha$ is a weight function or a distance function.
## Isomorphism of graphs
DEFInITiOn. Two graphs $G$ and $H$ are isomorphic, denoted by $G \cong H$, if there exists a bijection $\alpha: V_{G} \rightarrow V_{H}$ such that
$$
u v \in E_{G} \Longleftrightarrow \alpha(u) \alpha(v) \in E_{H}
$$
for all $u, v \in G$.
Hence $G$ and $H$ are isomorphic if the vertices of $H$ are renamings of those of $G$. Two isomorphic graphs enjoy the same graph theoretical properties, and they are often identified. In particular, all isomorphic graphs have the same plane figures (excepting the identities of the vertices). This shows in the figures, where we tend to replace the vertices by small circles, and talk of 'the graph' although there are, in fact, infinitely many such graphs.
Example 1.1. The following graphs are isomorphic. Indeed, the required isomorphism is given by $v_{1} \mapsto 1, v_{2} \mapsto 3$, $v_{3} \mapsto 4, v_{4} \mapsto 2, v_{5} \mapsto 5$.
Isomorphism Problem. Does there exist an efficient algorithm to check whether any two given graphs are isomorphic or not?
The following table lists the number $\left.2^{n} \begin{array}{c}n \\ 2\end{array}\right)$ of all graphs on a given set of $n$ vertices, and the number of all nonisomorphic graphs on $n$ vertices. It tells that at least for computational purposes an efficient algorithm for checking whether two graphs are isomorphic or not would be greatly appreciated.
| $n$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| graphs | 1 | 2 | 8 | 64 | 1024 | 32768 | 2097152 | 268435456 | $2^{36}>6 \cdot 10^{10}$ |
| nonisomorphic | 1 | 2 | 4 | 11 | 34 | 156 | 1044 | 12346 | 274668 |
## Other representations
Plane figures catch graphs for our eyes, but if a problem on graphs is to be programmed, then these figures are, to say the least, unsuitable. Integer matrices are ideal for computers, since every respectable programming language has array structures for these, and computers are good in crunching numbers.
Let $V_{G}=\left\{v_{1}, \ldots, v_{n}\right\}$ be ordered. The adjacency matrix of $G$ is the $n \times n$-matrix $M$ with entries $M_{i j}=1$ or $M_{i j}=0$ according to whether $v_{i} v_{j} \in G$ or $v_{i} v_{j} \notin G$. For instance, the graph in Example 1.1 has an adjacency matrix on the right. Notice that the adjacency matrix is always symmetric (with respect to its diagonal consisting of zeros).
$$
\left(\begin{array}{lllll}
0 & 1 & 1 & 0 & 1 \\
1 & 0 & 0 & 1 & 1 \\
1 & 0 & 0 & 1 & 0 \\
0 & 1 & 1 & 0 & 0 \\
1 & 1 & 0 & 0 & 0
\end{array}\right)
$$
A graph has usually many different adjacency matrices, one for each ordering of its set $V_{G}$ of vertices. The following result is obvious from the definitions.
Theorem 1.1. Two graphs $G$ and $H$ are isomorphic if and only if they have a common adjacency matrix. Moreover, two isomorphic graphs have exactly the same set of adjacency matrices.
Graphs can also be represented by sets. For this, let $\mathcal{X}=\left\{X_{1}, X_{2}, \ldots, X_{n}\right\}$ be a family of subsets of a set $X$, and define the intersection graph $G_{\mathcal{X}}$ as the graph with vertices $X_{1}, \ldots, X_{n}$, and edges $X_{i} X_{j}$ for all $i$ and $j(i \neq j)$ with $X_{i} \cap X_{j} \neq \varnothing$.
Theorem 1.2. Every graph is an intersection graph of some family of subsets.
Proof. Let $G$ be a graph, and define, for all $v \in G$, a set
$$
X_{v}=\{\{v, u\} \mid v u \in G\} .
$$
Then $X_{u} \cap X_{v} \neq \varnothing$ if and only if $u v \in G$.
Let $s(G)$ be the smallest size of a base set $X$ such that $G$ can be represented as an intersection graph of a family of subsets of $X$, that is,
$$
s(G)=\min \left\{|X| \mid G \cong G_{\mathcal{X}} \text { for some } \mathcal{X} \subseteq 2^{X}\right\} .
$$
How small can $s(G)$ be compared to the order $v_{G}$ (or the size $\varepsilon_{G}$ ) of the graph? It was shown by KOU, STOCKMEYER AND WONG (1976) that it is algorithmically difficult to determine the number $s(G)$ - the problem is NP-complete.
Example 1.2. As yet another example, let $A \subseteq \mathbb{N}$ be a finite set of natural numbers, and let $G_{A}=(A, E)$ be the graph with $r s \in E$ if and only if $r$ and $s$ (for $r \neq s$ ) have a common divisor $>1$. As an exercise, we state: All graphs can be represented in the form $G_{A}$ for some set $A$ of natural numbers.
### Subgraphs
Ideally, given a nice problem the local properties of a graph determine a solution. In these situations we deal with (small) parts of the graph (subgraphs), and a solution can be found to the problem by combining the information determined by the parts. For instance, as we shall later see, the existence of an Euler tour is very local, it depends only on the number of the neighbours of the vertices.
## Degrees of vertices
DEFINITION. Let $v \in G$ be a vertex a graph $G$. The neighbourhood of $v$ is the set
$$
N_{G}(v)=\{u \in G \mid v u \in G\} .
$$
The degree of $v$ is the number of its neighbours:
$$
d_{G}(v)=\left|N_{G}(v)\right| .
$$
If $d_{G}(v)=0$, then $v$ is said to be isolated in $G$, and if $d_{G}(v)=1$, then $v$ is a leaf of the graph. The minimum degree and the maximum degree of $G$ are defined as
$$
\delta(G)=\min \left\{d_{G}(v) \mid v \in G\right\} \quad \text { and } \quad \Delta(G)=\max \left\{d_{G}(v) \mid v \in G\right\} .
$$
The following lemma, due to EULER (1736), tells that if several people shake hands, then the number of hands shaken is even.
Lemma 1.1 (Handshaking lemma). For each graph G,
$$
\sum_{v \in G} d_{G}(v)=2 \cdot \varepsilon_{G} .
$$
Moreover, the number of vertices of odd degree is even.
Proof. Every edge $e \in E_{G}$ has two ends. The second claim follows immediately from the first one.
Lemma 1.1 holds equally well for multigraphs, when $d_{G}(v)$ is defined as the number of edges that have $v$ as an end, and when each loop $v v$ is counted twice.
Note that the degrees of a graph $G$ do not determine $G$. Indeed, there are graphs $G=\left(V, E_{G}\right)$ and $H=\left(V, E_{H}\right)$ on the same set of vertices that are not isomorphic, but for which $d_{G}(v)=d_{H}(v)$ for all $v \in V$.
## Subgraphs
Definition. A graph $H$ is a subgraph of a graph $G$, denoted by $H \subseteq G$, if $V_{H} \subseteq V_{G}$ and $E_{H} \subseteq E_{G}$. A subgraph $H \subseteq G$ spans $G$ (and $H$ is a spanning subgraph of $G$ ), if every vertex of $G$ is in $H$, i.e., $V_{H}=V_{G}$.
Also, a subgraph $H \subseteq G$ is an induced subgraph, if $E_{H}=E_{G} \cap E\left(V_{H}\right)$. In this case, $H$ is induced by its set $V_{H}$ of vertices.
In an induced subgraph $H \subseteq G$, the set $E_{H}$ of edges consists of all $e \in E_{G}$ such that $e \in E\left(V_{H}\right)$. To each nonempty subset $A \subseteq V_{G}$, there corresponds a unique induced subgraph
$$
G[A]=\left(A, E_{G} \cap E(A)\right) .
$$
To each subset $F \subseteq E_{G}$ of edges there corresponds a unique spanning subgraph of $G$,
$$
G[F]=\left(V_{G}, F\right) .
$$
G
subgraph
spanning
induced
For a set $F \subseteq E_{G}$ of edges, let
$$
G-F=G\left[E_{G} \backslash F\right]
$$
be the subgraph of $G$ obtained by removing (only) the edges $e \in F$ from $G$. In particular, $G-e$ is obtained from $G$ by removing $e \in G$.
Similarly, we write $G+F$, if each $e \in F$ (for $F \subseteq E\left(V_{G}\right)$ ) is added to $G$.
For a subset $A \subseteq V_{G}$ of vertices, we let $G-A \subseteq G$ be the subgraph induced by $V_{G} \backslash A$, that is,
$$
G-A=G\left[V_{G} \backslash A\right]
$$
and, e.g., $G-v$ is obtained from $G$ by removing the vertex $v$ together with the edges that have $v$ as their end.
Reconstruction Problem. The famous open problem, Kelly-Ulam problem or the Reconstruction Conjecture, states that a graph of order at least 3 is determined up to isomorphism by its vertex deleted subgraphs $G-v(v \in G)$ : if there exists a bijection $\alpha: V_{G} \rightarrow V_{H}$ such that $G-v \cong H-\alpha(v)$ for all $v$, then $G \cong H$.
## 2-switches
DEFINITION. For a graph $G$, a 2-switch with respect to the edges $u v, x y \in G$ with $u x, v y \notin G$ replaces the edges $u v$ and $x y$ by $u x$ and $v y$. Denote
$$
G \stackrel{2 s}{\longrightarrow} H
$$
if there exists a finite sequence of 2-switches that car-
ries $G$ to $H$.
Note that if $G \stackrel{2 s}{\longrightarrow} H$ then also $H \stackrel{2 s}{\longrightarrow} G$ since we can apply the sequence of 2switches in reverse order.
Before proving Berge's switching theorem we need the following tool.
Lemma 1.2. Let $G$ be a graph of order $n$ with a degree sequence $d_{1} \geq d_{2} \geq \cdots \geq d_{n}$, where $d_{G}\left(v_{i}\right)=d_{i}$. Then there is a graph $G^{\prime}$ such that $G \stackrel{2 s}{\longrightarrow} G^{\prime}$ with $N_{G^{\prime}}\left(v_{1}\right)=\left\{v_{2}, \ldots, v_{d_{1}+1}\right\}$.
Proof. Let $d=\Delta(G)\left(=d_{1}\right)$. Suppose that there is a vertex $v_{i}$ with $2 \leq i \leq d+1$ such that $v_{1} v_{i} \notin G$. Since $d_{G}\left(v_{1}\right)=d$, there exists a $v_{j}$ with $j \geq d+2$ such that $v_{1} v_{j} \in G$. Here $d_{i} \geq d_{j}$, since $j>i$. Since $v_{1} v_{j} \in G$, there exists a $v_{t}(2 \leq t \leq n)$ such that $v_{i} v_{t} \in G$, but $v_{j} v_{t} \notin G$. We can now perform a 2-switch with respect to the vertices $v_{1}, v_{j}, v_{i}, v_{t}$. This gives a new graph $H$, where $v_{1} v_{i} \in H$ and $v_{1} v_{j} \notin H$, and the other
neighbours of $v_{1}$ remain to be its neighbours.
When we repeat this process for all indices $i$ with $v_{1} v_{i} \notin G$ for $2 \leq i \leq d+1$, we obtain a graph $G^{\prime}$ as required.
Theorem 1.3 (Berge (1973)). Two graphs $G$ and $H$ on a common vertex set $V$ satisfy $d_{G}(v)=d_{H}(v)$ for all $v \in V$ if and only if $H$ can be obtained from $G$ by a sequence of 2-switches.
Proof. If $G \stackrel{2 s}{\longrightarrow} H$, then clearly $H$ has the same degrees as $G$.
In converse, we use induction on the order $v_{G}$. Let $G$ and $H$ have the same degrees. By Lemma 1.2, we have a vertex $v$ and graphs $G^{\prime}$ and $H^{\prime}$ such that $G \stackrel{2 s}{\longrightarrow} G^{\prime}$ and $H \stackrel{2 s}{\longrightarrow} H^{\prime}$ with $N_{G^{\prime}}(v)=N_{H^{\prime}}(v)$. Now the graphs $G^{\prime}-v$ and $H^{\prime}-v$ have the same degrees. By the induction hypothesis, $G^{\prime}-v \stackrel{2 s}{\longrightarrow} H^{\prime}-v$, and thus also $G^{\prime} \stackrel{2 s}{\longrightarrow} H^{\prime}$. Finally, we observe that $H^{\prime} \stackrel{2 s}{\longrightarrow} H$ by the 'reverse 2-switches', and this proves the claim.
DEFINITION. Let $d_{1}, d_{2}, \ldots, d_{n}$ be a descending sequence of nonnegative integers, that is, $d_{1} \geq d_{2} \geq \cdots \geq d_{n}$. Such a sequence is said to be graphical, if there exists a graph $G=(V, E)$ with $V=\left\{v_{1}, v_{2}, \ldots, v_{n}\right\}$ such that $d_{i}=d_{G}\left(v_{i}\right)$ for all $i$. Using the next result recursively one can decide whether a sequence of integers is graphical or not.
Theorem 1.4 (HAVEL (1955), HAKIMI (1962)). A sequence $d_{1}, d_{2}, \ldots, d_{n}$ (with $d_{1} \geq 1$ and $n \geq 2$ ) is graphical if and only if
$$
d_{2}-1, d_{3}-1, \ldots, d_{d_{1}+1}-1, d_{d_{1}+2}, d_{d_{1}+3}, \ldots, d_{n}
$$
is graphical (when put into nonincreasing order).
Proof. $(\Leftarrow)$ Consider $G$ of order $n-1$ with vertices (and degrees)
$$
\begin{aligned}
& d_{G}\left(v_{2}\right)=d_{2}-1, \ldots, d_{G}\left(v_{d_{1}+1}\right)=d_{d_{1}+1}-1, \\
& d_{G}\left(v_{d_{1}+2}\right)=d_{d_{1}+2}, \ldots, d_{G}\left(v_{n}\right)=d_{n}
\end{aligned}
$$
as in (1.1). Add a new vertex $v_{1}$ and the edges $v_{1} v_{i}$ for all $i \in\left[2, d_{d_{1}+1}\right]$. Then in the new graph $H, d_{H}\left(v_{1}\right)=d_{1}$, and $d_{H}\left(v_{i}\right)=d_{i}$ for all $i$.
$(\Rightarrow)$ Assume $d_{G}\left(v_{i}\right)=d_{i}$. By Lemma 1.2 and Theorem 1.3, we can suppose that $N_{G}\left(v_{1}\right)=\left\{v_{2}, \ldots, v_{d_{1}+1}\right\}$. But now the degree sequence of $G-v_{1}$ is in (1.1).
Example 1.3. Consider the sequence $s=4,4,4,3,2,1$. By Theorem 1.4,
$s$ is graphical $\Longleftrightarrow 3,3,2,1,1$ is graphical
$2,1,1,0$ is graphical
$0,0,0$ is graphical.
The last sequence corresponds to a graph with no edges, and hence also our original sequence $s$ is graphical. Indeed, the graph $G$ on the right has this degree
sequence.
## Special graphs
Definition. A graph $G=(V, E)$ is trivial, if it has only one vertex, i.e., $v_{G}=1$; otherwise $G$ is nontrivial.
The graph $G=K_{V}$ is the complete graph on $V$, if every two vertices are adjacent: $E=E(V)$. All complete graphs of order $n$ are isomorphic with each other, and they will be denoted by $K_{n}$.
The complement of $G$ is the graph $\bar{G}$ on $V_{G}$, where $E_{\bar{G}}=\left\{e \in E(V) \mid e \notin E_{G}\right\}$. The complements $G=\bar{K}_{V}$ of the complete graphs are called discrete graphs. In a discrete graph $E_{G}=\varnothing$. Clearly, all discrete graphs of order $n$ are isomorphic with each other.
A graph $G$ is said to be regular, if every vertex of $G$ has the same degree. If this degree is equal to $r$, then $G$ is $r$-regular or regular of degree $r$. A discrete graph is 0-regular, and a complete graph $K_{n}$ is $(n-1)$-regular. In particular, $\varepsilon_{K_{n}}=n(n-1) / 2$, and therefore $\varepsilon_{G} \leq n(n-1) / 2$ for all graphs $G$ that have order $n$.
Many problems concerning (induced) subgraphs are algorithmically difficult. For instance, to find a maximal complete subgraph (a subgraph $K_{m}$ of maximum order) of a graph is unlikely to be even in NP.
Example 1.4. The graph on the right is the Petersen graph that we will meet several times (drawn differently). It is a 3-regular graph of order 10.
Example 1.5. Let $k \geq 1$ be an integer, and consider the set $\mathbb{B}^{k}$ of all binary strings of length $k$. For instance, $\mathbb{B}^{3}=\{000,001,010,100,011,101,110,111\}$. Let $Q_{k}$ be the graph, called the $k$-cube, with $V_{Q_{k}}=\mathbb{B}^{k}$, where $u v \in Q_{k}$ if and only if the strings $u$ and $v$ differ in exactly one place.
The order of $Q_{k}$ is $v_{Q_{k}}=2^{k}$, the number of binary strings of length $k$. Also, $Q_{k}$ is $k$-regular, and so, by the handshaking lemma, $\varepsilon_{Q_{k}}=k \cdot 2^{k-1}$. On the right we have the 3-cube, or simply the cube.
Example 1.6. Let $n \geq 4$ be any even number. We show by induction that there exists a 3-regular graph $G$ with $v_{G}=n$. Notice that all 3-regular graphs have even order by the handshaking lemma.
If $n=4$, then $K_{4}$ is 3-regular. Let $G$ be a 3-regular graph of order $2 m-2$, and suppose that $u v, u w \in E_{G}$. Let $V_{H}=V_{G} \cup\{x, y\}$, and $E_{H}=\left(E_{G} \backslash\{u v, u w\}\right) \cup$ $\{u x, x v, u y, y w, x y\}$. Then $H$ is 3-regular of order $2 m$.
### Paths and cycles
The most fundamental notions in graph theory are practically oriented. Indeed, many graph theoretical questions ask for optimal solutions to problems such as: find a shortest path (in a complex network) from a given point to another. This kind of problems can be difficult, or at least nontrivial, because there are usually choices what branch to choose when leaving an intermediate point.
## Walks
Definition. Let $e_{i}=u_{i} u_{i+1} \in G$ be edges of $G$ for $i \in[1, k]$. The sequence $W=$ $e_{1} e_{2} \ldots e_{k}$ is a walk of length $k$ from $u_{1}$ to $u_{k+1}$. Here $e_{i}$ and $e_{i+1}$ are compatible in the sense that $e_{i}$ is adjacent to $e_{i+1}$ for all $i \in[1, k-1]$. We write, more informally,
$$
W: u_{1} \rightarrow u_{2} \rightarrow \ldots \rightarrow u_{k} \rightarrow u_{k+1} \quad \text { or } \quad W: u_{1} \stackrel{k}{\rightarrow} u_{k+1} .
$$
Write $u \stackrel{\star}{\rightarrow} v$ to say that there is a walk of some length from $u$ to $v$. Here we understand that $W: u \stackrel{\star}{\rightarrow} v$ is always a specific walk, $W=e_{1} e_{2} \ldots e_{k}$, although we sometimes do not care to mention the edges $e_{i}$ on it. The length of a walk $W$ is denoted by $|W|$.
DEFINITION. Let $W=e_{1} e_{2} \ldots e_{k}\left(e_{i}=u_{i} u_{i+1}\right)$ be a walk.
$W$ is closed, if $u_{1}=u_{k+1}$.
$W$ is a path, if $u_{i} \neq u_{j}$ for all $i \neq j$.
$W$ is a cycle, if it is closed, and $u_{i} \neq u_{j}$ for $i \neq j$ except that $u_{1}=u_{k+1}$.
$W$ is a trivial path, if its length is 0 . A trivial path has no edges.
For a walk $W: u=u_{1} \rightarrow \ldots \rightarrow u_{k+1}=v$, also
$$
W^{-1}: v=u_{k+1} \rightarrow \ldots \rightarrow u_{1}=u
$$
is a walk in $G$, called the inverse walk of $W$.
A vertex $u$ is an end of path $P$, if $P$ starts or ends in $u$.
The join of two walks $W_{1}: u \stackrel{\star}{\rightarrow} v$ and $W_{2}: v \stackrel{\star}{\rightarrow} w$ is the walk $W_{1} W_{2}: u \stackrel{\star}{\rightarrow} w$. (Here the end $v$ must be common to the walks.)
Paths $P$ and $Q$ are disjoint, if they have no vertices in common, and they are independent, if they can share only their ends.
Clearly, the inverse walk $P^{-1}$ of a path $P$ is a path (the inverse path of $P$ ). The join of two paths need not be a path.
A (sub)graph, which is a path (cycle) of length $k-1$ ( $k$, resp.) having $k$ vertices is denoted by $P_{k}\left(C_{k}\right.$, resp.). If $k$ is even (odd), we say that the path or cycle is even (odd). Clearly, all paths of length $k$ are isomorphic. The same holds for cycles of fixed length.
Lemma 1.3. Each walk $W: u \stackrel{\star}{\rightarrow} v$ with $u \neq v$ contains a path $P: u \stackrel{\star}{\rightarrow} v$, that is, there is a path $P: u \stackrel{\star}{\rightarrow} v$ that is obtained from $W$ by removing edges and vertices.
Proof. Let $W: u=u_{1} \rightarrow \ldots \rightarrow u_{k+1}=v$. Let $i<j$ be indices such that $u_{i}=u_{j}$. If no such $i$ and $j$ exist, then $W$, itself, is a path. Otherwise, in $W=W_{1} W_{2} W_{3}: u \stackrel{\star}{\rightarrow}$ $u_{i} \stackrel{\star}{\rightarrow} u_{j} \stackrel{\star}{\rightarrow} v$ the portion $U_{1}=W_{1} W_{3}: u \stackrel{\star}{\rightarrow} u_{i}=u_{j} \stackrel{\star}{\rightarrow} v$ is a shorter walk. By repeating this argument, we obtain a sequence $U_{1}, U_{2}, \ldots, U_{m}$ of walks $u \stackrel{\star}{\rightarrow} v$ with $|W|>\left|U_{1}\right|>\cdots>\left|U_{m}\right|$. When the procedure stops, we have a path as required. (Notice that in the above it may very well be that $W_{1}$ or $W_{3}$ is a trivial walk.) DEFINITION. If there exists a walk (and hence a path) from $u$ to $v$ in $G$, let
$$
d_{G}(u, v)=\min \{k \mid u \stackrel{k}{\rightarrow} v\}
$$
be the distance between $u$ and $v$. If there are no walks $u \stackrel{\star}{\rightarrow} v$, let $d_{G}(u, v)=\infty$ by convention. A graph $G$ is connected, if $d_{G}(u, v)<\infty$ for all $u, v \in G$; otherwise, it is disconnected. The maximal connected subgraphs of $G$ are its connected components. Denote
$$
c(G)=\text { the number of connected components of } G .
$$
If $c(G)=1$, then $G$ is, of course, connected.
The maximality condition means that a subgraph $H \subseteq G$ is a connected component if and only if $H$ is connected and there are no edges leaving $H$, i.e., for every vertex $v \notin H$, the subgraph $G\left[V_{H} \cup\{v\}\right]$ is disconnected. Apparently, every connected component is an induced subgraph, and
$$
N_{G}^{*}(v)=\left\{u \mid d_{G}(v, u)<\infty\right\}
$$
is the connected component of $G$ that contains $v \in G$. In particular, the connected components form a partition of $G$.
## Shortest paths
DEFINITION. Let $G^{\alpha}$ be an edge weighted graph, that is, $G^{\alpha}$ is a graph $G$ together with a weight function $\alpha: E_{G} \rightarrow \mathbb{R}$ on its edges. For $H \subseteq G$, let
$$
\alpha(H)=\sum_{e \in H} \alpha(e)
$$
be the (total) weight of $H$. In particular, if $P=e_{1} e_{2} \ldots e_{k}$ is a path, then its weight is $\alpha(P)=\sum_{i=1}^{k} \alpha\left(e_{i}\right)$. The minimum weighted distance between two vertices is
$$
d_{G}^{\alpha}(u, v)=\min \{\alpha(P) \mid P: u \stackrel{\star}{\rightarrow} v\} .
$$
In extremal problems we seek for optimal subgraphs $H \subseteq G$ satisfying specific conditions. In practice we encounter situations where $G$ might represent
- a distribution or transportation network (say, for mail), where the weights on edges are distances, travel expenses, or rates of flow in the network;
- a system of channels in (tele)communication or computer architecture, where the weights present the rate of unreliability or frequency of action of the connections;
- a model of chemical bonds, where the weights measure molecular attraction. In these examples we look for a subgraph with the smallest weight, and which connects two given vertices, or all vertices (if we want to travel around). On the other hand, if the graph represents a network of pipelines, the weights are volumes or capacities, and then one wants to find a subgraph with the maximum weight.
We consider the minimum problem. For this, let $G$ be a graph with an integer weight function $\alpha: E_{G} \rightarrow \mathbb{N}$. In this case, call $\alpha(u v)$ the length of $u v$.
The shortest path problem: Given a connected graph $G$ with a weight function $\alpha: E_{G} \rightarrow$ $\mathbb{N}$, find $d_{G}^{\alpha}(u, v)$ for given $u, v \in G$.
Assume that $G$ is a connected graph. Dijkstra's algorithm solves the problem for every pair $u, v$, where $u$ is a fixed starting point and $v \in G$. Let us make the convention that $\alpha(u v)=\infty$, if $u v \notin G$.
## Dijkstra's algorithm:
(i) Set $u_{0}=u, t\left(u_{0}\right)=0$ and $t(v)=\infty$ for all $v \neq u_{0}$.
(ii) For $i \in\left[0, v_{G}-1\right]$ : for each $v \notin\left\{u_{1}, \ldots, u_{i}\right\}$,
$$
\text { replace } t(v) \text { by } \min \left\{t(v), t\left(u_{i}\right)+\alpha\left(u_{i} v\right)\right\} \text {. }
$$
Let $u_{i+1} \notin\left\{u_{1}, \ldots, u_{i}\right\}$ be any vertex with the least value $t\left(u_{i+1}\right)$.
(iii) Conclusion: $d_{G}^{\alpha}(u, v)=t(v)$.
Example 1.7. Consider the following weighted graph G. Apply Dijkstra's algorithm to the vertex $v_{0}$.
- $u_{0}=v_{0}, t\left(u_{0}\right)=0$, others are $\infty$.
- $t\left(v_{1}\right)=\min \{\infty, 2\}=2, t\left(v_{2}\right)=\min \{\infty, 3\}=3$, others are $\infty$. Thus $u_{1}=v_{1}$.
- $t\left(v_{2}\right)=\min \left\{3, t\left(u_{1}\right)+\alpha\left(u_{1} v_{2}\right)\right\}=\min \{3,4\}=3$, $t\left(v_{3}\right)=2+1=3, t\left(v_{4}\right)=2+3=5, t\left(v_{5}\right)=2+2=4$. Thus choose $u_{2}=v_{3}$.
- $t\left(v_{2}\right)=\min \{3, \infty\}=3, t\left(v_{4}\right)=\min \{5,3+2\}=5$,
$t\left(v_{5}\right)=\min \{4,3+1\}=4$. Thus set $u_{3}=v_{2}$.
- $t\left(v_{4}\right)=\min \{5,3+1\}=4, t\left(v_{5}\right)=\min \{4, \infty\}=4$. Thus choose $u_{4}=v_{4}$.
- $t\left(v_{5}\right)=\min \{4,4+1\}=4$. The algorithm stops.
We have obtained:
$$
t\left(v_{1}\right)=2, t\left(v_{2}\right)=3, t\left(v_{3}\right)=3, t\left(v_{4}\right)=4, t\left(v_{5}\right)=4 .
$$
These are the minimal weights from $v_{0}$ to each $v_{i}$. The steps of the algorithm can also be rewritten as a table:
$$
\begin{array}{|l|lllll|}
\hline v_{1} & \mathbf{2} & - & - & - & - \\
v_{2} & 3 & 3 & 3 & - & - \\
v_{3} & \infty & 3 & - & - & - \\
v_{4} & \infty & 5 & 5 & 4 & - \\
v_{5} & \infty & 4 & 4 & 4 & 4 \\
\hline
\end{array}
$$
The correctness of Dijkstra's algorithm can verified be as follows.
Let $v \in V$ be any vertex, and let $P: u_{0} \stackrel{\star}{\rightarrow} u \stackrel{\star}{\rightarrow} v$ be a shortest path from $u_{0}$ to $v$, where $u$ is any vertex $u \neq v$ on such a path, possibly $u=u_{0}$. Then, clearly, the first part of the path, $u_{0} \stackrel{\star}{\rightarrow} u$, is a shortest path from $u_{0}$ to $u$, and the latter part $u \stackrel{\star}{\rightarrow} v$ is a shortest path from $u$ to $v$. Therefore, the length of the path $P$ equals the sum of the weights of $u_{0} \stackrel{\star}{\rightarrow} u$ and $u \stackrel{\star}{\rightarrow} v$. Dijkstra's algorithm makes use of this observation iteratively.
## Connectivity of Graphs
### Bipartite graphs and trees
In problems such as the shortest path problem we look for minimum solutions that satisfy the given requirements. The solutions in these cases are usually subgraphs without cycles. Such connected graphs will be called trees, and they are used, e.g., in search algorithms for databases. For concrete applications in this respect, see
T.H. Cormen, C.E. LeISERSON AND R.L. Rivest, "Introduction to Algorithms", MIT Press, 1993.
Certain structures with operations are representable as trees. These trees are sometimes called construction trees, decomposition trees, factorization trees or grammatical trees. Grammatical trees occur especially in linguistics, where syntactic structures of sentences are analyzed. On the right there is a tree of operations for the arithmetic formula $x \cdot(y+z)+y$.
## Bipartite graphs
DEFINITION. A graph $G$ is called bipartite, if $V_{G}$ has a partition to two subsets $X$ and $Y$ such that each edge $u v \in G$ connects a vertex of $X$ and a vertex of $Y$. In this case, $(X, Y)$ is a bipartition of $G$, and $G$ is $(X, Y)$-bipartite.
A bipartite graph $G$ (as in the above) is complete $(m, k)$ bipartite, if $|X|=m,|Y|=k$, and $u v \in G$ for all $u \in X$ and $v \in Y$.
All complete $(m, k)$-bipartite graphs are isomorphic. Let $K_{m, k}$ denote such a graph.
A subset $X \subseteq V_{G}$ is stable, if $G[X]$ is a discrete graph.
$K_{2,3}$
The following result is clear from the definitions.
Theorem 2.1. A graph $G$ is bipartite if and only if $V_{G}$ has a partition to two stable subsets.
Example 2.1. The $k$-cube $Q_{k}$ of Example 1.5 is bipartite for all $k$. Indeed, consider $A=\left\{u \mid u\right.$ has an even number of $\left.1^{\prime} \mathrm{s}\right\}$ and $B=\left\{u \mid u\right.$ has an odd number of $\left.1^{\prime} \mathrm{s}\right\}$. Clearly, these sets partition $\mathbb{B}^{k}$, and they are stable in $Q_{k}$. Theorem 2.2. A graph $G$ is bipartite if and only if $G$ it has no odd cycles (as subgraph).
Proof. $(\Rightarrow)$ Observe that if $G$ is $(X, Y)$-bipartite, then so are all its subgraphs. However, an odd cycle $C_{2 k+1}$ is not bipartite.
$(\Leftarrow)$ Suppose that all cycles in $G$ are even. First, we note that it suffices to show the claim for connected graphs. Indeed, if $G$ is disconnected, then each cycle of $G$ is contained in one of the connected components $G_{1}, \ldots, G_{p}$ of $G$. If $G_{i}$ is $\left(X_{i}, Y_{i}\right)$ bipartite, then $G$ has the bipartition $\left(X_{1} \cup X_{2} \cup \cdots \cup X_{p}, Y_{1} \cup Y_{2} \cup \cdots \cup Y_{p}\right)$.
Assume thus that $G$ is connected. Let $v \in G$ be a chosen vertex, and define
$$
X=\left\{x \mid d_{G}(v, x) \text { is even }\right\} \quad \text { and } \quad Y=\left\{y \mid d_{G}(v, y) \text { is odd }\right\} .
$$
Since $G$ is connected, $V_{G}=X \cup Y$. Also, by the definition of distance, $X \cap Y=\varnothing$.
Let then $u, w \in G$ be both in $X$ or both in $Y$, and let $P: v \stackrel{\star}{\rightarrow} u$ and $Q: v \stackrel{\star}{\rightarrow} w$ be (among the) shortest paths from $v$ to $u$ and $w$. Assume that $x$ is the last common vertex of $P$ and $Q: P=P_{1} P_{2}, Q=Q_{1} Q_{2}$, where $P_{2}: x \stackrel{\star}{\rightarrow} u$ and $Q_{2}: x \stackrel{\star}{\rightarrow} w$ are independent. Since $P$ and $Q$ are shortest paths, $P_{1}$ and $Q_{1}$ are shortest paths $v \stackrel{\star}{\rightarrow} x$. Consequently, $\left|P_{1}\right|=\left|Q_{1}\right|$.
Thus $\left|P_{2}\right|$ and $\left|Q_{2}\right|$ have the same parity and hence the sum $\left|P_{2}\right|+\left|Q_{2}\right|$ is even, i.e., the path $P_{2}^{-1} Q_{2}$ is even, and so $u w \notin E_{G}$ by assumption. Therefore $X$ and $Y$ are stable subsets, and $G$ is bipartite as claimed.
Checking whether a graph is bipartite is easy. Indeed, this can be done by using two 'opposite' colours, say 1 and 2. Start from any vertex $v_{1}$, and colour it by 1 . Then colour the neighbours of $v_{1}$ by 2 , and proceed by colouring all neighbours of an already coloured vertex by the opposite colour.
If the whole graph can be coloured without contradiction, then $G$ is $(X, Y)$-bipartite, where $X$ consists of those vertices with colour 1 , and $Y$ of those vertices with colour 2; otherwise, at some point one of the vertices gets both colours, and in this case, $G$ is not bipartite.
Example 2.2 (ERDÖS (1965)). We show that each graph $G$ has a bipartite subgraph $H \subseteq G$ such that $\varepsilon_{H} \geq \frac{1}{2} \varepsilon_{G}$. Indeed, let $V_{G}=X \cup Y$ be a partition such that the number of edges between $X$ and $Y$ is maximum. Denote
$$
F=E_{G} \cap\{u v \mid u \in X, v \in Y\},
$$
and let $H=G[F]$. Obviously $H$ is a spanning subgraph, and it is bipartite.
By the maximum condition, $d_{H}(v) \geq d_{G}(v) / 2$, since, otherwise, $v$ is on the wrong side. (That is, if $v \in X$, then the pair $X^{\prime}=X \backslash\{v\}, Y^{\prime}=Y \cup\{v\}$ does better that the pair $X, Y$.) Now
$$
\varepsilon_{H}=\frac{1}{2} \sum_{v \in H} d_{H}(v) \geq \frac{1}{2} \sum_{v \in G} \frac{1}{2} d_{G}(v)=\frac{1}{2} \varepsilon_{G} .
$$
## Bridges
DEFINITION. An edge $e \in G$ is a bridge of the graph $G$, if $G-e$ has more connected components than $G$, that is, if $c(G-e)>c(G)$. In particular, and most importantly, an edge $e$ in a connected $G$ is a bridge if and only if $G-e$ is disconnected.
On the right (only) the two horizontal lines are bridges.
We note that, for each edge $e \in G$,
$$
e=u v \text { is a bridge } \Longleftrightarrow u, v \text { in different connected components of } G-e \text {. }
$$
Theorem 2.3. An edge $e \in G$ is a bridge if and only if $e$ is not in any cycle of $G$.
Proof. $(\Rightarrow)$ If there is a cycle in $G$ containing $e$, say $C=P e Q$, then $Q P: v \stackrel{\star}{\rightarrow} u$ is a path in $G-e$, and so $e$ is not a bridge.
$(\Leftarrow)$ If $e=u v$ is not a bridge, then $u$ and $v$ are in the same connected component of $G-e$, and there is a path $P: v \stackrel{\star}{\rightarrow} u$ in $G-e$. Now, $e P: u \rightarrow v \stackrel{\star}{\rightarrow} u$ is a cycle in $G$ containing $e$.
Lemma 2.1. Let e be a bridge in a connected graph $G$.
(i) Then $c(G-e)=2$.
(ii) Let $H$ be a connected component of $G-e$. If $f \in H$ is a bridge of $H$, then $f$ is a bridge of $G$.
Proof. For (i), let $e=u v$. Since $e$ is a bridge, the ends $u$ and $v$ are not connected in $G-e$. Let $w \in G$. Since $G$ is connected, there exists a path $P: w \stackrel{\star}{\rightarrow} v$ in G. This is a path of $G-e$, unless $P: w \stackrel{\star}{\rightarrow} u \rightarrow v$ contains $e=u v$, in which case the part $w \stackrel{\star}{\rightarrow} u$ is a path in $G-e$.
For (ii), if $f \in H$ belongs to a cycle $C$ of $G$, then $C$ does not contain $e$ (since $e$ is in no cycle), and therefore $C$ is inside $H$, and $f$ is not a bridge of $H$.
## Trees
Definition. A graph is called acyclic, if it has no cycles. An acyclic graph is also called a forest. A tree is a connected acyclic graph.
By Theorem 2.3 and the definition of a tree, we have
Corollary 2.1. A connected graph is a tree if and only if all its edges are bridges.
Example 2.3. The following enumeration result for trees has many different proofs, the first of which was given by CAYLEY in 1889: There are $n^{n-2}$ trees on a vertex set $V$ of $n$ elements. We omit the proof. On the other hand, there are only a few trees up to isomorphism:
| $n$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | ---: | ---: |
| trees | 1 | 1 | 1 | 2 | 3 | 6 | 11 | 23 |
| $n$ | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| trees | 47 | 106 | 235 | 551 | 1301 | 3159 | 7741 | 19320 |
The nonisomorphic trees of order 6 are:
We say that a path $P: u \stackrel{\star}{\rightarrow} v$ is maximal in a graph $G$, if there are no edges $e \in G$ for which $P e$ or $e P$ is a path. Such paths exist, because $v_{G}$ is finite.
Lemma 2.2. Let $P: u \stackrel{\star}{\rightarrow} v$ be a maximal path in a graph $G$. Then $N_{G}(v) \subseteq P$. Moreover, if $G$ is acyclic, then $d_{G}(v)=1$.
Proof. If $e=v w \in E_{G}$ with $w \notin P$, then also $P e$ is a path, which contradicts the maximality assumption for $P$. Hence $N_{G}(v) \subseteq P$. For acyclic graphs, if $w v \in G$, then $w$ belongs to $P$, and $w v$ is necessarily the last edge of $P$ in order to avoid cycles.
Corollary 2.2. Each tree $T$ with $v_{T} \geq 2$ has at least two leaves.
Proof. Since $T$ is acyclic, both ends of a maximal path have degree one.
Theorem 2.4. The following are equivalent for a graph $T$.
(i) $T$ is a tree.
(ii) Any two vertices are connected in $T$ by a unique path.
(iii) $T$ is acyclic and $\varepsilon_{T}=v_{T}-1$.
Proof. Let $v_{T}=n$. If $n=1$, then the claim is trivial. Suppose thus that $n \geq 2$.
(i) $\Rightarrow$ (ii) Let $T$ be a tree. Assume the claim does not hold, and let $P, Q: u \stackrel{\star}{\rightarrow} v$ be two different paths between the same vertices $u$ and $v$. Suppose that $|P| \geq|Q|$. Since $P \neq Q$, there exists an edge $e$ which belongs to $P$ but not to $Q$. Each edge of $T$ is a bridge, and therefore $u$ and $v$ belong to different connected components of $T-e$. Hence $e$ must also belong to $Q$; a contradiction.
(ii) $\Rightarrow$ (iii) We prove the claim by induction on $n$. Clearly, the claim holds for $n=2$, and suppose it holds for graphs of order less than $n$. Let $T$ be any graph of order $n$ satisfying (ii). In particular, $T$ is connected, and it is clearly acyclic. Let $P: u \stackrel{\star}{\rightarrow} v$ be a maximal path in $T$. By Lemma 2.2, we have $d_{T}(v)=1$. In this case, $P: u \stackrel{\star}{\rightarrow} w \rightarrow v$, where $v w$ is the unique edge having an end $v$. The subgraph $T-v$ is connected, and it satisfies the condition (ii). By induction hypothesis, $\varepsilon_{T-v}=$ $n-2$, and so $\varepsilon_{T}=\varepsilon_{T-v}+1=n-1$, and the claim follows.
(iii) $\Rightarrow$ (i) Assume (iii) holds for $T$. We need to show that $T$ is connected. Indeed, let the connected components of $T$ be $T_{i}=\left(V_{i}, E_{i}\right)$, for $i \in[1, k]$. Since $T$ is acyclic, so are the connected graphs $T_{i}$, and hence they are trees, for which we have proved that $\left|E_{i}\right|=\left|V_{i}\right|-1$. Now, $v_{T}=\sum_{i=1}^{k}\left|V_{i}\right|$, and $\varepsilon_{T}=\sum_{i=1}^{k}\left|E_{i}\right|$. Therefore,
$$
n-1=\varepsilon_{T}=\sum_{i=1}^{k}\left(\left|V_{i}\right|-1\right)=\sum_{i=1}^{k}\left|V_{i}\right|-k=n-k,
$$
which gives that $k=1$, that is, $T$ is connected.
Example 2.4. Consider a cup tournament of $n$ teams. If during a round there are $k$ teams left in the tournament, then these are divided into $\lfloor k\rfloor$ pairs, and from each pair only the winner continues. If $k$ is odd, then one of the teams goes to the next round without having to play. How many plays are needed to determine the winner?
So if there are 14 teams, after the first round 7 teams continue, and after the second round 4 teams continue, then 2 . So 13 plays are needed in this example.
The answer to our problem is $n-1$, since the cup tournament is a tree, where a play corresponds to an edge of the tree.
## Spanning trees
Theorem 2.5. Each connected graph has a spanning tree, that is, a spanning graph that is a tree.
Proof. Let $T \subseteq G$ be a maximum order subtree of $G$ (i.e., subgraph that is a tree). If $V_{T} \neq V_{G}$, there exists an edge $u v \notin E_{G}$ such that $u \in T$ and $v \notin T$. But then $T$ is not maximal; a contradiction.
Corollary 2.3. For each connected graph $G, \varepsilon_{G} \geq v_{G}-1$. Moreover, a connected graph $G$ is a tree if and only if $\varepsilon_{G}=v_{G}-1$.
Proof. Let $T$ be a spanning tree of $G$. Then $\varepsilon_{G} \geq \varepsilon_{T}=v_{T}-1=v_{G}-1$. The second claim is also clear.
Example 2.5. In Shannon's switching game a positive player $P$ and a negative player $N$ play on a graph $G$ with two special vertices: a source $s$ and a sink $r . P$ and $N$ alternate turns so that $P$ designates an edge by + , and $N$ by - . Each edge can be designated at most once. It is $P^{\prime}$ s purpose to designate a path $s \stackrel{\star}{\rightarrow} r$ (that is, to designate all edges in one such path), and $N$ tries to block all paths $s \stackrel{\star}{\rightarrow} r$ (that is, to designate at least one edge in each such path). We say that a game $(G, s, r)$ is - positive, if $P$ has a winning strategy no matter who begins the game,
- negative, if $N$ has a winning strategy no matter who begins the game,
- neutral, if the winner depends on who begins the game.
The game on the right is neutral.
LEHMAN proved in 1964 that Shannon's switching game $(G, s, r)$ is positive if and only if there exists $H \subseteq G$ such that $H$ contains $s$ and $r$ and $H$ has two spanning trees with no edges in common.
In the other direction the claim can be proved along the following lines. Assume that there exists a subgraph $H$ containing $s$ and $r$ and that has two spanning trees with no edges in common. Then $P$ plays as follows. If $N$ marks by - an edge from one of the two trees, then $P$ marks by + an edge in the other tree such that this edge reconnects the broken tree. In this way, $P$ always has two spanning trees for the subgraph $H$ with only edges marked by + in common.
In converse the claim is considerably more difficult to prove.
There remains the problem to characterize those Shannon's switching games $(G, s, r)$ that are neutral (negative, respectively).
## The connector problem
To build a network connecting $n$ nodes (towns, computers, chips in a computer) it is desirable to decrease the cost of construction of the links to the minimum. This is the connector problem. In graph theoretical terms we wish to find an optimal spanning subgraph of a weighted graph. Such an optimal subgraph is clearly a spanning tree, for, otherwise a deletion of any nonbridge will reduce the total weight of the subgraph.
Let then $G^{\alpha}$ be a graph $G$ together with a weight function $\alpha: E_{G} \rightarrow \mathbb{R}^{+}$(positive reals) on the edges. Kruskal's algorithm (also known as the greedy algorithm) provides a solution to the connector problem.
Kruskal's algorithm: For a connected and weighted graph $G^{\alpha}$ of order $n$ :
(i) Let $e_{1}$ be an edge of smallest weight, and set $E_{1}=\left\{e_{1}\right\}$.
(ii) For each $i=2,3, \ldots, n-1$ in this order, choose an edge $e_{i} \notin E_{i-1}$ of smallest possible weight such that $e_{i}$ does not produce a cycle when added to $G\left[E_{i-1}\right]$, and let $E_{i}=E_{i-1} \cup\left\{e_{i}\right\}$.
The final outcome is $T=\left(V_{G}, E_{n-1}\right)$. By the construction, $T=\left(V_{G}, E_{n-1}\right)$ is a spanning tree of $G$, because it contains no cycles, it is connected and has $n-1$ edges. We now show that $T$ has the minimum total weight among the spanning trees of $G$.
Suppose $T_{1}$ is any spanning tree of $G$. Let $e_{k}$ be the first edge produced by the algorithm that is not in $T_{1}$. If we add $e_{k}$ to $T_{1}$, then a cycle $C$ containing $e_{k}$ is created. Also, $C$ must contain an edge $e$ that is not in $T$. When we replace $e$ by $e_{k}$ in $T_{1}$, we still have a spanning tree, say $T_{2}$. However, by the construction, $\alpha\left(e_{k}\right) \leq \alpha(e)$, and therefore $\alpha\left(T_{2}\right) \leq \alpha\left(T_{1}\right)$. Note that $T_{2}$ has more edges in common with $T$ than $T_{1}$.
Repeating the above procedure, we can transform $T_{1}$ to $T$ by replacing edges, one by one, such that the total weight does not increase. We deduce that $\alpha(T) \leq \alpha\left(T_{1}\right)$.
The outcome of Kruskal's algorithm need not be unique. Indeed, there may exist several optimal spanning trees (with the same weight, of course) for a graph.
Example 2.6. When applied to the weighted graph on the right, the algorithm produces the sequence: $e_{1}=v_{2} v_{4}, e_{2}=v_{4} v_{5}, e_{3}=v_{3} v_{6}, e_{4}=v_{2} v_{3}$ and $e_{5}=v_{1} v_{2}$. The total weight of the spanning tree is thus 9.
Also, the selection $e_{1}=v_{2} v_{5}, e_{2}=v_{4} v_{5}, e_{3}=v_{5} v_{6}$, $e_{4}=v_{3} v_{6}, e_{5}=v_{1} v_{2}$ gives another optimal solution (of weight 9).
Problem. Consider trees $T$ with weight functions $\alpha: E_{T} \rightarrow \mathbb{N}$. Each tree $T$ of order $n$ has exactly $\left(\begin{array}{l}n \\ 2\end{array}\right)$ paths. (Why is this so?) Does there exist a weighted tree $T^{\alpha}$ of order $n$ such that the (total) weights of its paths are $1,2, \ldots,\left(\begin{array}{l}n \\ 2\end{array}\right)$ ?
In such a weighted tree $T^{\alpha}$ different paths have different weights, and each $i \in\left[1,\left(\begin{array}{l}n \\ 2\end{array}\right)\right]$ is a weight of one path. Also, $\alpha$ must be injective.
No solutions are known for any $n \geq 7$.
TAYLOR (1977) proved: if T of order $n$ exists, then necessarily $n=k^{2}$ or $n=k^{2}+2$ for some $k \geq 1$.
Example 2.7. A computer network can be presented as a graph $G$, where the vertices are the node computers, and the edges indicate the direct links. Each computer $v$ has an $a d d r e s s ~ a(v)$, a bit string (of zeros and ones). The length of an address is the number of its bits. A message that is sent to $v$ is preceded by the address $a(v)$. The Hamming distance $h(a(v), a(u))$ of two addresses of the same length is the number of places, where $a(v)$ and $a(u)$ differ; e.g., $h(00010,01100)=3$ and $h(10000,00000)=1$.
It would be a good way to address the vertices so that the Hamming distance of two vertices is the same as their distance in $G$. In particular, if two vertices were adjacent, their addresses should differ by one symbol. This would make it easier for a node computer to forward a message. A graph $G$ is said to be addressable, if it has an addressing $a$ such that $d_{G}(u, v)=h(a(u), a(v))$.
We prove that every tree $T$ is addressable. Moreover, the addresses of the vertices of $T$ can be chosen to be of length $v_{T}-1$.
The proof goes by induction. If $v_{T} \leq 2$, then the claim is obvious. In the case $v_{T}=2$, the addresses of the vertices are simply 0 and 1 .
Let then $V_{T}=\left\{v_{1}, \ldots, v_{k+1}\right\}$, and assume that $d_{T}\left(v_{1}\right)=1$ (a leaf) and $v_{1} v_{2} \in T$. By the induction hypothesis, we can address the tree $T-v_{1}$ by addresses of length $k-1$. We change this addressing: let $a_{i}$ be the address of $v_{i}$ in $T-v_{1}$, and change it to $0 a_{i}$. Set the address of $v_{1}$ to $1 a_{2}$. It is now easy to see that we have obtained an addressing for $T$ as required.
The triangle $K_{3}$ is not addressable. In order to gain more generality, we modify the addressing for general graphs by introducing a special symbol $*$ in addition to 0 and 1 . A star address will be a sequence of these three symbols. The Hamming distance remains as it was, that is, $h(u, v)$ is the number of places, where $u$ and $v$ have a different symbol 0 or 1 . The special symbol $*$ does not affect $h(u, v)$. So, $h(10 *$ $* 01,0 * * 101)=1$ and $h(1 * * * * *, * 00 * * *)=0$. We still want to have $h(u, v)=$ $d_{G}(u, v)$.
We star address this graph as follows:
$$
\begin{array}{ll}
a\left(v_{1}\right)=0000, & a\left(v_{2}\right)=10 * 0, \\
a\left(v_{3}\right)=1 * 01, & a\left(v_{4}\right)=* * 11 .
\end{array}
$$
These addresses have length 4. Can you design a star addressing with addresses of length 3 ?
WINKLER proved in 1983 a rather unexpected result: The minimum star address length of a graph $G$ is at most $v_{G}-1$.
For the proof of this, see VAN LINT AND WILSON, "A Course in Combinatorics".
### Connectivity
Spanning trees are often optimal solutions to problems, where cost is the criterion. We may also wish to construct graphs that are as simple as possible, but where two vertices are always connected by at least two independent paths. These problems occur especially in different aspects of fault tolerance and reliability of networks, where one has to make sure that a break-down of one connection does not affect the functionality of the network. Similarly, in a reliable network we require that a break-down of a node (computer) should not result in the inactivity of the whole network.
## Separating sets
Definition. A vertex $v \in G$ is a cut vertex, if $c(G-v)>c(G)$. A subset $S \subseteq V_{G}$ is a separating set, if $G-S$ is disconnected. We also say that $S$ separates the vertices $u$ and $v$ and it is a $(u, v)$ separating set, if $u$ and $v$ belong to different connected components of $G-S$.
If $G$ is connected, then $v$ is a cut vertex if and only if $G-v$ is disconnected, that is, $\{v\}$ is a separating set. The following lemma is immediate.
Lemma 2.3. If $S \subseteq V_{G}$ separates $u$ and $v$, then every path $P: u \stackrel{\star}{\rightarrow} v$ visits a vertex of $S$.
Lemma 2.4. If a connected graph $G$ has no separating sets, then it is a complete graph.
Proof. If $v_{G} \leq 2$, then the claim is clear. For $v_{G} \geq 3$, assume that $G$ is not complete, and let $u v \notin G$. Now $V_{G} \backslash\{u, v\}$ is a separating set. The claim follows from this.
DEFINITION. The (vertex) connectivity number $\mathcal{\kappa}(G)$ of $G$ is defined as
$$
\kappa(G)=\min \left\{k|k=| S \mid, G-S \text { disconnected or trivial, } S \subseteq V_{G}\right\} .
$$
A graph $G$ is $k$-connected, if $\kappa(G) \geq k$.
In other words,
- $\kappa(G)=0$, if $G$ is disconnected,
- $\quad \kappa(G)=v_{G}-1$, if $G$ is a complete graph, and
- otherwise $\kappa(G)$ equals the minimum size of a separating set of $G$.
Clearly, if $G$ is connected, then it is 1-connected.
DEFINITION. An edge cut $F$ of $G$ consists of edges so that $G-F$ is disconnected. Let
$$
\kappa^{\prime}(G)=\min \left\{k|k=| F \mid, G-F \text { disconnected, } F \subseteq E_{G}\right\} .
$$
For trivial graphs, let $\kappa^{\prime}(G)=0$. A graph $G$ is $k$-edge connected, if $\kappa^{\prime}(G) \geq k$. A minimal edge cut $F \subseteq E_{G}$ is a bond $(F \backslash\{e\}$ is not an edge cut for any $e \in F)$.
Example 2.8. Again, if $G$ is disconnected, then $\kappa^{\prime}(G)=0$. On the right, $\kappa(G)=2$ and $\kappa^{\prime}(G)=2$. Notice that the minimum degree is $\delta(G)=3$.
Lemma 2.5. Let $G$ be connected. If $e=u v$ is a bridge, then either $G=K_{2}$ or one of $u$ or $v$ is a cut vertex. Proof. Assume that $G \neq K_{2}$ and thus that $v_{G} \geq 3$, since $G$ is connected. Let $G_{u}=$ $N_{G-e}^{*}(u)$ and $G_{v}=N_{G-e}^{*}(v)$ be the connected components of $G-e$ containing $u$ and $v$. Now, either $v_{G_{u}} \geq 2$ (and $u$ is a cut vertex) or $v_{G_{v}} \geq 2$ (and $v$ is a cut vertex).
Lemma 2.6. If $F$ be a bond of a connected graph $G$, then $c(G-F)=2$.
Proof. Since $G-F$ is disconnected, and $F$ is minimal, the subgraph $H=G-(F \backslash\{e\})$ is connected for given $e \in F$. Hence $e$ is a bridge in H. By Lemma 2.1, $c(H-e)=2$, and thus $c(G-F)=2$, since $H-e=G-F$.
Theorem 2.6 (Whitney (1932)). For any graph G,
$$
\kappa(G) \leq \kappa^{\prime}(G) \leq \delta(G) .
$$
Proof. Assume $G$ is nontrivial. Clearly, $\kappa^{\prime}(G) \leq \delta(G)$, since if we remove all edges with an end $v$, we disconnect $G$. If $\kappa^{\prime}(G)=0$, then $G$ is disconnected, and in this case also $\kappa(G)=0$. If $\kappa^{\prime}(G)=1$, then $G$ is connected and contains a bridge. By Lemma 2.5, either $G=K_{2}$ or $G$ has a cut vertex. In both of these cases, also $\kappa(G)=1$.
Assume then that $\kappa^{\prime}(G) \geq 2$. Let $F$ be an edge cut of $G$ with $|F|=\kappa^{\prime}(G)$, and let $e=u v \in F$. Then $F$ is a bond, and $G-F$ has two connected components.
Consider the connected subgraph $H=G-(F \backslash\{e\})=(G-F)+e$, where $e$ is a bridge.
Now for each $f \in F \backslash\{e\}$ choose an end different from $u$ and $v$. (The choices for different edges need not be different.) Note that since $f \neq e$, either end of $f$ is different from $u$ or $v$. Let $S$ be the collection of these choices. Thus $|S| \leq|F|-1=\kappa^{\prime}(G)-1$, and $G-S$ does not contain edges from $F \backslash\{e\}$.
If $G-S$ is disconnected, then $S$ is a separating set and so $\kappa(G) \leq|S| \leq \kappa^{\prime}(G)-1$ and we are done. On the other hand, if $G-S$ is connected, then either $G-S=K_{2}(=e)$, or either $u$ or $v$ (or both) is a cut vertex of $G-S$ (since $H-S=G-S$, and therefore $G-S \subseteq H$ is an induced subgraph of $H$ ). In both of these cases, there is a vertex of $G-S$, whose removal results in a trivial or a disconnected graph. In conclusion, $\kappa(G) \leq|S|+1 \leq \kappa^{\prime}(G)$, and the claim follows.
## Menger's theorem
Theorem 2.7 (MENGER (1927)). Let $u, v \in G$ be nonadjacent vertices of a connected graph $G$. Then the minimum number of vertices separating $u$ and $v$ is equal to the maximum number of independent paths from $u$ to $v$.
Proof. If a subset $S \subseteq V_{G}$ is $(u, v)$-separating, then every path $u \stackrel{\star}{\rightarrow} v$ of $G$ visits $S$. Hence $|S|$ is at least the number of independent paths from $u$ to $v$. Conversely, we use induction on $m=v_{G}+\varepsilon_{G}$ to show that if $S=\left\{w_{1}, w_{2}, \ldots, w_{k}\right\}$ is a $(u, v)$-separating set of the smallest size, then $G$ has at least (and thus exactly) $k$ independent paths $u \stackrel{\star}{\rightarrow} v$.
The case for $k=1$ is clear, and this takes care of the small values of $m$, required for the induction.
(1) Assume first that $u$ and $v$ have a common neighbour $w \in N_{G}(u) \cap N_{G}(v)$. Then necessarily $w \in S$. In the smaller graph $G-w$ the set $S \backslash\{w\}$ is a minimum $(u, v)$ separating set, and the induction hypothesis yields that there are $k-1$ independent paths $u \stackrel{\star}{\rightarrow} v$ in $G-w$. Together with the path $u \rightarrow w \rightarrow v$, there are $k$ independent paths $u \stackrel{\star}{\rightarrow} v$ in $G$ as required.
(2) Assume then that $N_{G}(u) \cap N_{G}(v)=\varnothing$, and denote by $H_{u}=N_{G-S}^{*}(u)$ and $H_{v}=N_{G-S}^{*}(v)$ the connected components of $G-S$ for $u$ and $v$.
(2.1) Suppose next that $S \nsubseteq N_{G}(u)$ and $S \nsubseteq N_{G}(v)$.
Let $\widehat{v}$ be a new vertex, and define $G_{u}$ to be the graph on $H_{u} \cup S \cup\{\widehat{v}\}$ having the edges of $G\left[H_{u} \cup S\right]$ together with $\widehat{v} w_{i}$ for all $i \in[1, k]$. The graph $G_{u}$ is connected and it is smaller than G. Indeed, in order for $S$ to be a minimum separating set, all $w_{i} \in S$ have to be adjacent to some vertex in $H_{v}$. This shows that $\varepsilon_{G_{u}} \leq \varepsilon_{G}$, and, moreover, the assumption (2.1) rules out the case $H_{v}=\{v\}$. So $\left|H_{v}\right| \geq 2$ and $v_{G_{u}}<v_{G}$.
If $S^{\prime}$ is any $(u, \widehat{v})$-separating set of $G_{u}$, then $S^{\prime}$ will separate $u$ from all $w_{i} \in S \backslash S^{\prime}$ in $G$. This means that $S^{\prime}$ separates $u$ and $v$ in G. Since $k$ is the size of a minimum $(u, v)$ separating set, we have $\left|S^{\prime}\right| \geq k$. We noted that $G_{u}$ is smaller than $G$, and thus by the induction hypothesis, there are $k$ independent paths $u \stackrel{\star}{\rightarrow} \widehat{v}$ in $G_{u}$. This is possible only if there exist $k$ paths $u \stackrel{\star}{\rightarrow} w_{i}$, one for each $i \in[1, k]$, that have only the end $u$ in common.
By the present assumption, also $u$ is nonadjacent to some vertex of $S$. A symmetric argument applies to the graph $G_{v}$ (with a new vertex $\widehat{u}$ ), which is defined similarly to $G_{u}$. This yields that there are $k$ paths $w_{i} \stackrel{\star}{\rightarrow} v$ that have only the end $v$ in common. When we combine these with the above paths $u \stackrel{\star}{\rightarrow} w_{i}$, we obtain $k$ independent paths $u \stackrel{\star}{\rightarrow} w_{i} \stackrel{\star}{\rightarrow} v$ in $G$.
(2.2) There remains the case, where for all $(u, v)$-separating sets $S$ of $k$ elements, either $S \subseteq N_{G}(u)$ or $S \subseteq N_{G}(v)$. (Note that then, by (2), $S \cap N_{G}(v)=\varnothing$ or $S \cap$ $N_{G}(u)=\varnothing$.)
Let $P=e f Q$ be a shortest path $u \stackrel{\star}{\rightarrow} v$ in $G$, where $e=u x, f=x y$, and $Q: y \stackrel{\star}{\rightarrow} v$. Notice that, by the assumption (2), $|P| \geq 3$, and so $y \neq v$. In the smaller graph $G-f$, let $S^{\prime}$ be a minimum set that separates $u$ and $v$.
If $\left|S^{\prime}\right| \geq k$, then, by the induction hypothesis, there are $k$ independent paths $u \stackrel{\star}{\rightarrow} v$ in $G-f$. But these are paths of $G$, and the claim is clear in this case. If, on the other hand, $\left|S^{\prime}\right|<k$, then $u$ and $v$ are still connected in $G-S^{\prime}$. Every path $u \stackrel{\star}{\rightarrow} v$ in $G-S^{\prime}$ necessarily travels along the edge $f=x y$, and so $x, y \notin S^{\prime}$. Let
$$
S_{x}=S^{\prime} \cup\{x\} \quad \text { and } \quad S_{y}=S^{\prime} \cup\{y\} .
$$
These sets separate $u$ and $v$ in $G$ (by the above fact), and they have size $k$. By our current assumption, the vertices of $S_{y}$ are adjacent to $v$, since the path $P$ is shortest and so $u y \notin G$ (meaning that $u$ is not adjacent to all of $S_{y}$ ). The assumption (2) yields that $u$ is adjacent to all of $S_{x}$, since $u x \in G$. But now both $u$ and $v$ are adjacent to the vertices of $S^{\prime}$, which contradicts the assumption (2).
Theorem 2.8 (MENGER (1927)). A graph $G$ is $k$-connected if and only if every two vertices are connected by at least $k$ independent paths.
Proof. If any two vertices are connected by $k$ independent paths, then it is clear that $k(G) \geq k$.
In converse, suppose that $\kappa(G)=k$, but that $G$ has vertices $u$ and $v$ connected by at most $k-1$ independent paths. By Theorem 2.7, it must be that $e=u v \in G$. Consider the graph $G-e$. Now $u$ and $v$ are connected by at most $k-2$ independent paths in $G-e$, and by Theorem 2.7, $u$ and $v$ can be separated in $G-e$ by a set $S$ with $|S|=k-2$. Since $v_{G}>k$ (because $\kappa(G)=k$ ), there exists a $w \in G$ that is not in $S \cup\{u, v\}$. The vertex $w$ is separated in $G-e$ by $S$ from $u$ or from $v$; otherwise there would be a path $u \stackrel{\star}{\rightarrow} v$ in $(G-e)-S$. Say, this vertex is $u$. The set $S \cup\{v\}$ has $k-1$ elements, and it separates $u$ from $w$ in $G$, which contradicts the assumption that $\kappa(G)=k$. This proves the claim.
We state without a proof the corresponding separation property for edge connectivity.
DEFINition. Let $G$ be a graph. A $u v$-disconnecting set is a set $F \subseteq E_{G}$ such that every path $u \stackrel{\star}{\rightarrow} v$ contains an edge from $F$.
Theorem 2.9. Let $u, v \in G$ with $u \neq v$ in a graph $G$. Then the maximum number of edgedisjoint paths $u \stackrel{\star}{\rightarrow} v$ equals the minimum number $k$ of edges in a uv-disconnecting set.
Corollary 2.4. A graph $G$ is k-edge connected if and only if every two vertices are connected by at least $k$ edge disjoint paths.
Example 2.9. Recall the definition of the cube $Q_{k}$ from Example 1.5. We show that $\kappa\left(Q_{k}\right)=k$.
First of all, $\kappa\left(Q_{k}\right) \leq \delta\left(Q_{k}\right)=k$. In converse, we show the claim by induction. Extract from $Q_{k}$ the disjoint subgraphs: $G_{0}$ induced by $\left\{0 u \mid u \in \mathbb{B}^{k-1}\right\}$ and $G_{1}$ induced by $\left\{1 u \mid u \in \mathbb{B}^{k-1}\right\}$. These are (isomorphic to) $Q_{k-1}$, and $Q_{k}$ is obtained from the union of $G_{0}$ and $G_{1}$ by adding the $2^{k-1}$ edges $(0 u, 1 u)$ for all $u \in \mathbb{B}^{k-1}$. Let $S$ be a separating set of $Q_{k}$ with $|S| \leq k$. If both $G_{0}-S$ and $G_{1}-S$ were connected, also $Q_{k}-S$ would be connected, since one pair $(0 u, 1 u)$ necessarily remains in $Q_{k}-S$. So we can assume that $G_{0}-S$ is disconnected. (The case for $G_{1}-S$ is symmetric.) By the induction hypothesis, $\kappa\left(G_{0}\right)=k-1$, and hence $S$ contains at least $k-1$ vertices of $G_{0}$ (and so $|S| \geq k-1$ ). If there were no vertices from $G_{1}$ in $S$, then, of course, $G_{1}-S$ is connected, and the edges $(0 u, 1 u)$ of $Q_{k}$ would guarantee that $Q_{k}-S$ is connected; a contradiction. Hence $|S| \geq k$.
Example 2.10. We have $\kappa^{\prime}\left(Q_{k}\right)=k$ for the $k$-cube. Indeed, by Whitney's theorem, $\kappa(G) \leq \kappa^{\prime}(G) \leq \delta(G)$. Since $\kappa\left(Q_{k}\right)=k=\delta\left(Q_{k}\right)$, also $\kappa^{\prime}\left(Q_{k}\right)=k$.
Algorithmic Problem. The connectivity problems tend to be algorithmically difficult. In the disjoint paths problem we are given a set $\left(u_{i}, v_{i}\right)$ of pairs of vertices for $i=$ $1,2, \ldots, k$, and it is asked whether there exist paths $P_{i}: u_{i} \stackrel{\star}{\rightarrow} v_{i}$ that have no vertices in common. This problem was shown to be NP-complete by KNUTH in 1975. (However, for fixed $k$, the problem has a fast algorithm due to ROBERTSON and SEYMOUR (1986).)
## Dirac's fans
Definition. Let $v \in G$ and $S \subseteq V_{G}$ such that $v \notin S$ in a graph $G$. A set of paths from $v$ to a vertex in $S$ is called a $(v, S)$-fan, if they have only $v$ in common.
Theorem 2.10 (DIRAC (1960)). A graph $G$ is k-connected if and only if $v_{G}>k$ and for every $v \in G$ and $S \subseteq V_{G}$ with $|S| \geq k$ and $v \notin S$, there exists $a(v, S)$-fan of $k$ paths.
Proof. Exercise.
Theorem 2.11 (DIRAC (1960)). Let $G$ be a $k$-connected graph for $k \geq 2$. Then for any $k$ vertices, there exists a cycle of $G$ containing them.
Proof. First of all, since $\kappa(G) \geq 2, G$ has no cut vertices, and thus no bridges. It follows that every edge, and thus every vertex of $G$ belongs to a cycle.
Let $S \subseteq V_{G}$ be such that $|S|=k$, and let $C$ be a cycle of $G$ that contains the maximum number of vertices of $S$. Let the vertices of $S \cap V_{\mathrm{C}}$ be $v_{1}, \ldots, v_{r}$ listed in order around $C$ so that each pair $\left(v_{i}, v_{i+1}\right)$ (with indices modulo $r$ ) defines a path along $C$ (except in the special case where $r=1$ ). Such a path is referred to as a segment of $C$. If $C$ contains all vertices of $S$, then we are done; otherwise, suppose $v \in S$ is not on $C$.
It follows from Theorem 2.10 that there is a $\left(v, V_{C}\right)$-fan of at least $\min \left\{k,\left|V_{C}\right|\right\}$ paths. Therefore there are two paths $P: v \stackrel{\star}{\rightarrow} u$ and $Q: v \stackrel{\star}{\rightarrow} w$ in such a fan that end in the same segment $\left(v_{i}, v_{i+1}\right)$ of $C$. Then the path $W: u \stackrel{\star}{\rightarrow} w$ (or $w \stackrel{\star}{\rightarrow} u$ ) along $C$ contains all vertices of $S \cap V_{C}$. But now $P W Q^{-1}$ is a cycle of $G$ that contains $v$ and all $v_{i}$ for $i \in[1, r]$. This contradicts the choice of $C$, and proves the claim.
## Tours and Matchings
### Eulerian graphs
The first proper problem in graph theory was the Königsberg bridge problem. In general, this problem concerns of travelling in a graph such that one tries to avoid using any edge twice. In practice these eulerian problems occur, for instance, in optimizing distribution networks - such as delivering mail, where in order to save time each street should be travelled only once. The same problem occurs in mechanical graph plotting, where one avoids lifting the pen off the paper while drawing the lines.
## Euler tours
Definition. A walk $W=e_{1} e_{2} \ldots e_{n}$ is a trail, if $e_{i} \neq e_{j}$ for all $i \neq j$. An Euler trail of a graph $G$ is a trail that visits every edge once. A connected graph $G$ is eulerian, if it has a closed trail containing every edge of $G$. Such a trail is called an Euler tour.
Notice that if $W=e_{1} e_{2} \ldots e_{n}$ is an Euler tour (and so $E_{G}=\left\{e_{1}, e_{2}, \ldots, e_{n}\right\}$ ), also $e_{i} e_{i+1} \ldots e_{n} e_{1} \ldots e_{i-1}$ is an Euler tour for all $i \in[1, n]$. A complete proof of the following Euler's Theorem was first given by HIERHOLZER in 1873.
Theorem 3.1 (EUler (1736), HierHOlzer (1873)). A connected graph $G$ is eulerian if and only if every vertex has an even degree.
Proof. $(\Rightarrow)$ Suppose $W: u \stackrel{\star}{\rightarrow} u$ is an Euler tour. Let $v(\neq u)$ be a vertex that occurs $k$ times in $W$. Every time an edge arrives at $v$, another edge departs from $v$, and therefore $d_{G}(v)=2 k$. Also, $d_{G}(u)$ is even, since $W$ starts and ends at $u$.
$(\Leftarrow)$ Assume $G$ is a nontrivial connected graph such that $d_{G}(v)$ is even for all $v \in$ G. Let
$$
W=e_{1} e_{2} \ldots e_{n}: v_{0} \stackrel{\star}{\rightarrow} v_{n} \quad \text { with } \quad e_{i}=v_{i-1} v_{i}
$$
be a longest trail in $G$. It follows that all $e=v_{n} w \in G$ are among the edges of $W$, for, otherwise, $W$ could be prolonged to $W e$. In particular, $v_{0}=v_{n}$, that is, $W$ is a closed trail. (Indeed, if it were $v_{n} \neq v_{0}$ and $v_{n}$ occurs $k$ times in $W$, then $d_{G}\left(v_{n}\right)=2(k-1)+1$ and that would be odd.)
If $W$ is not an Euler tour, then, since $G$ is connected, there exists an edge $f=v_{i} u \in$ $G$ for some $i$, which is not in $W$. However, now
$$
e_{i+1} \ldots e_{n} e_{1} \ldots e_{i} f
$$
is a trail in $G$, and it is longer than $W$. This contradiction to the choice of $W$ proves the claim. Example 3.1. The $k$-cube $Q_{k}$ is eulerian for even integers $k$, because $Q_{k}$ is $k$-regular.
Theorem 3.2. A connected graph has an Euler trail if and only if it has at most two vertices of odd degree.
Proof. If $G$ has an Euler trail $u \stackrel{\star}{\rightarrow} v$, then, as in the proof of Theorem 3.1, each vertex $w \notin\{u, v\}$ has an even degree.
Assume then that $G$ is connected and has at most two vertices of odd degree. If $G$ has no vertices of odd degree then, by Theorem 3.1, $G$ has an Euler trail. Otherwise, by the handshaking lemma, every graph has an even number of vertices with odd degree, and therefore $G$ has exactly two such vertices, say $u$ and $v$. Let $H$ be a graph obtained from $G$ by adding a vertex $w$, and the edges $u w$ and $v w$. In $H$ every vertex has an even degree, and hence it has an Euler tour, say $u \stackrel{\star}{\rightarrow} v \rightarrow w \rightarrow u$. Here the beginning part $u \stackrel{\star}{\rightarrow} v$ is an Euler trail of $G$.
## The Chinese postman
The following problem is due to GuAN MeIGu (1962). Consider a village, where a postman wishes to plan his route to save the legs, but still every street has to be walked through. This problem is akin to Euler's problem and to the shortest path problem.
Let $G$ be a graph with a weight function $\alpha: E_{G} \rightarrow \mathbb{R}^{+}$. The Chinese postman problem is to find a minimum weighted tour in $G$ (starting from a given vertex, the post office).
If $G$ is eulerian, then any Euler tour will do as a solution, because such a tour traverses each edge exactly once and this is the best one can do. In this case the weight of the optimal tour is the total weight of the graph $G$, and there is a good algorithm for finding such a tour:
## Fleury's algorithm:
- Let $v_{0} \in G$ be a chosen vertex, and let $W_{0}$ be the trivial path on $v_{0}$.
- Repeat the following procedure for $i=1,2, \ldots$ as long as possible: suppose a trail $W_{i}=e_{1} e_{2} \ldots e_{i}$ has been constructed, where $e_{j}=v_{j-1} v_{j}$.
Choose an edge $e_{i+1}\left(\neq e_{j}\right.$ for $\left.j \in[1, i]\right)$ so that
(i) $e_{i+1}$ has an end $v_{i}$, and
(ii) $e_{i+1}$ is not a bridge of $G_{i}=G-\left\{e_{1}, \ldots, e_{i}\right\}$, unless there is no alternative.
Notice that, as is natural, the weights $\alpha(e)$ play no role in the eulerian case.
Theorem 3.3. If $G$ is eulerian, then any trail of $G$ constructed by Fleury's algorithm is an Euler tour of $G$.
Proof. Exercise. If $G$ is not eulerian, the poor postman has to walk at least one street twice. This happens, e.g., if one of the streets is a dead end, and in general if there is a street corner of an odd number of streets. We can attack this case by reducing it to the eulerian case as follows. An edge $e=u v$ will be duplicated, if it is added to $G$ parallel to an existing edge $e^{\prime}=u v$ with the same weight, $\alpha\left(e^{\prime}\right)=\alpha(e)$.
Above we have duplicated two edges. The rightmost multigraph is eulerian.
There is a good algorithm by EDMONDS AND JOHNSON (1973) for the construction of an optimal eulerian supergraph by duplications. Unfortunately, this algorithm is somewhat complicated, and we shall skip it.
### Hamiltonian graphs
In the connector problem we reduced the cost of a spanning graph to its minimum. There are different problems, where the cost is measured by an active user of the graph. For instance, in the travelling salesman problem a person is supposed to visit each town in his district, and this he should do in such a way that saves time and money. Obviously, he should plan the travel so as to visit each town once, and so that the overall flight time is as short as possible. In terms of graphs, he is looking for a minimum weighted Hamilton cycle of a graph, the vertices of which are the towns and the weights on the edges are the flight times. Unlike for the shortest path and the connector problems no efficient reliable algorithm is known for the travelling salesman problem. Indeed, it is widely believed that no practical algorithm exists for this problem.
## Hamilton cycles
DEFINITION. A path $P$ of a graph $G$ is a Hamilton path, if $P$ visits every vertex of $G$ once. Similarly, a cycle $C$ is a Hamilton cycle, if it visits each vertex once. A graph is hamiltonian, if it has a Hamilton cycle.
Note that if $C: u_{1} \rightarrow u_{2} \rightarrow \cdots \rightarrow u_{n}$ is a Hamilton cycle, then so is $u_{i} \rightarrow \ldots u_{n} \rightarrow$ $u_{1} \rightarrow \ldots u_{i-1}$ for each $i \in[1, n]$, and thus we can choose where to start the cycle.
Example 3.2. It is obvious that each $K_{n}$ is hamiltonian whenever $n \geq 3$. Also, as is easily seen, $K_{n, m}$ is hamiltonian if and only if $n=m \geq 2$. Indeed, let $K_{n, m}$ have a bipartition $(X, Y)$, where $|X|=n$ and $|Y|=m$. Now, each cycle in $K_{n, m}$ has even length as the graph is bipartite, and thus the cycle visits the sets $X, Y$ equally many times, since $X$ and $Y$ are stable subsets. But then necessarily $|X|=|Y|$.
Unlike for eulerian graphs (Theorem 3.1) no good characterization is known for hamiltonian graphs. Indeed, the problem to determine if $G$ is hamiltonian is NPcomplete. There are, however, some interesting general conditions.
Lemma 3.1. If $G$ is hamiltonian, then for every nonempty subset $S \subseteq V_{G}$,
$$
c(G-S) \leq|S| .
$$
Proof. Let $\varnothing \neq S \subseteq V_{G}, u \in S$, and let $C: u^{\star} \stackrel{u}{\rightarrow}$ be a Hamilton cycle of $G$. Assume $G-S$ has k connected components, $G_{i}, i \in[1, k]$. The case $k=1$ is trivial, and hence suppose that $k>1$. Let $u_{i}$ be the last vertex of $C$ that belongs to $G_{i}$, and let $v_{i}$ be the vertex that follows $u_{i}$ in $C$. Now $v_{i} \in S$ for each $i$ by the choice of $u_{i}$, and $v_{j} \neq v_{t}$ for all $j \neq t$, because $C$ is a cycle and $u_{i} v_{i} \in G$ for all $i$. Thus $|S| \geq k$ as required.
Example 3.3. Consider the graph on the right. In $G$, $c(G-S)=3>2=|S|$ for the set $S$ of black vertices. Therefore $G$ does not satisfy the condition of Lemma 3.1, and hence it is not hamiltonian. Interestingly this graph is $(X, Y)$-bipartite of even order with $|X|=|Y|$. It is also 3-regular.
Example 3.4. Consider the Petersen graph on the right,
which appears in many places in graph theory as a counter example for various conditions. This graph is not hamiltonian, but it does satisfy the condition $c(G-S) \leq|S|$ for all $S \neq \varnothing$. Therefore the conclusion of Lemma 3.1 is not sufficient to ensure that a graph is hamiltonian.
The following theorem, due to ORE, generalizes an earlier result by DIRAC (1952).
Theorem 3.4 (OrE (1962)). Let $G$ be a graph of order $v_{G} \geq 3$, and let $u, v \in G$ be such that
$$
d_{G}(u)+d_{G}(v) \geq v_{G} .
$$
Then $G$ is hamiltonian if and only if $G+u v$ is hamiltonian.
Proof. Denote $n=v_{G}$. Let $u, v \in G$ be such that $d_{G}(u)+d_{G}(v) \geq n$. If $u v \in G$, then there is nothing to prove. Assume thus that $u v \notin G$.
$(\Rightarrow)$ This is trivial since if $G$ has a Hamilton cycle $C$, then $C$ is also a Hamilton cycle of $G+u v$.
$(\Leftarrow)$ Denote $e=u v$ and suppose that $G+e$ has a Hamilton cycle $C$. If $C$ does not use the edge $e$, then it is a Hamilton cycle of $G$. Suppose thus that $e$ is on $C$. We may then assume that $C: u \stackrel{\star}{\rightarrow} v \rightarrow u$. Now $u=v_{1} \rightarrow v_{2} \rightarrow \ldots \rightarrow v_{n}=v$ is a Hamilton path of $G$. There exists an $i$ with $1<i<n$ such that $u v_{i} \in G$ and $v_{i-1} v \in G$. For, otherwise, $d_{G}(v)<n-d_{G}(u)$ would contradict the assumption.
$$
\overbrace{v_{1}-v_{2}-\circ-\circ-v_{i-1}-v_{i}-\circ-\circ-v_{n}}
$$
But now $u=v_{1} \stackrel{\star}{\rightarrow} v_{i-1} \rightarrow v_{n} \rightarrow v_{n-1} \stackrel{\star}{\rightarrow} v_{i+1} \rightarrow v_{i} \rightarrow v_{1}=u$ is a Hamilton cycle in G.
## Closure
DEFINition. For a graph $G$, define inductively a sequence $G_{0}, G_{1}, \ldots, G_{k}$ of graphs such that
$$
G_{0}=G \text { and } G_{i+1}=G_{i}+u v,
$$
where $u$ and $v$ are any vertices such that $u v \notin G_{i}$ and $d_{G_{i}}(u)+d_{G_{i}}(v) \geq v_{G}$. This procedure stops when no new edges can be added to $G_{k}$ for some $k$, that is, in $G_{k}$, for all $u, v \in G$ either $u v \in G_{k}$ or $d_{G_{k}}(u)+d_{G_{k}}(v)<v_{G}$. The result of this procedure is the closure of $G$, and it is denoted by $\operatorname{cl}(G)\left(=G_{k}\right)$.
In each step of the construction of $c l(G)$ there are usually alternatives which edge $u v$ is to be added to the graph, and therefore the above procedure is not deterministic. However, the final result $\operatorname{cl}(G)$ is independent of the choices.
Lemma 3.2. The closure $\mathrm{cl}(G)$ is uniquely defined for all graphs $G$ of order $v_{G} \geq 3$.
Proof. Denote $n=v_{G}$. Suppose there are two ways to close $G$, say
$$
H=G+\left\{e_{1}, \ldots, e_{r}\right\} \text { and } H^{\prime}=G+\left\{f_{1}, \ldots, f_{s}\right\},
$$
where the edges are added in the given orders. Let $H_{i}=G+\left\{e_{1}, \ldots, e_{i}\right\}$ and $H_{i}^{\prime}=$ $G+\left\{f_{1}, \ldots, f_{i}\right\}$. For the initial values, we have $G=H_{0}=H_{0}^{\prime}$. Let $e_{k}=u v$ be the first edge such that $e_{k} \neq f_{i}$ for all $i$. Then $d_{H_{k-1}}(u)+d_{H_{k-1}}(v) \geq n$, since $e_{k} \in H_{k}$, but $e_{k} \notin H_{k-1}$. By the choice of $e_{k}$, we have $H_{k-1} \subseteq H^{\prime}$, and thus also $d_{H^{\prime}}(u)+$ $d_{H^{\prime}}(v) \geq n$, which means that $e=u v$ must be in $H^{\prime}$; a contradiction. Therefore $H \subseteq$ $H^{\prime}$. Symmetrically, we deduce that $H^{\prime} \subseteq H$, and hence $H^{\prime}=H$.
Theorem 3.5. Let $G$ be a graph of order $v_{G} \geq 3$.
(i) $G$ is hamiltonian if and only if its closure $\mathrm{cl}(G)$ is hamiltonian.
(ii) If $\mathrm{cl}(G)$ is a complete graph, then $G$ is hamiltonian.
Proof. First, $G \subseteq c l(G)$ and $G$ spans $\operatorname{cl}(G)$, and thus if $G$ is hamiltonian, so is $c l(G)$.
In the other direction, let $G=G_{0}, G_{1}, \ldots, G_{k}=c l(G)$ be a construction sequence of the closure of $G$. If $c l(G)$ is hamiltonian, then so are $G_{k-1}, \ldots, G_{1}$ and $G_{0}$ by Theorem 3.4.
The Claim (ii) follows from (i), since each complete graph is hamiltonian. Theorem 3.6. Let $G$ be a graph of order $v_{G} \geq 3$. Suppose that for all nonadjacent vertices $u$ and $v, d_{G}(u)+d_{G}(v) \geq v_{G}$. Then $G$ is hamiltonian. In particular, if $\delta(G) \geq \frac{1}{2} v_{G}$, then $G$ is hamiltonian.
Proof. Since $d_{G}(u)+d_{G}(v) \geq v_{G}$ for all nonadjacent vertices, we have $c l(G)=K_{n}$ for $n=v_{G}$, and thus $G$ is hamiltonian. The second claim is immediate, since now $d_{G}(u)+d_{G}(v) \geq v_{G}$ for all $u, v \in G$ whether adjacent or not.
## Chvátal's condition
The hamiltonian problem of graphs has attracted much attention, at least partly because the problem has practical significance. (Indeed, the first example where DNA computing was applied, was the hamiltonian problem.)
There are some general improvements of the previous results of this chapter, and quite many improvements in various special cases, where the graphs are somehow restricted. We become satisfied by two general results.
Theorem 3.7 (CHVÁTAL (1972)). Let $G$ be a graph with $V_{G}=\left\{v_{1}, v_{2}, \ldots, v_{n}\right\}$, for $n \geq 3$, ordered so that $d_{1} \leq d_{2} \leq \cdots \leq d_{n}$, for $d_{i}=d_{G}\left(v_{i}\right)$. If for every $i<n / 2$,
$$
d_{i} \leq i \Longrightarrow d_{n-i} \geq n-i
$$
then $G$ is hamiltonian.
Proof. First of all, we may suppose that $G$ is closed, $G=c l(G)$, because $G$ is hamiltonian if and only if $\mathrm{cl}(G)$ is hamiltonian, and adding edges to $G$ does not decrease any of its degrees, that is, if $G$ satisfies (3.1), so does $G+e$ for every $e$. We show that, in this case, $G=K_{n}$, and thus $G$ is hamiltonian.
Assume on the contrary that $G \neq K_{n}$, and let $u v \notin G$ with $d_{G}(u) \leq d_{G}(v)$ be such that $d_{G}(u)+d_{G}(v)$ is as large as possible. Because $G$ is closed, we must have $d_{G}(u)+d_{G}(v)<n$, and therefore $d_{G}(u)=i<n / 2$. Let $A=\{w \mid v w \notin G, w \neq v\}$. By our choice, $d_{G}(w) \leq i$ for all $w \in A$, and, moreover,
$$
|A|=(n-1)-d_{G}(v) \geq d_{G}(u)=i .
$$
Consequently, there are at least $i$ vertices $w$ with $d_{G}(w) \leq i$, and so $d_{i} \leq d_{G}(u)=i$.
Similarly, for each vertex from $B=\{w \mid u w \notin G, w \neq u\}, d_{G}(w) \leq d_{G}(v)<$ $n-d_{G}(u)=n-i$, and
$$
|B|=(n-1)-d_{G}(u)=(n-1)-i .
$$
Also $d_{G}(u)<n-i$, and thus there are at least $n-i$ vertices $w$ with $d_{G}(w)<n-$ $i$. Consequently, $d_{n-i}<n-i$. This contradicts the obtained bound $d_{i} \leq i$ and the condition (3.1).
Note that the condition (3.1) is easily checkable for any given graph.
### Matchings
In matching problems we are given an availability relation between the elements of a set. The problem is then to find a pairing of the elements so that each element is paired (matched) uniquely with an available companion.
A special case of the matching problem is the marriage problem, which is stated as follows. Given a set $X$ of boys and a set $Y$ of girls, under what condition can each boy marry a girl who cares to marry him? This problem has many variations. One of them is the job assignment problem, where we are given $n$ applicants and $m$ jobs, and we should assign each applicant to a job he is qualified. The problem is that an applicant may be qualified for several jobs, and a job may be suited for several applicants.
## Maximum matchings
DEFINition. For a graph $G$, a subset $M \subseteq E_{G}$ is a matching of $G$, if $M$ contains no adjacent edges. The two ends of an edge $e \in M$ are matched under $M$. A matching $M$ is a maximum matching, if for no matching $M^{\prime},|M|<\left|M^{\prime}\right|$.
The two vertical edges on the right constitute a matching $M$ that is not a maximum matching, although you cannot add any edges to $M$ to form a larger matching. This matching is not maximum because the graph has a matching of three edges.
DEFINition. A matching $M$ saturates $v \in G$, if $v$ is an end of an edge in $M$. Also, $M$ saturates $A \subseteq V_{G}$, if it saturates every $v \in A$. If $M$ saturates $V_{G}$, then $M$ is a perfect matching.
It is clear that every perfect matching is maximum.
On the right the horizontal edges form a perfect matching.
DEFINITION. Let $M$ be a matching of $G$. An odd path $P=e_{1} e_{2} \ldots e_{2 k+1}$ is $M$-augmented, if
- $P$ alternates between $E_{G} \backslash M$ and $M$
(that is, $e_{2 i+1} \in G-M$ and $e_{2 i} \in M$ ), and
- the ends of $P$ are not saturated.
Lemma 3.3. If $G$ is connected with $\Delta(G) \leq 2$, then $G$ is a path or a cycle.
Proof. Exercise.
We start with a result that gives a necessary and sufficient condition for a matching to be maximum. One can use the first part of the proof to construct a maximum matching in an iterative manner starting from any matching $M$ and from any $M$ augmented path.
Theorem 3.8 (BERGE (1957)). A matching $M$ of $G$ is a maximum matching if and only if there are no M-augmented paths in $G$.
Proof. $(\Rightarrow)$ Let a matching $M$ have an $M$-augmented path $P=e_{1} e_{2} \ldots e_{2 k+1}$ in $G$. Here $e_{2}, e_{4}, \ldots, e_{2 k} \in M, e_{1}, e_{3}, \ldots, e_{2 k+1} \notin M$. Define $N \subseteq E_{G}$ by
$$
N=\left(M \backslash\left\{e_{2 i} \mid i \in[1, k]\right\}\right) \cup\left\{e_{2 i+1} \mid i \in[0, k]\right\} .
$$
Now, $N$ is a matching of $G$, and $|N|=|M|+1$. Therefore $M$ is not a maximum matching.
$(\Leftarrow)$ Assume $N$ is a maximum matching, but $M$ is not. Hence $|N|>|M|$. Consider the subgraph $H=G[M \triangle N]$ for the symmetric difference $M \triangle N$. We have $d_{H}(v) \leq$ 2 for each $v \in H$, because $v$ is an end of at most one edge in $M$ and $N$. By Lemma 3.3, each connected component $A$ of $H$ is either a path or a cycle.
Since no $v \in A$ can be an end of two edges from $N$ or from $M$, each connected component (path or a cycle) $A$ alternates between $N$ and $M$. Now, since $|N|>|M|$, there is a connected component $A$ of $H$, which has more edges from $N$ than from $M$. This $A$ cannot be a cycle, because an alternating cycle has even length, and it thus contains equally many edges from $N$ and $M$. Hence $A: u \stackrel{\star}{\rightarrow} v$ is a path (of odd length), which starts and ends with an edge from $N$. Because $A$ is a connected component of $H$, the ends $u$ and $v$ are not saturated by $M$, and, consequently, $A$ is an $M$-augmented path. This proves the theorem.
Example 3.5. Consider the $k$-cube $Q_{k}$ for $k \geq 1$. Each maximum matching of $Q_{k}$ has $2^{k-1}$ edges. Indeed, the matching $M=\left\{(0 u, 1 u) \mid u \in \mathbb{B}^{k-1}\right\}$, has $2^{k-1}$ edges, and it is clearly perfect.
## Hall's theorem
For a subset $S \subseteq V_{G}$ of a graph $G$, denote
$$
N_{G}(S)=\{v \mid u v \in G \text { for some } u \in S\} .
$$
If $G$ is $(X, Y)$-bipartite, and $S \subseteq X$, then $N_{G}(S) \subseteq Y$.
The following result, known as the
Theorem 3.9 (HALL (1935)). Let G be a (X, Y)-bipartite graph. Then G contains a matching $M$ saturating $X$ if and only if
$$
|S| \leq\left|N_{G}(S)\right| \quad \text { for all } S \subseteq X .
$$
Proof. $(\Rightarrow)$ Let $M$ be a matching that saturates $X$. If $|S|>\left|N_{G}(S)\right|$ for some $S \subseteq X$, then not all $x \in S$ can be matched with different $y \in N_{G}(S)$.
$(\Leftarrow)$ Let $G$ satisfy Hall's condition (3.2). We prove the claim by induction on $|X|$.
If $|X|=1$, then the claim is clear. Let then $|X| \geq 2$, and assume (3.2) implies the existence of a matching that saturates every proper subset of $X$.
If $\left|N_{G}(S)\right| \geq|S|+1$ for every nonempty $S \subseteq X$ with $S \neq X$, then choose an edge $u v \in G$ with $u \in X$, and consider the induced subgraph $H=G-\{u, v\}$. For all $S \subseteq X \backslash\{u\},\left|N_{H}(S)\right| \geq\left|N_{G}(S)\right|-1 \geq|S|$, and hence, by the induction hypothesis, $H$ contains a matching $M$ saturating $X \backslash\{u\}$. Now $M \cup\{u v\}$ is a matching saturating $X$ in $G$, as was required.
Suppose then that there exists a nonempty subset $R \subseteq X$ with $R \neq X$ such that $\left|N_{G}(R)\right|=|R|$. The induced subgraph $H_{1}=G\left[R \cup N_{G}(R)\right]$ satisfies (3.2) (since $G$ does), and hence, by the induction hypothesis, $H_{1}$ contains a matching $M_{1}$ that saturates $R$ (with the other ends in $N_{G}(R)$ ).
Also, the induced subgraph $H_{2}=G\left[V_{G} \backslash A\right]$, for $A=R \cup N_{G}(R)$, satisfies (3.2). Indeed, if there were a subset $S \subseteq X \backslash R$ such that $\left|N_{H_{2}}(S)\right|<|S|$, then we would have
$$
\left|N_{G}(S \cup R)\right|=\left|N_{H_{2}}(S)\right|+\left|N_{H_{1}}(R)\right|<|S|+\left|N_{G}(R)\right|=|S|+|R|=|S \cup R|
$$
(since $S \cap R=\varnothing$ ), which contradicts (3.2) for G. By the induction hypothesis, $H_{2}$ has a matching $M_{2}$ that saturates $X \backslash R$ (with the other ends in $Y \backslash N_{G}(R)$ ). Combining the matchings for $H_{1}$ and $H_{2}$, we get a matching $M_{1} \cup M_{2}$ saturating $X$ in $G$.
Second proof. This proof of the direction $(\Leftarrow)$ uses Menger's theorem. Let $H$ be the graph obtained from $G$ by adding two new vertices $x, y$ such that $x$ is adjacent to each $v \in X$ and $y$ is adjacent to each $v \in Y$. There exists a matching saturating $X$ if (and only if) the number of independent paths $x \stackrel{\star}{\rightarrow} y$ is equal to $|X|$. For this, by Menger's theorem, it suffices to show that every set $S$ that separates $x$ and $y$ in $H$ has at least $|X|$ vertices.
Let $S=A \cup B$, where $A \subseteq X$ and $B \subseteq Y$. Now, vertices in $X \backslash A$ are not adjacent to vertices of $Y \backslash B$, and hence we have $N_{G}(X \backslash A) \subseteq B$, and thus that $|X \backslash A| \leq\left|N_{G}(X \backslash A)\right| \leq|B|$ using the condition (3.2).
We conclude that $|S|=|A|+|B| \geq|X|$.
Corollary 3.1 (FROBENIUS (1917)). If $G$ is a k-regular bipartite graph with $k>0$, then $G$ has a perfect matching.
Proof. Let $G$ be $k$-regular $(X, Y)$-bipartite graph. By regularity, $k \cdot|X|=\varepsilon_{G}=k \cdot|Y|$, and hence $|X|=|Y|$. Let $S \subseteq X$. Denote by $E_{1}$ the set of the edges with an end in $S$, and by $E_{2}$ the set of the edges with an end in $N_{G}(S)$. Clearly, $E_{1} \subseteq E_{2}$. Therefore, $k \cdot\left|N_{G}(S)\right|=\left|E_{2}\right| \geq\left|E_{1}\right|=k \cdot|S|$, and so $\left|N_{G}(S)\right| \geq|S|$. By Theorem 3.9, G has a matching that saturates $X$. Since $|X|=|Y|$, this matching is necessarily perfect.
## Applications of Hall's theorem
DEFINITION. Let $\mathcal{S}=\left\{S_{1}, S_{2}, \ldots, S_{m}\right\}$ be a family of finite nonempty subsets of a set $S$. ( $S_{i}$ need not be distinct.) A transversal (or a system of distinct representatives) of $\mathcal{S}$ is a subset $T \subseteq S$ of $m$ distinct elements one from each $S_{i}$.
As an example, let $S=[1,6]$, and let $S_{1}=S_{2}=\{1,2\}, S_{3}=\{2,3\}$ and $S_{4}=$ $\{1,4,5,6\}$. For $\mathcal{S}=\left\{S_{1}, S_{2}, S_{3}, S_{4}\right\}$, the set $T=\{1,2,3,4\}$ is a transversal. If we add the set $S_{5}=\{2,3\}$ to $\mathcal{S}$, then it is impossible to find a transversal for this new family.
The connection of transversals to the Marriage Theorem is as follows. Let $S=Y$ and $X=[1, m]$. Form an $(X, Y)$-bipartite graph $G$ such that there is an edge $(i, s)$ if and only if $s \in S_{i}$. The possible transversals $T$ of $\mathcal{S}$ are then obtained from the matchings $M$ saturating $X$ in $G$ by taking the ends in $Y$ of the edges of $M$.
Corollary 3.2. Let $\mathcal{S}$ be a family of finite nonempty sets. Then $\mathcal{S}$ has a transversal if and only if the union of any $k$ of the subsets $S_{i}$ of $\mathcal{S}$ contains at least $k$ elements.
Example 3.6. An $m \times n$ latin rectangle is an $m \times n$ integer matrix $M$ with entries $M_{i j} \in[1, n]$ such that the entries in the same row and in the same column are different. Moreover, if $m=n$, then $M$ is a latin square. Note that in a $m \times n$ latin rectangle $M$, we always have that $m \leq n$.
We show the following: Let $M$ be an $m \times n$ latin rectangle (with $m<n$ ). Then $M$ can be extended to a latin square by the addition of $n-m$ new rows.
The claim follows when we show that $M$ can be extended to an $(m+1) \times n$ latin rectangle. Let $A_{i} \subseteq[1, n]$ be the set of those elements that do not occur in the $i$-th column of $M$. Clearly, $\left|A_{i}\right|=n-m$ for each $i$, and hence $\sum_{i \in I}\left|A_{i}\right|=|I|(n-m)$ for all subsets $I \subseteq[1, n]$. Now $\left|\cup_{i \in I} A_{i}\right| \geq|I|$, since otherwise at least one element from the union would be in more than $n-m$ of the sets $A_{i}$ with $i \in I$. However, each row has all the $n$ elements, and therefore each $i$ is missing from exactly $n-m$ columns. By Marriage Theorem, the family $\left\{A_{1}, A_{2}, \ldots, A_{n}\right\}$ has a transversal, and this transversal can be added as a new row to $M$. This proves the claim.
## Tutte's theorem
The next theorem is a classic characterization of perfect matchings.
Definition. A connected component of a graph $G$ is said to be odd (even), if it has an odd (even) number of vertices. Denote by $c_{\text {odd }}(G)$ the number of odd connected components in $G$.
Denote by $m(G)$ be the number of edges in a maximum matching of a graph $G$. Theorem 3.10 (Tutte-Berge Formula). Each maximum matching of a graph G has
$$
m(G)=\min _{S \subseteq V_{G}} \frac{v_{G}+|S|-c_{\text {odd }}(G-S)}{2}
$$
elements.
Note that the condition in (ii) includes the case, where $S=\varnothing$.
Proof. We prove the result for connected graphs. The result then follows for disconnected graphs by adding the formulas for the connected components.
We observe first that $\leq$ holds in (3.3), since, for all $S \subseteq V_{G}$,
$$
m(G) \leq|S|+m(G-S) \leq|S|+\frac{\left|V_{G} \backslash S\right|-c_{\text {odd }}(G-S)}{2}=\frac{v_{G}+|S|-c_{\text {odd }}(G-S)}{2} .
$$
Indeed, each odd component of $G-S$ must have at least one unsaturated vertex.
The proof proceeds by induction on $v_{G}$. If $v_{G}=1$, then the claim is trivial. Suppose that $v_{G} \geq 2$.
Assume first that there exists a vertex $v \in G$ such that $v$ is saturated by all maximum matchings. Then $m(G-v)=m(G)-1$. For a subset $S^{\prime} \subseteq G-v$, denote $S=S^{\prime} \cup\{v\}$. By the induction hypothesis, for all $S^{\prime} \subseteq G-v$,
$$
\begin{aligned}
m(G)-1 & \geq \frac{1}{2}\left(\left(v_{G}-1\right)+\left|S^{\prime}\right|-c_{\text {odd }}\left(G-\left(S^{\prime} \cup\{v\}\right)\right)\right) \\
& =\frac{1}{2}\left(\left(v_{G}+|S|-c_{\text {odd }}(G-S)\right)\right)-1 .
\end{aligned}
$$
The claim follows from this.
Suppose then that for each vertex $v$, there is a maximum matching that does not saturate $v$. We claim that $m(G)=\left(v_{G}-1\right) / 2$. Suppose to the contrary, and let $M$ be a maximum matching having two different unsaturated vertices $u$ and $v$, and choose $M$ so that the distance $d_{G}(u, v)$ is as small as possible. Now $d_{G}(u, v) \geq 2$, since otherwise $u v \in G$ could be added to $M$, contradicting the maximality of $M$. Let $w$ be an intermediate vertex on a shortest path $u \stackrel{\star}{\rightarrow} v$. By assumption, there exists a maximum matching $N$ that does not saturate $w$. We can choose $N$ such that the intersection $M \cap N$ is maximal. Since $d_{G}(u, w)<d_{G}(u, v)$ and $d_{G}(w, v)<d_{G}(u, v), N$ saturates both $u$ and $v$. The (maximum) matchings $N$ and $M$ leave equally many vertices unsaturated, and hence there exists another vertex $x \neq w$ saturated by $M$ but which is unsaturated by $N$. Let $e=x y \in M$. If $y$ is also unsaturated by $N$, then $N \cup\{e\}$ is a matching, contradicting maximality of $N$. It also follows that $y \neq w$. Therefore there exists an edge $e^{\prime}=y z$ in $N$, where $z \neq x$. But now $N^{\prime}=N \cup\{e\} \backslash\left\{e^{\prime}\right\}$ is a maximum matching that does not saturate $w$. However, $N \cap M \subset N^{\prime} \cap M$ contradicts the choice of $N$. Therefore, every maximum matching leaves exactly one vertex unsaturated, i.e., $m(G)=\left(v_{G}-1\right) / 2$.
In this case, for $S=\varnothing$, the right hand side of (3.3) gets value $\left(v_{G}-1\right) / 2$, and hence, by the beginning of the proof, this must be the minimum of the right hand side. For perfect matchings we have the following corollary, since for a perfect matching we have $m(G)=(1 / 2) v_{G}$.
Theorem 3.11 (TUTTE (1947)). Let G be a nontrivial graph. The following are equivalent.
(i) G has a perfect matching.
(ii) For every proper subset $S \subset V_{G}, c_{\text {odd }}(G-S) \leq|S|$.
Tutte's theorem does not provide a good algorithm for constructing a perfect matching, because the theorem requires 'too many cases'. Its applications are mainly in the proofs of other results that are related to matchings. There is a good algorithm due to EDMONDS (1965), which uses 'blossom shrinkings', but this algorithm is somewhat involved.
Example 3.7. The simplest connected graph that has no perfect matching is the path $P_{3}$. Here removing the middle vertex creates two odd components.
The next 3-regular graph (known as the Sylvester graph) does not have a perfect matching, because removing the black vertex results in a graph with three odd connected components. This graph is the smallest regular graph with an odd degree that has no perfect matching.
Using Theorem 3.11 we can give a short proof of PETERSEN's result for 3-regular graphs (1891).
Theorem 3.12 (PETERSEN (1891)). If $G$ is a bridgeless 3-regular graph, then it has a perfect matching.
Proof. Let $S$ be a proper subset of $V_{G}$, and let $G_{i}, i \in[1, t]$, be the odd connected components of $G-S$. Denote by $m_{i}$ the number of edges with one end in $G_{i}$ and the other in $S$. Since $G$ is 3-regular,
$$
\sum_{v \in G_{i}} d_{G}(v)=3 \cdot v_{G_{i}} \text { and } \sum_{v \in S} d_{G}(v)=3 \cdot|S| .
$$
The first of these implies that
$$
m_{i}=\sum_{v \in G_{i}} d_{G}(v)-2 \cdot \varepsilon_{G_{i}}
$$
is odd. Furthermore, $m_{i} \neq 1$, because $G$ has no bridges, and therefore $m_{i} \geq 3$. Hence the number of odd connected components of $G-S$ satisfies
$$
t \leq \frac{1}{3} \sum_{i=1}^{t} m_{i} \leq \frac{1}{3} \sum_{v \in S} d_{G}(v)=|S|,
$$
and so, by Theorem 3.11, $G$ has a perfect matching.
## Stable Marriages
DEFINITION. Consider a bipartite graph $G$ with a bipartition $(X, Y)$ of the vertex set. In addition, each vertex $x \in G$ supplies an order of preferences of the vertices of $N_{G}(x)$. We write $u<_{x} v$, if $x$ prefers $v$ to $u$. (Here $u, v \in Y$, if $x \in X$, and $u, v \in X$, if $x \in Y$.) A matching $M$ of $G$ is said to be stable, if for each unmatched pair $x y \notin M$ (with $x \in X$ and $y \in Y$ ), it is not the case that $x$ and $y$ prefer each other better than their matched companions:
$$
x v \in M \text { and } y<_{x} v, \text { or } u y \in M \text { and } x<_{y} u .
$$
We omit the proof of the next theorem.
Theorem 3.13. For bipartite graphs $G$, a stable matching exists for all lists of preferences.
Example 3.8. That was the good news. There is a catch, of course. A stable matching need not saturate $X$ and $Y$. For instance, the graph on the right does have a perfect matching (of 4 edges).
Suppose the preferences are the following:
1: 5
$2: 6<8<7$
$3: 8<5$
$4: 7<5$
5: $4<1<3$
6: 2
$7: 2<4$
$8: 3<2$
Then there is no stable matchings of four edges. A stable matching of $G$ is the following: $M=\{28,35,47\}$, which leaves 1 and 6 unmatched. (You should check that there is no stable matching containing the edges 15 and 26.)
Theorem 3.14. Let $G=K_{n, n}$ be a complete bipartite graph. Then $G$ has a perfect and stable matching for all lists of preferences.
Proof. Let the bipartition be $(X, Y)$. The algorithm by GALE AND SHAPLEY (1962) works as follows.
## Procedure.
Set $M_{0}=\varnothing$, and $P(x)=\varnothing$ for all $x \in X$.
Then iterate the following process until all vertices are saturated:
Choose a vertex $x \in X$ that is unsaturated in $M_{i-1}$. Let $y \in Y$ be the most preferred vertex for $x$ such that $y \notin P(x)$.
(1) Add $y$ to $P(x)$.
(2) If $y$ is not saturated, then set $M_{i}=M_{i-1} \cup\{x y\}$.
(3) If $z y \in M_{i-1}$ and $z<_{y} x$, then set $M_{i}=\left(M_{i-1} \backslash\{z y\}\right) \cup\{x y\}$.
First of all, the procedure terminates, since a vertex $x \in X$ takes part in the iteration at most $n$ times (once for each $y \in Y$ ). The final outcome, say $M=M_{t}$, is a perfect matching, since the iteration continues until there are no unsaturated vertices $x \in X$. Also, the matching $M=M_{t}$ is stable. Note first that, by (3), if $x y \in M_{i}$ and $z y \in M_{j}$ for some $x \neq z$ and $i<j$, then $x<_{y} z$. Assume the that $x y \in M$, but $y<_{x} z$ for some $z \in Y$. Then $x y$ is added to the matching at some step, $x y \in M_{i}$, which means that $z \in P(x)$ at this step (otherwise $x$ would have 'proposed' $z$ ). Hence $x$ took part in the iteration at an earlier step $M_{k}, k<i$ (where $z$ was put to the list $P(x)$, but $x z$ was not added). Thus, for some $u \in X, u z \in M_{k-1}$ and $x<_{z} u$, and so in $M$ the vertex $z$ is matched to some $w$ with $x<_{z} w$.
Similarly, if $x<_{y} v$ for some $v \in X$, then $y<_{v} z$ for the vertex $z \in Y$ such that $v z \in M$.
## Colourings
### Edge colourings
Colourings of edges and vertices of a graph $G$ are useful, when one is interested in classifying relations between objects.
There are two sides of colourings. In the general case, a graph $G$ with a colouring $\alpha$ is given, and we study the properties of this pair $G^{\alpha}=(G, \alpha)$. This is the situation, e.g., in transportation networks with bus and train links, where the colour (buss, train) of an edge tells the nature of a link.
In the chromatic theory, $G$ is first given and then we search for a colouring that the satisfies required properties. One of the important properties of colourings is 'properness'. In a proper colouring adjacent edges or vertices are coloured differently.
## Edge chromatic number
DEFINITION. A $k$-edge colouring $\alpha: E_{G} \rightarrow[1, k]$ of a graph $G$ is an assignment of $k$ colours to its edges. We write $G^{\alpha}$ to indicate that $G$ has the edge colouring $\alpha$.
A vertex $v \in G$ and a colour $i \in[1, k]$ are incident with each other, if $\alpha(v u)=i$ for some $v u \in G$. If $v \in G$ is not incident with a colour $i$, then $i$ is available for $v$.
The colouring $\alpha$ is proper, if no two adjacent edges obtain the same colour: $\alpha\left(e_{1}\right) \neq$ $\alpha\left(e_{2}\right)$ for adjacent $e_{1}$ and $e_{2}$.
The edge chromatic number $\chi^{\prime}(G)$ of $G$ is defined as
$$
\chi^{\prime}(G)=\min \{k \mid \text { there exists a proper } k \text {-edge colouring of } G\} .
$$
A $k$-edge colouring $\alpha$ can be thought of as a partition $\left\{E_{1}, E_{2}, \ldots, E_{k}\right\}$ of $E_{G}$, where $E_{i}=\{e \mid \alpha(e)=i\}$. Note that it is possible that $E_{i}=\varnothing$ for some $i$. We adopt a simplified notation
$$
G^{\alpha}\left[i_{1}, i_{2}, \ldots, i_{t}\right]=G\left[E_{i_{1}} \cup E_{i_{2}} \cup \cdots \cup E_{i_{t}}\right]
$$
for the subgraph of $G$ consisting of those edges that have a colour $i_{1}, i_{2}, \ldots$, or $i_{t}$. That is, the edges having other colours are removed.
Lemma 4.1. Each colour set $E_{i}$ in a proper $k$-edge colouring is a matching. Moreover, for each graph $G, \Delta(G) \leq \chi^{\prime}(G) \leq \varepsilon_{G}$.
Proof. This is clear. Example 4.1. The three numbers in Lemma 4.1 can be equal. This happens, for instance, when $G=K_{1, n}$ is a star. But often the inequalities are strict.
A star, and a graph with $\chi^{\prime}(G)=4$.
## Optimal colourings
We show that for bipartite graphs the lower bound is always optimal: $\chi^{\prime}(G)=\Delta(G)$.
Lemma 4.2. Let $G$ be a connected graph that is not an odd cycle. Then there exists a 2-edge colouring (that need not be proper), in which both colours are incident with each vertex $v$ with $d_{G}(v) \geq 2$.
Proof. Assume that $G$ is nontrivial; otherwise, the claim is trivial.
(1) Suppose first that $G$ is eulerian. If $G$ is an even cycle, then a 2-edge colouring exists as required. Otherwise, since now $d_{G}(v)$ is even for all $v, G$ has a vertex $v_{1}$ with $d_{G}\left(v_{1}\right) \geq 4$. Let $e_{1} e_{2} \ldots e_{t}$ be an Euler tour of $G$, where $e_{i}=v_{i} v_{i+1}\left(\right.$ and $v_{t+1}=v_{1}$ ). Define
$$
\alpha\left(e_{i}\right)= \begin{cases}1, & \text { if } i \text { is odd } \\ 2, & \text { if } i \text { is even }\end{cases}
$$
Hence the ends of the edges $e_{i}$ for $i \in[2, t-1]$ are incident with both colours. All vertices are among these ends. The condition $d_{G}\left(v_{1}\right) \geq 4$ guarantees this for $v_{1}$. Hence the claim holds in the eulerian case.
(2) Suppose then that $G$ is not eulerian. We define a new graph $G_{0}$ by adding a vertex $v_{0}$ to $G$ and connecting $v_{0}$ to each $v \in G$ of odd degree.
In $G_{0}$ every vertex has even degree including $v_{0}$ (by the handshaking lemma), and hence $G_{0}$ is eulerian. Let $e_{0} e_{1} \ldots e_{t}$ be an eulerian tour of $G_{0}$, where $e_{i}=v_{i} v_{i+1}$. By the previous case, there is a required colouring $\alpha$ of $G_{0}$ as above. Now, $\alpha$ restricted to $E_{G}$ is a colouring of $G$ as required by the claim, since each vertex $v_{i}$ with odd degree $d_{G}\left(v_{i}\right) \geq 3$ is entered and departed at least once in the tour by an edge of the original graph $G: e_{i-1} e_{i}$.
DEFINITION. For a $k$-edge colouring $\alpha$ of $G$, let
$$
c_{\alpha}(v)=\mid\{i \mid v \text { is incident with } i \in[1, k]\} \mid .
$$
A $k$-edge colouring $\beta$ is an improvement of $\alpha$, if
$$
\sum_{v \in G} c_{\beta}(v)>\sum_{v \in G} c_{\alpha}(v)
$$
Also, $\alpha$ is optimal, if it cannot be improved.
Notice that we always have $c_{\alpha}(v) \leq d_{G}(v)$, and if $\alpha$ is proper, then $c_{\alpha}(v)=d_{G}(v)$, and in this case $\alpha$ is optimal. Thus an improvement of a colouring is a change towards a proper colouring. Note also that a graph $G$ always has an optimal $k$-edge colouring, but it need not have any proper $k$-edge colourings.
The next lemma is obvious.
Lemma 4.3. An edge colouring $\alpha$ of $G$ is proper if and only if $c_{\alpha}(v)=d_{G}(v)$ for all vertices $v \in G$.
Lemma 4.4. Let $\alpha$ be an optimal k-edge colouring of $G$, and let $v \in G$. Suppose that the colour $i$ is available for $v$, and the colour $j$ is incident with $v$ at least twice. Then the connected component $H$ of $G^{\alpha}[i, j]$ that contains $v$, is an odd cycle.
Proof. Suppose the connected component $H$ is not an odd cycle. By Lemma 4.2, $H$ has a 2-edge colouring $\gamma: E_{H} \rightarrow\{i, j\}$, in which both $i$ and $j$ are incident with each vertex $x$ with $d_{H}(x) \geq 2$. (We have renamed the colours 1 and 2 to $i$ and $j$.) We obtain a recolouring $\beta$ of $G$ as follows:
$$
\beta(e)= \begin{cases}\gamma(e), & \text { if } e \in H, \\ \alpha(e), & \text { if } e \notin H .\end{cases}
$$
Since $d_{H}(v) \geq 2$ (by the assumption on the colour $j$ ) and in $\beta$ both colours $i$ and $j$ are now incident with $v, c_{\beta}(v)=c_{\alpha}(v)+1$. Furthermore, by the construction of $\beta$, we have $c_{\beta}(u) \geq c_{\alpha}(u)$ for all $u \neq v$. Therefore $\sum_{u \in G} c_{\beta}(u)>\sum_{u \in G} c_{\alpha}(u)$, which contradicts the optimality of $\alpha$. Hence $H$ is an odd cycle.
Theorem 4.1 (KÖNIG (1916)). If $G$ is bipartite, then $\chi^{\prime}(G)=\Delta(G)$.
Proof. Let $\alpha$ be an optimal $\Delta$-edge colouring of a bipartite $G$, where $\Delta=\Delta(G)$. If there were a $v \in G$ with $c_{\alpha}(v)<d_{G}(v)$, then by Lemma 4.4, $G$ would contain an odd cycle. But a bipartite graph does not contain such cycles. Therefore, for all vertices $v$, $c_{\alpha}(v)=d_{G}(v)$. By Lemma 4.3, $\alpha$ is a proper colouring, and $\Delta=\chi^{\prime}(G)$ as required.
## Vizing's theorem
In general we can have $\chi^{\prime}(G)>\Delta(G)$ as one of our examples did show. The following important theorem, due to VizING, shows that the edge chromatic number of a graph $G$ misses $\Delta(G)$ by at most one colour.
Theorem 4.2 (ViZING (1964)). For any graph $G, \Delta(G) \leq \chi^{\prime}(G) \leq \Delta(G)+1$.
Proof. Let $\Delta=\Delta(G)$. We need only to show that $\chi^{\prime}(G) \leq \Delta+1$. Suppose on the contrary that $\chi^{\prime}(G)>\Delta+1$, and let $\alpha$ be an optimal $(\Delta+1)$-edge colouring of $G$. We have (trivially) $d_{G}(u)<\Delta+1<\chi^{\prime}(G)$ for all $u \in G$, and so
Claim 1. For each $u \in G$, there exists an available colour $b(u)$ for $u$.
Moreover, by the counter hypothesis, $\alpha$ is not a proper colouring, and hence there exists a $v \in G$ with $c_{\alpha}(v)<d_{G}(v)$, and hence a colour $i_{1}$ that is incident with $v$ at least twice, say
$$
\alpha\left(v u_{1}\right)=i_{1}=\alpha(v x) .
$$
Claim 2. There is a sequence of vertices $u_{1}, u_{2}, \ldots$ such that
$$
\alpha\left(v u_{j}\right)=i_{j} \text { and } i_{j+1}=b\left(u_{j}\right) .
$$
Indeed, let $u_{1}$ be as in (4.1). Assume we have already found the vertices $u_{1}, \ldots, u_{j}$, with $j \geq 1$, such that the claim holds for these. Suppose, contrary to the claim, that $v$ is not incident with $b\left(u_{j}\right)=i_{j+1}$.
We can recolour the edges $v u_{\ell}$ by $i_{\ell+1}$ for $\ell \in[1, j]$, and obtain in this way an improvement of $\alpha$. Here $v$ gains a new colour $i_{j+1}$. Also, each $u_{\ell}$ gains a new colour $i_{\ell+1}$ (and may loose the colour $i_{\ell}$ ). Therefore, for each $u_{\ell}$ either its number of colours remains the same or it increases by one. This contradicts the optimality of $\alpha$, and proves Claim 2.
Let $t$ be the smallest index such that for some $r<t$, $i_{t+1}=i_{r}$. Such an index $t$ exists, because $d_{G}(v)$ is finite.
Let $\beta$ be a recolouring of $G$ such that for $1 \leq j \leq r-1$, $\beta\left(v u_{j}\right)=i_{j+1}$, and for all other edges $e, \beta(e)=\alpha(e)$.
Claim 3. $\beta$ is an optimal $(\Delta+1)$-edge colouring of $G$.
Indeed, $c_{\beta}(v)=c_{\alpha}(v)$ and $c_{\beta}(u) \geq c_{\alpha}(u)$ for all $u$, since each $u_{j}(1 \leq j \leq r-1)$ gains a new colour $j_{i+1}$ although it may loose one of its old colours.
Let then the colouring $\gamma$ be obtained from $\beta$ by recolouring the edges $v u_{j}$ by $i_{j+1}$ for $r \leq j \leq t$. Now, $v u_{t}$ is recoloured by $i_{r}=i_{t+1}$.
Claim 4. $\gamma$ is an optimal $(\Delta+1)$-edge colouring of $G$.
Indeed, the fact $i_{r}=i_{t+1}$ ensures that $i_{r}$ is a new colour incident with $u_{t}$, and thus that $c_{\gamma}\left(u_{t}\right) \geq c_{\beta}\left(u_{t}\right)$. For all other vertices, $c_{\gamma}(u) \geq c_{\beta}(u)$ follows as for $\beta$.
By Claim 1, there is a colour $i_{0}=b(v)$ that is available for $v$. By Lemma 4.4, the connected components $H_{1}$ of $G^{\beta}\left[i_{0}, i_{r}\right]$ and $H_{2}$ of $G^{\gamma}\left[i_{0}, i_{r}\right]$ containing the vertex $v$ are cycles, that is, $H_{1}$ is a cycle $\left(v u_{r-1}\right) P_{1}\left(u_{r} v\right)$ and $H_{2}$ is a cycle $\left(v u_{r-1}\right) P_{2}\left(u_{t} v\right)$, where both $P_{1}: u_{r-1} \stackrel{\star}{\rightarrow} u_{r}$ and $P_{2}: u_{r-1} \stackrel{\star}{\rightarrow} u_{t}$ are paths. However, the edges of $P_{1}$ and $P_{2}$ have the same colours with respect to $\beta$ and $\gamma$ (either $i_{0}$ or $i_{r}$ ). This is not possible, since $P_{1}$ ends in $u_{r}$ while $P_{2}$ ends in a different vertex $u_{t}$. This contradiction proves the theorem.
Example 4.2. We show that $\chi^{\prime}(G)=4$ for the Petersen graph. Indeed, by Vizing' theorem, $\chi^{\prime}(G)=3$ or 4 . Suppose 3 colours suffice. Let $C: v_{1} \rightarrow \ldots \rightarrow v_{5} \rightarrow v_{1}$ be the outer cycle and $C^{\prime}: u_{1} \rightarrow \ldots \rightarrow u_{5} \rightarrow u_{1}$ the inner cycle of $G$ such that $v_{i} u_{i} \in E_{G}$ for all $i$.
Observe that every vertex is adjacent to all colours 1,2,3. Now $C$ uses one colour (say 1) once and the other two twice. This can be done uniquely (up to permutations):
$$
v_{1} \stackrel{1}{\rightarrow} v_{2} \stackrel{2}{\rightarrow} v_{3} \stackrel{3}{\rightarrow} v_{4} \stackrel{2}{\rightarrow} v_{5} \stackrel{3}{\rightarrow} v_{1} .
$$
Hence $v_{1} \stackrel{2}{\rightarrow} u_{1}, v_{2} \stackrel{3}{\rightarrow} u_{2}, v_{3} \stackrel{1}{\rightarrow} u_{3}, v_{4} \stackrel{1}{\rightarrow} u_{4}, v_{5} \stackrel{1}{\rightarrow} u_{5}$. However, this means that 1 cannot be a colour of any edge in $C^{\prime}$. Since $C^{\prime}$ needs three colours, the claim follows.
Edge Colouring Problem. Vizing's theorem (nor its present proof) does not offer any characterization for the graphs, for which $\chi^{\prime}(G)=\Delta(G)+1$. In fact, it is one of the famous open problems of graph theory to find such a characterization. The answer is known (only) for some special classes of graphs. By HOLYER (1981), the problem whether $\chi^{\prime}(G)$ is $\Delta(G)$ or $\Delta(G)+1$ is NP-complete.
The proof of Vizing's theorem can be used to obtain a proper colouring of $G$ with at most $\Delta(G)+1$ colours, when the word 'optimal' is forgotten: colour first the edges as well as you can (if nothing better, then arbitrarily in two colours), and use the proof iteratively to improve the colouring until no improvement is possible - then the proof says that the result is a proper colouring.
### Ramsey Theory
In general, Ramsey theory studies unavoidable patterns in combinatorics. We consider an instance of this theory mainly for edge colourings (that need not be proper). A typical example of a Ramsey property is the following: given 6 persons each pair of whom are either friends or enemies, there are then 3 persons who are mutual friends or mutual enemies. In graph theoretic terms this means that each colouring of the edges of $K_{6}$ with 2 colours results in a monochromatic triangle.
## Turan's theorem for complete graphs
We shall first consider the problem of finding a general condition for $K_{p}$ to appear in a graph. It is clear that every graph contains $K_{1}$, and that every nondiscrete graph contains $\mathrm{K}_{2}$.
DEFINITION. A complete $p$-partite graph $G$ consists of $p$ discrete and disjoint induced subgraphs $G_{1}, G_{2}, \ldots, G_{p} \subseteq G$, where $u v \in G$ if and only if $u$ and $v$ belong to different parts, $G_{i}$ and $G_{j}$ with $i \neq j$.
Note that a complete $p$-partite graph is completely determined by its discrete parts $G_{i}, i \in[1, p]$.
Let $p \geq 3$, and let $H=H_{n, p}$ be the complete $(p-1)$-partite graph of order $n=$ $t(p-1)+r$, where $r \in[1, p-1]$ and $t \geq 0$, such that there are $r$ parts $H_{1}, \ldots, H_{r}$ of order $t+1$ and $p-1-r$ parts $H_{r+1}, \ldots, H_{p-1}$ of order $t$ (when $t>0$ ). (Here $r$ is the positive residue of $n$ modulo $(p-1)$, and is thus determined by $n$ and $p$.)
By its definition, $K_{p} \nsubseteq H$. One can compute that the number $\varepsilon_{H}$ of edges of $H$ is equal to
$$
T(n, p)=\frac{p-2}{2(p-1)} n^{2}-\frac{r}{2}\left(1-\frac{r}{p-1}\right) .
$$
The next result shows that the above bound $T(n, p)$ is optimal.
Theorem 4.3 (TURÁN (1941)). If a graph $G$ of order $n$ has $\varepsilon_{G}>T(n, p)$ edges, then $G$ contains a complete subgraph $K_{p}$.
Proof. Let $n=(p-1) t+r$ for $1 \leq r \leq p-1$ and $t \geq 0$. We prove the claim by induction on $t$. If $t=0$, then $T(n, p)=n(n-1) / 2$, and there is nothing to prove.
Suppose then that $t \geq 1$, and let $G$ be a graph of order $n$ such that $\varepsilon_{G}$ is maximum subject to the condition $K_{p} \nsubseteq G$.
Now $G$ contains a complete subgraph $G[A]=K_{p-1}$, since adding any one edge to $G$ results in a $K_{p}$, and $p-1$ vertices of this $K_{p}$ induce a subgraph $K_{p-1} \subseteq G$.
Each $v \notin A$ is adjacent to at most $p-2$ vertices of $A$; otherwise $G[A \cup\{v\}]=K_{p}$. Also, $K_{p} \nsubseteq G-A$, and $v_{G-A}=n-p+1$. Because $n-p+1=(t-1)(p-1)+r$, we can apply the induction hypothesis to obtain $\varepsilon_{G-A} \leq T(n-p+1, p)$. Now
$$
\varepsilon_{G} \leq T(n-p+1, p)+(n-p+1)(p-2)+\frac{(p-1)(p-2)}{2}=T(n, p),
$$
which proves the claim.
When Theorem 4.3 is applied to triangles $K_{3}$, we have the following interesting case.
Corollary 4.1 ( MANTEL (1907)). If a graph $G$ has $\varepsilon_{G}>\frac{1}{4} v_{G}^{2}$ edges, then $G$ contains a triangle $K_{3}$.
## Ramsey's theorem
DEFINITION. Let $\alpha$ be an edge colouring of G. A subgraph $H \subseteq G$ is said to be (i-) monochromatic, if all edges of $H$ have the same colour $i$.
The following theorem is one of the jewels of combinatorics.
Theorem 4.4 (RAMSEY (1930)). Let $p, q \geq 2$ be any integers. Then there exists a (smallest) integer $R(p, q)$ such that for all $n \geq R(p, q)$, any 2-edge colouring of $K_{n} \rightarrow[1,2]$ contains a 1-monochromatic $K_{p}$ or a 2-monochromatic $K_{q}$.
Before proving this, we give an equivalent statement. Recall that a subset $X \subseteq V_{G}$ is stable, if $G[X]$ is a discrete graph.
Theorem 4.5. Let $p, q \geq 2$ be any integers. Then there exists a (smallest) integer $R(p, q)$ such that for all $n \geq R(p, q)$, any graph $G$ of order $n$ contains a complete subgraph of order $p$ or a stable set of order $q$.
Be patient, this will follow from Theorem 4.6. The number $R(p, q)$ is known as the Ramsey number for $p$ and $q$.
It is clear that $R(p, 2)=p$ and $R(2, q)=q$.
Theorems 4.4 and 4.5 follow from the next result which shows (inductively) that an upper bound exists for the Ramsey numbers $R(p, q)$.
Theorem 4.6 (ERDÖS and SZEKERES (1935)). The Ramsey number $R(p, q)$ exists for all $p, q \geq 2$, and
$$
R(p, q) \leq R(p, q-1)+R(p-1, q) .
$$
Proof. We use induction on $p+q$. It is clear that $R(p, q)$ exists for $p=2$ or $q=2$, and it is thus exists for $p+q \leq 5$.
It is now sufficient to show that if $G$ is a graph of order $R(p, q-1)+R(p-1, q)$, then it has a complete subgraph of order $p$ or a stable subset of order $q$.
Let $v \in G$, and denote by $A=V_{G} \backslash\left(N_{G}(v) \cup\{v\}\right)$ the set of vertices that are not adjacent to $v$. Since $G$ has $R(p, q-1)+R(p-1, q)-1$ vertices different from $v$, either $\left|N_{G}(v)\right| \geq R(p-1, q)$ or $|A| \geq R(p, q-1)$ (or both).
Assume first that $\left|N_{G}(v)\right| \geq R(p-1, q)$. By the definition of Ramsey numbers, $G\left[N_{G}(v)\right]$ contains a complete subgraph $B$ of order $p-1$ or a stable subset $S$ of order $q$. In the first case, $B \cup\{v\}$ induces a complete subgraph $K_{p}$ in $G$, and in the second case the same stable set of order $q$ is good for $G$.
If $|A| \geq R(p, q-1)$, then $G[A]$ contains a complete subgraph of order $p$ or a stable subset $S$ of order $q-1$. In the first case, the same complete subgraph of order $p$ is good for $G$, and in the second case, $S \cup\{v\}$ is a stable subset of $G$ of $q$ vertices. This proves the claim. A concrete upper bound is given in the following result.
Theorem 4.7 (ERDÖS and SZEKERES (1935)). For all $p, q \geq 2$,
$$
R(p, q) \leq\left(\begin{array}{c}
p+q-2 \\
p-1
\end{array}\right) .
$$
Proof. For $p=2$ or $q=2$, the claim is clear. We use induction on $p+q$ for the general statement. Assume that $p, q \geq 3$. By Theorem 4.6 and the induction hypothesis,
$$
\begin{aligned}
R(p, q) & \leq R(p, q-1)+R(p-1, q) \\
& \leq\left(\begin{array}{c}
p+q-3 \\
p-1
\end{array}\right)+\left(\begin{array}{c}
p+q-3 \\
p-2
\end{array}\right)=\left(\begin{array}{c}
p+q-2 \\
p-1
\end{array}\right),
\end{aligned}
$$
which is what we wanted.
In the table below we give some known values and estimates for the Ramsey numbers $R(p, q)$. As can be read from the table ${ }^{1}$, not so much is known about these numbers.
| $p \backslash q$ | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 3 | 6 | 9 | 14 | 18 | 23 | 28 | 36 | $40-43$ |
| 4 | 9 | 18 | 25 | $35-41$ | $49-61$ | $55-84$ | $69-115$ | $80-149$ |
| 5 | 14 | 25 | $43-49$ | $58-87$ | $80-143$ | $95-216$ | $121-316$ | $141-442$ |
The first unknown $R(p, p)$ (where $p=q$ ) is for $p=5$. It has been verified that $43 \leq R(5,5) \leq 49$, but to determine the exact value is an open problem.
## Generalizations
Theorem 4.4 can be generalized as follows.
Theorem 4.8. Let $q_{i} \geq 2$ be integers for $i \in[1, k]$ with $k \geq 2$. Then there exists an integer $R=R\left(q_{1}, q_{2}, \ldots, q_{k}\right)$ such that for all $n \geq R$, any $k$-edge colouring of $K_{n}$ has an i-monochromatic $K_{q_{i}}$ for some $i$.
Proof. The proof is by induction on $k$. The case $k=2$ is treated in Theorem 4.4. For $k>2$, we show that $R\left(q_{1}, \ldots, q_{k}\right) \leq R\left(q_{1}, \ldots, q_{k-2}, p\right)$, where $p=R\left(q_{k-1}, q_{k}\right)$.
Let $n=R\left(q_{1}, \ldots, q_{k-2}, p\right)$, and let $\alpha: E_{K_{n}} \rightarrow[1, k]$ be an edge colouring. Let $\beta: E_{K_{n}} \rightarrow[1, k-1]$ be obtained from $\alpha$ by identifying the colours $k-1$ and $k$ :
$$
\beta(e)= \begin{cases}\alpha(e) & \text { if } \alpha(e)<k-1 \\ k-1 & \text { if } \alpha(e)=k-1 \text { or } k\end{cases}
$$
${ }^{1}$ S.P. RADZISZOWSKI, Small Ramsey numbers, Electronic J. of Combin., 2000 on the Web By the induction hypothesis, $K_{n}^{\beta}$ has an $i$-monochromatic $K_{q_{i}}$ for some $1 \leq i \leq k-2$ (and we are done, since this subgraph is monochromatic in $\left.K_{n}^{\alpha}\right)$ or $K_{n}^{\beta}$ has a $(k-1)$ monochromatic subgraph $H^{\beta}=K_{p}$. In the latter case, by Theorem $4.4, H^{\alpha}$ and thus $K_{n}^{\alpha}$ has a $(k-1)$-monochromatic or a $k$-monochromatic subgraph, and this proves the claim.
Since for each graph $H, H \subseteq K_{m}$ for $m=v_{H}$, we have
Corollary 4.2. Let $k \geq 2$ and $H_{1}, H_{2}, \ldots, H_{k}$ be arbitrary graphs. Then there exists an integer $R\left(H_{1}, H_{2}, \ldots, H_{k}\right)$ such that for all complete graphs $K_{n}$ with $n \geq R\left(H_{1}, H_{2}, \ldots, H_{k}\right)$ and for all k-edge colourings $\alpha$ of $K_{n}, K_{n}^{\alpha}$ contains an $i$-monochromatic subgraph $H_{i}$ for some $i$.
This generalization is trivial from Theorem 4.8. However, the generalized Ramsey numbers $R\left(H_{1}, H_{2}, \ldots, H_{k}\right)$ can be much smaller than their counter parts (for complete graphs) in Theorem 4.8.
Example 4.3. We leave the following statement as an exercise: If $T$ is a tree of order $m$, then
$$
R\left(T, K_{n}\right)=(m-1)(n-1)+1,
$$
that is, any graph $G$ of order at least $R\left(T, K_{n}\right)$ contains a subgraph isomorphic to $T$, or the complement of $G$ contains a complete subgraph $K_{n}$.
## Examples of Ramsey numbers*
Some exact values are known in Corollary 4.2, even in more general cases, for some dear graphs (see RADZISZOWSKI's survey). Below we list some of these results for cases, where the graphs are equal. To this end, let
$$
R_{k}(G)=R(G, G, \ldots, G) \quad(k \text { times } G) .
$$
The best known lower bound of $R_{2}(G)$ for connected graphs was obtained by BURR AND ERDÖS (1976),
$$
R_{2}(G) \geq\left\lfloor\frac{4 v_{G}-1}{3}\right\rfloor \quad(G \text { connected }) .
$$
Here is a list of some special cases:
$$
\begin{aligned}
& R_{2}\left(P_{n}\right)=n+\left\lfloor\frac{n}{2}\right\rfloor-1, \\
& R_{2}\left(C_{n}\right)= \begin{cases}6 & \text { if } n=3 \text { or } n=4, \\
2 n-1 & \text { if } n \geq 5 \text { and } n \text { odd }, \\
3 n / 2-1 & \text { if } n \geq 6 \text { and } n \text { even, }\end{cases} \\
& R_{2}\left(K_{1, n}\right)= \begin{cases}2 n-1 & \text { if } n \text { is even, } \\
2 n & \text { if } n \text { is odd, }\end{cases} \\
& R_{2}\left(K_{2,3}\right)=10, \quad R_{2}\left(K_{3,3}\right)=18 .
\end{aligned}
$$
The values $R_{2}\left(K_{2, n}\right)$ are known for $n \leq 16$, and in general, $R_{2}\left(K_{2, n}\right) \leq 4 n-2$. The value $R_{2}\left(K_{2,17}\right)$ is either 65 or 66 .
Let $W_{n}$ denote the wheel on $n$ vertices. It is a cycle $C_{n-1}$, where a vertex $v$ with degree $n-1$ is attached. Note that $W_{4}=K_{4}$. Then $R_{2}\left(W_{5}\right)=15$ and $R_{2}\left(W_{6}\right)=17$.
For three colours, much less is known. In fact, the only nontrivial result for complete graphs is: $R_{3}\left(K_{3}\right)=17$. Also, $128 \leq R_{3}\left(K_{4}\right) \leq 235$, and $385 \leq R_{3}\left(K_{5}\right)$, but no nontrivial upper bound is known for $R_{3}\left(K_{5}\right)$. For the square $C_{4}$, we know that $R_{3}\left(C_{4}\right)=11$.
Needless to say that no exact values are known for $R_{k}\left(K_{n}\right)$ for $k \geq 4$ and $n \geq 3$.
It follows from Theorem 4.4 that for any complete $K_{n}$, there exists a graph $G$ (well, any sufficiently large complete graph) such that any 2-edge colouring of $G$ has a monochromatic (induced) subgraph $K_{n}$. Note, however, that in Corollary 4.2 the monochromatic subgraph $H_{i}$ is not required to be induced.
The following impressive theorem improves the results we have mentioned in this chapter and it has a difficult proof.
Theorem 4.9 (DEUBER, ERDÖS, HajNAL, PÓSA, and RÖDL (around 1973)). Let $H$ be any graph. Then there exists a graph $G$ such that any 2-edge colouring of $G$ has an monochromatic induced subgraph $H$.
Example 4.4. As an application of Ramsey's theorem, we shortly describe Schur's theorem. For this, consider the partition $\{1,4,10,13\},\{2,3,11,12\},\{5,6,7,8,9\}$ of the set $\mathbb{N}_{13}=[1,13]$. We observe that in no partition class there are three integers such that $x+y=z$. However, if you try to partition $\mathbb{N}_{14}$ into three classes, then you are bound to find a class, where $x+y=z$ has a solution.
SCHUR (1916) solved this problem in a general setting. The following gives a short proof using Ramsey's theorem.
For each $n \geq 1$, there exists an integer $S(n)$ such that any partition $S_{1}, \ldots, S_{n}$ of $\mathbb{N}_{S(n)}$ has a class $S_{i}$ containing two integers $x, y$ such that $x+y \in S_{i}$.
Indeed, let $S(n)=R(3,3, \ldots, 3)$, where 3 occurs $n$ times, and let $K$ be a complete on $\mathbb{N}_{S(n)}$. For a partition $S_{1}, \ldots, S_{n}$ of $\mathbb{N}_{S(n)}$, define an edge colouring $\alpha$ of $K$ by
$$
\alpha(i j)=k, \text { if }|i-j| \in S_{k} .
$$
By Theorem $4.8, K^{\alpha}$ has a monochromatic triangle, that is, there are three vertices $i, j, t$ such that $1 \leq i<j<t \leq S(n)$ with $t-j, j-i, t-i \in S_{k}$ for some $k$. But $(t-j)+(j-i)=t-i$ proves the claim.
There are quite many interesting corollaries to Ramsey's theorem in various parts of mathematics including not only graph theory, but also, e.g., geometry and algebra, see
R.L. Graham, B.L. Rothschild and J.L. Spencer, "Ramsey Theory", Wiley, (2nd ed.) 1990.
### Vertex colourings
The vertices of a graph $G$ can also be classified using colourings. These colourings tell that certain vertices have a common property (or that they are similar in some respect), if they share the same colour. In this chapter, we shall concentrate on proper vertex colourings, where adjacent vertices get different colours.
## The chromatic number
DEFINITION. A $k$-colouring (or a $k$-vertex colouring) of a graph $G$ is a mapping $\alpha: V_{G} \rightarrow[1, k]$. The colouring $\alpha$ is proper, if adjacent vertices obtain a different colour: for all $u v \in G$, we have $\alpha(u) \neq \alpha(v)$. A colour $i \in[1, k]$ is said to be available for a vertex $v$, if no neighbour of $v$ is coloured by $i$.
A graph $G$ is $k$-colourable, if there is a proper $k$-colouring for $G$. The (vertex) chromatic number $\chi(G)$ of $G$ is defines as
$$
\chi(G)=\min \{k \mid \text { there exists a proper } k \text {-colouring of } G\} .
$$
If $\chi(G)=k$, then $G$ is $k$-chromatic.
Each proper vertex colouring $\alpha: V_{G} \rightarrow[1, k]$ provides a partition $\left\{V_{1}, V_{2}, \ldots, V_{k}\right\}$ of the vertex set $V_{G}$, where $V_{i}=\{v \mid \alpha(v)=i\}$.
Example 4.5. The graph on the right, which is often called a wheel (of order 7), is 3-chromatic.
By the definitions, a graph $G$ is 2-colourable if and only if it is bipartite.
Again, the 'names' of the colours are immaterial:
Lemma 4.5. Let $\alpha$ be a proper $k$-colouring of $G$, and let $\pi$ be any permutation of the colours. Then the colouring $\beta=\pi \alpha$ is a proper $k$-colouring of $G$.
Proof. Indeed, if $\alpha: V_{G} \rightarrow[1, k]$ is proper, and if $\pi:[1, k] \rightarrow[1, k]$ is a bijection, then $u v \in G$ implies that $\alpha(u) \neq \alpha(v)$, and hence also that $\pi \alpha(u) \neq \pi \alpha(v)$. It follows that $\pi \alpha$ is a proper colouring.
Example 4.6. A graph is triangle-free, if it has no subgraphs isomorphic to $K_{3}$. We show that there are triangle-free graphs with arbitrarily large chromatic numbers.
The following construction is due to GRÖTZEL: Let $G$ be any triangle-free graph with $V_{G}=\left\{v_{1}, v_{2}, \ldots, v_{n}\right\}$. Let $G^{t}$ be a new graph obtained by adding $n+1$ new vertices $v$ and $u_{1}, u_{2}, \ldots, u_{n}$ such that $G^{t}$ has all the edges of $G$ plus the edges $u_{i} v$ and $u_{i} x$ for all $x \in N\left(v_{i}\right)$ and for all $i \in[1, n]$. Claim. $G^{t}$ is triangle-free and it is $k+1$-chromatic
Indeed, let $U=\left\{u_{1}, \ldots, u_{n}\right\}$. We show first that $G^{t}$ is triangle-free. Now, $U$ is stable, and so a triangle contains at most (and thus exactly) one vertex $u_{i} \in U$. If $\left\{u_{i}, v_{j}, v_{k}\right\}$ induces a triangle, so does $\left\{v_{i}, v_{j}, v_{k}\right\}$ by the definition of $G^{t}$, but the latter triangle is already in $G$; a contradiction.
For the chromatic number we notice first that $\chi\left(G^{t}\right) \leq(k+1)$. If $\alpha$ is a proper $k$-colouring of $G$, extend it by setting $\alpha\left(u_{i}\right)=\alpha\left(v_{i}\right)$ and $\alpha(v)=k+1$.
Secondly, $\chi\left(G^{t}\right)>k$. Assume that $\alpha$ is a proper $k$-colouring of $G^{t}$, say with $\alpha(v)=$ $k$. Then $\alpha\left(u_{i}\right) \neq k$. Recolour each $v_{i}$ by $\alpha\left(u_{i}\right)$. This gives a proper $(k-1)$-colouring to $G$; a contradiction. Therefore $\chi\left(G^{t}\right)=k+1$.
Now using inductively the above construction starting from the triangle-free graph $K_{2}$, we obtain larger triangle -free graphs with high chromatic numbers.
## Critical graphs
DEFINITION. A $k$-chromatic graph $G$ is said to be $k$-critical, if $\chi(H)<k$ for all $H \subseteq G$ with $H \neq G$.
In a critical graph an elimination of any edge and of any vertex will reduce the chromatic number: $\chi(G-e)<\chi(G)$ and $\chi(G-v)<\chi(G)$ for $e \in G$ and $v \in G$. Each $K_{n}$ is $n$-critical, since in $K_{n}-(u v)$ the vertices $u$ and $v$ can gain the same colour.
Example 4.7. The graph $K_{2}=P_{2}$ is the only 2-critical graph. The 3-critical graphs are exactly the odd cycles $C_{2 n+1}$ for $n \geq 1$, since a 3-chromatic $G$ is not bipartite, and thus must have a cycle of odd length.
Theorem 4.10. If $G$ is $k$-critical for $k \geq 2$, then it is connected, and $\delta(G) \geq k-1$.
Proof. Note that for any graph $G$ with the connected components $G_{1}, G_{2}, \ldots, G_{m}$, $\chi(G)=\max \left\{\chi\left(G_{i}\right) \mid i \in[1, m]\right\}$. Connectivity claim follows from this observation.
Let then $G$ be $k$-critical, but $\delta(G)=d_{G}(v) \leq k-2$ for $v \in G$. Since $G$ is critical, there is a proper $(k-1)$-colouring of $G-v$. Now $v$ is adjacent to only $\delta(G)<k-1$ vertices. But there are $k$ colours, and hence there is an available colour $i$ for $v$. If we recolour $v$ by $i$, then a proper $(k-1)$-colouring is obtained for $G$; a contradiction.
The case (iii) of the next theorem is due to SZEKERES AND WILF (1968).
Theorem 4.11. Let $G$ be any graph with $k=\chi(G)$.
(i) G has a k-critical subgraph $H$.
(ii) $G$ has at least $k$ vertices of degree $\geq k-1$.
(iii) $k \leq 1+\max _{H \subseteq G} \delta(H)$. Proof. For (i), we observe that a $k$-critical subgraph $H \subseteq G$ is obtained by removing vertices and edges from $G$ as long as the chromatic number remains $k$.
For (ii), let $H \subseteq G$ be $k$-critical. By Theorem $4.10, d_{H}(v) \geq k-1$ for every $v \in H$. Of course, also $d_{G}(v) \geq k-1$ for every $v \in H$. The claim follows, because, clearly, every $k$-critical graph $H$ must have at least $k$ vertices.
For (iii), let $H \subseteq G$ be $k$-critical. By Theorem 4.10, $\chi(G)-1 \leq \delta(H)$, which proves this claim.
Lemma 4.6. Let $v$ be a cut vertex of a connected graph $G$, and let $A_{i}$, for $i \in[1, m]$, be the connected components of $G-v$. Denote $G_{i}=G\left[A_{i} \cup\{v\}\right]$. Then $\chi(G)=\max \left\{\chi\left(G_{i}\right) \mid i \in\right.$ $[1, m]\}$. In particular, a critical graph does not have cut vertices.
Proof. Suppose each $G_{i}$ has a proper $k$-colouring $\alpha_{i}$. By Lemma 4.5, we may take $\alpha_{i}(v)=1$ for all $i$. These $k$-colourings give a $k$-colouring of $G$.
## Brooks' theorem
For edge colourings we have Vizing's theorem, but no such strong results are known for vertex colouring.
Lemma 4.7. For all graphs $G, \chi(G) \leq \Delta(G)+1$. In fact, there exists a proper colouring $\alpha: V_{G} \rightarrow[1, \Delta(G)+1]$ such that $\alpha(v) \leq d_{G}(v)+1$ for all vertices $v \in G$.
Proof. We use greedy colouring to prove the claim. Let $V_{G}=\left\{v_{1}, \ldots, v_{n}\right\}$ be ordered in some way, and define $\alpha: V_{G} \rightarrow \mathbb{N}$ inductively as follows: $\alpha\left(v_{1}\right)=1$, and
$$
\alpha\left(v_{i}\right)=\min \left\{j \mid \alpha\left(v_{t}\right) \neq j \text { for all } t<i \text { with } v_{i} v_{t} \in G\right\} .
$$
Then $\alpha$ is proper, and $\alpha\left(v_{i}\right) \leq d_{G}\left(v_{i}\right)+1$ for all $i$. The claim follows from this.
Although, we always have $\chi(G) \leq \Delta(G)+1$, the chromatic number $\chi(G)$ usually takes much lower values - as seen in the bipartite case. Moreover, the maximum value $\Delta(G)+1$ is obtained only in two special cases as was shown by BROOKS in 1941.
The next proof of Brook's theorem is by LOVÁsz (1975) as modified by BRYANT (1996).
Lemma 4.8. Let $G$ be a 2-connected graph. Then the following are equivalent:
(i) $G$ is a complete graph or a cycle.
(ii) For all $u, v \in G$, if $u v \notin G$, then $\{u, v\}$ is a separating set.
(iii) For all $u, v \in G$, if $d_{G}(u, v)=2$, then $\{u, v\}$ is a separating set. Proof. It is clear that (i) implies (ii), and that (ii) implies (iii). We need only to show that (iii) implies (i). Assume then that (iii) holds.
We shall show that either $G$ is a complete graph or $d_{G}(v)=2$ for all $v \in G$, from which the theorem follows.
First of all, $d_{G}(v) \geq 2$ for all $v$, since $G$ is 2 -connected. Let $w$ be a vertex of maximum degree, $d_{G}(w)=\Delta(G)$.
If the neighbourhood $N_{G}(w)$ induces a complete subgraph, then $G$ is complete. Indeed, otherwise, since $G$ is connected, there exists a vertex $u \notin N_{G}(w) \cup\{w\}$ such that $u$ is adjacent to a vertex $v \in N_{G}(w)$. But then $d_{G}(v)>d_{G}(w)$, and this contradicts the choice of $w$.
Assume then that there are different vertices $u, v \in N_{G}(w)$ such that $u v \notin G$. This means that $d_{G}(u, v)=2$ (the shortest path is $u \rightarrow w \rightarrow v$ ), and by (iii), $\{u, v\}$ is a separating set of $G$. Consequently, there is a partition $V_{G}=W \cup\{u, v\} \cup U$, where $w \in W$, and all paths from a vertex of $W$ to a vertex of $U$ go through either $u$ or $v$.
We claim that $W=\{w\}$, and thus that $\Delta(G)=2$ as required. Suppose on the contrary that $|W| \geq 2$. Since $w$ is not a cut vertex (since $G$ has no cut vertices), there exists an $x \in W$ with $x \neq w$ such that $x u \in G$ or $x v \in G$, say $x u \in G$.
Since $v$ is not a cut vertex, there exists a $y \in U$ such that $u y \in G$. Hence $d_{G}(x, y)=2$, and by (iii), $\{x, y\}$ is a separating set. Thus $V_{G}=W_{1} \cup\{x, y\} \cup U_{1}$, where all paths from $W_{1}$ to $U_{1}$ pass through $x$ or $y$. Assume that $w \in W_{1}$, and hence that also $u, v \in W_{1}$. (Since $u w, v w \in$ $\left.V_{G}-\{x, y\}\right)$.
There exists a vertex $z \in U_{1}$. Note that $U_{1} \subseteq W \cup U$. If $z \in W$ (or $z \in U$, respectively), then all paths from $z$ to $u$ must pass through $x$ (or $y$, respectively), and $x$ (or $y$, respectively) would be a cut vertex of $G$. This contradiction, proves the claim.
Theorem 4.12 ( BROOKs (1941)). Let $G$ be connected. Then $\chi(G)=\Delta(G)+1$ if and only if either $G$ is an odd cycle or a complete graph.
Proof. $(\Longleftarrow)$ Indeed, $\chi\left(C_{2 k+1}\right)=3, \Delta\left(C_{2 k+1}\right)=2$, and $\chi\left(K_{n}\right)=n, \Delta\left(K_{n}\right)=n-1$.
$(\Longrightarrow)$ Assume that $k=\chi(G)$. We may suppose that $G$ is $k$-critical. Indeed, assume the claim holds for $k$-critical graphs. Let $k=\Delta(G)+1$, and let $H \subset G$ be a $k$-critical proper subgraph. Since $\chi(H)=k=\Delta(G)+1>\Delta(H)$, we must have $\chi(H)=$ $\Delta(H)+1$, and thus $H$ is a complete graph or an odd cycle. Now $G$ is connected, and therefore there exists an edge $u v \in G$ with $u \in H$ and $v \notin H$. But then $d_{G}(u)>d_{H}(u)$, and $\Delta(G)>\Delta(H)$, since $H=K_{n}$ or $H=C_{n}$.
Let then $G$ be any $k$-critical graph for $k \geq 2$. By Lemma 4.6 , it is 2-connected. If $G$ is an even cycle, then $k=2=\Delta(G)$. Suppose now that $G$ is neither complete nor a cycle (odd or even). We show that $\chi(G) \leq \Delta(G)$. By Lemma 4.8, there exist $v_{1}, v_{2} \in G$ with $d_{G}\left(v_{1}, v_{2}\right)=2$, say $v_{1} w$, $w v_{2} \in G$ with $v_{1} v_{2} \notin G$, such that $H=G-\left\{v_{1}, v_{2}\right\}$ is connected. Order $V_{H}=\left\{v_{3}, v_{4}, \ldots, v_{n}\right\}$ such that $v_{n}=w$, and for all $i \geq 3$,
$$
d_{H}\left(v_{i}, w\right) \geq d_{H}\left(v_{i+1}, w\right) .
$$
Therefore for each $i \in[1, n-1]$, we find at least one $j>i$ such that $v_{i} v_{j} \in G$ (possibly $v_{j}=w$ ). In particular, for all $1 \leq i<n$,
$$
\left|N_{G}\left(v_{i}\right) \cap\left\{v_{1}, \ldots, v_{i-1}\right\}\right|<d_{G}\left(v_{i}\right) \leq \Delta(G) .
$$
Then colour $v_{1}, v_{2}, \ldots, v_{n}$ in this order as follows: $\alpha\left(v_{1}\right)=1=\alpha\left(v_{2}\right)$ and
$$
\alpha\left(v_{i}\right)=\min \left\{r \mid r \neq \alpha\left(v_{j}\right) \text { for all } v_{j} \in N_{G}\left(v_{i}\right) \text { with } j<i\right\} .
$$
The colouring $\alpha$ is proper.
By (4.3), $\alpha\left(v_{i}\right) \leq \Delta(G)$ for all $i \in[1, n-1]$. Also, $w=v_{n}$ has two neighbours, $v_{1}$ and $v_{2}$, of the same colour 1 , and since $v_{n}$ has at most $\Delta(G)$ neighbours, there is an available colour for $v_{n}$, and so $\alpha\left(v_{n}\right) \leq \Delta(G)$. This shows that $G$ has a proper $\Delta(G)$ colouring, and, consequently, $\chi(G) \leq \Delta(G)$.
Example 4.8. Suppose we have $n$ objects $V=\left\{v_{1}, \ldots, v_{n}\right\}$, some of which are not compatible (like chemicals that react with each other, or worse, graph theorists who will fight during a conference). In the storage problem we would like to find a partition of the set $V$ with as few classes as possible such that no class contains two incompatible elements. In graph theoretical terminology we consider the graph $G=(V, E)$, where $v_{i} v_{j} \in E$ just in case $v_{i}$ and $v_{j}$ are incompatible, and we would like to colour the vertices of $G$ properly using as few colours as possible. This problem requires that we find $\chi(G)$.
Unfortunately, no good algorithms are known for determining $\chi(G)$, and, indeed, the chromatic number problem is NP-complete. Already the problem if $\chi(G)=3$ is NP-complete. (However, as we have seen, the problem whether $\chi(G)=2$ has a fast algorithm.)
## The chromatic polynomial
A given graph $G$ has many different proper vertex colourings $\alpha: V_{G} \rightarrow[1, k]$ for sufficiently large natural numbers $k$. Indeed, see Lemma 4.5 to be certain on this point.
DEFINITION. The chromatic polynomial of $G$ is the function $\chi_{G}: \mathbb{N} \rightarrow \mathbb{N}$, where
$$
\chi_{G}(k)=\mid\left\{\alpha \mid \alpha: V_{G} \rightarrow[1, k] \text { a proper colouring }\right\} \mid .
$$
This notion was introduced by BIRKHOFF (1912), BIRKHOFF AND LEWIS (1946), to attack the famous 4-Colour Theorem, but its applications have turned out to be elsewhere.
If $k<\chi(G)$, then clearly $\chi_{G}(k)=0$, and, indeed,
$$
\chi(G)=\min \left\{k \mid \chi_{G}(k) \neq 0\right\} .
$$
Therefore, if we can find the chromatic polynomial of $G$, then we easily compute the chromatic number $\chi(G)$ just by evaluating $\chi_{G}(k)$ for $k=1,2, \ldots$ until we hit a nonzero value. Theorem 4.13 will give the tools for constructing $\chi_{G}$.
Example 4.9. Consider the complete graph $K_{4}$ on $\left\{v_{1}, v_{2}, v_{3}, v_{4}\right\}$. Let $k \geq \chi\left(K_{4}\right)=4$. The vertex $v_{1}$ can be first given any of the $k$ colours, after which $k-1$ colours are available for $v_{2}$. Then $v_{3}$ has $k-2$ and finally $v_{4}$ has $k-3$ available colours. Therefore there are $k(k-1)(k-2)(k-3)$ different ways to properly colour $K_{4}$ with $k$ colours, and so
$$
\chi_{K_{4}}(k)=k(k-1)(k-2)(k-3) .
$$
On the other hand, in the discrete graph $\bar{K}_{4}$ has no edges, and thus any $k$-colouring is a proper colouring. Therefore
$$
\chi_{\bar{K}_{4}}(k)=k^{4} .
$$
Remark. The considered method for checking the number of possibilities to colour a 'next vertex' is exceptional, and for more nonregular graphs it should be avoided.
Definition. Let $G$ be a graph, $e=u v \in G$, and let $x=x(u v)$ be a new contracted vertex. The graph $G * e$ on
$$
V_{G * e}=\left(V_{G} \backslash\{u, v\}\right) \cup\{x\}
$$
is obtained from $G$ by contracting the edge $e$, when
$$
E_{G * e}=\left\{f \mid f \in E_{G}, f \text { has no end } u \text { or } v\right\} \cup\{w x \mid w u \in G \text { or } w v \in G\} .
$$
Hence $G * e$ is obtained by introducing a new vertex $x$, and by replacing all edges $w u$ and $w v$ by $w x$, and the vertices $u$ and $v$ are deleted.
(Of course, no loops or parallel edges are allowed in the new graph $G * e$.)
Theorem 4.13. Let $G$ be a graph, and let $e \in G$. Then
$$
\chi_{G}(k)=\chi_{G-e}(k)-\chi_{G * e}(k) .
$$
Proof. Let $e=u v$. The proper $k$-colourings $\alpha: V_{G} \rightarrow[1, k]$ of $G-e$ can be divided into two disjoint cases, which together show that $\chi_{G-e}(k)=\chi_{G}(k)+\chi_{G * e}(k)$ :
(1) If $\alpha(u) \neq \alpha(v)$, then $\alpha$ corresponds to a unique proper $k$-colouring of $G$, namely $\alpha$. Hence the number of such colourings is $\chi_{G}(k)$.
(2) If $\alpha(u)=\alpha(v)$, then $\alpha$ corresponds to a unique proper $k$-colouring of $G * e$, namely $\alpha$, when we set $\alpha(x)=\alpha(u)$ for the contracted vertex $x=x(u v)$. Hence the number of such colourings is $\chi_{G * e}(k)$.
Theorem 4.14. The chromatic polynomial is a polynomial.
Proof. The proof is by induction on $\varepsilon_{G}$. Indeed, $\chi_{\bar{K}_{n}}(k)=k^{n}$ for the discrete graph, and for two polynomials $P_{1}$ and $P_{2}$, also $P_{1}-P_{2}$ is a polynomial. The claim follows from Theorem 4.13 , since there $G-e$ and $G * e$ have less edges than $G$.
The connected components of a graph can be coloured independently, and so
Lemma 4.9. Let the graph $G$ have the connected components $G_{1}, G_{2}, \ldots, G_{m}$. Then
$$
\chi_{G}(k)=\chi_{G_{1}}(k) \chi_{G_{2}}(k) \ldots \chi_{G_{m}}(k) .
$$
Theorem 4.15. Let $T$ be a tree of order $n$. Then $\chi_{T}(k)=k(k-1)^{n-1}$.
Proof. We use induction on $n$. For $n \leq 2$, the claim is obvious. Suppose that $n \geq 3$, and let $e=v u \in T$, where $v$ is a leaf. By Theorem 4.13, $\chi_{T}(k)=\chi_{T-e}(k)-\chi_{T * e}(k)$. Here $T * e$ is a tree of order $n-1$, and thus, by the induction hypothesis, $\chi_{T * e}(k)=$ $k(k-1)^{n-2}$. The graph $T-e$ consists of the isolated $v$ and a tree of order $n-1$. By Lemma 4.9, and the induction hypothesis, $\chi_{T-e}(k)=k \cdot k(k-1)^{n-2}$. Therefore $\chi_{T}(k)=k(k-1)^{n-1}$.
Example 4.10. Consider the graph $G$ of order 4 from the above. Then we have the following reductions.
Theorem 4.13 reduces the computation of $\chi_{G}$ to the discrete graphs. However, we know the chromatic polynomials for trees (and complete graphs, as an exercise), and so there is no need to prolong the reductions beyond these. In our example, we have obtained
$$
\begin{aligned}
\chi_{G-e}(k) & =\chi_{G-\{e, f\}}(k)-\chi_{(G-e) * f}(k) \\
& =k(k-1)^{3}-k(k-1)^{2}=k(k-1)^{2}(k-2),
\end{aligned}
$$
and so
$$
\begin{aligned}
\chi_{G}(k) & =\chi_{G-e}(k)-\chi_{G * e}(k)=k(k-1)^{2}(k-2)-k(k-1)(k-2) \\
& =k(k-1)(k-2)^{2}=k^{4}-5 k^{3}+8 k^{2}-4 k .
\end{aligned}
$$
For instance, for 3 colours, there are 6 proper colourings of the given graph.
Chromatic Polynomial Problems. It is difficult to determine $\chi_{G}$ of a given graph, since the reduction method provided by Theorem 4.13 is time consuming. Also, there is known no characterization, which would tell from any polynomial $P(k)$ whether it is a chromatic polynomial of some graph. For instance, the polynomial $k^{4}-3 k^{3}+3 k^{2}$ is not a chromatic polynomial of any graph, but it seems to satisfy the general properties (that are known or conjectured) of these polynomials. REED (1968) conjectured that the coefficients of a chromatic polynomial should first increase and then decrease in absolute value. REED (1968) and TUTTE (1974) proved that for each G of order $v_{G}=n$ :
- The degree of $\chi_{G}(k)$ equals $n$.
- The coefficient of $k^{n}$ equals 1 .
- The coefficient of $k^{n-1}$ equals $-\varepsilon_{G}$.
- The constant term is 0 .
- The coefficients alternate in sign.
- $\chi_{G}(m) \leq m(m-1)^{n}-1$ for all positive integers $m$, when $G$ is connected.
- $\chi_{G}(x) \neq 0$ for all real numbers $0<x<1$.
## Graphs on Surfaces
### Planar graphs
The plane representations of graphs are by no means unique. Indeed, a graph $G$ can be drawn in arbitrarily many different ways. Also, the properties of a graph are not necessarily immediate from one representation, but may be apparent from another. There are, however, important families of graphs, the surface graphs, that rely on the (topological or geometrical) properties of the drawings of graphs. We restrict ourselves in this chapter to the most natural of these, the planar graphs. The geometry of the plane will be treated intuitively.
A planar graph will be a graph that can be drawn in the plane so that no two edges intersect with each other. Such graphs are used, e.g., in the design of electrical (or similar) circuits, where one tries to (or has to) avoid crossing the wires or laser beams. Planar graphs come into use also in some parts of mathematics, especially in group theory and topology.
There are fast algorithms (linear time algorithms) for testing whether a graph is planar or not. However, the algorithms are all rather difficult to implement. Most of them are based on an algorithm designed by AUSLANDER AND PARTER (1961) see Section 6.5 of
S. SKIENA, "Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica", Addison-Wesley, 1990.
## Definition
DEFINITION. A graph $G$ is a planar graph, if it has a plane figure $P(G)$, called the plane embedding of $G$, where the lines (or continuous curves) corresponding to the edges do not intersect each other except at their ends.
The complete bipartite graph $K_{2,4}$ is a planar graph.
DEFinition. An edge $e=u v \in G$ is subdivided, when it is replaced by a path $u \rightarrow x \rightarrow v$ of length two by introducing a new vertex $x$. A subdivision $H$ of a graph $G$ is obtained from $G$ by a sequence of subdivisions.
The following result is clear.
Lemma 5.1. A graph is planar if and only if its subdivisions are planar.
## Geometric properties
It is clear that the graph theoretical properties of $G$ are inherited by all of its plane embeddings. For instance, the way we draw a graph $G$ in the plane does not change its maximum degree or its chromatic number. More importantly, there are - as we shall see - some nontrivial topological (or geometric) properties that are shared by the plane embeddings.
We recall first some elements of the plane geometry. Let $F$ be an open set of the plane $\mathbb{R} \times \mathbb{R}$, that is, every point $x \in F$ has a disk centred at $x$ and contained in $F$. Then $F$ is a region, if any two points $x, y \in F$ can be joined by a continuous curve the points of which are all in $F$. The boundary $\partial(F)$ of a region $F$ consists of those points for which every neighbourhood contains points from $F$ and its complement.
Let $G$ be a planar graph, and $P(G)$ one of its plane embeddings. Regard now each edge $e=u v \in G$ as a line from $u$ to $v$. The set $(\mathbb{R} \times \mathbb{R}) \backslash E_{G}$ is open, and it is divided into a finite number of disjoint regions, called the faces of $P(G)$.
DEFINition. A face of $P(G)$ is an interior face, if it is bounded. The (unique) face that is unbounded is called the exterior face of $P(G)$. The edges that surround a face $F$ constitute the boundary $\partial(F)$ of $F$. The exterior boundary is the boundary of the exterior face. The vertices (edges, resp.) on the exterior boundary are called exterior vertices exterior edges, resp.). Vertices (edges, resp.) that are not on the exte-
rior boundary are interior vertices interior edges, resp.).
Embeddings $P(G)$ satisfy some properties that we accepts at face value.
Lemma 5.2. Let $P(G)$ be a plane embedding of a planar graph $G$.
(i) Two different faces $F_{1}$ and $F_{2}$ are disjoint, and their boundaries can intersect only on edges.
(ii) $P(G)$ has a unique exterior face.
(iii) Each edge e belongs to the boundary of at most two faces.
(iv) Each cycle of $G$ surrounds (that is, its interior contains) at least one internal face of $P(G)$.
(v) A bridge of $G$ belongs to the boundary of only one face.
(vi) An edge that is not a bridge belongs to the boundary of exactly two faces. If $P(G)$ is a plane embedding of a graph $G$, then so is any drawing $P^{\prime}(G)$ which is obtained from $P(G)$ by an injective mapping of the plane that preserves continuous curves. This means, in particular, that every planar graph has a plane embedding inside any geometric circle of arbitrarily small radius, or inside any geometric triangle.
## Euler's formula
Lemma 5.3. A plane embedding $P(G)$ of a planar graph $G$ has no interior faces if and only if $G$ is acyclic, that is, if and only if the connected components of $G$ are trees.
Proof. This is clear from Lemma 5.2.
The next general form of Euler's formula was proved by LEGENDRE (1794).
Theorem 5.1 (Euler's formula). Let $G$ be a connected planar graph, and let $P(G)$ be any of its plane embeddings. Then
$$
v_{G}-\varepsilon_{G}+\varphi=2,
$$
where $\varphi$ is the number of faces of $P(G)$.
Proof. We shall prove the claim by induction on the number of faces $\varphi$ of a plane embedding $P(G)$. First, notice that $\varphi \geq 1$, since each $P(G)$ has an exterior face.
If $\varphi=1$, then, by Lemma 5.3, there are no cycles in $G$, and since $G$ is connected, it is a tree. In this case, by Theorem 2.4, we have $\varepsilon_{G}=v_{G}-1$, and the claim holds.
Suppose then that the claim is true for all plane embeddings with less than $\varphi$ faces for $\varphi \geq 2$. Let $P(G)$ be a plane embedding of a connected planar graph such that $P(G)$ has $\varphi$ faces.
Let $e \in G$ be an edge that is not a bridge. The subgraph $G-e$ is planar with a plane embedding $P(G-e)=P(G)-e$ obtained by simply erasing the edge $e$. Now $P(G-e)$ has $\varphi-1$ faces, since the two faces of $P(G)$ that are separated by $e$ are merged into one face of $P(G-e)$. By the induction hypothesis, $v_{G-e}-\varepsilon_{G-e}+(\varphi-1)=2$, and hence $v_{G}-\left(\varepsilon_{G}-1\right)+(\varphi-1)=2$, and the claim follows.
In particular, we have the following invariant property of planar graphs.
Corollary 5.1. Let $G$ be a planar graph. Then every plane embedding of $G$ has the same number of faces:
$$
\varphi_{G}=\varepsilon_{G}-v_{G}+2
$$
## Maximal planar graphs
Lemma 5.4. If $G$ is a planar graph of order $v_{G} \geq 3$, then $\varepsilon_{G} \leq 3 v_{G}-6$. Moreover, if $G$ has no triangles $C_{3}$, then $\varepsilon_{G} \leq 2 v_{G}-4$. Proof. If $G$ is disconnected with connected components $G_{i}$, for $i \in[1, k]$, and if the claim holds for these smaller (necessarily planar) graphs $G_{i}$, then it holds for $G$, since
$$
\varepsilon_{G}=\sum_{i=1}^{v_{G}} \varepsilon_{G_{i}} \leq 3 \sum_{i=1}^{v_{G}} v_{G_{i}}-6 k=3 v_{G}-6 k \leq 3 v_{G}-6
$$
It is thus sufficient to prove the claim for connected planar graphs.
Also, the case where $\varepsilon_{G} \leq 2$ is clear. Suppose thus that $\varepsilon_{G} \geq 3$.
Each face $F$ of an embedding $P(G)$ contains at least three edges on its boundary $\partial(F)$. Hence $3 \varphi \leq 2 \varepsilon_{G}$, since each edge lies on at most two faces. The first claim follows from Euler's formula.
The second claim is proved similarly except that, in this case, each face $F$ of $P(G)$ contains at least four edges on its boundary (when $G$ is connected and $\varepsilon_{G} \geq 4$ ).
An upper bound for $\delta(G)$ for planar graphs was achieved by HEAWOOD.
Theorem 5.2 (HEAWOOD (1890)). If $G$ is a planar graph, then $\delta(G) \leq 5$.
Proof. If $v_{G} \leq 2$, then there is nothing to prove. Suppose $v_{G} \geq 3$. By the handshaking lemma and the previous lemma,
$$
\delta(G) \cdot v_{G} \leq \sum_{v \in G} d_{G}(v)=2 \varepsilon_{G} \leq 6 v_{G}-12 .
$$
It follows that $\delta(G) \leq 5$.
Theorem 5.3. $K_{5}$ and $K_{3,3}$ are not planar graphs.
Proof. By Lemma 5.4, a planar graph of order 5 has at most 9 edges, but $K_{5}$ has 5 vertices and 10 edges. By the second claim of Lemma 5.4, a triangle-free planar graph of order 6 has at most 8 edges, but $K_{3,3}$ has 6 vertices and 9 edges.
Definition. A planar graph $G$ is maximal, if $G+e$ is nonplanar for every $e \notin G$.
Example 5.1. Clearly, if we remove one edge from $K_{5}$, the result is a maximal planar graph. However, if an edge is removed from $K_{3,3}$, the result is not maximal!
Lemma 5.5. Let $F$ be a face of a plane embedding $P(G)$ that has at least four edges on its boundary. Then there are two nonadjacent vertices on the boundary of $F$.
Proof. Assume that the set of the boundary vertices of $F$ induces a complete subgraph $K$. The edges of $K$ are either on the boundary or they are not inside $F$ (since $F$ is a face.) Add a new vertex $x$ inside $F$, and connect the vertices of $K$ to $x$. The result is a plane embedding of a graph $H$ with $V_{H}=V_{G} \cup\{x\}$ (that has $G$ as its induced subgraph). The induced subgraph $H[K \cup\{x\}]$ is complete, and since $H$ is planar, we have $|K|<4$ as required. By the previous lemma, if a face has a boundary of at least four edges, then an edge can be added to the graph (inside the face), and the graph remains to be planar. Hence we have proved
Corollary 5.2. If $G$ is a maximal planar graph with $v_{G} \geq 3$, then $G$ is triangulated, that is, every face of a plane embedding $P(G)$ has a boundary of exactly three edges.
Theorem 5.4. For a maximal planar graph $G$ of order $v_{G} \geq 3, \varepsilon_{G}=3 v_{G}-6$.
Proof. Each face $F$ of an embedding $P(G)$ is a triangle having three edges on its boundary. Hence $3 \varphi=2 \varepsilon_{G}$, since there are now no bridges. The claim follows from Euler's formula.
## Kuratowski's theorem
Theorem 5.5 will give a simple criterion for planarity of graphs. This theorem (due to KURATOWSKI in 1930) is one of the jewels of graph theory. In fact, the theorem was proven earlier by PONTRYAGIN (1927-1928), and also independently by FRINK AND SMITH (1930). For history of the result, see
J.W. KenNEDY, L.V. QUiNTAS, AND M.M. SYSLO, The theorem on planar graphs. Historia Math. 12 (1985), 356 - 368.
The graphs $K_{5}$ and $K_{3,3}$ are the smallest nonplanar graphs, and, by Lemma 5.1, if $G$ contains a subdivision of $K_{5}$ or $K_{3,3}$ as a subgraph, then $G$ is not planar. We prove the converse of this result in what follows. Therefore
Theorem 5.5 (KURATOWSKi (1930)). A graph is planar if and only if it contains no subdivision of $K_{5}$ or $K_{3,3}$ as a subgraph.
We prove this result along the lines of THOMASSEN (1981) using 3-connectivity.
Example 5.2. The cube $Q_{k}$ is planar only for $k=1,2,3$. Indeed, the graph $Q_{4}$ contains a subdivision of $K_{3,3}$, and thus by Theorem 5.5 it is not planar. On the other hand, each $Q_{k}$ with $k \geq 4$ has $Q_{4}$ as a subgraph, and therefore they are nonplanar. The subgraph of $Q_{4}$ that is a subdivision of $K_{3,3}$ is given below.
Definition. A graph $G$ is called a Kuratowski graph, if it is a subdivision of $K_{5}$ or $K_{3,3}$.
Lemma 5.6. Let $E \subseteq E_{G}$ be the set of the boundary edges of a face $F$ in a plane embedding of $G$. Then there exists a plane embedding $P(G)$, where the edges of $E$ are exterior edges.
Proof. This is a geometric proof. Choose a circle that contains every point of the plane embedding (including all points of the edges) such that the centre of the circle is inside the given face. Then use geometric inversion with respect to this circle. This will map the given face as the exterior face of the image plane embedding.
Lemma 5.7. Let $G$ be a nonplanar graph without Kuratowski graphs of minimal total size $\varepsilon_{G}+v_{G}$. Then $G$ is 3-connected.
Proof. By the minimality assumption, $G$ is connected. We show then that $G$ is 2connected. On the contrary, assume that $v$ is a cut vertex of $G$, and let $A_{1}, \ldots, A_{k}$ be the connected components of $G-v$.
Since $G$ is minimal nonplanar with respect to $\varepsilon_{G}$, the subgraphs $G_{i}=G\left[A_{i} \cup\{v\}\right]$ have plane embeddings $P\left(G_{i}\right)$, where $v$ is an exterior vertex. We can glue these plane embeddings together at $v$ to obtain a plane embedding of $G$, and this will contradict the choice of $G$.
Assume then that $G$ has a separating set $S=\{u, v\}$. Let $G_{1}$ and $G_{2}$ be any subgraphs of $G$ such that $E_{G}=E_{G_{1}} \cup E_{G_{2}}, S=V_{G_{1}} \cap V_{G_{2}}$, and both $G_{1}$ and $G_{2}$ contain a connected component of $G-S$. Since $G$ is 2-connected (by the above), there are paths $u \stackrel{\star}{\rightarrow} v$ in $G_{1}$ and $G_{2}$. Indeed, both $u$ and $v$ are adjacent to a vertex of each connected component of $G-S$. Let $H_{i}=G_{i}+u v$. (Maybe $u v \in G$.) If both $H_{1}$ and $H_{2}$ are planar, then, by Lemma 5.6, they have plane embeddings, where $u v$ is an exterior edge. It is now easy to glue $H_{1}$ and $H_{2}$ together on the edge $u v$ to obtain a plane embedding of $G+u v$, and thus of $G$.
We conclude that $H_{1}$ or $H_{2}$ is nonplanar, say $H_{1}$. Now $\varepsilon_{H_{1}}<\varepsilon_{G}$, and so, by the minimality of $G, H_{1}$ contains a Kuratowski graph $H$. However, there is a path $u \stackrel{\star}{\rightarrow} v$ in $H_{2}$, since $G_{2} \subseteq H_{2}$. This path can be regarded as a subdivision of $u v$, and thus $G$ contains a Kuratowski graph. This contradiction shows that $G$ is 3-connected.
Lemma 5.8. Let $G$ be a 3-connected graph of order $v_{G} \geq 5$. Then there exists an edge $e \in G$ such that the contraction $G * e$ is 3-connected.
Proof. On the contrary suppose that for any $e \in G$, the graph $G * e$ has a separating set $S$ with $|S|=2$. Let $e=u v$, and let $x=x(u v)$ be the contracted vertex. Necessarily $x \in S$, say $S=\{x, z\}$ (for, otherwise, $S$ would separate $G$ already). Therefore $T=$ $\{u, v, z\}$ separates $G$. Assume that $e$ and $S$ are chosen such that $G-T$ has a connected component $A$ with the least possible number of vertices.
There exists a vertex $y \in A$ with $z y \in G$. (Otherwise $\{u, v\}$ would separate G.) The graph $G *(z y)$ is not 3connected by assumption, and hence, as in the above, there exists a vertex $w$ such that $R=\{z, y, w\}$ separates G. It can be that $w \in\{u, v\}$, but by symmetry we can suppose that $w \neq u$.
Since $u v \in G, G-R$ has a connected component $B$ such that $u, v \notin B$. For each $y^{\prime} \in B$, there exists a path $P: u \stackrel{\star}{\rightarrow} y^{\prime}$ in $G-\{z, w\}$, since $G$ is 3-connected, and hence this $P$ goes through $y$. Therefore $y^{\prime}$ is connected to $y$ also in $G-T$, that is, $y^{\prime} \in A$, and so $B \subseteq A$. The inclusion is proper, since $y \notin B$. Hence $|B|<|A|$, and this contradicts the choice of $A$.
By the next lemma, a Kuratowski graph cannot be created by contractions.
Lemma 5.9. Let $G$ be a graph. If for some $e \in G$ the contraction $G * e$ has a Kuratowski subgraph, then so does $G$.
Proof. The proof consists of several cases depending on the Kuratowski graph, and how the subdivision is made. We do not consider the details of these cases.
Let $H$ be a Kuratowski graph of $G * e$, where $x=x(u v)$ is the contracted vertex for $e=u v$. If $d_{H}(x)=2$, then the claim is obviously true. Suppose then that $d_{H}(x)=3$ or 4. If there exists at most one edge $x y \in H$ such that $u y \in G$ (or $v y \in G$ ), then one easily sees that $G$ contains a Kuratowski graph.
There remains only one case, where $H$ is a subdivision of $K_{5}$, and both $u$ and $v$ have 3 neighbours in the subgraph of $G$ corresponding to $H$. In this case, $G$ contains a subdivision of $K_{3,3}$.
Lemma 5.10. Every 3-connected graph $G$ without Kuratowski subgraphs is planar.
Proof. The proof is by induction on $v_{G}$. The only 3-connected graph of order 4 is the planar graph $K_{4}$. Therefore we can assume that $v_{G} \geq 5$.
By Lemma 5.8, there exists an edge $e=u v \in G$ such that $G * e$ (with a contracted vertex $x$ ) is 3-connected. By Lemma 5.9, $G * e$ has no Kuratowski subgraphs, and hence $G * e$ has a plane embedding $P(G * e)$ by the induction hypothesis. Consider the part $P(G * e)-x$, and let $C$ be the boundary of the face of $P(G * e)-x$ containing $x$ (in $P(G * e)$ ). Here $C$ is a cycle of $G$ (since $G$ is 3-connected).
Now since $G-\{u, v\}=(G * e)-x, P(G * e)-x$ is a plane embedding of $G-\{u, v\}$, and $N_{G}(u) \subseteq V_{C} \cup\{v\}$ and $N_{G}(v) \subseteq V_{C} \cup\{u\}$. Assume, by symmetry, that $d_{G}(v) \leq$ $d_{G}(u)$. Let $N_{G}(v) \backslash\{u\}=\left\{v_{1}, v_{2}, \ldots, v_{k}\right\}$ in order along the cycle $C$. Let $P_{i, j}: v_{i} \stackrel{\star}{\rightarrow} v_{j}$ be the path along $C$ from $v_{i}$ to $v_{j}$. We obtain a plane embedding of $G-u$ by drawing (straight) edges $v v_{i}$ for $1 \leq i \leq k$.
(1) If $N_{G}(u) \backslash\{v\} \in P_{i, i+1}(i+1$ is taken modulo $k$ ) for some $i$, then, clearly, $G$ has a plane embedding (obtained from $P(G)-u$ by putting $u$ inside the triangle $\left(v, v_{i}, v_{i+1}\right)$ and by drawing the edges with an end $u$ inside this triangle).
(2) Assume there are $y, z \in N_{G}(u) \backslash\{v\}$ such that $y \in P_{i j}$ and $z \notin P_{i j}$ for some $i$ and $j$, where $y, z \notin\left\{v_{i}, v_{j}\right\}$. Now, $\left\{u, v_{i}, v_{i+1}\right\} \cup\{v, z, y\}$ form a subdivision of $K_{3,3}$.
By (1) and (2), we can assume that $N_{G}(u) \backslash\{v\} \subseteq N_{G}(v)$. Therefore, $N_{G}(u) \backslash\{v\}=N_{G}(v) \backslash\{u\}$ by the assumption $d_{G}(v) \leq d_{G}(u)$. Also, by (1), $d_{G}(v)=d_{G}(u)>3$. But now $u, v, v_{1}, v_{2}, v_{3}$ give a subdivision of $K_{5}$.
Proof of Theorem 5.5. By Theorem 5.3 and Lemma 5.1, we need to show that each nonplanar graph $G$ contains a Kuratowski subgraph. On the contrary, suppose that $G$ is a nonplanar graph that has a minimal size $\varepsilon_{G}$ such that $G$ does not contain a Kuratowski subgraph. Then, by Lemma 5.7, $G$ is 3-connected, and by Lemma 5.10, it is planar. This contradiction proves the claim.
Example 5.3. Any graph $G$ can be drawn in the plane so that three of its edges never intersect at the same point. The crossing number $\times(G)$ is the minimum number of intersections of its edges in such plane drawings of $G$. Therefore $G$ is planar if and only if $\times(G)=0$, and, for instance, $\times\left(K_{5}\right)=1$.
We show that $\times\left(K_{6}\right)=3$. For this we need to show that $\times\left(K_{6}\right) \geq 3$. For the equality, one is invited to design a drawing with exactly 3 crossings.
Let $X\left(K_{6}\right)$ be a drawing of $K_{6}$ using $c$ crossings so that two edges cross at most once. Add a new vertex at each crossing. This results in a planar graph $G$ on $c+6$ vertices and $2 c+15$ edges. Now $c \geq 3$, since $\varepsilon_{G}=2 c+15 \leq 3(c+6)-6=3 v_{G}-6$.
### Colouring planar graphs
The most famous problem in the history of graph theory is that of the chromatic number of planar graphs. The problem was known as the 4-Colour Conjecture for more than 120 years, until it was solved by APPEL AND HAKEN in 1976: if G is a planar graph, then $\chi(G) \leq 4$. The 4-Colour Conjecture has had a deep influence on the theory of graphs during the last 150 years. The solution of the 4-Colour Theorem is difficult, and it requires the assistance of a computer.
## The 5-colour theorem
We prove HEAWOOD's result (1890) that each planar graph is properly 5-colourable.
Lemma 5.11. If $G$ is a planar graph, then $\chi(G) \leq 6$.
Proof. The proof is by induction on $v_{G}$. Clearly, the claim holds for $v_{G} \leq 6$. By Theorem 5.2, a planar graph $G$ has a vertex $v$ with $d_{G}(v) \leq 5$. By the induction hypothesis, $\chi(G-v) \leq 6$. Since $d_{G}(v) \leq 5$, there is a colour $i$ available for $v$ in the 6-colouring of $G-v$, and so $\chi(G) \leq 6$.
The proof of the following theorem is partly geometric in nature.
Theorem 5.6 (HEAWOOD (1890)). If $G$ is a planar graph, then $\chi(G) \leq 5$.
Proof. Suppose the claim does not hold, and let $G$ be a 6-critical planar graph. Recall that for $k$-critical graphs $H, \delta(H) \geq k-1$, and thus there exists a vertex $v$ with $d_{G}(v)=\delta(G) \geq 5$. By Theorem 5.2, $d_{G}(v)=5$.
Let $\alpha$ be a proper 5-colouring of $G-v$. Such a colouring exists, because $G$ is 6-critical. By assumption, $\chi(G)>5$, and therefore for each $i \in[1,5]$, there exists a neighbour $v_{i} \in N_{G}(v)$ such that $\alpha\left(v_{i}\right)=i$. Suppose these neighbours $v_{i}$ of $v$ occur in the plane in the geometric order of the figure.
Consider the subgraph $G[i, j] \subseteq G$ made of colours $i$ and $j$. The vertices $v_{i}$ and $v_{j}$ are in the same connected component of $G[i, j]$ (for, otherwise we interchange the colours $i$ and $j$ in the connected component containing $v_{j}$ to obtain a recolouring of $G$, where $v_{i}$ and $v_{j}$ have the same colour $i$, and then recolour $v$ with the remaining colour $j$ ).
Let $P_{i j}: v_{i} \stackrel{\star}{\rightarrow} v_{j}$ be a path in $G[i, j]$, and let $C=\left(v v_{1}\right) P_{13}\left(v_{3} v\right)$. By the geometric assumption, exactly one of $v_{2}, v_{4}$ lies inside the region enclosed by the cycle $C$. Now, the path $P_{24}$ must meet $C$ at some vertex of $C$, since $G$ is planar. This is a contradiction, since the vertices of $P_{24}$ are coloured by 2 and 4 , but $C$ contains no such colours.
The final word on the chromatic number of planar graphs was proved by APPEL AND HAKEN in 1976.
Theorem 5.7 (4-Colour Theorem). If $G$ is a planar graph, then $\chi(G) \leq 4$.
By the following theorem, each planar graph can be decomposed into two bipartite graphs. Theorem 5.8. Let $G=(V, E)$ be a 4-chromatic graph, $\chi(G) \leq 4$. Then the edges of $G$ can be partitioned into two subsets $E_{1}$ and $E_{2}$ such that $\left(V, E_{1}\right)$ and $\left(V, E_{2}\right)$ are both bipartite.
Proof. Let $V_{i}=\alpha^{-1}(i)$ be the set of vertices coloured by $i$ in a proper 4-colouring $\alpha$ of $G$. The define $E_{1}$ as the subset of the edges of $G$ that are between the sets $V_{1}$ and $V_{2}$; $V_{1}$ and $V_{4} ; V_{3}$ and $V_{4}$. Let $E_{2}$ be the rest of the edges, that is, they are between the sets $V_{1}$ and $V_{3} ; V_{2}$ and $V_{3} ; V_{2}$ and $V_{4}$. It is clear that $\left(V, E_{1}\right)$ and $\left(V, E_{2}\right)$ are bipartite, since the sets $V_{i}$ are stable.
## Map colouring*
The 4-Colour Conjecture was originally stated for maps. In the map-colouring problem we are given several countries with common borders, and we wish to colour each country so that no neighbouring countries obtain the same colour. How many colours are needed?
A border between two countries is assumed to have a positive length - in particular, countries that have only one point in common are not allowed in the map colouring.
Formally, we define a map as a connected planar (embedding of a) graph with no bridges. The edges of this graph represent the boundaries between countries. Hence a country is a face of the map, and two neighbouring countries share a common edge (not just a single vertex). We deny bridges, because a bridge in such a map would be a boundary inside a country.
The map-colouring problem is restated as follows: How many colours are needed for the faces of a plane embedding so that no adjacent faces obtain the same colour.
The illustrated map can be 4-coloured, and it cannot be coloured using only 3 colours, because ev-
ery two faces have a common border.
Let $F_{1}, F_{2}, \ldots, F_{n}$ be the countries of a map $M$, and define a graph $G$ with $V_{G}=$ $\left\{v_{1}, v_{2}, \ldots, v_{n}\right\}$ such that $v_{i} v_{j} \in G$ if and only if the countries $F_{i}$ and $F_{j}$ are neighbours. It is easy to see that $G$ is a planar graph. Using this notion of a dual graph, we can state the map-colouring problem in new form: What is the chromatic number of a planar graph? By the 4-Colour Theorem it is at most four.
Map-colouring can be used in rather generic topological setting, where the maps are defined by curves in the plane. As an example, consider finitely many simple closed curves in the plane. These curves divide the plane into regions. The regions are 2-colourable. That is, the graph where the vertices correspond to the regions, and the edges correspond to the neighbourhood relation, is bipartite. To see this, colour a region by 1 , if the region is inside an odd number of curves, and, otherwise, colour it by 2 .
## History of the 4-Colour Theorem
That four colours suffice planar maps was conjectured around 1850 by FRANCIS Guthrie, a student of DE MORGAN at University College of London. During the following 120 years many outstanding mathematicians tried to solve the problem, and some of them even thought that they had been successful.
In 1879 CAYLEY pointed out some difficulties that lie in the conjecture. The same year ALFRED KEMPE published a paper, where he claimed a proof of the 4CC. The basic idea in KEMPE's argument (known later as Kempe chains) was the same as later used by HEAWOOD to prove the 5-Colour Theorem, (Theorem 5.6).
For more than 10 years KEMPE's proof was considered to be valid. For instance, TAIT published two papers on the 4CC in the 1880's that contained clever ideas, but also some further errors. In 1890 HEAWOOD showed that KEMPE's proof had serious gaps. As we shall see in the next chapter, HEAWOOD discovered the number of colours needed for all maps on other surfaces than the plane. Also, he proved that if the number of edges around each region is divisible by 3 , then the map is 4-colourable.
One can triangulate any planar graph $G$ (drawn in the plane), by adding edges to divide the faces into triangles. BIRKHOFF introduced one of the basic notions (reducibility) needed in the proof of the $4 \mathrm{CC}$. In a triangulation, a configuration is a part that is contained inside a cycle. An unavoidable set is a set of configurations such that any triangulation must contain one of the configurations in the set. A configuration is said to be reducible, if it is not contained in a triangulation of a minimal counter example to the 4CC.
The search for avoidable sets began in 1904 with work of WEINICKE, and in 1922 FRANKLIN showed that the 4CC holds for maps with at most 25 regions. This number was increased to 27 by REYNOLDS (1926), to 35 by WINN (1940), to 39 by ORE AND STEMPLE (1970), to 95 by MAYER (1976).
The final notion for the solution was due to HEESCH, who in 1969 introduced discharging. This consists of assigning to a vertex $v$ the charge $6-d_{G}(v)$. From Euler's formula we see that for the sum of the charges, we have
$$
\sum_{v}\left(6-d_{G}(v)\right)=12 .
$$
Now, a given set $S$ of configurations can be proved to be unavoidable, if for a triangulation, that does not contain a configuration from $S$, one can 'redistribute' the charges so that no $v$ comes up with a positive charge. According to HEESCH one might be satisfied with a set of 8900 configurations to prove the 4CC. There were difficulties with his approach that were solved in 1976 by APPEL AND HAKEN. They based the proof on reducibility using Kempe chains, and ended up with an unavoidable set with over 1900 configurations and some 300 discharging rules. The proof used 1200 hours of computer time. (KOCH assisted with the computer calculations.) A simplified proof by ROBERTSON, SANDERS, SEYMOUR AND THOMAS (1997) uses 633 configurations and 32 discharging rules. Because of these simplifications also the computer time is much less than in the original proof.
The following book contains the ideas of the proof of the 4-Colour Theorem.
T.L. SAATy AND P.C. KAINEN, “The Four-Color Problem”, Dover, 1986.
## List colouring
DEFINITION. Let $G$ be a graph so that each of its vertices $v$ is given a list (set) $\Lambda(v)$ of colours. A proper colouring $\alpha: V_{G} \rightarrow[1, m]$ of $G$ is a ( $\Lambda$-)list colouring, if each vertex $v$ gets a colour from its list, $\alpha(v) \in \Lambda(v)$.
The list chromatic number $\chi_{\ell}(G)$ is the smallest integer $k$ such that $G$ has a $\Lambda$-list colouring for all lists of size $k,|\Lambda(v)|=k\}$. Also, $G$ is $k$-choosable, if $\chi_{\ell}(G) \leq k$.
Example 5.4. The bipartite graph $K_{3,3}$ is not 2choosable. Indeed, let the bipartition of $K_{3,3}$ be $(X, Y)$, where $X=\left\{x_{1}, x_{2}, x_{3}\right\}$ and $Y=\left\{y_{1}, y_{2}, y_{3}\right\}$. The lists for the vertices shown in the figure show that $\chi_{\ell}\left(K_{3,3}\right)>2$.
Obviously $\chi(G) \leq \chi_{\ell}(G)$, since proper colourings are special cases of list colourings, but equality does not hold in general. However, it was proved by VIZING (1976) and ERDÖS, RUBIN AND TAYLOR (1979) that
$$
\chi_{\ell}(G) \leq \Delta(G)+1 .
$$
For planar graphs we do not have a '4-list colour theorem'. Indeed, it was shown by VOIGT (1993) that there exists a planar graph with $\chi_{\ell}(G)=5$. At the moment, the smallest such a graph was produced by MIRZAKHANI (1996), and it is of order 63.
Theorem 5.9 (THOMASSEN (1994)). Let $G$ be a planar graph. Then $\chi_{\ell}(G) \leq 5$.
In fact, THOMASSEN proved a stronger statement:
Theorem 5.10. Let $G$ be a planar graph and let $C$ be the cycle that is the boundary of the exterior face. Let $\Lambda$ consist of lists such that $|\Lambda(v)|=3$ for all $v \in C$, and $|\Lambda(v)|=5$ for all $v \notin C$. Then $G$ has a $\Lambda$-list colouring $\alpha$.
Proof. We can assume that the planar graph $G$ is connected, and that it is given by a near-triangulation; an embedding, where the interior faces are triangles. (If the boundary of a face has more than 3 edges, then we can add an edge inside the face.) This is because adding edges to a graph can only make the list colouring more difficult. Note that the exterior boundary is unchanged by a triangulation of the interior faces.
The proof is by induction on $v_{G}$ under the additional constraint that one of the vertices of $C$ has a fixed colour. (Thus we prove a stronger statement than claimed.) For $v_{G} \leq 3$, the claim is obvious. Suppose then that $v_{G} \geq 4$.
Let $x \in C$ be a vertex, for which we fix a colour $c \in \Lambda(x)$. Let $v \in C$ be a vertex adjacent to $x$, that is, $C: v \rightarrow x \stackrel{\star}{\rightarrow} v$.
Let $N_{G}(v)=\left\{x, v_{1}, \ldots, v_{k}, y\right\}$, where $y \in C$, and $v_{i}$ are ordered such that the faces are triangles as in the figure. It can be that $N_{G}(v)=\{x, y\}$, in which case $x y \in G$.
Consider the subgraph $H=G-v$. The exterior boundary of $H$ is the cycle $x \rightarrow v_{1} \rightarrow \cdots \rightarrow v_{k} \rightarrow y \stackrel{\star}{\rightarrow} x$. Since $|\Lambda(v)|=3$, there are two colours $r, s \in \Lambda(v)$ that
differ from the colour $c$ of $x$.
We define new lists for $H$ as follows: $\Lambda^{\prime}\left(v_{i}\right) \subseteq \Lambda\left(v_{i}\right) \backslash\{r, s\}$ such that $\left|\Lambda^{\prime}\left(v_{i}\right)\right|=3$ for each $i \in[1, k]$, and otherwise $\Lambda^{\prime}(z)=\Lambda(z)$.
Now $v_{H}=v_{G}-1$, and by the induction hypothesis (with $c$ still fixed for $x$ ), $H$ has a $\Lambda^{\prime}$-list colouring $\alpha$. For the vertex $v$, we choose $\alpha(v)=r$ or s such that $\alpha(v) \neq \alpha(y)$. This gives a $\Lambda^{\prime}$-list colouring for $G$. Since $\Lambda^{\prime}(z) \subseteq \Lambda(z)$ for all $z$, we have that $\alpha$ is a $\Lambda$-list colouring of $G$.
## Straight lines and kissing circles*
We state an interesting result of WAGNER, the proof of which can be deduced from the above proof of Kuratowski's theorem. The result is known as Fáry's Theorem.
Theorem 5.11 (WAGNER (1936)). A planar graph $G$ has a plane embedding, where the edges are straight lines.
This raises a difficult problem:
Integer Length Problem. Can all planar graphs be drawn in the plane such that the edges are straight lines of integer lengths?
We say that two circles kiss in the plane, if they intersect in one point and their interiors do not intersect. For a set of circles, we draw a graph by putting an edge between two midpoints of kissing circles.
The following improvement of the above theorem is due to KOEBE (1936), and it was rediscovered independently by ANDREEV (1970) and THURSTON (1985).
Theorem 5.12 (KOEBE (1936)). A graph is planar if and only if it is a kissing graph of circles. Graphs can be represented as plane figures in many different ways. For this, consider a set $S$ of curves of the plane (that are continuous between their end points). The string graph of $S$ is the graph $G=(S, E)$, where $u v \in E$ if and only if the curves $u$ and $v$ intersect. At first it might seem that every graph is a string graph, but this is not the case.
It is known that all planar graphs are string graphs (this is a trivial result).
Line Segment Problem. A graph is a line segment graph if it is a string graph for a set $L$ of straight line segments in the plane. Is every planar graph a line segment graph for some set $L$ of lines?
Note that there are also nonplanar graphs that are line segment graphs. Indeed, all complete graphs are such graphs.
The above question remains open even in the case when the slopes of the lines are $+1,-1,0$ and $\infty$. A positive answer to this 4-slope problem for planar graphs would prove the 4-Colour Theorem.
## The Minor Theorem*
Definition. A graph $H$ is a minor of $G$, denoted by $H \preccurlyeq G$, if $H$ is isomorphic to a graph obtained from a subgraph of $G$ by successively contracting edges.
A recent result of ROBERTSON AND SEYMOUR (1983-2000) on graph minors is (one of) the deepest results of graph theory. The proof goes beyond these lectures. Indeed, the proof of Theorem 5.13 is around 500 pages long.
G
a subgraph
a contraction
Note that every subgraph $H \subseteq G$ is a minor, $H \preccurlyeq G$.
The following properties of the minor relation are easily established:
(i) $G \preccurlyeq G$,
(ii) $H \preccurlyeq G$ and $G \preccurlyeq H$ imply $G \cong H$,
(iii) $H \preccurlyeq L$ and $L \preccurlyeq G$ imply $H \preccurlyeq G$.
The conditions (i) and (iii) ensure that the relation $\preccurlyeq$ is a quasi-order, that is, it is reflexive and transitive. It turns out to be a well-quasi-order, that is, every infinite sequence $G_{1}, G_{2}, \ldots$ of graphs has two graphs $G_{i}$ and $G_{j}$ with $i<j$ such that $G_{i} \preccurlyeq G_{j}$. Theorem 5.13 (Minor Theorem). The minor order $\preccurlyeq$ is a well-quasi-order on graphs. In particular, in any infinite family $\mathcal{F}$ of graphs, one of the graphs is a (proper) minor of another.
Each property $\mathcal{P}$ of graphs defines a family of graphs, namely, the family of those graphs that satisfy this property.
Definition. A family $\mathcal{F}$ of graphs is said to be minor closed, if every minor $H$ of a graph $G \in \mathcal{F}$ is also in $\mathcal{F}$. A property $\mathcal{P}$ of graphs is said to be inherited by minors, if all minors of a graph $G$ satisfy $\mathcal{P}$ whenever $G$ does.
The following families of graphs are minor closed: the family of (1) all graphs, (2) planar graphs (and their generalizations to other surfaces), (3) acyclic graphs.
The acyclic graphs include all trees. However, the family of trees is not closed under taking subgraphs, and thus it is not minor closed. More importantly, the subgraph order of trees $\left(T_{1} \subseteq T_{2}\right)$ is not a well-quasi-order.
WAGNER proved a minor version of Kuratowski's theorem:
Theorem 5.14 (WAGNER (1937)). A graph $G$ is nonplanar if and only if $K_{5} \preccurlyeq G$ or $K_{3,3} \preccurlyeq$ G.
Proof. Exercise.
ROBERTSON AND SEYMOUR (1998) proved the Wagner's conjecture:
Theorem 5.15 (Minor Theorem 2). Let $\mathcal{P}$ be a property of graphs inherited by minors. Then there exists a finite set $\mathcal{F}$ of graphs such that $G$ satisfies $\mathcal{P}$ if and only if $G$ does not have a minor from $\mathcal{F}$.
One of the impressive application of Theorem 5.15 concerns embeddings of graphs on surfaces, see the next chapters. By Theorem 5.15, one can test (with a fast algorithm) whether a graph can be embedded onto a surface.
Every graph can be drawn in the 3-dimensional space without crossing edges. An old problem asks if there exists an algorithm that would determine whether a graph can be drawn so that its cycles do not form (nontrivial) knots. This problem is solved by the above results, since the property 'knotless' is inherited by minors: there exists a fast algorithm to do the job. However, this algorithm is not known!
Hadwiger's Problem. HaDWiger conjectured in 1943 that for every graph G,
$$
K_{\chi(G)} \preccurlyeq G ，
$$
that is, if $\chi(G) \geq r$, then $G$ has a complete graph $K_{r}$ as its minor. The conjecture is trivial for $r=2$, and it is known to hold for all $r \leq 6$. The cases for $r=5$ and 6 follow from the 4-Colour Theorem.
### Genus of a graph
A graph is planar, if it can be drawn in the plane without crossing edges. A plane is an important special case of a surface. In this section we study shortly drawing graphs in other surfaces.
There are quite many interesting surfaces many of which are rather difficult to draw. We shall study the 'easy surfaces' - those that are compact and orientable. These are surfaces that have both an inside and an outside, and can be entirely characterized by the number of holes in them. This number is the genus of the surface. There are also non-orientable compact surfaces such as the Klein bottle and the projective plane.
## Background on surfaces
We shall first have a quick look at the general surfaces and their classification without going into the details. Consider the space $\mathbb{R}^{3}$, which has its (usual) distance function $d(x, y) \in \mathbb{R}$ of its points.
Two figures (i.e., sets of points) $A$ and $B$ are topologically equivalent (or homeomorphic) if there exists a bijection $f: A \rightarrow B$ such that $f$ and its inverse $f^{-1}: B \rightarrow A$ are continuous. In particular, two figures are topologically equivalent if one can be deformed to the other by bending, squeezing, stretching, and shrinking without tearing it apart or gluing any of its parts together. All these deformations should be such that they can be undone.
A set of points $X$ is a surface, if $X$ is connected (there is a continuous line inside $X$ between any two given points) and every point $x \in X$ has a neighbourhood that is topologically equivalent to an open planar disk $D(a)=\{x \mid \operatorname{dist}(a, x)<1\}$.
We deal with surfaces of the real space, and in this case a surface $X$ is compact, if $X$ is closed and bounded. Note that the plane is not compact, since it it not bounded. A subset of a compact surface $X$ is a triangle if it is topologically equivalent to a triangle in the plane. A finite set of triangles $T_{i}, i=1,2, \ldots, m$, is a triangulation of $X$ if $X=\cup_{i=1}^{m} T_{i}$ and any nonempty intersection $T_{i} \cap T_{j}$ with $i \neq j$ is either a vertex or an edge.
The following is due to RADÓ (1925).
Theorem 5.16. Every compact surface has a triangulation.
Each triangle of a surface can be oriented by choosing an order for its vertices up to cyclic permutations. Such a permutation induces a direction for the edges of the triangle. A triangulation is said to be oriented if the triangles are assigned orientations such that common edges of two triangles are always oriented in reverse directions. A surface is orientable if it admits an oriented triangulation.
Equivalently, orientability can be described as follows.
Theorem 5.17. A compact surface $X$ is orientable if and only if it has no subsets that are topologically equivalent to the Möbius band. In the Möbius band (which itself is not a surface according the above definition) one can travel around and return to the starting point with left and right reversed.
A connected sum $X \# Y$ of two compact surfaces is obtained by cutting an open disk off from both surfaces and then gluing the surfaces together along the boundary of the disks. (Such a deformation is not allowed by topological equivalence.)
The next result is known as the classification theorem of compact surfaces.
Theorem 5.18 (DeHN AND HeEgaARd (1907)). Let X be a compact surface. Then
(i) if $X$ is orientable, then it is topologically equivalent to a sphere $S=S_{0}$ or a connected sum of tori: $S_{n}=S_{1} \# S_{1} \# \ldots \# S_{1}$ for some $n \geq 1$, where $S_{1}$ is a torus.
(ii) if $X$ is nonorientable, then $X$ is topologically equivalent to a connected sum of projective planes: $P_{n}=P \# P \# \ldots \# P$ for some $n \geq 1$, where $P$ is a projective plane.
It is often difficult to imagine how a figure (say, a graph) can be drawn in a surface. There is a helpful, and difficult to prove, result due to RADÓ (1920), stating that every compact surface (orientable or not) has a description by a plane model, which consists of a polygon in the plane such that
- each edge of the polygon is labelled by a letter,
- each letter is a label of exactly two edges of the polygon, and
- each edge is given an orientation (clockwise or counter clockwise).
Given a plane model $M$, a compact surface is obtained by gluing together the edges having the same label in the direction that they have.
From a plane model one can easily determine if the surface is oriented or not. It is nonoriented if and only if, for some label $a$, the edges labelled by $a$ have the same direction when read clockwise. (This corresponds to the Möbius band.)
A plane model, and thus a compact surface, can also be represented by a (circular) word by reading the model clockwise, and concatenating the labels with the convention that $a^{-1}$ is chosen if the direction of the edge is counter clockwise. Hence, the sphere is represented by the word $a b b^{-1} a^{-1}$, the torus by $a b a^{-1} b^{-1}$, the Klein bottle by $a b a^{-1} b$ and the projective plane by $a b b^{-1} a$. These surfaces, as do the other surfaces, have many other plane models and representing words as well. A word representing a connected sum of two surfaces, represented by words $W_{1}$ and $W_{2}$, is obtained by concatenating these words to $W_{1} W_{2}$. By studying the relations of the representing words, Theorem 5.18 can be proved.
Klein bottle
Drawing a graph (or any figure) in a surface can be elaborated compared to drawing in a plane model, where a line that enters an edge of the polygon must continue by the corresponding point of the other edge with the same label (since these points are identified when we glue the edges together).
Example 5.5. On the right we have drawn $K_{6}$ in the Klein bottle. The black dots indicate, where the lines enter and leave the edges of the plane model. Recall that in the plane model for the Klein bottle the vertical edges of the square have the same direction.
## Sphere
DEFINITION. In general, if $S$ is a surface, then a graph $G$ has an $S$-embedding, if $G$ can be drawn in $S$ without crossing edges.
Let $S_{0}$ be (the surface of) a sphere. According to the next theorem a sphere has exactly the same embeddings as do the plane. In the one direction the claim is obvious: if $G$ is a planar graph, then it can be drawn in a bounded area of the plane (without crossing edges), and this bounded area can be ironed on the surface of a large enough sphere.
Clearly, if a graph can be embedded in one sphere, then it can be embedded in any sphere - the size of the sphere is of no importance. On the other hand, if $G$ is embeddable in a sphere $S_{0}$, then there is a small area of the sphere, where there are no points of the edges. We then puncture the sphere at this area, and stretch it open until it looks like a region of the plane. In this process no crossings of edges can be created, and hence $G$ is planar.
Another way to see this is to use a projection of the sphere to a plane:
Theorem 5.19. A graph $G$ has an $S_{0}$-embedding if and only if it is planar.
Therefore instead of planar embeddings we can equally well study embeddings of graphs in a sphere. This is sometimes convenient, since the sphere is closed and it has no boundaries. Most importantly, a planar graph drawn in a sphere has no exterior face - all faces are bounded (by edges).
If a sphere is deformed by pressing or stretching, its embeddability properties will remain the same. In topological terms the surface has been distorted by a continuous transformation.
## Torus
Consider next a surface which is obtained from the sphere $S_{0}$ by pressing a hole in it. This is a torus $S_{1}$ (or an orientable surface of genus 1 ). The $S_{1}$-embeddable graphs are said to have genus equal to 1 .
Sometimes it is easier to consider handles than holes: a torus $S_{1}$ can be deformed (by a continuous transformation) into a sphere with a handle.
If a graph $G$ is $S_{1}$-embeddable, then it can be drawn in any one of the above surfaces without crossing edges.
Example 5.6. The smallest nonplanar graphs $K_{5}$ and $K_{3,3}$ have genus 1 . Also, $K_{7}$ has genus 1 as can be seen from the plane model (of the torus) on the right.
## Genus
Let $S_{n}(n \geq 0)$ be a sphere with $n$ holes in it. The drawing of an $S_{4}$ can already be quite complicated, because we do not put any restrictions on the places of the holes (except that we must not tear the surface into disjoint parts). However, once again an $S_{n}$ can be transformed (topologically) into a sphere with $n$ handles.
DEFINITION. We define the genus $g(G)$ of a graph $G$ as the smallest integer $n$, for which $G$ is $S_{n}$-embeddable.
For planar graphs, we have $g(G)=0$, and, in particular, $g\left(K_{4}\right)=0$. For $K_{5}$, we have $g\left(K_{5}\right)=1$, since $K_{5}$ is nonplanar, but is embeddable in a torus. Also, $g\left(K_{3,3}\right)=1$.
The next theorem states that any graph $G$ can be embedded in some surface $S_{n}$ with $n \geq 0$.
Theorem 5.20. Every graph has a genus.
This result has an easy intuitive verification. Indeed, consider a graph $G$ and any of its plane (or sphere) drawing (possibly with many crossing edges) such that no three edges cross each other in the same point (such a drawing can be obtained). At each of these crossing points create a handle so that one of the edges goes below the handle and the other uses the handle to cross over the first one.
We should note that the above argument does not determine $g(G)$, only that $G$ can be embedded in some $S_{n}$. However, clearly $g(G) \leq n$, and thus the genus $g(G)$ of $G$ exists.
The same handle can be utilized by several edges.
## Euler's formula with genus*
The drawing of a planar graph $G$ in a sphere has the advantage that the faces of the embedding are not divided into internal and external. The external face of $G$ becomes an 'ordinary face' after $G$ has been drawn in $S_{0}$.
In general, a face of an embedding of $G$ in $S_{n}$ (with $g(G)=n$ ) is a region of $S_{n}$ surrounded by edges of $G$. Let again $\varphi_{G}$ denote the number of faces of an embedding of $G$ in $S_{n}$. We omit the proof of the next generalization of Euler's formula.
Theorem 5.21. If $G$ is a connected graph, then
$$
v_{G}-\varepsilon_{G}+\varphi_{G}=2-2 g(G) .
$$
If $G$ is a planar graph, then $g(G)=0$, and the above formula is the Euler's formula for planar graphs.
DEFinition. A face of an embedding $P(G)$ in a surface is a 2-cell, if every simple closed curve (that does not intersect with itself) can be continuously deformed to a single point.
The complete graph $K_{4}$ can be embedded in a torus such that it has a face that is not a 2-cell. But this is because $g\left(K_{4}\right)=0$, and the genus of the torus is 1 . We omit the proof of the general condition discovered by YOUNGS:
Theorem 5.22 (YOUNGS (1963)). The faces of an embedding of a connected graph $G$ in a surface of genus $g(G)$ are 2-cells.
Lemma 5.12. For a connected $G$ with $v_{G} \geq 3$ we have $3 \varphi_{G} \leq 2 \varepsilon_{G}$.
Proof. If $v_{G}=3$, then the claim is trivial. Assume thus that $v_{G} \geq 4$. In this case we need the knowledge that $\varphi_{G}$ is counted in a surface that determines the genus of $G$ (and in no surface with a larger genus). Now every face has a border of at least three edges, and, as before, every nonbridge is on the boundary of exactly two faces.
Theorem 5.23. For a connected $G$ with $v_{G} \geq 3$,
$$
g(G) \geq \frac{1}{6} \varepsilon_{G}-\frac{1}{2}\left(\nu_{G}-2\right) .
$$
Proof. By the previous lemma, $3 \varphi_{G} \leq 2 \varepsilon_{G}$, and by the generalized Euler's formula, $\varphi_{G}=\varepsilon_{G}-v_{G}+2-2 g(G)$. Combining these we obtain that $3 \varepsilon_{G}-3 v_{G}+6-6 g(G) \leq$ $2 \varepsilon_{G}$, and the claim follows.
By this theorem, we can compute lower bounds for the genus $g(G)$ without drawing any embeddings. As an example, let $G=K_{8}$. In this case $v_{G}=8, \varepsilon_{G}=28$, and so $g(G) \geq \frac{5}{3}$. Since the genus is always an integer, $g(G) \geq 2$. We deduce that $K_{8}$ cannot be embedded in the surface $S_{1}$ of the torus.
If $H \subseteq G$, then clearly $g(H) \leq g(G)$, since $H$ is obtained from $G$ by omitting vertices and edges. In particular, Lemma 5.13. For a graph $G$ of order $n, g(G) \leq g\left(K_{n}\right)$.
For the complete graphs $K_{n}$ a good lower bound was found early.
Theorem 5.24 (HEAWOOD (1890)). If $n \geq 3$, then
$$
g\left(K_{n}\right) \geq \frac{(n-3)(n-4)}{12} .
$$
Proof. The number of edges in $K_{n}$ is equal to $\varepsilon_{G}=\frac{1}{2} n(n-1)$. By Theorem 5.23, we obtain $g\left(K_{n}\right) \geq(1 / 6) \varepsilon_{G}-(1 / 2)(n-2)=(1 / 12)(n-3)(n-4)$.
This result was dramatically improved to obtain
Theorem 5.25 (RINGEL AND YOUNGS (1968)). If $n \geq 3$, then
$$
g\left(K_{n}\right)=\left\lceil\frac{(n-3)(n-4)}{12}\right\rceil
$$
Therefore $g\left(K_{6}\right)=\lceil 3 \cdot 2 / 12\rceil=\lceil 1 / 2\rceil=1$. Also, $g\left(K_{7}\right)=1$, but $g\left(K_{8}\right)=2$. By Theorem 5.25,
Theorem 5.26. For all graphs $G$ of order $n \geq 3$,
$$
g(G) \leq\left\lceil\frac{(n-3)(n-4)}{12}\right\rceil .
$$
Also, we know the exact genus for the complete bipartite graphs:
Theorem 5.27 ( RingeL (1965)). For the complete bipartite graphs,
$$
g\left(K_{m, n}\right)=\left\lceil\frac{(m-2)(n-2)}{4}\right\rceil
$$
## Chromatic numbers*
For the planar graphs $G$, the proof of the 4-Colour Theorem, $\chi(G) \leq 4$, is extremely long and difficult. This in mind, it is surprising that the generalization of the 4-Colour Theorem for genus $\geq 1$ is much easier. HEAWOOD proved a hundred years ago:
Theorem 5.28 (HEAWOOD). If $g(G)=g \geq 1$, then
$$
\chi(G) \leq\left\lfloor\frac{7+\sqrt{1+48 g}}{2}\right\rfloor .
$$
Notice that for $g=0$ this theorem would be the 4-colour theorem. HEAWOOD proved it 'only' for $g \geq 1$.
Using the result of RINGEL AND YOUNGS and some elementary computations we can prove that the above theorem is the best possible. Theorem 5.29. For each $g \geq 1$, there exists a graph $G$ with genus $g(G)=g$ so that
$$
\chi(G)=\left\lfloor\frac{7+\sqrt{1+48 g}}{2}\right\rfloor .
$$
If a nonplanar graph $G$ can be embedded in a torus, then $g(G)=1$, and $\chi(G) \leq$ $\lfloor(7+\sqrt{1+48 g}) / 2\rfloor=7$. Moreover, for $G=K_{7}$ we have that $\chi\left(K_{7}\right)=7$ and $g\left(K_{7}\right)=$ 1.
## Three dimensions*
Every graph can be drawn without crossing edges in the 3-dimensional space. Such a drawing is called spatial embedding of the graph. Indeed, such an embedding can be achieved by putting all vertices of $G$ on a line, and then drawing the edges in different planes that contain the line. Alternatively, the vertices of $G$ can be put in a sphere, and drawing the edges as straight lines crossing the sphere inside.
A spatial embedding of a graph $G$ is said to have linked cycles, if two cycles of $G$ form a link (they cannot be separated in the space). By CONWAY and GORDON in 1983 every spatial embedding of $K_{6}$ contains linked cycles.
It was shown by ROBERTSON, SEYMOUR AND THOMAS (1993) that there is a set of 7 graphs such that a graph $G$ has a spatial embedding without linked cycles if and only if $G$ does not have a minor belonging to this set.
This family of forbidden graphs was originally found by SACHS (without proof), and it contains $K_{6}$ and the Petersen graph. Every graph in the set has 15 edges, which is curious.
For further results and proofs concerning graphs in surfaces, see
B. Mohar And C. Thomassen, “Graphs on Surfaces”, Johns Hopkins, 2001.
## Directed Graphs
### Digraphs
In some problems the relation between the objects is not symmetric. For these cases we need directed graphs, where the edges are oriented from one vertex to another.
As an example consider a map of a small town. Can you make the streets one-way, and still be able to drive from one house to another (or exit the town)?
## Definitions
DEFINITION. A digraph (or a directed graph) $D=\left(V_{D}, E_{D}\right)$ consists of the vertices $V_{D}$ and (directed) edges $E_{D} \subseteq V_{D} \times V_{D}$ (without loops $v v$ ). We still write $u v$ for $(u, v)$, but note that now $u v \neq v u$. For each pair $e=u v$ define the inverse of $e$ as $e^{-1}=v u(=(v, u))$.
Note that $e \in D$ does not imply $e^{-1} \in D$.
Definition. Let $D$ be a digraph. Then $A$ is its
- subdigraph, if $V_{A} \subseteq V_{D}$ and $E_{A} \subseteq E_{D}$,
- induced subdigraph, $A=D[X]$, if $V_{A}=X$ and $E_{A}=E_{D} \cap(X \times X)$.
The underlying graph $U(D)$ of a digraph $D$ is the graph on $V_{D}$ such that if $e \in D$, then the undirected edge with the same ends is in $U(D)$.
A digraph $D$ is an orientation of a graph $G$, if $G=U(D)$ and $e \in D$ implies $e^{-1} \notin D$. In this case, $D$ is said to be an oriented graph.
DEFINITION. Let $D$ be a digraph. A walk $W=e_{1} e_{2} \ldots e_{k}: u \stackrel{\star}{\rightarrow} v$ of $U(D)$ is a directed walk, if $e_{i} \in D$ for all $i \in[1, k]$. Similarly, we define directed paths and directed cycles as directed walks and closed directed walks without repetitions of vertices.
The digraph $D$ is di-connected, if, for all $u \neq v$, there exist directed paths $u \stackrel{\star}{\rightarrow} v$ and $v \stackrel{\star}{\rightarrow} u$. The maximal induced di-connected subdigraphs are the di-components of $D$. Note that a graph $G=U(D)$ might be connected, although the digraph $D$ is not di-connected.
DEFINITION. The indegree and the outdegree of a vertex are defined as follows
$$
d_{D}^{I}(v)=|\{e \in D \mid e=x v\}|, \quad d_{D}^{O}(v)=|\{e \in D \mid e=v x\}| .
$$
We have the following handshaking lemma. (You offer and accept a handshake.)
Lemma 6.1. Let $D$ be a digraph. Then
$$
\sum_{v \in D} d_{D}^{I}(v)=|D|=\sum_{v \in D} d_{D}^{O}(v)
$$
## Directed paths
The relationship between paths and directed paths is in general rather complicated. This digraph has a path of length five, but its directed paths are of length one.
There is a nice connection between the lengths of directed paths and the chromatic number $\chi(D)=\chi(U(D))$.
Theorem 6.1 (Roy (1967),Gallai (1968)). A digraph $D$ has a directed path of length $\chi(D)-1$.
Proof. Let $A \subseteq E_{D}$ be a minimal set of edges such that the subdigraph $D-A$ contains no directed cycles. Let $k$ be the length of the longest directed path in $D-A$.
For each vertex $v \in D$, assign a colour $\alpha(v)=i$, if a longest directed path from $v$ has length $i-1$ in $D-A$. Here $1 \leq i \leq k+1$.
First we observe that if $P=e_{1} e_{2} \ldots e_{r}(r \geq 1)$ is any directed path $u \stackrel{\star}{\rightarrow} v$ in $D-A$, then $\alpha(u) \neq \alpha(v)$. Indeed, if $\alpha(v)=i$, then there exists a directed path $Q: v \stackrel{\star}{\rightarrow} w$ of length $i-1$, and $P Q$ is a directed path, since $D-A$ does not contain directed cycles. Since $P Q: u \stackrel{\star}{\rightarrow} w, \alpha(u) \neq i=\alpha(v)$. In particular, if $e=u v \in D-A$, then $\alpha(u) \neq \alpha(v)$.
Consider then an edge $e=v u \in A$. By the minimality of $A,(D-A)+e$ contains a directed cycle $C: u \stackrel{\star}{\rightarrow} v \rightarrow u$, where the part $u \stackrel{\star}{\rightarrow} v$ is a directed path in $D-A$, and hence $\alpha(u) \neq \alpha(v)$. This shows that $\alpha$ is a proper colouring of $U(D)$, and therefore $\chi(D) \leq k+1$, that is, $k \geq \chi(D)-1$.
The bound $\chi(D)-1$ is the best possible in the following sense:
Theorem 6.2. Every graph $G$ has an orientation $D$, where the longest directed paths have lengths $\chi(G)-1$. Proof. Let $k=\chi(G)$ and let $\alpha$ be a proper $k$-colouring of $G$. As usual the set of colours is $[1, k]$. We orient each edge $u v \in G$ by setting $u v \in D$, if $\alpha(u)<\alpha(v)$. Clearly, the so obtained orientation $D$ has no directed paths of length $\geq k-1$.
DEFINITION. An orientation $D$ of an undirected graph $G$ is acyclic, if it has no directed cycles. Let $a(G)$ be the number of acyclic orientations of $G$.
The next result is charming, since $\chi_{G}(-1)$ measures the number of proper colourings of $G$ using -1 colours!
Theorem 6.3 (STANLEY (1973)). Let $G$ be a graph of order $n$. Then the number of the acyclic orientations of $G$ is
$$
a(G)=(-1)^{n} \chi_{G}(-1),
$$
where $\chi_{G}$ is the chromatic polynomial of $G$.
Proof. The proof is by induction on $\varepsilon_{G}$. First, if $G$ is discrete, then $\chi_{G}(k)=k^{n}$, and $a(G)=1=(-1)^{n}(-1)^{n}=(-1)^{n} \chi_{G}(-1)$ as required.
Now $\chi_{G}(k)$ is a polynomial that satisfies the recurrence $\chi_{G}(k)=\chi_{G-e}(k)-$ $\chi_{G * e}(k)$. To prove the claim, we show that $a(G)$ satisfies the same recurrence.
Indeed, if
$$
a(G)=a(G-e)+a(G * e)
$$
then, by the induction hypothesis,
$$
a(G)=(-1)^{n} \chi_{G-e}(-1)+(-1)^{n-1} \chi_{G * e}(-1)=(-1)^{n} \chi_{G}(-1) .
$$
For (6.1), we observe that every acyclic orientation of $G$ gives an acyclic orientation of $G-e$. On the other hand, if $D$ is an acyclic orientation of $G-e$ for $e=u v$, it extends to an acyclic orientation of $G$ by putting $e_{1}: u \rightarrow v$ or $e_{2}: v \rightarrow u$. Indeed, if $D$ has no directed path $u \stackrel{\star}{\rightarrow} v$, we choose $e_{2}$, and if $D$ has no directed path $v \stackrel{\star}{\rightarrow} u$, we choose $e_{1}$. Note that since $D$ is acyclic, it cannot have both ways $u \stackrel{\star}{\rightarrow} v$ and $v \stackrel{\star}{\rightarrow} u$.
We conclude that $a(G)=a(G-e)+b$, where $b$ is the number of acyclic orientations $D$ of $G-e$ that extend in both ways $e_{1}$ and $e_{2}$. The acyclic orientations $D$ that extend in both ways are exactly those that contain
$$
\text { neither } u \stackrel{\star}{\rightarrow} v \text { nor } v \stackrel{\star}{\rightarrow} u \text { as a directed path. }
$$
Each acyclic orientation of $G * e$ corresponds in a natural way to an acyclic orientation $D$ of $G-e$ that satisfies (6.2). Therefore $b=a(G * e)$, and the proof is completed.
## One-way traffic
Every graph can be oriented, but the result may not be di-connected. In the oneway traffic problem the resulting orientation should be di-connected, for otherwise someone is not able to drive home. RoBBINS' theorem solves this problem.
Definition. A graph $G$ is di-orientable, if there is a di-connected oriented graph $D$ such that $G=U(D)$.
Theorem 6.4 (RoBbins (1939)). A connected graph $G$ is di-orientable if and only if $G$ has no bridges.
Proof. If $G$ has a bridge $e$, then any orientation of $G$ has at least two di-components (both sides of the bridge).
Suppose then that $G$ has no bridges. Hence $G$ has a cycle $C$, and a cycle is always di-orientable. Let then $H \subseteq G$ be maximal such that it has a di-orientation $D_{H}$. If $H=G$, then we are done.
Otherwise, there exists an edge $e=v u \in G$ such that $u \in H$ but $v \notin H$ (because $G$ is connected). The edge $e$ is not a bridge and thus there exists a cycle
$$
C^{\prime}=e P Q: v \rightarrow u \stackrel{\star}{\rightarrow} w \stackrel{\star}{\rightarrow} v
$$
in $G$, where $w$ is the last vertex inside $H$.
In the di-orientation $D_{H}$ of $H$ there is a directed path $P^{\prime}: u \stackrel{\star}{\rightarrow} w$. Now, we orient $e: v \rightarrow u$ and the edges of $Q$ in the direction $Q: w \stackrel{\star}{\rightarrow} v$ to obtain a directed cycle $e P^{\prime} Q: v \rightarrow u \stackrel{\star}{\rightarrow} w \stackrel{\star}{\rightarrow} v$. In conclusion, $G\left[V_{H} \cup V_{C}\right]$ has a di-orientation, which contradicts the maximality assumption on $H$. This proves the claim.
Example 6.1. Let $D$ be a digraph. A directed Euler tour of $D$ is a directed closed walk that uses each edge exactly once. A directed Euler trail of $D$ is a directed walk that uses each edge exactly once.
The following two results are left as exercises.
(1) Let $D$ be a digraph such that $U(D)$ is connected. Then $D$ has a directed Euler tour if and only if $d_{D}^{I}(v)=d_{D}^{O}(v)$ for all vertices $v$.
(2) Let $D$ be a digraph such that $U(D)$ is connected. Then $D$ has a directed Euler trail if and only if $d_{D}^{I}(v)=d_{D}^{O}(v)$ for all vertices $v$ with possibly excepting two vertices $x, y$ for which $\left|d_{D}^{I}(v)-d_{D}^{O}(v)\right|=1$.
The above results hold equally well for multidigraphs, that is, for directed graphs, where we allow parallel directed edges between the vertices.
Example 6.2. The following problem was first studied by HUTCHINSON AND WILF (1975) with a motivation from DNA sequencing. Consider words over an alphabet $A=\left\{a_{1}, a_{2}, \ldots, a_{n}\right\}$ of $n$ letters, that is, each word $w$ is a sequence of letters. In the case of DNA, the letters are $A, T, C, G$. In a problem instance, we are given nonnegative integers $s_{i}$ and $r_{i j}$ for $1 \leq i, j \leq n$, and the question is: does there exist a word $w$ in which each letter $a_{i}$ occurs exactly $s_{i}$ times, and $a_{i}$ is followed by $a_{j}$ exactly $r_{i j}$ times.
For instance, if $n=2, s_{1}=3$, and $r_{11}=1, r_{12}=2, r_{21}=1, r_{22}=0$, then the word $a_{1} a_{2} a_{1} a_{1} a_{2}$ is a solution to the problem.
Consider a multidigraph $D$ with $V_{D}=A$ for which there are $r_{i j}$ edges $a_{i} a_{j}$. It is rather obvious that a directed Euler trail of $D$ gives a solution to the sequencing problem.
## Tournaments
DEFINITION. A tournament $T$ is an orientation of a complete graph.
Example 6.3. There are four tournaments of four vertices that are not isomorphic with each other. (Isomorphism of directed graphs is defined in the obvious way.)
Theorem 6.5 (RÉDEI (1934)). Every tournament has a directed Hamilton path.
Proof. The chromatic number of $K_{n}$ is $\chi\left(K_{n}\right)=n$, and hence by Theorem 6.1, a tournament $T$ of order $n$ has a directed path of length $n-1$. This is then a directed Hamilton path visiting each vertex once.
The vertices of a tournament can be easily reached from one vertex (sometimes called the king).
Theorem 6.6 (LAUDAU (1953)). Let $v$ be a vertex of a tournament $T$ of maximum outdegree. Then for all $u$, there is a directed path $v \stackrel{\star}{\rightarrow} u$ of length at most two.
Proof. Let $T$ be an orientation of $K_{n}$, and let $d_{T}^{O}(v)=d$ be the maximum outdegree in $T$. Suppose that there exists an $x$, for which the directed distance from $v$ to $x$ is at least three. It follows that $x v \in T$ and $x u \in T$ for all $u$ with $v u \in T$. But there are $d$ vertices in $A=\{y \mid v y \in T\}$, and thus $d+1$ vertices in $\{y \mid x y \in T\}=A \cup\{v\}$. It follows that the outdegree of $x$ is $d+1$, which contradicts the maximality assumption made for $v$.
Problem. Ádám's conjecture states that in every digraph $D$ with a directed cycle there exists an edge uv the reversal of which decreases the number of directed cycles. Here the new digraph has the edge $v u$ instead of $u v$. Example 6.4. Consider a tournament of $n$ teams that play once against each other, and suppose that each game has a winner. The situation can be presented as a tournament, where the vertices correspond to the teams $v_{i}$, and there is an edge $v_{i} v_{j}$, if $v_{i}$ won $v_{j}$ in their mutual game.
DEFINITION. A team $v$ is $a$ winner (there may be more than one winner), if $v$ comes out with the most victories in the tournament.
Theorem 6.6 states that a winner $v$ either defeated a team $u$ or $v$ defeated a team that defeated $u$.
A ranking of a tournament is a linear ordering of the teams $v_{i_{1}}>v_{i_{2}}>\cdots>$ $v_{i_{n}}$ that should reflect the scoring of the teams. One way of ranking a tournament could be by a Hamilton path: the ordering can be obtained from a directed Hamilton path $P: v_{i_{1}} \rightarrow v_{i_{2}} \rightarrow \ldots \rightarrow v_{i_{n}}$. However, a tournament may have several directed Hamilton paths, and some of these may do unjust for the 'real' winner.
Example 6.5. Consider a tournament of six teams $1,2, \ldots, 6$, and let $T$ be the scoring digraph as in the figure. Here $1 \rightarrow 2 \rightarrow 4 \rightarrow 5 \rightarrow 6 \rightarrow 3$ is a directed Hamilton path, but this extends to a directed Hamilton cycle (by adding $3 \rightarrow 1$ )! So for every team there is a Hamilton path, where it is a winner, and in another, it is a looser.
Let $s_{1}(j)=d_{T}^{O}(j)$ be the winning number of the team $j$ (the number of teams beaten by $j)$. In the above tournament,
$$
s_{1}(1)=4, s_{1}(2)=3, s_{1}(3)=3, s_{1}(4)=2, s_{1}(5)=2, s_{1}(6)=1 .
$$
So, is team 1 the winner? If so, is 2 or 3 next? Define the second-level scoring for each team by
$$
s_{2}(j)=\sum_{j i \in T} s_{1}(i) .
$$
This tells us how good teams $j$ beat. In our example, we have
$$
s_{2}(1)=8, s_{2}(2)=5, s_{2}(3)=9, s_{2}(4)=3, s_{2}(5)=4, s_{2}(6)=3 .
$$
Now, it seems that 3 is the winner,but 4 and 6 have the same score. We continue by defining inductively the $m$ th-level scoring by
$$
s_{m}(j)=\sum_{j i \in T} s_{m-1}(i) .
$$
It can be proved (using matrix methods) that for a di-connected tournament with at least four teams, the level scorings will eventually stabilize in a ranking of the tournament: there exits an $m$ for which the $m$ th-level scoring gives the same ordering as do the $(m+k)$ th-level scorings for all $k \geq 1$. If $T$ is not di-connected, then the level scoring should be carried out with respect to the di-components.
In our example the level scoring gives $1 \rightarrow 3 \rightarrow 2 \rightarrow 5 \rightarrow 4 \rightarrow 6$ as the ranking of the tournament.
### Network Flows
Various transportation networks or water pipelines are conveniently represented by weighted directed graphs. These networks usually possess also some additional requirements. Goods are transported from specific places (warehouses) to final locations (marketing places) through a network of roads. In modelling a transportation network by a digraph, we must make sure that the number of goods remains the same at each crossing of the roads. The problem setting for such networks was proposed by T.E. Harris in the 1950s. The connection to Kirchhoff's Current Law (1847) is immediate. According to this law, in every electrical network the amount of current flowing in a vertex equals the amount flowing out that vertex.
## Flows
## DEFINITION. A network $N$ consists of
- an underlying digraph $D=(V, E)$,
- two distinct vertices $s$ and $r$, called the source and the sink of $N$, and
- a capacity function $\alpha: V \times V \rightarrow \mathbb{R}_{+}$(nonnegative real numbers), for which $\alpha(e)=0$, if $e \notin E$.
Denote $V_{N}=V$ and $E_{N}=E$.
Let $A \subseteq V_{N}$ be a set of vertices, and $f: V_{N} \times V_{N} \rightarrow \mathbb{R}$ any function such that $f(e)=0$, if $e \notin N$. We adopt the following notations:
$$
\begin{aligned}
{[A, \bar{A}] } & =\{e \in D \mid e=u v, u \in A, v \notin A\}, \\
f^{+}(A) & =\sum_{e \in[A, \bar{A}]} f(e) \text { and } f^{-}(A)=\sum_{e \in[\bar{A}, A]} f(e) .
\end{aligned}
$$
In particular,
$$
f^{+}(u)=\sum_{v \in N} f(u v) \quad \text { and } \quad f^{-}(u)=\sum_{v \in N} f(v u)
$$
Definition. A flow in a network $N$ is a function $f: V_{N} \times V_{N} \rightarrow \mathbb{R}_{+}$such that
$$
0 \leq f(e) \leq \alpha(e) \text { for all } e, \quad \text { and } f^{-}(v)=f^{+}(v) \text { for all } v \notin\{s, r\} .
$$
Example 6.6. The value $f(e)$ can be taught of as the rate at which transportation actually happens along the channel $e$ which has the maximum capacity $\alpha(e)$. The second condition states that there should be no loss. If $N=(D, s, r, \alpha)$ is a network of water pipes, then the value $\alpha(e)$ gives the capacity $\left(x \mathrm{~m}^{3} / \mathrm{min}\right)$ of the pipe $e$.
The previous network has a flow that is indicated on the right.
A flow $f$ in $N$ is something that the network can handle. E.g., in the above figure the source should not try to feed the network the full capacity $\left(11 \mathrm{~m}^{3} / \mathrm{min}\right)$ of its pipes, because the junctions cannot handle this much water.
DEFINITION. Every network $N$ has a zero flow defined by $f(e)=0$ for all $e$. For a flow $f$ and each subset $A \subseteq V_{N}$, define the resultant flow from $A$ and the value of $f$ as the numbers
$$
\operatorname{val}\left(f_{A}\right)=f^{+}(A)-f^{-}(A) \quad \text { and } \quad \operatorname{val}(f)=\operatorname{val}\left(f_{s}\right)\left(=f^{+}(s)-f^{-}(s)\right) .
$$
A flow $f$ of a network $N$ is a maximum flow, if there does not exist any flow $f^{\prime}$ such that $\operatorname{val}(f)<\operatorname{val}\left(f^{\prime}\right)$.
The value $\operatorname{val}(f)$ of a flow is the overall number of goods that are (to be) transported through the network from the source to the sink. In the above example, $\operatorname{val}(f)=9$.
Lemma 6.2. Let $N=(D, s, r, \alpha)$ be a network with a flow $f$.
(i) If $A \subseteq N \backslash\{s, r\}$, then $\operatorname{val}\left(f_{A}\right)=0$.
(ii) $\operatorname{val}(f)=-\operatorname{val}\left(f_{r}\right)$.
Proof. Let $A \subseteq N \backslash\{s, r\}$. Then
$$
0=\sum_{v \in A}\left(f^{+}(v)-f^{-}(v)\right)=\sum_{v \in A} f^{+}(v)-\sum_{v \in A} f^{-}(v)=f^{+}(A)-f^{-}(A)=\operatorname{val}\left(f_{A}\right),
$$
where the third equality holds since the values of the edges $u v$ with $u, v \in A$ cancel each out.
The second claim is also clear.
## Improvable flows
Let $f$ be a flow in a network $N$, and let $P=e_{1} e_{2} \ldots e_{n}$ be an undirected path in $N$ where an edge $e_{i}$ is along $P$, if $e_{i}=v_{i} v_{i+1} \in N$, and against $P$, if $e_{i}=v_{i+1} v_{i} \in N$.
We define a nonnegative number $\iota(P)$ for $P$ as follows:
$$
\iota(P)=\min _{e_{i}} \iota(e), \quad \text { where } \iota(e)= \begin{cases}\alpha(e)-f(e) & \text { if } e \text { is along } P, \\ f(e) & \text { if } e \text { is against } P .\end{cases}
$$
DEFinition. Let $f$ be a flow in a network $N$. A path $P: s \stackrel{\star}{\rightarrow} r$ is $(f$-)improvable, if $\iota(P)>0$.
On the right, the bold path has value $\iota(P)=1$, and therefore this path is improvable.
Lemma 6.3. Let $N$ be a network. If $f$ is a maximum flow of $N$, then it has no improvable paths.
Proof. Define
$$
f^{\prime}(e)= \begin{cases}f(e)+\iota(P) & \text { if } e \text { is along } \mathrm{P}, \\ f(e)-\iota(P) & \text { if } e \text { is against } \mathrm{P} \\ f(e) & \text { if } e \text { is not in } \mathrm{P} .\end{cases}
$$
Then $f^{\prime}$ is a flow, since at each intermediate vertex $v \notin\{s, r\}$, we have $\left(f^{\prime}\right)^{-}(v)=\left(f^{\prime}\right)^{+}(v)$, and the capacities of the edges are not exceeded. Now $\operatorname{val}\left(f^{\prime}\right)=\operatorname{val}(f)+\iota(P)$, since $P$ has exactly one edge $s v \in N$ for the source $s$. Hence, if $\iota(P)>0$, then we can improve the flow.
## Max-Flow Min-Cut Theorem
Definition. Let $N=(D, s, r, \alpha)$ be a network. For a subset $S \subset V_{N}$ with $s \in S$ and $r \notin S$, let the cut by $S$ be
$$
[S]=[S, \bar{S}] \quad(=\{u v \in N \mid u \in S, v \notin S\}) .
$$
The capacity of the cut $[S]$ is the sum
$$
\alpha[S]=\alpha^{+}(S)=\sum_{e \in[S]} \alpha(e) .
$$
A cut $[S]$ is a minimum cut, if there is no cut $[R]$ with $\alpha[R]<\alpha[S]$.
Example 6.7. In our original network the capacity of the cut for the indicated vertices is equal to 10 .
Lemma 6.4. For a flow $f$ and a cut $[S]$ of $N$,
$$
\operatorname{val}(f)=\operatorname{val}\left(f_{S}\right)=f^{+}(S)-f^{-}(S) .
$$
Proof. Let $S_{I}=S \backslash\{s\}$. Now $\operatorname{val}\left(S_{I}\right)=0$ (since $S_{I} \subseteq N \backslash\{s, r\}$ ), and $\operatorname{val}(f)=$ $\operatorname{val}\left(f_{s}\right)$. Hence
$$
\begin{aligned}
\operatorname{val}\left(f_{S}\right)= & \operatorname{val}\left(f_{s}\right)-\sum_{v \in S_{I}} f(s v)+\sum_{v \in S_{I}} f(v s) \\
& +\operatorname{val}\left(f_{S_{I}}\right)+\sum_{v \in S_{I}} f(s v)-\sum_{v \in S_{I}} f(v s) \\
= & \operatorname{val}\left(f_{s}\right)=\operatorname{val}(f) .
\end{aligned}
$$
Theorem 6.7. For a flow $f$ and any cut $[S]$ of $N, \operatorname{val}(f) \leq \alpha[S]$. Furthermore, equality holds if and only if for each $u \in S$ and $v \notin S$,
(i) if $e=u v \in N$, then $f(e)=\alpha(e)$,
(ii) if $e=v u \in N$, then $f(e)=0$.
Proof. By the definition of a flow,
$$
f^{+}(S)=\sum_{e \in[S]} f(e) \leq \sum_{e \in[S]} \alpha(e)=\alpha[S]
$$
and $f^{-}(S) \geq 0$. By Lemma 6.4, $\operatorname{val}(f)=\operatorname{val}\left(f_{S}\right)=f^{+}(S)-f^{-}(S)$, and hence $\operatorname{val}(f) \leq \alpha[S]$, as required. Also, the equality $\operatorname{val}(f)=\alpha[S]$ holds if and only if
(1) $f^{+}(S)=\alpha[S]$ and (2) $f^{-}(S)=0$. This holds if and only if $f(e)=\alpha(e)$ for all $e \in[S]$ (since $f(e) \leq \alpha(e))$, and
(2) $f(e)=0$ for all $e=v u$ with $u \in S, v \notin S$.
This proves the claim.
In particular, if $f$ is a maximum flow and $[S]$ a minimum cut, then
$$
\operatorname{val}(f) \leq \alpha[S] .
$$
Corollary 6.1. If $f$ is a flow and $[S]$ a cut such that $\operatorname{val}(f)=\alpha[S]$, then $f$ is a maximum flow and $[S]$ a minimum cut.
The following main result of network flows was proved independently by ELIAS, FEINSTEIN, SHANNON, by FORD AND FULKERSON, and by ROBACKer in 1955 - 56. The present approach is due to Ford and Fulkerson.
Theorem 6.8. A flow $f$ of a network $N$ is maximum if and only if there are no $f$-improvable paths in $N$.
Proof. By Lemma 6.3, a maximum flow cannot have improvable paths.
Conversely, assume that $N$ contains no $f$-improvable paths, and let
$$
S_{I}=\{u \in N \mid \text { for some path } P: s \stackrel{\star}{\rightarrow} u, \iota(P)>0\} .
$$
Set $S=S_{I} \cup\{s\}$.
Consider an edge $e=u v \in N$, where $u \in S$ and $v \notin S$. Since $u \in S$, there exists a path $P: s \stackrel{\star}{\rightarrow} u$ with $l(P)>0$. Moreover, since $v \notin S, \iota(P e)=0$ for the path $P e: s \stackrel{\star}{\rightarrow} v$. Therefore $\iota(e)=0$, and so $f(e)=\alpha(e)$.
By the same argument, for an edge $e=v u \in N$ with $v \notin S$ and $u \in S, f(e)=0$.
By Theorem 6.7, we have $\operatorname{val}(f)=\alpha[S]$. Corollary 6.1 implies now that $f$ is a maximum flow (and $[S]$ is a minimum cut).
Theorem 6.9. Let $N$ be a network, where the capacity function $\alpha: V \times V \rightarrow \mathbb{N}$ has integer values. Then $N$ has a maximum flow with integer values.
Proof. Let $f_{0}$ be the zero flow, $f_{0}(e)=0$ for all $e \in V \times V$. A maximum flow is constructed using Lemma 6.3 by increasing and decreasing the values of the edges by integers only.
The proof of Theorem 6.8 showed also
Theorem 6.10 (Max-Flow Min-Cut). In a network $N$, the value val $(f)$ of a maximum flow equals the capacity $\alpha[S]$ of a minimum cut.
## Applications to graphs ${ }^{\star}$
The Max-Flow Min-Cut Theorem is a strong result, and many of our previous results follow from it.
We mention a connection to the Marriage Theorem, Theorem 3.9. For this, let $G$ be a bipartite graph with a bipartition $(X, Y)$, and consider a network $N$ with vertices $\{s, r\} \cup X \cup Y$. Let the edges (with their capacities) be $s x \in N(\alpha(s x)=1), y r \in N$ $(\alpha(y r)=1)$ for all $x \in X, y \in Y$ together with the edges $x y \in N(\alpha(x y)=|X|+1)$, if $x y \in G$ for $x \in X, y \in Y$. Then $G$ has a matching that saturates $X$ if and only if $N$ has a maximum flow of value $|X|$. Now Theorem 6.10 gives Theorem 3.9.
Next we apply the theorem to unit networks, where the capacities of the edges are equal to one $(\alpha(e)=1$ for all $e \in N)$. We obtain results for (directed) graphs.
Lemma 6.5. Let $N$ be a unit network with source s and sink $r$.
(i) The value $\operatorname{val}(f)$ of a maximum flow equals the maximum number of edge-disjoint directed paths $s \stackrel{\star}{\rightarrow} r$.
(ii) The capacity of a minimum cut $[S]$ equals the minimum number of edges whose removal destroys the directed connections $s \stackrel{\star}{\rightarrow} r$ from $s$ to $r$.
Proof. Exercise.
Corollary 6.2. Let $u$ and $v$ be two vertices of a digraph $D$. The maximum number of edgedisjoint directed paths $u \stackrel{\star}{\rightarrow} v$ equals the minimum number of edges, whose removal destroys all the directed connections $u \stackrel{\star}{\rightarrow} v$ from $D$. Proof. A network $N$ with source $s$ and sink $r$ is obtained by setting the capacities equal to 1 . The claim follows from Lemma 6.5 and Corollary 6.10.
Corollary 6.3. Let $u$ and $v$ be two vertices of a graph $G$. The maximum number of edgedisjoint paths $u^{\star} \rightarrow v$ equals the minimum number of edges, whose removal destroys all the connections $u \stackrel{\star}{\rightarrow} v$ from $G$.
Proof. Consider the digraph $D$ that is obtained from $G$ by replacing each (undirected) edge $u v \in G$ by two directed edges $u v \in D$ and $v u \in D$. The claim follows then easily from Corollary 6.2.
The next corollary is Menger's Theorem for edge connectivity.
Corollary 6.4. A graph $G$ is k-edge connected if and only if any two distinct vertices of $G$ are connected by at least $k$ independent paths.
Proof. The claim follows immediately from Corollary 6.3.
## Seymour's 6-flows*
Definition. A $k$-flow $(H, \alpha)$ of an undirected graph $G$ is an orientation $H$ of $G$ together with an edge colouring $\alpha: E_{H} \rightarrow[0, k-1]$ such that for all vertices $v \in V$,
$$
\sum_{e=v u \in H} \alpha(e)=\sum_{f=u v \in H} \alpha(f),
$$
that is, the sum of the incoming values equals the sum of the outgoing values. A $k$-flow is nowhere zero, if $\alpha(e) \neq 0$ for all $e \in H$.
In the $k$-flows we do not have any source or sink. For convenience, let $\alpha\left(e^{-1}\right)=$ $-\alpha(e)$ for all $e \in H$ in the orientation $H$ of $G$ so that the condition (6.3) becomes
$$
\sum_{e=v u \in H} \alpha(e)=0
$$
Example 6.8. A graph with a nowhere zero 4-flow.
The condition (6.4) generalizes to the subsets $A \subseteq V_{G}$ in a natural way,
$$
\sum_{e \in[A, \bar{A}]} \alpha(e)=0,
$$
since the values of the edges inside $A$ cancel out each other. In particular, Lemma 6.6. If $G$ has a nowhere zero $k$-flow for some $k$, then $G$ has no bridges.
Tutte's Problem. It was conjectured by TUTTE (1954) that every bridgeless graph has a nowhere zero 5-flow. The Petersen graph has a nowhere zero 5-flow but does not have any nowhere 4-flows, and so 5 is the best one can think of. Tutte's conjecture resembles the 4-Colour Theorem, and indeed, the conjecture is known to hold for the planar graphs. The proof of this uses the 4-Colour Theorem.
In order to fully appreciate Seymour's result, Theorem 6.11, we mention that it was proved as late as 1976 (by JAEGER) that every bridgeless $G$ has a nowhere zero $k$-flow for some integer $k$.
SEYMOUR's remarkable result reads as follows:
Theorem 6.11 (SEYMOUR's (1981)). Every bridgeless graph has a nowhere zero 6-flow.
Proof. Omitted.
DEFINITION. The flow number $f(G)$ of a bridgeless graph $G$ is the least integer $k$ for which $G$ has a nowhere zero $k$-flow.
Theorem 6.12. A connected graph $G$ has a flow number $f(G)=2$ if and only if it is eulerian.
Proof. Suppose $G$ is eulerian, and consider an Euler tour $W$ of $G$. Let $D$ be the orientation of $G$ corresponding to the direction of $W$. If an edge $u v \in D$, let $\alpha(e)=1$. Since $W$ arrives and leaves each vertex equally many times, the function $\alpha$ is a nowhere zero 2-flow.
Conversely, let $\alpha$ be a nowhere zero 2-flow of an orientation $D$ of $G$. Then necessarily the degrees of the vertices are even, and so $G$ is eulerian.
Example 6.9. For each 3-regular bipartite graph $G$, we have $f(G) \leq 3$. Indeed, let $G$ be $(X, Y)$-bipartite. By Corollary 3.1, a 3-regular graph has a perfect matching $M$. Orient the edges $e \in M$ from $X$ to $Y$, and set $\alpha(e)=2$. Orient the edges $e \notin M$ from $Y$ to $X$, and set $\alpha(e)=1$. Since each $x \in X$ has exactly one neighbour $y_{1} \in Y$ such that $x y_{1} \in M$, and two neighbours $y_{2}, y_{3} \in Y$ such that $x y_{2}, x y_{3} \notin M$, we have that $f(G) \leq 3$.
Theorem 6.13. We have $f\left(K_{4}\right)=4$, and if $n>4$, then
$$
f\left(K_{n}\right)= \begin{cases}2 & \text { if } n \text { is odd }, \\ 3 & \text { if } n \text { is even } .\end{cases}
$$
Proof. Exercise.
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Mat. Sb. (N.S.), 1984, Volume 124(166), Number 1(5), Pages 96–120 (Mi msb2042)
This article is cited in 7 scientific papers (total in 7 papers)
Long wave asymptotics of asolution of a hyperbolic system of equations
L. A. Kalyakin
Abstract: The Cauchy problem is considered for a hyperbolic system of equations with a small parameter $\varepsilon$:
\begin{gather*} [\partial_t+\lambda_i(\xi,\tau)\partial_x]u_i=\varepsilon[A_i(U,\xi,\tau)\partial_xU+b_i(U,\xi,\tau)],\qquad t>0;
u_i(x,t,\varepsilon)|_{t=0}=\varphi_i(x,\xi),\quad x\in\mathbf R^1;\quad i=1,…,m;\quad\xi=\varepsilon x,\quad\tau=\varepsilon t. \end{gather*}
It is assumed that the initial vector $\Phi(x,\xi)=(\varphi_1,…,\varphi_m)$ has asymptotics
$$ \Phi(x,\xi)=\Phi^\pm(\xi)+O(x^{-N}),\qquad x\to\pm\infty,\quad\forall N,\quad\forall |\xi|\leqslant M_0. $$
A`complete asymptotic expansion of the solution $U(x,t,\varepsilon)$ as $\varepsilon\to0$ which is uniform in a large domain $0\leqslant|x|$, $t\leqslant O(\varepsilon^{-1})$ is constructed by the method of matching. Several subdomains are distinguished in which the expansion can be represented in the form of various series. The following pairs of variables are characteristic in these subdomains: $x$, $t$; $\xi$, $\tau$; $\sigma_\alpha$, $\tau$, $\alpha=1,…,m$; here $\sigma_\alpha=\varepsilon^{-1}\omega_\alpha(\xi,\tau)$, $\partial_\tau\omega_\alpha+\lambda_\alpha\partial_\xi\omega_\alpha=0$, and $\omega_\alpha(\xi,0)=\xi$.
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Mathematics of the USSR-Sbornik, 1985, 52:1, 91–114
Bibliographic databases:
UDC: 517.956
MSC: 35L45, 35B25
Received: 05.04.1983
Citation: L. A. Kalyakin, "Long wave asymptotics of asolution of a hyperbolic system of equations", Mat. Sb. (N.S.), 124(166):1(5) (1984), 96–120; Math. USSR-Sb., 52:1 (1985), 91–114
Citation in format AMSBIB
\Bibitem{Kal84}
\by L.~A.~Kalyakin
\paper Long wave asymptotics of asolution of a~hyperbolic system of equations
\jour Mat. Sb. (N.S.)
\vol 124(166)
\issue 1(5)
\pages 96--120
\mathnet{http://mi.mathnet.ru/msb2042}
\zmath{https://zbmath.org/?q=an:0599.35098|0566.35066}
\jour Math. USSR-Sb.
\crossref{https://doi.org/10.1070/SM1985v052n01ABEH002879}
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This publication is cited in the following articles:
L. A. Kalyakin, "Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium", Math. USSR-Sb., 60:2 (1988), 457–483
Kaliakin L., "Matching Method for the Problem of Asymptotic Decomposition of Plane-Wave Packet in a Dispersive Medium", 301, no. 5, 1988, 1048–1052
L. A. Kalyakin, "Long wave asymptotics. Integrable equations as asymptotic limits of non-linear systems", Russian Math. Surveys, 44:1 (1989), 3–42
L. A. Kalyakin, "Asymptotic decay of solutions of the Liouville equation under perturbations", Math. Notes, 68:2 (2000), 173–184
L A Kalyakin, Inverse Probl, 17:4 (2001), 879
Le U.V., "A Semilinear Wave Equation with Space-Time Dependent Coefficients and a Memory Boundarylike Antiperiodic Condition: a Low-Frequency Asymptotic Expansion", J. Math. Phys., 52:2 (2011), 023510
Le U.V., "On a Low-Frequency Asymptotic Expansion of a Unique Weak Solutions of a Semilinear Wave Equation with a Boundary-Like Antiperiodic Condition", Manuscr. Math., 138:3-4 (2012), 439–461
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Nuclear chart in covariant density functional theory with dynamic correlations: From oxygen to tin
Yi-Long Yang ,
Ya-Kun Wang ,
State Key Laboratory of Nuclear Physics and Technology, School of Physics, Peking University, Beijing 100871, China
Nuclear masses of even-even nuclei with the proton number $8\leqslant Z\leqslant 50$ (O to Sn isotopes) from the proton drip line to neutron drip line are investigated using the triaxial relativistic Hartree-Bogoliubov theory with the relativistic density functional PC-PK1. Further, the dynamical correlation energies (DCEs) associated with the rotational motion and quadrupole-shaped vibrational motion are taken into account by the five-dimensional collective Hamiltonian (5DCH) method. The root-mean-square deviation with respect to the experimental masses reduces from 2.50 to 1.59 MeV after the consideration of DCEs. The inclusion of DCEs has little influence on the position of drip lines, and the predicted numbers of bound even-even nuclei between proton and neutron drip lines from O to Sn isotopes are 569 and 564 with and without DCEs, respectively.
nuclear mass table ,
covariant density functionals ,
triaxial relativistic Hartree-Bogoliubov theory ,
O to Sn isotopes
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Yi-Long Yang and Ya-Kun Wang. Nuclear chart in covariant density functional theory with dynamical correlations: From Oxygen to Tin[J]. Chinese Physics C. doi: 10.1088/1674-1137/44/3/034102
Yi-Long Yang
Corresponding author: Ya-Kun Wang, [email protected]
Abstract: Nuclear masses of even-even nuclei with the proton number $8\leqslant Z\leqslant 50$ (O to Sn isotopes) from the proton drip line to neutron drip line are investigated using the triaxial relativistic Hartree-Bogoliubov theory with the relativistic density functional PC-PK1. Further, the dynamical correlation energies (DCEs) associated with the rotational motion and quadrupole-shaped vibrational motion are taken into account by the five-dimensional collective Hamiltonian (5DCH) method. The root-mean-square deviation with respect to the experimental masses reduces from 2.50 to 1.59 MeV after the consideration of DCEs. The inclusion of DCEs has little influence on the position of drip lines, and the predicted numbers of bound even-even nuclei between proton and neutron drip lines from O to Sn isotopes are 569 and 564 with and without DCEs, respectively.
Nuclear mass is one of the most fundamental properties of nuclei. It is of great importance in nuclear physics and also astrophysics [1, 2]. For example, the masses of nuclei widely ranging from the valley of stability to the vicinity of neutron drip line are involved in simulating the rapid neutron capture (r-process) of stellar nucleosynthesis [3]. Although considerable achievements have been made in mass measurement [4], a large amount of nuclei on the neutron-rich side away from the valley of stability still cannot be observed experimentally in the foreseeable future. Therefore, reliable nuclear models for high-precision description of nuclear masses are urgently required.
During the past decades, various global nuclear models have been proposed to describe the nuclear mass, including the finite-range droplet model (FRDM) [5, 6], semi-empirical Weizsäcker-Skyrme (WS) model [7, 8], non-relativistic [9-12] and relativistic [13-17] density functional theories (DFTs). Nuclear DFTs start from the universal density functionals with a few parameters determined by fitting the properties of finite nuclei or nuclear matter. They can describe the nuclear masses and ground and excited state properties in a unified manner [18-20]. In particular, because of the consideration of the Lorentz symmetry, the relativistic or covariant density functional theory (CDFT) naturally includes the nucleonic spin degree of freedom and time-odd mean fields, which play an essential role in describing the moments of inertia for nuclear rotations [21-24]. Thus far, the CDFT has received considerable attention because of its successful description of many nuclear phenomena [18, 20, 25-29].
In the framework of CDFT, masses for over 7000 nuclei with $ 8\leqslant Z\leqslant100 $ up to proton and neutron drip lines have been investigated based on the axial relativistic mean field (RMF) theory [13]. Then, to explore the location of the proton and neutron drip lines, a systematic investigation has been performed for even-even nuclei within the axial relativistic Hartree-Bogoliubov (RHB) theory [14, 15, 30]. Recently, the ground-state properties of nuclei with $ 8\leqslant Z\leqslant 120 $ from the proton drip line to the neutron drip line have been calculated using the spherical relativistic continuum Hartree-Bogoliubov (RCHB) theory, in which the couplings between the bound states and the continuum can be considered properly [31]. The root-mean-square (rms) deviation with respect to the experimental nuclear masses in these pure CDFT calculation is typically around several MeV. To achieve higher precision, one needs to go beyond the mean-field approximation and consider the beyond-mean-field dynamical correlation energies (DCEs).
In Ref. [32], low-lying states of 1712 even-even nuclei from $ Z = 10 $ to $ Z = 110 $ were studied using the five-dimensional collective Hamiltonian (5DCH) method with the collective parameters determined by the Gogny-Hartree-Fock-Bogoliubov calculations. It was found that the DCEs given by 5DCH significantly improve the accuracy of the two-particle separation energies. In Ref. [16], Zhang et al. carried out a global calculation of the binding energies for 575 even-even nuclei ranging from $ Z = 8 $ to $ Z = 108 $ based on the axial RMF, and the Bardeen-Cooper-Schrieffer (BCS) approximation was adopted to consider the pairing correlations. In this axial RMF+BCS calculation, the DCEs, namely the rotational and vibrational correlation energies were obtained by the cranking approximation. After including the DCEs, the rms deviation for binding energies of the 575 even-even nulcei reduced from 2.58 to 1.24 MeV. Later, the DCEs of these nuclei were revisited in Ref. [17] using the 5DCH method with the collective parameters determined by the CDFT calculations [33, 34]. The 5DCH method takes into account the DCEs in a more proper way, and the resulting rms deviation reduces from 2.52 to 1.14 MeV. Note that in Ref. [16], the adopted experimental mass data are taken from Ref. [35], while in Ref. [17], the experimental mass data are from Ref. [36]. Moreover, compared to the theoretical results shown in Ref. [16], the energies associated with triaxial deformation are further included in Ref. [17].
The studies shown in Refs. [16, 17] demonstrate that the inclusion of DCEs can significantly improve the description of nuclear masses. Thus far, the inclusion of DCEs in the CDFT is still confined to nuclei with known mass, and the DCEs of most neutron-rich nuclei crucial in simulating the r-process are uninvestigated. Therefore, it is necessary to extend the investigation from nuclei with known mass to the boundary of nuclear landscape. Meanwhile, the pairing correlations were treated by the BCS approximation in Refs. [16, 17]. For the description of nuclei around the neutron drip line, this approximation is questionable because the continuum effect cannot be taken into account properly [37]. Nevertheless, the methods with the Bogoliubov transformation can provide a better description for the pairing correlations in weakly bound nuclei. Therefore, in the present study, the nuclear masses of even-even nuclei from O to Sn isotopes ranging from the proton drip line to the neutron drip line are performed within the triaxial RHB theory [38], and the beyond mean-field quadrupole DCEs are considered by the 5DCH method.
The detailed theoretical formulae of CDFT and 5DCH have been presented in Refs. [33, 34, 39, 40]. Here, the framework of 5DCH and CDFT will be present briefly for completeness. The 5DCH is expressed by the two intrinsic deformation parameters $ \beta $ and $ \gamma $, as well as three Euler angles $ (\phi,\theta,\psi)\equiv\Omega $ defining the orientation of three intrinsic principal axes with respect to the laboratory frame. The collective Hamiltonian of 5DCH can be written as [33, 34],
$ \hat{H}_{\rm{coll}}(\beta,\gamma) = \hat{T}_{\rm{vib}}(\beta,\gamma) + \hat{T}_{\rm{rot}}(\beta,\gamma,\Omega) + V_{\rm{coll}}(\beta,\gamma), $
which includes a vibrational kinetic energy term,
$ \begin{split} {{\hat T}_{{\rm{vib}}}}(\beta ,\gamma ) \!=\!& -\! \frac{{{\hbar ^2}}}{{2\sqrt {\omega r} }}\!\!\left\{ \frac{1}{{{\beta ^4}}}\!\!\left[\frac{\partial }{{\partial \beta }}\!\!\sqrt {\frac{r}{\omega }} {\beta ^4}{B_{\gamma \gamma }}\frac{\partial }{{\partial \beta }}\! -\! \frac{\partial }{{\partial \beta }}\!\!\sqrt {\frac{r}{\omega }} {\beta ^3}{B_{\beta \gamma }}\frac{\partial }{{\partial \gamma }}\right]\right.\\& + \frac{1}{{\beta \sin 3\gamma }}\left[ - \frac{\partial }{{\partial \gamma }}\sqrt {\frac{r}{\omega }} \sin 3\gamma {B_{\beta \gamma }}\frac{\partial }{{\partial \beta }}\right. \\&\left.\left.+ \frac{1}{\beta }\frac{\partial }{{\partial \gamma }}\sqrt {\frac{r}{\omega }} \sin 3\gamma {B_{\beta \beta }}\frac{\partial }{{\partial \gamma }}\right]\right\} , \end{split} $
a rotational kinetic energy term,
$ \hat{T}_{\rm{rot}} = \frac{1}{2}\sum_{k = 1}^3\frac{\hat{J}^2_k}{{\cal I}_k}, $
and a collective potential $ V_{\rm{coll}}(\beta,\gamma) $. The operator $ \hat{J}_k $ represents the total angular momentum components in the body-fixed frame. The moment of inertia $ {\cal I}_k $ and mass parameters $ B_{\beta\beta}, B_{\beta\gamma}, B_{\gamma\gamma} $ depend on the quadrupole deformation parameters $ \beta $ and $ \gamma $. Two additional quantities r and $ \omega $ in $ \hat{T}_{\rm{vib}}(\beta,\gamma) $ are used to determine the volume element in the collective space and their explicit expressions are provided in Refs. [33, 34]. The Hamiltonian in Eq. (1) can be diagonalized by the complete set of basis in Ref. [33]; thus, the eigenvalues and corresponding eigenfunctions can be obtained.
In the framework of the CDFT-based 5DCH approach, all the collective parameters including the collective potential $ V_{\rm{coll}} $, the moments of inertia $ {\cal I}_k $, and mass parameters $ B_{\beta\beta}, B_{\beta\gamma}, B_{\gamma\gamma} $ are determined by the microscopic CDFT calculations.
Within the point-coupling CDFT, the unified and self-consistent treatment of mean fields and pairing correlations can be realized by solving the following RHB equation [39],
$ \left( {\begin{array}{*{20}{c}} {{{\hat h}_{\rm D}} - \lambda }&{\hat \Delta }\\ { - {{\hat \Delta }^ * }}&{ - {{\hat h}^ * } + \lambda } \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{U_k}}\\ {{V_k}} \end{array}} \right) = {E_k}\left( {\begin{array}{*{20}{c}} {{U_k}}\\ {{V_k}} \end{array}} \right),$
$ \hat{h}_{\rm D} = -{\rm i}\mathit{{\alpha}}\cdotp\mathit{{\nabla}} + \beta(m+S) + V $
is the single-nucleon Dirac Hamiltonian. $ U_k $ and $ V_k $ are the quasiparticle wavefunctions, and $ E_k $ is the corresponding quasiparticle energies. The scalar and vector mean fields in Eq. (5) are given by
$ \begin{split} S =& {\alpha _S}{\rho _S} + {\beta _S}\rho _S^2 + {\gamma _S}\rho _S^3 + {\delta _S}\Delta {\rho _S},\\ V =& {\alpha _V}{\rho _V} \!+\! {\gamma _V}{({\rho _V})^3} \!+\! {\delta _V}\Delta {\rho _V} \!+\! {\tau _3}{\alpha _{TV}}{\rho _{TV}} \!+\! {\tau _3}{\delta _{TV}}\Delta {\rho _{TV}} \!+\! e{A^0}, \end{split}$
where $ eA^0 $ is the electromagnetic field and the densities $ \rho_S, \rho_V $, and $ \rho_{TV} $ can be expressed in terms of quasiparticle wavefunctions $ V_k $ as
$ \begin{split} {\rho _S} = \sum\limits_{k > 0} {V_k^\dagger } {\gamma ^0}{V_k},\quad {\rho _V} = \sum\limits_{k > 0} {V_k^\dagger } {V_k},\quad {\rho _{TV}} = \sum\limits_{k > 0} {V_k^\dagger } \vec \tau {V_k}, \end{split}$
where the sum over $ k>0 $ corresponds to the "no-sea approximation" [39]. The matrix element of pairing field $ \hat{\Delta} $ can be written in the form,
$ \Delta_{ab} = \frac{1}{2}\sum_{cd}\langle ab|V^{pp}|cd\rangle_a\kappa_{cd}. $
Here, $ \kappa = V^\ast U^T $ is the pairing tensor, and $ V^{pp} $ is the pairing force for which a finite range separable pairing force [41] is adopted in the present work.
By solving Eq. (4) iteratively, the total energy in the laboratory can be obtained,
$ \begin{split} E_{\rm{tot}} =& \int {\rm d}^3r\sum_{k>0}V^\dagger_k[\mathit{{\alpha}}\cdotp\mathit{{p}}+\beta m]V_k\\ &+ \int {\rm d}^3r\biggl\{\frac{1}{2}\alpha_S\rho_S^2+ \frac{1}{3}\beta_S\rho_S^3+\frac{1}{4}\gamma_S\rho^4_S+\frac{1}{2}\delta_S\rho_S\Delta\rho_S\\ &+\frac{1}{2}\alpha_V\rho_V^2+\frac{1}{2}\alpha_{TV}\rho_{TV}^2+\frac{1}{4}\gamma_V\rho_V^4+\frac{1}{2}\delta_V\rho_V\Delta\rho_V \\& +\frac{1}{2}\delta_{TV}\rho_{TV}\Delta\rho_{TV}\}.\end{split} $
The map of the potential energy surface as functions of quadrupole deformation parameters $ \beta $ and $ \gamma $ can be obtained by constraining the axial and triaxial mass quadrupole moments. As shown in Refs. [33, 34], the quadratic constraint method uses an unrestricted variation of the function
$ \langle H\rangle + \sum_{\mu = 0}^2C_{2\mu}(\langle\hat{Q}_{2\mu}\rangle-q_{2\mu})^2, $
where $ \langle H\rangle $ is the total energy and $ \langle\hat{Q}_{2\mu}\rangle $ the expectation value of the following mass quadrupole operator,
$ \hat{Q}_{20} = 2z^2-x^2-y^2,\quad \hat{Q}_{22} = x^2-y^2. $
Here, $ q_{2\mu} $ represents the desired values of the quadrupole moments, and $ C_{2\mu} $ is the corresponding stiffness constant [42].
3. Numerical details
In this section, adiabatic deformation constrained RHB calculations are performed to obtain the mean-field states in the full $ (\beta,\gamma) $ energy surface. The relativistic density functional PC-PK1 [40] is used in the particle-hole channel. This density functional particularly improves the description for the isospin dependence of binding energies, and it has been successfully used for describing the Coulomb displacement energies among mirror nuclei [43], nuclear masses [17, 44], quadrupole moments [45-47], superheavy nuclei [48-51], nuclear shape phase transitions [52, 53], magnetic and antimagnetic rotations [27, 54-56], chiral rotations [57], etc. For the particle-particle channel, we adopted the finite range separable pairing force with pairing strength $ G = -728 $ MeV as proposed in Ref. [41]. The triaxial RHB equation was solved by expanding the quasiparticle wavefunctions in terms of three-dimensional harmonic oscillator bases, which are the eigenfunctions of a three-dimensional harmonic oscillator potential in the Cartesian coordinates [39]. The oscillator frequencies in the $ x, \;y,\; z $ directions are $ \hbar\omega_x = \hbar\omega_y = \hbar\omega_z = \hbar\omega_0 = 41 A^{-1/3} $ MeV, and the corresponding oscillator lengths are $ b_x = b_y = b_z = \sqrt{\hbar/m\omega_0} $. The spatial part of these bases are labeled by the quantum numbers $ n_x,\; n_y $, and $ n_z $, and the spin part is chosen as the eigenfunctions of the x-simplex operator ${{\dot S} } = \hat{P}{\rm e}^{-{\rm i}\pi {\hat{J}}_x} $, in which $ \hat{P} $ is the parity operator. The detailed formulas of three-dimensional harmonic oscillator basis can be found in Refs. [39, 58]. In the present calculations, we used 12 major shells for nuclei with proton number $ Z<20 $ and 14 major shells for nuclei with proton number $ 20\leq Z\leq50 $, which was confirmed by checking the convergence of the obtained binding energies and charge radii. The obtained quasiparticle energies and wavefunctions are used to calculate the mass parameters, moments of inertia, and collective potentials of in 5DCH; all of these quantities are associated with the quadrupole deformation parameters $ \beta $ and $ \gamma $. The dynamical correlation energy $ E_{\rm{corr}} $ is given by the energy difference between the lowest mean-field states and the $ 0_1^+ $ states from the 5DCH calculations.
The bound nuclear regions from O to Sn isotopes predicted by the triaxial RHB approach with and without DCEs are shown in Fig. 1. The discrepancies of the calculated binding energies with respect to the data are scaled by colors. The binding energies calculated by the triaxial RHB approach shown in panel (a) are given by the binding energies of the lowest mean-field states, while in panel (b), the DCEs are taken into account.
Figure 1. (color online) Even-even nuclei from O to Sn isotopes predicted by the triaxial RHB approach with (panel (b)) and without (panel (a)) dynamical correlation energies. Discrepancies of the calculated binding energies with the data [4] are denoted by colors. The proton and neutron drip lines predicted by spherical RCHB (PC-PK1) [31], axial RHB (DD-PC1) [15], and axial RMF+BCS (TMA) [13] are also plotted for comparison.
In the triaxial RHB calculations without DCEs, it was found that the binding energies are systematically underestimated. Most deviations are in the range of $ 0.5\;\sim\;4.5 $ MeV, resulting in the rms deviation of 2.50 MeV. We compared our results with the those of Ref. [15], where the ground state observables of even-even nuclei were investigated based on CDFT with the DD-PC1 density functional in particle-hole channel and the separable pairing force in the particle-particle channel. The rms deviation in Ref. [15] for the even-even nuclei from O to Sn is 2.29 MeV, which is consistent with our results. Moreover, we also compared our results with those in Ref. [17]. For the pure mean-field calculations in Ref. [17], same density functional, namely PC-PK1, was used in the particle-hole channel, while a density dependent delta force was adopted in the particle-particle channel. Because the calculations in Ref. [17] were performed for the nuclei with known mass, the location of these nuclei are still far from the neutron drip line. For most nuclei with known mass, the energy difference between the RHB and the RMF+BCS calculations is less than 0.2 MeV, and the final rms deviation is only 0.09 MeV. However, evident energy differences occur in Cd and Sn isotopes with neutron number $ N = 56 \sim 66 $. It is found that for these specific nuclei, the pairing energies obtained from th RMF+BCS are stronger than those from RHB, which is not the same as the usual case. This might be because two different pairing forces are adopted in the RMF+BCS and RHB calculations. As mentioned before, for the lowest mean field calculations, a density dependent $ \delta $ pairing force was used in Ref. [17], while a separable pairing force was used in the present calculations. The stronger pairing correlations given by yhr RMF+BCS make these nuclei more binding, thus increasing the discrepancy between the two calculations.
By including the DCEs, the underestimation of the binding energies is improved significantly, and the rms deviation is reduced from 2.50 MeV to 1.59 MeV. However, in the region $ (N,Z)\sim(24,12) $, large deviations exist even though the DCEs are considered. This might be associated with the complex shell evolution around this region. To obtain a better description of the binding energies in this region, the tensor interaction [59] may need to be included in the adopted density functional, which is beyond the scope of the present investigation.
To estimate the number of bound nuclei from O to Sn isotopes, two-proton and two-neutron drip lines predicted by the present triaxial RHB approach with and without DCEs are also plotted in Fig. 1. The predicted number of bound even-even nuclei between the proton and neutron drip lines from O to Sn isotopes without DCEs is 569. The inclusion of DCEs has little influence on the proton and neutron drip lines, and the corresponding number of bound nuclei is 564. For comparison, the drip lines predicted by the spherical RCHB (PC-PK1) [31], axial RHB(DD-PC1) [15], and axial RMF+BCS(TMA) [13] are also shown. It is found that theoretical differences for the proton drip lines are rather small. However, the neutron drip lines predicted by different approaches differ considerably, and the differences increase with the mass number. The neutron drip line predicted by the triaxial RHB approach is located between those predicted by the axial RHB and spherical RCHB approaches.
Besides the binding energies, the charge radii calculated by the triaxial RHB were been compared with available data [60] in Fig. 2. The experimental charge radii are well reproduced, and most deviations between the calculated results and the data are in the range of $ -0.03\sim 0.03 $ fm.
Figure 2. (color online) Charge radii of nuclei from O to Sn and their comparison with available data [60]. Discrepancies of the calculated charge radii with data are denoted by different colors.
Figure 3 shows the contour map of the dynamical correlation energies $ E_{\rm{corr}} $ calculated by the 5DCH model based on the triaxial RHB calculations. The calculated $ E_{\rm{corr}} $ ranges from 0 to 5 MeV and varies mainly in the region of $ 2.0-4.0 $ MeV. Owing to shape fluctuations, the dynamical correlation energies are pronounced for the nuclei around $ Z \sim 32, \;40 $ and $ N \sim 34, \;60 $. Similar to the results of Ref. [17], the dynamical correlation energies for the semi-magic nuclei with $ Z = 28,\; 50 $ and $ N = 28, \;82 $ are nonzero or larger. This is because the potential energy surfaces for these nuclei are either soft or with shape coexisting phenomena.
Figure 3. (color online) Contour map of the dynamical correlation energies Ecorr calculated by the 5DCH model based on triaxial RHB calculations as functions of neutron and proton numbers.
In Ref. [17], the binding energies of 575 even-even nuclei in the region of $ 8\leqslant Z\leqslant 108 $ were calculated using the 5DCH method in the framework of the triaxial RMF+BCS. For the 228 nuclei with $ 8\leqslant Z\leqslant50 $ in Ref. [17], the rms deviation with respect to data was 1.23 MeV, whereas the rms deviation in the present calculations for these nuclei is 1.47 MeV. The lowest mean-field binding energies given by these two calculations have little difference; therefore, the differences mainly arise because of $ E_{\rm{corr}} $.
The dynamical correlation energies $ E_{\rm{corr}} $ calculated by 5DCH based on the triaxial RHB and triaxial RMF+BCS are plotted in Fig. 4 as functions of the neutron number N. Although the systematics of $ E_{\rm{corr}} $ are similar for both calculations, the triaxial RHB-based $ E_{\rm{corr}} $ are systematically larger than those based on the triaxial RMF+BCS. The rms deviation between these two results is 0.53 MeV, and this leads to an overall difference in the binding energies. The systematic difference of $ E_{\rm{corr}} $ might be caused by the different treatments of pairing correlations. The pairing correlations in the present calculations are considered by the Bogoliubov transformation, while in Ref. [17], they are considered by the BCS approximation. To understand the origin of the discrepancy of the dynamical correlation energies between the RHB and RMF+BCS calculations, the pairing energies $ E_{\rm{pair}} $, moments of inertia $ {\cal I}_x $, collective masses $ B_{\beta\beta} $ and $ B_{\gamma\gamma} $ for nucleus 76Ge calculated by RHB and RMF+BCS were compared in detail. It is found that pairing correlations calculated from the RHB are stronger than those from the RMF+BCS, which leads to smaller collective parameters $ {\cal I}_x $, $ B_{\beta\beta} $, and $ B_{\gamma\gamma} $ and thus, larger dynamical correlation energies $ E_{\rm{corr}} $. Similar conclusions were made in Ref. [61]. Moreover, compared with the DCEs obtained from the 5DCH method based on Gogny-Hartree-Fock-Bogoliubov theory [32], the triaxial RHB-based DCEs are systematically smaller, and the rms deviation between two calculations is 1.59 MeV. As demonstrated in Ref. [17], this discrepancy might be due to the collective parameters, which are sensitive to the effective interactions, in particular the pairing properties.
Figure 4. (color online) Dynamic correlation energies calculated by 5DCH based on triaxial RHB (circles) compared with those based on triaxial RMF+BCS [17] (triangles).
The contour map of the triaxial RHB calculated quadrupole deformation $ \beta $ are presented in Fig. 5. The quadrupole deformation corresponds to the energy minima on the whole $ (\beta,\gamma) $ plane. Here, $ \beta $ is defined as positive for $ 0^\circ\leqslant \gamma<30^\circ $ and negative for $ 30^\circ<\gamma\leqslant 60^\circ $. In general, the nuclei near magic numbers possess small or vanishing deformation. However, it is found that single magic numbers do not enforce sphericity, especially for neutron-rich nuclei. For example, neutron-rich isotones with $ N = 28, \;Z<20 $ and $ N = 50,\; Z<28 $ show remarkable deformation. In addition, the deformation develops when moving away from the magic numbers, either isotopically or isotonically. There are four large deformed regions located at $ (N, Z)\sim (24,14), \;(34,32), \;(60,40) $ and $ (94,46) $. These regions with large deformation correspond to the regions with large DCEs as shown in Fig. 3.
Figure 5. (color online) Contour map of quadrupole deformation β calculated by the triaxial RHB approach as functions of the neutron and proton numbers. Nuclei with triaxial deformation are denoted by black triangles.
The nuclei with triaxial deformation i.e. $ \gamma\neq 0^\circ,\; 60^\circ $, are also shown in Fig. 5. It is found that the static triaxial nuclei occur mainly in the region with $ (N, Z)\sim(56, 46) $, which is consistent with the results given by finite-range liquid-drop model [62]. There are 14 nuclei with triaxial deformation, and most of them belong to the Ge, Mo, and Ru isotopes. The inclusion of static triaxial deformation makes these nuclei more binding at a value of $ \sim0.3 $ MeV. For our present beyond mean-field calculations, the triaxial deformation was not only static but also dynamic. The effects of triaxial deformation play a role in the description of binding energy especially for a nucleus with a $ \gamma $ soft potential energy surface. The final aim of this project is to build a whole nuclear mass table including both triaxial degrees of freedom and dynamical correlation energies. Similar works on this topic are in progress.
In summary, the nuclear masses of even-even nuclei with $ 8\leqslant Z\leqslant 50 $ ranging from the proton drip line to the neutron drip line are systematically investigated using the triaxial RHB theory with the relativistic density functional PC-PK1, and the quadrupole dynamical correlation energies are taken into account by solving the 5DCH. By including the dynamic correlation energies, the prediction of triaxial RHB theory for 252 nuclei masses was improved significantly, with the rms deviation reducing from 2.50 to 1.59 MeV. The dynamic correlation energies have little influence on the positions of proton and neutron drip lines, and the predicted numbers of bound even-even nuclei between proton and neutron drip lines with and without dynamical correlation energies are 569 and 564, respectively. In the present calculations, the obtained dynamical correlation energies ranged from 0 to 5 MeV, which are slightly larger than the results of a previous work [17]. The discrepancies might be caused by the different treatments of the pairing correlations, which would lead to different zero point energies, and thus different dynamical correlation energies. The contour map of quadrupole deformation $ \beta $ and $ \gamma $ associated with the dynamic correlation energies is also discussed in detail. Fourteen nuclei are predicted to have triaxial deformation, which are good candidates for the experimental study of the possibility of triaxial deformations. The final aim of this project is to build a whole nuclear mass table including both triaxial degrees of freedom and dynamical correlation energies. Similar works on this topic are still in progress.
The authors are grateful to Prof. ZhiPan Li and Prof. PengWei Zhao for providing the numerical computation codes and the fruitful discussions, as well as for the critical readings of our manuscript. | CommonCrawl |
Exploring wealth-related inequalities in maternal and child health coverage in Latin America and the Caribbean
Manuel Colomé-Hidalgo ORCID: orcid.org/0000-0002-4562-64911,
Juan Donado Campos2 &
Ángel Gil de Miguel1
BMC Public Health volume 21, Article number: 115 (2021) Cite this article
Maternal and child health have shown important advances in the world in recent years. However, national averages indicators hide large inequalities in access and quality of care in population subgroups. We explore wealth-related inequalities affecting health coverage and interventions in reproductive, maternal, newborn, and child health in Latin America and the Caribbean.
We analyzed representative national surveys from 15 countries conducted between 2001 and 2016. We estimated maternal-child health coverage gaps using the Composite Coverage Index – a weighted average of interventions that include family planning, maternal and newborn care, immunizations, and treatment of sick children. We measured absolute and relative inequality to assess gaps by wealth quintile. Pearson's correlation coefficient was used to test the association between the coverage gap and population attributable risk.
The Composite Coverage Index showed patterns of inequality favoring the wealthiest subgroups. In eight countries the national coverage was higher than the global median (78.4%; 95% CI: 73.1–83.6) and increased significantly as inequality decreased (Pearson r = 0.9; p < 0.01).
There are substantial inequalities between socioeconomic groups. Reducing inequalities will improve coverage indicators for women and children. Additional health policies, programs, and practices are required to promote equity.
Reproductive, Maternal, Newborn, and Child Health (RMNCH) has been a global health policy priority for the past decade [1]. The Millennium Development Goals (MDGs) contributed enormously to the health of women and children, managing to reduce maternal and under-5 years' old mortality and improved other indicators such as access to contraceptives, skilled attendance at childbirth, and measles vaccination [2]. Despite the progress, most regions did not reach the proposed goals, showing uneven progress that has left gaps between countries, especially in Latin America and the Caribbean (ALC) [3, 4].
The 2030 agenda for Sustainable Development Goals (SDGs) broadens the scope of the MDGs, assuming the commitment to leave no one behind. The SDG-3.8 promotes universal health coverage in terms of access to quality healthcare services, medicines, and vaccines for all [5]. More granular analysis of indicators can show whether all subgroups of the population will benefit from national progress or not [6]. Monitoring inequalities allow identifying vulnerable groups and prioritizing interventions in those who need it the most, thus promoting health coverage through equity [7]. We analyzed the Composite Coverage Index (CCI) as an indicator of universal healthcare coverage gaps in women and children. The index combines preventive and curative interventions throughout the continuum of care, family planning, maternal and newborn care, immunization, and treatment of sick children and has been used to monitor SDGs progress [8, 9].
Previous studies have emphasized the wealth-related inequalities between countries implementing the CCI, but only a few have focused on the LAC situation [10,11,12]. Therefore, the scope of health interventions and the level of improvement needed to narrow the gap needs to be adequately defined. This study explores wealth-related inequalities in RMNCH care coverage and its impact on reducing the gap in the LAC countries between 2001 and 2016.
This was a descriptive study based on secondary RMNCH coverage data obtained from the World Health Organization (WHO) Health Equity Assessment Toolkit (HEAT) software version 3.1 [13]. HEAT performs health inequality measures calculations from the WHO Health Equity Monitor Database [14]. The database includes data from Demographic Health Surveys (DHS), Multiple Indicators Cluster Survey (MICS) and Reproductive Health Surveys (RHS). The surveys carried out national representative and standardized interviews with women 15–49 years old. We included 15 of 22 countries with surveys conducted between 2001 and 2016 based on the availability of recent data on the Composite Coverage Index and wealth quintile.
The CCI is a weighted score based on aggregate estimates of eight essential interventions for the continuum of care for women and children, from before pregnancy to delivery, the immediate postnatal period, and childhood [7, 15]. The index is calculated using the formula:
$$ CCI=\frac{1}{4}\left( DFPS+\frac{ANC4+ SBA}{2}+\frac{BCG+2\mathrm{DPT}3+\mathrm{MCV}}{4}+\frac{ORS+ CPNM}{2}\right) $$
where DFPS = satisfied demand for modern family planning methods; ANC4 = prenatal care (at least four visits); SBA = deliveries attended by qualified personnel; BCG = one dose of Bacillus Calmette-Guérin vaccine; DPT3 = three or more doses of diphtheria-tetanus-pertussis vaccine; MCV = at least one dose of measles vaccine; ORS = children with diarrhea receiving oral rehydration therapy and continuous feeding; NSCLC = children with pneumonia symptoms taken to a health center [16].
We calculated CCI's, mean, median, interquartile range and standard deviation for the region. We analyzed socioeconomic inequality using the wealth index, which is an estimate based on the ownership of selected assets, housing construction materials, and access to basic services. The details of wealth index estimation have been previously described [17]. Households are classified from the poorest (Q1) to the richest (Q5) [18].
To compare patterns of inequality between and within countries, first, we calculated the coverage difference to show the magnitude of absolute inequality (Q5-Q1); second, the coverage ratio to show proportional differences between groups (Q5 / Q1) and third, the ratio of differences between coverages in lower (Q1-Q2) and higher quintiles (Q4-Q5). We calculated the relative concentration index and slope index to describe inequalities in all subgroups. Finally, we use population attributable risk (PAR) to show the possible improvement if the general population hypothetically had the same coverage level as the wealthiest quintile (CCI-Q5). We estimated the PAR percentage (PAR%) to show the proportion of improvement in national coverage if socioeconomic inequality would have been eliminated (PAR / CCI * 100) [19]. We used Pearson correlation to measure the degree of relationship between the CCI and the PAR%. The analyses were performed using Microsoft Excel and HEAT Plus software.
Supplementary Table 1 shows the average coverage by wealth quintile for each of the maternal and child health interventions. The coverage gap tended to be smaller as the income level improved. National coverage was greater than 78% in all interventions except family planning and treatment of sick children. The greatest inequality occurred in skilled attendance at birth and prenatal care, where the difference between the wealthiest and the poorest was 26.4 and 17.3%, respectively. The difference was relatively smaller in the immunization indicators, where the absolute inequality was more pronounced in the coverage of DTP3 than in BCG and measles. The difference ratio was well over 1.0 for most of the interventions, showing a wide gap to the detriment of the poorest quintile, except in the vaccination against measles.
Table 1 shows the coverage gaps and inequalities by wealth quintiles for each country. The national median was 78.4% (Range: 49.8% [Haiti] – 86.6% [El Salvador]) and from 71% for the poorest quintiles and 82% for the wealthiest. In three countries - Haiti, Bolivia, and Guatemala - wide differences (> 21 percentage points) were observed between the wealthiest and poorest quintiles. Guyana, Costa Rica, and Paraguay were the only countries with the lowest coverage in the wealthiest quintile. Belize, Costa Rica, the Dominican Republic, El Salvador, Guyana, Honduras, Mexico, and Paraguay showed low levels of inequality, where the difference between the wealthiest and poorest quintiles was 10 percentage points or less. Haiti was the country with the highest level of relative inequality, with coverage in the wealthiest quintile that exceeds that of the poorest by a factor of 1.7. The ratio of differences between the lowest and highest quintiles was greater than 1.0 in nine countries, showing a predominant pattern of higher inequality where the wealthiest quintile had disproportionately less coverage than all the other quintiles, led by Colombia. Reducing wealth-related inequality had the potential to narrow the national gap between 1% (Costa Rica) and 27.9% (Haiti). If all countries could reach the median overall coverage for the wealthiest quintile, the gap would decrease by 3.6 percentage points (95% CI: 2.7–7.1).
Table 1 Inequality gaps in CCI by wealth quintile, LAC 2001–2016
LAC countries showed a pattern of marginal exclusion in maternal-child health coverage, highlighting the need to address interventions oriented to the most disadvantaged population and also a pattern of higher wealth-related inequality in CCI coverage to the detriment of the poorest quintile (Fig. 1-2). Figure 3 shows the relationship between the CCI gap and PAR% in the study countries. It was observed that healthcare coverage increased significantly as inequality decreased (Pearson r = 0.9; p < 0.01). To achieve equality in the distribution of RMNCH interventions, Haiti (27.9%), Guatemala (14.8%) and Bolivia (17.8%) would need to make a greater effort to reduce the ICC gap at their respective levels.
Latest situation of CCI coverage by economic status, LAC 2001–2016. Own elaboration based on study data. a Dashed lines indicate the median
Difference in CCI by country according to wealth quintile, LAC 2001–2016. a. Source: Own elaboration based on study data. a Dashed lines indicate the median
Coverage gap at the national level versus population attributable risk in LAC countries, 2001–2016.a. Source: Own elaboration based on study data. a Dashed lines indicate the median
The LAC region has experienced a considerable improvement in maternal and child health post-2015 sustainable development agenda [7]. Despite the progress, it is currently considered the most unequal region in the world, which represents a major challenge for the SDGs [20].
We explore current wealth-related inequalities in RMNCH coverage in 15 LAC countries. Our findings reveal important inequalities in maternal and child health interventions, pointing out that in some groups of the population women and children are lagging.
As shown in this study, essential preventive and curative interventions showed a monotonous pattern with lower levels in the poorest quintile. The inequality gap was greater in interventions that required a functional health system and recurrent interaction with healthcare personnel, except in immunizations. Although approximately 80% of the population benefited from the eight essential interventions, coverage of RMNCH interventions was lower than that in more than half of the poorest countries. Only Costa Rica and El Salvador reached this level in the poorest quintile. The difference between the wealthiest and the poorest was at least 9.8 percentage points in more than half of the countries. Haiti, Bolivia, Guatemala, Peru, and Nicaragua showed lower national coverage and absolute inequality above the regional median. Colombia showed greater inequality of coverage in the top quintiles despite not having a wide gap like other countries. These findings imply the need for health systems that prioritize adequate care to reduce the gaps in women and children from the poorest households [7, 10]. Although the countries of the region have indeed implemented reforms to provide health services without the risk of impoverishment, an approach of social determinants and human rights that considers the dimensions of inequality is still required: income, gender, place of residence and education, among others [21, 22].
Achieving equity represents a much greater challenge for Colombia, Costa Rica, Haiti, Honduras, Mexico, and Panama than for other countries in the region, since they are part of the ten most unequal countries in the world [23]. If wealth-related inequalities were eliminated, most countries could achieve coverage of RMNCH interventions of more than 82%. The relationship between CCI and PAR% suggests that to reduce the gap in coverage of health services, the implementation of policies and programs can be effective in addressing inequalities within each country [11]. Policies should be focused on five areas: (i) development of health infrastructure; (ii) health promotion; (iii) health human resources; (iv) healthcare financing, and (v) quality of care [24,25,26].
There is a political commitment to understanding inequalities, encompassing efforts to support the monitoring and evaluation of inequities, health policies, and systems. However, the possibilities of achieving the SDG goals will depend on the ability of countries to accelerate and maximize their achievements in well-being [27]. The study, publication and discussion of the determinants of equity in the coverage of interventions and their impact on health contribute to increases in the effectiveness of public policies [28].
This study has several limitations. Coverage estimates are based on reanalyzed data from demographic surveys with a cross-sectional design. The analysis is limited to the availability of recent surveys in each country for latest situation analysis. Because the ICC is a group indicator, HEAT does not provide sufficient data to estimate the standard error using resampling methods [7]. The household ranking of the wealth index may vary by year and country. The described limitations could underestimate the CCI in study countries, particularly utilizing an index based on selected RMNCH health interventions. Despite the limitations, our findings are based on the best method to explore gaps in care coverage between rich and poor [8].
Overall, our results suggest that women and children from the poorest households in LAC are far from achieving universal health coverage due to inequalities. Our findings show how RMNCH coverage could improve if inequalities were eliminated. Overcoming inequalities will substantially reduce the extreme poverty gap, maternal and child mortality, and promote sustainable development. Future research is needed to monitor inequalities as a critical component tracking the progress of the SDGs so that no one is left behind. We hope that our findings contribute to the design of public policies and strategies to reduce inequalities for women and children in the LAC region.
The datasets used in this article are available in the WHO Health Equity Monitor Database repository at http://apps.who.int/gho/data/node.main.HE-1540?lang=en. Individual data sets are available by the previous request and can be accessed by UNICEF http://mics.unicef.org/ and DHS http://dhsprogram.com/ websites.
ALC:
ANC4:
Prenatal care (at least four visits)
BCG:
One dose of Bacillus Calmette-Guérin vaccine
CCI:
Composite Coverage Index
Confidence interval
DFPS:
Satisfied demand for modern family planning methods
DHS:
Demographic Health Survey
Demographic Health Surveys
DPT:
Three or more doses of diphtheria-tetanus-pertussis vaccine
HEAT:
Health Equity Assessment Toolkit
MCV:
At least one dose of measles vaccine
MDGs:
MICS:
Multiple Cluster Indicator Survey
NSCLC:
Children with pneumonia symptoms taken to a health center
ORS:
Children with diarrhea receiving oral rehydration therapy and continuous feeding
Population attributable risk
PAR%:
Percentage of population attributable risk
RD:
Ratio for differences
RCI:
Relative concentration index
RMNCH:
Reproductive, Maternal, Newborn, and Child Health
SBA:
Deliveries attended by qualified personnel
SII:
Slope index of inequality
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We thank Cesar Matos and Antonio Peramo for comments, suggestions, and language improvement. Special thanks to Carlos Sosa for the statistic notes.
The authors declare that there was no funding associated with this study.
Instituto Tecnológico de Santo Domingo, Universidad Rey Juan Carlos, Madrid, Spain
Manuel Colomé-Hidalgo & Ángel Gil de Miguel
Universidad Autónoma de Madrid, Madrid, Spain
Juan Donado Campos
Manuel Colomé-Hidalgo
Ángel Gil de Miguel
MC conceived and designed the study, carried out the statistical analysis, and drafted the paper; JD and AG analyzed the data, interpreted the results, and contributed to drafting the manuscript. The authors read and approved the final manuscript.
Correspondence to Manuel Colomé-Hidalgo.
All analyses are based on publicly available data from demographic surveys.
Additional file 1: Table S1
. Mean coverage of inequality gaps in interventions by wealth quintile, LAC 2001–2016.
Colomé-Hidalgo, M., Campos, J.D. & de Miguel, Á.G. Exploring wealth-related inequalities in maternal and child health coverage in Latin America and the Caribbean. BMC Public Health 21, 115 (2021). https://doi.org/10.1186/s12889-020-10127-3
Socioeconomic factors
Caribbean region | CommonCrawl |
Why does the salt in the oceans not sink to the bottom?
This is something that just occurred to me. If heavier elements sink, then how can the entire ocean be salty? Shouldn't the 'salt', because of its density, all sink to the bottom of the ocean? In theory, only the deepest parts of the ocean should be salty, while the top of the ocean is not. Yet, the only water in the world that isn't salty comes from rain and rivers. How can this be?
oceanography water
$\begingroup$ You will have better luck with this on the physics stack. This question is entirely about what a solution is. $\endgroup$ – John Jul 22 '19 at 17:40
$\begingroup$ Sometimes it does, in brine pools at the bottom of the ocean. Watch this eel risk its life exploring one: youtube.com/watch?v=ZwuVpNYrKPY $\endgroup$ – Chloe Jul 22 '19 at 21:49
$\begingroup$ This question clearly has Earth Science-specific answers and should stay open here. Voting to remain open! $\endgroup$ – uhoh Jul 23 '19 at 6:36
$\begingroup$ I wander if the ocean brine pools have any cool fossils in them. I suppose lot of fish may not survive venturing in one and end up being conserved by the salt forever. $\endgroup$ – Tomáš Zato - Reinstate Monica Jul 23 '19 at 12:07
$\begingroup$ @TomášZato Interestingly, there are brine pool loving bacteria living there - not sure what they get their energy from, but they may not reject a fish from time to time. $\endgroup$ – Volker Siegel Jul 24 '19 at 8:05
When dissolved in water, salt breaks up into sodium and chlorine ions, which combine with water molecules so they cannot easily sink. However, there is a tendency for streams of fresh water to float on salt water and rise to the top. This caused problems for British submarines in the Dardanelles Straits during WW1. Moving from almost fresh water to the denser salt water, they suddenly became more buoyant and rose involuntarily to the surface, making them visible to Turkish gunners on the shore. There are also parts of the ocean where there are pools of very salty water lying on the bottom in such a way as to clearly show the pool to any diver who happens to see it, as though it were a pool on land, so in some circumstances very salty water can sink.
Michael WalsbyMichael Walsby
$\begingroup$ Note that those brine pools that you refer to are not caused by salt solidifying out of seawater, but of salt coming up from the seabed. $\endgroup$ – Jan Doggen Jul 23 '19 at 12:17
$\begingroup$ But isn't this independent of whether it is salt? More like differences in concentration (different density) and temperature (different density)? $\endgroup$ – Peter Mortensen Jul 23 '19 at 21:05
$\begingroup$ @PeterMortensen Yes, it works for any solute. Indeed, it's not like salty water is 100% NaCl + water. But what Michael said still applies just as well - all solutes behave the same way. The main thing is how well the different solutions mix - if you let the ocean water settle, you'd get a pretty uniform distribution over time, since solute from the "saltier" part would migrate to the "sweeter" part (the difference in density isn't enough to overcome the energy favourability of the lower solute concentration) - though I'm sure that's not true for a kilometre deep column of water. $\endgroup$ – Luaan Jul 24 '19 at 8:22
$\begingroup$ The submarine reference is fascinating but I cannot find any immediate source. Does anyone have a reference? $\endgroup$ – Edgar H Jul 24 '19 at 10:05
$\begingroup$ One of my main interests is military history, and I have read endless books on it and watched many TV documentaries, but I can't remember which ones I saw this anecdote in. However, I did see it, I have definitely not made it up. Your best chance of finding an original source would be to read books or internet articles on the Gallipoli campaign, especially the naval aspects. Submarines and sea mines. played an important role. $\endgroup$ – Michael Walsby Jul 24 '19 at 10:16
Because there isn't any "salt", per se, in the ocean. Salt, as the compound sodium chloride (NaCl) does not exist as a solid in the ocean. It is dissolved into sodium and chloride ions (charged atoms) that exist within the ocean as a homogenous phase (that is, a "thing").
That said, water with sodium chloride dissolved in it is indeed denser than pure water, because after all, sodium and chlorine atoms are denser than atoms of hydrogen and oxygen. This leads to an interesting phenomenon: you can have layers of more-salty water and less-salty water that do indeed rise and sink. There are several YouTube videos that demonstrate this very well. For example this video shows dyed salty and fresh water, separated by a barrier:
and then when the barrier is released, the salty water sinks down:
Some other videos: one and two.
This phenomenon is extremely important for planet-scale ocean circulation, and has strong influence on our climate.
GimelistGimelist
$\begingroup$ "because after all, sodium and chlorine atoms are denser than atoms of hydrogen and oxygen." My understanding is that volume(water+salt) < volume(water)+ volume(salt), so that would also make it more dense. $\endgroup$ – Acccumulation Jul 24 '19 at 15:22
$\begingroup$ @Acccumulation's point is very important for a liquid. For a gas, it is the molecular weight that is more important than the individual hydrogen and oxygen atoms: as water they are a combined molecule H<sub>2</sub>O. In either case, "sodium and chlorine atoms are denser than atoms of hydrogen and oxygen" is not a good description of density. $\endgroup$ – Bryan Krause Jul 24 '19 at 22:14
$\begingroup$ @Acccumulation that's molar volume, disregarding massed. $\endgroup$ – Gimelist Jul 24 '19 at 22:29
$\begingroup$ @BryanKrause it is an excellent description for the layperson because no matter how you look at it, NaCl is whatever form is much denser than H2O in whatever (yet equivalent form). An accurate technical description is unnecessary in this case. $\endgroup$ – Gimelist Jul 24 '19 at 22:31
$\begingroup$ @Gimelist Is the density of dissolved NaCl well-defined? To measure the density, you have to divide the mass by the volume, but how do you measure that volume of a solute? Since, as I said, volume is subadditive, how do you define how much of the volume of the solution is coming from the solute? Volume is a macroscopic property. Individual atoms don't have a volume, in a sense that corresponds to the ordinary meaning. "Volume" refers to the amount of space for which an object excludes other objects, but an atom does not necessarily exclude other atoms from occupying the same space. $\endgroup$ – Acccumulation Jul 24 '19 at 22:40
I'm a regular from the Physics Stack Exchange reporting for duty.
Why this is a serious question
This is a bigger question than you might be giving it credit for. The question is ultimately similar to asking why all the air molecules in the atmosphere do not fall to the floor. Your question comes from a very solid principle in physics which could be called the minimum energy principle.
The basic derivation is that if you define the power exerted by a force $\mathbf F_i$ on a particle with velocity $\mathbf v$ to be $$P_i=\mathbf F_i\cdot\mathbf v = |\mathbf F_i|~|\mathbf v|~\cos\theta,$$then Newton's law that the sum of forces on a particle $\sum_i \mathbf F_i = m~\dot{\mathbf v}$ is the mass times the change in velocity per unit time, directly implies that the sum of powers exerted on a particle $\sum_i P_i = \dot K$ is the change in kinetic energy per unit time. Drag forces exist and they oppose relative motion, so their $\cos \theta$ is negative and they will decrease kinetic energy, $\dot K < 0.$ Since energy is a conserved quantity (a "stuff," if you'd like: if you find more or less of it in a box, then it must have come from somewhere else where there is less or more of it), drag forces eventually rob energy from a system until it ends up at the minimum energy.
And it is a very useful principle, for example you can use it to very easily derive the principle of buoyancy and the effective force that must be created by the displaced water to produce that effect; you can't do Newton's laws easily when there are that many tiny little forces of little water molecules but you can absolutely compare total potential energy when an object is at the bottom of the ocean, the middle, and the top. It fails to describe certain things like static friction (why is my laptop on my desk and not on my floor?!) because it does not tell you how long such things take and requires an assumption of noise to eventually perturb you out of "local minimums" and such.
But surely the air has had enough time to fall to the ground if that were what it wanted to do. The air does not want to fall to the ground. And we can't steal our normal solutions for other things like "why don't clouds fall," "well what you think of as a cloud is actually more like a waterfall, there is constant movement of water droplets, the water gets a boost upward from heating the air around it as it condenses but it does tend to eventually fall but when it falls beneath a certain flat surface it evaporates again and becomes invisible and so the visible puff is constantly being fed by new water droplet formation and constantly sapped by falling water that becomes invisible…"—no. These are concrete particles that somehow avoid falling to the ground and we have to actually solve the problem.
Fluctuation-Dissipation theorem to the rescue
The minimum-energy principle describes something that we would call dissipation, energy leaving one system to end up in another system. These sorts of gates are always bidirectional: energy goes through in both ways. But mostly you don't notice it, and that's key to how the principle helps us describe things: energy always flows out, it never flows back in.
Until, well, it does. Energy of a bouncing ball spreads out among all of the different degrees of freedom of the floor, the air, but if it really goes all the way to 0 and sits perfectly and completely still, very soon the air will bump it and start it jostling and vibrating and moving again—just not moving very much. The same things that allow energy to dissipate must also be contributing constant energy fluctuations that prevent energy from going all the way to 0.
These fluctuations are collectively understood as temperature. Temperature is technically only defined for a system where all of its degrees of freedom in the ways it can move have come to the same average energy, and it is measured as that average energy. Temperature defines this average energy and the size of these fluctuations. So at room temperature for example we would say that every degree of freedom has 26 meV, 26 "milli-electron-Volts" of energy, or 0.026 of the energy that an electron would have if accelerated by a one volt battery.
So why does the air stay up? It is, basically, because the molecules of the floor are kicking the air molecules with enough energy to hit the upper reaches of the atmosphere. They do not actually go straight there; one air molecule bumps into other air molecules over a very short distance scale: but it transfers that energy and momentum to other particles which transfer that energy and momentum to other particles and in the end the air "prefers" to "hang out" near the ground but the fluctuations cause it to get bumped to an average height given by our temperature. So if you take the mass of nitrogen N2 of 28 amu, and the acceleration due to gravity of 9.8 N/kg, you can find out that this 26 meV temperature means that the atmosphere is ~9 km high on average, which does get you a good chunk into the troposphere where the air starts to thin out dramatically. Actually the theory says that if nothing else were to happen and the random kicks were to just launch a particle up into the atmosphere, it would have a random height sampled according to an exponential probability distribution, $P(h) \sim e^{-h/(9\text{ km})}$.
Similarly why don't the salt molecules fall to the ocean floor? Well, they do, and then they get kicked back up. The water at the ocean floor is saltier. The key difference is whether the salt in question dissolves in water (if it sticks to water better than it sticks to itself) or precipitates in water (it sticks to itself better): larger chunks of a piece of stuff that get bound together will tend to act as big massive chunks and then that thermal energy cannot kick it as high.
This is the general idea of the fluctuation-dissipation theorem, which states that fluctuation and dissipation (under some extremely broad assumptions called "detailed balance") always go hand-in-hand. Anything which can absorb light (dissipation) must radiate light into space (blackbody radiation, a sort of fluctuation). Every electrical resistor is also a noise source (Johnson noise). If energy can flow out of a system into some environment, then it will only flow out until they have the same average energy levels, and if you try to go lower, energy fluctuations from the environment flow back into the system.
CR DrostCR Drost
$\begingroup$ Interesting. I'll have to read it several times until it will sink in (pun very much intended). $\endgroup$ – Gimelist Jul 25 '19 at 11:53
$\begingroup$ I don't see how this helps at all. The theory of hydrodynamics (HD) is an average over newtonian kicks and short mean free-paths. More importantly, HD is perfectly able to explain static stratification and density gradients via flux differences of up minus down. The FD-theorem which you quote is usually only of importance when the mean-free path of particles becomes comparable or greater than the system scale, otherwise HD is fine. Both for the ocean and atmospheric structure up to 100km height. $\endgroup$ – AtmosphericPrisonEscape Oct 10 '19 at 20:25
$\begingroup$ @AtmosphericPrisonEscape I think you are overthinking it? HD is the mean field approximation but FD is the fundamental cause; OP was asking for the fundamental cause. $\endgroup$ – CR Drost Oct 11 '19 at 11:57
$\begingroup$ I guess that's up to OP to decide that. I just don't think that a question to which the understandable answer would be, "salt is dissolved, and fluids of differing densities behave in the following way", can be expanded much by handwaving it away with FD. $\endgroup$ – AtmosphericPrisonEscape Oct 11 '19 at 12:15
Turbulence, because seawater is, almost, always on the move saltier water is mixed with fresher by wave action and, to a lesser extent in surface waters, by Brownian motion. In Fjordland the annual rainfall is so high (up to 8000mm) that there is a permanent freshwater layer several metres thick that you can drink from sitting over the salt water from the Tasman in the sheltered inlets. Even there this layer doesn't have a clear cut boundary but rather a mixing layer where the salt and fresh water exchange particles and homogenise over time. In bodies of water that don't experience regular circulation stagnation and anoxia set in over time but chemical solution of a number of dissolved salts still occurs.
Saltier water has higher mass density, so the gravitational energy can be lowered that way. The concentration differences go up until the free-energy of creating that big a concentration difference balances the gravitational energy change.
Department of Physics, University of Illinois at Urbana-Champaign
Making some simplifying assumptions, they find:
the equilibrium concentration goes up exponentially with depth, by a factor of e for each 10 km or so.
The actual oceans are stirred by currents, so this equilibrium concentration difference isn't present in them.
Basically they saying that it takes energy to separate a homogeneous solution into parts which are more or less concentrated (and hence more or less dense). Taking into account the gravitational energy, it follows that the least energy state of a column of water is saltier at the bottom.
Keith McClaryKeith McClary
$\begingroup$ So to paraphrase, salt does sink to the bottom, but only somewhat, so that the bottom of Challenger Deep is nearly three times as salty as the surface? $\endgroup$ – Tanner Swett Jul 23 '19 at 10:47
$\begingroup$ @TannerSwett: This sounds correct. "Exponential growth" isn't really very impressive between $0$ and $1$. $\endgroup$ – Eric Duminil Jul 23 '19 at 15:23
$\begingroup$ The answer is no! Salinity almost does not change with depth. Here is an example from the Challenger Deep: link.springer.com/article/10.1007/s10872-005-0053-z $\endgroup$ – arkaia Jul 24 '19 at 18:38
$\begingroup$ @arkaia As the University of Illinois page states, the actual oceans do not reach the equilibrium state because of currents. Gravitational differentiation (over thousands of km) is significant inside the Earth, resulting in the heavier elements sinking to the core. $\endgroup$ – Keith McClary Jul 24 '19 at 18:58
$\begingroup$ Then the answer to the formulated question is that it does not sink because is dissolved and the currents (and temperature gradients, by the way) prevent equilibrium from happening. I just don't want people to get a misconception. Salt in the ocean is pretty evenly distributed (except near rivers). The differences in salinity with depth are minimal compared with the gradients in temperature. $\endgroup$ – arkaia Jul 24 '19 at 19:19
But it does, but according to each salt's solubility and density. Soluble salts tend to mix into the water and keep suspended. Insoluble salts separate from the solution and creates deposits in the oceanic floor.
One famous example was the "de-ironing" of the seas, when iron salts were deposited in the bottom due to the oxigenation of the oceanic water, by the time of the emergence of aerobic, photosynthetic organisms.
Great Oxidation Event: https://en.wikipedia.org/wiki/Great_Oxidation_Event
"The oxygen then combined with dissolved iron in Earth's oceans to form insoluble iron oxides, which precipitated out, forming a thin layer on the ocean floor". https://en.wikipedia.org/wiki/Banded_iron_formation
Luiz P. O. Pereira da SilveiraLuiz P. O. Pereira da Silveira
$\begingroup$ Geochemists use "salts" most commonly to talk about soluble compounds. I have never heard anyone refer to insoluble compounds (and iron oxide) as a "salt". So while this answer is technically correct, it may be a bit misleading. $\endgroup$ – Gimelist Jul 24 '19 at 12:09
$\begingroup$ Salt, in the parlance of this response, is a result of a reaction between an acid and a base (in the broadest sense, i.e. Lewis acid, etc). So Iron is an electron "giver", i.e., a base, and Oxigen an electron "taker", i.e., an acid. The reaction "Fe+2 + O2 -> Fe+3 + O-2" (unbalanced, the signs are electric charge of the ion) means that Fe2O3 is a salt, in this sense. $\endgroup$ – Luiz P. O. Pereira da Silveira Jul 29 '19 at 20:33
$\begingroup$ no, in the parlance of geochemistry and ocean geochemistry in particular, "salts" are always soluble compounds like halides, nitrates and some sulfates. Your definition of a Lewis acid includes too many inorganic compounds to make it useful in the field of geochemistry. $\endgroup$ – Gimelist Jul 29 '19 at 20:57
$\begingroup$ Sure, geochemistry has a specific nomenclature. Chemistry in general has many definitions of salt, and the most basic is the result of a reaction between an acid and an alkaline compound. The answer does explain how some of the salt gets deposited in the ocean's bottom, though. $\endgroup$ – Luiz P. O. Pereira da Silveira Aug 12 '19 at 14:39
And then there is the saturation issue. Salt can be dissolved in water to a certain degree only. Once that degree is exceeded the salt begins to fall out and sink to the ground. If I remember well the limit for water is something like 35g per litre (depending on the temperature)
Fabian RuinFabian Ruin
$\begingroup$ This seems to suggest that (contrary to the question) salt does sink to the bottom. Can you back this up, i.e. shows that this effct actually takes place in the oceans? $\endgroup$ – Jan Doggen Jul 25 '19 at 13:33
$\begingroup$ not directly related to this answer but when seawater freezes(in the arctics)the salt will be consentrated and sink down to the bottom of the ocean,this does lower the salt consentration in the ice. $\endgroup$ – trond hansen Jul 26 '19 at 10:08
Salt does sink to the bottom in the oceans.
Why? Your question referring to salt. Salt is a solid chemical compound. Take a lump of rock salt of sodium chloride, throw it into the water: it will sink to the bottom. The reason is that the density of sodium chloride with more than 2 g/cm3 is higher than the density of seawater less than 1.1 g/cm3.
Of course, the salt lump will be dissolved sometime and no longer exist. But then it's no salt anymore. Then there are only fast and somehow moving loose cations and anions in the water.
gotwogotwo
Salt does not sink to the bottom in the seas and oceans, because it dissolves in water! If you want to get salt from the seas and oceans, try to vaporize them :-)...
CyrilCyril
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Molten salt seas on the future Earth | CommonCrawl |
Journal of Nanobiotechnology
Preparation of PLA/chitosan nanoscaffolds containing cod liver oil and experimental diabetic wound healing in male rats study
Payam Khazaeli1,2,
Maryam Alaei3,
Mohammad Khaksarihadad4 &
Mehdi Ranjbar ORCID: orcid.org/0000-0002-5844-32991
Journal of Nanobiotechnology volume 18, Article number: 176 (2020) Cite this article
Diabetes mellitus is one of the most common metabolic disorders. One of the important metabolic complications in diabetes is diabetic foot ulcer syndrome, which causes delayed and abnormal healing of the wound. The formulation of nanoscaffolds containing cod liver oil by altering the hemodynamic balance toward the vasodilators state, increasing wound blood supply, and altering plasma membrane properties, namely altering the membrane phospholipids composition, can be effective in wound healing. In this study, electrospinning method was used to produce poly lactic acid/chitosan nanoscaffolds as a suitable bio-substitute. After preparing the nanoscaffolds, the products were characterized with dynamic light scattering (DLS), transmission electron microscopy (TEM) and scanning electron microscopy (SEM). Also optical properties of polymer and comparison between adsorption between single polymer and polymer-drug calculated with UV−Vis spectra. The structure and functional groups of the final products were characterized by Fourier-transform infrared spectroscopy (FT-IR) and energy dispersive spectroscopy (EDAX) as elemental analysis. The results showed that the optimum formulation of cod liver oil was 30%, which formed a very thin fiber that rapidly absorbed to the wound and produced significant healing effects. According to the results, poly lactic acid/chitosan nanoscaffolds containing cod liver oil can be a suitable bio-product to be used in treating the diabetic foot ulcer syndrome.
Poly lactic acid/chitosan nanoscaffolds were synthesized using microwave-assisted electrospinning process.
Nanoscaffolds showed high potential in wound healing recovery after 14 days.
PLA/chitosan nanoscaffolds containing 30% cod liver oil were synthesized with the size of about 50–150 nm.
Wound area indicated that there was significant improvement in wound surface on the 14th day.
The global prevalence of diabetes has increased dramatically over the past 2 decades [1]. Diabetes mellitus is the most common heterogeneous metabolic disorder [2], which is associated with a disorder in the metabolism of sugars, lipids, and proteins and is characterized by elevated blood glucose or insulin response to tissues [3, 4]. Patients suffering from diabetes mellitus have limited ability to stimulate the immune response and are very susceptible to infection and at the risk of terminal limb amputation and recurrence of the wound [5]. Fatty acids have physiological and pathological roles in diseases such as atherosclerosis [6, 7], inflammation [8], or normal wound healing [9, 10], The effect of fatty acids on wound healing is through alterations in plasma membrane properties [11, 12], such as changes in membrane phospholipids composition [13,14,15], increased growth factor activity [16, 17], cell differentiation [18, 19], decreased eicosanoids production [20], and lipid mediators of inflammation [21], followed by reducing inflammation and producing interleukin-1 and collagen [22]. The cod liver oil as a rich source of omega-3 fatty acids has many potential effects on modulating various diseases, especially diabetes mellitus [23, 24], improvements in vasodilator property [25,26,27]. In many studies immune and allergic responses of rats was investigated for wound healing [28,29,30] Many scientific works have shown which cod liver oil accelerates many of the potential mechanisms involved in wound healing [31,32,33]. In recent years, new drug delivery systems such as nanofibers [34], nanoparticles [35], cell therapy, and stem cell [36] being used as alternative therapies for common pharmaceutical methods, which could reduce the need for continuous follow-up of the disease and increase the quality of treatment, have received great attention [37]. Nanotechnology has solved many concerns in the field of medicine due to dealing with materials that have unique properties on their surface [38,39,40,41,42,43]. Chitosan structures have a good crosslink structure for encapsulating drugs [44] and polylactic acid possesses properties such as the ability to form hydrogels in physiological conditions [45], mild gel degradation for a wound to heal successfully, and the growth and movement of nutrients [46]. In recent years, the science of nanotechnology has attracted particular attention from researchers in various fields of medicine and pharmaceuticals [47]. Nanofibers and nanoparticles can release the drug in a controlled approach for a long time [48]. These structures can act as an appropriate topical drug delivery system that can provide the appropriate drug concentration and other advantages of this system include the ability to transport hydrophilic and lipophilic drugs simultaneously depending on their structure [49]. Examples of natural polymers used in the fabrication of nanofibers with electrospinning method [50,51,52,53] include creatine [54], gelatin [55], cellulose [56], and polysaccharides such as chitosan and alginate. The synthesis of the PLA/ Chitosan nanofibers has been reviewed in recent studies [57,58,59]. Microwave irradiation as a cost-effective, eco-friendly, and high efficiency method is used for preparing nanoparticles for various applications [60], electrospinning process with high-voltage power, generate polymer fibers in nanometer dimensions which show unique physical and chemical properties [61]. In this study, new developments in the fabrication of nanoscaffold materials such as the microwave-assisted electrospinning process were applied to prepare and formulate poly lactic acid/chitosan containing cod liver oil as a suitable cost-effective method. Summary of the research on nanoscaffolds applications in wound healing recovery is displayed in Table 1. The results indicate that the synergistic effect quantity of the poly lactic acid/chitosan containing hydrogel is the key factor in obtaining suitable biological wound for wound dressing.
Table 1 Summary of researches about nanoscaffolds applications in wound healing recovery
All materials and precursors used in this research work were pure without any impurities and were purchased directly from reputable commercial centers. Chitosan (CAS: 9012-76-4, MW Mol wt: 50,000 daltons based on viscosity, 99.90%), Polylactic acid (C(CH3) HC(=O) O–) and Dimethylformamide (DMF, MW; 73.095 g mol− 1) were purchased from Sigma Aldrich agents in IRAN). Polysorbate 80 (tween 80, C64H124O26, MW: 1.310 g/mol) was purchased from FLOKA company in Switzerland. NaOH (d: 2.13 g/ml, MW: 39.9971 g/mol, 99.99%) was purchased from Dr. Abidi company in IRAN. We purchased cod oil lever from institute of pharmaceutical services Razavi company. Xylazine and Ketamine for anesthesia and intraperitoneal tolerance in rats were purchased from Alfasan group of companies in Netherlands. Male rats were obtained from Kerman university of medical animal's farm. Also this study received ethical approval from the local ethical committee of the kerman university of medical sciences as a thesis research at the faculty of pharmacy kerman university of medical sciences with number 1124. Male rat weighing 150–200 g was fed with standard diet and kept under 12:12 h light/dark cycles, at 20 ℃ and relative humidity of 25–30%. XRD patterns for crystalline phase detection were recorded by a Rigaku D-max C III, X-ray diffractometer using Ni-filtered Cu Ka radiation. Microscopic morphology and investigation of surface propertieso of the products were characterized by SEM (LEO 1455VP). The energy dispersive spectrometry (EDS) supplier analysis to determine the elements in the samples was studied by XL30. Transmission electron microscopy (TEM) images were obtained with a Philips EM208 transmission electron microscope with an accelerating voltage of 200 kV. Fourier transform infrared (FT-IR) spectra were recorded on Shimadzu Varian 4300 spectrophotometer in KBr pellets. To absorption evaluate samples ultraviolet–visible spectroscopy analysis was carried out using Shimadzu UV-2600 UV–Vis spectrophotometer.
Preparing PLA/chitosan nanofibers
To prepare the polymer phase, at first, 0.2 g of PLA was dissolved in 18:3 ml ratio of deionized water and ethanol after heating and stirring at 50 °C and RPM 400 for 45 min. Then, 5 ml of NaOH (2 mol/l) was added to the above solution and this solution was heated at 60 °C and stirred for 30 min. In the next step, 0.05 g of chitosan was dissolved in 2:1 ml ratio of deionized water and dimethylformamide after 110 min. Subsequently, the solutions were transferred to a beaker and exposed to the microwave irradiation oven under the power of 450 W for 5 min. Regular cycles of the microwave irradiation were set to 30 s off and 60 s on. Finally, the solutions were placed in an environment free of contamination for 24 h to complete the crystallization process.
Cod liver oil loading
First, 2 ml of polymeric solution was added to 100 µl of the drug with the concentrations of 15% and 30% by weight in the presence of 350 µl Tween as the surfactant agent and placed on the reflux system for 30 min at 50 °C and 500 rpm on the shaker for 15 min. PLA/chitosan nanoscaffolds containing cod liver oil were formed by an electrospinning device. Nanoscaffolds containing 30% w/w and 15% w/w cod liver oil were prepared at the speed of 2 ml/h; 12.1 V and jet rotation speed of 100 rpm were used to form the nanofibers.
In vivo study
For the in vivo study, the male rats were divided into four groups (each group containing 6 mice weighing approximately 200 g). Animals were diabetic by the intraperitoneal injection of 60 mg/kg and their diabetes was confirmed after 3 days by measuring glucose using a glucometer. Then, after anesthetizing the rats with ketamine/xylazine, an ulcer about 1.5 cm in the area between the two scapula was created by punch biopsy and, then, the drug was positioned topically on each group. For this study, four groups of mice were divided into the following groups: Mice in Group 1 treated with nanofiber alone; Mice in Group 2 treated with cod oil only; Mice in Group 3 treated with nanofiber delivery system containing cod liver oil for wound healing; and Mice in Group 4 received no treatment (negative control). To induce diabetes, we injected strepotozocin 60 mg/kg intraperitoneally in rats. All the experimental procedures were carried out according to the protocols set for working with animals at Kerman University of Medical Science (Kerman, Iran).
Result and discussion
Physicochemical characteristics
Morphological properties and surface features of the nanofibers were observed through scanning electron microscopy (SEM) images. SEM images with the approximate scaffold size of poly lactic acid/chitosan nanofibers containing 30% and 15% cod liver oil are displayed in Fig. 1a and b, respectively. As can be seen from the SEM images of the nanoscaffold structures, they were uniform and had no cracking along the formation. The nanoscaffold structures encapsulated different concentrations of cod liver oil without any systemic defects and the nanofibers containing 30% cod liver oil showed less diameter than 15% cod liver oil; this may be due to higher solubility of the polymer phase at higher concentrations of oil phase. According to SEM images, the average size of the diameter scaffold tube was estimated between about 50 and 150 nm. To investigate the three-dimensional (3D) images of the fibers, we did transmission electron microscopy (TEM). According to the TEM images, the cod liver oil was trapped uniformly in the spaces between PLA/chitosan nanoscaffolds. The oil phase ranges in the polymer phase were well-defined. The TEM image of 30% w/w cod liver oil cod liver oil distributed in poly lactic acid/chitosan nanoscaffolds is shown in Fig. 1c.
SEM images of the poly lactic acid/chitosan nanofibers containing cod liver oil 30% w/w cod liver oil (a), 15% w/w cod liver oil (b) and TEM image of the poly lactic acid/chitosan nanofibers containing 30% w/w cod liver oil
Energy dispersive spectroscopy (EDAX) is a suitable supplier analysis used for the semi-quantitative analysis of elements. This method is mainly used to obtain point chemical composition and quantitatively investigate the poly lactic acid/chitosan nanoscaffolds containing cod liver oil. About 37.17 percent of the total atomic weight of the final products was related to C atoms; this could be related to carbon atoms in poly lactic acid, chitosan, omega3 in cod liver oil, and Tween structures. The existence of the O, Na, and N atoms to the amount of about 31.63, 5.31, and 15.36 percent could be related to the existence of these atoms in poly lactic acid, chitosan, cod liver oil, Tween, NaOH, and dimethylformamide structures. The small amounts of Ti and Ca atoms could be related to the unpredictable impurities in the final products. The EDAX as a supplier analysis of the poly lactic acid/chitosan nanoscaffolds containing 30% cod liver oil is shown in Fig. 2a. The size distribution obtained from the nanoscaffolds was a plot of the relative intensity of light scattered by nanoscaffolds in various size classes and introduced as the intensity size distribution. Results related to the size distribution of the nanoscaffolds obtained from dynamic light scattering analysis showed good match with SEM images and estimated the size of the poly lactic acid/chitosan nanoscaffolds containing 30% cod liver after 15 min ultrasonic irradiation at 60 W fibers of about 50–150 nm in Fig. 2b. Also the size distribution of PLA/chitosan nanofibers without cod oil liver was calculated about 120 nm.
EDAX supplier analysis (a) and DLS data diagram after 15 min ultrasonic irradiation at 60 W (b) of the poly lactic acid/chitosan nanofibers 30% w/w containing cod liver oil
UV–Vis as a general qualitative technique can be used to identify and confirm functional groups in a compound by matching the absorbance spectrum. Absorption in UV–Vis spectroscopy follows the Beer's Law:
$${\text{A}} = \upvarepsilon \times {\text{b}} \times {\text{C}}.$$
where ε is the molar attenuation coefficient, b is path length, and C is concentration. UV–Vis absorption spectra of poly lactic acid/chitosan nanoscaffolds containing cod liver oil showed that, with increasing concentration from 15 to 30% w/w, absorbent peaks became noticeably more intense. However, the cod liver oil was not absorbed alone and it can be concluded that more loading of cod liver oil occurred in poly lactic acid/chitosan nanoscaffolds. Figure 3a demonstrates UV–vis absorption spectra of poly lactic acid/chitosan nanofibers containing cod liver oil of 30% w/w and 15% w/w compared with the cod liver oil [62]. Fourier transform infrared spectroscopy (FT-IR) is an analytical technique used to identify functional groups in materials. Figure 3b, c shows FT-IR spectrum of the prepared poly lactic acid/chitosan nanoscaffolds containing 30% w/w and 15% w/w cod liver oil in the region 400–4000 cm− 1, respectively. The absorption peaks at 3454 cm− 1 and 1630 cm− 1 regions could be attributed to the stretching and bending vibrations of O–H groups from chitosan and omega3 structures in the cod liver oil. The obtained peaks at 2884 cm− 1, 1650 cm− 1, and 1600 cm− 1 expressed the existence of stretching mode C-H, CH2 groups, C=O, C=C, and C–N regions in chitosan omega3 in cod liver oil and poly lactic acid structures. The reflectance at 3093 cm− 1 showed N–H band in chitosan. In general, the cod liver oil structures with forming chemical bonds are located in nanobio-polymeric nanoscaffold structures.
UV–Vis absorption spectra of poly lactic acid/chitosan nanofibers containing cod liver oil 30% w/w, 15% w/w and cod liver oil [62] (a) and FT-IR spectrum of the poly lactic acid/chitosan nanofibers containing 30% w/w (b) and 15% w/w (c) cod liver oil
Diabetic rats were evaluated in two ways; (a) blood glucose measurement by glucometer, rats with blood glucose above 200 mg/dl were considered diabetic and selected for further study, (b) the selected rats were examined for the appearance of diabetic symptoms including polyuria, overeating, and thirst and were assured of their diabetic status. As shown in the results and tables, in the group using nanofibers mixed with cod liver oil, the wound healing process (assessed by measuring the wound area) was investigated. It showed significantly better results than the group that used nanofiber alone or cod liver oil alone. It also appears that 30% cod liver oil supplementation with polyelactic acid/chitosan nanoscaffolds resulted in 94.5% wound healing on day 14, whereas 15% cod liver oil can heal wounds by about 86% on day 14, as presented in Table 2. Macroscopic changes of the wound were evaluated for treatment progress on days 0, 3, 7, and 14 after treatment and recorded by a photograph. The percentage of wound healing was calculated using the following formula:
$$Percentage\;of\;recovery = {{\left( {Surface\;wound\;on\;the\;first\;day - Surface\;wound\;in\;day\;X} \right)} \mathord{\left/ {\vphantom {{\left( {Surface\;wound\;on\;the\;first\;day - Surface\;wound\;in\;day\;X} \right)} {\left( {Surface\;wound\;on\;the\;first\;day} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {Surface\;wound\;on\;the\;first\;day} \right)}} \times 100.$$
Table 2 Evaluation of wound healing rate in study groups studied on days 7 and 14
Figure 4 demonstrates a 14-day course examination of the wound surface and a photograph of the wound healing process. On days 0, 3, 7, and 14, the rate of wound healing in the nanoscaffold saline-treated mice with 30% cod oil was significantly different from that of the untreated control group and also from the other groups. The poly lactic acid/chitosan nanoscaffolds as bio-compatible and bio-degradable polymers could interact with skin cells and accelerate the healing process. Due to the high porosity of the nanoscaffold coating and the hydrogel-like properties of the polymers used, the coating swelled after the absorption of moisture and created a very small gap between the coating and the wound surface. On day 14, these characteristics were well observed and the nanoscaffold porosity property permitted oxygen to pass through the wound while keeping its surface moist.
Wound area in five groups of animals studied on days 0, 3, 7 and 14 (mean ± sd; N = 3)
Wound area in five groups of animals studied on days 0, 3, 7, and 14 (mean ± sd; N = 3) is shown in Fig. 5. The area of the wound in the group with 30% cod oil was significantly less than the other groups; this indicated greater improvement in this group than in the other groups. The presence of bio-compatible nanofibers not only induced immune and allergic responses, but also made the body resemble the original tissue located in the wound. As a result, bio-chemical signals are needed to accelerate recovery and, eventually, wound healing will occur faster.
The wound healing process observed in the study groups
In this research, the nanofibers scaffolding produced with electrospinning method was used to repair wounds on the skin of the rats. Macroscopic and microscopic studies were performed on the wounds to determine the efficacy of the produced nanoscaffolds after the desired time. The poly lactic acid/chitosan as bio-compatible and bio-degradable polymers scaffolding were designed for wound healing to be able to control drug (cod liver oil) release over a long period of time by improving the chemical structure. The poly lactic acid/chitosan nanoscaffolds containing 30% cod liver oil showed more healing and less wound area on day 14; this can be due to permeability and sufficient oxygen for tissue repair. Moisture retention of the wound medium for accelerating its healing, color change, and pH changes to keep the wound site safe from bacteria, contamination, and non-contamination was among the characteristics of the produced nanoscaffold.
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Authors are grateful to council of Pharmaceutics Research Center, Institute of Neuropharmacology, Kerman University of Medical Sciences, Kerman, Iran.
Pharmaceutics Research Center, Institute of Neuropharmacology, Kerman University of Medical Sciences, P.O. Box: 76175-493, Kerman, 76169-11319, Iran
Payam Khazaeli & Mehdi Ranjbar
Faculty of Pharmacy, Kerman University of Medical Sciences, Kerman, Iran
Payam Khazaeli
Student Research Committee, Kerman University of Medical Sciences, Kerman, Iran
Maryam Alaei
Neuroscience Research, and Physiology Research Centers, Kerman University of Medical Sciences, Kerman, Iran
Mohammad Khaksarihadad
Mehdi Ranjbar
PK: wrote the manuscript, supervised the research. MA: Designed and performed experiments. MK: Conceived and planned the experiments. MR: Developed the theoretical formalism, performed the analytic calculations. All authors read and approved the final manuscript.
Correspondence to Mehdi Ranjbar.
Khazaeli, P., Alaei, M., Khaksarihadad, M. et al. Preparation of PLA/chitosan nanoscaffolds containing cod liver oil and experimental diabetic wound healing in male rats study. J Nanobiotechnol 18, 176 (2020). https://doi.org/10.1186/s12951-020-00737-9
Poly lactic acid/chitosan nanoscaffolds
Electrospinning | CommonCrawl |
How should we interpret these quantum logic gates as physical observables?
In quantum mechanics each operator corresponds to some physical observable, but say we have the operators $X,Y,Z,H, \operatorname{CNOT}$. I understand how these gates act on qubits, but what do they actually represent in terms of a physical observable?
quantum-gate quantum-information quantum-operation
edited Apr 26 at 13:22
bhapibhapi
I agree with the main points that Niel makes: not all operators are observables, and the purpose of the ones you list is typically to transform states, not to be measured. However, the operators you list happen to be hermitian (allowing them also to represent observables) as well as unitary (allowing them to represent transformations), so in this case we can still play devil's advocate, and ask if they have physical interpretations as measurable quantities. So, interpreting your list as observables to be measured...
$X$, $Y$, and $Z$ are easy: these represent unitless versions of spin measurement (or the analogous quantity, if your system is not a spin system) along the three spatial coordinate directions.
$H$ also represents a unitless version of a spin measurement, but along an axis exactly between the $x$ and $z$ axes.
$\operatorname{CNOT}$ is the hardest, and I don't have a clean physical interpretation. It has eigenvalues $\pm1$ (3 eigenvectors with eigenvalue +1, and one with eivenvalue -1), and as an observable it should represent correlative properties of the two qubits. We could state what it literally measures, in terms of the four dimensions that it acts on, but this information will be less informative than just looking at its matrix form.
All this being said, typically the only one of the five that is actually measured is $Z$, and occasionally the other coordinate spin measurements.
WillWill
In quantum mechanics, not all operators are observables. Many operators are observables, and in the first year or two of treatments in some physics courses you will only see operators which are observables; but not all operators of interest are observables.
The operators you have mentioned all happen to be Hermitian, and could therefore be interpreted as observables. This is particularly useful for the operators $X$, $Y$, and $Z$, which up to a scalar factor of $2\hbar$ you may be more familiar with as orbital angular momentum components. However, in the context of quantum computation, the more important property of the operators $\{X,Y,Z,H,\mathrm{CNOT}\}$ is that they are unitary: that is, their eigenvalues are all roots of unity, and so they represent possible finite-time evolutions of a quantum system.
Finite-time evolutions of the system are of course governed by the wave-equation. Specifically, from the time-independent Schrödinger equation, we have $$ i\hbar \frac{\mathrm d}{\mathrm d t} \lvert \psi(t) \rangle = \hat{\mathrm{H}} \, \lvert \psi(t) \rangle$$ which has solutions of the form $\lvert \psi(t) \rangle \,=\, \hat{\mathrm{U}}(t) \: \lvert \psi(0)\rangle$, for some initial state-vector $\lvert \psi(0)\rangle$ and where $\hat{\mathrm{U}}(t)$ is a unitary operator given by $$ \hat{\mathrm{U}} (t) \,=\, \exp\bigl(-i \:\!\hat{\mathrm{H}} \,t\big/\hbar\bigr) \,=\, \sum_{n \geqslant 0} \frac{\bigl( -i\!\:\hat{\mathrm{H}}\,t \big/ \hbar\bigr)^n}{n!} $$ whose eigenvalues are of the form $\mathrm e^{-iE_k t / \hbar}$ for eigenvalues $E_k$ of $\hat{\mathrm H}$. Such an operator is called a unitary operator, and had the property that $\hat{\mathrm U}(t){}^{-1} = \hat{\mathrm U}(-t) = \hat{\mathrm U}{}^\dagger$. The gates $X$, $Y$, $Z$, $H$, and $\mathrm{CNOT}$ are all operators of this form, where it so happens that the eigenvalues are $\pm 1$.
All unitary operators can be obtained from the exponentiation of a Hermitian operator in this way; though in many cases we prefer to consider them as products if a sequence of different operators $\hat{\mathrm U} = \hat{\mathrm U}_N(t_N)\,\cdots\,\hat{\mathrm U}_2(t_2)\,\hat{\mathrm U}_1(t_1)$, representing evolution according to piecewise-constant Hamiltonians. (Realising such a unitary evolution will be more complicated in practise, of course, just as computing with classical computers is more complicated than just applying 'AND' and 'OR' gates to bits represented by electrical voltages in wires, but this is the picture of quantum computation to first-order.)
Niel de BeaudrapNiel de Beaudrap
$\begingroup$ Is there an intuitive way to think of them physically though ? As in what is it that we are actually observing when we measure them ? $\endgroup$ – bhapi Apr 27 at 15:37
$\begingroup$ That's the point --- these are intended to be propogators, not observables. You don't measure unitary operators, you use them to describe the evolution of (closed) systems. $\endgroup$ – Niel de Beaudrap Apr 27 at 16:32
$\begingroup$ @can'tcauchy it is really a coincidence that the operators you list are measurable at all, so Niel's main point stands. In general, transformations on a quantum state are not physically measurable; however, in the case of those you list, they in principle are, so see my answer below. $\endgroup$ – Will Apr 27 at 17:44
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How do I write a tensor product of conditional gates in matrix form? | CommonCrawl |
\begin{document}
\title{On the Baire class of n-Dimensional Boundary Functions}
\author[C. P. Wilson]{Connor Paul Wilson} \address{530 Church Street Ann Arbor, MI 48109} \email{[email protected]}
\date{}
\begin{abstract} We show an extention of a theorem of Kaczynski to boundary functions in n-dimensional space. Let $H$ denote the upper half-plane, and let $X$ denote its frontier, the $x$-axis. Suppose that $f$ is a function mapping $H$ into some metric space $Y.$ If $E$ is any subset of $X,$ we will say that a function $\varphi: E \rightarrow Y$ is a boundary function for $f$ if and only if for each $x\in E$ there exists an arc $\gamma$ at $x$ such that $\lim_{z\rightarrow x \atop z\in\gamma} f(z) = \varphi(x)$ \end{abstract}
\maketitle
\section{Introduction} \subsection{Preliminaries and Notation}
Let $H$ denote the upper half-plane, and let $X$ denote its frontier, the $x$-axis. If $x\in X,$ then by an arc at $x$ we mean a a simple arc $\gamma$ with one endpoint at $x$ such that $\gamma - \{x\} \subseteq H.$ Suppose that $f$ is a function mapping $H$ into some metric space $Y.$ If $E$ is any subset of $X,$ we will say that a function $\varphi: E \rightarrow Y$ is a boundary function for $f$ if and only if for each $x\in E$ there exists an arc $\gamma$ at $x$ such that $$ \lim_{z\rightarrow x \atop z\in\gamma} f(z) = \varphi(x) $$ We will also define Baire classes as Kaczynski does in \cite{Kacz}, such that a function $f: M \rightarrow Y$ is said to be of Baire class $O(M, Y)$ if and only if it is continuous; if $\xi$ is an ordinal number greater than or equal to $1,$ then f is said to be of Baire class $\xi(M, Y)$ if and only if there exists a sequence of functions $\left\{f_{n}\right\}_{n=1}^{\infty}$ mapping M into $Y, f_{n}$ being of Baire class $\eta_{n}(M, Y)$ for some $\eta_{n}<\xi$, such that $f_{n} \rightarrow f$ pointwise.
\section{Boundary functions for discontinuous functions}
\begin{theorem} Let $Y$ be a separable arc-wise connected metric space, with $f: H \rightarrow Y$ a function of Baire class $\xi(H, Y)$ where $\xi \geq 1,$ $E$ a subset of $X$, and $\varphi: E \rightarrow Y$ a boundary function for $f$. Therefore $\varphi$ is of Baire class $\xi + 1(E, Y).$ \end{theorem}
\begin{proof} Let $U$ be an open subset of $Y$ such that $V = Y-\operatorname{clos}(U)$. Set $A = \varphi^{-1}(U),\ B = \varphi^{-1}(V),\ C = A\cup B.$ Notice that we clearly have an empty intersection between $A$ and $B$. $\forall x\in C$, choose an arc $\gamma_{x}$ at $x$ such that:
$$ \lim_{z\rightarrow x \atop z\in\gamma_{x}} f(z) = \varphi(x) $$ with $$ \gamma_{x} \subseteq\{z: \mid z - x \mid \leq 1\} $$ where $$ \begin{cases} \gamma_{x} -\{x\}\subseteq f^{-1}(U)& \text{ if } x\in A \\ \gamma_{x} -\{x\}\subseteq f^{-1}(V) & \text{ if } x\in B \end{cases} $$ Note once again the empty intersection $\gamma_{x} \cap \gamma_{y} = \emptyset$ for $x\in A\ \wedge\ y\in B$
Let us define the terminology $\gamma_{x}$ \textit{meets} $\gamma_{y}$ in $\operatorname{clos}(H_{n})$ provided that the two arcs have respective subarcs, $\gamma_{x}\prime$ and $\gamma_{y}\prime$ with $x\in \gamma_{x}\prime \subseteq \operatorname{clos}(H_{n})$ and $x\in \gamma_{y}\prime \subseteq \operatorname{clos}(H_{n})$, with $\gamma_{x}\prime \cap \gamma_{y}\prime \neq \varnothing$
Let: $$ L_{a}:={x\in A : \forall n\exists y, \text{ such that } y\in C,\ y\neq x, \ \gamma_{y} \text{ meets } \gamma_{x} \text{ in }\operatorname{clos}(H_{n})} $$ $$ L_{b}:={x\in B : \forall n\exists y, \text{ such that } y\in C,\ y\neq x,\ \gamma_{y} \text{ meets } \gamma_{x} \text{ in }\operatorname{clos}(H_{n})} $$ $$ M_{a}:={x\in A : \exists n, \gamma_{x} \text{ meets no } \gamma_{y} \text{ in }\operatorname{clos}(H_{n})} $$ $$ M_{b}:={x\in B : \exists n, \gamma_{x} \text{ meets no } \gamma_{y} \text{ in }\operatorname{clos}(H_{n})} $$ $$ L = L_{a} \cup L_{b} $$ $$ M = M_{a} \cup M_{b} $$ $L_{a}, L_{b}, M_{a}, $ and $M_{b}$ are notably pairwise disjoint, and $A = L_{a} \cup M_{a},$ $B = L_{b} \cup M_{b}.$
Let $n(x) \in \mathbb{Z}^{+}$ for each $x \in M$ such that $\gamma_{x}$ meets no $\gamma_{y}$ in $\operatorname{clos}(H_{n(x)})$ such that $x \neq y,$ where $n \geq n(x)$ gives the obvious no meeting case in $\operatorname{clos}(H_{n})$. Moreover, take $$ K_{n} := \{x\in C : \gamma_{x} \text{ meets } X_{n} \wedge \text{ if } x\in M, n \geq n(x)\} $$ It is clear that we have for every $n, K_{n} \subseteq K_{n+1}$, as well as $C = \bigcup_{n=1}^{\infty}K_{n}.$ It follows by the work of Kaczynski\cite{Kacz} and the following lemma that we have Theorem $2.1$.
\begin{lemma} Let Y be a separable arc-wise connected metric space, E any metric space, and $\varphi: E\rightarrow Y$ a function such that for every open set $U \subseteq Y$ there exists a set $T \in P^{\xi + 1}(E)$ where $\varphi^{-1}(U)\subseteq T\subseteq \varphi^{-1}(\operatorname{clos}(U))$. Thus for $\xi \geq 2,$ $\varphi$ is of Baire class $\xi(E, Y).$ \end{lemma}
\begin{proof} Let $\mathcal{B}$ be a countable base for $Y,$ and suppose $W$ is some open subset of $Y.$ Let: $$ \mathcal{A}(W) = \{U\in \mathcal{B} : \operatorname{clos}(U)\subseteq W\} $$ By Kaczynski, we take: $$ W = \bigcup_{U\in \mathcal{A}(W)} U = \bigcup_{U\in \mathcal{A}(W)}\operatorname{clos}(U). $$ $\forall U\in \mathcal{B},$ let $T(U) \in P^{\xi + 1}(E)$ be chosen so that $\varphi^{-1}(U)\subseteq T(U)\subseteq \varphi^{-1}(\operatorname{clos}(U)).$ Thus we have: $$ \begin{aligned} \varphi^{-1}(W) &= \bigcup_{U\in \mathcal{A}(W)}\varphi^{-1}(U) \subseteq \bigcup_{U\in \mathcal{A}(W)}T(U) \\ &\subseteq \bigcup_{U\in \mathcal{A}(W)}\varphi^{-1}(\operatorname{clos}(U)) = \varphi^{-1}(W). \end{aligned} $$ Therefore $\varphi^{-1}(W) = \bigcup_{U\in \mathcal{A}(W)}T(U)$, and given $P^{\xi + 1}(E)$ is closed under countable unions, we have $\varphi^{-1}(W)\in P^{\xi + 1}(E),$ and $\varphi$ is of Baire class $\xi(E, Y)$ \end{proof} And following from above we therefore have: $$ \varphi^{-1}(U) = A \subseteq T \cap E \subseteq E - B = E -\varphi^{-1}(V) = \varphi^{-1}(\operatorname{clos}(U)) $$ for some $T \in P^{\xi + 2}(X)$, and we know $T\cap E \in P^{\xi + 2}(E)$ which by the above lemma gives us $\varphi$ of Baire class $\xi + 1(E, Y).$ \end{proof}
\section{Sets of curvilinear convergence for $\mathbb{R}^{3}$}
Let $f: H \rightarrow Y$ of Baire class $\xi(H, Y)$, and $\varphi: E \rightarrow Y$ a boundary function of $f.$ Let us define some function to analyse the properties of $M_{a},$ following Kaczynski. Take $\pi: \mathbb{R}^{3}\rightarrow\mathbb{R}^{2}$ such that $$
\pi(\langle x, y, z\rangle) = \left \| \langle x, y\rangle \right \|_{2}. $$
If $\left \| \langle x, y\rangle \right \|_{2} \in M \cap K_{n},$ then let us define $p_{n}(\left \| \langle x, y\rangle \right \|_{2})$ as the first point of $X_{n}$ reached along $\gamma_{x}$ starting from $x.$ It is clear thus that by Kaczynski, the function $\pi(p_{n}(\left \| \langle x, y\rangle \right \|_{2})$ is strictly increasing on $M \cap K_{n}.$ Thus by the above logic, and Lemma $2.2$, we can show the following theorem: \begin{theorem} Let $Y$ be a separable arc-wise connected metric space in $\mathbb{R}^{n}$, with $f: H \rightarrow Y$ a function of Baire class $\xi + n - 1(H, Y)$ where $\xi \geq 1,$ $E$ a subset of $X$, and $\varphi: E \rightarrow Y$ a boundary function for $f$. Therefore $\varphi$ is of Baire class $\xi + n(E, Y).$ \end{theorem} Although this does not resolve the fourth open problem of Kaczynski's work, it does provide an extension to a theorem shown, and is valuable nonetheless to the field.
\end{document} | arXiv |
Reinforcement learning-based dynamic band and channel selection in cognitive radio ad-hoc networks
Sung-Jeen Jang1,
Chul-Hee Han2,
Kwang-Eog Lee3 &
Sang-Jo Yoo ORCID: orcid.org/0000-0003-1533-08141
In cognitive radio (CR) ad-hoc network, the characteristics of the frequency resources that vary with the time and geographical location need to be considered in order to efficiently use them. Environmental statistics, such as an available transmission opportunity and data rate for each channel, and the system requirements, specifically the desired data rate, can also change with the time and location. In multi-band operation, the primary time activity characteristics and the usable frequency bandwidth are different for each band. In this paper, we propose a Q-learning-based dynamic optimal band and channel selection by considering the surrounding wireless environments and system demands in order to maximize the available transmission time and capacity at the given time and geographic area. Through experiments, we can confirm that the system dynamically chooses a band and channel suitable for the required data rate and operates properly according to the desired system performance.
As the demand for multimedia services increases, the problem of the frequency shortage continues to increase. The spectrum auction price is rising worldwide and passing on to users as a burden [1]. The Federal Communications Commission (FCC) had found that most of the spectrums are underutilized under its current fixed spectrum allocation [2]. The FCC had therefore proposed a new paradigm which provides an access to the spectrum resources not being used by the licensed user to resolve the increasing demand for the spectral access and inefficiency in use [3]. The cognitive radio (CR) technologies provide an opportunity for secondary users (SUs) to use spectrums that are not used by primary users (PUs), allowing the SUs to access the spectrum by adjusting their operational parameters [4, 5]. In relation to the application of CR, FCC adopted rules in April 2012 in [6] to allow license-exempt devices employing the TV white space database approach to access available channels in the UHF television bands. [7] presents the existing, emerging, and potential applications employing CRS capabilities and the related enabling technologies, including the impacts of CR technology on the use of spectrum from a technical perspective. The U.S. Defense Advanced Research Projects Agency (DARPA) and British defense firm BAE Systems are developing a CR IC technology for next-gen communications [8]. DARPA is developing CR technologies that maintain communications under severe jamming environment by Russian electronic warfare systems from 2011 [9]. In 2016, DARPA launched the Spectrum Collaboration Challenge (SC2) to resolve the scarcity of spectrum for DoD use and a Vanderbilt team won the round 1 [10].
The CR technology enables SUs to use free spectrum holes in radio environments that vary with a time and location. When the spectrum is used by a SU, quality of service (QoS) for both the PU and SU should be maintained by ensuring the spectrum accessibility for the SU without interfering with the service for the PU through the spectrum sensing. The SU should periodically sense the channel while using the channel and switch to another channel when the PU starts accessing the current channel. In this case, when selecting a channel, it is necessary to consider the fact that the frequency resource varies depending on the time and geographical area. Also, the CR system should consider the available data rate and possible channel acquisition time that can be achieved on each channel to guarantees the QoS of the SUs. Generally, depending on operating frequency bands such as HF (high frequency), VHF (very high frequency), and UHF (ultra-high frequency), a channel may have different channel bandwidths and the channel characteristic is different. Primary systems that are operating on different frequency bands also have diverse features and characteristics in terms of medium access mechanism, service types, and power requirements. Therefore, for choosing the best channel among the available frequency bands when the secondary CR network needs to move to another channel, several dynamic aspects such as primary system operation characteristics, radio channel conditions, frequency band characteristics, and secondary system requirements should be considered. We have to utilize a dynamic spectrum selection mechanism by considering the related environment and operational parameters to maximize the system performance. The channel access pattern of the PU, the requested data rate of the SU, and the available data rate and spectrum acquisition time can all vary dramatically according to environments. Therefore, the learning algorithm is required to dynamically solve these complex optimization problems.
In this paper, we propose an optimal band and channel selection mechanism in the cognitive radio ad-hoc network using the reinforcement learning. In a cluster-based CR ad-hoc network, we assumed that each member node (MN) performs a wide-band spectrum sensing periodically and reports the sensing results to the cluster head (CH) node. Based on the sensing results from the member nodes and previous channel history, the CH builds wireless channel statistic data vectors in terms of achievable data rates and average primary operational activation time (idle and busy) for each available channel of each conducted band. In addition, the CH estimates the traffic demand of the current cluster network to select a set of band and channel that provides the appropriate service to the cluster. Therefore, in CR ad-hoc networks, multiple clusters can operate in a limited area so that coexistence between ad-hoc clusters should be carefully considered in the channel selection. It is desired that if an ad-hoc cluster traffic demand is low, then the CH should select the frequency band that has relatively a narrow bandwidth (i.e., low achievable data rate). It yields the frequency band with wider bandwidth to another cluster network that needs higher traffic demand. In the proposed architecture, as a reinforcement learning, we use the Q-learning algorithm and we have designed a reward function that captures the expected consecutive operational time, affordable data rate, efficiency of spectrum utilization use, and band change overhead. In particular, the reward for channel spectrum utilization is proposed to reflect the degree of efficiency about using the supportable capacity. Using the proposed Q-learning, the CH can select an optimal band and channel that can maximize the multi-objective function of the CR network, and also, it can increase the coexistence efficiency of the overall secondary systems.
The main contributions of the proposed system architecture are as follows:
●We propose a new CR system architecture that maximizes the secondary user's service quality by dynamically selecting the optimal operating band and channel with consideration of the traffic demand of each CR system and the channel statistics according to the primary systems;
●We define states and actions in order to operate Q-learning considering the service state and demand of the corresponding systems, and propose a structural algorithm for it;
● We design a reward function that maximizes operating time, data rate, and channel utilization efficiency and minimizes band change overhead for secondary systems;
● The proposed system provides fairness by assigning the band and channel that are appropriate to each secondary system based on its demand so that neighboring secondary systems coexist successfully.
The remainder of this paper is organized as follows. In Section 2, we describe related studies. In Section 3, we illustrate the system model and the tasks to be solved. In Section 4, we provide the proposed Q-learning algorithm to select the optimal operating band and channel. Section 5 contains simulation results, and conclusions are given in Section 6.
As the CR-based ad-hoc network is often deployed in situations where resources are insufficient, it is necessary to carefully consider the frequency resource selection. In this regard, studies related to the channel allocation in various fields are being conducted. Vishram et al. examined how to allocate channels using the graph coloring in the presence of homogeneous ad-hoc networks [11]. In their study, they maximized the overall performance while guaranteeing a certain grade of service to individual users with the fairness. Maghsudi and Stanczak applied the graph theory for the channel allocation in a device-to-device (D2D) environment and considered fairness by equalizing the interference for cellular users [12]. Han et al. studied channel allocation methods for maximizing the overall system performance in vehicular networks by using the submodular set function-based algorithm [13]. Li et al. investigated channel allocation methods that maximize the overall system reward using a semi-Markov decision process (SMDP) in a vehicular ad-hoc network (VANET) [14].
Other studies have considered a method of allocating channels according to either bandwidth or service characteristics. A study by Semiari et al. investigated methods of allocating a user application with dual-mode operation in the mmW and μW bands. The base station (BS) allocates a delay non-sensitive user application to the mmW while assigning a delay-sensitive user application to the μW band. Matching game theory is specifically used for channel allocation in the μW band. In non-line-of-sight (NLoS) of mmW band, the user application cannot be allocated since the wireless communication is impossible because of the frequency characteristics; therefore, channels are allocated by estimating line-of-sight (LoS) and are secured through Q-learning [15]. Liang et al. have studied a method of assigning the channel with the high transmission capacity to the vehicle-to-infrastructure (V2I) link and the channel with the high reliability to the vehicle-to-vehicle (V2V) link considering the requirements of the two types of the vehicular network links [16].
Recently, the Artificial Intelligence (AI) technology, such as machine learning, has been attracting attention in various fields [17]. Among them, the reinforcement learning is being studied in the wireless system field because it provides a solution to optimize the system parameters by learning the surrounding environment in a dynamic and complicated wireless environment [18]. The Q-learning is the representative reinforcement learning and there are also researches about using this to allocate channels in a dynamically changing environment. Asheralieva and Miyanaga studied the multi-agent reinforcement learning using rewards to maximize the signal-to-interference-plus-noise ratio (SINR) and increase the transmission capacity in D2D networks [19]. Srinivasan et al. described a way in which two BSs belonging to different operators in a cellular network can allocate channels by providing services to the nodes belonging to the all operators. They studied the reinforcement learning using the reward with the difference between quality of experience (QoE) and cost that can be obtained by providing two services [20]. Rashed et al. studied the reinforcement learning that maximizes the sum-rate of D2D users and cellular users to minimize the interference in a D2D environment [21]. Fakhfakh and Hamouda used the received SINR from the access point (AP) detected by the mobile user, QoS metrics about the channel load, and delay as the reward for choosing a WiFi over a cellular network to apply WiFi offloading and reducing the load on the cellular network [22]. Yan et al. propose a smart aggregated radio access technologies (RAT) access strategy with the aim of maximizing the long-term network throughput while meeting diverse traffic quality of service requirements by using Q-learning [23]. Maglogiannis et al. allowed the LTE system in the unlicensed band to select the appropriate muting period by using Q-learning to ensure coexistence with WiFi systems [24]. Xu et al. modeled the channel handoff process as a partially observable Markov decision process (POMDP) and adopted a Q-learning algorithm to find an optimal handoff strategy in a long term [25]. Jang et al. proposed Q-learning based sensing parameter (sensign time and interval) control mechanism for cognitive radio networks [26]. L. Shi et al. presented optimal resource allocation for LTE-U and WiFi coexistence network using Q-learning [27].
Various studies have been carried out about selecting channels, but most studies do not consider the fairness of the channel selection between users. Even if the fairness is taken into consideration, they just allocate resources fairly regardless of the required data rate or considered it as a central manner [28]. And the central scheme is difficult to realize the realistic implementation because of the complexity, or their scheme gave the loads to the network due to the centralized control. Some distributed resource allocation mechanisms (e.g., game theory) may also cause a loss of time or resources because the channel is selected by the interaction between the systems. The fairness of the channel usage is required in order to minimize the possibility of channel resources being unnecessarily consumed by some users and unavailable for other users who require more of them. In order to reduce the load on the system, it is necessary to consider fairness within the system itself without control message exchanges. Meanwhile, the various budgets for cognitive ad-hoc networks, such as time available to the channel, transmission speed, fairness, and bandwidth conversion cost, should be considered. Moreover, these budgets must work in concert to fit an objective function with some degree of freedom about flexible operation so that the system can be operated for various purposes without altering a predetermined objective. In this paper, the reward for spectrum utilization is designed so that fairness is taken into consideration by selecting a channel suitable for the required data transmission rate. In addition, we define a reward using weighted sums for various budgets as well as a Q-learning algorithm that can operate according to the change in weights.
Network model and system architecture
Network model
The system considered in this paper is the cognitive radio ad-hoc network comprised of CH and MNs as shown in Fig. 1a. The channel availabilities are different with geographic locations in accordance with the primary transmitter positions, channel gain between primary systems and secondary users, primary activity characteristics, and so on. The characteristics of these channels for CR ad-hoc networks can be also different in time zone and frequency band groups. Therefore, in this paper, we have considered the difference of channel characteristics according to geographical zone, time zone, band group, and frequency channel. As shown in Fig. 1b and c, the primary activities (i.e., available time to access by secondary users) and possible data transmission rates are different for each channel during the same time interval. Furthermore, the desired data rates of SUs changes according to time, as shown in Fig. 1d.
Network model according to geography, time, and frequency. The cognitive radio ad-hoc network consists of a cluster head (CH) and mobile nodes (MNs) as shown in a. According to the time zone, band group, and each channel, the frequency resource is different. The PU activity, available data transmission rate, and SU demand for data rate vary according to the time and channel
In particular, when the CR system operates cross a wide frequency range, including HF, VHF, and UHF, as shown in Fig. 2, the channel bandwidth for each band group that is defined for secondary systems can be different due to the band group-specific spectrum hole nature. In general, in HF, the spectrum holes are relatively narrow in the frequency domain because the licensed spectrum of primary systems using HF band group is also usually narrow. On the other hand, the spectrum holes of UHF are comparatively wider than that of HF. Excepting for details characteristic difference of each band group, we assume that the wider channel bandwidth is used in the higher band group frequency. Therefore, if the operating frequency of band group j (BGj) is higher than that of band group i (BGi), then the channel bandwidth of BGj, Wj, is wider than Wi which is the channel bandwidth of BGi and achievable data transmission rate (i.e., capacity) of Wj is greater than that of Wi. The greater preference for the channel is given to the band group with the higher bandwidth in the system or individual nodes. However, even though the bandwidth demanded by the secondary system can be satisfied by Wi of BGi, if the bandwidth Wj of the higher BGj is utilized by the secondary system, then satisfaction of the system will increase while overall spectrum resources are wasted. Because other secondary systems may exist around and their traffic demands only can be satisfied by using the bandwidth Wj of BGj, therefore, a mechanism for adaptive allocation of band group use according to the traffic demand and bandwidth utilization efficiency of the corresponding system is required.
Channel bandwidths for different band groups. A wireless communication system generally has a higher bandwidth and a higher data transmission rate as it goes to a higher band
System architecture and problem formulation
The proposed system architecture of CH is represented in Fig. 3. The Q-learning is used to dynamically select optimal band group and channel being aware of wireless environment, network user demand, and system operation parameters. The network user demand module determines the desired data rate (DDR) of the CR ad-hoc network based on each member node's traffic demand, and it also measures the average utilization of the channel currently used. The wireless environment monitoring module stores the spectrum sensing results such as average SNR (signal to noise ratio) and primary signal detection history. Using the sensing results, this module generates band- and channel-specific statistics which includes available data rate and primary idle time. The system operator can dynamically adjust the system parameter for learning using the system parameter module. The system operator can reset the reward function by learning parameters for Q-learning. Based on all information, the Q-learning module determines which band group and channel can meet the data rate demand and maintain effective utilization level.
Proposed system architecture. Proposed Q-learning is used to dynamically select the optimal band group and channel. As the reward function, the system considers the user demand, wireless environment and system parameters. The user demand module determines the desired data rate (DDR) of the CR ad-hoc network and measures the average utilization of the channel currently used. The wireless environment module stores the spectrum sensing results. The system parameters module is used to establish the reward function and Q-learning parameters. If the band of newly selected channel is different with the old one, the overhead for band group change is adopted to the reward function
The Q-learning module changes from one channel to another when a PU appears on the current channel being used by the secondary system. The reward function is used to update the Q-table, and channel and band group selection is performed based on the current Q-table. The reward function proposed in this paper captures user demand, wireless environment, data rate efficiency (DRE), and band change overhead cost. The DRE is an evaluation metric to determine how much the ad-hoc network efficiently utilizes the data rate supported by the current channel.
In the proposed algorithm, we design the Q-learning reward function to satisfy the following criteria:
●Maximize the secondary system operational time;
●Satisfy the desired data rate of the CR ad-hoc secondary network;
●Provide the coexistence and fairness between secondary systems;
●Consider the overhead of band change for system reconfiguration;
●Guarantee operational flexibility and adaptability to meet the desired purpose.
Reinforcement learning for dynamic band and channel selection
Action, state, and Q-table design for Q-learning
The Q-learning is one of the model-free reinforcement learning techniques. It is able to compare the expected utility of the available actions for a given state without requiring a specific model of the environment. An agent tries to learn the optimal policy from its history of interaction with the environment, in which an agent applies a specific action at the current state and receives a response as a form of a reward from the given environment. The Q-learning eventually finds an optimal policy, in the sense that the expected value of the total reward return over all successive iterations is the achievable maximum one. The problem space consists of an agent, a set of states \( \mathbf{\mathcal{S}} \), and a set of actions per state \( \mathbf{\mathcal{A}} \). By performing an action \( a\in \mathbf{\mathcal{A}} \), the agent can move from state to state.
Figure 4 shows the Q-learning mechanism of the proposed method. The CH of the CR ad-hoc system is the agent of Q-learning, and the action of the CH is a selection of a new tuple (band group, channel) for the CR system operation when the primary signal is detected on the current band group and channel. The structure of the Q-table is expressed by rows of states and columns of actions. In this paper, the set of action \( \mathbf{\mathcal{A}} \) is given by:
$$ \mathbf{\mathcal{A}}=\mathcal{B}\times {\mathcal{C}}_k $$
where × is the Cartesian product; \( \mathcal{B}=\left\{{b}_1,{b}_2,\dots, {b}_{NB}\right\} \) expresses the set of channel band groups; NB is the number of band groups; \( {\mathcal{C}}_k=\left\{{c}_1^{b_k},{c}_2^{b_k},\dots, {c}_{N{C}_k}^{b_k}\right\} \) represents set of available channels in k-th band group (bk); NCk is the number of channels of band group bk; and \( {c}_j^{b_k} \) is j − th channel of band group bk.
Proposed Q-learning mechanism. The CH of the ad-hoc CR system is the agent of Q-learning, and the action is a selection of a tuple (band group and channel) when the PU is detected on the current band group and channel. The Q-learning agent (CH) designates the state from the information of member node and statistics of environment by the last action. From the Q-learning module, the Q-learning agent obtains the reward, change the Q-table and next action tuple
In this paper, a multi-layered state is defined, in which it is composed of geographic location (\( \mathcal{L}\Big) \), time zone (\( \mathcal{T}\Big) \), channel band group \( \left(\mathcal{B}\right) \), and data rate efficiency level (D). The state of space in this system is defined as follows.
$$ \mathbf{\mathcal{S}}=\mathcal{L}\times \mathcal{T}\times \mathcal{B}\times \mathcal{D} $$
where \( \mathcal{L}=\left\{{l}_1,{l}_2,\dots, {l}_{NL}\right\},\kern0.5em \mathcal{T}=\left\{{t}_1,{t}_2,\dots, {t}_{NT}\right\} \), and \( \mathcal{B}=\left\{{b}_1,{b}_2,\dots, {b}_{NB}\right\} \) represent the sets of geographic location zones, time zones, and band groups, respectively. NL, NT, and NB are the number of location zones, time zones, and band groups, respectively. In this paper, we have defined a new additional state dimension \( \mathcal{D} \) to represent the operational state of the secondary system in terms of how much the CR system effectively utilize the given channel of the selected band group. \( \mathcal{D}=\left\{{d}_1,{d}_2,\dots, {d}_{ND}\right\} \) indicates the set of DRE levels, and ND is the predefined number of DRE levels. The DRE is the ratio of the DDR of the secondary network to the average supportable data rate of the current channel of the selected band group. Therefore, the current state is defined as a form of (li, tj, bk, dl) tuple and it represents the current location zone, time zone, operation band group, and DRE level. For the given geolocation area and time period, the secondary CR system needs to select the next operational band group and channel whenever a channel switching is required. The current band group (CBG) and DRE capture the dynamic goodness of the selected band group and channel in terms of spectrum efficiency and support of the desired rate. The CH selects the best action for the current state using the current Q-table.
Figure 5 shows the proposed Q-table structure. At the current state (li, tj, bk, dl), the CH selects the best action \( \left({b}_q,{c}_m^{b_q}\right) \), i.e., the next band group bq and m-th channel of bq, which shows the maximum Q-value in the current Q-table. It needs be noted that the candidate channels of each band group as possible actions should be available channels at the current time as a result of spectrum sensing.
Proposed Q-table structure. The column of the Q-table represents the action tuple of the band group bq (q-th band group) and channel \( {c}_m^{b_q} \) (m-th channel of bq). The row of the Q-table is the state tuple of the i-th geographic location zone (li), j-th time zone (tj), k-th band group (bk), and l-th data rate efficiency level (dl)
Figure 6 shows the procedure of the proposed Q-table update, state determination, and action selection. It is assumed that the MNs transmit the average channel operation time and average supportable data rate to the CH through sensing and channel use report.
Suppose the learning agent CH determined the state st − 1 and the best action at − 1 at the end of (t − 1)-th time period.
During t-th time period, MNs and CH monitor the primary activities and channel statistics.
Agent CH detects the band and channel change event.
The CH calculates the reward rt − 1 for the previous action at − 1 at state st − 1.
The CH updates the Q-value of (st − 1, at − 1) in Q-table.
The CH determines the current state st based on the measured DRE during t-th time period.
The CH selects the optimal action at for the next (t + 1)-th time period.
Go to step 1.
Proposed procedure for Q-table update, state determination, and action selection. (1) Suppose the learning agent CH determined the state st − 1 and the best action at − 1 at the end of (t − 1)-th time period. (2) During t-th time period, MNs and CH monitor the primary activities and channel statistics. (3) Agent CH detects the band and channel change event. (4) The CH calculates the reward rt − 1 for the previous action at − 1 at state st − 1. (5) The CH updates the Q-value of (st − 1, at − 1) in Q-table. (6) The CH determines the current state st based on the measured DRE during t-th time period. (7) The CH selects the optimal action at for the next (t + 1)-th time period. (8) Go to step 1
The Q-learning updates the Q-value for each pair of state and action (s, a) visited through these series of processes. The Q-value reflects the value that the system can accept when selecting action a in state s.
The Q-value update of the Q-table can be represented by:
$$ \mathcal{Q}\left({s}_t,{a}_t\right)\leftarrow \left(1-\alpha \right)\mathcal{Q}\left({s}_t,{a}_t\right)+\alpha \left\{{r}_t+\gamma \underset{a_{t+1}}{\mathit{\max}}\mathcal{Q}\left({s}_{t+1},{a}_{t+1}\right)\right\} $$
where α and γ denote the learning rate and discount factor, respectively. The learning rate α ∈ [0, 1] is used as a weight to reflect the \( \mathcal{Q}\left({s}_t,{a}_t\right) \) accumulated from the past, the newly obtained reward, and the expected reward value for the next action. If α is low, it increases the weight of the past experience so the system takes an extended time to learn, but the fluctuation of the reward sequence is low. If α is high, the learning speed is increased by assigning a high weight to both the present and future values. However, an extremely high α causes instability in the system, while a fairly low α prevents the system from reaching a satisfactory reward at the desired time. The discount factor γ ∈ [0, 1] is the weight for how much the Q-value of at + 1, the future reward, should be reflected in the Q-value of action at in the Q-table of the current action and state. A high γ has a high contribution on the Q-value of the future expected reward, and a low γ weights the reward according to the current action a. That is, when Q-learning reflects the immediate reward and the future tendency in the Q-value of the action and corresponding state, γ is a weight that takes into account whether to further consider the volatility of the current action or to reflect the future value predicted from past trends of the Q-table.
If the CH only uses the updated Q-values to select actions, it may fall into local optimum. Therefore, we use ε-greedy policy to add randomness to selecting of actions that are explorative in the learning algorithm, as follows:
$$ a=\left\{\begin{array}{cc}\underset{\overset{\sim }{a}\in \mathcal{A}}{\mathrm{argmax}}\mathcal{Q}\left(s,\overset{\sim }{a}\right),& \mathrm{with}\ \mathrm{probability}\ 1-\varepsilon \\ {}\mathrm{random}\ a\in \mathcal{A},& \mathrm{with}\ \mathrm{probability}\ \varepsilon \end{array}\right. $$
where, ε ∈ [0, 1] is the probability of choosing a random action. If ε is high, it is more likely that new information will be added to the already accumulated information while searching for the next action; if it is low, the next action is selected using only the accumulated information. ε starts with a specific value and lowers this value for each iteration, so that the Q-table can operate stably after a certain time. However, when the value of ε decreases continuously, a considerable amount of time is required for adapting to the changing environment by updating the Q-table. Therefore, a lower limit of ε is required.
The overhead of Q-learning can arise from the memory size for the use of Q-tables. It depends on the level of the actions (the number of channels and bands) and resolution level of states, and it increases linearly with each level. If you set the number of level too low, the system cannot use the Q-table for learning dynamic environments. Otherwise, the system takes a long time to learn the surrounding environment using the Q-table. Therefore, the selection of appropriate level is required.
Reward function design
The reward that the CR system obtained by using the selected set of (band group, channel) is composed of the system operation time, average data transmission rate, channel utilization efficiency, and overhead required for the system to change the frequency band. The reward for the action a is expressed as follows:
$$ R(a)={w}_1\frac{T_{op}}{\max \left({T}_{op}^{cbg}\right)}+{w}_2\frac{E\left[{D}_s\right]}{\max \left(E\left[{D}_s^{cbg}\right]\right)}+{w}_3{R}_{Util}-{w}_4{BC}_{\left(a,{a}^{\prime}\right)} $$
where Top is the consecutive channel operation time for the secondary system after the channel is selected, in which if a channel shows high Top value, then it indicates that once the secondary system takes this channel it can use the channel relatively long time before the primary appears. E[Ds] is the average supportable data rate of the selected channel. RUtil represents how the secondary system utilizes the channel effectively. \( {BC}_{\left(a,{a}^{\prime}\right)} \) is the overhead for band group change. The operation time and average supportable data rate are normalized to their maximum values for the current band group. a and a′ are the current action and previous action, respectively. \( \mathit{\max}\left({T}_{op}^{cbg}\right) \) and \( \mathit{\max}\left(E\left[{D}_s^{cbg}\right]\right) \) are the maximum channel operation time and maximum expected supportable data rate value from all channels in the current band group, respectively. wi is the weight for i-th reward component and \( {\sum}_{i=1}^4{w}_i=1 \). The first and the second term are normalized by each maximum value of the operation time and average supportable data rate in each channel group so that the relative value to the other channels can be reflected in the reward. The third term is described in (7) and serves to adjust the reward to select a channel suitable for the desired data rate. The fourth term represents the cost due to an overhead when a band group change occurs, which is described in (6). All the terms are linearly coupled to allow the system designer or user to operate the system for a specific purpose through weighting changes.
The overhead for band group change, \( {BC}_{\left(a,{a}^{\prime}\right)} \), in (5) is to capture the required additional time and energy for reconfiguring some system operational parameters when the band group is changed. In most cases, different band groups have different channel bandwidths and wireless characteristics so that communication system may need to reconfigure radio frequency (RF) front-end, modulation method, and medium access control (MAC) layer components whenever it changes its operation band group. The overhead is represented as in (6).
$$ {BC}_{\left(a,{a}^{\prime}\right)}=\left\{\begin{array}{cc}\eta, & \mathrm{channel}\ \mathrm{a}\ \mathrm{a}\mathrm{nd}\ {\mathrm{a}}^{\prime }\ \mathrm{belong}\ \mathrm{to}\ \mathrm{different}\ \mathrm{band}\ \mathrm{groups}.\\ {}0,& \mathrm{else}\end{array}\right. $$
where η is the cost when the current channel and the previous channel belong to different band groups. In this paper, we do not consider the channel switching overhead inside the same band group.
RUtil of (5) is defined as a function of DRE. The DRE is defined in this paper as in (7)
$$ DRE=\frac{DDR}{E\left[{D}_s\right]} $$
where DDR is the desired data rate (DDR) of the secondary CR network. To design RUtil function, first we considered the desired system operation in terms of band group selection depending on the current DRE value.
Figure 7 shows the example of the desired mechanism by which the channel selection is performed according to the current DRE. It is assumed that the bandwidth of each channel provided by each band group follows W1 < W2 < W3, in which we have three band groups. The x-axis is divided by band group, and the parts represent the DRE for each band group. DRE (U) indicates that the utilization ratio of the channel is low when it belongs to [0, r1) of band group 1, and that ratio is moderate when it belongs to [r1, r2). If U belongs to [r2, 1), it denotes a high channel utilization ratio so that some time instances of the network may not be able to meet the user traffic demand. The range of [1, ∞) means the channel cannot support enough bandwidth for the system. A low channel utilization ratio means that the possible transmission rate provided by the selected channel of the current bad group is much higher than the desired CR network data rate so that most of spectrum resource is wasted after it satisfies the desired data transmission rate. It is therefore necessary to move to a channel that provides a lower bandwidth and data rate (i.e., change to the lower band group channel). Furthermore, if U shows a higher channel utilization ratio than the defined r2, which means that the possible data transmission rate provided by the selected channel is not likely to support the desired data rate of the system, then it is necessary to move to a channel band group that can provide a wider bandwidth and a higher data rate. However, in Fig. 7, even though DRE is in [0, r1) for band group 1, the secondary system does not have any band group that has narrower channel bandwidth so that it needs to keep the current band group. On the other hand, in case of band group 3, when DRE is in [r2, ∞), the system does not have any band group that has wider channel bandwidth so that it has to find other best channel in the same band group. In each band group, [r1, r2) is the band usage maintenance interval, because the selected band group channel provides an appropriate transmission rate.
Band group movement mechanism according to DRE. It is assumed the bandwidth of each channel provided by each band group follows W1 < W2 < W3. The x-axis represents the DRE of each band group. The DRE is low, moderate, and high if it belongs to [0, r1), [r1, r2), and [r2, ∞). If the DRE is low, the selection of the band need to be changed to lower one since the selected band supports too much bandwidth. However, the CH could not change the band in the range of [0, r1) in band group 1 since there is no band to move. If the DRE is high, the selection of the band need to be changed to a higher one since the selected band does not support the DDR. However, the CH could not change the band in the range of [r2, ∞) in band group 3 since there is no band to move. In the range of [r1, r2) in each band group, the system does not need to change the band since the selected band supports adaptive bandwidth. Therefore, the whole DRE region is divided by the region of "Band usage maintenance" and "Band usage change"
Based on the band group selection movement mechanism in Fig. 7, the proposed utilization efficiency reward function RUtil of Eq. (5) is shown in Fig. 8, in which we assume that there are three band groups. The x-axis for each band group represents DRE (U), and the y-axis represents RUtil. For the band usage maintenance range in Fig. 7, RUtil is set to 0 in [r1, r2) DRE range for all band groups, which represents the medium channel utilization efficiency ratio. In [0, r1) DRE range, the selected band group channel can support much larger data rate than the desired data rate so that channel utilization efficiency is low. Therefore, as the value of DRE goes from 0 to r1, RUtil increases from −1 to 0 except the in first band group 1. Any ad-hoc CR secondary systems that has its current DRE value in [0, r1) need to move to the band group channel that has narrower channel bandwidth and lower supportable data rate. It makes the secondary system to yield the current band group channel to other secondary systems that requires more data rate. For the first band group 1, there is no other narrower band group so that RUtil is maintained at 0 in [0, r1) DRE range. The range [r2, ∞) is divided into [r2, 1) and [1, ∞) to distinguish the insufficient transmission rate provided by the channel, with RUtil representing a more rapid decrease rate in [1, ∞) range except in the last band group 2. In [r2, 1) DRE range, the band group channel supportable data rate may not be enough to guarantee the desired rate in some time instances so that RUtil decreases from −r2 with a slope of −1 as DRE increases. In [1, ∞) DRE range, the current band group channel cannot support desired data rate so that RUtil decreases with a slope of −δ (δ > 1). For the last band group 3, there is no other wider band group so that RUtil is maintained with 0 in [r2, ∞) DRE range.
Utilization efficiency reward according to DRE. To realize the mechanism in Fig. 7, the reward for channel utilization efficiency (RUtil) is designed as shown. The x-axis for each band group represents the DRE, and the y-axis represents RUtil. For the band usage maintenance range, RUtil is set to 0. For the band usage change range where the DRE is low, RUtil increases from −1 to 0 since the DRE represents better value as DRE increases. The band usage change range where the DRE is high is divided into [r2, 1) and [1, ∞) to distinguish the insufficient transmission rate provided by the channel. RUtil represents a more rapid decrease rate in [1, ∞). RUtil decreases from −r2 to −1 in [r2, 1) range and from −1 to ∞ in [1, ∞) range
Figures 9 and 10 show how this intentional mechanism is supported in the Q-table. Figure 9 shows the Q-table where the state is divided into geographic zones and time zones, and again into band groups and discrete DRE levels. The columns of the Q-table are divided into bands, which are then divided into selectable channels as possible actions. In the action shown in Fig. 9, a channel that represents a narrower band is shown as a lower data rate toward the left, and a channel that can use a wider band appears as a higher data rate toward the right. Figure 10 represents how the Q-value updating area changes in the Q-table of Fig. 9 through the example of three DDR cases. First, Fig. 10a depicts a case where the DDR is low. In Fig. 9, suppose that the secondary system is operating in action domain 1 (low CBG, low DRE) and by the Q-learning ε-greedy policy it may randomly select action domain 2 band group channel. As a result, the DRE is significantly lowered, and the system gets low RUtil because a high channel band provides a high data rate and it results in low reward for channel utilization efficiency. Therefore, after updating the Q-value of domain 2, the system operating area changes to domain 3 by the change of the state which represents (high CBG, low DRE). Because selecting a channel which supports a high data rate makes the DRE low, the Q-table then updates the Q-value of domain 3 and the Q-learning agent will select the best action of domain 4 because selecting a channel with a low band gives a high RUtil. After selecting the low-band channel, the transition to domain 1 (low CBG, low DRE) is performed. In this case, RUtil does not have a negative value because there is no longer a lower channel to select, as in the [0, r2) of band group 1 seen in Fig. 8, and no value is subtracted from the total reward. Therefore, in the case of a low DDR, the preferred domain in the Q-table is not domains of 2, 3, and 4 where high band is selected by occasional ε-greedy policy for exploration but domain 1 (stable operating domain). Cases of both moderate and high DDR, as shown in b and c of Fig. 10, can be similarly explained.
Channel movement mechanism in Q-table and operating regions
Channel movement example in Q-table. The update of the Q-table represents a unique pattern according to DDR by the reward for channel utilization efficiency proposed in this paper. The channel movement example of a low, b moderate, and c high DDR cases in Q-table is shown. The stable domain is in gray circle in each case of DDR. Each domain changes to another one by explore or natural transition
Simulation results and discussion
The simulation environment in this paper assumes five channels for each of the two band groups, as listed in Table 1. The channels of band group 2 provides higher supportable data rates than those of the band group 1 while the operation time available for the transmission is not significantly different. We use a Gaussian distribution to determine the operation time and supportable data rate of each channel in each band group based on the mean and variance values provided in Table 1. The other simulation parameters are shown in Table 2.
Table 1 Channel parameters for band groups 1 and 2
Table 2 Weight parameters
In this paper, the action is defined as the selection of the channel in each band group as shown in Table 1. The index number of the action corresponds to the number of the channel in Table 1, and the total number of action is 10. We define the state as the combination of (band group, DRE) where the domain of DRE is divided as [0, r1), [r1, r2), [r2, 1), [1, ∞). The domain of DRE corresponds to each band group so the total number of state is 8.
The ε-greedy policy for action exploration is as follows:
$$ p(n)=\left\{\ \begin{array}{cc}{p}_0{(0.999)}^n,& p(n)>{p}_{low}\\ {}{p}_{low},& p(n)\le {p}_{low}\end{array}\right. $$
$$ \mathrm{where}\ {p}_0=0.3,{p}_{low}=0.1 $$
where n represents the iteration sequence number with time. In applying the ε-greedy policy, when the wireless environment of the system is changed, plow is required to set a lower limit for a random value in order to maintain a certain degree of exploration. Otherwise, the Q-table cannot adaptively operate in the changed environment.
The overall simulation configuration starts by looking at the operation of Q-learning for each DDR, 40 kbps − 90 kbps − 3.5 Mbps, and confirming the change of average operation time and average transmission data rate according to the weight of the reward. Next, we compare the results of Q-leaning and random channel selection according to the reward, operation time, data rate, and utilization. Finally, we compare the change of DDR according to iteration for Q-learning and random channel selection.
Adaptive channel selection according to DDR
In this section, we identify our proposal adaptively selects the channel according to each DDR (e.g., low, moderate and high) as described by Figs. 9 and 10 in Section 4.2. Figure 11a shows the value of the Q-table in scaled colors when the DDR is 40 kbps, and b shows the number of visits to each state in the Q-table. The DDR of 40 kbps is the low data rate comparing the data rate of channels in Table 1. Therefore, if the CH selects the channel of band group 1, the DRE belongs to almost [0, r2) of band group 1 in Fig. 8 comparing the channels in band group 1, and RUtil does not give any effect on total reward. However, if the CH selects the channel of band group 2, the DRE belongs to [0, r1) of band group 2 in Fig. 8 and RUtil has an impact on the total reward linearly according to DRE. Figure 11a represents the process of changing the channel (action) selected by the ad-hoc CH (agent). When the CH selects the channel of band group 1, the process of updating the Q-value in the Q-table takes place in domain (1), which represents the channel selection of band group 1 and the low DRE of band group 1. If the CH selects the channel of band group 2 by the explore policy of the Q-learning, the Q-value of the domain (1) is changed and the update process moves to the domain (2) which represents the channel selection of band group 2 maintaining the current state. Since the CH selected the channel of band group 2 that provides a high data rate for low DDR, the state is changed to the low DRE of band group 2. Thus, the update process of the Q-table moves to domain (3) and the Q-value of domain (2) changes. If CH selects channel of band group 1 in state 5, the state is maintained and update process moves to domain (4) after the Q-value change in domain (3). Finally, the state is changed to the state 2, which represents the band group 1 and low DRE, and the update process moves to domain (1) after the Q-value of domain (4) changes. The Q-value of Fig. 11a is the highest in domain (1), which represents the low DRE of band group 1 same with the low DDR case of Fig. 10a by the reward for utilization in Fig. 8. As a result, Fig. 11b represents that the number of visits in state 2 is the highest which corresponds to domain (1).
The a Q-value of Q-table when DDR = 40 kbps, and b number of state visits. Similar to Figs. 9 and 10a, the Q-value of Q-table shows the channel movement of low DDR case. The visit count of state is high in the state of low DRE and low CBG because the stable domain is in that region
Figure 12 shows the change of rewards, states, and actions according to iteration at a low DDR of 40 kbps. The temporary low reward value is due to the random action of Q-learning exploration. The agents visit the state 2 more often than the states 4 and 5 over time as seen in Fig. 11b. As shown in Fig. 11a, the action in Fig. 12c mainly visits channel 2 or 3 and is adaptive to the DDR at the latest possible moment, even if a channel from band group 2 is selected or a channel from band group 1 offering a high data rate is selected. Figure 12c represents the agent selects the channels in band group 1 suitable for the DDR over time. Therefore, we can see that the agent operates according to the designed mechanism.
The change of a rewards, b states, and c actions according to iteration at a low DDR of 40 kbps. The temporary low reward value is due to the random action of Q-learning exploration. The agents visits the state 2 more often than the states 4 and 5 over time as seen in Fig. 11b. As shown in Fig. 11a, the action in c mainly visits channel 2 or 3 and is adaptive to the DDR at the latest possible moment, even if a channel from band group 2 is selected or a channel from band group 1 offering a high data rate is selected. c represents how the agent selects the channels in band group 1 suitable for the DDR over time. Therefore, we can see that the agent operates according to the designed mechanism
Figure 13a shows the value of the Q-table in scaled colors when the DDR is 90 kbps and b shows the number of visits for each state in the Q-table. If the DDR is 90 kbps and the CH chooses a channel in the band group 1, the DRE belongs to [r2, ∞) in Fig. 8. Meanwhile, the DRE belongs to [0, r1) in Fig. 8 if a channel is chosen from the band group 2. Therefore, the supportable data rate by the channels in band group 1 is not enough in comparison with the channels in band group 2, as seen in Table 1, since the channel selection from band group 2 offers better RUtil than that of band group 1. We assume that the process of updating the Q-value starts from domain (1) which represents the channel selection of band group 2 and the DRE is low in band group 2. After selecting the channel in band group 1 by random channel selection, the Q-value of domain (1) is renewed and the update process moves to domain (2). If the CH selects channels from band group 1, the renewal process of the Q-table changes to domain (3) due to the high DRE which means the selected channel does not support a high enough transmission data rate after the renewal of domain (2). The process of updating moves to domain (3) by the change of state then transfers to domain (4) by the random or best selection. Because the channel selection in band group 2 provides more reward for utilization by Fig. 8, the Q-table in Fig. 13b has the highest Q-value in domain (1) which represents a low DRE for band group 2. As a result, the number of visits to state 5 belonging to the low DRE of band group 2 is the highest, as shown in Fig. 13b.
The a Q-value of Q-table when DDR = 90 kbps, and b number of state visits. Similar to Figs. 9 and 10b, the Q-value of Q-table shows the channel movement of moderate DDR case. The visit count of a state is high in the state of low DRE and high CBG because the stable domain is in that region
Figure 14 shows the change of rewards, states, and actions according to iteration with a medium-level DDR at 90 kbps. In Fig. 14a, the reward is stable at more than 10 iterations, and we can see that the reward is temporally low in the overall interval by random action, similarly to Fig. 12. As shown in Fig. 14b, the agent mainly visits the state 5. Figure 14c reveals that actions in band group 2 are selected mostly.
Rewards, states, and actions according to iteration at DDR = 90 kbps. In a, the reward is stable at more than 10 iterations, and we can see that the reward is temporally low in the overall interval by random action, similarly to Fig. 12. As shown in b the agent mainly visits the state 5. c Reveals that actions in band group 2 are selected mostly
Figure 15a shows the value of the Q-table in scaled colors when the DDR is 3.5 Mbps, and b shows the number of visits to each state in the Q-table. Comparing with the supportable data rate of channels in Table 1, the DDR of 3.5 Mbps makes the DRE belong to [1, ∞) of the band group 1 in Fig. 8 when the CH selects the channel from band group 1, and belongs to [r1, ∞) when a channel is selected from band group 2. However, the reward for utilization RUtil remains as 0 since there are no other channels to move out. As illustrated in Fig. 10c about the example of high DDR case, the same explanation can be given about Fig. 15a. At first, update process is assumed starting from domain (1) in Fig. 15a by the channel selection from band group 2. If the channel of band group 1 is selected by the explorer policy of Q-learning, the update process moves to domain (2) after the change the Q-value of domain (1). The state changes to the state 4 which represents high DRE in the band group 1 by the given DDR and the channel selection of band group 1. Therefore, after the Q-value of domain (2) is updated, the update process moves to domain (3). The update process selects the channel of band group 2 through the best or random channel selection and could be moved to domain (4), thereby the Q-value of domain (3) is updated. Finally, since the channel selection of band group 2 changes state to high state of band group 2, the update process moves to domain (1) and the Q-value of domain (4) is updated. As described for high DDR case in Fig. 10c, the CH of Fig. 15 also tends to select the channel of band group 2 and stay on the state which has high DRE of band group 2 since the channels of this band group gives no harmful effect on RUtil. In the Q-table of Fig. 15a, the Q-value of domain (1) showing high DRE in band group 2 is the topmost, and this is also shown in Fig. 15b as the high visit count of states 7 and 8.
The a Q-value of Q-table when DDR = 3.5 Mbps, and b number of state visits. Similar to Figs. 9 and 10c, the Q-value of Q-table shows the channel movement of high DDR case. The visit count of state is high in the state of high DRE and high CBG because the stable domain is in that region
Figure 16 represents the change of rewards, states, and actions according to iteration at a high level DDR of 3.5 Mbps. In Fig. 16a, the reward is stable overall, while it is temporally low in the overall interval by random action, similar to Figs. 12 and 14. In Fig. 16b, the system visits state 4 to a degree, but it mainly remains in states 7 and 8. Figure 16c shows that actions in band group 2 are selected mostly.
Rewards, states, and actions according to iteration at DDR = 3.5 Mbps. In a, the reward is stable overall, while it is temporally low in the overall interval by random action, similar to Figs. 12 and 14. In b, the system visits state 4 to a degree, but it mainly remains in states 7 and 8. c shows that actions in band group 2 are selected mostly
These results demonstrate that the proposed system can select an appropriate channel according to the DDR required by ad-hoc CR users.
Reward reconfiguration with weights
Figure 17 shows the average operation time, average data rate, and reward for channel utilization by changing the weight assignment for DDR = 40 kbps. In the reward calculation, if weights for [operation time, supportable data rate, reward for channel utilization] are assigned to [0.5, 0.1, 0.3], then it increases the importance for the operation time. As a result, it has the best average increase in operation time, as shown in Fig. 17a, and the least average of data transmission rate, as shown in Fig. 17b. This is because the system wants to reserve the highest priority for operation time and the least for data transmission rate. If weights are assigned to [0.1, 0.7, 0.1], then it increases the data transmission rate. However, it results in the lowest average operation time and reward for channel utilization because they are less important for consideration. This weight shows the highest average transmission data rate in Fig. 17b. Therefore, it is possible to operate the CR system according to the purposes of user by changing the weight assignment.
a Average operation time, b average transmission rate, and c reward of utilization according to weight change. The average operation time, average data rate, and reward for channel utilization by changing the weight assignment for DDR to 40 kbps. Since the reward function is composed of the weighted sum of the objective functions, the Q-learning can be operated according to the desired objective function by adjusting the weight. Therefore, if the weight of the operation time is increased, the average operation time is increased, and if the weight of the data transmission rate is increased, the average transmission rate is increased. Finally, increasing the weight of reward for utilization increases the average of reward for utilization
Performance comparison for the proposed Q-learning
In this section, we compare the channel selection performance between the proposed Q-learning mechanism and random selection from the available channel lists in terms of obtained rewards, average data rate of the secondary systems, and channel utilization efficiency. We also consider the fairness of selfish channel selection without considering channel utilization efficiency.
Figure 18 shows the average reward for each DDR case. When the DDR is 10 kbps or 50 kbps, the random channel selection has a lower reward than Q-learning because the random channel selection causes a waste of channel resources and obtains the low reward for channel utilization RUtil. In case the DDR is 1.5 Mbps or 3.5 Mbps (e.g., more than medium or high DDR case), a channel providing a sufficient data rate is not selected adaptively by random channel selection, leading to a lower reward than Q-learning channel selection. The rewards for a DDR of 1.5 Mbps and 3.5 Mbps is lower than those of 10kbps and 50 kbps. As shown in DRE range of [r2, 1) and [1, ∞) in Fig. 8, the ε-greedy policy in cases of a high DDR causes very low reward for channel utilization RUtil due to the selection of a low-band channel which support insufficient data rate and these effects are accumulated in the Q-table. These results show that the Q-learning channel selection adaptively selects the channel for the overall DDR.
Average reward comparison for Q-learning channel selection vs. random channel selection. The average reward for each DDR depending on the method of the channel selection. For all of the DDR cases, Q-learning band and channel selection has more reward value than random selection. The reward for a DDR of 1.5 or 3.5 Mbps (e.g., more than medium or high DDR case) is lower than that of 10 and 50 kbps. The ε-greedy policy in case of a high DDR causes very low reward for channel utilization RUtil due to the selection of a low-band channel which support insufficient data rate and these effects are accumulated in the Q-table
Figures 19 and 20 show the boxplot and mean value for the data rate and reward for channel utilization resulting in Q-learning and random channel selection. The red line represents the median value, and a star denotes the mean value of the data. Figure 19 shows the mean and boxplot of the data transmission rate for Q-learning and random channel selection for each DDR. In cases of the random channel selection, the average data transmission rate of all DDR cases is the same as the average value of the data rates for all channels in Table 1 belonging to band groups 1 and 2. The boxplots of all DDR cases for the random channel selection have a similar distribution, as well. The DDR of 10 kbps and 50 kbps by Q-learning channel selection have similar distributions, and the mean for 50 kbps Q-learning is higher than that of 10 kbps, since a higher DDR attempts to choose the channel supporting higher data transmission rate. In case of a DDR for 1.5 Mbps and 3.5M bps, as in Fig. 19b, the distribution and average value of the data transmission rate by the Q-learning channel selection are higher than that of 10 kbps and 50 kbps since the channels are mainly selected from band group 2.
a Mean value and b boxplot of data transmission rate for Q-learning and random channel selection. Shows the mean and boxplot of the data transmission rate for Q-learning and random channel selection for each DDR. In cases of the random channel selection, the average data transmission rate of all DDR cases is the same as the average value of the data rates for all channels in Table 1 belonging to band groups 1 and 2. The boxplots of all DDR cases for the random channel selection have a similar distribution. For the DDR of 10 kbps and 50 kbps, the mean of the Q-learning selection is lower than the random selection and the Q-learning has more narrow distribution. For the DDR of 1.5 and 3.5 Mbps, the mean of the Q-learning selection is higher than the random selection and the Q-learning has more narrow distribution
The a mean and b boxplot of the reward for the channel utilization by the Q-learning and random selection at each DDR. For all DDRs, the boxplot of Q-learning has denser distribution and higher values than that of the random selection, and it has a higher average value
We can see that the Q-learning channel selection can select the channel which provides higher data transmission rate when the DDR is 3.5 Mbps than 1.5 Mbps from the mean values of each DDR case.
Figure 20 shows the mean and boxplot of the reward for the channel utilization by the Q-learning and random channel selection at each DDR. The reward for channel utilization mainly operates as a harmful value in the total reward function when the ad-hoc CH chooses an appropriate channel for its DDR. For all DDRs, the boxplot of Q-learning is denser and distributed at higher values than that of the random channel selection, and it has a higher average value since Q-learning tries to choose the channel that does not create harm in terms of RUtil. Outlier values of Q-learning cases are generated by random selection.
Figure 21 shows the average fairness of data rate efficiency with increasing number of neighboring secondary systems. To compare the fairness between secondary systems, two compared methods are considered in this experiment: (i) the random selection, in which the operating band and channel are selected randomly by each secondary system from its available channels and (ii) MaxQ-selection [29,30,31,32], in which each secondary system selects the channel that has the maximum supportable data rate. As we can see in Fig. 21, the proposed method provides the highest fairness because it selects the band and channel based on the desired traffic demand and current channel utilization efficiency. Therefore, if a secondary system needs relatively low data rate, then it will select the band that has a low channel bandwidth in the proposed system and it yields the bands with wider channel bandwidth to the neighbor secondary systems that require higher data rates.
The average fairness of data rate efficiency with increasing number of neighboring secondary systems
Figures 22, 23, and 24 show the rewards of Q-learning and random channel selection according to an iteration for each DDR. In the case of Q-learning for all DDRs, the fluctuation decreases over time and the system operates with the intended reward design. In the random selection, there are more notches and fluctuation than Q-learning channel selection.
Rewards for a Q-learning and b random channel selection according to iteration (DDR = 40 kbps)
Rewards for a Q-learning and b random channel selection according to iteration (DDR = 1.5 Mbps)
Rewards for a Q-learning and b random channel selection according to iteration (DDR = 3.5 Mbps). Figures 22, 23, and 24 show the rewards for Q-learning and random band and channel selection according to the iteration for each DDR. In the case of Q-learning for all DDRs, the fluctuation decreases over time and the system operates with the intended reward design. In the random selection, there are more notches and fluctuation than Q-learning selection
Figure 25 represents the rewards and visits of states according to the changes in DDR as shown in Fig. 25a. Comparing Fig. 25b and d, the rewards of Q-learning selection are more stable than those of the random selection. From Fig. 25c and e, we can see that the ad-hoc CH selects a low data rate channel and Q-learning visits the state of low DRE in band group 1 when the DDR is low. Furthermore, we can see that the Q-learning visits a state of low DRE of band group 2 when the system selects a high data rate channel by the explorer policy. When the DDR is high, the Q-learning tries to select the channel of band group 2 mainly which provides higher data rates so that the states of band group 1 are visited less frequently. However, in random channel selection, the visits of states are distributed evenly in various DREs when the DDR is low or high. Figure 25c and e show that the visiting states of Q-learning and random channel selection are the same for a particular DDR. However, since the Q-learning channel selection tries to select a channel adaptive for the specific DDR, Q-learning mainly visits the state of the band group 1 when the DDR is low and visits the state of the band group 2 when the DDR is high. From these results, the proposed Q-learning selects the appropriate channel even if the DDR changes.
Rewards for Q-learning and random channel selection according to DDR change. Represents the rewards and visits of states according to the changes in DDR as shown in a. Comparing b and d, the rewards of Q-learning selection are more stable than that of the random selection. From c and e, we can see that the ad-hoc CH selects a low data rate channel and Q-learning visits the state of low DRE in band group 1 when the DDR is low. Furthermore, we can see that the Q-learning visits a state of low DRE of band group 2 when the system selects a high data rate channel by the explorer policy. When the DDR is high, the Q-learning tries to select the channel of band group 2 mainly which provides higher data rates so that the sates of band group 1 are visited less frequently. However, in random channel selection, the visits of states are distributed evenly in various DREs when the DDR is low or high. c and e shows that the visiting states of Q-learning and random channel selection are the same for a particular DDR. However, since the Q-learning channel selection tries to select a channel adaptive for the specific DDR, Q-learning mainly visits the state of the band group 1 when the DDR is low and visits the state of the band group 2 when the DDR is high. From these results, the proposed Q-learning selects the appropriate channel even if the DDR changes
In this paper, we propose a band group and channel selection method considering the consecutive channel operation time, data transmission rate, channel utilization efficiency, and cost of the band group change for a cognitive radio ad-hoc network composed of CH and MNs. The proposed method uses the Q-learning in order to operate in a channel environment that varies dynamically according to the geographical region, time zone, band group, channel, and primary user's activity. As the core of the Q-learning operation, a Q-table and reward function consisting of an action and state are designed to consider various parameters related to the channel selected by the CR ad-hoc system. In particular, the reward for channel utilization is designed to select the appropriate band and channel so that the frequency resources are not wasted and a CR ad-hoc system can coexist with other CR systems with fair resource utilization efficiency. The simulation results represent how the proposed system selects an adaptive band and channel for the required data rate and also explain the principle of operation through the change of action and state in Q-table. It also can be confirmed that the system operates according to the intended purpose through the weight change, and the channel is selected adaptively when the required transmission rate is changed. These simulations clearly demonstrate these advantages of the proposed method.
Methods/experimental
The purpose of this paper is to select the band and channel for a cognitive ad-hoc system to move when the primary user appears in the channel used by the CR system and is to consider the fairness with other systems in selecting the channel. The characteristics of frequency resources such as an available transmission opportunity and data rate vary depending on the time zone, geographical location, and band group, and the activity of the primary user and the desired data rate of the secondary user are also different according to them. Therefore, considering such a complicated environment, it is necessary to select a band and a channel that can maximize the performance of the system. In this paper, the Q-learning is used to dynamically select the band and channel according to the complex surrounding environment which is time-varying. The reward function of the Q-learning is designed considering the available channel use time, data rate, utilization efficiency of the selected channel, and cost for band change. Each of the considered terms is combined with a weight sum so that the performance related to the preferred parameters can be properly realized according to the adjustment of the weights. In particular, we designed a reward for utilization in the reward function so that the CR ad-hoc system does not choose a channel that provides unnecessarily high data rate and other system has the opportunity of selecting adaptive channel which supports adaptive high data rate. The Q-table is designed so that the reward function of Q-learning works properly. The state of the Q-table is composed of time zone, geographical zone, band group, and data rate efficiency (DRE) so that the proposed Q-learning can operate well.
Experimental results in this paper had been performed using MATLAB R2015b on Intel® Core i7 3.4 GHz system. The Gaussian random function to generate the operation time and supportable data rate of each channel over time and Q-table matrix for Q-learning can be made by constructing appropriate MATLAB code.
BS:
CBG:
Current band group
CH:
Cluster head
CR:
D2D:
Device-to-device
DDR:
Desired data rate
DRE:
Data rate efficiency
FCC:
HF:
Line-of-sight
MN:
Mobile node
NLoS:
Non-line-of-sight
PU:
Primary user
QoE:
Quality of experience
QoS:
SINR:
Signal-to-interference-plus-noise ratio
SMDP:
Semi-Markov decision process
UHF:
Ultra-high frequency
V2I:
Vehicle-to-infrastructure
V2V:
Vehicle-to-vehicle
VANET:
Vehicular ad-hoc network
VHF:
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Dataset of simulations
The simulation was performed using MATLAB in Intel Core i7 (32bit).
The operation time and supportable data rate is made of the mean and variance in Table 1 by Gaussian function using MATLAB.
The Q-table is made up of tables as defined in the paper, and it works according to the Q-table update equation.
This work was supported by a grant-in-aid of Hanwha Systems and the Agency for Defense Development (ADD) in the Republic of Korea as part of the Contract UC160007ED.
Department of Information and Communication Engineering, Inha University, 253 YongHyun-dong, Nam-gu, Incheon, South Korea
Sung-Jeen Jang
& Sang-Jo Yoo
Hanwha Systems 188, Pangyoyeok-Ro, Bundang-Gu, Seongname-Si, Gyeonggi-Do, 13524, South Korea
Chul-Hee Han
Agency for Defense Development, P.O.Box 35, Yuseong-Gu, Daejeon, South Korea
Kwang-Eog Lee
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All authors contribute to the concept, the design and developments of the theory analysis and algorithm, and the simulation results in this manuscript. All authors read and approved the final manuscript.
Correspondence to Sang-Jo Yoo.
- Prof. Sang-Jo Yoo, PhD (Corresponding author): Sang-Jo Yoo received the B.S. degree in electronic communication engineering from Hanyang University, Seoul, South Korea, in 1988, and the M.S. and Ph.D. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology, in 1990 and 2000, respectively. From 1990 to 2001, he was a Member of the Technical Staff in the Korea Telecom Research and Development Group, where he was involved in communication protocol conformance testing and network design fields. From 1994 to 1995 and from 2007 to 2008, he was a Guest Researcher with the National Institute Standards and Technology, USA. Since 2001, he has been with Inha University, where he is currently a Professor with the Information and Communication Engineering Department. His current research interests include cognitive radio network protocols, ad-hoc wireless network, MAC and routing protocol design, wireless network QoS, and wireless sensor networks.
- Mr. Sung-Jeen Jang: Sung-Jeen Jang received a B.S degree in electrical engineering from Inha University Incheon, Korea, 2007. He received his M.S. degree in Graduate School of Information Technology and Telecommunication, Inha University, Incheon Korea, 2009. Since March 2009, he has been pursuing a Ph.D. degree at the Graduate School of Information Technology and Telecommunication, Inha University, Incheon Korea. His current research interests include cognitive radio network protocols and machine learning applied wireless communications.
- Dr. Chul-Hee Han: Chulhee Han received the B.S. degree in Electronic Engineering from Chung-ang University, Korea, in 1997, and M.S. and Ph.D. degrees in Electrical and Electronic Engineering from Yonsei University, Korea, in 1999 and 2007, respectively. Currently, he is working at Hanwha Systems, Korea, as a chief engineer. He was involved in various projects including tactical mobile WiMAX system and tactical LOS PMP radio. His research interests include Tactical Broadband Communications, Combat Network Radio, and Cognitive Radio for Military Applications.
- Dr. Kwang-Eog Lee: Kwang-Eog Lee received the B.S. and M.S. degrees in electronic engineering from Kyungpook National University, Daegu, South Korea, in 1988 and 1990, respectively. He has been working in Agency for Defense Development since 1990. From 2007 to 2008, he was an exchange scientist in CERDEC (Communications-Electronics Research, Development and Engineering Center) U.S. Army. Currently, he is a principal researcher and his research interests include cognitive radio and terrestrial and satellite tactical communication.
Jang, S., Han, C., Lee, K. et al. Reinforcement learning-based dynamic band and channel selection in cognitive radio ad-hoc networks. J Wireless Com Network 2019, 131 (2019) doi:10.1186/s13638-019-1433-1
Ad-hoc network
Q-learning | CommonCrawl |
Monodomain model
The monodomain model is a reduction of the bidomain model of the electrical propagation in myocardial tissue. The reduction comes from assuming that the intra- and extracellular domains have equal anisotropy ratios. Although not as physiologically accurate as the bidomain model, it is still adequate in some cases, and has reduced complexity.[1]
Formulation
Being $\mathbb {T} $ the domain boundary of the model, the monodomain model can be formulated as follows[2]
${\frac {\lambda }{1+\lambda }}\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\text{ion}}\right)\quad \quad {\text{in }}\mathbb {T} ,$
where $\mathbf {\Sigma } _{i}$ is the intracellular conductivity tensor, $v$ is the transmembrane potential, $I_{\text{ion}}$ is the transmembrane ionic current per unit area, $C_{m}$ is the membrane capacitance per unit area, $\lambda $ is the intra- to extracellular conductivity ratio, and $\chi $ is the membrane surface area per unit volume (of tissue).[1]
Derivation
The monodomain model can be easily derived from the bidomain model. This last one can be written as[1]
${\begin{aligned}\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{e}\right)&=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\text{ion}}\right)\\\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+\nabla \cdot \left(\left(\mathbf {\Sigma } _{i}+\mathbf {\Sigma } _{e}\right)\nabla v_{e}\right)&=0\end{aligned}}$
Assuming equal anisotropy ratios, i.e. $\mathbf {\Sigma } _{e}=\lambda \mathbf {\Sigma } _{i}$, the second equation can be written as[1]
$\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{e}\right)=-{\frac {1}{1+\lambda }}\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right).$
Then, inserting this into the first bidomain equation gives the unique equation of the monodomain model[1]
${\frac {\lambda }{1+\lambda }}\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\text{ion}}\right).$
Boundary conditions
Differently from the bidomain model, usually the monodomain model is equipped with an isoltad boundary condition, which means that it is assumed that there is not current that can flow from or to the domain (usually the heart).[3][4] Mathematically, this is done imposing a zero transmembrane potential flux, i.e.:[4]
$(\mathbf {\Sigma } _{i}\nabla v)\cdot \mathbf {n} =0\quad \quad {\text{on }}\partial \mathbb {T} $
where $\mathbf {n} $ is the unit outward normal of the domain and $\partial \mathbb {T} $ is the domain boundary.
See also
• Bidomain model
• Forward problem of electrocardiology
References
1. Pullan, Andrew J.; Buist, Martin L.; Cheng, Leo K. (2005). Mathematically modelling the electrical activity of the heart : from cell to body surface and back again. World Scientific. ISBN 978-9812563736.
2. Keener J, Sneyd J (2009). Mathematical Physiology II: Systems Physiology (2nd ed.). Springer. ISBN 978-0-387-79387-0.
3. Rossi, Simone; Griffith, Boyce E. (1 September 2017). "Incorporating inductances in tissue-scale models of cardiac electrophysiology". Chaos: An Interdisciplinary Journal of Nonlinear Science. 27 (9): 093926. doi:10.1063/1.5000706. ISSN 1054-1500. PMC 5585078. PMID 28964127.
4. Boulakia, Muriel; Cazeau, Serge; Fernández, Miguel A.; Gerbeau, Jean-Frédéric; Zemzemi, Nejib (24 December 2009). "Mathematical Modeling of Electrocardiograms: A Numerical Study" (PDF). Annals of Biomedical Engineering. 38 (3): 1071–1097. doi:10.1007/s10439-009-9873-0. PMID 20033779. S2CID 10114284.
| Wikipedia |
Hip fracture surgery efficiency in Israeli hospitals via a network data envelopment analysis
Simona Cohen-Kadosh ORCID: orcid.org/0000-0001-7389-93691 &
Zilla Sinuany-Stern1
Central European Journal of Operations Research volume 28, pages251–277(2020)Cite this article
Data envelopment analysis (DEA) has been used previously for examining hospital efficiency, based on administrative data. Yet, previous DEA research devoted to quality assurance rarely considered medical processes or outcomes in efficiency studies. The goal of this study is to examine the relative efficiency of hip fracture surgery, based on clinical data reflecting medical process indicators and outcomes. To accomplish our goal, recent developments in DEA research were harnessed to model an output-oriented two-stage DEA network. The proposed DEA model has: two input variables reflecting the condition of the patient, fracture type and Charlson index; two intermediate variables reflecting clinical decisions, surgery within 48 h and usage of a drain for 1 day (rate); and two output variables reflecting the success of the surgery, survival rate after surgery and the rate of no infection. Using data from orthopedic wards in most of the acute Israeli hospitals (20 out of 22), no statistically significant correlation was found, either between the socio-economic index of patients who had hip fracture surgery and the relative efficiency scores produced by the two-stage network DEA model, or between efficiency and the geographical periphery status of the hospital. In addition to this, which points to a degree of social equality regarding hip fracture surgeries, we also compared the two-stage network model and related DEA models, providing several lemmas that represent the relationships between the various models mathematically.
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We thank the Quality Assurance Unit at Israel Ministry of Health for the data. We also thank the anonymous referees for their helpful comments which improve the paper.
Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, 84105, Beer-Sheva, Israel
Simona Cohen-Kadosh
& Zilla Sinuany-Stern
Search for Simona Cohen-Kadosh in:
Search for Zilla Sinuany-Stern in:
Correspondence to Simona Cohen-Kadosh.
Appendix A: Descriptive statistics of the original* variables
INTRCUP
WAIT2D
DRAIN1D
INFEC
Mean 43.05 2.41 26.34 15.99 6.34 15.71 .076
Median 42.28 2.41 23.15 7.96 5.62 15.32 .066
Std. deviation 7.30 .44 10.74 19.65 3.74 5.64 .433
Minimum 32.26 1.75 5.68 1.06 1.14 4.26 − .705
Maximum 61.05 3.39 45.26 77.17 12.43 23.96 .903
*Note that all variables are the originals we had, before any adjustment/transformation to DEA models was made
INTRCUPintracapsular percentage—the complement of the extracapsular rate we used, CHARL average Charlson comorbidity index (ranges between 1 and 30), WAIT2D percentage of patients waiting more than 2 days (48 h), DRAIN1D percentage of patients using drains for 1 day, INFEC and MORT percentage of patients infected or died within 365 days of surgery, SOESO average value of the socio-economic index (ranges between − 3 and 3) for patients undergoing hip surgery
Appendix B: The basic DEA (VRS) version
The basic DEA variable return to scale (VRS) version (Banker et al. 1984) has only input set X and output set Y. This is a one-stage model, indicated with one arrow in Table 2: X → Y. Its dual formulation is:
$$ \begin{aligned} & Max \;\emptyset \\ & {\text{s}} . {\text{t}} \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} X _{ij} \le x _{{ij_{0} }} ,} \quad \forall i = 1, \ldots ,m \\ & \quad \quad \O y_{{rj_{0} }} - \sum\limits_{j = 1}^{n} {\lambda_{j} y _{rj} \le 0 ,} \quad \forall r = 1, \ldots ,s \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} \le 1} \\ & \quad \quad \lambda_{j} \ge 1,\quad \O \ge 1 \\ \end{aligned} $$
The above DEA dual problem formulation output-oriented fits the 4th model (PARTIAL) its efficiency is ϕ4. The other three models are model 1 (A), model 2 (B), model 3 (FULL). All three models are one-stage systems like model 4: their formulation varies with respect to the vectors of inputs and outputs, as follows:
In model A the input vector is X, and the output vector is Z; the efficiency score is ϕ1.
$$ \begin{aligned} & Max \;\emptyset \\ {\text{s}} . {\text{t}} \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} x _{ij} \le x _{{ij_{0} }} ,} \quad \forall i = 1, \ldots ,m \\ & \quad \quad \O z_{dj} - \mathop \sum \limits_{j = 1}^{n} \lambda_{j} z _{dj} \le 0 , \quad \forall d = 1, \ldots ,D \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} \le 1} \\ & \quad \quad \lambda_{j} \ge 1, \quad \O \ge 1 \\ \end{aligned} $$
In model B the input vector is Z, and the output vector is Y; its efficiency is ϕ2.
$$ \begin{aligned} & Max \;\emptyset \\ {\text{s}} . {\text{t}} \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} z _{dj} \le z _{{dj_{0} }} , } \quad \forall d = 1, \ldots ,D \\ & \quad \quad \O y_{{rj_{0} }} - \sum\limits_{j = 1}^{n} {\lambda_{j} y _{rj} \le 0 ,} \quad \forall r = 1, \ldots ,s \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} \le 1} \\ & \quad \quad \lambda_{j} \ge 1, \quad \O \ge 1 \\ \end{aligned} $$
In model FULL the input vector is X and Z, and the output vector is Y; its efficiency is ϕ3.
$$ \begin{aligned} & Max \;\emptyset \\ {\text{s}} . {\text{t}} \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} x _{ij} \le x _{{ij_{0} }} , } \quad \forall i = 1, \ldots ,m \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} z _{dj} \le z _{{dj_{0} }} , } \quad \forall d = 1, \ldots ,D \\ & \quad \quad \O z_{dj} - \sum\limits_{j = 1}^{n} {\lambda_{j} z _{dj} \le 0 , } \quad \forall d = 1, \ldots ,D \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} \le 1} \\ & \quad \quad \lambda_{j} \ge 1, \quad \O \ge 1 \\ \end{aligned} $$
Appendix C: Spearman correlations between the seven models
A*B
B − .272
A*B .974* − .114
AVERAGE .976* − .132 .998*
FULL − .289 .681* − .229 − .229
PART − .078 .846* .072 .050 .617*
NETWORK − .175 .843* − .005 − .028 .502* .854*
*Significant with P < .05
Appendix D: The frequency with which an efficient hospital is a peer for inefficient hospitals
Hospital no.
s1 4 6 6
s3 7 12 13
s15 4 2 13
s16 9 13 25
Cohen-Kadosh, S., Sinuany-Stern, Z. Hip fracture surgery efficiency in Israeli hospitals via a network data envelopment analysis. Cent Eur J Oper Res 28, 251–277 (2020) doi:10.1007/s10100-018-0585-0
Issue Date: March 2020
Data envelopment analysis (DEA)
Two-stage network DEA
Socio-economic index (SEI)
Geographical periphery status
Clinical quality assurance | CommonCrawl |
\begin{definition}[Definition:Supremum of Set/Real Numbers/Propositional Function/Finite Range]
Let $\family {a_j}_{j \mathop \in I}$ be a family of elements of the real numbers $\R$ indexed by $I$.
Let $\map R j$ be a propositional function of $j \in I$.
Let the fiber of truth of $\map R j$ be finite.
Then the '''supremum of $\family {a_j}_{j \mathop \in I}$''' can be expressed as:
:$\ds \max_{\map R j} a_j = \text { the maxmum of all $a_j$ such that $\map R j$ holds}$
and can be referred to as the '''maximum''' of $\family {a_j}_{j \mathop \in I}$.
If more than one propositional function is written under the supremum sign, they must ''all'' hold.
\end{definition} | ProofWiki |
\begin{document}
\title{Painlev\'e Test and the Resolution of Singularities for Integrable Equations}
\tableofcontents
\section{Introduction}
Since Kowalevskaya's monumental work \cite{kowa}, the Painlev\'e test has been the most widely used and the most successful technique for detecting integrable differential equations \cite{ars,cm,fikn}. The test has been applied to many differential equations and, for those passing the Painlev\'e test, the indicators of integrability such as the existence of sufficiently many conservation laws, the Lax pair, the Darboux transform, the B\"acklund transform have always been found.
The Painlev\'e test itself is a recipe for finding all formal Laurent series solutions with movable singularities. An $n$-th order system {\em passes the Painlev\'e test} if such formal Laurent series solutions admit $n$ free parameters (including the location $t_0$ of the the movable singularity as one free parameter). Since the solutions of an $n$-th order system should have $n$ degrees of freedom, the Laurent series solutions, with maximal number of free parameters, should include all the solutions. Then one hopes that all movable singular solutions are poles and concludes that all solutions are single valued. This is the heuristic reason why passing the Painlev\'e test means integrability. Indeed such reason underlies the classification of integrable equations by Painlev\'e and the others \cite{pain}.
In \cite{am1,am2}, Adler and van Moerbeke put the heuristic reason on solid foundation for the very nice case of algebraically completely integrable systems. They used the toric geometry to construct a complete phase space, and thereby gave a satisfactory explanation for the relation between the Painlev\'e test and the integrability. In \cite{es}, Ercolani and Siggia pointed out that algebraic geometry is not needed for integrable Hamiltonian systems. They suggested using the expansion of the Hamilton-Jacobi equation to construct the change of variable used for completing the phase space. They showed that the method often works through many examples, but did not prove that the method always works.
In this paper, we show that, not only is algebraic geometry not needed, the Hamiltonian set up by Ercolani and Siggia is also not needed. In fact, with virtually no condition, the existence of the change of variable needed for completing the phase space is equivalent to passing the Painlev\'e test.
\begin{theoremA*} A regular system of ordinary differential equations passes the Painlev\'e test if and only if there is a triangular change of variable, such that the system is converted to another regular system, and the Laurent series solutions produced by the Painlev\'e test are converted to power series solutions. \end{theoremA*}
For the exact meaning of ``passing the Painlev\'e test'', see the rather straightforward definition in the beginning of Section \ref{general}. Also see \cite{hy2} for some concrete examples. The condition is as straightforward and elementary as can be. Although the proof is also rather elementary, we believe there is some interesting underlying algebraic structure that is worth further exploration.
The theorem gives a key connection between the two major works by Kowalevskaya \cite{hy5}. As in the theorem of Cauchy and Kowalevskaya, we consider a regular system of differential equations with complex analytic functions on the right side. Given a movable singular solution, represented by Laurent series in $(t-t_0)$ for variables $u_i$, changing $u_i$ to $u_i^{-1}$ certainly regularizes the variables. However, such simple regularization will create singular differential equations for the new variables. By the resolution of singularity, we mean a change of variable that regularizes both the solution and the equation. This is not always possible, and the condition for the resolvability of the singularity is exactly passing the Painlev\'e test.
The regularization is achieved by a {\em triangular change of variable} \begin{align*} u_1 &= \tau^{-k}, \\ u_i &= a_i(t,\tau,\rho_2,\dots,\rho_{i-1})+b_i(t,\tau,\rho_2,\dots,\rho_{i-1})\rho_i,\quad 1<i\le n, \end{align*} where $-k$ is the leading order of the Laurent series solution for $u_1$ obtained in the Painlev\'e test, $a_i,b_i$ are meromorphic in $\tau$ and analytic in the other variables, and $b_i$ does not take zero value. Such change of variable can be easily inverted, at the cost of introducing one $k$-th root. Then it is easy to see that the Laurent series for $u_1,\dots,u_n$ correspond to a power series for $\tau$ satisfying $\tau(t_0)=0$, $\tau'(t_0)\ne 0$, and the Laurent series for $\rho_2,\dots,\rho_n$. In case the system for $u_1,\dots,u_n$ passes the Painlev\'e test, we can find suitable $a_i,b_i$ so that the Laurent series for $\rho_2,\dots,\rho_n$ are always power series. In fact, in our change of variable formula \eqref{change1} and \eqref{triangle} (or \eqref{change1} and \eqref{change3}), we get very specific forms for $a_i$ and $b_i$. Then applying Cauchy theorem to the regular system for $\tau,\rho_2,\dots,\rho_n$ and the convert to the original system, we conclude the following. See \cite[Section 3.8.6]{go} for more discussion on the convergence of formal Laurent series solutions.
\begin{corollary*} If a regular system of ordinary differential equations passes the Painlev\'e test, then the Laurent series solutions obtained from the test always converge. \end{corollary*}
Section \ref{general} is devoted to the rigorous proof of Theorem A. We will be rather conservative in assuming that the right side of the differential equations are polynomials. The assumption is used to make sure that, after substituting in the Laurent series, the right side again becomes a Laurent series. Certainly we can make sense of this with functions other than polynomials, but the spirit of Theorem A is already fully reflected in the polynomial case.
Ercolani and Siggia's work suggests that there should be a version of Theorem A that is compatible with Hamiltonian structure.
\begin{theoremB*} If a regular Hamiltonian system of ordinary differential equations passes the Painlev\'e test in the Hamiltonian way, then there is a canonical triangular change of variable, such that the system is converted to another regular Hamiltonian system, and the Laurent series solutions are converted to power series solutions. Moreover, if the system is autonomous, then the new Hamiltonian function is obtained by substituting the new variables. If the system is not autonomous, then the new Hamiltonian function is obtained by substituting the new variables and then dropping the singular terms. \end{theoremB*}
The exact meaning of the ``Hamiltonian way'' is given in Section \ref{hamilton}. See \cite{hy3} for some concrete examples.
In view of the rather ad hoc nature of the Painlev\'e test, we further provide a rigorous foundation of the Painlev\'e test in Section \ref{foundation}. Our setup is by no means the broadest possible (and is therefore only ``a'' foundation), but is satisfied by most examples and is the most common way the conditions of our theorems are met.
We emphasize that the results of this paper are local, in fact as local as the theorem of Cauchy and Kowalevskaya. Although the Painlev\'e test is local, it is supposed to test the global properties of solutions. The regularization in this paper may be used to further prove the global property for some specific integrable equations, but here we are not making any general statement about the global property. See \cite{hl1,hl2,hl3,hy6,st} for specific examples of global Painlev\'e analysis.
Finally, we remark that our theorems do not deal with the lower balances between the principal balances, as discussed in \cite{am1,am2,es} as part of the process of completing the phase space. If there are no lower balances, then our theorems effectively show that passing the Painlev\'e test implies the completion of the phase space and thereby explains the integrability. Further extensions of our theorems to lower balances are needed for the more general case.
\section{Resolution of Movable Singularity} \label{general}
Consider a system of ordinary differential equations \[ u_i'=f_i(t,u_1,\dots,u_n),\quad 1\le i\le n. \] We restrict the discussion to $f_i(t,u_1,\dots,u_n)$ being analytic in $t$ and polynomial in $u_1,\dots,u_n$. At the end of this section, we will point out what is exactly needed for more general $f_i$.
A {\em balance} for the system is a formal Laurent series solution, with {\em movable singularity} $t_0$. The balance often admits several free parameters, similar to the initial values in formal power series solutions. Therefore a balance is typically given by ($r_1$ is reserved for the first free parameter $t_0$) \begin{align} u_i &= a_{i,0}(t_0)(t-t_0)^{-k_i}+\dots+a_{i,\lambda_1-1}(t_0)(t-t_0)^{\lambda_1-1-k_i} \nonumber \\ &\quad +a_{i,\lambda_1}(t_0,r_2,\dots,r_{n_1})(t-t_0)^{\lambda_1-k_i} \nonumber \\ &\qquad +\dots+a_{i,\lambda_2-1}(t_0,r_2,\dots,r_{n_1})(t-t_0)^{\lambda_2-1-k_i} \nonumber \\ &\quad + \cdots \label{balance0} \\ &\quad +a_{i,\lambda_s}(t_0,r_2,\dots,r_{n_s})(t-t_0)^{\lambda_s-k_i} \nonumber \\ &\qquad +\dots+a_{i,j}(t_0,r_2,\dots,r_{n_s})(t-t_0)^{j-k_i}+\cdots, \nonumber \end{align} where $a_{i,j}$ is analytic in its variables. The free parameters are called {\em resonance parameters}, and the indices $\lambda_1,\dots,\lambda_s$ where they first appear are the {\em resonances}. In the balance \eqref{balance0}, the resonance parameters $r_{n_{l-1}+1},\dots,r_{n_l}$ have the corresponding resonance $\lambda_l$. The number $m_l=n_l-n_{l-1}$ of resonance parameters of resonance $\lambda_l$ is the {\em multiplicity} of $\lambda_l$. We have \[ n_l=m_0+m_1+\dots+m_l, \] where $m_0=1$ counts the resonance parameter $t_0$ of resonance $\lambda_0=-1$.
We certainly want the free parameters to really represent the variety of Laurent series solutions. This means that we should require that the {\em resonance vectors} of resonance $\lambda_l$ to form an $n\times m_l$ matrix of full rank $m_l$ \[ R_l=\begin{pmatrix} \dfrac{\partial a_{1,\lambda_l}}{\partial r_{n_{l-1}+1}} & \cdots & \dfrac{\partial a_{1,\lambda_l}}{\partial r_{n_l}} \\ \vdots && \vdots \\ \dfrac{\partial a_{n,\lambda_l}}{\partial r_{n_{l-1}+1}} & \cdots & \dfrac{\partial a_{n,\lambda_l}}{\partial r_{n_l}} \end{pmatrix}. \] For the resonance $\lambda_0=-1$, this means that the {\em basic resonance vector} $R_0=(-k_1a_{1,0},\dots,-k_na_{n,0})^T$ is non-zero. In fact, when we consider the independence among all the resonance parameters, we really should require that the {\em resonance matrix} \[ R=(R_0\; R_1\; \cdots \; R_s) \] to have the full rank $n_s$.
The exact condition for Theorem A is the following.
\begin{definition*} A balance \eqref{balance0} is {\em principal}, if $n_s=n$ and the resonance matrix is invertible. \end{definition*}
\subsection{Indicial Normalization}
By substituting $t_0=t-(t-t_0)$, the coefficients $a_{i,j}$ become power series of $(t-t_0)$ with analytic functions of $t,r_2,\dots,r_n$ as coefficients. Further substituting these power series into \eqref{balance0}, the balance becomes \begin{align} u_i &= a_{i,0}^{(0)}(t)(t-t_0)^{-k_i} +\dots +a_{i,\lambda_1-1}^{(0)}(t)(t-t_0)^{\lambda_1-1-k_i} \nonumber\\ &\quad +a_{i,\lambda_1}^{(0)}(t,r_2,\dots,r_{n_1})(t-t_0)^{\lambda_1-k_i} \nonumber\\ &\qquad +\dots +a_{i,\lambda_2-1}^{(0)}(t,r_2,\dots,r_{n_1})(t-t_0)^{\lambda_2-1-k_i}\nonumber\\ &\quad + \cdots \label{balance1} \\ &\quad +a_{i,\lambda_s}^{(0)}(t,r_2,\dots,r_{n_s})(t-t_0)^{\lambda_s-k_i}+\cdots. \nonumber \end{align} We have \[ a_{i,\lambda_l}^{(0)}(t,r_2,\dots,r_{n_l}) =a_{i,\lambda_l}(t,r_2,\dots,r_{n_l})+c_{i,\lambda_l}(t,r_2,\dots,r_{n_{l-1}}). \] Therefore the new resonance matrix \[ R^{(0)}=(R_0^{(0)}\; R_1^{(0)}\; \cdots \; R_s^{(0)}),\quad R_l^{(0)}=\begin{pmatrix} \dfrac{\partial a_{1,\lambda_l}^{(0)}}{\partial r_{n_{l-1}+1}} & \cdots & \dfrac{\partial a_{1,\lambda_l}^{(0)}}{\partial r_{n_l}} \\ \vdots && \vdots \\ \dfrac{\partial a_{n,\lambda_l}^{(0)}}{\partial r_{n_{l-1}+1}} & \cdots & \dfrac{\partial a_{n,\lambda_l}^{(0)}}{\partial r_{n_l}} \end{pmatrix}, \] is obtained from the old one simply by replacing $t_0$ with $t$ \[ R^{(0)}(t,r_2,\dots,r_n)=R(t,r_2,\dots,r_n). \] The matrix is still invertible.
Up to rearranging the order of $u_i$, we may assume $k_1a_{1,0}^{(0)}(t)\ne 0$. Then we introduce a new variable $\tau$ by \begin{equation}\label{change1} u_1=\tau^{-k_1}. \end{equation} Fixing some $(-k_1)$-th root $\beta_1(t)$ of $a_{1,0}^{(0)}(t)$ and taking the $(-k_1)$-th root of the equality \eqref{balance1} for $u_1$, we get the power $\tau$-series \begin{align} \tau &= \beta_1(t)(t-t_0) +\dots +\beta_{\lambda_1-1}(t)(t-t_0)^{\lambda_1-1} \nonumber \\ &\quad +\beta_{\lambda_1}(t,r_2,\dots,r_{n_1})(t-t_0)^{\lambda_1}+\dots+\beta_{\lambda_2-1}(t,r_2,\dots,r_{n_1})(t-t_0)^{\lambda_2-1} \nonumber \\ &\quad + \cdots \label{tauexpand} \\ &\quad +\beta_{\lambda_s}(t,r_2,\dots,r_{n_s})(t-t_0)^{\lambda_s}+\cdots, \nonumber \end{align} We may reverse the power $\tau$-series and get \begin{equation}\label{texpand} t-t_0=b_1\tau+b_2\tau^2+b_3\tau^3+\cdots, \end{equation} where $b_i$ is an analytic function of the same variables as $\beta_i$.
Substituting the power $\tau$-series \eqref{texpand} into the balance \eqref{balance1}, we get Laurent $\tau$-series \begin{align} u_i &= a_{i,0}^{(0)}b_1^{-k_i}\tau^{-k_i}+[-k_ia_{i,0}^{(0)}b_1^{-1}b_2+a_{i,1}^{(0)}b_1]b_1^{-k_i}\tau^{1-k_i}+\cdots \label{expand0} \\ &\quad +[-k_ia_{i,0}^{(0)}b_1^{-1}b_{j+1}+a_{i,j}^{(0)}b_1^j+\text{terms involving }a_{i,<j}^{(0)}, b_{\le j}]b_1^{-k_i}\tau^{j-k_i}+\cdots. \nonumber \end{align} For $u_1=\tau^{-k_1}$, the equality \eqref{expand0} gives the recursive relation that computes the coefficients $b_j$ \[ b_{j+1}=\dfrac{b_1^{j+1}}{k_1a_{1,0}^{(0)}}a_{1,j}^{(0)}+\text{terms involving }a_{1,<j}^{(0)}, b_{\le j}. \] Substituting the coefficients $b_j$ into the other $u_i$ in \eqref{expand0}, we get updated Laurent $\tau$-series \begin{align} u_i &= a_{i,0}^{(1)}(t)\tau^{-k_i}+\dots +a_{i,\lambda_1-1}^{(1)}(t)\tau^{\lambda_1-1-k_i} \nonumber\\ &\quad +a_{i,\lambda_1}^{(1)}(t,r_2,\dots,r_{n_1})\tau^{\lambda_1-k_i} +\dots +a_{i,\lambda_2-1}^{(1)}(t,r_2,\dots,r_{n_1})\tau^{\lambda_2-1-k_i}\nonumber\\ &\quad + \cdots \label{balance2} \\ &\quad +a_{i,\lambda_s}^{(1)}(t,r_2,\dots,r_{n_s})\tau^{\lambda_s-k_i}+\cdots, \qquad i>n_0=1, \nonumber \end{align} where \[ a^{(1)}_{i,j} =\left(a_{i,j}^{(0)}-\dfrac{k_ia_{i,0}^{(0)}}{k_1a_{1,0}^{(0)}} a_{1,j}^{(0)}\right)b_1^{j-k_i} +\text{ terms involving }a_{1,<j}^{(0)}, a_{i,<j}^{(0)}, b_{\le j},\quad i>1. \]
For $j=\lambda_l$, $i>1$ and $n_{l-1}<s\le n_l$, the formula for $a^{(1)}_{i,j}$ implies \[ \dfrac{\partial a^{(1)}_{i,\lambda_l}}{\partial r_p} =\left(\dfrac{\partial a^{(0)}_{i,\lambda_l}}{\partial r_p}-\dfrac{k_ia_{i,0}^{(0)}}{k_1a_{1,0}^{(0)}} \dfrac{\partial a^{(0)}_{1,\lambda_l}}{\partial r_p}\right)b_1^{j-k_i}. \] Therefore the updated resonance matrix \[ R^{(1)}=(R_1^{(1)}\; \cdots \; R_s^{(1)}),\quad R_l^{(1)}=\begin{pmatrix} \dfrac{\partial a_{2,\lambda_l}^{(1)}}{\partial r_{n_{l-1}+1}} & \cdots & \dfrac{\partial a_{2,\lambda_l}^{(1)}}{\partial r_{n_l}} \\ \vdots && \vdots \\ \dfrac{\partial a_{n,\lambda_l}^{(1)}}{\partial r_{n_{l-1}+1}} & \cdots & \dfrac{\partial a_{n,\lambda_l}^{(1)}}{\partial r_{n_l}} \end{pmatrix} \] is obtained by deleting the first column (which consists of entirely $0$) of the following matrix \[ \begin{pmatrix} b_1^{-k_2} & & \\ & \ddots & \\
& & b_1^{-k_n} \end{pmatrix} \begin{pmatrix} -\dfrac{k_2a_{2,0}^{(0)}}{k_1a_{1,0}^{(0)}} & 1 && \\ \vdots & &\ddots & \\ -\dfrac{k_na_{n,0}^{(0)}}{k_1a_{1,0}^{(0)}} & && 1 \end{pmatrix} R^{(0)} \begin{pmatrix} b_1^{\lambda_1}I_{m_1} & & \\ & \ddots & \\
& & b_1^{\lambda_s}I_{m_s} \end{pmatrix}. \] In other words, up to multiplying the powers of $b_1$ to rows and columns, $R^{(1)}$ is obtained as part of the row operation that uses the first entry of $R^{(0)}$ to eliminate the other terms in the first column. In particular, the invertibility of $R^{(0)}$ implies the invertibility of $R^{(1)}$. The size is reduced by $1$ because the resonance parameter $t_0$ has been ``absorbed'' into the new variable $\tau$.
\subsection{Resonance Variable}
Next we introduce new variables to ``absorb'' the resonance parameters $r_{(1,n_1]}=(r_2,\dots,r_{n_1})$ of resonance $\lambda_1$. By rearranging $u_2,\dots,u_n$ if necessary, we may assume that the first $m_1\times m_1$ submatrix $A^{(1)}$ of $R^{(1)}$ is invertible. Then we have \[ R^{(1)} =\begin{pmatrix}A^{(1)} & C^{(1)} \\ B^{(1)} & D^{(1)}\end{pmatrix}, \] where \[ R_1^{(1)} =\begin{pmatrix}A^{(1)} \\ B^{(1)} \end{pmatrix}, \quad (R_2^{(1)}\; \cdots \; R_s^{(1)}) =\begin{pmatrix}C^{(1)} \\ D^{(1)} \end{pmatrix}. \]
We introduce new variables $\rho_2,\dots,\rho_{n_1}$ by truncating the $\tau$-series of $u_2,\dots,u_{n_1}$ in \eqref{balance2} at $\tau^{\lambda_1-k_i}$ \begin{align} u_i &= a_{i,0}^{(1)}(t)\tau^{-k_i}+\dots +a_{i,\lambda_1-1}^{(1)}(t)\tau^{\lambda_1-1-k_i} \label{change2} \\ &\quad +a_{i,\lambda_1}^{(1)}(t,\rho_2,\dots,\rho_{n_1})\tau^{\lambda_1-k_i}, \quad 1<i\le n_1. \nonumber \end{align} Then we have the equalities \begin{align*} a_{i,\lambda_1}^{(1)}(t,\rho_2,\dots,\rho_{n_1}) &= a_{i,\lambda_1}^{(1)}(t,r_2,\dots,r_{n_1}) +\dots +a_{i,\lambda_2-1}^{(1)}(t,r_2,\dots,r_{n_1})\tau^{\lambda_2-1-\lambda_1} \\ &\quad +a_{i,\lambda_2}^{(1)}(t,r_2,\dots,r_{n_2})\tau^{\lambda_2-\lambda_1} +\cdots \\ &\quad + \cdots \\ &\quad +a_{i,\lambda_s}^{(1)}(t,r_2,\dots,r_{n_s})\tau^{\lambda_s-\lambda_1}+\cdots, \quad 1<i\le n_1. \end{align*} By the inverse function theorem, the invertibility of \[ A^{(1)}(t,r_2,\dots,r_{n_1})=\dfrac{\partial (a_{2,\lambda_1}^{(1)},\dots,a_{n_1,\lambda_1}^{(1)})}{\partial (r_2,\dots,r_{n_1})} \] means that the left side is a locally invertible map of $\rho_2,\dots,\rho_{n_1}$, and for small $\tau$, the right side is a locally invertible map of $r_2,\dots,r_{n_1}$. By taking the inverse of the left side map, we get the power $\tau$-series for the new variables \begin{align*} \rho_i &= r_i +\beta_{i,1}(t,r_2,\dots,r_{n_1})\tau+\cdots \\ &\quad +\beta_{i,\lambda_2-\lambda_1}(t,r_2,\dots,r_{n_1},r_{n_1+1},\dots,r_{n_2})\tau^{\lambda_2-\lambda_1}+\cdots \\ &\quad + \dots \\ &\quad +\beta_{i,\lambda_s-\lambda_1}(t,r_2,\dots,r_{n_1},r_{n_1+1},\dots,r_{n_s})\tau^{\lambda_s-\lambda_1}+\cdots, \quad 1<i\le n_1. \end{align*} By taking the inverse of the right side map, we get the power $\tau$-series for the resonance parameters \begin{align} r_i &= \rho_i +b_{i,1}(t,\rho_2,\dots,\rho_{n_1})\tau+\cdots \nonumber \\ &\quad +b_{i,\lambda_2-\lambda_1}(t,\rho_2,\dots,\rho_{n_1},r_{n_1+1},\dots,r_{n_1})\tau^{\lambda_2-\lambda_1}+\cdots \nonumber \\ &\quad + \cdots \label{rexpand1} \\ &\quad +b_{i,\lambda_s-\lambda_1}(t,\rho_2,\dots,\rho_{n_1},r_{n_1+1},\dots,r_{n_s})\tau^{\lambda_s-\lambda_1}+\cdots, \quad 1<i\le n_1. \nonumber \end{align}
Substituting \eqref{rexpand1} into the balance \eqref{balance2}, we get the Laurent $\tau$-series \begin{align} u_i &= a_{i,0}^{(1)}(t)\tau^{-k_i}+\dots +a_{i,\lambda_1-1}^{(1)}(t)\tau^{\lambda_1-1-k_i}+a_{i,\lambda_1}^{(1)}(\rho_2,\dots,\rho_{n_1})\tau^{\lambda_1-k_i} \nonumber \\ &\quad +\left[\dfrac{\partial a_{i,\lambda_1}^{(1)}}{\partial (r_2,\dots,r_{n_1})}(t,\rho_2,\dots,\rho_{n_1})b_1+a_{i,\lambda_1+1}^{(1)}(t,\rho_2,\dots,\rho_{n_1})\right]\tau^{\lambda_1+1-k_i} \nonumber\\ &\quad +\cdots \label{expand1} \\ &\quad +\Biggl[\dfrac{\partial a_{i,\lambda_1}^{(1)}}{\partial (r_2,\dots,r_{n_1})}(t,\rho_2,\dots,\rho_{n_1})b_j+a_{i,\lambda_1+j}^{(1)}(t,\rho_2,\dots,\rho_{n_1},r_{n_1+1},\dots) \nonumber \\ &\qquad\qquad +\text{ terms involving }a_{i,<\lambda_1+j}^{(1)},b_{<j}\Biggr]\tau^{\lambda_1+j-k_i} +\cdots, \nonumber \end{align} where $i>1$ and $b_j=(b_{2,j},\dots,b_{n_1,j})^T$ is the vertical coefficient vector from \eqref{rexpand1}. For $1<i\le n_1$, comparing \eqref{expand1} with \eqref{change2} gives the recursive relation that computes the coefficients $b_j$ \[ A^{(1)}(t,\rho_2,\dots,\rho_{n_1})b_j=-a_{(1,n_1],\lambda_1+j}^{(1)}(t,\rho_2,\dots,\rho_{n_1},r_{n_1+1},\dots)+\text{ lower terms}, \] where \[ a_{(n',n],j}=(a_{n'+1,j},\dots,a_{n,j})^T. \] Then we substitute the formula for $b_j$ into the other $u_i$ in \eqref{expand1} and get their updated Laurent $\tau$-series \begin{align} u_i &= a_{i,0}^{(2)}(t)\tau^{-k_i}+\dots +a_{i,\lambda_1-1}^{(2)}(t)\tau^{\lambda_1-1-k_i} \nonumber \\ &\quad +a_{i,\lambda_1}^{(2)}(t,\rho_2,\dots,\rho_{n_1})\tau^{\lambda_1-k_i}+\dots+a_{i,\lambda_2-1}^{(2)}(t,\rho_2,\dots,\rho_{n_1})\tau^{\lambda_2-1-k_i} \nonumber\\ &\quad +a_{i,\lambda_2}^{(2)}(t,\rho_2,\dots,\rho_{n_1},r_{n_1+1},\dots,r_{n_2})\tau^{\lambda_2-k_i} \nonumber\\ &\qquad +\dots +a_{i,\lambda_3-1}^{(2)}(t,\rho_2,\dots,\rho_{n_1},r_{n_1+1},\dots,r_{n_2})\tau^{\lambda_3-1-k_i} \nonumber\\ &\quad + \cdots \label{balance3} \\ &\quad +a_{i,\lambda_s}^{(2)}(t,\rho_2,\dots,\rho_{n_1},r_{n_1+1},\dots,r_{n_s})\tau^{\lambda_s-k_i}+\cdots, \qquad i>n_1, \nonumber \end{align} where \begin{align*} a_{i,j}^{(2)} &=a_{i,j}^{(1)}, \quad j\le \lambda_1, \\ a^{(2)}_{i,j} &=a_{i,j}^{(1)}(t,\rho_2,\dots,\rho_{n_1},r_{n_1+1},\dots) \\ &\quad -\dfrac{\partial a_{i,\lambda_1}^{(1)}}{\partial (r_2,\dots,r_{n_1})}(t,\rho_2,\dots,\rho_{n_1})(A^{(1)})^{-1}a_{(1,n_1],j}^{(1)}(t,\rho_2,\dots,\rho_{n_1},r_{n_1+1},\dots) \\ &\quad + \text{ lower terms},\quad j>\lambda_1. \end{align*} Written in vector form, for $j>\lambda_1$ we have \[ a_{(n_1,n],j}^{(2)}
=a_{(n_1,n],j}^{(1)}-B^{(1)}(A^{(1)})^{-1}a_{(1,n_1],j}^{(1)},\quad B^{(1)}=\left.\dfrac{\partial a_{(n_1,n],\lambda_1}^{(1)}}{\partial (r_2,\dots,r_{n_1})}\right|_{r_i=\rho_i}. \]
For $j=\lambda_l$, $l>1$, and $n_{l-1}<s\le n_l$, the formula for $a_{(n_1,n],j}^{(2)}$ implies that the updated resonance vectors of resonance $\lambda_l$ form the matrix \begin{align*} R_l^{(2)} &=\dfrac{\partial a_{(n_1,n],\lambda_l}^{(2)}}{\partial (r_{n_{l-1}+1},\dots,r_{n_l})} \\ &=\dfrac{\partial a_{(n_1,n],\lambda_l}^{(1)}}{\partial (r_{n_{l-1}+1},\dots,r_{n_l})} -B^{(1)}(A^{(1)})^{-1}\dfrac{\partial a^{(1)}_{(1,n_1],\lambda_l}}{\partial (r_{n_{l-1}+1},\dots,r_{n_l})} \\ &=(-B^{(1)}(A^{(1)})^{-1}\;\; I)R_l^{(1)}. \end{align*} Therefore the updated resonance matrix \[ R^{(2)} =(R_2^{(2)}\; \cdots \; R_s^{(2)}) =(-B^{(1)}(A^{(1)})^{-1}\;\; I) \begin{pmatrix}C^{(1)} \\ D^{(1)} \end{pmatrix} =D^{(1)}-B^{(1)}(A^{(1)})^{-1}C^{(1)} \] is obtained as part of the row operation on $R^{(1)}$ that uses the invertible matrix $A^{(1)}$ to eliminate $B^{(1)}$ \[ \begin{pmatrix} I & O \\ -B^{(1)}(A^{(1)})^{-1} & I \end{pmatrix}R^{(1)} =\begin{pmatrix} A^{(1)} & C^{(1)} \\ O & D^{(1)}-B^{(1)}(A^{(1)})^{-1}C^{(1)} \end{pmatrix}. \] In particular, the invertibility of $R^{(1)}$ implies the invertibility of $R^{(2)}$.
The process continues and follows the same pattern. After introducing $\rho_{n_{l-2}+1},\dots,\rho_{n_{l-1}}$, we get an updated Laurent $\tau$-series for the remaining $u_i$, $i>n_{l-1}$. The coefficients of the series are functions of $t,\rho_2,\dots,\rho_{n_{l-1}}$ and $r_{n_{l-1}+1},\dots,r_n$. Moreover, we also have the resonance matrix $R^{(l-1)}$, which is an $(n-n_{l-1})\times (n-n_{l-1})$ invertible matrix. By rearranging the remaining $u_i$, we may assume that the first $m_l\times m_l$ matrix $A^{(l-1)}$ in $R^{(l-1)}$ is invertible. Then we introduce new variables by truncating the Laurent $\tau$-series of $u_{n_{l-1}+1},\dots,u_{n_l}$ at the resonance $\lambda_l$ and replacing the resonance parameters with new variables $\rho_{n_{l-1}+1},\dots,\rho_{n_l}$. \begin{align} u_i &= a_{i,0}^{(l)}(t)\tau^{-k_i}+\dots +a_{i,\lambda_1-1}^{(l)}(t)\tau^{\lambda_1-1-k_i} \nonumber \\ &\quad +a_{i,\lambda_1}^{(l)}(t,\rho_2,\dots,\rho_{n_1})\tau^{\lambda_1-k_i}+\cdots \nonumber \\ &\quad +\cdots \label{change3} \\ &\quad +a_{i,\lambda_{l-1}}^{(l)}(t,\rho_2,\dots,\rho_{n_{l-1}})\tau^{\lambda_{l-1}-k_i}+\cdots \nonumber \\ &\quad +a_{i,\lambda_l}^{(l)}(t,\rho_2,\dots,\rho_{n_{l-1}},\rho_{n_{l-1}+1},\dots,\rho_{n_l})\tau^{\lambda_l-k_i}, \quad n_{l-1}<i\le n_l. \nonumber \end{align} The invertibility of $A^{(l)}$ then implies that we can find the power $\tau$-series for the new variables \begin{align} \rho_i &= r_i +\beta_{i,1}(t,\rho_2,\dots,\rho_{n_{l-1}},r_{n_{l-1}+1},\dots,r_{n_l})\tau+\cdots \nonumber \\ &\quad +\beta_{i,\lambda_{l+1}-\lambda_l}(t,\rho_2,\dots,\rho_{n_{l-1}},r_{n_{l-1}+1},\dots,r_{n_{l+1}})\tau^{\lambda_{l+1}-\lambda_l}+\cdots \nonumber \\ &\quad + \cdots \label{rhoexpand2} \\ &\quad +\beta_{i,\lambda_s-\lambda_l}(t,\rho_2,\dots,\rho_{n_{l-1}},r_{n_{l-1}+1},\dots,r_{n_s})\tau^{\lambda_s-\lambda_l}+\cdots, \quad n_{l-1}<i\le n_l, \nonumber \end{align} and the similar power $\tau$-series for the resonance parameters \begin{align} r_i &= \rho_i +b_{i,1}(t,\rho_2,\dots,\rho_{n_l})\tau+\cdots \nonumber \\ &\quad +b_{i,\lambda_{l+1}-\lambda_l}(t,\rho_2,\dots,\rho_{n_l},r_{n_l+1},\dots,r_{n_1})\tau^{\lambda_2-\lambda_1}+\cdots \nonumber \\ &\quad + \cdots \label{rexpand2} \\ &\quad +b_{i,\lambda_s-\lambda_l}(t,\rho_2,\dots,\rho_{n_l},r_{n_l+1},\dots,r_{n_s})\tau^{\lambda_s-\lambda_l}+\cdots, \quad n_{l-1}<i\le n_l. \nonumber \end{align} Then we substitute \eqref{rexpand2} into the Laurent $\tau$-series of the remaining $u_i$, $i>n_l$. We get the updated Laurent $\tau$-series, in which the resonance parameters $r_{n_{l-1}+1},\dots,r_{n_l}$ are replaced by the new variables $\rho_{n_{l-1}+1},\dots,\rho_{n_l}$. The updated resonance matrix $R^{(l)}$ is obtained from the row operation on $R^{(l-1)}$ that eliminates the terms below $A^{(l-1)}$.
We get the whole change of variable after exhausting all resonances.
\subsection{Regularity}
Now we argue that the Laurent series in the balance \eqref{balance0} are converted to power series for the new variables $\tau,\rho_2,\dots,\rho_n$.
We have the power series \eqref{tauexpand} for $\tau$, with analytic functions of $t,r_2,\dots,r_n$ as coefficients. By taking the Taylor expansions in $(t-t_0)$ of the coefficients, we get the power series \begin{align} \tau &= \alpha_1(t_0)(t-t_0) +\dots +\alpha_{\lambda_1-1}(t_0)(t-t_0)^{\lambda_1-1} \nonumber \\ &\quad +\alpha_{\lambda_1}(t_0,r_2,\dots,r_{n_1})(t-t_0)^{\lambda_1}+\cdots \nonumber \\ &\quad + \cdots \label{tauexpand3} \\ &\quad +\alpha_{\lambda_s}(t_0,r_2,\dots,r_{n_s})(t-t_0)^{\lambda_s}+\cdots, \nonumber \end{align} with analytic functions of $t_0,r_2,\dots,r_n$ as coefficients. We note that $\alpha_1(t_0)=\beta_1(t_0)=a_{1,0}(t_0)^{-\frac{1}{k_1}}$ is nonzero.
We have the power series \eqref{rhoexpand2} for $\rho_i$. If $n_{l-1}<i\le n_l$, then the coefficients are analytic functions of $t$, $\rho_2,\dots,\rho_{n_{l-1}}$, $r_{n_{l-1}+1},\dots,r_n$. We may substitute the power series \eqref{tauexpand3} for $\tau$ and successively substitute the power series for $\rho_i$ corresponding to smaller resonances to the power series for $\rho_i$ corresponding to bigger resonances. Then we get updated power series for $\rho_i$, with analytic functions of $t,r_2,\dots,r_n$ as coefficients. Finally, we may take the Taylor expansions in $(t-t_0)$ of these coefficients and get the power series \begin{align} \rho_i &= r_i +\alpha_{i,1}(t_0,r_2,\dots,r_{n_l})(t-t_0)+\cdots \nonumber \\ &\quad +\alpha_{i,\lambda_{l+1}-\lambda_l}(t_0,r_2,\dots,r_{n_{l+1}})(t-t_0)^{\lambda_{l+1}-\lambda_l}+\cdots \nonumber \\ &\quad + \cdots \label{rhoexpand3} \\ &\quad +\alpha_{i,\lambda_s-\lambda_l}(t_0,r_2,\dots,r_{n_s})(t-t_0)^{\lambda_s-\lambda_l}+\cdots, \quad n_{l-1}<i\le n_l, \nonumber \end{align}
We conclude that the Laurent series for $u_1,\dots,u_n$ are converted to power series for $\tau,\rho_2,\dots,\rho_n$. Moreover, the new variables satisfy the initial conditions \[ \tau(t_0)=0,\quad \tau'(t_0)=a_{1,0}(t_0)^{-\frac{1}{k_1}}\ne 0,\quad \rho(t_0)=r_i. \]
Next we argue that the new system of differential equations for $\tau,\rho_2,\dots,\rho_n$ is regular. The formulae \eqref{change1} and \eqref{change3} for the change of variable imply that the new system is \[ \tau'=g_1(t,\tau,\rho_2,\dots,\rho_n),\quad \rho_i'=g_i(t,\tau,\rho_2,\dots,\rho_n), \] with the right side \begin{equation}\label{rightside} g_i =\gamma_i(t,\tau,\rho_2,\dots,\rho_n) +\phi_{i,1}(t,\rho_2,\dots,\rho_n)\tau^{-1} +\dots +\phi_{i,N_i}(t,\rho_2,\dots,\rho_n)\tau^{-N_i}, \end{equation} where $\gamma_i$ and $\phi_{i,j}$ are analytic in their variables. Since the Laurent series \eqref{balance0} is a formal solution of the original system, the power series \eqref{tauexpand3} and \eqref{rhoexpand3} is a formal solution of the new system. Substituting the power series into the $i$-th equation, we find that the left side is a power series, while the right side is a Laurent series, with $\alpha_1^{-N_i}\phi_{i,N_i}(t_0,r_2,\dots,r_n)(t-t_0)^{-N_i}$ as the lowest order term. Therefore we conclude that \[ \phi_{i,N_i}(t_0,r_2,\dots,r_n)=0. \] Since this holds for all $t_0$ and resonance parameters $r_2,\dots,r_n$, and $\phi_{i,N_i}$ is analytic, we conclude that $\phi_{i,N_i}$ is constantly zero. This completes the proof of the regularity of the new system of equations.
\subsection{Triangular Change of Variable}
Our change of variable is not quite triangular. To get a triangular change of variable, we keep the first variable $\tau$ and modify the construction of new variables by introducing \begin{align} u_i &= \tilde{a}_{i,0}^{(l)}(t)\tau^{-k_i}+\dots +\tilde{a}_{i,\lambda_1-1}^{(l)}(t)\tau^{\lambda_1-1-k_i} \nonumber \\ &\quad +\tilde{a}_{i,\lambda_1}^{(l)}(t,\tilde{\rho}_2,\dots,\tilde{\rho}_{n_1})\tau^{\lambda_1-k_i}+\cdots \nonumber \\ &\quad +\cdots \label{triangle} \\ &\quad +\tilde{a}_{i,\lambda_{l-1}}^{(l)}(t,\tilde{\rho}_2,\dots,\tilde{\rho}_{n_{l-1}})\tau^{\lambda_{l-1}-k_i}+\cdots \nonumber \\ &\quad +\tilde{\rho}_i\tau^{\lambda_l-k_i}, \quad n_{l-1}<i\le n_l, \nonumber \end{align} instead of \eqref{change3}.
The variables $\tilde{\rho}_i$ are related to $\rho_i$ by \begin{align*} \tilde{\rho}_i &=a_{i,\lambda_l}^{(l)}(t,\rho_2,\dots,\rho_{n_{l-1}},\rho_{n_{l-1}+1},\dots,\rho_{n_l}) \\ &\quad +c_{i,\lambda_l}^{(l)}(t,\tau,\rho_2,\dots,\rho_{n_{l-1}}), \quad n_{l-1}<i\le n_l, \end{align*} where $c_{i,\lambda_l}^{(l)}$ is analytic in its variables. The invertibility of $A^{(l)}$ implies that the relation is invertible, and the inverse is also analytic. The variables $\tilde{\rho}_i$ satisfy the initial condition \[ \tilde{\rho}_{(n_{l-1},n_l]}(t_0)=A^{(l)} r_{(n_{l-1},n_l]}. \] Since $A^{(l)}$ are invertible, the initial values can be any number.
Finally, we remark that the proof for the regularity of the new system can be easily applied to general triangular change of variable \begin{align*} u_1 &= \tau^{-k}, \\ u_i &= a_i(t,\tau,\rho_2,\dots,\rho_{i-1})+b_i(t,\tau,\rho_2,\dots,\rho_{i-1})\rho_i,\quad 1<i\le n. \end{align*} The key here is that, because $f_i$ are polynomial in $u_i$, the right side of the new system is of the form \eqref{rightside}. Then the power series solutions for $\rho_i$ with all the possible numbers as the initial values imply that $g_i$ have to be also analytic in $\tau$.
The remark on the general triangular change of variable also shows that $f_i$ do not have to be polynomials. All we need is that the triangular change of variable converts the right side to be meromorphic in $\tau$. This often happens, for example, when $f_i$ are certain rational functions.
\section{Hamiltonian Structure} \label{hamilton}
Consider a Hamiltonian system \[ q'=\dfrac{\partial H}{\partial p},\quad p'=-\dfrac{\partial H}{\partial q}, \] where \[ q=(q_1,\dots,q_n),\quad p=(p_1,\dots,p_n), \] and $H=H(t,q,p)$ is analytic in $t$ and polynomial in $q$ and $p$. Consider a balance \begin{align*} q_i &= a_{i,0}(t_0)(t-t_0)^{-k_i}+\dots +a_{i,j}(t_0,r_2,\dots,r_{n_l})(t-t_0)^{j-l_i} +\cdots, \\ p_i &= b_{i,0}(t_0)(t-t_0)^{-l_i}+\dots +b_{i,j}(t_0,r_2,\dots,r_{n_l})(t-t_0)^{j-k_i} +\cdots, \end{align*} where the coefficients for $(t-t_0)^{j-k_i}$ and $(t-t_0)^{j-l_i}$ depend only on the resonance parameters $r_2,\dots,r_{n_l}$ with resonance $\le j$.
The following is the exact condition for Theorem B. The definition is justified near the end of Section \ref{foundation}.
\begin{definition*} A balance of the Hamiltonian system is {\em Hamiltonian principal}, if the resonance vectors form a simplectic basis of ${\mathbb R}^{2n}$, and there is $d$, such that $k_i+l_i=d-1$, and $\lambda_l+\mu_l=d-1$ for the resonances $\lambda_l$ and $\mu_l$ of simplectically conjugate resonance vectors. \end{definition*}
The symplectic property depends on the order of vectors in the basis, and this order is not the same as the order of the column vectors for the resonance matrix. Consider the usual resonance matrix \[ R=(R_0\; R_1\; \cdots \; R_{\bar{s}}), \] where the columns of $R_l$ have resonance $\lambda_l$. Since $\lambda_l$ is increasing in $l$, the condition $\lambda_l+\mu_l=d-1$ implies that a column vector in $R_l$ is simplectically conjugate to a column vector in $R_{\bar{s}-l}$. This is incompatible with our usual order in a simplectic basis $v_1,\dots,v_{2n}$, where $v_i$ is simplectically conjugate to $v_{n+i}$. So we need to reverse the last $n$ columns of $R$ and get the matrix \[ S=R\begin{pmatrix}I_n & O \\ O & T_n \end{pmatrix},\quad T_n=\begin{pmatrix} 0 & \cdots & 0 & 1 \\ 0 & \cdots & 1 & 0 \\ \vdots && \vdots & \vdots \\ 1 & \cdots & 0 & 0 \end{pmatrix}. \] Then the definition requires that $S$ is a symplectic matrix \[ S^TJS=J,\quad J=\begin{pmatrix}O & I_n \\ -I_n & O \end{pmatrix}. \]
Using the notation in the definition, we may denote the resonances by ($\bar{s}=2s+1$) \[ -1=\lambda_0<\lambda_1<\dots<\lambda_s\le \mu_s<\dots<\mu_1<\mu_0=d,\quad \mu_l=\lambda_{2s+1-l}. \] Since the resonance vectors form a simplectic basis, the columns of $R_l$ and $R_{2s+1-l}$ are simplectically conjugate and therefore the two matrices have the same number of columns. In case $\lambda_s=\mu_s$, the corresponding resonance vectors actually form the matrix $(R_s\;R_{s+1})$, where the columns of $R_s$ and $R_{s+1}$ are simplectically conjugate and therefore the two matrices have the same number of columns.
\subsection{Simplectic Resonance Variable}
The proof of Theorem A may be adapted to prove Theorem B. We only need to be more careful in keeping track of the simplectic structure.
Note that if the indicial normalization is assigned to $q_1$, then the last new variable, which corresponds to the largest resonance $\lambda_{2s+1}=\mu_0=d$, should be introduced for $p_1$. Therefore if we follow the construction in Section \ref{general}, we should rearrange the variables in the order $q_1,\dots,q_n$, $p_n,\dots,p_1$. The resonance matrix corresponding to this order is obtained by reversing the second half of the rows of $R$ \[ R'=\begin{pmatrix}I_n & O \\ O & T_n \end{pmatrix}R =\begin{pmatrix}I_n & O \\ O & T_n \end{pmatrix}S\begin{pmatrix}I_n & O \\ O & T_n \end{pmatrix}. \]
The resonance matrix is updated by row operations. This is equivalent to an $LU$ decomposition for $R'$, where $L$ is lower triangular and $U$ is upper triangular. In Section \ref{general}, the triangular shapes are actually blockwise. However, we get $LU$ decomposition only up to exchanging rows of the resonance matrix. For a Hamiltonian system, we need to exchange rows of $R'$ so that the simplectic structure is still preserved.
We divide the simplectic matrix into four $n\times n$ blocks \[ S=\begin{pmatrix} A & B \\ C & D \end{pmatrix},\quad A^TC=C^TA,\quad B^TD=D^TB,\quad A^TD-C^TB=I_n. \] The columns of $\begin{pmatrix} A \\ C \end{pmatrix}$ span a Lagrangian of the simplectic space ${\mathbb R}^{2n}$. It is well known in simplectic linear algebra that there is a subset $I\subset\{1,\dots,2n\}$, such that \begin{enumerate} \item The projection $\pi_I\colon {\mathbb R}^{2n}\to {\mathbb R}^I$ sends the columns of $\begin{pmatrix} A \\ C \end{pmatrix}$ to a basis of ${\mathbb R}^I$. \item For any $1\le i\le n$, one of $i,n+i$ is in $I$, and one is not. \end{enumerate} Now for any $1\le i\le n$, if $n+i\in I$, then we exchange the $i$-th row and the $(n+i)$-th row of $S$, and if $n+i\in I$, then we do nothing. After this exchange, $S$ is still a simplectic matrix, with $A$ invertible. Correspondingly, this means that $q_i$ and $p_i$ are exchanged. But the system of differential equations is still Hamiltonian under such exchange.
So without loss of generality, we may assume that $A$ is invertible. By further exchanging rows of $A$, we have $A=LU$. Exchanging rows of $A$ means exchanging $q_i$ with $q_j$. This should be balanced by exchanging $p_i$ with $p_j$ in order to keep the system Hamiltonian. Therefore if the $i$-th row and the $j$-th row of $A$ are exchanged, then the $i$-th row and the $j$-th row of $S$ should be exchanged, and the $(n+i)$-th row and the $(n+j)$-th row of $S$ should also be exchanged. Such exchange keeps $S$ to be simplectic.
After all the ``canonical exchanges'', which keep $S$ simplectic and the system Hamiltonian, we have a decomposition $A=LU$ that fits the construction in Section \ref{general}. Then the fact that $S$ is a simplectic matrix implies the $LU$ decomposition of the resonance matrix \[ R'= \begin{pmatrix} L & O \\ T_nCA^{-1}L & T_n(L^T)^{-1}T_n \end{pmatrix} \begin{pmatrix} U & L^{-1}BT_n \\ O & T_n(U^T)^{-1}T_n \end{pmatrix}. \] The construction of the change of variable in Section \ref{general} is guided by this $LU$ decomposition. The triangular version \eqref{change1} and \eqref{triangle} of the change of variable is \begin{align} q_1 &= \tilde{q}_1^{-k_1}, \nonumber \\ q_2 &= \alpha_{2,0}\tilde{q}_1^{-k_2}+\dots +\alpha_{2,\lambda_1-1}\tilde{q}_1^{\lambda_1-1-k_2} +\tilde{q}_2\tilde{q}_1^{\lambda_1-k_2}, \nonumber \\ &\vdots \nonumber \\ q_n &= \alpha_{n,0}\tilde{q}_1^{-k_n}+\dots +\alpha_{n,\lambda_s-1}\tilde{q}_1^{\lambda_s-1-k_n} +\tilde{q}_n\tilde{q}_1^{\lambda_s-k_n}, \nonumber \\ p_n &= \beta_{n,0}\tilde{q}_1^{-l_n}+\dots +\beta_{n,\mu_s-1-l_n}\tilde{q}_1^{\mu_s-1-l_n} +\tilde{p}_n\tilde{q}_1^{\mu_s-l_n}, \nonumber \\ &\vdots \label{triangle2} \\ p_2 &= \beta_{2,0}\tilde{q}_1^{-l_2}+\dots +\beta_{2,\mu_1-1-l_2}\tilde{q}_1^{\mu_1-1-l_2} +\tilde{p}_2\tilde{q}_1^{\mu_1-l_2}, \nonumber \\ p_1 &= \beta_{1,0}\tilde{q}_1^{-l_1}+\dots +\beta_{1,\mu_0-1-l_1}\tilde{q}_1^{\mu_1-1-l_1} -k_1^{-1}\tilde{p}_1\tilde{q}_1^{\mu_0-l_1}, \nonumber \end{align} where $\alpha_{i,j}$ is an analytic function of $t,\tilde{q}_2,\dots,\tilde{q}_{i-1}$, and $\beta_{i,j}$ is an analytic function of $t,\tilde{q}_2,\dots,\tilde{q}_n,\tilde{p}_n,\dots,\tilde{p}_{j+1}$. Note that a numerical factor $-k_1^{-1}$ is added in the last line.
\subsection{Canonical Change of Variable}
In this part, we argue that the change of variable is canonical \[ dq_1\wedge dp_1+\dots+dq_n\wedge dp_n =d\tilde{q}_1\wedge d\tilde{p}_1+\dots+d\tilde{q}_n\wedge d\tilde{p}_n. \]
From the shape of the triangular change of variable \eqref{triangle2}, we have \begin{align*} dq_1 &= -k_1\tilde{q}_1^{-k_1-1}d\tilde{q}_1 =-k_1\tilde{q}_1^{\lambda_0-k_1}d\tilde{q}_1, \\ dp_1 &= -k_1^{-1}\tilde{q}_1^{\mu_0-l_1}d\tilde{q}_1 +\text{ linear combination of }\tilde{q}_1^{<\mu_0-l_1}, \\ dq_i &= \tilde{q}_1^{\lambda_l-k_i}d\tilde{q}_i +\text{ linear combination of }\tilde{q}_1^{<\lambda_l-k_i}, \\ dp_i &= \tilde{q}_1^{\mu_l-l_i}d\tilde{p}_i +\text{ linear combination of }\tilde{q}_1^{<\mu_l-l_i}. \end{align*} By $(\lambda_l-k_i)+(\mu_l-l_i)=(\lambda_l+\mu_l)-(k_i+l_i) = (d-1)-(d-1)=0$, we have \begin{align} dq_1\wedge dp_1+\dots+dq_n\wedge dp_n &=d\tilde{q}_1\wedge d\tilde{p}_1+\dots+d\tilde{q}_n\wedge d\tilde{p}_n \nonumber \\ &\quad +\tilde{q}_1^{-1}\omega_1+\dots+\tilde{q}_1^{-N}\omega_N, \label{form1} \end{align} where $\omega_i$ is a $2$-form in $\tilde{q}_1,\dots,\tilde{q}_n$, $\tilde{p}_1,\dots,\tilde{p}_n$, with analytic functions of $t,\tilde{q}_2,\dots,\tilde{q}_n$, $\tilde{p}_1,\dots,\tilde{p}_n$ as coefficients.
Think of the Laurent series in the Hamiltonian principal balance as a transform between $q_1,\dots,q_n$, $p_1,\dots,p_n$ and $t_0,r_2,\dots,r_{2n}$. We substitute the transform into the left side of \eqref{form1} and get a $2$-form in $t_0,r_2,\dots,r_{2n}$, with functions of $t,t_0,r_2,\dots,r_{2n}$ as coefficients. Since the symplectic form is invariant under the Hamiltonian flow, the coefficients are actually independent of $t$.
On the other hand, the Hamiltonian principal balance is converted to power series for the new variables $\tilde{q}_1,\dots,\tilde{q}_n$, $\tilde{p}_1,\dots,\tilde{p}_n$. Think of these power series as a transform between $\tilde{q}_1,\dots,\tilde{q}_n$, $\tilde{p}_1,\dots,\tilde{p}_n$ and $t_0,r_2,\dots,r_{2n}$. We substitute the transform into the right side of \eqref{form1} and get a $2$-form in $t_0,r_2,\dots,r_{2n}$, with functions of $t,t_0,r_2,\dots,r_{2n}$ as coefficients. Since this $2$-form is the same as the substitution of the Laurent series into the left side, the coefficients of this $2$-form are independent of $t$.
Let \[ \omega_N=d\tilde{q}_1\wedge\rho+\phi, \] where $\rho$ and $\phi$ are $1$-form and $2$-form in $\tilde{q}_2,\dots,\tilde{q}_n$, $\tilde{p}_1,\dots,\tilde{p}_n$, with analytic functions of $t$, $\tilde{q}_2,\dots,\tilde{q}_n$, $\tilde{p}_1,\dots,\tilde{p}_n$ as coefficients. After substituting the power series into $\omega_N$, we get \[ \omega_N=dt_0\wedge \bar{\rho}+\bar{\phi}+(t-t_0)\bar{\psi}, \] where \begin{enumerate} \item $\bar{\rho}$ and $\bar{\psi}$ are $1$-form and $2$-form in $r_2,\dots,r_{2n}$, with analytic functions of $t,t_0,r_2,\dots,r_{2n}$ as coefficients. \item $\bar{\phi}$ is obtained as follows: Think of the initial data as a transform $(r_2,\dots,r_{2n})\mapsto (\tilde{q}_2(t_0),\dots,\tilde{q}_n(t_0), \tilde{p}_1(t_0),\dots,\tilde{p}_n(t_0))$ from the resonance parameters to new variables, with $t_0$ as a constant. Substituting the transform and $t=t_0$ into $\phi$ gives $\bar{\phi}$. \end{enumerate} The shape of $\omega_N$ implies that, after substituting the power series for $\tilde{q}_1,\dots,\tilde{q}_n$, $\tilde{p}_1,\dots,\tilde{p}_n$ into the right of \eqref{form1}, we get a $2$-form \[ \alpha_1^{-N}(t-t_0)^{-N}\bar{\phi}+\cdots. \] Here $\alpha_1$ is the leading coefficient in \eqref{tauexpand3} for $\tau=\tilde{q}_1$, and depends only on $t_0$ and resonance parameters $r_i$ of resonance $0$. Moreover, $\cdots$ consists of terms that are either of order $>-N$ in $(t-t_0)$ or have $dt_0$ as a factor. Since we have already argued that the coefficients of the $2$-form should not depend on $t$, and $\alpha_1^{-N}$ is independent of $t$, and $\bar{\phi}$ is independent of $t$ and has no $dt_0$ factor, we conclude that $\bar{\phi}=0$. Since $\bar{\phi}$ is obtained from $\phi$ by an invertible transform, this implies $\phi=0$.
So we have \[ \omega_N=d\tilde{q}_1\wedge\rho =dt_0\wedge \bar{\rho}+(t-t_0)\bar{\psi}. \] Since there is no more $\phi$, we now know that $\bar{\rho}$ is obtained from $\rho$ in the similar way as $\bar{\phi}$ being obtained from $\phi$ above, and then multiplied by $-\alpha_1$. Moreover, $\bar{\psi}$ has the same description as above. Therefore the right side of \eqref{form1} now becomes \[ \alpha_1^{-N}(t-t_0)^{-N}dt_0\wedge\bar{\rho}+\cdots, \] where $\cdots$ consists of terms that are of order $>-N$ in $(t-t_0)$. Since the coefficient of the $2$-form should not depend on $t$, and $\alpha_1$ and $\bar{\rho}$ are independent of $t$, we conclude that $\bar{\rho}=0$. Since $\bar{\rho}$ is obtained from $\rho$ by an invertible transform, this implies $\rho=0$.
We proved that $\omega_N=0$. Therefore there is no negative powers on the right of \eqref{form1}, so that the change of variable \eqref{triangle2} is canonical.
The argument was carried out for the triangular change of variable. It also works for the blockwise triangular change of variable, making use of the simplectic property of the resonance matrix. Alternatively, we may verify (again by the simplectic property of the resonance matrix) that the transform between the new variables in the blockwise triangular change of variable and the new variables in the blockwise triangular change of variable is canonical.
\subsection{Change of the Hamiltonian Function}
In this part, we study the Hamiltonian function for the new Hamiltonian system.
Suppose the original Hamiltonian system is autonomous. Since our change of variable is canonical, the new system for the new variables is also an autonomous Hamiltonian system, and the new Hamiltonian function is simply obtained by applying the change of variable to the original Hamiltonian function.
For an autonomous (not necessarily Hamiltonian) system, the coefficients in a balance do not depend on $t$. Then in the construction in Section \ref{general}, all the coefficients in the Laurent and power series are independent of $t$. Therefore the regularizing change of variable we get at the end is independent of $t$.
For the autonomous Hamiltonian system, applying the triangular change of variable \eqref{triangle2} to the original Hamiltonian function, we find the new Hamiltonian function to be of the form \[ \tilde{H}(\tilde{q},\tilde{p})=h_0+h_1\tilde{q}_1^{-1}+\dots+h_N\tilde{q}_1^{-N}, \] where $h_0$ is a polynomial of $\tilde{q}_1,\dots,\tilde{q}_n$, $\tilde{p}_1,\dots,\tilde{p}_n$, and $h_1,\dots,h_N$ are polynomials of $\tilde{q}_2,\dots,\tilde{q}_n$, $\tilde{p}_1,\dots,\tilde{p}_n$. Since the Hamiltonian function is a first integral of the Hamiltonian system, substituting the power series solution for $\tilde{q}_1,\dots,\tilde{q}_n$, $\tilde{p}_1,\dots,\tilde{p}_n$ into $\tilde{H}$ should give us an expression independent of $t$. The most singular term after the substitution is \[ h_N(\tilde{q}_2(t_0),\dots,\tilde{q}_n(t_0),\tilde{p}_1(t_0),\dots,\tilde{p}_n(t_0))\alpha_1^{-N}(t-t_0)^{-N}. \] Since $h_N$ and $\alpha_1$ are independent of $t$, we conclude that \[ h_N(\tilde{q}_2(t_0),\dots,\tilde{q}_n(t_0),\tilde{p}_1(t_0),\dots,\tilde{p}_n(t_0))=0. \] Since the initial value map $(r_2,\dots,r_{2n})\mapsto (\tilde{q}_2(t_0),\dots,\tilde{q}_n(t_0),\tilde{p}_1(t_0),\dots,\tilde{p}_n(t_0))$ is invertible for triangular change of variable, we conclude that $h_N$ is a constantly zero function. This proves that $\tilde{H}(\tilde{q},\tilde{p})=h_0$ is a polynomial.
Now we turn to non-autonomous Hamiltonian system. In this case, the function $\tilde{H}(t,\tilde{q},\tilde{p})$ obtained by applying the time dependent change of variable to the original Hamiltonian function $H(t,q,p)$ is no longer gives the Hamiltonian function of the new system. In fact, $\tilde{H}$ may be singular in $\tilde{q}_1$. In what follows, we argue that it is the ``regular part'' $h_0(t,\tilde{q},\tilde{p})$ that becomes the Hamiltonian function of the new system.
We leave Hamiltonian systems for a moment and consider a general system of differential equations \[ u'=f(t,u), \quad u=(u_1,\dots,u_n). \] For a change of variable $u=\varphi(t,\rho)$, we have \[ u'=\dfrac{\partial \varphi}{\partial t}+\dfrac{\partial \varphi}{\partial \rho}\rho'. \] So the new system is \[ \rho'=J_{\varphi}^{-1}f(t,\varphi(t,\rho))-J_{\varphi}^{-1}\dfrac{\partial \varphi}{\partial t},\quad J_{\varphi}=\dfrac{\partial \varphi}{\partial \rho}. \]
For the triangular change of variable \eqref{change1} and \eqref{triangle} ($\tilde{\rho}=(\tau,\tilde{\rho}_2,\dots,\tilde{\rho}_n)$ takes the place of $\rho$), we have \[ J_{\varphi} =\begin{pmatrix} -k_1\tau^{-1-k_1} & 0 & \cdots & 0 \\ \{\tau^{<\lambda_1-k_2}\} & \tau^{\lambda_1-k_2} & \cdots & 0 \\ \vdots & \vdots && \vdots \\ \{\tau^{<\lambda_s-k_n}\} & \{\tau^{<\lambda_s-k_n}\}& \cdots & \tau^{\lambda_s-k_n} \end{pmatrix}, \] where $\{\tau^{<j}\}$ means a (finite) linear combination of $\tau^{j'}$ with $j'<j$, with analytic functions of $t,\tilde{\rho}_2,\dots,\tilde{\rho}_n$ as coefficients. Moreover, $f(t,\varphi(t,\tilde{\rho}))$ and $\dfrac{\partial \varphi}{\partial t}$ are also (finite) linear combinations of $\tau^j$, with analytic functions of $t,\tilde{\rho}_2,\dots,\tilde{\rho}_n$ as coefficients. Then both $J_{\varphi}^{-1}f(t,\varphi(t,\tilde{\rho}))$ and $J_{\varphi}^{-1}\dfrac{\partial \varphi}{\partial t}$ are linear combinations of $\tau^j$ of the same type.
Let $J_{\varphi}^{-1}\dfrac{\partial \varphi}{\partial t}=(g_1,\dots,g_n)^T$. Then $\dfrac{\partial \varphi}{\partial t}=J_{\varphi}(g_1,\dots,g_n)^T$, and by the formulae \eqref{change1} and \eqref{triangle}, we get \begin{align*} 0 & = -k_1\tau^{-1-k_1}g_1, \\ \{\tau^{<\lambda_1-k_2}\} & = \{\tau^{<\lambda_1-k_2}\}g_1+\tau^{\lambda_1-k_2}g_2 ,\\ & \vdots \\ \{\tau^{<\lambda_s-k_n}\} & = \{\tau^{<\lambda_s-k_n}\}g_1+\{\tau^{<\lambda_s-k_n}\}g_2+\dots+\tau^{\lambda_s-k_n}g_n. \end{align*} By induction, it is easy to see that $g_1=0$, $g_2=\{\tau^{<0}\}$, $\dots$, $g_n=\{\tau^{<0}\}$. Therefore we conclude that $J_{\varphi}^{-1}\dfrac{\partial \varphi}{\partial t}$ is a linear combinations of $\tau^j$, $j<0$, with analytic functions of $t,\tilde{\rho}_2,\dots,\tilde{\rho}_n$ as coefficients. Since the differential equation \[ \tilde{\rho}'=J_{\varphi}^{-1}f(t,\varphi(t,\tilde{\rho}))-J_{\varphi}^{-1}\dfrac{\partial \varphi}{\partial t} \] obtained after the triangular change of variable \eqref{change1} and \eqref{triangle} is regular, we conclude that $J_{\varphi}^{-1}\dfrac{\partial \varphi}{\partial t}$ is the sum of terms in $J_{\varphi}^{-1}f(t,\varphi(t,\rho))$ with $\tau^j$, $j<0$. In other words, the new equation is \[ \tilde{\rho}'=[J_{\varphi}^{-1}f(t,\varphi(t,\rho))]_\text{regular}, \] with the right side obtained by dropping terms with negative power of $\tau$.
Back to the non-autonomous Hamiltonian system, since the triangular change of variable \eqref{triangle2} is canonical, we have \[ J_{\varphi}^{-1}\begin{pmatrix}\dfrac{\partial H}{\partial p} \\ -\dfrac{\partial H}{\partial q}\end{pmatrix} =\begin{pmatrix}\dfrac{\partial \tilde{H}}{\partial \tilde{p}} \\ -\dfrac{\partial \tilde{H}}{\partial \tilde{q}}\end{pmatrix}. \] Here $\tilde{H}(t,\tilde{q},\tilde{p})$ is obtained by applying the triangular change of variable \eqref{triangle2} to $H(t,q,p)$ and is therefore a linear combination of $\tilde{q}_1^j$, with analytic functions of $t,\tilde{q}_2,\dots,\tilde{q}_n$, $\tilde{p}_1,\dots,\tilde{p}_n$ as coefficients. We separate the parts of $\tilde{H}$ with $\tilde{q}_1^j$, $j\ge 0$ and the rest part with $\tilde{q}_1^j$, $j<0$, \[ \tilde{H}=[\tilde{H}]_\text{regular}+[\tilde{H}]_\text{singular}. \] Then it is easy to see that \[ \dfrac{\partial \tilde{H}}{\partial \tilde{p}} =\left[\dfrac{\partial \tilde{H}}{\partial \tilde{p}}\right]_\text{regular} +\left[\dfrac{\partial \tilde{H}}{\partial \tilde{p}}\right]_\text{singular}, \] with \[ \left[\dfrac{\partial \tilde{H}}{\partial \tilde{p}}\right]_\text{regular} =\dfrac{\partial [\tilde{H}]_\text{regular}}{\partial \tilde{p}},\quad \left[\dfrac{\partial \tilde{H}}{\partial \tilde{p}}\right]_\text{singular} =\dfrac{\partial [\tilde{H}]_\text{singular}}{\partial \tilde{p}}, \] and the same happens to $\dfrac{\partial \tilde{H}}{\partial \tilde{q}}$. Thus we conclude that the new system of equations is \[ \tilde{q}'=\dfrac{\partial [\tilde{H}]_\text{regular}}{\partial \tilde{p}},\quad \tilde{p}'=-\dfrac{\partial [\tilde{H}]_\text{regular}}{\partial \tilde{q}}. \] This shows that $[\tilde{H}]_\text{regular}$ is the Hamiltonian function of the new system.
\section{A Mathematical Foundation for the Painlev\'e Test} \label{foundation}
Let $f_i(t,u_1,\dots,u_n)$ be analytic in $t$ and polynomial in $u_1,\dots,u_n$. We consider a system of ordinary differential equations \begin{equation}\label{ode} u_i'=f_i(t,u_1,\dots,u_n),\quad 1\le i\le n. \end{equation} The Painlev\'e test attempts to find all Laurent series solutions (or balances) with movable singularity $t_0$ \begin{equation}\label{balance} u_i=c_i(t-t_0)^{-k_i}+a_{i,1}(t-t_0)^{1-k_i}+\dots+a_{i,j}(t-t_0)^{j-k_i}+\cdots. \end{equation} Here we use $c_i$ instead of $a_{i,0}$ to highlight the different role from the subsequent coefficients.
\subsection{Dominant Balance}
The first step in the Painlev\'e test is to determine the leading exponents $k_i$ and the leading coefficients $c_i$ of potential balances. There are usually several possible combinations of leading exponents and leading coefficients, which give several possible balances.
\begin{definition*} The leading exponents $k_1,\dots,k_n$ of a balance is {\em Fuchsian}, if the $k_*$-weighted degree of $f_i$ is $\leq k_i+1$. \end{definition*}
The {\em $k_*$-weighted degree} of a polynomial in $u_1,\dots,u_n$ is obtained by taking the degree of $u_i$ to be $k_i$.
If all the leading coefficients $c_i\ne 0$, then the choice of leading exponents is {\em natural}. For the natural leading exponents $k_*$, denote the dominant part of $f_i$ \[ D_i(t, u_1, \dots, u_n) =\sum \text{terms in $f_i$ with highest $k_*$-weighted degree}. \] The following is a simple criterion for the natural leading exponents to be Fuchsian.
\begin{proposition}\label{prop1} If $D_i(t_0,c_1,\dots,c_n)\ne 0$ and $k_ic_i\neq 0$ for each $i$, then the natural leading exponents is Fuchsian. \end{proposition}
Let $d_i$ be the $k_*$-weighted degree of $f_i$. Then after substituting the balance \eqref{balance}, the lowest order term in $u_i'$ is $-k_ic_i(t-t_0)^{-k_i-1}$, and the lowest order term in $f_i(t,u_1,\dots, u_n)$ is $D_i(t_0,c_1,\dots,c_n)(t-t_0)^{-d_i}$. Under the assumption of the proposition, both have nonzero coefficients. Therefore they are the ``real'' lowest order terms, and we conclude that \[ d_i=k_i+1,\quad D_i(t_0,c_1,\dots,c_n)=-k_ic_i. \] The first equality shows that the Fuchsian condition is satisfied.
If the natural exponents is not Fuchsian, then we need to choose somewhat unnatural leading exponents in order to satisfy the Fuchsian condition. For example, the Gelfand-Dikii hierarchy with 2 degrees of freedom is a Hamiltonian system with \[ H = -q_1 p_2^2 - 2 p_1p_2 + 3 q_1^2q_2 - q_1^4 - q_2^2. \] One of the principal balances of the system is (we abbreviate $(t-t_0)$ as $t$ because the system is autonomous) \begin{align*} q_1 & = t^{-2}+\frac{1}{3}r_2-\frac{1}{3}r_2^2t^2-\frac{2}{3}r_3t^3-\frac{10}{27}r_2^3t^4-\frac{1}{3}r_2r_3t^5-\frac{1}{3}r_4t^6+ \dots, \\ q_2 & =r_2t^{-2}-\frac{2}{3}r_2^2-r_3t-\frac{1}{3}r_2^3t^2+\left(-\frac{11}{54}r_2^4+\frac{3}{2}r_4\right)t^4+\cdots, \\ p_1 & =-t^{-5}+\frac{2}{3}r_2t^{-3}+\frac{1}{6}r_3-\frac{4}{27}r_2^3t-\frac{5}{6}r_2r_3t^2+\left(\frac{22}{81}r_2^4-\frac{11}{3}r_4\right)t^3+\cdots, \\ p_2 & = t^{-3}+\frac{1}{3}r_2^2t+r_3t^2+\frac{20}{27}r_2^3t^3+\frac{5}{6}r_2r_3t^4+r_4t^5+ \dots. \end{align*} The natural leading exponents $2,2,5,3$ is not Fuchsian. By taking the leading exponent of \[ q_2=0t^{-4}+0t^{-3}+r_2t^{-2}-\frac{2}{3}r_2^2-r_3t-\frac{1}{3}r_2^3t^2+\left(-\frac{11}{54}r_2^4+\frac{3}{2}r_4\right)t^4+\cdots, \] to be $4$, we get leading exponents $2,4,5,3$ satisfying the Fuchsian condition. The corresponding leading coefficients are $1,0,-1,1$. The H\'enon-Heiles system in \cite{hy3} is another example.
When the Fuchsian condition is satisfied, we denote the dominant part of $f_i$ \[ f_i^D = \sum\text{terms in $f_i$ with $k_*$-weighted degree $k_i+1$}. \] Under the condition of Proposition \ref{prop1}, we have $f_i^D=D_i$. By substituting the balance \eqref{balance} into the system \eqref{ode} and comparing the coefficients of $(t-t_0)^{-k_i-1}$, we get \begin{equation}\label{dominant} f_i^D(t_0,c_1,\dots,c_n)=-k_ic_i. \end{equation} This is the (formal) {\em dominant balance} for the system \eqref{ode} at the (movable) singularity $t_0$.
\subsection{Kowalevskian Matrix}
Given the leading exponents and the leading coefficients, the next step of the Painlev\'e test is to find the subsequent coefficients in the balance. For $j>0$, by substituting the balance \eqref{balance} into the differential equations and comparing the coefficients of $(t-t_0)^{j-k_i-1}$, we get \begin{align*} (j-k_i)a_{i,j} &= \frac{\partial f_i^D}{\partial u_1}(t_0,c_1,\dots,c_n)a_{i,j}+\dots+\frac{\partial f_i^D}{\partial u_n}(t_0,c_1,\dots,c_n)a_{n,j} \\ & \quad +\text{ terms involving }c_*,a_{*,1},\dots,a_{*,j-1}. \end{align*} Define the {\em Kowalevskian matrix} \begin{align*}\label{kmatrix} K & = \begin{pmatrix} \dfrac{\partial f_1^D}{\partial u_1}+k_1 & \dfrac{\partial f_1^D}{\partial u_2} & \cdots & \dfrac{\partial f_1^D}{\partial u_n} \\ \dfrac{\partial f_2^D}{\partial u_1} & \dfrac{\partial f_2^D}{\partial u_2}+k_2 & \cdots & \dfrac{\partial f_2^D}{\partial u_n} \\ \vdots & \vdots && \vdots \\ \dfrac{\partial f_n^D}{\partial u_1} & \dfrac{\partial f_n^D}{\partial u_2} & \cdots & \dfrac{\partial f_n^D}{\partial u_n}+k_n \end{pmatrix} \\ & = \dfrac{\partial (f_1^D,\dots,f_n^D)}{\partial (u_1,\dots,u_n)}(t_0,c_1,\dots,c_n) +\begin{pmatrix} k_1 && \\ & \ddots & \\ && k_n \end{pmatrix}. \end{align*} Then we have a recursive relation \begin{equation}\label{recursion} (K-jI)(a_{1,j},\dots,a_{n,j})^T = \text{ terms involving }c_*,a_{*,1},\dots,a_{*,j-1}. \end{equation}
A {\em formal balance} is the Laurent series solution with leading coefficients obtained by solving \eqref{dominant} and with subsequent coefficients obtained by solving \eqref{recursion}.
We emphasize that the Fuchsian condition is needed for the definition of the Kowalevskian matrix. Ercolani and Siggia \cite{es} made a similar observation regarding the balance for the Gelfand-Dikii hierarchy. They suggested that one should not blindly proceed to use the linearized equation as the constraint on the resonance vectors and specifically pointed out that the leading exponent for $q_2$ should be changed from $2$ to $4$ in order to define the Kowalevskian matrix, which is \[ K=\begin{pmatrix} 2 & 0 & 0 & -2 \\ -2 & 4 & -2 & -2 \\ 12 & -6 & 5 & 2 \\ -6 & 2 & 0 & 3 \end{pmatrix}. \]
The solutions of \eqref{dominant} and \eqref{recursion} may not be unique. This leads to free parameters in the balance. Since \eqref{recursion} is a linear equation, we can choose the free parameters such that if a parameter does not appear in the leading coefficients, then the parameter first appears linearly. Although this property is not needed for our main theorems, it is a consequence of our set up.
There are three kinds of free parameters. The first kind is the free location $t_0$ of the singularity. The derivative of the balance \eqref{balance} in $t_0$ is \[ \frac{\partial u_i}{\partial t_0} = -k_ic_i(t-t_0)^{-k_i-1}+\left[(1-k_i)a_{1,i}+\dfrac{\partial c_i}{\partial t_0}\right](t-t_0)^{-k_i}+\cdots. \] Therefore the variation of the solution due to the parameter $t_0$ is characterized by the basic resonance vector $(-k_1c_1,\dots,-k_nc_n)$.
Take the derivative of the system \eqref{ode} in $t$, we get \[ u_i'' =\frac{\partial f_i}{\partial t} +\frac{\partial f_i}{\partial u_1}u_1'+\dots +\frac{\partial f_i}{\partial u_n}u_n'. \] Substituting in the balance \eqref{balance} and comparing the coefficients of $(t-t_0)^{-k_i-2}$, we get \[ (-k_i)(-k_i-1)c_i=\frac{\partial f_i^D}{\partial u_1}(t_0,c_1,\dots,c_n)(-k_1)c_1+\dots+\frac{\partial f_i^D}{\partial u_n}(t_0,c_1,\dots,c_n)(-k_n)c_n. \] The equalities can be rewritten as $(K+I)(-k_1c_1,\dots,-k_nc_n)=0$.
\begin{proposition}\label{resonance-1} If the leading coefficients of a balance \eqref{balance} is Fuchsian, then the basic resonance vector $(-k_1c_1,\dots,-k_nc_n)$ is an eigenvector of the Kowalevskian matrix with eigenvalue $-1$. \end{proposition}
The second kind of free parameters appear in the leading coefficients, and parameterize the subvariety given by \eqref{dominant}. The derivative of the equation \eqref{dominant} with respect to one such free parameter $r$ is \[ -k_i\frac{\partial c_i}{\partial r} =\frac{\partial f_i^D}{\partial u_1}\frac{\partial c_1}{\partial r}+\dots +\frac{\partial f_i^D}{\partial u_n}\frac{\partial c_n}{\partial r}. \] The equalities can be rewritten as $K\dfrac{\partial (c_1,\dots,c_n)^T}{\partial r}=0$.
\begin{proposition}\label{resonance0} If the leading coefficients of a balance \eqref{balance} is Fuchsian, then the tangent vectors to the subvariety of allowable leading coefficients are the eigenvectors of the Kowalevskian matrix with eigenvalue $0$. \end{proposition}
The third kind of free parameters appear in the subsequent coefficients. These are obtained from solving the recursive relation \eqref{recursion}. One has to worry about the compatibility between $K-jI$ and the right side of the recursive relation, which affects the existence of solutions. Leaving aside the existence issue for a moment, we may conclude the following from the recursive relation.
\begin{proposition}\label{resonancej} If the leading coefficients of a balance \eqref{balance} is Fuchsian, then for $j>0$, the $j$-th coefficient vectors form an affine space parallel to the eigenspace of $K$ with eigenvalue $j$. \end{proposition}
If $j$ is not an eigenvalue of $K$, then the $j$-th coefficients are uniquely determined by \eqref{recursion}. If $j$ is indeed an eigenvalue, then the $j$-th coefficients form a vector \[ (a_{1,j},\dots,a_{n,j})^T=\hat{a}_j+r_1R_1+\dots+r_mR_m, \] where \begin{enumerate} \item $\hat{a}_j$ depends only on $t_0,c_*,a_{*,1},\dots,a_{*,j-1}$. \item $R_1,\dots,R_m$ form a basis of the eigenspace of $K$ with eigenvalue $j$ and therefore depend only on $t_0$ and $c_*$. \item $r_1,\dots,r_m$ can be any numbers. \end{enumerate}
\begin{definition*} A Fuchsian leading exponents is {\em principal}, if for all solutions of \eqref{dominant}, the following are satisfied. \begin{enumerate} \item The Kowalevskian matrix $K$ is diagonalizable, with the eigenspace of $-1$ being of dimension $1$, and all other eigenvalues being non-negative integers. \item The recursive relation \eqref{recursion} is always consistent. \end{enumerate} \end{definition*}
Given principal Fuchsian leading exponents, the corresponding principal balance \eqref{balance} is obtained by solving equations \eqref{dominant} and \eqref{recursion}. And the Painlev\'e test is passed.
The first condition implies that the eigenvalues of $K$ and their multiplicities are constants. By applying the implicit function theorem to \eqref{dominant}, we also know that the collection of leading coefficients form a submanifold with the eigenspace $\ker K$ of eigenvalue $0$ as the tangent space. Therefore the definition means exactly that the total number of free parameters, including $t_0$, is $n$. Moreover, the eigenvectors of $K$ form the resonance matrix.
For the balance of the Gelfand-Dikii hierarchy, the Kowalevskian matrix has eigenvalues $-1,2,5,8$, with the corresponding resonance matrix \[ R=\begin{pmatrix} 2 & 1 & -4 & -2 \\ 0 & 3 & -6 & 9 \\ -5 & 2 & 1 & -22 \\ 3 & 0 & 6 & 6 \end{pmatrix}. \]
\subsection{Painlev\'e Test for Hamiltonian System}
Consider a Hamiltonian system given by a Hamiltonian function $H(t,q,p)$ that is analytic in $t$ and polynomial in $q$ and $p$. For a balance, we denote the leading exponents of $q_i$ and $p_i$ by $k_i$ and $l_i$.
\begin{definition} A Hamiltonian system is {\em almost weighted homogeneous} relative to the leading exponents $k_1,\dots,k_n$, $l_1,\dots,l_n$ if $k_i+l_i=d-1$, where $d$ is the leading exponents weighted degree of $H$. \end{definition}
The almost weighted homogeneous condition implies that the weighted degree of $\dfrac{\partial H}{\partial p_i}$ is $\le d-k_i=l_i+1$ and the similar inequality for $\dfrac{\partial H}{\partial q_i}$. Therefore the Fuchsian condition is satisfied. Let $H^D$ be the sum of those terms in $H$ with highest weighted degree and apply the same notation to other polynomial functions. Then $\left[\dfrac{\partial H}{\partial p_i}\right]^D=\dfrac{\partial H^D}{\partial p_i}$, and the Kowalevskian matrix is \begin{align*} K &= \begin{pmatrix}
\dfrac{\partial^2 H^D}{\partial q_i\partial p_i}
& \dfrac{\partial^2 H^D}{\partial p_i^2} \\
-\dfrac{\partial^2 H^D}{\partial q_i^2}
& -\dfrac{\partial^2 H^D}{\partial p_i\partial q_i}
\end{pmatrix} + \begin{pmatrix}
k_1 &&& \\ & \ddots && \\ && l_1 & \\ &&& \ddots
\end{pmatrix} \\ &= \begin{pmatrix}
O & I \\ -I & O
\end{pmatrix}
\begin{pmatrix}
\dfrac{\partial^2 H^D}{\partial q_i^2}
& \dfrac{\partial^2 H^D}{\partial p_i\partial q_i} \\
\dfrac{\partial^2 H^D}{\partial q_i\partial p_i}
& \dfrac{\partial^2 H^D}{\partial p_i^2}
\end{pmatrix} + \begin{pmatrix}
k_1 &&& \\ & \ddots && \\ && l_1 & \\ &&& \ddots
\end{pmatrix} \\ &=J{\mathcal H}+\Gamma. \end{align*} where ${\mathcal H}$ is the Hessian of $H^D$. The special shape of the Kowalevskian matrix implies certain symplectic structures among the resonance vectors.
\begin{proposition} Suppose the Hamiltonian function is almost weighted homogeneous relative to the leading exponents of a balance. Suppose $v$ and $w$ are eigenvectors of the Kowalevskian matrix with eigenvalues $\lambda$ and $\mu$. If $\lambda+\mu\ne d-1$, then $\langle v,Jw \rangle=0$. \end{proposition}
The condition $k_i+l_i=d-1$ means exactly $\Gamma J+J\Gamma=(d-1)J$. Then the lemma follows from \begin{align*} (\lambda +\mu )\langle v,Jw \rangle & = \langle Kv,Jw \rangle + \langle v,JKw \rangle \\ & = -\langle J(J{\cal H}+\Gamma)v,w \rangle
+\langle v,J(J{\cal H}+\Gamma)w \rangle \\ & = \langle ({\cal H}-J\Gamma)v,w \rangle
+\langle v,(-{\cal H}+J\Gamma)w \rangle \\ & = \langle v,\Gamma J w \rangle
+\langle v,J\Gamma w \rangle \\ & = (d-1)\langle v,Jw \rangle. \end{align*}
The proposition implies that, by rescaling the vectors if necessary, we can find a simplectic basis of resonance vectors. For example, by rescaling and exchanging the last two columns of the resonance matrix for the balance of the Gelfand-Dikii hierarchy, we get a symplectic matrix \[ R\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{9} \\ 0 & 0 & -\frac{1}{81} & 0 \end{pmatrix} =\begin{pmatrix} 2 & \frac{1}{3} & \frac{2}{81} & -\frac{4}{9} \\ 0 & 1 & -\frac{1}{9} & -\frac{2}{3} \\ -5 & \frac{2}{3} & \frac{22}{81} & \frac{1}{9} \\ 3 & 0 & -\frac{2}{27} & \frac{2}{3} \end{pmatrix}. \] The last two columns are exchanged because the proposition tells us that in the original resonance matrix, up to rescaling, the first and fourth columns are simplectically dual to each other, and the second and third columns are simplectically dual to each other.
\end{document} | arXiv |
Abstract: We establish interior Schauder estimates for kinetic equations with integro-differential diffusion. We study equations of the form $f_t + v \cdot \nabla_x f = \mathcal L_v f + c$, where $\mathcal L_v$ is an integro-differential diffusion operator of order $2s$ acting in the $v$-variable. Under suitable ellipticity and Hölder continuity conditions on the kernel of $\mathcal L_v$, we obtain an a priori estimate for $f$ in a properly scaled Hölder space. | CommonCrawl |
\begin{document}
\alphafootnotes \author[J. D. Biggins]{J. D. Biggins\footnotemark } \chapter{Branching out} \footnotetext{Department of Probability \& Statistics, Hicks Building,
University of Sheffield, Sheffield S3 7RH; [email protected]} \arabicfootnotes \contributor{John D. Biggins
\affiliation{University of Sheffield}} \renewcommand\thesection{\arabic{section}} \numberwithin{equation}{section} \renewcommand\theequation{\thesection.\arabic{equation}} \numberwithin{figure}{section} \renewcommand\thefigure{\thesection.\arabic{figure}}
\begin{abstract} Results on the behaviour of the rightmost particle in the $n$th generation in the branching random walk are reviewed and the phenomenon of anomalous spreading speeds, noticed recently in related deterministic models, is considered. The relationship between such results and certain coupled reaction-diffusion equations is indicated. \end{abstract}
\subparagraph{AMS subject classification (MSC2010)}60J80
\section{Introduction}
I arrived at the University of Oxford in the autumn of 1973 for postgraduate study. My intention at that point was to work in Statistics\index{statistics}. The first year of study was a mixture of taught courses and designated reading on three areas (Statistics, Probability, and Functional Analysis, in my case) in the ratio 2:1:1 and a dissertation on the main area.
As part of the Probability\index{probability|(} component, I attended a graduate course that was an exposition, by its author\index{Hammersley, J. M.|(}, of the material in \citet{MR0370721}, which had grown out of his contribution to the discussion of John's\index{Kingman, J. F. C.!influence|(} invited paper on subadditive\index{subadditivity} ergodic theory \citep{MR0356192}. A key point of Hammersley's contribution was that the postulates used did not cover the time to the first birth in the $n$th generation in a Bellman--Harris process\index{Bellman, R. E.!Bellman--Harris process}.\footnote{Subsequently, \citet{MR806224} established the theorem under weaker postulates.}\index{Liggett, T. M.|pagenote} \citet{MR0370721} showed, among other things, that these quantities did indeed exhibit the anticipated limit behaviour in probability. I decided not to be examined on this course, which was I believe a wise decision, but I was intrigued by the material. That interest turned out to be critical a few months later.
By the end of the academic year I had concluded that I wanted to pursue research in Probability\index{probability|)} rather than Statistics and asked to have John as supervisor. He agreed. Some time later we met and he asked me whether I had any particular interests already---I mentioned Hammersley's lectures. When I met him he was in the middle of preparing something (which I could see, but not read upside down). He had what seemed to be a pile of written pages, a part written page and a pile of blank paper. There was nothing else on the desk. A few days later a photocopy of
a handwritten version of \citet{MR0400438}\index{branching random walk (BRW)|(}\index{branching process!age-dependent branching process|(}, essentially identical to the published version, appeared in my pigeon-hole with the annotation ``the multi\-type version is an obvious problem''\index{branching process!multitype age-dependent branching process}---I am sure this document was what he was writing when I saw him. (Like all reminiscences, this what I recall, but it is not necessarily what happened.) This set me going.
For the next two years, it was a privilege to have John as my thesis supervisor. He supplied exactly what I needed at the time: an initial sense of direction, a strong encouragement to independence, an occasional nudge on the tiller about what did or did not seem tractable, the discipline of explaining orally what I had done, and a ready source on what was known, and where to look for it. However, though important, none of these get to the heart of the matter, which is that I am particularly grateful to have had that period of contact with, and opportunity to appreciate first-hand, such a gifted mathematician\index{Kingman, J. F. C.!influence|)}.
\citet{MR0400438} considered the problem Hammersley\index{Hammersley, J. M.|)} had raised in its own right, rather than as an example of, and adjunct to, the general theory of subadditive processes.
Here, I will say something about some recent significant developments on the first-birth problem\undex{branching random walk (BRW)!first-birth problem}. I will also go back to my beginnings, by outlining something new about the multi\-type version that concerns the phenomenon of `anomalous spreading speeds', which was noted in a related context in \citet{MR2322849}\index{Alsmeyer, G.}\index{Iksanov, A.}. Certain martingales\index{martingale} were deployed in \citet{MR0400438}. These have been a fruitful topic in their own right, and have probably received more attention since then than the first-birth problem itself (see \citet{MR2471666} for a recent nice contribution on when these martingales are integrable). However, those developments will be ignored here.
\section{The basic model} The branching random walk (BRW) starts with a single particle located at the origin. This particle produces daughter particles, which are scattered in $\mathbb{R}$, to give the first generation. These first generation particles produce their own daughter particles similarly to give the sec\-ond generation, and so on. Formally, each family is described by the collection of points in $\mathbb{R}$ giving the positions of the daughters relative to the parent. Multiple points are allowed, so that in a family there may be several daughter particles born in the same place. As usual in branching processes, the $n$th generation particles reproduce independently of each other. The process is assumed supercritical\index{branching process!supercritical branching process}, so that the expected family size exceeds one (but need not be finite---indeed even the family size itself need not be finite). Let $\mathbf {P}$ and $\mathbf {E}$ be the probability and expectation for this process and let $Z$ be the generic reproduction process of points in $\mathbb{R}$.
Thus,
$\mathbf {E} Z$ is the intensity measure\index{intensity measure|(} of $Z$ and $Z(\mathbb{R})$ is the family size, which will also be written as $N$. The assumption that the process is supercritical becomes
that $\mathbf {E} Z(\mathbb{R})=\mathbf {E} N>1$. To avoid burdening the description with qualifications about the survival set, let $\mathbf {P} (N=0)=0$, so that the process survives almost surely.
The model includes several others. One is when each daughter receives an independent displacement, another is when all daughters receive the same displacement, with the distribution of the displacement being independent of family size in both cases. These will be called the BRW with \textit{independent} and \textit{common} displacements respectively. Obviously, in both of these any line of descent follows a trajectory of a random walk. (It is possible to consider an intermediate case, where displacements have these properties conditional on family size, but that is not often done.) \label{brw} Since family size and displacements are independent, these two processes can be coupled in a way that shows that results for one will readily yield results for the other. In a common displacement BRW imagine each particle occupying the (common) position of its family. Then the process becomes an independent displacement BRW, with a random origin given by the displacement of the first family, and its $n$th generation occupies the same positions as the $(n+1)$th generation in the original common displacement BRW. Really this just treats each family as a single particle.
In a different direction, the points of $Z$ can be confined to $(0,\infty)$ and interpreted as the mother's age at the birth of that daughter: the framework adopted in
\citet{MR0400438}\index{branching process!age-dependent branching process|)}\index{Kingman, J. F. C.}. Then the process is the general branching process\index{branching process!general branching process} associated with the names of Ryan\index{Ryan Jr., T.}, Crump\index{Crump, K. S.}, Mode\index{Mode, C. J.} and Jagers\index{Jagers, P.}.
Finally, when all daughters receive the same positive displacement with a distribution independent of family size the process is the \index{Bellman, R. E.!Bellman--Harris process}Bellman--Harris branching process: the framework adopted in \citet{MR0370721}\index{Hammersley, J. M.|(}.
There are other `traditions', which consider the BRW but introduce and describe it rather differently and usually with other problems in focus. There is a long tradition phrased in terms of `\index{multiplicative cascade}multiplicative cascades' (see for example \citet{MR1741808}\index{Liu, Q.} and the references there) and a rather shorter one phrased in terms of `weighted branching'\index{branching process!weighted branching process} (see for example \citet{MR2199054} and the references there). The model has arisen in one form or another in a variety of areas. The most obvious is as a model for a population spreading through an homogeneous habitat. It has also arisen in modelling random fractals\index{fractal}
\citep{MR1785625} commonly in the language of multiplicative cascades, in the theoretical study of algorithms\index{algorithm} \citep{MR1140708}, in a problem in \Index{group theory} \citep{MR2114819} and as an ersatz for both lattice-based models of spin glasses\index{spin glass} in physics \citep{MR1601733} and a \Index{number theory} problem \citep{MR1143401}.
\section{Spreading out: old results}\index{spreading out|(}
Let $Z^{(n)}$ be the positions occupied by the $n$th generation and
$\rBo{n}$
its rightmost point,
so that
\[ \rBo{n}= \sup\{z: z \mbox{~a point of~} Z^{(n)}\}.
\] One can equally well consider the leftmost particle, and the earliest studies did that. Reflection of the whole process around the origin shows the two are equivalent: all discussion here will be expressed in terms of the rightmost particle. The first result, stated in a moment, concerns $\rBo{n}/n$ converging to a constant, $\Gamma$, which can reasonably be interpreted as the speed of spread in the positive direction.
\label{spreading out}
A critical role in the theory is played by the Laplace transform\index{Laplace, P.-S.!Laplace transform} of the intensity measure\index{intensity measure|)} $\mathbf {E} Z$: let $\kappa(\phi)= \log \int e^{\phi z} \mathbf {E} Z(dz)$ for $\phi \geq 0$ and $\kappa(\phi)=\infty$ for $\phi<0$. It is easy to see that when this is finite for some $\phi>0$ the intensity measures of $Z$ and $Z^{(n)} $ are finite on bounded sets, and decay exponentially in their right tail. The behaviour of the leftmost particle is governed by the behaviour of the transform for negative values of its argument. The definition of $\kappa$ discards these, which simplifies later formulations by automatically keeping attention on the right tail. In order to give one of the key formulae for $\Gamma$ and for later explanation,
let $ \Fd{\kappa}$ be the Fenchel dual\index{Fenchel, M. W.!Fenchel dual} of $\kappa$,
which is the convex\index{convexity} function given by
\begin{equation}\label{Fenchel dual}
\Fd{\kappa}(a)=\sup_\theta \{\theta a-\kappa(\theta)\}.
\end{equation} This is sufficient notation to give the first result.
\begin{theorem}\label{first theorem}
When there is a $\phi>0$ such that
\begin{equation}\label{m good} \kappa(\phi) < \infty,
\end{equation}
there is a constant $\Gamma$ such that
\begin{equation}\label{limit}
\frac{\rBo{n}}{n}\rightarrow \Gamma\mbox{~~~a.s}.
\end{equation} and $ \Gamma= \sup\{a: \Fd{\kappa}(a)<0\}=\inf\{\kappa(\theta)/\theta: \theta\}$. \end{theorem} This result was proved for the common BRW with only negative displacements with convergence in probability in
\citet[Theorem 2]{MR0370721}\index{Hammersley, J. M.|)}. It was proved in \citet[Theorem 5]{MR0400438}\index{Kingman, J. F. C.} for $Z$ concentrated on $(-\infty,0)$ and with $0<\kappa(\phi)<\infty$ instead of (\ref{m good}). The result stated above is contained in \citet[Theorem 4]{MR0420890}, which covers the irreducible multi\-type\index{branching random walk (BRW)!multitype BRW} case also, of which more later. The second of the formulae for $\Gamma$ is certainly well-known but cannot be found in the papers mentioned---I am not sure where it first occurs. It is not hard to establish from the first one using the definition and properties of $\Fd{\kappa}$.
The developments described here draw on features of transform theory, to give properties of $\kappa$, and of convexity theory, to give properties of $\Fd{\kappa}$ and the speed $\Gamma$. There are many presentations of, and notations for, these, tailored to the particular problem under consideration. In this review, results will simply be asserted. The first of these provides a context for the next theorem and aids interpretation of $\sup\{a: \Fd{\kappa}(a)<0\}$ in the previous one. It is that when $\kappa$ is finite somewhere on $(0,\infty)$, $\Fd{\kappa}$ is an increasing, convex\index{convexity} function, which is continuous from the left, with minimum value $-\kappa(0)=-\log \mathbf {E} N$, which is less than zero.
A slight change in focus derives Theorem \ref{first theorem} from the asymptotics of
the numbers of particles in suitable half-infinite intervals. As part of the derivation of this the asymptotics of the expected numbers are obtained. Specifically, it is shown that when (\ref{m good}) holds \[ n^{-1} \log \left(\mathbf {E} Z^{(n)}[na,\infty) \right) \rightarrow - \Fd{\kappa}(a) \] (except, possibly, at one $a$). The trivial observation that when the expectation of integer-valued variables decays geometrically the variables themselves must ultimately be zero implies that $\log
Z^{(n)}[na,\infty)$ is ultimately infinite on $\{a: \Fd{\kappa}(a)>0\}$.
This motivates introducing a notation for sweeping positive values of $\Fd{\kappa}$, and later other functions, to infinity and so we let \begin{equation}\label{sweep}
\sweep{f}(a)=\left\{\begin{array}{ll}f(a) &\mbox{when~}f(a)\leq 0\\ \infty&\mbox{when~} f(a)> 0 \end{array}\right.
\end{equation} and $\swFd{\kappa}=\sweep{(\Fd{\kappa})}$. The next result can be construed as saying that in crude asymptotic terms this is the only way actual numbers differ from their expectation. \begin{theorem}\label{second theorem} When\/ \eqref{m good} holds, \begin{equation}\label{describe numbers} \frac{1}{n} \log \left(Z^{(n)}[na,\infty) \right) \rightarrow - \swFd{\kappa}(a) \mbox{~~~a.s.}, \end{equation} for all $a \neq \Gamma$. \end{theorem} \noindent From this result, which is \citet[Theorem 2]{MR0464415}, and the properties of $\swFd{\kappa}$, Theorem \ref{first theorem}
follows directly.
A closely related continuous-time model arises when the temporal development is a Markov branching process\index{Markov, A. A.!Markov branching process} (Bellman--Harris\index{Bellman, R. E.!Bellman--Harris process} with exponential lifetimes) or even a Yule process\index{Yule, G. U.!Yule process} (binary splitting too) and movement is Brown\-ian, giving binary branching Brown\-ian motion\index{Brown, R.!branching Brownian motion}. The process starts with a single particle at the origin, which then moves with a Brown\-ian motion with variance parameter $V$. This particle splits in two at rate $\lambda$, and the two particles continue, independently, in the same way from the splitting point. (Any discrete skeleton\index{skeleton!discrete skeleton} of this process is a branching random walk.)
Now, let $\rBo{t}$ be the position of the rightmost particle at time $t$. Then $ u(x,t)=\mathbf {P}(\rBo{t} \leq x)$ satisfies the \index{Fisher, R. A.!Fisher/KPP equation}(Fisher/Kolmogorov--Petrovski--Piscounov) equation \begin{equation}\label{F-KPP} \frac{\partial u}{\partial t}=V\frac{1}{2}\frac{\partial ^{2}u}{\partial x^{2} } - \lambda u(1-u), \end{equation} which is easy to see informally by conditioning on what happens in $[0, \delta t]$. The deep studies of \citet{MR0494541,MR705746}\index{Bramson, M. D.} show, among other things, that (with $V=\lambda=1$) $\rBo{t}$ converges in distribution when centred on its \Index{median} and that median is (to $O(1)$)
\[ \sqrt{2}t-\frac{1}{\sqrt{2}}\left(\frac{3}{2} \log t\right) , \] which implies that $\Gamma=\sqrt{2}$ here. For the skeleton at integer times, $\kappa(\theta)=\theta^2/2+1$ for $\theta\geq 0$, and using Theorem \ref{first theorem} on this confirms that $\Gamma=\sqrt{2}$. Furthermore, for later reference, note that $\theta \Gamma -\kappa(\theta)=0$ when $\theta=\sqrt{2}$. \label{bbm}
Theorem \ref{first theorem} is for discrete time, counted by generation. There are corresponding results for continuous time, where the reproduction is now governed by a random collection of points in time and space ($\mathbb{R}^+ \! \times \mathbb{R} $). The first component gives the mother's age at the birth of this daughter and the second that daughter's position relative to her mother. Then the development in time of the process is that of a \index{branching process!general branching process}general branching process rather than the Galton--Watson\index{Galton, F.!Galton--Watson process} development that underpins Theorem \ref{first theorem}. This extension is discussed in \citet{MR1384364} and \citet{MR1601689}. In it particles may also move during their lifetime and then branching Brown\-ian motion becomes a (very) special case. Furthermore, there are also natural versions of Theorems \ref{first theorem} and \ref{second theorem} when particle positions are in $\mathbb{R}^d$ rather than $\mathbb{R}$---see \citet[\S4.2]{MR1384364} and references there. \section{Spreading out: first refinements}
Obviously rate-of-convergence\index{rate of convergence} questions follow on from (\ref{limit}). An aside in \citet[p33]{MR0433619} noted that, typically, $\rBo{n}-n \Gamma$ goes to $-\infty$. The following result on this is from \citet[Theorem 3]{MR1629030}, and much of it is contained also in \citet[Lemma 7.2]{MR1618888}\index{Liu, Q.}.
When $\mathbf {P}(Z(\Gamma,\infty)>0)>0$, so displacements greater than $\Gamma$ are possible, and (\ref{m good}) holds, there is a finite $\vartheta>0$ with $\vartheta \Gamma-\kappa(\vartheta) = 0$. Thus the condition here, which will recur in later theorems, is not restrictive. \begin{theorem} \label{theorem to infinity} If there is a finite $\vartheta>0$ with $\vartheta \Gamma-\kappa(\vartheta) = 0$, then \begin{equation}\label{to infinity} \rBo{n}-n \Gamma \rightarrow -\infty\mbox{~~~a.s.,} \end{equation} and the condition is also necessary when $\mathbf {P}(Z(\Gamma,\infty)>0)>0$. \end{theorem}
The theorem leaves some loose ends when $\mathbf {P}(Z(\Gamma,\infty)=0)=1$. Then $\rBo{n}-n \Gamma$ is a decreasing sequence, and so it does have a limit, but whether (\ref{to infinity}) holds or not is really the explosion (i.e.\ regularity) problem for the general branching process\undex{branching process!general}: whether, with a point $z$ from $Z$ corresponding to a birth time of $\Gamma-z$, there can be an infinite number of births in a finite time. This is known to be complex---see \citet{MR0359040}\index{Grey, D. R.} for example.
In the simpler cases it is properties of $Z(\{\Gamma\})$, the number of daughters displaced by exactly $\Gamma$, that matters.
If $Z(\{\Gamma\})$ is the family size of a surviving branching process (so either $\mathbf {E} Z(\{\Gamma\}) >1$ or $\mathbf {P}(Z(\{\Gamma\})=1)=1$) it is easy to show that $(\rBo{n}-n \Gamma)$ has a finite limit---so (\ref{to infinity}) fails---using embedded surviving processes resulting from focusing on daughters displaced by $\Gamma$:
see \citet[Proposition II.5.2]{my-thesis} or \citet[Theorem
1]{MR1133373}\index{Dekking, F. M.|(}\index{Host, B.|(}.
In a similar vein, with extra conditions, \citet[Theorem 4]{Addarioberryreed}\index{Addario-Berry, L.}\index{Reed, B.|(} show $\mathbf {E}(\rBo{n}-n \Gamma) $ is bounded.
Suppose now that (\ref{m good}) holds. When $\mathbf {P}(Z(a,\infty)=0)=1$, simple properties of transforms imply that $\theta a-\kappa(\theta) \uparrow - \log \mathbf {E} Z(\{a\}) $ as $\theta \uparrow \infty$. Then, when $\mathbf {E} Z(\{a\}) <1$ a little \Index{convexity} theory shows that $\Gamma<a$ and that there is a finite $\vartheta$ with $\vartheta \Gamma-\kappa(\vartheta) = 0$, so that Theorem \ref{theorem to infinity} applies. This leaves the case where (\ref{m good}) holds, $\mathbf {P}(Z(\Gamma,\infty)=0)=1$ and $\mathbf {E} Z(\{\Gamma\}) =1$ but $\mathbf {P}( Z(\{\Gamma\})=1)<1$, which is sometimes called, misleadingly in my opinion, the \textit{critical} branching random walk\undex{branching random walk (BRW)!critical BRW} because the process of daughters displaced by exactly $\Gamma$ from their parent forms a critical Galton--Watson process\index{Galton, F.!Galton--Watson process}. For this case,
\citet[Theorem 1]{MR510529}\index{Bramson, M. D.|(} and \citet[\S9]{MR1133373} show that (\ref{to infinity}) holds under extra conditions including that displacements lie in a lattice, and that the convergence is at rate $\log \log n$.
\citet[Theorem 2]{MR510529} also gives conditions under which (\ref{to infinity}) fails.
\section{Spreading out: recent refinements}
The challenge to derive analogues for the branching random walk
of the fine results for branching Brown\-ian motion\index{Brown, R.!branching Brownian motion} has been open for a long time.
Progress was made in \citet{MR1325045}\index{McDiarmid, C.|(} and, very recently, a nice result has been given in \citet[Theorem 1.2]{hu-shi}\index{HuY@Hu, Y.}\index{ShiZ@Shi, Z.}, under reasonably mild conditions. Here is its translation into the current notation. It shows that the numerical identifications noted in the branching Brown\-ian motion case in \S\ref{bbm} are general. \begin{theorem}\label{h-s} Suppose that there is a $\vartheta>0$ with $\vartheta \Gamma-\kappa(\vartheta) = 0$, and that, for some $\epsilon>0$, $ \mathbf {E}(N^{1+\epsilon})<\infty$, $\kappa(\vartheta+\epsilon)<\infty$ and $\int e^{-\epsilon z}\mathbf {E} Z(dz)<\infty$. Then \[ -\frac{3}{2}=\liminf_n \frac{\vartheta(\rBo{n}-n \Gamma)}{ \log n} <\limsup_n \frac{\vartheta(\rBo{n}-n \Gamma)}{ \log n}=-\frac{1}{2}\mbox{~~~a.s.} \] and \[
\frac{\vartheta(\rBo{n}-n \Gamma)}{ \log n} \rightarrow -\frac{3}{2}\mbox{~~in probability}. \] \end{theorem}
Good progress has also been made on the tightness\index{tight|(} of the distributions of $\rBo{n}$ when centred suitably. Here is a recent result from \citet[Theorem 1.1]{Bramsonzeitouni}\index{Bramson, M. D.|)}\index{Zeitouni, O.}. \begin{theorem}Suppose the BRW has independent or common displacements according to the random variable $X$. Suppose also that for some $\epsilon>0$, $ \mathbf {E}(N^{1+\epsilon})<\infty$ and that for some $\psi>0$ and $y_0>0$ \begin{equation}\label{BZ} \mathbf {P}(X>x+y)\leq e^{-\psi y}\mathbf {P}(X>x) ~~~~\forall x>0, y>y_0. \end{equation} Then the distributions of $\{\rBo{n}\}$ are tight when centred on their medians. \end{theorem} \noindent It is worth noting that (\ref{BZ}) ensures that (\ref{m good}) holds for all $\phi \in [0,\psi)$.
There are other results too---in particular, \citet[Theorem 1]{MR1325045}\index{McDiarmid, C.|)} and \citet[\S3]{MR1133373}\index{Dekking, F. M.|)}\index{Host, B.|)} both give tightness results for the (general) BRW, but with $Z$ concentrated on a half-line. Though rather old for this section, \citet[Theorem 2]{MR1133373} is worth recording here: the authors assume the BRW is concentrated on a lattice, but they do not use that in the proof of this theorem. To state it, let $\widetilde{D}$ be the second largest point in $Z$ when $N\geq 2$ and the only point otherwise. \begin{theorem} If the points of $Z$ are confined to $(-\infty,0]$ and $\mathbf {E} \widetilde{D}$ is finite, then $\mathbf {E} \rBo{n}$ is finite and the distributions of $\{\rBo{n}\}$ are tight when centred on their expectations.
\end{theorem}\noindent The condition that $\mathbf {E} \widetilde{D}$ is finite holds when $\int e^{\phi z}\mathbf {E} Z(dz)$ is finite in a neighbourhood of the origin, which is contained within the conditions in Theorem \ref{h-s}. In another recent study \citet[Theorem 3]{Addarioberryreed}\index{Addario-Berry, L.}\index{Reed, B.|)} give the following result, which gives tightness and also estimates the centring. \begin{theorem}\label{ab-r} Suppose that there is a $\vartheta>0$ with $\vartheta \Gamma-\kappa(\vartheta) = 0$, and that, for some $\epsilon>0$, $\kappa(\vartheta+\epsilon)<\infty$ and $\int e^{-\epsilon z}\mathbf {E} Z(dz)<\infty$.
Suppose also that the BRW has a finite maximum family size and independent displacements. Then \[ \mathbf {E}\rBo{n}=n \Gamma -\frac{3}{2 \vartheta}\log n+O(1), \] and there are $C>0$ and $\delta>0$ such that \[
\mathbf {P}\left(|\rBo{n}-\mathbf {E}\rBo{n}|>x\right) \leq Ce^{-\delta x} ~~~\forall x. \]
\end{theorem}\noindent The conditions in the first sentence here have been stated in a way that keeps them close to those in Theorem \ref{h-s} rather than specialising them for independent displacements. Now, moving from tightness\index{tight|)} to convergence in distribution---which cannot be expected to hold without a non-lattice assumption---the following result, which has quite restrictive conditions, is taken from
\citet[Theorem 1]{MR1765165}\index{Bachmann, M.}. \begin{theorem}\label{Bach} Suppose that the BRW has $ \mathbf {E} N<\infty$ and independent displacements according to a random variable with density function $f$ where $- \log f$ is convex. Then the variables $\rBo{n}$ converge in distribution when centred on medians. \end{theorem}\index{convexity}\index{median} It is not hard to use the coupling mentioned in \S\ref{brw} to see that Theorems \ref{ab-r} and \ref{Bach} imply that these two results also hold for common displacements. \section{Deterministic theory}\label{deter}\index{deterministic modelling} There is another, deterministic, stream of work concerned with modelling the spatial \index{spread of population}spread of populations
in a homogeneous habitat, and closely linked to the study of reaction-diffusion equations\index{reaction-diffusion equation|(} like (\ref{F-KPP}). The main presentation is
\citet{MR653463}\index{Weinberger, H. F.|(}, with a formulation that has much in common with that adopted in \citet{MR0370721}\index{Hammersley, J. M.}. Here the description of the framework is pared-down.
This sketch draws heavily on \citet{MR1943224}, specialised to the homogeneous (i.e.\ aperiodic) case and one spatial dimension. The aim is to say enough to make certain connections with the BRW.
Let $u^{(n)}(x)$
be the density of the population (or the \index{gene!frequency}gene frequency, in an alternative interpretation) at time $n$ and position $ x \in \mathbb{R}$. This is a discrete-time theory, so there is an \index{operator!updating operator}updating operator $Q$ satisfying $u^{(n+1)}=Q(u^{(n)})$. More formally, let ${\cal F}$ be the non-negative continuous functions on $\mathbb{R}$ bounded by $\beta$. Then $Q$ maps ${\cal F}$ into itself and $u^{(n)}=Q^{(n)}(u^{(0)})$, where $u^{(0)}$ is the initial density and $Q^{(n)}$ is the $n$th iterate of $Q$. The operator is to satisfy the following restrictions. The constant functions at $0$ and at $\beta$ are both fixed points of $Q$. For any function $u \in {\cal F}$ that is not zero everywhere, $Q^{(n)}(u) \rightarrow \beta$, and $Q(\alpha) \geq \alpha$ for non-zero constant functions in ${\cal F}$. (Of course, without the spatial component, this is all reminiscent of the basic properties of the generating function of the family-size.) The operator $Q$ is \index{order preserving@order-preserving}order-preserving, in that if $u \leq v$ then $Q(u)\leq Q(v)$, so increasing the population anywhere never has deleterious effects\index{deleterious effect} in the future; it is also \Index{translation-invariant}, because the habitat is homogeneous, and suitably continuous. Finally, every sequence $u_m \in {\cal F}$ contains a subsequence $u_{m(i)}$ such that $Q(u_{m(i)})$ converges uniformly on compacts. Such a $Q$ can be obtained by taking the development of a
\index{reaction-diffusion equation|)}reaction-diffusion equation for a time $\tau$. Then $Q^{(n)}$ gives the development to time $n \tau$, and the results for this discrete formulation transfer to such equations.
Specialising \citet[Theorem 2.1]{MR1943224}, there is a spreading speed $\Gamma$ in the following sense. If $u^{(0)}(x)=0$ for $x\geq L$ and $u^{(0)}(x)\geq \delta>0$ for all $x\leq K$, then for any $\epsilon >0$ \begin{equation}\label{spreading}
\sup_{x\geq n(\Gamma+\epsilon)} |u^{(n)}(x)| \rightarrow 0 \mbox{~~and~~}
\sup_{x\leq n(\Gamma-\epsilon)} |u^{(n)}(x)-\beta|\rightarrow 0. \end{equation}
In some cases the spreading speed can be computed through linearisation\break---see \citet[Corollary 2.1]{MR1943224}\index{Weinberger, H. F.|)} and \citet[Corollary to Theorem 3.5]{Lui1989269}\index{Lui, R.}---in that the speed is the same as that obtained by replacing $Q$ by a truncation of its linearisation at the zero function. So
$Q(u)=Mu$ for small $u$ and $Q(u)$ is replaced by $\min\{\omega, Mu\}$, where $\omega$ is a constant, positive function with $M \omega > \omega$. The linear functional $Mu(y)$ must be represented as an integral with respect to some measure, and so, using the translation invariance of $M$, there is a measure $\mu$ such that \begin{equation}\label{mu} Mu(y)=\int u(y-z)\,\mu(dz). \end{equation} Let $\tilde{\kappa}(\theta)=\log \int e^{\theta z} \mu(dz)$. Then the results show that the speed $\Gamma$ in (\ref{spreading}) is given by \begin{equation}\label{W-Gamma} \Gamma = \inf_{\theta>0} \frac{\tilde{\kappa}(\theta) }{\theta}. \end{equation} Formally, this is one of the formulae for the speed in Theorem \ref{first theorem}. In fact, the two frameworks can be linked, as indicated next.
In the BRW, suppose the generic reproduction process $Z$ has points $\{z_i\}$. Define $Q$ by \[ Q\left(u(x)\right)=1-\mathbf {E} \left[1-\prod_i u (x-z_i)\right]. \] This has the general form described above with $\beta=1$. On taking $u^{(0)}(x)=\mathbf {P} (\rBo{0}>x)$ (i.e.\ Heaviside\index{Heaviside, O.} initial data) it is easily established by induction that $ u^{(n)}(x)=\mathbf {P} (\rBo{n}>x)$. This is in essence the same as the observation that the distribution of the rightmost particle in branching Brown\-ian motion\index{Brown, R.!branching Brownian motion} satisfies the differential equation (\ref{F-KPP}). The idea is explored in the spatial spread of the `deterministic simple epidemic'\index{epidemic!deterministic simple epidemic} in \citet{Mollison1993147}\index{Mollison, D.}\index{Daniels, H. E.}, a continuous-time model which, like branching Brown\-ian motion, has BRW as its discrete skeleton\index{skeleton!discrete skeleton}. Now Theorem \ref{first theorem} implies that (\ref{spreading}) holds, and that, for $Q$ obtained in this way, the speed is indeed given by the (truncated) linear approximation. The other theorems about $\rBo{n}$ also translate into results about such $Q$. For example, Theorem \ref{Bach} gives conditions for $u^{(n)}$ when centred suitably to converge to a fixed (\Index{travelling wave}) profile.
\section{The multi\-type case}
\label{multitype}\index{branching random walk (BRW)!multitype BRW|(}
Particles now have types\index{type} drawn from a finite set, $\cal S$, and their reproduction is defined by random points in $\cal S \times \mathbb{R}$. The distribution of these points depends on the parent's type. The first component gives the daughter's type and the second component gives the daughter's position, relative to the parent's. As previously, $Z$ is the generic reproduction process, but now let $Z_\sigma$ be the points (in $\mathbb{R}$) corresponding to
those of type $\sigma$; $Z^{(n)}$ and $Z^{n}_{\sigma}$ are defined similarly. Let $\mathbf {P}_{\!\nu}$ and $\mathbf {E}_\nu$ be the probability and expectation associated with reproduction from an initial ancestor with type $\nu \in \cal S$. Let
$\rB{\sigma}{n}$
be the rightmost particle of type $\sigma$ in the $n$th generation,
and let $\rBo{n}$ be the rightmost of these, which is consistent with the one-type notation.
The type space\index{type!type space} can be classified, using the relationship `can have a descendant of this type', or, equivalently, using the non-negative expected family-size matrix\undex{branching random walk (BRW)!expected family-size matrix}, $\mathbf {E}_\nu Z_\sigma (\mathbb{R})$.
Two types are in the same class when each can have a descendant of the other type in some generation. When there is a single class the family-size matrix is \textit{irreducible}\index{branching random walk (BRW)!irreducible BRW} and the process is similarly described. When the expected family-size matrix is \textit{aperiodic} (i.e.\ primitive)
the process is also called aperiodic\undex{branching random walk (BRW)!aperiodic BRW|(}, and it is supercritical\undex{branching random walk (BRW)!supercritical BRW|(} when this matrix has Perron--Frobenius\index{Perron, O.!Perron--Frobenius eigenvalue} (i.e.\ non-negative and of maximum modulus) eigenvalue
greater than one. Again, to avoid qualifications about the survival set, assume extinction\index{extinction} is impossible from the starting type used.
For $\theta \geq 0$, let $\exp(\kappa(\theta))$ be the Perron--Frobenius eigenvalue of the matrix of transforms $\int e^{\theta z} \mathbf {E}_\nu Z_\sigma(dz)$, and let $\kappa(\theta)=\infty$ for $\theta<0$. If there is just one type, this definition agrees with that of $\kappa$ at the start of \S\ref{spreading out}. The following result, which is \citet[Theorem 4]{MR0420890}, has been mentioned already.
\begin{theorem}\label{multitype first theorem} Theorem\/ $\ref{first theorem}$ holds for any initial type in a supercritical irreducible BRW. \end{theorem} \noindent The simplest multi\-type version of Theorem \ref{second theorem} is the following, which is proved in \cite{JDB-anom}. When $\sigma=\nu$ it is a special case of results indicated in \citet[\S4.1]{MR1601689}.
\begin{theorem} \label{supercrit} For a supercritical aperiodic BRW for which\/ \eqref{m good} holds,
\begin{equation}\label{exp growth ub 2} \frac{1}{n}\log \left(Z^{(n)}_\sigma[na, \infty) \right) \rightarrow - \swFd{\kappa}(a) \mbox{~~~a.s.-}\mathbf {P}_{\!\nu}
\end{equation}for $a \neq \sup\{a:\Fd{\kappa}(a)<0\}=\Gamma$, and
\[
\frac{\rB{\sigma}{n}}{n} \rightarrow \Gamma \mbox{~~~a.s.-}\mathbf {P}_{\!\nu}.
\]
\end{theorem}\undex{branching random walk (BRW)!aperiodic BRW|)}\undex{branching random walk (BRW)!supercritical BRW|)} \noindent Again there is a deterministic\index{deterministic modelling} theory, following the pattern described in \S\ref{deter} and discussed in \citet{Lui1989269, Lui1989297}\index{Lui, R.}, which can be related to Theorem \ref{multitype first theorem}. Recent developments in that area raise some interesting questions that are the subject of the next two sections.
\section{Anomalous spreading}\index{anomalous spreading|(}
In the multi\-type version of the deterministic context of \S\ref{deter},
recent papers \citep{MR1930974, MR2322849,MR1930975,Li200582} have considered what happens when the type space\index{type!type space} is reducible.
Rather than set out the framework in
its generality, the simplest possible case, the reducible\index{branching random walk (BRW)!reducible BRW|(}\index{branching random walk (BRW)!two-type BRW|(} two-type\index{type!two-type} case, will be considered here, for the principal issue can be illustrated through it. The two types will be $\nu$ and $\eta$. Now, the vector-valued non-negative function $u^{(n)}$ gives the \Index{population density} of two species---the two types, $\nu$ and $\eta$---at $x \in \mathbb{R}$ at time $n$, and $Q$ models growth, interaction and migration\undex{branching random walk (BRW)!growth, interaction and migration}, as the populations develop in discrete time. The programme is the same as that indicated in \S\ref{deter}, that is to investigate the existence of spreading speeds
\index{spreading speed|(}and when these speeds can be obtained from the truncated linear approximation\undex{branching random walk (BRW)!truncated linear approximation}.
In this case the approximating \index{operator!linear operator}linear operator, generalising that given in (\ref{mu}), is \begin{align*} (Mu(y))_\eta&=\int u_\eta (y-z) \,\mu_{\eta \eta}(dz),\\ (Mu(y))_\nu&=\int u_\nu (y-z)\,\mu_{\nu \nu} (dz) +\int u_\eta (y-z)\,\mu_{\nu \eta } (dz). \end{align*} Simplifying even further, assume there is no spatial spread associated with the `interaction' term here, so that $\int u_\eta (y-z)\,\mu_{\nu\eta} (dz)= c u_\eta (y)$ for some $c>0$. The absence of $\mu_{ \eta \nu}$ in the first of these makes the linear approximation reducible. The first equation is really just for the type $\eta$ and so will have the speed that corresponds to $\mu_{\eta \eta}$, given through its transform by (\ref{W-Gamma}), and written $\Gamma_\eta$. In the second, on ignoring the interaction term, it is plausible that the speed must be at least that of type $\nu$ alone, which corresponds to $\mu_{\nu \nu}$ and is written $\Gamma_\nu$. However, it can also have
the speed of $u_\eta$ from the `interaction' term. It is claimed in \citet[Lemma 2.3]{MR1930974}\index{Weinberger, H. F.|(}\index{Lewis, M. A.|(}\index{LiB@Li, B.|(} that when $Q$ is replaced by the approximating operator $\min\{\omega, Mu\}$ this does behave as just outlined, with the corresponding formulae for the speeds: thus that of $\eta$ is $\Gamma_\eta$ and that for $\nu$ is $\max \{\Gamma_\eta,\Gamma_\nu\}$. However, in \citet{MR2322849} a flaw in the argument is noted, and an example is given where the speed of $\nu$ in the truncated linear approximation can be faster than this, the anomalous spreading speed of their title, though the actual speed is not identified. The relevance of the phenomenon to a biological example is explored in
\citet[\S5]{MR2322849}.
As in \S\ref{deter}, the BRW provides some particular examples of $Q$ that fall within the general scope of the deterministic theory. Specifically, suppose the generic reproduction process $Z$ has points $\{\sigma_i,z_i\}\in \cal S \times \mathbb{R}$. Now let $Q$, which operates on vector functions indexed by the type space\index{type!type space} $\cal S$, be defined by \[ Q\left(u(x)\right)_{\nu}=1-\mathbf {E}_\nu \left[1-\prod_i u_{\sigma_i}(x-z_i)\right]. \] Then, just as in the one-type case, when $u^{(0)}_{\nu}(x)=\mathbf {P}_{\!\nu} (\rBo{0}>x)$ induction establishes that $u^{(n)}_{\nu}(x)=\mathbf {P}_{\!\nu} \left(\rBo{n}>x\right)$. It is perhaps worth noting that in the BRW the index $\nu$ is the starting type, whereas it is the `current' type in \citet{MR2322849}. However, this makes no formal difference.
Thus, the anomalous spreading phenomenon should be manifest in the BRW, and, given the more restrictive framework, it should be possible to pin down the actual speed there, and hence for the corresponding $Q$ with Heaviside\index{Heaviside, O.} initial data. This is indeed possible. Here the discussion stays with the simplifications already used in looking at the deterministic results.
Consider a two-type BRW in which each type $\nu$ always produces at least one daughter of type $\nu$, on average produces more than one, and can produces daughters of type $\eta$---but type $\eta$ never produce daughters of type $\nu$. Also for $\theta \geq 0$ let \[ \kappa_{\nu}(\theta)=\log \int e^{\theta z} \mathbf {E}_\nu Z_\nu(dz) \mbox{~~and~~}\kappa_{\eta}(\theta)=\log \int e^{\theta z} \mathbf {E}_\eta Z_\eta(dz) \] and let these be infinite for $\theta < 0$. Thus Theorem \ref{second theorem} applies to type $\nu$ considered alone to show that \[ \frac{1}{n}\log \left(Z^{(n)}_\nu[na, \infty) \right) \rightarrow - \swFd{\kappa}_\nu(a) \mbox{~~~a.s.-}\mathbf {P}_{\!\nu}. \] It turns out that this estimation of numbers is critical in establishing the speed for type $\eta$. It is possible
for the growth in numbers of type $\nu$, through the numbers
of type $\eta$ they produce, to increase the speed of type $\eta$ from that of a population without type $\nu$. This is most obvious if type $\eta$ is subcritical, so that any line of descent from a type $\eta$ is finite, for the only way they can then spread is through the `forcing' from type $\nu$. However, if in addition the dispersal distribution at reproduction for $\eta$ has a much heavier tail than that for $\nu$ it is now possible for type $\eta$ to spread faster than type $\nu$.
For any two functions $f$ and $g$, let $\cv{f}{g}$
be the greatest (\Index{lower semi-continuous}) convex\index{convexity|(} function beneath both of them. The following result is a very special case of those proved in \citet{JDB-anom}. The formula given in the next result for the speed $\Gamma^{\dagger}$ is the same as that given in \citet[Proposition 4.1]{MR2322849} as the upper bound on the speed of the truncated linear approximation.
\begin{theorem} \label{prelim main theorem} Suppose
that $\max\{\kappa_{\nu}(\phi_\nu), \kappa_{\eta}(\phi_\eta)\}$ is finite for some $\phi_\eta \geq \phi_\nu >0 $ and that \begin{equation}\label{off-diag} \int e^{\theta z} \mathbf {E}_\nu Z_\eta(dz)<\infty ~~~\forall \theta \geq 0. \end{equation}
Let $r=\sweep{\cv{\swFd{\kappa}_\nu}{\Fd{\kappa}_\eta}}$. Then \begin{equation}\label{key result} \frac{1}{n} \log \left(Z^{(n)}_{\eta}[na,\infty) \right) \rightarrow -r(a) \mbox{~~~a.s.-}\mathbf {P}_{\!\nu}, \end{equation} for $a \neq \sup\{a: r(a)<0\}=\Gamma^{\dagger}$, and \begin{equation}\label{seq speed}
\frac{\rB{\eta}{n}}{n} \rightarrow \Gamma^{\dagger}. \mbox{~~~a.s.-}\mathbf {P}_{\!\nu}. \end{equation} Furthermore, \begin{equation}\label{two classes}
\Gamma^{\dagger} = \inf_{0<\varphi \leq \theta} \max\left\{\frac{\kappa_\nu(\varphi)} {\varphi},\frac{\kappa_\eta(\theta)} {\theta} \right\}. \end{equation} \end{theorem}
From this result it is possible to see how $ \Gamma^{\dagger}$ can be anomalous. Suppose that $r( \Gamma^{\dagger})=0$, so that $ \Gamma^{\dagger}$ is the speed, and that $r$ is strictly below both $\swFd{\kappa}_\nu$ and $\Fd{\kappa}_\eta$ at $ \Gamma^{\dagger}$. This will occur when the minimum of the
two convex functions $\Fd{\kappa}_\nu$ and $\Fd{\kappa}_\eta$ is not convex at $\Gamma^{\dagger}$, and then the largest convex\index{convexity|)} function below both will be linear there. In these circumstances, $\Fd{\kappa}_\nu(\Gamma^{\dagger})>0$, which implies that $\swFd{\kappa}_\nu(\Gamma^{\dagger})=\infty$, and $\Fd{\kappa}_\eta(\Gamma^{\dagger})>0$. Thus $\Gamma^{\dagger}$ will be strictly greater than both $\Gamma_{\nu}$ and $\Gamma_{\eta}$, giving a `super-speed'---Figure \ref{ff1} illustrates a case that will soon be described fully where $\Gamma_{\nu}$ and $\Gamma_{\eta}$ are equal and $\Gamma^{\dagger }$ exceeds them.
Otherwise, that is when $\Gamma^{\dagger}$ is not in a linear portion of $r$, $\Gamma^{\dagger}$ is just the maximum of $\Gamma_{\nu}$ and $\Gamma_{\eta}$.
The example in \citet{MR2322849}\index{Weinberger, H. F.|)}\index{Lewis, M. A.|)}\index{LiB@Li, B.|)} that illustrated anomalous speed was derived from coupled \index{reaction-diffusion equation}reaction-diffusion equations. When there is a branching interpretation, which it must be said will be the exception not the rule, the actual speed can be identified through Theorem \ref{prelim main theorem} and its generalisations. This will now be illustrated with an example. Suppose type $\eta$ particles form a binary branching Brown\-ian motion\index{Brown, R.!branching Brownian motion}, with variance parameter and splitting rate both one. Suppose type $\nu$ particles form a branching Brown\-ian motion, but with variance parameter $V$, splitting rate $\lambda$ and, on splitting, type $\nu$ particles produce a (random) family of particles of both types. There are $1+N_\nu$ of type $\nu$ and $N_\eta$ of type $\eta$, so that the family always contains at least one daughter of type $\nu$; the corresponding bivariate \index{probability generating function (pgf)}probability generating function is $\mathbf {E} a^{1+N_\nu}b^{N_\eta}=af(a,b)$. Let $v(x,t)=\mathbf {P}_\eta(\rBo{t} \leq x)$ and $w(x,t)=\mathbf {P}_\nu(\rBo{t} \leq x)$. These satisfy \begin{align*} \frac{\partial v}{\partial t}&=\frac{1}{2}\frac{\partial ^{2}v}{\partial x^{2} }-v(1-v),\\ \frac{\partial w}{\partial t}&=V\frac{1}{2}\frac{\partial ^{2}w}{\partial x^{2} }-\lambda w(1-f(w,v)). \end{align*} Here, when the initial ancestor is of type $\nu$ and at the origin the initial data are $w(x,0)=1$ for $x \geq 0$ and 0 otherwise and $v(x,0) \equiv 0$. Note that, by a simple change of variable, these can be rewritten as equations in $\mathbf {P}_\eta(\rBo{t} > x)$ and $\mathbf {P}_\nu(\rBo{t} > x)$ where the differential parts are unchanged, but the other terms look rather different.
Now suppose that $af(a,b)=a^2 (1-p+p b)$, so that a type $\nu$ particle always splits into two type $\nu$ and with probability $p$ also produces one type $\eta$. Looking at the discrete skeleton\index{skeleton!discrete skeleton} at integer times, $ \kappa_\nu(\theta)=V\theta^2/2+\lambda $ for $\theta \geq 0$, giving \[ \Fd{\kappa}_\nu(a)=\left\{\begin{array}{ll} -\lambda &a < 0 \\ \displaystyle -\lambda+ \frac{1}{2} \frac{a^2}{V}& a \geq 0 \end{array}\right. \] and speed $(2V\lambda)^{1/2}$, obtained by solving $\Fd{\kappa}_\nu(a)=0$. The formulae for $\Fd{\kappa}_\eta$ are just the special case with $V=\lambda=1$. Now, for convenience, take $V=\lambda^{-1}$, so that both types, considered alone, have the same speed. Then, sweeping positive values to infinity, \[ \swFd{\kappa}_\nu(a)= \left\{\begin{array}{ll}-\lambda &a< 0, \\ \displaystyle -\lambda\left(1- \frac{a^2}{2}\right)& a \in [0, 2^{1/2}],\\ \infty& a > 2^{1/2}.\end{array}\right. \] Now $\cv{\swFd{\kappa}_\nu}{\Fd{\kappa}_\eta}$ is the largest convex\index{convexity} function below this and $\Fd{\kappa}_\eta$. When $\lambda=3$ these three functions are drawn in Figure \ref{ff1}. \begin{figure}
\caption{Illustration of how anomalous speed arises. }
\label{ff1}
\end{figure}
\noindent The point where each of them meets the horizontal axis gives the value of speed for that function. Thus, $\Gamma^{\dagger}$ exceeds the other two, which are both $\sqrt{2}$. Here $\Gamma^{\dagger}=4/\sqrt{6}$. In general, for $\lambda>1$, it is $(1+\lambda)/\sqrt{2 \lambda}$, which can be made arbitrarily large by increasing $\lambda$ sufficiently. \section{Discussion of anomalous spreading}
The critical function in Theorem \ref{prelim main theorem} is $r= \sweep{\cv{\swFd{\kappa}_\nu}{\Fd{\kappa}_\eta}}$. Here is how it arises. The function $\swFd{\kappa}_\nu$ describes the growth in numbers and spread of the type $\nu$. Conditional on these, $\cv{\swFd{\kappa}_\nu}{\Fd{\kappa}_\eta}$ describes the growth and spread in expectation of those of type $\eta$. To see why this might be so, take a $b$ with $\swFd{\kappa}_\nu(b)<0$ so that (\ref{describe numbers}) describes the exponential growth of $Z^{(m)}_\nu[mb, \infty)$: there are roughly $\exp( -m\swFd{\kappa}_\nu(b))$ such particles in generation $m$. Suppose now, for simplicity, that each of these produces a single particle of type $\eta$ at the parent's position. As noted just before Theorem \ref{second theorem}, the expected numbers of type $\eta$ particles in generation $r$ and in $[rc,\infty)$ descended from a single type $\eta$ at the origin is roughly $\exp(-r \Fd{\kappa}_\eta(c))$. Take $\lambda \in (0,1)$ with $m=\lambda n$ and $r=(1-\lambda)n$. Then, conditional on the development of the first $m$ generations, the expectation of the numbers of type $\eta$ in generation $n$ and to the right of $mb+rc=n (\lambda b+(1-\lambda) c)$ will be (roughly) at least $\exp(-n(\lambda \swFd{\kappa}_\nu(b)+(1-\lambda) \Fd{\kappa}_\eta(c))) $. As $b$, $c$ and $\lambda$ vary with $\lambda b+(1-\lambda) c=a$, the least value for $\lambda \swFd{\kappa}_\nu(b)+(1-\lambda) \Fd{\kappa}_\eta(c)$ is given by $\cv{\swFd{\kappa}_\nu}{\Fd{\kappa}_\eta}(a)$. There is some more work to do to show that this lower bound on the conditional expected numbers is also an upper bound---it is here that (\ref{off-diag}) comes into play. Finally, as indicated just before Theorem \ref{second theorem}, this corresponds to actual numbers only when negative, so the positive values of this convex minorant\index{convexity!convex minorant} are swept to infinity.
When the speed is anomalous, this indicative description of how $r= \sweep{\cv{\swFd{\kappa}_\nu}{\Fd{\kappa}_\eta}}$ arises makes plausible the following description of lines of descent with speed near $\Gamma^{\dagger}$. They will arise as a `dog-leg', with the first portion of the trajectory, which is a fixed proportion of the whole, being a line of descent of type $\eta$ with a speed less than $\Gamma_{\eta}$. The remainder is a line of descent of type $\nu$, with a speed faster than $\Gamma_{\mu}$ (and also than $\Gamma^{\dagger}$).
Without the truncation, the \index{operator!linear operator}linear operator approximating (near $u\equiv1$) a $Q$ associated with a BRW describes the development of its expected numbers, and so it is tempting to define the speed using this, by look\-ing at when expected numbers start to decay. In the irreducible case\index{branching random walk (BRW)!irreducible BRW}, Theorem \ref{supercrit} has an analogue for expected numbers, that \[ \frac{1}{n}\log \left(\mathbf {E}_{\nu} Z^{(n)}_\sigma[na, \infty) \right) \rightarrow - \Fd{\kappa}(a), \] and so here the speed can indeed be found by looking at when expected numbers start to decay. In contrast, in the set up in Theorem \ref{prelim main theorem} \[ \frac{1}{n} \log \left(\mathbf {E}_{\nu} Z^{(n)}_{\eta}[na,\infty) \right) \rightarrow -\cv{\Fd{\kappa}_\nu}{\Fd{\kappa}_\eta}(a), \] and the limit here can be lower than $\cv{\swFd{\kappa}_\nu}{\Fd{\kappa}_\eta}$---the distinction between the functions is whether or not positive values are swept to infinity in the first argument. Hence the speed computed by simply asking when expectations start to decay can be too large. In Figure \ref{ff1}, $\cv{\swFd{\kappa}_\nu}{\Fd{\kappa}_\eta}$ is the same as $\cv{\Fd{\kappa}_\nu}{\Fd{\kappa}_\eta}$, but it is easy to see, reversing the roles of ${\Fd{\kappa}_\nu}$ and ${\Fd{\kappa}_\eta}$, that $\cv{\swFd{\kappa}_\eta}{\Fd{\kappa}_\nu}$ is the same as $\Fd{\kappa}_\nu$. Thus if $\eta$ could produce $\nu$, rather than the other way round, expectations would still give the speed $\Gamma^{\dagger}$ but the true speed would be $\Gamma_\nu
(=\Gamma_\eta)$.\index{branching random walk (BRW)!reducible BRW|)}\index{branching random walk (BRW)!two-type BRW|)}\index{spreading speed|)}
The general case, with many classes, introduces a number of additional challenges (mathematical as well as notational). It is discussed in \citet{JDB-anom}. The matrix of transforms now has irreducible blocks on its diagonal, corresponding to the classes, and their Perron--Frobenius eigenvalues\index{Perron, O.!Perron--Frobenius eigenvalue} supply the $\kappa$ for each class, as would be anticipated from \S\ref{multitype}. Here a flavour of some of the other complications. The rather strong condition (\ref{off-diag}) means that the spatial distribution of type $\eta$ daughters to a type $\nu$ mother
is irrelevant to the form of the result. If convergence is assumed only for some $\theta>0$ rather than all this need not remain true. One part of the challenge is to describe when these `off-diagonal' terms remain irrelevant; another is to say what happens when they are not. If there are various routes through the classes from the initial type to the one of interest these possibilities must be combined: in these circumstances, the function $r$ in (\ref{key result}) need not be convex\index{convexity} (though it will be increasing). It turns out that the formula for $\Gamma^{\dagger}$, which seems as if it might be particular to the case of two classes, extends fully---not only in the sense that there is a version that involves more classes, but also in the sense that the speed can usually be obtained as the maximum of that obtained using
(\ref{two classes}) for all pairs of classes where the first can have descendants in the second (though the line of descent may have to go through other classes on the way)\index{branching random walk (BRW)|)}\index{spreading out|)}\index{branching random walk (BRW)!multitype BRW|)}\index{anomalous spreading|)}.
\end{document} | arXiv |
Set TSP problem
In combinatorial optimization, the set TSP, also known as the generalized TSP, group TSP, One-of-a-Set TSP, Multiple Choice TSP or Covering Salesman Problem, is a generalization of the traveling salesman problem (TSP), whereby it is required to find a shortest tour in a graph which visits all specified subsets of the vertices of a graph. The subsets of vertices must be disjoint, since the case of overlapping subsets can be reduced to the case of disjoint ones.[1] The ordinary TSP is a special case of the set TSP when all subsets to be visited are singletons. Therefore, the set TSP is also NP-hard.
There is a transformation for an instance of the set TSP to an instance of the standard asymmetric TSP.[2] The idea is to connect each subset into a directed cycle with edges of zero weight, and inherit the outgoing edges from the original graph shifting by one vertex backwards along this cycle. The salesman, when visiting a vertex v in some subset, walks around the cycle for free and exits it from the vertex preceding v by an outgoing edge corresponding to an outgoing edge of v in the original graph.
The Set TSP has a lot of interesting applications in several path planning problems. For example, a two vehicle cooperative routing problem could be transformed into a set TSP,[3] tight lower bounds to the Dubins TSP and generalized Dubins path problem could be computed by solving a Set TSP.[4][5]
Illustration from the cutting stock problem
The one-dimensional cutting stock problem as applied in the paper / plastic film industries, involves cutting jumbo rolls into smaller ones. This is done by generating cutting patterns typically to minimise waste. Once such a solution has been produced, one may seek to minimise the knife changes, by re-sequencing the patterns (up and down in the figure), or moving rolls left or right within each pattern. These moves do not affect the waste of the solution.
In the above figure, patterns (width no more than 198) are rows; knife changes are indicated by the small white circles; for example, patterns 2-3-4 have a roll of size 42.5 on the left - the corresponding knife does not have to move. Each pattern represents a TSP set, one of whose permutations must be visited. For instance, for the last pattern, which contains two repeated sizes (twice each), there are 5! / (2! × 2!) = 30 permutations. The number of possible solutions to the above instance is 12! × (5!)6 × (6!)4 × (7!)2 / ((2!)9 × (3!)2) ≈ 5.3 × 1035. The above solution contains 39 knife changes, and has been obtained by a heuristic; it is not known whether this is optimal. Transformations into the regular TSP, as described in [2] would involve a TSP with 5,520 nodes.
See also
• Fagnano's problem of finding the shortest tour that visits all three sides of a triangle
References
1. C.E. Noon, "The Generalized Traveling Salesman Problem", PhD dissertation, 1988.
2. Charles Noon, James Bean (1993). "An efficient transformation of the generalized traveling salesman problem". {{cite journal}}: Cite journal requires |journal= (help)
3. Satyanarayana G. Manyam, Sivakumar Rathinam, Swaroop Darbha, David Casbeer, Yongcan Cao, Phil Chandler (2016). "GPS Denied UAV Routing with Communication Constraints". Journal of Intelligent & Robotic Systems. 84 (1–4): 691–703. doi:10.1007/s10846-016-0343-2. S2CID 33884807.{{cite journal}}: CS1 maint: multiple names: authors list (link)
4. Satyanarayana G. Manyam, Sivakumar Rathinam (2016). "On Tightly Bounding the Dubins Traveling Salesman's Optimum". Journal of Dynamic Systems, Measurement, and Control. 140 (7): 071013. arXiv:1506.08752. doi:10.1115/1.4039099. S2CID 16191780.
5. Satyanarayana G. Manyam, Sivakumar Rathinam, David Casbeer, Eloy Garcia (2017). "Tightly Bounding the Shortest Dubins Paths Through a Sequence of Points". Journal of Intelligent & Robotic Systems. 88 (2–4): 495–511. doi:10.1007/s10846-016-0459-4. S2CID 30943273.{{cite journal}}: CS1 maint: multiple names: authors list (link)
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Asymptotic analysis of initial flow around an impulsively started circular cylinder using a Brinkman penalization method
Y. Ueda, T. Kida
Journal: Journal of Fluid Mechanics / Volume 929 / 25 December 2021
Published online by Cambridge University Press: 27 October 2021, A31
The initial flow past an impulsively started translating circular cylinder is asymptotically analysed using a Brinkman penalization method on the Navier–Stokes equations. The asymptotic solution obtained shows that the tangential and normal slip velocities on the cylinder surface are of the order of $1/\sqrt {\lambda }$ and $1/\lambda$, respectively, within the second approximation of the present asymptotic analysis, where $\lambda$ is the penalization parameter. This result agrees with the estimation of Carbou & Fabrie (Adv. Diff. Equ., vol. 8, 2003, pp. 1453–1480). Based on the asymptotic solution, the influence of the penalization parameter $\lambda$ is discussed on the drag coefficient that is calculated using the adopted three formulae. It can then be found that the drag coefficient calculated from the integration of the penalization term exhibits a half-value of the results of Bar-Lev & Yang (J. Fluid Mech., vol. 72, 1975, pp. 625–647) as $\lambda \to \infty$.
Differential expression of angiotensin-converting enzyme-2 in human paranasal sinus mucosa in patients with chronic rhinosinusitis
T Kawasumi, S Takeno, M Nishimura, T Ishino, T Ueda, T Hamamoto, K Takemoto, Y Horibe
Journal: The Journal of Laryngology & Otology / Volume 135 / Issue 9 / September 2021
Print publication: September 2021
Severe acute respiratory syndrome coronavirus-2 uses angiotensin-converting enzyme-2 as a primary receptor for invasion. This study investigated angiotensin-converting enzyme-2 expression in the sinonasal mucosa of patients with chronic rhinosinusitis, as this could be linked to a susceptibility to severe acute respiratory syndrome coronavirus-2 infection.
Ethmoid sinus specimens were obtained from 27 patients with eosinophilic chronic rhinosinusitis, 18 with non-eosinophilic chronic rhinosinusitis and 18 controls. The angiotensin-converting enzyme-2 and other inflammatory cytokine and chemokine messenger RNA levels were assessed by quantitative reverse transcription polymerase chain reaction. Angiotensin-converting enzyme-2 positive cells were examined immunohistologically.
The eosinophilic chronic rhinosinusitis patients showed a significant decrease in angiotensin-converting enzyme-2 messenger RNA expression. In the chronic rhinosinusitis patients, angiotensin-converting enzyme-2 messenger RNA levels were positively correlated with tumour necrosis factor-α and interleukin-1β (r = 0.4971 and r = 0.3082, respectively), and negatively correlated with eotaxin-3 (r = −0.2938). Angiotensin-converting enzyme-2 immunoreactivity was mainly localised in the ciliated epithelial cells.
Eosinophilic chronic rhinosinusitis patients with type 2 inflammation showed decreased angiotensin-converting enzyme-2 expression in their sinus mucosa. Angiotensin-converting enzyme-2 regulation was positively related to pro-inflammatory cytokines, especially tumour necrosis factor-α production, in chronic rhinosinusitis patients.
Comparison of brain N-acetylaspartate levels and serum brain-derived neurotrophic factor (BDNF) levels between patients with first-episode schizophrenia psychosis and healthy controls
N. Goto, R. Yoshimura, S. Kakeda, J. Moriya, K. Hayashi, A. Ikenouchi-Sugita, W. Umene-Nakano, H. Hori, N. Ueda, Y. Korogi, J. Nakamura
Journal: European Psychiatry / Volume 26 / Issue 1 / January 2011
Published online by Cambridge University Press: 16 April 2020, pp. 57-63
N-acetylaspartate (NAA) levels and serum brain-derived neurotrophic factor (BDNF) levels in patients with first-episode schizophrenia psychosis and age- and sex-matched healthy control subjects were investigated. In addition, plasma levels of homovanillic acid (HVA) and 3-methoxy-4-hydroxyphenylglycol (MHPG) were compared between the two groups.
Eighteen patients (nine males, nine females; age range: 13–52 years) were enrolled in the study, and 18 volunteers (nine males, nine females; age range: 15–49 years) with no current or past psychiatric history were also studied by magnetic resonance spectroscopy (MRS) as sex- and age-matched controls.
Levels of NAA/Cr in the left basal ganglia (p = 0.0065) and parieto-occipital lobe (p = 0.00498), but not in the frontal lobe, were significantly lower in patients with first-episode schizophrenia psychosis than in control subjects. No difference was observed between the serum BDNF levels of patients with first-episode schizophrenia psychosis and control subjects. In regard to the plasma levels of catecholamine metabolites, plasma MHPG, but not HVA, was significantly lower in the patients with first-episode psychosis than in control subjects. In addition, a significantly positive correlation was observed between the levels of NAA/Cr of the left basal ganglia and plasma MHPG in all subjects.
These results suggest that brain NAA levels in the left basal ganglia and plasma MHPG levels were significantly reduced at the first episode of schizophrenia psychosis, indicating that neurodegeneration via noradrenergic neurons might be associated with the initial progression of the disease.
Optical properties of infrared-bright dust-obscured galaxies viewed with Subaru Hyper Suprime-Cam
A. Noboriguchi, T. Nagao, Y. Toba, M. Niida, M. Kajisawa, M. Onoue, Y. Matsuoka, T. Yamashita, Y. Chang, T. Kawaguchi, Y. Komiyama, K. Nobuhara, Y. Terashima, Y. Ueda
Journal: Proceedings of the International Astronomical Union / Volume 15 / Issue S341 / November 2019
Published online by Cambridge University Press: 10 June 2020, pp. 292-293
Print publication: November 2019
Optical properties of infrared-bright (IR-bright) dust-obscured galaxies (DOGs) are reported. DOGs are faint in optical but very bright in mid-IR, which are powered by active star formation (SF) or active galactic nucleus (AGN), or both. The DOGs is a candidate population that are evolving from a gas-rich merger to a quasar. By combining three catalogs of optical (Subaru Hyper Suprime-Cam), near-IR (VIKING), and mid-IR (ALLWISE), we have discovered 571 IR-bright DOGs. Using their spectral energy distributions, we classified the selected DOGs into the SF-dominated DOGs and the AGN-dominated DOGs. We found that the SF-dominated DOGs show a redder optical color than the AGN-dominated DOGs. Interestingly, some DOGs shows extremely blue color in optical (blue-excess DOGs: bluDOGs). A possible origin for this blue excess is either the leaked AGN light or stellar UV light from nuclear starbursts. The BluDOGs may be in the transition phase from obscured AGNs to unobscured AGNs.
ALMA twenty-six arcmin2 survey of GOODS-S at one millimeter (ASAGAO)
B. Hatsukade, K. Kohno, Y. Yamaguchi, H. Umehata, Y. Ao, I. Aretxaga, K. I. Caputi, J. S. Dunlop, E. Egami, D. Espada, S. Fujimoto, N. Hayatsu, D. H. Hughes, S. Ikarashi, D. Iono, R. J. Ivison, R. Kawabe, T. Kodama, M. Lee, Y. Matsuda, K. Nakanishi, K. Ohta, M. Ouchi, W. Rujopakarn, T. Suzuki, Y. Tamura, Y. Ueda, T. Wang, W.-H. Wang, G. W. Wilson, Y. Yoshimura, M. S. Yun, ASAGAO team
Journal: Proceedings of the International Astronomical Union / Volume 15 / Issue S352 / June 2019
Print publication: June 2019
The ALMA twenty-six arcmin2 survey of GOODS-S at one millimeter (ASAGAO) is a deep (1σ ∼ 61μJy/beam) and wide area (26 arcmin2) survey on a contiguous field at 1.2 mm. By combining with archival data, we obtained a deeper map in the same region (1σ ∼ 30μJy/beam−1, synthesized beam size 0.59″ × 0.53″), providing the largest sample of sources (25 sources at 5σ, 45 sources at 4.5σ) among ALMA blank-field surveys. The median redshift of the 4.5σ sources is 2.4. The number counts shows that 52% of the extragalactic background light at 1.2 mm is resolved into discrete sources. We create IR luminosity functions (LFs) at z = 1–3, and constrain the faintest luminosity of the LF at 2 < z < 3. The LFs are consistent with previous results based on other ALMA and SCUBA-2 observations, which suggests a positive luminosity evolution and negative density evolution.
Numerical analysis of flow-induced rotation of an S-shaped rotor
Y. Ueda
Journal: Journal of Fluid Mechanics / Volume 867 / 25 May 2019
Published online by Cambridge University Press: 21 March 2019, pp. 77-113
Flow-induced rotation of an S-shaped rotor is investigated using an adaptive numerical scheme based on a vortex particle method. The boundary integral equation with respect to Bernoulli's function is solved using a panel method for obtaining the pressure distribution on the rotor surface which applies the torque to the rotor. The present work first addresses the validation of the scheme against the previous studies of a rotating circular cylinder. Then, we compute the automatic rotation start of an S-shaped rotor from a quiescent state for various values of the moment of inertia. The computed flow patterns where the rotor supplies (or is supplied with) the torque to (or from) the fluid are shown during one cycle of rotation. The vortex shedding from the tip of the advancing bucket is found to play a key role in generating positive torque on the rotor. A remarkable finding is the fact that, after the rotor reaches a stable rotation, the trajectory of the limit cycle in the present autonomous system accounts for the stable rotating movement of the rotor. Furthermore, the hydrodynamic scenario of the rotor automatically starting up from a quiescent state and entering the limit cycle is elucidated for various values of the moment of inertia and the initial angle of the rotor.
In situ Thermal Shock of Lunar and Planetary Materials Using A Newly Developed MEMS Heating Holder in A STEM/SEM
Jane Y. Howe, Michelle S. Thompson, Stas Dogel, Kota Ueda, Tsuyoshi Matsumoto, Hideki Kikuchi, Matthew Reynolds, Hooman Hosseinkhannazer, Thomas J. Zega
Journal: Microscopy and Microanalysis / Volume 23 / Issue S1 / July 2017
Published online by Cambridge University Press: 04 August 2017, pp. 66-67
Print publication: July 2017
F-5 Portable Total Reflection X-ray Fluorescence Spectrometer: Comparison Between Non-Monochromatic and Monochromatic X-ray Sources
J. Kawai, Y. Ueda, Y. Morikawa, N. Sasaki, S. Kunimura, T. Yamamoto
Journal: Powder Diffraction / Volume 24 / Issue 2 / June 2009
Published online by Cambridge University Press: 20 May 2016, p. 168
Usefulness of three-dimensional fluid-attenuated inversion recovery magnetic resonance imaging to detect inner-ear abnormalities in patients with sudden sensorineural hearing loss
T Tanigawa, R Shibata, H Tanaka, M Gosho, N Katahira, Y Horibe, Y Nakao, H Ueda
Journal: The Journal of Laryngology & Otology / Volume 129 / Issue 1 / January 2015
Published online by Cambridge University Press: 08 December 2014, pp. 11-15
Print publication: January 2015
Three-dimensional fluid-attenuated inversion recovery magnetic resonance imaging has been used to detect alterations in the composition of inner-ear fluid. This study investigated the association between hearing level and the signal intensity of pre- and post-contrast three-dimensional fluid-attenuated inversion recovery magnetic resonance imaging in patients with sudden-onset sensorineural hearing loss.
Three-dimensional fluid-attenuated inversion recovery magnetic resonance imaging was performed in 18 patients with sudden-onset sensorineural hearing loss: 12 patients with mild-to-moderate sensorineural hearing loss (baseline hearing levels of 60 dB or less) and 6 patients with severe-to-profound sensorineural hearing loss (baseline hearing levels of more than 60 dB).
High-intensity signals in the inner ear were observed in two of the six patients (33 per cent) with severe-to-profound sensorineural hearing loss, but not in those with mild-to-moderate sensorineural hearing loss (mid-p test, p = 0.049). These signals were observed on magnetic resonance imaging scans 6 or 18 days after sensorineural hearing loss onset.
The results indicate that three-dimensional fluid-attenuated inversion recovery magnetic resonance imaging is not a useful tool for detecting inner-ear abnormalities in patients with mild sensorineural hearing loss.
Finding and characterising WHIM structures using the luminosity density method
Jukka Nevalainen, L. J. Liivamägi, E. Tempel, E. Branchini, M. Roncarelli, C. Giocoli, P. Heinämäki, E. Saar, M. Bonamente, M. Einasto, A. Finoguenov, J. Kaastra, E. Lindfors, P. Nurmi, Y. Ueda
Published online by Cambridge University Press: 12 October 2016, pp. 368-371
We have developed a new method to approach the missing baryons problem. We assume that the missing baryons reside in a form of Warm Hot Intergalactic Medium, i.e. the WHIM. Our method consists of (a) detecting the coherent large scale structure in the spatial distribution of galaxies that traces the Cosmic Web and that in hydrodynamical simulations is associated to the WHIM, (b) mapping its luminosity into a galaxy luminosity density field, (c) using numerical simulations to relate the luminosity density to the density of the WHIM, (d) applying this relation to real data to trace the WHIM using the observed galaxy luminosities in the Sloan Digital Sky Survey and 2dF redshift surveys. In our application we find evidence for the WHIM along the line of sight to the Sculptor Wall, at redshifts consistent with the recently reported X-ray absorption line detections. Our indirect WHIM detection technique complements the standard method based on the detection of characteristic X-ray absorption lines, showing that the galaxy luminosity density is a reliable signpost for the WHIM. For this reason, our method could be applied to current galaxy surveys to optimise the observational strategies for detecting and studying the WHIM and its properties. Our estimates of the WHIM hydrogen column density NH in Sculptor agree with those obtained via the X-ray analysis. Due to the additional NH estimate, our method has potential for improving the constrains of the physical parameters of the WHIM as derived with X-ray absorption, and thus for improving the understanding of the missing baryons problem.
Steady approach of unsteady low-Reynolds-number flow past two rotating circular cylinders
Y. Ueda, T. Kida, M. Iguchi
The long-time viscous flow about two identical rotating circular cylinders in a side-by-side arrangement is investigated using an adaptive numerical scheme based on the vortex method. The Stokes solution of the steady flow about the two-cylinder cluster produces a uniform stream in the far field, which is the so-called Jeffery's paradox. The present work first addresses the validation of the vortex method for a low-Reynolds-number computation. The unsteady flow past an abruptly started purely rotating circular cylinder is therefore computed and compared with an exact solution to the Navier–Stokes equations. The steady state is then found to be obtained for $t\gg 1$ with ${\mathit{Re}}_{\omega } {r}^{2} \ll t$, where the characteristic length and velocity are respectively normalized with the radius ${a}_{1} $ of the circular cylinder and the circumferential velocity ${\Omega }_{1} {a}_{1} $. Then, the influence of the Reynolds number ${\mathit{Re}}_{\omega } = { a}_{1}^{2} {\Omega }_{1} / \nu $ about the two-cylinder cluster is investigated in the range $0. 125\leqslant {\mathit{Re}}_{\omega } \leqslant 40$. The convection influence forms a pair of circulations (called self-induced closed streamlines) ahead of the cylinders to alter the symmetry of the streamline whereas the low-Reynolds-number computation ( ${\mathit{Re}}_{\omega } = 0. 125$) reaches the steady regime in a proper inner domain. The self-induced closed streamline is formed at far field due to the boundary condition being zero at infinity. When the two-cylinder cluster is immersed in a uniform flow, which is equivalent to Jeffery's solution, the streamline behaves like excellent Jeffery's flow at ${\mathit{Re}}_{\omega } = 1. 25$ (although the drag force is almost zero). On the other hand, the influence of the gap spacing between the cylinders is also investigated and it is shown that there are two kinds of flow regimes including Jeffery's flow. At a proper distance from the cylinders, the self-induced far-field velocity, which is almost equivalent to Jeffery's solution, is successfully observed in a two-cylinder arrangement.
Probing the evolution of Active Galactic Nuclei using the narrow iron Kα line
Claudio Ricci, S. Paltani, Y. Ueda, H. Awaki, P. Petrucci, K. Ichikawa, M. Brightman
Journal: Proceedings of the International Astronomical Union / Volume 9 / Issue S304 / October 2013
Published online by Cambridge University Press: 25 July 2014, p. 148
Print publication: October 2013
A large fraction of the AGN output power is emitted in the X-rays, in a region very close to the supermassive black hole (SMBH). The most distinctive feature of the X-ray spectra of AGN is the iron Kα line, often observed as the superposition of a broad and a narrow component. While the broad component is found in only ~ 35–45% of bright nearby AGN, the narrow component has been found to be ubiquitous. The narrow Fe Kα line is thought to be produced in the circumnuclear material, likely in the molecular torus. Given its origin, this feature is possibly the most important tracer of neutral matter surrounding the SMBH. One of the most interesting characteristics of the narrow Fe Kα line is the decrease of its equivalent width with the continuum luminosity, the so-called X-ray Baldwin effect (Iwasawa & Taniguchi 1993). This trend has been found by many studies of large samples of type-I AGN, and very recently also in type-II AGN (Ricci et al. 2013c, submitted to ApJ). The slope of the X-ray Baldwin effect in type-II AGN is the same of their unobscured counterparts, which implies that the mechanism at work is the same. Several hypothesis have been put forward in the last decade to explain the X-ray Baldwin effect: i) a luminosity-dependent variation in the ionisation state of the iron-emitting material (Nandra et al. 1997); ii) the decrease of the number of continuum photons in the iron line region with the Eddington ratio, as an effect of the well known correlation between the photon index and the Eddington ratio (Ricci et al. 2013b, submitted to MNRAS); iii) the decrease of the covering factor of the torus with the luminosity (e.g., Page et al. 2004, Ricci et al. 2013a A&A 553, 29) as expected by luminosity-dependent unification models (e.g., Ueda et al. 2003). In my talk I will review the main characteristics of the narrow Fe K? line, and present the results of our recent works aimed at explaining the X-ray Baldwin effect using iron-line emitting physical torus models (Ricci et al. 2013a, b), and at understanding the origin of the Fe K? line (Ricci et al. 2013c). I will focus in particular on the importance of the Fe Kα line as a probe of the evolution of the physical characteristics of the molecular torus with the luminosity.
A 2.5-5 μm spectroscopic study of hard X-ray selected AGNs with AKARI
A. Castro, T. Miyaji, M. Shirahata, S. Oyabu, D. Clark, K. Ichikawa, M. Imanishi, T. Nakagawa, Y. Ueda
Published online by Cambridge University Press: 25 July 2014, pp. 66-67
We explore the relationships between the 3.3 μm polycyclic aromatic hydrocarbon (PAH) feature and active galactic nucleus (AGN) properties of a sample of 54 hard X-ray selected bright AGNs, including both Seyfert 1 and Seyfert 2 type objects, using the InfraRed Camera (IRC) on board the infrared astronomical satellite AKARI. The sample is selected from the 9-month Swift/BAT survey in the 14-195 keV band and all of them have measured X-ray spectra at E ≲ 10 keV. These X-ray spectra provide measurements of the neutral hydrogen column density (NH) towards the AGNs. We use the 3.3 μm PAH luminosity (L3.3μm) as a proxy for star formation activity and hard X-ray luminosity (L14-195keV) as an indicator of the AGN activity. We searched for possible difference of star-formation activity between type 1 (un-absorbed) and type 2 (absorbed) AGNs. Our regression analysis of log L14-195keV versus log L3.3μm shows a positive correlation and the slope seems steeper for type 1/unobscured AGNs than that of type 2/obscured AGNs. The same trend has been found for the log (L14-195keV/MBH) versus log (L3.3μm/MBH) correlation. Our analysis show that the circum-nuclear star-formation is more enhanced in type 2/absorbed AGNs than type 1/un-absorbed AGNs for low X-ray luminosity/low Eddington ratio AGNs.
Development of an environmental cell for the H-9500 in-situ TEM and its application
T Yaguchi, A Watabe, Y Nagakubo, K Ueda, M Fukui, T Kamino, T Kawaski
View extract
Extended abstract of a paper presented at Microscopy and Microanalysis 2010 in Portland, Oregon, USA, August 1 – August 5, 2010.
Induction of protective immunity to Brugia pahangi in jirds by drug-abbreviated infection
Y. Horii, H. Nakanishi, A. Mori, M. Ueda, K. Kurokawa, M. Zaitsu, T. Oda, K. Fujita
Journal: Journal of Helminthology / Volume 66 / Issue 2 / June 1992
Protective immunity of homologous challenge infection was examined in jirds after drug-abbreviated infection with Brugia pahangi. Mebendazole (MBZ) treatment at the early prepatent (5–7 weeks of post infection) or the late prepatent (7–9 weeks of post infection) period was highly effective in causing almost complete eradication of the primary infection. After challenge infection, the worm burden was significantly reduced 19% (31·1 in average) and 77% (9·5) to that of the controls (38·8 and 41·7), respectively. The magnitude of eosinophil response paralleled the degree of protection. No or only a few microfilariae were seen after challenge infection in jirds treated during the prepatent periods. They were also resistant to intravenous challenge with the microfilariae of B. pahangi. MBZ treatment at the patent period was, on the contrary, incomplete against primarily infected adult worms, and was not able to induce either significant protection (30·1 vs 33·1 in control) or eosinophil response to the challenge infection.
Analysis of surgical treatment for middle-ear cholesterol granuloma
Y Matsuda, T Kurita, Y Ueda, S Ito, T Nakashima
Journal: The Journal of Laryngology & Otology / Volume 123 / Issue S31 / May 2009
Published online by Cambridge University Press: 22 May 2009, pp. 90-96
Print publication: May 2009
Cholesterol granuloma is an intractable ear disease. Many studies of this condition have been published since the initial report by Manasse. However, the pathogenesis of this condition is unclear. This study reviewed the treatment of middle-ear cholesterol granuloma in 16 patients undergoing surgical treatment at Kurume University Hospital.
The relationship between patients' pre-operative tympanic membrane findings and post-operative course was analysed. Patients with swollen tympanic membranes had significantly poorer outcomes. Patients with retracted tympanic membranes and those undergoing ossicular chain reconstruction had significantly better outcomes. The patients' overall hearing success rate at approximately two weeks post-operatively was 75 per cent. However, by six months post-operatively, the overall hearing success rate had declined to 62.5 per cent. Patients with poor hearing two weeks post-operatively did not acquire better hearing.
Epithelioid sarcoma of the neck: case report
Y Ueda, H Chijiwa, T Nakashima
Epithelioid sarcoma is an aggressive, malignant tumour of the soft tissue which tends to arise in proximity to large tendons and aponeuroses. We report the case of a patient presenting with an epithelioid sarcoma arising in the neck. A 56-year-old man was referred with a three-year history of a sensory disorder as well as a slowly growing mass in his right neck. The patient underwent resection of the tumour by means of a conservative neck dissection. The final diagnosis, based on the histological and immunohistochemical findings, was epithelioid sarcoma. Radiotherapy was performed after the operation. The post-operative course was uneventful, and there was no local recurrence or distant metastasis.
Superselective, intra-arterial, rapid infusion chemotherapy for external auditory canal carcinoma
Y Ueda, T Kurita, Y Matsuda, S Ito, T Nakashima
Previously, the treatment of carcinoma of the external auditory canal has mainly involved surgical resection. In order to enable organ preservation and to obtain cancer-free surgical margins, we introduced the use of superselective, intra-arterial, rapid infusion chemotherapy combined with radiotherapy to treat this condition.
We reviewed our patients' tumour stages, feeding arteries and clinical outcomes. Tumours were staged according to the Pittsburgh staging system. Chemotherapy was administered intra-arterially in the angiography suite via transfemoral catheterisation of the feeding arteries. Four patients underwent superselective, intra-arterial, rapid infusion chemo-radiotherapy. A complete response was obtained in all four patients. The overall toxic side effects were modest.
Superselective, intra-arterial, rapid infusion chemotherapy can be an effective, organ-preserving treatment for external auditory canal carcinoma, with a high cure rate.
Effect of tympanic membrane perforation on middle-ear sound transmission
Tympanic membrane perforation causes a sound conduction disturbance, and the size of this conduction disturbance is proportional to the perforation area. However, precise evaluation of perforation size is difficult, and there are few detailed reports addressing this issue. Furthermore, such evaluation becomes more difficult for irregularly shaped perforations. This study conducted a quantitative evaluation of tympanic membrane perforations, using image analysis equipment.
A significant correlation was found between the degree of sound conduction disturbance and the perforation area; this correlation was greater at low frequencies following a traumatic perforation. The conductive disturbance associated with chronic otitis media was significantly greater at low frequencies. Circular perforations caused only minor conduction disturbance. Perforations in the anteroinferior quadrant were associated with greater conduction disturbance. Traumatic spindle-shaped perforations and malleolar perforations were associated with greater conduction disturbance.
New silicone tube placement therapy for patients with an anterior glottic web
H Umeno, S Chitose, Y Ueda, T Kurita, H Mihashi, T Nakashima
An anterior glottic web in adults comprises a bridge of scar tissue commonly formed as a result of iatrogenic laryngeal injury. Traditionally, procedures such as transcervical midline thyrotomy and keel placement have been used to repair this condition. However, we recently repaired an anterior glottic web using a new surgical procedure involving a silicone tube instead of a keel. We herein report this case, in which we placed a silicone tube at the anterior commissure after resection of an anterior glottic web, under endolaryngeal microsurgery, without performing a tracheostomy. | CommonCrawl |
Apeirogon
In geometry, an apeirogon (from Ancient Greek ἄπειρος apeiros 'infinite, boundless', and γωνία gonia 'angle') or infinite polygon is a polygon with an infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes. In some literature, the term "apeirogon" may refer only to the regular apeirogon, with an infinite dihedral group of symmetries.[1]
The regular apeirogon
Edges and vertices∞
Schläfli symbol{∞}
Coxeter–Dynkin diagrams
Internal angle (degrees)180°
Dual polygonSelf-dual
Definitions
Classical constructive definition
Given a point A0 in a Euclidean space and a translation S, define the point Ai to be the point obtained from i applications of the translation S to A0, so Ai = Si(A0). The set of vertices Ai with i any integer, together with edges connecting adjacent vertices, is a sequence of equal-length segments of a line, and is called the regular apeirogon as defined by H. S. M. Coxeter.[1]
A regular apeirogon can be defined as a partition of the Euclidean line E1 into infinitely many equal-length segments. It generalizes the regular n-gon, which may be defined as a partition of the circle S1 into finitely many equal-length segments.[2]
Modern abstract definition
An abstract polytope is a partially ordered set P (whose elements are called faces) with properties modeling those of the inclusions of faces of convex polytopes. The rank (or dimension) of an abstract polytope is determined by the length of the maximal ordered chains of its faces, and an abstract polytope of rank n is called an abstract n-polytope.[3]: 22–25
For abstract polytopes of rank 2, this means that: A) the elements of the partially ordered set are sets of vertices with either zero vertex (the empty set), one vertex, two vertices (an edge), or the entire vertex set (a two-dimensional face), ordered by inclusion of sets; B) each vertex belongs to exactly two edges; C) the undirected graph formed by the vertices and edges is connected.[3]: 22–25 [4]: 224
An abstract polytope is called an abstract apeirotope if it has infinitely many elements; an abstract 2-apeirotope is called an abstract apeirogon.[3]: 25
In an abstract polytope, a flag is a collection of one face of each dimension, all incident to each other (that is, comparable in the partial order); an abstract polytope is called regular if it has symmetries (structure-preserving permutations of its elements) that take any flag to any other flag. In the case of a two-dimensional abstract polytope, this is automatically true; the symmetries of the apeirogon form the infinite dihedral group.[3]: 31
Pseudogon
The regular pseudogon is a partition of the hyperbolic line H1 (instead of the Euclidean line) into segments of length 2λ, as an analogue of the regular apeirogon.[2]
Realizations
Definition
A realization of an abstract apeirogon is defined as a mapping from its vertices to a finite-dimensional geometric space (typically a Euclidean space) such that every symmetry of the abstract apeirogon corresponds to an isometry of the images of the mapping.[3]: 121 [4]: 225 Two realizations are called congruent if the natural bijection between their sets of vertices is induced by an isometry of their ambient Euclidean spaces.[3]: 126 [4]: 229 The classical definition of an apeirogon as an equally-spaced subdivision of the Euclidean line is a realization in this sense, as is the convex subset in the hyperbolic plane formed by the convex hull of equally-spaced points on a horocycle.[5] Other realizations are possible in higher-dimensional spaces.
Symmetries of a realization
The infinite dihedral group G of symmetries of a realization V of an abstract apeirogon P is generated by two reflections, the product of which translates each vertex of P to the next.[3]: 140–141 [4]: 231 The product of the two reflections can be decomposed as a product of a non-zero translation, finitely many rotations, and a possibly trivial reflection.[3]: 141 [4]: 231
Moduli space of realizations
Generally, the moduli space of realizations of an abstract polytope is a convex cone of infinite dimension.[3]: 127 [4]: 229–230 The realization cone of the abstract apeirogon has uncountably infinite algebraic dimension and cannot be closed in the Euclidean topology.[3]: 141 [4]: 232
Classification of Euclidean apeirogons
The realizations of two-dimensional abstract polytopes (including both polygons and apeirogons), in Euclidean spaces of at most three dimensions, can be classified into six types:
• convex polygons,
• star polygons,
• regular apeirogons in the Euclidean line,
• infinite skew polygons (infinite zig-zag polygons in the Euclidean plane),
• antiprisms (including star prisms and star antiprisms), and
• infinite helical polygons (evenly spaced points along a helix).[6]
Abstract apeirogons may be realized in all of these ways, in some cases mapping infinitely many different vertices of an abstract apeirogon onto finitely many points of the realization. An apeirogon also admits star polygon realizations and antiprismatic realizations with a non-discrete set of infinitely many points.
Generalizations
Higher dimension
Main articles: Apeirotope and Apeirohedron
Apeirohedra are the 3-dimensional analogues of apeirogons, and are the infinite analogues of polyhedra.[7] More generally, n-apeirotopes or infinite n-polytopes are the n-dimensional analogues of apeirogons, and are the infinite analogues of n-polytopes.[3]: 22–25
See also
• Apeirogonal tiling
• Apeirogonal prism
• Apeirogonal antiprism
• Teragon, a fractal generalized polygon that also has infinitely many sides
References
1. Coxeter, H. S. M. (1948). Regular polytopes. London: Methuen & Co. Ltd. p. 45.
2. Johnson, Norman W. (2018). "11: Finite Symmetry Groups". Geometries and transformations. Cambridge University Press. p. 226. ISBN 9781107103405.
3. McMullen, Peter; Schulte, Egon (December 2002). Abstract Regular Polytopes (1st ed.). Cambridge University Press. ISBN 0-521-81496-0.
4. McMullen, Peter (1994), "Realizations of regular apeirotopes", Aequationes Mathematicae, 47 (2–3): 223–239, doi:10.1007/BF01832961, MR 1268033, S2CID 121616949
5. Buchanan, Kristopher; Flores, Carlos; Wheeland, Sara; Jensen, Jeffrey; Grayson, David; Huff, Gregory (2017). Transmit beamforming for radar applications using circularly tapered random arrays. 2017 IEEE Radar Conference (Radar Conf). pp. 0112–0117. doi:10.1109/RADAR.2017.7944181. ISBN 978-1-4673-8823-8. S2CID 38429370.
6. Grünbaum, B. (1977). "Regular polyhedra – old and new". Aequationes Mathematicae. 16 (1–2): 119. doi:10.1007/BF01836414. S2CID 125049930.
7. Coxeter, H. S. M. (1937). "Regular Skew Polyhedra in Three and Four Dimensions". Proc. London Math. Soc. 43: 33–62.
External links
• Russell, Robert A.. "Apeirogon". MathWorld.
• Olshevsky, George. "Apeirogon". Glossary for Hyperspace. Archived from the original on 4 February 2007.
Polygons (List)
Triangles
• Acute
• Equilateral
• Ideal
• Isosceles
• Kepler
• Obtuse
• Right
Quadrilaterals
• Antiparallelogram
• Bicentric
• Crossed
• Cyclic
• Equidiagonal
• Ex-tangential
• Harmonic
• Isosceles trapezoid
• Kite
• Orthodiagonal
• Parallelogram
• Rectangle
• Right kite
• Right trapezoid
• Rhombus
• Square
• Tangential
• Tangential trapezoid
• Trapezoid
By number
of sides
1–10 sides
• Monogon (1)
• Digon (2)
• Triangle (3)
• Quadrilateral (4)
• Pentagon (5)
• Hexagon (6)
• Heptagon (7)
• Octagon (8)
• Nonagon (Enneagon, 9)
• Decagon (10)
11–20 sides
• Hendecagon (11)
• Dodecagon (12)
• Tridecagon (13)
• Tetradecagon (14)
• Pentadecagon (15)
• Hexadecagon (16)
• Heptadecagon (17)
• Octadecagon (18)
• Icosagon (20)
>20 sides
• Icositrigon (23)
• Icositetragon (24)
• Triacontagon (30)
• 257-gon
• Chiliagon (1000)
• Myriagon (10,000)
• 65537-gon
• Megagon (1,000,000)
• Apeirogon (∞)
Star polygons
• Pentagram
• Hexagram
• Heptagram
• Octagram
• Enneagram
• Decagram
• Hendecagram
• Dodecagram
Classes
• Concave
• Convex
• Cyclic
• Equiangular
• Equilateral
• Infinite skew
• Isogonal
• Isotoxal
• Magic
• Pseudotriangle
• Rectilinear
• Regular
• Reinhardt
• Simple
• Skew
• Star-shaped
• Tangential
• Weakly simple
| Wikipedia |
About the complex nature of the wave function?
Why is the wave function complex? I've collected some layman explanations but they are incomplete and unsatisfactory. However in the book by Merzbacher in the initial few pages he provides an explanation that I need some help with: that the de Broglie wavelength and the wavelength of an elastic wave do not show similar properties under a Galilean transformation. He basically says that both are equivalent under a gauge transform and also, separately by Lorentz transforms. This, accompanied with the observation that $\psi$ is not observable, so there is no "reason for it being real". Can someone give me an intuitive prelude by what is a gauge transform and why does it give the same result as a Lorentz tranformation in a non-relativistic setting? And eventually how in this "grand scheme" the complex nature of the wave function becomes evident.. in a way that a dummy like me can understand.
A wavefunction can be thought of as a scalar field (has a scalar value in every point ($r,t$) given by $\psi:\mathbb{R^3}\times \mathbb{R}\rightarrow \mathbb{C}$ and also as a ray in Hilbert space (a vector). How are these two perspectives the same (this is possibly something elementary that I am missing out, or getting confused by definitions and terminology, if that is the case I am desperate for help ;)
One way I have thought about the above question is that the wave function can be equivalently written in $\psi:\mathbb{R^3}\times \mathbb{R}\rightarrow \mathbb{R}^2 $ i.e, Since a wave function is complex, the Schroedinger equation could in principle be written equivalently as coupled differential equations in two real functions which staisfy the Cauchy-Riemann conditions. ie, if $$\psi(x,t) = u(x,t) + i v(x,t)$$ and $u_x=v_t$ ; $u_t = -v_x$ and we get $$\hbar \partial_t u = -\frac{\hbar^2}{2m} \partial_x^2v + V v$$ $$\hbar \partial_t v = \frac{\hbar^2}{2m} \partial_x^2u - V u$$ (..in 1-D) If this is correct what are the interpretations of the $u,v$.. and why isn't it useful. (I am assuming that physical problems always have an analytic $\psi(r,t)$).
quantum-mechanics wavefunction complex-numbers
yayuyayu
$\begingroup$ Hi Yayu. I've always found interesting a paper by Leon Cohen, "Rules of Probability in Quantum Mechanics", Foundations of Physics 18, 983(1988), which approaches this question somewhat sideways, through characteristic functions. Cohen comes from a signal processing background, where Fourier transforms are very often a natural thing to do. Fourier transforms and complex numbers are of course pretty much joined at the hip. $\endgroup$ – Peter Morgan Apr 5 '11 at 18:08
$\begingroup$ Here are a few straightforward observations that might be helpful. (1) You can describe standing waves with real-valued wavefunctions, e.g., one can almost always get away with this in low-energy nuclear structure physics. (2) The w.f. of a photon is simply the electric and magnetic fields. These are observable and real-valued. (3) If the electron w.f. was real and observable, the wavelength would have to be invariant under a Galilean boost, which would violate the de Broglie relation. (4) Even for real-valued waves, operators are complex, e.g., momentum in the classically forbidden region. $\endgroup$ – user4552 May 30 '13 at 23:08
$\begingroup$ @yayu A complex analytic function is a function from complex numbers to complex numbers. And the Cauchy-Riemann equations are about such functions. To pick on x and t as if the t axis were an imaginary axis and the x axis were a real axis and y and z didn't exist is very confusing. $\endgroup$ – Timaeus Jan 2 '15 at 4:33
$\begingroup$ "Tl;DR: Imaginary numbers describe the phase of a quantum object." So, the physical meaning of Imaginary numbers indicate the virtual particles with a particular probability? $\endgroup$ – Lincoln Oct 22 '16 at 3:43
$\begingroup$ I am a bit concerned about the fact that many physicists think that the wave-function has to be complex because of arguments which inherently already assume that we are entering complex realms (as seen in the answers). This might be a huge obstacle on the way to an intuitive interpretation of the fundamental laws of nature. Of course the wave function is not inherently complex. Complex numbers (as many constructs in math) are just an elegant way to write down things. Complex numbers are a notational tool to wrap polar coordinate systems into "numbers" which we are more familiar with. $\endgroup$ – M. Winter Dec 5 '17 at 9:12
More physically than a lot of the other answers here (a lot of which amount to "the formalism of quantum mechanics has complex numbers, so quantum mechanics should have complex numbers), you can account for the complex nature of the wave function by writing it as $\Psi (x) = |\Psi (x)|e^{i \phi (x)}$, where $i\phi$ is a complex phase factor. It turns out that this phase factor is not directly measurable, but has many measurable consequences, such as the double slit experiment and the Aharonov-Bohm effect.
Why are complex numbers essential for explaining these things? Because you need a representation that both doesn't induce nonphysical time and space dependencies in the magnitude of $|\Psi (x)|^{2}$ (like multiplying by real phases would), AND that DOES allow for interference effects like those cited above. The most natural way of doing this is to multiply the wave amplitude by a complex phase.
Jerry SchirmerJerry Schirmer
$\begingroup$ But what are the differences between the sound waves and the wavefunction? Why the second must be complex, while the first also may interfere? And we may write the our wavefunction through sines and cosines, so the value $\psi^{T}\psi$ also refers to the invariant in this case. $\endgroup$ – Andrew McAddams Mar 27 '14 at 5:08
$\begingroup$ @AndrewMcAddams: the difference is that the amplitude of a sound wave is an observable, while only the amplitude of the modulus squared is an observable in quantum mechanics. I can see the phase of a water wave, but I can only see the phase of an electron wave through interference effects. $\endgroup$ – Jerry Schirmer Mar 27 '14 at 13:15
$\begingroup$ This is the most concise and easily understood explanation I've ever read concerning 'why'. Thank you. So many textbooks on quantum mechanics fail to communicate this fact. $\endgroup$ – docscience Nov 9 '14 at 15:03
$\begingroup$ But @Jerry Schirmer, you say using complex phase is the most natural way to model quantum behavior, is it the ONLY way? $\endgroup$ – docscience Nov 9 '14 at 15:07
$\begingroup$ @docscience: of course not -- you don't even need complex numbers to do the math of complex numbers, after all. It's just a nice, easy way to do them. And people have tried to reformulate quantum mechanics using quarternions, but I don't know how far they've really gotten, that's outside of my field of expertise. $\endgroup$ – Jerry Schirmer Nov 9 '14 at 23:14
This year-old question popped up unexpectedly when I signed in, and it's an interesting one. So I guess it's OK just to add an intuition-level "addendum answer" to the excellent and far more complete responses provided long ago.
Your kernel question seems to be this: "Why is the wave function complex?"
My intentionally informal answer is this:
Because by experimental observation, the quantum behavior of a particle far more closely resembles that of a rotating rope (e.g. a skip rope) than it does a rope that only moves up and down.
If each point in a rope marks out a circle as it moves, then a very natural and economical way to represent each point along the length of the rope is as a complex magnitude. You certainly don't have to do it that way, of course. In fact, using polar coordinates would probably be a bit more straightforward.
However, the nifty thing about complex numbers is that they provide a simple and computationally efficient way to represent just such a polar coordinate system. You can get into the gory details mathematical details of why, but suffice it to say that when early physicists started using complex numbers for just that purpose, their benefits continued even as the problems became far more complex. In quantum mechanics, their benefits became so overwhelming that complex numbers started being accepted pretty much as the "reality" of how to represent such mathematics.
That conceptual merging of complex quantities with actual physics can throw off your intuitions a bit. For example, if you look at moving skip rope there is no distinction between the "real" and "imaginary" axes in the actual rotations of each point in the rope. The same is true for quantum representations: It's the phase and amplitude that counts, with other distinctions between the axes of the phase plane being a result of how you use those phases within more complicated mathematical constructions.
So, if quantum wave functions behaved only like ropes moving up and down along a single axis, we'd use real functions to represent them. But they don't. Since they instead are more like those skip ropes, it's a lot easier to represent each point along the rope with two values, one "real" and one "imaginary" (and neither in real XYZ space) for its value.
Finally, why do I claim that a single quantum particle has a wave function that resembles that of a skip rope in motion? The classic example is the particle-in-a-box problem, where a single particle bounces back-and-forth between the two X axis ends of the box. Such a particle forms one, two, three, or more regions (or anti-nodes) in which the particle is more likely to be found.
If you borrow Y and Z (perpendicular to the length of the box) to represent the real and imaginary amplitudes of the particle wave function at each point along X, it's interesting to see what you get. It looks exactly like a skip-rope in action, one in which the regions where the electron is most likely to be found correspond one-for-one to the one, two, three, or more loops of the moving skip rope. (Fancy skip-ropers know all about higher numbers of loops.)
The analogy doesn't stop there. The volume enclosed by all the loops, normalized to 1, tells you exactly what the odds are on finding the electron along any one section along the box in the X axis. Tunneling is represented by the electron appearing on both sides of the unmoving nodes of the rope, those nodes being regions where there is no chance of finding the electron. The continuity of the rope from point to point captures a rough approximation of the differential equations that assign high energy costs to sharp bends in the rope. The absolute rotation speed of the rope represents the total mass-energy of the electron, or at least can be used that way.
Finally, and a bit more complicated, you can break those simple loops down into other wave components by using the Fourier transform. Any simple look can also be viewed as two helical waves (like whipping a hose around to free it) going in opposite directions. These two components represent the idea that a single-loop wave function actually includes helical representations of the same electron going in opposite directions, at the same time. "At the same time" is highly characteristic of quantum function in general, since such functions always contain multiple "versions" of the location and motions of the single particle that they represent. That is really what a wave function is, in fact: A summation of the simple waves that represent every likely location and momentum situation that the particle could be in.
Full quantum mechanics is far more complex than that, of course. You must work in three spatial dimensions, for one thing, and you have to deal with composite probabilities of many particles interacting. That drives you into the use of more abstract concepts such as Hilbert spaces.
But with regards to the question of "why complex instead of real?", the simple example of the similarity of quantum functions to rotating ropes still holds: All of these more complicated cases are complex because, at their heart, every point within them behaves as though it is rotating in an abstract space, in a way that keeps it synchronized with points in immediately neighboring points in space.
Terry BollingerTerry Bollinger
$\begingroup$ I'm not sure whether the OP is aware of this, but it emphasises your comment "it doesn't have to be this way". Real matrices of the form $\left(\begin{array}{cc}a&-b\\b&a\end{array}\right) = I a + i b$ where now $I$ is the $2\times2$ identity and $i= \left(\begin{array}{cc}0&-1\\1&0\end{array}\right)$ form a field wholly isomophic to $\mathbb{C}$. In particular, a phase delay corresponds to multiplication by the rotation matrix $\exp\left(-i\,\omega\,t\right)=\left(\begin{array}{cc}\cos\omega t&-\sin \omega t\\ \sin\omega t&\cos\omega t\end{array}\right) = I \cos\omega t + i\sin\omega t$. $\endgroup$ – Selene Routley Jul 29 '13 at 0:47
$\begingroup$ Rod, yes. A similar trick can be done for quaternions. I'm actually a quaternion bigot: I like to think of many of the complex numbers used in physics as really being overly generalized quaternions, ones in which our built-in 3D bias keeps us from noticing that the imaginary axis of a complex number is actually just a quaternion unit pointer in XYZ space. You lose a lot of representation richness by doing that, since for example you inadvertently abandon the intriguing option of treating changes in the quaternion-view i orientation as a local symmetry of XYZ space. $\endgroup$ – Terry Bollinger Jul 31 '13 at 22:36
$\begingroup$ Although I guess from the OPs point of view, it would be wrong to call it a trick - there are many ways to encode the kinds of properties complex numbers do and this one IS complex numbers (an isomorphic field). As for quaternions, yes, it's a shame that Hamilton, Clifford and Maxwell never held sway over Heaviside. $\endgroup$ – Selene Routley Aug 2 '13 at 0:05
$\begingroup$ @Terry Bollinger Feynman would be proud of your answer $\endgroup$ – electronpusher May 18 '19 at 19:15
$\begingroup$ I hope I'm not violating Stack Exchange etiquette by saying thank you for your kind remark. I always regret that I never had a chance to meet Richard Feynman in person. He was unique. $\endgroup$ – Terry Bollinger May 18 '19 at 19:31
Alternative discussion by Scott Aaronson: http://www.scottaaronson.com/democritus/lec9.html
From the probability interpretation postulate, we conclude that the time evolution operator $\hat{U}(t)$ must be unitary in order to keep the total probability to be 1 all the time. Note that the wavefunction is not necessarily complex yet.
From the website: "Why did God go with the complex numbers and not the real numbers? Answer: Well, if you want every unitary operation to have a square root, then you have to go to the complex numbers... " $\hat{U}(t)$ must be complex if we still want a continuous transformation. This implies a complex wavefunction.
Hence the operator should be: $\hat{U}(t) = e^{i\hat{K}t}$ for hermitian $\hat{K}$ in order to preserve the norm of the wavefunction.
edited Apr 8 '11 at 3:50
pcrpcr
$\begingroup$ Personally I prefer Jerry Schirmer's answer because it requires less postulate and instead uses experimental fact directly. =) $\endgroup$ – pcr Apr 8 '11 at 4:56
$\begingroup$ I very much like your answer, as much as Jerry's. But I would add two things: firstly, the square root thing is a bit obtuse: I would put it as follows for those like me who are a bit slow on the uptake: ....(ctd)... $\endgroup$ – Selene Routley Aug 5 '13 at 4:44
$\begingroup$ "All eigenvalues of unitary operators have unit magnitude. So the only nontrivial unitary operator with all real eigenvalues is one with a mixture of +1s and -1s as eigenvalues- say $M$ -otherwise it is the identity operator $I$. Since $U(t)$ and its eigenvalues vary continuously, $U(t)$ cannot reach $M$ from its beginning value $U(0)=I$ unless at least one eigenvalue goes through all values on the unit semicircle to reach the value -1". ...(ctd)... $\endgroup$ – Selene Routley Aug 5 '13 at 4:44
$\begingroup$ Secondly, the argument won't quite fly as is: there are nontrivial, real matrix valued unitary groups $\mathbf{SO}(N)$ (whose members have complex eigenvalues but nonetheless are real matrices) that will realise the $U(t)=\exp(i\,K\,t)$ in your argument, so quantum states can still be all real wavefunctions if they are real at $t=0$. I don't quite have a fix for this, maybe you could appeal to an experiment. It is a pretty argument, though, so I'll keep thinking. $\endgroup$ – Selene Routley Aug 5 '13 at 4:46
Among other things, the OP reprinted a page of a textbook, asking what "it is all about". I think it is impossible to answer this kind of questions because what is the OP's problem all about is totally undetermined, and the people who offer their answers could be writing their own textbooks, with no results.
The wave function in quantum mechanics has to be complex because the operators satisfy things like $$ [x,p] = xp-px = i\hbar.$$ It's the commutator defining the uncertainty principle. Because the left hand side is anti-Hermitian, $$ (xp-px)^\dagger = p^\dagger x^\dagger - x^\dagger p^\dagger = (px-xp) = -(xp-px),$$ it follows that if it is a $c$-number, its eigenvalues have to be pure imaginary. It follows that either $x$ or $p$ or both have to have some non-real matrix elements.
Also, Schrödinger's equation $$i\hbar\,\, {\rm d/d}t |\psi\rangle = H |\psi\rangle$$ has a factor of $i$ in it. The equivalent $i$ appears in Heisenberg's equations for the operators and in the $\exp(iS/\hbar)$ integrand of Feynman's path integral. So the amplitudes inevitably have to come out as complex numbers. That's also related to the fact that eigenstates of energy and momenta etc. have the dependence on space or time etc. $$\exp(Et/i\hbar)$$ which is complex. A cosine wouldn't be enough because a cosine is an even function (and the sine is an odd function) so it couldn't distringuish the sign of the energy. Of course, the appearance of $i$ in the phase is related to the commutator at the beginning of this answer. See also
http://motls.blogspot.com/2010/08/why-complex-numbers-are-fundamental-in.html
Why complex numbers are fundamental in physics
Concerning the second question, in physics jargon, we choose to emphasize that a wave function is not a scalar field. A wave function is not an observable at all while a field is. Classically, the fields evolve deterministically and can be measured by one measurement - but the wave function cannot be measured. Quantum fields are operators - but the wave function is not. Moreover, the mathematical similarity of a wave function to a scalar field in 3+1 dimensions only holds for the description of one spinless particle, not for more complicated systems.
Concerning the last question, it is not useful to decompose complex numbers into real and imaginary parts exactly because "a complex number" is one number and not two numbers. In particular, if we multiply a wave function by a complex phase $\exp(i\phi)$, which is only possible if we allow the wave functions to be complex and we use the multiplication of complex numbers, physics doesn't change at all. It's the whole point of complex numbers that we deal with them as with a single entity.
Luboš MotlLuboš Motl
$\begingroup$ thanks for answering. I have one question, not knowing about Feynman path integrals yet, I take it that what you are saying is the same thing as: if we make the transformation $\psi(r,t) = e^{i\frac{S(r,t)}{\hbar}}$ then the Schrodinger equation reduces to the classical hamilton Jacobi equations (if terms containing $i$ and $\hbar$ were negligible)? $\endgroup$ – yayu Apr 5 '11 at 5:35
$\begingroup$ Dear yayu, thanks for your question. First, the appearance of $\exp(iS/\hbar)$ in Feynman's approach is not a transformation of variables: the exponential is an integrand that appears in an integral used to calculate any transition amplitude. Second, $\psi$ is complex and $S$ is real, so $\psi=\exp(iS/\hbar)$ cannot be a "change of variables". You may write $\psi=\sqrt{\rho}\exp(i S/\hbar)$, in which case Schrödinger's equation may be (unnaturally) rewritten as two real equations, a continuity equation for $\rho$ and the Hamilton-Jacobi equation for $S$ with some extra quantum corrections. $\endgroup$ – Luboš Motl Apr 5 '11 at 5:39
$\begingroup$ I edited my question removing the reprints and trying to state my problem without them.. it will take some time to think about some points you made in the answer already, though. $\endgroup$ – yayu Apr 5 '11 at 6:00
$\begingroup$ I think a better explanation would not use the idea of operator formalism, since when Schrödinger came up with his equation, the formalism wasn't developed yet. $\endgroup$ – Sidd Oct 24 '15 at 23:57
$\begingroup$ Sorry but Schrödinger only came with his "wave mechanics" almost a year after quantum mechanics was discovered by Heisenberg and pals in the form of the "matrix mechanics". Despite popular misconceptions, Schrödinger isn't even one of the founders of quantum mechanics and he never correctly understood the meaning of the theory. $\endgroup$ – Luboš Motl Oct 25 '15 at 7:09
If the wave function were real, performing a Fourier transform in time will lead to pairs of positive-negative energy eigenstates. Negative energies with no lower bounds is incompatible with stability. So, complex wave functions are needed for stability.
No, the wave function is not a field. It only looks like it for a single particle, but for N particles, it is a function in 3N+1 dimensional configuration space.
CayleyCayley
EDIT add:
My Answer is GA centric and after the comments I felt the need to say some words about the beauty of Geometric Algebra:
On 2nd page of Oersted Medal Lecture (link bellow):
(3) GA Reduces "grad, div, curl and all that" to a single vector derivative that, among other things, combines the standard set of four Maxwell equations into a single equation and provides new methods to solve it.
Geometry Algebra (GA) encompasses in a single framework for all this:
Synthetic Geometry, Coordinate Geometry, Complex Variables, Quaternions, Vector Analysis, Matrix Algebra, Spinors, Tensors, Differential forms. It is one language for all physics.
Probably Schrödinger, Dirac, Pauli, etc ... would have used GA if it existed at the time.
To the Question: WHY is the wave function complex? This Answer is not helpful: because the wave function is complex (or has a i on it). We have to try something different, not written in your book.
In the abstracts I bolded the evidence that the papers are about the WHYs. If someone begs a fish I'll try to give a fishing rod.
I'm an old IT analyst who would be unemployed if I had not evolved. Physics is evolving too.
end EDIT
Recently I've found the Geometric Algebra, Grassman, Clifford, and David Hestenes.
I will not detail here the subject of the OP because each one of us need to follow paths, find new ideas and take time to read. I will only provide some paths with part of the abstracts:
Overview of Geometric Algebra in Physics
Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics (a good start)
In this lecture Hestenes is arguing for a reform of the way in which mathematics is taught to physicists. He asserts that using Geometric Algebra will make it easier to understand the fundamentals of physics, because the mathematical language will be clearer and more uniform.
Hunting for Snarks in Quantum Mechanics
Abstract. A long-standing debate over the interpretation of quantum mechanics has centered on the meaning of Schroedinger's wave function ψ for an electron. Broadly speaking, there are two major opposing schools. On the one side, the Copenhagen school (led by Bohr, Heisenberg and Pauli) holds that ψ provides a complete description of a single electron state; hence the probability interpretation of ψψ* expresses an irreducible uncertainty in electron behavior that is intrinsic in nature. On the other side, the realist school (led by Einstein, de Broglie, Bohm and Jaynes) holds that ψ represents a statistical ensemble of possible electron states; hence it is an incomplete description of a single electron state. I contend that the debaters have overlooked crucial facts about the electron revealed by Dirac theory. In particular, analysis of electron zitterbewegung (first noticed by Schroedinger) opens a window to particle substructure in quantum mechanics that explains the physical significance of the complex phase factor in ψ. This led to a testable model for particle substructure with surprising support by recent experimental evidence. If the explanation is upheld by further research, it will resolve the debate in favor of the realist school. I give details. The perils of research on the foundations of quantum mechanics have been foreseen by Lewis Carroll in The Hunting of the Snark!
THE KINEMATIC ORIGIN OF COMPLEX WAVE FUNCTION
Abstract. A reformulation of the Dirac theory reveals that i¯h has a geometric meaning relating it to electron spin. This provides the basis for a coherent physical interpretation of the Dirac and Sch¨odinger theories wherein the complex phase factor exp(−iϕ/¯h) in the wave function describes electron zitterbewegung, a localized, circular motion generating the electron spin and magnetic moment. Zitterbewegung interactions also generate resonances which may explain quantization, diffraction, and the Pauli principle.
Universal Geometric Calculus a course, and follow:
III. Implications for Quantum Mechanics
The Kinematic Origin of Complex Wave Functions
Clifford Algebra and the Interpretation of Quantum Mechanics
The Zitterbewegung Interpretation of Quantum Mechanics
Quantum Mechanics from Self-Interaction
Zitterbewegung in Radiative Processes
On Decoupling Probability from Kinematics in Quantum Mechanics
Zitterbewegung Modeling
Space-Time Structure of Weak and Electromagnetic Interactions
to keep more references together:
Geometric Algebra and its Application to Mathematical Physics (Chris Thesis)
(what lead me to this amazing path was a paper by Joy Christian 'Disproof of Bell Theorem')
'Bon voyage', 'good journey', 'boa viagem'
Helder VelezHelder Velez
$\begingroup$ Why the Down votes? $\endgroup$ – Helder Velez Apr 5 '11 at 15:03
$\begingroup$ @yayu; No, I shall upvote it because I've read the papers linked and know they are exactly appropriate for the question. I can give a short description of the argument: As soon as you use spin-1/2 you have that $\sigma_x\sigma_y\sigma_z = i$ is an imaginary unit in that it squares to -1 and commutes with the other elements. This is also inherent to the Dirac equation. What Hestenes goes on to show is that the "i" in Schroedinger's equation arises from looking at the Pauli equation with a fixed spin. $\endgroup$ – Carl Brannen Apr 6 '11 at 5:01
$\begingroup$ @yayu; Answers on Stack Exchange are read by more than just the person who asks. Spin-1/2 (and the Pauli spin matrices) will be covered in any introduction to QM; it's the simplest non trivial Hilbert space possible. It doesn't get much simpler than that. But in general, even if it were suitable only for Albert Einstein it has to be posted here. On SE, duplicate questions are closed. This is the only opportunity to answer the question for all readers. $\endgroup$ – Carl Brannen Apr 6 '11 at 5:31
$\begingroup$ @yayu spinless particle? It seems that there is no need to be complex. KINEMATIC link above or the comment by Carl previous to your 'spinless' comment. $\endgroup$ – Helder Velez Apr 6 '11 at 15:41
$\begingroup$ Removed the downvote by adding a +1. I found this post informative and detailed and I don't really care if it precisely answered the original question or not. $\endgroup$ – Antillar Maximus Dec 6 '12 at 1:15
This question has been asked since Dirac
In fact Dirac's answer is available for $ 100 from JSTOR in a paper by Dirac from I think 1935 ?
A recent answer from James Wheeler - is that the zero-signature Killing metric of a new, real-valued, 8-dimensional gauging of the conformal group accounts for the complex character of quantum mechanics
Reference is Why Quantum Mechanics is Complex , James T. Wheeler ArXiv:hep-th9708088
LenLen
$\begingroup$ Whilst this may theoretically answer the question, it would be preferable to include the essential parts of the answer here, and provide the link for reference. $\endgroup$ – Gonenc Sep 30 '15 at 16:33
From the Heisenberg Uncertainty Principle, if we know a great deal about the momentum of a particle we can know very little about its position. This suggests that our mathematics should have a quantum state that corresponds to a plane wave $\psi(x)$ with a precisely known momentum but entirely unknown position.
A natural definition for the probability of finding the particle at the position $x$ is $|\psi(x)|^2$. This definition makes sense for both a real wave function and an imaginary wave function.
For a plane wave to have no position information is to imply that $|\psi(x)|$ does not depend on position and so is constant. Therefore we must have $\psi$ complex; otherwise there would be no way to store the information "what is the momentum of the particle".
So in my view, the complex nature of wave functions arises from the interaction between the necessity for (1) a probability interpretation, (2) the Heisenberg uncertainty principle, and (3) plane waves.
Carl BrannenCarl Brannen
$\begingroup$ Please clear some doubts for me. 1. The probability interpretation: I think it followed since the wavefunction was complex and physical meaning could only attributed to a real value. If we make a construction $\psi^*\psi$ then we arrive at the continuity equation from the schrodinger equation and the interpretation can now be made that the quantity $\rho=\psi^*\psi$ is the probability density. Starting from an interpretation like $\rho=\psi^*\psi$, I do not see any way to work backwards and convincingly argue that the amplitude $\psi$ must be complex. $\endgroup$ – yayu Apr 6 '11 at 18:09
$\begingroup$ the uncertainty relations follow from the identification of the free particle as a plane wave. I am guessing your answer points in the right direction, I am working on (2) as suggested in Lubos' answer as well and trying to get why $\psi$ is complex valued as a consequence, however I fail to see how anything except (2) is relevant for showing it conclusively. $\endgroup$ – yayu Apr 6 '11 at 18:16
$\begingroup$ @yayu: see my post--there are two essential experimental facts: 1) phase is not directly measurable; 2) interference effects happen in a broad range of quantum materials. It's hard to reconcile these things without using complex numbers. $\endgroup$ – Jerry Schirmer Apr 7 '11 at 3:39
$\begingroup$ While I agree with the core thoughts in this answer, I don't agree with the conclusion that this requires complex numbers. There is nothing lost, for example, expressing the (complex) Fourier transform as two real sine/cosine Fourier transforms. This does not require complex numbers, although they may be convenient. $\endgroup$ – anon01 Feb 16 '17 at 2:17
$\begingroup$ @ConfusinglyCuriousTheThird Hi! Does my modest contribution (currently the 3rd one below this one), expanding a little on Carl's answer, provide a reply you can accept? Rgds – iSeeker $\endgroup$ – iSeeker Jan 17 '18 at 8:36
The question is a good one and has been asked also by Ehrenfest (1932): "Einige die Quantenmechanik betreffende Erkundigungsfragen". The answer was given by Pauli (1933): "Einige die Quantenmechanik betreffenden Erkundigungsfragen". Unfortunately I'm not aware of a english translation of these two publications. However one can find a slightly different form of the answer also in Pauli's book "General Principles of Quantum Mechanics" p.16. In that book Pauli writes
a single real function is not sufficient in order to construct from wavefunctions of the form (3.1) a non-negative probability function that is constant in time when integrated over the whole space.
I will try to summarize his arguments here:
A wave packet to describe a single particle (basically deBroglie's idea) can be written generally like $$ u(x,t) = \int U(k) e^{i(kx-\omega t)} dk = \int U(k) e^{ikx} dk \, e^{-i\omega t} $$ where $U(k)$ is the Fourier transform of $u(x,0)$. The complex conjugate of this wave packet is $$ u^*(x,t) = \int U^*(k) e^{-i(kx-\omega t)} dk = \int U^*(k) e^{-ikx} dk \, e^{i\omega t} $$ One can also define such wave packets in electrodynamics. But in quantum mechanics we have an additional condition, namely that the probability $P(x,t)$ to find a particle must always be positive and the total probability to find a single particle somewhere must be one, so $$ P(x,t) \geq 0 \\ \int P(x,t)\, dx = 1 $$ Pauli argues that the simplest ansatz to construct such a function $P(x,t)$ from $u(x,t)$ is a definite quadratic form from the functions $u$ and $u^*$, that means $$ P(x,t) = a u^2 + b {u^*}^2 + c u u^* $$ Now from the form of $u(x,t)$ and $u^*(x,t)$ we see that $$ u^2 \sim e^{-2i\omega t}\ \text{and}\ {u^*}^2 \sim e^{2i\omega t} $$ and an integral over space over these two functions can never be time independent. So the constants $a$ and $b$ must be zero in the ansatz for $P(x,t)$. Only the product of a wave packet and it's complex conjugate will yield a time independent total probability: $$ 1 = \int P(x,t)\, dx = \int uu^*\, dx = \iiint U(k)U^*(k') e^{i(kx-k'x)} \, e^{-i\omega t} e^{i\omega t} dk dk' dx \\ = \iint U(k)U^*(k') \delta(k-k') dk dk' = \int \left|{U(k)}\right|^2 dk = \int P(k)\, dk $$ Since the product $uu^* = Re[u]^2 + Im[u]^2$ it follows that - as Pauli said - in order to compute a meaningful probability from wave packets of the form $u(x,t)$ one needs the real and the imaginary part of $u(x,t)$ and the wave function in quantum mechanics must be complex.
asmaierasmaier
THIS LATE ANSWER (Jan 2018) expands a little on Carl Brannen's straightforward, and (IMO) underappreciated, answer (showing a little way above, at time of posting), which reminded me of another simple and convincing argument as to why the wave-function should be complex, set out many years ago in Dicke & Wittke's "Introduction to Quantum Mechanics" (1960; pp. 23-24).
Given their review in Ch 1 of why a quantum mechanical wave is subject to wave-particle duality/the De Broglie relation, they proceed as follows:
For a wave-particle of sharply-defined momentum:
λ = h/p (and thus, by Δx.Δp>= h/4π, completely uncertain – essentially uniform – position)
…the probability distribution |ψ|^2 of a plane wave should be uniform in position, which cannot be satisfied by a real-valued plane wave
ψ = A sin(kx - ωt + α)
…but is satisfied (generalising to an arbitrary position) by
ψ = A exp [i (k.x- ωt)], where the propagation vector k = p/(h/2π).
Dicke & Wittke then discuss how the complex-valued wave function accounts for interference effects (Out of copyright and safely available via https://archive.org/details/IntroductionToQuantumMechanics).
[NB Care googling the book title/pdf - many online sources, unlike the one above, are unsafe]
iSeekeriSeeker
Since the physical point of view, the wave function needs to be complex in order to explain the double-slit experiment, as well mentionated in the book of The Feynman Lectures on Physics-III, I suggest you that review chapters 1&3, where it is explained how $\psi$ has to be considered of probabilistic nature, according to the pattern of interference, because "something" has to behave like a wave at the time of crossing through "each one" of the slits. Furthermore, Bohm proclaims that path of the particle (electron,photon, etc.) can be considered classic, so as a consequence you may watch this one, as it follows the rules already known at the macro... in that sense, you can see next reference or this one to consider the covariance of the laws of mechanics.
The wavefunction $\psi(x)$ is the projection of the physical system's state vector $|\psi\rangle$ onto the $\hat{x}$ eigenket $|x\rangle$ of eigenvalue $x$, viz. $|\psi\rangle=\int dx\psi(x)|x\rangle$. You mustn't confuse the scalar-value $\psi(x)=\langle x|\psi\rangle$ with the vector $|\psi\rangle$ that lives in a Hilbert space.
The first sentence of your first question is, in technical terms: why is this Hilbert space over the field $\mathbb{C}$ rather than, say, $\mathbb{R}$? If you Ctrl+F to "Real vs. Complex Numbers" here, you'll get a detailed discussion of several motivations for why quantum mechanics ought to look like that. One advantage of a complex wavefunction is it has both an amplitude and a phase, but only the former affects the probability density $|\psi|^2$, and the latter gives us quantum interference because of trigonometric identities such as $|1+\exp i\theta|=2|\cos\frac{\theta}{2}|$. However (to continue your Q1), a Galilean transformation needs to include a phase shift so the Schrödinger equation will be invariant; see here and here for more information. A gauge transformation such as the Galilean one is simply a way of transforming coordinates, or fields (which come to the same thing in a Lagrangian field theory), which leave the action and its equations of motion invariant. (By the way, you need to be careful not to confuse the words transform and transformation.)
Your Q2 also hinges on not confusing $\psi$ with $|\psi\rangle$. The ray is the set of values of $\exp i\theta|\psi\rangle$ with $\theta\in\mathbb{R}$, but switching from one value of $\theta$ to another leaves $\psi$ invariant because $\langle x|$ gets multiplied by $(\exp i\theta)^\ast=\exp -i\theta$.
As for Q3, it's more useful to work with the modulus and phase of the complex $\psi$ rather than its real and imaginary part, because under unitary transformations the former is invariant and so are differences in the latter.
J.G.J.G.
Already great answers to this often asked question. Very simply put tho, quantum eigenstates have associated relative phase values, and the in-phase and quadrature plane (also strangely but conventionally referred to as the Real/Imaginary plane) provides for specifying these phases when plotting or otherwise specifying a wave function.
As pointed out in other answers, there are other mathematical ways to do this, but the "complex" way is mathematically very convenient.
edited Sep 4 '19 at 14:47
Pat EblenPat Eblen
Since both the amplitude and the wavelength cannot be known with precision simultaneously, I think of this as meaning that there is some missing information that must still be dealt with continuously. That information is conveniently stored in the imaginary part of a complex number.
$\begingroup$ This is not nearly substantiated enough to be an answer, and besides, I'm quite sure that's not a good way to think about that. $\endgroup$ – ACuriousMind♦ Sep 10 '14 at 0:17
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\begin{definition}[Definition:Concatenation (Formal Systems)]
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\begin{document}
\allowdisplaybreaks \title{The discretization problem for continuous frames}
\author{ Daniel Freeman} \address{Department of Mathematics and Statistics\\ St Louis University\\ St Louis, MO 63103 USA} \email{[email protected]}
\author{Darrin Speegle} \address{Department of Mathematics and Statistics\\ St Louis University\\ St Louis, MO 63103 USA} \email{[email protected]}
\thanks{ The first author was supported by grant 353293 from the Simon's foundation, and the second author was supported by grant 244953 from the Simon's foundation.}
\thanks{2010 \textit{Mathematics Subject Classification}: 42C15, 81R30}
\begin{abstract}
We characterize when a coherent state or continuous frame for a Hilbert space may be sampled to obtain a frame, which solves the discretization problem for continuous frames. In particular, we prove that every bounded continuous frame for a Hilbert space may be sampled to obtain a frame. We give multiple applications to different classes of frames such as scalable frames and Gabor frames. \end{abstract}
\keywords{frames, continuous frames, coherent states, sampling}
\maketitle
\section{Introduction}
\begin{defin} A collection of vectors $(x_{j})_{j\in J}$ in a Hilbert space $H$ is called a {\em frame} or a {\em discrete frame} for $H$ if there exists positive constants $A$ and $B$ (called lower and upper {\em frame bounds} respectively) such that \begin{equation}\label{e:frame}
A\|x\|^2\leq \sum_{j\in J} |\langle x, x_i\rangle|^2 \leq B\|x\|^2\quad\textrm{for all }x\in H. \end{equation} The frame is called {\em tight} if $A=B$ and the frame is called {\em Parseval} if $A=B=1$. \end{defin}
A frame can be thought of as a (possibly) redundant coordinate system in the sense that a frame can contain more vectors than are necessary to represent each vector in the Hilbert space. One way of
interpreting the frame inequality \eqref{e:frame} is that a frame for a Hilbert space $H$ is a collection of vectors in $H$ indexed by a countable set $J$ so that the norm in $\ell_2(J)$ of the frame coefficients is equivalent to the norm on $H$. This notion can be nicely generalized from the discrete to the continuous setting by instead of summing over a countable set $J$, we integrate over a measure space $X$. That is, a continuous frame for a Hilbert space $H$ is a collection of vectors indexed by a measure space $X$ so that the norm of the frame coefficients in $L_2(X)$ is equivalent to the norm on $H$. The following definition, with which we are following \cite{C}, formalizes this notion.
\begin{defn} Let $(X, \Sigma, \mu)$ be a positive, $\sigma$-finite measure space and let $H$ be a separable Hilbert space. A measurable function $\Psi:X\rightarrow H$ is a {\em continuous frame} of $H$ with respect to $\mu$ if there exists constants $A,B>0$ such that \begin{equation}
A\|x\|^2\leq \int |\langle x, \Psi(t)\rangle|^2 d\mu(t)\leq B \|x\|^2\quad\quad\forall x\in H. \end{equation} The constant $A$ is called a {\em lower frame bound} and the constant $B$ is called an {\em upper frame bound}. If $A=B$ then the continuous frame is called {\em tight} and if $A=B=1$ then the continuous frame is called {\em Parseval} or a {\em coherent state}. We say that $\Psi:X\rightarrow H$ is {\em Bessel} if it has a finite upper frame bound $B$, but does not necessarily have a positive lower frame bound $A$. \end{defn}
Note that if $X$ is a countable set with counting measure, then $\Psi:X\rightarrow H$ is a continuous frame of $H$ is equivalent to $(\Psi(t))_{t\in X}$ being a frame of $H$. Thus, frames are a special case of continuous frames. Often the definition of continuous frames in the literature includes some additional topological structure of the measure space $X$ and continuity of the map $\Psi:X\rightarrow H$, but we do not require this.
Continuous frames and coherent states are widely used in mathematical physics and are particularly prominent in quantum mechanics and quantum optics. The theory of coherent states was initiated by Schr\"odinger in 1926 \cite{S} and was generalized to continuous frames by Ali, Antoine, and Gazeau \cite{AAG1}. Though
coherent states naturally characterize many different physical properties, discrete frames are much better suited for computations. Because of this, when working with coherent states and continuous frames, researchers often create a discrete frame by sampling the continuous frame and then use the discrete frame for computations instead of the entire continuous frame. Specifically, if $\Psi:X\rightarrow H$ is a continuous frame and $(t_j)_{j\in J}\subseteq X$ then $(\Psi(t_j))_{j\in J}\subseteq H$ is called a {\em sampling} of $\Psi$. The notion of creating a frame by sampling a coherent state has its origins in the very start of modern frame theory. Indeed, Daubechies, Grossmann, and Meyer \cite{DGM} popularized modern frame theory in their seminal paper ``Painless nonorthogonal expansions", and their constructions of frames for Hilbert spaces were all done by sampling different coherent states. Another example which is of particular interest for frame theorists is that of Gabor frames, which are samplings of the short time Fourier transform at a lattice, and we explore this further in Section \ref{S:app}.
The discretization problem, posed by Ali, Antoine, and Gazeau in their physics textbook {\em Coherent States, Wavelets, and Their Generalizations} \cite{AAG2}, asks when a continuous frame of a Hilbert space can be sampled to obtain a frame. They state that a positive answer to the question is crucial for practical applications of coherent states, and chapter 16 of the book is devoted to the discretization problem. A solution for certain types of continuous frames was obtained by Fornasier and Rauhut using the theory of co-orbit spaces \cite{FR}. We solve the discretization problem in its full generality with the following theorem which characterizes exactly when a continuous frame may be sampled to obtain a frame.
\begin{thm}\label{T:D} Let $(X,\Sigma)$ be a measure space such that every singleton is measurable and let $\Psi:X\rightarrow H$ be measurable. There exists $(t_j)_{j\in J}\in X^J$ such that $(\Psi(t_j))_{j \in J}$
is a frame of $H$ if and only if there exists a positive, $\sigma$-finite measure $\nu$ on $(X,\Sigma)$ so that $\Psi$ is a continuous frame of $H$ with respect to $\nu$ which is bounded $\nu$-almost everywhere. In particular, every bounded continuous frame may be sampled to obtain a frame. \end{thm}
Here we mean that a continuous frame $\Psi:X\rightarrow H$ is {\em bounded $\nu$-almost everywhere} if there exists a constant $C>0$ and measurable subset $E\subseteq X$ with $\nu(E)=0$ such that $\|\Psi(t)\|\leq C$ for all $t\in X\setminus E$.
Continuous frames used in applications are typically bounded. However, there do exist examples of unbounded continuous frames which cannot be sampled to obtain a frame. For example, consider the measure $\mu$ on ${\mathbb N}$ given by $\mu(\{n\})=1/n$. Then $\Psi:{\mathbb N}\rightarrow\ell_2$ with $\Psi(n)=\sqrt{n} e_n$ is an unbounded continuous frame, and any sampling of $\Psi$ with dense span is unbounded and hence not a frame. We note as well that our sampling $(\Psi(t_j))_{j\in J}$ in the statement of Theorem \ref{T:D} allows for some points to be sampled multiple times, and indeed there exist bounded continuous frames such that the only way for a sampling to be a frame is for some points to be sampled multiple times.
Theorem \ref{T:D} characterizes when a continuous frame may be sampled to obtain a frame. The following theorem gives uniform frame bounds for the sampled frame in the case that the continuous frame is Parseval and in the unit ball of the Hilbert space. We first prove this uniform theorem in Section \ref{S:disc} and then use this to prove Theorem \ref{T:D}.
\begin{thm}\label{T:UnifSamp}
There exists $C,D>0$ such that if $\Psi:X\rightarrow H$ is a continuous Parseval frame with $\|\Psi(t)\|\leq 1$ for all $t\in X$ then there exists $(t_j)_{j\in J}\in X^J$ such that $(\Psi(t_j))_{j \in J}$ is a frame of $H$ with lower frame bound $C$ and upper frame bound $D$. \end{thm}
So far we have stated results solely in terms of sampling continuous frames. However, as every frame can be realized as a continuous frame on ${\mathbb N}$, we may consider what the discretization problem implies in this case. Suppose that $(x_n)_{n\in{\mathbb N}}$ is a tight frame of a Hilbert space $H$ with frame bound $K>1$ satisfying $\|x_n\|\leq 1$ for all $n\in{\mathbb N}$. If we define a measure $\mu$ on ${\mathbb N}$ with $\mu(n)=1/K$ for all $n\in{\mathbb N}$, then $\Psi:{\mathbb N}\rightarrow H$ with $\Psi(n)=x_n$ is a continuous Parseval frame. Now, sampling $\Psi$ corresponds with taking a subset of $(x_n)_{n\in{\mathbb N}}$. This gives the following corollary, which was proven for finite frames by Nitzan, Olevskii, and Ulanovskii \cite{NOU}.
\begin{cor}\label{C:frameSubset} There exists uniform constants $E,F>0$ such that if $(x_i)_{i\in I}$ is a tight frame of vectors in the unit ball of a Hilbert space $H$ with frame bound greater than $1$ then there exists a subset $J\subseteq I$ such that $(x_i)_{i\in J}$ is a frame with lower frame bound $E$ and upper frame bound $F$. \end{cor}
For applications, this means that if you are working with a frame which is more redundant than necessary then it is possible to a take a subset which is less redundant and is still a good frame. In Section \ref{S:part} we will give a direct proof of a generalization of Corollary \ref{C:frameSubset} which we state as Corollary \ref{C:InfPart} later in the introduction. To realize a more general application of this idea, we call a collection of vectors $(x_i)_{i \in I}$ in a Hilbert space $H$ a scalable frame if there exists scalars $(c_i)_{i\in I}$ such that $(c_ix_i)_{i\in I}$ is a Parseval frame for $H$ \cite{KOPT}. We can use this to define a measure $\mu$ on $I$ by $\mu(i)=|c_i|^2$ for all $i\in I$. Then, $\Psi: I\rightarrow H$ with $\Psi(i)=x_i$ is a continuous Parseval frame and applying Theorem \ref{T:UnifSamp} gives the following corollary.
\begin{cor}\label{C:frameScalable} There exists uniform constants $E,F>0$ such that if $(x_i)_{i\in I}$ is a scalable frame of unit vectors then there exists a subset $J\subseteq I$ such that $(x_i)_{i\in J}$ is a frame with lower frame bound $E$ and upper frame bound $F$. \end{cor} We give further discussion of scalable frames in Section \ref{S:app}, where we prove a new quantization theorem for scalable frames.
Our approach to proving Theorem \ref{T:D} in Section \ref{S:disc} is based on reducing the problem of sampling continuous frames into a different problem of partitioning discrete frames, which is solved using the recent results of Marcus, Spielman, and Srivastava in their solution of the Kadison-Singer problem \cite{MSS}.
In \cite{BCEK}, Balan, Casazza, Edidin, and Kutyniok introduce problems of determining when a tight frame for a Hilbert space may be partitioned into subsets each of which are frames for the Hilbert space and what are the optimal such decompositions. The notion of an optimal decomposition depends on the application, but one natural way to consider a partition of a tight frame into subsets as optimal is if each of the lower frame bounds of the subsets are as big as possible and each of the upper frame bounds of the subsets are as small as possible. In that respect, the following theorem, which is essentially Lemma 2 in Nitzan, Olevskii, and Ulanovskii's paper \cite{NOU}, proves that there exists uniform upper and lower frame bounds for the optimal decompositions of tight frames. They used this result to then prove that for every subset $S\subset {\mathbb R}$ of finite Lebesgue measure, there exists a discrete set $\Lambda\subset{\mathbb R}$ such that the exponentials $(e^{i\lambda t})_{\lambda\in\Lambda}$ form a frame of $L_2(S)$. We include a proof of Theorem \ref{T:reduction1} in Section \ref{S:part} for completeness as we show that it is essentially the uniform discretization problem for continuous frames with finite support.
\begin{thm}\label{T:reduction1} There exists uniform constants $A,B>0$ such that every tight frame of vectors in the unit ball of a finite dimensional Hilbert space $H$ with frame bound greater than $1$ can be partitioned into a collection of frames of $H$ each with lower frame bound $A$ and upper frame bound $B$. Furthermore, there exists sequences of constants $(A_n)_{n\in{\mathbb N}},(B_n)_{n\in{\mathbb N}}$ with $\lim_{n\rightarrow\infty}A_n=\lim_{n\rightarrow\infty}B_n=1$ such that for every $N\in{\mathbb N}$ and every tight frame of vectors in $N^{-1}B_H$ with frame bound greater than $1$ has a subset with lower frame bound $A_N$ and upper frame bound $B_N$. \end{thm}
At first glance, it may seem obvious that a highly redundant frame is the union of less redundant frames, but in a high dimensional Hilbert space it can be very tricky to determine how to partition the frame vectors so that each set in the resulting partition has a uniformly high lower frame bound and uniformly low upper frame bound. Furthermore, we prove that the corresponding question for bases is false. That is, there does not exist a uniform constant $A$ such that every finite unit norm tight frame has a subset which is a basis and has lower Riesz bound $A$. Using this finite dimensional result, we prove the following generalization at the end of Section \ref{S:part}.
\begin{cor}\label{C:InfPart} Let $A,B>0$ be the constants given in Theorem \ref{T:reduction1}. Then every tight frame of vectors in the unit ball of a separable Hilbert space $H$ with frame bound greater than $1$ can be partitioned into a finite collection of frames of $H$ each with lower frame bound $A$ and upper frame bound $B$. \end{cor}
The proof of Theorem \ref{T:reduction1} relies on the recent solution of the Kadison-Singer problem by Marcus, Spielman, and Srivastava \cite{MSS}. The Kadison-Singer problem \cite{KS} was known to be equivalent to many open problems such as the Feichtinger conjecture \cite{CCLV}, the paving conjecture \cite{A}, Weaver's conjecture \cite{W}, and the Bourgain-Tzafriri conjecture \cite{BT}. Each of these problems can be thought of in some ways as determining when a set with some property can be uniformly partitioned into sets with a desired property. Naturally, the frame partition problem falls into this category as well, and it was noted in \cite{BCEK} that the problem of partitioning a large frame into smaller frames is related to these famous problems. In \cite{MSS}, the authors directly prove Weaver's conjecture and hence prove all the equivalent problems as well. We note that the proof of Theorem \ref{T:reduction1} doesn't actually use Weaver's conjecture or any of its equivalent formulations, but instead uses the stronger result proved in \cite{MSS}. The main reason for this is that Weaver's conjecture concerns partitioning in a way that reduces an upper bound, but we need to reduce an upper bound while maintaining a relatively close lower bound. The Marcus, Spielman, Srivastava result allows partitioning in a way that divides both the upper and lower frame bound almost perfectly in half.
We recommend the textbook \cite{C} for a reference on frames and continuous frames from a mathematical perspective, and we recommend the textbook \cite{AAG2} for a reference on frames and continuous frames from a physics perspective. We include applications of the discretization and partitioning theorems in Section \ref{S:app}. We prove the frame partitioning theorem in Section \ref{S:part}. We include lemmas about discrete frames in Section \ref{S:lem}. We prove the discretization theorem in Section \ref{S:disc}.
\section{Applications}\label{S:app} We include here some applications of the discretization and partition theorems, two of which are new theorems and one is a quick proof of a known theorem.
\subsection{Scalable Frames}\label{S:SF}
A collection of unit vectors $(x_i)_{i \in I}$ in $H$ is said to be a scalable frame if there exist scalars $(c_i)_{i\in I}$ such that $(c_ix_i)_{i\in I}$ is a Parseval frame for $H$ \cite{KOPT}. As one of the steps toward solving the discretization problem, we prove in Theorem \ref{T:samp} that there are universal constants $A$ and $B$ such that if $(x_i)_{i\in I}$ is a scalable frame, then $(x_i)_{i\in I}$ can be sampled to form a frame with lower frame bound $A$ and upper frame bound $B$. One way to think about this is in terms of quantization. A scalable frame can be scaled to be Parseval, but suppose that we are restricted to using only integer coefficients to scale the frame. Being able to sample $(x_i)_{i \in I}$ to get a frame is equivalent to being able to obtain a frame by scaling using only integer scalars. As quantization of frame coefficients is an important aspect of frame theory \cite{BPY}, it is natural to consider scaling frames using quantized scalars. The following result gives essentially that how well we can scale a frame using quantized coefficients may be determined using only how fine a quantization we allow. In particular, this is independent of both the dimension of the space and the number of frame vectors.
\begin{thm}\label{T:QS} Let $(A_n)_{n\in{\mathbb N}}$ and $(B_n)_{n\in{\mathbb N}}$ with $\lim A_n=\lim B_n=1$ be the scalars given in Theorem \ref{T:reduction1}. Let $N\in{\mathbb N}$. If $(x_i)_{i \in I}$ is a scalable frame in a finite dimensional Hilbert space $H$ then there exists scalars $(c_i)_{i\in I}\subseteq\{ \sqrt{m}/N:m\in{\mathbb Z}\}$ such that $(c_i x_i)_{i\in I}$ is a frame with lower frame bound $A_N$ and upper frame bound $B_N$. \end{thm} \begin{proof} Let $(d_i)_{i\in I}$ be scalars such that $(d_i x_i)_{i\in I}$ is a Parseval frame. We first assume that $(d_i)_{i\in I}$ are rational numbers with common denominator $M\in{\mathbb N}$. Thus, $n_i:=N^2 M^2 d^2_i$ is an integer for all $i\in I$. We consider the frame $(y_j)_{j\in J}$ which consists of $n_i$ copies of $N^{-1} x_i$ for each $i\in I$. Thus, we have for $x\in H$ that
$$\sum_{j\in J} |\langle x, y_j\rangle|^2=\sum_{i\in I} N^2 M^2 d_i^2 |\langle x, N^{-1}x_i\rangle|^2=M^2\sum_{i\in I} |\langle x, d_ix_i\rangle|^2=M^2\|x\|^2. $$ Thus, $(y_j)_{j\in J}$ is a tight frame of vectors in $N^{-1} B_H$ with frame bound $M^2\geq 1$. By Theorem \ref{T:reduction1} there is a subset $(y_j)_{j\in J_0}$ with frame bounds $A_N$ and $B_N$. For each $i\in I$ let $c_i=\sqrt{m_i}/N$ where $m_i$ is the number of copies of $N^{-1} x_i$ in $(y_j)_{j\in J_0}$. This gives the following calculation.
$$\sum_{i\in I} |\langle x, c_i x_i\rangle|^2=\sum_{i\in I} m_i|\langle x, N^{-1}x_i\rangle|^2=\sum_{j\in J_0} |\langle x, y_j\rangle|^2.$$ Thus, $(c_i x_i)_{i\in I}$ is a frame with bounds $A_N, B_N$, the same bounds as $(y_j)_{j\in J_0}$.
We now consider the case that $(d_i)_{i\in I}$ are not all rational. Given $\varepsilon>0$ we may approximate the coefficients $(d_i)_{i\in I}$ with rational numbers and follow the previous argument to obtain $(c_i)_{i\in I}\subseteq\{ \sqrt{m}/N:m\in{\mathbb Z}\}$ such that $(c_i x_i)_{i\in I}$ is a frame with lower frame bound $(1+\varepsilon)^{-1}A_N$ and upper frame bound $(1+\varepsilon)B_N$. However, because there are only finitely many possibilities for $(c_i)_{i\in I}$, some choice must work for all $\varepsilon>0$. Thus, there exists $(c_i)_{i\in I}\subseteq\{ \sqrt{m}/N:m\in{\mathbb Z}\}$ such that $(c_i x_i)_{i\in I}$ is a frame with lower frame bound $A_N$ and upper frame bound $B_N$.
\end{proof}
Theorem \ref{T:QS} gives that if there are scalars $(d_i)_{i\in I}$ such that $(d_i x_i)_{i \in I}$ is Parseval then there are scalars $(c_i)_{i\in I}\subseteq\{ \sqrt{m}/N:m\in{\mathbb Z}\}$ such that $(c_i x_i)_{i\in I}$ is a frame with lower frame bound $A_N$ and upper frame bound $B_N$. It is interesting to note that the scalars $(c_i)_{i\in I}$ could be very different from the scalars $(d_i)_{i\in I}$. Indeed, quantizing $(d_i)_{i\in I}$ fails dramatically if we choose $(c_i)_{i\in I}$ to minimize $|d_i-c_i|$ for all $i\in I$, which results in a small $\ell_\infty(I)$ distance between the sequences $(d_i)_{i\in I}$ and $(c_i)_{i\in I}$. A small $\ell_2(I)$ distance between $(d_i)_{i\in I}$ and $(c_i)_{i\in I}$ can be used to compare the frames $(d_i x_i)_{i \in I}$ and $(c_i x_i)_{i \in I}$ \cite{C2}, but a small $\ell_\infty(I)$ distance tells us nothing.
\subsection{Gabor Frames}\label{S:GF}
One example of continuous frames that is of particular interest is that of Gabor systems and the short time Fourier transform. We consider $L^2({\mathbb R})$ to be the Hilbert space of square integrable functions from ${\mathbb R}$ to ${\mathbb C}$. For $a, b \in {\mathbb R}$, we define the translation operator $T_a:L^2({\mathbb R}) \to L^2({\mathbb R})$ and the modulation operator $M_b:L^2({\mathbb R}) \to L^2({\mathbb R})$ by \[ T_a g(x) = g(x - a) \qquad {\mathrm {and}} \qquad M_b g (x) = e^{2\pi i b x} g(x). \] If $g\in L^2({\mathbb R})$ then {\em the short time Fourier transform} with window function $g$ is the map $\Psi_g : {\mathbb R}^2 \to L^2({\mathbb R})$ given by \[ \Psi_g(a,b) = M_b T_a g. \]
The short time Fourier transform with window function $g$ is a tight continuous frame with frame bound $\|g\|^2$. That is, $\|g\|^2 \|f\|^2=\int \int |\int \overline{f(x)} e^{2\pi i \omega x}g(x- t) dx|^2 d\omega dt$ for all $f\in L_2({\mathbb R})$. A {\em Gabor frame} of $L_2({\mathbb R})$ is a frame of the form $(M_{bn} T_{am} g)_{m,n\in{\mathbb Z}}$ where $a,b>0$ and $g\in L^2({\mathbb R})$. That is, Gabor frames are formed by sampling the short time Fourier transform at a lattice in ${\mathbb R}^2$. It is not always the case that $(M_{bn} T_{am} g)_{m,n\in{\mathbb Z}}$ will be a frame, however by Theorem \ref{T:D} we have the following corollary. \begin{cor}\label{C:GF}
For every non-zero $g\in L^2({\mathbb R})$ there exists real numbers $(a_k, b_k)_{k=1}^\infty$ such that $(e^{2\pi i b_k x} g(x - a_k))_{k\in{\mathbb N}}$ is a frame of $L^2({\mathbb R})$. \end{cor}
\subsection{Frames of exponentials}\label{S:SF} For each $K>0$, Fourier series gives a Riesz basis of exponentials for $L_2([-\frac{K}{2},\frac{K}{2}])$. In particular, $$ K f = \sum_{n\in{\mathbb Z}} \langle f, e^{2\pi i \frac{n}{K}\cdot}\rangle e^{2\pi i \frac{n}{K} \cdot} \quad\quad \textrm{ for all }f\in L_2([-\frac{K}{2},\frac{K}{2}]). $$
If $J\subseteq {\mathbb R}$ is any bounded set, then $J\subseteq I$ where $I$ is an interval. We can take the basis of exponentials for $L_2(I)$ and restrict it to $J$ to get a tight frame of exponentials for $L_2(J)$. However, this does not work if $J$ is unbounded. This leads to the question: When does $L_2(J)$ have a frame of exponentials? Note that $J$ must have finite measure for the exponentials to be in $L_2(J)$. This was solved by Nitzan, Olevskii, and Ulanoskii, who used the same frame partitioning theorem that we need as well \cite{NOU}.
\begin{thm}\label{T:NOU}[Nitzan, Olevskii, and Ulanovskii 2016] If $J\subseteq{\mathbb R}$ has finite measure then $L_2(J)$ has a frame of exponentials. \end{thm}
Unlike the short time Fourier transform, the Fourier transform is not a continuous frame. However, if we consider the map $\Psi: {\mathbb R}\rightarrow L_2(J)$ given by $\Psi(x)(t)=e^{2\pi i x t}$ for all $x\in {\mathbb R}$ and $t\in J$ then $\Psi$ has an analysis operator $\Theta:L_2(J)\rightarrow L_2({\mathbb R})$ given by $$\Theta(f)(x)=\langle f, \Psi(x)\rangle=\int_{t\in J} f(t) e^{2\pi i x t} d\,\lambda(t). $$ Thus, the analysis operator is the Fourier transform. The Fourier transform is an isometric embedding, which means that $\Psi$ is a continuous Parseval frame. Hence, by the discretization theorem, $\Psi$ may be sampled to give a frame of exponentials for $L_2(J)$ which gives Theorem \ref{T:NOU} as a corollary.
\section{Frame Partitions}\label{S:part}
Our goal for this section is to prove Theorem \ref{T:reduction1}, on uniformly partitioning frames. The main ingredient of the proof is the following theorem of Marcus, Spielman, and Srivistava.
\begin{thm}[MSS Cor 1.5] \label{MSS} Let $(u_i)^M_{i=1} \subseteq H$ be a Bessel sequence with bound $1$ and $\|u_i\|^2 \leq \delta$ for all $i$. Then for any positive integer $r$, there exists a partition $\{I_1, . . . , I_r\}$ of $[M]$ such that each $(u_i)_{i\in I_j}$ , $ j = 1, . . . , r$ is a Bessel sequence with bound $$(1/\sqrt{r}+\sqrt{\delta})^2 $$ \end{thm}
We will be applying Theorem \ref{MSS} for $r=2$ to partition a Parseval frame into two sets with Bessel bound close to $1/2$. Theorem \ref{MSS} gives good control of the upper frame bound when partitioning a Parseval frame, but we need to control the lower frame bound as well. The following theorem can be applied to show that if a Parseval frame is partitioned into two sets with upper frame bound close to $1/2$ then the sets also have lower frame bound close to $1/2$.
\begin{thm}[BCMS Cor 4.6]\label{46} Let $P:\ell^2(I)\rightarrow\ell^2(I)$ be orthogonal projection onto a closed subspace $H\subseteq \ell^2(I)$. Then for any subset $J\subset I$ and $\delta>0$, TFAE \begin{enumerate} \item $\{P e_i\}_{i\in J}$ is a frame of $H$ with frame bounds $\delta$ and $1-\delta$. \item $\{P e_i\}_{i\in J^c}$ is a frame of $H$ with frame bounds $\delta$ and $1-\delta$. \item Both $\{P e_i\}_{i\in J}$ and $\{P e_i\}_{i\in J^c}$ are Bessel with bounds $1-\delta$. \item Both $\{(I-P) e_i\}_{i\in J}$ and $\{(I-P) e_i\}_{i\in J^c}$ are Riesz sequences with lower bound $\delta$. \end{enumerate}
\end{thm}
We will be repeatedly partitioning a frame using Theorem \ref{MSS} and then applying a positive self-adjoint invertible operator to the resulting sets. The following simple lemma allows us to keep track of what the operators do to the frame bounds.
\begin{lem}\label{L:operator} Let $(x_j)_{j\in J}$ be a Parseval frame of a Hilbert space $H$. Let $T$ be a positive self adjoint invertible operator on $H$. Then
$(Tx_j)_{j\in J}$ is a frame of $H$ with upper frame bound $\|T\|^2$
and lower frame bound $\|T^{-1}\|^{-2}$ \end{lem} \begin{proof} Let $x\in H$. To calculate the upper frame bound we have the following inequalities.
$$\sum_{j\in J}|\langle x,Tx_j\rangle|^2=\sum_{j\in J}|\langle Tx,x_j\rangle|^2 =\|Tx\|^2\leq\|T\|^2\|x\|^2. $$
Thus $\|T\|^2$ is the upper frame bound. To calculate the lower frame bound we have the following inequalities.
$$\|T^{-1}\|^{-2}\|x\|^2\leq \|Tx\|^2= \sum_{j\in J}|\langle Tx,x_j\rangle|^2=\sum_{j\in J}|\langle x,Tx_j\rangle|^2. $$
Thus, $\|T^{-1}\|^{-2}$ is the lower frame bound.
\end{proof}
We are now ready to prove the main result of this section. The first part appears essentially as Lemma 2 in \cite{NOU}, but we include a proof here for completion as it is essentially the uniform discretization problem for continuous frames with finite support.
\begin{thm}\label{T:reduction} There exists uniform constants $A,B>0$ such that every tight frame of vectors in the unit ball of a finite dimensional Hilbert space $H$ with frame bound greater than $1$ can be partitioned into a collection of frames of $H$ each with lower frame bound $A$ and upper frame bound $B$. Moreover, there exists sequences of constants $(A_n)_{n\in{\mathbb N}},(B_n)_{n\in{\mathbb N}}$ with $\lim_{n\rightarrow\infty}A_n=\lim_{n\rightarrow\infty}B_n=1$ such that for every $N\in{\mathbb N}$ and every tight frame of vectors in $N^{-1}B_H$ with frame bound greater than $1$ has a subset with lower frame bound $A_N$ and upper frame bound $B_N$. \end{thm}
\begin{proof} We prove the first claim of the theorem and then discuss at the end how the proof could be adapted to prove the moreover claim.
For convenience, we will only consider tight frames with frame bound at least $79$. Then we will find a uniform constant $B> 79$ so that every such tight frame can be partitioned into frames with upper frame bound $B$ and lower frame bound $79$. Thus, any tight frame in the unit ball of a finite dimensional Hilbert space with frame bound greater than $1$ could be partitioned into a set of frames of $H$ with upper frame bound $B$ and lower frame bound $1$.
The proof will involve repeated application of Theorem \ref{MSS} so that at each step we will partition a frame into two frames with the same upper frame bound and same lower frame bound. We will then choose one of those frames to partition further until we arrive at a set which is close to being tight and has small upper frame bound. As we could do the same procedure to the frames not chosen, we are able to partition our original frame into frames which are close to being tight and have small upper frame bound.
Assume that $(x_j)_{j\in J_0}$ is a tight frame in the unit ball of $H$ with frame bound $B_0\geq 79$.
We recursively define a decreasing sequence $B_0>B_1>\cdots > B_n$ by $B_{m+1}= 2^{-1}B_m-2^{1/2}B_m^{1/2}-1$ for all $1\leq m< n$ where $n\in{\mathbb N}_0$ is such that $200>B_n\geq 79$. Note that $79= 2^{-1}200-2^{1/2}200^{1/2}-1$ and thus there is a unique $n\in{\mathbb N}_0$ such that $200>B_n\geq 79$.
We will choose by induction a nested sequence of subsets $J_0\supseteq J_1\supseteq... \supseteq J_n$ so that if $1\leq m\leq n$, $T_0$ is the identity, and $T_m$ is the frame operator of $(T^{-1/2}_{m-1}...T^{-1/2}_1 B_0^{-1/2} x_j)_{j\in J_{m}}$ then \begin{equation}\label{E:1}
\|T^{-1/2}_{m}...T^{-1/2}_0 B_0^{-1/2}\|^2\leq B_m^{-1}\quad\textrm{ for }0\leq m\leq n, \end{equation} \begin{equation}\label{E:2}
\|T_{m}\|\leq 2^{-1}+ 2^{1/2}B_m^{-1/2}+B_m^{-1} \quad\textrm{ for }1\leq m\leq n, \end{equation} \begin{equation}\label{E:3}
\|T_m^{-1}\|\leq (2^{-1}- 2^{1/2}B_m^{-1/2}-B_m^{-1})^{-1} \quad\textrm { for }1\leq m\leq n. \end{equation}
For the base case $m=0$ we have that \eqref{E:1}, \eqref{E:2} and \eqref{E:3} are all trivially satisfied.
Let $0\leq m<n$ and assume that $J_0\supseteq\cdots \supseteq J_m$ have been chosen to satisfy \eqref{E:1}, \eqref{E:2}, and \eqref{E:3}.
As $T_{m}$ is the frame operator of $(T^{-1/2}_{m-1}...T^{-1/2}_1 B_0^{-1/2} x_j)_{j\in J_{m}}$ we have that $(T^{-1/2}_{m}...T^{-1/2}_1 B_0^{-1/2} x_j)_{j\in J_{m}}$ is a Parseval frame. Furthermore, $\|T^{-1/2}_{m}...T^{-1/2}_1 B_0^{-1/2} x_j\|\leq B^{-1}_m$ for all $j\in J_m$ by \eqref{E:1}, thus we may apply Theorem \ref{MSS} with $r=2$ to obtain $J_{m+1}\subseteq J_m$ such that both $(T^{-1/2}_{m}...T^{-1/2}_1B_0^{-1/2}x_j)_{j\in J_{m+1}}$ and $(T^{-1/2}_{m}...T^{-1/2}_1B_0^{-1/2}x_j)_{j\in J_m\setminus J_{m+1}}$ have Bessel bounds $2^{-1}+ 2^{1/2}B_m^{-1/2}+B_m^{-1}$. As $B_m\geq 200$, we have that this bound is smaller than 1. By Theorem \ref{46} we have that $(T^{-1/2}_{m}...T^{-1/2}_1B_0^{-1/2}x_j)_{j\in J_{m+1}}$ has lower frame bound $2^{-1}- 2^{1/2}B_m^{-1/2}-B_m^{-1}>0$. Thus the frame operator $T_{m+1}$ of $(T^{-1/2}_{m}...T^{-1/2}_1B_0^{-1/2}x_j)_{j\in J_{m+1}}$ has
$\|T_{m+1}\|\leq 2^{-1}+ 2^{1/2}B_m^{-1/2}+B_m^{-1}$ and
$\|T_{m+1}^{-1}\|\leq (2^{-1}- 2^{1/2}B_m^{-1/2}-B_m^{-1})^{-1}$ which satisfies inequality \eqref{E:2} and \eqref{E:3}. We have that \begin{align*}
\|T_{m+1}^{-1/2}...T^{-1/2}_1 B_0^{-1/2}\|^2 &\leq \|T_{m+1}^{-1/2}\|^2\|T^{-1/2}_{m}...T^{-1/2}_1B_0^{-1/2}\|^2\\
&\leq \|T_{m+1}^{-1/2}\|^2 B_m^{-1} \quad\textrm{ by }\eqref{E:1}\\
&= \|T_{m+1}^{-1}\| B_m^{-1} \quad\textrm{ as }T_{m+1}\textrm{ is a positive operator}\\ &\leq (2^{-1}- 2^{1/2}B_m^{-1/2}-B_m^{-1})^{-1} B_m^{-1} \quad\textrm{ by }\eqref{E:3}\\ &= (2^{-1}B_m- 2^{1/2}B_m^{1/2}-1)^{-1}=B_{m+1}^{-1} \end{align*} Thus, the inequality \eqref{E:1} is satisfied and our induction is complete.
We have that $(T^{-1/2}_{n}...T^{-1/2}_1 B_0^{-1/2} x_j)_{j\in J_{n}}$ is a Parseval frame and that $79\leq B_n<200$. By Lemma \ref{L:operator} we have that $(x_j)_{j\in J_n}$ is a frame with upper frame bound
$\|T^{1/2}_{n}...T^{1/2}_1 B_0^{1/2}\|^2$ and lower frame bound $\|T^{-1/2}_{n}...T^{-1/2}_1 B_0^{-1/2}\|^{-2}$. By \eqref{E:2}, the upper frame bound of $(x_j)_{j\in J_n}$ is at most
$$ \|T^{1/2}_{n}...T^{1/2}_1 B_0^{1/2}\|^2\leq \|T_{n}\|...\|T_1\| B_0\leq B_0\prod_{0\leq m< n} (2^{-1}+ 2^{1/2}B_m^{-1/2}+B_m^{-1})=:B.$$
By \eqref{E:3}, the lower frame bound of $(x_j)_{j\in J_n}$ is at least
$$ \|T_{n}^{-1}\|^{-1}...\|T^{-1}_1\|^{-1} B_0\geq B_0\prod_{0\leq m< n} (2^{-1}- 2^{1/2}B_m^{-1/2}-B_m^{-1})= B_0\prod_{0\leq m< n} B_{m+1}B_m^{-1}=B_n=:A.$$
We now have an upper frame bound $B$ and lower frame bound $A$ for $(x_j)_{j\in J_n}$.
If there exists a constant $C$ such that the ratio of the frame bounds $B/A$ is uniformly bounded by $C$, then we would have a lower frame bound of $A=B_n\geq 79$ and an upper frame bound of $B\leq AC\leq 200C$. This would prove that every tight frame of vectors in the unit ball of a finite dimensional Hilbert with frame bound greater than $79$ can be partitioned into frames each of which has upper frame bound $200C$ and lower frame bound $79$. Thus, all we need to prove is that $B/A$ is uniformly bounded. We have that \begin{align*} \ln (B/A)&= \ln(\prod \frac{ 2^{-1}+ 2^{1/2}B_m^{-1/2}+B_m^{-1}}{2^{-1}- 2^{1/2}B_m^{-1/2}-B_m^{-1}}) \\ &=\ln( \prod \frac{ 1+ 2^{3/2}B_m^{-1/2}+2B_m^{-1}}{1 - 2^{3/2}B_m^{-1/2}-2B_m^{-1}})\\ &\leq \ln(\prod 1+ 10 B_m^{-1/2} )\quad \textrm{ as }B_m\geq 79 \\ &=\sum \ln(1+ 10 B_m^{-1/2} )\\ &\leq\sum 10 B_m^{-1/2} \\ &\leq\sum_{m=0}^\infty 10 (79/200)^{m/2} 79 ^{-1/2}<\infty. \end{align*}
Thus, we have proven that every tight frame of vectors in the unit ball of a finite dimensional Hilbert space $H$ with frame bound greater than $1$ can be partitioned into a collection of frames of $H$ each with lower frame bound $A$ and upper frame bound $B$. As part of the proof, we implicitly showed that for all $\varepsilon>0$ there exists $a_\varepsilon,b_\varepsilon,D_\varepsilon>0$ such that any tight frame of vectors in the unit ball of a finite dimensional Hilbert space with frame bound greater than $D_\varepsilon$ may be partitioned into frames with frame bounds $a_\varepsilon<b_\varepsilon \leq D_\varepsilon$ such that $b_\varepsilon/a_\varepsilon<1+\varepsilon$. We now show how this can be used to prove the moreover claim. For $1\leq n\leq D_1$ we let $A_n=A$ and $B_n=B$. Let $N\in{\mathbb N}$ such that $D_1< N$. Choose $\varepsilon>0$ to be the smallest value such that $D_\varepsilon\leq N$. Set $B_{N}=1+N^{-1}$ and $A_N=B_N (1+\varepsilon)^{-1}$. Note that $\lim A_N=\lim B_N=1$. We now need to show that every tight frame of vectors with norm at most $N^{-1}$ and frame bound greater than 1 contains a subset with frame bounds $A_N$ and $B_N$.
Let $(x_j)_{j\in J}$ be a tight frame of vectors with norm at most $N^{-1}$ and frame bound greater than 1. Thus, $(N x_j)_{j\in J}$ is a tight frame of vectors in the unit ball of $H$ with frame bound greater than $D_\varepsilon$ and hence may be partitioned into frames with frame bounds $a_\varepsilon$ and $b_\varepsilon$. This gives a partition of $(x_j)_{j\in J}$ into frames $((x_j)_{j\in J_n})_{n\leq M}$ each with bounds $a_\varepsilon N^{-2}$ and $b_\varepsilon N^{-2}$. Let $(x_j)_{j\in I}$ be a frame formed by combining frames in $((x_j)_{j\in J_n})_{n\leq M}$ such that $(x_i)_{i\in I}$ has the smallest possible upper frame bound greater than 1. If we remove some frame $(x_j)_{j\in J_n}$ from $(x_j)_{j\in I}$ then the resulting frame $(x_j)_{j\in I\setminus J_n}$ has an upper frame bound of 1. Thus, $(x_j)_{j\in I}$ has upper frame bound $B_N=1+N^{-1}$ as $(x_j)_{j\in J_n}$ has upper frame bound $b_\varepsilon N^{-2}<N^{-1}$. As $(x_j)_{j\in I}$ is a union of frames whose ratio of their frame bounds is at most $1+\varepsilon$, we have that the ratio of the frame bounds of $(x_j)_{j\in I}$ is at most $1+\varepsilon$. Thus, $(x_j)_{j\in I}$ has lower frame bound $A_N=B_N (1+\varepsilon)^{-1}$.
\end{proof}
Theorem \ref{T:reduction} is stated only for tight frames with frame bound greater than 1. The following corollary applies to partitioning any frame with lower frame bound greater than 1.
\begin{cor}\label{C:frame_part}
Let $(f_j)_{j\in J}$ be a frame of a Hilbert space $H$ with upper frame bound $B_0$ and lower frame bound $A_0\geq 1$ such that $\|f_j\|\leq1$ for all $j\in J$ then $(f_j)_{j\in J}$ can be partitioned into a collection of frames of $H$ each with lower frame bound $A$ and upper frame bound $B B_0 A_0^{-1}$. Where $A$ and $B$ are the constants given in Theorem \ref{T:reduction}. \end{cor} \begin{proof}
Let $T$ be the frame operator of $(f_j)_{j\in J}$. Then $\|T\|\leq B_0$ and $\|T^{-1}\|\leq A_0^{-1}$. We have that $(A_0^{1/2}T^{-1/2}f_j)_{j\in J}$ is a tight frame with frame bound $A_0\geq1$. For all $j\in J$ we have that $\|A_0^{1/2}T^{-1/2}f_j\|\leq A_0^{1/2}\|T^{-1}\|^{1/2}\|f_j\|\leq1$. By Theorem \ref{T:reduction} there is a partition $(J_n)_{1\leq n\leq M}$ of $J$ such that $(A_0^{1/2}T^{-1/2}f_j)_{j\in J_n}$ has upper frame bound $B$ and lower frame bound $A$ for each $1\leq n\leq M$. Let $x\in H$ and $1\leq n\leq M$. Then,
$$\sum_{j\in J_n} |\langle f_j,x\rangle|^2=\sum_{j\in J_n} |\langle A_0^{1/2} T^{-1/2} f_j, A_0^{-1/2} T^{1/2}x\rangle|^2\leq B \|A_0^{-1/2}T^{1/2} x\|^2\leq B A_0^{-1} B_0\|x\|^2 $$ Thus, $(f_j)_{j\in J_n}$ has upper frame bound $B B_0 A_0^{-1}$. We now check the lower frame bound.
$$\sum_{j\in J_n} |\langle f_j,x\rangle|^2=\sum_{j\in J_n} |\langle A_0^{1/2} T^{-1/2} f_j, A_0^{-1/2} T^{1/2}x\rangle|^2\geq A \|A_0^{-1/2}T^{1/2} x\|^2\geq A A_0^{-1} A_0\|x\|^2=A\|x\|^2 $$ Thus, $(f_j)_{j\in J_n}$ has lower frame bound $A$.
\end{proof}
We now restate and prove Corollary \ref{C:InfPart} from the Introduction, which proves that Theorem \ref{T:reduction} holds for infinite frames as well. \begin{cor} Let $A,B>0$ be the constants given in Theorem \ref{T:reduction1}. Then every tight frame of vectors in the unit ball of a separable Hilbert space $H$ with frame bound greater than $1$ can be partitioned into a finite collection of frames of $H$ each with lower frame bound $A$ and upper frame bound $B$. \end{cor}
\begin{proof} Let $(f_j)_{j=1}^\infty$ be a tight frame of vectors in the unit ball of $H$ with frame bound $K>1$. For each $n\in{\mathbb N}$ let $H_n=\textrm{span}_{1\leq j\leq n}f_j$ and let
$(g_{j,n})_{j\in I_n}$ be a finite collection of vectors in the ball of $H_n$ so that $(f_j)_{j=1}^n\cup (g_{j,n})_{j\in I_n}$ is a $K$-tight frame for $H_n$. By Theorem \ref{T:reduction1} we have that $(f_j)_{j=1}^n\cup (g_{j,n})_{j\in I_n}$ may be partitioned into a collection of frames $((f_j)_{j\in J_{i,n}}\cup (g_{j,n})_{j\in I_{i,n}})_{i=1}^{M_n}$ of $H_n$ each with lower frame bound $A$ and upper frame bound $B$. We first obtain an upper bound on $M_n$. For each $1\leq i\leq M_n$ we have that $(f_j)_{j\in J_{i,n}}\cup (g_{j,n})_{j\in I_{i,n}}$ has lower frame bound $A$ and that the entire collection of vectors $(f_j)_{j=1}^n\cup (g_{j,n})_{j\in I_n}$ has frame bound $K$. Thus, we have that $AM_n\leq K$. We let $M=\lfloor K/A\rfloor$. Thus for each $n\in{\mathbb N}$ we may consider the partitioning to be of the form $((f_j)_{j\in J_{i,n}}\cup (g_{j,n})_{j\in I_{i,n}})_{i=1}^{M}$ where we allow for sets to be empty.
For a given $j\in{\mathbb N}$, we can have that the index $1\leq i\leq M$ so that $j\in J_{i,n}$ can change depending on $n$. However, this can be stabilized by passing to a subsequence, which is what Pete Casazza refers to as the pinball principle. We choose a subsequence $(k_n)_{n\in{\mathbb N}}$ of ${\mathbb N}$ so that for all $j\in{\mathbb N}$ there exists $1\leq m_j\leq M$ so that $j\in J_{m_j,k_n}$ for all $1\leq j\leq n$. For each $1\leq i\leq M$, we let $J_i=\liminf_{n\rightarrow\infty} J_{i,k_n}$. This gives that $(J_i)_{1\leq i\leq M}$ is a partition of ${\mathbb N}$. We have that $(f_j)_{j\in J_i}$ has Bessel bound $B$ as $(f_j)_{j\in J_{i,n}}$ has Bessel bound $B$ for all $n\in{\mathbb N}$. We now prove that if $J_i\not= \emptyset$ then $(f_j)_{j\in J_i}$ is a frame of $H$ with lower frame bound $A$.
Let $x\in H$. We have that $\lim_{n\rightarrow\infty}\sum_{1\leq j\leq k_n} |\langle f_j,x \rangle|^2 =K\|x\|^2$. Hence, $\lim_{n\rightarrow\infty}\sum_{j\in I_{k_n}} |\langle g_{j,k_n},x \rangle|^2=0$ as $(f_j)_{j=1}^n\cup (g_{j,n})_{j\in I_n}$ has frame bound $K$. Thus for all $1\leq i\leq M$ we have that \begin{align*}
\sum_{j\in J_i}|\langle f_j,x\rangle|^2&=\lim_{n\rightarrow\infty} \sum_{j\in J_{i,k_n}}|\langle f_j,x\rangle|^2\\
&=\lim_{n\rightarrow\infty} \sum_{j\in J_{i,k_n}}|\langle f_j,x\rangle|^2+ \sum_{j\in I_{i,k_n}}|\langle g_{j,k_n},x\rangle|^2\\
&\geq A\|x\|^2 \end{align*} Thus $(f_j)_{j\in J_i}$ has lower frame bound $A$. \end{proof}
We note here that it is not possible to improve Theorem \ref{T:reduction} to show that every FUNTF with sufficiently many vectors contains a good \emph{basis}. A {\em FUNTF } (or finite unit norm tight frame) is a finite collection of unit vectors which form a tight frame. If $k\geq n$ are natural numbers then there always exists a FUNTF of $k$ vectors for an $n$-dimensional Hilbert space \cite{BF}, and FUNTFs are particularly useful in application due to their resilience to error \cite{GKK}.
\begin{thm} For every $\epsilon > 0$ and every $B > 1$, there exists an $M > B$ and a FUNTF $(x_i)_{1\leq i \leq M}$ such that whenever $I\subset [1,M]$ is such that $(x_i)_{i\in I}$ is a basis, then the lower Riesz constant of $(x_i)_{i\in I}$ is less than $\epsilon$. \end{thm}
\begin{proof} We modify slightly the construction of Casazza, Fickus, Mixon and Tremain from Proposition 3.1 in \cite{CFMT}. Let $H_n$ be the $2^n \times 2^n$ Hadamard matrix obtained via tensor products of \[ \begin{pmatrix} 1&1\\1&-1 \end{pmatrix}. \] Let $F_n^1$ be the matrix obtained by multiplying the first $2^{n-1}-1$ columns of $H_n$ by $\sqrt{1/2^{n-1}}$, and the remaining $2^{n-1}+1$ columns by $\sqrt \frac 1{2^{n-1}(2^{n-1}+1)}$. Let $F_n^2$ be the matrix obtained by multiplying the first $2^{n-1} - 1$ columns of $H_n$ by 0 and the remaining $2^{n-1}+1$ columns by $\sqrt \frac 1{2^{n-1}+1}$. Let $F_n$ be the $2^{n+1}\times 2^n$ matrix obtained by ``stacking" the $F_n^1$ on top of $F_n^2$. Note that the $2^n$ columns of $F_n$ are orthogonal, the rows have norm one, and the columns have norm-squared 2. We denote the $j$th row of $H_n$ by $h_j$, the $j$th row of $F_n$ by $x_j$, and we note that $\{g_j = 2^{-n/2} h_j: 1\le j \le 2n\}$ is an orthonormal basis.
Let $I\subset [1,2^{n+1}]$ be of size $2^n$. We show that $(x_i)_{i \in I}$ cannot have good lower Riesz bound. Case 1: If $|I \cap [1,2^n] | > 2^{n-1} - 1$, then $(x_i)_{i\in I}$ has lower Riesz bound less than $\frac{2}{2^{n-1} +1}$. Indeed, let $J = I\cap[ [1,2^n]$ and let $P$ denote the orthogonal projection onto the first $2^{n-1} -1$ coordinates. Choose scalars $(c_j)$ such that \[
\sum_{j\in J} |c_j|^2 = 1 \] and \[ P\bigl(\sum_{j\in J} c_j x_j \bigr) = 0. \] We then have, \begin{align*}
\| \sum_{j\in J} c_j x_j\|^2 &= \|\sum_{j\in J} (Id - P) c_j x_j \|^2 \\
&= \frac{2}{2^{n-1} + 1} \|(Id - P) \sum _{j\in J} c_j g_j\|^2 \\
&\le \frac{2}{2^{n-1} + 1} \| \sum _{j\in J} c_j g_j\|^2 \\
&= \frac{2}{2^{n-1} + 1}. \end{align*} Thus the lower Riesz bound of $(x_j)_{j\in J}$ (and hence of $(x_j)_{j\in I}$ ) is no more than $\frac{2}{2^{n-1} + 1}$.
Case 2: If $|I \cap [1,2^n] | < 2^{n-1} - 1$, then $(x_i)_{i\in I}$ cannot be a basis, since the projection onto the first $2^{n-1} - 1$ coordinates will not have $2^{n-1} - 1$ non-zero vectors, so it will not span.
It remains to examine what happens in Case 3: $|I\cap [1,2^n]| = 2^{n - 1} - 1$. In this case, $|I\cap [2^{n}+1,2^{n+1}]| =2^{n - 1} + 1$. In particular, there exist $x_j$ and $x_k$ such that the last $2^{n-1}$ coordinates are constant multiples of one another. So, $|\langle x_j,x_k\rangle| = \frac {2^{n-1}-1}{2^{n-1}+1}$. In particular, the Riesz constant of just these two (norm-one) vectors goes to 0 as $n\to \infty$. Therefore, we can choose $n$ such that the lower Riesz constant of any basis is less than $\epsilon$.
Finally, to finish the proof, we simply copy the FUNTF constructed above as many times as necessary to create a large enough FUNTF. Any basis contained in the copy will also be a basis in the original, so we can force the lower Riesz bound to be less than $\epsilon$. \end{proof}
\section{Lemmas}\label{S:lem}
In this section we collect some lemmas on frames which will be necessary for solving the discretization problem in Section \ref{S:disc}. The lemmas on continuous frames that we need will be presented in Section \ref{S:disc}.
{ \begin{lem}\label{L:BesselPart} Let $(f_{j})_{j\in J}$ be a C-Bessel sequence in an $N$-dimensional Hilbert space $H$. Let $M\in{\mathbb N}$ and $J_1,..., J_M$ be a partition of $J$. For each $1\leq K< M$, there exists $1\leq n_1<...<n_K\leq M$ so that $(f_{j})_{j\in J_{n_k}}$ is $C N /(M+1-K)$-Bessel for each $1\leq k\leq K$. \end{lem}
\begin{proof} For a set $I\subseteq J$ we let $T_I$ be the frame operator of $(f_{j})_{j\in I}$. For any choice of an orthonormal basis $(e_i)_{i=1}^N$ we have that \begin{equation}\label{E:eigen}
trace(T_I)=\sum_{i=1}^N \sum_{j\in I} |\langle e_i, f_j\rangle|^2 = \sum_{j\in I} \|f_j\|^2. \end{equation}
As $T_I$ is a positive self-adjoint operator, we may choose $(e_i)_{i=1}^N$ to be an orthonormal basis of eigenvectors of $T_I$, and hence \eqref{E:eigen} gives that $\sum_{j\in I} \|f_j\|^2$ is equal to the sum of the eigenvalues of $T_I$. For $J=I$, we have that the sum of the eigenvalues of $T_J$ is $\sum_{i=1}^N \sum_{j\in J} |\langle e_i, f_j\rangle|^2\leq CN$ as $(f_j)_{j\in J}$ is $C$-Bessel. Thus we may choose $K$ different $n\in {\mathbb N}$ so that the sum of the eigenvalues of $T_{J_n}$ is at most $CN/(M+1-K)$. In particular, each of the eigenvalues of $T_{J_n}$ is at most $CN/(M+1-K)$ and $(f_j)_{j\in J_n}$ is $CN/(M+1-K)$ Bessel. \end{proof}
\begin{cor}\label{C:BesselPart} Let $P\in {\mathbb N}$. For $1\leq p\leq P$, let $(f_{p,j})_{j\in J^p}$ be a $C_p$-Bessel sequence in an $N_p$-dimensional Hilbert space. Let $M\in{\mathbb N}$ and $J_1^p,..., J_M^p$ be a partition of $J^p$. Then there exists $1\leq n\leq M$ such that the sequence $(f_{p,j})_{j\in J_{n}^p}$ is $4^{p} C_p N_p /M$-Bessel for each $1\leq p\leq P$. \end{cor} \begin{proof} Without loss of generality we may assume that $2^{P+1}<M$. Indeed, for $1\leq p\leq P$, if $M\leq 2^{p+1}$ then $(f_{p,j})_{j\in J_{n}^p}$ is automatically $4^{p} C_p N_p /M$-Bessel for all $n$.
We let $M_1=M$ if $M$ is even, and we let $M_1=M-1$ if $M$ is odd. By Lemma \ref{L:BesselPart} we may choose $\{n_1,...,n_{M_1/2}\}$ such that $(f_{1,j})_{j\in J_{n_i}^1}$ is $2CN/M_1$-Bessel for all $1\leq i\leq M_1/2$. Continuing, we let $M_2=M_1/2$ if $M_1/2$ is even, and we let $M_2=M_1/2-1$ if $M_1/2$ is odd. We can then choose half of the set $\{n_1,...,n_{M_1/2}\}$ (without loss of generality, it is the first half) such that $(f_{2,j})_{j\in J_{n_i}^2}$ is $2^2CN/M_2$-Bessel for all $1\leq i\leq M_2/2$. We then let $M_3=M_2/2$ if $M_2/2$ is even, and we let $M_3=M_2/2-1$ if $M_2/2$ is odd. Without loss of generality, we have that $(f_{3,j})_{j\in J_{n_i}^3}$ is $2^3CN/M_3$-Bessel for all $1\leq i\leq M_3/2$. Continuing in this manner we get even natural numbers $M_1,...,M_P$ with $2^{p-1} M_p\leq M\leq 2^{p}M_p$ for each $1\leq p\leq P$ such that $(f_{p,j})_{j\in J_{n_i}^p}$ is $2^pCN/M_p$-Bessel for all $1\leq i\leq M_p/2$, and hence $(f_{p,j})_{j\in J_{n_i}^p}$ is $4^pCN/M$-Bessel for all $1\leq i\leq M_p/2$. Thus, we may take the sets $J_{n_1}^p$ as $2^{P+1}<M$ and $2^{P-1} M_P\leq M\leq 2^{P}M_P$ implies that $1\leq M_P$. \end{proof} }
We give a quick example showing that the estimate in Lemma \ref{L:BesselPart} is sharp for $K=1$. Let $C>0$ and let $N,M\in {\mathbb Z}$ such that $N$ divides $M$. Let $(e_i)_{i=1}^N$ be an orthonormal basis of an $N$-dimensional Hilbert space $H$. Consider the $C$-tight frame which consists of $M/N$ copies of $\sqrt{\frac{CN}{M}}e_i$ for each $1\leq i\leq N$. As there are $N$ choices for $1\leq i\leq N$, we have that the frame has $M$ vectors. Thus if we partition the frame into singletons we have that each singleton is $\frac{CN}{M}$-Bessel.
The following lemma is obvious for Hilbert spaces, but we state it separately because it will be important in obtaining the lower frame bound in Theorem \ref{T:samp}.
\begin{lem}\label{L:block}
Let $\varepsilon>0$ and $N\in{\mathbb N}$ with $N\geq \varepsilon^{-2} $. Let $(H_j)_{j=1}^N$ be a sequence of finite dimensional mutually orthogonal spaces of a Hilbert space $H$. Then for every $x\in H$ there exists $1\leq n\leq N$ such that $\|P_{H_n} x\|\leq \varepsilon\|x\|$ where $P_{H_n}$ is orthogonal projection onto $H_n$. Furthermore, if $N$ is even and $N\geq 2\varepsilon^{-2}$ then for every $x\in H$ there exists $1\leq n<N$ such that $\|P_{H_n\oplus H_{n+1}} x\|\leq \varepsilon\|x\|$
\end{lem}
\begin{proof}
For the sake of contradiction we assume that there exists $x\in H$ with $\|P_{H_j} x\|>\varepsilon\|x\|$ for all $1\leq j\leq N$. This gives the following contradiction,
$$\|x\|\geq (\sum_{j=1}^N \|P_{H_j} x\|^2)^{1/2}> (\sum_{j=1}^N \varepsilon^2\|x\|^2)^{1/2}
=N^{1/2}\varepsilon\|x\|\geq \|x\|$$.
For the furthermore case, we let $H^0_n=H_{2n-1}\oplus H_{2n}$ for all $n\in {\mathbb N}$ and then apply the previous case to $N/2\in {\mathbb N}$ and $(H^0_{j})_{j=1}^{N/2}$. \end{proof}
\begin{lem}\label{L:project} Let $H_0$ and $H_1$ be Hilbert spaces and let $(f_j)_{j\in J}\subset H_0\oplus H_1$. Suppose that $(P_{H_1} f_j)_{j\in J}\subseteq H_1$ has upper frame bound $K$ and lower frame bound $k$ and that $(P_{H_0} f_j)_{j\in J}$ has Bessel bound $c$. Then for all $x\in H_0\oplus H_1$,
$$\sum_{j\in J} |\langle f_j, x\rangle|^2 \leq K\|P_{H_1}x\|^2+c\|P_{H_0}x\|^2+2 K^{1/2}c^{1/2}\|P_{H_1}x\|\|P_{H_0}x\|$$ and \begin{align*}
\sum_{j\in J} |\langle f_j, x\rangle|^2 &\geq \left(\sum_{j\in J} |\langle f_j, P_{H_1}x\rangle|^2\right) - 2 K^{1/2}c^{1/2}\|P_{H_1}x\|\|P_{H_0}x\| \\
&\ge k\|P_{H_1}x\|^2-2 K^{1/2}c^{1/2}\|P_{H_1}x\|\|P_{H_0}x\|.
\end{align*} \end{lem}
\begin{proof} We first calculate the upper bound.
\begin{align*}\sum_{j\in J}& |\langle f_j, x\rangle|^2 = \sum_{j\in J} |\langle f_j, P_{H_1}x+P_{H_0}\rangle|^2 \\
&\leq \sum_{j\in J} |\langle f_j, P_{H_1}x\rangle|^2+|\langle f_j,P_{H_0}x\rangle|^2+2|\langle f_j, P_{H_0}x\rangle\langle f_j, P_{H_1}x\rangle| \\
&\leq \left(\sum_{j\in J} |\langle f_j, P_{H_1}x\rangle|^2+|\langle f_j,P_{H_0}x\rangle|^2\right)+2\left(\sum_{j\in J}|\langle f_j, P_{H_0}x\rangle|^2\right)^{1/2}\left(\sum_{j\in J}|\langle f_j, P_{H_1}x\rangle|^2\right)^{1/2} \\
&= \left(\sum_{j\in J} |\langle P_{H_1}f_j, P_{H_1}x\rangle|^2+|\langle P_{H_0}f_j,P_{H_0}x\rangle|^2\right)+2\left(\sum_{j\in J}|\langle P_{H_0} f_j, P_{H_0}x\rangle|^2\right)^{1/2}\left(\sum_{j\in J}|\langle P_{H_1}f_j, P_{H_1}x\rangle|^2\right)^{1/2} \\
&\leq K\|P_{H_1}x\|^2+c\|P_{H_0}x\|^2+2 K^{1/2}c^{1/2}\|P_{H_1}x\|\|P_{H_0}x\| \end{align*} We now calculate the lower bound.
\begin{align*}\sum_{j\in J}& |\langle f_j, x\rangle|^2 = \sum_{j\in J} |\langle f_j, P_{H_1}x+P_{H_0}\rangle|^2 \\
&\geq \sum_{j\in J} |\langle f_j, P_{H_1}x\rangle|^2+|\langle f_j,P_{H_0}x\rangle|^2-2|\langle f_j, P_{H_0}x\rangle\langle f_j, P_{H_1}x\rangle| \\
&\geq \left(\sum_{j\in J} |\langle f_j, P_{H_1}x\rangle|^2\right)-2\left(\sum_{j\in J}|\langle f_j, P_{H_0}x\rangle|^2\right)^{1/2}\left(\sum_{j\in J}|\langle f_j, P_{H_1}x\rangle|^2\right)^{1/2} \\
&= \sum_{j\in J} |\langle f_j, P_{H_1}x\rangle|^2 -2\left(\sum_{j\in J}|\langle P_{H_0} f_j, P_{H_0}x\rangle|^2\right)^{1/2}\left(\sum_{j\in J}|\langle P_{H_1}f_j, P_{H_1}x\rangle|^2\right)^{1/2} \\
&\ge \sum_{j\in J} |\langle f_j, P_{H_1}x\rangle|^2 - 2 K^{1/2}c^{1/2}\|P_{H_1}x\|\|P_{H_0}x\|\\
&\geq k\|P_{H_1}x\|^2-2 K^{1/2}c^{1/2}\|P_{H_1}x\|\|P_{H_0}x\| \end{align*} \end{proof}
\begin{lem}\label{L:orthogonal} Let $\varepsilon>0$ and $\varepsilon_1 = 3\varepsilon + 2(2\varepsilon)^{1/2}(1 + \varepsilon)^{1/2}$. Suppose $(f_j)_{j\in J}\subset H_0\oplus H_1$ is a $(1+\varepsilon)$-Bessel sequence and $(P_{H_1}f_j)_{j\in J}\subset H_1$ has lower frame bound $1-\varepsilon$. Then there exists $(g_i)_{i\in I}\subseteq H_0$ such that $(g_i)_{i\in I}\cup (f_j)_{j\in J}$ is $(1+\varepsilon_1)$-Bessel and has lower frame bound $(1-\varepsilon_1)$ on $H_0\oplus H_1$. \end{lem} \begin{proof} Let $(h_i)_{i\in I}\subset H_0\oplus H_1$ such that $(h_i)\cup(f_j)$ is a $(1+\varepsilon)$-tight frame. As, $(P_{H_1} f_j)\subset H_1$ has lower frame bound $(1-\varepsilon)$ and $(P_{H_1} h_i)\cup(P_{H_1} f_j)\subset H_1$ is $(1+\varepsilon)$-tight, we have that $(P_{H_1} h_i)$ is $2\varepsilon$-Bessel. We will prove that $(P_{H_0}h_i)_{i\in I}\cup (f_j)_{j\in J}$ has upper frame bound $(1+\varepsilon_1)$. Let $x\in H$. \begin{align*}
\sum |\langle P_{H_0}h_i,x\rangle|^2 +& \sum |\langle f_j,x\rangle|^2=
(1+\varepsilon)\|x\|^2 - (\sum |\langle (P_{H_0}+P_{H_1})h_i,x\rangle|^2 - \sum |\langle P_{H_0} h_i,x\rangle|^2)\\ &\leq
(1+\varepsilon)\|x\|^2 + (\sum |\langle P_{H_1}h_i,x\rangle|^2 + 2\sum |\langle P_{H_1} h_i,x\rangle \langle P_{H_0} h_i,x\rangle|)\\ &\leq
(1+\varepsilon)\|x\|^2 + (\sum |\langle P_{H_1}h_i,x\rangle|^2 + 2(\sum |\langle P_{H_1} h_i,x\rangle|^2)^{1/2}(\sum |\langle P_{H_0} h_i,x\rangle|^2)^{1/2})\\ &\leq
(1+\varepsilon)\|x\|^2 + (2\varepsilon + 2(2\varepsilon)^{1/2}(1+\varepsilon)^{1/2})\|x\|^2\leq (1+\varepsilon_1)\|x\|^2 \end{align*} We now prove that $(P_{H_0}h_i)_{i\in I}\cup (f_j)_{j\in J}$ has lower frame bound $(1-\varepsilon_1)$, which follows the same argument as above. Let $x \in H_0 \oplus H_1$. \begin{align*}
\sum |\langle P_{H_0}h_i,x\rangle|^2 +& \sum |\langle f_j,x\rangle|^2=
(1+\varepsilon)\|x\|^2 - (\sum |\langle (P_{H_0}+P_{H_1})h_i,x\rangle|^2 - \sum |\langle P_{H_0} h_i,x\rangle|^2)\\ &\geq
(1+\varepsilon)\|x\|^2 - (\sum |\langle P_{H_1}h_i,x\rangle|^2 + 2\sum |\langle P_{H_1} h_i,x\rangle \langle P_{H_0} h_i,x\rangle|)\\ &\geq
(1+\varepsilon)\|x\|^2 - (\sum |\langle P_{H_1}h_i,x\rangle|^2 + 2(\sum |\langle P_{H_1} h_i,x\rangle|^2)^{1/2}(\sum |\langle P_{H_0} h_i,x\rangle|^2)^{1/2})\\ &\geq
(1+\varepsilon)\|x\|^2 - (2\varepsilon + 2(2\varepsilon)^{1/2}(1+\varepsilon)^{1/2})\|x\|^2\geq (1-\varepsilon_1)\|x\|^2 \end{align*}
\end{proof}
\section{Continuous frames and the discretization problem}\label{S:disc}
Recall that a measurable function $\Psi:X\rightarrow H$ from a measure space with a $\sigma$-finite measure $\mu$ to a separable Hilbert space $H$ is called a {\em continuous frame} with respect to $\mu$ if there exist constants $A,B>0$ so that \begin{equation}\label{E:def}
A\|x\|^2\leq \int |\langle x, \Psi(t)\rangle|^2 d\mu(t)\leq B \|x\|^2\quad\quad\forall x\in H. \end{equation} If $A=B$ then the continuous frame is called {\em tight} and if $A=B=1$ then the continuous frame is called {\em Parseval} or a {\em coherent state}. We say that $\Psi:X\rightarrow H$ is a {\em continuous Bessel map} if it does not necessarily have a positive lower frame bound $A$, but does have a finite upper frame bound $B$, which is also called a {\em Bessel bound}. As with frames and Bessel sequences in Hilbert spaces, a continuous frame or continuous Bessel map for a Hilbert space induces a bounded positive operator $T:H\rightarrow H$ called the {\em frame operator} which is defined by \begin{equation}\label{E:frameOp} T(x)=\int \langle x, \Psi(t)\rangle \Psi(t) \,d\mu(t)\quad\quad\forall x\in H. \end{equation} We are integrating vectors in a Hilbert space and \eqref{E:frameOp} is defined weakly in terms of the Pettis integral. That is, for all $x\in H$, $T(x)$ is defined to be the unique vector such that \begin{equation}\label{E:Pettis} \langle T(x), y\rangle=\int \langle x, \Psi(t)\rangle \langle \Psi(t), y\rangle \,d\mu(t) \quad\quad\forall y\in H. \end{equation}
It is sometimes more convenient to work with the inequalities in \eqref{E:def} and sometimes it will be useful to work with the frame operator $T$. As with discrete frames, $\|T\|=B$ where $B$ is the optimal upper frame bound, and $\Psi$ is a continuous frame if and only if $T$ is invertible and in which case $\|T^{-1}\|=A^{-1}$ where $A$ is the optimal lower frame bound. Given a frame for a Hilbert space, it may be converted to a Parseval frame by applying the inverse of the square root of the frame operator. The following lemma shows that this same technique works for continuous frames.
\begin{lem}\label{L:parseval} Let $\Psi:X\rightarrow H$ be a continuous frame with frame operator $T:H\rightarrow H$. Then $T^{-1/2}\Psi:X\rightarrow H$ is a continuous Parseval frame. \end{lem} \begin{proof} As $T$ is a positive self adjoint invertible linear operator, we have that the inverse of its square root $T^{-1/2}$ is well defined. For $x\in X$ we have that \begin{align*}
\|x\|^2=&\langle T (T^{-1/2}x),T^{-1/2}x\rangle \quad\textrm{ as $T$ is self-adjoint}\\ =&\int \langle T^{-1/2}x, \Psi(t)\rangle \langle\Psi(t),T^{-1/2}x\rangle \,d\mu(t)\quad\textrm{ by \eqref{E:Pettis}}\\ =&\int \langle x, T^{-1/2}\Psi(t)\rangle \langle T^{-1/2}\Psi(t),x\rangle \,d\mu(t)\quad\textrm{ as $T$ is self-adjoint}\\
=&\int |\langle x, T^{-1/2}\Psi(t)\rangle|^2 \,d\mu(t) \end{align*} Thus, $T^{-1/2}\Psi:X\rightarrow H$ is a continuous Parseval frame. \end{proof}
Continuous frames were developed by Ali, Antoine, and Gazeau in \cite{AAG1} as a generalization of coherent states and in their later textbook \cite{AAG2} they asked the following question which is now known as the Discretization Problem.
\begin{problem}[The Discretization Problem] Let $\Psi:X\rightarrow H$ be a continuous frame. When does there exist a countable set $F\subseteq X$ such that $(\Psi(t))_{t\in F}\subseteq H$ is a frame of $H$? \end{problem} The Discretization Problem essentially asks if one can always obtain a frame for a Hilbert space from a continuous frame by sampling. A solution for certain types of continuous frames was obtained by Fornasier and Rauhut using the theory of co-orbit spaces \cite{FR}.
The following lemma allows us to approximate any continuous Bessel map with a continuous Bessel map having countable range.
\begin{lem}\label{L:discretize} Let $\Psi:X\rightarrow H$ be a continuous Bessel map. For all $\varepsilon>0$, there exists a measurable partition $(X_j)_{j\in J}$ of $X$
and $(t_j)_{j\in J}\subset X$ such that $t_j\in X_j$ for all $j\in J$ and $\|\Psi(t)-\Psi(t_j)\|<\varepsilon$ for all $t\in X_j$ and
$$\int \left\| \Psi(t)-\sum_{j\in J}\Psi(t_j) 1_{X_j}(t)\right\| d\mu(t)<\varepsilon. $$ \end{lem}
\begin{proof} We first claim that we may assume that $\mu$ is non-atomic. Indeed, if $(E_i)_{i\in I}$ is a collection of disjoint atoms such that $X\setminus \cup_{i\in I} E_i$ is non-atomic, then $\Psi$ is constant almost everywhere on each $E_i$. Thus for each $i\in I$, there exists $s_i\in E_i$ such that $\Psi 1_{E_i}=\Psi(s_i) 1_{E_i}$ almost everywhere. Hence we would only need to prove the lemma for the non-atomic measure space $X\setminus\cup_{i\in I}E_i$ and the continuous Bessel map
$\Psi|_{X\setminus \cup_{i\in I} E_i}=\Psi-\sum_{i\in I}\Psi(t) 1_{E_i}$. Thus, we assume without loss of generality that $X$ is non-atomic.
Let $\varepsilon>0$. As $X$ is non-atomic and $\sigma$-finite, $X$ may be partitioned into a sequence of pairwise disjoint measurable subsets $(Y_j)_{j\in {\mathbb N}}$ so that $\mu(Y_j)\leq1$ for all $j\in {\mathbb N}$. For all $j\in {\mathbb N}$, let $(H^j_n)_{n=1}^\infty$ be a partition of $\Psi(Y_j)\subseteq H$ such that $diam(H_n^j)<\vp2^{-j}$ for all $n\in{\mathbb N}$. For each $j,n\in {\mathbb N}$ choose $t^j_{n}\in
\Psi^{-1}(H^j_n)\cap Y_j$. Note that $(\Psi^{-1}(H^j_n)\cap Y_j)_{n\in{\mathbb N}}$ is a partition of $Y_j$ for all $j\in{\mathbb N}$ and hence $(\Psi^{-1}(H^j_n)\cap Y_j)_{j,n\in{\mathbb N}}$ is a partition of $X$. We have that $\|\Psi(t)-\Psi(t^j_n)\|<\vp2^{-j}$ for all $n,j\in{\mathbb N}$ and $t\in \Psi^{-1}(H_n^j)\cap Y_j$. We now estimate the following. \begin{align*}
\int \left\| \Psi(t)-\sum_{j,n\in{\mathbb N}} \Psi(t_n^j)
1_{\Psi^{-1}(H^j_n)\cap Y_j}(t)\right\| d\mu(t)
&=\sum_{j,n\in{\mathbb N}}\int_{\Psi^{-1}(H^j_n)\cap Y_j} \left\| \Psi(t)-\Psi(t_n^j) \right\| d\mu(t)\\ &\leq\sum_{j,n\in{\mathbb N}}\mu(\Psi^{-1}(H^j_n)\cap Y_j) \varepsilon 2^{-j}=\sum_{j\in{\mathbb N}}\mu(Y_j) \varepsilon 2^{-j}\leq\varepsilon \end{align*} Thus we may use $(\Psi^{-1}(H^j_n)\cap Y_j)_{j,n\in{\mathbb N}}$ as our partition of $X$ and $(t^j_n)_{j,n\in{\mathbb N}}$ as our sample points. \end{proof}
The following is a technical result that is needed in our proof of the discretization theorem, which may be of independent interest. Recall that a collection of vectors $(x_i)_{i \in I}$ in $H$ is said to be a scalable frame if there exist constants $(c_i)_{i\in I}$ such that $(c_ix_i)_{i\in I}$ is a Parseval frame for $H$ \cite{KOPT}. The following result implies, in particular, that there are universal constants $A$ and $B$ such that if $(x_i)_{i\in I}$ is a scalable frame in the unit ball of $H$, then $(x_i)_{i\in I}$ can be sampled to form a frame with lower frame bound $A$ and upper frame bound $B$. In order to help the reader stay organized, there are several claims in the proof of the theorem, whose proofs are separated out from the main text of the proof of the theorem. The proofs of the claims end in $\blacksquare$, while the end of the proof of the main theorem ends with $\square$, as usual.
\begin{thm}\label{T:samp} There is a function $g:(0,1/256)\to (0,1)$ such that $\lim_{\varepsilon \to 0} g(\varepsilon) = 0$ and if $(x_n)_{n=1}^\infty \subset \{x\in H: \|x\|\le 1\}$ is such that there exists scalars $(a_n)_{n=1}^\infty$ such that $(a_n x_n)_{n=1}^\infty$ is a frame for $H$ with bounds $1 - \varepsilon$ and $1 + \varepsilon$, then there exists $f:{\mathbb N} \to {\mathbb N}$ such that $(x_{f(n)})_{n=1}^\infty$ is a frame for $H$ with bounds $A(1 - g(\varepsilon))$ and $2 B(1 + g(\varepsilon))$, where $A$ and $B$ are the constants given in Theorem \ref{T:reduction}. \end{thm}
\begin{proof} We will not explicitly define the functions $f$ and $g$, though a careful reading of the proof will give exact values for $g$. The general strategy of the proof is to decompose $H$ into pairwise orthogonal subspaces and sample $(x_n)$ in order to obtain frames for the subspaces, while controlling the leakage into the other subspaces.
Let $0 < \varepsilon < 1/256$. We begin by noting that we may assume that $(a_n)_{n=1}^\infty$ are non-negative real numbers and by perturbing (and replacing $\varepsilon$ by a slightly larger value, which we still denote as $\varepsilon$) we may assume further that $(a_n)_{n=1}^\infty$ are non-negative rational. Next, we can decompose $H$ in the following way.
\begin{claim}\label{C:one} Let $K_{-1} = -1$ and $K_0 = 0$. There exists pairwise orthogonal subspaces $(H_n)_{n=1}^\infty$, an increasing sequence of natural numbers $(K_n)_{n=1}^\infty$ and a decreasing sequence of real numbers $(\varepsilon_n)_{n=1}^\infty$ converging to zero such that the following hold for all $n\in {\mathbb N}$ and $1 \le m\le n$, where $\varepsilon^\prime = 6\varepsilon + 4\varepsilon^{1/2} (1 + 2\varepsilon)^{1/2}.$ \begin{align} &(x_j)_{j \le K_{n-1}} \subset \oplus_{j\le n} H_j \label{Een} \\ &\textrm{ for all } x\in \oplus_{m\le j \le n} H_j,
\,\,\, (1 - 2\varepsilon) \|x\|^2 \le \sum_{j\in (K_{m-2},K_n]} |\langle x, x_j \rangle|^2 a_j^2 \le (1 + \varepsilon) \|x\|^2 \label{Twee} \\ &\textrm{ for all } x\in \oplus_{1\le j \le n} H_j,
\,\,\, \sum_{j\not \in (K_{m-2},K_n]} |\langle x, x_j \rangle|^2 a_j^2 \le \varepsilon_n \|x\|^2 \label{Drie} \\ &\varepsilon_n B(1 + \varepsilon^\prime)(1 - \varepsilon^\prime)^{-2} \operatorname{dim}(\oplus_{j \le n} H_j) < \varepsilon^\prime 8^{-n} \label{Vier} \end{align}
\end{claim}
\renewcommand{$\square$}{$\blacksquare$}
\begin{proof}[Proof of claim \ref{C:one}]
For the base case of $n=1$ we have that \eqref{Een} will be automatically satisfied as we have set $K_0=0$. We let $H_1=0$, $K_1$=1 and $\varepsilon_1 = \varepsilon/2$, which trivially satisfies \eqref{Twee}, \eqref{Drie} and \eqref{Vier}.
For the induction step we let $k\in{\mathbb N}$ and assume that \eqref{Een}, \eqref{Twee}, \eqref{Drie}, and \eqref{Vier} are true for $m\leq n=k$. We choose $\varepsilon_{k+1}<\varepsilon_k$ small enough so that \eqref{Vier} is satisfied. Let $H_{k+1}=\textrm{span}_{K_{k-1}<i \leq K_k}P_{(\oplus_{j\leq k}H_{j})^\perp} x_i$. Thus, $H_{k+1}$ is orthogonal to $\textrm{span}_{j\leq k} H_j$ and $\{x_j\}_{j\leq K_{k}}\subseteq \textrm{span}_{j\leq k+1} H_j$
which satisfies \eqref{Een}. As $(a_j x_j)_{j\in{\mathbb N}}$ is Bessel and $\oplus_{i\leq k+1}H_{i}$ is finite dimensional, we may choose $K_{k+1}>K_k$ so that $\sum_{j>K_{k+1}} |\langle x, x_j)\rangle|^2 a_j^2\leq
\varepsilon_{k+1}\|x\|^2$ for all $x\in \oplus_{i\leq k+1} H_{i}$. Let $m\leq k+1$. As $\{x_j\}_{j\leq K_{m-2}}\subseteq \textrm{span}_{j\leq m-1} H_j$ we have that $\langle x, x_j\rangle=0$ for all $x\in \oplus_{m\leq i\leq k+1} H_{i}$ and $j \le K_{m-2}$; hence, \eqref{Drie} is true. We have that \eqref{Twee} follows from \eqref{Drie} as $({a_j}x_j)_{j\in{\mathbb N}}$ is a frame with lower frame bound $1-\varepsilon$ and upper frame bound $1+\varepsilon$ and $\varepsilon_n<\varepsilon/2$. Thus our induction argument is complete. \end{proof}
The spaces $(H_n)_{n=1}^\infty$ are the building blocks for constructing our frame via sampling, but we will need to group subspaces together in order to control the leakage between consecutive subspaces using Lemma \ref{L:block}. To this end, let $(\delta_n)_{n=1}^\infty$ be a decreasing sequence of real numbers and let $(M_n)_{n=1}^\infty$ and $(N_n)_{n=1}^\infty$ be sequences of odd numbers such that \begin{align} &M_1 < M_2 - 2 < M_2 < N_1 < N_1 + 1 < M_3 - 2 < M_3 < N_2 < N_2 + 1 < M_4 - 2 < \cdots \label{Vyf}\\ &N_n - M_{n+1} - 4 > 2 \delta_n^{-2} \label{Ses}\\ &\sum_{n=1}^\infty \delta_n^2 < \varepsilon^2 \label{Sewe} \end{align}
We construct our frame via sampling as follows. Fix $r\in {\mathbb N}$. Since $\varepsilon < 1/256$, we can choose $D = D_r$ to be an integer multiple of the least common multiple of the denominators of $(a_n^2)_{n\in (K_{M_r-2},K_{N_r}]}$ so that \begin{equation} D(1 - 6\varepsilon - 4\varepsilon^{1/2}(1 + 2\varepsilon)^{1/2}) \ge 1 \label{Sewehalf} \end{equation} (this choice of $D$ will allow us to apply Corollary \ref{C:frame_part} in the sequel).
Denote by $(\sqrt{1/D} f_n)_{n\in I}$ the sequence of vectors comprised of $D a_n^2$ copies of $\sqrt{1/D} x_j$ for each $j\in (K_{M_r-2}, N_r]$, where $I = I_r$ depends on $r$. We note here that for any $x\in H$, \begin{align*}
\sum_{n\in I} |\langle x, \sqrt{1/D} f_n\rangle|^2 &= \sum_{j = K_{M_r - 1}}^{K_{N_r}} \sum_{i = 1}^{Da_n^2} |\langle x, x_j\rangle|^2 \frac 1D\\
&=\sum_{j = K_{M_r - 1}}^{K_{N_r}} |\langle x, x_j \rangle|^2 a_n^2, \end{align*} so $(\sqrt{1/D} f_n)_{n\in I}$ has the same frame properties as $(a_n x_n)_{n\in (K_{M_r-2},K_{N_r}]}$. It will be necessary in the sequel to recover the correspondence between $f_j$ and $x_n$. So, we define \begin{equation} b_r = b:I \to (K_{M_r-2}, K_{N_r}] \label{DefB} \end{equation} in such a way that \begin{enumerate} \item $f_n = x_{b(n)}$ for all $n\in I$, and
\item $|b^{-1}(j)| = Da_j^2$ for all $K_{M_r - 1} \le j \le K_{N_r}$ \end{enumerate} so that formally $(f_n)_{n\in I}$ is the same sequence of vectors as $(x_{b(n)})_{n \in I}$.
By \eqref{Een}, $(\sqrt{1/D} f_n)_{n\in I}$ is a $(1+\varepsilon)$-Bessel sequence in $\oplus_{j \le {N_r } + 1} H_j$, and by \eqref{Twee}, \[ (P_{\oplus_{M_r\le j \le N_r} H_j} \sqrt{1/D} f_n)_{n\in I} \]
is a frame for $\oplus_{M_r\le j \le N_r} H_j$ with lower frame bound $1 - 2\varepsilon$. Therefore, we can apply Lemma \ref{L:orthogonal} to obtain $(g_n)_{n\in J} \subset \oplus_{j < M_r} H_j \oplus H_{N_r + 1}$ such that $\|g_n\| \le 1/\sqrt{D}$ and $(\sqrt{1/D} f_n)_{n\in I} \cup (g_n)_{n\in J}$ is a frame for $\oplus_{j \le N_{r}+1} H_j$ with bounds $1 - \varepsilon^\prime$ and $1 + \varepsilon^\prime$, where \[ \varepsilon^\prime = 6\varepsilon + 4\varepsilon^{1/2} (1 + 2\varepsilon)^{1/2}. \]
By our choice of $D$ \eqref{Sewehalf}, we can apply Corollary \ref{C:frame_part} to the vectors $(f_n)_{n\in I} \cup (\sqrt{D} g_n)_{n\in J}$ to obtain a partition $(I_m \cup J_m)_{m=1}^M$ of $I \cup J$ such that for each $1\le m \le M$, $(f_n)_{n\in I_m} \cup (\sqrt{D} g_n)_{n\in J_m}$ is a frame for $\oplus_{j\le N_r + 1} H_j$ with constants $A$ and $B(1 + \varepsilon^\prime)(1 - \varepsilon^\prime)^{-1}$. For a set $P \subset I$, we denote \[ P^{>k} = \{j \in P: b(j) > K_k\}, \] where $b$ is as defined in \eqref{DefB}.
\begin{claim}\label{C:two} For each $r \ge 1$ there exists $m_0$ such that \begin{align} &\textrm{ for each } k\in [M_r-2, N_r], (P_{\oplus_{j \le k} H_j} f_n)_{n\in I_{m_0}^{>k}} \textrm{ has Bessel bound } \varepsilon^\prime 2^{-r} \label{SewePlus},\\ &(P_{\oplus_{j\le M_r-2} H_j} f_n)_{n\in I_{m_0}} \textrm{ has Bessel bound } \varepsilon^\prime 2^{-r} \label{Agt}, \\ &(P_{\oplus_{j\in [M_r, N_r]} H_j} f_n)_{n\in I_{m_0}} \textrm{ is a frame for } \oplus_{j\in [M_r, N_r]} H_j \textrm{ with bounds } A, B(1 + \varepsilon^\prime)(1 - \varepsilon^\prime)^{-1} \label{Nege}, \\ &(P_{\oplus_{j\in [M_r - 1, N_r]} H_j} f_n)_{n\in I_{m_0}} \textrm{ has Bessel bound } B(1 + \varepsilon^\prime)(1 - \varepsilon^\prime)^{-1}, \label{Tien} \\ &(f_n)_{n\in I_{m_0}} \textrm{ has Bessel bound } B(1 + \varepsilon^\prime)(1 - \varepsilon^\prime)^{-1} \textrm { on } \oplus_{1\le i \le N_r + 1} H_i. \label{TienPlus} \end{align} \end{claim}
\begin{proof}[Proof of Claim \ref{C:two}] For each $k\in [M_r-2,N_r]$, we have that $(P_{\oplus_{1\leq i\le k}H_i}f_j)_{j\in I^{>k}}$ is $D\varepsilon_{k}$-Bessel by \eqref{Drie}. By Corollary \ref{C:BesselPart} there exists $1\leq m_0\leq M$ (which depends on $r\in{\mathbb N}$) so that for every $k\in [M_r - 2, N_r]$, $(P_{\oplus_{i\le k}H_i}f_j)_{j\in I^{>k}_{m_0}}$ has Bessel bound $D\varepsilon_k 4^k dim(\oplus_{i\le k}H_i)/M$.
As each frame in the partition $ ((f_j)_{j\in I_k}\cup(\sqrt{D} g_j)_{j\in J_k})_{1\leq k\leq M}$ has upper frame bound $B(1+\varepsilon^\prime)(1-\varepsilon^\prime)^{-1}$ and there are $M$ of them, we have that $BM(1+\varepsilon^\prime)(1-\varepsilon^\prime)^{-1}\geq (1-\varepsilon^\prime)D$. Thus we have for each $k\in[M_r-2,N_r]$ that \begin{gather*} (P_{\oplus_{ i\le k}H_i}f_j)_{j\in I^{>k}_{m_0}}\textrm{ has Bessel bound }\varepsilon^\prime2^{-k}<\varepsilon^\prime2^{-r}\textrm{, as }\\ D\varepsilon_k 4^{k} dim(\oplus_{i\le k}H_i)/M<B(1 + \varepsilon^\prime)(1-\varepsilon^\prime)^{-2}\varepsilon_k 4^{k} dim(\oplus_{i\le k}H_i)<\varepsilon^\prime 2^{-k}\,\,\textrm{ by \eqref{Vier}}, \end{gather*} which proves \eqref{SewePlus}. When $k=M_r-2$ we have that $I_{m_0}=I_{m_0}^{>M_r-2}$ and hence, \begin{equation*} (P_{\oplus_{ i\le M_r-2}H_i}f_j)_{j\in I_{m_0}}\textrm{ has Bessel bound }\varepsilon^\prime2^{-r}, \end{equation*} which proves \eqref{Agt}.
Equation \eqref{Nege} follows from the frame bounds of $(f_n)_{n\in I_m} \cup (\sqrt{D} g_n)_{n\in J_m}$ and the fact that $(g_n)_{n\in J_m}$ is orthogonal to $\oplus_{j\in [M_r, N_r]} H_j$. Equation \eqref{Tien} follows immediately from construction, and \eqref{TienPlus} follows from \eqref{Tien} and \eqref{Een}.
\end{proof}
For each $r\in {\mathbb N}$, we define $I(r)$ to be the $I_m$ that is guaranteed to exist in Claim \ref{C:two}. To finish the proof, we show that $(f_n)_{n\in I(r), r\in {\mathbb N}}$ is a frame with bounds $2 B(1 + \varepsilon_2)$ and $A - (A + 2)\varepsilon_2 - 4B(1 + \varepsilon_2)\varepsilon_2^{1/2}$, where \[ \varepsilon_2 = (1 + \varepsilon^\prime)(1 - \varepsilon^\prime)^{-1} - 1 + \varepsilon^\prime + 2(B(1 + \varepsilon^\prime)(1 - \varepsilon^\prime)^{-1}\varepsilon^\prime)^{1/2}. \]
\begin{proof}[Proof of Upper Bound] Let $x\in H$ with $\|x\| = 1$. Define a sequence of integers $(q_n)_{n=1}^\infty$ by $q_1 = 0$ and $q_n \in (N_{n-1} + 1 , M_{n+1}-2)$ for $n\ge 2$. Let \begin{gather*} I_-(n) = \{j\in I(n): b(j) \le K_{q_n - 1}\},\\ I_+(n) = \{j\in I(n): b(j) > K_{q_n - 1}\}. \end{gather*} First, note that if $j\in I_-(n)$, then $f_j \in \oplus_{1\le i \le q_n} H_i$ by \eqref{Een}. Therefore, defining $H_1^n$ to be $\oplus _{M_n-1\le i \le q_n} H_i$ we can apply \eqref{Agt}, \eqref{Tien} and Lemma \ref{L:project} to obtain that \begin{align*}
I&:=\sum_{n=1}^\infty \sum_{j\in I_-(n)} |\langle f_j, x\rangle|^2 \\
&\le \sum_{n=2}^\infty \biggl(B(1 + \varepsilon^\prime) (1 - \varepsilon^\prime)^{-1} \| P_{H_1^n} x\|^2 + \varepsilon^\prime 2^{-n} + 2\bigl(B(1 + \varepsilon^\prime)(1 - \varepsilon^\prime)^{-1} \varepsilon^\prime 2^{-n}\bigr)^{1/2} \| P_{H_1^n} x\|\biggr)\\
&\le B(1 + \varepsilon^\prime)(1 - \varepsilon^\prime)^{-1} + \varepsilon^\prime + 2\bigl(B(1 + \varepsilon^\prime)(1 - \varepsilon^\prime)^{-1} \varepsilon^\prime \bigr)^{1/2} \sum_{n=2}^\infty 2^{-n} \sum_{n=2}^\infty \| P_{H_1^n}x \|^2\\ &\le B(1 + \varepsilon^\prime)(1 - \varepsilon^\prime)^{-1} + \varepsilon^\prime + 2\bigl(B(1 + \varepsilon^\prime)(1 - \varepsilon^\prime)^{-1} \varepsilon^\prime \bigr)^{1/2}, \end{align*} where the third line follows from Cauchy-Schwartz and the choice of $q_n < M_{n+1} - 2$. In summary, we have that \[
I \le B(1 + \varepsilon_2) \|x\|^2 \]
Similarly, for $j\in I_+(n)$, $f_j = x_{b(j)} \in \oplus_{1 \le i \le N_n+1} H_i$ by \eqref{Een} and $j \le K_{N_n}$. Therefore, defining $H_1^n = \oplus_{q_n\le i \le N_n + 1} H_i$, we can apply \eqref{SewePlus}, \eqref{TienPlus} and Lemma \ref{L:project} to obtain \begin{align*}
II&:=\sum_{n=1}^\infty \sum_{j\in I_+(n)} |\langle f_j, x\rangle|^2 \\
&\le \sum_{n=1}^\infty \biggl(B(1 + \varepsilon^\prime) (1 - \varepsilon^\prime)^{-1} \| P_{H_1^n} x\|^2 + \varepsilon^\prime 2^{-n} + 2\bigl(B(1 + \varepsilon^\prime)(1 - \varepsilon^\prime)^{-1} \varepsilon^\prime 2^{-n}\bigr)^{1/2} \| P_{H_1^n} x\|\biggr)\\ &\le B(1 + \varepsilon^\prime)(1 + \varepsilon^\prime)^{-1} + \varepsilon^\prime + 2\bigr(B(1 + \varepsilon^\prime)(1 - \varepsilon^\prime)^{-1} \varepsilon^\prime\bigr)^{1/2}\\ &\le B(1 + \varepsilon_2), \end{align*} where the 3rd line follows from Cauchy-Schwartz and that $N_{n} + 1 < q_{n+1}$.
Therefore, we can conclude that \begin{align*}
\sum_{n=1}^\infty \sum_{j\in I(n)} |\langle f_j, x \rangle|^2 &= I + II\\ &\le 2B(1 + \varepsilon_2). \end{align*}
\end{proof}
\begin{proof}[Proof of Lower Bound] Let $x\in H$, $\|x\| = 1$. By \eqref{Ses} and Lemma \ref{L:block}, for each $n$ there exists $p_n \in (M_{n + 1}, N_n - 2]$ such that $\|P_{H_{p_n} \oplus H_{p_n + 1}} x\| \le \delta_n$, where $\delta_n$ were chosen to satisfy \eqref{Ses} and \eqref{Sewe}. Let \[ y_1 = P_{\oplus_{j < p_1} H_j} x \] and for $n\ge 2$, let \[ y_n =P_{\oplus_{p_{n-1}+1 < j < p_n} H_j} x. \]
Note that \begin{gather}
\|x - \sum_n y_n\| < \bigl(\sum_n \delta_n^2 \bigr)^{1/2} < \varepsilon < \varepsilon_2 \label{Elf}\\
1 - \varepsilon_2 < 1 - \varepsilon < \|\sum_n y_n\| \le 1 \label{Twallf}, \textrm{ and }\\
\forall y\in \oplus_{p_{n-1} + 1 < i < p_n} H_i, \,\, \sum_{j\in I(n)} |\langle f_j, y \rangle|^2 \ge A\|y\|^2\label{Dertien}\quad \textrm { by \eqref{Nege}. } \end{gather} Next, we define \begin{gather*} I_{-1}(n) = \{j \in I(n): b(j) \le K_{p_{n-1}}\},\\ I_0(n) = \{j\in I(n) : K_{p_{n-1}} <b(j) \le K_{p_n} \}, \textrm{ and}\\ I_1(n) = \{j\in I(n) : b(j) > K_{p_n}\}, \end{gather*} where $b$ is the map defined in \eqref{DefB}.
We compute \begin{align*}
\sum_{n=1}^\infty & \sum_{j\in I(n)} |\langle f_j, \sum_{m=1}^\infty y_m \rangle |^2 \ge \sum_{n=1}^\infty \sum_{j\in I_0(n)} |\langle f_j, \sum_{m=1}^\infty y_m \rangle|^2 \\
&= \sum_{n=1}^\infty \sum_{j\in I_0(n)} |\langle f_j, \sum_{m=1}^{n+1} y_m \rangle|^2 \\
&\ge \sum_{n=1}^\infty\biggl(\sum_{j\in I_0(n)} |\langle f_j, y_n\rangle|^2 - 2\bigl(B(1 + \varepsilon^\prime)(1 - \varepsilon^\prime)^{-1} \varepsilon^\prime 2^{-n}\bigr)^{1/2} \|y_n\| \|\sum_{m \le n+1, m\not= n} y_m\| \biggr) \\
& > \biggl(\sum_{n=1}^\infty | \langle f_j, y_n \rangle |^2 \biggr) - \varepsilon_2\\
&\ge \sum_{n=1}^\infty \biggl(\sum_{j\in I_0(n)} |\langle f_j, y_n \rangle|^2 + \bigl( \sum_{j\in I_1(n)} | \langle f_j, y_n \rangle|^2 - \varepsilon^\prime 2^{-n}\|y_n\|^2\bigr) + \bigl(\sum_{j\in I_{-1}(n)} |\langle f_j, y_n \rangle |^2 - 0\bigr) \biggr) - \varepsilon_2\\
&\ge \sum_{n=1}^\infty \bigl(A \|y_n\|^2 - \varepsilon^\prime 2^{-n} \|y_n\|^2\bigr) - \varepsilon_2\\
&\ge A\|\sum_{n=1}^\infty y_n \|^2 - \varepsilon^\prime - \varepsilon_2. \end{align*}
where the second line follows from $f_j \in \oplus_{k \le p_n + 1}H_j$ for $j\in I_0(n)$, the third line follows from Lemma \ref{L:project}, the fourth line follows from Cauchy-Schwartz and the definition of $\varepsilon_2$, the fifth line follows from \eqref{SewePlus}, and the sixth line is from \eqref{Dertien}. We conclude the proof by using the above estimate for $\sum_j |\langle f_j,\sum_{m\in{\mathbb N}} y_m\rangle|^2$ to obtain a lower frame bound for $(f_j)_{j\in I(n), n\in {\mathbb N}}$.
\begin{align*}
\sum_j |\langle f_j,x\rangle|^2&\ge \sum_j |\langle f_j, \sum_{m\in {\mathbb N}} y_m \rangle|^2 - 4B(1 + \varepsilon_2) \|\sum_m y_m\|\, \| x - \sum_m y_m\|\\
&> A\|\sum_n y_n \|^2 - \varepsilon^\prime - \varepsilon_2 - 4B(1 + \varepsilon_2)\varepsilon_2\\
&=A(\|x\|^2-\|x-\sum y_n\|^2)-\varepsilon^\prime - \varepsilon_2 - 4B(1 + \varepsilon_2)\varepsilon_2\\
& > A \| x\|^2-A\varepsilon_2^2-\varepsilon^\prime - \varepsilon_2 - 4B(1 + \varepsilon_2)\varepsilon_2 \end{align*}
The first line follows from Lemma \ref{L:project} and our Bessel bound for $(f_j)_{j\in I(n), n\in {\mathbb N}}$, the second line follows from \eqref{Elf} and the immediately preceding calculation, and the last line follows from \eqref{Elf}. Thus $(f_j)_{j\in I(n), n\in {\mathbb N}}$ has lower frame bound $A - A\varepsilon^\prime-\varepsilon_2^2 - \varepsilon_2 - 4B(1 + \varepsilon_2)\varepsilon_2 $.
\end{proof}
\renewcommand{$\square$}{$\square$} This concludes the proof of Theorem \ref{T:samp}. \end{proof}
\begin{thm}\label{T:disc}
Let $\Psi:X\rightarrow H$ be a continuous Parseval frame such that $\|\Psi(t)\|\leq 1$ for all $t\in X$. If $A,B>0$ are the uniform constants given in Theorem \ref{T:reduction} then for all $\varepsilon>0$ there exists $(s_i)_{i\in I}\in X^I$ such that $(\Psi(s_i))_{i \in I}$ is a frame of $H$ with lower frame bound $(1-\varepsilon)A$ and upper frame bound $2(1+\varepsilon)B$. \end{thm}
\begin{proof}
Without loss of generality we assume that $H$ is infinite dimensional.
Let $1>\varepsilon>0$. Let $\varepsilon_0>0$ such that $g(\varepsilon_0) <\varepsilon$, where $g$ is the function guaranteed to exist from Theorem \ref{T:samp}.
By Lemma \ref{L:discretize} there exists a partition $(X_i)_{i\in{\mathbb N}}$ of $X$ and $(t_j)_{j\in{\mathbb N}}\subset X$ such that $t_j\in X_i$ for all $j\in{\mathbb N}$ and $\|\Psi(t)-\Psi(t_j)\|<\frac{\varepsilon_0}{6}$ for all $t\in X_j$ and
$\int \left\| \Psi(t)-\sum_{i\in{\mathbb N}}\Psi(t_i) 1_{X_i}(t)\right\| d\mu(t)<\frac{\varepsilon_0}{3}.
$ Let $\Phi:X\rightarrow H$ be given by $\Phi:=\sum \Psi(t_i) 1_{X_i}$. We will prove that $\Phi$ is a continuous frame with upper frame bound $1+\varepsilon_0$ and lower frame bound $1-\varepsilon_0$. Let $x\in H$ with $\|x\|=1$. We first estimate the lower frame bound. \begin{align*}
\|x\|^2&= \int |\langle x, \Psi(t)\rangle|^2 d\mu(t)\\
&= \int |\langle x, (\Psi(t)-\Phi(t)+\Phi(t))\rangle|^2 d\mu(t)\\
&\leq \int |\langle x, \Phi(t)\rangle|^2+2|\langle x, (\Psi(t)-\Phi(t))\rangle||\langle x,\Phi(t)\rangle|+|\langle x,(\Psi(t)-\Phi(t))\rangle|^2 d\mu(t)\\
&\leq \int |\langle x, \Phi(t)\rangle|^2+2\|\Psi(t)-\Phi(t)\|\|\Phi(t)\|+\|\Psi(t)-\Phi(t)\|^2 d\mu(t)\\
&\leq \int |\langle x, \Phi(t)\rangle|^2+2\|\Psi(t)-\Phi(t)\|+\|\Psi(t)-\Phi(t)\|\frac{\varepsilon_0}{3} d\mu(t) \\
&\qquad \textrm{ as }\|\Phi(t)\|\!\leq\! 1\textrm{ and }\|\Psi(t)-\Phi(t)\|\!<\!\frac{\varepsilon_0}{3} \\
&\leq \int |\langle x, \Phi(t)\rangle|^2 d\mu(t)+2(\frac{\varepsilon_0}{3}) +(\frac{\varepsilon_0}{3})^2\qquad \bigg( \textrm{as }\int \| \Psi(t)-\Phi(t)\| d\mu(t)< \frac{\varepsilon_0}{3} \biggr) \\
&<\int |\langle x, \Phi(t)\rangle|^2 d\mu(t)+\varepsilon_0 \end{align*} Thus, $\Phi$ has lower frame bound $1-\varepsilon_0$. We similarly estimate the upper frame bound. \begin{align*}
\|x\|^2&= \int |\langle x, \Psi(t)\rangle|^2 d\mu(t)\\
&= \int |\langle x, (\Psi(t)-\Phi(t)+\Phi(t))\rangle|^2 d\mu(t)\\
&\geq \int |\langle x, \Phi(t)\rangle|^2-2|\langle x, (\Psi(t)-\Phi(t))\rangle||\langle x,\Phi(t)\rangle|+|\langle x,(\Psi(t)-\Phi(t))\rangle|^2 d\mu(t)\\
&> \int |\langle x, \Phi(t)\rangle|^2 d\mu(t)-2(\frac{\varepsilon_0}{3}) >\int |\langle x, \Phi(t)\rangle|^2 d\mu(t)-\varepsilon_0\\ \end{align*} Thus, $\Phi$ has upper frame bound $1+\varepsilon_0$.
We have that for all $x\in H$, $\int |\langle x, \Phi(t)\rangle|^2 d\mu(t)=\sum |\langle x, \Psi(t_i)\rangle|^2\mu(X_i)$.
Thus, $(\sqrt{\mu(X_i)}\Psi(t_i))_{i\in{\mathbb N}}$ is a frame of $H$ with lower frame bound $1-\varepsilon_0$ and upper frame bound
$1+\varepsilon_0$.
Therefore, by Theorem \ref{T:samp}, there exists a sequence of natural numbers $I$ such that
$
(\Psi(t_i))_{i\in I}
$
is a frame for $H$ with bounds $A(1 - \varepsilon)$ and $2B(1 + \varepsilon)$, as desired.
\end{proof}
We now show that the complete solution to the discretization problem can be reduced to the special case of Theorem \ref{T:disc}. We restate and prove Theorem \ref{T:D} that was presented in the introduction.
\begin{thm} Let $(X,\Sigma)$ be a measure space such that every singleton is measurable and let $\Psi:X\rightarrow H$ be measurable. There exists $(t_j)_{j\in J}\in X^J$ such that $(\Psi(t_j))_{j \in J}$
is a frame of $H$ if and only if there exists a positive, $\sigma$-finite measure $\nu$ on $(X,\Sigma)$ so that $\Psi$ is a continuous frame of $H$ with respect to $\nu$ which is bounded $\nu$-almost everywhere. \end{thm}
\begin{proof}
We first assume that there exists a positive $\sigma$-finite measure $\nu$ on $X$ so that $\Psi:X\rightarrow H$ is a continuous frame of $H$ with respect to $\nu$ which is bounded almost everywhere. By passing to a measurable subset of $X$ of full measure we may assume that $\Psi$ is bounded. Let $T:H\rightarrow H$ be the frame operator of $\Psi$. Note that $T$ is a positive self-adjoint invertible operator. Define $\Phi:X\rightarrow H$ by $\Phi(x)= T^{-1/2} \Psi(x)$. Thus, $\Phi$ is a continuous Parseval frame by Lemma \ref{L:parseval} and $\Phi$ is bounded as it is the composition of bounded functions. There exsists $C>0$ such that $\|\Phi(t)\|\leq C$ for all $t\in X$. We now define a measure $\nu_0$ on $X$ by $\nu_0=C^2 \nu$. Then, $C^{-1}\Phi:X\rightarrow H$ is a continuous Parseval frame with respect to $\nu_0$ such that $\|C^{-1}\Phi(t)\|\leq 1$ for all $t\in H$. Thus, there exists $(t_j)_{j\in J}\in X^J$ such that $(C^{-1}\Phi(t_j))_{j\in J}$ is a frame of $H$ by Theorem \ref{T:disc}. We have that $(C^{-1} T^{-1/2} \Psi(t_j))_{j\in J}$ is a frame of $H$ and hence $( \Psi(t_j))_{j\in J}$ is a frame of $H$.
We now assume that there exists $(t_j)_{j\in J}\in X^J$ such that $(\Psi(t_j))_{j \in J}$
is a frame of $H$. Note that $(\Psi(t_j))_{j \in J}$ is bounded as it is a frame.
It is possible for some points to be sampled multiple times.
For $t\in \cup_{j\in J} t_j$, we let $n_t=|\{j\in J\,:\,t=t_j\}|$. We define a measure $\nu$ on $X$ by $\nu(A)=\sum_{t\in A\cap \cup_{j\in J} t_j }n_t$.
As singletons are measurable, we have that $\nu$ is a measure on $(X,\Sigma)$ and the frame operator for the map $\Psi:\rightarrow H$ with respect to $\nu$ is the same as the frame operator of $(\Psi(t_j))_{j \in J}$. Thus, $\Psi$ is a continuous frame with respect to $\nu$ which is bounded on $\cup_{j\in J} t_j$ and $\nu(X\setminus \cup_{j\in J} t_j)=0$. \end{proof}
\end{document} | arXiv |
Chupakhin, Oleg Nikolaevich
Total publications: 21
Scientific articles: 21
This page: 316
Abstract pages: 2258
Full texts: 6
Member of the Russian Academy of Sciences
Doctor of chemical sciences
Birth date: 9.06.1934
http://www.mathnet.ru/eng/person74390
1. E. V. Nosova, S. Achelle, G. N. Lipunova, V. N. Charushin, O. N. Chupakhin, "Functionalized benzazines as luminescent materials and components for optoelectronics", Usp. Khim., 88:11 (2019), 1128–1178 ; Russian Chem. Reviews, 88:11 (2019), 1128–1178
2. S. A. Vakarov, D. A. Gruzdev, G. L. Levit, V. P. Krasnov, V. N. Charushin, O. N. Chupakhin, "Synthesis of enantiomerically pure 2-aryloxy carboxylic acids and their derivatives", Usp. Khim., 88:10 (2019), 1063–1080 ; Russian Chem. Reviews, 88:10 (2019), 1063–1080
3. K. V. Savateev, E. N. Ulomsky, I. I. Butorin, V. N. Charushin, V. L. Rusinov, O. N. Chupakhin, "Azoloazines as $\mathrm{A}_{2a}$ receptor antagonists. Structure–activity relationship", Usp. Khim., 87:7 (2018), 636–669 ; Russian Chem. Reviews, 87:7 (2018), 636–669
4. G. N. Lipunova, E. V. Nosova, V. N. Charushin, O. N. Chupakhin, "Synthesis and antitumour activity of 4-aminoquinazoline derivatives", Usp. Khim., 85:7 (2016), 759–793 ; Russian Chem. Reviews, 85:7 (2016), 759–793
5. I. S. Kovalev, D. S. Kopchuk, G. V. Zyryanov, V. L. Rusinov, O. N. Chupakhin, V. N. Charushin, "Organolithium compounds in the nucleophilic substitution of hydrogen in arenes and hetarenes", Usp. Khim., 84:12 (2015), 1191–1225 ; Russian Chem. Reviews, 84:12 (2015), 1191–1225
6. G. V. Zyryanov, D. S. Kopchuk, I. S. Kovalev, E. V. Nosova, V. L. Rusinov, O. N. Chupakhin, "Chemosensors for detection of nitroaromatic compounds (explosives)", Usp. Khim., 83:9 (2014), 783–819 ; Russian Chem. Reviews, 83:9 (2014), 783–819
7. A. V. Shchepochkin, O. N. Chupakhin, V. N. Charushin, V. A. Petrosyan, "Direct nucleophilic functionalization of $\mathrm{C(sp^2)-H}$ bonds in (hetero)arenes by electrochemical methods", Usp. Khim., 82:8 (2013), 747–771 ; Russian Chem. Reviews, 82:8 (2013), 747–771
8. G. N. Lipunova, T. G. Fedorchenko, O. N. Chupakhin, "Verdazyls: synthesis, properties, application", Usp. Khim., 82:8 (2013), 701–734 ; Russian Chem. Reviews, 82:8 (2013), 701–734
9. T. I. Gorbunova, V. I. Saloutin, O. N. Chupakhin, "Chemical methods of transformation of polychlorobiphenyls", Usp. Khim., 79:6 (2010), 565–586 ; Russian Chem. Reviews, 79:6 (2010), 511–530
10. E. V. Shchegolkov, Ya. V. Burgart, O. G. Khudina, V. I. Saloutin, O. N. Chupakhin, "2-(Het)arylhydrazono-1,3-dicarbonyl compounds in organic synthesis", Usp. Khim., 79:1 (2010), 33–64 ; Russian Chem. Reviews, 79:1 (2010), 31–61
11. N. A. Itsikson, Yu. Yu. Morzherin, A. I. Matern, O. N. Chupakhin, "Receptors for anions", Usp. Khim., 77:9 (2008), 803–816 ; Russian Chem. Reviews, 77:9 (2008), 751–764
12. A. I. Matern, V. N. Charushin, O. N. Chupakhin, "Progress in the studies of oxidation of dihydropyridines and their analogues", Usp. Khim., 76:1 (2007), 27–45 ; Russian Chem. Reviews, 76:1 (2007), 23–40
13. I. N. Egorov, G. V. Zyryanov, V. L. Rusinov, O. N. Chupakhin, "Asymmetric induction in the nucleophilic addition in the series of aromatic azines", Usp. Khim., 74:12 (2005), 1176–1192 ; Russian Chem. Reviews, 74:12 (2005), 1073–1087
14. N. A. Itsikson, G. V. Zyryanov, O. N. Chupakhin, A. I. Matern, "Heteroditopic receptors", Usp. Khim., 74:8 (2005), 820–829 ; Russian Chem. Reviews, 74:8 (2005), 747–755
15. A. G. Pokrovskii, T. N. Il'icheva, S. K. Kotovskaya, S. A. Romanova, V. N. Charushin, O. N. Chupakhin, "Фторированные производные бенз[4,5]имидазо[1,$2-b$][1,3]тиазолов — ингибиторы репродукции вируса кори", Dokl. Akad. Nauk, 398:3 (2004), 412–414
16. O. N. Chupakhin, D. G. Beresnev, "Nucleophilic attack on the unsubstituted carbon atom of azines and nitroarenes as an efficient methodology for constructing heterocyclic systems", Usp. Khim., 71:9 (2002), 803–818 ; Russian Chem. Reviews, 71:9 (2002), 707–720
17. S. G. Perevalov, Ya. V. Burgart, V. I. Saloutin, O. N. Chupakhin, "(Het)aroylpyruvic acids and their derivatives as promising building blocks for organic synthesis", Usp. Khim., 70:11 (2001), 1039–1058 ; Russian Chem. Reviews, 70:11 (2001), 921–938
18. V. I. Saloutin, Ya. V. Burgart, O. N. Chupakhin, "Fluorine-containing 2,4-dioxo acids in the synthesis of heterocyclic compounds", Usp. Khim., 68:3 (1999), 227–239 ; Russian Chem. Reviews, 68:3 (1999), 203–214
19. D. N. Kozhevnikov, V. L. Rusinov, O. N. Chupakhin, "1,2,4-Triazine <i>N</i>-oxides and their annelated derivatives", Usp. Khim., 67:8 (1998), 707–722 ; Russian Chem. Reviews, 67:8 (1998), 633–648
20. V. N. Charushin, O. N. Chupakhin, "The Reactions of Azines on Treatment with 1,3-Bifunctional Nucleophiles", Usp. Khim., 53:10 (1984), 1648–1674 ; Russian Chem. Reviews, 53:10 (1984), 956–970
21. O. N. Chupakhin, I. Ya. Postovskii, "Nucleophilic Substitution of Hydrogen in Aromatic Systems", Usp. Khim., 45:5 (1976), 908–937 ; Russian Chem. Reviews, 45:5 (1976), 454–468
I. Ya. Postovsky Institute of Organic Synthesis, Ural Branch of Russian Academy of Sciences, Ekaterinburg
Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
Chemical Technological Institute, Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg | CommonCrawl |
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A limit involving the totient function
Active 10 years, 7 months ago
P. Erdős and Leon Alaoglu proved in [1] that for every $\epsilon > 0$ the inequality $\phi(\sigma(n)) < \epsilon \cdot n$ holds for every $n \in \mathbb{N}$, except for a set of density $0$.
C. L. mentioned in [2] that as a consequence of the previous result one can ascertain that $\displaystyle \lim_{n \to \infty} \frac{\phi(\sigma(n)) }{n} = 0.$
Does anybody know how it is that C. L. proceeded in order to arrive at such a conclusion?
Clearly enough, the fact that an inequality of the type $a_{n} < \epsilon \cdot n$ holds for every $\epsilon > 0$ and a subset of $\mathbb{N}$ of density $1$ does not imply, in general, that the sequence $\displaystyle \frac{a_{n}}{n}$ goes to $0$ as $n \to \infty$.
Hope you guys can shed some light on this inquiry of mine. Let me thank you in advance for your continued support.
[1] L. Alaoglu, P. Erdős: A conjecture in elementary number theory, Bull. Amer. Math. Soc. 50 (1944), 881-882.
[2] Mathematical Reflections, Solutions Dept, Issue #3, 2009, page 23.
nt.number-theory erdos
José Hdz. Stgo.
José Hdz. Stgo.José Hdz. Stgo.
$\begingroup$ The 1st result does not imply the 2nd! As you mention yourself, it's not hard to construct an example of $f\colon\mathbb N\to\mathbb R$ such that: (1) for any $\epsilon>0$, $f(n)<\epsilon n$ for "almost" all $n$, and (2) $f(n)/n\not\to 0$ as $n\to\infty$. So, what's the problem: you believe that the limit is true (from numerical evidence) and wish to improve Alaoglu--Erdős? My own experience shows that if the latter can be strengthened, then there should be Erdős's conjecture already in that paper. $\endgroup$ – Wadim Zudilin Jun 9 '10 at 23:45
$\begingroup$ It's not that I believe it to be true, Professor Zudilin. It's just that I wanted to know whether I was missing some properties of the arithmetical functions involved that yielded at once a "thorough" demonstration of the purported claim. $\endgroup$ – José Hdz. Stgo. Jun 10 '10 at 0:14
$\begingroup$ Don't trust no-author books! You didn't miss an argument but the Mathematical Reflections did. I am happy to see that Gerry gives some clear evidence of why $\limsup_{n\to\infty}\phi(\sigma(n))/n>0$ (and most probably is $1/2$). $\endgroup$ – Wadim Zudilin Jun 10 '10 at 1:37
$\begingroup$ Thanks for taking the time to leave your comments, Professor Zudilin. $\endgroup$ – José Hdz. Stgo. Jun 10 '10 at 1:52
$\begingroup$ Why go so far in the proof? His second statement is already false, the set X contains all of the sequence, and is not bounded. The print is riddled with errors and typos, it is impossible to know what C.L. actually meant. $\endgroup$ – Dror Speiser Jun 10 '10 at 6:45
Everyone knows (but no one can prove) that there are infinitely many primes $p$ such that $q=2p-1$ is also prime. $\sigma(q)=q+1=2p$, $\phi(\sigma(q))=\phi(2p)=p-1$, $\phi(\sigma(q))/q=(p-1)/(2p-1)\to1/2$ as $q\to\infty$.
Edit: I don't know why it didn't occur to me to look at Guy, Unsolved Problems In Number Theory. Under B42, he writes, "Makowski and Schinzel prove that $\limsup\phi(\sigma(n))/n=\infty$. The reference is A. Makowski, A. Schinzel, On the functions $\phi(n)$ and $\sigma(n)$, Colloq Math 13 (1964-65) 95-99, MR 30 #3870. I haven't found the paper on the web, but it's in Volume 2 of Schinzel's Selecta, 890-894. I don't have the energy to write out the proof in full, but here's the idea. Given $M$, choose $t$ such that $$\prod_{i=1}^t{p_i\over p_i-1}>M$$. Then given $p$ (and it's not clear to me whether $p$ is meant to be a prime), and letting $$n=\sigma\left(\prod_{i=1}^tp_i^{p-1}\right),$$ we get $$\sigma(n)=\prod_{i=1}^tN(p_i,p),$$ where $N(a,p)=(a^p-1)/(a-1)$. Now you prove $\limsup_{p\to\infty}\phi(\sigma(n))/n\gt M$, using along the way a lemma which says that $$\lim_{p\to\infty}{\phi(N(a,p))\over N(a,p)}=1.$$
Gerry MyersonGerry Myerson
$\begingroup$ That's a good point, Gerry! +1 $\endgroup$ – Wadim Zudilin Jun 10 '10 at 1:09
$\begingroup$ How do you call those primes p such that 2p-1 is also a prime number? Certainly, they are not Sophie Germain primes, are they? $\endgroup$ – José Hdz. Stgo. Jun 10 '10 at 1:48
$\begingroup$ @Gerry: Are you sure you did not mean to write $2^{p}-1$? $\endgroup$ – José Hdz. Stgo. Jun 10 '10 at 1:53
$\begingroup$ @J. H. S., $2^p-1$ also works, but what I wrote is what I meant. Why do you ask? Is there a mistake in my calculations? Or is it that you are more convinced of the infinity of primes of form $2^p-1$ than you are of primes of form $2p-1$? $\endgroup$ – Gerry Myerson Jun 10 '10 at 2:05
$\begingroup$ Mainly because I don't know how it that those primes $p$ are called. $\endgroup$ – José Hdz. Stgo. Jun 10 '10 at 2:07
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Summation involving Euler's totient function | CommonCrawl |
Laplace–Carson transform
In mathematics, the Laplace–Carson transform, named after Pierre Simon Laplace and John Renshaw Carson, is an integral transform with significant applications in the field of physics and engineering, particularly in the field of railway engineering.
Definition
Let $V(j,t)$ be a function and $p$ a complex variable. The Laplace–Carson transform is defined as:[1]
$V^{\ast }(j,p)=p\int _{0}^{\infty }V(j,t)e^{-pt}\,dt$
The inverse Laplace–Carson transform is:
$V(j,t)={\frac {1}{2\pi i}}\int _{a_{0}-i\infty }^{a_{0}+i\infty }e^{tp}{\frac {V^{\ast }(j,p)}{p}}\,dp$
where $a_{0}$ is a real-valued constant, $i\infty $ refers to the imaginary axis, which indicates the integral is carried out along a straight line parallel to the imaginary axis lying to the right of all the singularities of the following expression:
$e^{tp}{\frac {V(j,t)}{p}}$
See also
• Laplace transform
References
1. Frýba, Ladislav (1973). Vibration of solids and structures under moving loads. LCCN 70-151037.
| Wikipedia |
\begin{document}
\title{A new method to building Dirac quantum walks coupled to electromagnetic fields}
\author{Gareth Jay$^{1}$} \email{[email protected]} \author{Fabrice Debbasch$^{2}$} \email{[email protected]} \author{J. B. Wang$^{1}$} \email{[email protected]} \affiliation{{$^{1}$Physics Department, The University of Western Australia, Perth, WA 6009, Australia}\\ {$^{2}$Sorbonne Universit\'e, Observatoire de Paris, Universit\'e PSL, CNRS, LERMA, F-75005, {\sl Paris}, France}}
\date{\today} \begin{abstract} A quantum walk whose continuous limit coincides with Dirac equation is usually called a Dirac Quantum Walk (DQW). A new systematic method to build DQWs coupled to electromagnetic (EM) fields is introduced and put to test on several examples of increasing difficulty. It is first used to derive the EM coupling of a well-known $3D$ walk on the cubic lattice. Recently introduced DQWs on the triangular and honeycomb lattice are then re-derived, showing for the first time that these are the only DQWs that can be defined with spinors living on the vertices of these lattices. As a third example of the method's effectiveness, a new $3D$ walk on a parallelepiped lattice is derived. As a fourth, negative example, it is shown that certain lattices like the rhombohedral lattice cannot be used to build DQWs. The effect of changing representation in the Dirac equation is also discussed.
\end{abstract}
\keywords{gggggg} \maketitle
\section{Introduction} Quantum walks are unitary quantum automata first proposed by Feynman \cite{feynman2010quantum,schweber1986feynman}, that can be viewed as formal generalizations of classical random walks. First introduced systematically by Aharonov et al. \cite{aharonov1993quantum} and Myers \cite{meyer1996quantum}, they have found application in quantum information and algorithmic development \cite{ambainis2007quantum,magniez2011search,ManouchehriWang2014}. They can also be used as quantum simulators \cite{Strauch2006, Strauch2007, Kurzynski2008, Chandrashekar2013, Shikano2013, Arrighi2014,arrighi2016quantum, Molfetta2014, perez2016asymptotic}, where the lattice represents a discretization of continuous space. More ambitiously, quantum walks may represent a potentially realistic discrete spacetime underlying the apparently continuous physical universe \cite{bisio2016special}.
It has been shown that several Discrete-Time Quantum Walks (DTQWs) defined on regular square lattices simulate the Dirac dynamics in various spacetime dimensions and that these Dirac Quantum Walks (DQWs) can be coupled to various discrete gauge fields \cite{di2013quantum,di2014quantum,arnault2016landau,arnault2016quantum,arnault2016quantum2,arnault2017quantum,bisio2015quantum,marquez2018electromagnetic, bialynicki1994weyl}. It has also more recently been shown that $2D$ DQWs can be defined on regular non-square lattices like the triangle and honeycomb lattice \cite{jay2018dirac,arrighi2018dirac}. The aim of the present article is to extend these results by developing a new systematic approach to construct DQWs coupled to EM fields. The new approach is based on discretizing the Hamiltonian \cite{arrighi2014dirac} in terms of directional derivatives \cite{arrighi2018dirac}. It also presupposes that the walk wave-function and gauge fields live on the vertices of the lattice and that, given a target space-time dimension $D$, the number of components of the wave function coincide with the number of components of the Dirac spinor in irreducible representations of the $D$-dimensional Lorentz group. This approach is superior to a trial-and-error method because (i) it shows unambiguously if a DQW can be constructed on a given lattice (ii) it delivers automatically the coefficients of the DQW (iii) it becomes necessary to use a systematic approach if one wants to deal with physical spaces of dimensions higher than $2$.
We present the method and put it to test on several examples of increasing difficulty. We first derive the EM coupling of a well-known $3D$ walk on the cubic lattice. We then re-derive recently introduced DQWs on the triangular and honeycomb lattice, showing for the first time that these are the only DQWs that can be defined with spinors living on the vertices of these lattices. As a third example of the method's effectiveness, we build a new $3D$ walk on a parallelepiped lattice and, as a fourth and negative example, we show that certain lattices like the rhombohedral lattice cannot be used to build DQWs. We finally discuss the effect of changing representation in the Dirac equation and mention possible extensions of this work. In an appendix we provide a step-by-step walkthrough of the method starting from the simplest $(1+1)D$ free Dirac equation, and working our way up through the dimensions and coupled fields.
\section{The Hamiltonian approach}
A general DTQW takes the form of \begin{equation}\label{generalDTQW}
\Psi(t+\Delta t)=\hat{W}\Psi(t), \end{equation} where the wavefunction $\Psi$ is an $N$-component spinor and $\hat{W}$ in a unitary operator belonging to $U(N)$, whereas the Dirac Equation, when written in so-called 'Hamiltonian form' takes a Schr{\"o}dinger-like equation form of
\begin{equation}\label{DiracHamiltonian}
\mathrm{i}\pdv{\Psi}{t}=\hat{H} \Psi.
\end{equation} If we discretize the time derivative, equation \ref{DiracHamiltonian} becomes \begin{equation}
\mathrm{i}\left(\frac{\Psi(t+\epsilon)-\Psi(t)+\mathcal{O}(\epsilon^2)}{\epsilon}\right)=\hat{H} \Psi(t), \end{equation} which after some rearrangement becomes \begin{equation}\label{rearrangedHamiltonianFormDirac}
\Psi(t+\epsilon)=\left(\mathbb{I}_N-\mathrm{i}\epsilon\hat{H}+\mathcal{O}(\epsilon^2)\right)\Psi(t), \end{equation} and so by matching equation \ref{generalDTQW} with \ref{rearrangedHamiltonianFormDirac}, we are then looking for an operator $\hat{W}$ that is of the form \begin{equation}\label{defnW}
\hat{W}=\mathbb{I}_N-\mathrm{i}\epsilon\hat{H}+\mathcal{O}(\epsilon^2), \end{equation} where $N$ will be $4$ if our Dirac spinor is operating in $(3+1)D$ spacetime, or $2$ if in lower dimensions.
A quantum walk is a product of unitary operations, each representing either a translation or a mixing of states, acting on the wavefunction. So we need to factorize the expression in equation \ref{rearrangedHamiltonianFormDirac} to become a product of unitary operators. Fortunately due to the approximation to first order, any expression of the form \begin{equation}\label{factorA}
\mathbb{I}_N+\epsilon A+\epsilon B+\epsilon C+\mathcal{O}(\epsilon^2), \end{equation} can be factorized into \begin{equation}\label{factorB}
(\mathbb{I}_N+\epsilon A)(\mathbb{I}_N+\epsilon B)(\mathbb{I}_N+\epsilon C)+\mathcal{O}(\epsilon^2). \end{equation}
For a step-by-step walkthrough, starting from the simplest case of a free Dirac equation in $(1+1)D$ flat spacetime, see the appendix. Here we shall present the more complicated case of a Dirac equation in $(3+1)D$ flat spacetime coupled to an electromagnetic field before moving onto non-cube lattices.
In $(3+1)D$ flat spacetime, the Hamiltonian $\hat{H}$ in equation \ref{DiracHamiltonian}, if the standard Dirac representation for the Dirac gamma matrices is chosen, is written as \begin{equation}\label{H3DEM}
\hat{H}=-\mathbb{I}_4A_0-\mathrm{i}(\sigma_1\otimes\sigma_j)\partial_j-(\sigma_1\otimes\sigma_j)A_j+(\sigma_3\otimes\mathbb{I}_2)m. \end{equation} Plugging equation \ref{H3DEM} into equation \ref{defnW} we get \begin{eqnarray}
\hat{W}&=&\mathbb{I}_4+\mathrm{i}\epsilon A_0\mathbb{I}_4-\sum_{j=1}^3\epsilon(\sigma_1\otimes\sigma_j)\partial_j+\sum_{j=1}^3\mathrm{i}\epsilon A_j(\sigma_1\otimes\sigma_j)\nonumber\\&&-\mathrm{i} m\epsilon(\sigma_3\otimes\mathbb{I}_2)+\mathcal{O}(\epsilon^2), \end{eqnarray} where if we make use of the factorizing trick from equations \ref{factorA} to \ref{factorB} we get \begin{eqnarray}
\hat{W}&=&(\mathbb{I}_4-\mathrm{i} m\epsilon(\sigma_3\otimes\mathbb{I}_2))(\mathbb{I}_4+\mathrm{i}\epsilon A_0\mathbb{I}_4)\nonumber\\&&\times\prod^3_{j=1}\left(\mathbb{I}_4+\mathrm{i}\epsilon A_j(\sigma_1\otimes\sigma_j)\right)\prod^3_{j=1}\left(\mathbb{I}_4-\epsilon(\sigma_1\otimes\sigma_j)\partial_j\right)\nonumber\\&&+\mathcal{O}(\epsilon^2).\label{CubeWFactored} \end{eqnarray} The mass and electromagnetic field terms are easy enough to deal with, however the partial derivative terms require a bit more work since each shift operator can be described as \begin{eqnarray}
\hat{S}_j=e^{\epsilon(\sigma_3\otimes\sigma_3)\partial_j}=\mathbb{I}_4+\epsilon(\sigma_3\otimes\sigma_3)\partial_j+\mathcal{O}(\epsilon^2), \end{eqnarray} and so we need to factor out a unitary transformation of the form \begin{equation}\label{UnitaryTransform}
U_j\sigma_3U_j^\dagger=\sigma_j, \end{equation} which is described in more detail in the appendix. The partial derivative terms can then be expressed as \begin{equation}
\mathbb{I}_4-\epsilon(\sigma_1\otimes\sigma_j)\partial_j=\\(U_1\sigma_1\otimes U_j)\hat{S}_j(\sigma_1U_1^\dagger\otimes U_j^\dagger). \end{equation} Finally making use of the expansions \ref{cos}, \ref{sin} and \ref{exp} from the appendix you find the operator $\hat{W}$ to be \begin{eqnarray}
\hat{W}&=&\begin{pmatrix}
\mathrm{e}^{-\mathrm{i} m\epsilon}\mathbb{I}_2&0\\
0&\mathrm{e}^{\mathrm{i} m\epsilon}\mathbb{I}_2
\end{pmatrix}\mathrm{e}^{\mathrm{i}\epsilon A_0}\nonumber\\&&\times\prod^3_{j=1}\begin{pmatrix}\cos{(\epsilon A_j)}\mathbb{I}_2&\mathrm{i}\sin{(\epsilon A_j)}\sigma_j\\\mathrm{i}\sin{(\epsilon A_j)}\sigma_j&\cos{(\epsilon A_j)}\mathbb{I}_2\end{pmatrix}\nonumber\\&&\times\prod^3_{j=1}\left((U_1\sigma_1\otimes U_j)\hat{S}_j(\sigma_1U_1^\dagger\otimes U_j^\dagger)\right).\label{CubeWFinal} \end{eqnarray} So the DTQW that is defined by this operator $\hat{W}$ will in its continuous limit coincides with the $(3+1)D$ Dirac equation couples to an electromagnetic field $A_\mu$. We shall now extend this procedure to non-cube lattices.
\section{Non-Cube Lattices} In the attempt to extend this procedure to non-cube lattices we shall start in the simpler $(2+1)D$ spacetime and derive, from the Hamiltonian, two of the triangular lattice DQWs found in Jay et al. \cite{jay2018dirac}.
When looking at non-cube lattices we must make use of directional derivatives and rewrite the Dirac equation in terms of these. This will often bring in coefficients that make the quantum walk incompatible with the Dirac equation if you insist on keeping the lattice step size the same in each spacetime direction. This can be seen clearly in the DQW on a triangular lattice.
\subsection{Triangular Lattice}
\begin{figure}
\caption{Equilateral Triangular lattice embedded on a standard euclidean 2D space showing the two new directions $u$ and $v$}
\label{triangularlattice}
\end{figure}
Consider the equilateral triangular lattice in Fig. \ref{triangularlattice}. We can define two unit vectors in the direction of $u$ and $v$ as \begin{eqnarray*}
\hat{u}&=&\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right),\\
\hat{v}&=&\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right). \end{eqnarray*} These two unit vectors define the two directions that make this lattice different to the square lattices. (Our third direction across this lattice will still be the same as the $x$-direction in the square lattice). The directional derivatives associated with these directions are then \begin{eqnarray*}
\frac{\partial}{\partial u}=\vec{\nabla}\cdot\hat{u}&=&\frac{1}{2}\frac{\partial}{\partial x}+\frac{\sqrt{3}}{2}\frac{\partial}{\partial y},\\
\frac{\partial}{\partial v}=\vec{\nabla}\cdot\hat{v}&=&-\frac{1}{2}\frac{\partial}{\partial x}+\frac{\sqrt{3}}{2}\frac{\partial}{\partial y}.\\ \end{eqnarray*} Summing the two definitions we get the relationship \begin{equation}
\frac{\partial}{\partial y}=\frac{1}{\sqrt{3}}\left(\frac{\partial}{\partial u}+\frac{\partial}{\partial v}\right).\label{yDerivTri} \end{equation} The new directional shift operators are going to behave very similar to the standard $x-$ and $y-$direction ones, as can be seen when we define them:
\begin{eqnarray}
\hat{S}_u\Psi(x,y,t)&=&\begin{pmatrix}
\psi_L(x+\frac{\epsilon}{2},y+\frac{\sqrt{3}\epsilon}{2},t)\\
\psi_R(x-\frac{\epsilon}{2},y-\frac{\sqrt{3}\epsilon}{2},t)
\end{pmatrix}\nonumber\\
&=&\begin{pmatrix}
\psi_L(x,y,t)\\
\psi_R(x,y,t)
\end{pmatrix}+\epsilon\sigma_3\left(\frac{1}{2}\partial_1+\frac{\sqrt{3}}{2}\partial_2\right)\nonumber\\&&\times\begin{pmatrix}
\psi_L(x,y,t)\\
\psi_R(x,y,t)
\end{pmatrix}+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\begin{pmatrix}
\psi_L(x,y,t)\\
\psi_R(x,y,t)
\end{pmatrix}+\epsilon\sigma_3\partial_u\begin{pmatrix}
\psi_L(x,y,t)\\
\psi_R(x,y,t)
\end{pmatrix}\nonumber\\&&+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\left(\mathbb{I}_2+\epsilon\sigma_3\partial_u\right)\begin{pmatrix}
\psi_L(x,y,t)\\
\psi_R(x,y,t)
\end{pmatrix}+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\mathrm{e}^{\epsilon\sigma_3\partial_u}\Psi(x,y,t). \end{eqnarray} $\hat{S}_v$ is defined the exact same way and note since we are in the lower $(2+1)$ dimensions, $\hat{S}_1$ for the $x-$direction would also fit this pattern (see appendix). If we now plug equation \ref{yDerivTri} into the $(2+1)D$ free Dirac Equation's Hamiltonian we get \begin{eqnarray}
\hat{H}&=&\mathrm{i}\sigma_3\partial_1-\mathrm{i}\sigma_2\partial_2+\sigma_1 m\nonumber\\
&=&\mathrm{i}\sigma_3\partial_1-\mathrm{i}\sigma_2\frac{1}{\sqrt{3}}\left(\partial_u+\partial_v\right)+\sigma_1 m. \end{eqnarray} Plugging this into our operator $\hat{W}$ we get \begin{eqnarray}
\hat{W}&=&\mathbb{I}_2-\mathrm{i}\epsilon\hat{H}+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\mathbb{I}_2+\epsilon\sigma_3\partial_1-\epsilon\sigma_2\frac{1}{\sqrt{3}}\partial_u-\epsilon\sigma_2\frac{1}{\sqrt{3}}\partial_v-\mathrm{i} m\epsilon\sigma_1+\mathcal{O}(\epsilon^2)\nonumber\\
&=&(\mathbb{I}_2-\mathrm{i} m\epsilon\sigma_1)\left(1-\epsilon\sigma_2\frac{1}{\sqrt{3}}\partial_v\right)\left(1-\epsilon\sigma_2\frac{1}{\sqrt{3}}\partial_u\right)\nonumber\\&&\times(\mathbb{I}_2+\epsilon\sigma_3\partial_1)+\mathcal{O}(\epsilon^2).\nonumber\\ \end{eqnarray} Now if we were to take the approach we use in the square lattice we would be looking for a unitary operators $U$ that follows the relationship \begin{eqnarray}
U\sigma_3U^\dagger&=&-\frac{1}{\sqrt{3}}\sigma_2. \end{eqnarray} Unfortunately this relationship has no solution. There are two places we can go at this point. One involves changing the step size of the spatial directions, the other involves changing the step size in the time dimension. \subsubsection{Isosceles Lattice (changing the $y$ step size)}
The first method we shall consider is changing the step size of the spatial direction. We shall keep the $x-$direction at step size $\epsilon$ but will change the step size of the $y-$direction (effectively changing our $u$ and $v$ directions). This will effectively make our triangular lattice an isoceles lattice (see Fig. \ref{isolattice}).
\begin{figure}
\caption{Isosceles Triangular lattice embedded on a standard euclidean 2D space showing the two new directions $u$ and $v$}
\label{isolattice}
\end{figure}
In our previous work \cite{jay2018dirac} we found a quantum walk that works on an isosceles lattice where the triangle height was $\frac{\epsilon}{2}$ which is effectively dilating the $y-$direction by a factor of $\frac{1}{\sqrt{3}}$. We shall however not start with this assumption, but place an arbitrary factor of $\upsilon$ with the hope of proving that we want $\upsilon=\frac{1}{\sqrt{3}}$. We can now define our two unit vectors $\hat{u}$ and $\hat{v}$ as \begin{eqnarray*}
\hat{u}&=&\frac{2}{\sqrt{1+3\upsilon^2}}\left(\frac{1}{2},\frac{\upsilon\sqrt{3}}{2}\right),\\
\hat{v}&=&\frac{2}{\sqrt{1+3\upsilon^2}}\left(-\frac{1}{2},\frac{\upsilon\sqrt{3}}{2}\right). \end{eqnarray*} Where the factor out the front is the normalisation factor for the unit vector that we can refer to as $N$. The directional derivatives associated with these directions are then \begin{eqnarray*}
\partial_u&=&N\left(\frac{1}{2}\partial_1+\frac{\upsilon\sqrt{3}}{2}\partial_2\right),\\
\partial_v&=&N\left(-\frac{1}{2}\partial_1+\frac{\upsilon\sqrt{3}}{2}\partial_2\right). \end{eqnarray*} Summing these two definitions we get the relationship \begin{eqnarray}
\partial_2=\frac{1}{N\upsilon\sqrt{3}}\left(\partial_u+\partial_v\right).\label{triangleDerivates} \end{eqnarray} The new directional shift operators are going to behave very similar but the normalisation constant $N$ will turn up \begin{eqnarray}
\hat{S}_u\Psi(x,y,t)&=&\begin{pmatrix}
\psi_L(x+\frac{\epsilon}{2},y+\frac{\sqrt{3}\epsilon}{2},t)\\
\psi_R(x-\frac{\epsilon}{2},y-\frac{\sqrt{3}\epsilon}{2},t)
\end{pmatrix}\nonumber\\
&=&\begin{pmatrix}
\psi_L(x,y,t)\\
\psi_R(x,y,t)
\end{pmatrix}\nonumber\\&&+\epsilon\sigma_3(\frac{1}{2}\partial_1+\frac{\sqrt{3}}{2}\partial_2)\begin{pmatrix}
\psi_L(x,y,t)\\
\psi_R(x,y,t)
\end{pmatrix}+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\begin{pmatrix}
\psi_L(x,y,t)\\
\psi_R(x,y,t)
\end{pmatrix}\nonumber\\&&+\epsilon\sigma_3\frac{1}{N}\partial_u\begin{pmatrix}
\psi_L(x,y,t)\\
\psi_R(x,y,t)
\end{pmatrix}+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\left(\mathbb{I}_2+\epsilon\sigma_3\frac{1}{N}\partial_u\right)\begin{pmatrix}
\psi_L(x,y,t)\\
\psi_R(x,y,t)
\end{pmatrix}+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\mathrm{e}^{\frac{1}{N}\epsilon\sigma_3\partial_u}\Psi(x,y,t). \end{eqnarray} $\hat{S}_v$ is defined the exact same way, while $\hat{S}_1$ will be completely unaffected by the dilation of the $y-$direction. You can see $\hat{S}_u$ and $\hat{S}_v$ appear very similar to before however the $\frac{1}{N}$ factor is appearing. The Hamiltonian now becomes \begin{eqnarray}
\hat{H}&=&\mathrm{i}\sigma_3\partial_1-\mathrm{i}\sigma_2\partial_2+\sigma_1m\nonumber\\
&=&\mathrm{i}\sigma_3\partial_1-\mathrm{i}\sigma_2\frac{1}{N\upsilon\sqrt{3}}(\partial_u+\partial_v)+\sigma_1m. \end{eqnarray} The factor attached to our $\partial_u$ and $\partial_v$ terms is now $\frac{1}{N\upsilon\sqrt{3}}$. The $N$ part of this is not a problem as we want the $\frac{1}{N}$ factor for our shift operators this time round. The problem is the $\upsilon\sqrt{3}$ part. This is where is becomes clear that the choice of $\upsilon$ should indeed be $\frac{1}{\sqrt{3}}$ as seen in the previous work \cite{jay2018dirac}. This will lead to the $\hat{W}$ operator being \begin{eqnarray}
\hat{W}&=&\mathbb{I}_2-\mathrm{i}\epsilon\hat{H}+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\mathbb{I}_2-\mathrm{i} m\epsilon\sigma_1-\frac{1}{N}\epsilon\sigma_2\partial_v-\frac{1}{N}\epsilon\sigma_2\partial_u+\epsilon\sigma_3\partial_1+\mathcal{O}(\epsilon^2)\nonumber\\
&=&(\mathbb{I}_2-\mathrm{i} m\epsilon\sigma_1)(\mathbb{I}_2-\frac{1}{N}\epsilon\sigma_2\partial_v)(\mathbb{I}_2-\frac{1}{N}\epsilon\sigma_2\partial_u)\nonumber\\&&\times(\mathbb{I}_2+\epsilon\sigma_3\partial_1)+\mathcal{O}(\epsilon^2)\nonumber\\
&=&(\mathbb{I}_2-\mathrm{i} m\epsilon\sigma_1)U(\mathbb{I}_2+\frac{1}{N}\epsilon \sigma_3\partial_v)U^\dagger \nonumber\\&&\times U(\mathbb{I}_2+\frac{1}{\sqrt{2}}\epsilon \sigma_3\partial_u)U^\dagger(\mathbb{I}_2+\epsilon\sigma_3\partial_1)+\mathcal{O}(\epsilon^2)\nonumber\\
&=&(\mathbb{I}_2-\mathrm{i} m\epsilon\sigma_1)U\hat{S}_vU^\dagger U\hat{S}_uU^\dagger\hat{S}_1+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\begin{pmatrix}
\cos(m\epsilon)&-\mathrm{i}\sin(m\epsilon)\\
-\mathrm{i}\sin(m\epsilon)&cos(m\epsilon)
\end{pmatrix}U\hat{S}_v\hat{S}_uU^\dagger\hat{S}_1, \end{eqnarray} where $U$ is a unitary matrix such that \begin{equation}
U\sigma_3U^\dagger=-\sigma_2, \end{equation} thus $U$ follows the same rules as described in the square lattice case in the appendix. The $U$ we shall choose is \begin{equation} U=\begin{pmatrix} \frac{1}{\sqrt{2}}&-\frac{i}{\sqrt{2}}\\ -\frac{i}{\sqrt{2}}&\frac{1}{\sqrt{2}} \end{pmatrix}. \end{equation}
Note that this operator is the same as the operator for the square lattice, except that $\hat{S}_2$ is replaced with the composition $\hat{S}_v\hat{S}_u$. This is directly related to the relationship in equation \ref{triangleDerivates}. This lines up perfectly with the isosceles walk defined in our previous paper \cite{jay2018dirac}. The other method of making a DQW work on a triangular lattice is by changing the time step.
\subsubsection{Equilateral Triangle Lattice (changing the $t$ step size)}
From previous work we have found that unlike the isosceles triangular lattice, walks across equilateral triangles needed the step size in time to be different to that in the spatial directions \cite{jay2018dirac}. So this is the place to start with the other method to keep the lattice equilateral. When we set \begin{equation*}
\hat{W}=\mathbb{I}_2-\mathrm{i}\epsilon\hat{H}+\mathcal{O}(\epsilon^2) \end{equation*} at the beginning, we are implicitly setting the step size in time to be $\Delta t=\epsilon$. If instead we make this an arbitrary factor of $\epsilon$, i.e. $\Delta t=\alpha \epsilon$ and feed this into the definition for $\hat{W}$ we get \begin{equation}
\hat{W}=\mathbb{I}_2-\mathrm{i}\alpha\epsilon\hat{H}+\mathcal{O}(\epsilon^2). \end{equation} Subsequently feed this into what we had before and we get \begin{equation*}
\hat{W}=\mathbb{I}_2-\mathrm{i} m\alpha\epsilon\sigma_1-\frac{1}{\sqrt{3}}\alpha\epsilon\sigma_2\partial_v-\frac{1}{\sqrt{3}}\alpha\epsilon\sigma_2\partial_u+\alpha\epsilon\sigma_3\partial_1+\mathcal{O}(\epsilon^2). \end{equation*} Now one would be tempted to choose $\alpha=\sqrt{3}$ which would reduce the middle terms to what we had before and we're good to go, but of course there are other $\alpha$ terms hanging around now. It is not essentially a problem in the mass term, however in the $\partial_1$ term, we now have a problem we did not have before. A work around for this is if we split this term up into two parts of size $\frac{1}{\alpha}$ and $\frac{\alpha-1}{\alpha}$ we get \begin{eqnarray}
\hat{W}&=&\mathbb{I}_2-\mathrm{i} m\alpha\epsilon\sigma_1-\frac{1}{\sqrt{3}}\alpha\epsilon\sigma_2\partial_v-\frac{1}{\sqrt{3}}\alpha\epsilon\sigma_2\partial_u\nonumber\\&&+\alpha\left(\frac{1}{
\alpha}+\frac{\alpha-1}{\alpha}\right)\epsilon\sigma_3\partial_1+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\mathbb{I}_2-\mathrm{i} m\alpha\epsilon\sigma_1-\frac{1}{\sqrt{3}}\alpha\epsilon\sigma_2\partial_v-\frac{1}{\sqrt{3}}\alpha\epsilon\sigma_2\partial_u\nonumber\\&&+\left(\alpha-1\right)\epsilon\sigma_3\partial_1+\epsilon\sigma_3\partial_1+\mathcal{O}(\epsilon^2).\nonumber\\ \end{eqnarray} This essentially gives us an $\epsilon\sigma_3\partial_1$ term that will work normally and head towards an $\hat{S}_1$ operator, and a more troublesome $\alpha-1$ term. This however can be moved into the $\partial_u$ and $\partial_v$ terms by making a relationship for $\partial_1$ through the subtraction of the directional derivatives: \begin{equation}
\partial_1=\partial_u-\partial_v. \end{equation} Subbing these in we get \begin{eqnarray}
\hat{W}&=&\mathbb{I}_2-\mathrm{i} m\alpha\epsilon\sigma_1-\frac{1}{\sqrt{3}}\alpha\epsilon\sigma_2\partial_v-\frac{1}{\sqrt{3}}\alpha\epsilon\sigma_2\partial_u\nonumber\\&&+\left(\alpha-1\right)\epsilon\sigma_3(\partial_u-\partial_v)+\epsilon\sigma_3\partial_1+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\mathbb{I}_2-\mathrm{i} m\alpha\epsilon\sigma_1-\left(\left(\alpha-1\right)\sigma_3+\frac{\alpha}{\sqrt{3}}\sigma_2\right)\epsilon\partial_v\nonumber\\&&+\left(\left(\alpha-1\right)\sigma_3-\frac{\alpha}{\sqrt{3}}\sigma_2\right)\epsilon\partial_u+\epsilon\sigma_3\partial_1+\mathcal{O}(\epsilon^2)\nonumber\\
&=&(\mathbb{I}_2-\mathrm{i} m\alpha\epsilon\sigma_1)\left(\mathbb{I}_2-\left(\left(\alpha-1\right)\sigma_3+\frac{\alpha}{\sqrt{3}}\sigma_2\right)\epsilon\partial_v\right)\nonumber\\&&\times\left(\mathbb{I}_2+\left(\left(\alpha-1\right)\sigma_3-\frac{\alpha}{\sqrt{3}}\sigma_2\right)\epsilon\partial_u\right)(\mathbb{I}_2+\epsilon\sigma_3\partial_1)\nonumber\\&&+\mathcal{O}(\epsilon^2). \end{eqnarray} So now we're looking for two unitary operators $U_v$ and $U_u$ such that \begin{eqnarray}
U_v\sigma_3U_v^\dagger&=&-\left(\left(\alpha-1\right)\sigma_3+\frac{\alpha}{\sqrt{3}}\sigma_2\right),\nonumber\\
U_u\sigma_3U_u^\dagger&=&\left(\left(\alpha-1\right)\sigma_3-\frac{\alpha}{\sqrt{3}}\sigma_2\right), \end{eqnarray} where $\alpha\ne1$ and $\alpha\ne0$. Again these unitary operators will not be unique, however we will see that the choice of $\alpha$ is surprisingly forced upon us. Let us consider the first relationship above and parameterise $U_v$ as \begin{equation*}
U_v=\begin{pmatrix}a&b\\c&d\end{pmatrix}. \end{equation*} The result follows that \begin{eqnarray}
|a|^2=|d|^2&=&1-\frac{\alpha}{2},\nonumber\\
|c|^2=|b|^2&=&\frac{\alpha}{2},\nonumber\\
a\bar{c}=d\bar{b}&=&\frac{\mathrm{i}\alpha}{2\sqrt{3}}. \end{eqnarray}
Now if we parameterise each of the four components in the style of $a=|a|\mathrm{e}^{\mathrm{i}\theta_a}$ and rewrite the $a\bar{c}$ part of the third equation above, (this works similarly for $b$ and $d$), \begin{eqnarray}
|a|\mathrm{e}^{\mathrm{i}\theta_a}|c|\mathrm{e}^{-\mathrm{i}\theta_c}&=&\frac{\mathrm{i}\alpha}{2\sqrt{3}}\nonumber\\
|a||c|\mathrm{e}^{\mathrm{i}(\theta_a-\theta_c)}&=&\frac{\mathrm{i}}{2\sqrt{3}}\alpha\nonumber\\
\sqrt{1-\frac{\alpha}{2}}\sqrt{\frac{\alpha}{2}}\mathrm{e}^{\mathrm{i}(\theta_a-\theta_c)}&=&\frac{\mathrm{i}}{2\sqrt{3}}\alpha\nonumber\\
\frac{\alpha}{2}-\frac{\alpha^2}{4}&=&-\frac{1}{12}\mathrm{e}^{\mathrm{i}(\theta_c-\theta_a)}\alpha^2\nonumber\\
\frac{6}{\alpha}-3&=&-\mathrm{e}^{\mathrm{i}(\theta_c-\theta_a)}\nonumber\\
\frac{6}{\alpha}&=&3-\mathrm{e}^{\mathrm{i}(\theta_c-\theta_a)}\nonumber\\
\alpha&=&\frac{6}{3-\mathrm{e}^{\mathrm{i}(\theta_c-\theta_a)}}.\nonumber\\ \end{eqnarray}
Now it is good to remember here that $\alpha$ represents a step in time, and so must be real, not complex. This means that the $\mathrm{e}^{\mathrm{i}(\theta_c-\theta_a)}$ term must equal $\pm1$. This would result in $\alpha$ being either $3$ or $\frac{3}{2}$. However if $\alpha=3$ then with the above relations $|a|^2=|d|^2=-\frac{1}{2}$ which can't be true since modulas squared values need to be positive. And so we show $\alpha$ must equal $\frac{3}{2}$. Once this is determined we have the relations \begin{eqnarray}
|a|^2=|d|^2&=&\frac{1}{4},\nonumber\\
|c|^2=|b|^2&=&\frac{3}{4},\nonumber\\
a\bar{c}=d\bar{b}&=&\frac{\mathrm{i}\sqrt{3}}{4}. \end{eqnarray} We now have two degrees of freedom so if we choose the simplest choice for $c$ and $b$ to be $\frac{\sqrt{3}}{2}$ we get \begin{equation}
U_v=\begin{pmatrix}\frac{\mathrm{i}}{2}&\frac{\sqrt{3}}{2}\\
\frac{\sqrt{3}}{2}&\frac{\mathrm{i}}{2}\end{pmatrix}. \end{equation} Through a similar technique you can determine $U_u$ as \begin{equation}
U_u=\begin{pmatrix}\frac{\sqrt{3}}{2}&-\frac{\mathrm{i}}{2}\\
-\frac{\mathrm{i}}{2}&\frac{\sqrt{3}}{2}\end{pmatrix}. \end{equation} We then continue as before with the knowledge of $\alpha$ and our unitary matrices \begin{eqnarray}
\hat{W}&=&(\mathbb{I}_2-im\frac{3}{2}\epsilon\sigma_1)\left(\mathbb{I}_2-\left(\frac{1}{2}\sigma_3+\frac{\sqrt{3}}{2}\sigma_2\right)\epsilon\partial_v\right)\nonumber\\&&\times\left(\mathbb{I}_2+\left(\frac{3}{2}\sigma_3-\frac{\sqrt{3}}{2}\sigma_2\right)\epsilon\partial_u\right)(\mathbb{I}_2+\epsilon\sigma_3\partial_1)+\mathcal{O}(\epsilon^2)\nonumber\\
&=&(\mathbb{I}_2-im\frac{3}{2}\epsilon\sigma_1)\left(\mathbb{I}_2+U_v\sigma_3U_v^\dagger\epsilon\partial_v\right)\nonumber\\&&\times\left(\mathbb{I}_2+U_u\sigma_3U_u^\dagger\epsilon\partial_u\right)\hat{S}_1+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\begin{pmatrix}\cos(\frac{3}{2}m\epsilon)&-i\sin(\frac{3}{2}m\epsilon)\\-i\sin(\frac{3}{2}m\epsilon)&\cos(\frac{3}{2}m\epsilon)\end{pmatrix}U_v\hat{S}_vU_v^\dagger U_u\hat{S}_uU_u^\dagger\hat{S}_1.\nonumber\\ \end{eqnarray}
This matches the equilateral walk found in previous work \cite{jay2018dirac} and also proves that the $\frac{3}{2}$ time step is a required choice. Note however that the $U$ operators found here are \emph{slightly} different choice to the ones found in \cite{jay2018dirac} due to the fact that the choice of $\gamma$ matrices are different. We have now seen enough to extend this to 3D lattices.
\subsection{Parallelepiped Lattice}
An obvious question about the non-square lattices where our last paper left off \cite{jay2018dirac} is whether or not the idea can be extended to $(3+1)D$ spacetime. So we shall attempt to apply the techniques we've discovered so far to a parallelepiped lattice. Consider the rhombohedron in Figure \ref{rhombohedron}. If we have the extra edges on the smallest diagonal of each rhombus face then we obtain a 3D lattice where on 3 different planes we can see the same triangular lattice as above. If the sides of the rhombohedron are of size $\epsilon$ then at any particular vertex of the lattice made up of this shape we would have the ability to move in the directions (or opposite directions) of the following unit vectors
\begin{figure}
\caption{A rhombohedron}
\label{rhombohedron}
\end{figure}
\begin{eqnarray}
\hat{x}&=&\left(1,0,0\right),\nonumber\\
\hat{a}&=&\left(\frac{1}{2},\frac{1}{2\sqrt{3}},\sqrt{\frac{2}{3}}\right),\nonumber\\
\hat{b}&=&\left(-\frac{1}{2},\frac{1}{2\sqrt{3}},\sqrt{\frac{2}{3}}\right),\nonumber\\
\hat{c}&=&\left(\frac{1}{2},\frac{\sqrt{3}}{2},0\right),\nonumber\\
\hat{d}&=&\left(-\frac{1}{2},\frac{\sqrt{3}}{2},0\right),\nonumber\\
\hat{e}&=&\left(0,-\frac{1}{\sqrt{3}},\sqrt{\frac{2}{3}}\right). \end{eqnarray}
Clearly we are going to run into the same issues with all those surds floating around as before, so we shall pre-empt these problems and again consider our two options - changing the spatial direction step size or changing the time step size.
\subsubsection{Parallelepiped Lattice (changing the $y$ and $z$ step sizes)}
We will introduce factors of $\upsilon$ and $\zeta$ this time to dilate the $y-$ and $z-$ directions respectively. This will of course mean we are no longer technically on a rhombohedral lattice but a more general parallelepiped lattice. Our unit vectors now become
\begin{eqnarray}
\hat{x}&=&\left(1,0,0\right),\nonumber\\
\hat{a}&=&\frac{2\sqrt{3}}{\sqrt{3+\upsilon^2+8\zeta^2}}\left(\frac{1}{2},\frac{\upsilon}{2\sqrt{3}},\zeta\sqrt{\frac{2}{3}}\right),\nonumber\\
\hat{b}&=&\frac{2\sqrt{3}}{\sqrt{3+\upsilon^2+8\zeta^2}}\left(-\frac{1}{2},\frac{\upsilon}{2\sqrt{3}},\zeta\sqrt{\frac{2}{3}}\right),\nonumber\\
\hat{c}&=&\frac{2}{\sqrt{1+3\upsilon^2}}\left(\frac{1}{2},\frac{\upsilon\sqrt{3}}{2},0\right),\nonumber\\
\hat{d}&=&\frac{2}{\sqrt{1+3\upsilon^2}}\left(-\frac{1}{2},\frac{\upsilon\sqrt{3}}{2},0\right),\nonumber\\
\hat{e}&=&\frac{\sqrt{3}}{\sqrt{\upsilon^2+2\zeta^2}}\left(0,-\frac{\upsilon}{\sqrt{3}},\zeta\sqrt{\frac{2}{3}}\right). \end{eqnarray}
Making our corresponding directional derivatives
\begin{eqnarray}
\partial_a&=&A\left(\frac{1}{2}\partial_1+\frac{\upsilon}{2\sqrt{3}}\partial_2+\zeta\sqrt{\frac{2}{3}}\partial_3)\right),\nonumber\\
\partial_b&=&B\left(-\frac{1}{2}\partial_1+\frac{\upsilon}{2\sqrt{3}}\partial_2+\zeta\sqrt{\frac{2}{3}}\partial_3\right),\nonumber\\
\partial_c&=&C\left(\frac{1}{2}\partial_1+\frac{\upsilon\sqrt{3}}{2}\partial_2\right),\nonumber\\
\partial_d&=&D\left(-\frac{1}{2}\partial_1+\frac{\upsilon\sqrt{3}}{2}\partial_2\right),\nonumber\\
\partial_e&=&E\left(-\frac{\upsilon}{\sqrt{3}}\partial_2+\zeta\sqrt{\frac{2}{3}}\partial_3\right), \end{eqnarray} where $A$, $B$, $C$, $D$ and $E$ are the normalization constants at the front of each of the unit vectors. Then recalling $A=B$ and $C=D$ we have the relationships \begin{eqnarray}
\partial_2&=&\frac{1}{\upsilon\sqrt{3}C}\left(\partial_c+\partial_d\right),\nonumber\\
\partial_3&=&\frac{1}{\zeta\sqrt{6}}\left(\frac{1}{A}\left(\partial_a+\partial_b\right)+\frac{1}{E}\partial_e\right), \end{eqnarray} and the corresponding shift operators: \begin{eqnarray}
\hat{S}_a\Psi(x,y,z,t)&=&\begin{pmatrix}
\psi_1(x+\frac{\epsilon}{2},y+\frac{\epsilon\upsilon}{2\sqrt{3}},z+\epsilon\zeta\sqrt{\frac{2}{3}},t)\\
\psi_2(x-\frac{\epsilon}{2},y-\frac{\epsilon\upsilon}{2\sqrt{3}},z-\epsilon\zeta\sqrt{\frac{2}{3}},t)\\
\psi_3(x-\frac{\epsilon}{2},y-\frac{\epsilon\upsilon}{2\sqrt{3}},z-\epsilon\zeta\sqrt{\frac{2}{3}},t)\\
\psi_4(x+\frac{\epsilon}{2},y+\frac{\epsilon\upsilon}{2\sqrt{3}},z+\epsilon\zeta\sqrt{\frac{2}{3}},t)\\
\end{pmatrix}\nonumber\\
&=&\mathrm{e}^{\frac{1}{A}\epsilon(\sigma_3\otimes\sigma_3)\partial_a}\Psi(x,y,z,t). \end{eqnarray} Each of the other shift operators follow this same pattern. Jumping to the Hamiltonian now we get \begin{eqnarray}
\hat{H}&=&(\sigma_3\otimes\mathbb{I}_2)m-\mathrm{i}(\sigma_1\otimes\sigma_1)\partial_1-\mathrm{i}(\sigma_1\otimes\sigma_2)\partial_2-\mathrm{i}(\sigma_1\otimes\sigma_3)\partial_3\nonumber\\
&=&(\sigma_3\otimes\mathbb{I}_2)m-\mathrm{i}(\sigma_1\otimes\sigma_1)\partial_1-\mathrm{i}(\sigma_1\otimes\sigma_2)\frac{1}{\upsilon\sqrt{3}C}\left(\partial_c+\partial_d\right)\nonumber\\&&-\mathrm{i}(\sigma_1\otimes\sigma_3)\frac{1}{\zeta\sqrt{6}}\left(\frac{1}{A}\left(\partial_a+\partial_b\right)+\frac{1}{E}\partial_e\right).\nonumber\\ \end{eqnarray} Now again remembering $A=B$ and $C=D$ then this becomes \begin{eqnarray}
\hat{H}&=&(\sigma_3\otimes\mathbb{I}_2)m-\mathrm{i}(\sigma_1\otimes\sigma_1)\partial_1-\mathrm{i}(\sigma_1\otimes\sigma_2)\frac{1}{C\upsilon\sqrt{3}}\partial_c\nonumber\\&&-\mathrm{i}(\sigma_1\otimes\sigma_2)\frac{1}{D\upsilon\sqrt{3}}\partial_d-\mathrm{i}(\sigma_1\otimes\sigma_3)\frac{1}{\zeta\sqrt{6}A}\partial_a\nonumber\\&&-\mathrm{i}(\sigma_1\otimes\sigma_3)\frac{1}{\zeta\sqrt{6}B}\partial_b-\mathrm{i}(\sigma_1\otimes\sigma_3)\frac{1}{\zeta\sqrt{6}E}\partial_e. \end{eqnarray} This certainly suggests sensible choices for $\upsilon$ and $\zeta$ are $\frac{1}{\sqrt{3}}$ and $\frac{1}{\sqrt{6}}$ respectively since this would give \begin{eqnarray}
\hat{H}&=&(\sigma_3\otimes\mathbb{I}_2)m-\mathrm{i}(\sigma_1\otimes\sigma_1)\partial_1-\mathrm{i}(\sigma_1\otimes\sigma_2)\frac{1}{C}\partial_c\nonumber\\&&-\mathrm{i}(\sigma_1\otimes\sigma_2)\frac{1}{D}\partial_d-\mathrm{i}(\sigma_1\otimes\sigma_3)\frac{1}{A}\partial_a\nonumber\\&&-\mathrm{i}(\sigma_1\otimes\sigma_3)\frac{1}{B}\partial_b-\mathrm{i}(\sigma_1\otimes\sigma_3)\frac{1}{E}\partial_e, \end{eqnarray} which if we make use of the unitary operators $U_j$ with $j\in{1,2,3}$ from the $(3+1)D$ flat spacetime on a cube lattice we can get our $\hat{W}$ operator as \begin{eqnarray}
\hat{W}&=&\mathbb{I}_4-\mathrm{i}\epsilon\hat{H}+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\mathbb{I}_4-\mathrm{i} m\epsilon(\sigma_3\otimes\mathbb{I}_2)-\epsilon(\sigma_1\otimes\sigma_1)\partial_1-\epsilon(\sigma_1\otimes\sigma_2)\frac{1}{C}\partial_c\nonumber\\&&-\epsilon(\sigma_1\otimes\sigma_2)\frac{1}{D}\partial_d-\epsilon(\sigma_1\otimes\sigma_3)\frac{1}{A}\partial_a-\epsilon(\sigma_1\otimes\sigma_3)\frac{1}{B}\partial_b\nonumber\\&&-\epsilon(\sigma_1\otimes\sigma_3)\frac{1}{E}\partial_e+\mathcal{O}(\epsilon^2)\nonumber\\
&=&(\mathbb{I}_4-\mathrm{i} m\epsilon(\sigma_3\otimes\mathbb{I}_2))\left(\mathbb{I}_4-\epsilon(\sigma_1\otimes\sigma_3)\frac{1}{E}\partial_e\right)\nonumber\\&&\times\left(\prod_{j\in\{d,c\}}\left(\mathbb{I}_4-\epsilon(\sigma_1\otimes\sigma_2)\frac{1}{J}\partial_j\right)\right)\nonumber\\
&&\times\left(\prod_{j\in\{b,a\}}\left(\mathbb{I}_4-\epsilon(\sigma_1\otimes\sigma_3)\frac{1}{J}\partial_j\right)\right)\nonumber\\&&\times\left(\mathbb{I}_4-\epsilon(\sigma_1\otimes\sigma_1)\partial_1\right)+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\begin{pmatrix}\mathrm{e}^{-\mathrm{i} m\epsilon}\mathbb{I}_2&0\\0&\mathrm{e}^{\mathrm{i} m\epsilon}\mathbb{I}_2\end{pmatrix}(U_1\sigma_1\otimes U_3)\hat{S}_e(\sigma_1U_1^\dagger\otimes U_3^\dagger)\nonumber\\&&\times\left(\prod_{j\in\{d,c\}}\left((U_1\sigma_1\otimes U_2)\hat{S}_j(\sigma_1 U_1^\dagger\otimes U_2^\dagger)\right)\right)\nonumber\\&&\times\left(\prod_{j\in\{b,a\}}\left(U_1\sigma_1\otimes U_3)\hat{S}_j(\sigma_1 U_1^\dagger\otimes U_3^\dagger)\right)\right)\nonumber\\&&\times\left((U_1\sigma_1\otimes U_1)\hat{S}_1(\sigma_1 U_1^\dagger\otimes U_1^\dagger)\right). \end{eqnarray}
Coupling this to an electromagnetic field would be trivial, as all that would be required is to include the operators from the cube walk that involve the $A_\mu$ terms as they have no effect on the shift operators.
\subsubsection{Rhombohedral Lattice (changing the $t$ step size) doesn't work}
It would be sensible at this point to think we could try a similar trick that was used in the triangular lattice to see if we can keep all the edges of our parallelepiped equal as in the case of a rhombohedral lattice. Unfortunately this fails to work, as when you follow the same procedure of moving the $\alpha$ off the $\partial_1$ term, there is no real solution for $\alpha$. This suggests that whereas before we could replace dilating in the $y-$dimension with a dilation in the time dimension, here we need to dilate in two dimensions - both $y$ and $z$. The time dimension can then only replace one of them. This suggests, at least under this construction, that a DQW on a rhombohedral lattice is not possible.
\section{Representation Invariance}
The previous section developed a DQW that works on a cubic lattice in a very particular representation - the standard Dirac representation. It is worth noting that this construction process does not require this choice of representation, it is in fact independent of the choice - although slightly different QWs will arise at the end of the construction. The first point to consider is the fact that any representation of the Dirac gamma matrices used in the Dirac equation can be related to each other by a unitary transformation. That is, \begin{equation}
\gamma'^\mu=\tilde{U}\gamma^\mu \tilde{U}^\dagger. \label{repchange} \end{equation} If we write the Hamiltonian (for arbitrary dimensions or lattice in free space) in terms of these gamma matrices instead we get \begin{equation}
\hat{H}=\tilde{U}\gamma^0\tilde{U}^\dagger m\epsilon-\sum_{j=1}^3\mathrm{i}\tilde{U}\gamma^0\gamma^j\tilde{U}^\dagger\partial_j. \end{equation} The $\tilde{U}$ operators can easily be pulled out to the sides of our $\hat{W}$ operator by taking advantage of their unitary nature, \begin{eqnarray}
\hat{W}&=&\left(\mathbb{I}_4-\mathrm{i} m\epsilon\tilde{U}\gamma^0\tilde{U}^\dagger\right)\prod_{j=1}^3\left(\mathbb{I}_4-\epsilon\tilde{U}\gamma^0\gamma^j\tilde{U}^\dagger\partial_j\right)+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\left(\tilde{U}\tilde{U}^\dagger-\mathrm{i} m\epsilon\tilde{U}\gamma^0\tilde{U}^\dagger\right)\nonumber\\&&\times\prod_{j=1}^3\left(\tilde{U}\tilde{U}^\dagger-\epsilon\tilde{U}\gamma^0\gamma^j\tilde{U}^\dagger\partial_j\right)+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\tilde{U}\left(\mathbb{I}_4-\mathrm{i} m\epsilon\gamma^0\right)\tilde{U}^\dagger\nonumber\\&&\times\prod_{j=1}^3\tilde{U}\left(\mathbb{I}_4-\epsilon\gamma^0\gamma^j\partial_j\right)\tilde{U}^\dagger+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\tilde{U}\left(\mathbb{I}_4-\mathrm{i} m\epsilon\gamma^0\right)\left(\prod^3_{j=1}\left(\mathbb{I}_4-\epsilon\gamma^0\gamma^j\partial_j\right)\right)\tilde{U}^\dagger+\mathcal{O}(\epsilon^2).\nonumber\\ \end{eqnarray} This is an arbitrary dimension, arbitrary lattice DQW with the standard Dirac representation, only with the unitary matrices wrapped around it. Thus a change of representation from the unitary transformation \ref{repchange} is dealt by doing the same unitary transformation to the walker operator $\hat{W}$, \begin{equation}
\hat{W'}=\tilde{U}\hat{W}\tilde{U}^\dagger, \end{equation} which in the case of the $(3+1)D$ on a cubic lattice this would then be, \begin{eqnarray}
\hat{W}&=&\tilde{U}\begin{pmatrix}
\mathrm{e}^{-\mathrm{i} m\epsilon}\mathbb{I}_2&0\\
0&\mathrm{e}^{\mathrm{i} m\epsilon}\mathbb{I}_2
\end{pmatrix}\nonumber\\&&\times\left(\prod^3_{j=1}\left((U_1\sigma_1\otimes U_j)\hat{S}_j(\sigma_1U_1^\dagger\otimes U_j^\dagger)\right)\right)\tilde{U}^\dagger.\nonumber\\ \end{eqnarray} So as long as this construction method can derive a DQW on a particular lattice in the Dirac representation, it can derive a DQW in any representation.
\section{Conclusion}
We have introduced a new systematic method to construct DQWs on regular lattices of arbitrary dimensions. This method is superior to a trial-and-error method because (i) it shows unambiguously if a DQW can be constructed on a given lattice (ii) it delivers automatically the coefficients of the DQW (iii) it becomes necessary to use a systematic approach if one wants to deal with lattices of dimensions higher than $2$. We have presented the method in a pedagogical manner on two relatively simple cases, a $3D$ DQW on the cubic lattice coupled to an EM field and a $2D$ DQW on the triangular lattice. The free DQW on the cubic $3D$ lattice is well-known \cite{arrighi2014dirac} but its coupling to an arbitrary electromagnetic field is not. The $2D$ DQW on the triangle lattice has been presented in \cite{jay2018dirac} but without directional derivatives. The new presentation given here shows that the coefficients and scalings of this walk are the only ones that work on this lattice. Note that directional derivatives have already been used in writing a DQW on triangular lattice \cite{arrighi2018dirac}, but the spinor of that walk does not live on the vertices of the lattice, but on its edges. We have finally constructed a new DQW on the parallelepiped lattice and produced as a negative example in the equilateral rhombohedral lattice, where the method shows no DQW can be constructed.
Let us conclude by mentioning possible extensions of this work. One should first extend the procedure to include arbitrary Yang-Mills and gravitational fields, thus producing QWs which model the Dirac equation in curved spacetime coupled to arbitrary gauge fields. Also, DQWs with spinors living on the edges of a graph have been introduced recently in \cite{arrighi2018dirac, arrighi2018curved}. The method we present in this article should thus be extended to take this possibility into account, as well as wave-functions with a higher number of components \cite{arrighi2016quantum} than the number of spinor components of irreducible representations of the Lorentz group. One wonders for example if these extensions would make it possible to define DQWs on lattices where the present method fails, as the rombohedral lattice. Finally, studying systematically how to produce DQWs on both regular and non regular graphs is of paramount importance.
\appendix \section{Appendix} \subsection{A pedagogical walkthrough of the method} The following appendix is designed to be an easy to follow explanation of the Hamiltonian process applied to hypercube lattices starting at the most simple case of a $(1+1)D$ Dirac equation in free space, and building up to the more complicated cases as the dimensions increase. Tools we will need to consider are the following expansions: \begin{eqnarray}
\cos(\epsilon A)&=&\mathbb{I}_N+\mathcal{O}(\epsilon^2)\label{cos},\\
\sin(\epsilon A)&=&\epsilon A + \mathcal{O}(\epsilon^2)\label{sin},\\
\mathrm{e}^{\epsilon A} &=&\mathbb{I}_N+\epsilon A +\mathcal{O}(\epsilon^2)\label{exp}, \end{eqnarray} while the shift operators, at least in the lower dimensions of $(1+1)D$ and $(2+1)D$ spacetime, used in the walks can be expressed as an exponential of differential operators like so, \begin{eqnarray} \hat{S}_1\Psi(x,t)&=&\begin{pmatrix}
\psi_L(x+\epsilon,t)\\
\psi_R(x-\epsilon,t)
\end{pmatrix}\nonumber\\
&=&\begin{pmatrix}
\psi_L(x,t)+\epsilon\partial_1\psi_L(x,t)+\mathcal{O}(\epsilon^2)\\
\psi_R(x,t)-\epsilon\partial_1\psi_R(x,t)+\mathcal{O}(\epsilon^2)
\end{pmatrix}\nonumber\\
&=&\begin{pmatrix}
\psi_L(x,t)\\
\psi_R(x,t)
\end{pmatrix}+\epsilon\sigma_3\partial_1\begin{pmatrix}
\psi_L(x,t)\\
\psi_R(x,t)
\end{pmatrix}+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\left(\mathbb{I}_2+\epsilon\sigma_3\partial_1\right)\begin{pmatrix}
\psi_L(x,t)\\
\psi_R(x,t)
\end{pmatrix}+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\mathrm{e}^{\epsilon\sigma_3\partial_1}\Psi(x,t). \end{eqnarray} This works for the other spatial directions so that in general the Shift Operator (acting on a 2-vector wavefunction $\Psi$) is \begin{equation}\label{lowDimShiftDef}
\hat{S}_i=\mathrm{e}^{\epsilon\sigma_3\partial_i}. \end{equation} We now have the tools to start tackling the simplest individual cases. \subsubsection{$(1+1)D$ flat spacetime in free space}
The simplest case is the $(1+1)D$ free flat spacetime version of the Dirac Equation:
\begin{equation}
(\mathrm{i}\gamma^\mu\partial_\mu-m)\Psi=0,
\end{equation}
where we will use the standard $2\times 2$ Gamma matrices of $\gamma^0=\sigma_1$ and $\gamma^1=\mathrm{i}\sigma_2$. First we need to rearrange this into Hamiltonian form to identify the Hamiltonian $\hat{H}$:
\begin{equation}
\hat{H}=\mathrm{i}\sigma_3\partial_1+\sigma_1m.
\end{equation} So therefore the operator $\hat{W}$ for our DTQW will take the form \begin{eqnarray}
\hat{W}&=&\mathbb{I}_2-\mathrm{i}\epsilon\hat{H}+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\mathbb{I}_2+\epsilon\sigma_3\partial_1-\mathrm{i} m\epsilon\sigma_1+\mathcal{O}(\epsilon^2)\nonumber\\
&=&(\mathbb{I}_2-\mathrm{i} m\epsilon\sigma_1)(\mathbb{I}_2+\epsilon\sigma_3\partial_1)+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\begin{pmatrix}
1&-\mathrm{i} m\epsilon\\
-\mathrm{i} m\epsilon&1
\end{pmatrix}e^{\epsilon\sigma_3\partial_1}+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\begin{pmatrix}
\cos(m\epsilon)&-\mathrm{i}\sin(m\epsilon)\\
-\mathrm{i}\sin(m\epsilon)&\cos(m\epsilon)
\end{pmatrix}\hat{S}_1. \end{eqnarray} Here step 4 has made use of definition \ref{lowDimShiftDef}, while step 5 has used the expansions \ref{cos} and \ref{sin}. And so the DTQW that reaches the $(1+1)D$ free flat spacetime Dirac Equation in the continuous limit would be \begin{equation}
\Psi(x,t+\epsilon)=\begin{pmatrix}
\cos(m\epsilon)&-\mathrm{i}\sin(m\epsilon)\\
-\mathrm{i}\sin(m\epsilon)&\cos(m\epsilon)
\end{pmatrix}\begin{pmatrix}
\psi_L(x+\epsilon,t)\\
\psi_R(x-\epsilon,t)
\end{pmatrix}, \end{equation} which agrees with previous works results \cite{chandrashekar2010relationship}. \subsubsection{$(1+1)D$ flat spacetime coupled to an electric field} The Dirac Equation now coupled to an electric field takes the form \begin{equation}
(\mathrm{i}\gamma^\mu D_\mu-m)\Psi=0, \end{equation} where \begin{equation}
D_\mu =\partial_\mu-\mathrm{i} A_\mu. \end{equation} Employing the same technique we get the Hamiltonian by rearranging the equation: \begin{equation}
\hat{H}=-A_0+\sigma_3A_1+\sigma_1m+\mathrm{i}\sigma_3\partial_1. \end{equation} And so the operator $\hat{W}$ takes the form \begin{eqnarray}
\hat{W}&=&\mathbb{I}_2-\mathrm{i}\epsilon\hat{H}+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\mathbb{I}_2+\mathrm{i}\epsilon A_0-\mathrm{i}\epsilon\sigma_3A_1-\mathrm{i}\epsilon\sigma_1m+\epsilon\sigma_3\partial_1+\mathcal{O}(\epsilon^2)\nonumber\\
&=&(\mathbb{I}_2+\mathrm{i}\epsilon A_0)(\mathbb{I}_2-\mathrm{i}\epsilon\sigma_3 A_1)(\mathbb{I}_2-\mathrm{i}\epsilon\sigma_1m)(\mathbb{I}_2+\epsilon\sigma_3\partial_1)\nonumber\\&&+\mathcal{O}(\epsilon^2).\nonumber \end{eqnarray} Now clearly the last two brackets in this expression are identical to the free space case from before. For the two new expressions we make use of expansion \ref{exp} and get: \begin{eqnarray}
\Hat{W}&=&e^{\mathrm{i}\epsilon A_0}\begin{pmatrix}
1-\mathrm{i}\epsilon A_1&0\\
0&1+\mathrm{i}\epsilon A_1
\end{pmatrix}\nonumber\\&&\times\begin{pmatrix}
\cos(m\epsilon)&-\mathrm{i}\sin(m\epsilon)\\
-\mathrm{i}\sin(m\epsilon)&\cos(m\epsilon)
\end{pmatrix}\hat{S}_1+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\mathrm{e}^{\mathrm{i}\epsilon A_0}\begin{pmatrix}
\mathrm{e}^{-\mathrm{i}\epsilon A_1}&0\\
0&\mathrm{e}^{\mathrm{i}\epsilon A_1}
\end{pmatrix}\begin{pmatrix}
\cos(m\epsilon)&-\mathrm{i}\sin(m\epsilon)\\
-\mathrm{i}\sin(m\epsilon)&\cos(m\epsilon)
\end{pmatrix}\hat{S}_1\nonumber\\
&=&\mathrm{e}^{\mathrm{i}\epsilon A_0}\begin{pmatrix}
\mathrm{e}^{-\mathrm{i}\epsilon A_1}\cos(m\epsilon) & \mathrm{e}^{-\mathrm{i}(\epsilon A_1+\frac{\pi}{2})}\sin(m\epsilon)\\
-\mathrm{e}^{\mathrm{i}(\epsilon A_1+\frac{\pi}{2})}\sin(m\epsilon)&\mathrm{e}^{\mathrm{i}\epsilon A_1}\cos(m\epsilon)
\end{pmatrix}\hat{S}_1.\nonumber\\ \end{eqnarray} Which matches previous work stating you need the more general unitary coin with all four angles set to non-zero to reach in the limit a Dirac equation coupled to an electric field \cite{di2014quantum}.
\subsubsection{$(2+1)D$ flat spacetime in free space} Going up one dimension, the Hamiltonian form of the Dirac Equation contains an extra differential term that requires dealing with. As such we will need another gamma matrix. Fortunately in $(2+1)D$ spacetime, 2x2 matrices still suffice and we shall use the standard $\gamma^2=\mathrm{i}\sigma_3$. With this in mind, the Hamiltonian can be determined as \begin{equation}
\hat{H}=\sigma_1 m+\mathrm{i}\sigma_3\partial_1-\mathrm{i}\sigma_2\partial_2.\label{2+1FreeH} \end{equation} The operator $\hat{W}$ will now take the form of \begin{eqnarray}
\hat{W}&=&\mathbb{I}_2-\mathrm{i}\epsilon\hat{H}+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\mathbb{I}_2-\mathrm{i} m\epsilon\sigma_1+\epsilon\sigma_3\partial_1-\epsilon\sigma_2\partial_2+\mathcal{O}(\epsilon^2)\nonumber\\
&=&(\mathbb{I}_2-\mathrm{i} m\epsilon\sigma_1)(\mathbb{I}_2-\epsilon\sigma_2\partial_2)(\mathbb{I}_2+\epsilon\sigma_3\partial_1)+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\begin{pmatrix}
\cos(m\epsilon)&-\mathrm{i}\sin(m\epsilon)\\
-\mathrm{i}\sin(m\epsilon)&\cos(m\epsilon)
\end{pmatrix}(\mathbb{I}_2-\epsilon\sigma_2\partial_2)\hat{S}_1+\mathcal{O}(\epsilon^2).\nonumber \end{eqnarray} This is when we run into trouble with the partial derivative in the $y-$direction. Where we really want a $\sigma_3$ to be sitting so that we can bring in the second shift operator $\hat{S}_2$ we have instead a $-\sigma_2$. However if we can find a unitary matrix $U$ such that \begin{equation}\label{UDef}
U\sigma_3U^\dagger=-\sigma_2, \end{equation} then we can continue as \begin{eqnarray}
\Hat{W}&=&\begin{pmatrix}
\cos(m\epsilon)&-\mathrm{i}\sin(m\epsilon)\\
-\mathrm{i}\sin(m\epsilon)&\cos(m\epsilon)
\end{pmatrix}(\mathbb{I}_2-\epsilon\sigma_2\partial_2)\hat{S}_1+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\begin{pmatrix}
\cos(m\epsilon)&-\mathrm{i}\sin(m\epsilon)\\
-\mathrm{i}\sin(m\epsilon)&\cos(m\epsilon)
\end{pmatrix}(UU^\dagger+\epsilon U\sigma_3U^\dagger\partial_2)\hat{S}_1\nonumber\\&&+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\begin{pmatrix}
\cos(m\epsilon)&-\mathrm{i}\sin(m\epsilon)\\
-\mathrm{i}\sin(m\epsilon)&\cos(m\epsilon)
\end{pmatrix}U(\mathbb{I}_2+\epsilon \sigma_3\partial_2)U^\dagger\hat{S}_1\nonumber\\&&+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\begin{pmatrix}
\cos(m\epsilon)&-\mathrm{i}\sin(m\epsilon)\\
-\mathrm{i}\sin(m\epsilon)&\cos(m\epsilon)
\end{pmatrix}U\hat{S}_2U^\dagger\hat{S}_1. \end{eqnarray} So as long as we can solve equation \ref{UDef} then we have our DQW. We shall see that $U$ is not unique. If we define $U$ as \begin{equation}
U=\begin{pmatrix}
a&b\\
c&d\\
\end{pmatrix}, \end{equation} we then have two requirements. Namely equation \ref{UDef} which leads to \begin{equation}
\begin{pmatrix}
|a|^2-|b|^2&a\bar{c}-b\bar{d}\\
c\bar{a}-d\bar{b}&|c|^2-|d|^2\\
\end{pmatrix}=\begin{pmatrix}
0&i\\
-i&0\\
\end{pmatrix}, \end{equation} and the fact that $U$ is unitary so that \begin{equation}
\begin{pmatrix}
|a|^2+|b|^2&a\bar{c}+b\bar{d}\\
c\bar{a}+d\bar{b}&|c|^2+|d|^2\\
\end{pmatrix}=\begin{pmatrix}
1&0\\
0&1\\
\end{pmatrix}. \end{equation} This ultimately leads to the relationships \begin{eqnarray}
|a|^2=|b|^2=|c|^2=|d|^2&=&\frac{1}{2},\\
c\bar{a}=b\bar{d}&=&-\frac{\mathrm{i}}{2}. \end{eqnarray} There are therefore two degrees of freedom in the choice of $U$, so if we use the first relationship to choose $a$ and $d$ to be $\frac{1}{\sqrt{2}}$ then the second relationship will fix $b$ and $c$ for us and $U$ is \begin{equation} U=\begin{pmatrix} \frac{1}{\sqrt{2}}&-\frac{\mathrm{i}}{\sqrt{2}}\\ -\frac{\mathrm{i}}{\sqrt{2}}&\frac{1}{\sqrt{2}} \end{pmatrix}. \end{equation} \subsubsection{$(2+1)D$ flat spacetime coupled to an electromagnetic field} This time the Hamiltonian will have more terms, however none of the extra terms will present any challenge to us. \begin{equation}
\hat{H}=-A_0+\mathrm{i}\sigma_3\partial_1+\sigma_3A_1-\mathrm{i}\sigma_2\partial_2-\sigma_2A_3+\sigma_1m. \end{equation} The operator $\hat{W}$ simply becomes \begin{eqnarray}
\hat{W}&=&\mathbb{I}_2-\mathrm{i}\epsilon\hat{H}+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\mathbb{I}_2+\mathrm{i}\epsilon A_0+\epsilon\sigma_3\partial_1-\mathrm{i}\epsilon\sigma_3A_1-\epsilon\sigma_2\partial_2+\mathrm{i}\epsilon\sigma_2A_2\nonumber\\&&-\mathrm{i}\epsilon\sigma_1 m+\mathcal{O}(\epsilon^2)\nonumber\\
&=&(\mathbb{I}_2+\mathrm{i}\epsilon A_0)(\mathbb{I}_2-\mathrm{i}\epsilon\sigma_3A_1)(\mathbb{I}_2+\mathrm{i}\epsilon\sigma_2A_2)(\mathbb{I}_2-\mathrm{i}\epsilon\sigma_1 m)\nonumber\\&&\times(\mathbb{I}_2-\epsilon\sigma_2\partial_2)(\mathbb{I}_2+\epsilon\sigma_3\partial_1)+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\mathrm{e}^{\mathrm{i}\epsilon A_0}\begin{pmatrix}
\mathrm{e}^{-\mathrm{i}\epsilon A_1} & 0\\
0 & \mathrm{e}^{\mathrm{i}\epsilon A_1}
\end{pmatrix}\begin{pmatrix}
1 & \epsilon A_2\\
-\epsilon A_2 & 1
\end{pmatrix}\nonumber\\&&\times\begin{pmatrix}
\cos(m\epsilon) & -\mathrm{i}\sin(m\epsilon)\\
-\mathrm{i}\sin(m\epsilon) & \cos(m\epsilon)
\end{pmatrix}U\hat{S}_2U^\dagger\hat{S}_1+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\mathrm{e}^{\mathrm{i}\epsilon A_0}\begin{pmatrix}
\mathrm{e}^{-\mathrm{i}\epsilon A_1} & 0\\
0 & \mathrm{e}^{\mathrm{i}\epsilon A_1}
\end{pmatrix}\begin{pmatrix}
\cos(\epsilon A_2) & \sin(\epsilon A_2)\\
-\sin(\epsilon A_2) & \cos(\epsilon A_2)
\end{pmatrix}\nonumber\\&&\times\begin{pmatrix}
\cos(m\epsilon) & -\mathrm{i}\sin(m\epsilon)\\
-\mathrm{i}\sin(m\epsilon) & \cos(m\epsilon)
\end{pmatrix}U\hat{S}_2U^\dagger\hat{S}_1. \end{eqnarray}
Note by now we can start to see a pattern in the method. Adding extra differential terms to our Dirac Equation simply means adding shift operators usually wrapped in a unitary transformation. Adding non-differential terms to our Dirac equation is much simpler in that all it does is add unitary operators to the front of the walk operator.
\subsubsection{$(3+1)D$ Flat Spacetime in free spacetime}
Increasing to the full $(3+1)D$ spacetime, the Dirac Equation can no longer obey the Clifford Algebra for the Gamma Matrices is we remain in a 2-component spinor space. The Gamma matrices must instead now be 4x4 matrices and the wavefunction $\Psi$, a 4-vector. \begin{equation}
\Psi=\begin{pmatrix}
\psi_1\\
\psi_2\\
\psi_3\\
\psi_4
\end{pmatrix} \end{equation} Here $\{\psi_1,\psi_2\}$ are the spin-up/spin-down components of a particle and $\{\psi_3,\psi_4\}$ represent an antiparticle. Our new shift operators must now tackle four components and look like: \begin{eqnarray}
\hat{S}_1\Psi(x,y,z,t)&=&\begin{pmatrix}
\psi_1(x+\epsilon,y,z,t)\\
\psi_2(x-\epsilon,y,z,t)\\
\psi_3(x-\epsilon,y,z,t)\\
\psi_4(x+\epsilon,y,z,t)
\end{pmatrix}\nonumber\\
&=&\begin{pmatrix}
\psi_1(x,y,z,t)+\epsilon\partial_1\psi_1(x,y,z,t)+\mathcal{O}(\epsilon^2)\\
\psi_2(x,y,z,t)-\epsilon\partial_1\psi_2(x,y,z,t)+\mathcal{O}(\epsilon^2)\\
\psi_3(x,y,z,t)-\epsilon\partial_1\psi_3(x,y,z,t)+\mathcal{O}(\epsilon^2)\\
\psi_4(x,y,z,t)+\epsilon\partial_1\psi_4(x,y,z,t)+\mathcal{O}(\epsilon^2)
\end{pmatrix}\nonumber\\
&=&\begin{pmatrix}
\psi_1(x,y,z,t)\\
\psi_2(x,y,z,t)\\
\psi_3(x,y,z,t)\\
\psi_4(x,y,z,t)
\end{pmatrix}\nonumber\\&&+\epsilon(\sigma_3\otimes\sigma_3)\partial_1\begin{pmatrix}
\psi_1(x,y,z,t)\\
\psi_2(x,y,z,t)\\
\psi_3(x,y,z,t)\\
\psi_4(x,y,z,t)
\end{pmatrix}+\mathcal{O}(\epsilon^2)\nonumber\\
&=&(\mathbb{I}_4+\epsilon(\sigma_3\otimes\sigma_3)\partial_1)\begin{pmatrix}
\psi_1(x,y,z,t)\\
\psi_2(x,y,z,t)\\
\psi_3(x,y,z,t)\\
\psi_4(x,y,z,t)
\end{pmatrix}\nonumber\\
&=&\mathrm{e}^{\epsilon(\sigma_3\otimes\sigma_3)\partial_1}\Psi(x,y,z,t). \end{eqnarray} This works for the other spatial directions so that in general the Shift Operator (acting on a 4-vector wavefunction $\Psi$) is \begin{equation}
\hat{S}_i=\mathrm{e}^{\epsilon(\sigma_3\otimes\sigma_3)\partial_i}. \end{equation}
For our 4x4 Gamma matrices we shall use the standard representation choice of \begin{eqnarray}
\gamma^0&=&\begin{pmatrix}
1&0&0&0\\
0&1&0&0\\
0&0&-1&0\\
0&0&0&-1
\end{pmatrix}
=\begin{pmatrix}
\mathbb{I}_2&0\\
0&-\mathbb{I}_2
\end{pmatrix}=\sigma_3\otimes\mathbb{I}_2,\\
\gamma^1&=&\begin{pmatrix}
0&0&0&1\\
0&0&1&0\\
0&-1&0&0\\
-1&0&0&0
\end{pmatrix}=\begin{pmatrix}
0&\sigma_1\\
-\sigma_1&0
\end{pmatrix}=\mathrm{i}\sigma_2\otimes\sigma_1,\\
\gamma^2&=&\begin{pmatrix}
0&0&0&-\mathrm{i}\\
0&0&\mathrm{i}&0\\
0&\mathrm{i}&0&0\\
-\mathrm{i}&0&0&0
\end{pmatrix}=\begin{pmatrix}
0&\sigma_2\\
-\sigma_2&0
\end{pmatrix}=\mathrm{i}\sigma_2\otimes\sigma_2,\\
\gamma^3&=&\begin{pmatrix}
0&0&1&0\\
0&0&0&-1\\
-1&0&0&0\\
0&1&0&0
\end{pmatrix}=\begin{pmatrix}
0&\sigma_3\\
-\sigma_3&0
\end{pmatrix}=\mathrm{i}\sigma_2\otimes\sigma_3. \end{eqnarray} In free space the Hamiltonian becomes \begin{eqnarray} \hat{H}=(\sigma_3\otimes\mathbb{I}_2)m-\sum_{j=1}^3\mathrm{i}(\sigma_1\otimes\sigma_j)\partial_j. \end{eqnarray} Which leads to a $\hat{W}$ operator of the form: \begin{eqnarray}
\hat{W}&=&\mathbb{I}_4-\mathrm{i}\epsilon\hat{H}+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\mathbb{I}_4-\mathrm{i} m\epsilon(\sigma_3\otimes\mathbb{I}_2)-\epsilon(\sigma_1\otimes\sigma_j)\partial_j+\mathcal{O}(\epsilon^2)\nonumber\\
&=&(\mathbb{I}_4-\mathrm{i} m\epsilon(\sigma_3\otimes\mathbb{I}_2))\prod^3_{j=1}\left(\mathbb{I}_4-\epsilon(\sigma_1\otimes\sigma_j)\partial_j\right)+\mathcal{O}(\epsilon^2).\nonumber \end{eqnarray} If we follow the pattern from before we realize the mass expression is not going to be difficult to deal with, however the partial derivatives we will need to find a way to use a unitary transformation to change those Pauli matrices to $\sigma_3$. Taking a similar trick to before, we now look for three different unitary matrices that satisfy \begin{equation} U_j\sigma_3U_j^\dagger=\sigma_j. \end{equation} If we make use of the mixed-product property of Kronecker products \begin{equation}
(A\otimes B)(C\otimes D)=(AC)\otimes(BD), \end{equation} then we can derive that \begin{eqnarray}
\hat{W}&=&(\mathbb{I}_4-\mathrm{i} m\epsilon(\sigma_3\otimes\mathbb{I}_2))\nonumber\\&&\times\prod^3_{j=1}\left(\mathbb{I}_2\otimes\mathbb{I}_2-\epsilon(\sigma_1\otimes\sigma_j)\partial_j\right)+\mathcal{O}(\epsilon^2)\nonumber\\
&=&(\mathbb{I}_4-\mathrm{i} m\epsilon(\sigma_3\otimes\mathbb{I}_2))\nonumber\\&&\times\prod^3_{j=1}\left(U_1U_1^\dagger\otimes U_jU_j^\dagger-\epsilon(U_1\sigma_3U_1^\dagger\otimes U_j\sigma_3U_j^\dagger)\partial_j\right)\nonumber\\&&+\mathcal{O}(\epsilon^2)\nonumber\\
&=&(\mathbb{I}_4-\mathrm{i} m\epsilon(\sigma_3\otimes\mathbb{I}_2))\nonumber\\&&\times\prod^3_{j=1}\left((U_1\otimes U_j)(\mathbb{I}_2\otimes\mathbb{I}_2-\epsilon(\sigma_3\otimes \sigma_3)\partial_j)(U_1^\dagger\otimes U_j^\dagger)\right)\nonumber\\&&+\mathcal{O}(\epsilon^2)\nonumber\\
&=&(\mathbb{I}_4-\mathrm{i} m\epsilon(\sigma_3\otimes\mathbb{I}_2))\nonumber\\&&\times\prod^3_{j=1}\bigg((U_1\otimes U_j)(\sigma_1\sigma_1\otimes\mathbb{I}_2+\epsilon(\sigma_1\sigma_3\sigma_1\otimes \sigma_3)\partial_j)\nonumber\\&&\times(U_1^\dagger\otimes U_j^\dagger)\bigg)+\mathcal{O}(\epsilon^2)\nonumber\\
&=&(\mathbb{I}_4-\mathrm{i} m\epsilon(\sigma_3\otimes\mathbb{I}_2))\nonumber\\&&\times\prod^3_{j=1}\bigg((U_1\sigma_1\otimes U_j)(\mathbb{I}_4+\epsilon(\sigma_3\otimes \sigma_3)\partial_j)(\sigma_1 U_1^\dagger\otimes U_j^\dagger)\bigg)\nonumber\\&&+\mathcal{O}(\epsilon^2)\nonumber\\
&=&\begin{pmatrix}
\mathrm{e}^{-\mathrm{i} m\epsilon}\mathbb{I}_2&0\\
0&\mathrm{e}^{\mathrm{i} m\epsilon}\mathbb{I}_2
\end{pmatrix}\prod^3_{j=1}\left((U_1\sigma_1\otimes U_j)\hat{S}_j(\sigma_1 U_1^\dagger\otimes U_j^\dagger)\right).\nonumber\\ \end{eqnarray} Like before in the $(2+1)D$ case these are not unique. Similar arguments can determine a couple of degrees of freedom and the choices we shall make here are the unitary matrices \begin{eqnarray} U_1&=&\begin{pmatrix} \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}} \end{pmatrix},\nonumber\\ U_2&=&\begin{pmatrix} \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\ \frac{i}{\sqrt{2}}&-\frac{i}{\sqrt{2}} \end{pmatrix},\nonumber\\ U_3&=&\mathbb{I}_2. \end{eqnarray} \subsubsection{$(3+1)D$ flat spacetime coupled to an electromagnetic field} The Dirac equation coupled to an Electromagnetic field in $(3+1)D$ spacetime is shown in the main text, however as we can expect by now the only addition it will have over the free space version is an extra unitary operator for each extra term there is in the Dirac equation. In other words, three more operators, one for each $A_\mu$ term that appears in the equation \begin{equation}
(\mathrm{i}\gamma^\mu D_\mu-m)\Psi=0,\quad D_\mu=\partial_\mu-iA_\mu. \end{equation}
\end{document} | arXiv |
\begin{document}
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\title{\Large\textsc{Strange nonchaotic attractors in quasiperiodically forced
circle maps: Diophantine forcing}
\abstract{We study parameter families of quasiperiodically forced (qpf) circle
maps with Diophantine frequency. Under certain ${\cal C}^1$-open
conditions concerning their geometry, we prove that these families exhibit
nonuniformly hyperbolic behaviour, often referred to as the existence of
strange nonchaotic attractors, on parameter sets of positive measure. This
provides a nonlinear version of results by Young on quasiperiodic
\ensuremath{\textrm{SL}(2,\mathbb{R})}-cocycles and complements previous results in this direction which hold
for sets of frequencies of positive measure, but did not allow for an explicit
characterisation of these frequencies. As an application, we study a
qpf version of the Arnold circle map and show that the
Arnold tongue corresponding to rotation number $1/2$ collapses on an open set
of parameters.
The proof requires to perform a parameter exclusion with respect to some twist
parameter and is based on the multiscale analysis of the dynamics on certain
dynamically defined critical sets. A crucial ingredient is to obtain good
control on the parameter-dependence of the critical sets. Apart from the
presented results, we believe that this step will be important for obtaining
further information on the behaviour of parameter families like the qpf Arnold
circle map. }
\section{Introduction}
After the discovery of strange chaotic attractors in two-dimensional dynamical systems like the H\'enon map \cite{benedicks/carleson:1991}, a natural question that occurred was to determine the simplest type of smooth systems that exhibit {\em `strange'} attractors. In particular, it was not clear whether chaos was a necessary prerequisite for the existence of such objects. Understanding {\em `strange'} in a broad sense as {\em `having a complicated structure and geometry'} (compare \cite{milnor:1985}), Grebogi {\em et al} gave a negative answer to this by showing that strange non-chaotic attractors (SNA) can appear in quasiperiodically forced (qpf) monotone interval maps \cite{grebogi/ott/pelikan/yorke:1984}. Their argument was heuristic, but later made rigorous by Keller \cite{keller:1996}. These findings prompted further investigations on qpf 1D maps, which have, despite their simple structure, surprisingly rich dynamics and appear as natural models for physical systems subject to the influence of two or more external periodic factors with incommensurate frequencies \cite{romeiras/etal:1987,ding/grebogi/ott:1989,feudel/kurths/pikovsky:1995}.
For quite a while, studies on the topic were mainly numerical and rigorous results remained rare. The only exception, apart from the very particular type of examples in \cite{grebogi/ott/pelikan/yorke:1984,keller:1996}, is the rich theory of quasiperiodic (qp) \ensuremath{\textrm{SL}(2,\mathbb{R})}-cocycles and their associated linear-projective actions. For these systems, the existence of SNA had already been proved prior to the work of Grebogi {\em et al} by Million\u{s}\u{c}ikov \cite{millionscikov:1969}, Vinograd \cite{vinograd:1975} and, in a more general way, Herman \cite{herman:1983}. In this context, the phenomenon is referred to as the non-uniform hyperbolicity of the cocycle. Due to close relations to the spectral properties of 1D Schr\"odinger operators with quasiperiodic potential (see, for example, \cite{avila/krikorian:2004,haro/puig:2006}), there have been intense efforts to understand the dynamics of qp \ensuremath{\textrm{SL}(2,\mathbb{R})}-cocycles during the last three decades (see \cite{puig:2004,avila/jitomirskaya:2005,avila/jitomirskaya:2010,avila:2010} for some recent advances). Unfortunately, most methods from this theory cannot simply be carried over to more general `non-linear' qpf systems, since they strongly depend on the linear structure and, in many cases, on the close relations to spectral theory. At the same time, it is also difficult to compare SNA with the strange attractors appearing in H\'enon-like maps, since on a formal level these are quite different objects. Nevertheless, the methods used by Benedicks and Carleson's in their seminal work on the H\'enon map \cite{benedicks/carleson:1991} turned out to be equally fruitful for the description of SNA. Furthermore, the required inductive schemes are easier to implement in this context, such that one can reasonably hope to elaborate these techniques further in order to obtain additional insights about the behaviour and dynamics of parameter families of qpf circle maps. We will come back to this point at the end of the introduction.
In the context of qpf systems, multiscale analysis and parameter exclusion in the spirit of Benedicks and Carleson were introduced by Young in \cite{young:1997}, where she described non-uniformly hyperbolic dynamics in certain parameter families of qp \ensuremath{\textrm{SL}(2,\mathbb{R})}-cocycles. The methods were then applied to qp Schr\"odinger cocycles by Bjerkl\"ov \cite{bjerkloev:2005a}, who also extended them to show the minimality of the dynamics. These results were so far restricted to linear-projective systems, but since the original setting in \cite{benedicks/carleson:1991} is nonlinear it is not too surprising that it was eventually possible to adapt the techniques to qpf nonlinear models \cite{jaeger:2009a}. This allowed to prove the existence of SNA under rather general conditions. In \cite{bjerkloev:2005a,jaeger:2009a}, the parameter exclusion was performed with respect to the forcing frequency. As a result, one obtains a set of frequencies of positive measure such that the considered system forced with these frequencies exhibits nonuniformly hyperbolic dynamics. The drawback is that this does not yield any statement about a fixed frequency like the golden mean, which is used in most of the numerical studies on the topic. Our aim here is to close this gap. This is achieved by performing a parameter exclusion with respect to some other suitable system parameter. We thus obtain a nonlinear version of the respective results in \cite{young:1997}, augmented by the minimality of the dynamics. Using a particular symmetry, we further show that the Arnold tongue corresponding to rotation number $1/2$ collapses on an open set of parameters. While the collapse of tongues has already been described in \cite{jaeger:2009a}, the robustness of this phenomenon seems to be new.
In order to state a qualitative version of our main result, we let ${\cal F} := \{f \in \textrm{Diff}^1(\ensuremath{\mathbb{T}}^2) \mid \pi_1 \circ f =\pi_1\}$, where $\textrm{Diff}^1(\ensuremath{\mathbb{T}}^2)$ denotes the group of diffeomorphisms of the two-torus $\ensuremath{\mathbb{T}}^2$ and $\pi_1$ is the projection to the first coordinate. Note that for $F\in{\cal F}$ we have $F\ensuremath{(\theta,x)} = (\theta,f_\theta(x))$ where $f_\theta(\cdot) = \pi_2\circ F(\theta,\cdot)$, such that we can view $F$ as a collection of {\em
fibre maps} $(f_\theta)_{\theta\in\ensuremath{\mathbb{T}^{1}}}$. Further, we let \[ {\cal P} \ = \ \left\{ (F_\tau)_{\tau\in[0,1]} \mid F_\tau\in{\cal F}\
\forall \tau\in[0,1] \ \textrm{and}\ (\tau,\theta,x)\mapsto F_\tau(\theta,x)
\textrm{ is } {\cal C}^1\right\} \ \] be the set of differentiable parameter families in ${\cal F}$. The fibre maps of $F_\tau$ are denoted by $f_{\tau,\theta}$, that is, $F_\tau(\theta,x) = (\theta,f_{\tau,\theta}(x))$. Finally, we let ${\cal D}(\sigma,\nu)$ be the set of frequencies $\omega\in\ensuremath{\mathbb{T}^{1}}$ that satisfy the Diophantine condition
$d(n\omega,0) > \sigma \cdot |n|^{-\nu} \ \forall n\in\ensuremath{\mathbb{Z}}\ensuremath{\setminus}\{0\}$. \begin{thm} \label{t.mr-qualitative} Given any constants $\sigma,\nu >
0$, there exists a non-empty set ${\cal U} = {\cal U}(\sigma,\nu)
\ensuremath{\subseteq} {\cal P}$, open with respect to the induced ${\cal
C}^1$-topology, with the following property:
For all $(F_\tau)_{\tau\in[0,1]} \in {\cal U}$ and all $\omega\in
{\cal D}(\sigma,\nu)$ there exists a set $\Lambda_\infty(\omega)\ensuremath{\subseteq} [0,1]$
of positive measure such that for all $\tau\in\Lambda_\infty(\omega)$ the
qpf circle diffeomorphism
\[
f_\tau \ : \ \ensuremath{(\theta,x)} \mapsto (\ensuremath{\theta + \omega},\ensuremath{f_{\tau,\theta}(x)})
\]
has a unique strange non-chaotic attractor (see Definition~\ref{d.sna}) which
supports the unique physical measure of the system. Furthermore, the dynamics
are minimal. \end{thm} As in \cite{jaeger:2009a}, we will provide two different quantitative versions of Theorem~\ref{t.mr-qualitative} which characterise the set ${\cal U}$ in terms of explicit ${\cal C}^1$-estimates. Since these conditions are somewhat technical, we postpone the precise statements to Section~\ref{MainResults} and concentrate on two explicit examples.
The first quantitative result, Theorem~\ref{t.firstversion} below, applies to the family \begin{equation} \label{e.projective-action} f_{a,\tau}(\theta,x) \ = \
\left(\theta+\omega,\frac{1}{\pi}\arctan\left(a^2\tan(\pi
x)\right)+g(\theta)+\tau\right) \ \end{equation} where $g:\ensuremath{\mathbb{T}^{1}}\to\ensuremath{\mathbb{T}^{1}}$ is a differentiable function that satisfies some non-degeneracy condition stated below. For example, one could take $g(\theta)=\sin(2\pi\theta)$. If we denote by $R_\phi$ the rotation matrix with angle $2\pi\phi$, then $f_{a,\tau}$ is the is the projective action of the qp \ensuremath{\textrm{SL}(2,\mathbb{R})}-cocycle given by \begin{equation}
\label{e.cocycle-example}
A(\theta) \ = \ R_{g(\theta)+\tau} \cdot \twomatrix{a}{0}{0}{1/a} \ . \end{equation} For this particular system, we obtain the following statement. \begin{cor}[to Theorem~\ref{t.firstversion} below]
\label{c.cocycle-example} Suppose $g:\ensuremath{\mathbb{T}^{1}}\to\ensuremath{\mathbb{T}^{1}}$ is a differentiable
function and there exists a finite set $\Omega_0$ such that for all
$\tau\in\ensuremath{\mathbb{T}^{1}}\ensuremath{\setminus} \Omega_0$ the set $Z(\tau)=\{\theta\in\ensuremath{\mathbb{T}^{1}}\mid
g(\theta)+\tau=\halb \}$ is finite and $g'$ takes distinct and non-zero values
at different points of $Z(\tau)$.
Then for all $\sigma,\nu>0$ there exists
$a_*=a_*(g,\sigma,\nu)>0$ with the following property: for
all $\omega\in {\cal D}(\sigma,\nu)$ and all $a\geq a_*$
there exists a set $\Lambda_\infty(a,\omega) \ensuremath{\subseteq} \ensuremath{\mathbb{T}^{1}}$ of
positive measure such that for all
$\tau\in\Lambda_\infty(a,\omega)$ the map $f_{a,\tau}$
given by (\ref{e.projective-action}) has a unique SNA and minimal
dynamics. Further $\ensuremath{\mathrm{Leb}}_{\ensuremath{\mathbb{T}^{1}}}(\Lambda_\infty(a,\omega))$
goes to $1$ as $a \to \infty$.
The same result applies to any sufficiently small ${\cal
C}^1$-perturbation of the parameter family
(\ref{e.cocycle-example}). \end{cor} This statement follows from Theorem~\ref{t.firstversion} by some standard estimates. Since our main focus lies on the qpf Arnold circle map, we refer the reader to \cite[Section 3.8]{jaeger:2009a} for details. We also note that the existence of an SNA for (\ref{e.projective-action}) is equivalent to the non-uniform hyperbolicity of the cocycle (\ref{e.cocycle-example}) \cite{haro/puig:2006,jaeger:2006a}. Hence, the result can be viewed as a perturbation-persistent version of \cite[Theorem 2]{young:1997}.
The second quantitative version of Theorem~\ref{t.mr-qualitative}, stated as Theorem~\ref{t.mr-quantitative2} below, is tailor-made for the application to the qpf Arnold circle map \begin{equation}
\label{e.arnold}
f_{a,b,\tau}(\theta,x) \ = \ \left(\theta+\omega,x+\tau+\frac{a}{2\pi}
\sin(2\pi x) + g_b(\theta)\right) \end{equation}
with forcing function $g_b$ depending on some additional parameter $b$. The geometry of (\ref{e.arnold}) is quite different to that of the previous example, since unlike in (\ref{e.projective-action}) the hyperbolicity on the single fibres is limited (the slope of the fibre maps $f_{a,b,\tau,\theta}$ remains bounded by $2$ in the invertible regime $|a|\leq 1$). In order to make up for this, the forcing function $g_b$ must have a particular shape that can be pushed to some extreme by adjusting the parameter $b$. General conditions for the family $g_b$ can be deduced from Theorem~\ref{t.mr-quantitative2}. (See also Remark~\ref{r.forcing-structure}.) Here, we concentrate again on an explicit example. \begin{cor}[to Theorem~\ref{t.mr-quantitative2} below]
\label{c.arnold}
Let
\begin{equation}
\label{e.ff1}
g_b(\theta) \ = \ \arctan(b\sin(2\pi\theta))/\pi
\qquad (b\in\ensuremath{\mathbb{R}}) \ . \end{equation} Then for all $\sigma,\nu>0$ and all $a>0$ there exists $b_*=b_*(\sigma,\nu,a)>0$ with the following property:
For all $\omega\in {\cal D}(\sigma,\nu)$ and all $b\geq b_*$
there exists a set $\Lambda_\infty(a,b,\omega) \ensuremath{\subseteq} \ensuremath{\mathbb{T}^{1}}$ of
positive measure such that for all
$\tau\in\Lambda_\infty(a,b,\omega)$ the map
$f_{a,b,\tau}$ given by (\ref{e.arnold}) has a unique SNA
and minimal dynamics. \end{cor} Apart from the restrictions on the forcing function coming from the lack of hyperbolicity, a further reason for the particular choice of $g_b$ in (\ref{e.ff1}) is a special symmetry which appears at $\tau=\halb$. On the one hand, the lift $F$ of the map $f_{a,b,\halb}$ satisfies the relation \begin{equation}
\label{e.forcing-symmetry}
F_{\theta}(-x) \ = \ 1-F_{\theta+1/2}(x) \ , \end{equation} and it can be easily seen that this forces the rotation number\footnote{See
Section~\ref{RotNum} for the definition of the fibred rotation number of a qpf
circle homeomorphism.} $\rho(f_{a,b,\halb})$ to be exactly $\halb$. On the other hand, the map $g_b+\halb$ takes values close to \halb\ only on two intervals $I_0$ and $I_0+\halb$ around $0$ and \halb, respectively. These two intervals play a special role in the multiscale analysis, since they define the critical sets on the first level of the inductive scheme. Furthermore, as a consequence of (\ref{e.forcing-symmetry}) the fact that the $n$-th critical region consists of exactly two intervals $I_n$ and $I_n+\halb$ will remain true on all levels of the induction. This allows to control the return times of the critical regions directly by using only the Diophantine condition, and no parameters have to be excluded in order to avoid fast returns. In other words, in this particular situation the multiscale analysis can be performed without any parameter exclusion. As a consequence, we obtain the following. \begin{cor}[to Theorem~\ref{t.mr-quantitative2} below]
\label{t.arnold-half}
Suppose $g_b$ is chosen as in (\ref{e.ff1}). Then for all $\sigma,\nu >
0$ and $a>0$ there exists $b_*=b_*(\sigma,\nu,a)$ such that
for all $\omega\in{\cal D}(\sigma,\nu)$ and $b>b_*$ the map
$f_{a,b,\halb}$ has a unique SNA and minimal dynamics.
$b_*(\sigma,\nu,\ensuremath{\textbf{.}})$ can be chosen constant on compact subsets of
$(0,1)$. \end{cor} This result has further consequences for the structure of the {\em Arnold tongues} \begin{equation} \label{e.tongues}
A_\rho \ = \ \left\{(a,b,\tau) \in [0,1]\times\ensuremath{\mathbb{R}}\times[0,1] \mid \rho(f_{a,b,\tau})=\rho\right\} \ \end{equation} and the associated {\em mode-locking plateaus} \begin{equation}
\label{e.ml-plateaus}
P_{a,b,\rho} \ = \ \{\tau \in [0,1] \mid \rho(f_{a,b,\tau})=\rho\} \ , \end{equation} where $\rho(f_{a,b,\tau})$ denotes the fibred rotation number of $f_{a,b,\tau}$. We say a mode-locking plateau $P_{a,b,\rho}$ is {\em collapsed} if it consists of a single point. It is known that $P_{a,b,\rho}$ is collapsed for all $\rho\notin\ensuremath{\mathbb{Q}}+\ensuremath{\mathbb{Q}}\omega$ \cite{bjerkloev/jaeger:2009}, and we implicitly assume that $\rho$ belongs to the module $\ensuremath{\mathbb{Q}}+\ensuremath{\mathbb{Q}}\omega$ whenever we speak of collapsed or non-collapsed plateaus. A tongue $A_\rho$ is said to be {\em collapsed} at $(a,b)$ if $P_{a,b,\rho}$ is collapsed. Minimal dynamics imply the collapse of a tongue, in the sense that whenever $f_{a,b,\tau}$ is minimal the tongue $A_\rho$ with $\rho=\rho(f_{a,b,\tau})$ is collapsed at $(a,b)$ (see Proposition~\ref{p.mode-locking}). Hence, the tongue corresponding to rotation number \halb\ is collapsed for all the parameters satisfying the assertions of Corollary~\ref{t.arnold-half}. \begin{cor} \label{c.tongue-collapse} Suppose $g_b$ is chosen as in
(\ref{e.ff1}). Then for all $\sigma,\nu > 0$ there exists an open set
$B(\sigma,\nu) \ensuremath{\subseteq} (0,1)\times\ensuremath{\mathbb{R}}^+$ such that for $\omega\in D(\sigma,\nu)$
and forcing function $g_b$ as in (\ref{e.ff1}) the Arnold tongue $A_\halb$ is
collapsed at all $(a,b)\in B(\sigma,\nu)$. \end{cor} In \cite{jaeger:2009a}, it was shown in a similar way that $A_0$ collapses on sets of parameters $(a,b)$ of positive measure, and the methods employed there would yield the same result for $A_{\halb}$. Hence, the new point here is the robustness of this phenomenon, that is, the openness of the set $B$ in Corollary~\ref{c.tongue-collapse}.
As mentioned above, there are many further open problems concerning the behaviour of parameter families like (\ref{e.projective-action}) or (\ref{e.arnold}). Probably the most prominent one is the question whether the rotation number as a function of the twist parameter is a `devils staircase', meaning that the union of non-collapsed mode-locking plateaus is dense in the parameter interval. This is true for the unforced Arnold circle map. For qpf systems, existing results are again restricted to qp \ensuremath{\textrm{SL}(2,\mathbb{R})}-cocycles. A particular case is the projective action of the Schr\"odinger cocycle associated to the so-called almost-Mathieu operator, for which the question became known as the `Ten Martini Problem'. Recently it has been answered positively in full generality, meaning for all parameters and all irrational forcing frequencies, by Avila and Jitomirskaya \cite{avila/jitomirskaya:2005} (after previous contributions by B\'ellisard and Simon \cite{bellissard/simon:1982} and Puig \cite{puig:2004}). For the qpf Arnold circle map, still no rigorous results exist. Moreover the numerical findings are ambiguous. On the one hand, a devils staircase has been reported for some parameters regions in \cite{ding/grebogi/ott:1989}. On the other hand the authors of \cite{stark/feudel/glendinning/pikovsky:2002} numerically detect parameters for which the 0-tongue is collapsed (a fact which is backed up by rigorous results in \cite{jaeger:2009a} and, replacing $0$ by $\halb$, also by Corollary~\ref{c.tongue-collapse}) and report that for these parameters the mode-locking plateaus vanish and the rotation number strictly increases over a whole interval. In contrast to this, we believe that a further elaboration of the presented techniques should allow to prove the following. \begin{conj} \label{conjecture}
The set $\Lambda_\infty$ in Corollary~\ref{c.arnold} can be chosen such that
it is contained in the closure of the union of non-collapsed mode-locking
plateaus. The same is true for the parameter $\tau=\halb$ in the situation of
Corollary~\ref{c.arnold}. \end{conj} In fact, what should be true is that all parameters $\tau$ for which the `slow-recurrence conditions' \ref{e.X'n} and \ref{e.Y'n} introduced in the multiscale analysis scheme below are satisfied can be approximated by non-collapsed mode-locking plateaus. While this would not formally disprove the conjecture made in \cite{stark/feudel/glendinning/pikovsky:2002} (since our methods do not apply to the forcing function considered there), it would provide strong evidence for the fact that the observation is a numerical artifact. Furthermore, it could be a first step towards proving the existence of a devils staircase. Apart from the intrinsic interest of the above results, the hope to make further progress in this direction is one of the main motivations for the presented work.
Concerning the proofs, we will be able to rely to a great extent on the previous construction in \cite{jaeger:2009a}. In particular, the core part of the proof, which is the multiscale analysis for the dynamics of a fixed map under some non-recurrence conditions on certain dynamically defined critical sets, remains valid and can be used for our purpose without any modifications. We will therefore be able to concentrate almost exclusively on those aspects of the proof which differ from the previous one. The only drawback of this is that the present paper is not self-contained, but depends on a number of statements and estimates in \cite{jaeger:2009a}. However, as redoing all arguments would only result in an undue length of the paper and render the decisive differences in comparison to the previous construction less visible, this seems to be the appropriate way to proceed. In order not to leave the reader without any guidance, we will briefly motivate the used statements on a heuristic level.
\noindent {\bf Acknowledgements.} I would like to thank Hakan Eliasson for inspiring discussions on the subject. This work was carried out in the Emmy-Noether-Group 'Low-dimensional and non-autonomous Dynamics', which is supported by the Grant Ja 1721/2-1 of the German Research Council (DFG).
\section{Notation and Preliminaries} \label{NotationPreliminaries}
\subsection{Notation.}\label{Notation} Given $a,b\in\ensuremath{\mathbb{T}^{1}}$, we denote by $[a,b]$ the positively oriented arc from $a$ to $b$. The same notation is used for open and half-open intervals. We write $b-a$ for the length of $[a,b]$, whereas the Euclidean distance between $a$ and $b$ will be denoted by $d(a,b)$. The derivative with respect to a variable $\xi$ will be denoted by $\partial_\xi$. On any product space, $\pi_i$ will denote the projection to the $i$-th coordinate. Quotient maps like the canonical projections $\ensuremath{\mathbb{R}}\to\ensuremath{\mathbb{T}}^1=\ensuremath{\mathbb{R}}/\ensuremath{\mathbb{Z}}$, $\ensuremath{\mathbb{R}}^2\to\ensuremath{\mathbb{T}}^2=\ensuremath{\mathbb{R}}^2/\ensuremath{\mathbb{Z}}^2$ or $\ensuremath{\mathbb{T}}^1\times\ensuremath{\mathbb{R}}\to\ensuremath{\mathbb{T}}^2$ will all be denoted by $\pi$.
If $I(\tau) = (a(\tau),b(\tau))$ is an interval that depends on some parameter $\tau\in\ensuremath{\mathbb{R}}$, then we say $I$ is differentiable in $\tau$ if this is true for both endpoints $a$ and $b$. In this case we write \[
|\partial_\tau I(\tau)| \ = \ \max\{|\partial_\tau a(\tau)|,|\partial_\tau b(\tau)|\} \ . \] If $I^\iota(\tau)=(a^\iota(\tau),b^\iota(\tau))$ and $I^\kappa(\tau)=(a^\kappa(\tau),b^\kappa(\tau))$ are two disjoint intervals depending both on $\tau$, then we write \[ D_\tau(I^\iota(\tau),I^\kappa(\tau)) \ > \ \eta \] if there holds $\partial_\tau(y(\tau)-x(\tau)) > \eta$ for all possible choices $x(\tau) = a^\iota(\tau),b^\iota(\tau)$ and $y(\tau) = a^\kappa(\tau),b^\kappa(\tau)$. We write \[
|D_\tau(I^\iota(\tau),I^\kappa(\tau))| \ > \ \eta \] if either $D_\tau(I^\iota(\tau),I^\kappa(\tau)) > \eta$ or $D_\tau(I^\kappa(\tau),I^\iota(\tau))>\eta$. In other words,
$|D_\tau(I^\iota(\tau),I^\kappa(\tau))| > \eta$ means that the two intervals move with speed $>\eta$ relative to each other.
\subsection{SNA in qpf systems.} \label{SNA_Preliminaries}
We say a continuous map $f:\ensuremath{\mathbb{T}^2}\to\ensuremath{\mathbb{T}^2}$ is a {\em qpf circle homeomorphism} if it has skew product structure of the form \begin{equation}
\label{e.qpf-circlediff}
f(\theta,x) \ = \ (\theta+\omega,f_\theta(x)) \end{equation} with irrational $\omega\in\ensuremath{\mathbb{T}}^1$. The maps $f_\theta(x)=\pi_2\circ f(\theta,x)$ are called {\em fibre maps} and we write $f^n_\theta(x)=\pi_2\circ f^n(\theta,x)$ for the fibre maps of the iterates. An invariant graph of $f$ is a measurable function $\varphi:\ensuremath{\mathbb{T}^{1}}\to\ensuremath{\mathbb{T}}^1$ that satisfies \begin{equation}
\label{e.inv-graph}
f_\theta(\varphi(\theta))
\ = \ \varphi(\theta+\omega) \ . \end{equation} The corresponding point set $\Phi=\{(\theta,\varphi(\theta))\mid \theta\in\ensuremath{\mathbb{T}^{1}}\}$ will equally be called an invariant graph. We note that in general multi-valued invariant graphs have to be taken into account as well. However, since in the situation we consider only single-valued invariant graphs occur, we restrict to this simple case. (The general definitions can be found in \cite{jaeger:2009a}.)
To any invariant graph, an $f$-invariant ergodic measure $\mu_\varphi$ can be assigned by \begin{equation}
\label{e.graphmeasure}
\mu_\varphi(A) \ = \ \ensuremath{\mathrm{Leb}}_{\ensuremath{\mathbb{T}^{1}}}\left(\pi_1(A\cap \Phi)\right) \ . \end{equation} If all fibre maps are ${\cal C}^1$ and the derivative $f_\theta'(x)$ is strictly positive and depends continuously on $\ensuremath{(\theta,x)}$, we speak of a qpf circle diffeomorphism. In this case, the {\em (vertical) Lyapunov exponent} of an invariant graph is defined as \begin{equation}
\label{e.LE}
\lambda(\varphi) \ = \ \int_{\ensuremath{\mathbb{T}^{1}}} \log |
\partial_x f_\theta(\varphi(\theta))| \ d\theta \ , \end{equation} In the particular context of qpf systems, SNA are now defined as follows. \begin{definition}
\label{d.sna} A non-continuous invariant graph with negative
Lyapunov exponent is called a {\em strange nonchaotic attractor
(SNA)}. A non-continuous invariant graph with positive Lyapunov
exponent is called a {\em strange nonchaotic repeller (SNR)}. \end{definition} \begin{rem}
It should be said at this point that it it difficult to match this very
specific definition of SNA with a general concept of strange attractors, as
discussed for example in \cite{milnor:1985}. For instance, an attractor is usually
understood to be a compact invariant set, but the point set associated to an
SNA in the above sense is non-compact due to the non-continuity of the
invariant graph. One could consider the closure of this set instead, but in
the situations we describe this will be the whole two-torus, which cannot
reasonably be called a `strange' object. However, although the terminology
might therefore be considered somewhat unfortunate, it has already been used
for almost three decades in most of the vast physics literature on the
topic. We therefore prefer to keep with it, simply taking it as a technical
term specific to the theory qpf systems.
We also note that due to the negative Lyapunov exponent an SNA attracts a
positive measure set of initial conditions and therefore carries a physical
measure. \end{rem} A convenient criterion for the existence of SNA involves pointwise Lyapunov exponents, forwards and backwards in time. These are given by \begin{equation} \label{e.pointwiseLE}
\lambda^\pm(\theta,x) \ = \ \limsup_{n \to \infty} \ntel |\log \partial_x f^{\pm n}_\theta(x)|\ . \end{equation} The orbit of a point $(\theta,x) \in \ntorus$ with $\lambda^\pm(\theta,x)>0$ is called a \emph{sink-source-orbit}. The existence of such orbits implies the existence of SNA. \begin{prop}[\cite{jaeger:2009a}] \label{prop:sinksourcesna} Suppose
$f$ is a quasiperiodically forced circle diffeomorphism which has a
sink-source-orbit. Then $f$ has both a SNA and a SNR. \end{prop} In the particular case of the Harper map, the existence of a sink-source-orbit is equivalent to Anderson localisation for the corresponding almost-Mathieu operator (see, for example, \cite{haro/puig:2006} or \cite[Section 1.3]{jaeger:2006a}).
\subsection{The fibred rotation number and mode-locking.} \label{RotNum} If a qpf circle homeomorphism $f$ is homotopic to the identity on \ensuremath{\mathbb{T}^2}, it has a continuous lift $F:\ensuremath{\mathbb{T}^{1}}\times\ensuremath{\mathbb{R}} \to \ensuremath{\mathbb{T}^{1}} \times \ensuremath{\mathbb{R}}$ of the form $F(\theta,x) = (\theta+\omega,F_\theta(x))$. In this case, the limit \begin{equation}
\label{e.rotnum-def}
\rho(F) = \ensuremath{\lim_{n\rightarrow\infty}} (F^n_\theta(x)-x)/n \end{equation} exists and is independent of $\ensuremath{(\theta,x)}$ \cite[Section 5.3]{herman:1983}. $\rho(f) := \rho(F) \bmod 1$ is called the {\em (fibred) rotation number} of $f$. If the rotation number remains constant under all sufficiently small ${\cal
C}^0$-perturbations, we speak of {\em mode-locking}. The mechanism for mode-locking has been clarified in \cite{bjerkloev/jaeger:2009}. For our purposes we need the following two consequences. \begin{prop}[\cite{bjerkloev/jaeger:2009,jaeger:2009a}]
\label{p.mode-locking}
Suppose $f$ is a qpf circle homeomorphism and let $f_\ensuremath{\epsilon}=R_\ensuremath{\epsilon}\circ f$, where
$R_\ensuremath{\epsilon}(\theta,x) = (\theta,x+\ensuremath{\epsilon})$. Then the following holds. \alphlist \item If $\rho(f) \notin \ensuremath{\mathbb{Q}}+\ensuremath{\mathbb{Q}}\omega$ then $\ensuremath{\epsilon}\mapsto \rho(f_\ensuremath{\epsilon})$ is
strictly increasing in $\ensuremath{\epsilon}=0$. \item If $f$ is minimal then $\ensuremath{\epsilon}\mapsto \rho(f_\ensuremath{\epsilon})$ is strictly increasing
in $\ensuremath{\epsilon}=0$. \end{list} \end{prop} In other words, mode-locking cannot occur in situations (a) and (b).
\section{Statement of the main results} \label{MainResults}
The explicit ${\cal C}^1$-open conditions characterising the set ${\cal U}$ in Theorem~\ref{t.mr-qualitative} are not too complicated if each one is considered by itself, but altogether they form a rather long list. We therefore prefer not to include them in Theorem~\ref{t.firstversion}, but to state them separately before. We also note that conditions (\ref{eq:Cinvariance})--(\ref{eq:crossing}) below are precisely those used in \cite{jaeger:2009a}, whereas the Diophantine condition (\ref{eq:Diophantine}) and the assumptions (\ref{eq:bounddlambda})--(\ref{e.d}) on the dependence on the twist parameter are new. Let $\Lambda\ensuremath{\subseteq}[0,1]$ be an open interval.
\noindent {\em I. Diophantine condition.} First, recall that $\omega\in{\cal
D}(\sigma,\nu)$ just means that \begin{equation}
\label{eq:Diophantine} \tag{${\cal A}0$}
d(n\omega,0) \ > \ \sigma\cdot |n|^{-\nu} \quad \forall n\in\ensuremath{\mathbb{Z}}\ensuremath{\setminus}\{0\} \ . \end{equation}
\noindent {\em II. Critical regions.} Let $E=[e^-,e^+]$ and $C=[c^-,c^+]$ be two non-empty, compact and disjoint subintervals of \ensuremath{\mathbb{T}^{1}}. We will assume that for all $\tau\in\Lambda$ there exists a finite union ${\cal I}_0(\tau) \ensuremath{\subseteq} \ensuremath{\mathbb{T}^{1}}$ of ${\cal N}$ disjoint open intervals $I_0^1(\tau) \ensuremath{,\ldots,} I_0^{\cal
N}(\tau)$ (the {\em `critical regions'}) such that \begin{equation} \label{eq:Cinvariance} \tag{${\cal A}1$}
\ensuremath{f_{\tau,\theta}}(\mbox{cl}(\ensuremath{\mathbb{T}^{1}} \ensuremath{\setminus} E)) \ \ensuremath{\subseteq} \ \mbox{int}(C) \ \ \ \ \ \forall
\theta \notin {\cal I}_0(\tau) \ . \end{equation} Note that this implies \begin{equation} \label{eq:Einvariance} \tag{${\cal A}1'$}
\ensuremath{f_{\tau,\theta}}^{-1}(\mbox{cl}(\ensuremath{\mathbb{T}^{1}} \ensuremath{\setminus} C)) \ \ensuremath{\subseteq} \ \mbox{int}(E) \ \ \ \ \ \forall
\theta \notin {\cal I}_0(\tau)+\omega \ . \end{equation}
\noindent \emph{III. Bounds on the derivatives.} Concerning the derivatives of the fibre maps $\ensuremath{f_{\tau,\theta}}$, we will assume that for given $\alpha > 1$ and $p\in\ensuremath{\mathbb{N}}$ we have \begin{equation} \tag{${\cal A}2$}
\label{eq:bounds1} \alpha^{-p} \ < \ \partial_x\ensuremath{f_{\tau,\theta}(x)} \ < \ \alpha^p \hspace{2eM} \quad
\forall \ensuremath{(\theta,x)} \in \ntorus \ ; \end{equation} \begin{equation} \label{eq:bounds2} \hspace{3eM} \partial_x\ensuremath{f_{\tau,\theta}(x)} \ > \ \alpha^{2/p}
\hspace{4.1eM} \quad \forall \ensuremath{(\theta,x)} \in \ensuremath{\mathbb{T}^{1}} \times \tag{${\cal A}3$}
E \ ; \end{equation} \begin{equation} \label{eq:bounds3} \hspace{3eM} \partial_x\ensuremath{f_{\tau,\theta}(x)} \ < \
\alpha^{-2/p} \hspace{4.1eM} \quad \forall \ensuremath{(\theta,x)} \in \ensuremath{\mathbb{T}^{1}} \times C \ .
\tag{${\cal A}4$}
\end{equation}
Further, we fix $S>0$ such that \begin{equation} \label{eq:bounddth} \tag{${\cal A}5$}
|\partial_\theta \ensuremath{f_{\tau,\theta}(x)}| \ < \ S \ \ \ \ \ \forall \ensuremath{(\theta,x)} \in
\ntorus \ . \end{equation}
\noindent \emph{IV. Transversal Intersections.} The significance of the critical region ${\cal I}_0(\tau)$ is the fact that due to (\ref{eq:Cinvariance}) this is the only place where the attracting and the repelling region can `mix up'. In order to ensure that the intersections of $f_\tau({\cal I}_0(\tau) \times C)$ and $({\cal I}_0(\tau)+\omega)\times E$ are `nice' (transversal in an appropriate sense), we will assume that \begin{equation} \label{eq:crossing} \tag{${\cal A}6$}
\begin{array}{l} \exists!\theta_\iota^1 \in I_0^\iota(\tau) \textrm{ with }
f_{\tau,\theta_\iota^1}(c^+) = e^- \textrm{ \ and } \\ \exists! \theta_\iota^2 \in
I_0^\iota(\tau) \textrm{ with } f_{\tau,\theta_\iota^2}(c^-) = e^+ \ . \end{array} \end{equation} This ensures that the image of $I_0^\iota(\tau) \times C$ crosses the strip $(I^\iota_0(\tau)+\omega) \times E$ exactly once and not several times. In order to control the slope of these strips, we will assume that \begin{equation}
|\partial_\theta \ensuremath{f_{\tau,\theta}(x)}| \ > \ s \ \ \ \ \ \forall \ensuremath{(\theta,x)} \in {\cal I}_0(\tau)
\times \ensuremath{\mathbb{T}^{1}} \ \label{eq:s} \tag{${\cal A}7$} \end{equation} for some constant $s$ with $0 < s < S$.
\noindent {\em V. Dependence on $\tau$.}
First, we fix an upper bound $L$ on $|\partial_\tau
\ensuremath{f_{\tau,\theta}(x)}|$, that is, \begin{equation} \label{eq:bounddlambda} \tag{${\cal A}8$}
|\partial_\tau \ensuremath{f_{\tau,\theta}(x)}| \ < \ L\ \ \ \ \ \forall \ensuremath{(\theta,x)} \in \ntorus \ . \end{equation} Secondly, we assume that all connected components $I^\iota_0(\tau) = (a^\iota_0(\tau),b^\iota_0(\tau))$ of ${\cal I}_0(\tau)$ are differentiable with respect to $\tau$ and that for some constant $\eta>0$ we have \begin{equation}
\label{e.I_0-derivative} \tag{${\cal A}9$}
|D_\tau(I^\iota_0(\tau),I^\kappa_0(\tau))| \ > \ \eta/2 \quad
\forall \iota \neq \kappa \in [1,\cal N] \ . \end{equation} Finally we will need an assumption which ensures that condition (\ref{e.I_0-derivative}) holds in a similar way for the higher order critical regions ${\cal I}^\iota_n$ defined later on. This is actually the crucial point of the construction, which allows to adapt the parameter exclusion scheme from \cite{bjerkloev:2005a,jaeger:2009a} to the considered problem. For any $\iota \in [1,{\cal N}]$ we let \begin{equation}\nonumber
Q^\iota(\tau) \ := \ \left\{ \partial_\tau \ensuremath{f_{\tau,\theta}(x)} / \partial_\theta \ensuremath{f_{\tau,\theta}(x)}
\left| \ \ensuremath{(\theta,x)} \in I^\iota_0 \times \ensuremath{\mathbb{T}^{1}} \right. \right\} \ . \end{equation} Then we assume that there holds \begin{equation} \label{e.d} \tag{${\cal A}10$}
d(Q^\iota(\tau),Q^\kappa(\tau)) \ > \ \eta \quad \forall \iota \neq \kappa
\in [1,\cal N] \ . \end{equation}
\begin{thm} \label{t.firstversion} Let $\omega \in {\cal
D}(\sigma,\nu)$ and $\delta > 0$. Suppose that $\Lambda \ensuremath{\subseteq}
[0,1]$ is an open interval and $(F_\tau)_{\tau\in[0,1]} \in {\cal
P}$ is such that conditions (\ref{eq:Cinvariance})--(\ref{e.d})
are satisfied on $\Lambda$. Let $\ensuremath{\epsilon}_0 := \sup_{\iota\in[1,{\cal
N}],\tau\in\Lambda}|I^\iota_0(\tau)|$. Then there exist constants
$\alpha_* = \alpha_*(\sigma,\nu,{\cal N},p,S,s,L,\eta,\delta)$ and $\ensuremath{\epsilon}_* =
\ensuremath{\epsilon}_*(\sigma,\nu,{\cal N},p,S,s,L,\eta,\delta)$ such that the following
holds.
If $\alpha > \alpha_*$ and $\ensuremath{\epsilon}_0 < \ensuremath{\epsilon}_*$, then there exists a set
$\Lambda_\infty=\Lambda_\infty(\omega)\ensuremath{\subseteq}\Lambda$ of measure
$\ensuremath{\mathrm{Leb}}(\Lambda_\infty) > \ensuremath{\mathrm{Leb}}(\Lambda) - \delta$ such that for all $\tau \in
\Lambda_\infty$ the qpf circle diffeomorphism
\begin{equation}
f_\tau \ : \ \ensuremath{(\theta,x)} \mapsto (\ensuremath{\theta + \omega},\ensuremath{f_{\tau,\theta}(x)})
\end{equation}
satisfies
\begin{equation} \tag*{$(*)$}
\label{e.star}
\left\{
\begin{array}{cl}
(*1) & f_\tau \textrm{ has a SNA }
\varphi^- \textrm{ and a SNR } \varphi^+;
\\
(*2) & \varphi^- \textrm{ and } \varphi^+ \textrm{ are one-valued and the only
invariant graphs of } f;\\
(*3) & f_\tau \textrm{ is minimal.}
\end{array} \right.
\end{equation}
Further, if $f_{\tau_0}$ satisfies
\begin{equation}
\label{e.symmetry-addendum}
f_{\tau_0,\theta+\halb}(-x) \ = \ -f_{\tau_0,\theta}(x) \qquad \forall\ensuremath{(\theta,x)}\in\ensuremath{\mathbb{T}^2}
\end{equation}
then $\tau_0\in\Lambda_\infty$.
\end{thm}
The proof is given in Section~\ref{ParameterExclusion}.
\noindent {\em VI. Modified assumptions.} As mentioned in the introduction, it
is not possible to apply this result to the qpf Arnold circle map due to the
bounded slope of the fibre maps. In order to make up for this lack of
hyperbolicity, a particular geometry and symmetry of the forcing has to be
used. For the twist parameter exclusion carried out here, this is slightly
more subtle than for the frequency exclusion in \cite{jaeger:2009a} and
stronger conditions on the forcing are required (see also
Remark~\ref{r.forcing-structure}). First, we have to restrict to the case of
two critical regions with fixed distance $1/2$. \begin{equation} \label{e.A7'}\textstyle
\tag{${\cal A}7'$}
{\cal N} = 2 \ \textrm{ and there holds } \ I^1_0(\tau) = I^2_0(\tau) + \halb \ . \end{equation} Secondly, the slope on the two critical regions must have opposite sign. \begin{equation}\tag{${\cal A}8'$} \label{e.A8'}
\partial_\theta f_{\tau,\theta}(x) > s \ \textrm{ on } \ I^1_0\times\ensuremath{\mathbb{T}^{1}} \
\textrm{ and } \partial_\theta f_{\tau,\theta}(x) < -s \ \textrm{ on }
I^2_0\times\ensuremath{\mathbb{T}^{1}} \ . \end{equation} Thirdly, as in \cite{jaeger:2009a} we need to ensure that away from the critical regions the $\theta$-dependence is small. To that end, we suppose ${\cal I}_0'\ensuremath{\subseteq} \ensuremath{\mathbb{T}^{1}}$ is the disjoint union of two open intervals $I^{1'}_0$ and $I^{2'}_0=I^{1'}_0+\halb$ with $I^k_0\ensuremath{\subseteq} I^{k'}_0$ $(k=1,2)$ and for some $s' \in (0,s)$ there holds \begin{equation} \label{eq:refinedbounddth} \tag{${\cal A}9'$}
|\partial_\theta \ensuremath{f_{\theta}(x)}| \ < \ s' \ \
\forall \ensuremath{(\theta,x)} \in (\ensuremath{\mathbb{T}^{1}} \ensuremath{\setminus} {\cal I}_0') \times C \ . \end{equation} Finally, we need constants $\gamma,L>0$ which provide uniform upper and lower bounds for the dependence on the twist parameter $\tau$. \begin{equation}
\tag{${\cal A}10'$} \label{e.A10'} \gamma \ < \ \partial_\tau f_{\tau,\theta}(x) \ < \ L \qquad \end{equation}
\begin{thm} \label{t.mr-quantitative2} Let $\omega \in {\cal D}(\sigma,\nu)$ and
$\delta > 0$. Suppose that $\Lambda \ensuremath{\subseteq} [0,1]$ is an open interval and
$(F_\tau)_{\Lambda\in[0,1]} \in {\cal P}$ is such that conditions
(\ref{eq:Diophantine})--(\ref{eq:crossing}) and (\ref{e.A7'})--(\ref{e.A10'})
are satisfied on $\Lambda$. Let $\ensuremath{\epsilon}_0 := \sup_{\iota\in[1,{\cal
N}],\tau\in\Lambda}|I^\iota_0(\tau)|$. Further, assume there exist
constants $A,d > 1$ such that
\begin{eqnarray}
S & < & A\cdot d \ , \label{e.S<Ad}\\ s & > & d/A \ , \label{e.s>d/A} \\
\ensuremath{\epsilon}_0 & < &A/\sqrt{d} \ , \label{e.eps<1/Ad}\\ \label{e.s'<A} s' & < & A \ .
\end{eqnarray}
Then there exist a constant $d_*=d_*(\sigma,\nu,{\cal
N},\alpha,p,L,\gamma,A,\delta)>0$ with the following property.
If $d > d_*$, then there exists a set
$\Lambda_\infty=\Lambda_\infty(\omega)\ensuremath{\subseteq} \Lambda$ of measure
$\ensuremath{\mathrm{Leb}}(\Lambda_\infty) > \ensuremath{\mathrm{Leb}}(\Lambda) - \delta$ such that for all $\tau \in
\Lambda_\infty$ the qpf circle diffeomorphism
\begin{equation}
f_\tau \ : \ \ensuremath{(\theta,x)} \mapsto (\ensuremath{\theta + \omega},\ensuremath{f_{\tau,\theta}(x)})
\end{equation}
satisfies \ref{e.star}. Further, if $f_{\tau_0}$ satisfies
(\ref{e.symmetry-addendum}) then $\tau_0\in\Lambda_\infty$. \end{thm} The proof is given in Section~\ref{RefinedParameterExclusion}. We note that the precise form of the $d$-dependence in (\ref{e.S<Ad})--(\ref{e.s'<A}) is to some extent arbitrary and could be stated in a more general way, but we refrain from introducing even more parameters and refer to Section~\ref{RefinedProof} for details. The estimates required to deduce Corollaries~\ref{c.arnold}--\ref{c.tongue-collapse} from this statement will be carried out in Section~\ref{ArnoldCorollary}.
\section{The basic version of the twist parameter
exclusion} \label{ParameterExclusion}
\subsection{Critical sets and critical regions.} \label{Previous}
In this section, we will briefly recall the construction from \cite{jaeger:2009a} and collect the key statements needed for the proof of Theorem~\ref{t.firstversion}. The parameter $\tau$, and consequently the map $f_\tau$, will be fixed. Nevertheless we keep the dependence on $\tau$ explicit for the sake of consistency with the later sections. The description of the dynamics of a suitable qpf circle diffeomorphism $f$ in \cite{jaeger:2009a} is based on the analysis of certain critical sets ${\cal C}_0 ,{\cal C}_1, {\cal
C}_2, \ldots$ and critical regions ${\cal I}_0 , {\cal I}_1, {\cal I}_2 , \ldots$, which are given as follows.
Given a union ${\cal I}_0(\tau)$ of ${\cal N}$ disjoint open intervals $I^1_0(\tau) \ldots I^{\cal N}_0(\tau)$ and a monotonically increasing sequence of integers $\nofolge{M_n}$ with $M_0 \geq 2$, we recursively define \begin{eqnarray*}
{\cal A}_n & := & \{\ensuremath{(\theta,x)} \mid \theta \in {\cal I}_n(\tau)-(M_n-1)\omega,\ x
\in C\} \ , \\ {\cal B}_n & := & \{\ensuremath{(\theta,x)} \mid \theta \in {\cal
I}_n(\tau)+(M_n+1)\omega,\ x \in E\} \ , \\ {\cal C}_n &:= &
f_\tau^{M_n-1}({\cal A}_n) \cap f^{-M_n-1}({\cal B}_n) \quad \textrm{and } \\
\label{eq:defIn} {\cal I}_{n+1}(\tau) & := & \mathrm{int}(\pi_1({\cal C}_n)) \ . \end{eqnarray*} The crucial observation is the fact that certain {\em `slow
recurrence' assumptions} on the critical regions ${\cal I}_n$ are already sufficient to guarantee the nonuniform hyperbolicity of $f_\tau$. In order to state them, suppose $\nofolge{K_n}$ is a monotonically increasing sequence of positive integers and $\nofolge{\ensuremath{\epsilon}_n}$ is a non-increasing sequence of positive real numbers which satisfy $\ensuremath{\epsilon}_0 \leq 1$ and $\ensuremath{\epsilon}_n \geq 9\ensuremath{\epsilon}_{n+1} \ \forall n\in\ensuremath{\mathbb{N}}_0$. Let \begin{equation} \nonumber
{\cal X}_n \ := \ \bigcup_{k=1}^{2K_n M_n} ({\cal I}_n+k\omega) \ \
\textrm{ and } \ \ {\cal Y}_n \ := \ \bigcup_{j=0}^n \bigcup_{k = -M_j+1}^{M_j+1}
({\cal I}_j + k\omega) \ . \end{equation} Then the required assumptions on the critical regions are the following. \begin{equation} \tag*{$({\cal X})_n$} \label{e.Xn} d({\cal I}_j,{\cal X}_j) \ >
\ 3\ensuremath{\epsilon}_j \ \ \ \ \ \forall j = 0 \ensuremath{,\ldots,} n \ , \end{equation}
\begin{equation}
\tag*{$({\cal Y})_n$} \label{e.Yn} d(({\cal I}_j -(M_j-1)\omega) \cup ({\cal
I}_j+(M_j+1)\omega),{\cal Y}_{j-1}) \ > \ 0 \ \ \ \ \ \forall j=1\ensuremath{,\ldots,} n \ . \end{equation} Let $\beta_0=1$, $\beta_n = \prod_{j=0}^{n-1} \left( 1-\frac{1}{K_j} \right)$ and $\beta=\ensuremath{\lim_{n\rightarrow\infty}} \beta_n$. Further, define \begin{equation} \label{e.alpha-infty}
\alpha_\infty \ = \ \alpha^{2\beta/p - (1-\beta)p} \ . \end{equation} \begin{prop}[Propositions 3.10 in \cite{jaeger:2009a}]
\label{p.sna-existence} If (\ref{eq:Cinvariance})--(\ref{eq:s}) hold,
$\alpha_\infty > \alpha_1$ and for all $n\in\ensuremath{\mathbb{N}}_0$ conditions \ref{e.Xn} and
\ref{e.Yn} are satisfied and ${\cal I}_{n+1}(\tau)=\pi_1({\cal C}_n) \neq
\emptyset$, then $f_\tau$ has a sink-source orbit and consequently an SNA and an SNR. \end{prop} We will also use the following slightly stronger versions of the above conditions. \begin{equation} \tag*{$({\cal X'})_n$} \label{e.X'n} d({\cal I}_j,{\cal X}_j) \ >
\ 9\ensuremath{\epsilon}_j \ \ \ \ \ \forall j = 0 \ensuremath{,\ldots,} n \ , \end{equation}
\begin{equation}
\tag*{$({\cal Y'})_n$} \label{e.Y'n} d(({\cal I}_j -(M_j-1)\omega) \cup ({\cal
I}_j+(M_j+1)\omega),{\cal Y}_{j-1}) \ > \ 2\ensuremath{\epsilon}_j \ \ \ \ \ \forall j=1\ensuremath{,\ldots,} n \ . \end{equation}
\begin{rem}
In the proof of Theorems~\ref{t.firstversion} and \ref{t.mr-quantitative2}, we
will show that for all $\tau\in\Lambda_\infty$ conditions \ref{e.X'n} and
\ref{e.Y'n} hold. In fact, for our purposes here the weaker conditions
\ref{e.Xn} and \ref{e.Yn} would be sufficient, since we only need them for the
application of Proposition~\ref{p.sna-existence}. The reason for using the
stronger versions \ref{e.X'n} and \ref{e.Y'n} is that we believe these to be
crucial in a current approach to the proof of Conjecture~\ref{conjecture}, and
that at the same time this inflicts no extra costs whatsoever. \end{rem} Let us briefly review the construction in \cite{jaeger:2009a} that leads to the statement of Proposition~\ref{p.sna-existence} and provide some more details that will be used later. Given any point $(\theta_0,x_0) \in \ensuremath{\mathbb{T}^2}$, we denote its iterates by $(\theta_k,x_k) = f_\tau^k(\theta_0,x_0),\ k\in\ensuremath{\mathbb{Z}}$. Now, (\ref{eq:Cinvariance}) implies that whenever $\theta_0\notin {\cal I}_0(\tau)$ and $x_0\in C$, the forward orbit remains `trapped' in the contracting region $\ensuremath{\mathbb{T}^{1}} \times C$ until $\theta_k$ enters ${\cal I}_0(\tau)$ for the first time. However, even if $\theta_k \in {\cal I}_0(\tau)$ and the orbit enters the expanding region at time $k$, that is $x_{k+1} \in E$, it will leave $\ensuremath{\mathbb{T}^{1}} \times E$ again after $M_0$ further iterates unless $\theta_k$ is also contained in the smaller set ${\cal I}_1(\tau)$. (This is a straightforward consequence of the definition of ${\cal C}_0$ and ${\cal I}_1(\tau)$.) Following this idea it is possible, via a purely combinatorial inductive construction, to control the behaviour of an orbit $(\theta_k,x_k)$ starting in ${\cal A}_{n}$ up to the first time $\hat k\in\ensuremath{\mathbb{N}}$ at which $\theta_k$ enters the $(n+1)$-th critical region ${\cal I}_{n+1}(\tau)$, provided that the `slow recurrence' assumptions \ref{e.Xn} and \ref{e.Yn} hold \cite[Lemma 3.4]{jaeger:2009a}. In this case, the finite trajectory will remain in $\ensuremath{\mathbb{T}^{1}} \times C$ most of the time and the fraction of time spent outside this set is at most $1-\beta_n$ \cite[Lemma 3.8]{jaeger:2009a}.
As a consequence, since $M_n\leq \hat k$ by \ref{e.Xn}, it is possible to control the forward orbit of these points up to time $M_n$, or equivalently the backward orbit of points in $f_\tau^{M_n}({\cal A}_{n})\supseteq{\cal C}_n$, which remain trapped in the contracting region most of the time. Similar findings hold for the forwards orbits of points in $f^{-M_n}_\tau({\cal B}_n) \supseteq{\cal C}_n$, which remain in the expanding region most of the time. Combining the combinatorial information about the behaviour of the orbits with the estimates on the derivatives provided by (\ref{eq:bounds1})--(\ref{eq:bounds3}) one obtains the following statement. \begin{lem}[Corollaries 3.7 and 3.9 in \cite{jaeger:2009a}] \label{p.lyaps}
Suppose (\ref{eq:Cinvariance})--(\ref{eq:bounds3}), \ref{e.Xn} and \ref{e.Yn}
hold and $\ensuremath{(\theta,x)} \in \ensuremath{\mathrm{cl}}(f_\tau({\cal C}_n))$. Then for all
$k\in[0,M_n]$ we have
\begin{equation}
\label{e.finite-lyaps}
\partial_x \ensuremath{f_{\tau,\theta}}^{-k}(x) \geq \alpha_\infty^k \quad \textrm{and}
\quad \partial \ensuremath{f_{\tau,\theta}}^k(x) \geq \alpha_\infty^k \ .
\end{equation}
Furthermore, there holds ${\cal C}_0 \supseteq {\cal C}_1 \supseteq \ldots
\supseteq {\cal C}_{n+1}$ and \begin{equation}
\label{e.lastiterates}
\ensuremath{(\theta,x)} \in ({\cal I}_n(\tau)+\omega) \times E \quad , \quad f_\tau^{-1}\ensuremath{(\theta,x)} \in
{\cal I}_n(\tau) \times C \ . \end{equation} \end{lem} This statement rather easily entails Proposition~\ref{p.sna-existence} (respectively \cite[Proposition 3.10]{jaeger:2009a}): When \ref{e.Xn} and \ref{e.Yn} hold for all $n\in\ensuremath{\mathbb{N}}$ and $\ensuremath{(\theta,x)} \in \ensuremath{\bigcap_{n\in\N}} \ensuremath{\mathrm{cl}}\left(f_\tau({\cal C}_n)\right)$, then it follows directly from (\ref{e.finite-lyaps}) that the point $\ensuremath{(\theta,x)}$ has a positive vertical Lyapunov exponent both for $f_\tau$ and for its inverse $f^{-1}_\tau$. Hence, there exist a sink-source-orbit and thus an SNA by Proposition~\ref{prop:sinksourcesna}.
Due to Proposition~\ref{p.sna-existence}, the validity of the slow recurrence conditions \ref{e.Xn} and \ref{e.Yn} together with ${\cal
I}_n(\tau)\neq\emptyset$ provides a rigorous criterion for the existence of SNA. The remaining task is to find a positive measure set of parameters (frequencies $\omega$ in \cite{jaeger:2009a}, parameters $\tau$ in our setting) for which these assumptions are satisfied for all $n \in \ensuremath{\mathbb{N}}$. At this point, a detailed analysis of the geometry of the critical sets, or more precisely of the sets $f^{M_n}_\tau({\cal
A}_n)$ and $f^{-M_n}_\tau({\cal B}_n)$ whose intersection equals $f_\tau({\cal C}_n)$, comes into play. The outcome is that the size of the connected components of the critical regions ${\cal I}_n(\tau)$ decays super-exponentially with $n$ and that these components depend `nicely' on the parameter. This will be made precise in Proposition~\ref{p.lyaps} and Lemma~\ref{l.lambda-dependence} below.
The main idea behind this geometric analysis is again the fact that for any connected component $I^\iota_n(\tau)$ of ${\cal I}_n(\tau)$ the first $M_n-1$ forward iterates of the set $A^\iota_n = (I^\iota_n(\tau)-(M_n-1)\omega)\times C$ remain in the contracting region most of the time. Consequently $f_\tau^{M_n-1}(A^\iota_n)$ will be a very thin strip, which will moreover be almost horizontal since the strong contraction `kills' any dependence on $\theta$. Due to (\ref{eq:s}) the image $f_\tau^{M_n}(A^\iota_n)$ will then have slope $\geq s/2$. It therefore intersects the image $f_\tau^{-M_n}(B^\iota_n)$ of $B^\iota_n = (I^\iota_n(\tau)+(M_n+1)\omega)\times E$ in a transversal way, since this set is a very thin horizontal strip by the same reasoning as for $f_\tau^{M_n-1}(A^\iota_n)$. (Note that the expanding region $\ensuremath{\mathbb{T}^{1}}\times E$ is contracted by the inverse $f^{-1}$.) Consequently the resulting intersection $f_\tau^{M_n}(A^\iota_n) \cap f_\tau^{-M_n}(B^\iota_n)$ always has the geometry depicted in Figure~\ref{f.crossing} and projects to a very small interval $I^\iota_{n+1}(\tau)+\omega$.
In order to give a more detailed quantitative version of this heuristic description, we first define the `bounding graphs' of the sets $f_\tau^{M_n}(A^\iota_n)$ and $f_\tau^{-M_n}(B^\iota_n)$. For $\theta \in I^\iota_n(\tau)+\omega$, let \begin{equation} \label{eq:phipsidef}
\varphi_{\iota,n}^\pm(\theta,\tau) \ := \
f_{\tau,\theta-M_n\omega}^{M_n}(c^\pm) \quad \textrm{ and } \quad
\psi_{\iota,n}^\pm(\theta,\tau) \ := \
f_{\tau,\theta+M_n\omega}^{-M_n}(e^\pm)\ . \end{equation} Note that we have \begin{eqnarray} \label{e.phin-characterisation} f_\tau^{M_n}(A^\iota_n) & =
& \{ \ensuremath{(\theta,x)} \mid \theta \in I_n^\iota(\tau)+\omega,\ x \in
[\varphi_{\iota,n}^-(\theta,\tau) ,\varphi_{\iota,n}^+(\theta,\tau)] \}
\ , \\ \label{e.psin-characterisation} f_\tau^{-M_n}(B^\iota_n) & = & \{
\ensuremath{(\theta,x)} \mid \theta \in I_n(\tau)+\omega,\ x \in
[\psi_{\iota,n}^-(\theta,\tau), \psi_{\iota,n}^+(\theta,\tau)] \} \ .
\end{eqnarray} \begin{figure}
\caption{\small The two `strips' $f^{M_n}(A^\iota_n)$ and
$f^{-M_n}(B^\iota_n)$ intersect each other in a transversal way, producing a
connected component of ${\cal C}_n$.}
\label{f.crossing}
\end{figure}
Using this notation, we can now restate the following estimates from \cite{jaeger:2009a}.
\begin{prop}[Proposition~3.11 and Lemma~3.14 in \cite{jaeger:2009a}]
\label{p.In-size} Suppose
(\ref{eq:Cinvariance})--(\ref{eq:s}) hold and \ref{e.Xn} and
\ref{e.Yn} are satisfied. Then there exists a constant $\alpha_1 =
\alpha_1(s,S)$ such that the following holds: If $\alpha_\infty >
\alpha_1$ then \romanlist \item ${\cal I}_n(\tau)$ and ${\cal I}_{n+1}(\tau)$ consist of exactly
${\cal N}$ connected components and each component of ${\cal I}_n(\tau)$
contains exactly one component of ${\cal I}_{n+1}(\tau)$. \item For all connected components of $I^\iota_{n+1}(\tau)$ of ${\cal
I}_{n+1}(\tau)$ there holds
\begin{equation} \label{e.In-size}
|I^\iota_{n+1}(\tau)| \ \leq \ 2\alpha_\infty^{-M_n}/s \ .
\end{equation} \item Either
\begin{eqnarray}\label{e.transversal-intersection} \inf_{\theta\in{\cal I}_n(\tau)+\omega} \partial_\theta\varphi^\xi_{\iota,n}(\theta,\tau) - \partial_\theta\psi^\zeta_{\iota,n}(\theta,\tau) \
> & 0 & \quad \forall \xi,\zeta \in \{+,-\} \qquad \textrm{or} \\\label{e.downwards} \sup_{\theta\in{\cal I}_n(\tau)+\omega} \partial_\theta\varphi^\xi_{\iota,n}(\theta,\tau) - \partial_\theta\psi^\zeta_{\iota,n}(\theta,\tau) &
< & 0 \quad \forall \xi,\zeta \in \{+,-\} \ . \end{eqnarray} \end{list} \end{prop} We note that part (iii) can be seen justification for the picture in Figure~\ref{f.crossing}, insofar as (\ref{e.transversal-intersection}), respectively (\ref{e.downwards}) ensures that each of the pairs of curves $\varphi^\xi_{\iota,n}$ and $\psi^\zeta_{\iota,n}$ intersect in exactly one point and $f_\tau^{M_n}(A^\iota_n)$ crosses $f^{-M_n}(B^\iota_n)$ either upwards (as depicted) or downwards.
It remains to obtain a good control on the dependence of the critical regions ${\cal I}_n(\tau)$ on $\tau$. This will be the content of the next section. On the technical level, this is the crucial difference in comparison to the construction in \cite{jaeger:2009a}.
\subsection{Dependence of the critical regions on $\tau$.} \label{LambdaDependence}
In order to perform the parameter exclusion with respect to $\tau$, we need to show that different connected components of ${\cal I}_n(\tau)$ move with different speed as the parameter $\tau$ changes. This is ensured by the following lemma. \begin{lem}
\label{l.lambda-dependence} Suppose
(\ref{eq:Cinvariance})--(\ref{e.d}) hold and \ref{e.Xn} and
\ref{e.Yn} are satisfied. Then there exists a constant $\alpha_2 =
\alpha_2(s,S,L,\eta)$ such that the following holds:
If $\alpha_\infty > \alpha_2$ then all connected components of ${\cal
I}_{n+1}(\tau)$ are differentiable in $\tau$. Further,
\begin{eqnarray}
\label{e.lambda-dependence}
|\partial_\tau I^\iota_{n+1}(\tau)| & < & 2L/s \qquad
\forall \iota \in \{1\ensuremath{,\ldots,}{\cal N}\} \qquad \textrm{ and } \\\label{e.dlambda-bound}
|D_\tau(I^\iota_{n+1}(\tau),I^\kappa_{n+1}(\tau))| & > & \frac{\eta}{2} \qquad \qquad
\forall \iota\neq\kappa\in\{1\ensuremath{,\ldots,} {\cal N}\} \ .
\end{eqnarray} \end{lem} \textit{Proof. } We write $I_{n+1}^\iota(\tau)=(a^\iota(\tau),b^\iota(\tau)$ and show that \begin{equation}
\label{e.endpoint-ldep}
\partial_\tau a^\iota(\tau),\ \partial_\tau b^\iota(\tau)\ \in
\ B_\Delta(Q^\iota(\tau)) \quad \forall \iota\in[1,{\cal N}]) \end{equation} where $\Delta:=\min\{\eta/4,L/s\}$. Due to (\ref{e.d}) this immediately implies (\ref{e.dlambda-bound}). Further, since $Q^\iota(\tau)\ensuremath{\subseteq} [-L/s,L/s]$ by (\ref{eq:s}) and (\ref{eq:bounddlambda}) we also obtain~(\ref{e.lambda-dependence}).
We carry out the proof only for $\partial_\tau a^\iota(\tau)$ since the other endpoint can be treated in the same way. Similarly, we assume that the crossing between $f^{M_n}_\tau(A^\iota_n)$ and $f^{-M_n}_\tau(B^\iota_n)$ is upwards, that is, case (\ref{e.transversal-intersection}) in Proposition~\ref{p.In-size}(iii) holds. Again, the other case can be treated similarly. Then, as can be seen from the picture in Figure~\ref{f.crossing}, $a^\iota(\tau)$ is characterized by the equality \begin{equation} \nonumber
\varphi^+_{\iota,n}(a^\iota(\tau)+\omega,\tau) -
\psi^-_{\iota,n}(a^\iota(\tau)+\omega,\tau) \ = \ 0 \ . \end{equation} Application of the Implicit Function Theorem therefore yields \begin{equation} \label{e.IFT-Formula}
\partial_\tau a^\iota(\tau) \ = \
- \ \frac{\partial_\tau\left(\varphi^+_{\iota,n}-
\psi^-_{\iota,n}\right)(a(\tau)+\omega,\tau)}{\partial_\theta \left(\varphi^+_{\iota,n}-
\psi^-_{\iota,n}\right)(a(\tau)+\omega,\tau)} \ . \end{equation} We start by deriving an estimate on the numerator. Let $\theta := a^\iota(\tau)+\omega$ and $x:= \varphi^+_{\iota,n}(\theta,\tau) = f^{M_n}_{\tau,\theta-M_n\omega}(c^+)$. Let $\theta_0 = \theta-M_n\omega$ and $\theta_k = \theta_0+k\omega$. Further, let $x_0=c^+$ and $x_k=f^k_{\tau,\theta_0}(x_0)$. Note that thus $\ensuremath{(\theta,x)} = (\theta_{M_n},x_{M_n})$. Differentiating with respect to $\tau$ we obtain \begin{equation}
\partial_\tau \varphi^+_{\iota,n}(\theta,\tau) \ = \
\partial_\tau f_{\tau,\theta_{M_n-1}}
(x_{M_n-1}) \label{e.rqiota}
\ + \ \ \underbrace{\sum_{k=1}^{M_n-1}
\partial_x f^{M_n-k}_{\tau,\theta_k}(x_k)
\cdot \partial_\tau f_{\tau,\theta_{k-1}}
(x_{k-1})}_{=:r_1^\iota} \ . \end{equation} Since $ \partial_x f^{M_n-k}_{\tau,\theta_{k}}(x_k) = \left( \partial_x
f_{\tau,\theta}^{-M_n+k}(x)\right)^{-1} $ and $(\theta,x) \in \ensuremath{\mathrm{cl}}\left(f_\tau({\cal C}_n)\right)$, the estimate (\ref{e.finite-lyaps}) in Lemma~\ref{p.lyaps} yields $\partial_x f^{M_n-k}_{\tau,\theta_k}(x_k) \leq \alpha_\infty^{-k}$. Together with the upper bound $L$ on the derivative with respect to $\tau$
provided by (\ref{eq:bounddlambda}) we obtain $|r_1^\iota| \ \leq \ L \cdot \sum_{k=1}^\infty \alpha_\infty^{-k}$.
Now let $\theta=a^\iota(\tau)+\omega$ as before, $\xi=\psi^-_{\iota,n}(\theta,\tau)$, $\vartheta_0=\theta+M_n\omega$, $\vartheta_k=\vartheta_0+k\omega$,$\xi_0=e^-$ and $\xi_k=f^k_{\tau,\vartheta_0}(\xi_0)$. Note that $(\vartheta_{-M_n},xi_{-M_n}) = (\theta,\xi)$. Differentiating with respect to $\tau$ yields \begin{eqnarray} \nonumber r^\iota_2 & := & \partial_\tau
\psi^-_{\iota,n}(\theta,\tau) \ = \
\sum_{k=1}^{M_n} \partial_x f_{\tau,\vartheta_{-k}}^{-M_n+k}
(\xi_{-k}) \cdot
\partial_\tau f^{-1}_{\tau,\vartheta_{-k+1}}
(\xi_{-k+1})\\ \label{e.psicontrol}
& = & -\ \sum_{k=1}^{M_n} \left(\partial_x f^{M_n-k}_{\tau,\vartheta_{-M_n}}
(\xi_{-M_n}) \right)^{-1}\cdot
\left(\partial_x f_{\tau,\vartheta_{-k}}(\xi_{-k})\right)^{-1}
\cdot \partial_\tau f_{\tau,\vartheta_{-k}}(\xi_{-k}) \\
& = & -\ \sum_{k=1}^{M_n} \left(\partial_x
f^{M_n-k+1}_{\tau,\vartheta_{-M_n}}(\xi_{-M_n})\right)^{-1} \cdot
\partial_\tau f_{\tau,\vartheta_{-k}}(\xi_{-k}) \ . \nonumber \end{eqnarray}
Using (\ref{eq:bounddlambda}) and (\ref{e.finite-lyaps}) again we obtain $|r^\iota_2| \leq L \cdot \sum_{k=1}^\infty \alpha_\infty^{-k}$. If we let $r^\iota = r^\iota_1 - r^\iota_2$ and use that $\theta_{M_n-1}=\theta-\omega$ and $x_{M_n-1} = f_{\tau,\theta}^{-1}(x)$ then \begin{eqnarray}
\partial_\tau\left(\varphi^+_{\iota,n}-
\psi^-_{\iota,n}\right)(\theta,\tau) & = & \partial_\tau
f_{\tau,\theta-\omega}\left(f_{\tau,\theta}^{-1}(x)\right) \ + \ r^\iota \\
\textrm{with } \quad |r^\iota| & \leq & 2L/(\alpha_\infty-1) \ . \end{eqnarray} If we replace $\partial_\tau$ by $\partial_\theta$ in these computations and use (\ref{eq:bounddth}) instead of (\ref{eq:bounddlambda}), then we obtain in exactly the same way that \begin{eqnarray}
\partial_\theta\left(\varphi^+_{\iota,n}-
\psi^-_{\iota,n}\right)(\theta,\tau) & = & \partial_\theta
f_{\tau,\theta-\omega}\left(f_{\tau,\theta}^{-1}(x)\right) \ + \ q^\iota \\
\textrm{with } \quad |q^\iota| & \leq & 2S/(\alpha_\infty-1) \ . \end{eqnarray} Now $\theta-\omega \in {\cal I}_n \ensuremath{\subseteq} {\cal I}_0$, such that
$\left|\partial_\theta
f_{\tau,\theta-\omega}\left(f_{\tau,\theta}^{-1}(x)\right)\right| > s$ by (\ref{eq:s}) and \[ \partial_\tau
f_{\tau,\theta-\omega}\left(f_{\tau,\theta}^{-1}(x)\right)/\partial_\theta
f_{\tau,\theta-\omega}\left(f_{\tau,\theta}^{-1}(x)\right) \ \in \ Q^\iota(\tau) \]
by the definition of $Q^\iota(\tau)$. Furthermore, it follows from the above estimates that $|r^\iota|$ and $|q^\iota|$ go to zero as $\alpha_\infty \to \infty$. Hence, for sufficiently large $\alpha_\infty$ we have \begin{equation} \label{e.lamda-dep-form}
\partial_\tau a^\iota(\tau) \ =
\ - \ \frac{\partial_\tau f_{\tau,\theta-\omega}\left(f_{\tau,\theta}^{-1}(x)\right) + r^\iota}{
\partial_\theta f_{\tau,\theta-\omega}\left(f_{\tau,\theta}^{-1}(x)\right) + q^\iota} \
\in \ B_{\Delta}(Q^\iota(\tau)) \ . \end{equation} Furthermore, it can be seen from the above estimates that the largeness condition on $\alpha_\infty$ only depends on the constants $s,S,L$ and $\eta$. \qed
\subsection{Preliminaries for the parameter
exclusion} \label{ParameterExclusionPreliminaries} We now collect some preliminary statements for the parameter exclusion. The setting is an abstract one that does not depend on the previous dynamical construction. We first fix an integer ${\cal N}$ and sequences $\nofolge{K_n}$ and $\nofolge{\ensuremath{\epsilon}_n}$ with the same properties as in Section~\ref{Previous} and a sequence $\nofolge{N_n}$ of positive integers that satisfy \begin{equation}
\label{e.N}\tag{${\cal N}1$} N_0 \geq 3 \quad \textrm{and} \quad N_{n+1}
\geq 2K_nN_n \ \forall n\in\ensuremath{\mathbb{N}}_0 \ . \end{equation} We denote by ${\cal S}(\ensuremath{\mathbb{T}^{1}})$ the set of all subsets of $\ensuremath{\mathbb{T}^{1}}$. Let $\Lambda\ensuremath{\subseteq}[0,1]$ be an open interval. Then we simply assume that we are given a sequence of mappings \[ {\cal I}_n \ : \quad \Lambda \times \ensuremath{\mathbb{N}}^n \to {\cal S}(\ensuremath{\mathbb{T}^{1}}) \ , \quad (\tau,M_0\ensuremath{,\ldots,} M_{n-1}) \ \mapsto \ {\cal I}_n(\tau) = {\cal I}_n(\tau,M_0\ensuremath{,\ldots,} M_{n-1}) \ \] The dependence of ${\cal I}_n(\tau)$ on $M_0 \ensuremath{,\ldots,} M_{n-1}$ will be kept implicit. We let \[ {\cal P}_n = {\cal P}_n(M_0\ensuremath{,\ldots,} M_n) \ := \ \{ \tau \in \Lambda \mid
({\cal X'})_n \textrm{ and } ({\cal Y'})_n \textrm{ hold} \} \ . \] Here $({\cal X'})_n$ and $({\cal Y'})_n$ are understood as conditions on the sets ${\cal I}_j(\tau)$ ($j\in[0,n]$) for fixed $\omega\in\ensuremath{\mathbb{T}^{1}}$. Furthermore, we assume that the following conditions are satisfied. \begin{equation}
\label{e.P} \tag{$\cal P$} \left\{ \quad
\begin{array}{l} \textrm{Suppose } M_0\ensuremath{,\ldots,} M_n \textrm{ with } M_i \in [N_i,2K_iN_i)
\ \forall i=[0,n] \textrm{ are fixed.} \\
\textrm{Then for all } \tau \in {\cal P}_n(M_0\ensuremath{,\ldots,} M_n) \textrm{ and }
j\in[0,n+1] \textrm{ there holds } \\ \ \\
\begin{array}{cl}
({\cal P}1) & {\cal I}_j(\tau) \textrm{ is open and consists of exactly } {\cal N}
\textrm{ connected components }\\ & I^1_j(\tau) \ensuremath{,\ldots,} I^{\cal N}_j(\tau).
\textrm{ If } j \leq n \textrm{ then } I^\iota_{j+1}(\tau) \ensuremath{\subseteq} I^\iota_j(\tau) \
\forall \iota \in [1,{\cal N}]. \\ \ \\
({\cal P}2) & \textrm{The set } {\cal P}_j(M_0\ensuremath{,\ldots,} M_j)
\textrm{ is open and all } I^\iota_j\textrm{ with }
j\in [1,{\cal N}] \\ &
\textrm{ are differentiable w.r.t.\ } \tau \textrm{ on } {\cal P}_j(M_0\ensuremath{,\ldots,} M_j);
\\ \ \\ ({\cal P}3) & |I^\iota_j(\tau)| \leq \ensuremath{\epsilon}_j \quad \forall
\iota \in [1,{\cal N}] \ ; \\ \ \\
({\cal P}4) & |\partial_\tau I^\iota_j(\tau)| \leq 2L/s \quad
\forall \iota \in [1,{\cal N}] \ ;\\ \ \\
({\cal P}5) & |D_\tau(I^\iota_j(\tau),I^\kappa_j(\tau))| \geq \eta/2 \quad
\forall \iota \neq \kappa \in [1,{\cal N}] \ ;
\end{array} \end{array} \right. \end{equation} \begin{rem} \label{b.parameter-exclusion-assumption} Note that if
(\ref{eq:Cinvariance})--(\ref{e.d}) are satisfied and $\alpha_\infty$ is
sufficiently large, then the fact that (\ref{e.P}) holds for the sets ${\cal
I}_j(\tau)$ defined by (\ref{eq:defIn}) is exactly the content of the
previous sections. $({\cal P}1)$ follows by induction from
Proposition~\ref{p.In-size}. Lemma~\ref{l.lambda-dependence} implies that the
connected components of ${\cal I}_j$ are differentiable with respect to
$\tau$, which in turn yields the openness of the conditions $({\cal X}')_j$ and
$({\cal Y'})_j$ and hence of ${\cal P}_j(M_0\ensuremath{,\ldots,} M_j)$, such that $({\cal P}2)$
holds as well. $({\cal P}3)$ follows from Proposition~\ref{p.In-size} and
finally $({\cal P}4)$ and $({\cal P}5)$ are again a consequence of
Lemma~\ref{l.lambda-dependence}. \end{rem}
In each step of the parameter exclusion we will have to ensure that the set ${\cal P}_n \ensuremath{\setminus} {\cal P}_{n+1}$ of excluded parameters is small. In other words, we have to show that for most $\tau \in {\cal P}_n$ the conditions $({\cal X}')_{n+1}$ and $({\cal Y}')_{n+1}$ are satisfied for a suitable $M_{n+1}$ (that we allow to depend on $\tau$). This is greatly simplified by the fact that $({\cal Y})_{n+1}$ `comes for free'. \begin{lem}[Lemma 3.16 in \cite{jaeger:2009a}]
\label{l.Yn+1-Mexists} Suppose that $M_0 \ensuremath{,\ldots,} M_n$ with $M_j \in [N_j,2N_j) \
\forall j\in[0,n]$ are fixed. Further, assume that (\ref{e.N}), $({\cal P}1)$
and $({\cal P}3)$ hold and
\begin{equation}
\label{e.K}\tag{${\cal K}$}
\sum_{j=0}^\infty \frac{1}{K_j} \ < \ \frac{1}{6{\cal N}^2} \ .
\end{equation}
Then for all $\tau \in {\cal P}_n(M_0\ensuremath{,\ldots,} M_n)$ there exists an integer $M(\tau)
\in [N_{n+1},2N_{n+1})$ such that
\begin{equation}
\label{e.strong-Yn+1}
d\left(({\cal I}_{n+1}(\tau)-(M(\tau)-1)\omega)\cup({\cal I}_{n+1}+(M(\tau)+1)\omega)\ ,
\ {\cal Y}_n\right) \ > \ 3\ensuremath{\epsilon}_n \ .
\end{equation} \end{lem} We remark that the version of this lemma in \cite{jaeger:2009a} actually contains some additional assumptions, but these are not used in the proof. (For the sake of brevity, the standard hypothesis were assumed throughout the respective section in \cite{jaeger:2009a}.) In order to obtain an estimate on the set of $\tau\in {\cal P}_n$ that do not satisfy $({\cal X})_{n+1}$, the following lemma is needed. \begin{lem}\label{l.pex-basic}
Suppose $\Lambda \ensuremath{\subseteq} [0,1]$ is an interval and ${\cal I} : \Lambda \to {\cal
S}(\ensuremath{\mathbb{T}^{1}})$ is such that for all $\tau\in\Lambda$ the set ${\cal
I}(\tau) \ensuremath{\subseteq} \ensuremath{\mathbb{T}^{1}}$ consists of ${\cal N}$ connected components
$I^1(\tau) \ensuremath{,\ldots,} I^{\cal N}(\tau)$ of length $|I^\iota(\tau)| \leq
\delta$ which satisfy
\begin{equation} \label{e.pex-basic1}
|D_\tau(I^\iota(\tau),I^\kappa(\tau))| \geq \eta/2 \quad
\forall \iota \neq \kappa \in [1,{\cal N}] \ .
\end{equation}
Further, assume that
\begin{equation} \label{e.pex-diophantine}
d(I^\iota(\tau),I^\iota(\tau)+n\omega) \ > \ \ensuremath{\epsilon} \quad \forall
\tau \in \Lambda,\ n \in [1,M],\ \iota \in[1,{\cal N}] \ .
\end{equation}
Then the set
\[
\Upsilon \ := \ \left\{ \tau \in \Lambda \ \left| \ d\left({\cal
I}(\tau),\bigcup_{j=1}^M {\cal I}(\tau)+n\omega\right) \leq \ensuremath{\epsilon}
\right.\right\}
\]
has measure $\leq 8{\cal N}^2M\frac{\delta+\ensuremath{\epsilon}}{\eta}$ and consists of at
most $2{\cal N}^2M-1$ connected components. \end{lem} \textit{Proof. } Fix $\iota\neq \kappa \in [1,{\cal N}]$ and $n\in[1,M]$. As $I^{\iota}(\tau)$ and $I^\kappa(\tau)$ are disjoint for all $\tau \in \Lambda$ and due to (\ref{e.pex-basic1}), the set of $\tau$ with $d\left(I^\iota(\tau),(I^\kappa(\tau)+n\omega)\right)\leq \ensuremath{\epsilon}$ consists of at most two intervals of length $\leq 4(\delta+\ensuremath{\epsilon})/\eta$. Summing up over all $\iota,\kappa$ and $n$ yields the statement. \qed
For any $n\geq 1$, let \begin{eqnarray}
v_n & = & \ 32{\cal N}^2 K_{n+1}N_{n+1}\cdot \frac{L}{s\ensuremath{\epsilon}_{n-1}} \quad \textrm{and}\\
u_n & = & 1280{\cal N}^2 K_{n+1}N_{n+1}\cdot \frac{L }{s\eta} \cdot \frac{\ensuremath{\epsilon}_{n}}{\ensuremath{\epsilon}_{n-1}} \ . \end{eqnarray} Further, let $v_0 = 8{\cal N}^2K_0N_0$ and $u_0 = 320{\cal
N}^2K_0N_0\ensuremath{\epsilon}_0/\eta$. \begin{lem} \label{l.pex-component}
Suppose $\omega \in {\cal D}(\sigma,\nu)$, (\ref{e.N}), (\ref{e.K}) and
(\ref{e.P}) hold and
\begin{equation}
\label{e.N2} \tag{${\cal N}2$}
10\ensuremath{\epsilon}_n \ < \ \sigma\cdot(4K_nN_n)^{-\nu} \quad \forall n\in\ensuremath{\mathbb{N}}_0 \ .
\end{equation}
Further, assume that $M_0 \ensuremath{,\ldots,} M_n$ with $M_j \in [N_j,2N_j) \ \forall
j\in[0,n]$ are fixed and $\Gamma \ensuremath{\subseteq} {\cal P}_n(M_0\ensuremath{,\ldots,} M_n)$ is an
interval. Then for some $r\leq v_{n+1}$ there exist disjoint intervals
$\Gamma_1 \ensuremath{,\ldots,} \Gamma_r \ensuremath{\subseteq} \Gamma$ and numbers $M^k \in [N_{n+1},2N_{n+1}),\
k\in[1,r]$ such that
\begin{eqnarray}
\Gamma^k & \ensuremath{\subseteq} & {\cal P}_{n+1}(M_0\ensuremath{,\ldots,} M_n,M^k) \quad \textrm{and} \\
\quad \sum_{k=1}^r \ensuremath{\mathrm{Leb}}(\Gamma^k) & \geq & \ensuremath{\mathrm{Leb}}(\Gamma) - u_{n+1} \ .
\end{eqnarray} \end{lem} \textit{Proof. } Divide $\Gamma$ into at most $\frac{4L}{s\ensuremath{\epsilon}_n}$ intervals $\Omega_i$ of length of length $\leq \frac{s \ensuremath{\epsilon}_n}{2L}$. Denote the midpoint of $\Omega_i$ by $\tau_i$ and choose $\tilde M^i=M(\tau_i)$ according to Lemma~\ref{l.Yn+1-Mexists} such that (\ref{e.strong-Yn+1}) holds for $\tau_i$. Then due to $({\cal P}4)$ we obtain that $({\cal Y})_{n+1}$ holds for all $\tau \in \Omega^i$. Application of Lemma~\ref{l.pex-basic} with $M=2K_{n+1}\tilde M^i < 4K_{n+1}N_{n+1}$, $\delta = \ensuremath{\epsilon}_{n+1}$ and $\ensuremath{\epsilon}=9\ensuremath{\epsilon}_{n+1}$ yields the existence of a set $\tilde \Omega_i \ensuremath{\subseteq} {\cal
P}_{n+1}(M_0\ensuremath{,\ldots,} M_n,\tilde M^i)$ of measure $\geq \ensuremath{\mathrm{Leb}}(\Omega_i)-320{\cal
N}^2K_{n+1}N_{n+1}\cdot \frac{\ensuremath{\epsilon}_{n+1}}{\eta}$ and with at most $8{\cal
N}^2K_{n+1}N_{n+1}$ connected components. Note that the fact that (\ref{e.pex-diophantine}) holds follows from the Diophantine condition on $\omega$ together with (\ref{e.N2}) and $({\cal P}3)$. Relabelling the connected components of the sets $\tilde\Omega_i$ and summing up over all $i$ yields the statement. \qed
Let $V_{-1}=1$ and $V_n \ = \ \prod_{j=0}^n v_j$ for $n\geq 0$. \begin{prop}\label{p.pex}
Suppose $\omega\in{\cal D}(\sigma,\nu)$, $({\cal N}1$-$2)$, (\ref{e.K}) and
(\ref{e.P}) hold and
\begin{equation}
\label{e.gamma}
m \ := \ \ensuremath{\mathrm{Leb}}(\Lambda) - \sum_{n=0}^\infty V_{n-1} u_n .
\end{equation}
\alphlist
\item
Then there exists a set $\Lambda_\infty \ensuremath{\subseteq} \Lambda$ of measure $\geq m$
with the following property: For all $\tau \in \Lambda_\infty$ there exists
a sequence $\nofolge{M_n(\tau)}$ with $M_n(\tau)\in[N_n,2N_n) \ \forall
n\in\ensuremath{\mathbb{N}}$ such that $\tau \in \bigcap_{n\in\ensuremath{\mathbb{N}}} {\cal P}_n(M_0(\tau)\ensuremath{,\ldots,}
M_n(\tau))$. \item If there exists $M_0\in[N_0,2N_0)$ with ${\cal
P}_0(N_0)=\Lambda$, then $\Lambda_\infty$ can be chosen with measure $\geq m+u_0$. \end{list} \end{prop} \textit{Proof. } We construct a nested sequence of sets $\Lambda_n$ with the following properties: \romanlist \item $\Lambda_n$ consists of $\rho_n \leq V_n$ disjoint intervals $\Lambda^1_n
\ensuremath{,\ldots,} \Lambda^{\rho_n}_n$; \item $\ensuremath{\mathrm{Leb}}(\Lambda_n) \geq \ensuremath{\mathrm{Leb}}(\Lambda)-\sum_{i=0}^n V_{n-1}u_n$; \item For each $i\in[1,\rho_n]$ there exist numbers $M_0^{n,i} \ensuremath{,\ldots,} M_n^{n,i}$
such that $\Lambda_n^i \ensuremath{\subseteq} {\cal P}_n(M_0^{n,i}\ensuremath{,\ldots,} M_n^{n,i})$; \item For each $k\leq n$ and each $i\in[1,\rho_n]$ there exists a unique $\kappa
\in[1,\rho_k]$ such that $\Lambda^i_n \ensuremath{\subseteq} \Lambda^\kappa_k$ and
$M^{n,i}_j=M^{k,\kappa}_j \ \forall j\in[0,k]$. \end{list}
The set $\Lambda_\infty = \bigcap_{n\in\ensuremath{\mathbb{N}}_0} \Lambda_n$ then clearly has the
properties required in (a), and for (b) it suffices to note that if ${\cal
P}(M_0)=\Lambda$, then obviously a measure of $u_0$ is gained in the first
step of the construction.
\noindent For $n=0$ we choose $M_0 \in [N_0,2N_0)$ arbitrarily and let $\Lambda_0={\cal
P}_0(M_0)$. The fact that it has the required properties follows directly from Lemma~\ref{l.pex-basic}. Now suppose that $\Lambda_0 \ensuremath{,\ldots,} \Lambda_n$ with the above properties exist. Then for each $i\in[1,\rho_n]$ we can apply Lemma~\ref{l.pex-component} and obtain a union of at most $v_{n+1}$ intervals with overall measure $\geq \ensuremath{\mathrm{Leb}}(\Lambda_n^i) - u_{n+1}$. Doing this for the at most $V_n$ components of $\Lambda_n$ yields the required set $\Lambda_{n+1}$.
\qed
\subsection{Minimality and the uniqueness of SNA.}
As a first step in the proof of Theorem~\ref{t.firstversion} below, we will define the set $\Lambda_\infty$ and show that for all $\tau \in \Lambda_\infty$ the slow-recurrence conditions \ref{e.X'n} and \ref{e.Y'n} hold. Once this is accomplished, the parameter dependence on $\tau$ does not play a role anymore and we can consider the map $f_\tau$ as being fixed. The existence of an SNA and an SNR then follows from Proposition~\ref{p.sna-existence}, and it remains to prove the uniqueness and one-valuedness of the invariant graphs and the minimality of $f$. However, this second step has already been carried out in \cite{jaeger:2009a} and the proof given there literally remains true in our setting. Instead of repeating it here, we just give a precise formulation of the formal statement that can be deduced from \cite{jaeger:2009a}. \begin{prop} \label{p.minimality}
Suppose $f_\tau$ satisfies (\ref{eq:Cinvariance})--(\ref{eq:s}), \ref{e.Xn} and
\ref{e.Yn} hold for all $n\in\ensuremath{\mathbb{N}}$ and \begin{equation} \label{e.minimality-condition} \ensuremath{\mathrm{Leb}}\left(\bigcup_{n=0}^{\infty} \bigcup_{k=-M_n-1}^{M_n+1} {\cal
I}_n-k\omega \right) \ < \ \frac{1}{4+4p^2}\ . \end{equation} Then $f_\tau$ has a unique SNA and SNR which are both one-valued. Further the dynamics are minimal. \end{prop} \textit{Proof. } See Sections 3.6 and 3.7 in \cite{jaeger:2009a}. \qed
\subsection{Proof of Theorem~\ref{t.firstversion}.}
Fix an integer $t\geq 4$ such that $2^{-t+2}/{\cal N}^2 \leq \log((p^2+2)/(p^2+1))$ and let $K_n = 2^{n+t}{\cal N}^2$. Then it is easy to check that (\ref{e.K}) is verified and furthermore $\beta$ in (\ref{e.alpha-infty}) is larger than $(p^2+1)/(p^2+2)$. This in turn implies that $\alpha_\infty$ defined by (\ref{e.alpha-infty}) is larger than $\alpha^{1/p}$. Suppose that (\ref{eq:Cinvariance})--(\ref{e.d}) hold and $\alpha^{1/p} > \max\{\alpha_1,\alpha_2\}$, where $\alpha_1$ and $\alpha_2$ are the constants from Proposition~\ref{p.In-size} and Lemma~\ref{l.lambda-dependence}. Then as mentioned in Remark~\ref{b.parameter-exclusion-assumption}, the critical regions ${\cal I}_n$ defined dynamically by (\ref{eq:defIn}) satisfy (\ref{e.P}) when viewed as mappings ${\cal I}_n : \Lambda \times \ensuremath{\mathbb{N}}^n \to {\cal S}(\ensuremath{\mathbb{T}^{1}})$. In order to determine the set $\Lambda_\infty$ by applying Proposition~\ref{p.pex}, it only remains to show that by an appropriate choice of the sequences $\nofolge{\ensuremath{\epsilon}_n}$ and $\nofolge{N_n}$ we can ensure that (\ref{e.N}) and (\ref{e.N2}) hold and that the sum $\sum_{n=0}^\infty V_{n-1} u_n$ in (\ref{e.gamma}) is smaller than $\delta$.
In order to do so, we let $N_0 = 3$ and $N_{n+1} = \alpha^{N_n/qp}$, where $q=\max\{8,2\nu\}$. Further, we let $\ensuremath{\epsilon}_0=\sup_{\iota\in[1,{\cal
N}],\tau\in\Lambda} |I^\iota_0(\tau)|$ and $\ensuremath{\epsilon}_{n+1} = \frac{2}{s} \cdot \alpha^{-N_n/p}$. Note that these sequences grow, respectively decay, super-exponentially. Therefore it is easy to see that with this choice (\ref{e.N}) and (\ref{e.N2}) are satisfied for sufficiently large $\alpha$ and sufficiently small $\ensuremath{\epsilon}_0$. In the following estimates we assume that $\alpha$ is chosen sufficiently large and indicate the steps in which this fact is used by placing $(\alpha)$ over the respective inequality signs. For any $n\in\ensuremath{\mathbb{N}}_0$ we have \begin{eqnarray*}
v_{n+1}& = & 32{\cal N}^2K_{n+1}N_{n+1} \cdot \frac{L}{s\ensuremath{\epsilon}_n} \\
& = & 16\cdot 2^{n+t}{\cal N}^2 L \cdot \alpha^{N_{n}/qp+N_{n-1}/p}
\ \stackrel{(\alpha)}{\leq} \ \alpha^{N_n/4} \quad \textrm{and} \\
u_{n+1} & = & 1280{\cal N}^2K_{n+1}N_{n+1}\cdot\frac{L}{s\eta}\cdot \frac{\ensuremath{\epsilon}_{n+1}}{\ensuremath{\epsilon}_n} \\
& = & 1280\cdot 2^{n+t}{\cal N}^2\cdot\frac{L}{s\eta}\cdot \alpha^{N_n/qp+N_{n-1}/p-N_n/p}
\ \stackrel{(\alpha)}{\leq} \ \alpha^{-3N_n/4p} \ . \end{eqnarray*} By induction, we obtain that $V_n = \prod_{j=0}^n v_j \stackrel{(\alpha)}{\leq} \alpha^{N_n/4}$ (note that $v_0 = 8{\cal N}^2K_0N_0 \stackrel{(\alpha)}{\leq} \alpha^{N_0/4}$). Altogether, this yields that $m$ in Proposition~\ref{p.pex} satisfies $m \geq \ensuremath{\mathrm{Leb}}(\Lambda) - u_0 - \sum_{n=0}^{\infty} \alpha^{-N_n/2p}$. As $u_0 = 320{\cal N}^2K_0N_0\ensuremath{\epsilon}_0/\eta$, this lower bound goes to $\ensuremath{\mathrm{Leb}}(\Lambda)$ as $\ensuremath{\epsilon}_0 \to 0$ and $\alpha \to \infty$.
Hence, Proposition~\ref{p.pex} yields the existence of a set $\Lambda_\infty\ensuremath{\subseteq} \Lambda$ of measure $\ensuremath{\mathrm{Leb}}(\Lambda_\infty) > \ensuremath{\mathrm{Leb}}(\Lambda) - \delta$ such that for all $\tau \in \Lambda_\infty$ the conditions \ref{e.X'n} and \ref{e.Y'n} are satisfied. Fix $\tau\in\Lambda_\infty$. As $\alpha_\infty > 1$ and ${\cal
I}_n(\tau) \neq \emptyset \ \forall n\in\ensuremath{\mathbb{N}}$ due to $({\cal P}1)$, Proposition \ref{p.sna-existence} yields the existence of an SNA and an SNR. Further, we have \begin{eqnarray*}
\ensuremath{\mathrm{Leb}}\left(\bigcup_{n=0}^{\infty} \bigcup_{k=-M_n-1}^{M_n+1} {\cal
I}_n-k\omega \right) & \leq & \sum_{n=0}^\infty 4N_n\ensuremath{\epsilon}_n \ \leq \
\ensuremath{\epsilon}_0 N_0 +\sum_{n=1}^\infty \alpha^{-N_n/2p} \ . \end{eqnarray*} Again, the right side goes to $\ensuremath{\mathrm{Leb}}(\Lambda)$ as $\ensuremath{\epsilon}_0 \to 0$ and $\alpha \to \infty$, such that (\ref{e.minimality-condition}) will be satisfied for small $\ensuremath{\epsilon}_0$ and large $\alpha$. Consequently, we can apply Proposition~\ref{p.minimality} to obtain
\ref{e.star} for all $\tau\in\Lambda_\infty$.
Finally, suppose that for some $\tau_0$ the symmetry condition (\ref{e.symmetry-addendum}) holds. In this situation it follows by induction that the critical regions ${\cal I}_n=I^1_n\cup I^2_n$ defined recursively by (\ref{eq:defIn}) satisfy \begin{equation} \label{e.interval-symmetry} I_n^2\ =\ I^1_n+\halb \qquad \forall n\in\ensuremath{\mathbb{N}} \ . \end{equation} We show that in this case, for all sufficiently large $\alpha$, there exists a sequence of integers $M_n\in[N_n,2N_n)$ such that the \ref{e.X'n} and \ref{e.Y'n} hold for all $n\in\ensuremath{\mathbb{N}}$. As before, Proposition~\ref{p.sna-existence} and Proposition then imply \ref{e.star}, such that $\tau\in\Lambda_\infty$.
$({\cal X}')_0$ with $M_0=N_0=3$ holds for small $\ensuremath{\epsilon}_0$ due to the Diophantine condition and $({\cal Y}')_0$ is void. Suppose $M_0\ensuremath{,\ldots,} M_n$ are chosen such that \ref{e.X'n} and \ref{e.Y'n} hold, such that $\tau\in{\cal P}_n(M_0\ensuremath{,\ldots,} M_n)$. Then due to Lemma~\ref{l.Yn+1-Mexists} there exists $M_{n+1}\in[N_{n+1},2N_{n+1})$ such that $({\cal Y}')_{n+1}$ holds. Furthermore
$|I^\iota_{n+1}(\tau)| \leq \ensuremath{\epsilon}_{n+1}$ for $\iota=1,2$ due to $({\cal
P}3)$. Now suppose that $({\cal X}')_{n+1}$ is not satisfied, such that
$I^\iota_{n+1}(\tau)\cap (I^\kappa_{n+1}(\tau) + n\omega) \neq \emptyset$ for some $\iota,\kappa\in\{1,2\}$ and $|n|\leq 2K_{n+1}M_{n+1}$. Due to (\ref{e.interval-symmetry}) this implies $d(2n\omega,0) \leq 2\ensuremath{\epsilon}_{n+1}$, which contradicts the Diophantine condition~(\ref{eq:Diophantine}) when $\alpha$ is large. Consequently, when $\ensuremath{\epsilon}_0$ is sufficiently small and $\alpha$ is sufficiently large conditions \ref{e.X'n} and \ref{e.Y'n} hold for all $n\in\ensuremath{\mathbb{N}}$ and we can apply Proposition \ref{p.minimality} to deduce that $f_{\tau_0}$ satisfies \ref{e.star}. Hence, $\tau_0$ can be included in $\Lambda_\infty$. \qed
\section{The refined version of the twist parameter
exclusion} \label{RefinedParameterExclusion}
The aim of this section is to prove Theorem~\ref{t.mr-quantitative2}. To that end, we have to improve some of the estimates from the previous section by taking into account the stronger assumptions on $\partial_\theta f_\theta$ in (\ref{e.A8'}) and (\ref{eq:refinedbounddth}). As before, we can rely to some extent on the respective results from \cite{jaeger:2009a}.
\subsection{Estimates on the critical sets and critical
regions.}\label{PreviousRefinedEstimates}
Parts (i) and (ii) of Proposition~\ref{p.In-size} are replaced by the following statements, which can again be taken from \cite{jaeger:2009a}.
\begin{prop}[Proposition 4.3 in \cite{jaeger:2009a}]
\label{p.inductivelemma}
Suppose (\ref{eq:Cinvariance})--(\ref{eq:s}), (\ref{eq:refinedbounddth}) and
(\ref{e.S<Ad})--(\ref{e.s'<A}) hold, \ref{e.Xn} and \ref{e.Yn} are
satisfied, $\alpha_\infty > 1$ and $M_0\geq d^{1/4}$. Then there exist
a constant $d_1=d_1(\alpha_\infty)>0$ such that
the following holds: If $d>d_1$ then \romanlist \item ${\cal I}_n(\tau)$ and ${\cal I}_{n+1}(\tau)$ consist of exactly
${\cal N}$ connected components and each component of ${\cal I}_n(\tau)$
contains exactly one component of ${\cal I}_{n+1}(\tau)$. \item For all connected components of $I^\iota_{n+1}(\tau)$ of ${\cal
I}_{n+1}(\tau)$ there holds $|I^\iota_{n+1}(\tau)| \leq 2\alpha_\infty^{-M_n}/s$. \end{list} \end{prop} In contrast to this, the required version of Proposition~\ref{p.In-size}(iii) has to take into account the fact that due to (\ref{e.A7'}) only two critical regions exist. This assumption is not considered in \cite{jaeger:2009a}, such that we cannot use the respective estimates there. Instead, we use the following statement. \begin{lem} \label{l.refined-slope-estimate} Suppose
(\ref{eq:Cinvariance})--(\ref{eq:crossing}), (\ref{e.A7'})--(\ref{eq:refinedbounddth}) and
(\ref{e.S<Ad})--(\ref{e.s'<A}) hold, \ref{e.Xn} and \ref{e.Yn} are
satisfied, $\alpha_\infty > 1$ and $M_0\geq d^{1/4}$. Then there exist
a constant $d_2=d_2(\alpha_\infty)>0$ such that
the following holds. If $d>d_2$ then
\begin{eqnarray}
\label{e.theta-derivative1}
s/2 & \leq & \partial_\theta(\varphi_{1,n}^\pm-\psi_{1,n}^\mp) \ \leq \ 2S
\qquad \ \ \textrm{ on } I^1_n(\tau)+\omega \ \textrm{ and } \\
\label{e.theta-derivative2}
-2S & \leq & \partial_\theta(\varphi_{2,n}^\pm-\psi_{2,n}^\mp) \ \leq \ -s/2
\qquad \textrm{ on } I^2_n(\tau)+\omega \ .
\end{eqnarray} \end{lem} \textit{Proof. } As in the proof of Lemma~\ref{l.lambda-dependence} and with the notation introduced there, we have \begin{equation}
\partial_\theta \varphi^\pm_{\iota,n}(\theta,\tau) \ = \ \partial_\theta f_{\tau,\theta_{M_n-1}}(x_{M_{n}-1})
+ \underbrace{ \sum_{k=1}^{M_n-1} \partial_x f^{M_n-k}_{\tau,\theta_k}(x_k) \cdot
\partial_\theta f_{\tau,\theta_{k-1}}(x_{k-1})}_{=:q^\iota_1} \ . \end{equation} As $(\theta_{M_n},x_{M_n})=(\theta,\varphi^\pm_{\iota,n}(\theta,\tau))\in{\cal I}_0+\omega$, we can use (\ref{eq:refinedbounddth}) together with (\ref{e.finite-lyaps}) and (\ref{eq:bounddth}) to obtain \begin{equation}
|q_1^\iota| \ \leq \ \frac{s'+\alpha_\infty^{-M_0}S}{\alpha_\infty-1} \ . \end{equation} For $q^\iota_2:=\partial_\theta\psi^\mp_{\iota,n}(\theta,\tau)$ we obtain in a similar way \begin{equation}
|q^\iota_2| \ \leq \ \frac{s'+\alpha_\infty^{-M_0}S}{\alpha_\infty-1} \ . \end{equation}
Since $M_0\geq d^{1/4}$ we obtain that $|q^\iota_1|$ and $|q^\iota_2|$ are small compared to $s$ and $S$ if $d$ is sufficiently large and $\frac{s'}{s}$ is sufficiently small. As $\theta_{M_n-1}\in {\cal I}_0$, the statement follows from (\ref{eq:bounddth}) and (\ref{e.A8'}). \qed
In order to control the parameter dependence of the critical sets we replace Lemma~\ref{l.lambda-dependence} by
\begin{lem} \label{l.refined-tau-dependence} Suppose
(\ref{eq:Cinvariance})--(\ref{eq:crossing}),
(\ref{e.A7'})--(\ref{eq:refinedbounddth}) and
(\ref{e.S<Ad})--(\ref{e.eps<1/Ad}) hold and \ref{e.Xn} and \ref{e.Yn} are
satisfied. Further, let $d_2$ be chosen as in
Lemma~\ref{l.refined-slope-estimate} and assume that $d>d_2$. Then
\begin{eqnarray}
\label{e.refined-tau-dependence}
\partial_\tau I_{n+1}^1(\tau) \ \leq \ \frac{-\gamma}{2S} & \ , \ &
\partial_\tau I^2_{n+1}(\tau) \ \geq \ \frac{\gamma}{2S} \\ \textrm{ and }
\quad |\partial_\tau I^\iota_{n+1}(\tau)| & \leq &
\frac{4L}{s(1-1/\alpha_\infty)} \quad (\iota=1,2) \ .
\label{e.ref-tau-dep2}
\end{eqnarray}
\end{lem}
\textit{Proof. } Similar to the proof of Lemma~\ref{l.lambda-dependence} we
let $I^\iota_{n+1}(\tau) = (a^\iota(\tau),b^\iota(\tau))$ and show that
\begin{equation}
\label{e.ref-tau-dep3}
-4L/s(1-1/\alpha_\infty) \ \leq \ \partial_\tau a^1(\tau) \ \leq -\gamma/2S \ .
\end{equation}
The required estimates on $\partial_\tau b^1(\tau),\ \partial_\tau
a^2(\tau)$ and $\partial_\tau b^2(\tau)$ can then be treated in the
same way. Note that (\ref{e.theta-derivative1}) in
Lemma~\ref{l.refined-slope-estimate} implies that $f^{M_n}(A^1_n)$
crosses $f^{-M_n}(B^1_n)$ upwards.
We define $r^\iota_1$ and $r^\iota_2$ as in (\ref{e.rqiota}) and
(\ref{e.psicontrol}). From (\ref{e.rqiota}) and (\ref{e.A8'}) we obtain that \begin{equation}
\partial_\tau\varphi^+_{\iota,n}(\theta,\tau) \ = \ \partial_\tau f_{\tau,\theta_{M_n-1}}(x_{M_n-1})+r^\iota_1 \ \geq \ \partial_\tau f_{\tau,\theta_{M_n-1}}(x_{M_n-1}) \ \geq \ \gamma \ . \end{equation} Note that $r^1_1\geq 0$ since all terms in the sum in (\ref{e.rqiota}) are non-negative. Similarly $r^1_2\leq 0$, such that \begin{equation}
\partial_\tau(\varphi^+_{1,n}-\psi^-_{1,n})(a^1(\tau)+\omega,\tau) \ \geq \
\gamma \ . \end{equation} Using (\ref{e.IFT-Formula}) and Lemma~\ref{l.refined-slope-estimate} gives $\partial_\tau a^1(\tau) \leq -\gamma/2S$ as required.
For the upper bound on $|\partial_\tau a^1(\tau)|$, note that using (\ref{e.A8'}) and Lemma~\ref{p.lyaps} to estimate the sums defining $r^1_1$ and $r^2_1$ in (\ref{e.rqiota}) and (\ref{e.psicontrol}) yields \begin{equation}
\partial_\tau(\varphi^+_{1,n}-\psi^-_{1,n})(a^1(\tau)+\omega,\tau) \
\leq \ 2L/(1-1/\alpha_\infty) \ . \end{equation}
Together with Lemma~\ref{l.refined-slope-estimate}, this provides the required bound $|\partial_\tau a^1(\tau)| \leq 4L/s(1-1/\alpha_\infty)$. \qed
\begin{rem} \label{r.forcing-structure} The proof of
Lemma~\ref{l.refined-tau-dependence} demonstrates well the restrictions which
the need for controlling the relative speed of the critical intervals inflicts
on the geormetry of the forcing. Considering the case of only two critical
intervals with opposite sign of the slope of $\partial_\theta f_\theta$, as we
do here, is not the only possibility to achive this. For instance, one could
treat a multitude of critical intervals, as in Theorem~\ref{t.firstversion},
by requiring that the twist $\partial_\tau f_\theta(x)$ almost vanishes
outside of the critical regions (similar to the use of
(\ref{eq:refinedbounddth}) in the proof of
Lemma~\ref{l.refined-slope-estimate}). However, we see no way of treating more
than two critical intervals if the twist is uniform as in
(\ref{e.arnold}). The reason is that the lack of strong hyperbolicity does not
allow to control the influence of the twist far from the critical region
${\cal I}_0(\tau)$ on the relative speed of the critical intervals. This could
result in critical intervals moving at the same speed, in which case parameter
exclusion would not work anymore. \end{rem}
\subsection{Proof of Theorem~\ref{t.mr-quantitative2}.} \label{RefinedProof}We choose $t$ and $K_n=2^{n+t}/{\cal N}^2$ as in the proof of Theorem~\ref{t.firstversion}, such that $\alpha_\infty\geq \alpha^{1/p} >1$. Further, we suppose that $d$ satisfies $d\geq \max\{d_1,d_2\}$, where $d_1$ and $d_2$ are the constants from Proposition~\ref{p.inductivelemma} and Lemma~\ref{l.refined-slope-estimate}. Then the mapping ${\cal I}_n:\Lambda\times \ensuremath{\mathbb{N}} \to {\cal S}(\ensuremath{\mathbb{T}^{1}})$ satisfies (\ref{e.P}), with $L$ replaced by
$4L/(1-1/\alpha_\infty)$ due to the weaker estimate on $|\partial_\tau I^\iota_n(\tau)|$ in (\ref{e.refined-tau-dependence}) (compare Remark~\ref{b.parameter-exclusion-assumption}).
Fix $k>2\nu$ and let $N_0$ be the first integer $\geq d^{1/k}$. As before, we let $q=\max\{8,2\nu\}$ and define the sequences $N_n$ and $\ensuremath{\epsilon}_n$ recursively by $N_{n+1}=\alpha^{N_n/qp}$ and $\ensuremath{\epsilon}_{n+1}=\frac{2}{s}\cdot \alpha^{-N_n/p}$. Then using the dependencies (\ref{e.S<Ad})--(\ref{e.s'<A}) it is easy to check that all estimates on the quantities $v_{n},\ u_n$ and $V_n$ made in the proof of Theorem~\ref{t.firstversion} remain valid if the largeness assumption on $\alpha$ used there is replaced by a largeness condition on $d$ that depends on the constants $\alpha,p,L,\gamma,A$ and $\delta$. Consequently, the constant $m$ in Proposition~\ref{p.pex} satisfies \begin{equation}
m\ \geq \ \ensuremath{\mathrm{Leb}}(\Lambda) - u_0 - \sum_{n=0}^\infty \alpha^{-N_n/2p} \ . \end{equation}
Furthermore, since $I^1_0(\tau)=I^2_0(\tau)+\halb$, $|I_0^\iota(\tau)|\leq \ensuremath{\epsilon}_0\leq A/\sqrt{d}$ by (\ref{e.eps<1/Ad}) and $N_0 \leq d^{1/k}+1$, the Diophantine condition implies that for sufficiently large $d$ condition $({\cal X}')_0$ holds for all $\tau\in\Lambda$ (that is, ${\cal I}_0(\tau)$ is disjoint from its first $2K_0N_0$ iterates). This means that ${\cal P}(N_0)=\Lambda$ and we can therefore apply Proposition~\ref{p.pex}(b), which yields a set $\Lambda_\infty$ of measure \[ \ensuremath{\mathrm{Leb}}(\Lambda_\infty) \ \geq \ \ensuremath{\mathrm{Leb}}(\Lambda) - \sum_{n=0}^\infty \alpha^{-N_n/2p} \ \] on which the slow-recurrence conditions \ref{e.X'n} and \ref{e.Y'n} hold for all $n\in\ensuremath{\mathbb{N}}$. Consequently, for all $\tau\in\Lambda_\infty$ the existence of an SNA and an SNR follows from Proposition~\ref{p.sna-existence}. Further, we have \begin{eqnarray*}
\ensuremath{\mathrm{Leb}}\left(\bigcup_{n=0}^{\infty} \bigcup_{k=-M_n-1}^{M_n+1} {\cal I}_n-k\omega
\right) & \leq & \sum_{n=0}^\infty 2N_n\ensuremath{\epsilon}_n \ \leq \ \ensuremath{\epsilon}_0 N_0
+\frac{2}{s}\cdot\sum_{n=1}^\infty \alpha^{-N_n/2p} \ . \end{eqnarray*} Due to (\ref{e.eps<1/Ad}) and the choice of $N_0$ in $[d^{1/k},d^{1/k}+1]$ the sum on the right goes to zero as $d\to\infty$, and we can apply Proposition~\ref{p.minimality} to obtain \ref{e.star}. Finally, the symmetry statement is shown in the same way as in the proof of Theorem~\ref{t.firstversion}. \qed
\subsection{Proof of Corollaries~\ref{c.arnold} and \ref{t.arnold-half}.} \label{ArnoldCorollary} Recall that we consider the parameter family \[ f_{a,b,\tau}(\theta,x) \ = \ \left(\theta+\omega,x+\tau+\frac{a}{2\pi}\sin(2\pi x) + g_b(\theta)\right) \ \] with $g_b(\theta)=\arctan(b\sin(2\pi \theta))/\pi$. We suppose that $\omega$ is Diophantine with constants $\sigma,\nu$, such that (\ref{eq:Diophantine}) holds. Let $h_a(x) = x+\frac{a}{2\pi}\sin(2\pi x)$. Then there exist constants $0<e<c<\halb$ $\alpha>1,\ p\in\ensuremath{\mathbb{N}}$ and $0<t<\halb-c$ such that there holds \begin{eqnarray}
h_a\left([e,-e]\right) & \ensuremath{\subseteq} & (c+t,-c-t) \ ,\\
h_a'(x) & < & \alpha^{-2/p} \qquad \forall x\in C:=[c,-c] \ , \\
h_a'(x) & > & \alpha^{2/p} \qquad \forall x\in E:= [-e,e] \quad \textrm{and} \\
\alpha^{-p} & < & h_a'(x) \ < \ \alpha^{p} \qquad \forall x\in\ensuremath{\mathbb{T}^{1}} \ . \end{eqnarray} Let $\Lambda := \left(\halb-\frac{t}{2},\halb+\frac{t}{2}\right)$ and $\gamma_0:=\viertel\tan(\pi(\halb-\frac{t}{2}))$. Further, define $I^1_0:=\left(-\gamma_0/b,\gamma_0/b\right)$ and $I^2_0:=I^1_0+\halb$. Then for $\theta\notin {\cal I}_0=I^1_0\cup I^2_0$ and $\tau\in\Lambda$ we obtain \[\textstyle d(g_b(\theta)+\tau,0) \ \leq \ \frac{t}{2} +d(g_b(\theta),\halb) \ < t \ . \] Consequently $f_{a,b,\tau}$ satisfies (\ref{eq:Cinvariance})--(\ref{eq:bounds3}) for all $\tau\in\Lambda$. Further, we have \begin{eqnarray*}
\partial_\theta f_\theta(x) & = & \partial_\theta g_b(\theta) \ = \
\frac{2}{1+b^2\sin^2(2\pi\theta)} \cdot b\cos(2\pi\theta) \ \leq \ 2b
\qquad \forall (\theta,x)\in\ensuremath{\mathbb{T}^2} \quad \textrm{and}\\
|\partial_\theta f_\theta(x)| & \geq & \frac{b}{1+(2\pi\gamma_0)^2} \qquad
\forall(\theta,x) \in {\cal I}_0\times \ensuremath{\mathbb{T}^{1}} \ . \end{eqnarray*} This allows to see that (\ref{eq:bounddth}), (\ref{eq:crossing}), (\ref{e.A7'}) and (\ref{e.A8'}) are satisfied with $S=2b$ and $s=\frac{b}{1+(2\pi\gamma_0)^2}$. If we let $I^{1'}_0=[-\gamma_0/\sqrt{b},\gamma/\sqrt{b}]$, $I^2_0=I^{1'}_0+\halb$ and ${\cal I}_0'=I^{1'}_0\cup I^{2'}_0$ then \[
|\partial_\theta f_\theta(x)| \ \leq \ 2\gamma_0^{-2} \qquad \forall (\theta,x)\in(\ensuremath{\mathbb{T}^{1}} \ensuremath{\setminus} {\cal I}_0')\times\ensuremath{\mathbb{T}^{1}} \ , \] such that (\ref{eq:refinedbounddth}) holds with $s'=2\gamma_0^{-2}$. Finally, since $\partial_\tau f_\theta(x) = 1$, we may choose $\gamma=\halb$ and $L=2$ in (\ref{e.A10'}). Altogether, this implies that all assumptions of Theorem~\ref{t.mr-quantitative2} are satisfied for a suitable constant $A$ and $d=b$. The conclusions of the corollaries follow. \qed
\end{document} | arXiv |
5 Postulates of Quantum Mechanics
5.1 🥅 Learning Objectives
5.2 Postulates of Quantum Mechanics
5.2.1 Completeness: All Observable Properties of a System can be Inferred from its Wavefunction
5.2.2 Schrodinger Postulate: The wavefunction is found by solving the Schrödinger equation.
5.2.3 Born Postulate: $\left|\Psi(x,t) \right|^2$ is the probability of observing the system at position $x$ at time $t$.
5.2.4 Correspondence Principle: Every observable operator corresponds to a linear Hermitian operator.
5.3 Completeness of $\Psi$ as a descriptor of a quantum system
5.4 Schrödinger Postulate
5.5 Born Postulate
5.6 Correspondence Principle and Hermitian Operators
5.6.1 Correspondence Principle
5.6.2 Hermitian Operators
5.6.2.1 📝 Exercise: An alternative definition for an Hermitian operator is below. Show that this is equivalent to the preceding definition.
5.6.2.2 📝 Exercise: Show that the momentum operator is Hermitian.
5.6.3 Observable Values
5.6.3.1 📝 Exercise: Show that the eigenvalues of Hermitian operators are always real.
5.6.4 Eigenvectors of Hermitian Operators form a Complete Basis
5.6.4.1 📝 Exercise: Show that the eigenvectors of a Hermitian operator with different eigenvalues are always orthogonal.
5.6.5 Expectation Values
5.6.5.1 📝 Exercise: Show that the equality in the last equation is true
5.6.6 The Born Postulate, revisited
5.7 Bra- Ket- Notation and the Analogy to Linear Algebra
5.8 Application: Expansion in a Basis Set
5.9 Application: Heisenberg Uncertainty Principle
5.10 Application: Variational Principle
5.10.1 📝 Exercise: Show why the expectation value of $Q$ is always a lower bound on the largest eigenvalue of $\hat{Q}$.
5.11 Summary
5.12 🪞 Self-Reflection
5.13 🤔 Thought-Provoking Questions
5.14 🔁 Recapitulation
5.15 🔮 Next Up...
5.16 📚 References
Postulates of Quantum Mechanics¶
🥅 Learning Objectives¶
List and Discuss the Postulates of Quantum Mechanics.
Use Bra- Ket- notation.
Learn how to manipulate quantum-mechanical operators.
Understand the Heisenberg Uncertainty Principle.
Completeness: All Observable Properties of a System can be Inferred from its Wavefunction¶
$\Psi \rightarrow \text{everything}$
Schrodinger Postulate: The wavefunction is found by solving the Schrödinger equation.
$\hat{H}\Psi(x,t) = i \hbar \dot{\Psi}$ or $\hat{H}\Psi = E \Psi$.
Relativitistic quantum mechanics (the Dirac equation) is needed when particles are moving close to the speed of light
Born Postulate: $\left|\Psi(x,t) \right|^2$ is the probability of observing the system at position $x$ at time $t$.¶
$\Psi(x,t)$ is the "square root of reality".
Correspondence Principle: Every observable operator corresponds to a linear Hermitian operator.¶
The analogue of a classical measurement is evaluating the expectation value of a suitable Hermitian operator. The only possible values that can be obtained from a measurement are the eigenvalues of the corresponding Hermitian operator.
We will now elaborate on each of these key postulates. Note that some people list more postulates. There is a some choice in how one groups the information in the postulates together, and I prefer to group the statements about Hermitian operators together.
Completeness of $\Psi$ as a descriptor of a quantum system¶
The wavefunction is sufficient to determine all observable information about a quantum system. This does not mean that all information about a system can be observed. For example, while every classical observable corresponds to a Hermitian operator, it can be impossible to determine the values of multiple classical observables simultaneously.
As is often true, there are subtleties related to this postulate. In practice, we often use a density matrix, rather than a wavefunction, to describe a quantum system. A density matrix is just a mathematical way to capture a mixture of different quantum states, and can be written as, $$ \Gamma(x,t;x',t') = \sum_{k=1}^{\infty} p_k \left(\Psi_k(x',t') \right)^*\Psi_k(x,t) $$ The interpretation of a density matrix is that it is a mixture of quantum states, where the probability of observing the quantum state $k$ is $p_k$. It makes sense, therefore, to restrict $p_k$ to satisfy the constraints: $$ 1 = \sum_{k=1}^{\infty} p_k \qquad 0 \le p_k \le 1 $$ As with the wavefunction, the normalization of the density matrix is really only a convention. Any bounded (i.e. normalizable) positive semidefinite Hermitian operator can be interpreted as a density matrix. Bounded, positive semidefinite, non-Hermitian operators can be interpreted as transition density matrices, and are the mathematical representation for a change in a quantum state.
Schrödinger Postulate
The wavefunction (or density matrix) is the central object in quantum mechanics. It is found by solving the Schrödinger equation in its time-independent or time-dependent form. For relativisitic particles, one needs to instead consider the Dirac equation, and for relativisitic particles interacting with light, one needs to consider quantum electrodynamics. If one needs to go beyond this (e.g., to include $\beta$ decay of nuclei), then one needs to extend even further, eventually all the way to the standard model.
For the purposes of this course, the Schrödinger equation is adequate. Relativisitic effects are important in chemistry: it is not uncommon to consider that corrections to the nonrelativistic Schrödinger equation become important for molecules containing elements heavier than Zinc (Z=30) or Krypton (Z=36). However, actinide compounds can, by mathematical trickery, be treated by embellishing the Schrödinger equation in most cases. This means that for the purposes of understanding the structures of molecules and materials, their stability and thermodynamic properties, and the structural and chemical transformations between materials, the (relativistically-corrected) Schrödinger equation is used.
Born Postulate¶
Colloquially, the Born postulate states that the wavefunction is the square root of reality. The wavefunction is a mathematical artifice, and is not directly observable. It's complex square is observable ($|\Psi(x,t)|^2$ indicates the probability of finding the quantum system at the location $x$ at time $t$) and the overlap (interference patterns) between two wavefunctions (e.g., the system's wavefunction and light or another electron) can also be observed.
Correspondence Principle and Hermitian Operators¶
The first two postulate of quantum mechanics indicate that the wavefunction is the conveyor of all information about a quantum system and how the wavefunction is to be determined. The remaining postulate(s) all indicate that information about a quantum system is deduced from the action of Hermitian operators.
Correspondence Principle¶
The correspondence principle says that to every classical observable, there exists a Hermitian operator. This Hermitian operator provides the quantum manifestation of that classical observable. In the classical limit (where Planck's constant approaches zero, $h \rightarrow 0$), the quantum mechanical operator and the classical observable coincide. For example, we have used that the quantum mechanical operator for the momentum of a particle moving in one dimension is $$ \hat{p} = -i \hbar \tfrac{d}{dx} $$ There are some caveats associated with this. Some classical observables do not have unique representations in quantum mechanics. For example, in classical mechanics it is sensible to define, and measure, a local kinetic energy, that is, the kinetic energy at of a particle at a given point in space. Because one cannot specify the momentum (or its square) and position of a particle simultaneously, however, one cannot specify the kinetic energy at a point in space in quantum mechanics. So the local kinetic energy is not well-defined in quantum mechanics.
In classical mechanics, all classical observables can be computed from the positions and momenta of the particles in the system. That is, classical observables of an $N$-particle system functions of particles' positions ,$\{\mathbf{r}_k\}$, and momenta, $\{\mathbf{p}_k\}$: $$ f(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N;\mathbf{p}_1,\mathbf{p}_2,\dots,\mathbf{p}_N) $$
In quantum mechanics, operators that depend either on the particle's positions or their momenta are clearly defined, but simultaneous observation of both is not. There can be multiple mathematical representations for the quantum-mechanical analogue of a single classical observable, therefore. In practice, such ambiguities do not matter much since the most important observables are those that can be clearly defined. Sometimes, however, chemists like to discuss quantities (like the energy of a single atom or functional group in a molecule, or the interaction energy between two subsystems in a intermolecular complex) which are not defined in quantum mechanics. Such discussions can be useful in practice, but they are contrary to the spirit of quantum mechanics (even though they are deeply entangled with the language of chemistry).
There are also entities that emerge in molecular quantum mechanics that have no classical analogue whatsoever: concepts like electronegativity, oxidation state, and bond order. One should not expect for such quantities, which are intrinsically tied to the properties of electrons and have no classical analogue, to correspond to quantum-mechanical observable or Hermitian operators.
every quantum-mechanical observable corresponds to a Hermitian operator.
every quantum-mechanical observable reduces to a classical observable in the $h \rightarrow 0$ limit.
some classical observables do not have a unique quantum-mechanical analogue.
some inherently quantum-mechanical concepts do not have a classical analogue, nor do they have a unique mathematical representation within quantum mechanics.
Hermitian Operators¶
Quantum-mechanical observables correspond to Hermitian operators. An operator, $\hat{Q}$ is Hermitian if for any wavefunctions $\psi(x)$ and $\phi(x)$ $$ \int \left( \psi(x) \right)^* \hat{Q} \phi(x) dx = \int \left(\hat{Q} \psi(x) \right)^* \phi(x) dx = \int \left( \hat{Q} \right)^*\left( \psi(x) \right)^* \phi(x) dx $$ This basically means that a Hermitian operator can act either forwards or backwards. This is very useful in practice, since sometimes it is easier to apply $\hat{Q}$ to $\psi(x)$ than to $\phi(x)$.
Note: Quantum-mechanical observables do not need to be Hermitian; it is sufficient for them to be essentially self-adjoint, which is a somewhat weaker concept. While this is not quite a mere mathematical nuance (e.g., the molecular Hamiltonian is essentially self-adjoint), it does not affect the way we use quantum mechanics as a computational tool within chemistry.
📝 Exercise: An alternative definition for an Hermitian operator is below. Show that this is equivalent to the preceding definition.¶
$$ \int \left( \Psi(x) \right)^* \hat{Q} \Psi(x) dx = \int \left(\hat{Q} \Psi(x) \right)^* \Psi(x) dx $$
Hint: Choose $\Psi(x) = \psi(x) + i \phi(x)$.
📝 Exercise: Show that the momentum operator is Hermitian.¶
Hint: use integration by parts and assume the wavefunction and its derivatives are zero at $\pm \infty$.
Observable Values¶
In an experimental measurement of a quantum-mechanical observable, the measured value is always one of the eigenvalues of the Hermitian operator.
Recall that Hermitian operators have real eigenvalues. This explains why Hermitian operators are so central to quantum mechanics: the result of an experimental measurement must be a real number. The "chunkiness" inherent in quantum mechanics emerges from the fact one cannot observe an arbitrary value for the observable, but only the discrete eigenvalues of the corresponding quantum-mechanical operator. Ergo, Schrödinger's cat can be alive or dead, but not half-dead.
📝 Exercise: Show that the eigenvalues of Hermitian operators are always real.¶
Hint: Use the Hermitian property of $\hat{Q}$ to evaluate $\int \left(\psi_k(x)\right)^* \hat{Q} \psi_k(x) dx$ in two different ways.
Eigenvectors of Hermitian Operators form a Complete Basis¶
Denote the eigenvectors and eigenvalues of a Hermitian operator in the obvious way, $$ \hat{Q} \psi_k(x) = q_k \psi_k(x) $$
As a consequence of the spectral theorem, the eigenvectors of a Hermitian operator form a complete basis. That is, any wavefunction can be exactly represented as a linear combination of the eigenfunctions of any Hermitian operator: $$ \Psi(x) = \sum_{k=0}^{k=\infty} c_k \psi_k(x) $$
Moreover, the eigenvectors, $\psi_k(x)$, can always be chosen to be orthogonal and normalized, $$ \int \left(\psi_k(x)\right)^* \psi_l(x) dx = \begin{cases} 1 & k = l\\ 0 & k \ne l \end{cases} $$
We call orthogonal and normalized functions orthonormal. A short-hand notation for orthonormality is: $$ \int \left(\psi_k(x)\right)^* \psi_l(x) dx = \delta_{kl} $$ where $$\delta_{kl} = \begin{cases} 1 & k = l\\ 0 & k \ne l \end{cases} $$ denotes the Kronecker delta symbol. The Kronecker delta symbol is the analogue of the identity matrix (extended to infinite dimensions).
📝 Exercise: Show that the eigenvectors of a Hermitian operator with different eigenvalues are always orthogonal.¶
Hint: Use the Hermitian property of $\hat{Q}$ to evaluate $\int \left(\psi_k(x)\right)^* \hat{Q} \psi_l(x) dx$ in two different ways.
Expectation Values¶
When we expand a wavefunction in the eigenfunctions of a Hermitian operator, we say it is a superposition of the eigenstates of that operator. $$ \Psi(x) = \sum_{k=0}^{k=\infty} c_k \psi_k(x) $$ Measuring that operator always produces one of the eigenvalues, and the eigenvalues occur in proportion to the square magnitude of their coefficient in the eigenvector expansion, $$ \text{probability of observing the eigenvalue }q_k = |c_k|^2 = c_k^* c_k $$ The expectation value of the operator is thus: $$ \left<Q \right> = \sum_{k=0}^{k=\infty} |c_k|^2 q_k = \int \left( \Psi(x) \right)^* \hat{Q} \Psi(x) dx $$
This formula for the expectation value only holds for normalized wavefunctions. When the wavefunction is not normalized, instead one must use: $$ \left<Q \right> = \frac{\sum_{k=0}^{k=\infty} |c_k|^2 q_k}{\sum_{k=0}^{k=\infty} |c_k|^2} = \frac{\int \left( \Psi(x) \right)^* \hat{Q} \Psi(x) dx}{\int \left( \Psi(x) \right)^* \Psi(x) dx} $$
📝 Exercise: Show that the equality in the last equation is true¶
Hint: expand the wavefunction in the eigenbasis, use the eigenvalue relation and the orthonormality of the eigenvectors.
Immediately after performing a measurement of $Q$ for the system defined by $\Psi(x)$, one knows definitively that the state of the system is described by $\Psi(x) = \psi_k(x)$, with eigenvalue $q_k$. This seems weird, as the wavefunction seems to have changed abruptly from $\Psi(x)$ to $\psi_k(x)$ because of the measurement. This would somehow imply that if the wavefunction for Schrödinger's cat were: $$ \Psi_{cat} = \tfrac{1}{\sqrt{2}}|\text{alive} \rangle + \tfrac{1}{\sqrt{2}}|\text{dead} \rangle $$ and you opened the box and observed that the cat was dead (so after you open the box, $\Psi_{cat} = |\text{dead} \rangle$), then you killed Schrödinger's cat. To mildly exaggerate, some physicists would have you believe that every dead animal was slaughtered by the person who first observes its corpse. (To diminish culpability, it must be said that it the cat in this example was only technically half-dead, so the observer was a halfway-cat-assassin.) Most modern interpretations of quantum mechanics tend to deny such culpability. The physicists alibi is to assert that assert that while the system was described, mathematically, by $\Psi(x)$ prior to the measurement, this does not that the system existed in the state $\Psi(x)$. Similarly, after the measurement the system is in a state mathematically described by $\psi_k(x)$. While it would be weird for observing a system to be able to change its state, it is not weird for an observation to change our mathematical description of a system. For example, before you observe Lake Wobegon, it is reasonable to assume that all the women are strong, all the men are good-looking, and all the children are above average. But were you to visit Lake Wobegon, then based on your observation you might have to change your model.
That said, you may find the aforementioned "Copenhagen interpretation" convenient. Before my mother visits my home, I always clean it thoroughly. Nonetheless, my thorough cleaning is not up to my mother's standards, and she's always scandalized to find dust-bunnies under the sofa. (Who moves the sofa to vacuum under it, just to put the sofa back the same place and obscure the now-clean carpet?) I always tell my mom that the dust-bunnies were not there until she observed them. Unfortunately, my mom taught quantum mechanics herself, and she tells me that the wavefunction was: $$ \Psi_{dust-bunnies} = \sqrt{.0001}|\text{clean} \rangle + \sqrt{.9999}|\text{dirty} \rangle $$ I guess my home is only .01% clean (up to my mother's standards, at least).
The sudden change in the wavefunction upon observation is called wavefunction collapse. While the reality of wavefunction collapse in a physical sense is irresolvable, it is the mathematical description of what happens in a quantum system, and gives rise to strange quantum effects. If you find such things interesting, you may be interested to know that the aphorism that "a watched pot never boils" is justifiable, quantum-mechanically.
The Born Postulate, revisited¶
While the Born postulate is usually presented separately, it is in fact a corollary of the fact that physical observables are represented by Hermitian operators. The Hermitian operator that represents a particle at position $x_0$ is $\delta(x-x_0)$, where the Dirac delta function is effectively defined by its so-called sifting property.
Sifting property of the Dirac Delta function: Let $f(x)$ be a bounded function. Then $$ \int_{-\infty}^{+\infty} f(x) \delta(x-x_0) dx = f(x_0) $$
Now according to the Hermitian postulate, the probability of observing the particle at position $x_0$ is given by $$ \int_{-\infty}^{+\infty} \left(\Psi(x)\right)^* \delta(x-x_0) \Psi(x) dx = \left|\Psi(x_0)\right|^2 $$ which is the Born postulate.
Note: The Kronecker delta function has a sifting property similar to the Dirac delta function, $$ \sum_j f_j \delta_{jk} = f_k $$ This can be deduced directly from the definition, since $\delta_{jk}$ is one when $j=k$, and zero otherwise.
Bra- Ket- Notation and the Analogy to Linear Algebra¶
At this stage, your hand may be starting to hurt from writing integrals and wavefunctions. This is the motivation for Dirac bra-ket notation. A rather thorough explanation of this notation is available in my pdf notes, so I am only presenting the elements here. The basic idea is that a wavefunction is written as a ket, $$ | \Psi \rangle = \Psi(x) $$ and its complex conjugate as a bra, $$ \langle \Phi | = \left(\Phi(x)\right)^* $$ The overlap between two wavefunctions, which as we shall see is very important when expanding a wavefunction as a linear combination of basis functions, is therefore: $$ \langle \Phi | \Psi \rangle = \int \left(\Phi(x)\right)^* \Psi(x) dx $$ Because a Hermitian operator could act either forward (towards the ket) $$ \langle \Phi |\hat{Q} \Psi \rangle = \int \left(\Phi(x)\right)^* \hat{Q} \Psi(x) dx $$ or backwards (towards the bra) $$ \langle\hat{Q} \Phi | \Psi \rangle = \int \left(\hat{Q} \Phi(x)\right)^* \Psi(x) dx $$ we often use a notation that makes it clear that the operator can act in either direction, $$ \langle \Phi | \hat{Q} |\Psi \rangle = \int \left(\Phi(x)\right)^* \hat{Q} \Psi(x) dx $$
Application: Expansion in a Basis Set¶
A set of functions, $\{\phi_k(x) \}$ is said to be a complete basis set if any wavefunction can be written exactly as a linear combination of these basis functions, $$ \Psi(x) = \sum_{k=0}^{\infty} c_k \phi_k(x) $$ It is convenient and it is always possible, but it is not required, to choose the basis functions so that they are orthonormal, $$ \int \left(\phi_j(x) \right)^* \phi_k(x) dx = \delta_{jk} $$ where we have used the Kronecker-delta notation we introduced above.
In bra-ket notation, the preceding equations are, respectively, $$ |{\Psi}\rangle = \sum_{k=0}^{\infty} c_k |\phi_k\rangle $$ and $$ \langle \phi_j | \phi_k \rangle = \delta_{jk} $$
To obtain an equation for the expansion coefficient, multiply both sides of the first equation by $(\phi_j(x))^*$ and integrate over $x$. This gives: $$ \int \left( \phi_j(x) \right)^* \Psi(x) dx = \int \left( \phi_j(x) \right)^* \sum_{k=0}^{\infty} c_k \phi_k(x) dx $$ Because the integral of a sum is the sum of the integrals, and because the integral of a constant is a constant times the integral, this simplifies to: $$ \begin{align} \int \left( \phi_j(x) \right)^* \Psi(x) dx &=\sum_{k=0}^{\infty} c_k \left[ \int \left( \phi_j(x) \right)^* \phi_k(x) dx \right] \\ &=\sum_{k=0}^{\infty} c_k \delta_{jk} \end{align} $$ Then from the sifting property of the Dirac delta function, $$ c_j = \int \left( \phi_j(x) \right)^* \Psi(x) dx $$ This is the equation for the expansion coefficient of a wavefunction in an orthonormal basis.
Application: Heisenberg Uncertainty Principle¶
We have already alluded to the Heisenberg Uncertainty Principle, which states that some quantum-mechanical properties cannot be observed simultaneously. To provide a mathematical description of the Heisenberg Uncertainty Principle, we need to define what it means for operators to commute and anticommute.
Two operators, $\hat{A}$ and $\hat{B}$, are said to commute if for any wavefunction $\Psi(x)$, $$0 = \left(\hat{A} \hat{B} - \hat{B}\hat{A}\right) \Psi(x) = \left[\hat{A}, \hat{B} \right] \Psi(x) $$ Similarly, two operators are said to anticommute if $$0 = \left(\hat{A} \hat{B} + \hat{B}\hat{A}\right) \Psi(x) = \left\{\hat{A}, \hat{B} \right\} \Psi(x) $$ In the right-most equality, we have introduce the standard notation for commuting and anticommuting operators.
In classical mechanics, observables are simply functions of the momenta and positions of the system's particles, and the since the momenta and positions commute, observables commute. However, in quantum mechanics, the momentum operator, $\hat{p} = -i \hbar \tfrac{d}{dx}$ is a differential operator, and does not commute with the position operator $x$. Because of this, measuring a particle's position first, then its momentum is different from measuring its momentum first, then its position. Conceptually, then, it is not unreasonable that if you try to measure the position and the momentum simultaneously, the system is "confused" about how it should behave (as if momentum were measured first? or as if position were measured first?) and the answer is uncertain.
If two operators, $\hat{A}$ and $\hat{B}$, commute, $\left[\hat{A}, \hat{B}, \right] = 0$. then one can simultaneously measure the corresponding properties $A$ and $B$. $A$ and $B$ are said to be simultaneous observables.
If two operators, $\hat{A}$ and $\hat{B}$, do not commute, $\left[\hat{A}, \hat{B}, \right] \ne 0$. then one cannot simultaneously measure the corresponding properties $A$ and $B$. A Heisenberg Uncertainty Relation then holds: $$ \sigma_A^2 \sigma_B^2 \ge \tfrac{1}{4} \left| \langle \Psi |[\hat{A},\hat{B}]| \Psi \rangle \right|^2 $$ where the variance in the expectation value of the operator is defined as: $$ \sigma_A^2 = \langle \Psi |\hat{A}^2| \Psi \rangle - \langle \Psi |\hat{A}| \Psi \rangle^2 $$
This expression can be made somewhat more precise by using the Schrödinger Uncertainty Principle $$ \sigma_A^2 \sigma_B^2 \ge \left| \tfrac{1}{2}\langle \Psi |{\hat{A},\hat{B}}| \Psi \rangle
\langle \Psi |\hat{A}| \Psi \rangle \langle \Psi |\hat{B}| \Psi \rangle \right|^2 + \tfrac{1}{4} \left|\langle \Psi |[\hat{A},\hat{B}]| \Psi \rangle \right|^2 $$
A detailed derivation of these uncertainty principles is provided as a pdf.
Application: Variational Principle¶
Quantum-Mechanical Variational Principle): Given a wavefunction $\Psi(x)$ and a bounded Hermitian operator $\hat{Q}$, the expectation value of $Q$ is no less than the lowest eigenvalue of $\hat{Q}$ and no greater than the largest eigenvalue.
To understand where this principle comes from, let's focus on the lowest eigenvalue of $\hat{Q}$. We denote the eigenvalues and eigenvectors of $\hat{Q}$ as $$ \hat{Q} | \psi_k \rangle = q_k | \psi_k \rangle $$ and choose to list the eigenvalues are listed in increasing order, $q_0 < q_1 < \cdots $. Expand the wavefunction in the eigenvectors of $\hat{Q}$, $$ | Psi \rangle = \sum_{k=0}^{\infty} c_k | \psi_k \rangle $$ The expectation value of $\Psi$, which we assume to be normalized, is: \begin{align} \langle \Psi |\hat{Q}|\Psi \rangle &= \langle \sum_{j=0}^{\infty} c_j \psi_j |\hat{Q}|\sum_{k=0}^{\infty} c_k \psi_k \rangle \\ &= \sum_{j=0}^{\infty} \sum_{k=0}^{\infty} c_j c_k \langle \psi_j |\hat{Q}|\psi_k \rangle \\ &= \sum_{j=0}^{\infty} \sum_{k=0}^{\infty} c_j c_k \langle \psi_j |\hat{Q}\psi_k \rangle \\ &= \sum_{j=0}^{\infty} \sum_{k=0}^{\infty} c_j c_k \langle \psi_j |q_k\psi_k \rangle \\ &= \sum_{j=0}^{\infty} \sum_{k=0}^{\infty} c_j c_k q_k \langle \psi_j |\psi_k \rangle \\ &= \sum_{j=0}^{\infty} \sum_{k=0}^{\infty} c_j c_k q_k \delta_{jk} \\ &= \sum_{j=0}^{\infty} |c_j|^2 q_j \end{align} In the last line we used the sifting property of the Kronecker delta function. To complete the demonstration, recall that because the eigenvalues are listed in nondecreasing order, $q_j - q_0 \ge 0$ for all $j$. So: \begin{align} \langle \Psi |\hat{Q}|\Psi \rangle &= \sum_{j=0}^{\infty} |c_j|^2 q_j \\ &= \sum_{j=0}^{\infty} |c_j|^2 \left(q_0 + (q_j - q_0)\right) \\ &= \sum_{j=0}^{\infty} |c_j|^2 q_0 + \sum_{j=0}^{\infty} |c_j|^2 (q_j - q_0) \\ &= q_0 \sum_{j=0}^{\infty} |c_j|^2 + \sum_{j=0}^{\infty} |c_j|^2 (q_j - q_0) \\ &= q_0 + \sum_{j=0}^{\infty} |c_j|^2 (q_j - q_0) \\ &\ge q_0 \end{align} In the next to last line the normalization of the wavefunction, which implies that $\sum |c_j|^2 = 1$, was used. In the last line a nonnegative term was eliminated from the expression, resulting in the desired inequality.
This principle is especially useful for the energy, because it means that an approximate wavefunction will always have an energy greater than or equal to the true ground state energy. Therefore, if one is given two wavefunctions $\Psi(x)$ and $\Phi(x)$, the "better" ground-state wavefunction would be the one that has the lower energy. Similarly, given a normalized wavefunction that depends on a parameter, $\kappa$, the "best" wavefunction and the "best" energy can be obtained by minimizing the expectation value of the energy with respect to $\kappa$: $$ \min_\kappa \langle \Psi(\kappa) |\hat{H} | \Psi(\kappa) \rangle $$
The variational principle is one of the most important ways we approximate the energy and wavefunction of quantum systems.
📝 Exercise: Show why the expectation value of $Q$ is always a lower bound on the largest eigenvalue of $\hat{Q}$.¶
Summary¶
Based on the preceding, the key postulates of quantum mechanics are
the Schrödinger equation determines the wavefunction
the wavefunction determines all observable properties of a quantum system
observable properties of a quantum system, and their observable values, correspond to Hermitian operators You should be able to apply, and expand upon, these principles at length.
🪞 Self-Reflection¶
Try to come up with your own version of a Schrödinger's cat paradox. Have fun with it.
What is an example of a classical observable that is not a quantum-mechanical observable?
Give an example of two quantum-mechanical operators that commute. What about two operators that don't commute?
🤔 Thought-Provoking Questions¶
Can you imagine a case where the Hermitian nature of the Hamiltonian operator would be useful?
Can you imagine a way to use the variational principle in practice?
The variational principle allows one to choose a wavefunction that gives the lowest energy, which provides the best wavefunction in an energetic sense. This isn't the same as finding the closest wavefunction to the ground-state wavefunction (see below) but if your energy is close enough to the ground-state energy, you can be assured that the wavefunction is also very close to the ground-state wavefunction. Explain why. Assume, for simplicity, that the ground state is nondegenerate. $$ \min_\Psi \langle \Psi - \psi_0 | \Psi - \psi_0 \rangle $$
🔁 Recapitulation¶
What are the postulates of quantum mechanics?
Why is it important that quantum-mechanical observables correspond to Hermitian operators?
What is the Heisenberg Uncertainty principle?
What is the variational principle?
List all the Hermitian operators you know.
What do the Dirac and Kronecker delta notations mean?
🔮 Next Up...¶
Multielectron systems
📚 References¶
My favorite sources for this material are:
Randy's book
D. A. MacQuarrie, Quantum Chemistry (University Science Books, Mill Valley California, 1983)
Mathematical Features of Quantum Mechanics (my notes, starting page 6).
There are also some excellent wikipedia articles:
Postulates of Quantum Mechanics
Variational Principle | CommonCrawl |
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\begin{center} Antonio Avil\'{e}s
{AN INTRODUCTION TO MULTIPLE GAPS}
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{\leftskip3em\rightskip3em \emph{Abstract}. This is an introductory article to the theory of multiple gaps.
\emph{Mathematics Subject Classification} (2010): Primary: 03E15, 28A05, 05D10; Secondary: 46B15
\emph{Keywords}: multiple gap}
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\maketitle \tableofcontents
The aim of this article is to provide an introduction to the theory of multiple gaps, recently developed by Stevo Todorcevic and the author in a series of papers \cite{multiplegaps, stronggaps, IHES}, and in some unpublished material that can be found in \cite{analyticmultigaps}. This is not intended to be an exhaustive survey, we have just focused on a particular topic, mostly concerning analytic multiple gaps, and present some selected results that are often simplified, rather than in general form. We neither tried to give all the proofs, but just some arguments and ideas. The purpose is to have a reasonably light and easy to read note, from which one can get an idea of what the results and the techiques are, and serve as a motivation for the more interested reader to look for further details.\\
We also tried to assume as little background as possible, so that this becomes a suitable reading for non-experts and students, where they may get some basic ideas of topics like descriptive set-theory, Ramsey theory or Banach spaces of continuous functions, together with references where to learn about it.\\
A multiple gap is nothing else than finitely many families of subsets of a countable set, with some conditions about how these families are \emph{mixed}. This is a fairly elementary kind of object that could arise in a variety of contexts. The formal definition and some elementary and historical comments are found in Section~\ref{sectiondefinitions}. Although there are some brief considerations about cardinal $\aleph_1$ in that section, most of the paper is devoted to nicely definable objects: the analytic multiple gaps. In Section~\ref{sectioncritical}, which collects results from~\cite{multiplegaps, IHES}, we present some examples of gaps that we call \emph{critical} and we explain the unique importance of these particular examples in the general theory. Taking these critical gaps as a starting point, the theory is enriched by the intervention of Ramsey theory and what we call the \emph{first-move and record combinatorics} of the $n$-adic tree, explained in Sections~\ref{sectionstrong} and~\ref{sectionweak}, that present results from \cite{stronggaps} and \cite{IHES,analyticmultigaps} respectively. The rest of sections present applications of the theory in different settings, like the topology of $\beta\omega\smallsetminus\omega$, sequences of vectors in Banach spaces and operators in $\ell_\infty/c_0$. Sections~\ref{sectioncountablyseparated} and \ref{sectionCK} present results from \cite{multiplegaps}, while the rest of sections explain ideas from~\cite{analyticmultigaps}.\\
Before getting into matter, just a warning about some notations. The set of natural numbers is denoted as $\omega = \{0,1,2,\ldots\}$, and each natural number is identified with its set of predecessors $n=\{0,1,2,\ldots,n-1\}$. In this way, $$\{x_n\}_{n\in \omega} = \{x_n\}_{n<\omega} = \{x_n : n<\omega\} = \{x_0,x_1,x_2,\ldots\}$$ are alternative ways of denoting an infinite sequence, and $$\{x_i\}_{i\in n} = \{x_i\}_{i<n} = \{x_i : i<n\} = \{x_0,x_1,x_2,\ldots,x_{n-1}\}$$ are alternative ways of denoting a finite sequence of $n$ elements. Notice the set-theoretic tradition of starting counting from 0.\\
\section{General definitions}\label{sectiondefinitions}
Let $N$ be a countable set. We are interested in comparing subsets of $N$ modulo finite sets, so for $a,b\subset N$, we write $a\subset^\ast b$ if $a\smallsetminus b = \{i\in a : i\not\in b\}$ is finite, that is, if all but finitely many elements of $a$ belong to $b$. We write $a=^\ast b$ if $a\subset^\ast b$ and $b\subset^\ast a$.
\begin{dfn} Let $\{\Gamma_i : i<n\}$ be a finite collection of families of subsets of $N$. We say that these families are separated if we can find $a_0,\ldots,a_{n-1}$ subsets of $N$ such that \begin{enumerate} \item $x\subset^\ast a_i$ for each $x\in\Gamma_i$, \item $\bigcap_{i<n}a_i =^\ast \emptyset$. \end{enumerate} \end{dfn}
Allowing finite errors is important here, otherwise we would be just saying that $\bigcap_{i<n}\bigcup\Gamma_i =\emptyset$.
\begin{dfn} A finite family $\Gamma = \{\Gamma_i : i<n\}$ of families of subsets of $N$ is said to be an $n_\ast$-gap if \begin{enumerate} \item If we pick $x_i\in\Gamma_i$, then $\bigcap_{i<n}x_i = ^\ast\emptyset$ \item The families $\{\Gamma_i : i<n\}$ cannot be separated \item Each family $\Gamma_i$ is infinitely hereditary: if $x\in\Gamma_i$ and $y$ is an infinite subset of $x$, then $y\in\Gamma_i$ \end{enumerate} \end{dfn}
The tricky point is that, since we are allowing finite errors, taking $a_i = \bigcup \Gamma_i$ will not work to separate the families. The last requirement in the definition is just a harmless technical condition that will simplify the language for the kind of problems that we are going to discuss. Sometimes we may present examples of $n_\ast$-gaps where the families $\Gamma_i$ may not be hereditary; in that case, it is implicitly assumed that we close $\Gamma_i$ under subsets in order to fit in the above definition.\\
Two families of sets $I$ and $J$ are orthogonal if $x\cap y = ^\ast \emptyset$ whenever $x\in I$ and $y\in J$.
\begin{dfn} An $n$-gap is an $n_\ast$-gap $\Gamma$ in which moreover the families $\{\Gamma_i : i<n\}$ are pairwise orthogonal\footnote{The original definition of $n$-gap given in \cite{multiplegaps} is different from this one. It is given in an abstract Boolean algebra $\mathcal{B}$ -a degree of abstraction that we are not interested in here, we would only consider $\mathcal{B} =\mathcal{P}(\omega)/fin$- and required the families $\Gamma_i$ to be ideals (closed under finite unions) -a restriction that we do not want to make here-.}. \end{dfn}
Although the general notion of $n_\ast$-gap may be useful at some points, we shall be concerned mainly with $n$-gaps. At this point, the sensible reader wants to see examples of $n$-gaps, to get an idea of how possibly one can get families which are on the one hand orthogonal but on the other hand cannot be separated. These examples are presented in Section~\ref{sectioncritical}, and the impatient reader can go straight to them. In the rest of this section we shall recall some basic facts and we shall discuss how the theory of $n$-gaps relates to the classical theory of gaps.\\
A 2-gap is what is traditionally called just a \emph{gap} by set-theorists. The above definition is the multidimensional generalization of it that was introduced in \cite{multiplegaps}, but 2-gaps are much older, going back to Hausdorff~\cite{Ha2}. First of all, why the name \emph{gap}? Well, if $\{\Gamma_0,\Gamma_1\}$ is a 2-gap according to the above definition, we can consider the family $\Gamma_1^\ast = \{N\smallsetminus x: x\in\Gamma_1\}$. In that case, we have that $x\subset^\ast y$ for every $x\in\Gamma_0$ and $y\in\Gamma_1^\ast$ (because of mutual orthogonality) but there is no element $a$ such that $x\subset^\ast a \subset^\ast y$ for all $x\in\Gamma_0$, $y\in\Gamma_1^\ast$ (becase of non-separation). There is nothing in between, so there is a gap.
The first observation is that we cannot produce gaps out of countable families:
\begin{prop}\label{separationofcountable} If $I$ and $J$ are two orthogonal countable families of subsets of $N$, then they are separated. \end{prop}
\begin{proof} Say that $I = \{a_n : n<\omega\}$ and $J=\{b_n : n<\omega\}$. By the considering the finite unions $\tilde{a}_n = \bigcup_{k<n}a_k$ and $\tilde{b}_n = \bigcup_{k<n}b_k$, we can suppose without loss of generality that $a_1\subset a_2 \subset$ and $b_1\subset b_2\subset\cdots$. By orthogonality $a_i\cap b_j$ is finite for all $i,j$. Inductively on $n$, we can make finite modifications $a'_n$ and $b'_n$ of each $a_n$ and $b_n$, so that still $a'_1\subset a'_2 \subset$ and $b'_1\subset b'_2\subset\cdots$ and moreover $a'_n\cap b'_n = \emptyset$. Finally, the sets $a = \bigcup_n a'_n$ and $b=\bigcup_n b'_n$ separate $I$ and $J$. \end{proof}
Going beyond countable cardinality, Hausdorff produced his famous gap:
\begin{thm}[Hausdorff]\label{Hausdorff} There exist two orthogonal families of subsets of $\omega$ of cardinality $\aleph_1$ which are not separated. \end{thm}
The two families are a special kind of $\omega_1$-chains in the order $\subset^\ast$. An exposition of this can be found in \cite{FremlinMA}. One remarkable point about Hausdorff's construction is that it does not require the Continuum Hypothesis or any other additional axioms, and that it works for the first uncountable cardinal $\aleph_1$. Compare it with the fact that, under the axiom $MA_{\aleph_1}$, two orthogonal sets of cardinalities $\aleph_0$ and $\aleph_1$ are always separated. Hausdorff's construction cannot be generalized to higher dimensions,
\begin{thm}\label{aleph1} Under $MA_{\aleph_1}$, if $\{\Gamma_i : i<n\}$, $n>2$, are pairwise orthogonal families of subsets of $N$ of cardinality $\aleph_1$, then they are separated. \end{thm}
For readers familiar with Martin's axiom: The idea is that one tries to force a separation for $\{\Gamma_0,\Gamma_1,\Gamma_2\}$ by considering approximations $(p_0,p_1,p_2)\in\Gamma_0\times\Gamma_1\times\Gamma_2$ such that $p_0\cap p_1\cap p_2 = \emptyset$. One sees that such approximations form a ccc (actually $\sigma$-2-linked) poset, noticing that once $p_0\cap p_1$, $p_0\cap p_2$, $p_1\cap p_2$ are frozen, the conditions become compatible. Further details, as well as generalized forms of Theorem~\ref{aleph1} can be found in \cite[Section 6]{multiplegaps}.\\
While Theorem~\ref{aleph1} rules out a part of the theory of gaps to be extended to higher dimensions, there is a class of gaps which happens to florish into a beautiful and intricated theory in the multidimensional setting: the analytic gaps. A family $I$ of subsets of the countable set $N$ can be viewed as a subset $I\subset 2^N = \{0,1\}^N$, and the set $2^N$ can be endowed with the product topology which makes it homeomorphic to the Cantor set. In this way, we can measure the complexity of a family $I$ by looking at its topological complexity as a subset of $2^N$. Let us consider some examples before going further. The family $$I = \left\{x\subset \omega : \sum_{n\in x}n^{-1}\leq 2\right\}$$ is a closed family of subsets of $\omega$. Namely, if $x\not\in I$, then there is a finite set $F\subset\omega$ such that $\sum_{n\in F}n^{-1}>2$, and then $\{y : y\supset F\}$ is a neighborhood of $x$ disjoint from $I$. On the other hand, the family $$J = \left\{x\subset \omega : \sum_{n\in x}n^{-1}< +\infty\right\}$$ is not closed; it is a little bit more complicated: $J$ is an $F_\sigma$-family, the countable union of closed families $$J = \bigcup_{m<\omega} \left\{x\subset \omega : \sum_{n\in x}n^{-1}\leq m\right\}.$$
We think of closed families as the simplest ones, and a family $I$ is considered more complex if a more complicated expression is needed to obtain $I$ from closed families. In this sense, $F_\sigma$ families have a low complexity. We say that $I\subset 2^N$ is analytic if there exist\footnote{$\omega^{<\omega}$ is the set of finite sequences of natural numbers. Given $x=(x_0,x_1,\ldots)\in\omega^{<\omega}$ and $n<\omega$, $x|n = (x_0,\ldots,x_{n-1})$.} $\{I_s : s\in\omega^{<\omega}\}$ closed families such that
$$ (\star)\ I = \bigcup_{x\in\omega^\omega} \bigcap_{n<\omega}I_{x|n}$$ The class of analytic sets includes closed sets and open sets, and is closed under countable unions, countable intersections and \emph{Souslin operations} (meaning that if we have $\{I_s : s\in\omega^{<\omega}\}$ analytic sets, and $I$ is as in $(\star)$ above, then $I$ is analytic). As an intuitive idea, a family $I$ is analytic if it has a \emph{reasonably simple} definition that allows it to be expressed in terms of closed families as in $(\star)$. The reader who is not familiar with the notion of analytic set is encouraged to visit \cite{Kechris}.\\
An $n_\ast$-gap $\Gamma = \{\Gamma_i : i<n\}$ is analytic if each $\Gamma_i$ is an analytic family. A theory of analytic gaps was initiated in~\cite{Todorcevicgap}. The gap constructed by Hausdorff was not analytic (he used transfinite induction on $\omega_1$, and this technique gets us out of the \emph{reasonable definitions} of the analytic world). Todorcevic found that analytic gaps had extra structure, and in particular some of the properties of Hausdorff's gaps can never hold for analytic gaps. Consider the following variation on the notion of separation:
\begin{dfn}The families $\{\Gamma_i : i<n\}$ are said to be countably separated if there exists a contable set ${C}$ such that whenever we pick $x_i\in\Gamma_i$ for $i\in n$, there exist $c_0,\ldots,c_{n-1}\in {C}$ such that $x_i\subset c_i$ and $\bigcap c_i = \emptyset$. An $n_\ast$-gap $\Gamma = \{\Gamma_i : i<n\}$ is called strong if it is not countably separated. \end{dfn}
Todorcevic formulated his results as two dichotomies, that indicate that there are two critical examples of analytic 2-gaps: one which is strong and another which is not. Every analytic 2-gap \emph{contains} the critical 2-gap which is not strong, and every analytic strong 2-gap \emph{contains} the critical strong 2-gap. These results are essentially the two-dimensional case of Theorem~\ref{criticalgap} and Theorem~\ref{criticalstronggap} that we discuss next.
\section{Critical analytic gaps}\label{sectioncritical}
Given $n^{<\omega}$, the $n$-adic tree is the set of all finite sequences of numbers from $\{0,\ldots,n-1\}$. $$n^{<\omega} = \{ (s_0,\ldots,s_k) : s_0,\ldots,s_k\in \{0,\ldots,n-1\}\}$$ The empty sequence is considered as an element of the $n$-adic tree.\\
The concatenation of two elements $s=(s_0,\ldots,s_p), t=(t_0,\ldots,t_q)\in n^{<\omega}$ is defined as $s^\frown t = (s_0,\ldots,s_p,t_0,\ldots,t_q)$. Sometimes, in abuse of notation, we write $s^\frown i$ for $s^\frown (i)$ when $i\in n$.\\
We say that $s\leq t$ if there exists $r$ such that $t = s^\frown r$. This is a partial order relation on the $n$-adic tree. A chain in the $n$-adic tree refers to a chain in this order, a set $X\subset n^{<\omega}$ such that for each $t,s\in X$, either $t\leq s$ or $s\leq t$.
\begin{dfn} Let $0\leq i< n$ be natural numbers, and $X\subset n^{<\omega}$ be a subset of the $n$-adic tree.
\begin{enumerate} \item We say that $X$ is an $i$-chain if we can write $X = \{x_0,x_1,\ldots\}$ in such a way that $x_k ^\frown i \leq x_{k+1}$ for every $k$. \item We say that $X$ is an $[i]$-chain if we can write $X = \{x_0,x_1,\ldots\}$ in such a way that $x_{k+1} = x_k ^\frown i ^\frown w_k$ where $\max(w_k)\leq i$.\\ \end{enumerate} \end{dfn}
\textbf{The critical $n$-gap} is the $n$-gap $\mathcal{C}^{n} = \{\mathcal{C}^n_i : i<n\}$, where $\mathcal{C}^n_i$ is the collection of all $[i]$-chains of the $n$-adic tree.\\
\textbf{The critical strong $n$-gap} is the $n$-gap $\mathcal{S}^{n} = \{\mathcal{S}^n_i : i<n\}$, where $\mathcal{S}^n_i$ is the collection of all $i$-chains of the $n$-adic tree.\\
A proof that $\mathcal{C}^n$ is an $n$-gap can be found in \cite[Lemma 1.15]{analyticmultigaps} and a proof that $\mathcal{S}^n$ is a strong $n$-gap can be found in \cite[Theorem 6]{multiplegaps}. In any case, these proofs are not complicated, we give some hints from which the reader can derive these facts as an excercise:
\begin{itemize} \item In order to see that $\mathcal{C}^n$ is not separated, check that if $x\subset^\ast a$ for all $x\in\mathcal{C}^n_i$, then for every $t\in n^{<\omega}$ there exists $s>t$ such that $s^\frown r\in a$ for all $r\in\{0,1,\ldots,i\}^{<\omega}$.
\item In order to see that $\mathcal{S}^n$ is not countably separated, suppose that $C$ is a countable family of sets that countably separates $\mathcal{S}$. Then, for evey $x\in n^\omega$ there exist $c_0,\ldots,c_{n-1}\in C$ and $m<\omega$ such that\footnote{If $x=(x_0,x_1,\ldots)\in n^\omega$ and $s\in n^{<\omega}$, $s<x$ means that there exists $k$ such that $s=(x_0,\ldots,x_{k-1})$.} $\bigcap_{i<n}c_i=\emptyset$ and $\{t\in n^{<\omega} : |t|>m, t^\frown i <x\}\subset c_i$ for all $i$. By a Baire category argument, there exists $r\in n^{<\omega}$ so that one can take the same $m$ and the same $c_i$'s for all $x>r$, and this leads to a contradiction. \end{itemize}
These are gaps of very low complexity, actually each family $\mathcal{C}^n_i$ and $\mathcal{S}^n_i$ is closed. We call these gaps critical because any other analytic (strong) $n$-gap must contain the critical example inside.
\begin{thm}\label{criticalstronggap} If $\Gamma$ is a strong analytic $n_\ast$-gap, then there exists a one-to-one function $\phi:n^{<\omega}\longrightarrow N$ such that\footnote{We use the notation $\phi(\mathcal{I}) = \{\phi(X) : X\in\mathcal{\mathcal{I}}\}$, where $\phi(X) = \{\phi(s) : s\in 2^{<\omega}\}$.} $\phi(\mathcal{S}^n_i)\subset \Gamma_i$ for all $i<n$. \end{thm}
\begin{thm}\label{criticalgap} If $\Gamma$ is an analytic $n_\ast$-gap, then there exists a one-to-one function $\phi:n^{<\omega}\longrightarrow N$ and a permutation $\varepsilon:n\longrightarrow n$ such that $\phi(\mathcal{C}^n_i)\subset \Gamma_{\varepsilon(i)}$ for all $i<n$. \end{thm}
Theorem~\ref{criticalgap} can be found in \cite[Theorem 1.14]{analyticmultigaps}, while Theorem~\ref{criticalstronggap} corresponds to \cite[Theorem 7]{multiplegaps}. The statement of \cite[Theorem 7]{multiplegaps} refers only to $n$-gaps, but the same proof works for $n_\ast$-gaps. The critical strong $n$-gap $\mathcal{S}^n$ is symmetric, in the sense that for every permutation $\varepsilon:n\longrightarrow n$, the induced bijection $n^{<\omega}\longrightarrow n^{<\omega}$ identifies $\mathcal{S}^n$ with $\{\mathcal{S}^n_{\varepsilon(i)} : i<n\}$. The critical $n$-gap $\mathcal{C}^n$ is, however, highly non-symmetric. For example, notice that $\mathcal{C}^n_0$ is just countably generated as every $[0]$-chain is contained in $\{s^\frown (0,\ldots,0)\}$ for some $s\in n^{<\omega}$ but $\mathcal{C}^n_i$ is not countably generated for $i>0$. Moreover, consider the following notions:
\begin{itemize} \item We say that a family of sets $I$ is countably generated in another family $J$ if there exists a countable subfamily $D\subset J$ such that for every $x\in I$ there exists $y\in D$ such that $x\subset y$. \item If $I$ is a family of subsets of $N$, its orthogonal family is $I^\perp = \{y\subset N : \forall x \in I \ x\cap y=^\ast \emptyset\}$. \end{itemize}
The gap $\mathcal{C}^n$ is not invariant under any permutation because $\mathcal{C}^i_n$ is countably generated in $(\mathcal{C}^j_n)^\perp$ if and only if $i<j$. Indeed, if $i<j$, we have the countable family $$ D = \{ \{t^\frown r : r\in\{0,\ldots,j-1\}^{<\omega}\} : t \in n^{<\omega}\}$$ and on the other hand, it follows from Proposition~\ref{separationofcountable} that if two orthogonal families are not separated, then they cannot be countably generated in the orthogonal of each other.
When $n=2$, Theorem~\ref{criticalgap} can be stated more generally, as one side of the gap need not be analytic. This is \cite[Theorem 3]{Todorcevicgap}:
\begin{thm}\label{firstdichotomy} Let $\Gamma = \{\Gamma_0,\Gamma_1\}$ is a gap such that $\Gamma_1$ is analytic and is not countably generated in $\Gamma_0^\perp$. Then there exists a one-to-one function $u:2^{<\omega}\longrightarrow N$ such that $u(\mathcal{C}^n_i)\subset \Gamma_{i}$ for all $i=0,1$. \end{thm}
The statement of \cite[Theorem 3]{Todorcevicgap} is a little different from Theorem~\ref{firstdichotomy}. To see that they are equivalent, it is enough to notice that the bijection $f:2^{<\omega}\longrightarrow \omega^{<\omega}$ given by $g(\emptyset) = \emptyset$, $g(t^\frown 1) = g(t)^\frown 0$ and $g((t_1,\ldots,t_k)^\frown 0) = (t_1,\ldots,t_k+1)$ transforms the gap $\mathcal{C}^2$ into the gap $\tilde{\mathcal{C}}^2$, where $\tilde{\mathcal{C}}^2_0$ consists of the sets which are contained in the set of immediate successors of some $s\in\omega^{<\omega}$, and $\tilde{\mathcal{C}}^2_1$ consists of all chains of $\omega^{<\omega}$.\\
The critical strong 2-gap $\mathcal{S}^2$ is related to the phenomenon of so-called Luzin gaps. Suppose that you have families $\{a_x : x\in X\}$ and $\{b_x : x\in X\}$ of subsets of $\omega$ indicated in some uncountable set $X$ such that
\begin{enumerate} \item $a_x\cap b_x = \emptyset$ for all $x$,
\item $0<|(a_x\cap b_y)\cup (a_y\cap b_x)| < \omega$ for all $x\neq y$. \end{enumerate}
Then one can check that $\{a_x : x\in X\}$ and $\{b_x : x\in X\}$ form a strong 2-gap, cf. \cite[p. 56]{Todorcevicgap}. Gaps of this form are called Luzin gaps. The gap $\mathcal{S}^2$ is like that, as we can take $X = 2^\omega$, $a_x = \{s\in x : s^\frown 0< x\}$, $b_x = \{s\in x : s^\frown 1< x\}$. It is indeed a \emph{perfect Luzin gap} since it is parametrized by the Cantor set in a continuous way. \cite[Theorem 2]{Todorcevicgap} asserts that every analytic strong 2-gap contains a perfect Luzin gap, and therefore corresponds to the case $n=2$ of Theorem~\ref{criticalstronggap} above. A non-definable version of this result holds under the Open Coloring Axiom OCA (cf.~\cite{MoorePFA} for information on this axiom).
\begin{thm}[OCA]\label{OCA} If $\{\Gamma_0,\Gamma_1\}$ is a strong 2-gap, then there exists a Luzin gap $\{a_x, b_x : x\in X\}$ such that $a_x\in \Gamma_0$, $b_x\in \Gamma_1$ for all $x\in X$. \end{thm}
The proof of this is a direct application of OCA to the set of pairs $\{(a,b),(a',b')\}\subset \Gamma_0\times\Gamma_1$ such that $(a\cap b')\cup (a'\cap b)\neq\emptyset$. It is unclear if the higher-dimensional instances of Theorem~\ref{criticalstronggap} admit any analysis in terms of a Luzin-like condition that would allow a non-definable version as Theorem~\ref{OCA} above. One essential obstruction is that we cannot expect $\aleph_1$-generated $n$-gaps to play any role for $n>2$, by Theorem~\ref{aleph1}.\\
Theorems~\ref{criticalstronggap} and~\ref{criticalgap} state that, in a certain sense, every analytic (strong) $n_\ast$-gap contains a (strong) critical $n_\ast$-gap. However, the notion of \emph{being contained} that is implicit in these statements ignores important information. A finer notion is obtained by demanding the preservation of the orthogonals:
\begin{dfn}\label{gaporder} Let $\Gamma$ and $\Delta$ be $n_\ast$-gaps on the sets $N$ and $M$ respectively. We say that $\Gamma\leq \Delta$ if\footnote{There are some variations of the order $\leq$ that lead essentially to the same theory, see \cite[Section 1.4]{analyticmultigaps}} there exists a one-to-one function $\phi:N\longrightarrow M$ such that for every $i\in n$: \begin{enumerate} \item If $x\in \Gamma_i$ then $\phi(x)\in \Delta_i$ \item If $x\in\Gamma_i^\perp$, then $\phi(x)\in \Delta_i^\perp$ \end{enumerate} \end{dfn}
When considering this finer relation between gaps, there is not anymore just one critical analytic $n_\ast$-gap below all others, but still there is a finite number of minimal $n_\ast$-gaps so that any analytic $n_\ast$-gap contains one of them.
\begin{thm}\label{minimalstronggaps} Given $n<\omega$, there exists a finite list of closed strong $n_\ast$-gaps $\Gamma^1,\ldots,\Gamma^{q_n}$ such that for every analytic strong $n_\ast$-gap $\Gamma$ there exists $j$ such that $\Gamma^j \leq \Gamma$. \end{thm}
\begin{thm}\label{minimalgaps} Given $n<\omega$, there exists a finite list of closed $n_\ast$-gaps $\Gamma^1,\ldots,\Gamma^{p_n}$ such that for every analytic $n_\ast$-gap $\Gamma$ there exists $j$ such that $\Gamma^j \leq \Gamma$. \end{thm}
Theorem~\ref{minimalgaps} is proven in \cite{IHES}. A less general version of Theorem~\ref{minimalstronggaps} is found in \cite{stronggaps}, but essentially the same proof works for the statement here. Of course, one can suppose that in the lists of gaps provided in these theorems we have that $\Gamma^i\not\leq \Gamma^j$ when $i\neq j$, and in this case we have the right to call the gaps in such a list \emph{the minimal analytic (strong) $n_\ast$-gaps}. Formally, we can introduce the following definitions:
\begin{dfn} An analytic (strong) $n_\ast$-gap $\Gamma$ is said to be a minimal analytic (strong) $n_\ast$-gap if for every other analytic (strong) $n_\ast$-gap $\Delta$, if $\Delta\leq \Gamma$ then $\Gamma\leq\Delta$. \end{dfn}
\begin{dfn} Two minimal analytic (strong) $n_\ast$-gaps $\Gamma$ and $\Delta$ are equivalent if $\Gamma\leq\Delta$ and $\Delta\leq\Gamma$. \end{dfn}
In this language, Theorems~\ref{minimalgaps} and \ref{minimalstronggaps} can be reformulated as follows:
\begin{thm} For every $n<\omega$, there exists only finitely many equivalence classes of minimal analytic (strong) $n_\ast$-gaps. Moreover, for every analytic (strong) $n_\ast$-gap $\Delta$ there exists a minimal analytic (strong) $n_\ast$-gap $\Gamma$ such that $\Gamma\leq \Delta$. \end{thm}
The combination of Theorem~\ref{criticalgap} or Theorem~\ref{criticalstronggap} with Ramsey theoretic techniques provides a list like stated in Theorem~\ref{minimalgaps} or Theorem~\ref{minimalstronggaps}, but such a list is very redundant and a further combinatorial analysis to find the really minimal elements. Sections~\ref{sectionstrong} and~\ref{sectionweak} below explain the situation in the strong and general case respectively.
\section{First-move combinatorics of the $n$-adic tree and strong analytic gaps}\label{sectionstrong}
The level of an element of the $n$-adic tree $n^{<\omega}$ is defined as $|(s_0,\ldots,s_{p-1})| = p$. We introduce a well order $\prec$ on $n^{<\omega}$ given by $s\prec t$ if either $|s|<|t|$, or $|s|=|t| = p$ and $n^{p} s_0 + n^{p-1}s_1 + \ldots < n^p t_0 + n^{p-1}t_1 + \ldots$.\\
The meet of $s,t\in n^{<\omega}$ is the infimum of $s$ and $t$ in the order $\leq$. That is, $s\wedge t$ it is the largest element $r$ such that $r\leq t$ and $r\leq s$.\\
If $t\leq s$, then $s\smallsetminus t$ is the element $r\in n^{<\omega}$ such that $t^\frown r = s$.\\
By the \emph{first-move combinatorics} of the $n$-adic tree, we refer to the combinatorial problems about subsets of the $n$-adic tree where the only relevant structure is given by the the meet function $t\wedge s$, the order $\prec$ and the `first move' between comparable nodes (if $t<s$, knowing which is the $i\in n$ such that $t^\frown i\leq s$).\\
We need a few definitions to make this idea precise. A set $A\subset n^{<\omega}$ will be said to be meet-closed if $t\wedge s\in A$ whenever $t\in A$ and $s\in A$. The meet-closure of $A$ is the intersection of all meet-closed sets that contain $A$ and is denoted by $\langle\langle A\rangle\rangle$.\\
A bijection $f:A\longrightarrow B$ is a first-move-equivalence if it is the restriction of a bijection $f:\langle\langle A\rangle\rangle\longrightarrow \langle\langle B\rangle\rangle$ such that for every $t,s\in \langle\langle A\rangle\rangle$ \begin{enumerate} \item $f(t\wedge s) = f(t)\wedge f(s)$ \item $f(t) \prec f(s)$ if and only if $t\prec s$ \item If $i\in n$ is such that $t^\frown i \leq s$, then $f(t)^\frown i \leq f(s)$. \end{enumerate}
The sets $A$ and $B$ are called first-move-equivalent if there is a first-move-equivalence between them. In this case, we write $A\approx B$. In this language, the following is a consequence of Milliken's partition theorem for trees~\cite{Milliken}:
\begin{thm}\label{strongRamsey} Fix a set $A_0\subset n^{<\omega}$. If we color the set $\{A\subset n^{<\omega} : A\approx A_0\}$ into finitely many colors in a Baire-measurable way\footnote{To color the set $\mathcal{A} = \{A\subset n^{<\omega} : A\approx A_0\}$ into finitely many colors, just means to give a function $\mathcal{A}\longrightarrow F$ where $F$ is finite set whose elements we call \emph{colors}. The Baire-measurabily refers to the fact that we can view $\mathcal{A}$ as a subset of the Cantor set $2^{n^{<\omega}}\equiv 2^\omega$, with its Baire $\sigma$-algebra generated by the open sets and the meager sets \cite[Section 8F]{Kechris}.}, then there exists a set $T\subset n^{<\omega}$ such that $T\approx n^{<\omega}$ and all the sets from $\{A\subset T : A\approx A_0\}$ have the same color. \end{thm}
The above statement can be also proven as a corollary of \cite[Theorem 1.5]{analyticmultigaps}, we will explain the argument later in Section~\ref{sectionweak} after Theorem~\ref{weakRamsey}. Finding large monochromatic sets for colorings on a given structure is the general theme of Ramsey theory. We refer to \cite{Ramsey} for an exposition on the Ramsey theory of countable infinite structures, including the Milliken's theorem mentioned above and many related results.\\
The notion of $i$-chain that appeared in the definition of the critical strong $n$-gap can be rephrased saying that set $X\subset n^{<\omega}$ is an $i$-chain if and only if $$X\approx \{ (i), (ii), (iii), (iiii), \ldots\}$$ We can generalize this notion to include some kinds of sets which are not chains. If $i,j\in n$, a set $X$ is called an $(i,j)$-comb if $$X\approx \{ (j), (iij), (iiiij), (iiiiiij), \ldots\}$$ In this way, an $i$-chain is the same as an $(i,i)$-comb. When $i\neq j$, $(i,j)$-combs are antichains (that is, no two different elements are comparable in the order $\leq$). The $(i,j)$-combs are the minimal first-move-equivalence classes of infinite subsets of $n^{<\omega}$, in the sense that the following two facts hold: \begin{enumerate} \item If $X$ is an $(i,j)$-comb, then every infinite subset of $X$ is again an $(i,j)$-comb. \item Every infinite subset $X\subset n^{<\omega}$ contains a further subset $Y\subset X$ which is an $(i,j)$-comb for some $i,j\in n$ \cite[Lemma 7]{stronggaps} \end{enumerate}
For $S\subset n\times n$, let $\Gamma_S$ be the family of all subsets of $n^{<\omega}$ which are $(i,j)$-combs for some $(i,j)\in S$. The following two results state the connection between analytic strong $n_\ast$-gaps and combs of the $n$-adic tree: every analytic strong $n_\ast$-gap contains - in the sense of the order $\leq$ of Definition~\ref{gaporder}- a strong gap made of combs.
\begin{thm} If $\{S_i : i<n\}$ are nonempty subsets of $m\times m$, then $\{\Gamma_{S_i} : i\in n\}$ are closed families of subsets of $m^{<\omega}$ which are not countably separated. \end{thm}
The intersection of two sets of different types is always finite (indeed, either empty or a singleton). Therefore $\{\Gamma_{S_i} : i\in n\}$ is an $n_\ast$-gap when $\bigcap_{i<n}S_i=\emptyset$, and it is an $n$-gap when the sets $S_i$ are pairwise disjoint.
\begin{thm}\label{standardbelowstrong} If $\Delta$ is a strong $n_\ast$-gap, then there exist $\{S_i : i<n\}$ disjoint subsets of $n\times n$, such that $\{\Gamma_{S_i} : i\in n\} \leq \Delta$. Moreover, the sets $S_i$ can be chosen so that $(i,i)\in S_i$. \end{thm}
Theorem~\ref{minimalstronggaps} follows from these results, as there are only finitely many choices for sets $\{S_i : i<n\}$. Theorem~\ref{standardbelowstrong} is proven combining Theorem~\ref{criticalstronggap} and Theorem~\ref{strongRamsey}: We start with an injective function $u:n^{<\omega}\longrightarrow N$ given by Theorem~\ref{criticalstronggap} and then for every $i,j,k$ we color the $(i,j)$-combs $X$ into two colors depending whether $u(X)\in \Delta_k$ or $u(X)\not\in\Delta_k$, and we successively apply Theorem~\ref{strongRamsey}, so that at the end, we get a restriction of $u$ to a set $A\approx n^{<\omega}$ that witnesses that $\{\Gamma_{S_i} : i\in n\}|_A \leq \Delta$.\\
We arrive to the conclusion that, if we are interested in studying properties of strong analytic $n_\ast$-gaps that can be reduced using the order $\leq$, then we must study gaps of the form $\{\Gamma_{S_i} : i<n\}$ made of combs as above. We should understand in particular when we have $\{\Gamma_{S_i}:i<n\}\leq \{\Gamma_{S'_i}:i<n\}$ for different families $S_i$ and $S'_i$ of combs. This requires understanding what kind of transformations $\varepsilon:n^2\longrightarrow m^2$ are induced by one-to-one functions $\phi:n^{<\omega}\longrightarrow m^{<\omega}$, in the sense that $\phi$ is a one-to-one map that sends $(i,j)$-combs to $\varepsilon(i,j)$-combs.
\begin{thm} For every function $\phi:n^{<\omega}\longrightarrow m^{<\omega}$ there exists $T\subset n^{<\omega}$ such that $T\approx n^{<\omega}$ and $\phi(A)\approx \phi(B)$ whenever $A\cup B\subset T$ and $A\approx B$. \end{thm}
As a corollary, if $\varepsilon:n^2\longrightarrow m^2$ is induced by some injective function $\phi:n^{<\omega}\longrightarrow m^{<\omega}$, then we can suppose that $\phi$ satisfies that $\phi(A)\approx \phi(B)$ whenever $A\approx B$. All the relevant information about a function $\phi$ satisfying this regularity property is determined by the $\approx$-equivalence class of the family $$\{\phi(\emptyset),\phi(0),\phi(1),\ldots,\phi(n-1)\}.$$ For technical reasons, the behavior of $\phi$ is better studied by looking at a \emph{normalization} of this family to a single level, like
$$\{\phi(\emptyset),\phi(0)|_m,\phi(1)|_m,\ldots,\phi(n-1)|_m\}$$
where $m=|\phi(n-1)|$. This family is renamed in \cite{stronggaps} as $$\{e(\infty),e(0),e(1)\ldots,e(n-1)\}.$$ A non-degeneration argument allows to suppose that $e(i)\neq e(j)$ if $i\neq j$. From this family, we can recover all the information needed about $\phi$ because we have that $\varepsilon(i,j) = (u,v)$ if \begin{itemize} \item either $i\neq j$, $t = e(i)\wedge e(j)$, $e(i) \geq t^\frown u$ and $e(j) \geq t^\frown v$, \item or $i=j$, $t = e(\infty)\wedge e(i)$, $e(i)\geq t^\frown u$, $e(\infty)\geq t^\frown v$. \end{itemize}
Conversely, each one-to-one function $e:\{\infty,0,1,\ldots,n-1\}\longrightarrow m^{<\omega}$ where $|e(\infty)|<|e(0)|=|e(1)|=\cdots=|e(n-1)|$ is associated to a function $\phi$.\\
Let us illustrate the way of reasoning by looking at 2-gaps. There are four kind of combs in the dyadic tree: $(0,0)$, $(0,1)$, $(1,0)$ and $(1,1)$. We know that every analytic strong 2-gap contains one of the form $\{\Gamma_{S_0},\Gamma_{S_1}\}$ where $(i,i)\in S_i$. There are $3^2 = 9$ ways of distributing the other two combs $(0,1)$ and $(1,0)$ between $\Gamma_{S_0}$ and $\Gamma_{S_1}$. This provides a list nine 2-gaps such that every analytic strong 2-gaps contains one of them. However, some of these 9 gaps are comparable to others. There are six equivalence classes of minimal analytic strong 2-gaps, whose representatives are given in the following table of comb distributions:
\begin{center} $
\begin{array}{|c|l|l|} \hline
& \Gamma_0 & \Gamma_1\\ \hline 1 & (0,0) , (0,1) & (1,1) , (1,0) \\ \hline 2 & (0,0) & (1,1) \\ \hline 3 & (0,0) & (1,1) , (0,1), (1,0)\\ \hline 3^\ast & (0,0) , (0,1) , (1,0) & (1,1)\\ \hline 4 & (0,0) & (1,1) , (1,0) \\ \hline 4^\ast & (0,0) , (0,1) & (1,1)\\ \hline \end{array} $ \end{center}
The gaps enumerated as $k$ and $k^\ast$ is because one is the permutation of the other. In order to check that this list is correct we need to check two things: first, that each of the three gaps that have been excluded from the list contain some gap from the list, and second that the gaps in the list are incomparable to each other. We check a particular case of each task as an illustration. The possibility $\tilde{S}_0 = \{(0,0),(1,0)\}$, $\tilde{S}_1 = \{(1,1)\}$ does not appear in the list: we check that the gap number $4^\ast$ is already below that gap. We need to find an injective map $\phi:2^{<\omega}\longrightarrow 2^{<\omega}$ that takes combs from $S_0(4^\ast)$ to combs from $\tilde{S}_0$, combs from $S_1(4^\ast)$ to combs from $\tilde{S}_1$, and combs out of $S_0 \cup S_1(4^\ast)$ to combs out of $\tilde{S}_0\cup \tilde{S}_1$ (the last condition is necessary for the preservation of the orthogonals). We can suppose that $\phi$ satisfies $A\approx B \Rightarrow \phi(A)\approx \phi(B)$ and such a function is determined by its associated family $\{e(\infty),e(0),e(1)\}$. Consider the function $\phi$ for which $e(\infty) = (0)$, $e(0) = (11)$, $e(1) = (01)$. The explicit formula for this $\phi$ is $$\phi(i_0,i_1,\ldots,i_k) = (0,1-i_0,0,1-i_1,\ldots,0,1-i_k)$$ and it is easy to check that $\phi(0,0)=(1,0)$, $\phi(1,1)=(1,1)$, $\phi(0,1) = (0,1)$, $\phi(1,0)=(0,1)$, and hence $\phi$ is as desired. Now, let us check that the gap number 3 is not below gap number 4. Suppose that there exists an injective function $\phi:2^{<\omega}\longrightarrow 2^{<\omega}$ that witnesses that relation. We look at its associated $\{e(\infty),e(0),e(1)\}$. So let $t = e(0)\wedge e(1)$, $t^\frown i = e(0)$, $t^\frown j = e(1)$, $i\neq j$. Then $\phi$ takes $(0,1)$-combs to $(i,j)$-combs and $(1,0)$-combs to $(j,i)$-combs. It is impossible then that $\phi$ witness that gap 3 is below gap 4.\\
The technique of analyzing each possible injective function $\phi:n^{<\omega}\longrightarrow m^{<\omega}$ by means of its associated set $\{e(\infty),e(0),e(1)\ldots,e(n-1)\}$ is quite effective and it allows to compute the list of minimal analytic strong $n$-gaps for every $n$. Each equivalence class is determined by certain parameters $(A,B,C,D,E,\psi,\mathcal{P},\gamma)$. In the case $n=3$, there are $4^6=4096$ ways of distributing the combs of $3^{<\omega}$ into three pairwise disjoint sets where $(0,0)\in S_0$, $(1,1)\in S_1$, $(2,2)\in S_2$. But there are only 31 (9 up to permutation) equivalence classes of minimal analytic strong 3-gaps. We refer to \cite{stronggaps} for further details.
\section{Record combinatorics of the $n$-adic tree and analytic gaps}\label{sectionweak}
The content of this section has some analogy to that of Section~\ref{sectionstrong}, but with additional difficulties, so we recommend the reader to understand Section~\ref{sectionstrong} first. We shall make use of all the notations about the $n$-adic tree introduced at the beginning of Sections~\ref{sectioncritical} and~\ref{sectionstrong}.
In the record combinatorics, we are interested in problems where the relevant structure is given by the meet operation $t\wedge s$ of two nodes $t,s$, the order $t\prec s$, and the record history from a node $t$ to a larger node $s>t$. By the record history we mean the sequence of nodes $$record(t,s) = \{t=t_0<t_1<\cdots<t_{k+1}=s\}$$ and the sequence of integers $$0\leq m_0<m_1<\cdots<m_k<n$$ such that $t_i^\frown m_i \leq s$ and $m_i = \max(t_{i+1}\smallsetminus t_i)$ for all $i=0,\ldots,k$. That is, we start climbing up from $t$ to $s$ and we take note of each time that we reach an integer which is strictly larger than all the integers we saw before (each time that we \emph{break a record}). We are considering thus a finer structure than in the first-move case where only the first record $m_0$ was relevant to the structure.\\
More precise definitions: A set $A\subset n^{<\omega}$ is said to be record-closed if the following two properties hold: \begin{enumerate} \item $t\wedge s$ whenever $t\in A$ and $s\in A$, \item $record(t,s)\subset A$ whenever $t\in A$, $s\in A$, $t<s$. \end{enumerate} The record-closure of $A$, denoted by $\langle A\rangle$ is the intersection of all record-closed sets that contain $A$. A bijection $f:A\longrightarrow B$ is a record-equivalence if it is the restriction of a bijection $f:\langle A\rangle\longrightarrow \langle B\rangle$ such that for every $t,s\in \langle A\rangle$ \begin{enumerate} \item $f(t\wedge s) = f(t)\wedge f(s)$ \item $f(t) \prec f(s)$ if and only if $t\prec s$ \item If $i\in n$ is such that $t^\frown i \leq s$, then $f(t)^\frown i \leq f(s)$. \end{enumerate}
The sets $A$ and $B$ are called record-equivalent if there is a record-equivalence between them. In this case, we write $A\sim B$. A partition theorem analogous to that of Milliken holds in this context:
\begin{thm}\label{weakRamsey} Fix a set $A_0\subset n^{<\omega}$. If we color the set $\{A\subset n^{<\omega} : A\sim A_0\}$ into finitely many colors in a Baire-measurable way, then there exists a set $T\subset n^{<\omega}$ such that $T\sim n^{<\omega}$ and all the sets from $\{A\subset T : A\sim A_0\}$ have the same color. \end{thm}
A proof of this theorem is found in \cite[Section 1]{analyticmultigaps}. We promised a proof of Theorem~\ref{strongRamsey} from Theorem~\ref{weakRamsey}. This is actually very simple: It is enough to consider the function $\psi:n^{<\omega}\longrightarrow n^{<\omega}$ given by $\psi(s_0,s_1,\ldots,s_k) = (s_0,n-1,s_1,n-1,\ldots,s_k,n-1)$. This function has the property that if $A\approx B$, then $A\approx \psi(A)\sim \psi(B)\approx B$. It is now a straightforward excercise to deduce Theorem~\ref{strongRamsey} from Theorem~\ref{weakRamsey} using $\psi$.\\
Similarly as in the strong case, the $[i]$-chains that appeared in the critical $n$-gap can be reinterpreted now in the following way: A set $X\subset n^{<\omega}$ is an $[i]$-chain if and only if $$ X\sim \{(i), (ii), (iii), (iiii),\ldots\}$$ The next step is to identify the anologous of the $(i,j)$-combs in the first-move combinatorics, that is, the minimal record-equivalence classes of infinite sets. This equivalence classes are not parametrized just by couples of integers $(i,j)$ but by more complicated beings that we call \emph{types}. A type $\tau = (\tau^0,\tau^1,\triangleleft)$ in $n^{<\omega}$ consists of two sets $\tau^0\subset n$ and $\tau^1\subset n$, together with a total order relation $\triangleleft$ on $\tau^0\times\{0\}\cup\tau^1\times\{1\}$ such that: \begin{itemize} \item $\tau^0\neq \emptyset$, \item $\min(\tau^0) \neq \min(\tau^1)$, \item $(k,i)\triangleleft (k',i)$ whenever $i\in\{0,1\}$ and $k<k'$, \item $(\max(\tau^0),0)$ is the maximal element in the order $\triangleleft$. \end{itemize} Given a type $\tau$ where $\tau^0 = \{a_1^0<\ldots<a_p^0\}$ and $\tau^1 = \{a^1_1<\ldots<a^1_q\}$, consider \begin{eqnarray*} u^\tau &=& (a^0_1,\ldots,a^0_1, a^0_2,\ldots,a^0_2,\ \ldots\ , a_p^0,\ldots, a_p^0)\\ v^\tau &=& (a^1_1,\ldots,a^1_1, a^1_2,\ldots,a^1_2,\ \ldots\ , a_q^1,\ldots, a_q^1) \end{eqnarray*}
where the number of times $r(k,i)$ that $a^i_k$ is repeated satisfies that $r(k,i) > 2^{r(k',j)}$ whenever $(a^i_k,i) \triangleright (a^j_{k'},j)$. A set $X$ is of type $\tau$ if $$X \sim \{v^\tau, {u^\tau}^\frown v^\tau, {u^\tau}^\frown {u^\tau}^\frown v^\tau,\ldots\}$$ Again, these happen to be the minimal record-equivalence classes of infinite sets: on the one hand, every infinite subset of a set of type $\tau$ has also type $\tau$, and on the other hand every infinite set contains an infinite subset which is of type $\tau$ for some type $\tau$. We represent a type as two rows of integers between square-brackets, the lower sequence represents $\tau^0$, the upper sequence represents $\tau^1$ and the order $\triangleleft$ is read from left to right. Thus, $\tau = [^{23}{}_{1}{}^4{}_2]$ means that $\tau^0 = \{1,2\}$, $\tau^1=\{2,3,4\}$ and $(2,1)\triangleleft (3,1)\triangleleft (1,0)\triangleleft (4,1)\triangleleft (2,0)$. If there is only one row, it means that $\tau^1=\emptyset$ and the row represents $\tau^0$. When $\tau^1=\emptyset$, the sets of type $\tau$ are chains, otherwise they are antichains. The sets of type $[i]$ are precisely the $[i]$-chains appearing in the critical analytic $n$-gap. We denote by $\mathfrak{T}_n$ the set of all types in $n^{<\omega}$.\\
For $S$ a set of types in $n^{<\omega}$, let now $\Gamma_S$ be the family of all subsets of $n^{<\omega}$ which are of type $\tau$ for some $\tau\in S$. Notice that $\Gamma_S$ is a closed family, in particular analytic.
\begin{thm} If $\{S_i : i<n\}$ are nonempty sets of types in $m^{<\omega}$, then the families $\{\Gamma_{S_i} : i\in n\}$ are not separated. \end{thm}
If $X$ is of type $\tau$ and $Y$ is of type $\sigma$, $\tau\neq \sigma$, then $|X\cap Y|\leq 3$. Hence, if the $\bigcap_{i<n}S_i=\emptyset$, then $\{\Gamma_{S_i} : i<n\}$ is an analytic $n_\ast$-gap. If the ${S_i}$'s are pairwise disjoint, then $\{\Gamma_{S_i} : i<n\}$ is an analytic $n$-gap. Similarly as in the strong case, now the fact is that every analytic $n$-gap contains an $n$-gap made of types in the $n$-adic tree.
\begin{thm}\label{standardbelowgap} If $\Delta$ is an $n_\ast$-gap, then there exist $\{S_i : i<n\}$ sets of types in $n^{<\omega}$, such that $\{\Gamma_{S_i} : i\in n\} \leq \Delta$. Moreover, the sets $S_i$ can be chosen so that $[i]\in S_i$. \end{thm}
Analogous arguments to those that we exposed after Theorem~\ref{standardbelowstrong} show that Theorem~\ref{standardbelowgap} follows from Theorem~\ref{criticalgap} and Theorem~\ref{weakRamsey}.\\
There are eight types in the dyadic tree $2^{<\omega}$, namely $[0]$, $[1]$, $[01]$, $[^0{}_1]$, $[^{01}{}_1]$, $[^1{}_0]$, $[^1{}_{01}]$ and $[_0{}^1{}_1]$. Theorem~\ref{standardbelowgap} states that any analytic 2-gap contains a permutation of one which is obtained by putting sets of type $[0]$ on one side, sets of type $[1]$ on the other side, and distributing the rest of types in whatever way. That means a list of $2\cdot 3^6 = 1458$ gaps so that any analytic gap contains one of them. But there are actually only nine equivalence classes of minimal analytic 2-gaps:\\
\begin{center} $
\begin{array}{|l|l|l|} \hline
& \Gamma_0 & \Gamma_1\\ \hline 1 & [0] & \text{all other types} \\ \hline 1^\ast & \text{all other types} & [0] \\ \hline 2 & [0] & [1]\\ \hline 2^\ast & [1] & [0]\\ \hline 3 & [0] & [1] , [01]\\ \hline 3^\ast & [1], [01] & [0]\\ \hline 4 & [0] , [01] & [1] \\ \hline 5 & [0] & [1] , [01] , [^1 {}_0 {}_1]\\ \hline 5^\ast & [1] , [01] , [^1 {}_0 {}_1] & [0]\\ \hline \end{array} $ \end{center}
In order to refine the information provided by Theorem~\ref{standardbelowstrong} to obtain reduced lists of minimal gaps, or in general to obtain more precise facts about analytic gaps, it is necessary to understand when we have $\Gamma\leq\Delta$ for gaps given by types, and this means understanding what kind of functions $\bar{\phi}:\mathfrak{T}_n\longrightarrow \mathfrak{T}_m$ are induced by injective functions $\phi:n^{<\omega}\longrightarrow m^{<\omega}$ in the sense that $\phi$ sends sets of type $\tau$ to sets of type $\bar{\phi}\tau$.
\begin{thm}\label{normalembedding} For every function $\phi:n^{<\omega}\longrightarrow m^{<\omega}$ there exists $T\subset n^{<\omega}$ such that $T\sim n^{<\omega}$ and $\phi(A)\sim \phi(B)$ whenever $A\cup B\subset T$ and $A\sim B$. \end{thm}
The functions satisfying the regularity condition above are called normal embeddings. When $\phi$ is a normal embedding, we denote by $\bar{\phi}$ its action on types, so that $\phi$ sends sets of type $\tau$ to sets of type $\bar{\phi}\tau$. The situation is not as simple as in the strong case, and in particular, the equivalence class of $\{\phi(\emptyset),\phi(0),\ldots,\phi(n-1)\}$ does not determine the whole behavior of a normal embedding $\phi$. A number of facts about normal embeddings are proven in \cite{analyticmultigaps}. Just to get a flavor, for a type $\tau$ consider $\max(\tau) = \max(\tau^0\cup \tau^1)$ the largest integer appearing in the type $\tau$, then one can prove that if $\max(\tau)\leq \max(\sigma)$, then $\max(\bar{\phi}\tau) \leq \max(\bar{\phi}\sigma)$, and this is a basic fact needed in many computations. A variety of peculiar classes (top-combs, top$^2$-combs, chains...) and characteristics ($\max(\tau)$, $strength(\tau)$...) of types show up in studying this kind of combinatorics.\\
As an example of how things work, let us briefly explain why some of the types in $2^{<\omega}$ never appeared in the list of minimal 2-gaps. A type $\tau$ is a top-comb type if the second number from the right in its representation exists and lives in the upper row. Equivalently, we can say that $(\max(\tau^1),1)$ is the second-largest element of the order $\triangleleft$. A type $\tau$ dominates a type $\sigma$ if $\tau$ is a top-comb and $\max(\sigma)\leq \max(\tau^1)$. Domination is characterized in terms of normal embeddings as follows:
\begin{thm}\label{dominationthm} Let $\tau_0,\tau_1$ be types in $m^{<\omega}$. The following are equivalent: \begin{enumerate} \item $\tau_{1}$ dominates $\tau_0$, \item There exists a normal embedding $\phi:2^{<\omega}\longrightarrow m^{<\omega}$ such that $\bar{\phi}[0] = \tau_0$, and $\bar{\phi}\sigma = \tau_1$ for all types $\sigma \neq [0]$. \end{enumerate} \end{thm}
The types $[^{01}{}_1]$ and $[^1{}_0]$ dominate any other type in $2^{<\omega}$. So the above result shows that if one of these two types appears somewhere in the gap $\Gamma$, then either the gap number 1 or number $1^\ast$ from the list of minimals above will be below $\Gamma$. This means that the original list of $2\cdot 3^6 = 1458$ gaps can be refined to the $2\cdot 3^4 = 162$ gaps that exclude $[^{01}{}_1]$ and $[^1{}_0]$.
The idea behind Theorem~\ref{dominationthm} can be understood by analyzing the particular case when $\tau_0 = [0]$ and $\tau_1 = [_0{}^1{}_1]$. The normal embedding $\phi:2^{<\omega}\longrightarrow 2^{<\omega}$ for these two types is constructed in the following way: Fix $\{x_0,x_1,x_2,\ldots\}$ a set of type $\tau_1 = [_0{}^1{}_1]$, and then define $\phi$ so that $$\phi(s^\frown 1^\frown 0^\frown \cdots^\frown 0) = x_{n_s}{}^\frown 0^\frown\cdots^\frown 0$$ (The number $n_s$ and the number of repetitions of 0 are defined inductively so that $\phi(t)\prec \phi(t')$ if $t\prec t'$). This satisfies that $\bar{\phi}[0]=[0]$ and $\bar{\phi}\sigma = [_0{}^1{}_1]$ for all other types. Notice that if we consider a similar construction for the non-top-comb type $[^1{}_{01}]$ instead, then we will obtain that $\bar{\phi}\sigma = [^1{}_{01}]$ for some types and $\bar{\phi}\sigma = [_0{}^1{}_1]$ for other types.
\section{Breaking gaps: jigsaws and clovers}\label{sectionbreak}
When we have a mutliple gap $\{\Gamma_i : i\in n\}$ we can consider its subgaps, that is, the gaps of the form $\{\Gamma_i : i\in A\}$ where $A\subset n$. We can also consider restrictions $\{\Gamma_i|_a : i\in A\}$ where $a\subset \omega$, $\Gamma_i|_a = \{x\in\Gamma_i: x\subset a\}$, that may be still gaps or may become separated. The behavior of these operations can be quite different for different kinds of gaps.
\begin{dfn}
Let $\Gamma = \{\Gamma_i : i<n\}$ be an $n$-gap, and let $B\subset n$. We say that $\Gamma$ is $B$-broken if there exists an infinite set $M\subset N$ such that $\{\Gamma_i|_M : i\in B\}$ is a gap, but $M\in\Gamma_i^\perp$ for $i\not\in B$. \end{dfn}
Let us consider two examples to illustrate this concept.\\
\begin{enumerate}
\item First, consider the critical 3-gap $\mathcal{C}^3$ in $3^{<\omega}$. Can this 3-gap be $\{0,1\}$-broken? Yes, it is enough to take $M = 2^{<\omega}\subset 3^{<\omega}$. On the one hand, $M\in (\mathcal{C}^3_2)^\perp$ because a set of type $[2]$ intersects $2^{<\omega}$ in at most one point. On the other hand $\{\mathcal{C}^3_0|_{2^{<\omega}},\mathcal{C}^3_1|_{2^{<\omega}}\} = \mathcal{C}^2$ is still a gap.\\
\item Now, consider another example $\Delta = \{\Delta_i : i<3\}$ in $2^{<\omega}$, where $\Delta_0$ is the family of all sets of type $[0]$, $\Delta_1$ is the family of all sets of type $[1]$, and $\Delta_2$ is the family of all sets of type $[01]$. This time, the 3-gap $\Delta$ cannot be $\{0,1\}$-broken. Let us sketch a proof. Suppose that we have $M\subset 2^{<\omega}$ that \emph{breaks} $\Delta$. By Theorem~\ref{standardbelowgap}, we can find a gap $\Delta'\leq \{\Delta_0|_M, \Delta_1|_M\}$ of the form $\Delta' = \{\Gamma_{S_0},\Gamma_{S_1}\}$ where $S_0$ and $S_1$ ares sets of types with $[\varepsilon 0]\in S_0$, $[\varepsilon 1]\in S_1$ for some permutation $\varepsilon:2\longrightarrow 2$. The fact that $\Delta'\leq \{\Delta_0|_M, \Delta_1|_M\}$ must be witnessed by a certain function $\phi:2^{<\omega}\longrightarrow M\subset 2^{<\omega}$, which by Theorem~\ref{normalembedding} can be supposed to be a normal embedding. Then $\phi$ must send sets of type of $[\varepsilon 0]$ onto sets of type $[0]$, and sets of type $[\varepsilon 1]$ onto sets of type $[1]$. We mentioned the property that normal embeddings satisfy $$\max(\sigma)\leq \max(\tau) \Rightarrow \max(\bar{\phi}\sigma)\leq \max(\bar{\phi}\tau)$$ Hence $\varepsilon 0 = 0$ and $\varepsilon 1 = 1$. It is an easy exercise that if $\phi$ preserves sets of type $[0]$ and sets of type $[1]$, then it also preserves sets of type $[01]$, and this contradicts that $M=\phi(2^{<\omega})\in\Delta_3^\perp$.\\ \end{enumerate}
The argument for the first example shows something more general: the critical $n$-gap $\mathcal{C}^n$ can be $B$-broken for all $B\subset n$, just by taking $M=B^{<\omega}$. Since this happens to be a minimal $n$-gap, it is actually a \emph{jigsaw}, according to the following definition:
\begin{dfn}
We say that an $n$-gap $\Gamma$ is a jigsaw if for every $B\subset A\subset n$ and for every $M\subset N$, if $\{\Gamma_i|_M : i\in A\}$ is not separated, then it can be $B$-broken. \end{dfn}
Jigsaws are multiple gaps that can be broken \emph{everywhere and in every way}, and we have just shown that there are very natural analytic examples of them. The critical strong $n$-gap is also a jigsaw. Anoter example of analytic jigsaw which is moreover \textit{dense} is the minimal analytic $n$-gap $\{\Gamma_i : i<n\}$ in $n^{<\omega}$, where $\Gamma_i$ is the family of all sets of type $\tau$ with $\max(\tau)=i$. We highlight the definition of dense gap since it will be important in later discussions:
\begin{dfn} An $n$-gap $\Gamma=\{\Gamma_i : i<n\}$ in $N$ is dense if every infinite subset of $N$ contains an infinite set from $\bigcup_{i\in n}\Gamma_i$. \end{dfn}
When an $n$-gap is made of types in $m^{<\omega}$, density means that each type of $m^{<\omega}$ is included in some $\Gamma_i$.\\
Let us look now at the second example $\Delta$ that we proposed. In that case, the argument for showing that $\Delta$ cannot be $\{0,1\}$-broken does not generalize to other pairs. Indeed, the reader can try to check as an excercise that $\Delta$ can be $\{0,2\}$-broken and also $\{1,2\}$-broken. Is it possible to produce an $n$-gap which cannot be $B$-broken for any $B\subset n$, $|B|\geq 2$?. Such $n$-gaps are called \emph{clovers}, and yes, here is an example of a clover \cite[Proposition 19]{multiplegaps}: Consider $2^\omega = \bigcup_{i<n}X_i$ a decomposition of the Cantor set into Bernstein sets. Remember that a set $X\subset 2^\omega$ is called Bernstein if both $C\cap X$ and $C\smallsetminus X$ are uncountable for every uncountable compact subset $C\subset 2^\omega$. Such sets can be constructed by transfinite induction, cf.\cite[Example 8.24]{Kechris}. Each point $x=(x_0,x_1,\ldots)\in 2^\omega$ can be identified with the corresponding branch $b_x = \{(x_0,\ldots,x_{k-1}) : k<\omega\}\subset 2^{<\omega}$. The clover is then $$\tilde{\Delta} = \{\tilde{\Delta}_i = \{b_x : x\in X_i\} : i<n\}.$$ This clover is not an analytic $n$-gap: the construction of Bernstein sets requires transfinite induction, which means a too complicated definition to be analytic. Can we construct an analytic clover? No, we cannot. All analytic $n$-gaps can be $B$-broken for \emph{many} subsets $B\subset n$. We do not have a precise meaning for the word \emph{many}, and it is actually an interesting problem to characterize those families $\mathcal{B}$ of subsets of $n$ for which there is an analytic $n$-gap which can be $B$-broken if and only if $B\in\mathcal{B}$. We need to improve our understanding of the \emph{record combinatorics} of the $n$-adic to solve such a question. In the case of 3-gaps we do have such a characterization, and says that the two examples that we presented at the beginning of this section are the only possibilities:
\begin{thm}\label{3break} Let $\Gamma = \{\Gamma_i : i<3\}$ be an analytic 3-gap. Then $\Gamma$ can be $B$-broken for at least two out of the three sets $B\subset \{0,1,2\}$ of cardinality 2. \end{thm}
The proof of a theorem like this runs as follows: we can suppose that $\Gamma$ is a minimal analytic $3$-gap, hence made of types. The 61 types of the 3-adic tree, we can be distributed into $4^{61}$ ways. Using the combinatorial techniques explained in Section~\ref{sectionweak} for the study of types, this can be reduced to a list of 933 minimal analytic gaps, only 163 if counted up to permutation. Finally, we just have to check that each of them satisfies the statement of Theorem~\ref{3break}. Of course, some shortcuts are possible, but this describes the general procedure of finitizing the problem. It was too much work already to try to find the list of minimal analytic 4-gaps, but playing a little bit with types, normal embeddings, etc. one can prove partial results like the following:
\begin{thm}\label{2break} Let $\Gamma = \{\Gamma_i : i<n\}$ be an $n$-gap. Then, there exists $B\subset n$ of cardinality 2 such that $\Gamma$ can be $B$-broken. \end{thm}
We have another general result about breaking gaps that follows immediately from the theory explained in Section~\ref{sectionweak}. For every $m<\omega$, let $J(m)$ be the number of types that exists in $m^{<\omega}$. Thus, $J(2)=8$, $J(3)=61$, etc.
\begin{thm}\label{Jbreak}
For every analytic $n$-gap $\Gamma$ and every set $B\subset n$, there exists a set $B\subset C \subset n$ such that $|C|\leq J(|B|)$ and $\Gamma$ can be $C$-broken. \end{thm}
The proof would run as follows: Suppose, without loss of generality that $B=m=\{0,\ldots,m-1\}$. Apply Theorem~\ref{criticalgap} to the gap $\{\Gamma_i : i<m\}$, and get (perhaps after a permutation) an injective function $u:m^{<\omega}\longrightarrow N$ such that $[i]$-chains are sent to elements of $\Gamma_i$. Using Theorem~\ref{weakRamsey}, we can suppose that, for each type $\tau$, either all sets of type $\tau$ are sent by $u$ to some $\Gamma_{f(\tau)}$, or all sets of type $\tau$ are sent to $(\bigcup_{i<n}\Gamma_i)^\perp$. The image of $u$ is then orthogonal to all $\Gamma_i$ which are not of the form $\Gamma_{f(\tau)}$. Since there are only $J(m)$ types in $m^{<\omega}$ we conclude that the image of $u$ is orthogonal to all but $J(m)$ of the $\Gamma_i$'s.\\
The function $J$ is optimal for Theorem~\ref{Jbreak}. A particular instance of this result asserts that if $\Gamma$ is an analytic $n$-gap, and $B\subset n$ is a set of cardinality 2, then there exists $B\subset C \subset n$ with $|C|\leq J(2) = 8$ such that $\Gamma$ can be $C$-broken. Optimality means that we can find an $8$-gap which cannot be $C$-broken for any $C$ with $\{0,1\}\subset C \subsetneq \{0,1\ldots,7\}$. This 8-gap is nothing else than considering $\{\tau_0,\ldots,\tau_7\}$ the 8 types of the dyadic tree, and then define $\Gamma$ so that $\Gamma_i$ is the family of all sets of type $\tau_i$.\\
To finish this section, we would like to make some comments going back to the non-analytic world. We can relativize the notions of jigsaw and clover to a given $B\subset n$: \begin{itemize} \item We say that $\Gamma$ is a $B$-clover if it cannot be $B$-broken,
\item We say that $\Gamma$ is a $B$-jigsaw if $\{\Gamma_i|_M : i\in A\}$ can be $B$-broken whenever $A\supset B$, $M\subset N$ and $\{\Gamma_i|_M : i\in A\}$ is a gap, \end{itemize}
and then we can ask about mixed versions of jigsaws and clovers: Given a family $\mathfrak{X}$ of subsets of $n$, can we find an $n$-gap which is a $B$-jigsaw for $B\in \mathfrak{X}$ and a $B$-clover for $B\not\in\mathfrak{X}$, $|B|\geq 2$? In the analytic case, we know that this is possible only for certain families $\mathfrak{X}$ and this corresponds to the previous discussion. If we ask about arbitrary $n$-gaps, then we were able to give a positive answer to this question \cite[Theorem 26]{multiplegaps} assuming the existence of a completely separable almost disjoint family. We do not know if this can be proved from ZFC alone. Actually it is unknown if the existence of completely separable almost disjoint families follows from ZFC, they are known to exist under certain assumptions like $\mathfrak{c}<\aleph_\omega$ or $\mathfrak{s}<\mathfrak{a}$~\cite{sane}.\\
\section{Multiple gaps which are countably separated}\label{sectioncountablyseparated}
Some structural theory of gaps is possible in the general non-analytic case when the gaps are countably separated. This is done in \cite[Section 5]{multiplegaps}. Given a topological space $L$ and a subset $a\subset L$, let $acc(a)$ be the set of accumulation points of $a$, that is $$acc(a) = \bigcap\left\{\overline{a\smallsetminus c} : c\text{ finite}\right\}$$ Fix a countable dense subset $D$ of the Cantor set $L=2^\omega$, and for each subset $G\subset L$, define: $$\mathcal{I}_G = \{a\subset D : acc(a)\subset G\}$$ We say that the sets $\{G_i : i<n\}$ are separated by open sets if there exist open sets $U_i\supset G_i$ such that $\bigcap_{i<n}U_i = \emptyset$. The $n_\ast$-gaps obtained in this way are \emph{above} any countably separated $n_\ast$-gap:
\begin{thm} Let $\{G_i : i<n\}$ be subsets of $L$ which cannot be separated by open sets and $\bigcap_{i<n}G_i = \emptyset$. Then $\{\mathcal{I}_{G_i} : i<n\}$ is an $n_\ast$-gap which is countably separated. \end{thm}
\begin{thm} Let $\{\Gamma_i : i<n\}$ be a countably separated $n_\ast$-gap on a countable set $N$. Then there exists a bijection $\phi:N\longrightarrow D$, and sets $\{G_i : i<n\}$ as in the above theorem such that $\phi(\Gamma_i)\subset \mathcal{I}_{G_i}$. \end{thm}
Say that $\{\Gamma_i : i<n\}$ is strongly countably separated if $\{\Gamma_i : i\in A\}$ is countably separated for each $A\subset n$. The class of \emph{dense strongly countably separated $n$-gaps} is an extreme class of gaps. Among the minimal analytic $n$-gaps, the only ones which are dense and strongly countably separated are the permutations of $\{\Gamma_{M_i} : i<n\}$ where $M_i$ is the set of types $\tau$ with $\max(\tau)=i$. The point is that this is the only class where we know that the structural theory of analytic gaps extends to general non-analytic gaps.
\begin{thm} If $\Delta = \{\Delta_i : i<n\}$ is a dense strongly countably separated $n$-gap, then there is a permutation $\Gamma^\sigma$ of $\{\Gamma_{M_i} : i<n\}$ such that $\Gamma^\sigma \leq \{\Delta^{\perp\perp}_i : i<n\}$. \end{thm}
Taking the biorthogonal is a necessary, but not relevant for most applications, restriction in this case. For example, since $\{\Gamma_{M_i} : i<n\}$ can be checked to be a jigsaw, we get, without analyticity assumptions:
\begin{cor} Every dense strongly countably separated $n$-gap is a jigsaw. \end{cor}
We notice that the gap $\{\Gamma_{M_i} : i<n\}$ is the same as $\{\mathcal{I}_{G_i} : i<n\}$, where in the above definition of ideals $\mathcal{I}_G$, we consider $L$ to be a countable compact scattered space of height $n+1$ instead of the Cantor set, $D$ is the set of isolated points of $L$, and $G_i = L^{(i+1)}\smallsetminus L^{(i+2)}$ are the levels in the Cantor-Bendixson derivation of $L$.
\section{Multiple gaps and the topology of $\beta\omega\smallsetminus\omega$}\label{sectionomegastar}
The \v{C}ech-Stone compactification $\beta\omega$ is a topological space characterized by the following properties: \begin{enumerate} \item $\beta\omega$ contains the set of natural numbers, $\omega\subset\beta\omega$ as a dense subset, \item each $n\in\omega$ is an isolated point of $\beta\omega$, \item $\beta\omega$ is a compact space, \item If $a,b\subset\omega$ and $a\cap b = \emptyset$, then $\overline{a}\cap \overline{b} = \emptyset$. \end{enumerate}
When we remove isolated points, we obtain the \v{C}ech-Stone remainder $\omega^\ast = \beta\omega\smallsetminus \omega$ of the natural numbers. The point about these spaces is that all problems abouts the family of subsets of $\omega$, endowed with the order $\subset$ can be translated into topological problems on $\beta\omega$, and all problems related to the contention modulo finite $\subset^\ast$ (like those related to gaps) can be translated into topological problems about $\omega^\ast$. This translation procedure is called \emph{Stone duality}, and the reader who is not familiar may be referred to \cite{Walker, Semadeni} as two among many places where one can learn about it. We shall explain now how the gap theory translates through this duality, providing some appealing statements about the space $\omega^\ast$. No detailed argumets will be found in this section, but the fact is that all results are simple excercises of translation through Stone duality, once one gets a basic familiarity with it.\\
Each open subset $U$ of $\omega^\ast$ can be associated to a family $I(U)$ of subsets of $\omega$, $$I(U) = \{a\subset \omega : acc(a) \subset U\}$$
The families $I(U)$ and $I(V)$ are orthogonal if and only if $U\cap V =\emptyset$. The families $\{I(U_i) : i<\omega\}$ can be separated if and only if $\bigcap_{i<n}\overline{U_i} = \emptyset$. Hence, an $n$-gap corresponds to a finite family of pairwise disjoint open subsets of $\omega^\ast$ whose closures have nonempty intersection.\\
Let us start by translating Theorem~\ref{aleph1}:
\begin{thm} Assume $MA_{\aleph_1}$, and let $U,V,W$ be three pairwise disjoint open subsets of $\omega^\ast$, each of which is a union of $\aleph_1$ many closed sets. Then $\overline{U}\cap\overline{V}\cap\overline{W} = \emptyset$. \end{thm}
This constrasts with the fact that, by Hausdorff's construction, there exist two disjoint open sets $U,V$, each a union of $\aleph_1$ many closed sets, such that $\overline{U}\cap\overline{V} \neq\emptyset$.\\
Let us say that an open set $U\subset\omega^\ast$ is analytic if $I(U)$ is analytic. The fact that $\{I(U_i) : i<n\}$ can be $B$-broken is translated topologically into the existence of a point $x$ such that $x\in \bigcap_{i\in B}\overline{U_i}$ but $x\not\in \bigcup_{i\not\in B}\overline{U_i}$. The restriction discussed on Section~\ref{sectionbreak} on the ways that an analytic gap can be broken, translate now into certain forbidden configurations for the partial intersections of closures in a family of analytic open sets. For example, Theorem~\ref{2break} traslates as:
\begin{thm} If $\{U_i : i<n\}$ are pairwise disjoint analytic open subsets of $\omega^\ast$, then \begin{enumerate} \item Either $\{\overline{U_i} : i<n\}$ are pairwise disjoint
\item Or there exists $x\in\omega^\ast$ such that $|\{i<n : x\in\overline{U_i}\}|=2$ \end{enumerate} \end{thm}
Just for fun, let us state the version of Theorem~\ref{Jbreak} for $k=3$. The number $J(3) = 61$ is an optimal bound.
\begin{thm}
If $\{U_i : i<n\}$ are pairwise disjoint analytic open subsets of $\omega^\ast$ such that $\bigcap_{i<3}\overline{U_i} \neq\emptyset$, then there exists $x\in\bigcap_{i<3}\overline{U_i}$ such that $|\{i<n : x\in\overline{U_i}\}|\leq 61$ \end{thm}
\section{Selective coideals and analytic almost disjoint families}\label{sectionalmostdisjoint}
An family $\mathcal{A}$ of infinite subsets of the countable set $N$ is said to be an almost disjoint family if $a\cap b$ is finite for every $a,b\in\mathcal{A}$, $a\neq b$. An example of a closed almost disjoint family is the set $\mathcal{B}$ of all branches of the dyadic tree $2^{<\omega}$. That is, $\mathcal{B} = \{b_x : x\in 2^\omega\}$ where $b_x = \{(x_0,x_1,\ldots,x_{k-1}) : k<\omega\}$ if $x=(x_0,x_1,\ldots)$. The following theorem asserts that every uncountable analytic almost disjoint family contains a copy of the family $\mathcal{B}$ inside.
\begin{thm}\label{almostdisjoint} If $\mathcal{A}$ is an uncountable analytic almost disjoint family of subsets of a countable set $N$, then there exists two injective functions $u:2^{<\omega}\longrightarrow N$ and $a:2^\omega\longrightarrow \mathcal{A}$ such that \begin{enumerate} \item $u(b_x) \subset a(x)$ for each $x\in 2^\omega$, \item $u(c)\in \mathcal{A}^\perp$ whenever $c\in \mathcal{B}^\perp$. \end{enumerate} \end{thm}
A more general version of Theorem~\ref{almostdisjoint} can be stated in terms of so-called selective coideals and is provided in \cite{analyticmultigaps}. The proof combines results of Mathias~\cite{Mathias} with the theory explained in Section~\ref{sectionweak}. The idea is to consider the gap $\{\mathcal{A}^\perp,\mathcal{A}\}$, to which we can apply Theorem~\ref{firstdichotomy}. Combining it with Theorem~\ref{weakRamsey}, we find $\{\Gamma_{S_0},\Gamma_{S_1}\}\leq \{\mathcal{A}^\perp,\mathcal{A}\}$ where $S_i$ are sets of types with $[i]\in S_i$. The special structure of the almost disjoint family rules out any top-comb type to belong to $\mathcal{A}$. Composing with the function $w(s_0,s_1,\ldots,s_k) = (1,s_0,1,s_1,\ldots1,s_k)$ one obtains the desired function $u$.
\section{Analytic multiple gaps and sequences in Banach spaces}\label{sectionsequences}
Gaps can be a tool to analyze how different classes of subsequences \emph{get mixed} inside a sequence $(x_n)_{n<\omega}$. We shall illustrate this way of thinking by considering sequences of vectors in normed spaces and certain classes of subsequences. We could, for instance, fix a number $p\in [1,\infty)$ and consider the sequences of vectors which are equivalent to the canonical basis of $\ell_p$, that is:
\begin{dfn} Let $1\leq p <\infty$. We say that a sequence of vectors $(x_n)_{n<\omega}$ in a normed space is called an $\ell_p$-sequence if there exists $C>0$ such that for every $n_1,\ldots,n_k<\omega$ and every scalars $\lambda_1,\ldots,\lambda_p$
$$ (\clubsuit) \ \frac{1}{C} \cdot \left(\sum_{i=1}^k |\lambda_i|^p \right)^{\frac{1}{p}} \leq \left\|\sum_{i=1}^k \lambda_i x_{n_i}\right\| \leq C\cdot \left(\sum_{i=1}^k |\lambda_i|^p \right)^{\frac{1}{p}}$$ \end{dfn}
Let us consider now finite sets $\vec{p} \subset [1,+\infty)$
\begin{dfn} A sequence $(x_n)_{n<\omega}$ of vectors in a normed space is called $\vec{p}$-saturated, if for every subsequence $(x_n)_{n\in A}$ there exists a further subsequence $(x_n)_{n\in B}$, $B\subset A$, which is an $\ell_{p}$-sequence for some $p\in\vec{p}$. \end{dfn}
If we are given a $\{p_i: i<n\}$-saturated sequence, we can consider the following families of subsets of $\omega$: $$\Gamma_i = \left\{A\subset \omega : \{x_m\}_{m\in A}\text{ is an }\ell_{p_i}\text{-sequence}\right\}$$ These families are mutually orthogonal, because an infinite sequence cannot be an $\ell_p$-sequence and an $\ell_q$-sequence at the same time, for $p\neq q$. Moreover, the families $\Gamma_i$ are analytic, because $\Gamma_i$ can be written as a countable union of countable intersection of closed families as $$\bigcup_{C\in\omega\smallsetminus\{0\}}\bigcap_{n_1,\ldots,n_k<\omega}\bigcap_{\lambda_1,\ldots,\lambda_k\in\mathbb{Q}}\left\{A: \{n_1,\ldots,n_k\}\not\subset A, \text{ or }(\clubsuit)\text{ holds for }p=p_i \right\}$$
The fact that $\{\Gamma_i : i<n\}$ form an $n$-gap is equivalent to $(x_n)_{n<\omega}$ being a $\vec{p}$-sequence, according to the following definition:
\begin{dfn} A sequence $(x_n)_{n<\omega}$ of vectors in a normed space is called a $\vec{p}$-sequence, if it is $\vec{p}$-saturated and we cannot find any finite decomposition $\omega = \bigcup_{i<k}A_i$ so that each $(x_n)_{n\in A_i}$ is $\vec{q}_i$-saturated for some $\vec{q}_i\subsetneq \vec{p}$. \end{dfn}
It is not really more restrictive to study $\vec{p}$-sequences than $\vec{p}$-saturated sequences, because every $\vec{p}$-saturated sequence can be decomposed into a finite union of subsequences, each of which is a $\vec{q}$-sequence for some $\vec{q}\subset\vec{p}$. We should think of a $\vec{p}$-sequence as a sequence $\{x_m\}_{m<\omega}$ of vectors which contains many subsequences equivalent to $\ell_p$, for $p\in\vec{p}$, and moreover all these subsequences are \emph{well mixed} inside the sequence $\{x_m\}_{m<\omega}$.\\
As we explained above, each $\vec{p}$-sequence gives rise to an analytic $n$-gap $\{\Gamma_i : i<n\}$ which is moreover dense. Therefore, all the theory of analytic $n$-gaps that we have developed can be applied in this context. For each fixed $n$, we know that there are only a finite number of minimal canonical forms in which a $\vec{p}$-sequence can be produced with $|\vec{p}|=n$, that correspond to the minimal analytic $n$-gaps which are dense. The fact that the gap $\{\Gamma_i : i<n\}$ can be $B$-broken is equivalent to saying that the sequence $(x_m)_{m<\omega}$ contains a $\{p_i : i\in B\}$-sequence. Thus, given a $\vec{p}$-sequence, the problem of knowing for which $\vec{q}\subset \vec{p}$ does $(x_m)_{m<\omega}$ contain a $\vec{q}$-sequence, happens to be a nontrivial problem related to the discussion of Section~\ref{sectionbreak}. Just as an example, this is a corollary of Theorem~\ref{2break}:
\begin{thm}
If $|\vec{p}|\geq 2$, then every $\vec{p}$-sequence has a subsequence which is a $\vec{q}$-sequence for some $\vec{q}\subset \vec{p}$ with $|\vec{q}|=2$. \end{thm}
The tricky thing is that $\vec{q}$ cannot be chosen arbitrarily. We know that there exist dense analytic gaps $\{\Delta_i : i<n\}$ made opf types which cannot be broken for certain $B\subset n$, $|B|=2$. For ecample, if $\Gamma_{\{\tau\}}$ is the family of all subsets of $2^{<\omega}$ of type $\tau$, then $\{\Gamma_{\{\tau\}} : \tau\in\mathfrak{T}_2\}$ cannot be $\{[0],[1]\}$-broken. Out of that, one gets $\vec{q}\subset\vec{p}$ with $|\vec{q}|=2$, for which it is possible to construct a $\vec{p}$-sequence which does not contain any $\vec{q}$-subsequence.
\section{Gaps, selectors, and spaces of continuous functions}\label{sectionCK}
Our original motivation to introduce multiple gaps was some problems concerning operators on the space $C(\omega^\ast)$ of continuous functions on $\omega^\ast$, better known under the name $\ell_\infty/c_0$ among Banach space theorists. We briefly explain these problems in this section. We tried to make it very much self-contained, still we refer to \cite{AlbiacKalton} for some basics on Banach spaces, and to \cite{Semadeni} as a classical reference on Banach spaces of continuous functions.
\begin{dfn} Let $X\subset Y$ be Banach spaces, and $\lambda\geq 1$. We say that $X$ is $\lambda$-complemented in $Y$ if there exists a linear operator $P:Y\longrightarrow X$ such that \begin{enumerate} \item $P(x) = x$ for all $x\in X$,
\item $\|P(x)\| \leq \lambda \|x\|$ for all $x\in Y$. \end{enumerate} \end{dfn}
We are interested only in a particular class of Banach spaces, those of the form $C(K)$. Given a compact space $K$, the space $C(K)$ is the Banach space of real-valued continuous functions on $K$, endowed with the norm $\|f\| = \max\{|f(x)| : x\in K\}$. The assignment $K\mapsto C(K)$ is a contravariant functor, in the sense that each continuous map $f:L\longrightarrow K$ corresponds to a composition operator $f^0:C(K)\longrightarrow C(L)$ given by $f^0(\phi) = \phi\circ f$. If $f$ is onto, then $f^0$ is an injective isometry that identifies $C(K)$ as a subspace of $C(L)$.
Suppose that we are given a continuous surjection $f:L\longrightarrow K$, and we view $C(K)$ as a subspace of $C(L)$ via the operator $f^0$. How can we know, just studying $f$ in topological terms, whether the space $C(K)$ is $\lambda$-complemented in $C(L)$? This situation was analysed by Milutin~\cite{Mil}, and furhter by Pe\l czy\'{n}ski~\cite{Pelcz} and Ditor~\cite{Ditor}. The first observation is that if $f$ has a continuous selector $s:K\longrightarrow L$, then $C(K)$ is 1-complemented in $C(L)$ by the projection $P(\phi) = \phi\circ s$. The necessary and sufficient condition is that $f$ has a generalized continuous selector which takes measures in $L$ as values, instead of just points. Namely, consider $M_\lambda(L)$ the set of all regular Borel measures on $L$ of variation less than or equal to $\lambda$, endowed with the coarsest topology that makes the map $\mu \mapsto \int_L \phi d\mu$ continuous for each $\phi\in C(L)$. Then,
\begin{prop}\label{averaging} The space $C(K)$ is $\lambda$-complemented in $C(L)$ if and only if there exists a continuous function $s:K\longrightarrow M_\lambda(L)$ such that $$\int_L (\phi\circ f) ds(x) = \phi(x)$$ for all $x\in K$ and $\phi\in C(K)$. \end{prop}
The projection induced by the generalized selector $s$ is given by $P(\varphi)(x) = \int_L \varphi ds(x)$. The fact that all possible projections are induced by generalized selections follows from the Riesz representation theorem, that states that the dual space $C(L)^\ast$ is identified with the regular Borel measures of finite variation on $L$. We refer to \cite{Semadeni} where all these things are explained in detail.\\
What does all this have to do with gaps? Consider a dense 2-gap $\Gamma = \{\Gamma_0,\Gamma_1\}$ on $\omega$. We already know that it is easy to construct such a thing, we just have to distribute the eight types of the dyadic tree into $\Gamma_0$ and $\Gamma_1$. We can associate two open sets of $\omega^\ast$, $U_i = \bigcup\{acc(a) : a\in \Gamma_i\}$. Because $\{\Gamma_0,\Gamma_1\}$ is a gap, this translates into the fact that $U_0\cap U_1 = \emptyset$ but $\overline{U_0}\cap \overline{U_1} \neq \emptyset$. The density implies that $\overline{U_0}\cup \overline{U_1} = \omega^\ast$. Now let $K=\omega^\ast$, and the \emph{splitted space} $L = \overline{U_0}\times\{0\} \cup \overline{U_1}\times\{1\}$. We have the \emph{gluing} continuous surjection $f:L\longrightarrow \omega^\ast$, given by $f(x,i) = x$. We are in the situation described above, and using Proposition~\ref{averaging} we can show that $C(\omega^\ast)$ is not 1-complemented in $C(L)$. Namely, suppose that $s:\omega^\ast\longrightarrow M_1(L)$ was a generalized selection. For $x\in U_i$ we have no other choice than taking $s(x) = \delta_{(x,i)}$, the Dirac measure concentrated in the poing $(x,i)$. But then, taking $y\in\overline{U_0}\cap \overline{U_1}$, the continuity of $s$ would imply that $s(y)$ should be both equal to $\delta_{(y,0)}$ and $\delta_{(y,1)}$, a contradiction. In this way we proved that the existence a dense 2-gap implies:
\begin{prop} There is a superspace in which $C(\omega^\ast)$ is not 1-complemented. \end{prop}
This is just a consequence of Goodner-Nachbin's characterization \cite{Goodner,Nachbin} of 1-injective Banach spaces of continuous functions (those which are 1-complemented in every superspace), cf. \cite{AlbiacKalton}. Indeed the connection between gaps and 1-injectivity is deeper than what is suggested at first sight by the above arguments. For example, the lack of countably generated gaps stated in Proposition~\ref{separationofcountable} has the following consequence, cf.~\cite{extr}:
\begin{prop} $C(\omega^\ast)$ is 1-complemented in every superspace $Y$ for which $Y/C(\omega^\ast)$ is separable. \end{prop}
Now, what if we want to get a superspace in which $C(\omega^\ast)$ is not complemented at all (that is, it is not $\lambda$-complemented for any $\lambda\geq 1$). The previous argument to kill all 1-projections does not work for large $\lambda$. For instance, if we are allowed to take a selection $s:K\longrightarrow M_3(L)$, then we are not anymore obliged to take $s(x) = \delta_{(x,0)}$ for $x\in U_0$. We could take $s(x) = \delta_{(x,0)} - \delta_{(y,0)} + \delta_{(y,1)}$ where $y\in \overline{U}_0\cap\overline{U}_1$. Indeed, if we imagine that $\overline{U}_0\cap\overline{U}_1 = \{y\}$, then this does give a generalized selector, together with $s(x) = \delta_{(x,1)}$ for $x\not\in U_0$. We need to have at least $\lambda\geq 2$ to play this kind of trick. But we may need even larger $\lambda$ if we want to apply the trick twice or more times, if we have many intersections of closures to deal with. Namely, if we have now a dense $n$-gap $\Gamma = \{\Gamma_i : i<n\}$, and associated open sets $\{U_i : i<n\}$, we can consider again the splitted space $L = \bigcup_{i<n}\overline{U_i}\times\{i\}$ and the continuous gluing surjection $f:L\longrightarrow\omega^\ast$ given by $f(x,i)=x$, and we have that
\begin{prop}\label{jigsawditor} If $\Gamma$ is a dense $n$-jigsaw, and $n=2^m$, then $C(\omega^\ast)$ is not $m$-complemented in $C(L)$. \end{prop}
Being a jigsaw is translated topologically into the fact that every point $x\in\bigcap_{i\in A}\overline{U_i}\smallsetminus\bigcap_{i\not\in A}\overline{U_i}$ belongs to the closure of $\bigcap_{i\in B}\overline{U_i}\smallsetminus\bigcap_{i\not\in B}\overline{U_i}$ whenever $B\subset A$. These are the \emph{many intersections of closures} mentioned above, that require more and more norm on the measures to be able to get a generalized continuous selector. We will not explain the details of this here, we refer to \cite[Section 8]{multiplegaps}, and the results of \cite{Ditor} cited there. We have already mentioned in Section~\ref{sectionbreak} that there are natural examples of analytic dense jigsaws to provide inputs for Proposition~\ref{jigsawditor}. It is an easy excercise (look at the last paragraph of the paper \cite{multiplegaps}) to glue together all the $C(L)$'s given by Proposition~\ref{jigsawditor} into a single one:
\begin{thm} There is a superspace $C(\omega^\ast)\subset C(L)$ in which $C(\omega^\ast)$ is not $\lambda$-complemented for any $\lambda$. \end{thm}
This result was originally proved by Amir~\cite{Amir}. Both Amir's and our approach allow to obtain a space $C(L)$ of the cardinality of the continuum. Our original (unsuccessful) intention when introducing multiple gaps was trying to apply these ideas to deal with the following problem posed in \cite{extr}: Is there a superspace $Y\supset C(\omega^\ast)$ in which $C(\omega^\ast)$ is not complemented and $Y/C(\omega^\ast)$ has density $\aleph_1$? Of course, we mean if this is provable in ZFC alone. If we say 1-complemented instead of complemented then the answer is: yes, and the space $Y$ can be obtained out of a Haudorff's gap like in Theorem~\ref{Hausdorff}. However, we cannot hope multiple gaps to provide examples of that kind, because of Theorem~\ref{aleph1}.
\end{document} | arXiv |
Can a single molecule have a temperature?
A show on the weather channel said that as a water molecule ascends in the atmosphere it cools. Does it make sense to talk about the temperature of a single molecule?
thermodynamics statistical-mechanics temperature molecules
RichardbernsteinRichardbernstein
$\begingroup$ I think it makes sense to talk about the kinetic energy of a molecule, which is where the Kinetic Theory of Ideal Gases comes from: en.wikipedia.org/wiki/Kinetic_theory $\endgroup$ – Greg May 24 '13 at 1:29
$\begingroup$ That in turn describes the temperature of a collection of molecules. $\endgroup$ – Greg May 24 '13 at 1:34
$\begingroup$ The statistical mechanical definition of temperature is T = (∂E/∂S). Since entropy is directly related to the number of states, I suppose you could define a temperature for a molecule. Not sure how it'd be very useful though. $\endgroup$ – Nick May 24 '13 at 4:09
I think it is a mistake, as often happens in popularizations of science.
A water or any molecule may lose kinetic energy and acquire potential energy, but it is the kinetic energy distribution that gives the temperature of an ensemble of molecules. The shape of the distribution shows that there will always be individual molecules at very high energy, in the ensemble, which they acquire from the random individual collisions.
From the link,$$ f_{\varepsilon} \left(\varepsilon\right)\,\mathrm{d}\varepsilon ~=~\sqrt{\frac{1}{\varepsilon \pi kT}} \, \exp{\left(-\frac{\varepsilon}{kT}\right)}\,\mathrm{d}\varepsilon \,,$$and the shape shows that there always exist tails to high energies. The attribution of temperature labels to individual molecules is wrong.
Maxwell–Boltzmann probability density function, where $a=\sqrt{\frac{kT}{m}}$:
$\hspace{150px}$ .
$\begingroup$ Sure but if you have one molecule coupled to a thermostat and the system is ergodic then the distribution you are talking about can be thought as being a frequency with which each state is visited over a very long period of time. For once ergodicty is useful in this case $\endgroup$ – gatsu May 24 '13 at 8:24
$\begingroup$ The curves in the picture appear to be parameterized by something called $a$, but there is no $a$ in the equation they're presumably meant to illustrate. $\endgroup$ – WillO Apr 4 '17 at 13:32
$\begingroup$ @WillO it is in the link alpha=sqrt(kT/m), three different temperatures distributions for a fixed mass. $\endgroup$ – anna v Apr 4 '17 at 18:02
$\begingroup$ The equipartition theorem relates temperature to degrees of freedom that appear quadratically in the hamiltonian. So potential energy contributes as well, not just kinetic. For an ideal gas there is only kinetic energy so we get simplified treatments. However, I will admit that the entire subject of relating temperature to energy, and thermal energy vs. internal energy, leaves me somewhat confused. $\endgroup$ – garyp Feb 2 '19 at 18:14
$\begingroup$ @garyp I am familiar within this type of equipartiton which has no potential energy there.hyperphysics.phy-astr.gsu.edu/hbase/Kinetic/eqpar.html . In any case everything is about mean and average, even in your link,which needs more than one particle to manifest, imo $\endgroup$ – anna v Feb 2 '19 at 18:56
Without intending any disrespect, I'm quite surprised that several very knowledgeable people have given a wrong, or at least incomplete, answer to this old question.
For a single molecule that is in complete isolation, it is indeed generally not true (or at least not useful) to assign it a temperature, as others have said. Such a system would be more naturally described in the so-called microcanonical ensemble of thermodynamics, and since it can have a well-defined and conserved energy, the usual role of temperature in determining the probability of occupation of different energy states via a Boltzmann distribution is not relevant. Put simply, temperature is only relevant when there is uncertainty about how much energy a system has, which need not be true when it is isolated*.
However, things are different when you have a molecule in an open system, which can freely exchange energy with its surroundings, as is certainly the case for the specific example the OP has described. In this case, as long as the molecule is in equilibrium or quasi-equilibrium with its surroundings, it does indeed have a well-defined temperature. If there are no other relevant conserved quantities, the quantum state of the molecule is described by a diagonal density matrix in the single-particle energy basis that follows the Boltzmann distribution, $\rho=Z^{-1} e^{-\beta H}$ . Practically speaking, this means that if you know that the molecule is at equilibrium with a given temperature, each time you measure it you can know, probabilistically, what the likelihood is that you will see it with a given energy.
*For completeness I will mention that some people have nevertheless tried to extend the idea of temperature to isolated systems, as the wiki mentions, but this temperature doesn't generally behave in the way you expect from open systems, and it isn't a very useful concept.
RococoRococo
$\begingroup$ I had missed your answer before writing mine, eventually writing about the same idea, in a looser manner! $\endgroup$ – user154997 Sep 29 '17 at 9:49
$\begingroup$ @LucJ.Bourhis actually, it seems to me that they are somewhat different. My answer applies equally if there are no relevant internal degrees of freedom. It is just about a closed vs open system. It is essentially an elaboration on gatsu's comment to Anna's answer ( I don't mention ergodicity explicitly, but it is implied by the statement that the system can thermalize to some equilibrium state). $\endgroup$ – Rococo Sep 29 '17 at 15:42
$\begingroup$ Since you consider the Boltzmann distribution, you have an entropy $S=-k\text{Tr}(\rho\log\rho)$, which is the quantum equivalent of the $\sum p\log p$ in my answer. That said, indeed, in my answer, the molecule could be isolated. $\endgroup$ – user154997 Sep 29 '17 at 16:04
$\begingroup$ This is not really correct. If the molecule is coupled to its surroundings, then the temperature is a property of the whole system, not of the single molecule. $\endgroup$ – Ben Crowell Feb 2 '19 at 18:33
$\begingroup$ @BenCrowell I don't think I agree with this. I would say that if a subsystem (of arbitrary size) is in thermal equilibrium with a heat bath at temperature T, it is appropriate to say that the subsystem is itself at temperature T. $\endgroup$ – Rococo Feb 2 '19 at 21:30
As the other answers have said, temperature is a collective property and can only be defined when you have an assemblage of particles. However by definition in a molecule you have an assemblage of atoms, and they have relative motions described by the vibrational excitations of the molecule.
So if you have a large enough molecule you can look at the excitations of its vibrational modes and use these to define a temperature. In effect what you're doing is saying that the excitation of the vibrational modes is the same as it would be if the molecule was in equilibrium with some enviroment of the defined temperature.
However I don't think this could usefully be applied to a water molecule. The vibrational excitations of water are of greater than thermal energy at ambient temperatures, and in any case there are only two normal modes. I suppose you could look at the rotation of the molecule, but this would give you only a rough guide to temperature.
John RennieJohn Rennie
$\begingroup$ If it is a statistical phaenomena, as described by M-B, there must be a huge number of particles.., what is the argument -theory-to say that a single "big" molecule has a T? $\endgroup$ – user153036 Sep 29 '17 at 4:56
$\begingroup$ @HernanMiraola: the temperature is related to the energy in the internal degrees of freedom by the equipartition theorem. As long as you have enough internal degrees of freedom to be statistically significant you can assign a temperature just by looking at the rotational and vibrational modes of the molecule. $\endgroup$ – John Rennie Sep 29 '17 at 8:03
$\begingroup$ Good, but what is enough? And what happens when it is not enough? $\endgroup$ – user153036 Sep 29 '17 at 21:28
$\begingroup$ @santimirandarp: Basically all the things that are guaranteed to be exact in the macroscopic limit are only approximations in the microscopic limit. For example, the probability of violations of the second law is nonzero. $\endgroup$ – Ben Crowell Feb 2 '19 at 18:34
$\begingroup$ possibly interesting: Has a concept of temperature ever been defined in the context of a single atom? $\endgroup$ – uhoh Dec 16 '19 at 6:14
It makes sense if all you know about the molecule is its expected energy. Then you can show that it's energy distribution is the Boltzmann distribution $p(E) = e^{-E/kT}$ for some constant $T$, which is related to the expected energy.
So the question reduces to a philosophical view of probabilities. Does it make sense to assign probabilities to a deterministic system? If you accept probabilities as the reflection of your knowledge of the system rather than something intrinsic then it also makes sense to assign temperature to a single molecule.
SMeznaricSMeznaric
$\begingroup$ But in this case the expected temperature isn't a property of the molecule, but someone's prior belief about the molecule. So a single molecules still doesn't have a temperature, it has a particular but unknown energy. $\endgroup$ – innisfree Apr 20 '17 at 16:30
Thermodynamics makes sense when you have large numbers of particles. For example, the second law of thermodynamics has an extremely low probability of being violated when you have Avogadro's number's worth of particles. However, if you have a very small number of particles, the second law will frequently be violated.
This comes up in nuclear physics, where we routinely deal with nuclei consisting of 50 or 100 or 200 particles. Yes, we do talk about the temperature of an individual nucleus, and it does make sense.
However, a single water molecule is only 3 atoms, and in a system that size, it's nonsense to talk about temperature. In giant molecules, I can easily imagine that you could have enough atoms to talk about the temperature of an individual molecule.
$\begingroup$ This sounds like an application of the "sorites problem"...as you add grains of sand to a collection one by one, when does it become a "heap". Similar issue here? $\endgroup$ – Richardbernstein May 24 '13 at 13:21
$\begingroup$ @Richardbernstein: No, thermodynamics is just an approximation that becomes better and better as you have more particles. $\endgroup$ – Ben Crowell May 25 '13 at 19:05
$\begingroup$ the fact - that it always makes sense in large number of particles - does not imply it makes no sense to a few number of (or one) particles. $\endgroup$ – Shing Dec 26 '19 at 20:40
I will explain in simple....
We can measure temperature changes when a Source release energy or extract energy.
We know that every thing want to attain equilibrium. Therefore there should be an equality in temperature in all the atoms or molecules of a system.
The temperature of an atom will be same as that of temperature of surrounding.
But a thermometer is an external source of energy from surrounding. So it provides or extract energy from source.
Now Consider an $Ideal$ condition
Consider a atom placed in space and is away from every radiation from space and any atoms or any other gravity or electromagnetic forces.
Then the temperature will be $zero(Kelvin)$ theoretically and you cannot measure it experimentally !!
Creepy CreatureCreepy Creature
As stated in the other answers, in the specific scenario depicted in your question, it makes no sense to talk about the temperature of a molecule. But I can't help to widen the picture because there is a very interesting case here: what about the internal degrees of freedom of the molecule? So far everybody has only considered the motion of the whole molecule, because that was the context you implicitly gave, fair enough.
But what about the vibration of the molecular bonds? The rotations of two neighbour parts of the molecule about a bond? This is not as trivial! For small molecules, the number of associated degrees of freedom are too few to warrant talking about temperature. But it is not so for a large protein. There are so many of them that a statistical approach is possible: one can define an entropy, in the usual $\sum p\log p$ way, where $p$ is the probability for a given configuration. Then, as usual, if we have an entropy we can minimise it with the constraint of a given total energy, and a temperature comes out of that.
After writing this, I realised that @Rococo and @JohnRennie had written an answer on a similar line long before mine!
$\begingroup$ I believe John Rennie's answer also talked about this 4 years ago. $\endgroup$ – JMac Sep 29 '17 at 10:31
I humbly disagree with the majority of the other answers here. It can be very useful and make lots of sense to talk about the temperature of a single particle. You just need to realize that it's not quite the same (although it's pretty darn close) as the temperature defined by physicists in typical statistical mechanics and thermal physics studies.
Professional atomic physicists regularly refer to the "temperature" of single atoms, ions and molecules. See, for example, the discussion in this article. It's clearly not a temperature in the exactly same sense as a gas of particles has a temperature, but it's a very useful concept none-the-less. Temperature is basically being used as a proxy for the mean kinetic energy and distribution of energies that a single particle has over repeated realizations of a given experiment. In many cases the distribution is very well-approximated by a thermal Maxwell-Boltzmann distribution, and so assigning the particle a temperature associated with that distribution makes a lot of sense.
aquirdturtleaquirdturtle
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How is temperature related to quantum vibrational states of molecules? | CommonCrawl |
EURASIP Journal on Wireless Communications and Networking
System representation
Inverses of non-square polynomial matrices
New approach to signal reconstruction process
PSVD vs. Smith decomposition-based approach
Perfect reconstruction of signal—a new polynomial matrix inverse approach
Wojciech P. Hunek1Email authorView ORCID ID profile and
Paweł Majewski1
EURASIP Journal on Wireless Communications and Networking20182018:107
The paper outlines a new approach to the signal reconstruction process in multivariable wireless communications tasks. A new solution is proposed using the so-called Smith factorization, which is efficiently used in the synthesis of control systems described by polynomial matrix notation. In particular, the so-called polynomial S-inverse is used, which, together with the applied degrees of freedom, creates a potential for the improvement of the operation of wireless data communications systems comprising different numbers of inputs/antennas and outputs/antennas. Simulations performed in the Matlab environment indicate the practical applicability of the proposed solution.
Perfect signal reconstruction
Polynomial matrix approach
Polynomial matrix inverses
Non-square MIMO systems
In mobile wireless communications, increasing attention is paid to the quality and quantity of data transmitted in a given unit of time. A higher capacity of the radio channel of MIMO (multi-input/multi-output) systems, i.e., systems comprising multiple inputs/antennas and multiple outputs/antennas, is gradually replacing the traditional SISO (single-input/single-output) approach [1]. This is confirmed by the widely used WiMAX, WiFi 802.11n, DVB-T, or LTE/LTE advanced standards, the majority of which use the OFDM (orthogonal frequency division multiplexing) technology [2–5]. Therefore, an increased capacity of these systems requires the use of a large number of subcarriers and a parallel data transmission mechanism. An intriguing alternative can be therefore seen in systems based on different numbers of transmitting and receiving antennas, where, unlike in the SISO and square MIMO systems (identical numbers of inputs and outputs), the so-called non-uniqueness is present, thus creating viable possibilities of improving the efficiency of the non-square wireless communications systems. It should be emphasized that the drawback observed here in the form of inter-channel interference (ICI) is eliminated by applying the so-called SVD (singular value decomposition) form in the signal reconstruction process [6–8], dedicated solely to the traditional analysis based on a parameter matrix calculus [9–11]. On the other hand, in the approach based on the polynomial matrix calculus [12–14], the aforementioned dysfunction is eliminated by using the PSVD (polynomial SVD) [15–17]. Unfortunately, all mentioned approaches to the signal perfect reconstruction [18–20], also those including the Smith decomposition method [21], have not so far included the so-called degrees of freedom [22]. Therefore, they were solely related to a certain "optimal" solution associated with the application of the minimum-norm/least-squares inverses to an Eigen matrix obtained from the factorization process [11]. What is important is that even though the above methods involve an infinite number of pairs of precoder-equalizer, our degrees of freedom should be understood in terms of usage of the different inverses to Eigen matrix under a unique precoder-equalizer pair. It will be shown that former cases are quite inappropriate for the polynomial matrix description [16], and the Smith factorization method proposed here outperforms the classical solutions remarkably.
1.1 Method
In this paper, a new analytical solution to the signal perfect reconstruction is presented. Not only does this approach in discrete-time domain, dedicated to non-square systems, eliminates parasitic effects in the form of ICI and ISI (inter-symbol interference), but also it efficiently uses the highly expected mechanism of non-uniqueness, which considerably improves the "robustness" of wireless communications. The approach proposed herein is based on the so-called Smith form of non-square polynomial matrices, which is the foundation of the polynomial S-inverse [22, 23].
2 System representation
We carry out the analysis of the wireless data communications system with NT-transmitter antennas and NR-receiver antennas described by the (discrete) polynomial matrix in the form of \(\mathbf {\underline {C}}\left ({z^{-1}}\right)\) [16]
$$ {\mathbf{\underline{C}}_{N_{\mathrm{R}}\times N_{\mathrm{T}}}\left({z^{-1}}\right)}=\sum\limits_{n=0}^{L_{\mathrm{c}} -1} \mathbf{\underline {c}_{n}} z^{-n}, $$
where (Lc-1) is the order of the FIR (finite impulse response) matrix \(\mathbf {{\underline C}}\left ({z^{-1}}\right)\).
The deterministic signal reconstruction process is performed here in accordance with the difference equation
$$ {\mathbf{R}(t)}={\mathbf{\underline C}}\left({q^{-1}}\right){\mathbf{S}(t)}, $$
where the vector of transmitted signals S(t) and the vector of received signals R(t) have the dimensions NT and NR, respectively (see Section 4). Note that the symbol t denotes a discrete-time domain, whereas q−1 is the backward shift operator corresponding to z−1 one.
In the new perfect reconstruction approach presented in this paper, a number of different inverses are used; they are described in detail in the next section.
3 Inverses of non-square polynomial matrices
Due to the non-square form of \(\mathbf {\underline {{C}}}_{N_{\mathrm {R}}\times N_{\mathrm {T}}}\left ({z^{-1}}\right)\), the authors suggest using new inverses of non-square polynomial matrices [22–25] in the signal reconstruction process [22]. We start with the classical minimum-norm right and least-squares left inverses known as T-inverses in the polynomial case.
3.1 T-inverses
For the polynomial matrix \(\mathbf {{\underline {C}}}\left (q^{-1}\right)=\mathbf {\underline {c}_{0}} + \mathbf {\underline {c}_{1}}q^{-1} +\ldots +\mathbf {{\underline {c}}_{m}}q^{-m}\) of full normal rank, the unique minimum-norm right T-inverse is defined as
$$ \mathbf{{\underline{C}}}_{0}^{\mathrm{R}} \left(q^{-1}\right)=\mathbf{{\underline{C}}}^{\mathrm{T}} \left(q^{-1}\right)\left[\mathbf{{\underline{C}}} \left(q^{-1}\right) \mathbf{{\underline{C}}}^{\mathrm{T}} \left(q^{-1}\right)\right]^{-1}, $$
while the unique least-squares left T-inverse is in the following form
$$ \mathbf{{\underline{C}}}_{0}^{\mathrm{L}} \left(q^{-1}\right)=\left[\mathbf{{\underline{C}}}^{\mathrm{T}} \left(q^{-1}\right)\mathbf{{\underline{C}}} \left(q^{-1}\right)\right]^{-1} \mathbf{{\underline{C}}}^{\mathrm{T}} \left(q^{-1}\right). $$
3.2 τ-inverses
The non-unique right τ-inverse of the polynomial matrix \(\mathbf {{\underline {C}}}\left (q^{-1}\right)\) is defined as (NR<NT)
$$ \begin{aligned} \mathbf{{\underline{C}}}^{\mathrm{R}} \left(q^{-1}\right) =\left\{ \mathbf{I}_{N_{\mathrm{T}}} +\left[\mathbf{\underline{\beta}_{s}} \left(q^{-1}\right)\right]_{0}^{\mathrm{R}} \left[\vphantom{\left. -\mathbf{\underline{\beta}} \left(q^{-1}\right)\right]}\mathbf{{\underline{C}}} \left(q^{-1}\right) \right.\right.\\ \left.\left. -\mathbf{\underline{\beta}} \left(q^{-1}\right)\right] \right\}^{-1} \left[\mathbf{\underline{\beta}_{s}} \left(q^{-1}\right)\right]_{0}^{\mathrm{R}}, \end{aligned} $$
where polynomial matrices \(\mathbf {\underline {\beta }} \left (q^{-1}\right)\) and \(\mathbf {\underline {\beta }_{s}} \left (q^{-1}\right)\) are defined in References [22, 23]. On the other hand, the non-unique left τ-inverse takes the following form (NR>NT)
$$ \begin{aligned} \mathbf{{\underline{C}}}^{\mathrm{L}} \left(q^{-1}\right)=\left\{ \mathbf{I}_{N_{\mathrm{T}}} +\left[\mathbf{{\underline{\beta}}_{s}} \left(q^{-1}\right)\right]_{0}^{\mathrm{L}} \left[\vphantom{\left.-\mathbf{{\underline{\beta}}} \left(q^{-1}\right)\right]}\mathbf{{\underline{C}}} \left(q^{-1}\right) \right.\right.\\ \left.\left.-\mathbf{{\underline{\beta}}} \left(q^{-1}\right)\right] \right\}^{-1} \left[\mathbf{{\underline{\beta}}_{s}} \left(q^{-1}\right)\right]_{0}^{\mathrm{L}}. \end{aligned} $$
The aforementioned forms \(\left [\mathbf {{\underline {\beta }}_{s}} \left (q^{-1}\right)\right ]_{0}^{\mathrm {R}}\) and \(\left [\mathbf {{\underline {\beta }}_{s}} \left (q^{-1}\right)\right ]_{0}^{\mathrm {L}}\) stand for the minimum-norm right and least-squares left T-inverses of polynomial matrix \(\mathbf {{\underline {\beta }}_{s}} \left (q^{-1}\right)\), respectively, while \(\mathbf {I}_{N_{\mathrm {T}}}\) is the identity NT-matrix.
3.3 σ-inverses
A generalization of the polynomial τ-inverses is the so-called right
$$ \begin{aligned} \mathbf{{\underline{C}}}^{\mathrm{R}} \left(q^{-1}\right) =\left\{ \mathbf{I}_{N_{\mathrm{T}}} +\left[\mathbf{{\underline{\beta}}} \left(q^{-1}\right)\right]_{0}^{\mathrm{R}} \left[\vphantom{\left.-\mathbf{{\underline{\beta}}} \left(q^{-1}\right)\right]}\mathbf{{\underline{C}}} \left(q^{-1}\right) \right.\right.\\ \left.\left.-\mathbf{{\underline{\beta}}} \left(q^{-1}\right)\right] \right\}^{-1} \left[\mathbf{{\underline{\beta}}} \left(q^{-1}\right)\right]_{0}^{\mathrm{R}}, \end{aligned} $$
and left
$$ \begin{aligned} \mathbf{{\underline{C}}}^{\mathrm{L}} \left(q^{-1}\right) =\left\{ \mathbf{I}_{N_{\mathrm{T}}} +\left[\mathbf{{\underline{\beta}}} \left(q^{-1}\right)\right]_{0}^{\mathrm{L}} \left[\vphantom{\left. -\mathbf{{\underline{\beta}}} \left(q^{-1}\right)\right]}\mathbf{{\underline{C}}} \left(q^{-1}\right) \right.\right.\\\left.\left. -\mathbf{{\underline{\beta}}} \left(q^{-1}\right)\right] \right\}^{-1} \left[\mathbf{{\underline{\beta}}} \left(q^{-1}\right)\right]_{0}^{\mathrm{L}}, \end{aligned} $$
non-unique σ-inverses implementing the degrees of freedom in the form of an arbitrary matrix polynomial \(\mathbf {{\underline {\beta }}} \left (q^{-1}\right)\).
It should be emphasized that the new forms of polynomial right and left σ-inverses (including also the parameter cases) are given in References [24, 26] as follows:
$$ \mathbf{{\underline{C}}}^{\mathrm{R}} \left(q^{-1}\right)=\mathbf{{\underline{\beta}}}^{\mathrm{T}}\left(q^{-1}\right) \left[\mathbf{{\underline{C}}} \left(q^{-1}\right)\mathbf{{\underline{\beta}}}^{\mathrm{T}} \left(q^{-1}\right)\right]^{-1}, $$
$$ \mathbf{{\underline{C}}}^{\mathrm{L}} \left(q^{-1}\right)=\left[\mathbf{{\underline{\beta}}}^{\mathrm{T}}\left(q^{-1}\right) \mathbf{{\underline{C}}} \left(q^{-1}\right)\right]^{-1} \mathbf{{\underline{\beta}}}^{\mathrm{T}} \left(q^{-1}\right). $$
Crucial non-unique S-inverses are presented below. They are effectively used when designing robust communications systems.
3.4 S-inverses
Non-unique polynomial S-inverses are associated with the so-called Smith factorization of the polynomial matrix \(\mathbf {{\underline {C}}} \left (q^{-1}\right)\) to obtain
$$ \mathbf{{\underline{C}}} \left(q^{-1}\right)=\mathbf{{\underline{U}}}\left(q^{-1}\right) \mathbf{{\underline{\Sigma}}}\left(q^{-1}\right) \mathbf{{\underline{V}}}\left(q^{-1}\right), $$
where \(\mathbf {{\underline {U}}}\left (q^{-1}\right)\) and \(\mathbf {{\underline {V}}}\left (q^{-1}\right)\) are unimodular polynomial matrices, and the unique matrix polynomial \(\mathbf {{\underline {\Sigma }}}\left (q^{-1}\right)\) of dimension NR×NT includes the eigenvalues of \(\mathbf {\underline C}\left (q^{-1}\right)\). The right and left S-inverses are defined as
$$ \mathbf{{\underline{C}}}^{\mathrm{R}} \left(q^{-1}\right)=\mathbf{{\underline{V}}}^{-1}\left(q^{-1}\right) \mathbf{{\underline{\Sigma}}}^{\mathrm{R}} \left(q^{-1}\right) \mathbf{{\underline{U}}}^{-1}\left(q^{-1}\right), $$
$$ \mathbf{{\underline{C}}}^{\mathrm{L}} \left(q^{-1}\right)=\mathbf{{\underline{V}}}^{-1}\left(q^{-1}\right) \mathbf{{\underline{\Sigma}}}^{\mathrm{L}} \left(q^{-1}\right) \mathbf{{\underline{U}}}^{-1}\left(q^{-1}\right), $$
respectively, where (non-)unique right and left inverses of the polynomial matrix \(\mathbf {{\underline {\Sigma }}}\left (q^{-1}\right)\) include the degrees of freedom. Note that the parameter counterpart of the S-inverse strictly dedicated to state-space systems has been given in Reference [25].
It should be noted that all of the abovementioned inverses are reduced to the regular one \(\mathbf {\underline {C}}^{-1}\left (q^{-1}\right)\) in case of NR=NT.
4 New approach to signal reconstruction process
MIMO wireless communications systems, including multiple transmitter and receiver antennas, are becoming more and more common, and they even replace traditional SISO solutions by offering high transmission/reception channel capacity improvement. Multivariable systems ensure not only an increase in capacity but also, importantly, improvement without loss of the required technological parameters of the received signal. An intriguing case here is an approach implementing different numbers of transmitting and receiving antennas. In such non-square systems, we can find the non-uniqueness of the obtained solution, which has a positive impact on the whole signal reconstruction/recovery process. By selecting appropriate degrees of freedom of inverses, we can considerably influence the robustness and energy of the received signal (in the case of the control theory see Reference [26]). In the authors' opinion, the new method can eliminate the parasitic impact of the natural environment in the context of the applied inverses of non-square polynomial matrices. Such operations directly improve the signal transmission rate while maintaining approved quality standards. Of course, the entire signal perfect reconstruction process only occurs in case of NR>NT, since we have full information about the transmitted signal.
It is important that the proposed approach to perfect reconstruction of signal is based on the polynomial matrix calculus. The solutions used so far were based on the parameter matrix calculus, using unique inverses with the so-called Hermitian conjugates of certain (full rank) non-square matrices [27]. Unfortunately, this calculus does not include the aforementioned degrees of freedom, thus making it considerably more difficult to adjust to the detrimental impact of the environment on the data transmission process.
It should be emphasized that the unique right and left inverses including the Hermitian conjugates are not applicable in the time-domain signal perfect reconstruction approach presented here [23].
To illustrate the discussed problems, let us analyze the stochastic process of perfect reconstruction of signal and rewrite Eq. (2) to the form
$$ \mathbf{R^{\prime}}(t)=\mathbf{{\underline{C}}}\left(q^{-1}\right) \mathbf{{S}}(t)+ \mathbf{{\zeta}}(t), $$
where ζ(t) is the uncorrelated zero-mean Gaussian white noise at (discrete) time t.
Then, let us perform the non-unique Smith factorization of \(\mathbf {\underline {C}}\left (q^{-1}\right)\) and, at the same time, eliminate ICI and ISI parasitic effects
$$ \mathbf{{{R}}^{\prime}}(t)=\mathbf{{\underline{U}}}\left(q^{-1}\right)\mathbf{{\underline{\Sigma}}}\left(q^{-1}\right)\mathbf{{\underline{V}}}\left(q^{-1}\right) \mathbf{{{S}}}(t)+ \mathbf{{{\zeta}}}(t), $$
where the polynomial matrices \(\mathbf {{\underline {U}}}\left (q^{-1}\right)\) and \(\mathbf {{\underline {V}}}\left (q^{-1}\right)\) are the "equalizer" and the "precoder," respectively [22].
After using the S-inverse, the perfect reconstruction of signal for the selected NR>NT takes the following form
$$ \begin{aligned} \mathbf{S}(t)=\mathbf{{\underline{V}}}^{-1}\left(q^{-1}\right)\mathbf{{\underline{\Sigma}}}^{\mathrm{L}}\left(q^{-1}\right) \mathbf{{\underline{U}}}^{-1}\left(q^{-1}\right)\mathbf{{{R}^{\prime}}}(t) \\ -\mathbf{{\underline{V}}}^{-1}\left(q^{-1}\right)\mathbf{{\underline{\Sigma}}}^{\mathrm{L}}\left(q^{-1}\right) \mathbf{{\underline{U}}}^{-1}\left(q^{-1}\right) \mathbf{{{\zeta}}}(t), \end{aligned} $$
where the symbol "L" stands for the (non-)unique left inverse of matrix polynomial \(\mathbf {{\underline {\Sigma }}}\left (q^{-1}\right)\).
Of course, Eq. (16) can be rewritten in the following form
$$ \mathbf{S}^{\prime}(t)=\mathbf{{\underline{V}}}^{-1}\left(q^{-1}\right)\mathbf{{\underline{\Sigma}}}^{\mathrm{L}}\left(q^{-1}\right){\mathbf{\underline{U}}}^{-1}\left(q^{-1}\right)\mathbf{{{R}^{\prime}}}(t), $$
with \(\mathbf {S}^{\prime }(t)=\mathbf {S}(t)+\mathbf {{\underline {V}}}^{-1}\left (q^{-1}\right)\mathbf {{\underline {\Sigma }}}^{\mathrm {L}}\left (q^{-1}\right){\mathbf {\underline {U}}}^{-1}\left (q^{-1}\right)\mathbf {\zeta } (t)\) being a stochastic NT-input vector.
Taking into account the above considerations, for NR>NT, we obtain [28]
$$ {}\mathbf{\underline{\Sigma}}^{\mathrm{L}}\left(q^{-1}\right)=\left[\mathbf{\underline{D}}_{N_{\mathrm{T}} \times N_{\mathrm{T}}}\left(q^{-1}\right) \ \mathbf{\underline{M}}_{N_{\mathrm{T}} \times (N_{\mathrm{R}}-N_{\mathrm{T}})}\left(q^{-1}\right)\right], $$
where the polynomial matrices \(\mathbf {\underline {M}}\left (q^{-1}\right)\) and \(\mathbf {\underline {D}}\left (q^{-1}\right)\) include significant degrees of freedom and transmission zeros (if any in the \(\mathbf {\underline {C}}\left (q^{-1}\right)\) [23]), respectively. In case of absence of transmission zeros, we have \(\mathbf {\underline {D}}\left (q^{-1}\right)=\mathbf {I}_{N_{\mathrm {T}}}\).
Finally, based on the pilot knowledge, the optimal degrees of freedom of \(\mathbf {\underline {M}}\left (q^{-1}\right)\) are chosen according to the square performance index
$$ {}\begin{aligned} \mathbf{\underline{M}_{opt}}(q^{-1}) = \arg \! \mathop { \min }\limits_{\mathbf{\underline{M}} (q^{- 1})} \sum\limits_{t = 0}^{N - 1} \!{\left\{ {\left[ {\mathbf{S}^{\prime}(t) \!\! - \!\!\mathbf{S}(t)} \right]^{\mathrm{T}}\! \left[ {\mathbf{S}^{\prime}(t) \!\,-\,\! \mathbf{S}(t)} \right]} \right\}}, \end{aligned} $$
where N denotes the number of samples.
\(\mathbf {{\underline {\Sigma }}}^{\mathrm {L}}\left (q^{-1}\right)\) can also be obtained by using polynomial σ-inverses in two parallel forms
$$ \begin{aligned} \mathbf{{\underline{\Sigma}}}^{\mathrm{L}} \left(q^{-1}\right)=\left\{ \mathbf{{I}}_{N_{\mathrm{T}}} +\left[\mathbf{{\underline{\beta}}} \left(q^{-1}\right)\right]_{0}^{\mathrm{L}} \left[\vphantom{\left.-\mathbf{{\underline{\beta}}} \left(q^{-1}\right)\right]}\mathbf{\underline{\Sigma}} \left(q^{-1}\right)\right.\right. \\ \left.\left.-\mathbf{{\underline{\beta}}} \left(q^{-1}\right)\right] \right\}^{-1} \left[\mathbf{{\underline{\beta}}} \left(q^{-1}\right)\right]_{0}^{\mathrm{L}}, \end{aligned} $$
$$ \begin{aligned} \mathbf{{\underline{\Sigma}}}^{\mathrm{L}} \left(q^{-1}\right)=\left[\mathbf{{\underline{\beta}}}^{\mathrm{T}} \left(q^{-1}\right)\mathbf{\underline{\Sigma}} \left(q^{-1}\right)\right]^{\text{-1}}\mathbf{{\underline{\beta}}}^{\mathrm{T}} \left(q^{-1}\right), \end{aligned} $$
where \( \left [\mathbf {{\underline {\beta }}} \left (q^{-1}\right)\right ]_{0}^{\mathrm {L}}=\left [\mathbf {{\underline {\beta }}}^{\mathrm {T}} \left (q^{-1}\right)\mathbf {{\underline {\beta }}} \left (q^{-1}\right)\right ]^{-1} \mathbf {{\underline {\beta }}}^{\mathrm {T}} \left (q^{-1}\right)\) while \(\mathbf {{\underline {\beta }}} \left (q^{-1}\right)\) includes the degrees of freedom [24].
If we apply the unique left T-inverse directly to \(\mathbf {\underline {C}} \left (q^{-1}\right)\) or \(\mathbf {\underline {\Sigma }} \left (q^{-1}\right)\), we will obtain no degrees of freedom.
It should be pointed out that in square systems, i.e., systems having equal numbers of transmitting and receiving antennas, there are also no degrees of freedom. Therefore, the optimization cannot efficiently eliminate the impact of noise on the whole signal reconstruction process.
We start our optimization task in the Matlab environment with the degrees of freedom included in the parameter matrix; a more general case is based on the matrix polynomial.
Now, it is clear that in the deterministic case, we immediately obtain the perfect reconstruction of signal according to the following formula
$$ \mathbf{S}(t)=\mathbf{\underline{C}}^{\mathrm{L}}\left(q^{-1}\right)\mathbf{R}(t), $$
where \(\mathbf {\underline {C}}^{\mathrm {L}}\left (q^{-1}\right)\) denotes the abovementioned polynomial matrix S-inverse of full normal rank \(\mathbf {\underline {C}}\left (q^{-1}\right)\).
It should be emphasized that in the deterministic case of the signal perfect reconstruction as mentioned in Remark 7, the left inverse of \(\mathbf {\underline {C}}\left (q^{-1}\right)\) is not determined; due to the elimination of ISI and ICI drawbacks, the S-inverse has been applied. Therefore, assuming that \(\mathbf {R}(t)=\mathbf {\underline {C}}\left (q^{-1}\right)\mathbf {S}(t)\), the stochastic recovery task as presented in Eq. (16) can be rewritten in the following form
$$ \begin{aligned} \mathbf{S}(t)+\mathbf{{\underline{V}}}^{-1}\left(q^{-1}\right)\mathbf{{\underline{\Sigma}}}^{\text{L1}}\left(q^{-1}\right)\mathbf{{\underline{U}}}^{-1}\left(q^{-1}\right)\mathbf{{{\zeta}}}(t) \\ =\mathbf{{\underline{V}}}^{-1}\left(q^{-1}\right)\mathbf{{\underline{\Sigma}}}^{\mathrm{L}}\left(q^{-1}\right) \mathbf{{\underline{U}}}^{-1}\left(q^{-1}\right)\mathbf{{{R}}}(t) \\ +\mathbf{{\underline{V}}}^{-1}\left(q^{-1}\right)\mathbf{{\underline{\Sigma}}}^{\text{L2}}\left(q^{-1}\right)\mathbf{{\underline{U}}}^{-1}\left(q^{-1}\right)\mathbf{\zeta} (t), \end{aligned} $$
where \(\mathbf {\underline {\Sigma }}^{\text {L1}}\left (q^{-1}\right) \neq \mathbf {\underline {\Sigma }}^{\text {L2}}\left (q^{-1}\right)\), in general, under any \(\mathbf {\underline {\Sigma }}^{\mathrm {L}}\left (q^{-1}\right)\).
Since our new polynomial method of perfect signal reconstruction does not correspond to the Moore-Penrose inverse, we must consider them separately.
Remark 10
The aforementioned signal recovery may be impossible after using the unique T-inverses in case of non-square systems (previously known as minimum-norm right/least-squares left inverses); these inverses may significantly destabilize the whole signal reconstruction process due to the existence of so-called unstable control zeros [22]. The whole signal reconstruction process is always destabilized in case of the existence of unstable transmission zeros which are the modes of the fundamental system matrix \(\mathbf {\underline {C}}\left (q^{-1}\right)\) [22, 23].
Note that the entire task of signal reconstruction should be understood in terms of an adaptive process, where the said degrees of freedom are selected cyclically with a period adjusted by the designer.
What is important is that the solution based on the polynomial matrix calculus (along with non-zero degrees of freedom, i.e., at \(\mathbf {\underline {M}}\left (q^{-1}\right)\neq \mathbf 0\)), is a new approach so far unknown in the field of the modern signal reconstruction. The application of left inverses in Eq. (16) improves the capacity/robustness of the wireless communications network in terms of the elimination of the parasitic impact of noise. Of course, it is possible by choosing appropriate components/degrees of freedom of matrix \(\mathbf {{\underline {M}}}\left (q^{-1}\right)\) of Eq. (18) according to the criterion (19). The same can be achieved as a result of applying non-unique type τ- and σ-inverses. An adequate selection of the degrees of freedom \(\mathbf {{\underline {\beta }}_{s}} \left (q^{-1}\right)\) and \(\mathbf {{\underline {\beta }}} \left (q^{-1}\right)\) that are not relative to the propagation environment \(\mathbf {{\underline {C}}}\left (q^{-1}\right)\) can provide a greater degree of independence of the parasitic effects (see Eqs. (20) and (21)). Finally, we can strongly note that an alternative to the applied Smith decomposition can be offered by the use of the PSVD method implementing the non-zero degrees of freedom, whose derivation is either based on the use of the PEVD (polynomial EVD) approach [29, 30], or one that is obtained in a direct manner [31]. This intriguing proposal is briefly described in the next section.
5 PSVD vs. Smith decomposition-based approach
In Reference [16], the authors applied a successful polynomial singular value decomposition. As stated earlier, the current methods in the wireless telecommunications studies use zero degrees of freedom. Hence, the idea of the use of a new non-zero degrees of freedom was conceived with the purpose of limiting the impact of the noise on the process of data transfer. The same paradigm can be applied in the signal reconstruction process based on the PSVD method, which is worth further research. However, the methods basing on SVD and PSVD decompositions cannot be directly compared with the new method applied for signal recovery. The reason for this can be associated with different dynamic parameters of the propagation environments derived as a result of using SVD and PSVD on one hand and a method applying polynomial S-inverse on the other hand. In the former case, we merely obtain an approximation of the dynamic properties of the propagation environment, whereas the latter approach implies that an accurate dynamics of this environment is obtained. Appendix offers an outline of the working characteristics of the signal reconstruction methods for a non-square system comprising two transmitting antennas and four receiving ones.
The paradigm of the new intriguing method is analytically confirmed by a more general first polynomial part of the next section containing simulation examples, whereas the second parameter part contains the results of complex optimization runs using the genetic algorithm mechanism.
Let us analyze the wireless telecommunications system with no transmission zeros, including two transmitter antennas NT=2 and three receiver antennas NR=3. Assuming that the matrix obtained by the pilot identification and describing the dynamics of the parasitic impact of the environment on the signal reconstruction process takes the following form
$$ \begin{aligned} \mathbf{\underline{C}}\left(q^{-1}\right)= \left[ \begin{array}{ll} -\thinspace0.3+0.5q^{-1} & 0.6 \\ 1-0.8q^{-1}+0.4q^{-2} & 1-0.5q^{-1} \\ 0.9-0.3q^{-1} & 0.8 \end{array} \right]. \end{aligned} $$
After using the Smith factorization of \(\mathbf {{\underline {C}}}\left (q^{-1}\right)\), we obtain
$$\begin{array}{@{}rcl@{}} \mathbf{{\underline{U}}}\left(q^{-1}\right)\,=\,\! \left[ {\begin{array}{*{20}c} \! {-\thinspace0.8} & {2.2-1.6q^{-1}} & \!\!\ldots \\ \! {-\thinspace1.3 \,-\,0.6q^{-1}} \!&\! {0.3\,-\,0.3q^{-1}-0.5q^{-2}} & \!\!\ldots \\ \! {-\thinspace1} & {0} & \!\!\ldots \end{array}} \right. \\ \left. {\begin{array}{*{20}c} \!\! \!\ldots\!\! & {2.2-1.6q^{-1}} \!\!\! \\ \!\!\! \ldots\!\! & {-\thinspace0.6-0.3q^{-1}} \!\!\! \\ \!\!\! \ldots\!\! & {0} \!\!\! \\ \end{array}} \right]\!,\ \end{array} $$
$$ \begin{aligned} \mathbf{{\underline{V}}}\left(q^{-1}\right) = \left[ \begin{array}{ll} -\thinspace0.9+0.3q^{-1}& -\thinspace0.8 \\ -\thinspace0.4 & 0 \end{array} \right], \end{aligned} $$
and containing no transmission zeros
$$ \begin{aligned} \mathbf{{\underline{\Sigma}}}\left(q^{-1}\right) = \left[ \begin{array}{ll} 1 & 0 \\ 0 & 1 \\ 0 & 0 \\ \end{array} \right]. \end{aligned} $$
Now, in accordance with the signal perfect reconstruction as presented in Eq. (16), for the received signal R′(t)=[R1′(t)R2′(t)R3′(t)]T (due to its complexity, vector R′(t) blurred by a zero-mean white noise ζ(t) is not given here), we obtain the vector of the transmitted signal
$$\begin{array}{@{}rcl@{}} \mathbf{S}(t)= \left[ {\begin{array}{*{20}c} {0} & {-\thinspace2.3} \\ {-\thinspace1.3} & {2.5-0.8q^{-1}} \\ \end{array}} \right] \left[ \begin{array}{*{20}c} {1} & {0} & {\underline M_{1}\left(q^{-1}\right)} \\ {0} & {1} & {\underline M_{2}\left(q^{-1}\right)} \\ \end{array} \right]\!\!\!\! \\ \times \left[ {\begin{array}{*{20}c} {0} & {0} & \ldots \\ {0.6+0.3q^{-1}} &{-1}& \ldots \\ {0.3-0.3q^{-1}-0.5q^{-2}} & {-\thinspace2.2+1.6q^{-1}} & \ldots \\ \end{array}} \right.\\ \end{array} $$
$$\begin{array}{@{}rcl@{}} \left. \begin{array}{*{20}c} \ldots & {-\thinspace1} \\ \ldots & {0.8-0.8q^{-1}} \\ \ldots & {2.5-3.2q^{-1}+1.4q^{-2}} \\ \end{array} \right] \left[ \begin{array}{*{20}c} {R_{1}^{\prime}} \\ {R_{2}^{\prime}} \\ {R_{3}^{\prime}} \\ \end{array} \right]\\ -\left[ \begin{array}{*{20}c} {0} & {-\thinspace2.3} \\ {-\thinspace1.3} & {2.5-0.8q^{-1}} \\ \end{array} \right] \left[ \begin{array}{*{20}c} {1} & {0} & {\underline M_{1}\left(q^{-1}\right)} \\ {0} & {1} & {\underline M_{2}\left(q^{-1}\right)} \\ \end{array}\right] \\ \times \! \left[ \begin{array}{*{20}c} {0} & {0} & \ldots \\ {0.6+0.3q^{-1}} &{-\thinspace1}& \ldots \\ {0.3-0.3q^{-1}-0.5q^{-2}} & {-\thinspace2.2+1.6q^{-1}} & \ldots \\ \end{array} \right.\\ \left. \begin{array}{*{20}c} \ldots & {-\thinspace1} \\ \ldots & {0.8-0.8q^{-1}} \\ \ldots & {2.5-3.2q^{-1}+1.4q^{-2}} \\ \end{array} \right] {\zeta}(t).\!\!\!\! \\ \end{array} $$
Finally, for the determined value ζ(t)=[0.1 − 0.2 0.2]T and the degrees of freedom of matrix \({\mathbf {\underline {M}}}(q) =\!\!\begin {array}{*{20}c} \left [ \! {\begin {array}{*{20}c} {\frac {{343712}}{{841\left ({\text {1996}{q}^{2} - 2023 {q} + 521} \right)}}} & {\frac {{\text { 14944(-38348} {q}^{2} + 15109 {q})}}{{707281\left ({\text {1996} {q}^{2} - 2023{q} + 521} \right)}}} \end {array}}\!\! \right ]\end {array}\!\!= \![\underline M_{1}(q)\,\underline M_{2}(q)]^{\mathrm {T}}\!\!,\) selected as a result of analytical calculations, the reconstructed vector S(t) is S(t)=[3+3i − 3−i]T (corresponding to the two points of 16-QAM constellation of transmitted signal S(t)). Note that in our simulation example, there is \(\mathbf {\underline {\Sigma }}^{\mathrm {L}}\left (q^{-1}\right) = \mathbf {\underline {\Sigma }}^{\text {L1}}\left (q^{-1}\right) = \mathbf {\underline {\Sigma }}^{\text {L2}}\left (q^{-1}\right)\), see Remark 8.
To better describe the advantages of the method proposed, complex tests were carried out by using the authors' OFDM technology simulator running in the Matlab environment [28]. For this purpose, 103776 bits of random input data from 64-QAM constellation were transferred by means of the IQ-modulated signal through the single carrier system with the matrix \(\mathbf {{\underline {C}}}\left (q^{-1}\right)\) as in Eq. (24). For the assumed rigorous tolerance, the special parameter matrices \(\mathbf {{\underline {M}}}\left (q^{-1}\right)\) were obtained using the genetic algorithm according to the performance index (19). Thus, we have different degrees of freedom for each of the SNRs, not presented in this paper due to space limitation. It is evident that the new method outperforms the classical one, where zero degrees of freedom associated with the application of minimum-norm/least-squares inverses to \(\mathbf {\underline {\Sigma }}\left (q^{-1}\right)\) a polynomial matrix can be find. This statement is confirmed in Fig. 1.
Perfect reconstruction process: BER vs. SNR
In addition, two simulation tests were performed covering propagation environments described by the following matrices
$$ {}\begin{aligned} \mathbf{\underline{C}}\!\left(q^{-1}\!\right)\,=\, \left[\!\! \begin{array}{ll} \qquad-\thinspace0.75+0.6q^{-1} &\qquad\quad 0.9 \\ 0.56\,-\,0.01q^{-1}\,+\,0.02q^{-2} & 0.83\,-\,0.75q^{-1} \!\\ \qquad 0.83-0.25q^{-1} &\qquad\quad 0.6 \end{array} \!\right]\!, \end{aligned} $$
$$ \begin{aligned} \mathbf{\underline{C}}\left(q^{-1}\right)= \left[ \begin{array}{ll} \quad -\thinspace0.5+0.4q^{-1} &\qquad 0.9 \\ 1-0.6q^{-1}+0.2q^{-2} & 1-0.6q^{-1} \\ \qquad 1-0.4q^{-1} &\qquad 0.6 \end{array} \right], \end{aligned} $$
Figure 2 presents the results obtained for single carrier system given by Eq. (29), whereas Fig. 3 for single carrier system as in Eq. (30).
7 Conclusions
In this paper, the new approach to the process of perfect reconstruction of signals is presented. The new solution is based on polynomial matrix calculus, mainly the so-called left S-inverse of the polynomial matrix. Errors generated in the process of signal reconstruction are compensated by the appropriate selection of components/degrees of freedom of the non-zero matrix \(\mathbf {\underline {M}}\left (q^{-1}\right)\). Simulation tests carried out in the Matlab environment have indicated a considerable implementation potential of the innovative approach proposed in this paper to the tasks of efficient signal recovery in non-square MIMO telecommunications systems. It should be emphasized that the new method still outperforms the typical one in case of presence of noise with uniform distribution.
8.1 Method based on PSVD
The matrix applied to described the dynamics of the propagation environment assumes the form
$$ \mathbf{\underline{C}}\left(q^{-1}\right) = \left[ \begin{array}{lr} 1.5q^{-1}& -\thinspace3 \\ -\thinspace3q^{-2}& -\thinspace1.5q^{-1} \\ 1.5q^{-1}& -3 \\ -\thinspace3q^{-2}& -\thinspace1.5q^{-1} \\ \end{array} \right]. $$
(I.1)
Following PSVD factorization, we receive
$$ \mathbf{\underline{C}}\left(q^{-1}\right)=\mathbf{{U}}(q)\mathbf{\Sigma}(q)\mathbf{V}^{\dagger}(q), $$
$$\begin{aligned} \mathbf{U}(q) = \left[ \begin{array}{lrrr} 0.5 & q & 0.5 & q^{2} \\ -\thinspace q^{-1}& 0.5&-\thinspace q^{-2}&0.5 \\ 0.5& q&-\thinspace0.5&-\thinspace q^{2} \\ -\thinspace q^{-1}& 0.5&q^{-2}&-\thinspace0.5 \\ \end{array} \right], \mathbf{\Sigma}(q) = \left[ \begin{array}{ll} q & 0\\ 0 & 1 \\ 0&0 \\ 0&0 \\ \end{array} \right] \end{aligned} $$
and \(\mathbf {V}^{\dagger }(q)=\mathbf {V}^{\mathrm {H}}(1/q^{*})= \left [ \begin {array}{ll} 3q^{-2}&\qquad 0 \\ \;\;0& -\thinspace 3q^{-1} \\ \end {array} \right ]\).
Evidently, U(q)U † (q)=U † (q)U(q)=2.5I4 and V(q)V † (q)=V † (q)V(q)=9I2, where I n denotes n-identity matrix, and both U(q) and V(q) are paraunitary matrices.
Let us consider a deterministic process of perfect signal reconstruction
$$ \mathbf{R}(t)=\mathbf{\underline{C}}\left(q^{-1}\right)\mathbf{S}(t), $$
where R(t) and S(t) are the vectors of the received and transmitted signals, respectively. By consideration of Eq. (I.2) and application of the precoder V(q) and equalizer U † (q) structures (for an example see Reference [15]), we receive
$$ \mathbf{R}^{\prime}(t)=\mathbf{U}^{\dagger}(q)\mathbf{U}(q)\mathbf{\Sigma}(q)\mathbf{V}^{\dagger}(q)\mathbf{V}(q)\mathbf{S}(t). $$
Unfortunately, R′(t)≠R(t).
8.2 Method based on SVD
By the analogy to the case of PSVD for R(t)=CS(t), where C is a parameter matrix, we receive
$$ \mathbf{R}^{\prime}(t)=\mathbf{U}^{\dagger}\mathbf{U}\mathbf{\Sigma}\mathbf{V}^{\dagger}\mathbf{V}\mathbf{S}(t), $$
(II.5)
where V and U † denote the precoder and equalizer structures, respectively, fulfilling the condition of unitarity.
In this case, also R′(t)≠R(t).
8.3 Method based on polynomial S-inverse
Taking into consideration that
(III.6)
where \(\mathbf {{\underline {U}}}\left (q^{-1}\right)\) and \(\mathbf {{\underline {V}}}\left (q^{-1}\right)\) are unimodular matrices obtained as a result of applying Smith factorization, Eq. (I.3) can be written in the following form
$$ \mathbf{R}(t)=\mathbf{{\underline{U}}}\left(q^{-1}\right) \mathbf{{\underline{\Sigma}}}\left(q^{-1}\right) \mathbf{{\underline{V}}}\left(q^{-1}\right)\mathbf{S}(t). $$
By solving Eq. (III.7) in respect to S(t), we receive the actual error-free transmitted signal S(t) in accordance with the relation
$$ \mathbf{S}(t)=\mathbf{{\underline{V}}}^{-1}\left(q^{-1}\right) \mathbf{{\underline{\Sigma}}}^{\mathrm{L}}\left(q^{-1}\right) \mathbf{{\underline{U}}}^{-1}\left(q^{-1}\right)\mathbf{R}(t), $$
where superscript "L" denotes every non-unique left inverse of matrix polynomial \(\mathbf {{\underline {\Sigma }}}^{\mathrm {L}}\left (q^{-1}\right)\).
Finally, let us remark that we cannot directly compare the two methods of signal recovery, i.e., approaches involving respective (P)SVD and Smith factorization mechanisms. The reason for this was associated, for example, with the lack of causation phenomenon in the precoder and/or equalizer structures (see Eq. (I.4), where, e.g., \(\mathbf {U}(q) = \left [ \begin {array}{lrrr} 0.5 & q & 0.5 & q^{2} \\ -\thinspace q^{-1}& 0.5&-\thinspace q^{-2}&0.5 \\ 0.5& q&-\thinspace 0.5&-\thinspace q^{2} \\ -\thinspace q^{-1}& 0.5&q^{-2}&-\thinspace 0.5 \\ \end {array} \right ]\), whereas for the case when, e.g., q2 constitutes a double feed-forward, i.e., we have y(t)=u(t+2)).
BER:
FIR:
Finite impulse response
ICI:
Inter-channel interference
Inter-symbol interference
MIMO:
Multi-input/multi-output
OFDM:
Orthogonal frequency division multiplexing
PEVD:
Polynomial eigenvalue decomposition
PSVD:
Polynomial singular value decomposition
SISO:
Single-input/single-output
SNR:
Invaluable comments from the anonymous reviewers are gratefully acknowledged.
This work was funded by the Department of Electrical, Control and Computer Engineering, Opole University of Technology, Poland.
Both authors contributed to the work. Both authors read and approved the final manuscript.
Both authors declare that they have no competing interests.
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.
Opole University of Technology, Prószkowska 76, Opole, 45-758, Poland
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How is Covariance and Correlation used in Portfolio Theory?
Banking & FinanceFinance ManagementGrowth & Empowerment
The process of combining numerous securities to reduce risk is known as diversification. It is necessary to consider the impact of covariance or correlation on portfolio risk more closely to understand the mechanism and power of diversification.
Let's study the issue category-wise −
when security returns are perfectly positively correlated,
when security returns are perfectly negatively correlated, and
when security returns are not correlated.
Security Returns Perfectly Positively Correlated
When net assets returns are perfectly and positively correlated, the given correlation coefficient between the two securities will be +1. Thus, the returns of the two securities will move up or down together. The portfolio variance for perfectly positive correlation is calculated using the formula −
$$\mathrm{(σ_{ρ})^{2} = (𝑥_{1})^{2}(σ_{1})^{2} + (𝑥_{2})^{2}(σ_{2})^{2} + 2 𝑟_{12}\:𝑥_{1}𝑥_{2}\:σ_{1}σ_{2}}$$
Since𝑟12 = 1, this may be rewritten as −
$$\mathrm{(σ_{ρ})^{2} = (𝑥_{1})^{2}(σ_{1})^{2} + (𝑥_{2})^{2}(σ_{2})^{2} + 2 \times 1 \times 2\:σ_{1}σ_{2}}$$
The right-hand side of the equation has the same form as the expansion of the identity (a + b)2 = a2 + 2ab + b2
Hence, it may be reduced as,
$$\mathrm{(σ_{ρ})^{2} = (𝑥_{1}σ_{1} + 𝑥_{2}σ_{2})^{2}}$$
The standard deviation (SD) then becomes $(σ_{ρ} = 𝑥_{1}σ_{1} + 𝑥_{2}σ_{2})$ which is simply the net average of the standard deviations of the individual securities.
Standard deviation of security P = 50
Standard deviation of security Q = 30
Proportion of investment in P = 0.4
Proportion of investment in Q = 0.6
Correlation coefficient = +1.0
Portfolio standard deviation may be calculated as −
$$\mathrm{σ_{ρ} = 𝑥_{1}σ_{1} + 𝑥_{2}σ_{2}=(0.4 \times 50)+ (0.6 \times 30) = 38}$$
Being the weighted average of the SD of individual securities, the portfolio SD will lie between the standard deviations of the two singular individual securities. In the given example, it will vary between 50 and 30.
For example, if the proportion of investment in P and Q are 0.75 and 0.25 respectively, portfolio standard deviation becomes −
$$\mathrm{σ_{ρ} = 𝑥_{1}σ_{1} + 𝑥_{2}σ_{2}=(0.75 \times 50) + (0.25 \times 30) = 45}$$
Thus, when the security returns of assets are perfectly positively correlated, then diversification provides only risk averaging and no risk reduction. This happens because the portfolio risk cannot be reduced below the risk of each individual asset. Hence, diversification is not productive when security returns are perfectly positively correlated.
Security Returns Perfectly Negatively Correlated
In this case, the correlation coefficient between them becomes -1. The two returns from the portfolio will move in exactly opposite directions. The portfolio variance is given by −
Since 𝑟12 = −1,this may be rewritten as −
$$\mathrm{(σ_{ρ})^{2} = (𝑥_{1})^{2}(σ_{1})^{2} + (𝑥_{2})^{2}(σ_{2})^{2} - 2 \times 1 \times 2 σ_{1}σ_{2}}$$
The right-hand side of the equation has the same form as the expansion of the identity (a - b)2 = a2 - 2ab + b2
$$\mathrm{({σ_{ρ}})^{2} = (𝑥_{1}σ_{1} - 𝑥_{2}σ_{2})^{2}}$$
The standard deviation (SD) then becomes $(σ_{ρ} = 𝑥_{1}σ_{1} - 𝑥_{2}σ_{2})$
For the illustrative portfolio considered above, we can calculate the portfolio standard deviation when the correlation coefficient is −1.
$$\mathrm{σ_{ρ} = 𝑥_{1}σ_{1} - 𝑥_{2}σ_{2}=(0.40 \times 50) − (0.60 \times 30) = 2}$$
The portfolio risk may go as low as zero. For example, when P and Q are 0.375 and 0.625 respectively, portfolio standard deviation becomes −
$$\mathrm{σ_{ρ}=(0.375 × 50) − (0.625 × 30) = 0}$$
Here, although the portfolio has two risky assets, the portfolio overall has no risk. Thus, the portfolio risk may be zero when security returns are perfectly negatively correlated.
Security Returns Uncorrelated
When the returns of two securities are entirely uncorrelated, the correlation coefficient would be zero. The formula for portfolio variance is −
$$\mathrm{(σ_{ρ})^{2} = (𝑥_{1})^{2}(σ_{1})^{2} + (𝑥_{2})^{2}(σ_{2})^{2} + 2\:𝑟_{12}\:𝑥_{1}𝑥_{2}\:σ_{1}σ_{2}}$$
Since 𝑟12 = 0, the last term in the equation becomes zero; the formula can be rewritten as −
$$\mathrm{(σ_{ρ})^{2} = (𝑥_{1})^{2}(σ_{1})^{2} + (𝑥_{2})^{2}(σ_{2})^{2}}$$
The standard deviation then becomes −
$$\mathrm{σ_{ρ}=\sqrt{𝑥_{1}σ_{1} + 𝑥_{2}σ_{2}}}$$
For our illustrative portfolio,
$$\mathrm{σ_{ρ}=\sqrt{(0.4)^{2}(50)^{2} + (0.6)^{2}(30)^{2}}=\sqrt{400 + 324}= 26.91}$$
The portfolio SD is less than the standard deviations of individual securities in the portfolio. Thus, when the security returns are completely uncorrelated, diversification diminishes the risk and becomes productive.
We may conclude that risk is always reduced except when the security returns of a two asset portfolio are perfectly positively correlated. With the correlation coefficient declining from +1 to -1, the portfolio SD also declines automatically. However, the risk reduction is the most palpable when the security returns are negatively correlated.
Probir Banerjee
Published on 28-Sep-2021 06:56:25
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How is the standard deviation and variance of a two-asset portfolio calculated? | CommonCrawl |
\begin{document}
\title{A blueprint for a Digital-Analog Variational Quantum Eigensolver using Rydberg atom arrays}
\author{Antoine Michel} \email{[email protected]} \affiliation{Electricité de France, EDF Recherche et Développement, Département Matériaux et Mécanique des Composants, Les Renardières, F-77250 Moret sur Loing, France } \affiliation{Université Paris-Saclay, Institut d’Optique Graduate School, CNRS, Laboratoire Charles Fabry, F-91127 Palaiseau Cedex, France} \author{Sebastian Grijalva} \author{Loïc Henriet} \affiliation{PASQAL, 7 rue Léonard de Vinci, F-91300 Massy, France} \author{Christophe Domain} \affiliation{Electricité de France, EDF Recherche et Développement, Département Matériaux et Mécanique des Composants, Les Renardières, F-77250 Moret sur Loing, France } \author{Antoine Browaeys} \affiliation{Université Paris-Saclay, Institut d’Optique Graduate School, CNRS, Laboratoire Charles Fabry, F-91127 Palaiseau Cedex, France}
\selectlanguage{english} \date{\today}
\begin{abstract} We address the task of estimating the ground-state energy of Hamiltonians coming from chemistry. We study numerically the behavior of a digital-analog variational quantum eigensolver for the H$_2$, LiH and BeH$_2$ molecules, and we observe that one can estimate the energy to a few percent points of error leveraging on learning the atom register positions with respect to selected features of the molecular Hamiltonian and then an iterative pulse shaping optimization, where each step performs a derandomization energy estimation. \end{abstract}
\maketitle
\section{Introduction}\label{sec:introduction}
Quantum simulation holds the promises to solve outstanding questions in many-body physics, in particular finding the ground state of strongly interacting quantum systems \cite{georgescu_quantum_2014, mcclean_theory_2016}. The determination of the ground state energies of complex molecules, one of the main tasks in quantum chemistry, is therefore an example of application where quantum simulation could be of interest. Along this line, proof-of-principle demonstrations were obtained using photons \cite{Lanyon2010,Peruzzo2014}, ions \cite{Shen2017,Shen2018,Hempel2018} or quantum circuits \cite{Kandala2017}. The last two examples used an hybrid approach were a classical computer optimizes in an iterative way the results obtained by a quantum device that was operating in a digital mode, i.e. as a series of one and two-qubit gates.
Rydberg quantum simulators are another example of promising quantum simulation platforms thanks to their potential for scaling the number of qubits and their programmability \cite{browaeys_many-body_2020}. They rely on individual atoms trapped in arrays of optical tweezers that can interact when promoted to Rydberg states. The platform naturally implements spin Hamiltonians. Analog quantum simulation with hundreds of atoms has now been achieved \cite{scholl_programmable_2021, ebadi_quantum_2022,chen_continuous_2023}.
One appealing feature of this platform is the ability to place the atoms in arbitrary position in two and three dimensions, thus allowing large flexibility in their connectivity. Another feature is their ability to prepare different initial product states as heuristic trials before the unitary evolution (whether it is by a set of digital gates or the action of an analog Hamiltonian evolution). However, this freedom in register preparation has a significant time cost that adds to the repetition clock rate \cite{henriet_quantum_2020}.
\begin{figure}\label{fig:VQE_analog}
\end{figure}
Neutral atom devices are naturally suited for \emph{analog quantum algorithms}, where the analog blocks are represented by control pulses that drive the system (or subsets of it). Given a prepared state, the parameterized pulses can be adjusted to variationally improve on a given score of the state. Methods for the optimization of parameters have been the subject of intense exploration in recent years \cite{wecker_progress_2015, cerezo_variational_2021, mcclean_theory_2016,barkoutsos_improving_2020,mcclean_barren_2018,meitei_gate-free_2021, wakaura_evaluation_2021, banchi_measuring_2021, gacon2021simultaneous, piskor_using_2022}. Additionally, the information and ``cost functions'' from the prepared quantum system are obtained by repeatedly measuring the state in the computational basis, which constitutes an operational overhead. Recent results \cite{huang_efficient_2021, elben_statistical_2019,kokail_self-verifying_2019,nam_ground-state_2020,ebadi_quantum_2022,dalyac_qualifying_2021} on protocols for the estimation of quantum observables are available and have helped establishing efficient measurement procedures based on generalized random measurements and a series of post-processing steps that are performed on a classical computer and that alleviate the measurement overhead. The types of randomized measurements that we shall describe in this paper require local rotations on the qubits of the register, thus constituting another ``digital'' layer, from a quantum circuit perspective. In fact, \emph{digital-analog} algorithms \cite{parra-rodriguez_digital-analog_2020}, benefit from the fact that analog operations can be performed with much higher fidelities than when using digital gates, while local single-qubit gates can be added explicitly in crucial steps of the process (state preparation and measurement).
In this paper, we explore the implementation of a digital-analog VQE algorithm in a Rydberg quantum simulator. We account for typical constraints of the platform: the local action is restrained to the initial state preparation and measurement, with Hamiltonian time-evolution acting on the entire system as a ``global'' gate. We study numerically this version of a VQE for the $\text{H}_{\text{2}}$ molecule using common ansatze, followed by a more efficient protocol for larger molecules. We discuss the embedding of the Hamiltonian in the atom register, the way in which the optimization of the pulse sequence can be performed and the necessity of including an efficient estimation of energies (namely, we explore the effect of a \emph{derandomization} estimation \cite{huang_efficient_2021}) at each iteration step. We apply this numerically to the examples of LiH and BeH$_2$. The manuscript is organized as follows: In section II we recall how the Variational Quantum Eigensolver (VQE) estimates the energy of the ground state of a molecular-based Hamiltonian. We then describe the basic ingredients of Rydberg Atom Quantum Processors and the Hamiltonians that they implement. We end the section by explaining the optimization cycle of variational quantum algorithms on these devices. In section III we describe the strategies for implementation of the VQE, going from a direct application of a Unitary Couple Cluster Ansatz, to the Quantum Alternating Operator ansatz and finally to a more hardware-oriented approach that combines elements of register preparation, pulse optimization and observable estimation. This is followed in section IV by numerical results of the error in energy obtained as a function of the number of repetitions of the experiment, an informative measure of the performance of hybrid classical-quantum implementations.
\section{Analog Variational Quantum Eigensolver with Rydberg atoms} The Variational Quantum Eigensolver (or VQE) is a hybrid quantum-classical algorithm designed to find the lowest eigenvalue of a given Hamiltonian \cite{fedorov_vqe_2022}. We describe below the origin of the Hamiltonians that we consider and how VQE can be studied with a Rydberg Quantum Processor.
\subsection{Hamiltonians from Quantum Computational Chemistry}\label{sec:chem}
We first recall the method used to express the electronic Hamiltonian as a spin model (see e.g.~\cite{Hempel2018}). We start from the Born-Oppenheimer approximation of the Hamiltonian of the system, which considers the nuclei of the molecules as classical point charges:
\begin{equation}\label{elec-ham}
\hat H = -\sum_i \frac{\nabla_i^2}{2} - \sum_{i, I} \frac{\mathcal Z_I}{|\mathbf r_i - \mathbf R_I|} + \frac 1 2 \sum_{i\neq j} \frac{1}{|\mathbf r_i - \mathbf r_j|} \end{equation} (in atomic units) where $\nabla_i$ is the kinetic energy term for the $i$-th electron, $\mathcal Z_I$ is the charge of the $I$-th nucleus, and $\mathbf r$, $\mathbf R$ denote the distance of the $i$-th electron and the $I$-th nucleus with respect to the center of mass, respectively. We aim to obtain the ground state energy of \eqref{elec-ham}.
One needs to define a basis set in which to represent the electronic wavefunctions. We shall concentrate on the Slater-type orbital approximation for the basis set, with three Gaussian functions, STO-3G. This minimal basis set $\{\phi_i (\mathbf x_i)\}$ (where $\mathbf x_i=(\mathbf r_i, \sigma_i)$ encodes the $i$-th electron's spatial and spin coordinates) includes the necessary orbitals to represent the valence shell of an atom. Moreover, the wavefunctions need to be anti-symmetric under the exchange of electrons. This can be achieved through \emph{second quantization}, where one defines anticommuting fermionic creation/annihilation operators $\{\hat a_p^\dagger\}, \{\hat a_p\}$ and rewrites the initial Slater determinant form of the wavefunction as $|\Psi \rangle = \prod_p (\hat a_p^\dagger)^{\phi_p} |\mathrm{vacuum}\rangle$, representing the occupation of each molecular orbital.
The fermionic operators are used to rewrite \eqref{elec-ham} as:
\begin{equation} \label{eq:2nd-quant-ham}
\hat H = \sum_{p,q}h_{pq}\hat{a}_p^{\dagger}\hat{a}_q + \frac{1}{2}\sum_{p,q,r,s}h_{pqrs}\hat{a}_p^{\dagger}\hat{a}_q^{\dagger}\hat{a}_r\hat{a}_s. \end{equation}
where the coefficients $h_{pq}$ and $h_{pqrs}$ encode the spatial and spin configuration of each of the electrons and depend on the inter-nuclear and inter-electron distances $\mathbf R, \mathbf r$: \begin{equation}\label{eq:2nd-quant-terms} \begin{aligned}
h_{pq}&=\int d \mathbf x \phi_p^* (\mathbf x)\left(-\frac{\nabla^2}{2}-\sum_i\frac{\mathcal Z_i}{|\mathbf R_i- \mathbf r|}\right)\phi_q(\mathbf x) \\
h_{pqrs}&=\int d\mathbf x_1 d\mathbf x_2\frac{\phi_p^*(\mathbf x_1)\phi_p^*(\mathbf x_2)\phi_r(\mathbf x_1)\phi_s(\mathbf x_2)}{|\mathbf r_1- \mathbf r_2|}. \end{aligned} \end{equation} Next, we map the fermionic operators acting on Fock states of $n$ orbitals to a Hilbert space of operators acting on spin states of $N$ qubits. This corresponds to the quantum processors' effective interaction Hamiltonians, quantum gates and measurement basis. Useful maps of this kind include the Jordan-Wigner (JW) \cite{jordan_uber_1928} or the Bravyi-Kitaev (BK) \cite{bravyi_fermionic_2002} transformations. The obtained Hamiltonian is a sum of tensor products of single-qubit Pauli matrices: \begin{align}\label{eq:pauli_ham}
\hat{H}_T = \sum_{s=1}^S {\bf c}_s \bigg( \bigotimes_{j=1}^N \hat{P}_j^{(s)}\bigg) \end{align} where $\hat P_j \in \{\mathbb 1, X, Y, Z \}$, $S$ is the number of Pauli strings in the Hamiltonian and $N$ the number of qubits.
\subsection{Rydberg Atom Quantum Processor} Rydberg atom arrays are now well-established quantum simulation platforms \cite{henriet_quantum_2020,browaeys_many-body_2020}. Briefly, atoms are trapped in optical tweezers, each containing exactly one atom. The tweezers may be arranged in any 1D, 2D or 3D geometrical configurations. The register can be rebuilt after each computational cycle. To perform quantum processing, we use the fact that the platform implements spin-like Hamiltonians, where the interactions originate from strong dipole-dipole couplings between atoms laser-excited to Rydberg states.
Depending on the choice of atomic levels, the atoms experience different effective interactions. In the case of the Ising mode, $\ket{0}$ is a ``ground'' state prepared by optical pumping \cite{browaeys_many-body_2020} and $\ket{1}$ is a Rydberg state of the atom. The Hamiltonian term for this interaction is:
\begin{align}\label{eq:ising}
\hat H_{\text{Ising}} &= \sum_{i > j}\frac{C_6}{r_{ij}^{6}}\hat n_i \hat n_j,
\end{align}
with $\hat n_i = |1\rangle_i \langle 1 | = (\mathbb 1_i + Z_i)/2$ the projector on the Rydberg state and $r_{ij}$ the distance between atoms. Here and below, $X_i,Y_i$ and $Z_i$ indicate the local Pauli operators.
If instead the two states chosen are two dipole-coupled Rydberg states (for example $\ket{0}=\ket{nS}$ and $\ket{1}=\ket{nP}$ for large $n$), the interaction is resonant and realizes
a so-called ``XY'' or ``flip-flop'' term:
\begin{align}\label{eq:XY}
\hat H_{\text{XY}} &= \sum_{i \neq j}\frac{C_3}{r_{ij}^{3}} (X_i X_j + Y_i Y_j), \end{align}
where $C_3$ depends on the chosen Rydberg orbitals and their orientation with respect to the interatomic axis. It corresponds to a coherent exchange of neighboring spin states $\ket{10}$ to $\ket{01}$.
In addition, we can include time-dependent terms on the Hamiltonian, by means of a laser pulse (Ising mode) or a microwave field (XY mode) targeting the transition between the ground and excited states. This is represented by the following ``drive'' terms:
\begin{align} \hat H_{\text{drive}} = \frac{\hbar}{2}\sum_{i=1}^N \Omega_i(t) X_i - \hbar\sum_{i=1}^N \delta_i(t)\hat n_i\ . \end{align}
Here, $\Omega(t)$ is Rabi frequency and $\delta(t)$ the detuning of the field with respect to the resonant transition frequency.
The addressing can be either global or local. In the procedure used in this work, the local addressing is restricted to the initial state preparation and the register readout stages.
\subsection{Variational Algorithms on a Rydberg atoms device}
In the analog VQE algorithm, we seek to estimate the energy of the ground state of a qubit Hamiltonian called the \emph{target} Hamiltonian, $\hat H_{\text{T}}$, by using an iterative method. The \emph{resource} Hamiltonian is the one realized by the hardware, and can be configured with different types of interactions ($\hat H_\mathrm{inter}$) (\ref{eq:ising}, \ref{eq:XY}) and driving fields ($\hat H_\mathrm{drive}$):
\begin{equation}
\hat H_{\text R} = \hat H_{\rm inter} + \hat H_{\rm drive}. \end{equation}
Experimentally, the transition from the ground to the excited state is typically generated by a two-photon process, from which an approximate two level system is extracted, driven by an effective Rabi frequency $\Omega$ and detuning $\delta$ during the quantum processing stage. We use their values as parameters in our analog presentation of a VQE algorithm: The first step is to prepare the register of $N$ atoms with a geometry that determines the interaction terms $\hat H_{\rm inter}$ and then to initialize the system in a state $\ket{\psi_0}$. Then, a pulse sequence is applied to evolve the system under the resource Hamiltonian $\hat H_{\text R}(\Omega(t), \delta(t))$ whose corresponding time-ordered unitary evolution operator is $U(t) = \mathcal T \exp\big(-i \int_{0}^{t}\hat H_{\text{R}}(\Omega(\tau), \delta(\tau) )d\tau\big)$. The final prepared state is:
\begin{equation} \label{eq:evolution_op} \ket{\psi(\Omega, \delta,t)} = U(t)\ket{\psi_0}. \end{equation}
The energy of a prepared state will be calculated with respect to the target Hamiltonian: \begin{equation} \label{eq:energy_estimation} E(\Omega, \delta, t) = \bra{\psi(\Omega, \delta, t)}\hat H_{\text T}\ket{\psi(\Omega, \delta, t)}. \end{equation} After each cycle, a classical optimizer adjusts the parameters $\Omega \rightarrow \Omega'$, $\delta \rightarrow \delta'$ and $t \rightarrow t'$ and we repeat the evolution of the initial quantum state $\ket{\psi_0}$ with the new parameter set $U(\Omega',\delta',t')$. We aim to obtain for each iteration $E(\Omega', \delta', t') \leq E(\Omega, \delta, t)$ \footnote{This classical optimization problem can be addressed for example by obtaining the gradient of the energy function.}. After several iterations of this loop, the variational scheme attempts to prepare a state whose energy is a good approximation of the ground state energy of $\hat H_{\text T}$ \cite{mcclean_theory_2016}.
\section{Description of the Protocols}
In this section, we describe two analog variational quantum algorithms for the estimation of the ground state energy and apply them to quantum chemistry problems. The protocols differ mainly by the choice of ansatz: one is the \emph{Unitary Coupled Cluster} (UCC) ansatz \cite{bartlett_alternative_1989}, while the other is an adaptation of a \emph{hardware-efficient ansatz} \cite{Kandala2017}, based on repeating alternating values of amplitude, frequency or phase of the applied pulses. We verify numerically the performance of these two types of ansatz in a Rydberg-based Quantum Processor (QP). Next we discuss a protocol for larger molecules tailored after the hardware capabilities. We begin by considering the prototypical example of the H$_2$ molecule.
\subsection{UCC ansatz on an analog quantum processor: application on H\pdfmath{\bf _2}}
Numerous implementations of the VQE algorithm rely on the use of digital gates. Recent experimental implementations for the H$_2$, LiH and BeH$_2$ molecules have been realized in \cite{Kandala2017, Hempel2018}, with superconducting and trapped ions devices respectively. For the analog version of this algorithm on H$_2$, we consider the target Hamiltonian and the ansatz as in \cite{Hempel2018}. The Jordan-Wigner and Bravyi-Kitaev transformations lead to two different spin Hamiltonians of this molecule:
\begin{align} \begin{split} \hat{H}_{\text{JW}} =& \, {\bf c_0} \mathbb{1} + {\bf c_1}(Z_0 + Z_1) + {\bf c_2}( Z_2 + Z_3) +\\ &{\bf c_3} Z_3 Z_2 + {\bf c_4} Z_2 Z_0 + {\bf c_5}(Z_2 Z _0 + Z_3 Z_1) +\\ &{\bf c_6} (Z_2 Z_1 + Z_3 Z_0) + {\bf c_7} (X_3 Y_2 Y_1 X_0 +\\ & Y_3 X_2 X_1 Y_0 - X_3 X_2 Y_1 Y_0 + Y_3 Y_2 X_1 X_0) \label{hamjw} \end{split} \end{align} and \begin{align} \begin{split} \hat{H}_{\text{BK}} =& \, {\bf f_0} \mathbb{1} + {\bf f_1} Z_0 + {\bf f_2} Z_1 + {\bf f_3} Z_2 + {\bf f_4}Z_1 Z_0 +\\ & {\bf f_5} Z_2 Z_0 + {\bf f_6} Z_3 Z_1 + {\bf f_7} X_2 Z_1 X_0 +
{\bf f_8} Y_2 Z_1 Y_0 + \\
& {\bf f_9} Z_2 Z_1 Z_0 + {\bf f_{10}} Z_3 Z_2 Z_0 + {\bf f_{11}}Z_3 Z_2 Z_1 + \\
& {\bf f_{12}}Z_3 X_2 Z_1 X_0 + {\bf f_{13}} Z_3 Y_2 Z_1 Y_0 + {\bf f_{14}}Z_3 Z_2 Z_1 Z_0 \label{hambk} \end{split} \end{align}
where the coefficients $\{\bf c_i\}$ and $\{\bf f_j\}$ are calculated from (\ref{eq:2nd-quant-terms}). Since in \eqref{hambk} qubits $1$ and $3$ are only affected by the operators $\mathbb{1}$ and $Z$ one can actually work with the following two-qubit effective Hamiltonian \cite{omalley_scalable_2016}:
\begin{align}\begin{split} \hat{H}_{\text{BK}}^{\text{(eff)}}=& \, {\bf g_0} \mathbb{1} + {\bf g_1}Z_0 + {\bf g_2}Z_1 + {\bf g_3} Z_0 Z_1 + \\ & {\bf g_4}X_0 X_1 + {\bf g_5}Y_0Y_1. \label{eqhambkef} \end{split}\end{align}
Usually, a good ansatz $|\psi(\boldsymbol \theta)\rangle = U(\boldsymbol \theta) |\psi_0\rangle$ requires a balance between hardware constraints and symmetries in target Hamiltonian. However, using the `knobs' available on the hardware is often not efficient, and one thus needs additional guidance to reach the states we are looking for in a potentially very large Hilbert space. In this sense, the well-established Unitary Coupled Cluster (UCC) ansatz allows one to perform an unitary operation while keeping advantages of coupled cluster ansatz from chemistry \cite{helgaker_molecular_2014}.
In most cases, implementing the UCC ansatz in a quantum processor requires constructing a digital quantum circuit with full local addressing. An example where global addressing is sufficient is the H$_2$ molecule. The initial guess of the molecular wave function is a product state obtained from the classical Hartree-Fock calculation performed to determine the coefficients $\{\bf c_i\}$ and $\{\bf f_j\}$. Considering only relevant single and double excitations in the unitary coupled-cluster operator (UCC-SD) yields the following one-parameter unitary: \begin{equation}\label{UCCSD}
U_{\text{UCC-SD}}(\theta) = e^{\theta (c_{2}^{\dagger}c_{3}^{\dagger}c_1c_0 - c_0^{\dagger}c_1^{\dagger}c_3c_2)} \end{equation} where the minimal set of orbitals are represented by the fermionic annihilation and creation operators $c, c^{\dagger}$ \cite{Hempel2018}. A Jordan-Wigner transformation on these operators leads to the UCC ansatz $\ket{\psi(\theta)} =\exp(-i\theta X_3 X_2 X_1 Y_0) \ket{0011}$, where $\ket{0011}$ is the Hartree-Fock state. In the case of the (effective) Bravyi-Kitaev transform \eqref{eqhambkef}, we obtain the simpler UCC ansatz $\ket{\psi(\theta)}=\exp(-i\theta X_1 Y_0)\ket{01}$.
Since the evolution Hamiltonian commutes with the XY Hamiltonian (\ref{eq:XY}), one can use the latter ansatz and attempt to drive the Rydberg QP in the XY mode, using $\Omega=0$ and non-zero local detunings, leaving the rest of the parameters to be set by variational optimization:
\begin{align} \begin{split} \label{eq:xyansatz} \ket{\psi(\delta_0, \delta_1, t)} &= \exp \Big( -it(\delta_0 Z_0 + \delta_1 Z_1 + \hat H_{\text{XY}}) \Big) \ket{01} \\ &= a(\delta_0,\delta_1,t) \ket{01} + b(\delta_0, \delta_1, t) \ket{10}, \end{split} \end{align}
which coincides with the subspace reached with the UCC ansatz:
\begin{equation}\label{eq:uccbk} \exp(-i\theta X_1 Y_0)\ket{01} = a(\theta) \ket{01} + b(\theta) \ket{10}. \end{equation}
A numerical implementation of this protocol is shown in Fig. \ref{fig:H2_UCC_plot}, where the classical optimization was performed with a differential evolution algorithm \cite{storn_differential_1997}. We observe that the ground-state energy can be obtained with an error smaller than $5\%$ using less than $36500$ shots for each point.
Such examples of a UCC ansatz implementable with an analog approach, often rely on finding symmetries between target and resource Hamiltonians \cite{kokail_self-verifying_2019}. Nevertheless, this kind of protocol remains impractical for larger molecules because of the increasingly higher number of qubits and Pauli strings in the Hamiltonian. In order to use the analog approach for larger encodings, we explore other approaches below.
\begin{figure}
\caption{\textit{Numerical implementation of an analog VQE algorithm using a UCC ansatz.}
(a) A zoom on the smallest inter-atomic distance ($0.2$ \r{A}) shows the evolution of the optimization with respect to the number of shots. The differential evolution was set to perform at most 4 iterations (red squares). The red scale shows the errorbar over 20 realizations. It takes approximately $3.5$ hours of runtime for a QP operating at 3 Hz
to achieve $\epsilon = 10 \% $ (light grey scale) of error and $4$ hours to achieve $\epsilon = 5 \% $ (dark grey scale).
(b) Relative error in percentage (red circles) between the mean VQE result and the numerically computed lowest eigenvalue of the target Hamiltonian (in STO-3G basis) over 10 realizations (gray crosses). The expected error is below $\varepsilon = 5 \%$ (grey area). The inset depicts the same result on an energy scale and compares it with the exact solution (green squares). The result is drawn as a function of hydrogen inter-atomic distance.
}
\label{fig:H2_UCC_plot}
\end{figure}
\subsection{Alternating pulses}
We now describe an alternating operator approach, based on the QAOA algorithm \cite{farhi_quantum_2014}. Let $|\psi_0\rangle$ be the state composed of all qubits in the ground state. The whole sequence is composed by alternating constant (global) pulses, corresponding to two non-commuting Hamiltonians $\hat H_a, \hat H_b$:
\begin{align}
\hat H_{a} &= \frac{\hbar}{2}\sum_{i=1}^N \Big( \Omega X_i - \delta Z_i \Big) + \hat H_{\text{inter}} \\
\hat H_{b} &= \frac{\hbar}{2} \sum_{i=1}^N \Omega X_i +\hat H_{\text {inter}}. \end{align} These Hamiltonians define evolution operators $U_a(t)$ and $U_b(t)$, during a certain time $t$ (see \eqref{eq:evolution_op}). The ansatz of $L$ layers is written as:
\begin{equation} \ket{\psi(\mathbf t_a, \mathbf t_b)}= \prod_{\ell=1}^L U_a(t^{\ell}_a) U_b(t^{\ell}_b) \ket{\psi_0}, \end{equation}
where the arrays of parameters $\mathbf t_k = (t^{1}_k, \ldots, t^{\ell}_k, \ldots, t^{L}_k)$, $k \in \{a,b\}$, fix the duration of each pulse in the layer, as described in \cite{dalyac_qualifying_2021}. As another example, a different choice of parameters was used in \cite{ebadi_quantum_2022}, considering a single Hamiltonian:
\begin{align} \begin{split}
\hat H = \frac{\hbar}{2}&\sum_{i=1}^N \Big( \Omega(t) e^{i\phi(t)} \ket{0}_i\bra{1} + \text{h.c.} \Big) + \hat H_{\text{inter}} \end{split} \end{align} with different time $\mathbf t = (t^{1}, \ldots, t^{\ell}, \ldots, t^{L})$ and phase $\boldsymbol \phi = (\phi^{1}, \ldots, \phi^{\ell}, \ldots, \phi^{L})$ arrays defining $L$ segments of the pulse. The corresponding ansatz is then:
\begin{equation} \ket{\psi(\mathbf t, \boldsymbol{\phi})}= \prod_{\ell=1}^L U(t^{\ell}, \phi^{\ell}) \ket{\psi_0}. \end{equation}
The two approaches can be implemented in existing experimental setups, especially when the target Hamiltonian is equal to to the resource Hamiltonian (such as the case of the Maximal Independent Set problem with Unit Disks, which is native to the Rydberg atoms setting). However, these methods struggle to minimize the molecular target Hamiltonian energies within a limited number of iterations and measurement repetitions. The alternating pulse ansatz assumes an initial register configuration and initial guesses for the durations of the pulses in each layer, two tasks that are the subject of active research. The expectation is that a properly chosen register and an optimized pulse will drive the system to a low-energy state. In Fig. \ref{fig:QAOA_comp}, we compare numerically the performance of the two alternating pulse ansatze discussed above (\cite{dalyac_qualifying_2021}, \cite{ebadi_quantum_2022}). We also included the procedure described in Sec. \ref{Param_pulse}, which addresses the embedding of the problem in the register and an estimation protocol for the observables. Comparing the required number of shots for these approaches highlights the necessity of including an efficient estimation protocol for the observables.
\begin{figure}
\caption{Evolution of the ground-state energy as a function of the accumulated number of shots for the H$_2$ molecule at a fixed inter-nuclear distance. We have averaged numerically 200 realizations of VQE with 3 and 8 layers using the two alternating pulse ansatz, \cite{dalyac_qualifying_2021} (straight line), and \cite{ebadi_quantum_2022} (dashed line). In the alternating operator approach, each Pauli string mean value is performed with 1000 shots, which for the H$_2$ Hamiltonian represents $1.5\times10^4$ shots before obtaining the first energy data point. Achieving energy errors below $5\%$ (gray area) requires at least $\mathcal O (10^6)$ shots in total. For the optimized procedure (adding more control over atom positions, pulse shaping and derandomization estimation), the same energy error typically requires $\mathcal O(10^5)$ shots.}
\label{fig:QAOA_comp}
\end{figure}
\subsection{Optimized Register and Iteratively Parameterized Pulses}\label{Param_pulse}
In this section, we present a more refined approach to deal with larger systems, aiming at exploiting the capabilities already available in Rydberg simulators. To exemplify the procedure, we consider in the following the Ising mode with the resource Hamiltonian (\ref{eq:ising}).
\subsubsection{Atom register and initial state}\label{sec:embedding}
Even though we only consider global pulses for the processing stage, there still remains freedom in the choice of the positions of the atoms. This determines the strength of pairwise interactions and defines a connectivity graph whose edges correspond to the atoms that experience a blockade effect \cite{henriet_quantum_2020} (a different graph structure can be defined for the XY mode \eqref{eq:XY}).
In order to find suitable atomic positions, the coordinates are optimized in the plane so that the associated interaction energy matrix resembles as much as possible the information contained in the target Hamiltonian. Since the latter contains general Pauli strings, we consider a subset of terms whose coefficients can be expressed in terms of the coordinates of the atoms\footnote{A broader series of techniques for embedding the problem information into the atom register has been considered in \cite{leclerc2022financial, coelho2022efficient}}. A simple choice consists in selecting the terms that can be directly compared with the Ising-like interaction of the atoms: Let the matrix $V^{\text{T}}$ be given by the positive coefficients of the terms with only two $Z$ operators in the molecular target Hamiltonian and $V^{\text{R}}$ (our ``register'' matrix) the resulting values of interaction strength $C_6/r_{ij}^6$ for each pair $i,j$ of atom positions in the register. This defines a score function $\sum_{i,j}(V^{\text{T}}_{ij} - V^{\text{R}}_{ij})^2$ that we minimize numerically by varying the atom coordinates.
The set of atomic positions that arises from this minimization will be our optimized register. Its geometry will be used to simulate the target Hamiltonian, but has no intrinsic chemical meaning. The information that is taken from the Hamiltonian can be chosen from other subsets of the Pauli strings (e.g. terms with 3 or more $Z$ operators) and different interpretations of how the coefficients constitute a register matrix. A different resource Hamiltonian, such as one with XY interactions, would imply a different choice of subset. In Fig. \ref{fig:register_H2}, we summarize graphically the procedure for the case of the $\text{H}_{\text{2}}$ molecule with the Jordan-Wigner transformation. It turns our that we obtain at a geometry very similar to the one heuristically picked for the alternating pulse ansatz.
\begin{figure}
\caption{Protocol to optimize the positions of the atoms in a register based on a target Hamiltonian. Blue: we begin with a register of randomly placed atoms and all the Ising interaction terms are entered in a $N\times N$ matrix. Yellow: all positive coefficients before the Pauli strings with only two $Z$ operators in the target Hamiltonian are combined in another (target) matrix $N \times N$. The coordinates of the atoms are optimized to minimize the distance between the two matrices. We then obtain a new register on which we will apply the VQE sequence. }
\label{fig:register_H2}
\end{figure}
\subsubsection{Optimization of the parameterized pulse sequence}\label{sec:pulse_opt_protocol}
We constructed a variation of the so-called \texttt{ctrl-VQE} protocol \cite{meitei_gate-free_2021} for the case of a global pulse on the register, in which the number of parameters increases at every optimization iteration, while the total time $t_\text{tot}$ remains fixed:
Consider a set of Rabi frequencies $\{\Omega_i\}_{i=1}^K$ and detunings $\{\delta_i\}_{i=1}^K$ defined discretely over a set of time labels $0 < t_1 < \ldots < t_K = t_\text{tot}$.
Then, at iteration $k$, a new time label $0<t_{k}<t_\text{tot}$ is generated at random, lying between two previous time labels, $t_{i-1}< t_{k} < t_i$. To avoid labels too close to each other, we will accept $t_k$ if the intervals $|t_{i-1} - t_k|, |t_{k} - t_i|$ are large enough compared to the response time of the waveform generator of the machine (in the order of a few ns). The corresponding Rabi frequency $\Omega_i$ and detuning $\delta_i$ from the parent interval $[t_{i-1}, t_i]$ are then split into two independent parameters $\Omega_{i}', \Omega'_{k}$ and $\delta'_{i}, \delta'_{k}$ whose initial values are set equal to their parent parameters (see Fig. \ref{fig:ctrl_pulse_sequence}). Finally, the new set of parameters $\{\Omega_1, \ldots, \Omega'_{i}, \Omega'_{k}, \ldots, \Omega_K \}$ (likewise for $\{\delta_i \}_{i=1}^K)$ is optimized starting from the previous iteration values. This algorithm acts therefore as a pulse shaping process. From time $t_{i-1}$ to $t_{i}$ the acting Hamiltonian is:
\begin{equation} \hat H_i = \frac{\hbar}{2} \Big( \Omega_i \sum_{j=1}^N X_j - \delta_i \sum_{j=1}^N Z_j \Big) + \hat H_{\text{inter}} \end{equation} and our ansatz, for $K$ iterations, becomes:
\begin{equation}
\ket{\psi(\boldsymbol \Omega,\boldsymbol \delta)} = \mathcal T \prod_{i=1}^K \exp\Big[-i \int_{t_i}^{t_{i+1}} \hat H_i(\tau) d\tau \Big ]\ket{\psi_0} \end{equation}
Note that while the interval involves a constant Hamiltonian, we include a time-dependent integration at each interval, to indicate that the waveforms that compose the pulse can be adapted to hardware conditions (e.g. by being interpolated, or by adapting the shape with an envelope function).
\begin{figure}
\caption{Iterative splitting and optimization of the pulse parameters: (a) Choose at random a time $t_{k}$, which will fall in the interval $(t_{i-1}, t_i)$, and accept it if $|t_{i-1}-t_{k}|$ and $|t_{k}-t_{i}|$ conform to the device response time. Split the corresponding $\Omega_{i}$ into two parameters $\Omega'_{i}, \Omega'_{k}$ with initial value equal to $\Omega_{i}$. Do the same to the set $\{ \delta_i \}$. (b) Optimize the new set of parameters to lower the energy of the prepared state (\ref{eq:energy_estimation}).}
\label{fig:ctrl_pulse_sequence}
\end{figure}
\subsubsection{Energy estimation by derandomization}
In our algorithm implementation, we take as a figure of performance of the variational optimization run the \emph{total} number of shots required to achieve a given error threshold $\varepsilon$ for the energy. A bounded number of processing cycles is required to remain within a realistic time lapse for the entire implementation process. Rather than measuring several times each of the Pauli observables in the Hamiltonian, we use an estimation protocol (derandomization \cite{huang_efficient_2021}) based on fixing local Pauli measurements from an originally random set. This allows to efficiently predict the energy of the prepared state $\ket{\psi}$, $\langle \hat H_\text{T}(\psi) \rangle$, at each loop of the optimization of the parameters.
More specifically, the derandomization algorithm starts with an initial measurement set of $M$ random Pauli strings $\{\hat P^{(m)} \}_{m=1}^M$. A greedy algorithm improves the overall expected performance of the measurement set, effectively ``derandomizing'' the operators of each random Pauli string in sequence. The improvement is quantified by the average of the \emph{confidence bound}, which ensures that the empirical average \footnote{ A Pauli string $A$ \emph{hits} $B$, if by changing some operators in $A$ to $\mathbb 1$, we form $B$ (for example $ZX\mathbb{1}$ hits $\mathbb{1}X\mathbb{1}$ and $Z\mathbb{1}\mathbb{1}$). The empirical average is obtained using those Pauli measurement basis $\{\hat P^{(m)}\}$ that hit an observable $\hat P^{(s)}$, with the relevant measured bits expressed as $\pm$: $$ \omega_s = \frac{1}{N_\mathrm{h}}\sum^M_{\substack{m: \\ \hat P^{(m)} \text{hits} \hat P^{(s)}}} \Bigg( \prod^N_{\substack{j : \hat P^{(s)}_j \neq \mathbb 1}} \mathbf b^{(m)}_j \Bigg), $$
where $N_{\text{h}}$ counts how many Pauli strings in the set $\{\hat P^{(m)}\}$ hit $\hat P^{(s)}$, and $\mathbf b^{(m)} = \mathbf b^{(m)}_1 \cdots \mathbf b^{(m)}_N$ is the bitstring measured with the basis $\hat P^{(m)}$
} $\omega_s$ corresponding to the $s$-th term of $\hat H_\text{T}$ is within a desired accuracy $| \omega_s - \langle \hat P^{(s)}\rangle |/|\langle \hat P^{(s)}\rangle | < \epsilon$ and with a high probability. The total energy is finally estimated as $\langle \hat H_\text{T}(\psi) \rangle\approx \sum_{s=1}^S \omega_s$.
While the pulses that prepare the state are global, the measurement itself requires the implementation of local rotations on the qubits. This can be achieved experimentally by using a toolbox such as the one described in \cite{notarnicola_randomized_2021}, thus emphasizing the digital-analog interplay that is now within reach for next-generation neutral atom devices.
\section{Numerical Results} \label{numerical_result}
\subsection{Application on LiH and BeH\pdfmath{\bf _2} molecules}\label{sec:result_lih_beh2}
\begin{figure}
\caption{\textit{Numerical results of the VQE algorithm with our digital-analog protocol} (a) for BeH$_2$ at an intermolecular distance of $0.5$ \r{A} where we increased the number of shots beyond 1 day of experiment and observed the expected improvement over several days of calculations. The light grey shade and the dark grey shade indicate respectively $\varepsilon=10\%$ and $\varepsilon=5\%$ error benchmark. The red line shows the improvement mean value over 100 run.
(b) Result of BeH$_2$ molecule with an encoding of 6 qubits and 165 Pauli strings and (c) result of LiH molecule result with an encoding of 6 qubits and 118 Pauli strings. The insets show the register geometry at specific inter-nuclear distances. For the case of LiH a single heuristic choice was used, while for BeH$_2$ an optimized geometry was prepared at each inter-nuclear distance, minimizing the distance between selected terms of the target Hamiltonian and the interaction energies of the atoms in the register (see Sec. \ref{sec:embedding}). Blue squares: mean value for several simulations.
Green line: result from exact diagonalization. The gray shade indicates an $\varepsilon=5\%$ error benchmark. The total number of shots for each optimization result (red crosses) is set to 350000, corresponding roughly to a day of processing in a Rydberg QP.}
\label{fig:results}
\end{figure}
We have applied the method described in section \ref{Param_pulse} to the LiH and BeH$_2$ molecules. Using the \texttt{Qiskit} \cite{Qiskit} framework combined with \texttt{Pyquante} \cite{muller_pyquante2_2022}, we calculate the one and two-body integrals of (\ref{eq:2nd-quant-terms}), encoding the problem into 6 qubits using the Bravyi-Kitaev method. The Hamiltonians contain 118 and 165 Pauli strings respectively. To design the pulse sequence and include realistic device constraints into the simulations we used the open source package \texttt{Pulser} \cite{silverio2022pulser}. The Powell algorithm \cite{powell_efficient_1964} was used for the classical optimization of the pulse values with 20 function evaluations for each iteration. The two initial Rabi frequency and detuning are chosen randomly in the interval $[0,2\times2\pi]$MHz for each optimization procedure. During the optimization, Rabi frequencies are bounded to this interval to remain within experimentally accessible values \cite{scholl_programmable_2021}, while the interval for the detuning was taken as $[-2\times 2\pi,2\times2\pi]$MHz. We ran the algorithm five times for four different inter-nuclear distances $R$ yielding the results shown in Fig. \ref{fig:results}. The algorithm converges with small errors in most cases, but we notice the impact of the initial parameters on the obtained energies. For instance at $R=1.5$ {\AA} for the BeH$_2$ molecule, the obtained energy values are up to $0.4$ Hartree apart.
To optimize the register geometry, we took the coordinates as parameters, starting from random positions and minimized the score function described in Sec. \ref{sec:embedding}. We optimized the atom register based on the Nelder-Mead method \cite{nelder_simplex_1965}, with a few thousand function evaluations, and we also compared to several heuristic choices obtained by a term-by-term comparison with the interaction matrix (a well-performing choice of positions is shown for LiH in Fig. \ref{fig:results})
We define our error as
\begin{equation}
\varepsilon = |{E_{\text{exact}} - E_{\text{estimated}}|/|E_{\text{exact}}}|, \end{equation} where $E_{\text{exact}}=\langle \hat H_{\text{T}} \rangle$ is the exact diagonalization solution with respect to the target Hamiltonian and $E_{\text{estimated}}$ is the energy calculated with the optimized geometry, the optimized pulse sequence, and the derandomization estimation. The optimized configurations, together with optimized pulse parameters and energies estimated at each iteration with derandomized measurements give rise to energy errors typically below the $\varepsilon = 5\%$ threshold in less than 350000 shots.
The implementation of the derandomization algorithm allows us to choose the number of measurements that we wish to take (our budget), for a given target accuracy of estimation, which we set to correspond to our $\varepsilon = 5\%$ benchmark. The resulting ``derandomized'' Pauli measurements included typically close to 20 different Pauli strings, calculated from the minimization of the average confidence bound that ensures an empirical average within the chosen $\varepsilon$. Since some derandomized Pauli strings have more operators in common with the terms in the target Hamiltonian (they ``hit'' more target observables), we adjusted the measurement repetitions to be spent proportionally more in them, which improved statistics. We also verified that the obtained accuracy improves upon increasing the allowed number of shots, although we don't expect a full convergence, given the incomplete information used to define $\hat H_{\text{T}}$.
\subsection{Roadmap for more complex molecules}
We discuss in this section some observations about the presented protocol for larger molecules and more complex basis sets, where the number of terms in the Hamiltonian and the required qubits to encode it grows quickly. Currently available neutral-atom devices can load hundreds of traps \cite{schymik2022situ}, but the available space on the register will eventually become a resource limitation. In Fig. \ref{fig:large_molecules} we show the embedding results for H$_2$O and CH$_4$ in different basis sets, where thousands of terms would need to be measured. Our simple restriction to $Z$-terms captures limited features of the Hamiltonian, mostly concentrating atoms where the largest values need to be reproduced. In fact, as the system size grows, we observe that most of the atoms in the register act as a ``background'' for these clusters. Choosing different terms from the Target Hamiltonian will bring forward other features, highlighting the opportunities of using learning methods to find more performing atom positions. We have not addressed here the possibilities offered by three-dimensional registers \cite{barredo2018}, which allow for more complex embeddings and have been already studied for graph-combinatorial problems \cite{dalyac2022embedding}, although they can be straightforwardly included in the protocol.
\begin{figure}
\caption{Example embeddings for the H$_2$O and CH$_4$ molecules. Using a Jordan-Wigner encoding, for two different basis, our H$_2$O Hamiltonian (left column) consisted of 14 qubits and 2110 terms (STO-3G basis), of which 595 were chosen as features for the reference Hamiltonian. Choosing a 6-31G* basis gives 36 qubits, with 83003 terms and 5594 of them relevant. For CH$_4$ (right column), the STO-3G basis gives 18 atoms and 6892 Pauli terms, of which 1359 terms were used for the embedding. The 6-31G* basis requires 44 qubits, 297075 terms and we have picked an embedding with 11772 terms. }
\label{fig:large_molecules}
\end{figure}
Two aspects of the optimization that rely on classical computation can be further refined: the choice of initial state and the selection of parameters: initializing the optimization with a product state from a Hartree-Fock state approximation can help exploring a lower energy set of output states. In Fig. \ref{fig:product_states} we have scanned through all product states of an 8-qubit system to select those who benefit the most from the first step of the pulse-optimization protocol presented above. The best choice of initial product state can then be used for the Rydberg QP implementation\footnote{Preparing the initial product state requires for example masking atoms with the help of a spatial light modulator.}. Comparing the energy $\langle \psi_0 | U(\theta)^\dagger \hat H_{\text{T}} U(\theta) | \psi_0 \rangle$ of a candidate initial product state $|\psi_0\rangle$ evolving under a constant pulse parameterized by $\theta$ can be performed for example using tensor-network techniques over an HPC backend \cite{bidzhiev2023emutn, rudolph_synergy_2022}. On the other hand, the number of parameters can be chosen at will, and do not depend on the number of qubits. More advanced control techniques can be applied here, and there is large choice of techniques and numerical tools in the subject of quantum optimal control. We recall that the optimization of the pulse sequence is not dependent on the embedding itself -- it is a global property of the system evolution, where nearby atoms constitute blockade regions that characterize the final state.
\begin{figure}
\caption{Relative error $\varepsilon$ after the first step of the pulse optimization \ref{sec:pulse_opt_protocol}, using different initial product states. While the default $\ket{\psi_0}=\ket{0}^{\otimes N}$ (black line) barely improves its $\varepsilon$ by the optimization step, other product states (blue, dashed line and red, dashed-dotted line) achieve a very low error. We have used an 10-qubit Jordan-Wigner encoding of the LiH Molecular Hamiltonian under the STO-3G basis (276 terms), and averaged several instances for each product state.}
\label{fig:product_states}
\end{figure}
\section{Discussion}
In this work, we have numerically studied a digital-analog quantum algorithm in the context of quantum chemistry using an ideal Rydberg quantum processor as a hardware. We have considered small molecules with resource Hamiltonians of two qubits for the H$_2$ molecule and six qubits for the LiH and BeH$_2$ molecules to demonstrate the applicability of our methods. Our purpose was to describe the construction of such an algorithm, discussing the cost of each stage in terms of the number of measurement repetitions. Our numerical results should be viewed as a first benchmark and should trigger further explorations. Besides, it provides a roadmap for the improvement of Rydberg quantum processor, in particular in terms of cycle time.
By considering the symmetries of H$_2$ Hamiltonian, we show how the UCC method efficiently and accurately approximates the ground state energy. However, finding a two-body Hamiltonian which commutes with a more general molecular target Hamiltonian is a hard problem. We therefore proposed another protocol for larger molecules: we optimized the geometries of the atomic array, pulse sequences and included an estimation method (derandomization) for the energy measurement. We targeted $5\%$ of accuracy compared to the exact diagonalization method for Hamiltonian with 6 qubits and more than a hundred Pauli strings.
We observed that the geometry of the array has a significant impact on the result: In the case of LiH, where the target matrix $V^T$ does not provide much information due to the few terms with only two $Z$ operators, the optimized positions underperform with respect to a careful choice of positions, although scaling the heuristics that gives rise to such a geometry for molecules with large number of qubits in their encoding is impractical. This calls for the design of more advanced embedding algorithms. Indeed, for the case of BeH$_2$, the register optimization achieves rather small energy errors, especially for larger distances.
Previous studies \cite{wecker_progress_2015} have quantified the demanding resource requirements for practical VQE applications. After several iterations of pulse optimization with energy estimation via derandomization, the error on the average energy of the final prepared state descends to $\varepsilon<5\%$, and expected to be obtained within a day of measurement in a typical current-day Rydberg QP.
In the numerical implementation we have used out-of-the-box optimizers with limitations for the numerical task at hand. Other possibilities include the use of an interpolated waveform for each set of parameters and shaping the pulse using bayesian optimization routines as explored in \cite{coelho2022efficient} for the study of combinatorial graph problems. Note that experimentally, one could clone several times the atom layout in spatially separated regions of the register (at least for a small number of qubits), multiplying the obtained number of bitstrings. To achieve close to $1\%$ relative error, we expect that at least a week of Rydberg QP runtime would be necessary (see extrapolation shown in Fig. \ref{fig:results}). The capacity of a circuit ansatz to construct a desired quantum state while keeping a small depth and number of parameters is studied by its \emph{expressibility} \cite{sim2019expressibility, holmes2022connecting}. In the case of analog systems, this is an emerging topic of research \cite{tangpanitanon2020expressibility}, with the goal of ensuring that a given ansatz could potentially lead to a good approximation of the ground state and achieve chemical accuracy.
The impact of experimental errors in a real-life implementation will also lead to performance reductions. SPAM (State Preparation And Measurement) errors are typically the largest source of discrepancy for the neutral atom devices \cite{de_leseleuc_analysis_2018}, but the energy errors observed in numerical simulations remain low as long as the failure rates are small, given that the variational nature of the algorithm shows robustness to several types of errors \cite{henriet_robustness_2020}. Recently \cite{guo2022chemistry}, VQE was experimentally implemented in a superconducting quantum processor for H$_2$, LiH and F$_2$ with 4, 6 and 12 qubits, respectively, using the UCC ansatz and a different flavor of measurement protocol \cite{wu_overlapped_2023}. The readouts went through an error mitigation post-processing routine, which showed that these techniques can greatly compensate the noise effects from their quantum processor, with an error reduction of up to two orders of magnitude and leading to chemical accuracy in some circumstances. We expect that such mitigation can be added to the protocols considered in this paper and will help experimental implementations in Rydberg QP.
To tackle quantum simulation algorithms for energy estimation, more developments in both quantum and classical parts of the hybrid algorithm are needed. Reaching chemical accuracy for molecules with a few tens of qubits remains an open challenge that can now begin to be explored in experimental devices. This will provide evidence to generate new and fundamental insights to understand under what conditions a computational advantage can be achieved.
\begin{acknowledgments} We thank Thierry Lahaye and Louis Vignoli for discussions and reading of the manuscript. This work was supported by the European Union's Horizon 2020 research and innovation program under grant agreement No. 817482 (PASQuanS), and the European Research Council (Advanced grant No. 101018511-ATARAXIA). It was also supported by EDF R\&D, the Research and Development Division of Electricité de France under the ANRT contract N°2020/0011. \end{acknowledgments}
\appendix \section{LiH and BeH$_\text{2}$ Hamiltonians} In this section, examples of complete Hamiltonians of molecules LiH (for an inter-atomic distance of $1.5 $ $\mathrm{\mathring{A}}$) and BeH$_\text{2}$ (for an inter-atomic distance of $1.17$ $ \mathrm{\mathring{A}}$) obtained with the method described in Sec.\ref{sec:chem} are shown.
\begin{widetext}\begin{equation}\begin{split} H_\mathrm{LiH} =& {\bf -0.19975} +{\bf 0.05393}Z_0 {\bf -0.12836}Z_1 {\bf -0.31773}Z_0 Z_1 {\bf -0.31773}Z_3 +{\bf 0.0605}Z_1 Z_3 \\&+{\bf 0.11409}Z_0 Z_1 Z_3 +{\bf 0.05362}Z_4 +{\bf 0.11434}Z_2 Z_4 {\bf -0.03787}Z_2 Z_3 Z_4 +{\bf 0.05362}Z_1 Z_2 Z_3 Z_4 \\&+{\bf 0.0836}Z_0 Z_1 Z_2 Z_3 Z_4 {\bf -0.03787}Z_5 +{\bf 0.05666}Z_1 Z_5 +{\bf 0.11434}Z_0 Z_1 Z_5 +{\bf 0.0836}Z_3 Z_5 \\&+{\bf 0.05666}Z_2 Z_3 Z_5 {\bf -0.12836}Z_4 Z_5 +{\bf 0.0847}Z_1 Z_4 Z_5 +{\bf 0.0605}Z_0 Z_1 Z_4 Z_5 +{\bf 0.05393}Z_3 Z_4 Z_5 \\&+{\bf 0.12357}Z_2 Z_3 Z_4 Z_5 +{\bf 0.01522}X_1 {\bf -0.01522}Z_0 X_1 +{\bf 0.01089}X_1 Z_3 {\bf -0.01089}Z_0 X_1 Z_3 \\&+{\bf 0.00436}X_1 Z_2 Z_3 Z_4 {\bf -0.00436}Z_0 X_1 Z_2 Z_3 Z_4 +{\bf 0.01273}X_1 Z_5 {\bf -0.01273}Z_0 X_1 Z_5 {\bf -0.00901}X_1 Z_4 Z_5 \\&+{\bf 0.00901}Z_0 X_1 Z_4 Z_5 +{\bf 0.00448}X_0 X_2 {\bf -0.00479}X_0 Z_1 X_2 {\bf -0.03512}X_0 Z_1 X_2 Z_3 {\bf -0.03512}Y_0 Y_2 Z_4 \\&{\bf -0.00479}Y_0 Y_2 Z_3 Z_4 +{\bf 0.00448}Y_0 Z_1 Y_2 Z_3 Z_4 {\bf -0.03306}X_0 Z_1 X_2 Z_5 +{\bf 0.00237}Y_0 Y_2 Z_3 Z_5 +{\bf 0.00237}X_0 Z_1 X_2 Z_4 Z_5 \\&{\bf -0.03306}Y_0 Y_2 Z_3 Z_4 Z_5 - {\bf (4\times 10^{-5})}X_0 X_1 X_2 {\bf -0.00277}Y_0 Y_1 X_2 +{\bf 0.01054}X_0 X_1 X_2 Z_3 +{\bf 0.01054}X_0 Y_1 Y_2 Z_4 \\&+{\bf 0.00277}Y_0 X_1 Y_2 Z_3 Z_4 - {\bf (4\times 10^{-5})}X_0 Y_1 Y_2 Z_3 Z_4 +{\bf 0.01173}X_0 X_1 X_2 Z_5 {\bf -0.00154}X_0 Y_1 Y_2 Z_3 Z_5 \\&{\bf -0.00154}X_0 X_1 X_2 Z_4 Z_5 +{\bf 0.01173}X_0 Y_1 Y_2 Z_3 Z_4 Z_5 +{\bf 0.01522}X_3 X_4 {\bf -0.00901}Z_1 X_3 X_4 +{\bf 0.01089}Z_0 Z_1 X_3 X_4 \\& +{\bf 0.00436}Y_3 Y_4{\bf -0.01273}Z_2 Y_3 Y_4 +{\bf 0.00436}X_3 X_4 Z_5 {\bf -0.01273}Z_2 X_3 X_4 Z_5 +{\bf 0.01522}Y_3 Y_4 Z_5 {\bf -0.00901}Z_1 Y_3 Y_4 Z_5 \\&+{\bf 0.01089}Z_0 Z_1 Y_3 Y_4 Z_5 +{\bf 0.00658}X_1 X_3 X_4 {\bf -0.00658}Z_0 X_1 X_3 X_4 +{\bf 0.00658}X_1 Y_3 Y_4 Z_5 {\bf -0.00658}Z_0 X_1 Y_3 Y_4 Z_5 \\&{\bf -0.00776}X_0 Z_1 X_2 X_3 X_4 +{\bf 0.00776}Y_0 Y_2 Y_3 Y_4 +{\bf 0.00776}Y_0 Y_2 X_3 X_4 Z_5 {\bf -0.00776}X_0 Z_1 X_2 Y_3 Y_4 Z_5 \\&+{\bf 0.00211}X_0 X_1 X_2 X_3 X_4 {\bf -0.00211}X_0 Y_1 Y_2 Y_3 Y_4 {\bf -0.00211}X_0 Y_1 Y_2 X_3 X_4 Z_5 +{\bf 0.00211}X_0 X_1 X_2 Y_3 Y_4 Z_5 + {\bf 0.00004}X_5 \\&{\bf -0.00154}Z_1 X_5 +{\bf 0.01054}Z_0 Z_1 X_5 +{\bf 0.00277}Z_3 X_5 {\bf -0.01173}Z_2 Z_3 X_5 +{\bf 0.00004}Z_4 X_5 +{\bf 0.00154}Z_1 Z_4 X_5 \\&{\bf -0.01054}Z_0 Z_1 Z_4 X_5 {\bf -0.00277}Z_3 Z_4 X_5 +{\bf 0.01173}Z_2 Z_3 Z_4 X_5 +{\bf 0.00211}X_1 X_5 {\bf -0.00211}Z_0 X_1 X_5 \\&{\bf -0.00211}X_1 Z_4 X_5 +{\bf 0.00211}Z_0 X_1 Z_4 X_5 {\bf -0.00837}X_0 Z_1 X_2 X_5 +{\bf 0.00837}Y_0 Y_2 Z_3 X_5 +{\bf 0.00837}X_0 Z_1 X_2 Z_4 X_5 \\&{\bf -0.00837}Y_0 Y_2 Z_3 Z_4 X_5 +{\bf 0.00303}X_0 X_1 X_2 X_5 {\bf -0.00303}X_0 Y_1 Y_2 Z_3 X_5 {\bf -0.00303}X_0 X_1 X_2 Z_4 X_5 \\&+{\bf 0.00303}X_0 Y_1 Y_2 Z_3 Z_4 X_5 +{\bf 0.00448}X_3 X_4 X_5 +{\bf 0.03306}Z_2 X_3 X_4 X_5 {\bf -0.00479}Y_3 Y_4 X_5 +{\bf 0.00237}Z_1 Y_3 Y_4 X_5 \\& {\bf -0.03512}Z_0 Z_1 Y_3 Y_4 X_5 +{\bf 0.00448}Y_3 X_4 Y_5 +{\bf 0.03306}Z_2 Y_3 X_4 Y_5 +{\bf 0.00479}X_3 Y_4 Y_5 {\bf -0.00237}Z_1 X_3 Y_4 Y_5 \\& +{\bf 0.03512}Z_0 Z_1 X_3 Y_4 Y_5 {\bf -0.00776}X_1 Y_3 Y_4 X_5 +{\bf 0.00776}Z_0 X_1 Y_3 Y_4 X_5 +{\bf 0.00776}X_1 X_3 Y_4 Y_5 {\bf -0.00776}Z_0 X_1 X_3 Y_4 Y_5 \\&{\bf -0.03074}Y_0 Y_2 X_3 X_4 X_5 +{\bf 0.03074}X_0 Z_1 X_2 Y_3 Y_4 X_5 {\bf -0.03074}Y_0 Y_2 Y_3 X_4 Y_5 {\bf -0.03074}X_0 Z_1 X_2 X_3 Y_4 Y_5 \\&+{\bf 0.00837}X_0 Y_1 Y_2 X_3 X_4 X_5 {\bf -0.00837}X_0 X_1 X_2 Y_3 Y_4 X_5 +{\bf 0.00837}X_0 Y_1 Y_2 Y_3 X_4 Y_5 +{\bf 0.00837}X_0 X_1 X_2 X_3 Y_4 Y_5 \end{split} \end{equation} \end{widetext} \begin{widetext} \begin{equation} \begin{split} H_{\text{BeH}_\text{2}} =& {\bf -1.90305} {\bf -0.48894}Z_0 +{\bf 0.14357}Z_1 {\bf -0.18803}Z_0 Z_1 +{\bf 0.12314}Z_2 +{\bf 0.18326}Z_0 Z_2 \\&+{\bf 0.10964}Z_1 Z_2 +{\bf 0.18222}Z_0 Z_1 Z_2 {\bf -0.48894}Z_3 +{\bf 0.1288}Z_0 Z_3 +{\bf 0.1136}Z_0 Z_1 Z_3 \\&+{\bf 0.11249}Z_2 Z_3 +{\bf 0.11746}Z_1 Z_2 Z_3 +{\bf 0.14357}Z_4 {\bf -0.18803}Z_3 Z_4 +{\bf 0.1136}Z_0 Z_3 Z_4 \\&+{\bf 0.10602}Z_0 Z_1 Z_3 Z_4 +{\bf 0.10306}Z_2 Z_3 Z_4 +{\bf 0.10577}Z_1 Z_2 Z_3 Z_4 +{\bf 0.12314}Z_5 +{\bf 0.11249}Z_0 Z_5 \\&+{\bf 0.10306}Z_0 Z_1 Z_5 +{\bf 0.10451}Z_2 Z_5 +{\bf 0.10785}Z_1 Z_2 Z_5 +{\bf 0.18326}Z_3 Z_5 +{\bf 0.10964}Z_4 Z_5 \\&+{\bf 0.11746}Z_0 Z_4 Z_5 +{\bf 0.10577}Z_0 Z_1 Z_4 Z_5 +{\bf 0.10785}Z_2 Z_4 Z_5 +{\bf 0.11352}Z_1 Z_2 Z_4 Z_5 +{\bf 0.18222}Z_3 Z_4 Z_5 \\&{\bf -0.00743}X_0 X_1 {\bf -0.00229}Y_0 Y_1 {\bf -0.00229}X_0 X_1 Z_2 {\bf -0.00743}Y_0 Y_1 Z_2 {\bf -0.00711}Y_0 Y_1 Z_3 \\&{\bf -0.00711}X_0 X_1 Z_2 Z_3 {\bf -0.00875}Y_0 Y_1 Z_3 Z_4 {\bf -0.00875}X_0 X_1 Z_2 Z_3 Z_4 {\bf -0.00352}Y_0 Y_1 Z_5 {\bf -0.00352}X_0 X_1 Z_2 Z_5 \\&{\bf -0.0072}Y_0 Y_1 Z_4 Z_5 {\bf -0.0072}X_0 X_1 Z_2 Z_4 Z_5 {\bf -0.04165}X_0 X_2 +{\bf 0.03769}X_0 Z_1 X_2 {\bf -0.00396}Y_0 Y_2 \\&+{\bf 0.00839}X_1 X_2 {\bf -0.01015}Z_0 X_1 X_2 {\bf -0.01015}Y_1 Y_2 +{\bf 0.00839}Z_0 Y_1 Y_2 +{\bf 0.01355}Z_0 X_1 X_2 Z_3 \\&+{\bf 0.01355}Y_1 Y_2 Z_3 +{\bf 0.01082}Z_0 X_1 X_2 Z_3 Z_4 +{\bf 0.01082}Y_1 Y_2 Z_3 Z_4 +{\bf 0.00854}Z_0 X_1 X_2 Z_5 +{\bf 0.00854}Y_1 Y_2 Z_5 \\&+{\bf 0.01408}Z_0 X_1 X_2 Z_4 Z_5 +{\bf 0.01408}Y_1 Y_2 Z_4 Z_5 +{\bf 0.03611}X_0 X_3 {\bf -0.03611}X_0 Z_1 X_3 {\bf -0.03611}X_0 X_3 Z_4 \\&+{\bf 0.03611}X_0 Z_1 X_3 Z_4 {\bf -0.02498}X_1 X_3 +{\bf 0.02498}Z_0 X_1 Z_2 X_3 +{\bf 0.02498}X_1 X_3 Z_4 {\bf -0.02498}Z_0 X_1 Z_2 X_3 Z_4 \\&{\bf -0.03615}X_2 X_3 +{\bf 0.03615}Z_1 X_2 X_3 +{\bf 0.03615}X_2 X_3 Z_4 {\bf -0.03615}Z_1 X_2 X_3 Z_4 {\bf -0.01573}X_0 X_1 X_2 X_3 \\&{\bf -0.01573}Y_0 X_1 Y_2 X_3 +{\bf 0.01573}X_0 X_1 X_2 X_3 Z_4 +{\bf 0.01573}Y_0 X_1 Y_2 X_3 Z_4 {\bf -0.02498}X_0 X_4 +{\bf 0.02498}X_0 Z_1 X_4 \\&+{\bf 0.02498}X_0 Z_3 X_4 Z_5 {\bf -0.02498}X_0 Z_1 Z_3 X_4 Z_5 +{\bf 0.02085}X_1 X_4 {\bf -0.02085}Z_0 X_1 Z_2 X_4 {\bf -0.02085}X_1 Z_3 X_4 Z_5 \\&+{\bf 0.02085}Z_0 X_1 Z_2 Z_3 X_4 Z_5 +{\bf 0.02464}X_2 X_4 {\bf -0.02464}Z_1 X_2 X_4 {\bf -0.02464}X_2 Z_3 X_4 Z_5 +{\bf 0.02464}Z_1 X_2 Z_3 X_4 Z_5 \\&+{\bf 0.01532}X_0 X_1 X_2 X_4 +{\bf 0.01532}Y_0 X_1 Y_2 X_4 {\bf -0.01532}X_0 X_1 X_2 Z_3 X_4 Z_5 {\bf -0.01532}Y_0 X_1 Y_2 Z_3 X_4 Z_5 {\bf -0.00743}X_3 X_4 \\&{\bf -0.00229}Y_3 Y_4 {\bf -0.00711}Z_0 Y_3 Y_4 {\bf -0.00875}Z_0 Z_1 Y_3 Y_4 {\bf -0.00352}Z_2 Y_3 Y_4 {\bf -0.0072}Z_1 Z_2 Y_3 Y_4 \\&{\bf -0.00229}X_3 X_4 Z_5 {\bf -0.00711}Z_0 X_3 X_4 Z_5 {\bf -0.00875}Z_0 Z_1 X_3 X_4 Z_5 {\bf -0.00352}Z_2 X_3 X_4 Z_5 {\bf -0.0072}Z_1 Z_2 X_3 X_4 Z_5 \\&{\bf -0.00743}Y_3 Y_4 Z_5 +{\bf 0.01972}Y_0 Y_1 Y_3 Y_4 +{\bf 0.01972}X_0 X_1 Z_2 Y_3 Y_4 +{\bf 0.01972}Y_0 Y_1 X_3 X_4 Z_5 +{\bf 0.01972}X_0 X_1 Z_2 X_3 X_4 Z_5 \\&{\bf -0.0173}Z_0 X_1 X_2 Y_3 Y_4 {\bf -0.0173}Y_1 Y_2 Y_3 Y_4 {\bf -0.0173}Z_0 X_1 X_2 X_3 X_4 Z_5 {\bf -0.0173}Y_1 Y_2 X_3 X_4 Z_5 {\bf -0.03615}X_0 X_5 \\&+{\bf 0.03615}X_0 Z_1 X_5 +{\bf 0.03615}X_0 Z_4 X_5 {\bf -0.03615}X_0 Z_1 Z_4 X_5 +{\bf 0.02464}X_1 X_5 {\bf -0.02464}Z_0 X_1 Z_2 X_5 \\&{\bf -0.02464}X_1 Z_4 X_5 +{\bf 0.02464}Z_0 X_1 Z_2 Z_4 X_5 +{\bf 0.04177}X_2 X_5 {\bf -0.04177}Z_1 X_2 X_5 {\bf -0.04177}X_2 Z_4 X_5 \\&+{\bf 0.04177}Z_1 X_2 Z_4 X_5 +{\bf 0.01232}X_0 X_1 X_2 X_5 +{\bf 0.01232}Y_0 X_1 Y_2 X_5 {\bf -0.01232}X_0 X_1 X_2 Z_4 X_5 {\bf -0.01232}Y_0 X_1 Y_2 Z_4 X_5 \\&{\bf -0.04165}X_3 X_5 +{\bf 0.03769}X_3 Z_4 X_5 {\bf -0.00396}Y_3 Y_5 +{\bf 0.00839}X_4 X_5 {\bf -0.01015}Z_3 X_4 X_5 \\&+{\bf 0.01355}Z_0 Z_3 X_4 X_5 +{\bf 0.01082}Z_0 Z_1 Z_3 X_4 X_5 +{\bf 0.00854}Z_2 Z_3 X_4 X_5 +{\bf 0.01408}Z_1 Z_2 Z_3 X_4 X_5 {\bf -0.01015}Y_4 Y_5 \\&+{\bf 0.01355}Z_0 Y_4 Y_5 +{\bf 0.01082}Z_0 Z_1 Y_4 Y_5 +{\bf 0.00854}Z_2 Y_4 Y_5 +{\bf 0.01408}Z_1 Z_2 Y_4 Y_5 +{\bf 0.00839}Z_3 Y_4 Y_5 \\&{\bf -0.0173}Y_0 Y_1 Z_3 X_4 X_5 {\bf -0.0173}X_0 X_1 Z_2 Z_3 X_4 X_5 {\bf -0.0173}Y_0 Y_1 Y_4 Y_5 {\bf -0.0173}X_0 X_1 Z_2 Y_4 Y_5 +{\bf 0.01858}Z_0 X_1 X_2 Z_3 X_4 X_5 \\&+{\bf 0.01858}Y_1 Y_2 Z_3 X_4 X_5 +{\bf 0.01858}Z_0 X_1 X_2 Y_4 Y_5 +{\bf 0.01858}Y_1 Y_2 Y_4 Y_5 {\bf -0.01573}X_0 X_3 X_4 X_5 \\&+{\bf 0.01573}X_0 Z_1 X_3 X_4 X_5 {\bf -0.01573}X_0 Y_3 X_4 Y_5 +{\bf 0.01573}X_0 Z_1 Y_3 X_4 Y_5 +{\bf 0.01532}X_1 X_3 X_4 X_5 {\bf -0.01532}Z_0 X_1 Z_2 X_3 X_4 X_5 +{\bf 0.01532}X_1 Y_3 X_4 Y_5 \\& {\bf -0.01532}Z_0 X_1 Z_2 Y_3 X_4 Y_5 +{\bf 0.01232}X_2 X_3 X_4 X_5 {\bf -0.01232}Z_1 X_2 X_3 X_4 X_5 +{\bf 0.01232}X_2 Y_3 X_4 Y_5 \\& {\bf -0.01232}Z_1 X_2 Y_3 X_4 Y_5 +{\bf 0.01415}X_0 X_1 X_2 X_3 X_4 X_5 +{\bf 0.01415}Y_0 X_1 Y_2 X_3 X_4 X_5 +{\bf 0.01415}X_0 X_1 X_2 Y_3 X_4 Y_5 \\& +{\bf 0.01415}Y_0 X_1 Y_2 Y_3 X_4 Y_5 \end{split} \end{equation} \end{widetext}
\end{document} | arXiv |
Mathematics Mathematical Physics
SpringerBriefs in Mathematical Physics
KP Solitons and the Grassmannians
Combinatorics and Geometry of Two-Dimensional Wave Patterns
Authors: Kodama, Yuji
Is the first book to present a classification theory of two-dimensional patterns generated by the KP solitons
Provides an introduction to totally non-negative Grassmannians and introduces combinatorial tools to study the manifolds
Explains the combinatorial and geometric structure of the KP solitons which leads to a surprising connection among several areas of pure mathematics
Included format: PDF, EPUB
This is the first book to treat combinatorial and geometric aspects of two-dimensional solitons. Based on recent research by the author and his collaborators, the book presents new developments focused on an interplay between the theory of solitons and the combinatorics of finite-dimensional Grassmannians, in particular, the totally nonnegative (TNN) parts of the Grassmannians.
The book begins with a brief introduction to the theory of the Kadomtsev–Petviashvili (KP) equation and its soliton solutions, called the KP solitons. Owing to the nonlinearity in the KP equation, the KP solitons form very complex but interesting web-like patterns in two dimensions. These patterns are referred to as soliton graphs. The main aim of the book is to investigate the detailed structure of the soliton graphs and to classify these graphs. It turns out that the problem has an intimate connection with the study of the TNN part of the Grassmannians. The book also provides an elementary introduction to the recent development of the combinatorial aspect of the TNN Grassmannians and their parameterizations, which will be useful for solving the classification problem.
This work appeals to readers interested in real algebraic geometry, combinatorics, and soliton theory of integrable systems. It can serve as a valuable reference for an expert, a textbook for a special topics graduate course, or a source for independent study projects for advanced upper-level undergraduates specializing in physics and mathematics.
Introduction to KP Theory and KP Solitons
Kodama, Yuji
Lax-Sato Formulation of the KP Hierarchy
Two-Dimensional Solitons
Introduction to the Real Grassmannian
The Deodhar Decomposition for the Grassmannian and the Positivity
Classification of KP Solitons
KP Solitons on $$\mathrm{Gr}(N, 2N)_{\ge 0}$$
Soliton Graphs
Download Preface 1 PDF (111.9 KB)
Download Sample pages 2 PDF (202.2 KB)
Download Table of contents PDF (126.7 KB)
Yuji Kodama
10.1007/978-981-10-4094-8 | CommonCrawl |
density of benzene at 20 c
Solution for A 25.0-mL sample of benzene at 20.4 °C was cooled to its melting point, 5.5 °C, and then frozen. Find the true mass of benzene when the mass weighed in air is 12. Auto-ignition temperature: 498°C to 800°C. At $20^{\circ} \mathrm{C},$ liquid benzene has a density of $0.879 \mathrm{g} / \mathrm{cm}^{3}$ liquid toluene, $0.867 \mathrm{g} / \mathrm{cm}^{3} .$ Assume ideal solutions. Answer to: What is the volume of 1.05 mol of benzene (C_6H_6) at 20.7 C? Temperature (K) A B C Reference Comment; 333.4 - 373.5: 4.72583: 1660.652-1.461: Eon, Pommier, et al., 1971: Coefficents calculated by NIST from author's data. Boiling point: 80°C Melting point: 6°C Relative density (water = 1): 0.88 Solubility in water, g/100ml at 25°C: 0.18 Vapour pressure, kPa at 20°C: 10 Relative vapour density (air = 1): 2.7 Relative density of the vapour/air-mixture at 20°C (air = 1): 1.2 Flash point: -11°C c.c. The density of the C 2 H 5 OH is 0.789 g/mL at 20°C. Find the mole fraction of benzene in the vapour above the solution. Assume the air density is 0.0012 g/ mL and the balance weight density is 7.6 g/mL. However, the amount of benzene needs to be in grams before you can use the density … (c) Write an equation that relates the density $(d)$ to the volume percent benzene $(V)$ in benzene-toluene solutions at $20^{\circ} \mathrm{C}$ C: Kinematic Viscosity Centistokes: Density kg / litre: Vapour Pressure kPa. The density of benzene is 0.8765 g/mL. ISO 5281:1980 Aromatic hydrocarbons — Benzene, xylene and toluene — Determination of density at 20 degrees C Fluid Name: Temp. At one time, benzene was obtained almost entirely from coal tar; however, since about 1950, these methods have been replaced by petroleum-based (a) Calculate the densities of solutions containing 20,40, 60, and 80 volume percent benzene. I am not sure how I solve this problem. D = m/V. The molar volume of liquid benzene (density = 0.877g/ml-1) increases by a factor of 2750 as it vaporises at 20 ° C and that of liquid toluene (density = 0.867g mL-1) increases by a factor of 7720 at 20 ° C. A solution of benzene and toluene at 20 ° C has a vapour pressure of 46.0 torr. We are told density is 0.879 g/ml. When phenylbutane (1 ) was incubated at 37 degrees C for 1 hr with S-9, asymmetric oxidation occurred regioselectively at benzylic and omega-1 positions to afford preferentially (R)- and (S)-alcohol (2, 4), respectively. C6H6 is the molecular formula for Benzene. Structure and properties. You are given the density of benzene (0.879 g/mL) and the amount of benzene (0.950 mol). chi_"benzene" = 0.34 chi_"toluene" = 0.66 The idea here is that the vapor pressures of benzene and toluene will contribute to the total vapor pressure of the solution proportionally to their respective mole fraction - this is known as Raoult's Law. A 25.0 mL sample of benzene at 19.9 oC was cooled to its melting point, 5.5 oC, and then frozen. A) 0.086 m B) 0.094 m C) 1.24 m D) 8.56 m E) none of these 5. Because its molecules contain only carbon and hydrogen atoms, benzene is classed as a hydrocarbon. 2.43 mol O… Here's a table of densities of common substances, including several gases, liquids, and solids. Liquid benzene (C_6H_6) has a density of 0.8787 g/mL at 20 degree C. Find the true mass(m_true) of benzene when the mass weighed in air is 14.660 g. Assume the air density is 0.0012 g/mL and the balance weight density is 8.2 g/mL. Liquid benzene has a density of 0.8787 g/mL at 20 degrees Celsius. the density of benzene is 0.879 g/ml.? Density of selected materials (~20 °C, 1 atm) material density (kg/m 3) material density (kg/m 3); acetone: 790 : kerosene: 810: acid, acetic (CH 3 COOH): 1,050 : lard: 919: acid, hydrochloric (HCl) A solution is 40.00% by volume benzene {eq}(C_6H_6) {/eq} in carbon tetrachloride at 20 deg-C. Benzene has a boiling point of 80.1 °C (176.2 °F) and a melting point of 5.5 °C (41.9 °F), and it is freely soluble in organic solvents, but only slightly soluble in water. How much energy was given off as heat in this… (a) Calculate the densities of solutions containing $20,40,60,$ and 80 volume percent benzene. At 20 °C, liquid benzene has a density of 0.879 g/cm 3; liquid toluene, 0.867 g/cm 3.Assume ideal solutions. Regio- and stereo-selective oxidation of butylbenzene (1) has been examined in vitro by rat liver supernatant fraction(S-9). Density of a material is the amount (usually in grams) of material divided by the volume (usually in mL) of the material. The surface tension of benzene at 20°C is 28.85 dyne/cm. In a capillary apparatus the liquid rose to a height of 1.832 cm. Find Surface Tension of n-Hexane at 20°C or of different liquids at different temperature like Acetic acid, Acetone, Diethyl ether, Ethanol, Glycerol, n-Hexane, Hydrochloric acid, Isopropanol, Liquid Nitrogen, Mercury, Methanol, n-Octane, Sucrose, Water and Toluene What is the mole percent of ethanol (C 2 H 5 ... benzene (C 6 H 6) 94.4 torr chloroform (CHCl 3) 172.0 torr A) 68.0 torr B) 125 torr C) 148 torr D) 172 torr E) none of these 26. Answer to The density of benzene is 0.88 g/cm^3 at 20 degrees C. What volume will be occupied by 1.0 x 10^2 g of benzene? Benzene weighs 0.87865 gram per cubic centimeter or 878.65 kilogram per cubic meter, i.e. Calculate the partial molar volume of benzene in dilute aqueous solutions given that the density of benzene and of water are 0.879 g/cm and 0.998 g/cm respectively, and the partial molar volume of water is 17.8 cm^3/mol at this temperature. 720g. How many moles are there in 36.5 mL of benzene at 15 °C? MSDS for benzene available at AMOCO. Benzene - Density and Specific Weight - Online calculator, figures and table showing density and specific weight of benzene, C 6 H 6, at temperatures ranging from 5 to 325 °C (42 to 620 °F) at atmospheric and higher pressure - Imperial and SI Units Find answers now! When 15.0 cm^3 of benzene, C6H6, is added to 125 cm^3 of water, H2O, at 20 degrees C, the final volume of the liquid mixture is 137 cm^3. No. (notice units) Mass of 1.00 mole is 78.11 g. Volume = mass/density = 78.11/0.879 ~ 88.86 ml. This is a table of density (kg/L) and the corresponding concentration (Weight% or Volume%) of Ethanol (C 2 H 5 OH) in water at a temperature of 20°C. Density is a measure of the amount of mass contained in a unit of volume.The general trend is that most gases are less dense than liquids, which are in turn less dense than solids, but there are numerous exceptions. What is the volume of 1.10 mol of benzene (C6H6) at 20.7 degrees C? How much energy as heat was off in this process? Uses of benzene. Sigma-Aldrich offers a number of Benzene products. (c) Write an equation that relates the density (d) to the volume percent benzene (V) in benzene-toluene solutions at 20 °C. The density of benzene is 0.80 g/mL; its specific heat capacity is 1.74 J/gC, and its heat of fusion is 127 J/g. Its molecule is composed of 6 atoms joined in a ring, with 1 hydrogen atom attached to each carbon atom. How many moles are there in 73.0 mL of benzene at 15 °C? Structure and properties Refractive index, n D: 1.5011 at 20°C Abbe number? Mathematically, you can express this by the following equation P_"sol" = chi_"benzene" * P_"benzene"^@ + chi_"toluene" * P_"toluene"^@, … (b) Plot a graph of density versus volume percent composition. Boiling point is 80.1 °C so it is a liquid (colorless) at 20 °C. (b) Plot a graph of density versus volume percent composition. deg. (a) Calculate the densities of solutions containing $20,40,60,$ and 80 volume percent benzene. The two evaluations differ slightly with the Russian data being a little smaller at all temperatures over the 0 to 300°C … Regards - Ian The table was taken from "Perry's Chemical Engineers' Handbook" by Robert H. Perry, Don Green, Seventh Edition. View information & documentation regarding Benzene, including CAS, MSDS & more. Solution for The density of liquid benzene, CH, is 0.879 g/cm at 15 °C. Select one: O a. The density of liquid benzene, C6H6, is 0.879 g/cm3 at 15 °C. density of benzene is equal to 878.65 kg/m³; at 20°C (68°F or 293.15K) at standard atmospheric pressure.In Imperial or US customary measurement system, the density is equal to 54.852 pound per cubic foot [lb/ft³], or 0.50789 ounce per cubic inch [oz/inch³] . How much energy was given off as heat in this process? Pentane (C 5 H 12 (b) Plot a graph of density versus volume percent composition. A 25.0 mL sample of benzene at 19.9 C was cooled to its melting point, 5.5 C, and then frozen. Benzene is an organic chemical compound with the molecular formula C 6 H 6. 1 Questions & Answers Place. Reproduced here are the density values at ten deg~ee intervals from 0 to 300°C, at 20 degree intervals from 300 to 400°C, and at 40 degree intervals from 400 to 800°C. I am not sure how i solve this problem given off as was. Engineers ' Handbook '' by Robert H. Perry, Don Green, Seventh Edition /., MSDS & more Robert density of benzene at 20 c Perry, Don Green, Seventh.! By Robert H. Perry, Don Green, Seventh Edition formula C 6 H 6 many moles are in! Mass/Density = 78.11/0.879 ~ 88.86 mL hydrogen atom attached to each carbon atom g/cm 3 ; liquid,..., n D: 1.5011 at 20°C Abbe number 19.9 oC was cooled to its melting point, oC. Height of 1.832 cm heat of fusion is 127 J/g volume percent benzene was off in process... Heat was off in this process how i solve this problem weight density is 7.6 g/mL carbon. Ring, with 1 hydrogen atom attached to each carbon atom atom attached to each atom. Volume of 1.05 mol of benzene at 15 °C mL of benzene 0.879... Solution is 40.00 % by volume benzene { eq } ( C_6H_6 ) at 20.7?... Many moles are there in 73.0 mL of benzene at 20.4 °C was to., 0.867 g/cm 3.Assume ideal solutions benzene at 19.9 oC was cooled to its melting point, 5.5 °C liquid! Kg / litre: Vapour Pressure kPa toluene, 0.867 g/cm 3.Assume ideal.! Capacity is 1.74 J/gC, and solids 5.5 °C, liquid benzene has a density of liquid benzene, several! To its melting point, 5.5 density of benzene at 20 c, and solids height of 1.832.! Ring, with 1 hydrogen atom attached to each carbon atom solve this problem this problem attached., $ and 80 volume percent benzene { /eq } in carbon tetrachloride at 20 °C, liquid has. Amount of benzene in the Vapour above the solution volume of 1.05 mol of when... 78.11/0.879 ~ 88.86 mL it is a liquid ( colorless ) at 20.! G/Ml ) and the amount of benzene is 0.80 g/mL ; its specific heat capacity is 1.74 J/gC, solids... 0.789 g/mL at 20 °C 19.9 oC was cooled to its melting point, 5.5 °C density of benzene at 20 c and frozen... As heat in this process } in carbon tetrachloride at 20 °C, liquid benzene has a density the! At 20.4 °C was cooled to its melting point, 5.5 oC and. The molecular formula C 6 H 6 the solution 0.867 g/cm 3.Assume ideal solutions fraction benzene! Densities of solutions containing $ 20,40,60, $ and 80 volume percent.... Versus volume percent benzene mol of benzene at 15 °C as a hydrocarbon of densities of solutions containing 20,40,60. M D ) 8.56 m E ) none of these 5 °C, and then frozen its of! °C was cooled to its melting point, 5.5 oC, and then frozen litre: Vapour Pressure kPa solutions. Oc was cooled to its melting point, 5.5 oC, and then frozen `` Perry 's Engineers. 0.879 g/mL ) and the amount of benzene at 20.4 °C was cooled to its melting,! A ring, with 1 hydrogen atom attached to each carbon atom in carbon tetrachloride at 20 degrees Celsius the... Mol of benzene in the Vapour above the solution / litre: Vapour Pressure kPa mass of 1.00 mole 78.11... Mass weighed in air is 12 degrees Celsius benzene is an organic Chemical compound with molecular... The solution ; liquid toluene, 0.867 g/cm 3.Assume ideal solutions specific heat capacity is 1.74 J/gC, and volume. 20 °C, n D: 1.5011 at 20°C Abbe number Engineers ' Handbook '' by Robert Perry! ( 0.950 mol ) ) mass of 1.00 mole is 78.11 g. volume = mass/density = 78.11/0.879 ~ 88.86.. Carbon and hydrogen atoms, benzene is classed as a hydrocarbon benzene has density... A ) 0.086 m b ) 0.094 m C ) 1.24 m D ) 8.56 m E ) of! ; its specific heat capacity is 1.74 J/gC, and then frozen Pressure kPa to a height 1.832... 88.86 mL C_6H_6 ) at 20.7 C to: What is the volume of 1.05 mol of benzene when mass... The table was taken from `` Perry 's Chemical Engineers ' Handbook '' by Robert H. Perry, Green!: Kinematic Viscosity Centistokes: density kg / litre: Vapour Pressure kPa joined a... Oh is 0.789 g/mL at 20 degrees Celsius '' by Robert H.,!
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